$~500~µJ is t$_\\textnormal{n}~>$~2ms, whereas for the 105~µm fiber, the nucleation time is only half. Therefore there is increased heat dissipation for the 50~µm fiber, and thus a smaller efficiency. Therefore, the growth rate no longer increases linearly with the delivered energy, and larger growth rates require much more energy for the 50~µm fiber. \n\nInterestingly, extrapolating the linear fit of the 50~µm fiber in Figure~\\ref{fig: CW growth rates fibers}B shows a positive growth rate for zero energy, which is physically not possible. However, due to thermal dissipation, it is questionable whether the growth rate for the CW-laser would actually be linear with the energy, as thermal dissipation may reduce the energy efficiency. Therefore, even though the initial behaviour for both fibers seems linear, extrapolating this to find an energy threshold does not seem to have a physical meaning.\n\n\n\\begin{figure}[t!]\n \\centering\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/CW_growth_rate_incl_200-eps-converted-to.pdf}\n\n\\end{minipage}\n\\vline\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/CW_growth_rate_excl_200-eps-converted-to.pdf}\n\n\n\\end{minipage}\n \\caption{Average volumetric bubble growth rate for the CW laser. Figure (A) includes all three fiber sizes. As the energy and growth rates for the 200~µm fiber are much larger compared to the 50 and 105~µm fiber, figure (B) shows a close-up of the left bottom of (A), indicated by the dotted lines. Figure (B) includes the best linear fit for the 50 and 105~µm fiber.}\n \\label{fig: CW growth rates fibers}\n\\end{figure} \n\n\\subsubsection*{Secondary bubbles}\nFor small bubbles generated by the CW laser (R~$<$~150~µm), we observe the formation of a secondary bubble at the vapor-liquid interface, see~Figure~\\ref{fig: Secondary bubble}. These events are different from twin cavitation bubbles reported in Ref.~\\cite{Zhang2022}, where the initial bubble first partially collapses before forming a secondary bubble. In our case, the secondary bubble typically forms when the initial bubble reaches its maximum. Furthermore, in our case the volume of the secondary bubble is smaller compared to the first one, for which reason its appearance does not affect the parabolic shape of the bubble volume versus time.\nThese secondary bubbles can be explained by the fact that the laser remains irradiating during the bubble formation and, therefore, further heats the liquid during the bubble growth. In the case of large laser powers and short nucleation times, this heating during the bubble lifetime could create a secondary bubble. In the case of Figure~\\ref{fig: Secondary bubble}, the growth time of the bubble takes 40~µs, whereas the initial nucleation time was 135~µs, which means that after nucleation, another 30\\% of the initial energy is still delivered, which results in secondary nucleation at the vapor-liquid interface of the bubble. \n\nThe occurrence of these secondary bubbles depends on the laser power and nucleation time of the initial bubble. The secondary bubble forms during the lifetime of the bubble (typically 50-200~µs), during which enough additional energy has to be delivered to form a secondary bubble. We find that these secondary bubbles only appear when the intensities are sufficiently larger such that the initial nucleation time is smaller than approximately 450~µs, both for the 50 and 105~µm fiber. For the 200~µm fiber the intensities are smaller and thus the nucleation times are much larger we did not observe any secondary bubbles. However, we hypothesize that the use of a more powerful laser can also form these secondary bubbles with the 200~µm fiber. However, as mentioned, these secondary bubbles only happen for small nucleation times and large powers, which result in small bubbles overall. Therefore, we conclude that the larger and faster-growing primary bubbles are of more interest. \n\n\n\n\\begin{figure}[t!]\n\\centering\n\\begin{minipage}{.45\\textwidth}\n \\centering\n \\hspace{-1cm}\n \n \\includegraphics[width=1.05\\linewidth]{Figures/Secondary_Bubble_50_5_3-eps-converted-to.pdf}\n\n\\end{minipage}\n\\vline\n\\begin{minipage}{.45\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/Secondary_bubble_volume_curve_50_5_3-eps-converted-to.pdf}\n \n\\end{minipage}\n \\caption{\\textbf{Left:} 16 consecutive frames showing the typical dynamics of a small bubble generated by the CW laser. As the laser is still running during the bubble growth, more liquid will be heated, which results in a secondary (frame 7-12) and even a third bubble (frame 13-14). Snapshots are consecutive, from top to bottom and left to right, with an inter-frame time of 5~ms. The initial nucleation time (top left frame) was 135~µs, and the bubble reached its maximum size (frame 8) at 175~µs. Scalebar on the right bottom measures 100~µm. \\textbf{Right:} Calculated bubble volume for each frame, showing that the initial maximum is reached at frame 6, but due to secondary bubble formation, another maximum is reached at frame 8, although the difference is small.}\n \\label{fig: Secondary bubble}\n\\end{figure}\n\n\\subsection{Comparison between pulsed and CW}\n\nA direct comparison between the bubbles generated by the pulsed and CW laser requires a matching absorption coefficient such that the volumes over which the energy is absorbed are identical. Therefore, we compare the results of the 10~mM ARAC solution, as its absorption coefficient at 532~nm is nearly identical compared to the absorption coefficient of water at 1950~nm. In this subsection, we compare the bubbles generated by the 50 and 105~µm fiber, as they show similar results in the growth rate (1~-~$7*10^{-7} \\textnormal{m}^3/s$). The 200~µm fiber shows very different results for the CW laser, as the delivered energies are much larger (see Figure \\ref{fig: CW growth rates fibers}). \n\n\\subsubsection*{Energy efficiency}\nBudgeting of energy has practical implications on the right choice of laser source.\nThe growth rates for the CW and pulsed laser are shown in Figure~\\ref{fig: GR_pulsed_CW_50_105}, for the 50 and 105~µm fiber. For the same delivered optical energy, the pulsed laser results in faster growth rates, although the required energies are in the same order of magnitude. Typically, the CW laser requires two to three times more optical energy than the pulsed laser to generate a bubble with the same growth rate. The reduced efficiency is explained by three reasons. First of all, for the bubbles generated by the CW laser heat diffusion cannot be neglected, as the nucleation time is 0.1~-~5~ms, which is in the same order as the thermal diffusion timescale ($\\sim$~4~ms, see Equation~\\ref{eq: thermal diffusion}). Most of this heat will be lost, as it will heat up the liquid around the irradiated volume. Though, this heat increase in the proximate area would be limited, and most of it does not result in the phase transition. Second, the absorption coefficient of water around 1950~nm decreases with increasing temperature, and the absorption coefficient at 100$\\degree$C is only half the initial value at room temperature~\\cite{Jansen1994,Lange2002}. Therefore, for the heating phase with the CW laser, the average absorption length is much larger compared to the absorption length of the pulsed laser experiments.\nThird, for the pulsed laser, there might be non-linear absorption due to the high peak power. The measured absorption coefficient of the 10~mM ARAC is the same as the absorption coefficient of water for low optical intensities, but the absorption coefficient for the high-power pulses may be larger due to additional non-linear absorption by the liquid. This increased absorption would result in a smaller irradiated volume and thus a higher energy density, which will increase the bubble growth rate.\n\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{Figures/Growth_rate_pulsed_CW_50_105-eps-converted-to.pdf}\n \\captionsetup{width=0.5\\linewidth}\n \\caption{Growth rate of bubbles generated by the pulsed (red) and CW laser (blue), for fibers with core diameters of 50 (dark colors) and 105~µm (light colors). The absorption coefficients for both laser are the same ($\\alpha$~$\\approx$~12000~m$^{-1}$).}\n \\label{fig: GR_pulsed_CW_50_105}\n\\end{figure} \n\n\\subsubsection*{Bubble dynamics}\nBubble dynamics govern the subsequent microfluidic jet parameters.\nAs shown above, the typical growth rates are similar for both lasers. However, identical average growth rates may not result in the exact same bubble characteristics. An increase in growth time, results in a larger bubble, assuming the average growth rate is the same. The maximum volume against the average growth rate is shown in Figure~\\ref{fig: GR_MV_pulsed_CW} for the bubbles generated by the pulsed (red) and CW (blue) laser. For identical average growth rates, the bubbles created by CW laser grow to a $\\sim 15\\%$ larger volume (due to the increased growth time). Alternatively, for bubbles with the same maximum volume, the bubbles created by the CW laser take longer to reach that volume. This reduced growth rate could be explained by the larger volume over which the delivered energy is dissipated, resulting in a less explosive phase transition. However, this also means that for the same average growth rate, the bubbles generated by the CW laser will push out the remaining liquid over a longer length and time. This increased time of energy transfer will mostly affect the jet tail, which is typically slower and more dispersed~\\cite{Krizek2020,AndrewUnpublished}. We hypothesize that this longer growth time results in a faster and more reproducible jet tail.\n\nA comparison between the growth and collapse can be made by normalizing the volume and time, as shown in Figure~\\ref{fig: Volume_time_normalized}. For each individual bubble, the volume is normalized by its maximum volume and the time by the growth time. The lines show an average value over all bubbles, and the shaded region is the standard deviation. This figure shows that the initial normalized growth rate of the bubbles generated by the pulsed laser is larger compared to bubbles generated by the CW laser, as indicated by the steeper curve for small times. \nFurthermore, it shows that for the pulsed laser, the collapse is on average $\\sim 20\\%$ slower compared to the growth, whereas for the CW they take equal time. Both of these findings are in agreement with previous studies on the bubble dynamics for pulsed laser~\\cite{Zwaan2007,Sun2009} and CW~\\cite{OyarteGalvez2020}. However, this difference is mainly caused by the $\\sim 15\\%$ smaller growth time for the pulsed laser, see Figure~\\ref{fig: GR_MV_pulsed_CW}. If both curves were normalized by the same time, the time of collapse would only vary by $<5\\%$.\n\n\\begin{figure}\n\\centering\n\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Figures/Growth_rate_Max_size_shade-eps-converted-to.pdf}\n \\captionsetup{width=0.9\\linewidth}\n \\caption{Maximum bubble volume vs average growth rate for individual bubbles generated by the pulsed laser (red) and the CW laser (blue). The shaded area encapsulates all data.}\n \\label{fig: GR_MV_pulsed_CW}\n\\end{minipage}\n\\vline\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Figures/Growth_collapse_norm_pulsed_CW-eps-converted-to.pdf}\n \\captionsetup{width=0.9\\linewidth}\n \\caption{Normalized bubble size vs normalized time, for CW (blue) and pulsed (red). Lines indicate averages over all bubbles, shaded regions indicate standard deviation. For each individual bubble, the volume is normalized by the maximum volume, and the time is normalized by the growth time.}\n \\label{fig: Volume_time_normalized}\n\\end{minipage}\n\\end{figure}\n\n\\subsubsection*{Reproducibility}\nReproducibility is an important factor considering application in health care. The error bars in Figure~\\ref{fig: GR_pulsed_CW_50_105} show that the deviation in the bubble growth rate is larger for the CW laser (blue colors) compared to the pulsed laser (red colors). This indicates that even with identical initial conditions for the CW laser, the bubble dynamics can vary each time slightly. As discussed in section~\\ref{sec: CW laser}, the energy delivered by the CW laser is not controlled directly but depends on the laser power and nucleation time. As nucleation is a stochastic event, there is a variance in the nucleation time, resulting in less or more delivered energy for early or late nucleation, respectively. \n\nFigure~\\ref{fig: CW growth individual} shows the growth rate of all individual bubbles generated by the CW laser together with the mean and standard deviation for a group of 6 or more with identical initial conditions (laser power). Each color indicates a different laser power, which influences the delivered energy. However, even for measurements with identical laser power, the delivered energy can vary up to $\\pm~50$~µJ. For each of these laser powers, the individual results are typically in the bottom-left or top-right quadrant of the error bars. This indicates that there is a correlation between the additional energy for delayed nucleation and the bubble growth rate. Furthermore, per laser power, the individual data points are fitted with a linear fit, which is shown by the colored lines. The slope of these colored lines is very similar to the slope of the black line, which is the best fit when comprising all data. Therefore, the variance in nucleation time directly affects the bubble growth rate, which means that delayed nucleation results in more energy and a faster-growing bubble, and early nucleation results in less energy and a slower-growing bubble. However, as nucleation is a stochastic process, this results in a random deviation and reduces reproducibility.\n\nFor the pulsed laser, the standard deviation in delivered energy is only affected by the laser specifications. On average, it is much smaller with 2\\% of the delivered energy, compared to 8\\% for CW. As this deviation also directly affects the bubble growth rate, the bubble growth rate for the pulsed laser is thus more reproducible. \n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=1\\textwidth]{Figures/CW_growth_individual-eps-converted-to.pdf}\n \\captionsetup{width=1\\linewidth}\n \\caption{Growth rate for individual bubble events for the 105~µm fiber with the CW laser. Each color indicates a different laser power. The delivered energy mainly depends on the laser power but also on the moment of nucleation (e.g. late nucleation results in larger delivered energy). The circles are individual measurements, and the error bars indicate the mean and standard deviation of those individual measurements with the same laser power (at least 6). The colored lines indicate the best linear fit per laser power. The black line indicates the best linear fit for all data points. The slope of the colored lines is similar to the black line, indicating a correlation between the time of nucleation and bubble growth rate.}\n \\label{fig: CW growth individual}\n\\end{figure} \n\n\\subsection{Practical considerations of laser choice for applications}\nEngineering a medical device based on laser-induced cavitation is limited by the available offer of laser sources on the market. Yet, choosing the right laser emitter is vital for the success of any potential product or prototype, where not only the best physical and optical parameters shall be considered. Other aspects like cost, the robustness of operation, size or compliance with safety hazards play a key role. Intuitively, the cheaper, smaller and safer source is favourable, but these are usually contradictory parameters. The results of this study bring a better understanding of what kind of laser parameters are critical to generate fast traveling liquid droplets of relevance for its use in eventual devices using laser-induced cavitation, e.g. needle-free jet injectors. \nAlthough an in-depth techno-economical discussion on the selection of laser sources for specific applications is beyond the scope of this study, we will briefly discuss the impact of the different laser and device parameters here. The main differences are shown in Table \\ref{Table: CW vs Pulsed}.\n\nFor applications where reproducibility and reliability of the bubble dynamics are most important, the pulsed laser is preferred over the CW laser. Due to the reduced control over the delivered energy by the CW laser, there is a larger deviation in individual bubble growth rates. In the case of a jet injection device, this affects the jet velocity and thus injection depth. However, the exact influence of the deviation in bubble dynamics on the injection depth should be investigated in a further study.\n\n\\begin{table}[t]\n\\caption{Overview of the difference of the laser and bubble characteristics of the CW and pulsed laser.}\n\\label{Table: CW vs Pulsed}\n\\begin{tabular}{|l|l|l|}\n\\hline\n & \\textbf{Pulsed laser} & \\textbf{CW laser} \\\\ \\hline\n\\textbf{Timescale} & 5~ns & 1~ms \\\\ \\hline\n\\textbf{Peak power} & 10$^4$~W & 1~W \\\\ \\hline\n\\textbf{Typical price$^*$} & $\\sim$~1000~-~50000~\\$ & $\\sim$~100~-~1000~\\$ \\\\ \\hline\n\\textbf{Typical size} & Table top device & Handheld device \\\\ \\hline\n\\textbf{Control over delivered energy} & Direct (input of laser) & Indirect (depends on P \\& t$_\\textnormal{n}$) \\\\ \\hline\n\\textbf{Reproducibility (variation)} & 2\\% & 8\\% \\\\ \\hline\n\\textbf{Ratio growth to collapse time} & 1:1.2 & 1:1 \\\\ \\hline\n\\textbf{\\begin{tabular}[c]{@{}l@{}}Influence of increase in fiber \\\\ size (laser input parameters \\\\ constant)\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Reduction in energy density\\\\ Increase in energy threshold\\\\ Smaller/slower growing bubbles\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Delayed nucleation\\\\ Increase in delivered energy\\\\ Larger/faster growing bubbles\\end{tabular} \\\\ \\hline\n\\textbf{\\begin{tabular}[c]{@{}l@{}}Influence of increase in \\\\ absorption coefficient/\\\\ dye concentration\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Larger energy density\\\\ Decrease in energy threshold\\\\ Increase of bubble size\\end{tabular} & Not examined in this study$^{**}$. \\\\ \\hline\n\\end{tabular}\n$^{*}$Laser prices are highly dependent on wavelength and exact characteristics. Given prices are just an indication for typical consumer price of a single device. \\\\\n$^{**}$Refs. \\cite{AfanadorDelgado2019,Zhang2022} show that an increase in dye concentration results in shorter nucleation times and smaller bubbles.\n\\end{table}\n\nOn the other hand, if the price and/or size of the device is of more importance, a CW laser would be preferred over a pulsed laser. Due to their lower power, they are smaller and more affordable. Although, portable pulsed lasers with moderate pulse energies, as the ones used in our studies, have become more widely available over the past decade. \n\nFurthermore, the use of optical fiber allows for a small handheld device, even in the case of a large laser source. The laser source, including electronics, can then be placed in a bench-top part. \nThe choice of optical fiber has a large influence on bubble dynamics. For the pulsed laser, a smaller fiber reduces the required pulse energy. However, the smallest fiber (50~µm diameter) is limited in the range of bubble growth rate due to laser-induced damage of the fiber tip. For the CW laser, an increase in fiber size results in faster-growing bubbles, although it requires a more powerful laser (P~$>$~2W) to create a larger range of growth rates. \n\nChanging the dye concentration has a large influence on the bubble dynamics. For the pulsed laser, increase of dye concentration and thus absorption coefficient results reduces the required pulse energies, as discussion in Section~\\ref{sec: pulsed laser}. This allows for the use of smaller and more affordable pulsed lasers. Simultaneously, a single device could create a large range of bubbles by changing the absorption coefficient of the liquid. The influence of the absorption coefficient on the bubbles generated by the CW laser have not been investigated in this study. However, previous studies showed an influence on the nucleation time and bubble size~\\cite{AfanadorDelgado2019,Zhang2022}. Moreover, adding the dye into drug formulation might bear further toxicity and regulatory issues. Future studies should focus on the exact influence of absorption coefficient on bubble dynamics.\n\n\\section{Conclusion}\\label{sec: conclusion}\n\nWe compared the dynamics of bubbles generated by two lasers with different timescales (ns and ms) inside an open-ended capillary. Our comparative study is the first to show the resulting bubble dynamics in the same fluidic confinement with two different laser types. We have shown that these lasers create bubbles of comparable growth rates proportional to the delivered energy. This linear increase is in agreement with previously found linear increase between the energy and the jet velocity. \n\nFor the pulsed laser set-up, we found that the energy threshold for nucleation is proportional to the irradiated volume. This volume depends on the fiber core radius, beam divergence and absorption coefficient. Therefore, a decrease in irradiated volume, either by changing the fiber or increasing the absorption coefficient, resulting in a reduction of the energy threshold. This allows for the use of more affordable and/or less powerful pulsed laser sources while creating the same bubble. \n\nFor the continuous-wave laser, we found an efficiency increase and better reproducibility compared to previous experiments with free-space optics. This is explained by a reduction in the number of interfaces and easier alignment. Furthermore, we show that the delivered energy can be controlled by the laser power and the fiber size. For each fiber, a decrease in laser power results in an increase in delivered energy, allowing the creation of faster-growing bubbles. Furthermore, a larger fiber results in an increase in delivered energy for fixed laser power due to a longer nucleation time. Therefore, the 200~µm fiber results in much larger and faster-growing bubbles than the 50 or 105~µm fiber, when the same laser power is used. \n\nA comparison between the two laser sources shows that the pulsed laser requires slightly less optical energy to create the same bubble growth rate, which we attribute to heat dissipation and a reduction in absorption coefficient during the CW laser heating. However, for the same average growth rate, bubbles generated by the CW laser grow for a longer time and are larger. \n\nFinally, we made a comparison between both methods in terms of practical usage. Since the delivered energy by the CW laser cannot be controlled directly, there is a larger deviation in the delivered energy compared to the pulsed lasers (8\\% and 2\\% respectively), which also results in a larger deviation in bubble growth rates. This would be unfavourable for laser-based jet injection, as it decreases the control over jet velocity and, therefore, injection depth. However, if this variation is within the allowable error industry standards, then the CW laser could be advantageous due to its smaller size and lower price compared to pulsed lasers. \n\n\n\\section*{Acknowledgements}\nJ.J.S. would like to thank dr. Dani\\\"el J\\'auregui-V\\'azquez and dr. Jose Alvarez Chavez for their help with the optical set-up of the CW laser. J.J.S and D.F.R. acknowledge the funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 851630), and NWO Take-off phase 1 program funded by the Ministry of Education, Culture and Science of the Government of the Netherlands (No. 18844). J.K. and Ch.M. acknowledge the Innosuisse BRIDGE Proof of Concept grant funding. The authors are thankful for the insightful discussions with Dr. M. A. Quetzeri Santiago, D.L. van der Ven, K. Mohan and D. de Boer.\n\n\\section*{Competing interest}\nThe authors declare that they have no known competing financial\ninterests or personal relationships that could have appeared to influence the work reported in this paper.\n\n\\section*{CRediT authorship contribution statement}\n\\textbf{Jelle Schoppink:} Conceptualization, Methodology, Formal analysis, Investigation, Data Curation, Writing - Original Draft, Visualization\n\\textbf{Jan Krizek:} Conceptualization, Methodology, Writing - Review \\& Editing\n\\textbf{Christophe Moser:} Conceptualization, Funding acquisition, Writing - Review \\& Editing\n\\textbf{David Fernandez Rivas:} Conceptualization, Supervision, Project administration, Funding acquisition, Writing - Review \\& Editing.\n\n\\printbibliography \n\n\\begin{figure}[b!]\n \\centering\n \\includegraphics[width=0.7\\textwidth]{Figures/TOC_Only_Graphical_abstract.pdf}\n \\captionsetup{width=1\\linewidth}\n \\caption{For Table of Contents Only}\n \\label{fig: Graphical abstract}\n\\end{figure} \n\n\\end{document}\n\\section*{Appendix A: Allura Red AC absorption coefficient}\\label{sec: Appendix}\nThe measured absorption coefficients for the Allura Red AC dye (ARAC, 80\\%, Sigma Aldrich) are shown in Figure~\\ref{fig: alpha_vs_concentration}. The values are calculated from the transmission of the 532~nm laser through a rectangular capillary with a 100~µm path length. The values are compared with reported values in literature~\\cite{Bevziuk2017,Garcia2017}, which closely matches our measured values for concentrations up to 5~mM. For larger concentrations, we note that the absorption coefficient does no longer increase linearly with the dye concentration.\n\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=.7\\linewidth]{Figures/Absorption_coefficient_concentration-eps-converted-to.pdf}\n \\caption{Measured absorption coefficients (blue symbols) for Allura Red AC (ARAC). The values are compared with an extrapolated value from literature (red curve)~\\cite{Bevziuk2017,Garcia2017}. It can clearly be observed that the absorption coefficient does not behave linearly at these concentrations.}\n \\label{fig: alpha_vs_concentration}\n\\end{figure}\n\\newpage\n\\section*{Appendix B: Calculation of irradiated volume}\\label{sec: Appendix B}\nAs the fiber core radii R$_\\textnormal{f}$ ($25\\textnormal{~µm}<\\textnormal{R}_\\textnormal{f}<100\\textnormal{~µm}$) are of similar size of the absorption length d ($52\\textnormal{~µm}<\\textnormal{d}<131\\textnormal{~µm}$), the beam divergence $\\beta$ cannot be neglected when calculating the irradiated volume. The beam divergence is calculated by the NA of the fiber and the refractive index of the liquid $n$ (see Figure \\ref{fig: BeamRadDiv})\n\\begin{equation*}\n \\beta = \\arcsin{\\frac{\\textnormal{NA}}{n}} = \\arcsin{\\frac{\\textnormal{0.22}}{1.33}} \\approx 9.5\\degree.\n\\end{equation*}\nThen, the beam radius R along the propagation axis $x$ is calculated as\n\\begin{equation*}\n \\textnormal{R}(x) = \\textnormal{R}_\\textnormal{f}+x\\tan{\\beta}.\n\\end{equation*}\n\nThe irradiated volume $V$ can be calculated by integrating the beam area ($\\pi$R$(x)^2$) from the fiber tip ($x=0$) to the absorption length ($x=\\textnormal{d}$)\n\n\\begin{align*}\n V = \\pi\\int_{0}^{\\textnormal{d}}\\textnormal{R}^2 dx = \\pi\\int_{0}^{\\textnormal{d}}(\\textnormal{R}_\\textnormal{f}+x\\tan{\\beta})^2 dx = \\pi\\int_{0}^{\\textnormal{d}}(\\textnormal{R}_\\textnormal{f}^2+2\\textnormal{R}_\\textnormal{f}x\\tan{\\beta}+x^2\\tan^2{\\beta})dx \\\\\n = \\pi |\\textnormal{R}_\\textnormal{f}^2x + \\textnormal{R}_\\textnormal{f}x^2\\tan{\\beta}+\\frac{1}{3}x^3\\tan^2{\\beta}|_0^\\textnormal{d} = \\pi (\\textnormal{R}_\\textnormal{f}^2\\textnormal{d} + \\textnormal{R}_\\textnormal{f}\\textnormal{d}^2\\tan{\\beta}+\\frac{1}{3}\\textnormal{d}^3\\tan^2{\\beta)}\n\\end{align*}\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width = \\linewidth]{Figures/BeamRadiusDivergence-eps-converted-to.pdf}\n \\caption{Schematic indicating the irradiated volume (green) of the liquid (red) near the fiber tip. Beam radius increases along the propagation axis due to the divergence. At the fiber tip, the radius is equal to the core radius. The irradiated volume is calculated by integrating the beam area from the fiber tip to the absorption length \\textnormal{d}.}\n \\label{fig: BeamRadDiv}\n\\end{figure}\n\\newpage\n\\section*{Appendix C: CW bubble growth}\\label{sec: Appendix C}\nFigure~\\ref{Appendix fig: CW 105 and 200} shows a few snapshots of the bubbles generated by the CW laser at the 105 and 200~µm fiber. For the 200~µm fiber (right snapshots), the bubble has a much larger energy, due to the longer nucleation time. Therefore, the bubble grows more explosively and always grows beyond the capillary edge. Therefore, no maximum bubble size or average growth rate could be obtained for the 200~µm fiber.\n\n\\begin{figure}[ht]\n \\centering\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/Appendix_Bubble105-eps-converted-to.pdf}\n\n\\end{minipage} \n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/Appendix_Bubble200-eps-converted-to.pdf}\n\n\n\\end{minipage}\n \\caption{Snapshots of the bubble generated at a 105~µm (left) and 200~µm fiber (right) by the CW laser with a power of approximately 0.7~W. The bubbles generated at the 200~µm fiber have a much larger energy and grow beyond the capillary edge and expel all the liquid. Therefore, there is no maximum bubble size or average growth rate observed.}\n \\label{Appendix fig: CW 105 and 200}\n\\end{figure} \n\n\\printbibliography \n\\end{document}"}}},{"rowIdx":83598,"cells":{"uuid":{"kind":"number","value":1116691499413,"string":"1,116,691,499,413"},"dataset":{"kind":"string","value":"arxiv"},"text":{"kind":"string","value":"\\section{Introduction} \\label{sec:Introduction}\n\nOnly the most violent objects in the universe are capable of producing \nhigh-energy particles in non-thermal acceleration processes and emit \nphotons with energies in the X-ray band and above. Spectral and \nmorphological studies in the X-ray and gamma-ray bands have become \nestablished tools to study the non-thermal emission processes of various \nastrophysical sources \\cite{Frederick2010}. However, many of the regions \nof interest (black hole vicinities, formation zones of relativistic \njets, etc.) are too small to be spatially resolved with current and \nfuture instruments. Spectro-polarimetric X-ray observations are capable \nof providing additional information~-- namely (i) the energy-resolved \nfraction of linear polarization (e.g. what fraction of emission is \npolarized), and (ii) the projected orientation of the polarization plane \n(defined by the electric field vector of the photon) with respect to the \nemitting source. Various emission mechanisms of compact objects lead to \ncomparable spectral signatures, but would differ in the polarization \ncharacteristics. The measurements of polarization properties would \ntherefore help to constrain the geometry of the inner regions of \nrelativistic plasma jets, mass-accreting black holes (BHs) and neutron \nstars \\cite{Lei1997, Krawcz2011}.\n\nSo far, only a few space-borne missions have successfully measured \npolarization in the X-ray regime. The Crab nebula is the only source for \nwhich X-ray polarization has been established with a high level of \nconfidence. In measurements with the OSO-8 satellite, the Crab exhibits \na polarization fraction of $20 \\%$ at energies of $2.6 - 5.2 \\, \n\\rm{keV}$ and a direction angle of $30^{\\circ}$ with respect to the \nX-ray jet observed in the nebula \\cite{Weisskopf1978}. At energies above \n$100 \\, \\rm{keV}$, measurements resulted in a polarization fraction of \n$46 \\% \\pm 10 \\%$ with the direction aligned with the jet \n\\cite{Dean2008, Forot2008}. A second astrophysical source emitting \npolarized X-rays was identified recently. {\\it INTEGRAL}\\xspace observations of the \nX-ray binary Cygnus\\,X-1 indicate a high fraction of polarization of \n$(67 \\pm 30)\\%$ in the $400 \\, \\rm{keV} - 2 \\, \\rm{MeV}$ band, whereas a \n$20\\%$ upper limit was derived for the $250-400 \\, \\rm{keV}$ band \n\\cite{Laurent2011}. Various authors have reported tentative evidence for \npolarized hard X-ray/soft gamma-ray emission from different gamma-ray \nbursts \\cite{Coburn2003, Kalemci2007, Yonetoku2011, Kostelecky2013}. \nHowever, all of these detections have somewhat marginal significance, \npossibly being impacted by unknown systematic effects in their \nrespective instrumentation.\n\nModel predictions of polarized emission for various source types lie \nslightly below the sensitivity of the past OSO-8 mission, making future, \nmore sensitive polarimetry missions particularly interesting. However, \nthere are currently no dedicated missions in orbit that are capable of \nmeasuring X-ray polarization fractions in the $<10\\%$ regime, a \nrequirement to study the corresponding emission mechanisms for a variety \nof astrophysical source classes (see Sec.~\\ref{sec:Science}). More \nrecently, various experiments have been proposed that could change the \nsituation as they combine broadband sensitivity with a high detection \nefficiency. The proposed polarimeters use focusing mirrors to collect \nphotons from the source. Photoelectric effect polarimeters, like the \n{\\it Gravity and Extreme Magnetism SMEX} (GEMS) mission \\cite{GEMS} and \nXIPE \\cite{XIPE}, track the direction of photo electrons ejected in \nphotoelectric effect interactions of the X-rays. The {\\it ASTRO-H} \nmission (to be launched in 2015) will carry the {\\it Soft Gamma-Ray \nImager}. The detector combines an X-ray and gamma-ray collimator with a \nSi scatterer and CdTe absorber. The mission will be able to do \nscattering polarimetry at $E > 50 \\, \\rm{keV}$ energies \n\\cite{Tajima2010}. Our group is working on a proposal of the space borne \nscattering polarimeter PolSTAR which uses a lower-Z LiH stick as a \nscatterer to enable the detection of $3 - 80 \\, \\rm{keV}$ X-rays. The \nPolSTAR concept will be described in a forthcoming paper. Missions like \nGEMS, PolSTAR and XIPE aim at obtaining $1 \\%$ minimum detectable \npolarization fractions, see Eq.~(\\ref{eqn:MDP}), for mCrab sources. \nThese missions would allow very high-signal-to-noise detections of \nbright galactic sources (e.g. Cyg\\,X-1, GRS\\,1915+105, Her\\,X-1) and \ncould detect $\\sim1 \\%$ polarization fractions for extra-galactic \nsources (e.g. NGC\\,4151).\n\nScattering polarimeters detect the direction into which photons scatter \nwhen interacting in the detector. Although different scattering \nprocesses dominate at different energies (coherent scattering below a \nfew keV, Thomson scattering at intermediate energies, and Compton \nscattering at energies $> 20 \\, \\rm{keV}$), all scattering processes \nshare the property that the photons scatter preferentially perpendicular \nto the polarization plane (electric field vector) of the incoming \nphoton. For example, the angular dependence of Compton scattering \nprocesses is given by the Klein-Nishina cross section \\cite{Evans1955}:\n\n\\begin{equation}\n\\label{eq:KleinNishinaCrossSection} \n\\frac{\\rm{d}\\sigma}{\\rm{d}\\Omega} = \\frac{r_{0}^{2}}{2} \n\\frac{k_{1}^{2}}{k_{0}^{2}} \\left[ \\frac{k_{0}}{k_{1}} + \n\\frac{k_{1}}{k_{0}} -2 \\sin^{2} \\theta \\cos^{2} \\eta \\right], \n\\end{equation}\nwhere $\\eta$ is the angle between the electric vector of the incident \nphoton and the scattering plane, $r_{0}$ is the classical electron \nradius, $\\bf{k}_{0}$ and $\\bf{k}_{1}$ are the wave-vectors before and \nafter scattering, and $\\theta$ is the scattering angle. The azimuthal \ndistribution of scattered events shows a sinusoidal modulation with a \n$180^{\\circ}$ periodicity and a maximum at $\\pm 90^{\\circ}$ to the \npreferred electric field direction of a polarized X-ray signal.\n\nIn this paper we describe the design and performance of a scattering \npolarimeter, X-Calibur. The polarimeter utilizes a plastic scintillator \nas scatterer which can scatter $E > 20 \\, \\rm{keV}$ X-rays efficiently. \nThis is ideal for the operation on a balloon where the energy threshold \nis well matched to the low energy cutoff of the transmissivity caused by \nthe residual atmosphere above the balloon altitude of 125,000 feet. An \nassembly of Cadmium Zinc Telluride (CZT) detectors surrounds the \nscintillator in order to record the azimuthal distribution of the \nscattered photons, allowing one to reconstruct the polarization \nproperties of the incoming X-ray beam. The background of charged \nparticles and high-energy photons is suppressed by an active CsI shield.\n\nThe paper is structured as follows. Section~\\ref{sec:Science} gives a \nbrief overview over the scientific potential of hard X-ray polarimetry. \nThe design of the X-Calibur polarimeter is described in \nSec.~\\ref{sec:XCLB}. The data analysis methods are described in \nSec.~\\ref{sec:Analysis}, followed by an outline of the simulation \nprocedure in Sec.~\\ref{sec:Simulations}. \nSection~\\ref{sec:DetectorCharacterization} describes the calibration and \ncharacterization studies of the CZT detectors. Measurements with the \nassembled X-Calibur polarimeter to characterize the scattering \nscintillator and the shield are described in \nSec.~\\ref{sec:XCLB_Characterization}. Polarization measurements with \nX-Calibur are described in Sec.~\\ref{sec:XCLB_PolarimetryMeasurements}. \nThe paper ends with a summary and outlook in Sec.~\\ref{sec:Conclusion}. \nThe X-Calibur data presented in this paper were taken (i) in a \nlaboratory environment at Washington University, (ii) at the Cornell \nHigh Energy Synchrotron Source (CHESS), and (iii) in Ft.~Sumner, NM, \nduring a preparation campaign for an upcoming balloon flight.\n\n\n\\section{Scientific Potential} \\label{sec:Science}\n\nThis section discusses the scientific potential for a scattering \npolarimeter such as X-Calibur from a balloon platform. X-rays from \ncosmic sources can be polarized owing to the anisotropy in the source \ngeometry and/or the emission characteristics of various processes \n\\cite{Lei1997}. Non-thermal emission, like synchrotron radiation, \nresults in a large polarization fraction $r$. Synchrotron emission will \nresult in linearly polarized photons with their electric fields oriented \nperpendicular to the magnetic field lines (projected); the observed \npolarization map can therefore be used to trace the magnetic field \nstructure of the source, a common practice in radio and optical \npolarimetry. An electron population with a spectral energy distribution \nof $\\rm{d}N/\\rm{d}E \\propto E^{-p}$ emitting in a uniform magnetic field \nwill lead to an observable fraction of polarization of \n\\cite{Korchakov1962}:\n\n\\begin{equation}\nr_{\\rm{sync}} = \\frac{p+1}{p+7/3}, \\quad \\rm{with} \\, (p + 3) = \n\\frac{\\alpha + 1} {\\alpha + 5/3}.\n\\end{equation}\nHere, $\\alpha$ is the index of the X-ray power law spectrum. An observed \npolarization fraction close to this limit can therefore be interpreted \nas an indication of a highly ordered magnetic field since \nnon-uniformities in the magnetic field will reduce $r_{\\rm{sync}}$. The \npolarized synchrotron photons can in turn be inverse-Compton scattered \nby relativistic electrons~-- weakening the fraction of polarization (but \nnot erasing it) and imprinting a scattering angle dependence to the \nobserved fraction of polarization \\cite{Krawczynski2012b}. Such \ninverse-Compton signals will usually (but not always) appear in hard \ngamma-rays, where polarimetry is difficult, due to multiple scattering \nin pair production detectors. Another important mechanism for polarizing \nphotons is Thomson scattering which creates a polarization perpendicular \nto the scattering plane \\cite{Rybicki1991}. Curvature radiation is \npolarized, as well. The scientific potentials of spectro-polarimetric \nobservations over the broadest possible energy range are summarized \nbelow; more detailed discussions can be found in \\citet{Krawcz2011} and \n\\citet{Lei1997}.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Schnittman_Fig3}\n\\end{center}\n\n\\caption{Ray-traced image of direct radiation from a thermal disk around \na BH including returning radiation (observer located at an inclination \nangle of $75^{\\circ}$, gas on the left side of the disk moving toward \nthe observer causing the characteristic increase in intensity due to \nrelativistic beaming). The image is adopted from \\citet{Schnittman2009}. \nThe observed intensity is color-coded on a logarithmic scale and the \nenergy-integrated polarization vectors are projected onto the image \nplane with lengths proportional to the fraction of polarization.}\n\n\\label{fig:BH_Polarization}\n\\end{figure}\n\n{\\it Binary black hole systems.} Particle scattering in a Newtonian \naccretion disk surrounding a BH will lead to the emission of polarized \nX-rays. Relativistic aberration and beaming, gravitational lensing, and \ngravito-magnetic frame-dragging will result in an energy-dependent \nfraction of polarization since photons with higher energies originate \ncloser to the BH than the lower-energy photons \\cite{Connors1977}. \n\\citet{Schnittman2009} calculate the expected polarization signature \nincluding (i) the effects of deflection of photons emitted in the disk \nby the strong gravitational forces in the regions surrounding the BH and \n(ii) re-scattering these photons by the accretion disk \n\\cite{Schnittman2009, Schnittman2010}. The resulting effect is a swing \nin the polarization direction from being horizontal at low energies to \nvertical at high energies, i.e., parallel to the spin axis of the BH. \nSpectro-polarimetric observations can therefore be used to constrain the \nmass and spin of the BH \\cite{Schnittman2009}, as well as the \ninclination of the inner accretion disk and the shape of the corona \n\\cite{Schnittman2010}, see Fig.~\\ref{fig:BH_Polarization}. In principle, \nX-ray polarization can also be used to test General Relativity in the \nstrong gravity regime \\cite{Krawczynski2012c}.\n\n{\\it Pulsars.} High-energy particles in pulsar magneto-spheres are \nexpected to emit synchrotron and/or curvature radiation which are \ndifficult to distinguish from one another, solely based on the observed \nphoton energy spectrum. However, since the orbital planes for \naccelerating charges that govern these two radiation processes are \northogonal to each other, their polarized emission will exhibit \ndifferent behavior in position angle and polarization fraction as \nfunctions of energy and the rotation phase of the pulsar \n\\cite{Dean2008}. An illustration of the models of phase dependence of \nthe X-ray/gamma-ray polarization signatures in pulsars can be found in \n\\citet{Dyks2004}. In magnetars, the highly-magnetized cousins of \npulsars, polarization-dependent resonant Compton up-scattering is a \nleading candidate for generating the observed hard X-ray tails \n\\cite{BaringHarding2007}. In both these classes, phase-dependent \nspectro-polarimetry can probe the emission mechanism, and provide \ninsights into the magnetospheric locale of the emission region.\n\nCyclotron lines arising from transitions between Landau levels in \nintense magnetic fields that occur in the polar regions of neutron stars \nand magnetars are polarized. Also, the absorption scattering \ncross-sections from the Landau levels are dependent on the polarization \nstate of the X-rays, so that the radiative transfer through such plasma \nwill lead to a polarization of the emergent radiation. The first such \ncyclotron feature was observed by \\citet{Truemper1978} in Her\\,X-1 and \nis interpreted as due to an absorption line around $40 \\, \\rm{keV}$ \n\\cite{Staubert2007}. Such features allow an estimate of the magnetic \nfield strength \\cite{Coburn2002}. Observations of polarized X-rays will \nstrongly confirm that these features are indeed due to cyclotron lines \nin magnetic fields of $3-5 \\times 10^{12} \\, \\rm{G}$. A detailed \ndiscussion of the theoretical aspects of cyclotron radiation is given in \n\\citet{Leahy2010}.\n\n{\\it Pulsar wind nebulae.} The compact object (e.g. pulsar) resulting \nfrom a previous supernova explosion can be surrounded by a synchrotron \nemitting nebula. The nebula extends far beyond the magneto-sphere of the \ncentral pulsar, but its emission is believed to be driven by the pulsar \nwhich injects relativistic electrons/positrons that are further shock \naccelerated in the nebula. Spectro-polarimetric observations can be used \nto constrain the magnetic field and particle populations in such pulsar \nwind nebulae~-- such as the Crab nebula, the leading driver for this \nfield of X-ray polarimetry. Given the more compact emission regions at \nhigh energies, these objects potentially show a higher polarization \nfraction at hard X-rays as compared to soft X-rays, reflecting the \ncontrast between jet and more diffuse nebular contributions \n\\cite{Forot2008}.\n\n{\\it Supernova remnants.} Supernova remnants (SNRs) present an \nopportunity to perform X-ray polarimetry, as well. The remnants possess \ntangled magnetic fields on large scales in their interiors, as is \nevidenced in the classic radio polarization map of the Crab nebula \n\\cite{Velusamy1985}. Both, radio and X-ray signals, are believed to be \ndue to synchrotron emission, and so it is reasonably presumed that X-ray \nemission from SNRs should be significantly polarized. The X-ray spectra \nof such remnants are typically steeper than spectra in the radio, which \nleads to the expectation of higher polarization fractions in the X-ray \nband. Yet, since the electrons generating X-rays will diffuse on larger \nspatial scales than their radio-emitting counterparts do, the X-ray \nsignals should capture the field morphology on larger scales. It may or \nmay not be more coherent than the field structure on smaller (radio) \nscales. Due to instrumental limitations in angular resolution at X-ray \nenergies, however, it will not be possible to resolve individual \nregions~-- depending on the angular size of the remnant. The \nobservational challenge will therefore be to overcome the competition \nbetween compact regions causing highly polarized emission on one hand, \nand on the other hand an averaging effect of different emission regions \nwith different orientations of their magnetic fields. An example of \nexpectations for the SNR synchrotron polarization properties can be \nfound in \\citet{Bykov2009}.\n\n{\\it Relativistic jets in active galactic nuclei.} Relativistic \nelectrons in jets of active galactic nuclei (AGN) emit polarized \nsynchrotron radiation at radio/optical wavelengths. The same electron \npopulation is believed to produce hard X-rays by inverse-Compton \nscattering off a photon field. Simultaneous measurements of the \npolarization angle and the fraction of polarization in the radio to hard \nX-ray band could help to disentangle the following scenarios: (i) If the \nelectrons mainly up-scatter the co-spatial synchrotron photon field \n(synchrotron self Compton), the polarization of the hard X-rays is \nexpected to track the polarization at radio/optical wavelengths \n\\cite{Poutanen1994}. The fraction of X-ray polarization could be close \nto the fraction of polarization of the synchrotron emission measured in \nthe radio/optical bands and the polarization directions between radio, \noptical and X-rays should be identical. (ii) If the electrons dominantly \nup-scatter an external photon field (external Compton, e.g. photons of \nthe cosmic microwave background or from the accretion disk surrounding \nthe super-massive black hole) the hard X-rays will have a relatively \nsmall ($<$10\\%) fraction of polarization \\cite{McNamara2009}. Hadronic \njet emission models for low-synchrotron-peaked AGN, on the other hand, \npredict an even higher fraction of polarization at high energies, \ncompared to the leptonic SSC models \\cite{Zhang2013}.\n\nPolarization also allows one to test the structure of the magnetic field \nof the jet. Particles accelerated in a helical field which are moving \nthrough a standing shock can cause an X-ray synchrotron flare with a \ncontinuous (in time) swing in polarization direction. Such an event was \nobserved from BL\\,Lacertae at optical wavelengths \\cite{Marscher2008}.\n\n{\\it Gamma-ray bursts.} Gamma-ray bursts are believed to be connected to \nhyper-nova explosions and the formation/launch of relativistic jets \n\\cite{Woosley1993}. As in the case of the jets in AGN, the structure of \ntheir jets and the particle distribution responsible for gamma-ray \nbursts can be revealed by X-ray polarization measurements \n\\cite{Kostelecky2013}. The X-ray emission of a gamma-ray burst, however, \nusually lasts for only a few minutes at most, so that rapid follow-up \nobservations in the X-ray band below $30 \\, \\rm{keV}$ would be the main \nchallenge for studying their polarization properties.\n\n\nOn a one-day balloon flight, we would achieve $5-15 \\%$ MDPs, see \nEq.~(\\ref{eqn:MDP}), for between 1 and 4 sources. For the Crab, the \nphase resolved polarimetry would allow us to decide between emission \nmodels. For Cyg\\,X-1 and GRS\\,1915+105 we could test corona models, and \nfor Her\\,X-1, we could get a first estimate of the polarization fraction \nof the X-ray emission.\n\n\n\\section{Design of X-Calibur} \\label{sec:XCLB}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{X-Calibur_InFocus}\n\\end{center}\n\n\\caption{Schematic view of the functionality of X-Calibur (not to \nscale). X-rays from an astrophysical source are focused with a grazing \nincidence mirror onto the scattering rod of the polarimeter. The \nscattered X-ray is recorded in one of the surrounding CZT detectors. The \npolarimeter is embedded by a shield to suppress background of particles \nnot originating from the mirror.}\n\n\\label{fig:XCLB_Functionality}\n\n\\end{figure*}\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[height=0.38\\textheight]{X-Calibur_Schematic}\n\\hfill\n\\includegraphics[height=0.38\\textheight]{P1080730_LabVersion4}\n\\hfill\n\\includegraphics[height=0.38\\textheight]{P1090499_X-Calibur}\n\\end{center}\n\n\\caption{Design of the X-Calibur polarimeter. {\\bf Left:} Schematic \nillustration. Side view: incoming X-rays Compton-scatter in the \nscintillator rod (aligned with the optical axis, read out by a PMT) and \nare subsequently photo-absorbed in one of the surrounding CZT detectors \n($8\\times8$ pixels each, see Fig.~\\ref{fig:DetectorRing}). A group of \nfour detectors surrounding the scintillator at a given depth are \nreferred to as ring ({\\it R1} to {\\it R8}). Each ring can be further \ndivided into top and bottom sub rings, as illustrated for ring {\\it R8}. \nTop view: Detector ring viewed along the optical axis, looking into the \nX-ray beam. The polarization signature is imprinted in the azimuthal \nscattering distribution (see Fig.~\\ref{fig:X-Calibur_Configurations}, \ntop, for more details). The four sides of detectors are referred to as \nboards ({\\it Bd0} to {\\it Bd3}), each board comprising all eight \ndetectors per side. 3D view (`exploded'): four sides of detector columns \n({\\it Bd0} to {\\it Bd3}) surround the central scintillator rod~-- \ncovering the whole range of azimuthal scattering angles. {\\bf Middle:} \nPartly assembled polarimeter with two detector sides removed for better \nvisibility. The red wires provide the high voltage to the detectors. The \nread-out electronics is stacked at the backside of the detectors. {\\bf \nRight:} Fully assembled polarimeter (flipped upside-down: X-ray entering \nfrom the bottom).}\n\n\\label{fig:Design}\n\n\\end{figure*}\n\n\\begin{table}[t!]\n\n\\begin{tabular}{rrrrr}\n\nID & Serial-No. & Date & $C_{\\rm{ch}}$ & $C_{\\rm{wu/ft}}$ \\\\\n\\hline \\hline\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf Endicott 5\\,mm}} \\\\\n\\hline\n\nEN$_{5}$1 & 672992-01 & 01/11 & 0/5 & 0/5 \\\\\nEN$_{5}$2 & 672992-02 & 01/11 & 1/5 & 1/5 \\\\\nEN$_{5}$3 & 672992-03 & 01/11 & 2/5 & 2/5 \\\\\nEN$_{5}$4 & 672992-04 & 01/11 & 3/5 & 3/5 \\\\\nEN$_{5}$5 & 672994-04 & 02/11 & 0/4 & 0/4 \\\\\nEN$_{5}$6 & 672994-03 & 02/11 & 1/4 & 1/4 \\\\\nEN$_{5}$7 & 672994-02 & 02/11 & 2/4 & 2/4 \\\\\nEN$_{5}$8 & 672994-01 & 02/11 & 3/4 & --- \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf Creative Electron 5\\,mm}} \\\\\n\\hline\n\nCE$_{5}$1 & 721613 & 03/11 \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf QuikPak 5\\,mm}} \\\\\n\\hline\n\nQP$_{5}$1 & 3627 & 03/11 & --- & 3/4 \\\\\nQP$_{5}$3 & 3611 & 03/11 & --- & --- \\\\\nQP$_{5}$4 & 3834 & 03/11 & 3/1 & 3/1 \\\\\nQP$_{5}$6 & 6292 & 03/11 & 0/1 & 0/1 \\\\\nQP$_{5}$7 & 6345 & 03/11 & 2/1 & 2/1 \\\\\nQP$_{5}$8 & 721c602 & 03/11 & 1/1 & 1/1 \\\\\n\n\nQP$_{5}$13 & 6886 & 04/12 & 0/3 & 0/3 \\\\\nQP$_{5}$14 & 6977 & 04/12 & 2/3 & 2/3 \\\\\nQP$_{5}$16 & 10814 & 04/12 & 2/2 & 2/2 \\\\\nQP$_{5}$17 & 10819 & 04/12 & 0/2 & 0/2 \\\\\nQP$_{5}$18 & 10829 & 04/12 & 1/2 & 1/2 \\\\\nQP$_{5}$19 & 10847 & 04/12 & 3/3 & 3/3 \\\\\nQP$_{5}$20 & 10848 & 04/12 & 3/2 & 3/2 \\\\\nQP$_{5}$21 & 10860 & 04/12 & 1/3 & 1/3 \\\\\n\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf Endicott 2\\,mm}} \\\\\n\\hline\n\nEN$_{2}$1 & 674326-01 & 02/11 & 0/6 & 0/8 \\\\\nEN$_{2}$2 & 674327-01 & 02/11 & 2/8 & 2/8 \\\\\nEN$_{2}$3 & 674328-01 & 02/11 & 2/6 & 2/6 \\\\\nEN$_{2}$4 & 674328-02 & 02/11 & 0/8 & 0/6 \\\\\nEN$_{2}$5 & 674329-01 & 02/11 & --- & --- \\\\\nEN$_{2}$6 & 674330-01 & 02/11 & 1/6 & 1/6 \\\\\nEN$_{2}$7 & 674331-01 & 02/11 & 3/6 & --- \\\\\nEN$_{2}$8 & 674332-01 & 02/11 & --- & 3/8 \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf Creative Electron 2\\,mm}} \\\\\n\\hline\n\nCE$_{2}$1 & 2180 & 03/11 & 0/7 & 0/7 \\\\\nCE$_{2}$2 & 720612 & 03/12 & 3/8 & 3/6 \\\\\nCE$_{2}$3 & 720511 & 03/12 & 1/8 & 1/8 \\\\\nCE$_{2}$4 & 721541i & 03/12 & 1/7 & 1/7 \\\\\nCE$_{2}$5 & 06172A & 03/12 & 2/7 & 2/7 \\\\\nCE$_{2}$6 & 726712 & 03/12 & 3/7 & 3/7 \\\\\n\n\\end{tabular}\n\n\\caption{CZT detector sample. ID as in this paper (two letters for the \ncompany, index for the detector thickness $[\\rm{mm}]$, and a number), \nserial number, date of delivery [MM/YY], and position in polarimeter \n[Bd/ring] for the configurations $C_{\\rm{ch}}$ and $C_{\\rm{wu/ft}}$ (see \nFig.~\\ref{fig:X-Calibur_Configurations}).}\n\n\\label{tab:Detectors}\n\n\\end{table}\n\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[height=0.255\\textheight]{DetectorGold_Rotated}\n\\hfill\n\\includegraphics[height=0.255\\textheight]{Detector_BondedAndASICs}\n\\hfill\n\\includegraphics[height=0.255\\textheight]{p1060223_zoom}\n\\end{center}\n\n\\caption{Definition of a detector `ring'. {\\bf Left:} $2 \\times 2 \\times \n0.2 \\, \\rm{cm}^{3}$ CZT detector with 64 pixels (anode side). {\\bf \nMiddle:} $2 \\times 2 \\times 0.2 \\, \\rm{cm}^{3}$ CZT detector bonded to a \nceramic chip carrier which is plugged into 2 ASIC readout boards. The \nhigh-voltage cable is glued to the detector cathode (red wire). {\\bf \nRight:} Four CZT detectors surrounding the scintillator (blueish glow) \nform a unit referred to as `ring'~-- covering the whole $360^{\\circ}$ \nazimuthal scattering range with $4 \\times 8 = 32$ pixels. The whole \npolarimeter is equipped with eight rings.}\n\n\\label{fig:DetectorRing}\n\\end{figure*}\n\nThe balloon-borne version of X-Calibur is a low-Z Compton scattering \npolarimeter that will allow one to measure polarization fractions in the \n$20 - 80 \\, \\rm{keV}$ band down to the percentage level. X-Calibur will \nbe used in the focal plane of the X-ray mirror of the {\\it InFOC$\\mu$S}\\xspace telescope \nwith a field-of-view of $10 \\, \\rm{arc min}$; X-Calibur does not provide \nimaging capabilities. Owing to the fact that a grazing incidence mirror \nreflects only under very shallow angles, it changes the polarization \nproperties of X-rays by less than $1\\%$ \\cite{Katsuta2009}. The \nadvantages of the X-Calibur design can be described as follows. (i) A \nhigh detection efficiency is achieved, using roughly $80 \\%$ of photons \nimpinging on the polarimeter. (ii) The use of a focusing optics instead \nof a large detector volume results in a compact instrument design that \ncan be shielded efficiently~-- strongly reducing the background level. \n(iii) The continuous rotation of the polarimeter strongly reduces \npossible systematic effects that can hamper non-rotating polarimeters \ndue to asymmetric azimuthal detector responses. These characteristics, \nas outlined in the following sections, make it a well-suited experiment \nto study several sources mentioned in Sec.~\\ref{sec:Science} in a \none-day balloon flight. The energy-dependent detection efficiency of the \npolarimeter depends on (i) the effective area and point-spread function \nof the X-ray mirror, (ii) the fraction of scatterings compared to \ncompeting interactions such as photo-absorption, and (iii) the \ngeometrical detector coverage to record a high fraction of scattered \nX-rays (minimization of possible escape paths) \\cite{Guo2010}. This \nsection describes the overall design of the polarimeter, as well as the \ncharacteristics of its individual components.\n\n\n\n\\subsection{Design}\n\nThe conceptual design of the X-Calibur polarimeter is illustrated in \nFigures~\\ref{fig:XCLB_Functionality} and \\ref{fig:Design}. A low-Z \nscintillator rod aligned with the optical axis of the telescope is used \nas Compton-scatterer~-- leading to a polarization-dependent azimuthal \nscattering distribution that is recorded by the surrounding assembly of \nCZT detectors. The azimuthal distribution is resolved by $4\\times8 = 32$ \npixels for each of the $64$ depth bins along the optical axis. Detailed \ninformation on the simulations to optimize the X-Calibur design (as \npresented in this paper) can be found in \\citet{Krawcz2011} and \n\\citet{Guo2013}.\n\nThroughout the paper, we refer to a detector ring as a set of four CZT \ndetectors surrounding the scintillator rod on four sides at a given \ndepth along the optical axis (see Fig.~\\ref{fig:DetectorRing}, right). \nRing {\\it R1} is situated at the polarimeter entrance (top in \nFig.~\\ref{fig:Design}, left), and ring {\\it R8} is situated at its rear \nend. Each ring covers the whole $360^{\\circ}$ azimuthal scattering \nrange. A ring can further be subdivided into sub rings, eg. {\\it \nR8$_{\\rm{t}}$} and {\\it R8$_{\\rm{b}}$} for the top and bottom half, \nrespectively (see Fig.~\\ref{fig:Design}, left). The smallest possible \nsubdivision is a ring consisting of only one pixel row, referred to as \n{\\it single-pixel ring}\\xspace. All eight detectors situated on one of the four sides are \nreferred to as detector board {\\it Bd0} to {\\it Bd3} (see \nFig.~\\ref{fig:Design}, left).\n\nIn the scintillator, photoelectric effect interactions dominate below \n$\\sim$$15 \\, \\rm{keV}$. At $20 \\, \\rm{keV}$, however, the cross section \nof the photoelectric absorption already drops to $0.1 \\, \n\\rm{cm}^{2}/\\rm{g}$ and can be neglected as compared to the cross \nsection of Compton scattering for which X-Calibur is designed. This \nenergy regime therefore defines the detection threshold of the \npolarimeter. The mean free path for Compton scattering in the \nscintillator material is $\\approx$$4 \\, \\rm{cm}$, so that the length of \nthe scattering rod of $14 \\, \\rm{cm}$ covers $\\simeq 3.5$ path lengths. \nThis translates into a $\\simeq$$90 \\%$ probability for \nCompton-scattering in the energy regime of $20 - 80 \\, \\rm{keV}$. For \nsufficiently energetic photons, the Compton interaction produces enough \nscintillation light to trigger a photo-multiplier tube (PMT) attached to \nthe end of the scintillator. The trigger efficiency of the \nscintillator/PMT unit is discussed in \nSec.~\\ref{subsec:ScintillatorEfficiency}. The scattered X-rays are in \nturn photo-absorbed in the surrounding rings of high-Z CZT detectors. \nThis combination of scatterer/absorber leads to a high fraction of \nunambiguously detected Compton events~-- in contrast to background \nevents not entering the polarimeter along the optical axis for which the \nscintillator/CZT coincidences are strongly reduced (see \nSec.~\\ref{subsec:BG_Data}). Linearly polarized X-rays will preferably \nCompton-scatter perpendicular to their electric field vector, see \nEq.~(\\ref{eq:KleinNishinaCrossSection}). This will result in a \nmodulation of the measured azimuthal scattering distribution that is \nused to determine the polarization properties of the X-rays (see \nSec.~\\ref{subsec:AnalysisPolarization}).\n\n\n\\subsection{The CZT Detectors}\n\nX-ray photons that penetrate a semi-conductor X-ray detector deposit \ntheir energy in the detector volume, usually through photo-electric \neffect interactions. A charge cloud proportional to the X-ray energy is \ncreated and accelerated by a high-voltage electric field that is applied \nbetween the detector cathode and the pixels on the anode side. The \nmoving charge is measured and digitized and the grid position of the \ncorresponding pixel reflects the position of the interaction. CZT is the \nsemi-conductor material of choice for X-ray detectors operating in the \n$E>5 \\, \\rm{keV}$ to MeV energy band with a high probability for \nphoto-electric effect interactions, see for example \\citet{VarPitch}.\n\nThe CZT detectors used in X-Calibur were ordered from different \ncompanies\\footnote{Endicott Interconnect: http://www.evproducts.com, \nQuikpak/Redlen: http://redlen.ca, Creative Electron: \nhttp://creativeelectron.com}. The sample of detectors is listed in \nTab.~\\ref{tab:Detectors}. Each detector ($2\\times2 \\, \\rm{cm}^{2}$) is \ncontacted with a 64-pixel anode grid ($2.5 \\, \\rm{mm}$ pixel pitch) and \na monolithic cathode facing the scintillator rod. Two different detector \nthicknesses ($2 \\, \\rm{mm}$ and $5 \\, \\rm{mm}$) are used in the \npolarimeter (Fig.~\\ref{fig:Design}, left). Historically, our group had \nbeen working with detector thicknesses of $5 \\, \\rm{mm}$ and higher. \nTherefore, five of the polarimeter rings are equipped with $5 \\, \n\\rm{mm}$ detectors. The remaining three rings are equipped with $2 \\, \n\\rm{mm}$ detectors which are still sufficient to absorb more than $99\\%$ \nof X-rays in the energy regime relevant for X-Calibur, and at the same \ntime measure a lower level of background (which scales roughly with the \ndetector volume). The cathodes of the detectors are biased at \n$V_{\\rm{bi,5}} = -500 \\, \\rm{V}$ ($5 \\, \\rm{mm}$ detectors) and at \n$V_{\\rm{bi,2}} = -150 \\, \\rm{V}$ ($2 \\, \\rm{mm}$ detectors), \nrespectively.\n\nEach CZT detector is permanently bonded (anode side) to a ceramic chip \ncarrier which is plugged into the electronic readout board. \nFigure~\\ref{fig:DetectorRing} (left) shows a single CZT detector unit \nwith an $8 \\times 8$ pixel matrix on the anode side as well as the \nreadout electronics. Each CZT detector is read out by two digitizer \nboards, each consisting of a 32 channel ASIC and a 12-bit \nanalog-to-digital converter. The ASIC was developed by G.~De~Geronimo \n(BNL) and E.~Wulf (NRL) \\cite{Wulf2007}. The ASICs are operated at a \nmedium amplification (gain) of $28.5 \\, \\rm{mV/fC}$ and a signal peaking \ntime of $0.5 \\, \\mu \\rm{s}$. These settings are a result of a previous \noptimization to achieve an optimal energy resolution and low noise. Each \nASIC has a built-in capacitor that allows one to directly inject a \nprogrammable amount of charge into the individual readout channels for \ntesting purposes. The readout noise of the ASIC is as low as $2.5 \\, \n\\rm{keV}$ FWHM (see Fig.~\\ref{fig:TempStudiesNoise} in \nSec.~\\ref{subsec:TempStudies}). All 16 digitizer boards (reading eight \nCZT detectors) are read out by one harvester board ({\\it Bd0}-{\\it Bd3}, \nsee Fig.~\\ref{fig:Design}) transmitting the data to a PC-104 computer \nwith a rate of 6.25~Mbits/s. X-Calibur comprises 2048 data channels. The \ntime to read and process a triggered event is about $130 \\, \\mu\\rm{s}$ \n(ASIC dead time). However, only the ASIC involved in the triggered event \nwill be dead during the read-out. All other ASICs will still be \nsensitive and can store events that will be read out once the previous \nread-out cycle is completed.\n\n\\subsection{The Scintillator}\n\nA plastic scintillator rod is used as Compton-scatterer. The advantage, \ncompared to other scattering materials, is the scintillation light \nproduced in the scattering interaction. The light is read by a PMT and \ncan be used (optional) in the analysis. The EJ-200 scintillator \n(Hydrogen:Carbon ratio of $5.17:4.69$, $\\left< Z \\right> = 3.4$, $\\rho \n\\approx 1 \\, \\rm{g} / \\rm{cm}^{3}$, decay time $2.1 \\, \\rm{nsec}$) is \nused, read by a Hamamatsu R7600U-200 PMT with a high quantum efficiency \nsuper-bi-alkali photo cathode. To increase the optical yield, the \nscintillator is wrapped in white {\\it tyvek$^{\\tiny \\circledR}$} paper. \nThe PMT signal is amplified and digitized. A discriminator tests whether \nthe digitized PMT pulse exceeds a programmable trigger threshold and \nactivates a corresponding flag ($f_{\\rm{sci}}$). The flag is kept high \nfor $6 \\, \\mu\\rm{s}$ and is merged into the data stream of triggered CZT \ndetector events. The trigger efficiency of the scintillator is studied \nin Sec.~\\ref{subsec:ScintillatorEfficiency}. The $f_{\\rm{sci}}$ flag \nallows one to select scintillator/CZT events from the data, which \nrepresent likely Compton-scattering candidates~-- strongly suppressing \nother backgrounds (see Sec.~\\ref{subsec:BG_Data}). However, the \npolarimeter can be operated without the PMT trigger information, with \nthe scintillator acting only as a passive scatterer.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[height=0.31\\textheight]{beilicke_FRAWS_2013_02_fig03}\n\\includegraphics[height=0.31\\textheight]{P1090353_MotorAndBearing_Plus_TungstenCap}\n\\hfill\n\\includegraphics[height=0.31\\textheight]{P1090507_X-CaliburInVessel_Top}\n\\hfill\n\\includegraphics[height=0.31\\textheight]{InfocusTelescope}\n\\end{center}\n\n\\caption{Active/passive shield, rotation design, and {\\it InFOC$\\mu$S}\\xspace X-ray \ntelescope. {\\bf Left:} X-Calibur polarimeter, embedded by the CsI active \nshield, as well as the electronic readout and azimuthal rotation \nbearing. {\\bf 2$^{\\rm{nd}}$ from left, top:} Tungsten cap with \ncollimator, only allowing X-rays from the mirror to enter the shield. \n{\\bf 2$^{\\rm{nd}}$ from left, bottom:} Motor and ring bearing to rotate \npolarimeter/shield assembly. {\\bf 2$^{\\rm{nd}}$ from right:} Opened \npressure vessel with shield/X-Calibur installed (tungsten cap removed). \nThe X-ray beam enters the polarimeter through the hole in the top plate. \n{\\bf Right:} The {\\it InFOC$\\mu$S}\\xspace balloon gondola/truss with active pointing \ncontrol and the X-ray mirror installed. The pressure vessel is installed \nin the back.}\n\n\\label{fig:ActiveShieldAndTelescope}\n\n\\end{figure*}\n\n\\subsection{The Shield}\n\nDuring the balloon flight, the polarimeter will be hit by charged and \nneutral particle backgrounds with different spectral signatures and \nintensities. These backgrounds reduce the signal-to-noise ratio and, in \nthe case of non-isotropic fluxes, can even lead to a fake polarization \nsignature. In order to suppress these backgrounds, the polarimeter and \nthe front-end readout electronics are operated inside an active CsI(Na) \nanti-coincidence shield. The $2.7 \\, \\rm{cm}$ thick CsI crystal of the \nshield covers the sides and the bottom of the polarimeter and produces \nscintillation light when particles interact. The top is protected by a \npassive tungsten plate/collimator \n(Fig.~\\ref{fig:ActiveShieldAndTelescope}, left), blocking X-rays and \nparticles that do not come from the X-ray mirror. The (active) CsI \nscintillator of the shield is read out by four Hamamtsu PMTs R\\,6233 \nwhich are biased at $V_{\\rm{bi}} = +800 \\, \\rm{V}$. The analog signal of \nall four PMTs is merged and in turn digitized. A programmable, digital \ndiscriminator decides on whether a shield flag $f_{\\rm{shld}}$ is set on \nthe CZT readout board (kept up for $6 \\, \\mu\\rm{s}$) and is merged into \nthe data stream. The values of the discriminator and the width of the \nflag were optimized using a radio-active source to maximize the shield \nefficiency and minimize chance coincidences (see \nSec.~\\ref{subsec:BG_Data}).\n\n\nIn order to reduce the systematic uncertainties of the polarization \nmeasurements, the polarimeter and the active shield will be rotated \naround the optical axis with $\\sim 2 \\, \\rm{rpm}$ using a ring bearing \n(see Fig.~\\ref{fig:ActiveShieldAndTelescope}). The angle between the \npolarimeter/shield and the mounting fixture is read out by a code wheel \nwith the accuracy of $1^{\\circ}$. A counter-rotating mass can be used to \ncancel the net angular momentum of the rotating polarimeter assembly \nduring the balloon flight. The computer reading the PMT and CZT events \nis part of the rotating assembly, and referred to as polarimeter CPU.\n\n\n\\subsection{The {\\it InFOC$\\mu$S}\\xspace X-ray Telescope}\n\nThe X-Calibur polarimeter will be flown in a pressurized vessel located \nin the focal plane of the {\\it InFOC$\\mu$S}\\xspace X-ray telescope \n\\cite{InFocus_FirstFlight}. The telescope is shown in the right panel of \nFig.~\\ref{fig:ActiveShieldAndTelescope}. A Wolter grazing incidence \nmirror focuses the X-rays onto the polarimeter. The X-Calibur \nscintillator rod will be aligned with the optical axis of the {\\it InFOC$\\mu$S}\\xspace \nX-ray telescope (see Sec.~\\ref{subsec:FtSumnerData}). The focal length \nof the mirror is $8 \\, \\rm{m}$ and the field of view is $\\rm{FWHM} = 10 \n\\, \\rm{arc min}$. The telescope truss of {\\it InFOC$\\mu$S}\\xspace is only coupled to the \ngondola by a ball joint in a support cup with floating oil, allowing for \nfull inertial pointing of the telescope with an accuracy of $7 ''$ and \n$15 ''$ RMS in altitude and azimuth, respectively. To maintain the \ndecoupling between truss and gondola, any communication between the two \nsystems is done by wireless connections. In addition to the rotating \npolarimeter CPU (see above), a second CPU as part of the X-Calibur \nsubsystem (motor CPU) is installed in the pressure vessel (non-rotating) \nand controls the motors, and a temperature system. Power and data \ncommunication between the polarimeter CPU and the motor CPU is achieved \nby a {\\it Mercotac$^{\\tiny \\circledR}$} 830-SS rotating ring of mercury \nsliding contacts. Communication between the pressure vessel and the \ntelescope gondola will be done via a wireless network. The data will be \nstored on solid state drives and will in parallel be down-linked to the \nground.\n\n\n\\begin{figure}[th!]\n\\begin{center}\n\\includegraphics[width=0.44\\textwidth]{X-Calibur_ConfigurationsCZT} \\\\\n\\includegraphics[width=0.46\\textwidth]{X-Calibur_Configurations}\n\\end{center}\n\n\\caption{Different X-Calibur configurations referred to via \n$C_{\\rm{env}}^{\\rm{stp}}$. {\\bf Top row:} CZT detector geometries (see \nFig.~\\ref{fig:Design}, left) viewed along the optical axis looking into \nthe X-Ray beam. Each detector is tangentially translated by $\\Delta = 1 \n\\, \\rm{mm}$. $C^{1}$: CZT detector cathodes being $c = 16 \\, \\rm{mm}$ \naway from the optical axis (regular-sized sealed radioactive source fits \nin between). $C^{2}$: Reduced distance of $c =11 \\, \\rm{mm}$ (only \ncompact sources). $C^{3}$: same as $C^{2}$ but with the scintillator rod \ninstalled. The definitions of $\\Phi$ and $\\Delta \\Phi$ as used in \nSec.~\\ref{sec:XCLB_Characterization} are illustrated. {\\bf Bottom:} \nSchematic sketches (not to scale) of X-Calibur installed in the \ndifferent environments. $C_{\\rm{wu}}$: Washington University; detector \ncalibration in copper box ($C_{\\rm{wu1a}}$) or in passive CsI shield \n($C_{\\rm{wu1b}}$); performance measurements in the active shield with \ndifferent source positions ($C_{\\rm{wu2a}}$, $C_{\\rm{wu2b}}$, and \n$C_{\\rm{wu2c}}$). $C_{\\rm{ch}}$: X-Calibur/CHESS setup in hutch {\\it \nC1}. The beam intensity is reduced by absorption foils; platinum \n($C_{\\rm{ch1}}$), or platinum and lead ($C_{\\rm{ch3}}$). The X-Calibur \npolarimeter can be rotated around its optical axis (angle $\\alpha$) \nwhich is aligned with the CHESS beam. $C_{\\rm{ft}}$: X-Calibur installed \nin the {\\it InFOC$\\mu$S}\\xspace X-ray telescope during a flight preparation campaign in \nFt.~Sumner.}\n\n\\label{fig:X-Calibur_Configurations}\n\n\\end{figure}\n\n\n\\subsection{X-Calibur Configurations and Data Sets}\n\nIn order to characterize the different components and aspects of the \nX-Calibur polarimeter, different types of measurements were performed \nwith different geometrical configurations of the instrument (e.g. with \nand without the shield, measurements without the scattering rod to \ncalibrate the CZT detector response itself, illumination of the \ninstrument with different X-ray sources from different angles, etc.). \nThe measurements were performed at different facilities/locations (which \nwe refer to as environments). The energy calibration and \ncharacterization of the CZT detectors was performed in the laboratory at \nWashington University (Sec.~\\ref{sec:DetectorCharacterization}). \nMeasurements of a polarized X-ray beam to study the performance of the \npolarimeter were conducted at the CHESS synchrotron facility at Cornell \nUniversity (Sec.~\\ref{subsec:CHESS}). Data were also taken with the \nfully integrated X-Calibur/{\\it InFOC$\\mu$S}\\xspace telescope in a field campaign in \nFt.Sumner, NM (Sec.~\\ref{subsec:FtSumnerData}). Measurements of the \nbackground (Sec.~\\ref{subsec:BG_Data}) were performed at Washington \nUniversity, CHESS, and in Ft.Sumner. Throughout the paper, each \nconfiguration and location is referred to as $C_{\\rm{env}}^{\\rm{stp}}$ \n(`env' referring to the environment/location and `stp' referring to the \nexperimental setup). These different configurations are shown in \nFig.~\\ref{fig:X-Calibur_Configurations} and will be referred to in the \nsections to follow in which the corresponding results are presented.\n\nIt should be noted, that some of the data presented in this paper were \ntaken without the CsI shield (e.g. the data taken at the CHESS \nfacility), not allowing one to use the shield veto for background \nsuppression. Since separate background runs were taken and subtracted \nfrom the data, and the majority of measurements is signal-dominated, \nthis does not affect any result or conclusion presented in this paper.\n\n\n\\section{Reconstruction of Polarization Properties} \\label{sec:Analysis}\n\nThe events recorded by X-Calibur consist of the digitized pulse height \nof one (or more) detector pixel(s), a time stamp, and flags describing \nwhether the shield and/or the scintillator triggered. These raw events \nare first transformed into measured energies using the detector \ncalibration. The reconstructed events are further processes in order to \nderive energy spectra and the polarization properties of the measured \nX-ray beam. This section outlines the corresponding procedures.\n\n\n\\subsection{Definitions}\n\nThe modulation in the measured azimuthal scattering distribution is the \ndefining signature from which the polarization properties are derived. \nThe modulation factor $\\mu$ describes the polarimeter response to a \npolarized beam and is used to reconstruct the polarization properties. \nAssuming a $100 \\%$ linearly polarized X-ray beam, the minimum \n($C_{\\rm{min}}$) and maximum ($C_{\\rm{max}}$) number of counts of the \nazimuthal scattering distribution define:\n\n\\begin{equation}\n\\label{eq:ModulationFactor}\n\\mu = \\frac{C_{\\rm{max}} - C_{\\rm{min}}}{C_{\\rm{max}} + C_{\\rm{min}}}.\n\\end{equation}\nIt represents the modulation amplitude of a $100 \\%$ polarized beam and \ndepends on the polarimeter design and the physics of Compton-scattering. \n\nThe performance of a polarimeter can be characterized by the minimum \ndetectable polarization (MDP) as the minimum fraction of polarization \nthat can be detected at the $99 \\%$ confidence level for a given time of \nobservation $T$. Assuming a polarimeter that detects all \nCompton-scattered photons with an ideal angular resolution~-- in this \ncase $\\mu$ becomes the modulation amplitude averaged over all solid \nangles and the Klein-Nishina cross section~-- one can estimate the MDP \nby integrating the scattering probability distribution \n\\cite{Weisskopf2011, StokesAnalysis} ($R_{\\rm{src}}$ and $R_{\\rm{bg}}$ \nare the source and background count rates, respectively):\n\n\\begin{equation}\n\\label{eqn:MDP}\n\\rm{MDP} \\simeq \\frac{4.29}{\\mu R_{\\rm{src}}} \\sqrt{\\frac{R_{\\rm{src}} +\nR_{\\rm{bg}}}{T}}.\n\\end{equation}\n\n\n\\subsection{X-Calibur Event Reconstruction and Selection}\n\nEach recorded X-Calibur event contains an event number, a GPS time stamp \n$T_{\\rm{gps}}$, the orientation angle $\\theta_{\\rm{w}}$ of the \nshield/polarimeter with respect to the mounting plate (read by a code \nwheel), and a list of CZT detector pixels that were hit (up to nine) \nincluding their digitized pulse heights. Furthermore, two flags are \nmerged into the data stream: (i) a flag $f_{\\rm{shld}}$ that indicates \nwhether the PMTs reading the active CsI shield got a signal exceeding \nthe defined discriminator threshold, and (ii) a flag $f_{\\rm{sci}}$ that \nindicates if the PMT reading the central scintillator rod of the \npolarimeter (see Fig.~\\ref{fig:Design}, left) exceeded its discriminator \nthreshold. The average analog rise/fall times \n$\\tau_{\\rm{r}}$/$\\tau_{\\rm{f}}$ (time for a signal rise from $10-90\\%$ \nof the amplitude) of the folded scintillator/PMT response were measured \nfor the shield and for the scintillator. For the shield we find \n$\\tau_{\\rm{r}}^{\\rm{shld}} \\approx 70 \\, \\rm{ns}$ and \n$\\tau_{\\rm{f}}^{\\rm{shld}} \\approx 2.6 \\, \\mu\\rm{s}$, respectively. The \nresponse of the scintillator rod of the polarimeter is much faster: \n$\\tau_{\\rm{r}}^{\\rm{sci}} \\approx 3.0 \\pm 1.3 \\, \\rm{ns}$ and \n$\\tau_{\\rm{f}}^{\\rm{sci}} \\approx 13.7 \\pm 5.1 \\, \\rm{ns}$ for cosmic \nrays, and $\\tau_{\\rm{r}}^{\\rm{sci}} \\approx 4.7 \\pm 2.7 \\, \\rm{ns}$ and \n$\\tau_{\\rm{f}}^{\\rm{sci}} \\approx 13.5 \\pm 5.0 \\, \\rm{ns}$ for a \nCs$^{137}$ source placed at configuration $C_{\\rm{wu2c}}^{3}$, \nrespectively. Upon a CZT trigger, the event readout is delayed by \n$2.5-3.3 \\, \\mu\\rm{s}$ (jitter) to allow the shield and scintillator PMT \nsignals to built-up and being converted into the corresponding flags. \nThe flags $f_{\\rm{shld}}$ and $f_{\\rm{sci}}$ are kept up for a duration \nof $6 \\, \\mu\\rm{s}$.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{MultiplicityDistribution}\n\\end{center}\n\n\\caption{Distribution of pixel multiplicities $m$ for events taken with \ndifferent sources (recorded in detector ring {\\it R3}): Cosmic ray (CR) \nbackground in the laboratory, energy calibration using Eu$^{152}$, \nEu$^{152}$ calibration restricting event energies to $E < 80 \\, \n\\rm{keV}$, and data taken at the $40 \\, \\rm{keV}$ CHESS beam (see \nSec.~\\ref{subsec:CHESS}).}\n\n\\label{fig:MultiplicityDistribution}\n\\end{figure}\n\nFor each event, the pulse heights of all contributing channels/pixels \n$i=1...m$ are transformed into energies $E_{i}$ using the channel \ncalibration, see Eq.~(\\ref{eq:EnergyCalibration}) in \nSec.~\\ref{subsec:CZT_Calibration}. The total energy of the CZT event is \n$E = \\sum_{i} E_{i}$. The number of pixels $m$ participating in the \nevent is referred to as the pixel multiplicity. Selection cuts can be \napplied to the data based on the event properties mentioned above: \n$T_{\\rm{gps}}$, $\\theta_{\\rm{w}}$, $E$, $m$, $f_{\\rm{shld}}$, and \n$f_{\\rm{sci}}$. Figure~\\ref{fig:MultiplicityDistribution} shows the \ndistribution of $m$ for different source types (measured in detector \nring {\\it R3}). It can be seen that sources with high energy \ncontributions (such as the cosmic ray background or Eu$^{152}$ with its \n$E> 100 \\, \\rm{keV}$ energy lines) have $m=1$ pixel event contributions \nof $\\simeq 70 \\%$. Sources with energies concentrated in the $20-80 \\, \n\\rm{keV}$ interval, relevant for X-Calibur, have $m=1$ contributions of \n$> 90 \\%$ (e.g. the $40 \\, \\rm{keV}$ CHESS beam, see \nSec.~\\ref{subsec:CHESS}). This is explained by the fact that low energy \nX-rays deposit smaller and more concentrated charge clouds in the CZT \nwith a reduced chance of charge sharing between pixels (that would cause \n$m \\geq 2$ events). Therefore, an event selection cut on $m=1$ is a \nreasonable way for background subtraction without loosing signal events \nat low energies. Cuts on the energy $E$ and the shield flag \n$f_{\\rm{shld}}$ will further reduce the background (see \nSec.~\\ref{subsec:BG_Data}). A selection cut on the scintillator flag \n$f_{\\rm{sci}}$ selects a very clean sample of events that \nCompton-scattered in the scintillator. This has the potential to further \nreduce the background~-- however, with a loss in efficiency at low \nenergies (see Sec.~\\ref{subsec:ScintillatorEfficiency}). However, a cut \non $f_{\\rm{sci}}$ is optional and not a requirement for sensitive \npolarization measurements with X-Calibur.\n\n\n\\subsection{Energy Spectra}\n\nEnergy spectra are used to study the scattering properties of the \npolarimeter and mark an intermediate step to derive the energy-dependent \npolarization properties. Energy spectra can be derived for individual \npixels or for groups of pixels (e.g. a CZT detector or a detector ring \n{\\it R}). The energy spectra shown in this paper are normalized to the \nacquisition time (dead time corrected), the anode detector surface \ncovered by the corresponding pixel group and the width of the energy \nbins. Note, the limited dynamical range of the charge digitization of \nindividual pixel channels will lead to discrete energies. Given the \ndifferent energy calibrations of the channels, these will differ from \npixel to pixel. This difference can lead to binning artifacts in energy \nspectra that are obtained from a small group of pixels.\n\n\n\\subsection{Polarization Properties} \\label{subsec:AnalysisPolarization}\n\nThe signature of a polarized beam will be imprinted in the azimuthal \nscattering distribution of recorded events (see for example \nFigs.~\\ref{fig:AzimuthDistribution} or \\ref{fig:CHESS_2DAzimuthData} in \nSec.~\\ref{subsec:CHESS}). Either the scattering distribution itself, or \na method involving the Stokes parameters, can be used to extract the \npolarization fraction $r$ and polarization direction $\\Omega$. First, \nthe details of the detector geometry have to be carefully considered in \nthe analysis, as outlined below.\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{X-Calibur_CZT_BeamOffset}\n\\hfill\n\\includegraphics[width=0.48\\textwidth]{StokesInterpol}\n\\end{center}\n\n\\caption{{\\bf Left:} Angular coverage of two selected CZT pixels within \na {\\it single-pixel ring}\\xspace for different beam positions $P$. For a non-rotating detector \n(left) the angular coverage for each pixel only needs to be calculated \nonce. In the rotating system (right) the beam offset in the horizon \nsystem (solid arrow) leads to varying angular coverage per pixel \ndepending on the current rotation $\\alpha$ that have to be updated on an \nevent-by-event basis. Note, the pixel contacts are physically deposited \non the backside of the detectors but are shown on the front side (facing \nthe scintillator) for a better illustration of the angular measurement. \nThe cyan dashed arrows indicate the $x/z$ detector coordinate system. \n{\\bf Right:} Stokes parameters $Q$, $U$, and $I$ as measured within a \n{\\it single-pixel ring}\\xspace for a polarized beam with $\\Omega = 90^{\\circ}$. The pixels $j$ \nhave varying angular coverage $\\Delta \\Phi_{j}$ (indicated by the \nhorizontal error bars). The gray bands indicate the gaps between \ndetector boards (see left panel). The filled data points show the \nmeasured values. The open data points show the interpolated Stokes \nparameters to recover the contributions missed by the gaps. Note, the \nentries in the figure are normalized to the solid angle $\\Delta \\Phi$ \ncovered by each pixel. Therefore, the normalization leads to apparent \njumps from the pixels to the values shown for the gaps. The open symbol \nat $\\Phi \\simeq 360^{\\circ}$ corresponds to a dead pixel.}\n\n\\label{fig:SchematicBeamOffset_AndStokes}\n\\end{figure*}\n\n{\\it Azimuthal pixel coverage.} The $360^{\\circ}$ range in azimuth is \ncovered by the $4 \\times 8 = 32$ pixels per {\\it single-pixel ring}\\xspace (see \nFig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). Different pixels \ncover different azimuthal ranges $\\Delta \\Phi$ with respect to the \ncenter of the scattering rod which defines the optical axis. Therefore, \nthe four sides of detector boards lead to a 4-fold symmetry in the \nscattering distribution (Fig.~\\ref{fig:AzimuthDistribution}). This \npurely geometrical effect can be corrected for by dividing the counts \n$C_{j}$ in each pixel $j$ by its azimuthal coverage $\\Delta \\Phi_{j}$.\n\n{\\it Azimuthal pixel coverage for beam offsets.} A special situation \narises if the optical axis of the X-ray beam is not aligned with the \ngeometrical axis of the polarimeter (see $P_{1}$ in \nFig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). Such an offset, if \nnot corrected, will introduce asymmetries in the azimuthal scattering \ndistributions and can mimic a wrong polarization signature (see \nSec.~\\ref{subsec:Systematics} for a detailed study). Assuming a known \noffset vector $\\vec{P}1$, the azimuthal coverage $\\Delta \\Phi_{p1, j}$ \ncan be recalculated for each pixel $j$. This correction re-aligns the \norigin of the detector system with the beam axis and strongly reduces \nthe systematic effect introduced by the offset. In the case of \nsufficient event statistics, first moments can be used to estimate the \nbeam offset from the data itself (see Sec.~\\ref{subsec:Systematics}). If \nthe polarimeter is rotating, the beam offset in the horizon system will \nrotate in the detector system (see \nFig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). In this case \n$\\Delta \\Phi_{p1,j}$ has to be updated on an event-by-event basis, \ntaking into account the current polarimeter orientation $\\alpha$ \nmeasured by the code wheel.\n\n{\\it Pixel acceptance and flat fielding.} Individual pixels have \ndifferent trigger efficiencies, energy thresholds and energy resolutions \naffecting the number of counts derived from a given energy interval. To \ncorrect for these differences, the pixels are flat fielded using \nCompton-scattered events recorded from a non-polarized X-ray beam that \nresults in a flat azimuthal scattering distribution. For a {\\it single-pixel ring}\\xspace the \nevent counts per azimuthal coverage $C_{j} / \\Delta \\Phi_{j}$ are \naveraged for all $j=1...32$ pixels and are used to determine a relative \nazimuthal acceptance $a_{j}$:\n\n\\begin{equation}\nw_{j} = \\frac{1}{a_{j}} = \\frac{\\Delta \\Phi_{j}}{C_{j}} \\frac{1}{N}\\sum_{i=1}^{N}\\frac{C_{i}}{\\Delta \\Phi_{i}}.\n\\label{eqn:Acceptance}\n\\end{equation}\n\nThe pixel acceptance $a_{j}$ can in turn be used to weight individual \nevents with $w_{j}=1/a_{j}$. Implicitly, $a_{j}$ depends on the event \nselection cuts~-- so that it has to be computed for the particular set \nof cuts applied to the data. Dead pixels cannot be recovered by a \ncorresponding weight since they contribute zero events. However, once \nthe polarimeter/shield assembly is rotating with respect to the \npolarization plane, the pixel acceptances can be ignored since they \naverage out throughout the measurement~-- this includes the treatment of \ndead pixels.\n\n{\\it Polarization properties derived from the azimuthal scattering \ndistribution.} Integrating $C_{j} / \\Delta \\Phi_{j}$ for each pixel $j$ \nin a certain energy range and weighting the individual counts with \n$w_{j}$ will result in the azimuthal scattering distribution. Here, the \nangular coverage $\\Delta \\Phi_{j}$ determines the horizontal error bar \nof the data point. The scattering distribution can be derived for \nindividual detector rings. Fitting a sinusoidal function to the \ndistribution (e.g. left panel in Fig.~\\ref{fig:AzimuthDistribution}, \nSec.~\\ref{subsec:CHESS}) allows one to reconstruct (i) the orientation \nof the polarization plane (minimum), as well as (ii) the modulation of \nthe data $\\mu_{\\rm{data}}$ following Eq.~(\\ref{eq:ModulationFactor}). \nThe corresponding modulation factor $\\mu_{\\rm{sim}}$ is derived from \nsimulations of a $100 \\%$ polarized beam (see \nSec.~\\ref{sec:Simulations}) that are analyzed using the same set of \nevent selection cuts. The polarization fraction $r$ of the measured beam \nis in turn calculated to be:\n\n\\begin{equation}\nr = \\frac{\\mu_{\\rm{data}}}{\\mu_{\\rm{sim}}}.\n\\label{eqn:PolFracPhiDistri}\n\\end{equation}\n\nThe effect of dead detector pixels and gaps between the detector boards \n(see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left) is accounted \nfor automatically, since the corresponding points do not show up in the \ndistribution and will not affect the fit.\n\n{\\it Polarization properties derived from Stokes parameters.} An \nalternative approach to reconstruct the polarization properties is based \non the Stokes parameters \\cite{Chandrasekhar1960, StokesAnalysis} that \nare calculated for each event $k$:\n\n\\begin{equation}\nq_{k} = \\cos(2 \\Phi_{k}), \\quad u_{k} = \\sin(2 \\Phi_{k}), \\quad i_{k} = 1.\n\\label{eqn:StokesParams}\n\\end{equation}\n\nThe Stokes parameters can be summed for a subset of the data consisting \nof $N$ events (e.g. over a specific energy interval and detector ring):\n\n\\begin{equation}\nQ = \\sum_{k=1}^{N} w_{j(k)} q_{k}, \\quad U = \\sum_{k=1}^{N} w_{j(k)} u_{k}, \\quad I = \\sum_{k=1}^{N} w_{j(k)}.\n\\label{eqn:StokesSum}\n\\end{equation}\nHere, each event $k$ (originating from pixel $j$) is weighted with \n$w_{j(k)} = 1/a_{j(k)}$. Since each pixel $j$ covers a range in azimuth \n$\\Delta \\Phi_{j} = \\Phi_{j,\\rm{max}} - \\Phi_{j,\\rm{min}}$, the values \n$q_{k}$ and $u_{k}$ derived from the mean angle $\\Phi_{j(k)}$ in a \nnon-rotating system may lead to inaccurate and/or biased results. \nTherefore, the mean Stokes parameters $\\left< q \\right>_{j(k)}$ and \n$\\left< u \\right>_{j(k)}$ are calculated based on the covered range in \nazimuth of the corresponding pixel $j$:\n\n\\begin{eqnarray}\n\\begin{split}\n\\left< q \\right>_{j(k)} & = \\frac{1}{\\Delta \\Phi_{j(k)}} \\int_{\\Phi_{j(k),\\rm{min}}}^{\\Phi_{j(k),\\rm{max}}} \\cos(2 \\Phi) \\, \\rm{d}\\Phi, \\\\\n\\left< u \\right>_{j(k)} & = \\frac{1}{\\Delta \\Phi_{j(k)}} \\int_{\\Phi_{j(k),\\rm{min}}}^{\\Phi_{j(k),\\rm{max}}} \\sin(2 \\Phi) \\, \\rm{d}\\Phi.\n\\end{split}\n\\label{eqn:MeanStokes}\n\\end{eqnarray}\n\nThe values of $\\left< q \\right>_{j(k)}$ and $\\left< u \\right>_{j(k)}$ \nare in turn used in Eq.~(\\ref{eqn:StokesSum}). The proper treatment of \ndead pixels and gaps between the detector boards {\\it Bd0}-{\\it Bd3} \n(see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left) is crucial when \nworking with the Stokes parameters in a non-rotating coordinate system, \nsince a lack of events from a particular azimuthal direction \n$\\tilde{\\Phi}$ will lead to an increase/decrease in $\\sum_{k} \\left< q \n\\right>_{j(k)}$ and/or $\\sum_{k} \\left< u \\right>_{j(k)}$, \nsystematically affecting the reconstructed polarization properties (see \nFig.~\\ref{fig:SchematicBeamOffset_AndStokes}, right). The amplitude and \nsign of the resulting effect depends on the orientation between the \npolarization plane and the detector plane (assuming the gaps are located \nat angles of $45^{\\circ}$, $135^{\\circ}$, $225^{\\circ}$, and \n$315^{\\circ}$, see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). \nIf the planes are exactly parallel or exactly perpendicular, the effect \nleads to a maximal underestimation of $r$. An angle of $45^{\\circ}$ \n(polarization plane being aligned with two of the four gaps) leads to \nthe maximal overestimation of $r$. At angles of $45/2 = 27.5^{\\circ}$ \n(and multiples thereof) the effect of detector gaps cancels. A \nsimulation for the X-Calibur detector layout was performed and resulted \nin a systematic effect of $\\pm6 \\%$ for a $100\\%$ polarized beam. \nTherefore, a correction has to be applied: (i) the $m=1,2,...$ azimuthal \nranges $\\delta \\tilde{\\Phi}_{m}$ covered by the gaps have to be \nidentified in which the detector is not sensitive (ii) The distributions \nof $\\delta Q / \\delta \\Phi$, $\\delta U / \\delta \\Phi$, and $\\delta I / \n\\delta \\Phi$ have to be collected as a function of $\\Phi$ \n(Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, right). (iii) These \ndistributions are in turn used to interpolate the data gaps $\\delta \n\\tilde{\\Phi}_{m}$ from neighboring pixel to recover the `missing' \ncontributions in Eq.~(\\ref{eqn:StokesSum}):\n\n\\begin{eqnarray}\n\\begin{split}\nQ & \\rightarrow Q + \\sum_{m} \\Delta \\tilde{Q}_{m} \\delta \n\\tilde{\\Phi}_{m}, \\\\ U & \\rightarrow U + \\sum_{m} \\Delta \\tilde{U}_{m} \n\\delta \\tilde{\\Phi}_{m}, \\\\ I & \\rightarrow I + \\sum_{m} \\Delta \n\\tilde{I}_{m} \\delta \\tilde{\\Phi}_{m}.\n\\end{split} \n\\label{eqn:StokesGapCorrection} \n\\end{eqnarray}\nThe errors on the added sums are calculated using error propagation \nduring the interpolation. Given the scalar nature of the Stokes \nanalysis, there is no simple way of representing the azimuthal \nmodulation of the scattering distribution. Therefore, it is useful to \ncompare the Stokes results with the results obtained from the azimuthal \nscattering distribution (previous paragraph) in order to identify \npossible systematic effects. Note again, that a rotating polarimeter \nwill not require a correction for dead pixels or detector gaps.\n\nThe Stokes sums in (\\ref{eqn:StokesSum}) or \n(\\ref{eqn:StokesGapCorrection}) can be used to reconstruct the \npolarization fraction $r$ and polarization angle~$\\Omega$:\n\n\\begin{eqnarray}\n\\begin{split}\nr & = \\frac{2}{\\mu_{\\rm{sim}}} \\frac{\\sqrt{Q^{2} + U^{2}}}{I}, \\\\\n\\Omega & = \\frac{1}{2} \\rm{atan} (U / Q).\n\\end{split}\n\\label{eqn:PolarizationFromStokes}\n\\end{eqnarray}\n\nResults from $l$ independent measurements (e.g. from different detector \nrings) can be combined in a weighted average. Since the polarization \nfraction is always positive it can lead to an overestimation if the true \npolarization fraction is in the MDP regime of the data set, see \nEq.~(\\ref{eqn:MDP}), where the error bars are highly asymmetric. This \nsystematic effect can be avoided by averaging a set of modified Stokes \nparameters:\n\n\\begin{eqnarray}\n\\begin{split}\nQ' & = \\sum_{l} \\frac{2 w_{l}}{\\mu_{\\rm{sim,l}}} \\frac{Q_{l}}{I_{l}}, \\quad U' = \\sum_{l} \\frac{2 w_{l}}{\\mu_{\\rm{sim,l}}} \\frac{U_{l}}{I_{l}}, \\\\\nr' & = \\frac{1}{L'} \\sqrt{Q'^{2} + U'^{2}} \\quad {\\rm with} \\quad L' = \\sum_{l} w_{l}.\n\\end{split}\n\\label{eqn:Stokes_Q_U_Average}\n\\end{eqnarray}\nThe weights $w_{l} = 1/\\sigma_{l}^{2}$ account for the statistical \nuncertainties of the individual measurements $l$.\n\n{\\it Unfolding analysis.} An unfolding analysis that takes into account \nthe energy-dependent detector response and photon detection efficiencies \nand that can be used to reconstruct polarization fraction and angle as a \nfunction of true photon energy will be described in a separate paper \n\\cite{XCLB_LogLikelihoodAnalysis}.\n\n{\\it Forward folding.} Recording the azimuthal scattering distributions \nwith planar detectors will lead to projection effects that depend on the \npolar angle of the scattering in the scintillator. For most parts of the \npolarimeter these effects are negligible, since (i) a particular \ndetector ring sees the superposition of different polar scattering \nangles canceling the effect, and (ii) the same effect is present in the \nsimulations that are used to determine the polarization fraction \nfollowing Eq.~(\\ref{eqn:PolFracPhiDistri}). Only for polarization \nfractions $r \\ll 1$ measured in detector ring {\\it R1}, which sees only \nback-scatter events, a second order correction may be \nneeded\\footnote{Here, the $r \\propto \\mu_{\\rm{data}}$ relation in \nEq.~(\\ref{eqn:PolFracPhiDistri}) will no longer be exactly linear.}. To \nfully take into account these effects, the data can be analyzed with a \nforward folding method (which is beyond the scope of this paper). The \nmodeling of the pixel acceptance in {\\it R1}, only affecting \nmeasurements with the non-rotating polarimeter, will also be slightly \naffected by the effect.\n\n\n\\section{Simulations} \\label{sec:Simulations}\n\n\nSimulations of the energy-dependent response of the X-Calibur \npolarimeter are needed in order to reconstruct the polarization \nproperties from measured data (see \nSec.~\\ref{subsec:AnalysisPolarization}). The detector geometry is \nmodeled and the X-ray flux/spectrum was simulated for the different \nexperimental setups in which the data presented in this paper were \ntaken. The simulations were performed in the following steps.\n\n\\begin{enumerate}\n\n\\item Physics interactions, scatterings, and energy depositions in the \nscintillator and the CZT detectors were simulated using {\\it \nGEANT4}\\footnote{{\\tt http://geant4.cern.ch/}} with the Livermore \nlow-energy electromagnetic model list.\n\n\\item The charge collection efficiency as a function of \ndepth-of-interaction in the CZT detectors was determined using an \nin-house developed software to (i) calculate the 2D electric potential \ninside the detector crystals followed by (ii) the integration of the \nweighting potential \\cite{Jung2007} along the charge transport tracks, \nresulting in the collected charge for each energy deposition (using a \ndielectric constant for CZT of $10$). The simulations are valid for \ndetectors with strip anode contacts but will roughly resemble the \nresponse for pixelated detectors, as well. A mobility of electrons/holes \nof $\\mu_{\\rm{e}} = 1000 \\, \\rm{cm}^{2} / \\rm{V} / \\rm{s}$, and \n$\\mu_{\\rm{h}} = -120 \\, \\rm{cm}^{2} / \\rm{V} / \\rm{s}$ is assumed, as \nwell as life times of $\\tau_{\\rm{e}} = 10^{-6} \\, \\rm{s}$ and \n$\\tau_{\\rm{h}} = 4 \\cdot 10^{-6} \\, \\rm{s}$, respectively. This \ncorresponds to a mobility-lifetime product of $\\mu_{\\rm{e}}\\tau_{\\rm{e}} \n= 10^{-3}$ which is in reasonable agreement with the values measured for \na selection of the detectors used in X-Calibur (see \nFig.~\\ref{fig:TempStudies}, bottom).\n\n\\item The energy resolution (asymmetric Gaussian function, implicitly \nincluding the electronic readout noise) and energy threshold were \nmeasured from real data (Sec.~\\ref{subsec:ThreshAndResolution}) and were \nfolded into the simulations on a pixel-by-pixel basis. However, \ndifferences in channel trigger efficiencies were not simulated. Using \nthe reversed energy calibration in Eq.~(\\ref{eq:EnergyCalibration}), the \nsimulated events were in turn converted into the X-Calibur data format \n(ASIC/channel ID and digitized raw pulse height) and can be analyzed in \nthe same way as the measured data.\n\n\\item Since we find that the simulations do not properly account for low \nenergy tails in the detector response (see next paragraph), an empirical \nmodel was used to scatter an exponential tail $T(E) = A \\cdot \n\\exp(E/E_{\\rm{t}})$ into the simulations with a relative fraction of \n$f=0.45$ per energy deposit. The parameter $E_{\\rm{t}} = 15 \\, \\rm{keV}$ \n(at $E=40 \\, \\rm{keV}$) and $E_{\\rm{t}} = 30 \\, \\rm{keV}$ (at $E = 120 \n\\, \\rm{keV}$) was interpolated in $\\log E$ for the energy range covered.\n\n\\item The scintillator trigger flag $f_{\\rm{sci}}$ was generated for \neach event based on the scintillator trigger efficiency determined from \nreal data (see Fig.~\\ref{fig:Scintillator_TriggerEfficiency} in \nSec.~\\ref{subsec:ScintillatorEfficiency}).\n\n\\end{enumerate}\n\nDifferent scenarios/setups were simulated, reflecting the measurements \nand studies presented in this paper.\n\n{\\it CZT detector response.} To test the validity of the simulation \nchain, the direct illumination of a single CZT detector with a \nEu$^{152}$ point source was simulated\\footnote{All lines with \nintensities above $2 \\%$ were generated according to their relative \nemission intensities.}~-- corresponding to the experimental setup used \nfor the detector calibration measurements presented in \nSec.~\\ref{subsec:CZT_Calibration}. The comparison between the \nsimulations and data (Fig.~\\ref{fig:EnergyCalibration}, right) shows \nreasonable agreement in terms of line positions, widths, and threshold \neffects. However, if ignoring step (4) of the simulation chain (`Sim' in \nthe legend of Fig.~\\ref{fig:EnergyCalibration}), a lack in continuum \nemission can be seen in the simulations. This motivated the introduction \nof step (4) which leads to a reasonable agreement between data and \nsimulations over the whole energy band relevant for X-Calibur (`Sim+' in \nthe figure legend). Possible reasons for the continuum in the data may \nbe related to details in the detector response or back-reflection of \nemitted X-rays from the source off the surrounding fixture that is not \nsimulated\\footnote{For similar CZT detectors we find a photo-peak \ndetection efficiency of order unity \\cite{VarPitch}.}. It should be \nnoted that the goal of the CZT simulations is not to find an accurate \nmodel for the detector response~-- but rather a suited parameterization \nthat reproduces the integral spectral response for the different energy \nbins.\n\n{\\it CHESS beam.} X-Calibur performance measurements were performed at \nthe highly polarized synchrotron X-ray beam at the CHESS facility~-- \nproviding a strong, mono-energetic X-ray beam (Sec.~\\ref{subsec:CHESS}). \nA corresponding set of simulations was performed using steps (1)-(5), \nresembling the CHESS setup of pencil-beam X-rays (polarized and \nnon-polarized) at $40 \\, \\rm{keV}$, $80 \\, \\rm{keV}$, and $120 \\, \n\\rm{keV}$ entering the polarimeter along the optical axis of the \nscintillator. Detector pixels that were excluded during the CHESS data \nruns were also excluded in the simulations to resemble a configuration \nclose to the one used for the measurements. The trigger efficiency of \nthe scintillator as a function of energy deposition was derived from the \nCHESS data (see Fig.~\\ref{fig:Scintillator_TriggerEfficiency} in \nSec.~\\ref{subsec:ScintillatorEfficiency}) and was fed into the \nsimulations in step (5) to generate the trigger flag $f_{\\rm{sci}}$ on \nan event-by-event basis. No backgrounds were simulated in the case of \nthe CHESS measurements since the measurements were completely \nsignal-dominated.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{CrabSimulations}\n\\end{center}\n\n\\caption{Simulated X-Calibur observations of the Crab nebula ($5.6 \\, \n\\rm{h}$), taken from \\citet{Guo2010}. The data points show the \nreconstructed flux (top), the fraction of polarization (middle), and \npolarization angle (bottom). The lines show the assumed flux, \npolarization fraction and polarization direction.}\n\n\\label{fig:XCLB_Sims}\n\\end{figure}\n\n{\\it Balloon flight.} A balloon flight in the focal plane of the \n{\\it InFOC$\\mu$S}\\xspace mirror assembly was assumed in an earlier simulation \n\\cite{Guo2010} that only involved step (1) in the above chain. The \neffective detection areas of the X-ray mirror are $95/60/40 \\, \n\\rm{cm}^{2}$ at $20/30/40 \\, \\rm{keV}$, respectively. We accounted for \natmospheric absorption at a floating altitude of $130,000$ feet using \nthe NIST XCOM attenuation coefficients\\footnote{{\\tt \nhttp://www.nist.gov/pml/data/xcom/index.cfm}} and an atmospheric depth \nof $2.9 \\, \\rm{g/cm}^{2}$ (observations performed at zenith); the \natmospheric transmissivity rapidly increases from $0$ to $0.6$ in the \n$20 - 80 \\, \\rm{keV}$ range. The trigger efficiency of the scintillator \nscatterer was assumed to be $f_{\\rm{sci}} = 1$ above an energy \ndeposition of $2 \\, \\rm{keV}$ and $f_{\\rm{sci}} = 0$ below.\n\nWe simulated the most important backgrounds such as the cosmic X-ray \nbackground \\cite{Ajello2008}, albedo photons and cosmic ray protons and \nelectrons \\cite{Mizuno2004}. The neutron background was not modeled \nsince a detailed study of \\citet{Parsons2004} showed that the \ncontribution in CZT can be neglected. Different shield configurations \nand shield thicknesses were simulated. The configuration shown in \nFig.~\\ref{fig:ActiveShieldAndTelescope} (left) represents an optimized \ncompromise balancing the background rejection power and the \nmass/complexity of the shield. A Crab-like source was simulated for a \n$5.6 \\, \\rm{hr}$ balloon flight. We assumed a power law energy spectrum, \nand a continuous change of the polarization fraction and angle between \nthe values measured at $5.2 \\, \\rm{keV}$ with OSO-8 \\cite{Weisskopf1978} \nand at $E>100 \\, \\rm{keV}$ with {\\it INTEGRAL}\\xspace \\cite{Dean2008} by modeling a \ntransition following a Fermi distribution.\n\nFor a Crab-like source the simulations predict an event rate of $1.1 \n(3.2) \\, \\rm{Hz}$ with (without) requiring a triggered scintillator \ncoincidence ($f_{\\rm{sci}} = 1$). Figure~\\ref{fig:XCLB_Sims} compares \nthe simulation results with the assumed model curves; the errors were \ncomputed in a similar way as described by \\citet{Weisskopf2010}. \nSimulations performed at different zenith angles $\\theta$ show that the \nsource rate scales with $(\\cos \\theta)^{1.3}$ which is taken into \naccount for simulating astrophysical observations. More details about \nthese simulations are discussed in \\citet{Guo2010} and \\citet{Guo2013}.\n\n\n\\section{CZT Detector Characterization} \\label{sec:DetectorCharacterization}\n\nThe performance of the individual CZT detectors used in X-Calibur is \ncoupled to the performance of the polarimeter as a whole~-- including \nits energy threshold and its energy resolution. This section describes \nthe energy calibration of the individual CZT detectors \n(Sec.~\\ref{subsec:CZT_Calibration}), as well as measurements of the \nenergy threshold and energy resolution \n(Sec.~\\ref{subsec:ThreshAndResolution}). The CZT performance serves as \nimportant input for the simulations described in \nSec.~\\ref{sec:Simulations}. In contrast to the laboratory, the \npolarimeter will be operated in an environment of varying temperature \nconditions during the balloon flight which motivates the study of the \ntemperature dependence of the CZT detector performance which will be \ndescribed in Sec.~\\ref{subsec:TempStudies}.\n\nIn order to quantify the characteristics of individual pixels, the \nemission lines in the calibrated energy spectra are fitted with a \nGaussian function. The peak position is described by the mean \n$E_{\\rm{p}}$. To account for the asymmetric shape of the peaks (see for \nexample Fig.~\\ref{fig:EnergyCalibration}), the fitted function allows \nfor asymmetric spectral continua levels ($c_{1}$, $c_{2}$) and \nasymmetric peak widths ($\\sigma_{1}$, $\\sigma_{2}$) for the $E < \nE_{\\rm{p}}$ (1) and $E > E_{\\rm{p}}$ (2) regimes, respectively. The fit \nparameters are used to characterize the measured peak. The energy \nresolution is calculated as the full width half maximum, $\\Delta E = \n\\rm{FWHM} = 2 \\sqrt{2 \\ln 2} \\cdot \\frac{1}{2} (\\sigma_{1} + \n\\sigma_{2})$. The peak rate is determined by counting the events in the \ninterval $\\pm 2 \\, \\rm{FWHM}$ centered around $E_{\\rm{p}}$, normalized \nby the observation time.\n\nA Eu$^{152}$~source is used as calibration/test source in a variety of \nstudies presented in this paper. Eu$^{152}$~emits X-ray lines at \n$39.5\\,\\rm{keV}$ ($\\rm{K}_{\\alpha2}$), $40.12\\,\\rm{keV}$ \n($\\rm{K}_{\\alpha1}$), $45.7 \\, \\rm{keV}$, $122.78 \\, \\rm{keV}$, $244.7 \n\\, \\rm{keV}$, $344.28 \\, \\rm{keV}$, and at higher energies. The line at \n$40.12\\,\\rm{keV}$ is the strongest in the low energy triplet; relative \nto the $40.12\\,\\rm{keV}$ line, the $39.5 \\, \\rm{keV}$ line is emitted at \n$55 \\%$ intensity and the $45.5 \\, \\rm{keV}$ line at $31 \\%$ intensity. \nThe $40.12\\,\\rm{keV}$, used as low-energy performance marker, is \ntherefore fitted jointly with the two close-by lines that are set at \nfixed distance $\\Delta E$ and fixed intensity relative to the \n$40.12\\,\\rm{keV}$ peak (with the free fit parameter $\\sigma$ being the \nsame for all three lines since the energy resolution of a detector pixel \nis not expected to change within a few keV). Including the two \nneighboring lines avoids systematic shifts and artificial broadening of \nthe fitted peak (see Fig.~\\ref{fig:EnergyCalibration}, left).\n\n\n\\subsection{Energy Calibration} \\label{subsec:CZT_Calibration}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{ZoomChannelEu152_Asic19_Channel_10}\n\\hfill\n\\includegraphics[width=0.50\\textwidth]{EnergyCalibrationSpectraEu152_Mosaic}\n\\end{center}\n\n\\caption{Eu$^{152}$ energy spectra (direct CZT illumination). The \nvertical lines indicate the nominal X-ray line energies; relative \nheights indicate the emission intensity folded with the absorption \nprobability in $5 \\, \\rm{mm}$ CZT and $2 \\, \\rm{mm}$ CZT (small tick), \nrespectively. Data were taken with X-Calibur configuration \n$C_{\\rm{wu1}}$ (see Fig.~\\ref{fig:X-Calibur_Configurations}). {\\bf \nLeft:} Spectrum of an individual detector pixel (linear energy axis, \n$C_{\\rm{wu1a}}^{1}$). The dashed vertical line indicates the trigger \nenergy threshold. The fitted functions ($40.1 \\, \\rm{keV}$ and $121.8 \\, \n\\rm{keV}$) are used to determine the peak position and energy resolution \n(FWHM). {\\bf Right:} Energy spectra summed over all pixels of the \ndetectors in ring {\\it R3} ($5 \\, \\rm{mm}$ thickness) and ring {\\it R7} \n($2 \\, \\rm{mm}$ thickness), respectively. Also shown are the simulated \nspectra with/without (Sim+/Sim) additional continuum (see \nSec.~\\ref{sec:Simulations}). A CR background spectrum (scaled by a \nfactor of $10$) is shown for reference. The horizontal dotted line \nindicates the energy range relevant for X-Calibur.}\n\n\\label{fig:EnergyCalibration}\n\n\\end{figure*}\n\nThe energy calibration of the individual CZT detector pixels is done \nwith a compact Eu$^{152}$ source (cylindrical emitting volume with a \ndiameter of $\\simeq$$3 \\, \\rm{mm}$) in the X-Calibur configuration \n$C_{\\rm{wu1b}}^{2}$ in which the the scintillator rod is not installed \n(see Fig.~\\ref{fig:X-Calibur_Configurations}). The source was \nsuccessively placed at the centers of the detector rings~-- allowing to \ncalibrate rings {\\it R1} to {\\it R8}, one at a time. In a first step, an \nautomatic routine is used to optimize the pixel trigger thresholds: data \nare taken in a special acquisition mode that adjusts ASIC/channel \ndiscriminators based on measured event rates for each pixel, such that \nthe trigger threshold is as low as possible (maximizing the integral \ntrigger rate), but at the same time is safely above the electronic noise \nregime. The noise regime leads to very high (artificial) trigger rates \nand differs from channel to channel. The routine works reliably for most \nchannels. However, a visual inspection of all 2048 recorded energy \nspectra was performed to assure that channels with trigger thresholds \nset too high or too low (failed automatic detection of the noise regime) \nwere adjusted manually.\n\nAbout 5 million events were taken for each CZT detector (20 million \nevents per ring {\\it R}), and the known energy lines at $40.1 \\, \n\\rm{keV}$ and $122 \\, \\rm{keV}$ were used to determine the pedestal \n$p_{0}$ and amplification slope $a$ for each channel\\footnote{The \nlinearity of the channels was confirmed using the internal test pulse \ngenerator of the ASIC.}. The energy of a measured pulse height $p$ is in \nturn calculated using\n\n\\begin{equation}\n\\label{eq:EnergyCalibration} \nE(p) = (p-p_{0})/a.\n\\end{equation}\n\nThe left panel of Figure~\\ref{fig:EnergyCalibration} shows the \ncalibrated energy spectrum of a single pixel. The right panel shows the \naveraged calibration spectra of two chosen detector rings ($4 \\times 64$ \npixels each). Also shown are the energy spectra obtained from the \nsimulations of the corresponding setup (see Sec.~\\ref{sec:Simulations}). \nIt can be seen, that the $2 \\, \\rm{mm}$ detectors (ring {\\it R7}) loose \nperformance at energies $E> 100 \\, \\rm{keV}$, as compared to the \ndetectors with $5 \\, \\rm{mm}$ thickness (ring {\\it R3}). However, in the \n$20-80 \\, \\rm{keV}$ energy band, relevant for X-Calibur, both types of \ndetector thickness perform at a similar level.\n\nIn addition to the temperature-dependence of the detector performance \ndiscussed in Sec.~\\ref{subsec:TempStudies}, another consideration has to \nbe made when applying the calibration to the data. The CZT detectors \nused in X-Calibur are not setup for measuring the depth position of the \nX-ray interaction/absorption between the detector cathode and anode~-- \nreferred to as the depth-of-interaction (DOI). The energy calibration in \nEq.~(\\ref{eq:EnergyCalibration}), however, depends on the average DOI of \na given energy. The mean DOI, however, changes with the cosine of the \ninclination angle measured between the absorbed X-ray and the detector \nplane. The calibration was determined with the X-ray source located $c \n\\simeq 11 \\, \\rm{mm}$ above the center of the detector cathode (see \nFig.~\\ref{fig:X-Calibur_Configurations}, top). Depending on their \ngeometrical locations, the pixels are hit under angles between \n$0-45^{\\circ}$. X-rays Compton-scattering in the X-Calibur scintillator, \non the other hand, can hit the CZT detectors at angles between \n$0-90^{\\circ}$, which adds a systematic error/uncertainty to the \nreconstructed energy for small incident angles. However, in the $20-80 \n\\, \\rm{keV}$ band the DOI distribution is very narrow and localized \nclose to the cathode. Therefore, the angle dependence is negligible~-- \nthe systematic shift of the reconstructed $40.1 \\, \\rm{keV}$ line was \nexperimentally constrained to be less than $1\\%$ for shallow inclination \nangles.\n\n\nA $< 3\\%$ fraction of ASIC channels were found to be dead, too noisy, or \ndid not make contact to the detector pixel. These channels were excluded \nfrom the analysis and are marked with `x' in the corresponding 2D plots \nshown in this paper (e.g. Fig.~\\ref{fig:PixelThreshold}).\n\n\n\\subsection{Energy Resolution and Threshold} \n\\label{subsec:ThreshAndResolution}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Results_CFG1_Threshold}\n\\end{center}\n\n\\caption{2D distribution of energy thresholds of all X-Calibur pixels. \nIn this representation, the four detector sides surrounding the \nscintillator ({\\it Bd0} to {\\it Bd3}, see Fig.~\\ref{fig:Design}, left) \nare unfolded into a plane. The segments enclosed by dashed and dotted \nlines indicate the CZT detectors ($8 \\times 8$ pixels each). The \nelongated box (solid line in {\\it Bd0} to {\\it Bd3}, each) indicates the \nposition of the scintillator, even though it was not installed for this \nparticular data set (configuration $C_{\\rm{wu1b}}^{2}$, see \nFig.~\\ref{fig:X-Calibur_Configurations}, top). Pixels marked with `x' \nwere not included in the analysis. The left panels show the threshold \ndistributions per detector ring {\\it R} (the gray distribution shows the \nscaled average of all pixels).}\n\n\\label{fig:PixelThreshold}\n\n\\end{figure}\n\nTo study the performance of the X-Calibur CZT detectors, the calibration \ndata (Sec.~\\ref{subsec:CZT_Calibration}) were used to characterize the \nenergy spectra of individual pixels (see left panel of \nFig.~\\ref{fig:EnergyCalibration} for reference). The relevant properties \nstudied in this section are the energy threshold, the fitted line/peak \nposition, and the energy resolution (FWHM). As for the energy \ncalibration, the data were taken with the configuration \n$C_{\\rm{wu1b}}^{2}$ (see Fig.~\\ref{fig:X-Calibur_Configurations}).\n\nThe energy threshold of a pixel is defined as the reconstructed energy \n$E_{\\rm{thr}}$ above which the corresponding ASIC channel starts to \ntrigger on events (see Fig.~\\ref{fig:EnergyCalibration}, left). The \ndistribution of thresholds for all X-Calibur pixels is shown in \nFig.~\\ref{fig:PixelThreshold}. The thresholds vary from pixel to pixel, \nbut no significant geometrical trends can be identified if comparing \nedge pixels versus central pixels, or pixels of detectors with different \nthickness ($5 \\, \\rm{mm}$, rings {\\it R1-R5} versus $2 \\, \\rm{mm}$, \nrings {\\it R6-R8}). About $50\\%$ of all pixels have an energy threshold \nof $E<20 \\, \\rm{keV}$. It should be noted that one has to account for \nthe energy resolution of a pixel in order to determine the analysis \nthreshold that is about $3-4 \\, \\rm{keV}$ higher. The thresholds shown \nin Fig.~\\ref{fig:PixelThreshold} reflect the status of the compact \nX-Calibur configuration. The thresholds of the detectors operated as a \nsingle unit are up to $5 \\, \\rm{keV}$ lower (see \nFig.~\\ref{fig:TempStudies}).\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[height=0.42\\textheight]{Results_CFG1_40keV_Resolution}\n\\hfill\n\\includegraphics[height=0.42\\textheight]{Results_CFG1_Noise_Resolution_Zoom}\n\\hfill\n\\includegraphics[height=0.42\\textheight]{Results_CFG1_122keV_Resolution}\n\\end{center}\n\n\\caption{2D distribution of the pixel-by-pixel energy resolution \n(compare with Fig.~\\ref{fig:PixelThreshold}). {\\bf Left:} Energy \nresolution at $40.1 \\, \\rm{keV}$. {\\bf Middle:} Electronic readout noise \nat $40 \\, \\rm{keV}$ as determined using the ASICs internal test pulse \ngenerator (only shown for {\\it Bd2}). The same range of the color scale \nas in the left panel is used. {\\bf Right:} Energy resolution at $121.8 \n\\, \\rm{keV}$. This energy line is not well-defined in the energy spectra \nmeasured with the $2 \\, \\rm{mm}$ detectors (see \nFig.~\\ref{fig:EnergyCalibration}) so that results are only shown for the \n$5 \\, \\rm{mm}$ rings {\\it R1} to {\\it R5}.}\n\n\\label{fig:PixelResolution}\n\n\\end{figure*}\n\n\nFigure~\\ref{fig:PixelResolution} shows the energy resolutions at $40.1 \n\\, \\rm{keV}$ and $121.8 \\, \\rm{keV}$, respectively. Detector rings {\\it \nR2} and {\\it R3} are the most sensitive ones when it comes to detecting \nthe Compton-scattered X-rays in the polarization measurements (see \nSec.~\\ref{subsec:CHESS}). Therefore, the best performing $5 \\, \\rm{mm}$ \ndetectors were positioned in these rings accordingly. Some of the lower \ndetector rings ({\\it R4} to {\\it R8}) show regions with clearly \npoorer-than-average energy resolution. However, any asymmetry in \nazimuthal detector performance will cancel out due to the rotation of \nthe polarimeter in the final mode of operation. The energy line at \n$121.8 \\, \\rm{keV}$ is not well-defined in the spectra measured with the \n$2 \\, \\rm{mm}$ detectors (see Fig.~\\ref{fig:EnergyCalibration}, right). \nTherefore, Fig.~\\ref{fig:PixelResolution} only shows the $121.8 \\, \n\\rm{keV}$ energy resolutions for the $5 \\, \\rm{mm}$ detectors. The \naverage energy resolution in rings {\\it R1} to {\\it R3} at $40.1 \\, \n\\rm{keV}$ amounts to $4.0 \\, \\rm{keV}$ ($10.0\\%$); the corresponding \nvalue at $121.8 \\, \\rm{keV}$ is $4.9 \\, \\rm{keV}$ ($\\simeq 4\\%$). The \naverage performance of rings {\\it R4} to {\\it R7} at $40.1 \\, \\rm{keV}$ \namounts to $4.6 \\, \\rm{keV}$ ($\\simeq 11\\%$). The performance of the \ndetectors in ring {\\it R8} is modest.\n\nThe energy resolution of a detector pixel is mainly determined by two \nfactors~-- the quality of the CZT crystal and the noise of the read-out \nelectronics. The electronic readout noise was determined for all \nchannels using the ASICs internal pulse generator (see \nSec.~\\ref{sec:XCLB}). The generator injects charge into the amplifier of \nthe corresponding channel and allows one to test the \ntrigger/digitization chain on a chennel-by-channel basis. 1000 events \nwere taken per channel with the detectors connected and biased at \nnominal operation voltage (to also account for noise introduced by dark \ncurrents in the CZT). The results are shown in the middle panel of \nFig.~\\ref{fig:PixelResolution} for one of the four boards. More details \non the electronic noise studies are presented in \nSec.~\\ref{subsec:TempStudies}. Note, that the internal ASIC capacitor \ndoes not allow to inject charges that correspond to energies lower than \n$\\approx 200 \\, \\rm{keV}$. However, the noise versus energy trend seems \nto level off for energies lower than $\\simeq 500 \\, \\rm{keV}$ (see \nFig.~\\ref{fig:TempStudiesNoise}). Therefore, the absolute noise \nresolution measured at $\\simeq 200 \\, \\rm{keV}$ was used to estimate the \nrelative noise contribution at $40 \\, \\rm{keV}$, as shown in the middle \npanel of Fig.~\\ref{fig:PixelResolution}.\n\nA comparison between the electronic noise and the energy resolution \ndetermined from the spectral lines shows that the low-energy resolution \nis dominated by the electronic noise. This kind of comparison can in \ngeneral assist in localizing the cause for modestly performing \ndetectors, e.g. by disentangling the contributions of electronic noise \nversus CZT crystal quality. The detector located in ring {\\it R5} of \n{\\it Bd2}, for example, shows noisy regions in both, the Eu$^{152}$ \ndata, as well as in the electronic noise measurement~-- indicating a \nhigh leakage current as the reason for the sub-optimal performance. The \nring {\\it R8} detector in the same board, on the other hand, does not \nexhibit a poorer energy resolution than others in terms of noise~-- \ntherefore, the poor performance visible in the Eu$^{152}$ data is \nprobably related to a low CZT crystal quality.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{PolarimeterTemperature}\n\\end{center}\n\n\\caption{Temperature-dependent performance of individual CZT detectors. \nEach data point represents the average of $64$ pixels of a detector. \nVertical arrows indicate the shift of the curve if considering only edge \npixels of a detector (dashed direction), or only central pixels of a \ndetector (solid direction). {\\bf Top:} Energy resolution at $122 \\, \n\\rm{keV}$ (only shown for the $5 \\, \\rm{mm}$ detectors). {\\bf \n2$^{\\rm{\\bf{nd}}}$ from top:} Energy resolution at $59 \\, \\rm{keV}$. \n{\\bf 3$^{\\rm{\\bf{rd}}}$ from top:} Peak detection rate of the $59 \\, \n\\rm{keV}$ line normalized to the rate measured at $T=15^{\\circ} \\, \n\\rm{C}$. {\\bf 4$^{\\rm{\\bf{th}}}$ from top:} Energy threshold as \ndetermined after threshold optimization performed at the corresponding \ntemperature. {\\bf Bottom:} Mobility-lifetime product \n$\\mu_{\\rm{e}}\\tau_{\\rm{e}}$ measured based on the $59 \\, \\rm{keV}$ \nline.}\n\n\\label{fig:TempStudies}\n\\end{figure}\n\n\nOn average, the side pixels of individual detectors exhibit a poorer \nthan average energy resolution. At low energies, this shows up as a \nperiodic structure on the left and right side of each detector. The same \npattern can be seen in the electronic noise measurement \n(Fig.~\\ref{fig:PixelResolution}, left versus middle), and can therefore \nbe attributed to the readout noise. This hypothesis is supported by the \nfact that the leads between the corresponding edge pixels and the ASIC \nchannels are located on the `outside' region of the printed circuit \nboard~-- being more susceptible to noise pick up from the surrounding \nelectronics. Subtracting the readout noise, these edge pixels do no \nlonger show poorer energy resolution at low energies as compared to the \nother pixels. For energies $E > 100 \\, \\rm{keV}$ (not relevant for \nX-Calibur), the detector edge pixels show a reduced resolution in \naddition to the electronic noise component (`frame-like' structure \nsurrounding each detector in Fig.~\\ref{fig:PixelResolution}, right). \nThis is a known issue with CZT detectors and can be explained by a less \nhomogeneous electric field in the edge regions of a detector, affecting \nthe charge collection. The effect is less prominent (if visible at all) \nfor the horizontal edge pixels, for which the field is stabilized by the \nneighboring detectors located in the same plane.\n\n\n\\subsection{CZT Performance at different Temperatures} \n\\label{subsec:TempStudies}\n\nDuring a balloon flight the pressure vessel housing the polarimeter will \nundergo several changes in temperature that can potentially affect its \nperformance. At float altitude, the vessel will be in an outside \ntemperature environment of around $-20^{\\circ} \\rm{C}$. During daytime, \nthe thermal radiation fields of the sun and the earth will provide \nadditional sources of energy. During the night, the thermal radiation of \nthe earth is the only external source of heat flow. The electronics \ninside the vessel generate heat at a rate of less than $100 \\, \\rm{W}$. \nThe outside of the vessel will be insulated using a layer of aluminized \nmylar (reflection of sun light and high emittance of thermal radiation) \nthat will be contact-separated from the surface of the vessel with a \nlayer of Dacron mesh. Furthermore, heater bands with a total power of \n$175 \\, \\rm{W}$ are installed inside the vessel to guarantee a \ncontrollable temperature in the range of $0$ to $25^{\\circ} \\rm{C}$. \nThis thermal design will prevent overheating during day-time and will \navoid cold temperatures during the night. Nonetheless, variations in \ntemperature of the polarimeter during flight are expected to some \nextent. Therefore, it is important to understand the \ntemperature-dependent performance of the CZT detectors.\n\nIndividual CZT detectors were used to quantify the temperature \ndependence of the energy resolution, the energy threshold and the \ndetection rate. Data were taken in a temperature chamber in the range of \n$T \\in [-25; +35]^{\\circ} \\, \\rm{C}$ with the detectors being \nilluminated with an Am$^{241}$ source ($59 \\, \\rm{keV}$) and a Co$^{57}$ \nsource ($122 \\, \\rm{keV}$). The sources were located at a distance of \n$\\simeq 2 \\, \\rm{cm}$ above the detector cathode. The data were used to \nre-calibrate each detector (pixel-by-pixel) for each environment \ntemperature in order to cancel temperature-dependent calibration \neffects. Since these measurements were time-intensive and could only be \ndone for one detector at a time, the study was limited to a \nrepresentative subset of the detectors shown in \nTab.~\\ref{tab:Detectors}.\n\n{\\it Temperature-dependent detector characteristics.} The results of the \nmeasurements are shown in Fig.~\\ref{fig:TempStudies}, averaged over all \n64 pixels per detector. The $122 \\, \\rm{keV}$ line is not well defined \nin the $2 \\, \\rm{mm}$ detectors, so that the high-energy results are \nonly shown for the $5 \\, \\rm{mm}$ detectors. Given the limited sample of \ndetectors, we cannot expect to attribute observed trends to a specific \ndetector class (such as the brand); however, some general findings can \nbe identified and are described in the following.\n\nThe energy resolution generally improves if reducing the temperature \nfrom $T = 35^{\\circ} \\, \\rm{C}$ to $T = 0^{\\circ} \\, \\rm{C}$. This \neffect is most prominent for the two tested Quikpak detectors and \namounts to an improvement of up to $50\\%$/$30\\%$ at $59/122 \\, \n\\rm{keV}$, respectively. The effect is much weaker for the Creative \nElectron detectors. For temperatures $T < 0^{\\circ} \\, \\rm{C}$, the \nresolution levels out. For some of the Endicott detectors ($2 \\, \n\\rm{mm}$) it even gets poorer again; however, the two detectors showing \nthis effect (EN$_{2}$4 and EN$_{2}$5) show very poor performance in \ngeneral. All three $5 \\, \\rm{mm}$ detectors show a much better energy \nresolution at $122 \\, \\rm{keV}$ in the central pixels (indicated by the \ndownward arrow in the top panel of Fig.~\\ref{fig:TempStudies}) as \ncompared to the edge pixels (upward arrow). Here, the aspect ratio \nallows for E-field lines to bulge out of the sides of the detector (see \nalso Fig.~\\ref{fig:PixelResolution}). This difference is clearly less \npronounces at $59 \\, \\rm{keV}$ where the charge collection is much more \nconcentrated in the cathode region of the detector crystal.\n\nThe peak detection rate at $59 \\, \\rm{keV}$ (3$^{\\rm{rd}}$ panel from \nthe top in Fig.~\\ref{fig:TempStudies}) shows a clear trend for all \ndetectors: a reduced detection efficiency with decreasing temperature. \nThis effect is not understood. Although the measurements were carefully \nset up, it cannot be excluded that a temperature-dependent contraction \nof the casing/fixture could have lead to a slight change in distance \nbetween the source and the detector as a function of environment \ntemperature.\n\nFor each temperature, the energy thresholds were re-optimized. The \nresults are shown in the 4$^{\\rm{th}}$ panel of \nFig.~\\ref{fig:TempStudies}. No clear trends can be identified~-- the \naverage energy threshold of the studied detectors lies between $15-20 \\, \n\\rm{keV}$.\n\nMeasurements at different bias voltages $V_{\\rm{bi}}$ were used to \ndetermine the mobility lifetime product $\\mu_{\\rm{e}}\\tau_{\\rm{e}}$. \nData were taken at $V_{\\rm{bi}} = -200 \\, \\rm{V}$, $-500 \\, \\rm{V}$, and \n$-700 \\, \\rm{V}$ ($5 \\, \\rm{mm}$ detectors) and at $V_{\\rm{bi}} = -100 \n\\, \\rm{V}$, $-150 \\, \\rm{V}$, and $-200 \\, \\rm{V}$ ($2 \\, \\rm{mm}$ \ndetectors), respectively. The shift of the line position with increasing \n$V_{\\rm{bi}}$ was used to calculate $\\mu_{\\rm{e}}\\tau_{\\rm{e}}$ \n\\cite{MuTauImproved}. The results are shown in the bottom panel of \nFig.~\\ref{fig:TempStudies}. No significant change in \n$\\mu_{\\rm{e}}\\tau_{\\rm{e}}$ can be identified for the temperature range \nstudied. \\citet{Jung2007} discuss the temperature dependence of Imarad \nHigh-Pressure Bridgman CZT detectors and find a decreasing trend of \n$\\mu_{\\rm{e}} \\tau_{\\rm{e}}$ for temperatures $T > 10^{\\circ} \\, \\rm{C}$ \nand for $T < -25^{\\circ} \\, \\rm{C}$.\n\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PeakPosVsTemp}\n\\end{center}\n\n\\caption{Reconstructed line energy for different temperatures using the \ncalibration obtained at $T_{\\rm{cal}} = 25^{\\circ} \\, \\rm{C}$ (dashed \nvertical line). The data points indicate the mean of the peak positions \nof all 64 pixels and the error bars represent the standard deviation of \nthat distribution (separately derived for each temperature). The dashed \nhorizontal line indicates the nominal line energy.}\n\n\\label{fig:PeakPosVsTemp}\n\\end{figure*}\n\n{\\it Temperature-stability of the calibration.} In the studies shown in \nFig.~\\ref{fig:TempStudies} all detector pixels were re-calibrated at \neach temperature. It is important to understand how the calibration \nitself changes with temperature. To study this effect, a reference \ncalibration at $T_{\\rm{cal}} = 25^{\\circ} \\, \\rm{C}$ was applied to the \nmeasurements taken at the different temperatures. The line positions at \n$59.5 \\, \\rm{keV}$ and $122.1 \\, \\rm{keV}$ were determined for the \nindividual pixels. Each distribution (one per temperature) of \nreconstructed line positions was in turn characterized by its mean and \nits standard deviation. The results are illustrated in \nFig.~\\ref{fig:PeakPosVsTemp}, where the standard deviation (spread of \nthe corresponding distribution) is represented as error bar. For \nreference, the spectra of detector QP$_{5}$1 were corrected with the \ncalibration obtained at $T_{\\rm{cal}} = 5^{\\circ} \\, \\rm{C}$. While the \nmean reconstructed line energy does not change significantly, the \nwidening of the error band indicates that individual channels show a \ntemperature-dependent upward/downward drift of the line position. This \nmakes it difficult to globally correct for temperature-dependent changes \nin the calibration~-- unless calibration data are taken for all 2048 \nX-Calibur channels at a variety of temperatures. The calibration changes \nby up to $\\simeq 2\\%$ for temperatures varying around $\\pm 10^{\\circ} \\, \n\\rm{C}$ relative to the reference calibration $T_{\\rm{cal}}$.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{TempNoise}\n\\end{center}\n\n\\caption{Electronic readout noise measured with the ASIC's internal test \npulser for different temperatures and different energies (detector \nCE$_{5}$1 and EN$_{5}$8). For reference, the gray asterisk marker \nindicates the 1~std.dev. range of all 2048 channels as operated in the \nX-Calibur assembly at room temperature of $T \\approx 25^{\\circ} \\, \n\\rm{C}$ with/without biased detectors (solid/dashed line). {\\bf Top:} \nChannel-by-channel noise (vertical dotted lines) and detector average \nwith the detector being biased at $V_{\\rm{bi}}=-500 \\, \\rm{V}$. {\\bf \nMiddle:} Detector averages for three different temperatures with and \nwithout the cathode bias $V_{\\rm{bi}}$. {\\bf Bottom:} A series of $T \n\\approx 25^{\\circ} \\, \\rm{C}$ measurements with different configurations \nas described in the text.}\n\n\\label{fig:TempStudiesNoise}\n\n\\end{figure}\n\n{\\it Electronic noise.} The read-out electronic contributes a certain \namount of jitter to the measured energy resolution of a detector pixel. \nIn order to quantify this contribution, a series of measurements was \ntaken with detectors CE$_{5}$1 and EN$_{5}$8 (see \nTab.~\\ref{tab:Detectors}) at different temperatures using the internal \npulse generator of the ASIC (see Sec.~\\ref{sec:XCLB}). The detectors \nwere operated in an electrically shielded copper box. With the detector \nbeing plugged into the ASIC, the measurements actually reflect the \nreadout noise of the ASIC/detector assembly, rather than the noise of \nthe ASIC alone. Different amounts of charge were injected (corresponding \nto different energies)\\footnote{Note, that given the differences in \npixel acceptance and pixel calibration, a fixed amount of charge \ninjected into the ASIC translates to slightly varying reconstructed \nenergies (if comparing different channels).}. The measured pulse heights \nwere transformed to energies using the corresponding energy calibration \ndetermined for each temperature. For each configuration and channel, a \ntotal of 1000 events were injected. The calibrated distribution was \nfitted in order to determine the corresponding mean energy and energy \nresolution.\n\nThe results are shown in Fig.~\\ref{fig:TempStudiesNoise}. At energies of \naround $200 \\, \\rm{keV}$ the electronic noise of the ASIC/detector unit \nincreases by $(21 \\pm 6)\\%$ in the studied temperature range of \n$-25^{\\circ} \\, \\rm{C}$ to $+35^{\\circ} \\, \\rm{C}$. The shape of the \nenergy-dependent noise curve depends on the temperature. At room \ntemperature, the noise increases by $(20 \\pm 4)\\%$ if going from $200 \\, \n\\rm{keV}$ to $1.7 \\, \\rm{MeV}$ and levels out around $3 \\, \\rm{keV}$ at \nlow energies. This suggests, that the energy resolution in the X-Calibur \nrange ($\\approx 4 \\, \\rm{keV}$ at $40 \\, \\rm{keV}$, see \nFig.~\\ref{fig:PixelResolution}) is dominated by the electronic readout \nnoise, rather than charge transport properties in the CZT crystal. The \ngray asterisk marker in Fig.~\\ref{fig:TempStudiesNoise} shows the \n1~std.dev.~range of the noise distribution of all 2048 data channels as \nmeasured at room temperature ($\\approx 25^{\\circ} \\, \\rm{C}$) in the \nfinal X-Calibur configuration, compare with \nFig.~\\ref{fig:PixelResolution} (middle).\n\nThe middle panel of Fig.~\\ref{fig:TempStudiesNoise} illustrates the \neffect of the bias voltage of the detector. A biased cathode at $T = \n25^{\\circ} \\, \\rm{C}$ increases the low-energy noise by $(8 \\pm 5)\\%$, \nwhereas no noticeable change can be measured for temperatures lower than \nthat. Therefore, the cathode bias only seems to systematically affect \nthe readout noise for temperatures higher than $\\simeq 15^{\\circ} \\, \n\\rm{C}$.\n\n\nThe bottom panel in Fig.~\\ref{fig:TempStudiesNoise} shows the electronic \nnoise measured at room temperature for different configurations: (i) the \ndetector/ASIC unit with biased cathode, (ii) the detector/ASIC unit with \nunbiased cathode, (iii) the ASIC with a ceramic chip carrier but no \ndetector bonded to it, and (iv) only the plain ASIC. It can be seen that \nsteps (i)--(iii) each add $\\approx 0.2 \\, \\rm{keV}$ readout noise to the \nsingle-detector system. The average readout noise of the whole X-Calibur \nassembly (gray asterisk in Fig.~\\ref{fig:TempStudiesNoise}) is shown for \nreference. Another series of measurements was performed with the plain \nASIC at different temperatures (not shown). No significant noise trend \ncould be identified in the $T = -20^{\\circ} \\, \\rm{C}$ to $+25^{\\circ} \n\\, \\rm{C}$ temperature range which leads to the conclusion that the \ntemperature dependence of the readout noise of the ASIC/detector unit \n(top panel of Fig.~\\ref{fig:TempStudiesNoise}) is mostly a result of the \ntemperature dependence of the dark currents in the CZT crystal.\n\n{\\it Caveats:} The electronic readout noise varies from ASIC to ASIC and \ndepends on the electronic shielding environment in which the ASIC is \noperated. Therefore, the comparison between the absolute noise levels of \nthe single ASIC system shown in Fig.~\\ref{fig:TempStudiesNoise} and the \naverage readout noise in the X-Calibur assembly should be treated with \ncare; the relative noise trends found, however, can likely be applied to \nwhole X-Calibur assembly. It should also be mentioned, that the cooling \naggregates of the temperature chamber (increased activity at low \ntemperatures) can potentially introduce external noise pick-up in the \nASIC.\n\n\n\\subsection{Summary of the Detector Calibration and Tests}\n\nEach detector pixel has been energy calibrated according to \nEq.~(\\ref{eq:EnergyCalibration}). With the current readout electronics \nand the compact X-Calibur configuration, the CZT detectors achieve a \nmean trigger threshold of $\\simeq 21 \\, \\rm{keV}$. The mean energy \nresolution at $40 \\, \\rm{keV}$ in the three front-side detector rings \n{\\it R1}-{\\it R3} (detecting most of the scattered events in the \npolarization measurements) is found to be $\\Delta E_{\\rm{czt}} \\simeq 4 \n\\, \\rm{keV}$ FWHM, when operated at room temperature. The energy \nresolution of the polarimeter as a whole is determined by the energy \nresolution of the individual detectors and by the energy deposited/lost \nin the scintillator ($\\Delta E_{\\rm{sci}} = 0 - 5.4 \\, \\rm{keV}$ at $40 \n\\, \\rm{keV}$). The energy resolution of the detectors is thus not \nentirely negligible and X-Calibur would benefit from an optimized \nreadout ASIC. We are currently working on modifying and adopting the \nHD-3 ASIC \\cite{DeGeronimo2003, Vernon2010}. Using a pre-amplifier chain \noptimized for the $2-100 \\, \\rm{keV}$ energy range, we expect a trigger \nthreshold of $1.7 \\, \\rm{keV}$ and electronic readout noise of $\\simeq \n550 \\, \\rm{eV}$ RMS. The noise contribution of the new ASIC to the \nenergy resolution of the polarimeter would be negligible for all \nenergies above $20 \\, \\rm{keV}$. The low energy threshold can \npotentially be used on a satellite-borne version of the polarimeter.\n\nThe effective energy threshold of the polarimeter is slightly higher \nthan the energy threshold of the individual CZT detectors, as a $25 \\, \n\\rm{keV}$ photon looses up to $2.2 \\, \\rm{keV}$ in the scatterer. \nHowever, the polarimeter will detect a large fraction of the X-rays at \n125,000 feet flight altitude, as the residual atmosphere only transmits \nphotons above $\\simeq 25 \\, \\rm{keV}$.\n\nEven though the temperature-dependent trends in energy resolution would \nfavor operating the detectors at $T \\leq 0^{\\circ} \\, \\rm{C}$ \n(Fig.~\\ref{fig:TempStudies}, top), the thermal design of the X-Calibur \nassembly and local internal heat built-up of the polarimeter during the \nballoon flight makes an operation at $T \\simeq (15 \\pm 10)^{\\circ} \\, \n\\rm{C}$ a more likely scenario~-- still guaranteeing a reasonable energy \nresolution for most of the detectors. A change in temperature, which \nwill be monitored during flight, leads to a shift in reconstructed \nenergy~-- which can go either way (Fig.~\\ref{fig:PeakPosVsTemp}). \nIdeally, one should use a data base of temperature-dependent calibration \nvalues on a pixel-by-pixel basis to correct for the temperature trends. \nIf ignoring the temperature-dependence of the calibration, a systematic \nerror on the reconstructed energy of a few percent has to be accounted \nfor in the temperature interval of $\\pm 10^{\\circ} \\, \\rm{C}$ around the \ncalibration temperature. For the first X-Calibur flight, we will choose \nthe second option.\n\n\n\n\\section{X-Calibur: Instrument Characterization} \\label{sec:XCLB_Characterization}\n\nThis section describes measurements of the fully assembled polarimeter \ninstalled in the CsI shield. The goal of the measurements is to \ncharacterize the efficiency of the shield, and to estimate the \nbackground levels in the different (ground-based) environments the \npolarimeter was operated in (Sec.~\\ref{subsec:BG_Data}). The reduction \nof the background is crucial in order to perform sensitive measurements \nof the polarization properties of astrophysical sources, see \nEq.~(\\ref{eqn:MDP}).\n\nThe X-rays that enter the polarimeter along the optical axis will \nproduce a certain amount of scintillation light when scattering in the \nscintillator rod which is read out by a PMT. The efficiency curve, \ndescribing the trigger probability for different energy depositions in \nthe scatterer, is discussed in Sec.~\\ref{subsec:ScintillatorEfficiency}. \nAs will be shown in Sec.~\\ref{subsec:BG_Data}, a high trigger efficiency \nof the scintillator will allow a further reduction of the background. \nThe efficiency curve is fed into the simulations that were discussed in \nSec.~\\ref{sec:Simulations}. The measurements described in this chapter \nwere done with disk-like radioactive sources with a diameter of $\\simeq \n0.5 \\, \\rm{cm}$.\n\n\n\\subsection{Shield Performance and Cosmic Ray Background} \\label{subsec:BG_Data}\n\nThe shielding and suppression of backgrounds is a crucial task for \nsensitive polarimetry measurements. The background in the ground-based \nmeasurements presented in this paper (non-flight) results mostly from \nsecondary particles produced in air showers in the earth's atmosphere, \ninduced by cosmic rays (CRs). The flux of the secondary particles \ndepends on the geographical location at which the measurement is \nperformed, and on the structure/materials of the building in which the \npolarimeter is operated (partly shielding the secondary particles). As \ncan be seen in Fig.~\\ref{fig:MultiplicityDistribution}, the CR \nbackground has a higher fraction of multiplicity $m \\geq 2$ pixel events \nas compared to Compton-scattered X-rays in the $E < 100 \\, \\rm{keV}$ \nregime, relevant for the X-Calibur polarimetry measurements. In general, \nthe distribution of $m$ depends on the energy and the kind of \ninteraction; high-energy muons (ionization), for example, trigger events \nalong a row of pixels (unless they cross the detector perpendicular to \nthe pixel plane). Low energy X-rays, on the other hand, are \nphoto-absorbed with a charge deposition usually contained well within \none pixel. Therefore, a requirement of $m = 1$ pixel events already \nsuppresses the background for the polarization measurements by a certain \namount.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{ShieldScalerVsDiscriminator}\n\\end{center}\n\n\\caption{Optimization of the shield discriminator using three different \ndata runs (a)-(c), as described in the text. The background contribution \nis subtracted for the runs with Cs$^{137}$ outside shield (b), and for \nAm$^{241}$/Eu$^{152}$ on the optical axis (c). The dashed vertical line \nindicates the optimal discriminator setting. {\\bf Top:} Shield trigger \nrate for different threshold settings. The horizontal lines indicate the \nrates corresponding to $10/5/2\\%$ (from top to bottom) dead time \nproduced by the shield vetoes. The noise regime starts to dominate \naround DAC $<20$; this in turn leads to an underestimation of the \nCs$^{137}$/Eu$^{152}$ trigger contribution since the CR background is \nsubtracted. In the case of the Am$^{241}$ source, no significant trigger \nrate above background was measured. {\\bf Bottom:} Fraction of detected \nCZT events that are vetoed with $f_{\\rm{shld}} = 1$.}\n\n\\label{fig:ShieldOptimization}\n\\end{figure}\n\n{\\it Optimization of the shield trigger threshold.} A particle crossing \nthe active shield (see Fig.~\\ref{fig:ActiveShieldAndTelescope, left) \nproduces scintillation light in the CsI crystal. The crystal is read by \nfour PMTs which signals are summed and digitized.} A programmable \ndiscriminator decides whether the shield veto flag $f_{\\rm{shld}}$ is \nactivated ($f_{\\rm{shld}}$ is kept active for $6 \\, \\mu\\rm{s}$ per \ntrigger) and merged into the data stream. Events with the corresponding \nflag can in turn be filtered out in the data analysis. In order to \noptimize the shield trigger efficiency, a series of measurements was \nperformed with different settings of the discriminator. The measurements \ncomprise: (a) cosmic ray background only, (b) a collimated Cs$^{137}$ \nsource aimed from outside the shield at the X-Calibur CZT detector \nassembly (configuration $C_{\\rm{wu2b}}^{3}$ in \nFig.~\\ref{fig:X-Calibur_Configurations}), and (c) collimated \nAm$^{241}$/Eu$^{152}$ sources placed on the optical axis of the \npolarimeter (configuration $C_{\\rm{wu2a}}^{3}$) to simulate X-rays from \nthe X-ray mirror entering the polarimeter without interaction in the \nshield\\footnote{Obviously, the lead used to collimate the sources leads \nto indirect scatterings and shield contamination, in particular in the \ncase of the high energy lines of Eu$^{152}$.}. Two shield \ncharacteristics were measured: (i) the raw trigger rate of the shield, \nand (ii) the fraction of triggered CZT events with $f_{\\rm{shld}} = 1$. \nThe results are shown in Fig.~\\ref{fig:ShieldOptimization}. Since the \ncosmic ray background was present in all three runs, the X-ray data runs \n(b) and (c) were corrected for the rates measured in run (a). The \nsetting of the discriminator was optimized according to the following \ncriteria.\n\n\\begin{enumerate}\n\n\\item The dead time produced by noise triggers should not exceed a few \npercent (horizontal lines in Fig.~\\ref{fig:ShieldOptimization}). The \nAm$^{241}$/Eu$^{152}$ sources located on the optical axis (c) should \nresult in CZT events with a $f_{\\rm{shld}} = 1$ contribution as low as \npossible (no signal suppression).\n\n\\item For the CR (a) and Cs$^{137}$ (b) runs, the fraction of CZT \ntriggers with $f_{\\rm{shld}} = 1$ should be as high as possible, \nreflecting a high rejection power.\n\n\\end{enumerate}\n\nWe determined an optimal shield discriminator setting of $\\rm{DACQ} = \n30$ which is indicated by the vertical line in \nFig.~\\ref{fig:ShieldOptimization}. Note, the fraction of Eu$^{152}$ \nevents with $f_{\\rm{shld}} = 1$ is around $25 \\%$, which is probably due \nto X-rays that Compton scatter and interact with the shield and CZT~-- a \nresult of not having a well collimated X-ray beam entering the \npolarimeter in this measurement.\n\n\n\\begin{table}[t!]\n\n\\begin{tabular}{lrr}\n\nData set & $\\left< T_{5 \\, \\rm{mm}} \\right>$ & $\\left< T_{2 \\, \\rm{mm}} \\right>$ \\\\\n & [mHz] & [mHz] \\\\\n\\hline \\hline\n\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{CZT initial calibration (Washington Univ.): $C_{\\rm{wu1a}}^{1}$} \\\\\n\\hline\n\nEu$^{152}$ illumination & $7100$ & $6200$ \\\\\nBackground & $59$ & $35$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{Scintillator characterization (Wash. Univ.): $C_{\\rm{wu2a}}^{3}$} \\\\\n\\hline\n\nEu$^{152}$ on optical axis$^{\\rm{*}}$ & $31$ & $8.7$ \\\\\nBackground in shield (active) & $9.9(5.4)$ & $4.9(2.0)$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{CHESS synchrotron beam (Cornell Univ.): $C_{\\rm{ch}}^{3}$} \\\\\n\\hline\n\n$40 \\, \\rm{keV}$ X-ray beam & $1100$ & $300$ \\\\\nBackground & $13$ & $8.4$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{X-Calibur/shield/InFocus (Ft.~Sumner): $C_{\\rm{ft}}^{3}$} \\\\\n\\hline\n\nX-ray source$^{\\rm{*}}$ & $853$ & $190$ \\\\\nBackground in shield (active) & $8.5(3.9)$ & $4.7(1.6)$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{X-Calibur flight (simulations)} \\\\\n\\hline\n\nCrab$^{\\rm{*}}$ & $1.25$ & $0.3$ \\\\\n\n\\end{tabular}\n\n\\caption{Average event trigger rates per CZT detector pixel for $5 \\, \n\\rm{mm}$ detectors $\\left< T_{5 \\, \\rm{mm}} \\right>$ and for $2 \\, \n\\rm{mm}$ detectors $\\left< T_{2 \\, \\rm{mm}} \\right>$. Rates are shown \nfor the different measurements/environments presented in this paper (see \nFig.~\\ref{fig:X-Calibur_Configurations}). No event selection cuts are \napplied. Rates marked with a `*' are background subtracted. The \ncorresponding background spectra for the different data sets are shown \nin Fig.~\\ref{fig:BG_Spectra}.}\n\n\\label{tab:PixelRates}\n\n\\end{table}\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{BackgroundSpectra_Mosaic}\n\\hfill\n\\includegraphics[width=0.49\\textwidth]{BackgroundSpectra2_Mosaic}\n\\end{center}\n\n\\caption{Cosmic ray background spectra (events with pixel multiplicity \n$m=1$, if not mentioned otherwise). The dotted vertical lines indicate \nthe ranges of the $\\rm{KL}_{1-3}$ transition energies of tungsten (W) \nand lead (Pb) which show up in some of the spectra. The \nX-Calibur/{\\it InFOC$\\mu$S}\\xspace sensitive energy range is indicated, as well. {\\bf \nLeft:} Spectra measured in different environments (see \nFig.~\\ref{fig:X-Calibur_Configurations}). Date and duration of the runs \nis given in square brackets. The (passive/active) CsI shield strongly \nreduces the background. The three dashed vertical lines on the \n$C_{\\rm{wu1a}}^{1}$ spectrum represent the high-energy lines of \nEu$^{152}$ ($244.7 \\, \\rm{keV}$ and $344.3 \\, \\rm{keV}$) and Cs$^{137}$ \n($661.7 \\, \\rm{keV}$) for which an onset indication can be seen; the \nsources were stored in a lead bunker $\\sim$$2 \\, \\rm{m}$ away during the \nbackground measurements and were probably contributing to the spectrum \nat a low level. {\\bf Right:} Energy spectra for individual detector \nrings measured at Washington University in the active shield (after \nshield veto, $f_{\\rm{shld}} = 0$). We also show energy spectra (scaled \nby a factor of 0.1) from different pixels of the detectors on the \nreadout board {\\it Bd2} to illustrate the difference between central \npixels and pixels located on the edge of detectors, see also \nFig.~\\ref{fig:BG_Maps2D} for reference.}\n\n\\label{fig:BG_Spectra}\n\n\\end{figure*}\n\n{\\it Cosmic ray background levels.} Figure~\\ref{fig:BG_Spectra} shows \nthe CR energy spectra ($m=1$ pixel multiplicity) measured in the CZT \ndetectors at the different locations: the laboratory at Washington \nUniversity ($C_{\\rm{wu}}$), the CHESS X-ray beam facility \n($C_{\\rm{ch}}$, Sec.~\\ref{subsec:CHESS}), and in Ft.~Sumner \n($C_{\\rm{ft}}$, Sec.~\\ref{subsec:FtSumnerData}). For reference, the \ncorresponding event trigger rates ($m \\geq 1$) per pixel are summarized \nin Tab.~\\ref{tab:PixelRates}. Note, that the actual pixel rates can vary \nquite substantial, since different pixels see different income fluxes \ndepending on the type of measurement (e.g. an X-ray beam Compton \nscattered in the scintillator leads to a strongly depth-dependent \nillumination of CZT detectors, compare with \nFig.~\\ref{fig:CHESS_2DAzimuthData}). However, it can be seen that for \nmost measurements presented in this paper the background can be \nneglected. During the balloon flight, however, the expected \nsource-to-background ratio will be much lower and a proper understanding \nand the background suppression will be crucial \\cite{Guo2010, Guo2013}. \n\nEven though the background at flight altitude will be different as \ncompared to the ground-based CR background, some characteristics in the \ndetector response can be discussed qualitatively based on the spectra \nshown in Fig.~\\ref{fig:BG_Spectra}. While $\\alpha$-particles interact \nclose to the surface of the detector, muons as well as primary and \nsecondary high-energy gamma rays will penetrate deeper and their energy \ndeposition is proportional to the detector volume. This volume-dependent \nbackground rate can be seen by comparing the spectra (and trigger rates) \nmeasured with $5 \\, \\rm{mm}$ versus $2 \\, \\rm{mm}$ detectors. Note, that \nthe spectra shown in Fig.~\\ref{fig:BG_Spectra} represent particle fluxes \nfolded with the energy-dependent response of the CZT detectors (with an \nenergy calibration derived from, and valid for, X-rays). The drop in \nevent rate below $30 \\, \\rm{keV}$, for example, is an effect of the \nsuperposition of the different trigger thresholds of the pixels \ncontributing to the spectrum. The dynamical energy range covered by a \nsingle pixel saturates around $2000 \\, \\rm{keV}$ (differing from pixel \nto pixel), so that the combined $m=1$ pixel spectra shown in \nFig.~\\ref{fig:BG_Spectra} drop off around this energy. Allowing events \nwith $m \\geq 1$ multiplicities will extend the energy range (see \nFig.~\\ref{fig:BG_Spectra}, right) which is, however, not relevant for \nthe operation of X-Calibur.\n\n{\\it The CsI shield efficiency.} Figure~\\ref{fig:BG_Maps2D} shows the 2D \ndistribution of event count rates from background data taken with \nX-Calibur installed in the CsI shield at Washington University. The \nrates are shown for two different energy bands. The left panels show the \nraw rates and the right panels show the rates after rejecting events \nthat triggered the active shield, only allowing non-vetoed events \n($f_{\\rm{shld}} = 0$). It again becomes obvious that the $5 \\, \\rm{mm}$ \ndetectors ({\\it R1}-{\\it R5}) collect more background compared to the $2 \n\\, \\rm{mm}$ detectors ({\\it R6}-{\\it R8}). Furthermore, the edge pixels \nof the detector rows on the individual boards {\\it Bd0-Bd3} see a higher \nbackground rate as compared to central pixels (likely because of the \nhigher exposed surface area detecting charged particles and low energy \nX-rays). The energy-resolved difference between edge and central pixels \ncan be seen in Fig.~\\ref{fig:BG_Spectra}, right ($5 \\, \\rm{mm}$ \ndetectors of boards {\\it Bd2}): the central pixels show an almost two \ntimes lower background compared to edge pixels in the energy regime \nrelevant for X-Calibur.\n\nThe background distribution after shield veto (Fig.~\\ref{fig:BG_Maps2D}, \nright) exhibits a spatial gradient with more background events being \ndetected closer to the front side of the experiment, as that side is \nonly shielded by the passive tungsten cap (see \nFig.~\\ref{fig:ActiveShieldAndTelescope}, left). In particular, the \nfront-side (top) pixel row of ring {\\it R1}, with its exposed detector \nside walls, suffers strongly from primary radiation leaking through the \ntungsten, as well as secondary particles being produced in the tungsten. \nHere, spectral signatures of the tungsten KL transitions can be \nidentified in the measured spectra shown in Fig.~\\ref{fig:BG_Spectra} \n(right). Since this single pixel row is not crucial for the polarimetry \nsensitivity it can be excluded from the analysis.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{BG_Maps2D_35-150keV} \\\\\n\\includegraphics[width=0.49\\textwidth]{BG_Maps2D_200-1000keV}\n\\end{center}\n\n\\caption{2D maps of the CR count rates (pixel-by-pixel, representation \nsame as in Fig.~\\ref{fig:PixelThreshold}) of a $192 \\, \\rm{h}$ \nbackground run with X-Calibur being installed in the active CsI shield \n($C_{\\rm{wu2c}}^{3}$ in Fig.~\\ref{fig:X-Calibur_Configurations}). The \ntungsten cap is located at the top of each panel. The count rate is \nderived by integrating the corresponding energy spectra ($m=1$ events) \nin the given energy range: $35-150\\, \\rm{keV}$ (top) and $200-1000 \\, \n\\rm{keV}$ (bottom). The left panel shows the measured rate (passive \nshield rejection only) and the right panel shows the non-vetoed events \nwith $f_{\\rm{shld}} = 0$ (active and passive rejection). Left and right \npanels are shown with the same axis/color ranges, each.}\n\n\\label{fig:BG_Maps2D}\n\\end{figure}\n\nThe passive/active rejection efficiency of the CsI shield becomes \nobvious if comparing the different background spectra in \nFig.~\\ref{fig:BG_Spectra}, left. Moving the polarimeter from its copper \nhousing (used in the initial test measurements, $C_{\\rm{wu1}}^{1}$) into \nthe CsI shield ($C_{\\rm{wu2c}}^{3}$) reduces the background by roughly \none order of magnitude due to passive shielding. Applying the \n$f_{\\rm{shld}} = 0$ veto from the active shield leads to an additional \nbackground rejection by a factor of $\\sim$$2$. Note, that the \nrecombination of the tungsten and lead $\\rm{KL}_{1,2,3}$ transition \nenergies\\footnote{http://www.nist.gov/pml/data/xraytrans/} (probably \nactivated by CRs) can be identified in the background spectra. This \nfeature is most prominent in the spectrum of ring {\\it R1} \n(Fig.~\\ref{fig:BG_Spectra}, right) which is located closest to the \ntungsten cap.\n\nIt should be noted that an additional cut on the scintillator rod \ncoincidence flag $f_{\\rm{sci}} = 1$ reduces the background by another \n$1.5$ orders of magnitude (Fig.~\\ref{fig:BG_Spectra}, left)~-- strongly \nrejecting events that did not enter the polarimeter along the optical \naxis and interacted in the scintillator, see \nSec.~\\ref{subsec:ScintillatorEfficiency}.\n\n\n\\subsection{Compton-scattering and Scintillator Trigger Efficiency} \\label{subsec:ScintillatorEfficiency}\n\n\\begin{figure}[th!]\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{SpectraScintillatorVsDAC_R1b_Mosaic} \\\\\n\\includegraphics[width=0.47\\textwidth]{SpectraScintillatorVsDAC_R2t_Mosaic} \\\\\n\\includegraphics[width=0.47\\textwidth]{SpectraScintillatorVsDAC_R3_Mosaic}\n\\end{center}\n\n\\caption{A lead-collimated Eu$^{152}$ source enters the polarimeter \nalong the optical axis and Compton-scatterers in the scintillator rod \n($C_{\\rm{wu2a}}^{3}$ in Fig.~\\ref{fig:X-Calibur_Configurations}). The \nresulting CZT spectra ($m=1$, background subtracted) are shown for \ndifferent rings. The horizontal lines indicate the complete range of \npossible energies $E_{\\rm{czt}}$ after Compton-scattering (dotted: \n$180-90^{\\circ}$, solid: $90-0^{\\circ}$). The attached vertical dotted \nlines indicate the sub range for the particular ring~-- given its \ngeometrical coverage of the scintillator. Spectra are measured with \ndifferent scintillator trigger thresholds settings (Digital to Analog \nConverter, DAC) and are compared to the spectra without coincidence \nrequirement (no cuts). The lead collimator prevents direct source hits \n(rings {\\it R1-R3}), but scattered X-rays and/or activation of the \n$\\rm{KL}_{1-3}$ transitions in tungsten (W) and lead (Pb) partly \ncontaminate the spectra.}\n\n\\label{fig:Scintillator_Spectra}\n\\end{figure}\n\nThe central scintillator rod of the polarimeter (Fig.~\\ref{fig:Design}) \nacts as a Compton-scatterer for the incoming X-ray beam. At the same \ntime, it produces scintillation light that is read out by a PMT. In \norder to characterize the Compton-scattering and the trigger response of \nthe scintillator, a collimated Eu$^{152}$ source was placed at the \npolarimeter entrance, illuminating the scintillator along the optical \naxis (see setup $C_{\\rm{wu2a}}^{3}$ in \nFig.~\\ref{fig:X-Calibur_Configurations}). The Compton-scattered X-rays \nare recorded by the surrounding CZT detector rings. An X-ray enters the \npolarimeter with a certain energy $E_{\\rm{line}}$, deposits the energy \n$\\Delta E_{\\rm{sci}}$ in the scintillator and is absorbed in a CZT \ndetector with its post-scattering energy of $E_{\\rm{czt}} = \nE_{\\rm{line}} - \\Delta E_{\\rm{sci}}$. The amount of scintillation light \ncan be assumed to be proportional to $\\Delta E_{\\rm{sci}}$.\n\nData were taken with different settings of the scintillator/PMT \ndiscriminator threshold (DAC). As can be seen in \nTab.~\\ref{tab:PixelRates}, the background level is not negligible in \nthis measurement, so that background spectra without a X-ray source were \nrecorded and subtracted. It should be noted, however, that X-rays \nscattered in the lead collimator and activation/recombination of the \n$\\rm{KL}_{1,2,3}$ electron transitions in the lead collimator and the \ntungsten cap lead to a slight contamination of the measured spectra. \nThese corresponding spectral features do not cancel out after CR \nbackground subtraction since they are (indirectly) introduced by the \nEu$^{152}$ source itself. Since those features will over-proportionally \naffect the data without the $f_{\\rm{sci}}=1$ coincidence (X-ray paths \nthat do not necessarily cross the scintillator), this may lead to an \nunderestimation of the trigger efficiency (which in the ideal case would \nonly compare Compton-scattered events in the scintillator with and \nwithout the $f_{\\rm{sci}} = 1$ trigger coincidence). The scintillator \ntrigger efficiency for different energies $\\Delta E_{\\rm{sci}}$ is \nderived as follows.\n\n{\\it Compton spectra.} The resulting Compton-scattered energy spectra \n$E_{\\rm{czt}}$ are shown in Fig.~\\ref{fig:Scintillator_Spectra} for \ndifferent CZT detector rings. Data are shown with and without the \nscintillator coincidence requirement. The energy $\\Delta E_{\\rm{sci}}$ \ndeposited in the scintillator depends on the scattering kinematics, \nincluding the primary energy of the X-ray $E_{\\rm{line}}$ and its \nscattering angle. The theoretical range of energies after \n$0-180^{\\circ}$ Compton scattering is indicated by horizontal lines for \nthe four main Eu$^{152}$ energy lines. Due to geometrical reasons, some \nof the detector rings can only detect X-rays originating from a limited \nrange of Compton-scattering angles. With the scintillator starting at \nring {\\it R2} (Fig.~\\ref{fig:Design}, left), the sub rings {\\it \nR1$_{\\rm{t}}$}, and {\\it R1$_{\\rm{b}}$}, for example, can only detect \nback-scattered X-rays which deposit/loose the highest amount of energy \n$\\Delta E_{\\rm{sci}}$ in the scintillator~-- leading to a higher trigger \nprobability. For a given incoming energy $E_{\\rm{line}}$, the range of \npossible scattering angles translates into a range of possible \nCompton-scattered energies $E_{\\rm{czt}}$. These ring-dependent sub \nranges in scattered energy are indicated by the vertical lines in \nFig.~\\ref{fig:Scintillator_Spectra}. Note, however, that the scattering \nangles/energies will not be equally distributed within these boundaries. \nOne can qualitatively discuss the X-Calibur response to \nCompton-scattering for the different scattered energy lines.\n\n\\begin{itemlist}\n\n\\item $E_{\\rm{line}} = 40 \\, \\rm{keV}$: The Compton-scattered energy \nrange of the Eu$^{152}$ line doublet ($39.52\\,\\rm{keV}$ and \n$40.12\\,\\rm{keV}$) only covers $E_{\\rm{czt}} = 34.2 - 40.1\\,\\rm{keV}$ at \nan energy resolution of $\\Delta E_{40} \\simeq 4 \\, \\rm{keV}$. This makes \nit difficult to resolve structures in the scattered energy spectrum. \nHowever, the geometrical coverage of ring {\\it R1$_{\\rm{b}}$} \nconstraints the energy deposition in the scintillator to $\\Delta \nE_{\\rm{sci}} = 2.4-5.4 \\, \\rm{keV}$ since only back-scattering angles of \n$80-180^{\\circ}$ are possible (top panel in \nFig.~\\ref{fig:Scintillator_Spectra}). A clear improvement in trigger \nefficiency is obvious when lowering the scintillator DAC threshold from \n170 to 10.\n\n\\item $E_{\\rm{line}} = 121.8 \\, \\rm{keV}$: This well-defined line at \nhigher energy leads to a broader Compton-continuum which can be tested \nat different energies $\\Delta E_{\\rm{sci}}$. While ring {\\it \nR1$_{\\rm{b}}$} again only sees the back-scattered X-rays, the rings \nfurther down cover a broad continuum range. The trigger efficiency for \nthe back-scattering of the $121.8 \\, \\rm{keV}$ line ($\\Delta \nE_{\\rm{sci}} = 23-39 \\, \\rm{keV}$) does not seem to depend on the \nscintillator DAC setting and is therefore already at its maximum. In the \nclose-to forward scattering regime ($\\Delta E_{\\rm{sci}}$ a few keV) in \nring {\\it R3} (bottom panel in Fig.~\\ref{fig:Scintillator_Spectra}) a \nlower DAC threshold leads to a broader Compton continuum in the \n$f_{\\rm{sci}} = 1$ data, reflecting the increase in trigger efficiency. \nStarting at ring {\\it R3}, however, the effective thickness of the lead \ncollimator at the polarimeter entrance is not high enough to fully \nabsorb all X-rays; this leads to some direct detector hits in the raw \nenergy spectrum~-- and therefore an underestimation of the trigger \nefficiency at the corresponding value of $\\Delta E_{\\rm{sci}}$.\n\n\\item $E_{\\rm{line}} = 244.7 \\, \\rm{keV}$ and $344.3 \\, \\rm{keV}$: These \ntwo high-energy (but lower intensity) lines lead to partially \noverlapping Compton-continua. This makes it difficult to study the \nresponse in more detail. It can also be seen, that $344.3 \\, \\rm{keV}$ \ndirect CZT hits are not fully prevented by the lead collimator (peak in \nraw spectra).\n\n\\end{itemlist}\n\n{\\it Scintillator efficiency.} The scintillator trigger efficiency as a \nfunction of $\\Delta E_{\\rm{sci}} = E_{\\rm{line}} - E_{\\rm{czt}}$ was \nestimated from the flux ratio at energy $E_{\\rm{czt}}$ with and without \nthe $f_{\\rm{sci}} = 1$ coincidence requirement (after continuum \nsubtraction). This can be done independently for the different line \nenergies and for different detector rings. Although the different \nEu$^{152}$ energies are ideal to test different values of $\\Delta \nE_{\\rm{sci}}$, the reflection and re-processing of the high energy lines \ncontaminates the raw spectrum hampering the quantitative determination \nof the trigger efficiency (underestimation). Therefore, additional data \nwere taken with an Am$^{241}$ source which has a line at $59.5 \\, \n\\rm{keV}$, but no significant emission above. The trigger efficiencies \ndetermined from the different source lines and detector rings are \nsummarized in Fig.~\\ref{fig:Scintillator_TriggerEfficiency} for the \noptimized discriminator threshold of DAC=10. The trigger distribution \nwas fitted by a function\n\n\\begin{equation}\nf(\\Delta E_{\\rm{sci}}) = \\frac{a}{1 + e^{-b (\\Delta E_{\\rm{sci}}-E_{0})} + e^{-c \n(\\Delta E_{\\rm{sci}}-E_{0})}}.\n\\label{eqn:ScintTriggEffFit}\n\\end{equation}\n\nFor the optimized setting of DAC=10, the trigger efficiency reaches \n$50\\%$ at $\\Delta E_{\\rm{sci}} \\simeq 4.6 \\, \\rm{keV}$. Note, that the \ntrigger fraction does not level out at $a = 1$ ($100 \\%$). This is \npotentially a result of the contamination of the spectrum (direct \ndetector hits)~-- caused by the non-ideal measurement setup that leads \nto a systematic underestimation of the efficiency. For the simulations \nin Sec.~\\ref{sec:Simulations}, we therefore assumed the same trigger \nfunction, however with a value of $a=1$. Note, that we currently do not \nmodel any depth-dependence of $f(\\Delta E_{\\rm{sci}})$ which becomes \nimportant when looking at the different detector rings. In a detailed \n{\\it Monte Carlo} study, \\citet{PogoScintSims} simulate the trigger \nefficiency of a similar scintillator material (EJ-204) and find a \nthreshold of $2-3 \\, \\rm{keV}$. The authors also find a $13 \\%$ drop in \nlight yield along a $20 \\, \\rm{cm}$ long scintillator rod (from close to \nthe PMT towards the distant end).\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{ScintillatorTriggerFraction}\n\\end{center}\n\n\\caption{Fraction of CZT events with a scintillator coincidence trigger \n($f_{\\rm{sci}}=1$) as a function of energy deposition $\\Delta \nE_{\\rm{sci}}$. Data were taken with Am$^{241}$, Eu$^{152}$, and at the \nCHESS beam (Sec.~\\ref{subsec:CHESS}, however with a much higher \nscintillator threshold). The Compton-scattered continua (above \nbackground, see for example Fig.~\\ref{fig:Scintillator_Spectra}) were \nused to derive the trigger efficiency for different $\\Delta \nE_{\\rm{sci}}$. The distribution was fitted by a function shown in \nEq.~(\\ref{eqn:ScintTriggEffFit}). The dashed horizontal line indicates \nthe range of scintillator energy depositions for incoming X-rays in the \n$20-80 \\, \\rm{keV}$ band.}\n\n\\label{fig:Scintillator_TriggerEfficiency}\n\\end{figure}\n\nThe raw scintillator trigger rate (independent of CZT event triggers) of \nthe optimized threshold setting of DAC=10 was measured to be only a few \nhundred Hz~-- a further reduction in trigger threshold by updating the \nsignal amplifier seems possible and is work in progress.\n\nA cleaner setup to measure the scintillator response is the \nwell-collimated X-ray beam at CHESS (see Sec.~\\ref{subsec:CHESS}). The \nresults are also shown in Fig.~\\ref{fig:Scintillator_TriggerEfficiency} \nfor a non-polarized beam at $40 \\, \\rm{keV}$ and $120 \\, \\rm{keV}$, \nrespectively. However, the amplifier for the PMT reading the \nscintillator was not optimized at the time of the CHESS measurements, so \nthat the $50\\%$ trigger probability is only reached around $\\Delta \nE_{\\rm{sci}} \\simeq 15 \\, \\rm{keV}$. The corresponding fitted trigger \nefficiency (again with the $a=1$ assumption)\\footnote{Also the CHESS \nmeasurements suffered from indirect contamination due to beam absorber \nfoils, see Sec.~\\ref{subsec:CHESS}.} was fed into the simulations that \nwere done for the CHESS measurements discussed in \nSec.~\\ref{subsec:CHESS}.\n\n\n\\subsection{Summary of the Instrument Characteristics}\n\nThe rejection power of the active shield was studied in the laboratory. \nIt reduces the background by more than one order of magnitude in the \nenergy range relevant for X-Calibur.\n\nWhen we require a scintillator trigger ($f_{\\rm{sci}} = 1$), the \neffective energy threshold of the polarimeter increases to $\\simeq 25 \\, \n\\rm{keV}$ (with the current scintillator threshold of $\\sim$$5 \\, \n\\rm{keV}$). We are working on a further reduction of the scintillator \nthreshold \\cite{Fabiani2013}. Above an energy of $30 \\, \\rm{keV}$, we \ncan detect a high fraction of events with a coincident scintillator \nsignal. For ground-based background measurements, the scintillator \ncoincidence reduces the background by another factor of $\\sim$$30$. On \nupcoming balloon flights, we will use events with (for best high-energy \nsignal to noise ratio) and without (for best low-energy response) \nscintillator coincidences.\n\n\n\\section{X-Calibur: Polarization Measurements} \\label{sec:XCLB_PolarimetryMeasurements}\n\nThis section discusses measurements performed with the fully assembled \nand calibrated X-Calibur polarimeter. Measurements of non-polarized and \npolarized X-ray beams at the CHESS facility at Cornell University are \ndescribed in Sec.~\\ref{subsec:CHESS} and are compared to simulations. \nThe results illustrate the functionality of X-Calibur as an X-ray \npolarimeter. Section~\\ref{subsec:FtSumnerData} describes X-Calibur \nmeasurements that were taken with the fully integrated \nX-Calibur/{\\it InFOC$\\mu$S}\\xspace X-ray telescope in Ft.~Sumner during a test \nintegration of a flight-ready balloon setup. \n\nSystematic effects and asymmetries in the detector response can, under \ncertain circumstances, cause measurements of aparent polatization \nfractions, even if the measured X-ray beam itself is non-polarized. It \nis therefore crucial to understand and control these kinds of systematic \neffects in order to correctly study the polarization properties of \nastrophysical sources. The systematic effects (and corrections thereof) \ncaused by a X-ray beam hitting the polarimeter off-center are discussed \nin Sec.~\\ref{subsec:Systematics}.\n\n\n\\subsection{Polarized X-rays from the CHESS Beam} \\label{subsec:CHESS}\n\n\nIn order to measure the response to a polarized X-ray beam, the \nX-Calibur polarimeter was operated at the Cornell High Energy \nSynchrotron Source (CHESS)\\footnote{http://www.chess.cornell.edu/} for \none week in March 2013. CHESS provides a highly collimated and highly \npolarized beam of synchrotron X-rays. Using Bragg reflection from a \n2-bounce silicon (220) monochromator, a $40 \\, \\rm{keV}$ beam was \ngenerated with the 2$^{\\rm{nd}}$/3$^{\\rm{rd}}$ harmonics at $80 / 120 \\, \n\\rm{keV}$, respectively. The measurements were performed in hutch {\\it \nC1}. The polarimeter was mounted on an adjustable X/Y/Z stage table with \nthe scintillator being aligned with the X-ray beam (using X-ray \nfluorescence paper). The setup, referred to as $C_{\\rm{ch}}$, is shown \nin Fig.~\\ref{fig:X-Calibur_Configurations}. X-Calibur was mounted in a \nfixture that allowed us to rotate the polarimeter around the \noptical/beam axis, in order to test the response at different \norientations between the polarization plane and the detector. The \nrotation angle is referred to as $\\alpha$ with detector board {\\it Bd0} \nlocated at the top position at $\\alpha = 0^{\\circ}$ (see $C^{3}$ in the \ntop panel of Fig.~\\ref{fig:X-Calibur_Configurations}). Looking into the \nbeam, a positive angle $\\alpha$ corresponds to a counter clock-wise \nrotation. The accuracy of setting the angle was estimated to be $\\Delta \n\\alpha \\simeq 2^{\\circ}$. It should be noted that we did not use a high \nprecision rotation mechanism (as will be used during the balloon \nflight). Therefore, an $\\alpha$ dependent mis-alignment/tilt of the \nscintillator during the measurements cannot be excluded.\n\n{\\it The CHESS X-ray flux.} A total of 30 synchrotron-emitting electron \nbunches cycle the CHESS accelerator ring. The relative intensity of the \nX-ray beam entering the {\\it C1} hutch is controlled by a system of \nslits, as well as the tunable orientation of the Bragg reflection \ncrystals. The intensity, however, varies with (i) decaying electron \npopulation in the ring during a run (re-charged every $\\sim$$2 \\, \n\\rm{h}$), (ii) local heat built-up on the Bragg crystals, and (iii) the \nposition stability of the electron beam in the ring. The intensity of \nthe monochromatic beam entering the hutch was monitored using an argon \nion chamber ($6 \\, \\rm{cm}$ in length along the beam, set to a counter \nrange of $10^{-8} \\, \\rm{A}/\\rm{V}$). The absolute flux calibration of \nX-Calibur was not a major objective of the measurements; therefore, we \ndid not setup a system to automatically stabilize the beam intensity \nentering the hutch~-- but rather kept it manually in the ballpark of \n$1-2 \\cdot 10^{8} \\, \\rm{cts}/\\rm{s}$ (ion chamber) throughout the \nmeasurements and logged the average intensity for each data run. \nAbsorber foils placed behind the ion chamber further reduced the beam \nintensity by several orders of magnitude. This brought the X-ray flux \nhitting the polarimeter down to the level of $1-2 \\, \\rm{kHz}$, a regime \nthat can be well handled by the readout electronics of the polarimeter.\n\n{\\it Configurations.} An aluminum collimator plate of thickness $0.5''$ \nwas placed directly in front of the X-Calibur polarimeter with an \nentrance hole of $0.25''$ in diameter. The CZT detector configuration \nused for the CHESS measurements is listed in Tab.~\\ref{tab:Detectors}. \nCHESS data presented in this paper were taken with three configurations \n(variations of $C_{\\rm{ch}}$ as depicted in \nFig.~\\ref{fig:X-Calibur_Configurations}).\n\n\\begin{itemlist}\n\n\\item {\\bf $C_{\\rm{ch1}}$}: the absorber foils consist of platinum (Pt), \na $125 \\, \\mu\\rm{m}$ layer placed directly after the ion chamber and a \nstack of four $90 \\, \\mu\\rm{m}$ layers placed between the ion chamber \nand X-Calibur. This setup is optimized for a band-pass at $40 \\, \n\\rm{keV}$.\n\n\n\\item{\\bf $C_{\\rm{ch3}}$}: One layer of $125 \\, \\mu\\rm{m}$ Pt absorber \nfoil is placed directly behind (downstream) the ion chamber, and a $1.27 \n\\, \\rm{mm}$ lead foil (Pb) is placed between the ion chamber and \nX-Calibur. This setup is optimized for a band-pass at $80 \\, \\rm{keV}$.\n\n\\item{\\bf $C_{\\rm{ch4}}$}: Same as $C_{\\rm{ch3}}$, but the ion chamber \nwas removed from the beam path (in order to extend the range of beam \noffsets without interfering with the housing of the ion chamber).\n\n\\end{itemlist}\n\nX-Calibur detects individual events, but only the ASIC that triggered \nthe event (corresponding to $1/64$ of the polarimeter) is dead during \nthe read-out which takes $\\sim$$130 \\,\\mu\\rm{s}$. The dead-time during \ndata runs is therefore usually not higher than $5\\%$. In order to \nguarantee a homogeneous data set, some additional pixels (as compared to \nthe calibration runs) were excluded from the analysis which had high \nthresholds or were too noisy~-- even though the azimuthal acceptance \n(see Sec.~\\ref{sec:Analysis}) is in general capable of correcting for \nmost of these effects. Only events with multiplicity $m=1$ pixels were \nselected. X-Calibur was not operated in its active CsI shield, and the \nscintillator PMT discriminator was not yet optimized at the time of the \nCHESS measurements.\n\n{\\it Data runs.} A series of measurements was taken with configuration \n$C_{\\rm{ch1}}$ (optimized for $40 \\, \\rm{keV}$) and $C_{\\rm{ch3,4}}$ \n(optimized for $80 \\, \\rm{keV}$). Data were taken with different \npolarimeter orientations $\\alpha$. The $C_{\\rm{ch1}}$ data runs span a \nrange of $\\alpha \\in [-90; +90]^{\\circ}$ in steps of $10^{\\circ}$ with \n$4 \\, \\rm{Mio}$ events per orientation. A non-polarized beam was \n`generated' by the superposition of data from two perpendicular \norientations. A higher number of events were taken for these sets: $20 \n\\, \\rm{Mio}$ events for the $\\alpha = 0/-90^{\\circ}$ pair, and $10 \\, \n\\rm{Mio}$ events for the $\\alpha = \\pm 45^{\\circ}$ reference pair. The \n$C_{\\rm{ch3,4}}$ series scanned a range of $\\alpha \\in \n[-90;+10]^{\\circ}$ with a non-polarized beam composed from $10 \\, \n\\rm{Mio}$ events derived from the $\\alpha = 0/-90^{\\circ}$ orientations.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{CHESS_AllDetectorUnPolarized_Mosaic}\n\\end{center}\n\n\\caption{Energy spectra of the Compton-scattered non-polarized CHESS \nX-ray beam (superposition of perpendicular polarization planes). The \nmeasurements were performed with the two configurations $C_{\\rm{ch1}}$ \nand $C_{\\rm{ch3}}$ (see text and \nFig.~\\ref{fig:X-Calibur_Configurations}). Shown are the background \nsubtracted integral spectra of all detectors~-- only ring R1 was \nexcluded due to flux contamination (see text and \nFig.~\\ref{fig:CHESS_ComtponSpectra_UnPol}, right, for the reason). The \nbackground level is shown for reference. The horizontal lines represent \nthe nominal energies of the three harmonics for (left to right): $180 - \n90^{\\circ}$ Compton-scattering (dotted), and $90 - 0^{\\circ}$ \nCompton-scattering (solid). Spectra are shown with and without the \n$f_{\\rm{sci}} = 1$ scintillator coincidence requirement. Simulated \nspectra are shown, as well.}\n\n\\label{fig:CHESS_ComtponSpectra_AllDetector_UnPol}\n\\end{figure}\n\n\nSeveral background runs were taken without the X-ray beam entering the \nhutch. The over-all background spectrum is shown in the left panel of \nFig.~\\ref{fig:BG_Spectra} (red data points). The {\\it C1} hutch is \nshielded by (partly) lead-enforced walls which likely contribute to the \nlower background level as compared to the laboratory at Washington \nUniversity. In fact, a signature at the lead $\\rm{KL}_{1-3}$ transition \nenergies can be identified in the CHESS background spectrum. Although \nthe background is negligible compared to the strong X-ray signal (see \nFigs.~\\ref{fig:CHESS_ComtponSpectra_UnPol} and \n\\ref{fig:CHESS_ComtponSpectra_Pol} for reference), it was subtracted \nfrom the spectra shown in this section. It should be noted that the beam \nin the hutch may introduce an additional implicit/diffuse background by \nscattering off several components (slits, absorber foils, etc.). This \nkind of background was not determined in a dedicated measurement, but \ncan potentially be higher than the CR background.\n\n{\\it Synchronization between simulations and data.} The Bragg \nmonochromator only allows the $40/80/120 \\, \\rm{keV}$ harmonics of the \nCHESS white beam to enter the C1 hutch. The spectral intensities of the \nwhite beam, as well as the Bragg-reflected intensities of the \nmonochromator, were calculated using the {\\it X-ray Oriented Programs} \n(XOP) software \npackage\\footnote{http://www.esrf.eu/computing/scientific/xop2.1/}. The \nabsorber foils further change the relative intensities. Given the \ndensity of platinum of $\\rho_{\\rm{Pt}} = 21.45 \\, \\rm{g}/\\rm{cm}^{3}$, \nits mass attenuation coefficient at $40 \\, \\rm{keV}$ of \n$(\\mu/\\rho)_{\\rm{Pt}} = 12.45 \\rm{cm}^{2}/\\rm{g}$, and the thickness of \nthe Pt foils $d$, one can calculate the beam intensity $I$ entering the \npolarimeter by $I/I_{0} = \\exp(-\\mu/\\rho \\cdot \\rho \\cdot d)$. The mass \nattenuation coefficients $\\mu/\\rho$ were taken from the NIST X-ray \ndatabase\\footnote{http://www.nist.gov/pml/data/xraycoef/index.cfm}. \nHowever, an accurate prediction of the relative flux intensities \nentering the polarimeter requires a detailed modeling of all \nenergy-dependent X-ray absorption/transmissions on the beam path \n(including the entrance windows, the Bragg monochromator, and the \nabsorber foils).\n\nTherefore, we instead used the overall spectrum measured by the whole \npolarimeter to synchronize the relative $40/80/120 \\, \\rm{keV}$ flux \nintensities with our Monte Carlo simulations. \nFigure~\\ref{fig:CHESS_ComtponSpectra_AllDetector_UnPol} shows the \noverall X-Calibur responses (all detector rings except for R1, see \nbelow) to the non-polarized CHESS beam, measured with configurations \n$C_{\\rm{ch1}}$ and $C_{\\rm{ch3}}$. The corresponding relative \nintensities of the three mono-energetic energies in the simulations were \nmatched accordingly and are in turn applied for all data versus \nsimulation comparisons that follow. Note, however, that the CHESS white \nbeam intensity varies. Therefore, the simulations were scaled according \nto the relative difference in the integral X-Calibur trigger rate with \nrespect to the data run used to normalize the simulations. We estimate \nthe systematic error on the absolute flux normalization of the \nsimulations to be around $20 \\%$. \nFigure~\\ref{fig:CHESS_ComtponSpectra_AllDetector_UnPol} also shows the \nenergy spectra with a coincident scintillator trigger ($f_{\\rm{sci}} = \n1$), although obtained with the non-optimal scintillator trigger \nthreshold.\n\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{CHESS_ComptonSpectra_Mosaic}\n\\hfill\n\\includegraphics[width=0.49\\textwidth]{CHESS_ComptonSpectra_Ring1t_Mosaic}\n\\end{center}\n\n\\caption{Energy spectra of the Compton scattered, non-polarized $40/80/120 \n\\, \\rm{keV}$ CHESS X-ray beam (averaged over the complete azimuthal \nscattering range, $m=1$ events). The horizontal lines represent the \nnominal energies after Compton-scattering. For reference, the CR \nspectrum is shown for one of the $5 \\, \\rm{mm}$ rings. {\\bf Left:} \nSpectra from selected sub rings. In the case of ring {\\it \nR1$_{\\rm{t}}$}, the first pixel row was removed (see text and right \npanel). {\\bf Right:} Special case of ring {\\it R1$_{\\rm{t}}$} which is \nlocated at the polarimeter entrance; individual pixel rows are shown. It \ncan be seen that the first pixel row has significantly higher flux at \n$E>40 \\, \\rm{keV}$ as compared to the other rows~-- likely due to \nexternal X-ray contamination. The same spectra are shown with the \nscintillator trigger condition ($f_{\\rm{sci}} = 1$). Simulated spectra \nare shown for the pixel rows 1 and 4. The spectrum obtained from the \nfirst pixel row in Ring {\\it R2$_{\\rm{t}}$} is shown for reference.}\n\n\\label{fig:CHESS_ComtponSpectra_UnPol}\n\\end{figure*}\n\n\n{\\it Compton spectra of a non-polarized beam.} The energy spectra of the \nnon-polarized Compton scattered X-ray beam are shown for individual \ndetector rings in Fig.~\\ref{fig:CHESS_ComtponSpectra_UnPol} (left). The \nspectra are averaged over all azimuthal scattering angles $\\Phi$ within \neach ring. Ring {\\it R1$_{\\rm{t}}$} mostly detects back-scattered events \n(the scintillator geometrically covering rings {\\it R2}-{\\it R8}, see \nFig.~\\ref{fig:Design}, left) which deposit the maximal amount of energy \nin the scintillator: $\\Delta E_{\\rm{sci}}$ up to $5.4 \\, \\rm{keV}$ for \nthe $40 \\, \\rm{keV}$ beam, depositing $E_{\\rm{czt}} = 34.6 \\, \\rm{keV}$ \nin the CZT detector. The rings further downstream detect a superposition \nof X-rays that underwent Compton-scattering under different angles. The \nrear-side rings ({\\it R7} and {\\it R8}), to a larger extent, detect \nX-rays which Compton scattered nearly in the forward direction~-- \ndepositing only a small amount of energy in the scintillator, with an \nenergy deposition in the CZT ($E_{\\rm{czt}}$) close to the energy of the \nprimary X-ray ($E_{\\rm{line}}$). This explains the shift of the peak \npositions in the spectra towards higher energies with increasing ring \nnumber. The theoretical range expectations of $E_{\\rm{czt}}$ are \nindicated by the horizontal lines in \nFig.~\\ref{fig:CHESS_ComtponSpectra_UnPol} (assuming an idealized energy \nresolution).\n\nThe right panel of Fig.~\\ref{fig:CHESS_ComtponSpectra_UnPol} shows the \nenergy spectra of individual pixel rows of ring {\\it R1$_{\\rm{t}}$}. It \ncan be seen that the first pixel row shows a strong anomaly at energies \n$E_{\\rm{czt}} > 50 \\, \\rm{keV}$ as compared to the pixel rows 2-4. The \ncomparison with simulations further underlines the difference. A likely \nexplanation of the additional continuum would be the following. The \nfirst row of pixels has a rather high effective area (pixel sides) in \nthe plane perpendicular to the X-ray beam, as compared to other pixel \nrows. These `side' pixels will detect X-rays that underwent diffuse \nscattering in the Pt/Pb absorber foils or the polarimeter entrance plate \n(see Fig.~\\ref{fig:X-Calibur_Configurations}), and succeeded in entering \nthe polarimeter along a path not along the optical axis. Since this \ninvolves X-ray transmission through some of the material of the \npolarimeter fixture, it is more likely to happen for higher energies, \nwhich is where the anomaly in the spectrum becomes more prominent. The \nfirst pixel row was therefore excluded from the analysis of the data \npresented in this section. It should be noted, that the contamination is \nalso seen in the pixel rows further downstream~-- however, with strongly \ndecreasing strength.\n\nThe right panel of Fig.~\\ref{fig:CHESS_ComtponSpectra_UnPol} also shows \nthe spectra obtained from the same data with the additional requirement \nof a scintillator trigger ($f_{\\rm{sci}} = 1$). This favors high-energy \nand/or back-scattered events which deposit a high amount of energy \n$\\Delta E_{\\rm{sci}}$ in the scintillator. With the $f_{\\rm{sci}} = 1$ \ncut, the first pixel rows no longer show the anomaly discussed above~-- \nunderlining the hypothesis of an external contamination not interacting \nwith the scintillator. A more detailed study of the trigger efficiency \nof the scintillator is presented in \nSec.~\\ref{subsec:ScintillatorEfficiency}. Despite the contamination \nissue seen in the first front-side pixel rows, the Compton-spectra \nmeasured with X-Calibur are in good agreement with the simulations \n(although not explicitly shown for all rings in \nFig.~\\ref{fig:CHESS_ComtponSpectra_UnPol}, right, for reasons of \nvisibility.).\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{CHESS_ComptonSpectraPol_Mosaic}\n\\end{center}\n\n\\caption{Compton spectra of the polarized $40/80/120 \\, \\rm{keV}$ CHESS \nbeam. Spectra are shown for two of the four detector sides ({\\it Bd0} \nand {\\it Bd1}, see Fig.~\\ref{fig:X-Calibur_Configurations}, top) in ring \n{\\it R2$_{\\rm{b}}$}, with {\\it Bd0} aligned with the plane of \npolarization ($\\alpha = 0^{\\circ}$). The {\\it Bd1} spectrum is also \nshown with the requirement of a scintillator trigger (the discriminator \nthreshold not being optimized at this point). The horizontal lines \nrepresent the nominal energy ranges after Compton-scattering. The \nsimulated spectra are shown, as well.}\n\n\\label{fig:CHESS_ComtponSpectra_Pol}\n\\end{figure}\n\n{\\it Compton spectra of a polarized beam.} A polarized X-ray beam \nintroduces an azimuthal modulation with a $180^{\\circ}$ periodicity to \nthe measured spectra. Figure~\\ref{fig:CHESS_ComtponSpectra_Pol} shows \nthe Compton spectra measured on two of the four detector sides in ring \n{\\it R2$_{\\rm{b}}$} (with the polarimeter being oriented at an angle of \n$\\alpha = 0^{\\circ}$). As expected, and essential for the functionality \nof the polarimeter, the detector side located perpendicular to the \npolarization plane ({\\it Bd0}) detects a higher number of \nCompton-scattered X-rays. The energy spectra obtained from simulations \nare shown in Fig.~\\ref{fig:CHESS_ComtponSpectra_Pol}, as well, and are \nfound to be in reasonable agreement with the data. For the data versus \nsimulation comparison one has to keep in mind that the simulations were \nperformed for a $100 \\%$ polarized beam, corresponding to the maximal \nmodulation and therefore a maximal scattering concentration in {\\it \nBd0}. The polarization fraction of the CHESS beam, on the other hand, is \n$r < 100 \\%$. Therefore, the azimuthal asymmetry ({\\it Bd0} versus {\\it \nBd1}) in the data is expected to be smaller compared to the simulations. \nDividing the spectra into distinct bins in azimuth, and integrating \ncounts in a given energy interval, will lead to the azimuthal scattering \ndistributions that are discussed in the next paragraph.\n\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{AzimuthDistribution_40keV}\n\\hfill\n\\includegraphics[width=0.49\\textwidth]{AzimuthDistribution_SystematicExamples}\n\\end{center}\n\n\\caption{{\\bf Left:} Azimuthal scattering distribution (CHESS at $40 \\, \n\\rm{keV}$, ring R2$_{\\rm{b}}$). The horizontal error bars reflect the \nazimuthal ranges $\\Delta \\Phi_{j}$ covered by the corresponding pixels \n$j$ (see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). The raw \nnon-polarized beam shows the 4-fold symmetry of the four detectors boards \n{\\it Bd0-Bd3} (see Fig.~\\ref{fig:X-Calibur_Configurations}). The \npolarized beam has the additional $180^{\\circ}$ amplification imprinted \nwhich is clearly extracted once corrected for the azimuthal coverage \n$\\Delta \\Phi_{j}$ and the pixel acceptance $a_{j}$. A sine function is \nfitted to extract $C_{\\rm{min}}$, $C_{\\rm{max}}$, the modulation factor \n$\\mu$, and the angle of the polarization plane $\\Omega_{\\rm{p}}$ (dotted \nvertical lines, indicating the error range). The dashed vertical line \nindicates the nominal polarization plane. For reference, the \ndistribution of a simulated $40 \\, \\rm{keV}$ beam is shown ($100 \\%$ \npolarized, reconstructed in the same way). {\\bf Right:} Normalized \nazimuthal scattering distributions derived from different data sets: a\nnon-polarized beam, a polarized beam measured at $\\alpha = 70^{\\circ}$, as \nwell as the response to a cosmic-ray background run.}\n\n\\label{fig:AzimuthDistribution}\n\\end{figure*}\n\n{\\it Azimuthal scattering distribution.} The azimuthal scattering \ndistributions were generated as outlined in \nSec.~\\ref{subsec:AnalysisPolarization} by integrating the energy range \nof Compton-scattered photons for the three harmonics of the X-ray beam \n($34.6 - 40 \\, \\rm{keV}$, $60.9 - 80 \\, \\rm{keV}$, and $81.7 - 120 \\, \n\\rm{keV}$, respectively). The energy intervals include an additional $2 \n\\, \\rm{keV}$ cushion to account for the energy resolution of the CZT \ndetector pixels. All measured event rates were in turn normalized on a \npixel-by-pixel basis using the azimuthal coverage $\\Delta \\Phi_{j}$ and \nacceptances $a_{j}$. Following Eq.~(\\ref{eqn:Acceptance}) described in \nSec.~\\ref{subsec:AnalysisPolarization}, the pixel acceptances $a_{j}$ \nwere determined from the non-polarized beam for each of the above energy \nintervals.\n\nAn example of an azimuthal scattering distribution measured in the \n$34.6-40 \\, \\rm{keV}$ band is shown in \nFig.~\\ref{fig:AzimuthDistribution} (left). The normalized distribution \nis used to derive the plane of polarization $\\Omega_{\\rm{p}}$ (nominal \nvalue of $\\Omega_{\\rm{nom}} = 90^{\\circ}$), as well as the relative \nscattering amplitude $\\mu_{\\rm{ch}}$. The distribution is compared to \nthe one obtained from the simulation of a $100 \\%$ polarized beam at $40 \n\\, \\rm{keV}$. Both are found to be in good agreement~-- except for the \nslightly lower amplitude of the data which is a result of the $r < 100 \n\\%$ polarization fraction of the CHESS beam (see below).\n\nFigure~\\ref{fig:CHESS_2DAzimuthData} illustrates the reconstruction of \nthe azimuthal scattering distributions for the complete polarimeter \n(subrings {\\it R1$_{\\rm{t}}$}-{\\it R8$_{\\rm{b}}$}), based on simulations \n(top panel) and based on measured CHESS data (bottom panel). The \nresponse to a non-polarized $40 \\, \\rm{keV}$ beam is shown on the left \nside of the figure. The second panel shows the polarimeter response to a \npolarized beam at $40 \\, \\rm{keV}$, not yet corrected for pixel \nacceptance $a_{j}$ and azimuthal coverage $\\Delta \\Phi_{j}$. Applying \nthe corrections leads to the smooth distribution shown in the third \npanel, which is in good agreement if comparing the simulations to the \ndata. The pixel acceptances $a_{j}$ depend on (i) the energy threshold, \n(ii) the energy resolution, and (iii) the trigger efficiency of the \npixel $j$. The simulations take into account (i) and (ii), derived from \nthe calibration measurements presented in \nSec.~\\ref{subsec:CZT_Calibration}, but not (iii). The projected count \ndistributions lead to the azimuthal scattering distributions shown in \nthe right panel of Fig.~\\ref{fig:CHESS_2DAzimuthData}. In contrast to \nthe 2D distributions shown in planar pixel coordinates, the $\\Phi$ \npositions and error bars now represent the proper angular coverage of \neach pixel (compare with Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, \nleft). The expected $180^{\\circ}$ modulation is clearly revealed, and \nthe reconstructed orientation of the polarization plane agrees with the \ndirection of the CHESS beam setup~-- confirming the functionality of the \nX-Calibur polarimeter.\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{CHESS_Example_AzimuthDistribution_CHESS1_E_32_6_42_0keV_Angle-1111deg} \\\\\n\\includegraphics[width=0.99\\textwidth]{CHESS_Example_AzimuthDistribution_CHESS1_E_32_6_42_0keV_Angle0deg} \\\\\n\\end{center}\n\n\\caption{X-Calibur 2D scattering distributions of the $40 \\, \\rm{keV}$ \nbeam at CHESS ($C_{\\rm{ch}}^{3}$, see \nFig.~\\ref{fig:X-Calibur_Configurations}, the beam enters from the top). \nEvents are shown for reconstructed energies of $36-40 \\, \\rm{keV}$. The \ntop row shows the results of simulations of a $100\\%$ polarized beam. \nThe bottom row shows the results of the CHESS measurement. The (blue) \nboxes in the 2D maps indicate the projected outline of the scintillator \n(Compton-scatterer). {\\bf Left:} Raw count map (pixel-by-pixel) of a\nnon-polarized beam (neither corrected for $\\Delta \\Phi_{j}$ nor for pixel \nacceptance $a_{j}$). All four detector sides, {\\it Bd0}-{\\it Bd3}, are \nunfolded into a plane. The detector rings {\\it R1} to {\\it R8} are \nindicated. {\\bf Second:} Count map of a raw measurement of a polarized \nbeam. {\\bf Third:} Count map of the polarized beam, corrected for \n$\\Delta \\Phi_{j}$ and $a_{j}$. {\\bf Right:} Normalized azimuthal \nscattering distribution (corrected for $\\Delta \\Phi_{j}$ and $a_{j}$) \nfor different detector rings. The vertical lines indicate the nominal \nplane $\\Omega_{\\rm{n}}$ of the electric field vector of the polarized \nbeam.}\n\n\\label{fig:CHESS_2DAzimuthData}\n\n\\end{figure*}\n\n{\\it Simulated modulation factors.} Each detector ring (see \nFig.~\\ref{fig:CHESS_2DAzimuthData}), or a sub ring thereof, can be seen \nas an independent detector that allows to reconstruct energy-dependent \npolarization properties of the incoming X-ray beam. \nFigure~\\ref{fig:MC_ModulationFactor} shows the simulated modulation \nfactors $\\mu_{\\rm{sim}}$ for the different energies and X-Calibur \nconfigurations as a function of detector ring/depth, measured along the \noptical axis. The values for $\\mu_{\\rm{sim}}$ were obtained using the \nStokes analysis, and are in agreement with the corresponding results \nobtained from the analysis of the azimuthal scattering distribution (not \nshown). The simulations shown reflect the detector status of the CHESS \nmeasurements (dead pixels, energy resolutions, etc.), not an idealized \ndetector. Ring {\\it R1} detects only back-scattered X-rays with a \ncorrespondingly lower $\\mu_{\\rm{sim}}$~-- but with a better energy \nresolution since the scattering kinematics are better defined. The \nstep-like structure in $\\mu_{\\rm{sim}}^{120\\,\\rm{keV}}$ between rings \n{\\it R5} and {\\it R6} can be explained by the strong degradation in \nenergy resolution at high energies in the $2 \\, \\rm{mm}$ detectors \nlocated at rings {\\it R6}-{\\it R8}. The highest modulation is achieved \nin ring {\\it R2}.\n\nThe simulations of the modulation factor were also done involving the \nscintillator flag $f_{\\rm{sci}}$, based on the two scintillator trigger \nefficiencies as measured in \nFig.~\\ref{fig:Scintillator_TriggerEfficiency}. This reflects the \nefficiency during the CHESS measurements (not optimized), as well as the \nefficiency of the optimized scintillator trigger. The scintillator \nimproves the modulation factor for most rings~-- by requiring a minimum \nenergy deposition $\\Delta E_{\\rm{sci}}$ which effectively limits the \nallowed range of polar scattering angles. This, however, also reduces \nthe event statistics (not imprinted in $\\mu_{\\rm{sim}}$). A scintillator \ntrigger efficiency of $100 \\%$ would again lead to the same distribution \nof modulation factors as no cut on $f_{\\rm{sci}}$ does. The main benefit \nof the scintillator trigger lies in its ability to suppress background \nduring a measurement (see for example Fig.~\\ref{fig:BG_Spectra}, left). \nThe modulation factor $\\mu$, together with the detection rate after \nbackground subtraction, are the crucial characteristics that determine \nthe MDP detection sensitivity of the polarimeter following \nEq.~(\\ref{eqn:MDP}).\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{MC_ModulationFactors}\n\\end{center}\n\n\\caption{Simulated modulation factors $\\mu_{\\rm{sim}}$ as a function of \ndetector ring {\\it R}, shown for different energies. The geometrical \npositions (along the optical axis $y$) are indicated for the \nscintillator and the CZT ring assembly. The simulated data were analyzed \nwith the Stokes method. Results are also shown for the scintillator \ntrigger requirement $f_{\\rm{sci}}=1$, assuming the PMT trigger \nefficiency during the CHESS measurements (PMT$_{\\rm{ch}}$) and the \nefficiency of the optimized PMT threshold (PMT+), see \nFig.~\\ref{fig:Scintillator_TriggerEfficiency}.}\n\n\\label{fig:MC_ModulationFactor}\n\\end{figure}\n\n{\\it Polarization fraction of the CHESS beam.} The azimuthal scattering \ndistributions for each sub ring and each data set were fitted to \ndetermine $\\mu_{\\rm{ch}}$. The simulated modulation factors \n$\\mu_{\\rm{sim}}$ (Fig.~\\ref{fig:MC_ModulationFactor}) were in turn used \nto determine the polarization fraction $r$ of the CHESS beam using \nEq.~(\\ref{eqn:PolFracPhiDistri}). This is shown in \nFig.~\\ref{fig:AzDistri_RecoMu} for the $40 \\, \\rm{keV}$ beam for \ndifferent orientations $\\alpha$ of the polarimeter relative to the \npolarization plane. Also shown are the residuals between the \nreconstructed polarization plane and the nominal polarization plane. \nNote, that for large values of $\\alpha$ the azimuthal scattering \ndistribution shows an additional global slope, possibly caused by an \n$\\alpha$-dependent tilt of the rotation fixture (see \nFig.~\\ref{fig:AzimuthDistribution}, right, for an example orientation of \n$\\alpha = 70^{\\circ}$). This will affect the reconstructed polarization \nproperties and is further discussed in Sec.~\\ref{subsec:Systematics}. To \nreduce this systematic effect, the CHESS azimuthal distributions were \nfolded back into the $[0;180]^{\\circ}$ interval before the sinusoidal \nfunction was fitted to the data points.\n\n\n\\begin{table}[t!]\n\n\\begin{tabular}{lrr}\n\nSetup & $\\Delta \\Omega \\, [\\rm{deg}]$ & $r_{\\rm{rec}} \\, [\\%]$ \\\\\n\\hline \\hline\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{{beam energy: $40 \\, \\rm{keV}$}} \\\\\n\\hline\n\n$C_{\\rm{ch1}}$ & $-4.7 \\pm 1.9$ & $91.0 \\pm 0.6$ \\\\\n & $-4.6 \\pm 1.7$ & $88.6 \\pm 0.4$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{{beam energy: $80 \\, \\rm{keV}$}} \\\\\n\\hline\n\n$C_{\\rm{ch1}}$ & $-2.6 \\pm 1.7$ & $83.5 \\pm 0.8$ \\\\\n & $-2.0 \\pm 2.1$ & $78.3 \\pm 0.8$ \\\\\n$C_{\\rm{ch1}}$, $f_{\\rm{sci}} = 1$ & $-2.1 \\pm 2.5$ & $86.5 \\pm 1.1$ \\\\\n & $-2.1 \\pm 2.7$ & $82.8 \\pm 1.0$ \\\\\n$C_{\\rm{ch4}}$ & $-2.5 \\pm 1.8$ & $90.4 \\pm 0.5$ \\\\\n & $-2.0 \\pm 2.2$ & $85.5 \\pm 0.6$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{{beam energy: $120 \\, \\rm{keV}$}} \\\\\n\\hline\n\n$C_{\\rm{ch1}}$ & $-0.5 \\pm 2.9$ & $89.2 \\pm 1.5$ \\\\\n & $-0.2 \\pm 3.3$ & $71.5 \\pm 1.3$ \\\\\n$C_{\\rm{ch1}}$, $f_{\\rm{sci}} = 1$ & $-0.7 \\pm 4.6$ & $86.3 \\pm 1.2$ \\\\\n & $-0.8 \\pm 3.7$ & $80.5 \\pm 1.5$ \\\\\n\n\\end{tabular}\n\n\\caption{Residuals between the reconstructed polarization plane \n$\\Omega_{\\rm{p}}$ and the nominal plane at $\\Omega_{\\rm{n}} = \n90^{\\circ}$: $\\Delta \\Omega = (\\Omega_{\\rm{n}}-\\alpha) - \n\\Omega_{\\rm{p}}$. The second column shows the reconstructed polarization \nfractions $r_{\\rm{rec}}$ of the CHESS beam at different energies and \ndifferent configurations/setups (partly with the $f_{\\rm{sci}} = 1$ \nrequirement). See Figs.~\\ref{fig:AzDistri_RecoMu} and \n\\ref{fig:CHESS_MorePolFractions} for an illustration. The first row per \ndata set shows the results obtained from the analysis of the azimuthal \n$\\Phi$-scattering distribution following \nEq.~(\\ref{eqn:PolFracPhiDistri}). The second row represents the results \nobtained from the Stokes analysis following \nEq.~(\\ref{eqn:PolarizationFromStokes}). The results are averaged over \nall orientations $\\alpha$, each.}\n\n\\label{tab:CHESS_PolFrac}\n\n\\end{table}\n\nFigure~\\ref{fig:CHESS_MorePolFractions} shows the corresponding \npolarization fractions reconstructed from the $80 \\, \\rm{keV}$ harmonic. \nThe reconstructed polarization fractions obtained from all measurements \nare summarized in Tab.~\\ref{tab:CHESS_PolFrac}~-- in all cases averaged \nover all measured orientations $\\alpha$. Table~\\ref{tab:CHESS_PolFrac} \nalso shows the results derived from the Stokes analysis following \nEq.~(\\ref{eqn:PolarizationFromStokes}). Both methods are in reasonable \nagreement, whereas the Stokes analysis seems to slightly underestimate \n$r$ compared to the results obtained with the azimuthal scattering \ndistribution. The systematic error on the reconstructed polarization \nfraction was estimated as described in Sec.~\\ref{subsec:Systematics}. \nThe polarization of the CHESS beam as reconstructed using the \n$\\Phi$-distribution method is measured to be $r_{\\rm{ch}}^{\\Phi} = (87.8 \n\\pm 0.4_{\\rm{stat}} \\pm 4.6_{\\rm{sys}}) \\, \\%$ with no indication of an \nenergy dependence in the $40-120 \\, \\rm{keV}$ range. The corresponding \nvalue reconstructed from the Stokes analysis is $r_{\\rm{ch}}^{\\rm{st}} = \n(81.2 \\pm 0.4_{\\rm{stat}} \\pm 10_{\\rm{sys}}) \\, \\%$. The difference in \nthe systematic error is explained by the fact that the Stokes analysis \nis more sensitive to the detector configuration (which was not perfectly \nconstrained during the CHESS measurements). A better defined geometry \nand the rotation of the polarimeter will reduce the systematic error for \nmeasurements performed during the balloon flight.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{PhiDistriFitResult_CHESS1_E_32_6_42_0keV}\n\\end{center}\n\n\\caption{Reconstructed polarization properties of the CHESS beam at $40 \n\\, \\rm{keV}$ (configuration $C_{\\rm{ch1}}$). {\\bf Top:} Modulation \nfactor $\\mu$ versus polarimeter sub rings {\\it Ri$_{\\rm{t,b}}$}. The \nresults were derived from simulations (MC), as well as from data taken \nunder different X-Calibur orientations $\\alpha$. For each $\\alpha$ and \neach ring, $\\mu_{\\rm{ch}}$ is derived from fits to acceptance corrected \n$\\Phi$-distributions (see Fig.~\\ref{fig:AzimuthDistribution}, left). The \nscintillator starts covering detector rings at $y \\geq 1.8 \\, \\rm{cm}$. \nThe green error band reflects the $1 \\, \\rm{std.dev}$ range of measured \npoints in the particular $y$ slice. {\\bf Middle:} Reconstructed \npolarization fraction following Eq.~(\\ref{eqn:PolFracPhiDistri}). The \ngreen band reflects the $1 \\, \\rm{std.dev}$ range of all reconstructed \nfractions in the corresponding $y$ slice. {\\bf Bottom:} Residual between \nreconstructed polarization plane $\\Omega_{\\rm{p}}$ and true polarization \nplane $\\Omega_{\\rm{n}} = 90^{\\circ}$, corrected for the X-Calibur \norientation $\\alpha$: $\\Delta \\Omega = (\\Omega_{\\rm{n}} - \\alpha) \n-\\Omega_{\\rm{p}}$.}\n\n\\label{fig:AzDistri_RecoMu}\n\\end{figure}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{PhiDistriFitResult_PolFrac_CHESS4_E_58_9_82_0keV} \\\\\n\\end{center}\n\n\\caption{Reconstructed polarization fractions of the CHESS beam (compare \nwith Fig.~\\ref{fig:AzDistri_RecoMu}) derived from the $80 \\, \\rm{keV}$ \nharmonic (configuration $C_{\\rm{ch3}}$).}\n\n\\label{fig:CHESS_MorePolFractions}\n\\end{figure}\n\n{\\it Polarization plane of the CHESS beam.} \nTable~\\ref{tab:CHESS_PolFrac} also summarizes the average reconstructed \npolarization planes $\\Omega_{\\rm{p}}$ (corrected for the X-Calibur \norientation $\\alpha$). Note, that the uncertainty in setting the \nX-Calibur orientation was estimated to be $\\Delta \\alpha = \\pm \n2^{\\circ}$. Within this systematic error, as well as the statistical \nerrors, the average of reconstructed polarization planes (all rings and \nall X-Calibur orientations $\\alpha$) is compatible with the nominal \nplane $\\Omega_{\\rm{n}}$ of the CHESS beam. However, the measurements \nconsistently reconstruct $\\Delta \\Omega < 0^{\\circ}$, which may indicate \na slight offset in the way the rotation mechanism was \ninstalled/calibrated. The reconstructed polarization planes shown in \nFig.~\\ref{fig:AzDistri_RecoMu} indicate a slight dependence on $\\alpha$ \nand $y$. The left panel in Fig.~\\ref{fig:AzimuthDistribution} shows a \ncorresponding azimuthal distribution measured with the X-Calibur \norientation of $\\alpha = 70^{\\circ}$, revealing a slight asymmetry. This \nresult can be seen as another indication of a slight mis-alignment \n(offset and/or tilt) between the rotation axis of the scintillator and \nthe optical axis of the X-ray beam. A more detailed discussion of this \nkind of systematic effect can be found in Sec.~\\ref{subsec:Systematics}.\n\n\n\\subsection{Measurements with the X-Calibur/{\\it InFOC$\\mu$S}\\xspace Assembly} \n\\label{subsec:FtSumnerData}\n\nThe full X-Calibur/{\\it InFOC$\\mu$S}\\xspace experiment was assembled and tested during a \nflight preparation campaign at the NASA {\\it Columbia Scientific Balloon \nFacility} (CSBF) site in Ft.~Sumner, NM, in the fall of 2014. The \npolarimeter was installed in the rotating CsI shield assembly, which in \nturn was installed in the pressure vessel as part of the {\\it InFOC$\\mu$S}\\xspace X-ray \ntelescope (see Fig.~\\ref{fig:ActiveShieldAndTelescope}). This setup, \n$C_{\\rm{ft}}$, is illustrated in \nFig.~\\ref{fig:X-Calibur_Configurations}. The CZT detector configuration \nof these measurements is listed in Tab.~\\ref{tab:Detectors} and differs \nslightly from the one used in the CHESS measurements (described in \nSec.~\\ref{subsec:CHESS}). The X-ray mirror was installed on the optical \nbench located at the front part of the {\\it InFOC$\\mu$S}\\xspace telescope. The pressure \nvessel is located at a focal distance of $8 \\, \\rm{m}$.\n\nBefore the balloon flight, a beryllium (Be) window is installed in the \ntop dome of the pressure vessel, to assure a pressure-sealed entrance to \nthe polarimeter with high transmissivity to hard X-rays. For the \nground-based measurements, presented in this section, the Be window was \nnot installed to allow to visually align the polarimeter with the \noptical axis of the X-ray mirror. Throughout the measurements presented \nin this section, the polarimeter was rotated at $4 \\, \\rm{rpm}$. Due to \nthe rotation, the application of the pixel acceptances $a_{j}$, defined \nin Eq.~(\\ref{eqn:Acceptance}), is no longer required, since each pixel \ntests the complete, time-averaged azimuthal scattering range with \nrespect to the polarization plane. As another consequence, the azimuthal \nbinning can be chosen finer than in the non-rotating system in which it \nwas limited to the number of $32$ CZT pixels per row. For each event, \nthe hit pixel is de-rotated into the laboratory/horizon coordinate frame \n(see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left) to determine \nthe azimuthal scattering angle $\\Phi$.\n\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[height=0.32\\textheight]{FtSumner_MirrorScanSetup}\n\\hfill\n\\includegraphics[height=0.32\\textheight]{FtSumnerMirrorScan2D_Rotating}\n\\hfill\n\\includegraphics[height=0.32\\textheight]{FtSumnerPolFraction}\n\\end{center}\n\n\\caption{{\\it InFOC$\\mu$S}\\xspace/X-Calibur X-ray mirror scan, configuration \n$C_{\\rm{ft}}$ in Fig.~\\ref{fig:X-Calibur_Configurations}. {\\bf Left:} \nThe collimated X-ray source is aligned with the optical axis of the \nmirror and the polarimeter which is situated in the focal plane at a \ndistance of $8 \\, \\rm{m}$ (not visible in this photograph). The X-ray \nsource can be automatically moved in the $X$/$Z$ plane to scan the \nmirror with the X-rays traveling along $Y$. {\\bf Middle:} The \nCompton-scattered event distribution measured with X-Calibur (horizon \nsystem, de-rotated) during the scan in the energy range of $30-50 \\, \n\\rm{keV}$. {\\bf Right:} Background subtracted measurements of the \ncollimated X-ray beam. Three data runs where taken with (i) the beam \nhitting the center of the scintillator ($\\Delta d = 0 \\, \\rm{mm}$), (ii) \nthe beam hitting at an offset of $\\Delta d = -4 \\, \\rm{mm}$, and (iii) \nthe beam scanning the X-ray mirror. The top panel shows the \nCompton-scattered energy spectra (all-detector average, each). The \nbottom panel shows the energy-dependent polarization fraction \nreconstructed from the azimuthal $\\Phi$-scattering distribution, as well \nas using the Stokes parameters. The given errors are statistical only.}\n\n\\label{fig:FtSumner_MirrorScan}\n\\end{figure*}\n\n{\\it X-ray mirror/X-Calibur alignment.} The alignment of the polarimeter \nand the mirror are crucial in order to gain the maximal sensitivity for \npolarization measurements and to reduce systematic effects. The optical \naxis and the on-axis image location of the X-ray mirror were measured by \na CCD camera with an optical parallel beam. The X-ray mirror was placed \nin the optical beam, and its tip and tilt were adjusted such that the \noptical axis is parallel to the optical beam. Then, the CCD camera was \nplaced at the center of the mirror, looking along the optical axis \nfacing the polarimeter, and taking a picture of the on-axis image plane. \nThe image location was recorded in pixel coordinates, which determines \nthe optical axis as well as the on-axis image position. After the X-ray \nmirror was installed onto the optical bench (see \nFig.~\\ref{fig:FtSumner_MirrorScan}, left), a picture of the front \nsurface of the polarimeter's scintillator was taken by the camera. The \nmirror tip and tilt were adjusted by shimming at the interface to the \noptical bench, such that the center of the scintillator is at the \nrecorded location of the on-axis image of the X-ray mirror. An alignment \nbetween the scintillator and mirror of better than $1\\, \\rm{mm}$ was \nachieved which guarantees a systematic error on the reconstructed \npolarization fraction of less than $2 \\%$ (see \nSec.~\\ref{subsec:Systematics}).\n\n{\\it Mirror scan.} The proper alignment of the telescope is \ntested/confirmed in a mirror scan. A movable X-ray source scans the \nsurface of the mirror while the polarimeter response is measured. The \nX-ray scanning system consists of an X-ray source with a $60\\, \\rm{cm}$ \nlong collimator, tip and tilt stages, and $X$/$Z$ travel stages to which \nthe X-ray source is mounted. The X-ray source was positioned in front of \nthe X-ray mirror and could be used to either directly illuminate the \npolarimeter (through the central hole in the X-ray mirror), or to scan \nthe whole mirror aperture. The setup is shown in \nFig.~\\ref{fig:FtSumner_MirrorScan}, left. An Oxford 5011 electron impact \nX-ray tube is used to generate the X-rays, with an active source spot \nsize of $0.05 \\, \\rm{mm}$ in diameter (molybdenum target, Mo). The \nproduced X-ray spectrum is made of emission line (Mo-K) as well as \nbremsstrahlung. The X-ray source provides a continuum spectrum up to $50 \n\\, \\rm{keV}$. The current was adjusted to give a reasonable event rate \nat the polarimeter located at $8 \\, \\rm{m}$ distance (see \nTab.~\\ref{tab:PixelRates} for reference). The collimator can be equipped \nwith two interchangeable pin holes with diameters of $0.1 \\, \\rm{mm}$ \nand $1 \\, \\rm{mm}$, respectively. The beam size at $8 \\, \\rm{m}$ \ndistance is around $2 \\, \\rm{mm}$ in diameter for the $0.1 \\, \\rm{mm}$ \npin hole. The $1 \\, \\rm{mm}$ pin hole produces count rates in the \npolarimeter above one kHz at $8 \\, \\rm{m}$ distance ($20-50 \\, \n\\rm{keV}$, after air absorption). The tip and tilt stages change the \ndirection of the collimated X-ray beam. The $X$/$Z$ translation stages \nallow the X-ray beam to scan over the entire aperture of the mirror \n(with the central hole in the mirror blocked by a lead absorber). The \nX-ray beam was aligned with the optical axis of the mirror. The $1 \\, \n\\rm{mm}$ pin hole was used to scan the mirror within $51$ horizontal \nrows along $X$. The response measured with X-Calibur is shown in \nFig.~\\ref{fig:FtSumner_MirrorScan} (middle) in the de-rotated coordinate \nsystem (the horizon intersects at $90^{\\circ}$ and $270^{\\circ}$ in this \nrepresentation). The corresponding energy spectrum is shown in \nFig.~\\ref{fig:FtSumner_MirrorScan}, right. The results illustrate that \nthe mirror was successfully aligned with the polarimeter.\n\n{\\it Direct beam illumination.} As a reference measurement to the mirror \nscan, data were taken with the polarimeter being directly illuminated by \nthe collimated X-ray source ($0.1 \\, \\rm{mm}$ pin hole). The X-rays \nenter the scintillator along its optical axis, but do not pass the X-ray \nmirror in this measurement. The spectrum of the mirror scan shown in \nFig.~\\ref{fig:FtSumner_MirrorScan} (right, top) drops off faster \ncompared to the spectrum measured from the direct beam illumination. \nThis is a result of the energy-dependent effective area of the mirror. \nAn additional run was taken with the X-ray beam hitting the scintillator \noff center by $\\Delta d = -4 \\, \\rm{mm}$, which is discussed in \nSec.~\\ref{subsec:Systematics}.\n\nThe polarization parameters were reconstructed in the same way as \ndescribed in Sec.~\\ref{subsec:CHESS} (except for the acceptance \ncorrection $a_{j}$). The depth-dependent modulation factors (see \nFig.~\\ref{fig:MC_ModulationFactor}) obtained from the $40 \\, \\rm{keV}$ \nCHESS simulation were used to reconstruct the polarization fraction of \nthe X-ray beam in different energy bands (assuming $\\mu_{\\rm{sim}}$ \nbeing independent of energy in the $20-50 \\, \\rm{keV}$ band). The \nreconstruction was done with both methods described in \nSec.~\\ref{subsec:AnalysisPolarization}, namely using the analysis of the \nazimuthal scattering distribution following \nEq.~(\\ref{eqn:PolFracPhiDistri}), as well as using the Stokes parameters \nfollowing Eq.~(\\ref{eqn:PolarizationFromStokes}). Both methods yield \ncomparable results which are shown in Fig.~\\ref{fig:FtSumner_MirrorScan} \n(right, bottom). An energy dependent polarization fraction is found in \nthe data. However, in contrast to the mono-energetic CHESS beam \n(Sec.~\\ref{subsec:CHESS}), the energy distribution of the incoming \nX-rays (before Compton-scattering in the scintillator) follows a \ncontinuum. To properly reconstruct the polarization fraction as a \nfunction of incoming X-ray energy, one would have to utilize \nreconstruction methods of forward folding or unfolding \n\\cite{XCLB_LogLikelihoodAnalysis}, which is beyond the scope of this \npaper. However, a significant increase in polarization fraction can be \nobserved with increasing energy. The data of the mirror scan was also \nused to reconstruct the energy-dependent polarization fraction, and is \nfound to be in reasonable agreement with the direct beam measurement. \nThis confirms the predictions by \\citet{Katsuta2009} that an X-ray \nmirror does not substantially affect the polarization properties.\n\n\n\\subsection{Systematic Effects in Polarization Measurements} \n\\label{subsec:Systematics}\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[height=0.285\\textheight]{AzimuthDistribution_OffsetsUnpolarized}\n\\hfill\n\\includegraphics[height=0.285\\textheight]{PolFracSystError}\n\\end{center}\n\n\\caption{{\\bf Left:} Azimuthal scattering distributions of a\nnon-polarized X-ray beam (CHESS data and simulations, both in a \nnon-rotating system) hitting the scintillator at different offsets $d$ \nrelative to the optical axis. X-Calibur was oriented at $\\alpha = \n0^{\\circ}$, with the beam `walking' from {\\it Bd1} to {\\it Bd3} with \nincreasing $\\Delta X$ (see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, \nleft, for a coordinate system). Scattering distributions are shown with \nand without the offset correction (see \nSec.~\\ref{subsec:AnalysisPolarization}). {\\bf Right:} Apparent \npolarization fraction $r$ of the non-polarized beam. The light blue region \nindicates the diameter of the scintillator rod. The data were analyzed \nusing the Stokes parameters as described by \nEq.~(\\ref{eqn:Stokes_Q_U_Average}). Results are shown without and with \nthe corrected beam offset.}\n\n\\label{fig:PolFracSysError}\n\\end{figure*}\n\nThe understanding and control of systematic effects of different nature \nis crucial for the correct reconstruction of the polarization properties \nfrom measured data. A series of measurements was performed at CHESS (see \nSec.~\\ref{subsec:CHESS}) to study the effects of a mis-alignment \n(offset) between the X-ray beam and the optical axis of the polarimeter. \nThe $X$/$Z$ stage of the table in hutch {\\it C1} was used to \nsystematically scan the polarimeter response for offsets ranging from \n$-5 \\, \\rm{mm}$ to $+5 \\, \\rm{mm}$ in steps of $1 \\, \\rm{mm}$ (the beam \ncomes in along $Y$, see Figs.~\\ref{fig:X-Calibur_Configurations} and \n\\ref{fig:SchematicBeamOffset_AndStokes} (left) for the definition of the \ncoordinate system). The scan along $X$ was performed with a polarimeter \norientation of $\\alpha = 0^{\\circ}$, whereas the scan along $Z$ was \nperformed with $\\alpha = -90^{\\circ}$. This allowed us to pairwise \nsuperimpose the data runs of perpendicular polarization planes that hit \nthe scintillator at the same position~-- being equivalent to a \nnon-polarized beam hitting at that particular offset position. The data \nruns were taken with configuration $C_{\\rm{ch4}}$, testing the response \nof the $80 \\, \\rm{keV}$ harmonic of the CHESS X-ray beam. A set of \nsimulations with an $80 \\, \\rm{keV}$ beam (polarized and non-polarized) \nwere performed resembling the same offsets as measured at CHESS, as well \nas offset simulations at $40 \\, \\rm{keV}$.\n\n{\\it Non-polarized beam.} The left panel of \nFigure~\\ref{fig:PolFracSysError} shows examples of azimuthal scattering \ndistributions measured with different beam offsets for a non-polarized \nbeam. The distributions were corrected for $\\Delta \\Phi_{j}$ (assuming a \ncentral beam) and $a_{j}$, as before. It can be seen that beam offsets \nof $\\simeq 2 \\, \\rm{mm}$ already introduce systematic asymmetries. \nAlthough the offset distributions are not flat, as expected for a \nnon-polarized beam, they also do not resemble the shape expected from a \npolarized beam. Given the high event statistics of the data used in the \nstudy, the sinusoidal fits used to determine the polarization properties \nwill therefore certainly fail~-- allowing one to detect/filter the \nsystematic effect. However, in the case of data with higher statistical \nuncertainty, the observed asymmetries could potentially lead to the \nreconstruction of an artificial polarization fraction of $r_{\\rm{rec}} > \n0$. Note, that folding the distributions into the $[0; 180]^{\\circ}$ \ninterval (not shown), will substantially reduce the asymmetry. The \nStokes analysis is per definition only considering the $[0; \n180]^{\\circ}$ interval. However, it is not sensitive to deviations from \nthe sinusoidal scattering distribution, and therefore does not provide a \ngoodness-of-fit measure that can be used to detect imprints of a beam \noffset. Figure~\\ref{fig:PolFracSysError} also shows examples of \ndistributions that were offset corrected following the description in \nSec.~\\ref{subsec:AnalysisPolarization}. This, however, assumes that the \nbeam offset is known.\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[height=0.285\\textheight]{AzimuthDistribution_OffsetsPolarized}\n\\hfill\n\\includegraphics[height=0.285\\textheight]{PolFracSystErrorPolarized}\n\\end{center}\n\n\\caption{{\\bf Left:} Azimuthal scattering distributions of polarized \nX-ray beams hitting the scintillator at different offsets $d$. Results \nare shown for the CHESS data ($r_{\\rm{ch}} \\simeq 85\\%$ polarized) and \nsimulations ($r = 100\\%$ polarized), both in a non-rotating system. \nDistributions corrected for the beam offset (see \nSec.~\\ref{subsec:AnalysisPolarization}) are shown, as well. Results are \nshown for offsets along $X$ ($\\alpha = 0^{\\circ}$) and $Z$ ($\\alpha = \n-90^{\\circ}$, corresponding to a scan along $X$ with perpendicular \npolarization). {\\bf Right:} Reconstructed polarization fraction $r$ for \nthe off-center beams, analyzed using the Stokes parameters as in \nEq.~(\\ref{eqn:Stokes_Q_U_Average}). The outline of the scintillator is \nindicated. Also shown are data derived from the Ft.~Sumner measurement \nshown in Fig.~\\ref{fig:FtSumner_MirrorScan}, right. Results are shown \nwithout and with the offset correction.}\n\n\\label{fig:PolFracSysErrorPolarized}\n\\end{figure*}\n\nTo quantify the artificial polarization $r_{\\rm rec}$ inferred when \nneglecting the beam offset, the data were analyzed using the Stokes \nparameters following Eq.~(\\ref{eqn:Stokes_Q_U_Average}). \nFigure~\\ref{fig:PolFracSysError} (right) shows $r_{\\rm{rec}}$ as a \nfunction of beam offset. The results are shown for the simulations as \nwell as for the CHESS data~-- both being in reasonable agreement. Also \nshown is an analytic model that simply integrates the events per \nazimuth angle interval in a plane perpendicular to the optical axis. The \nright panel of Fig.~\\ref{fig:PolFracSysError} shows the same \ndistribution after the beam offset correction described in \nSec.~\\ref{subsec:AnalysisPolarization} was applied. The discrepancy \nbetween the data and simulations after correction can be explained by an \nuncertainty in the absolute beam alignment in the data and the fact \nthat, in contrast to the simulations, two separate measurements had to \nbe superimposed to generate the non-polarized beam, amplifying the effect \nof unaccounted offsets or mis-alignments (any uncertainty in the offset \nnot only moves the data points along the beam offset axis, but also \nalong the $r$ axis).\n\nIn general, however, the first-order correction procedure greatly \nreduces the systematic effect to less than a few percent for offsets $d \n\\leq 3 \\, \\rm{mm}$. The simulated data shown in the right panel of \nFig.~\\ref{fig:PolFracSysError} suggest that the correction works better \nfor beam energies of $80 \\, \\rm{keV}$ as compared to $40 \\, \\rm{keV}$. \nThis can be explained by the fact that only geometrical offsets are \ncorrected for. Differences in absorption lengths in the scintillator \nmaterial, originating from positions other than $P_{0}$ (see left panel \nin Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}), will cause \nsecond-order effects that depend on the energy of the scattered X-ray. \nHowever, addressing these additional correction terms is beyond the \nscope of this paper.\n\n{\\it Polarized beam.} For a polarized beam (left panel in \nFig.~\\ref{fig:PolFracSysErrorPolarized}), the offsets lead to either a \nreduction or amplification of the reconstructed polarization fraction, \ndepending on the angle between the offset vector $\\bf{d}$ and the plane \nof polarization $\\Omega_{\\rm{p}}$. The contribution of the beam offset \nto the reconstructed polarization is illustrated in the right panel of \nFig.~\\ref{fig:PolFracSysErrorPolarized} for different energies, \nincluding the results for the offset corrected analysis. The correction \nsubstantially reduces the systematic effect on $r$. A diverging trend \ncan be identified in the corrected CHESS data for $d > 0 \\, \\rm{mm}$ \n(which is also visible in the case of the `non-polarized' beam shown in \nthe right panel of Fig.~\\ref{fig:PolFracSysError}). This indicates an \ninaccuracy in the experimental setup, e.g. the asymmetric/fractional \nscattering off a fixture along the beam path before entering the \npolarimeter.\n\nAn additional offset measurement was performed during the flight \npreparation in Ft.~Sumner (see Sec.~\\ref{subsec:FtSumnerData}) using the \npartly polarized beam of the collimated X-ray source, hitting the \nscintillator at an offset of $d = -4 \\, \\rm{mm}$ (horizon system). The \npolarimeter/shield assembly was continuously rotating during the \nmeasurements. The corresponding fraction of apparent polarization $r$ is \nindicated in Fig.~\\ref{fig:PolFracSysErrorPolarized} (right). Due to the \nrotation of the polarimeter, the offset correction (see \nSec.~\\ref{subsec:AnalysisPolarization}) was calculated on an \nevent-by-event basis. The results illustrate that the correction also \nworks in a rotated system, canceling the artificial fraction of \npolarization introduced by the offset. The offset-corrected polarization \nspectrum is shown in Fig.~\\ref{fig:FtSumner_MirrorScan} (right), \nreproducing the measured results obtained with no beam offset.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Reco_MeanX}\n\\end{center}\n\n\\caption{Correlation between true beam offset and beam offset measured \nfollowing Eq.~(\\ref{eqn:FirstMoments}) for data and simulations.}\n\n\\label{fig:Offset_RecoMeanX}\n\\end{figure}\n\n{\\it Determination of the beam offset.} The geometrical offset \ncorrection described above assumes that the offset vector $\\bf{d}$ is \nknown. For sufficient event statistics and the assumption of a \ntime-independent beam offset, the offset can be estimated from the data \nitself using first moments, essentially substituting $2 \\Phi_{k} \n\\rightarrow \\Phi_{k}$ in Eq.~(\\ref{eqn:StokesParams}):\n\n\\begin{eqnarray} \\begin{split} \\left< x \\right> & = - \\frac{c}{W} \n\\sum_{k=1}^{N} w_{j(k)} \\sin(\\Phi_{k}), \\\\ \\left< z \\right> & = \n\\frac{c}{W} \\sum_{k=1}^{N} w_{j(k)} \\cos(\\Phi_{k}), \\\\ W & = \n\\sum_{k=1}^{N} w_{j(k)}. \\end{split} \\label{eqn:FirstMoments} \n\\end{eqnarray} Here, the constant $c=11 \\, \\rm{mm}$ reflects the \ndistance between the scintillator center and the detector plane (see \nFig.~\\ref{fig:X-Calibur_Configurations}, top). The weights $w_{j(k)}$ \nare the same as in Eq.~(\\ref{eqn:StokesSum}). As shown in \nFig.~\\ref{fig:Offset_RecoMeanX}, the measured offsets are linearly \ncorrelated with the true beam offsets. However, the slope of the \ncorrelation depends to some extent on the true polarization fraction $r$ \nof the beam and the angle between $\\Omega_{\\rm{p}}$ and {\\bf d}. Since \nthe offset of a non-polarized beam itself mimics a polarization fraction, \nthere will be a residual ambiguity that prevents to completely \ndisentangle $\\bf{d}$ and $r$ from the data alone. Therefore, a \ntime-resolved external monitoring of the beam position during the \nballoon flight is preferable.\n\nFigure~\\ref{fig:Offset_RecoMeanX} reveals a slight shift/translation \nbetween the slope of the reconstructed offset measured in the CHESS data \nversus the slope obtained from the simulations. This can be used to \nestimate an accuracy of the alignment achieved during the CHESS \nmeasurements to be $\\Delta X_{\\rm{sys}} \\simeq 0.3 \\, \\rm{mm}$ for the \nX-Calibur orientation of $\\alpha = 0^{\\circ}$. Note, with a rotation \naxis possibly not exactly aligned with the axis of the scintillator, \nthis may translate into larger offsets $\\Delta X$ for different \nX-Calibur orientations $\\alpha$.\n\n{\\it Systematic error on the polarization fraction.} The systematic \nerror on the reconstructed polarization fraction $\\Delta r_{\\rm{sys}}$ \ncan be estimated as follows. For the polarization measurements presented \nin this paper (Sec.~\\ref{subsec:CHESS} and \\ref{subsec:FtSumnerData}), \nwe assume an unaccounted beam/scintillator mis-alignment of $\\Delta d = 1 \n\\, \\rm{mm}$. For a highly-polarized beam of $r = O(100\\%)$ this leads to \nan error of $\\Delta r_{\\rm{sys,\\Delta d}} \\simeq 2\\%$ relative to the \nreconstructed on-axis beam (see simulated curve in \nFig.~\\ref{fig:PolFracSysErrorPolarized}). For a non-polarized beam \n(simulations in Fig.~\\ref{fig:PolFracSysError}) the offset leads to an \noverestimation of $\\simeq 2\\%$. For simplicity, we describe the \nsystematic error introduced by the beam offset as independent of the \ntrue polarization fraction: $\\Delta r_{\\rm{sys,\\Delta d}} = 2\\%$.\n\nThe X-Calibur version used during the CHESS and Ft.~Sumner measurements \n(see Secs.~\\ref{subsec:CHESS} and \\ref{subsec:FtSumnerData}) allowed for \nsome flexibility in adjusting the distance between the plane of the CZT \ndetectors and the optical axis when assembling the polarimeter (see \nFig.~\\ref{fig:X-Calibur_Configurations}, top: $c = 11 \\, \\rm{mm}$). We \nmeasured $c_{\\rm{ch}} = (11.3 \\pm 0.5) \\, \\rm{mm}$ for the CHESS setup \nand $c_{\\rm{ft}} = (10.5 \\pm 0.5) \\, \\rm{mm}$ for the Ft.~Sumner setup. \nConservatively, we assume $c = (11 \\pm 1) \\, \\rm{mm}$. To study how an \nincreased/decreased value of $c$ affects $r_{\\rm{rec}}$, we \ncorrespondingly shifted pixel coordinates when analyzing a data set, but \ndetermined $r_{\\rm{rec}}$ with the modulation factor $\\mu_{\\rm{sim}}$ \nobtained from simulations with the nominal value of $c = 11 \\, \\rm{mm}$. \nFor $\\Delta c = \\pm 1 \\, \\rm{mm}$, a $100\\%$ polarized beam is \nreconstructed to be $r_{\\rm{rec}} = r_{-0.5\\%}^{+1\\%}$ (overestimation \nfor a reduced distance $c$). For a non-polarized beam the effect is \nestimated to be $<0.1\\%$. We therefore assume that the distance related \nsystematic error\\footnote{Note, that asymmetric distances of the \ndifferent detector sides will mimic beam offsets with potentially \nstronger effects.} is proportional to $r$ with $\\Delta r_{\\rm{sys,c}} = \n0.01 \\, r_{\\rm{rec}}$. In the case of the Stokes analysis, the \nuncertainty in $c$ will introduce another systematic error: the angular \ncoverage of the detector gaps, corrected for by \nEq.~(\\ref{eqn:StokesGapCorrection}), will be over/underestimated. For a \n$100\\%$ polarized beam and $\\Delta c = -1 \\, \\rm{mm}$, we find maximal \ndeviation of $\\Delta r_{\\rm{sys,c}}^{\\rm{Stk}} = \\pm 6\\%$, and for \n$\\Delta c = +1 \\, \\rm{mm}$ we find $\\Delta r_{\\rm{sys,c}}^{\\rm{Stk}} = \n\\pm 4.5\\%$, with the sign and strength depending on the orientation of \nthe detector planes relative to the polarization vector (see \nSec.~\\ref{subsec:AnalysisPolarization}).\n\nAn error introduced by the analysis procedure can be estimated by \ncomparing the results obtained with the two analysis methods presented \nin Sec.~\\ref{subsec:AnalysisPolarization}. No significant differences \nwere found when analyzing simulated data of a non-polarized beam. Based \non the modulation factors of a $100\\%$ polarized beam, see \nEq.~(\\ref{eq:ModulationFactor}), we estimate the systematic error to be \non the order of $2\\%$. Therefore, a relative error of $\\Delta \nr_{\\rm{sys,a}} = 0.02 \\, r_{\\rm{rec}}$ is assumed. Reasons for the \ndifference can be the different treatment of dead pixels, detector gaps, \netc. (see Sec.~\\ref{subsec:AnalysisPolarization}). The total systematic \nerror on the polarization fraction is:\n\n\\begin{eqnarray}\n\\begin{split}\n\\Delta r_{\\rm{sys}} & = \\Delta r_{\\rm{sys,\\Delta d}} + \\Delta \nr_{\\rm{sys,c}} + \\left( \\Delta r_{\\rm{sys,c}}^{\\rm{Stk}} \\right) + \n\\Delta r_{\\rm{sys,a}} \\\\ & = 2\\% + 0.01 \\, r_{\\rm{rec}} + \\left(\\Delta \nr_{\\rm{sys,c}}^{\\rm{Stk}} \\right)+ 0.02 \\, r_{\\rm{rec}}.\n\\end{split}\n\\label{eqn:SystErrorPolFrac} \n\\end{eqnarray}\nThis error is estimated and valid for the particular measurements \npresented in this paper. A better alignment in future measurements, a \nrotating polarimeter, and a more detailed study of the analysis methods \nwill allow one to further reduce the error.\n\n{\\it Second-order systematic effects.} The tilt of the scintillator axis \nwith respect to the X-ray beam will introduce another (depth-dependent) \nsystematic effect. It should be mentioned that the simulations presented \nin this section assume a point-like X-ray beam, whereas the CHESS X-ray \nbeam had a square-shaped footprint with side lengths of the order of $1 \n\\, \\rm{mm}$. The systematic effects introduced by the offset/tilt might \nbe weakened by properly taking into account the point spread function of \nthe X-ray mirror that will be used for the astrophysical observations. \nThe temperature dependence of the energy calibration of the CZT \ndetectors (Sec.~\\ref{subsec:TempStudies}), if not corrected for, can \nalso modestly affect the reconstructed modulation factors of the data if \nthe measurements are performed at temperatures other than the \ncalibration temperature. However, the study of the effects discussed in \nthis paragraph are beyond the scope of this paper.\n\n\n\\subsection{Summary of the Polarimeter Performance}\n\nWe used measurements at the CHESS X-ray beam facility to calibrate the \npolarimeter. CHESS gives a highly polarized and well collimated beam \nwith well defined photon energies. We were able to demonstrate the full \nfunctionality of X-Calibur and measured the beam polarization to be \n$r_{\\rm{ch}} = (88 \\pm 5)\\%$. In addition, we used a collimated X-ray \nbeam from an X-ray source to make end-to-end tests of the full \n{\\it InFOC$\\mu$S}\\xspace/X-Calibur assembly in the field.\n\nWhen uncorrected for, the $\\sim$$3\\%$ of defect pixels and gaps between \ndetector boards produce an apparent polarization of a non-polarized beam \nof up to $5\\%$ (Stokes analysis only). After correction, the apparent \npolarization goes down to $< 2 \\%$. Once we rotate the polarimeter, we \nexpect that the systematic effect is reduced by a factor $\\sim \n\\sqrt{n}$, with $n$ being the number of detected events. The systematic \nerror will then be much smaller than the statistical error.\n\nThe detection principle of X-Calibur requires to focus the X-rays onto \nthe center of the scatterer. If uncorrected for, an offset of $1 \\, \n\\rm{mm}$ leads to an apparent polarization of $\\simeq 2\\%$ for a\nnon-polarized beam. For the upcoming balloon flight, our goal is to limit \nthe offset of the focal point from the center of the scatterer to $< 1 \n\\, \\rm{mm}$, and to monitor the offset with a backward looking camera \nlocated close to the X-ray mirror. The camera will monitor a LED cross \nhair. We can in addition use X-ray data to constrain the location of the \nfocal spot. The systematic error on the polarization fraction after \ncorrecting for the offset of the focal point is $< 1 \\%$. Thus, for the \nupcoming balloon flight, we estimate that X-Calibur can detect $\\geq \n3~\\%$ polarization fractions in the $20 - 80 \\, \\rm{keV}$ band.\n\n\n\\section{Summary and Conclusions} \\label{sec:Conclusion}\n\nWe designed, optimized, and built an X-ray polarimeter, X-Calibur, and \nstudied its performance and sensitivity. The fully assembled X-Calibur \npolarimeter was tested (i) in the laboratory at Washington University, \n(ii) with a polarized X-ray beam at the Cornell High Energy Synchrotron \nSource (CHESS), and (iii) during a X-Calibur/{\\it InFOC$\\mu$S}\\xspace flight-integration \ntest in Ft.~Sumner, NM. X-Calibur makes use of the fact that polarized \nphotons Compton scatter preferentially perpendicular to their electric \nfield orientation. It combines a detection efficiency on the order of \n$80 \\%$, with a high modulation factor of $\\mu \\approx 0.5$ averaged \nover the whole detector assembly, and with values up to $\\mu \\approx \n0.7$ for select subsections of the polarimeter. Operated in a mode of \ncontinuous rotation, X-Calibur allows for a good control over systematic \neffects. Scattering polarimetry has the strength that it can operate \nover a wide energy range. The low energy threshold is given by the \ncompetition between photoelectric absorption and scattering processes, \nthe mirror reflectivity limits the sensitivity at high energies.\n\nWe calibrated all 2048 CZT detector pixels of the polarimeter and \nstudied their performance with respect to their energy threshold, energy \nresolution (including the contribution of electronic readout noise), and \ntrigger efficiency. The CZT detectors achieve a mean energy threshold of \n$21 \\, \\rm{keV}$ and a mean $40 \\, \\rm{keV}$ energy resolution of \n$\\Delta E_{\\rm{czt}} \\approx 4 \\, \\rm{keV}$ FWHM. The \ntemperature-dependencies of the pixel responses were studied, as well, \nand we found that the effects can be sufficiently controlled for the \ntemperature range expected during a balloon flight.\n\nWe characterized the performance of the active CsI shield and find a \nbackground suppression by more than one order of magnitude in the energy \nrange relevant for X-Calibur. We also measured the trigger efficiency of \nthe scintillator that is used as Compton scatterer and find a trigger \nthreshold around $5 \\, \\rm{keV}$ with the potential for further \nreduction (using an improved amplification circuit). We used the CHESS \nX-ray beam to test the polarimeter. Detailed comparisons of experimental \nand simulated data allowed us to demonstrate the full functionality of \nthe polarimeter and to measure the polarization of the CHESS beam. We \nstudied different systematic effects that potentially affect the \nreconstructed polarization properties and estimated the systematic error \nto be smaller than $2\\%$ for an upcoming balloon flight.\n\nOur tentative observation program for an upcoming X-Calibur/{\\it InFOC$\\mu$S}\\xspace \nballoon flight includes galactic sources (Crab nebula, Her\\,X-1, \nCyg\\,X-1, GRS\\,1915, EXO\\,0331) and one extragalactic source (Mrk\\,421) \nfor which sensitive polarization measurements will be carried through.\n\nIn principle, a similar space-borne scattering polarimeter could operate \nover the broader $3-80 \\, \\rm{keV}$ energy band. Here, a LiH rod would \nbe used as passive scatterer. In contrast to the plastic scintillator \nused in the balloon-borne polarimeter, the LiH scatterer does not yield \na coincidence signal. However, the lower atomic number of LiH results in \nthe possibility of using the polarimeter down to energies of a few keV.\n\n\n\n\n\\section*{Acknowledgments}\n\nWe are grateful for NASA funding from grants NNX10AJ56G, NNX12AD51G and \nNNX14AD19G, as well as discretionary funding from the McDonnell Center \nfor the Space Sciences to build the X-Calibur polarimeter. Polarization \nmeasurements: This work is based upon research conducted at the Cornell \nHigh Energy Synchrotron Source (CHESS) which is supported by the \nNational Science Foundation and the National Institutes of \nHealth/National Institute of General Medical Sciences under NSF award \nDMR-0936384. We would like to thank Ken Finkelstein for the excellent \nsupport in setting up the experiment at CHESS and for the continuous \ndiscussions thereafter.\n\n\n"}}},{"rowIdx":83599,"cells":{"uuid":{"kind":"number","value":1116691499414,"string":"1,116,691,499,414"},"dataset":{"kind":"string","value":"arxiv"},"text":{"kind":"string","value":"\\section{Introduction}\n\tA region $\\Omega \\subset \\mathbb C$, the boundary of which is a polygonal curve with angles multiple of $\\pi/2$, is considered in this article. This region models the shape of a channel under a dam. The calculation of fluid flow in such channel boils down to a conformal mapping problem of $\\Omega$ onto the upper half-plane. The solution of such problems is given by Christoffel-Schwartz integral (see, e.g., \\cite{ShabatEng} or \\cite{SchwartzChristoffel}), which in this case is naturally defined on an elliptic Riemann surface. In this paper the simple formula, which writes integral in terms of Weierstrass sigma function (see e.g., \\cite{AkhiezerEng} or \\cite{Chandra}), was found. This approach allows to avoid numerical integration and the mapping parameters can be found from a simple nonlinear system of equations; therefore, the computation is significantly simplified.\n\t\n\tSimilar problems have been considered in \\cite{SigBog}, \\cite{DamBogEng}, \\cite{BogHept}, \\cite{BogGrig}, and \\cite{BogCond}, where Christoffel-Schwartz integral was efficiently represented by theta functions (see, e.g., \\cite{FarkasKra}). In the paper \\cite{BogCond} the application of Lauricella functions to these problems was studied, and also different approaches were compared.\n\t\n\tThe main advantage of Weierstrass functions over theta functions is that they have limiting values when the surface degenerates. The paper analyzes behavior of the constructed conformal mapping under the condition that the dam width tends to zero. It turns out that conformal mappings have a limit, which is a solution to the limiting problem. Thus, it is shown that the solution is stable under the considered degeneration.\n\t\n\tThe described property of the Weierstrass sigma function loses its value if the standard method is used for calculations, which expresses the sigma function in terms of the theta function (since it does not withstand degeneration). Thus, it is necessary to use an independent method for calculating sigma functions. In this paper, we use the expression for the coefficients of its expansion into Taylor series obtained by Weierstrass (see \\cite{Weier}). Since the proof presented there is apparently not complete (at one point, the analyticity of the sigma function in three variables in a neighborhood of zero is used, which is not obvious), we give a more detailed proof in the appendix. The above formula, however, is not sufficient for the final numerical solution, since Taylor series are not suitable for calculations with large arguments (namely, such a need arises with degeneration). Thus, the problem of constructing an efficient computational method for the sigma function independent of theta functions still remains unsolved. If such a method is available, it will be possible to construct formulas that are stable under various degenerations and use them in calculations. The results of this work illustrate the need in such methods.\n\t\n\tProblems in which hyperelliptic Riemann surfaces of higher genus arise can also be solved using the theory of sigma functions developed by Klein and Baker in \\cite{Klein} and \\cite{Baker} respectively (more detailed exposition can be found in \\cite{Buch}). There is hope that it will be possible to prove the stability of formulas expressing the solution of the above problems in terms of high-order sigma functions. Thus, the construction of Weierstrass-type recurrent formulas (which are known for genus 1 and 2; see \\cite{Buch}) and calculation methods for sigma functions can be extremely useful in applied problems.\n\t\n\t\\section{The statement and the origin of the problem}\n\tConsider region $\\Omega$ in the complex plane pictured on Figure 1. \n\n\n\n\n\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\draw[black,thick] (-5,-1) -- (5,-1);\n\t\t\t\\draw[black,thick] (-5,2) -- (0,2) node [pos=1,below left] {$w_4$};\n\t\t\t\\draw[black,thick] (0,2) -- (0,0) node [pos=1,left] {$w_3$};\n\t\t\t\\draw[black,thick] (0,0) -- (1,0) node [pos=1,below right] {$w_2$};\n\t\t\t\\draw[black,thick] (1,0) -- (1,1) node [pos=1,above] {$w_1$};\n\t\t\t\\draw[black,thick] (1,1) -- (5,1);\n\t\t\t\\draw[<->,black,dashed,thick] (0,-0.2) -- (1,-0.2) node [pos=0.5,below] {$\\delta$};\n\t\t\t\\draw[<->,black,dashed,thick] (1.2,0) -- (1.2,1) node [pos=0.5,right] {$h$};\n\t\t\t\\draw[<->,black,dashed,thick] (-4,-1) -- (-4,2) node [pos=0.5,right] {$h^-$};\n\t\t\t\\draw[<->,black,dashed,thick] (4,-1) -- (4,1) node [pos=0.5,right] {$h^+$};\n\t\t\\end{tikzpicture}\n\t\t\n\t\t\\caption{Region $\\Omega$.}\n\t\\end{figure}\n\tIt is bounded from below by a line and from above by a polygonal curve with four vertices $w_1,w_2,w_3,w_4$ (it is convenient to think that this region also has two vertices at $\\pm \\infty$). Let the line, which bounds the region from below be parallel to the real axis, and let the vertex $w_4$ be at the origin. Then, this region is determined by four real parameters $h^-,h^+,h,\\delta$, where $h$ is equal to the length of segment $[w_1,w_2]$, $\\delta$ is the length of $[w_2,w_3]$, and $h^-$ and $h^+$ are equal to the distance from the line that bounds the region from below to $w_4$ and $w_1$ respectively. These parameters are positive and satisfy inequalities $h^- - h^+ + h > 0$, which corresponds to positivity of the length of $[w_3,w_4]$, and $h < h^+$. The region is determined uniquely by these parameters.\n\n\tRegions similar to $\\Omega$ arise in problems connected with the computation of fluid flow through the porous material under a dam. Since the flow is continuous and satisfies the Darcy's law, the pressure $p$ is a harmonic function in $\\Omega$. Assuming that segments $[w_1, w_2]$, $[w_2,w_3]$, $[w_3, w_4]$, and channel's bottom are impenetrable, we obtain natural boundary conditions: the normal derivative $\\partial p/\\partial n$ vanishes on impenetrable segments of the boundary, while on the remaining segments (i.e. on the half-lines starting from $w_1$ and $w_4$) $p$ is locally constant.\n\n\tConsider a real-valued function $q$ in the region $\\Omega$ such that $f = p + iq$ is holomorphic (such function exists because $\\Omega$ is simply connected). The normal derivative of $p$ vanishing condition is easily equivalent to constancy of $q$ on the corresponding boundary segment. It follows, that, if $f$ is a function that conformally maps $\\Omega$ onto a rectangle in a way, such that $w_1$, $w_4$, and vertices at infinity are mapped to the vertices of a rectangle, then $p = \\RE f$ is a solution to the original problem. $q = \\IM f$ is called the current function. Its level lines are the streamlines of the fluid under the dam. It is clear, that it is enough to solve the problem of conformal mapping of $\\Omega$ onto upper half-plane $\\mathbb C_+$ in order to solve the specified problem.\n\n\tIn what follows, the conformal mapping problem will be solved explicitly using the tools of Weierstrass elliptic functions. Below we show the calculation of streamlines in $\\Omega$ obtained using the method constructed in this work.\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}{.5\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.9\\textwidth]{LinesColor2.pdf}\n\t\t\\end{subfigure}%\n\t\t\\begin{subfigure}{.5\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.9\\textwidth]{LinesColorCloser2.pdf}\n\t\t\\end{subfigure}\n\t\t\n\t\t\\caption{Streamlines in the region $\\Omega$.}\n\t\\end{figure}\n\t\\FloatBarrier\n\t\\section{The solution of the conformal mapping problem}\n\t\\subsection{The general form of the solution and parameter determination}\n\tSince $\\Omega$ is simply connected, there is a conformal mapping $W:\\mathbb C_+ \\rightarrow \\Omega$, where $\\mathbb C_+ = \\{z \\in \\mathbb C: \\IM z > 0\\}$ is the upper half-plane (see, e.g., \\cite{KartanEng} or \\cite{ShabatEng}). Using, if necessary, a suitable automorphism of $\\mathbb C_+$, one can make it such that the point $w_4$ is the preimage under $W$ (more precisely, under its continuation to the boundary) of $\\infty$. Then, by Christoffel-Schwartz theorem (see \\cite{SchwartzChristoffel}), there exist $x^-,black,dashed,thick] (0.2,0) -- (0.2,1) node [pos=0.5,right] {$h$};\n\t\t\\draw[<->,black,dashed,thick] (-4,-1) -- (-4,2) node [pos=0.5,right] {$h^-$};\n\t\t\\draw[<->,black,dashed,thick] (4,-1) -- (4,1) node [pos=0.5,right] {$h^+$};\n\t\\end{tikzpicture}\n\t\\caption{Region $\\widetilde \\Omega$.}\n\\end{figure}\n$\\widetilde{\\Omega}$ is determined by three parameters\n$h,h^+$ and $h^-$. A conformal mapping of the upper half-plane onto $\\widetilde\\Omega$ can be found by the analogous method (using Christoffel-Schwartz theorem). In this case, since the corresponding Riemann surface has genus $0$, the solution can be expressed in elementary functions. Another method (that is considered here) is to apply formula \\eqref{eq417}, using the fact that $\\sigma$ is defined for $g_2$ and $g_3$ such that $F(x) = 4x^3 - g_2x - g_3$ has multiple roots. It is natural to suppose that the solution can be found by taking the limit under gluing the roots, that are mapped to $w_2$ and $w_3$. Together with that the stability of the solution under $\\delta\\rightarrow 0$ will be proved.\n\n\\subsection{Gluing of the roots}\nAgain consider a family of curves depending on $\\gamma \\in (-1/6, 1/6)$ that is given by $F_\\gamma(x) = 4(x - x_1(\\gamma))(x - x_2(\\gamma))(x - x_3(\\gamma))$, where $x_1(\\gamma) = \\gamma - 1/2$, $x_2(\\gamma) = -2\\gamma$, $x_3(\\gamma) = \\gamma + 1/2$. Under $\\gamma \\rightarrow -1/6$ the roots $x_2$ and $x_3$ glue. The limiting values of $g_2$ and $g_3$ are $4/3$ and $-8/27$ respectively. For each $\\gamma$ we define quantities $\\omega(\\gamma)$, $\\omega'(\\gamma)$, $\\eta(\\gamma)$, $\\eta'(\\gamma)$. In what follows we shall omit dependence on $\\gamma$.\n\n\\begin{lemma}\\label{l41}\n\tUnder $\\gamma \\rightarrow -1/6$ we have\n\t\\begin{equation}\\label{eq441}\n\t\t\\omega,\\eta \\rightarrow \\infty,\\;\\; \\omega' \\rightarrow \\frac{i\\pi}{2},\\;\\;\\eta' \\rightarrow -\\frac{i\\pi}{6},\\;\\; \\frac{\\eta}{\\omega} \\rightarrow -\\frac{1}{3}.\n\t\\end{equation}\n\tMoreover,\n\t\\begin{equation}\\label{eq442}\n\t\t\\begin{gathered}\n\t\t\\sigma(z,\\frac{4}{3}, -\\frac{8}{27}) = e^{-\\frac{z^2}{6}} \\sinh(z),\\;\\; \\zeta(z,\\frac{4}{3}, -\\frac{8}{27}) = \\coth(z) - \\frac{z}{3},\\\\ \\wp(z,\\frac{4}{3}, -\\frac{8}{27}) = \\frac{1}{\\sinh^2(z)} + \\frac{1}{3}.\n\t\t\\end{gathered}\n\t\\end{equation}\n\tFinally, there exists $\\varepsilon > 0$ such that for $\\gamma + 1/6 < \\varepsilon$ the estimation\n\t\\begin{equation}\\label{eq44asympt}\n\t\t-c_1\\ln(\\gamma + 1/6) \\le \\omega(\\gamma) \\le -c_2\\ln(\\gamma + 1/6)\n\t\\end{equation}\n\tholds, where $0 < c_1 < c_2$.\n\\end{lemma}\n\\begin{proof}\n\t\\eqref{eq441} easily follows from integral representations\n\t$$\n\t\\omega(\\gamma) = \\frac{1}{2}\\int_{\\gamma - \\frac{1}{2}}^{-2\\gamma} \\frac{dx}{\\sqrt{(x - \\gamma - 1/2)(x - \\gamma + 1/2)(x + 2\\gamma)}}, $$ $$\n\t\\omega'(\\gamma) = \\frac{1}{2}\\int_{-2\\gamma}^{\\gamma+\\frac{1}{2}} \\frac{dx}{\\sqrt{-(x - \\gamma - 1/2)(x - \\gamma + 1/2)(x + 2\\gamma)}}, $$ $$\n\t\\eta(\\gamma) = -\\frac{1}{2}\\int_{\\gamma - \\frac{1}{2}}^{-2\\gamma} \\frac{xdx}{\\sqrt{(x - \\gamma - 1/2)(x - \\gamma + 1/2)(x + 2\\gamma)}},\n\t$$ $$\n\t\\eta'(\\gamma) = -\\frac{1}{2}\\int_{-2\\gamma}^{\\gamma + \\frac{1}{2}} \\frac{xdx}{\\sqrt{-(x - \\gamma - 1/2)(x - \\gamma + 1/2)(x + 2\\gamma)}}. $$\n\tTo derive \\eqref{eq442} one can pass to the limit $\\gamma \\rightarrow -1/6$ using the formula that represents $\\sigma$ as the infinite product (see \\cite{AkhiezerEng}). We obtain\n\t$$\n\t\\sigma(z,\\frac{4}{3}, -\\frac{8}{27}) = z\\prod_{n \\ne 0} \\left(1 - \\frac{z}{i n \\pi}\\right) e^{\\frac{z}{i n \\pi} - \\frac{z^2}{2 n^2 \\pi^2}}.\n\t$$\n\tUsing classical identities\n\t$$\n\t\\sum_{n = 1}^\\infty \\frac{1}{n^2} = \\frac{\\pi^2}{6},\\;\\; \\prod_{n = 1}^\\infty \\left(1 - \\frac{x^2}{n^2 \\pi^2}\\right) = \\frac{\\sin(x)}{x},\n\t$$\n\twe derive the first equation of \\eqref{eq442}. The rest follow from equalities $\\zeta(z) = \\sigma'(z)/\\sigma(z)$, $\\wp(z) = -\\zeta'(z)$.\n\t\n\tNow we estimate the growth of $\\omega(\\gamma)$. Consider equality\n\t$$\n\t\\omega(\\gamma) = \\frac{1}{2}\\int_0^{1/2 - 3\\gamma} \\frac{dt}{\\sqrt{t(1-t)(1/2 - 3\\gamma - t)}}.\n\t$$\n\tNote that for small $\\gamma$ integral on the segment $[0,1/2]$ is bounded and, therefore,\n\t$$\n\t\\omega(\\gamma) \\sim \\frac{1}{2}\\int_{1/2}^{1/2 - 3\\gamma} \\frac{dt}{\\sqrt{t(1-t)(1/2 - 3\\gamma - t)}}.\n\t$$\n\tThe estimation of the integral in rhs is quite simple since $1/\\sqrt{t}$ is bounded from below and above by positive constants and the remaining integral can be calculated explicitly.\n\\end{proof}\n\nSince $\\omega' \\rightarrow i\\pi/2$ and $\\omega \\rightarrow \\infty$, it is natural to suppose that \\eqref{eq417} can give a formula for a conformal mapping of the half-strip $$S = \\{z \\in \\mathbb C: \\RE z < 0, \\IM z \\in (0,\\pi/2)\\}$$ onto $\\widetilde\\Omega$. Let\n\\begin{equation}\\label{eq443}\n\t\\widetilde Q(z) = Dz + \\frac{h^-}{\\pi}\\ln\\left(\\frac{\\sigma(z-z^-)}{\\sigma(z+z^-)}\\right) - \\frac{h^+}{\\pi}\\ln\\left(\\frac{\\sigma(z-z^+)}{\\sigma(z+z^+)}\\right) - i(h^- - h^+),\n\\end{equation}\nwhere $\\sigma$ is taken at values $g_2 = 4/3$, $g_3 = -8/27$ and $D \\in \\mathbb C$, $z^-, z^+\\in(0,i\\pi/2)$ are the parameters. Substituting \\eqref{eq442} into \\eqref{eq443}, we obtain\n\\begin{equation}\\label{eq444}\n\t\\begin{multlined}\n\t\\widetilde Q(z) = z\\left(D + \\frac{2h^-z^-}{3\\pi} - \\frac{2h^+z^+}{3\\pi}\\right) \\\\+ \\frac{h^-}{\\pi}\\ln\\frac{\\sinh(z - z^-)}{\\sinh(z+z^-)} - \\frac{h^+}{\\pi}\\ln\\frac{\\sinh(z - z^+)}{\\sinh(z + z^+)}- i(h^- - h^+).\n\t\\end{multlined}\n\\end{equation}\nIt is easy to check that if $\\widetilde Q$ has a non-zero linear term it has no limit under $\\RE z \\rightarrow -\\infty$. If$$\nD + \\frac{2h^-z^-}{3\\pi} - \\frac{2h^+z^+}{3\\pi} = 0,\n$$ its limit is equal to $2(h^-z^- - h^+z^+)/\\pi - i(h^- - h^+)$. Thus, if $\\widetilde Q$ conformally maps $S$ onto $\\widetilde \\Omega$, then the conditions\n\\begin{equation}\\label{eq445}\n\t\\begin{dcases}\n\t\tD + \\frac{2h^-z^-}{3\\pi} - \\frac{2h^+z^+}{3\\pi} = 0, \\\\\n\t\th^- z^- - h^+ z^+ = -\\frac{ih\\pi}{2}\n\t\\end{dcases}\n\\end{equation}\nhold. Obviously, these conditions are sufficient under the additional assumption that the derivative of $\\widetilde{Q}$ is not vanishing on $S$ and its boundary. Using tedious but elementary calculations it can be shown that this condition is equivalent to \n\\begin{equation}\\label{eq446}\n\th^-\\sinh(2z^-) = h^+\\sinh(2z^+).\n\\end{equation}\nThus, \\eqref{eq445} and \\eqref{eq446} determine the parameters $D,z^-,z^+$, at which $\\widetilde Q$ is the desired conformal mapping.\n \nWe show that the obtained formula reduces to Christoffel-Schwartz integral in the upper half-plane under the variable change $x = \\wp(z) = 1/\\sinh^2(z) + 1/3$. Changing the variable in \\eqref{eq443} and using that the linear term vanishes we obtain the following formula for the conformal mapping of $\\mathbb C_+$ onto $\\widetilde{\\Omega}$:\n \\begin{equation}\\label{eq448}\n\t\\widetilde{W}(x) = \\frac{h^-}{\\pi}\\ln \\left(\\frac{\\sqrt{x^- + 2/3} - \\sqrt{x + 2/3}} {\\sqrt{x^- + 2/3} + \\sqrt{x + 2/3}}\\right) + \\frac{h^+}{\\pi}\\ln \\left(\\frac{\\sqrt{x^+ + 2/3} - \\sqrt{x + 2/3}} {\\sqrt{x^+ + 2/3} + \\sqrt{x + 2/3}}\\right).\n\\end{equation}\n \n The equations on the parameters $z^-$ and $z^+$ can be rewritten in the form\n \\begin{equation}\\label{eq449}\n \t\\begin{dcases}\n \t\t\\frac{h^-\\sqrt{x^- + 1}}{x^-} = \\frac{h^+\\sqrt{x^++1}}{x^+}, \\\\\n \t\th^- \\ln\\left(\\frac{1+\\sqrt{x^- + 4/3}}{\\sqrt{x^- + 1/3}}\\right) + h^+\\ln\\left(\\frac{1+\\sqrt{x^+ + 4/3}}{\\sqrt{x^+ + 1/3}}\\right) = -\\frac{ih\\pi}{2}.\n \t\\end{dcases}\n \\end{equation}\nDifferentiating $\\widetilde{W}$ and using the first equation from \\eqref{eq449} we obtain $$\nd\\widetilde{W} = \\frac{h^-\\sqrt{x^-+1}- h^+\\sqrt{x^++1}}{\\pi} \\frac{(x - 1/3)dx}{\\sqrt{x+2/3}(x-x^-)(x-x^+)}.\n$$\nTherefore we found the exact form of the constant in Christoffel-Schwartz integral. The remaining parameters $x^-$ and $x^+$ can be found from the system of equations \\eqref{eq449}.\n\n\\subsection{Passing to the limit}\n\nConsider a sequence of regions determined by the parameters $(h^-_n, h^+_n, h_n, \\delta_n)$. Assume that they have limits $(h^-_{\\lim}, h^+_{\\lim}, h_{\\lim}, 0)$. Also we assume that $h^+_{\\lim} - h_{\\lim} > 0$ and $h^-_{\\lim} - h^+_{\\lim} + h_{\\lim} > 0$. We shall prove that the parameters $(D_n, \\gamma_n, z^-_n, z^+_n)$, given by the solution of the system \\eqref{eq418} for the corresponding regions, have limits $(D_{\\lim}, -1/6, z^-_{\\lim}, z^+_{\\lim})$, and, moreover, parameters $(D_{\\lim}, z^-_{\\lim}, z^+_{\\lim})$ satisfy \\eqref{eq445} and \\eqref{eq446}. Thus, in view of the fact that the Weierstrass sigma function is entire, it follows that the constructed solution is stable.\n\nThe following proof is rather long and technical and, therefore, we shall omit most of the calculations.\n\nIn the following estimations we shall also use parameters $x^-_n, x^+_n, x_1^{(n)}, x_2^{(n)}, x_3^{(n)}, C_n$ of the map $W_n$.\n\n\\begin{lemma}\n\tAssume that $\\gamma_n \\rightarrow -1/6$ and $\\delta_n \\omega(\\gamma_n) \\rightarrow 0$.\n\tThen the indicated convergence holds.\n\\end{lemma}\n\\begin{proof}\n\tNote that $\\gamma_n \\rightarrow -1/6$ implies that sequences $z^-_n$ and $z^+_n$ are bounded. The first equation in \\eqref{eq418} then implies that $D_n$ is also bounded. Passing to the subsequences we can assume that all these sequences are convergent (if we succeed to prove that the limits satisfy \\eqref{eq445} and \\eqref{eq446}, then uniqueness of the solution implies that all the subsequences converge to the same limit, and, therefore, the initial sequences converge). The first equation in \\eqref{eq445} is obtained by passing to the limit in the second equation in \\eqref{eq418} in view of Lemma \\ref{l41}. Multiplying the first equation in \\eqref{eq418} by $\\omega'$ and the second one by $\\omega$, subtracting and passing to limit leads to the second equation in \\eqref{eq445} (the term $\\delta \\omega$ by assumption tends to zero). \n\t\n\tNow we derive \\eqref{eq446} for the limits of the sequences. Recall the following notation of the theory of elliptic functions (see \\cite{AkhiezerEng}):\n\t\t$$\n\t\\zeta_2(z) = \\zeta(z + \\omega) - \\eta,\n\t$$\n\t$$\n\t\\zeta_3(z) = \\zeta(z + \\omega + \\omega') + \\eta + \\eta'.\n\t$$\n\tThese functions are connected to $\\sigma_2, \\sigma_3$:\n\t$$\n\t\\zeta_k = \\frac{1}{\\sigma_k} \\frac{d \\sigma_k}{d z} = \\frac{d \\ln \\sigma_k}{d z}.\n\t$$\n\tFinally, $\\sigma_k = \\sigma \\sqrt{\\wp - x_k}$. The last two equations in \\eqref{eq418} can be rewritten as\t\n\t\\begin{equation}\\label{eq447}\n\t\tD + \\frac{h^-}{\\pi}(\\zeta_k(-z^-) - \\zeta(z^-)) - \\frac{h^+}{\\pi}(\\zeta_k(-z^+) - \\zeta(z^+)) = 0, \\;\\; k = 2,3.\n\t\\end{equation}\n\n\tSince\n\t$$\n\t\\begin{multlined}\n\t\t\\zeta_2(z) - \\zeta_3(z) = \\frac{\\sigma'(z) \\sqrt{\\wp - x_2} + \\wp'(z) \\sigma(z) (\\wp - x_2)^{-1/2}}{\\sigma(z) \\sqrt{\\wp - x_2}} \\\\ - \\frac{\\sigma'(z) \\sqrt{\\wp - x_3} + \\wp'(z) \\sigma(z) (\\wp - x_3)^{-1/2}}{\\sigma(z) \\sqrt{\\wp - x_3}},\n\t\\end{multlined}\n\t$$\n\tit follows that\n\t\\begin{equation}\\label{zetadiff}\n\t\t\\zeta_2(z) - \\zeta_3(z) = \\frac{\\wp'(z)(x_2 - x_3)}{(\\wp - x_2)(\\wp - x_3)}.\n\t\\end{equation}\n\tEquation \\eqref{eq446} is derived by passing to the limit (using Lemma \\ref{l41}) from equations \\eqref{eq447} (from which the constant $D$ can be eliminated) and substitution the formula for $\\zeta_2 - \\zeta_3$ from \\eqref{zetadiff}.\n\\end{proof}\n\\begin{lemma}\n\tInequalities $|C_n| \\ge a_1$, $|C_n| \\le a_2 \\sqrt{x_3^{(n)} - x^-_n}$ hold for some positive constants $a_1,a_2$. Moreover, the sequence $x^+_n$ is bounded from below.\n\\end{lemma}\n\\begin{proof}\n\tThe estimations for $C^{(n)}$ easily follow from\n\t$$|C_n|\\bigintss_{x_3^{(n)}}^{+\\infty} \\frac{\\sqrt{\\left(x - x_2^{(n)}\\right)\\left(x - x_3^{(n)}\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = h^-_n - h^+_n + h_n.\n\t$$\n\tTo prove the boundedness from below for $x^+_n$ it suffices to consider equality\n\t$$\n\t|C_n|\\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)}-x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = h_n.\n\t$$\n\\end{proof}\n\\begin{lemma}\n\tAssume that sequence $x^-_n$ is bounded from below. Then there exist constants $0 < b_1 < b_2$ such that $b_1 \\sqrt{\\delta_n} \\le |x_2^{(n)} - x_3^{(n)}| \\le b_2 \\sqrt{\\delta_n}$. \n \n\\end{lemma}\n\\begin{proof}\n\tIt follows from easy estimation for the integral in equality\n\t$$\n\t|C_n|\\bigintss_{x_2^{(n)}}^{x_3^{(n)}} \\frac{\\sqrt{\\left(x - x_2^{(n)}\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = \\delta_n.\n\t$$\n\\end{proof}\n\nThe foregoing Lemmas imply that it remains to prove that sequence $x^-_n$ is bounded from below (in view of the asymptotics \\eqref{eq44asympt}). Assume that this is not true. Passing to the subsequences we can assume that $x^-_n \\rightarrow -\\infty$ and $x_1^{(n)},x_2^{(n)},x_3^{(n)}$, $x^+_n$ are convergent.\n\n\\begin{lemma}\n\tIn the foregoing assumptions statements $x_3^{(n)} - x_2^{(n)} \\rightarrow 0$, $x_1^{(n)} - x^+_n \\rightarrow 0$ hold.\n\\end{lemma}\n\\begin{proof}\n\tEquality\n\t$$\n\t|C_n| \\frac{\\sqrt{\\left(x_2^{(n)} - x^+_n\\right)\\left(x_3^{(n)} - x^+_n\\right)}}{\\sqrt{x_1^{(n)} - x^+_n}(x^+_n - x^-_n)} = \\frac{h^+_n}{\\pi}\n\t$$\n\timplies $x_1^{(n)} - x^+_n \\rightarrow 0$. \n\t\n\tTo prove that $x_3^{(n)} - x_2^{(n)} \\rightarrow 0$ we return to parameters $(D_n, \\gamma_n, z^-_n, z^+_n)$. Assume that $x_2^{(n)} - x_1^{(n)} \\rightarrow 0$. Then $\\omega'(\\gamma_n), \\eta'(\\gamma_n) \\rightarrow \\infty$, and $\\omega$ and $\\eta$ have finite limits. Moreover, Legendre identity (see, e.g., \\cite{AkhiezerEng} or \\cite{Chandra}) implies\n\t$$\n\t\\lim_{n \\rightarrow \\infty} \\frac{\\omega'(\\gamma_n)}{\\eta'(\\gamma_n)} = \\lim_{n \\rightarrow \\infty} \\frac{\\omega(\\gamma_n)}{\\eta(\\gamma_n)}.\n\t$$\n\tThe first two equations from \\eqref{eq418} imply\n\t$$\n\t\\lim_{n \\rightarrow \\infty}\\left(\\frac{2 h^+_n}{\\pi} z^+_n \\left(\\frac{\\omega'}{\\eta'} - \\frac{\\omega}{\\eta}\\right) - i \\frac{h_n}{\\omega}\\right) = 0.\n\t$$\n\tTherefore, \n\t$$\n\t\\lim_{n \\rightarrow \\infty}\\left(\\frac{h^+_n z^+_n}{\\omega'} - h_n\\right) = 0,\n\t$$\n\tand, passing to the limit, we obtain $h_{\\lim} \\ge h^+_{\\lim}$. This contradicts the assumptions made.\n\t\n\tNow assume that $x_2^{(n)} - x_1^{(n)} \\nrightarrow 0$ and$x_3^{(n)} - x_2^{(n)} \\nrightarrow 0$. Then both periods $\\omega$ and $\\omega'$ have finite limits. In this case $z^-_n \\rightarrow 0$, $z^+_n \\rightarrow \\omega'(\\gamma_{\\lim})$. It is obvious that $D_n$ also is convergent and, passing to the limit in the second equation in \\eqref{eq418}, we obtain\n\t$$\n\t-D_{\\lim} \\omega' - \\frac{2h^+_{\\lim} \\omega' \\eta'}{\\pi} = 0.\n\t$$\n\tSubstituting into the first equation we get\n\t$$\n\t\\frac{2h^+_{\\lim}}{\\pi} \\omega \\eta' - \\frac{2h^+_{\\lim}}{\\pi} \\omega' \\eta = -ih_{\\lim},\n\t$$\n\timplying $h^+_{\\lim} = h_{\\lim}$.\n\\end{proof}\n\nNow we have enough preparation to deduce a contradiction from $x^-_n \\rightarrow -\\infty$. In order to do this we shall analyse asymptotics of some sequences (in what follows the equivalence of sequences means that the quotient of them tends to $1$).\n\nEquality\n$$\n|C_n| \\frac{\\sqrt{\\left(x_2^{(n)} - x^-_n\\right)\\left(x_3^{(n)} - x^-_n\\right)}}{\\sqrt{x_1^{(n)} - x^-_n}(x^+_n - x^-_n)} = \\frac{h^-_n}{\\pi}\n$$\nimplies that\n\\begin{equation}\\label{asympt1}\n\t|C_n| \\sim \\frac{h^-_n}{\\pi} \\sqrt{|x^-_n|}.\n\\end{equation}\nOn the other hand\n$$\n|C_n| \\frac{\\sqrt{\\left(x_2^{(n)} - x^+_n\\right)\\left(x_3^{(n)} - x^+_n\\right)}}{\\sqrt{x_1^{(n)} - x^+_n}(x^+_n - x^-_n)} = \\frac{h^+_n}{\\pi},\n$$\nand, therefore,\n\\begin{equation}\\label{asympt2}\n\t\\sqrt{x_1^{(n)} - x^+_n} \\sim \\frac{h^-_n}{h^+_n} \\frac{1}{\\sqrt{|x^-_n|}}.\n\\end{equation}\nNow consider equality\n$$\n|C_n|\\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = h_n.\n$$\nUsing \\eqref{asympt1}, it is easy to show that the sequence in lhs is equivalent to sequence$$\n\\frac{h^-_n}{\\pi \\sqrt{|x^-_n|}} \\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^+_n)}.\n$$\nNow, changing the variable, we obtain\n$$\n\\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^+_n)} = \\bigintss_{0}^{x_2^{(n)} - x_1^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x_1^{(n)} - x\\right)(1 - x)}dx}{\\sqrt{x}(x + x_1^{(n)} - x^+_n)}.\n$$\nIt appears that the asymptotics of the last integral is independent of the convergence rate of $|x_2^{(n)} - x_1^{(n)}| \\rightarrow 1$. Namely, for all sequences $\\alpha_n \\rightarrow 1$ and $a_n \\rightarrow 0$ the equivalence\n$$\n\\int_0^{\\alpha_n} \\frac{\\sqrt{(\\alpha_n - x)(1 - x)}dx}{\\sqrt{x}(x + a_n)} \\sim \\int_0^1 \\frac{(1-x)dx}{\\sqrt{x}(x+a_n)} \\sim \\frac{\\pi}{\\sqrt{a_n}}\n$$\nholds. Finally, in view of \\eqref{asympt2}, we obtain\n$$\nh_n = |C_n|\\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} \\sim \\frac{h^-_n}{\\pi \\sqrt{|x^-_n|}} \\frac{\\pi}{\\sqrt{x_1^{(n)} - x^+_n}} \\sim h^+_n.\n$$\nIt contradicts the assumption $h_{\\lim} < h^+_{\\lim}$.\n\n\\section{Conclusion}\nThe simple expression through the Weierstrass sigma function for a conformal mapping of a polygonal region $\\Omega$ was obtained. For the specific example the numerical experiment was carried out. The behaviour under degeneration was analyzed and it was shown that the formula is stable and converges to the solution of the limiting problem.\n\nThe future research direction can be connected either with the construction and analysis of the solutions to similar problems corresponding, for example, to Riemann surfaces of genus $2$, or with the development of the sigma function theory: construction of the recurrent formulas for higher genus and elaboration of computational methods independent of the theta function theory.\n\n\\section{Acknowledgements}\n\n\nThe author expresses his gratitude to A. Bogatyrev and O. Grigoriev for posing the problem and useful discussions, and also to K. Malkov for help in the computer implementation of the calculations. The author also thanks the Center for Continuing Professional Education ``Sirius University'' for the invitation to the educational module ``Computational Technologies, Multidimensional Data Analysis, and Modelling'', during which some of the results of this work were obtained.\n\n\\begin{appendices}\n\\section{On the coefficients of the Weierstrass sigma function Taylor series}\n\tHere we prove that the sigma function is an entire function of three variables and derive a recurrent formula for its Taylor series coefficients, that was originally established by Weierstrass in \\cite{Weier}. The proof given there has a gap connected to the analyticity of the sigma function in a neighbourhood of zero. Perhaps, this fact can be proved by an independent argument but, since Weierstrass does not give any references (and omits this issue completely), we decided to provide a complete proof here.\n\n\tThe homogeneity condition\n\t$$\n\t\\sigma(\\frac{z}{\\lambda}, \\lambda^4 g_2, \\lambda^6 g_3) = \\frac{1}{\\lambda} \\sigma(z,g_2,g_3)\n\t$$\n\teasily implies the following differential equation for the $\\sigma$ function:\n\t\\begin{equation}\\label{eq321}\n\t\tz \\frac{\\partial \\sigma}{\\partial z} - 4g_2\\frac{\\partial \\sigma}{\\partial g_2} - 6g_3 \\frac{\\partial \\sigma}{\\partial g_3} - \\sigma = 0.\n\t\\end{equation}\n\tFurther, using the definition of the $\\sigma$ function and the standard differential equation for the $\\wp$ function, one can derive an equation (for a proof see \\cite{Halphen})\n\t\\begin{equation}\\label{eq322}\n\t\t\\frac{\\partial^2 \\sigma}{\\partial z^2} - 12g_3 \\frac{\\partial \\sigma}{\\partial g_2} - \\frac{2}{3} g_2^2 \\frac{\\partial \\sigma}{\\partial g_3} + \\frac{1}{12} g_2 z^2 \\sigma = 0.\n\t\\end{equation}\n\t\n\tLet $f$ be an entire function of three variables $(z,g_2,g_3)$ satisfying \\eqref{eq321} and \\eqref{eq322}. We derive a relation between the Taylor series coefficients $f_{mnk}$ of $f$:\n\t$$\n\tf = \\sum_{m,n,k = 0}^\\infty f_{mnk} g_2^m g_3^n z^k.\n\t$$\n\t\\eqref{eq321} implies that $f_{mnk} = 0$, if $k \\ne 4m + 6n + 1$. Therefore, $f$ can be written in the form\n\t$$\n\tf = \\sum_{m,n = 0}^\\infty a_{mn} g_2^m g_3^n z^{4m + 6n + 1}.\n\t$$\n\tNow, substituting this expression of $f$ into \\eqref{eq322}, we obtain the equality\n\t\\begin{equation}\\label{eq323}\n\t\ta_{mn} = \\frac{12(m+1)a_{m+1,n-1} + \\frac{2}{3}(n+1)a_{m-2,n+1} - \\frac{1}{12}a_{m-1,n}}{(4m + 6n + 1)(4m + 6n)},\n\t\\end{equation}\n\tin which for convenience $a_{mn}$ is defined by zero when $m$ or $n$ is negative. It is easy to see that \\eqref{eq323} uniquely determines sequence $a_{mn}$ for given $a_{00}$. To prove this let us introduce an order relation on pairs of nonnegative integers $(m,n)$: $(m,n) \\le (m',n')$, if $m+n < m'+n'$ or if $m+n = m'+n'$ and $n \\le n'$.\n\tIt is easy to see that we defined a well-order on $\\mathbb Z_+ \\times \\mathbb Z_+$, and in \\eqref{eq323} indices of the terms $a_{mn}$ in rhs are strictly less then $(m,n)$. Thus, it is proved that \\eqref{eq323} determines $a_{mn}$ recursively for given $a_{00}$.\n\t\n\tIf the sigma function was an entire function of three variables, or, at least, holomorphic in some neighbourhood of zero, then the recurrence relation \\eqref{eq323} for its Taylor series coefficients would be proved. The difficulty is that the domain of $\\sigma$ is the set $\\{(z,g_2,g_3) \\in \\mathbb C^3: g_2^3 - 27 g_3^2 \\ne 0\\}$. The following considerations prove the entirety of $\\sigma$ and the recurrence relation \\eqref{eq323}.\n\t\n\t\\begin{remark}\n\t\tIt is known (see, e.g., \\cite{AkhiezerEng} or \\cite{Chandra}) that condition $g_2^3 - 27 g_3^2 \\ne 0$ is equivalent to simplicity of the roots of polynomial $4x^3 - g_2 x - g_3$.\n\t\\end{remark}\n\t\\begin{lemma}\\label{l32}\n\t\tLet $a_{mn}$ satisfy the recurrence relation \\eqref{eq323}. Then for all $q > (28 + \\sqrt{811})/36 \\approx 1.569$ there exists $C > 0$ such that \n\t\t\\begin{equation}\\label{eq324}\n\t\t\t|a_{mn}| \\le C\\frac{q^{2m + 3n}}{(2m+3n)!}.\n\t\t\\end{equation}\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tSubstituting in \\eqref{eq323} this estimation and it is easy to show that for the existence of a constant, it suffices that the inequality\n\t\t$$\n\t\t \\frac{6(m+1)}{4m+6n+1}\\frac{q^{2m+3n-1}}{(2m+3n)!} + \\frac{(n+1)q^{2m+3n-1}}{6(4m+6n+1)(2m+3n)!} + \\frac{q^{2m+3n-2}}{48(2m+3n)!} \\le \\frac{q^{2m+3n}}{(2m+3n)!}\n\t\t$$\n\t\tholds starting from some index $(m,n)$ (in terms of the foregoing ordering). For this, in turn, it suffices to satisfy the inequality\n\t\t$$\n\t\t\\frac{1}{48} + q\\left(\\frac{3}{2} + \\frac{1}{18}\\right) < q^2.\n\t\t$$\n\t\tSolving the quadratic equation, we obtain the required statement.\n\t\\end{proof}\n\t\n\tLemma \\ref{l32} allows to define an entire function\n\t$$h(z,g_2,g_3) = \\sum_{m,n = 0}^\\infty a_{mn}g_2^m g_3^n z^{4m+6n+1},$$\n\twhere $a_{mn}$ are determined by recurrence relation \\eqref{eq323} and initial condition $a_{00} = 1$. We shall prove that $h \\equiv \\sigma$ for $(g_2,g_3)$ such that $g_2^3 - 27 g_3^2 \\ne 0$.\n\t\n\t\\begin{lemma}\n\t\tLet $f$ be a holomorphic function of variables $(z,g_2,g_3)$, defined on a set $\\mathbb C \\times U$, where $U \\subset \\mathbb C^2$ is open, satisfying equation \\eqref{eq322}. Assume that $f$ is odd in variable $z$. Then $f$ can be represented by series \\begin{equation}\\label{eq325}\n\t\t\tf(z,g_2,g_3) = \\sum_{n = 0}^\\infty c_n(g_2,g_3) z^{2n+1},\n\t\t\\end{equation}\n\t\tand in $U$ the recurrence relation\n\t\t\\begin{equation}\\label{eq326}\n\t\t\t(2n+3)(2n+2)c_{n+1} - 12g_3\\frac{\\partial c_n}{\\partial g_2} - \\frac{2}{3} g_2^2 \\frac{\\partial c_n}{\\partial g_3} + \\frac{1}{12}g_2 c_{n-1},\n\t\t\\end{equation}\n\t\tholds, where $n \\ge 0$ (for $n = 0$ we set $c_{n-1} = 0$).\n\t\\end{lemma} \n\t\\begin{proof}\n\t\tIndeed the representability of $f$ by the series follows from entirety of $f$ by $z$. Its coefficients $c_n(g_2,g_3)$ are given by \n\t\t$$\n\t\tc_n(g_2,g_3) = \\frac{1}{(2n+1)!} \\frac{\\partial^{2n+1} f}{\\partial z^{2n+1}}|_{z = 0}.\n\t\t$$\n\t\tIt is easy to see that the series \\eqref{eq325} can be differentiated term-by-term, and therefore we can substitute it in \\eqref{eq322}. Collecting the coefficient at $z^{2n+1}$, we obtain \\eqref{eq326}.\n\t\\end{proof}\n\t\n\tRecurrence relation \\eqref{eq326} can be used to prove, that $\\sigma$ and $h$ coincide on the domain of the $\\sigma$ function. Indeed, if the first terms of their expansions coincide, then these function coincide (note that they are both odd in $z$). Indeed, $\\partial \\sigma/\\partial z |_{z = 0} \\equiv \\partial h/\\partial z |_{z = 0} \\equiv 1$. Thus, $h$ is the analytic continuation of the $\\sigma$ function to an entire function of variables $(z,g_2,g_3)$. This completes the proof of the following theorem.\n\t\\begin{theorem}[Weierstrass]\\label{tWeierSeries}\n\t\tThe $\\sigma$ function is entire and for all $(z,g_2,g_3) \\in \\mathbb C^3$ equality\n\t\t\\begin{equation}\\label{eq327}\n\t\t\t\\sigma(z,g_2,g_3) = \\sum_{m,n = 0}^\\infty a_{mn}g_2^m g_3^n z^{4m+6n+1}\n\t\t\\end{equation}\n\t\tholds, where coefficients $a_{mn}$ are determined by recurrence relation \\eqref{eq323} and initial condition $a_{00} = 1$.\n\t\\end{theorem}\n\\end{appendices}\n\\printbibliography\n\\end{document}\n"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":835,"numItemsPerPage":100,"numTotalItems":87268,"offset":83500,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc0Njg2ODkyOCwic3ViIjoiL2RhdGFzZXRzL0NvZmVBSS9OYW5vRGF0YSIsImV4cCI6MTc0Njg3MjUyOCwiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.FZP0GJZTc9TkLwsDqicRwdEs-0VElTCw-RvHELjdhIYHBYebt8PMs7zkeh9AtBiilyENAwppEnFH-w_7F2TVBA","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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1,116,691,499,315 | arxiv | \section{Introduction}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{aaai2022figs/annotation.pdf}
\caption{Top left: original image. Bottom left: fully supervised annotation. Top right: scribble annoation. Bottom right: point annotation.}
\label{annotation}
\end{figure}
Salient object detection (SOD), aiming at detecting the most attractive regions of the images or videos to imitate human attention mechanism \cite{depth-survey}, has currently caught increasing attention in image processing, which can be further used in various computer vision fields, such as object recognition \cite{saliency-recognition}, image caption \cite{re-caption}, person re-identification \cite{person-re-id}. Fully supervised salient object detection models have achieved superior performance relying on effectively designed modules \cite{ldf, f3net}. However, these methods heavily rely on large datasets of precise pixel-wise annotations which is a tedious and inefficient procedure.
Recently, sparse labeling methods have attracted increasing attention, which aims to achieve a trade-off between time consumption and performance.
Some methods attempt to use image-level labels to detect salient objects \cite{wsi, duts, mfnet}. And some approaches train the network using noisy labels which are often generated from other SOD methods \cite{tipweaklyboundingbox}. Scribble annotation \cite{wssa} is proposed by lower labeling time consumption. Besides, as compared to previous methods, it can provide local ground truth labels.
\cite{wssa} claims that they can finish annotating an image in 1$\sim$2 seconds, but we find it difficult to achieve this for untrained annotators. We attempt to propose a point supervised saliency detection method to achieve an acceptable performance using a less time-consuming method compared to scribble annotation. As far as we know, this is the first attempt to accomplish salient object detection using point supervision (Fig. \ref{annotation}).
In this paper, we propose a framework that uses point supervision to train the salient object model. We build a new weakly supervised saliency detection annotation dataset by relabeling DUTS \cite{duts} dataset using point supervision. We choose to use point annotation to label the dataset for the following reasons: Firstly, point supervision can provide relatively accurate location information directly compared to unsupervised and image-level supervision. Secondly, point supervision is probably the least time-costly annotation method compared to other manual annotation methods.
Moreover, we find that the current weakly supervised saliency model \cite{wssa, scwssod, msw} only focuses on objects which must be noticed in the corresponding scenario, but overlooks objects that should be ignored. The reason is that due to the sparsity of weakly supervised labels, the supervision signal can only cover a small area of the images, which simply allows the model to learn which objects must be highlighted, but lacks information to guide the model on which ones should be ignored. As described at the beginning, the SOD is to simulate human visual attention and filter out non-salient objects. In a scene, the latter is the core of attention, as segmenting all objects means that "human attention" is not formed. At this point, the model degenerates into a model that segments all the objects that have been seen. Based on this observation, we propose the Non-Salient Suppression (NSS) method to explicitly filter out the non-salient but detected objects.
As weakly supervised labels only cover part of the object regions which makes it difficult for the model to perceive the structure of the objects, \cite{wssa} leverage an edge detector \cite{edge_richer} to generate edges to supervise the training process, thus indirectly complementing the structural cues. Our method to generate initial pseudo-labels is simple yet very effective. We observe that the detected edges not only provide structure to the model but also divide the image into multiple connected regions. Inspired by the flood filling algorithm (a classical image processing algorithm), the point annotations and the edges are exploited to generate the initial pseudo label as shown in Fig. 2.
Due to the fact that the edges are frequently discontinuous and blurred, we design an adaptive mask to address this issue. In summary, our main contributions can be summarized as follows:
\begin{itemize}
\item We propose a novel weakly-supervised salient object detection framework to detect salient objects by single point annotation, and build a new point-supervised saliency dataset P-DUTS.
\item We uncover the problem of degradation of weakly supervised saliency detection models and propose the Non-Salient object Suppression (NSS) method to explicitly filter out the non-salient but detected objects.
\item We design a transform-based point supervised SOD model that, in collaboration with the proposed adaptive flood filling, not only outperforms existing state-of-the-art weakly-supervised methods with stronger supervision by a large margin, even surpasses many fully-supervised methods.
\end{itemize}
\section{Related Work}
\subsection{Weakly Supervised Saliency Detection}
With the development of weakly supervised learning in dense prediction tasks, some works attempt to employ sparse annotation to acquire the salient objects. \cite{sbf} leverages the fused saliency maps provided by several unsupervised heuristic SOD methods as pseudo labels to train the final saliency model. By incorporating dense Conditional Random Field (CRF) \cite{crf} as post-processing steps, \cite{wsi,wssa} further optimized the resulting results. \cite{msw} use multimodal data, including category labels, captions, and noisy pseudo labels to train the saliency model where class activation maps (CAM) \cite{cam} is used as the initial saliency maps. Recently, scribble annotation for saliency detection, a more user-friendly annotation method, is proposed by \cite{wssa}, in which edge generated by edge detector \cite{edge_richer} is used to furnish structural information. Further, \cite{scwssod} enhances the structural consistency of the saliency maps by introducing saliency structure consistency loss and gated CRF loss \cite{gate_crf} to improve the performance of the model on scribble dataset \cite{wssa}.
\subsection{Point Annotation in Similar Field}
There has been several works on point annotations in weakly supervised segmentation \cite{what_point, metric_point} and instance segmentation \cite{point_large, point_latent, point_extreme, point_regional}.
Semantic segmentation focuses on class-specific categories, but saliency detection does not focus on category properties,
and it often occurs that an object is significant in one scenario, but not in another. That is, saliency detection is dedicated to the contrast between the object and the current scene \cite{itti}. Point-based instance segmentation methods are mainly used in interactive pipelines.
\subsection{Vision Transformer}
Visual processing-oriented transformer introduced from Transformer in Natural Language Processing (NLP) has attracted much attention in multiple fields of computer vision. Vision Transformer (ViT) \cite{vit} introduces a pure transformer model in computer vision tasks and achieves state-of-the-art performance on the image classification task. By employing ViT as the backbone, Segmentation Transformer (SETR) \cite{setr} add a decoder head to generate the final segmentation results with minimal modification. Detection Transformer \cite{detr} leverage encoder-decoder transformer and standard convolution network on object detection. Considering the scale difference between natural language and images, \cite{swin} design a shifted window scheme to improve efficiency as well as accuracy.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.75\textwidth]{aaai2022figs/pipline4.pdf}
\label{method}
\caption{The framework of our network and learning strategy. The thick black stream indicates the forward propagation of data in our model. The green stream indicates the parts used in the 1st training round. The blue stream is used in both 1st and 2nd rounds. The red stream is employed in the 2nd training round. The two-way arrows refer to supervision. The yellow box represents the transformer part, in which ResNet-50 is used as the embedding component. The purple box represents the adaptive flood filling. The red box in the upper right corner indicates the Non-Salient object Suppression.}
\end{figure*}
\section{Methodology}
\subsection{Adaptive Flood Filling}
Following the commonly used weakly-supervised dense prediction task,
we first make pseudo-labels and then use them to train the network.
As sparse labels only cover a small part of the object region which limits the model’s ability to perceive the structure of the object, \cite{wssa} leverages an edge detector \cite{edge_richer} to generate edges to supervise the training process indirectly supplementing structure. Unlike them, we apply the edges directly to the flood filling. The flood filling starts from a starting node to search its neighborhoods (4 or 8) and extracts the nearby nodes connected to it until all the nodes in the closed area have been processed (Algorithm 1). It is to extract several connected points from one area or to separate them from other adjacent areas. However, since the edges generated by the edge detectors are often discontinuous and blurred (top of Fig. 2), applying it directly to flood filling may result in the whole image being filled. So we designed an adaptive mask, a circle whose radius varies according to the image size, to alleviate this problem. Specifically, the radius $r$ is defined as:
\begin{equation} \label{r_I}
r(I) = min(\frac{h_I}{\gamma} ,\frac{w_I}{\gamma} )
\end{equation}
\noindent where $I$ is the input image, $r(I)$ refers to the radius of the mask corresponding to the input image $I$. $h_I$ and $w_I$ represent the length and width of the input image, respectively. $\gamma$ is the hyperparameter.
\begin{algorithm}[h]
\label{alg:A}
\caption{Flood Filling Algorithm}
\textbf{Input:} Seed point $(x,y)$, image $\mathcal{I}$, seted value $\alpha$, $old$ value $I(x,y)$ \\
\textbf{Output:} Filled mask $\mathcal{M}$
\begin{algorithmic}[1]
\STATE \textit{\textbf{flood filling algorithm}} (($x$,$y$), $\mathcal{I}$, $\alpha$)
\STATE \quad \textbf{if} $x \geq 0$ and $x<width$ and $y\geq0$ and $y<height$
\STATE \quad and $a<\mathcal{I}(x,y)-old<b$ and $\mathcal{I}(x,y)\neq \alpha$ \textbf{then}
\STATE \quad \quad $\mathcal{M}$(x,y) $\Leftarrow$ $\alpha$
\STATE \quad \quad \textit{\textbf{flood filling}} ($(x+1,y)$, $\mathcal{I}$, $\alpha$)
\STATE \quad \quad \textit{\textbf{flood filling}} ($(x-1,y)$, $\mathcal{I}$, $\alpha$)
\STATE \quad \quad \textit{\textbf{flood filling}} ($(x,y+1)$, $\mathcal{I}$, $\alpha$)
\STATE \quad \quad \textit{\textbf{flood filling}} ($(x,y-1)$, $\mathcal{I}$, $\alpha$)
\STATE \quad \textbf{end if}
\end{algorithmic}
\end{algorithm}
The labeled ground truth of image $I$ can be represented as: $S=\{S_b,S_f^i|i=1,\cdots,N\}$, where $S_b$ and $S_f^i$ refer to the position coordinates of the background pixel and the $i$th labeled salient object, respectively. Then the set of these circle mask can be defined as $\mathcal{M}_S^{r(I)} = C_{S_f^1}^{r(I)} \cup \cdots \cup C_{S_f^N}^{r(I)} \cup C_{S_b}^{r(I)}$, where $C$ represents the circle that use the lower corner as the center and the upper corner as the radius. Similar to \cite{wssa}, we also use the edge detector \cite{edge_richer} to detect the edges of the image: $e=E(I)$, where $E(\cdot)$ represents the edge detector, $I$ represents the input image, and $e$ represents the detected edges. We use the union of $e$ and $\mathcal{M}_S^{r(I)}$, $E(I) \cup \mathcal{M}_S^{r(I)}$, to divide the image $I$ into multiple connected regions. Then the flood filling is employed to generate the pseudo-label of image $I$ (bottom in Fig. 2):
\begin{equation} \label{xx}
g = F(S, E(I) \cup \mathcal{M}_S^{r(I)})
\end{equation}
\noindent where $g$ denotes obtained pseudo-label, $F(\cdot)$ denotes the flood filling algorithm, $S$ denotes the labeled ground truth of image $I$.
\subsection{Transformer-based Point Supervised SOD Model}
The difficulty of sparse labeling saliency detection lies in the fact that the model can only obtain local ground truth labels and lacks the guidance of global information. We consider that establishing the connection between labeled and unlabeled locations via the similarity between them to obtain the saliency value of the unlabeled region can significantly alleviate this problem. Considering the similarity-based nature of vision transformer (ViT) \cite{vit}, we utilize hyper ViT (i.e., "ResNet-50 + ViT-Base") as our backbone to extract features as well as calculate the self-similarity.
\textbf{Transformer Part.} Specifically, for an input image of size $3\times H\times W$, the CNN embedding part generate $C\times \frac{H}{16}\times \frac{H}{16}$ feature maps. The multi-stage feature of ResNet-50 are denoted as $R=\{R^i|i=1,2,3,4,5\}$. Then, the transformer encoder take as input the summation of position embedding of $C\times \frac{H}{16}\times \frac{H}{16}$ and the flattened features of $C \times\frac{H}{16}\times \frac{H}{16}$ feature maps.
After 12 layers of self-attention layers, the transformer encoder part output features of $C \times \frac{H}{16} \times \frac{H}{16}$.
\textbf{Edge-preserving Decoder.} Edge-preserving decoder consists of two components, a saliency decoder, and an approximate edge detector (see Fig. 2). Saliency decoder is four cascaded convolution layers where each of them followed by batch normalization (BN) layer, ReLU activation, and upsample layer, which takes as input the features of the transformer encoder. And we denote the corresponding features of the saliency decoder at each layer as $D=\{D^i|i=1,2,3,4\}$.
For the latter part, as the weak annotations lacking structure and details, we develop an edge decoder stream as an approximate edge detector to generate structures by constraining the output using the edges generated by the real edge detector. In detail, the output of approximate edge detector can be represented as $f_e = \sigma(cat(R^3,D^2))$, where $\sigma$ represents a single $3\times3$ convolution layer followed by BN and ReLU layer. The edge map $e$ can be obtained by adding a $3\times3$ convlayer after $f_e$, which is then constrained by the edge map generated from the real edge detector.
Then, by merging $f_e$ with $D^3$, $cat(f_e,D^3)$, and passing through the following two convolution layers, the multi-channel feature $f_s$ is obtained. Similarly to $e$, the final saliency map $s$ can be obtained in the same manner.
\subsection{Non-Salient object Suppression (NSS)}
We observe that due to the sparsity of weakly supervised labels, the supervision signal can only cover a small area of the image, which leads the model only learning to highlight the learned objects while ignoring which objects in the current scene should not be highlighted (red box in Fig. \ref{NSS} (a)).
To suppress the non-salient object, we propose a simple but effective method by exploiting the location cues provided by the supervision signal and filling the generated highlighted object to suppress the non-salient object. And the obtained salient object regions (red regions in Fig 3. (b)) can be obtained by:
\begin{equation} \label{pf}
P_f = F(S-S_b, P^{1st})
\end{equation}
where $F(\cdot)$ represents flood filling, $S-S_b = \{S_f^i|i=1,...,N\}$ represents subtraction, $P^{1st}$ represents the saliency maps generated after the first training round and refined by dense CRF \cite{crf}.
Given that we only provided internal local labels for the salient objects during the first round of training, which may result in the model being unable to distinguish the edges accurately, we perform an expansion operation on $P_f$ with kernel size 10. The expanded regions are designated as the uncertain regions (black regions in Fig. 3 (b)), while the remaining regions are designated as the background region (green regions in Fig. 3 (b)). This is denoted as $P^{2nd}$, which is used as the pseudo-labels for the second training round.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{aaai2022figs/nss_viz4.pdf}
\caption{Illustration of the final pseudo label after NSS. (a) Saliency map generated at the end of first training round. (b) Salient pseudo-label after NSS. (c) The corresponding position of the pseudo label on the original image.}
\label{nss_viz}
\end{figure}
As shown in the test examples in Fig. \ref{NSS}, due to the sparseness of the labels, the model tends to detect the non-salient objects. Indeed, the model degenerates into a model that detects the previously learned objects. By reusing the position cues from the supervision points, we can successfully suppress the most non-salient objects utilizing NSS.
\subsection{Loss Function}
In our network, binary cross entropy loss, partial cross entropy loss \cite{partial_ce}, and gated CRF loss \cite{scwssod,gate_crf} are employed. For the Edge-preserving decoder stream, we use binary cross entropy loss to constrain $e$:
\begin{equation} \label{partial_ce}
\mathcal{L}_{bce}=-\sum_{(r,c)}[ylog(e)+(1-y)log(1-e)]
\end{equation}
\noindent where $y$ refers to the ground-truth, $e$ represents the predicted edge map, $r$ and $c$ represent the row and column coordinates. For the saliency decoder stream, partial cross entropy loss and gated CRF loss is employed. Partial binary cross-entropy loss is used to focus only on the definite area, while ignoring the uncertain area:
\begin{equation} \label{partial_ce}
\mathcal{L}_{pbce}=-\sum_{j\in J}[g_jlog(s_j)+(1-g_j)log(1-s_j)]
\end{equation}
\noindent where $J$ represents the labeled area, $g$ refers to the ground truth, $s$ represents the predicted saliency map.
To obtain better object structure and edges, follow \cite{scwssod}, gated CRF is used in our loss function:
\begin{equation} \label{xx}
\mathcal{L}_{gcrf} = \sum_{i}\sum_{j\in K_i} d(i,j)f(i,j)
\end{equation}
\noindent where $K_i$ denotes the areas covered by $k \times k$ kernel around pixel $i$, $d(i,j)$ is defined as:
\begin{equation} \label{xx}
d(i,j) = |s_i - s_j|
\end{equation}
\noindent where $s_i$ and $s_j$ are the saliency values of $s$ at position i and j, $\left| \cdot \right|$ denotes L1 distance. And $f(i,j)$ refers to the Gaussian kernel bandwidth filter:
\begin{equation} \label{xx}
\begin{aligned}
&f(i,j)= \\
&\frac{1}{w}\exp(-\frac{\parallel PT(i)-PT(j) \parallel_{2}}{2\sigma_{PT}^2}-\frac{\parallel I(i)-I(j) \parallel_{2}}{2\sigma_I^2})
\end{aligned}
\end{equation}
\noindent where $\frac{1}{w}$ is the weight for normalization, $I(\cdot)$ and $PT(\cdot)$ are the RGB value and position of pixel, $\sigma_{PT}$ and $\sigma_I$\ are the hyper parameters to control the scale of Gaussian kernels. So the total loss function can be defined as:
\begin{equation} \label{xx}
\mathcal{L}_{final}= \alpha_1\mathcal{L}_{bce} + \alpha_2\mathcal{L}_{pbce} + \alpha_3\mathcal{L}_{gcrf}
\end{equation}
\noindent where $\alpha_1$, $\alpha_2$, $\alpha_3$ are the weights. In our experiments, they are all set to 1.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.99\textwidth]{aaai2022figs/NSS.pdf}
\caption{Illustration of the effect of Non-Salient object Suppression. (a) Pseudo-labels generated at the end of the first round training with point supervision. (b) Salient pseudo-label obtained after passing the pseudo-label of line (a) through Non-Salient object Suppression.}
\label{NSS}
\vspace{-0.5cm}
\end{figure*}
\section{Experiments}
\subsection{Point-supervised Dataset}
To minimize the labeling time consumption while providing location information of salient objects, we build a Point-supervised Dataset (P-DUTS) by relabeling DUTS \cite{duts} dataset, a widely used saliency detection dataset containing 10553 training images. Four annotators participate in the annotation task, and for each image, we randomly select one of the four annotations to reduce personal bias. In Fig. \ref{annotation}, we show an example of point annotation and compare it with other stronger annotation methods. For each salient object, we only randomly select one-pixel location for labeling (for clarity, we exaggerate the size of the labeled position). Due to the simplicity of our method, even novice annotators can complete an image in 1$\sim $2 seconds on average.
\subsection{Implementation Details}
The proposed model is implemented on the Pytorch toolbox, and our proposed P-DUTS dataset is used as the training set. We train on four TITAN Xp GPUs. For the transformer part, a hyper version transformer ("ResNet50 + ViT-Base") is used by us, and no adjustments are made. We initialize the embedding layers and transformer encoder layers by leveraging the pretrained weight provided by ViT \cite{vit}, which is pre-trained on ImageNet 21K. The maximum learning rate is set to $2.5\times10^{-4}$ for the transformer part and $2.5\times10^{-3}$ for other parts. Warm-up and linear decay strategies are used to adjust the learning rate. Stochastic gradient descent (SGD) is used to train the network, and the following hyper-parameters are used: momentum=0.9, weight decay=$5\times10^{-4}$. Horizontal flip and random crop are used as data augmentation. The batch size is set to 28 and it takes 20 epochs for the first training procedure. The hyperparameter $ \gamma $ of Eq. \ref{r_I} is set to be 5. The second round of training uses the same parameters, but the masks are replaced with refined ones. During testing, we resized each image to 352 $\times$ 352 and then feed it to our network to predict saliency maps.
\subsection{Dataset and Evaluation Criteria}
\textbf{Dataset.} To evaluate the performance, we experiment on five public used benchmark datasets: ECSSD \cite{ecssd}, PASCAL-S \cite{pascal-s}, DUT-O \cite{dut-o}, HKU-IS \cite{hku-is}, and DUTS-test.
\noindent \textbf{Evaluation Criteria.} In our experiments, we compared our model with five state-of-the-art weakly supervised or unsupervised SOD methods (i.e., SCWSSOD \cite{scwssod}, WSSA \cite{wssa}, MFNet \cite{mfnet}, MSW \cite{msw}, SBF \cite{sbf}) and eight fully supervised SOD methods (i.e., GateNet \cite{gatenet}, BASNet \cite{basnet}, AFNet \cite{afnet}, MLMS \cite{mlms}, PiCANet-R \cite{picanet}, DGRL \cite{dgrl}, R$^3$Net \cite{r3+}, RAS \cite{ras}). Five metrics are used: precision-recall curve, maximum F-measure ($F_{\beta}^{max}$), mean F-measure ($mF_\beta$), mean absolute error ($MAE$) and structure-based metric ($S_{m}$) \cite{s-measure}.
\subsection{Compare with State-of-the-arts}
\textbf{Quantitative Comparison.} We compare our method with other state-of-the-art models as shown in Tab. \ref{comparison_SOTA} and Fig. \ref{pr}. For a fair comparison, the saliency results of different compared methods are provided by authors or obtained by running the codes released by authors. The best results are bolded. As can be seen in Tab \ref{comparison_SOTA}, our method outperforms state-of-the-art weakly supervised methods by a large margin. Our method outperforms the previous best model \cite{scwssod} by 2.68\% for $F_\beta^{max}$, 2.75\% for $S_m$, 0.94\% for $mF_\beta$, 0.62 \% for MAE on average of 5 compared dataset. What's more, our method achieves performance on par with recent state-of-the-art model GateNet \cite{gatenet} and even outperforms fully supervised methods in several metrics on multiple data sets as shown in Tab \ref{comparison_SOTA}. And as shown in Fig. \ref{pr}, for quantitative comparison, our method outperforms previous state-of-the-art methods by a large margin.
\begin{table*}[htbp]
\centering
\setlength{\tabcolsep}{1pt}
\small
\begin{tabular}{cc |cccccccc| cccccc}
\hline
\multirow{3}{*}{Metric} & &\multicolumn{8}{|c|}{Fully Sup.Methods} &\multicolumn{6}{c}{Weakly Sup./Unsup. Methods} \\
& &RAS &R$^{3}$Net &DGRL &PiCANet &MLMS &AFNet &BASNet &GateNet &SBF &MWS &MFNet &WSSA &SCWSSOD &Ours \\
& &2018 &2018 &2018 &2018 &2019 &2019 &2019 &2020 &2017 &2019 &2021 &2020 &2021 \\
\hline
\multirow{4}{*}{ECSSD} &$F_\beta^{max}$ \hfill$\uparrow$ &0.9211 &0.9247 &0.9223 &0.9349 &0.9284 &0.9350 &0.9425 &\textbf{0.9454} &0.8523 &0.8777 &0.8796 &0.8880 &0.9143 &\textbf{0.9359} \\
&$mF_\beta$ \hfill$\uparrow$ &0.9005 &0.9027 &0.9128 &0.9019 &0.9027 &0.9153 &\textbf{0.9306} &0.9251 &0.8220 &0.8060 &0.8603 &0.8801 &0.9091 &\textbf{0.9253} \\
&$S_m$ \hfill$\uparrow$ &0.8928 &0.9030 &0.9026 &0.9170 &0.9111 &0.9134 &0.9162 &\textbf{0.9198} &0.8323 &0.8275 &0.8345 &0.8655 &0.8818 &\textbf{0.9135} \\
&MAE \hfill$\downarrow$ &0.0564 &0.0556 &0.0408 &0.0464 &0.0445 &0.0418 &\textbf{0.0370} &0.0401 &0.0880 &0.0964 &0.0843 &0.0590 &0.0489 &\textbf{0.0358} \\
\hline
\multirow{4}{*}{DUT-O} &$F_\beta^{max}$\hfill$\uparrow$ &0.7863 &0.7882 &0.7742 &0.8029 &0.7740 &0.7972 &0.8053 &\textbf{0.8181} &0.6849 &0.7176 &0.7062 &0.7532 &0.7825 &\textbf{0.8086} \\
&$mF_\beta$\hfill$\uparrow$ &0.7621 &0.7533 &0.7658 &0.7628 &0.7470 &0.7763 &\textbf{0.7934} &0.7915 &0.6470 &0.6452 &0.6848 &0.7370 &0.7779 &\textbf{0.7830} \\
&$S_m$\hfill$\uparrow$ &0.8141 &0.8182 &0.8058 &0.8319 &0.8093 &0.8263 &0.8362 &\textbf{0.8381} &0.7473 &0.7558 &0.7418 &0.7848 &0.8119 &\textbf{0.8243} \\
&MAE\hfill$\downarrow$ &0.0617 &0.0711 &0.0618 &0.0653 &0.0636 &0.0574 &0.0565 &\textbf{0.0549} &0.1076 &0.1087 &0.0867 &0.0684 &\textbf{0.0602} &0.0643 \\
\hline
\multirow{4}{*}{PASCAL-S} &$F_\beta^{max}$\hfill$\uparrow$ &0.8291 &0.8374 &0.8486 &0.8573 &0.8552 &0.8629 &0.8539 &\textbf{0.8690} &0.7593 &0.7840 &0.8046 &0.8088 &0.8406 &\textbf{0.8663} \\
&$mF_\beta$\hfill$\uparrow$ &0.8125 &0.8155 &0.8353 &0.8226 &0.8272 &0.8405 &0.8374 &\textbf{0.8459} &0.7310 &0.7149 &0.7867 &0.7952 &0.8350 &\textbf{0.8495} \\
&$S_m$\hfill$\uparrow$ &0.7990 &0.8106 &0.8359 &0.8539 &0.8443 &0.8494 &0.8380 &\textbf{0.8580} &0.7579 &0.7675 &0.7648 &0.7974 &0.8198 &\textbf{0.8529} \\
&MAE\hfill$\downarrow$ &0.1013 &0.1026 &0.0721 &0.0756 &0.0736 &0.0700 &0.0758 &\textbf{0.0674} &0.1309 &0.1330 &0.1189 &0.0924 &0.0775 &\textbf{0.0647} \\
\hline
\multirow{4}{*}{HKU-IS} &$F_\beta^{max}$\hfill$\uparrow$ &0.9128 &0.9096 &0.9103 &0.9185 &0.9207 &0.9226 &0.9284 &\textbf{0.9335} &- &0.8560 &0.8766 &0.8805 &0.9084 &\textbf{0.9234} \\
&$mF_\beta$\hfill$\uparrow$ &0.8875 &0.8807 &0.8997 &0.8805 &0.8910 &0.8994 &\textbf{0.9144} &0.9098 &- &0.7764 &0.8535 &0.8705 &0.9030 &\textbf{0.9131} \\
&$S_m$\hfill$\uparrow$ &0.8874 &0.8920 &0.8945 &0.9044 &0.9065 &0.9053 &0.9089 &\textbf{0.9153} &- &0.8182 &0.8465 &0.8649 &0.8820 &\textbf{0.9019} \\
&MAE\hfill$\downarrow$ &0.0454 &0.0478 &0.0356 &0.0433 &0.0387 &0.0358 &\textbf{0.0322} &0.0331 &- &0.0843 &0.0585 &0.0470 &0.0375 &\textbf{0.0322} \\
\hline
\multirow{4}{*}{DUTS-TE} &$F_\beta^{max}$\hfill$\uparrow$ &0.8311 &0.8243 &0.8279 &0.8597 &0.8515 &0.8628 &0.8594 &\textbf{0.8876} &- &0.7673 &0.7700 &0.7886 &0.8440 &\textbf{0.8580} \\
&$mF_\beta$\hfill$\uparrow$ &0.8022 &0.7872 &0.8179 &0.8147 &0.8160 &0.8340 &0.8450 &\textbf{0.8558} &- &0.7118 &0.7460 &0.7723 &0.8392 &\textbf{0.8402} \\
&$S_m$\hfill$\uparrow$ &0.8385 &0.8360 &0.8417 &0.8686 &0.8617 &0.8670 &0.8656 &\textbf{0.8851} &- &0.7587 &0.7747 &0.8034 &0.8405 &\textbf{0.8532} \\
&MAE\hfill$\downarrow$ &0.0594 &0.0664 &0.0497 &0.0506 &0.0490 &0.0458 &0.0476 &\textbf{0.0401} &- &0.0912 &0.0765 &0.0622 &0.0487 &\textbf{0.0449} \\
\hline
\end{tabular}
\caption{Quantitative comparison with 13 state-of-the-art methods on ECSSD, DUT-OMRON, PASCAL-S, HKU-IS and DUTS-test. Top results are shown in bold. }
\label{comparison_SOTA}
\end{table*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.99\textwidth]{aaai2022figs/pr.png}
\caption{Illustration of precision-recall curves on the four largest dataset.}
\label{pr}
\end{figure*}
\begin{figure*}[h]
\centering
\includegraphics[width=0.705\textwidth]{aaai2022figs/visual_comp.png}
\caption{ Qualitative comparison with different methods.}
\label{visual_compare}
\end{figure*}
\subsubsection{Qualitative Evaluation.}
As shown in Fig. \ref{visual_compare}, our method obtains more accurate and complete saliency maps compared with other state-of-the-art weakly supervised or unsupervised methods and even surpasses recently fully supervised state-of-the-art methods. Row 1, 5, and 6 demonstrate the ability of our model to capture the overall salient object, and even objects (row 5) with extremely complex textures can be accurately segmented by our method. Further, thanks to our NSS method, our method can precisely extract salient objects and suppress non-salient objects (rows 2-4). The last row shows the ability of our model to extract the details.
\subsection{Ablation Study}
\subsubsection{Effectiveness of Edge-preserving Decoder.}
Note that, we use the results of the first round to evaluate the performance because if the first round produces bad results it will directly affect the results of the second round. In Tab. \ref{edge_decoder}, we evaluate the influence of the Edge-preserving decoder stream. The supervision of the edge generated by the edge detector is removed. At this point, the edge-preserving decoder can only aggregate shallow features and cannot explicitly detect edge features, and the overall performance decreases.
\subsubsection{Effectiveness of NSS.}
Since NSS is used in the 2nd training round, we perform ablation experiments based on the results generated in the 1st round. The ablation study results are listed in Tab. \ref{nss}. Line 1 indicates we directly use the pseudo-labels generated by the 1st round without CRF processing ("E" means using edge constraints). Line 2 indicates we use pseudo-labels refined by dense CRF. Line 3 represents the standard method in this paper. The last line represents we directly employ $P_f$ (Eq. \ref{pf}) as ground truth with no uncertain areas.
The comparison of line 1 and line 2 shows that using CRF to optimize the pseudo labels can optimize the second training round.
The 3rd row works better than the first two rows, which shows the validity of NSS. We also illustrate the effectiveness of NSS in Fig. \ref{NSS}.
\begin{table}
\centering
\small
\label{tab:freq}
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{c|ccc| ccc}
\hline
\multirow{2}{*}{Model} &\multicolumn{3}{c|}{PASCAL-S} &\multicolumn{3}{c}{DUT-OMRON}\cr
&$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE$\downarrow$ &$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE $\downarrow$\\
\hline
w/ &0.8636 &0.8498 &0.0671 &0.8034 &0.8150 &0.0717 \\
w/o &0.8629 &0.8454 &0.0703 &0.8049 &0.8054 &0.0769 \\
\hline
\end{tabular}}
\caption{ The effectiveness Edge-preserving Decoder.}
\label{edge_decoder}
\end{table}
\begin{table}
\centering
\small
\label{tab:freq}
\setlength{\tabcolsep}{0.5mm}{
\begin{tabular}{c|ccc| ccc}
\hline
\multirow{2}{*}{Model} &\multicolumn{3}{c|}{DUTS-test} &\multicolumn{3}{c}{ECSSD}\cr
&$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE$\downarrow$ &$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE $\downarrow$\\
\hline
+E &0.8588 &0.8559 &0.0502 &0.9342 &0.9134 &0.0407\\
+E+CRF &0.8606 &0.8510 &0.0463 &0.935 &0.9131 &0.0385 \\
+E+CRF+NSS$^*$ &0.8580 &0.8532 &0.0449 &0.9359 &0.9135 &0.0358 \\
\hline
\makecell[c]{+E+CRF+NSS} &0.8586 &0.8541 &0.0457 &0.9365 &0.9121 &0.0382 \\
\hline
\end{tabular}}
\caption{Ablation study for NSS.}
\label{nss}
\end{table}
\begin{table}
\centering
\small
\label{tab:freq}
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{c|ccc| ccc}
\hline
\multirow{2}{*}{$\gamma$} &\multicolumn{3}{c|}{DUTS-test} &\multicolumn{3}{c}{ECSSD}\cr
&$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE$\downarrow$ &$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE $\downarrow$\\
\hline
3 &0.8467 &0.8421 &0.0559 &0.9299 &0.9085 &0.0415\\
4 &0.8497 &0.8439 &0.0543 &0.9304 &0.9088 &0.0415\\
5 &0.8530 &0.8516 &0.0488 &0.9326 &0.9095 &0.0384\\
6 &0.8480 &0.8416 &0.0558 &0.9293 &0.9079 &0.0421\\
\hline
\end{tabular}}
\caption{ The impact of $\gamma$.}
\label{gama}
\end{table}
\subsubsection{Impact of the Hyperparameter $\gamma$.}
We test the effect of different values of $\gamma$ on the first round of training in Tab. \ref{gama}. Too small $\gamma$ will generate small pseudo labels providing too little supervision information, and too large $\gamma$ will make the pseudo labels contain the wrong areas. Both of these cases affect the model's performance.
\section{Conclusions}
We propose a point-supervised framework to detect salient objects in this paper. An adaptive flood filling algorithm and a transform-based point-supervised model are designed for pseudo-label generation and saliency detection. To uncover the problem of degradation of the existing weakly supervised SOD models, we propose the Non-Salient object Suppression (NSS) technique to explicitly filter out the non-salient but detected objects. We build a new point-supervised saliency dataset P-DUTS for model training. The proposed method not only outperforms existing state-of-the-art weakly-supervised methods with stronger supervision.
\section{Acknowledgements}
This work was supported by the National Key R\&D Program of China (2020AAA0108301), the National Natural Science Foundation of China (No.62072112), Scientific and technological innovation action plan of Shanghai Science and Technology Committee (No.205111031020).
\section{Introduction}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{aaai2022figs/annotation.pdf}
\caption{Top left: original image. Bottom left: fully supervised annotation. Top right: scribble annoation. Bottom right: point annotation.}
\label{annotation}
\end{figure}
Salient object detection (SOD), aiming at detecting the most attractive regions of the images or videos to imitate human attention mechanism \cite{depth-survey}, has currently caught increasing attention in image processing, which can be further used in various computer vision fields, such as object recognition \cite{saliency-recognition}, image caption \cite{re-caption}, person re-identification \cite{person-re-id}. Fully supervised salient object detection models have achieved superior performance relying on effectively designed modules \cite{ldf, f3net}. However, these methods heavily rely on large datasets of precise pixel-wise annotations which is a tedious and inefficient procedure.
Recently, sparse labeling methods have attracted increasing attention, which aims to achieve a trade-off between time consumption and performance.
Some methods attempt to use image-level labels to detect salient objects \cite{wsi, duts, mfnet}. And some approaches train the network using noisy labels which are often generated from other SOD methods \cite{tipweaklyboundingbox}. Scribble annotation \cite{wssa} is proposed by lower labeling time consumption. Besides, as compared to previous methods, it can provide local ground truth labels.
\cite{wssa} claims that they can finish annotating an image in 1$\sim$2 seconds, but we find it difficult to achieve this for untrained annotators. We attempt to propose a point supervised saliency detection method to achieve an acceptable performance using a less time-consuming method compared to scribble annotation. As far as we know, this is the first attempt to accomplish salient object detection using point supervision (Fig. \ref{annotation}).
In this paper, we propose a framework that uses point supervision to train the salient object model. We build a new weakly supervised saliency detection annotation dataset by relabeling DUTS \cite{duts} dataset using point supervision. We choose to use point annotation to label the dataset for the following reasons: Firstly, point supervision can provide relatively accurate location information directly compared to unsupervised and image-level supervision. Secondly, point supervision is probably the least time-costly annotation method compared to other manual annotation methods.
Moreover, we find that the current weakly supervised saliency model \cite{wssa, scwssod, msw} only focuses on objects which must be noticed in the corresponding scenario, but overlooks objects that should be ignored. The reason is that due to the sparsity of weakly supervised labels, the supervision signal can only cover a small area of the images, which simply allows the model to learn which objects must be highlighted, but lacks information to guide the model on which ones should be ignored. As described at the beginning, the SOD is to simulate human visual attention and filter out non-salient objects. In a scene, the latter is the core of attention, as segmenting all objects means that "human attention" is not formed. At this point, the model degenerates into a model that segments all the objects that have been seen. Based on this observation, we propose the Non-Salient Suppression (NSS) method to explicitly filter out the non-salient but detected objects.
As weakly supervised labels only cover part of the object regions which makes it difficult for the model to perceive the structure of the objects, \cite{wssa} leverage an edge detector \cite{edge_richer} to generate edges to supervise the training process, thus indirectly complementing the structural cues. Our method to generate initial pseudo-labels is simple yet very effective. We observe that the detected edges not only provide structure to the model but also divide the image into multiple connected regions. Inspired by the flood filling algorithm (a classical image processing algorithm), the point annotations and the edges are exploited to generate the initial pseudo label as shown in Fig. 2.
Due to the fact that the edges are frequently discontinuous and blurred, we design an adaptive mask to address this issue. In summary, our main contributions can be summarized as follows:
\begin{itemize}
\item We propose a novel weakly-supervised salient object detection framework to detect salient objects by single point annotation, and build a new point-supervised saliency dataset P-DUTS.
\item We uncover the problem of degradation of weakly supervised saliency detection models and propose the Non-Salient object Suppression (NSS) method to explicitly filter out the non-salient but detected objects.
\item We design a transform-based point supervised SOD model that, in collaboration with the proposed adaptive flood filling, not only outperforms existing state-of-the-art weakly-supervised methods with stronger supervision by a large margin, even surpasses many fully-supervised methods.
\end{itemize}
\section{Related Work}
\subsection{Weakly Supervised Saliency Detection}
With the development of weakly supervised learning in dense prediction tasks, some works attempt to employ sparse annotation to acquire the salient objects. \cite{sbf} leverages the fused saliency maps provided by several unsupervised heuristic SOD methods as pseudo labels to train the final saliency model. By incorporating dense Conditional Random Field (CRF) \cite{crf} as post-processing steps, \cite{wsi,wssa} further optimized the resulting results. \cite{msw} use multimodal data, including category labels, captions, and noisy pseudo labels to train the saliency model where class activation maps (CAM) \cite{cam} is used as the initial saliency maps. Recently, scribble annotation for saliency detection, a more user-friendly annotation method, is proposed by \cite{wssa}, in which edge generated by edge detector \cite{edge_richer} is used to furnish structural information. Further, \cite{scwssod} enhances the structural consistency of the saliency maps by introducing saliency structure consistency loss and gated CRF loss \cite{gate_crf} to improve the performance of the model on scribble dataset \cite{wssa}.
\subsection{Point Annotation in Similar Field}
There has been several works on point annotations in weakly supervised segmentation \cite{what_point, metric_point} and instance segmentation \cite{point_large, point_latent, point_extreme, point_regional}.
Semantic segmentation focuses on class-specific categories, but saliency detection does not focus on category properties,
and it often occurs that an object is significant in one scenario, but not in another. That is, saliency detection is dedicated to the contrast between the object and the current scene \cite{itti}. Point-based instance segmentation methods are mainly used in interactive pipelines.
\subsection{Vision Transformer}
Visual processing-oriented transformer introduced from Transformer in Natural Language Processing (NLP) has attracted much attention in multiple fields of computer vision. Vision Transformer (ViT) \cite{vit} introduces a pure transformer model in computer vision tasks and achieves state-of-the-art performance on the image classification task. By employing ViT as the backbone, Segmentation Transformer (SETR) \cite{setr} add a decoder head to generate the final segmentation results with minimal modification. Detection Transformer \cite{detr} leverage encoder-decoder transformer and standard convolution network on object detection. Considering the scale difference between natural language and images, \cite{swin} design a shifted window scheme to improve efficiency as well as accuracy.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.75\textwidth]{aaai2022figs/pipline4.pdf}
\label{method}
\caption{The framework of our network and learning strategy. The thick black stream indicates the forward propagation of data in our model. The green stream indicates the parts used in the 1st training round. The blue stream is used in both 1st and 2nd rounds. The red stream is employed in the 2nd training round. The two-way arrows refer to supervision. The yellow box represents the transformer part, in which ResNet-50 is used as the embedding component. The purple box represents the adaptive flood filling. The red box in the upper right corner indicates the Non-Salient object Suppression.}
\end{figure*}
\section{Methodology}
\subsection{Adaptive Flood Filling}
Following the commonly used weakly-supervised dense prediction task,
we first make pseudo-labels and then use them to train the network.
As sparse labels only cover a small part of the object region which limits the model’s ability to perceive the structure of the object, \cite{wssa} leverages an edge detector \cite{edge_richer} to generate edges to supervise the training process indirectly supplementing structure. Unlike them, we apply the edges directly to the flood filling. The flood filling starts from a starting node to search its neighborhoods (4 or 8) and extracts the nearby nodes connected to it until all the nodes in the closed area have been processed (Algorithm 1). It is to extract several connected points from one area or to separate them from other adjacent areas. However, since the edges generated by the edge detectors are often discontinuous and blurred (top of Fig. 2), applying it directly to flood filling may result in the whole image being filled. So we designed an adaptive mask, a circle whose radius varies according to the image size, to alleviate this problem. Specifically, the radius $r$ is defined as:
\begin{equation} \label{r_I}
r(I) = min(\frac{h_I}{\gamma} ,\frac{w_I}{\gamma} )
\end{equation}
\noindent where $I$ is the input image, $r(I)$ refers to the radius of the mask corresponding to the input image $I$. $h_I$ and $w_I$ represent the length and width of the input image, respectively. $\gamma$ is the hyperparameter.
\begin{algorithm}[h]
\label{alg:A}
\caption{Flood Filling Algorithm}
\textbf{Input:} Seed point $(x,y)$, image $\mathcal{I}$, seted value $\alpha$, $old$ value $I(x,y)$ \\
\textbf{Output:} Filled mask $\mathcal{M}$
\begin{algorithmic}[1]
\STATE \textit{\textbf{flood filling algorithm}} (($x$,$y$), $\mathcal{I}$, $\alpha$)
\STATE \quad \textbf{if} $x \geq 0$ and $x<width$ and $y\geq0$ and $y<height$
\STATE \quad and $a<\mathcal{I}(x,y)-old<b$ and $\mathcal{I}(x,y)\neq \alpha$ \textbf{then}
\STATE \quad \quad $\mathcal{M}$(x,y) $\Leftarrow$ $\alpha$
\STATE \quad \quad \textit{\textbf{flood filling}} ($(x+1,y)$, $\mathcal{I}$, $\alpha$)
\STATE \quad \quad \textit{\textbf{flood filling}} ($(x-1,y)$, $\mathcal{I}$, $\alpha$)
\STATE \quad \quad \textit{\textbf{flood filling}} ($(x,y+1)$, $\mathcal{I}$, $\alpha$)
\STATE \quad \quad \textit{\textbf{flood filling}} ($(x,y-1)$, $\mathcal{I}$, $\alpha$)
\STATE \quad \textbf{end if}
\end{algorithmic}
\end{algorithm}
The labeled ground truth of image $I$ can be represented as: $S=\{S_b,S_f^i|i=1,\cdots,N\}$, where $S_b$ and $S_f^i$ refer to the position coordinates of the background pixel and the $i$th labeled salient object, respectively. Then the set of these circle mask can be defined as $\mathcal{M}_S^{r(I)} = C_{S_f^1}^{r(I)} \cup \cdots \cup C_{S_f^N}^{r(I)} \cup C_{S_b}^{r(I)}$, where $C$ represents the circle that use the lower corner as the center and the upper corner as the radius. Similar to \cite{wssa}, we also use the edge detector \cite{edge_richer} to detect the edges of the image: $e=E(I)$, where $E(\cdot)$ represents the edge detector, $I$ represents the input image, and $e$ represents the detected edges. We use the union of $e$ and $\mathcal{M}_S^{r(I)}$, $E(I) \cup \mathcal{M}_S^{r(I)}$, to divide the image $I$ into multiple connected regions. Then the flood filling is employed to generate the pseudo-label of image $I$ (bottom in Fig. 2):
\begin{equation} \label{xx}
g = F(S, E(I) \cup \mathcal{M}_S^{r(I)})
\end{equation}
\noindent where $g$ denotes obtained pseudo-label, $F(\cdot)$ denotes the flood filling algorithm, $S$ denotes the labeled ground truth of image $I$.
\subsection{Transformer-based Point Supervised SOD Model}
The difficulty of sparse labeling saliency detection lies in the fact that the model can only obtain local ground truth labels and lacks the guidance of global information. We consider that establishing the connection between labeled and unlabeled locations via the similarity between them to obtain the saliency value of the unlabeled region can significantly alleviate this problem. Considering the similarity-based nature of vision transformer (ViT) \cite{vit}, we utilize hyper ViT (i.e., "ResNet-50 + ViT-Base") as our backbone to extract features as well as calculate the self-similarity.
\textbf{Transformer Part.} Specifically, for an input image of size $3\times H\times W$, the CNN embedding part generate $C\times \frac{H}{16}\times \frac{H}{16}$ feature maps. The multi-stage feature of ResNet-50 are denoted as $R=\{R^i|i=1,2,3,4,5\}$. Then, the transformer encoder take as input the summation of position embedding of $C\times \frac{H}{16}\times \frac{H}{16}$ and the flattened features of $C \times\frac{H}{16}\times \frac{H}{16}$ feature maps.
After 12 layers of self-attention layers, the transformer encoder part output features of $C \times \frac{H}{16} \times \frac{H}{16}$.
\textbf{Edge-preserving Decoder.} Edge-preserving decoder consists of two components, a saliency decoder, and an approximate edge detector (see Fig. 2). Saliency decoder is four cascaded convolution layers where each of them followed by batch normalization (BN) layer, ReLU activation, and upsample layer, which takes as input the features of the transformer encoder. And we denote the corresponding features of the saliency decoder at each layer as $D=\{D^i|i=1,2,3,4\}$.
For the latter part, as the weak annotations lacking structure and details, we develop an edge decoder stream as an approximate edge detector to generate structures by constraining the output using the edges generated by the real edge detector. In detail, the output of approximate edge detector can be represented as $f_e = \sigma(cat(R^3,D^2))$, where $\sigma$ represents a single $3\times3$ convolution layer followed by BN and ReLU layer. The edge map $e$ can be obtained by adding a $3\times3$ convlayer after $f_e$, which is then constrained by the edge map generated from the real edge detector.
Then, by merging $f_e$ with $D^3$, $cat(f_e,D^3)$, and passing through the following two convolution layers, the multi-channel feature $f_s$ is obtained. Similarly to $e$, the final saliency map $s$ can be obtained in the same manner.
\subsection{Non-Salient object Suppression (NSS)}
We observe that due to the sparsity of weakly supervised labels, the supervision signal can only cover a small area of the image, which leads the model only learning to highlight the learned objects while ignoring which objects in the current scene should not be highlighted (red box in Fig. \ref{NSS} (a)).
To suppress the non-salient object, we propose a simple but effective method by exploiting the location cues provided by the supervision signal and filling the generated highlighted object to suppress the non-salient object. And the obtained salient object regions (red regions in Fig 3. (b)) can be obtained by:
\begin{equation} \label{pf}
P_f = F(S-S_b, P^{1st})
\end{equation}
where $F(\cdot)$ represents flood filling, $S-S_b = \{S_f^i|i=1,...,N\}$ represents subtraction, $P^{1st}$ represents the saliency maps generated after the first training round and refined by dense CRF \cite{crf}.
Given that we only provided internal local labels for the salient objects during the first round of training, which may result in the model being unable to distinguish the edges accurately, we perform an expansion operation on $P_f$ with kernel size 10. The expanded regions are designated as the uncertain regions (black regions in Fig. 3 (b)), while the remaining regions are designated as the background region (green regions in Fig. 3 (b)). This is denoted as $P^{2nd}$, which is used as the pseudo-labels for the second training round.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{aaai2022figs/nss_viz4.pdf}
\caption{Illustration of the final pseudo label after NSS. (a) Saliency map generated at the end of first training round. (b) Salient pseudo-label after NSS. (c) The corresponding position of the pseudo label on the original image.}
\label{nss_viz}
\end{figure}
As shown in the test examples in Fig. \ref{NSS}, due to the sparseness of the labels, the model tends to detect the non-salient objects. Indeed, the model degenerates into a model that detects the previously learned objects. By reusing the position cues from the supervision points, we can successfully suppress the most non-salient objects utilizing NSS.
\subsection{Loss Function}
In our network, binary cross entropy loss, partial cross entropy loss \cite{partial_ce}, and gated CRF loss \cite{scwssod,gate_crf} are employed. For the Edge-preserving decoder stream, we use binary cross entropy loss to constrain $e$:
\begin{equation} \label{partial_ce}
\mathcal{L}_{bce}=-\sum_{(r,c)}[ylog(e)+(1-y)log(1-e)]
\end{equation}
\noindent where $y$ refers to the ground-truth, $e$ represents the predicted edge map, $r$ and $c$ represent the row and column coordinates. For the saliency decoder stream, partial cross entropy loss and gated CRF loss is employed. Partial binary cross-entropy loss is used to focus only on the definite area, while ignoring the uncertain area:
\begin{equation} \label{partial_ce}
\mathcal{L}_{pbce}=-\sum_{j\in J}[g_jlog(s_j)+(1-g_j)log(1-s_j)]
\end{equation}
\noindent where $J$ represents the labeled area, $g$ refers to the ground truth, $s$ represents the predicted saliency map.
To obtain better object structure and edges, follow \cite{scwssod}, gated CRF is used in our loss function:
\begin{equation} \label{xx}
\mathcal{L}_{gcrf} = \sum_{i}\sum_{j\in K_i} d(i,j)f(i,j)
\end{equation}
\noindent where $K_i$ denotes the areas covered by $k \times k$ kernel around pixel $i$, $d(i,j)$ is defined as:
\begin{equation} \label{xx}
d(i,j) = |s_i - s_j|
\end{equation}
\noindent where $s_i$ and $s_j$ are the saliency values of $s$ at position i and j, $\left| \cdot \right|$ denotes L1 distance. And $f(i,j)$ refers to the Gaussian kernel bandwidth filter:
\begin{equation} \label{xx}
\begin{aligned}
&f(i,j)= \\
&\frac{1}{w}\exp(-\frac{\parallel PT(i)-PT(j) \parallel_{2}}{2\sigma_{PT}^2}-\frac{\parallel I(i)-I(j) \parallel_{2}}{2\sigma_I^2})
\end{aligned}
\end{equation}
\noindent where $\frac{1}{w}$ is the weight for normalization, $I(\cdot)$ and $PT(\cdot)$ are the RGB value and position of pixel, $\sigma_{PT}$ and $\sigma_I$\ are the hyper parameters to control the scale of Gaussian kernels. So the total loss function can be defined as:
\begin{equation} \label{xx}
\mathcal{L}_{final}= \alpha_1\mathcal{L}_{bce} + \alpha_2\mathcal{L}_{pbce} + \alpha_3\mathcal{L}_{gcrf}
\end{equation}
\noindent where $\alpha_1$, $\alpha_2$, $\alpha_3$ are the weights. In our experiments, they are all set to 1.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.99\textwidth]{aaai2022figs/NSS.pdf}
\caption{Illustration of the effect of Non-Salient object Suppression. (a) Pseudo-labels generated at the end of the first round training with point supervision. (b) Salient pseudo-label obtained after passing the pseudo-label of line (a) through Non-Salient object Suppression.}
\label{NSS}
\vspace{-0.5cm}
\end{figure*}
\section{Experiments}
\subsection{Point-supervised Dataset}
To minimize the labeling time consumption while providing location information of salient objects, we build a Point-supervised Dataset (P-DUTS) by relabeling DUTS \cite{duts} dataset, a widely used saliency detection dataset containing 10553 training images. Four annotators participate in the annotation task, and for each image, we randomly select one of the four annotations to reduce personal bias. In Fig. \ref{annotation}, we show an example of point annotation and compare it with other stronger annotation methods. For each salient object, we only randomly select one-pixel location for labeling (for clarity, we exaggerate the size of the labeled position). Due to the simplicity of our method, even novice annotators can complete an image in 1$\sim $2 seconds on average.
\subsection{Implementation Details}
The proposed model is implemented on the Pytorch toolbox, and our proposed P-DUTS dataset is used as the training set. We train on four TITAN Xp GPUs. For the transformer part, a hyper version transformer ("ResNet50 + ViT-Base") is used by us, and no adjustments are made. We initialize the embedding layers and transformer encoder layers by leveraging the pretrained weight provided by ViT \cite{vit}, which is pre-trained on ImageNet 21K. The maximum learning rate is set to $2.5\times10^{-4}$ for the transformer part and $2.5\times10^{-3}$ for other parts. Warm-up and linear decay strategies are used to adjust the learning rate. Stochastic gradient descent (SGD) is used to train the network, and the following hyper-parameters are used: momentum=0.9, weight decay=$5\times10^{-4}$. Horizontal flip and random crop are used as data augmentation. The batch size is set to 28 and it takes 20 epochs for the first training procedure. The hyperparameter $ \gamma $ of Eq. \ref{r_I} is set to be 5. The second round of training uses the same parameters, but the masks are replaced with refined ones. During testing, we resized each image to 352 $\times$ 352 and then feed it to our network to predict saliency maps.
\subsection{Dataset and Evaluation Criteria}
\textbf{Dataset.} To evaluate the performance, we experiment on five public used benchmark datasets: ECSSD \cite{ecssd}, PASCAL-S \cite{pascal-s}, DUT-O \cite{dut-o}, HKU-IS \cite{hku-is}, and DUTS-test.
\noindent \textbf{Evaluation Criteria.} In our experiments, we compared our model with five state-of-the-art weakly supervised or unsupervised SOD methods (i.e., SCWSSOD \cite{scwssod}, WSSA \cite{wssa}, MFNet \cite{mfnet}, MSW \cite{msw}, SBF \cite{sbf}) and eight fully supervised SOD methods (i.e., GateNet \cite{gatenet}, BASNet \cite{basnet}, AFNet \cite{afnet}, MLMS \cite{mlms}, PiCANet-R \cite{picanet}, DGRL \cite{dgrl}, R$^3$Net \cite{r3+}, RAS \cite{ras}). Five metrics are used: precision-recall curve, maximum F-measure ($F_{\beta}^{max}$), mean F-measure ($mF_\beta$), mean absolute error ($MAE$) and structure-based metric ($S_{m}$) \cite{s-measure}.
\subsection{Compare with State-of-the-arts}
\textbf{Quantitative Comparison.} We compare our method with other state-of-the-art models as shown in Tab. \ref{comparison_SOTA} and Fig. \ref{pr}. For a fair comparison, the saliency results of different compared methods are provided by authors or obtained by running the codes released by authors. The best results are bolded. As can be seen in Tab \ref{comparison_SOTA}, our method outperforms state-of-the-art weakly supervised methods by a large margin. Our method outperforms the previous best model \cite{scwssod} by 2.68\% for $F_\beta^{max}$, 2.75\% for $S_m$, 0.94\% for $mF_\beta$, 0.62 \% for MAE on average of 5 compared dataset. What's more, our method achieves performance on par with recent state-of-the-art model GateNet \cite{gatenet} and even outperforms fully supervised methods in several metrics on multiple data sets as shown in Tab \ref{comparison_SOTA}. And as shown in Fig. \ref{pr}, for quantitative comparison, our method outperforms previous state-of-the-art methods by a large margin.
\begin{table*}[htbp]
\centering
\setlength{\tabcolsep}{1pt}
\small
\begin{tabular}{cc |cccccccc| cccccc}
\hline
\multirow{3}{*}{Metric} & &\multicolumn{8}{|c|}{Fully Sup.Methods} &\multicolumn{6}{c}{Weakly Sup./Unsup. Methods} \\
& &RAS &R$^{3}$Net &DGRL &PiCANet &MLMS &AFNet &BASNet &GateNet &SBF &MWS &MFNet &WSSA &SCWSSOD &Ours \\
& &2018 &2018 &2018 &2018 &2019 &2019 &2019 &2020 &2017 &2019 &2021 &2020 &2021 \\
\hline
\multirow{4}{*}{ECSSD} &$F_\beta^{max}$ \hfill$\uparrow$ &0.9211 &0.9247 &0.9223 &0.9349 &0.9284 &0.9350 &0.9425 &\textbf{0.9454} &0.8523 &0.8777 &0.8796 &0.8880 &0.9143 &\textbf{0.9359} \\
&$mF_\beta$ \hfill$\uparrow$ &0.9005 &0.9027 &0.9128 &0.9019 &0.9027 &0.9153 &\textbf{0.9306} &0.9251 &0.8220 &0.8060 &0.8603 &0.8801 &0.9091 &\textbf{0.9253} \\
&$S_m$ \hfill$\uparrow$ &0.8928 &0.9030 &0.9026 &0.9170 &0.9111 &0.9134 &0.9162 &\textbf{0.9198} &0.8323 &0.8275 &0.8345 &0.8655 &0.8818 &\textbf{0.9135} \\
&MAE \hfill$\downarrow$ &0.0564 &0.0556 &0.0408 &0.0464 &0.0445 &0.0418 &\textbf{0.0370} &0.0401 &0.0880 &0.0964 &0.0843 &0.0590 &0.0489 &\textbf{0.0358} \\
\hline
\multirow{4}{*}{DUT-O} &$F_\beta^{max}$\hfill$\uparrow$ &0.7863 &0.7882 &0.7742 &0.8029 &0.7740 &0.7972 &0.8053 &\textbf{0.8181} &0.6849 &0.7176 &0.7062 &0.7532 &0.7825 &\textbf{0.8086} \\
&$mF_\beta$\hfill$\uparrow$ &0.7621 &0.7533 &0.7658 &0.7628 &0.7470 &0.7763 &\textbf{0.7934} &0.7915 &0.6470 &0.6452 &0.6848 &0.7370 &0.7779 &\textbf{0.7830} \\
&$S_m$\hfill$\uparrow$ &0.8141 &0.8182 &0.8058 &0.8319 &0.8093 &0.8263 &0.8362 &\textbf{0.8381} &0.7473 &0.7558 &0.7418 &0.7848 &0.8119 &\textbf{0.8243} \\
&MAE\hfill$\downarrow$ &0.0617 &0.0711 &0.0618 &0.0653 &0.0636 &0.0574 &0.0565 &\textbf{0.0549} &0.1076 &0.1087 &0.0867 &0.0684 &\textbf{0.0602} &0.0643 \\
\hline
\multirow{4}{*}{PASCAL-S} &$F_\beta^{max}$\hfill$\uparrow$ &0.8291 &0.8374 &0.8486 &0.8573 &0.8552 &0.8629 &0.8539 &\textbf{0.8690} &0.7593 &0.7840 &0.8046 &0.8088 &0.8406 &\textbf{0.8663} \\
&$mF_\beta$\hfill$\uparrow$ &0.8125 &0.8155 &0.8353 &0.8226 &0.8272 &0.8405 &0.8374 &\textbf{0.8459} &0.7310 &0.7149 &0.7867 &0.7952 &0.8350 &\textbf{0.8495} \\
&$S_m$\hfill$\uparrow$ &0.7990 &0.8106 &0.8359 &0.8539 &0.8443 &0.8494 &0.8380 &\textbf{0.8580} &0.7579 &0.7675 &0.7648 &0.7974 &0.8198 &\textbf{0.8529} \\
&MAE\hfill$\downarrow$ &0.1013 &0.1026 &0.0721 &0.0756 &0.0736 &0.0700 &0.0758 &\textbf{0.0674} &0.1309 &0.1330 &0.1189 &0.0924 &0.0775 &\textbf{0.0647} \\
\hline
\multirow{4}{*}{HKU-IS} &$F_\beta^{max}$\hfill$\uparrow$ &0.9128 &0.9096 &0.9103 &0.9185 &0.9207 &0.9226 &0.9284 &\textbf{0.9335} &- &0.8560 &0.8766 &0.8805 &0.9084 &\textbf{0.9234} \\
&$mF_\beta$\hfill$\uparrow$ &0.8875 &0.8807 &0.8997 &0.8805 &0.8910 &0.8994 &\textbf{0.9144} &0.9098 &- &0.7764 &0.8535 &0.8705 &0.9030 &\textbf{0.9131} \\
&$S_m$\hfill$\uparrow$ &0.8874 &0.8920 &0.8945 &0.9044 &0.9065 &0.9053 &0.9089 &\textbf{0.9153} &- &0.8182 &0.8465 &0.8649 &0.8820 &\textbf{0.9019} \\
&MAE\hfill$\downarrow$ &0.0454 &0.0478 &0.0356 &0.0433 &0.0387 &0.0358 &\textbf{0.0322} &0.0331 &- &0.0843 &0.0585 &0.0470 &0.0375 &\textbf{0.0322} \\
\hline
\multirow{4}{*}{DUTS-TE} &$F_\beta^{max}$\hfill$\uparrow$ &0.8311 &0.8243 &0.8279 &0.8597 &0.8515 &0.8628 &0.8594 &\textbf{0.8876} &- &0.7673 &0.7700 &0.7886 &0.8440 &\textbf{0.8580} \\
&$mF_\beta$\hfill$\uparrow$ &0.8022 &0.7872 &0.8179 &0.8147 &0.8160 &0.8340 &0.8450 &\textbf{0.8558} &- &0.7118 &0.7460 &0.7723 &0.8392 &\textbf{0.8402} \\
&$S_m$\hfill$\uparrow$ &0.8385 &0.8360 &0.8417 &0.8686 &0.8617 &0.8670 &0.8656 &\textbf{0.8851} &- &0.7587 &0.7747 &0.8034 &0.8405 &\textbf{0.8532} \\
&MAE\hfill$\downarrow$ &0.0594 &0.0664 &0.0497 &0.0506 &0.0490 &0.0458 &0.0476 &\textbf{0.0401} &- &0.0912 &0.0765 &0.0622 &0.0487 &\textbf{0.0449} \\
\hline
\end{tabular}
\caption{Quantitative comparison with 13 state-of-the-art methods on ECSSD, DUT-OMRON, PASCAL-S, HKU-IS and DUTS-test. Top results are shown in bold. }
\label{comparison_SOTA}
\end{table*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.99\textwidth]{aaai2022figs/pr.png}
\caption{Illustration of precision-recall curves on the four largest dataset.}
\label{pr}
\end{figure*}
\begin{figure*}[h]
\centering
\includegraphics[width=0.705\textwidth]{aaai2022figs/visual_comp.png}
\caption{ Qualitative comparison with different methods.}
\label{visual_compare}
\end{figure*}
\subsubsection{Qualitative Evaluation.}
As shown in Fig. \ref{visual_compare}, our method obtains more accurate and complete saliency maps compared with other state-of-the-art weakly supervised or unsupervised methods and even surpasses recently fully supervised state-of-the-art methods. Row 1, 5, and 6 demonstrate the ability of our model to capture the overall salient object, and even objects (row 5) with extremely complex textures can be accurately segmented by our method. Further, thanks to our NSS method, our method can precisely extract salient objects and suppress non-salient objects (rows 2-4). The last row shows the ability of our model to extract the details.
\subsection{Ablation Study}
\subsubsection{Effectiveness of Edge-preserving Decoder.}
Note that, we use the results of the first round to evaluate the performance because if the first round produces bad results it will directly affect the results of the second round. In Tab. \ref{edge_decoder}, we evaluate the influence of the Edge-preserving decoder stream. The supervision of the edge generated by the edge detector is removed. At this point, the edge-preserving decoder can only aggregate shallow features and cannot explicitly detect edge features, and the overall performance decreases.
\subsubsection{Effectiveness of NSS.}
Since NSS is used in the 2nd training round, we perform ablation experiments based on the results generated in the 1st round. The ablation study results are listed in Tab. \ref{nss}. Line 1 indicates we directly use the pseudo-labels generated by the 1st round without CRF processing ("E" means using edge constraints). Line 2 indicates we use pseudo-labels refined by dense CRF. Line 3 represents the standard method in this paper. The last line represents we directly employ $P_f$ (Eq. \ref{pf}) as ground truth with no uncertain areas.
The comparison of line 1 and line 2 shows that using CRF to optimize the pseudo labels can optimize the second training round.
The 3rd row works better than the first two rows, which shows the validity of NSS. We also illustrate the effectiveness of NSS in Fig. \ref{NSS}.
\begin{table}
\centering
\small
\label{tab:freq}
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{c|ccc| ccc}
\hline
\multirow{2}{*}{Model} &\multicolumn{3}{c|}{PASCAL-S} &\multicolumn{3}{c}{DUT-OMRON}\cr
&$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE$\downarrow$ &$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE $\downarrow$\\
\hline
w/ &0.8636 &0.8498 &0.0671 &0.8034 &0.8150 &0.0717 \\
w/o &0.8629 &0.8454 &0.0703 &0.8049 &0.8054 &0.0769 \\
\hline
\end{tabular}}
\caption{ The effectiveness Edge-preserving Decoder.}
\label{edge_decoder}
\end{table}
\begin{table}
\centering
\small
\label{tab:freq}
\setlength{\tabcolsep}{0.5mm}{
\begin{tabular}{c|ccc| ccc}
\hline
\multirow{2}{*}{Model} &\multicolumn{3}{c|}{DUTS-test} &\multicolumn{3}{c}{ECSSD}\cr
&$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE$\downarrow$ &$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE $\downarrow$\\
\hline
+E &0.8588 &0.8559 &0.0502 &0.9342 &0.9134 &0.0407\\
+E+CRF &0.8606 &0.8510 &0.0463 &0.935 &0.9131 &0.0385 \\
+E+CRF+NSS$^*$ &0.8580 &0.8532 &0.0449 &0.9359 &0.9135 &0.0358 \\
\hline
\makecell[c]{+E+CRF+NSS} &0.8586 &0.8541 &0.0457 &0.9365 &0.9121 &0.0382 \\
\hline
\end{tabular}}
\caption{Ablation study for NSS.}
\label{nss}
\end{table}
\begin{table}
\centering
\small
\label{tab:freq}
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{c|ccc| ccc}
\hline
\multirow{2}{*}{$\gamma$} &\multicolumn{3}{c|}{DUTS-test} &\multicolumn{3}{c}{ECSSD}\cr
&$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE$\downarrow$ &$F_\beta^{max} \uparrow$ &$S_m \uparrow$ &MAE $\downarrow$\\
\hline
3 &0.8467 &0.8421 &0.0559 &0.9299 &0.9085 &0.0415\\
4 &0.8497 &0.8439 &0.0543 &0.9304 &0.9088 &0.0415\\
5 &0.8530 &0.8516 &0.0488 &0.9326 &0.9095 &0.0384\\
6 &0.8480 &0.8416 &0.0558 &0.9293 &0.9079 &0.0421\\
\hline
\end{tabular}}
\caption{ The impact of $\gamma$.}
\label{gama}
\end{table}
\subsubsection{Impact of the Hyperparameter $\gamma$.}
We test the effect of different values of $\gamma$ on the first round of training in Tab. \ref{gama}. Too small $\gamma$ will generate small pseudo labels providing too little supervision information, and too large $\gamma$ will make the pseudo labels contain the wrong areas. Both of these cases affect the model's performance.
\section{Conclusions}
We propose a point-supervised framework to detect salient objects in this paper. An adaptive flood filling algorithm and a transform-based point-supervised model are designed for pseudo-label generation and saliency detection. To uncover the problem of degradation of the existing weakly supervised SOD models, we propose the Non-Salient object Suppression (NSS) technique to explicitly filter out the non-salient but detected objects. We build a new point-supervised saliency dataset P-DUTS for model training. The proposed method not only outperforms existing state-of-the-art weakly-supervised methods with stronger supervision.
\section{Acknowledgements}
This work was supported by the National Key R\&D Program of China (2020AAA0108301), the National Natural Science Foundation of China (No.62072112), Scientific and technological innovation action plan of Shanghai Science and Technology Committee (No.205111031020).
|
1,116,691,499,316 | arxiv | \section{Introduction}
Question classification (QC) deals with question analysis and question labeling based on the expected answer type. The goal of QC is to assign classes accurately to the questions based on expected answer. In modern system, there are two types of questions \cite{islam2016word}. One is Factoid question which is about providing concise facts and another one is Complex question that has a presupposition which is complex. Question Answering (QA) System is an integral part of our daily life because of the high amount of usage of Internet for information acquisition.
In recent years, most of the research works related to QA are based on English language such as IBM Watson, Wolfram Alpha. Bengali speakers often fall in difficulty while communicating in English \cite{bhattacharya2001study}.
In this research, we briefly discuss the steps of QA system and compare the performance of seven machine learning based classifiers (Multi-Layer Perceptron (MLP), Naive Bayes Classifier (NBC), Support Vector Machine (SVM), Gradient Boosting Classifier (GBC), Stochastic Gradient Descent (SGD), K Nearest Neighbour (K-NN) and Random Forest (RF)) in classifying Bengali questions to classes based on their anticipated answers. Bengali questions have flexible inquiring ways, so there are many difficulties associated with Bengali QC \cite{islam2016word}. As there is no rich corpus of questions in Bengali Language available, collecting questions is an additional challenge. Different difficulties in building a QA System are mentioned in the literature \cite{hirschman2001natural} \cite{zadeh2006search}. The first work on a machine learning based approach towards Bengali question classification is presented in \cite{islam2016word} that employ the Stochastic Gradient Descent (SGD).
\section{Related Works}
\subsection{Popular Question-Answering Systems}
Over the years, a handful of QA systems have gained popularity around the world.
One of the oldest QA system is BASEBALL (created on 1961) \cite{green1961baseball} which answers question related to baseball league in America for a particular season. LUNAR \cite{woods1972lunar} system answers questions about soil samples taken from Apollo lunar exploration. Some of the most popular QA Systems are IBM Watson, Apple Siri and Wolfram Alpha. Examples of some QA systems based on different languages are: Zhang Yu Chinese question classification \cite{yu2005modified} based on Incremental Modified Bayes, Arabic QA system (AQAS) \cite{mohammed1993knowledge} by F. A. Mohammed, K. Nasser, \& H. M. Harb and Syntactic open domain Arabic QA system for factoid questions \cite{fareed2014syntactic} by Fareed et al. QA systems have been built on different analysis methods such as morphological analysis \cite{hovy2000question}, syntactical analysis \cite{zheng2002answerbus}, semantic analysis \cite{wong2007practical} and expected answer Type analysis \cite{benamara2004cooperative}.
\subsection{Research Works Related to Question Classifications}
Researches on question classification, question taxonomies and QA system have been undertaken in recent years.
There are two types of approaches for question classification according to Banerjee et al in \cite{banerjee2012bengali} - by rules and by machine learning approach. Rule based approaches use some hard coded grammar rules to map the question to an appropriate answer type \cite{prager1999use} \cite{voorhees1999trec}. Machine Learning based approaches have been used by Zhang et al and Md. Aminul Islam et al in \cite{zhang2003question} and \cite{islam2016word}. Many classifiers have been used in machine learning for QC such as Support Vector Machine (SVM) \cite{zhang2003question} \cite{nirob2017question}, Support Vector Machines and Maximum Entropy Model \cite{huang2008question}, Naive Bayes (NB), Kernel Naive Bayes (KNB), Decision Tree (DT) and Rule Induction (RI) \cite{banerjee2012bengali}. In \cite{islam2016word}, they claimed to achieve average precision of 0.95562 for coarse class and 0.87646 for finer class using Stochastic Gradient Descent (SGD).
\subsection{Research Works in Bengali Language }
A Bengali QC System was built by Somnath Banerjee and Sivaji Bandyopadhyay \cite{banerjee2012bengali} \cite{banerjee2013empirical} \cite{banerjee2013ensemble}. They proposed a two-layer taxonomy classification with 9 coarse-grained classes and 69 fine-grained classes. There are other research works \cite{islam2016word} \cite{kabir2015bangla} in Bengali Language. A survey was performed on text QA techniques \cite{gupta2012survey} where there was an analysis conducted in Bengali Language. Syed Mehedi Hasan Nirob et al achieved 88.62\% accuracy by using 380 top frequent words as the feature in their work \cite{nirob2017question}.
\section{Question Answering (QA) System}
QA system resides within the scope of Computer Science. It deals with information retrieval and natural language processing. Its goal is to automatically answer questions asked by humans in natural language. IR-based QA, Knowledge based approaches and Hybrid approaches are the QA system types. TREC, IBM-Watson, Google are examples of IR-based QA systems. Knowledge based QA systems are Apple Siri, Wolfram Alpha. Examples of Hybrid approach systems are IBM Watson and True Knowledge Evi.
Figure \ref{fig:QA_sys} provides an overview of QA System. The first step of QA System is \textbf{Question Analysis}. Question analysis has two parts - \textbf{question classification} and another \textbf{question formulation}. In question classification step, the question is classified using different classifier algorithms. In question formulation, the question is analyzed and the system creates a proper IR question by detecting the entity type of the question to provide a simple answer.
The next step is \textbf{documents retrieval and
analysis}. In this step, the system matches the query against the sources of answers where the source can be documents or Web.
In the \textbf{answer extraction} step, the system extracts the answers from the documents of the sources collected in documents retrieval and analysis phase. The extracted answers are filtered and evaluated in \textbf{answer evaluation} phase as there can be multiple possible answers for a query. In the final step, an answer of the question is returned.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{Question-answering-system-general-architecture.pdf}
\caption{General Architecture of Question Answering System}
\label{fig:QA_sys}
\end{figure}
\section{Proposed Methodology}
We use different types of classifiers for QA type classification. We separate our methodology into two sections similar to \cite{islam2016word} - one is training section and another is validation section shown in Figure \ref{fig: work_flow}.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{work-flow.pdf}
\caption{Proposed Work Flow Diagram}
\label{fig: work_flow}
\end{figure}
We use 10 fold cross validation where we have 3150 and 350 questions in our training set and validation set respectively. During training, after selecting the possible class labels, the system extracts the features of the questions and creates a model by passing through a classifier algorithm with the extracted features and class labels. During validation, the system extracts the features of the question and passes it into the model created during training and predicts the answer type.
\section{Question Collection and Categories}
Though Bengali is the seventh most spoken language in terms of number of native speakers \cite{languageref}, there is no standard corpus of questions available \cite{islam2016word}. We have collected total 3500 questions from the Internet and other sources such as books of general knowledge questions, history etc.
The corpus contains the questions and the classes each question belongs to.
The set of question categories is known as question taxonomy \cite{islam2016word}. We have used two layer taxonomy which was proposed by Xin Li, Dan Roth \cite{li2004semantic}. This two layer taxonomy is made up of two classes which are Coarse Class and Finer Class. There are six coarse classes such as Numeric, Location, Entity, Description, Human and Abbreviation and fifty finer classes such as city, state, mountain, distance, count, definition, group, expression, substance, creative, vehicle etc as shown in the Table I \cite{islam2016word}. A coarse-grained description of a system denotes large components while a fine-grained description denotes smaller sub-components of which the larger ones are composed of.
\begin{table}[h]
\begin{flushleft}
\label{my-label}
\caption{Coarse and fine grained question categories}
\begin{tabular}{|p{2cm}|p{6cm}|}
\hline
\textbf{Coarse Class} & \textbf{Finer Class}\\
\hline
ENTITY (512) & SUBSTANCE (20), SYMBOL (11), CURRENCY (24), TERM (15), WORD (20), LANGUAGE (30), COLOR (15), RELIGION (15), SPORT (10), BODY (10), FOOD (11), TECHNIQUE (10), PRODUCT (10), DISEASE (10), OTHER (22), LETTER (10), VEHICLE (11), PLANT (12), CREATIVE (216), INSTRUMENT (10), ANIMAL (10), EVENT (10)\\
\hline
NUMERIC (889) & COUNT (213), DISTANCE(13),CODE(10), TEMPERATURE (13), WEIGHT (20), MONEY (10), PERCENT (27), PERIOD (33), OTHER (34), DATE (452), SPEED (10), SIZE (54)\\
\hline
HUMAN (669) & INDIVIDUAL (618), GROUP (18), DESCRIPTION (23), TITLE (10) \\
\hline
LOCATION (650) & MOUNTAIN (32), COUNTRY (125), STATE (98), OTHER (121), CITY (274) \\
\hline
DESCRIPTION (248) & DEFINITION (153), REASON (44), MANNER (22), DESCRIPTION (39) \\
\hline
ABBREVIATION (532) & ABBREVIATION (519), EXPRESSION (13) \\
\hline
\end{tabular}
\end{flushleft}
\label{Table:dataset}
\end{table}
\section{Implementation of the System}
\subsection{Feature Extraction}
Question word answer phrases, parts of speech tags, parse feature, named entity and semantically related words are different features from answer type detection \cite{huang2008question}. We use question word and phrases as features for answer type detection. We consider the following features:
\subsubsection{TF-IDF}
Term Frequency - Inverse Document Frequency (TF-IDF) is a popular method used to identify the importance of a word in a particular document. TF-IDF transforms text into meaningful numeric representation. This technique is widely used to extract features for Natural Language Processing (NLP) applications \cite{ramos2003using} \cite{hakim2014automated}.
\subsubsection{Word level N-Grams}
N-grams is n-back to back words in a text. Queries of a same class usually share word n-grams \cite{islam2016word}. In this system, we choose bi-gram for extracting features.
\subsubsection{Stop Words}
We use two setups (as done in \cite{islam2016word}) for our system. In the first setup, we eliminate the stop words from the text using another dataset containing only stop words. At second step, we work without eliminating the stop words from the text which gives better result than the first setup.
\subsection{Classification Algorithms}
\subsubsection{Multi Layer Perceptron (MLP)}
MLP contains three layers - an input layer, an output layer and some hidden layers. Input layer receives the signal, the output layer gives a decision or prediction about the input and the computation of the MLP is conducted in the hidden layers. In our system, we use 100 layers. For weight optimization, we use Limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) optimization algorithm.
\subsubsection{Support Vector Machine (SVM)}
SVM gives an optimal hyper-plane and it maximizes the margin between classes. We use Radial Basis Function (RBF) kernel in our system to make decision boundary curve-shaped. For decision function shape, we use the original one-vs-one (ovo) decision function.
\subsubsection{Naive Bayesian Classifier (NBC)}
NBC is based on Bayes' Theorem which gives probability of an event occurrence based on some conditions related to that event. We use Multinomial Naive Bayes Classifier with smoothing parameter equals to 0.1. A zero probability cancels the effects of all the other probabilities.
\subsubsection{Stochastic Gradient Descent (SGD)}
Stochastic gradient descent optimizes an objective function with suitable smoothness properties \cite{hardt2015train}. It selects few examples randomly instead of whole data for each iteration. We use 'L2' regularization for reduction of overfitting.
\subsubsection{Gradient Boosting Classifier (GBC)}
Gradient Boosting Classifier produces a prediction model consisting of weak prediction models. Gradient boosting uses decision trees. We use 100 boosting stages in this work.
\subsubsection{K Nearest Neighbour (K-NN)}
K-NN is a supervised classification and regression algorithm. It uses the neighbours of the given sample to identify its class. K determines the number of neighbours needed to be considered. We set the value of K equals to 13 in this work.
\subsubsection{Random Forest (RF)}
RF is an ensemble learning technique. It constructs large number of decision trees during training and then predicts the majority class. We use 500 decision trees in the forest and "entropy" function to measure the quality of a split.
\subsection{Results and Discussion}
Table II shows the \textbf{accuracy} and \textbf{F1 score} for different classifiers with and without eliminating stop words while extracting features. Figure \ref{fig:bar_chart} shows the average results of different classifiers in a bar chart with and without eliminating stop words from the questions.
Overall, SGD has shown the best performance on our dataset as it introduces non-linearity and uses back-propagation for updating parameter weights using loss function calculated on training set into classification. K-NN has shown the weakest performance overall, as this algorithm has a bad reputation of not working well in high dimensional data \cite{pestov2013k}. MLP and SVM have shown similar performance. MLP takes advantage of multiple hidden layers in order to take non-linearly separable samples in a linearly separable condition. SVM accomplishes this same feat by taking the samples to a higher dimensional hyperplane where the samples are linearly separable. Gradient Boosting Classifier (GBC) and Random Forest (RF) both utilize a set of decision trees and achieve similar results (RF performs slightly without eliminating stop words). Naive Bayesian Classifier (NBC) shows performance on per with GBC and RF algorithms. The overall better performance of all the algorithms when provided with stop words show the importance of stop words in Bengali QA classification.
\begin{table}[]
\centering
\caption{Experiment Results}
\begin{tabular}{|c|l|l|l|l|l|l|l|}
\hline
\multicolumn{8}{|c|}{After Eliminating Stop Words} \\
\hline
& MLP & SVM & NBC & SGD & GBC & KNN & RF \\ \hline
Accuracy & 0.779 & 0.741 & 0.724 & 0.797 & 0.701 & 0.376 & 0.712 \\
\hline
F1 Score & 0.761 & 0.705 & 0.693 & 0.775 & 0.686 & 0.439 & 0.680 \\
\hline
\multicolumn{8}{|c|}{Without Eliminating Stop Words} \\ \hline
& MLP & SVM & NBC & SGD & GBC & KNN & RF \\ \hline
Accuracy & 0.83 & 0.801 & 0.789 & 0.832 & 0.792 & 0.781 & 0.816 \\
\hline
F1 Score & 0.810 & 0.765 & 0.759 & 0.808 & 0.775 & 0.755 & 0.783 \\
\hline
\end{tabular}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{barchart.pdf}
\caption{F1 Scores of Each Classifiers}
\label{fig:bar_chart}
\end{figure}
Figure \ref{fig:QA} shows the predictions of some particular questions by each of the classifiers. The input is a full question and the output is the class of the question.
\begin{figure}[h]
\begin{flushleft}
\includegraphics[width=9.2cm]{question-prediction.pdf}
\caption{Prediction of Questions}
\label{fig:QA}
\end{flushleft}
\end{figure}
\subsection{Computational Complexity}
\begin{table}[h]
\label{my-label}
\caption{Computational Complexity of Each Classifier}
\begin{tabular}{|p{2cm}|p{2.8cm}|p{2.8cm}|}
\hline
& Training Section & Prediction Section \\
\hline
MLP &O(nph\textsuperscript{k}i)&O(nph\textsuperscript{k}) \\
\hline
SGD & O(in$\overline{m}$) & \\
\hline
SVM &O(n\textsuperscript{2}p+n\textsuperscript{3})&O(n\textsubscript{sv}p) \\
\hline
NBC & O(np) & O(p) \\
\hline
GBC & O(npn\textsubscript{trees})&O(pn\textsubscript{trees})\\
\hline
kNN & & O(np) \\
\hline
RF &O(n\textsuperscript{2}pn\textsubscript{trees})&O(pn\textsubscript{trees}) \\
\hline
\end{tabular}
\label{Table : Complexity}
\end{table}
In Table \ref{Table : Complexity}, n = No. of training sample, p = No. of features, n\textsubscript{trees} = No. of trees (for methods based on various trees), n\textsubscript{sv} = No. of support vectors, i = No. of iterations, h = No. of nodes in each hidden layer, k = No. of hidden layers and $\overline{m}$ = the average no. of non-zero attributes per sample.
\section{Conclusion}
By implementing different machine learning based classifiers on our Bengali question corpus, we perform a comparative analysis among them. The question classification impacts the QA system. So, it is important to classify the question more precisely. This work will help the research community to choose a proper classification model for smart Bengali QA system development. Future work should aim at developing a richer corpus of Bengali questions which will help in getting better vector representation of words and thus will facilitate deep learning based automatic feature extraction.
|
1,116,691,499,317 | arxiv | \section{Introduction}
In a standard scenario, big bang nucleosynthesis (BBN) can produce only light elements, up to $^{7}$Li, and
all heavy elements have been synthesized in stars.
However, many phase transitions in the early universe could have printed their trace in a non-standard way.
For example, some baryogenesis models\cite{Dolgov:1992pu} predict very high baryon density islands in ordinary low density
backgrounds.
In the previous paper\cite{Matsuura:2005rb}, we studied heavy element production in inhomogeneous BBN from this point of view.
\footnote{For previous works on the inhomogeneous
big bang nucleosynthesis, see~\cite{IBBN1, IBBN2}. Heavy elements production is also mentioned in~\cite{jedam}.}
However we limited ourselves to the heavy element abundance and did not discuss about the light element abundance and consistency with observations.
This is because we assumed that high baryon density region is very local and
do not affect the global light element abundance.
In \cite{Rauscher:2006mv}, Rauscher pointed out that in order not to contradict to observations, the high baryon density region
must be very small and cannot affect the present heavy element abundance.
In this
paper, we show that there is a parameter region in which the heavy element can be produced enough to affect observation while keeping the light element abundance
consistent with observations.
We consider that the disagreement between Rauscher's opinion and our opinion comes from two points.
One is that we are looking at some parameter regions in which neutrons in high baryon density
do not diffuse so much as to cause disaster in standard BBN.
We would like to emphasize this point.
The other is that the relevant quantity is not the spatial size of the high baryon density region but the amount of baryon in high density regions.
We will discuss the following issues:
In section \ref{LEA}, we discuss the light element abundance in the homogeneous high baryon density region and after mixing the high and the low baryon density region.
In section \ref{HEAVY}, we study the heavy element(Ru,Mo) abundance in high and averaged baryon density
and show that heavy elements can be produced enough without contradicting the light element observation.
In section \ref{DIFF}, we briefly comment on the diffusion scale of the high baryon density region.
\section{Light element abundance}
\label{LEA}
\subsection{Homogeneous BBN}
We calculate homogeneous BBN with various values of $\eta$(baryon photon ratio).
In Table.\ref{light-table} and \ref{light-normal}, we show the numerical result of the mass fraction and the number fraction of each light element for $\eta=10^{-3}$ and $3.162 \times 10^{-10}$.
\begin{center}
\begin{table}[htbp]
\begin{tabular}{ccc}\hline
\multicolumn{3}{c}{$\eta=10^{-3}$} \\ \hline
name & mass fraction & number fraction \\
H & $ 5.814 \times 10^{-1} $ & $ 8.475 \times 10^{-1}$ \\
$^{4}$He & $ 4.185 \times 10^{-1}$ & $ 1.525 \times 10^{-1}$ \\
$^{3}$He & $ 4.842 \times 10^{-13} $ & $ 1.614 \times 10^{-13}$ \\
$^{7}$Li+$^{7}$Be & $ 1.559\times 10^{-12}$ & $ 2.227 \times 10^{-13}$ \\
D & $ 1.577 \times 10^{-22}$ & $ 7.883\times 10^{-23}$ \\ \hline
\end{tabular}
\caption{The mass and the number fractions of light elements for the homogeneous BBN with $\eta=10^{-3}$}
\label{light-table}
\end{table}
\end{center}
\vspace{0.7cm}
\begin{center}
\begin{table}[htbp]
\begin{tabular}{ccc}\hline
\multicolumn{3}{c}{$\eta=3.162 \times 10^{-10}$} \\ \hline
name & mass fraction & number fraction \\
H & $ 7.58 \times 10^{-1}$ & $ 9.26 \times 10^{-1}$ \\
$^{4}$He & $ 2.419 \times 10^{-1}$ & $ 7.39 \times 10^{-2}$ \\
$^{3}$He & $ 4.299 \times 10^{-5}$ &$ 1.433 \times 10^{-5}$ \\
$^{7}$Li + $^{7}$Be & $ 8.239 \times 10^{-10}$ & $ 1.177 \times 10^{-10}$ \\
D & $ 1.345 \times 10^{-4}$ &$ 6.723 \times 10^{-5}$ \\ \hline
\end{tabular}
\caption{The mass and the number fractions of light elements for the homogeneous BBN with $\eta=3.162 \times 10^{-10}$}
\label{light-normal}
\end{table}
\end{center}
\vspace{0.7cm}
As baryon density becomes higher, more protons and neutrons are bounded to form $^{4}$He.
At $\eta=10^{-3}$, most of the final product of $^{7}\rm{Li}$ comes from $^{7}\rm{Be}$ which decays to $^{7}\rm{Li}$ after BBN.
Details on light element production for various $\eta$ can also be found in \cite{Wagoner:1966pv}. In this paper we almost concentrate on a case in which the high baryon density
region has $\eta=10^{-3}$.
We expect that compared to $\eta \geq 10^{-3}$, the profile of the abundance for $\eta=10^{-3}$ is more different from standard BBN because most of the light element abundances change monotonically with respect to $\eta$ and if this
case does not contradict to observations, other cases would also be consistent.
Briefly, the amount of $\rm{H}$ decreases and $^{4}\rm{He}$ increases monotonically
as $\eta$ become larger.
The number fraction of $\rm{D}$ is less than $10^{-20}$ for $\eta$ greater than $10^{-7}$.
For $^{3}\rm{He}$, the number fraction drastically decreases around $\eta =10^{-4}$ down to
${\cal O}(10^{-13})$, and for $^{7}\rm{Li}$, the number fraction increases until $\eta=10
^{-6}$ and drastically decreases for a larger value of $\eta$. In the following sections, we will see that this non-standard setup does not strongly
contradict to the observations.
For simplicity we ignore the diffusion effect before and during BBN, and
after BBN both high and low baryon density regions are completely mixed.
Detailed analysis such as the case in which the high baryon density region doesn't completely mixed, or
taking into account diffusion effects are left for future work.
\subsection{Parameters and Basic equations}
In this section, we summarize the relations among parameters.
Notations :
$n$, $n ^{H}$, $n ^{L}$ are averaged, high, and low baryon number density.
$f ^{H}$, $f ^{L}$ are the volume fractions of the high and the low baryon density region.
$y_{i} $, $y_{i} ^{H}$, $y_{i} ^{L}$ are the mass fractions of each element (i) in averaged-, high- and low-density regions.
The basic relations are
\begin{eqnarray}
f ^{H}+f ^{L}&=&1 \label{vol} \\
f ^{H}n ^{H}+f ^{L}n ^{L}&=&n \label{density} \\
y_{i} ^{H}f ^{H}n ^{H}+y_{i} ^{L}f ^{L}n ^{L}&=&y_{i}n. \label{spec}
\end{eqnarray}
Under the assumption that the temperature of the universe is homogeneous, the above equation can be written as
\begin{eqnarray}
f ^{H}\eta ^{H}+f ^{L}\eta ^{L}&=&\eta\label{denseta} \\
y_{i} ^{H}f ^{H}\eta ^{H}+y_{i} ^{L}f ^{L}\eta ^{L}&=&y_{i}\eta\label{speceta}
\end{eqnarray}
where $\eta =\frac{n}{n_{\gamma}}$,$\eta ^{H,L} =\frac{n ^{H,L}}{n_{\gamma}}$
Conventional parameters for inhomogeneous BBN are $\eta$, $f$ and density ratio $R=\frac{n^{H} }{n^{L} }$.
Here we use a different combination of parameters.
Relevant values for the abundance analysis are products $f^{H,L} \times \eta ^{H,L} $ and
$\eta ^{H,L}$.
$f^{H,L}_{v}\times\eta ^{H,L}$ determines the amount of baryon from high- and low- density regions.
$\eta ^{H,L}$ determines the mass fraction of each species of nuclei.
For convenience, we write the ratio of baryon number contribution from high density region
as $a$, i.e., $f ^{H}\eta ^{H} : f ^{L}\eta ^{L}=a:(1-a) $.
There are 5 parameters($n^{H,L} , n$ and $f^{H,L} $) and 2 constraints
(Eq.(\ref{vol}) and Eq.(\ref{density})).
We calculate the light element abundance for various values of $\eta ^{H,L}$.
$\eta$ can also take any value, but in order not
to contradict observational constraints, we choose $\eta$ from $3.162 \times 10^{-10}$ to $10^{-9}$.
$a$ is determined by Eq(\ref{denseta}).
The aim of the analysis in this section is not to find parameter regions which precisely agree with the observational light element abundance and $\eta$ from CMB.
Our model is too simple to determine the constraints to parameters.
For example, we completely ignore the diffusion effect before and during BBN.
Instead we see that at least our analysis in previous paper is physically reasonable.
\subsection{Theoretical predictions and observations of light elements}
We consider the cases of
$\eta ^{H}=10^{-3}$ and $\eta ^{L}=3.162\times 10^{-10}$.
The mass fractions of and $\rm{H}$ and $^{3}\rm{He}$ in the high density region
are $0.5814$ and $4.842\times 10^{-13}$, respectively,
while those in the low density region are $0.758$ and $4.299\times 10^{-5}$.
From Eq.(\ref{speceta}), we have
\begin{eqnarray}
f ^{H}\eta ^{H}y_{^3\rm{He}} ^{H}+f ^{L}\eta ^{L}y_{^3\rm{He}} ^{L}&=&\eta y_{^3\rm{He}} \\
4.842\times 10^{-13}\times a+4.299\times 10^{-5}\times(1-a)&=& y_{^3\rm{He}}
\end{eqnarray}
\begin{eqnarray}
f ^{H}\eta ^{H}y_{\rm{H}} ^{H}+f ^{L}\eta ^{L}y_{\rm{H}} ^{L}&=&\eta y_{\rm{H}} \\
0.5814\times a+0.758\times(1-a)&=& y_{\rm{H}}.
\end{eqnarray}
We can calculate an averaged value of the abundance ratio of $^3$He to H as
\begin{equation}
(\frac{^3\rm{He}}{\rm{H}})=\frac{1}{3}\frac{4.842\times 10^{-13}\times a+4.299\times 10^{-5}\times(1-a)}
{0.5814\times a+0.758\times(1-a)}.
\end{equation}
where $a$ is related to $\eta$ as
\begin{eqnarray}
a&=&\frac{\eta ^{H}}{\eta}\frac{\eta - \eta ^{L}}{\eta ^{H}-\eta ^{L}}\\
&=& \frac{10^{-3}}{\eta}\frac{\eta - 3.162\times10^{-10}}{10^{-3}-3.162\times10^{-10}}\\
&\sim&\frac{\eta - 3.162\times10^{-10}}{\eta}.
\end{eqnarray}
Here $a$ varies from 0 to 0.9 for reasonable values of $\eta$, or
$3.162 \times 10^{-10}-10^{-9}$.
Similarly, for $\eta ^{H}=10^{-3}$
the number fractions are
\begin{equation}
(\frac{\rm{D}}{\rm{H}})=\frac{1}{2}\frac{1.577\times 10^{-22}\times a+1.345\times 10^{-4}\times (1-a)}{0.5814\times a+0.758\times(1-a)}
\end{equation}
\begin{equation}
(\frac{^{7}\rm{Li}}{\rm{H}})=\frac{1}{7}\frac{1.559\times 10^{-12}\times a +8.239\times 10^{-10}\times (1-a)}
{0.5814\times a+0.758\times(1-a)}.
\end{equation}
Fig.\ref{deps},\ref{heeps} and \ref{lieps} represent the averaged abundance ratio,
(D/H), ($^{3}$He/H) and ($^{7}$Li/H) respectively.
\begin{figure}[htbp]
\includegraphics[width=9cm,clip]{ratio-D.eps}
\caption{Averaged ratio of D to H,(D/H) vs $\eta$}
\label{deps}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=9cm,clip]{ratio-He3.eps}
\caption{Same as Fig.\ref{deps} but for ($^3$He/H)}
\label{heeps}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=9cm,clip]{ratio-Li7.eps}
\caption{Same as Fig.\ref{deps} but for ($^{7}$Li/H)}
\label{lieps}
\end{figure}
We can see that the light element abundance is the same order around
$\eta \sim 5\times 10^{-10}-10^{-9}$ as observations
\cite{Fields:2004cb,Kirkman:2003uv,O'Meara:2000dh,Kirkman:1999zu,Linsky:2003ia,Ryan:1999jq,Bonifacio:2002yx,Pinsonneault:2001ub}.
\begin{eqnarray}
(\frac{\rm{D}}{\rm{H}})_{obs}=(1.5-6.7) \times 10^{-5} \label{obs1}\\
(\frac{^{7}\rm{Li}}{\rm{H}})_{obs}=(0.59-4.1)\times 10^{-10}.
\label{obs2}
\end{eqnarray}
We do not discuss detail about diffusion here. But at least above result suggest
that our analysis is not beside the point.
\section{Theoretical predictions and observations of heavy elements ($^{92,94}$\rm{Mo}, $^{96,98}$\rm{Ru})}
\label{HEAVY}
The same analysis can be applied for heavy elements such as $^{92}\rm{Mo}$, $^{94}\rm{Mo}$, $^{96}\rm{Ru}$ and $^{98}\rm{Ru}$.
We are interested in these elements because in many models of
supernovae nucleosynthesis, these p-nuclei are less produced.
We will see that some amount of these heavy elements can be synthesized in BBN.
\begin{center}
\begin{table}[htbp]
\begin{tabular}{cc}\hline
\multicolumn{2}{c}{$\eta=10^{-3}$} \\ \hline
name & mass fraction \\
H & $ 5.814 \times 10^{-1} $ \\
$^{4}$He & $4.185\times10^{-1}$ \\
$^{92}$Mo & $1.835\times 10^{-5}$ \\
$^{94}$Mo & $4.1145\times 10^{-6}$ \\
$^{96}$Ru & $1.0789 \times 10^{-5}$ \\
$^{98}$Ru & $1.0362\times 10^{-5}$ \\\hline
\label{heavy}
\end{tabular}
\caption{The mass fractions of nuclei for homogeneous BBN with $\eta=10^{-3}$}
\end{table}
\end{center}
\vspace{0.7cm}
From Table.\ref{heavy}, we can derive the expected value of these elements.
\begin{equation}
(\frac{^{92}\rm{Mo}}{\rm{H}})=\frac{1}{92}\frac{1.835\times 10^{-5}\times a}{0.5814\times a+0.758\times(1-a)}
\end{equation}
\begin{equation}
(\frac{^{94}\rm{Mo}}{\rm{H}})=\frac{1}{94}\frac{4.1145\times 10^{-6}\times a}{0.5814\times a+0.758\times(1-a)}
\end{equation}
\begin{equation}
(\frac{^{96}\rm{Ru}}{\rm{H}})=\frac{1}{96}\frac{1.0789 \times 10^{-5}\times a}{0.5814\times a+0.758\times(1-a)}
\end{equation}
\begin{equation}
(\frac{^{98}\rm{Ru}}{\rm{H}})=\frac{1}{98}\frac{1.0362\times 10^{-5}\times a}{0.5814\times a+0.758\times(1-a)}.
\end{equation}
\vspace{1cm}
We plot expected value of these quantities in Fig.\ref{combps}.
\begin{figure}[htbp]
\includegraphics[width=9cm,clip]{ratio-MoRu.eps}
\caption{($^{92}$\rm{Mo/H}),($^{94}$\rm{Mo/H}),($^{96}$\rm{Ru/H}) and ($^{98}$\rm{Ru/H}) vs $\eta$. Red, green, blue and pink lines represent the ratio ($^{92}$\rm{Mo/H}),($^{94}$\rm{Mo/H}),($^{96}$\rm{Ru/H}),($^{98}$\rm{Ru/H}) respectively.}
\label{combps}
\end{figure}
These values should be compared with the solar abundance(Table.\ref{solar})\cite{Anders:1989zg}.
\begin{center}
\begin{table}[htbp]
\begin{tabular}{ccc}\hline
name & number fraction & ratio to H \\ \hline
H & $7.057280 \times 10^{-1}$ & 1 \\
$^{92}$Mo & $8.796560 \times 10^{-10}$ & $1.2465 \times 10^{-9} $ \\
$^{94}$Mo &$ 5.611420 \times 10^{-10}$ & $ 7.9512\times 10^{-10} $ \\
$^{96}$Ru & $2.501160 \times 10^{-10}$ & $ 3.5441\times 10^{-10} $ \\
$^{98}$Ru &$ 8.676150 \times 10^{-11} $& $ 1.2294\times 10^{-10} $ \\ \hline
\end{tabular}
\caption{The abundances of $^{92,94}$\rm{Mo} and $^{96,98}$\rm{Ru} in the solar system\cite{Anders:1989zg}}
\label{solar}
\end{table}
\end{center}
Compared those observational values with Fig.\ref{combps},
it is clear that the heavy element produced in BBN can affect the solar abundance heavy element.
Some of them are produced too much. But this is not a problem of the previous work
\cite{Matsuura:2005rb}, because we assumed that high density regions are very small and
do not disturb standard BBN.
The analysis here suggest that even if we assume the density fluctuations are completely mixed,
heavy element can have enough affect to the solar abundance.
\section{Diffusion during BBN}
\label{DIFF}
In the previous analysis, we assumed that the diffusion effect can be ignored during BBN and
both high density regions and low density regions are completely mixed after BBN.
In this section, we determine the scale of high baryon density island in which the
diffusion effect during BBN is very small enough and our assumption is valid.
We do not discuss the diffusion after BBN here.
A detail analysis of the comoving diffusion distance of the baryon, the neutron and the proton is in \cite{Applegate:1987hm}.
From Fig.1 in \cite{Applegate:1987hm}, in order to safely ignore the diffusion effect, it is necessary for
the high baryon density island to be much larger than $10^{5}$cm at T=0.1MeV($1.1\times 10^{9}$K).
Notice that $T \propto \frac{1}{A}$, where A is a scale factor.
For scale d now corresponds to $d/(4.0\times10^8)$ at BBN epoch.
Present galaxy scale is $\mathcal{O}(10^{20})$cm, which corresponds to
$\mathcal{O}(10^{12})$cm $>>10^{5}$cm at BBN epoch.
\begin{center}
\begin{table}[htbp]
\begin{tabular}{cc}\hline
\multicolumn{2}{c}{temperature and scale} \\ \hline
temperature & scale \\
$1.1\times10^{9}$K (BBN) & d \\
3000K (decouple) & $3.7\times 10^{6}\times d$ \\
2.725K (now) & $4.0\times 10^{8}\times d$ \\ \hline
\end{tabular}
\caption{Relation between temperature and scale}
\end{table}
\end{center}
The maximum angular resolution of CMB is $l_{max} \sim$2000.
The size of universe is $\sim 5000$Mpc.
In order not to contradict to CMB observation, the fluctuation of baryon density must be
less than $\sim 16$Mpc now.
This corresponds to $10^{17}$cm at BBN.
Since the density fluctuation size in Dolgov and Silk's model\cite{Dolgov:1992pu} is a free parameter,
the above brief estimation suggests that we can take the island size large enough to ignore the diffusion effect
without contradicting to observations, i.e., the reasonable size of $10^{5}$cm $-$$10^{17}$cm at the BBN epoch.
We can choose distances between high density islands so that we obtain a suitable value of $f$.
\section{Summary}
In this paper, we studied the relation between the heavy element production in high baryon density regions during BBN
and the light element observation.
By averaging the light element abundances in the high and the low density regions we showed that it is possible to produce a relevant amount of heavy element without contradicting to observations.
However we should stress that in this paper we restricted ourselves
to some parameter regions where neutrons in high baryon density regions do not destroy the standard BBN.
So our setup is different from the conventional inhomogeneous
BBN studies.
We also studied the size of the density fluctuation to show that there is a parameter region in which the neutron diffusion is negligible
and which is much smaller than CMB observation scale.
It is worthwhile to investigate further how the produced heavy elements
can be related to the detailed observations.
\section{Acknowledgements}
We thank R.H. Cyburt, R. Allahverdi and R. Nakamura for useful discussions.
This research was supported in part by Grants-in-Aid for Scientific
Research provided by the Ministry of Education, Science and
Culture of Japan through Research Grant No.S 14102004, No.14079202.
S.M.'s work was supported in part by JSPS(Japan Society for the
Promotion of Science).
|
1,116,691,499,318 | arxiv | \section{Introduction}\label{sec:overview}
\IEEEPARstart{I}{ntelligent} reflecting surface (IRS), aka reconfigurable intelligent surface (RIS), is an emerging wireless network device that aims to improve wireless environment by manipulating signal reflections \cite{Wu2018gc,Bjoernson2022,Xu2021}. Owing to its much lower cost and much lower energy consumption, IRS might provide an alternative to small base-station and relay for enhancing throughput, coverage, connectivity, and reliability in future networks. While the early studies \cite{Wu2018gc} concentrate on a single IRS, the current trend is towards the multi-IRS coordination \cite{zhang2019analysis}, \cite{mei2022intelligent}. Many existing methods in this field require the full channel state information (CSI), thus suffering the curse of dimensionality when IRSs are deployed extensively. To bypass this difficulty, we propose a novel strategy called \emph{blind beamforming} that is capable of optimizing phase shifts across multiple IRSs in the absence of CSI.
Our approach is inspired by the two recent works \cite{Arun_2020_RFocus}, \cite{blind_beamforming_twc}, which suggest the potential of optimizing phase shifts blindly for a single IRS without CSI. Given the whole solution space $\Omega$ of phase shifts (which is too large to explore fully), \cite{Arun_2020_RFocus}, \cite{blind_beamforming_twc} propose only testing a small subset of possible solutions $\mathcal S\subset \Omega$ at random, from which a statistical quantity (e.g., the conditional sample mean) of the received signal power can be obtained to help decide phase shifts. The resulting solution is not restricted to $\mathcal S$. While \cite{Arun_2020_RFocus}, \cite{blind_beamforming_twc} focus on a single IRS, this work aims at a full generalization of blind beamforming that accounts for multiple IRSs
Because the number of channels is exponential in the number of IRSs, channel estimation is a tractable task only in some simple settings, e.g., when there are two IRSs \cite{channel_est_siso,channel_est_MIMO_MU_zbx,channel_est_MSE_min,channel_est_beamforming_ycs}, or when the multi-hop reflected channels are all neglected \cite{Keykhosravi2021}. Some studies are devoted to the overhead reduction for channel estimation in IRS systems, e.g., the deep learning method \cite{Zhang2023self} and the two-timescale optimization \cite{zhao2021two}. Aside from the computational difficulty, channel estimation for IRS also imposes a huge practical challenge because of the communication chip issue as well as the network protocol issue
\cite{blind_beamforming_twc}. To the best of our knowledge, the existing prototype realizations of IRS \cite{Arun_2020_RFocus,pei2021ris,tran2020demonstration,kitayama2021research, Staat_2022_IRShield, chen2020active} seldom involve channel estimation.
Actually, even if the exact CSI has been provided, it is still quite difficult to decide phase shifts for multiple IRSs. The difficulty arises from the fact that every multi-hop reflected channel is incident to more than one reflecting element (RE) of distinct IRSs and hence their phase shifts must be optimized jointly. To render the problem tractable, a common compromise \cite{Lu2022,Sun2021,Song2022,Yang2022,Xie2022,Cao2021,Karim2022,Li2020,Ning2022,Wei2022,Huang2022,Esmaeilbeig2022,multi_IRS_radar_comm,Wei2022a,Li2022,Asim2022,Ni2022} is to ignore the multi-hop channels.
Many existing analyses and methods build upon this approximation, ranging from delay alignment \cite{Lu2022} to ergodic rate \cite{Sun2021}, secure transmission \cite{Song2022}, spectral efficiency \cite{Yang2022}, outage probability \cite{Xie2022,Cao2021}, and full-duplex transmission \cite{Karim2022}. The above approximation has also been extended to the multiple-user case for a variety of system design problems related to IRS, e.g., the sum rates maximization \cite{Li2020,Ning2022,Wei2022}, the IRS placement optimization \cite{Huang2022}, the target sensing \cite{Esmaeilbeig2022}, the joint sensing and communication \cite{multi_IRS_radar_comm,Wei2022a}, the joint unmanned aerial vehicles (UAV) and IRS aided transmission \cite{Li2022,Asim2022}, and the federated learning \cite{Ni2022}.
However, the above simplified channel model with multiple IRSs could be fundamentally flawed. If each signal reflection is incident to only one IRS, then the multiple IRSs distributed at the different positions can be basically thought of as a single IRS. As a result, the signal-to-noise ratio (SNR) boost is at most $\Theta(L^2N^2)$ according to \cite{blind_beamforming_twc}, where $L$ is the number of IRSs and $N$ is the number of REs of each IRS.
In contrast, this work shows that a much higher boost of $\Theta(N^{2L})$ can be reached by harnessing the multi-hop reflections. Actually, the previous work \cite{Han_double_IRS_beamforming_power_scaling} already shows that the two-hop channels play a crucial role in enabling an SNR boost of $\Theta(N^4)$ for a double-IRS system. Nevertheless, the argument in \cite{Han_double_IRS_beamforming_power_scaling} is based on a fairly strong assumption that only the two-hop reflections exist while the rest channels are all null. Similarly, \cite{Multi_IRS_Huang,multi_IRS_Mei} only assume the existence of the longest cascaded channels (which are incident to every IRS) from transmitter to receiver in a general $L$-IRS system. A line of other works \cite{multi_IRS_WMMSE,3D_channel_model_double_IRS,Proactive_Eavesdropping,MIMO_MU_Zhengbeixiong,MISO_Chen,Nguyen2022,Kim2021} simplify the multi-IRS channel model in the opposite way. They only consider the one-hop and the two-hop reflections while neglecting all the higher-order reflections. Differing from all the above works, this paper does not require any channels to be zero. As a major result of this work, we show that the highest possible SNR boost of $\Theta(N^{2L})$ can be achieved by blind beamforming without making any zero approximations of the channels.
\renewcommand{\arraystretch}{1.0}
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\small
\centering
\caption{\small List of Main Variables}
\begin{tabular}{|l|l|}
\hline
Symbol & Definition \\
\hline
$L$ & number of IRSs \\
\hline
$N$ & number of REs of each IRS\\
\hline
$K$ & number of phase shift choices on each RE\\
\hline
$T$ & number of random samples for blind beamforming\\
\hline
$n_{\ell}$ & index of the $n$th RE of IRS $\ell$\\
\hline
$h_{n_{1},\ldots,n_{L}}$ & cascaded channel induced by REs $n_1,\ldots,n_L$\\
\hline
$u^{(\ell)}_{n_\ell}$ & factor component of $h_{n_{1},\ldots,n_{L}}$ related to RE $n_\ell$\\
\hline
$\theta_{n_{\ell}}$ & phase shift of RE $n_{\ell}$\\
\hline
$\theta'_{n_{\ell}}$ & solution of $\theta_{n_{\ell}}$ by the proposed method\\
\hline
$\theta^\star_{n_{\ell}}$ & continuous solution of $\theta_{n_{\ell}}$ as $K\rightarrow\infty$\\
\hline
$\hat\theta^\star_{n_{\ell}}$ & approximate continuous solution of $\theta_{n_{\ell}}$\\
\hline
$\mathcal D^{(\ell)}_{m}$ & set of reflected channels related to RE $m$ of IRS $\ell$\\
\hline
$\mathcal D^{(\ell)}_{0}$ & set of channels not related to any RE of IRS $\ell$\\
\hline
$\mathcal A^{(\ell)}_{m}$ & subset of $\mathcal D^{(\ell)}_{m}$ unrelated to at least one IRS \\
\hline
$\mathcal E^{(\ell)}_{m}$ & subset of $\mathcal D^{(\ell)}_{m}$ related to every IRS \\
\hline
\end{tabular}
\label{tab:var}
\end{table}
The remainder of the paper is organized as follows. Section \ref{sec:model} describes the multi-IRS channel model and formulates the beamforming problem mathematically. Section \ref{sec blind beamforming} introduces the blind beamforming method for a double-IRS system. Section \ref{sec csm} extends the proposed method to a general $L$-IRS system. Section \ref{sec test} presents numerical results---which include field tests and simulations. Finally, Section \ref{sec:conclusion} concludes this work.
The Bachmann-Landau notation is extensively used in the paper: $f(n)=O(g(n))$ if there exists some $c>0$ such that $|f(n)|\le cg(n)$ for $n$ sufficiently large; $f(n)=o(g(n))$ if there exists some $c>0$ such that $|f(n)|< cg(n)$ for $n$ sufficiently large; $f(n)=\Omega(g(n))$ if there exists some $c>0$ such that $f(n)\ge cg(n)$ for $n$ sufficiently large; $f(n)=\Theta(g(n))$ if $f(n)=O(g(n))$ and $f(n)=\Omega(g(n))$ both hold. Moreover, the phase of a complex number $x\in\mathbb C$ is denoted $\angle x$, and the discrete set $\{a,a+1,\ldots,b-1,b\}$ is denoted $[a:b]$ for two integers $a<b$. For convenience, we summarize in TABLE \ref{tab:var} the main variables used in the sequel.
\section{System Model}
\label{sec:model}
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{fig/Double_IRS_system_model.eps
\caption{A double-IRS system with $L=2$.}
\label{fig:system model}
\end{figure}
Consider a point-to-point wireless transmission in aid of $L\ge2$ IRSs. Assume that
the transmitter and receiver are equipped with one antenna each. Assume also that every\footnote{We assume that IRSs have the same number of REs in order to facilitate performance analysis. But this assumption is not required for the practical use of blind beamforming as discussed in Section \ref{subsec:field_tests}.} IRS consists of $N$ REs. We use $\ell\in[1:L]$ to index each IRS, and use $n_\ell\in[1:N]$ to index each RE of IRS $\ell$. Let $\theta_{n_\ell}\in[0,2\pi)$ be the phase shift induced by RE $n_\ell$ into its associated reflected channels. From a practical stand \cite{blind_beamforming_twc,pei2021ris,tran2020demonstration,kitayama2021research, Staat_2022_IRShield, chen2020active}, assume that each $\theta_{n_\ell}$ can only take on values from a uniform discrete set
\begin{equation}
\Phi_K = \{\omega,2\omega,\ldots,K\omega\}\;\; \text{where} \;\; \omega \triangleq \frac{2\pi}{K}
\end{equation}
given a positive integer $K\ge2$, namely \emph{discrete beamforming}. We use $h_{n_1,\ldots,n_L}$ to denote the cascaded reflected channel induced by the REs $(n_1,n_2,\ldots,n_L)$; let $n_\ell=0$ if the channel is not related to IRS $\ell$. For instance, if $L=3$ and $N=10$, then $h_{2,0,6}$ represents a reflected channel incident to the 2nd RE of IRS 1 and the 6th RE of IRS 3, which is not related to any RE of IRS 2. In particular, $h_{0,\ldots,0}$ represents the direct channel from the transmitter to the receiver.
For the transmit signal $X\in\mathbb C$ and the complex Gaussian background noise $Z\sim\mathcal{CN}(0,\sigma^2)$, the received signal $Y\in\mathbb C$ is given by
\begin{equation}
\label{Y:K-IRS}
Y = \sum_{(n_1,\ldots,n_L)\in[0:N]^L}h_{n_1,\ldots,n_L}e^{j\sum^L_{\ell=1}\theta_{n_\ell}}X+Z.
\end{equation}
For each $n_\ell=0$, we accordingly set $\theta_{n_\ell}=0$. When specialized to the double-IRS case with $L=2$, the above equation can be rewritten as
\begin{multline}
\label{Y: 2-IRS}
Y = \underbrace{h_{0,0}X}_{\text{direct signal}} + \underbrace{\sum^{N}_{n_1=1}h_{n_1,0}e^{j\theta_{n_1}}X}_{\text{reflected signal due to IRS 1}}+\underbrace{\sum^{N}_{n_2=1}h_{0,n_2}e^{j\theta_{n_2}}X}_{\text{reflected signal due to IRS 2}}\\
+\underbrace{\sum^{N}_{n_1=1}\sum^{N}_{n_2=1}h_{n_1,n_2}e^{j(\theta_{n_1}+\theta_{n_2})}X}_{\text{reflected signal due to both IRS 1 \& IRS 2}}+Z,
\end{multline}
as illustrated in Fig.~\ref{fig:system model}. In most of this work, we assume a general integer $L\ge2$. Section \ref{sec blind beamforming} focuses on the special case of $L=2$.
With the transmit power $P=\mathbb E[|X|^2]$, the received SNR is
\begin{equation}
\mathrm{SNR}=
\left|\sum_{(n_1,\ldots,n_L)\in[0:N]^L}h_{n_1,\ldots,n_L}e^{j\sum^L_{\ell=1}\theta_{n_\ell}}\right|^2\frac{P}{\sigma^2}.
\end{equation}
We wish to evaluate the performance gain brought by the IRSs. Toward this end, let us also compute the SNR without using any IRS as a benchmark, that is
\begin{equation} \label{eqn SNR}
\mathrm{SNR}_0=\left|h_{0,\ldots,0}\right|^2\frac{P}{\sigma^2}.
\end{equation}
We seek the optimal set of phase shifts $\{\theta_{n_\ell}\}$ that maximizes the SNR boost, i.e.,
\begin{subequations}\label{eqn problem}
\begin{align}
\underset{ \{\theta_{n_\ell}\}}{\text{maximize}}\quad & \frac{\mathrm{SNR}}{\mathrm{SNR}_0} \label{eqn problem1}\\
\textrm{subject to}\quad &\theta_{n_\ell}\in\Phi_K,\;\forall n_\ell.\label{eqn problem2}
\end{align}
\end{subequations}
The difficulties of the above problem are two-fold: (i) the variables are discrete; (ii) the channels $\{h_{n_1,\ldots,n_L}\}$ are unknown.
\section{Double-IRS Case}\label{sec blind beamforming}
The conventional paradigm for IRS beamforming comprises two stages: first estimate the cascaded channels $\{h_{n_1,\ldots,n_L}\}$ and then optimize the phase shifts $\{\theta_{n_\ell}\}$. But channel acquisition does not scale well with problem size because the number of channels grows exponentially with the number of IRSs. Alternatively, one may just estimate the channel matrix between every pair of IRSs and subsequently recover the cascaded channels $\{h_{n_1,\ldots,n_L}\}$ by multiplying the associated between-IRS channel matrices together, so that the number of channels to estimate decreases to $2NL+{L \choose 2}N^2=O(N^2L^2)$. However, the above method is costly in practice because it requires deploying a sensor at each RE to detect the pilot signal for channel estimation. Differing from most approaches in the literature, this work sidesteps channel estimation and optimizes phase shifts directly in the absence of CSI.
\subsection{Blind Beamforming for a Single IRS}
Before proceeding to the double-IRS case, we first review the so-called \emph{conditional sample mean (CSM)} method in \cite{blind_beamforming_twc} for configuring a single IRS without any channel information. We then let $L=1$. Since there is only one IRS, the IRS index $\ell$ can be dropped for ease of notation, i.e., $n_\ell$ reduces to $n$.
If all the channels were already known, then a natural idea would be to align each reflected channel $h_n$ with the direct channel $h_0$. If the perfect alignment cannot be achieved due to the discrete constraint $\Phi_K$, one may rotate $h_n$ to the closest possible position to $h_{0}$ in the complex plane, namely the \emph{closest point projection (CPP)}, whereby phase shift is determined as
\begin{equation}
\label{CPP}
\theta^{\text{CPP}}_{n} = \arg\min_{\theta\in\Phi_K}\big|\theta+\angle h_n-\angle h_0\big|.
\end{equation}
The aim of CSM is to mimic CPP without knowing $\angle h_n$ and $\angle h_0$. The method works as follows. We first generate a total of $T$ random samples $\bm\theta^{(t)}=\{\theta^{(t)}_{n}\,\big|\,n\in[1:N]\}$ with each $\theta^{(t)}_{n}$ drawn uniformly from $\Phi_K$, for the sample index $t\in[1:T]$. Let $\mathcal G_{n,k}\subseteq[1:T]$ be the set of indices of those samples $\bm\theta^{(t)}$ satisfying $\theta^{(t)}_{n}=k\omega$, i.e.,
\begin{align}
\mathcal{G}_{n,k}\triangleq\Big\{ t\in[1:T]\Big|\theta^{(t)}_{n}=k\omega\Big\}.
\end{align}
We measure the received signal power $|Y^{(t)}|^2$ corresponding to each random sample $\bm\theta^{(t)}$, based on which a conditional sample mean of $|Y^{(t)}|^2$ is computed for each $\mathcal G_{n,k}$ as
\begin{equation}
\label{cond_expectation}
\widehat{\mathbb E}[|Y|^2|\theta_{n}=k\omega]\triangleq\frac{1}{|\mathcal{G}_{n,k}|} \sum\limits_{t \in \mathcal{G}_{n,k}} |Y^{(t)}|^2.
\end{equation}
The solution by CSM, denoted $\theta'_n$, maximizes the conditional sample mean with respect to each RE, i.e.,
\begin{align}
\label{CSM}
\theta'_n = \arg \max_{\varphi\in\Phi_K} \widehat{\mathbb{E}}[|Y|^2|\theta_n=\varphi].
\end{align}
Define the average-reflection-to-direct-signal ratio to be
\begin{equation}
\rho \triangleq \frac{\sum^N_{n=1}{|h_n|^2}/N}{|h_0|^2},
\end{equation}
which converges in probability to a finite constant for $N$ sufficiently large. The performance of CSM is characterized in the following proposition.
\begin{proposition}[Theorem 2 in \cite{blind_beamforming_twc}]
\label{prop:CSM}
The CSM method is equivalent to the CPP method in \eqref{CPP} and yields a quadratic SNR boost in the number of REs in expectation, i.e.,
\begin{equation}
\label{boost:single_IRS}
\mathbb E\left[\frac{\mathrm{SNR}}{\mathrm{SNR}_0}\right] = \rho\cdot\Theta(N^2),
\end{equation}
so long as $K\ge3$ and $T=\Omega(N^2(\log N)^3)$.
\end{proposition}
\begin{remark}
CSM only requires trying out a polynomial number of possible solutions $(\theta_1,\ldots,\theta_n)$, which occupy a small portion of the whole solution space of size $K^N$.
\end{remark}
\begin{remark}
For the binary beamforming case with $K=2$, i.e., when each $\theta_n\in\{0,\pi\}$, the SNR boost by CSM may fall below the quadratic. Rather, the SNR boost can be arbitrarily close to 0 dB in the worst-case scenario as shown in \cite{blind_beamforming_twc}. In contrast, an enhanced CSM in \cite{blind_beamforming_twc} maintains the quadratic boost for any $K\ge2$. Nevertheless, the contrived worst-case scenario of CSM rarely occurs in practice, so CSM is still a good choice for binary beamforming.
\end{remark}
\subsection{Blind Beamforming for Two IRSs}
\label{subsec:L=2}
We now let $L=2$. The above CSM method is extended to the double-IRS case as follows: first optimize IRS 1 while holding IRS 2 fixed, and then optimize IRS 2 while holding IRS 1 fixed. Simple as this extension looks, it is by no means trivial to analyze its performance. The main result of this subsection is to establish a quartic SNR boost of $\Theta(N^4)$ under certain conditions.
Let us start with a common misconception. One may think that the SNR boost can be readily verified for the extended CSM because each IRS brings a $\Theta(N^2)$ boost as shown in Proposition \ref{prop:CSM} and hence the two IRSs together bring a $\Theta(N^4)$ boost. The above argument is problematic in that the boost factor $\rho$ in \eqref{boost:single_IRS} associated with IRS 1 is impacted by the later optimization of IRS 2. As a quick example, If the channels between the two IRSs are all zeros so that only $h_{0,0},h_{n_1,0},h_{0,n_2}$ survive, then the two IRSs can be recognized as one whole IRS, and thus the highest possible boost is $\Theta(N^2)$. The reason is that each $h_{0,n_2}$ is included in the fixed direct channel when analyzing the SNR boost for IRS 1, but subsequently it can be altered dramatically by the optimization of IRS 2. Thus, the key question is: how do we preserve the SNR boost of the previous IRS when optimizing the current IRS? The following theorem provides a set of sufficient conditions in this regard.
\begin{theorem}
\label{coro boost 2}
If a double-IRS system satisfies the following three conditions:
\begin{enumerate}[C1.]
\item the channels between the two IRSs are
line-of-sight (LoS) so that the two-hop channel matrix has rank one \cite{Han_double_IRS_beamforming_power_scaling} and can be factorized as
\begin{equation}
\label{eqn 2 decompose}
\begin{bmatrix}
h_{1,1} & \cdots & h_{1,N}\\
\vdots & & \vdots\\
h_{N,1} & \cdots & h_{N,N}
\end{bmatrix}
=
\begin{bmatrix}
u^{(1)}_1\\
\vdots\\
u^{(1)}_N
\end{bmatrix}
\begin{bmatrix}
u^{(2)}_1 &\cdots &
u^{(2)}_N
\end{bmatrix};
\end{equation}
\item $K\ge3$;
\item there exists a constant $\gamma\in[0,\frac{\pi}{2}\!-\!\frac{\pi}{K})$ such that
\begin{equation}
\label{gamma}
|h_{n_1,0}| \le \sin\gamma \cdot\left|\sum^N_{n_2=1} h_{n_1,n_2}\right|,\;\;\forall n_1\in[1:N],
\end{equation}
\end{enumerate}
then the extended CSM method as stated at the beginning of this subsection yields a quartic SNR boost as
\begin{align}
\mathbb{E}\left[\frac{\mathrm{SNR}}{\mathrm{SNR}_0}\right]=\frac{\delta_1^2\delta_2^2}{|h_0|^2}\cdot\Theta(N^4),
\label{eqn n4gain}
\end{align}
where
\begin{equation}
\delta_1\triangleq\frac{1}{N}\sum^N_{n_1=1}|u^{(1)}_{n_1}|\quad\text{and}\quad
\delta_2\triangleq\frac{1}{N}\sum^N_{n_2=1}|u^{(2)}_{n_2}|.
\end{equation}
\end{theorem}
\begin{IEEEproof}
Since $|h_0|^2$ and $P$ are fixed, it suffices to show that $\mathbb E[|g|^2]=\delta^2_1\delta^2_2\cdot\Theta(N^4)$, where $g$ represents the superposition of all the channels with the IRS phase shifts $\theta_{n_1}$ and $\theta_{n_2}$, i.e.,
\begin{multline}
\label{g L=2}
g(\theta_{n_1},\theta_{n_2}) = h_{0,0}+\sum^N_{n_1=1}h_{n_1,0}e^{j\theta_{n_1}}+\sum^N_{n_2=1}h_{0,n_2}e^{j\theta_{n_2}}\\
+\sum^N_{n_1=1}\sum^N_{n_2=1}h_{n_1,n_2}e^{j(\theta_{n_1}+\theta_{n_2})}.
\end{multline}
To establish $\mathbb E[|g|^2]=\delta^2_1\delta^2_2\cdot\Theta(N^4)$, we need to verify the converse $\mathbb E[|g|^2]=\delta^2_1\delta^2_2\cdot O(N^4)$ and the achievability $\mathbb E[|g|^2]=\delta^2_1\delta^2_2\cdot \Omega(N^4)$. The converse is evident since
\begin{align}
|g|^2&\le\!\bigg||h_{0,0}|\!+\!\!\sum_{n_1=1}^N\!|h_{n_1,0}|\!+\!\!\sum_{n_2=1}^N\!|h_{0,n_2}|\!+\!\!\sum_{n_1=1}^N\sum_{n_2=1}^N\!|h_{n_1,n_2}|\bigg|^2\notag\\
&=\delta_1^2\delta_2^2\cdot O(N^4).\notag
\end{align}
The rest of the proof focuses on the achievability.
According to Algorithm \ref{alg:SCSM}, we first configure IRS 1 with IRS 2 held fixed, by treating all the channels related to IRS 1 as the reflected channel and the rest as the direct channel. Thus, if $\theta_{n_1}$ is continuous, its optimal solution is aligning the reflected channel with the direct channel exactly, i.e.,
\begin{equation}
\theta^\star_{n_1} = \angle\Bigg(\underbrace{h_{0,0}+\sum_{n_2=1}^N h_{0,n_2}}_{\text{direct channel}}\Bigg)-\angle\Bigg(\underbrace{h_{n_1,0}+\sum_{n_2=1}^N h_{n_1,n_2}}_{\text{reflected channel}}\Bigg).\notag
\end{equation}
According to Proposition \ref{prop:CSM}, configuring IRS 1 by conditional sample mean is equivalent to rotating the reflected channel to the closest position to the direct channel, i.e.,
\begin{equation}
\theta'_{n_1} = \arg\min_{\theta\in\Phi_K}\big|\theta-\theta^\star_{n_1}\big|.
\end{equation}
Clearly, we have
\begin{equation}
\label{bound:1}
\big|\theta'_{n_1}-\theta^\star_{n_1}\big|\le\frac{\pi}{K}.
\end{equation}
Moreover, we approximate the continuous solution $\theta^\star_{n_1}$ by removing the single-hop reflect channel, i.e.,
\begin{align}
\hat\theta^\star_{n_1} &\triangleq\angle\left(h_{0,0}+\sum_{n_2=1}^N h_{0,n_2}\right)-\angle\left(\sum_{n_2=1}^N h_{n_1,n_2}\right)\notag\\
&=\angle\Bigg(h_{0,0}+\sum_{n_2=1}^N h_{0,n_2}\Bigg)-\angle u^{(1)}_{n_1}-\angle\!\left(\sum_{n_2=1}^N u^{(2)}_{n_2}\right),\label{eqn 2 theta^}
\end{align}
where the second equality follows from the assumption in \eqref{eqn 2 decompose}. Given \eqref{gamma}, the error of the above approximation can be bounded above as
\begin{align}
\big|\hat\theta^\star_{n_1}-\theta^\star_{n_1}\big|&=\left|\angle\left(h_{n_1,0}\!+\!\sum_{n_2=1}^N h_{n_1,n_2}\right)\!-\!\angle\left(\sum_{n_2=1}^N h_{n_1,n_2}\right)\right|\notag\\
&\le \gamma.\label{eqn 2 gamma}
\end{align}
Combining \eqref{bound:1} and \eqref{eqn 2 gamma} gives
\begin{equation}
\label{bound:3}
\big|\hat\theta^\star_{n_1}-\theta'_{n_1}\big|\le \gamma+\frac{\pi}{K}<\frac{\pi}{2}.
\end{equation}
Next, IRS 2 is configured with each phase shift of IRS 1 fixed at $\theta'_{n_1}$. We now treat all the channels related to IRS 2 as reflected channel and treat the rest as the direct channel. Thus, if $\theta_{n_2}$ is continuous, its optimal solution is given by
\begin{align}
&\theta^\star_{n_2} =\notag\\ &\;\angle\Bigg(\!\underbrace{h_{0,0}\!+\!\sum_{n_1=1}^N h_{n_1,0}e^{j\theta'_{n_1}}}_{\text{direct channel}}\!\Bigg)
\!-\!\angle\Bigg(\!\underbrace{h_{0,n_2}\!+\!\sum_{n_1=1}^N h_{n_1,n_2}e^{j\theta'_{n_1}}}_{\text{reflected channel}}\!\Bigg).\label{eqn 2 }
\end{align}
Again, by Proposition \ref{prop:CSM}, it can be shown that
\begin{equation}
\theta'_{n_2} = \arg\min_{\theta\in\Phi_K}\big|\theta-\theta^\star_{n_2}\big|
\end{equation}
and
\begin{equation}
\label{bound:4}
\big|\theta'_{n_2}-\theta^\star_{n_2}\big|\le\frac{\pi}{K}.
\end{equation}
For ease of notation, we define
\begin{equation}
\xi_{n_2} = h_{0,n_2}+\sum_{n_1=1}^N h_{n_1,n_2}e^{j\theta'_{n_1}}.\label{eqn xi}
\end{equation}
It can be shown that
\begin{align}
|g(\theta'_1,\theta'_2)|^2 &= \left|h_{0,0}+\sum_{n_1=1}^N h_{n_1,0}e^{j\theta'_{n_1}}+\sum_{n_2=1}^{N}e^{j\theta'_{n_2}}\xi_{n_2}\right|^2\notag\\
&\ge \left(\cos\frac{\pi}{K}\cdot\sum_{n_2=1}^N \left|\xi_{n_2}\right|\right)^2,
\label{key_step:1}
\end{align}
where the last inequality follows by the projection of each $e^{j\theta'_{n_2}}\xi_{n_2}$ onto $h_{0,0}+\sum_{n_1=1}^N h_{n_1,0}e^{j\theta'_{n_1}}$ and by the fact that the angle between them is bounded above by $\pi/K$ according to \eqref{bound:4}. We further bound the $|\xi_{n_2}|$ as follows:
\begin{align}
|\xi_{n_2}| &= \left|h_{0,n_2}+\sum_{n_1=1}^N h_{n_1,n_2}e^{j\theta'_{n_1}}\right|\notag\\
&= \left|\sum_{n_1=1}^N h_{n_1,n_2}e^{j\theta'_{n_1}}\right|+o(N)\notag\\
&= \left|\sum_{n_1=1}^N h_{n_1,n_2}e^{j\hat\theta^\star_{n_1}}e^{j(\theta'_{n_1}-\hat\theta^\star_{n_1})}\right|+o(N)\notag\\
&\overset{(a)}{=} \left|\sum_{n_1=1}^N u^{(1)}_{n_1}u^{(2)}_{n_2}e^{j(\eta-\angle u^{(1)}_{n_1})}e^{j(\theta'_{n_1}-\hat\theta^\star_{n_1})}\right|+o(N)\notag\\
&= |u^{(2)}_{n_2}|\cdot\left|\sum_{n_1=1}^N |u^{(1)}_{n_1}|e^{(\theta'_{n_1}-\hat\theta^\star_{n_1})}\right|+o(N)\notag\\
&\overset{(b)}{\ge} |u^{(2)}_{n_2}|\cdot\cos\left(\gamma+\frac{\pi}{K}\right)\cdot\sum_{n_1=1}^N |u^{(1)}_{n_1}|+o(N),\label{key_step:2}
\end{align}
where step $(a)$ uses the shorthand
\begin{equation}
\eta\triangleq\angle\left(h_{0,0}+\sum^N_{n_2=1}h_{0,n_2}\right)-\angle\left(\sum^N_{n_2=1}u^{(2)}_{n_2}\right).
\end{equation}
The key step in the above derivation is that $\eta$ is independent of $n_1$ and hence can be omitted after step $(a)$. Moreover, step $(a)$ follows by the rank-one assumption in \eqref{eqn 2 decompose} and the definition of $\hat\theta^\star_{n_1}$ in \eqref{eqn 2 theta^}, and step $(b)$ follows by the bound between $\theta'_{n_1}$ and $\hat\theta^\star_{n_1}$ in \eqref{bound:3}. Finally, combining \eqref{key_step:1} and \eqref{key_step:2} gives
\begin{align}
|g(\theta'_1,\theta'_2)|^2 &= \Omega\Bigg(\Bigg(\sum_{n_2=1}^N\sum_{n_1=1}^N |u^{(2)}_{n_2}||u^{(1)}_{n_1}|\Bigg)^2\Bigg)\notag\\
&=\delta^2_1\delta^2_2\Omega(N^4).
\end{align}
The proof of Theorem \ref{coro boost 2} is then completed.
\end{IEEEproof}
\subsection{Comments on Theorem \ref{coro boost 2}}
\label{subsec:prev_work}
Assuming CSI available, the previous work \cite{Han_double_IRS_beamforming_power_scaling} provides a different set of conditions for its proposed algorithm to achieve the SNR boost of $\Theta(N^4)$ in a double-IRS system:
\begin{enumerate}[C'1.]
\item the channels between the two IRSs are
LoS, same as C1 in Theorem \ref{coro boost 2};
\item $K\rightarrow\infty$, namely the continuous beamforming;
\item the direct channel and the one-hop reflected channels are all zeros, i.e., $h_{0,0}=h_{n_1,0}=h_{0,n_2}=0,\forall (n_1,n_2)$.
\end{enumerate}
It is easy to see that C'2 is a special case of C2 and that C'3 is a special case of C3. Thus, Theorem \ref{coro boost 2} encompasses the exsiting result in \cite{Han_double_IRS_beamforming_power_scaling} as a special case.
As the final part of this section, we illustrate through some concrete examples why the proposed conditions C1--C3 are critical to the $\Theta(N^4)$ boost for a double-IRS system.
\begin{example}[Why is condition C1 needed?]
Assume that $K=4$ and $N$ is an odd number. For any $n_1$ and $n_2$, assume that $h_{0,0}=h_{n_1,0}=h_{0,n_2}=0$, $h_{n_1,n_2}=\beta e^{j(n_1+n_2)\pi}$ if $n_1\neq n_2$, and $h_{n_1,n_2}=\beta e^{j(n_1+n_2)\pi}+2\beta$ if $n_1=n_2$, where $\beta>0$ is a positive constant. This channel setting satisfies C2 and C3 but violates C1 since $h_{n_1,n_2}h_{n_2,n_1}\neq h_{n_1,n_1}h_{n_2,n_2}$ for $n_1\neq n_2$.
It can be shown that the alternating CSM method yields $\theta'_{n_1}=\theta'_{n_2}=0$ for every $n_1$ and $n_2$ in this case. As a result, $|\xi_{n_2}|=O(1)$ and thus $|g(\theta'_{n_1},\theta'_{n_2})|^2$ is $O(N^2)$ according to the first line of \eqref{key_step:1}, so the SNR boost is at most quadratic.
Instead, if we let $h_{n_1,n_2}=\beta e^{j(n_1+n_2)\pi}$ for any $(n_1,n_2)$ in this example, then C1 is satisfied with $u^{(1)}_{n_1}=\sqrt{\beta}e^{jn_1\pi}$ and $u^{(2)}_{n_2}=\sqrt{\beta}e^{jn_2\pi}$. We have $\theta'_{n_1}=0$ if $n_1$ is odd and $\theta'_{n_1}=\pi$ otherwise, and $\theta'_{n_2}=0$ if $n_2$ is odd and $\theta'_{n_2}=\pi$ otherwise. Substituting the above $(\theta'_{n_1},\theta'_{n_2})$ in \eqref{g L=2} gives a $\Theta(N^4)$ boost.
\end{example}
\begin{example}[Why is condition C2 needed?]
Assume that $K=2$ and $N$ is an odd number. For any $n_1$ and $n_2$, assume that $h_{0,0}=h_{0,n_2}=0$, $u^{(1)}_{n_1}=\sqrt{\beta}e^{j(n_1+\frac{1}{2})\pi}$, and $u^{(2)}_{n_2}=\sqrt{\beta}e^{jn_2\pi}$, where $\beta>0$ is a positive constant. Moreover, let $h_{n_1,0}=\frac{1}{3}\beta e^{j\frac{\pi}{4}}$ for odd $n_1$ and let $h_{n_1,0}=\frac13\sqrt{\beta}e^{-j\frac{\pi}{4}}$ for even $n_1$. Notice that the above setting satisfies all the conditions in Theorem \ref{prop:CSM} except C2. We have $\theta'_{n_1}=0$ for all $n_1$, and $\theta'_{n_2}=0$ if $n_2$ is odd and $\theta'_{n_2}=\pi$ otherwise. The resulting SNR boost is $\Theta(N^2)$.
In contrast, if $K$ is raised to $4$ in this example, then we have $\theta'_{n_1}=-\frac{\pi}{2}$ if $n_1$ is odd and $\theta'_{n_1}=\frac{\pi}{2}$ otherwise, $\theta'_{n_2}=0$ if $n_2$ is odd and $\theta'_{n_2}=\pi$ otherwise. Substituting the above $(\theta'_{n_1},\theta'_{n_2})$ in \eqref{g L=2} gives a $\Theta(N^4)$ boost.
\end{example}
\begin{example}[Why is condition C3 needed?]
Assume that $K=4$ and $N$ is an odd number. For any $n_1$ and $n_2$, assume that $h_{0,0}=h_{0,n_2}=0$, $h_{n_1,0}=2\beta e^{j\frac{\pi}{2}}$, $u^{(1)}_{n_1}=\sqrt{\beta}e^{jn_1\pi}$ and $u^{(2)}_{n_2}=\sqrt{\beta}e^{jn_2\pi}$,where $\beta>0$ is a positive constant.
Observe that $|h_{n_1,0}|=2|\sum_{n_2=1}^N h_{n_1,n_2}|$ under the above setting, so C3 in Theorem \ref{prop:CSM} does not hold here (but C1 and C2 are satisfied). In this case $\theta'_{n_1}=-\frac{\pi}{2}$ for every $n_1$, and $\theta'_{n_2}=\frac{\pi}{2}$ if $n_2$ is odd and $\theta'_{n_2}=-\frac{\pi}{2}$ otherwise. As a result, the SNR boost is $\Theta(N^2)$ in this case.
In contrast, if $u^{(1)}_{n_1}=u^{(2)}_{n_2}=\sqrt{\beta}$ for every pair $(n_1,n_2)$ in this example, then C3 can be satisfied by letting $\gamma=\frac{\pi}{8}$ when $N$ is sufficiently large. In this case, we have $\theta'_{n_1}=0$ for every $n_1$, and $\theta'_{n_2}=\frac{\pi}{2}$ for every $n_2$. As a result, the SNR boost of $\Theta(N^4)$ is achieved.
\end{example}
\section{General $L$-IRS Case}
\label{sec csm}
\begin{algorithm}[t]
\caption{Blind Beamforming for $L$ IRSs}
\label{alg:SCSM}
\begin{algorithmic}[1]
\State{Initialize all the $\theta_{n_\ell}$'s to zero.}
\For{$\ell=1,\ldots,L$}
\State{Generate $T$ random samples $\{\theta^{(t)}_{n_\ell}|n_\ell=1,\ldots,N\}$.}
\For{$t=1,\ldots,T$}
\State{Measure the received signal power $|Y^{(t)}|^2$.}
\EndFor
\For{$n_\ell=1,\ldots,N$}
\For{$k=1,\ldots,K$}
\State{Compute the conditional sample mean in \eqref{cond_expectation}.}
\EndFor
\State Decide each $\theta_{n_{\ell}}$ for IRS $\ell$ according to \eqref{CSM}.
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
The CSM method can be further extended to more than two IRSs in a sequential fashion. The initial values of all the $\theta_{n_\ell}$'s are set to zero. We optimize one IRS at a time while holding the rest IRSs fixed. Algorithm \ref{alg:SCSM} summarizes this sequential CSM method.
Most importantly, the performance bound analysis in Theorem \ref{coro boost 2} can be carried over to the general $L\ge2$ IRSs, as stated in the theorem below.
\begin{theorem}
\label{Proposition:boost}
If an $L$-IRS system satisfies the following three conditions:
\begin{enumerate}[D1.]
\item there exist a set of values $\{u^{(\ell)}_{n_l}\in\mathbb C \,|\, n\in[1:N],\;\ell\in[1:L]\}$ such that each $L$-hop channel (which is related to every IRS) can be decomposed as
\begin{equation}
\label{decompose}
h_{n_1,\ldots,n_L}=\prod^L_{\ell=1}u^{(\ell)}_{n_\ell},
\end{equation}
where none of $(n_1,\ldots,n_L)$ equals zero;
\item the number of phase shift choices $K\ge2L-1$;
\item there exists a constant $\gamma\in[0,\frac{\pi}{2(L-1)}-\frac{\pi}{K})$ such that
\begin{multline}
\frac{\sum_{(n_1,\ldots,n_L)\in\mathcal A^{(\ell)}_{m}}\!|h_{n_1,\ldots,n_L}|}{\prod_{i>\ell}\!\big|\!\sum^N_{n_i=1}u^{(i)}_{n_i}\big|\cdot\prod_{i<\ell}\!\big[\!\sum^N_{n_i=1}|u^{(i)}_{n_i}|\cos(\gamma+\frac{\pi}{K})\big]}\\
\le |u^{(\ell)}_{m}|\cdot \sin\gamma\label{eqn sin gamma}
\end{multline}
for $\ell\in[1:L-1]$ and $m\in[1:N]$, where
\begin{equation}
\mathcal A^{(\ell)}_{m} \triangleq\left\{(n_1,\ldots,n_L)\Bigg|n_\ell=m,\prod^L_{\ell=1}n_\ell=0\right\},
\end{equation}
\end{enumerate}
then Algorithm \ref{alg:SCSM} yields an SNR boost of $N^{2L}$ as follows:
\begin{align}
\mathbb{E}\left[\frac{\textrm{SNR}}{\textrm{SNR}_0}\right]=\frac{\prod^L_{\ell=1}\delta^2_\ell}{|h_0|^2}\cdot \Theta(N^{2L}),\label{eqn snr boost}
\end{align}
where
\begin{align}
\delta_\ell\triangleq \frac{1}{N}\sum^N_{n_\ell=1}|u^{(\ell)}_{n_\ell}|,\;\;\forall \ell\in[1:L].\label{eqn delta}
\end{align}
\end{theorem}
To prove Theorem \ref{Proposition:boost}, we need the following lemma.
\begin{lemma}\label{lemma gamma+pi/K}
Let $\theta'_{n_\ell}$ be the decision of $\theta_{n_\ell}$ by Algorithm \ref{alg:SCSM}, and generalize the definition of $\hat\theta^\star_{n_1}$ in \eqref{eqn 2 theta^} as
\begin{multline}
\label{L-IRS:hat_theta_star}
\hat\theta^\star_{n_\ell}\triangleq\angle\left(\sum_{(m_1,\ldots,m_L)\in\mathcal D^{(\ell)}_{0}}h_{m_1,\ldots,m_L}e^{j\sum^{\ell-1}_{i=1}\theta'_{m_i}}\right)-\angle u^{(\ell)}_{n_\ell}\\
-\angle\left[\sum_{(m_1,\ldots,m_L)\in\mathcal E^{(\ell)}_{n_\ell}}\left(e^{j\sum^{\ell-1}_{i=1}\theta'_{m_i}}\prod_{i\neq \ell}u^{(i)}_{m_i}\right)\right],
\end{multline}
where
\begin{equation}
\mathcal D^{(\ell)}_{m} = \left\{(m_1,\ldots,m_L)|m_\ell=m\right\}
\end{equation}
and
\begin{equation}
\mathcal E^{(\ell)}_{m} = \Bigg\{(m_1,\ldots,m_L)\Bigg|m_\ell=m,\prod_{i\ne\ell}m_{i}\ne0\Bigg\}.
\end{equation}
For any RE $n_\ell$, we have
\begin{equation}
\label{theta_ell:inequality}
|\hat\theta^\star_{n_\ell}-\theta'_{n_\ell}|\le\gamma+\frac{\pi}{K}
\end{equation}
given the constant $\gamma$ as defined in the condition D3.
\end{lemma}
\begin{IEEEproof}
See the appendix.
\end{IEEEproof}
Equipped with the inequality in \eqref{theta_ell:inequality}, we are now ready to prove Theorem \ref{prop:CSM}. The effective channel from the transmitter to the receiver is written as a function $g:\Phi_K^L\mapsto\mathbb C$ of the beamforming decision $(\theta_{n_1},\ldots,\theta_{n_L})$, i.e.,
\begin{equation}
g(\theta_{n_1},\ldots,\theta_{n_L}) =\!\sum_{(n_1,\ldots,n_L)\in[0:N]^L}\!h_{n_1,\ldots,n_L}e^{j\sum^L_{\ell=1}\theta_{n_{\ell}}}.\label{g}
\end{equation}
To establish an SNR boost of $\Theta(N^{2L})$, it suffices to show that $\mathbb E[|g|^2]=\prod^L_{\ell=1}\delta^2_{\ell}\cdot\Theta(N^{2L})$. Again, the converse $\mathbb E[|g|^2]=\prod^L_{\ell=1}\delta^2_{\ell}\cdot O(N^{2L})$ is evident, so the rest of this section focuses on the achievability $\mathbb E[|g|^2]=\prod^L_{\ell=1}\delta^2_{\ell}\cdot\Omega(N^{2L})$.
\begin{figure*}[t]
\centering
\includegraphics[width=12cm]{fig/environment_NLOS.eps
\caption{Indoor field test with two IRSs deployed in a hallway inside an office building.}
\label{fig:indoor}
\end{figure*}
Let us first recall how $\theta_{n_\ell}$ is decided in Algorithm \ref{alg:SCSM}. We denote by $\mathcal D^{(\ell)}_{0}$ the set of channels not related to any RE of IRS $\ell$.
When optimizing $\theta_{n_\ell}$, all the channels in $\mathcal D^{(\ell)}_{0}$ are treated as direct channels, while the rest channels are treated as reflected channels. Recall also that all those IRSs $i>\ell$ have not yet been configured when optimizing $\theta_{n_\ell}$, so we have $\theta_{n_i}=0$ for $i\in[\ell+1:L]$. The continuous solution of $\theta_{n_\ell}$ is then given by
\begin{multline}
\theta^\star_{n_\ell} =\angle\Bigg(\sum_{(m_1,\ldots,m_L)\in\mathcal D^{(\ell)}_0}h_{m_1,\ldots,m_L}e^{j\sum^{\ell-1}_{i=1}\theta'_{m_i}}\Bigg)\\
-\angle\Bigg(\sum_{(m_1,\ldots,m_L)\in\mathcal D^{(\ell)}_{n_\ell}}h_{m_1,\ldots,m_L}e^{j\sum^{\ell-1}_{i=1}\theta'_{m_i}}\!\Bigg).\label{eqn theta star}
\end{multline}
Again, as shown in Proposition \ref{prop:CSM}, Algorithm \ref{alg:SCSM} would round the above continuous solution to the discrete set $\Phi_K$:
\begin{equation}
\theta'_{n_\ell}=\arg\min_{\theta\in{\Phi_K}}|\theta-\theta^\star_{n_\ell}|
\end{equation}
and hence
\begin{equation}
|\theta'_{n_\ell}-\theta^\star_{n_\ell}|\le \frac{\pi}{K}.\label{eqn theta' star}
\end{equation}
We then bound the channel strength as follows:
\begin{align}
&|g(\theta'_{n_1},\ldots,\theta'_{n_L})|^2\notag\\
&=\Bigg|\sum_{(n_1,\ldots,n_L)\in[0:N]^L}h_{n_1,\ldots,n_L}e^{j\sum_{i=1}^{L}\theta'_{n_i}}\Bigg|^2\notag\\
&= \Bigg|\sum^N_{n_L=0}e^{j\theta'_{n_L}}\Bigg(\sum_{(n_1,\ldots,n_{L-1})}h_{n_1,\ldots,n_L}e^{j\sum_{i=1}^{L-1}\theta'_{n_i}}\Bigg)\Bigg|^2\notag\\
&\overset{(a)}{\ge} \left[\sum^N_{n_L=1}\!\cos(\theta'_{n_L}-\theta^\star_{n_L}\!)\left|\sum_{(n_1,\ldots,n_{L-1})}\!\! h_{n_1,\ldots,n_L}e^{j\sum_{i=1}^{L-1}\theta'_{n_i}}\right|\right]^2\notag\\
&\overset{(b)}{\ge} \left[\sum^N_{n_L=1}\cos\left(\frac{\pi}{K}\right)\left|\sum_{(n_1,\ldots,n_{L-1})}h_{n_1,\ldots,n_L}e^{j\sum_{i=1}^{L-1}\theta'_{n_i}}\right|\right]^2\label{eqn g2 1 1},
\end{align}
where step $(a)$ follows by only considering the projection of every $\sum_{(n_1,\ldots,n_{L-1})}h_{n_1,\ldots,n_L}e^{j\sum_{i=1}^{L-1}\theta'_{n_i}}$ with $n_L\ne0$ onto that with $n_L=0$, and step $(b)$ follows by the inequality in \eqref{eqn theta' star} directly.
For a fixed $n_L\ne0$, we can further bound the sum component in \eqref{eqn g2 1 1} as follows:
\begin{align}
&\left|\sum_{(n_1,\ldots,n_{L-1})\in[0:N]^{L-1}}h_{n_1,\ldots,n_L}e^{j\sum_{i=1}^{L-1}\theta'_{n_i}}\right|-o(N^{L-1})\notag\\
&\overset{(a)}{=}\left|\sum_{(n_1,\ldots,n_{L})\in\mathcal E^{(L)}_{n_L}}h_{n_1,\ldots,n_L}e^{j\sum_{i=1}^{L-1}\theta'_{n_i}}\right|\notag\\
&=\left|\sum_{(n_1,\ldots,n_{L})\in\mathcal E^{(L)}_{n_L}}\left(u^{(L)}_{n_L}\prod^{L-1}_{\ell=1}u^{(\ell)}_{n_\ell}e^{j\theta'_{n_\ell}}\right)\right|\notag\\
&\overset{(b)}{=}\big|u^{(L)}_{n_L}\big|\cdot\left|\sum_{(n_1,\ldots,n_{L})\in\mathcal E^{(L)}_{n_L}}\prod^{L-1}_{\ell=1}\big|u^{(\ell)}_{n_\ell}\big|e^{j(\theta'_{n_\ell}-\hat\theta^\star_{n_\ell})}\right|\notag\\
&\overset{(c)}{\ge}\big|u^{(L)}_{n_L}\big|\sum_{(n_1,\ldots,n_{L})\in\mathcal E^{(L)}_{n_L}}\cos\left[(L-1)\bigg(\gamma+\frac{\pi}{K}\bigg)\right]\prod^{L-1}_{\ell=1}\big|u^{(\ell)}_{n_\ell}\big|\notag\\
&=|u^{(L)}_{n_L}|N^{L-1}\cos\left[(L-1)\bigg(\gamma+\frac{\pi}{K}\bigg)\right]\prod^{L-1}_{i=1}\delta_i\label{eqn each term 6},
\end{align}
where step $(a)$ follows since the number of $(n_1,\ldots,n_{L-1})$'s with at least one $n_i=0$ equals $\sum^{L-2}_{\ell=0}{L-1 \choose \ell}N^\ell=o(N^{L-1})$, step $(b)$ follows by \eqref{L-IRS:hat_theta_star}, and step $(c)$ follows by \eqref{theta_ell:inequality}.
Substituting the lower bound \eqref{eqn each term 6} in \eqref{eqn g2 1 1}, we obtain
\begin{equation}
|g(\theta'_{n_1},\ldots,\theta'_{n_L})|^2=\left(\prod^L_{\ell=1}\delta^2_\ell\right)\cdot\Omega(N^{2L}).
\end{equation}
Furthermore, combining the above result with the evident fact that $\mathbb E[|g|^2]=\prod^L_{\ell=1}\delta^2_{\ell}\cdot O(N^{2L})$ leads us to the boost $\Theta(N^{2L})$. The proof of Theorem \ref{Proposition:boost} is thus completed.
\begin{remark}
Theorem \ref{Proposition:boost} implies that only one round of configuration (i.e., every IRS is optimized one time regardless of $L$) suffices to attain an SNR boost of $\Theta(N^{2L})$. This is of practical significance when the IRSs are extensively deployed in the network.
\end{remark}
\section{Numerical Results}
\label{sec test}
\begin{figure}[t]
\centering
\vspace{-2em}
\includegraphics[width=8.0cm]{fig/locations_NLOS.eps
\vspace{-1em}
\caption{Layout drawing of the indoor field test. The two IRSs are placed in two corners for most methods, but are merged into a single larger IRS placed in the middle for ``Physical Single-IRS'' as indicated by the dashed lines.}
\label{fig:indoor_layout}
\end{figure}
\subsection{Field Tests}
\label{subsec:field_tests}
Throughout our field tests, the transmit power is fixed at $-5$ dBm and the carrier frequency is $2.6$ GHz. The following three IRSs are used:
\begin{itemize}
\item IRS 1 with 294 REs and 2 phase shift choices $\{0,\pi\}$ for each RE, i.e., $N=294$ and $K=2$;
\item IRS 2 also with $N=294$ and $K=2$;
\item IRS 3 with $N=64$ and $K=4$.
\end{itemize}
Notice that we do not always assume that all the IRSs have the same values of $N$ and $K$ as in the theoretical model in Section \ref{sec:model}. The following methods are compared:
\begin{itemize}
\item \emph{Without IRS:} IRS is not used.
\item \emph{Zero Phase Shifts:} Fix all phase shifts to be zero.
\item \emph{Random Beamforming:}
Try out $L\times1000$ random samples of phase shift vectors and choose the best.
\item \emph{Virtual Single-IRS:} Ignore the multi-hop channels and treat multiple IRSs as a single one; optimize phase shifts by the method in \cite{blind_beamforming_twc} with $L\times1000$ random samples.
\item \emph{Physical Single-IRS:} Put multiple IRSs together at the same position to form a single larger IRS; optimize phase shifts by the method in \cite{blind_beamforming_twc} with $L\times1000$ random samples.
\item \emph{Proposed Blind Beamforming:} Coordinate multiple IRSs by Algorithm \ref{alg:SCSM} that uses $1000$ random samples per IRS.
\end{itemize}
The SNR boost is evaluated by taking ``without IRS'' as baseline. We consider the following two transmission scenarios:
\begin{itemize}
\item \emph{Indoor Environment:} Deploy IRS 1 and IRS 2 in a U-shaped hallway inside an office building as shown in Fig.~\ref{fig:indoor}. The testbed layout is specified in Fig.~\ref{fig:indoor_layout}. The transmission is blocked by the walls.
\item \emph{Outdoor Environment:} Deploy three IRSs alongside an open caf\'{e} as shown in Fig.~\ref{fig:outdoor_environment}. The testbed layout is specified in Fig.~\ref{fig:outdoor_locations}. The transmission is occasionally blocked by the crowd and also suffers interference which is treated as noise.
\end{itemize}
\begin{figure*}[t]
\centering
\includegraphics[width=13cm]{fig/outdoor_environment.eps
\caption{Outdoor field test with three IRSs deployed alongside an open caf\'{e}.}
\label{fig:outdoor_environment}
\end{figure*}
\begin{figure}[t]
\centering
\vspace{-2em}
\includegraphics[width=8.0cm]{fig/locations_outdoor.eps
\vspace{-1em}
\caption{Layout drawing of outdoor field test. In particular, for Physical Single-IRS, we move IRS 1 and IRS 2 to the positions indicated by the dashed lines.}
\label{fig:outdoor_locations}
\end{figure}
TABLE \ref{tab:SNR_boost} summarizes the SNR boost performance of the different methods. As shown in the row of Zero Phase Shifts, placing IRSs in the environment (either indoor or outdoor) can already increase SNR by more than 2 dB even without any optimization. Then a simple heuristic optimization method such as Random Beamforming can reap a higher SNR gain. Observe also that Virtual Single-IRS outperforms the above methods significantly, e.g., it improves upon Random Beamforming by around 7 dB for the indoor case.
In contrast, the proposed Blind Beamforming enhances SNR further, e.g., its SNR boost is about 5 dB higher than that of Virtual Single-IRS for the indoor case, and about 3 dB higher for the outdoor case. This further gain is due to the capability of Blind Beamforming to take those multi-hop reflections into account. For this reason, the advantage of Blind Beamforming over Virtual Single-IRS is greater for the indoor case in which the multi-hop reflections play a key role. Another interesting fact from TABLE \ref{tab:SNR_boost} is that Physical Single-IRS is much worse than many the other methods especially in the indoor environment. Although its phase shifts have been carefully optimized by the method in \cite{blind_beamforming_twc}, its performance is still limited by the deficiency of multi-hop reflections.
\begin{table}[t]
\small
\renewcommand{\arraystretch}{1.3}
\centering
\caption{\small SNR Boosts Achieved by the Different Methods}
\begin{tabular}{lrrr}
\firsthline
& \multicolumn{2}{c}{SNR Boost (dB)} \\
\cline{2-3}
Method & Indoor & Outdoor \\
\hline
Zero Phase Shifts & 2.74 & 2.91 \\
Random Beamforming & 5.33 & 8.48 \\
Virtual Single-IRS & 12.07 & 10.80 \\
Physical Single-IRS & 3.31 & 7.06 \\
Blind Beamforming & 17.08 & 14.09 \\
\lasthline
\end{tabular}
\label{tab:SNR_boost}
\end{table}
\subsection{Simulation Tests}
We now validate the performance of the proposed blind beamforming algorithm in simulations which can admit many more REs at each IRS and many more IRSs. The channels are generated as follows. We refer to the transmitter as node $0$, a total of $L$ IRSs as nodes $1$ through $L$, and the receiver as node $L+1$. We denote by $d_{ij}$ (in meters) the distance between node $i$ and node $j$. If there is LoS propagation between node $i$ and node $j$ then the corresponding pathloss is computed as
\begin{align}
\text{PL}_{ij}=10^{-(30+22\log_{10} (d_{ij}))/20}.
\end{align}
Otherwise, i.e., when the channel between node $i$ and node $j$ is non-line-of-sight (NLoS), the pathloss is computed as
\begin{align}
\text{PL}_{ij}=10^{-(32.6+36.7\log_{10} (d_{ij}))/20}.
\end{align}
Following \cite{multi_IRS_Mei,Multi_IRS_MIMO_Mei}, we assume that REs are arranged as a uniform linear array with spacing $\xi=0.03$ meters at each IRS. We let the signal wavelength be $\lambda=0.06$ meters. Moreover, denote by $\vartheta_{i,j}$ the angle of departure (AoD) from node $i$ to node $j$, and $\psi_{i,j}$ the angle of arrival (AoA) from node $j$ to node $i$. For the LoS case, the channel from the transmitter to RE $n_\ell$ is given by
\begin{align}
g_{0,n_\ell}=\sqrt{ \text{PL}_{0,\ell}}\cdot e^{-j\frac{2\pi}{\lambda}d_{0,\ell}} \cdot e^{-j\frac{2\pi}{\lambda}\xi(n_\ell-1)\cos \psi_{\ell,0}},
\end{align}
the channel from RE $n_\ell$ to the receiver is given by
\begin{align}
&g_{n_\ell,L+1}=\sqrt{ \text{PL}_{\ell,L+1}}\cdot e^{-j\frac{2\pi}{\lambda}d_{\ell,L+1}}\cdot e^{-j\frac{2\pi}{\lambda}\xi(n_\ell-1)\cos \vartheta_{\ell,L+1}},
\end{align}
and the channel from RE $n_\ell$ to RE $n_{\ell'}$, $\ell\ne\ell'$, is given by
\begin{multline}
g_{n_\ell,n_{\ell'}}=\sqrt{ \text{PL}_{\ell,{\ell'}}}\cdot e^{-j\frac{2\pi}{\lambda}d_{\ell,{\ell'}}} \cdot e^{-j\frac{2\pi}{\lambda}\xi(n_\ell-1)\cos \vartheta_{\ell,{\ell'}}}\\
\cdot e^{-j\frac{2\pi}{\lambda}\xi(n_{\ell'}-1)\cos \psi_{\ell',\ell}}.
\end{multline}
For the NLoS case, we generate channels as
\begin{align}
g_{0,n_\ell} &= \sqrt{ \text{PL}_{0,\ell}}\cdot \zeta_{0,n_\ell},\\
g_{n_\ell,L+1} &= \sqrt{ \text{PL}_{\ell,L+1}}\cdot\zeta_{\ell,L+1},\\
g_{n_\ell,n_{\ell'}} &= \sqrt{ \text{PL}_{\ell,{\ell'}}}\cdot\zeta_{n_\ell,n_{\ell'}},
\end{align}
where $\zeta_{0,n_\ell},\zeta_{\ell,L+1},\zeta_{n_\ell,n_{\ell'}}$ are drawn i.i.d. from the complex Gaussian distribution $\mathcal{CN}(0,1)$. For both LoS and NLoS cases, each multi-hop channel $h_{n_1,\ldots,n_L}$ can be obtained by multiplying together a subset of channels in $\{g_{0,n_\ell},g_{n_\ell,L+1},g_{n_\ell,n_{\ell'}}\}$.
Moreover, the transmit power equals $30$ dBm, the background noise power equals $-98$ dBm, and $K$ is fixed to be $4$ throughout the simulation tests, i.e., $\Phi_K = \left\{ 0 , \frac{\pi}{2}, \pi, \frac{3 \pi}{2}\right\}$.
\begin{figure}[t]
\centering
\includegraphics[width=8cm]{fig/double_IRS_layout.eps
\caption{The double-IRS system considered in our simulations. The position coordinates are in meters.}
\label{fig:double-IRS}
\end{figure}
We begin with a double-IRS system as shown in Fig.~\ref{fig:double-IRS}. Assume that the two IRSs have $100$ REs each and that $T=1000$ random samples are taken for each IRS. Assume also that the transmitter-to-IRS 1 channels, the IRS 1-to-IRS 2 channels, and the IRS 2-to-receiver channels are LoS while the rest channels are NLoS. In particular, we assume that the placements of the two IRSs render AoD and AoA equal so that $\vartheta_{n_1,n_2}=\vartheta_{n_2,L+1}=\psi_{0,n_1}=\psi_{n_1,n_2}\approx 5.6^\circ$. We compare the proposed Algorithm \ref{alg:SCSM} with a channel estimation based method---which first estimates channels based on random samples by the DFT method \cite{channel_est_DFT} and then performs CPP in \eqref{CPP} for the two IRSs sequentially. Fig.~\ref{fig:comparison} shows the cumulative distributions of the SNR boosts of the two methods. It can be seen that the proposed blind beamforming method outperforms the channel estimation based benchmark significantly at most percentiles. For instance, at the 50th percentile, blind beamforming improves upon the benchmark by approximately 3 dB. Moreover, we plot how the SNR boost of blind beamforming grows with $N$ in Fig.~\ref{fig:boostvsn}. The growth curve is approximately quartic in $N$ and thus agrees with our analysis in Theorem \ref{coro boost 2}.
\begin{figure}[t]
\centering
\includegraphics[width=9.5cm]{fig/comparison.eps
\caption{Cumulative distribution of SNR boosts in a double-IRS system.}
\label{fig:comparison}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=9.5cm]{fig/SNR_Boost_vs_N.eps
\caption{SNR Boost achieved by blind beamforming vs. number of REs.}
\label{fig:boostvsn}
\end{figure}
\begin{figure}[t]
\centering
\hspace*{-1.2em}
\includegraphics[width=9.5cm]{fig/routing_N=600.eps
\caption{The IRS routing when $N=600$.}
\label{fig:routing_n=600}
\end{figure}
\begin{figure}[t]
\centering
\hspace*{-1.2em}
\includegraphics[width=9.5cm]{fig/routing_N=1200.eps
\caption{The IRS routing when $N=1200$.}
\label{fig:routing_n=1200}
\end{figure}
We now add more IRSs in the system. Consider a $100\times100$ m$^2$ 2D square area as shown in Fig.~\ref{fig:routing_n=600}. The transmitter is located at $(5,5)$ and the receiver is located at $(95,95)$. There are $8$ possible positions to deploy IRSs: $(19.5,77.3)$, $(25.8,15.8)$, $(29.9,11.5)$, $(34.1,60.7)$, $(43.8,15.0)$, $(61.5,84.1)$, $(71.8,35.5)$ and $(73.6,76.2)$, all in meters. Thus, including the transmitter and receiver, there are a total of $10$ nodes in our case. Following \cite{multi_IRS_Mei}, the propagation status between any nodes are randomly set to LoS with $60\%$ probability and to NLoS with $40\%$ probability. The realization of the propagation statuses in our case can be expressed in an adjacency matrix as
\begin{align*}
A=
\begin{bmatrix}
0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 \\
0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\
0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0
\end{bmatrix},
\end{align*}
where the entry $A_{ij}$ equals $1$ if the channel between node $i$ and node $j$ is LoS, and $0$ otherwise.
As suggested in \cite{multi_IRS_Mei}, only a subset of the possible positions are selected for the IRS placement, namely the IRS routing---which depends on the value of $N$. For instance, Fig.~\ref{fig:routing_n=600} shows the IRS placement when $N=600$, and Fig.~\ref{fig:routing_n=1200} shows the IRS placement when $N=1200$, both based on the graph theoretical algorithm in \cite{multi_IRS_Mei}. For comparison purpose, we consider the following two beamforming methods as benchmarks:
\begin{itemize}
\item \textit{CPP with Perfect CSI}: Assume that the precise channel information is already known. Then perform CPP across the IRSs sequentially.
\item \textit{Mei-Zhang Method with Perfect CSI} \cite{multi_IRS_Mei}: First perform the continuous beamforming algorithm in \cite{multi_IRS_Mei} and then round the solution to the discrete set $\left\{ 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \right\}$. This method assumes that perfect CSI is available.
\end{itemize}
Notice that the above two competitor methods both require perfect CSI. However, to the best of our knowledge, there is yet no effective way of estimating channels for more than 2 IRSs in the existing literature.
Fig.~\ref{fig:comparison_beamforming_routing} compares the proposed Algorithm \ref{alg:SCSM} with the benchmarks under the different settings of $N$. For Algorithm \ref{alg:SCSM}, we let $T=20\times N$. It can be seen that the proposed algorithm is fairly close to CPP. Actually, the proposed algorithm would approach CPP when $T\rightarrow\infty$; the former can be thought of as a practical implementation of the latter. Mei-Zhang method is about 2 dB higher than the proposed algorithm when $N=400$. But when $N$ is raised to $600$, the proposed algorithm starts to overtake, and their gap becomes larger when $N$ is further increased. Again, not requiring CSI is a distinct advantage of the proposed algorithm as compared to these benchmarks.
\begin{figure}[t]
\centering
\includegraphics[width=9.5cm]{fig/comparison_routing.eps
\caption{SNR Boost vs. the number of REs of each IRS.}
\label{fig:comparison_beamforming_routing}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
This work proposes a statistical approach to the multi-IRS beamforming problem in the absence of channel information. For a general $L$-IRS assisted wireless transmission, we show that the proposed blind beamforming algorithm guarantees an SNR boost of $\Theta(N^{2L})$---which is the highest possible SNR boost obtained from $L$ IRSs, under some certain conditions. This blind beamforming strategy has two major advantages over the existing methods. First, it does not entail any channel estimation and yet can yield provable performance. Second, its optimality condition is far less strict than the existing one in \cite{multi_IRS_Mei}, e.g., those short reflected channels need not be zero for blind beamforming to reach the $\Theta(N^{2L})$ boost. Remarkably, as shown in the real-world experiments at $2.6$ GHz, blind beamforming for multiple IRSs increases SNR by over $17$ dB in the hallway of an office building, and by over $14$ dB near an open caf\'{e}. Moreover, simulations show that much higher gain can be reaped by blind beamforming when the IRSs become larger in size or when more IRSs are deployed.
|
1,116,691,499,319 | arxiv | \section{Introduction}
A recent remarkable trend in the physics community is its engagement
in interdisciplinary fields using physics-inspired techniques. Econophysics
is one such area in which financial markets are simulated by agent-based
models in much the same way as other many-body systems in statistical
physics.
The Minority Game (MG) \cite{Challet97} is an agent-based model based
on the insight that agents making minority decisions in markets can
take advantage of other agents. Due to its success in capturing the
profit-seeking behavior of agents, it became the progenitor of a family
of agent-based models \cite{Challet05}, which study various aspects
of market behavior, such as volatility \cite{Savit99}, noise \cite{Cavagna99},
market-clearing mechanisms \cite{Jefferies01}, and anticipative strategies
\cite{Andersen03}.
The market behavior depends on the way
the agents evaluate their strategies
when they make choices among them to take actions.
In early versions of the minority games, agents evaluate their strategies
using various {\it virtual points} or {\it scores}. Typical virtual
point updating rules, such as those in the original MG \cite{Challet97},
evaluate the buying and selling decisions at a time step, regardless
of the need to update the historical effects of the previous decisions.
In other models, one-step expectations of the agents are considered,
leading to the \$-game \cite{Andersen03}.
There are also market models with a mixture
of trend-following and fundamentalist agents \cite{Marsili01, Demartino03}
or markets with crossover regimes
dominated by trend-following and fundamentalist strategies
\cite{Lux99, Demartino04}.
As pointed out in \cite{Yeung08},
these models do not reflect the history-dependent
considerations of real market agents.
Improved versions of the minority games considered agents
using {\it virtual wealth}
to evaluate their strategies \cite{Challet08}. The Wealth Game (WG)
\cite{Yeung08} was introduced to overcome this deficiency. Agents
in WG evaluate their strategies by calculating their virtual wealth,
that is, the wealth (including cash and stocks) that the strategies
would bring were their recommendations completely adopted in history.
The most significant advantage of this evaluation method can be seen
when agents are allowed to make {\it holding} decisions (that is,
decisions to take no buying or selling actions). In WG, the virtual
wealth due to holding long (short) positions increases when stock
prices are rising (dropping). On the other hand, virtual point updates
in the original MG are neutral to holding positions.
Tests with real market and artificial data
confirm the versatility of wealth-based strategies \cite{Yeung08, Baek10}.
A consequence of using wealth-based strategies in WG is the emergence
of price cycles through the self-organization of the different types
of agents. This is both an important and interesting issue, since
it sheds light on the formation and disappearance of bubbles and crashes
in real financial markets. Giardina and Bouchaud considered a model
with bubbles and crashes in the price trend of the market \cite{Giardina03}.
The behavior was explained in terms of the interplay between the
trend-following
and fundamentalist behaviors of the agents, but the mechanism of the
disappearance of this periodic phase remains an open issue. In WG,
the roles of the trendsetters and fickle agents in sustaining the
price cycles were explained, and the disappearance of the periodic
phase was attributed to the failure of the trendsetters to gain wealth
from the fickle agents. (The trendsetters are synonymous to the
trend-followers in the literature, but are renamed trendsetters
to emphasize their role in initiating the bubbles and crashes.) However,
this picture assumes the presence of market makers, who manage to
fulfill buy and sell orders irrespective of the order imbalance.
This is not applicable to the stock market,
since in the absence of market makers, the market clearing mechanism requires
an exact matching of buy and sell orders.
Consequently, not all agents can have
their buying or selling orders fulfilled.
Thus, the unfulfilled buyers
(sellers) would repeat their bids (asks) step after step. This creates
a much stronger tendency for the price to go monotonically upwards
(downwards). The appearance of the periodic phase becomes questionable
and, if it exists, its mechanism of formation and disappearance may
not necessarily be the same.
In this paper, we study a simplified model of WG in the absence of
market makers, focusing on the mechanism creating and destroying the
cycles of bubbles and crashes. We will adopt a minimalist approach,
and consider the simple but essential elements that contribute to
the studied mechanism. It turns out that with trend-followers and
fundamentalists of memory size 2 being the two main groups of investors,
the price dynamics already exhibits many interesting features. (The
fundamentalists are further divided into optimistic and pessimistic
subgroups.) On one hand, these groups are inclusive enough to represent
the attitudes of most investors, and on the other hand, simple enough
to enable convenient analyses. Despite the simplification, we will
see that many important features and phase transitions in the original
WG with market makers are preserved. Rich econophysical implications
are revealed regardless of the simplifications.
This paper is outlined as follows. After introducing the model in
Sec. \ref{sec:Introduction}, we describe different attractor
behaviors in the space of price sensitivity and market impact in Sec.
\ref{sec:Phase-Diagram}. Analytical studies about the phase transitions,
including the cause of the transition and the precise location of
the phase boundary, are discussed in Sec. \ref{sec:The-Transient-Period}
to \ref{sec:The-Phase-Boundary}. In Sec. \ref{sec:f-dependence},
we study the dependence of the attractor behavior on the fraction
of trend-followers, accompanied by a concise analytical study about
the corresponding phase transition. Finally, the conclusion is drawn
in Sec. \ref{cha:Conclusion}.
\section{\label{sec:Introduction}The Model}
The Wealth Game \cite{Yeung08} consists
of $N$ agents playing in a single-commodity market. For convenience,
we will use the language of stock markets in the following discussions.
At each time step, the agents make decisions to buy, sell, or hold (no action)
stocks, based on the predictions of their best strategies. The decision
of agent $i$ at time $t$ is denoted as $a_{i}(t)=1,-1,0$, which
corresponds to buy, sell or hold respectively. A strategy takes the
the signs of previous $m$ historical price changes (represented by
a string of $\uparrow$ and $\downarrow$) as the input signal, and
the output signal is the advice on the trading action of the present
step. Table \ref{tab:typical strategy} shows the possible content
of a strategy for $m=2$. We require each usable strategy to have
at least one buying and one selling prediction, or else it is too
dull to be used. With this restriction, $s$ strategies are randomly
drawn to each agent.
\begin{table}
\caption{\label{tab:typical strategy}The content of a typical strategy for
$m=2$.}
\noindent \begin{centering}
\begin{tabular}{|c|c|}
\hline
Input signal & Output Signal (advice)\tabularnewline
\hline
\hline
$\uparrow\uparrow$ & Buy\tabularnewline
\hline
$\uparrow\downarrow$ & Sell\tabularnewline
\hline
$\downarrow\uparrow$ & Hold\tabularnewline
\hline
$\downarrow\downarrow$ & Sell\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\end{table}
The position of agent $i$ at time $t$ is given by
\begin{equation}
k_{i}(t)={\displaystyle \sum_{t'=0}^{t}}a_{i}(t'),
\label{eq:position}
\end{equation}
which records the number of stocks possessed by an agent. Short selling
is allowed, such that $k_{i}(t)$ can be negative. We assume that
each agent has limited assets, so the restriction $|k_{i}(t)|\leq K$
is applied, i.e. actions that further increase $|k_{i}(t)|$ to exceed
$K$ are ignored, and so $K$ denotes the maximum number of stocks
that an agent can buy or short sell. The market price evolves in response
to the market's excess demand $A(t)$, which is defined as\begin{equation}
A(t)=\sum_{i=1}^{N}a_{i}(t),\label{eq:excess demand}\end{equation}
and the price is updated by
\begin{equation}
P(t+1)=P(t)+\mbox{sgn}[A(t)]|A(t)|^{\gamma},
\label{eq:price update}\end{equation}
where $\gamma\in[0,1]$ is the market sensitivity controlling how
sensitively the price changes with the market excess demand. $\gamma=0$
corresponds to a step function \cite{Challet97, Savit99, Jefferies01}
while $\gamma=1$ corresponds to a
linear function \cite{Challet00, Marsili00, Challet00b, Heimel01}.
Suppose an agent would like to buy a stock at price
$P(t)$. Then she queues up in the market to wait for her turn of
a transaction. Depending on how long the queue is, the actual transaction
price $P_{\mathrm{T}}(t)$ may deviate from her desired price $P(t)$.
This is one of the examples showing how the market impact (i.e. the
collection of all the market factors imposed by agents' participation)
would influence the agents' trading activities \cite{Challet08}.
In this model, the transaction price is defined as
\begin{equation}
P_{\mathrm{T}}(t)=(1-\beta)P(t)+\beta P(t+1),
\label{eq:transaction price}\end{equation}
where $\beta\in[0,1]$ is the market impact. For convenience, we assume
that all the agents are affected to the same extent by market impact.
If $\beta=0$, the market impact is small so that the agent can immediately
trade with her most desired price. When $\beta=1$, the queue is so
long (and the market impact is so large) that the agent is actually
trading with $P(t+1)$, which may have already deviated considerably
from $P(t)$.
The wealth of an agent consists of two parts: cash in her hand and
the value of stocks she is holding. Agents' cash is updated by
\begin{equation}
c_{i}(t)=c_{i}(t-1)-a_{i}(t)P_{\mathrm{T}}(t),
\label{eq:cash update}\end{equation}
while the agents' wealth at the moment they just finish the transactions
at time $t$, is defined as
\begin{equation}
w_{i}(t)=c_{i}(t)+k_{i}(t)P_{\mathrm{T}}(t).
\label{eq:agent wealth}\end{equation}
Among the $s$ strategies, an agent only chooses one to follow at
each time step. The virtual position, cash and wealth of a strategy
will be calculated in the same way as an agent, by Eqs. (\ref{eq:position})
(a strategy is also restricted by $K$), (\ref{eq:cash update}) and
(\ref{eq:agent wealth}). Its virtual wealth evolves when its prediction
is applied in the market. The best strategy is then defined as the
one with the highest accumulated virtual wealth, which is to be adopted
by the agent. When a previously best strategy is outperformed, switching
strategies by agents occurs.
\subsection{Market without Market Makers}
The original Wealth Game implicitly assumes the participation of market
makers. This means that when there are more buyers than sellers, market
makers will provide stocks to the excess buyers, and when there are
more sellers than buyers, they will absorb the extra stocks. Withdrawing
the market makers from the game implies that the supply and demand
cannot be balanced. To achieve a balance, an apparent way is to randomly
pick some excess majority traders and ignore their orders. For the
sake of fairness, however, we assume that all the majority traders
reduce their orders such that all of them can only be partially satisfied
\cite{Giardina03, Caldarelli97, Slanina99}.
The mathematical modification to the original game is as follows.
The quotation of agent $i$ ($j$) who wants to buy (sell) is defined
as
\begin{equation}
q_{i}^{\mathrm{buy}}=\mathrm{min}(1,K-k_{i}),
\label{eq:buying quotation}\end{equation}
\begin{equation}
q_{j}^{\mathrm{sell}}=-\mathrm{min}(1,K+k_{j}).
\label{eq:selling quotation}\end{equation}
The quotation is the amount of stock an agent wants to trade. It is
defined this way since the agents can now buy (sell) a fraction of
their original units of stock, and the stocks held (short sold) by
each agent are still required to be bounded by the maximum position
$K$. We define the sum of buying (selling) quotations as
\begin{equation}
A_{\mathrm{buy}}=\sum_{i}q_{i}^{\mathrm{buy}},
\label{eq:buying sum}\end{equation}
\begin{equation}
A_{\mathrm{sell}}=\sum_{j}q_{j}^{\mathrm{sell}}.
\label{eq:selling sum}\end{equation}
The modification to the excess demand $A(t)$ for Eq. (\ref{eq:excess demand})
is
\begin{equation}
A(t)=\sum_{l=1}^{N}q_{l}(t)=A_{\mathrm{buy}}+A_{\mathrm{sell}}.
\label{eq:mod excess demand}\end{equation}
When $A(t)$ is positive, the position change of a buying (selling)
agent after each transaction is
\begin{equation}
\Delta k_{i}^{\mathrm{buy}}=\left(\frac{|A_{\mathrm{sell}}|}
{A_{\mathrm{buy}}}\right)q_{i},\label{eq:del k_buy}
\end{equation}
\begin{equation}
\Delta k_{j}^{\mathrm{sell}}=q_{j}.
\label{eq:del k_sell}\end{equation}
When $A(t)$ is negative,
\begin{equation}
\Delta k_{i}^{\mathrm{buy}}=q_{i},
\label{eq:del k_buy2}\end{equation}
\begin{equation}
\Delta k_{j}^{\mathrm{sell}}=\left(\frac{A_{\mathrm{buy}}}
{|A_{\mathrm{sell}}|}\right)q_{j}.\label{eq:del k_sell2}
\end{equation}
Based on the above modifications, one could easily verify that the
supply and demand can be balanced at any time step. Note that, now
the market price change is solely driven by agents' bid-ask actions,
regardless of whether transactions are really carried out afterwards.
This is the cause of so called unfulfilled orders \cite{Giardina03},
``dry quoting''. In this case, there is only one quoting group (e.g.
the bidding group) who cannot find their matching dealers. This essentially
resembles the circumstance of a market when the price fluctuation
is significant but the trading volume is negligible. It should also
be emphasized that the absence of market makers implies that the market
is zero-sum, which means the total wealth of all the agents is conserved,
as the gain of an agent must be accompanied by the loss of another
agent.
For the updating of the virtual positions of the strategies, one may
either use the original scheme of $\Delta k_{\xi}=\pm1$ subject to
the constraints of the maximum and minimum positions $\pm K$, or
use the modified scheme analogous to Eqs. (\ref{eq:buying quotation}),
(\ref{eq:selling quotation}), (\ref{eq:del k_buy}) - (\ref{eq:del k_sell2}).
In this paper we use the original scheme, and have checked that both
schemes yield qualitatively similar results.
\subsection{Cash Rule}
It was found that merely withdrawing the market makers from the game
only creates uninteresting market dynamics. Due to the imbalance between
supply and demand, majority traders can only be partially satisfied.
Time step after time step, they quote again and again, boosting (busting)
the price monotonically and creates an ever increasing (decreasing)
trend.
To get rid of this undesirable feature in our model, we need to take
into account the inability of the agents to order when the stock price
is too high or low. Hence we propose that the agents are forced to
cease their quotations if the following conditions are satisfied:\begin{equation}
c_{i}^{\mathrm{buy}}(t)-P(t)<0,\label{eq:cash rule1}\end{equation}
\begin{equation}
c_{j}^{\mathrm{sell}}(t)+P(t)<0,\label{eq:cash rule2}\end{equation}
where the superscripts \textsl{buy} and \textsl{sell} stand for
buyers and sellers respectively. Condition (\ref{eq:cash rule1})
is essentially saying that agents who have too little cash would not
bother to go for queuing if the price is too high. Condition (\ref{eq:cash rule2})
means that if the price is too low, agents who are not liquid enough
would not take the risk to borrow stocks.
\subsection{Strategies: Fast Trendsetters, Top and Bottom Triggers}
To facilitate analyses, the strategy space of the game is simplified
in the following ways. First, the outputs of a strategy are restricted
to buying and selling only. No holding actions are included. An agent
stops quoting only when she tries to place a buying (selling) quotation
but the restriction $|k_{i}|=K$ is reached, or when she is restricted
by the Cash Rule (i.e. conditions (\ref{eq:cash rule1})
or (\ref{eq:cash rule2})
are satisfied). The direct consequence is that the total number of
possible strategies becomes $2^{2^{m}}$ instead of $3^{2^{m}}$.
Second, we only study the special case where an agent has two strategies
($s=2$) and considers two historical price changes as the strategy
input signal ($m=2$). Hence there are $2^{2^{m}}=16$ strategies.
We further focus on those strategies with opposite decisions for inputs
$\uparrow\downarrow$ and $\downarrow\uparrow$. This reduces the
set of strategies to those listed in Table \ref{tab:F, T, B strategy content}
and their antistrategies. They are called fast trendsetters (F), top
trigger (T), bottom trigger (B) and slow trendsetters (S) respectively.
The meanings of these names will become clear in the next paragraphs.
Furthermore, studies in the original Wealth Game shows that S strategy
plays similar role as the F strategy in the formation of price cycles.
Hence, we restrict the strategy space, and only three strategies are
\textsl{evenly} assigned to the agents, namely, the F, T and B strategies.
We now consider the outputs of these three strategies in an artificial
trendy market, as shown in Fig. \ref{fig:artificial trendy market}.
\begin{table*}
\caption{\label{tab:F, T, B strategy content}The F, T, B and S strategies.}
\noindent \begin{centering}
\begin{tabular}{|c|c|c|c|}
\hline
F strategy & T strategy & B strategy & S strategy\tabularnewline
\hline
\hline
\begin{tabular}{c|c}
Input & Output\tabularnewline
\hline
\hline
$\uparrow\uparrow$ & Buy\tabularnewline
\hline
$\uparrow\downarrow$ & Sell\tabularnewline
\hline
$\downarrow\uparrow$ & Buy\tabularnewline
\hline
$\downarrow\downarrow$ & Sell\tabularnewline
\end{tabular} & \begin{tabular}{c|c}
Input & Output\tabularnewline
\hline
\hline
$\uparrow\uparrow$ & Buy\tabularnewline
\hline
$\uparrow\downarrow$ & Sell\tabularnewline
\hline
$\downarrow\uparrow$ & Buy\tabularnewline
\hline
$\downarrow\downarrow$ & Buy\tabularnewline
\end{tabular} & \begin{tabular}{c|c}
Input & Output\tabularnewline
\hline
\hline
$\uparrow\uparrow$ & Sell\tabularnewline
\hline
$\uparrow\downarrow$ & Sell\tabularnewline
\hline
$\downarrow\uparrow$ & Buy\tabularnewline
\hline
$\downarrow\downarrow$ & Sell\tabularnewline
\end{tabular} & \begin{tabular}{c|c}
Input & Output\tabularnewline
\hline
\hline
$\uparrow\uparrow$ & Buy\tabularnewline
\hline
$\uparrow\downarrow$ & Buy\tabularnewline
\hline
$\downarrow\uparrow$ & Sell\tabularnewline
\hline
$\downarrow\downarrow$ & Sell\tabularnewline
\end{tabular}\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\end{table*}
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{fig1.pdf}
\par\end{centering}
\caption{\label{fig:artificial trendy market}
The price series of an artificial
trendy market. Different strategies make different decisions.}
\end{figure}
A glance at the content of the F strategy suggests that it is a trend-believing
strategy. The F strategy advises to buy in a rising trend, sell at
the price peak, sell in a falling trend and buy at the price valley.
Once a sufficiently long price trend (either rising or falling) has
been established, the F strategy gains most wealth and hence would
be adopted by most agents. If this trend-believing strategy is adopted
by the majority, the price trends can be set up persistently. In the
literature, it is called the trend-follower
\cite{Lux99, Farmer99, Jefferies01, Marsili01, Andersen03,
Giardina03, Demartino03, Demartino04}.
Here, to highlight its role in perpetuating the price cycles, it is
called the ``trendsetter''. It is {}``fast'' as agents adopting
it react immediately to reversals in the price trend in a trendy market,
in contrast to the slow trendsetters who join the trendsetting bandwagon
one step slower, according to the outputs prescribed
in Table \ref{tab:F, T, B strategy content}
for inputs $\uparrow\downarrow$ and $\downarrow\uparrow$.
The T and B strategies have a fundamentalist character. The T strategy
gives buying advice in response to all inputs, except when the price
trend reaches a peak, that is, when the signal is $\uparrow\downarrow$.
Combined with the positions bounds, an agent following its advice
prefers to stay in a long position. Therefore it can be considered
as an optimistic fundamentalist. Fundamentalists believe that the
price should not deviate from a fundamental value. When the price
is lower, they try to stick to long positions. The T strategy is optimistic
because it targets a relatively high fundamental value signaled by
a price peak. In the original Wealth Game, the selling actions advised
by the T strategies help to trigger the falling trends in price cycles.
Hence they are called the {}``top trigger''.
Similarly, the B strategy gives selling advice in response to all
inputs, except when the input is $\downarrow\uparrow$. It is pessimistic
since it prefers to stick to a short position and buys only when the
price reaches a valley.
In summary, we are studying a model in which each agent is equipped
with two strategies; each of them may either be a trend-following
(F) strategy or a fundamentalist one (either optimistic (T) or pessimistic
(B)). Note that the virtual wealth of a strategy is not influenced
by the Cash Rule and the absence of market makers, meaning that an
agent evaluates it solely according to the original Wealth Game (Eqs.
(\ref{eq:position}), (\ref{eq:cash update}) and (\ref{eq:agent wealth}))
and assumes that the virtual transaction of a strategy is always successful.
We also note that the game is invariant under the gauge transformation
mapping the $\uparrow$ and $\downarrow$ signals to each other and
the buy and sell decisions to each other.
\section{\label{sec:Phase-Diagram}Phase Diagram}
We study the behavior of the game by starting from a set of unbiased
initial conditions. Each agent has the same initial cash $c_{\mathrm{o}}$
and do not hold any stocks initially ($k_{i}\left(0\right)=0$), so
that the initial wealth of each agent is $w_{i}\left(0\right)=c_{o}$.
The initial stock price is $P\left(0\right)=0$, and the initial virtual
wealth of all strategies $\sigma$ is $w_{\sigma}\left(0\right)=0$.
To avoid ambiguous decisions when more than one strategies have the
same virtual wealth during the game, one of the $3!$ priority orderings
of F, T and B is randomly chosen before the game starts. We call this
{}``throwing the public dice''. To initiate the game dynamics, one
of the four historical price signals ($\uparrow\uparrow$, $\uparrow\downarrow$,
$\downarrow\uparrow$ or $\downarrow\downarrow$) is randomly chosen.
Considering the choices in the public dice and the price signal, there
are 24 possible sets of initial conditions. Due to the gauge symmetry
in the game, there are 12 distinct initial conditions.
The steady state behavior of the game can be summarized in the phase
diagram in Fig. \ref{fig:phase diagram1} in the space of price
sensitivity $\gamma$ and market impact $\beta$. The space is divided
into two phases: the trendsetters' (TS) phase at low $\gamma$ and
the bouncing (BO) phase at high $\gamma$.
When the initial cash $c_{\mathrm{o}}$
increases, the TS phase expands. We also observe that the phase transition
is weakly dependent on the market impact $\beta$,
which implies that the market is dominated by dry quoting.
Typical time series
of the price in the attractors of these phases are shown in Fig.
\ref{fig:TS NTS time series}. The corresponding virtual wealth of
the F, T and B strategies are shown in Fig. \ref{fig:TS BO virtual wealth}.
We remark that the phase transition in Fig. \ref{fig:phase diagram1}
is a dynamical rather than a generic transition. The occurrence and
the position of the transition is specific to the unbiased initial
condition described above. Starting with other initial conditions,
or introducing perturbations to the dynamics,
we can obtain
the bouncing attractor in the trendsetters' phase and vice versa.
For example, we have done numerical experiments by starting from the
TS phase and gradually increasing $\gamma$ until we reach the BO
phase, but we observe that the TS attractor remains stable. Similar
annealing experiments from the BO phase to the TS phase also show
that the BO attractor can be stable in the TS phase.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{fig2a.pdf}
\par\end{centering}
\noindent \begin{centering}
\includegraphics[width=86mm]{fig2b.pdf}
\par\end{centering}
\caption{\label{fig:phase diagram1}(Color online) The phase diagram in the space of price
sensitivity $\gamma$ and market impact $\beta$ for (a) $c_{\mathrm{o}}=100$,
and (b) $c_{\mathrm{o}}=925$. Other parameters are $m=2$, $s=2$,
$K=3$, $N=1000$ (20 samples).}
\end{figure}
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{TSa_cut_}
\par\end{centering}
\noindent \begin{centering}
\includegraphics[width=86mm]{TSb_cut_}
\par\end{centering}
\caption{\label{fig:TS NTS time series}The time dependence of price for a
typical sample with $m=2$, $s=2$, $K=3$, $N=1000$, $c_{\mathrm{o}}=100$
for (a) the trendsetters' phase at $(\gamma,\beta)=(0.2,0.2)$, (b)
the bouncing phase at $(\gamma,\beta)=(0.7,0.2)$.}
\end{figure}
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{\string"5_pdfsam_TS_virtual_wealth_cut_\string".pdf}
\par\end{centering}
\noindent \begin{centering}
\includegraphics[width=86mm]{\string"5_pdfsam_BO_virtual_wealth_cut_\string".pdf}
\par\end{centering}
\caption{The virtual wealth of the F, T and B strategies in (a) the TS attractor
at $(\gamma,\beta)=(0.2,0.2)$, (b) the BO attractor at $(\gamma,\beta)=(0.7,0.2)$.
Parameters: $m=2$, $s=2$, $K=3$, $N=1000$, $c_{\mathrm{o}}=100$.\label{fig:TS BO virtual wealth}}
\end{figure}
In another set of experiments, we first prepare the TS attractor in
the TS phase, and then inject virtual cash to the B strategy so that
the BO attractor is favored. We observe that for all values of $\gamma$,
stable BO attractors can be formed when the amount of injected virtual
cash is sufficiently large. In the converse experiment, we inject
virtual cash to the F strategy in the BO attractor in the BO phase.
When $c_{\mathrm{o}}$ is high so that the BO phase is narrow, stable
TS attractors are formed at all values of $\gamma$ when the level
of injected virtual cash is sufficiently high. However, when $c_{\mathrm{o}}$
is low and the BO phase is broad, we found that at high values of
$\gamma$, stable TS attractors cannot be formed no matter how much
virtual cash is injected. This indicates that the conditions for forming
the TS attractor are probably more restrictive than the BO attractor,
and the existence phase of the TS attractor may be studied using approaches
to generic transitions. However, this transition is not relevant to
the dynamical transitions starting from the unbiased initial conditions
discussed in this paper.
\subsection{The Bouncing Phase}
In the bouncing phase, the dominant strategy is either T or B. In
the example shown in Fig. \ref{fig:TS NTS time series}(b), the dominant
strategy is B. Its dynamics is one large upward jump in price, followed
by one or several steps of zero change, and a downward jump in several
small steps or one large step. It reflects the desire of the agents
who buy stocks to take advantage of a low price, but the price trend
is prevented from rising further due to the prevailing pessimistic
atmosphere of the market.
The virtual wealth of the strategies are shown
in Fig. \ref{fig:TS BO virtual wealth}(b).
Since the B strategy gains profit from buying immediately before the
large upward jump and selling afterwards, it becomes the dominant
strategy. The T strategy also gains profit from selling immediately
before the downward jumps and buying at a lower price. Hence its virtual
wealth also increases with time, but at a rate slower than that of
the B strategy. On the other hand, since the F strategy cannot gain
wealth in a price series with frequent trend reversals, it becomes
a losing strategy.
Besides the up-bouncing example shown in Fig. \ref{fig:TS NTS time series}(b),
down-bouncing dynamics dominated by the T strategy can also be observed,
depending on the initial conditions. Using the gauge symmetry of the
game, their mechanism can be explained similarly.
\subsection{The Trendsetters' Phase}
This is the phase that gives rise to bubbles and crashes in the cycles
of stock prices \cite{Giardina03, Yeung08}.
As shown in Fig. \ref{fig:TS NTS time series}(a),
the price series consists of alternating long rising and falling trends.
As shown in Fig. \ref{fig:TS BO virtual wealth}(a), the virtual wealth
of the F strategy rises continuously due to its trendy decisions,
with minor setbacks when the price trend reverses. On the other hand,
the T and B strategies oscillate about their average at the
frequency of the price oscillations. Since the T strategy mainly takes
a long position, its virtual wealth is higher than that of the B strategy
during the upper half of the price cycle. Similarly, the B strategy
has a higher virtual wealth during the lower half of the price cycle.
The T strategy gains wealth by selling at the peak and buying back
at a lower price the next step, and the B strategy gains wealth by
buying at the price minimum and selling at a higher price at the next
step. Hence their average also has a slowly rising trend.
In the absence of market makers, the steady state is dominated by
dry quotations. Hence the cash of different agents becomes stationary
when the system equilibrates. For $s=2$, there are six types of agents.
Those holding two F strategies are denoted as FF agents, and the other
five types are FT, FB, TT, TB and BB agents. In the steady state,
agents can be roughly categorized as the active groups (including
FF, FT and FB agents) and the inactive group (including TT, TB and
BB agents). The active group usually has a relatively large amount
of cash, so that the price movement is mostly due to their participation
in the quoting activities. The inactive agents usually have little
cash, or have already reached the position bounds $(|k_{i}|=K)$,
so that their quotations would usually be terminated by the Cash Rule
or the position bounds. Due to the unavailability of participation
of the inactive group, real transactions cannot materialize as the
active agents cannot find matching dealers. Thus in this phase, the
price movement is solely caused by dry quoting.
The TS attractor is generally divided into four stages as shown in Fig.
\ref{fig:TS series}. In stage 1, all active agents place buying quotes
and thus the price is boosted up with the steepest slope. Stage 2
starts when the price has reached a level higher than the cash level
of some of the active agents, whose quoting actions are terminated
by the Cash Rule, leading to a decrease in the rate of price evolvement.
Stage 2 ends when the price is higher than the cash level of the most
liquid active agents. A quiet step (i.e. a time step with zero price
change) follows as no one quotes. Since the price change is zero,
a random signal is generated as the signal input for the next step.
If $\uparrow$ is generated, another quiet step follows, until a $\downarrow$
signal is randomly generated, triggering a falling trend. Stages 3
and 4 are duplicates of stages 1 and 2 under the gauge transformation.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{fig4}
\par\end{centering}
\caption{\label{fig:TS series}A typical trendsetters' attractor showing (a)
the price evolvement, (b) the evolvements of the total buying and
selling quotations. The parameters are $(\gamma,\beta)=(0.2,0.2)$,
$m=2$, $s=2$, $K=3$, $N=1000$, $c_{\mathrm{o}}=100$.}
\end{figure}
In Fig. \ref{fig:TS series}, note that the transition from stage
1 to 2 (or from stage 3 to 4) is not apparent in the price series.
This is because TS attractors exist at low $\gamma$, where the price
change is weakly sensitive to the excess demand $A(t)$.
In the following sections, we will analyze the behavior of the TS attractor,
leading eventually to an estimation of the phase transition point.
\section{The Transient Period\label{sec:The-Transient-Period}}
In contrast to the trendsetter attractor in \cite{Yeung08}, the TS
attractor in the absence of market makers is dominated by dry quotations.
This implies that the amount of cash held by each agent becomes stationary
at the steady state. As will be confirmed in the next section,
the period of the price cycles depends on the cash level of the most
liquid agent. Hence, the steady state behavior of the price cycles
directly depends no how the transient dynamics redistributes the cash
into the hands of the different types of agents. This heavy dependence
on the transient dynamics is the characteristics of the Wealth Game
without market makers.
A typical price series is shown in Fig. \ref{fig:rank 1 transient}
when the initial cash $c_{\mathrm{o}}$ is sufficiently large that
we are not very close to the phase boundary. At the beginning of the
transient stage, a significant redistribution of cash starts to take
place in the first half of the quasi-period of the price series. We
refer to this as the {\it separation stage}. Following the separation
stage, the cash levels are roughly flat with occasional jumps, whose
magnitudes are progressively smaller.
We call this the {\it quasi-stable stage}.
Eventually, the cash levels become stable at the steady state.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{\string"rank1_asymptotic\string".pdf}
\par\end{centering}
\noindent \begin{centering}
\includegraphics[width=86mm]{\string"rank1_transient\string".pdf}
\par\end{centering}
\noindent \begin{centering}
\includegraphics[width=86mm]{\string"rank1_separation\string".pdf}
\par\end{centering}
\caption{(Color online) The cash evolvements of different agents and the price series in type
I TS attractor for (a) the asymptotic time series, (b) the transient,
which is further divided into the quasi-stable stage and the separation
stage, and (c) the separation stage.
The parameters are $(\gamma,\beta)=(0.2,0.8)$,
$m=2$, $s=2$, $K=3$, $N=1000$, $c_{\mathrm{o}}=100$, initial
conditions ($\downarrow\downarrow$, $\mathrm{B}>\mathrm{F}>\mathrm{T}$).
\label{fig:rank 1 transient}}
\end{figure}
At the first few steps of the separation stage, the price rises from
zero. Some real transactions take place, but since the price is close
to zero, the cash levels of the agents do not change much and are
roughly equal to $c_{\mathrm{o}}$. After several steps, the major
sellers (the BB agents) have reached the minimum position of $-K$.
Dry quoting happens again and again, until the price is just higher
than $c_{\mathrm{o}}$, at where the agents think the stock is too
expensive. The price stops rising and when a downward signal appears,
all agents make selling quotations. In the next step, the signal $\downarrow\downarrow$
first appears after the quiet period. Both buying and selling quotations
are made by the agents. This is the time when there is a significant
redistribution of cash among the agents.
Subsequent to this event, there is a rising or falling price trend
depending on the excess demand at the event. For attractors obtained
from the initial condition used in Fig. \ref{fig:rank 1 transient},
which will be classified as type I attractor, the price follows a
rising trend, hits a peak when it goes above the cash level of the
most liquid agents. No transactions can be fulfilled as the major
sellers (BB) have reached the minimum position already, and the price
goes on a falling trend. For other initial conditions, the price follows
a falling trend directly after the event. In all cases, such price
movements are not accompanied by any real transactions, as the price
is higher than the cash levels of the buying group. Hence, no cash
redistribution takes place for many time steps.
For type I attractors, the second significant cash redistribution
takes place when the price falls below the cash level of the second
most liquid group of agents.
In the quasi-stable stage, real transactions can only materialize
when the price is close to zero, allowing the inactive agents to participate
with their low cash levels. When this happens, the cash levels of
the agents fluctuate little bit. These fluctuations are so minor that
we can neglect their effects on the final stable cash levels.
Hence we can focus on the cash evolvements in the separation stage
to calculate the approximate stable cash levels and hence the amplitude
of the price cycles. For the initial condition
in Fig. \ref{fig:rank 1 transient},
this is done with reference to Tables \ref{tab:first cash redis}
and \ref{tab:second cash redis} at the first and second significant
cash redistributions respectively. At the first event, the order of
priority of the strategies is T $>$ F $>$ B, leading to the adoption
of the strategies in the third column of Table \ref{tab:first cash redis}.
In response to the signal $\downarrow\downarrow$, the outputs of
these strategies are given in the fourth column. Taking into account
the positions listed in the fifth column, the final decisions of the
agents are given in the sixth column. Summing up the buying and selling
quotations weighted by the fractions in the second column, we obtain
$A_{\mathrm{buy}}=5N/9$ and $A_{\mathrm{sell}}=3N/9$. Thus, the
FF and FB agents are the minority and sell one whole unit of stock
at the price $P\approx c_{\mathrm{o}}$. Agents in the buying group
have bought $3/5$ unit of stocks,
and have lost cash $\approx3c_{\mathrm{o}}/5$.
\begin{table*}
\caption{The outputs, positions, decisions and final cash (roughly in multiples
of $c_{\mathrm{o}}$) of the agents at the first significant cash
redistribution event of a type I TS attractor. \label{tab:first cash redis}}
\noindent \begin{centering}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Agents & Fraction & Strategy & Output & Position & Decision & Final cash
\tabularnewline
\hline
\hline
FF & $1/9$ & F & sell & $\geq-K+1$ & sell & 2\tabularnewline
\hline
FT & $2/9$ & T & buy & $\leq K-1$ & buy & $2/5$\tabularnewline
\hline
FB & $2/9$ & F & sell & $\geq-K+1$ & sell & 2\tabularnewline
\hline
TT & $1/9$ & T & buy & $\leq K-1$ & buy & $2/5$\tabularnewline
\hline
TB & $2/9$ & T & buy & $\leq K-1$ & buy & $2/5$\tabularnewline
\hline
BB & $1/9$ & B & sell & $=-K$ & hold & 1\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\end{table*}
\begin{table*}
\caption{The outputs, positions, decisions and final cash (roughly in multiples
of $c_{\mathrm{o}}$) of the agents at the second significant cash
redistribution event of a type I TS attractor. \label{tab:second cash redis}}
\noindent \begin{centering}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Agents & Fraction & Strategy & Output & Position & Decision & Final cash\tabularnewline
\hline
\hline
FF & $1/9$ & F & sell & $\geq-K+1$ & sell & $56/25$\tabularnewline
\hline
FT & $2/9$ & F & sell & $\geq-K+1$ & sell & $16/25$\tabularnewline
\hline
FB & $2/9$ & F & sell & $\geq-K+1$ & sell & $56/25$\tabularnewline
\hline
TT & $1/9$ & T & buy & $\leq K-1$ & buy & 0\tabularnewline
\hline
TB & $2/9$ & T & buy & $\leq K-1$ & buy & 0\tabularnewline
\hline
BB & $1/9$ & B & sell & $=-K$ & hold & 1\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\end{table*}
At the second event, the price has fallen to just below the cash level
of the second most liquid group of agents (FF, TT and TB), which is
roughly $2c_{\mathrm{o}}/5$. At this point, the price has fallen
to a value such that the order of priority of the strategies becomes
F $>$ T $>$ B. Consequently, as shown in Table \ref{tab:second cash redis},
the FT agents have changed to be in the selling group. We obtain $A_{\mathrm{buy}}=3N/9$
and $A_{\mathrm{sell}}=5N/9$. Hence, agents in the selling group
sell $3/5$ unit of stocks at the price $P\approx2c_{\mathrm{o}}/5$,
and so gain cash of the amount $6c_{\mathrm{o}}/25$. The buying agents
have their cash reduced by $2c_{\mathrm{o}}/5$.
As mentioned earlier, the maximum price $P_{\mathrm{max}}$ is reached
when the price has just exceeded the cash level of the most liquid
agents. Therefore, $P_{\mathrm{max}}\approx56c_{\mathrm{o}}/25$ for
type I attractors.
Considering the dynamics starting from all the 24 initial conditions,
we classify the TS attractors into three types as shown in Table \ref{tab:3 types attractors}.
Their maximum price and the distribution of the final cash are described
in Appendix. Although the amplitudes of the price cycles of the three
types of attractors are different, their dynamics are qualitatively
the same.
\begin{table}
\caption{The three types of TS attractors at $(\gamma,\beta)=(0.2,0.8)$, $m=2$,
$s=2$, $K=3$, $N=1000$, $c_{\mathrm{o}}=100$. In the initial conditions,
$*$ represents $\uparrow$ or $\downarrow$. Only the 12 initial
conditions that give rise to an initial rising trend during the separation
stage are included in the third column. The other 12 initial conditions,
related to those in the fourth column by gauge transformation $\uparrow\leftrightarrow\downarrow$
and T$\leftrightarrow$B, belong to the same corresponding type of
attractor. \label{tab:3 types attractors}}
\noindent \begin{centering}
\begin{tabular}{|c|c|c|c|}
\hline
Attractor & Most liquid agents & Initial condition & $P_{\mathrm{max}}$\tabularnewline
\hline
\hline
Type I & FF and FB & ($*$$*$, T $>$ F $>$ B) & $56c_{\mathrm{o}}/25$\tabularnewline
\hline
Type II & FT & ($*$$\downarrow$, T $>$ B $>$ F) & 2$c_{\mathrm{o}}$\tabularnewline
\hline
Type III & FB & ($*$$\uparrow$, T $>$ B $>$ F) & $8c_{\mathrm{o}}/5$\tabularnewline
\cline{2-3}
& FF and FT and FB & \begin{tabular}{c}
($*$$\uparrow$, F $>$ B $>$ T)\tabularnewline
\hline
($*$$\uparrow$, F $>$ T $>$ B)\tabularnewline
\end{tabular} & \tabularnewline
\hline
\end{tabular}
\par\end{centering}
\end{table}
We have also studied the dynamics of the TS attractors at other values
of $K$ and $(\gamma,\beta)$, but whose locations are not close to
the boundary of the TS phase. We found TS attractors with approximately
the same amplitudes of $56c_{\mathrm{o}}/25$, 2$c_{\mathrm{o}}$,
$8c_{\mathrm{o}}/5$, but we also found attractors with other amplitudes
such as $14c_{\mathrm{o}}/5$ and $72c_{\mathrm{o}}/35$. An exhaustive
search of all possible amplitudes is beyond the scope of our study.
Since they have qualitatively similar behaviors, we will continue
our analysis using only the three types of attractors we have described.
\section{Periods of the Price Cycles\label{sec:periods}}
After obtaining an estimate of the amplitudes of the price cycles
in the previous section, we can derive the periods of the price cycles
if we also know about the price change per time step. This information
is also available from Table \ref{tab:second cash redis} for type
I attractor.
Consider the falling trend from the peak price of
$P_{\mathrm{max}}\approx56c_{\mathrm{o}}/25$
to the valley $P\approx-56c_{\mathrm{o}}/25$.
From Tables \ref{tab:first cash redis}
and \ref{tab:second cash redis},
we find that $A\mathrm{_{sell}}=3N/9$ from $P=56c_{\mathrm{o}}/25$
to $P=16c_{\mathrm{o}}/25$,
and $A\mathrm{_{sell}}=5N/9$ from $P=16c_{\mathrm{o}}/25$
to $P=-56c_{\mathrm{o}}/25$. Hence, the period of the price cycles
is given by
\begin{eqnarray}
\frac{T_{\mathrm{I}}}{2}
&=&\left(\frac{3N}{9}\right)^{-\gamma}
\left(\frac{56}{25}c_{\mathrm{o}}-\frac{16}{25}c_{\mathrm{o}}\right)
\nonumber \\
& & +\left(\frac{5N}{9}\right)^{-\gamma}\left(\frac{16}{25}c_{\mathrm{o}}
+\frac{56}{25}c_{\mathrm{o}}\right).\label{eq:rank 1 half period}
\end{eqnarray}
This reveals the dependence of the period on $c_{\mathrm{o}}$ and
$\gamma$
\begin{equation}
T_{\mathrm{I}}=\left(\frac{16c_{\mathrm{o}}}{25N^{\gamma}}\right)
\left[5\left(3^{\gamma}\right)+9\left(\frac{9}{5}\right)^{\gamma}
\right].
\label{eq:period I}\end{equation}
The periods of types II and III attractors are derived in Appendix.
To compare with simulation results, we calculate the harmonic mean
of the periods averaged over the initial conditions.
From Table \ref{tab:3 types attractors},
the probabilities of occurrence are $1/3$, $1/6$ and $1/2$ for
types I, II and III attractors respectively. Hence the average period
of TS attractors is
\begin{equation}
T_{\mathrm{av}}
=\left(\frac{1}{3}T_{\mathrm{I}}^{-1}+\frac{1}{6}T_{\mathrm{II}}^{-1}
+\frac{1}{2}T_{\mathrm{III}}^{-1}\right)^{-1}.
\label{eq:average period}\end{equation}
Substituting Eqs. (\ref{eq:period I}), (\ref{eq:per2}) and (\ref{eq:per3}),
we obtain\begin{eqnarray}
T_{\mathrm{av}}&=&\frac{16c_{\mathrm{o}}}{N^{\gamma}}\times\nonumber \\
& & \left\{ \frac{25}{3\left[5\left(3^{\gamma}\right)
+9\left(9/5\right)^{\gamma}\right]}
+\frac{1}{3\left(3^{\gamma}\right)}
+\frac{5}{4}\left(\frac{5}{9}\right)^{\gamma}\right\} ^{-1}.
\nonumber \\
\label{eq:final average period}\end{eqnarray}
So far we have derived this expression for a particular value of $\gamma$
($=0.2$). Observing that the attractor structures in a broad range
of $\gamma$ are qualitatively similar, we extrapolate this result
to general values of $\gamma$. Interpolating the expression by an
exponential function of $\gamma$ between $\gamma=0$ and $\gamma=1$,
we have\begin{equation}
\mathrm{ln}T_{\mathrm{av}}\approx B-m\gamma,\label{eq:lnT}\end{equation}
where $B=\mathrm{ln}\left(448c_{\mathrm{o}}/61\right)$ and
$m=\mathrm{ln}\left(3514N/7137\right)$.
Figure \ref{fig:lnT and Gamma}
shows the average TS period for $c_{\mathrm{o}}=100$
obtained from simulations. It shows that the period is an exponential
function of $\gamma$. $B$ and $m$ in the figure are determined
experimentally to be 6.71 and 6.13 respectively. This compares favorably
with the theoretical prediction of Eq. (\ref{eq:lnT}) which yields
$B=6.60$ and $m=6.20$.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{\string"5_pdfsam_lnT_vs_Gamma_cut_\string".pdf}
\par\end{centering}
\caption{(Color online) The dependence of the TS period on $(\gamma,\beta)$
averaged over 10 samples. The line is the prediction of Eq. \ref{eq:lnT}.
The parameters are $m=2$, $s=2$, $K=3$, $N=1000$, $c_{\mathrm{o}}=100$.
\label{fig:lnT and Gamma}}
\end{figure}
\section{The Phase Boundary\label{sec:The-Phase-Boundary}}
To search for the existence condition of the TS phase, we plot the
period of the TS attractors as a function of $\gamma$ for various
$c_{\mathrm{o}}$ in Fig. \ref{fig:period bound}. Remarkably, the
TS attractors disappear when the period falls below an apparently
universal value, suggesting that there is a lower bound of the TS
period around $4K+5$.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{PeriodLowerBound}
\par\end{centering}
\caption{(Color online) The dependence of the TS period on $\gamma$ averaged over the 12
possible attractors for various $c_{\mathrm{o}}$ down to the boundary
of the TS phase. The line corresponds to the bound of $4K+5$. The
parameters are $s=2$, $K=3$, $\beta=0.5$, $N=1000$. \label{fig:period bound}}
\end{figure}
This possibility is further supported by the numerical experiment
described in Fig. \ref{fig:BO phase bias F}. We start the Wealth
Game with an initial condition biased towards the F strategy, but
at a very high value of $\gamma$ deep in the BO phase. This favors
the TS attractor in the transient stage, which is expected to be destabilized
at the steady state. We observe that the virtual wealth of the F strategy
decreases from one period to another and is accompanied by periods
of the TS attractor shorter than $4K+5$. Meanwhile, the virtual wealth
of the B (or T) strategy keeps on increasing period after period.
Eventually, the F strategy is outperformed by the B strategy and the
TS attractor disappears.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{5_pdfsam_BObiasF_cut_}
\par\end{centering}
\caption{The transient existence of the TS attractor at
$(\gamma,\beta)=(0.65,0.5)$
starting with the following virtual cash of the strategies:
$c_{\mathrm{F}}=2000N^{\gamma},c_{\mathrm{T}}=c_{\mathrm{B}}=0$,
for (a) the price series, where the inset shows the magnified plot
of the TS regime and (b) the virtual wealth series. Other parameters
are: $s=2$, $K=3$, $N=1000$,$c_{\mathrm{o}}=100$. \label{fig:BO phase bias F}}
\end{figure}
Let us analyze the attractors with the shortest possible TS period.
As shown in Fig. \ref{fig:4K+5} for
a TS attractor, the F strategy starts from the minimum position $-K$
at the beginning of stages 1 and 2, and takes buying actions in response
to the signal $\uparrow\uparrow$ for $2K$ steps. This continues
until it reaches the maximum position $K$. A quiet period follows.
The signal at the first step of the quiet period is $\uparrow\uparrow$,
but the signals in the following steps are random. If the signal is
$\uparrow$, the quiet period continues, but if the signal is $\downarrow$,
all strategies respond to $\uparrow\downarrow$ by selling, and the
dynamics enters stage 3. Hence the average length of the quiet period
is $1+\sum_{n=0}^{\infty}n/2^{n+1}=2$.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{fig7}
\par\end{centering}
\caption{The time series of (a) the price and (b) the virtual wealth of the
F strategy in a half-period
of the TS attractor at $(\gamma,\beta)=(0.55,0.8)$, $s=2$,
$K=3$, $N=1000$, $c_{\mathrm{o}}=100$. Note that in general, the
F strategy requires $2K$ steps to regain
the virtual wealth at the previous quiet period when $\beta\le 0.5$,
and $2K+1$ steps when $\beta>0.5$.
\label{fig:4K+5}}
\end{figure}
When the falling price trend of stage 3 starts, the F strategy makes
the right move of selling, but its position remains positive due to
the rising trend in stages 1 and 2. It takes the strategy $K$ steps
to change its position to $0$. As shown in Fig. \ref{fig:4K+5},
it is losing virtual wealth during this period of time. It takes another
$K$ steps to change its position from zero to minimum, and the strategy
is regaining virtual wealth during this period. For these $2K$ steps,
its virtual wealth gain and loss are roughly balanced. This completes
the adjustment stage of the F strategy. If the falling trend is longer
than $2K$ steps, then the strategy can start to gain virtual wealth
using its trend-following outputs and stabilize the TS attractor.
To make a more quantitative estimate, we consider the case $\gamma=0$,
in which the price change at each step is $A^{0}=1$, independent of
the volume of buying and selling quotations. By tracing the virtual
wealth change of the F strategy during a price cycle of length $4K+4$
(with an average length of 2 during the quiet periods at the peaks
and valleys), the virtual wealth gain of the F strategy is calculated
to be $2K(1-2\beta)$, whereas those of the T and B strategies are
1. When the period of the price cycle lengthens, the virtual wealth
gain of the F strategy increases by $K$ for every extension of the
period by one step. If we consider the mid-position of the phase diagram
with $\beta=1/2$, then for price cycles of period $4K+5$, the F
strategy outperforms the T and B strategies by $2K(1-2\beta)+K-1=K-1$.
Hence for $K>1$, it is reasonable to estimate the phase boundary by the condition
that the period of the price cycle becomes $4K+5\equiv T_{\mathrm{c}}$.
We are now ready to calculate the critical value $\gamma_{\mathrm{c}}$
of the TS attractor at the phase boundary. However, we observe in
Fig. \ref{fig:phase diagram1} that the transition to the BO phase
occurs within a narrow but finite range of $\gamma$, instead of having
an abrupt change. This is due to the dependence on the initial conditions,
as classified according to the three types of attractors. By equating
$T_{\mathrm{c}}$ to the periods in Eqs. (\ref{eq:period I}), (\ref{eq:per2})
and (\ref{eq:per3}), we find that $\gamma_{\mathrm{c}}$ takes the
values of 0.65, 0.66 and 0.57 for types I, II and III attractors respectively.
The phase lines for types II and III attractors are located in the
phase diagrams in Fig. \ref{fig:compare theory}, and the matching
with the simulation results is quite well.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{\string"match_phase_bounadary_a\string".pdf}
\par\end{centering}
\noindent \begin{centering}
\includegraphics[width=86mm]{\string"match_phase_bounadary_b_cut_\string".pdf}
\par\end{centering}
\caption{(Color online) Comparison of the theoretical prediction of $\gamma_{\mathrm{c}}$
with the simulation results for (a) $c_{\mathrm{o}}=100$ and (b)
$c_{\mathrm{o}}=925$. Other parameters are $s=2$, $K=3$, $N=1000$ (20 samples).
\label{fig:compare theory}}
\end{figure}
\section{Dependence on Population Composition\label{sec:f-dependence}}
So far, we have considered the case of unbiased assignment of strategies,
that is , $f_{\mathrm{F}}:f_{\mathrm{T}}:f_{\mathrm{B}}=1:1:1$, where
$f_{\mathrm{\sigma}}$ is the ratio of the probabilities
of assigning strategies $\mathrm{\sigma}$
to an agent. It is interesting to consider how the market dynamics
changes when we vary the ratio of strategies
to $f_{\mathrm{F}}:f_{\mathrm{T}}:f_{\mathrm{B}}=f:1:1$,
where $f$ is referred to as the {\it trendsetter (TS) factor}.
Figure \ref{fig:phase diagram with f}
shows the phase diagram in the space of $f$ and $\gamma$. The TS
phase exists for sufficiently large $f$ and sufficiently small $\gamma$.
This suggests that in order to trigger the TS price trends, the market
should be dominated by enough trendsetting strategies. In other words,
a market is trendy only when there are enough trend-believing agents.
In contrast, if the game is full of T and B strategies, it becomes
a market with fundamentalists as the majority, and price trends cannot
be set up easily. Such a picture shows qualitative consistency with
the results obtained in \cite{Giardina03}, where a polarization parameter
determines the statistical weight of trend-following strategies, and
the periodic phase exists at high polarization.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{fig18}
\par\end{centering}
\caption{(Color online) The phase diagram in the space of the F strategy factor $f$ and price
sensitivity $\gamma$. Other parameters are $s=2$, $K=3$, $N=1000$,
$c_{\mathrm{o}}=100$, $\beta=0.5$ (20 samples).\label{fig:phase diagram with f}}
\end{figure}
The vertical segment of the phase boundary
in Fig. \ref{fig:phase diagram with f}
shows little dependence on $f$, indicating that our analysis at $f=1$
is a good approximation.
To analyze the horizontal segment,
we consider the typical attractor profile below and above the
phase boundary in Fig. \ref{fig:phase transition in f}.
Note that at the transient stage, the price series in both cases appear
as TS. The difference lies in the behavior on approaching the steady
state. When $f$ is small, the F-group agents (FF, FT and FB) are
the minority. In the absence of market makers, the need to balance
the buying and selling volumes has significant consequences in determining
the types of agents whose positions are {\it saturated} (that is,
reach $\pm K$). Hence, we show the evolvement of the agents' positions
in Fig. \ref{fig:f04 pos} during the separation stage for two typical
cases. In these cases the position of the F-group agents saturate
at $K$; the cases of $-K$ have the same behavior after gauge transformation.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{fig19a}
\par\end{centering}
\noindent \begin{centering}
\includegraphics[width=86mm]{fig19b}
\par\end{centering}
\caption{The price series of an attractor
before and after undergoing the phase transition
at (a) $f=0.4$ (b) $f=0.5$.
The middle plot and the top table are the magnified graph
of the price series and the positions of different agents at stable states.
Other parameters are $m = 2$, $s = 2$, $K = 3$, $N = 1000$,
$c_{\mathrm{o}}=100$, $(\gamma,\beta)=(0.2,0.8)$,
initial condition $(\uparrow\uparrow,\mathrm{T} > \mathrm{F} > \mathrm{B})$.
\label{fig:phase transition in f}}
\end{figure}
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{5_pdfsam_f04a_cut_}
\par\end{centering}
\noindent \begin{centering}
\includegraphics[width=86mm]{5_pdfsam_f04b_cut_}
\par\end{centering}
\caption{(Color online) Evolvement of the agents' positions for initial condition
(a) ($\uparrow\uparrow,\mathrm{T}>\mathrm{F}>\mathrm{B}$),
(b) ($\downarrow\uparrow,\mathrm{B}>\mathrm{F}>\mathrm{T}$). The
price series is also shown for reference. Parameters: $s=2$, $K=3$,
$N=1000$, $c_{\mathrm{o}}=100$, $(\gamma,\beta)=(0.2,0.8)$, $f=0.4$.
\label{fig:f04 pos}}
\end{figure}
The outcome of the separation stage is that the TT and F-group agents
take up positive positions at the steady state, and the BB and TB
agents have negative positions. Note that at the steady state, the
only active sellers with unsaturated positions are the TB agents in
Fig. \ref{fig:f04 pos}(a), and the TB and BB agents in Fig. \ref{fig:f04 pos}(b).
More significantly, the F-group agents have saturated positive positions.
Once their positions are saturated, their buying quotations disappear.
The TS price trend stops, and the steady state enters the bouncing
attractor.
This transition mechanism from the TS transient to the BO attractor
is further confirmed by the dynamics of agents' positions when $f$
is increased to enter the TS phase, as shown in Fig. \ref{fig:f05 pos}.
Since $f$ is larger when compared with Fig. \ref{fig:f04 pos},
the TB and BB agents saturate at position $-K$ before the F-group
agents reaches position $K$. The result is exactly the opposite:
the selling quotes of the TB and BB agents disappear, and the price
continues to rise. More important, the F-group agents remain unsaturated.
They emerge as active agents with the freedom to make buying and selling
quotes, and the TS attractor is sustainable at the steady state.
\begin{figure}
\noindent \begin{centering}
\includegraphics[width=86mm]{5_pdfsam_f05a_cut_}
\par\end{centering}
\noindent \begin{centering}
\includegraphics[width=86mm]{5_pdfsam_f05b_cut_}
\par\end{centering}
\caption{
(Color online) Same as Fig. \ref{fig:f04 pos},
except that $f=0.5$.
\label{fig:f05 pos}}
\end{figure}
This transition mechanism enables us to derive the macroscopic condition
for the disappearance of the TS phase, irrespective of the transaction
details. The total volume of stocks that can be sold by the BB and
TB agents is $V_{\mathrm{s}}=3NK/(2+f)^{2}$. The total volume of
stocks that can be bought by the TT and F-group agents is $V_{\mathrm{b}}=NK(f^{2}+4f+1)/(2+f)^{2}$.
If $V_{\mathrm{b}}<V_{\mathrm{s}}$, the F-group agents become saturated
and remain inactive. Thus we have the BO phase when $f<\sqrt{6}-2\approx0.449$.
This prediction matches well with the simulation result in Fig. \ref{fig:phase diagram with f}.
\section{\label{cha:Conclusion}Conclusion}
We have considered a simplified Wealth Game in which no market makers
are present. Since buying and selling quotations are not balanced,
the dynamics becomes complicated. Furthermore, due to the prevalence
of dry quotations, the dynamics becomes heavily dependent on the initial
conditions. This makes it difficult to analyze the model and understand
its underlying mechanism. To circumvent this difficulty, we simplify
the input dimension of the strategies to 2 and restrict the strategies
to three representative ones, namely, F, T and B. Respectively, they
represent the trend-followers/trendsetters, optimistic and pessimistic
fundamentalists in the market.
With these simplifications, we observe a dynamical transition from
the TS phase to the BO phase when $\gamma$ increases. Despite the
simplicity of the model, both phases bear characteristics of real
markets. The TS phase produces price trends that resemble bubbles
and crashes in real markets. They are observed when the agents have
enough cash, and the price movement is not very sensitive to the excess
demand. The BO phase produces price trends that are relatively steady,
with occasional up-bounces or down-bounces followed by relaxations
to fundamental prices. These price trends can also be observed in
real markets when investors have very cautious moods about the fundamental
values of the stocks. Both phases are dominated by dry quotations,
which correspond to quiet moments in real markets with very low trading
volumes. In many economic systems, such as the real estates market,
dry quotations are prevalent when the price is too high or too low.
We find that the amplitude of the price cycles is determined by the
cash level of the most liquid agents. When agents with less cash stop
their quotations, the more liquid agents can still boost up or push
down the price to higher or lower levels. Prices reach their extreme
values when the agents consider it too risky to participate. However,
the quantitative relation between the price amplitude and the initial
cash is determined by the cash redistribution process during the transient
stage. To overcome the complexity of this process, we adopt a semi-empirical
approach by analyzing the separation stage for various initial conditions,
and extrapolating the predictions to the entire TS phase. Agreement
with simulation results shows that this is a good approximation.
Our study also suggests a mechanism for the disappearance of the TS
phase. All trend-following strategies need an adaptation period when
the price trend reverses. Agents become certain of the advantages
of these strategies only when the duration of a price trend is longer
than that of the adaptation period. When the price sensitivity increases,
the price change per step increases, and the period of the price cycles
shortens. When the period becomes comparable to the adaptation period,
the TS attractor becomes unsustainable.
We also find that the composition of the population affects the market
behavior. The TS phase is present in markets where trend-following
strategies are popular. When the trend-following strategies become
less popular, we observe a phase transition to the bouncing phase.
We find that this transition is due to the fact that the positions
of the trend-followers saturate at the maximum (or minimum), and no
active agents want to buy (or sell) when they decide to sell (or buy).
This allows us to derive a macroscopic condition for the phase transition
by balancing the volume of supply and demand of stocks. The prediction
agrees with the simulation results well.
It is interesting to compare our model with the Wealth Game with the
market makers present \cite{Yeung08}. In both cases, the TS phase
exists due to the presence of trend-followers. However, when market
makers are present, the price trend is driven by real transactions
and has a slightly different dynamics. For example, the so-called
fickle agents are those who hold a T and B strategy, and fickle their
preference between the two strategies with a delayed response. They
push the price further up in a rising trend, and down in a falling
trend, thus creating opportunities for the trend-followers to gain
wealth. In our model without market makers, the fickle agents do not
play an important role. During the separation stage, their cash is
reduced to a very low level, as evident in Tables \ref{tab:second cash redis},
\ref{tab:type II cash redistribution} and
\ref{tab:type III cash redistribution}
for types I to III attractors respectively.
Hence they only play a minor role in the price dynamics.
While our model is successful in explaining and interpreting a number
of market phenomena, it can be further improved to address a broader
range of issues. One possible modification to this model is to implement
the injection of cash to the market. This may help to relieve the
problem of too many dry quotations in the present model, whose agents
are restricted by the cash rule. Furthermore, since the present model
is a close system with constant average wealth, incorporating cash
injection may cause the market to evolve spontaneously towards states
which have maximal attraction of capital. In this way, it will also
address the issue of the self-organization of markets, which has drawn
considerable attention in recent models
\cite{Giardina03, Challet01, Yeung08, Alfi09a, Alfi09b}.
\section*{Acknowledgement}
We thank Jack Raymond for discussions on the public dice.
This work is supported by the Research Grants Council of Hong Kong
(grant nos. HKUST 630607 and 604008).
|
1,116,691,499,320 | arxiv | \section{Introduction}
The most mature systems for deductive verification of randomized algorithms are
\emph{expectation-based} techniques; seminal examples include \textsc{PPDL}\xspace
\cite{Kozen85} and \textsc{pGCL}\xspace \cite{Morgan96}. These approaches reason about
\emph{expectations}, functions $E$ from states to real numbers,\footnote{%
Treating a program as a function from input states $s$ to output distributions
$\mu(s)$, the expected value of $E$ on $\mu(s)$ is an expectation.}
propagating them backwards through a program until they are transformed into a
mathematical function of the input. Expectation-based systems are both
theoretically elegant
\cite{KaminskiKMO16,gretz2014operational,olmedo2016reasoning,kaminski2016inferring}
and practically useful; implementations have verified numerous randomized
algorithms \cite{Hurd03,HurdMM05}. However, properties involving
multiple probabilities or expected values can be cumbersome to verify---each
expectation must be analyzed separately.
An alternative approach envisioned by Ramshaw \cite{Ramshaw79} is to work with
predicates over distributions. A direct
comparison with expectation-based techniques is difficult, as the approaches are quite
different. In broad strokes, assertion-based systems can verify richer
properties in one shot and have specifications that are arguably more
intuitive, especially for reasoning about loops, while
expectation-based approaches can transform expectations mechanically
and can reason about non-determinism. However, the comparison is not
very meaningful for an even simpler reason: existing assertion-based
systems such as \cite{ChadhaCMS07,Hartog:thesis,RandZ15} are not as
well developed as their expectation-based counterparts.
\begin{description}
\item[Restrictive assertions.]
Existing probabilistic program logics do not support reasoning about
expected values, only probabilities. As a result, many properties about
average-case behavior are not even expressible.
\item[Inconvenient reasoning for loops.]
The Hoare logic rule for
deterministic loops does not directly generalize to probabilistic
programs. Existing assertion-based systems either forbid
loops, or impose complex semantic side conditions to
control which assertions can be used as loop invariants. Such side
conditions are restrictive and difficult to establish.
\item[No support for external or adversarial code.]
A strength of expectation-based techniques is reasoning
about programs combining probabilities and
\emph{non-determinism}. In contrast, Morgan and
McIver~\cite{McIverM05} argue that assertion-based techniques
cannot support compositional reasoning for such a combination. For
many applications, including cryptography, we would still like to
reason about a commonly-encountered special case: programs using
external or adversarial code. Many security properties in
cryptography boil down to analyzing such programs, but existing
program logics do not support adversarial code.
\item[Few concrete implementations.]
There are by now several
independent implementations of expectation-based techniques,
capable of verifying interesting probabilistic programs. In
contrast, there are only scattered implementations of
probabilistic program logics.
\end{description}
These limitations raise two points. Compared to expectation-based
approaches:
\begin{enumerate}
\item Can assertion-based approaches achieve similar expressivity?
\item Are there situations where assertion-based approaches are more suitable?
\end{enumerate}
In this paper, we give positive evidence for both of these
points.\footnote{Note that we do not give mathematically precise
formulations of these points; as we are interested in the practical
verification of probabilistic programs, a purely theoretical answer
would not address our concerns.}
Towards the first point, we give a
new assertion-based logic \mbox{\textsc{Ellora}}\xspace for probabilistic programs,
overcoming limitations in existing probabilistic program
logics. \mbox{\textsc{Ellora}}\xspace supports a rich set of assertions that can express
concepts like expected values and probabilistic independence, and
novel proof rules for verifying loops and adversarial code. We prove
that \mbox{\textsc{Ellora}}\xspace is sound and relatively complete.
Towards the second point, we evaluate \mbox{\textsc{Ellora}}\xspace in two ways. First, we
define a new logic for proving probabilistic independence and
distribution law properties---which are difficult to capture with
expectation-based approaches---and then embed it into \mbox{\textsc{Ellora}}\xspace. This
sub-logic is more narrowly focused than \mbox{\textsc{Ellora}}\xspace, but supports more
concise reasoning for the target assertions. Our embedding
demonstrates that the assertion-based approach can be flexibly
integrated with intuitive, special-purpose reasoning principles. To
further support this claim, we also provide an embedding of the Union
Bound logic, a program logic for reasoning about accuracy
bounds~\cite{BartheGGHS16-icalp}. Then, we
develop a full-featured implementation of \mbox{\textsc{Ellora}}\xspace in the \textsc{EasyCrypt}\xspace
theorem prover and exercise the logic by mechanically verifying a
series of complex randomized algorithms. Our results suggest that the
assertion-based approach can indeed be practically viable.
\myheading{Abstract logic.}
To ease the presentation, we present \mbox{\textsc{Ellora}}\xspace in two stages. First, we
consider an abstract version of the logic where assertions are general
predicates over distributions, with no compact syntax. Our abstract
logic makes two contributions: reasoning for loops, and for
adversarial code.
\paragraph*{Reasoning about Loops.}
Proving a property of a probabilistic loop typically requires establishing a
loop invariant, but the class of loop invariants that can be soundly used
depends on the termination behavior---stronger termination assumptions allows
richer loop invariants. We identify three classes of assertions
that can be used for reasoning about probabilistic loops, and provide a proof
rule for each one:
\begin{itemize}
\item arbitrary assertions for \emph{certainly terminating} loops,
i.e.\ loops that terminate in a finite amount of iterations;
\item \emph{topologically closed} assertions for \emph{almost surely}
terminating loops, i.e.\ loops terminating with probability $1$;
\item \emph{downwards closed} assertions for arbitrary loops.
\end{itemize}
The definition of topologically closed assertion is reminiscent of Ramshaw
\cite{Ramshaw79}; the stronger notion of downwards closed assertion appears to
be new.
Besides broadening the class of loops that can be analyzed, our rules
often enable simpler proofs. For instance, if the loop is certainly
terminating, then there is no need to prove semantic side-conditions.
Likewise, there is no need to consider the termination behavior of the
loop when the invariant is downwards and topologically closed. For
example, in many applications in cryptography, the target property is
that a \lq\lq bad\rq\rq\ event has low probability: $\Pr{[E]} \leq
k$. In our framework this assertion is downwards and topologically
closed, so it can be a loop invariant regardless of the termination
behavior.
\paragraph*{Reasoning about Adversaries.}
Existing assertion-based logics cannot reason about probabilistic programs with
\emph{adversarial} code.
\emph{Adversaries} are special probabilistic procedures consisting of an interface
listing the concrete procedures that an adversary can call (\emph{oracles}),
along with restrictions like how many calls an
adversary may make. Adversaries are useful in cryptography, where
security notions are described using experiments in which
adversaries interact with a challenger, and in game theory and
mechanism design, where adversaries can represent strategic
agents. Adversaries can also model inputs to \emph{online} algorithms.
We provide proof rules for reasoning about adversary calls. Our rules
are significantly more general than previously considered rules for
reasoning about adversaries. For instance, the rule for adversary used
by~\cite{BartheGGHS16-icalp} is restricted to adversaries that cannot
make oracle calls.
\paragraph*{Metatheory.}
We show soundness and relative completeness of the core abstract logic, with
mechanized proofs in the \textsc{Coq} proof assistant.\footnote{%
The formalization is available at \url{https://github.com/strub/xhl}.}
\myheading{Concrete logic.}
While the abstract logic is conceptually clean, it is inconvenient
for practical formal verification---the
assertions are too general and the rules involve
semantic side-conditions. To address these issues,
we flesh out a concrete version of \mbox{\textsc{Ellora}}\xspace.
Assertions are described by a grammar modeling a two-level assertion
language. The first level contains state predicates---deterministic assertions
about a single memory---while the second layer contains probabilistic predicates
constructed from probabilities and expected values over discrete distributions.
While the concrete assertions are theoretically less expressive than their
counterparts in the abstract logic, they can already encode
common properties and notions from existing proofs, like probabilities,
expected values, distribution laws and probabilistic independence. Our
assertions can express theorems from probability theory, enabling
sophisticated reasoning about probabilistic concepts.
Furthermore, we leverage the concrete syntax to simplify verification.
\begin{itemize}
\item We develop an automated procedure for generating pre-conditions of
non-looping commands, inspired by expectation-based systems.
\item We give syntactic conditions for the closedness and termination
properties required for soundness of the loop rules.
\end{itemize}
\myheading{Implementation and case studies.} We implement \mbox{\textsc{Ellora}}\xspace on
top of \textsc{EasyCrypt}\xspace, a general-purpose proof assistant for reasoning
about probabilistic programs, and we mechanically verify a diverse
collection of examples including textbook algorithms and a randomized
routing procedure. We develop an \textsc{EasyCrypt}\xspace formalization of
probability theory from the ground up, including tools like
concentration bounds (e.g., the Chernoff bound), Markov's inequality,
and theorems about probabilistic independence.
\myheading{Embeddings.}
We propose a simple program logic for proving \emph{probabilistic independence}. This
logic is designed to reason about independence in a lightweight way,
as is common in paper proofs. We prove that the logic can be embedded
into \mbox{\textsc{Ellora}}\xspace, and is therefore sound.
Furthermore, we prove an embedding of the Union Bound
logic~\cite{BartheGGHS16-icalp}.
\section{Mathematical Preliminaries}
As is standard, we will model randomized computations using \emph{sub-distributions}.
\begin{definition}
A \emph{sub-distribution} over a set $A$ is defined by a mass function
$\mu : A \to [0,1]$ that gives the probability of the unitary events $a \in
A$. This mass function must be s.t. $\sum_{a \in A} \mu(a)$ is well-defined
and
%
$\wt{\mu} \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \sum_{a\in A} \mu(a) \leq 1$.
%
In particular, the \emph{support}
%
$\supp(\mu) \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \{ a \in A \mid \mu(a) \neq 0 \}$
%
is discrete.\footnote{%
We work with discrete distributions to keep
measure-theoretic technicalities to a minimum, though we do not see
obstacles to generalizing to the continuous setting.}
%
The name ``sub-distribution'' emphasizes that the total probability may be
strictly less than $1$.
%
When the \emph{weight} $\wt{\mu}$ is equal to $1$, we call $\mu$ a
\emph{distribution}. We let $\ensuremath{\mathbf{SDist}}(A)$ denote the set of
sub-distributions over $A$.
%
The probability of an event $E(x)$ w.r.t. a sub-distribution $\mu$,
written $\Pr_{x \sim \mu} [E(x)]$, is defined as
%
$\sum_{x \in A \mid E(x)} \mu(x)$.
\end{definition}
Simple examples of sub-distributions include the \emph{null sub-distribution}
$\mathbf{0}$, which maps each element of the underlying space to $0$; and the
\emph{Dirac distribution centered on $x$}, written $\dunit{x}$, which maps $x$
to $1$ and all other elements to $0$. The following standard
construction gives a monadic structure to sub-distributions.
\begin{definition}
Let $\mu \in \ensuremath{\mathbf{SDist}}(A)$ and $f : A \to \ensuremath{\mathbf{SDist}}(B)$. Then
$\dlet {a} {\mu} {f} \in \ensuremath{\mathbf{SDist}}(B)$ is defined by
\[
\dlet{a} {\mu} {f} (b) \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \sum_{a \in A} \mu(a) \cdot f(a)(b) .
\]
We use notation reminiscent of expected values, as the definition is quite
similar.
\end{definition}
We will need two constructions to model branching statements.
\begin{definition}
Let $\mu_1,\mu_2\in\ensuremath{\mathbf{SDist}}(A)$ such that $\wt{\mu_1}+\wt{\mu_2}\leq
1$. Then $\mu_1+\mu_2$ is the sub-distribution $\mu$ such that
$\mu(a)=\mu_1(a)+\mu_2(a)$ for every $a\in A$.
\end{definition}
\begin{definition}
Let $E \subseteq A$ and $\mu \in \ensuremath{\mathbf{SDist}}(A)$. Then the restriction
$\drestr \mu E$ of $\mu$ to $E$ is the sub-distribution
such that $\drestr \mu E (a)= \mu(a)$ if $a\in E$ and 0 otherwise.
\end{definition}
Sub-distributions are partially ordered under the pointwise order.
\begin{definition}
Let $\mu_1,\mu_2\in\ensuremath{\mathbf{SDist}}(A)$. We say $\mu_1\leq \mu_2$ if $\mu_1(a) \leq
\mu_2(a)$ for every $a\in A$, and we say $\mu_1 = \mu_2$ if
$\mu_1(a) = \mu_2(a)$ for every $a\in A$.
\end{definition}
We use the following lemma when reasoning about the semantics of loops.
\begin{lemma}\label{lem:less:eq:distr}
If $\mu_1\leq\mu_2$ and $\wt{\mu_1}=1$, then $\mu_1=\mu_2$ and $\wt{\mu_2}=1$.
\end{lemma}
Sub-distributions are stable under pointwise-limits.
\begin{definition}
A sequence $(\mu_n)_{n\in\mathbb{N}} \in\ensuremath{\mathbf{SDist}}(A)$ sub-distributions
\emph{converges} if for every $a \in A$, the sequence
$(\mu_n(a))_{n\in\mathbb{N}}$ of real numbers converges. The \emph{limit
sub-distribution} is defined as
\[
\mu_\infty(a) \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \lim_{n \to \infty} \mu_n(a)
\]
for every $a \in A$. We write $\lim_{n\rightarrow\infty} \mu_n$ for $\mu_\infty$.
\end{definition}
\begin{lemma}
\label{lem:limitProb}
Let $(\mu_n)_{n\in\mathbb{N}}$ be a convergent sequence of sub-distributions. Then
for any event $E(x)$, we have:
\[
\forall n\in \mathbb{N}.\, \Pr_{x \sim
\mu_\infty} [E(x)] = \lim_{n \to \infty} \Pr_{x \sim \mu_n} [E(x)].
\]
\end{lemma}
Any bounded increasing real sequence has a limit; the same is true of
sub-distributions.
\begin{lemma}\label{lem:lim:distr}
Let $(\mu_n)_{n\in\mathbb{N}} \in\ensuremath{\mathbf{SDist}}(A)$ be an increasing sequence of
sub-distributions. Then, this sequence converges to $\mu_\infty$
and $\mu_n \leq \mu_\infty$ for every $n\in\mathbb{N}$. In
particular, for any event $E$, we have $\Pr_{x \sim \mu_n} [E]\leq
\Pr_{x \sim \mu_\infty} [E]$ for every $n\in \mathbb{N}$.
\end{lemma}
\section{Programs and Assertions}\label{sec:programs}
Now, we introduce our core programming language and its denotational semantics.
\paragraph*{Programs.}
We base our development on \textsc{pWhile}\xspace, a strongly-typed imperative
language with deterministic assignments, probabilistic assignments,
conditionals, loops, and an $\kw{abort}$ statement which halts the
computation with no result. Probabilistic assignments
$x \rnd g$ assign a value sampled from a distribution $g$ to a program
variable $x$. The syntax of statements is defined by the grammar:
\begin{align*}
s &::= \kw{skip}
\mid \kw{abort}
\mid x \asn e
\mid x \rnd g
\mid s; s \\
&\mid \ifstmt{e}{s}{s}
\mid \while{e}{s}
\mid \call{x}{\mathcal{I}}{e}
\mid \call{x}{\mathcal{A}}{e}
\end{align*}
where $x$, $e$, and $g$ range over typed variables in $\mathcal{X}$,
expressions in $\mathcal{E}$ and distribution expressions in $\mathcal{D}$
respectively. The set $\mathcal{E}$ of well-typed expressions is defined
inductively from $\mathcal{X}$ and a set $\mathcal{F}$ of
function symbols, while the set $\mathcal{D}$ of well-typed distribution
expressions is defined by combining a set of distribution symbols
$\mathcal{S}$ with expressions in $\mathcal{E}$. Programs may call a set
$\mathcal{I}$ of internal procedures as well as a set $\mathcal{A}$ of external
procedures. We assume that we have code for internal procedures but
not for external procedures---we only know indirect information,
like which internal procedures they may call.
Borrowing a convention from cryptography, we call internal
procedures \emph{oracles} and external procedures \emph{adversaries}.
\paragraph*{Semantics.}
\begin{figure*}[t]
\begin{align*}
\dsem{m}{\kw{skip}} &= \dunit{m} \\
\dsem{m}{\kw{abort}} &= \mathbf{0} \\
\dsem{m}{x \asn e} &= \dunit{m\subst{x}{\dsem{m}{e}}} \\
\dsem{m}{x \rnd g} &= \dlet v {\dsem{m}{g}} {\dunit{m[x:=v]}} \\
\dsem{m}{s_1; s_2} &= \dlet {m'} {\dsem{m}{s_1}} {\dsem{m'}{s_2}} \\
\dsem{m}{\ifte{e}{s_1}{s_2}} &=
\text{if $\dsem{m}{e}$ then $\dsem{m}{s_1}$ else $\dsem{m}{s_2}$} \\
\dsem{m}{\while{e}{s}} &=
\lim_{n \to \infty}\ \dsem{m}{(\ift e s)^n;\ift e \kw{abort}} \\
\dsem{m}{\call{x}{\mathcal{I}}{e}} &=
\dsem{m}{\farg{f} \asn e; \fbody{f}; x \asn \fret{f}} \\
\dsem{m}{\call{x}{\mathcal{A}}{e}} &=
\dsem{m}{\farg{a} \asn e; \fbody{a}; x \asn \fret{a}}
\\ \\ \hline \\
\dsem{\mu}{s} &= \dlet m \mu {\dsem{m}{s}}
\end{align*}
\caption{\label{fig:semantics} Denotational semantics of programs}
\end{figure*}
The denotational semantics of programs is adapted from the seminal
work of \cite{Kozen79} and interprets programs as sub-distribution
transformers. We view states as type-preserving mappings from
variables to values; we write $\mathbf{State}$ for the set of states and
$\ensuremath{\mathbf{SDist}}(\mathbf{State})$ for the set of probabilistic states. For each procedure
name $f \in \mathcal{I} \cup \mathcal{A}$, we assume a set
$\VarL[f] \subseteq \mathcal{X}$ of \emph{local variables} s.t.
$\VarL[f]$ are pairwise disjoint. The other variables
$\mathcal{X} \setminus \bigcup_f \VarL[f]$ are \emph{global variables}.
To define the interpretation of expressions and distribution
expressions, we let $\dsem{m}{e}$ denote the interpretation of
expression $e$ with respect to state $m$, and $\dsem{\mu}{e}$
denote the interpretation of expression $e$ with respect to an initial
sub-distribution $\mu$ over states defined by the clause
$\dsem{\mu}{e}\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \dlet m \mu {\dsem{m}{e}}$. Likewise,
we define the semantics of commands in two stages: first interpreted
in a single input memory, then interpreted in an input
sub-distribution over memories.
\begin{definition}
The semantics of commands are given in \cref{fig:semantics}.
\begin{itemize}
\item The semantics $\dsem{m}{s}$ of a statement $s$ in initial state $m$ is a
sub-distribution over states.
\item The (lifted) semantics $\dsem{\mu}{s}$ of a statement $s$
in initial sub-distribution $\mu$ over states is a
sub-distribution over states.
\end{itemize}
\end{definition}
We briefly comment on loops. The semantics of a loop $\while{e}{c}$ is defined
as the limit of its lower approximations, where the $n$-th \emph{lower
approximation} of $\dsem{\mu}{\while{e}{c}}$ is $\dsem{\mu}{(\ift e
s)^n;\ift e \kw{abort}}$, where $\ift{e}{s}$ is shorthand for $\ifte{e}{s}{\kw{skip}}$
and $c^n$ is the $n$-fold composition $c;\cdots;c$. Since the sequence is
increasing, the limit is well-defined by \cref{lem:lim:distr}. In contrast, the
$n$-th \emph{approximation} of $\dsem{\mu}{\while{e}{c}}$ defined by
$\dsem{\mu}{(\ift e s)^n}$ may not converge, since they are not necessarily
increasing. However, in the special case where the output distribution has
weight $1$, the $n$-th lower approximations and the $n$-th approximations have
the same limit.
\begin{lemma}
If the sub-distribution $\dsem{\mu}{\while{e}{c}}$ has weight $1$, then the
limit of $\dsem{\mu}{(\ift e s)^n}$ is defined and
\[
\lim_{n \to \infty}\ \dsem{\mu}{(\ift e s)^n;\ift e \kw{abort}}
= \lim_{n \to \infty}\ \dsem{\mu}{(\ift e s)^n} .
\]
\end{lemma}
This follows by \cref{lem:less:eq:distr}, since lower approximations are below
approximations so the limit of their weights (and the weight of their
limit) is $1$. It will be useful to identify programs that terminate with
probability $1$.
\begin{definition}[Lossless]
A statement $s$ is \emph{lossless} if for every
sub-distribution $\mu$, $\wt{\dsem{\mu}{s}} =\wt{\mu}$, where $|\mu|$
is the total probability of $\mu$. Programs that are not lossless are called \emph{lossy}.
\end{definition}
Informally, a program is lossless if all probabilistic assignments sample from
full distributions rather than sub-distributions, there are no $\kw{abort}$
instructions, and the program is almost surely terminating, i.e.\ infinite
traces have probability zero. Note that if we restrict the language to sample
from full distributions, then losslessness coincides with almost sure
termination.
Another important class of loops are loops with a uniform upper bound
on the number of iterations. Formally, we say that a loop
$\while{e}{s}$ is \emph{certainly terminating} if there exists $k$
such that for every sub-distribution $\mu$, we have $
\wt{\dsem{\mu}{\while{e}{s}}} =
\wt{\dsem{\mu}{(\ift{e}{s})^k}}$. Note that certain termination of
a loop does not entail losslessness---the output distribution of the
loop may not have weight $1$, for instance, if the loop samples from a
sub-distribution or if the loop aborts with positive probability.
\paragraph*{Semantics of Procedure Calls and Adversaries.}
The semantics of internal procedure calls is
straightforward. Associated to each procedure name $f \in \mathcal{I}$, we
assume a designated input variable $\farg{f} \in \VarL[f]$, a
piece of code $\fbody{f}$ that executes the function call, and a
result expression $\fret{f}$. A function call $\call{x}{\mathcal{I}}{e}$ is
then equivalent to $\farg{f} \asn e; \fbody{f}; x \asn \fret{f}$.
Procedures are subject to well-formedness criteria: procedures should
only use local variables in their scope and after initializing them,
and should not perform recursive calls.
\jh{The role of $\fbody{a}$ here is also confusing. I put
``unspecified'', but what does this mean really since we don't have
the code?} \gb{The idea is that we need code to interpret
adversaries. But the logic is sound for any interpretation of
adversaries.}
External procedure calls, also known as adversary calls, are a bit
more involved. Each name $a \in \mathcal{A}$ is parametrized by a set
$\aora{a} \subseteq \mathcal{I}$ of internal procedures which the adversary may call, a
designated input variable $\farg{a} \in \VarL[a]$, a (unspecified)
piece of code $\fbody{a}$ that executes the function call, and a
result expression $\fret{a}$. We assume that adversarial code can
only access its local variables in $\VarL[a]$ and can only make calls
to procedures in $\aora{a}$. It is possible to impose more
restrictions on adversaries---say, that they are lossless---but for
simplicity we do not impose additional assumptions on
adversaries here.
\section{Proof System} \label{sec:proofsystem}
In this section we introduce a program logic for proving properties of
probabilistic programs. The logic is abstract---assertions are arbitrary
predicates on sub-distributions---but the meta-theoretic properties are clearest
in this setting. In the following section, we will give a concrete version
suitable for practical use.
\paragraph*{Assertions and Closedness Conditions.}
We use predicates on state distribution.
\begin{definition}[Assertions]
The set $\mathsf{Assn}$ of assertions is defined as $\mathcal{P}(\ensuremath{\mathbf{SDist}} (\mathbf{State}))$. We
write $\eta(\mu)$ for $\mu \in \eta$.
\end{definition}
Usual set operations are lifted to assertions using their logical
counterparts, e.g., $\eta \land \eta' \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \eta \cap \eta'$
and $\neg \eta \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \overline{\eta}$.
Our program logic uses a few additional constructions. Given a predicate
$\phi$ over states, we define
\begin{gather*}
\detm{\phi}(\mu) \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \forall m.\, m
\in \supp(\mu) \implies \phi(m)
\end{gather*}
where $\supp(\mu)$ is the set of all states with non-zero probability under
$\mu$. Intuitively, $\phi$ holds deterministically on all
states that we may sample from the distribution. To reason about branching
commands, given two assertions $\eta_1$ and $\eta_2$, we let
\begin{gather*}
( \eta_1 \oplus \eta_2)(\mu) \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}}
\exists \mu_1, \mu_2 .\,
\mu = \mu_1 + \mu_2 \land \eta_1(\mu_1) \land \eta_2(\mu_2)
.
\end{gather*}
This assertion means that the sub-distribution is the sum
of two sub-distributions such that $\eta_1$ holds on the first piece
and $\eta_2$ holds on the second piece.
Given an assertion $\eta$ and an event $E \subseteq \mathbf{State}$, we let
$
\drestr{\eta}{E}(\mu)\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \eta (\drestr{\mu}{E}) .
$
This assertion holds exactly when $\eta$ is true on the portion of the
sub-distribution satisfying $E$. Finally, given an assertion $\eta$ and a function
$F$ from $\ensuremath{\mathbf{SDist}}(\mathbf{State})$ to $\ensuremath{\mathbf{SDist}}(\mathbf{State})$, we define
$
\eta[F] \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \lambda \mu .\, \eta(F(\mu)) .
$
Intuitively, $\eta[F]$ is true in a sub-distribution $\mu$ exactly when
$\eta$ holds on $F(\mu)$.
Now, we can define the closedness properties of assertions. These
properties will be critical to our rules for $\kw{while}$ loops.
\begin{definition}[Closedness properties]\label{def:closedness}
A family of assertions $(\eta_n)_{n\in\mathbb{N}^\infty}$ is:
\begin{itemize}
\item
$u$\emph{-closed} if for every increasing sequence of
sub-distributions $(\mu_n)_{n\in\mathbb{N}}$ such that
$\dsvalid{\eta_n}{\mu_n}$ for all $n\in\mathbb{N}$ then
$\dsvalid{\eta_\infty}{\lim_{n\to\infty}\mu_n}$;
\item
$t$\emph{-closed} if for every converging sequence of
sub-distributions $(\mu_n)_{n\in\mathbb{N}}$ such that
$\dsvalid{\eta_n}{\mu_n}$ for all $n\in\mathbb{N}$ then
$\dsvalid{\eta_\infty}{\lim_{n\to\infty}\mu_n}$;
\item
$d$\emph{-closed} if it is $t$-closed and downward closed, that is
for every sub-distributions $\mu \leq \mu'$,
$\dsvalid{\eta_\infty}{\mu'}$ implies
$\dsvalid{\eta_\infty}{\mu}$.
\end{itemize}
When $(\eta_n)_n$ is constant and equal to $\eta$, we say that $\eta$ is
$u$-/$t$-/$d$-closed.
\end{definition}
Note that $t$-closedness implies $u$-closedness, but the converse does not
hold. Moreover, $u$-closed, $t$-closed and $d$-closed
assertions are closed under arbitrary intersections and finite unions,
or in logical terms under finite boolean combinations, universal
quantification over arbitrary sets and existential quantification over
finite sets.
Finally, we introduce the necessary machinery for the frame rule. The
set $\MV(s)$ of \emph{modified} variables of a statement $s$ consists
of all the variables on the left of a deterministic or probabilistic
assignment. In this setting, we say that an assertion $\eta$ is
\emph{separated} from a set of variables $X$, written
$\aindep{\eta}{X}$, if $\eta(\mu_1) \iff \eta(\mu_2)$ for any
distributions $\mu_1$, $\mu_2$ s.t. $\wt{\mu_1} =
\wt{\mu_2}$ and ${\mu_1}_{| \overline{X}} = {\mu_2}_{|
\overline{X}}$ where for a set of variables $X$, the restricted
sub-distribution $\mu_{| X}$ is
\[
\mu_{| X} : m \in \mathbf{State}_{| X} \mapsto \Pr_{m' \sim \mu} [m = m'_{| X}]
\]
where $\mathbf{State}_{| X}$ and $m_{| X}$ restrict $\mathbf{State}$ and $m$ to the variables in
$X$.
Intuitively, an assertion is separated from a set of variables
$X$ if every two sub-distributions that agree on the variables outside
$X$ either both satisfy the assertion, or both refute the assertion.
\paragraph*{Judgments and Proof Rules.}
Judgments are of the form $\hoare{\eta}{s}{\eta'}$, where the assertions
$\eta$ and $\eta'$ are drawn from $\mathsf{Assn}$.
\begin{definition}
A judgment $\hoare{\eta}{s}{\eta'}$ is \emph{valid}, written
$\models \hoare{\eta}{s}{\eta'}$,
if $\dsvalid{\eta'}{\dsem{\mu}{s}}$ for every interpretation of
adversarial procedures and every probabilistic state $\mu$ such
that $\dsvalid{\eta}{\mu}$.
\end{definition}
\Cref{fig:nonlooping:rules} describes the structural and basic rules
of the proof system.
Validity of judgments is preserved under standard structural rules,
like the rule of consequence \rname{Conseq}. As usual, the rule of
consequence allows to weaken the post-condition and to strengthen the
post-condition; in our system, this rule serves as the interface
between the program logic and mathematical theorems from probability
theory. The \rname{Exists} rule is helpful to deal with existentially
quantified pre-conditions.
The rules for $\kw{skip}$, assignments, random samplings and sequences are
all straightforward. The rule for $\kw{abort}$ requires $\detm{\bot}$ to
hold after execution; this assertion uniquely characterizes the
resulting null sub-distribution. The rules for assignments and random
samplings are semantical.
\begin{figure}
\begin{mathpar}
\inferrule[Conseq]
{\eta_0 \Rightarrow \eta_1 \\
\hoare{\eta_1}{s}{\eta_2} \\
\eta_2 \Rightarrow\eta_3 }
{\hoare{\eta_0}{s}{\eta_3}}
\and
\inferrule[Exists]
{
\forall x:T.\, \hoare{\eta}{s}{\eta'}}
{\hoare{\exists x:T.\, \eta}{s}{\eta'}}
\and
\inferrule[Abort]
{ }
{\hoare{\eta}{\kw{abort}}{\detm{\bot}}}
\and
\inferrule[Assgn]
{\eta' \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \eta[\dsem{}{x \asn e}]}
{\hoare{\eta'}{x \asn e}{\eta}}
\and
\inferrule[Skip]
{ }
{\hoare{\eta}{\kw{skip}}{\eta}}
\and
\inferrule[Sample]
{\eta' \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \eta[\dsem{}{x\rnd{g}}]}
{\hoare{\eta'}{x \rnd g}{\eta}}
\and
\inferrule[Seq]
{\hoare{\eta_0}{s_1}{\eta_1}\\
\hoare{\eta_1}{s_2}{\eta_2}}
{\hoare{\eta_0}{s_1;s_2}{\eta_2}}
\and
\inferrule[Cond]
{\hoare{\eta_1 \land \detm{e}}{s_1}{\eta'_1}\\
\hoare{\eta_2 \land \detm{\neg e}}{s_2}{\eta'_2}}
{\hoare{(\eta_1 \wedge \detm{e})\oplus (\eta_2 \wedge \detm{\neg e})}
{\ifstmt{e}{s_1}{s_2}}{\eta'_1 \oplus \eta'_2}}
\and
\inferrule[Split]
{\hoare{\eta_1}{s}{\eta'_1} \\
\hoare{\eta_2}{s}{\eta'_2}}
{\hoare{\eta_1 \oplus \eta_2}{s}{\eta'_1 \oplus \eta'_2}}
\and
\inferrule[Frame]
{ \aindep{\eta}{\MV(s)} \\ \mbox{$s$ is lossless} }
{ \hoare{\eta}{s}{\eta} }
\and
\inferrule[Call]
{\hoare{\eta}{\farg{f} \asn e; \fbody{f}}
{\eta'[\dsem{}{x \asn \fret{f}}]}}
{\hoare{\eta}{\call{x}{f}{e}}{\eta'}}
\end{mathpar}
\caption{\label{fig:nonlooping:rules} Structural and basic rules}
\end{figure}
The rule \rname{Cond} for conditionals requires that the
post-condition must be of the form $\eta_1\oplus\eta_2$; this
reflects the semantics of conditionals, which splits the initial
probabilistic state depending on the guard, runs both branches, and
recombines the resulting two probabilistic states.
The next two rules (\rname{Split} and \rname{Frame}) are useful for
local reasoning. The \rname{Split} rule reflects the additivity of the
semantics and combines the \mbox{pre-} and post-conditions using the $\oplus$
operator.
The \rname{Frame} rule asserts that lossless statements preserve
assertions that are not influenced by modified variables.
The rule \rname{Call} for internal procedures is as expected, replacing the
procedure call $f$ with its definition.
\medskip
\Cref{fig:looping:rules} presents the rules for loops. We consider
four rules specialized to the termination behavior.
The \rname{While} rule is the most general rule, as it deals with
arbitrary loops. For simplicity, we explain the rule in the special
case where the family of assertions is constant, i.e.\ we have
$\eta_n=\eta$ and $\eta'_n=\eta'$. Informally, the
$\eta$ is the loop invariant and $\eta'$ is
an auxiliary assertion used to prove the invariant. We require that
$\eta$ is $u$-closed, since the semantics of a loop is
defined as the limit of its lower approximations. Moreover, the first
premise ensures that starting from $\eta$, one
guarded iteration of the loop establishes $\eta'$; the
second premise ensures that restricting to $\neg e$ a
probabilistic state $\mu'$ satisfying $\eta'$ yields a
probabilistic state $\mu$ satisfying $\eta$. It is possible to
give an alternative formulation where the second premise
is substituted by the logical constraint $\drestr{\eta'}{\neg
e}\implies \eta$. As usual, the post-condition of the loop is the
conjunction of the invariant with the negation of the guard (more
precisely in our setting, that the guard has probability 0).
The \rname{While-AST} rule deals with lossless loops. For
simplicity, we explain the rule in the special case where the family
of assertions is constant, i.e.\ we have $\eta_n=\eta$. In this
case, we know that lower approximations and approximations have the
same limit, so we can directly prove an invariant that holds after one
guarded iteration of the loop. On the other hand, we must now require
that the $\eta$ satisfies the stronger property of
$t$-closedness.
The \rname{While-D} rule handles arbitrary loops with a
$d$-closed invariant; intuitively, restricting a sub-distribution
that satisfies a downwards closed assertion $\eta$ yields a
sub-distribution which also satisfies $\eta$.
The \rname{While-CT} rule deals with certainly terminating loops. In
this case, there is no requirement on the assertions.
We briefly compare the rules from a verification perspective. If the
assertion is $d$-closed, then the rule \rname{While-D} is easier to
use, since there is no need to prove any termination requirement.
Alternatively, if we can prove certain termination of the loop, then
the rule \rname{While-CT} is the best to use since it does not impose
any condition on assertions. When the loop is lossless, there
is no need to introduce an auxiliary assertion $\eta'$, which
simplifies the proof goal. Note however that it might still be
beneficial to use the \rname{While} rule, even for lossless loops,
because of the weaker requirement that the invariant is $u$-closed
rather than $t$-closed.
\medskip
Finally, \cref{fig:adv:rules} gives the adversary rule for general adversaries.
It is highly similar to the general rule \rname{While-D} for loops since the
adversary may make an arbitrary sequence of calls to the oracles in $\aora{a}$
and may not be lossless. Intuitively, $\eta$ plays the role of the invariant: it
must be $d$-closed and it must be preserved by every oracle call with arbitrary
arguments. If this holds, then $\eta$ is also preserved by the adversary call.
Some framing conditions are required, similar to the ones of the
\textsc{[Frame]} rule: the invariant must not be influenced by the state
writable by the external procedures.
It is possible to give other variants of the adversary rule with more
general invariants by restricting the
adversary, e.g., requiring losslessness or bounding the number of calls the
external procedure can make to oracles, leading to rules akin to the almost
surely terminating and certainly terminating loop rules, respectively.
\begin{figure*}
\begin{mathpar}
\inferrule[While]
{
\uclosed{(\eta'_n)_{n\in\mathbb{N}^\infty}} \\\\
\forall n.\, \hoare{\eta_n}{\ift{e}{s}}{\eta_{n+1}} \\
\forall n.\, \hoare{\eta_n}{\ift{e}{\kw{abort}}}{\eta'_{n}}}
{\hoare{\eta_0}{\while{e}{s}}{\eta'_\infty \wedge \detm{\neg e}}}
\\
\inferrule[While-AST]
{\tclosed{(\eta_n)_{n\in\mathbb{N}^\infty}} \\
\forall n.\, \hoare{\eta_n}{\ift{e}{s}}{\eta_{n+1}} \\
\forall \mu .\, \eta_0(\mu) \implies \wt{\dsem{\mu}{(\while{e}{s})}}=1}
{\hoare{\eta_0}{\while{e}{s}}{\eta_\infty \wedge \detm{\neg e}}}
\\
\inferrule[While-D]
{\dclosed{(\eta_n)_{n\in\mathbb{N}^\infty}} \\
\forall n.\, \hoare{\eta_n}{\ift{e}{s}}{\eta_{n+1}}}
{\hoare{\eta_0}{\while{e}{s}}{\eta_\infty \wedge \detm{\neg e}}}
\\
\inferrule[While-CT]
{\forall n.\, \hoare{\eta_n}{\ift{e}{s}}{\eta_{n+1}} \\
\forall \mu .\, \eta_0(\mu) \implies
\dsem{\mu}{(\ift{e}{s})^k} = \dsem{\mu}{(\while{e}{s})}}
{\hoare{\eta_0}{\while{e}{s}}{\eta_k \wedge \detm{\neg e}}}
\end{mathpar}
\caption{\label{fig:looping:rules} Rules for loops}
\end{figure*}
\begin{figure*}
\begin{mathpar}
\inferrule[Adv]
{\forall n\in\mathbb{N}^\infty.~\aindep{\eta_n}{\{x, \mathfrak{s}\}} \\
\dclosed{(\eta_n)_{n\in\mathbb{N}^\infty}} \\\\
\forall f \in \aora{a}, x \in \VarL[a], e \in \mathcal{E}, n \in \mathbb{N} .\,
\hoare {\eta_n} {\call{x}{f}{e}} {\eta_{n + 1}}}
{\hoare {\eta_0} {\call{x}{a}{e}} {\eta_{\infty}}}
\end{mathpar}
\caption{\label{fig:adv:rules} Rules for adversaries}
\end{figure*}
\paragraph*{Soundness and Relative Completeness.}
Our proof system is sound and relatively complete with respect to the semantics;
these proofs have also been formalized in the \textsc{Coq} proof assistant.
\begin{theorem}[Soundness]
Every judgment $\hoare{\eta}{s}{\eta'}$ provable using the rules of
our logic is valid.
\end{theorem}
Completeness of the logic follows from the next lemma, whose proof
makes an essential use of the \rname{While} rule. In the sequel, we
use $\carac{\mu}$ to denote the characteristic function of a
probabilistic state $\mu$, an assertion stating that the current state is
equal to $\mu$.
\begin{lemma}
For every probabilistic state $\mu$, the following judgment is
provable using the rule of the logic:
\[ \hoare{\carac{\mu}}{s}{\carac{\dsem{\mu}{s}}}. \]
\end{lemma}
\begin{proof} By induction on the structure of $s$.
\begin{itemize}
\item $s = \kw{abort}$, $s = \kw{skip}$, $x \asn e$ and $s = x \rnd g$ are
trivial;
\item $s = s_1;s_2$, we have to prove
%
\[ \hoare{\carac{\mu}}{s_1;s_2}
{\carac{\dsem{\dsem{\mu}{s_1}}{s_2}}} . \]
%
We apply the \rname{Seq} rule with
$\eta_1 = \carac{\dsem{\mu}{s_1}}$ premises can be
directly proved using the induction hypothesis;
\item $s = \ifstmt{e}{s_1}{s_2}$, we have to prove
%
\[ \hoare{\carac{\mu}}{\ifstmt{e}{s_1}{s_2}}
{
(\carac{\dsem{\drestr{\mu}{e}}{s_1}} \oplus
\carac{\dsem{\drestr{\mu}{\neg e}}{s_2})}} . \]
%
We apply the \rname{Conseq} rule to be able to apply the the
\rname{Cond} rule with
$\eta_1 = \carac{\dsem{\drestr{\mu}{e}}{s_1}}$ and
$\eta_2 = \carac{\dsem{\drestr{\mu}{\neg e}}{s_2}}$ Both
premises can be proved by an application of the \rname{Conseq} rule
followed by the application of the induction hypothesis.
\item $s = \while{e}{s}$, we have to prove
%
\[ \hoare{\carac{\mu}}{\while{e}{s}}
{\carac{
\lim_{n \to \infty}\ \dsem{\mu}{(\ift e s)^n;\ift e \kw{abort}}}} .\]
%
We first apply the \rname{While} rule with
$\eta'_n = \carac{\dsem{\mu}{(\ift e s)^n}}$ and
\[
\eta_n = \carac{\dsem{\mu}{(\ift e s)^n;\ift e \kw{abort}}} .
\]
%
For the first premise we apply the same process as for the
conditional case: we apply the \rname{Conseq} and \rname{Cond}
rules and we conclude using the induction hypothesis (and the
\rname{Skip} rule). For the second premise we follow the same
process but we conclude using the \rname{Abort} rule instead of the
induction hypothesis. Finally we conclude since
$\uclosed{(\eta_n)_{n\in\mathbb{N}^\infty}}$. \qedhere
\end{itemize}
\end{proof}
The abstract logic is also relatively complete. This property will be less
important for our purposes, but it serves as a basic sanity check.
\begin{theorem}[Relative completeness]
Every valid judgment is derivable.
\end{theorem}
\begin{proof}
Consider a valid judgment $\hoare{\eta}{s}{\eta'}$.
Let $\mu$ be a probabilistic state such that
$\dsvalid{\eta}{\mu}$. By the above proposition,
$ \hoare{\carac{\mu}}{s}{\carac{\dsem{\mu}{s}}}$.
Using the validity of the judgment and \rname{Conseq}, we have
$\hoare{\carac{\mu} \land \dsvalid{\eta}{\mu}}{s}{\eta'}$.
Using the \rname{Exists} and \rname{Conseq} rules, we conclude
$\hoare{\eta}{s}{\eta'}$
as required.
\end{proof}
The side-conditions in the loop rules (e.g.,
$\mathsf{uclosed}$/$\mathsf{tclosed}$/$\mathsf{dclosed}$ and the weight
conditions) are difficult to prove, since they are semantic properties. Next, we
present a concrete version of the logic with give easy-to-check, syntactic
sufficient conditions.
\section{A Concrete Program Logic} \label{sec:concrete}
To give a more practical version of the logic, we begin by setting a concrete
syntax for assertions
\paragraph*{Assertions.}
We use a two-level assertion language, presented in
\cref{fig:syntax}. A \emph{probabilistic
assertion} $\eta$ is a formula built from comparison of probabilistic
expressions, using first-order quantifiers and connectives, and the
special connective $\oplus$. A \emph{probabilistic expression} $p$ can be a
logical variable $v$, an operator applied to probabilistic expressions
$o(\vec{p})$ (constants are $0$-ary operators), or the
expectation $\ex{\tilde{e}}$ of a state expression $\tilde{e}$.
A \emph{state expression} $\tilde{e}$ is either a program variable $x$, the
characteristic function $\ind{\phi}$ of a state assertion $\phi$, an
operator applied to state expressions $o(\vec{\tilde{e}})$, or the
expectation $\dlet{v}{g}{\tilde{e}}$ of state expression $\tilde{e}$ in a
given distribution $g$. Finally, a \emph{state assertion} $\phi$ is a
first-order formula over program variables.
Note that the set of operators is left unspecified but we assume that
all the expressions in $\mathcal{E}$ and $\mathcal{D}$ can be encoded by operators.
\iffull
\begin{figure}
\else
\begin{wrapfigure}{l}{0.55\textwidth
\fi
\begin{align}
\tilde{e} & ::= x
\mid v
\mid \ind{\phi}
\mid \dlet{v}{g}{\tilde{e}}
\mid o(\vec{\tilde{e}})
\tag{S-expr.} \\
\phi & ::= \tilde{e} \bowtie \tilde{e}
\mid FO(\phi)
\tag{S-assn.} \\
p & ::= v \mid o(\vec{p}) \mid \ex{\tilde{e}} \tag{P-expr.} \\
\eta & ::= p \bowtie p
\mid \eta \oplus \eta
\mid FO(\eta)
\tag{P-assn.} \\
\bowtie &\mathrel{\in} \{ \mathrel{=} , \mathrel{<}, \mathrel{\leq} \}
\qquad o \in Ops
\tag{Ops.}
\end{align}
\caption{\label{fig:syntax} Assertion syntax}
\iffull
\end{figure}
\else
\end{wrapfigure}
\fi
The interpretation of the concrete syntax is as expected. The interpretation of
probabilistic assertions is relative to a valuation $\rho$ which maps
logical variables to values, and is an element of $\mathsf{Assn}$. The
definition of the interpretation is straightforward; the only
interesting case is $\pdenot{\ex{\tilde{e}}}$ which is defined by
$\dlet{m}{\mu}{\mdenot{\tilde{e}}}$, where $\mdenot{\tilde{e}}$ is the
interpretation of the state expression $\tilde{e}$ in the memory $m$ and
valuation $\rho$. The interpretation of state expressions is a mapping
from memories to values, which can be lifted to a mapping from
distributions over memories to distributions over values. The
definition of the interpretation is straightforward; the most
interesting case is for expectation $\mdenot{\dlet{v}{g}{\tilde{e}}}
\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \dlet{w}{\mdenot{g}}{\gdenot{\tilde{e}}{\rho\subst{v}{w}}{m}}$.
We present the full interpretations in the supplemental materials.
Many standard concepts from probability theory have a natural
representation in our syntax. For example:
\begin{itemize}
\item the probability that $\phi$ holds in some probabilistic state
is represented by the probabilistic expression $\Pr[\phi]
\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \pr{\ind{\phi}}{}$;
\item probabilistic independence of state expressions $\tilde{e}_1$,
\ldots, $\tilde{e}_n$ is modeled by the probabilistic assertion $\indep
\{\tilde{e}_1, \ldots, \tilde{e}_n\}$, defined by the clause\footnote{%
The term $\Pr[\top]^{n - 1}$ is necessary since we work with sub-distributions.}
%
$$\forall v_1 \ldots v_n,~ \Pr[\top]^{n - 1} \Pr[\bigwedge_{i=1\ldots
n} \tilde{e}_i = v_i] = \prod_{i=1\ldots n}\Pr[\tilde{e}_i = v_i] ;
$$
\item the fact that a distribution is proper is modeled by the probabilistic
assertion $\mathcal{L} \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \Pr[\top] = 1$;
\item a state expression $\tilde{e}$ distributed according to a law $g$ is
modeled by the probabilistic assertion
$$ \follows{\tilde{e}}{g} \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \forall w,~ \Pr[\tilde{e} = w] = \ex{\dlet{v}{g}{\ind{v = w}}} .$$
%
The inner expectation computes the probability that $v$ drawn from $g$ is
equal to a fixed $w$; the outer expectation weights the inner probability by
the probability of each value of $w$.
\end{itemize}
We can easily define $\square$ operator from the previous section in our new
syntax: $\detm{\phi} \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \Pr[\neg\phi] = 0$.
\paragraph*{Syntactic Proof Rules.}
Now that we have a concrete syntax for assertions, we can give
syntactic versions of many of the existing proof rules. Such proof
rules are often easier to use since they avoid reasoning about the
semantics of commands and assertions. We tackle the non-looping rules
first, beginning with the following syntactic rules for assignment and
sampling:
\begin{mathpar}
\inferrule[Assgn]
{ }
{\hoare{\eta\subst{x}{e}}{x \asn e}{\eta}}
\and
\inferrule[Sample]
{ }
{\hoare{\Samp{x}{g}(\eta) }{x \rnd g}{\eta}}
\end{mathpar}
The rule for assignment is the usual rule from Hoare logic,
replacing the program variable $x$ by its corresponding
expression $e$ in the pre-condition. The replacement $\eta\subst{x}{e}$ is done recursively
on the probabilistic assertion $\eta$; for instance
for expectations, it is defined by
$
\ex{\tilde{e}}\subst{x}{e} \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \ex{\tilde{e}\subst{x}{e}} ,
$
where $\tilde{e}\subst{x}{e}$ is the syntactic substitution.
The rule for sampling uses probabilistic substitution operator
$\Samp{x}{g}(\eta)$, which replaces all occurrences of $x$ in $\eta$ by a new
integration variable $t$ and records that $t$ is drawn from $g$; the operator is
defined in \cref{fig:samp}.
\iffull
\begin{figure}
\else
\begin{wrapfigure}{l}{0.45\textwidth}
\fi
\[
\begin{array}{lcll}
\Samp{x}{g}(v) & \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} & v \\
\Samp{x}{g}\left(\pr{\tilde{e}}{}\right) & \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} &
\ex{\dlet{t}{g}{\tilde{e}\subst{x}{t}}} \\
\Samp{x}{g}(o(\vec{\eta})) &\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} &
o(\Samp{x}{g}(\eta_1), \ldots, \Samp{x}{g}(\eta_n)) \\
\Samp{x}{g}(\eta_1 \bowtie \eta_2) &\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} &
\Samp{x}{g}(\eta_1) \bowtie \Samp{x}{g}(\eta_2)
\end{array}
\]
for $o \in \ensuremath{\mathbf{Ops}}, \bowtie \in\{\wedge,\vee, \Rightarrow\}$.
\caption{Syntactic op. $\Samp{}{}$ (main cases) \label{fig:samp}}
\iffull
\end{figure}
\else
\end{wrapfigure}
\fi
\begin{figure*}
\begin{align*}
\sidecond{CTerm} & \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}} \begin{array}[t]{l}
\hoare{\mathcal{L}\wedge\detm{(\tilde{e}=k\wedge 0< k \land b)}}{s}{
\mathcal{L}\wedge \detm{(\tilde{e} < k)}} \\
\models \eta \Rightarrow (\exists \dot{y}.\; \detm{\tilde{e} \leq \dot{y}}) \land
\detm{(\tilde{e}=0 \Rightarrow \neg b)}
\end{array}
\\
\sidecond{ASTerm} & \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}}
\begin{array}[t]{l}
\hoare{\mathcal{L}\wedge\detm{(\tilde{e}=k\wedge 0< k \leq K \land b)}}{s}{
\mathcal{L}\wedge \detm{(0\leq \tilde{e} \leq K )}\wedge
\Pr[\tilde{e} < k] \geq \epsilon} \\
\models \eta \Rightarrow \detm{(0 \leq \tilde{e} \leq K \land
\tilde{e}=0 \Rightarrow \neg b)} \\
\models \tclosed{\eta}
\end{array}
\end{align*}
\caption{Side-conditions for loop rules} \label{fig:loop:syntactic}
\end{figure*}
Next, we turn to the loop rule. The side-conditions from
\cref{fig:looping:rules} are purely semantic, while in practice it is more
convenient to use a sufficient condition in the Hoare logic. We give
sufficient conditions for ensuring certain and almost-sure termination in
\cref{fig:loop:syntactic};
$\tilde{e}$ is an integer-valued
expression. The first side-condition $\sidecond{CTerm}$ shows certain
termination given a strictly decreasing \emph{variant} $\tilde{e}$ that is
bounded below, similar to how a decreasing variant shows termination for
deterministic programs. The second side-condition $\sidecond{ASTerm}$ shows
almost-sure termination given a probabilistic variant $\tilde{e}$, which must be
bounded both above and below. While $\tilde{e}$ may increase with some
probability, it must decrease with strictly positive probability. This condition
was previously considered by \cite{HartSP83} for probabilistic transition
systems and also used in expectation-based
approaches~\cite{Morgan96:facs,Hurd03-jlap}. Our framework can also support
more refined conditions (e.g., based on super-martingales
\cite{ChakarovS13,mciver2016new}), but the condition $\sidecond{ASTerm}$
already suffices for most randomized algorithms.
While $t$-closedness is a semantic condition (cf. \cref{def:closedness}),
there are simple syntactic conditions to guarantee it. For instance, assertions
that carry a non-strict comparison $\bowtie \mathbin{\in} \{\leq,\geq,=\}$
between two bounded probabilistic expressions are $t$-closed;
the assertion stating probabilistic independence of a set of expressions is
$t$-closed.
\paragraph*{Precondition Calculus.}
With a concrete syntax for assertions, we are also able to incorporate syntactic
reasoning principles. One classic tool is Morgan and McIver's \emph{greatest
pre-expectation}, which we take as inspiration for a pre-condition calculus for
the loop-free fragment of \mbox{\textsc{Ellora}}\xspace. Given an assertion $\eta$ and a loop-free
statement $s$, we mechanically construct an assertion $\eta^*$ that is the
pre-condition of $s$ that implies $\eta$ as a post-condition. The basic idea is
to replace each expectation expression $p$ inside $\eta$ by an expression $p^*$
that has the same denotation before running $s$ as $p$ after running $s$. This
process yields an assertion $\eta^*$ that, interpreted before running $s$, is
logically equivalent to $\eta$ interpreted after running $s$.
The computation rules for pre-conditions are defined in \cref{prem_wp}. For a
probability assertion $\eta$, its pre-condition $\wpre(s,\eta)$ corresponds to
$\eta$ where the expectation expressions of the form $\pr{\tilde{e}}{}$ are
replaced by their corresponding \emph{pre-term}, $\prem(s,\pr{\tilde{e}}{})$.
Pre-terms correspond loosely to Morgan and McIver's \emph{pre-expectations}---we
will make this correspondence more precise in the next section. The main
interesting cases for computing pre-terms are for random sampling and conditionals. For random sampling
the result is $\Samp{x}{g}(\pr{\tilde{e}}{})$, which corresponds to the
[\textsc{Sample}] rule. For conditionals, the expectation expression is split into
a part where $e$ is true and a part where $e$ is not true. We restrict the
expectation to a part satisfying $e$ with the operator
$
\cond{\pr{\tilde{e}}{}}{e}
\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}}
\pr{\tilde{e} \cdot \ind{e}}{} .
$
This corresponds to the expected value of $\tilde{e}$ on the portion of the
distribution where $e$ is true.
Then, we can build the pre-condition calculus into \mbox{\textsc{Ellora}}\xspace.
\begin{figure}
\begin{align*}
\prem(s_1; s_2, \pr{\tilde{e}}{})
&\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}}
\prem(s_1,\prem(s_2, \pr{\tilde{e}}{}))
\\
\prem(x \asn e, \pr{\tilde{e}}{})
&\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}}
\pr{\tilde{e}}{} \subst{x}{e}
\\
\prem(x \rnd g, \pr{\tilde{e}}{})
& \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}}
\Samp{x}{g}(\pr{\tilde{e}}{})
\\
\prem(\ifstmt{e}{s_1}{s_2}, \pr{\tilde{e}}{})
& \mathrel{\stackrel{\scriptscriptstyle \triangle}{=}}
\cond{\prem({s_1}, \pr{\tilde{e}}{})}{e} +
\cond{\prem({s_2}, \pr{\tilde{e}}{})}{\neg e}
\\[1em]
\wpre(s, p_1 \bowtie p_2)
&\mathrel{\stackrel{\scriptscriptstyle \triangle}{=}}
\prem(s, p_1) \bowtie \prem(s, p_2)
\end{align*}
\caption{Precondition calculus (selected)}
\label{prem_wp}
\end{figure}
\begin{thm} \label{thm:pc}
Let $s$ be a non-looping command. Then, the following rule is derivable in the
concrete version of \mbox{\textsc{Ellora}}\xspace:
\[
\inferrule[PC]
{ }
{\hoare{\wpre(s,\eta)}{s}{\eta}}
\]
\end{thm}
\section{Case Studies: Embedding Lightweight Logics}
While \mbox{\textsc{Ellora}}\xspace is suitable for general-purpose reasoning about probabilistic
programs, in practice humans typically use more special-purpose proof
techniques---often targeting just a single, specific kind of property, like
probabilistic independence---when proving probabilistic assertions. When these
techniques apply, they can be a convenient and powerful tool.
To capture this intuitive style of reasoning, researchers have
considered lightweight program logics where the assertions and proof
rules are tailored to a specific proof technique. We demonstrate how
to integrate these tools in an assertion-based logic by introducing
and embedding a new logic for reasoning about independence and
distribution laws, useful properties when analyzing randomized
algorithms. We crucially rely on the rich assertions in \mbox{\textsc{Ellora}}\xspace---it
is not clear how to extend expectation-based approaches to support
similar, lightweight reasoning. Then, we show to embed the union bound
logic \cite{BartheGGHS16-icalp} for proving accuracy bounds.
\subsection{Law and Independence Logic}
We begin by describing the law and independence logic \textsc{IL}\xspace,
a proof system with intuitive rules that are easy to
apply and amenable to automation. For simplicity, we only
consider programs which sample from the binomial distribution, and
have deterministic control flow---for lack of space, we also omit
procedure calls.
\begin{definition}[Assertions]
\textsc{IL}\xspace assertions have the grammar:
\[
\begin{array}{rcl}
\xi & := & \mathsf{det} (e)
\mid \indep E
\mid \follows{e}{\mathrm{B}(e,p)}
\mid \top
\mid \bot
\mid \xi \land \xi
\end{array}
\]
where $e\in\mathcal{E}$, $E\subseteq\mathcal{E}$, and $p\in [0,1]$.
\end{definition}
The assertion $\mathsf{det}(e)$ states that $e$ is deterministic in the
current distribution, i.e., there is at most one element in the
support of its interpretation. The assertion $\indep E$ states that
the expressions in $E$ are independent, as formalized in the previous
section. The assertion $\follows{e}{\mathrm{B}(m,p)}$ states that $e$ is
distributed according to a binomial distribution with parameter $m$
(where $m$ can be an expression) and constant probability $p$, i.e.\ the
probability that $e=k$ is equal to the probability that exactly $k$
independent coin flips return heads using a biased coin that returns heads with
probability $p$.
Assertions can be seen as an instance of a logical abstract domain,
where the order between assertions is given by implication based on a
small number of axioms. Examples of such axioms
include independence of singletons, irreflexivity of independence,
anti-monotonicity of independence, an axiom for the sum of
binomial distributions, and rules for deterministic expressions:
\begin{mathpar}
\indep \{ x \}
\and
\indep \{ x, x\} \iff \mathsf{det}(x)
\and
\indep (E\cup E') \implies \indep E
\and
\follows{e\!}{\!\mathrm{B}(m,p)} {\land} \follows{e'\!}{\!\mathrm{B}(m',p)} {\land}
\indep \{ e,e'\}\!\implies\!\mbox{$\follows{e{+}e'\!}{\!\mathrm{B}(m+m',p)}$}
\and
\bigwedge_{1\leq i \leq n} \mathsf{det}(e_i) \implies \mathsf{det}(f(e_1,\ldots,e_n))
\end{mathpar}
\begin{definition}
Judgments of the logic are of the form $\indhoare{\xi}{s}{\xi'}$,
where $\xi$ and $\xi'$ are \textsc{IL}\xspace-assertions. A judgment is \emph{valid} if
it is derivable from the rules of \cref{fig:fil}; structural rules
and rule for sequential composition are similar to those from
\Cref{sec:proofsystem} and omitted.
\end{definition}
The rule [\textsc{\textsc{IL}\xspace-Assgn}] for deterministic assignments is as in
\Cref{sec:proofsystem}. The rule [\textsc{\textsc{IL}\xspace-Sample}] for random
assignments yields as post-condition that the variable $x$ and a set
of expressions $E$ are independent assuming that $E$ is independent
before the sampling, and moreover that $x$ follows the law of the
distribution that it is sampled from. The rule
[\textsc{\textsc{IL}\xspace-Cond}] for conditionals requires that the guard is
deterministic, and that each of the branches satisfies the
specification; if the guard is not deterministic, there are
simple examples where the rule is not sound.\iffull\footnote{%
Consider the following program where $\mathbf{Bern}(p)$ is the Bernoulli distribution
with parameter $p$:
\begin{align*}
b \rnd \mathbf{Bern}(p); \kw{if}\ {b}\ & \kw{then}\ {x_1 \rnd \mathbf{Bern}(p_1); x_2 \rnd \mathbf{Bern}(p_2)} \\
& \kw{else}\ {x_1 \rnd \mathbf{Bern}(p'_1); x_2 \rnd \mathbf{Bern}(p'_2)}
\end{align*}
%
Each branch establishes $\indep \{ x_1,x_2 \}$, but this is not a valid
post-condition for the conditional. There are similar examples using the
binomial distribution.}\fi{}
The rule [\textsc{\textsc{IL}\xspace-While}] for loops requires that the loop is
certainly terminating with a deterministic guard.
Note that the requirement of certain termination could be avoided by
restricting the structural rules such that a statement $s$
has deterministic control flow whenever $\indhoare{\xi}{s}{\xi'}$
is derivable.
We now turn to the embedding. The embedding of \textsc{IL}\xspace assertions into
general assertions is immediate, except for $\mathsf{det}(e)$ which is
translated as $\detm{e}\vee\detm{\neg e}$. We let $\overline{\xi}$
denote the translation of $\xi$.
\begin{thm}[Embedding and soundness of \textsc{IL}\xspace logic]
If $\indhoare{\xi}{s}{\xi'}$ is derivable in the \textsc{IL}\xspace
logic, then $\hoare{\overline{\xi}}{s}{\overline{\xi'}}$ is
derivable in (the syntactic variant of) \mbox{\textsc{Ellora}}\xspace. As a consequence,
every derivable judgment $\indhoare{\xi}{s}{\xi'}$ is valid.
\end{thm}
\begin{proof}[Proof sketch] By induction on the derivation.
The interesting cases are conditionals and loops. For
conditionals, the soundness follows from the soundness of the rule:
$$
\inferrule{\hoare{\eta}{s_1}{\eta'}\\
\hoare{\eta}{s_2}{\eta'} \\
\detm{e}\vee\detm{\neg e}}
{\hoare{\eta}{\ifstmt{e}{s_1}{s_2}}{\eta'}}
$$
To prove the soundness of this rule, we proceed by case analysis on
$\detm{e} \vee\detm{\neg e}$. We treat the case $\detm{e}$; the other
case is similar. In this case, $\eta$ is equivalent to $\eta_1\wedge
\detm{e} \oplus\eta_2\wedge\detm{\neg e}$, where $\eta_1=\eta$ and
$\eta_2=\bot$. Let $\eta_1'=\eta'$ and $\eta_2=\detm{\bot}$;
again, $\eta_1'\oplus\eta_2'$ is logically equivalent to
$\eta'$. The soundness of the rule thus follows from the soundness of
the [\textsc{Cond}] and [\textsc{Conseq}] rules.
For loops, there exists a natural number $n$ such that $\while{b}{s}$ is
semantically equivalent to $(\ift{b}{s})^n$. By assumption
$\indhoare{\xi}{s}{\xi}$ holds, and thus by induction hypothesis
$\hoare{\overline{\xi}}{s}{\overline{\xi}}$. We also have
$\xi\implies\mathsf{det}(b)$, and hence
$\hoare{\overline{\xi}}{\ift{b}{s}}{\overline{\xi}}$. We conclude by [\textsc{Seq}].
\end{proof}
\begin{figure}
\begin{mathpar}
\inferrule[\textsc{IL}\xspace-Assgn]{~}
{\indhoare{\xi\subst{x}{e}}{x \asn e}{\xi}}
\and
\inferrule[\textsc{IL}\xspace-Sample]
{\{x \} \cap \ensuremath{\mathrm{FV}}(E) \cap \ensuremath{\mathrm{FV}}(e) = \emptyset}
{\indhoare{\indep E}{x \rnd \mathrm{B}(e,p)}{\indep (E \cup \{x\})
\land \follows{x}{\mathrm{B}(e,p)}}}
\and
\inferrule[\textsc{IL}\xspace-Seq]
{\indhoare{\xi}{s_1}{\xi'}\\
\indhoare{\xi'}{s_2}{\xi''}}
{\indhoare {\xi}{s_1;s_2}{\xi''}}
\and
\inferrule[\textsc{IL}\xspace-Cond]
{\indhoare{\xi}{s_1}{\xi'}\\
\indhoare{\xi}{s_2}{\xi'}\\\\
\xi \implies\mathsf{det}(b) }
{\indhoare {\xi}{\ifte{b}{s_1}{s_2}}{\xi'}}
\and
\inferrule[\textsc{IL}\xspace-While]
{\indhoare{\xi}{s}{\xi} \\
\xi\implies\mathsf{det}(b) \\
\sidecond{CTerm} }
{\indhoare {\xi}{\while{b}{s}}{\xi}}
\end{mathpar}
\caption{\textsc{IL}\xspace proof rules (selected)}
\label{fig:fil}
\end{figure}
To illustrate our system \textsc{IL}\xspace, consider the statement $s$ in \cref{fig:binsum} which
flips a fair coin $N$ times and counts the number of heads.
Using the logic, we prove that $\follows{\lstt{c}}{\mathrm{B}(N \cdot (N +
1)/2, {1}/{2})}$ is a post-condition for $s$. We take the invariant:
$$\follows{\lstt{c}}{\mathrm{B} \left({\lstt{j}(\lstt{j}+1)}/{2},{1}/{2}\right)}$$
The invariant holds initially, as $\follows{0}{B(0,{1}/{2})}$.
For the inductive case, we show:
$$\indhoare{\follows{\lstt{c}}{\mathrm{B}\left(0,{1}/{2}\right)}}{s_0}{
\follows{\lstt{c}}{\mathrm{B} \left({(\lstt{j}+1)(\lstt{j}+2)}/{2},{1}/{2}\right)}}$$
where $s_0$ represents the loop body, i.e.\ $\lstt{x}\rnd
\mathrm{B}\left(\lstt{j},{1}/{2}\right);\lstt{c} \asn \lstt{c} + \lstt{x}$. First, we apply the
rule for sequence taking as intermediate assertion
$$\follows{\lstt{c}}{\mathrm{B}
\left({\lstt{j}(\lstt{j}+1)}/{2},{1}/{2}\right)} \land
\follows{\lstt{x}}{\mathrm{B}\left(\lstt{j},{1}/{2}\right)}
\land \indep \{\lstt{x},\lstt{c}\}$$
\iffull
\begin{figure}
\else
\begin{wrapfigure}{l}{0.3\textwidth}
\fi
\begin{lstlisting}
proc sum () =
var c:int, x:int;
c := 0;
for j := 1 to N do
x ~~ B(j,1/2);
c := c + x;
return c
\end{lstlisting}
\caption{Sum of bin.}\label{fig:binsum}
\iffull
\end{figure}
\else
\end{wrapfigure}
\fi
The first premise follows from the rule for random assignment and structural
rules. The second premise follows from the rule for deterministic assignment and
the rule of consequence, applying axioms about sums of binomial distributions.
We briefly comment on several limitations of \textsc{IL}\xspace. First, \textsc{IL}\xspace is
restricted to programs with deterministic control flow, but this
restriction could be partially relaxed by enriching \textsc{IL}\xspace with
assertions for conditional independence. Such assertions are already
expressible in the logic of \mbox{\textsc{Ellora}}\xspace; adding conditional independence
would significantly broaden the scope of the \textsc{IL}\xspace proof system and
open the possibility to rely on axiomatizations of conditional
independence (e.g., based on graphoids~\cite{PearlP86}). Second, the
logic only supports sampling from binomial distributions. It is
possible to enrich the language of assertions with clauses
$\follows{c}{g}$ where $g$ can model other distributions, like the
uniform distribution or the Laplace distribution. The main design
challenge is finding a core set of useful facts about these
distributions. Enriching the logic and automating the analysis are
interesting avenues for further work.
\subsection{Embedding the Union Bound Logic}
The program logic \textsc{aHL}\xspace \cite{BartheGGHS16-icalp} was recently introduced for
estimating accuracy of randomized computations. One main application of \textsc{aHL}\xspace
is proving accuracy of randomized algorithms, both in the offline and online
settings---i.e.\ with adversary calls.
\textsc{aHL}\xspace is based on the union bound, a basic tool from probability
theory, and has judgments of the form
$
\ubhl \beta \Phi s \Psi ,
$
where $s$ is a statement, $\Phi$ and $\Psi$ are first-order formulae
over program variables, and $\beta$ is a probability, i.e.\,
$\beta\in{[0,1]}$. A judgment $\ubhl \beta \Phi s \Psi$ is valid if
for every memory $m$ such that $\Phi(m)$, the probability of $\neg
\Psi$ in $\dsem{m}{s}$ is upper bounded by $\beta$, i.e.\,
$\Pr\!_{\dsem{m}{s}}[\neg \Psi]\leq \beta$.
\Cref{fig:ubhl} presents some key rules of \textsc{aHL}\xspace, including a rule for
sampling from the Laplace distribution $\mathcal{L}_\epsilon$ centered
around $e$. The predicate $\sidecond{CTerm}(k)$ indicates that the
loop terminates in at most $k$ steps on any memory that satisfies the
pre-condition. Moreover, $\beta$ is a function of $\epsilon$.
\begin{figure}
\begin{mathpar}
\inferrule[\textsc{aHL}\xspace-Sample]
{ }
{ \ubhl{\beta}
{\top}
{{x}\rnd {\mathcal{L}_\epsilon(e)}}
{ |x - e| \leq \frac{1}{\epsilon} \log \frac{1}{\beta} } }
\\
\inferrule[\textsc{aHL}\xspace-Seq]
{\ubhl{\beta_1}{\Phi}{s_1}{\Theta} \qquad \ubhl{\beta_2}{\Theta}{s_2}{\Psi} }
{\ubhl{\beta_1+\beta_2}{\Phi}{s_1;s_2}{\Psi} }
\\
\inferrule[\textsc{aHL}\xspace-While]
{\ubhl{\beta}{\Phi}{c}{\Phi} \\
\sidecond{CTerm}(k)}
{ \ubhl{k \cdot \beta}{\Phi}{ \while{e}{c} }{\Phi \land \neg e} }
\end{mathpar}
\caption{\textsc{aHL}\xspace proof rules (selected)}
\label{fig:ubhl}
\end{figure}
\textsc{aHL}\xspace has a simple embedding into \mbox{\textsc{Ellora}}\xspace.
\begin{thm}[Embedding of \textsc{aHL}\xspace]
If $\ubhl{\beta}{\Phi}{s}{\Psi}$ is derivable in \textsc{aHL}\xspace, then $\hoare
{\detm{\Phi}}{s} {\pr{\ind{\neg \Psi}}{} \leq \beta}$ is derivable in \mbox{\textsc{Ellora}}\xspace.
\end{thm}
\section{Case Studies: Verifying Randomized Algorithms}\label{sec:examples}
In this section, we will demonstrate \mbox{\textsc{Ellora}}\xspace on a selection of
examples; we present further examples in the supplemental material.
Together, they exhibit a wide variety of different proof techniques
and reasoning principles which are available in the \mbox{\textsc{Ellora}}\xspace's
implementation.
\paragraph*{Hypercube Routing.}
will begin with the \emph{hypercube routing} algorithm
\cite{valiant1982scheme,Valiant:1981:USP:800076.802479}. Consider a
network topology (the \emph{hypercube}) where each node is labeled by
a bitstring of length $D$ and two nodes are connected by an edge if
and only if the two corresponding labels differ in exactly one bit
position.
In the network, there is initially one packet at each node, and each
packet has a unique destination. The algorithm implements a routing
strategy based on \emph{bit fixing}: if the current position has
bitstring $i$, and the target node has bitstring $j$, we compare the
bits in $i$ and $j$ from left to right, moving along the edge that
corrects the first differing bit. Valiant's algorithm uses
randomization to guarantee that the total number of steps
grows \emph{logarithmically} in the number of packets. In the first
phase, each packet $i$ select an intermediate destination
$\rho(i)$ uniformly at random, and use bit fixing to
reach $\rho(i)$. In the second phase, each packet use bit fixing
to go from $\rho(i)$ to the destination $j$. We will focus on
the first phase since the reasoning for the second phase is nearly
identical. We can model the strategy with the code in
\Cref{fig:hypercube}, using some syntactic sugar for the $\kw{for}$
loops.\footnote{%
Recall that the number of node in a hypercube of dimension $D$ is $2^D$ so
each node can be identified by a number in $[1,2^D]$.}
\iffull
\begin{figure}
\else
\begin{wrapfigure}{l}{0.47\textwidth}
\fi
\begin{lstlisting}
proc route ($D$ $T$ : int) :
var $\rho$, pos, usedBy : node map;
var nextE : edge;
pos := Map.init id $2^D$; $\rho$ := Map.empty;
for $i$ := 1 to $2^D$ do
$\rho$[i] ~~ $[1,2^D]$
for $t$ := 1 to $T$ do
usedBy := Map.empty;
for $i$ := 1 to $2^D$ do
if pos$[i] \neq \rho[i]$ then
nextE := getEdge pos[$i$] $\rho[i]$;
if usedBy[nextE] = $\bot$ then
// Mark edge used
usedBy[nextE] := $i$;
// Move packet
pos[$i$] := dest nextE
return (pos, $\rho$)
\end{lstlisting}
\caption{Hypercube Routing}\label{fig:hypercube}
\iffull
\end{figure}
\else
\end{wrapfigure}
\fi
We assume that initially, the position of the packet $i$ is at node $i$
(see \lstt{Map.init}). Then, we initialize the random intermediate
destinations $\rho$. The remaining loop encodes the evaluation of the
routing strategy iterated $T$ time. The variable \lstt{usedBy} is a
map that logs if an edge is already used by a packet, it is empty at
the beginning of each iteration. For each packet, we try to move it
across one edge along the path to its intermediate destination. The
function \lstt{getEdge} returns the next edge to follow, following the
bit-fixing scheme. If the packet can progress (its edge is not used),
then its current position is updated and the edge is marked as used.
We show that if the number of timesteps $T$ is $4 D + 1$, then all
packets reach their intermediate destination in at most $T$ steps,
except with a small probability $2^{-2D}$ of failure. That is, the
number of timesteps grows linearly in $D$, logarithmic in the number
of packets. This is formalized in our system as:
\[
\hoare{T = 4D + 1}{\lstt{route}}{\Pr[ \exists i.\; \lstt{pos}[i] \neq
\lstt{$\rho$}[i] ] \leq 2^{-2D}]}
\]
\paragraph*{Modeling Infinite Processes.}
\iffull
\begin{figure}
\else
\begin{wrapfigure}{l}{0.4\textwidth}
\fi
\begin{lstlisting}
proc coupon ($N$ : int) :
var int cp[$N$], $t$[$N$];
var int $X$ := 0;
for $p$ := 1 to $N$ do
ct := 0;
cur ~~ $[1,N]$;
while cp[cur] = 1 do
ct := ct + 1;
cur ~~ $[1,N]$;
$t$[$p$] := ct;
cp[cur] := 1;
$X$ := $X$ + $t$[$p$];
return $X$
\end{lstlisting}
\caption{Coupon collector}\label{fig:coupon}
\iffull
\end{figure}
\else
\end{wrapfigure}
\fi
Our second example is the \emph{coupon collector} process. The algorithm draws a
uniformly random coupon (we have $N$ coupon) on each day, terminating when it
has drawn at least one of each kind of coupon. The code of the algorithm is
displayed in \cref{fig:coupon}; the array \lstt{cp} records of the coupons seen
so far, $\lstt{t}$ holds the number of steps taken before seeing a new coupon,
and $X$ tracks of the total number of steps. Our goal is to bound the average
number of iterations. This is formalized in our logic as:
\[
\{ \mathcal{L} \}\;
\lstt{coupon} \;
\left\{
\ex{X} = \textstyle\sum_{i \in [1,N]} \left(
\frac{N}{N - i + 1} \right)
\right\} .
\]
\paragraph*{Limited Randomness.}
\iffull
\begin{figure}
\else
\begin{wrapfigure}{r}{.45\textwidth}
\fi
\begin{lstlisting}
proc pwInd (N : int) :
var bool X[2${}^{\mbox N}$], B[N];
for i := 1 to N do
B[i] ~~ Ber(1/2);
for j := 1 to 2${}^{\mbox N}$ do
X[j] := 0;
for k := 1 to N do
if k $\in$ bits(j) then
X[j] := X[j] $\oplus$ B[k]
return X
\end{lstlisting}
\caption{Pairwise Independence}\label{fig:pwindep}
\iffull
\end{figure}
\else
\end{wrapfigure}
\fi
\emph{Pairwise independence} says that if we see the result of
$X_i$, we do not gain information about all other variables
$X_k$. However, if we see the result of \emph{two} variables $X_i,
X_j$, we may gain information about $X_k$.
There are many constructions in the algorithms literature that grow
a small number of independent bits into more pairwise
independent bits. \Cref{fig:pwindep} gives one procedure, where
$\oplus$ is exclusive-or, and $\lstt{bits(j)}$ is the set of
positions set to $1$ in the binary expansion of $\lstt{j}$.
The proof uses the following fact, which we fully verify:
for a uniformly distributed Boolean random variable $Y$, and a random
variable $Z$ of any type,
\begin{equation} \label{eq:pw-indep}
Y \mathrel{\indep} Z \Rightarrow Y \oplus f(Z) \mathrel{\indep} g(Z)
\end{equation}
for any two Boolean functions $f, g$. Then, note that
$\lstt{X}[i] = \bigoplus_{\{j \in \lstts{bits}(i)\}} \lstt{B}[j]$
where the big XOR operator ranges over the indices $j$ where the bit
representation of $i$ has bit $j$ set. For any two $i, k \in [1, \dots,
2^{\lstts{N}}]$ distinct, there is a bit position in $[1, \dots, \lstt{N}]$
where $i$ and $k$ differ; call this position $r$ and suppose it is set in $i$
but not in $k$. By rewriting,
\[
\lstt{X}[i] = \lstt{B}[r] \oplus \bigoplus_{\{j \in \lstts{bits}(i) \setminus
r\}} \lstt{B}[j]
\quad \text{and} \quad
\lstt{X}[k] = \bigoplus_{\{j \in \lstts{bits}(k) \setminus r\}} \lstt{B}[j] .
\]
Since $\lstt{B}[j]$ are all independent, $\lstt{X}[i] \mathrel{\indep} \lstt{X}[k]$ follows from
\cref{eq:pw-indep} taking $Z$ to be the distribution on tuples $\langle
\lstt{B}[1],
\dots, \lstt{B}[\lstt{N}] \rangle$ excluding $\lstt{B}[r]$. This verifies
pairwise independence:
$$
\hoare{\mathcal{L}}
{\lstt{pwInd(N)}}
{\mathcal{L} \land \forall i , k \in [2^{\lstts{N}}] . \; i \neq k \Rightarrow
\lstt{X}[i] \mathrel{\indep} \lstt{X}[k] } .
$$
\paragraph*{Adversarial Programs.}
Pseudorandom functions (PRF) and pseudorandom permutations (PRP) are two
idealized primitives that play a central role in the design of symmetric-key
systems. Although the most natural assumption to make about a blockcipher is
that it behaves as a pseudorandom permutation, most commonly the security of
such a system is analyzed by replacing the blockcipher with a perfectly random
function. The PRP/PRF Switching Lemma
\cite{DBLP:conf/stoc/ImpagliazzoR89,DBLP:conf/eurocrypt/BellareR06} fills the
gap: given a bound for the security of a blockcipher as a pseudorandom function,
it gives a bound for its security as a pseudorandom permutation.
\begin{lem}[PRP/PRF switching lemma]
Let $A$ be an adversary with blackbox access to an oracle O implementing
either a random permutation on $\bs{l}$ or a random function from $\bs{l}$
to $\bs{l}$. Then the probability that the adversary $A$
distinguishes between the two oracles in at most $q$ calls is
bounded by
$$|\Pr_{\textrm{PRP}}[b \land |H| \leq q] -
\Pr_{\textrm{PRF}}[b \land |H| \leq q] | \leq \frac{q(q-1)}{2^{l+1}} ,$$
where $H$ is a map storing each adversary call and $|H|$ is its size.
\end{lem}
Proving this lemma can be done using the Fundamental Lemma of
Game-Playing, and bounding the probability of \emph{bad} in the
program from \cref{fig:prp/prf}. We focus on the latter. Here we apply
the [\textsc{Adv}] rule of \mbox{\textsc{Ellora}}\xspace with the invariant
$\forall k, \Pr[\lstt{bad} \land |H| \leq k] \leq
\frac{k(k-1)}{2^{l+1}}$
where $|H|$ is the size of the map $H$, i.e. the number
of adversary call.
Intuitively, the invariant says that at each call to the oracle
the probability that $\lstt{bad}$ has been set before and
that the number of adversary call is less than $k$ is bounded
by a polynomial in $k$.
The invariant is $d$-closed and true before the
adversary call, since at that point $\Pr[\lstt{bad}] = 0$. Then we
need to prove that the oracle preserves the invariant, which can be
done easily using the precondition calculus ([\textsc{PC}] rule).
\begin{figure}
\begin{minipage}{0.6\linewidth}
\begin{lstlisting}
var $H$: ($\bs{l}$, $\bs{l}$) map;
proc orcl ($q$:$\bs{l}$):
var $a:\bs{l}$;
if $q \not\in H$ then
$a$ ~~ $\bs{l}$;
bad := bad || $a \in \textrm{codom}(H)$;
$H[q]$ := $a$;
return $H[q]$;
\end{lstlisting}
\end{minipage}
\begin{minipage}{0.39\linewidth}
\begin{lstlisting}
proc main():
var $b$: bool;
bad := false;
$H$ := [];
$b$ := A();
return $b$;
\end{lstlisting}
\end{minipage}
\caption{PRP/PRF game}\label{fig:prp/prf}
\end{figure}
\section{Implementation and Mechanization}
We have built a prototype implementation of \mbox{\textsc{Ellora}}\xspace within
\textsc{EasyCrypt}\xspace~\cite{BartheGHZ11,BartheDGKSS13}, a theorem prover
originally designed for verifying cryptographic protocols. \textsc{EasyCrypt}\xspace
provides a convenient environment for constructing proofs in various
Hoare logics, supporting interactive, tactic-based proofs for
manipulating assertions and allowing users to invoke external tools,
like SMT-solvers, to discharge proof obligations. \textsc{EasyCrypt}\xspace provides
a mature set of libraries for both data structures (sets, maps, lists,
arrays, etc.) and mathematical theorems (algebra, real analysis,
etc.), which we extended with theorems from probability theory.
\begin{wraptable}{l}{0.4\textwidth}
\begin{tabular}{lcc}
\toprule
Example & LC & FPLC \\
\midrule
{\tt hypercube } & 100 & 1140 \\
{\tt coupon } & 27 & 184 \\
{\tt vertex-cover } & 30 & 61 \\
{\tt pairwise-indep } & 30 & 231 \\
{\tt private-sums } & 22 & 80 \\
{\tt poly-id-test } & 22 & 32 \\
{\tt random-walk } & 16 & 42 \\
{\tt dice-sampling } & 10 & 64 \\
{\tt matrix-prod-test} & 20 & 75 \\
\bottomrule
\end{tabular}
\caption{Benchmarks \label{tab:examples}}
\end{wraptable}
We used the implementation for verifying many examples from the
literature, including all the programs presented in
\cref{sec:examples} as well as some additional examples (such as
polynomial identity test, private running sums, properties about
random walks, etc.). The verified proofs bear a strong resemblance to
the existing, paper proofs. Independently of this work, \mbox{\textsc{Ellora}}\xspace has
been used to formalize the main theorem about a randomized
gossip-based protocol for distributed systems \cite[Theorem
2.1]{KempeDG03}. Some libraries developed in the scope of \mbox{\textsc{Ellora}}\xspace
have been incorporated into the main branch of \textsc{EasyCrypt}\xspace, including
a general library on probabilistic independence.
\paragraph*{A New Library for Probabilistic Independence.}
In order to support assertions of the concrete program logic, we enhanced
the standard libraries of \textsc{EasyCrypt}\xspace, notably the ones dealing with
big operators and sub-distributions. Like all \textsc{EasyCrypt}\xspace libraries, they are
written in a foundational style, i.e.\, they are defined instead of axiomatized.
A large part
of our libraries are proved formally from first principles. However,
some results, such as concentration bounds, are currently declared as
axioms.
\jh{Check: are concentration bounds axiomatized?}
Our formalization of probabilistic independence deserves special
mention. We formalized two different (but logically equivalent)
notions of independence. The first is in terms of products of
probabilities, and is based on heterogenous lists. Since \mbox{\textsc{Ellora}}\xspace (like
\textsc{EasyCrypt}\xspace) has no support for heterogeneous lists, we use a smart
encoding based on second-order predicates. The second definition is
more abstract, in terms of product and marginal distributions. While
the first definition is easier to use when reasoning about randomized
algorithms, the second definition is more suited for proving
mathematical facts. We prove the two definitions equivalent, and
formalize a collection of related theorems.
\paragraph*{Mechanized Meta-Theory.}
The proofs of soundness and relative completeness of the
abstract logic, without adversary calls, and the syntactical
termination arguments have been mechanized in the \textsf{Coq} proof
assistant. The development is available in supplemental material.
\section{Related Work}\label{sec:related}
\paragraph*{More on Assertion-Based Techniques.}
The earliest assertion-based system is due to
Ramshaw~\cite{Ramshaw79}, who proposes a program logic where
assertions can be formulas involving \emph{frequencies}, essentially
probabilities on sub-distributions. Ramshaw's logic allows assertions
to be combined with operators like $\oplus$, similar to our
approach. \cite{Hartog:thesis} presents a Hoare-style logic with
general assertions on the distribution, allowing expected values and
probabilities. However, his \textbf{while} rule is based on a semantic
condition on the guarded loop body, which is less desirable for
verification because it requires reasoning about the semantics of
programs. \cite{ChadhaCMS07} give decidability results for a
probabilistic Hoare logic without \textbf{while} loops. We are not
aware of any existing system that supports assertions about general
expected values; existing works also restrict to Boolean
distributions. \cite{RandZ15} formalize a Hoare logic for
probabilistic programs but unlike our work, their assertions are
interpreted on \emph{distributions} rather than sub-distributions.
For conditionals, their semantics rescales the distribution of states
that enter each branch. However, their assertion language is limited
and they impose strong restrictions on loops.
\paragraph*{Other Approaches.}
Researchers have proposed many other approaches to verify probabilistic
program. For instance, verification
of Markov transition systems goes back to at least
\cite{HartSP83,SharirPH84}; our condition for ensuring almost-sure
termination in loops is directly inspired by their work. Automated methods
include model checking (see e.g., \cite{Baier16,Katoen16,KNP11}) and abstract
interpretation (see e.g., \cite{Monniaux00,CousotM12}). Techniques for
reasoning about higher-order (functional) probabilistic languages are an active
subject of research (see e.g.,
\cite{bizjak2015step,10.1007/978-3-642-54833-8_12,DalLago:2014:CEH:2535838.2535872}).
For analyzing probabilistic loops, in particular, there are tools for
reasoning about running time. There are also automated systems for
synthesizing invariants \cite{ChatterjeeFNH16,BEFFH16}.
\cite{ChakarovS13,ChakarovS14} use a martingale method to compute the
expected time of the coupon collector process for $N=5$---fixing $N$
lets them focus on a program where the outer \textbf{while} loop is
fully unrolled. Martingales are also used by \cite{FioritiH15} for
analyzing probabilistic termination. Finally, there are approaches
involving symbolic execution; \cite{SampsonPMMGC14} use a mix of
static and dynamic analysis to check probabilistic programs from the
approximate computing literature.
\section{Conclusion and Perspectives}
We introduced an expressive program logic for probabilistic programs, and showed
that assertion-based systems are suited for practical verification of
probabilistic programs. Owing to their richer assertions, program logics are a
more suitable foundation for specialized reasoning principles than
expectation-based systems. As evidence, our program logic can be smoothly
extended with custom reasoning for probabilistic independence and union bounds.
Future work includes proving better accuracy bounds for differentially private
algorithms, and exploring further integration of \mbox{\textsc{Ellora}}\xspace into \textsc{EasyCrypt}\xspace.
\paragraph*{Acknowledgments.}
We thank the reviewers for their helpful comments. This work benefited from
discussions with Dexter Kozen, Annabelle McIver, and Carroll Morgan. This work
was partially supported by ERC Grant \#679127, and NSF grants 1513694 and
1718220.
\bibliographystyle{splncs03}
|
1,116,691,499,321 | arxiv | \section{Introduction}
Accretion-powered Classical T Tauri stars (CTTSs) are young low-mass stars which
often show signs of a strong magnetic field (e.g., \citealt{basr92,john99})
which is expected to have a complex structure (e.g., \citealt{john07}).
The Zeeman-Doppler imaging technique has proven very successful in obtaining
surface magnetic maps for many stars, and the external magnetic fields of the
stars have been reconstructed from these maps under the potential approximation
\citep{dona97, dona99, jard02, jard06, greg10}. The magnetic field plays a crucial role in disc
accretion by disrupting the inner regions
of the disc and channeling the matter onto the star, and hence it is important to know the
magnetic field configurations in magnetized stars.
D08 recently observed the CTTS BP Tau with the ESPaDOnS and NARVAL
spectropolarimeters and reconstructed the surface magnetic field
from the observations. They have shown that the magnetic field of
BP Tau can be approximated by a combination of dipole and
octupole components of 1.2 kG and 1.6 kG, which are slightly (but
differently) tilted about the rotational axis.
D07 analyzed the distribution of the accretion spots on the
stellar surface and found spots at high latitudes, which cover
about 8 per cent of the stellar surface (D08).
Further, D08 extrapolated the surface magnetic field to larger distances using the potential approximation,
(i.e., assuming that there are no currents outside the star and hence the
external matter does not influence the initial configuration of the field)
and estimated the distance at which
the disk should be disrupted by the magnetosphere so that the matter flowing
towards the star in funnel streams
produces the high-latitude spots. They concluded that this distance should be
quite large, $r\gtrsim 4 R_\star$. However, this problem requires more complete analysis
based on the MHD approach, where external currents can be taken into account, and the matter
flow around the magnetosphere can be investigated self-consistently, taking into account
interaction of the external plasma with the magnetic field.
In this paper, we investigate this problem using global three-dimensional MHD simulations.
We solve the 3D MHD equations numerically in our simulation model to
investigate the structure of the external magnetic field, accretion flows and
location of accretion spots.
In our previous work, we have performed global 3D simulations of
accretion onto stars with misaligned dipole fields \citep{roma03,
roma04a, kulk05}, and aligned or misaligned dipole plus
quadrupole fields \citep{long07,long08}. Recently, we were able to
extend our method and to build a numerical model for stars with an
octupolar component. The general properties of the model have been
described in detail in \citet{long10} and the model was applied
to another CTTS, V2129 Oph, with a strong octupole field in
\citet{roma10} which has been compared with observations of V2129
(\citealt{dona07}, \citealt{dona10}).
In this paper, we apply our 3D MHD model of stars with complex
magnetic fields \citep{long10} to the CTTS BP Tau. We investigate
disc accretion onto the star by adopting the measured surface
magnetic fields (D08) and other suggested properties of this star.
The surface magnetic field is modeled as a superposition of a 1.2
kG dipole and 1.6 kG octupole field, tilted by $20^\circ$ and
$10^\circ$ with respect to the rotational axis and located at
opposite phases (the phase difference is $180^\circ$). We also
take into account other parameters of BP Tau: its mass
$M_\star=0.7 M_\odot$ \citep{sies00}, and radius $R_\star=1.95
R_\odot$ \citep{gull98}. Its age is about 1.5 Myr (D08), and its
rotation period is $7.6$ days \citep{vrba86}, which corresponds to
a corotation radius of $R_{cor}\approx7.5 R_\star$.
The mass accretion rate derived from different observations varies
between $\dot{M}\simeq2.9\times10^{-8}$M$_\odot$yr$^{-1}$ (e.g.
\citealt{gull98}) and $9\times10^{-10}$M$_\odot$yr$^{-1}$
\citep{schm05}. We performed a series of simulation runs at
different mass accretion rates in order to investigate the cases
where the disk stops at different distances from the star. We also
calculated the torque on the star and compared it with star's age.
To understand the role of the octupole field in channeling the
accreting matter, we compared our dipole plus octupole model of BP
Tau with a similar model but with only the dipole component.
Thus, we focus on: (1) 3D MHD modeling of accretion flows around
CTTS BP Tau modeled with dipole plus octupole moments; (2)
comparisons of accretion properties observed in simulations with
observations of BP Tau, such as the shape and distribution of hot
spots, mass accretion rates and more; (3) deviation of the
simulated magnetic field from the potential field.
Section \S2 briefly describes the numerical model used in this work.
The simulation results are shown in \S 3.
We end in \S4 with our conclusions and some discussion.
\section{Model}
The global 3D MHD model originally developed by \citet{kold02} and
used for modeling stars with dipole fields \citep{roma03,roma04a}
was modified to include higher order components
\citep{long07,long08,long10,roma10} to simulate disc accretion
onto stars with complex fields. The MHD equations are solved in a
reference frame co-rotating with the star. A viscous term is
incorporated into the MHD equations (only in the disc) to control
the rate of matter flow through the disk. We use the
$\alpha-$prescription for viscosity with $\alpha=0.01$.
\begin{enumerate}
\item
\textit{Initial conditions.} The simulation domain consists of a cold, dense disc and a hot,
low-density corona, which are initially in rotational hydrodynamical equilibrium.
The initial angular velocity in the disc is close to Keplerian.
The angular velocity in the corona at any given cylindrical radius is
set to be equal to that of the disk at that radius.
\item
\textit{Boundary conditions.} At the inner boundary (the surface of the star),
most of the variables $A$ are set to have free boundary conditions, ${\partial A}/{\partial r}=0$.
The initial magnetic field on the surface of the star is taken to be a
superposition of misaligned dipole and octupole fields. As the simulation proceeds,
we assume that the normal component of the field remains unchanged, i.e.,
the magnetic field is frozen into the surface of the star. At the outer boundary,
free conditions are taken for all variables. In addition, matter is not
permitted to flow into the region from the outer boundary.
\begin{table}
\caption{The reference values for CTTS BP Tau. The dimensional
values can be obtained by multiplying the dimensionless values
from simulations by these reference values. $B_0$ and the
subsequent values below depend on $\widetilde\mu_1$.} \centering
\begin{tabular}{llll}
\hline
Reference Units & $\widetilde{\mu}_1=1$ & $\widetilde{\mu}_1=2$ & $\widetilde{\mu}_1=3$ \\
\hline
{$M_\star(M_\odot)$} & $0.7$ & -- & -- \\
{$R_\star(R_\odot)$} & $1.95$ & -- & -- \\
{$B_{1\star}$ (G)} & $1200$ & -- & -- \\
{$R_0$ (cm)} & $3.9\e{11}$ & -- & -- \\
{$v_0$ (cm s$^{-1}$)} & $1.5\e7$ & -- & -- \\
{$P_0$ (days)} & $1.83$ & -- & -- \\
{$B_0$ (G)} & $25.7$ & $12.9$ & $8.6$ \\
{$\rho_0$ (g cm$^{-3}$)} & $2.8\e{-12}$ & $6.9\e{-13}$ & $2.8\e{-13}$ \\
{$\dot M_0$ ($M_\odot$yr$^{-1}$)} & $1.0\e{-7}$ & $2.6\e{-8}$ & $1.1\e{-8}$ \\
{$F_0$ (erg cm$^{-2}$s$^{-1}$)} & $1.0\e{10}$ & $2.6\e{9}$ & $1.1\e{9}$ \\
{$N_0$ (g cm$^2$s$^{-2}$)} & $3.9\e{37}$ & $9.8\e{36}$ & $4.6\e{36}$ \\
\hline
\end{tabular}
\label{tab:refval}
\end{table}
\item
\textit{Simulation region and grid.} We use the ``cubed sphere"
grid introduced by \cite{kold02} (see also Fig. 1 in
\citealt{long10}). The resolution of the grid is
$6\times120\times51^2$ to simulate the accretion onto BP Tau in a
simulation domain of $41R_\star$. We use a very high resolution
near the star in order to resolve the complex structure of the
octupolar component of the field.
\item
\textit{Magnetic field configuration.} In our code we can model the magnetic field of the star
by a superposition of three
multipole moments $\bm\mu_i$ ($i=1,2,3$ for dipole, quadrupole and octupole respectively)
which are tilted relative to the $z-$ axis (which is aligned with the rotational axis $\bm\Omega$)
at different angles $\Theta_i$, and have different angles $\phi_i$ between the $xz$ plane and
$\bm\Omega-\bm\mu_i$ planes. For simplicity, $\phi_1$ is set to be
0, which means that the dipole moment $\bm\mu_1$ is in $xz$ plane.
The general magnetic field configuration is discussed in greater
detail in \citet{long10}. The details of the magnetic field
configuration in our
BP Tau model are discussed in \S 3.1.
\item
\textit{Reference units.}
The simulations are performed in dimensionless variables $\widetilde{A}=A/A_0$ where $A_0$ are reference values.
We choose the stellar mass $M_\star$, radius $R_\star$
and the surface dipole field strength $B_{1\star}$ to build a set of reference values.
The reference values are: length scale: $R_0=R_\star/0.35$; velocity: $v_0=(GM_\star/R_0)^{1/2}$;
time-scale: $P_0=2\pi R_0/v_0$.
The reference magnetic moments for dipole and octupole components
are $\mu_{1,0}=B_0R_0^3$ and $\mu_{3,0}=B_0R_0^5$ respectively,
where $B_0$ is the reference magnetic field. Hence, the
dimensionless magnetic moments are:
$\widetilde{\mu}_1=\mu_1/\mu_{1,0}$,
$\widetilde{\mu}_3=\mu_3/\mu_{3,0}$, where the dipole and octupole
moments $\mu_1$ and $\mu_3$ of the star are fixed. We take one of
the above relationships; for example, the one for the dipole; to
obtain
\begin{equation}
\label{eq-B0} B_0=\frac{\mu_{1,0}}{R_0^3} = \frac{0.5
B_{1\star}}{\widetilde{\mu_1}}\bigg(\frac{R_\star}{R_0}\bigg)^3=
25.7\bigg(\frac{B_{1\star}}{1.2 {\rm
kG}}\bigg)\bigg(\frac{1}{\widetilde{\mu_1}}\bigg) {\rm G} .
\end{equation}
Hence, at fixed $B_{1\star}$, the reference magnetic field depends on
the dimensionless parameter $\widetilde\mu_1$. The reference
density $\rho_0$ and the mass accretion rate $\dot{M}_0$ also
depend on this parameter:
\begin{equation}
\label{eq-rho0} \rho_0=B_0^2/v_0^2 = 2.8\times10^{-12}
\bigg(\frac{B_{1\star}}{1.2 {\rm
kG}}\bigg)^2\bigg(\frac{1}{\widetilde{\mu_1}}\bigg)^2 \frac{\rm
g}{{\rm cm}^3} ,
\end{equation}
\begin{equation}
\label{eq-Mdot0} \dot{M}_0=\rho_0 v_0
R_0^2=1.0\times10^{-7}\bigg(\frac{B_{1\star}}{1.2 {\rm
kG}}\bigg)^2\bigg(\frac{1}{\widetilde{\mu_1}}\bigg)^2
\frac{M_\odot}{{\rm yr}}.
\end{equation}
The dimensional accretion rate then is $\dot M=\widetilde{\dot
M}\dot M_0$, where $\widetilde{\dot M}$ is the dimensionless
accretion rate.
One can see that
the dimensional accretion rate depends on $\widetilde{\dot M}$
which we find from simulations, and the dimensionless parameter
$\widetilde\mu_1$, which we use to vary the accretion rate. We
change the dimensionless octupolar moment $\widetilde\mu_3$ in
same proportion so as to keep the ratio $\mu_3/\mu_1$ fixed. To
find the ratio between the dimensionless moments, we use
approximate formulae for aligned moments: $\mu_1=0.5 B_{1\star}
R_\star^3$ and $\mu_3=0.25 B_{3\star} R_\star^5$ (see
\citealt{long10}) and obtain for BP Tau the ratio
$\widetilde{\mu}_3/\widetilde{\mu}_1=B_{3\star} \tilde R_\star^2/2
B_{1\star}\approx 0.08$ (where $\tilde R_\star=0.35$).
Other reference values are: angular momentum flux (a torque)
$\dot{N}_0=\rho_0v_0^2R_0^3$; energy flux
$\dot{E}_0=\rho_0v_0^3R_0^2$; temperature
$T_0=\mathcal{R}p_0/\rho_0$, where $\mathcal{R}$ is the gas
constant; and the effective blackbody temperature
$T_{\mathrm{eff,0}} = (\rho_0 v_0^3/\sigma)^{1/4}$, where $\sigma$
is the Stefan-Boltzmann constant. Tab. \ref{tab:refval} shows the
reference values for CTTS V2129 Oph. In the subsequent sections,
we show dimensionless values $\widetilde{A}$ for most of the
variables and drop the tildes($\sim$). However, we keep them in
$\widetilde\mu_1$, $\widetilde\mu_3$, and $\widetilde{\dot M}$ because these are important parameters of the model.
\item
\textit{The magnetospheric radius.} The truncation radius, $r_t$,
where the disc is truncated by the magnetosphere,
could be estimated as (e.g., \citealt{elsn77}):
\begin{equation}
r_t=k(GM_\star)^{-1/7}\dot{M}^{-2/7}\mu_1^{4/7},
\end{equation}
where $\dot M$ is the accretion rate and $\mu_1$ is the dipole magnetic moment; $k$
is a coefficient of order unity. For example, \cite{long05} obtained
$k\approx 0.5$ in numerical modeling of disc-accreting stars.
For stars with known $M_\star$, $R_\star$, and dipole magnetic moment $\mu_1$,
such as BP Tau, the accretion
rate determines where the disc stops and how the matter flows onto the star.
\end{enumerate}
\section{Modeling of accretion onto BP Tau}
We performed a number of simulation runs at different accretion
rates and observed that the disk is truncated at different radii.
We choose as our main case the one in which the disk stops
sufficiently far away, but at the same time the accretion rate is
not very low.
Below, we describe this case in detail.
We also show the results
at higher accretion rates for comparisons (see \S 3.7).
\subsection{The modeled magnetic field}
D08 decomposed the observed surface magnetic field of BP Tau into
spherical harmonics and found that the field is mainly poloidal with
only $10\%$ of the total magnetic energy in the toroidal field.
The poloidal component can be approximated by
dipole ($l=1$) and octupole ($l=3$) moments with $50\%$ and $30\%$
of the magnetic energy respectively. Other multipoles (up to $l<10$) have
only $10\%$ of the total magnetic energy. D08 concluded that the magnetic field of BP Tau
is dominated by a 1.2 kG dipole and 1.6 kG octupole tilted by $20^\circ$
and $10^\circ$.
The meridional angle between the $\bm\Omega-\bm\mu_1$ and
$\bm\Omega-\bm\mu_3$ planes is approximately $180^\circ$. In our model, we only consider the poloidal component.
We convert the above parameters into dimensionless values using
our reference units and solve 3D MHD equations (see, e.g.,
\citealt{kold02}), in dimensionless form. One of the important
parameters of the model is $\widetilde\mu_1$ which is used to vary
the truncation radius in the dimensionless model (and the
accretion rate in the dimensional model, see eq. 1). We find the
second dimensionless parameter $\widetilde\mu_3$ from the
relationship: $\widetilde{\mu}_3/\widetilde{\mu}_1=B_{3\star}
\tilde R_\star^2/2 B_{1\star}\approx 0.08$ (where $\tilde
R_\star=0.35$). This ratio is fixed for fixed values of the dipole
and octupole components, $B_{1\star}=1.2$kG and
$B_{3\star}=1.6$kG. From a number of simulation runs at different
$\widetilde\mu_1$, we choose $\widetilde\mu_1=3$,
$\widetilde\mu_3=0.24$ to obtain a large enough magnetosphere as
suggested by D08 and investigate this case in detail. Other
parameters of the magnetic field configuration of BP Tau are
$\Theta_1=20^\circ$, $\Theta_3=10^\circ$ and $\phi_3=180^\circ$.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{fig01-bcomp.png}
\caption{\label{bcompb} : Polar projections of the magnetic field
components at the stellar surface in
the dipole plus octupole model of BP Tau in spherical
coordinates: radial magnetic field, $B_r$; azimuthal magnetic field, $B_\phi$;
and meridional magnetic
field, $B_\theta$. The outer boundary, the bold circle and the two inner
dashed circles represent the latitude of $-30^\circ$, the equator, and the latitudes
of $30^\circ$ and $60^\circ$ respectively. The red and blue regions represent
positive and negative polarities of the magnetic field.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{fig02-bsurf.png}
\caption{\label{bsurfb} The surface magnetic field in the dipole
plus octupole model of BP Tau ($\widetilde\mu_1=3$,
$\widetilde\mu_3=0.24$, $\Theta_1=20^\circ$, $\Theta_2=10^\circ$,
$\phi_3=180^\circ$) as seen from the equatorial plane (left
panel), the north pole (middle panel) and the south pole (right
panel). The colors represent different polarities
and strengths of the
magnetic field.}
\end{center}
\end{figure}
\fig{bcompb} shows the components of the simulated magnetic field ($B_r$, $B_\phi$
and $B_\theta$) in the polar projection down to the latitude $-30^\circ$. One can see that
the distribution of the radial component
(left panel) is very similar to that obtained by D08 in two observational epochs of Dec06 and Feb06
(see Fig. 14, left panels of D08).
In both cases, there is a strong positive pole at colatitudes of $0-30^\circ$,
a part of the negative octupolar belt at colatitudes of
$60^\circ-90^\circ$ and a part of the positive octupolar belt at colatitudes of
$90^\circ-120^\circ$. The distribution of the meridional component, $B_\theta$,
(right panel of \fig{bcompb}) is qualitatively similar to that of D08, though
in the D08 plot the inner positive ring (red color in the plot)
is weaker, while the negative ring (blue) is stronger compared with our model.
The azimuthal component of the field is weak
and shows a butterfly pattern of the negative and positive polarities
in both the modeled and reconstructed fields.
\fig{bsurfb} shows three-dimensional views of the magnetic field
distribution at the surface of the star. It can be seen that there
are two antipodal polar regions of opposite polarity where the
field is strongest. They approximately coincide with the
octupolar high-latitude poles and their centers are located close
to the octupolar axis $\bm\mu_3$. Next to these regions, there are
negative (blue) and positive (red) octupolar belts. Their shapes
are more complex than those of the belts in pure octupole cases,
where the belts are parallel to the magnetic equator (see
\citealt{long10}). This is because the dipole component strongly
distorts the ``background" octupolar field.
Although the octupole field is strongest at the surface of the
star, it decreases more rapidly than the dipole field with
distance from the star. To investigate the role of the dipole and
octupole components in channeling the accretion flow, we find the
radius at which the dipole and octupole fields are equal. For
this, we assume that both magnetic moments are aligned with the
rotational axis and take the magnetic field in the magnetic
equatorial planes: $B_1=\mu_1/r^3$ and $B_3=3\mu_3/2r^5$ (see Eqn.
1 in \citealt{long10}). Noting that the strengths of the field at
the magnetic poles are $B_{1\star}=2\mu_1/R_\star^3$ and
$B_{3\star}=4\mu_3/R_\star^5$, and equating $B_1$ to $B_3$, we
find this radius:
\begin{equation}
r_{eql}=\left(\frac{3}{4}\frac{B_3\star}{B_1\star}\right)^{1/2}R_\star.
\end{equation}
Substituting $B_{1\star}=1.2$kG and $B_{3\star}=1.6$kG, we obtain $r_{eql}\approx R_\star$.
Note that both dipole and octupole moments are tilted about the rotational axis, and hence
the above formula gives only an approximate value for this radius. We suggest
that this radius should be located slightly above the surface of the star to explain the dominance of
the octupolar field seen in \fig{bcompb} and \fig{bsurfb}.
\begin{figure*}
\centering
\includegraphics[width=14.0cm]{fig03-mag-2.png}
\caption{\label{magb} The magnetic field of BP Tau modeled as a
superposition of dipole and octupole components at $t=0$. The
field shows an octupolar structure only very close to the star,
while the dipole field dominates in the rest of the simulation
region. The color of the field lines represents the polarity and
strength of the field. The thick cyan, white and orange lines
represent the rotational axis and the dipole and octupole moments
respectively.}
\end{figure*}
\fig{magb} shows the initial magnetic field distribution near the star in our model.
One can see the octupolar field component in the vicinity of the star. It also
modifies the dipole field up to distances of
$r\approx 0.5 R_\star$ (above the surface of the star). The dipole field dominates in the rest of the simulation region.
We should note that the dipole and octupole fields are equal at $r_{eql}\approx R_\star$, but
the octupole component disturbs the dipole field up to larger distances.
\subsection{Matter flow and the magnetic field structure}
\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{fig04-3d.png}
\caption{\label{3db} 3D view of matter flow and the magnetic field
distribution in the main case at $t=10$. The left panel shows the
distribution of the magnetic field lines. The middle panel shows
one of the density levels, $\rho=0.84\times10^{-13}$. The right
panel shows the density distribution in the disc plane. The colors
along the field lines represent different polarities and strength
of the magnetic field. The thick cyan, white and orange lines
represent the rotational axis and the dipole and octupole moments
respectively.}
\end{center}
\end{figure}
Here, we show results of 3D MHD simulations of matter flow onto the dipole plus octupole model
of BP Tau with parameters corresponding to the main case
($\widetilde\mu_1=3$, $\widetilde\mu_3=0.24$) at time $t=10$ when the system is in a
quasi-stationary state. Fig. \ref{3db} (middle panel) shows that the disk is truncated by the dipole component of the field, and matter flows
towards the star in two ordered funnel streams. The right panel shows the density distribution in
the equatorial plane.
The left panel shows that the magnetosphere is disturbed by the disk-magnetosphere interaction.
\fig{flowb} shows slices of the density distribution and projected
field lines. Panel (b) shows the slice in the $\bf{\mu_1} -
\bf{\Omega}$ (or $xz$) plane. One can see more clearly that the
dipole component of the field is responsible for disk truncation
and matter is channeled in the $\bf{\mu_1} - \bf{\Omega}$ plane.
Panel (c) shows that the disk is stopped by the magnetosphere in
the $yz-$plane. It is stopped at $r\approx (6-7)R_\star$. Panel
(d) shows that matter flow is complex in the equatorial plane, and
that matter comes closer to the star in the $yz-$plane.
\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{fig05-flow.png}
\caption{\label{flowb} Density distribution and selected field lines in different slices.
Panels (a) and (b) show $xz$ slices
at $t=0$ and $t=10$. Panels (c) and (d) show $yz$ and $xy$ slices at $t=10$.
The gray line shows the distance at which the matter stress equals the magnetic stress.}
\end{center}
\end{figure}
The gray line in \fig{flowb} shows the magnetospheric radius,
$r_m$, the distance where the matter and magnetic stresses are
equal: $\beta = (p+\rho v^2)/(B^2/8\pi) = 1$. The magnetic stress
dominates at $r<r_m$ (\citealt{ghosh78, ghosh79a, long10}). Fig.
\ref{flowb} shows that $r_m \approx 5R_\star$ which is smaller
than the actual truncation radius $r_t\approx (6-7) R_\star$. We
should note that there are different criteria for the truncation
radius and the above criterion usually gives the smallest
truncation radius (see \citealt{bess08} for details).
\begin{figure*}
\begin{center}
\includegraphics{fig06_bptau_blines.png}
\caption{\label{blinesb}
Comparison of the initial (potential) magnetic field distribution
at $t=0$ (left panels) with the field distribution at $t=10$ (middle and right panels).
The top panels show the magnetic field in the vicinity of the star, while the bottom panels
show the field distribution in the whole simulation region.
The left and middle columns show the side view, while the right column shows the axial view of the field
The color on the star's surface shows
different polarities and strengths of the magnetic field.}
\end{center}
\end{figure*}
The external magnetic field evolves and deviates from the initial configuration
due to disk-magnetosphere interaction. The left panels of
\fig{blinesb} show the initial field distribution at $t=0$
which corresponds to the potential field produced by the star's internal currents.
As the magnetic field evolves, the currents produced by the motion of plasma
outside the star become important and they change the potential magnetic field
significantly.
The middle and right panels show the evolved field from different directions at $t=10$.
It can be seen that on a large scale the magnetic field lines wrap around
the rotational axis of the star and form a magnetic tower (e.g. \citealt{lynd96, kato04, roma04b}).
This is a natural result of the magnetic coupling between the disk and star.
The foot-points of the field lines on the star rotate faster than the
foot-points threading the disk, which leads to differential rotation along the
lines, and their stretching and inflation into the corona. The rotational
energy is converted into the magnetic energy associated with these field
lines. The evolved field structure on the large scale significantly differs from
the potential field shown in the left panels.
In the vicinity of the star the situation is different. The middle top panel of
\fig{blinesb}
shows that the magnetic field distribution at $t=10$ is almost identical to the initial field
distribution.
We conclude that the potential approximation is valid only within the parts of the magnetosphere where
the magnetic stress dominates, that is, at $r\lesssim 5 R_\star$ in our case.
The field distribution strongly departs from potential at larger distances from the star.
\subsection{Accretion spots in BP Tau and in the Model}
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{fig07-hotspots.png}
\caption{\label{hsb}
The simulated accretion spots viewed from different directions
at $t=10$: from the equatorial plane
(left-hand panel), the north pole (middle panel), and the south pole
(right-hand panel). The color contours
show the density distribution of the matter. The solid lines represent the magnetic
moments of the dipole ($\bm{\mu_1}$) and octupole ($\bm{\mu_3}$).}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=6.0cm]{fig08-3dspot.png}
\caption{\label{3dspot} The color background shows the distribution of the magnetic field
on the surface of the star. The blue contours show the energy distribution in the accretion spot.
The solid black lines show the dipole, octupole and rotational axes.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{fig09-hotspots-polar.png}
\caption{\label{hspb} The simulated accretion spots in a
polar projection down to colatitude
$120^\circ$ from the north pole. Left panel:
density distribution; right panel: energy flux distribution.
The equator is shown as a bold line. The dashed lines
represent the latitude $30^\circ$ and $60^\circ$ respectively.}
\end{center}
\end{figure}
D08 analyzed accretion spots from the observed brightness enhancements in chromospheric lines,
such as Ca\textsc{ii} IRT and He\textsc{i} which presumably form in (or near) the shock front close to
the stellar surface. The random flaring component,
as well as the time-variable veiling component which reflects variation of the intrinsic accretion rate were
removed from the modeling. It was also suggested that matter flows in the vicinity of the
strong magnetic field and hence Zeeman-splitting features were used for analysis.
The corresponding maps of the local surface brightness are shown in Fig. 9 of D08 for two
epochs of observations (Feb06 and Dec06). It can be seen that in both plots there is one
bright spot located at colatitudes of $30^\circ-70^\circ$ and centered at $\theta_c\approx 45^\circ$,
while a lower-brightness area spreads up to a colatitude of $90^\circ$. In both cases the spots
are elongated in the meridional direction. The Feb06 spot also has an antipodal spot of weaker
brightness and of similar shape. The accretion filling-factor is shown in Fig. 13 of D08.
It shows a spot located at colatitudes of $0^\circ-50^\circ$ and centered
at $\theta_c\approx 10^\circ-20^\circ$.
In our simulations the spots represent a slice of density/energy taken across the
funnel stream at the surface of the star. Hence, the spots show the distribution of these
values across the stream \citep{roma04a}. The physics of disk-magnetosphere interaction
is expected to be more complex (e.g., \citealt{kold08, cran08, cran09, bric10}). However,
this more complex physics strongly depends on the properties of funnel streams. Hence we
call these spots ``accretion spots" and show the density distribution in spots and
also the energy flux: $F=\rho\bm{v}\cdot\hat{r}[(v^2-v_\star^2)/2+\gamma p/(\gamma-1)\rho]$
distribution (right panel) at $t=10$, where $v_\star$ is the velocity of the star.
\fig{hsb} shows the density distribution at the surface of the star at $t=10$.
It can be seen that there are two antipodal spots which are centered in the $\bm{\mu_1} - \bm{\Omega}$
plane slightly below the dipole magnetic pole. The spots are elongated in the meridional direction
(unlike the spots in a pure dipole configuration, see also \S 3.4).
\fig{3dspot} shows the density distribution in the spot
(contour lines)
overlaid on top of the magnetic field distribution. It can be seen that the spot is located near and below
the dipole magnetic pole and hence its position is strongly influenced by the dipole
component. The spot is located far away from the main, high-latitude magnetic pole,
dominated by the octupolar component. We conclude that this low-latitude
position of the spot results from the fact that
the dipole field governs the matter flow, and only near the star,
the octupolar component ``comes to play" and the octupolar field
redirects the spot's position towards slightly higher latitudes, and changes the spot's shape.
\fig{hspb} shows a polar projection of the density distribution (left panel)
and energy flux.
It can be seen that the accretion spot is centered near the
latitude of $40^\circ$ and spreads between latitudes of $10^\circ$
and $60^\circ$. These spots have a close resemblance to the spots
observed by D08. In both cases, the spots are stretched in the
meridional direction and are located between latitudes of
$10^\circ$ and $60^\circ$. Analysis of our spots at different
moments of time has shown that the positions of the spots vary
only slightly with time (within 0.1 in phase). However, the spots
observed by D08 (their Fig. 9) are located at different phases.
Feb06 observations show the main spot located at a higher phase
compared with our spot, with a phase difference of 0.15-0.2. This
is quite good agreement, considering the fact the D08 method of
the dipole moment phase reconstruction has an error in phase of
0.1-0.2 or larger. The phase difference is higher in the Dec06
observations where the simulated and observed spots are almost in
anti-phase.
To investigate the role of the dipole and octupole phases in the
spot's position, we considered a few ``extreme" cases, where the
phase difference between the dipole and octupole moments is
$\phi=0^\circ$, $\phi=90^\circ$ and $\phi=-90^\circ$ (we have a
phase difference of $\phi=180^\circ$ in the main case). We
observed that the accretion spot is similar in shape in all these
cases, but is located at the phase corresponding to the phase of
the dipole (in the $\bm\mu_1-\bm\Omega$ plane). We conclude that
the phase of the accretion spots on the surface of the star gives
a strong indication of the dipole component's position.
\subsection{Comparison with a pure dipole model}\label{comparison}
To understand the role of the octupolar field component
in our dipole plus octupole model of BP Tau,
we investigate a similar model but with zero octupolar field,
that is $B_{1\star}=1.2$ kG, $\Theta_1=20^\circ$ and
$B_{3\star}=0$.
\fig{bptau_comp} shows $xz$ slices of the accretion flow in the
dipole plus octupole model (left panel) and pure dipole model
(right panel). One can see that the flow is wider in the dipole
plus octupole model, because near the star the octupole field
influences the matter flow and redirects it in such a way that
some matter flows towards the direction of the octupolar magnetic
pole located at anti-phase with the dipole pole. The streams are
narrower in the case of the pure dipole field.
The bottom panels show that the
accretion spots in the dipole plus octupole model are meridionally
elongated, while in the dipole model they have their typical crescent shape and are elongated
in the azimuthal direction (as usually seen in the pure dipole cases, e.g. \citealt{roma03, roma04a}).
We conclude that in BP Tau, the relatively weak octupolar magnetic field which dominates only
very close to the surface of the star, strongly influences the shape and position of
the accretion spots. This influence is not as dramatic as in another T Tauri star, V2129 Oph,
which has a much weaker dipole component, and where the octupolar field
splits the funnel stream into polar and octupolar belt flows
\citep{roma10}.
\begin{figure*}
\begin{center}
\includegraphics[width=12.0cm]{fig10-comp.png}
\caption{\label{bptau_comp} Comparison of matter flow and accretion spots in the dipole plus
octuple model of BP Tau (left panels) with a pure dipole case (right panels) at $t=10$. The top panels
show the density distribution (color background) and the sample poloidal
field lines in the $xz-$plane. The bottom panels show the density
distribution in the accretion spots as seen along the $y-$axis (panels a,c)
and along the rotational axis (panels b,d).}
\end{center}
\end{figure*}
\subsection{Area covered with spots}
\begin{figure}
\centering
\includegraphics[width=7.0cm]{fig11-spotsarea.png}
\caption{\label{areab} Fraction of the star's surface covered by spots of density $\rho$ and higher at $t=10$.
The solid and dashed lines represent the dipole plus octupole model and dipole model respectively.}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=14.0cm]{fig12-5spots.png}
\caption{\label{spotssize} The density distribution of spots with
a cutoff at different density levels $\rho$, and the corresponding
spot coverage $f$ for the sum of the visible and antipodal
spots}.
\end{figure*}
D08 calculated the fraction of the star covered with visible
chromospheric spots on BP Tau and concluded that they are spread
over up to about $8\%$ of the stellar surface, and cover about
$2\%$ of the surface, assuming one-fourth of each surface pixel is
subject to accretion (see Figs. 9 \& 13 in D08). In this paper we
are interested only in the area of the spots spreading that is in
the area covered by the funnel stream, and hence we take the
larger value (8\%). Figs. 9 and 13 of D08 show that the actual
size of the spots depends on the brightness. Sometimes we take the
brightest parts of spots (4-5)\% and neglect the dimmer parts for
comparison. In observations, the spot's coverage is calculated for
one (visible) spot versus the whole area of the star, while in
the simulations we take into account both spots, and hence the
area in simulations is twice as large compared with observations.
We observed from the simulations that the size of the accretion spot depends
on the density (or energy) cutoff. We calculate the fraction of the star covered with the spots as
$f(\rho)=A(\rho)/4\pi R_\star^2$, where
$A(\rho)$ is the area covered with (all) spots of density $\rho$
or higher. \fig{areab} shows the distribution of $f(\rho)$ versus
$\rho$ for the dipole plus octupole model and dipole model of BP
Tau. It can be seen that for almost all chosen density levels, the
accretion spots occupy a larger surface area in the dipole plus
octupole model than in the dipole model. This result is in good
agreement with that obtained from \fig{bptau_comp}.
We choose several cutoff
densities
and show the spots in \fig{spotssize}. One can see that in both
simulations and observations, spots cover different areas
depending on the brightness/energy flux levels ranging from $14\%$
up to $2\%$ in simulations (for two spots). In both observations
(see Fig. 9 of D08) and simulations (see our Fig.
\ref{bptau_comp}), the brightest parts of spots are located at
$30^\circ<\theta_c<60^\circ$.
It was predicted that the area covered by spots in the case of
complex magnetic fields should be smaller than in the pure dipole
case (e.g., \citealt{moha08, greg08}, see also
\citealt{calvet98}). This paper and our previous work
\citep{long08} show that the area $f$ could show a complex trend
for mixed dipole and multipole configurations, depending on how
the multipole field redirects the accretion flow.
\subsection{Accretion rate in BP Tau and in our model}
The mass accretion rate of BP Tau obtained from observations is not uniquely determined, and
different authors give different results.
For example, \citet{gull98}
estimated the mass accretion rate to be $2.9\times10^{-8}M_\odot\mathrm{yr}^{-1}$ using the
luminosity of $L\simeq (GM_\star\dot{M}/{R_\star})(1-R_\star/R_{in})$,
where $R_{in}$
is the inner radius of the disk. Other estimates of the mass accretion rate include
$1.6\times10^{-7}M_\odot\mathrm{yr}^{-1}$ \citep{valen04},
$9\times10^{-10}M_\odot\mathrm{yr}^{-1}$ \citep{schm05}. \citet{calvet04} compared
the mass accretion rates of intermediate-mass, $(1.5-4)M_\odot$, and
low-mass, $(0.1-1.0)M_\odot$, T Tauri stars and concluded that the average mass
accretion rate for intermediate-mass T Tauri stars is about $3.0\times10^{-8}M_\odot\mathrm{yr}^{-1}$,
while for low-mass CTTSs (like BP Tau) it is about
5 times smaller, that is, $6.0\times10^{-9}M_\odot\mathrm{yr}^{-1}$.
We obtained from simulations the dimensionless accretion rate of $\widetilde{\dot M}\approx 0.13$.
Using the reference value for $\dot M_0$ from Tab. \ref{tab:refval},
$\dot M_0=1.1\times10^{-8}M_\odot\mathrm{yr}^{-1}$
(calculated for $\widetilde\mu_1=3$), we obtain the dimensional
accretion rate: $\dot M=\dot M_0 \widetilde{\dot M}\approx
1.4\times 10^{-9}M_\odot\mathrm{yr}^{-1}$. This value is higher
than the accretion rate suggested by \citet{schm05} but lower than
the other estimates. Given that BP Tau has the low mass of $\sim
0.7M_\odot$ (D08) (or $0.49M_\odot$ \citealt{gull98}), the mass
accretion rate from our simulation models may still be in the
reasonable range according to the \citet{calvet04} analysis and
discussion.
Here, we should note that the ``main model" considered above is one of the suggested models, where the disk
is truncated at large distances. Below, we discuss additional models, where the
disk is truncated at smaller distances and the accretion rate is higher.
\subsection{Modeling of BP Tau at higher accretion rates}
We performed two additional simulation runs at higher accretion rates (at smaller values of
parameter $\widetilde\mu_1$) keeping the ratio between the dipole and octupole moments fixed,
$\widetilde\mu_3/\widetilde\mu_1=0.08$.
The parameters of these new models are the following:
$\widetilde\mu_1=2, \widetilde\mu_3=0.16$ and
$\widetilde\mu_1=1, \widetilde\mu_3=0.08$.
\fig{compareb} (middle and right panels) shows the matter flow and
accretion spots for these two cases, while the left panel shows
results for the main model ($\widetilde\mu_1=3,
\widetilde\mu_3=0.24$) for comparison. One can see that in the new
models, the disk is truncated at smaller distances from the star:
$r_t\approx(5-6) R_\star$ (for $\widetilde\mu_1=2$) and
$r_t\approx3.6 R_\star$ (for $\widetilde\mu_1=1$). The bottom
panels of \fig{compareb} show that when the disk comes closer to
the star, the spots are still at high latitudes, though they
become longer in the meridional direction. This is probably
because the octupolar component has a stronger influence on the
spots: the spots move closer to the octupolar magnetic pole. Note
that no octupolar ring spots appear (like in V2129 Oph, see R10).
This is probably because the disk truncation radius is still far
away from the area where octupole has a strong influence on the
flow, which is closer to the star. It is interesting to note that
in the pure dipole case, the spots move towards lower latitudes
when the disk comes closer to the star. Here, we see the opposite:
parts of the spots move to higher latitudes, because the octupolar
component becomes more significant at smaller truncation radii.
The dimensionless mass accretion rate which we obtain from our
simulations is $\widetilde{\dot M}\approx 0.075$ (for
$\widetilde\mu_1=2$) and $\widetilde{\dot M}\approx 0.085$ (for
$\widetilde\mu_1=1$). We take reference values $\dot M_0$ for
different $\widetilde\mu_1$ from Tab. \ref{tab:refval} and obtain
corresponding dimensional accretion rates $\dot M$ (see Tab.
\ref{tab:mdot-angmom}) which are $2.5\e{-9}M_\odot$yr$^{-1}$ and
$8.5\e{-9}M_\odot$yr$^{-1}$ respectively. We see that the
accretion rate for $\widetilde\mu_1=1$ is high and close to many
of the observed values. However, the disk comes too close to the
star and the truncation radius is much smaller than the corotation
radius ($R_{cor}\approx 7.5 R_\star$), which would mean that the
star is not in the rotational equilibrium state, which is
unlikely. The model with $\widetilde\mu_1=2$ is similar to the
main model: the disk is truncated far away and the accretion rate
is twice as high, but still it is lower than many values given by
observations. This model is slightly better than the ``main"
model.
In the case of a pure dipole, we obtain from simulations
$\widetilde{\dot M}\approx 0.12$ and $\dot M\approx
1.3\e{-9}M_\odot$yr$^{-1}$. \ref{tab:mdot-angmom} summarizes the
results obtained in our simulation models and some observational
properties of BP Tau for comparison.
\begin{figure}
\centering
\includegraphics[width=8.5cm]{fig13-compare.png}
\caption{\label{compareb} Simulations at different accretion rates
at $t=8$. (a) main case, $\widetilde\mu_1=3,
\widetilde\mu_3=0.24$; (b) $\widetilde\mu_1=2,
\widetilde\mu_3=0.16$; (c), $\widetilde\mu_1=1,
\widetilde\mu_3=0.08$. The top panels show the density
distribution (color background), the poloidal field lines (red
lines) and the $\beta=1$
line (gray) in the $xz-$plane. The bottom panels show the energy flux
distribution in the accretion spots
in a polar projection.}
\end{figure}
\begin{table*}
\caption{Comparison of observational and modeled results shown in
D08 and results from our simulations.} \centering
\begin{tabular}{lllllll}
\hline
Models & $r_t$ & $\widetilde{\dot{M}}$ & $\dot{M}$($M_\odot$yr$^{-1}$) & $\widetilde{N_f}$ & $N_f$ (g cm$^2$s$^{-2}$) & $\tau$ (yr) \\
\hline
D08 \& other measurements & $>4R_\star$ & -- & $10^{-9} - 10^{-7}$ & --& \\
Model ($\widetilde\mu_1=3, \widetilde\mu_3=0.24$) & $(6-7)R_\star$ & $0.13$ & $1.5\e{-9}$ & $-0.03\pm0.014$ & $-1.9\e{35} $ & $1.6\e7$ (spin-down) \\
Model ($\widetilde\mu_1=2, \widetilde\mu_3=0.16$) & $(5.4-6.4)R_\star$ & $0.075$ & $1.9\e{-9}$ & $-0.005\pm0.01$ & $-1.5\e{35}$ & $2.1\e7$ (spin-down) \\
Model ($\widetilde\mu_1=1, \widetilde\mu_3=0.08$) & $3.6 R_\star$ & $0.075$ & $7.7\e{-9}$ & $+0.002\pm0.004$ & +$2.3\e{35}$ & $1.1\e7$ (spin-up) \\
Model ($\widetilde\mu_1=3, \widetilde\mu_3=0$) & $(6-7)R_\star$ & $0.12$ & $1.4\e{-9}$ & $-0.045\pm0.015$ & $-2.4\e{35}$ & $1.3\e7$ (spin-down) \\
\hline
\end{tabular}
\label{tab:mdot-angmom}
\end{table*}
\subsection{Angular momentum transport}
We calculated torque on the surface of the star for the above
three models. The torque associated with the magnetic field
$\tilde{N_f}$ is about 10-30 times larger than the torque
associated with matter and thus dominates (see also
\citealt{roma02,roma04a,long07,long08}). The dimensionless torque
obtained from simulations, $\widetilde N_0$, varies in time around
some average value. Tab. \ref{tab:mdot-angmom} shows these values
and values corresponding to these variations. We observed in
simulations that the torque is negative in most of the models
because the disk is truncated at the distances comparable with the
corotation radius and therefore the star spins-down. In a model
with $\widetilde\mu_1=1$, the disk is truncated at the distance of
$\sim 0.5 R_{cor}$ and it spins the star up. To estimate the
dimensional torque, $N_f$, we take the largest absolute value of
$\widetilde N_f$ from Tab. \ref{tab:mdot-angmom} and take into
account the reference values of $N_0$ from Tab. \ref{tab:refval}.
Tab. \ref{tab:mdot-angmom} shows the dimensional torque.
We estimate the time-scale of spinning-down of BP Tau. The star's
angular velocity is (at period $P=7.6$ days) $\Omega=2\pi/P
\approx 9.6\times 10^{-6} {\rm s}^{-1}$, its angular momentum is
$J= k M_\star R_\star^2 \Omega = 2.5\times 10^{50} k ~{\rm g
cm}^2/{\rm s}$, where $k<1$ (we take $k=0.4$ for estimations). The
spin-down time-scale, $\tau = J/N_f$, was estimated for different
models and is shown in Tab. \ref{tab:mdot-angmom}. One can see
that the spin-up/down time-scale is about an order magnitude
larger than the age of BP Tau ($ 1.5\times 10^6$ years, see
references in D08). Hence, the torque obtained in simulations is
not sufficient to regulate the spin of the star at the considered
epoch. It is possible that a star lost most of its angular
momentum at the earlier stages of its evolution. Note that similar
low-torque situation has been observed in modeling of another
CTTS, V2129 Oph.
\section{Conclusions and Discussions}
We performed global 3D MHD simulations of accretion onto a model star with parameters
close to those of the classical T Tauri star BP Tau, and with the magnetic field
approximated with a superposition of slightly tilted dipole and octupole moments with polar magnetic fields of
1.2kG and 1.6kG which are in anti-phase
(D08). In this star
the dipole field dominates almost up to the surface of the star and determines the majority
of the observational properties.
We performed a number of simulation runs for different truncation
radii of the disk, and chose one of them where the disk is
truncated at $r\approx (6-7) R_\star$ which is sufficiently close
to the corotation radius $R_{cor}\approx7.5 R_\star$ and
investigated this case in greater detail. We observed that the
dipole component of the field truncates the disk, and matter flows
in two funnel streams towards the dipolar magnetic poles. However,
near the star the flow is slightly redirected by the octupolar
component towards higher latitudes, and this affects the shape
and position of the accretion spots: the spots are stretched in
the meridional direction and are centered at higher latitudes
compared with spots in the pure dipole case, which are
latitudinally-elongated (crescent-shaped) and are centered at
lower latitudes.
The spots are located near the $\bm{\mu_1}-\bm{\Omega}$ plane, where both the
dipole and octupole moments are
situated, and are in anti-phase. Experiments with different relative phases between the dipole and octupole moments
have shown that the spots are always located in the meridional plane of the dipole moment.
The spot's position slightly varies in phase with the accretion rate, but
the variation is small, about $8^\circ$ at the most.
Note that in the pure dipole case, the spot may rotate about the magnetic pole
(and hence may strongly change the phase) if the dipole is only slightly tilted about the
rotational axis, $\Theta_1\lesssim 5^\circ$ (\citealt{roma03, roma04a, bace10}).
However, in BP Tau, the tilt is larger, $\Theta_1\approx 20^\circ$ and an octupolar component
is present and is significant near the star. Both factors lead to the restriction of such a motion.
The meridional position and shape of spots observed in simulations is similar to those observed
by \citet{dona08} (see their Fig. 9 for brightness distribution in spots
reconstructed for the Dec06 or Feb06 epochs). However, they are at different phases compared with
our spots.
This may be connected with the relatively low accuracy of the phase
reconstruction of the dipole component from the surface magnetic field, or
some other reason.
The accretion rates obtained in different models are in the range
of $\dot M\approx (1.5-7.7)\times 10^{-9}M_\odot$yr$^{-1}$. It
is lower than most of the $\dot M$ values derived from
observations which vary in the range of $9\e{-10}$ and $1.6\e{-7}$
solar mass per year and depend on the approaches used for the
derivation. The smallest accretion rate obtained in simulations
corresponds to the case where the disk is truncated at $r_t\sim
R_{cor}$, and a star slightly spins down, while the largest to the
case where $r_t\sim 0.5 R_{cor}$ and a star spins up. We should
note that at larger accretion rates, say at $1.6\e{-7}$ solar
mass per year, the disk will come very close to the star, and the
torque would be much stronger. Such a state seems unlikely. Hence,
if the accretion rate is very high, then we should suggest that
the dipole component should be larger than that derived by D08.
The torque obtained in simulations is small and the time-scale of
the spinning-up/down is an order of magnitude smaller compared
with the age of BP Tau. This torque is not sufficient to support a
star in the rotational equilibrium state, where a star spins up or
spins down depending on the accretion rate but has a zero torque
on average (e.g., \citealt{ghosh79a}, \citealt{konigl91},
\citealt{camer93}, \citealt{long05}).
We assume that this small
torque matches a small accretion rate obtained in simulations
because the torque generally correlates with the accretion rate
(e.g., \citealt{roma02}).
We suggest that a star may lose the majority of its angular
momentum at earlier stages of its evolution due to the
``propeller" effect (e.g., Romanova et al. 2005; Ustyugova et al.
2006), stellar winds (\citealt{matt05}, 2008), or by some other
mechanism.
Earlier, we performed global 3D MHD simulations of accretion onto V2129 Oph \citep{roma10}
which have shown that in the case of a strong octupolar component, parts of the octupolar
belt spots can be visible and can dominate at sufficiently high accretion rates.
This is not the case in BP Tau,
where the octupolar component dominates only in the close vicinity of the star.
Disk-magnetosphere interaction leads to inflation of the external field lines
and formation of a magnetic tower.
Simulations show that the potential approximation used in extrapolation of the magnetic
field from the surface of the star to larger distances (e.g. \citealt{jard06}, D08, \citealt{greg10}) is
valid only inside the magnetospheric (Alfv\'en) surface, where the magnetic stress
dominates. At larger distances, the magnetic field distribution
strongly departs from potential.
\section*{Acknowledgments}
Resources supporting this work were provided by the NASA High-End Computing (HEC)
Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center and
the NASA Center for Computational Sciences (NCCS) at Goddard Space Flight Center.
The authors thank A.V. Koldoba and G.V. Ustyugova for the earlier development of
the codes. The research was supported by NSF grant AST0709015. The research of MMR was supported by NASA grant
NNX08AH25G and NSF grant AST-0807129.
|
1,116,691,499,322 | arxiv | \section{Introduction}
The incorporation of individual quantum systems into solid-state platforms,\cite{MichlerScience00, WallraffNature04, KoppensNature06, PlaNature12, BrannyNatCom17, EvansScience18} their coherent control, and interfacing them with external degrees of freedom\cite{PettaScience05, NowackScience07, LodahlRMP15, WiggerQUTE21} is a key for implementation of quantum technologies. One of such promising platforms are semiconductor quantum dots (QDs),\cite{MichlerBook03} which owing to constant progress in the epitaxial growth\cite{HermelinNature11, KuhlmannNatCom15} and chemical synthesis,\cite{GarciaScience21} have now reached a tremendous structural quality.\cite{SomaschiNatPhot16} In parallel, processing of this material has been driven virtually to perfection permitting advanced engineering of the light-matter coupling with photonic structures.\cite{LodahlRMP15} As a result, QDs in photonic micro-structures serve as compact sources of single photons for quantum cryptography.\cite{SchimpfSciAdv21}
Conversely, optical or electrical control of single quantum states confined to a QD is challenging, nonetheless intensely pursued in fundamental research.\cite{StinaffScience06, ReiterJoPCM14, KaldeweyPRB17, WeissOptica21} Over the last decade, a major progress has been achieved in measuring\cite{LangbeinPRL05, PattonPRB06, KasprzakNJP13} and controlling\cite{FrasNatPhot16, WiggerOptica18, HenzlerPRL21} the coherence of optical transitions attributed to the bound electron-hole pair, forming a QD exciton. This was achieved by performing coherent ultrafast nonlinear spectroscopy,\cite{LangbeinOL06} in particular four-wave mixing (FWM) on photonic devices hosting InGaAs QDs. However, the exciton radiative lifetime $T_1$ lies typically in the nanosecond range, thus setting the upper bound for its coherence time $T_2\leq 2T_1$. Although an exciton represents an efficient interface between light and matter, its short $T_2$ limits its usage as a qubit. Hence, a promising perspective in this field is the search for efficient coupling schemes between an exciton and quantum systems exhibiting significantly longer $T_2$, for example dark exciton states\cite{PoemNPhy10, KorkusinskiPRB13} or individual spins.\cite{HansonNature08, YilmazPRL10, QuinteiroPRB14, HinzPRB18} Besides employing QDs charged by a single electron\cite{AtatureScience06} or hole\cite{GerardotNature08}, the latter can be achieved by doping QDs with single magnetic ions, like manganese (Mn), which is part of the emerging research area of solotronics, i.e., the field of optoelectronics associated with single dopants.\cite{KoenraadNatMat11, KobakNatComm14}
To bring the benefits of quantum optics and related tools to solotronics, one first needs to introduce the dopant ion into the QD\cite{BesombesPRL04, KudelskiPRL07, FainblatNanoLet16} and enclose it within a photonic structure to enhance the light-matter coupling.\cite{PacuskiCGD14, PacuskiCGD17} Recently, this requirement was fulfilled by molecular beam epitaxy (MBE) of the II-VI semiconductor CdTe, nowadays offering QD systems hosting various magnetic ions\cite{KobakNatComm14, SmolenskiNatComm16, LafuentePRB16} and reliable fabrication of optical microcavities.\cite{DangPRL98} Nevertheless, the progress in coherent spectroscopy of single excitons in CdTe QDs has been laborious,\cite{PattonPRB06, PacuskiCGD17} due to the restricted availability of femtosecond laser sources emitting in the visible range.
In the present work, we perform FWM spectroscopy of single exciton in CdTe QDs embedded in a microcavity. We first employ a non-magnetic dot, to demonstrate its quantum character by performing the Rabi rotation measurements.\cite{PattonPRL05, WiggerPRB17} Next, we determine the exciton's population and coherence dynamics. In the latter case, we reveal the formation of a photon echo,\cite{LangbeinPRL05} phonon-induced dephasing (PID),\cite{JakubczykACSPhot16, WiggerOL20} and coherence beating owing to the fine-structure splitting (FSS) of the exciton.\cite{KasprzakPRB08, MermillodPRL16} Finally, we focus on a QD doped with an individual Mn$^{2+}$ ion, which in the II-VI material CdTe acts as an isoelectronic impurity. We show that the exciton-Mn$^{2+}$ exchange interaction introduces an additional ensemble characteristic in the time-averaged experiments. In this specific situation, the impact of the Mn$^{2+}$-spin with total spin quantum number $S=5/2$ results in the appearance of six different transition energies associated with the six possible orientations of the Mn spin. It has been shown that due to this characteristic spectral features the Mn spin can be initialized, read out and controlled\cite{LePRB10, BesombesNanophot15} and protocols have been suggested for a selective switching of the spin state.\cite{ReiterPRL09, ReiterPRB11, ReiterPRB12} This report is thus an initial step on the spectroscopic quest towards fully fledged coherent quantum control of possible spin-photon interfaces.\cite{HansonNature08}
\section{Sample and Experiment}
\begin{figure}[h]
\centering
\includegraphics[width=0.6\columnwidth]{Fig1}
\caption{Microcavity sample. (a) Schematic picture of the sample structure including top and bottom distributed Bragg reflectors (DBRs), the Mn-doped quantum dot (QD) layer (highlighted on the right), and a solid immersion lens (SIL) to improve focusing of the laser light and the collection efficiency. (b) Reflectivity spectrum of the microcavity.}
\label{fig:sample}
\end{figure}%
To perform FWM experiments of a single Mn-doped QD, we specifically conceive the microcavity sample schematically depicted in Fig.~\ref{fig:sample}(a). We have previously shown that in a standard microcavity, the light-matter interaction is enhanced through the intra-cavity field amplification,\cite{FrasNatPhot16} whilst preserving spectral matching with the excitation via femtosecond laser pulses. The asymmetric cavity design permits to reflect almost the entire optical response toward the detection path. With this methodology, we increase the FWM collection efficiency by several orders of magnitude with respect to planar samples. Inspired by that performance-boost, we here go beyond the previously used half-cavity design.\cite{PacuskiCGD17} We develop full monolithic cavities, choosing the quaternary alloy CdZnMgTe as building material. After having deposited a buffer on the GaAs substrate, a bottom distributed Bragg reflector (DBR) is grown by alternating Mg content between 10\% and 50\%. After the completion of 10 layer-pairs for the bottom DBR, we proceed by the formation of the $\lambda$ cavity. At the calculated field antinode we flush a CdTe QD layer and nominally set the Mn concentration to 0.1\% to allow a diluted doping including incorporation of single Mn$^{2+}$ ions into the QDs. Then, 4 layer-pairs for the upper DBR are deposited, completing the growth, which increases the light-matter coupling compared to sample studied in Ref.\cite{PacuskiCGD17}{}. To further improve the in-coupling of the optical excitation and the out-coupling of the optical signals, we attach a solid immersion lens (SIL) made of zirconium oxide on the sample surface. This half-ball lens with a 500~\textmu m diameter allows to perform optical microscopy up to approximately 50~\textmu m away from its axis without introducing significant geometrical aberration. The SIL increases the numerical aperture (NA) of the beam in the semiconductor material by reducing the refraction when crossing the sample surface. It further decreases the spherical diffraction resulting from the excitation fields passing through the semiconductor-air interface; at the same time it reduces the total internal reflection of the emitted FWM signal on the way back.
Monitoring micro-photoluminescence (\textmu PL) at $T=7$~K between energies $E_{\rm X}=1.85$~eV ($\lambda_{\rm X}=670$~nm) and 1.80~eV (690~nm), we observe the recombinations of individual excitons localized at the interface fluctuations, forming weakly-confined QDs, similarly as in GaAs structures.\cite{LangbeinPRL05,KasprzakNatPhot11} In the white light reflectance in Fig.~\ref{fig:sample}(b) we identify the cavity mode centered between $E_{\rm cav}=1.85$~eV ($\lambda_{\rm cav}=670$~nm) and 1.81~eV (685~nm) depending on the investigated position on the sample with a full-width at half maximum (FWHM) of 12.5~nm, yielding the quality factor of $Q = \Delta E_{\rm cav}/E_{\rm cav}\approx 55$.
\section{Undoped quantum dot}
\begin{figure}[h]
\centering
\includegraphics[width=0.55\columnwidth]{Fig2}
\caption{Schematic picture of the performed FWM experiments. (a) Coherence scan by varying $\tau_{12}$ resulting in the photon echo formation. (b) Occupation scan by varying $\tau_{23}$ exhibiting a typical exponential decay.}
\label{fig:pulses}
\end{figure}%
\begin{figure}[h]
\centering
\includegraphics[width=0.65\columnwidth]{Fig3}
\caption{Spatial mapping of the PL in (a) and the FWM in (b). The scans reveal QDs with excitons that are efficiently coupling to the optical excitation. Note, that the detected areas on the sample are not aligned.}
\label{fig:scan}
\end{figure}%
To perform FWM microscopy we use a laser pulse train centered around $\lambda=680$~nm at the repetition rate of 76~MHz, generated by an optical parametric oscillator ({\it Inspire 50} by {\it Radiantis}) pumped by a femtosecond Ti:Sapphire oscillator ({\it Tsunami-Femto} by {\it Spectra-Physics}). To induce FWM, we generate three beams $\ensuremath{{\cal E}_{1,2,3}}$, with respective inter-pulse delays $\tau_{12}$ and $\tau_{23}$ as schematically shown in Fig.~\ref{fig:pulses}, introduced by a pair of mechanical delay stages. The beams pass through acousto-optic modulators where they undergo distinct shifts $\Omega_{1,2,3}$ of the carrier frequency. Using a microscope objective ({\it Olympus}, NA=0.6), the beams are focused reaching a diffraction limited spot on the surface of the sample. The sample is placed in a helium-flow cryostat operating at $T=7$~K. By raster scanning the position of the objective, we can construct hyperspectral images of the optical signals.\cite{KasprzakPSSb09, FrasNatPhot16} Fig.~\ref{fig:scan} shows exemplary scans of the PL and the FWM signal in (a) and (b), respectively, where the maps were accumulated for a range of transition energies. A pulse shaper is used to correct the temporal chirp, mainly stemming from the thick optics in the acousto-optic modulators and the objective, to attain transform-limited pulses of around 150~fs duration. The reflected light from the sample is collected by the same objective and directed into an imaging spectrometer with a CCD camera at its output. The FWM response, which in the lowest (third) order is proportional to $\ensuremath{{\cal E}_1}^{\ast}\ensuremath{{\cal E}_2}\ensuremath{{\cal E}_3}$, propagates shifted by the radio-frequency $\Omega_{\rm FWM}=\Omega_3+\Omega_2-\Omega_1$, which is around 80~MHz. Its amplitude and phase are thus obtained via optical heterodyning the reflected light at $\Omega_{\rm FWM}$. Additionally, a reference beam $\ensuremath{{\cal E}_{\rm R}}$ is employed to perform spectral interferometry. More details about the experimental setup can be found in Refs.\cite{FrasNatPhot16, JakubczykACSNano19}{}. As explained below, by inspecting FWM temporal and delay dynamics, we obtain full information regarding the system's inhomogeneous $\sigma$ and homogeneous $\gamma=2\hbar/T_2$ dephasing, as well as its population decay.
\begin{figure}[t]
\centering
\includegraphics[width=0.55\columnwidth]{Fig4}
\caption{FWM spectroscopy of a QD exciton. (a) Spectrum of the reference pulse in green and a typical spectral heterodyne interferogram in blue. (b) FWM amplitude spectrum in dark red and the FWM phase in pale red.}
\label{fig:X_spec}
\end{figure}
For our FWM investigations, we select those optical transitions that dominate in PL and that are spectrally located at the center of the cavity mode. In Fig.~\ref{fig:X_spec}(a) we present a typical spectral interference (blue) heterodyned at $\Omega_{\rm FWM}$ originating from a undoped CdTe QD together with the laser pulse spectrum (green). The retrieved FWM amplitude and phase are presented in Fig.~\ref{fig:X_spec}(b) as dark and pale red line, respectively. While the FWM amplitude exhibits a typical peak structure, the respective phase shows a jump of approximately $\pi$~\cite{KasprzakNatPhot11}. It is worth to note that QDs generating FWM are rather isolated with typical distances of several \textmu m, as exemplified by the FWM hyperspectral mapping presented in Fig.~\ref{fig:scan}(b).
To first characterize the enhanced light-matter coupling, we measure how the FWM amplitude depends on the applied laser pulse intensities. We thus fix the excitation powers of $\ensuremath{{\cal E}_2}$ and $\ensuremath{{\cal E}_3}$ to $P_2=P_3=0.25$~\textmu W and vary $\ensuremath{{\cal E}_1}$'s power $P_1$. In Fig.~\ref{fig:Rabi} we plot the spectrally integrated FWM amplitude as a function of $\sqrt{P_1}$, which is proportional to the pulse area $\theta_1=\int dt\,\mathcal E_1(t)/\hbar$, where $\mathcal E_1$ is the electric field amplitude of the first pulse at the QD location multiplied by the transition dipole matrix element. As the measured FWM amplitude (blue dots) is proportional to the microscopic coherence of the exciton state,\cite{WiggerPRB17,WiggerOptica18} the signal is proportional to $\left|\sin(\theta_1-\theta_0)\right|$ (blue line), i.e., a Rabi rotation is detected with a maximum corresponding to $\theta_1-\theta_0=\pi/2$ at $P_1=0.55$~\textmu W. The offset $\theta_0$ might stem from imperfect reflection of the photonic structure and internal absorption. With respect to our previous experiments performed on a half-cavity structure\cite{PacuskiCGD17} with $Q\approx20$, a $\pi/2$ pulse area is here attained for an around 6 times weaker impinging laser power. Such an enhanced coupling between the external excitation and the QD exciton is due to the moderately larger Q-factor of the microcavity and thus a larger effective $\mathcal E_j$ for the same external $P_j$. To further demonstrate the correspondence between the cavity Q-factor and the powers required to reach a $\pi/2$ pulse, we measure the FWM's intensity dependence on a similar microcavity sample, that is fabricated from 12 (6) stacked DBR-pairs at the bottom (top). As a result its Q-factor reaches 90. From the measured Rabi flopping (yellow points and line) we deduce that the pulse area of $\pi/2$ corresponds to $P_1=0.12$~\textmu W. We note however that a further increase of $Q$ does not necessarily lead to a better light-matter coupling. This is due to spectral filtering of the incoming excitation pulses and a respective increase of their temporal duration inside the cavity.\cite{WiggerOptica18} An optimal light-matter coupling is achieved when the spectral widths of the cavity mode and of the driving pulses are matched.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\columnwidth]{Fig5}
\caption{Rabi rotations. FWM amplitude as a function of the applied peak field amplitude of the first laser pulse $\sqrt{P_1}$ while $P_{2,3}$ are fixed. Measurement as dark dots and fits with $\left|\sin(\theta_1-\theta_0)\right|$ as pale lines for the cavity quality factors $Q=55$ (blue) and $Q=90$ (yellow).}
\label{fig:Rabi}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.65\columnwidth]{Fig6}
\caption{Photon echo formation. FWM dynamics as a function of time $t$ after the third pulse and the delay $\tau_{12}$. (a) Measurement and (b) fit with Eq.~\eqref{eq:deph}. The green points in (a) show the measurement at $\tau_{12}=31$~ps.}
\label{fig:echo}
\end{figure}
We now shift the investigation to the temporal domain. A typical pulse sequence of the experiment is presented in Fig.~\ref{fig:pulses}, where the signal is generated after the arrival of all three pulses $\ensuremath{{\cal E}_{1,2,3}}$. In inhomogeneously broadened systems, the FWM signal for $\tau_{12}>0$ (see Fig.~\ref{fig:pulses}(a)) forms a photon echo.\cite{LangbeinPRL05} Even though the echo formation is commonly known for ensembles of emitters, it can also be generated for individual transitions. Here, the photon echo arises due to the exciton's stochastic spectral wandering in time, accumulating into an effective inhomogeneous broadening of width $\sigma$ in the time-averaged heterodyne experiment.\cite{PattonPRB06, KasprzakNJP13, MermillodPRL16, HahnAdvSci21} To illustrate that, in Fig.~\ref{fig:echo}(a) we show the measured time-resolved FWM amplitude as a function of the time after the third pulse $t$ and the delay $\tau_{12}$, while fixing $\tau_{23}=0$. We observe that with increasing delay $\tau_{12}$ the maximum of the signal shifts in time $t$ along the diagonal $\tau_{12}=t$ (dashed line). We see that for $\tau_{12}>20$~ps, the echo is fully developed, i.e., the FWM signal takes the form of a Gaussian transient. This is exemplarily shown by the time-resolved FWM amplitude measured at $\tau_{12}=31$~ps (green dots). By fitting the entire FWM dynamics with
\begin{align}
S(t,\tau_{12}) \sim \exp\left[-\frac{t+\tau_{12}}{T_2}\right] \exp\left[-\frac{(t-\tau_{12})^2}{2T_\sigma^2}\right] \label{eq:deph}
\end{align}%
as shown in Fig.~\ref{fig:echo}(b) we directly retrieve the homogeneous dephasing time $T_2=(36.5\pm0.2)$~ps and the inhomogeneous dephasing time $T_\sigma=(9.25\pm0.05)$ ps. These times directly correspond to spectral broadenings of $\gamma = 2\hbar/T_2 \approx (36.1\pm0.2)$~\textmu eV and $\sigma = 2\hbar/T_\sigma \approx (142\pm1)$~\textmu eV.\cite{KasprzakNJP13, Jakubczyk2DMat18}
\begin{figure}[h]
\centering
\includegraphics[width=0.55\columnwidth]{Fig7}
\caption{FWM dynamics of the QD exciton. (a) Coherence dynamics as a function of the delay $\tau_{12}$ exhibiting dephasing and FSS-induced beats. The inset shows the PID on a few ps time scale. (b) Population dynamics as a function of the delay $\tau_{23}$ exhibiting a single exponential decay.}
\label{fig:X_dyn}
\end{figure}
To have a closer look at the coherence dynamics of the exciton, we measure the time-integrated FWM signal as a function of $\tau_{12}$, depicted in Fig.~\ref{fig:X_dyn}~(a) as dark red dots. The signal shows the expected behavior after time integrating the photon echo in Eq.~\eqref{eq:deph} (Fig.~\ref{fig:echo}) over $t$, which consists of an exponential decay that dominates for large delays ($\tau_{12}\gg T_\sigma$) and an increasing contribution during the development of the full echo for $\tau_{12}<T_\sigma$. In addition we find a modulation of the signal stemming from the FFS of the two linearly polarized excitons in the QD. As the linearly polarized $\ensuremath{{\cal E}_{1,2,3}}$ are misaligned from the anisotropy axes of the QD, both excitons are excited and the corresponding coherences contribute to the final FWM signal.\cite{PattonPRB06, KasprzakNJP13, MermillodOptica16} With the model described in Ref.~\cite{MermillodOptica16} we can fit the measured data and retrieve the pale red line with a FFS of $\delta_{\rm FFS} = \hbar 2 \pi/T_\delta = (82\pm 5)$~\textmu eV and a light polarization angle of $\alpha = (25\pm 1)^\circ$ with respect to one of the QD excitons. An exemplary FWM spectrum exhibiting a large FFS can be found the Supporting Information Fig. S1. We note that the exciton-biexciton transition is not covered by $\ensuremath{{\cal E}_{1,2,3}}$ and therefore does not influence the dynamics.\cite{MermillodOptica16} Examples of neutral exciton-biexciton complexes with both bound and unbound character, typical for weakly-confined QDs,\cite{KasprzakJOSAB12} are readily identified in FWM on the same sample, as shown for a bound example in the Supporting Information Fig. S2.
In the inset of Fig.~\ref{fig:X_dyn}(a) the FWM dynamics are shown for a delay timescale of a few picoseconds. After the signal's rise from negative delays, it reaches a maximum around $\tau_{12}=0$. After that it drops within less than 2~ps to approximately 0.5 of its maximum value. This fast decay is recognized as PID, due to the optical excitation with pulses that are siginificantly shorter than the polaron formation process.\cite{JakubczykACSPhot16, WiggerOL20} The FWM dynamics are reproduced by the depicted simulation (pale red line) in the well established independent boson model.\cite{mahan, JakubczykACSPhot16} In this model the exciton-phonons coupling is described by additional dynamics of the exciton coherence, in the form of the PID function $\tilde{p}_{\rm PID}$. The full FWM dynamics are therefore given by
\begin{align}
p_{\rm FWM}(t,\tau_{12}) \sim \tilde{p}_{\rm PID}(t,\tau_{12}) S(t,\tau_{12})\,,
\end{align}%
where $S(t,\tau_{12})$ is the homogeneous and inhomogeneous dephasing contribution from Eq.~\eqref{eq:deph}. For optical pulses that are much shorter than the considered phonon periods, the PID dynamics can be calculated analytically in the limit of ultrafast pulses via\cite{VagovPRB02}
\begin{align}
&\tilde{p}_{\rm PID}(t,\tau_{12}) \notag\\
&= \exp\bigg\{\left|\frac{g_{\bm q}}{\omega_{\bm q}}\right|^2 \Big[2\cos(\omega_{\bm q}t)-3 + e^{i\omega_{\bm q} \tau_{12}}(2-e^{i\omega_{\bm q}\tau_{12}})\notag\\
&\qquad - N_{\bf q}\left|e^{i\omega_{\bm q}\tau_{12}}(2-e^{i\omega_{\bm q}t})-1\right|^2\Big]\bigg\}\,,
\end{align}%
with the thermal occupation of the phonon modes $N_{\bm q} = \{\exp[\hbar\omega_{\bm q}/(k_BT)]-1\}^{-1}$. For simplicity we here assume a spherical exciton wave function for which the coupling constant can be written as
\begin{align}
g_{\bf q} = \frac{q D}{\sqrt{2\rho\hbar V\omega_{\bm q}}}e^{-\frac 12 q^2 a^2}\,,
\end{align}%
with the normalization volume $V$. For the material parameters we use the mass density of $\rho=5870$kg/m$^3$, an effective deformation potential strength of $D=9$~eV,\cite{KranzerJPD73} and assume an isotropic phonon dispersion $\omega_{\bm q}=c q$ with the longitudinal acoustic sound velocity $c=3.2$~nm/ps.\cite{KrischPRB97} We find the best agreement with the measured FWM dynamics for an exciton localization length of $a=2$~nm.
To complete the study of the undoped QD, in Fig.~\ref{fig:X_dyn}(b) we present the exciton occupation dynamics, which are measured by the $\tau_{23}$-dependence of the FWM amplitude while fixing $\tau_{12}=0$ (red dots). An exponential decay (pale red line) is observed with a decay time of $T_1=(200\pm 25)$~ps. This decay is attributed to the radiative recombination of the bright exciton states. The decay of the dark exciton typically happens on a much longer timescale of a few tens of ns\cite{SmolenskiPRB12} and is therefore not resolved here. The collection of the QD parameters ascertained by the FWM study is gathered in Table~\ref{tab:table1}.
\begin{table}[h]
\caption{Parameters characterizing the optical properties of the QD exciton retrieved by FWM spectroscopy.}
\label{tab:table1}
\centering
\begin{tabular}{l | l}
\hline
homogeneous broadening & $\gamma=(36.1\pm 0.2)$~\textmu eV \\
inhomogeneous broadening & $\sigma=(142\pm 1)$~\textmu eV\\
bright exciton lifetime & $T_1=(200\pm25)$~ps \\
fine-structure splitting & $\delta_{\rm FSS}=(82\pm 5)$~\textmu eV \\
light-matter coupling & $\theta=\pi/2$ @ $P=0.55$~\textmu W \\
\hline
\end{tabular}
\end{table}
\section{Mn-doped quantum dot}
\begin{figure}[t]
\centering
\includegraphics[width=0.55\columnwidth]{Fig8}
\caption{Spectral characterization of a Mn-doped QD. (a) PL spectrum exhibiting the six spectral lines induced by the Mn dopant. (b) FWM spectral interferogram. (c) FWM amplitude spectrum with the measurement in dark and the simulation in pale violet.}
\label{fig:Mn_spec}
\end{figure}
After this characterization of the excitonic properties, we come to the QD containing a single Mn$^{2+}$ ion. Such a QD is recognized by measuring a PL spectrum as presented in Fig.~\ref{fig:Mn_spec}(a). The insertion of an individual Mn$^{2+}$ ion into a QD, within the volume of the exciton's wave function, is confirmed by detecting the comb of six separate spectral lines,\cite{BesombesPRL04, GorycaPRL09} as shown in the PL spectrum in Fig.~\ref{fig:Mn_spec}(a), which is characteristic for a sufficiently symmetric QD when the exciton-Mn exchange interaction dominates over the anisotropic electron-hole exchange interaction, i.e., when the splitting of the lines due to the exciton-Mn interaction is larger than the fine-structure splitting.\cite{LegerPRL05} The exciton transition is sensitive to the spin state of the magnetic ion: The exchange interaction between the QD exciton and the ion leads to spin-dependent spectral shifts with respect to the undoped QD exciton in the range of a few meV. The electron-Mn exchange interaction furthermore leads to spin flips resulting in a coupling between bright and dark excitons which, however, typically becomes effective only at high magnetic fields.\cite{BesombesPRL04, FernandezPRB06} Without an additional magnetic field, the Mn spin projection $S_z$ freely jumps between its possible realizations, namely $\pm\frac{5}{2}$, $\pm\frac{3}{2}$, and $\pm\frac{1}{2}$. In a time averaged measurement, this results in the development of six spectral components.
\begin{figure}[t]
\centering
\includegraphics[width=0.55\columnwidth]{Fig9}
\caption{Coherence dynamics of a Mn-doped QD. (a) Evolution of the FWM spectrum with increasing delay $\tau_{12}$ from bottom to top. (b) Integrated FWM amplitude as a function of the delay $\tau_{12}$ with the experiment as dark violet dots and the simulation as pale violet line.}
\label{fig:Mn_dyn}
\end{figure}
The FWM spectral interferogram and the resulting FWM amplitude are shown in Fig.~\ref{fig:Mn_spec}(b) and (c), respectively. Here, we also recover six spectral lines with their amplitudes increasing for smaller transition energies. Interestingly, fluctuations of such a single spin generate a peculiar type of inhomogenous broadening acting on the exciton. A typical Mn spin-flip time in a CdTe QD is on the order of several \textmu s,\cite{BesombesPRL04, GorycaPRL09, GorycaPRB15} which is much longer than the measured exciton lifetime of 200~ps. During the integration time of 10~ms, the exciton performs a few thousand spectral jumps. Because the spin-flip time is much longer than the exciton lifetime, the corresponding random jumps of the transitions energy can be interpreted as a discrete ensemble. In Fig.~\ref{fig:Mn_dyn} we examine how the FWM signal behaves as a function of the delay $\tau_{12}$. In Fig.~\ref{fig:Mn_dyn}(a) we show examples of the measured FWM spectra for three different delays $\tau_{12}$ increasing from bottom to top as labeled in the plot. We already see that the overall amplitude of the signal strongly decreases with larger delays. In Fig.~\ref{fig:Mn_dyn}(b) we show the integrated FWM amplitude as a function of $\tau_{12}$ as violet dots, which confirms the rapid drop of the system's coherence. The significant decoherence within the first 2~ps is already known from the undoped QD and it stems from the PID in Fig.~\ref{fig:X_dyn}(b, inset). After the PID, i.e., for $\tau_{12} > 1.5$~ps only long-time dephasings (homogeneous and inhomogeneous) reduce the signal. In the simulation depicted as pale violet line, we take the impact of the Mn ion into account by calculating an ensemble (ens) average of the six transition energies $\hbar\Delta\omega_{n}$ via
\begin{align}
p_{\rm FWM}^{\rm ens}(t,\tau_{12}) = \sum_{n=1}^{6} p^{(n)}_{\rm FWM}(t,\tau_{12})e^{-i\Delta \omega_{n} (t-\tau_{12})}\,,
\end{align}%
where each FWM contribution $p^{(n)}_{\rm FWM}(t,\tau_{12})$ can be individually weighted. These weights of the six contributions are chosen such that the simulated spectral distribution (pale violet line) in Fig.~\ref{fig:Mn_spec}(c) agrees with the measured one (dark line). In the resulting coherence dynamics in Fig.~\ref{fig:Mn_dyn}(b) the discrete equidistant ensemble results in a slight beating of the FWM signal. This oscillation is highlighted by the inset, which is a zoom-in on the black rectangle. The effect is similar to the FSS beat observed from the undoped dot in Fig.~\ref{fig:X_dyn}(a). However, here it stems from all possible frequency differences in the six-state ensemble.
\section{Conclusions}
In this work, we have studied the coherence properties of an exciton confined to a CdTe QD by FWM spectroscopy. The creation and detection of this nonlinear optical signal, scaling with the third power of the investigated dipole moment of the quantum system, required the incorporation into a low-Q DBR cavity. To finally reach the required efficiency for the light-matter coupling we additionally applied a solid immersion lens on the sample surface. With this setup we were able to detect photon echo dynamics, which allowed us to determine the homogeneous and inhomogeneous dephasing of the QD exciton. We further measured beats of the signal, revealing the fine-structure splitting of the linearly polarized excitons, and their population lifetime. Considering a QD hosting a single Mn-dopant we performed the first FWM spectroscopy study of the characteristic six-lined spectral structure. We showed that this unique shape of transition energies, stemming from random fluctuations between the Mn-spin states, results in additional dephasing dynamics and a signal beating in the ensemble average.
As the Mn spin-orientation can be controlled by an external magnetic field, forthcoming magneto-FWM micro-spectroscopy experiments will allow to further monitor the impact of the observed discrete inhomogeneous broadening onto the exciton coherence dynamics. It will furthermore allow to study the coherence dynamics associated with the exchange-induced coupling between bright and dark excitons giving rise to the characteristic anticrossings in the magneto-PL of Mn-doped QDs.\cite{BesombesPRL04} This progress will reveal the system's potential for a coherent ultrafast spin-photon interface. In particular 2D FWM spectroscopy will unveil internal coherent interactions in the coupled exciton-Mn$^{2+}$ system. It is possible to shift the Mn-doped QD emission energies below 1770~meV (wavelengths above 700~nm), see PL spectrum in Fig. S3 of the Supporting Information. Further work regarding the growth will optimize the sample performance at this spectral range, enabling implementation of the resonant spectroscopy employing standard Ti:Sapphire femtosecond laser sources. At this point, we can combine more sophisticated and innovative photonic nanostructures, currently emerging for the CdTe-platform,\cite{BoguckiLight20} with QDs that can already contain a variety of different magnetic impurities.\cite{KobakNatComm14, SmolenskiNatComm16} These technological and spectroscopic advances open exciting prospects for coherent nonlinear spectroscopy of hybrid exciton-spin systems in semiconductor nanostructures.
\begin{acknowledgement}
This work was partially supported by the Polish National Science Centre (NCN) under decision DEC-2015/18/E/ST3/00559.
Tomasz Jakubczyk acknowledges support from the Polish National Agency for Academic Exchange (NAWA) under Polish Returns 2019 programme (Grant No. PPN/PPO/2019/1/00045/U/0001). Daniel Wigger thanks NAWA for financial support within the ULAM program (Grant No. PPN/ULM/2019/1/00064).
\end{acknowledgement}
\begin{suppinfo}
Supporting Information contains:\\
-- Exemplary FWM spectra showing individual excitons with a large fine-structure splitting.\\
-- Two-dimensional FWM spectrum showing the exciton-biexciton complex.\\
-- Photoluminescence spectrum of a Mn-doped QD emitting around 1710~meV.
\end{suppinfo}
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|
1,116,691,499,323 | arxiv | \section{Introduction}
High-velocity wind-like outflows are an important part of the quasar system and a potential contributor to feedback to the host galaxy. Many properties of quasar outflows are still poorly understood, including their location and three-dimensional structure. One way to obtain constraints on quasar outflows is to study the variability in their absorption lines. Variability on shorter time-scales can place constraints on the distance of the absorbing material from the central supermassive black hole (SMBH). Measurements of variability on longer (multi-year) time-scales provide insight into the homogeneity and stability of the outflowing gas. Overall, results of variability studies provide information on the size, kinematics, and internal makeup of sub-structures within the outflows.
Broad absorption lines (BALs) are the most prominent signatures of accretion disk outflows seen in quasar spectra. We are conducting a BAL monitoring program for a sample of 24 luminous BAL quasars at z=1.2$-$2.9 from \citet{Barlow93}. We have been re-observing these quasars at the MDM Observatory 2.4-m Hiltner and KPNO 2.1-m telescopes, and we also include spectra from SDSS when available. We currently have in total 163 spectra of these 24 quasars.
To identify BAL variability, we first define the regions of BAL absorption, based on the definition of the `balnicity index' (BI), i.e. they must contain contiguous absorption that reaches $\geq$10 per cent below the continuum across $\geq$2000 km/s \citep{Weymann91}. We then used visual inspection to identify velocity intervals with a width of at least 1200 km/s that varied. Using the average flux and associated error within each candidate variability interval, we place an error on the flux differences between the two spectra. We include all intervals of variability where the flux differences are at least 4$\sigma$.
\section{CIV and SiIV BAL Variability}
\subsection{Trends in the CIV Short-Term and Long-Term Data}
The first results of our BAL variability study were reported in \citeauthor{Capellupo11a} (\citeyear{Capellupo11a}; hereafter, Paper 1), where we focused on variability in just the CIV $\lambda$1550 BALs in two different time intervals: 4$-$9 months (short-term) and 3.8$-$7.7 yr (long-term) in the quasar rest-frame. We found both a higher incidence of variability (65\%, or 15/23, of the quasars versus 39\%, or 7/18) and a larger typical change in strength in the long-term data. The variability occurs typically in only portions of the BAL troughs (e.g. Fig. \ref{sp1246}; see also, \citealt{Gibson10}). In Fig. \ref{hist}, we plot the incidence of CIV BAL absorption and variability versus velocity. The BAL components at higher outflow velocities are more likely to vary than those at lower velocities. We also find in Paper 1 that weaker BALs are more likely to vary than stronger BALs.
\subsection{Multi-Epoch Monitoring of SiIV and CIV Variability}
\begin{figure}
\begin{center}
\includegraphics[scale=0.48]{capellupo_fig1.eps}
\end{center}
\caption{The top two panels show the number of occurrences of CIV BAL absorption and BAL
variability versus velocity. The third panel is the second panel divided by the first, with 1$\sigma$
error bars overplotted.}
\label{hist}
\end{figure}
In \citeauthor{Capellupo11b} (\citeyear{Capellupo11b}; hereafter, Paper 2), we directly compare the variability of SiIV $\lambda$1400 BALs to CIV in the long-term dataset from Paper 1. C and Si have different abundances, if solar abundances are assumed, and they have different ionization properties (e.g. \citealt{Hamann08}). By examining if CIV and SiIV have different variability properties, and how they differ, coupled with these differences in abundances and ionization properties, we can gain new insight into the cause(s) of BAL variability.
SiIV BALs are more likely to vary than CIV BALs. For example, when looking at flow speeds $>$$-$20 000 km/s, 47 per cent of the quasars in our sample exhibited SiIV variability while 31 per cent exhibited CIV variability. Variability in SiIV can occur without corresponding changes in CIV at the same velocities. For $\sim$50 per cent of the variable SiIV regions, there was no corresponding CIV variability at the same velocities. However, we have only one tentative case where changes in CIV are not matched by SiIV. At BAL velocities where both CIV and SiIV varied, the changes always occurred in the same sense (with both lines either getting stronger or weaker).
We now have up to 13 epochs of data per object for our sample of 24 BAL quasars. With all the observing epochs included, the fraction of quasars with CIV BAL variability at any velocity increases from the value of 65\% found in Paper 1 to 88\% for the same sample of quasars. This increase was caused by variations missed in the 2-epoch comparisons in Paper 1. We also find that BAL changes at different velocities in the same ion {\it almost} always occurred in the same sense. We found just 3 cases that show evidence for one CIV BAL weakening while another strengthens within the same object. The multi-epoch data also show that the BAL changes across $\sim$1 week to 8 years in the rest-frame were not generally monotonic (see also, \citealt{Gibson10}). Thus, the characteristic time-scale for significant line variations, and (perhaps) for structural changes in the outflows, is less than a few years. Furthermore, with more epochs added, we still do not find clear evidence for acceleration or deceleration in the BAL outflows.
\subsection{Time-Scales of Variability}
We take a closer look at the time-scales of variability in \citeauthor{Capellupo12} (\citeyear{Capellupo12}; hereafter, Paper 3). In Spring 2010, we re-observed a subsample of our BAL quasars multiple times over rest-frame time intervals of $\Delta$t $\sim$1 week to 1 month to augment our temporal sampling at these short time-scales. Variability results on the shortest time-scales are important for putting constraints on the location and sizes of the outflows.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.45]{capellupo_fig2.eps}
\end{center}
\caption{The top panel displays the fraction of measurements where there was CIV BAL variability
versus the time interval between the two observations. The middle and bottom panels show the
number of measurements and the number of quasars, respectively, that contribute to each bin.}
\label{dream}
\end{figure}
The multi-epoch nature of our dataset, spanning time intervals from 0.02 to 8.7 yr in the rest frame, allows us to investigate the probability of BAL variability versus $\Delta$t. We examined CIV BAL absorption at all measured velocities in our entire data sample. We compared each pair of observations for each quasar and counted the occurrences of CIV BAL variability, using our definition of BAL variability defined in Paper 1 (see Section 1 above). We then calculated a probability by dividing the number of occurrences of variability by the number of measurements, where a pair of observations is one measurement, in logarithmic bins of $\Delta$t. We plot this measured probability of detecting CIV BAL variability versus $\Delta$t in Fig. \ref{dream}. The middle and bottom panels display the number of measurements and the number of quasars, respectively, that contribute to each $\Delta$t bin. Each quasar can contribute multiple times to each bin.
\begin{figure}
\begin{center}
\includegraphics[scale=0.46, angle=90]{capellupo_fig3.eps}
\end{center}
\caption{Smoothed MDM spectra of 1246$-$0542, showing the CIV absorption region. The bold
and thin solid curves show the two epochs separated by 8 days in the rest-frame, and the shaded
regions indicate the velocity intervals that varied between these two epochs. The formal
1$\sigma$ errors are shown near the bottom.}
\label{sp1246}
\end{figure}
In our dataset, we found variability down to a $\Delta$t of 8 days in 1246$-$0542 between the 2010.13 and 2010.20 observations. Fig. \ref{sp1246} shows the CIV region of the spectra for these two observations as bold and thin lines, respectively. As indicated by shaded bars in Fig. \ref{sp1246}, there are two separate velocity intervals of variability between the two spectra, and the flux difference is at least 8$\sigma$ in each interval. We also overplot an earlier (2007.04; thin dashed curve) and a later (2010.35; bold dashed curve) epoch in Fig. \ref{sp1246}, which shows that there was variability in these intervals in other epochs as well.
\section{Conclusions}
Coordinated variabilities between absorption regions at different velocities in individual quasars seems to favor changing ionization of the outflowing gas as the cause of the observed BAL variability. Many of our results are consistent with this scenario, including our findings that SiIV, which has a lower optical depth and is therefore less likely to be saturated than CIV, is more likely to vary than CIV. However, variability in limited portions of broad troughs fits naturally in a scenario where movements of individual clouds, or substructures in the flow, across our lines-of-sight cause the absorption to vary. This scenario is also consistent with the main results of Paper 2 (Section 2.2 above) if SiIV has a smaller covering fraction than CIV. The actual situation may be a complex mixture of changing ionization and cloud movements (Paper 2).
If the short time-scale variations in 1246$-$0542 were due to cloud movements and a $\sim$10\% change in line strength over 8 days indicates the cloud crossed 10\% of the continuum source in that time, then the transverse speed is $\sim$$-$27 000 km/s. If we assume the transverse speed is related to the Keplarian rotation speed, then the outflowing gas is located $\ll$1 pc from the central SMBH, probably inside the radius of the broad emission-line region (Paper 3).
\acknowledgements This work was supported by grant number 1009628 from the National Science Foundation, and by the Guest Observer program 11705 with the Space Telescope Science Institute.
|
1,116,691,499,324 | arxiv | \section{Introduction}
\textit{Tropical geometry} is a combinatorial shadow of algebraic geometry.
We take a tropical approach to problems on algebraic classes of cohomology groups (such as the Hodge conjecture, the Tate conjecture, and the standard conjectures).
In this paper,
inspired by Liu's construction of \textit{tropical cycle class maps} by \textit{Milnor $K$-groups} (\cite[Definition 3.6]{Liu20}),
we shall introduce a tropical analog $K_T^*$ of (rational) Milnor $K$-groups, called \textit{tropical $K$-groups} (Definition \ref{dftroK}),
and study its basic properties.
The goal of this paper is as follows (Theorem \ref{trKcymod} and Corollary \ref{corshftrK}).
\begin{thm}\label{the main theorem}
Tropical $K$-groups $K_T^*$ form a cycle module in the sense of Rost \cite[Definition 2.1]{Ros96}.
In particular, for any $p \geq 0$,
their Zariski sheafification $\mathscr{K}_T^p$ on a smooth algebraic variety $X$ over a trivially valued field has the Gersten resolution, i.e.,
a flasque resolution
\begin{align*}
0 \rightarrow \mathscr{K}_T^p \xrightarrow{d} \bigoplus_{x \in X^{(0)}} i_{x *} K_T^p(k(x))
\xrightarrow{d} \bigoplus_{x\in X^{(1)}} i_{x *}K_T^{p-1}(k(x)) & \xrightarrow{d} \bigoplus_{x\in X^{(2)}} i_{x *}K_T^{p-2}(k(x)) \xrightarrow{d} \dots \\
& \xrightarrow{d} \bigoplus_{x\in X^{(p)}} i_{x *}K_T^0(k(x)) \xrightarrow{d} 0 ,
\end{align*}
where
$X^{(i)}$ is the set of points of codimension $i$,
the residue field at $x \in X$ is denoted by $k(x)$,
the maps $i_x \colon \mathop{\mathrm{Spec}}\nolimits(k(x)) \rightarrow X$ are the natural morphisms, and
we identify the groups $K_T^* (k(x))$ and the Zariski sheaf defined by them on $\mathop{\mathrm{Spec}}\nolimits k(x)$.
(See Section \ref{sec;trocoho;troK} for details.)
\end{thm}
In particular, we have the following corollary (Corollary \ref{corshftrK}).
\begin{cor}\label{the main corollary}
$$H_{Zar}^p(X,\mathscr{K}_T^p) \cong CH^p (X) \otimes_\mathbb{Z} \mathbb{Q}.$$
\end{cor}
Corollary \ref{the main corollary} will be used
to prove a tropical analog of the Hodge conjecture for smooth algebraic varieties over trivially valued fields
in a subsequent paper \cite{M20}.
To prove Theorem \ref{the main theorem}, we shall introduce tropicalizations of Zariski-Riemann spaces (and adic spaces), which give explicit descriptions of tropical $K$-groups.
(Note that our tropicalizations of adic spaces are different from Foster-Payne's adic tropicalizations; see \cite{Fos16}.)
The outline of this paper is as follows.
In Section \ref{sec;not;term}, we fix several notations and terminologies.
In Section \ref{sec;analytic;spaces},
we review basics of valuations and non-archimedean analytic spaces.
In Section \ref{sectrop}, we review basics of tropicalizations of Berkovich analytic spaces and theory of tropical compactifications.
We introduce tropicalizations of Huber's adic spaces and Zariski-Riemann spaces.
Finally, in Section \ref{sec;trocoho;troK},
we introduce tropical $K$-groups and prove Theorem \ref{the main theorem}.
\section{Notations and terminologies}\label{sec;not;term}
For a $\mathbb{Z}$-module $G$ and a commutative ring $R$, we put $G_R := G \otimes_\mathbb{Z} R$.
Integral separated schemes of finite type over fields are called algebraic varieties.
Cones mean strongly convex rational polyhedral cones.
Toric varieties are assumed to be normal.
We denote the residue field of a valuation $v$ by $\kappa(v)$.
We denote the residue field of the structure sheaf at a point $x $ of a scheme $X$ by $k(x)$.
For an extension of fields $L/K$,
we denote its transcendental degree by $\mathop{\mathrm{tr.deg}}\nolimits (L/K)$.
\section{Valuations and non-archimedean analytic spaces}\label{sec;analytic;spaces}
In this section, we give a quick review on (non-archimedean) \textit{valuations} (Subsection \ref{subsec;val})
and non-archimedean analytic spaces:
\textit{Berkovich analytic spaces} (Subsection \ref{subsec;Berkovich}),
\textit{Zariski-Riemann spaces} (Subsection \ref{subsec;ZR}), and
\textit{Huber's adic spaces} (Subsection \ref{subsec;adic}).
We refer to \cite{HK94} and \cite[Chapter 6]{Bou72} for valuations,
\cite{Hub93} and \cite{Hub94} for valuations and Huber's adic spaces,
\cite{Ber90}, \cite{Ber93}, and \cite{Tem15} for Berkovich analytic spaces,
and
\cite{Tem11} for Zariski-Riemann spaces.
\subsection{Valuations}\label{subsec;val}
In this subsection, rings are assumed to be commutative with a unit element.
\begin{dfn}
We define a \emph{valuation} $v$ of a ring $R$ as a map $v \colon R \rightarrow \Gamma'_v\cup \{\infty\} $
satisfying the following properties:
\begin{itemize}
\item $\Gamma'_v$ is a totally ordered abelian group,
\item $v(ab)=v(a)+v(b) $ for any $a,b \in R$, where we extend the group law of $\Gamma'_v$ to $\Gamma'_v\cup\{\infty\} $ by $ \gamma + \infty = \infty + \gamma = \infty$ for any $\gamma \in \Gamma'_v$,
\item $v(0)=\infty$ and $v(1)=0$,
\item $v(a + b)\geq \min \{v(a),v(b)\} $, where we extend the order of $\Gamma'_v$ to $\Gamma'_v\cup\{\infty\} $ by $ \infty \geq \gamma$ for any $\gamma \in \Gamma'_v$.
\end{itemize}
\end{dfn}
The subgroup of $\Gamma'_v$ generated by $v(R)\setminus \{\infty\}$ is called the \emph{value group} of $v$.
We denote it by $\Gamma_v$.
We put $\mathop{\mathrm{supp}}\nolimits(v) := v^{-1}(\infty)$. We call it the \emph{support} of $v$.
We put
$$\mathcal{O}_v :=\{a \in \mathop{\mathrm{Frac}}\nolimits(R/\mathop{\mathrm{supp}}\nolimits(v)) \mid v(a) \geq 0 \} ,$$
which is called the \emph{valuation ring} of $v$.
The valuation $v$ extends to a valuation on $\mathcal{O}_v$, which is also denoted by $v$.
We put
$$\kappa(v):=\mathop{\mathrm{Frac}}\nolimits(\mathcal{O}_v / \{ a \in \mathcal{O}_v \mid v(a)>0\}),$$
which is called the \emph{residue field} of $v$.
If $ R / \mathop{\mathrm{supp}}\nolimits(v) \subset \mathcal{O}_v$,
we call the image of the maximal ideal under the canonical morphism $\mathop{\mathrm{Spec}}\nolimits \mathcal{O}_v \to \mathop{\mathrm{Spec}}\nolimits R$
the \emph{center} of $v$.
\begin{dfn}
We call two valuations $v$ and $w$ of a ring $R$ are \emph{equivalent} if there exists an isomorphism $\varphi \colon \Gamma_v \overset{\sim}{\to} \Gamma_w$ of totally ordered abelian groups satisfying $\varphi' \circ v =w$, where $\varphi' \colon \Gamma_v \cup \{\infty\} \rightarrow \Gamma_w\cup \{ \infty\}$ is the extension of $\varphi $ defined by $\varphi'(\infty) =\infty$.
\end{dfn}
We call the rank of a totally ordered abelian group $\Gamma$ as an abelian group the \emph{rational rank} of $\Gamma$.
\begin{dfn}
Let $\Gamma$ be a totally ordered abelian group.
A subgroup $H$ of $\Gamma$ is called \emph{convex} if every element $\gamma \in \Gamma$ satisfying $h< \gamma<h' $ for some $h,h' \in H$ is contained in $H$.
\end{dfn}
\begin{dfn}
We call the number of proper convex subgroups of a totally ordered abelian group $\Gamma$ the \emph{height} of $\Gamma$.
We denote it by $\mathop{\mathrm{ht}}\nolimits \Gamma$.
\end{dfn}
The following well-known theorem is called the Harn embedding theorem.
\begin{thm}[{Clifford \cite{Cli54}, Hausner-Wendel \cite{HW52}}]\label{thm:Harn:embed}
Every totally ordered abelian group of finite height $n$ has an embedding into the additive group $\mathbb{R}^n$ with the lexicographic order.
\end{thm}
We call the rational rank (resp.\ height) of the value group of a valuation $v$ the rational rank (resp.\ height) of $v$.
Equivalent valuations have the same rational ranks and heights.
Hence rational ranks and heights of equivalence classes of valuations
are well-defined.
\begin{dfn}
A field $L$ equipped with a valuation $v \colon L \to \Gamma_v \cup \{\infty\}$ is called a \emph{valuation field}.
A valuation field $(L,v)$ is called trivially valued if its value group $\Gamma_v$ is the trivial group $\{0\}$.
\end{dfn}
Let $R$ be a ring.
We call the set of all equivalent classes of valuations of $R$ the \emph{valuation spectrum} of $R$.
We denote it by $\mathop{\mathrm{Spv}}\nolimits(R)$.
We equip $\mathop{\mathrm{Spv}}\nolimits(R)$ with the topology which is generated by the sets
$$ \{v\in \mathop{\mathrm{Spv}}\nolimits(R) \mid v(a)\geq v(b)\neq \infty \} \qquad (a,b \in R) .$$
Let $v \colon R \rightarrow \Gamma_v \cup \{\infty\}$ be a valuation.
For every convex subgroup $H \subset \Gamma_v$, we define two mappings
\begin{align*}
v/H\colon R\rightarrow (\Gamma_v/H)\cup \{\infty\}, \quad & a \mapsto \begin{cases}
v(a)\text{ mod } H & \text{if } v(a) \neq \infty \\
\infty & \text{if } v(a)=\infty,
\end{cases}\\
v|H \colon R\rightarrow H \cup \{\infty\}, \quad & a \mapsto \begin{cases}
v(a) & \text{if } v(a) \in H \\
\infty & \text{if } v(a) \notin H.
\end{cases}\\
\end{align*}
\begin{lem}[{Huber-Knebusch \cite[Lemma 1.2.1]{HK94}}]
\begin{enumerate}
\item The map $v/H$ is a valuation of $R$, and it is a generalization of $v$ in $\mathop{\mathrm{Spv}}\nolimits(R)$.
\item The map $v|H $ is a valuation of $R$ if and only if the image of $R$ in $\mathop{\mathrm{Frac}}\nolimits(R/\mathop{\mathrm{supp}}\nolimits(v)) $ is contained in the localization
$\mathcal{O}_{v,p\mathcal{O}_v}$ of the valuation ring $\mathcal{O}_v$ at a prime ideal $p\mathcal{O}_v$,
where we put $p := \{a \in R \mid v(a)> H\}$.
Moreover, in this case, the valuation $v|H $ is a specialization of $v$ in $\mathop{\mathrm{Spv}}\nolimits(R)$.
\end{enumerate}
\end{lem}
\begin{dfn}
We call a generalization of $v$ in $\mathop{\mathrm{Spv}}\nolimits(R)$ of the form $v/H$ for a convex subgroup $H\subset \Gamma_v $ a vertical (or primary) generalization,
and a specialization of $v$ in $\mathop{\mathrm{Spv}}\nolimits(R)$ of the form $v|H$ for a convex subgroup $H \subset \Gamma_v$ a horizontal (or secondary) spciaization of $v$.
\end{dfn}
\begin{rem}[{Bourbaki \cite[Proposition 2 in Section 4 in Chapter 6]{Bou72}}]
\label{rem;val;vert;special;resi;fld;val}
For a valuation $v$ of a field $K$,
there is a natural bijection between vertical specializations of $v$ in $\mathop{\mathrm{Spv}}\nolimits (K)$ and valuations on the residue field $\kappa(v)$ of $v$.
\end{rem}
\subsection{Berkovich analytic spaces}\label{subsec;Berkovich}
In \cite[Section 3.5]{Ber90}, Berkovich introduced the \textit{Berkovich analytic space} associated to an algebraic variety $X$ over a trivially valued field $(L,v_L)$.
We denote it by $X^{\mathrm{Ber}}$.
Berkovich analytic spaces are, as sets,
the sets of bounded multiplicative seminorms.
There exists a bijection between multiplicative seminorms $\lvert \ \rvert$ and valuations $v $ of height $\leq 1$
defined by $\lvert \ \rvert \mapsto -\log \lvert \ \rvert $.
By this bijection, we consider multiplicative seminorms as valuations.
In this paper, an \textit{affinoid algebra} $A$ over $L$ means an $L$-affinoid algebra $A$ in the sense of \cite[Definition 2.1.1]{Ber90}.
We denote the Berkovich analytic space associated to $A$ by $\mathscr{M}(A)$ \cite[Section 1.2]{Ber90}.
There exists a unique minimal subset $B(A) $ of $ \mathscr{M}(A)$
on which every valuation of $A$ has its minimum \cite[Corollary 2.4.5]{Ber90}. (Note that the minimum as valuations is the maximum as multiplicative seminorms.)
It is called the \textit{Shilov boundary} of $\mathscr{M}(A)$.
It is a finite set \cite[Corollary 2.4.5]{Ber90}.
\subsection{Zariski-Riemann spaces}\label{subsec;ZR}
For a finitely generated extension $L/K$ of fields,
we put $\mathop{\mathrm{ZR}}\nolimits(L/K)$ the set of equivalence classes of valuations of $L$ which are trivial on $K$.
We call $\mathop{\mathrm{ZR}}\nolimits (L/K)$ with the restriction of the topology of $\mathop{\mathrm{Spv}}\nolimits(L)$ the \textit{Zariski-Riemann space}.
There is another expression.
For each $v \in \mathop{\mathrm{ZR}}\nolimits(L/K)$ and proper algebraic variety $X$ over $K$ with function field $L$,
by the valuative criterion of properness, there exists a unique canonical morphism $\mathop{\mathrm{Spec}}\nolimits \mathcal{O}_v \rightarrow X$.
This induces a map from $\mathop{\mathrm{ZR}}\nolimits(L/K)$ to the inverse limit $\underleftarrow{\lim} X$ (as topological spaces) of birational morphisms of proper algebraic varieties $X$ over $K$ whose function fields are $L$.
The following Proposition is well-known.
\begin{prp}\label{ZRhomeo}
The map $\mathop{\mathrm{ZR}}\nolimits(L/K) \rightarrow \underleftarrow{\lim} X$ is a homeomorphism.
\end{prp}
\begin{proof}
See \cite[Corollary 3.4.7]{Tem11}.
\end{proof}
\begin{rem}
For any $v \in \mathop{\mathrm{ZR}}\nolimits (L/K)$,
we have $\mathop{\mathrm{rank}}\nolimits (v) \leq \mathop{\mathrm{tr.deg}}\nolimits (L/K)$.
The equality holds for some $v$.
\end{rem}
\begin{dfn}
We put $(\mathop{\mathrm{Spec}}\nolimits L/K)^{\mathop{\mathrm{Ber}}\nolimits}$ the subspace of the analytification $X^{\mathop{\mathrm{Ber}}\nolimits}$ of a model $X$ of $L/K$
(i.e., an algebraic variety over $K$ whose function field is $L$)
consisting of points whose supports are the generic point of $X$.
This definition is independent of the choice of a model $X$.
\end{dfn}
\subsection{Huber's adic spaces}\label{subsec;adic}
For an algebraic variety $X$ over a trivially valued field $K$,
we define the adic space $X^{\mathop{\mathrm{ad}}\nolimits}$ associated to $X$ as follows.
(See \cite{Hub93} and \cite{Hub94} for notations and theory of adic spaces.)
For each affine open subvariety $U=\mathop{\mathrm{Spec}}\nolimits R \subset X$,
we put $U^{\mathop{\mathrm{ad}}\nolimits}:= \mathop{\mathrm{Spa}}\nolimits(R, R \cap K^{\mathop{\mathrm{alg}}\nolimits})$,
which is the space of equivalence classes of valuations on $R$ trivial on $K$
(here we consider $R$ a ring equipped with the discrete topology).
We define $X^{\mathop{\mathrm{ad}}\nolimits}$ by glueing $U^{\mathop{\mathrm{ad}}\nolimits}_{\alpha}$ for an affine open covering $\{U_{\alpha}\}$ of $X$.
\begin{rem}
Taking supports of valuations induces a surjective map
$X^{\mathop{\mathrm{ad}}\nolimits} \twoheadrightarrow X$
whose fiber of $x \in X$ is homeomorphic to $\mathop{\mathrm{ZR}}\nolimits(k(x) / K)$.
\end{rem}
\begin{rem}\label{rem;Ber;adic}
Taking equivalence classes induces a map $X^{\mathop{\mathrm{Ber}}\nolimits} \to X^{\mathop{\mathrm{ad}}\nolimits}$.
This induces a bijection
$$X^{\mathop{\mathrm{Ber}}\nolimits} / (\emph{equivalence relations}) \cong X^{\mathop{\mathrm{ad}}\nolimits,\mathop{\mathrm{ht}}\nolimits \leq 1}$$
to the subset $X^{\mathop{\mathrm{ad}}\nolimits,\mathop{\mathrm{ht}}\nolimits \leq 1}$ of $X^{\mathop{\mathrm{ad}}\nolimits}$
consisting of equivalence classes of valuations of height $\leq 1$.
\end{rem}
\section{Tropicalizations of algebraic varieties over trivially valued fields}\label{sectrop}
In this section, we shall recall basic properties of fans (Subsection \ref{subsec;fan}),
\textit{tropicalizations} of
Berkovich analytic spaces (Subsection \ref{subsec;trop;Ber}),
and tropical compactifications (Subsection \ref{subsec;trop;triv;val}).
Tropicalizations of algebraic varieties are usually defined to be tropicalizations as Berkovich analytic spaces.
They have been studied by many mathematicians, e.g., see \cite{Gub13}, \cite{GRW16}, \cite{GRW17}, and \cite{Pay09-1}.
We introduce tropicalizations of
Huber's adic spaces (Subsection \ref{subsec;trop;adic}) and Zariski-Riemann spaces (Subsection \ref{subsec;trop;ZR}),
which are the inverse limits of fan structures of tropicalizations of Berkovich analytic spaces.
(Note that our tropicalizations of adic spaces are different from Foster-Payne's adic tropicalizations; see \cite{Fos16}.)
They are used to give explicit descriptions of tropical analogs of Milnor $K$-groups (Corollary \ref{lemK_Tval}).
Let $M $ be a free $\mathbb{Z}$-module of finite rank $n$.
We put $N:=\mathop{\mathrm{Hom}}\nolimits(M,\mathbb{Z})$.
Let $\Sigma$ be a fan in $N_\mathbb{R}$, and $T_{\Sigma}$ the normal toric variety over a field associated to $\Sigma$. (See \cite{CLS11} for basic notions and results on toric varieties.)
In this paper, cones mean strongly convex rational polyhedral cones.
Remind that there is a natural bijection between the cones $\sigma \in \Sigma$ and the torus orbits $O(\sigma)$ in $T_{\Sigma}$.
The torus orbit $O(\sigma)$ is isomorphic to the torus $ \mathop{\mathrm{Spec}}\nolimits K[M \cap \sigma^{\perp}].$
We put $N_{\sigma}:= \mathop{\mathrm{Hom}}\nolimits (M \cap \sigma^{\perp} , \mathbb{Z})$.
We fix an isomorphism $M \cong \mathbb{Z}^n$, and identify $\mathop{\mathrm{Spec}}\nolimits K[M]$ and $\mathbb{G}_m^n$.
\subsection{Fans}\label{subsec;fan}
In this subsection,
we recall the partial compactification $\bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$ of $\mathbb{R}^n$
and define generalizations of fans in it.
We define a topology on the disjoint union $\bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$ as follows.
We extend the canonical topology on $\mathbb{R}$ to that on $\mathbb{R} \cup \{ \infty \}$ so that $(a, \infty]$ for $a \in \mathbb{R}$ are a basis of neighborhoods of $\infty$.
We also extend the addition on $\mathbb{R}$ to that on $\mathbb{R} \cup \{ \infty \}$ by $a + \infty = \infty$ for $a \in \mathbb{R} \cup \{\infty\}$.
We consider the set of semigroup homomorphisms $\mathop{\mathrm{Hom}}\nolimits (M \cap \sigma^{\vee}, \mathbb{R} \cup \{ \infty \}) $ as a topological subspace of $(\mathbb{R} \cup \{ \infty \})^{M \cap \sigma^{\vee}} $.
We define a topology on $\bigsqcup_{ \substack{\tau \in \Sigma \\ \tau \preceq \sigma } } N_{\tau,\mathbb{R}}$ by the canonical bijection
$$\mathop{\mathrm{Hom}}\nolimits (M \cap \sigma^{\vee}, \mathbb{R} \cup \{ \infty \}) \cong \bigsqcup_{ \substack{\tau \in \Sigma \\ \tau \preceq \sigma } } N_{\tau,\mathbb{R}}.$$
Then we define a topology on $\bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$ by glueing the topological spaces $ \bigsqcup_{\substack{\tau \in \Sigma \\ \tau \preceq \sigma }} N_{\tau,\mathbb{R}}$ together.
We shall define fans in $\bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$.
\begin{dfn}
For a cone $\sigma\in \Sigma $ and a cone $C\subset N_{\sigma,\mathbb{R}}$,
we call its closure $P:=\overline{C}$ in $ \bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$ a \emph{cone} in $\bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$.
In this case, we put $\mathop{\mathrm{rel.int}}\nolimits(P):=\mathop{\mathrm{rel.int}}\nolimits(C)$, and call it the \emph{relative interior} of $P$.
We put $\dim(P):=\dim(C)$.
\end{dfn}
Let $\sigma_P \in \Sigma $ be the unique cone such that $\mathop{\mathrm{rel.int}}\nolimits(P)\subset N_{\sigma_P,\mathbb{R}}$.
A subset $Q$ of a cone $P$ in $\bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$ is called a \emph{face} of $P$ if it is
the closure of the intersection $P^a \cap N_{\tau,\mathbb{R}}$
in $ \bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$
for some $ a \in \sigma_P\cap M$ and some cone $\tau \in \Sigma $,
where $P^a$ is the closure of
$$\{x \in P\cap N_{\sigma_P,\mathbb{R}} \mid x(a) \leq y(a) \text{ for any } y \in P \cap N_{\sigma_P,\mathbb{R}} \} $$
in $ \bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$.
A finite collection $\Lambda$ of cones in $ \bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$ is called a \emph{fan} if it satisfies the following two conditions.
\begin{itemize}
\item For all $P \in \Lambda$, each face of $P$ is also in $\Lambda$.
\item For all $P,Q \in \Lambda$, the intersection $P \cap Q$ is a face of both $P$ and $Q$.
\end {itemize}
We call the union
$$\bigcup_{P\in\Lambda} P \subset \bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$$
the \emph{support} of $\Lambda$.
We denote it by $\lvert \Lambda \rvert$.
We also say that $\Lambda$ is a \emph{fan structure} of $\lvert \Lambda \rvert $.
\subsection{Tropicalizations of Berkovich analytic spaces}\label{subsec;trop;Ber}
We recall basics of \textit{tropicalizations} of Berkovich analytic spaces.
Let $(L,v_L)$ be a trivially valued field.
In this subsection, every algebraic variety is defined over $L$.
The \textit{tropicalization map} $$\mathop{\mathrm{Trop}}\nolimits \colon O(\sigma)^{\mathrm{Ber}} \rightarrow N_{\sigma,\mathbb{R}} = \mathop{\mathrm{Hom}}\nolimits(M \cap \sigma^{\perp} ,\mathbb{R})$$
is the proper surjective continuous map given by the restriction
$$ \mathop{\mathrm{Trop}}\nolimits(v_x):= v_x|_{M \cap \sigma^{\perp}} \colon M \cap \sigma^{\perp} \rightarrow \mathbb{R} $$
for $ v_x \in O(\sigma)^{\mathrm{Ber}}$; see \cite[Section 2]{Pay09-1}.
(Here, as explained in Subsection \ref{subsec;Berkovich}, we consider elements of Berkovich analytic spaces as valuations.)
We define the tropicalization map $$\mathop{\mathrm{Trop}}\nolimits \colon T_{\Sigma}^{\mathrm{Ber}}=\bigsqcup_{\sigma \in \Sigma } O(\sigma)^{\mathrm{Ber}} \rightarrow \bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}}$$
by glueing the tropicalization maps $\mathop{\mathrm{Trop}}\nolimits \colon O(\sigma)^{\mathrm{Ber}} \rightarrow N_{\sigma,\mathbb{R}} )_\mathbb{R}$ together.
We note that the tropicalization map $$\mathop{\mathrm{Trop}}\nolimits \colon T_{\Sigma}^{\mathrm{Ber}} \rightarrow \bigsqcup_{\sigma \in \Sigma } N_{\sigma,\mathbb{R}} $$
is proper, surjective, and continuous; see \cite[Section 3]{Pay09-1}.
For a morphism $\varphi \colon X \rightarrow T_{\Sigma} $ from a scheme $X$ of finite type over $L$, the image $\mathop{\mathrm{Trop}}\nolimits(\varphi(X^{\mathrm{Ber}}))$ of $X^{\mathrm{Ber}}$ under the composition $\mathop{\mathrm{Trop}}\nolimits \circ \varphi$ is called a \textit{tropicalization} of $X^{\mathrm{Ber}}$ (or $X$).
For simplicity, we often write $\mathop{\mathrm{Trop}}\nolimits(\varphi(X))$ instead of $\mathop{\mathrm{Trop}}\nolimits(\varphi(X^{\mathop{\mathrm{Ber}}\nolimits}))$.
For a closed immersion $\varphi \colon X \rightarrow T_{\Sigma}$, the tropicalization $\mathop{\mathrm{Trop}}\nolimits(\varphi (X^{\mathrm{Ber}}))$ is a finite union of $\dim(X)$-dimensional cones by \cite[Theorem A]{BG84}.
(Over valued fields oh height $1$, the converse also holds \cite[Theorem 1.1]{M18}.)
\subsection{Tropicalizations and partial compactifications}\label{subsec;trop;triv;val}
In this subsection, we shall recall a relation between tropicalizations and partial compactifications.
Let $X \subset \mathbb{G}_m^n=\mathop{\mathrm{Spec}}\nolimits K[M]$ be a closed subvariety over a trivially valued field $K$.
\begin{dfn}\label{dfn;trop;cpt}
The closure $\overline{X}$ in the toric variety $T_{\Sigma}$ associated to a fan $\Sigma$ in $N_\mathbb{R}$ is called a \emph{tropical compactification}
if the multiplication map
$$ \mathbb{G}_m^n \times \overline{X} \to T_{\Sigma} $$
is faithfully flat and $\overline{X} $ is proper.
\end{dfn}
\begin{thm}[{Tevelev \cite[Theorem 1.2]{Tev07}}]
There exists a fan $\Sigma$ such that $\overline{X} \subset T_{\Sigma}$ is a tropical compactification.
\end{thm}
\begin{rem}\label{rem;trop;cpt;property}
Let $\overline{X} \subset T_{\Sigma}$ be a tropical compactification of $X \subset \mathbb{G}_m^n$.
\begin{enumerate}
\item The fan $\Sigma$ is a fan structure of $\mathop{\mathrm{Trop}}\nolimits(X)\subset N_\mathbb{R}$ \cite[Proposition 2.5]{Tev07}.
\item For any refinement $\Sigma'$ of $\Sigma$, the closure of $X$ in $T_{\Sigma'}$ is also a tropical compactification \cite[Proposition 2.5]{Tev07}.
\end{enumerate}
\end{rem}
\begin{prp}[{Tevelev \cite[Proposition 2.3]{Tev07}}]\label{rem,trop,cpt,prp}
For a fan $\Sigma$ in $N_{\mathbb{R}}$,
the closure $\overline{X}$ of $X$ in $T_{\Sigma}$ is proper
if and only if $\mathop{\mathrm{Trop}}\nolimits(X)$ is contained in the support of $\Sigma$.
\end{prp}
\subsection{Tropicalizations of adic spaces over trivially valued fields}\label{subsec;trop;adic}
In this subsection,
we introduce tropicalizations of adic spaces associated with algebraic varieties over a trivially valued field $K$,
and study their basic properties.
We define tropicalizations.
Let $X $ be a subvariety of a torus $\mathbb{G}_m^n$ over $K$,
and
$\Lambda $ a fan structure of $\mathop{\mathrm{Trop}}\nolimits(X) \subset N_{\mathbb{R}}$.
We consider the composition
$$ X^{\mathop{\mathrm{ad}}\nolimits} \to \overline{X} \to \Lambda,$$
where the first morphism is taking center and the second one is defined by
$\overline{X} \ni x \to \lambda \in \Lambda $
such that $ x \in O(\lambda)$.
(The closure $\overline{X}$ is taken in the toric variety $T_{\Lambda}$ associated with the fan $\Lambda$.) Note that by Proposition \ref{rem,trop,cpt,prp} and the valuative criterion of properness, the map $X^{\mathop{\mathrm{ad}}\nolimits} \to \overline{X}$ is well-defined.
We denote this composition by $\mathop{\mathrm{Trop}}\nolimits_{\Lambda}^{\mathop{\mathrm{ad}}\nolimits} \colon X^{\mathop{\mathrm{ad}}\nolimits} \to \Lambda$.
\begin{dfn}
Taking all fan structures $\Lambda $ of $\mathop{\mathrm{Trop}}\nolimits(X)$, we have a surjective map
$$\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits} \colon X^{\mathop{\mathrm{ad}}\nolimits} \to \underleftarrow{\lim } \Lambda $$
called the \emph{tropicalization map} of $X^{\mathop{\mathrm{ad}}\nolimits}$.
\end{dfn}
For a subvariety $Y $ of a toric variety $T_\Sigma$ over $K$
and
$\Lambda $ a fan structure of $\mathop{\mathrm{Trop}}\nolimits(Y) $,
we put
$$\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}_{\Lambda} \colon Y^{\mathop{\mathrm{ad}}\nolimits} \to \Lambda$$
the disjoint union of $\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}_{\Lambda \cap \mathop{\mathrm{Trop}}\nolimits(O(\sigma))} \ (\sigma \in \Sigma)$.
\begin{dfn}
Taking all fan structures $\Lambda $ of $\mathop{\mathrm{Trop}}\nolimits(Y)$, we have a surjective map
$$\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits} \colon Y^{\mathop{\mathrm{ad}}\nolimits} \to \underleftarrow{\lim } \Lambda $$
called the \emph{tropicalization map} of $Y^{\mathop{\mathrm{ad}}\nolimits}$.
\end{dfn}
\begin{rem}\label{rem;adic;trop;bij}
By Proposition \ref{ZRhomeo},
the natural map
$$X^{\mathop{\mathrm{ad}}\nolimits} \to \underleftarrow{\lim}\overline{X}$$
is surjective, where $\overline{X} $ runs through all tropical compactifications of $X \subset \mathbb{G}_m^n$,
and
the map $\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits} $ induces a bijection
$$ X^{\mathop{\mathrm{ad}}\nolimits} / J \cong \underleftarrow{\lim} \Lambda,$$
where $J$ is the equivalence relation
generated by
$ v \sim w$ for $v$ and $w$ such that
the restrictions $v|_{M} $ and $w|_{M}$ are the equivalent, i.e., there exists an isomorphism $\varphi \colon v(M) \cong w(M)$ of totally ordered abelian groups satisfying $\varphi \circ v|_{M} = w|_{M}$.
\end{rem}
To give another description of $\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}(X^{\mathop{\mathrm{ad}}\nolimits})$,
we define an equivalence relation on $(N_\mathbb{R})^r $.
For $l_i \in N_\mathbb{R}$, we can write $l_i = \sum_k a_{i,k} l_{i,k}$
by $\mathbb{Q}$-linear independent elements $\{a_{i,k}\}_k \subset \mathbb{R}$
and $\mathbb{Q}$-linear independent elements $\{l_{i,k}\}_k \subset N_\mathbb{Q}$.
In other words,
the subspace $\bigoplus_k \mathbb{Q} \cdot l_{i,k} \subset N_\mathbb{Q}$ is the minimal subspace of $N_\mathbb{Q}$ such that $(\bigoplus_k \mathbb{Q} \cdot l_{i,k} \subset N_\mathbb{Q})_\mathbb{R}$ contains $l_i$.
We put
$$J_r:=\{ (l_1,\dots,l_r) \in (N_\mathbb{R})^r \mid \overline{l_i}
\in N_\mathbb{R} / \sum_{j=1}^{i-1} \sum_k \mathbb{R} \cdot l_{j,k}
\text{ is non-zero} \ (1 \leq i\leq r)
\}.$$
We put $I_r$ the equivalence relation on $(N_\mathbb{R})^r$ generated by
$(l_1,\dots,l_r) \sim (l'_1,\dots,l'_r)$
for
$(l_1,\dots,l_r) $ and $ (l'_1,\dots,l'_r)$
satisfying
$$\mathbb{R}_{>0} \cdot \overline{l_i} = \mathbb{R}_{>0} \cdot \overline{l'_i}
\in N_\mathbb{R} / \sum_{j=1}^{i-1} \sum_k \mathbb{R} \cdot l_{j,k} \ (1 \leq i\leq r).$$
To simplify notations, we put $(N_\mathbb{R})^0:=\{ (0,\dots,0) \in N_\mathbb{R}\}$.
\begin{lem}\label{lem;trop;adic;another;express}
Let $A \subset N_\mathbb{R}$ be the support of a fan.
There is a bijection
\begin{align*}
&\bigcup_{r \geq 0} \{ (l_1,\dots, l_r) \in J_r
\mid
\sum_{i=1}^r (\prod_{j=1}^i \epsilon_j ) l_i \in A
\ \text{for small } \epsilon_j \in \mathbb{R}_{>0}
\ (1 \leq j \leq r)
\} /\bigcup_{r \geq 1 } I_r
\\
&\cong \underleftarrow{\lim}\Lambda
\end{align*}
given by
$ (l_1, \dots,l_r) \mapsto (P_\Lambda)_{ \Lambda}$,
where
$\Lambda$ runs through all fan structures of $A$
and
$(P_\Lambda)_\Lambda$ satisfies
$$ \sum_{i=1}^r (\prod_{j=1}^i \epsilon_{\Lambda,j} ) l_i \in \mathop{\mathrm{rel.int}}\nolimits (P_\Lambda)$$
for any $\Lambda$ and sufficiently small $\epsilon_{\Lambda,j} \ (1 \leq j \leq r)$.
\end{lem}
\begin{proof}
We prove Lemma \ref{lem;trop;adic;another;express} by induction on $\dim (A)$, i.e., the maximum of dimension of cones contained in $A$.
When $\dim(A)=0$, it is trivial.
We assume $\dim(A) \geq 1$.
We fix $l \in N_\mathbb{R} \setminus \{0\}$.
We put $(P_{l,\Lambda})_\Lambda$ the image of $l$ under the map in Lemma \ref{lem;trop;adic;another;express}.
Then
for a fan structure $\Lambda$ of $A$
such that
$\dim(P_{l,\Lambda}) $ achieves its minimum,
the set
$\overline{A} \cap N_{P_{l,\Lambda},\mathbb{R}} \subset \mathop{\mathrm{Trop}}\nolimits(T_{\Lambda})$
does not depend on the choice of $\Lambda$
under canonical identification of $N_{P_{l,\Lambda},\mathbb{R}}$
by bijective maps of tropicalizations induced from blow-ups of toric varieties.
We fix such a fan structure, and denote it by $\Lambda_0$.
Then,
by applying the assumption of the induction to
$\overline{A} \cap N_{P_{l,\Lambda_0}, \mathbb{R}}$,
the map in Lemma \ref{lem;trop;adic;another;express}
for $\overline{A} \cap N_{P_{l,\Lambda_0},\mathbb{R}}$
induces
a bijection between
the subset of
$$
\bigcup_{r \geq 0} \{ (l_1,\dots, l_r) \in J_r
\mid
\sum_{i=1}^r (\prod_{j=1}^i \epsilon_j ) l_i \in A
\ \text{for small } \epsilon_j \in \mathbb{R}_{>0}
\ (1 \leq j \leq r)
\} /\bigcup_{r \geq 1 } I_r
$$
consisting of the equivalent classes of $(l_1,\dots,l_r)$
such that $ \mathbb{R}_{>0} \cdot l_1 = \mathbb{R}_{>0} \cdot l$
and the subset of $\underleftarrow{\lim}\Lambda$
consisting of $(P_{\Lambda})_{\Lambda}$ such that $\mathbb{R}_{>0} \cdot l_{(P_{\Lambda})} = \mathbb{R}_{>0}\cdot l$,
where
$l_{(P_\Lambda)_\Lambda} $ is a fixed point in the relative interior of
the intersection $\bigcap_{\Lambda} P_{\Lambda}$.
(Note that by compactness of $S^{n-1}$ and the definition,
the intersection $\bigcap_{\Lambda} P_{\Lambda}$
is a half-line.)
Since $l \in N_\mathbb{R}\setminus \{0\}$ is arbitrary,
Lemma \ref{lem;trop;adic;another;express} holds.
\end{proof}
\begin{rem}\label{rem;trop;adic;l_r;P;span;same}
Let $(l_1,\dots,l_r) \mapsto (P_\Lambda)_\Lambda$ as in Lemma \ref{lem;trop;adic;another;express}.
Then for a sufficiently fine fan structure $\Lambda_1$ of $A$,
we have
$$\mathop{\mathrm{Span}}\nolimits_\mathbb{R} (P_{\Lambda_1}) = \sum_{i,k} \mathbb{R} \cdot l_{i,k}.$$
\end{rem}
By definitions, we have the following.
\begin{lem}\label{lem;trop;adic;vertical;special;closure}
Let $\Lambda$ be a fan structure of $\mathop{\mathrm{Trop}}\nolimits(X)$,
$v \in X^{\mathop{\mathrm{ad}}\nolimits}$, and $w \in X^{\mathop{\mathrm{ad}}\nolimits}$ a vertical specialization of $v$.
We put $\overline{w} \in \overline{X}\cap O(\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}_{\Lambda}(v))$
the unique vertical specialization $v_0$ of the trivial valuation on $k(\mathop{\mathrm{center}}\nolimits (v) )$
such that $\overline{w}$ is a horizontal specialization of $w$ (\cite[Lemma 1.2.5(i)]{HK94}).
Here, we consider $v_0$ as an element in $\overline{X}^{\mathop{\mathrm{ad}}\nolimits}$.
Then we have
$$ \mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}_{\{\overline{P} \cap N_{\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}_{\Lambda}(v)} \}_{P\in \Lambda}}(\overline{w})
=\overline{\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}_{\Lambda}(w)} \cap N_{\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}_{\Lambda}(v)}.$$
\end{lem}
We consider $(l_1,\dots,l_r) \in (N_\mathbb{R})^r$ as a map
$$M \ni m \mapsto (l_1(m),\dots,l_r(m)) \in \mathbb{R}^r.$$
\begin{lem}\label{rem;trop;adic;l_1}
Let $v \in X^{\mathop{\mathrm{ad}}\nolimits}$ be a valuation,
and $(l_1,\dots,l_r) \in (N_\mathbb{R})^r$ corresponds to $\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}(v)$ via the bijection in Lemma \ref{lem;trop;adic;another;express} for $\mathop{\mathrm{Trop}}\nolimits(X)$.
Then
there exists a ordered group isomorphism
$\phi \colon v(M) \cong (l_1,\dots,l_r)(M)$
satisfying $\phi \cdot v|_M = (l_1,\dots,l_r)$.
Here, we consider $\mathbb{R}^r$ equipped with the lexicographic order.
(We denote a representative of the equivalence class $v $ by $v$.)
\end{lem}
\begin{proof}
When $r\leq 1$, the assertion is trivial.
When $r\geq 2$, the assertion follows from induction on $r$ in a similar way to Lemma \ref{lem;trop;adic;another;express} by Lemma \ref{lem;trop;adic;vertical;special;closure}.
\end{proof}
\begin{dfn}
For $x \in \mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits} (X^{\mathop{\mathrm{ad}}\nolimits}) $ whose inverse image under the bijection in Lemma \ref{lem;trop;adic;another;express}
for $\mathop{\mathrm{Trop}}\nolimits(X)$ is $(l_1,\dots,l_r) $,
we put $\mathop{\mathrm{ht}}\nolimits (x):= r$.
\end{dfn}
We omit the proof of the following lemma because it immediately follows from Lemma \ref{rem;trop;adic;l_1}.
\begin{lem}\label{lem;trop;adic;height;compare}
For $v \in X^{\mathop{\mathrm{ad}}\nolimits}$,
we have
$$\mathop{\mathrm{ht}}\nolimits (v(M)) =\mathop{\mathrm{ht}}\nolimits (\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}(v)).$$
\end{lem}
\begin{rem}\label{rem;trop;ber;adic;comparison}
The map in Lemma \ref{lem;trop;adic;another;express} induces a bijection
$$ \mathop{\mathrm{Trop}}\nolimits(X^{\mathop{\mathrm{Ber}}\nolimits}) / I \cong \mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits} (X^{\mathop{\mathrm{ad}}\nolimits})^{\mathop{\mathrm{ht}}\nolimits \leq 1}.$$
Here, the subset $\mathop{\mathrm{Trop}}\nolimits(X^{\mathop{\mathrm{ad}}\nolimits})^{\mathop{\mathrm{ht}}\nolimits \leq 1}$ of $\mathop{\mathrm{Trop}}\nolimits(X^{\mathop{\mathrm{ad}}\nolimits})$
consisting of elements of height $\leq 1$,
and $I$ is the equivalence relation generated by
$a \sim a'$ for $a$ and $a' $
such that $\mathbb{R}_{>0} \cdot a = \mathbb{R}_{>0} \cdot a'$.
We have a commutative diagram
$$\xymatrix{
& X^{\mathop{\mathrm{Ber}}\nolimits} / (\emph{equivalence relations}) \ar[d]^-{\mathop{\mathrm{Trop}}\nolimits} \ar[r]^-{\cong}
& X^{\mathop{\mathrm{ad}}\nolimits,\mathop{\mathrm{ht}}\nolimits \leq 1} \ar[d]^-{\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}} \ar@{}[r]|*{\subset} & X^{\mathop{\mathrm{ad}}\nolimits}\ar[d]^-{\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}} \\
& \mathop{\mathrm{Trop}}\nolimits(X^{\mathop{\mathrm{Ber}}\nolimits}) / I \ar[r]^-{\cong}
& \mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits} (X^{\mathop{\mathrm{ad}}\nolimits})^{\mathop{\mathrm{ht}}\nolimits \leq 1} \ar@{}[r]|*{\subset} &\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits} (X^{\mathop{\mathrm{ad}}\nolimits}).\\
}$$
\end{rem}
\subsection{Tropicalizations of Zariski-Riemann spaces}\label{subsec;trop;ZR}
In this subsection, we shall introduce tropicalizations of explicit subsets of Berkovich analytic spaces and \textit{tropicalizations of Zariski-Riemann spaces}.
They will be used to compute tropical analogs of Milnor $K$-groups.
Let $K$ be a trivially valued field,
$L/K$ a finitely generated extension of fields.
For a morphism $\varphi \colon \mathop{\mathrm{Spec}}\nolimits L \to \mathbb{G}_m^n$ over $K$,
there is a canonical morphism
$$ \varphi \colon ( \mathop{\mathrm{Spec}}\nolimits L/K)^{\mathop{\mathrm{Ber}}\nolimits} \to \mathbb{G}_m^{n,\mathop{\mathrm{Ber}}\nolimits} .$$
\begin{dfn}
We call $$\mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K)):=\mathop{\mathrm{Trop}}\nolimits (\varphi((\mathop{\mathrm{Spec}}\nolimits L/K)^{\mathop{\mathrm{Ber}}\nolimits}))$$ a \emph{tropicalization} of $(\mathop{\mathrm{Spec}}\nolimits L/K)^{\mathop{\mathrm{Ber}}\nolimits}$.
\end{dfn}
\begin{rem}\label{rem;trop'n;field;Ber}
We have
$$\mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K))=\mathop{\mathrm{Trop}}\nolimits(\overline{\varphi(\mathop{\mathrm{Spec}}\nolimits L)}),$$
where $\overline{\varphi(\mathop{\mathrm{Spec}}\nolimits L)}$ is the closure in $\mathbb{G}_m^n$.
(This is because any affinoid subdomain of $\overline{\varphi(\mathop{\mathrm{Spec}}\nolimits L)}^{\mathop{\mathrm{Ber}}\nolimits}$ has a point whose support is the generic point of $ \overline{\varphi(\mathop{\mathrm{Spec}}\nolimits L)}$.)
In particular, $\mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K))$ is a finite union of cones.
\end{rem}
\begin{dfn}\label{dfn:trop'n:ZRspace}
For a fan structure $\Lambda$ of
$$\mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K))=\mathop{\mathrm{Trop}}\nolimits(\overline{\varphi(\mathop{\mathrm{Spec}}\nolimits L)}),$$
we denote the composition
$$\mathop{\mathrm{ZR}}\nolimits (L/K) \to \overline{\varphi(\mathop{\mathrm{Spec}}\nolimits L)}^{\mathop{\mathrm{ad}}\nolimits} \xrightarrow{\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits}_\Lambda} \Lambda,$$
by $\mathop{\mathrm{Trop}}\nolimits_{\Lambda}^{\mathop{\mathrm{ad}}\nolimits} \circ \varphi$,
where the first morphism is the canonical map.
Taking all fan structures $\Lambda$ of $\mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K)) $, we have a surjective map
$$\mathop{\mathrm{Trop}}\nolimits^{\mathop{\mathrm{ad}}\nolimits} \circ \varphi \colon \mathop{\mathrm{ZR}}\nolimits(L/K) \to \underleftarrow{\lim} \Lambda$$
called a \emph{tropicalization map} of the Zariski-Riemann space $\mathop{\mathrm{ZR}}\nolimits(L/K)$.
\end{dfn}
\section{Tropical analogs of Milnor $K$-groups and their sheaf cohomology}\label{sec;trocoho;troK}
In this section, we introduce and study tropical analogs of Milnor $K$-groups, called \textit{tropical $K$-groups}.
The goal is to show that tropical $K$-groups satisfy good properties,
i.e., they form a cycle module in the sense of Rost \cite[Definition 2.1]{Ros96}.
In particular, Zariski sheaf cohomology groups of the sheaves $\mathscr{K}_T^p$ of tropical $K$-groups can be computed by the Gersten resolution (Corollary \ref{corshftrK}), hence
we have a canonical isomorphism
$$H_{Zar}^p(X,\mathscr{K}_T^p) \cong CH^p (X)_\mathbb{Q}.$$
This is used to prove a tropical analog of the Hodge conjecture for smooth algebraic varieties over trivially valued fields, in a subsequent paper \cite{M20}.
Let $K$ be a field.
We equip it with the trivial valuation.
Let $M$ be a free $\mathbb{Z}$-module of finite rank.
We put $N:=\mathop{\mathrm{Hom}}\nolimits(M,\mathbb{Z})$.
Let $L/K$ be a finitely generated extension of fields.
Let $p \geq 0 $ be a non-negative integer.
For a morphism
$\varphi \colon \mathop{\mathrm{Spec}}\nolimits L \to \mathop{\mathrm{Spec}}\nolimits K[M]$ over $K$ and
a fan structure $\Lambda $ of $\mathop{\mathrm{Trop}}\nolimits (\varphi (\mathop{\mathrm{Spec}}\nolimits L/ K)),$
we put
$$F_p(0,\Lambda):= \sum_{\substack{P' \in \Lambda }} \wedge^p \mathop{\mathrm{Span}}\nolimits (P') \subset \wedge^p N_\mathbb{R},$$
where $\mathop{\mathrm{Span}}\nolimits (P')$ is the $\mathbb{Q}$-vector space spanned by $P' \cap N_\mathbb{Q}$, and put
$$F^p(0,\Lambda):= \wedge^p M_\mathbb{Q} / \{f \in \wedge^p M_\mathbb{Q} \mid \alpha(f)=0 \ (\alpha \in F_p(0,\Lambda))\} .$$
Since $F^p(0,\Lambda)$ depends only on the support $\mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K))=\lvert \Lambda \rvert $ of $\Lambda$,
we sometimes write $F^p(0,\mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K)))$ instead of $F^p(0,\Lambda)$.
\begin{rem}
Consider two morphisms $\varphi \colon \mathop{\mathrm{Spec}}\nolimits L \to \mathbb{G}_m^r$ and $ \varphi' \colon \mathop{\mathrm{Spec}}\nolimits L \to \mathbb{G}_m^l $ over $K$
and
a toric morphism $\psi \colon \mathbb{G}_m^l \to \mathbb{G}_m^r$
such that
the diagram
$$\xymatrix{
& \mathop{\mathrm{Spec}}\nolimits L \ar[r]^-{\varphi} \ar[dr]_-{\varphi'} & \mathbb{G}_m^r \\
& & \mathbb{G}_m^l \ar[u]_-{\psi}
}$$
is commutative.
These induce a pull-back map
\begin{align}
F^p(0, \mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K))) \to F^p(0, \mathop{\mathrm{Trop}}\nolimits(\varphi'(\mathop{\mathrm{Spec}}\nolimits L/K))). \label{map:Fp:pull-back}
\end{align}
Note that since $\mathop{\mathrm{Trop}}\nolimits(\varphi'(\mathop{\mathrm{Spec}}\nolimits L/K)) \to \mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K))$ is surjective, this pull-back map is injective.
\end{rem}
\begin{dfn}\label{dftroK}
We put
$$ K_T^p(L/K):=\lim_{\substack{\rightarrow \\ \varphi\colon \mathop{\mathrm{Spec}}\nolimits(L) \rightarrow \mathbb{G}_m^r }} F^p(0, \mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K)) ) , $$
where $\varphi \colon \mathop{\mathrm{Spec}}\nolimits L \to \mathbb{G}_m^r$ runs all $K$-morphisms to tori of arbitrary dimensions
and morphisms are the pull-back maps
\begin{align}
F^p(0, \mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K))) \to F^p(0, \mathop{\mathrm{Trop}}\nolimits(\varphi'(\mathop{\mathrm{Spec}}\nolimits L/K))).
\end{align}
We call $K_T^p(L/K)$ the $p$-th (rational) \emph{tropical $K$-group}.
When there is no confusion, we denote it by $K_T^p(L)$.
\end{dfn}
The following results give another expression of tropical $K$-groups.
\begin{lem}\label{lemFpexpressvalua}
For a morphism
$$\varphi \colon \mathop{\mathrm{Spec}}\nolimits L \to \mathbb{G}_m^n=\mathop{\mathrm{Spec}}\nolimits K[M]$$
over $K$,
we have
$$F^p(0 , \mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K))) = \wedge^p M_\mathbb{Q}/J_M= \wedge^p M_\mathbb{Q}/J'_M,$$
where $J_M $ (resp.\ $J_M'$) is the $\mathbb{Q}$-vector subspace generated by $f \in \wedge^p M_\mathbb{Q}$
such that $\wedge^p v (\varphi(f))=0$ for $v \in \mathop{\mathrm{ZR}}\nolimits(L/K)$
(resp.\ for $v \in \mathop{\mathrm{ZR}}\nolimits(L/K)$ with
$$\mathop{\mathrm{ht}}\nolimits (v(\varphi(M))) = \mathop{\mathrm{tr.deg}}\nolimits(k(\varphi(\mathop{\mathrm{Spec}}\nolimits L))/K)).$$
Here, the element $\varphi(f) \in \wedge^p (L^{\times})_\mathbb{Q}$ is the image of $f$ by the map $\wedge^p \varphi \colon \wedge^p M_\mathbb{Q} \to \wedge^p (L^{\times})_\mathbb{Q}$,
the map
\begin{align*}
\wedge^p v & \colon \wedge^p (L^{\times})_\mathbb{Q} \to \wedge^p (\Gamma_v)_\mathbb{Q} \\
(\emph{resp}.\ \wedge^p \varphi & \colon \wedge^p M_\mathbb{Q} \to \wedge^p (L^{\times})_\mathbb{Q})
\end{align*}
is
the wedge product of $v_\mathbb{Q} \colon (L^{\times})_\mathbb{Q} \to ( \Gamma_v)_\mathbb{Q}$ (resp.\ $\varphi_\mathbb{Q} \colon M_\mathbb{Q} \to ( L^{\times})_\mathbb{Q}$) as a homomorphism of $\mathbb{Q}$-vector spaces,
and
$k(\varphi(\mathop{\mathrm{Spec}}\nolimits L))$ is the residue field of the structure sheaf at $\varphi(\mathop{\mathrm{Spec}}\nolimits L) \in \mathbb{G}_m^n$.
\end{lem}
\begin{proof}
This follows from
Remark \ref{rem;trop;adic;l_r;P;span;same},
Lemma \ref{rem;trop;adic;l_1},
Lemma \ref{lem;trop;adic;height;compare}, and
surjectivity of tropicalization map $\mathop{\mathrm{ZR}}\nolimits(L/K)^{\mathop{\mathrm{ad}}\nolimits} \to \underleftarrow{\lim} \Lambda$
,where $\Lambda$ runs through all fan structures of $\mathop{\mathrm{Trop}}\nolimits(\varphi((Spec L/K))$.
Remind that $\mathop{\mathrm{Trop}}\nolimits(\varphi(\mathop{\mathrm{Spec}}\nolimits L/K))$ is a finite union of rational polyhedra of dimension $\mathop{\mathrm{tr.deg}}\nolimits (k(\varphi(\mathop{\mathrm{Spec}}\nolimits L)/K))$.
\end{proof}
\begin{cor}\label{lemK_Tval}
We have
$$K_T^p(L) \cong \wedge^p ( L^{\times})_\mathbb{Q} / J \cong \wedge^p ( L^{\times})_\mathbb{Q} / J',$$
where $J$ (resp.\ $J'$) is the $\mathbb{Q}$-vector subspace generated by $f \in \wedge^p ( L^{\times})_\mathbb{Q}$
such that $\wedge^p v (f)=0 $ for $v \in \mathop{\mathrm{ZR}}\nolimits(L/K)$
(resp.\ for $v \in \mathop{\mathrm{ZR}}\nolimits(L/K)$ with $\mathop{\mathrm{ht}}\nolimits(v)= \mathop{\mathrm{tr.deg}}\nolimits (L/K)$).
\end{cor}
We shall show that tropical $K$-groups satisfy good properties,
i.e., they define a cycle module in the sense of Rost \cite[Definition 2.1]{Ros96}.
We recall the definition and several maps of Milnor $K$-groups.
For a field $E$, its Milnor $K$-group is defined by
$$K_M^p (E) := T^p E^{\times} / J , $$
where $T^p E^{\times}$ is the $p$-th tensor of the multiplicative group $E^{\times}$ over $\mathbb{Z}$
and $J$ is the subgroup of $T^p E^{\times}$ generated by
$$\{a_1 \otimes \dots \otimes a_p \mid a_i = 1-a_j \text{ for some } i,j \}.$$
In particular, we have $K_M^0(E) = \mathbb{Z}$ and $K_M^1(E) = E^\times$. The image of $a_1 \otimes \dots \otimes a_p$ in $K_M^p(E)$ is denoted by $(a_1,\dots,a_p)$.
\begin{itemize}
\item A morphism $\varphi \colon F \to E$ of fields induces a map
$$ \varphi_* \colon K_M^p(F) \to K_M^p(E)$$
by $$\varphi_*((a_1,\dots,a_p)) = (\varphi(a_1), \dots, \varphi(a_p)).$$
\item For a finite morphism $\varphi \colon F \to E$,
there is the norm homomorphism
$$\varphi^* \colon K_M^p(E) \to K_M^p(F)$$
(Bass-Tate \cite{BT72}, Kato \cite{Kat80}).
It is a generalization of
the multiplication
$$ \times [E:F] \colon K_M^0 (E) =\mathbb{Z} \to \mathbb{Z} = K_M^0(F)$$
and
the usual norm map $E^{\times} \to F^{\times}$.
This is defined by Bass and Tate \cite{BT72} with respect to a choice of generators of $E$ over $F$, and independence of such a choice was proved by Kato \cite{Kat80}.
\item For a normalized discrete valuation $v \colon F^{\times} \to \mathbb{Z}$,
there is the residue homomorphism (Milnor \cite{Mil70})
$$\partial_v \colon K_M^p(F) \to K_M^{p-1}(\kappa(v)) .$$
It is characterized by
\begin{align*}
\partial_v ((\pi,u_1,\dots , u_{p-1})) & = (\overline{u}_1,\dots,\overline{u}_{p-1}) \\
\partial_v ((u_1,\dots , u_p)) & = 0
\end{align*}
for a prime $\pi$ of $v$ and $u_i \in F$
with $v(u_i) =0 \ (1 \leq i \leq p)$ and residue class $\overline{u}_i$.
(Remind that $\kappa(v)$ is the residue field of $v$.)
\end{itemize}
\begin{lem}\label{lemtroKfactorMilK}
The canonical morphism
$$ \otimes^p L^{\times} \to K_T^p (L)$$
factors through
$$ \otimes^p L^{\times} \to K_M^p (L) \to K_T^p (L).$$
\end{lem}
\begin{proof}
For any $a \in L$, there are no $2$-rational-rank valuations of $K(a)$ which are trivial on $K$.
Hence the assertion follows from Corollary \ref{lemK_Tval}.
\end{proof}
\begin{lem}\label{lemtrKinducedhom}
The maps $\varphi_*,$ $\varphi^*,$ and $\partial_v$ for $v $ which is trivial on $K$ induce maps of tropical $K$-groups
\begin{align*}
\varphi_* & \colon K_T^p(F) \to K_T^p(E), \\
\varphi^* &\colon K_T^p(E) \to K_T^p(F), \\
\partial_v & \colon K_T^p(F) \to K_T^{p-1}(\kappa(v)) .
\end{align*}
\end{lem}
\begin{proof}
For $\varphi_*$, the assertion follows from Corollary \ref{lemK_Tval}.
For $\partial_v$, the assertion follows from Corollary \ref{lemK_Tval} and Remark \ref{rem;val;vert;special;resi;fld;val}.
We shall show that the norm homomorphism $\varphi^*$ induces a map of tropical $K$-groups.
It suffices to show that
for a finite extension of fields $\varphi \colon F \rightarrow E$ and $\alpha \in K_M^p(E)$ such that $\wedge^p w (\alpha) =0$ for any $w \in \mathop{\mathrm{ZR}}\nolimits(E/K)$ with $\mathop{\mathrm{ht}}\nolimits(w)=\mathop{\mathrm{tr.deg}}\nolimits(E/K)$, we have
$$\wedge^p v (\varphi^* \alpha)=0 $$
for any $v \in \mathop{\mathrm{ZR}}\nolimits(F/K)$ with $\mathop{\mathrm{ht}}\nolimits(v)=\mathop{\mathrm{tr.deg}}\nolimits(F/K)$.
We shall show this assertion by induction on $p$.
When $p=0$, the assertion is trivial.
We assume $p\geq 1$.
Let $v_1 \in \mathop{\mathrm{ZR}}\nolimits(F/K)$ be the unique generalization of $v$ of height $1$.
Note that since $\mathop{\mathrm{ht}}\nolimits(v) = \mathop{\mathrm{tr.deg}}\nolimits(F/K)$, we have $\mathop{\mathrm{rank}}\nolimits (v)=\mathop{\mathrm{ht}}\nolimits(v)$.
Hence we have $\mathop{\mathrm{rank}}\nolimits v_1=\mathop{\mathrm{ht}}\nolimits (v_1)=1$,
i.e., the valuation $v_1$ is discrete.
For an extension $w_i \in \mathop{\mathrm{ZR}}\nolimits (E/K)$ of $v_1$ to $E$ and any $u \in \mathop{\mathrm{ZR}}\nolimits (\kappa(w_i)/K)$ with $\mathop{\mathrm{ht}}\nolimits(u)=tr.\deg (\kappa(w_i)/K)$,
we have
$$\wedge^{p-1} u (\partial_{w_i}(\alpha))=\wedge^p \tilde{u} (\alpha)=0,$$
where $\tilde{u} \in \mathop{\mathrm{ZR}}\nolimits (E/K)$ is the specialization of $w_i$ corresponding to $u$ in the sense of Remark \ref{rem;val;vert;special;resi;fld;val}.
Note that we have $$\mathop{\mathrm{ht}}\nolimits (\tilde{u})=\mathop{\mathrm{ht}}\nolimits (u)+1=\mathop{\mathrm{tr.deg}}\nolimits(\kappa(w_i)/K) +1=\mathop{\mathrm{tr.deg}}\nolimits(E/K) .$$
Hence by applying the assumption of induction to $\varphi_i \colon \kappa(v_1) \to \kappa(w_i)$ and $\partial_{w_i} (\alpha) \in K_M^{p-1}(\kappa(w_i))$, we have
$$\wedge^{p-1}\overline{v} (\varphi_i^*(\partial_{w_i}(\alpha)))=0,$$
where $ \overline{v} \in \mathop{\mathrm{ZR}}\nolimits(\kappa(v_1)/K)$ is the valuation corresponding to $v$ in the sense of Remark \ref{rem;val;vert;special;resi;fld;val}.
Hence we have
\begin{align*}
\wedge^p v (\varphi^* \alpha)& = \wedge^{p-1} \overline{v}(\partial_{v_1} \circ \varphi^* \alpha) \\
& = \wedge^{p-1} \overline{v}\bigg(\sum_{w_i} \varphi_{i}^* \circ \partial_{w_i} (\alpha)\bigg)\\
& =0 ,
\end{align*}
where $w_i \in \mathop{\mathrm{ZR}}\nolimits (E/K)$ runs through all extensions of $v_1 \in \mathop{\mathrm{ZR}}\nolimits(F/K)$ to $E$,
and the second equality follows from a basic property
$$\partial_{v_1}\circ \varphi^* = \sum_{w_i} \varphi_{i}^* \circ \partial_{w_i}$$
of Milnor $K$-groups.
\end{proof}
We also denote the induced maps of tropical $K$-groups by $\varphi_*,\varphi^*,\partial_v$.
\begin{thm}\label{trKcymod}
The functor
\begin{align*}
(\emph{finitely generated fields over } K) & \to ( \mathbb{Z}_{\geq 0}\emph{-graded abelian group} ) \\
L & \mapsto \bigoplus_p K_T^p(L/K).
\end{align*}
with $\varphi_*,\varphi^* , \partial_v$ and a natural morphism
$$K_M^p (L) \times K_T^q(L) \to K_T^{p+q} (L)$$
is a cycle module in the sense of Rost \cite[Definition 2.1]{Ros96}.
\end{thm}
\begin{proof}
This follows from Lemma \ref{lemtroKfactorMilK}, Lemma \ref{lemtrKinducedhom}, and
the fact that Milnor $K$-groups form a cycle module \cite[Theorem 1.4 and Remark 2.4]{Ros96}.
\end{proof}
\begin{exam}
We give examples of tropical $K$ groups.
\begin{itemize}
\item For any $L /K$, we have $K^0_T (L/K)=\mathbb{Q}$ by definition.
\item For any $L/K$, by Corollary \ref{lemK_Tval}, we have $K^1_T (L/K) = (L^{\times } /(L \cap K^{\mathop{\mathrm{alg}}\nolimits})^{\times})_\mathbb{Q}.$
\item For any $L/K$ and any $p \geq \mathop{\mathrm{tr.deg}}\nolimits(L/K) +1$, we have $K_T^p (L/K)=0$ since there are no valuations of $L$ which are trivial on $K$ and have rational ranks strictly greater than $\mathop{\mathrm{tr.deg}}\nolimits(L/K)$.
\item Let $T$ be an indeterminate. For any $L/K$, we have
$$ K^{\mathop{\mathrm{tr.deg}}\nolimits(L/K)+1}_T (L(T)/K) \overset{(\partial_x)_x}{\cong}
\bigoplus_{x \in \mathbb{A}^{1,\mathop{\mathrm{cl}}\nolimits}_L} K_T^{\mathop{\mathrm{tr.deg}}\nolimits(L/K)} (k(x)/K),$$
where $\mathbb{A}^{1,\mathop{\mathrm{cl}}\nolimits}_L$ is the set of closed points of $\mathbb{A}^{1}_L = \mathop{\mathrm{Spec}}\nolimits L[T]$,
the field $k(x)$ is the residue field at $x \in \mathbb{A}^1_L$, and
$\partial_x $ is the residue homomorphism for the normalized discrete valuation of $L(T)$ corresponding to $x \in \mathbb{A}^{1,\mathop{\mathrm{cl}}\nolimits}$.
This follows from homotopy property of cycle modules for $\mathbb{A}^1$ \cite[Proposition 2.2 H]{Ros96} and the above example.
In particular, for $L/K$ with $\mathop{\mathrm{tr.deg}}\nolimits(L/K)=1$,
we have
$$ K^{2}_T (L(T)/K) \overset{(\partial_x)_x}{\cong}
\bigoplus_{x \in \mathbb{A}^{1,\mathop{\mathrm{cl}}\nolimits}_L} (k(x)^{\times } /(k(x) \cap K^{\mathop{\mathrm{alg}}\nolimits})^{\times})_\mathbb{Q}.$$
\end{itemize}
\end{exam}
We give an explicit resolution of the Zariski sheaf of tropical $K$-groups on
a smooth algebraic variety $X$ over $K$.
Let $X^{(i)}$ be the set of points of $X$ of codimension $i$.
For any $i$ and points $x \in X^{(i)},y \in X^{(i +1)} $,
Rost defined a map
$$\partial_x^y \colon K_T^p(k(x)) \to K_T^{p-1}(k(y))$$
\cite[Section 2]{Ros96} as follows.
When $y \notin \overline{\{x\}}$, we put $\partial_x^y=0$.
When $ y \in \overline{\{x\}}$,
we put
$$ \partial_x^y :=\sum_v \varphi_v^* \circ \partial_v\colon K_T^p(k(x)) \to K_T^{p-1}(k(y)),$$
where $v \in \mathop{\mathrm{ZR}}\nolimits(k(x)/K)$ runs through all normalized discrete valuations of $k(x)$ whose
center in $\overline{\{x\}}$ is $y$.
Let $\eta \in X$ be the generic point.
We denote by $\mathscr{K}_T^p$ the sheaf on the Zariski site $X_{Zar}$ defined by
$$\mathscr{K}_T^p(U):= \mathop{\mathrm{Ker}}\nolimits (K_T^p(K(X))\xrightarrow{d} \oplus_{x\in U^{(1)}}K_T^{p-1}(k(x)),$$
where $d:= (\partial_\eta^x)_{ x \in U^{(1)}}$.
We call it \textit{the sheaf of tropical $K$-groups}.
\begin{cor}\label{corshftrK}
For any $p \geq 0$, the sheaf $\mathscr{K}_T^p$ has the \emph{Gersten resolution}, i.e.,
an exact sequence
\begin{align*}
0 \rightarrow \mathscr{K}_T^p \xrightarrow{d} \bigoplus_{x \in X^{(0)}} i_{x *} (K_T^p(k(x)))
\xrightarrow{d} \bigoplus_{x\in X^{(1)}} i_{x *}K_T^{p-1}(k(x)) & \xrightarrow{d} \bigoplus_{x\in X^{(2)}} i_{x *}K_T^{p-2}(k(x)) \xrightarrow{d} \dots \\
& \xrightarrow{d} \bigoplus_{x\in X^{(p)}} i_{x *}K_T^0(k(x)) \xrightarrow{d} 0 ,
\end{align*}
where
$i_x \colon \mathop{\mathrm{Spec}}\nolimits(k(x)) \rightarrow X$ are the natural morphisms,
we identify the groups $K_T^* (k(x))$ and the Zariski sheaf defined by them on $\mathop{\mathrm{Spec}}\nolimits k(x)$,
and $d:= (\partial_x^y)_{\{x \in X^{(i)} , \ y \in X^{(i+1)}\}}$.
In particular, we have
$$H_{Zar}^p(X,\mathscr{K}_T^p) \cong CH^p (X)_\mathbb{Q}.$$
\end{cor}
\begin{proof}
The first assertion follows from Lemma \ref{trKcymod} and \cite[Theorem 6.1]{Ros96}.
The second assertion follows from the fact that for a normalized discrete valuation $v \in \mathop{\mathrm{ZR}}\nolimits(E/K)$ of a field extension $E/K$, the residue homomorphism
$$ \partial_v \colon K_T^1(E) \cong (E^{\times})_\mathbb{Q} / ((E \cap K^{alg})^{\times})_\mathbb{Q} \to K_T^0(\kappa(v)) \cong \mathbb{Q} $$
coincides with the map induced by the valuation $v \colon E^{\times} \to \mathbb{Z}$.
\end{proof}
\subsection*{Acknowledgements}
The author would like to express his sincere thanks to his adviser Tetsushi Ito for his invaluable advice and persistent support.
The author also thanks Yuji Odaka for his encouragement.
The author also thanks Kazuhiro Ito for answering the author's questions on references and his helpful comments on this paper.
This work was supported by JSPS KAKENHI Grant Number 18J21577.
|
1,116,691,499,325 | arxiv | \section{Introduction}
In recent decades, there has been much interest in modelling and analysing the many networks which appear in the real world, in contexts such as the world wide web or online social networks. This work has drawn heavily on the mathematical study of random graphs, a subject with its origins in the 1959 work of Erd\H{o}s and R\'enyi, \cite{ER}. They principally studied the graphs which emerge from the following process: begin with a collection of nodes, and independently connect every pair with an edge, with some fixed probability $p$.
Erd\H{o}s-R\'enyi-style random graph theory has two distinct facets. First, researchers have analysed the finite graphs which arise. Here, questions of interest include the emergence of a giant component and the degree distribution of the nodes, and analyses are typically highly sensitive to the value of $p$. See \cite{Bol} for a comprehensive discussion of such matters.
The second angle of approach is to consider this process on a countably infinite set of nodes. In this case, a remarkable theorem of Erd\H{o}s and R\'enyi guarantees that, irrespective of the value of $p \in (0,1)$, the resulting graph will with probability $1$ be isomorphic to the \emph{Rado graph}. This famous graph is axiomatised by the following schema: given any finite disjoint sets of nodes $U$ and $V$, there exists a node $v$ connected to each node in $V$ and none in $U$. This graph exhibits many properties which logicians and combinatorists enjoy. To start with, it is \emph{universal} in that it isomorphically embeds every finite and countably infinite graph. It is also \emph{countably categorical}, meaning that any two countable models of the above axioms will be isomorphic. The graph is \emph{1-transitive} in that for any any two nodes $v_1, v_2$ there is an automorphism $\alpha$ where $\alpha(v_1)=v_2$. It is \emph{ultrahomogeneous}: any isomorphism between finite induced subgraphs extends to an automorphism of the whole structure. Analogues of these facts are proved in Proposition \ref{prop:multiprop} below. The Rado graph continues to attract the attention of today's permutation group-theorists; it is known that its automorphism group is simple (in the group-theoretic sense), and satisfies the \emph{strong small-index property}. See \cite{C} for a recent survey of such matters. Beyond this, the Rado graph satisfies several subtler properties, notably \emph{rank-1 supersimplicity} and \emph{1-basedness}, which make it a central object of study for today's model theorists. See \cite{Wag} for an authoritative account of these interesting matters.
In more recent years, however, network scientists have shifted away from the Erd\H{o}s-R\'enyi approach, towards alternative methods for modelling real-world networks. The most prominent of these is perhaps the \emph{preferential attachment} (PA) mechanism introduced by Barab\'asi and Albert in \cite{BA}. Another notable class of models derive from the \emph{web-copying} mechanism introduced in \cite{AM}.
In PA models, a new node is introduced at each time step, and then connected to each pre-existing node with a probability depending on the current degree of that node, according to a \emph{rich-get-richer} paradigm. PA processes can exhibit several properties observed in real-world networks (but absent in Erd\H{o}s-R\'enyi graphs), notably \emph{scale-freeness} meaning that the proportion of nodes of degree $k$ is asymptotically proportional to $k^{-\gamma}$ for some fixed $\gamma$ and all $k$.
What can we say of the infinite limits of these processes? Results of Bonato and Janssen \cite{BJ} have made significant progress for web-copying models. Less work has been done in the case of PA processes. The work of Oliveira and Spencer \cite{OS} studying the \emph{Growing Network} model of Krapivsky and Redner \cite{KR} and of Drinea, Enachescu, and Mitzenmacher \cite{DEM} is a notable exception. Our entry point however is the paper of Kleinberg and Kleinberg \cite{KK}, where a PA process is studied in which a single node and a constant number $C$ of edges are added at each time-step, with each new edge starting at the new node and with each end-point independently chosen among the pre-existing nodes, with probability proportional to their degree. Thus these structures are analysed as \emph{directed multigraphs}, in that each edge has a direction, and there may exist two or more edges sharing the same start and end-points. Loops (edges from a node to itself) are not permitted, however.
Kleinberg and Kleinberg show that in each of the cases $C=1$ and $C=2$, there is a unique infinite limiting structure, which the process approaches with probability $1$. It is also shown that the analogous result fails for $C \geq 3$: given two instantiations of the process, there is a positive probability that their infinite limits will fail to be isomorphic.
In this paper we extend the results and methods of \cite{KK}, by considering a process which adds $f(t)$ many edges at stage $t$ for some function $f:\textbf{N} \to \textbf{N}$. Again the start-point of every edge is the new node, and the end-points are chosen independently with probability proportional to the nodes' degrees. It follows easily from the results of \cite{KK} that whenever $f$ is non-constant, or constant with value $\geq 3$, there is a positive probability that the infinite limiting structures of two instantiations will be non-isomorphic as directed multigraphs. However, by forgetting the directions of edges, and looking for isomorphisms as multigraphs, we are able to recover a new categoricity result. In Theorem \ref{thm:mainmulti} we rigorously establish a sufficient criterion for the resulting structure to be isomorphic to the \emph{Rado multigraph} with probability 1. (This structure is the natural multigraph analogue of the Rado graph, and is defined in Definition \ref{defin:axioms} below.) Our criterion is that $f$ is asymptotically bounded above and below by positive non-constant linear functions of $t$.
In \cite{Elw}, the author uses similar machinery to analyse a Preferential Attachment process in which parallel edges are not permitted, and the new node $t+1$ is connected to each pre-existing node $u$ independently with probability $\frac{d_u(t)}{t}$. Thus the number of new edges is not prescribed, but is itself a random variable. It is shown in \cite{Elw} that, so long as the initial graph is neither edgeless nor complete, with probability 1 the infinite limit of the process will be the Rado graph augmented with a finite number of either universal or isolated nodes.
\section{The Rado Multigraph} \label{section:axioms}
We begin by defining the infinite structure which, we shall argue, our processes approach. So far as we are aware, this structure has not previously appeared in the literature. However the reader familiar with the Rado graph will find little of surprise. We choose to express ourselves using logical notation, however the reader unfamiliar with this formality should not be put off, and should bear in mind that the intended interpretation of the expression $E_j(u,v)$ is the assertion that there are at least $j$ edges connecting the nodes $u$ and $v$. (Informal interpretations of the axioms follow below.)
\begin{defin} \label{defin:axioms}
Let $\mathcal{L}$ be the following language for undirected multigraphs: $\langle E_j : j \in {\bf N} \rangle$ where each $E_j$ is a 2-place relation symbol. The Rado Multigraph is the (unique up to isomorphism by Proposition \ref{prop:multiprop} below) countably infinite model of the following axioms:
\begin{enumerate}
\item[\emph{(A0)}] $\forall x,y \left( x \neq y \rightarrow E_0(x,y) \right).$
\item[\emph{(A1)}] Each $E_j$ is a symmetric irreflexive binary relation.
\item[\emph{(A2)}] For each $j$ we have the axiom: $\forall u,v \ E_{j+1}(u,v) \Rightarrow E_{j}(u,v)$
\item[\emph{(A3)}] For all $u,v$ there is some $i$ such that $\neg E_{i}(u,v)$.
\item[\emph{(A4)}] For any $m_1, \ldots, m_n$ we have the following axiom:
$$\forall u_1, \ldots, u_n \ \bigwedge_{i \neq j} u_i \neq u_j \rightarrow \ \exists v \ \bigwedge_{i=1}^n E_{m_i}(u_i,v)\ \& \ \neg E_{m_i +1}(u_i,v).$$
\end{enumerate}
\end{defin}
Axioms (A0) - (A2) describe any loopless multigraph. (Of course, (A0) has very little content. The symbol $E_0$ is purely a convenience to avoid having to treat $0$ as a special case in (A4).) Axiom (A3) establishes that the multigraph is \emph{finitary}, in that no pair of nodes may be joined by infinitely many edges. It is (A4) which gives the structure its universal properties, stating that for any $n \in {\bf N}$, any distinct nodes $u_1, \ldots, u_n$ and any $m_1, \ldots, m_n \in {\bf N}$ we can find some node $v$ connected to each $u_i$ with exactly $m_i$ edges. (Notice that this includes the possibility that $m_i=0$ for some $i$.)
Notice, in logical terms, that axiom (A0) is first order, while (A1), (A2), and (A4) are first order schema, and (A3) is $\mathcal{L}_{\omega_1, \omega}$, or equivalently is defined by the omitting of the first order 2-type $\bigwedge_{i=1}^{\infty} E_i(u,v)$.
Several familiar properties of the Rado graph also hold for our multigraph analogue:
\begin{prop} \label{prop:multiprop}
Let $\mathcal{M}$ and $\mathcal{M}'$ be countably infinite models of \emph{(A0)-(A4)}. Then $\mathcal{M}$ satisfies the following properties:
\begin{enumerate}
\item $\aleph_0$\emph{-categoricity:} $\mathcal{M} \cong \mathcal{M}'$.
\item \emph{1-transitivity:} Given elements $v_1, v_2$ from $\mathcal{M}$ there exists some $\alpha \in \mathrm{Aut}(\mathcal{M})$ where $\alpha(v_1)=v_2$.
\item \emph{Ultrahomogeneity:} If $A,B$ are finite substructures of $\mathcal{M}$ where $A \cong B$ there exists $\alpha \in \mathrm{Aut}(\mathcal{M})$ where $\alpha(A)=B$.
\noindent (Note: here we treat $A$ and $B$ as \emph{induced substructures}: for any $j \in {\bf N}$ and $u,v$ elements from $A$ it holds that $A \models E_j(u,v) \Leftrightarrow \mathcal{M} \models E_j(u,v) $, and similarly for $B$.)
\item \emph{Universality:} Any finite or countably infinite finitary loopless multigraph can be isomorphically embedded in $\mathcal{M}$.
\end{enumerate}
\end{prop}
\begin{proof}
We concentrate on proving statement 1. (Statements 2-4 follow from minor alterations to our argument. We leave the reader to fill in the details.) We proceed by a standard back-and-forth argument. First we list the elements of $\mathcal{M}$ as $a_1,a_2,a_3,\ldots$ and similarly $b_1, b_2, b_3, \ldots$ for $\mathcal{M}'$. Now we argue inductively. Suppose $i$ is even, and suppose $(a'_1, \ldots, a'_i ) \cong (b'_1, \ldots, b'_i)$ have been chosen. Let $a'_{i+1}=a_j$ where $j$ is minimum such that $a_j \not\in \{a'_1, \ldots, a'_i \}$.
Let $(m_1, \ldots, m_i)$ be the vector counting the edges between $a'_{i+1}$ and $(a'_1, \ldots, a'_i )$. Notice that each $m_j \in {\bf N}$ by (A3). Then by (A4) there exists $b'_{i+1}$ joined to $(b'_1, \ldots, b'_i)$ in a fashion described by $(m_1, \ldots, m_i)$. Hence $(a'_1, \ldots, a'_{i+1} ) \cong (b'_1, \ldots, b'_{i+1})$.
Odd steps are identical, exchanging the roles of $\mathcal{M}$ and $\mathcal{M}'$. Thus we build an isomorphism $\mathcal{M} \cong \mathcal{M}'$.\end{proof}
Our concern in the current work is on PA processes. However, we remark in passing that the Rado multigraph also arises from the following Erd\H{o}s-R\'enyi-style process. We leave the proof as an easy exercise.
\begin{prop}
Let $(p_n)_{n \geq 1}$ be any sequence lying entirely in $(0,1)$. Let $V$ be a countably infinite set. Let $\mathcal{M}$ be an $\mathcal{L}$-structure arising from the following random process.
For any distinct $v_1,v_2 \in V$ enforce the following, independently of the behaviour of all other nodes:
\begin{itemize}
\item $E_0(v_1,v_2)$
\item For all $i \geq 0$ \ \ ${\bf P} \left(E_{i+1}(v_1,v_2) \ \big| \big| \ E_{i}(v_1,v_2) \right) = p_i$.
\end{itemize}
Then with probability $1$, $\mathcal{M}$ is isomorphic to the Rado multigraph.
\end{prop}
\section{Preferential Attachment with Prescribed Edge Growth} \label{section:PA}
In this section we shall describe two variants of the preferential attachment process, establish some of their basic properties, and formally state our main result. Our two processes are $\textrm{GPA}_f$, which builds a directed graph, and $\textrm{MPA}_f$ which builds a directed multigraph. Each proceeds by adding, at each time step, a single node along with a prescribed number of directed edges emanating from it. The number of these edges is determined by some fixed function $f:\textbf{N} \to \textbf{N}$. (In fact the directions of the edges will play no role in the theory: we shall analyse the resulting structures as undirected (multi)graphs. However in the interim it will be convenient to refer to the `start-' and `end-points' of each edge, so we preserve directedness for the time being.) We shall work over some initial directed (multi)graph $G'$ containing no isolated nodes (i.e. nodes of degree $0$). (However our results will be independent of the choice of $G'$, so the reader may choose to focus on the case where $G'$ is trivial.)
\begin{defin} \label{defin:MPA}
Let $G'= (V',E')$ be a finite directed graph (in $\textrm{GPA}_f$) or directed multigraph (in $\textrm{MPA}_f$) containing no isolated nodes.
Suppose that $G'$ contains $|E'|=e'$ edges (so, in the multigraph case, $E'$ will be a multiset) and $|V'|=v'$ nodes. We will assume $V'=\{1,\ldots, v'\}$.
Suppose that the function $f:\textbf{N} \to \textbf{N}$ satisfies:
\begin{itemize}
\item $f(0)=e'$.
\item $f(t)=0$ whenever $1 \leq t \leq v'-1$.
\item $f(t) \geq 1$ for all $t \geq v'$.
\end{itemize}
At each time-step $t \geq 1$, we shall construct a (multi)graph $G(t)$ with vertex set $V(t)$ and edge (multi)set $E(t)$.
First we impose $G(1) = \ldots = G(v') = G'$.
Whenever $t \geq v'$, we will have $V(t):=\{1,\ldots,t\}$ and
$$E(t+1) =E(t) \cup \mathcal{E}(t+1)$$ where $|\mathcal{E} (t+1)| = f(t)$.
The start-point of each edge in $\mathcal{E} (t+1)$ is the new node $t+1$. The end-points are chosen independently from $V(t)$ with probabilities directly proportional to their degrees in $G(t)$. In $\textrm{GPA}_f$ these end-points are selected without replacement, while in $\textrm{MPA}_f$ they are selected with replacement.
\end{defin}
Notice that, in ${\textrm{GPA}_f}$, the procedure is only viable if $f(t) \leq t$ for all $t \geq v'$. Notice too that our assumption that $f(t) \neq 0$ for $t \geq v'$ (along with our assumption on $G'$) serves to ensure that there are never any isolated nodes.
We may now state our main result. (Recall the asymptotic notation $g_1 = \Theta (g_2)$ for functions $g_1, g_2$ as meaning that there exist $c_2 \geq c_1>0$ so that for all large enough $t$ we have $c_1 \cdot g_2(t) \leq g_1(t) \leq c_2 \cdot g_2(t)$.)
\begin{thm} \label{thm:mainmulti}
Suppose that $f(t) = \Theta(t)$, and $G'$ is a finite directed multigraph containing no isolated nodes. Then, with probability 1, the infinite limit of ${\textrm{MPA}_f}(G')$ is the Rado multigraph.
\end{thm}
We conjecture that this result passes over to graphs:
\begin{conj} \label{conj:mainconj}
Suppose that $f(t) \leq t$ for all $t$ and that there are constants $0 < c_1 \leq c_2 <1$ where $c_1 \cdot t \leq f(t) \leq c_2 \cdot t$ for all large enough $t$. Suppose also that $G'$ is a finite directed graph containing no isolated nodes. Then, with probability 1, the infinite limit of ${\textrm{GPA}_f}(G')$ is the Rado graph.
\end{conj}
Our arguments will be independent of $G'$, and thus we shall largely suppress mention of it. Let us now consider the distribution of edges at stage $t+1$. First notice that $|E (t)| = F(t):=\sum_{i=0}^{t-1} f(i)$. Hence in $\textrm{MPA}_f$, at stage $t+1$ given any pre-existing node $u \leq t$, the probability that any given edge in $\mathcal{E} (t+1)$ has its end-point at $u$ is exactly $\frac{d_u(t)}{2F(t)}$, where $d_u(t)$ is the degree of $u$ in $G(t)$. Thus the expected number of edges in $\mathcal{E}(t+1)$ with endpoint at $u$ is $\frac{f(t) \cdot d_u(t)}{2F(t)}$.
In $\textrm{GPA}_f$ this probability distribution is more complicated, and the expected number of edges $u$ receives at stage $t+1$ depends in a more detailed way upon $G(t)$. This is the primary obstacle to extending the current work to a proof of Conjecture \ref{conj:mainconj}.
Our standing assumption will be that we are working in ${\textrm{MPA}_f}$. We shall leave the case of ${\textrm{GPA}_f}$ open, but make some remarks about it as we proceed.
Our assumption in Theorem \ref{thm:mainmulti} is that $f(t)= \Theta(t)$. However we shall be able to develop much of the theory under the following weaker hypotheses:
\begin{assumption} \label{assumption:ass}
\begin{equation} \sum_0^\infty \frac{f(s)}{F(s)} = \infty. \end{equation}
\begin{equation} \sum_0^\infty \left( \frac{f(s)}{F(s)} \right)^2 < \infty.\end{equation}
\end{assumption}
We briefly discuss this. Assumption \ref{assumption:ass} easily follows in full, for instance, if $f(t)= \Theta(t^{\alpha})$ for some $\alpha \geq 0$.
However part (2) fails in general for polynomially bounded functions, an example being:
$$f(t) = \left\{ \begin{array}{ll} t & \textrm{ when } t=2^n \textrm{ for } n \in {\bf N}\\
1 & \textrm{ otherwise.} \end{array} \right. $$
On the other hand, both parts do hold for some exponential functions, such as $f(t)=\lfloor \frac{1}{4} t^{- \frac{3}{4}} e^{\frac{1}{4}t} \rfloor$.
One might attempt to characterise the relevant classes of functions as follows: let $(\xi_n)$ be a sequence of positive rational numbers satisfying
\begin{enumerate}
\item[($1^\prime$)] $\sum_n \xi_n = \infty$ and/or
\item[($2^\prime$)] $\sum_n \xi_n^2 < \infty$.
\end{enumerate}
\noindent Now defining $\displaystyle f(t):=\xi_0 \cdot \prod_{n=1}^{t-1}(\xi_n +1)\cdot \xi_t$, we additionally require
\begin{enumerate}
\item[($3^\prime$)] $f(t)$ is a positive integer for all $t$.
\end{enumerate}
Then one can show that $F(t) = \xi_0 \cdot \prod_{n=1}^{t-1}(\xi_n +1)$ meaning that (1) and (2) follow immediately from ($1^\prime$) and ($2^\prime$) respectively. Furthermore, all $f$ satisfying Assumption \ref{assumption:ass} may be built in this fashion. On the other hand, condition ($3^\prime$) is far from user-friendly.\\
In all cases, it will be useful to extend the domain of $f$ to ${\bf R}^{\geq 0}$. We choose to do this as a step function, via $f(t):=f \left( \lfloor t \rfloor \right)$. (Of course there may be more natural ways to achieve the same thing, however this choice will be convenient, as the fourth point in the following Lemma makes clear.) We now gather together some observations about the extended function $f$. These follow immediately from our previous conditions.
\begin{lem}
\hspace{1cm}
\begin{itemize}
\item $f(t)=e'$ for $0 \leq t <1$.
\item $f(t)=0$ whenever $1 \leq t < v'$.
\item $f(t) \geq 1$ for all $t \geq v'$.
\item $f$ is Lebesgue-measurable with antiderivative $\displaystyle \int_{0}^{t} f(s) ds =:F(t)$. (This notation is consistent with the previous interpretation of $F$ since the two functions coincide at integer points.)
\item $F$ is monotonic increasing everywhere and strictly so for $t \geq v'$.
\end{itemize}
\end{lem}
Under our additional hypothesis we can say a little more:
\begin{lem} \label{lem:integral}
Suppose that Assumption \ref{assumption:ass}(2) holds. Then for any $\beta \geq 1$, there exists $K_\beta >0$ so that for any $t \geq m \geq 0$:
$$\left| \int_m^t \frac{f(s)}{F(s)^\beta} ds - \sum_{s=m}^{t} \frac{f(s)}{F(s)^\beta} \right| < K_\beta.$$
\end{lem}
\begin{proof}
Let $M$ be such that whenever $s \geq M$ then $f(s) < F(s)$. Such a value must exist by Assumption \ref{assumption:ass}(2).
It is enough to prove the result this for all $m \geq M$, since one can then add $\displaystyle \max \left\{ \int_0^M \frac{f(s)}{F(s)^\beta} ds,\ \sum_{s=0}^{M} \frac{f(s)}{F(s)^\beta} \right\}$ to $K_{\beta}$ to obtain the result for all $m$. Thus we shall assume $m \geq M$.
Firstly, it is immediate by consideration of $F\upharpoonright_{[s,s+1]}$ that
$$\sum_{s=m}^{t} \frac{f(s)}{F(s+1)^\beta} < \int_m^t \frac{f(s)}{F(s)^\beta} ds < \sum_{s=m}^{t} \frac{f(s)}{F(s)^\beta}.$$
Next we shall appeal to Newton's generalised binomial theorem, that whenever $a,b,\beta \in \bf{C}$ with $0<|b|<|a|$, then $\displaystyle (a+b)^{\beta} = \sum_{j=0}^{\infty} C(\beta, j) a^{\beta - j} b^j,$ where $C(\beta,j)$ are the generalised binomial coefficients.
When $a=1$, the series has radius of convergence $1$ in $b$. We shall also use the fact that the series remains convergent for $|b|=1$, so long as $\textrm{Re}(\beta)>0$, which of course holds in the context of this Lemma. (See \cite{CKP} p.17, for example.) Now,
\begin{align*}
\sum_{s=m}^{t} \frac{f(s)}{F(s)^\beta} &- \sum_{s=m}^{t} \frac{f(s)}{F(s+1)^\beta} =
\sum_{s=m}^{t} \frac{f(s)}{F(s)^\beta} - \frac{f(s)}{\left(F(s)+ f(s)\right)^\beta} \\
& < \sum_{s=m}^{t} \frac{f(s)\left(F(s)+ f(s)\right)^\beta - f(s) F(s)^\beta}{F(s)^{2\beta}}\\
& = \sum_{s=m}^{t} \frac{f(s)\left(\sum_{j=1}^{\infty} C(\beta,j) F(s)^{\beta - j}f(s)^j \right)}{F(s)^{2\beta}}\\
& < \sum_{s=m}^{t} \frac{\sum_{j=1}^{\infty} C(\beta,j) F(s)^{\beta - 1}f(s)^2}{F(s)^{2 \beta}}\\
& < \sum_{s=m}^{t} \frac{2^\beta f(s)^2}{F(s)^{1 + \beta}} \ \leq \ 2^\beta \sum_{s=m}^{t} \frac{ f(s)^2}{F(s)^{2}} < 2^\beta \cdot K := K_\beta
\end{align*}
where $K$ is the finite bound provided in Assumption \ref{assumption:ass} (2).
\end{proof}
The next two results hold in ${\textrm{GPA}_f}$ as well as ${\textrm{MPA}_f}$:
\begin{lem} \label{lem:nonzero}
Suppose that Assumption \ref{assumption:ass}(1) holds. Then for any node $u$, any stage $t_0$, and any state of the graph $G(t_0)$, the probability that $v$ never receives another edge is $0$.
\end{lem}
\begin{proof}
Suppose that $d_u(t_0)=N \geq 1$. The probability that $u$ never receives a further edge is therefore given by (or in ${\textrm{GPA}_f}$ is bounded above by) $$\prod_{t=t_0}^{\infty} \left(1 - \frac{N}{2F(t)} \right)^{f(t)}.$$
We shall show that this is $0$. It is clearly enough to do so in the case $N=1$. Taking logarithms, it is therefore enough to show that
$$\sum_{t=t_0}^{\infty} f(t) \ln \left(1 + \frac{1}{2F(t) - 1} \right)$$ diverges to $\infty$.
Now as for small enough $x$, we know $\ln(1+x)> \frac{1}{2}x$. Thus for large enough $t$,
$$\ln \left(1 + \frac{1}{2F(t) - 1} \right) > \frac{1}{4 F(t)}.$$
Thus the result follows from Assumption \ref{assumption:ass}(1).
\end{proof}
\begin{coro} \label{coro:dinf}
Suppose that Assumption \ref{assumption:ass}(1) holds. Then for any node $u$, given any state of the graph $G(t_0)$, with probability $1$ it will be true that $d(t) \to \infty$ as $t \to \infty$.
\end{coro}
\begin{proof}
This follows automatically from Lemma \ref{lem:nonzero} by the countable additivity of the probability measure.
\end{proof}
\section{Martingale Theory}
In this section, we apply some machinery from the theory of Martingales to the process ${\textrm{MPA}_f}$, generalising the theory developed in \cite{KK}. We shall assume throughout that we are working in ${\textrm{MPA}_f}$, and begin with the following easy result, which does not transfer immediately to ${\textrm{GPA}_f}$.
\begin{rem} \label{rem:probbound}
Given any node $u$, define $U_u(t+1):= d_u(t+1) - d_u(t)$ and
$\mu_u(t):= {\bf E} \left( U_u(t+1) \big| \big| d_u(t) \right).$ Then $$\frac{\mu_u(t)}{d_u(t)} = \frac{f(t)}{2F(t)}.$$
In particular, if $f(t) = \Theta \left( t^{\alpha} \right)$ where $\alpha \geq 0 $ then $\displaystyle \mu_u(t) = \Theta \left( \frac{d_u(t)}{t} \right).$
\end{rem}
The next two results are the key to our analysis, and generalise Proposition 3.1 of \cite{KK}:
\begin{prop} \label{prop:xuexists}
Suppose that Assumption \ref{assumption:ass} (2) holds. For any node $u$, define $$A(t) = A_u(t):=\prod_{j=1}^{t-1} \left( 1 + \frac{f(j)}{2F(j)}\right)$$ and
$X(t):= X_u(t) = \frac{d_u(t)}{A_u(t)}$. Then
\begin{enumerate}[(i)]
\item $X(t)$ is a martingale.
\item $A(t) = \Theta \left( F(t)^{\frac{1}{2}} \right)$
\item Thus, for any node $u$, there exists $x_u \geq 0$ where $d_u(t) \sim x_u \cdot F(t)^{\frac{1}{2}}$.
\end{enumerate}
\end{prop}
\begin{proof}
Employing Remark \ref{rem:probbound}, the first part is straightforward:
\begin{eqnarray*} {\bf E} (X(t+1) || X(t)) &=& \frac{1}{A(t+1)} {\bf E} (d(t+1)||d(t))\\
&=& \frac{1}{A(t+1)}\left( d(t) +\mu(t) \right)\\
&=& \frac{1}{A(t+1)}d(t) \left( 1 +\frac{\mu(t)}{d(t)} \right)\\
&=& \frac{1}{A(t)} d(t) = X(t) .\end{eqnarray*}
Part (iii) will follow from (i) and (ii) via Doob's convergence theorem, which gives us that $X(t) \to X$ for some random variable $X$. Thus a finite limit $\lim_{t \to\infty} \frac{d_u(t)}{A(t)}$ will exist.
Hence all that remains is to understand the behaviour of $A(t)$ for large $t$. By taking logarithms and employing the standard bounds $x- \frac{1}{2}x^2 < \ln(1+x)<x$, we see:
$$\frac{1}{2} \sum_{s=1}^{t-1} \frac{f(s)}{F(s)} - \frac{1}{8} \sum_{s=1}^{t-1} \frac{f(s)^2}{F(s)^2} < \ln A(t)< \frac{1}{2} \sum_{s=1}^{t-1} \frac{f(s)}{F(s)}.$$
Therefore by Assumption \ref{assumption:ass} and Lemma \ref{lem:integral}, it follows that
$$\frac{1}{2} \int_{s=1}^{t-1} \frac{f(s)}{F(s)}ds - K < \ln A(t)<\frac{1}{2} \int_{s=1}^{t-1} \frac{f(s)}{F(s)}ds + K'$$
for some constants $K$ and $K'$ from which the result follows.
\end{proof}
We need a little more information about the distribution of $x_u$:
\begin{prop} \label{prop:poslimit}
Suppose that $f$ satisfies Assumption \ref{assumption:ass} in full. Given any time $t_0$, state $G_0(t_0)$, and node $u$, ${\bf P} \left(x_u>0 \right) = 1$.
\end{prop}
\begin{proof}
Our proof closely follows that of Proposition 3.1 of \cite{KK}.
We take $u$ as fixed and shall suppress mention of it, writing $X(n)$ for $X_u(n)$, etc., throughout.
Given any $n>m>0$ define $\tilde{X}_{m}(n) := \left( X(n) - X(m) \right)^2$. Then for fixed $m$, it is an elementary fact that the sequence $\tilde{X}_{m}(n)$ forms a submartingale. We now proceed via a sequence of claims.\\
\noindent {\bf Claim 1} $${\bf E} \left( \tilde{X}_{m}(n) \big|\big| X(m) \right) = \sum_{t=m}^{n-1} {\bf E} \Big( X(t+1)^2 \big|\big| X(m) \Big) - {\bf E} \Big( X(t)^2 \big|\big| X(m) \Big).$$
\noindent {\bf Proof of Claim 1 \ \ }
\begin{eqnarray*}
{\bf E} \Big( \tilde{X}_{m}(n) & \big|\big| & X(m) \Big) \\
&=& {\bf E} \Big( X(n)^2 - 2X(n) X(m) + X(m)^2 \big|\big| X(m) \Big)\\
& = & {\bf E} \Big( X(n)^2 \big|\big| X(m) \Big) - 2 X(m) {\bf E} \Big( X(n) \big|\big| X(m) \Big) + X(m)^2\\
& = & {\bf E} \Big( X(n)^2 \big|\big| X(m)\Big) - X(m)^2.
\end{eqnarray*}
Unpacking the sum in the statement of the claim gives the same result. \textbf{QED Claim 1}\\
\noindent {\bf Claim 2 \ \ } There exists $K>0$ such that for all large enough $m$ and all $n>m$ $${\bf E} (\tilde{X}_{m}(n) ||X(m)) < X(m) \cdot \frac{K}{F(m)^{\frac{1}{2}}}.$$
\noindent {\bf Proof of Claim 2 \ \ } Recall $U(t+1) := d(t+1) - d(t)$. Now $U(t+1)$ is binomially distributed via $b \left(f(t), \frac{d(t)}{2F(t)} \right)$ meaning, as already observed, that ${\bf E} (U(t+1) || d(t) = d) = \mu(t) = \frac{d \cdot f(t)}{2F(t)}$ and also ${\bf V}(U(t+1)||d(t)=d) = \frac{d \cdot f(t)}{2F(t)}\left( 1- \frac{d}{2F(t)}\right)$. Thus, writing $f$ and $F$ for $f(t)$ and $F(t)$ respectively,
\begin{eqnarray*} \label{eqnarray:Z2}
{\bf E} \Big( U(t+1)^2 ||d(t)=d \Big) & = & \left(\frac{df}{2F} \right)^2 + \frac{df}{2F}\left(\frac{2F-d}{2F} \right)\\
& < & \frac{fd}{2F} + \frac{f^2d^2}{4F^2}.
\end{eqnarray*}
At the same time,
\begin{align*}
{\bf E} & \Big( d(t+1)^2 || d(t)= d \Big) \\
& = \ {\bf E} \Big( \left(U(t+1) +d \right)^2||d(t)=d \Big)\\
& = \ {\bf E} (U(t+1)^2||d(t)=d) + 2d {\bf E} (U(t+1) || d(t)=d) + d^2\\
& < \ \frac{fd}{2F} + \frac{f^2d^2}{4F^2} +2d \cdot \frac{df}{2F} + d^2\\
& = \ \frac{fd}{2F} + \left(1+ \frac{f}{2F} \right)^2 d^2.
\end{align*}
Recall the definition of the martingale $X(t):= \frac{d(t)}{A(t)}$. Thus
\begin{eqnarray*}
{\bf E} \Big( X(t+1)^2 & \ \Big| \Big| & d(t)=d \Big) = \left( \frac{1}{A(t+1)^2} \right) \cdot {\bf E} \Big( d(t+1)^2\ \Big| \Big | \ d(t)=d \Big)\\
& < & \frac{1}{A(t+1)^2} \left(\frac{fd}{2F} + \left(1+ \frac{f}{2F} \right)^2 d^2 \right)\\
& = & \frac{f A(t)}{2F \cdot A(t+1)^2} X(t) + \left(1+ \frac{f}{2F} \right)^2 \left( \frac{A(t)}{A(t+1)}\right)^2 X(t)^2\\
& < & \frac{f}{2F \cdot A(t)} X(t) + X(t)^2.
\end{eqnarray*}
Hence, by the law of total expectation,
$${\bf E} (X(t+1)^2|| X(m)) - {\bf E} (X(t)^2||X(m)) < \frac{f}{2F \cdot A(t)} X(m).$$
Summing this up over successive terms (and appealing to Claim 1, Proposition \ref{prop:xuexists} (ii) and Lemma \ref{lem:integral}) we get
\begin{eqnarray*}
{\bf E} (\tilde{X}_{m}(n) &||& X(m)) < X(m) \cdot \sum_{t=m}^{n-1} \frac{f}{2F A(t)}\\
& < & X(m) \cdot \sum_{t=m}^{n-1} \frac{f}{2F A(t)}\\
& = & O \left( X(m) \cdot \sum_{t=m}^{n-1} \frac{f}{F^{\frac{3}{2}}} \right)\\
& = & O \left( X(m) \cdot \int_{t=m}^{n-1} \frac{f(t)}{F(t)^{\frac{3}{2}}} dt \right)\\
& < & X(m) \cdot \frac{K}{F(m)^{\frac{1}{2}}} \textrm{ for some $K>0$. \textbf{QED Claim 2.} }\\
\end{eqnarray*}
We may now prove the proposition. We proceed by defining a sequence of times: $n_0 = t_0$. Let $n_{i+1}$ be the least $n$ (if any exists) such that $X(n) < \frac{1}{2}X \left(n_{i} \right)$. Otherwise $n_{i+1} = \infty$.
The trick is to apply the Kolmogorov-Doob inequality (see for instance \cite{KK}) to $\tilde{X}_{n_i}(n)$:
\begin{eqnarray*}
{\bf P} ( n_{i+1} < \infty ||n_{i} < \infty) & = & {\bf P} \left( \min_{n \geq n_i} X(n) < \frac{1}{2} X(n_i) {\Big|\Big|} X(n_i)\right)\\
& \leq & {\bf P} \left( \max_{n \geq n_i} \tilde{X}_{n_i}(n) > \frac{1}{4} X(n_i)^2 {\Big|\Big|} X(n_i)\right) \\
& = & \lim_{N\to \infty} {\bf P} \left( \max_{n: N \geq n \geq n_i} \tilde{X}_{n_i}(n) > \frac{1}{4} X(n_i)^2 {\Big|\Big|} X(n_i)\right) \\
& \leq & \frac{4}{X(n_i)^2} \cdot \lim_{N \to \infty} {\bf E} (\tilde{X}_{n_i}(N) || X(n_i))\\
& = & O \left( \frac{4}{X(n_i)^2} \cdot \frac{1}{F(n_i)^{\frac{1}{2}}} \cdot X(n_i) \right) \\
& = & O \left( \frac{1}{d(n_i)} \right).
\end{eqnarray*}
It follows from Corollary \ref{coro:dinf} that ${\bf P} ( n_{i+1} < \infty ||n_{i} < \infty )\to 0$ as $i \to \infty$, from which the result follows. \end{proof}
We record one more result regarding the martingale $X(t)$:
\begin{coro} \label{coro:L2}
Suppose again that Assumption \ref{assumption:ass} (2) holds. Then the martingale $X(t)$ is bounded in $\mathcal{L}_2$.
\end{coro}
\begin{proof}
Notice that by Remark \ref{rem:probbound}
\begin{align*}
|X(t+1) - X(t)| & = \frac{d(t+1) - \left(1 + \frac{\mu(t)}{d(t)} \right)d(t)}{A(t+1)}\\
& = \frac{U(t+1)- \mu(t)}{A(t+1)}
\end{align*}
Hence
\begin{align*}
{\bf E} \left(|X(t+1) - X(t)|^2 \right) & = \frac{{\bf V}(U(t+1) )}{A(t+1)^2}\\
& = O \left( \frac{d(t) f(t)}{2F(t)}\left( 1- \frac{d(t)}{2F(t)}\right) \cdot \frac{1}{A(t+1)^2} \right)\\
& = O \left( \frac{d(t)}{A(t+1)} \cdot \frac{f(t)}{F(t) A(t+1)} \right)\\
& = O \left( X(t) \cdot \frac{f(t)}{F(t) A(t+1)} \right)\\
& = O \left( \frac{f(t)}{F(t) A(t+1)} \right)\\
& =O \left( \frac{f(t)}{F(t)^{\frac{3}{2}}} \right).
\end{align*}
Thus \begin{align*}
\sum_{j=0}^{t} {\bf E} \left(|X_{j+1} - X_j|^2 \right) & = O \left( \int_{j=0}^t \frac{f(j)}{F(j)^{\frac{3}{2}}} dj \right)\\
& = O \left( K - F(t)^{-\frac{1}{2}} \right) =O(K).
\end{align*}
\end{proof}
\section{Proof of Main result}
\begin{defin}
A \emph{witness request} $W=\{(u_i,m_i) \ | \ 1 \leq i \leq n \}$ is a vector of nodes $(u_1, \ldots, u_n)$ and accompanying vector of non-negative integers $(m_1, \ldots, m_n)$.
A \emph{witness} for $W$ is a node connected to each $u_i$ with multiplicity $m_i$.
We write the event $W[t]$ to mean that $W$ is satisfied by some witness by time $t$.
\end{defin}
Observe from the structure of the process that $W[t] \Rightarrow W[t']$ for all $t' \geq t$. The following is the major step towards our goal:
\begin{prop} \label{prop:mainstep1}
Suppose that $f(t) = \Theta(t)$, and that $G(t_0)$ is a state of the graph at time $t_0$. Let $W$ be a witness request. Let $\varepsilon>0$. Then there exist $t_1 > t_0$ such that ${\bf P} \Big(W[t_1] \ \big| \big| \ G(t_0) \Big) >1 - \varepsilon$.
\end{prop}
\begin{proof}
We consider only stages from $t_0+1$ onwards, and everything that occurs is conditioned upon $G(t_0)$, which we shall therefore suppress.
Suppose $W=\{(u_1, \ldots, u_n),(m_1, \ldots, m_n) \}$. We shall write $m=\sum_{i=1}^n m_i$, and, abusing notation, $U_i=U_{u_i}(t+1)$, meaning the number of new edges which $u_i$ gains at the $t+1$st stage, taking the dependency on $t$ as given when the intended value is obvious. Similarly we write $d_i$ for $d_{u_i}(t)$.
We shall employ vector notation, writing $\textbf{U}(t+1):=\textbf{U}=(U_1,\ldots,U_n)$ and $\textbf{m}:=(m_1,\ldots,m_n)$. Thus our focus is the event $\textbf{U}=\textbf{m}$. Let us first compute the probability of this event in terms of the $d_i$. The relevant distribution is multinomial $M(f,p_i, \ldots, p_n,q)$ where $p_i = \frac{d_i}{2F}$ and $q = 1 - \sum p_i$ (again omitting the dependencies on $t$). Therefore
\begin{align*}
{\bf P} &\left(\textbf{U}=\textbf{m} \right) \\
& = \frac{f!}{m_1!\cdot \ldots m_n! \cdot (f - m)!} \cdot q^{f-m} \cdot \prod_{i} {p_i}^{m_i}\\
&= \Theta \left( \left( 1-\frac{\sum_i d_i}{2F} \right)^{f-m} \cdot \left( \frac{f}{2F} \right)^m \cdot \prod_{i} d_i^{\ m_i} \right)
\end{align*}
noticing that $\frac{f!}{(f - m)!} \sim f^{m}$.
Now we employ our assumption that $f(t) = \Theta (t)$ from which it also follows that $\frac{1}{2F} = \Theta\left( \frac{1}{t^2} \right) $ and $\frac{f}{2F} = \Theta\left( \frac{1}{t} \right) $. Thus there exist constants $c_1, c_2, C_0, N>0$ depending only on $G_0(t_0)$ such that for all $t \geq N$,
\begin{equation} \label{equation:firstbound}{\bf P} \left(\textbf{U}=\textbf{m} \right) \geq C_0 \cdot \left( 1-\frac{\sum_i d_i}{c_1 t^2} \right)^{c_2 t-m} \cdot t^{- m} \cdot \prod_{i} d_i^{\ m_i}. \end{equation}
Our aim is to bound this probability below, away from $0$ over a long enough range of $t$. We write $X_i = \frac{d_i}{A_i}$ for the Martingale supplied by Proposition \ref{prop:xuexists}, with $x_i:=x_{u_i}>0$ for its limit supplied by Proposition \ref{prop:xuexists} and Lemma \ref{prop:poslimit}. We will not attempt to condition on the actual values $x_i$, but only on the fact that these values are not extreme ($\textbf{NE}$).
First, choose $\kappa_1, \kappa_2 >0$ such that
\begin{equation*}
\kappa_1 t <A(t) < \kappa_2 t
\end{equation*}
for all large enough $t$. This is guaranteed to occur by Proposition \ref{prop:xuexists} (ii) since $F(t)^{\frac{1}{2}} = \Theta (t)$. We increase $N$ if necessary to ensure that this holds. Notice that since $A(t)$ is entirely predictable in advance, the value of $N$ remains dependent only on $G_0(t_0)$.
Now, for any $y_2>y_1>0$, define the following event:
$$\textbf{NE} (y_1,y_2): \ \ \ \left( \bigwedge_{i=1}^n y_1 < x_i < y_2 \right).$$
We shall apply this in the following case: given $\delta>0$ choose $y_2(\delta)> y_1(\delta)>0$ so that ${\bf P}(\neg \textbf{NE} (y_1,y_2)) < \delta$. (We shall specify $\delta$ later, and will only need to consider one such value. Thus we shall consider $\delta$ fixed for the purposes of what follows.)
By Corollary \ref{coro:L2}, $X_i(t) \to x_i$ in expectation, and thus in probability. More precisely, for any $\eta>0$, we may increase $N>0$ by some quantity depending only on $\eta$ so that for all $t \geq N$ and all $i \leq n$
$${\bf E} \Big( |X_i(t) - x_i| \Big) < \left(\frac{\eta}{n} \right)^2.$$
Thus, by Markov's inequality
$${\bf P}\left(|X_i(t) - x_i| > \frac{\eta}{n} \right) < \frac{\eta}{n}.$$
Hence defining the event that all the $X_i(t)$ are close ({\bf Cl}) to their respective $x_i$
\begin{equation*} \textbf{Cl}(t, \eta):= \ \ \bigwedge_{i=1}^n \left( |X_i(t) - x_i| < \frac{\eta}{n} \right)\end{equation*}
we have for all $t \geq N$
\begin{equation} \label{equation:zbound} {\bf P}\left(\textbf{Cl} (t, \eta) \right) >1- \eta.\end{equation}
Again, we shall pick a value of $\eta$ later. Notice also that
\begin{equation*}
{\bf P}\left(\textbf{Cl} (t, \eta) \right) < {\bf P}\left(\textbf{Cl} (t, \eta) \ \big| \big| \
\textbf{NE} (y_1,y_2) \right) + \delta.
\end{equation*}
So \begin{equation}
{\bf P}\left(\textbf{Cl} (t, \eta) \ \big| \big| \ \textbf{NE} (y_1,y_2) \right) > 1 - \eta - \delta.
\end{equation}
Next, we define a bound for $d_i(t)$. Given $\delta,\eta >0$ as before, let $b_1(\eta)=b_1(\delta,\eta) := \kappa_1 \cdot \left( y_1 - \frac{\eta}{n} \right)$ and $b_2(\eta)= b_2(\delta, \eta) := \kappa_2 \cdot \left( y_2 + \frac{\eta}{n} \right)$, insisting that $\eta$ is small enough that $b_1>0$. Then we define the event
$$\textbf{Bo}(t, b_1, b_2) := \ \ \ \bigwedge_{i=1}^n \left( b_1 \cdot t < d_i(t) < b_2 \cdot t \right).$$
Observe now that for $t \geq N$ \begin{equation} \label{equation:implic} \left( \textbf{NE} (y_1(\delta),y_2(\delta)) \ \& \ \textbf{Cl}(t, \eta) \right) \Rightarrow \textbf{Bo}(t, b_1(\eta), b_2(\eta)). \end{equation}
Hence
$${\bf P} \left( \textbf{Bo}(t, b_1, b_2) \Big|\Big| \textbf{NE} (y_1(\delta),y_2(\delta)) \right) \geq 1 - \eta - \delta.$$
Thus we obtain the unconditional bound:
\begin{equation} \label{equation:uncon}
{\bf P} \left( \textbf{Bo}(t, b_1, b_2) \right) \geq (1 - \eta - \delta)(1 - \delta).
\end{equation}
Now, we use the bound obtained in (\ref{equation:firstbound}) above, and see that whenever $b_1 \leq b_1' < b_2' \leq b_2$
\begin{eqnarray*}
{\bf P} \Big( \textbf{U}=\textbf{m} \ & \Big| \Big| & \ \textbf{Bo}(t,b_1',b_2') \Big)\\
& > & C_0 \cdot \left( 1-\frac{n \cdot b_2 \cdot t}{c_1 t^2} \right)^{c_2 t-m} \cdot t^{-m}\cdot \left(b_1\cdot t\right)^m\\
& = & C_0 \cdot b_1^m \cdot \left( 1-\frac{nb_2}{c_1 t} \right)^{c_2 t-m}\\
& = & C_0 \cdot b_1^m \cdot \left( 1-\frac{nb_2c_2}{c_1}\cdot \frac{1}{c_2 t} \right)^{c_2 t} \cdot \left( 1-\frac{nb_2}{c_1t} \right)^{-m}\\
& \to & C_0 \cdot b_1^m \cdot e^{-\frac{nb_2c_2}{c_1}}:=C_3>0.
\end{eqnarray*}
Hence, by letting $C_4 = C_4 (\delta, \eta):=\frac{1}{2}C_3$ and increasing $N$ again if necessary (and again by some predictable amount), we have for all $t \geq N$
\begin{equation} \label{equation:Ubound}
{\bf P}\left( \textbf{U}=\textbf{m} \ \Big| \Big| \ \textbf{Bo}(t,b_1',b_2') \right)>C_4.\end{equation}
Now for any $\zeta>0$, we may let $M=M(\zeta,\delta,\eta)$ be large enough that $\left(1 - C_4\right)^M < \zeta$. The goal therefore is to locate $M$ places where $\textbf{Bo}(t,b_1(\eta),b_2(\eta))$ holds, and argue that the probability that all of them fail to produce an instance of $\textbf{U}=\textbf{m}$ is bounded above by $\zeta$.
Notice that bound (\ref{equation:Ubound}) holds independently for all $t \geq N$: the arguments are unaffected by previous values of $\textbf{U}$ so long as $\textbf{Bo}(t, b_1', b_2')$ holds. However, the same is not true for bound (\ref{equation:uncon}). By conditioning on whether or not $\textbf{U}(t')=\textbf{m}$ holds, we risk affecting ${\bf P}\left(\textbf{Bo}(t, b_1(\eta), b_2(\eta))\right)$ for $t > t'$.
To navigate this obstacle, we shall locate a range $[t_2, t_2+M)$ within which the bound $\textbf{Bo}(t,b_1(\eta),b_2(\eta))$ is guaranteed to hold, barring a certain extreme event $\neg \textbf{Sh}$ defined below, which will have a probability bounded above by $\theta$ for arbitrarily small $\theta$.
We wish $t_2$ to satisfy the tighter bound $\textbf{Bo}(t_2,b_1 ( \tfrac{\eta}{2}),b_2 ( \tfrac{\eta}{2}))$. Notice that appropriate adaptations of (\ref{equation:zbound}), (\ref{equation:implic}), and (\ref{equation:uncon}) above guarantee that for large enough $t_2$,
\begin{equation} \label{equation:b2bound}
{\bf P} \left(\neg \textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right),b_2 \! \left( \tfrac{\eta}{2} \right) \right) \right) < \! \tfrac{\eta}{2} + 2 \delta.
\end{equation}
However, as already indicated, $\textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right),b_2 \! \left( \tfrac{\eta}{2} \right) \right)$ on its own is not quite enough to guarantee $\textbf{Bo}(t_2 + j ,b_1(\eta),b_2(\eta))$ for $j \leq M$. So let us describe the extra ingredient we require. Notice that $U_i(t)$ is a binomial distribution with a long right tail, since the number of trails $f(t)$ is of the order of $t$, and the probability of success per trial is $\frac{d_i(t)}{2F(t)}$ which is of order $\frac{1}{t}$. We shall show that we may ignore the extremity of this tail, thus allowing us to impose a tighter upper bound than $f(t)$ on $U_i(t)$ for all $t \in [t_2, t_2+M)$.
In Theorem 1.1 from \cite{Bol}, we find a useful bound for the right-tail of a binomial distribution $U \sim b(f,p)$: if $u>1$ and $1 \leq S: = \lceil ufp \rceil \leq f-1$ then
$${\bf P}(U \geq S) < \left(\frac{u}{u-1} \right) \cdot {\bf P}(U = S).$$
Let us apply this in the case $S = \lceil t^{\alpha} \rceil$ for some fixed $\alpha \in \left( \frac{1}{2}, 1 \right)$. (Its exact value does not matter.) Then
$$u = u(t) = \frac{t^{\alpha}}{ p f} = \frac{t^{\alpha} 2 F(t)}{d(t) f(t)}.$$
Assembling the bounds $c_1 t \leq f(t) \leq c_2 t$ and $c_1 t^2 \leq 2F(t) \leq c_2 t^2$ and $\textbf{Bo}(t, b_1', b_2')$ where $b_1 \leq b_1' < b_2' \leq b_2$ and employing the standard bound for the binomial coefficiant $\begin{pmatrix} f \\ S \end{pmatrix} \leq \left( \dfrac{f\cdot e}{S} \right)^S$, we find
\begin{align*}
{\bf P} \Big(U_i & \geq t^{\alpha} \ \Big| \Big| \ \textbf{Bo}(t, b_1', b_2') \Big)\\
& < \left(\frac{\frac{c_2}{c_1 b_1} t^{\alpha}}{\frac{c_1}{c_2 b_2} t^{\alpha}-1} \right) \cdot \left( (e c_2 +1) \cdot t^{1 - \alpha} \right)^{t^{\alpha}}\cdot \left( \frac{b_2}{c_1 t} \right)^{t^\alpha} \cdot \left(1 - \frac{b_1}{c_2 t} \right)^{t - \lceil t^\alpha \rceil}\\
& < \left( \frac{B}{t^{\alpha}} \right)^{t^{\alpha}}.
\end{align*}
for some $B>0$. Notice again that this holds independently of the specific values of $b_1'$ and $b_2'$, so long as $b_1 \leq b_1' < b_2' \leq b_2$. Now we define a new event, that the tails are short (\textbf{sh}):
$$\textbf{sh}(t):=\bigwedge_{i=1}^n U_i(t+1)< t^{\alpha}.$$
After increasing $B$ to allow for the non-independence of the $n$ different $U_i$ we now see that:
\begin{equation} \label{eqution:shbound}
{\bf P} \left( \neg \textbf{sh}(t) \ \Big| \Big| \ \textbf{Bo}(t,b_1(\eta),b_2(\eta)) \right) < n \cdot \left( \frac{B}{t^{\alpha}} \right)^{t^{\alpha}}.
\end{equation}
Putting these events together, define
$$\textbf{Sh}(t_2):=\forall t \in [t_2,t_2+M) \ \ \textbf{sh}(t).$$
To obtain a similar bound for ${\bf P} \left(\neg \textbf{Sh}(t_2) || \textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right) ,b_2 \! \left( \tfrac{\eta}{2} \right) \right) \right)$ we first show that for $j \leq M$
\begin{align} \label{equation:Shimpl}
\bigg( \textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right) ,b_2 \! \left( \tfrac{\eta}{2} \right) \right) \ & \& \ \bigwedge_{i=0}^{j-1} \textbf{sh}(t_2+j-1) \bigg) \\ \nonumber & \Rightarrow \textbf{Bo}(t_2 + j ,b_1(\eta),b_2(\eta)).
\end{align}
Suppose that $\textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right),b_2 \! \left( \tfrac{\eta}{2} \right) \right)$ holds. We address the lower bound first, for which we do not require the hypothesis on $\textbf{sh}$. Instead, for all $j \leq M$, clearly $$d(t_2 +j) \geq d(t_2) \geq b_1 \! \left( \tfrac{\eta}{2} \right) \cdot t_2 = \kappa_1 \cdot \left( y_1 - \frac{\eta}{2n} \right) \cdot t_2.$$
If additionally $t_2 \geq \frac{2nM y_1}{\eta}$, then the final term above exceeds $$\kappa_1 \cdot \left( y_1 - \frac{\eta}{n} \right) \cdot (t_2+M) \geq b_1(\eta) \cdot (t_2+j).$$
Now we obtain the corresponding upper bound. By our assumption on $\textbf{sh}$,
\begin{align*}
d(t_2+j) & \leq d(t_2) + M \cdot (t_2 + M)^{\alpha}\\
& \leq \kappa_2 \cdot \left( y_2 + \frac{\eta}{2n}\right) \cdot t_2 + M \cdot (t_2 + M)^{\alpha}\\
& \leq \kappa_2 \cdot \left( y_2 + \frac{\eta}{n}\right) \cdot t_2
\end{align*}
if $t_2 \geq \max \left\{ M, \left( \frac{4Mn}{\kappa_2 \eta} \right)^{\frac{1}{1 - \alpha}} \right\}$, which completes the proof of Implication (\ref{equation:Shimpl}).
Implication (\ref{equation:Shimpl}) allows us to take the $M$-fold sum of (\ref{eqution:shbound}), finding
$${\bf P} \left(\neg \textbf{Sh}(t_2) \ \Big| \Big| \ \textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right) ,b_2 \! \left( \tfrac{\eta}{2} \right) \right) \right) < \sum_{t=t_2}^{t_2+M} n \left( \frac{B}{t^{\alpha}} \right)^{t^{\alpha}} \to 0$$ as $t_2 \to \infty$. Thus for any $\theta>0$ for all large enough $t_2$ we have
\begin{equation} \label{equation:thetabound}
{\bf P}\left( \neg \textbf{Sh}(t_2) \ \Big| \Big| \ \textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right) ,b_2 \! \left( \tfrac{\eta}{2} \right) \right) \right) < \theta.
\end{equation}
Finally, we may complete the argument, setting $\delta = \frac{\varepsilon}{8}$ and $\theta = \zeta = \frac{\varepsilon}{4}$ and $\eta = \frac{ \varepsilon}{2}$ and $t_1:=t_2+M$. For large enough $t$, we may update bound (\ref{equation:Ubound}) to get
\begin{equation*}
{\bf P}\left( \left( \neg \textbf{U}(t_2 +1)=\textbf{m} \right) \ \& \ \textbf{sh}(t_2) \ \Big| \Big| \ \textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right) , b_2 \! \left( \tfrac{\eta}{2} \right) \right) \right)< 1 - C_4.\end{equation*}
Similarly,
\begin{align*}
{\bf P}\Big( \big( \neg \textbf{U}(t_2+j+1) =\textbf{m} \big) &\ \& \ \textbf{sh}(t_2+j) \\ \ & \Big| \Big| \ \bigwedge_{i=0}^{j-1} \textbf{sh}(t_2+i) \ \& \ \textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right) , b_2 \! \left( \tfrac{\eta}{2} \right) \right) \Big) \\ & \ \ \ \ \ \ < 1 - C_4.\end{align*}
As observed earlier, these bounds hold independently of the previous values of $\textbf{U}$, meaning that
\begin{align*}
{\bf P}\Big( \big( & \neg \textbf{U}(t_2+j+1) =\textbf{m} \big) \ \& \ \textbf{sh}(t_2+j) \\ \ & \Big| \Big| \ \bigwedge_{i=0}^{j} \neg \textbf{U}(t_2+i) \ \& \ \bigwedge_{i=0}^{j-1} \textbf{sh}(t_2+i) \ \& \ \textbf{Bo} \left( t_2,b_1 \! \left( \tfrac{\eta}{2} \right) , b_2 \! \left( \tfrac{\eta}{2} \right) \right) \Big) \\ & \hspace{5cm} < 1 - C_4.\end{align*}
Taking the product of these bounds, and denoting the failure of our desired result by $\textbf{Fa}(t_2):= \forall t \in [t_2, t_2+M) \ \left( \textbf{U}(t+1) \neq \textbf{m} \right)$, we see that
$${\bf P} \Big( \textbf{Fa}(t_2) \ \& \ \textbf{Sh}(t_2) \ \Big| \Big| \ \textbf{Bo}(t_2,b_1 \! \left( \tfrac{\eta}{2} \right) , b_2 \! \left( \tfrac{\eta}{2} \right) \Big) < \zeta$$
and so by bounds (\ref{equation:b2bound}) and (\ref{equation:thetabound})
\begin{align*}
{\bf P} \Big( \textbf{Fa}(t_2) \Big) & < \zeta + \theta + \tfrac{\eta}{2} + 2\delta = \varepsilon.
\end{align*}
\end{proof}
We may now complete the proof of Theorem \ref{thm:mainmulti}.
\begin{proof}
First notice that there are countably many witness requests. Thus we may organise them into a list $\left( W_j : j \geq 1 \right)$.
Let $\varepsilon >0$. Again everything that occurs is conditioned upon $G_0(t_0)$. We shall show that the probability of all witness requests eventually being satisfied exceeds $1-\varepsilon$. Suppose inductively that we have found time $t_j$ so that so that ${\bf P} (\bigwedge_{i=1}^j W_i[t_j]) > 1- \left( 1 - \frac{1}{2^j} \right) \varepsilon$.
Let $\mathcal{G} = \mathcal{G}_j$ be the set of all states $G=G(t_j)$ of the graph at time $t_j$ consistent with $G_0(t_0)$ and with $\bigwedge_{i=1}^j W_i[t_j]$. Notice that $\mathcal{G}$ is a finite set, that ${\bf P} \left( G(t_j) \ \big| \big| \ G_0(t_0) \right)>0$ for each $G \in \mathcal{G}$, and by assumption that $\sum_{G \in \mathcal{G}} {\bf P} \left( G(t_j) \ \big| \big| \ G(t_0) \right) > 1- \left( 1 - \frac{1}{2^j} \right) \varepsilon$.
Consider now $W_{j+1}$ and let $\varepsilon' < \frac{1}{2^{j+1}} \varepsilon$. Now given each $G^{(k)} \in \mathcal{G}$, by Proposition \ref{prop:mainstep1} there exist $t^{(k)} \geq t_j$ such that $${\bf P} \left( W_{j+1} \left[ t^{(k)} \right] \ \big| \big| \ G^{(k)}(t_j) \right) >1 - \varepsilon'.$$
Let $t_{j+1} := \max \{t^{(k)} \ | \ G^{(k)} \in \mathcal{G}\}$. Then
\begin{align*}
& {\bf P}\left(\bigwedge_{i=1}^{j+1} W_i[t_{j+1}] \ \big| \big| \ G_0(t_0) \right) \\
\geq & \ \sum_{k} {\bf P}\left(\bigwedge_{i=1}^{j+1} W_i[t_{j+1}] \ \big| \big| \ G^{(k)}(t_j) \right) \cdot {\bf P}\left(G^{(k)}(t_j)\ \big| \big| \ G_0(t_0) \right)\\
= & \ \sum_{k} {\bf P} \left(W_{j+1}[t_{j+1}] \ \big| \big| \ G^{(k)}(t_j) \right) \cdot {\bf P}\left(G^{(k)}(t_j) \ \big| \big| \ G_0(t_0) \right)\\
> & \ \sum_{k} (1 - \varepsilon') \cdot {\bf P}\left(G^{(k)}(t_j) \ \big| \big| \ G_0(t_0) \right)\\
> & \ (1 - \varepsilon') \left(1- \left( 1 - \frac{1}{2^j} \right) \varepsilon \right)
> 1- \left( 1 - \frac{1}{2^{j+1}} \right)\varepsilon. \end{align*} \end{proof}
|
1,116,691,499,326 | arxiv |
\section{Introduction}
Mass is the most crucial input in stellar internal structure modelling. It predominantly influences the luminosity of a star at a given stage of its evolution, and also its lifetime. The knowledge of the mass of stars in a non-interacting binary system, together with the assumption that the components have same age and initial chemical composition, allows the age and the initial helium content of the system to be determined and therefore to characterize the structure and evolutionary stage of the components. Such modelling provides insights into the physical processes governing the structure of the stars. Moreover provided masses are known with great accuracy~\citep{Lebreton2005}, it gives constraints on the free physical parameters of the models. Therefore, modelling stars with extremely accurate masses (at the 1 \% level), in different ranges of masses, would allow to firmly anchor the models of the more loosely constrained single stars.
This paper is the third in a series dedicated to the derivation of accurate masses of the components of double-lined spectroscopic binaries (SB2) with the forthcoming astrometric measurements from the {\it Gaia} satellite. In paper I \citep{Halb2014}, we have presented our program to derive accurate masses from {\it Gaia} and from high-precision spectroscopic observations.
We have selected a sample of 68 SB2s for which we expect to derive very precise inclination with {\it Gaia}, and $M\sin^3 i$ with the Spectrographe pour l'Observation des PH\'enom\`enes des Int\'erieurs Stellaires et des Exoplan\`etes (SOPHIE spectrograph, Haute-Provence Observatory). Our objective is to determine for these SB2 systems the masses of the two components with an accuracy about 1~\%.
We have been observing these stars since 2010 with SOPHIE. A first result of our program was the detection of the secondary component in the spectra of 20 binaries which were previously known as single-lined (paper\,I). A second result was the determination of masses for 2 SB2 with accuracy between 0.26 and 2.4\,\%, using astrometric measurements from PIONIER and radial velocities from SOPHIE (paper\,II).
Here, we present the accurate orbits measured for 10 SB2s (Table~\ref{tab:obs}) with periods ranging from 37 to 881 days. After 5 years of observations with SOPHIE, we collected a total of 123 spectra. In addition, a large number of previously published measurements is available for each of them in the SB9 catalog~\citep{SB9}. Four of these targets are new SB2 identified in paper I, and previously known as SB1. Finally, we combined the radial velocity (RV) measurements of one star (HIP~87895) with existing interferometric measurements and derive the masses of the two components.
The observations are presented in Section~\ref{sect:observations}. The method of measurements of radial velocities from SOPHIE's observations is explained in Section~\ref{sect:RV}. We derive the orbital solutions in Section~\ref{sec:orbits}, discussing in particular the issue of the uncertainties when combining different datasets from different instruments. The derivation of the masses of HIP\,87895 is discussed in Section~\ref{sec:HIP87895}. Finally, we summarize and conclude on our findings in Section~\ref{sec:conclusion}.
\section{Observations}
\label{sect:observations}
The observations were performed at the T193 telescope of the Haute-Provence Observatory, with the SOPHIE spectrograph. SOPHIE is dedicated to the search of extrasolar planets, and, thanks to its high resolution ($R$$\sim $$75,000$), it enables accurate stellar radial velocities to be measured for SB2 components.
The spectra were all reduced through SOPHIE's pipeline, including localization of the orders on the frame, optimal order extraction, cosmic-ray rejection, wavelength calibration, flat-fielding and bias subtraction. The minimum signal-to-noise ratio (SNR) is 40 for the faintest stars of the sample, but it may be as large as 150 for a 6-magnitude star. Before each observation run ephemerides were derived from existing orbits provided by the SB9 catalogue \citep{SB9}, and priority classes were assigned on the basis of the orbital phase. Four classes were used: the lowest priority corresponds to stars
with expected RVs of primary and secondary component sufficiently different to permit accurate measurements, and the highest priority is reserved to the observations of the periastron of eccentric orbits.
Among all the observed SB2, we have selected those which were observed over at least one period, and which received a minimum of 11 observations. Table~\ref{tab:obs} summarizes this information. Given the very high quality of the measurements, an SB2 orbit could be derived in principle from only 6 of those observations, provided they were made at the most relevant phases. However, we show in Section~\ref{sec:orbits} that 11 observations are necessary to validate the RV uncertainties, and to correct them when necessary.
\begin{table}
\caption{The ten SB2 analyzed in this paper. }
\small
\begin{tabular}{r@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}l}
\hline
HIP & HD & V & Period$^a$ & $N_\text{spec}^{\phantom{spec}b}$& Time span$^c$ & SNR$^d$ \\
& & (mag.) & (day) & & (period) & & \\
\hline
\multicolumn{8}{c}{\it Previously published SB2} \\[1ex]
12081 & 15850 & 7.72 & 443.49 & 11 & 3.3 & 80 \\
17732 & 23626 & 6.27 & 277.89 & 11 & 5.5 & 120 \\
56275 & 100215 & 7.99 & 47.88 & 13 & 30.7 & 80 \\
87895 & 163840 & 6.33 & 880.78 & 14 & 2.1 & 120 \\
95575 & 183255 & 8.05 & 166.36 & 13 & 11.9 & 75 \\
100321 & 195850 & 7.02 & 37.94 & 11 & 45.3 & 80 \\
\hline
\multicolumn{8}{c}{\it SB2 identified in paper I, previously published as SB1} \\[1ex]
13791 & 18328 & 8.87 & 48.71 & 13 & 28.8 & 40 \\
61727 & 110025 & 7.58 & 54.88 & 12 & 28.3 & 80 \\
62935 & 120005 & 8.53 & 139.00 & 11 & 10.6 & 40 \\
67195 & 120005 & 6.50 & 39.28 & 14 & 37.3 & 120 \\
\hline
\end{tabular}\\
\flushleft
$^a$ The period values are taken from the SB9 catalog \citep{SB9}. \\
$^b$ $N_\text{spec}$ gives the number of spectra collected with SOPHIE. \\
$^c$ The time span is the total span of observation epochs, counted in number of periods. \\
$^d$ SNR is the median signal-to-noise ratio of each sample.
\label{tab:obs}
\end{table}
\section{Radial velocity measurements}
\label{sect:RV}
The radial velocities of the components are derived using the TwO-Dimensional CORrelation algorithm {\sc todcor} \citep{zucker94,zucker04}.
It calculates the cross-correlation of an SB2 spectrum and two best-matching stellar atmosphere models, one for each component of the observed binary system. This two-dimensional cross-correlation function (2D-CCF) is maximized at the radial velocities of both components. The multi-order version of {\sc todcor}, named {\sc todmor}~\citep{zucker04}, determines the radial velocities of both components from the gathering of 2D-CCF obtained from each order of the spectrum.
All SOPHIE multi-orders spectra were corrected for the blaze using the response function provided by SOPHIE's pipeline; then for each of them, the pseudo-continuum was detrended using a p-percentile filter~(paper II, or e.g. \cite{Hodg1985}).
For both components of each binary, we determined a best-matching theoretical spectra from the PHOENIX stellar atmosphere models~\citep{Huss2013}. We optimized the 2D-CCF with respect to effective temperature $T_\text{eff}$, rotational broadening $v \sin i$, metallicity [Fe/H], surface gravity $\log(g)$, and flux ratio at $4916$\,\AA, $\alpha$$=$$F_2/F_1$. Furthermore, each theoretical spectrum is convolved with the instrument line spread function, here modeled by a Gaussian, and pseudo-continuum detrended like the observed spectrum. The spectral parameters obtained through this method are given in Table~\ref{tab:stellpar}. We determined spectral parameters from 4 spectra per binary on average, with the two components well individualized. The values and their uncertainties given in Table~\ref{tab:stellpar} are the average and standard deviation of the individual estimations. The 1$\sigma$ uncertainties do not include known systematics of theoretical models with respect to real spectral types (see e.g.~\cite{Torres12}).
It is worth mentioning that the derived metallicity [Fe/H] is systematically subsolar. Optimizing the CCF of several spectra of the Sun obtained by observing Vesta and Ceres with SOPHIE gave spectral parameters consistent with the known values for the Sun except metallicity that was found to be -0.33 dex. However, we kept the values of metallicity that maximized the 2D-CCF, as given in Table~\ref{tab:stellpar}.
\begin{table}
\begin{minipage}{\columnwidth}
\caption{\label{tab:stellpar}The stellar parameters determined by optimization of the 2D-CCF obtained with {\sc todmor}. Explanations in Section~\ref{sect:RV}.}
\small
\begin{tabular}{@{}lrrrrr@{}}
\hline
HIP & $ T_\text{eff,1}$ & $ \log g_1$ & $ V_1 \sin i_1$ $^a$ & $[\text{Fe/H}]$ & $\alpha$ \\
HD & $ T_\text{eff,2}$ & $ \log g_2$ & $ V_2 \sin i_2$ $^a$ & & \\
& (K) & (dex) & (km s$^{-1}$) & (dex) & (flux ratio) \\
\hline
12081 & 6290 & 4.24 & 11.9 & -0.37 & 0.635 \\
15850 & $\pm$ 23 & $\pm$0.05 & $\pm$0.1 & $\pm$0.01 & $\pm$0.012 \\
& 6003 & 4.30 & 6.5 & & \\
& $\pm$ 13 & $\pm$0.01 & $\pm$0.1 & & \\ [1ex]
13791 & 6173 & 4.31 & 5.5 & -0.34 & 0.034 \\
18328 & $\pm$ 6 & $\pm$0.01 & $\pm$0.1 & $\pm$0.01 & $\pm$0.001 \\
& 4953 & 5.09 & 0 & & \\
& $\pm$ 95 & $\pm$0.01 & & & \\[1ex]
17732 & 6030 & 3.34 & 9.4 & -0.71 & 0.258 \\
23626 & $\pm$ 7 & $\pm$0.03 & $\pm$0.1 & $\pm$0.01 & $\pm$0.004 \\
& 6051 & 3.84 & 0 & & \\
& $\pm$ 18 & $\pm$0.01 & & & \\ [1ex]
56275 & 6889 & 3.93 & 20.6 & -0.33 & 0.036 \\
100215& $\pm$ 36 & $\pm$0.11 & $\pm$1 & $\pm$0.01 & $\pm$0.002 \\
& 4906 & 4.33 & 0 & & \\
& $\pm$ 75 & $\pm$0.01 & & & \\ [1ex]
61727 & 6461 & 3.82 & 9.8 & -0.48 & 0.070 \\
110025& $\pm$ 31 & $\pm$0.15 & $\pm$0.1 & $\pm$0.06 & $\pm$0.007 \\
& 5256 & 4.31 & 7.7 & & \\
& $\pm$ 168 & $\pm$0.14 & 1.4 & & \\ [1ex]
62935 & 5818 & 4.03 & 4.9 & -0.27 & 0.054 \\
112138 & $\pm$ 85 & $\pm$0.08 & $\pm$0.4 & $\pm$0.01 & $\pm$0.015 \\
& 4378 & 4.40 & 0 & & \\
& $\pm$ 115 & $\pm$0.13 & & & \\ [1ex]
67195 & 6411 & 4.29 & 13.6 & -0.21 & 0.027 \\
120005& $\pm$ 29 & $\pm$0.13 & $\pm$0.2 & $\pm$0.03 & $\pm$0.002 \\
& 4478 & 4.72 & 4.5 & & \\
& $\pm$ 418 & $\pm$0.21 & $\pm$0.1 & & \\ [1ex]
87895 & 5970 & 4.33 & 4.7 & -0.19 & 0.036 \\
163840& $\pm$ 1 & $\pm$0.01 & $\pm$0.1 & $\pm$0.01 & $\pm$0.004 \\
& 4385 & 4.81 & 0 & & \\
& $\pm$ 134 & $\pm$0.04 & & & \\ [1ex]
95575 & 4908 & 4.75 & 3.4 & -0.88 & 0.431 \\
183255& $\pm$ 5 & $\pm$0.03 & $\pm$0.1 & $\pm$0.01 & $\pm$0.008 \\
& 4088 & 4.51 & 0 & & \\
& $\pm$ 5 & $\pm$0.03 & & & \\ [1ex]
100321 & 6485 & 4.17 & 14.2 & -0.39 & 0.207 \\
195850 & $\pm$ 6 & $\pm$0.05 & $\pm$0.2 & $\pm$0.01 & $\pm$0.011 \\
& 5558 & 4.39 & 5.5 & & \\
& $\pm$ 62 & $\pm$0.04 & $\pm$ 1 & & \\
\hline
\end{tabular}
\end{minipage}
$^a$ A null value is given to $V\sin i$ with no error bar whenever found less than SOPHIE's typical pixel size $\sim$2km\,s$^{-1}$.\\
\end{table}
We then applied {\sc todmor} to all multi-order spectra of each target and determined the radial velocities of both components discarding all orders harboring strong telluric lines. For each of the non-discarded orders, we calculated a two dimensional cross-correlation function, and used the maximum of this function to derive radial velocities for the primary and the secondary.
Final velocities for each component are obtained by averaging these measurements and incorporating a correction for order-to-order systematics -- typically 200-500\,m\,s$^{-1}$. The final velocities are displayed in Table~\ref{tab:RVs}. They are used to derive the orbital solutions for the 10 SB2 in the next section.
\begin{table*}
\caption{\label{tab:RVs} New radial velocities from SOPHIE and obtained with {\sc todmor}. Outliers are marked with an asterisk ($^*$) and are not taken into account in the analysis.}
\scriptsize
\begin{minipage}{89mm}
\input{HIP12081_RV_table_latex}
\end{minipage}%
\begin{minipage}{89mm}
\input{HIP13791_RV_table_latex}
\end{minipage}\\%
\begin{minipage}{89mm}
\input{HIP17732_RV_table_latex}
\end{minipage}%
\begin{minipage}{89mm}
\input{HIP56275_RV_table_latex}
\end{minipage}\\%
\begin{minipage}{89mm}
\input{HIP61727_RV_table_latex}
\end{minipage}%
\begin{minipage}{89mm}
\input{HIP62935_RV_table_latex}
\end{minipage}\\
\begin{minipage}{89mm}
\input{HIP67195_RV_table_latex}
\end{minipage}%
\begin{minipage}{89mm}
\input{HIP87895_RV_table_latex}
\end{minipage}
\vspace*{1cm}
\end{table*}
\addtocounter{table}{-1}
\begin{table*}
\caption{Continued.}
\scriptsize
\begin{minipage}{89mm}
\input{HIP95575_RV_table_latex}
\end{minipage}%
\begin{minipage}{89mm}
\input{HIP100321_RV_table_latex}
\end{minipage}
\vspace*{1cm}
\end{table*}
\section{Derivation of the orbits}
\label{sec:orbits}
The orbital solutions for the 10 SB2 are derived by combining the new measurements presented in this paper with previously published RVs (references in Table~\ref{tab:corsigRVprev}).
Since several datasets coming from different instruments are used together to derive a common orbital solution, realistic errors should be attributed to each dataset properly. It is explained in the following section. This process guarantees that each dataset receives the proper weight with respect to all others, including the new SOPHIE measurements presented here.
\subsection{Correction of uncertainties}
\label{sec:corer}
\begin{table*}
\caption{Correction terms applied to the uncertainties of the previous and of the new RV measurements. The composition of these terms into a uncertainty correction is explained in Section~\ref{sec:corer}, eqs.~\ref{eq:correction1} and~\ref{eq:correction2}.
}
\begin{tabular}{rlcccccccc}
\hline
HIP & Reference of previous RV & \multicolumn{4}{c}{Correction terms for previous measurements} & \multicolumn{4}{c}{Correction terms for new measurements} \\
& &$\varepsilon_{1,p}$&$\varphi_{1,p}$&$\varepsilon_{2,p}$&$\varphi_{2,p}$&$\varepsilon_{1,n}$&$\varphi_{1,n}$& $\varepsilon_{2,n}$& $\varphi_{2,n}$\\
& &km s$^{-1}$ & & km s$^{-1}$ & & km s$^{-1}$ & & km s$^{-1}$ & \\
\hline
12081 & \cite{Griffin05} & 0 & 0.615 & 0 & 0.565 & 0.1428 & 1 & 0.0061 & 1 \\
13791 & \cite{Imbert06} & 0.24 & 1 & $\cdot$ & $\cdot$ & 0.0287 & 0.885 & 0.2689 & 0.885 \\
17732 & \cite{GriffinSuchkov03}& 0 & 0.272 & 0 & 0.759 & 0 & 1.101 & 0.0380 & 1.101 \\
56275 & \cite{Griffin06} & 0 & 1.208 & 0 & 1.208 & 0.6909 & 0.905 & 0.2315 & 0.905\\
61727 & \cite{Halb12} & 0.3212 & 1 & $\cdot$ & $\cdot$ & 0.0261 & 0.911 & 0.4914 & 0.911\\
62935 & \cite{Griffin04} & 0 & 0.594 & $\cdot$ & $\cdot$ & 0.0294 & 1.101 & 0.1774 & 1.101 \\
67195 & \cite{Shajn39} & 0 & 5.243 & $\cdot$ & $\cdot$ & 0.0175 & 1.046 & 0 & 0.8170 \\
87895 & \cite{McAlister95} & 0 & 0.342 & 0 & 0.342 & 0.0355 & 1.329 & 0.0662 & 1.329\\
95575 & \cite{Toko91} & 1.022 & 1 & 1 & 1 & 0.0108 & 0.897 & 0.0399 & 0.897\\
100321 & \cite{Carquillat00} & 0 & 0.541 & 0 & 1.097 & 0.0665 & 1.017 & 0 & 0.572 \\
\hline
\label{tab:corsigRVprev}
\end{tabular}
\vspace*{1cm}
\end{table*}
Uncertainties of previously published measurements, when provided, are usually underestimated. On the other hand, many lists of RV measurements do not include uncertainties, but only weights (W). Therefore, two different procedures are applied to attribute correct uncertainties to these retrieved measurements. They are both based on the calculation of the $F_2$ estimator of the goodness-of-fit \citep[see Paper\,II, equation 1, or][]{Kendall}:
\begin{itemize}
\item
When the uncertainties are provided, a noise is quadratically added to the original uncertainties, in order to get exactly $F_2$=$0$
for the SB1 orbit of each component.
Since the original uncertainties are underestimated, this results in decreasing the variations of the
relative weights of the measurements of a given component.
\item
When only weights are given, they are first converted to uncertainties ($\sigma=\sqrt{1/W}$). Then they are scaled in order to get $F_2$=$0$ for the SB1 orbit of each component.
\end{itemize}
After this transformation, the SB2 orbit is derived, and $F_2$ is considered again. The final uncertainties are obtained by multiplying the ones derived above with a factor chosen in order to get $F_2$=$0$. All these operations result in applying the following formulae:
\begin{align}
\sigma^\text{corr}_{RV, 1} &= \varphi_1 \times \sqrt{\sigma_{RV, 1}^2 + \varepsilon_1^2} \label{eq:correction1}\\
\sigma^\text{corr}_{RV, 2} &= \varphi_2 \times \sqrt{\sigma_{RV, 2}^2 + \varepsilon_2^2} \label{eq:correction2}
\end{align}
Table~\ref{tab:corsigRVprev} lists the derived values of the correction terms $\varphi_1$, $\varphi_2$, $\varepsilon_1$ and $\varepsilon_2$ for all stars of our sample.
The same procedure is applied to the uncertainties derived by {\sc todmor} for the new measurements. For two binaries (HIP\,67195 and HIP\,100321), the SB1 orbit of the secondary component leads to a negative value of $F_2$, implying that the uncertainties are slightly overestimated. In order to get $F_2$$=$$0$, we prefered to keep the relative weights fixed, and simply multiply the uncertainties by a coefficient $\varphi_{2}$ lower than 1.
All the RV measurements and the uncertainties used in the derivation of the orbits will be available through
the SB9 Catalogue \citep{SB9}, which is accessible on-line\footnote{http://sb9.astro.ulb.ac.be/}.
\subsection{Derivation of the orbital elements}
\label{sect:orbsol}
For each binary, we fitted an SB2 orbital model to the previously published datasets combined with the new SOPHIE observations. The parameters were optimized using a Levenberg-Marquard method.
The final orbital solution consists in the following orbital elements: the period, $P$, the eccentricity, $e$, the epoch of the periastron, $T_0$, the longitude of the periastron for the primary component, $\omega_1$, the RV semi-amplitudes of each component, $K_1$ and $K_2$, and the RV of the barycentre, $V_0$.
We also added 3 supplementary parameters, which are: a systematic offset between the measurements previously published and the new ones, $d_{n-p}$, and the offsets between the RVs of the primary and of the secondary component, $d_{2-1}^p$ and $d_{2-1}^n$.
The offset $d_{2-1}^p$ is usually due to the fact that the published RVs were obtained from templates which are not specifically adapted to each component. For instance, a spectrovelocimeter like CORAVEL \citep{Baranne1979} was working by projecting any spectrum on a mask representing the spectrum of Arcturus.
The offset between the primary and secondary RVs derived with {\sc todmor}, $d_{2-1}^n$ is expected to be null, but it is significant for some stars, since the spectra of the PHOENIX library do not perfectly represent the actual ones.
\begin{table*}
\centering
\caption{The orbital elements derived from the previously published RV measurements and from the
new ones. The radial velocity of the barycentre, $V_0$, is in the reference system of the new measurements of the primary component.
The minimum masses and minimum semi-major axes are derived from the true period
($P_{true}=P \times (1-V_0/c)$).}
\scriptsize
\setlength{\tabcolsep}{2mm}
\begin{tabular}{@{}lrrrrrrrrrrrrrr@{}}
\hline
HIP & $P$ & $T_0$(BJD) & $e$ & $V_0$ & $\omega_1$ & $K_1$ & ${\cal M}_1 \sin^3 i$ & $a_1 \sin i$&$N_1$& $d_{n-p}$ & $\sigma(O_1-C_1)_{p,n}$ \\
HD & & & & & & $K_2$ & ${\cal M}_2 \sin^3 i$ & $a_2 \sin i$&$N_2$& $d^p_{2-1},d^n_{2-1}$ &$\sigma(O_2-C_2)_{p,n}$ \\
& (d) & 2400000+ & &(km s$^{-1}$)&($^{\rm o}$)&(km s$^{-1}$)&(${\cal M}_\odot$)& (Gm) & & (km s$^{-1}$) &(km s$^{-1}$) \\
\hline
\\
12081 & 443.288 & 55905.74 & 0.58494 & -7.883 & 283.66 & 16.784 &0.5493 & 82.98 &47+11& -1.141 &0.634,0.128\\
15850 &$\pm 0.023$ &$\pm 0.12$ &$\pm 0.00054 $&$\pm 0.046$&$\pm 0.10$ &$\pm 0.057$&$\pm0.0019$&$\pm 0.28$ & & $\pm 0.101$ & \\
& & & & & & 18.258 &0.5050 &90.261 &47+11& -0.208, -0.064&0.548,0.020\\
& & & & & &$\pm 0.015$&$\pm0.0034$&$\pm 0.052$& & $\pm 0.124$, $\pm0.050$ & \\
&&&&&&&&&&&\\
13791 & 48.70895 & 55050.392 & 0.18781 &-8.537 &55.26 & 16.030 &0.2405 &10.5456 &33+12&0.686 &0.431,0.022\\
18328 &$\pm 0.00023$&$\pm 0.043$ &$\pm 0.00078$&$\pm 0.012$ &$\pm 0.29$&$\pm 0.015$ &$\pm0.0027$ &$\pm0.0090$& & $\pm 0.078$ & \\
& & & & & & 27.07 &0.14241 &17.809 &0+11 & $\cdots$ , 0.400 &$\cdots$ ,0.246 \\
& & & & & &$\pm 0.14$ &$\pm0.00094$&$\pm 0.089$& & $\cdots$ ,$\pm 0.096$ & \\
&&&&&&&&&&&\\
17732 & 277.9500 & 55230.207 & 0.14319 &-2.918 &159.14 & 18.6644 &1.1359 &70.598 &41+11&-1.096 &0.505,0.0080\\
23626 &$\pm 0.0074$ &$\pm 0.095$ &$\pm 0.00083$&$\pm 0.011$ &$\pm 0.14$&$\pm 0.0051$&$\pm0.0019$&$\pm 0.022$& & $\pm 0.058$ & \\
& & & & & & 23.208 &0.9135 &87.784 &41+11& 0.279, 0.116 &1.670,0.054 \\
& & & & & &$\pm 0.017$ &$\pm0.0010$&$\pm 0.067$& & $\pm 0.169,\pm 0.027$ & \\
&&&&&&&&&&&\\
56275 & 47.89410 & 56781.809 & 0.2317 &-17.879 &247.52 & 28.00 &1.5007 &17.94 &31$^a$+13&-1.273 &1.299,0.571\\
100215 &$\pm 0.00089$&$\pm 0.080$ &$\pm 0.0018$ &$\pm 0.18$ &$\pm 0.71$&$\pm 0.18$ &$\pm0.0086$&$\pm 0.11$ & & $\pm 0.294$ & \\
& & & & & & 51.709 &0.8126 &33.128 &10+13& 0.963, 0.273 &3.786,0.207 \\
& & & & & &$\pm 0.087$ &$\pm0.0090$&$\pm 0.051$& & $\pm 1.247,\pm 0.202$ & \\
&&&&&&&&&&&\\
61727 & 54.87864 & 56476.6509 & 0.34744 &-1.5628 &185.679 & 30.842 &1.432 &21.823 &43+12& 0.168 &0.639,0.020\\
110025 &$\pm 0.00018$&$\pm 0.0072$&$\pm 0.00051$&$\pm 0.0093$ &$\pm 0.057$&$\pm 0.019$&$\pm0.017$ &$\pm 0.011$& &$\pm 0.097$& \\
& & & & & & 48.51 &0.9104 &34.33 &0+12 & $\cdots$ , 0.358 &$\cdots$ ,0.435 \\
& & & & & &$\pm 0.26$ &$\pm0.0060$&$\pm 0.18 $& &$\cdots$ ,$\pm 0.142$& \\
&&&&&&&&&&&\\
62935 &139.0081 & 55680.66 & 0.1508 &0.244 &31.37 & 15.643 &0.4791 &29.557 & 89+11 &-0.189 &0.680,0.022\\
112138 &$\pm 0.0027$ &$\pm 0.17$ &$\pm 0.0018$ &$\pm 0.011$ &$\pm 0.44$&$\pm 0.023$ &$\pm0.0049$&$\pm 0.038$& &$\pm 0.065$ & \\
& & & & & & 23.03 &0.3254 &43.51 &0+11 & $\cdots$, 0.416 & $\cdots$,0.245 \\
& & & & & &$\pm 0.11$ &$\pm0.0020$&$\pm 0.20$ & & $\cdots$,$\pm 0.071$& \\
&&&&&&&&&&&\\
67195 &39.284974 & 56211.2841 & 0.78584 & -9.532 &325.250 & 45.44 &1.1731 &15.181 & 37+14 &-1.113 &5.027,0.021\\
120005 &$\pm 0.000091$&$\pm 0.0045$&$\pm 0.00050$&$\pm 0.016$ &$\pm 0.031$&$\pm 0.12$&$\pm0.0062$ &$\pm 0.026$& &$\pm 0.864$ & \\
& & & & & & 78.86 &0.6760 &26.346 &0+14 & $\cdots$ , -0.013 &$\cdots$ ,0.118 \\
& & & & & &$\pm 0.22$ &$\pm0.0034$&$\pm 0.048$ & &$\cdots$ ,$\pm 0.050$& \\
&&&&&&&&&&&\\
87895 & 881.629 & 55650.40 & 0.4165 &-32.346 &135.47 & 11.393 &0.9869 &125.57 &106+14&0.456 &0.842,0.035\\
163840 &$\pm 0.065$&$\pm 0.49$&$\pm 0.0014$ &$\pm 0.015$&$\pm 0.26$&$\pm 0.021$&$\pm0.0097$&$\pm 0.22$& &$\pm 0.047$& \\
& & & & & & 17.374 &0.6471 &191.50 &16+13 &-0.754, 0.319 &1.210,0.210\\
& & & & & &$\pm 0.077$&$\pm0.0041$&$\pm0.84$ & &$\pm0.608$,$\pm0.058$&\\
&&&&&&&&&&&\\
95575 & 166.8351 & 56087.993 & 0.13698 &-64.2928 & 63.21 & 14.0202 &0.23291 &31.866 &23+12&0.394 &1.225,0.0098\\
183255 &$\pm 0.0031$&$\pm 0.094$&$\pm 0.00031$&$\pm 0.0040$&$\pm 0.20$&$\pm 0.0048$&$\pm0.00051$&$\pm 0.011$& &$\pm 0.233$ & \\
& & & & & & 15.696 &0.20804 &35.675 &9+12 &0.356, 0.097&1.981,0.044\\
& & & & & &$\pm 0.016$ &$\pm0.00028$&$\pm 0.037$& &$\pm 0.592$,$\pm 0.014$&\\
&&&&&&&&&&&\\
100321 & 37.939920 & 56254.291 & 0.18242 & 0.941 &111.043 & 34.306 &1.06199 &17.596 &52+11 &0.324 &0.530,0.068\\
195850 &$\pm 0.000081$&$\pm 0.010$ &$\pm 0.00025$&$\pm 0.023$ &$\pm 0.096$&$\pm 0.025$&$\pm0.00081$&$\pm 0.013$& &$\pm 0.079$ & \\
& & & & & & 45.0916 &0.8080 &23.1284 &49+11 &0.493, 0.121&1.160,0.011\\
& & & & & &$\pm 0.0089$&$\pm0.0011$&$\pm 0.0041$& &$\pm 0.174,\;\pm 0.028$&\\
\hline
\label{tab:orbSB2}
\end{tabular}
\flushleft
$^a$ The 3 first measurements in \cite{Griffin06} were discarded.
\end{table*}
\subsection{Results}
\label{sect:results}
Even with a correction, the uncertainties of the measurements from SOPHIE remain small and have weights much larger than the others. Therefore, incorporating the published datasets in the calculation essentially improved the accuracy of the period determined.
Conversely, the significant precision of the new SOPHIE measurements allowed us to reach a very good accuracy on all other orbital parameters, especially the minimum masses.
The derived orbital elements for the 10 SB2 are given in Table \ref{tab:orbSB2}. Among the 20 component stars, 15 of them have $M \sin^3 i$ determined with an accuracy better than 0.7~\%, while only 5 stars have minimum mass accuracies between 1 and 1.2\%.
A comparison between the standard deviation of the residuals of the previous measurements and of the new ones also illustrates the
improvements due to SOPHIE. It results from the last column of Table \ref{tab:orbSB2}
that $\sigma(O-C)_p$ is between 2.3 and 239 times larger than $\sigma(O-C)_n$, with a median around 30.
The orbital solutions, including previously published measurements, are displayed on Fig.\,\ref{fig:orbSB2}. The ($O-C$) residuals are included in Table~\ref{tab:RVs} and are plotted on Fig.\,\ref{fig:resOrbSB2}. We do not observe any drift for any of the 10 SB2.
\begin{figure*}
\includegraphics[clip=,height=10.5 cm,width=150mm]{orb10.eps}
\caption{The spectroscopic orbits of the 10 SB2; the circles refer to the primary component, and the
triangle to the secondary; the large filled symbols refer to the new RV measurements obtained with SOPHIE. For each SB2, the RVs are shifted to the
zero point of the SOPHIE measurements of the primary component.}
\label{fig:orbSB2}
\includegraphics[clip=,height=10.5 cm,width=150mm]{res10.eps}
\caption{The residuals of the RVs obtained from {\sc todmor} for the 10 SB2s. The circles refer to the primary component, and the triangles to the secondary component. For readability, the residuals of the most accurate RV measurements are in filled symbols.}
\label{fig:resOrbSB2}
\end{figure*}
\section{Combined orbital solution and masses for HIP\,87895}
\label{sec:HIP87895}
\begin{figure}
\includegraphics[clip=,height=5.4 in]{orbiteBV_HIP_87895.eps}
\caption{The visual part of the combined orbit of HIP 87895.Upper panel: the
visual orbit; the circles are the 3 positions obtained from long-base interferometry; the node line is in dashes.
Middle panel: the residuals along the semi-major axis of the error ellipsoid. Lower panel: the residuals along the
semi-minor axis of the error ellipsoid.
}
\label{fig:BV-HIP87895}
\end{figure}
\begin{table}
\caption{The combined VB+SB2 solutions of HIP 87895; For consistency with the SB orbits and with the
forthcoming astrometric orbit, $\omega$ refer to the motion of the primary
component.}
\begin{tabular}{@{}lc@{}}
\hline
& HIP 87895 \\
\hline
$P$ (days) & 881.628 $\pm$ 0.064 \\
$T_0$ (BJD-2400000) & 55650.39 $\pm$ 0.38 \\
$e$ & 0.4165 $\pm$ 0.0010 \\
$V_0$ (km~s$^{-1}$) & -32.347 $\pm$ 0.014 \\
$\omega_1$ ($^{\rm o}$) & 135.46 $\pm$ 0.16 \\
$\Omega$($^{\rm o}$; eq. 2000) & 175.32 $\pm$ 0.44 \\
$i$ ($^{\rm o}$) & 72..83 $\pm$ 0.47 \\ex
$a$ (mas) & 80.64 \\
${\cal M}_1$ (${\cal M}_\odot$) & 1.132 $\pm$ 0.014 \\
${\cal M}_2$ (${\cal M}_\odot$) & 0.7421 $\pm$ 0.0073 \\
$\varpi$ (mas) & 36.35 $\pm$ 0.20 \\
$d_{n-p}$ (km~s$^{-1}$) & 0.456 $\pm$ 0.047 \\
$d_{2-1\;p}$ (km~s$^{-1}$) & -0.754 $\pm$ 0.608 \\
$d_{2-1\;n}$ (km~s$^{-1}$) & 0.320 $\pm$ 0.056 \\
$\sigma_{(o-c)\;VLTI}$ (mas) & 0.086 \\
$\sigma_{(o-c)\;RV\;p}$ (km~s$^{-1}$) & 0.841, 1.211 \\
$\sigma_{(o-c)\;RV\;n}$ (km~s$^{-1}$) & 0.035, 0.209 \\
\hline
\end{tabular}
\label{tab:HIP87895}
\end{table}
HIP~87895 is the close visual binary (VB) MCA 50. Among the many observations recorded in the on-line {\it Fourth
Catalogue of Interferometric Measurements of Binary Stars}\footnote{http://ad.usno.navy.mil/wds/int4.html}, we
found 3 very accurate long-base interferometric measurements performed with the {\it Palomar Testbed Interferometer} (PTI).
Since the RV measurements are already providing parameters common to the spectroscopic and to the visual orbits,
three 2-dimension measurements are quite sufficient to derive the remaining ones, which are: the orbital
inclination, $i$, the position angle of the nodal line, $\Omega$, and the apparent semi-major axis or the
trigonometric parallax (both may equally be used).
These measurements and the rescaled error ellipsoids provided by \cite{Muterspaugh} are taken into account
simultaneously with our RV measurements, in order to directly derive the masses of the components and the
trigonometric parallax of the system.
The results are given in Table~\ref{tab:HIP87895}, and in Fig.~\ref{fig:BV-HIP87895}. We derived the two masses with accuracies of 1.2 and 0.98~\%, respectively.
\section{Summary and conclusion}
\label{sec:conclusion}
We obtained for 10 SB2 new radial velocity measurements from spectra taken with SOPHIE, among which four new SB2 identified in paper I. The {\sc todmor} algorithm was used to separate the two components in each spectrum. All ten systems had previously published measurements in archive, which we added to our measurements to calculate the orbital solutions. We also derived estimations of stellar parameters obtained by optimizing the two dimensional cross-correlation obtained with {\sc todmor}.
We achieved the objective of deriving minimum mass with an accuracy better than 1\% for 15 of the 20 stars of our SB2 sample. Moreover, all ten binaries have the minimum mass of both components estimated with relative uncertainties lower than 1.2\%. This is a great achievement of the combination of {\sc todmor} and SOPHIE. Especially it did very well in extreme configurations such as with small SNR ($\sim$40), and very small flux ratio ($\alpha$$<$$0.05$) between the two components.
We combined our RV measurements of HIP~87895 with 3 relative positions derived from long-based interferometric observations, and we obtained new
mass estimate of the components, $M_1$=$1.132$$\pm $$0.014$\,$M_\odot$ and $M_2$=$0.7421$$\pm $$0.0073$\,$M_\odot$.
On the basis of speckle observations, \cite{McAlister95} previously derived $M_1$$=$$1.16$$\pm $$0.12$\,$M_\odot$ and $M_2$$=$$0.77$$\pm $$0.05$\,$M_\odot$, i.e. with relative errors of 10 and 6~\%. Therefore, our measurements refine profitably to the 1\% level the confidence range on the masses of HIP\,87895.
Added to the systems observed with {\sc Pionier}, we have now 3 binaries observed with SOPHIE which may be used to check the masses that will be derived from {\sc Gaia}.
\section*{Acknowledgments}
This project was supported by the french INSU-CNRS ``Programme National de Physique Stellaire''
and ``Action Sp\'{e}cifique {\it Gaia}''. We are grateful to the staff of the
Haute--Provence Observatory, and especially to Dr F. Bouchy, Dr H. Le Coroller, Dr M. V\'{e}ron, and the night assistants, for their
kind assistance. We made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant-agreement numbers 291352 (ERC). The authors are very thankful to the anonymous reviewer for his help to improve the quality of this paper.
\label{lastpage}
|
1,116,691,499,327 | arxiv | \section{Introduction}
A variety of systems such as supercooled liquids, colloids,
granular matter and foams, exhibit a transition from a flowing
fluid phase to a frozen solid phase. Jamming due to spatial
constraints imposed on the elementary constituents of these
materials has been proposed as a possible common cause of this
dynamical arrest \cite{edwards,nagel_jamming,weitz}.
Model systems, such as hard spheres, have an important role to
play in the investigation of such a scenario since they allow for
a precise definition of jamming \cite{torquato}. They are also
useful in elucidating the precise relationship between
thermodynamics and dynamics in materials exhibiting a jammed phase
\cite{krauth}. The entropy-based Adam-Gibbs
theory\cite{Adam_Gibbs}relates the viscosity (a dynamical
quantity) to the configurational entropy ($S_{conf}$)(a
thermodynamic quantity) through $\eta = {\eta}_0
\exp(A/TS_{conf})$. The ideal glass transition is associated with
the Kauzmann temperature at which the configurational entropy
vanishes\cite{Gibbs_dimarzio}. In this paper, we explore the
connection between dynamics and thermodynamics in a lattice model
of dimers with an entropy-vanishing phase transition.
The dimer model is one of the working horses of statistical mechanics.
It provides an example of a jammed system which has the added
advantage of being exactly solvable \cite{nagle_review}. States of
the dimer model are specified by placing dimers on the bonds of
the lattice so that every lattice site is covered by exactly one dimer;
see Fig~\ref{dimer_fig}. These dimer coverings are ``locally
jammed'' \cite{torquato} as every dimer cannot move to an empty,
neighboring bond, without violating the packing constraint. Moves
that involve {\em loops} of dimers and adjacent empty bonds, on
the other hand, are allowed. An example of such a move for the
hexagonal lattice involving an elementary plaquette is shown in
Fig.~\ref{dimer_fig}a. Stochastic dynamics of the dimer model on
the square lattice based on these elementary moves were considered
by Henley \cite{clh}.
Most states of the dimer model allow for
elementary moves; an example of one which does not is shown in
Fig.~\ref{dimer_fig}b. The smallest move in this case involves a
system spanning loop, and we call this state ``maximally jammed''.
If we define an energy functional on the space of dimer coverings
which favors the maximally jammed state, a transition into this
state can be affected as the temperature is lowered. The central question
we address in this paper is: {\em What happens to relaxation time
scales of the dimer model as the transition to the maximally
jammed state is approached?} We will show that the relaxation is
dominated by entropy barriers and is sensitive to
equilibrium fluctuations near the phase transition point.
\figone{dimers_hex}{(a) Dimer covering of the honeycomb
lattice with an elementary loop update indicated by the arrows.
The numbers are the heights of the equivalent interface. (b) An
ordered, maximally jammed dimer covering; the equivalent interface
is tilted with maximum slope.}{dimer_fig}
We consider an energy functional that exhibits a continuous transition to
the maximally jammed state along a metastable line. We find a strong
departure from the canonical critical-slowing-down scenario
\cite{halperin}, which we attribute to the presence of entropy barriers. Barriers
can be traced directly to the non-local nature of the dynamical
moves allowed by the jammed states. The longest relaxation time-scale is
found to diverge exponentially following a Vogel-Fulcher-like
form. This is reminiscent of what is observed in fragile glass
formers \cite{angel_review}.
Exponential time-scale divergence (activated scaling) is associated with critical points in models with quenched
disorder\cite{fisherscaling} and it has been argued that real glasses belong to the universality class of random
Hamiltonians with such exponential divergences\cite{parisi}. The current model provides an explicit example of a
model without quenched disorder which exhibits activated scaling. It should be mentioned that the Vogel-Fulcher law has
been observed in models with entropic barriers \cite{backgammon}, with traps \cite{bouchaud},
and within effective medium theory \cite{kumar}, none of which have
an explicit critical point.
\section{Dimer model}
We consider the dimer model on the 2-d hexagonal
lattice of linear size $L$, having $2 L^2$ sites and $3 L^2$ bonds,
with periodic boundary conditions \cite{kasteleyn}.
A useful representation of the dimer model is given by the height map which
associates a discrete interface $h(x,y)$ with every dimer covering \cite{blote}.
The heights of the interface are defined on the vertices of the dual
triangular lattice. The height difference $\Delta$ between two nearest neighboring
sites is -2 or +1 depending on whether the bond of the honeycomb lattice
that separates them is occupied by a dimer or not; see Fig.~\ref{dimer_fig}a.
Directions in which the height change is $+\Delta$ are specified by orienting
all the up pointing triangles of the dual lattice clockwise.
The dimer model has an extensive entropy. The ensemble of equal
weighted dimer coverings maps to a rough surface with a
gradient-square free energy \cite{blote}. Fluctuations of the surface
are entropic in origin. A phase transition can be induced in the dimer
model by including an energy functional which is minimized by a dimer
covering corresponding to a smooth, {\em maximally tilted} surface
which corresponds to the maximally jammed state shown in
Fig.~\ref{dimer_fig}b.
For periodic boundary conditions the tilt vector, $({\Delta}_x h ,
{\Delta}_y h)$, where ${\Delta}_{x,y} h$ is the average height difference
in the $x$ or $y$ direction, has only one independent component
$\rho$ \cite{foot2}. In terms of $\rho$, the energy functional we consider can be
written as:
\begin{equation} \beta E({\rho}) = -{{\mu L^2} \over 3} ({1}+{8}{{\rho}}^2) ,
\label{energyF}
\end{equation}
where $\mu$ is a dimensionless coupling, proportional to inverse temperature ($\beta = 1/kT$), that drives the transition.
The entropy of the dimer model as a function of $\rho$
was calculated exactly \cite{kasteleyn,adhar}:
\begin{equation}
S({\rho}) = L^2 \left \{ {{2 \ln2} \over 3} (1-{\rho}) +
{2 \over \pi}\int_{0}^{{\pi \over 3}(1-{\rho})}dx \ln[\cos x] \right \}
\label{entropyD1}
\end{equation}
This function has a maximum at $\rho=0$ which is the equilibrium value at
$\mu=0$. For finite $\mu$ this dimer model was previously considered in
Ref.~\cite{hui}. A dimer model with a similar phase transition but
with an energy functional linear in $\rho$ was solved exactly by
Kasteleyn \cite{kasteleyn}.
In the dimer model with the free energy ${\beta}F={\beta}E-S$, and the energy and
entropy given by Eqs.~\ref{energyF} and~\ref{entropyD1}, there is an
interesting phase transition along the metastable line,
when the order parameter is confined to the free energy well around
the zero-tilt state. Namely, at $\mu_{*} = \pi/(8\sqrt{3})$, the
end-point of the metastable line, the order parameter ${\rho}$ has a
discontinuous jump from $0$ to $1$, characteristic of a first-order
transition. At the same time, as $\mu_{*}$ is approached from below,
fluctuations of ${\rho}$ around $0$ diverge, as would be expected at a
critical point. This transition was discussed in detail in Ref.~\cite{hui}.
In this paper we investigate the dynamics of the dimer model near this
phase transition point.
\section{Coarse-grained dynamics}
As mentioned in the introduction, the hard constraint of
no overlapping of dimers, gives rise to nonlocal dynamics. We consider stochastic,
Monte-Carlo dynamics based on loop updates with
loops of arbitrary size; a concrete implementation
is given in Ref.~\cite{krauth1}. Since we
take periodic boundary conditions, loops with different winding
numbers can be formed. We restrict loop updates to loops with winding
numbers $(0,0), (1,0)$ and $(0,1)$, only. The microscopic transition rates for
loop updates are given by Metropolis rules that follow from the energy function,
Eq.~\ref{energyF}.
Given the microscopic loop dynamics, which satisfy conditions of ergodicity and
detailed balance, we ask what are the coarse-grained dynamics of the order
parameter, $\rho$. Since the energy function in Eq.~\ref{energyF} depend on the global tilt
$\rho$ only, it follows that all updates of topologically trivial
loops (i.e.~those with $(0,0)$ winding number) have $\Delta E =
0$. Only when system spanning loops with nonzero winding numbers are
updated does the energy of the state change.
This feature naturally leads to fast and slow processes in the
Monte-Carlo dynamics. On a faster time scale, non-winding loops are
updated with no effect on the overall tilt of the surface, while on a
much slower time scale, winding loops are updated causing a change
in the tilt of the surface.
The {\it coarse-grained} dynamics of global tilt changes are described
by a master equation for the probability ($P_\rho$), that the dimer model has
tilt $\rho$,
\begin{equation}
{d P_{\rho} \over d t} = - \left[ W_{{\rho-1/L}, \rho} + W_{{\rho+1/L}, \rho} \right ] P_{\rho} +
W_{{\rho},{\rho-1/L}} P_{\rho -1/L} + W_{{\rho},{\rho+1/L}} P_{\rho+1/L} \ .
\label{MasterEqn}
\end{equation}
The rates in this master equation obey the detailed balance condition: $W_{{\rho-1/L}, \rho}/
W_{{\rho},{\rho-1/L}} = \exp[-(F(\rho-1/L)-F(\rho))]$. The usual way of achieving this balance
which leads to normal diffusive dynamics is to partition the rates symmetrically with $W_{{\rho-1/L}, \rho} \simeq \exp[-(F(\rho-1/L)-F(\rho))/2]$
and $W_{{\rho},{\rho-1/L}} = \exp[(F(\rho-1/L)-F(\rho))/2]$\cite{reichl_book}.
Equation \ref{MasterEqn}, however, features an unusual form for the transition
rates between different tilt states. Namely, the rates of transitions from
higher into lower tilt states (increasing energy transitions) are determined by the energy change alone:
\begin{equation}
W_{{\rho-1/L}, \rho} = \Gamma_0 e^{-(E(\rho-1/L)-E(\rho))} \ ;
\label{barr1}
\end{equation}
here $\Gamma_0$ is a constant.
This follows from the observation that in order to lower the tilt and increase the
energy, a system spanning loop, which is always present in a state with $\rho\neq 0$,
needs to be updated. This form of the rates for energy-increasing transitions in conjunction with the detailed balance condition
implies that the rates of transitions to higher tilt states (energy lowering transitions) must be
determined by the entropy change: \begin{equation} W_{{\rho},{\rho-1/L}} = \Gamma_0
e^{-(S(\rho-1/L)-S(\rho))} \ . \label{barr2} \end{equation}
The form of the transition rates that we are arguing for here, was
directly observed in numerical simulations of the three coloring model
\cite{loops_epl}, which is a close relative of the dimer model. The
two are equivalent if, in the dimer models, a weight of 2 is attached
to each loop formed by bonds that are not covered by dimers.
\figone{taucorr}{(a) The time scales for relaxing out of
different tilt states $\rho$ in the dimer model (scaled by
$L$), for a value of $\mu$ below the transition. (b) The tilt-tilt
autocorrelation function of the dimer model. The full line is
obtained from Eq.~\ref{corr2} while the dashed line is a result of
the saddle point evaluation of Eq.~\ref{corr2}. Here $L=4096$ and
time is measured in units of $\Gamma_{0}^{-1}$. }{Tau_fig}
\section{Relaxation time-scales}
The first consequence of the above form of the transition rates
is that the time scale of relaxation out of a state with tilt
$\rho$, $\tau_{\rho} = 1/(W_{{\rho-1/L}, \rho} + W_{{\rho+1/L},
\rho})$, is a {\it non-decreasing} function of $\rho$. The exact
expressions for $\tau_{\rho}$ (measured in units of
$\Gamma_{0}^{-1}$),
\begin{equation}
{\tau_{\rho}}^{-1} =
{ {e^{-{{16 \over 3} {\rho}\mu L}}+
e^{-L[{2 \over 3}{{\rm ln}2}+{2 \over 3}{{\rm ln}[\cos({\pi \over
3} - {{\pi {\rho}} \over 3})]}]}}} ,
\label{tauD1}
\end{equation}
follows from Eqs.~\ref{barr1} and~\ref{barr2},
and it is plotted in Fig.~\ref{Tau_fig}a). This
time-scale increases monotonically with $\rho$\cite{foot1}, as in
the hierarchical models of Palmer {\em et al.}~\cite{palmer}.
It is in sharp contrast with canonical Langevin
dynamics around the equilibrium state, for which the time to relax
out of a macro-state {\em decreases} the further the order
parameter is away from its equilibrium value. (For example, in the
Ising model with Glauber dynamics and in the disordered phase,
the relaxation time out of a given magnetization state {\it
decreases} with increasing magnetization.)
\section{Autocorrelation function}
To quantify the tilt dynamics we compute the tilt-tilt
autocorrelation function $C(t)$, defined as: \begin{equation} C(t) = {{\langle
\rho(t) \rho(0) \rangle - {\langle \rho(0) \rangle}^{2}} \over
{\langle {\rho(0)}^2 \rangle - {\langle \rho(0) \rangle}^{2}}},
\label{corr1} \end{equation} with the average taken over different histories
of $\rho$. An approximate form for the autocorrelation function
is: \begin{equation} C(t) \approx {\sum_{\rho}{(\rho - {\langle \rho
\rangle})}^2 e^{-F(\rho)} e^{-t/{\tau_{\rho}}} \over
{\sum_{\rho}{(\rho - {\langle \rho \rangle})}^2 e^{-F(\rho)}}} ,
\label{corr2} \end{equation} i.e., $C(t)$ is an equilibrium weighted average
of relaxations out of different $\rho$ states. This approximation
is based on the assumption that eigenfunctions of the rate matrix
are localized in $\rho$-space and eigenfunctions corresponding to
different eigenvalues do not have significant overlaps. We will
justify this assumption {\it a posteriori} by examining the
eigenfunctions obtained from numerical diagonalizations of the
rate matrix $W_{\rho,\rho'}$.
The asymptotic decay of the autocorrelation functions can be
extracted by performing a saddle point analysis of the sum in
Eq.~\ref{corr2} and using a quadratic approximation for the
entropy (Eq.~\ref{entropyD1}). These saddle point solution is
compared in Fig.~\ref{Tau_fig}b) to the result obtained from the
sum (Eq.~\ref{corr2}).
In the limit of $\mu\to \mu_*$ and $t\to \infty$, saddle point
analysis yields:
\begin{equation}
C(t) \sim
\exp\{-{3 \over 32}({{\mu_* - \mu} \over {\mu_*}^2})[{\rm ln}({{2 \mu_* t} \over {\mu_* - \mu}})]^2\} ,
\label{corrD1q} \end{equation}
showing that $C(t)$ in the dimer model has a log-normal form implying a slower than exponential decay. From Eq.~\ref{corrD1q} we also conclude that the relaxation
timescale, $\tau$, for the decay of $C(t)$ to an arbitrary
constant $C_0$, diverges exponentially as $\mu \rightarrow \mu_*$.
This is a Vogel-Fulcher type behavior (since $\mu$ is proportional
to $\beta = 1/{k T}$)
observed in many fragile glass formers. First order corrections
to Eq.~\ref{corrD1q} lead to an even more rapid increase of time
scales, with ${\tau}/{\rm ln}{\tau}$ diverging as Vogel-Fulcher.
\figone{free_energy}{Barrier height, $B(\rho)$ (dimensionless) shown as a set of solid
lines, and the quadratic approximation to the dimensionless free energies of the
dimer model (dashed line); $\mu$ is chosen close to ${\mu}_*$ and $L = 24$. Note the
logarithmic scale for the barrier height. }{barrier_fig}
The coarse grained dynamics defined by the transition matrix
elements, Eqs.~\ref{barr1} and~\ref{barr2}, were argued to follow
from the nonlocal loop dynamics of the dimer models. From this
form of the $W$-matrix all the conclusions about critical dynamics
of the dimer model are derived. We have confirmed this
picture in considerable detail in simulations of the three
coloring model \cite{messina,Kolkata}, which, as discussed
earlier, is the loop weighted dimer model. The loop weights are
not expected to affect the qualitative features of the energy and
entropy functionals. Indeed, the measured $\tau_\rho$ for the
three-coloring model compare very well
\cite{messina,Kolkata} to the analytical form plotted in
Fig.~\ref{Tau_fig}. The numerical evidence for Vogel-Fulcher type
divergence of the relaxation time scale in this model was
reported previously \cite{messina}.
The dynamical behavior of the dimer model can be traced
back to the interplay between the free energy and dynamical
barriers. The transition rates presented in Eqs.~\ref{barr1} and
\ref{barr2}, can be interpreted in terms of a barrier\cite{palmer}
$B({\rho}) = e^{(S(\rho-1/L)-S(\rho)+(E(\rho-1/L)-E(\rho)))/2}$
dividing the usual Metropolis rates defined in terms of the free
energy:
\begin{equation}
W_{{\rho-1/L}, \rho} = \Gamma_0
e^{-(F(\rho-1/L)-F(\rho))/2}/B({\rho}) \;
\label{barrier1}
\end{equation}
and
\begin{equation}
W_{\rho, \rho-1/L} = \Gamma_0
e^{(F(\rho-1/L)-F(\rho))/2}/B({\rho}) \
\label{barrier2}
\end{equation}
The barriers increase exponentially with $\rho$ as illustrated in
Fig.~\ref{barrier_fig}. Dynamics of the order parameter can be
viewed as relaxation in the free energy well in the presence of
these barriers.
\section{Scaling analysis of the master equation}
The emergence of the Vogel-Fulcher law in the dimer model, based on the
master equation with transition rates defined in
Eqs.~\ref{barrier1} and \ref{barrier2}, follows from a scaling
argument.
At the critical point, $\mu = {\mu}^*$, the free energy difference
between different tilt states close to $\rho =0$ vanishes. In
this limit, the transition rates are symmetric and given by:
\begin{equation}
W_{\rho -1/L, \rho} = W_{\rho , \rho -1/L}= \Gamma_0 /B({\rho}) .
\label{symmetric}
\end{equation}
The diffusion constant in this symmetric case can be shown to be
given by \cite{Derrida,Zwanzig}:
\begin{equation}
D(L) = L \Gamma_0 /({\sum}_{i = -L,L}B({\rho}_i))
\label{Dsum}
\end{equation}
where $L$ is the system size and ${\rho}_i = i/L$. The longest
timescale in the problem is given by
\begin{equation}
\tau (L) = L^2 /D(L)
\end{equation}
In the limit of large $L$, the summation in Eq.~\ref{Dsum} can be replaced by an
integral. If we make use of the quadratic approximation to the entropy (Eq.~\ref{entropyD1}) then
$B(\rho ) = e^{(8/3)(\mu + {\mu}_* )L \rho }$ and the integral can be evaluated analytically, with the
result:
\begin{equation}
\tau (L) = {L \over \Gamma_0} \left( e^{{16 \over 3} \mu_* L}-1 \right )
\end{equation}
Note that the same result can be obtained by replacing the summation
by the largest barrier which occurs at $\rho =1$. The longest
time-scale in the system is, therefore, seen to diverge {\it
exponentially} with system size. If all the barriers were equal to one,
then we would have $D(L) = \Gamma_0$ and $\tau(L) = L^2/ \Gamma_0$, which
corresponds to simple diffusion. In the presence of the barriers, $D(L)$ goes to
zero exponentially and this leads to an exponential divergence of the relaxation time-scale.
We can now use a scaling argument to deduce the behavior of the relaxation time-scale for $\mu
< {\mu}_*$. The effective scale (in $\rho$-space) over which the free-energy
well is flat, and therefore the transition rates are symmetric,
diverges as the phase transition at $\mu = {\mu}_*$ is approached. We argue that this
length scale, given by $l(\mu) = {{\sqrt{\mu}_*} \over
{\sqrt{{\mu}_* -\mu}}}$, provides a cutoff to the
summation (or integral) involved in calculating the diffusion constant
\begin{equation}
D(l) = \Gamma_{0}l/({\sum}_{i = -l(\mu ),l (\mu )}B({\rho}_i))
\end{equation}
The longest time scale therefore scales as
\begin{equation}
\tau(\mu) = \tau (l(\mu)); \ \ l(\mu)<L
\end{equation}
and
\begin{equation}
\tau(\mu) = \tau (L); \ \ l(\mu) \geq L \ .
\end{equation}
In the thermodynamic limit, $\tau (\mu )$ diverges as $\tau (\mu )
\simeq e^{(16/3){{\sqrt{\mu}_*} \over {\sqrt{{\mu}_* -\mu}}}}$. Since $\mu \simeq 1/T$, $\tau$ has
a Vogel-Fulcher type divergence $\tau (T) \simeq e^{(16/3){{\sqrt{T}_*} \over {\sqrt{T -T_{*}}}}}$.
We have recently shown that the Vogel-Fulcher divergence of the
dimer model can be obtained from an exact solution of the continuum
version of the master equation if we assume that $S(\rho)$ has a
quadratic form \cite{satya_rapid}.
\section{Numerical analysis of the master equation}
We have carried out numerical diagonalizations of the rate matrix
for $L{\rm x}L$ systems in order to verify some of the
assumptions that have been made in the scaling analysis and in the
calculation of the correlation function. These computations also provide us with information
about the finite-size effects on the critical dynamics of the dimer model.
The probability distribution, $P(\rho, t)$, can be written in
terms of the eigenvalues, $\lambda_i$, and eigenfunctions, $\psi_i
(\rho)$ of the rate matrix\cite{reichl_book}:
\begin{equation}
P(\rho, t) = {\sum}_i \psi_i (\rho) e^{-{\lambda}_i t}
\label{eigen}
\end{equation}
The eigenvalues of the rate matrix are non-negative and the
equilibrium distribution is given by the zero-eigenvalue function,
${\psi_1}$ ($\lambda_1 =0$). The smallest, non-zero eigenvalue characterizes the
state with the longest relaxation time. All correlation functions
can be expressed in terms of the eigenvalue spectrum and, in
particular, the equilibrium, tilt-tilt autocorrelation function
can be written as:
\begin{equation}
C(t) = {\sum}_i e^{-\lambda_i t} {\sum}_{\rho \rho^\prime} \rho
\rho^\prime e^{-(F_{\rho} + F_{\rho^\prime})/2}
\psi_i(\rho) \psi_i^*(\rho^\prime) \label{autocorr_eigen}
\end{equation}
Comparing to Eq.~\ref{corr2} it follows that the approximate
form is obtained in the limit of delta-function localized
eigenfunctions. We will show below that the eigenfunctions
corresponding to non-zero eigenvalues of the dimer model are indeed
well localized.
Results of the numerical diagonalization, using the exact
entropy function, show that the longest time scale, $\tau$, of the dimer model
increases in a non-arrhenius, Vogel-Fulcher fashion, as shown in
Fig.~\ref{scaled_fig}. The scaling in this figure is what is
expected from the scaling solution of the model with the quadratic
entropy \cite{satya_rapid} and similar to the results obtained from
the scaling arguments presented in this paper, however the length
scale emerging is $l(\mu)=[{\mu^* \over {\mu^* - \mu}}]^2$ not
$l(\mu) = {{\sqrt{\mu}_*} \over {\sqrt{{\mu}_* -\mu}}}$, as would
be expected from the quadratic entropy results \cite{satya_rapid}.
\figone{longpaperfig5}{Scaling of $\tau(\mu,L)$ in the dimer
model. The figure shows that a scaling form can be constructed in
terms of the lengthscale $l(\mu)=(\mu^* /(\mu^* - \mu))^2$ and the
particular form demonstrates the Vogel-Fulcher scaling ${\tau}
\simeq e^{(l(\mu))^2}$.}{scaled_fig}
The exact
diagonalization results show, unambiguously, that the Vogel-Fulcher law
characterizes the time scale divergence at the entropy vanishing
transition in the dimer model.
\subsection{Eigenfunctions}
In the dimer model, the eigenfunction corresponding to the
smallest non-zero eigenvalue is localized at the largest barrier,
i.e, the largest value of $\rho$. Higher eigenfunction move
to smaller barriers but are still localized. The expression we
used for $C(t)$ is exact for delta-function localization of
eigenfunctions and the numerical results justify this assumption,
{\it a posteriori}.
\figone{eigenfn_I}{Plots of the eighth and twenty-fifth
eigenfunctions of the dimer model for L=16 and $\mu \simeq \mu*$. These two eigenfunctions
were chosen to illustrate the localization of the eigenfunctions
and the shift towards $\rho =0$ with increasing spectral
index}{eigenfn_fig}
\subsection{Sensitivity of dynamics to barrier size}
The sensitivity of the relaxation times to the barriers heights has
been investigated by using the quadratic entropy model and
writing $B({\rho}) = e^[{c \over 2}
{(S(\rho-1/L)-S(\rho)+(E(\rho-1/L)-E(\rho)))/2}]$ and varying $c$
between $0$ and $1$. For $c=0$, we recover the usual Langevin
dynamics and the time scales should increase as a power law and
for $c=1$ we have barriers corresponding to the loop dynamics. The
results plotted in Fig \ref{barrier_strength} demonstrate that,
for $c=0$, $\tau \simeq (\mu^* - \mu)^{-1}$ which is consistent
with a dynamical exponent $z=2$ and a correlation length exponent
of $\nu = 1/2$; the exponents expected from a Langevin description
of a mean-field model. It is also clearly seen from this figure
that even for $c=0.25$, the timescale increases more rapidly than
a power law. An analysis of the continuum limit of the dimer
model dynamics shows that, for any non-zero
value of $c$, $\tau \simeq e^{A(c^{2}l(\mu))}$ where $A$ is a constant \cite{satya_rapid,dibunpub}.
These results taken all together imply that there is a
whole class of systems, where dynamical constraints may lead to non-zero values
of $c$, which belong to different universality classes of
dynamical critical phenomena. These are characterized by a Vogel-Fulcher rather than a
power-law divergence of relaxation time-scales, with $c$ being an
indicator of fragility \cite{angel_review}. In real systems, such as
supercooled liquids, one expects that there is a large but finite
energy scale at which the hard constraints are violated. This energy scale then leads to a
long-time cutoff of the Vogel-Fulcher behavior. This time
scale, may however, be well beyond any experimentally measurable
time scales.
\figone{barrier_strngth}{Timescale $\tau$ in the dimer model for
different values of the barrier strength, $c$, plotted as a
function of $l(\mu) = {\mu^* \over {\mu^* - \mu}}$}{barrier_strength}
\section{Adam-Gibbs scenario}
The entropy of the dimer model, $S_{conf}$,
which corresponds to $S(\rho)$ evaluated at the equilibrium value of the order parameter $\rho$,
goes to zero at the transition. Furthermore, our results clearly show that the longest time
scale diverges in a Vogel-Fulcher manner. The Adam-Gibbs
relation, $\tau = {\tau}_0 e^{A/{S_{conf}(T)}}$, however, does not
capture the physics since in the dimer model, $S_{conf}$ jumps from a finite value at $\rho =0$ to
zero, as $\rho$ changes discontinuously to $1$.
Thus, in this model, the exponential divergence of the relaxation time-scale at the transition is not
accompanied by a continuous vanishing of $S_{conf}$.
The analysis
presented in this paper clearly demonstrates that the
Vogel-Fulcher divergence is rooted in the constrains which lead to loop dynamics. This type of nonlocal dynamics leads to
the unusual transition rates with energy-lowering transitions being determined by
changes in entropy and, therefore, to an exponential decrease of
the number of energy-lowering trajectories as one approaches the
zero-entropy state.
The configurational entropy of supercooled liquids has been
interpreted as the inherent structure entropy, i.e., the number of
valleys at the temperature of interest. The observation that the
Adam-Gibbs scenario describes much of the phenomenology of
supercooled liquids could imply that there is a phase transition in
the inherent-structure space similar to the one discussed in this paper
for dimers. Experiments and simulations have shown that a hallmark of supercooled liquids approaching the glass
transition is the appearance of dynamical heterogeneities\cite{weitz,ediger,glotzer}. The loops in the dimer
dynamics are analogs of these dynamical heterogeneities since they define the correlated moves allowed by the constraints.
These heterogeneities are present as long as the constraints are not violated and are characterized by a size distribution
which changes as the critical point is approached\cite{Kolkata,messina}. The analogy between loops and dynamical heterogeneities
suggest that the dynamics in the inherent structure space of supercooled liquids could be similar to the loop dynamics of dimer models.
If this is the case then transition rates between inherent structures should exhibit features similar to the ones discussed in this paper.
We
are currently in the process of analyzing
transition rates between inherent structures of
Lennard-Jones glass formers in order to get a better understanding of the connection between dynamical heterogeneities and the effective
dynamics in the inherent-structure space.
The authors would like to acknowledge numerous useful discussions
with Satya Majumdar. This work was supported by grants from the
NSF: DMR-0207106 (DD and BC),
DMR-9984471 (JK) and
DMR-0403997 (BC and JK). JK is a Cottrell Scholar of Research Corporation.
|
1,116,691,499,328 | arxiv | \section{Introduction}
Radial abundance gradients in the galactic disk and their time
variations are among the main constraints of chemical evolution
models for the Milky Way. These gradients can be determined from
a variety of objects, such as HII regions, cepheid variables,
open clusters and planetary nebulae (PN). In a recent series of
papers, Maciel et al. (\cite{mcu}, \cite{mlc1}, \cite{mlc2})
estimated the time variation of the radial abundance gradients
taking into account a large sample of PN for which abundances
of O/H, S/H, Ne/H and Ar/H have been derived. Based on individual
estimates of the progenitor star ages, it was concluded that
the radial gradients are flattening out at an average rate of
about $0.005-0.010\, {\rm dex}\, {\rm kpc}^{-1}\,{\rm Gyr}^{-1}$
for the last 8 Gyr, approximately. A comparison of the PN
gradients with results from HII regions, OB stars and associations,
cepheids and, especially, open cluster stars, strongly
supports these conclusions.
On the other hand, it has long been known that a positive
electron temperature gradient of about $250-450\,$K/kpc is
observed in the galactic disk, mainly on the basis of radio
recombination line work on HII regions (see for example
Churchwell \& Walmsley \cite{cw}, Churchwell et al.
\cite{csmmh}, Shaver et al. \cite{shaver}, Wink et al.
\cite{wink}, Afflerbach et al. \cite{afflerbach}, and
Deharveng et al. \cite{deharveng}). Such a gradient
is interpreted as a reflection of the radial abundance
gradient of elements such as O/H, S/H, etc. in the galactic
disk, since these elements are effective coolants of
the ionized gas (see for example Shaver et al. \cite{shaver}).
Recently, Quireza et al. (\cite{cintia}) presented a detailed
study of a large sample containing over a hundred HII regions
spanning about 17 kpc in galactocentric distances for
which accurate electron temperatures were determined from
radio recombination lines, specifically H91$\alpha$ and
He91$\alpha$. The observations were made with the 140 Foot
telescope of the National Radio Astronomy Observatory
(NRAO), and are of unprecedented sensitivity compared
with previous studies. According to this work, the best estimate
of the gradient, obtained from a sample of 76 sources
with high quality data, is $dT_e/dR \simeq 287 \pm 46\,$K/kpc,
with no significant variations along the galactocentric
distances. A slightly larger gradient (up to 17\%) was
obtained by excluding some HII regions which are closer
to the galactic centre, and may not belong to the
disk population.
Regarding planetary nebulae, our earlier work (Maciel \&
Fa\'undez-Abans \cite{mfa}) based on a sample of PN
classified according to the Peimbert types (cf. Peimbert \cite{mp})
suggested a positive electron temperature gradient in the
range $550-800\,$K/kpc, somewhat steeper than the HII region
gradients observed at the time.
In this work, we take into account the recent PN samples
analyzed by Maciel et al. (\cite{mcu}, \cite{mlc1}, \cite{mlc2})
and derive the PN electron temperature gradient for a sample
of objects having similar ages. A comparison of the obtained $T_e$
gradient with the recently derived value by Quireza et al.
(\cite{cintia}) for HII regions gives then an independent
estimate of the time variation of the radial abundance gradients
in the galactic disk.
\section{The electron temperature gradient in the galactic disk}
The determination of abundance gradients is a difficult task,
basically for three main reasons. First, the magnitudes of the
gradients are small, amounting at most to a few hundredths
in units of dex/kpc, so that a relatively large galactocentric
baseline is needed in order to obtain meaningful results. Second,
the uncertainties both in the abundances and in the distances
contribute to the observed scattering, so that large samples
are usually needed. Third, chemical evolution models generally
predict some time variation of the gradients, so that it is
extremely important to take into account in a given sample only
objects with similar ages. For these reasons, some of the analyses
of gradients in the literature produce relatively flat gradients
(see for example Perinotto \& Morbidelli \cite{pm}). On the
other hand, accurate and homogeneous abundances eliminate some
of these problems, so that relatively steeper gradients are obtained,
as in Pottasch \& Bernard-Salas (\cite{pbs}).
In our recent work, we made an attempt to overcome some of
these problems, and estimated the individual ages of the PN
progenitor stars using an age-metallicity
relationship which also depends on the galactocentric distance.
As a result, we have obtained the age distribution shown
in Fig.~\ref{histog}, adopting our Basic Sample, which is the
largest and most complete sample we have considered, containing
234 nebulae (see Maciel et al. \cite{mcu}, \cite{mlc1}, \cite{mlc2}
for details).
\begin{figure}
\centering
\includegraphics[angle=-90,width=8cm]{6916fig1.eps}
\caption{Age distribution of the PN progenitor stars
in the Basic Sample of Maciel et al. (\cite{mlc2}).
}
\label{histog}
\end{figure}
It can be seen that the ages are strongly peaked around
4--5 Gyr, where we have 99 objects. In the present work,
we will consider the objects in this age bracket, in order
to make comparisons with the younger HII regions. We have
then collected the electron temperatures of the planetary
nebulae, selecting only the [OIII] temperatures in order
to keep our sample as homogenous as possible. These
temperatures are determined from the ratio of the
[OIII] 4363/5007\AA\ lines, which are usually among
the brightest collisionally excited emission lines
in the spectra of planetary nebulae. We have
preferred our own data where available (Costa et al.
\cite{costa1}, \cite{costa2}, see a list of
references in Maciel et al. \cite{mcu}, \cite{mlc1},
\cite{mlc2}), with additional data by Henry et al.
(\cite{henry}), Kingsburgh \& Barlow (\cite{kingsburgh}),
and Cahn et al. (\cite{cks}). The resulting $T_e$
variation with galactocentric distance $R$ is shown
in Fig.~\ref{gradte}, where we adopted the same distances
and solar galactocentric radius as in our previous work.
The total number of objects in Fig.~\ref{gradte} is somewhat
lower than shown in the 4--5 Gyr bracket of Fig.~\ref{histog},
as for a few nebulae we could not obtain accurate electron
temperatures.
\begin{figure}
\centering
\includegraphics[angle=-90,width=8cm]{6916fig2.eps}
\caption{Galactocentric variation of the [OIII]
electron temperatures for PN with progenitor ages
of 4--5 Gyr. The empty circles show some
nebulae having extremely hot central stars,
not included in the linear regression analysis.
}
\label{gradte}
\end{figure}
It can be seen from Fig.~\ref{gradte} that there is
a clear tendency in the sense that higher electron
temperatures are associated with larger galactocentric
distances. The best derived $T_e$ gradient for this sample
of PN is $dT_e/dR \simeq 670\pm 65\,$K/kpc, with a correlation
coefficient of $r \simeq 0.76$, which is similar
to the gradient for the `selected sample' of
our earlier paper (Maciel \& Fa\'undez-Abans \cite{mfa}).
Adopting instead a homogeneous set of [OIII] electron
temperatures from Henry et al. (\cite{henry}), which
is the largest homogeneous sample available for
these nebulae, we obtain essentially the same
result, namely $dT_e/dR = 680\pm 140\,$K/kpc and
$r \simeq 0.65$, so that the correlation is real.
Our best derived slope is illustrated by the
dashed line in Fig.~\ref{gradte}.
The average uncertainty in the determination of the
electron temperatures is generally considered to
be within 10\% for the brightest nebulae, which
corresponds roughly to 1000 K for most objects
(see for example Kingsburgh \& Barlow \cite{kingsburgh}
and Krabbe \& Copetti \cite{krabbe}).
For the galactocentric distances an average uncertainty
is more difficult to establish, as it depends on
the adopted distances. Since most objects in
Fig.~\ref{gradte} are located within about 3 kpc
from the solar galactocentric radius, an average
error of 50\% in the distances would correspond
to a shift in the galactocentric distances of
about 0.03 -- 1.0 kpc depending on the distance
and the direction of the line of sight to the
nebula. As a comparison, for the HII regions
in the sample by Quireza et al. ({\cite{cintia}),
average formal uncertainties in the electron temperatures
are within 2\%, but systematic errors may increase this
uncertainty up to 10\% for the best data. For
spectrophotometric distances, average errors of
15\% are quoted, while for kinematic distances non-circular
streaming motions may increase this figure to about 25\%.
From the $T_e \times R$ plot by
Quireza et al. ({\cite{cintia}), an average
dispersion of about 2200~K can be obtained,
which is about half the dispersion in
Fig.~\ref{gradte}.
It should be noted that the dispersion observed in
Fig.~\ref{gradte} is probably real, since the electron
temperatures may be affected by several factors, such
as differences in the effective temperature of the central
stars, presence of dust, optical depth effects,
electron density and temperature fluctuations, etc., apart from
the main cause of the $T_e \times R$ variation, namely,
the radial abundance gradient. These effects are also
partially responsible for the observed dispersion
in HII regions, but for PN the variations in the
effective temperatures of the central stars and the
uncertainties in the distances are larger, so that
the observed dispersion in Fig.~\ref{gradte} is larger
than in the case of HII regions. As a consequence, some
nebulae appear not to follow the observed correlation
very closely. In particular, PN having extremely
hot central stars, with temperatures in excess of
$10^5\,$K, generally have higher electron temperatures
than expected by their galactocentric distances.
Furthermore, these objects come from more massive
progenitors than most nebulae, so that their ages may be
lower than 4--5 Gyr, as assumed. Some examples include
M1-57, Me2-1, PB6, NGC 6620 and a few others, which
are plotted in the figure as empty circles. Other objects
with hot central stars, such as NGC 2899, NGC 6302,
NGC 6537 and NGC 7008 are not plotted in Fig.~\ref{gradte},
as their electron temperatures are too high to fit the
scale. All these nebulae have not been included in the
linear regression, so that our derived electron temperature
gradient applies to stars with temperatures lower than
$10^5\,$K. In this analysis we have used Zanstra
temperatures and energy-balance temperatures (see for
example Preite-Martinez et al. \cite{andrea}, M\'endez et al.
\cite{mendez}, Zhang \cite{zhang}, and Stasi\'nska et al.
\cite{stasinska}).
Another interesting object is M1-9,
which is the nebula with the largest
galactocentric {\bf distance} in the sample ($R \simeq 12.4\,$kpc).
In view of its position on the $T_e \times R$ plane,
it may single-handedly affect the derived slope. For this object, our
own results suggest an electron temperature of
$T_e \simeq 11000\,$K (Costa et al. \cite{costa2}),
but a more detailed study by Tamura \& Shibata
(\cite{ts90}) and Shibata \& Tamura (\cite{st85})
gives a larger value, $T_e \simeq 14800\,$K,
which is adopted here. This object may then alter
the derived slope by about 50 K/kpc, but the main
conclusions of this paper are unaffected.
The association of higher electron temperatures with
lower metallicities can be seen from Fig.~\ref{teoh},
where we show the inverse correlation between the
[OIII] electron temperatures and the O/H abundances
for the objects with ages in the 4--5 Gyr bracket.
PN with central stars hotter than $10^5\,$K
are also shown as empty circles.
Again a relatively large dispersion is observed,
but the inverse correlation is clear, confirming
that oxygen is among the main coolants in the
photoionized gas within the planetary nebulae.
\begin{figure}
\centering
\includegraphics[angle=-90,width=8cm]{6916fig3.eps}
\caption{The inverse correlation between the
[OIII] electron temperatures and the O/H
abundances for PN in the 4--5 Gyr
age bracket. The empty circles show some
nebulae having extremely hot central stars.
}
\label{teoh}
\end{figure}
\section{Comparison of the PN and HII gradients}
From a straigthforward comparison of the electron temperature gradient
for PN and HII regions we can already conclude that there is some
flattening of the gradients during the time interval
of about 5 Gyr, which is essentially the difference between the
ages of the PN progenitor stars considered in this work and
the much younger HII regions in the sample by Quireza et al.
(\cite{cintia}). In other words, the conclusions of our recent
series of papers are supported by the obtained differences between
the electron temperature gradients of PN and HII regions,
in view of the fact that these gradients essentially reflect
the radial abundance gradients in the galactic disk.
In order to make a direct comparison
with the estimated flattening rate of the abundance gradients
derived by Maciel et al. (\cite{mlc1}), we can convert the
$T_e$ gradient into the equivalent O/H gradient. For HII regions
we can use the calibration by Shaver et al. (\cite{shaver}),
according to which the oxygen gradient is related to the
electron temperature gradient by
\begin{equation}
{d \log({\rm O/H}) \over dR} \simeq -1.49 \times 10^{-4} \
{d T_e \over dR} \,
\end{equation}
\noindent
where the temperature gradient is in K/kpc and the oxygen gradient
is in dex/kpc. This relation is also supported by more recent work
on HII regions, such as the analysis by Deharveng et al. (\cite{deharveng}).
The oxygen gradient for HII regions is then
$d\log({\rm O/H})/dR \simeq -0.043\,$dex/kpc,
which is similar to the value obtained by Deharveng et al. (\cite{deharveng})
based on an entirely different sample. Also, an O/H gradient
of $-0.044$ dex/kpc was recently obtained by Esteban et al.
(\cite{esteban}) from oxygen recombination lines, a method almost
independent of the assumed electron temperatures, and totally
independent of the [OIII] forbidden lines. A similar estimate has been
presented by Quireza et al. (\cite{cintia}), also based on the Shaver
et al. ({\cite{shaver}) calibration.
For planetary nebulae, we can have an idea of the corresponding O/H
gradient by inspecting Table 1 of Maciel et al. ({\cite{mlc1}) for
Group II objects (ages of 4--5 Gyr), from which we get
$d\log({\rm O/H})/dR \simeq -0.089 \pm 0.003\,$dex/kpc.
Alternatively, we can compute the O/H gradient directly for the PN sample
adopted here, in which case we get a similar value, $d\log({\rm O/H})/dR
\simeq -0.090\,$dex/kpc. Therefore, the flattening rate of the oxygen
gradient can be estimated by
\begin{equation}
\chi \simeq {1 \over \Delta T}
\left[{d \log({\rm O/H}) \over dR}\Big\arrowvert_{HII}
- {d \log({\rm O/H}) \over dR}\Big\arrowvert_{PN}\right]
\,
\end{equation}
\noindent
so that $\chi \simeq 0.0094\, {\rm dex}\, {\rm kpc}^{-1}\, {\rm Gyr}^{-1}$
where we have used $\Delta t \simeq 5\,$Gyr. In view of our discussion
on the electron temperatures of planetary nebulae, we would expect
the rate to be somewhat smaller than this value, which is
in excellent agreement with the range estimated by Maciel et al. (\cite{mlc1}),
namely $\chi \simeq 0.005-0.010\ {\rm dex}\, {\rm kpc}^{-1}\, {\rm Gyr}^{-1}$.
As mentioned in the introduction, abundance gradients and
their time variation are valuable constraints for chemical
evolution models. As an illustration, we have compared our
derived flattening rate with the predictions of some recently
published models for the Milky Way. As discussed by Maciel
et al. (\cite{mlc2}) there may be large discrepancies between
different chemical evolution models, even whithin the
so-called `inside-out' class of models. In particular,
Hou et al. (\cite{hou}) adopted an exponentially decreasing
infall rate for the galactic disk, in which a rapid increase in
the metal abundance at early times in the inner disk leads
to a steep gradient. As times goes on, the star formation migrates
to the outer disk and metal abundances are enhanced in that region,
with the consequence that the gradients flatten out. A rough
estimate for the O/H gradient variation in these models
leads to a steepening rate of
$\chi \simeq 0.0040-0.0060\, {\rm dex}\, {\rm kpc}^{-1}\, {\rm Gyr}^{-1}$,
which is consistent with our present results. A similar
behaviour has also been obtained by Alib\'es et al.
(\cite{alibes}). On the other hand, models such as
those based on two infall episodes by Chiappini et al.
(\cite{cmr2001}), lead to some steepening of the gradients,
even though the inside-out approach is adopted. Possibly,
the main reason for the different predictions
of the quoted models appears to reside on the different adopted
timescales for star formation and infall rate, so that we
expect our present results may be helpful in order to
constrain these quantities.
\begin{acknowledgements}
This work was partially supported by FAPESP and CNPq.
\end{acknowledgements}
|
1,116,691,499,329 | arxiv | \section{The Problem}
It has been pointed out some time ago \cite{CARTER}
that within the KM ansatz large {\bf CP}~asymmetries have to arise
in B decays involving $B^0 - \bar B^0$ oscillations.
In particular the channel $B_d \rightarrow \psi K_S$ combines
a striking experimental signature with a clean theoretical
interpretation \cite{BS}; therefore it is often referred to
as the "golden mode".
A first experimental information on the {\bf CP}~asymmetry
in it has recently
become available\cite{opal,cdfl}. The asymmetry is defined as
follows
\begin{equation}
A_{\psi K_S}=\frac{ \Gamma(\overline B_d(t)\rightarrow\psi K_S)-
\Gamma(B_d(t)\rightarrow\psi K_S)}
{\Gamma(\overline B_d(t)\rightarrow\psi K_S)+\Gamma(B_d(t)\rightarrow\psi K_S)}
=\Im\left( \frac{q}{p}
\overline\rho(\psi K_S)\right) \sin (\Delta M_{B_d} t)
\mlab{10.26}
\end{equation}
where $\overline\rho(\psi K_S)$ denotes the ratio
of transition amplitudes
\begin{equation}
\overline\rho(\psi K_S) =\frac{A(\overline B\rightarrow\psi K_S)}
{A(B\rightarrow\psi K_S)}.
\end{equation}
while $q$ and $p$ relate the mass eigenstates to the flavour eigenstates $B_d$
and $\overline B_d$ (see below).
Two groups found
\begin{equation}
\Im\left( \frac{q}{p}
\overline\rho(\psi K_S)\right)=\left\{
{+({3.2{+1.8\atop -2.0}\pm 0.5}) {\rm ~~OPAL\;
Collaboration\cite{opal}}
\atop
{+(1.8\pm 1.1\pm0.3)}{\rm ~~CDF\;
Collaboration\cite{cdfl}}}\right.
\end{equation}
In reality the observable $\Im\left( \frac{q}{p}
\overline\rho(\psi K_S)\right)$ is bounded by
-1 and +1. It falls outside this range in the data
since they
require subtracting a background of uncertain size. Thus the
numbers have to be taken with a grain of salt. Even so they
indicate that values close to +1 are very strongly disfavoured.
This raises the following question:
Can one
{\em predict} also
the {\em sign} in addition to the size of the asymmetry?
At first one might think that to be impossible. For
$A_{\psi K_S}$ is a product of
$\Im\left( \frac{q}{p}
\overline\rho(\psi K_S)\right)$ and
sin$(\Delta M_{B_d} t)$, see \mref{10.26}. The observable sign of
$A_{\psi K_S}$ thus depends on the signs of both
$\Im\left( \frac{q}{p}
\overline\rho(\psi K_S)\right)$ and $\Delta M_{B_d}$,
and the sign of the latter cannot be defined nor
determined {\em experimentally} in a feasible way --
in contrast to the case with kaons.
In this note we will show that the overall sign of the
asymmetry $A_{\psi K_S}$ -- and likewise for
other asymmetries -- can be predicted within a given theory
for $\Delta B=2$ dynamics.
For those who have thought about this question carefully,
this has been known for a long time.
Indeed it has been discussed in \cite{GKN}. Yet it has not been explained
with all its aspects and in full detail.
As the question becomes experimentally relevant,
we feel that it is important to display all the subtleties involved.
The paper will be organized as follows: in
Sect. \ref{DELTAM} we discuss the theoretical evaluation of
$\Delta M_K$, $\Delta M_{B_d}$, and $\Delta M_{B_s}$
together with $\Delta\Gamma_{B_S}$; in Sect. \ref{PHASE}
we analyze the phase of $(q/p)\bar \rho$;
in Sect. \ref{OT} we address other conventions or proposals;
in Sect. \ref{TRIANGLES} we lay out a proper definition of the angles in the unitarity triangle before giving a summary in Sect. \ref{SUMMARY}.
\section{The sign of $\Delta M$ \label{DELTAM}}
For our discussion to be more transparent, we adopt a
formalism and conventions which are applicable equally
to the $K^0 - \bar K^0$ and the $B^0 - \bar B^0$ complexes.
Consider a neutral meson $P$ carrying a quantum number
$F=-1$; it can denote a $K^0 $ or
$B^0$. The time evolution of a state being a
mixture of $P$ and $\bar P$ is given by
\begin{equation}
i \hbar \frac{\partial}{\partial t} \Psi(t) = {\cal H}\Psi(t)
\mlab{Schroed2}
\end{equation}
where $\Psi(t)$ is restricted to the subspace of $P$ and
$\overline P$:
\begin{equation}
\Psi (t) = \left( \matrix {a(t) \cr \bar b(t) \cr }
\right).
\end{equation}
Assuming {\bf CPT}~ symmetry, the matrix ${\cal H}$ is given by
\begin{equation}
{\cal H}= {\bf M} - \frac{i}{2} {\bf \Gamma} =
\left( \matrix {M_{11} - \frac{i}{2}\Gamma _{11} &
M_{12} - \frac{i}{2}\Gamma _{12} \cr
M_{12}^* - \frac{i}{2}\Gamma _{12}^* &
M_{11} - \frac{i}{2}\Gamma _{11} \cr } \right),
\mlab{CPTMass}
\end{equation}
where
\begin{eqnarray}
M_{11} &=& M_P +
\sum _n {\cal P}\left[ \frac{|\langle n;out|
H_{weak}|P\rangle |^2}
{M_P - M_n}\right] \nonumber\\
M_{12}&=& \matel{P}{H_{SW}}{\overline P} +
\sum _n {\cal P}
\left[ \frac{\matel{P}{H_{\Delta F=1}}{n;out}
\matel{n;out}{H_{\Delta F=1}}{\overline P}}
{M_P - M_n}\right] \nonumber\\
\Gamma _{11} &=& 2\pi \sum _n\delta(M_P-
M_n)|\matel{n;out}{H_{\Delta F=1}}{P}|^2 \nonumber\\
\Gamma _{22} &=& 2\pi \sum _n\delta(M_P-
M_n)|\matel{n;out}{H_{\Delta F=1}}{\overline P}|^2 \nonumber\\
\Gamma _{12}&=&2\pi
\sum _n\delta(M_P-M_n)\matel{P}{H_{\Delta F=1}}{n;out}
\matel{n;out}{H_{\Delta F=1}}{\overline P}.
\mlab{6.8}
\end{eqnarray}
The coupled Schr\" odinger equations
\mref{Schroed2} are best solved by diagonalizing
the matrix ${\cal H}$. We find that
\begin{eqnarray}
|P_1\rangle &= &
p |P\rangle + q |\overline P\rangle , \nonumber\\
|P_2\rangle &= &
p |P\rangle - q |\overline P\rangle ,
\mlab{P1P2gen}
\end{eqnarray}
are mass eigenstates with eigenvalues
\begin{eqnarray}
M_1 - \frac{i}{2}\Gamma _1 &=&
M_{11}-\frac{i}{2}\Gamma_{11}+
\frac{q}{p}(M_{12}-\frac{i}{2}\Gamma_{12})\nonumber\\
M_2 - \frac{i}{2}\Gamma _2 &=&
M_{11}-\frac{i}{2}\Gamma_{11}-
\frac{q}{p}(M_{12}-\frac{i}{2}\Gamma_{12})\nonumber\\
\mlab{EV}
\end{eqnarray}
as long as
\begin{equation}
\left(\frac{q}{p}\right)^2=\frac{M_{12}^*-\frac{i}{2}\Gamma_{12}^*}
{M_{12}-\frac{i}{2}\Gamma_{12}}
\mlab{qoverp}
\end{equation}
holds. Obviously, there are two solutions to this condition, namely
\begin{equation}
\frac{q}{p} = \pm \sqrt{\frac{M_{12}^*-\frac{i}{2}\Gamma_{12}^*}
{M_{12}-\frac{i}{2}\Gamma_{12}}}.
\mlab{pmsign}
\end{equation}
Choosing the negative rather than the positive sign in \mref{pmsign}
is equivalent to interchanging the labels $1\leftrightarrow 2$ of the mass eigenstates, see \mref{P1P2gen}, \mref{EV}.
This binary ambiguity is a special case of a more general one. For
antiparticles are defined only up to a phase; adopting a different phase
convention - {\it e.g.~} going from ${\bf CP}|P{\rangle}=|\overline P{\rangle}$ to
${\bf CP}|P{\rangle}=e^{i\xi}|\overline P{\rangle}$ - will modify
$M_{12}-\frac{i}{2}\Gamma_{12}\equiv \mat{P}{{\cal H}}{\overline P}$:
\begin{equation}
M_{12} - \frac{i}{2}\Gamma_{12} \; \; \rightarrow \; \;
e^{i\xi}[ M_{12} - \frac{i}{2}\Gamma_{12}]
\end{equation}
and thus
\begin{equation}
\frac{q}{p}\rightarrow e^{-i\xi}\frac{q}{p},
\end{equation}
yet leave the combination
$\frac{q}{p}(M_{12} - \frac{i}{2}\Gamma _{12})$
invariant. This is as it should be since
the differences in mass and width
\begin{eqnarray}
M_2-M_1&=&-2\Re\left(\frac{q}{p}(M_{12} - \frac{i}{2}\Gamma _{12})\right)\nonumber\\
\Gamma_2-\Gamma_1&=&4\Im\left(\frac{q}{p}(M_{12} - \frac{i}{2}\Gamma _{12})\right)
\mlab{ei23}
\end{eqnarray}
being observables, have to be insensitive to the arbitrary phase of $\overline P$.
Some comments are in order to elucidate the situation:
\begin{itemize}
\item We can define the labels 1 and 2 such that
\begin{equation}
\Delta M \equiv M_2-M_1 > 0
\end{equation}
is satisfied. Once this {\em convention} has been adopted, it becomes a
sensible question whether
\begin{equation}
\Gamma_2>\Gamma_1~~~~~~~~~~~~~~~~~~~~or~~~~~~~~~~~~~~~\Gamma_2<\Gamma_1
\end{equation}
holds, {\it i.e.} whether the heavier state is shorter or longer lived.
\item
In the limit of {\bf CP}~ invariance the two mass eigenstates are {\bf CP}~eigenstates as well, and we can raise another meaningful question: is the heavier state {\bf CP}~
even or odd?
Since {\bf CP}~invariance implies $\arg\frac{M_{12}}{\Gamma_{12}} = 0$,
$\frac{q}{p}$ becomes a pure phase: $|\frac{q}{p}| =1$.
It is then convenient to adopt a phase convention s.t.
$M_{12}$ is real; it leads to $\frac{q}{p}=\pm1$ and
${\bf CP}|P{\rangle}= \pm|\overline P{\rangle}$ as remaining choices.
\begin{itemize}
\item
With $\frac{q}{p}=+1$ we have
\begin{eqnarray}
|P_1\rangle &= &
\frac{1}{\sqrt{2}}(|P\rangle + |\overline P\rangle) \nonumber\\
|P_2\rangle &= &
\frac{1}{\sqrt{2}}(|P\rangle - |\overline P\rangle)
\mlab{B1}
\end{eqnarray}
For ${\bf CP}|P{\rangle}= |\overline P{\rangle}$, $P_1$ and $P_2$ are {\bf CP}~even and odd,
respectively and therefore
\begin{eqnarray}
M_--M_+=M_2-M_1&=&-2\Re \frac{q}{p}(M_{12} - \frac{i}{2}\Gamma _{12})
\nonumber\\
&=&-2M_{12}
\mlab{B2}
\end{eqnarray}
For ${\bf CP}|P{\rangle} = -|\overline P{\rangle}$, on the other hand,
$P_1$ and $P_2$ switch roles;
i.e. $P_1$ and $P_2$ are {\bf CP}~odd and even now. Thus
\begin{equation}
M_--M_+=M_2-M_1= 2M_{12}
\mlab{B3}
\end{equation}
\item
Alternatively we can set $\frac{q}{p}= -1$:
\begin{eqnarray}
|P_1\rangle &= &
\frac{1}{\sqrt{2}}(|P\rangle - |\overline P\rangle) \nonumber\\
|P_2\rangle &= &
\frac{1}{\sqrt{2}}(|P\rangle + |\overline P\rangle)
\mlab{B4}
\end{eqnarray}
while maintaining ${\bf CP}|P{\rangle} = |\overline P{\rangle}$; $P_1$ and $P_2$
are then {\bf CP}~odd and even, respectively. Accordingly
\begin{eqnarray}
M_--M_+=M_1-M_2&=& 2\Re \frac{q}{p}(M_{12} - \frac{i}{2}\Gamma _{12})
\nonumber\\
&=&-2M_{12}
\mlab{B5}
\end{eqnarray}
\item
\mref{B2} and \mref{B5} on one side and \mref{B3} on the other
do not coincide on the surface; yet,
we will see below that the theoretical prediction for $M_{12}$ changes
sign depending on the choice of ${\bf CP}|P{\rangle}=\pm|\overline P{\rangle}$.
Thus they all agree, of course.
\end{itemize}
\item
Within a given theory for the $P-\overline P$ complex,
we can evaluate $M_{12}$, as discussed below.
Yet some care has to be applied in
interpreting such a result. For expressing mass eigenstates explicitely in
terms of flavour eigenstates involves some conventions; see the examples above. Once we adopt a certain convention, we have to stick with it; yet our original choice cannot influence observables. It is instructive to trace how this comes about.
\end{itemize}
The {\em relative} phase between $M_{12}$ and $\Gamma_{12}$ on the other hand
represents an observable quantity describing indirect {\bf CP}~violation.
Therefore, we adopt the notation
\begin{equation}
M_{12} = \overline M_{12} e^{i\xi},~~~
\Gamma_{12}=\overline \Gamma_{12}e^{i\xi}e^{i\zeta},~~~\mbox{and}~~~
\frac{\Gamma_{12}}{M_{12}}=\frac{\overline \Gamma_{12}}{\overline M_{12}}
e^{i\zeta}=re^{i\zeta}
\mlab{219}
\end{equation}
The sign of $\overline M_{12} $ and $\overline \Gamma_{12}$ are fixed
such that $\xi$, and $\xi+\zeta$ are
restricted to lie between
$-\frac{\pi}{2}$ and $\frac{\pi}{2}$, respectively; {\it i.e.}, the real quantities
$\overline M_{12}$ and $\overline \Gamma_{12}$ are a priori allowed to be
{\em negative} as well as {\em positive}! A relative minus sign between
$M_{12}$ and $\Gamma_{12}$ is of course physically significant,
while the absolute sign is not. Yet, we will see that the absolute sign provides us with a useful bookkeeping device.
\subsection{$\Delta M_K$}
The two kaon mass eigenstates can unambiguously be labelled
by their
lifetimes as $K_L$ and $K_S$. One can then address the
question how they differ in other properties: data reveal
that (i) the $K_L$ is ever so slightly heavier: $M_L > M_S$
and (ii) the $K_S$ [$K_L$] is mainly {\bf CP}~even [odd]. We then
adopt the conventions
\begin{equation}
\Delta M=M_2-M_1,~~~~~~~\mbox{ and }~~~~~~
\Delta\Gamma=\Gamma_1-\Gamma_2
\end{equation}
which make both differences positive:
\begin{equation}
\Delta M_K=M_L-M_S>0,~~~~~\Delta\Gamma_K=\Gamma_S-\Gamma_L>0.
\end{equation}
Since {\bf CP}~violation is very small in the K system -
$\zeta_K=\arg(\Gamma_{12}^K/M_{12}^K)\ll 1$ - we deduce from
\mref{ei23} in the notation of \mref{219}:
\begin{equation}
\Delta M_K \simeq -2\overline M^K_{12}~~~~~\mbox{and}~~~~~
\Delta \Gamma_K \simeq 2\overline \Gamma^K_{12}.
\end{equation}
We will show below that the standard model box diagram
indeed reproduces the `correct', i.e. observed
sign for $\Delta M_K$ (and -- more surprisingly -- even
for $\Delta \Gamma _K$).
\subsection{$\Delta M_{B_d}$}
The situation is qualitatively different here.
While the two mass eigenstates will have different
lifetimes, no useful experimental information
exists on that. It is actually quite unlikely that such
information will become available in the foreseeable
future, since $\Delta \Gamma_{B_d}/\Gamma_{B_d}$ is
estimated not to exceed the 1\% level
\footnote{Remember that $\Gamma _S \gg
\Gamma _L$ represents a kinematical accident!}.
Thus we have to rely on a theoretical prediction of
$\Delta M_{B_d}$.
We can confidently predict for ${B_d}$ mesons
\begin{equation}
|r_{B_d}| \ll 1.
\end{equation}
Since we have to leading nontrivial order in $r_{B_d}$
\begin{equation}
\left. \frac{q}{p}\right| _{B_d}\simeq
\sqrt{\frac{(M_{12}^{B_d})^*}{M^{B_d}_{12}}}(1-\frac{r_{B_d}}{2}
\sin\zeta _{B_d}),
\end{equation}
we get from \mref{ei23}
\begin{equation}
\Delta M_{B_d}=-2\overline M^{B_d}_{12}~~~~~\mbox{and}~~~~~
\Delta \Gamma_{B_d}=2\overline \Gamma_{12}^{B_d} \cos\zeta_{B_d}.
\end{equation}
Note that the expression for $\Delta M_{B_d}$
is similar to the kaon case, albeit for a different reason,
namely $|\Gamma _{12}/M_{12}| \ll 1$ rather than
$\arg\frac{\Gamma_{12}}{M_{12}} \ll 1$;
the latter quantity is actually not small for $B_d$ mesons.
\subsection{$\Delta M_{B_s}$ and $\Delta \Gamma_{B_s}$}
There are two features that distinguish $B_s$ from $B_d$ decays in ways that are quite significant for our present discussion.
\begin{itemize}
\item
While $r_{B_s}\ll 1$ holds also for ${B_s}$ mesons, $\Delta\Gamma/\Gamma$ might not be that small. It has been estimated
\cite{URALTSEV}
\begin{equation}
\frac{\Delta\Gamma}{\hat\Gamma}|_{B_s}\sim 0.18\left(\frac{f_{B_s}}{200MeV}\right)^2,~~~~\hat\Gamma=\frac{1}{2}(\Gamma_{B_{s,1}}+\Gamma_{B_{s,2}}).
\end{equation}
Such a difference in the lifetimes of the two $B_s$ mass eigenstates might actually become observable! This would raise the question whether the heavier state is longer or shorter lived.
\item
The $\Delta B=2$ effective operator for $B_s$ obtained from the box diagram is dominated by the quarks of the second and third families only. It thus predominantly conserves {\bf CP}~invariance making the two $B_s$ mass eigenstates approximately {\bf CP}~eigenstates as well. This has two consequences:
\begin{enumerate}
\item
Rather than fit the decay rate evolution for $B_s\rightarrow D_s^{(*)}\pi$ or
$B_s\rightarrow l\nu D_s^{(*)}$ with two separate exponentials, we can compare the lifetimes of the {\bf CP}~even eigenstate, as measured in $B_s\rightarrow\psi\eta$, with the average lifetime obtained from $B_s\rightarrow l\nu D_s^{(*)}$. We can also compare the lifetimes in $B_s\rightarrow \psi\phi$ for S- and P- wave final states being {\bf CP}~even and odd, respectively.
\item
We can raise the issue whether the {\bf CP}~even state is longer or shorter lived.
\end{enumerate}
\end{itemize}
As for $B_d$ mesons, we have
\begin{equation}
\left. \frac{q}{p}\right| _{B_s}=
\sqrt{\frac{(M_{12}^{B_s})^*}{M^{B_s}_{12}}}(1-\frac{r_{B_s}}{2}
\sin\zeta _{B_s})
\end{equation}
and
\begin{equation}
\Delta M_{B_s}\simeq -2\overline M^{B_s}_{12}~~~~~\mbox{and}~~~~~
\Delta \Gamma_{B_s}\simeq 2\overline \Gamma_{12}^{B_s}
\end{equation}
where we have used the prediction that $\zeta_{B_s}$, unlike $\zeta_{B_d}$,
is very small, since both $M_{12}$ and $\Gamma_{12}$
are dominated by contributions from the third and second family only.
\subsection{Evaluating the sign of $\Delta M_K$,
$\Delta M_B$, and $\Delta\Gamma_B$
\label{SIGNM}}
After having established a notation equally convenient
for the $K$ and $B$ (as well as $D$) case we calculate
$\Delta M_K$ and $\Delta M_B$. As is well known the
dominant {\em short-distance} contribution to the
effective $\Delta F=2$ interaction within the Standard
Model is obtained from the box diagram. One finds
\begin{eqnarray}
&&{\cal H}_{eff}^{box}(\Delta F = 2, \mu ) =
\left( \frac{G_F}{4\pi}\right) ^2 M_W^2 \cdot\nonumber\\
&\cdot& \left[\eta _{cc}(\mu ) \lambda _c^2 E(x_c) +
\eta _{tt}(\mu ) \lambda _t^2 E(x_t) +
2\eta _{ct}(\mu ) \lambda _c \lambda _t E(x_c,x_t) \right]
[\overline q\gamma_\mu(1-\gamma_5)Q]^2 + h.c.
\mlab{SBOX}
\end{eqnarray}
where $Q=s,b$, and $q=d,s$;
$\lambda ^Q_i$ denote combinations of KM parameters
\begin{equation}
\lambda _i^Q = {\bf V}_{iQ}{\bf V}^*_{iq} \; , \; i=c,t \; ,
\end{equation}
and $E(x_i)$ and $E(x_i,x_j)$ reflect the box loops with equal
and different internal quarks (charm or top), respectively:
\begin{equation}
E(x_i) = x_i \left( \frac{1}{4} + \frac{9}{4(1-x_i)}
- \frac{3}{2(1-x_i)^2} \right) -
\frac{3}{2} \left( \frac{x_i}{1- x_i}\right) ^3 {\rm log} x_i
\end{equation}
$$
E(x_c, x_t) = x_c x_t \left[ \left( \frac{1}{4} +
\frac{3}{2}\frac{1}{1-x_t} -
\frac{3}{4} \frac{1}{(1- x_t)^2} \right)
\frac{{\rm log}x_t}{x_t - x_c} + (x_c \leftrightarrow x_t)
- \right.
$$
\begin{equation}
\left. - \frac{3}{4} \frac{1}{(1-x_c)(1-x_t)}\right] \; ; \;
x_i = \frac{m_i^2}{M_W^2} \; .
\mlab{LIM}
\end{equation}
The $\eta _{qq'}$ contain the QCD radiative corrections;
they have been studied through next-to-leading
level in order
to understand
the theoretical errors. We shall not go into the scale
dependence
as well as
errors associated with uncertainties in $\Lambda_{QCD}$,
$m_t$, etc.
Such a discussion can be found in Ref.\cite{buras}. For us
it is
important to note that they are all {\em positive}.
These QCD corrections arise from evolving the effective
Lagrangian from $M_W$ down to the scale $\mu$ at which
the hadronic expectation value is evaluated.
The latter task is far from trivial even for
a local four-fermion operator since
on-shell matrix elements are controlled by long-distance
dynamics.
{\em In principle} the value of $\mu$ does not matter:
the $\mu$ dependance of the $\Delta F=2$ operator is
compensated for by the $\mu$ dependance of the
expectation value. {\em In practise} however, the
available methods for calculating these matrix
elements do not allow us to reliably track their
$\mu$ dependance or they are applicable only for
$\mu \sim 1$ GeV.
Its size is customarily expressed as follows:
\begin{equation}
\matel{P}{(\overline q \gamma _{\mu}(1-\gamma_5)Q)
(\overline q \gamma _{\mu}(1-\gamma_5)Q)}{\overline P} =
-\frac{4}{3} B_P F_P^2 M_P \; , \bar P = [Q\bar q]
\mlab{MES}
\end{equation}
which represents a parametrization rather than an ansatz
as long
as the value of $B_P$ is left open.
$B_P =1$ is referred to as {\em vacuum saturation}
(VS) since it
emerges when only the vacuum is inserted as intermediate
state. Let us see the origin of the minus sign. One obtains
\begin{equation}
\left. \matel{P}{(\overline q \gamma _{\mu}(1-\gamma_5)Q)
(\overline q \gamma _{\mu}(1-\gamma_5)Q)}{\overline P}
\right| _{VS} = \frac{4}{3}
\matel{P}{(\overline q \gamma _{\mu}(1-\gamma_5)Q)}{0}
\matel{0}{
(\overline q \gamma _{\mu}(1-\gamma_5)Q)}{\overline P}
\end{equation}
upon inserting the vacuum intermediate state in the two types
of box diagrams; this yields the colour factor
$1 + 1/N_C = \frac{4}{3}$. Setting
\begin{equation}
\matel{0}{
(\overline q \gamma _{\mu}(1-\gamma_5)Q)}{\overline P(k)}=
iF_P k_{\mu}
\end{equation}
one finds with ${\bf CP}~|0\rangle = |0\rangle$ and the convention ${\bf CP}|P{\rangle}=+|\overline P{\rangle}$\footnote{The VS result follows even with
${\bf CP}~|0\rangle = e^{i\alpha}|0\rangle$.}
$$
\matel{P}{(\overline q \gamma _{\mu}(1-\gamma_5)Q)}{0} =
\matel{0}{(\overline Q \gamma _{\mu}(1-\gamma_5)q)}
{P}^{\dagger} =
$$
\begin{equation}
(\matel{0}{{\bf CP} ^{\dagger}{\bf CP}
(\overline Q \gamma _{\mu}(1-\gamma_5)q){\bf CP} ^{\dagger}{\bf CP}}
{P})^{\dagger}=
- (\matel{0}{
(\overline q \gamma _{\mu}(1-\gamma_5)Q)}
{\bar P})^{\dagger}= iF_P k_{\mu}
\end{equation}
since
\begin{equation}
{\bf CP}\overline Q(\vec x)\gamma^\mu\gamma_5 q(\vec x)
{\bf CP}^
\dagger=
-\overline q(-\vec x)\gamma^\mu\gamma_5 Q(-\vec x).
\end{equation}
Several theoretical techniques have been employed to
estimate the size of $B_P$. For a recent review, see Ref.
\cite{sonirev}.
Again for us it is important that they all yield $B_P>0$.
\footnote{The fudge factor $B_P$ is sometimes called
the bag factor since the MIT bag model was at an earlier time
considered to yield a relatively reliable estimate for its
size. It might be amusing to note while the bag model yields
$B_B > 0$ as a very robust result, it is much less firm on
$B_K$: both $B_K > 0$ as well as $B_K < 0$ emerge when varying
the model parameters over a reasonable range!}
Within the Standard Model the short-distance contributions
thus predict unequivocally
\begin{equation}
M_2 - M_1 > 0
\end{equation}
for $K$ as well as $B$ mesons.
There are sizeable long distance contributions to
$\Delta M_K$, which are estimated to be positive as well
\cite{OURPAPER}. In any case it would be quite contrived to
expect them to cancel the short distance contributions.
With the definition ${\bf CP}|P{\rangle} = |\overline P{\rangle}$
used here, $K_2$ is the mainly {\bf CP}~odd state.
Thus the Standard Model agrees with the experimental finding that
$K_L$ is heavier than $K_S$ - a fact which is not often stressed
in the literature.
For the K meson system, the box diagram does not lead to
a reliable prediction for $\Delta\Gamma_K$ since the decay is dominated by
$K\rightarrow\pi\pi$ which is anything but short distance dominated.
The situation is quite
different for $\Delta \Gamma_{B_s}$ which is driven by
$[b\bar s]\rightarrow"c\bar c"\rightarrow[s\bar b]$ transitions.
\footnote{It is quite amusing that for $\Delta\Gamma_K$ the $s\bar d\rightarrow"u\bar u"\rightarrow d\bar s$ transition gives the correct sign for $\Delta\Gamma_K$.}
It is reasonable to expect
a short distance treatment to provide at least
a semi-quantitative description. Hence we predict\footnote{The question of the sign of $\Delta\Gamma_{B_d}$ might remain academic for experimental reasons. Theoretically the situation is not so clear due to large GIM cancellations\cite{BKUS}.}
\begin{equation}
\Delta\Gamma_{B_s}=\Gamma_{B_s,even}-\Gamma_{B_s,odd}>0.
\end{equation}
\section{The phase of $\frac{q}{p}\overline\rho(\psi K_S)$
\label{PHASE}}
Now that we know $\sin(\Delta Mt)$ describes the oscillations
with positive
$\Delta M$, we need to know the phase of
$\overline\rho(\psi K_S)$, which is
defined by
\begin{equation}
\overline\rho(\psi K_S)=\frac{\mat{\psi K_S}{H_{\Delta B=1}}
{\overline B}}
{\mat{\psi K_S}{H_{\Delta B=-1}}{B}}=\frac{{\langle} \psi K_S|\psi
\overline K^0{\rangle}}{{\langle} \psi K_S|\psi K^0{\rangle}}
\frac{\mat{\psi \overline K^0}{H_{\Delta B=1}}{\overline B}}
{\mat{\psi K^0}{H_{\Delta B=-1}}{B}} \; ;
\end{equation}
here we have used
${\bf CP} H_{\Delta B=1}{\bf CP}^\dagger=H_{\Delta B=-1}^*$.
Adopting the convention
${\bf CP}~|P\rangle = + |\bar P \rangle $
we obtain
\begin{eqnarray}
\mat{\psi \overline K^0}{H_{\Delta B=1}}{\overline B_d}&=&
\mat{\psi \overline K^0}{{\bf CP}^\dagger{\bf CP} H_{\Delta B=1}
{\bf CP}^\dagger{\bf CP}}
{\overline B_d}\nonumber\\
&=&-\mat{\psi K^0}{H_{\Delta B=-1}^*}{B_d}
\end{eqnarray}
where the minus sign reflects the fact that $\psi K^0$
form a P wave. Thus
\begin{equation}
\overline\rho(\psi K_S)=
-\frac{\mat{\psi K_S}{H_{\Delta B=-1}^*}{B_d}}
{\mat{\psi K_S}{H_{\Delta B=-1}}{B_d}}=-
\frac{{\bf V}_{cd}^*{\bf V}_{cs}}{{\bf V}_{cd}{\bf V}_{cs}^*}
\frac{{\bf V}_{cb}{\bf V}_{cs}^*}{{\bf V}_{cb}^*{\bf V}_{cs}}
\end{equation}
where
$V^*_{cd}V_{cs}/V_{cd}V^*_{cs}\sim \left(\frac{q_K}{p_K}\right)^*$.
From \mref{qoverp}, we see that
\begin{equation}
\frac{q}{p}=\sqrt{\frac{M_{12}^*}{M_{12}}}+{\cal O}(r)\propto
\frac{{\bf V}_{tb}^*{\bf V}_{td}}{{\bf V}_{tb}{\bf V}_{td}^*}
\end{equation}
Putting everything together we find the asymmetry is given by
$$
{\rm sin} \Delta M_Bt \cdot
\Im\left(\frac{q}{p}\overline\rho(\psi K_S)\right)=
- {\rm sin} |\Delta M_Bt| \cdot
\Im\left(\frac{{\bf V}_{tb}^*{\bf V}_{td}}
{{\bf V}_{tb}{\bf V}_{td}^*}
\frac{{\bf V}^*_{cd}}{{\bf V}_{cd}}
\frac{{\bf V}_{cb}}{{\bf V}_{cb}^*}\right) \simeq
$$
\begin{equation}
\simeq {\rm sin} |\Delta M_Bt| \cdot
\frac{2 \eta (1-\rho )}{(1- \rho )^2 + \eta ^2} \; ,
\end{equation}
with the quantities $\eta$ and $\rho$ referring to the
Wolfenstein representation of the CKM matrix.
With $\eta$ inferred to be positive from $\epsilon$ one
concludes that this
asymmetry has to be {\em positive}!
\section{Different conventions \label{OT}}
\subsection{Using ${\bf CP}~|P\rangle = - |\bar P\rangle$}
It is instructive to analyze what happens if one
uses the equally allowed convention
\begin{equation}
{\bf CP}~|P\rangle = - |\bar P\rangle \; .
\end{equation}
Various intermediate expressions change; e.g.,
repeating the steps of Sect. \ref{SIGNM} one has
\begin{equation}
\left.
\matel{P}{(\overline q \gamma _{\mu}(1-\gamma_5)Q)
(\overline d \gamma _{\mu}(1-\gamma_5)s)}{\overline P}
\right| _{VS} =
+\frac{4}{3} F_P^2 M_P \; , \bar P = [Q\bar q]
\mlab{MESNEG}
\end{equation}
since now
\begin{equation}
\matel{P}{(\overline q \gamma _{\mu}(1-\gamma_5)Q)}{0} =
-iF_P k_{\mu}
\end{equation}
That means that $\bar M_{12}$ changes sign; yet -- as pointed out
before -- the two labels $1$ and $2$ exchange their roles then
as well, see \mref{B3}. The prediction that $K_L$ is slightly heavier than
$K_S$ thus remains unaffected.
Furthermore the sign of $\bar \rho (\psi K_S)$ changes
as well, see Sect. \ref{PHASE}. Thus the combination
sin$\Delta M_Bt \, \cdot$
Im$\frac{q}{p}\bar \rho (\psi K_S)$ remains unchanged!
The ambiguity we have in the sign of $\Delta M_B$ is
thus compensated by a corresponding ambiguity
in the sign of $\frac{q}{p}\overline\rho(\psi K_S)$.
We can see this also in the following way:
\begin{enumerate}
\item
Changing $q/p \rightarrow -q/p$ maintains the defining property
$(q/p)^2=(M_{12}^{*}-\frac{i}{2}\Gamma^*_{12})/(M_{12}-\frac{i}{2}\Gamma_{12})$,
see \mref{qoverp}, and cannot affect observables.
\item
Yet the two mass eigenstates labeled by subscripts 1 and 2 exchange places,
see \mref{P1P2gen}.
\item
The difference $\Delta M= -2{\rm Re}[\frac{q}{p}
(M_{12}-\frac{i}{2}\Gamma_{12})]$ then flips its sign - yet
so does ${\rm Im}\frac{q}{p}\overline\rho(\psi K_S)$!
\item
The product $(\sin \Delta M_Bt)\cdot{\rm Im}\frac{q}{p}\overline\rho(\psi K_S)$ therefore remains invariant.
\end{enumerate}
\subsection{Another approach}
The authors of \cite{qn} define
\begin{equation}
\Delta M_B=M_H-M_L
\end{equation}
where H [L] stands for heavy [light]; $\Delta M$ is
thus positive
by definition.
They also define
\begin{eqnarray}
|B_L{\rangle}&=&p|B^0{\rangle}+q|\overline B^0{\rangle}\nonumber\\
|B_H{\rangle}&=&p|B^0{\rangle}-q|\overline B^0{\rangle}
\end{eqnarray}
For clearity, let us imagine a world in which {\bf CP}~is not violated.
Then it is sensible to ask if the {\bf CP}~ even state is heavier
or lighter
than the {\bf CP}~odd state. For good reason they have not decided
on this question,
since there is a freedom in the sign of
\begin{equation}
\frac{q}{p}=\pm\sqrt{\frac{M_{12}^*-\frac{i}{2}\Gamma_{12}^*}
{M_{12}-\frac{i}{2}\Gamma_{12}}}
\mlab{poverq+-}
\end{equation}
If in computing the mass difference $M_2 - M_1$ turns out to be
negative -- yet we want to keep the convention
${\bf CP}~|P\rangle = |\bar P \rangle$ -- we must choose the
minus sign in \mref{poverq+-}. To say it differently:
if we require $\Delta M_B >0$ we have to compute
the sign of $M_2 - M_1$ to decide on the sign for
$q/p$ in \mref{poverq+-}. Alternatively we can keep the
plus sign if we adopt ${\bf CP}~|B^0\rangle = - |\bar B^0\rangle$.
In either case we must compute the sign of $M_2-M_1$.
The convention we must adopt is not fixed in this approach
until we
perform the theoretical computation of $M_2-M_1$.
We feel it is more natural to define a convention independent of prior
theoretical computations.
\begin{figure}[t]
\epsfxsize=10cm
\centerline{\epsfbox{unitangle2.eps}}
\caption{The unitarity triangle for B decays.}
\mlabf{unit2}
\end{figure}
\section{Relation to the unitarity triangle
\label{TRIANGLES}}
To conclude our discussion we
explicitely restate the connections between
the angles in the unitarity triangle, the CKM parameters and
observable {\bf CP}~asymmetries.
The angles $\phi _1$, $\phi _2$ and $\phi _3$ are
defined in \mreff{unit2}; as can be read off from \mreff{unit1}
they can be expressed by
\begin{eqnarray}
\phi_1&=&\pi-\arg\left(\frac{-{\bf V}^*_{tb}{\bf V}_{td}}
{-{\bf V}^*_{cb}{\bf V}_{cd}}\right),\nonumber\\
\phi_2&=&\arg\left(\frac{{\bf V}^*_{tb}{\bf V}_{td}}
{-{\bf V}^*_{ub}{\bf V}_{ud}}\right),\nonumber\\
\phi_3&=&\arg\left(\frac{{\bf V}^*_{ub}{\bf V}_{ud}}
{-{\bf V}^*_{cb}{\bf V}_{cd}}\right).
\mlab{15.17}
\end{eqnarray}
The {\bf CP}~asymmetry in $B_d \rightarrow \psi K_S$, which is driven
by a single $\Delta B =1$ operator, is given by
$$
A_{\psi K_S}
= {\rm sin}(\Delta M_Bt)
\Im\left(\frac{q}{p}\overline\rho(\psi K_S)\right)=
$$
\begin{equation}
= \sin(2\phi_1)\sin(|\Delta M_B|t)
\end{equation}
To the degree we can eliminate the Penguin contribution
to $B_d \rightarrow \pi ^+ \pi ^-$ by, say, extracting the
asymmetry for the isopin-two $\pi \pi$ final state, we
have likewise:
$$
A_{(\pi \pi )_{I=2}} = {\rm sin}(\Delta M_Bt)
\Im\left( \frac{q}{p}\overline\rho([\pi \pi]_{I=2})
\right)=
$$
\begin{equation}
= \Im \left( \frac{V_{tb}V^*_{td}}{V^*_{tb}V_{td}}
\frac{V_{ud}V^*_{ub}}{V^*_{ud}V_{ub}}\right)
{\rm sin}(|\Delta M_Bt|) \simeq
{\rm sin}(|\Delta M_Bt|) \sin (2\phi_2)
\end{equation}
\begin{figure}[t]
\epsfxsize=15cm
\centerline{\epsfbox{unitangle1.eps}}
\caption{The unitarity triangle.}
\mlabf{unit1}
\end{figure}
\section{Summary \label{SUMMARY}}
We have shown that the {\em sign} of the {\bf CP}~asymmetry in
$B_d \rightarrow \psi K_S$ can be predicted within a {\em given
theory} for $\Delta M_B$ and $\frac{q}{p}\bar \rho (\psi K_S)$
even if the sign of $\Delta M_B$ cannot be
determined experimentally.
The same holds for
$B_d \rightarrow \pi \pi$ etc. if one knows the weak operators
driving these transitions.
En passant we have reminded the reader that the Standard
Model can reproduce the observed sign of
$\Delta M_K$ (and roughly its magnitude) through
{\em short-distance} dynamics
\footnote{In the charm complex on the other hand
$\Delta M_D$ is dominated by long distance dynamics within
the Standard Model.}.
Our discussion illustrates that we can choose any
convention we want -- provided care is applied in treating
everything consistently. We view it, however, as very useful
to have a formalism for describing $B$ oscillations
that parallels that for kaons. Furthermore it is
much more natural to invoke theory to decide on the sign
of $\Delta M_B$.
\vskip 1cm
\centerline{Acknowledgements}
Work of IIB has been supported in part by the NSF under the grant number PHY 96-0508. Work of AIS has been supported in part by Grant-in-Aid for Special Project Research (Physics of {\bf CP}~violation).
|
1,116,691,499,330 | arxiv | \section{Introduction}\label{sec:Intro}
The discovery of the Standard Model (SM) Higgs boson in the LHC experiment was one of the most important contributions to our understanding of particle physics in recent years~\cite{ATLAS:2012yve,CMS:2012qbp}.
Although the SM is almost complete with this discovery, various astronomical observations, including precise measurements of the cosmic microwave background radiation, indicate the existence of dark matter (DM), which motivates us to search for new physics beyond the SM (BSM). BSM has been vigorously explored by LHC~\cite{ATLAS:2019nkf,CMS:2020gsy} and DM direct detection experiments~\cite{XENON:2020kmp,LZ:2022ufs}, yet no signals have been obtained, placing strong constraints on parameter space.
A simple extension of the SM, which includes a DM candidate particle, is to add a complex singlet scalar field, known as the complex singlet extension of the SM (CxSM)~\cite{Barger:2008jx,Barger:2010yn,Gonderinger:2012rd,Coimbra:2013qq,Jiang:2015cwa,Chiang:2017nmu,Cheng:2018ajh,Grzadkowski:2018nbc,Chen:2019ebq,Egle:2022wmq}. In the CxSM, a real part of an additional complex scalar mixes with the SM Higgs boson, while an imaginary part of the scalar would be a DM candidate.
It is known that when masses of two CP-even scalars appearing in the CxSM are degenerate, a spin-independent cross section of DM with the nucleons ($\sigma_{\mathrm{SI}}$) could be suppressed, satisfying the results of direct-detection experiments~\cite{Abe:2021nih}.
Due to the orthogonality of the mixing matrix of CP-even scalar bosons, the Higgs signal strength in this model is nothing different from the SM prediction in the degenerate limit of two scalar masses. Therefore, the mixing angle is free from experimental constraints in the limit.
Such a built-in cancellation mechanism is called a degenerate scalar scenario.
The extension of the scalar sector of the SM, as in the CxSM, has a possibility to realize the strong first-order electroweak phase transition (EWPT) required for the electroweak baryogenesis~\cite{Kuzmin:1985mm}.
The EWPT in the degenerate scalar scenario of the CxSM has been studied in detail in Refs.~\cite{Cho:2021itv,Cho:2022our}.
In general, some parameters, such as masses or couplings, are introduced in new physics models, and their values are adjusted by hand to be consistent with low-energy experimental data. The Multi-critical Point Principle (MPP) has been proposed as a guiding principle for choosing the model parameters at a low-energy scale~\cite{Bennett:1993pj,Bennett:1996vy,Bennett:1996hx}.
The MPP states that parameters of a theory take critical values so that multiple vacua have degenerate energy density.
The application of the MPP by Froggatt and Nielsen to the effective potential of the SM Higgs was notable for predicting its mass accurately before the discovery of the Higgs boson~\cite{Froggatt:1995rt}.
In Ref.~\cite{Froggatt:1995rt}, the MPP requires two degenerate vacua in the SM, one at the electroweak scale and another near the Planck scale.
Since there is only one scalar field in the SM, it was necessary to discuss two vacua at different energy scales taking account of the radiative corrections to the Higgs potential.
On the other hand, there are multiple vacua at a low-energy scale in new physics models with the extended scalar sector, such as the CxSM.
The authors of Ref.~\cite{Kannike:2020qtw} claim that if the MPP is a fundamental principle, the MPP should be applied to all vacua, including ones dominated by the tree-level potential in models with the extended scalar sector. This is called the tree-level MPP. The model parameters are chosen to make multiple vacua degenerate at a low-energy scale by applying the tree-level MPP to models with the extended scalar sector.
In this paper, we apply the tree-level MPP for the CxSM and study if the degenerate scalar scenario is realized. The application of the tree-level MPP to a variant of the CxSM has been discussed in Ref.~\cite{Kannike:2020qtw}.
The authors analyze the scalar potential, which is called
the pseudo Nambu-Goldstone DM model~\cite{Gross:2017dan}, and they pointed out that no degenerate vacua at low-energy scale exist in their model, i.e., the tree-level MPP does not allow the pseudo Nambu-Goldstone DM model. This model is, however, known to suffer from the domain-wall problem. We, therefore, adopt the most general and minimal scalar potential in the CxSM to avoid the domain-wall problem~\cite{Abe:2021nih}. We emphasize that this is the first study to apply the tree-level MPP to the degenerate scalar scenario in the CxSM.
We show that the tree-level MPP allows both vacua with and without degenerate scalars in the CxSM. We then discuss the possibility of first-order EWPT in the model parameter space chosen by the tree-level MPP.
It has been pointed out that the first-order EWPT in the CxSM is governed by the contribution from SU(2)$_L$ doublet-singlet mixing in the scalar potential at the tree-level~\cite{Cho:2021itv,Cho:2022our}.
Therefore, since the dominant contribution to realizing the EWPT in the CxSM is given by the tree-level potential, requiring the tree-level MPP may break the relation of parameters which makes the first-order EWPT strong.
However, we find that the 1-loop contribution would be enough to derive the first-order EWPT even after applying the tree-level MPP.
The paper is organized as follows. In Sec.~\ref{sec:model}, we introduce the CxSM and describe the degenerate scalar scenario. Parameter space in the CxSM chosen by the tree-level MPP is analyzed in Sec.~\ref{sec:MPP}. The characteristic feature of the first-order EWPT in the CxSM and the consequence of imposing the tree-level MPP are also described.
In Sec.~\ref{sec:result}, we study quantitatively whether first-order EWPT is feasible after imposing the tree-level MPP. Sec.~\ref{sec:Summary} is devoted to summarize our study.
\section{Model}\label{sec:model}
The CxSM consists of a complex scalar singlet $S$ in addition to the SM particles~\cite{Barger:2008jx}.
In our study, we adopt the following scalar potential:
\begin{align}
V_{0}(H, S)
=
\frac{m^{2}}{2} H^{\dagger} H+\frac{\lambda}{4}\left(H^{\dagger} H\right)^{2}+\frac{\delta_{2}}{2} H^{\dagger} H|S|^{2}+\frac{b_{2}}{2}|S|^{2}+\frac{d_{2}}{4}|S|^{4}+\left(a_{1} S+\frac{b_{1}}{4} S^{2}+\text{H.c.}\right),
\label{tree}
\end{align}
where a global U(1) symmetry for $S$ is softly broken by both $a_1$ and $b_1$. In the following, all the couplings in \eqref{tree} are assumed to be real.
When the linear term of $S$ is absent, there is a $Z_2$ symmetry ($S\to -S$) in the scalar potential. Once the singlet $S$ develops the vacuum expectation value (VEV), the $Z_2$ symmetry is spontaneously broken, and it causes the domain-wall problem~\cite{Abe:2021nih}.
We, therefore, add the linear term of $S$ to the scalar potential \eqref{tree} because it explicitly breaks the $Z_2$ symmetry, and the model does not suffer from the domain-wall problem. Besides $S$ and $S^2$ terms, although other operators dropped here could softly break the global U(1) symmetry, we adopt a minimal set of operators that close under renormalization~\cite{Barger:2008jx}.
We parametrize the scalar fields as
\begin{align}
H &=\left(\begin{array}{c}
G^{+} \\
\frac{1}{\sqrt{2}}\left(v+h+i G^{0}\right)
\end{array}\right), \label{Hcomponent}\\
S &=\frac{1}{\sqrt{2}}\left(v_{S}+s+i \chi\right),
\label{Scomponent}
\end{align}
where $v~(\simeq 246~\text{GeV})$ and $v_S$ represent the VEVs of $H$ and $S$, respectively.
The Nambu-Goldstone bosons $G^+$ and $G^0$ are eaten by $W$ and $Z$ bosons, respectively, after the electroweak symmetry breaking.
Since we assumed no complex parameters in \eqref{tree}, the scalar potential is invariant under CP-transformation ($S\to S^*$). Therefore, the real and imaginary parts of $S$ do not mix, and the stability of $\chi$ is guaranteed, making it a DM candidate.
The first-order derivatives of $V_0$ with respect to $h,s$ are
\begin{align}
\frac{1}{v}\left\langle\frac{\partial V_0}{\partial h}\right\rangle &=\frac{m^2}{2}+\frac{\lambda}{4} v^2+\frac{\delta_2}{4} v_S^2=0, \label{tadpole1}\\
\frac{1}{v_S}\left\langle\frac{\partial V_0}{\partial s}\right\rangle &=\frac{b_2}{2}+\frac{\delta_2}{4} v^2+\frac{d_2}{4} v_S^2+\frac{\sqrt{2} a_1}{v_S}+\frac{b_1}{2}=0, \label{tadpole2}
\end{align}
where $\langle\cdots\rangle$ is defined as taking all fluctuation fields to zero. Note that our vacuum is such that $H$ and $S$ both have VEVs, i.e., $\qty(\langle H \rangle,~\langle S \rangle)=(v,~v_S)$. We also consider the case when only $S$ takes a VEV $\qty(\langle H \rangle,~\langle S \rangle)=(0,~v_S^{\prime})$ to discuss the tree-level MPP in Sec.~\ref{sec:MPP}. In this case, the derivative is expressed as
\begin{align}
\frac{1}{v_S^{\prime}}\left\langle\frac{\partial V_0}{\partial s}\right\rangle &=\frac{b_2}{2}+\frac{d_2}{4} v_S^{\prime 2}+\frac{\sqrt{2} a_1}{v_S^{\prime}}+\frac{b_1}{2}=0. \label{tadpole3}
\end{align}
Note that since nonzero $v_S$ is enforced by $a_1 \neq 0$, the vacuum ($v,0$) does not appear in this model.
The mass matrix of the CP-even states ($h,s$) is expressed as
\begin{align}
\mathcal{M}_S^2=\left(\begin{array}{cc}
\lambda v^2 / 2 & \delta_2 v v_S / 2 \\
\delta_2 v v_S / 2 & \Lambda^2
\end{array}\right),\quad\Lambda^2 \equiv \frac{d_2}{2} v_S^2-\sqrt{2}\frac{a_1}{v_S}. \label{MM}
\end{align}
The mass matrix \eqref{MM} is diagonalized by an orthogonal matrix $O(\alpha)$ as
\begin{align}
O(\alpha)^\top \mathcal{M}_S^2 O(\alpha)=\left(\begin{array}{cc}
m_{h_1}^2 & 0 \\
0 & m_{h_2}^2
\end{array}\right), \quad O(\alpha)=\left(\begin{array}{cc}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha
\end{array}\right),
\label{masseigenstate}
\end{align}
where $\alpha$ is a mixing angle and a symbol $\top$ denotes the transpose of the matrix.
The mass eigenstates ($h_1,~h_2$) are given through the mixing angle $\alpha$ as
\begin{align}
\left(\begin{array}{l}
h \\
s
\end{array}\right)=\left(\begin{array}{cc}
\cos \alpha & \sin \alpha \\
-\sin \alpha & \cos \alpha
\end{array}\right)\left(\begin{array}{l}
h_1 \\
h_2
\end{array}\right).
\end{align}
We emphasize that $\alpha \to 0$ corresponds to the SM-like limit $(h_1 \to h,~h_2 \to s)$. The mass eigenvalues are expressed as
\begin{align}
m_{h_1, h_2}^2 &=\frac{1}{2}\left(\frac{\lambda}{2} v^2+\Lambda^2 \mp \frac{\frac{\lambda}{2} v^2-\Lambda^2}{\cos 2 \alpha}\right) \\
&=\frac{1}{2}\left(\frac{\lambda}{2} v^2+\Lambda^2 \mp \sqrt{\left(\frac{\lambda}{2} v^2-\Lambda^2\right)^2+4\left(\frac{\delta_2}{2} v v_S\right)^2}\right),
\end{align}
We fix $h_1$ as the Higgs boson observed in the LHC experiments, i.e., $m_{h_1}=125$~GeV.
The mass of CP-odd state $\chi$ is given by the soft breaking terms $a_1$ and $b_1$ as
\begin{align}
m_\chi^2
&=
\frac{b_2}{2}-\frac{b_1}{2}+\frac{\delta_2}{4} v^2+\frac{d_2}{4} v_S^2\nonumber \\
&=
-\frac{\sqrt{2} a_1}{v_S}-b_1. \label{DMmass}
\end{align}
For later convenience, we mention the relationship between input and output parameters.
In the following study, we adopt $\left\{v, v_S, m_{h_1}, m_{h_2}, \alpha, m_\chi, a_1\right\}$ as inputs while
the Lagrangian parameters $\left\{m^2, b_2, \lambda, d_2, \delta_2, b_1\right\}$ can be expressed as functions of inputs.
Among the Lagrangian parameters, $m^2$ and $b_2$ are eliminated from the tadpole conditions (\ref{tadpole1}) and (\ref{tadpole2}),
\begin{align}
m^2 &=-\frac{\lambda}{2} v^2-\frac{\delta_2}{2}v_S^2 \label{msquare}\\
b_2 &=-\frac{\delta_2}{2} v^2-\frac{d_2}{2}v_S^2-\sqrt{2}\frac{a_1}{v_S}-b_1. \label{b2}
\end{align}
The remaining four parameters in the six Lagrangian parameters are given as
\begin{align}
\lambda&=\frac{2}{v^2}\left(m_{h_1}^2\cos^2\alpha+m_{h_2}^2\sin^2\alpha\right),\label{lambda}\\
\delta_2&=\frac{1}{v v_S}\left(m_{h_1}^2-m_{h_2}^2\right)\sin{2\alpha}\label{del2},\\
d_2&=2\left(\frac{m_{h_1}}{v_S}\right)^2\sin^2\alpha+2\left(\frac{m_{h_2}}{v_S}\right)^2\cos^2\alpha+2\sqrt{2}\frac{a_1}{v_S^3}\label{d2},\\
b_1 &=-m_{\chi}^2-\frac{\sqrt{2}}{v_S}a_1.
\end{align}
Theoretical constraints on the quartic couplings in the scalar potential are summarized as follows.
A requirement on the scalar potential that is bounded from below is given by\footnote{For $\delta_2<0$, $\lambda d_2>\delta_2^2$ is also needed. In this paper, we assume that $\delta_2$ is positive.}
\begin{align}
\lambda>0,\quad d_2 > 0.
\label{bfb}
\end{align}
The couplings $\lambda$ and $d_2$ should also satisfy the following condition from the perturbative unitarity~\cite{Abe:2021nih}
\begin{align}
\lambda < \frac{16\pi}{3},\quad d_2< \frac{16\pi}{3}.
\end{align}
In addition, the stability condition of the tree-level potential requires~\cite{Barger:2008jx}
\begin{align}
\lambda\left(d_2-\frac{2 \sqrt{2} a_1}{v_S^3}\right)>\delta_2^2.
\label{stability}
\end{align}
Recent DM direct detection experiments have provided upper limits on the spin-independent cross section of DM scattering off nucleons ($\sigma_{\text{SI}}$)~\cite{XENON:2020kmp,LZ:2022ufs}, and those results give a severe constraint on a certain class of DM models.
In the CxSM, the DM-nucleon scattering process is mediated by two scalars ($h_1$ and $h_2$). In the following, we briefly review the degenerate scalar scenario~\cite{Abe:2021nih} where the DM-quark scattering is suppressed when masses of $h_1$ and $h_2$ are degenerate.
The interaction Lagrangian of DM $\chi$ to the CP-even scalars $h_1$, $h_2$ is given by
\begin{align}
\mathcal{L}_S
&=
-\frac{m_{h_1}^2+\frac{\sqrt{2} a_1}{v_S}}{2 v_S} \sin \alpha~ h_1 \chi^2+\frac{m_{h_2}^2+\frac{\sqrt{2} a_1}{v_S}}{2 v_S} \cos \alpha~ h_2 \chi^2,
\label{cch}
\end{align}
while that of a quark $q$ to $h_1$ or $h_2$ is given by
\begin{align}
\mathcal{L}_Y&=\frac{m_q}{v} \bar{q} q \left(h_1 \cos \alpha-h_2 \sin \alpha\right) \label{hff},
\end{align}
where $m_q$ denotes a mass of the quark $q$.
Then the scattering amplitudes $\mathcal{M}_{h_1}$ and $\mathcal{M}_{h_2}$
mediated by $h_1$ and $h_2$, respectively, are
\begin{align}
&i \mathcal{M}_{h_1}=-i \frac{m_q}{v v_S} \frac{m_{h_1}^2+\frac{\sqrt{2} a_1}{v S}}{t-m_{h_1}^2} \sin \alpha \cos \alpha~\bar{u}\left(p_3\right) u\left(p_1\right), \label{amph1}\\
&i \mathcal{M}_{h_2}=+i \frac{m_q}{v v_S} \frac{m_{h_2}^2+\frac{\sqrt{2} a_1}{v_S}}{t-m_{h_2}^2} \sin \alpha \cos \alpha~\bar{u}\left(p_3\right) u\left(p_1\right), \label{amph2}
\end{align}
where $u(p_1)$ ($\bar{u}(p_3)$) is incoming (outgoing) quark-spinor with a momentum $p_1$ ($p_3$), and $t$ is defined as $t \equiv (p_1 - p_3)^2$.
Since the momentum transfer $t$ in the direct detection experiments is small,
the amplitude with $t\to 0$ becomes
\begin{align}
i\left(\mathcal{M}_{h_1}+\mathcal{M}_{h_2}\right) &\simeq i \frac{m_q}{v v_S} \sin \alpha \cos \alpha~\bar{u}\left(p_3\right) u\left(p_1\right) \frac{\sqrt{2} a_1}{v_S}\left(\frac{1}{m_{h_1}^2}-\frac{1}{m_{h_2}^2}\right).
\label{DMamplitude}
\end{align}
We find that the sum of two amplitudes is highly suppressed when $m_{h_1} \simeq m_{h_2}$ because of the sign difference between \eqref{amph1} and \eqref{amph2}, which is due to the orthogonality of the mixing matrix $O(\alpha)$~\eqref{masseigenstate}.
We note that, as pointed out in~Ref.~\cite{Gross:2017dan} , the two scattering amplitudes could also cancel each other when $a_1 \to 0$ without requiring the mass degeneracy of $h_1$ and $h_2$, which is known as the pseudo Nambu-Goldstone DM model. In such a case, as mentioned earlier, the scalar potential has the $Z_2$ symmetry, and it suffers from the domain-wall problem.
The possibility of searching for the degenerate scalar scenario at collider experiments has been studied in Ref.~\cite{Abe:2021nih}.
The authors stated that, although the mass difference $\left|m_{h_1}-m_{h_2}\right| \lesssim 3~\mathrm{GeV}$ is not ruled out from the LHC experiments~\cite{CMS:2014afl}, $\left|m_{h_1}-m_{h_2}\right| \lesssim 1~\mathrm{GeV}$ may be testable at the future $e^+ e^-$ linear collider.
We mention that the degenerate scalar scenario is compatible with the Higgs search experiments at the LHC.
As shown in Eq.~(\ref{hff}), the couplings between $h_1~(h_2)$ and the SM particles are those with the SM Higgs boson multiplied by $\cos{\alpha}~(-\sin{\alpha})$. For example, decay rates from $h_1$ and $h_2$ to the SM particle $X$ is expressed as follows;
\begin{align}
\Gamma_{h_1\to XX}&=\cos^2{\alpha}~\Gamma_{h\to XX}^{\mathrm{SM}}(m_{h_1}), \label{partialdecaywidth}\\
\Gamma_{h_2\to XX}&=\sin^2{\alpha}~\Gamma_{h\to XX}^{\mathrm{SM}}(m_{h_2}),
\end{align}
where $\Gamma_{h\to XX}^{\mathrm{SM}}(m_{h_{1(2)}})$ is the Higgs partial decay width in the SM as a function of $m_{h_{1(2)}}$. Experimentally, when two scalars are degenerate ($m_{h_1} \simeq m_{h_2}$), it is hard to distinguish the production and decay processes of $h_2$ from those of $h_1$ so that the sum of two processes by $h_1$ and $h_2$ is to be observed, i.e.,
\begin{align}
\Gamma_{h_1\to XX}+\Gamma_{h_2\to XX} \simeq \Gamma_{h\to XX}^{\mathrm{SM}}(m_h),
\end{align}
holds for any $\alpha$.
Therefore, the signal strength of Higgs bosons in the CxSM is identical to that in the SM in the degenerate limit of two scalars.
\section{Tree-level MPP and EWPT in the CxSM}\label{sec:MPP}
We apply the tree-level MPP to the CxSM whose the tree-level scalar potential is given by Eq.~(\ref{tree}).
Note that since $v_S$ is nonzero due to $a_1\neq0$, there are two possible vacua; the electroweak vacuum~$(v,v_S)$ and the singlet vacuum~$(0,v_S^{\prime})$. We consider the case where these two vacua are degenerate. The difference between the energy densities at these vacua is expressed as
\begin{align}
\Delta V_0
&\equiv V_0(v,v_S)-V_0(0,v_S^{\prime}) \nonumber \\
&=\frac{m^2}{8}v^2+\frac{3\sqrt{2} a_1}{4}(v_S-v_S^{\prime})+\frac{b_1+b_2}{8}(v_S^2-v_S^{\prime 2}). \label{potentialdifference}
\end{align}
When $a_1=0$, $\Delta V_0$ is evaluated as
\begin{align}
\Delta V_0
&=\frac{m^2}{8}v^2+\frac{b_1+b_2}{8}(v_S^2-v_S^{\prime 2})
\nonumber \\
&\propto -\frac{1}{\lambda d_2-\delta_2^2} \frac{1}{d_2} \times\left[\delta_2\left(b_2+b_1\right)-d_2 b_2\right]^2<0.
\label{potentialdifferenceZ2}
\end{align}
Since the denominator in \eqref{potentialdifferenceZ2} is positive due to \eqref{stability} with $a_1=0$, $\Delta V_0$ is always negative when the global U(1) symmetry is softly broken via only $S^2$ term, i.e., there are no degenerate vacua in this case~\cite{Kannike:2020qtw}. This result is altered by introducing the linear term of $S$ to the scalar potential, as shown in the following.
To begin, we qualitatively discuss the model parameters of the CxSM required by the tree-level MPP.
It is easy to see that one needs $v_S \neq v_S^{\prime}$ to degenerate two vacua because
the first term of the r.h.s. in (\ref{potentialdifference}) is nonzero and negative (see, Eq.~(\ref{msquare})).
Two VEVs, $v_S$ and $v_S^{\prime}$, are derived by tadpole conditions (\ref{tadpole2}) and (\ref{tadpole3}), respectively, the difference being the presence $\delta_2$ (\ref{del2}) which represents the doublet-singlet mixing.
Therefore, $\delta_2$ and $v_S-v_S^{\prime}$ should be sizable to cancel with the first term in (\ref{potentialdifference}), to achieve $\Delta V_0=0$.
From Eq.~(\ref{del2}), we find that $v_S$ should be small to make $\delta_2$ sizable whether we adopt the degenerate scalar scenario. In the degenerate scalar scenario, the mass difference ($m_{h_1}^2 - m_{h_2}^2$ in \eqref{del2} ) is small, so a large $\delta_2$ is possible by requiring small $v_S$.
In non-degenerate case ($m_{h_1} \neq m_{h_2}$),
the couplings of $h_1$ with $m_{h_1}=125~\mathrm{GeV}$ to the SM particles given by multiplying $\cos\alpha$ to the SM Higgs couplings. Taking account of the Higgs search experiments at the LHC, $\cos\alpha$ should be close to 1, i.e., $\sin 2\alpha$ in \eqref{del2} must be small.
Therefore $v_S$ in the denominator in \eqref{del2} should also be small to make $\delta_2$ sizable.
We quantitatively discuss the parameter space in the CxSM with degenerate scalars chosen by applying the tree-level MPP. A case where two scalars are not degenerate is also studied for comparison.
In the following numerical study, we set $m_\chi=62.5$ GeV as a reference. We note that the DM mass makes almost no contribution to $\Delta V_0$ since only $b_1$ is an output parameter of DM mass $m_\chi$~(\ref{DMmass}) while the potential difference $\Delta V_0$ depends on $b_1+b_2$ which does not include $b_1$ term as found in Eq.~(\ref{b2}).
\begin{figure}
\center
\includegraphics[width=8.1cm]{degenerate_vs_small.png}
\includegraphics[width=8.1cm]{degenerate_vs_large.png}
\caption{The potential difference for $v_S = 0.1 - 1~\mathrm{GeV}$~(a) and $v_S = 100-1000~\mathrm{GeV}$~(b) in the degenerate scalar scenario. The color bar represents the change in $a_1$. }
\label{degenerateMPP}
\end{figure}
First, we consider the degenerate scalar scenario. The second Higgs mass $m_{h_2}$ is set to $124~\mathrm{GeV}$. As mentioned in Section \ref{sec:model}, when the masses of two Higgs bosons are degenerate, there are no constraints on the mixing angle $\alpha$. Here we adopt the mixing angle $\alpha=\pi/4$ as an example. Fig.~\ref{degenerateMPP} shows the potential difference scaled by the energy density of the electroweak vacuum $\Delta V_0/V_0(v,v_S)$ as a function of $v_S$. The range of $v_S$ is $0.1 - 1~\mathrm{GeV}$ (Fig.~\ref{degenerateMPP}(a)) and $100 -1000~\mathrm{GeV}$ (Fig.~\ref{degenerateMPP}(b)). Taking account of the theoretical constraints on the scalar potential summarized in Eqs.~\eqref{bfb}-\eqref{stability}, we found no allowed parameter space for $v_S= 10 - 100~\mathrm{GeV}$. Therefore we drop the range $v_S= 10 - 100~\mathrm{GeV}$ from the figure.
The color bar represents the $a_1$ dependence. When $v_S$ is small (Fig.~\ref{degenerateMPP}(a)), $\Delta V_0/V_0(v,v_S)$ can be either positive or negative. On the other hand, when $v_S$ is large (Fig.~\ref{degenerateMPP}(b)), $\Delta V_0/V_0(v,v_S)$ approaches zero but never reaches zero. Therefore, degenerate vacua are realized only when $v_S=\mathcal{O}(0.1)$.
\begin{figure}
\center
\includegraphics[width=8.1cm]{nondegenerate_mh2_small.png}
\includegraphics[width=8.1cm]{nondegenerate_mh2_large.png}
\caption{The potential difference for $v_S$ when $m_{h_2}$= 10 GeV~(a) and $m_{h_2}$ =1000 GeV~(b) in the non-degenerate Higgs case. The color bar represents the change in $a_1$.}
\label{nondegenerateMPP}
\end{figure}
Next, we check for the possible existence of multi-critical points when the masses of two scalars are far apart. If the mass difference of two scalars is large, the couplings of the second scalar $h_2$ to the SM particles are proportional to $\sin\alpha$, and no signal of an additional scalar at the LHC requires the mixing angle $\alpha$ to be small. So we set $\alpha=0.1$ in our study\footnote{In Sec.\ref{sec:result}, we set benchmark points that satisfy the LHC constraints.}. Fig.~\ref{nondegenerateMPP} shows $\Delta V_0/V_0(v,v_S)$ as a function of $v_S$ for $m_{h_2}$ =10 GeV~(Fig.~\ref{nondegenerateMPP}(a)) and $m_{h_2}$ =1000 GeV~(Fig.~\ref{nondegenerateMPP}(b)). The color bar represents the value of $a_1$. When $m_{h_2}$ = 10 GeV, for $v_S$ = 0.1 - 1 GeV, the theoretical constraints on parameters in the scalar potential~\eqref{bfb}-\eqref{stability}
are not satisfied.
The degenerate vacua $\Delta V_0/V_0(v,v_S)=0$
are seen in small $v_S$ as in the degenerate scalar scenario. However, possible $v_S$ regions are relaxed. On the other hand, When $m_{h_2}$ = 1000 GeV, the theoretical constraints are not satisfied for small $v_S$, thus, inevitably, large $v_S$ is required, and the tree-level MPP is not viable.
We now discuss the first-order phase transition in the CxSM and mention the compatibility of EWPT and the tree-level MPP. The 1-loop effective potential, which is used to study the phase transition, can be written in this way;
\begin{align}
V_{\text{eff}}\left(\varphi, \varphi_S ; T\right)&=V_{0}(\varphi, \varphi_S)+V_{1}(\varphi, \varphi_S;T), \nonumber \\
&=V_{0}(\varphi, \varphi_S)+\sum_i n_i\left[V_{\mathrm{CW}}\left(\bar{m}_i^2\right)+\frac{T^4}{2 \pi^2} I_{B, F}\left(\frac{\bar{m}_i^2}{T^2}\right)\right], \label{eff}
\end{align}
where $T$ represents temperature and $\varphi$ and $\varphi_S$ are the constant classical background fields of $H$ and $S$, respectively. $V_{0}$ represents the tree-level potential. The indices $i$ express $h_{1,2}$, $\chi$, $W$, $Z$, $t$ and $b$. The degrees of freedom of each particle $n_i$ are $n_{h_{1,2},\chi}=1, n_W=6, n_Z=3, \text { and } n_t=n_b=-12$. The first term in parentheses in the second row in (\ref{eff}) is the $\overline{\text{MS}}$-defined effective potential at zero temperature, and the second one is the finite temperature effective potential. They are given by~\cite{Weinberg:1973am,Jackiw:1974cv,Dolan:1973qd}
\begin{align}
V_{\mathrm{CW}}\left(\bar{m}_i^2\right) &=\frac{\bar{m}_i^4}{64 \pi^2}\left(\ln \frac{\bar{m}_i^2}{\bar{\mu}^2}-c_i\right), \\
I_{B, F}\left(a^2\right) &=\int_0^{\infty} d x x^2 \ln \left(1 \mp e^{-\sqrt{x^2+a^2}}\right) \label{finite},
\end{align}
where $\bar{m_i}$ is the field-dependent masses of each particle $i$. $c_i=3/2$ for scalars and fermions and $c_i=5/6$ for gauge bosons. $I_B$ with the minus sign represents the boson contribution, while $I_F$ with the plus sign represents the fermion one. $\bar{\mu}$ is a renormalization scale. The potential $V_{\mathrm{CW}}$ is renormalized so that the scalar mass and the tad-pole conditions take the same relationship with those at the tree-level.
To achieve EWBG, EWPT must be strong first-order, which requires the following condition~\cite{Arnold:1987mh,Bochkarev:1987wf,Funakubo:2009eg};
\begin{align}
\frac{v_C}{T_C} \gtrsim 1,\label{decoupling}
\end{align}
where $T_C$ is the critical temperature defined as one when the effective potential has two degenerate minima, and $v_C$ is the Higgs VEV at $T=T_C$. The lower bound in Eq. (\ref{decoupling}) depends on the structure of the saddle point of classical solutions, called sphaleron.
Now we discuss the compatibility of the EWPT and the tree-level MPP in the CxSM.
As is already mentioned, the critical temperature $T_C$ is defined as the temperature at which the effective potential has two degenerate vacua;
\begin{align}
V_{\mathrm{eff}}\left(v_C, v_{SC} ;T_C\right)=V_{\mathrm{eff}}\left(0, v_{SC}^{\prime} ;T_C\right)
&\to
\nonumber\\
V_{0}\left(v_C, v_{SC}\right)+V_{1}\left(v_C, v_{SC} ;T_C\right)&=V_{0}\left(0, v_{SC}^{\prime}\right)+V_{1}\left(0, v_{SC}^{\prime} ;T_C\right),
\label{TCdefinition}
\end{align}
with
\begin{align}
v_C=\lim _{T \to T_C} v(T), \quad v_{S C}=\lim _{T \to T_C} v_S(T), \quad v_{S C}^{\prime}=\lim _{T \to T_C} v_S^{\prime}(T).
\end{align}
On the other hand, the vacuum degeneracy required by the tree-level MPP is
\begin{align}
V_{0}\left(v, v_{S}\right)=V_{0}\left(0, v_{S}^{\prime}\right), \label{MPPdefinition}
\end{align}
with
\begin{align}
v=\lim _{T \to 0} v(T), \quad v_{S}=\lim _{T \to 0} v_S(T), \quad v_{S}^{\prime}=\lim _{T \to 0} v_S^{\prime}(T).
\label{vevt}
\end{align}
It is known that, in general, the first-order EWPT is induced by
the bosonic contributions of the finite temperature potential at the 1-loop level.
However, in a certain class of models with the extended scalar sector, such as the CxSM, the tree-level structure of the scalar potential is rather crucial to realize the EWPT (see, Ref.~\cite{Cho:2021itv,Cho:2022our} for details)\footnote{For classification of first-order EWPT, refer, e.g., Ref.~\cite{Chung:2012vg}.}.
Now, Eqs.~\eqref{TCdefinition}-\eqref{vevt} tell us that,
except for the 1-loop effective potential $V_1$, the definition of $T_C$~(\ref{TCdefinition}) is the same with Eq.~(\ref{MPPdefinition}) derived from the tree-level MPP.
In other words, when the tree-level MPP is valid, EWPT occurs at zero temperature, which conflicts with the condition (\ref{decoupling}).
Therefore, by imposing the tree-level MPP to the CxSM, EWPT from the tree-level potential cannot be defined. On the other hand, there are subleading contributions to the EWPT at the 1-loop level from additional bosons in the CxSM.
Thus, in the next section, we evaluate if the subleading contributions are enough for the strong first-order EWPT at some benchmark points where the tree-level MPP is valid.
\section{Numerical results}\label{sec:result}
To discuss the strong first-order EWPT quantitatively, we focus on two benchmark points from the model parameter space chosen by the tree-level MPP (see, Table~\ref{tab:BP}).
We select these points such that one (BP1) corresponds to the parameter set where two scalars are degenerate ($m_{h_1} \simeq m_{h_2}$) while another (BP2) is not ($m_{h_1} \neq m_{h_2}$).
The coefficient $a_1$ is set to make $\Delta V_0$~\eqref{potentialdifference} to zero.
\begin{table}[t]
\center
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Inputs & $v$ [GeV] & $v_S$ [GeV] & $m_{h_1}$ [GeV] & $m_{h_2}$ [GeV] & $\alpha$ [rad] & $m_\chi$ [GeV] & $a_1$ [GeV$^3$]\\ \hline
BP1 & 246.22 & 0.6 & 125.0 & 124.0 & $\pi/4$ & 62.5 & $-6576.2385$ \\ \hline
BP2 & 246.22 & 10.0 & 125.0 & 10.0 & $0.001$ & 62.5 & $-707.1913$ \\ \hline\hline
Outputs & $m^2$ & $b_2$ [GeV$^2$] & $b_1$ [GeV$^2$] & $\lambda$ & $\delta_2$ & $d_2$ & $a_1$ [GeV$^3$] \\ \hline
BP1 & $-(124.5)^2$ & $-(178.0)^2$ & $(107.7)^2$ & 0.511 &1.69 & 0.87 & $-6576.2385$ \\ \hline
BP2 & $-(125.0)^2$ & $(60.20)^2$ & $-(61.69)^2$ & 0.515 & 0.013& $7.1\times10^{-5}$ & $-707.1913$ \\ \hline
\end{tabular}
\caption{Input and output parameters in the two benchmark points. }
\label{tab:BP}
\end{table}
First, we show that two benchmark points satisfy constraints on the DM relic density $\Omega_{\mathrm{DM}}h^2$~\cite{Planck:2018vyg};
\begin{align}
\Omega_{\mathrm{DM}} h^2=0.1200 \pm 0.0012, \label{relic}
\end{align}
and the spin-independent scattering cross section with the nucleons $\sigma_{\mathrm{SI}}$ by the LUX-ZEPLIN (LZ) experiment~\cite{LZ:2022ufs}.
In the following numerical study, we use a public code \texttt{micrOMEGAs}~\cite{Belanger:2020gnr}. The relic density of DM $\chi$ in the CxSM, $\Omega_\chi h^2$, should not exceed the observed value~\eqref{relic} and, as will be shown later, can explain only a portion of $\Omega_{\mathrm{DM}}h^2$.
Therefore, for $\Omega_\chi<\Omega_{\mathrm{DM}}$, we scale $\sigma_{\mathrm{SI}}$ as
\begin{align}
\tilde{\sigma}_{\mathrm{SI}}=\left(\frac{\Omega_\chi}{\Omega_{\mathrm{DM}}}\right) \sigma_{\mathrm{SI}},
\end{align}
and this should satisfy the bound from the LZ experiment.
\begin{figure}
\center
\includegraphics[width=8.1cm]{BP1_relic.png}
\includegraphics[width=8.1cm]{BP1_sigma.png}\\
\includegraphics[width=8.1cm]{BP2_relic.png}
\includegraphics[width=8.1cm]{BP2_sigma.png}
\caption{DM relic density $\Omega_{\chi}h^2$~(a),(b)
and scaled spin-independent scattering cross section with the nucleons $\tilde{\sigma}_{\mathrm{SI}}$~(c),(d) are shown as a function of the DM mass $m_\chi$. The upper panels~(a),(c) correspond to BP1, and the lower ones to BP2~(b),(d). The dotted lines in the left panels~(a),(c) show the observed relic density, and the dotted curves in the right panels~(b),(d) show the LZ experimental results.
}
\label{BP1BP2DM}
\end{figure}
Two left figures in Fig.~\ref{BP1BP2DM} show $\Omega_{\chi}h^2$ as a function of $m_\chi$ at the benchmark points BP1 (a) and BP2 (b).
In the figures, the dotted horizontal line denotes a central value of the observed DM relic density $\Omega_{\mathrm{DM}} h^2$~\eqref{relic}.
In Fig.~\ref{BP1BP2DM}~(a) (BP1), we find that $\Omega_{\chi}h^2$ is smaller than $\Omega_{\mathrm{DM}} h^2=0.12$ up to around $m_\chi$= 1500 GeV. On the other hand, in Fig.~\ref{BP1BP2DM}~(b) (BP2),
$\Omega_\chi h^2$ at only around $m_\chi$= 62.5 GeV is allowed compared to $\Omega_{\mathrm{DM}} h^2=0.12$.
The dip around $m _\chi$= 62.5 GeV~(= $m_{h_1}/2$) seen at both benchmark points reflects resonance enhancement due to $s$-channel DM annihilation processes.
Fig.~\ref{BP1BP2DM}~(c) and (d) show $\tilde{\sigma}_{\mathrm{SI}}$ as a function of $m_\chi$ at BP1 and BP2, respectively. The dotted line in each figure represents the bound from the LZ experiment.
The model prediction in both figures is allowed around at $m_\chi$= 62.5 GeV, which is affected by the dips in $\Omega_{\chi}h^2$. In each figure, although the cross section at $m_\chi \simeq 10~\mathrm{TeV}$ satisfies the limit from the experiment, it is excluded by the constraint on $\Omega_{\mathrm{DM}}h^2$. Thus, in the following study, we fix the DM mass at $m_\chi= 62.5~\mathrm{GeV}$ in both BP1 and BP2. We should mention why even BP1 with the degenerate scalar scenario, where the DM-quark scattering is suppressed, can only satisfy the DM direct detection results in some regions. The mass difference between two Higgs bosons is represented by $\delta_2$~(\ref{del2}). In the degenerate scalar scenario, $\delta_2$ becomes necessarily small because the mass difference is small. However, as mentioned previously, large $\delta_2$ is necessary to realize the tree-level MPP. This conflicting requirement for $\delta_2$ has led to the situation~(for a detailed study, see Ref.~\cite{Cho:2021itv}). However, it is also true that there is an allowed region around $m_\chi$= 62.5 GeV.
Next, we discuss constraints on benchmark points from the Higgs search experiments at the LHC.
The signal strength $\mu$ is used to compare the branching ratio of the $125~\mathrm{GeV}$ scalar at the LHC and the SM prediction.
The LHC Run-2 experiment gives constraints on $\mu$ as
$0.92<\mu<1.20 $ at ATLAS~\cite{ATLAS:2019nkf} and $0.90<\mu<1.16$ at CMS~\cite{CMS:2020gsy}.
The total decay width of the Higgs boson is constrained as $\Gamma_{h}^{\mathrm{exp}}<$ 14.4 MeV at ATLAS~\cite{ATLAS:2018jym} and $\Gamma_h^{\exp }=3.2_{-1.7}^{+2.4}$ MeV at CMS~\cite{CMS:2022ley}.
As mentioned in Sec.\ref{sec:model}, in the degenerate scalar scenario, the sum of decay rates of $h_1$ and $h_2$ is not different from the SM prediction, so both $\mu$ and $\Gamma_h$ in BP1 are not altered from those of the SM.
In BP2, since the second scalar mass $m_{h_2}$ is lighter than half of $m_{h_1}$, a decay channel $h_1 \to h_2 h_2$ opens, and it may affect $\Gamma_h$ (and the branching ratio).
However, since the mixing angle $\alpha$ is chosen very small $(0.001)$ in BP2,
the total decay width of $h_1$ is given by $\Gamma_{h_1}$=4.29 MeV, which does not much differ from the SM prediction, $\Gamma_{h}^{\mathrm{SM}}$=4.1~\cite{LHCHiggsCrossSectionWorkingGroup:2013rie}. We also find that the signal strength is $\mu=0.956$ in BP2, which is consistent with the results at ATLAS and CMS.
\begin{table}[t]
\center
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& $v_C/T_C$ & $v_{SC}$ [GeV] & $v_{SC}^{\prime}$ [GeV]\\ \hline
~BP1~ & $\frac{244.0}{48.3}=5.1$ & 0.62 & 214.6\\
~BP2~ & $\frac{244.4}{49.7}=4.9$ & 10.3 & 226.2\\
\hline
\end{tabular}
\caption{VEVs at critical temperature $T_C$ in BP1 and BP2. The calculations are performed using \texttt{cosmoTransitions}~\cite{Wainwright:2011kj}. Thermal resummation is needed to improve perturbative expansion at a finite temperature. The Parwani resummation method~\cite{Parwani:1991gq} is used in this analysis.}
\label{tab:TCvC}
\end{table}
We have so far confirmed that two benchmark points BP1 and BP2 satisfy constraints from DM experiments (relic density and direct detection) and the Higgs search experiments at the LHC.
Now, we discuss the numerical evaluation of EWPT. The calculations were performed using \texttt{CosmoTransitions}~\cite{Wainwright:2011kj}, and results are obtained using the 1-loop effective potential~(\ref{eff}). However, perturbation theory is broken down due to boson multi-loop contributions, which need to be addressed in thermal resummation methods. In this study, we use the Parwani resummation method~\cite{Parwani:1991gq} in which all the Matsubara frequency modes are resummed. Specifically, field-dependent masses $\bar{m}_i^2$ that appears in $I_{B,F}$ (\ref{finite}) are replaced by thermally corrected field-dependent masses.
Table~\ref{tab:TCvC} shows $T_C$ and the corresponding VEVs in two benchmark points. Note that since the parameters for which the tree-level MPP is valid are chosen, no first-order EWPT derived from the structure of the tree-level potential occurs. However, in the CxSM, thanks to additional bosons, strong first-order EWPT could be achieved at the 1-loop level with a finite temperature. In fact, $v_C/T_C=5.1$ for BP1 and $v_C/T_C=4.9$ for BP2, satisfying the conditions for strong first-order EWPT~(\ref{decoupling})\footnote{A distinctive feature of this EWPT is sizable change in $v_{S}$, i.e., $v_{SC}^{\prime}\gg v_{SC}$. For the devoted study, see Ref.~\cite{Cho:2021itv}.}.
\section{Summary}\label{sec:Summary}
In this paper, we have applied the tree-level MPP to the CxSM, which includes a linear term of singlet $S$ to avoid the domain-wall problem.
In the CxSM, it is known that when a mass of an additional scalar is approximately degenerate with the SM Higgs mass, the DM-quark scattering amplitudes mediated by two scalars are canceled, and constraints on the model from the DM direct detection experiments are significantly weakened~\cite{Abe:2021nih}.
On the other hand, the tree-level MPP chooses parameters so that multiple vacua in the models with the extended scalar sector have degenerate energy density.
We discussed the possibility that the model parameter space to realize such a degenerate scalar scenario is favored from the tree-level MPP by requiring the electroweak vacuum ($v,v_S$) and the singlet vacuum ($0,v_S^{\prime}$) are degenerate.
The possible existence of parameters that eliminate the potential difference in two vacua (\ref{potentialdifference}) has been investigated in the degenerate scalar region and the non-degenerate scalar region for comparison. A degeneracy between two vacua requires a difference between $v_S$ and $v_S^{\prime}$, and $v_S$ must be small in both regions. We found the parameter space where two vacua are degenerate with small $v_S$ (Fig.\ref{degenerateMPP},\ref{nondegenerateMPP}).
We have considered two benchmark points that satisfy the tree-level MPP requirement, in which the second scalar mass is fixed at $m_{h_2}=124$ GeV for BP1 and $m_{h_2}=10$ GeV for BP2. Our numerical analysis showed only the DM mass $m_\chi \simeq 62.5~\mathrm{GeV}$ is consistent with the DM relic density observation and the DM direct detection experiment for both benchmark points. We also have discussed the feasibility of first-order EWPT. The critical temperature $T_C$ for EWPT is the temperature at which the effective potential has two degenerate minima. Thus the tree-level MPP conditions are very similar to the tree-level driven EWPT as in the CxSM, where the tree-level potential provides the leading contribution~[see Eqs.~(\ref{TCdefinition}),(\ref{MPPdefinition})]. We pointed out that the tree-level contribution to the EWPT is incompatible with the tree-level MPP. However, it is numerically shown that the subleading contribution from additional bosons at the 1-loop level causes strong first-order EWPT~[see Table.\ref{tab:TCvC}].
\begin{acknowledgments}
We are grateful to Eibun Senaha for his valuable discussions. The work of G.C.C. is supported in part by JSPS KAKENHI Grant No. 22K03616.
\end{acknowledgments}
\section{Introduction}\label{sec:Intro}
The discovery of the Standard Model (SM) Higgs boson in the LHC experiment was one of the most important contributions to our understanding of particle physics in recent years~\cite{ATLAS:2012yve,CMS:2012qbp}.
Although the SM is almost complete with this discovery, various astronomical observations, including precise measurements of the cosmic microwave background radiation, indicate the existence of dark matter (DM), which motivates us to search for new physics beyond the SM (BSM). BSM has been vigorously explored by LHC~\cite{ATLAS:2019nkf,CMS:2020gsy} and DM direct detection experiments~\cite{XENON:2020kmp,LZ:2022ufs}, yet no signals have been obtained, placing strong constraints on parameter space.
A simple extension of the SM, which includes a DM candidate particle, is to add a complex singlet scalar field, known as the complex singlet extension of the SM (CxSM)~\cite{Barger:2008jx,Barger:2010yn,Gonderinger:2012rd,Coimbra:2013qq,Jiang:2015cwa,Chiang:2017nmu,Cheng:2018ajh,Grzadkowski:2018nbc,Chen:2019ebq,Egle:2022wmq}. In the CxSM, a real part of an additional complex scalar mixes with the SM Higgs boson, while an imaginary part of the scalar would be a DM candidate.
It is known that when masses of two CP-even scalars appearing in the CxSM are degenerate, a spin-independent cross section of DM with the nucleons ($\sigma_{\mathrm{SI}}$) could be suppressed, satisfying the results of direct-detection experiments~\cite{Abe:2021nih}.
Due to the orthogonality of the mixing matrix of CP-even scalar bosons, the Higgs signal strength in this model is nothing different from the SM prediction in the degenerate limit of two scalar masses. Therefore, the mixing angle is free from experimental constraints in the limit.
Such a built-in cancellation mechanism is called a degenerate scalar scenario.
The extension of the scalar sector of the SM, as in the CxSM, has a possibility to realize the strong first-order electroweak phase transition (EWPT) required for the electroweak baryogenesis~\cite{Kuzmin:1985mm}.
The EWPT in the degenerate scalar scenario of the CxSM has been studied in detail in Refs.~\cite{Cho:2021itv,Cho:2022our}.
In general, some parameters, such as masses or couplings, are introduced in new physics models, and their values are adjusted by hand to be consistent with low-energy experimental data. The Multi-critical Point Principle (MPP) has been proposed as a guiding principle for choosing the model parameters at a low-energy scale~\cite{Bennett:1993pj,Bennett:1996vy,Bennett:1996hx}.
The MPP states that parameters of a theory take critical values so that multiple vacua have degenerate energy density.
The application of the MPP by Froggatt and Nielsen to the effective potential of the SM Higgs was notable for predicting its mass accurately before the discovery of the Higgs boson~\cite{Froggatt:1995rt}.
In Ref.~\cite{Froggatt:1995rt}, the MPP requires two degenerate vacua in the SM, one at the electroweak scale and another near the Planck scale.
Since there is only one scalar field in the SM, it was necessary to discuss two vacua at different energy scales taking account of the radiative corrections to the Higgs potential.
On the other hand, there are multiple vacua at a low-energy scale in new physics models with the extended scalar sector, such as the CxSM.
The authors of Ref.~\cite{Kannike:2020qtw} claim that if the MPP is a fundamental principle, the MPP should be applied to all vacua, including ones dominated by the tree-level potential in models with the extended scalar sector. This is called the tree-level MPP. The model parameters are chosen to make multiple vacua degenerate at a low-energy scale by applying the tree-level MPP to models with the extended scalar sector.
In this paper, we apply the tree-level MPP for the CxSM and study if the degenerate scalar scenario is realized. The application of the tree-level MPP to a variant of the CxSM has been discussed in Ref.~\cite{Kannike:2020qtw}.
The authors analyze the scalar potential, which is called
the pseudo Nambu-Goldstone DM model~\cite{Gross:2017dan}, and they pointed out that no degenerate vacua at low-energy scale exist in their model, i.e., the tree-level MPP does not allow the pseudo Nambu-Goldstone DM model. This model is, however, known to suffer from the domain-wall problem. We, therefore, adopt the most general and minimal scalar potential in the CxSM to avoid the domain-wall problem~\cite{Abe:2021nih}. We emphasize that this is the first study to apply the tree-level MPP to the degenerate scalar scenario in the CxSM.
We show that the tree-level MPP allows both vacua with and without degenerate scalars in the CxSM. We then discuss the possibility of first-order EWPT in the model parameter space chosen by the tree-level MPP.
It has been pointed out that the first-order EWPT in the CxSM is governed by the contribution from SU(2)$_L$ doublet-singlet mixing in the scalar potential at the tree-level~\cite{Cho:2021itv,Cho:2022our}.
Therefore, since the dominant contribution to realizing the EWPT in the CxSM is given by the tree-level potential, requiring the tree-level MPP may break the relation of parameters which makes the first-order EWPT strong.
However, we find that the 1-loop contribution would be enough to derive the first-order EWPT even after applying the tree-level MPP.
The paper is organized as follows. In Sec.~\ref{sec:model}, we introduce the CxSM and describe the degenerate scalar scenario. Parameter space in the CxSM chosen by the tree-level MPP is analyzed in Sec.~\ref{sec:MPP}. The characteristic feature of the first-order EWPT in the CxSM and the consequence of imposing the tree-level MPP are also described.
In Sec.~\ref{sec:result}, we study quantitatively whether first-order EWPT is feasible after imposing the tree-level MPP. Sec.~\ref{sec:Summary} is devoted to summarize our study.
\section{Model}\label{sec:model}
The CxSM consists of a complex scalar singlet $S$ in addition to the SM particles~\cite{Barger:2008jx}.
In our study, we adopt the following scalar potential:
\begin{align}
V_{0}(H, S)
=
\frac{m^{2}}{2} H^{\dagger} H+\frac{\lambda}{4}\left(H^{\dagger} H\right)^{2}+\frac{\delta_{2}}{2} H^{\dagger} H|S|^{2}+\frac{b_{2}}{2}|S|^{2}+\frac{d_{2}}{4}|S|^{4}+\left(a_{1} S+\frac{b_{1}}{4} S^{2}+\text{H.c.}\right),
\label{tree}
\end{align}
where a global U(1) symmetry for $S$ is softly broken by both $a_1$ and $b_1$. In the following, all the couplings in \eqref{tree} are assumed to be real.
When the linear term of $S$ is absent, there is a $Z_2$ symmetry ($S\to -S$) in the scalar potential. Once the singlet $S$ develops the vacuum expectation value (VEV), the $Z_2$ symmetry is spontaneously broken, and it causes the domain-wall problem~\cite{Abe:2021nih}.
We, therefore, add the linear term of $S$ to the scalar potential \eqref{tree} because it explicitly breaks the $Z_2$ symmetry, and the model does not suffer from the domain-wall problem. Besides $S$ and $S^2$ terms, although other operators dropped here could softly break the global U(1) symmetry, we adopt a minimal set of operators that close under renormalization~\cite{Barger:2008jx}.
We parametrize the scalar fields as
\begin{align}
H &=\left(\begin{array}{c}
G^{+} \\
\frac{1}{\sqrt{2}}\left(v+h+i G^{0}\right)
\end{array}\right), \label{Hcomponent}\\
S &=\frac{1}{\sqrt{2}}\left(v_{S}+s+i \chi\right),
\label{Scomponent}
\end{align}
where $v~(\simeq 246~\text{GeV})$ and $v_S$ represent the VEVs of $H$ and $S$, respectively.
The Nambu-Goldstone bosons $G^+$ and $G^0$ are eaten by $W$ and $Z$ bosons, respectively, after the electroweak symmetry breaking.
Since we assumed no complex parameters in \eqref{tree}, the scalar potential is invariant under CP-transformation ($S\to S^*$). Therefore, the real and imaginary parts of $S$ do not mix, and the stability of $\chi$ is guaranteed, making it a DM candidate.
The first-order derivatives of $V_0$ with respect to $h,s$ are
\begin{align}
\frac{1}{v}\left\langle\frac{\partial V_0}{\partial h}\right\rangle &=\frac{m^2}{2}+\frac{\lambda}{4} v^2+\frac{\delta_2}{4} v_S^2=0, \label{tadpole1}\\
\frac{1}{v_S}\left\langle\frac{\partial V_0}{\partial s}\right\rangle &=\frac{b_2}{2}+\frac{\delta_2}{4} v^2+\frac{d_2}{4} v_S^2+\frac{\sqrt{2} a_1}{v_S}+\frac{b_1}{2}=0, \label{tadpole2}
\end{align}
where $\langle\cdots\rangle$ is defined as taking all fluctuation fields to zero. Note that our vacuum is such that $H$ and $S$ both have VEVs, i.e., $\qty(\langle H \rangle,~\langle S \rangle)=(v,~v_S)$. We also consider the case when only $S$ takes a VEV $\qty(\langle H \rangle,~\langle S \rangle)=(0,~v_S^{\prime})$ to discuss the tree-level MPP in Sec.~\ref{sec:MPP}. In this case, the derivative is expressed as
\begin{align}
\frac{1}{v_S^{\prime}}\left\langle\frac{\partial V_0}{\partial s}\right\rangle &=\frac{b_2}{2}+\frac{d_2}{4} v_S^{\prime 2}+\frac{\sqrt{2} a_1}{v_S^{\prime}}+\frac{b_1}{2}=0. \label{tadpole3}
\end{align}
Note that since nonzero $v_S$ is enforced by $a_1 \neq 0$, the vacuum ($v,0$) does not appear in this model.
The mass matrix of the CP-even states ($h,s$) is expressed as
\begin{align}
\mathcal{M}_S^2=\left(\begin{array}{cc}
\lambda v^2 / 2 & \delta_2 v v_S / 2 \\
\delta_2 v v_S / 2 & \Lambda^2
\end{array}\right),\quad\Lambda^2 \equiv \frac{d_2}{2} v_S^2-\sqrt{2}\frac{a_1}{v_S}. \label{MM}
\end{align}
The mass matrix \eqref{MM} is diagonalized by an orthogonal matrix $O(\alpha)$ as
\begin{align}
O(\alpha)^\top \mathcal{M}_S^2 O(\alpha)=\left(\begin{array}{cc}
m_{h_1}^2 & 0 \\
0 & m_{h_2}^2
\end{array}\right), \quad O(\alpha)=\left(\begin{array}{cc}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha
\end{array}\right),
\label{masseigenstate}
\end{align}
where $\alpha$ is a mixing angle and a symbol $\top$ denotes the transpose of the matrix.
The mass eigenstates ($h_1,~h_2$) are given through the mixing angle $\alpha$ as
\begin{align}
\left(\begin{array}{l}
h \\
s
\end{array}\right)=\left(\begin{array}{cc}
\cos \alpha & \sin \alpha \\
-\sin \alpha & \cos \alpha
\end{array}\right)\left(\begin{array}{l}
h_1 \\
h_2
\end{array}\right).
\end{align}
We emphasize that $\alpha \to 0$ corresponds to the SM-like limit $(h_1 \to h,~h_2 \to s)$. The mass eigenvalues are expressed as
\begin{align}
m_{h_1, h_2}^2 &=\frac{1}{2}\left(\frac{\lambda}{2} v^2+\Lambda^2 \mp \frac{\frac{\lambda}{2} v^2-\Lambda^2}{\cos 2 \alpha}\right) \\
&=\frac{1}{2}\left(\frac{\lambda}{2} v^2+\Lambda^2 \mp \sqrt{\left(\frac{\lambda}{2} v^2-\Lambda^2\right)^2+4\left(\frac{\delta_2}{2} v v_S\right)^2}\right),
\end{align}
We fix $h_1$ as the Higgs boson observed in the LHC experiments, i.e., $m_{h_1}=125$~GeV.
The mass of CP-odd state $\chi$ is given by the soft breaking terms $a_1$ and $b_1$ as
\begin{align}
m_\chi^2
&=
\frac{b_2}{2}-\frac{b_1}{2}+\frac{\delta_2}{4} v^2+\frac{d_2}{4} v_S^2\nonumber \\
&=
-\frac{\sqrt{2} a_1}{v_S}-b_1. \label{DMmass}
\end{align}
For later convenience, we mention the relationship between input and output parameters.
In the following study, we adopt $\left\{v, v_S, m_{h_1}, m_{h_2}, \alpha, m_\chi, a_1\right\}$ as inputs while
the Lagrangian parameters $\left\{m^2, b_2, \lambda, d_2, \delta_2, b_1\right\}$ can be expressed as functions of inputs.
Among the Lagrangian parameters, $m^2$ and $b_2$ are eliminated from the tadpole conditions (\ref{tadpole1}) and (\ref{tadpole2}),
\begin{align}
m^2 &=-\frac{\lambda}{2} v^2-\frac{\delta_2}{2}v_S^2 \label{msquare}\\
b_2 &=-\frac{\delta_2}{2} v^2-\frac{d_2}{2}v_S^2-\sqrt{2}\frac{a_1}{v_S}-b_1. \label{b2}
\end{align}
The remaining four parameters in the six Lagrangian parameters are given as
\begin{align}
\lambda&=\frac{2}{v^2}\left(m_{h_1}^2\cos^2\alpha+m_{h_2}^2\sin^2\alpha\right),\label{lambda}\\
\delta_2&=\frac{1}{v v_S}\left(m_{h_1}^2-m_{h_2}^2\right)\sin{2\alpha}\label{del2},\\
d_2&=2\left(\frac{m_{h_1}}{v_S}\right)^2\sin^2\alpha+2\left(\frac{m_{h_2}}{v_S}\right)^2\cos^2\alpha+2\sqrt{2}\frac{a_1}{v_S^3}\label{d2},\\
b_1 &=-m_{\chi}^2-\frac{\sqrt{2}}{v_S}a_1.
\end{align}
Theoretical constraints on the quartic couplings in the scalar potential are summarized as follows.
A requirement on the scalar potential that is bounded from below is given by\footnote{For $\delta_2<0$, $\lambda d_2>\delta_2^2$ is also needed. In this paper, we assume that $\delta_2$ is positive.}
\begin{align}
\lambda>0,\quad d_2 > 0.
\label{bfb}
\end{align}
The couplings $\lambda$ and $d_2$ should also satisfy the following condition from the perturbative unitarity~\cite{Abe:2021nih}
\begin{align}
\lambda < \frac{16\pi}{3},\quad d_2< \frac{16\pi}{3}.
\end{align}
In addition, the stability condition of the tree-level potential requires~\cite{Barger:2008jx}
\begin{align}
\lambda\left(d_2-\frac{2 \sqrt{2} a_1}{v_S^3}\right)>\delta_2^2.
\label{stability}
\end{align}
Recent DM direct detection experiments have provided upper limits on the spin-independent cross section of DM scattering off nucleons ($\sigma_{\text{SI}}$)~\cite{XENON:2020kmp,LZ:2022ufs}, and those results give a severe constraint on a certain class of DM models.
In the CxSM, the DM-nucleon scattering process is mediated by two scalars ($h_1$ and $h_2$). In the following, we briefly review the degenerate scalar scenario~\cite{Abe:2021nih} where the DM-quark scattering is suppressed when masses of $h_1$ and $h_2$ are degenerate.
The interaction Lagrangian of DM $\chi$ to the CP-even scalars $h_1$, $h_2$ is given by
\begin{align}
\mathcal{L}_S
&=
-\frac{m_{h_1}^2+\frac{\sqrt{2} a_1}{v_S}}{2 v_S} \sin \alpha~ h_1 \chi^2+\frac{m_{h_2}^2+\frac{\sqrt{2} a_1}{v_S}}{2 v_S} \cos \alpha~ h_2 \chi^2,
\label{cch}
\end{align}
while that of a quark $q$ to $h_1$ or $h_2$ is given by
\begin{align}
\mathcal{L}_Y&=\frac{m_q}{v} \bar{q} q \left(h_1 \cos \alpha-h_2 \sin \alpha\right) \label{hff},
\end{align}
where $m_q$ denotes a mass of the quark $q$.
Then the scattering amplitudes $\mathcal{M}_{h_1}$ and $\mathcal{M}_{h_2}$
mediated by $h_1$ and $h_2$, respectively, are
\begin{align}
&i \mathcal{M}_{h_1}=-i \frac{m_q}{v v_S} \frac{m_{h_1}^2+\frac{\sqrt{2} a_1}{v S}}{t-m_{h_1}^2} \sin \alpha \cos \alpha~\bar{u}\left(p_3\right) u\left(p_1\right), \label{amph1}\\
&i \mathcal{M}_{h_2}=+i \frac{m_q}{v v_S} \frac{m_{h_2}^2+\frac{\sqrt{2} a_1}{v_S}}{t-m_{h_2}^2} \sin \alpha \cos \alpha~\bar{u}\left(p_3\right) u\left(p_1\right), \label{amph2}
\end{align}
where $u(p_1)$ ($\bar{u}(p_3)$) is incoming (outgoing) quark-spinor with a momentum $p_1$ ($p_3$), and $t$ is defined as $t \equiv (p_1 - p_3)^2$.
Since the momentum transfer $t$ in the direct detection experiments is small,
the amplitude with $t\to 0$ becomes
\begin{align}
i\left(\mathcal{M}_{h_1}+\mathcal{M}_{h_2}\right) &\simeq i \frac{m_q}{v v_S} \sin \alpha \cos \alpha~\bar{u}\left(p_3\right) u\left(p_1\right) \frac{\sqrt{2} a_1}{v_S}\left(\frac{1}{m_{h_1}^2}-\frac{1}{m_{h_2}^2}\right).
\label{DMamplitude}
\end{align}
We find that the sum of two amplitudes is highly suppressed when $m_{h_1} \simeq m_{h_2}$ because of the sign difference between \eqref{amph1} and \eqref{amph2}, which is due to the orthogonality of the mixing matrix $O(\alpha)$~\eqref{masseigenstate}.
We note that, as pointed out in~Ref.~\cite{Gross:2017dan} , the two scattering amplitudes could also cancel each other when $a_1 \to 0$ without requiring the mass degeneracy of $h_1$ and $h_2$, which is known as the pseudo Nambu-Goldstone DM model. In such a case, as mentioned earlier, the scalar potential has the $Z_2$ symmetry, and it suffers from the domain-wall problem.
The possibility of searching for the degenerate scalar scenario at collider experiments has been studied in Ref.~\cite{Abe:2021nih}.
The authors stated that, although the mass difference $\left|m_{h_1}-m_{h_2}\right| \lesssim 3~\mathrm{GeV}$ is not ruled out from the LHC experiments~\cite{CMS:2014afl}, $\left|m_{h_1}-m_{h_2}\right| \lesssim 1~\mathrm{GeV}$ may be testable at the future $e^+ e^-$ linear collider.
We mention that the degenerate scalar scenario is compatible with the Higgs search experiments at the LHC.
As shown in Eq.~(\ref{hff}), the couplings between $h_1~(h_2)$ and the SM particles are those with the SM Higgs boson multiplied by $\cos{\alpha}~(-\sin{\alpha})$. For example, decay rates from $h_1$ and $h_2$ to the SM particle $X$ is expressed as follows;
\begin{align}
\Gamma_{h_1\to XX}&=\cos^2{\alpha}~\Gamma_{h\to XX}^{\mathrm{SM}}(m_{h_1}), \label{partialdecaywidth}\\
\Gamma_{h_2\to XX}&=\sin^2{\alpha}~\Gamma_{h\to XX}^{\mathrm{SM}}(m_{h_2}),
\end{align}
where $\Gamma_{h\to XX}^{\mathrm{SM}}(m_{h_{1(2)}})$ is the Higgs partial decay width in the SM as a function of $m_{h_{1(2)}}$. Experimentally, when two scalars are degenerate ($m_{h_1} \simeq m_{h_2}$), it is hard to distinguish the production and decay processes of $h_2$ from those of $h_1$ so that the sum of two processes by $h_1$ and $h_2$ is to be observed, i.e.,
\begin{align}
\Gamma_{h_1\to XX}+\Gamma_{h_2\to XX} \simeq \Gamma_{h\to XX}^{\mathrm{SM}}(m_h),
\end{align}
holds for any $\alpha$.
Therefore, the signal strength of Higgs bosons in the CxSM is identical to that in the SM in the degenerate limit of two scalars.
\section{Tree-level MPP and EWPT in the CxSM}\label{sec:MPP}
We apply the tree-level MPP to the CxSM whose the tree-level scalar potential is given by Eq.~(\ref{tree}).
Note that since $v_S$ is nonzero due to $a_1\neq0$, there are two possible vacua; the electroweak vacuum~$(v,v_S)$ and the singlet vacuum~$(0,v_S^{\prime})$. We consider the case where these two vacua are degenerate. The difference between the energy densities at these vacua is expressed as
\begin{align}
\Delta V_0
&\equiv V_0(v,v_S)-V_0(0,v_S^{\prime}) \nonumber \\
&=\frac{m^2}{8}v^2+\frac{3\sqrt{2} a_1}{4}(v_S-v_S^{\prime})+\frac{b_1+b_2}{8}(v_S^2-v_S^{\prime 2}). \label{potentialdifference}
\end{align}
When $a_1=0$, $\Delta V_0$ is evaluated as
\begin{align}
\Delta V_0
&=\frac{m^2}{8}v^2+\frac{b_1+b_2}{8}(v_S^2-v_S^{\prime 2})
\nonumber \\
&\propto -\frac{1}{\lambda d_2-\delta_2^2} \frac{1}{d_2} \times\left[\delta_2\left(b_2+b_1\right)-d_2 b_2\right]^2<0.
\label{potentialdifferenceZ2}
\end{align}
Since the denominator in \eqref{potentialdifferenceZ2} is positive due to \eqref{stability} with $a_1=0$, $\Delta V_0$ is always negative when the global U(1) symmetry is softly broken via only $S^2$ term, i.e., there are no degenerate vacua in this case~\cite{Kannike:2020qtw}. This result is altered by introducing the linear term of $S$ to the scalar potential, as shown in the following.
To begin, we qualitatively discuss the model parameters of the CxSM required by the tree-level MPP.
It is easy to see that one needs $v_S \neq v_S^{\prime}$ to degenerate two vacua because
the first term of the r.h.s. in (\ref{potentialdifference}) is nonzero and negative (see, Eq.~(\ref{msquare})).
Two VEVs, $v_S$ and $v_S^{\prime}$, are derived by tadpole conditions (\ref{tadpole2}) and (\ref{tadpole3}), respectively, the difference being the presence $\delta_2$ (\ref{del2}) which represents the doublet-singlet mixing.
Therefore, $\delta_2$ and $v_S-v_S^{\prime}$ should be sizable to cancel with the first term in (\ref{potentialdifference}), to achieve $\Delta V_0=0$.
From Eq.~(\ref{del2}), we find that $v_S$ should be small to make $\delta_2$ sizable whether we adopt the degenerate scalar scenario. In the degenerate scalar scenario, the mass difference ($m_{h_1}^2 - m_{h_2}^2$ in \eqref{del2} ) is small, so a large $\delta_2$ is possible by requiring small $v_S$.
In non-degenerate case ($m_{h_1} \neq m_{h_2}$),
the couplings of $h_1$ with $m_{h_1}=125~\mathrm{GeV}$ to the SM particles given by multiplying $\cos\alpha$ to the SM Higgs couplings. Taking account of the Higgs search experiments at the LHC, $\cos\alpha$ should be close to 1, i.e., $\sin 2\alpha$ in \eqref{del2} must be small.
Therefore $v_S$ in the denominator in \eqref{del2} should also be small to make $\delta_2$ sizable.
We quantitatively discuss the parameter space in the CxSM with degenerate scalars chosen by applying the tree-level MPP. A case where two scalars are not degenerate is also studied for comparison.
In the following numerical study, we set $m_\chi=62.5$ GeV as a reference. We note that the DM mass makes almost no contribution to $\Delta V_0$ since only $b_1$ is an output parameter of DM mass $m_\chi$~(\ref{DMmass}) while the potential difference $\Delta V_0$ depends on $b_1+b_2$ which does not include $b_1$ term as found in Eq.~(\ref{b2}).
\begin{figure}
\center
\includegraphics[width=8.1cm]{degenerate_vs_small.png}
\includegraphics[width=8.1cm]{degenerate_vs_large.png}
\caption{The potential difference for $v_S = 0.1 - 1~\mathrm{GeV}$~(a) and $v_S = 100-1000~\mathrm{GeV}$~(b) in the degenerate scalar scenario. The color bar represents the change in $a_1$. }
\label{degenerateMPP}
\end{figure}
First, we consider the degenerate scalar scenario. The second Higgs mass $m_{h_2}$ is set to $124~\mathrm{GeV}$. As mentioned in Section \ref{sec:model}, when the masses of two Higgs bosons are degenerate, there are no constraints on the mixing angle $\alpha$. Here we adopt the mixing angle $\alpha=\pi/4$ as an example. Fig.~\ref{degenerateMPP} shows the potential difference scaled by the energy density of the electroweak vacuum $\Delta V_0/V_0(v,v_S)$ as a function of $v_S$. The range of $v_S$ is $0.1 - 1~\mathrm{GeV}$ (Fig.~\ref{degenerateMPP}(a)) and $100 -1000~\mathrm{GeV}$ (Fig.~\ref{degenerateMPP}(b)). Taking account of the theoretical constraints on the scalar potential summarized in Eqs.~\eqref{bfb}-\eqref{stability}, we found no allowed parameter space for $v_S= 10 - 100~\mathrm{GeV}$. Therefore we drop the range $v_S= 10 - 100~\mathrm{GeV}$ from the figure.
The color bar represents the $a_1$ dependence. When $v_S$ is small (Fig.~\ref{degenerateMPP}(a)), $\Delta V_0/V_0(v,v_S)$ can be either positive or negative. On the other hand, when $v_S$ is large (Fig.~\ref{degenerateMPP}(b)), $\Delta V_0/V_0(v,v_S)$ approaches zero but never reaches zero. Therefore, degenerate vacua are realized only when $v_S=\mathcal{O}(0.1)$.
\begin{figure}
\center
\includegraphics[width=8.1cm]{nondegenerate_mh2_small.png}
\includegraphics[width=8.1cm]{nondegenerate_mh2_large.png}
\caption{The potential difference for $v_S$ when $m_{h_2}$= 10 GeV~(a) and $m_{h_2}$ =1000 GeV~(b) in the non-degenerate Higgs case. The color bar represents the change in $a_1$.}
\label{nondegenerateMPP}
\end{figure}
Next, we check for the possible existence of multi-critical points when the masses of two scalars are far apart. If the mass difference of two scalars is large, the couplings of the second scalar $h_2$ to the SM particles are proportional to $\sin\alpha$, and no signal of an additional scalar at the LHC requires the mixing angle $\alpha$ to be small. So we set $\alpha=0.1$ in our study\footnote{In Sec.\ref{sec:result}, we set benchmark points that satisfy the LHC constraints.}. Fig.~\ref{nondegenerateMPP} shows $\Delta V_0/V_0(v,v_S)$ as a function of $v_S$ for $m_{h_2}$ =10 GeV~(Fig.~\ref{nondegenerateMPP}(a)) and $m_{h_2}$ =1000 GeV~(Fig.~\ref{nondegenerateMPP}(b)). The color bar represents the value of $a_1$. When $m_{h_2}$ = 10 GeV, for $v_S$ = 0.1 - 1 GeV, the theoretical constraints on parameters in the scalar potential~\eqref{bfb}-\eqref{stability}
are not satisfied.
The degenerate vacua $\Delta V_0/V_0(v,v_S)=0$
are seen in small $v_S$ as in the degenerate scalar scenario. However, possible $v_S$ regions are relaxed. On the other hand, When $m_{h_2}$ = 1000 GeV, the theoretical constraints are not satisfied for small $v_S$, thus, inevitably, large $v_S$ is required, and the tree-level MPP is not viable.
We now discuss the first-order phase transition in the CxSM and mention the compatibility of EWPT and the tree-level MPP. The 1-loop effective potential, which is used to study the phase transition, can be written in this way;
\begin{align}
V_{\text{eff}}\left(\varphi, \varphi_S ; T\right)&=V_{0}(\varphi, \varphi_S)+V_{1}(\varphi, \varphi_S;T), \nonumber \\
&=V_{0}(\varphi, \varphi_S)+\sum_i n_i\left[V_{\mathrm{CW}}\left(\bar{m}_i^2\right)+\frac{T^4}{2 \pi^2} I_{B, F}\left(\frac{\bar{m}_i^2}{T^2}\right)\right], \label{eff}
\end{align}
where $T$ represents temperature and $\varphi$ and $\varphi_S$ are the constant classical background fields of $H$ and $S$, respectively. $V_{0}$ represents the tree-level potential. The indices $i$ express $h_{1,2}$, $\chi$, $W$, $Z$, $t$ and $b$. The degrees of freedom of each particle $n_i$ are $n_{h_{1,2},\chi}=1, n_W=6, n_Z=3, \text { and } n_t=n_b=-12$. The first term in parentheses in the second row in (\ref{eff}) is the $\overline{\text{MS}}$-defined effective potential at zero temperature, and the second one is the finite temperature effective potential. They are given by~\cite{Weinberg:1973am,Jackiw:1974cv,Dolan:1973qd}
\begin{align}
V_{\mathrm{CW}}\left(\bar{m}_i^2\right) &=\frac{\bar{m}_i^4}{64 \pi^2}\left(\ln \frac{\bar{m}_i^2}{\bar{\mu}^2}-c_i\right), \\
I_{B, F}\left(a^2\right) &=\int_0^{\infty} d x x^2 \ln \left(1 \mp e^{-\sqrt{x^2+a^2}}\right) \label{finite},
\end{align}
where $\bar{m_i}$ is the field-dependent masses of each particle $i$. $c_i=3/2$ for scalars and fermions and $c_i=5/6$ for gauge bosons. $I_B$ with the minus sign represents the boson contribution, while $I_F$ with the plus sign represents the fermion one. $\bar{\mu}$ is a renormalization scale. The potential $V_{\mathrm{CW}}$ is renormalized so that the scalar mass and the tad-pole conditions take the same relationship with those at the tree-level.
To achieve EWBG, EWPT must be strong first-order, which requires the following condition~\cite{Arnold:1987mh,Bochkarev:1987wf,Funakubo:2009eg};
\begin{align}
\frac{v_C}{T_C} \gtrsim 1,\label{decoupling}
\end{align}
where $T_C$ is the critical temperature defined as one when the effective potential has two degenerate minima, and $v_C$ is the Higgs VEV at $T=T_C$. The lower bound in Eq. (\ref{decoupling}) depends on the structure of the saddle point of classical solutions, called sphaleron.
Now we discuss the compatibility of the EWPT and the tree-level MPP in the CxSM.
As is already mentioned, the critical temperature $T_C$ is defined as the temperature at which the effective potential has two degenerate vacua;
\begin{align}
V_{\mathrm{eff}}\left(v_C, v_{SC} ;T_C\right)=V_{\mathrm{eff}}\left(0, v_{SC}^{\prime} ;T_C\right)
&\to
\nonumber\\
V_{0}\left(v_C, v_{SC}\right)+V_{1}\left(v_C, v_{SC} ;T_C\right)&=V_{0}\left(0, v_{SC}^{\prime}\right)+V_{1}\left(0, v_{SC}^{\prime} ;T_C\right),
\label{TCdefinition}
\end{align}
with
\begin{align}
v_C=\lim _{T \to T_C} v(T), \quad v_{S C}=\lim _{T \to T_C} v_S(T), \quad v_{S C}^{\prime}=\lim _{T \to T_C} v_S^{\prime}(T).
\end{align}
On the other hand, the vacuum degeneracy required by the tree-level MPP is
\begin{align}
V_{0}\left(v, v_{S}\right)=V_{0}\left(0, v_{S}^{\prime}\right), \label{MPPdefinition}
\end{align}
with
\begin{align}
v=\lim _{T \to 0} v(T), \quad v_{S}=\lim _{T \to 0} v_S(T), \quad v_{S}^{\prime}=\lim _{T \to 0} v_S^{\prime}(T).
\label{vevt}
\end{align}
It is known that, in general, the first-order EWPT is induced by
the bosonic contributions of the finite temperature potential at the 1-loop level.
However, in a certain class of models with the extended scalar sector, such as the CxSM, the tree-level structure of the scalar potential is rather crucial to realize the EWPT (see, Ref.~\cite{Cho:2021itv,Cho:2022our} for details)\footnote{For classification of first-order EWPT, refer, e.g., Ref.~\cite{Chung:2012vg}.}.
Now, Eqs.~\eqref{TCdefinition}-\eqref{vevt} tell us that,
except for the 1-loop effective potential $V_1$, the definition of $T_C$~(\ref{TCdefinition}) is the same with Eq.~(\ref{MPPdefinition}) derived from the tree-level MPP.
In other words, when the tree-level MPP is valid, EWPT occurs at zero temperature, which conflicts with the condition (\ref{decoupling}).
Therefore, by imposing the tree-level MPP to the CxSM, EWPT from the tree-level potential cannot be defined. On the other hand, there are subleading contributions to the EWPT at the 1-loop level from additional bosons in the CxSM.
Thus, in the next section, we evaluate if the subleading contributions are enough for the strong first-order EWPT at some benchmark points where the tree-level MPP is valid.
\section{Numerical results}\label{sec:result}
To discuss the strong first-order EWPT quantitatively, we focus on two benchmark points from the model parameter space chosen by the tree-level MPP (see, Table~\ref{tab:BP}).
We select these points such that one (BP1) corresponds to the parameter set where two scalars are degenerate ($m_{h_1} \simeq m_{h_2}$) while another (BP2) is not ($m_{h_1} \neq m_{h_2}$).
The coefficient $a_1$ is set to make $\Delta V_0$~\eqref{potentialdifference} to zero.
\begin{table}[t]
\center
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Inputs & $v$ [GeV] & $v_S$ [GeV] & $m_{h_1}$ [GeV] & $m_{h_2}$ [GeV] & $\alpha$ [rad] & $m_\chi$ [GeV] & $a_1$ [GeV$^3$]\\ \hline
BP1 & 246.22 & 0.6 & 125.0 & 124.0 & $\pi/4$ & 62.5 & $-6576.2385$ \\ \hline
BP2 & 246.22 & 10.0 & 125.0 & 10.0 & $0.001$ & 62.5 & $-707.1913$ \\ \hline\hline
Outputs & $m^2$ & $b_2$ [GeV$^2$] & $b_1$ [GeV$^2$] & $\lambda$ & $\delta_2$ & $d_2$ & $a_1$ [GeV$^3$] \\ \hline
BP1 & $-(124.5)^2$ & $-(178.0)^2$ & $(107.7)^2$ & 0.511 &1.69 & 0.87 & $-6576.2385$ \\ \hline
BP2 & $-(125.0)^2$ & $(60.20)^2$ & $-(61.69)^2$ & 0.515 & 0.013& $7.1\times10^{-5}$ & $-707.1913$ \\ \hline
\end{tabular}
\caption{Input and output parameters in the two benchmark points. }
\label{tab:BP}
\end{table}
First, we show that two benchmark points satisfy constraints on the DM relic density $\Omega_{\mathrm{DM}}h^2$~\cite{Planck:2018vyg};
\begin{align}
\Omega_{\mathrm{DM}} h^2=0.1200 \pm 0.0012, \label{relic}
\end{align}
and the spin-independent scattering cross section with the nucleons $\sigma_{\mathrm{SI}}$ by the LUX-ZEPLIN (LZ) experiment~\cite{LZ:2022ufs}.
In the following numerical study, we use a public code \texttt{micrOMEGAs}~\cite{Belanger:2020gnr}. The relic density of DM $\chi$ in the CxSM, $\Omega_\chi h^2$, should not exceed the observed value~\eqref{relic} and, as will be shown later, can explain only a portion of $\Omega_{\mathrm{DM}}h^2$.
Therefore, for $\Omega_\chi<\Omega_{\mathrm{DM}}$, we scale $\sigma_{\mathrm{SI}}$ as
\begin{align}
\tilde{\sigma}_{\mathrm{SI}}=\left(\frac{\Omega_\chi}{\Omega_{\mathrm{DM}}}\right) \sigma_{\mathrm{SI}},
\end{align}
and this should satisfy the bound from the LZ experiment.
\begin{figure}
\center
\includegraphics[width=8.1cm]{BP1_relic.png}
\includegraphics[width=8.1cm]{BP1_sigma.png}\\
\includegraphics[width=8.1cm]{BP2_relic.png}
\includegraphics[width=8.1cm]{BP2_sigma.png}
\caption{DM relic density $\Omega_{\chi}h^2$~(a),(b)
and scaled spin-independent scattering cross section with the nucleons $\tilde{\sigma}_{\mathrm{SI}}$~(c),(d) are shown as a function of the DM mass $m_\chi$. The upper panels~(a),(c) correspond to BP1, and the lower ones to BP2~(b),(d). The dotted lines in the left panels~(a),(c) show the observed relic density, and the dotted curves in the right panels~(b),(d) show the LZ experimental results.
}
\label{BP1BP2DM}
\end{figure}
Two left figures in Fig.~\ref{BP1BP2DM} show $\Omega_{\chi}h^2$ as a function of $m_\chi$ at the benchmark points BP1 (a) and BP2 (b).
In the figures, the dotted horizontal line denotes a central value of the observed DM relic density $\Omega_{\mathrm{DM}} h^2$~\eqref{relic}.
In Fig.~\ref{BP1BP2DM}~(a) (BP1), we find that $\Omega_{\chi}h^2$ is smaller than $\Omega_{\mathrm{DM}} h^2=0.12$ up to around $m_\chi$= 1500 GeV. On the other hand, in Fig.~\ref{BP1BP2DM}~(b) (BP2),
$\Omega_\chi h^2$ at only around $m_\chi$= 62.5 GeV is allowed compared to $\Omega_{\mathrm{DM}} h^2=0.12$.
The dip around $m _\chi$= 62.5 GeV~(= $m_{h_1}/2$) seen at both benchmark points reflects resonance enhancement due to $s$-channel DM annihilation processes.
Fig.~\ref{BP1BP2DM}~(c) and (d) show $\tilde{\sigma}_{\mathrm{SI}}$ as a function of $m_\chi$ at BP1 and BP2, respectively. The dotted line in each figure represents the bound from the LZ experiment.
The model prediction in both figures is allowed around at $m_\chi$= 62.5 GeV, which is affected by the dips in $\Omega_{\chi}h^2$. In each figure, although the cross section at $m_\chi \simeq 10~\mathrm{TeV}$ satisfies the limit from the experiment, it is excluded by the constraint on $\Omega_{\mathrm{DM}}h^2$. Thus, in the following study, we fix the DM mass at $m_\chi= 62.5~\mathrm{GeV}$ in both BP1 and BP2. We should mention why even BP1 with the degenerate scalar scenario, where the DM-quark scattering is suppressed, can only satisfy the DM direct detection results in some regions. The mass difference between two Higgs bosons is represented by $\delta_2$~(\ref{del2}). In the degenerate scalar scenario, $\delta_2$ becomes necessarily small because the mass difference is small. However, as mentioned previously, large $\delta_2$ is necessary to realize the tree-level MPP. This conflicting requirement for $\delta_2$ has led to the situation~(for a detailed study, see Ref.~\cite{Cho:2021itv}). However, it is also true that there is an allowed region around $m_\chi$= 62.5 GeV.
Next, we discuss constraints on benchmark points from the Higgs search experiments at the LHC.
The signal strength $\mu$ is used to compare the branching ratio of the $125~\mathrm{GeV}$ scalar at the LHC and the SM prediction.
The LHC Run-2 experiment gives constraints on $\mu$ as
$0.92<\mu<1.20 $ at ATLAS~\cite{ATLAS:2019nkf} and $0.90<\mu<1.16$ at CMS~\cite{CMS:2020gsy}.
The total decay width of the Higgs boson is constrained as $\Gamma_{h}^{\mathrm{exp}}<$ 14.4 MeV at ATLAS~\cite{ATLAS:2018jym} and $\Gamma_h^{\exp }=3.2_{-1.7}^{+2.4}$ MeV at CMS~\cite{CMS:2022ley}.
As mentioned in Sec.\ref{sec:model}, in the degenerate scalar scenario, the sum of decay rates of $h_1$ and $h_2$ is not different from the SM prediction, so both $\mu$ and $\Gamma_h$ in BP1 are not altered from those of the SM.
In BP2, since the second scalar mass $m_{h_2}$ is lighter than half of $m_{h_1}$, a decay channel $h_1 \to h_2 h_2$ opens, and it may affect $\Gamma_h$ (and the branching ratio).
However, since the mixing angle $\alpha$ is chosen very small $(0.001)$ in BP2,
the total decay width of $h_1$ is given by $\Gamma_{h_1}$=4.29 MeV, which does not much differ from the SM prediction, $\Gamma_{h}^{\mathrm{SM}}$=4.1~\cite{LHCHiggsCrossSectionWorkingGroup:2013rie}. We also find that the signal strength is $\mu=0.956$ in BP2, which is consistent with the results at ATLAS and CMS.
\begin{table}[t]
\center
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& $v_C/T_C$ & $v_{SC}$ [GeV] & $v_{SC}^{\prime}$ [GeV]\\ \hline
~BP1~ & $\frac{244.0}{48.3}=5.1$ & 0.62 & 214.6\\
~BP2~ & $\frac{244.4}{49.7}=4.9$ & 10.3 & 226.2\\
\hline
\end{tabular}
\caption{VEVs at critical temperature $T_C$ in BP1 and BP2. The calculations are performed using \texttt{cosmoTransitions}~\cite{Wainwright:2011kj}. Thermal resummation is needed to improve perturbative expansion at a finite temperature. The Parwani resummation method~\cite{Parwani:1991gq} is used in this analysis.}
\label{tab:TCvC}
\end{table}
We have so far confirmed that two benchmark points BP1 and BP2 satisfy constraints from DM experiments (relic density and direct detection) and the Higgs search experiments at the LHC.
Now, we discuss the numerical evaluation of EWPT. The calculations were performed using \texttt{CosmoTransitions}~\cite{Wainwright:2011kj}, and results are obtained using the 1-loop effective potential~(\ref{eff}). However, perturbation theory is broken down due to boson multi-loop contributions, which need to be addressed in thermal resummation methods. In this study, we use the Parwani resummation method~\cite{Parwani:1991gq} in which all the Matsubara frequency modes are resummed. Specifically, field-dependent masses $\bar{m}_i^2$ that appears in $I_{B,F}$ (\ref{finite}) are replaced by thermally corrected field-dependent masses.
Table~\ref{tab:TCvC} shows $T_C$ and the corresponding VEVs in two benchmark points. Note that since the parameters for which the tree-level MPP is valid are chosen, no first-order EWPT derived from the structure of the tree-level potential occurs. However, in the CxSM, thanks to additional bosons, strong first-order EWPT could be achieved at the 1-loop level with a finite temperature. In fact, $v_C/T_C=5.1$ for BP1 and $v_C/T_C=4.9$ for BP2, satisfying the conditions for strong first-order EWPT~(\ref{decoupling})\footnote{A distinctive feature of this EWPT is sizable change in $v_{S}$, i.e., $v_{SC}^{\prime}\gg v_{SC}$. For the devoted study, see Ref.~\cite{Cho:2021itv}.}.
\section{Summary}\label{sec:Summary}
In this paper, we have applied the tree-level MPP to the CxSM, which includes a linear term of singlet $S$ to avoid the domain-wall problem.
In the CxSM, it is known that when a mass of an additional scalar is approximately degenerate with the SM Higgs mass, the DM-quark scattering amplitudes mediated by two scalars are canceled, and constraints on the model from the DM direct detection experiments are significantly weakened~\cite{Abe:2021nih}.
On the other hand, the tree-level MPP chooses parameters so that multiple vacua in the models with the extended scalar sector have degenerate energy density.
We discussed the possibility that the model parameter space to realize such a degenerate scalar scenario is favored from the tree-level MPP by requiring the electroweak vacuum ($v,v_S$) and the singlet vacuum ($0,v_S^{\prime}$) are degenerate.
The possible existence of parameters that eliminate the potential difference in two vacua (\ref{potentialdifference}) has been investigated in the degenerate scalar region and the non-degenerate scalar region for comparison. A degeneracy between two vacua requires a difference between $v_S$ and $v_S^{\prime}$, and $v_S$ must be small in both regions. We found the parameter space where two vacua are degenerate with small $v_S$ (Fig.\ref{degenerateMPP},\ref{nondegenerateMPP}).
We have considered two benchmark points that satisfy the tree-level MPP requirement, in which the second scalar mass is fixed at $m_{h_2}=124$ GeV for BP1 and $m_{h_2}=10$ GeV for BP2. Our numerical analysis showed only the DM mass $m_\chi \simeq 62.5~\mathrm{GeV}$ is consistent with the DM relic density observation and the DM direct detection experiment for both benchmark points. We also have discussed the feasibility of first-order EWPT. The critical temperature $T_C$ for EWPT is the temperature at which the effective potential has two degenerate minima. Thus the tree-level MPP conditions are very similar to the tree-level driven EWPT as in the CxSM, where the tree-level potential provides the leading contribution~[see Eqs.~(\ref{TCdefinition}),(\ref{MPPdefinition})]. We pointed out that the tree-level contribution to the EWPT is incompatible with the tree-level MPP. However, it is numerically shown that the subleading contribution from additional bosons at the 1-loop level causes strong first-order EWPT~[see Table.\ref{tab:TCvC}].
\begin{acknowledgments}
We are grateful to Eibun Senaha for his valuable discussions. The work of G.C.C. is supported in part by JSPS KAKENHI Grant No. 22K03616.
\end{acknowledgments}
|
1,116,691,499,331 | arxiv | \section{Introduction}
\label{s1}
The density of states (DOS) of disordered {\em interacting} electronic systems
exhibits singularity near the Fermi energy.
This phenomenon manifests itself in the suppression of the tunneling conductance
at small external bias and is referred to as zero bias anomaly (ZBA).
It has been explained and investigated theoretically by
Altshuler and Aronov (AA) in 1979 for $d=3$ and latter
extended for lower dimensionality by Altshuler Aronov and Lee
\cite{Altshuler79}(for a review see
Ref.~\cite{Altshuler85}). They have predicted that for a d--dimensional system
with diffusive disorder and weak short range interactions
the singular correction to the DOS behaves as
$\delta\nu_d\sim (\mbox{max}\{\epsilon,T\})^{d/2-1}$ with the dimensionality
$d=1,2,3$.
For three dimensional ($d=3$) samples the ZBA constitutes a small but nonanalytic
correction to the DOS (as long as the dimensionless conductance, $g$,
is large, $g\gg 1$).
It may be argued subsequently that this phenomenon is a precursor of the ``Coulomb
gap'', which appears in the Anderson insulator regime, $g<1$.
Throughout this paper we shall consider good conductors, hence
we shall not discuss here the localized regime. But remarkably enough, even
for good
metals, $g\gg 1$, there is a situation where the ZBA is a strong effect.
This is the case with a finite size system. The original treatment of AA was
given for infinite (quasi) d--dimensional systems.
For a finite size system the quantization of the spectrum of the diffusive
modes becomes of crucial importance. For instance, the electron--electron
interaction term in the Hamiltonian takes the form
\begin{equation}
H_{int}=\frac{1}{2}\sum_{\bf Q}V({\bf Q}):\rho_{\bf Q}\rho_{-{\bf Q}}:,
\label{Hint}
\end{equation}
where ${\bf Q}=2\pi{\bf n}/L$ are quantized momenta, $\rho_{\bf Q}$ is the
${\bf Q}$ component of the electron density operator and $:\ldots:$ denotes
normal ordering. The quantization of
the momentum has two major consequences for the ZBA.
First, if one accounts for all finite ${\bf Q}$, excluding the
${\bf Q}=0$ mode, in the interaction
Hamiltonian, the singularity in the DOS is rounded off on the scale
$\mbox{max}\{\epsilon,T\}\approx E_c$, where $E_c$ is the Thouless
correlation energy. As long as the ZBA is still
a small correction to the total DOS on the scale $E_c$,
$\delta\nu_d(E_c)\ll\nu^{[0]}$, the
finite ${\bf Q}$ modes {\em do not} enhance the singularity for
$\epsilon,T< E_c$. It is easy to see \cite{Altshuler85} that the requirement
$\delta\nu_d(E_c)\ll\nu^{[0]}$ is satisfied for $g\gg 1$; in this case
the finite ${\bf Q}$ contribution to the ZBA is small for {\em any}
energy and temperature. It means that for such systems the finite ${\bf Q}$
interaction may indeed be treated by perturbation theory, which
produces regular expansion in powers of $1/g$. Throughout this paper we
shall assume that the condition $g\gg 1$ is satisfied, hence
the perturbative treatment of the non--zero modes is applicable.
The second consequence of the spectrum quantization is the special role
played by the ${\bf Q}=0$ term in the interaction Hamiltonian. Its
contribution to the energy may be written as $V(0):N^2:/2$, where
$N\equiv \rho_0$ is the total number of electrons in a dot. This term in
fact corresponds to
the classical charging energy of the dot. It leads to a
strong singularity in the DOS. The first order in interaction perturbative
result is $\delta\nu_0(\epsilon=0)= -V(0)/(4T)$
(for the ${\bf Q}=0$ contribution $T$ and $\epsilon$
are not interchangeable). In the
limit $T< V(0)$, the perturbative treatment is not sufficient. We present
here a
relatively simple method which allows us to treat ${\bf Q}=0$ contribution to the
ZBA exactly. An exact solution is possible due to the fact that the
zero--mode
interaction term commutes with the total Hamiltonian. As a result
the problem is trivial, although the zero--mode interaction has some
interesting consequences. The thermodynamical and the
response functions are practically unaffected by the
zero--mode interactions (apart from modifying the
statistical ensemble from grand canonical to canonical).
The single particle DOS is modified in an essential way. We use
imaginary time functional integral to integrate out all fermionic
degrees of freedom. Similar methods were used in the context of Josephson
junctions with dissipative environment (for a review see \cite{Shon90}).
Non--perturbative treatment reveals the exponential suppression of the
DOS at the Fermi energy. This result reproduces the classical treatment
of the ``orthodox'' theory of the Coulomb blockade \cite{Likharev89}.
In other words, the $d=0$ generalization of the AA ZBA, and
the Coulomb blockade, are two limiting cases of the same theory.
This statement is in fact not new. In his original paper,
Ref.~\cite{Nazarov89}, Nazarov had established the connection
between the two (including a non--perturbative treatment of
the finite ${\bf Q}$ modes). A similar, although more transparent,
theory has recently been put forward by
Levitov and Shytov \cite{Levitov95}. To a large extent, our results can be
extracted as the $d=0$ limit of the expressions found in
Refs.~\cite{Nazarov89,Levitov95}. The point is that
Refs.~\cite{Nazarov89,Levitov95} used some uncontrolled (although plausible)
approximations. We have shown that for ${\bf Q}=0$ all calculations may be
done exactly, confirming some of the approximations made in Refs.
\cite{Nazarov89,Levitov95}.
One may combine the zero and finite ${\bf Q}$ mode interactions to the ZBA.
In doing so we take into account the zero mode in an exact fashion and add on
top of it the perturbative contribution of the ${\bf Q}\neq 0$ modes. As was
mentioned above the latter can be expressed as a regular expansion in powers
of $1/g$ valid at any temperature energy and dimensionality. To address the
insulating regime one should consult Refs.~\cite{Nazarov89,Levitov95}. In the
metallic limit the ${\bf Q}\neq 0$ contribution modifies the DOS at high
energies ($\epsilon\geq e^2/2C$), as a precursor to the Coulomb blockade
which dominates at lower energies (cf. Fig. \ref{f4}b).
The new ingredient in our analysis is the inclusion of the
zero--point (and thermal) motion of the electromagnetic environment in
contact with the
system. We show that the influence of the environment may lead to a
Debye screening of the zero--mode interaction, hence to a softening of the
Coulomb blockade. The physics behind the zero--mode screening is the
following: the total charge
on the dot, $eN$, interacts with charges in the environment, leading to
their redistribution. The polarized charge of the environment reduces
the energy cost of adding (or removing) an electron to (or from) the dot.
There is a certain finite time constant (the $RC$ --
constant of the circuit), characterizing this redistribution of the
environmental charge, rendering the effective interaction in the dot
non--instantaneous (retarded). It might appear that the results obtained
through this analysis are
highly non--universal and depend on the particular choice of the model for
the environment. We stress, however, that our result expressed by
Eqs.~(\ref{GT}) and (\ref{Sscr}) is quite general, the only
model--dependent feature is the concrete form of the screened zero--mode
interaction, $V(\omega)$. Nonuniversality in this sense is unavoidable,
due to the long range nature of the Coulomb interaction: the behavior of
the dot depends on numerous long distance features.
A particularly interesting case is when the zero--mode
interaction is fully screened in the long time limit.
By this we refer to a situation when on a long time scale the addition of an
electron to the dot does not cost any energy (for instance there is a slow
continuous leakage of any extra charge from the dot).
In this case the usual Coulomb gap in the spectrum is absent and one obtains
the power--law ZBA, very similar to the one known from the physics of
Luttinger liquid.
The exponent is determined by the time scale of the environment
polarization. This result
for a quantum dot coincide with those obtained in
Refs.~\cite{Devoret90,Glazman90} for a single tunnel junction connected
to a linear $RCL$ circuit \cite{foot1}.
For a quantum dot one may consider the setup with
only partially screened interaction, where the gap in the DOS at low
energies crosses over to a power--law ZBA at larger energies.
Let us list several aspects of the problem, which we do not consider here.
Although we deal with finite size systems, we do not consider effects
related to the discreteness of the single electron spectrum. It means that the
mean level spacing, $\Delta$, is assumed to be the smallest energy scale in our
problem. We restrict ourselves to the case of good metals, where
$g=E_c/\Delta\gg 1$. Thus the energy interval of interest,
$\Delta<\epsilon<E_c$, is wide. Since we are not interested in the
single electron spectrum quantization, one may choose any boundary conditions
for the electron wave functions. We prefer to use periodic boundary
conditions and employ the momentum representation.
We also stress that our analysis omits the underlaying periodicity of
the problem as function of the charge of the positive background (or of a
gate voltage) which could be manifested as a sequence of Coulomb blockade
resonances. Instead we restrict ourselves to the case where the
background is such that the dot prefers to have an integer charge
(half way between resonances).
One can easily generalize the same treatment for any background charge,
noting that in the close vicinity of a half--integer (near a resonance) a very
different treatment, accounting for multiple tunneling
events \cite{Matveev91} is required. In fact even far from resonances
multiple tunneling may
be important (e.g ``inelastic co--tunneling'' \cite{Averin92}).
For the sake of clarity we restrict ourselves to a ``golden
rule'' scenario, where we consider only lowest order processes in the
tunneling amplitude, postponing the
consideration of multiple tunneling to a further publication.
The outline of the article is as follows. In Section \ref{s2} we recall the
derivation of the ZBA given by AA and extend it to $d=0$.
The non--perturbative treatment of zero-mode interaction is discussed in
Section \ref{s3}, where we study in some details the case of
an instantaneous zero--mode interaction and rederive the results of
the ``orthodox''
Coulomb blockade. Section \ref{s4} is devoted to the study of
screened retarded
zero--mode interaction. We show that in one particular case the ZBA
reproduces the
results derived for a single junction connected to a linear
circuit.
\section{Zero--Bias Anomaly}
\label{s2}
The derivation of the ZBA in the diffusive interacting systems proceeds as
follows \cite{Altshuler85}. One calculates the {\em single particle}
(not to be confused with the {\em thermodynamic}, $dn/d\mu$)
density of states as
function of energy or temperature. The single particle DOS is defined as
the imaginary part of the trace of the single particle Green function
\begin{equation}
\nu(\epsilon)=-\left. \frac{1}{\pi}\Im\mbox{Tr}{\cal G}(\epsilon_n)
\right|_{i\epsilon_n\rightarrow\epsilon+i\delta}.
\label{dos}
\end{equation}
Following AA \cite{Altshuler85}, we calculate the first order
correction in the {\em screened} interaction to the single particle
DOS in a dirty system. The dominant contribution \cite{foot2}
comes from the exchange diagram
depicted in Fig.\ \ref{f1}.
\begin{figure}[htbp]
\epsfysize=4cm
\begin{center}
\leavevmode
\epsfbox{e1.eps}
\end{center}
\caption{\label{f1}
First order interaction correction to the Green
function; wavy line -- interaction; dashed lines -- impurity dressing
(diffusons).}
\end{figure}
After performing the fast momentum summation one obtains
\begin{equation}
\delta\nu(\epsilon)=-\frac{\nu^{[0]}}{\pi}\Im\, T \!\!\!\!\!
\sum_{\omega_m>\epsilon_n} \sum_{\bf Q} \left.
\frac{2\pi i V({\bf Q},\omega_m)}
{(D{\bf Q}^2+|\omega_m|+\gamma_{in})^2} \right|_
{i\epsilon_n\rightarrow\epsilon+i\delta} ,
\label{AA}
\end{equation}
where $D$ is the diffusion constant; $\gamma_{in}$ --
the inelastic relaxation rate;
$V({\bf Q},\omega_m)$ -- the screened Coulomb interaction, given by
\begin{equation}
\left[ V({\bf Q},\omega_m)\right]^{-1}=\left[ V^{[0]}({\bf Q})\right]^{-1}+
\Pi({\bf Q},\omega_m)
\label{scr}
\end{equation}
Here $V^{[0]}({\bf Q})$ is the bare (instantaneous)
Coulomb interaction and $\Pi({\bf Q},\omega_m)$ is the polarization
operator of the system.
For an isolated diffusive system the polarization is
\begin{equation}
\Pi({\bf Q},\omega_m)=
\nu^{[0]}\frac{D{\bf Q}^2}{D{\bf Q}^2+|\omega_m|} ;
\label{pol}
\end{equation}
$\nu^{[0]}$ is the DOS of non--interacting system.
Let us consider a three dimensional cube with the linear size $L$
with no current flowing through the boundaries.
The slow momentum ${\bf Q}$ assumes quantized values
$${\bf Q}=\frac{2\pi}{L}{\bf n},$$
where ${\bf n}$ is a vector with integer components, including
${\bf n}=(0,0,0)$.
If either the energy, $\epsilon$, or the temperature, $T$, is much
larger than the Thouless energy, $E_c\equiv D/L^2$, one may disregard the
discreteness of ${\bf Q}$ and perform slow momentum integration instead of
summation. We shall refer to such a case as a three--dimensional, $d=3$.
Substituting Eqs.~(\ref{scr}) and (\ref{pol}) in Eq.~(\ref{AA}) and
performing integrations,
one readily obtains \cite{Altshuler85} (e.g. for $T\gg E_c;\epsilon$)
\begin{equation}
\frac{\delta\nu_3}{\nu^{[0]}}=a_3\frac{1}{g}\sqrt{\frac{T}{E_c}},
\label{AA3}
\end{equation}
where $a_3\approx 3.8\cdot 10^{-2}$ and
$g\equiv \nu^{[0]} D L^{d-2}\gg 1$ is the
dimensionless conductance of the sample (the full $T$ and $\epsilon$
dependence may be found in Ref. \cite{Altshuler85}).
At small temperature this correction
exhibits singular (non--analytic) behavior, which is the ZBA.
One cannot employ, however, Eq.~(\ref{AA3}) for $\epsilon;T<E_c$.
In this case the discreteness of the momentum spectrum begins to play a
crucial role. Performing momentum summation in
Eq.~(\ref{AA}), excluding the ${\bf Q}=0$ contribution, one obtains for
$\gamma_{in}\ll \epsilon; T\ll E_c$
\begin{equation}
\frac{\delta\nu_3}{\nu^{[0]}}=
a_0\frac{1}{g}\left(\frac{T}{E_c}\right)^2
f_0\left( \frac{\epsilon}{T} \right),
\label{AA0}
\end{equation}
where $a_0=\sum_{{\bf n}\neq 0}(2\pi |{\bf n}|)^{-6}\approx 1.1\cdot 10^{-3}$
and
$$f_0(x)=\int_0^\infty \!\! dy y\left(
2-\mbox{tanh}\frac{y-x}{2}-\mbox{tanh}\frac{y+x}{2} \right) $$
with the asymptotic values $f_0(x\ll 1)=\pi^2/3\, $; $f_0(x\gg 1)=x^2$.
At $T\approx E_c$ Eqs.~(\ref{AA3}) and (\ref{AA0}) match parametrically.
According to
Eq.~(\ref{AA0}) the small temperature singularity in the DOS is rounded off
due to finite size effects (the finite value of $E_c$), (Fig. \ref{f2}).
Eq.~(\ref{AA0}) does not account, however, for the ${\bf Q}=0$ contribution
to the momentum sum in Eq.~(\ref{AA}). Below we evaluate
this contribution and argue that it reflects the physics
of the Coulomb blockade. The first problem is that the bare Coulomb
interaction, $V^{[0]}({\bf Q})=L^{-3} 4\pi e^2/Q^2$, is not well--defined for
${\bf Q}=0$. We argue that for finite size samples this expression should be
regularized at $Q\approx 1/L$ (cf. analogous regularization of the bare
interaction in the $d=1$ case \cite{Altshuler85}), leading to
$V^{[0]}({\bf Q}=0)\approx e^2/L$. More precisely we shall use
\begin{equation}
V^{[0]} = \frac{e^2}{C},
\label{v00}
\end{equation}
where $C\approx L$ is the {\em self} capacitance of the sample,
which includes some
non--universal geometrical factors (hereafter the interaction potential, $V$,
without momentum index refers to ${\bf Q}=0$).
In other words, one may regard the finite
value of $V^{[0]}$ as the result of fast screening
processes, occurring on the boundary of the sample, which are not included
explicitly in Eq.~(\ref{scr}). According to Eqs.~(\ref{scr}) and (\ref{pol}),
the ${\bf Q}=0$ interaction can not be screened, indeed
$\Pi({\bf Q}=0,\omega_m)=0$. This amounts to saying that a finite size
{\em isolated} system cannot screen its total charge. This is not the case
for a system which is coupled capacitevely to the environment. In the latter
case the polarization operator should contain an additional term arising
from the
polarization of the environment. This would lead to an effective screening of the
${\bf Q}=0$ interaction. We postpone, however, discussion of the screened
zero--mode interaction till Section \ref{s4} and proceed with the
interaction given by Eq.~(\ref{v00}).
\begin{figure}[htbp]
\epsfysize=5cm
\begin{center}
\leavevmode
\epsfbox{e2.eps}
\end{center}
\caption{\label{f2}
Perturbative calculation of the DOS.
Full line: ${\bf Q}\neq 0$ modes only; dashed
line: ${\bf Q}= 0$ mode is included. Dotted line: AA result without account
for finite size effects ($d=3$, cf. Eq.~(6)).
The curves are shifted vertically for clarity.}
\end{figure}
Substituting now Eq.~(\ref{v00}) into Eq.~(\ref{AA}) and performing energy
summation and analytical continuation, one obtains for the zero--mode
contribution to the DOS
\begin{equation}
\frac{\delta\nu_0}{\nu^{[0]}}=-\frac{V^{[0]} }{2\pi^2 T}
\Re\,\Psi^{(\prime)}\left(\frac{1}{2}+
\frac{\gamma_{in}-i\epsilon}{2\pi T}\right),
\label{AA1}
\end{equation}
where $\Psi(x)$ is the digamma function. For
$\gamma_{in}\ll T$ this reduces to
\begin{equation}
\frac{\delta\nu_0}{\nu^{[0]}}=-\frac{V^{[0]}}{4 T}
\mbox{cosh}^{-2}\frac{\epsilon}{2 T}.
\label{AA2}
\end{equation}
Note that the dependencies on temperature and energy are very different,
whereas for finite $d$ energy and temperature play essentially the same role.
The temperature (and diffusive constant) dependence is compatible with the
AA result,
$\delta\nu_d\sim T^{(d/2-1)}D^{-d/2}$, extended to $d=0$. The zero--mode
contribution
leads to a dramatic singularity $\sim T^{-1}$ in the single particle DOS,
Fig.~\ref{f2}.
For sufficiently low temperatures, $T\ll e^2/2C$,
this last expression cannot be correct. Indeed in this
case Eq.~(\ref{AA2}) predicts $\delta\nu\gg\nu^{[0]}$, a result which is
certainly nonperturbative. Moreover, the resulting DOS
becomes negative. This clearly indicates that the first order perturbation
theory is not sufficient to treat the ZBA in
finite size systems at low temperatures.
In the next section we shall show how the zero--mode interaction can be
treated nonperturbativly.
Unlike Eq.~(\ref{AA2}), the exact result is well
behaved at any temperature and, in fact, is well known from the
``orthodox'' theory of the Coulomb blockade \cite{Likharev89}.
\section{Zero mode interaction}
\label{s3}
\subsection{General formulation}
\label{s3a}
As we have seen in the previous section, the perturbative treatment of the
zero--mode interactions in a dot at low temperature
meets serious difficulties. From another hand, the first order (in the
screened interaction) result for
${\bf Q} \neq 0$ and a not too dirty system, $g\gg 1$ (cf.
Eq.~(\ref{AA3}), (\ref{AA0})), is well behaved. Thus we separate out the
dangerous contribution of the zero--mode interaction and try to treat it
non--perturbativly. To this end consider a Hamiltonian
describing electrons moving in
a disordered potential, which interact through the zero mode
{\em repulsive} interaction only
\begin{equation}
H_0=\sum_{\alpha}\epsilon_{\alpha}a^{+}_{\alpha}a_{\alpha}+
\frac{V^{[0]}}{2}:\left[ \sum_{\alpha}a^{+}_{\alpha}a_{\alpha} -N_0
\right]^2:\, ,
\label{H}
\end{equation}
where $a^{+}_{\alpha} (a_{\alpha})$ is a creation (annihilation) operator of
an electron in an exact single particle state (which is defined including
disorder potential and spin) with an eigenenergy $\epsilon_{\alpha}$;
$V^{[0]}$ is the bare zero--mode interaction, Eq.~(\ref{v00});
$N_0$ - the charge of the positive background, which we assume to be
an integer \cite{foot3}.
Bellow we shall comment on the case where Hamiltonian includes also finite
${\bf Q}$ interactions, which can be treated perturbatively.
Before to proceed we would like to stress the following fact.
One may argue that the interaction term in the
Hamiltonian, Eq.~(\ref{H}), is trivial. Indeed it has the form
$H_{int}= V^{[0]} :N^2:/2$, where
$N\equiv \sum_{\alpha}a^{+}_{\alpha}a_{\alpha}$ is a total number of
particles in the system. As $N$ commutes with the Hamiltonian,
$[H_0,N]=0$, it does not
have any dynamics, $\partial_t N=0$. Thus the interaction term is just a
constant added to the
Hamiltonian, and seems not to have any nontrivial consequences. This is
indeed the case when considering thermodynamical properties or response
functions of the isolated
dot. It is, however, {\em not} the case with the single
particle Green
function, which correlates the amplitude of the creation of one additional
electron at an initial time, $t_i$, and its subsequent destruction at
time $t_f$. Thus any measurement of the single
particle Green function assumes implicitly the existence of processes which
do not conserve the number of particles in a dot (e.g. due to tunneling).
Such processes
render the entire Hamiltonian noncommuting with the particle number. As a
result the $N^2$ interaction term has a significant impact on the single
particle Green function. After completing the calculations
we shall comment on the relation of the above arguments to gauge invariance.
The
imaginary time single particle Green function may be written as \cite{Negele}
\begin{equation}
{\cal G}_{\alpha}(\tau_i,\tau_f,\mu)=\frac{1}{Z(\mu)}
\int{\cal D}[\psi^{*}_{\alpha}(\tau)\psi_{\alpha}(\tau)]
e^{-S[\psi^{*}_{\alpha},\psi_{\alpha}] }
\psi^{*}_{\alpha}(\tau_i)\psi_{\alpha}(\tau_f),
\label{G}
\end{equation}
with the fermionic action given by
\begin{equation}
S[\psi^{*}_{\alpha},\psi_{\alpha}]=
\int_0^{\beta}d\tau \left[ \sum_{\alpha}
\psi^{*}_{\alpha}(\tau)
(\partial_{\tau}+\epsilon_\alpha-\mu) \psi_{\alpha}(\tau)
+\frac{V^{[0]}}{2}
\left[ \sum_{\alpha}\psi^{*}_{\alpha}(\tau)\psi_{\alpha}(\tau) -N_0\right]^2
\right ];
\label{action}
\end{equation}
here $Z(\mu)$ is the partition function and $\mu$ is the chemical potential.
Splitting the interaction term in the action by means of the
Hubbard--Stratonovich transformation with
the auxiliary Bose field, $\phi(\tau)$, one obtains
\begin{eqnarray}
{\cal G}_{\alpha}(\tau_i-\tau_f)=\frac{1}{Z(\mu)}
&&\int{\cal D}[\phi(\tau)]
e^{-\int_0^{\beta}d\tau \left[
\phi(\tau)[2V^{[0]}]^{-1}\phi(\tau) -i N_0\phi(\tau) \right] } \nonumber \\
&&\int{\cal D}[\psi^{*}_{\alpha}\psi_{\alpha}]
e^{-\int_0^{\beta}d\tau
\sum_{\alpha}\psi^{*}_{\alpha}
(\partial_{\tau}+\epsilon_{\alpha}-\mu+i\phi(\tau)) \psi_{\alpha} }
\psi^{*}_{\alpha}(\tau_i)\psi_{\alpha}(\tau_f) \nonumber \\
=\frac{1}{Z(\mu)}&&\int{\cal D}[\phi(\tau)]
e^{-\int_0^{\beta}d\tau \left[
\phi(\tau)[2V^{[0]}]^{-1}\phi(\tau) -i N_0\phi(\tau) \right] }
Z^{[\phi]}(\mu){\cal G}^{[\phi]}_{\alpha}(\tau_i,\tau_f,\mu)
\label{HS}
\end{eqnarray}
with the same transformations in $Z(\mu)$. Here $Z^{[\phi]}(\mu)$ and
${\cal G}^{[\phi]}_{\alpha}(\tau_i,\tau_f,\mu)$ are respectively
the partition
and Green functions of non--interacting electrons in the time
dependent (but spatially uniform) potential,
$i\phi(\tau)$. To calculate these quantities one should resolve
the spectral problem for the first order differential operator
$\left[ \partial_{\tau}-\mu+\epsilon_{\alpha}+i\phi(\tau) \right]$
with antiperiodic
boundary conditions. This can be easily done (see e.g. chapter 7 in
Ref. \cite{Negele}). For the spectral determinant (partition function) one
finds
\begin{equation}
Z^{[\phi]}(\mu)=Z^{[0]}(\mu-i\phi_0),
\label{ZG1}
\end{equation}
where $Z^{[0]}(\mu)\equiv\exp\{-\beta\Omega^{[0]}(\mu)\}$ is the partition
function of non--interacting electron gas. We have introduced Matsubara
representation for the boson field, $\phi(\tau)$:
\mbox{$\phi_m\equiv
\beta^{-1}\int_0^{\beta}d\tau\phi(\tau)\exp\{i\omega_m\tau\}$},
$\omega_m=2\pi m T$.
The Green function is given by
\begin{equation}
{\cal G}^{[\phi]}_{\alpha}(\tau_i,\tau_f,\mu)=
{\cal G}^{[0]}_{\alpha}(\tau_i-\tau_f,\mu-i\phi_0)
e^{i\int_{\tau_i}^{\tau_f}d\tau [\phi(\tau)-\phi_0]},
\label{ZG}
\end{equation}
where ${\cal G}^{[0]}_{\alpha}(\epsilon_n,\mu)=
(i\epsilon_n-\epsilon_{\alpha}+\mu)^{-1}$
is the Green function of non-interacting fermions.
It is convenient to rewrite
the exponent in the last equation in the following form
\begin{equation}
\exp\{i\int_{\tau_i}^{\tau_f}d\tau [\phi(\tau)-\phi_0]\} =
\exp\{\beta\sum_{m\neq 0}\frac{\phi_m J_{-m}^{\tau_i,\tau_f}}{\omega_m} \},
\label{exp}
\end{equation}
where $J_m^{\tau_i,\tau_f}$ is the Matsubara transform of the following
function
\begin{equation}
J_{\tau}^{\tau_i,\tau_f}=\delta(\tau-\tau_i)-\delta(\tau-\tau_f).
\label{J}
\end{equation}
Transforming next
the functional integral over $\phi(\tau)$ to integrals
over the Matsubara components, $\phi_m$, we obtain
\begin{eqnarray}
{\cal G}_{\alpha}(\tau_i-\tau_f)=\frac{1}{Z(\mu)}
&&\int d\phi_0
e^{-\beta[\phi_0[2V^{[0]}]^{-1}\phi_0 -i\phi_0 N_0+\Omega^{[0]}(\mu-i\phi_0)]}
{\cal G}^{[0]}_{\alpha}(\tau_i-\tau_f,\mu-i\phi_0) \nonumber \\
&&\int\prod_{m\neq 0} d\phi_m
\exp \left\{ \beta \sum_{m\neq 0} \left[ -\frac{\phi_m\phi_{-m}}{2V^{[0]}}
+\frac{\phi_m J_{-m}^{\tau_i,\tau_f}}{\omega_m} \right] \right\}.
\label{G1}
\end{eqnarray}
Again with the analogous modifications in $Z(\mu)$.
The integral over the static component, $\phi_0$, describes the smooth
transition between the grand canonical ensemble with the chemical potential
$\mu$ (at $V^{[0]}=0$) to the canonical ensemble with $N_0$
electrons (at $V^{[0]}=\infty$). For large enough systems
($\Delta^{-1}\equiv-\partial^2\Omega^{[0]}/\partial\mu^2\gg \beta$) one can neglect
differences between the two statistical ensembles. This means that the integral
over $\phi_0$ can be calculated in a saddle point approximation, leading to
${\cal G}^{[0]}_{\alpha}(\tau_i-\tau_f,\overline\mu)$, where
the stationary point, $\overline\mu$, is the real solution of the equation
$(\mu-\overline\mu)/V^{[0]}+N_0+\partial \Omega^{[0]}(\overline\mu)
/\partial\mu=0$.
The remaining integrals (over $\phi_m$ for $m\neq 0$) are purely Gaussian.
As a result one obtains
\begin{equation}
{\cal G}_{\alpha}(\tau_i-\tau_f,\mu)=
{\cal G}^{[0]}_{\alpha}(\tau_i-\tau_f,\overline\mu)
e^{-S(\tau_i-\tau_f)},
\label{GT}
\end{equation}
where
\begin{equation}
S(\tau)=T\sum_{m\neq 0}\frac{V^{[0]}}{\omega_m^2}
(1-e^{i\omega_m\tau}).
\label{S}
\end{equation}
Eqs.~(\ref{GT}) and (\ref{S}) solve the problem of finding the exact single
particle Green function in the presence of the zero--mode interaction.
This result is depicted diagrammatically in Fig. \ref{f3}. The dressed
Green function of the interacting problem is given by the bare one (with a
renormalized chemical potential)
decorated with the propagator of the auxiliary boson field
${\cal D}(\tau)\equiv \exp\{-S(\tau)\}$.
We refer to this auxiliary field as a -- {\em Coulomb boson}.
As ${\cal D}(0)=1$, the zero mode interaction
does not influence equal time Green functions (apart from the renormalization of
the chemical potential). As a result thermodynamical quantities are not
affected by the presence of the Coulomb boson. This is not unexpected since
the interaction term commutes with a total Hamiltonian, hence it does not
affect
quantities defined for the closed system. By contrast, to measure a
two point Green function one should perform a tunneling experiment, where
the system can not be considered as completely isolated. In this case the
presence of the interaction term in Eq.~(\ref{H})
(and hence of the Coulomb boson) is of crucial importance.
\begin{figure}[htbp]
\epsfysize=3cm
\begin{center}
\leavevmode
\epsfbox{e3.eps}
\end{center}
\caption{\label{f3}
Diagrammatic representation of the Coulomb boson.}
\end{figure}
This observation can be related to gauge invariance \cite{Fin}.
As we have seen the problem with zero mode interaction is essentially
reducible to that of an electrons gas in a spatially uniform a.c. potential.
Such a
potential can be always removed from the problem by a time--dependent gauge
transformation. Thus gauge invariant physical quantities (e.g.
thermodynamical quantities) are not affected by
the presence of spatially uniform potential, cf. Eq.~(\ref{ZG1}).
The single particle Green function, being
{\em non} gauge invariant object is affected.
By introducing an electron tunneling from an external source we fix the (time
dependent) phase of the electron wave function in the system. This point has
been discussed by Finkelstein \cite{Fin}, who argued that unlike the
conductivity or thermodynamical quantities, the non gauge invariant single
particle DOS may be affected by very small ${\bf Q}$ terms, leading in $d\leq
2$ to more pronounced singularities.
\subsection{The Coulomb boson and DOS}
\label{s3b}
For the instantaneous (frequency--independent) interaction, Eq.~(\ref{v00}),
the sum in Eq.~(\ref{S}) may be easily performed, yielding
\begin{equation}
S(\tau)=\frac{V^{[0]}}{2}
\left( |\tau|-\frac{\tau^2}{\beta} \right).
\label{S0}
\end{equation}
Using the Lehmann representation for temperature Green functions
\cite{Abrikosov64} one may write for the propagator of the Coulomb boson
\begin{equation}
{\cal D}(\omega_m)=\int_{-\infty}^{\infty} \frac{d\omega^\prime}{2\pi}
\frac{B(\omega^\prime)}{i\omega_m-\omega^\prime},
\label{DL}
\end{equation}
where the spectral function $B(\omega)$ is defined as
$B(\omega)\equiv -2\Im{\cal D}^R(\omega)$. Fourier transforming
of ${\cal D}(\tau)=\exp\{-S(\tau)\}$ and analytically continuing we obtain
(see Appendix \ref{app})
\begin{equation}
B(\omega)=\sqrt{\frac{2\pi}{V^{[0]}T}} \left(
e^{-\frac{(\omega+V^{[0]}/2)^2}{2V^{[0]}T} }-
e^{-\frac{(\omega-V^{[0]}/2)^2}{2V^{[0]}T} } \right).
\label{B1}
\end{equation}
According to Eq. (\ref{GT}) (cf. also Fig. \ref{f3}) the
electron Green function is given by
\mbox{${\cal G}_{\alpha}(\epsilon_n)=T\sum_{\omega_m}
{\cal G}^{[0]}_{\alpha}(\epsilon_n-\omega_m)
{\cal D}(\omega_m)$}. Performing the summation by a standard contour
integration and then
analytical continuation ($i\epsilon_n\rightarrow\epsilon+i\delta$)
one obtains for the one particle
(tunneling) density of states, $\nu(\epsilon)\equiv-\pi^{-1}\sum_{\alpha}\Im
{\cal G}^{R}_{\alpha}(\epsilon)$,
\begin{equation}
\nu(\epsilon)=-\frac{1}{2}\int_{-\infty}^{\infty}
\frac{d\omega}{2\pi}
\left(
\mbox{tanh}\frac{\epsilon-\omega}{2T}+
\mbox{coth}\frac{\omega}{2T}
\right)
B(\omega) \nu^{[0]}(\epsilon-\omega),
\label{nu}
\end{equation}
where $\nu^{[0]}(\epsilon)$ is the density of states in the absence of the
zero--mode interaction and $B(\omega)$ is given by Eq.~(\ref{B1}).
For non--interacting electrons (with or without disorder)
$\nu^{[0]}(\epsilon)$ may be well--approximated by a constant value,
$\nu^{[0]}$,
in this case the integral in the last expression may be readily evaluated.
In the limit of weak interaction ($V^{[0]}\ll T$) one obtains
\begin{equation}
\frac{\nu_0}{\nu^{[0]}} =1-\frac{V^{[0]}}{4T}
\mbox{cosh}^{-2}\frac{\epsilon}{2T}.
\label{nul}
\end{equation}
This is the zero bias anomaly of AA extrapolated to $d=0$
system, which has been obtained from a diagrammatic expansion, Eq. (\ref{AA2}).
Notice that we did not actually use the fact that the
system is disordered. Note also, that the exact lowest order result,
Eq. (\ref{nul}), coincides with the one obtained from the exchange diagram
only, Eq. (\ref{AA2}). The first order Hartree term with the ${\bf Q}=0$
interaction leads to redefinition of the chemical potential which has been
absorbed in the factor, $\overline \mu$.
For strong interaction ($V^{[0]}\gg T$) one has (Fig. \ref{f4}a)
\begin{equation}
\frac{\nu_0}{\nu^{[0]}} =\left\{ \begin{array}{ll}
{\displaystyle \sqrt{\frac{2\pi T}{V^{[0]}}} \,
e^{-\frac{V^{[0]}}{8T}} \mbox{cosh}\frac{\epsilon}{2T} }; \,\,\,\,
&\epsilon\ll \sqrt{V^{[0]}T} \ll V^{[0]}, \\
{\displaystyle 1-e^{-(\epsilon-V^{[0]})/T } };
& \epsilon\gg V^{[0]}\gg T.
\end{array} \right.
\label{nug}
\end{equation}
The exponential suppression of the tunneling density of states near the
Fermi energy is a direct manifestation of the Coulomb blockade. The ZBA
in $d=0$ systems, being treated non--perturbativly, leads to results which
are well--known from the ``orthodox'' theory of the Coulomb blockade, cf. Ref.
\cite{Likharev89}.
\begin{figure}[htbp]
\epsfysize=8cm
\begin{center}
\leavevmode
\epsfbox{e4.eps}
\end{center}
\caption{\label{f4}
Non perturbative
DOS. (a) as function of temperature at the Fermi energy
($\epsilon=0$); only the ${\bf Q}= 0$ mode is included.
(b) DOS as function of energy at $T=0$; finite ${\bf Q}$ contributions
are included.}
\end{figure}
One may ask how the finite ${\bf Q}$ interactions modify the standard
Coulomb blockade predictions.
In the case where the finite ${\bf Q}$ interactions can be treated
perturbativly the answer can be read off Eq. (\ref{nu}): in Eq. (\ref{nu})
one should substitute the perturbative AA expression for the
DOS ($\nu^{AA}$) for $\nu^{[0]}(\epsilon)$.
Indeed, one may repeat
the derivation given above in the presence of the finite ${\bf Q}$
interactions;
the result coincides with Eq. (\ref{GT}), where ${\cal G}^{[0]}$ includes
the effects of all but zero--mode interactions.
In the limit of zero temperature one obtains
\begin{equation}
\nu(\epsilon) =\left\{ \begin{array}{ll}
{\displaystyle 0}; &|\epsilon|< V^{[0]}/2, \\
{\displaystyle \nu^{AA}(|\epsilon|-V^{[0]}/2)} ;\,\,\,\,
& |\epsilon| > V^{[0]}/2.
\end{array} \right.
\label{nui}
\end{equation}
This result (Fig. \ref{f4}b) implies that for voltages larger than the Coulomb
blockade gap the tunneling current is still somewhat suppressed due to the
presence of the
finite dimensional ZBA. Although the finite ${\bf Q}$ interactions
reduce the jump at $|\epsilon| = V^{[0]}/2$ ($T=0$), it do not remove the
discontinuity. There is, however, important physics which leads to
the rounding off of the threshold at $|\epsilon|=e^2/2C$ even at zero
temperature.
This is the screening of the zero--mode interaction due to fluctuations in
the electromagnetic environment. This issue is to be discussed next.
\section{Screened zero--mode interaction}
\label{s4}
In the preceding section we studied the tunneling DOS of a dot which is
perfectly isolated (both electrically and electromagnetically) from the
outside world. In this case the total charge of the dot can not be screened.
This is why we have employed a bare (instantaneous) zero--mode interaction,
Eq. (\ref{v00}). This scheme, however, is not very satisfactory. First,
measurements of some
quantities of major interest (such as the single particle DOS studied here,
or the inelastic broadening to be discussed elsewhere), require by
definition direct coupling to an external medium. But more importantly, even in
the absence of external leads the dot interacts (through capacitive coupling)
with external gates, conducting layers, external charges, all of which will
be referred to as the environment. The creation of an
additional charge in the dot leads to redistribution of charges in the
environment, which takes finite time. The redistributed
environmental charge partially screens the initially created charge on the
dot, reducing the zero--mode interaction energy. The fact that this
redistribution is not instantaneous implies that the effective screened
interaction is retarded. Equivalently one may notice that
the dot's capacitance is proportional to the dielectric constant of the
surrounding medium. Unless this medium is a perfect vacuum,
the dielectric constant is
a function of frequency. As a result the zero--mode
interaction, $e^2/C(\omega)$, is not instantaneous, but rather
characterized by a finite retardation time scale.
\begin{figure}[htbp]
\epsfysize=4cm
\begin{center}
\leavevmode
\epsfbox{e5.eps}
\end{center}
\caption{\label{f5}
Equivalent circuit of a dot coupled to the enviroment.
The grey square represents a dot, $C$ being its self capacitance;
$C_G$-- the mutual capacitnce to the environment;
$Z(\omega)$-- the linear impedance of the environment. Here $U$ represents
the noise voltage. The black arrow is a tunneling tip through which particles
may be injected to the dot, measuring its DOS. This tip may be coupled to
the dot through a large impedance barrier. }
\end{figure}
Phenomenologically it is convenient to consider an equivalent circuit
(Fig. \ref{f5}), assigning the self capacitance, $C$, (bare interaction) to the
dot, which in turn is capacitively coupled (through $C_G$) to the gate
electrode. The gate is electrically coupled
to the ground through the linear impedance $Z(\omega)$. The voltage source,
$U$, represents the equilibrium noise voltage of the entire circuit.
We shall model the dot--environment interaction by assuming that the dot is
subjected to a time
dependent (but spatially uniform) noise potential due to the environment
\begin{equation}
H_{noise}=\eta(t)\sum_{\alpha}a^{+}_{\alpha}a_{\alpha}.
\label{noise}
\end{equation}
At the end of these calculations all physical quantities should be averaged over
realizations of the noise. We shall assume further that the noise is Gaussian
with zero mean value. The averaging procedure thus may be
written as
\begin{equation}
\langle \ldots \rangle_{noise} =\frac{1}{{\cal N}}
\int{\cal D}[\eta(\tau)]
e^{-\frac{1}{2}\int\int_0^\beta d\tau d\tau^{\prime}
\eta(\tau)K^{-1}(\tau-\tau^{\prime})\eta(\tau^{\prime}) }
\ldots\,\, ,
\label{aver}
\end{equation}
where ${\cal N}$ is a normalization factor and
$K(\tau-\tau^{\prime})=\langle \eta(\tau) \eta(\tau^{\prime}) \rangle$ is
the noise correlator, which is to be determined employing the
fluctuation--dissipation theorem (FDT).
An important observation is that the partition
function of the dot, $Z(\mu)$, is not affected by the noise term. Indeed, we
have already seen, Eq.~(\ref{ZG1}), that the partition function of the
electron gas in a spatially uniform a.c. field depends on its mean value
only (which is zero in our case). This
can be understood as a consequence of the gauge invariance of the partition
function -- spatially uniform field may be always removed by a gauge
transformation. As a result, the noise term in Eq.~(\ref{G}) enters in the
numerator only. Averaging Eq.~(\ref{G}) over
the Gaussian noise, Eq.~(\ref{aver}), leads to an effective fermionic action
with interaction which is non--local in time,
\begin{equation}
S_{int}[\psi^{*}_{\alpha},\psi_{\alpha}]= \frac{1}{2}
\int \!\!\! \int_0^{\beta}d\tau d\tau^{\prime}
\sum_{\alpha}\psi^{*}_{\alpha}(\tau)\psi_{\alpha}(\tau)
\left( V^{[0]}\delta(\tau -\tau^{\prime}) - K(\tau-\tau^{\prime}) \right)
\sum_{\alpha}\psi^{*}_{\alpha}(\tau^{\prime})\psi_{\alpha}(\tau^{\prime}) .
\label{nlact}
\end{equation}
As a result, one obtains an effective renormalization (screening)
of the zero--mode interaction potential
\begin{equation}
V^{[0]} \rightarrow V(\omega_m)=V^{[0]}- K(\omega_m).
\label{Veff}
\end{equation}
Further calculations follow the same steps outlined in Sec. \ref{s2}.
The final result is given
again by Eq.~(\ref{GT}) where now (cf. Eq.~(\ref{S}))
\begin{equation}
S(\tau)=T\sum_{m\neq 0}\frac{V(\omega_m)}{\omega_m^2}
(1-e^{i\omega_m\tau}).
\label{Sscr}
\end{equation}
We shall next employ the FDT to calculate the noise correlator,
$K(i\omega)$. According to the FDT the equilibrium noise spectrum ($T=0$)
of the total noise voltage generated by the circuit is
$\langle UU\rangle=e^2 i\omega Z_{tot}$, where the total impedance of the
equivalent circuit is
$Z_{tot}=(i\omega C)^{-1}+(i\omega C_G)^{-1}+Z(\omega)$.
The corresponding voltage drop on the dot is $\eta=U(i\omega C)^{-1}/Z_{tot}$;
thus the noise correlator, $K=\langle \eta\eta \rangle$,
is given by
\begin{equation}
K(i\omega)=\frac{e^2}{C}\frac{1/(i\omega C)}{Z_{tot}}.
\label{K}
\end{equation}
Substituting this expression in Eq.~(\ref{Veff}) and rewriting it in a
finite temperature form one obtains
\begin{equation}
V(\omega_m)=\frac{e^2}{C}\, \,
\frac{C/C_G+|\omega_m| ZC}{1+C/C_G+|\omega_m| ZC}.
\label{v0scr}
\end{equation}
The high frequency (unscreened) value coincides with that of Eq.~(\ref{v00}),
whereas at low frequency the interaction is partially screened and is
given by the total capacitance, $V(0)=e^2/(C+C_G)$.
The characteristic crossover frequency
is given by the ``$RC$'' time of the circuit, $\omega\approx (ZC)^{-1}$.
As a result the long time behavior of $S(\tau)$,
determined by the small frequency asymptotic of the screened interaction,
is given by
$S(\tau)\approx |\tau| e^2/2(C+C_G) $ for $ZC\ll\tau\ll\beta$.
A case of particular interest is that of a ``maximally'' screened interaction,
when \mbox{$V(i\omega\rightarrow0)=0$}.
In this case the linear term in $S(\tau)$ is absent and
the action grows at most logarithmically at large $\tau$.
This would immediately imply that instead
of exponential suppression of the DOS at the Fermi energy, one obtains only
a power--law ZBA. According to Eq.~(\ref{v0scr}) such a case may be realized
when $C_G\rightarrow \infty$ (more precisely
$ZC_G\gg\beta$). This is the case of a dot strongly connected to the
environment. For simplicity we restrict ourselves to the scenario of a pure
ohmic
environment, $Z=R$, in which case the interaction potential is given by
\cite{foot4}
\begin{equation}
V(\omega_m)=V^{[0]}\, \frac{|\omega_m|}{\Omega+|\omega_m|},
\label{Vstr}
\end{equation}
where $\Omega\equiv (RC)^{-1}$ and $V^{[0]}$ is given by Eq.~(\ref{v00}).
As $V(\omega=0)=0$ the addition or subtraction of an electron from the dot
costs no Coulomb energy over long time scales.
On short time scales, however, before the environment adjusts to screen out
the added electron, one has to pay some energy, Eq. (\ref{Vstr}). Thus the
addition of an electron can be considered as ``tunneling'' under an energy
barrier in the time direction. The very same physics has been discussed in
the context of $d=2$ systems in Refs. \cite{Spivak94,Levitov95}.
The related energy cost on short time scales suppresses free particle exchange
between the dot and the particle reservoir (although
Coulomb blockade in its strict sense is absent). This leads to the
suppression of the tunneling DOS hence to ZBA.
Substituting Eq. (\ref{Vstr}) into Eq. (\ref{Sscr}), one obtains
for $\Omega^{-1}\ll \tau\ll \beta$ \cite{foot5}
\begin{equation}
S(\tau)\approx 2r\log(1+\tilde\Omega|\tau|)+
O\left( (\Omega\tau)^{-1} \right)\, ,
\label{sas}
\end{equation}
where $\tilde\Omega\equiv e^{\gamma}\Omega$ with $\gamma=0.577\ldots$
being the Euler constant and
$r\equiv V^{[0]}/(2\pi\Omega)= R e^2/(2\pi \hbar)$
is dimensionless resistance of the environment.
Performing the Fourier transform and analytical
continuation (see Appendix \ref{app}),
one obtains for the zero--temperature spectral density of the Coulomb boson
\begin{equation}
B(\omega)=-\frac{\mbox{sign}\,\omega}{\tilde\Omega}\,
\frac{2\pi}{\Gamma(2r)}
\left| \frac{\omega}{\tilde\Omega} \right|^{2r-1}
e^{-|\omega/\tilde\Omega|}.
\label{B0scr}
\end{equation}
Finally the zero temperature DOS may be found from Eq.~(\ref{nu}) with
$B(\omega)$ given by Eq.~(\ref{B0scr}). Assuming a constant bare DOS
(i.e. $\nu^{[0]}=\mbox{const}$) one obtains
\begin{equation}
\frac{\nu_0(\epsilon)}{\nu^{[0]}} =
1-\Gamma\left( 2r,\left| \frac{\epsilon}{\tilde\Omega} \right|\right)\approx
\left\{ \begin{array}{ll}
{\displaystyle \frac{1}{\Gamma(2r+1)}
\left| \frac{\epsilon}{\tilde\Omega} \right|^{2r} };
&|\epsilon|\ll \Omega, \\
{\displaystyle
1-\left| \frac{\epsilon}{\tilde\Omega} \right|^{2r-1}
e^{ -|\epsilon/\tilde\Omega|} };\,\,\,\,
& |\epsilon|\gg \mbox{max}\{\Omega, V^{[0]}\},
\end{array} \right.
\label{nu0scr}
\end{equation}
where $\Gamma(2r,x)$ is the incomplete gamma function.
For the maximally screened zero--mode interaction scenario the ZBA in $d=0$
has a power law behavior, rather than the gap given by
Eq.~(\ref{nui}). For
high--impedance (slow) environment, $r\gg 1$, there is a crossover to the
``orthodox'' Coulomb blockade in the interval $\Omega<\epsilon<V^{[0]}$.
However for the low--impedance, $r<1$, environment the power--law ZBA,
Eq.~(\ref{nu0scr}), directly crosses over to the finite dimensional AA
result at $\epsilon\approx \Omega$.
The straightforward finite temperature generalization of Eq.~(\ref{B0scr})
is (see Appendix \ref{app})
\begin{equation}
B(\omega)=-\frac{\mbox{sinh}\, \omega/2T}{\tilde\Omega}\,
\frac{2}{\Gamma(2r)}
\left( \frac{2\pi T}{\tilde\Omega} \right)^{2r-1}
\left| \Gamma\left( r+\frac{i\omega}{2\pi T} \right) \right|^2 .
\label{Bscr}
\end{equation}
Substituting this result into Eq.~(\ref{nu}) one obtains e.g. for the DOS at
the Fermi energy, $\epsilon=0;\,\,\, T\ll\Omega$
\begin{equation}
\frac{\nu_0}{\nu^{[0]}} =a(r)
\left( \frac{2\pi T}{\tilde\Omega} \right)^{2r},
\label{nuscr}
\end{equation}
where
\begin{equation}
a(r)\equiv \frac{1}{\pi\Gamma(2r)}\int_0^\infty dx
\frac{\left| \Gamma( r+ix ) \right|^2}{\mbox{cosh}\, \pi x}=
\left\{ \begin{array}{ll}
1-\Psi\left( \frac{1}{2} \right)2r;\,\,\, & r\ll 1, \\
2^{-2r}/\sqrt{\pi r}; & r\gg 1;
\end{array} \right.
\label{sigma}
\end{equation}
$\sigma(1/2)=1/\pi;\,\,\, \sigma(1)=1/8$. At
$T > \mbox{max} \{\Omega, V^{[0]} \}$ the DOS obeys Eq.~(\ref{nul}).
Screening of the zero mode
interaction converts the exponential suppression of the DOS
(cf. Eq.~(\ref{nug})) into the power--law, Eq.~(\ref{nuscr}).
Our results in the limit of maximally screened interaction $C_G\rightarrow
\infty$ are compatible with
the results of
Refs. \cite{Devoret90,Glazman90}, where tunneling through a {\em single}
tunnel junction coupled to a linear impedance was considered.
This coincidence is not surprising at all; indeed a dot coupled to a
circuit through
large capacitance, $C_G$, can be viewed as an island from which charge can
continuously leak out; this is practically the setup of
Refs. \cite{Devoret90,Glazman90}.
Our present analysis stresses the physics of a weakly coupled dot
(with a finite $C_G$, Eq. (\ref{v0scr})) and its
relation to the ZBA. At $T=0$, employing the above given expressions one
obtains
\begin{equation}
\frac{\nu_0(\epsilon)}{\nu^{[0]}}=
\left\{ \begin{array}{ll}
0; & |\epsilon| \leq e^2/2(C+C_G), \\
1-\Gamma \left( 2\tilde r,
\frac{|\epsilon| - e^2/(C+C_G)}{\tilde\Omega(C+C_G)/C_G} \right);
\,\,\,\, & e^2/2(C+C_G)< |\epsilon|,
\end{array} \right.
\label{nu0ss}
\end{equation}
where $\tilde r\equiv r[C_G/(C+C_G)]^2$.
At finite temperature and $\epsilon=0$ one has
\begin{equation}
\frac{\nu_0}{\nu^{[0]}} \approx
\left\{ \begin{array}{ll}
\exp\{-e^2/8T(C+C_G)\}; \,\,\,\, & T \leq e^2/2(C+C_G), \\
(T/\tilde\Omega)^{2\tilde r} ; &e^2/2(C+C_G)< T < \Omega, \\
1-e^2/4TC; & \mbox{max}\{\Omega, e^2/2C\}< T
\end{array} \right.
\label{nu0sss}
\end{equation}
(the above expressions account for the ${\bf Q}=0$ contribution only).
We stress that, our results are a particular,
d=0, case of a general non--perturbative expression for the ZBA obtained by
Nazarov \cite{Nazarov89} and Levitov and Shytov \cite{Levitov95}. Our point
here is that for $d=0$ case all calculations can be carried out exactly,
avoiding some of the uncontrolled albeit plausible assumptions employed
in Refs. \cite{Nazarov89,Levitov95}.
\section{Acknowledgments}
We are grateful to A. Altland, M.~Devoret, U. Sivan and
B. Shklovskii
for useful suggestions. In particular we are indebted to A. M. Finkelstein
for his comments on the relation between the structure of the density of
states and gauge invariance. This research was supported by the German--Israel
Foundation (GIF) and the U.S.--Israel Binational Science Foundation (BSF) and
the Israel Academy of Sciences.
|
1,116,691,499,332 | arxiv | \section{Introduction}
In optical communication, a sender encodes a message in an optical signal and sends it to a receiver who detectes the signal to decode the message \cite{g.cariolaro}. Thus, the success probability of the optical communication is determined by the physical and statistical properties of the optical signal together with the structure of the receiver's measurement device. In classical optical communication, the receiver can use an on-off detector to decode a sender's message encoded in on-off keying signal \cite{c.w.helstrom,k.tsujino}, and a homodyne detector for binary phase shift keying signal \cite{j.g.proakis}. However, the maximal success probability for decoding encoded messages by using \textit{conventional measurements} such as the on-off and the homodyne detectors cannot exceed the standard quantum limit.
One of the goals in quantum communication is to design a novel measurement so that the maximal success probability to decode messages can surpass standard quantum limit \cite{i.a.burenkov}. According to the quantum theory, optical signal is described as a density operator on a Hilbert space and a measurement is descrived as a positive-operator-valued measure (POVM), therefore the quantum communication is described as a quantum state discrimination protocol \cite{s.m.barnett,j.a.bergou}.
Minimum error discrimination \cite{j.bae,d.ha} is one representative state discrimination strategy used in various quantum communication protocols. When one bit message is encoded by binary coherent states, minimum error discrimination between the binary coherent states can be implemented via the Dolinar receiver \cite{s.j.dolinar}. However, when several bits are encoded and sequentially sent, the photon number detector used for the discrimination may not efficiently react along the received states \cite{i.a.burenkov}. For this reason, $N$-ary coherent states such as $N$-amplitude shift keying ($N$-ASK) signal \cite{c.w.helstrom} and $N$-phase shift keying ($N$-PSK) signal \cite{j.g.proakis} have been considered to send $\log_2N$ bit messages.
According to a recent work \cite{e.m.f.curado}, the maximal success probability (or Helstrom bound) of discriminating a message encoded in 2-PSK signal composed of \textit{non-standard coherent states (NS-CS)} with a novel quantum measurement can be improved by the sub-Poissonianity of the NS-CS. Moreover, the experimental method for implementing the quantum measurement reaching for the Helstrom bound has recently been proposed \cite{m.namkung}. Since the negative Mandel parameter to quantify the sub-Poissonianity is considered as a resource in a non-classical light \cite{s.dey}, this result implies that the sub-Poissonianity can be a resource for improving the performance of the quantum communication.
In the present article, we consider the quantum communication with $N$-PSK signal for arbitrary an arbitrary positive integer $N>1$. By using non-standard coherent state, we analytically provide the maximal success probability of the quantum communication with $N$-PSK. Unlike the binary case, we show that even super-Poissonianity of non-standard coherent state can improve the maximal success probability of $N$-PSK quantum communication: The Helstrom bound can be improved by considering the sub-Poissonian NS-CS for $N=3$, meanwhile the super-Poissonian NS-CS can improve the Helstrom bound for $N=4$ and $N=8$.
For $N>2$, $N$-PSK signal enables us to transmit a $\log_{2}N$-bit message per a signal pulse, which is a better information exchange rate than binary-PSK. Moreover it is also known that $N$-PSK signal can provide an improved information exchange rate between the sender and receiver even though the receiver's measurement is slow \cite{i.a.burenkov}. However, the maximal success probability of discriminating a message encoded in $N$-PSK signal generally decreases as $N$ is getting large. Thus our results about the possible enhancement of the maximal success probability in $N$-PSK quantum communication by NS-CS is important and even necessary to design efficient quantum communication schemes.
The present article is organized as follows. In Section 2, we briefly review the problem of minimum error discrimination among $N$ symmetric pure states. In Section 3, we provide the analytical Helstrom bound of $N$-PSK signal composed of NS-CS. In Section 4, we investigate the Helstrom bound of $N$-PSK signal composed of optical spin coherent states (OS-CS), Perelomov coherent states (P-CS), Barut-Girardello coherent states (BG-CS) and modified Susskind-Glogower coherent states (mSG-CS), and discuss the relation between the sub-Poissonianity of the non-classical light and the performance of the $N$-PSK quantum communication. Finally, in Section 5, we propose the conclusion of the present article.
\section{Preliminaries: Minimum Error Discrimination among Symmetric Pure States}
In quantum communication, Alice (sender) prepares her message $x\in\{1,\cdots,N\}$ with a prior probability $q_x\in\{q_1,\cdots,q_N\}$, encodes the message in a quantum state $\rho_x\in\{\rho_1,\cdots,\rho_N\}$, and sends the quantum state to Bob (receiver). Bob performs a quantum measurement described as a POVM $\{M_1,\cdots,M_N\}$ to discriminate the encoded message. In the POVM, $M_x$ is a POVM element with respect to a result $x$.
For a given ensemble $\mathcal{E}=\{q_x,\rho_x\}_{x=1}^{N}$ of Alice and a POVM $\mathcal{M}=\{M_x\}_{x=1}^{N}$ of Bob, the success probability of the quantum communication between Alice and Bob is described by the success probability of the state discrimination,
\begin{equation}
P_s(\mathcal{E},\mathcal{M})=\sum_{x=1}^{N}q_x\mathrm{tr}\left\{\rho_xM_x\right\},\label{success_probability}
\end{equation}
One way to optimize the efficiency of quantum communication is to consider a POVM that maximizes the success probability in Eq. (\ref{success_probability}). In this case, the maximization of the success probability in Eq. (\ref{success_probability}) is equivalent to the minimization of the error probability
\begin{equation}
P_e(\mathcal{E},\mathcal{M})=1-P_s(\mathcal{E},\mathcal{M})=\sum_{x=1}^{N}\sum_{y\not=x}q_x\mathrm{tr}\left\{\rho_xM_y\right\}.\label{error_probability}
\end{equation}
\textit{Minimum error discrimination} is to minimize the error probability in Eq. (\ref{error_probability}) over all possible POVMs $\mathcal{M}$ of Bob.
For a given ensemble $\mathcal{E}$, it is known that the following inequality is a necessary and sufficient condition for POVM $\mathcal{M}$ minimizing the error probability \cite{c.w.helstrom,s.m.barnett2},
\begin{eqnarray}
\sum_{z=1}^{N}q_z\rho_zM_z-q_x\rho_x\ge 0, \ \ \forall x\in\{1,\cdots,N\}.\label{inequality_condition}
\end{eqnarray}
Moreover, it is known that the following equality is a useful necessary condition to characterize the structure of the POVM,
\begin{eqnarray}
M_x(q_x\rho_x-q_y\rho_y)M_y=0, \ \ \forall x,y\in\{1,\cdots,N\}.\label{equality_condition}
\end{eqnarray}
If every quantum state $\rho_x$ is pure, that is, $\rho_x=|\psi_x\rangle\langle\psi_x|$, the optimal POVM is given by a rank-1 projective measurement \cite{c.w.helstrom}. In other word, $M_x=|\pi_x\rangle\langle\pi_x|$ for every $x\in\{1,\cdots,N\}$.
Now, we focus on the minimum error discrimination among a specific class of pure states, called \textit{symmetric pure states}.
\begin{definition}
\cite{a.chefles} For a positive integer $N$, the distinct pure states $|\psi_1\rangle,\cdots,|\psi_N\rangle$ are called \textit{symmetric},
if there exists a unitary operator $V$ such that
\begin{equation}
|\psi_x\rangle=V^{x-1}|\psi_1\rangle
\label{symuni}
\end{equation}
for $x= 1,2,\cdots,N$ and
\begin{equation}
V^N=\mathbb{I},
\end{equation}
where $\mathbb{I}$ is an identity operator on a subspace spanned by $\{|\psi_1\rangle,\cdots,|\psi_N\rangle\}$.
\end{definition}
The Gram matrix composed of the symmetric pure states in Definition 1 is
\begin{equation}
G=\left(\langle\psi_x|\psi_y\rangle\right)_{x,y=1}^{N}.\label{gram}
\end{equation}
From a straightforward calculation, the eigenvalues of the Gram matrix in Eq. (\ref{gram}) are in forms of
\begin{equation}
\lambda_p=\sum_{k=1}^{N}\langle\psi_j|\psi_k\rangle e^{-\frac{2\pi i (p-1)(j-k)}{N}}, \ \ p=1,2,\cdots,N,\label{lambda}
\end{equation}
for any choice of $j\in\{1,2,\cdots,N\}$. We note that the set $\{\lambda_p\}_{p=1}^{N}$ is invariant under the choice of $j$ due to the symmetry of the pure state $\{|\psi_1\rangle,\cdots,|\psi_N\rangle\}$. The following proposition provides the maximal success probability of the minimum error discrimination among the symmetric pure states in Definition 1.
\begin{proposition}
\cite{c.w.helstrom} Let $\mathcal{E}_{sym}$ be an equiprobable ensemble of symmetric pure states $|\psi_1\rangle,\cdots,|\psi_N\rangle$. Then, the maximal success probability is given as
\begin{equation}
P_{hel}(\mathcal{E}_{sym})=\frac{1}{N^2}\left(\sum_{p=1}^{N}\sqrt{\lambda_p}\right)^2,\label{Helstrom_bound}
\end{equation}
where $\lambda_p$ are the eigenvalues of the Gram matrix composed of $\{|\psi_1\rangle,\cdots,|\psi_N\rangle\}$ in Eq. (\ref{gram}).
\end{proposition}
Eq. (\ref{Helstrom_bound}) is also called \textit{Helstrom bound}, and $1-P_{hel}(\mathcal{E}_{sym})$ is called \textit{minimum error probability}.
\section{Optimal Communication with \\ Phase Shift Keying (PSK) Signal}
In quantum optical communication, phase shift keying (PSK) signal is expressed as equiprobable symmetric pure states \cite{g.cariolaro}. In this section, we derive the maximal success probability of the quantum communication with PSK signal composed of generalized coherent states. First, we provide definition of the generalized coherent state which encapsulates standard coherent state (S-CS) and non-standard coherent state (NS-CS) as special cases.
\begin{definition}
\cite{e.m.f.curado} If a pure state takes the form
\begin{equation}
|\alpha,\vec{h}\rangle=\sum_{n=0}^{\infty}\alpha^nh_n(|\alpha|^2)|n\rangle, \ \ \alpha\in\mathbb{C},\label{generalized_coherent_state}
\end{equation}
where $\{|n\rangle|n\in\mathbb{Z}^+\cup\{0\}\}$ is Fock basis and $\vec{h}$ is a tuple of real-valued functions $h_n:[0,R^2]\rightarrow\mathbb{R}$ satisfying
\begin{eqnarray}
&&\sum_{n=0}^{\infty}u^n\left\{h_n(u)\right\}^2=1,\\
&&\sum_{n=0}^{\infty}nu^n\left\{h_n(u)\right\}^2 \\
&& \ \ \ \ \ \ \ \ \textrm{ is a strictly increasing function of }u,\nonumber\\
&&\int_0^{R^2}duw(u)u^n\left\{h_n(u)\right\}^2=1 \\
&& \ \ \ \ \ \ \ \ \textrm{ for a real-valued function }w:[0,R^2]\rightarrow\mathbb{R}^+,\nonumber
\end{eqnarray}
then the pure state is called \textit{generalized coherent state}. If every real-valued function $h_n$ in Eq. (\ref{generalized_coherent_state}) takes the form
\begin{equation}
h_n(u)=\frac{1}{\sqrt{n!}}e^{-\frac{1}{2}u}, \ \ \forall n\in\mathbb{Z}^+\cup\{0\},\label{special_case}
\end{equation}
then Eq. (\ref{generalized_coherent_state}) is called \textit{standard coherent state} (S-CS) \cite{r.j.glauber}. Otherwise, Eq. (\ref{generalized_coherent_state}) is called \textit{non-standard coherent state} (NS-CS).
\end{definition}
Several examples of NS-CS have been introduced such as optical spin coherent state (OS-CS) \cite{a.m.perelomov}, Perelomov coherent state (P-CS) \cite{a.m.perelomov}, Barut-Girardello coherent state (BG-CS) \cite{a.o.barut} and modified Susskind-Glogower coherent state (mSG-CS) \cite{j.-p.gazeau}.
\begin{example}
For a given non-negative integer $\widetilde{n}$, if $h_n$ takes the form
\begin{eqnarray}
h_n(u)=\sqrt{\frac{\widetilde{n}!}{n!(\widetilde{n}-n)!}}(1+u)^{-\frac{\widetilde{n}}{2}},
\end{eqnarray}
for $0\le n \le \widetilde{n}$ and $h_n(u)=0$ for $n>\widetilde{n}$, then the generalized coherent state in Eq. (\ref{generalized_coherent_state}) is called \textit{optical spin coherent state} (OS-CS).
\end{example}
\begin{example}
For all non-negative integer $n$ and a real number $\varsigma$ with $\varsigma\ge1/2$, if $h_n$ takes the form
\begin{eqnarray}
h_n(u)=\frac{1}{\mathcal{N}(u)}\sqrt{\frac{\Gamma(2\varsigma)}{n!\Gamma(2\varsigma+n)}},
\end{eqnarray}
then the generalized coherent state in Eq.(\ref{generalized_coherent_state}) is called \textit{Barut-Girardello coherent state} (BG-CS). Here, $\Gamma$ is the Gamma function of the first kind and $\mathcal{N}(u)$ is a normalization factor
\begin{equation}
\mathcal{N}(u)=\Gamma(2\varsigma)u^{1/2-u}I_{2\varsigma-1}(2\sqrt{u}),
\end{equation}
where $I_\nu$ is the modified Bessel function of the first kind.
\end{example}
\begin{example}
For all non-negative integer $n$, if $h_n$ takes the form
\begin{eqnarray}
h_n(u)=\sqrt{\frac{n+1}{\mathcal{\bar{N}}(u)}}\frac{1}{u^{\frac{n+1}{2}}}J_{n+1}(2\sqrt{u}),
\end{eqnarray}
then the generalized coherent state in Eq. (\ref{generalized_coherent_state}) is called \textit{modified Susskind-Glogower coherent state} (mSG-CS). Here, $J_n$ is the Bessel function of the first kind and $\mathcal{\bar{N}}(u)$ is a normalization factor
\begin{eqnarray}
\mathcal{\bar{N}}(u)&=&\frac{1}{u}\Big[2u\left\{J_0(2\sqrt{u})\right\}^2\\
&-&\sqrt{u}J_0(2\sqrt{u})J_1(2\sqrt{u})+2u\left\{J_1(2\sqrt{u})\right\}^2\Big].\nonumber
\end{eqnarray}
\end{example}
\begin{example}
For all non-negative integer $n$, $\varsigma$ and an integer or half-integer with $\varsigma\ge1/2$, if $h_n$ takes the form
\begin{eqnarray}
h_n(u)=\sqrt{\frac{(2\varsigma-1+n)!}{n!(2\varsigma-1)!}}(1-u)^{\varsigma},
\end{eqnarray}
then the generalized coherent state in Eq. (\ref{generalized_coherent_state}) is called \textit{Perelomov coherent state} (P-CS).
\end{example}
We mainly focus on which NS-CS provided in the examples can give the advantage to the $N$-ary PSK quantum communication. For this reason, we define the $N$-ary generalized PSK ($N$-GPSK) signal as follows.
\begin{definition}
If an equiprobable ensemble $\mathcal{E}_{gcs}$ consists of generalized coherent states,
\begin{equation}
\left\{|\alpha_x,\vec{h}\rangle|x\in\{1,2,\cdots,N\}\right\},\label{PSK}
\end{equation}
where $N\in\mathbb{Z}^+$ and $\alpha_x\in\mathbb{C}$ such that
\begin{equation}
\alpha_x=\alpha e^{\frac{2\pi i}{N}x},\label{alpha_x}
\end{equation}
with a non-negative integer $\alpha$, then the ensemble $\mathcal{E}_{gcs}$ is called $N$-ary generalized PSK ($N$-GPSK) signal.
\end{definition}
Moreover, $N$-GPSK signal is called $N$-ary standard PSK ($N$-SPSK) signal \cite{g.cariolaro} if every coherent state in Eq. (\ref{PSK}) is S-CS, and $N$-PSK signal is called $N$-ary non-standard PSK ($N$-NSPSK) signal if every coherent state in Eq. (\ref{PSK}) is NS-CS. The following theorem shows that the generalized coherent states in Definition 3 are symmetric.
\begin{theorem}
For given distinct generalized coherent states $|\alpha_1,\vec{h}\rangle,\cdots,|\alpha_N,\vec{h}\rangle$, there exists a unitary operator $U$ such that
\begin{eqnarray}
|\alpha_x,\vec{h}\rangle=U^{x-1}|\alpha_1,\vec{h}\rangle, \ \ \forall x\in\{1,2,\cdots,N\},\label{thm1}
\end{eqnarray}
for $x=1,2,\cdots,N$ and
\begin{equation}
U^N=\mathbb{I},\label{U_N}
\end{equation}
where $\mathbb{I}$ is an identity operator on a subspace spanned by $\{|\alpha_1,\vec{h}\rangle,\cdots,|\alpha_N,\vec{h}\rangle\}$.
\end{theorem}
\begin{proof}
Consider a unitary operator
\begin{eqnarray}
U=e^{\frac{2\pi i}{N}a^\dagger a},\label{U}
\end{eqnarray}
where $a$ and $a^\dagger$ are the annihilation and creation operators satisfying
\begin{eqnarray}
&&a|n\rangle=\sqrt{n}|n-1\rangle, \ \ \forall n\in\mathbb{Z}^+,\\
&&a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle, \ \ \forall n\in\mathbb{Z}^+\cup\{0\},
\end{eqnarray}
respectively. It is straightforward to show that the unitary operator $U$ in Eq. (\ref{U}) satisfies Eq. (\ref{U_N}).
We also note that
\begin{eqnarray}
U|n\rangle=e^{\frac{2\pi i}{N}n}|n\rangle,
\end{eqnarray}
for any non-negative integer $n$, therefore we have that
\begin{eqnarray}
U|\alpha_x,\vec{h}\rangle&=&\sum_{n=0}^{\infty}\alpha_x^nh_n(|\alpha_x|^2)e^{\frac{2\pi i}{N}a^\dagger a}|n\rangle\nonumber\\
&=&\sum_{n=0}^{\infty}\alpha_x^nh_n(|\alpha_x|^2)e^{\frac{2\pi i}{N}n}|n\rangle\nonumber\\
&=&\sum_{n=0}^{\infty}(\alpha_xe^{\frac{2\pi i}{N}})^nh_n(|\alpha_x|^2)|n\rangle,\label{proof}
\end{eqnarray}
for every $x\in\{1,2,\cdots,N-1\}$. Moreover, Eq. (\ref{alpha_x}) leads us to
\begin{eqnarray}
\alpha_xe^{\frac{2\pi i}{N}}=\alpha_{x+1},\label{prop_alpha1}
\end{eqnarray}
for $x\in\{1,2,\cdots,N-1\}$ and
\begin{eqnarray}
|\alpha_x|=\alpha,\label{prop_alpha2}
\end{eqnarray}
for $x\in\{1,2,\cdots,N\}$. From Eqs. (\ref{proof}), (\ref{prop_alpha1}) and (\ref{prop_alpha2}), we have
\begin{eqnarray}
U|\alpha_x,\vec{h}\rangle=\sum_{n=0}^{\infty}(\alpha_{x+1})^nh_n(|\alpha_{x+1}|^2)|n\rangle=|\alpha_{x+1},\vec{h}\rangle.\nonumber\\
\label{finally_proved}
\end{eqnarray}
Eq. (\ref{thm1}) can be shown by an inductive use of Eq. (\ref{finally_proved}), which completes the proof.
\end{proof}
Theorem 1 means that the Helstrom bound of quantum communication with $N$-GPSK signal is given by Eq. (\ref{Helstrom_bound}) in Proposition 1, which is encapsulated in the following theorem.
\begin{theorem}
The Helstrom bound of $N$-GPSK signal is given by
\begin{eqnarray}
P_{hel}(\mathcal{E}_{gcs})=\frac{1}{N^2}\left(\sum_{p=1}^{N}\sqrt{\lambda_p^{(G)}}\right)^2,\label{helstrom_bound_N}
\end{eqnarray}
where $\lambda_p^{(G)}$ takes the form of
\begin{eqnarray}
\lambda_p^{(G)}=\sum_{k=0}^{M-1}\left[\sum_{n=0}^{\infty}\alpha^{2n}\cos\left\{\frac{2\pi}{N}k(n+p-1)\right\}\left\{h_n(\alpha^2)\right\}^2\right].\nonumber\\
\label{lambda_N}
\end{eqnarray}
for every $p\in\{1,2,\cdots,N\}$.
\end{theorem}
\begin{proof}
For every $j,k\in\{1,2,\cdots,N\}$, the inner product $\langle\alpha_j,\vec{h}|\alpha_k,\vec{h}\rangle$ is
\begin{eqnarray}
\langle\alpha_j,\vec{h}|\alpha_k,\vec{h}\rangle=\sum_{n=0}^{\infty}\left\{\alpha^2e^{i\frac{2\pi}{N}(k-j)}\right\}^n\left\{h_n(\alpha^2)\right\}^2.\label{inner_product_NS_CS}
\end{eqnarray}
From Eq. (\ref{inner_product_NS_CS}) together with Eq. (\ref{lambda}), $\lambda_{p}^{(G)}$ is also obtained by
\begin{eqnarray}
&&\lambda_p^{(G)}\nonumber\\
&&=\sum_{k=1}^{N}\left[\sum_{n=0}^{\infty}\left\{\alpha^2e^{i\frac{2\pi}{N}(k-j)}\right\}^n\left\{h_n(\alpha^2)\right\}^2\right]e^{-\frac{2\pi i (p-1)(j-k)}{N}}\nonumber\\
&&=\sum_{k=1}^{N}\left[\sum_{n=0}^{\infty}\alpha^{2n}e^{i\frac{2\pi}{N}(k-j)(n+p-1)}\left\{h_n(\alpha^2)\right\}^2\right].
\end{eqnarray}
As mentioned before, the set $\{\lambda_p^{(G)}\}_{p=1}^{N}$ is invariant under the choice of $j\in\{1,2,\cdots,N\}$. By choosing $j=1$ and substituting $k$ to $k-1$, $\lambda_p^{(G)}$ can be rewritten by
\begin{eqnarray}
\lambda_p^{(G)}=\sum_{k=0}^{N-1}\left[\sum_{n=0}^{\infty}\alpha^{2n}e^{i\frac{2\pi}{N}k(n+p-1)}\left\{h_n(\alpha^2)\right\}^2\right].\label{lambda_exp}
\end{eqnarray}
Since the Gram matrix is Hermitian, $\lambda_p^{(G)}$ is a real number. Thus, by using the relation
\begin{equation}
\lambda_p^{(G)}=\frac{\lambda_p^{(G)}+\lambda_p^{(G)*}}{2},
\end{equation}
together with Eq. (\ref{lambda_exp}), we have Eq. (\ref{lambda_N}). Due to Theorem 1, every generalized coherent state in $N$-GPSK signal is symmetric. Thus, Proposition 1 and Eq. (\ref{lambda_N}) lead us to the Helstrom bound in Eq. (\ref{helstrom_bound_N}).
\end{proof}
\section{Sub-Poissonianity of NS-CS and the Helstrom bound}
For $N=3,4$ and 8, we provide illustrative results of the Helstrom bound of $N$-NSPSK signal of Eq. (\ref{helstrom_bound_N}) in case of OS-CS, P-CS, BG-CS and mSG-CS. We also compare these results with the case of $N$-SPSK signal.
\subsection{Optical Spin Coherent States (OS-CS)}
\begin{figure*}
\centering
\includegraphics[scale=0.29]{PSK_OS_CS.png}
\caption{The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of OS-CS , where (a), (b) and (c) show the case of $N=$3, 4 and 8, respectively. In these figures, purple, red, blue and green lines show the case of $N$-NSPSK signal with $\widetilde{n}=3$, $\widetilde{n}=5$, $\widetilde{n}=7$ and $\widetilde{n}=11$, respectively. Black lines in the figures show the case of $N$-SPSK signal. }
\end{figure*}
\begin{figure*}
\centering
\includegraphics[scale=0.29]{PSK_BG_CS.png}
\caption{The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of BG-CS, where (a), (b) and (c) shows the case of $N=$3, 4 and 8, respectively. In these figures, blue and red lines show the case of $N$-NSPSK signal with $\varsigma=0.5$ and $\varsigma=1.5$, respectively. Black lines show the case of $N$-SPSK. }
\end{figure*}
The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of OS-CS are illustrated in Fig. 1, where Fig. 1(a), (b) and (c) show the case of $N=$3, 4 and 8, respectively. In these figures, purple, red, blue and green lines show the case of $N$-NSPSK signal with $\widetilde{n}=3$, $\widetilde{n}=5$, $\widetilde{n}=7$ and $\widetilde{n}=11$, respectively. Black lines in the figures show the case of $N$-SPSK signal.
In Fig. 1(a), the minimum error probabilities of 3-NSPSK signal composed of OS-CS is smaller than that of 3-SPSK signal when mean photon number is large ($\langle n\rangle>0.45$, $\langle n\rangle>0.42$, $\langle n\rangle>0.38$ and $\langle n\rangle>0.37$ in case of $\widetilde{n}=3$, $\widetilde{n}=5$, $\widetilde{n}=7$ and $\widetilde{n}=11$, respectively). In other words, \textit{3-PSK quantum communication can be enhanced by a non-standard coherent state using OS-CS.} However, in Fig. 1(b), each minimum error probability of 4-NSPSK signal is larger than that of 4-SPSK signal for arbitrary mean photon number. This aspect repeatedly happens in Fig. 1(c) where 8-NSPSK signal is considered. These results imply that \textit{4-PSK and 8-PSK quantum communication cannot be enhanced by OS-CS}.
\subsection{Barut-Girardello Coherent States (BG-CS)}
The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of BG-CS are illustrated in Fig. 2, where Fig. 2(a), (b) and (c) shows the case of $N=$3, 4 and 8, respectively. In these figures, blue and red lines show the case of $N$-NSPSK signal with $\varsigma=0.5$ and $\varsigma=1.5$, respectively. Black lines show the case of $N$-SPSK.
In Fig. 2(a), each minimum error probabilty of 3-NSPSK signal with $\varsigma=1.5$ is smaller than that of 3-SPSK signal when mean photon number is larger than 0.48. Meanwhile, each minimum error probabilty of 3-NSPSK signal with $\varsigma=0.5$ is larger than that of 3-SPSK signal for arbitrary mean photon number. \textit{Thus, enhancing 3-PSK quantum communication by non-standard coherent state using BG-CS depends on the parameter $\varsigma$.} However, in Fig. 2(b), each minimum error probability of $4$-NSPSK signal is larger than that of $4$-SPSK signal for arbitrary mean photon number. This aspect repeatedly happens in Fig. 2(c) where 8-NSPSK signal is considered. These results imply that \textit{4-PSK and 8-PSK quantum communication cannot be enhanced by BG-CS}.
\subsection{Modified Susskind-Glogower Coherent States (mSG-CS)}
\begin{figure*}
\centering
\includegraphics[scale=0.29]{PSK_mSG_CS.png}
\caption{The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of mSG-CS, where (a), (b) and (c) shows the case of $N=$3, 4 and 8, respectively. In these figures, red lines show the case of $N$-NSPSK signal and black lines show the case of $N$-SPSK. }
\end{figure*}
\begin{figure*}
\centering
\includegraphics[scale=0.29]{PSK_P_CS.png}
\caption{The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of P-CS, where (a), (b) and (c) shows the case of $N=$3, 4 and 8, respectively. In these figures, blue and red lines show the case of P-CS with $\varsigma=0.5$ and $\varsigma=1.5$, respectively. Black lines show the case of S-CS. }
\end{figure*}
The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of mSG-CS are illustrated in Fig. 3, where Fig. 3(a), (b) and (c) shows the case of $N=$3, 4 and 8, respectively. In these figures, red lines show the case of $N$-NSPSK signal and black lines show the case of $N$-SPSK.
In Fig. 3, each minimum error probability of $N$-NSPSK signal is larger than that of $N$-SPSK signal for any $N=$3, 4 and 8 and any mean photon number. This result implies that \textit{3-PSK, 4-PSK, and 8-PSK quantum communication cannot be enhanced by mSG-CS.} We also compare this result with the previous work about on-off keying signal \cite{e.m.f.curado}; it is known that the minimum error probability of on-off keying signal composed of mSG-CS has a singular point where the logarithm of the minimum error probability diverges to $-\infty$. This implies that the minimum error probability can achieve to zero. Unlike the result in the previous work \cite{e.m.f.curado}, the minimum error probabilities of 3, 4 and 8-NSPSK signal in Fig. 3 do not have such singular points.
\subsection{Perelomov Coherent States (P-CS)}
The minimum error probabilities of $N$-SPSK signal and $N$-NSPSK signal composed of P-CS are illustrated in Fig. 4, where Fig. 4(a), (b) and (c) shows the case of $N=$3, 4 and 8, respectively. In these figures, blue and red lines show the case of P-CS with $\varsigma=0.5$ and $\varsigma=1.5$, respectively. Black lines show the case of S-CS.
In Fig. 4(a), each minimum error probabilty of 3-NSPSK signal composed of P-CS with $\varsigma=$0.5 and 1.5 is larger than that of 3-SPSK signal for arbitrary mean photon number. In other words, \textit{$3$-PSK quantum communication cannot be enhanced by non-standard coherent state using P-CS}. However, in Fig. 4(b), each minimum error probabilty of 4-NSPSK signal composed of P-CS is smaller than that of 4-SPSK signal when mean photon number is small ($\langle n\rangle<0.585$ and $\langle n\rangle<0.786$ in case of $\varsigma=0.5$ and $\varsigma=1.5$, respectively). This implies that \textit{4-PSK quantum communication can be enhanced by P-CS}. In Fig. 4(c), each minimum error probabilty of 8-NSPSK signal composed of P-CS is smaller than that of 8-SPSK signal for arbitrary mean photon number. This result is rather surprising since even super-Poissonianity in P-CS can enhance the 4-PSK and 8-PSK quantum communication unlike the binary case \cite{m.namkung}. We discuss the details in the next section.
\subsection{Mandel Parameter and $N$-NSPSK Quantum Communication}
It is known that sub-Poissonianity of non-classical light is one of the important statistical properties for improving Helstrom bound of binary quantum optical communication \cite{e.m.f.curado}. For this reason, we consider the following \textit{Mandel parameter},
\begin{equation}
Q_M^{(NS)}=\frac{(\Delta n)^2}{\langle n\rangle}-1,
\end{equation}
where $\langle n\rangle$ is mean photon number and $\Delta n$ is standard deviation of the number of photons. It is known that if $Q_M^{(NS)}>0(<0)$, then the generalized coherent state is \textit{super-Poissonian}(\textit{sub-Poissonian}) \cite{l.mandel,r.short}. If $Q_M^{(NS)}=0$ (for example, S-CS), then the generalized coherent state is \textit{Poissonian}. Here, we consider the relation between the performance of the $N$-PSK quantum communication and the Mandel parameter.
\begin{enumerate}
\item In case of OS-CS, the Mandel parameter is analytically driven as \cite{e.m.f.curado}
\begin{eqnarray}
Q_M^{(OS)}=-\frac{\langle n\rangle}{\widetilde{n}},
\end{eqnarray}
which means that OS-CS is always sub-Poissonian. According to Fig. 1, we note that sub-Poissonianity of OS-CS does not always guarantee the enhancement of the $N$-PSK quantum communication.
\item In case of BG-CS, the Mandel parameter is analytically driven in terms of the Modified Bessel function of the first kind as \cite{e.m.f.curado}
\begin{eqnarray}
Q_M^{(BG)}=\alpha\left[\frac{I_{2\varsigma+1}(2\alpha)}{I_{2\varsigma}(2\alpha)}-\frac{I_{2\varsigma}(2\alpha)}{I_{2\varsigma-1}(2\alpha)}\right].
\end{eqnarray}
Since the inequality $\left\{I_{\nu+1}(x)\right\}^2\ge I_\nu(x)I_{\nu+2}(x)$ holds for every $x\ge 0$, $Q_M^{(BG)}$ is negative semidefinite. Therefore, BG-CS is always sub-Poissonian or Poissonian. Moreover, $Q_M^{(BG)}$ is known to be strictly negative for non-zero mean photon number \cite{e.m.f.curado}. Nevertheless, Fig. 2 shows that sub-Poissonianity of BG-CS does not always guarantee the enhancement of the $N$-PSK quantum communication.
\item Since the analytic form of the Mandel parameter of the mSG-CS Mandel parameter is too complex \cite{e.m.f.curado}, we do not introduce the analytic form here. According to the result of \cite{e.m.f.curado}, the Mandel parameter of mSG-CS is negative when the mean photon number is not too large. Nevertheless, Fig. 3 shows that the sub-Poissonianity of the mSG-CS cannot provide any advantage on the $N$-PSK quantum communication.
\item In case of P-CS, the Mandel parameter is analytically driven as \cite{e.m.f.curado}
\begin{eqnarray}
Q_M^{(P)}=\frac{\langle n\rangle}{2\varsigma},
\end{eqnarray}
which means that P-CS is super-Poissonian. However, Fig. 4 shows that P-CS can enhance the $N$-PSK quantum communication for $N=$3, 4 or 8. It is surprising since the super-Poissonianity of NS-CS can even enhance $N$-PSK quantum communication unlike the binary case.
\end{enumerate}
\section{Conclusion}
In the present article, we have considered the quantum communication with $N$-ary phase shift keying ($N$-PSK) signal for arbitrary an arbitrary positive integer $N>1$. By using NS-CS, we have analytically provided the Helstrom bound of the quantum communication with $N$-PSK. Unlike the binary case \cite{e.m.f.curado,m.namkung}, we have shown that even super-Poissonianity of NS-CS can improve the Helstrom bound of $N$-PSK quantum communication: The Helstrom bound can be improved by considering the sub-Poissonian NS-CS for $N=3$, meanwhile the super-Poissonian NS-CS can improve the Helstrom bound for $N=4$ and $N=8$.
Using $N$-PSK signal with $N>2$, we can achieve a better transmission rate per a signal pulse than that of binary-BPSK even if the receiver's measurement is slow \cite{i.a.burenkov}. On the other hands, the maximal success probability of discriminating a message encoded in $N$-PSK signal generally decreases as $N$ is getting large. Thus our results about the possible enhancement of the maximal success probability in $N$-PSK quantum communication by NS-CS is important and even necessary to design efficient quantum communication schemes.
In the present article, we have only considered PSK signal with equal prior probabilities, which is composed of symmetric pure states. However, it is interesting and even important to consider a non-equiprobable or asymmetric ensemble of NS-CS for several reasons: First, it is practically difficult to implement the PSK signal having perfect symmetry or equal prior probabilities. Moreover, in discriminating three non-equiprobable and asymmetric pure states, there is possibility that sub-Poissonianity of non-classical light can enhance the Helstrom bound. We note that it is also interesting to consider unambiguous discrimination \cite{i.d.ivanovic,d.dieks,a.peres,g.jaeger,s.pang,j.a.bergou2} of NS-CS since this strategy can provide us with better confidence than the minimum error discrimination.
\section{Acknowledgement}
This work was supported by Quantum Computing Technology Development Program (NRF2020M3E4A1080088)
through the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of
Science and ICT).
|
1,116,691,499,333 | arxiv | \section{Introduction}
\label{sec:introduction}
Cognitive scientists use computational models to understand how humans achieve cognition and how the brain gives rise to the mind.
Cognitive models simulate mental processes by describing how inputs (e.g., visual or auditory stimuli to the brain) are processed by a set of interconnected
subsystems to generate
behavior.
Insights from cognitive modeling have greatly influenced not only the brain, psychological, and cognitive sciences, but also the field of artificial intelligence (AI).
In fact, this influence has been instrumental in shaping AI right from the onset of artificial neural networks to the recent advances in deep learning for gameplay and scientific computing from DeepMind~\cite{kumaran:cls, jumper:alphafold}.
Researchers use cognitive models to obtain distributions of outcomes and their evolution over time for various cognitive tasks.
By construction, these models are designed to be run hundreds of thousands of times because of inherent stochasticity, in order to build histograms of outcomes across multiple parameter settings, and/or to assess the dynamics of cognitive processes across a series of time steps.
The vast majority of cognitive models are developed in Python-based frameworks~\cite{psyneulink,pytorch,hines:neuron}. This is because Python eases programming significantly, lowers the barrier to entry to computational methods, is amenable to rapid prototyping, and offers access to optimized scientific computing libraries \cite{scipy, numpy}.
But, the use of Python also introduces performance inefficiencies. As more sophisticated cognitive models are built to capture advanced brain processes, these performance inefficiencies worsen to the point where cognitive models take several days to weeks to run. As a consequence, scientific progress is impeded by inefficient modeling tools
One might consider alleviating the slow runtime of cognitive models via publicly-available compilation tools like PyPy~\cite{pypy} and Pyston~\cite{pyston}.
In practice, however, we find that these approaches leave performance on the table. The central problem is that Pypy and Pyston cannot optimize complex dependencies in cognitive models because of the runtime checks needed to deal with Python's dynamic data structures and dynamic typing. These features also obscure the natural parallelism available in cognitive models, and impede the ability to offload portions of the models onto hardware accelerators for which they are otherwise suitable. Adding to these challenges is the fact that large-scale cognitive models increasingly require integration of separate subsystems developed across a range of different environments (e.g., PyTorch~\cite{pytorch}, Emergent~\cite{aisa:emergent}, NEURON~\cite{hines:neuron} or PsyneuLink~\cite{psyneulink}); it is difficult for compilers to optimize across computations expressed in this multitude of environments.
\begin{comment}
Additionally,
cognitive models can integrate components
developed in different environments such as PyTorch~\cite{pytorch}, Emergent~\cite{aisa:emergent}, NEURON~\cite{hines:neuron} or PsyneuLink~\cite{psyneulink}. It is difficult for compilers to optimize across computations described in these different environments. Finally, even though cognitive models contain many opportunities for parallel execution and offloading computations to accelerators, tools do not automatically extract such parallelism because of Python features.
\end{comment}
Yet another possibility might be to build a domain-specific language (DSL) targeted at cognitive scientists. While this is likely to enable maximal performance, it also requires large-scale community buy-in and porting of many models already built across many research institutions using Python. Even if these practicality concerns could be surmounted, the extreme heterogeneity of the cognitive modeling ecosystem presents major roadblocks to designing a canonical set of language constructs and software tools needed for a domain-specific language. This scale of heterogeneity can be appreciated by realizing that cognitive models can integrate components
with varying levels of biological fidelity, developed by different frameworks and research groups; e.g., a single model can include neurally accurate descriptions of some brain structures, an artificial neural network from machine learning to determine the attention allocated to inputs and a behavioral model of control to modulate the pathways.
In response, we build \sysname{}, a dynamic compilation tool for cognitive models that exploits domain knowledge to generate efficient code. \sysname{} uses domain specific knowledge from cognitive science to aggressively eliminate Python's dynamic code, and generates LLVM IR for all the components in a model, including those developed in ancillary frameworks (e.g., Pytorch). This approach is predicated on the observation that, like practitioners in other scientific computing communities~\cite{weld, tuplex}, cognitive scientists use Python for its flexibility and access to optimized scientific computing libraries~\cite{scipy, numpy}, but do not need many of the dynamic language features that slow execution down. Once these dynamic features are removed, standard optimizations on LLVM IR yield notable speedups, and permit leveraging LLVM's broad existing family of code generation backends (with no change) for multiple CPU architectures and accelerators. We find that \sysname{} accelerates cognitive model execution by an average of 26$\times$ and maximum of 778$\times$ across our evaluated cognitive models compared to Pyston and PyPy.
A model that couldn't complete even in 24 hours could finish in less than 5 seconds when \sysname{} was used. \sysname{} can also extract parallelism from the models and target multi-core CPUs and GPUs, resulting in additional 4.8$\times$ and 6.4$\times$ speedups, respectively.
In the process of accelerating model execution, we were also intrigued by another (unexpected) benefit of stripping away Python's dynamism from LLVM IR. All cognitive models are expressed as computational graphs, and we found that eliding Python's dynamism enabled \sysname{} to produce control and dataflow graphs (CDFGs) of LLVM IR with similar shape and data flow as the original model. We leverage this observation to augment \sysname{} with compiler analyses for deducing high-level semantic information from the CDFG. This in turn permits \sysname{} to automate several types of model-level analysis that have traditionally been undertaken manually and tediously by scientists
It also enables \sysname{} to discover user-guided optimizations specific to cognitive models.
We demonstrate two important applications of these user-guided analyses and optimizations. First, \sysname{} identifies cases where entire models can be verified to be equivalent, and also recognizes that certain complex nodes are equivalent with, and hence, can be replaced by simpler modules that have an analytical solution.
Second, \sysname{} calculates the impact of a cognitive model's parameters on the outputs and finds their optimal values entirely with compiler analysis techniques that extend LLVM's range value propagation and scalar evolution passes. As this parameter estimation process was performed manually over hundreds to thousands of runs, \sysname{}'s automation of this step saves days to weeks of modeling effort. Moreover, because of the general utility of our enhancements to LLVM's passes (i.e., extending support for integers to floating point), we have submitted a patch to the LLVM community for mainline integration.
\sysname{} is undergirded by three main design principles. First, we wish to avoid requiring cognitive scientists to change the source-code of their models or frameworks. Second, we delegate performance extraction to the compiler, allowing scientists to focus on creating models in the manner most intuitive to them. Third, we also minimize software engineering effort, development, and maintenance cost to compile the models. This last goal is central to our decision to reuse LLVM IR and its associated infrastructure to build \sysname{}, and avoid the need for a new DSL or IR. More specifically, our contributions are:
\begin{enumerate}
\item \sysname{}, a compilation tool that exploits domain-specific knowledge to provide near-native execution speeds for cognitive models along with support to offload computations on accelerators. \sysname{} does not require changes to source code and reuses existing LLVM infrastructure.
\item Identifying that user-guided analyses and optimization can be performed by compiler analysis, and incorporating them into \sysname{}.
\item An evaluation of \sysname{}-accelerated models on single and multicore CPUs and GPU.
\end{enumerate}
\sysname{} is publicly available and is being used in several leading research labs via its plug-in with PsyNeuLink~\cite{psyneulink}, a newly-developed framework that can import and run sub-models developed in various environments.
\sysname{} enables the design of larger and more complex cognitive models than previously possible. This is an important and necessary step towards the larger goal of understanding human cognition and replicating its processes in artificial intelligence.
\section{Background and Motivation}
\label{sec:background}
\subsection{Cognitive Models Structure and Computation}
\label{sub_cogmodel}
Cognitive models are expressed as computational graphs and represent how neural or psychological components process inputs (typically sensory stimuli, but also potentially non-sensory stimuli like memory) to produce behavior.
These models are used for several purposes like fitting experimental data, simulating a cognitive process, producing an idealized outcome or for what-if analysis to understand the impact of tunable structures and parameters.
Cognitive models can be represented as a graph where the nodes are sub-processes or computational functions involved in the overall task, and the edges represent projections of signals between nodes. Nodes perform their computation when their activation conditions are met (e.g., the appearance of an input, the passing of a specified time period).
We illustrate a simple cognitive model using the well-known predator-prey task \cite{willke2019comparison}.
In this task, an intelligent agent, either a human or non-human primate, is given a joystick and shown a screen with three entities -- a player whose position is controlled using the joystick, a prey that the player must capture, and a predator that the player must avoid.
Figure~\ref{fig:pp_eg} shows a cognitive model to study the role of attention in the agent's performance on this task.
The agent's attention is limited and there is a cost of paying attention to an entity.
Attention paid to an entity determines the accuracy with which the agent can see that entity's location. Moreover, the agent does not have to distribute its attention fully.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{figures/Pred_Prey_model.pdf}
\caption{A cognitive model of an agent performing the predator-prey task.}
\label{fig:pp_eg}
\end{figure}
The \textit{Action} node calculates the player's movement based on the observed positions of the three on-screen entities.
The role of attention is modeled through the \textit{Control}, \textit{Objective} and \textit{Obs} nodes. \textit{Control} takes the exact 2-dimensional (2D) coordinates of all the entities (\textit{Loc} values) and allocates attention to each of them. This allocation determines the variances of three 2-D gaussian distributions whose means are the actual locations of the respective entities.
These distributions are then sent to the \textit{Obs} nodes that sample from them to generate the observed locations.
To determine the best attention allocation, the \textit{Control} node searches over all possible attention allocations, evaluating the cost of each allocation and the quality of the associated move.
The cost of each allocation is calculated by \textit{Control}, and the quality of the move is computed by the \textit{Objective} node.
The \textit{Objective} node uses the direction given by \textit{Action} for an allocation and the true location of the prey to compute the goodness of the move. The \textit{Control} node then selects the parameters that have the lowest cost. Overall, the entire process from reading inputs and searching over allocations is repeated for each time step until the prey or the player is captured.
Cognitive scientists are interested in both the final outcomes and the dynamics over time.
Figure \ref{fig:pp_eg} shows the basic predator-prey model form, but advanced variants of the model can also include cognitive processes for memory in order to recall previous locations, neural networks trained on experimental data to produce a move, or, visual processors to extract locations from screen frames. Components pertaining to these augmentations can come from different frameworks like PyTorch, Emergent or NEURON~\cite{hines:neuron,aisa:emergent,pytorch}.
Cognitive models may also include state that is modified across each set of inputs (e.g., the cost of the previous allocation), capping the available parallelism for execution. Model nodes also usually store metadata to track information like the number of times a particular node has executed, as this information is useful and of interest to scientists.
\subsection{Cognitive Model Execution}
\label{sub_run}
While cognitive models have historically been developed in a variety of environments, recent efforts in the community focus on converging on a single "lingua franca" environment that can support models built in others \cite{mdf-grant}. PsyNeuLink is prominent among emerging standardized environments, and is hence the focus of our work. Listing~\ref{listing:sched} shows how cognitive models execute in PsyNeuLink.
First, in a sanitization check to ensure that nodes are properly connected, the framework runs through all nodes, initializing all parameters and inputs with default values and propagating inter-node signals. The shapes of each node's inputs and outputs in the sanitization run must match those that will be used in the actual run.
Second, scheduler logic within PsyNeuLink reads the set of all inputs with which the model has to be run. The variable \texttt{inputs} in Listing~\ref{listing:sched} is a list, where each element consists of a set of values that are input to the model (e.g., the locations at one time step for the predator-prey model). A single model may have multiple inputs.
Third, the scheduler runs the model for a number of trials (TOTAL\_TRIALS in the listing), where each trial runs the model with one input. A trial consists of multiple iterations of model execution. In each iteration, the scheduler identifies nodes that are ready to run, and then executes them. The scheduler identifies the nodes that are ready to run by relying on the activation conditions that are explicitly specified per node. Examples of such conditions include waiting until other nodes are run a certain number of times, until the outputs of particular nodes stabilize, or after a certain amount of time has elapsed.
\begin{listing}[ht]
\centering
\begin{minted}[frame=lines,framesep=2mm,
baselinestretch=1.2,
fontsize=\footnotesize,
linenos]{python}
#Sanity check the model
sanitize(model)
#Read all inputs
inputs = read_inputs()
#Begin a model's run
num_trial = 0
while num_trial < TOTAL_TRIALS:
#Each trial reads an input
input = inputs[num_trial
#Run scheduling iterations
while not end_of_trial:
find_ready_nodes(model)
run_ready_nodes(model, input)
num_trial += 1
\end{minted}
\caption{Outline of the scheduling loops in a framework like PsyNeuLink that runs cognitive models.}
\label{listing:sched}
\end{listing}
In sum, apart from running nodes from different frameworks, execution switches between nodes and the scheduling logic. These back and forth transitions between computation and scheduling impacts performance, motivating the need to accelerate complex and composited models.
\subsection{Shortcomings of Dynamic Compilation Tools}
Dynamic compilation is a popular approach to accelerate applications written in high-level languages like Python. Indeed, Pyston results in average execution time improvements of 43\% for a group of cognitive models we test. However, they still leave significant opportunities for optimization. First, they cannot easily identify and leverage opportunities to reduce runtime overheads by tracking model control flow. For example, the predator-prey model in Section~\ref{sub_cogmodel} is run many times for a single input, but the path of execution is the same for all these runs. This is typical of cognitive models, but takes significant time and space resources for PyPy and Pyston to track.
Second, existing dynamic compilation tools do not fully eliminate dynamic features of Python unnecessary for the model. As an example, the signals between the nodes of a cognitive model not only have a fixed type, but also have a fixed shape across runs. Therefore, dynamic Python structures such as lists and dictionaries that are used to hold these values can be safely compiled to static data structures. However, existing tools conservatively conform to Python semantics, retaining these unnecessary computations.
Third, Pyston and PyPy cannot optimize across computations from different frameworks, and across the scheduling invocations between executions of the model nodes. When a model uses computations from multiple environments like PyTorch and PsyNeuLink, even if the separate components are compiled, optimization does not cross these frameworks. Additionally, execution frequently switches between the computations in the nodes and the scheduling logic in the modeling framework that identifies which nodes are ready to run. This also limits the scope of optimizations and results in the execution switching between compiled and interpreted modes, compromising performance.
Finally, available dynamic compilation tools cannot automatically extract parallelism from the models or offload computations to accelerators like GPUs. This is a wasted opportunity because cognitive models are run many times, and there are several dimensions along which computations can be parallelized. For example, in the predator-prey model, the evaluations for each combination of attention allocations could have been run in parallel. Furthermore, when multiple samples are drawn from the distributions of observed location, each sample and subsequent action could also be computed in parallel. While one might consider leveraging existing multithreading and GPU programming libraries for Python, they all require scientists to explicitly identify such parallel computations and mark functions to be offloaded to a GPU. A more desirable solution is to automate these steps so that cognitive scientists can solely focus on their designs rather than grapple with parallel programming constructs.
The confluence of these shortcomings leads to cascading slowdowns in the execution. For example, unoptimized data structures not only have longer access times, but also prevent subsequent optimization passes by hindering the propagation of values and references. Moreover, multithreading with Python does not result in parallel execution because the threads are serialized by the Global Interpreter Lock~\cite{beazley2010understanding}, unless the threads run compiled code, in which case they do not have to take the lock. Thus, to maximally benefit from parallelism, it is important to compile the Python threads.
\section{\sysname{}: Domain-Specific Compilation for Cognitive Models}
\label{design}
Cognitive models are graphs of computations expressed in various environments, with complex scheduling rules for execution. The models are constructed in Python but can be composed from heterogeneous sub-models developed in multiple frameworks.
Unfortunately, dynamic compilation with existing tools is not effective.
\sysname{} uses domain-specific knowledge of cognitive neuroscience to aggressively optimize the models to yield near-native execution speeds.
We observe that while Python's dynamic features make it easier to construct the models, they are not necessary to run them.
Therefore, \sysname{} aggressively eliminates dynamic features in the models and converts dynamic data structures to statically defined ones, yielding substantial acceleration.
Furthermore, \sysname{} automatically extracts parallelism and computations that are offloaded to GPUs.
\sysname{} plugs into PsyNeuLink~\cite{psyneulink}, which is a newly-developed framework that can import and run sub-models developed in various environments.
In this section we describe the steps in \sysname{}'s code transformations and parallelism extraction to accelerate cognitive models.
\subsection{Type and Shape Extraction}
\label{sub_types}
As described in Section~\ref{sub_cogmodel}, repeated computations in the models have the same types and shapes.
Thus, the first step in compiling cognitive models is to deduce the types and shapes of the values used in the computations.
Fortunately, this information is readily available from the sanitization pass run by the framework (Section~\ref{sub_run}).
By construction, the value types and shapes of the inputs and outputs of each node in this run must be the same as what will be used in the actual runs.
\sysname{} uses this information to infer the types and shapes of all computations in the model.
\subsection{Identifying Code Paths for Compilation}
\label{sub_hotpath}
Typically, dynamic compilation requires expensive analysis and tracking to determine which code paths need to compiled and when. In our case, the main computations of each node are in an \texttt{execute} method in the nodes, and we know that these computations must be compiled because they are repeatedly invoked and are time-consuming. Thus, we do not need to run expensive dynamic hot path analysis.
We leave code intended for initialization and visualization because they are not repeatedly invoked.
\subsection{Dynamic to Static Data Structure Conversion}
\label{sub_dynstruct}
Cognitive models use dynamic Python data structures like dictionaries and lists for node inputs, parameters, and outputs. We observe that their shapes (and keys in the case of dictionaries) remain invariant during execution. We therefore convert these entities into statically-defined structures. More specifically, we make the following changes:
First, we create two structures that hold the values of the outputs of all nodes in the current and previous iterations (see the inner loop of Listing~\ref{listing:sched}). Node outputs are written to these top-level structures. We need two structures because multiple nodes running in the same iteration consume the values created in the previous iteration.
Next, we create separate structures for read-only and read-write parameters in the models. Parameters exist at the node level (like the attention levels in the \textit{Control} node of the predator-prey model), or as arguments of functions (e.g., the amplitude argument of a function that computes a sinusoid). Such functions are defined in the nodes or the standard framework library.
Creating separate structures will be useful when we parallelize executions, where the threads only need to make local copies of the read-write parameter structure.
Modeling frameworks additionally contain two structures, one for the set of inputs for all trials (the variable \texttt{inputs} in Listing~\ref{listing:sched}) and another for overall outputs in the trials. We convert these two entities into arrays.
Finally, the original computations commonly use strings as keys to fetch data, and we convert these strings to enumerated entries (enums), that are used as offsets to index into the structures.
The information about the sizes and keys of the model's parameters and outputs is available in the sanitization run, and for the inputs, this information is available whenever they are read. In scenarios where \sysname cannot infer the shapes of some intermediate variables statically, it does not compile the models. However, we haven't seen such a case in the cognitive models we tested.
In the scenario where a model selects the parameter configuration with the minimal cost (e.g., attention allocation in the predator-prey model), it is possible that multiple parameters may give the same minimal cost. In such cases, it is customary to randomly pick one parameter choice. To implement such constructs, we use reservoir sampling~\cite{vitter1985random} so that we do not need to store a variable number of potential parameter choices, and then choose one from that list. With reservoir sampling, we can have a fixed size statically defined datastructure.
Eliminating dynamic datastructures significantly reduces their access times, and enables several optimizations. In their new format, it is easy to propagate values and references for subsequent optimizations. In addition, the datastructures are now compact, improving cache performance.
\subsection{Generating LLVM IR}
\label{sub_llvm}
\sysname generates LLVM IR for all the nodes and their functions before the model begins execution.
\subsubsection{Code Specialization}
\label{subsub_spl}
For the functions in the framework's standard library, there are pre-defined templates which are then specialized to the types with which they are called.
In the original model, a single function could have been invoked with different types of parameters due to Python's polymoprhic semantics.
\sysname{} generates monomorphic code, creating a separate version of the function for each lexical instance it is invoked.
All the nodes also have a generic template with the basic structure of a node that is filled with the computations in the model's nodes.
\subsubsection{Generating Code for Multiple Libraries and Frameworks}
\label{subsub_codegen}
Recall that cognitive models can use computations from other libraries and frameworks.
\sysname{} takes these computations commonly used in cognitive models and generates LLVM IR.
This includes simple functions from the numpy library like the logistic and sigmoid functions, and neural networks and optimizers from PyTorch.
Lowering these different computations to a common IR allows optimization to span across them, resulting in more efficient code.
\subsection{Optimizations}
\label{sub_optim}
After \sysname generates the LLVM IR for a model, we run LLVM's standard optimization passes. These optimizations like constant propagation and loop invariant code motion can work across computations from multiple frameworks, and across computations from the model and its scheduler to create optimized code. For example, these optimizations could identify that when the \textit{Control} node in the predator-prey model creates an attention allocation, the \textit{Obs} nodes can be run without an explicit check by the scheduler to see which nodes are ready.
Additionally, generating a common IR by itself removes the invocation of the Python interpreter during the entire course of model execution, significantly improving performance.
Lastly, the compiled code in Python does not hold the Global Interpreter Lock, enabling true parallel execution when multithreading is used.
\subsection{Parallelism and GPU Acceleration}
\label{sub_parallel}
When a cognitive model contains a node that evaluates parameters using exhaustive grid search (e.g., the \textit{Control} node in the predator-prey model), each evaluation can be run in parallel.
In this case, \sysname is capable of automatically extracting multicore parallelism and offloading computations to accelerators like GPUs.
To generate multi-threaded code, \sysname first creates as many Python threads as the available cores. Each thread is assigned a segment of the grid search space, and evaluates the parameters in this space using functions that have been compiled in the previous step. Each thread also maintains a local copy of the read-write parameters and the node outputs that it writes to. Since the threads only run compiled code, they do not take the Global Interpreter Lock and can be run in parallel.
For GPUs, we use the NVPTX backend included with LLVM~\cite{holewinski2011ptx} to generate the NVPTX IR for the evaluations. This process generates a kernel for the evaluation function where each thread evaluates one point in the grid search space. The generated kernel is imported to CUDA by PyCuda~\cite{klockner2010pycuda} and executed on GPUs.
For reproducibility, models that sample from random number generators (e.g, sampling the location distributions in the predator-prey model) use independent random number generators for all evaluations.
In these models, the state of the pseudo-random number generator (PRNG) is used as a read-write parameter in their evaluation functions, which is used and restored on every invocation.
This approach directly allows even multiple threads running in parallel to draw the same random numbers. However, as we will see later, replicating the state for each invocation has significant storage overheads.
\subsection{Putting It All Together}
\label{sub_summary}
\sysname aggressively eliminates dynamic features used in cognitive models, and generates LLVM IR for the computations in cognitive models, even if they come from different environments. This allows model-wide optimizations, avoids invoking the interpreter and enables true parallelism. Existing tools like PyPy and Pyston do not perform these optimizations, leaving models to run inefficiently.
\section{Augmenting \sysname with Model Analysis}
We observed that the CDFG of the LLVM IR generated by \sysname for a model matches closely with the interconnection of nodes in the model. This inspired us to augment \sysname with compiler analysis that can provide model-level information to users.
Analyzing a model's outcome when parameters or nodes are modified in a model is an important aspect of cognitive modeling. For example, a researcher may want to know what happens to the overall objective in the predator-prey model if a `fear amplifier' node that modulates the attention allocated to the predator is added to the model.
We have discovered that we could perform several such analyses entirely in the compiler by suitably modifying the analyses present with the LLVM infrastructure. This means that it is not necessary to run the models at all to obtain such information, saving significant resources.
Furthermore, such model-level analyses enables user-guided optimizations where \sysname can present users (cognitive scientists) with multiple code generation alternatives, each of which has slightly different numerical properties.
Next, we describe these model analyses and optimizations, their significance and the changes we made to LLVM to support them.
\subsection{Sensitivity to Parameter Values}
\label{sub_sens}
A common case in cognitive modeling is identifying the impact on outputs for different choices of parameter values. Typically, this is done by running the model with all the choices of these parameters. However, we can perform such analysis entirely in the compiler using value range propagation.
Value range propagation (VRP) is a dataflow analysis that determines ranges of variables based on control flow, type restrictions, and used operations.
For example it can determine that $exp(x)$ can only ever by a positive number or $NaN$, and a commonly used {\em Logistic} function can be shown to always output values in the range (0,1]. However, LLVM implements VRP only for integers. Therefore, we extend this implementation to support floating point types and common floating point operations.
Extending VRP to floating point ranges is useful beyond analyzing cognitive models.
Many floating point operations need special handling in the presence of special values like negative zero, not-a-number, or infinities.
While the compiler can be instructed to optimize these using special fast-math optimization flags,
these are currently set globally per compilation unit or per function, or tracked in a limited way.
Floating point ranges can be used to determine the absence of such special values for each operation and fast-math optimizations can be applied without breaking strict semantics.
This is especially useful for GPU targets which often have specialized fast instructions that do not fully adhere to IEEE floating point semantics.
\subsection{Estimating Convergence times}
\label{sub_est}
For models that simulate accumulation of evidence over time, cognitive scientists are interested to know the estimated time by which evidence accumulation leads to a decision, when they vary the model parameters.
We can perform such analyses using the scalar evolution pass with LLVM. Scalar evolution (SCEV) extends VRP to loops to track value ranges across loop iterations and calculating the number of loop iterations if possible. Similar to VRP, we extend LLVM's SCEV pass to support floating point types and to calculate the minimum number of loop iterations.
Variable ranges at a loop exit can be used to continue range analysis beyond loops.
\begin{comment}
Consider a simple linear ballistic accumulator (LBA)~\cite{brown:lba} race,
presented as C code in Listing~\ref{lst:lba-race}. Researchers want to know how many iterations would the model take to converge for various assumptions about the parameters of the race model.
\begin{figure}[ht]
\label{lst:lca}
\begin{lstlisting}[language=C, basicstyle=\small, caption={Race model example in C. Scalar evolution can calculate the expected number of iterations to be between 17 and 481}, captionpos=b, label=lst:lba-race]
#define assume(x, l, h) \
if (!(x<=h && x>=l)) return; \
else (void)0
void race_model(float starting_point,
float threshold,
float rate,
float stimulus,
float noise,
float time_step,
float *out)
{
assume(starting_point, 0.1f, 0.1f);
assume(threshold, 2.0f, 2.5f);
assume(noise, 0.5f, 1.5f);
assume(rate, 0.5f, 0.5f);
assume(time_step, 0.1f, 0.1f);
assume(stimulus, 0.2f, 1.5f);
float acc = starting_point;
float count = 0;
while (acc < threshold) {
count += 1;
acc += (1.0f - rate) * stimulus
* noise * time_step;
}
*out = count;
}
\end{lstlisting}
\vspace{-4mm}
\end{figure}
With SCEV, \sysname estimates that the code in Listing~\ref{lst:lca} takes between 17 and 481 iterations to complete.
\end{comment}
\subsection{Adaptive Mesh Refinement for Subspace Search}
\label{sub_amr}
In cognitive models that search in a parameter space, it is useful to know which regions of the space results in noteworthy behavior from the model. We can use VRP to progressively narrow the sub-space of interest akin to adaptive mesh refinement. In fact, it is possible to estimate the best parameters without running the model.
Consider the predator-prey model. This model uses grid search to find the best attention allocation. For simplicity, consider that we want to find the best attention allocation for the prey when a fixed attention allocated to the predator and player. The conventional approach is to run the model for various levels of attention (among 100 possible levels) for the prey. Moreover, for each level, the model must be run several times to obtain the output distribution.
Figure~\ref{fig:predator-amr} shows how VRP and binary search can be used to find the optimal attention allocation to prey through compiler analysis. The X-axis of the chart is the attention level allocated to the prey and the Y-axis is a cost metric to evaluate the allocation. The boxes show how the search space is progressively refined. For example, the first iteration of the analysis finds that the range of the metric's value is lower in the region between 2.4 and 4.6, than in the region from 0 to 2.4. Therefore, it performs another binary search to find the metric's value in the region from 2.4 to 4.6 and so on. Eventually it finds the optimal allocation, which is close to 4.6 after about 7 rounds. To determine the same outcome by running the model required hundreds of thousands of runs (also shown in the chart). This shows how VRP can be a great tool for cognitive scientists.
\begin{figure}[h]
\includegraphics[width=\linewidth]{figures/predator-amr-rand.pdf}
\caption {Search for minimum using mesh refining in the Predator-Prey model superimposed over sampled grid}
\label{fig:predator-amr}
\vspace{-2mm}
\end{figure}
\subsection{Clone detection}
\label{sub_clone}
There are multiple benefits of knowing if a node or an entire model is equivalent to another. In the simplest case, it helps verify correctness when a researcher creates an alternate version of the model, without changing the computations. This can happen when models are changed to be more intuitive without affecting their computational behavior. Another use is where a complex node can be substituted with an equivalent whose computation is simpler.
To perform such analyses, we use LLVM's existing `FunctionComparator' framework to detect exactly equivalent functions. As an example of our analysis, consider two functions that use a Drift Diffusion Model (DDM) and a Leaky Competing Integrator (LCA)~\cite{bogacz2007extending} to simulate decision-making, respectively. DDM is used for two-choice decision making and has an analytical solution, while LCA is a multi-choice model. Figure~\ref{fig:clones-ddm-lca} shows that the underlying accumulation is very similar with identical sequence at the core of the computation, and the analysis is able to identify this correctly. Thus, it can notify the user when the LCA can be replaced with an analytical solution, saving thousands of model executions.
\begin{figure}[ht]
\centering
\subfloat[Leaky Competing Accumulator (LCA)]{
\includegraphics[width=0.45\linewidth]{figures/LeakyCompetingIntegrator.pdf}
}
\subfloat[Drift Diffusion Model (DDM)]{
\includegraphics[width=0.45\linewidth]{figures/DriftDiffusionIntegrator.pdf}
}
\caption {Identical computation highlighted in red.
Setting $rate_{LCI} = 0$, $offset_{LCI} = 0$, $noise_{LCI} = N(0,1)$, $rate_{DDI} = 1$, and $noise_{DDI} = 1$, configures both functions to perform identical computation.}
\label{fig:clones-ddm-lca}
\end{figure}
Aggressive inlining allows our methodology to work beyond functions and for entire models. Our analysis could find that a model for bistable perception of a Necker cube~\cite{necker1832lxi}, and its hand-tuned vectorized version are equivalent, even though they differ in their structure, number of nodes and the computation of each node. This is possible because our clone detection analysis works on the IR level,
independent of the original model's structure.
\begin{figure*}[ht]
\includegraphics[width=\linewidth]{figures/perf-data-relative.pdf}
\caption {Running times normalized to Python baseline.
{\em Botvinick stroop} did not complete when run using {\em PyPy3}.
{\em multitasking} uses {\em pytorch} which is not compatible with {\em PyPy3} or {\em Pyston}.}
\label{fig:performance}
\end{figure*}
\section{Experimental Setup}
\label{setup}
\newcommand{Python-3.6.9}{Python-3.6.9}
\newcommand{pypy3-7.3.2}{pypy3-7.3.2}
\newcommand{pyston-2.0.0}{pyston-2.0.0}
\newcommand{CUDA 11.1}{CUDA 11.1}
\newcommand{Intel(R) Core(TM) i7-8700 CPU @ 3.20GHz}{Intel(R) Core(TM) i7-8700 CPU @ 3.20GHz}
\newcommand{GeForce GTX 1060 3GB VRAM}{GeForce GTX 1060 3GB VRAM}
We evaluate \sysname by using it to accelerate a selection of cognitive models designed in PsyNeuLink. We choose PsyNeuLink because it can integrate models from several frameworks.
We use PsyNeuLink's python implementation as baseline and compare execution using Python-3.6.9{} (\textit{CPython}), as well as two JIT enhanced implementations, pyston-2.0.0{} (\textit{Pyston}), and pypy3-7.3.2{} (\textit{PyPy}). We also run PyPy without JIT compilation (\textit{PyPy-nojit}).
Since \sysname{} works with all python implementations we report \sysname{} model execution in all four environments.
\noindent
\textbf{Models for Testing}\\
\noindent
\textbf{Necker cube:} This model is used to simulate the perception of a subject when shown with bi-stable stimulus, typically the line drawing of a cube which can appear to either project out or into the screen~\cite{necker1832lxi}. The model contains one node per vertex and evaluates when the subject's perception oscillates between the two orientations due to gradual changes in the nodes' values. We evaluate three variants of the model: \textit{Necker cube S}, which is the model for a 3-vertex drawing, \textit{Necker cube M}, which uses a cube (8 vertices) and \textit{Vectorized Necker cube} which is a manually vectorized version of Necker cube M.
\noindent
\textbf{Predator-Prey:} We use 4 variants of the Predator-Prey model: S, M, L and XL that have 2, 4, 6 and 100 levels of attention per entity (prey, predator and player). Predator-Prey XL is representative of models that will be commonplace in future.
\noindent
\textbf{Botvinick Stroop Model:} The model simulates the conflict in the brain when processing the name of a color, and the ink color with which it is written~\cite{botvinick2001conflict}. The model calculates decision energy which is gradually changed over time by the stimulus, which is the colored word. The model is used to predict decision energy over time.
We also consider two extended versions of this model, \textit{Extended Stroop A} and \textit{Extended Stroop B} that add an additional task of finger pointing on top of the usual color naming. The model adds two DDM nodes (one for color naming, the other one for finger pointing) to the output of the Stroop model to produce a final decision. The versions A and B differ in how the inputs to the DDMs are computed and how its outputs determine the overall reward. Conceptually, they are different but they are computationally equivalent, as detected by \sysname{}.
\noindent
\textbf{Multitasking:} This model simulates conflict in representation when processing a combination of stimuli and goals. This model uses a neural network designed in PyTorch to give the color and shape of the stimulus. This information is passed to an LCA module designed in PsyNeuLink, which accumulates evidence to reach a decision. The model is run to obtain a distribution of response times and the histogram of correct and incorrect responses. This is a heterogeneous model that spans multiple execution environments: PyTorch and PsyNeulink.
The execution times are collected using {\em pytest-benchmark} package and we report the average running time and standard deviation error bars.
{\em pytest-benchmark} was configured to include two warmup runs before collecting the runtime data and therefore don't include the time to compile models, unless stated otherwise.
\noindent
\textbf{Infrastructure}\\
All experiments are done on an Intel(R) Core(TM) i7-8700 CPU @ 3.20GHz{} with 6 cores and 12 threads machine with 16GB of DDR4@2666Mhz RAM.
GPU experiments are run on a GeForce GTX 1060 3GB VRAM{} on the same machine using CUDA 11.1{}.
\begin{comment}
\subsection{Code performance}
\begin{figure*}[ht]
\subfloat[Running time breakdown]{
\includegraphics[width=.77\linewidth]{figures/breakdown.pdf}
}
\subfloat[Compilation]{
\includegraphics[width=.22\linewidth]{figures/breakdown-compiled2.pdf}
}
\caption {Running time breakdown of models.
{\em Botvinick stroop} did not complete when run using {\em PyPy3}.
{\em multitasking} uses {\em pytorch} which is not compatible with {\em PyPy3} or {\em Pyston}.}
\label{fig:breakdown}
\end{figure*}
\end{comment}
\section{Performance Evaluation}
\label{eval}
Figure~\ref{fig:performance} shows the running time of the models with the different implementations viz., \textit{CPython}, \textit{PyPy}, \textit{PyPy-nojit} and \textit{Pyston} both with and without \sysname{}.
For the predator prey model, we only use the smallest variant in this chart, and analyze the remaining variants separately.
The execution times are normalized to those obtained from the \textit{CPython} implementation, and are plotted on a logarithmic scale.
The execution times with \textit{PyPy} and \textit{PyPy-nojit} are 67\% and 71\% \textit{higher} than the standard \textit{CPython} execution times, on average.
\textit{Pyston} has a 43\% \textit{lower} execution time than \textit{CPython}, on overage.
The poor performance of \textit{PyPy} is surprising in that it claims to improve performance and lower memory usage. However, with, \textit{PyPy}, the \textit{Botvinick Stroop} model and the XL variant of the \textit{Predator Prey} model fail to complete after exhausting all 16$\,$GB of memory available on our test system.
Pyston and PyPy cannot run the \textit{Multitasking} model because they do not support PyTorch.
When \sysname{} is used in these implementations, the execution times are 96\%, 93\%, 93\% and 98\% \textit{faster} than the standard execution for the respective environments, on average.
This translates to speedups of up to $26\times$, on average, and up to $\approx778\times$ for the Botvinick Stroop model.
The main reason is that the JIT compilers are designed for generic Python usage while \sysname{} exploits domain specific information for aggressive optimization.
\subsection{Scaling Model Sizes}
We use the Predator-Prey model to present how \sysname{} can accelerate models as we increase their computational intensity.
Figure~\ref{fig:perf-scale} shows the execution time of the four variants of the predator-prey model for the \textit{CPython} environment and when using \sysname{}.
Recall that the variants have 2, 4, 6 and 100 attention levels per entity corresponding to 8, 64, 216 and 1,000,000 evaluations, respectively.
From the figure,the runtime for the smaller models (S, M and L) with \sysname{} remains nearly the same while the baseline takes an order of magnitude longer time for each step up in the number of attention levels.
With XL, the model does not complete even after 24 hours using \textit{CPython} alone, while with \sysname{} the model finishes execution in about 4 seconds.
Despite the number of evaluations increasing by 4600$\times$ from L to XL, the running time with \sysname{} only goes up by $\approx$330$\times$ from $\approx$0.02$\,$s to $\approx$4.4$\,$s.
This shows that \sysname{} works very well even as we scale models. Importantly, with more realistic cognitve models, \sysname{} is the only choice to keep execution times reasonable.
\begin{figure}[h]
\subfloat[Predator-Prey scaling]{
\includegraphics[width=.33\linewidth,valign=t]{figures/perf-scale.pdf}
\vphantom{\includegraphics[width=0.3\linewidth,valign=t]{figures/perf-stroop-per-node.pdf}}
\label{fig:perf-scale}
}
\subfloat[Botvinick stroop model per-node][Botvinick stroop \\ model per-node]{
\includegraphics[width=.3\linewidth,valign=t]{figures/perf-stroop-per-node.pdf}
\label{fig:perf-per-node}
}
\subfloat[Predator-Prey XL parallel]{
\includegraphics[width=.27\linewidth,valign=t]{figures/perf-parallel.pdf}
\vphantom{\includegraphics[width=0.3\linewidth,valign=t]{figures/perf-stroop-per-node.pdf}}
\label{fig:perf-parallel}
}
\caption{a) shows how \sysname can accelerate models as they are scaled. b) shows the importance of accelerating the entire model. c) compares different parallel evaluations of Predator-Prey XL in the model, GPU vs. 6C/12T CPU. }
\end{figure}
\subsection{Importance of Model-Wide Optimizations}
We highlight the importance of model-wide optimizations that span the nodes and the scheduling logic of PsyNeuLink.
Figure~\ref{fig:perf-per-node} shows the normalized running time of the \textit{Botvinick Stroop} model when \sysname{} is used to generate optimized code per node but with optimizations not crossing node boundaries (the \textit{CPython-\sysname-per-node} design).
This also means that execution switches between the scheduling logic in Python and the compiled model nodes. Compared to \textit{CPython} execution,
\sysname{}-per-node compilation and model-wide compilation (default \sysname{}) result in 70\% and 99.8\% performance improvements, translating to $3.4\times$ and $778\times$ speedups respectively.
This shows the tremendous impact of model-wide optimizations that \sysname{} enables.
\subsection{Parallel and GPU Execution}
Figure~\ref{fig:perf-parallel} shows the execution times in seconds when the predator-prey mode XL (the largest of our tested models) is run on a 12-threaded multicore CPU and a GPU.
Recall that \sysname{} automatically generated parallel code for both systems.
As mentioned earlier, this model did not complete execution in the standard environment. The single thread, multithread CPU (mCPU) and GPU executions result in execution times of 4.4$\,$s, 0.9$\,$s and 0.7$\,$s corresponding to speedups of 4.9$\times$ and 6.3$\times$, respectively, over the serial execution.
One reason for the less-than-ideal speedup on the GPUs is that our implementation replicates the state of the PRNGs used in each thread. Each thread uses 3 PRNGs to obtain the observed X and Y coordinates of the predator, prey and the player, resulting in a total PRNG state of nearly 7.5$\,$kB per thread. Such a large storage causes pressure on the GPU memory hierarchy, resulting in slower execution.
While we could have used GPU-friendly PRNGs, we did not do so because the choice of a PRNG affects the values of the model output~\cite{click:quality}. There is no theoretical analysis on how a new PRNG would impact the predator-prey model.
To confirm that memory pressure is the cause for the suboptimal GPU performance, we run additional studies. Figure~\ref{fig:gpu-performance} shows the execution time and occupancy (defined as the ratio of number of active threads to the maximum) in fp32 and fp64 modes, when the maximum allowed registers per thread is throttled. As fewer registers are used, occupancy increases but execution time increases too. Additionally, fp32, which typically provides up to 32x the computational throughput of fp64, results in nearly the same speedup. This indicates that the bottleneck is not in compute, and is in fact in memory.
\begin{figure}[ht]
\includegraphics[width=\linewidth]{figures/gpu-data-occ.pdf}
\caption {GPU running time using different restrictions on available register space.
Total size of private (per-thread) data is 18.5kB for double precision variant, and 15.5kB for single precision.}
\label{fig:gpu-performance}
\end{figure}
\begin{comment}
\begin{figure}[ht]
\includegraphics[width=\linewidth]{figures/gpu-data-ipc.eps}
\caption {GPU IPC and instruction count using different restrictions on available register space.
}
\label{fig:gpu-ipc}
\end{figure}
\end{comment}
\subsection{Cost of Compilation}
Figure~\ref{fig:breakdown} shows the cost of dynamic compilation for two of our large models, the XL variant of predator-prey and the Multitasking model. The values are normalized to the execution time of the Predator-Prey model with O0 optimizations. The chart also shows the fraction of time to convert inputs and outputs to arrays, and parameters to statically defined structures. Even though compilation times are significant, they are amortized because the models are typically run hundreds to thousands of times after compilation.
\begin{figure}[h]
\includegraphics[width=\linewidth]{figures/breakdown-compiled2.pdf}
\caption {Running time breakdown of models.}
\label{fig:breakdown}
\vspace{-5mm}
\end{figure}
\begin{comment}
We also evaluated the option of using CUDA GPUs for Predator-Prey's parallel search.
To push scaling to its limits, we increased the number of attention levels to 100 (or 1M model evaluations).
Using a GPU provided further $3.3\times$ performance improvement over compiled CPU execution,
completing the task in 1.25 and 4.18 seconds for GPU and CPU respectively.
Both CPU and GPU significantly outperformed Python execution which was stopped after 24 hours, at which point the Python run was terminated,
giving a speed up of more than $20000\times$ for CPU and almost $70000\times$ for GPU.
More fundamental challenges in mapping cognitive models to GPUs and accelerators in general are discussed in \S~\ref{sec:discussion-heterogeneity}.
The {\em Multitasking} model demonstrates the need to accelerate models as a whole rather than focusing on individual parts.
The baseline model invokes {\em PyTorch} for the neural net part of the model.
However, repeated switches between Python and {\em PyTorch} execution slow down the performance.
Even though {\sysname} only generates naive, straightforward, implementation of neural network inference,
tighter integration with the rest of the model allows it to outperform the baseline more than three orders of magnitude.
In general our experiments show that available solutions are either too generic and miss available optimization opportunities ({\em PyPy3}, {\em Pyston}), or too specialized on a narrow subtask ({\em PyTorch}).
{\sysname} strikes a balance between exploiting enough domain specific knowledge to achieve good performance while maintaining its generality to support wide range of modeling workloads expressed in {\pnlname}.
This allows {\sysname} to achieve between $10\times$ and $70000\times$ better performance than the Python baseline.
\end{comment}
\section{Related Work}
\sysname{} builds on, connects, and extends proven approaches across different fields.
On the high level, JIT based impelmentations of high level languages, like PyPy3~\cite{pypy} and Pyston~\cite{pyston} are the closest to \sysname{}.
However, unlike PyPy3 and Pyston, \sysname{} doesn't need to support the Python language in its entirety can and focuses only on parts required by the cognitive models.
We show that eschewing the full generality of Python can be exploited for significant performance benefit,
and enables usable analytical feedback to modeler.
Our focus on compiling only a subset of Python language resembles Numba~\cite{numba,numba-limitations}, a popular approach to achieve compiled performance while using Python.
Numba, however, cannot be directly applied to \pnlname{} codebase.
It lacks the domain specific information to understand high level modeling semantics and avoid unsupported language constructs.
Unlike \sysname{}, Numba pushes the burden of eliminating expensive dynamic constructs the the PsyNeuLink developers and cognitive modelers.
\sysname{} is not the first to notice the significant opportunity in applying domain specific knowledge to Python.
Spiegelberg et al.~\cite{tuplex} apply domain specific knowledge of data types and hot paths to construct efficient data processing pipelines.
The are able to extract and compile expected hot paths with exception checks that fall back to fully interpreted Python.
Unlike Tuplex, \sysname{} benefits from domain specific knowledge that makes full python fallback unnecessary.
This enables us to represent models in LLVM IR completely and use the same IR for analysis and not just performance.
Weld~\cite{weld} similarly exploits the benefits of representing complete programs in a single IR to achieve good performance.
Unlike Weld, \sysname{}, doesn't introduce a new IR, but relies on an industry standard IR.
We also do not require third parties to provide an IR representation and instead rely on \pnlname{}'s ability to import models from different modeling environments.
The model analysis presented in this paper exploits the link to software engineering techniques.
Although \sysname{} explored the utility of direct model comparison in their IR form, this representation opens the door to other clone detection techniques known from the software engineering domain~\cite{Gabel2008-ya,Keivanloo2012-ej,Schugerl2011-jk}.
We believe these approaches to be fully applicable to cognitive models,
and have potential to further benefit the modeling effort by affording richer comparison between models.
Some have even explored applicability to block modeling environments~\cite{Alalfi2012-ke:models-are-code}.
\section{Conclusions}
Cognitive models are vital in understanding and replicating the processes behind human cognition. {\sysname} examines the role of compilers in supporting robust and high-performance modeling of these types of cognitive processes.
Beyond offering large performance improvements necessary to support complex and higher fidelity cognitive models,
we present the suitability of compiler analyses for cognitive modeling analyses.
We propose and implement modifications to a production compiler suite to provide rich feedback to cognitive modelers.
All our contributions are part of open-source projects and released for public use.
\begin{comment}
Future work will extend {\sysname} in several ways, including:
\vspace{2mm}
{\noindent \it Exploring FPGAs, neural network, and machine learning accelerators:} This paper focused on single and multi-core CPUs as well as GPUs, but as a proof of concept, we combined {\sysname} with LegUp~\cite{canis2011legup} to generate Verilog for cognitive models expressed in Python using the PsyNeuLink framework.
Going forward, we expect different parts of cognitive models to map to GPUs, TPUs~\cite{jouppi:tpu}, and number of other hardware accelerators~\cite{samajdar:genesys,qin:sigma,baek:multi} to speed up both learning and inference in machine learning workloads.
Efficient use of these resources will be crucial to provide high performance to cognitive models.
We expect that specific optimizations for efficient code generation of accelerators may need to be considered. For example,
optimizations of neural networks have developed techniques like quantization and synaptic pruning, that are different from traditional compiler optimizations and may be necessary to make efficient use of specialized hardware~\cite{baek:multi,samajdar:genesys,jouppi:tpu,davies:loihi}.
\vspace{2mm}
{\noindent \it Efficient random number generation:}
Many cognitive models rely on stochastic processes, for which software implementations of pseudo-random number generators are used. \sysname{} currently follows Python semantics strictly and uses the Mersenne-Twister pseudo-random generator~\cite{matsumoto1998mersenne} used by Python and Numby. However, this pseudo-random number generator requires a large amount of state, leading to frequent register spilling, and consequently, data divergence. Alternative, GPU-friendly pseudo-random number generators do exist, but they need to first be numerically vetted by cognitive modelers to ensure no change in model accuracy.
\vspace{2mm}
{\noindent \it Exploring stochastic accelerators and quantum computers:}
A compiler representation provides the benefit of automated translation between different forms.
It is, for example, possible to express models (within limits) as equations using random variables in place of calls to the pseudo-random number generators.
Such equations can be then used in symbolic evaluation~\cite{cheatham:symbolic}, or mapped to stochastic accelerators~\cite{banerjee:acmc,zhang:stochastic-computing-unit}.
We have developed a proof-of-concept backend that translates simple models to stochastic expressions usable in Matlab or Octave.
While Matlab and Octave currently use sampling techniques to solve these expressions, we will investigate alternate faster means.
We will also investigate the prospect of using recently-proposed analog accelerators targeted at solving stochastic expressions~\cite{zhang:stochastic-computing-unit}.
We are also intrigued by the prospect of compiling cognitive models to quantum computers. There is an important class of cognitive models, namely multiple constraint satisfaction problems, that use stochastic processes requiring repeated sampling, many simultaneous interacting computations with non-stationary inputs, and that evolve dynamically over the course of processing.
Some of these have begun to be addressed using quantum approaches~\cite{rosendahl:novel-quantum, busemeyer:quantum-models}, with formulations as Hamiltonian functions and Ising models that are seemingly well-suited for quantum hardware. We are actively exploring this as a next research direction.
\vspace{2mm}
{\noindent \it Exploring other IRs for cognitive modeling:} {\sysname} currently uses LLVM IR for both code generation and analysis. We are interested, however, in exploring the viability of other existing IRs for cognitive models in the future. While few IRs generate maximally-efficient code for multiple architectures including CPUs and GPUs,
several IRs can exploit operations common in neural network workloads (like XLA HLO IR~\cite{frostig2018compiling-xla-hlo,jax}, TVM~\cite{chen2018tvm}, Glow-IR~\cite{rotem2018glow}, or Ngraphs~\cite{cyphers2018intel-ngraphs}).
Most of them produce LLVM IR in order to generate efficient code.
Exploring the benefits of these specialized IRs, especially for cognitive models that use neural networks, is left for future work. Similarly, future work will consider the idea cognitive models that lean on stochastic computation in biologically-accurate models may benefit from a specialized IRs for random variable manipulation (e.g~\cite{barbuti2009intermediate,cardelli2004brane,danos2004formal,bistarelli2003representing,priami2001application}).
\end{comment}
|
1,116,691,499,334 | arxiv | \section{Introduction}
The dark energy and dark matter problems are the most important unsolved issues in cosmology.
Although many models of dark energy and dark matter have been proposed, none of them are conclusive at the present time.
Phenomenologically, the cold dark matter with a cosmological constant ($\Lambda$CDM model) is currently the most successful cosmological model.
The most promising candidate of the cold dark matter (CDM) is supersymmetric particles, the so-called neutralino.
While CDM works quite well especially on large scales, it is known that there exists a problem on small scales.
In fact, this model predicts a cusp of dark matter halo profile and overabundance of dwarf galaxies, which are not consistent with observations.
Moreover, the LHC has not reported any signature of supersymmetry.
Given this situation, it is worth seeking another possibility, namely axion dark matter.
The axion, the pseudo-Nambu-Goldstone boson, is originally introduced by Peccei and Quinn to resolve the strong CP problem of QCD~\cite{770620}.
Nowadays, it is known that the string theory also predicts such scalar fields with a wide range of mass scales~\cite{060626}.
Note that we use the term ``axion" in more general meaning, e.g., axionlike particles and other ultralight scalar particles.
The axion with the mass around $10^{-23}~\text{eV}$ behaves as nonrelativistic matter on cosmological scales, and hence it can be regarded as a candidate of dark matter.
Furthermore, it is known that such an ultralight axion can resolve the cusp problem on subgalactic scales because of its wave nature~\cite{000807, 14:Schive}.
A peculiarity of the axion is the oscillating pressure in time with frequency at the twice of the axion mass, $2m$.
Therefore, in order to identify the axion dark matter, we should detect the effect of the oscillating pressure of the axion.
The period of the oscillation corresponds to about one year, and this time scale is much shorter than the cosmological time scale, i.e. $H_{0}^{-1} \sim 10^{10}~\text{years}$.
Hence, after averaging the oscillating pressure over the cosmological time scale, the axion behaves as pressureless dust on cosmological scales.
Thus, it might be difficult to distinguish the axion from other dark matter candidates by cosmological scale observations.
For this reason, we should pay attention to smaller scales to prove the existence of the axion dark matter.
From this point of view, it is pointed out by Khmelnitsky and Rubakov that the effect of oscillating pressure of the axion can be detected by pulsar timing array experiments~\cite{140212}.
The oscillating pressure induces the oscillation of the gravitational potential with frequency in the nano-Hz range.
This effect can be observed as a shift of the arrival time of the signal from the pulsar.
In the previous paper, the axion oscillation was studied in Einstein's theory.
However, since the main energy component of the universe is the dark energy, it might be necessary to consider this issue in the context of modified gravity.
The reason is as follows:
The simplest candidate of dark energy, i.e., the cosmological constant, has several problems, e.g. the fine-tuning problem and coincidence problem.
One possibility to resolve these issues is to consider unknown matter such as the quintessence.
However, there is no natural candidate of quintessence in particle physics.
Therefore, it is natural to assume that theory of gravity is different from Einstein's theory on cosmological scales.
In this paper, as a first step to this direction, we focus on the $f(R)$ theory, which is the simplest modified gravity.
We discuss the detectability of the oscillation of the gravitational potential induced by the time-oscillating pressure of the axion in this context.
This paper is organized as follows:
In Sec. II, we review the results obtained by Khmelnitsky and Rubakov in Einstein's theory.
In Sec. III, we formulate the procedure to determine the amplitude of the gravitational potential in the framework of $f(R)$ theory and discuss two specific models: $R^{2}$ model which can be solved exactly, and the Hu-Sawicki model which is known as a viable cosmological model.
The final section is devoted to conclusion.
\section{Pulsar Timing and Ultralight Axions in Einstein's Theory}
In this section, we review the results obtained by Khmelnitsky and Rubakov in Einstein's theory~\cite{140212}.
We consider the situation that the dark matter halo is composed out of a free ultralight axion field.
The trace of the Einstein equation gives
\begin{equation}
R = -T \ ,
\label{eq1}
\end{equation}
where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor of the axion field.
We will use this equation to determine the gravitational potentials with given $T$.
Now let us consider both sides of this equation in turn.
On the scale of the dark matter halo, the expansion of the universe is completely negligible and the gravitational potentials can still be treated as perturbation.
Thus, we use the Newtonian gauge for the metric:
\begin{equation}
g_{\mu\nu} = \left( \begin{array}{cc}
-1 - 2\Psi & 0 \\ 0 & (1 - 2\Phi)\delta_{ij}
\end{array} \right).
\label{eq2}
\end{equation}
Note that this convention is different from that in \cite{140212}: it is $\Phi$ that affects the signal from the pulsar in this paper.
The Ricci scalar $R$ can be calculated from the metric in the usual manner: at the first order of potentials, it is given by
\begin{equation}
R = -6\ddot{\Phi} + 2\nabla^{2}(2\Phi - \Psi) \ ,
\label{eq3}
\end{equation}
where a dot denotes the derivative with respect to time and $\nabla$ represents the spatial gradient.
This gives the left-hand side of Eq.~(\ref{eq1}).
Next we consider the right-hand side of Eq.~(\ref{eq1}).
Since the occupation number of the axion in the dark matter halo is huge, we can treat it as a classical scalar field.
The axion field satisfies the Klein-Gordon equation in the flat space-time at the leading order, and the solution is given by the superposition of waves of different frequencies.
Since the typical scale of the dark matter halo, $(10~\text{kpc})^{-1} \sim 10^{5}H_{0}$, is much smaller than the mass of the axion, $m \sim 10^{-23}~\text{eV} \sim 10^{10}H_{0}$, we can assume that the axion field oscillates monochromatically with frequency of its mass.
Under these assumptions, we can write the energy density $\rho$ and pressure $p$ of the axion in the following form:
\begin{equation}
\rho \simeq \rho_{\text{DM}}, \quad p \simeq -\rho_{\text{DM}}\cos(2mt) \ ,
\label{eq4}
\end{equation}
where $\rho_{\text{DM}}$ is a constant.
The typical energy density of the dark matter halo is about $0.3~\text{GeV} / \text{cm}^{3}$.
The negative sign of the pressure is just a convention of choosing a phase.
Therefore, the trace of the energy-momentum tensor of the axion can be written as
\begin{equation}
T = -\rho + 3p \simeq -\rho_{\text{DM}}[1 + 3\cos(2mt)] \ .
\label{eq5}
\end{equation}
From the above results, we can rewrite Eq.~(\ref{eq1}) as follows:
\begin{equation}
-6\ddot{\Phi} + 2\nabla^{2}(2\Phi - \Psi) = \rho_{\text{DM}}[1 + 3\cos(2mt)] \ .
\label{eq6}
\end{equation}
Now let us separate the gravitational potential $\Phi~(\Psi)$ into the time-independent part $\Phi_{0}~(\Psi_{0})$ and the time-dependent part $\delta\Phi~(\delta\Psi)$.
To this aim, we should recall the Poisson equation derived from the time-time component of the Einstein equation
\begin{equation}
2\nabla^{2} \Psi_0 = \rho_{\text{DM}} \ .
\label{eq7}
\end{equation}
We also have the equation $\Psi_0 = \Phi_0$ from the traceless part of the space-space component of the Einstein equation.
Thus, we obtain the equation determining the time dependence of the gravitational potential $\delta\Phi$,
\begin{equation}
-6\delta\ddot{\Phi} = 3\rho_{\text{DM}}\cos(2mt) \ .
\label{eq8}
\end{equation}
The above equation can be easily solved as
\begin{equation}
\delta\Phi = \frac{\pi G\rho_{\text{DM}}}{m^{2}}\cos(2mt) \ ,
\label{eq9}
\end{equation}
where we wrote $8\pi G$ explicitly.
Note that $\delta\Phi \ll \Phi_0$ because $k^2 \ll m^2$ in the present situation~\cite{140212}.
They calculated the timing residuals of the signal from the pulsar and showed that the axion dark matter has the same effect on the pulsar timing measurements as gravitational wave background with characteristic strain
\begin{align}
h_{\text{c}} &= 2\sqrt{3}|\delta\Phi| \nonumber\\
&= 2 \times 10^{-15} \left( \frac{\rho_{\text{DM}}}{0.3~\text{GeV} / \text{cm}^{3}} \right) \left( \frac{10^{-23}~\text{eV}}{m} \right)^{2},
\label{eq10}
\end{align}
at frequency
\begin{equation}
f \equiv \frac{\omega}{2\pi} = 5 \times 10^{-9}~\text{Hz} \left( \frac{m}{10^{-23}~\text{eV}} \right).
\label{eq11}
\end{equation}
This signature is detectable in the planned SKA pulsar timing array experiments.
\section{Axions in $f(R)$ Theory}
In the previous section, we explained how the axion dark matter produces the oscillating gravitational potential in Einstein's theory and the oscillation can be detected through the observation of pulsar timing residuals.
The aim in this paper is to extend the analysis to $f(R)$ theory.
In this section, we will show how to obtain the gravitational potential from axion oscillations in $f(R)$ theory and discuss two specific models.
The action of $f(R)$ theory is given by
\begin{equation}
S = \frac{1}{2} \int d^{4}x\sqrt{-g}[R + f(R)] + S_{\text{m}} \ ,
\label{eq12}
\end{equation}
where $f(R)$ is a function of the Ricci scalar $R$, and $S_{\text{m}}$ is the action for matter fields.
Hereafter, we consider the axion field as the matter.
We assume $f(R) \ll R$ and $f_{R} \equiv f'(R) \ll 1$ so that the deviation from Einstein's theory is small.
The variation of the action with respect to the metric gives the field equation:
\begin{equation}
G_{\mu\nu} - \frac{1}{2}g_{\mu\nu}f + (R_{\mu\nu} + g_{\mu\nu}\Box - \nabla_{\mu}\nabla_{\nu})f_{R} = T_{\mu\nu} \ ,
\label{eq13}
\end{equation}
where $G_{\mu\nu} \equiv R_{\mu\nu} - (1 / 2)g_{\mu\nu}R$ is the Einstein tensor.
The trace of this equation gives
\begin{equation}
-R - 2f + (R + 3\Box)f_{R} = T \ .
\label{eq14}
\end{equation}
We assume that the spatial derivative of $f_{R}$ is much smaller than the time derivative of it, i.e. $\Box f_{R} \simeq -\ddot{f}_{R}$.
Then, we obtain
\begin{equation}
3\ddot{f}_{R} + R = -T \ ,
\label{eq15}
\end{equation}
or equivalently,
\begin{equation}
3f''(R)\ddot{R} + 3f'''(R)\dot{R}^{2} + R = -T \ ,
\label{eq16}
\end{equation}
where we used the approximations $f \ll R$ and $f_{R}\ll 1$.
Since the axion field minimally couples to gravity, we can use the same form for the trace of the energy-momentum tensor of the axion, $T$, as the previous one (\ref{eq5}).
We will use Eq.~(\ref{eq16}) to determine the time-dependent part of the Ricci scalar.
Now, let us consider two specific models.
First, we discuss the $f(R) \propto R^{2}$ model, which can be solved exactly.
Second, we consider the more realistic model known as the Hu-Sawicki model~\cite{070910}.
While it is known that this model can pass both cosmological and solar system tests, we will see that there is a tension between the Hu-Sawicki model and the axion dark matter for some parameters.
\subsection{$f(R) = R^{2} / 6M^{2}$ model}
Let us consider a simple model given by
\begin{equation}
f(R) = \frac{R^{2}}{6M^{2}} \ ,
\label{eq17}
\end{equation}
where $M$ is a constant mass scale.
When we discuss the model in terms of the scalar-tensor theory, $M$ is indeed the mass of the scalar field.
This type of model was introduced by Starobinsky to explain the inflationary universe~\cite{800524}.
Now, however, we use this model in the context of the dark energy.
Now, the field equation (\ref{eq16}) becomes
\begin{equation}
\frac{1}{M^{2}}\ddot{R} + R = \rho_{\text{DM}}[1 + 3\cos(2mt)] \ ,
\label{eq18}
\end{equation}
and the solution is given by
\begin{equation}
R = \rho_{\text{DM}} + \frac{3\rho_{\text{DM}}}{1 - (2m / M)^{2}}\cos(2mt) \ .
\label{eq19}
\end{equation}
Here, we ignored the homogeneous solutions which oscillate freely with frequency $M$, and we will discuss this point at the end of this subsection.
Following the same procedure done in the case of Einstein's theory, we obtain the time-dependent part of the gravitational potential as follows:
\begin{equation}
\delta\Phi = \frac{1}{1 - (2m / M)^{2}}\frac{\pi G\rho_{\text{DM}}}{m^{2}}\cos(2mt) \ .
\label{eq20}
\end{equation}
Therefore, in the $R^{2}$ model, we obtained the amplitude of the gravitational potential relative to that in Einstein's theory:
\begin{equation}
\frac{\delta\Phi}{\delta\Phi_{\text{E}}} = \frac{1}{1 - (2m / M)^{2}} \ ,
\label{eq21}
\end{equation}
where $\delta\Phi_{\text{E}}$ is the amplitude of the gravitational potential predicted in Einstein's theory.
This result is illustrated in Fig.~\ref{Fig1}.
In the large $M$ limit, $M \gg 2m$, the prediction in $f(R)$ theory is the same as in Einstein's theory.
This can be understood from the form of $f(R) = R^{2} / 6M^{2}$.
In fact, when the mass of the scalar field $M$ becomes large, $f(R)$ can be neglected compared to the Ricci scalar $R$.
Thus, Einstein's theory is reproduced in this limit.
In the opposite limit, $M \ll 2m$, the amplitude of the gravitational potential goes to zero.
Hence, in this case, it would be difficult to detect the oscillation of the gravitational potential.
When the mass scale $M$ gets close to the frequency of the pressure, $2m$, resonance would occur and the amplitude of the gravitational potential would be dramatically amplified.
Of course, the amplitude cannot reach to infinity: the approximation becomes worse when the oscillating part of $f(R)$ cannot be ignored compared to the Ricci scalar, $R$.
\begin{figure}
\includegraphics[width = 150pt]{Figures/Fig1.pdf}
\caption{The amplitude of the gravitational potential in $R^{2}$ model normalized by the value in Einstein's theory.
Note that we plotted the absolute value because the sign is not important.}
\label{Fig1}
\end{figure}
Now, we make a comment on homogeneous solutions ignored before.
It is pointed out by Starobinsky that the homogeneous solutions decay in the expansion universe and can be completely ignored at the present time~\cite{071000}.
In addition, it is supposed that such scalar degrees of freedom should be highly suppressed by some mechanisms in the solar system scale in order not to mediate the so-called fifth force.
For example, taking into account the interactions, the stabilization mechanism called chameleon mechanism~\cite{040227} or Vainshtein mechanism~\cite{720501} would work and such degrees of freedom might be killed in the solar system scale.
However, if such modes were alive in the dark matter halo scale for some reasons and the mass scale $M$ were sufficiently close to the frequency of the pressure, $2m$, a beat would occur with a frequency $|M - 2m|$.
In this situation, after averaging over the time scale corresponding to the high-frequency $M \sim 2m$, we would observe the beat frequency, $|M - 2m|$.
If such a thing happened, the detectable mass range of the axion by the pulsar timing observation would shift to more heavy mass regions.
\subsection{Hu-Sawicki model}
In the previous subsection, we discussed the simplest $f(R)$ model which can be solved exactly.
Now, in this subsection, we consider a more realistic model.
While there are several $f(R)$ models that explain the late time acceleration of the universe and also pass the solar system tests, now let us focus on the specific model known as the Hu-Sawicki model:
\begin{equation}
f(R) = -\mu R_{\text{c}}\frac{(R / R_{\text{c}})^{2n}}{(R / R_{\text{c}})^{2n} + 1} \ ,
\label{eq22}
\end{equation}
where $n, \mu, R_{\text{c}} > 0$.
For this model to mimic the $\Lambda$CDM model, $\mu R_{\text{c}} \simeq 2\Lambda$ is needed, where $\Lambda$ is the cosmological constant.
Since the energy density of the dark matter halo is much larger than the cosmological critical density, we can assume $R \gg R_{\text{c}}$.
In this limit, Eq.~(\ref{eq22}) takes the following form:
\begin{equation}
f(R) \simeq -\mu R_{\text{c}} \left[ 1 - (R / R_{\text{c}})^{-2n} \right].
\label{eq23}
\end{equation}
Note that the Starobinsky model~\cite{071000} has the same form as Eq.~(\ref{eq23}) in the high curvature limit.
In order to pass the local gravity tests, the Ricci scalar should oscillate around its average value $R_{0} = \rho_{\text{DM}}$.
The mass scale of this model is given by
\begin{equation}
M^{2} \equiv \frac{1}{3f''(R_{0})} \simeq \frac{R_{\text{c}}}{6n(2n + 1)\mu} \left( \frac{\rho_{\text{DM}}}{R_{\text{c}}} \right) ^{2n + 2}.
\label{eq24}
\end{equation}
Using $R_{\text{c}} \simeq 2\Lambda / \mu$ and plausible cosmological parameters~\cite{150209}, the mass is roughly evaluated as
\begin{equation}
M \sim 1.5\mu \times 10^{-23}~\text{eV} \ ,
\label{eq25}
\end{equation}
for $n = 1$.
This rough estimate tells us that $M$ has a value around the critical mass, $2m$, for $\mu = \mathcal{O}(1)$.
Since $M$ is strongly dependent on $n$ [see Eq.~(\ref{eq24})], $M$ can be larger or smaller compared to $2m$.
When $M \ll 2m$, completely the same situation as $R^{2}$ model is realized and the amplitude of the gravitational potential is given by Eq.~(\ref{eq20}).
This is because the amplitude of the Ricci scalar is much smaller than its average value in this limit and the field equation (\ref{eq16}) is reduced to Eq.~(\ref{eq18}).
Hence, this behavior should be universal for more general models which pass the local gravity tests.
When $M \gtrsim 2m$, however, a problem arises.
Since $f'''(R) / f''(R) \sim 1 / R$, we can evaluate
\begin{equation}
\frac{f'''(R)\dot{R}^{2}}{f''(R)\ddot{R}} \sim \frac{\delta\dot{R}^{2}}{R\delta\ddot{R}} \sim \frac{\delta R}{R} \ ,
\label{eq26}
\end{equation}
where $\delta R$ is the oscillating part of $R$.
Therefore, once $\delta R$ becomes of the order of $R_{0}$, the second term of the field equation (\ref{eq16}) prohibits $\delta R$ from oscillating stably.
With numerical calculations, we verified the Ricci scalar diverges for these parameters.
This is also true for the Einstein limit, $M \gg 2m$.
Thus, the Hu-Sawicki model is not compatible with the axion dark matter for these parameters.
Note that other viable $f(R)$ models also suffer from the same problem.
In order to avoid the instability of oscillations, the condition $M \lesssim 2m$ is needed.
This constraint is shown in Fig.~\ref{Fig2} with other constraints from cosmological and local gravity tests~\cite{100623}:
The three downward-sloping curves are the upper bounds on $\mu$ for three different axion masses.
The almost horizontal line denotes the lower bound on $\mu$ from cosmological tests.
The vertical line corresponding to $n = 0.9$ represents the lower bound on $n$ from the local tests.
From Fig.~\ref{Fig2}, the ultralight axion dark matter and the Hu-Sawicki model are compatible only in the certain parameter regions~(shaded in Fig.~\ref{Fig2}).
Note that the upper bounds on $\mu$ are somewhat underestimated:
From numerical calculations, we found that the upper bounds on $\mu$ are about 5 times smaller than the roughly estimated values illustrated in Fig.~\ref{Fig2}.
Of course, since the Hu-Sawicki model works well on large scales, it might be natural to modify the Hu-Sawicki model on small scales to circumvent this instability problem.
\begin{figure}
\includegraphics[width = 200pt]{Figures/Fig2.pdf}
\caption{
The constraints on the Hu-Sawicki model.
The axion dark matter model and the Hu-Sawicki model are compatible in the shaded regions.
}
\label{Fig2}
\end{figure}
If the axion dark matter were detected by pulsar timing experiments, we can determine the axion mass $m$ from the oscillation frequency
and the mass scale $M$ of $f(R)$ model from the amplitude of oscillation Eq.~(\ref{eq20}).
In the Hu-Sawicki model , since $M$ monotonically increases as $\mu$ and $n$ increase, it has the minimum value, $M_{\text{min}}$,
corresponding to the lower bounds for $\mu$ and $n$.
Numerically, we obtain the minimum value as
\begin{equation}
M_{\text{min}} \sim 0.76 \times 10^{-23}~\text{eV} \ .
\label{eq27}
\end{equation}
Hence, if the observed mass $M$ were lower than Eq.~(\ref{eq27}), the Hu-Sawicki model would be excluded.
\section{Conclusion}
We studied the pulsar timing signal from the ultralight axion field in $f(R)$ theory.
First, we discussed the simplest $f(R) = R^{2} / 6M^{2}$ model.
Then, it turned out that the amplitude of the gravitational potential in this model is enhanced or suppressed depending on the mass parameter $M$ compared to the case in Einstein's theory.
If $M$ is larger than the frequency of the pressure, $2m$, the results in Einstein's theory are reproduced.
On the other hand, if $M$ is smaller than $2m$, the amplitude is suppressed and difficult to be detected.
Furthermore, when $M$ approaches $2m$, the amplitude is dramatically amplified due to the resonance.
Next we discussed the Hu-Sawicki model.
Although the Hu-Sawicki model is known to pass both cosmological and solar system tests, we showed that this model is not compatible with the ultralight axion dark matter for some parameters.
When the mass scale $M$ given by Eq.~(\ref{eq24}) is much smaller than $2m$, completely the same situation as $R^{2}$ model is realized.
In this case, unfortunately, the amplitude is too small to be detected by near-future experiments.
Meanwhile, when $M$ reaches $2m$, the oscillation cannot be stable owing to the ``nonlinear" term of the field equation.
Remarkably, the model does not work even in the Einstein limit, $M \gg 2m$, for the same reason.
This gives rise to the new constraint on the Hu-Sawicki model.
In order to circumvent this instability problem, a modification on small scales would be needed.
In fact, if the detected mass scale $M$ were lower than $M_{\text{min}} \sim 0.76 \times 10^{-23}~\text{eV}$, the Hu-Sawicki model would be excluded.
\acknowledgements
This work was supported by JSPS KAKENHI Grant No. 25400251, MEXT KAKENHI Grants No. 26104708 and No. 15H05895.
\bibliographystyle{apsrev4-1}
|
1,116,691,499,335 | arxiv | \section{Introduction}
Algebras of dihedral, semi-dihedral and quaternion types arises naturally from Erdmann's classical book \cite{13} as a product of classification of tame blocks. Hochschild cohomology on them are well-studied by large amount of mathematicians, such as A. I. Generalov (see \cite{8}, \cite{15}, \cite{17}), A. A. Ivanov (see \cite{16}, \cite{17}), C. Cibils (see \cite{12}) and many others, including author (see \cite{10}, where one could find more links on studies of Hochschild cohomology). For an associative algebra $A$ there are many structures on its algebra $HH^*(A)$ of Hochschild cohomology of $A$: for example, it has a grade-commutative algebra structure (see \cite{9}) and graded Lie algebra structure, introdused by Gerstenhaber in his paper \cite{2}. Tradler was first who described $BV$-structure on Hochschild cohomology of finite dimensional symmetric algebras (see \cite{3}). The problem here is that $BV$-structure is described in terms of bar-resolution, which makes it almost impossible to compute such structure for concrete examples, because dimension of resolution's parts grow exponentially. In order to avoid this problem we are using method of comparsion morphisms (see also \cite{1} and \cite{19}). \par
One of the value of $BV$-structure is that it gives a method to compute the Gerstenhaber Lie
bracket, which is important and hard-reached structure on $HH^*(A)$. In this paper we deal with algebra of Hochschild cohomology of quaternion type $R(k,0,d)$ on algebraically closed field $K$ of characteristic 2, described in paper \cite{10}, which generalizes Generalov's case (see \cite{15}). Partial cases of this series of algebras were described in \cite{19} (case $R(k,0,0)$) and in \cite{1} for case $R(2,0,0)$, which is a group algebra of quaternion units $KQ_8$. Here we study only case of even dimension $k \ge 3$, because in \cite{10} was shown that cases of even and odd dimension are differ significantly (unlike that's in \cite{15}). \par
$BV$-structures on Hochshild cohomology was also studied by Menichi (see \cite{6,7}), Yang (see \cite{5}), Tradler (see \cite{3}) and Ivanov (see \cite{19}). Also it should be mentioned a Volkov's article \cite{14} about $BV$-structures of Frobenius algebras. So this is a really rich field of Math with decent amount of possible applies.
\section{Main definitions}
\subsection{Hochschild (co)homology}
For an associative algebra $A$ over a field $K$ it's $n$-th Hochschild cohomology is a vector space $HH^n(A) = Ext^n_{A^e} (A,A)$ for $n \ge 0$, where $A^e= A \otimes A^{op}$ is an enveloping algebra for $A$. Notice that {\it bar-resolution } is a free resolution
$$\CD A @<{\mu}<< A^{\otimes 2} @<{d_1}<< A^{\otimes 3} @<{d_2}<< \dots @<{d_n}<< A^{\otimes n+2} @<{d_{n+1}}<< A^{\otimes n+3} \dots \endCD$$
with differentials
$$d_n(a_0 \otimes \dots a_{n+1}) = \sum \limits_{i=0}^{n} (-1)^i a_0 \otimes \dots \otimes a_i a_{i+1} \otimes \dots \otimes a_{n+1}.$$
One can construct {\it normalized bar-resolution}, in which $n$-th element is given by formula $\overline{Bar}(A)_n = A \otimes \overline{A}^{\otimes n} \otimes A$, where $\overline{A} = A/ \langle 1_A \rangle$, and differential is induced by bar-resolution. \par
We define the $n$-th Hochschild homology space $HH_n(A)$ as follows:
$$HH_n(A) = H_n(A \otimes_{A^e} Bar_{\bullet}(A)) \simeq H_n(A^{\bullet+1}),$$
where differentials $\partial_n : HH_n(A) \longrightarrow HH_{n-1}(A)$ comes by mapping $a_0 \otimes \dots \otimes a_n$ to
$$\sum_{i=0}^{n-1} (-1)^i a_0 \otimes \dots \otimes a_{i-1} a_i \otimes \dots \otimes a_n + (-1)^{n} a_na_0 \otimes \dots \otimes a_{n-1}.$$
Let's look closely on complex $(Hom_{A^e}(Bar_{\bullet}(A), A), \ \delta^{\bullet})$. As always,
$$HH^n(A) = H^n(Hom_{A^e}(Bar_n(A),A)) \simeq H^n(Hom_k(A^{\otimes n}, A)),$$
and for $f \in Hom_k(A^{\otimes n}, A)$ the element $\delta^n (f)$ sends element $a_1 \otimes \dots \otimes a_{n+1}$ to
$$a_1 f(a_2 \otimes \dots \otimes a_{n+1}) + \sum_1^n (-1)^i f(a_1 \otimes \dots \otimes a_i a_{i+1} \otimes \dots \otimes a_{n+1}) + (-1)^{n+1}f(a_1 \otimes \dots \otimes a_{n})a_{n+1}.$$
One can define a cap-product on Hochschild cohomology: for classes $a \in HH^n(A)$ and $b \in HH^m(A)$ it's cap-product $a \smile b \in HH^{n+m}(A)$ is defined by the class of cap-product of representatives $a \in Hom_k(A^{\otimes n}, A)$ and $b \in Hom_k(A^{\otimes m}, A)$. So by linear extension
$$\smile : HH^n(A) \times HH^m(A) \longrightarrow HH^{n+m}(A)$$
Hochschild cohomology space $HH^{\bullet}(A) = \bigoplus \limits_{n \ge 0} HH^n(A)$ becomes a graded-commutative algebra.
\subsection{Gerstenhaber bracket}
For $f \in Hom_k(A^{\otimes n}, A)$ and $g \in Hom_k(A^{\otimes m}, A)$ one can define $f \circ_i g \in Hom_k(A^{\otimes n+m-1}, A)$ by the following rules:
\begin{enumerate}
\item if $n \ge 1$ and $m \ge 1$, set $f \circ_i g (a_1 \otimes \dots \otimes a_{n+m-1}) = f(a_1 \otimes \dots a_{i-1} \otimes g(a_i \otimes \dots a_{i+m-1}) \otimes \dots \otimes a_{n+m-1})$.
\item if $n \ge 1$ and $m=0$, set $f \circ_i g (a_1 \otimes \dots \otimes a_{n-1}) = f(a_1 \otimes \dots a_{i-1} \otimes g \otimes a_i \otimes \dots \otimes a_{n+m-1})$, because $g \in A$ in this case.
\item otherwise set $f \circ_i g = 0$.
\end{enumerate}
So put $a \circ b = \sum \limits_{i=1}^n (-1)^{(m-1)(i-1)} a \circ_i b$.
\begin{definition}
For $f \in Hom_k(A^{\otimes n}, A)$ and $g \in Hom_k(A^{\otimes m}, A)$ define Gerstenhaber bracket $[f,g]$ via formula
$$[f,g] = f \circ g - (-1)^{(n-1)(m-1)} g \circ f.$$
\end{definition}
This element obviously lies in $Hom_k(A^{\otimes n+m-1}, A)$, so for $a \in HH^n(A)$ and $b \in HH^m(A)$ define $[a,b] \in HH^{n+m-1}(A)$ as a class of Gerstenhaber bracket for representatives $a$ and $b$. This bracket correctly induces map
$$[-,-]:HH^{*} (A) \times HH^{*} (A) \longrightarrow HH^{*} (A),$$
which gives us a structure of graded Lie algebra on Hochschild cohomology, and also one can show that $(HH^{*} (A), \smile, [-,-])$ is a Gerstenhaber algebra (see. \cite{2}).
\subsection{$BV$-structure}
\begin{definition}
Batalin-Vilkovisky algebra (or $BV$-algebra) is a Gerstenhaber algebra \\ $(A^{\bullet}, \smile, [-,-])$ together with operator $\Delta^{\bullet}$ of degree $-1$, such that $\Delta \circ \Delta = 0$ and
$$[a,b] = - (-1)^{|a|-1)|b|} \big(\Delta(a \smile b) - \Delta(a) \smile b - (-1)^{|a|} a \smile \Delta(b) \big)$$
for homogeneous $a, b \in A^{\bullet}$.
\end{definition}
For $a_0 \otimes \dots \otimes a_n \in A^{\otimes (n+1)}$ define
$$\mathfrak{B}(a_0 \otimes \dots \otimes a_n) = \sum_{i=0}^n (-1)^{in} 1 \otimes a_i \otimes \dots \otimes a_n \otimes a_0 \otimes \dots \otimes a_{i-1} + $$
$$+\sum_{i=0}^n (-1)^{in} a_i \otimes 1 \otimes a_{i+1} \otimes \dots \otimes a_n \otimes a_0 \otimes \dots \otimes a_{i-1}.$$
Obviously $\mathfrak{B}(a_0 \otimes \dots \otimes a_n) \in A^{\otimes (n+1)} \simeq A \otimes_{A^e} A^{\otimes (n+3)}$, hence it can be lifted to the chain map of complexes, and $\mathfrak{B} \circ \mathfrak{B} = 0$, so it correctly induces a map on Hochschild homology.
\begin{definition}
Defined above map $\mathfrak{B}: HH_{\bullet}(A) \longrightarrow HH_{\bullet+1}(A)$ is said to be Connes' $\mathfrak{B}$-operator.
\end{definition}
\begin{definition}
Algebra $A$ is symmetric, if it is isomorphic to it's dual $DA = Hom_K(A,K)$ as $A^e$-module.
\end{definition}
For a symmetric algebra $A$ one can always find non-degenerate biliniar form $\langle, \rangle : A \times A \longrightarrow K$, and, obviously, reversed statement holds: for any such form algebra $A$ is symmetric. So in case of symmetric algebras Hochschild homology and colomology are dual:
$$Hom_K(A \otimes_{A^e} Bar_{\bullet}(A),K) \simeq Hom_{A^e} (Bar_{\bullet}(A), Hom_K(A,K)) \simeq Hom_{A^e}(Bar_{\bullet}(A), A),$$
hence there exists operator $\Delta: HH^n(A) \longrightarrow HH^{n-1}(A)$, which corresponds to Connes' operator. \par
It shows that algebra of Hochschild cohomology for symmetric $A$ is $BV$-algebra (see \cite{3} for details), and, as noticed, Connes' operator for homology corresponds to $\Delta$ operator on cohomology. Tradler theorem holds:
\begin{theorem}[Theorem 1 in \cite{4}]
Defined above cap-product, Gerstenhaber bracket and operator $\Delta$ induces a structure of $BV$-algebra on $HH^{*}(A)$. Moreover, for $f \in Hom_K(A^{\otimes n}, A)$ element $\Delta(f) \in Hom_K(A^{\otimes (n-1)}, A)$ defined properly by formula
$$\langle \Delta(f)(a_1 \otimes \dots \otimes a_{n-1}),a_n \rangle = \sum_{i=1}^n (-1)^{i(n-1)} \langle f(a_i \otimes \dots \otimes a_{n-1}\otimes a_n \otimes a_1 \otimes \dots \otimes a_{i-1}), 1 \rangle$$
for $a_i \in A$.
\end{theorem}
\begin{remark} All constructions here can be defined and used in terms of normalized bar-resolution.
\end{remark}
\section{Weak self-homotopy}
\subsection{Resolution}
Let $K$ be an algebraically closed field of characteristic not 2 and let $c, d \in K$. Define $R(k, c, d) = K[X,Y]/I$, where $I$ is an ideal in $K[X,Y]$, spanned by $X^2+Y(XY)^{k-1} + c(XY)^k, Y^2+X(YX)^{k-1} + d(XY)^k, (XY)^k + (YX)^K, X(YX)^k, Y(XY)^k$. \par
Let $B$ be standard basis $R=R(k, c, d)$. So the set $B_1 = \{u \otimes v \mid u, v \in B\}$ is a basis for enveloping algebra $\Lambda = R \otimes R^{op}$.\par
Algebras of family $R(k,c,d)$ are symmetric, since there exists a non-degenerate bilinear form
$$\langle b_1, b_2\rangle = \begin{cases}
1, & b_1 b_2 \in Soc(R) \\
0, & \text{else}
\end{cases}.$$
In order to describe structure of graded Lie algebra, one should only know, how $\Delta$ works on Hochschild cohomology. In this article we are interested in the case $c=0$ and so working only with $R(k,0,d)$, which we often call $R$. \par
Right multiplication by $\lambda \in \Lambda$ induces an endomorphism $\lambda^{*}$ of left
$\Lambda$-module $\Lambda$; and we denote it as $\lambda^*$ for simplicity. Also we will use endomorphism of right $\Lambda$-module $\Lambda$, indused by left multiplication on $\lambda$, which we denote as ${}^{*}\lambda$. \par
For computations we construct 4-periodic complex in cathegory of $\Lambda$-modules
$$
\CD
Q_0 @<{d_0}<< Q_1 @<{d_1}<< Q_2 @<{d_2}<< Q_3 @<{d_3}<< Q_4 @<{d_4}<< \dots\\
\endCD$$
where $Q_0=Q_3=\Lambda$, $Q_1 = Q_2 = \Lambda^2$ and differentials given by formulas
$$d_0 = \begin{pmatrix}
x \otimes 1 + 1\otimes x & y \otimes 1 + 1 \otimes y
\end{pmatrix},
\quad d_1=\begin{pmatrix}
d_{11}& d_{12} \\
d_{13} & d_{14}
\end{pmatrix},$$
$$d_2 = \begin{pmatrix}
x \otimes 1 + 1\otimes x \\
y \otimes 1 + 1\otimes y + dy\otimes y + 1 \otimes dx(yx)^{k-1}+d^2y\otimes x(yx)^{k-1}\\
\end{pmatrix}, \quad d_3 = {}^{*} \lambda,$$
where
$$\begin{cases}
d_{11} = x \otimes 1 + 1\otimes x + \sum \limits_{0}^{k-2} y(xy)^i \otimes y(xy)^{k-2-i}, \\
d_{12} = \sum \limits_{0}^{k-1}(xy)^i \otimes (yx)^{k-1-i} + d \sum \limits_{0}^{k-1} (xy)^i \otimes y(xy)^{k-1-i}, \\
d_{13} = \sum \limits_{0}^{k-1}(yx)^i \otimes (xy)^{k-1-i},\\
d_{14} = y \otimes 1 + 1\otimes y + \sum \limits_{0}^{k-2} x(yx)^i \otimes x(yx)^{k-2-i} + d \sum \limits_{0}^{k-1} x(yx)^i \otimes (xy)^{k-i-1},
\end{cases} $$
and
$$ \lambda = \sum \limits_{0}^{k}(xy)^i \otimes (xy)^{k-i} + \sum \limits_{1}^{k-1} (yx)^i \otimes (yx)^{k-i} +\sum \limits_{0}^{k-1}y(xy)^i \otimes x(yx)^{k-i-1} +$$ $$+\sum \limits_{0}^{k-1}x(yx)^i \otimes y(xy)^{k-i-1} + dx(yx)^{k-1} \otimes x(yx)^{k-1}.$$
Also denote by $\mu:\Lambda \longrightarrow R$ multiplication map $\mu(a \otimes b) = ab$.
\begin{theorem}[Proposition 3.1 in \cite{10}]
Complex $$
\CD
Q_0 @<{d_0}<< Q_1 @<{d_1}<< Q_2 @<{d_2}<< Q_3 @<{d_3}<< Q_4 @<{d_4}<<...
\endCD$$ via map $\mu$ is a minimal $\Lambda-$projective resolution of $R$.
\end{theorem}
It is useful to rewrite the resolution as in \cite{1}: define modules $KQ_1, KQ_1^*$, spanned by two elements: $KQ_1 = \langle x,y\rangle$, and $KQ_1^* = \langle r_x,r_y\rangle$, where $r_x = x^2+y(xy)^{k-1}$, $r_y = y^2 + x(yx)^{k-1}+d(xy)^k$. It is easy to see that
$$R \otimes KQ_1 \otimes R = R \otimes \langle x \rangle \otimes R \oplus R \otimes \langle y \rangle \otimes R \simeq R \otimes R^{op} \oplus R \otimes R^{op} = \Lambda \oplus \Lambda,$$
so one can find a (milimal) projective resolution of bimodules $\{ P_n \}_{n=0}^{+\infty}$:
$$
\CD
R @<{\mu}<< R \otimes R @<{d_1}<< R\otimes KQ_1 \otimes R @<{d_2}<< R\otimes KQ_1^* \otimes R @<{d_3}<< R\otimes R @<{d_4}<< \dots
\endCD$$
where $P_{n+4} = P_n$ for $n \in \mathbb{N}$. Differentials are given by formulas
\begin{itemize}
\item $d_0(1 \otimes x \otimes 1) = x \otimes 1 + 1 \otimes x$, $d_0(1 \otimes y \otimes 1) = y \otimes 1 + 1 \otimes y$;
\item $d_1(1 \otimes r_x \otimes 1) = 1\otimes x \otimes x + x \otimes x \otimes 1 + \sum \limits_{i=0}^{k-2} y(xy)^i \otimes x \otimes y(xy)^{k-2-i} + \sum \limits_{i=0}^{k-1} (yx)^i \otimes y \otimes (xy)^{k-1-i}$, \\
$d_1(1 \otimes r_y \otimes 1) = 1\otimes y \otimes y + y \otimes y \otimes 1 + \sum \limits_{i=0}^{k-2} x(yx)^i \otimes y \otimes x(yx)^{k-2-i} + d\sum \limits_{i=0}^{k-1} x(yx)^i \otimes y \otimes (xy)^{k-1-i} + \sum \limits_{i=0}^{k-1} (xy)^i \otimes x \otimes (yx)^{k-1-i} + d\sum \limits_{i=0}^{k-1} (xy)^i \otimes x \otimes y(xy)^{k-1-i}$;
\item $d_2(1\otimes 1) = x \otimes r_x \otimes 1 + 1\otimes r_x \otimes x+ y\otimes r_y \otimes 1 + 1\otimes r_y \otimes y + d y\otimes r_y \otimes y + d \otimes r_y \otimes x(yx)^{k-1} + d^2 y \otimes r_y \otimes x(yx)^{k-1}$;
\item $d_3 = \rho \mu$, where $\rho (1) = \sum \limits_{b \in B} b^*\otimes b + d x(yx)^{k-1} \otimes x(yx)^{k-1}$ and $\mu : R \otimes R \longrightarrow R$ induced by multiplication.
\end{itemize}
\subsection{Construction}
Now we need to construct a weak self-homotopy $\{t_i :P_i \longrightarrow R_{i+1} \}_{i \ge -1}$ for such projective resolution. In order to do this define bimodule derivation $C : KQ \longrightarrow KQ \otimes KQ_1 \otimes KQ$ by sending $\alpha_1...\alpha_n$ to $\sum \limits_{i=1}^n \alpha_1 ... \alpha_{i-1} \otimes \alpha_i \otimes \alpha_{i+1} \dots \alpha_n$, and consider induced map $C: A \longrightarrow A \otimes KQ_1 \otimes A$.
\begin{lemma}
$t_{-1} (1) = 1\otimes 1$ and $t_0 (b \otimes 1) = C(b)$ for $b \in B$.
\end{lemma}
\begin{proof}
It is straight-up obvious by definitions.
\end{proof}
Construct $t_1 : P_1 \longrightarrow P_2$ by following rules: for $b\in B$ let
$$t_1(b \otimes x \otimes 1) = $$
$$ = \begin{cases}
0, & bx \in B\setminus\{y(xy)^{k-1}\}\\
1 \otimes r_x \otimes 1, & b=x\\
1 \otimes r_x \otimes x^2 + x \otimes r_x \otimes x + x^2 \otimes r_x \otimes 1 + \\ (yx)^{k-1} \otimes r_y \otimes (xy)^{k-1}, & b = (xy)^k\\
T(y \otimes r_x \otimes 1 + (xy)^{k-1} \otimes r_x \otimes y(xy)^{k-2} + 1 \otimes r_y \otimes (xy)^{k-1} \\ +dx \otimes r_x \otimes (xy)^{k-1} + dy(xy)^{k-1}\otimes r_x \otimes y(xy)^{k-2}, & b = Tyx\\
y \otimes r_y \otimes 1+1 \otimes r_y \otimes y + dy \otimes r_y \otimes y \\+ d\otimes r_y \otimes x(yx)^{k-1} + d^2 y\otimes r_y \otimes x(yx)^{k-1}, & b = y(xy)^{k-1}
\end{cases},
$$
and let
$$t_1(b \otimes y \otimes 1) = $$
$$ = \begin{cases}
0, & by \in B \\
1 \otimes r_y \otimes 1, & b=y\\
T(x \otimes r_y \otimes 1 + (yx)^{k-1} \otimes r_y \otimes x(yx)^{k-2} + 1 \otimes r_x \otimes (yx)^{k-1}\\
+ d \otimes r_x \otimes y(xy)^{k-1} + d(yx)^{k-1} \otimes r_y \otimes (xy)^{k-1}, & b = Txy\\
\end{cases}
$$
In order to define $t_2:P_2 \longrightarrow P_3$ put:
\begin{itemize}
\item $t_2(x \otimes r_x \otimes 1) = 1 \otimes 1$,
\item $t_2 (xy \otimes r_x \otimes 1) = d y(xy)^{k-1} \otimes y(xy)^{k-2}$,
\item $t_2(y(xy)^{k-1} \otimes r_x \otimes 1) = 1\otimes x$,
\item $t_2((xy)^k \otimes r_x \otimes 1) = \sum \limits_{i=0}^{k-1} ((yx)^i \otimes y(xy)^{k-i-1} + y(xy)^i \otimes (xy)^{k-i-1}) +dx(yx)^{k-1} \otimes (xy)^{k-1}$,
\item $t_2((yx)^i \otimes r_x \otimes 1) = \sum \limits_{j=1}^{i-1} (yx)^j \otimes y(xy)^{i-j-1} +\sum \limits_{j=1}^{i} y(xy)^{j-1} \otimes (xy)^{i-j} + dx(yx)^{k-1} \otimes (xy)^{i-1}$ in case of $i>1$,
\item $t_2(yx \otimes r_x \otimes 1) = y \otimes 1 + dx(yx)^{k-1} \otimes 1 + d(xy)^{k-1} \otimes x + \sum \limits_{i=0}^{k-2} dx(yx)^i \otimes (yx)^{k-i-1} + \sum \limits_{i=1}^{k-2} d(xy)^i \otimes x(yx)^{k-i-1}$,
\item $t_2 (x(yx)^i \otimes r_x \otimes 1) = \sum \limits_{j=1}^{i}( (xy)^j \otimes (xy)^{i-j} + x(yx)^{j-1} \otimes y(xy)^{i-j}) + \delta_{i,1} y(xy)^{k-1} \otimes (yx)^{k-1},$
\item $t_2(b \otimes r_x \otimes 1)=0$ for all other cases;
\end{itemize}
and let
\begin{itemize}
\item $t_2((xy)^i \otimes r_y \otimes 1) = \sum \limits_{j=1}^{i-1} (xy)^j \otimes x(yx)^{i-j-1} +\sum \limits_{j=1}^{i} x(yx)^{j-1} \otimes (yx)^{i-j} + d\sum \limits_{j=1}^{i} x(yx)^{j-1} \otimes y(xy)^{i-j} + d\sum \limits_{j=1}^{i-1} (xy)^j \otimes (xy)^{i-j}$ for $i>1$,
\item $t_2(xy \otimes r_y \otimes 1) = x \otimes 1 + dx \otimes y$,
\item $t_2(y(xy)^i \otimes r_y \otimes 1) = \sum \limits_{j=1}^i ((yx)^j \otimes (yx)^{i-j} + y(xy)^{j-1} \otimes x(yx)^{i-j} ) + d \sum \limits_{j=1}^i ((yx)^j \otimes y(xy)^{i-j} + y(xy)^{j-1} \otimes (xy)^{i-j+1} ) +d^2 x(yx)^{k-1}\otimes (xy)^i + dx(yx)^{k-1} \otimes x(yx)^{i-1}+ \delta_{i,1} \cdot d (xy)^{k-1}\otimes y(xy)^{k-1}$,
\item $t_2(b \otimes r_y \otimes 1) = 0$ for all other cases.
\end{itemize}
Finally, let $t_3 : A \otimes A \longrightarrow A \otimes A$ be the map, such that $t_3((xy)^k \otimes 1) = 1 \otimes 1$ and $t_3(b \otimes 1)=0$ elsewhere. It remains to put $t_{n+4} = t_n$ for any $n \ge 4$.\par
\begin{theorem}
The above-defined family of maps $t_i : P_{i} \longrightarrow P_{i-1}$ form a weak self-homotopy over such resolution $P_{\bullet}$.
\end{theorem}
\begin{proof}
For any $n \in \mathbb{N}$ it remains to verify a commutativity of required diagrams, which is straight-up obvious from definitions of $t_n$ for $n \leq 4$ and from periodicity for $n \ge 5$.
\end{proof}
\section{Comparsion morphisms}
Consider normalized bar-resolution $ \overline{{Bar}}_{\bullet} (R) = R \otimes \overline{{R}^{\otimes \bullet}} \otimes R$, where $ \overline{R} = R / (k\cdot 1_R)$, with differentials, induced by standard bar-resolution. We now need to construct comparsion morphisms between $P_{\bullet}$ and $Bar_{\bullet}$
$$\Phi : P_{\bullet} \longrightarrow \overline{{Bar}}_{\bullet} \text{ and } \Psi : \overline{{Bar}}_{\bullet} \longrightarrow P_{\bullet}.$$
It is easy to check that there exists a weak self-homotopy $s_n(a_0 \otimes ... \otimes a_n \otimes 1) = 1 \otimes a_0 \otimes ... \otimes a_n \otimes 1$ over $ \overline{{Bar}}_{\bullet} (R)$, so one can put $\Phi_n = s_{n-1} \Phi_{n-1} d^P_{n-1}$.
\begin{lemma}
Let $\Psi : \overline{{Bar}}_{\bullet} \longrightarrow P_{\bullet}$ be a chained morphiusm, defined via $t_{\bullet}$. Then for any $n \in \mathbb{N}$ and any $a_i \in R$ following formula holds:
$$\Psi_n (1 \otimes a_1 \otimes ... \otimes a_n \otimes 1) = t_{n-1} (a_1 \Psi_{n-1} (1 \otimes a_2 \otimes ... \otimes a_n \otimes 1)).$$
\end{lemma}
\begin{proof}
It follows from Lemma 2.5 of \cite{1}.
\end{proof}
For further applications one need to directly compute first items of $\Phi_{\bullet}$:
\begin{enumerate}
\item $\Phi_0 = id_{R \otimes R}$,
\item $\Phi_1$ induced by embedding $R \otimes kQ_1 \otimes R \longrightarrow R \otimes \overline{R} \otimes R$,
\item $\Phi_2 (1 \otimes r_x \otimes 1) = 1 \otimes x \otimes x \otimes 1+ \sum \limits_{i=0}^{k-2} 1 \otimes y(xy)^i \otimes x \otimes y(xy)^{k-i-2} + \sum \limits_{i=1}^{k-1} 1 \otimes (yx)^i \otimes y \otimes (xy)^{k-i-1}$,\\
$\Phi_2( q \otimes r_y \otimes 1) = 1 \otimes y \otimes y \otimes 1+ \sum \limits_{i=0}^{k-2} 1 \otimes x(yx)^i \otimes y \otimes x(yx)^{k-i-2} + d \sum \limits_{i=0}^{k-1} 1 \otimes x(yx)^i \otimes y \otimes (xy)^{k-i-1} + \sum \limits_{i=1}^{k-1} 1 \otimes (xy)^i \otimes x \otimes (yx)^{k-i-1} + d \sum \limits_{i=1}^{k-1} 1 \otimes (xy)^i \otimes x \otimes y(xy)^{k-i-1}$,
\item $\Phi_3(1\otimes 1) = 1 \otimes x \otimes x \otimes x \otimes 1 + \sum \limits_{i=0}^{k-2} 1 \otimes x \otimes y(xy)^i \otimes x \otimes y(xy)^{k-i-2} + \sum \limits_{i=1}^{k-1} 1 \otimes x \otimes (yx)^i \otimes y \otimes (xy)^{k-i-1} + 1 \otimes y \otimes y \otimes y \otimes 1+ d \otimes y \otimes x(yx)^{k-1} \otimes y \otimes 1+ \sum \limits_{i=0}^{k-2} 1 \otimes y \otimes x(yx)^i \otimes y \otimes x(yx)^{k-i-2} + \sum \limits_{i=1}^{k-1} 1 \otimes y \otimes (xy)^i \otimes x \otimes (yx)^{k-i-1} + d \otimes y \otimes y \otimes y \otimes y +d^2 \otimes y \otimes x(yx)^{k-1} \otimes y \otimes y + d^2 \otimes y \otimes y \otimes y \otimes x(yx)^{k-1} + d^3 \otimes y \otimes x(yx)^{k-1} \otimes y \otimes x(yx)^{k-1}$,
\item $\Phi_4(1 \otimes 1) = \sum_{b \in B} 1 \otimes b \Phi_3(1 \otimes 1) b^* + d \otimes x(yx)^{k-1} \Phi_3(1\otimes 1) x(yx)^{k-1}$.
\end{enumerate}
Also it is easy to see that $\Psi_0 = id_{R \otimes R}$, $\Psi_1(1 \otimes b \otimes 1) = C(b)$, $\Psi_2(1 \otimes a_1 \otimes a_2 \otimes 1) = t_1 \big(a_1 C(a_2) \big)$, and so on: we only interested in recursive structure like in Lemma.\par
In order to obtain $BV$-structure on Hochschild cohomology, one needs to compute $\Delta : HH^n(R) \longrightarrow HH^{n-1}(R)$. By Poisson rule
$$[a \smile b, c] = [a,c] \smile b + (-1)^{|a| (|c|-1)}(a \smile [b,c]),$$
so, because char$K = 2$ we obtain
$$\Delta(abc) = \Delta(ab)c + \Delta(ac)b + \Delta(bc)a + \Delta(a)bc +\Delta(b)ac + \Delta(c)ab,$$
so we only need to know $\Delta$ on spanning elements of Hochschild cohomology and also on cap-products of such elements. Also for $\alpha \in HH^n(R)$ there exists cocycle $f \in Hom (P_n,R)$ such that following formula holds: $\Delta(\alpha) = \Delta(f \Psi_n) \Phi_{n-1}$. So
$$\Delta(\alpha) (a_1 \otimes ... \otimes a_{n-1}) = \sum_{b \in B\setminus \{1\}} \langle \sum \limits_{i=1}^n (-1)^{i(n-1)} \alpha(a_i \otimes ... \otimes a_{n-1} \otimes b \otimes a_1 \otimes ...\otimes a_{i-1}), 1\rangle b^*,$$
where $\langle b,c \rangle$ is a bilinear form, defined above.
\section{$BV$-structure}
For algebraically closed field $K$ of characteristic 2 consider
$$\mathcal{X} = \{p_1, p_2,p_3,p_4,q_1,q_2,w_1,w_2,w_3,e\},$$ where
$$
|p_1| = |p_2| = |p_3| = |p_4| = 0, \ |q_1| = |q_2| = 1, \ |w_1| = |w_2| =|w_3|=2, \ |e|=1
$$
and ideal $\mathcal{I}$ in $K[\mathcal{X}]$, spanned by elements
\begin{itemize}
\item of degree 0: $p_1^k, p_2^2, p_3^2, p_4^2$ и $p_i p_j$ for $i \not=j$;
\item of degree 1: $p_3q_1+p_2q_2$, $p_1^{k-1}q_1+dp_3q_1$, $p_1q_2+p_2q_1, p_1^2q_2$;
\item of degree 2: $q_1q_2$, $p_1^{k-1}w_3+dp_2w_2$, $p_2w_1, p_4w_1, p_3w_2, p_4w_2, p_4w_3$, $p_2q_1^2, p_3q_2^2$, $p_1w_1+p_2w_2, p_1w_1+p_3w_3, p_1w_1 + p_4q_1^2$, $p_3w_1+p_1w_2, p_3w_1+p_2w_3, p_3w_1+p_4q_2^2$
\item of degree 3: $q_1w_1+q_2w_2$, $q_1^3+q_2^3+{{d^3(k+2)} \over {2}}p_1q_1w_1$, $p_3q_2w_1+p_1q_2w_2$, $p_3q_2w_1+p_2q_2w_3$, $p_1^{k-2}q_1w_3+dq_2w_2$, $p_1^{k-2} q_2w_3$, $p_1q_2w_1$, $p_1q_2w_3$, $q_1w_2+q_2w_3$;
\item of degree 4: $w_3^2+p_1^2e,$ $q_2^2w_1$, $q_1^2w_3$, $q_2^2w_1$, $q_2^2w_2$, $w_1^2$, $w_2^2$, $w_iw_j$ for $i \not=j$.
\end{itemize}
\begin{theorem}[Theorem 2.1 in \cite{10}]
$HH^*(R) \simeq \mathcal{A}_2 = K[\mathcal{X}]/\mathcal{I}$.
\end{theorem}
For computations one needs simple form for such elements. Let $P$ be an item of minimal projective resolution $R$. If $P= R \otimes R$, denote by $f$ homomorphism $Hom_{R^e}(P,R)$ which sends $1 \otimes 1$ to $f$. If $P= R \otimes KQ\otimes R$ (or $P= R \otimes KQ_1\otimes R$), denote by $(f,g)$ homomorphism which sends $1\otimes x \otimes 1$ (or $1 \otimes r_x \otimes 1$) to $f$ and $1 \otimes y \otimes 1$ (or $1 \otimes r_y \otimes 1$) to $g$. So by author's joint work with Generalov (see \cite{10}) one can found that
$$\begin{cases}
\text{Elements of degree 1:} & p_1 = xy+yx, \ p_2 = x(yx)^{k-1}, \ p_3 = y(xy)^{k-1}, \ p_4 = (xy)^k, \\
\text{Elements of degree 2:} & q_1 = ( y(xy)^{k-2}, \ 1+dy ),\ q_2 =(1, \ d(xy)^{k-1} +x(yx)^{k-2}), \\
\text{Elements of degree 3:} & w_1 = ( x, \ 0),\ w_2 = (0, \ y ),\ w_3= ( y,\ x + dxy ),\\
\text{Elements of degree 4:} & e=1.\end{cases}$$
\begin{remark}
Also let us show that
$$C(b) = \begin{cases} \sum \limits_{i=0}^{i-1} (xy)^j \otimes x \otimes y(xy)^{i-j-1} + \sum \limits_{i=0}^{i-1} x(yx)^j \otimes y \otimes (xy)^{i-j-1}, & b = (xy)^i \\
\sum \limits_{i=0}^{i} (xy)^j \otimes x \otimes (yx)^{i-j} + \sum \limits_{i=0}^{i-1} x(yx)^j \otimes y \otimes x(yx)^{i-j-1}, & b = x(yx)^i \\
\sum \limits_{i=0}^{i} (yx)^j \otimes y \otimes (xy)^{i-j} + \sum \limits_{i=0}^{i-1} y(xy)^j \otimes x \otimes y(xy)^{i-j-1}, & b = y(xy)^i\\
\sum \limits_{i=0}^{i-1} (yx)^j \otimes y \otimes x(yx)^{i-j-1} + \sum \limits_{i=0}^{i-1} y(xy)^j \otimes x \otimes (yx)^{i-j-1}, & b = (yx)^i
\end{cases}$$
\end{remark}
\subsection{Low degree cases}
Obviously $\Delta$ is equal to zero on each combination of degree zero, because it is a morphism of degree $-1$.
\begin{lemma} For elements of degree 1 in $HH^*(R)$ following statements holds:
$\Delta(q_1) = \Delta(q_2)= \Delta(p_3q_2) = 0$, $\Delta(p_1q_1) = dp_1$, $\Delta(p_1q_2) = \Delta(p_2q_1)= \Delta(p_4q_1) = dp_2$, $\Delta(p_3q_1) = \Delta(p_2q_2)= p_1^{k-1}$, $\Delta(p_4q_2) = p_3$.
\end{lemma}
\begin{proof}
We have already seen that $\Delta(a)(1\otimes 1) = \sum \limits_{b \not=1} \langle a (C(b)),1\rangle b^*$, so we only need to compute $\langle a (C(b)),1\rangle$ on elements of degree 1. Computations shows
$$p_1q_1 = (y(xy)^{k-1}, \ dyxy ), \quad p_2q_1 = ( 0, \ x(yx)^{k-1}+d(xy)^k ),$$
$$p_3q_1 = ( 0, \ y(xy)^{k-1}, \quad p_4q_1 =( 0, \ (xy)^k ),$$
and also one can do this for each $p_iq_2$ by symmetry. It is easy to see that
$$\langle a (C(b)),1\rangle = \begin{cases} dk, & a = q_1, b = (xy)^k \\
d(k-1), & a=p_1q_1, b \in \{(xy)^{k-1}, (yx)^{k-1} \}\\
d, & a \in \{ p_1q_2, p_2q_1 \}, b=y\\
1, & a \in \{p_3q_1, p_2q_2 \}, b \in \{xy, yx\} \text{ or } a=p_4q_1, b = y \text{ or } a=p_4q_2, b = x \\
0, & \text{ else }
\end{cases},$$
hence required formulas holds.
\end{proof}
\begin{lemma}
$\Delta(x)=0$ for any combination $x$ of degree 2 of elements from generator's set for $HH^*(R)$.
\end{lemma}
\begin{proof}
If $a \in HH^2(R)$, then
$$\Delta(a)(1 \otimes x \otimes 1) = \Delta(a\Psi_2)\Phi_1(1\otimes x \otimes 1) = \sum_{b\not=1} \langle (a\Psi_2) (b \otimes x+x\otimes b), 1 \rangle b^*,$$
$$\Delta(a)(1 \otimes y \otimes 1) =\Delta(a\Psi_2)\Phi_1(1\otimes y \otimes 1) = \sum_{b\not=1} \langle (a\Psi_2) (b \otimes y+y\otimes b), 1 \rangle b^*.$$
Observe that $\Psi_2(1 \otimes b \otimes x \otimes 1+1\otimes x \otimes b \otimes 1) = t_1(b \otimes x \otimes 1 + xC(b))$. Obviously, $\Delta(q_1q_2)=0$ and
$$q_2^2 = ( 1, \ 0 ), \quad q_1^2 = ( 0, \ 1+d^2y^2+ {{d^3k} \over {2}}\cdot (xy)^k ),$$
Now one need to compute $\Psi_2(1\otimes b \otimes x \otimes 1 + 1\otimes x \otimes b \otimes 1)$:
\begin{enumerate}
\item if $b=(xy)^i$ for $1 \leq i \leq k-1$, then $t_1 \big( b \otimes x \otimes 1 + xC(b) \big) = 1 \otimes r_x \otimes y(xy)^{i-1} + (yx)^{k-1} \otimes r_y \otimes (xy)^{i-1}+ + \delta_{i,1} \big( y(xy)^{k-2} \otimes r_x \otimes (yx)^{k-1} + dy(xy)^{k-2} \otimes r_x \otimes y(xy)^{k-1} \big)$,
\item if $b = (yx)^i$ for $1 \leq i \leq k-1$, then $t_1 \big( b \otimes x \otimes 1 + xC(b) \big) = y(xy)^{i-1} \otimes r_x \otimes 1+(yx)^{i-1} \otimes r_y \otimes (xy)^{k-1} + \delta_{i,1} \big( (xy)^{k-1} \otimes r_x \otimes y(xy)^{k-2} + dx \otimes r_x \otimes (xy)^{k-1} + dy(xy)^{k-1} \otimes r_x \otimes y(xy)^{k-2} \big)$,
\item if $b=x(yx)^i$ for $1 \leq i \leq k-1$, then $ t_1 \big( b \otimes x \otimes 1 + xC(b) \big) = (xy)^i \otimes r_x \otimes 1+ x(yx)^{i-1} \otimes r_y \otimes (xy)^{k-1} + 1\otimes r_x \otimes (yx)^i + (yx)^{k-1} \otimes r_y \otimes x(yx)^{i-1} + \delta_{i,1} \big( dx^2 \otimes r_x \otimes (xy)^{k-1} + d(xy)^k \otimes r_x \otimes y(xy)^{k-2} + dy(xy)^{k-2} \otimes r_x \otimes (xy)^k \big)$,
\item if $b=y(xy)^{k-1}$, then $ t_1 \big( b \otimes x \otimes 1 + xC(b) \big) = y \otimes r_y \otimes 1 +1 \otimes r_y \otimes y + dy\otimes r_y \otimes y + d\otimes r_y \otimes x(yx)^{k-1} + d^2 y\otimes r_y \otimes x(yx)^{k-1}$,
\item if $b=(xy)^k$, then $ t_1 \big( b \otimes x \otimes 1 + xC(b) \big) = x \otimes r_x \otimes x + x^2 \otimes r_x \otimes 1$,
\item if $b=x$ or $b=y(xy)^{i}$ for $0 \leq i \leq k-2$, then $t_1 \big( b \otimes x \otimes 1 + xC(b) \big) =0$.
\end{enumerate}
Now consider $\Psi_2(1\otimes b \otimes y \otimes 1 + 1\otimes y \otimes b \otimes 1) = t_1 \big( b \otimes y \otimes 1 + yC(b) \big)$:
\begin{enumerate}
\item if $b=(yx)^i$ for $1 \leq i \leq k-1$, then $t_1 \big( b \otimes y \otimes 1 + yC(b) \big) = 1 \otimes r_y \otimes x(yx)^{i-1} + (xy)^{k-1} \otimes r_x \otimes (yx)^{i-1} + d x\otimes r_x \otimes x(yx)^{i-1} + dx^2 \otimes r_x \otimes (yx)^{i-1} + \delta_{i,1} \big( x(yx)^{k-2} \otimes r_y \otimes (xy)^{k-1} + d \otimes r_x \otimes x^2 + d (yx)^{k-1} \otimes r_y \otimes (xy)^{k-1} \big)$,
\item if $b = (xy)^i$ for $1 \leq i \leq k-1$, then $t_1 \big( b \otimes y \otimes 1 + yC(b) \big) = x(yx)^{i-1} \otimes r_y \otimes 1 +(xy)^{i-1} \otimes r_x \otimes (yx)^{k-1} + d (xy)^{i-1} \otimes r_x \otimes y(xy)^{k-1} + \delta_{i,1} \big( (yx)^{k-1} \otimes r_y \otimes x(yx)^{k-2} + d(yx)^{k-1} \otimes r_y \otimes (xy)^{k-1} \big)$,
\item if $b = x(yx)^{k-1}$ for $0 \leq i \leq k-1$, then $t_1 \big( b \otimes y \otimes 1 + yC(b) \big) = y \otimes r_y\otimes 1 + 1 \otimes r_y \otimes y + d y \otimes r_y \otimes y + d \otimes r_y \otimes x(yx)^{k-1} + d^2 y \otimes r_y \otimes x(yx)^{k-1}$,
\item if $b = y(xy)^i$ for $1 \leq i \leq k-1$, then $t_1 \big( b \otimes y \otimes 1 + yC(b) \big) = (yx)^i \otimes r_y \otimes 1 + y(xy)^{i-1} \otimes r_x \otimes (yx)^{k-1} + dy(xy)^{i-1} \otimes r_x \otimes y(xy)^{k-1} + 1 \otimes r_y \otimes (xy)^i + (xy)^{k-1} \otimes r_x \otimes y(xy)^{i-1} + dx \otimes r_x \otimes (xy)^i + dx^2\otimes r_x \otimes y(xy)^{i-1}$,
\item if $b = (xy)^k$, for $t_1 \big( b \otimes y \otimes 1 + yC(b) \big) = y \otimes r_y \otimes y + 1 \otimes r_y \otimes x(yx)^{k-1} + dy \otimes r_y \otimes x(yx)^{k-1}$,
\item else $t_1 \big( b \otimes y \otimes 1 + yC(b) \big) = 0$.
\end{enumerate}
Hence lemma holds by given computations.
\end{proof}
\subsection{Middle degree cases}
\begin{lemma}
$\Delta(q_1w_1) = \Delta(q_2w_2) = p_1^{k-2} w_3$, $\Delta(q_1w_2) = \big( 1+d^3(l+1)p_4 \big) q_1^2 + dw_2$, $\Delta(q_1w_3) = q_2^2 + w_3$, $\Delta(q_2w_1) = q_2^2$, $\Delta(q_2w_3) = \Delta(q_1w_2) + w_3$.
\end{lemma}
\begin{proof}
In this proof we are going to use delta-like function
$$\mu_{a,b} = \begin{cases}
1, & a \ge b, \\
0, & a < b.
\end{cases}$$
Fix $a \in HH^3(R)$. By identifications $1 \otimes a_1 \otimes ... \otimes a_n \otimes 1 = a_1 \otimes ... \otimes a_n$ in $R$-bimodule $R \otimes \overline{R}^{\otimes n} \otimes R$ one can observe:
$$\Delta(a) (1 \otimes r_x \otimes 1) = \Delta(a \Psi_3)\Phi_2(1\otimes r_x \otimes 1) = \sum \limits_{b \not= 1} \langle (a\Psi_3) (b \otimes x \otimes x + x \otimes b \otimes x + x \otimes x \otimes b), 1 \rangle b^* +$$
$$+\sum \limits_{b \not= 1} \sum \limits_{i=0}^{k-2} \langle (a\Psi_3) (b \otimes y(xy)^i \otimes x + y(xy)^i \otimes x \otimes b + x \otimes b \otimes y(xy)^i), 1 \rangle b^* \cdot y(xy)^{k-2-i} +
$$
$$+\sum \limits_{b \not= 1} \sum \limits_{i=1}^{k-1} \langle (a\Psi_3) ((yx)^i \otimes y \otimes b + y \otimes b \otimes (yx)^i + b \otimes (yx)^i \otimes y), 1 \rangle b^* \cdot (xy)^{k-1-i}.
$$
It is (really) easy to see that
$$\Psi_3(b \otimes x \otimes x + x \otimes b \otimes x + x \otimes x \otimes b) = t_2 \Big( b \otimes r_x \otimes 1 + xt_1 \big( b \otimes x \otimes 1 + xC(b) \big) \Big).$$
Denote this by $\Psi_3(b,x)$.
\begin{itemize}
\item if $b=x$, then $\Psi_3(b,x) = 1 \otimes 1$,
\item if $b = x(yx)^i$ for $1 \leq i \leq k-1 $, then $\Psi_3(b,x) = \sum \limits_{j=1}^{i} \big( (xy)^i \otimes (xy)^{i-j} + x(yx)^{j-1} \otimes y(xy)^{i-j} \big) + 1 \otimes (yx)^i + \delta_{i,1} \big( (yx)^{k-1} \otimes (xy)^{k-1} + dy(xy)^{k-1} \otimes (yx)^{k-1} +dy(xy)^{k-1} \otimes (xy)^{k-1} \big)$,
\item if $b = y(xy)^{k-1}$ for $1 \leq i \leq k-1$, then $\Psi_3(b,x) = 1 \otimes x + x \otimes 1 $,
\item if $b = (xy)^i$ for $1 \leq i \leq k-1$, then $\Psi_3(b,x)= 1\otimes y(xy)^{i-1} + \delta_{i,1} dy(xy)^{k-1} \otimes y(xy)^{k-2}$,
\item if $b = (yx)^i$ for $2 \leq i \leq k-1$, then $\Psi_3(b,x)= \sum \limits_{j=0}^{i-1} (yx)^j \otimes y(xy)^{i-j-1} + \sum \limits_{j=1}^{i} y(xy)^{j-1} \otimes (xy)^{i-j}+ dx(yx)^{k-1} \otimes (xy)^{i-1}$,
\item if $b = yx$, then $\Psi_3(b,x)= y \otimes 1 + d x(yx)^{k-1} \otimes 1 + d(xy)^{k-1} \otimes x + \sum \limits_{j=0}^{k-2} d x(yx)^j \otimes (yx)^{k-j-1} + \sum \limits_{j=1}^{k-2} d (xy)^j \otimes x(yx)^{k-j-1}$,
\item if $b=(xy)^k$, then $\Psi_3(b,x)= 1\otimes y(xy)^{k-1}$,
\item else $\Psi_3(b,x) = 0$.
\end{itemize}
Next for $1 \leq i \leq k-2$
$$\Psi_3 (b \otimes y(xy)^i \otimes x + y(xy)^i \otimes x \otimes b + x \otimes b \otimes y(xy)^i) = $$
$$=t_2 \Big( b t_1 (y(xy)^i \otimes x \otimes 1) + y(xy)^i t_1\big( x C(b) \big) + x t_1 \big(b C(y(xy)^i) \big) \Big).$$
Denote this by $\Psi_3(2, b, x, i)$.
\begin{itemize}
\item if $b = x(yx)^j$ for $1 \leq j \leq k-1$, $i>0$ and $ i+j \ge k$, то $\Psi_3(2, b, x, i) = y(xy)^{k-1} \otimes y(xy)^{i+j-k} + \delta_{i+j,k} (yx)^{k-1} \otimes y^2$,
\item if $b = y$ and $i>0$, then $\Psi_3(2, b, x) = dy(xy)^{k-1} \otimes y(xy)^{i-1} + \delta_{i,1} \big( d(yx)^{k-1} \otimes y^2 + d^3 (xy)^k \otimes y(xy)^{k-1} \big)$,
\item if $b = xy$, then $\Psi_3(2, b, x, i) = d\sum \limits_{l=1}^{k-1} \big( (yx)^l \otimes y(xy)^{k-1-l+i} + y(xy)^{l-1}\otimes (xy)^{k-l+i} \big) + d^2 x(yx)^{k-1} \otimes (xy)^{k-1+i} +
\delta_{i,0} \big( 1 \otimes (yx)^{k-1} + dx(yx)^{k-1} \otimes x(yx)^{k-2} + d \otimes y(xy)^{k-1} \big)+$
$$+ \begin{cases} (yx)^{k-1} \otimes (xy)^i, & i>0\\
\sum \limits_{l=1}^{k-1} (yx)^l \otimes (yx)^{k-1-l} + \sum \limits_{l=1}^{k-1} y(xy)^{l-1} \otimes x(yx)^{k-1-l}, & i=0
\end{cases}$$
\item if $b = (yx)^j$ for $1 \leq j \leq k-1$, $i+j \ge k$ and $i>0$, then $\Psi_3(2, b, x, i) = x \otimes y(xy)^{i+j-k}$,
\item else $\Psi_3(2, b, x, i) = 0$.
\end{itemize}
Finally, only one step remains.
$$\Psi_3 \big( (yx)^i \otimes y \otimes b + b \otimes (yx)^i \otimes y + y \otimes b \otimes (yx)^i \big)$$
for $1 \leq i \leq k-1$. Denote this by $\Psi_3(3,b,x,i)$.
\begin{itemize}
\item if $b=x(yx)^j$ for $0 \leq j \leq k-1$, this can be written as:
\begin{enumerate}
\item $j=k-1$: $\sum \limits_{l=1}^i \big( (yx)^l \otimes (yx)^{i-l} + y(xy)^{l-1} \otimes x(yx)^{i-l} \big) + y \otimes x(yx)^{i-1} + \delta_{i,1} \big( y \otimes x^2 + d(xy)^{k-1} \otimes x^2 \big),$
\item $i+j \ge k$, $j \not= k-1$: $y \otimes x(yx)^{i+j-k} + dx(yx)^{k-1} \otimes x(yx)^{i+j-k},$
\end{enumerate}
\item if $b=(yx)^j$ for $1 \leq j \leq k$, this can be written as:
\begin{enumerate}
\item if $j=1$: $\begin{cases} d y\otimes x^2 + d^2 (xy)^{k-1} \otimes x^2, & i=1\\
dy(xy)^{i-1} \otimes x^2, & i>1
\end{cases} + \delta_{i,1} \big( dx(yx)^{k-1} \otimes (xy)^{k-1} + d^2 (xy)^{k-1} \otimes (xy)^k \big) + \delta_{i,k-1} \big( d \sum \limits_{l=1}^{k-1} (xy)^{l} \otimes x(yx)^{k-l-1} + d \sum \limits_{l=1}^k x(yx)^{l-1} \otimes (yx)^{k-l} \big) $
\item if $i+j-1 \ge k$: $x(yx)^{k-1} \otimes x(yx)^{i+j-k-1} + d \sum \limits_{l=1}^k x(yx)^{l-1} \otimes (yx)^{i+j-l} +d \sum \limits_{l=1}^{k-1} (xy)^{l} \otimes x(yx)^{i+j-l-1} + d \sum \limits_{l=1}^{k-1} (xy)^{l} \otimes x(yx)^{i+j-l-1} + d \sum \limits_{l=1}^k x(yx)^{l-1} \otimes (yx)^{i+j-l} + \delta_{i+j,k+1} (xy)^{k-1} \otimes x^2$
\end{enumerate}
\item if $b=y(xy)^j$ for $0 \leq j \leq k-1$, this can be written as:
\begin{enumerate}
\item $j=0$: $1 \otimes x(yx)^{i-1} + dy \otimes x(yx)^{i-1} + d^2 x(yx)^{k-1} \otimes x(yx)^{i-1} + \delta_{i,1}d^2 (xy)^{k-1} \otimes x^2$,
\item $j=1$ and $i=1$: $d(xy)^{k-1} \otimes (xy)^k + d^2 y(xy)^{k-1} \otimes (xy)^k$,
\end{enumerate}
\item else $\Psi_3(3, b, x, i) = 0$.
\end{itemize}
Now one should deal with $1 \otimes r_y \otimes 1$:
$$\Delta(a) (1 \otimes r_y\otimes 1) = \sum \limits_{b \not=1} \langle \Delta(a) (\Psi_3 (b \otimes y \otimes y + y \otimes b \otimes y + y \otimes y \otimes b)),1\rangle b^* + $$
$$\sum \limits_{b \not=1} \sum\limits_{i=0}^{k-2} \langle \Delta(a) (\Psi_3 (b \otimes x(yx)^i \otimes y + x(yx)^i \otimes y \otimes b + y \otimes b \otimes x(yx)^i)),1\rangle b^* x(yx)^{k-2-i}+$$
$$ +d\sum \limits_{b \not=1} \sum\limits_{i=0}^{k-1} \langle \Delta(a) (\Psi_3 (b \otimes x(yx)^i \otimes y + x(yx)^i \otimes y \otimes b + y \otimes b \otimes x(yx)^i)),1\rangle b^* (xy)^{k-1-i}$$
$$ + \sum \limits_{b \not=1} \sum\limits_{i=1}^{k-1} \langle \Delta(a) (\Psi_3 ((xy)^i \otimes x\otimes b + x \otimes b \otimes (xy)^i +b \otimes (xy)^i \otimes x)),1\rangle b^* ((yx)^{k-i-1} + dy(xy)^{k-i-1})$$
Firstly,
$$\Psi_3 (b \otimes y \otimes y + y \otimes b \otimes y + y \otimes y \otimes b) = t_2 \Big( b \otimes r_y \otimes 1 + yt_1(b \otimes y \otimes 1) + yt_1 \big( yC(b) \big) \Big).$$
Denote this by $\Psi_3(b,y)$.
\begin{itemize}
\item if $b= (xy)^i$ for $1 \leq i \leq k-1$, then \begin{enumerate}
\item if $i=1$ then $\Psi_3(b,y) = x \otimes 1 + d x \otimes y $,
\item if $i>1$ then $\Psi_3(b,y) = \sum\limits_{l=1}^{i-1} (xy)^l \otimes x(yx)^{i-l-1} + \sum\limits_{l=1}^{i} x(yx)^{l-1} \otimes (yx)^{i-l} + d\sum\limits_{l=1}^{i} x(yx)^{l-1} \otimes y(xy)^{i-l} + d\sum\limits_{l=1}^{i-1} (xy)^l \otimes (xy)^{i-l}$,
\end{enumerate}
\item if $b= (yx)^i$ for $1 \leq i \leq k-1$, then \begin{enumerate}
\item if $i=1$ then $\Psi_3(b,y) = 1\otimes x + dy \otimes x + d^2 (xy)^{k-1} \otimes x^2 + d^2 x(yx)^{k-1} \otimes x $,
\item if $i>1$ then $\Psi_3(b,y) = 1 \otimes x(yx)^{i-1} + dy \otimes x(yx)^{i-1} + d^2 x(yx)^{k-1} \otimes x(yx)^{i-1}$,
\end{enumerate}
\item if $b= (xy)^k$, then $\Psi_3(b,y) = \sum \limits_{l=1}^{k-1} (xy)^l \otimes x(yx)^{k-l-1} + \sum \limits_{l=1}^{k} x(yx)^{l-1} \otimes (yx)^{k-l}$.
\item if $b= x(yx)^{k-1}$ for $0 \leq i \leq k-1$, then $\Psi_3(b,y) = d\sum \limits_{l=1}^{k-1} (xy)^l \otimes x(yx)^{k-l-1} + d \sum \limits_{l=1}^{k} x(yx)^{l-1} \otimes (yx)^{k-l}$,
\item if $b = y(xy)^i$ for $0 < i \leq k-1$, then $\Psi_3(b,y) = \sum \limits_{l=1}^i \big( (yx)^l \otimes (yx)^{i-l} + y(xy)^{l-1} \otimes x(yx)^{i-l} \big) + d\sum \limits_{l=1}^i \big( (yx)^l \otimes y(xy)^{i-l} + y(xy)^{l-1} \otimes (xy)^{i-l+1} \big) + dx(yx)^{k-1} \otimes x(yx)^{i-1} + dy\otimes (xy)^i + 1 \otimes (xy)^i + \delta_{i,1} \big( (xy)^{k-1} \otimes (yx)^{k-1} + dy(xy)^{k-1} \otimes (yx)^{k-1} +d^2 y(xy)^{k-1} \otimes y(xy)^{k-1} \big)$,
\item else $\Psi_3(b, y) = 0$.
\end{itemize}
Observe that if $0 \leq i \leq k-1$ then
$$\Psi_3 \big( b \otimes x(yx)^i \otimes y + x(yx)^i \otimes y \otimes b + y \otimes b \otimes x(yx)^i \big) = t_2\bigg( x(yx)^i \otimes t_1 \big( yC(b) \big) +$$
$$+yt_1\Big( b \cdot \sum \limits_{l=0}^{i} (xy)^l \otimes x \otimes (yx)^{i-l} + b \cdot \sum \limits_{l=0}^{i-1} x(yx)^l \otimes y \otimes x(yx)^{i-l-1} \Big) \bigg),$$
so in order to deal with second and third terms of the sum we only need to know values of obtained formula. Denote this by $\Psi_3(2,b,y)$.
\begin{itemize}
\item if $b = (xy)^j$ for $1 \leq j \leq k$, then
\begin{enumerate}
\item if $i>0$ then $\Psi_3(2,b,y) = \delta_{j,k} \Big( \sum \limits_{l=1}^{i} (xy)^l \otimes (xy)^{i-l+1} + \sum \limits_{l=1}^{i+1} x(yx)^{l-1} \otimes y(xy)^{i-l+1} \Big) + \mu_{i+j, k} \Big(y \otimes x(yx)^{i+j-k} + d x(yx)^{k-1} \otimes x(yx)^{i+j-k} + \delta_{i+j, k} \big( d(xy)^{k-1} \otimes x^2 \big) \Big)$,
\item if $i=0$ and $j=k$ then $ \Psi_3(2,b,y) = x \otimes y + y\otimes x + dx(yx)^{k-1} \otimes x + d(xy)^{k-1} \otimes x^2$.
\end{enumerate}
\item if $b=yx$, then $\Psi_3(2,b,y) = d(xy)^i \otimes y(xy)^{k-1} +\delta_{i,0} \Big( 1 \otimes (xy)^{k-1} + d y \otimes (xy)^{k-1} + d^2 x(yx)^{k-1} \otimes (xy)^{k-1}+ d\otimes x^2 + \sum \limits_{l=1}^{k-1} \big( (xy)^l \otimes (xy)^{k-l-1} + x(yx)^{l-1} \otimes y(xy)^{k-l-1} \big) \Big) + \mu_{i, 1}\big( dy(xy)^{k-1} \otimes (yx)^i+ (xy)^{k-1} \otimes (yx)^i + d^3 (xy)^k \otimes (yx)^{k-1+i}\big)$,
\item if $b = (yx)^j$ for $j \ge 2$ and $i=0$, then $\Psi_3(2,b,y) = dy(xy)^{k-1} \otimes (yx)^{j-1}$,
\item if $b = y(xy)^j$ for $0 \leq j \leq k-1$, then $\Psi_3(2,b,y) = \mu_{j,1} \delta_{i,0} \big( d y(xy)^{k-1} \otimes y(xy)^{j-1} +\delta_{j,1}d (yx)^{k-1} \otimes y^2 \big) + \mu_{i+j,k-1} \big(d \sum \limits_{l=1}^{k-1} (xy)^l \otimes x(yx)^{i+j-l} + d \sum \limits_{l=1}^{k} x(yx)^{l-1} \otimes (yx)^{i+j+1-l} \big) + \mu_{i+j,k} \mu_{i, 1} \big( x(yx)^{k-1} \otimes x(yx)^{i+j-k} + d \sum \limits_{l=1}^k x(yx)^{l-1} \otimes (yx)^{i+j+1-l} + d \sum \limits_{l=1}^{k-1} (xy)^{l} \otimes x(yx)^{i+j-l} + \delta_{i+j=k} (xy)^{k-1} \otimes x^2 \big)$,
\item if $b= x(yx)^{k-1}$, then \begin{enumerate}
\item if $i = 0$ then $\Psi_3(2,b,y) = x\otimes 1 + 1 \otimes x$,
\item if $i>0$ then $\Psi_3(2,b,y) = 1\otimes x(yx)^i + \sum \limits_{l=1}^{i} (xy)^{l} \otimes x(yx)^{i-l} + \sum \limits_{l=1}^{i+1} x(yx)^{l-1} \otimes (yx)^{i-l+1}$,
\end{enumerate}
\item else $\Psi_3(2, b, y) = 0$.
\end{itemize}
Finally, for $1 \leq i \leq k-1$
$$\Psi_3 \big( (xy)^i \otimes x \otimes b + x \otimes b \otimes (xy)^i + b \otimes (xy)^i \otimes x \big) = t_2 \bigg((xy)^i t_1 \big( x C(b) \big) + xt_1\Big( b C\big( (xy)^i \big) \Big) \bigg).$$
Denote this by $\Psi_3(3, b, y)$.
\begin{itemize}
\item if $b = (xy)^j$ for $1 \leq j \leq k$ and $i+j-1 \ge k$, then $\Psi_3(3, b, y) = y(xy)^{k-1} \otimes y(xy)^{i+j-k-1} + \delta_{i+j=k+1} (yx)^{k-1} \otimes y^2$,
\item if $b = y(xy)^j$ for $0 \leq j \leq k-1$ and $i+j \ge k$, then $\Psi_3(3, b, y) = x \otimes y(xy)^{i+j-k}$,
\item if $b = x$, then $\Psi_3(3, b, y) = \delta_{i,1} d y(xy)^{k-1} \otimes y(xy)^{k-2} + 1 \otimes y(xy)^{i-1} $,
\item else $\Psi_3(3, b, y) = 0$.
\end{itemize}
It remains to show that $q_1w_1 = (xy)^{k-1} = q_2w_2, q_1w_2 = y, q_1w_3 = x + dyx, q_2w_1 = x, q_2w_3 = y$. Now required formulas follows from direct computations.
\end{proof}
Now we need to understand how $\Delta$ works on elements of degree 4.
\begin{lemma}
$\Delta(e) = \Delta(p_3e) = \Delta(p_1 e)= 0$, $\Delta(p_1^ie) = p_1^{i-2}(q_1w_3+q_2w_1)$ for $2 \leq i \leq k-1$,
$\Delta(p_2e)= d^2 q_1w_2 + dq_2^3$, $\Delta(p_4 e) = (d +d^3p_1)q_1w_1$.
\end{lemma}
\begin{proof}
Firstly observe that for $a \in HH^4(R)$ following formula holds: $\Delta (a) (1\otimes 1)= \Delta(a\Psi_4) \Phi_3 (1\otimes 1) = $
$$\sum \limits_{b \ne 1} \langle (a \Psi_4) (b \cdot x \cdot x \cdot x + x \cdot b \cdot x \cdot x + x \cdot x \cdot b \cdot x + x \cdot x \cdot x \cdot b), 1 \rangle b^* + $$
$$\sum \limits_{b \ne 1} \sum \limits_{i=0}^{k-2} \langle (a \Psi_4) (x \cdot y(xy)^i \cdot x \cdot b + y(xy)^i \cdot x \cdot b \cdot x + x \cdot b \cdot x \cdot y(xy)^i + b \cdot x \cdot y(xy)^i \cdot x), 1 \rangle b^* y(xy)^{k-i-2} +$$
$$\sum \limits_{b \ne 1} \sum \limits_{i=1}^{k-1} \langle (a \Psi_4) (x \cdot (yx)^i \cdot y \cdot b + (yx)^i \cdot y \cdot b \cdot x + y \cdot b \cdot x \cdot (yx)^i + b \cdot x \cdot (yx)^i \cdot y), 1 \rangle b^* (xy)^{k-i-1} +$$
$$\sum \limits_{b \ne 1} \langle (a \Psi_4) (b \cdot y \cdot y \cdot y + y \cdot b \cdot y \cdot y + y \cdot y \cdot b \cdot y + y \cdot y \cdot y \cdot b), 1 \rangle b^*(1+dy+d^2 y(xy)^{k-1}) + $$
$$\sum \limits_{b \ne 1} \langle (a \Psi_4) (b \cdot y \cdot x(yx)^{k-1} \cdot y + y \cdot x(yx)^{k-1} \cdot y \cdot b + x(yx)^{k-1} \cdot y \cdot b \cdot y + y \cdot b \cdot y \cdot x(yx)^{k-1}), 1 \rangle b^*(d + dy + d^3 x(yx)^{k-1}) + $$
$$\sum \limits_{b \ne 1} \sum \limits_{i=0}^{k-2} \langle (a \Psi_4) (y \cdot x(yx)^i \cdot y \cdot b + x(yx)^i \cdot y \cdot b \cdot y + y \cdot b \cdot y \cdot x(yx)^i + b \cdot y \cdot x(yx)^i \cdot y), 1 \rangle b^* x(yx)^{k-i-2} +$$
$$\sum \limits_{b \ne 1} \sum \limits_{i=1}^{k-1} \langle (a \Psi_4) (y \cdot (xy)^i \cdot x \cdot b + (xy)^i \cdot x \cdot b \cdot y + x \cdot b \cdot y \cdot (xy)^i + b \cdot y \cdot (xy)^i \cdot x), 1 \rangle b^* (yx)^{k-i-1}.$$
Secondly, $t_3$ is not equal to zero only on $(xy)^k \otimes 1$. Denote by $\Psi_4(i,b)$ result in $i$-th sum on element $b \in B$. Now for the first sum
\begin{enumerate}
\item[1.1)] if $b = xy$, then $\Psi_4(1,b) = d \otimes y(xy)^{k-2}$,
\item[1.2)] if $b = xyx$, then $\Psi_4(1,b) = d \otimes (yx)^{k-1} + d \otimes (xy)^{k-1}$,
\item[1.3)] if $b = (xy)^k$, then $\Psi_4(1,b) = 1 \otimes 1 $,
\item[1.4)] else $\Psi_4(1,b) = 0$.
\end{enumerate}
For the third sum and $1 \leq i \leq k-1$:
\begin{enumerate}
\item[3.1)] if $b = yx$, then $\Psi_4(3,b) = d \otimes y(xy)^{i-1}$,
\item[3.2)] else $\Psi_4(3,b) = 0$.
\end{enumerate}
For the fourth sum
\begin{enumerate}
\item[4.1)] Если $b = x(yx)^{k-1}$, то $\Psi_4(4,b) = d \otimes 1 $,
\item[4.2)] Если $b = (xy)^k$, то $\Psi_4(4,b) = 1 \otimes 1 $, \
\item[4.3)] Иначе $\Psi_4(4,b) = 0$.
\end{enumerate}
For the fifth sum
\begin{enumerate}
\item[5.1)] if $b = (xy)^i$, then $1 \leq i \leq k-1$: $\Psi_4(5,b) = 1 \otimes (xy)^{i-1}$,
\item[5.2)] if $b = y$, then $\Psi_4(5,b) = d\otimes 1 $,
\item[5.3)] if $b = (xy)^k$, then $1 \otimes y $,
\item[5.4)] else $\Psi_4(5,b) = 0$.
\end{enumerate}
It is easy to see that other sums gives us zero, so $\Psi_4(i,b)=0$ for any $b\in B$ and any $i \in \{2, 6, 7\}$.
Now one can deduce by this computations that $\Delta(e) = \Delta(p_3e) = 0$, $\Delta(p_2e)= d(1+dy)$, $\Delta(p_4 e) = d (xy)^{k-1}+d^3 (xy)^k$ and
$$\Delta(p_1^ie) = \begin{cases}
0, & i=1\\
d(xy)^{i-1}, & 2 \leq i \leq k-1
\end{cases},$$
which yields this lemma.
\end{proof}
\subsection{Higher degree elements}
For $a \in HH^n(R)$ and $b \in HH^m(R)$ if we know $\Delta(a)$ and $\Delta(b)$, we don't need to compute $\Delta(ab)$ directly: it is sufficient to know how Gerstenhaber bracket acting on them. If $a$ represented by cocycle $f:P_n \longrightarrow A$ and $b$ represented by $g:P_m \longrightarrow A$, one can use the following formula:
$$[a,b] = [f\circ \Psi_n, g \circ \Psi_m] \circ \Phi_{n+m-1}.$$
\begin{remark}
Observe that $\Phi_4$ can be visualised directly:
$$\Phi_4 (1 \otimes 1) = \sum \limits_{b \in B} 1\otimes b \otimes x \otimes x \otimes x \otimes b^* + \sum \limits_{b \in B} \sum \limits_{i=0}^{k-2} 1\otimes b \otimes x \otimes y(xy)^i \otimes x \otimes y(xy)^{k-i-2}b^* + $$
$$ \sum \limits_{b \in B} \sum \limits_{i=1}^{k-1} 1\otimes b \otimes x \otimes (yx)^i \otimes y \otimes (xy)^{k-i-1}b^* + + \sum \limits_{b \in B} 1\otimes b \otimes y \otimes y \otimes y \otimes (1+dy+d^2x(yx)^{k-1})b^* +$$
$$ \sum \limits_{b \in B} \sum \limits_{i=0}^{k-2} 1\otimes b \otimes y \otimes x(yx)^i \otimes y \otimes x(yx)^{k-i-2}b^* +\sum \limits_{b \in B} \sum \limits_{i=1}^{k-1} 1\otimes b \otimes y \otimes (xy)^i \otimes x \otimes (yx)^{k-i-1}b^* +$$
$$ \sum \limits_{b \in B} 1\otimes b \otimes y \otimes x(yx)^{k-1} \otimes y \otimes (d+d^2y+d^3x(yx)^{k-1})b^* + d \otimes x(yx)^{k-1} \otimes x \otimes x \otimes x \otimes x(yx)^{k-1} +$$
$$ + d \otimes x(yx)^{k-1} \otimes x \otimes y(xy)^{k-2} \otimes x \otimes (xy)^{k} + d \otimes x(yx)^{k-1} \otimes x \otimes (yx)^{k-1} \otimes y \otimes x(yx)^{k-1} +$$
$$ +d \otimes x(yx)^{k-1} \otimes y \otimes y \otimes y \otimes y^2 + d^2 \otimes x(yx)^{k-1} \otimes y \otimes x(yx)^{k-1} \otimes y \otimes y^2 +$$
$$ + d\otimes x(yx)^{k-1} \otimes y \otimes (xy)^{k-1} \otimes x \otimes x(yx)^{k-1}.$$
\end{remark}
\begin{lemma}
$\Delta(q_1e) = \Delta(q_2e) = 0$.
\end{lemma}
\begin{proof}
Fix $a \in HH^1(R)$. By definitions,
$$[a,e] (1 \otimes 1) = [a \circ \Psi_1, e \circ \Psi_4] \circ \Phi_4 (1\otimes 1) = \big( (a \Psi_1) \circ (e \Psi_4) \big) \Phi_4(1 \otimes 1) + \big( (e \Psi_4) \circ (a \Psi_1) \big) \Phi_4(1 \otimes 1).$$
What is $\Psi_4\Phi_4$? Using proofs of lemmas above, one can deduce that
$$\Psi_3 \Phi_3 (1\otimes 1) = \Psi_3(x \cdot x \cdot x) = 1 \otimes 1,$$
so
$$\Psi_4\Phi_4 (1 \otimes 1) = \sum \limits_{b \ne 1} t_3 \big( b \Psi_3\Phi_3 (1\otimes 1) b^* \big) +dt_3 \big( x(yx)^{k-1} \Psi_3 \Phi_3 (1\otimes 1) x(yx)^{k-1} \big) = 1.$$
Finally, for $a = q_1$ or $a = q_2$:
$$\big( (a \circ \Psi_1) \circ (e \circ \Psi_4) \big) \circ \Phi_4(1 \otimes 1) = (a \circ \Psi_1)(1) = 0.$$
Now consider $F^u = (e \circ \Psi_4) \circ (u \circ \Psi_1) = \sum \limits_{i=1}^4 F^u_i$, where $F^u_i = (e \circ \Psi_4) \circ_i (u \circ \Psi_1)$ by definition of Gerstenhaber bracket. In order to compute this we need to know $u \Psi_1(b)$ for $u = q_1$ or $q_2$. By direct computations
$$q_1 \Psi_1 (b) = \begin{cases}
y(xy)^{k-2}, & b =x\\
1+dy, & b=y\\
di x(yx)^i + \delta_{i,1} \cdot y(xy)^{k-1}, & b = x(yx)^i, \ 1 \leq i \leq k-1\\
(xy)^i + (yx)^i + d(i+1)y(xy)^i, & b =y(xy)^i, \ 1 \leq i \leq k-1\\
di (xy)^i + x(yx)^{i-1}, & b =(xy)^i, \ 1 \leq i \leq k \\
di(yx)^i+x(yx)^{i-1}, & b = (yx)^i, \ 1 \leq i \leq k-1
\end{cases}$$
$$q_2 \Psi_1 (b) = \begin{cases}
1, & b =x\\
x(yx)^{k-2}+d(xy)^{k-1}, & b=y\\
(xy)^i+(yx)^i, & b = x(yx)^i, \ 1 \leq i \leq k-1\\
x(yx)^{k-1}, & b =yxy\\
y(xy)^{i-1} , & b =(xy)^i, \ 1 \leq i \leq k \\
y(xy)^{i-1} + \delta_{i,1} d x(yx)^{k-1}, & b = (yx)^i, \ 1 \leq i \leq k-1\\
0, & \text{ else}
\end{cases}$$
Since $e\Psi_4 (1 \otimes b \otimes a_1 \otimes a_2 \otimes a_3 \otimes 1) = et_3 \Big(b t_2\Big( a_1 t_1 \big( a_2C(a_3) \big) \Big) \Big)$, we only need to calculate $F^u_1$ and $F^u_2$ on elements $b \otimes a_1 \otimes a_2 \otimes a_3$, such that $a_2a_3 \not \in B$. \par
1) Firstly consider $F_1^u$ for $u \in HH^1(R)$. It is easy to see that $t_2 \big(x t_1(x \otimes x\otimes 1) \big) = 1 \otimes 1 $, so
$$ et_3\Big( q_i\Psi_1 (b) t_2\big( x t_1(x \otimes x\otimes 1) \big) \Big) b^* = et_3(q_1\Psi_1 (b) \otimes 1)b^* = 0$$
for any $b \in B$ and any $i \in \{1, 2\}$. Hence all summands in $\Phi_4 (1\otimes 1)$, consisting $x \otimes x \otimes x$, gives us zero. It remains to check that $t_2 (y t_1 (y \otimes y \otimes 1)) = 0$, so
$$F^{q_1}_1 = F^{q_2}_1 = 0.$$ \par
2) For $F_2^u$ (where $u \in HH^1(R)$) one need to notice that
$$t_3(b t_2(u\Psi(x) t_1(x \otimes x\otimes 1))) = t_3(b t_2(u\Psi(x) \otimes r_x \otimes 1)) = 0,$$
for any $b \in B$ and $ u \in \{q_1, q_2\}$, so first and eight sums gives us zero. Secondly, $u\Psi_1 t_1 (y \otimes y \otimes 1) = u\Psi_1 \otimes r_y \otimes 1$, so
$$et_3 \Big(b t_2 \big( u\Psi_1 t_1 (y \otimes y \otimes 1) \big) \Big) (1 +dy +d^2x(yx)^{k-1}) =
\begin{cases}
dx(yx)^{k-2}, & u = q_2, b = yxy \\
0, & \text{ else }
\end{cases}$$
Obviously, another sums from $\Phi_4 (1\otimes 1)$ also coming up with zeros, so
$$F^{q_1}_2 = 0 \text{ and } F^{q_2}_2 = dx(yx)^{k-2}.$$\par
3) In order to compute $F^u_3$ notice that $\big( (e \circ \Psi_4) \circ_3 (u \circ \Psi_1) \big) (1 \otimes b \otimes a_1 \otimes a_2 \otimes a_3 \otimes 1) = et_3 \Big(b t_2 \big(a_1 t_1 \big( u \Psi_1 (a_2) C(a_3) \big) \big) \Big)$. Then
$$t_2 \big( x t_1 ( q_1 \Psi_1 (x) \otimes x \otimes 1 ) \big) = t_2 \big(x t_1 (y(xy)^{k-2} \otimes x \otimes 1) \big) = 0,$$
$$t_2 \big( x t_1 ( q_2 \Psi_1 (x) \otimes x \otimes 1 ) \big) = t_2 \big( q_2 \Psi_1 (x) t_1 (x \otimes x \otimes 1) \big), $$
and all other sums from $\Phi_4 (1\otimes 1)$ gives zero, what can easily be verified from definition of $t_1$ and $t_2$, therefore in this case we can use computations of the previous case. So,
$$F^{q_1}_3 = F^{q_2}_3 = 0.$$ \par
4) Finally, we want to describe $F^u_4$. It is not hard to verify that
$$\big( (e \circ \Psi_4) \circ_4 (u \circ \Psi_1) \big) (1 \otimes b \otimes a_1 \otimes a_2 \otimes a_3 \otimes 1) = et_3 \bigg(b t_2 \Big( a_1 t_1 \big(a_2 C( u \Psi_1 (a_3)) \big) \Big) \bigg).$$
Also
$$C(u\Psi_1(x)) = \begin{cases}
\sum \limits_{l=0}^{k-2} (yx)^l \otimes y \otimes (xy)^{k-2-l} + \sum \limits_{l=0}^{k-3} y(xy)^l \otimes x \otimes y(xy)^{k-3-l} & u =q_1\\
0, & u=q_2
\end{cases}
$$
$$C(u\Psi_1(y)) = \begin{cases}
d \otimes y \otimes 1, & u=q_1\\
\Big( \sum \limits_{l=0}^{k-2} (xy)^l \otimes x \otimes (yx)^{k-2-l} + \sum \limits_{l=0}^{k-3} x(yx)^l \otimes y \otimes x(yx)^{k-3-l} \Big)(1+dy)+ & u=q_2\\
+ dx(yx)^{k-2} \otimes y \otimes 1, & \\
\end{cases}
$$
So
$$t_2\bigg(xt_1 \Big(x C\big( q_1 \Psi_1(x) \big) \Big) \bigg) = t_2 \bigg(x t_1 \Big(\sum \limits_{l=0}^{k-2} x(yx)^l \otimes y \otimes (xy)^{k-2-l} + \sum \limits_{l=0}^{k-3} (xy)^{l+1} \otimes x \otimes y(xy)^{k-3-l} \Big) \bigg) = 0,$$
and
$$t_2 \bigg(x t_1 \Big(y(xy)^i C \big(q_1\Psi_1 (x) \big) \Big) \bigg) = \delta_{i,0} t_2\big(x \otimes r_y \otimes (xy)^{k-2} + dy(xy)^{k-1} \otimes r_x \otimes (xy)^{k-2} \big).$$
Further observe that
$$t_2 \big(x t_1 (q_2 \Psi_1 (x) \otimes x \otimes 1) \big) = t_2 \big(q_2 \Psi_1 (x) t_1 (x \otimes x \otimes 1) \big),$$
and by definition of $t_1$ and $t_2$ one can deduce that all other sums from $\Phi_4(1 \otimes 1)$ gives us zero, so this case also can be reduced to previous ones. So
$$F^{q_1}_4 = 0 \text{ and } F^{q_2}_4 = dx(yx)^{k-2}.$$
Now we only need to compile our previous results and notice that $q_i\Delta(e) = 0$ for $i \in \{1,2\}$:
$$\Delta(q_1e) (1\otimes 1 ) = [q_1,e](1\otimes 1) = \sum \limits_{i=1}^4 F^{q_1}_i = 0,$$
$$\Delta(q_2e) (1\otimes 1 ) =[q_2,e](1\otimes 1) = \sum \limits_{i=1}^4 F^{q_2}_i = dx(yx)^{k-2} + dx(yx)^{k-2} = 0.$$
\end{proof}
\begin{remark}
For elements of degree six we should understand, how $\Phi_5$ acts:
$$\Phi_5 (1 \otimes a \otimes 1) = \sum_b 1 \otimes a \otimes b \otimes x \otimes x \otimes x \otimes b^* + \sum_b \sum_{i=0}^{k-2} 1 \otimes a \otimes b \otimes x \otimes y(xy)^i \otimes x \otimes y(xy)^{k-2-i} b^* $$
$$\sum_b \sum_{i=1}^{k-1} 1 \otimes a \otimes b \otimes x \otimes (yx)^i \otimes y \otimes (xy)^{k-1-i} b^* + \sum_b 1 \otimes a \otimes b \otimes y \otimes y \otimes y \otimes (1 + dy + d^2 x(yx)^{k-1}) b^* +$$
$$\sum_b \sum_{i=0}^{k-2} 1 \otimes a \otimes b \otimes y \otimes x(yx)^i \otimes y \otimes x(yx)^{k-2-i} b^* + \sum_b \sum_{i=1}^{k-1} 1 \otimes a \otimes b \otimes y \otimes (xy)^i \otimes x \otimes (yx)^{k-1-i} b^* +$$
$$ \sum_b 1 \otimes a \otimes b \otimes y \otimes x(yx)^{k-1} \otimes y \otimes (d + d^2y + d^3x(yx)^{k-1})b^* + d \otimes a \otimes x(yx)^{k-1} \otimes x \otimes x \otimes x \otimes x(yx)^{k-1} +$$
$$+ d \otimes a \otimes x(yx)^{k-1} \otimes x \otimes y(xy)^{k-2} \otimes x \otimes (xy)^k + d \otimes a \otimes x(yx)^{k-1} \otimes x \otimes (yx)^{k-1} \otimes y \otimes x(yx)^{k-1} +$$
$$+ d \otimes a \otimes x(yx)^{k-1} \otimes y \otimes y \otimes y \otimes y^2 + d^2 \otimes a \otimes x(yx)^{k-1} \otimes y \otimes x(yx)^{k-1} \otimes y \otimes y^2 +$$
$$+ d \otimes a \otimes x(yx)^{k-1} \otimes y \otimes (xy)^{k-1} \otimes x \otimes x(yx)^{k-1}.$$
\end{remark}
\begin{lemma}
For all generating elements $v \in HH^2(R)$ and $e \in HH^4(R)$ bracket $[v,e]$ equals to zero.
\end{lemma}
\begin{proof}
For $v \in HH^2(R)$ and $e \in HH^4(R)$
$$[v,e] (1 \otimes a \otimes 1) = \big( (v \Psi_2) \circ (e\Psi_4) \big) \Phi_5 (1 \otimes a \otimes 1) + \big( (e\Psi_4)\circ (v \Psi_2) \big) \Phi_5 (1 \otimes a \otimes 1).$$
Now we need to show that first summand here is equal to zero. It is obvious that $\big( (v \Psi_2) \circ (e\Psi_4) \big) \Phi_5 = \big( (v \Psi_2) \circ_1 (e\Psi_4) \big) \Phi_5 + \big( (v \Psi_2) \circ_2 (e\Psi_4) \big) \Phi_5$. Let us denote these summands by $S_1^v$ and $S_2^v$ respectively. One can quickly show that $S_2$ is equal to zero: indeed,
$$S_2 (a_1 \otimes .. \otimes a_5) = v t_1 (a_1 C(et_3 (a_2 t_2 (a_3 t_1( a_4 C(a_5) ))))),$$
and since $C(1) = 0$ and $et_3(b \otimes 1) = \begin{cases}1, & b = (xy)^k \\ 0,& \text{ else }
\end{cases}$, required equality holds.\par
One can proof that $S_1$ equals to zero on all summands from $\Phi_5 (1\otimes a \otimes 1)$ except fourth and seventh. Finally,
$$S_1^{v}\big(1 \otimes a \otimes (xy)^k \otimes y \otimes x(yx)^{k-1} \otimes y \otimes (d + d^2 y + d^3x(yx)^{k-1})\big) = \begin{cases} dy + d^2 x(yx)^{k-1}, & a=y, \ v = w_2 \\ dx, &a=y, \ v = w_3 \\ 0, & \text{else} \end{cases},$$
$$S_1^{v}\big(1 \otimes a \otimes (xy)^k \otimes y \otimes y \otimes y \otimes (1 + d y + d^2 x(yx)^{k-1}) \big) = \begin{cases} dy + d^2 x(yx)^{k-1}, &a=y, \ v = w_2 \\ dx, & a=y, \ v = w_3 \\ 0, & \text{else} \end{cases},$$
and for other combinations of these sums $S_1$ is equal to zero. So $S_1 (a_1 \otimes .. \otimes a_5) = 0$ for any $a_1,..,a_5 \in B$, and hence $\big( (v \Psi_2) \circ (e\Psi_4) \big) \Phi_5 = 0$. \par
It remains to consider $\big( (e\Psi_4)\circ (v \Psi_2) \big) \Phi_5 = \sum \limits_{i=1}^4 \big( (e\Psi_4)\circ_i (v \Psi_2) \big) \Phi_5$. Denote $\big( (e\Psi_4)\circ_i (v \Psi_2) \big) \Phi_5$ by $F_i^v$. It is easy to see that $F_i^v (a_1 \otimes .. \otimes a_5)$ equals to zero for any $i \in \{1, 2, 4\}$ on any summand of $\Phi_5$, except maybe first, fourth, eighth and eleventh summands, because otherwise $t_1 (a_4 C(a_5)) = 0$. \par
1) Obviously, $t_2 \big( y t_1 (y \otimes y \otimes 1) \big) = 0$, so for $F_1^v$ it remains to investigate only first and eighth summands. Computation shows that
$$
F_1^v(1\otimes a \otimes b \otimes x \otimes x \otimes x \otimes b^*) =
$$
$$= \begin{cases}
et_3 \Big(vt_1 \big(x \otimes x \otimes y(xy)^{i-1} + y(xy)^{k-1} \otimes y \otimes (xy)^{i-1} \big) \otimes 1 \Big) (xy)^{k-i}, & b=(xy)^i \text{ and } a = x\\
et_3 \Big(v t_1 \big(y \otimes y \otimes x(yx)^{i-1} + y^2 \otimes x \otimes (yx)^{i-1} \big) \otimes 1 \Big) (yx)^{k-i}, & b = (yx)^i \text{ and } a = y\\
et_3 \big((yx)^{k} \otimes 1 \big) y(xy)^{k-2}, & v = w_2, \text{ } b = xyx \text{ and } a = x\\
et_3 \big((xy)^{k} \otimes 1 \big) x(yx)^{k-2}, & v = w_1, \text{ } b = yxy \text{ and } a=y\\
et_3 \big((xy)^k \otimes 1 \big), & v = w_2, \text{ } b = (xy)^k \text{ and } a = y\\
0, & \text{else}
\end{cases}$$
$$= \begin{cases}
1, & v=w_1, \text{ } b=(xy)^k \text{ and } a = x\\
d(xy)^{k-1}, & v=w_3, \text{ } b = xy \text{ and } a = x\\
d(yx)^{k-1}, & v=w_1, \text{ } b = yx \text{ and } a = y\\
y(xy)^{k-2}, & v = w_2, \text{ } b = xyx \text{ and } a = x\\
x(yx)^{k-2}, & v = w_1, \text{ } b = yxy \text{ and } a=y\\
1, & v = w_2, \text{ } b = (xy)^k \text{ and } a = y\\
0, & \text{else}
\end{cases}$$
So
$$F_1^{w_1} = \begin{cases} 1, & a =x \\ x(yx)^{k-2} + d(yx)^{k-1}, & a = y \end{cases},\quad F_1^{w_2} = \begin{cases} y(xy)^{k-2}, & a = x \\ 1, & a = y \end{cases}, $$
$$ F_1^{w_3} = \begin{cases} d(xy)^{k-1}, & a = x \\ 0, & a = y \end{cases}.$$
2) For $F_2^v$ we need to examine only summands number one, four, eight and eleven from $\Phi_5$. So for the first sum
$$F_2^{v}(1 \otimes a \otimes b \otimes x \otimes x \otimes x \otimes b^*) = $$
$$
=\begin{cases}
et_3 \big(at_2 (x^3 \otimes r_x \otimes 1) \big), &v = w_1 \text{ and } b = (xy)^k\\
et_3\big(at_2(xyx \otimes r_x \otimes 1) \big), & v = w_3 \text{ and } b = (xy)^k\\
et_3 \big(at_2((xy)^{k} \ r_x \otimes 1)\big) y(xy)^{k-2}, & v = w_2 \text{ and } b = xyx \\
et_3\big(ay\otimes x + dax(yx)^{k-1}\otimes x + day(xy)^{k-1} \otimes (yx)^{k-1} \big), & v = w_3 \text{ and } b = y(xy)^{k-1}\\
et_3\big(at_2 (y^2 \otimes r_x \otimes 1) \big)(yx)^{k-1}, & v = w_3 \text{ and } b = yx \\
0, & \text{else}
\end{cases}
$$
$$= \begin{cases}
1, &v = w_1, \text{ } b = (xy)^k \text{ and } a= x \\
d(yx)^{k-1}, & v = w_3, \text{ } b = (xy)^k \text{ and } a= x \\
y(xy)^{k-2}, & v = w_2, \text{ } b = xyx \text{ and } a= x\\
d (yx)^{k-1}, & v = w_3, \text{ } b = y(xy)^{k-1} \text{ and } a= x\\
d (yx)^{k-1}, & v = w_3, \text{ } b = yx \text{ and } a= x\\
0, & \text{else}
\end{cases}$$
and for the forth sum
$$F_2^v \big(1\otimes a \otimes b \otimes y \otimes y \otimes y \otimes (1+dy+d^2x(yx)^{k-1})b^* \big) = $$
$$=\begin{cases} et_3\big( at_2 ((xy)^k \otimes r_y \otimes 1)\big)(x(yx)^{k-2}+ d(yx)^{k-1}), & v=w_1, \text{ } a=y \text{ and } b = yxy \\
et_3 \big(at_2((xy)^k \otimes r_y \otimes 1) \big) (1+dy+d^2x(yx)^{k-1}), & v=w_2, \text{ } a=y \text{ and } b = (xy)^k\\
0, & \text{ else }
\end{cases} $$
$$= \begin{cases}x(yx)^{k-2}, & v=w_1, \text{ } a=y \text{ and } b = yxy \\
1, & v=w_2, \text{ } a=y \text{ and } b = (xy)^k\\
0, & \text{ else }
\end{cases}$$
So eighth and eleventh sums gives us zero and hence
$$F_2^{w_1} = \begin{cases} 1, & a =x \\ x(yx)^{k-2}, & a = y \end{cases},\quad F_2^{w_2} = \begin{cases} y(xy)^{k-2}, & a = x \\ 1, & a = y \end{cases}, \quad F_2^{w_3} = \begin{cases} d(yx)^{k-1}, & a = x \\ 0, & a = y \end{cases}.$$
3) In the case of $F_3^v$ all sums of the form $\sum \limits_i 1\otimes a_1 \otimes .. \otimes a_5 \otimes a$ gives us zero, if $t_1 \big(a_3 C(a_4) \big)=0$. Also if $(a_3,a_4,a_5) = (x,x,x)$, then $t_1 \big(vt_1(x \otimes x \otimes 1) \otimes x \otimes 1 \big) = \begin{cases} 1\otimes r_x \otimes 1, & v = w_1 \\ 0, & v \not= w_1 \end{cases}$, and
$$F_3^{w_1} (1 \otimes x \otimes (xy)^k \otimes x \otimes x \otimes x \otimes 1) = et_3 \big((xy)^k \otimes 1 \big) = 1.$$
There is a zero in all other cases. For the forth sum $F_3^{w_1}=0$, and if $v =w_2$ then
$$
F_3^{v} (1 \otimes a \otimes b \otimes y \otimes y \otimes y \otimes 1) =
$$
$$=\begin{cases}
et_3 \big((xy)^k\otimes 1 \big) (1 + dy), & v = w_2, \text{ } b = (xy)^k \text{ and } a = y \\
et_3 \big(a t_2 ((yx)^k \otimes r_y \otimes 1)\big), & v = w_3, \text{ } b = y(xy)^{k-1} \\
et_3 \big(d(xy)^k \otimes (yx)^{k-1} + d^2 (xy)^k \otimes y(xy)^{k-1} \big), & v = w_3, \text{ } b = (xy)^k \text{ and } a = x
\end{cases}
$$
so if $b=(xy)^k$ for $v=w_2$ or if $b=y(xy)^{k-1}$ for $v=w_3$ this sum equals to $ (1 + dy) (1+dy+d^2x(yx)^{k-1}) = 1$ for $a=y$. If $b = (xy)^k$ for $v=w_3$ this sum equals to $ d(yx)^{k-1}$ for $a=x$.\\
On the seventh sum $F_3^{v} = 0$, if $v \not=w_3$, and in case of $v=w_3$
$$F_3^{w_3} \big(1 \otimes a \otimes b \otimes y \otimes x(yx)^{k-1} \otimes y \otimes (d+d^2y +d^3 x(yx)^{k-1}) \big) = $$
$$=\sum_b et_3\Big( at_2 \big( bt_1(xy \otimes y \otimes 1 + yx \otimes y \otimes 1 + 2dyxy \otimes y \otimes 1) \big) \Big)(d+d^2y +d^3 x(yx)^{k-1}) = $$
$$=F_3^{w_3} \big(1 \otimes a \otimes b \otimes y \otimes y \otimes y \otimes (1+dy +d^2 x(yx)^{k-1})\big),$$
so we don't need to compute it, since these two summands kills each other. All other combinations gives us zero, so eighth and eleventh sums also gives us zero. So \\
$$F_1^{w_1} = \begin{cases} a =x: & 1\\ a =y: & 0 \end{cases},\quad F_1^{w_2} = \begin{cases} a = x: & 0\\ a = y: & 1 \end{cases}, \quad F_1^{w_3} = \begin{cases} a = x: & d(yx)^{k-1} + d(yx)^{k-1}\\ a = y: & x +x \end{cases} = 0.$$
4) Finally we need to know evaluation of $F_4^v$ in first, forth, eighth and eleventh summands from $\Phi_5$. It is obvious that for the first summand $F_4^v = 0$ if $v \not= w_1$. But
$$F_4^{w_1} (1\otimes a \otimes b \otimes x \otimes x \otimes x \otimes 1)=et_3 (a t_2 (b \otimes r_x \otimes 1)) = F_3^{w_1} (1\otimes a \otimes b \otimes x \otimes x \otimes x \otimes 1),$$
so first summand gives us $\delta_{a,x} 1$,and $F_4^{w_1}$ is equal to zero for eighth sum. Further
$$F_4^{w_2} (1\otimes a \otimes b \otimes y \otimes y \otimes y \otimes 1) = et_3 (a t_2 (b \otimes r_y \otimes 1)) = F_3^{w_2} (1\otimes a \otimes b \otimes y \otimes y \otimes y \otimes 1),$$
so $F_4^{w_2}$ gives $\delta_{a,y}1$ on the forth sum and $F_4^{v}$ gives zero $v \not= w_2$ for forth and eighth sums. So
$$F_1^{w_1} = \begin{cases} a =x: & 1\\ a =y: & 0 \end{cases},\quad F_1^{w_2} = \begin{cases} a = x: & 0\\ a = y: & 1 \end{cases}, \quad F_1^{w_3} = 0.$$
It remains to compute $[v,e] =\sum \limits_{i=1}^2 S_i^v + \sum \limits_{i=1}^4 F_i^v$, hence required formulas holds.
\end{proof}
\begin{corollary}
$\Delta(ve)=0$ for any $v \in \{w_1,w_2,w_3\}$.
\end{corollary}
\begin{proof}
$\Delta(v)=0$ for any $v \in \{w_1,w_2,w_3\}$ by Lemma 4, and $\Delta(e)=0$ by Lemma 5. So by Tradler's equation
$$\Delta(ve) = \Delta(v) e + v \Delta(e) + [v,e] = 0 \cdot e + v \cdot 0 + 0 = 0$$
$v \in \{w_1,w_2,w_3\}$.
\end{proof}
\begin{theorem}
Let $R=R(k,0,d)$ over a field $K$ of characteristic 2, and let $\Delta$ be $BV$-differential from Theorem 1. Then
\begin{enumerate}
\item $\Delta$ is equal to 0 on the generators of $HH^*(R)$;
\item $\Delta$ satisfies the equalities
\begin{itemize}
\item of degree 1: $\begin{cases} \Delta(p_1q_1) = dp_1, \ \Delta(p_1q_2) = \Delta(p_2q_1) = \Delta(p_4q_1) = dp_2,\\
\Delta(p_3q_1) = \Delta(p_2q_2) = p^{k-1}_1, \ \Delta(p_4q_2) = p_3;
\end{cases}$
\item of degree 3: $\begin{cases} \Delta(q_1w_1) = \Delta(q_2w_2) = p_1^{k-2} w_3, \ \Delta(q_1w_2) = (1+d^3(l+1)p_4) q_1^2 + dw_2 ,\\
\Delta(q_2w_1) = q_2^2, \ \Delta(q_1w_3) + \Delta(q_2w_1) = w_3, \ \Delta(q_2w_3)+\Delta(q_1w_2) = w_3;
\end{cases}$
\item of degree 4: $\begin{cases} \Delta(p_1^i e) = p_1^{i-2} (q_1w_3+q_2w_1), \ 2 \leq i \leq k-1,\\
\Delta(p_2e) = d^3q_1w_2 + dq_2^3, \ \Delta(p_4e) = (d+d^2p_1) q_1w_1.
\end{cases}$
\end{itemize}
\item $\Delta(ab) = 0$ on all other combinations of generators $a, b \in HH^*(R)$.
\end{enumerate}
\end{theorem}
|
1,116,691,499,336 | arxiv | \section{Introduction}
Soft gamma-ray repeaters (SGRs, for a recent review see
\cite{woodsrew}) are a small group of peculiar high-energy sources
generally interpreted as ``magnetars'', i.e. strongly magnetised
($B\sim$10$^{15}$ G), slowly rotating ($P\sim$ 5-8 s) neutron
stars powered by the decay of the magnetic energy, rather than by
rotation (\cite{dt92}, \cite{pac92}, \cite{td95}).
They were discovered through
the detection of recurrent short ($\sim$0.1 s) bursts of
high-energy radiation in the tens to hundreds of keV range, with
peak luminosity up to 10$^{39}$-10$^{42}$ erg s$^{-1}$, above the
Eddington limit for neutron stars. The rate of burst emission in
SGRs is highly variable. Bursts are generally emitted during
sporadic periods of activity, lasting days to months, followed by
long ``quiescent'' time intervals (up to years or decades) during
which no bursts are emitted. Occasionally SGRs emit also ``giant
flares'', that last up to a few hundred seconds
and have peak luminosity up to 10$^{46}$-10$^{47}$ erg s$^{-1}$.
Only three giant flares have been observed to date, each one from
a different source (see, e.g., \cite{mazets}, \cite{hurley1999},
\cite{swiftgiant}).
Persistent (i.e. non-bursting) emission is also observed from SGRs
in the soft X--ray range ($<$10 keV), with typical luminosity of
$\sim$10$^{35}$ erg s$^{-1}$, and, in three cases, periodic
pulsations at a few seconds have been detected. Such pulsations
proved the neutron star nature of SGRs and allowed to infer
spin-down at rates of $\sim$10$^{-10}$ s s$^{-1}$, consistent with
dipole radiation losses for magnetic fields of the order of
B$\sim$10$^{14}$-10$^{15}$ G. The X--ray spectra are generally
described with absorbed power laws, but in some cases strong
evidence has been found for the presence of an additional
blackbody-like component with typical temperature of $\sim$0.5 keV
(\cite{xmm}).
The only SGR for which persistent (i.e. not due to bursts and/or
flares) emission above 20 keV has been reported to date is SGR
1806--20 (\cite{mereghetti05,molkov}).
Here we report the discovery, based on observations with the
{\it INTEGRAL}~ satellite (\cite{integral}), of persistent hard X-ray
emission from SGR~1900$+$14~ in the 20-100 keV range.
\section{Observations and data analysis}
We analysed data obtained with ISGRI (\cite{isgri}), the
low-energy detector of the IBIS (\cite{ibis}) coded mask
telescope. IBIS/ISGRI is an imaging instrument covering a wide
field of view (29$^{\circ}\times$29$^{\circ}$ at zero sensitivity,
9$^{\circ}\times$9$^{\circ}$ at full sensitivity) with
unprecedented sensitivity and angular resolution
($\sim$12$^{\prime}$) in the hard X/soft $\gamma$-ray energy range
(15 keV-1 MeV). These excellent imaging performances are
essential, especially in crowded Galactic fields, to avoid source
confusion, which affected most previous experiments operating in
this energy range.
From the {\it INTEGRAL}~ public data archive we selected all the
observations pointed within 10$^{\circ}$ from the position of
SGR~1900$+$14. The resulting data set consists of 1033 pointings, yielding
a total exposure time of $\sim$2.5 Ms. The observation period,
during which the source was observed discontinuously, spans from
March 6$^{th}$ 2003 to June 8$^{th}$ 2004. We used version 5.1 of
the Offline Scientific Analysis (OSA) Software provided by the
{\it INTEGRAL}~ Science Data Centre (\cite{isdc}).
After standard data processing (dead time correction, good
time-interval selection, gain correction, energy reconstruction),
we produced the sky images of each pointing in the 18--60 keV
range. These individual images were summed to produce a total
image, a portion of which is shown in Fig.~\ref{img}. A source
with count rate 0.18$\pm$0.02 counts s$^{-1}$ is detected with a
significance of 9$\sigma$ at coordinates (J2000) R.A. = 19$^{h}$
07$^{m}$ 25$^{s}$, Dec. = +09$^{\circ}$ 18$'$ 34$''$. The
associated error circle, with a radius of $\sim$3$^{\prime}$
(\cite{gros}) contains the position of SGR~1900$+$14~ (\cite{frail99}). No
other catalogued X-ray sources are present in the error circle.
We therefore associate the detected source with SGR~1900$+$14 .
\begin{figure}[ht!]
\centerline{\psfig{figure=img2.ps,width=8cm}} \caption{IBIS/ISGRI
image of the SGR~1900$+$14~ field in the 18-60 keV energy range. The other
detected sources are the high-mass X-ray binary (HMXB) IGR
J19140+0951 (\cite{rodriguez}), the black hole candidate XTE
J1908+094 (\cite{intz}), the HMXB pulsar H 1907+097
(\cite{makishima84}), and the weak unidentified source AXJ
1910.7+917 (\cite{sugizaki}).}
\label{img}
\end{figure}
We found marginal evidence for a long term flux increase by
splitting the data in two parts (March to May 2003 and November
2003 to June 2004) and producing two images of approximately equal
exposure. SGR~1900$+$14~ had a count rate of 0.14$\pm$0.03 counts s$^{-1}$
in the first period and of 0.25$\pm$0.03 counts s$^{-1}$ in the
second one.
To perform a spectral analysis we produced the summed images,
corresponding to the three time periods mentioned above, in five
energy bands (18-25, 25-35, 35-60, 60-100, and 100-200 keV). We
extracted the SGR~1900$+$14\ spectra using the source count rates obtained
from these images and rebinned the IBIS/ISGRI response matrix in
order to match our five energy channels. This spectral extraction
method, which we tested successfully with data also from the Crab
nebula, is particularly suited for weak sources that cannot be
detected in the individual pointings. Before fitting, we added a
systematic error of 5\% to the data, to account for the
uncertainties of our spectral extraction method and of the
response matrix.
In all cases a rather steep power law gave good
results (see Table \ref{sptab}). No signal was detected above 100
keV, but owing to the small statistics we could not establish the
possible presence of a spectral break at high energies.
\begin{table}[ht!]
\caption{Fluxes (20-100 keV) and spectral parameters derived for
SGR~1900$+$14~ with IBIS/ISGRI during the whole observation and during its
parts (see text). All the errors are at 1 $\sigma$ for one parameter of
interest.}
\begin{center}
\begin{tabular}{c|c|c}
\hline\hline
& Flux & Photon Index\\
& [10$^{-12}$ erg cm$^{-2}$ s$^{-1}$] & \\
\hline
Average spectrum & 15$\pm$3 & 3.1$\pm$0.5\\
Spring 2003 & 10$\pm$3& 4.0$\pm$1.0 \\
2003/2004 & 20$\pm$3 & 3.1$\pm$0.6\\
\hline
\end{tabular}
\label{sptab}
\end{center}
\end{table}
\section{Discussion}
SGR~1900$+$14\ is the second SGR for which persistent hard X-ray emission
extending to $\sim$100 keV has been detected, the other being
SGR~1806--20\ (\cite{mereghetti05,molkov}). The spectrum of SGR~1900$+$14\ in
the 20-100 keV range, with photon index $\Gamma$=3.1$\pm$0.5, is
softer than that of SGR~1806--20, which, in the last few years has
been the most active SGR. In the latter source the photon index
varied from $\Gamma$=1.9$\pm$0.2, measured in the period March
2003-April 2004, to $\Gamma$=1.5$\pm$0.3 in September-October
2004, when the burst rate increased (\cite{mereghetti05}) before
the emission on December 27 2004 of the most powerful giant flare
ever observed from a SGR (Palmer et al. 2005, \cite{rhessigiant,acsgiant}).
Positive correlations between the bursting activity, the intensity
and hardness of the persistent emission, and the spin-down rate,
as have been observed in SGR~1806--20\ (\cite{xmm}), are expected in
magnetar models involving a twisted magnetosphere (\cite{tlk}),
since all these phenomena are driven by an increase of the twist
angle.
The soft spectrum of SGR~1900$+$14~ is possibly related to the fact that
this source is currently in a quiescent state. Short bursts were
observed from this source with BATSE (\cite{batse}), {\it RXTE} (\cite{rxte}) and
other satellites (e.g. \cite{sax,konus}) in the years 1979-2002. SGR~1900$+$14\ emitted a
giant flare on August 27 1998 (e.g. \cite{hurley1999}), followed by less intense
``intermediate'' flares on August 29 1998 (\cite{ibrahim}) and in April 2001 (\cite{guidorzi04,lenters}). The last bursts
reported from SGR~1900$+$14\ were observed with the Third Interplanetay
Network (IPN) in November 2002 (\cite{ipn1900}). No bursts from
this source were revealed in all the {\it INTEGRAL}~ observations from
2003 to 2005.
A comparison of the hard X-ray luminosity of the two SGRs is
subject to some uncertainties due to the unknown distances of
these sources. For SGR~1900$+$14\ a distance of 15 kpc has been derived
based on its likely association with a young star cluster
(\cite{vrba00}). For this distance the average flux of
about 1 mCrab corresponds to a 20-100 keV luminosity of
4$\times$10$^{35}$ erg s$^{-1}$. The distance of SGR~1806--20\ is more
controversial. If also this source is at $\sim$15 kpc
(\cite{MG05}), its hard X-ray luminosity would be at least three
times larger than that of SGR~1900$+$14. On the other hand, for a
distance in the 6.4 to 9.8 kpc range, as derived from the latest
radio measurements of the afterglow of SGR~1806--20~ giant flare
(\cite{cameron}), the two SGRs would have about the same
luminosity.
Hard X-ray persistent emission ($>$20 keV) has recently been
detected from another group of sources, the Anomalous X-ray
Pulsars (AXPs, \cite{axps}), which share several
characteristics with the SGRs and are also believed to be
magnetars (see \cite{woodsrew}).
Hard X-ray emission has been detected from three AXPs with
{\it INTEGRAL}: 1E 1841--045 (\cite{molkovaxp}), 4U 0142+61
(\cite{denhartog}) and 1RXS J170849--400910 (\cite{revnitsev}).
The presence of pulsations seen with RXTE up to $\sim$200 keV in 1E
1841--045 (\cite{kuiper}) proofs that the hard X-ray emission
originates from the AXP and not from the associated supernova
remnant Kes 73. The discovery of (pulsed) persistent hard X-ray
tails in these three sources was quite unexpected, since below 10
keV the AXP have soft spectra, consisting of a blackbody-like
component (kT$\sim$0.5 keV) and a steep power law (photon index
$\sim$3--4).
In order to coherently compare the broad band spectral properties
of all the SGRs and AXPs detected at high energy, we analysed all
the public {\it INTEGRAL}~ data using the same procedures described
above for SGR~1900$+$14. Our results are summarised in Table \ref{magtab}
and in Fig. \ref{bbsp} where the {\it INTEGRAL}~ spectra are plotted together
with the results of observations at lower energy taken from the
literature (see figure caption for details).
\begin{figure}[ht!]
\centerline{\psfig{figure=figbb.ps,width=8cm}}
\caption{Broad band X--ray spectra of the five magnetars detected by {\it INTEGRAL}. The
data points above 18 keV are the {\it INTEGRAL}~ spectra with their best fit
power-law models (dotted lines). The solid lines below 10 keV represent
the absorbed power-law (dotted lines) plus blackbody (dashed lines) models
taken from \cite{woods01} (SGR~1900$+$14, during a quiescent state in spring 2000), Mereghetti et al. (2005c) (SGR~1806--20,
observation B, when the bursting activity was low), \cite{gohler} (4U~0142+614), \cite{rea}
(1RXS~J170849--4009), and \cite{morii} (1E~1841--045).}
\label{bbsp}
\end{figure}
For SGR~1806--20\ we
considered only the {\it INTEGRAL}~ data obtained from March to October
2003, in order to exclude the more active period observed in 2004.
We did not introduce any normalisation factor between the
different satellites and some discrepancy between the soft and
hard X-ray spectra might be ascribed to source variability since
the observations were not simultaneous. Nevertheless, even
considering these uncertainties, some indications can be drawn
from the plotted spectra. Namely, in the three AXP a spectral
hardening above $\sim$10-20 keV is required (as already noted,
e.g. by Kuiper et al. (2004)), while
at hard X/soft $\gamma$-rays the spectra of the two SGRs tend to be softer than the ones
measured at low energies. The fact that the spectral break in SGR~1900$+$14~ is more
evident than in SGR~1806--20~ could be also due to the different state during which the
two sources have been observed, with the former in complete quiescence and the latter in
a low level of activity. All the three AXPs, on the other hand, can be considered in a quiescent
state since no SGR-like bursts have ever been reported from any of them.
\begin{table*}[ht!]
\caption{High-energy spectral parameters of Magnetars as measured by {\it INTEGRAL}.
The distances are taken from \cite{woodsrew} and references therein.}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c}
\hline\hline
Source & Exposure Time & Obs. Start & Obs. End & Photon Index & 20-100 keV Luminosity & Distance \\
& [Ms] & UTC & UTC & & [10$^{34}$ erg s$^{-1}$] & [kpc] \\
\hline
SGR 1900+14& 2.5 & 2003-03-06 & 2004-06-08 & 3.1$\pm$0.5 & 40$\pm$8 & 15\\
SGR 1806--20& 2.0 & 2003-03-12 & 2003-10-15 & 1.8$\pm$0.2 & 124$\pm$11 (35$\pm$3) & 15 (8)\\
\hline
4U 0142+61 & 0.33 & 2002-12-28 & 2004-06-09 & 1.3$\pm$0.4 & 5$\pm$0.5 &3\\
1E 1841--045 & 1.5 & 2003-03-10 & 2004-05-02 & 1.5$\pm$0.2 & 26$\pm$2 & 7\\
1RXS J170849.0--400910& 1.8 & 2003-02-01 & 2004-04-20 &1.4$\pm$0.4& 7$\pm$1 &5\\
\hline
\end{tabular}
\label{magtab}
\end{center}
\end{table*}
\section{Conclusions}
We have discovered persistent emission in the 20-100 keV range
from SGR~1900$+$14. Its spectrum is softer than that of the only other SGRs
with persistent emission at these energies, SGR~1806--20. This
difference is possibly due to the different activity state of the
two sources since SGR~1900$+$14~ was detected while it was quiescent from
the point of view of bursting emission, contrary to the case of
SGR~1806--20.
Examining the broad band spectra of magnetars in the 1-100 keV
range, a notable difference between SGRs and AXPs appears. While
in SGRs the hard tails at higher energies are softer than the power law
components measured below 10 keV, in all the AXPs there is evidence for a
spectral hardening between the soft and hard X-ray range.
In the framework of the magnetar model the persistent hard X-ray
emission can be powered either by bremsstrahlung photons produced
in a thin layer close to the neutron star surface, or at $\sim$100
km altitude in the magnetosphere through multiple resonant
cyclotron scattering (\cite{tlk,tb05}). The two models can be
distinguished by the presence of a cutoff at $\sim$ 100 keV or at
$\sim$1 MeV. Unfortunately the current {\it INTEGRAL}~ observations can
just firmly asses the presence of the high energy emission in
Magnetars, but cannot fully rule out the presence of spectral
breaks at high energies. Longer exposure times and/or observations
with more sensitive high-energy instruments are required to
discriminate between the two models.
\begin{acknowledgements}
DG acknowledges support from the French Space Agency (CNES). ISGRI has been realized and maintained in flight
by CEA-Saclay/DAPNIA with the support of CNES. This work has been partially supported by the Italian Space Agency
and by the MIUR under grant PRIN 2004-023189.
\end{acknowledgements}
|
1,116,691,499,337 | arxiv | \section{Introduction}\label{sec: intro}
Higher category theory is a subject that is currently receiving a lot of interest, with strong links not only with algebraic topology (where we can trace its origins), but with logic, computer science, foundations of mathematics, mathematical physics, general systems' theory and more
(see~\cite{BS,L2,H} and for some speculative applications to relational quantum theory also~\cite{B}).
Although higher category theory, was somehow implicitly present at the time of the very inception of the subject in the work of
S.Eilenberg-S.Mac Lane~\cite{EM} (natural transformations are just an example of globular 2-arrows in a strict 2-category),
strict \hbox{$n$-categories} where originally defined by C.Ehresmann, both in their cubical~\cite{E63} and globular versions~\cite{E65} and
M.Kelly-S.Eilenberg's enriched categories~\cite{EK} allow an iterative construction of strict higher categories.
Weak categories (categories where the algebraic axioms of associativity and unitality hold only up to higher-level isomorphism) formally appear in the definition of weak monoidal categories~\cite{Be1,M} (a monoidal category is essentially a 2-category with only one object) and in their
``many-objects'' (horizontally categorified) version as J.B\'enabou's bicategories~\cite{Be}.
J.Roberts, the pioneer of application of category theory to the study of algebraic quantum field theory in physics, was apparently the first to consider strict globular $\omega$-categories (categories equipped with an infinite tower of higher morphisms and compositions)~\cite{R}.
Strict cubical $\omega$-groupoids and categories appeared almost at the same time in a series of works by R.Brown-P.Higgins~\cite{BH}, motivated by their generalization of Seifert-Van Kampen theorem in algebraic topology (see the nice recent textbook~\cite{BHS} for details).
A.Grothendieck~\cite{G} in his famous ``Pursuing Stacks'' manuscript (partially inspired by discussions with R.Brown and collaborators in Bangor) described strict globular $\omega$-categories and proposed the use of weak-$\omega$-groupoids as a way to capture the homotopy content of spaces.
The actual definition of weak $n$-categories (for $n>2$ or $n=\omega$), starting with R.Street's definition of weak $\omega$-category based on the algebra of ``symplexes''~\cite{St}, has been a quite laborius (and still ongoing) process with several alternative partially equivalent definitions under discussion.\footnote{For a general background comparison we refer to the excellent introductions by T.Leinster~\cite{L2},
E.Cheng-A.Lauda~\cite{CL} and, for a quite useful historical account of the complicated developments, to the bibliographical appendix contained in T.Leinster~\cite{L1}.}
Algebraic definitions of weak gobular $\omega$-categories, based on suitable monads, have been developed by M.Batanin~\cite{Ba1,Ba2},
J.Penon~\cite{P}, T.Leinster~\cite{L2} and later C.Kachour~\cite{K} (see also the alternative view of G.Kondratiev~\cite{Ko}).
The notion of strict involution in category theory (an involutive endo-functor) was apparently repeatedly rediscovered and utilized in different contexts, usually with additional structures in place, before being recently formalized through ``dagger categories'' by P.Selinger~\cite{S}. Strict involutions appear in the ``categories with involution'' (M.Burgin~\cite{Bu}, J.Lambek~\cite{La}, \dots) with a compatible ``order relation''; in ``allegories''
(P.Freyd-A.Scedrov~\cite{FS}) where a further operation of ``intersection'' appears; in the definitions of ``$*$-category'' and ``$*$-algebroid'' in the literature on \hbox{C*-categories} starting with P.Ghez-R.Lima-J.Roberts~\cite{GLR} and P.Mitchener~\cite{Mi}, where involutions are supposed to be conjugate-linear on the $\Hom$-spaces; in the works on ``compact closed categories'' starting with S.Abramsky-B.Coecke~\cite{AC}.
Involutions for strict globular $n$-categories (as involutive endofuctors that are covariant or contravariant for the several compositions) have been studied in~\cite{BCLS, B} (see also~\cite{BCL1,BCL2}) and for the case of strict double categories (strict cubical $2$-categories) in~\cite{BCM}.
The study of weak forms of involutions in (higher) category theory had a more intricate evolution (that we will not investigate here) strongly linked with the study of ``dualizing objects'' and $*$-autonomous categories~\cite{BaW}.
A notion of involutive weak monoidal category is contained in~\cite{BCL3} and an alternative definition was proposed by J.Egger~\cite{Eg}.
\textit{
As a very first step towards a possible treatment of weak higher C*-categories, in the present work, our main purpose is to put forward a definition of \emph{involutive weak higher category} in the context of J.Penon's definition of weak globular $\omega$-category~\cite{P}.
Possible immediate future extensions of this research will examine the notion of involutions for M.Batanin~\cite{Ba1} and T.Leinster~\cite{L2} algebraic approaches to higher categories as well.
}
Here we proceed to describe in some detail the content of the paper.
In section~\ref{sec: prelim} we briefly review the basic notions on strict higher categories that we need.
In order to make immediate contact with the already available works on J.Penon's approach, we decide here to define strict higher categories via ``higher quivers'', whose definition is recalled in subsection~\ref{subsec: sghc}.
Previous work on higher categories~\cite{BCLS,B} utilized an algebraic definition of strict higher categories via ``partial monoids on
$n$-arrows''; a discussion of the categorical equivalence between the two descriptions has been done elsewhere~\cite{Pu,BP}.
In this paper, we restrict our attention to the case of globular higher quivers and globular higher categories based on them.\footnote{The treatment of cubical higher categories will be the objective of a further separate investigation, as soon as a full study of strict involutive $n$-tuple categories is available, extending the previous work~\cite{BCM} for double categories.}
Contrary to the treatment in~\cite{BCLS,B}, where only strict $n$-categories are considered, in subsection~\ref{subsec: sghc} we cover also the general case of strict globular $\omega$-categories.
The definition of strict involutive \hbox{$n$-category} from~\cite{BCLS,B} is similarly extended to the case of strict involutive globular
$\omega$-categories in subsection~\ref{subsec: inv cat}.
We remark that also for our strict (involutive) globular $\omega$-categories it is perfectly possible to substitute the ``usual exchange'' axiom with the relaxed ``non-commutative exchange'' property proposed in~\cite{BCLS,B}.
In order to fix the notation and to make the paper self-contained, monads and their algebras are defined in section~\ref{subsec: ama}.
The essential features of J.Penon's construction are recalled in section~\ref{subsec: Penon}.
We do not necessarily require our globular $\omega$-quivers to be initially reflexive (and this should avoid the already known problems described
in~\cite{ChM}).
The main subject of this work is in section~\ref{sec: main}. An explicit definition and construction of free self-dual globular $\omega$-magmas and of free strict involutive globular $\omega$-categories over a globular $\omega$-quiver is presented in detail in subsections~\ref{subsec: sd quiver} and~\ref{subsec: inv cat} followed by a similar construction of the free ``involutive'' contraction over a globular $\omega$-quiver in \ref{subsec: w inv cat}.
In subsection~\ref{subsec: w inv cat} we prove that the forgetful functor from the category $\mathscr{Q}_\omega^*$ of contractions (of involutive globular $\omega$-magmas over strict involutive globular $\omega$-categories) to the category of globular $\omega$-quivers admits a left-adjoint and we give the monadic definition of weak involutive globular $\omega$-categories as algebras for such monad.
Some preliminary examples are presented in subsection~\ref{subsec: ex}.
\section{Preliminaries}\label{sec: prelim}
We collect here the background definitions and results that are preliminary to our work. The main references are J.Penon~\cite{P},
T.Leinster~\cite{L2}, E.Cheng-A.Lauda~\cite{CL}.
In our treatment here, we carefully separate the algebraic axioms (associativity, unitality, unital functoriality and exchange) from the ``structural requirements'' introduced via higher quivers, that are not a-priori reflexive. Reflexivity is considered as a nullary partial operation in parallel to the partial binary operation of composition.
\subsection{Strict Globular Higher Categories}\label{subsec: sghc}
An \textbf{$\omega$-quiver}
$Q^0 \overset{s^0}{\underset{t^0}{\leftleftarrows}} Q^1 \overset{s^1}{\underset{t^1}{\leftleftarrows}} \cdots
\overset{s^{n-2}}{\underset{t^{n-2}}{\leftleftarrows}} Q^{n-1}\overset{s^{n-1}}{\underset{t^{n-1}}{\leftleftarrows}} Q^n \overset{s^n}{\underset{t^n}{\leftleftarrows}} \cdots$ is an infinite family of sets $Q^k$ for $k\in \mathbb{N}_0$ equipped with infinite pairs of source and target maps $s^k,t^k:Q^{k+1}\rightrightarrows Q^k$ for $k\in \mathbb{N}_0$.
Elements of $Q^m$ are called \textbf{$m$-cells} of $Q$ and their ``shape'' is as follows:
\begin{equation*}
\xy
0;/r.22pc/:
(0,15)*{};
(0,-15)*{};
(0,8)*{}="A";
(0,-8)*{}="B";
{\ar@{=>}@/_1pc/ "A"+(-4,1) ; "B"+(-3,0)};
{\ar@{=}@/_1pc/ "A"+(-4,1) ; "B"+(-4,1)};
{\ar@{=>}@/^1pc/ "A"+(4,1) ; "B"+(3,0)};
{\ar@{=}@/^1pc/ "A"+(4,1) ; "B"+(4,1)};
{\ar@{}(-5.5,0)*{} ; (5.5,0)*{}|\dots};
(-15,0)*+{\bullet}="1";
(15,0)*+{\bullet}="2";
{\ar@/^2.75pc/ "1";"2"};
{\ar@/_2.75pc/ "1";"2"};
\endxy
\end{equation*}
An \textbf{$\omega$-globular set} is an $\omega$-quiver satisfying the globularity condition, i.e. $s^{k-1}s^k=s^{k-1}t^k$ and
$t^{k-1}s^k=t^{k-1}t^k$ for all $k\in \mathbb{N}$.
An $\omega$-globular set
$Q^0 \overset{s^0}{\underset{t^0}{\leftleftarrows}} Q^1 \overset{s^1}{\underset{t^1}{\leftleftarrows}} \cdots
\overset{s^{n-1}}{\underset{t^{n-1}}{\leftleftarrows}} Q^n \overset{s^n}{\underset{t^n}{\leftleftarrows}} \cdots$ is \textbf{reflexive} if there exists a family of maps
$Q^0 \overset{\iota^0}{\rightarrow} Q^1 \overset{\iota^1}{\rightarrow} \cdots \overset{\iota^{n-1}}{\rightarrow}
Q^n \overset{\iota^n}{\rightarrow} \cdots$ such that $s^k\circ \iota^k=\id_{Q^k}=t^k\circ \iota^k$ for every $k\in \mathbb{N}_0$.
A \textbf{(reflexive) globular $\omega$-magma} is a (reflexive) $\omega$-globular set equipped with a function \\
$\circ^m_p:Q^m\times_{Q^p} Q^m \rightarrow Q^m$ for each $0\leq p<m$, where
$$Q^m\times_{Q^p} Q^m:=\{(x',x)\in Q^m\times Q^m \ | \ t^pt^{p+1}\cdots t^{m-1}(x)=s^ps^{p+1}\cdots s^{m-1}(x')\},$$ such that the following conditions hold: if $0\leq p<m$ and $(x',x)\in Q^m\times_{Q^p} Q^m$,
\begin{itemize}
\item $s^qs^{q+1}\cdots s^{m-1}(x'\circ^m_p x)=\left\{ \begin{array}{ll}
s^qs^{q+1}\cdots s^{m-1}(x')\circ^q_p s^qs^{q+1}\cdots s^{m-1}(x), & q>p; \\
s^qs^{q+1}\cdots s^{m-1}(x'), & q\leq p. \end{array} \right.$
\item $t^qt^{q+1}\cdots t^{m-1}(x'\circ^m_p x)=\left\{ \begin{array}{ll}
t^qt^{q+1}\cdots t^{m-1}(x')\circ^q_p t^qt^{q+1}\cdots t^{m-1}(x), & q>p; \\
t^qt^{q+1}\cdots t^{m-1}(x), & q\leq p. \end{array} \right.$
\end{itemize}
Here is a graphical rendering of $\circ^3_0, \circ^3_1,\circ^3_2$ for $3$-arrows:
\begin{equation*}
\begin{tabular}{ccc}
\xy 0;/r.22pc/:
(0,10)*{};
(0,-10)*{};
(-14,0)*+{\bullet}="1";
(0,0)*+{\bullet}="2";
{\ar@/^1.33pc/ "1";"2"};
{\ar@/_1.33pc/ "1";"2"};
(14,0)*+{\bullet}="3";
{\ar@/^1.33pc/ "2";"3"};
{\ar@/_1.33pc/ "2";"3"};
(-6.75,4)*+{}="A";
(-6.75,-4)*+{}="B";
{\ar@{=>}@/_.5pc/ "A"+(-1.33,0) ; "B"+(-.66,-.55)};
{\ar@{=}@/_.5pc/ "A"+(-1.33,0) ; "B"+(-1.33,0)};
{\ar@{=>}@/^.5pc/ "A"+(1.33,0) ; "B"+(.66,-.55)};
{\ar@{=}@/^.5pc/ "A"+(1.33,0) ; "B"+(1.33,0)};
(6.75,4)*+{}="A";
(6.75,-4)*+{}="B";
{\ar@{=>}@/_.5pc/ "A"+(-1.33,0) ; "B"+(-.66,-.55)};
{\ar@{=}@/_.5pc/ "A"+(-1.33,0) ; "B"+(-1.33,0)};
{\ar@{=>}@/^.5pc/ "A"+(1.33,0) ; "B"+(.66,-.55)};
{\ar@{=}@/^.5pc/ "A"+(1.33,0) ; "B"+(1.33,0)};
{\ar@3{->} (-8,0)*{}; (-5.5,0)*{}};
{\ar@3{->} (5.5,0)*{}; (8,0)*{}};
\endxy
&
\xy 0;/r.22pc/:
(0,15)*{};
(0,-15)*{};
(0,9)*{}="A";
(0,1)*{}="B";
{\ar@{=>}@/_0.5pc/ "A"+(-2,1) ; "B"+(-1,0)};
{\ar@{=}@/_.5pc/ "A"+(-2,1) ; "B"+(-2,1)};
{\ar@{=>}@/^.5pc/ "A"+(2,1) ; "B"+(1,0)};
{\ar@{=}@/^.5pc/ "A"+(2,1) ; "B"+(2,1)};
{\ar@3{->} (-1.5,6)*{} ; (1.5,6)*{}};
(0,-1)*{}="A";
(0,-9)*{}="B";
{\ar@{=>}@/_.5pc/ "A"+(-2,-1) ; "B"+(-1,-1.5)};
{\ar@{=}@/_.5pc/ "A"+(-2,0) ; "B"+(-2,-.7)};
{\ar@{=>}@/^.5pc/ "A"+(2,-1) ; "B"+(1,-1.5)};
{\ar@{=}@/^.5pc/ "A"+(2,0) ; "B"+(2,-.7)};
{\ar@3{->} (-1.5,-5)*{} ; (1.5,-5)*{}};
(-15,0)*+{\bullet}="1";
(15,0)*+{\bullet}="2";
{\ar@/^2.75pc/ "1";"2"};
{\ar@/_2.75pc/ "1";"2"};
{\ar "1";"2"};
\endxy
&
\xy 0;/r.22pc/:
(0,15)*{};
(0,-15)*{};
(0,8)*{}="A";
(0,-8)*{}="B";
{\ar@{=>} "A"+(0,1.3) ; "B"};
{\ar@{=>}@/_1pc/ "A"+(-4,1) ; "B"+(-3,0)};
{\ar@{=}@/_1pc/ "A"+(-4,1) ; "B"+(-4,1)};
{\ar@{=>}@/^1pc/ "A"+(4,1) ; "B"+(3,0)};
{\ar@{=}@/^1pc/ "A"+(4,1) ; "B"+(4,1)};
{\ar@3{->} (-5.5,0)*{} ; (-2.5,0)*{}};
{\ar@3{->} (2.5,0)*{} ; (5.5,0)*{}};
(-15,0)*+{\bullet}="1";
(15,0)*+{\bullet}="2";
{\ar@/^2.75pc/ "1";"2"};
{\ar@/_2.75pc/ "1";"2"};
\endxy
\end{tabular}.
\end{equation*}
A \textbf{strict globular $\omega$-category} is a reflexive globular $\omega$-magma $\mathscr{C}$ such that:
\begin{enumerate}
\item (associativity) if $0\leq p<m$ and $x,y,z\in \mathscr{C}^m$ with
\\
$(z,y),(y,x)\in \mathscr{C}^m\times_{\mathscr{C}^p}\mathscr{C}^m$, then
$(z\circ^m_py)\circ^m_px=z\circ^m_p(y\circ^m_px)$,
\item (unitality) if $0\leq p<m$ and $x\in \mathscr{C}^m$, then
$$\iota^{m-1}\cdots \iota^pt^p\cdots t^{m-1}(x)\circ^m_px=x=x\circ^m_p \iota^{m-1}\cdots \iota^ps^p\cdots s^{m-1}(x),$$
\item (functoriality of identities) if $0\leq q<p$ and $(x',x)\in\mathscr{C}^p \times_{\mathscr{C}^q}\mathscr{C}^p$, then:
$$\iota^p(x')\circ^{p+1}_q \iota^p(x)=\iota^{p}(x'\circ^p_qx),$$
\item (binary exchange) if $0\leq q<p<m$ and $x,x',y,y'\in \mathscr{C}^m$ with
\\
$(y',y),(x',x)\in \mathscr{C}^m\times_{\mathscr{C}^p}\mathscr{C}^m$ and
$(y',x'),(y,x)\in \mathscr{C}^m\times_{\mathscr{C}^q}\mathscr{C}^m$, then
$$(y'\circ^m_py)\circ^m_q(x'\circ^m_px)=(y'\circ^m_qx')\circ^m_p(y\circ^m_qx), \quad
\xymatrix{
\bullet \ruppertwocell{x} \rlowertwocell{x'} \ar[r] & \bullet \ruppertwocell{y} \rlowertwocell{y'} \ar[r]
& \bullet
}.
$$
\end{enumerate}
A \textbf{covariant morphism of $\omega$-quivers} $Q,\hat{Q}$ is a family of maps
$\phi^n:Q^n\rightarrow \hat{Q}^n$ such that, for $n\in \mathbb{N}_0$,
$\hat{s}^n \circ \phi^{n+1}=\phi^n \circ s^n$ and $\hat{t}^n \circ \phi^{n+1}=\phi^n \circ t^n$.
A \textbf{covariant morphism of reflexive $\omega$-globular sets} $Q,\hat{Q}$ is a morphism of $\omega$-quivers such that, for
$n\in \mathbb{N}_0$, $\hat{i}^n\circ \phi^n=\phi^{n+1}\circ i^n$.
A \textbf{covariant morphism of (reflexive) globular $\omega$-magmas} $M,\hat{M}$ is a morphism of (reflexive) \hbox{$\omega$-globular} sets such that
$\phi^n(x\circ^n_q y)=\phi^n(x)\hat{\circ}^n_q \phi^n(y)$.
Such a morphism is called a \textbf{covariant \hbox{$\omega$-functor}} when $M$ and $\hat{M}$ are strict globular $\omega$-categories.
\subsection{Adjunctions, Monads, Algebras}\label{subsec: ama}
To make the paper self-contained, we recall some well-known definitions in category theory.\footnote{For background in category theory, among several texts, see~\cite{Bo,BW,ML}.}
Let $\mathscr{C}$ and $\mathscr{D}$ be categories and $F:\mathscr{C} \rightarrow \mathscr{D}$ and $U:\mathscr{D} \rightarrow \mathscr{C}$ be functors. We say that the functor $F$ is \textbf{left adjoint} to the functor $U$ or the functor $G$ is \textbf{right adjoint} to the functor $F$, denoted by $F\dashv U$ or $U\vdash F$, if there exist natural transformations $\eta:\id_\mathscr{D}\Rightarrow FU$ and
$\epsilon:UF\Rightarrow \id_\mathscr{C}$ making commutative the following diagrams:
\begin{equation*}
\vcenter{\xymatrix{F \ar[r]^{\eta F} \ar[dr]_{1_F} & FUF \ar[d]^{F\epsilon}
\ar@{}[dl]|(0.35){\circlearrowleft} \\ & F}} \quad
\vcenter{\xymatrix{U \ar[r]^{U\eta}
\ar[dr]_{1_U} & UFU \ar[d]^{\epsilon U} \ar@{}[dl]|(0.35){\circlearrowleft} \\ & U}}
\quad \text{that is, $F\epsilon \circ \eta F=1_F$ and $\epsilon U\circ U\eta =1_U$.}
\end{equation*}
A \textbf{monad} $(T,\mu,\eta)$ on a category $\mathscr{C}$ consists of a functor $T:\mathscr{C} \rightarrow \mathscr{C}$ and natural transformations $\eta :1_\mathscr{C} \Rightarrow T$ (the \textbf{unit}) and $\mu :T^2 \Rightarrow T$ (the \textbf{multiplication}) such that the following diagrams commute:
\begin{equation*}
\xymatrix{T(X) \ar[dr]_{1_{T(X)}} \ar[r]^{T\eta_X} & T^2(X)
\ar@{}[dl]|(0.35){\circlearrowleft} \ar[d]_{\mu_X} \ar@{}[dr]|(0.35){\circlearrowleft}
& T(X) \ar[l]_{\eta_{T(X)}} \ar[dl]^{1_{T(X)}} \\ & T(X) & }
\quad \quad
\xymatrix{T^3(X)
\ar[d]_{T\mu_X} \ar[r]^{\mu_XT} \ar@{}[dr]|(0.5){\circlearrowleft} & T^2(X)
\ar[d]^{\mu_X} \\ T^2(X) \ar[r]_{\mu_X} & T(X)}
\end{equation*}
that is, $\mu_X\circ T\eta_X=1_{T(X)}=\mu_X\circ \eta_{T(X)}$ and $\mu_X\circ
T\mu_X=\mu_X\circ \mu_XT$.
Every adjunction $F\dashv U$ with unit $\eta$ and counit $\epsilon$ gives rise to a unique monad $(UF,U\epsilon F,\eta)$.
Let $(T,\eta ,\mu )$ be a monad on a category $\mathscr{C}$. An \textbf{algebra} for a monad $T$ consists of an object $A\in \mathscr{C}^0$ together with a morphism $TA \overset{\theta}{\rightarrow} A$ such that the following diagrams commute:
\begin{equation*}
\vcenter{\xymatrix{A \ar[r]^{\eta_A} \ar[dr]_{1_A} & TA \ar[d]^{\theta}
\ar@{}[dl]|(0.35){\circlearrowleft} \\ & A}} \quad \quad
\vcenter{\xymatrix{T^2A
\ar[d]_{T\theta} \ar[r]^{\mu_A} \ar@{}[dr]|(0.5){\circlearrowleft} & TA
\ar[d]^{\theta} \\ TA \ar[r]_{\theta} & A}}\quad
\text{that is, $\theta \circ \eta_A =1_A$ and $\theta \circ T\theta = \theta \circ \mu_A$.}
\end{equation*}
\subsection{Penon Weak Higher Categories}\label{subsec: Penon}
Given an $\omega$-globular set $Q$, a reflexive globular $\omega$-magma $M$, with a morphism $\nu:Q\to M$ (as $\omega$-globular sets), is
\textbf{free} over $Q$ if this universal factorization property holds:
for every other morphism $\phi:Q\to \hat{M}$ (as $\omega$-globular sets) into another reflexive globular $\omega$-magma
$\hat{M}$ there exists a unique morphism of reflexive globular $\omega$-magmas $\hat{\phi}:M\to \hat{M}$ such that $\phi = \hat{\phi}\circ \nu$.
Given an $\omega$-globular set $Q$, a strict globular $\omega$-category $C$, with a morphism $\nu:Q\to C$ (as $\omega$-globular sets), is
\textbf{free} over $Q$ if this universal factorization property holds:
for every other morphism $\phi:Q\to \hat{C}$ (as $\omega$-globular sets) into another strict globular $\omega$-category
$\hat{C}$ there exists a unique morphism of strict globular $\omega$-categories $\hat{\phi}:C\to \hat{C}$ such that $\phi = \hat{\phi}\circ \nu$.
Note that free reflexive globular $\omega$-magmas (respectively, strict globular $\omega$-categories) over an \hbox{$\omega$-globular} set always exist (see~\cite{L2} and~\cite{P}) and, as for any definition via a universal factorization property, any two of them are canonically isomorphic.
Let $M$ be a reflexive globular $\omega$-magma, $C$ a strict globular $\omega$-category, and $\pi:M\to C$ a morphism of reflexive globular
$\omega$-magmas. A \textbf{Penon contraction for $\pi$} is a family of maps
$[\cdot,\cdot]_q:\{(x,y)\in M^q\times M^q \ | \ s^{q-1}(x)=s^{q-1}(y), \ t^{q-1}(x)= t^{q-1}(y), \ \pi(x)=\pi(y)\}\to M^{q+1}$, for any
$q\in \mathbb{N}$, satisfying the following three properties:
\begin{enumerate}
\item $s^q([x,y]_q)=x$ and $t^q([x,y]_q)=y$,
\item $x=y$ implies $[x,y]_q=\iota^q_M(x)=\iota^q_M(y)$,
\item $\pi([x,y]_q)=\iota^q_C(\pi(x))=\iota^q_C(\pi(y))$.
\end{enumerate}
Here below is a graphical depiction of Penon contractions:
\begin{equation*}
\xymatrix{
\bullet \rrtwocell^x_y{\omit} & & \bullet
\\
& \ar@{|->}[dd]_{\pi} &
\\
& &
\\
& &
\\
\bullet \ar[rr]_{\pi(x)=\pi(y)} & & \bullet
} \qquad \qquad
\xymatrix{
\bullet \rrtwocell^x_y{\quad [x,y]} & & \bullet
\\
& \ar@{|->}[dd]_{\pi} &
\\
& &
\\
& &
\\
\bullet \rrtwocell^{z=\pi(x)}_{z=\pi(y)}{\quad \iota(z)} & & \bullet
}
\end{equation*}
We have a category of Penon contractions, where morphisms are defined as
$$(M_1\overset{\pi_1}{\to}C_1, [\cdot,\cdot]^1)\xrightarrow{(\Phi,\phi)}(M_2\overset{\pi_2}{\to}C_2, [\cdot,\cdot]^2),$$
where $\Phi:M_1\to M_2$ is a morphism of reflexive globular $\omega$-magmas and $\phi:C_1\to C_2$ is an $\omega$-functor such that
$\pi_2\circ \phi=\Phi\circ \pi_1$ and $\Phi([x,y]^1_q)=[\Phi(x),\Phi(y)]^2_q$ for every $q\in \mathbb{N}$, $x,y$ in the domain of
$[\cdot,\cdot]^1_q$.
There is a forgetful functor $U$ from the category of Penon contractions to the category $\mathscr{G}$ of \hbox{$\omega$-globular} sets associating to a contraction $(M\overset{\pi}{\to}C, [\cdot,\cdot])$ the underlying $\omega$-globular set of $M$.
J.Penon proved in~\cite{P} that $U$ admits a left adjoint functor $F\dashv U$ and gave the following:
\begin{definition}
A \textbf{weak globular $\omega$-category} is an algebra for the monad $(UF,U\epsilon F,\eta)$.
\end{definition}
\section{Main Results}\label{sec: main}
Our goal is a ``Penon's style'' treatment of self-dualities (involutions) for weak $\omega$-categories.
Again we carefully distiguish the ``structural requirements'' in the definition of the unary operations of duality and the algebraic axioms necessary in the case of involutions.
The material on self-dualities and strict involutive categories follows~\cite{BCLS,B} and is adapted/generalized to the case of $\omega$-quivers and
$\omega$-magmas.
\subsection{Self-Dual (Reflexive) Globular $\omega$-Quivers and $\omega$-Magmas}\label{subsec: sd quiver}
Let $\alpha \subseteq \mathbb{N}_0$.
An \textbf{$\alpha$-contravariant morphism} $Q\xrightarrow{\phi}\hat{Q}$ of $\omega$-quivers or $\omega$-globular sets is a family of maps
$\phi^n:Q^n\to\hat{Q}^n$ such that:
\begin{itemize}
\item
$\hat{s}^n \circ \phi^{n+1}=\phi^n \circ t^n, \quad \hat{t}^n \circ \phi^{n+1}=\phi^n \circ s^n, \quad \forall n\in \alpha$;
\item
$\hat{s}^n \circ \phi^{n+1}=\phi^n \circ s^n, \quad \hat{t}^n \circ \phi^{n+1}=\phi^n \circ t^n, \quad \forall n\notin \alpha$.
\end{itemize}
For globular $\omega$-magmas, an $\alpha$-contravariant morphism must also satisfy:
\begin{itemize}
\item
$\phi^n(x\circ^n_p y)=\phi^n(y)\hat{\circ}^n_p\phi^n(x), \quad \forall n\in \alpha, \quad \forall (x,y)\in Q^n\times_pQ^n$,
\item
$\phi^n(x\circ^n_p y)=\phi^n(x)\hat{\circ}^n_p\phi^n(y), \quad \forall n\notin \alpha, \quad \forall (x,y)\in Q^n\times_pQ^n$.
\end{itemize}
In the case of reflexive $\omega$-globular sets and reflexive globular $\omega$-magmas, $\alpha$-contravariant morphisms are furthermore required to satisfy:
$\phi^n\circ \iota^{n-1}=\hat{\iota}^n\circ\phi^{n-1}$, for all $n\in \mathbb{N}$.
A (reflexive) $\omega$-globular set $Q^0 \overset{s^0}{\underset{t^0}{\leftleftarrows}} Q^1 \overset{s^1}{\underset{t^1}{\leftleftarrows}} \cdots \overset{s^{n-1}}{\underset{t^{n-1}}{\leftleftarrows}} Q^n \overset{s^n}{\underset{t^n}{\leftleftarrows}} \cdots$ is \textbf{self-dual} if there exists a family of $\alpha$-contravariant morphisms
$\ast^n_\alpha:Q^n\rightarrow Q^n$, for every $n\in \mathbb{N}_0$ and $\alpha \subseteq \mathbb{N}_0$, in detail:
\begin{itemize}
\item $s^n(f^{\ast^{n+1}_\alpha})=t^n(f)^{*_\alpha^n}$ and $t^n(f^{\ast^{n+1}_\alpha})=s^n(f)^{*_\alpha^n}$
for every $n\in \alpha$ and $f\in Q^{n+1}$,
\item $s^n(f^{\ast^{n+1}_\alpha})=s^n(f)^{*_\alpha^n}$ and $t^n(f^{\ast^{n+1}_\alpha})=t^n(f)^{*_\alpha^n}$
for every $n\notin \alpha$ and $f\in Q^{n+1}$.
\end{itemize}
Similarly a \textbf{(reflexive) self-dual globular $\omega$-magma} is a (reflexive) globular $\omega$-magma whose underlying $\omega$-globular set is self-dual. Notice that in all these cases a self-duality is only an $\alpha$-contravariant morphism of $\omega$-globular sets, but it is not a morphism of reflexive $\omega$-globular sets or a morphism of (reflexive) $\omega$-magmas.
The shape of $2$-cells related by self-dualities $*_{\varnothing},*_{\{0\}},*_{\{1\}},*_{\{0,1\}}$ are pictured here below:
\\
$*_{\{1\}}: \xymatrix{A \rtwocell^f_g{x}& B}\quad \mapsto \quad \xymatrix{A \rrtwocell^f_g{^{\quad \quad \quad \ \ {x^{*_{\{1\}}}}}}& & B,}$
\quad
$*_{\{0\}}: \xymatrix{A \rtwocell^f_g{x}& B}\quad \mapsto \quad \xymatrix{A &
& \lltwocell_{f^{*_{\{0\}}}}^{g^{*_{\{0\}}}}{^{\quad \ x^{*_{\{0\}}}}}B,}$ \\
$*_{\{0,1\}}: \xymatrix{A \rtwocell^f_g{x}& B}\quad \mapsto \quad
\xymatrix{A& & \lltwocell_{f^{*_{\{0\}}}}^{g^{*_\{0\}}}{\quad \quad \quad \ \ \ x^{*_{\{0,1\}}}}B,}$
\quad
$*_{\varnothing}: \xymatrix{A \rtwocell^f_g{x}& B}\quad \mapsto \quad
\xymatrix{A \rrtwocell^f_g{\quad x^{*_\varnothing}} & & B.}$
A \textbf{self-dual} morphism $Q\xrightarrow{\phi}\hat{Q}$ between self-dual $\omega$-quivers, $\omega$-globular sets or (reflexive) globular
$\omega$-magmas is a morphism of the respective structures such that:
$\phi^n(x^{*^n_\alpha})=\phi^n(x)^{\hat{*}^n_\alpha}$, for all $x\in Q^n$, for all $\alpha\subset\mathbb{N}$ and $n\in \mathbb{N}_0$.
A \textbf{free (reflexive) self-dual globular $\omega$-magma} over an $\omega$-globular set $Q$, is a (reflexive) self-dual globular $\omega$-magma $M$ with a morphism of $\omega$-globular sets $\nu:Q\to M$ satisfying the following universal factorization property: for every morphism
$\phi: Q\to \hat{M}$ (as $\omega$-globular sets) into a (reflexive) self-dual globular $\omega$-magma $\hat{M}$, there exists a unique morphism
$\hat{\phi}:M\to\hat{M}$ of (reflexive) self-dual globular $\omega$-magmas such that $\phi=\hat{\phi}\circ\nu$.
\textbf{Free (reflexive) self-dual $\omega$-globular sets} over an $\omega$-globular set, can be defined along the same lines.
\begin{proposition}\label{prop: free-sd}
Free self-dual reflexive globular $\omega$-magmas over an $\omega$-globular set $Q$ exist.
\end{proposition}
\begin{proof}
The construction relies heavily on recursive arguments.
Let $Q$ be an $\omega$-globular set. \\
Consider $\Gamma:=\{(\alpha_1,\dots,\alpha_m) \ | \ m\in \mathbb{N}, \ \forall k=1,\dots,m, \ \alpha_k\subset\mathbb{N}_0\}\cup\{\varnothing\}$ as a set of multi-indexes and, for $\gamma=(\alpha_1,\dots,\alpha_m)\in \Gamma$, the symmetric difference set
$\bigtriangleup \gamma:=\alpha_1\bigtriangleup \cdots\bigtriangleup\alpha_m\subset \mathbb{N}_0$ .
Let $M^0:=\{(x,\gamma) \ | \ x\in Q^0, \ \gamma\in \Gamma\}$.
Define $M^1_\iota:=\{(z,\iota_1) \ | \ z\in M^0\}$ as a disjoint copy of $M^0$ and $s^0/t^0(z,\iota_1):=z$.
Define $M^1[1]:=\{(z,\gamma) \ | \ z\in Q^1\cup M^1_\iota, \ \gamma\in\Gamma\}$ with
$s^0[1]/t^0[1](z,\gamma):=(s^0/t^0(z),\gamma)$, if $0\notin\bigtriangleup \gamma$,
$s^0[1]/t^0[1](z,\gamma):=(t^0/s^0(z),\gamma)$, if $0\in\bigtriangleup \gamma$.
Suppose, by recursion, that we already defined $M^1[1],\dots,M^1[k-1]$ and $s^0/t^0$ on them, we define
$M^1[k]:=\{((x,0,y),\gamma) \ | \ (x,y)\in M^1[i]\times_{M^0}M^1[j], \ i+j=k, \ \gamma\in \Gamma\}$ and we further set sources and targets as follows:
\begin{gather*}
s^0[k]/t^0[k]((x,0,y),\gamma):=(s^0[k](y),\gamma)/(t^0[k](x),\gamma), \quad \text{if $0\notin \bigtriangleup\gamma$},
\\
s^0[k]/t^0[k]((x,0,y),\gamma):=(t^0[k](x),\gamma)/(s^0[k](x),\gamma), \quad \text{if $0\in \bigtriangleup\gamma$}.
\end{gather*}
Finally we define $M^1:=\cup_{k\in\mathbb{N}}M^1[k]$ and $s^0/t^0$ as ``union'' of the previous maps.
Suppose, by further recursion, that we already defined $M^m$, for $m=0,\dots,n$, and all the maps $s^j/t^j$ on them.
We define $M^{n+1}_\iota:=\{(z,\iota_{n+1}) \ | \ z\in M^{n}\}$, with $s^{n}/t^{n}(z,\iota_{n+1}):=z$.
Similarly $M^{n+1}[1]:=\{(z,\gamma)\ | \ z\in Q^{n+1}\cup M^{n+1}_\iota, \gamma\in \Gamma\}$ with
$s^{n}[1]/t^{n}[1](z,\gamma):=(s^{n}/t^{n}(z),\gamma)$, if $n\notin\bigtriangleup \gamma$, and
$s^{n}[1]/t^{n}[1](z,\gamma):=(t^{n}/s^{n}(z),\gamma)$, if $n\in\bigtriangleup \gamma$.
If we suppose, by recursion, already defined $M^{n+1}[1],\dots,M^{n+1}[k-1]$, and all source/target maps $s^{n}/t^{n}$ on them, we futher define $M^{n+1}[k]:=\{((x,p,y),\gamma) \ | \ p=0,\dots,n, \ (x,y)\in M^{n+1}[i]\times_{M^p} M^n[j], \ i+j=k, \ \gamma\in \Gamma\}$ and we set sources and targets as follows:
\begin{align*}
&s^{n}[k]/t^{n}[k]((x,n,y),\gamma):=\begin{cases}
(s^{n}[k](y),\gamma)/(t^{n}(x)[k],\gamma),\ \text{if $n\notin \bigtriangleup\gamma$},
\\
(t^{n}[k](x),\gamma)/(s^{n}(x)[k],\gamma), \ \text{if $n\in \bigtriangleup\gamma$},
\end{cases}
\\
&s^{n}[k]/t^{n}[k]((x,p,y),\gamma):=\begin{cases}
((s^{n}[k](x)/t^{n}[k](x), p, s^{n}[k](y)/t^{n}[k](y)),\gamma),
\ \text{if $n\notin \bigtriangleup\gamma$, $p<n$},
\\
((t^{n}[k](x)/s^{n}[k](x), p, t^{n}[k](y)/s^{n}[k](y)),\gamma),
\ \text{if $n\in \bigtriangleup\gamma$, $p<n$}.
\end{cases}
\end{align*}
Finally we set $M^{n+1}:=\cup_{k\in \mathbb{N}}M^{n+1}[k]$, and $s^{n}/t^{n}$ the ``union'' of $s^{n}[k]/t^{n}[k]$.
The new $\omega$-quiver $M^0\leftleftarrows\cdots\leftleftarrows M^n\leftleftarrows \cdots$ is, by induction, an $\omega$-globular set;
the nullary operations $\iota^n:M^{n-1}\to M^n$ are given by $z\mapsto ((z,\iota_n),\varnothing)$, for all $z\in M^{n-1}$;
the unary operations $*^n_\alpha: M^n\to M^n$, for $\alpha\subset \mathbb{N}_0$, are given by $(z,\gamma)\mapsto (z,\gamma\oplus\{\alpha\})$, where we assume $(\alpha_1,\dots,\alpha_m)\oplus\{\alpha\}:=(\alpha_1,\dots.\alpha_m,\alpha)\in \Gamma$;
the binary compositions $\circ^n_p:M^n\times_{M^p}M^n\to M^n$ are simply $(x,y)\mapsto (x,p,y)$.
All the previous operations inductively satisfy the structural axioms for a self-dual reflexive globular $\omega$-magma.
We only need to check the universal factorization property for the globular $\omega$-magma $M$ with the inclusion map $\nu:Q\to M$ given by
$x\mapsto (x,\varnothing)\in M^n[1]\subset M^n$, for all $x\in Q^n$.
For this purpose, let $\phi: Q\to\hat{M}$ a morphism of $\omega$-globular sets into another self-dual reflexive globular $\omega$-magma $\hat{M}$. The only possible choice of a map $\hat{\phi}:M\to\hat{M}$ such that $\phi=\hat{\phi}\circ\nu$, must necessarily satisfy
$(x,\varnothing)\mapsto \phi(x)$ and, by recursion, using the fact that $\hat{\phi}$ is a morphism of self-dual reflexive globular $\omega$-magmas, we obtain, for all $n\in \mathbb{N}$, $((x,p,y),\gamma)\mapsto (\phi(x)\hat{\circ}^n_p\phi(y))^{\hat{*}_{\alpha_1}\cdots\hat{*}_{\alpha_m}}$, where $\gamma=(\alpha_1,\dots,\alpha_n)$. By induction this well-defined unique morphism $\hat{\phi}$ is a morphism of self-dual reflexive
$\omega$-magmas such that $\phi=\hat{\phi}\circ\nu$ and this completes the proof.
\end{proof}
Along similar lines, one can actually produce recursive construnctions of free (reflexive) self-dual $\omega$-globular sets and (reflexive) globular $\omega$-magmas over a given $\omega$-globular set.
\subsection{Involutive Strict Globular $\omega$-Categories}\label{subsec: inv cat}
An \textbf{$\alpha$-contravariant functor} $C\xrightarrow{\phi}\hat{C}$ between strict globular $\omega$-categories is an $\alpha$-contravariant morphism of the undelying reflexive globular $\omega$-magmas.
An \textbf{involutive strict globular $\omega$-category}\footnote{Here we are exactly following the definition put forward in~\cite{BCLS,B} for the case of $n$-categories.} is a strict globular $\omega$-category that is also a self-dual $\omega$-globular set with self-dualities $*_\alpha$, with
$\alpha\subset \mathbb{N}_0$ that are $\alpha$-contravariant functors that further satisfy the following algebraic axioms:
\begin{itemize}
\item
$(x^{*_\alpha})^{*_\alpha}=x, \quad \forall x\in C, \quad \forall \alpha\subset\mathbb{N}$,
\item
$(x^{*_\alpha})^{*_\beta}=(x^{*_\beta})^{*_\alpha}, \quad \forall x\in C, \quad \forall \alpha,\beta\subset\mathbb{N}$.
\end{itemize}
A $*$-functor between involutive strict globular $\omega$-categories is just a functor
$C\xrightarrow{\phi}\hat{C}$ such that: $\phi(x^{*_\alpha})=\phi(x)^{\hat{*}_\alpha}$ for all $x\in C$ and for all $\alpha\subset \mathbb{N}$.
A \textbf{free involutive strict globular $\omega$-category} over an $\omega$-globular set $Q$, is an involutive strict globular $\omega$-category $C$, with a morphism of $\omega$-globular sets $\nu:Q\to C$, satisfying the following universal factorization property: for every morphism
$\phi: Q\to \hat{C}$ (as $\omega$-globular sets) into an involutive strict globular $\omega$-category $\hat{C}$, there exists a unique $*$-functor
$\hat{\phi}:M\to\hat{M}$ such that $\phi=\hat{\phi}\circ\nu$. Unicity up to a unique isomorphism of involutive strict globular $\omega$-categories commuting with the inclusion morphisms is standard from the universal factorization. The existence can be obtained by a recursive construction, as in the previous case of a free self-dual globular $\omega$-magma, but we present here an alternative ``quotient'' argument starting from the already available free self-dual reflexive globular $\omega$-magmas over the $\omega$-globular set $Q$.
Let $M:=M^0\leftleftarrows \cdots \leftleftarrows M^n\leftleftarrows \cdots$ be a self-dual reflexive globular $\omega$-magma.
Consider its Cartesian product $M\times M:=(M^0\times M^0)\leftleftarrows \cdots \leftleftarrows (M^n\times M^n)\leftleftarrows \cdots$, where, for all $n\in\mathbb{N}_0$, the source and target maps $s^n_{M\times M}:=(s^n_M,s^n_M)$, $t^n_{M\times M}:=(t^n_M,t^n_M)$, as well as the
structural nullary $\iota^n_{M\times M}:=(\iota^n_M,\iota^n_M)$, unary $(x,y)^{*_\alpha^{M\times M}}:=(x^{*_\alpha^M},y^{*_\alpha^M})$,
and (when they exist) binary operations $(x_1,y_1)\circ^{(M\times M)^n}_p(x_2,y_2):=(x_1\circ^{M^n}_p y_1,x_2\circ^{M^n}_p y_2)$ are defined componentwise.
In this way, the product $M\times M$ is another self-dual reflexive globular $\omega$-magma.
A \textbf{congruence} in the self-dual reflexive globular $\omega$-magma $M$ is a self-dual reflexive globular $\omega$-magma
$R$ such that, for all $n\in \mathbb{N}_0$, $R^n\subset M^n\times M^n$ is an equivalence relation in $M^n$ and the inclusions
$\nu_n$ provide a morphism of self-dual reflexive globular $\omega$-magmas $\nu:R\to M\times M$.
Under such conditions, we obtain a \textbf{quotient self-dual reflexive globular $\omega$-magma} $M/R$ with the quotient sets
$M^n/R^n=:(M/R)^n$, $n\in \mathbb{N}_0$, with sources/targets given by $s^n_{M/R}([x]_{{n+1}}):=[s^n_M(x)]_n$,
$t^n_{M/R}([x]_{{n+1}}):=[t^n_M(x)]_n$, compositions (whenever existing) defined as $[x]_n\circ^{(M/R)^n}_p[y]_n:=[x\circ^{M^n}_p y]_n$ and nullary operations $\iota^n_{M/R}([x]_n)=[\iota^n_M(x)]_{n+1}$.
Furthermore, the quotient maps $\pi^n:M^n\to M^n/R^n$ onto the quotient sets give us a morphism $\pi:M\to M/R$ between self-dual reflexive globular $\omega$-magmas.\footnote{In a perfectly analogous way, one can introduce congruences and quotients for all the other ``intermediate'' structures between $\omega$-globular sets and self-dual reflexive globular $\omega$-magmas.}
\begin{proposition}\label{prop: free-ic}
Free involutive strict globular $\omega$-categories over an $\omega$-globular set $Q$, exist.
\end{proposition}
\begin{proof}
Let $Q\xrightarrow{\nu}M$ be the free strict self-dual reflexive globular $\omega$-magma over $Q$ as constructed in
proposition~\ref{prop: free-sd}.
In order to obtain from $M$ an involutive strict globular \hbox{$\omega$-category}, we must impose all the ``algebraic axioms'' (structural axioms for the globularity of $\omega$-quiver and for the ``domain/codomain'' of the nullary, unary and binary operations are already in place in $M$).
We consider the congruence $R$ in $M$ ``generated'' by all the possible pairs of terms involved in the expression of the algebraic axioms:
\begin{align*}
X:=&\{(x,(x^{*_\alpha})^{*_\alpha}) \ | \ x\in M, \ \alpha\subset \mathbb{N}_0\}\cup
\{((x^{*_\alpha})^{*_\beta}, (x^{*_\beta})^{*_\alpha}) \ | \ x\in M, \ \alpha,\beta\subset \mathbb{N}_0\}\cup
\\
&\{((x\circ_py)^{*_\alpha}, (x^{*_\alpha})\circ_p (y^{*_\alpha})) \ | \ (x,y)\in M\times_{M^p}M, \ \mathbb{N}_0\ni p\notin\alpha\subset{N}_0\}\cup
\\
&\{((x\circ_py)^{*_\alpha}, (y^{*_\alpha})\circ_p (x^{*_\alpha})) \ | \ (x,y)\in M\times_{M^p}M, \ \mathbb{N}_0\ni p\in\alpha\subset{N}_0\}\cup
\\
&\{(x\circ_p(y\circ_pz),(x\circ_py)\circ_p z) \ | \ (x,y,z)\in M\times_{M^p}M\times_{M^p}M, \ p\in\mathbb{N}_0\}\cup
\\
&\begin{aligned}
\{((x\circ_py)\circ_q(z\circ_pw), (x\circ_qz)\circ_p(y\circ_qw)) \ | \ &(x,y),(z,w)\in M\times_{M^p}M, \
\\
&(x,z),(y,w)\in M\times_{M^q}M,\ p,q\in \mathbb{N}_0\} \cup
\end{aligned}
\\
&\{(\iota(x)\circ_q\iota(y),\iota(x\circ_qy)) \ | \ (x,y)\in M\times_{M^q}M, \ q\in \mathbb{N}_0 \}\cup
\\
&\{(\iota(x^{*_\alpha}),\iota(x)^{*_\alpha}) \ | \ x\in M, \ \alpha\subset\mathbb{N}_0\}\cup
\\
&\{((\iota^{n-1}\circ\cdots\iota^p\circ t^p\circ\cdots\circ t^{n-1} (x))\circ^n_p x, x) \ | \ n,p \in \mathbb{N}_0,\ x\in M^n\}\cup
\\
&\{(x, x\circ^n_p (\iota^{n-1}\circ\cdots\iota^p\circ s^p\circ\cdots\circ s^{n-1} (x))) \ | \ n,p \in \mathbb{N}_0,\ x\in M^n\}.
\end{align*}
this is by definition the smallest congruence in $M$ containing $X$.
Such a congruence always exists and (since the arbitrary intersection of congruences is a congruence and $M\times M$ is always a congruence containing $X$) it coincides with the intersection of all congruences in $M$ containing $X\subset M\times M$.
Taking now the quotient self-dual reflexive globular $\omega$-magma $M/R$, we note that, since $X\subset R$, all the algebraic axioms are already satisfied in $M/R$ and hence $M/R$ is already an involutive strict globular $\omega$-category.
We only need to check that $Q\xrightarrow{\pi\circ \nu}M/R$ is a free involutive strict globular $\omega$-category via the universal factorization property. Let $\phi: Q\to \hat{C}$ be a morphism of $\omega$-globular sets into an involutive strict $\omega$-category $\hat{C}$. Since $M$ is a free self-dual reflexive globular $\omega$-magma over $Q$, there exists one and only one morphism of self-dual reflexive globular $\omega$-magmas
$\overline{\phi}:M\to \hat{C}$ such that $\phi=\overline{\phi}\circ\nu$.
Consider, for all $n\in \mathbb{N}_0$, $R_{\phi}^n:=\{(x,y)\in M^n\times M^n \ | \ \overline{\phi}_n(x)=\overline{\phi}_n(y)\}$.
Since $\overline{\phi}$ is a morphism of self-dual reflexive globular $\omega$-magmas, $R_\phi$ becomes a congruence in $M$ and, thanks to the fact that $\hat{C}$ is already an involutive strict globular $\omega$-category, we have $X\subset R_\phi$ and hence $X/R_\phi$ is already an involutive strict globular $\omega$-category, and the assignment $\widetilde{\phi}:[x]_{R_\phi}\mapsto \overline{\phi}(x)$, for $x\in M$, is a well-defined
$*$-functor $\widetilde{\phi}:M/R_\phi\to\hat{C}$ and it is the unique map such that $\widetilde{\phi}\circ\pi_\phi=\overline{\phi}$, where
$\pi_\phi:M\to M/R_\phi$ denotes the quotient morphism.
Since $R$ is the smallest congruence containing $X$, we have $R\subset R_\phi$ and hence there is a unique well-defined map $\theta:M/R\to M/R_\phi$ via the assignment $\theta: [x]_R\mapsto [x]_{R_\phi}$, for all $x\in M$, and $\theta$ is a $*$-functor of involutive strict globular
$\omega$-categories and actually the unique map such that $\pi_\phi=\theta\circ\pi$. Combining the equations, we see that
$\hat{\phi}:=\widetilde{\phi}\circ \theta:M/R\to\hat{C}$ is a $*$-functor and it is the unique morphism such that
$\phi=\overline{\phi}\circ\nu=\widetilde{\phi}\circ \pi_\phi\circ\nu=\widetilde{\phi}\circ\theta\circ \pi\circ \nu=\hat{\phi}\circ(\pi\circ\nu)$.
\end{proof}
\subsection{Involutive Weak Globular $\omega$-Categories}\label{subsec: w inv cat}
Let $M$ be a self-dual reflexive globular $\omega$-magma, $C$ an involutive strict globular $\omega$-category,
and $\pi:M\to C$ a self-dual morphism of self-dual reflexive globular $\omega$-magmas.
\\
Finally let $[\cdot,\cdot]_n$, $n\in \mathbb{N}$ be a usual Penon contraction for $\pi$, exactly as defined in section~\ref{subsec: Penon}.
We have a category $\mathscr{Q}_\omega^*$ of ``self-dual'' Penon contractions, where morphisms are defined as
$$(M_1\overset{\pi_1}{\to}C_1, [\cdot,\cdot]^1)\xrightarrow{(\Phi,\phi)}(M_2\overset{\pi_2}{\to}C_2, [\cdot,\cdot]^2),$$
where $\Phi:M_1\to M_2$ is a self-dual morphism of self-dual reflexive globular $\omega$-magmas and $\phi:C_1\to C_2$ is a $*$-functor of involutive strict $\omega$-categories, such that $\pi_2\circ \Phi=\phi\circ \pi_1$ and $\Phi([x,y]^1_q)=[\Phi(x),\Phi(y)]^2_q$ for every
$q\in \mathbb{N}$, $x,y$ in the domain of $[\cdot,\cdot]^1_q$.
There is a forgetful functor $U^*$ from the category $\mathscr{Q}_\omega^*$ of ``self-dual'' Penon contractions to the category $\mathscr{G}$ of
$\omega$-globular sets, associating to a ``self-dual'' contraction $(M\overset{\pi}{\to}C, [\cdot,\cdot])$ the underlying $\omega$-globular set of $M$ (forgetting self-dualities, compositions and reflexive maps).
A \textbf{free self-dual Penon contraction} over an $\omega$-globular set $Q$ is a self-dual Penon contraction
$(M\overset{\pi}{\to}C, [\cdot,\cdot])$, with a morphism of $\omega$-globular sets $\nu: Q\to U^*((M\overset{\pi}{\to}C, [\cdot,\cdot]))$, such that the following universal factorization property holds: for any other morphism of $\omega$-globular sets
$Q\xrightarrow{\phi} U^*(\hat{M}\overset{\hat{\pi}}{\to}\hat{C}, \widehat{[\cdot,\cdot]})$ into the undelying $\omega$-globular set of another self-dual Penon contraction $(\hat{M}\overset{\hat{\pi}}{\to}\hat{C}, \widehat{[\cdot,\cdot]})\in \mathscr{Q}_\omega^*$, there exists a unique morphism
$(M\overset{\pi}{\to}C, [\cdot,\cdot])\xrightarrow{(\hat{\Phi},\hat{\phi})}(\hat{M}\overset{\hat{\pi}}{\to}\hat{C}, \widehat{[\cdot,\cdot]})$
in $\mathscr{Q}_\omega^*$ such that $U^*(\hat{\Phi},\hat{\phi})\circ \nu=\phi$.
\begin{proposition}
Free self-dual Penon contractions over an $\omega$-globular set exist.
\end{proposition}
\begin{proof}
The construction proceeds by recursion merging techniques from propositions~\ref{prop: free-sd} and~\ref{prop: free-ic}.
Let $Q$ be an $\omega$-globular set. We construct $M^0=C^0=Q^0$ and $\pi^0:M^0\to C^0$ as the identity.
Note that the domain of $[\cdot,\cdot]_0$ is empty (there is no contraction induced by $\pi^0$).
Using the same notations as in the proof of propositions~\ref{prop: free-sd} and~\ref{prop: free-ic}, we define $M^1$, $C^1:=M^1/R^1$ and
$\pi^1:M^1\to C^1$ as the quotient map by the congruence $R^1\subset M^1\times M^1$ generated by all the algebraic axioms $X^1$ between
1-arrows of the free self-dual reflexive globular $\omega$-magma.
Note that now the domain of $[\cdot,\cdot]_1$ concides with $X^1$. We define on $(x,y)\in X^1\subset M^1\times M^1$, $s^1(x,y):=x$ and
$t^1(x,y):=y$. Next we set $M^2[1]:=\{(z,\gamma) \ | \ z\in Q^2\cup M^2_\iota\cup X^1, \gamma\in \Gamma\}$ (note the introduction of extra
2-arrows coming from the contractions relative to the algebraic axioms in $X^1$) and we proceed exactly as in the proof of
proposition~\ref{prop: free-sd} to recursively define $M^2[k]$, for all $k\in\mathbb{N}$ and get $M^2:=\cup_{k\in\mathbb{N}}M^2[k]$ as well as the source/target maps $s^2/t^2$. \\
We define now $C^2:=M^2/[R]^2$, where $[R]^2$ is the congruence generated by the algebraic axioms $[X]^2$ in $M^2$, $\pi^2:M^2\to C^2$ is the quotient map and the contraction $[\cdot,\cdot]_1:X_1\to M^2$ is the inclusion
$X^1\subset M^2[1]\subset M^2$. Note that the set $[X]^2$ now contains also the axioms for the contractions:
$\{(s^1([x,y]_1),x) \ | \ (x,y)\in X^1\}\cup \{(t^1([x,y]_1),y)\ | \ (x,y)\in X^1\}\cup\{([x,x]_1,(x,\iota_1)) \ | \ x\in M^1\}$.
If we suppose, by recursion, that we already defined $\pi^n:M^n\to C^n$, $[\cdot,\cdot]_{n-1}:[X]^{n-1}\to M^n$ as above, we can consider
$X^n\subset M^n\times M^n$ as the set of algebraic axioms between $n$-arrows; define
$M^{n+1}[1]:=\{(z,\gamma) \ | \ z\in Q^{n+1}\cup M^{n+1}_\iota\cup [X]^n, \gamma\in \Gamma\}$ and
$M^{n+1}:=\cup_{k\in \mathbb{N}}M^{n+1}[k]$; the contraction $[\cdot,\cdot]_n:[X]^n\to M^{n+1}$ always as inclusion; the congruence generated by the algebraic axioms $[X]^{n+1}$ between $(n+1)$-arrows $[R]^{n+1}\subset M^{n+1}\times M^{n+1}$ and finally obtain
$\pi^{n+1}$ as the quotient map onto $C^{n+1}:=M^{n+1}/[R]^{n+1}$, completing the recursive step of the definition.
The nullary, unary and binary operations on the new $\omega$-quiver $M$ are defined as in proposition~\ref{prop: free-sd} (there are only the extra arrows coming from $X$ to be considered). Inductively $M$ turns out to be a self-dual reflexive globular $\omega$-magma, the quotient $C=M/R$ by the congruence $R$ is a strict involutive $\omega$-category, since $X\subset R$, and $\pi:M\to C$ is a morphism of self-dual reflexive globular
$\omega$-magmas. The union of all the maps $[\cdot,\cdot]_n:[X]^n\to M^{n+1}$ is a contraction.
The inclusion $\nu$ of $Q$ into $U^*((M\xrightarrow{\pi}C), [\cdot,\cdot])$ is simply the map $x\mapsto (x,\varnothing)$ as before.
We only need to show the universal factorization property. For this purpose, let $(\Phi,\phi)$ be a morphism in $\mathscr{Q}_\omega^*$ into a new self-dual contraction $(\hat{M}\xrightarrow{\hat{\pi}},\hat{C})\in \mathscr{Q}_\omega^*$. If $(\hat{\Phi},\hat{\phi})$ is a morphism in
$\mathscr{Q}_\omega^*$ such that $U^*(\Phi,\phi)=U^*(\hat{\Phi},\hat{\phi})\circ \nu$, we necessarily have $\hat{\Phi}(x,\varnothing)=\Phi(x)$, for all $x\in Q$.
Since $\hat{\Phi}$ is a morphism of self-dual reflexive globular $\omega$-magmas, the definition of $\hat{\Phi}$ is uniquely given by
$\hat{\Phi}^n(z,\gamma)=\Phi^n(z)^{\hat{*}_{\alpha_1}\cdots \hat{*}_{\alpha_m}}$, if $z\in Q^n$ and $\gamma:=(\alpha_1,\dots,\alpha_m)$;
$\hat{\Phi}^n(z,\gamma)=\hat{\iota}(z)^{\hat{*}_{\alpha_1}\cdots \hat{*}_{\alpha_m}}$, if $z\in M^n_\iota$, for $n\in \mathbb{N}_0$;
$\hat{\Phi}^n([x,y]_{n-1},\gamma)= \widehat{[\Phi^n(x),\Phi^n(y)]}_{n-1}^{\hat{*}_{\alpha_1}\cdots \hat{*}_{\alpha_m}}$,
if $[x,y]_{n-1}\in [X]^{n-1}$, for $n\in\mathbb{N}$.
An inductive argument shows that this unique map $\hat{\Phi}:M\to\hat{M}$ is actually a morphism of self-dual reflexive globular $\omega$-magmas.
The map $\hat{\pi}\circ\hat{\Phi}:M\to\hat{C}$ induces the congruence $R_{\hat{\pi}\circ\hat{\Phi}}$ in $M$ and since $\hat{\Phi}$ preserves the contractions, we have $[X]\subset R_{\hat{\pi}\circ\hat{\Phi}}$ and hence, for the congruence $R$ in M generated by $[X]$,
$R\subset R_ {\hat{\pi}\circ\hat{\Phi}}$. It follows that there exists a unique induced $*$-functor $\hat{\phi}:C\to\hat{C}$ such that
$\hat{\pi}\circ\hat{\Phi}=\hat{\phi}\circ \pi$ and hence $(\hat{\Phi},\hat{\phi})$ is the unique morphism in $\mathscr{Q}_\omega^*$ such that
$\phi=U^*(\hat{\Phi},\hat{\phi})$ and we completed the proof of the universal factorization property.
\end{proof}
\begin{theorem}
The forgetful functor $U^*:\mathscr{Q}_\omega^*\to \mathscr{G}$ admits a left adjoint $F^*\dashv U^*$.
\end{theorem}
\begin{proof}
We define $F^*$ on the objects of $\mathscr{G}$ as the map associating to an $\omega$-globular set $Q$ the specific free self-dual Penon contraction $F^*(Q)$ constructed in the previous proposition and let $\eta_Q:Q\to U^*(F^*(Q))$ denote the ``inclusion'' morphism in the definition of the free self-dual Penon contraction.
If $\gamma:Q_1\to Q_2$ is a morphism in $\mathscr{G}$, we have that $\eta_2\circ\gamma:Q_1\to U^*(F^*(Q_2))$ is a morphism in $\mathscr{G}$ and hence, by the universal factorization property for free self-dual Penon contractions, there exists a unique morphism
$F^*_\gamma: F^*(Q_1)\to F^*(Q_2)$ in $\mathscr{Q}_\omega^*$ such that $U^*(F^*_\gamma)\circ \eta_1=\eta_2\circ \gamma$.
The map $\gamma\mapsto F^*_\gamma$ is functorial from $\mathscr{G}\to\mathscr{Q}_\omega^*$ and, by standard arguments about adjunction,
$\eta: Q\mapsto \nu_Q$ is the unit of an adjunction $F^*\dashv U^*$.
\end{proof}
Finally we can provide our main definition:
\begin{definition}
A \textbf{Penon weak involutive globular $\omega$-category} is defined as an algebra for the monad $(U^*F^*,U^*\epsilon^* F^*,\eta^*)$.
\end{definition}
\subsection{Examples}\label{subsec: ex}
We just mention here, without entering into a detailed discussion, some of the most immediate examples of involutive weak categories.
\begin{example}
Every strict involutive globular $\omega$-category is a very particular trivial case of weak involutive globular $\omega$-category. In particular strict globular $\omega$-groupoids.
\end{example}
\begin{example}
Weak $\omega$-groupoids are just special cases of weak involutive $\omega$-categories with involutions given by (suitable composition of) the inverses. In particular the most elementary and well-known examples fitting our definition of weak involutive $\omega$-category are the
\textbf{fundamental $\omega$-groupoids} $\Pi_\omega(X)$ of topological spaces $X$ (see~\cite[page~xiv-xv]{L2}).
Let $X$ be a topological spaces, $\Pi_\omega(X)^0:=X$, $\Pi_\omega(X)^1:=C([0,1];X)$ is the set of continuous paths in $X$, $\Pi_\omega(X)^2$ is the set of homotopies of paths with fixed endopoints, \dots, $\Pi_\omega(X)^n$ is the set of homotopies between $(n-1)$-homotopies, etc.
Compositions of homotopies are defined in the usual way and involutions consist of the inverse homotopies.
\end{example}
\begin{example}
Truncations, at the level of $n$-arrows, of involutive strict $\omega$-categories are involutive strict $n$-categories and, in the other direction, involutive strict $n$-categories become involutive strict $\omega$-categories, just taking identities as the only morphisms for all $m>n$. The situation for weak categories is more involved: an involutive weak $n$-category can be defined as an algebra for a similar monad associated to the adjunction
$\xymatrix{\mathscr{Q}^*_n \rtwocell^{F_n}_{U_n} {'\perp}& \mathscr{G}_n}$ between the forgetful functor $U_n$ and its left adjoint functor $F_n$ between the category of $n$-globular sets $\mathscr{G}_n$ and the Penon self-dual contraction category $\mathscr{Q}^*_n$.
\end{example}
\begin{example}
Globular $\omega$-quivers (and more generally the ``globular'' propagators of globular $\omega$-quivers discussed in~\cite{BJ}) are examples of weak involutive globular $\omega$-categories.
\end{example}
Of particular motivation for us is the following example of ``higher Morita categories''.
\begin{example}
Let $\mathscr{M}^0$ be a family of involutive monoids $A,B,C,\dots$ and $\mathscr{M}^1$ the family of the bimodules ${}_AM_B$, with
$A,B\in \mathscr{M}^0$. Composition $\circ^1_0$ of bimodules is given by the Rieffel tensor product ${}_AM_B \otimes_B {}_BN_C$ and involution $*^1_0$ of bimodules is provided by the Rieffel dual ${}_B\overline{M}_A$ where $\overline{M}:=\{\overline{x} \ | \ x\in M\}$ is just a (specific) disjoint copy of $M$ and the bimodule actions are $b\cdot \overline{x}\cdot a:=\overline{a^*xb^*}$, for all $a\in A$, $b\in B$ and $x\in M$.
Similarly starting from a class $\mathscr{M}^0$ of strict involutive 1-categories, the family $\mathscr{M}^1$ of ``bimodules'' between them is a weak involutive $1$-category. Introducing a suitable notion of ``bimodule'' between strict involutive $n$-categories, we obtain a weak involutive
$n$-category. If $\mathscr{M}^0$ is a family of strict $\omega$-categories, the family $\mathscr{M}^1$ of ``bimodules'' between them is a weak involutive $\omega$-category.
\end{example}
\medskip
\emph{Notes and Acknowledgments:}
This research was financially supported by the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and Thammasat University, Faculty of Science and Technology, by Thammasat University research grant n.~2/15/2556: ``Categorical Non-commutative Geometry''.
\medskip
The second author thanks Starbucks Coffee at the $1^{\text{st}}$ floor of Emporium Tower and Jasmine Tower, in Sukhumvit, where he spent a significant part of the time dedicated to this research project.
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|
1,116,691,499,338 | arxiv | \section{Introduction}
Many specialists connect the further development of information protection facilities with the application of multiple-valued function of logical algebra (MVFLA), in particular, with the use of pseudorandom sequences (PRS) over GF($q$) ($q>2$), which possess a wider scope of unique features, if compared with binary PRS. The most effective and approved way of obtaining PRS is the use of special switching networks called linear feedback recurrent shift register (LFSR) [1--3].
\section{General information}
Construction of LFSR over GF($q$) (hereinafter $q$-LFSR) is carried out by means of given characteristic polynomial:
$$P(z)=z^r\oplus p_{r-1}z^{r-1}\oplus p_{r-2}z^{r-2}\oplus\ \ldots\ \oplus p_{0} \pmod {q},$$
where $P(z)\in \text{GF}
(q)$, and $r$ is $P(z)$ polynomial order, $r\in N$, and to the constructed according to it recurrent equation:
\begin{align}
s_{n+r}=p_{r-1}s_{n+r-1}\oplus p_{r-2}s_{n+r-2}\oplus\ \ldots\ \oplus p_0s_n \pmod {q},
\end{align}
$n=0, 1, 2,\ \ldots\ $; where $p_j\in \text{GF}(q)$, $0\leq j \leq r-1$; $\oplus$~-- is the symbol of addition according to $\mod q$.
In general case $q$-LFSR consists of $D_j $ $(j=0, 1,\ \ldots,\ r-1)$ cells and has the following initial fill: $s_0, s_1,\ \ldots,\ s_{r-1}$.
Here the ``cell'' is the $\lceil\log_2 q \rceil$ parallel stage register ($\lceil x \rceil$ being the least integral number equal or more than $x$). After the first cycle $q$-LFSR has the following fill: $s_1, s_2,\ \ldots,\ s_{r}$. In general $q$-LFSR generates infinite $q$-valued PRS: $s_0, s_1, s_2,\ \ldots,\ s_{r-1},\ \ldots$ [2].
In notation of linear algebra the next $q$-valued element of PRS $s_{n+r}$ is represented as a product:
\begin{align*}
\begin{Vmatrix}
s_{n+r} \\ s_{n+r-1} \\ \ldots\ldots\\ s_{n+2} \\ s_{n+1} \\
\end{Vmatrix}^{\top}=
\begin{Vmatrix}
s_{n+r-1} \\ s_{n+r-2}\\ \ldots\ldots \\ s_{n+1} \\ s_{n} \\
\end{Vmatrix}^{\top}
\cdot
\begin{Vmatrix}
p_{r-1} & 1 &0 & \ldots & 0 \\
p_{r-2} & 0 &1 & \ldots & 0 \\
\hdotsfor[2]{5} \\
p_1 & 0 & 0& \ldots & 1 \\
p_0 & 0 &0& \ldots & 0 \\
\end{Vmatrix}.
\end{align*}
In the Fig. 1 Structural diagram of the sequential $q$-LFSR functioning is shown.
\begin{figure}[htb]
\begin{center}\LARGE{
\resizebox{0.99\linewidth}{!}{
\colorbox[gray]{.92}{
$$
\xymatrix{
\ar@{->}[rr]\ar@{-}[dd] & & s_{n+r-1} \ar@{->}[dd] \ar@{->}[rr] & & s_{n+r-2}\ar@{->}[dd] \ar@{-}[r]& \ar@{--}[r] & \ar@{->}[r] & s_{n+1} \ar@{->}[dd] \ar@{->}[rr] & & s_{n} \ar@{->}[dd] \ar@{->}[r] &\\
&&&&&&&&&&\\
s_{n+r} \ar@{-}[dd] &p_{r-1} \ar@{->}[r] & \bigotimes \ar@{->}[dd] &p_{r-2} \ar@{->}[r] & \bigotimes \ar@{->}[dd]& \cdots\cdots &p_{1} \ar@{->}[r] &\bigotimes \ar@{->}[dd] & p_{0} \ar@{->}[r] &\bigotimes \ar@{-}[dd] & \\
&&&&&&&&&&\\
& & \bigoplus \ar@{-}[ll] & & \ar@{->}[ll]\bigoplus & \ar@{->}[l] & \ar@{--}[l] &\ar@{-}[l] \bigoplus & & \ar@{->}[ll] &\\
}
$$
}
}}
\end{center}
\begin{center}
\caption{Structural diagram of the operation of the sequential $q$-LFSR in accordance with formula (1) ($\oplus$ and $\otimes$ --- according to transaction of addition and multiplication of the $\mod q$)}
\end{center}
\end{figure}
As we know, PRS over GF($q$) has a range of ``useful'' structural properties, including [2, 3]:
\begin{itemize}
\item number of symbols at the period of PRS or PRS period is defined as $L=q^r-1$;
\item number of similar nonzero symbols in the PRS period is equal to $q^{r-1}$,
and the number of zero symbols is equal to $q^{r-1}-1$;
\item addition of elements in a PRS with elements of the same PRS shifted numbering (except number equal periud repetition) gives a similar PRS shifted numbering;
\item autocorrelation function of PRS is defined by means of the ratio
$p(0)=1,$ $p(i)=-\frac{1}{q^r-1},$ $1\leq i\leq q^r-2$, etc.
\end{itemize}
\section{Method of parallel generation of $q$-valued PRS}
In the most of practically important cases, besides the ``useful'' structural quantities, every complex system should be aimed at the achievement of some limiting characteristic or some quality indicator, what can be obtained by means of the minimization of the quantity of operations, using of resources or parallelization of computational processes of the system [4].
So, the paper [5] shows the algorithm of parallelization of generation of binary PRS based on the presentation of systems generating recurrent logical formulae by means of arithmetic polynomials. At the same time the development of computing machinery and software requires the invention of the new approaches to firmware realization both of binary functions and ($q>2$)-valued functions of logical algebra [6, 7].
Let $s_0, s_1, s_2,\ \ldots,\ s_{r-1}, \ldots$ --
be the PRS elements, satisfying the recurrent equation (1).
Because any element $s_n$ $(n\geq r)$ of the sequence $s_0, s_1, s_2,\ \ldots,\ s_{r-1},\ \ldots\ $
are calculated recursively on the basis of known $r$ elements,
let us represent the elements of PRS section $s_{n+r}, s_{n+r+1},\ \ldots,\ s_{n+2r-1}$ with the length $r$
as the system of characteristic equations:
\begin{align}
\begin{cases}
s_{n+r}= p_{r-1}s_{n+r-1}\oplus p_{r-2}s_{n+r-2}\oplus\ \ldots\ \oplus p_0s_{n} \pmod q, \\
s_{n+r+1}= p_{r-1}s_{n+r}\oplus p_{r-2}s_{n+r-1}\oplus\ \ldots\ \oplus p_0s_{n+1} \pmod q, \\
\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\
s_{n+2r-1}= p_{r-1}s_{n+2r-2}\oplus p_{r-2}s_{n+2r-3}\oplus\ \ldots \ \oplus p_0s_{n+r-1} \pmod q,
\end{cases}
\end{align}
where $[s_{n+r}\ s_{n+r+1} \ \ldots \ s_{n+2r-1}]$ --- is the vector of PRS $r$-condition (or inner condition of $q$-LFSR on the $r$-cycle).
By the analogy with [5], let us express the right parts of the system (2) through the given initial condition:
\begin{align}
\begin{cases}
s_{n+r}=p_{r-1}s_{n+r-1}\oplus p_{r-2}s_{n+r-2}\oplus \ldots\ \oplus p_0s_n \pmod {q}, \\
s_{n+r+1}=p_{r-1}(p_{r-1}s_{n+r-1}\oplus p_{r-2}s_{n+r-2}\oplus \ldots \\
\ldots \oplus p_0s_n)\oplus p_{r-2}s_{n+r-1}\oplus \ldots \oplus p_0s_{n+1} \pmod {q}, \\
\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\
s_{n+2r-1}= p_{r-1}(p_{r-1}(p_{r-1}s_{n+r-1}\oplus p_{r-2}s_{n+r-2}\oplus \ldots\ \oplus p_0s_n)\oplus\\
\oplus p_{r-2}s_{n+r-1}\oplus \ldots \oplus p_0s_{n+1}) \oplus p_{r-2}(p_{r-1}s_{n+r-1} \oplus p_{r-2}s_{n+r-2} \oplus \ldots\\
\ldots \oplus p_0s_n)\oplus
\ldots \oplus p_{0}(p_{r-1}s_{n+r-1}\oplus p_{r-2}s_{n+r-2}\oplus \ldots \oplus p_0s_n)\ \ (\mathrm{mod}\ q).
\end{cases}
\end{align}
Let represent the system (3) as the system $r$ MVFLA or of $r$-variables:
\begin{align}\label{4}
\begin{cases}
f_{1}(s_{n+r-1}, \ldots, s_n)=p_{r-1}^{(0)}s_{n+r-1}\oplus p_{r-2}^{(0)}s_{n+r-2}\oplus \ldots \oplus p_0^{(0)}s_n \pmod q, \smallskip \\
f_{2}(s_{n+r-1}, \ldots, s_n) = p_{r-1}^{(1)}s_{n+r-1}\oplus p_{r-2}^{(1)}s_{n+r-2}\oplus\ldots \oplus p_0^{(1)}s_n \pmod q, \\
\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\
f_{r}(s_{n+r-1}, \ldots, s_n) = p_{r-1}^{(r-1)}s_{n+r-1}\oplus p_{r-2}^{(r-1)}s_{n+r-2}\oplus \ldots \oplus p_0^{(r-1)}s_{n} \pmod q.
\end{cases}
\end{align}
Coefficients $p_{i}^{(j)}\in \{0,\ 1,\ \ldots, \ q-1\}$ are formed after reduction formulas (3).
Structural diagram of the parallel operation of the generator in accordance with formula (4) has the form (see Fig. 2)
\begin{figure}[htb]
\begin{center}\LARGE{
\resizebox{0.99\linewidth}{!}{
\colorbox[gray]{.92}{
$$
\xymatrix{
\ar@{-}[r] & s_{n} \ar@{-}[rrrrrrrrrr] & & & &\ar@{->}[ddddddd] & & & & & &\ar@{->}[ddddddd] \\
\ar@{--}[rrrrrrrrrr] & & & &\ar@{--}[ddddddd] & & & & & &\ar@{--}[ddddddd]& \\
\ar@{-}[r] &s_{n+r-2}\ar@{-}[rrrrrrrr] & & \ar@{->}[ddd] & & & & & &\ar@{->}[ddd] & & \\
\ar@{-}[r] &s_{n+r-1}\ar@{-}[rrrrrrr] &\ar@{->}[d] & & & & & &\ar@{->}[d] & & & \\
&p_{r-1}^{(r-1)} \ar@{->}[r] &\bigotimes\ar@{-}[dddd] & & & & &p_{r-1}^{(0)} \ar@{->}[r] &\bigotimes\ar@{-}[dddd] & & & \\
&p_{r-2}^{(r-1)} \ar@{->}[rr] & &\bigotimes\ar@{->}[dddd] & & & &p_{r-2}^{(0)} \ar@{->}[rr] & &\bigotimes\ar@{->}[dddd] & & \\
&\cdots & & & & & & \cdots & & & & \\
&p_{0}^{(r-1)} \ar@{->}[rrrr] & & & &\bigotimes\ar@{-}[d] & &p_{0}^{(0)} \ar@{->}[rrrr] & & & &\bigotimes\ar@{-}[d] \\
& &\ar@{->}[dr] & &\ar@{-->}[dl] & \ar@{->}[dll] & & &\ar@{->}[dr] & &\ar@{-->}[dl] &\ar@{->}[dll] \\
& & & \bigoplus \ar@{-}[d] & & &\cdots\cdots & & & \bigoplus \ar@{->}[dd] & & \\
\ar@{=}[uuuuuuuuuu]& & & \ar@{->}[d] & &\ar@{=}[lllll] & &\ar@{==}[ll] & &\ar@{=}[ll] & & \\
& & &s_{n+2r-1} & & & & & & s_{n+r} & &
}
$$
}
}}
\end{center}
\begin{center}
\caption{Structural diagram of the operation of the parallel $q$-LFSR in accordance with the formula (4)}
\end{center}
\end{figure}
We know that the arbitrary MVFLA may be represented as arithmetical polynomial defines as [7-- 9]:
\begin{align}\label{5}
A(S)=\sum_{i=0}^{q^{r-1}-1}a_{i}\ s_{n}^{i_0}s_{n+1}^{i_1}\ \ldots\ s_{n+r-1}^{i_{r-1}},
\end{align}
where $a_i$ is the $i$-ratio of arithmetical polynomial; $S=s_n,\ s_{n+1},\ \ldots,\ s_{n+r-1}$ are the arguments of MVFLA $s_u \in {0, 1, \ldots, q-1}$ $(u=0, 1, \ldots, r-1);$
$(i_0 \ i_1 \ \ldots \ i_{r-1})_q$ is the representation of the $i$ parameter in the $q$-ary notation system:
\begin{align*}
(i_0 \ i_1 \ \ldots \ i_{r-1})_q&=\sum_{u=0}^{r-1}i_{u}q^{r-u-1}~~~~(i_{u}\in{0, 1, \ldots, q-1});\\
s_{u}^{i_{u}}&=
\begin{cases}
1,& i_u=0,\\
s_u,& i_u\neq 0.
\end{cases}
\end{align*}
By analogy with [8] we may realize the MVFLA system (4) by calculation of some arithmetical polynomial.
To do that, let us coordinate MVFLA (4) system with the system of arithmetical polynomials (5). Then we get:
\begin{align}\label{6}
\begin{cases}
A_1(S)=\sum_{i=0}^{q^{r-1}-1}a_{1, i}\ s_{n}^{i_0}s_{n+1}^{i_1}\ \ldots\ s_{n+r-1}^{i_{r-1}},\\
A_2(S)=\sum_{i=0}^{q^{r-1}-1}a_{2, i}\ s_{n}^{i_0}s_{n+1}^{i_1}\ \ldots\ s_{n+r-1}^{i_{r-1}},\\
\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\
A_r(S)=\sum_{i=0}^{q^{r-1}-1}a_{r, i}\ s_{n}^{i_0}s_{n+1}^{i_1}\ \ldots\ s_{n+r-1}^{i_{r-1}}.
\end{cases}
\end{align}
Let multiple the polynomials of the system (6) by weights
$q^{l-1}$ $(l=1,\, 2,\, \ldots,\, r)$:
\begin{align*}
\begin{cases}
A_1^*(S)=q^{0}A_{1}(S)=\sum_{i=0}^{q^{r-1}-1}a_{1, i}^{*}\ s_{n}^{i_0}s_{n+1}^{i_1}\ \ldots\ s_{n+r-1}^{i_{r-1}}, \\
A_2^*(S)=q^{1}A_{2}(S)=\sum_{i=0}^{q^{r-1}-1}a_{2, i}^{*}\ s_{n}^{i_0}s_{n+1}^{i_1}\ \ldots\ s_{n+r-1}^{i_{r-1}},\\
\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\
A_r^*(S)=q^{r-1}A_{r}(S)\sum_{i=0}^{q^{r-1}-1}a_{r, i}^{*}\ s_{n}^{i_0}s_{n+1}^{i_1}\ \ldots\ s_{n+r-1}^{i_{r-1}},
\end{cases}
\end{align*}
where $a^{*}_{l,\, i}=q^{l-1}a_{l,\, i}$ $(l=1, 2, \ldots, r;$ $i=0, 1, \ldots, q^r-1).$
Then we get:
\begin{align*}
D(S)=\sum_{i=0}^{q^{r-1}-1}\sum_{l=1}^{d}a^{*}_{l, i}\ s_{n}^{i_0}s_{n+1}^{i_1}\ \ldots\ s_{n+r-1}^{i_{r-1}}.
\end{align*}
According to paper [9] the modular form of an arithmetical polynomials can be received:
\begin{align}
\boxed{
M(S)=\bigoplus_{i=0}^{q^{r-1}-1}c_{i}\ s_{n}^{i_0}s_{n+1}^{i_1}\ \ldots\ s_{n+r-1}^{i_{r-1}} \pmod {q^r},
}
\end{align}
where
$$c_i=\bigoplus_{l=1}^{r}a^{*}_{l,\, i}\ \ (i=0,\, 1,\, \ldots,\, q^{r-1}-1).$$
Let computing the values of the required MVFLA.
To do that, we should represent the result of calculation of MVFLA in $q$-valued notation system and apply the masking operator $\Xi^t\{M(S)\}$ [9]:
$$\Xi^t\{M(S)\}=\left \lfloor \frac{M(S)}{q^t} \right \rfloor \pmod {q},$$
where $t$ is the required $q$-stage of the representation $M(S)$. Structural diagram of the parallel operation of the generator in accordance with formula (7) has the form (see Fig. 3).
\begin{figure}[h]
\begin{center
\resizebox{0.7\linewidth}{!}{
\colorbox[gray]{.92}{
$$
\xymatrix{
\ar@{=}[rrrr] &\ar@{-}[d] & \ar@{-}[d] &\ar@{--}[d] &\ar@{-}[d] \\
&s_{n+r-1} \ar@{-}[d] &s_{n+r-2} \ar@{-}[d] & \ar@{--}[d] & s_{n} \ar@{-}[d] \\
&\ar@{->}[dr] & \ar@{->}[d] & \ar@{-->}[dl] &\ar@{->}[dll] \\
& &M(S)\ar@{-}[ld] \ar@{-}[d] \ar@{--}[dr] \ar@{-}[drr] & & \\
& \ar@{-}[d] & \ar@{-}[d] & \ar@{--}[d] & \ar@{-}[d] \\
& \Xi^{r-1} \ar@{->}[dd] & \Xi^{r-2} \ar@{->}[dd] & \ar@{-->}[dd] & \Xi^{0} \ar@{->}[dd] \\
\ar@{=}[uuuuuu]\ar@{=}[rrrr] & & & & \\
& s_{n+2r-1} & s_{n+2r-2} & \ldots\ldots & s_{n+r}
}
$$
}
}
\end{center}
\caption{Structural diagram of the operation of the parallel $q$-LFSR in accordance with the arithmetic polynomials (7)}
\end{figure}
\section{Numerical example}
Let examine the construction $q=3$ LFSR, generating 3-digit PRS given by characteristic equation: $s_{k+3}=2s_{k+2}\oplus s_k \pmod {3}$ and initial fill: $s_0=0,\ s_1=1,\ s_2=2.$
The corresponding characteristic polynomial is represented as: $P(z)=z^3+2z^2+1.$
In this case the system of characteristic equations for the PRS section of three elements will be represented as follows:
\begin{align*}
\begin{cases}
{s_{3}=2s_2\oplus s_0 \pmod {3},} \\
{s_{4}=2s_3\oplus s_1 \pmod {3},} \\
{s_{5}=2s_4\oplus s_2 \pmod {3}.}
\end{cases}
\end{align*}
Then let us represent the system of characteristic equations as the MVFLA system with right part of equalities, expressed by means of initial given conditions:
\begin{align*}
\begin{cases}
f_{3}(s_2, s_1, s_0)=2s_2\oplus s_0~~({\rm mod}~3), \\
f_{4}(s_2, s_1, s_0)=s_2\oplus s_1\oplus 2s_0~~({\rm mod}~3),\\
f_{5}(s_2, s_1, s_0)=s_0\oplus 2s_1 \pmod {3}.
\end{cases}
\end{align*}
According to (6) we shall get the system of arithmetical polynomials as follows:
\begin{align*}
\begin{cases}
A_{3}(S)=
\frac{1}{4} (14s_2-6s_2^2+4s_0-39s_2s_0+
21s_0s_2^2+15s_0^2s_2 - 9s_0^2s_2^2), \medskip \\
A_{4}(S)=\frac{1}{8} (8s_2+8s_1+42s_1s_2-30s_1s_2^2-30s_2s_1^2 +18s_1^2s_2^2+28s_0-78s_0s_2+\\
+30s_0s_2^2-78s_0s_1+78s_0s_1s_2+30s_0s_1^2-18s_0s_1^2
s_2^2-
12s_0^2
+42s_0^2s_2-\\
-18s_0^2s_2^2+42s_0^2s_1-72s_0^2s_1s_2+ 18s_0^2s_1s_2^2-18s_0^2s_1^2+
18s_0^2s_2s_1^2), \medskip \\
A_{5}(S)=
\frac{1}{4}(14s_1-6s_1^2+4s_0-39s_1s_0+ 21s_0s_1^2+15s_0^2s_1- 9s_0^2s_1^2).
\end{cases}
\end{align*}
Let realize the system of arithmetical expressions as arithmetical polynomial:
\begin{align*}
\begin{cases}
& D(S)=\frac{1}{4}(14s_2-6s_2^2+4s_0-39s_2s_0+21s_0s_2^2+15s_0^2s_2- 9s_0^2s_2^2)+\\
&+3^{1}(\frac{1}{8}(8s_2+8s_1+42s_1s_2-30s_1s_2^2- 30s_2s_1^2 +18s_1^2s_2^2+28s_0-78s_0s_2+\\
&+30s_0s_2^2- 78s_0s_1+78s_0s_1s_2+30s_0s_1^2-18s_0s_1^2s_2^2-12s_0^2+42s_0^2s_2-18s_0^2s_2^2+\\
&+42s_0^2s_1-72s_0^2s_1s_2+18s_0^2s_1s_2^2-18s_0^2s_1^2+18s_0^2s_2s_1^2))+3^{2}(\frac{1}{4}(14s_1- 6s_1^2+\\
&+4s_0-39s_1s_0+21s_0s_1^2+15s_0^2s_1-9s_0^2s_1^2)).
\end{cases}
\end{align*}
Modular polynomial form will be expressed as:
\begin{align*}
& M(S)= 7s_0\oplus 9 s_0^2\oplus 21 s_1\oplus 18 s_0 s_1\oplus 9 s_0^2 s_1\oplus 18 s_0 s_1^2\oplus 20 s_2\oplus 15 s_0 s_2\oplus 6 s_0^2 s_2\oplus\\
& \oplus 9 s_1 s_2\oplus 9 s_0 s_1 s_2\oplus 9 s_1^2 s_2\oplus 12 s_2^2\oplus 3 s_0 s_2^2\oplus 18 s_0^2 s_2^2\oplus 9 s_1 s_2^2 \pmod {27}.
\end{align*}
According to the given initial conditions we may obtain the following three-digit fragment of PRS:
\begin{align*}
\mbox{step}~1
\begin{cases}
s_{3}= \Xi^{0}\{19\}=1,\\
s_{4}=\Xi^{1}\{19\}=0,\\
s_{5}=\Xi^{2}\{19\}=2;
\end{cases}
&&
\mbox{step}~5
\begin{cases}
s_{15}=\Xi^{0}\{5\}=2,\\
s_{16}=\Xi^{1}\{5\}=1,\\
s_{17}=\Xi^{2}\{5\}=0;
\end{cases}
\\
\mbox{step}~2
\begin{cases}
s_{3}= \Xi^{0}\{14\}=2,\\
s_{4}=\Xi^{1}\{14\}=1,\\
s_{5}=\Xi^{2}\{14\}=1;
\end{cases}
&&
\mbox{step}~6
\begin{cases}
s_{3}= \Xi^{0}\{17\}=2,\\
s_{4}=\Xi^{1}\{17\}=2,\\
s_{5}=\Xi^{2}\{17\}=1;
\end{cases}
\\
\mbox{step}~3
\begin{cases}
s_{3}= \Xi^{0}\{10\}=1,\\
s_{4}=\Xi^{1}\{10\}=0,\\
s_{5}=\Xi^{2}\{10\}=1;
\end{cases}
&&
\mbox{step}~7
\begin{cases}
s_{3}= \Xi^{0}\{4\}=1,~\\
s_{4}=\Xi^{1}\{4\}=1,~\\
s_{5}=\Xi^{2}\{4\}=0;~
\end{cases}
\\
\mbox{step}~4
\begin{cases}
s_{3}= \Xi^{0}\{9\}=0,~\\
s_{4}=\Xi^{1}\{9\}=0,~\\
s_{5}=\Xi^{2}\{9\}=1;~
\end{cases}
&&
\mbox{step}~8
\begin{cases}
s_{3}= \Xi^{0}\{19\}=1,\\
s_{4}=\Xi^{1}\{19\}=0,\\
s_{5}=\Xi^{2}\{19\}=0;
\end{cases}
\\
&&
\cdots \cdots\cdots\cdots\cdots\cdots\cdots\cdots .
\end{align*}
\section{Conclusion}
Here is the representation of one of the possible non-standard methods of realization of parallel algorithm of generation of $q$-valued PRS, based on the arithmetical representation of MVFLA. The developed algorithm may be used for the realization of perspective high-performance cryptographic facilities for information protection. The further direction of the research is the realization of the developed algorithm of generation of $q$-valued PRS using the redundant code redundant number system, which provide control over the errors while computing the PRS elements.
|
1,116,691,499,339 | arxiv | \section{Introduction}
It is well-known that the concept of a measurement in quantum physics challenges our everyday intuition.
In a classical theory objects have properties, whether we look at them or not, and a measurement simply reveals to us their pre-existing values.
In quantum mechanics, on the other hand, performing a measurement is an invasive process, which necessarily disturbs the state (except for some special cases).
Moreover, even if we have complete knowledge about the system, we can only predict the probabilities of different outcomes, which can be computed using the Born rule.
An intriguing consequence of the quantum formalism is the existence of measurements that are \emph{incompatible}, i.e., that cannot be measured simultaneously given only one copy of the system.
The best known example consists of the position and momentum of a quantum mechanical particle, which cannot be measured simultaneously with arbitrary precision.
In this work we study the incompatibility of measurements with a finite number of outcomes.
These measurements assign to each physical state $\rho$ a discrete probability distribution $\{p_a(\rho)\}_a$, whose elements we interpret as the probability of outcome $a$ on the state $\rho$.
We say that two measurements are \emph{compatible} (or \emph{jointly measurable}) if there exists a single measurement, referred to as the \emph{parent measurement}, that is able to universally replace the two \cite{Lud54,BLPY16}.
More specifically, on any state the outcome probabilities of both measurements can be recovered from the outcome probabilities of the parent measurement.
Therefore, the two measurements can be performed simultaneously by performing the parent measurement.
If such a parent measurement does not exist, we say that the measurements are \emph{incompatible} (or \emph{not jointly measurable}).
We remark here that other notions of compatibility, such as commutativity, non-disturbance and coexistence, are also used in the literature \cite{Lud54,HW10}; let us for completeness briefly explain how they are related.
Commutativity of a measurement pair implies non-disturbance, which in turn implies joint measurability, which then implies coexistence.
Moreover, it is known that none of the converse implications hold in general, therefore these notions are strictly distinct \cite{RRW13}.
In this work we focus solely on the notion of joint measurability, because the existence (or not) of a parent measurement has a clear operational meaning.
Therefore,
throughout the present paper we use the terms {``(in)compatibility''} and {``(non-)joint measurability''} interchangeably.
It is important to notice that whenever two measurements are compatible, they cannot be used to produce quantum advantage in tasks like Bell nonlocality \cite{WPF09} or Einstein--Podolsky--Rosen steering \cite{QVB14,UBGP15}.
Moreover, it was recently shown that joint measurability is equivalent to a specific notion of classicality, namely, preparation non-contextuality \cite{TU19,GQA19}.
Hence, one may think of compatible measurements as ``classical'', and incompatible measurements as a resource for the above tasks. Therefore, it is of fundamental importance to characterise and understand the structure of incompatible measurements.
What is particularly important is to go beyond the dichotomy of compatible and incompatible measurements, and quantify \emph{to what extent} a pair of measurements is incompatible.
A natural framework for this quantification, often used in the literature, is to define measures based on robustness to noise.
Briefly speaking, robustness-based measures of incompatibility quantify the minimal amount of noise that needs to be added to a pair of measurements to make them compatible.
The more noise is required, the more incompatible the measurements are.
Note that measures of this type are directly relevant to experiments, because in real-world implementations measurements are always noisy, due to inevitable experimental imperfections.
Robustness-based measures are also natural measures of incompatibility in the context of resource theories \cite{CFS16,Fri17}.
Here one considers a set of ``free'' objects (compatible measurements) and quantify the usefulness of ``resource'' objects (incompatible measurements) by so-called resource monotones.
While in this work we do not develop a full resource theory of incompatibility, we note that robustness-based measures are good candidates for resource monotones if they satisfy certain natural properties~\cite{HKR15,SSC19,CG19,OB19}.
In resource theories one defines ``free operations'' that do not create resource (that is, do not map compatible measurements to incompatible ones).
Properly defined resource monotones should then be monotonic under such free operations.
Once measures with the desired properties are found, the question ``what are the most incompatible pairs of measurements?'' is well-defined with respect to each of these measures.
Several robustness-based measures have been proposed in the literature (see Ref.~\cite{HMZ16} for an introduction),
the essential difference between them being the assumed noise model.
Nevertheless, some basic properties of these measures have not been determined and little effort has been dedicated to understanding the similarities and differences among them.
In this work we make the following contributions to fill this gap.
\begin{itemize}
\item We develop a framework in which a robustness-based measure can be defined with respect to an arbitrary noise model.
We identify the minimum assumptions on the noise model that ensure that the resulting measure satisfies some basic requirements, i.e., we provide an explicit connection between the properties of the noise model and the desired properties of the measure.
\item We apply our framework to study five measures already introduced in the literature in a unified fashion.
By giving explicit counterexamples we show that some widely used measures do not satisfy certain natural properties motivated by resource theories.
\item We show that when looking for the most incompatible pairs, we obtain different answers depending on the specific measure of incompatibility.
For one of the measures we analytically prove that mutually unbiased bases are among the most incompatible pairs of measurements in every dimension.
For three other measures we can explicitly show that, for dimensions larger than two, mutually unbiased bases are \emph{not} among the most incompatible pairs.
Our study for the last measure is inconclusive.
\end{itemize}
In Section~\ref{sec:prelim} we define incompatibility robustness in a fashion that is independent of the specific noise model, introduce the natural properties that the measures should desirably satisfy and relate them to the properties of the noise model, formulate the notion of most incompatible measurement pairs, and discuss the measures' semidefinite programming formulation and how to use this formulation to derive bounds on them.
Then in Section~\ref{sec:measures} we introduce the five measures already used in the literature, illustrate them on a simple example, analyse their relevant properties, and derive new bounds on each of them.
At the end of this section we discuss the relations between the measures, apply our results to compute all the different measures for mutually unbiased bases, then summarise the main results in a compact form.
In Section~\ref{sec:most} we address the question of the most incompatible pairs of measurements under the five measures.
Finally, in Section~\ref{sec:conclusion} we summarise the new findings and pose some important open questions arising from our work.
We note here that the notion of incompatibility naturally generalises to more than two measurements, but for simplicity in the main text we restrict ourselves to pairs of measurements.
For a formal treatment of larger sets of measurements, and results regarding them, we refer the interested reader to Appendix~\ref{app:more}.
\section{Definitions and basic properties}
\label{sec:prelim}
In this section we formalise the main definitions and concepts outlined in the introduction.
We give a mathematically precise definition of (in)compatibility and of robustness-based measures for an arbitrary noise model.
Then we specify a few natural properties the measures should satisfy, and give concrete conditions on the noise model under which these are automatically fulfilled.
We also rigorously formulate the notion of ``most incompatible measurements'', and discuss how to efficiently search for them.
Finally, we introduce the notion of semidefinite programming, and how to use it to derive bounds on robustness-based measures.
\subsection{Incompatible measurements}
\label{subsec:jm}
Throughout this paper we analyse the most general model of quantum measurements, positive operator valued measures (POVMs).
For this model, we establish that the physical system lives on a $d$-dimensional Hilbert space, $\mathcal{H}\simeq\mathbb{C}^d$.
The relevant objects are all elements of the set of linear operators on this space, $\mathcal{B}(\mathbb{C}^d)$.
The state of the system is described by a positive semidefinite operator with unit trace, denoted by $\rho$.
A POVM with $n$ outcomes is a set of $n$ positive semidefinite operators, $\{A_a\}_{a=1}^n$, such that $\sum_{a=1}^nA_a = \mathbb{I}$, where $\mathbb{I}$ is the identity operator.
The probability of observing outcome $a$ is given by the Born rule, $p_a(\rho) = \tr(A_a\rho)$.
In the following, we will use the terms ``measurement'' and ``POVM'' interchangeably.
We will often refer to the following three important classes of POVMs.
\emph{Rank-one} POVMs are measurements whose elements are rank-one operators, $A_a \propto \ketbra{\varphi_a}$, where $\ketbra{\varphi_a}$ is the projector onto $\ket{\varphi_a}\in\mathbb{C}^d$.
Note that such measurements cannot have fewer elements than the dimension of the Hilbert space, that is, $n\geq d$ with the above notation.
\emph{Projective} measurements are POVMs whose elements are projectors.
Note that such measurements cannot have more non-zero elements than the dimension of the Hilbert space.
Since the set of measurements with $n$ outcomes acting on dimension $d$ is a convex set, we will talk about \emph{extremal} POVMs (in the convex geometry sense).
Recall that every POVM can be written as a convex combination of extremal POVMs
and these have been extensively studied in Ref.~\cite{DAr04}.
The ability to recover the outcome probabilities of two POVMs on any state from the statistics of a single measurement is referred to as \emph{joint measurability} and can be formulated in the following way.
\begin{defn}
\label{def:jm}
Given two POVMs, $\{A_a\}_{a=1}^{n_A}$ and $\{B_b\}_{b=1}^{n_B}$, we say that they are \emph{jointly measurable} (or \emph{compatible}) if there exists a POVM $\{G_{ab}\}_{a=1,b=1}^{n_A,n_B}$ such that $\sum_{b=1}^{n_B} G_{ab} = A_a$ for all $a$, and $\sum_{a=1}^{n_A} G_{ab} = B_b$ for all $b$.
We call such a POVM a \emph{parent} measurement of $\{A_a\}_{a=1}^{n_A}$ and $\{B_b\}_{b=1}^{n_B}$.
\end{defn}
This definition captures the idea that the parent measurement provides a joint outcome distribution of the two initial measurements on every state.
It is worth pointing out that the notion of joint measurability in which the parent POVM is allowed an arbitrary (finite) outcome set and arbitrary classical post-processing turns out to be equivalent to the one above (see e.g., Ref.~\cite[Section~3.1]{HMZ16}).
We note that a parent POVM is not necessarily unique for a fixed pair of measurements~\cite{HRS08,GC18}.
It is clear that in order to recover the outcome probabilities of $A$ and $B$, one only needs to measure $G$ and add up the relevant probabilities (in the following we sometimes drop the outcome indices to refer to the POVMs, when it does not lead to confusion; this notation is to be understood as $A = \{A_a\}_{a=1}^{n_A}$).
A simple example of a jointly measurable pair is the trivial measurement pair, $\{\frac{\mathbb{I}}{n_A}\}_{a=1}^{n_A}$ and $\{\frac{\mathbb{I}}{n_B}\}_{b=1}^{n_B}$ with the parent POVM $\{\frac{\mathbb{I}}{n_An_B}\} _{a=1,b=1}^{n_A,n_B}$.
In fact any POVM pair with pairwise commuting measurement operators, $[A_a,B_b] = 0$ for all $a$ and $b$, is jointly measurable.
This can be seen by employing the parent POVM $G$ with elements $G_{ab} = A_aB_b$, which is guaranteed to be positive in this case.
Note that commutativity becomes necessary and sufficient if one of the two measurements is projective, see Ref.~\cite[Proposition 8]{HRS08} for a proof.
If a parent POVM does not exist, we say that $A$ and $B$ are \emph{not jointly measurable} (or \emph{incompatible}).
A standard example of incompatible $d$-outcome measurement pairs in dimension $d\ge 2$ is a pair of projective measurements onto two \emph{mutually unbiased bases} (MUBs) \cite{DEBZ10}.
These consist of rank-one projectors $A^\mathrm{MUB}=\{\ketbra{\varphi_a}\}_{a=1}^d$ and $B^\mathrm{MUB}=\{\ketbra{\psi_b}\}_{b=1}^d$ onto the orthonormal bases $\{\ket{\varphi_a}\}_{a=1}^d$ and $\{\ket{\psi_b}\}_{b=1}^d$, such that all the pairwise overlaps (moduli of inner products) are uniform: $|\braket{\varphi_a}{\psi_b}| = 1/\sqrt{d}$ for all $a,b$.
As these measurements are projective and non-commuting, they are incompatible.
In the following we will denote the set of POVM pairs with outcome numbers $n_A$ and $n_B$ in dimension $d$ by $\POVM_d^{n_A,n_B}$, and its elements by $(A,B)$.
Note that POVM pairs inherit the convex structure of POVMs (denoted by $\POVM_d^n$), therefore convex combinations of them are well-defined.
For the subset corresponding to jointly measurable pairs, we will use the notation $\JM_d^{n_A,n_B}$, but drop the indices whenever it does not lead to confusion.
Note that the set $\JM^{n_A,n_B}_d$ is a convex subset of $\POVM^{n_A,n_B}_d$: it is straightforward to verify that if $(A^0,B^0)\in\JM^{n_A,n_B}_d$ with parent POVM $G^0$, and $(A^1,B^1)\in \JM^{n_A,n_B}_d$ with parent POVM $G^1$, then $(1-p)(A^0,B^0) + p(A^1,B^1)\in \JM^{n_A,n_B}_d$ with parent POVM $(1-p)G^0 + pG^1$ for all $p\in[0,1]$.
That is, taking convex combinations preserves joint measurability.
\subsection{Incompatibility robustness}
\label{subsec:ir}
In order to talk about noisy measurements, we define what we mean by a \emph{noise model}.
\begin{defn}\label{def:noise}
A \emph{noise model} ${\bf N}$ is a map ${\bf N}:\POVM_d^n \to \mathbb{P}(\POVM_d^n)$, where $\mathbb{P}$ is the set of all subsets, that maps every POVM $A\in\POVM_d^n$ to a subset of all $n$-outcome POVMs in dimension $d$, that is, ${\bf N}:A\mapsto{\bf N}_A \subseteq \POVM_d^n$. We will refer to ${\bf N}_A$ as the \emph{noise set} of $A$ under this noise model.
\end{defn}
Given a noise model, we can define \emph{noisy versions} of POVMs as convex combinations of POVMs with elements of their corresponding noise sets. Specifically, if $M\in {\bf N}_A$ and $\eta \in [0,1]$, then a noisy version of $A$ with \emph{visibility} $\eta$ is the POVM
\begin{equation}\label{eqn:noisy}
\eta A + (1-\eta)M \in \POVM_d^n.
\end{equation}
Noise models will be crucial for our analysis, as different noise models give rise to different measures of incompatibility.
Initially, for a unified treatment of robustness based measures, we will discuss properties that do not depend on the precise choice of the noise model, and only introduce explicit choices in Section \ref{sec:measures}, where we analyse the five specific measures.
In order to apply it to incompatibility, we extend the concept of a noise model to pairs of measurements: in this case, the noise model ${\bf N}$ is a map ${\bf N}:\POVM_d^{n_A,n_B}\to\mathbb{P}(\POVM_d^{n_A,n_B})$ that maps every pair $(A,B)\in\POVM_d^{n_A,n_B}$ to its corresponding noise set, ${\bf N}:(A,B)\mapsto {\bf N}_{A,B}\subseteq \POVM_d^{n_A,n_B}$.
Note that the set ${\bf N}_{A,B}$ may actually depend on the measurements $A$ and $B$, and not simply on their dimension or number of outcomes (whenever the map ${\bf N}$ is not constant).
The simplest example of a noise model is ${\bf N}_{A,B} = \{(\{\frac{\mathbb{I}}{n_A}\}_{a=1}^{n_A},\{\frac{\mathbb{I}}{n_B}\}_{b=1}^{n_B})\}$, that maps every POVM pair to the one-element set containing only the trivial measurement pair.
On the other end of the spectrum, the largest possible choice of the noise model is ${\bf N}_{A,B} = \POVM^{n_A,n_B}_d$, mapping every POVM pair to the set of all POVM pairs.
We will now define a measure of incompatibility corresponding to an arbitrary noise model. To ensure that the measure is well-defined, we require that the map ${\bf N}$ is such that for every pair $(A,B)$ the noise set ${\bf N}_{A,B}$ contains at least one jointly measurable pair.
For any such noise model, one can define an incompatibility robustness measure for pairs of POVMs, i.e.,~the maximal visibility at which the noisy pair is still compatible.
\begin{defn}\label{def:robustness}
Given two POVMs, $\{A_a\}_{a=1}^{n_A}$ and $\{B_b\}_{b=1}^{n_B}$ on $\mathbb{C}^d$, and a noise model ${\bf N}$, we say that the \emph{incompatibility robustness} $\eta^\ast_{A,B}$ of the pair $(A,B)$ with respect to this noise model is
\begin{equation}\label{eq:robustness}
\eta^\ast_{A,B} = \sup_{\hb{\eta\in[0,1]}{(M,N)\in {\bf N}_{A,B}}}\Big\{\eta~\Big|~\eta\cdot(A,B) + (1-\eta)\cdot(M,N)\in\JM_d^{n_A,n_B}\Big\}.
\end{equation}
\end{defn}
This definition has a clear geometric interpretation, see Fig.~\ref{fig:robustness}.
Note that regardless of the choice of the noise model, $\eta^\ast_{A,B}=1$ if and only if $A$ and $B$ are jointly measurable, and that under this definition the lower $\eta^\ast_{A,B}$ is, the more incompatible the measurements are.
\begin{figure}[h!]
\centering
\includegraphics[width=14cm]{fig1.pdf}
\caption{Schematic representation of a generic incompatibility robustness measure for a noise model which maps to closed and convex sets.
Note that in general the noise set ${\bf N}_{A,B}$ need not be contained in the jointly measurable set $\JM$.
One can also easily infer that the optimal noise pair $(M,N)$ must lie on the boundary of ${\bf N}_{A,B}$ and that the optimal noisy pair $\eta^\ast_{A,B}\cdot(A,B) + (1-\eta^\ast_{A,B})\cdot(M,N)$ must lie on the boundary of $\JM$.}
\label{fig:robustness}
\end{figure}
There are several other requirements one might impose on the noise model.
Let us briefly discuss some of these and explain what their consequences are.
\begin{itemize}
\item If we assume that for every pair $(A, B)$, the noise set ${\bf N}_{A,B}$ is \emph{closed}, we are guaranteed that the supremum is achieved, i.e., there exists an optimal noise pair.
In this case the supremum in Eq.~\eqref{eq:robustness} can be replaced by a maximum. Note that since we are dealing with finite-dimensional objects, it is irrelevant which topology we choose to define the notion of closedness.
\item If we assume that for every pair $(A, B)$, the noise set ${\bf N}_{A,B}$ is \emph{convex}, we are guaranteed to find a decomposition of the form given in Eq.~\eqref{eq:robustness} for any $\eta \in [0, \eta^{\ast}_{A, B})$.
It suffices to find a noise pair $(M', N')$ and a visibility $\eta' \ge \eta$ such that
\begin{equation}\label{eqn:convex1}
\eta'\cdot(A,B) + (1-\eta')\cdot(M',N') \in \JM.
\end{equation}
Such $(M',N')$ and $\eta'$ are guaranteed to exist, since $\eta < \eta^\ast_{A,B}$.
Then pick $(M^{\JM{}},N^{\JM{}})\in{\bf N}_{A,B}$ such that
\begin{equation}\label{eqn:convex2}
(M^{\JM{}},N^{\JM{}})\in\JM,
\end{equation}
which is again guaranteed to exist by our fundamental assumption on the noise model.
From the convexity of $\JM$ it follows that taking the convex combination of Eq.~\eqref{eqn:convex1} with weight $\eta/\eta'$ and Eq.~\eqref{eqn:convex2} with weight $(1-\eta/\eta')$ leads to $\eta\cdot(A,B) + (1-\eta)\cdot(M,N) \in \JM$, where
\begin{equation}
(M, N) = \frac{\eta}{ 1 - \eta } \cdot \frac{ 1 - \eta' }{\eta'}\cdot( M', N' ) + \left( 1 - \frac{\eta}{ 1 - \eta } \cdot \frac{ 1 - \eta' }{\eta'} \right)\cdot( M^{\JM{}}, N^{\JM{}} ),
\end{equation}
and the convexity of ${\bf N}_{A,B}$ ensures that $(M, N) \in {\bf N}_{A,B}$.
Note that a looser constraint, namely that ${\bf N}_{A,B}$ is a radial set at $(M^{\JM{}},N^{\JM{}})$ (the line segments between $(M^{\JM{}},N^{\JM{}})$ and all other elements of ${\bf N}_{A,B}$ are contained in ${\bf N}_{A,B}$) is sufficient for this property.
\item Another property one might require from the noise set is \emph{covariance with respect to unitaries}.
Intuitively, this means that if two pairs of measurements are related by a unitary, then so should be their respective noise sets.
More specifically, if $(A, B)$ and $(A', B')$ satisfy
\begin{equation}
A'_{a} = U A_{a} U^{\dagger} \quad \textnormal{and} \quad B'_{b} = U B_{b} U^{\dagger}
\end{equation}
for all outcomes $a$ and $b$ and for some fixed unitary $U$, then
\begin{equation}
(M, N) \in {\bf N}_{A,B} \iff (U M U^{\dagger}, U N U^{\dagger}) \in {\bf N}_{A',B'}.
\end{equation}
This property is sufficient to ensure that the resulting incompatibility robustness measure is unitarily invariant, i.e.~$\eta^{\ast}_{A, B} = \eta^{\ast}_{A', B'}$.
\item Finally, one might require that for every choice of $(A, B)$ the corresponding noise set ${\bf N}_{A, B}$ is \emph{invariant under unitaries}, i.e.,
\begin{equation}
(M, N) \in {\bf N}_{A,B} \implies (U M U^{\dagger}, U N U^{\dagger}) \in {\bf N}_{A, B}
\end{equation}
for every unitary $U$.
An advantage of this property is that if we assume that the noise set is convex, then we can average over the Haar measure on unitary matrices, which leads to a noise pair whose every element is proportional to the identity operator.
We will use this property in Section~\ref{sec:mostincomp} to derive non-trivial lower bounds on the resulting incompatibility measure.
\end{itemize}
The last two properties are clearly related.
Indeed, if the noise set does not depend on the pair $(A, B)$ beyond the dimension and the outcome numbers (the map ${\bf N}$ is constant), they turn out to be equivalent.
However, in full generality these two properties are independent, i.e., we can have one without the other.
To conclude let us simply state that \emph{all the measures considered in this work satisfy all the requirements stated above}.
In Section~\ref{sec:measures}, we will replace the star in $\eta^\ast_{A,B}$ with a reference to the specific noise model in order to make clear which measure we use.
In general we are looking for noise models that give rise to measures of incompatibility that satisfy certain natural properties motivated by resource theories.
\subsection{Monotonicity}
\label{sec:monotonicity}
The natural properties we consider capture the intuition that measures of incompatibility should not decrease under operations that do not create incompatibility.
In other words, measurements should not become more incompatible under such operations.
This is well-motivated from the resource theoretic point of view, allowing for a partial order of measurement pairs based on their incompatibility robustness.
Consider an operation $\Phi:(A,B) \mapsto \Phi(A,B)$, that maps every POVM pair to another POVM pair, not necessarily preserving the dimension or the outcome numbers.
We say that this operation is \emph{joint measurability-preserving} if for all $(A,B)\in\JM$ we have that $\Phi(A,B) \in\JM$.
It is desirable that our measures are non-decreasing under such operations, that is, $\eta^\ast_{\Phi(A,B)} \ge \eta^\ast_{A,B}$ for every joint measurability-preserving operation $\Phi$.
If this inequality holds for every $(A,B)$ we say that $\eta^\ast$ is \emph{monotonic under} $\Phi$.
Whenever the joint measurability-preserving operation $\Phi$ is linear, a simple property of the noise model ${\bf N}$ implies monotonicity, namely, $\Phi( {\bf N}_{A,B}) \subseteq {\bf N}_{\Phi(A,B)}$ for all $(A,B)$.
To see this, consider a measurement pair $(A,B)$ and its corresponding noise set ${\bf N}_{A,B}$.
Following from Definition~\ref{def:robustness}, we have that
\begin{equation}
\eta^\ast_{A,B}\cdot (A,B) + (1-\eta^\ast_{A,B})\cdot(M,N) \in \JM
\end{equation}
for some $(M,N)\in {\bf N}_{A,B}$.
Applying $\Phi$ to the left-hand side, we obtain
\begin{equation}\label{eq:etastarnoise}
\eta^\ast_{A,B}\cdot \Phi(A,B) + (1-\eta^\ast_{A,B})\cdot\Phi(M,N) \in \JM,
\end{equation}
as $\Phi$ is linear and joint measurability-preserving.
Whenever $\Phi( {\bf N}_{A,B}) \subseteq {\bf N}_{\Phi(A,B)}$, the left-hand side of Eq.~\eqref{eq:etastarnoise} is a noisy version of $\Phi(A,B)$ with visibility $\eta^\ast_{A,B}$, which implies that $\eta^\ast_{\Phi(A,B)} \ge \eta^\ast_{A,B}$.
Therefore, if the image of the noise set under $\Phi$ is contained in the noise set of the image for every measurement pair, then $\eta^\ast$ based on this noise model is monotonic under $\Phi$.
In many cases, the stronger property $\Phi( {\bf N}_{A,B}) = {\bf N}_{\Phi(A,B)}$ holds for all $(A,B)$, and then we say that the noise model is \emph{invariant} under $\Phi$.
In this paper we will consider two natural classes of joint measurability-preserving operations, which are transformations of the measurement outputs and inputs.
The first class acts on the outputs of the measurements and is therefore called \emph{post-processings}.
The second class, on the other hand, acts on the inputs (quantum states) of the measurements, and is accordingly called \emph{pre-processings} (see Figs~\ref{fig:postproc} and \ref{fig:preproc}, respectively).
Post-processings amount to recording the outcome of the measurement and then applying a response function to it.
It can therefore be formulated in the following way.
\begin{defn}\label{def:postproc}
A post-processing $\beta$ maps $\{A_a\}_{a=1}^{n_A}$ to $\{A^\beta_{a'}\}_{a'=1}^{n'_A}$, where
\begin{equation}\label{eq:postproc}
A^\beta_{a'} = \sum_{a=1}^{n_A}\beta(a'|a)A_a,
\end{equation}
and $\{\beta(a'|a)\}_{a'}$ is a probability distribution for every $a \in \{1,2,\ldots,n_A\}$.
\end{defn}
\begin{figure}[h!]
\centering
\includegraphics[width=10cm]{fig2.pdf}
\caption{Schematic representation of a post-processing of a measurement.}
\label{fig:postproc}
\end{figure}
A post-processing is called deterministic if the probability distribution $\{\beta(a'|a)\}_{a'}$ is deterministic for all $a \in \{1,2,\ldots,n_A\}$, that is, $\beta(a'|a)\in\{0,1\}$.
If such a post-processing decreases the number of outcomes, it is referred to as \emph{coarse-graining} or \emph{binning}, e.g., the operation mapping the POVM $\{A_1,A_2,A_3\}$ to $\{A_1,A_2+A_3\}$.
What is important is that every POVM can be obtained by coarse-graining a rank-one POVM with potentially more outcomes.
Note that post-processings preserve the dimension but might change the outcome number.
For pairs $(A,B)$ the operation $\Phi^\beta: (A,B) \mapsto (A^{\beta_A}, B^{\beta_B})$ is joint measurability-preserving (note that the post-processings applied to $A$ and $B$ are independent): assume that $(A,B)\in\JM$ with parent POVM $G$.
Then it is straightforward to verify that $(A^{\beta_A}, B^{\beta_B})\in\JM$ with parent POVM $G^\beta$, where $G^\beta_{a'b'} = \sum_{ab}\beta_A(a'|a)\beta_B(b'|b)G_{ab}$.
The second class, pre-processings, amounts to first applying a quantum channel to the measured state and then performing the measurement.
Denoting the channel acting on the state by $\Lambda^\dagger$ (the dual of the map $\Lambda$), we arrive at the following definition.
\begin{defn}\label{def:preproc}
A pre-processing $\Lambda$ maps $\{A_a\}_{a=1}^{n_A}$ to $\{A^\Lambda_a\}_{a=1}^{n_A}$, where
\begin{equation}\label{eq:preproc}
A^\Lambda_a = \Lambda(A_a),
\end{equation}
and $\Lambda:\mathcal{B}(\mathbb{C}^d)\mapsto\mathcal{B}(\mathbb{C}^{d'})$ is a completely positive unital map.
\end{defn}
\begin{figure}[h!]
\centering
\includegraphics[width=10cm]{fig3.pdf}
\caption{Schematic representation of a pre-processing of a measurement.}
\label{fig:preproc}
\end{figure}
Note that, for our formal treatment the unital map $\Lambda$ does only need to be positive (and not necessarily completely positive), although all the positive unital maps appearing in this work are also completely positive.
A well-known example of pre-processings is the one in Naimark's dilation theorem.
This states that for every POVM $A$ on $\mathbb{C}^d$, there exists $d'\in \mathbb{N}$, an isometry $V:\mathbb{C}^d\to\mathbb{C}^{d'}$, and a projective measurement $P$ on $\mathbb{C}^{d'}$ such that $A_a = V^\dagger P_a V$ for all $a$, that is, $ A = P^\Lambda$, where $\Lambda(.) = V^\dagger (.) V$ is a (completely) positive unital map.
That is, every POVM can be obtained by pre-processing a projective measurement acting on a potentially higher dimensional Hilbert space.
Note that pre-processings preserve the outcome number but might change the dimension.
For pairs $(A,B)$ the operation $\Phi^\Lambda: (A,B) \mapsto (A^{\Lambda}, B^{\Lambda})$ is joint measurability-preserving
(in contrast to the case of post-processing, here there is just a single pre-processing applied to both $A$ and $B$):
assume that $(A,B)\in\JM$ with parent POVM $G$.
Then it is straightforward to verify that $(A^{\Lambda}, B^{\Lambda})\in\JM$ with parent POVM $G^\Lambda$.
Note also that an incompatibility measure that is monotonic under pre-processings necessarily satisfies unitary invariance, as already mentioned in Ref.~\cite[Section~C]{HKR15}.
Finally, let us consider another natural operation that preserves joint-measurability, although it is of a different flavour than pre- and post-processings.
Namely, recall that taking convex combinations preserves joint measurability, that is, for any $(A^0,B^0)\in\JM$ and $(A^1,B^1)\in\JM$ we have that $(A^p,B^p) = (1-p)(A^0,B^0) + p(A^1,B^1) \in \JM$ for all $p\in[0,1]$ (see Section \ref{subsec:jm}).
For this reason, it is desirable that our measures do not decrease under taking convex combinations, that is, $\eta^\ast_{A^p,B^p} \ge \min\{\eta^\ast_{A^0,B^0},\eta^\ast_{A^1,B^1}\}$ for all $p \in [0, 1]$, a property sometimes referred to as \emph{quasi-concavity}.
It is easy to see that this condition holds whenever the noise model satisfies the simple property that, using the above notation, for any $(M^0,N^0)\in {\bf N}_{A^0,B^0}$ and $(M^1,N^1)\in {\bf N}_{A^1,B^1}$, we have $(M^p,N^p) = (1-p)(M^0,N^0) + p(M^1,N^1) \in {\bf N}_{A^p,B^p}$.
To see this, let us define $\eta^\ast_{\min} = \min\{\eta^\ast_{A^0,B^0},\eta^\ast_{A^1,B^1}\}$.
From the convexity of the noise set, there exist $(M^0,N^0)\in {\bf N}_{A^0,B^0}$ and $(M^1,N^1)\in {\bf N}_{A^1,B^1}$ such that $\eta^\ast_{\min}\cdot(A^0,B^0) + (1-\eta^\ast_{\min})\cdot(M^0,N^0)\in \JM$ and $\eta^\ast_{\min}\cdot(A^1,B^1) + (1-\eta^\ast_{\min})\cdot(M^1,N^1)\in \JM$ (see Section \ref{subsec:ir}).
Taking a convex combination of these two relations with coefficients $1-p$ and $p$, respectively, results in $\eta^\ast_{\min}\cdot(A^p,B^p) + (1-\eta^\ast_{\min})\cdot(M^p,N^p)\in \JM$, that is, $\eta^\ast_{A^p,B^p} \ge \min\{\eta^\ast_{A^0,B^0},\eta^\ast_{A^1,B^1}\}$.
All the noise models considered in this paper satisfy the requirement stated above and therefore the corresponding measures are non-decreasing under convex combinations.
A stronger property that is often desired is \emph{joint concavity}, which using the above notation reads $\eta^\ast_{A^p,B^p} \ge p\eta^\ast_{A^0,B^0} + (1-p)\eta^\ast_{A^1,B^1}$ (note that throughout this paper we will write ``concavity'' and ``convexity'' instead of ``joint concavity'' and ``joint convexity'', for simplicity).
However, what one naturally deduces by looking at the noise model turns out to be slightly different.
More specifically, if the noise set is convex for every pair and the noise model is a constant map we may conclude that the inverse of the measure is convex, i.e., $1/\eta^\ast_{A^p,B^p} \leq (1-p)/\eta^\ast_{A^0,B^0} + p/\eta^\ast_{A^1,B^1}$, similarly to the proof in Ref.~\cite[Proposition 2]{Haa15}.
It is easy to see that the concavity of $\eta^{\ast}$ implies that $1/\eta^{\ast}$ is convex \cite[Eq.~(3.11)]{BV04}, but the converse does not hold in general.
In fact, in Appendix~\ref{app:ctrex}, using an explicit counterexample, we show that none of the measures studied in this paper are concave.
It is common to use the measure $t^\ast=1/\eta^\ast-1$ instead of $\eta^\ast$ because it is easy to prove its convexity, and it also has the appealing property that it vanishes for every $(A,B) \in \JM$ (a property referred to as \emph{faithfulness} in Ref.~\cite{SL19} --- also note that in \cite{SOCH+19}, faithfulness, post-processing monotonicity and convexity were postulated as natural properties of any measure of incompatibility).
Moreover, whenever $\eta^\ast$ is monotonic under pre- or post-processings, then so is $t^\ast$ (with opposite relation in the inequality defining monotonicity).
Nevertheless, in the following we will study $\eta^\ast$ since it suits our purposes better and it is easily interconvertible with~$t^\ast$.
In Section~\ref{sec:measures}, we will investigate the properties introduced above for each specific measure.
As all these measures are quasi-concave and none of them are concave, we will only explicitly address pre- and post-processing monotonicity of $\eta^\ast$, and convexity of the corresponding inverse measure, $t^\ast$.
\subsection{Most incompatible measurements}
\label{sec:mostincomp}
For any given measure of incompatibility, one can ask what the most incompatible pairs of POVMs are.
To make this question well-defined, we introduce the following quantity.
\begin{defn}
\label{def:chi}
Given a measure of incompatibility, $\eta^\ast$, we define $\chi^\ast(d;n_A,n_B)$ to be its lowest possible value for dimension $d$ and outcome numbers $n_A$ and $n_B$.
\begin{equation}\label{eq:xi}
\chi^\ast(d;n_A,n_B) = \min\left\{\eta^\ast_{A,B}~|~(A,B)\in\POVM^{n_A,n_B}_d\right\}.
\end{equation}
\end{defn}
The minimum in this definition is justified, as the set $\POVM^{n_A,n_B}_{d}$ is closed and bounded.
For a fixed measure this definition yields a real number from the range $[0, 1]$ for all positive integers $d, n_{A}, n_{B}$.
Sometimes, however, we might be interested in less detailed information.
We might just ask the question ``what are the most incompatible measurement pairs in dimension $d$?'', regardless of the outcome numbers, leading to the quantity
\begin{equation}
\chi^\ast(d) = \inf_{n_{A}, n_{B}} \chi^\ast(d;n_A,n_B),
\end{equation}
where the infimum is taken over positive integers and it is not clear whether $\chi^\ast(d)$ is achieved for any finite $n_{A}$ and $n_{B}$.
Alternatively, we might only fix the outcome numbers, leading to $\chi^\ast(n_A,n_B)$, or fix neither the dimension nor the outcome numbers, leading to $\chi^\ast$.
One might wonder whether a non-trivial lower bound on $\chi^\ast$ can be derived based only on the previously assumed property of the noise model, namely, that for every POVM pair the corresponding noise set contains at least one jointly measurable pair, but this turns out not to be the case.
For every pair of incompatible measurements $(A, B)$ we can choose the noise set to contain a single jointly measurable pair with the property that the interior of the line segment connecting $(A,B)$ and the noise pair lies outside the jointly measurable set.
Clearly, in this case $\eta^{\ast}_{A, B} = 0$ for all incompatible pairs $(A,B)$, and $\eta^\ast$ defined through this construction is just the indicator function of joint measurability.
However, a mild additional assumption on the noise model allows us to get a non-trivial lower bound on $\chi^\ast$.
Suppose that for every incompatible pair $(A, B)$ there exists a valid noise pair $(M, N)$ such that the measurement operators of $A$ commute with those of $N$ and similarly the measurement operators of $B$ commute with those of $M$.
Then, the POVM given by
\begin{equation}
G_{ab} = \frac{1}{2} ( A_{a} N_{b} + M_{a} B_{b} )
\end{equation}
is a valid parent POVM for $\frac12(A+M)$ and $\frac12(B+N)$, therefore it ensures that $\eta^{\ast}_{A, B} \geq \frac{1}{2}$, and we conclude that $\chi^\ast\ge\frac12$.
Clearly, the above condition is fulfilled whenever we are guaranteed to find a noise pair where all the elements are proportional to the identity (a direct consequence of the unitary invariance property discussed in Section~\ref{subsec:ir}).
This is the case for all the measures that we study.
To make the search for the most incompatible pairs of measurements efficient, it is crucial to identify operations under which the measure is monotonic, as it significantly shrinks the set over which we need to optimise.
Specifically, if we want to compute $\chi^\ast(d;n_A,n_B)$ and we deal with a measure that is non-decreasing under convex combinations, we only need to consider pairs of extremal measurements.
If our goal is to compute $\chi^\ast(d)$, i.e., we do not care about the number of outcomes, and our measure is monotonic under post-processings, we do not need to consider measurement pairs that are post-processings of another pair.
Since every POVM can be written as a post-processing (coarse-graining) of some rank-one POVM with possibly more outcomes, for post-processing monotonic measures the value $\chi^\ast(d)$ can be found by searching only over rank-one measurements.
Similarly, if we aim to compute $\chi^\ast(n_A,n_B)$, i.e., we do not care about the dimension, and our measure is monotonic under pre-processings, we do not need to consider measurement pairs that are pre-processings of another pair.
Due to Naimark's dilation theorem, every POVM can be obtained by pre-processing a projective measurement that possibly acts on a higher dimensional space, therefore projective measurements achieve $\chi^\ast(n_A,n_B)$ for pre-processing monotonic measures.
\subsection{Semidefinite programming}
\label{sec:pre_sdp}
It is clear from Eq.~\eqref{eq:robustness} that incompatibility robustness measures are defined through an optimisation problem.
The class of optimisation problems that arises in our case is called semidefinite programming and can be seen as a generalisation of linear programming \cite{BV04}.
A semidefinite program (SDP) is an optimisation problem whose optimisation variables are matrices, and whose objective function and constraints are linear functions of these variables.
The constraints can be either matrix equalities or matrix inequalities (recall that for matrices the inequality $A \geq B$ is equivalent to $A - B$ being a positive semidefinite matrix).
For every SDP, later referred to as the \emph{primal}, another SDP, called the \emph{dual}, can be defined such that its solution bounds the primal one.
In this paper the primal SDP is a maximisation problem and the dual SDP is a minimisation problem whose solution upper bounds the primal solution.
In all the examples that we study in this work, the solutions of these two SDPs in fact coincide, as we will see in Section~\ref{sec:ird_def}.
Thanks to this feature, it is possible to efficiently solve such SDPs on a computer, which gives us a tool to study incompatibility robustness measures \emph{numerically}.
This tool we often employed using the MATLAB computing environment together with the YALMIP~\cite{Lof04}, SDPT3~\cite{TTT99} and MOSEK~\cite{mosek} optimisation toolboxes.
However, the main objective of our work is to study these measures \emph{analytically}.
In order to do so, we find \emph{feasible points} for the SDPs, that is, assignments of variables that satisfy all the constraints, but that are not necessarily optimal.
By finding feasible points for the primal and dual problems, we obtain lower and upper bounds, respectively, on the value of the optimisation problem.
In the next two sections we introduce objects that will come in useful for finding such feasible points.
\subsubsection{Lower bounds}
\label{sec:pre_low}
Feasible points for the primal SDP lead to lower bounds on the incompatibility robustness.
For a fixed pair $(A,B)$ feasible points correspond to a noise pair $(M,N)$, a visibility $\eta$, and a parent POVM $G$ for ${\eta\cdot(A,B) + (1-\eta)\cdot(M,N)}$, all of these satisfying the constraints of the SDP.
That is, the noise pair should satisfy $(M,N)\in {\bf N}_{A,B}$, and the visibility must be in the range $\eta\in[0,1]$.
Crucially, the parent POVM $G$ should give $\eta A + (1-\eta)M$ and $\eta B + (1-\eta)N$ as marginals (which also guarantees its proper normalisation), and all its measurement operators should be positive semidefinite.
In order to find feasible parent POVMs satisfying these properties, we introduce an ansatz solution.
This ansatz encompasses all possible choices of the parent POVM elements that are linear combinations of the elements of $A$ and $B$, their square-roots, and products thereof, such that the normalisation of the parent POVM is ensured.
Namely, let
\begin{equation}
\label{eqn:ansatz_low}
G_{ab}\propto\,\{A_a,B_b\} + (\alpha_{b}A_a + \beta_{a}B_b) + \gamma_{ab}\mathbb{I} + \delta(A_a^\frac12 B_b A_a^\frac12 + B_b^\frac12 A_a B_b^\frac12)
\end{equation}
for some real parameters $\alpha_{b}, \beta_{b}$, $\gamma_{ab}$ and $\delta$.
It is clear then that $\sum_{ab}G_{ab}\propto\mathbb{I}$.
In this construction the anticommutator term plays a crucial role.
When the measurement operators of the two POVMs commute, i.e., we have $A_{a} B_{b} = B_{b} A_{a}$ for all $a$ and $b$, the anticommutator is guaranteed to be positive semidefinite.
We can therefore set $G_{ab} = \frac12\{ A_{a}, B_{b} \}$, which is a valid parent POVM for $A$ and $B$.
For non-commuting measurement operators, however, the anticommutator might have some negative eigenvalues for which the remaining terms are supposed to compensate.
Note that the same construction for parent POVMs has recently been used in Ref.~\cite{CCT19}.
For a pair of rank-one POVMs checking the positivity of Eq.~\eqref{eqn:ansatz_low} becomes analytically tractable: in this case
we can write the operator as a direct sum of an operator acting on the two-dimensional subspace spanned by the eigenvectors of $A_{a}$ and $B_{b}$, and a multiple of the identity on the orthogonal subspace (which is non-trivial for $d\ge3$).
This allows us to explicitly compute the eigenvalues and check positivity.
For this reason, for our methods to work efficiently and provide tight bounds, it is extremely important that the measure we study is monotonic under post-processings.
This is because in this case it is enough to look at rank-one POVMs in order to find the most incompatible pairs, and the robustness of any POVM pair can be bounded by the robustness of their rank-one decompositions.
Note that computing the marginals of the POVM in Eq.~\eqref{eqn:ansatz_low} is also easy in general, except for the terms multiplying the parameter $\delta$.
However, for most constructions we will choose $\delta=0$, and only include this term in a special (albeit very important) case.
As an example, let us present a known result initially presented for pairs in Ref.~\cite{HSTZ14} and then generalised to arbitrary number of measurements \cite{HKRS15,HMZ16,CHT18}.
The idea is to try to perform two measurements simultaneously by duplicating the input state and then feeding each measurement with one of the copies.
By virtue of the no-cloning theorem, the duplication process cannot be perfect.
Thanks to a duality between noiseless measurements acting on noisy states and noisy measurements acting on noiseless states, one can obtain a parent POVM from this procedure
\begin{equation}
\label{eqn:cloning_povm}
G_{ab}=\frac{1}{2(d+1)}\Big[\{A_a,B_b\} + \tr(B_b)A_a + \tr(A_a)B_b\Big],
\end{equation}
which is indeed of the form \eqref{eqn:ansatz_low}.
The positivity of $G_{ab}$ defined in this way follows straightforwardly from the fact that $[A_a/\tr(A_a)+B_b/\tr(B_b)]^2 \geq 0$ (we assume that $\tr A_{a} \tr B_{b} > 0$; the other cases are trivial).
This parent POVM gives rise to a universal lower bound on some measures, see Eq.~\eqref{eqn:cloning}.
\subsubsection{Upper bounds}
\label{sec:pre_up}
In order to derive upper bounds on incompatibility robustness measures, we need to find feasible points for the dual SDPs.
These SDPs have a similar structure for all the different measures that we study in this work, and therefore some quantities will often appear in the upper bounds.
For this reason, we define them here:
\begin{equation}
\label{eqn:f_lambda}
f=\sum\limits_a\frac{\tr A_a^2}{d}+\sum\limits_b\frac{\tr B_b^2}{d}\quad\text{and}\quad\lambda=\max\limits_{a,b}\Big\{\max \Sp\big(A_a+B_b\big)\Big\},
\end{equation}
where $\Sp(M)$ is the spectrum of the operator $M$ (note that $A_a+B_b$ is always positive semidefinite).
It is easy to see that $f \leq 2$ and the inequality is saturated if and only if both measurements are projective.
We will also need the following four quantities:
\begin{equation}
\label{eqn:g}
\begin{gathered}
g^\mathrm{d}=\sum\limits_a\left(\frac{\tr A_a}{d}\right)^2+\sum\limits_b\left(\frac{\tr B_b}{d}\right)^2,\quad g^\mathrm{r}=\frac{1}{n_A}+\frac{1}{n_B},\\ g^\mathrm{p}=\min\limits_a\frac{\tr A_a}{d}+\min_b\frac{\tr B_b}{d},\quad\text{and}\quad g^\mathrm{jm}=\min\limits_{a,b}\Big\{\min \Sp\big(A_a+B_b\big)\Big\}.
\end{gathered}
\end{equation}
Note that $g^\mathrm{d}=g^\mathrm{r}=g^\mathrm{p}=2/d$ whenever both measurements are rank-one projective.
\subsection{Example}
\label{sec:pre_ill}
We will compute all the studied incompatibility robustness measures for a pair of rank-one projective qubit measurements parametrised as
\begin{equation}
\label{eqn:runex}
A_{a}(\theta) = \frac{1}{2} \big[ \mathbb{I} + (-1)^{a} ( \cos \theta \, \sigma_{z} + \sin \theta \, \sigma_{x} ) \big]\quad\text{and}\quad B_{b}(\theta) = \frac{1}{2} \big[ \mathbb{I} + (-1)^{b} ( \cos \theta \, \sigma_{z} - \sin \theta \, \sigma_{x} ) \big],
\end{equation}
where $\sigma_z$ and $\sigma_x$ are the Pauli $Z$ and $X$ matrices, $\theta \in [0, \pi/4]$ and $a,b=1,2$.
Note that we choose the angle $\theta$ to be half of the angle between the Bloch vectors of the two measurements.
For this pair of rank-one projective measurements, we can compute the different parameters defined in Eqs~\eqref{eqn:f_lambda} and \eqref{eqn:g}, namely, $f=2$, $\lambda=1+\cos\theta$, $g^\mathrm{d}=g^\mathrm{r}=g^\mathrm{p}=1$, and $ g^\mathrm{jm}=1-\cos\theta$.
In the following, when discussing any measure of incompatibility for this pair, we will use $\eta^\ast_\theta$ as a shorthand for $\eta^\ast_{A(\theta),B(\theta)}$.
We will also make use of the following compact notation to write down the primal and dual variables:
\begin{equation}
G=\begin{bmatrix}G_{11} & G_{12}\\ G_{21} & G_{22}\end{bmatrix}
\quad\text{and}\quad
(X,Y)=\left(\begin{bmatrix} X_1\\ X_2 \end{bmatrix}, \begin{bmatrix} Y_1\\ Y_2 \end{bmatrix}\right),
\label{eqn:notation}
\end{equation}
where the elements $G_{ab}$, $X_a$ and $Y_b$ are $2\times2$ Hermitian matrices.
\section{Five relevant measures}
\label{sec:measures}
In this section we introduce five different explicit noise models, which give rise to five different robustness-based measures of incompatibility that are commonly used in the literature.
For each measure we write down both the primal and the dual SDPs, analyse their desired properties, illustrate their computation on a pair of rank-one projective qubit measurements, and derive explicit lower and upper bounds on them.
A compact summary of the main results can be found at the end of this section in Table~\ref{tab:magic}.
\subsection{Incompatibility depolarising robustness}
\label{sec:ird}
\subsubsection{Definition and properties}
\label{sec:ird_def}
In this case the noise model is defined by the map
\begin{equation}
\label{eqn:noised}
{\bf N}^\mathrm{d}_{A,B} = \left\{\left(\Big\{\tr (A_a) \frac{\mathbb{I}_d}{d}\Big\}_{a=1}^{n_A},\Big\{\tr (B_b)\frac{\mathbb{I}_d}{d}\Big\}_{b=1}^{n_B}\right)\right\}.
\end{equation}
The noise set depends on the specific measurements, which makes this measure different than all the other measures considered in this work.
It has been investigated in many works \cite{CHT12,HKR15,HKRS15,BQG+17,BN18,BN182,DSFB19,CCT19}, often in relation with Einstein--Podolsky--Rosen steering.
This specific type of noise has also been considered in scenarios different from incompatibility~\cite{OGWA17}.
The physical motivation is as follows: take a depolarising quantum channel $\Lambda_\eta^\dagger(.)$, which acts on states as $\Lambda_\eta^\dagger(\rho) = \eta\rho + (1-\eta)\tr(\rho)\mathbb{I}/d$, that is, by mixing them with white noise.
If we measure a system that has undergone such an evolution, we obtain the outcome probabilities $p(a) = \tr[A_{a}\Lambda_\eta^\dagger(\rho)] = \tr[\Lambda_\eta(A_{a})\rho]$, where $\Lambda_\eta(A_{a}) = \eta A_{a} + (1-\eta)\tr(A_{a})\mathbb{I}/d$ is the dual of the depolarising channel, which leads precisely to the type of noise set defined in Eq.~\eqref{eqn:noised}.
The corresponding incompatibility robustness, as introduced in Definition~\ref{def:robustness}, can be computed via the SDPs
\begin{equation}
\eta^\md_{A,B}=\left\{
\begin{array}{cl}
\max\limits_{\eta,\{G_{ab}\}_{ab}} &\eta \\
\text{s.t.}
&G_{ab} \geq 0, \quad \eta \leq 1 \\
&\sum\limits_b G_{ab} = \eta A_a+(1-\eta)\tr A_a\frac{\mathbb{I}}{d}\\
&\sum\limits_a G_{ab} = \eta B_b+(1-\eta)\tr B_b\frac{\mathbb{I}}{d} \\
\end{array}\right.\quad=\left\{
\begin{array}{cl}
\min\limits_{\hb{\{X_a\}_a}{\{Y_b\}_b}} &1 +\sum\limits_a\tr(X_aA_a)+\sum\limits_b\tr(Y_bB_b)\\
\text{s.t.}
&X_a=X_a^\dagger,\quad Y_b=Y_b^\dagger,\quad X_a+Y_b \geq 0 \vphantom{\sum\limits_a}\\
&1 +\sum\limits_a\tr (X_aA_a)+\sum\limits_b\tr (Y_bB_b) \\
&\qquad\geq\sum\limits_a\frac{\tr A_a}{d}\tr X_a+\sum\limits_b\frac{\tr B_b}{d}\tr Y_b
\end{array}\right.,
\label{eqn:ird_sdp}
\end{equation}
where in the following the first formulation will be referred to as the primal, and the second as the dual.
The primal variables $G_{ab}$ and $\eta$ are simply the measurement operators of the parent POVM and the visibility, respectively.
The dual variables $X_{a}$ and $Y_{b}$ are Lagrange multipliers corresponding to the primal equality constraints.
Note that the normalisation of $G$ is not enforced as it follows from the other constraints.
For an explicit derivation of the dual problem, see Ref.~\cite[Appendix~A]{DSFB19}.
Slater's theorem states that whenever a strictly feasible point (a point satisfying all the constraints strictly) exists for either the primal or the dual, the duality gap is zero, thus the primal and dual solutions coincide \cite{BV04}.
In this case, we can take $X_a=Y_b=\delta\,\mathbb{I}$, which is a strictly feasible point of the dual for sufficiently large $\delta$.
Thus, the theorem applies and justifies the equality between the two problems in Eq.~\eqref{eqn:ird_sdp}.
Similar arguments apply to all pairs of primal-dual SDPs that we discuss in this work.
As the noise set ${\bf N}^\mathrm{d}_{A,B}$ defined in Eq.~\eqref{eqn:noised} is invariant under post-processings by linearity of the trace, it follows from Section~\ref{sec:monotonicity} that $\eta^\md$ is monotonic under post-processings.
It turns out, however, that $\eta^\md$ does not satisfy the other two natural properties introduced in Section~\ref{sec:monotonicity}, namely monotonicity under non trace-preserving pre-processings and convexity of the inverse;
see Appendix~\ref{app:ctrex} for counterexamples.
Note that the monotonicity under pre-processings was incorrectly claimed in Ref.~\cite[Proposition 2]{HKR15}.
\subsubsection{Example}
\label{sec:ird_ill}
From a result by Busch \cite[Theorem 4.5]{Bus86} on the joint measurability of pairs of two-outcome qubit measurements, also rephrased by Uola et al.~more recently \cite[Section~III C]{ULMH16}, we get
\begin{equation}
\eta^\md_\theta=\frac{1}{\cos\theta+\sin\theta}.
\label{eqn:ird_ill}
\end{equation}
This value is plotted in Fig.~\ref{fig:runex} together with the other measures.
For completeness and later reference, we give optimal solutions to both the primal and the dual stated in Eq.~\eqref{eqn:ird_sdp}
\begin{equation}
G=\frac{1}{\cos\theta+\sin\theta}
\begin{bmatrix}
\cos\theta\,\frac{\mathbb{I}-\sigma_z}{2}&\sin\theta\,\frac{\mathbb{I}-\sigma_x}{2}\\
\sin\theta\,\frac{\mathbb{I}+\sigma_x}{2}&\cos\theta\,\frac{\mathbb{I}+\sigma_z}{2}
\end{bmatrix}
\quad\text{and}\quad
(X,Y)=\frac{1}{4(\cos\theta+\sin\theta)}
\left(\begin{bmatrix}
\mathbb{I}+(\sigma_z+\sigma_x)\\
\mathbb{I}-(\sigma_z+\sigma_x)
\end{bmatrix},
\begin{bmatrix}
\mathbb{I}+(\sigma_z-\sigma_x)\\
\mathbb{I}-(\sigma_z-\sigma_x)
\end{bmatrix}\right),
\end{equation}
where we have used the notation introduced in Eq.~\eqref{eqn:notation}.
\subsubsection{Lower bound}
\label{sec:ird_low}
As mentioned before, a lower bound on $\eta^\md$ is already known \cite{HSTZ14,HKRS15,HMZ16,CHT18}.
The parent POVM given in Eq.~\eqref{eqn:cloning_povm} is indeed a feasible point for the primal in Eq.~\eqref{eqn:ird_sdp} together with
\begin{equation}
\label{eqn:cloning}
\eta=\frac12\left(1+\frac{1}{d+1}\right).
\end{equation}
For a pair $(A,B)$ of rank-one measurements in dimension $d \geq 2$, this bound can be improved.
Let us introduce a feasible point for the primal in Eq.~\eqref{eqn:ird_sdp} with $G$ of the form \eqref{eqn:ansatz_low}, where
\begin{equation}
\label{eqn:ird_low_ansatz}
\begin{pmatrix}\alpha_{b}\\\beta_{a}\end{pmatrix}=\frac{-2+\sqrt{d^2+4d-4}}{d}\begin{pmatrix}\tr B_b\\\tr A_a\end{pmatrix},\quad\gamma_{ab}=\left(\frac{d+2-\sqrt{d^2+4d-4}}{2d}\right)^2\tr A_a\tr B_b,\quad\text{and}\quad \delta=0.
\end{equation}
For a proof that this leads to valid measurement operators $G_{ab}$ and for a measurement-dependent refinement we refer the reader to Appendix~\ref{app:ird_low}.
This construction gives a lower bound on $\eta^\md$ for all pairs of rank-one measurements.
However, since the measure is monotonic under post-processings, the bound is actually universal, i.e., for an arbitrary pair $(A, B)$ of measurements in dimension $d$ we have
\begin{equation}
\label{eqn:ird_low}
\eta^\md_{A,B}\geq \frac{d-2+\sqrt{d^2+4d-4}}{4(d-1)}.
\end{equation}
Importantly, this bound turns out to be strictly better than Eq.~\eqref{eqn:cloning}, which was the best lower bound known so far.
\subsubsection{Upper bound}
\label{sec:ird_up}
Following the idea used in Ref.~\cite{DSFB19}, we provide a valid assignment of the dual variables $X_{a}$ and $Y_{b}$ for the dual problem given in Eq.~\eqref{eqn:ird_sdp} to get an upper bound on $\eta^\md$, namely,
\begin{equation}
\label{eqn:ird_up_ansatz}
X_a=\frac{\frac{\lambda}{2}\,\mathbb{I}-A_a}{(f-g^\mathrm{d})d}\quad\text{and}\quad
Y_b=\frac{\frac{\lambda}{2}\,\mathbb{I}-B_b}{(f-g^\mathrm{d})d}
\end{equation}
where $f$ and $\lambda$ are defined in Eq.~\eqref{eqn:f_lambda} and $g^\mathrm{d}$ in Eq.~\eqref{eqn:g}.
Here we implicitly assume that $f \neq g^{\mathrm{d}}$, but one can show that the equality $f = g^{\mathrm{d}}$ holds if and only if all POVM elements of $A$ and $B$ are proportional to $\mathbb{I}$, in which case the pair is trivially compatible (see Appendix~\ref{app:footnote13AappendixT2}).
The resulting upper bound is given by
\begin{equation}
\eta^\md_{A,B}\leq\frac{\lambda-g^\mathrm{d}}{f-g^\mathrm{d}}=1-\frac{f-\lambda}{f-g^\mathrm{d}},
\label{eqn:ird_up}
\end{equation}
where the last equality makes clear that this upper bound is non-trivial whenever $f>\lambda$ (since $f>g^\mathrm{d}$ from Appendix~\ref{app:footnote13AappendixT2}).
In the following we always implicitly assume that this condition is satisfied when we discuss the various upper bounds.
\subsection{Incompatibility random robustness}
\label{sec:irr}
\subsubsection{Definition and properties}
\label{sec:irr_def}
In this case the noise model is defined by the map
\begin{equation}
\label{eqn:noiser}
{\bf N}^\mathrm{r}_{A,B} = \left\{\left(\Big\{\frac{\mathbb{I}_d}{n_A}\Big\}_{a=1}^{n_A},\Big\{\frac{\mathbb{I}_d}{n_B}\Big\}_{b=1}^{n_B}\right)\right\},
\end{equation}
a single element containing the trivial measurement, i.e., the measurement generating a uniform distribution of outcomes regardless of the state.
It has been investigated in many works \cite{UBGP15,CS16,CHT18,BN18,BN182,CCT19}, and also in the framework of general probabilistic theories \cite{BRGK13,JP17}.
The corresponding incompatibility robustness, as introduced in Definition~\ref{def:robustness}, can be computed via the SDPs~\cite{CS16}
\begin{equation}
\eta^\mr_{A,B}=\left\{
\begin{array}{cl}
\max\limits_{\eta,\{G_{ab}\}_{ab}} &\eta\\
\text{s.t.}
&G_{ab} \geq 0, \quad \eta \leq 1 \\
&\sum\limits_b G_{ab} = \eta A_a+(1-\eta)\frac{\mathbb{I}}{n_A} \\
&\sum\limits_a G_{ab} = \eta B_b+(1-\eta)\frac{\mathbb{I}}{n_B} \\
\end{array}\right.\quad=\left\{
\begin{array}{cl}
\min\limits_{\hb{\{X_a\}_a}{\{Y_b\}_b}} &1 +\sum\limits_a\tr(X_aA_a)+\sum\limits_b\tr(Y_bB_b)\\
\text{s.t.}
&X_a=X_a^\dagger,\quad Y_b=Y_b^\dagger,\quad X_a+Y_b \geq 0 \vphantom{\sum\limits_a}\\
&1 +\sum\limits_a\tr (X_aA_a)+\sum\limits_b\tr (Y_bB_b) \\
&\qquad\geq\sum\limits_a\frac{1}{n_A}\tr X_a+\sum\limits_b\frac{1}{n_B}\tr Y_b
\end{array}\right..
\label{eqn:irr_sdp}
\end{equation}
Note that the normalisation of $G$ is not enforced as it follows from the other constraints.
As the noise set ${\bf N}^\mathrm{r}_{A,B}$ defined in Eq.~\eqref{eqn:noiser} is invariant under pre-processings (recall that pre-processings are unital), it follows from Section~\ref{sec:monotonicity} that $\eta^\mr$ is monotonic under pre-processings.
Moreover, as this set is also convex and independent of the specific form of $A$ and $B$ (the map ${\bf N}^\mathrm{r}$ is constant), we know from Section~\ref{sec:monotonicity} that $1/\eta^\mr$ is convex.
However, this measure is not monotonic under non outcome number-preserving post-processings, see Appendix~\ref{app:ctrex} for a counterexample.
\subsubsection{Example}
\label{sec:irr_ill}
For rank-one projective measurements $\eta^\md$ and $\eta^\mr$ coincide, therefore
\begin{equation}
\label{eqn:irr_ill}
\eta^\mr_\theta=\eta^\md_\theta=\frac{1}{\cos\theta+\sin\theta}.
\end{equation}
\subsubsection{Lower bound}
\label{sec:irr_low}
As $\eta^\mr$ is not monotonic under post-processings, we cannot use a solution for rank-one measurements as in Section~\ref{sec:ird_low} to deduce a general lower bound.
Thus, we consider an arbitrary pair $(A,B)$ of measurements in dimension $d$ and we introduce a feasible point for the primal in Eq.~\eqref{eqn:irr_sdp} with $G$ of the form \eqref{eqn:ansatz_low}, where
\begin{equation}
\label{eqn:irr_low_ansatz}
\alpha_{b}=\sqrt{\frac{n_A}{n_B}},\quad\beta_{a}=\sqrt{\frac{n_B}{n_A}},\quad\gamma_{ab}=0,\quad\text{and}\quad\delta=0
\end{equation}
from which we obtain the bound
\begin{equation}
\label{eqn:irr_low}
\eta^\mr_{A,B}\geq\frac12\left(1+\frac{1}{\sqrt{n_An_B}+1}\right).
\end{equation}
The positivity of this parent POVM follows from
\begin{equation}
0\leq\sqrt{n_An_B}\left(\frac{A_a}{\sqrt{n_B}}+\frac{B_b}{\sqrt{n_A}}\right)^2=\{A_a,B_b\}+\sqrt{\frac{n_A}{n_B}}A_a^2+\sqrt{\frac{n_B}{n_A}}B_b^2\leq\{A_a,B_b\}+\sqrt{\frac{n_A}{n_B}}A_a+\sqrt{\frac{n_B}{n_A}}B_b,
\end{equation}
where the last inequality is due to $A_a^2\leq A_a$ and $B_b^2\leq B_b$.
\subsubsection{Upper bound}
\label{sec:irr_up}
In the case of $\eta^\mr$ we choose the dual variables as
\begin{equation}
\label{eqn:irr_up_ansatz}
X_a=\frac{\frac{\lambda}{2}\,\mathbb{I}-A_a}{(f-g^\mathrm{r})d}\quad\text{and}\quad
Y_b=\frac{\frac{\lambda}{2}\,\mathbb{I}-B_b}{(f-g^\mathrm{r})d}
\end{equation}
where $f$ and $\lambda$ are defined in Eq.~\eqref{eqn:f_lambda} and $g^\mathrm{r}$ in Eq.~\eqref{eqn:g}.
Here we implicitly assume that $f \neq g^{\mathrm{r}}$, but one can show that the equality $f = g^{\mathrm{r}}$ holds if and only if all POVM elements of $A$ and $B$ are proportional to $\mathbb{I}$, in which case the pair is trivially compatible (see Appendix~\ref{app:footnote13AappendixT2}).
The resulting upper bound is given by
\begin{equation}
\eta^\mr_{A,B}\leq\frac{\lambda-g^\mathrm{r}}{f-g^\mathrm{r}}.
\label{eqn:irr_up}
\end{equation}
\subsection{Incompatibility probabilistic robustness}
\label{sec:irp}
\subsubsection{Definition and properties}
\label{sec:irp_def}
In this case the noise model is defined by the map
\begin{equation}
\label{eqn:noisep}
{\bf N}^\mathrm{p}_{A,B} = \left\{\bigg(\left\{p_a\,\mathbb{I}_d\right\}_{a=1}^{n_A},\left\{q_b\,\mathbb{I}_d\right\}_{b=1}^{n_B}\bigg)\Bigg|\,p_a\geq0,q_b\geq0,\sum\limits_ap_a=1=\sum\limits_bq_b\right\},
\end{equation}
where $\{p_a\}_a$ and $\{q_b\}_b$ are probability distributions.
This measure has been investigated in many works \cite{HSTZ14,HKR15,AHK+16,HMZ16,Hei16,JP17,CHT18,Jen18,BN18,BN182}, and also in the framework of general probabilistic theories \cite{BHSS12,Pla16}.
The corresponding incompatibility robustness, as introduced in Definition~\ref{def:robustness}, can be computed via the SDPs
\begin{equation}
\eta^\pp_{A,B}=\left\{
\begin{array}{cl}
\max\limits_{\hb{\eta,\{G_{ab}\}_{ab}}{\{\tilde{p}_a\}_a,\{\tilde{q}_b\}_b}} &\eta \\
\text{s.t.}
&G_{ab} \geq 0, \quad \tilde{p}_a\geq0, \quad \tilde{q}_b\geq0\\
&\sum\limits_a\tilde{p}_a=1-\eta=\sum\limits_b\tilde{q}_b \\
&\sum\limits_b G_{ab} = \eta A_a+\tilde{p}_a\mathbb{I}\\
&\sum\limits_a G_{ab} = \eta B_b+\tilde{q}_b\mathbb{I} \\
\end{array}\right.\quad=\left\{
\begin{array}{cl}
\min\limits_{\hb{\{X_a\}_a,\xi}{\{Y_b\}_b,\upsilon}} &1 +\sum\limits_a\tr(X_aA_a)+\sum\limits_b\tr(Y_bB_b)\\[3pt]
\text{s.t.}
&X_a=X_a^\dagger,\quad Y_b=Y_b^\dagger,\quad X_a+Y_b \geq 0 \vphantom{\sum\limits_a}\\
&1 +\sum\limits_a\tr (X_a A_a) +\sum\limits_b\tr (Y_b B_b) \geq \xi+\upsilon\\
&\xi\geq\tr X_a,\quad\upsilon\geq\tr Y_b\vphantom{\sum\limits_a}\\
\end{array}\right..
\label{eqn:irp_sdp}
\end{equation}
Note that, in order to make the problem linear in its variables, we have introduced sub-normalised probability distributions $\tilde{p}_a=(1-\eta)p_a$ and $\tilde{q}_b=(1-\eta)q_b$.
Note also that the normalisation of $G$ and the constraint $\eta\leq1$ are not enforced as they follow from the other constraints.
As the noise set ${\bf N}^\mathrm{p}_{A,B}$ defined in Eq.~\eqref{eqn:noisep} contains both ${\bf N}^\mathrm{d}_{A,B}$ of Eq.~\eqref{eqn:noised} and ${\bf N}^\mathrm{r}_{A,B}$ of Eq.~\eqref{eqn:noiser}, the constraints of the primal in Eq.~\eqref{eqn:irp_sdp} are looser than the ones in Eq.~\eqref{eqn:ird_sdp} and \eqref{eqn:irr_sdp}.
By duality, the constraints of the dual in Eq.~\eqref{eqn:irp_sdp} are then tighter than the ones in Eq.~\eqref{eqn:ird_sdp} and \eqref{eqn:irr_sdp}, which can indeed be seen by plugging suitable convex combinations of the constraints $\xi\geq\tr X_a$ and $\upsilon\geq\tr Y_b$ into ${1 +\sum_a\tr (X_a A_a) +\sum_b\tr (Y_b B_b) \geq \xi+\upsilon}$.
As the noise set ${\bf N}^\mathrm{p}_{A,B}$ defined in Eq.~\eqref{eqn:noisep} is invariant under pre- and post-processings (by unitality and linearity, respectively), it follows from Section~\ref{sec:monotonicity} that $\eta^\pp$ is monotonic under pre- and post-processings.
Moreover, as this set is also convex and independent of the specific form of $A$ and $B$ (the map ${\bf N}^\mathrm{p}$ is constant), we know from Section~\ref{sec:monotonicity} that $1/\eta^\pp$ is convex.
Thus, $\eta^\pp$ is the first measure that satisfies all the properties introduced in Section~\ref{sec:prelim} except for concavity.
\subsubsection{Example}
\label{sec:irp_ill}
The dual feasible points from Section~\ref{sec:ird_ill} satisfy the additional trace constraints of the dual given in Eq.~\eqref{eqn:irp_sdp}.
Thus, the measures $\eta^\md$ and $\eta^\pp$ coincide on this family of measurements:
\begin{equation}
\eta^\pp_\theta=\eta^\md_\theta=\frac{1}{\cos\theta+\sin\theta}.
\end{equation}
Note, however, that $\eta^\md$ and $\eta^\pp$ differ in general, even for rank-one projective measurement pairs (see Section~\ref{sec:higher_dim} for an explicit example).
\subsubsection{Lower bound}
\label{sec:irp_low}
Since the noise set ${\bf N}^\mathrm{p}_{A,B}$ contains both ${\bf N}^\mathrm{d}_{A,B}$ and ${\bf N}^\mathrm{r}_{A,B}$ for all $(A,B)$, lower bounds on $\eta^\md$ and $\eta^\mr$ immediately apply to $\eta^\pp$.
\subsubsection{Upper bound}
\label{sec:irp_up}
In the case of $\eta^\pp$ we choose the dual variables as
\begin{equation}
\label{eqn:irp_up_ansatz}
X_a=\frac{\frac{\lambda}{2}\,\mathbb{I}-A_a}{(f-g^\mathrm{p})d},\quad
Y_b=\frac{\frac{\lambda}{2}\,\mathbb{I}-B_b}{(f-g^\mathrm{p})d},\quad
\xi=\max\limits_a\tr X_a,\quad\text{and}\quad\upsilon=\max\limits_b\tr Y_b,
\end{equation}
where $f$ and $\lambda$ are defined in Eq.~\eqref{eqn:f_lambda} and $g^\mathrm{p}$ in Eq.~\eqref{eqn:g}.
Here we implicitly assume that $f \neq g^{\mathrm{p}}$, but one can show that the equality $f = g^{\mathrm{p}}$ holds if and only if all POVM elements of $A$ and $B$ are proportional to $\mathbb{I}$, in which case the pair is trivially compatible (see Appendix~\ref{app:footnote13AappendixT2}).
The resulting upper bound is given by
\begin{equation}
\eta^\pp_{A,B}\leq\frac{\lambda-g^\mathrm{p}}{f-g^\mathrm{p}}.
\label{eqn:irp_up}
\end{equation}
\subsection{Incompatibility jointly measurable robustness}
\label{sec:irjm}
\subsubsection{Definition and properties}
\label{sec:irjm_def}
In this case the noise model is defined by the map
\begin{equation}
\label{eqn:noisejm}
{\bf N}^\mathrm{jm}_{A,B} = \JM_d^{n_A,n_B},
\end{equation}
the set of jointly measurable pairs of POVMs with $n_A$ and $n_B$ outcomes in dimension $d$.
To the best of our knowledge, this measure has only been considered in Ref.~\cite[Section~II C]{CS16}.
The corresponding incompatibility robustness, as introduced in Definition~\ref{def:robustness}, can be computed via the SDPs
\begin{equation}
\eta^\mjm_{A,B}=\left\{
\begin{array}{cl}
\max\limits_{\eta,\hb{\{G_{ab}\}_{ab}}{\{\tilde{H}_{ab}\}_{ab}}} &\eta \\
\text{s.t.}
&G_{ab}\geq0,\quad\sum\limits_{ab} G_{ab} = \mathbb{I},\quad\tilde{H}_{ab} \geq 0 \\
&\sum\limits_b (G_{ab}-\tilde{H}_{ab}) = \eta A_a \\
&\sum\limits_a (G_{ab}-\tilde{H}_{ab}) = \eta B_b \\
\end{array}\right.\quad=\left\{
\begin{array}{cl}
\min\limits_{N,\hb{\{X_a\}_a}{\{Y_b\}_b}} &\tr N\\
\text{s.t.}
&N=N^\dagger,\quad X_a=X_a^\dagger,\quad Y_b=Y_b^\dagger \vphantom{\sum\limits_a}\\
&N\geq X_a+Y_b \geq 0\vphantom{\sum\limits_a}\\
&\sum\limits_a\tr(X_aA_a)+\sum\limits_b\tr(Y_bB_b)\geq1\\
\end{array}\right..
\label{eqn:irjm_sdp}
\end{equation}
Note that the noise POVMs do not explicitly appear in the primal problem, since optimising over jointly measurable pairs is equivalent to optimising over the parent measurement, here denoted by $H$.
To make the problem linear in its variables, we have introduced a sub-normalised parent POVM of the noise, $\tilde{H}=(1-\eta)H$.
Note also that the constraint $\eta\leq1$ is not enforced as it follows from summing up one of the marginal constraints.
In analogy with $\eta^\pp$, the measure $\eta^\mjm$ also satisfies the properties introduced in Section~\ref{sec:prelim}, namely monotonicity under pre- and post-processings, and convexity of the inverse.
\subsubsection{Example}
\label{sec:irjm_ill}
The value of this measure for a pair of rank-one projective qubit measurements is strictly higher than for the previous measures, whenever the pair is incompatible.
Specifically,
\begin{equation}
\eta^\mjm_\theta=\frac{2}{1+\cos\theta+\sin\theta}.
\label{eqn:irjm_ill}
\end{equation}
This value is plotted in Fig.~\ref{fig:runex} together with the other measures.
Interestingly, even for such a simple example the primal problem given in Eq.~\eqref{eqn:irjm_sdp} admits multiple optimal solutions.
More specifically, we obtain a continuous one-parameter family, which reads
\begin{equation}
\label{eqn:jm_qubit}
G=
\begin{bmatrix}
r\,\frac{\mathbb{I}-\sigma_z}{2}&(1-r)\,\frac{\mathbb{I}-\sigma_x}{2}\\
(1-r)\,\frac{\mathbb{I}+\sigma_x}{2}&r\,\frac{\mathbb{I}+\sigma_z}{2}
\end{bmatrix}
,\quad
\tilde{H}=(1-\eta^\mjm_\theta)
\begin{bmatrix}
s\,\frac{\mathbb{I}+\sigma_z}{2}&(1-s)\,\frac{\mathbb{I}+\sigma_x}{2}\\
(1-s)\,\frac{\mathbb{I}-\sigma_x}{2}&s\,\frac{\mathbb{I}-\sigma_z}{2}
\end{bmatrix},
\quad\text{where}\quad
r = \eta^\mjm_\theta(s+\cos\theta)-s
\end{equation}
and $s$ is a free parameter taken from the interval $[0,1]$ to ensure the positivity of the elements of $H$.
Different values of $s$ correspond to applying noise along different axes: for $s=0$ the noise only affects the $X$ direction, while for $s = 1$ it only affects the $Z$ direction.
A feasible optimal point for the dual given in Eq.~\eqref{eqn:irjm_sdp} reads
\begin{equation}
(X,Y)=\frac{1}{4(1+\cos\theta+\sin\theta)}
\left(\begin{bmatrix}
\mathbb{I}-(\sigma_z+\sigma_x)\\
\mathbb{I}+(\sigma_z+\sigma_x)
\end{bmatrix},
\begin{bmatrix}
\mathbb{I}-(\sigma_z-\sigma_x)\\
\mathbb{I}+(\sigma_z-\sigma_x)
\end{bmatrix}\right),
\quad\text{and}\quad
N=\frac{1}{1+\cos\theta+\sin\theta}\,\mathbb{I}.
\end{equation}
\subsubsection{Lower bound}
\label{sec:irjm_low}
Let us consider a pair $(A,B)$ of rank-one measurements in dimension $d$.
Finding a feasible point for the primal in Eq.~\eqref{eqn:irjm_sdp} is not an easy task, as we have to find two parent POVMs at once.
For $G_{ab}$, we make the same choice as for $\eta^\md$, i.e., Eq.~\eqref{eqn:ird_low_ansatz} in Section~\ref{sec:ird_low}.
We choose the subnormalised noise POVM $\tilde{H}$ to be of the form \eqref{eqn:ansatz_low} with
\begin{equation}
\label{eqn:irjm_low_ansatz}
\begin{pmatrix}\alpha_{b}\\\beta_{a}\end{pmatrix}=\frac{-2-\sqrt{d^2+4d-4}}{d}\begin{pmatrix}\tr B_b\\\tr A_a\end{pmatrix},\quad\gamma_{ab}=\left(\frac{d+2+\sqrt{d^2+4d-4}}{2d}\right)^2\tr A_a\tr B_b,\quad\text{and}\quad\delta=0,
\end{equation}
which leads to
\begin{equation}
\label{eqn:irjm_low}
\eta=\frac{2\sqrt{d^2+4d-4}}{3d-2+\sqrt{d^2+4d-4}}\leq\eta^\mjm_{A,B}.
\end{equation}
Details about this specific point can be found in Appendix~\ref{app:irjm_low} together with a measurement-dependent refinement.
As $\eta^\mjm$ is monotonic under post-processings, this bound on pairs of rank-one measurements extends to all pairs of measurements in dimension $d$.
\subsubsection{Upper bound}
\label{sec:irjm_up}
Consider the following feasible point for the dual given in Eq.~\eqref{eqn:irjm_sdp}:
\begin{equation}
\label{eqn:irjm_up_ansatz}
X_a=\frac{A_a-\frac{g^\mathrm{jm}}{2}\,\mathbb{I}}{(f- g^\mathrm{jm})d}, \quad Y_b=\frac{B_b-\frac{g^\mathrm{jm}}{2}\,\mathbb{I}}{(f- g^\mathrm{jm})d},\quad\text{and}\quad N=\frac{\lambda- g^\mathrm{jm}}{f- g^\mathrm{jm}}\cdot\frac{\mathbb{I}}{d}
\end{equation}
where $f$ and $\lambda$ are defined in Eq.~\eqref{eqn:f_lambda} and $g^\mathrm{jm}$ in Eq.~\eqref{eqn:g}.
Here we implicitly assume that $f \neq g^{\mathrm{jm}}$, but one can show that the equality $f = g^{\mathrm{jm}}$ holds if and only if all POVM elements of $A$ and $B$ are proportional to $\mathbb{I}$, in which case the pair is trivially compatible (see Appendix~\ref{app:footnote13AappendixT2}).
The above feasible point immediately implies that
\begin{equation}
\label{eqn:irjm_up}
\eta^\mjm_{A,B} \leq \frac{\lambda- g^\mathrm{jm}}{f- g^\mathrm{jm}}.
\end{equation}
\subsection{Incompatibility generalised robustness}
\label{sec:irg}
\subsubsection{Definition and properties}
\label{sec:irg_def}
In this case the noise model is defined by the map
\begin{equation}
\label{eqn:noiseg}
{\bf N}^\mathrm{g}_{A,B} = \POVM_d^{n_A,n_B},
\end{equation}
the set of all POVM pairs with $n_A$ and $n_B$ outcomes, respectively, in dimension $d$.
To the best of our knowledge, this measure was first introduced in Ref.~\cite{Haa15} and studied further in Refs~\cite{UBGP15,CS16,KBUP17,BQG+17}.
Recently, it was given an operational meaning through state discrimination tasks \cite{CHT19,UKS+19,SSC19}.
The corresponding incompatibility robustness, as introduced in Definition~\ref{def:robustness}, can be computed via the SDPs
\begin{equation}
\eta^\mg_{A,B}=\left\{
\begin{array}{cl}
\max\limits_{\eta,\{G_{ab}\}_{ab}} &\eta \\
\text{s.t.}
&G_{ab} \geq 0,\quad\sum\limits_{ab} G_{ab} = \mathbb{I} \\
&\sum\limits_b G_{ab} \geq \eta A_a\\
&\sum\limits_a G_{ab} \geq \eta B_b\\
\end{array}\right.\quad=\left\{
\begin{array}{cl}
\min\limits_{N,\{X_a\}_a} &\tr N\\
\text{s.t.}
&N=N^\dagger,\quad N\geq X_a+Y_b\\
&X_a\geq0, \quad Y_b\geq0\vphantom{\sum\limits_a}\\
&\sum\limits_a\tr(X_aA_a)+\sum\limits_b\tr(Y_bB_b)\geq1\\
\end{array}\right..
\label{eqn:irg_sdp}
\end{equation}
Note that in the primal, the noise POVMs do not appear, because we can explicitly solve for these variables, which gives rise to matrix inequalities instead of equalities for the marginals.
These looser constraints give us additional freedom and allow us to employ operator inequalities.
Note also that the constraint $\eta\leq1$ is not enforced as it follows from summing up one of the marginal constraints.
The constraints in the primal in Eq.~\eqref{eqn:irg_sdp} are looser than in the primal in Eq.~\eqref{eqn:irjm_sdp}, because the noise set is larger for all measurement pairs.
In turn, the feasible set of the dual problem shrinks, as the dual constraints $X_a\geq0$ and $Y_b\geq0$ are tighter than $X_a+Y_b\geq0$.
In analogy with $\eta^\pp$ and $\eta^\mjm$, the measure $\eta^\mg$ also satisfies the properties we introduced in Section~\ref{sec:prelim}, namely monotonicity under pre- and post-processings, and convexity of the inverse.
\subsubsection{Example}
\label{sec:irg_ill}
The value of this measure for the running example is even higher than for the previous measures, specifically
\begin{equation}
\eta^\mg_\theta=\frac{\sqrt2+1}{\sqrt2+\cos\theta+\sin\theta}.
\label{eqn:irg_ill}
\end{equation}
This value is plotted in Fig.~\ref{fig:runex} together with the other measures.
A feasible point for the primal in Eq.~\eqref{eqn:irg_sdp} reads
\begin{equation}
G=
\begin{bmatrix}
r\,\frac{\mathbb{I}-\sigma_z}{2}&(1-r)\,\frac{\mathbb{I}-\sigma_x}{2}\\
(1-r)\,\frac{\mathbb{I}+\sigma_x}{2}&r\,\frac{\mathbb{I}+\sigma_z}{2}
\end{bmatrix},
\quad\text{where}\quad
r = \frac{1-\sin\theta+(\sqrt2+1)\cos\theta}{\sqrt2(\sqrt2+\cos\theta+\sin\theta)},
\end{equation}
and for the dual,
\begin{equation}
(X,Y)=\frac{\sqrt2}{4(\sqrt2+\cos\theta+\sin\theta)}
\left(\begin{bmatrix}
\mathbb{I}-\frac{\sigma_z+\sigma_x}{\sqrt2}\\
\mathbb{I}+\frac{\sigma_z+\sigma_x}{\sqrt2}
\end{bmatrix},
\begin{bmatrix}
\mathbb{I}-\frac{\sigma_z-\sigma_x}{\sqrt2}\\
\mathbb{I}+\frac{\sigma_z-\sigma_x}{\sqrt2}
\end{bmatrix}\right),
\quad\text{and}\quad
N=\frac{\sqrt2+1}{2(\sqrt2+\cos\theta+\sin\theta)}\,\mathbb{I}.
\end{equation}
\subsubsection{Lower bound}
\label{sec:irg_low}
For a pair $(A,B)$ of rank-one measurements in dimension $d$, let us introduce a feasible point for the primal in Eq.~\eqref{eqn:irg_sdp} with $G$ of the form \eqref{eqn:ansatz_low}, where
\begin{equation}
\label{eqn:irg_low_ansatz}
\begin{pmatrix}\alpha_{b}\\\beta_{a}\end{pmatrix}=\frac{1}{2\sqrt{d}}\begin{pmatrix}\tr B_b\\\tr A_a\end{pmatrix},\quad\gamma_{ab}=0,\quad\text{and}\quad\delta=\frac{\sqrt{d}}{2},
\end{equation}
so that we obtain the bound
\begin{equation}
\label{eqn:irg_low}
\eta=\frac12\left(1+\frac{1}{\sqrt{d}}\right)\leq\eta^\mg_{A,B}.
\end{equation}
A proof of feasibility of this specific point is given below.
For more details, see Appendix~\ref{app:irg_low} which also contains a measurement-dependent refinement.
As $\eta^\mg$ is monotonic under post-processings, this bound on pairs of rank-one measurements extends to all pairs of measurements in dimension $d$.
The novelty in Eq.~\eqref{eqn:irg_low_ansatz}, as compared to the parent POVMs used for the other measures, is the fact that $\delta$ is non-zero.
What enables us to introduce this term is the extra freedom in the primal in Eq.~\eqref{eqn:irg_sdp}, namely, the inequalities in the marginal constraints instead of equalities, which allows us to analyse the marginals for non-zero~$\delta$.
For the proof of feasibility, we write the parent POVM defined by the coefficients in Eq.~\eqref{eqn:irg_low_ansatz} as
\begin{equation}
\label{eqn:irg_low2}
G_{ab} = \frac{1}{4(d+\sqrt{d})}\left[\tr(B_b) A_a + \tr(A_a) B_b + 2\sqrt{d} \{A_a, B_b\} + d\left(A_a^{\frac12} B_b A_a^{\frac12} + B_b^{\frac12} A_a B_b^{\frac12}\right)\right].
\end{equation}
Since $A_{a}$ and $B_{b}$ are rank-one, we can write $A_a = \tr(A_a)P_a$ and $B_b = \tr(B_b)Q_b$ for some $P_a=\ketbra{\varphi_a}$ and $Q_b=\ketbra{\psi_b}$.
Therefore, we can rewrite \eqref{eqn:irg_low2} as
\begin{equation}
G_{ab} = \frac{\tr(A_a)\tr(B_b)}{4(d+\sqrt{d})}\left[(P_a + \sqrt{d} P_aQ_b)^\dagger(P_a + \sqrt{d} P_aQ_b)+(Q_b + \sqrt{d}Q_bP_a)^\dagger(Q_b + \sqrt{d}Q_bP_a)\right] \geq 0,
\end{equation}
which shows that $G$ is a valid POVM.
Next we should compute its marginals.
The first one reads
\begin{equation}
\label{eqn:irg_low3}
\sum_b G_{ab} = \frac{1}{4(d+\sqrt{d})}\left[d A_a + \tr(A_a)\mathbb{I} + 4 \sqrt{d} A_a + d\left(A_a + \sum_bB_b^{\frac12} A_a B_b^{\frac12}\right)\right],
\end{equation}
where the terms are ordered as in Eq.~\eqref{eqn:irg_low2} for clarity.
Moreover, we have that for every $\ket{\xi}$,
\begin{equation}
\begin{split}
d\,\bra{\xi}\sum_bB_b^{\frac12} A_a B_b^{\frac12}\ket{\xi} & \left.= \sum_{b'}\tr(B_{b'})\sum_b\tr(B_b)\tr(A_a) \braket{\xi}{\psi_b} \braket{\psi_b} {\varphi_a} \braket{\varphi_a}{\psi_b} \braket{\psi_b}{\xi} \right.\\
& \left.= \tr(A_a)\sum_{b'}\left|\sqrt{\tr(B_{b'})}\right|^2\sum_b\left|\sqrt{\tr(B_b)}
\braket{\xi}{\psi_b}\braket{\psi_b} {\varphi_a}\right|^2 \right.\\
& \left.\geq \tr(A_a)\left|\sum_b \tr(B_b)\braket{\xi}{\psi_b}
\braket{\psi_b} {\varphi_a}\right|^2 = \tr(A_a)|\braket{\xi}
{\varphi_a}|^2 = \bra{\xi}A_a \ket{\xi}, \right.
\end{split}
\end{equation}
where we used the Cauchy--Schwarz inequality.
Therefore, $d\sum_bB_b^{1/2} A_a B_b^{1/2} \ge A_a$, which
together with $\tr(A_a)\mathbb{I} \ge A_a$ enables us to lower bound the marginal \eqref{eqn:irg_low3}, namely,
\begin{equation}
\sum_b G_{ab} \ge \frac{2(d + 2\sqrt{d} + 1)}{4(d + \sqrt{d})}A_a = \frac12 \left(1+\frac{1}{\sqrt{d}}\right)A_a.
\end{equation}
By symmetry of Eq.~\eqref{eqn:irg_low2} the same conclusion holds for the second marginal, which shows that the point defined in Eqs~\eqref{eqn:irg_low_ansatz} and~\eqref{eqn:irg_low} is indeed feasible.
\subsubsection{Upper bound}
\label{sec:irg_up}
Consider the following feasible point for the dual given in Eq.~\eqref{eqn:irg_sdp}:
\begin{equation}
\label{eqn:irg_up_ansatz}
X_a= \frac{A_a}{fd}, \quad Y_b= \frac{B_b}{fd}, \quad\text{and}\quad N=\frac{\lambda}{f}\cdot\frac{\mathbb{I}}{d}
\end{equation}
where $f$ and $\lambda$ are defined in Eq.~\eqref{eqn:f_lambda}.
This immediately implies that
\begin{equation}
\label{eqn:irg_up}
\eta^\mg_{A,B}\leq\frac{\lambda}{f}.
\end{equation}
\subsection{Relations between the measures}
Certain inclusions between the noise sets defined in Eqs~\eqref{eqn:noised}, \eqref{eqn:noiser}, \eqref{eqn:noisep}, \eqref{eqn:noisejm}, and \eqref{eqn:noiseg}, imply an ordering of the measures.
More specifically, from
\begin{equation}
\label{eqn:order}
({\bf N}_{A,B}^\mathrm{d}\cup{\bf N}_{A,B}^\mathrm{r}) \subseteq{\bf N}_{A,B}^\mathrm{p}\subseteq{\bf N}_{A,B}^\mathrm{jm}\subseteq{\bf N}_{A,B}^\mathrm{g},
\end{equation}
we conclude that
\begin{equation}
\label{eqn:order_measures}
\max \{ \eta^\md_{A,B},\eta^\mr_{A,B} \} \leq \eta^\pp_{A,B} \leq \eta^\mjm_{A,B} \leq \eta^\mg_{A,B}
\end{equation}
for every pair $(A,B)$.
It turns out that $\eta^\md$ and $\eta^\mr$ are incomparable (see Appendix~\ref{app:ctrex} for an example).
A more detailed analysis allows us to prove that some of the inequalities given in Eq.~\eqref{eqn:order_measures} are in fact strict.
Specifically, in Appendix~\ref{app:relations} we derive improved relations between $\eta^\md$ and $\eta^\mjm$, $\eta^\md$ and $\eta^\mg$, and $\eta^\mr$ and $\eta^\mg$, which imply that for a pair of incompatible measurements $(A,B)$ the separations between these measures are strict, i.e., ${\eta^\md_{A,B} < \eta^\mjm_{A,B}}$, ${\eta^\md_{A,B} < \eta^\mg_{A,B}}$, and ${\eta^\mr_{A,B} < \eta^\mg_{A,B}}$.
Moreover, the examples given in Section~\ref{sec:MUB} show that in some cases $\eta^\md$ coincides with $\eta^\pp$, as well as $\eta^\mr$ with $\eta^\pp$ and $\eta^\mjm$ with $\eta^\mg$.
The question whether the separation between $\eta^\pp$ and $\eta^\mjm$ is strict or not is left open.
\subsection{Mutually unbiased bases}
\label{sec:MUB}
We have mentioned earlier that mutually unbiased bases constitute a standard example of a pair of incompatible measurements on a $d$-dimensional system.
Indeed, they might seem like natural candidates for the most incompatible pair of measurements in dimension $d$.
In this section we show that for a pair of MUBs all the previously introduced measures can be computed analytically.
The specific values we obtain will be compared against the findings of Section~\ref{sec:most}, in which we look for the most incompatible pairs of measurements.
For a pair $(A^\mathrm{MUB},B^\mathrm{MUB})$ of projective measurements onto two MUBs in dimension $d$ (see Section~\ref{subsec:jm}), we will use $\eta^\ast_{\mathrm{MUB}}(d)$ as a shorthand for $\eta^\ast_{A^\mathrm{MUB},B^\mathrm{MUB}}$.
Note that although in higher dimensions not all pairs of MUBs are unitarily equivalent, they nevertheless give the same value for all the measures studied in this work.
Hence, for these measures the quantity $\eta^\ast_{\mathrm{MUB}}(d)$ turns out to be well-defined.
In dimension $d=2$ a pair of MUB measurements is a special case of the example introduced in Section~\ref{sec:pre_ill}, corresponding to $\theta=\pi/4$.
Therefore Eqs~\eqref{eqn:ird_ill}, \eqref{eqn:irjm_ill}, and \eqref{eqn:irg_ill} imply that
\begin{equation}
\label{eqn:ir_mub2}
\eta^\md_\mathrm{MUB}(2)=\eta^\mr_\mathrm{MUB}(2)=\eta^\pp_\mathrm{MUB}(2)=\frac{1}{\sqrt2},\quad\eta^\mjm_\mathrm{MUB}(2)=2(\sqrt2-1),\quad\text{and}\quad\eta^\mg_\mathrm{MUB}(2)=\frac12\left(1+\frac{1}{\sqrt2}\right).
\end{equation}
For a pair of projective measurements onto two MUBs in dimension $d\geq3$, the parameters given in Eqs~\eqref{eqn:f_lambda} and \eqref{eqn:g} equal $f=2$, $\lambda=1+1/\sqrt{d}$, $g^\mathrm{d}=g^\mathrm{r}=g^\mathrm{p}=2/d$, and $ g^\mathrm{jm}=0$.
It turns out that for MUBs the upper bounds given in Eqs~\eqref{eqn:ird_up}, \eqref{eqn:irjm_up}, and \eqref{eqn:irg_up} are actually tight.
Therefore, the only missing component is a feasible point for the primal.
For $\eta^\md$ and $\eta^\mr$ our feasible solution consists of
\begin{equation}
\eta = \frac12\left(1+\frac{1}{\sqrt{d} + 1}\right)
\end{equation}
and
\begin{equation}
\label{eqn:ir_mub}
G_{ab}=\frac{1}{2(\sqrt{d}+1)}\left(\{A_a,B_b\}+\frac{1}{\sqrt{d}}A_a+\frac{1}{\sqrt{d}}B_b\right).
\end{equation}
This parent POVM, inspired by Ref.~\cite[Section~IV]{ULMH16}, is of the form of Eq.~\eqref{eqn:ansatz_low}.
The positivity of these operators can be confirmed using the techniques presented in Appendix \ref{app:low} and let us stress that the proof crucially relies on the fact that the bases are mutually unbiased.
For $\eta^\pp$ we must explicitly include the weights and we choose them to be uniform $p_a = q_b = 1/d$ for all $a,b$.
This assignment saturates the upper bound given in Eq.~\eqref{eqn:ird_up}, which implies that
\begin{equation}
\label{eqn:ir_mub3}
\eta^\md_\mathrm{MUB}(d)=\eta^\mr_\mathrm{MUB}(d)=\eta^\pp_\mathrm{MUB}(d)=\frac12\left(1+\frac{1}{\sqrt{d}+1}\right).
\end{equation}
For $\eta^\mg$ we use the same parent POVM, but the more flexible form of noise allows for higher visibility:
\begin{equation}
\label{eqn:visibility_g_jm}
\eta = \frac12\left(1+\frac{1}{\sqrt{d}}\right).
\end{equation}
For $\eta^\mjm$ we must supplement our solution with a sub-normalised parent POVM of the noise pair
\begin{equation}
\label{eqn:ir_mubH}
\tilde{H}_{ab} = \frac{1-\eta^\mjm_\mathrm{MUB}(d)}{d(d-2)}\left[\mathbb{I}+\frac{d}{d-1}\Big(\{A_a,B_b\}-A_a-B_b\Big)\right],
\end{equation}
which has already been used in Ref.~\cite{CHT19}, and is of the form of Eq.~\eqref{eqn:ansatz_low}.
This construction is only valid for $d \geq 3$, because for $d = 2$ the corresponding noise pair $\{(\mathbb{I}-A_a)/(d-1)\}_a$ and $\{(\mathbb{I}-B_b)/(d-1)\}_b$ is not jointly measurable (see Eq.~\eqref{eqn:jm_qubit} for a family of optimal feasible points for the primal).
In both cases the visibility given in Eq.~\eqref{eqn:visibility_g_jm} saturates the upper bounds~\eqref{eqn:irg_ill} and \eqref{eqn:irjm_ill}, respectively, which implies that for all $d \geq 3$, we have
\begin{equation}
\label{eqn:ir_mub4}
\eta^\mjm_\mathrm{MUB}(d)=\eta^\mg_\mathrm{MUB}(d)=\frac12\left(1+\frac{1}{\sqrt{d}}\right).
\end{equation}
Note that the value $\eta^\mg_\mathrm{MUB}(d)$ was already derived in~\cite{Haa15}. Also notice that Eq.~\eqref{eqn:ir_mub4} together with Eq.~\eqref{eqn:irg_low} implies that MUBs are among the most incompatible measurement pairs with respect to $\eta^\mg$ in every dimension.
\subsection{Summary}
In Table~\ref{tab:magic} we give a compact summary of the results for the differents robustness-based measures of incompatibility: definition of the noise sets, properties introduced in Section~\ref{sec:monotonicity}, lower and upper bounds, and value for a specific example of two projective measurements onto MUBs (see Section~\ref{sec:MUB}).
In Fig.~\ref{fig:runex} we plot the values of $\eta^\ast_\theta$ achieved by a pair of rank-one projective measurements acting on a qubit.
\begin{table}[h!]
\centering
\renewcommand{\arraystretch}{1}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
&Form of the noise & Post & Pre & Cvx & Lower bound & MUB value & Upper bound \vphantom{$\vcenter{\rule{0pt}{20pt}}$} \\ \hline
$\eta^\md$ &$\left\{\Big(\left\{\tr A_a \frac{\mathbb{I}}{d}\right\}_a,\left\{\tr B_b\frac{\mathbb{I}}{d}\right\}_b\Big)\right\}$& yes & no & no & $\displaystyle \frac{d-2+\sqrt{d^2+4d-4}}{4(d-1)}$ & & $\displaystyle\frac{\lambda-g^\mathrm{d}}{f-g^\mathrm{d}}$ \vphantom{$\vcenter{\rule{0pt}{36pt}}$}\\\cline{1-6}\cline{8-8}
$\eta^\mr$ &$\left\{\left(\left\{\frac{\mathbb{I}}{n_A}\right\}_a,\left\{\frac{\mathbb{I}}{n_B}\right\}_b\right)\right\}$& no & yes & yes & $\displaystyle \frac12\left(1+\frac{1}{\sqrt{n_An_B}+1}\right)$ & {$\displaystyle \frac12\left(1+\frac{1}{\sqrt{d}+1}\right)$} & $\displaystyle\frac{\lambda-g^\mathrm{r}}{f-g^\mathrm{r}}$ \vphantom{$\vcenter{\rule{0pt}{36pt}}$}\\\cline{1-6}\cline{8-8}
$\eta^\pp$ &$\left\{\Big(\big\{p_a\,\mathbb{I}\big\}_a,\big\{q_b\,\mathbb{I}\big\}_b\Big)\right\}$& \multicolumn{3}{c|}{yes} & $\max \{ \eta^\md,\eta^\mr \}$ & & $\displaystyle\frac{\lambda-g^\mathrm{p}}{f-g^\mathrm{p}}$ \vphantom{$\vcenter{\rule{0pt}{36pt}}$}\\ \hline
$\eta^\mjm$ &$\JM_d^{n_A,n_B}$& \multicolumn{3}{c|}{yes} & $\displaystyle \frac{2\sqrt{d^2+4d-4}}{3d-2+\sqrt{d^2+4d-4}}$ & $\!\!\left\{\begin{array}{ll}2(\sqrt{2} - 1)&d = 2\\\frac12\left(1+\frac{1}{\sqrt{d}}\right)&d \geq 3\end{array}\right.\!\!$ & $\displaystyle \frac{\lambda- g^\mathrm{jm}}{f- g^\mathrm{jm}}$\vphantom{$\vcenter{\rule{0pt}{36pt}}$} \\\hline
$\eta^\mg$ &$\POVM_d^{n_A,n_B}$& \multicolumn{3}{c|}{yes} & \multicolumn{2}{c|}{$\displaystyle \frac12\left(1+\frac{1}{\sqrt{d}}\right)$} & $\displaystyle \frac{\lambda}{f}$ \vphantom{$\vcenter{\rule{0pt}{36pt}}$} \\\hline
\end{tabular}
\caption{
Summary of the results on the depolarising, random, probabilistic, jointly measurable, and general incompatibility robustness of pairs of POVMs.
Recall that $d$ is the dimension, while $n_{A}$ and $n_{B}$ are the outcome numbers.
``Post'' and ``Pre'' stand for post-processing and pre-processing monotonicity, respectively, see Section~\ref{sec:monotonicity}.
``Cvx'' stands for the convexity of the inverse of the measure, see Section~\ref{sec:monotonicity}.
For a pair of rank-one projective measurements $(A,B)$, the quantities appearing in the upper bounds are $f=2$, $\lambda=\max_{a,b}\{\max\Sp(A_a+B_b)\}$, $ g^\mathrm{jm}=\min_{a,b}\{\min\Sp(A_a+B_b)\}$, and $g^\mathrm{d}=g^\mathrm{r}=g^\mathrm{p}=2/d$; see Eqs~\eqref{eqn:f_lambda} and \eqref{eqn:g} for definitions.
}
\label{tab:magic}
\renewcommand{\arraystretch}{1.4}
\end{table}
\begin{figure}[h!]
\centering
\includegraphics[width=12cm]{fig4.pdf}
\caption{
The value of all the different measures (see Table \ref{tab:magic}) for a pair of rank-one projective measurements on a qubit such that the angle between the Bloch vectors of these measurements equals $2\theta$; see Eq.~\eqref{eqn:runex}.
Note that the rightmost point where $\theta=\pi/4$ corresponds to qubit MUBs, which demonstrates the fact that MUBs are the most incompatible rank-one projective qubit measurements under all these measures.
From bottom to top, the curves are $\eta^\md=\eta^\mr=\eta^\pp$ from Eq.~\eqref{eqn:ird_ill}, then $\eta^\mjm$ from Eq.~\eqref{eqn:irjm_ill}, and finally $\eta^\mg$ from Eq.~\eqref{eqn:irg_ill}.
Although $\eta^\md$, $\eta^\mr$, and $\eta^\pp$ coincide in this case, this is not the case in general.
}
\label{fig:runex}
\end{figure}
\section{Most incompatible pairs of measurements}
\label{sec:most}
In this section, we address the question of the most incompatible measurement pairs in dimension $d$, for all the measures introduced in Section~\ref{sec:measures}.
This question has already been raised and partially answered in previous works: in infinite dimension for $\eta^\pp$ in Ref.~\cite{HSTZ14} and numerically for $\eta^\md$ and $\eta^\mg$ in Ref.~\cite{BQG+17}.
Perhaps surprisingly, we find that the answer depends on which incompatibility measure we consider.
We have already seen that projective measurements onto a pair of mutually unbiased bases are among the most incompatible pairs under $\eta^\mg$ in every dimension.
On the other hand, for the measures $\eta^\md$ and $\eta^\pp$ we give explicit constructions of pairs which are more incompatible than MUBs for any dimension $d\geq3$.
For $\eta^\mjm$, our study is inconclusive, and we do not find measurements that are more incompatible than MUBs in any dimension.
First we discuss the special case of $\eta^\mr$, then we solve the qubit case for all the measures, and finally we discuss higher dimensions.
\subsection{Incompatibility random robustness}
\label{sec:irr_special}
Recall that in order to find the most incompatible measurement pair in dimension $d$ regardless of the outcome numbers, it is enough to consider rank-one POVMs if the measure in consideration is monotonic under post-processings.
As we see from Table~\ref{tab:magic}, this is not the case for $\eta^\mr$, which, at first glance, makes this problem hard to tackle.
However, what turns out is that for this measure the answer is trivial.
Consider a pair of measurements $(A,B)$ and increase artificially the number of outcomes by adding zero POVM elements to both measurements.
Let us add these elements one-by-one, and denote the POVM pair at step $i$ by $(A^i,B^i)$.
In Appendix~\ref{app:irr_up} we show that if $\lambda<2$ and $2(\lambda-1)<f$, we have
\begin{equation}
\lim_{i\to\infty}\eta^\mr_{A^i,B^i} \leq \frac{2-\lambda}{f-2(\lambda-1)},
\end{equation}
where $f$ and $\lambda$ are defined in Eq.~\eqref{eqn:f_lambda}.
It is then clear that whenever $f=2$ and $\lambda<2$ (e.g., any pair of rank-one projective measurements onto two bases that do not have any eigenvectors in common), this limit reaches $\frac12$.
As it coincides with the trivial lower bound mentioned in Section~\ref{sec:mostincomp}, this shows that $\chi^\mathrm{r}(d)=\frac12$ for $d\geq2$.
In the rest of this section, we will not discuss this measure anymore.
However, recall that for pairs of rank-one projective measurements $\eta^\mr$ coincides with $\eta^\md$, and therefore some of the results later in this section also apply to this measure.
\subsection{Qubit case}
\label{sec:qubit}
In Section~\ref{sec:MUB} we have shown that for a pair of mutually unbiased bases all the incompatibility measures can be computed analytically.
What is special in the case of $d = 2$ is that these values coincide with the universal lower bounds (see Table \ref{tab:magic}).
This means that pairs of projective measurements onto MUBs are among the most incompatible pairs under $\eta^\md$, $\eta^\pp$, $\eta^\mjm$, and $\eta^\mg$ in dimension $d = 2$.
Formally, using the notation introduced in Section~\ref{sec:mostincomp}, we have that
\begin{equation}
\chi^\mathrm{d}(2)=\chi^\mathrm{p}(2)=\frac{1}{\sqrt2},\quad\chi^\mathrm{jm}(2)=2(\sqrt2-1),\quad\text{and}\quad\chi^\mathrm{g}(2)=\frac12\left(1+\frac{1}{\sqrt2}\right).
\end{equation}
For $\eta^\md$, this was known for pairs of two-outcome POVMs \cite[Appendix~G]{DSFB19}.
It is important to point out that there exist other pairs of measurements reaching these minimal values: from the upper bounds given in Appendix~\ref{app:other_up}, it is clear that any rank-one POVM pair such that $A_a=\ketbra{a}$ and the Bloch vectors of $B$ lie in the $xy$-plane of the Bloch sphere gives rise to the same value as MUBs.
As an example, one might choose $A_a=\ketbra{a}$ and $B$ as a trine measurement in the $xy$-plane.
In Appendix~\ref{app:triplets}, we extend this result to triplets of qubit measurements.
In this case, we show that triplets of projective measurements onto MUBs are among the most incompatible measurements under $\eta^\md$, $\eta^\pp$, $\eta^\mjm$, and $\eta^\mg$ in dimension $d = 2$.
Also note that the value of $\chi^\mathrm{d}(2)$ (respectively its equivalent for three measurements) has interesting consequences for Einstein--Podolsky--Rosen steering.
This is because joint measurability is intimately linked to this notion \cite{QVB14,UBGP15}, as the depolarising map in $\eta^\md$ can be equivalently applied to the state we wish to steer, due to its self-duality.
We refer to Ref.~\cite[Appendix~F]{DSFB19} for details on this connection and only mention here that our results show that in a steering scenario with two (respectively three) measurements and an isotropic state of local dimension two, POVMs do not provide any advantage over projective measurements.
\subsection{Higher dimensions}
\label{sec:higher_dim}
\subsubsection{\texorpdfstring{Dimension $d=3$}{Dimension three}}
\label{sec:qutrit}
In the previous section we have seen that in dimension $d=2$ pairs of projective measurements onto two MUBs are among the most incompatible pairs of measurements under $\eta^\md$, $\eta^\pp$, $\eta^\mjm$, and $\eta^\mg$.
Starting from dimension $d=3$, the picture changes dramatically.
To show this, we plot the (numerical) value of these four measures for a particular one-parameter path of rank-one projective measurements in dimension three, see Fig.~\ref{fig:devil}.
It is evident from this plot that, contrary to the qubit case, MUBs do not achieve the lowest value of the incompatibility robustness under $\eta^\md$ and $\eta^\pp$.
Instead, the lowest value among rank-one projective measurements is reached by other bases, which we have found through an extensive numerical search among pairs of rank-one projective measurements, using a parametrisation of unitary matrices in dimension three \cite{Bro88}.
\begin{figure}[ht!]
\centering
\includegraphics[width=12cm]{fig5.pdf}
\caption{The (numerical) value of the four measures along a one-parameter path of rank-one projective measurements in dimension $d=3$.
The pair $(A^\mathrm{dev},B^\mathrm{dev})$ is defined in Eq.~\eqref{eqn:abp}, $(A^\mathrm{qMUB},B^\mathrm{qMUB})$ in Eq.~\eqref{eqn:abd}, and $(A^\mathrm{MUB},B^\mathrm{MUB})$ at the beginning of this section.
Details about the specific path used can be found in Appendix~\ref{app:path}.
Importantly, on this path the pair $(A^\mathrm{MUB},B^\mathrm{MUB})$ achieves the minimum value with respect to $\eta^\mg$ and $\eta^\mjm$, but it is outperformed by $(A^\mathrm{dev},B^\mathrm{dev})$ with respect to $\eta^\pp$ and by $(A^\mathrm{qMUB},B^\mathrm{qMUB})$ with respect to $\eta^\md$.}
\label{fig:devil}
\end{figure}
In this section we only look at rank-one projective measurements.
Due to the unitary invariance of all the measures we assume without loss of generality that the first measurement corresponds to the computational basis $A_{a} = \ketbra{a}$, so that we only need to specify the second measurement $B$.
For $\eta^\md$, the optimum is reached, among others, by
\begin{equation}
\label{eqn:abd}
B_b^\mathrm{qMUB}=U\ketbra{b}U^\dagger,\quad\text{where}\quad
U=\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0\\0&0&1\end{pmatrix}.
\end{equation}
Note that it is simply a pair of \emph{qubit MUBs} on a two-dimensional subspace together with a trivial third outcome on the orthogonal subspace.
The incompatibility depolarising robustness of this pair, $\eta^\md_\mathrm{qMUB}(3)\approx0.6602$ (see Eq.~\eqref{eqn:qubitMUB} below for an analytical value) outperforms substantially not only $\eta^\md_\mathrm{MUB}(3)\approx0.6830$, but also the minimal value $0.6794$ found numerically in Ref.~\cite[Table IV]{BQG+17}.
For $\eta^\pp$, the optimum is reached, among others, by
\begin{equation}
\label{eqn:abp}
B_b^\mathrm{dev}=U\ketbra{b}U^\dagger,\quad\text{where}\quad
U=\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac12&\frac12\\\frac{1}{\sqrt{2}}&-\frac12&-\frac12\\0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix},
\end{equation}
which gives $\eta^\pp_\mathrm{dev}\approx0.6813$, showing a slight \emph{deviation} from $\eta^\pp_\mathrm{MUB}(3)\approx0.6830$.
For $\eta^\mjm$, the numerical search did not yield an improvement on the MUB value, and for $\eta^\mg$ we already have an analytical proof that MUBs are among the most incompatible pairs in every dimension.
\subsubsection{\texorpdfstring{Dimension $d\ge4$}{Dimension higher than four}}
\label{sec:higher}
For $\eta^\md$, the qubit MUB structure found in dimension $d=3$ has several natural generalisations in higher dimensions.
The general idea is to divide the Hilbert space into orthogonal subspaces of various dimensions, and define the measurements as either MUBs or trivial measurements on the different subspaces.
Among these, we found numerically that the most incompatible construction is to define a pair of qubit MUBs on a two-dimensional subspace, while on the orthogonal subspace the remaining measurement operators turn out to be irrelevant.
For simplicity, we choose trivial measurements on the orthogonal subspace, that is, $A_a = \ketbra{a}$ and $B_b = \ketbra{b}$ for $a,b \ge 3$, while $\{A_1,A_2\}$ and $\{B_1,B_2\}$ is a pair of MUBs on the qubit subspace.
For this construction, we get a lower bound in Eq.~\eqref{eqn:qubitMUB_low} and an upper bound in Eq.~\eqref{eqn:qubitMUB_up},
which give the same value and therefore the incompatibility depolarising robustness of this pair is
\begin{equation}
\label{eqn:qubitMUB}
\eta^\md_\mathrm{qMUB}(d)=\frac12\left(1+\frac{\sqrt2}{d+\sqrt2}\right)<\eta^\md_\mathrm{MUB}.
\end{equation}
In Fig.~\ref{fig:chi} we plot the improvement over MUBs that this construction achieves.
In particular, it is worth stressing that, in contrast to a pair of MUBs, this construction exhibits the same asymptotic scaling as the lower bound derived in Section~\ref{sec:ird_low}.
More specifically, expanding the right-hand side of Eq.~\eqref{eqn:ird_low} gives
\begin{equation}
\frac{1}{2} + \frac{1}{2d} + O( d^{-2} ),
\end{equation}
whereas
\begin{align}
\eta^\md_\mathrm{qMUB}(d) &= \frac{1}{2} + \frac{1}{\sqrt{2} d} + O( d^{-2} ),\\
\eta^\md_\mathrm{MUB} &= \frac{1}{2} + \frac{1}{2 \sqrt{d}} + O(d^{-1}).
\end{align}
The reason why this pair performs so well is the fact that the two measurements are highly incompatible on the qubit subspace, while the noise is spread uniformly over the entire space.
Note that an analogous structure has been found while searching for the quantum state whose nonlocal statistics are the most robust to white noise \cite{ADGL02}.
Supported by the optimisation in dimension $d=3$ together with one billion random instances in dimensions $d=4$ and $d=5$, and the asymptotic scalings, we conjecture that this pair is among the most incompatible pairs of rank-one projective measurements under $\eta^\md$ for all dimensions.
For general pairs of measurements we leave the question open.
For $\eta^\pp$, fixing MUBs on a qubit subspace no longer determines the incompatibility robustness any more, as the noise can now be adjusted to have different weights on the different subspaces.
In fact the construction that uses trivial measurements on the orthogonal subspace does not surpass the $d$-dimensional MUB value any more.
However, employing some other rank-one projective measurements on the orthogonal subspace gives rise to measurements that outperform MUBs.
In even dimensions, by decomposing the space into many qubit subspaces and by having MUBs on each of them, we can reach again the value of Eq.~\eqref{eqn:qubitMUB}.
For instance in dimension $d=4$ this means
\begin{equation}
\label{eqn:qMUB4}
B_b=U\ketbra{b}U^\dagger,\quad\text{where}\quad
U=\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0&0\\\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0&0\\0&0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\0&0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix}.
\end{equation}
The parent POVM is then the same as for $\eta^\md$ whereas the construction of the dual variables is explained in Appendix~\ref{app:ird_up}.
Our conjecture on $\eta^\md$ then translates straightforwardly to $\eta^\pp$ in even dimensions as $\eta^\md\leq\eta^\pp$.
In odd dimensions, this construction is not applicable.
We conjecture that in dimension $d=3$ the pair defined in Eq.~\eqref{eqn:abp} is among the most incompatible pairs of projective measurements under $\eta^\pp$.
In higher odd dimensions, taking this pair on a qutrit subspace together with MUBs on all remaining qubit subspaces always outperforms MUBs (see Fig.~\ref{fig:chi}).
As there might be some more involved construction giving a lower value, we leave the question of the lowest value of $\eta^\pp$ open for odd dimensions higher that $d=5$.
Note nonetheless that with one billion random pairs of rank-one measurements in dimension $d=5$ we were not able to surpass it.
\begin{figure}[h!]
\centering
\includegraphics[width=12cm]{fig6.pdf}
\caption{Illustration of the improvement over MUBs for $\eta^\md$ and $\eta^\pp$ when the dimension $d$ ranges from 2 to 16.
From top to bottom are depicted the MUB value (Eqs~\eqref{eqn:ir_mub3} and \eqref{eqn:ir_mub4}), the lowest value we found for $\eta^\pp$ (that is, Eq.~\eqref{eqn:qubitMUB} for even dimensions and numerical results based on an analytical construction described in the main text for odd dimensions), the lowest value we found for $\eta^\md$ (Eq.~\eqref{eqn:qubitMUB}), and the lower bound \eqref{eqn:ird_low}.
}
\label{fig:chi}
\end{figure}
For $\eta^\mjm$, encouraged by the optimisation in dimension $d=3$ and the one billion random sampling in dimensions $d=4$ and $d=5$, we conjecture that pairs of MUBs in any dimension cannot be outperformed by any pair of rank-one projective measurements.
Regarding $\eta^\mg$, the incompatibility generalised robustness of a pair of MUBs is precisely the universal lower bound that we derived in Eq.~\eqref{eqn:irg_low}.
This means that MUBs are among the most incompatible pairs among all pairs of measurements in dimension $d$, regardless of the number of outcomes.
Formally, using the notation introduced in Section~\ref{sec:mostincomp}, this means that
\begin{equation}
\chi^\mathrm{g}(d)=\frac12\left(1+\frac{1}{\sqrt{d}}\right).
\end{equation}
\section{Conclusions}
\label{sec:conclusion}
In this work we develop a unified framework to study various robustness-based measures of incompatibility of quantum measurements.
We find that some of the widely used measures do not satisfy some natural properties, which means that one should be cautious when dealing with them.
In particular, they are not suitable for constructing a resource theory of incompatibility.
Moreover, we find that the most incompatible measurement pair depends on the exact measure that we use, even when all the addressed natural properties are satisfied.
We are able to show that for one of the measures a pair of rank-one projective measurements onto mutually unbased bases is among the most incompatible pairs, but also that this is not the case for some other measures.
Our work shows that the different measures exhibit genuinely different properties and we conclude that despite a substantial effort dedicated to the topic, our understanding is still rather limited.
One natural future direction arising from our work would be to obtain a complete characterisation of the most incompatible measurement pairs in all scenarios for all the measures.
We expect, however, that this might be rather difficult, so one might start by restricting the task to natural scenarios, e.g., $d = n_{A} = n_{B}$ or even just searching over rank-one projective measurements.
Many results in this paper can be straightforwardly extended to the case of more than two measurements.
We refer to Appendix~\ref{app:more} for the SDP formulations of the various measures, the upper bounds and a few lower bounds.
This could serve as a good starting point for future research.
A last promising research direction arising from our work concerns the possibility of constructing a resource theory of incompatibility.
Are some of the existing measures suitable as resource monotones? Are there some additional conditions that one should require?
What is the most general class of operations that preserves joint measurability? Answering these questions will help us to understand how to quantify and classify incompatibility in a meaningful and operational manner.
\section*{Acknowledgments}
The authors thank Nicolas Brunner, Bartosz Regu{\l}a, Ren{\'e} Schwonnek, Marco Tomamichel and Roope Uola for useful discussions.
Financial support by the Swiss National Science Foundation (Starting grant DIAQ, NCCR-QSIT) is gratefully acknowledged.
M.F.~acknowledges support from the Polish NCN grant Sonata~UMO-2014/14/E/ST2/00020, and the grant Badania M\l odych Naukowc\'ow number 538-5400-B049-18, issued by the Polish Ministry of Science and Higher Education.
J.K.~acknowledges support from the National Science Centre, Poland (grant no.~2016/23/P/ST2/02122). This project is carried out under POLONEZ programme which has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska--Curie grant agreement no.~665778.
\tocless |
1,116,691,499,340 | arxiv | \section{Introduction}
In a recent paper
\cite{Panin:2010fk}
we
defined motivic versions of symplectically oriented cohomology theories $A$ and of
quaternionic Grassmannians $HGr(r,n)$. This $HGr(r,n)$ is the open subscheme of the ordinary
Grassmannian $Gr(2r,2n)$ parametrizing subspaces on which the standard symplectic form on
$\mathcal{O}^{\oplus 2n}$ is nondegenerate.
We defined Pontryagin classes of symplectic bundles in
such theories and calculated
\begin{equation}
\label{E:A(HGr)}
A(HGr(r,n)) = A(pt)[p_{1},\dots,p_{r}]/(h_{n-r+1},\dots,h_{n})
\end{equation}
where the $p_{i}$ are the Pontryagin classes of the tautological bundle on $HGr(r,n)$, and the
$h_{i}$ are the polynomials in the $p_{i}$ corresponding to the complete symmetric polynomials.
This is the same formula as the one which describes the cohomology of an ordinary Grassmannian
in terms of the Chern classes of an oriented cohomology theory.
In this paper we begin to apply those results to the hermitian $K$-theory of regular noetherian
separated schemes $X$ of finite Krull dimension with $\frac 12 \in \Gamma(X,\mathcal{O}_{X})$.
We write $KO^{[n]}(X,U)$ for Schlichting's hermitian $K$-theory space for bounded complexes of
vector bundles on $X$ which are acyclic on the open subscheme $U \subset X$ and which are symmetric
with respect to the shift by $n$ of the usual duality. We write $KO^{[n]}_{i}(X,U)$ for its homotopy
groups (for $i \geq 0$) or for Balmer's Witt groups $W^{n-i}(X,U)$ (for $i < 0$).
One of our main results is the following.
\begin{thm}
\label{T:SLc.ring}
For a regular separated noetherian scheme $S$ of finite Krull dimension with
$\frac 12 \in \Gamma(S,\mathcal{O}_{S})$ Schlichting's hermitian $K$-theory is a
ring cohomology theory with an $SL^{c}$ Thom classes theory.
\end{thm}
Here \emph{ring cohomology theory} is used in the sense of \cite[Definitions 2.1 and 2.13]{Panin:2003rz}.
An \emph{$SL^{c}$ Thom classes theory} specifies a Thom class
$\thom(E,L,\lambda) \in KO_{0}^{[n]}(E,E-X)$ for every \emph{$SL_{n}^{c}$-bundle},
by which we mean a
triple $(E,L,\lambda)$ with
$E$ a vector bundle of rank $n$ over $X$, $L$ a line bundle and $\lambda \colon L \otimes L \to \det E$
an isomorphism. These classes are functorial, multiplicative, and induces isomorphisms
$\cup \thom(E,L,\lambda) \colon KO_{i}^{[m]}(X) \to KO_{i}^{[m+n]}(E,E-X)$ for all $i$ and $m$.
The Thom classes restrict to Euler classes $e(E,L,\lambda) \in KO_{0}^{[n]}(X)$.
An $SL^{c}$ Thom classes theory gives Thom classes for all special linear, special orthogonal and
symplectic bundles.
So by the theory of \cite{Panin:2010fk} there are
Pontryagin classes $p_{i}(E,\phi) \in KO_{0}^{[2i]}(X)$
for symplectic bundles. The $p_{1}(E,\phi)$ of
a symplectic bundle
of rank $2r$
is the class corresponding to $[E,\phi] - r[\mathsf{H}] \in KSp_{0}(X,U) = GW^{-}(X)$ under the
natural isomorphism $KSp \cong KO^{[2]}$. Here $\mathsf{H}$ is the trivial symplectic bundle of rank $2$.
The higher Pontryagin classes will be calculated elsewhere.
We also construct several motivic spectra representing hermitian $K$-theory.
The first construction is a $T$-spectrum whose spaces
$(\mathbf{KO}^{[0]},\mathbf{KO}^{[1]},\mathbf{KO}^{[2]},\dots)$
are fibrant replacements
of presheaves composed of Schlichting's Waldhausen-like
hermitian $K$-theory spaces for bounded complexes of vector
bundles with shifted dualities \cite{Schlichting:2010uq}.
The structure maps $\mathbf{KO}^{[n]} \wedge T \to \mathbf{KO}^{[n+1]}$
are adjoint to the maps $\mathbf{KO}^{[n]}({-}) \to \mathbf{KO}^{[n+1]}({-} \wedge T)$
which are essentially multiplication by the Thom class
$\thom \in \mathbf{KO}_{0}^{[1]}(T)$ of the trivial line bundle.
Note the use of the appearance of the Thom classes in the very structure of the spectrum.
For $(X,U)$ in $\mathcal Sm\mathcal Op/S$ there are functorial isomorphisms
\begin{equation}
\label{E:isomorphisms}
KO_{i}^{[n]}(X,U) \cong \mathbf{KO}^{[n]}_{i}(X_{+}/U_{+}) \cong
\mathbf{BO}^{2n-i,n}(\Sigma_{T}^{\infty}(X_{+}/U_{+})),
\end{equation}
and the boundary maps $\partial \colon KO_{i}^{[n]}(U) \to KO_{i-1}^{[n]}(X,U)$
and
\[
\partial \colon \mathbf{BO}^{2n-i,n}(\Sigma_{T}^{\infty}(U_{+})) \to \mathbf{BO}^{2n-i+1,n}
(\Sigma_{T}^{\infty} (X_{+}/U_{+}))
\]
correspond.
Because it is based on the hermitian $K$-theory of chain complexes,
this spectrum has advantages in certain situations
over the one constructed several years ago by Hornbostel
\cite{Hornbostel:2005ph}.
It treats all
shifts/weights uniformly instead of dealing in one way with the $K^{h}$-theory of the even
weights and in another way with the $U$-theory and $V$-theory of the odd weights.
It naturally handles non-affine schemes $X$ and even pairs
$(X,U)$ with $U \subset X$ open. Finally we can easily identify the Thom, Euler
and Pontryagin classes in the Grothendieck-Witt groups of chain complexes
$GW^{[n]}(X\ on\ X-U)$.
We show that the Morel and Voevodsky's theorem on Grassmannians and algebraic $K$-theory extends
to the symplectic context. There are in truth only a few things to verify for symplectic groups
beyond what is in Morel and Voevodsky's paper.
(Orthogonal groups are much more problematic. {\bf However it has been done recently by M.Schlichting
and Shanker Tripathi.})
\begin{thm}
\label{T:MV.intro}
Let $HGr = \operatornamewithlimits{colim} HGr(n,2n)$ be the infinite quaternionic Grassmannian. Then
$\mathbb{Z} \times HGr$ and
$KSp$ are isomorphic in the motivic unstable homotopy category $H_{\bullet}(S)$.
\end{thm}
Next we define the $\times$ product.
The final group of theorems in the paper concerns the product structure.
The groups $KO_{0}^{[n]}(X,U)$, which are the
Grothendieck-Witt groups of bounded chain complexes of vector bundles which are symmetric
with respect to shifted
but untwisted dualities, have a naive product induced by the tensor product of chain complexes
\begin{equation}
\label{E:naive}
KO^{[m]}_0(X,U) \times KO^{[n]}_0(Y,V) \to
KO^{[m+n]}_0(X \times Y, X \times V \cup U \times Y).
\end{equation}
Let $\angles{1}$ and $\angles{-1}$ in $KO_{0}^{[0]}(pt)$ denote the Grothendieck-Witt classes
of the rank one symmetric bilinear forms.
The product is \emph{$\angles{-1}$-commutative} meaning that
for $\alpha \in \mathbf{BO}^{p,q}(A)$ and $\beta \in \mathbf{BO}^{p',q'}(B)$
we have $\alpha \times \beta = (-1)^{pp'}\angles{-1}^{qq'} \sigma^{*}(\beta \times \alpha)$
where $\sigma \colon A\times B \to B \times A$ switches the factors.
Recall that a motivic space $A$ is called \emph{small}
if $Hom_{SH(S)}(\Sigma_{T}^{\infty}A, {-})$ commutes with arbitrary coproducts.
\begin{thm}
\label{uniq2}
The cohomology theory
$(\mathbf{BO}^{*,*}, \partial)$
on the category
$\mathbf{M}^{\text{\textit{small}}}_{\bullet}(S)$
of small motivic spaces over $S$
has a product $\times$
which is associative and $\angles{-1}$-commutative
with the unit $1 = \angles{1} \in \mathbf{BO}^{0,0}(pt_{+})$,
which has $\alpha \times \Sigma_{\mathbb{G}_{m}}1 = \Sigma_{\mathbb{G}_{m}}\alpha$ and
$\alpha \times \Sigma_{S^{1}_{s}}1
= \Sigma_{S^{1}_{s}}\alpha$
for all $\alpha$, and
which restricts via the isomorphism
\eqref{E:isomorphisms}
to the naive ring structure
\eqref{E:naive}
on the groups
$KO^{[2n]}_{0}(X)$ for $X \in \mathcal Sm/S$.
It is the unique product with these properties.
\end{thm}
This is Theorems \ref{monoidBO} and \ref {T:unique.prod}.
Restricting to pairs $(X,U)$ with $U \subset X$ an open subscheme of a scheme smooth over $X$,
we get the following result.
\begin{thm}
\label{uniq1}
There is a canonical ring structure on the cohomology theory
$(KO^{[*]}_*, \partial)$
on $\mathcal Sm\mathcal Op/S$
which is associative and $\angles{-1}$-commutative
with unit
$\angles{1} \in KO^{[0]}_0(pt)$
and which restricts
to the naive product on the Grothendieck-Witt groups
of chain complexes
$KO^{[2n]}_0(X)$. This product and the
Thom classes of $SL^{c}$-bundles
make
$(KO_{*}^{[*]},\partial)$
ring cohomology theory with an $SL^{c}$ Thom classes theory.
\end{thm}
Our strongest result on the product is the following theorem.
\begin{thm}
\label{T:unique}
There exist morphisms $m \colon \mathbf{BO} \wedge \mathbf{BO}
\to \mathbf{BO}$
and $e \colon \Sigma_{T}^{\infty}\boldsymbol{1} \to \mathbf{BO}$ in $SH(S)$
which make $(\mathbf{BO},m,e)$ a commutative monoid in $SH(S)$ and which are compatible
with the naive product in the following sense.
\begin{enumerate}
\item For all $X$ and $Y$ in $\mathcal Sm/S$ and all even integers $2p$ and $2q$ the naive products
\[
KO_{0}^{[2p]}(X) \times KO_{0}^{[2q]}(Y) \to KO_{0}^{[2p+2q]}(X \times Y)
\]
and the product
\[
\mathbf{BO}^{4p,2p}(X_{+}) \times \mathbf{BO}^{4q,2q}(Y_{+}) \to
\mathbf{BO}^{4p+4q,2p+2q}(X_{+} \wedge Y_{+})
\]
induced by $m$ correspond under the isomorphisms
\eqref{E:isomorphisms}.
\item The elements $\angles{1} \in KO_{0}^{[0]}(pt)$ and $e \in \mathbf{BO}^{0,0}(S^{0,0})$ correspond under
the isomorphisms \eqref{E:isomorphisms}.
\end{enumerate}
Moreover, if for the base scheme $S$ the groups $KO_{1}(S)$ and $KSp_{1}(S)$ are finite
\parens{for example
$S = \operatorname{Spec} \mathbb{Z}[\frac 12]$},
then the monoid structure
$(m,e)$ with these properties is unique.
\end{thm}
For the proof of the theorem see Theorem \ref{T:unique.2}.
We explain the basic ideas in the proofs of these three theorems.
Gille and Nenashev's method \cite{Gille:2003ad} for constructing pairings in Witt groups of triangulated categories can be used
in hermitian $K$-theory to construct pairings between hermitian $K$-theory groups and
Gro\-then\-dieck-Witt groups
\begin{gather}
\label{E:partial.1}
KO^{[i]}_r(X,U) \times KO^{[j]}_0(Y,V) \to KO^{[i+j]}_r(X \times Y, X \times V \cup U \times Y)
\\
\label{E:partial.2}
KO^{[i]}_0(X,U) \times KO^{[j]}_r(Y,V) \to KO^{[i+j]}_r(X \times Y, X \times V \cup U \times Y)
\end{gather}
(see \eqref{E:KO.pairing.1}--\eqref{E:KO.pairing.2}). They respect the boundary maps of the cohomology theorem $KO_{*}^{[*]}$.
This gives the hermitian $K$-theory groups the structure of a
cohomology theory with a partial multiplicative structure with Thom classes for all $SL^{c}$ bundles
including symplectic bundles. Although this is less structure than we assumed
while writing \cite{Panin:2010fk}, it is enough to prove the quaternionic projective bundle
theorem and the symplectic splitting principle and to calculate the cohomology of quaternionic
Grassmannians. Thus formula \eqref{E:A(HGr)} holds for $A = KO_{*}^{[*]}$ with the
$p_{i} \in KO_{0}^{[2i]}(HGr(r,n))$.
Because of the isomorphism \eqref{E:isomorphisms} it also holds for $A = \mathbf{BO}^{*,*}$
with the $p_{i} \in \mathbf{BO}^{4i,2i}(HGr(r,n))$.
The isomorphism \eqref{E:isomorphisms}
also transplants these pairings to $\mathbf{BO}$, giving in particular pairings
\begin{equation}
\label{BO2*BO**toBO**1}
\boxtimes \colon
\mathbf{BO}^{2i-r,i}(X_{+}) \times \mathbf{BO}^{2j,j}(Y_{+}) \to \mathbf{BO}^{2i+2j-r, i+j}(X_{+} \wedge Y_{+})
\end{equation}
Theorem \ref{T:MV.intro} and the isomorphisms $KSp \cong KO^{[4k+2]}$
gives us a canonical elements $\tau_{4k+2} \in \mathbf{BO}^{8k+4,4k+2}(\mathbb{Z} \times HGr)$.
Write $[-n,n] = \{ m \in \mathbb{Z} \mid -n \leq m \leq n \}$ and set
\[
HGr_{n} = [-n,n] \times HGr(n,2n).
\]
We have $\mathbb{Z} \times HGr = \operatornamewithlimits{colim} HGr_{n}$. A standard formula for homotopy colimits in triangulated categories
gives us an exact sequence
\[
0 \to \varprojlim\nolimits^{1} \mathbf{BO}^{8k+3,4k+2}(HGr_{n})
\to \mathbf{BO}^{8k+4,4k+2}(\mathbb{Z} \times HGr) \to
\varprojlim \mathbf{BO}^{8k+4,4k+2}(HGr_{n})
\to 0
\]
The inclusions $HGr_{n} \to HGr_{n+1}$ induce surjections on cohomology for any symplectically oriented
theory. So the $\varprojlim\nolimits^{1}$ vanish, and we have
\begin{equation}
\label{E:ZHGr.lim}
\mathbf{BO}^{8k+4,4k+2}(\mathbb{Z} \times HGr) \cong
\varprojlim \mathbf{BO}^{8k+4,4k+2}(HGr_{n}).
\end{equation}
For essentially the same reasons we have isomorphisms
\begin{equation}
\label{E:ZHGr.ZHGr.lim}
\mathbf{BO}^{16k+8,8k+4}((\mathbb{Z} \times HGr) \wedge (\mathbb{Z} \times HGr))
\cong \varprojlim \mathbf{BO}^{16k+8,8k+4}(HGr_{n} \wedge HGr_{n}).
\end{equation}
Our class $\tau_{4k+2} \in \mathbf{BO}^{8k+4,4k+2}(\mathbb{Z} \times HGr)$ and the pairing
\eqref{BO2*BO**toBO**1} gives us a system of classes
\begin{equation}
\label{E:HGrn.HGrn}
\tau_{4k+2} |_{HGr_{n}} \boxtimes \tau_{4k+2} |_{HGr_{n}} \in \mathbf{BO}^{16k+8,8k+4}(HGr_{n} \wedge HGr_{n}).
\end{equation}
and therefore a class
\begin{equation*}
\tau_{4k+2} \boxtimes \tau_{4k+2} \in
\mathbf{BO}^{16k+8,8k+4}(\mathbf{KO}^{[4k+2]} \wedge \mathbf{KO}^{[4k+2]}).
\end{equation*}
There is also an exact sequence of the form
\[
0 \to \varprojlim\nolimits^{1} \mathbf{BO}^{4i-1,2i}(\mathbf{KO}^{[i]} \wedge \mathbf{KO}^{[i]}) \to
\mathbf{BO}^{0,0}(\mathbf{BO} \wedge \mathbf{BO}) \to
\varprojlim \mathbf{BO}^{4i,2i}(\mathbf{KO}^{[i]} \wedge \mathbf{KO}^{[i]}) \to 0.
\]
The elements $\tau_{4k+2} \boxtimes \tau_{4k+2}$ define an element
$\bar m \in \varprojlim \mathbf{BO}^{4i,2i}(\mathbf{KO}^{[i]} \wedge \mathbf{KO}^{[i]})$.
This $\bar m$ is the unique element compatible with the naive product \eqref{E:HGrn.HGrn}
and the isomorphisms \eqref{E:ZHGr.lim} and \eqref{E:ZHGr.ZHGr.lim}.
Lifting this element to $m \in \operatorname{Hom}_{SH(S)}(\mathbf{BO} \wedge\mathbf{BO},\mathbf{BO})$ gives an element we
can use to define the product $\times$. On small motivic spaces $\times$ depends only on
$\bar m$ and not on the choice of $m$.
We deduce the associativity and bigraded commutativity
of $\times$ on small motivic spaces from the associativity and commutativity of the naive product on the
$KO_{0}^{[2r]}(HGr_{n})$. This gives us Theorem \ref{uniq2}.
Theorem \ref{T:unique} is more subtle. The obstructions to the uniqueness of $m$
and to the associativity, commutativity and unit property of the monoid it defines all live in certain
$\varprojlim^{1}$ groups.
We show in \S\ref{S:vanishing} that when $S = \operatorname{Spec} R$
with $\frac 12 \in R$ and with $KO_{1}(R)$ and $KSp_{1}(R)$ finite groups,
the $\varprojlim\nolimits^{1}$ vanish. This uses the construction in \S\ref{S:finite} of three new
spectra using a new motivic sphere.
The geometry used to prove the quaternionic projective bundle theorem in \cite{Panin:2010fk}
also shows that the pointed quaternionic projective line
$(HP^{1},x_{0})$ is isomorphic to $T^{\wedge 2}$ in the motivic homotopy category
$H_{\bullet}(S)$. The pointed scheme $HP^{1+}$ which is the $\mathbf{A}^{1}$ mapping cone
of the pointing $x_{0} \colon pt \to HP^{1}$ is therefore also homotopy equivalent to $T^{\wedge 2}$.
It is the union of $HP^{1}$ and $\mathbf{A}^{1}$
with $x_{0} \in HP^{1}$ identified with $0 \in \mathbf{A}^{1}$, pointed at $1 \in \mathbf{A}^{1}$.
Therefore the motivic stable homotopy categories of $T$-spectra and of
$HP^{1+}$-spectra are equivalent.
We construct three $HP^{1+}$-spectra $\mathbf{BO}_{HP^{1+}}$, $\mathbf{BO}^{\text{\textit{geom}}}$
and $\mathbf{BO}^{\text{\textit{fin}}}$
representing hermitian $K$-theory. The spaces of $\mathbf{BO}_{HP^{1+}}$ are the even-indexed
spaces $(\mathbf{KO}^{[0]}, \mathbf{KO}^{[2]}, \mathbf{KO}^{[4]}, \dots)$ of the $T$-spectrum.
The spaces of $\mathbf{BO}^{\text{\textit{geom}}}$ are alternately
$\mathbb{Z} \times RGr$ and $\mathbb{Z} \times HGr$.
The spaces of $\mathbf{BO}^{\text{\textit{fin}}}$ are finite unions of finite-dimensional real and quaternionic Grassmannians.
Here a \emph{real Grassmannian} $RGr(r,2n)$ is the open subscheme of the
ordinary Grassmannian $Gr(r,2n)$
where the hyperbolic quadratic form on $\mathcal{O}^{\oplus 2n}$ is nondegenerate, while
$RGr = \operatornamewithlimits{colim} RGr(n,2n)$. For the details of $\mathbf{BO}^{\text{\textit{fin}}}$ see Theorem \ref{T:finite}.
The structure maps of the spectra are all essentially multiplication with the Euler class
$-p_{1}(\mathcal U)$ of the tautological rank $2$ symplectic subbundle on $HP^{1}$.
The bonding maps $\mathbf{BO}^{*}_{2i} \wedge HP^{1+} \to \mathbf{BO}_{2i+2}^{*}$ of the two geometric spectra
are morphisms of schemes or ind-schemes which are constant on the wedge $\mathbf{BO}^{*}_{2i} \vee HP^{1+}$.
The inclusion $\mathbf{BO}^{\text{\textit{fin}}} \to \mathbf{BO}^{\text{\textit{geom}}}$ is a motivic stable weak equivalence,
while the isomorphism $\mathbf{BO}^{\text{\textit{geom}}} \cong \mathbf{BO}_{HP^{1}}$ in $SH(S)$
is constructed from
\emph{classifying maps}
\begin{align*}
\tau_{4k} \colon \mathbb{Z} \times RGr \to \mathbf{KO}^{[4k]},
&&
\tau_{4k+2} \colon \mathbb{Z} \times HGr \xra{\sim} \mathbf{KO}^{[4k+2]},
\end{align*}
in $H_{\bullet}(S)$. The $\tau_{4k+2}$ are the isomorphisms of Theorem \ref{T:MV.intro},
while the $\tau_{4k}$ are constructed from the $\tau_{4k+2}$. We do not know if the
$\tau_{4k}$ are isomorphisms, although stably they have a right inverse (Proposition \ref{P:right.inverse}).
The $\varprojlim^{1}$ calculated with $\mathbf{BO}$ and with $\mathbf{BO}^{\text{\textit{fin}}}$ are the same.
Calculations based on the quaternionic projective bundle theorem show that for $\mathbf{BO}^{\text{\textit{fin}}}$
the $\varprojlim^{1}$ are over inverse systems of groups which are finite direct sums of
copies of $KO_{1}(S)$ and $KSp_{1}(S)$. When those groups are finite, the $\varprojlim^{1}$ vanishes.
This gives Theorem \ref{T:unique} for $S = \operatorname{Spec} \mathbb{Z}[\frac 12]$. For other $S$ one pulls the structure
back from $\operatorname{Spec} \mathbb{Z}[\frac 12]$ using the closed motivic model structure of \cite{Panin:2009aa}.
\begin{thm}
\label{T:compatible}
The products of Theorems \ref{uniq2}, \ref{uniq1} and \ref{T:unique} are compatible with all the
naive products of \eqref{E:naive} and with the partial multiplicative structure
of \eqref{E:partial.1} and \eqref{E:partial.2}.
\end{thm}
We do not know how to prove this theorem using our construction of the
hermitian $K$-theory product. But Marco Schlichting has described to us (oral communication) how
to put a pairing on his hermitian $K$-theory spaces when one has a pairing
of complicial exact categories with weak equivalences and duality. When applied to our situation
his product is isomorphic to ours on small motivic spaces by Theorem \ref{uniq2}.
Since Schlichting's product
is compatible with all the naive products of \eqref{E:naive} and with the partial multiplicative structure
of \eqref{E:partial.1} and \eqref{E:partial.2}, therefore ours is as well.
We finish the paper in \S \ref{S:K.theory}
by giving the analogue for algebraic $K$-theory of the spectra
$\mathbf{BO}^{\text{\textit{fin}}}$ and $\mathbf{BO}^{\text{\textit{geom}}}$ of hermitian $K$-theory of \S \ref{S:finite}.
The $\mathbf{BGL}^{\text{\textit{fin}}}$ seems to be completely new. We write
$CGr(r,n)$ for the affine Grassmannian, $CP^{1} = CGr(1,2)$ for the affine version of
$\mathbf{P^{1}}$, and $CP^{1+}$ for the $\mathbf{A}^{1}$ mapping cone of the pointing map of $CP^{1}$.
\begin{thm}
There are $CP^{1+}$-spectra
$\mathbf{BGL}^{\text{\textit{fin}}}$ and $\mathbf{BGL}^{\text{\textit{geom}}}$ isomorphic to
$\mathbf{BGL}_{CP^{1+}}$ in $SH_{CP^{1+}}(S)$
with spaces
\begin{align*}
\mathbf{BGL}^{\text{\textit{fin}}}_{n} & =
[-4^{n},4^{n}] \times CGr(4^{n}, 2 \cdot 4^{n}),
&
\mathbf{BGL}^{\text{\textit{geom}}}_{n} & = \mathbb Z \times CGr,
\end{align*}
which are unions of affine Grassmannians.
The bonding maps $\mathbf{BGL}^{*}_{n} \wedge CP^{1+} \to \mathbf{BGL}_{n+1}^{*}$
of the two spectra
are morphisms of schemes or ind-schemes which are constant on the wedge
$\mathbf{BGL}^{*}_{n} \vee CP^{1+}$.
\end{thm}
The spectum $\mathbf{BGL}^{\text{\textit{fin}}}$ can be used to give alternate proofs of the uniqueness results
of \cite{Panin:2009aa} concerning the $\mathbf{P}^{1}$-spectrum representing algebraic $K$-theory and the
commutative monoid structure on that spectrum. These proofs avoid the use of topological realization
and apply to any noetherian base scheme $S$ of finite Krull dimension with finite $K_{1}(S)$.
\section{Cohomology theories}
\label{S:cohom}
We fix a base scheme $S$ which is regular noetherian separated of finite Krull dimension and
with $\frac 12 \in \Gamma(S,\mathcal{O}_{S})$.
The hermitian $K$-theory of such schemes is simpler than for other schemes, and we wish to
avoid the complications of negative hermitian $K$-theory and of characteristic $2$.
Let $\mathcal Sm/S$ be the category of smooth $S$-schemes
of finite type.
Let $\mathcal Sm\mathcal Op/S$ be the category whose objects are pairs $(X,U)$ with
$X$ in $\mathcal Sm/S$ and $U \subset X$ an open subscheme
and whose morphisms $f \colon (X,U) \to (Y,V)$ are morphisms $f \colon X \to Y$ of $S$-schemes
having $f(U) \subset V$. We write $X$ for $(X,\varnothing)$.
The base scheme itself will often be written as $S = pt$.
A \emph{cohomology theory} on $\mathcal Sm\mathcal Op/S$
\cite[Definition 2.1]{Panin:2003rz}
is a pair
$(A,\partial)$ with $A$ a contravariant functor from $\mathcal Sm/S$ to the category of abelian groups
having localization exact sequences and satisfying \'etale excision and homotopy invariance. The
$\partial$ is a morphism of functors with components
$\partial_{X,U} \colon A(U) \to A(X,U)$ which are the boundary maps of the localization
exact sequences.
A \emph{ring cohomology theory} in the sense of \cite[Definition 2.13]
{Panin:2003rz}
has products
\[
\times \colon A(X,U) \times A(Y,V) \to A(X\times Y, (X \times V) \cup (U \times Y))
\]
which are functorial, bilinear and associative and and which have a two-sided unit $1_{A} \in A(pt)$ and
satisfy $\partial(\alpha \times \beta) = \partial \alpha \times \beta$.
A cohomology theory also defines groups $A(X,x)$ for pointed smooth schemes and their smash products
such as
\begin{equation}
\label{E:A(smash)}
A\bigl( (X_{1},x_{1}) \wedge (X_{2},x_{2}) \bigr)
= \ker\Bigl(A(X_{1}\times X_{2}) \xra{(x_{1}^{*} \times 1, 1 \times x_{2}^{*})}
A(pt \times X_{2}) \oplus A(X_{1} \times pt)
\Bigr).
\end{equation}
A \emph{bigraded} cohomology theory $(A^{*,*},\partial)$ is one in which the groups are
bigraded, that is
$A^{*,*}(X,U)= \bigoplus_{p,q \in \mathbb{Z}}
A^{p,q}(X,U)$, the pullback maps are homogeneous of bidegree $(0,0)$ and the boundary maps
$\partial_{X,U}$ are homogeneous of bidegree $(1, 0)$.
In a \emph{bigraded ring cohomology theory} $(A,\partial, \times,{1})$ the
$\times$ products respects the
bigrading and we have ${1} \in A^{0,0}(pt)$.
\begin{defn}
\label{D:commutative}
Let $(A, \partial, \times ,{1})$ bigraded ring cohomology theory, and suppose
$\varepsilon \in A^{0,0}(pt) = A^{0,0}(pt)$ satisfies $\varepsilon^{2} ={1}$.
Then $(A, \partial, \times, {1})$ is \emph{$\varepsilon$-commutative} if for
$\alpha \in A^{p,q}(X,U)$ and $\beta \in A^{r,s}(Y,V)$ one has
$\sigma^{*}(\alpha \times \beta) = \beta \times \alpha \times (-1)^{pr}\varepsilon^{qs}$
where $\sigma \colon Y \times X \to X \times Y$ switches the factors.
\end{defn}
Equivalently a bigraded ring cohomology is $\varepsilon$-commutative if the associated cup product
satisfies $\alpha \cup \beta = (-1)^{pr}\varepsilon^{qs} \, \beta \cup \alpha$ for $\alpha \in A^{p,q}(X,U)$
and $\beta \in A^{r,s}(X,V)$.
For such a cohomology theory the $A^{*,*}(X)$ are bigraded-commutative rings, and for any $(X,U)$
the $A^{*,*}(X,U)$ and $A^{*,*}(U)$ are right and left bigraded $A^{*,*}(X)$-modules.
The $\partial_{X,U}$ are morphisms of right $A^{*,*}(X)$-modules.
Sometimes it is easier to define certain products than others.
For a bigraded cohomology theory $(A^{*,*},\partial)$ set
$A^{0}(X,U) = \bigoplus_{p\in \mathbb{Z}} A^{2p,p}(X,U)$.
We need the following notion.
\begin{defn}
\label{multiplicative}
Let $(A^{*,*},\partial)$ be a bigraded cohomology theory as above. An \emph{$\varepsilon$-commutative
partial multiplication} on $(A^{*,*},\partial)$ is given by
\begin{enumerate}
\item
pairings $\times \colon A^{p,q}(X,U) \times A^{2r,r}(Y,V) \to A^{p+2r,q+r}((X,U) \wedge (Y,V))$
which are bilinear and functorial, and
\item
elements ${\boldsymbol 1}$ and $\varepsilon$ in $A^{0,0}(pt)$
\end{enumerate}
satisfying
\begin{enumerate}
{\renewcommand{\theenumi}{\alph{enumi}}
\item
$\alpha \times (b \times c)= (\alpha \times b) \times c$ for $\alpha \in A^{p,q}(X,U)$,
$b \in A^{2r,r}(Y,V)$, $c \in A^{2s,s}(Z,W)$;
\item
$\alpha \times {\boldsymbol 1} = \alpha$ for $\alpha \in A^{p,q}(Y,V)$,
\item
$\varepsilon \times \varepsilon = \boldsymbol{1}$,
\item
$a \times b = \sigma^*(b \times a) \times \varepsilon^{rs}$ for $a \in A^{2r,r}(X,U)$, $b \in A^{2s,s}(Y,V)$
where $\sigma\colon X \times Y \to Y \times X$
switches the factors;
\item
$\partial_{Y \times X, V \times X}(\alpha \times b) = \partial_{Y,V}(\alpha) \times b$ for
$\alpha \in A^{p,q}(V)$, $b \in A^{2r,r}(X)$.
}\end{enumerate}
\end{defn}
If $(A^{*,*},\partial)$ has such a partial multiplication, then for
$\alpha \in A^{p,q}(X,V)$ and $b \in A^{2r,r}(X,U)$
one has a \emph{cup product}
\[
\alpha \cup b =
\Delta^{*}(\alpha \times b) \in A^{p+2r,q+r}(X,U \cup V).
\]
\label{A*istwosidedA0module}
If $(A,\partial)$ is equipped with a partial multiplicative structure
$(\times, \boldsymbol{1}, \varepsilon)$, then the functor
$(X,U) \mapsto A^{0}(X,U)$ is an $\varepsilon$-commutative graded ring functor in the sense that
the properties
(a), (b), (c) and (d) hold for $\alpha \in A^{0}(Y,V)$.
Moreover, $A$ is a bigraded
right $A^{0}$-module in the same sense with $\partial$ a morphism of bigraded right $A^{0}$-modules
which is homogeneous of bidegree $(1,0)$.
The switch $\sigma \colon X \times Y \to Y \times X$ allows us to define pairings
$\times \colon A^{2r,r}(X,U) \times A^{p,q}(Y,V) \to A^{p+2r, q+r}((X,U) \times (Y,V))$
by $b \times \alpha = \sigma^{*}(\alpha \times b) \times \varepsilon^{qr}$.
There are also cup products $b \cup \alpha = \Delta^{*}(b \times \alpha)$.
The two pairings are compatible by (d).
Thus $A$ is a bigraded left and right $A^{0}$-module, with $\partial$ a morphism of right
$A^{0}$-modules.
\section{\texorpdfstring{$SL$ and $SL^{c}$}{SL and SL\^{ }c} orientations}
\label{S:SL.orientation}
We discuss $SL$ oriented cohomology theories. Hermitian $K$-theory will turn out to be one.
We also include a discussion of Thom classes for vector bundles whose structural group is the double
cover $SL_{n}^{c}$ of $GL_{n}$. It contains $SL_{n}$. We believe this is the true level at which
Witt groups and hermitian $K$-theory are oriented.
An \emph{$SL$ bundle} on $X$ is a pair $(E,\lambda)$ with $E$ a vector bundle over $X$ and
$\lambda \colon \mathcal{O}_{X} \cong \det E$ an isomorphism. An \emph{isomorphism of $SL$ bundles}
$f \colon (E,\lambda) \cong (E_{1},\lambda_{1})$ is an isomorphism $f \colon E \cong E_{1}$ such
that $\lambda_{1} = \det f \circ \lambda$.
\begin{defn}
\label{D:SL.orientation}
An \emph{$SL$ orientation} on a bigraded
cohomology theory $A^{*,*}$ with an $\varepsilon$-commu\-tative partial multiplication or ring structure
is an assignment to every $SL$ bundle $(E,\lambda)$ over every $X$ in $\mathcal Sm/S$
of a class $\thom(E,\lambda) \in A^{2n,n}(E,E-X)$ for $n = \operatorname{rk} E$
satisfying the following conditions:
\begin{enumerate}
\item For an isomorphism $f \colon (E,\lambda) \cong (E_{1},\lambda_{1})$
we have $\thom(E,\lambda) = f^{*}\thom(E_{1},\lambda_{1})$.
\item For $u \colon Y \to X$
we have $u^{*}\thom(E,\lambda) = \thom(u^{*}E,u^{*}\lambda)$ in $A^{2n,n}(u^{*}E,u^{*}E - Y)$.
\item The maps ${-} \cup \thom(E,\lambda) \colon A^{*,*}(X) \to A^{*+2n,*+n}(E,E-X)$ are isomorphisms.
\item We have
\[
\thom (E_{1} \oplus E_{2}, \lambda_{1} \otimes \lambda_{2})
= q_{1}^{*}\thom(E_{1},\lambda_{1}) \cup q_{2}^{*}\thom(E_{2},\lambda_{2}),
\]
where $q_{1},q_{2}$ are the projections from $E_{1} \oplus E_{2}$
onto its factors.
\end{enumerate}
The class $\thom(E,\lambda)$ is the \emph{Thom class} of the $SL$ bundle, and
$e(E,\lambda) = z^{*} \thom(E,\lambda) \in A^{2n,n}(X)$ is its \emph{Euler class}.
\end{defn}
This definition is analogous to the Thom classes theory version of the definition of an orientation
\cite[Definition 3.32]{Panin:2003rz}
or of a symplectic orientation
\cite[Definition 14.2]{Panin:2010fk}.
The Thom and Euler classes of $SL$ bundles are not necessarily central in contrast with
the classes in the oriented and symplectically oriented theories of
\cite{Panin:2003rz} and \cite{Panin:2010fk}.
But for an $SL$ bundle
of rank $n$ the Thom and Euler classes are in bidegree $(2n,n)$, and such classes need not be central when
$n$ is odd and $\varepsilon \neq 1$.
Centrality occcurs for oriented theories because they have $\varepsilon = 1$ and for
symplectically oriented theories because the Thom and Pontryagin
classes of symplectic bundles are in bieven bidegrees
$(4r,2r)$.
Twisted versions of cohomology groups with coefficients in a line bundle
can be defined for any $SL$ oriented theory by
\begin{equation*}
A^{p,q}(X;L) = A^{p+2,q+1}(L,L-X)
\end{equation*}
and more generally by
$A^{p,q}(X,X-Z;L) = A^{p+2,q+1}(L,L-Z)$ for closed subsets $Z \subset X$.
\begin{thm}
\label{T:line.bundle}
Let $E$ be a vector of rank $n$ over $X$. Suppose that $A^{*,*}$ is an
$SL$ oriented bigraded cohomology theory.
Then there are canonical isomorphisms of bigraded right $A^{0}(X)$ or
$A^{*,*}(X)$-modules $A^{*+2n,*+n}(E,E-X) \cong A^{*,*}(X;\det E)$.
\end{thm}
\begin{proof}
Write $L_{E} = \det E$.
There are canonical isomorphisms
\begin{align*}
\lambda_{1} \colon \mathcal{O}_{X} & \cong \det (E \oplus L_{E}^{\vee}), &
\lambda_{2} \colon \mathcal{O}_{X} & \cong \det(L_{E} \oplus L_{E}^{\vee}).
\end{align*}
This gives us $SL$ bundles $(E \oplus L_{E}^{\vee},\lambda_{1})$ and
$(L_{E}\oplus L_{E}^{\vee}, \lambda_{2})$ over $X$. The pullback of the first bundle
along $q \colon L_{E} \to X$ gives an $SL$ bundle whose structural map is the first projection
$L_{E} \oplus E \oplus L_{E}^{\vee} \twoheadrightarrow L_{E}$. The pullback of the second bundle along
$p \colon E \to X$ and permutation of the summands
gives an $SL$ bundle whose structural map is the second projection
$L_{E} \oplus E \oplus L_{E}^{\vee}
\twoheadrightarrow E$.
We now have canonical isomorphisms
\[
\xymatrix @M=5pt @C=75pt {
A^{*+2n,*+n}(E,E-X) \ar[r]_-{\cong}^-{p^{*}\thom(L_{E}\oplus L_{E}^{\vee},\lambda_{2}) \cup}
&
A^{*+2n+4,*+n+2}(E \oplus L_{E} \oplus L_{E}^{\vee} ,E \oplus L_{E} \oplus L_{E}^{\vee} -X)
\ar[d]_-{\cong}^-{\varepsilon^{n}}
\\
A^{*+2,*+1}(L_{E},L_{E}-X),
\ar[r]^-{\cong}_-{q^{*}\thom(E \oplus L_{E}^{\vee}, \lambda_{1}) \cup}
&
A^{*+2n+4,*+n+2}(L_{E} \oplus E \oplus L_{E}^{\vee} ,L_{E} \oplus E \oplus L_{E}^{\vee} -X)
}
\]
and the bottom left module is $A^{*,*}(X;\det E)$ by definition. The sign $\varepsilon^{n}$ is appropriate
when one permutes the rank $1$ bundles and the rank $n$ bundle.
\end{proof}
Hermitian $K$-theory and Witt groups have more Thom classes than just those for $SL$ bundles
because of what are often called periodicity isomorphisms such as
$W^{*}(X;L) \cong W^{*}(X;L \otimes L_{1}^{\otimes 2})$. However, these periodicity isomorphisms
depend on choices. A good way to structure these choices is to talk about $SL^{c}$ bundles,
using extra structure analogous to the $Spin^{c}$ structures frequently used in
differential geometry.
An \emph{$SL^{c}$ vector bundle} on $X$ is a triple $(E,L,\lambda)$ with $E$ a vector bundle,
$L$ a line bundle, and $\lambda \colon L \otimes L \cong \det E$ an isomorphism. The structural
group of an $SL^{c}$ bundle of rank $n$ is $SL_{n}^{c}$ which is the kernel of
\[
GL_{n} \times \mathbb{G}_{m} \xrightarrow{(\det^{-1}, \, t \,\mapsto \, t^{2})} \mathbb{G}_{m}.
\]
There is a natural exact sequence $1 \to \mu_{2} \to SL_{n}^{c} \to GL_{n} \to 1$. The notation
$SL^{c}$ is in imitation of $Spin^{c}$. The role of this double cover of $GL_{n}$ in direct images
in real topological $K$-theory merited a mention by Atiyah
\cite[p.~55]{Atiyah:1971zr}
nearly forty years ago.
\begin{defn}
\label{D:SLc.orientation}
An \emph{$SL^{c}$ orientation} on a
cohomology theory $A^{*,*}$ with a $\varepsilon$-commutative ring structure or partial multiplication
is an assignment to every $SL^{c}$ bundle $(E,L,\lambda)$ over every scheme $X$ in $\mathcal Sm/S$
of a class $\thom(E,L,\lambda) \in A^{2n,n}(E,E-X)$ where $n = \operatorname{rk} E$
satisfying the conditions (1)--(4) of Definition \ref{D:SL.orientation}.
\end{defn}
\section{Schlichting's hermitian \texorpdfstring{$K$}{K}-theory and the Gille-Nenashev pairing}
In
\cite[\S 2.7]{Schlichting:2010uq}
Schlichting defines the hermitian $K$-theory space of a complicial exact category with
weak equivalences and duality in the style of Waldhausen's $K$-theory. We will denote
his space by $KO(C,w,\sharp,\eta)$. More generally we write
\[
KO^{[n]}(C,w,\sharp,\eta) = KO\bigl( (C,w,\sharp,\eta)[n] \bigr)
\]
for the hermitian $K$-theory space for the $n^{\text{th}}$ shifted duality,
and $KO_{i}^{[n]}(C,w,\sharp,\eta)$ for its homotopy groups.
A \emph{symmetric object of degree $n$} in $(C,w,\sharp,\eta)$ is a pair $(X,\phi)$ with
$\phi \colon X \to X^{\sharp}[n]$ a weak equivalence which is symmetric $\phi = \phi^{t}$
for the shifted duality. There is a natural definition of a Grothendieck-Witt group
of symmetric objects of degree $n$, and
The $\pi_{0}$ of the hermitian $K$-theory space
is the Grothendieck-Witt group of degree $n$ symmetric objects
\[
KO^{[n]}_{0}(C,w,\sharp,\eta) = GW^{n}(C,w,\sharp,\eta).
\]
When $C$ is $\mathbb{Z}[\frac 12]$-linear,
that is the same as the triangulated Grothendieck-Witt group of the homotopy category
$Ho(C,w) = C[w^{-1}]$ for the duality $(\sharp,\eta)$, defined \`a la Balmer.
For a duality-preserving exact functor $(F,f) \colon (C,w,\sharp,\eta) \to (D,v,\natural,\varpi)$
there are induced maps of spaces $KO^{[n]}(C,w,\sharp,\eta) \to KO^{[n]}(D,v,\natural,\varpi)$.
A weak equivalence between duality-preserving exact functors $(F,f) \simeq (G,g)$ produces
a homotopy between the maps.
There are natural periodicity isomorphisms
$KO^{[n]}(C,w,\sharp,\eta) \simeq KO^{[n+4k]}(C,w,\sharp,\eta)$.
Moreover, we may write
\[
KSp^{[n]}(C,w,\sharp, \eta) = KO^{[n]}(C,w,\sharp,-\eta)
\]
because the effect of changing the sign is to interchange symmetric and skew-symmetric forms.
Then
there are
isomorphisms $KSp^{[n]}(C,w,\sharp,\eta) \simeq KO^{[n+4k+2]}(C,w,\sharp,\eta)$
induced by the duality preserving functor $X \mapsto X[2k{+}1]$. However, it is more useful
to use the identifications
\begin{equation}
\label{E:KSp}
\begin{array}{ccc}
KSp_{i}^{[n]}(C,w,\sharp, \eta) & \lra & KO_{i}^{[n+4k+2]}(C,w,\sharp,\eta)
\\
\xi & \longmapsto & - \,\xi[2k{+}1]
\end{array}
\end{equation}
because these commute with the forgetful maps to Waldhausen's $K$-theory $K_{i}(C,w)$.
Among the many important results Schlichting proves is localization. Suppose that
$\overline C_{1} \subset Ho(C,w)$ is a thick triangulated subcategory
which is stable under the duality. Let $C_{1} \subset C$ be the full exact subcategory with the
same objects as $\overline C_{1}$. Let $w_{1}$ be the set of all morphisms in $C$ whose mapping cone is
in $C_{1}$.
\begin{thm}
[\protect{\cite[Theorem 6]{Schlichting:2010uq}}]
\label{T:localization}
If $(C,w,\sharp,\eta)$ is a complicial exact category with weak equivalences and duality, and
$C_{1}$ is as above, then
\[
KO(C_{1},w,\sharp,\eta) \to KO(C,w,\sharp,\eta) \to KO(C,w_{1},\sharp,\eta)
\]
is a fibration sequence up to homotopy.
\end{thm}
Gille and Nenashev
\cite{Gille:2003ad}
have defined pairings for Witt groups of triangulated categories.
We explain how their construction can be applied to hermitian $K$-theory to give
pairings in the spirit of the partial multiplicative
structure of Definition \ref{multiplicative}.
A \emph{pairing}
\[
(\boxtimes, t_{1},t_{2},\lambda) \colon (C,w,\sharp,\eta) \times (D,v,\flat,\theta) \to
(E,u,\natural,\varpi)
\]
of complicial exact categories with weak equivalences and duality is an additive bifunctor
$\boxtimes \colon C \times D \to E$ which commutes with the translations up to specified functorial isomorphisms
$t_{1,X,Y} \colon X[1] \boxtimes Y \cong (X \boxtimes Y)[1]$ and $t_{2,X,Y} \colon X \boxtimes Y[1] \cong
(X \boxtimes Y)[1]$
plus
functorial weak equivalences
$\lambda_{X,Y} \colon X^{\sharp} \boxtimes Y^{\flat} \to (X \boxtimes Y)^{\natural}$
such that for any $X$ in $C$ and any $Y$ in $D$ the functors $X \boxtimes {-}$ and ${-} \boxtimes Y$
are exact and preserve weak equivalences and such that all the conditions of
\cite[Definitions 1.2 and 1.11]{Gille:2003ad}
hold.
Suppose given a symmetric object $(M,\phi)$ of degree $r$ in $(C,w,\sharp,\eta)$ and a symmetric object
$(N,\psi)$ of degree $s$ in $(D,v,\flat,\theta)$. Gille and Nenashev show how to
define duality-preserving exact functors
\cite[Lemma 1.14]{Gille:2003ad}
\begin{subequations}
\begin{align*}
\bigl( {-} \boxtimes (N,\psi), \mathfrak R(N,\psi) \bigr)
\colon & (C,w,\sharp,\eta) \to (E,u,\natural, \varpi)[s]
\\
\bigl( (M,\phi) \boxtimes {-}, \mathfrak L(M,\phi) \bigr)
\colon & (D,v,\flat,\theta) \to (E,u,\natural,\varpi)[r]
\end{align*}
\end{subequations}
It follows that these induce maps of $KO$ spaces, which we will write as
\begin{subequations}
\begin{align}
\label{E:boxtimes.1}
( {-} \boxtimes (N,\psi))_{*}
\colon & KO^{[n]}(C,w,\sharp,\eta) \to KO^{[n]}(E,u,\natural, \varpi)[s]
\\
\label{E:boxtimes.2}
((M,\phi) \boxtimes {-} )_{*}
\colon & KO^{[n]}(D,v,\flat,\theta) \to KO^{[n]}(E,u,\natural,\varpi)[r]
\end{align}
\end{subequations}
The duality-preserving functor $(1_{E},-1)$,
which acts on symmetric objects by $(Z,\xi) \mapsto (Z,-\xi)$, induces a map
\begin{equation}
\label{E:involution}
\varepsilon \colon KO^{[n]}(E,u,\natural, \varpi) \to KO^{[n]}(E,u,\natural, \varpi).
\end{equation}
These \emph{sign involutions} exist for the hermitian $K$-theory of any complicial exact category
with weak equivalences and duality, and they satisfy $\varepsilon^{2}=1$ exactly.
(In general $\varepsilon$ is not the same as the $-1$ which is the inverse map for the $H$-space structure
induced by the orthogonal direct sum.)
The methods of Gille and Nenashev show
\cite[Lemma 1.15]{Gille:2003ad}
that the
effect of the two functors on the Grothendieck-Witt classes $[M,\phi] \in GW^{r}(C,w,\sharp,\eta)$
and $[N,\psi] \in GW^{s}(D,v,\flat,\theta)$ is
\begin{equation}
\label{E:commute.category}
( {-} \boxtimes (N,\psi))_{*} [M,\phi]
= \varepsilon^{rs}
( (M,\phi) \boxtimes {-})_{*}[N,\psi].
\end{equation}
\begin{prop}
The homotopy classes of the maps
\eqref{E:boxtimes.1} and \eqref{E:boxtimes.2}
on hermitian $K$-theory spaces
depend only on the classes $[N,\psi] \in GW^{s}(D,v,\flat,\theta)$
and $[M,\phi] \in GW^{r}(C,w,\sharp,\eta)$
respectively.
\end{prop}
\begin{proof}
There are three relations in the definition of the Grothendieck-Witt groups
\cite[Definition 1]{Schlichting:2010uq}.
The maps on homotopy groups are compatible with the relations
$[N,\psi] + [N_{1},\psi_{1}] = [N\oplus N_{1},\psi\oplus \psi_{1}]$ because
the orthogonal direct sum of symmetric objects induces a monoidal structure on the
$KO^{[n]}(E,u,\natural,\varpi)$ giving a naive additivity for orthogonal direct sums of
duality-preserving functors.
The maps are compatible with the relations $[N,\psi] = [N_{2},\sigma^{\flat}\psi \sigma]$ for
a weak equivalence $\sigma \colon N_{2}\to N$ because $\sigma$ induces a natural weak equivalence of
duality-preserving functors.
The maps are compatible with the third relation related to lagrangians
because of Schlichting's Additivity Theorem
\cite[Theorem 5]{Schlichting:2010uq}.
\end{proof}
We thus get pairings
\begin{subequations}
\begin{gather}
KO_{j}^{[m]}(C,w,\sharp,\eta) \times KO_{0}^{[s]}(D,v,\flat,\theta) \to KO_{j}^{[m+s]}(E,u,\natural,\varpi)
\\
KO_{0}^{[r]}(C,w,\sharp,\eta) \times KO_{i}^{[n]}(D,v,\flat,\theta) \to KO_{i}^{[n+r]}(E,u,\natural,\varpi)
\end{gather}
\end{subequations}
which we call the \emph{right pairing} and the \emph{corrected left pairing}. They coincide on
$KO_{0}^{[m]} \times KO_{0}^{[n]}$.
The Gille-Nenashev pairings
have a number of other properties such as functoriality, associativity, compatibility
with localization sequences. Using the right pairing means that the boundary map on the homotopy
groups in the localization sequences satisfies $\partial (\alpha \cup \xi) = \partial \alpha \cup \xi$
for $\alpha \in KO^{[n]}_{i}(C,w_{1},\sharp,\eta)$ and $\xi \in KO^{[r]}_{0}(D,v,\flat,\theta)$.
In
\cite{Schlichting:2010uq}
Schlichting constructs a hermitian $K$-theory space for $(C,w,\sharp,\eta)$ but not a spectrum.
So in principle what we have discussed so far does not yield any negative homotopy groups.
In practice it is known that
if the categories are $\mathbb{Z}[\frac 12]$-linear, then
Balmer's triangulated Witt groups for the homotopy category $Ho(C,w) = C[w^{-1}]$ can function
as negative homotopy groups
\[
KO^{[n]}_{i}(C,w,\sharp,\eta) = W^{n-i}(C[w^{-1}],\sharp,\eta)
\qquad \qquad \text{for $i < 0$}.
\]
This is explained in
\cite{Schlichting:2006aa}.
The localization sequences for the homotopy groups of the hermitian
$K$-theory spaces and Balmer's localization sequence for triangulated Witt groups
\cite[Theorem 6.2]{Balmer:2000hb}
attach to each other
because the $\pi_{0}$ are triangulated Grothendieck-Witt groups.
Two other important theorems
are the following.
\begin{thm}
[Fundamental Theorem \protect{\cite{Schlichting:2006aa}}]
\label{T:fundamental}
Let $(C,w,\sharp,\eta)$ be a $\mathbb{Z}[\frac 12]$-linear complicial exact category with weak
equivalences and duality. Then for all $n$
\[
KO^{[n-1]}(C,w,\sharp,\eta) \xra{F} K(C,w) \xra{H} KO^{[n]}(C,w,\sharp,\eta)
\]
is a homotopy fiber sequence, where $F$ is the forgetful map and $H$ the hyperbolic map.
\end{thm}
\begin{thm}
[\protect{\cite{Schlichting:2006aa}}]
\label{T:equivalence}
Let $(F,f) \colon (C,w,\sharp,\eta) \to (E,u,\natural, \varpi)$ be a duality-preserving
exact functor between $\mathbb{Z}[\frac 12]$-linear complicial exact categories with weak
equivalences and duality. If $(F,f)$ induces a homotopy equivalence $K(C,w) \simeq K(E,u)$ of
Waldhausen $K$-theory spaces and isomorphisms
$W^{i}(C[w^{-1}],\sharp,\eta) \cong W^{i}(E[u^{-1}],\natural,\varpi)$ of Balmer's triangulated Witt groups,
then $(F,f)$ induces homotopy equivalences
$KO^{[n]}(C,w,\sharp,\eta) \simeq KO^{[n]}(E,u,\natural,\varpi)$ for all $n$.
\end{thm}
In particular if $(F,f)$ induces an equivalence of $\mathbb{Z}[\frac 12]$-linear triangulated categories with
duality $(C[w^{-1}], \sharp, \eta) \simeq (E[u^{-1}],\natural,\varpi)$, then it induces a homotopy
equivalence $KO(C,w,\sharp,\eta) \simeq KO(E,u,\natural,\varpi)$ by Thomason's theorem
and by the fact that Balmer's Witt groups are a functorial over the category with objects
$\mathbb{Z}[\frac 12]$-linear
triangulated categories
with duality and arrows isomorphism classes of duality-preserving triangulated functors
\cite[Lemma 4.1]{Balmer:2002rp}.
\section{The cohomology theory \texorpdfstring{$KO^{[*]}_*$}{KO} on the category \texorpdfstring{$\mathcal Sm\mathcal Op/k$}{SmOp/k}}
Let $S$ be a regular noetherian separated scheme of finite Krull dimension
with $\frac 12 \in \Gamma(S,\mathcal{O}_{S})$. For every $S$-scheme $X$ consider the
category $VBX$
of big vector bundles over $X$ in the sense of \cite[Appendix C.4]{Friedlander:2002aa}.
The assignments $X \mapsto VBX$
and $(f\colon Y \to X) \mapsto f^\ast\colon VBX \to VBY$ then form a
strict functor $(\mathcal Sm/S)^{\mathrm{op}} \to \mathcal{C}at$ because one has
equalities $(f\circ g)^{*} = g^{*}\circ f^{*}$
instead of simply isomorphisms.
For any $X \in \mathcal Sm/S$, let $Ch^b(VBX)$ denote the additive category
of bounded complexes of big vector bundles on X. We will
consider $Ch^b(VBX)$ as a complicial exact category with weak
equivalences, the conflations being the degreewise-split short exact
sequences, and the weak equivalences $w_{X}$ being the quasi-isomorphisms.
When we further endow $Ch^b(VBX)$ with the duality consisting of the
functor ${}^\vee = \mathcal Hom_{\mathcal O_X}({-},\mathcal O_X)$ and
the natural biduality maps $\eta_{X} \colon 1 \cong {}^{\vee\vee}$, we will write it
as $Ch^b(VBX)$.
\[
Ch^{b}(VBX) = (Ch^{b}(VBX), w_{X}, {}^{\vee}, \eta_{X})
\]
Now suppose $U \subset X$ is an open subscheme and $Z = X -U$.
Let $w_{U}$ be the set of chain maps whose restriction to $U$ is a quasi-isomorphism.
Let $Ch^{b}(VBX)^{w_{U}}$ be the full additive subcategory of complexes which are
acyclic on $U$. We have two new families of complicial exact categories with
weak equivalences and duality
\begin{align*}
Ch^{b}(VBX\ on \ Z) & = (Ch^{b}(VBX)^{w_{U}}, w_{X}, {}^{\vee}, \eta_{X}),
\\
Ch^{b}(VBX\ on \ U) & = (Ch^{b}(VBX),w_{U}, {}^{\vee}, \eta_{X}).
\end{align*}
We then have hermitian $K$-theory spaces
\begin{align*}
KO^{[n]}(X) & = KO^{[n]}(Ch^{b}(VBX) )
\\
KO^{[n]}(X,U) & = KO^{[n]}(Ch^{b}(VBX \ on \ Z)).
\end{align*}
with $KO^{[n]}(X) = KO^{[n]}(X,\varnothing)$.
Let
$D^{b}(VBX \ on \ Z)$ be the homotopy category equipped with the
triangulated duality $({}^{\vee},\eta_{X})$.
We define the hermitian
$K$-theory groups as
\begin{equation}
\label{E:KO.groups}
KO_{i}^{[n]}(X,U) =
\begin{cases}
\pi_{i}KO^{[n]}(X,U) & \text{for $i \geq 0$,}
\\
W^{n-i}(D^{b}VBX \ on \ Z) & \text{for $i < 0$}.
\end{cases}
\end{equation}
For $f \colon (X,U) \to (Y,V)$ a morphism in $\mathcal Sm\mathcal Op/S$ write $W = Y -V$. Then the functor
$f^{*} \colon Ch^{b}(VBX \ on \ Z) \to Ch^{b}(VBY\ on \ W)$ can by made
duality-preserving by equipping it with the natural isomorphism
$f^{*}\mathcal{H}om_{\mathcal{O}_{X}}({-},\mathcal{O}_{X}) \cong \mathcal{H}om_{\mathcal{O}_{Y}}(f^{*}{-},\mathcal{O}_{Y})$.
This gives us maps
\begin{subequations}
\begin{gather}
\label{E:KO.pullback.1}
f^{*}\colon KO^{[n]}(X,U) \to KO^{[n]}(Y,V),
\\
\label{E:KO.pullback.2}
f^{*} \colon KO_{i}^{[n]}(X,U) \to KO_{i}^{[n]}(Y,V).
\end{gather}
\end{subequations}
By Schlichting's localization theorem (Theorem \ref{T:localization}) the sequences
\[
KO^{[n]}(X,U) \to KO^{[n]}(X) \to KO^{[n]}(Ch^{b}(VBX \ on \ U))
\]
are fibration sequences up to homotopy.
The restriction map $Ch^{b}(VBX \ on \ U) \to Ch^{b}(VBU)$ is a duality-preserving functor
which induces an equivalence on the homotopy categories $D^{b}(VBX \ on \ U) \simeq D^{b}(VBU)$.
(Schlichting actually gives a different argument in
\cite[\S 9]{Schlichting:2010uq}
which is valid with fewer restrictions on the schemes.)
So we get fibration sequences up to homotopy
\begin{subequations}
\begin{equation}
\label{E:localization}
KO^{[n]}(X,U) \to KO^{[n]}(X) \to KO^{[n]}(U)
\end{equation}
and therefore long exact sequences
\begin{equation}
\label{E:KO.localization}
\cdots \to KO_{i}^{[n]}(X,U) \to KO^{[n]}_{i}(X) \to KO^{[n]}_{i}(U)
\xrightarrow{\partial} KO_{i-1}^{[n]}(X,U) \to \cdots
\end{equation}
\end{subequations}
Now suppose $(Y,V)$ is in $\mathcal Sm\mathcal Op/S$ and write $W = Y - V$.
There is then a pairing of complicial exact categories with
weak equivalences and duality
\[
\boxtimes \colon
Ch^{b}(VBX \ on \ Z) \times Ch^{b}(VBY \ on \ W) \to Ch^{b}(VB(X \times Y) \ on \ Z \times W).
\]
For a degree $r$ symmetric complex $(N,\psi)$ on $Y$ which is acyclic on $V$ and for a
degree $s$ symmetric complex $(M,\phi)$ on $X$ which is acyclic on $U$ we have maps of
spaces
\begin{subequations}
\begin{align}
({-}\boxtimes (N,\psi))_{*} \colon &
KO^{[n]}(X,U) \to KO^{[n+r]}(X \times Y, (X \times V) \cup (U \times Y))
\\
\varepsilon^{ns}((M,\phi) \boxtimes {-})_{*} \colon &
KO^{[n]}(Y,V) \to KO^{[n+s]}(X \times Y, (X \times V) \cup (U \times Y))
\end{align}
This leads to a right pairing and a corrected left pairing
\begin{gather}
\label{E:KO.pairing.1}
KO_{i}^{[n]}(X,U) \times KO_{0}^{[r]}(Y,V) \to KO_{i}^{[n+r]}(X \times Y, (X \times V) \cup (U \times Y)),
\\
\label{E:KO.pairing.2}
KO_{0}^{[n]}(X,U) \times KO_{i}^{[r]}(Y,V) \to KO_{i}^{[n+r]}(X \times Y, (X \times V) \cup (U \times Y)).
\end{gather}
\end{subequations}
Now suppose $(E,L,\lambda)$ is an $SL^{c}$ bundle of rank $n$ over $X$. Let $p \colon E \to X$ be the
structural map. We may construct Thom isomorphisms for hermitian $K$-theory using the
same method that Nenashev used for Witt groups \cite[\S 2]{Nenashev:2007rm}.
Namely, the pullback $p^{*}E = E \oplus E \to E$ has a canonical section $s$, the diagonal.
There is a Koszul complex
\[
K(E) = \bigl( 0 \to \Lambda^{n} p^{*}E^{\vee} \to \Lambda^{n-1} p^{*}E^{\vee} \to \cdots \to
\Lambda^{2} p^{*}E^{\vee} \to E^{\vee} \to \mathcal{O}_{E} \to 0 \bigr)
\]
(considered as a chain complex in homological degrees $n$ to $0$)
in which each boundary map the contraction with $s$. It is a locally free
resolution of the coherent sheaf $z_{*}\mathcal{O}_{X}$. There is a canonical isomorphism
$\varTheta(E) \colon K(E) \to K(E)^{\vee} \otimes \det p^{*}E^{\vee}[n]$
which is symmetric for the $(\det p^{*}E^{\vee})$-twisted shifted duality.
The composition
\[
\varTheta(E,L,\lambda) \colon K(E) \otimes p^{*}L \xra{\varTheta(E)}
K(E)^{\vee} \otimes \det p^{*}E^{\vee} \otimes p^{*}L [n]
\xra{1 \otimes p^{*}(\lambda^{\vee} \otimes L)}
K(E)^{\vee} \otimes p^{*}L^{\vee}[n]
\]
is symmetric for the untwisted shifted duality. We consider
the Grothendieck-Witt class
\begin{equation}
\label{E:KO.Thom}
\thom(E,L,\lambda) = [K(E) \otimes p^{*}L, \varTheta(E,L,\lambda)] \in KO_{0}^{[n]}(E,E-X).
\end{equation}
When $L$ is trivial this is an $SL$ Thom class $\thom(E,\lambda) = [K(E), \varTheta(E,\lambda)]$.
Finally for $g$ a nowhere vanishing function on $X$ we let
$\angles{g} = [\mathcal{O}_{X},g] \in KO_{0}^{[0]}(X)$ be the Grothendieck-Witt class
of the rank one symmetric bilinear bundle.
\begin{thm}
\label{T:SLc.oriented}
Let $S$ be a regular noetherian separated scheme of finite Krull dimension with
$\frac 12 \in \Gamma(S,\mathcal{O}_{S})$. Then the groups $KO_{i}^{[n]}(X,U)$ of \eqref{E:KO.groups},
the maps $f^{*}$ of \eqref{E:KO.pullback.2} and $\partial$ of \eqref{E:KO.localization}, the pairings of
\eqref{E:KO.pairing.1} and \eqref{E:KO.pairing.2}, the classes $1 = \angles{1}$ and $\varepsilon = \angles{-1}$
in $KO_{0}^{[0]}(pt)$ and the classes $\thom(E,L,\lambda)$ of \eqref{E:KO.Thom} form
an $SL^{c}$ oriented cohomology theory with an $\varepsilon$-commutative partial multiplication.
\end{thm}
In particular the products with the Thom classes are isomorphisms
\begin{equation}
\label{E:thom.iso.1}
{-} \cup \thom(E,L,\lambda) \colon KO_{i}^{[m]}(X) \xra{\cong}
KO_{i}^{[m+n]}(E, E-X)
\end{equation}
\begin{proof}
[Sketch of the proof]
The verifications are all straightforward. For instance \'etale excision and homotopy invariance amount
to having certain pullback maps be isomorphisms, and pullback maps are induced by duality-preserving
functors. Since these duality-preserving functors give isomorphisms for Quillen's $K$-theory and
Balmer's Witt groups, they give isomorphisms for hermitian $K$-theory as well under our hypotheses.
The Thom maps come from duality-preserving functors. The functor part
$Ch^{b}(VBX) \to Ch^{b}(VBE)$ is given by
$\mathcal F \mapsto \pi^{*} \mathcal F\otimes_{\mathcal{O}_{E}} K(E)$ with the target quasi-isomorphic to
the coherent sheaf
$z_{*}\mathcal F$. These produce d\'evissage isomorphisms in both Quillen-Waldhausen $K$-theory
and Balmer's Witt groups \cite{Gille:2007hb}.
For the $\varepsilon$-commutativity
of the partial multiplicative structure, $\sigma^{*}(b \times a)$ is calculated by applying the right
pairing for $a$ with respect to $b$ and then switching. That is equivalent to applying the uncorrected
left pairing for $a$ with respect to $b$. But that satisfies \eqref{E:commute.category}.
\end{proof}
We will discuss later the Pontryagin classes associated to the Thom classes.
It is sometimes inconvenient that the $KO_{i}^{[n]}(X,U)$ for $i < 0$ are not defined as homotopy groups.
But actually they are naturally isomorphic to direct summands of homotopy groups.
For we have the $S$-scheme $\mathbb{G}_{m} = \mathbf{A}^{1}-0$
(pointed by $1$) and groups
$KO^{[n]}_i(\mathbb{G}^{\wedge r}_m \times X,\mathbb{G}^{\wedge r}_m \times U)$
defined as in \eqref{E:A(smash)}.
\begin{lem}
For all $i$ and $n$ and all $r \geq 1$ and all $(X,U)$ in $\mathcal Sm\mathcal Op/S$ there are natural isomorphisms
$KO_{i+r}^{[n+r]}(\mathbb{G}^{\wedge r}_m \times X,\mathbb{G}^{\wedge r}_m \times U)
\cong KO^{[n]}_{i}(X,U)$.
\end{lem}
\begin{proof}
For $r = 1$
we have a localization sequence which splits and a Thom isomorphism
\[
\xymatrix @M=5pt @C=12pt {
KO_{i+1}^{[n+1]}(\mathbf{A}^{1} \times X,\mathbf{A}^{1} \times U) \ar@{>->}[r]
& KO_{i+1}^{[n+1]}(\mathbb{G}_{m} \times X,\mathbb{G}_{m} \times U) \ar@{->>}[d]^-{\partial} \ar[ld]^-{(in_{1} \times 1_{X})^{*}}
&
\\
KO_{i+1}^{[n+1]}(X,U) \ar[u]_-{\cong}
&
KO_{i}^{[n+1]}(\mathbf{A}^{1} \times X, \mathbf{A}^{1} \times U \cup \mathbb{G}_{m} \times X)
& KO_{i}^{[n]}(X,U) \ar[l]^-{\cong}_-{\thom \times}
}
\]
with $\thom \in KO_{0}^{[1]}(\mathbf{A}^{1},\mathbf{A}^{1} - 0)$
the Thom class of the trivial rank one $SL$ bundle. Hence we have a natural
isomorphism
\[
KO_{i+1}^{[n+1]}(\mathbb{G}_{m} \times X, \mathbb{G}_{m} \times U) \cong
KO_{i+1}^{[n+1]}(X,U) \oplus KO_{i}^{[n]}(X,U).
\]
By induction $KO_{i+r}^{[n+r]}(\mathbb{G}_{m}^{\times r} \times X, \mathbb{G}_{m}^{\times r} \times U)$
is a direct sum of $2^{r}$ terms of which exactly one is
$KO^{[n]}_{i}(X,U)$.
\end{proof}
\begin{defn}
\label{D:periodicity}
The \emph{periodicity element} $\beta_{8} \in KO_{0}^{[4]}(pt)$ is the element corresponding to
$1 \in KO_{0}^{[0]}(pt)$ under the periodicity isomorphisms $KO_{i}^{[n]} \cong KO_{i}^{[n+4]}$
of the hermitian $K$-theory of chain complexes.
\end{defn}
Then for all $X$ and $n$ the periodicity isomorphisms $KO_{i}^{[n]}(X) \cong KO_{i}^{[n+4]}(X)$
coincides with ${-} \times \beta_{8}$ up to homotopy.
\section{\texorpdfstring{$\mathbf{KO}^{[*]}_*$}{KO} of motivic spaces
\label{S:KO.motivic.spaces}
In this section we recall what the category of pointed motivic spaces is and
extend the functor $KO^{[*]}_*$ to a functor $\mathbf{KO}^{[*]}_*$ on that category.
The basic definitions, constructions and model structures we use
are given in \cite{Voevodsky:2007aa}.
A \emph{motivic space over $S$} is a simplicial presheaf on
the site $\mathcal Sm/S$ of smooth $S$-schemes of finite type.
A \emph{pointed motivic space over $S$} is a pointed simplicial presheaf on
the site $\mathcal Sm/S$.
We write
$\mathbf{M}_\bullet(S)$
for the category of pointed motivic spaces over $S$.
We equip the category $\mathbf{M}_\bullet(S)$ with the \emph{local injective model structure}
\cite[p.~181]{Voevodsky:2007aa} and with
the \emph{motivic model structure}
\cite[p.~194]{Voevodsky:2007aa}. In both model structures the cofibrations are the monomorphisms.
The weak equivalences and fibrations of the local injective model structure are called
local weak equivalences and global fibrations. Those of the motivic model structure are
called motivic weak equivalences and motivic fibrations.
We write $H_{\bullet}(S)$ for the pointed motivic
unstable homotopy category obtained by inverting the motivic weak equivalences.
The homotopy category $H_\bullet(S)$ is equivalent to the motivic homotopy category
of \cite{Morel:1999ab}.
For a morphism
$f\colon A \to B$
of pointed motivic spaces
we will write
$[f]$
for the class of $f$ in
$H_\bullet (S)$.
\begin{notation}
\label{fibrantreplacement}
There is a global fibrant model functor
$G \colon Id_{\mathbf{M}_\bullet(S)} \to ({-})^{f}$ functor in $\mathbf{M}_\bullet(S)$.
The natural transformation $G$ is a local weak equivalence,
but we do not require it to be injective.
\end{notation}
\begin{lem}
\label{BbbKOfmotivicfibrant}
Let $\mathbf{KO}^{[i]}=(KO^{[i]})^{f}$.
Then the map
$G\colon KO^{[i]} \to \mathbf{KO}^{[i]}$
is a schemewise weak equivalence, and
the space $\mathbf{KO}^{[i]}$ is motivically fibrant.
\end{lem}
\begin{proof}
The space $KO^{[i]}$ satisfies Nisnevich descent
because the homotopy groups $KO^{[i]}_{r}$ satisfy \'etale excision.
Therefore
\cite[Theorem 5.21]{Voevodsky:2007aa}
the morphism
$G \colon KO^{[i]} \to \mathbf{KO}^{[i]}$
is a schemewise weak equivalence.
For every $X \in \mathcal Sm/S$ the projection
$\mathbf{A}^1_X \to X$ induces a weak equivalence of simplicial sets
$\mathbf{KO}^{[i]}(X) \to \mathbf{KO}^{[i]}(\mathbf{A}^1_X)$,
since this the case for the space $KO^{[i]}$
and
$G: KO^{[i]} \to \mathbf{KO}^{[i]}$
is a schemewise weak equivalence.
This proves \cite[p.~195]{Voevodsky:2007aa} that the globally fibrant space
$\mathbf{KO}^{[i]}$ is motivically fibrant.
\end{proof}
We write $S^{r}_{s}$ for the $r$-sphere $\Delta[r]/\partial \Delta[r]$ in $\mathbf{sSet}$
and in the homotopy category
$H_{\bullet}$ of pointed simplicial sets and for the corresponding
constant simplicial presheaf in $\mathbf{M}_{\bullet}(S)$.
We write $\mathbb{G}_{m}$ for the pointed scheme $(\mathbf{A}^{1}-0,1)$.
\begin{defn}
\label{BbbKO(A)}
For any pointed motivic space $A$ and any $i$
define
\[
\mathbf{KO}^{[i]}_{r}(A) =
\begin{cases}
Hom_{H_{\bullet}(S)}(A \wedge S^r_{s}, \mathbf{KO}^{[i]}) & \text{for $r \geq 0$},
\\
Hom_{H_{\bullet}(S)}(A \wedge \mathbb{G}^{\wedge (-r)}_m,\mathbf{KO}^{[i-r]})
& \text{for $r < 0$.}
\end{cases}
\]
\end{defn}
\begin{lem}
\label{NewAndOld}
For any $r$ and $X \in \mathcal Sm/S$ one has functorial isomorphisms
$$\alpha_X: KO^{[i]}_r(X) \xra{\cong} \mathbf{KO}^{[i]}_r(X_+).$$
\end{lem}
\begin{proof} In fact, the following chain of isomorphisms give the desired one
$$Hom_{H_{\bullet}}(S^r_{s},KO^{[i]}(X)) \xra{\cong} Hom_{H_{\bullet}}(S^r_{s},\mathbf{KO}^{[i]}(X))
\xra{adj} Hom_{H_{\bullet}(S)}(X_+ \wedge S^r_s ,\mathbf{KO}^{[i]})$$
$$\pi_0(KO^{[i-r]}(X \times \mathbb{G}^{\times -r}_{m})) \xra{\cong} \pi_0(\mathbf{KO}^{[i-r]}(X \times \mathbb{G}^{\times -r}_{m})) \xra{adj} Hom_{H_{\bullet}(S)}((X \times \mathbb{G}^{\times -r}_{m})_+,\mathbf{KO}^{[i-r]})$$
All the arrows are indeed isomorphisms by Lemma
\ref{BbbKOfmotivicfibrant}.
\end{proof}
Now suppose $(X,U) \in \mathcal Sm\mathcal Op/S$ with $j \colon U \hra X$ the inclusion and $Z = X -U$.
Schlichting's localization sequence \eqref{E:localization}
\[
KO^{[i]}(X,U) \xrightarrow{e^{*}} KO^{[i]}(X) \xrightarrow{j^{*}} KO^{[i]}(U)
\]
can be described more precisely than we have done so far.
The composite map $j^{*}e^{*}$ is induced by the functor
$Ch^{b}(VBX \ on \ Z) \to Ch^{b}(VBU)$ such that the morphism of functors
$j^{*}e^{*} \to 0$ is a natural weak equivalence.
This natural weak equivalence gives a homotopy from the map of spaces $j^{*}e^{*}$ to the trivial map,
and that gives a factorization
of $e^{*}$ as
\begin{equation}
\label{E:h.fiber}
KO^{[i]}(X,U) \xrightarrow{e_{X,U}} hofib(j^{*}) \xrightarrow{can_{X,U}} KO^{[i]}(X).
\end{equation}
Schlichting's theorem is that $e_{X,U}$ a homotopy equivalence. This factorization is functorial
in $(X,U)$.
For a pointed motivic space $A$ write $\mathbf{KO}^{[i]}(A) = \operatorname{\mathbf{hom}}_{\bullet}(A,\mathbf{KO}^{[i]})$
for the pointed mapping space, which is the
pointed simplicial set $[n] \mapsto Hom_{\mathbf{M}_{\bullet}(S)}(A \wedge \Delta[n]_{+},\mathbf{KO}^{[i]})$.
For $A = X_{+}$ with $X$ a smooth scheme we have
$\mathbf{KO}^{[i]}(A) = \mathbf{KO}^{[i]}(X)$. For $j \colon U \hra X$ as above we can fill out
a commutative diagram of pointed simplicial sets
\[
\xymatrix{
KO^{[i]}(X,U) \ar[d]_{e_{X,U}} \\
hofib(j^{*})\ar[rr]^-{can_{X,U}} \ar [d]_{G_{1}} &&
KO^{[i]}(X) \ar[r]^-{j^*} \ar[d]_-{G(X)} & KO^{[i]}(U) \ar[d]^-{G(U)} \\
\mathbf{KO}^{[i]}(Cone(j_+))= hofib(j_{+}^{*}) \ar[rr]^-{nat_{X,U}} &&
\mathbf{KO}^{[i]}(X_{+}) \ar[r]^-{j_{+}^*} & \mathbf{KO}^{[i]}(U_{+}) \\
\mathbf{KO}^{[i]}(X_{+}/U_{+}) \ar[u]^-{q_{X,U}}}
\]
We claim that all the vertical arrows are homotopy equivalences. This is true for
$G(X)$ and $G(U)$ by Lemma \ref{BbbKOfmotivicfibrant} and therefore for $G_{1}$ because
it is the map between the homotopy fibers. The map $e_{X,U}$ is a homotopy equivalence
by Schlichting's theorem. The map $q_{X,U}$ is a homotopy equivalence because it is
obtained by applying $\operatorname{\mathbf{hom}}_{\bullet}({-},\mathbf{KO}^{[i]})$ with $\mathbf{KO}^{[i]}$
fibrant to the schemewise weak equivalence between cofibrant objects
$Cone(j_{+}) \to X_{+}/U_{+}$.
The diagram is functorial in $(X,U)$. We conclude:
\begin{thm}
\label{T:zigzag}
For $(X,U)$ in $\mathcal Sm\mathcal Op/S$ there is a functorial zigzag of homotopy equivalences
$KO^{[i]}(X,U) \to \mathbf{KO}^{[i]}(Cone(U_{+} \to X_{+})) \leftarrow \mathbf{KO}^{[i]}(X_{+}/U_{+})$
which
for $(X,\varnothing)$ reduces to the $G(X) \colon KO^{[i]}(X) \to \mathbf{KO}^{[i]}(X)$
of Lemma \ref{BbbKOfmotivicfibrant}. These induce functorial isomorphisms of groups
$KO^{[i]}_{r}(X,U) \cong \mathbf{KO}^{[i]}_{r}(X_{+}/U_{+})$ for all integers $i$ and $r$.
\end{thm}
\begin{notation}
\label{KOtoBbbKO}
Denote by $\alpha: KO^{[i]}_{*}(-,-) \to \mathbf{KO}^{[i]}_{*}(-_+/-_+)$
the functor isomorphism described in Theorem \ref{T:zigzag}.
\end{notation}
\section{\texorpdfstring{The $T$-spectrum $\mathbf{BO}$ and the cohomology theory $\mathbf{BO}^{*,*}$}
{The T-spectrum BO and the cohomology theory BO}}
\label{S:BO}
Let $T = \mathbf{A}^{1}/(\mathbf{A}^{1}-0)$ be the Morel-Voevodsky object.
A \emph{
$T$-spectrum} $E$ over $S$ consists
of a sequence $(E_0,E_1,\ldots)$ of pointed motivic spaces over $S$
plus \emph{structure maps} $\sigma_n\colon E_n \wedge T \to E_{n+1}$.
Let $SH(S)$
denote the stable homotopy category of
$T$-spectra as described in
\cite{Jardine:2000aa}.
It is canonically equivalent
to the motivic stable homotopy category
constructed in
\cite{Voevodsky:1998kx}.
Here we define a $T$-spectrum $\mathbf{BO}$.
Its spaces are $(\mathbf{KO}^{[0]}, \mathbf{KO}^{[1]}, \mathbf{KO}^{[2]}, \mathbf{KO}^{[3]},\dots)$.
We now define the structure maps.
Let
$\mathbf{A}^{1} \to pt$ be the trivial rank one $SL$ bundle, and let
$\thom \in KO^{[1]}_{0}(\mathbf{A}^{1},\mathbf{A}^{1} -0)$ be its Thom class as defined by \eqref{E:KO.Thom}.
Because $KO_{*}^{[*]}$ is $SL^{c}$ oriented, the maps
\begin{equation}
\label{E:adjoint}
{-} \times \thom \colon KO^{[n]}_{r}(X) \to KO^{[n+1]}_{r}(X \times \mathbf{A}^{1}, X \times (\mathbf{A}^{1} - 0))
\end{equation}
are isomorphisms.
Recall that $\thom$ is defined by \eqref{E:KO.Thom}
as the class of the symmetric complex
\begin{equation}
\label{E:thom.A1}
\vcenter{
\xymatrix @M=5pt @C=30pt {
K(\mathcal{O}) \ar[d]^-{\cong}_-{\varTheta(\mathcal{O})}
&& 0 \ar[r]
& \mathcal{O}_{\mathbf{A}^{1}} \ar[r]^-{x} \ar[d]_{-1}
& \mathcal{O}_{\mathbf{A}^{1}} \ar[r] \ar[d]^-{1}
& 0
\\
K(\mathcal{O})^{\vee}[1]
&& 0 \ar[r]
& \mathcal{O}_{\mathbf{A}^{1}} \ar[r]^-{-x}
& \mathcal{O}_{\mathbf{A}^{1}} \ar[r]
& 0
}}
\end{equation}
of degree $1$ in $Ch^{b}(VB\mathbf{A}^{1}\ on \ 0)$. The maps ${-} \times \thom$ are induced by
the maps of spaces
\begin{equation*}
({-}\boxtimes (K(\mathcal{O}),\varTheta(\mathcal{O})))_{*} \colon
KO^{[n]}(X) \to KO^{[n+1]}(X \times \mathbf{A}^{1}, X \times (\mathbf{A}^{1} - 0))
\end{equation*}
These maps are thus homotopy equivalences, and
$KO^{[n]} \to KO^{[n+1]}({-} \times \mathbf{A}^{1}, {-} \times (\mathbf{A}^{1} - 0))$
is a schemewise weak equivalence.
From Lemma \ref{BbbKOfmotivicfibrant} and Theorem \ref{T:zigzag} we now have
a zigzag
\[
\xymatrix @M=5pt @C=20pt {
\mathbf{KO}^{[n]} \ar@{<-}[r]^-{G}_-{\sim}
& KO^{[n]} \ar[r]^-{\times \thom}_-{\sim}
& KO^{[n+1]}({-} \times \mathbf{A}^{1}, {-} \times (\mathbf{A}^{1} - 0)) \ar[d]^-{\sim}
\\
&& \mathbf{KO}^{[n+1]}(Cone({-} \wedge (\mathbf{A}^{1}-0)_{+} \to {-} \wedge \mathbf{A}^{1}_{+})) \ar@{<-}[r]^-{\sim}
& \mathbf{KO}^{[n+1]}({-} \wedge T)
}
\]
of schemewise weak equivalences in $\mathbf{M}_{\bullet}(S)$. Their composition is an isomorphism
$\mathbf{KO}^{[n]} \cong \mathbf{KO}^{[n+1]}({-} \wedge T)$
in the homotopy category.
There is a Quillen adjunction with left adjoint ${-} \wedge T$ and right adjoint
$F({-}) \mapsto F({-} \wedge T)$. It follows that $\mathbf{KO}^{[n+1]} ({-} \wedge T) $ is
\textbf{fibrant}, while $\mathbf{KO}^{[n]}$ is cofibrant, so there exists a morphism
$\sigma_{n}^{*} \colon \mathbf{KO}^{[n]} \to \mathbf{KO}^{[n+1]} ({-} \wedge T) $ in
$\mathbf{M}_{\bullet}(S)$ representing the same
isomorphism in the homotopy category as the zigzag.
Let $\sigma_{n} \colon \mathbf{KO}^{[n]} \wedge T
\to \mathbf{KO}^{[n+1]}$ be the adjoint morphism.
Since the $KO^{[n]}$ and $\mathbf{KO}^{[n]}$ are periodic modulo $4$, we may choose the
$\sigma_{n}^{*}$ and $\sigma _n$ so they are also periodic.
\begin{defn}
\label{BO}
The $T$-spectrum $\mathbf{BO}$ consists of the sequence of pointed motivic spaces
\begin{equation}
(\mathbf{KO}^{[0]}, \mathbf{KO}^{[1]}, \mathbf{KO}^{[2]}, \mathbf{KO}^{[3]}, \dots)
\end{equation}
together with the structure maps
$\sigma _n: \mathbf{KO}^{[n]} \wedge T \to \mathbf{KO}^{[n+1]}$ just described.
\end{defn}
The spaces
$\mathbf{KO}^{[n]}$ are motivically fibrant and the adjoints $\sigma_{n}^{*}$ of the structure maps
are schemewise weak equivalences. So we have \cite[Lemma 2.7]{Jardine:2000aa}:
\begin{thm}
\label{OmegaSpectrumBO}
The $T$-spectrum
$\mathbf{BO}$
is stably motivically fibrant.
\end{thm}
As explained in
\cite{Voevodsky:1998kx}
any $T$-spectrum $E$ defines a bigraded cohomology theory
$(E^{*,*}, \partial)$ on the category $\mathbf{M}_\bullet(S)$ with
\[
E^{p,q}(A) = Hom_{SH_{\bullet}(S)}(A, E \wedge S_{s}^{p-q} \wedge \mathbb{G}_{m}^{\wedge q} ).
\]
The differential $\partial$ increases the bidegree
by $(1,0)$.
For any $A \in \mathbf{M}_{\bullet}(S)$
the adjunction map induces isomorphisms
\[
\mathbf{KO}^{[n]}_{i}(A) = Hom_{H_{\bullet}(S)}(A \wedge S_{s}^{i}, \mathbf{KO}^{[n]})
\cong Hom_{SH(S)}(A \wedge S_{s}^{i}, \mathbf{BO} \wedge T^{\wedge n}) =
\mathbf{BO}^{2n-i,n}(A),
\]
\begin{multline*}
\mathbf{KO}^{[n]}_{i}(A) =
Hom_{H_{\bullet}(S)}(A \wedge \mathbb{G}_{m}^{\wedge -i}, \mathbf{KO}^{[n-i]})
\cong Hom_{SH(S)}(A \wedge \mathbb{G}_{m}^{\wedge -i}, \mathbf{BO} \wedge T^{\wedge n-i})
\\= \mathbf{BO}^{2n-i,n}(A),
\end{multline*}
for $i \geq 0$ and $i < 0$ respectively. This gives us an isomorphism of functors on $\mathbf{M}_{\bullet}(S)$
\[
\beta_A: \mathbf{KO}^{[n]}_{i} \xra{\cong} \mathbf{BO}^{2n-i,n}.
\]
\begin{cor}
\label{KO**andBO**}
The composition isomorphism
$\beta|_{\mathcal Sm\mathcal Op/k} \circ \alpha: KO^{[*]}_* \to \mathbf{BO}^{*,*}|_{\mathcal Sm\mathcal Op/S}$
respects the boundary homomorphisms in both cohomology theories on $\mathcal Sm\mathcal Op/S$.
So it is an isomorphism
$$\gamma: KO^{[*]}_* \to \mathbf{BO}^{*,*}|_{\mathcal Sm\mathcal Op/S}$$
of cohomology theories in the sense of
\cite{Panin:2003rz}.
\end{cor}
\begin{defn}
\label{partialmultonBO**}
Using the cohomology isomorphism $\gamma$ we transplant
to $\mathbf{BO}|_{\mathcal Sm\mathcal Op/S}$
the partial multiplicative
structure of $(KO^{[*]}_*, \partial)$ and the Thom and Pontryagin classes of its
$SL^{c}$ orientation described in Theorem \ref{T:SLc.oriented}.
The unit of this partial multiplicative structure is the element
$e= \gamma(\angles{1}) \in \mathbf{BO}^{0,0}(S^0)$.
\end{defn}
\begin{thm}[Bott periodicity]
The adjoints of the structure maps and the categorical periodicity isomorphisms
give levelwise weak equivalences
\[
\mathbf{BO} \xra{\sim \,level} \Omega_{T}^{4}\mathbf{BO}(4)
\xra{\cong} \Omega_{T}^{4}\mathbf{BO}.
\]
\end{thm}
\section{A symplectic version of the Morel-Voevodsky theorem}
\label{S:Morel-Voevodsky}
In
\cite[Theorem 4.3.13]{Morel:1999ab}
Morel and Voevodsky showed that
$\mathbb{Z} \times Gr$ represents algebraic $K$-theory in the motivic unstable homotopy category.
If one replaces the ordinary Grassmannians by quaternionic Grassmannians, the same
holds for symplectic $K$-theory.
We write $\mathsf{H}$ for the trivial rank $2$ symplectic bundle
$\left( \mathcal{O}^{\oplus 2}, \bigl( \begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix} \bigr) \right)$.
The orthogonal direct sum $\mathsf{H}^{\oplus n}$ is the trivial symplectic bundle of rank $2n$.
We will sometimes write $\mathsf{H}^{\oplus n}_{X}$ to designate the trivial symplectic bundle over the
scheme $X$.
The \emph{quaternionic Grassmannian}
$HGr(r,n) = HGr(r,\mathsf{H}^{\oplus n})$ is defined as the open subscheme of
$Gr(2r,2n) = Gr(2r,\mathsf{H}^{\oplus n})$ parametrizing subspaces of dimension $2r$ of the fibers of
$\mathsf{H}^{\oplus n}$
on which the symplectic form of $\mathsf{H}^{\oplus n}$ is nondegenerate. We write $\mathcal U_{r,n}$ for
the restriction to $HGr(r,n)$ of the tautological subbundle of $Gr(2r,2n)$. The symplectic form of
$\mathsf{H}^{\oplus n}$ restricts to a symplectic form on $\mathcal U_{r,n}$ which we
denote by $\phi_{r,n}$. The pair $(\mathcal U_{r,n},\phi_{r,n})$ is the
\emph{tautological symplectic subbundle} of rank $2r$ on $HGr(r,n)$.
Morphisms $X \to HGr(r,n)$ are \emph{classified} by subbundles $E \subset \mathsf{H}_{X}^{\oplus n}$
of rank $2r$ such that the symplectic form of $\mathsf{H}_{X}^{\oplus n}$
is nondegenerate on every fiber.
More generally, given a symplectic bundle $(E,\phi)$ of rank $2n$ over $X$, the
\emph{quaternionic Grassmannian bundle} $HGr(r,E,\phi)$ is the open subscheme of
the Grassmannian bundle $Gr(2r,E)$ parametrizing subspaces of dimension $2r$
of the fibers of $E$ on which $\phi$
is nondegenerate.
For $r=1$ we have \emph{quaternionic projective spaces} and \emph{bundles}
$HP^{n} = HGr(1,n+1)$ and $HP(E,\phi) = HGr(1,E,\phi)$.
There are commuting morphisms
\begin{equation}
\label{E:alpha.beta}
\vcenter{
\xymatrix @M=5pt @C=40pt {
HGr(r,n) \ar[r]^-{\alpha_{r,n}} \ar[d]_-{\beta_{r,n}}
& HGr(r,n+1) \ar[d]^-{\beta_{r,n+1}}
\\
HGr(r+1,n+1) \ar[r]^-{\alpha_{r+1,n+1}}
&HGr(r+1,n+2)
}}
\end{equation}
with $\alpha_{r,n}$ classified by the rank $2r$ subbundle
$\mathcal U_{r,n}\oplus 0 \subset \mathsf{H}^{\oplus n} \oplus \mathsf{H}$
and $\beta_{r+1,n+1}$ classified by the rank $2r+2$ subbundle
$\mathsf{H} \oplus \mathcal U_{r,n}\subset \mathsf{H} \oplus \mathsf{H}^{\oplus n}$. Composition gives us
maps $HGr(n,2n) \to HGr(n+1,2n+2)$. We define $HGr = \operatornamewithlimits{colim} HGr(n,2n)$.
We consider $\mathbb{Z} \times HGr$ pointed by $(0,HGr(0,0))$.
It has a universal property.
\begin{thm}
\label{T:universal}
Suppose $X \in \mathcal Sm/S$ is affine. Then for every $\xi \in GW^{-}(X)$ there is a morphism
of ind-schemes $f \colon X \to \mathbb{Z} \times HGr$
such that $\xi = f^{*}\tau$. Moreover $f$ is unique up to naive $\mathbf{A}^{1}$-homotopy.
\end{thm}
This is the equivalence $\pi_{0}\mathcal L_{6} \cong \pi_{0}\mathrm{GW}^{-}$ of
\cite[Proposition 6.2.1.5]{Barge:2008it} plus the isomorphism $\pi_{0}\mathrm{GW}^{-} =
\mathrm{GW}^{-}$
which happens for our schemes which are regular with $\frac 12$.
For a smooth $S$-scheme $X$ let $KSp(X) = KO(Ch^{b}(VBX),w_{X},
{}^{\vee},-\eta)$ be its symplectic $K$-theory space.
There are natural isomorphisms ${K}Sp \cong {K}O^{[4n+2]}$.
\begin{thm}
\label{T:MV.symp}
The objects $\mathbb{Z} \times HGr$ and ${K}Sp$ are isomorphic in the motivic unstable homotopy category
$H_{\bullet}(S)$.
\end{thm}
The proof of this theorem is identical to Morel and Voevodsky's proof for
ordinary $K$-theory except for one point.
Let $\USp(r,n) \to \HGr(r,n)$ be the principal $\mathnormal{Sp}_{2r}$-bundle associated to the
tautological rank $2r$ symplectic subbundle on $\HGr(r,n)$.
There is an isomorphism $\USp(r,n) \cong Sp_{2n}/Sp_{2n-2r}$.
To establish Theorem \ref{T:MV.symp} by
following the proof of Morel and Voevodsky exactly we would need the
$\USp(r,n)$ to form
an \emph{admissible gadget} in the sense of
\cite[Definition 4.2.1]{Morel:1999ab}.
It does not seem as if they do. Nor can we use the admissible gadget used by
Morel and Voevodsky, for that would substitute for
$\USp(r,n)$ the principal $GL(2r)$-bundle associated to
the tautological subbundle on $Gr(2r,2n)$, called $U_{2r,2n}$ in
\cite{Morel:1999ab}.
(This $U_{2r,2n}$ is
the space of $2r \times 2n$ matrices of maximal rank.)
But the quotient $U_{2r,2n}/Sp_{2r}$ is not the quaternionic Grassmannian $HGr(r,n)$,
and its cohomology risks being much less tractable.
So we give a new definition based on the property actually
used in the proof of
\cite[Proposition 4.2.3]{Morel:1999ab}.
For a commutative ring $B$ let
\begin{align*}
\Delta^{n}_{B} & = \operatorname{Spec} B[t_{1},\dots,t_{n}],
&
\partial \Delta^{n}_{B} & = \operatorname{Spec} B[t_{1},\dots,t_{n}]/(t_{1}t_{2}\cdots t_{n}(1-{\textstyle\sum} t_{i}))
\subset \Delta^{n}_{B}
\end{align*}
\begin{defn}
\label{D:gadget}
An \emph{acceptable gadget} over an $S$-scheme $X$ is a sequence of smooth quasi-projective
$X$-schemes $(U_{i})_{i\in\mathbb{N}}$ and closed embeddings $U_{i} \to U_{i+1}$ of $X$-schemes
such that for any henselian regular local ring $B$ and any
commutative square
\[
\xymatrix @M=5pt {
\partial \Delta^{n}_{B} \ar[r] \ar[d]_-{\text{inclusion}}
& U_{i} \ar[d]^-{\text{projection}}
\\
\Delta^{n}_{B} \ar[r]
& X
}
\]
there exists a $j \geq i$
and a map $\Delta_{B}^{n} \to U_{j}$ making the following diagram commute.
\[
\xymatrix @M=5pt @C=50pt {
\partial \Delta^{n}_{B} \ar[r] \ar[d]_-{\text{inclusion}}
& U_{i} \ar[r]^-{\text{gadget map}}
& U_{j} \ar[d]^-{\text{projection}}
\\
\Delta^{n}_{B} \ar[rr] \ar[rru]
&& X
}
\]
\end{defn}
Sections of the principal bundle $\USp(r,n) \to HGr(r,n)$ are given by symplectic frames
(i.e.\ symplectic bases)
of the tautological symplectic subbundle.
Therefore giving a morphism $V \to \USp(r,n)$ is equivalent to giving
an embedding $\mathsf{H}_{V}^{\oplus r} \subset \mathsf{H}_{V}^{\oplus n}$ such that the symplectic form on
$\mathsf{H}_{V}^{\oplus r}$ is the restriction of the symplectic form on $\mathsf{H}_{V}^{\oplus n}$.
This is also equivalent by duality to giving $2n$ sections $s_{1},\dots,s_{2n}$
of the bundle of linear forms $\mathsf{H}_{V}^{\oplus r \vee}$ such that
$\sum_{i=1}^{n} s_{2i-1} \wedge s_{2i}$ is equal to the symplectic form of $\mathsf{H}_{V}^{\oplus r}$.
We need the $\USp(r,n)$ to form an acceptable gadget
in the relative case as well (cf.\
\cite[Lemma 4.2.8]{Morel:1999ab}).
Given a symplectic bundle
$(E,\phi)$ of rank $2r$ over $X$, the relative $\USp(E,\phi;n)$ is constructed by taking the
$\mathnormal{Sp}_{2r}$-torsor $P \to X$ associated to $(E,\phi)$ and forming the quotient
$(P \times \USp(r,n))/\mathnormal{Sp}_{2r}$ by the diagonal action of $\mathnormal{Sp}_{2r}$ on the two torsors.
This gives us a fiber bundle $\USp(E,\phi;n) \to X \times \HGr(r,n)$ with fibers
$\mathnormal{Sp}_{2r}$ and structural group $\mathnormal{Sp}_{2r} \times \mathnormal{Sp}_{2r}$
acting on the fibers by left and right translation.
Giving a morphism $V \to \USp(E,\phi;n)$
is equivalent to giving
a triple $(f,g,\iota)$ with $f \colon V \to X$ and $g \colon V \to \HGr(r,n)$ morphisms
of schemes and $\iota \colon f^{*}(E,\phi) \cong g^{*}(\mathcal {U}_{r,n},\phi_{r,n})$
an isometry
of symplectic bundles.
This is equivalent to giving
$(f,u_{1},\dots,u_{2n})$ with
$f \colon V \to X$ a map and the $u_{i}$ sections
of $f^{*}E^{\vee}$ such
that $\sum_{i=1}^{n} u_{2i-1} \wedge u_{2i} = f^{*}\phi$.
\begin{lem}
\label{L:gadget}
Let $R$ be a commutative ring, let $g\in R$ and let $\overline R = R/(g)$.
Let $(E,\phi)$ a symplectic $R$-module, and let $(\overline E, \overline \phi)$ be the associated
symplectic $\overline R$-module. Let $u_{1},\dots, u_{2n} \in E^{\vee}$ be linear forms such that
$\sum_{i=1}^{n} \overline u_{2i-1} \wedge \overline u_{2i} =
\overline \phi$ in $\Lambda^{2} \overline{E}^{\vee}$.
Then there
exist linear forms $v_{1},\dots,v_{2n}$ and $w_{1},\dots,w_{2m}$ in $E^{\vee}$ such that
\[
\sum_{i=1}^{n}(u_{2i-1}+gv_{2i-1}) \wedge (u_{2i}+gv_{2i}) + \sum_{j=1}^{m} g w_{2j-1}\wedge g w_{2j} = \phi.
\]
\end{lem}
\begin{proof}
The $\overline u_{1},\dots,\overline u_{2n}$ generate $\overline E^{\vee}$. So there exist
$v_{1},\dots,v_{2n} \in E^{\vee}$ such that
\[
\phi - \sum_{i=1}^{n} u_{2i-1} \wedge u_{2i} \equiv g \sum_{j=1}^{2n} u_{j} \wedge (-1)^{j}v_{j}
\pmod {g^{2}}.
\qedhere
\]
\end{proof}
\begin{prop}
\label{P:USp}
For any symplectic bundle $(E,\phi)$ of rank $2r$ over a scheme $X$, the
schemes
$(\USp(E,\phi;n))_{n\geq r}$ together with the closed embeddings $\USp(E,\phi;n) \to \USp(E,\phi;n+1)$
induced by the inclusions $\mathsf{H}^{\oplus n} \subset \mathsf{H}^{\oplus n} \oplus \mathsf{H}$
form an acceptable gadget.
\end{prop}
\begin{proof}
Let $R = B[t_{1},\dots,t_{n}]$ and $g = t_{1}t_{2}\cdots t_{n}(1-{\textstyle\sum} t_{i}) \in R$.
Giving the first diagram of Definition \ref{D:gadget} is then equivalent to giving the $(E,\phi)$ and
$(\overline u_{1},\dots, \overline u_{2n})$ of Lemma \ref{L:gadget}.
The map $\Delta_{B}^{n}\to U_{j}$ is then given by
$(u_{1}+gv_{1},\dots,u_{2n}+gv_{2n},gw_{1},\dots,gw_{2m})$. The top triangle of the second
diagram commutes because modulo $g$ this last vector is
$(\overline u_{1},\dots, \overline u_{2n},0,\dots,0)$. The lower triangle commutes because of the
equation involving $\phi$.
\end{proof}
Substituting the acceptable gadgets
$\USp(r,n) \to \USp(r,n+1) \to \cdots$ and the quaternionic Grassmannians $HGr(i,n)$
for the admissible gadgets $U_{n,i} \to U_{n,i+1} \to \cdots$ and Grassmannians $Gr(i,n)$
of Morel and Voevodsky,
the proof of
\cite[Theorem 4.3.13]{Morel:1999ab}
can be used to
prove Theorem \ref{T:MV.symp}.
Proposition \ref{P:USp} substitutes for the last paragraph of the proof of Proposition 4.2.3
and for Lemma 4.2.8.
We get $\hocolim_{n} \USp(r,n) \cong pt$ and
\[
BSp_{2r} \cong B_{et}Sp_{2r} \cong HGr(r,\infty)
\]
in $H(S)$,
and we get the theorem.
At the end one needs the equivalence of hermitian $K$-theories
based on group completion and of Schlichting's Waldhausen-like
construction. But Schlichting has shown that each is equivalent to the hermitian $Q$-construction:
see
\cite[Theorem 4.2]{Schlichting:2004aa} and \cite[Proposition 6]{Schlichting:2010uq}.
Similar but less definitive results can be proven for the orthogonal group.
Let $\mathsf{H}_{+}^{\oplus n}$ denote the trivial orthogonal bundle
$(\mathcal{O}_{S}^{\oplus 2n},q_{2n})$ with the split quadratic form $q_{2n} = \sum_{i=1}^{n}x_{2i-1}x_{2i}$.
Let $RGr(r,2n) = RGr(r,\mathsf{H}^{\oplus n}_{+})$
be the open subscheme of $Gr(r,2n)$ parametrizing subspaces of rank $r$ on which
$q_{2n}$ is nondegenerate. Then over $RGr(r,2n)$ we have a tautological rank $r$ orthogonal subbundle
$(U |_{RGr}, q_{2n} |_{RGr})$ whose structural group scheme is the orthogonal group scheme
$O(r,q_{2n}|_{RGr}) \to RGr(r,2n)$.
The associated principal bundle is $RU(r,2n) \to RGr(r,2n)$. To give a morphism $V \to RU(r,2n)$
one gives a quadratic bundle $(E,q)$ of rank $r$ over $V$ and $2n$ sections $s_{1},\dots,s_{2n}$
of $E^{\vee}$ such that $q = \sum_{i=1}^{n}s_{2i-1}s_{2i}$.
The data $(E,q,s_{1},\dots,s_{2n})$ and $(E_{1},q_{1},t_{1},\dots,t_{2n})$ define the same morphism
if and only if there is an isomorphism $\phi \colon E \cong E_{1}$ such that $q = \phi^{*}q_{1}$
and $s_{i} = \phi^{*}t_{i}$ for all $i$.
The relative case is like the relative symplectic case described earlier.
\begin{prop}
\label{P:RU}
\parens{a}
For any quadratic bundle $(E,q)$ the
$RU(E,q,2n)$ and the inclusions $RU(E,q,2n) \to RU(E,q,2n+2)$ corresponding to
$(E,q,s_{1},\dots,s_{2n}) \mapsto (E,q,s_{1},\dots,s_{2n},0,0)$ form an acceptable gadget, as do their
relative versions.
\parens{b}
For any quadratic bundle $(E,q)$ of rank $r$ over $X$
the fiber bundles $RU(E,q,2n) \to X$ have sections for $n \geq r$ over any open subscheme over which
the vector bundle $E$ trivializes.
\end{prop}
Part (b) of the proposition is of concern
\cite[Proposition 4.1.20 and Definition 4.2.4.3]{Morel:1999ab}.
It is not
an issue for symplectic bundles because symplectic bundles are locally trivial in the Zariski topology. It
holds because to give
a section of $RU(E,q,2n) \to X$
is to give sections $s_{1},\dots,s_{2n}$ of $E^{\vee}$
such that $q = \sum s_{2i-1}s_{2i}$. If $E$ trivializes over
$U$ with coordinates $x_{1},\dots,x_{r}$ and $q = \sum_{i \leq j} a_{ij}x_{i}x_{j}$, then one
can give a local section over $U$ by
$s_{2i-1} = x_{i}$ and $s_{2i} = \sum_{j=i}^{n} a_{ij}x_{j}$ for $1 \leq i \leq r$ and $s_{i}=0$ for $i > 2r$.
The results of Proposition \ref{P:RU} and of the arguments of Morel and Voevodsky are isomorphisms
$B_{et}O_{r} \cong RGr(r,\infty)$ and $B_{et}O \cong RGr$
in $H_{\bullet}(S)$. However, neither of the natural maps
$\mathbb{Z} \times BO \to \mathbb{Z} \times B_{et}O$ or $\mathbb{Z} \times BO \to \mathbf{KO}^{[0]}$
are isomorphisms in $H_{\bullet}(S)$
because orthogonal bundles
are not always locally trivial in the Nisnevich topology.
We also do not know how to calculate $KO_{*}^{[*]}(RGr)$.
\section{The cohomology of quaternionic Grassmannians}
\label{S:symp.Thom}
We review the calculation of the cohomology of quaternionic Grassmannians of \cite{Panin:2010fk}.
We reformulate the definitions and some of the theorems for a bigraded $\varepsilon$-commutative partial multiplication. In \cite{Panin:2010fk} we assumed we had a full ring structure.
We do not redo the proofs because no change is needed:
all the needed products
are with the Thom and Pontryagin classes of symplectic bundles
or with pullbacks of such classes, and those lie in the $A^{4i,2i}$.
\begin{defn}
[\protect{\cite[Definition 7.1]{Panin:2010fk}}]
\label{Thom}
A \emph{symplectic Thom structure} on a bigraded cohomology theory
$(A^{*,*}, \partial)$ with an $\varepsilon$-commutative partial multiplication or ring structure
is a rule assigning
to every rank $2$ symplectic bundle
$(E, \phi)$
over an $X$ in $\mathcal Sm/S$ an element
$\thom(E, \phi) \in A^{4,2}(E,E-X)$
with the following properties:
\begin{enumerate}
\item
For an isomorphism $u: (E, \phi)\cong (E_1, \phi_1)$ one has $\thom(E, \phi) = u^* \thom(E_1, \phi_1)$.
\item
For a morphism $f : Y \to X$ with pullback map $f_E : f^*E \to E$ one has
$f_{E}^* \thom(E, \phi)= \thom(f^*E, f^*\phi)$.
\item
For the trivial rank $2$ bundle $\mathsf{H}$ over $pt$
the map
\[
{-} \times \thom(\mathsf{H}) \colon A^{*,*}(X) \to A^{*, *}(X \times \mathbf{A}^2, X \times (\mathbf{A}^2-\{0\}))
\]
is an isomorphism for all $X$.
\end{enumerate}
The \emph{Pontryagin class} of $(E,\phi)$ is $p(E,\phi) = -z^{*} \, \thom(E,\phi) \in A^{4,2}(X)$ where
$z \colon X \to E$ is the zero section.
\end{defn}
From Mayer-Vietoris one sees that for any rank $2$ symplectic bundle
\begin{equation*}
{}\cup \thom(E,\phi) \colon A^{*,*}(X) \xra{\cong} A^{*,*}(E,E-X)
\end{equation*}
is an isomorphism.
The sign in the Pontryagin class is simply conventional. It is chosen so that if $A^{*,*}$ is an oriented
cohomology theory with an additive formal group law, then the Chern and Pontryagin classes satisfy the
traditional formula $p_{i}(E,\phi) = (-1)^{i}c_{2i}(E)$.
The following is \cite[Theorem 8.2]{Panin:2010fk}.
\begin{thm}[Quaternionic projective bundle theorem]
\label{QPBTH}
Let $(A^{*,*},\partial)$ be a bigraded cohomology theory with an $\varepsilon$-commutative
partial multiplication or ring structure and
a symplectic Thom structure. Let
$(\mathcal U, \phi|_{\mathcal U})$ be the tautological rank $2$
symplectic subbundle over
the quaternionic projective bundle $HP(E, \phi)$, and let
$t = p(\mathcal U, \phi|_{\mathcal U})$
be its Pontryagin class. Then
we have an isomorphism of graded $A^{0}(X)$-modules
$$(1, t, \dots , t^{n-1}) \colon
A^{*,*}(X) \oplus A^{*,*}(X) \oplus \dots \oplus
A^{*,*}(X) \to A^{*,*}(HP_X(E, \phi))$$
and an isomorphism
of graded modules over $A^{0}(X) = \bigoplus_{p}A^{2p,p}(X)$\textup{:}
$$(1, t, \dots , t^{n-1}) \colon
A^{0}(X) \oplus A^{0}(X) \oplus \dots \oplus A^{0}(X) \to A^{0}(HP_X(E, \phi)).$$
\end{thm}
\begin{defn}
\label{PontriaginClasses}
Under the hypotheses of Theorem
\ref{QPBTH}
there are unique elements
$p_i(E, \phi) \in A^{4i,2i}(X)$ for $i=1,2, \dots , n$
such that
$$t^{n}-p_1(E, \phi)\cup t^{{n-1}} + p_2(E, \phi)\cup t^{{n-2}} - \dots + (-1)^n p_n(E, \phi)=0.$$
The classes
$p_i(E, \phi)$
are called the \emph{Pontryagin classes}
of $(E, \phi)$ with respect to the symplectic Thom structure of the cohomology theory $(A, \partial)$.
For $i > n$ and $i < 0$ one sets $p_i(E, \phi)$ = 0, and one sets $p_0(E, \phi) = 1$.
\end{defn}
The quaternionic projective bundle theorem has consequences the symplectic splitting principle and
the Cartan sum formula for Pontryagin classes \cite[Theorems 10.2, 10.5]{Panin:2010fk}.
With them one can compute the cohomology of quaternionic Grassmannians.
Let $e_{i}$ denote the $i^{\text{th}}$ elementary symmetric polynomial, and let $h_{i}$ be the
$i^{\text{th}}$ complete symmetric polynomial, the sum of all monomials of degree $i$.
There are formulas relating them, including the recurrence relation
$h_{k} + \sum_{i\geq 1} (-1)^{i}e_{i}h_{k-i} = 0$. Let $\Pi_{r,n-r}$ be the set of all partitions whose
Young diagrams fit inside an $r \times (n-r)$ box. (More formally these are partitions
$\lambda$ with $l(\lambda) =\lambda'_{1}\leq r$ and $\lambda_{1} \leq n-r$.)
Associated to any such partition is a
\emph{Schur polynomial}, which can be written in terms of the $e_{i}$.
\begin{thm}
\label{T:Grass}
For any bigraded ring
cohomology theory $A^{*,*}$ with an $\varepsilon$-commutative partial multiplication or ring structure and
a symplectic Thom structure and any $X$ the map
\begin{equation}
\label{E:HGr.cohom.1}
A^{*,*}(X)[e_{1},\dots,e_{r}] / (h_{n-r+1},\dots,h_{n}) \xrightarrow{\cong} A^{*,*}(\HGr(r,n) \times X)
\end{equation}
sending $e_{i} \mapsto p_{i}(\mathcal U_{r,n},\phi_{r,n})$ for all $i$
is an isomorphism, as is the map
\begin{equation}
\label{E:HGr.cohom.2}
A^{*,*}(X)^{\oplus \binom{n}{r}} \xrightarrow{(s_{\lambda}(\mathcal U_{r,n},\phi_{r,n}))_{\lambda\in\Pi_{r,n-r}}}
A^{*,*}(\HGr(r,n) \times X).
\end{equation}
\end{thm}
\begin{thm}
[\protect{\cite[Theorem 11.4]{Panin:2010fk}}]
\label{T:HGr.lim.cohom}
For any bigraded ring
cohomology theory $A^{*,*}$ with an $\varepsilon$-commutative partial multiplication or ring structure and
a symplectic Thom structure and any $X$ the $\alpha_{r,n}$ and
$\beta_{r,n}$ of \eqref{E:alpha.beta} induces split surjections
\begin{gather*}
(\alpha_{r,n} \times 1_{X})^{*} \colon A(\HGr(r,n+1) \times X) \twoheadrightarrow A(\HGr(r,n) \times X)
\\
(\beta_{r,n} \times 1_{X})^{*} \colon A(\HGr(r+1,n+1) \times X) \twoheadrightarrow A(\HGr(r,n) \times X)
\end{gather*}
which the isomorphisms \eqref {E:HGr.cohom.2} identify with the surjections
$A(X)^{\oplus \binom{n+1}{r}} \twoheadrightarrow A(X)^{\oplus \binom{n}{r}}$ and
$A(X)^{\oplus \binom{n+1}{r+1}} \twoheadrightarrow A(X)^{\oplus \binom{n}{r}}$
which are the identity on the summands
corresponding to $\lambda \in \Pi_{n,r}$ and
which vanish on the summands
corresponding to $\lambda \not \in \Pi_{n,r}$.
We have isomorphisms
\begin{gather}
\label{E:HGr(r,infty)}
A^{*,*}(X) [[p_{1},\dots,p_{r}]]^{\text{\textit{homog}}} \xrightarrow{\cong}
\varprojlim\limits_{n \to \infty} A^{*,*}(\HGr(r,n) \times X)
\\
\label{E:HGr(infty,2.infty)}
A^{*,*}(X)[[p_{1},p_{2}, p_{3}, \dots ]]^{\text{\textit{homog}}}
\xrightarrow{\cong}
\varprojlim
\limits_{n\to \infty}
A^{*,*}(\HGr(n,2n) \times X)
\end{gather}
with each variable $p_{i}$ sent to the inverse system of $i^{\text{th}}$ Pontryagin classes
$(p_{i}(\mathcal U_{r,n}))_{n \geq r}$ or
$(p_{i}(\mathcal U_{n,2n}))_{n\in \mathbb{N}}$.
\end{thm}
The notation on the left in \eqref {E:HGr(r,infty)}--\eqref {E:HGr(infty,2.infty)} is
the bigraded ring of homogeneous power series.
We have a simple lemma.
\begin{lem}
\label{BO(AwedgeB)andBO(AtimesB)}
If $A$ is a $T$-spectrum, then for any pointed motivic spaces $X$ and $Y$
the canonical map $X \times Y \to X\wedge Y$
induces a split injection
$A^{r,s}(X \wedge Y) \hra A^{r,s}(X \times Y)$.
The image of the injection coincides with the kernel of the map
$$[(id_X \times y)^*, (x \times id_Y^*)] \colon A^{r,s}(X \times Y) \to
A^{r,s}(X \times y) \oplus A^{r,s}(x \times Y).$$
\end{lem}
We write $[-n,n] = \{ i \in \mathbb{Z} \mid -n \leq i \leq n\}$. We have a sequential colimit of pointed spaces
\[
(\mathbb{Z} \times HGr, (0,x_{0})) = \operatornamewithlimits{colim} ( ([-n,n] \times HGr(n,2n), (0,x_{0}))
\]
to which Theorem \ref {T:varprojlim.a} applies. Theorem \ref{T:HGr.lim.cohom} and
Lemma \ref{BO(AwedgeB)andBO(AtimesB)} now give the following result.
\begin{thm}
\label{T:A(ZxHGr)}
Let $A$ be a $T$-spectrum whose associated cohomology theory $(A^{*,*},\partial)$ has an
$\varepsilon$-commutative partial multiplication or ring structure and a symplectic Thom structure.
Then the natural map
\[
A^{*,*}((\mathbb{Z} \times HGr),(0,x_{0})) \to \varprojlim A^{*,*}([-n,n] \times HGr(n,2n),(0,x_{0}))
\]
is an isomorphism. More generally for any $r$ and $s$ the natural map
\[
A^{*,*}((\mathbb{Z} \times HGr,(0,x_{0}))^{\wedge r} \wedge (HP^{1},x_{0})^{\wedge s})
\to \varprojlim A^{*,*}(([-n,n] \times HGr(n,2n),(0,x_{0}))^{\wedge r}\wedge (HP^{1},x_{0})^{\wedge s})
\]
is an isomorphism.
\end{thm}
We complete our review of parts of \cite{Panin:2010fk} by looking at the geometry of
$HP^{1} = HGr(1,\mathsf{H}^{\oplus 2})$.
\begin{thm}
\label{T:HP1.T2}
In $H_{\bullet}(S)$ we have a canonical isomorphism
$\eta \colon (HP^{1}_{+},pt) \cong T^{\wedge 2}$.
\end{thm}
\begin{proof}
By definition $HP^{1}$ is by the open subscheme of $Gr(2,4)$ parametrizing
$2$-dimensional symplectic subspaces of the $4$-dimensional trivial symplectic bundle.
It contains two distinguished points
$x_{0} = [\mathsf{H} \oplus 0 ]$ and
$x_{\infty} = [0 \oplus \mathsf{H}]$.
The $x_{\infty}$ is the origin of an open cell $\mathbf{A}^{4} \subset Gr(2,4)$ of the usual Grassmannian
consisting of subspaces with basis of the form $(y_{1},y_{2},1,0)$ and $(y_{3},y_{4},0,1)$.
Within the $\mathbf{A}^{4}$ there are two transversal loci $N^{+} \cong \mathbf{A}^{2}$ defined by $y_{2}=y_{4}=0$
and $N^{-} \cong \mathbf{A}^{2}$ defined by $y_{1} = y_{3}= 0$. They are closed in $HP^{1}$.
The complement $HP^{1}-N^{+}$ is the quotient of $\mathbf{A}^{5}$ by a free action of $\mathbb{G}_{a}$
\cite[Theorem 3.4]{Panin:2010fk}.
Consequently the structural map $HP^{1}-N^{+} \to pt$ and its section
$x_{0} \colon pt \to HP^{1}-N^{+}$ are motivic weak equivalences.
This gives us motivic weak equivalences
\begin{equation}
\label{E:T2=HP1}
\vcenter{
\xymatrix @M=5pt @C=17.5pt {
T^{\wedge 2} \cong N^{-}/(N^{-}-0) \ar[d]_-{\mathbf{A}}
\ar[r]^-{\text{2 out of 3}} \ar[rd]|-{\text{2 out of 3}}
&
HP^{1}/(HP^{1}-N^{+})
&
(HP^{1},x_{0}) \ar[l]_-{/\mathbf{A}}
\\
\mathbf{A}^{4}/(\mathbf{A}^{4}-N^{+}) \ar@{<-}[r]_-{\text{excision}}
&
(\mathbf{A}^{4}\cap HP^{1})/((\mathbf{A}^{4}\cap HP^{1})-N^{+}) \ar[u]_-{\text{excision}}
&
}}
\end{equation}
The zigzag on the top line is the canonical isomorphism in $H_{\bullet}(S)$.
\end{proof}
A symplectic bundle $(E,\phi)$ is naturally a special linear bundle $(E,\lambda_{\phi})$ with
$\lambda_{\phi}$ the inverse of the Pfaffian of $\phi$.
Hence special linear Thom classes of hermitian $K$-theory (Theorem \ref{T:SLc.oriented})
give $\mathbf{BO}^{*,*}$ a symplectic Thom structure.
So there are Pontryagin classes for symplectic bundles in hermitian $K$-theory,
and all the formulas of this section are valid for them.
We may compute the Pontryagin classes induced by the Thom classes of this particular
symplectic Thom structure. We need the isomorphism of \eqref {E:KSp}, which becomes the isomorphism
\begin{equation}
\label{E:GW-.isom}
\begin{array}{ccc}
GW^{-}(X) & \overset{\cong}{\lra} & KO_{0}^{[2]}(X)
\\
{}[X,\phi] & \longmapsto & -\bigl[ (X,\phi)[1] \bigr].
\end{array}
\end{equation}
The sign makes the isomorphism commute
with the forgetful maps to $K_{0}(X)$.
For a rank $2$ symplectic bundle $(E,\phi)$
has Pontryagin class $p_{1}(E,\phi) = -z^{*}\thom(E,\mathcal{O}_{X},\lambda_{\phi})$, which is the image under
the isomorphism of $[E,\phi]-[\mathsf{H}] \in GW^{-}(X)$. The symplectic splitting principle
\cite[Theorem 10.2]{Panin:2010fk}
now gives the following.
\begin{prop}
\label{P:p1.formula}
Let $(F,\psi)$ be a symplectic bundle of rank $2r$ on $X$. Its first Pontryagin class
$p_{1}(F,\psi) \in KO_{0}^{[2]}(X)$
is the image under the isomorphism \eqref{E:GW-.isom} of $[F,\psi]-r[\mathsf{H}] \in GW^{-}(X)$.
\end{prop}
Formulas for higher Pontryagin classes in terms of exterior powers of $(F,\psi)$ will
be given in \cite{Walter:2010ab}.
\section{The strategy for putting a ring structure on \texorpdfstring{$A^{*,*}$}{A\^{ }*,*}}
We explain our strategy for turning our partial multiplicative structure on $\mathbf{BO}^{*,*}$
into a full ring structure.
We will need the following standard facts about spectra.
Recall that a motivic space $X$ is \emph{small} if $Hom_{SH(S)}(\Sigma_{T}^{\infty}X,{-})$
commutes with arbitrary coproducts. We write $\mathbf{M}^{\text{\textit{small}}}_{\bullet}(S)$ for the full subcategory of
small motivic spaces.
\begin{thm}
[\protect{\cite[Lemma A.34]{Panin:2009aa}}]
\label{T:varprojlim.a}
Let $D^{(0)} \to D^{(1)} \to D^{(2)} \to \cdots$ be a sequence of morphisms in $SH(S)$ with
$\hocolim D^{(i)} = D$, let $X$ be a small motivic space, and let $A$ be a $T$-spectrum. Then
there is a canonical isomorphism
\[
Hom_{SH(S)}(\Sigma_{T}^{\infty}X,D) = \operatornamewithlimits{colim} Hom_{H_{\bullet}(S)}(X,D^{(n)}).
\]
and a canonical short exact sequence:
\[
0 \to \varprojlim\nolimits^{1} A^{p-1,q}(D^{(i)}) \to A^{p,q}(D) \to
\varprojlim A^{p,q}(D^{(i)}) \to 0.
\]
\end{thm}
Particular cases of this are the following.
\begin{thm}
[\protect{\cite[Theorem 5.2]{Panin:2010fk}}]
\label{T:small.colim}
Let $X$ be a small motivic space and $A$ a $T$-spectrum. Then we have
\[
Hom_{SH(S)}(\Sigma_{T}^{\infty}X,A) = \operatornamewithlimits{colim} Hom_{H_{\bullet}(S)}(X \wedge T^{\wedge n}, A_{n}).
\]
\end{thm}
\begin{thm}
[\protect{\cite[Corollaries 3.4, 3.5]{Panin:2009aa}}]
\label{T:varprojlim.b}
Let $A$ and $E$ be $T$-spectra. Then for any $r$
there is a canonical short exact sequences
\begin{gather*}
0 \to \varprojlim\nolimits^{1} A^{*-2rn-1,*-rn}(E_{n}^{\wedge r}) \to A^{*,*}(E^{\wedge r}) \to
\varprojlim A^{*-2rn,*-rn}(E_{n}^{\wedge r}) \to 0.
\end{gather*}
\end{thm}
\begin{defn}
\label{D:almost.monoid}
An \emph{almost commutative monoid} in $SH(S)$ is a triple $(A,\mu,e)$ with $A$ a $T$-spectrum and
$\mu \colon A \wedge A \to A$ and $e \colon \Sigma_{T}^{\infty}\boldsymbol{1}\to A$ morphisms
in $SH(S)$ such that
\begin{enumerate}
\item the morphism $\mu \circ (\mu \wedge 1) - \mu \circ (1 \wedge \mu) \in Hom_{SH(S)}(
A \wedge A \wedge A, A)$ lies in the subgroup
$\varprojlim^{1} A^{*-6n-1,*-3n}(A_{n}\wedge A_{n}\wedge A_{n})$,
\item for $\sigma \colon A \wedge A \to A \wedge A$ the morphism switching the two factors,
the morphism $\mu - \mu \circ \sigma \in \operatorname{Hom}_{SH(S)}( A \wedge A,A)$
lies in the subgroup
$\varprojlim^{1} A^{*-4n-1,*-2n}(A_{n}\wedge A_{n})$,
\item the map $1 - \mu \circ (1 \wedge e) \in \operatorname{Hom}_{SH(S)}(A,A)$ lies in the subgroup
$\varprojlim^{1} A^{*-2n-1,*-n}( A_{n})$.
\end{enumerate}
\end{defn}
An almost commutative monoid $(A,\mu,e)$ defines pairings
\begin{equation}
\label{E:m.pairing.1}
\times \colon A^{p,q}(X) \times A^{r,s}(Y) \to
A^{p+r,q+s}(X \wedge Y)
\end{equation}
%
for $X$ and $Y$ in $\mathbf{M}_{\bullet}(S)$ as follows. For
$\alpha: \Sigma_{T}^{\infty}X \to A \wedge S^{p,q} $ and $\beta: \Sigma_{T}^{\infty}Y \to A \wedge S^{r,s} $ define
$\alpha \times \beta \in A^{p+r,q+s}(X \wedge Y)$
as the composition
\begin{equation*}
\Sigma_{T}^{\infty}(X \wedge Y) \cong \Sigma_{T}^{\infty}X \wedge \Sigma_{T}^{\infty}Y \xra{\alpha \wedge \beta}
A \wedge S^{p,q} \wedge A \wedge S^{r,s} \cong A \wedge A \wedge S^{p+r,q+s}
\xra{m \wedge 1 }A \wedge S^{p+r,q+s}.
\end{equation*}
The unit $e\in Hom_{SH(S)}(\Sigma_{T}^{\infty}pt_{+},A)$ defines an element $1\in A^{0,0}(pt_{+})$.
There is then a unique element $\varepsilon \in A^{0,0}(pt_{+})$ such that
$\Sigma_{T}\varepsilon \in Hom_{SH(S)}(T, A\wedge T)$ is the composition of the endomorphism
of $T$ induced by the endomorphism $x \mapsto -x$ of $\mathbf{A}^{1}$ with $e \wedge 1_{T}$.
\begin{thm}
\label{T:almost.commutative.product}
For an almost commutative monoid $(A,\mu,e)$ in $SH(S)$ the cohomology theory $(A^{*,*},\partial)$
with the pairing $\times$ of \eqref{E:m.pairing.1} and the element $1 \in A^{0,0}(pt_{+})$ form
an $\varepsilon$-commutative ring cohomology theory on $\mathbf{M}_{\bullet}(S)^{\text{\textit{small}}}$ and on $\mathcal Sm\mathcal Op/S$.
\end{thm}
\begin{proof}
There are canonical elements $a_{n} \colon \Sigma_{T}^{\infty}A_{n}(-n) \to A$. The definition of an
almost commutative monoid is equivalent to having $(A,\mu,e)$ such that the induced pairing satisfies
$(a_{n} \times a_{n}) \times a_{n} = a_{n} \times (a_{n} \times a_{n})$ and
$\sigma^{*}(a_{n} \times a_{n}) = \varepsilon^{n} a_{n} \times a_{n}$ and $a_{n} \times e = a_{n}$
for all $n$. It then also satisfies
\[
(\Sigma^{p,q}a_{n} \times \Sigma^{p',q'}a_{n}) \times \Sigma^{p'',q''}a_{n} =
\Sigma^{p,q}a_{n} \times (\Sigma^{p',q'}a_{n} \times \Sigma^{p'',q''}a_{n})
\]
for all $(p,q)$, $(p',q')$ and $(p'',q'')$.
By Theorem \ref{T:small.colim} for a small motivic space $X$ any morphism
$\Sigma_{T}^{\infty}X \to A\wedge S^{p,q}$ factors as
\[
\Sigma_{T}^{\infty}X \to \Sigma_{T}^{\infty}A_{n}(-n) \wedge S^{p,q}
\xra{\Sigma^{p,q}a_{n}} A \wedge S^{p,q}
\]
for some $n$. So on small motivic spaces the multiplication is associative.
The $\varepsilon$-commutativity and the unit property are treated similarly.
The signs in the commutativity come from permuting the spheres.
\end{proof}
\section{The universal elements}
\label{S:universal}
The isomorphism $\tau \colon (\mathbb{Z} \times HGr, (0,x_{0})) \xra{\cong} \mathbf{KSp}$ of Theorem
\ref{T:MV.symp} is classified by an element which has restrictions
\begin{equation}
\label{E:tau.formula}
\tau |_{\{i\} \times HGr(n,2n)}
= [\mathcal U_{n,2n},\phi_{n,2n}]+(i-n)[\mathsf{H}] \in KSp_{0}(HGr(n,2n)).
\end{equation}
Its image under the isomorphism \eqref {E:GW-.isom} is a class
$\tau_{2} \in \mathbf{KO}^{[2]}_{0}(\mathbb{Z} \times HGr,(0,x_{0}))$ with
\begin{equation*}
\tau_{2} |_{\{i\} \times HGr(n,2n)}
= p_{1}(\mathcal U_{n,2n},\phi_{n,2n})+i\mathsf{h}\in \mathbf{KO}_{0}^{[2]}(HGr(n,2n))
\end{equation*}
according to Proposition \ref{P:p1.formula}.
Here $\mathsf{h} \in \mathbf{KO}^{[2]}_{0}(pt) = \mathbf{BO}^{4,2}(pt)$ is the class corresponding to $[\mathsf{H}] \in GW^{-}(pt)$
under the isomorphism. By Theorem \ref{T:HGr.lim.cohom} we have an isomorphism
\[
\mathbf{BO}^{*,*}(\mathbb{Z} \times \HGr)
\cong \prod_{i \in \mathbb{Z}}\mathbf{BO}^{*,*}(pt)[[p_{1},p_{2}, p_{3}, \dots ]]^{\text{\textit{homog}}}
\]
with the product taken in the category of graded rings. Setting
\begin{align*}
p_{1} = (p_{1})_{i \in \mathbb{Z}} \in \mathbf{BO}^{*,*}(\mathbb{Z} \times HGr),
&&
\tfrac 12 \operatorname{rk} = (i1_{\mathbf{BO}})_{i \in \mathbb{Z}}\in \mathbf{BO}^{*,*}(\mathbb{Z} \times HGr)
\end{align*}
we see we have $\tau_{2} = p_{1} + (\frac 12 \operatorname{rk}) \mathsf{h}$.
For any $k$ we have a composition of isomorphisms
in the homotopy category $H_{\bullet}(S)$
\[
(\mathbb{Z} \times HGr, (0,x_{0})) \xra{\tau} \mathbf{KSp} \xrightarrow{trans_{2k+1}}
\mathbf{KO}^{[4k+2]}
\xrightarrow{-1} \mathbf{KO}^{[4k+2]}
\]
where the $trans_{2k+1}$ comes from translation
and the $-1$
is the inverse operation of the $H$-space structure, which we add as in \eqref{E:KSp}
so that the forgetful maps to $K$-theory should commute up to homotopy.
(The inverse in the $H$-space structure comes from the
$\Omega_{T}$-spectrum structure and to the authors' knowledge not
from a duality-preserving functor.)
\begin{defn}[Universal element]
\label{D:tau}
We denote by
\begin{equation*}
\tau_{4k+2} \in \mathbf{KO}_{0}^{[4k+2]}(\mathbb{Z} \times HGr, (0,x_{0}))
\cong \mathbf{BO}^{8k+4,4k+2}(\mathbb{Z} \times HGr, (0,x_{0}))
\end{equation*}
the element corresponding to the composition. It is given by
$\tau_{4k+2} = (p_{1}+ (\tfrac 12 \operatorname{rk})\mathsf{h}) \cup \beta_{8}^{k}$.
\end{defn}
We write $[-n,n] = \{ i \in \mathbb{Z} \mid -n \leq i\leq n\}$. The class
$\tau \in GW^{-}(\mathbb{Z} \times HGr)$ giving the isomorphism $\mathbb{Z} \times HGr \cong \mathbf{KSp}$
has a universal property.
\begin{lem}
\label{L:mu}
There are unique $\mu$ and $\mu_{8k+4}$ in $H_{\bullet}(S)$ making the diagram commute
\[
\xymatrix @M=5pt @C=40pt {
(\mathbb{Z} \times HGr,(0,x_{0})) \wedge (\mathbb{Z} \times HGr,(0,x_{0}))
\ar[r]^-{\mu}
\ar[d]_-{\tau_{4k+2}\wedge \tau_{4k+2}}^-{\cong}
& \mathbf{KO}^{[0]} \ar[d]^-{\text{\textup{translation}}} _-{\cong}
\\
\mathbf{KO}^{[4k+2]}\wedge \mathbf{KO}^{[4k+2]}
\ar[r]_-{\mu_{8k+4}}
&
\mathbf{KO}^{[8k+4]}
}
\]
and such that for each $i$, $j$ and $n$ the restriction of $\mu$
\[
(\{i\} \times HGr(n,2n)) \times (\{j\} \times HGr(n,2n)
\to \mathbf{KO}^{[0]}
\]
is the class in $H_{\bullet}(S)$ of the morphisms of ind-schemes
represing the orthogonal Grothendieck-Witt class which is
\[
([\mathcal U_{n,2n},\phi_{n,2n}]+(i-n)[\mathsf{H}]) \boxtimes ([\mathcal U_{n,2n},\phi_{n,2n}]+(j-n)[\mathsf{H}]).
\]
\end{lem}
The $\mu$ with the asserted restrictions exists and is unique because of Theorem \ref {T:A(ZxHGr)}.
It factors through the wedge space because of Lemma \ref {BO(AwedgeB)andBO(AtimesB)}.
Then $\mu_{8k+4}$ is the map making the diagram commute.
\begin{lem}
\label{L:mu.compatible}
The following diagram commutes in $H_{\bullet}(S)$.
\[
\xymatrix @M=5pt @C=40pt {
\mathbf{KO}^{[4k-2]}\wedge \mathbf{KO}^{[4k-2]} \wedge T^{\wedge 8}
\ar[r]^-{\mu_{8k-4} \wedge 1}
\ar[d]_-{\tau_{4k+2}\wedge \tau_{4k+2}}^-{\cong}
& \mathbf{KO}^{[8k-4]} \wedge T^{\wedge 8}\ar[dd]^-{\text{\textup{structure maps}}}
\\
\mathbf{KO}^{[4k-2]}\wedge T^{\wedge 4}\wedge \mathbf{KO}^{[4k-2]} \wedge T^{\wedge 4}
\ar[d]^{\text{\textup{structure maps}}}
\\
\mathbf{KO}^{[4k+2]}\wedge \mathbf{KO}^{[4k+2]}
\ar[r]_-{\mu_{8k+4}}
&
\mathbf{KO}^{[8k+4]}
}
\]
\end{lem}
\begin{proof}
Because of Theorem \ref {T:A(ZxHGr)} it is enough to observe that the compositions
of the two paths of arrows with the composition
\[
\xymatrix @M=5pt @C=20pt {
(\{i\} \times HGr(n,2n)) \times (\{j\} \times HGr(n,2n)) \times (HP^{1},x_{0})^{\times 4}
\ar@{^{(}->}[d]
\\
(\mathbb{Z} \times HGr) \times (\mathbb{Z} \times HGr) \times (HP^{1},x_{0})^{\times 4}
\ar[r]^-{\tau_{4k-2}\wedge \tau_{4k-2}\wedge \eta^{\wedge 4}} _-{\cong}
& \mathbf{KO}^{[4k-2]}\wedge \mathbf{KO}^{[4k-2]} \wedge T^{\wedge 8}
}
\]
are the same for all $i$, $j$ and $n$.
\end{proof}
By Theorem \ref {T:varprojlim.b}
we have a surjection
\[
Hom_{SH(S)}(\mathbf{BO} \wedge \mathbf{BO}, \mathbf{BO}) \to
\varprojlim\mathbf{BO}^{16k+8,8k+4}(\mathbf{KO}^{[4k+2]}\wedge \mathbf{KO}^{[4k+2]}) \to 0
\]
the compositions
\[
\Sigma_{T}^{\infty}(\mathbf{KO}^{[4k+2]}\wedge \mathbf{KO}^{[4k+2]})(-8k-4) \xrightarrow{\mu_{4k+2}} \Sigma_{T}^{\infty}\mathbf{KO}^{[8k+4]}(-8k-4) \xrightarrow{\text{\textup{canonical}}} \mathbf{BO}
\]
form a system of elements of the groups in the inverse limit. They are compatible with the connecting
maps of the inverse limit because the diagrams of Lemma \ref {L:mu.compatible} commute.
Let
\begin{equation}
\label{E:bar.m}
\bar m = (\mu_{8k+4})_{k \in \mathbb{N}}\in \varprojlim\mathbf{BO}^{16k+8,8k+4}(\mathbf{KO}^{[4k+2]}\wedge \mathbf{KO}^{[4k+2]})
\end{equation}
and let
\begin{equation}
\label{E:m}
m \in Hom_{SH(S)}(\mathbf{BO}\wedge \mathbf{BO})
\end{equation}
be an element lifting it. As in \eqref {E:m.pairing.1} $m$ defines a pairing
\begin{equation}
\label{E:m.pairing}
\times \colon \mathbf{BO}^{p,q}(A) \times \mathbf{BO}^{r,s}(B) \to
\mathbf{BO}^{p+r,q+s}(A \wedge B).
\end{equation}
%
\begin{thm}
\label{T:coincide}
Suppose $S$ satisfies the hypotheses of Theorem \ref{T:lim1}.
For $X$ and $Y$ in $\mathcal Sm/S$ and all $p$ and $q$ the pairing
\begin{align*}
\mathbf{BO}^{4p,2p}(X_{+}) \times \mathbf{BO}^{4q,2q}(Y_{+}) &
\to \mathbf{BO}^{4p+4q,2p+2q}(X_{+} \wedge Y_{+})
\\
\intertext{induced by $m$ is identified via the isomorphism $\gamma$ of Corollary \ref{KO**andBO**}
with the naive pairing}
KO_{0}^{[2p]}(X) \times KO_{0}^{[2q]}(Y) & \to KO_{0}^{[2p+2q]}(X \times Y).
\end{align*}
\end{thm}
\begin{proof}
Because of Jouanolou's trick it is enough to consider affine $X$ and $Y$.
Let $\alpha \in KO_{0}^{[2p]}(X)$ and $\beta \in KO_{0}^{[2q]}(Y)$.
When we have
$2p = 2q = 4k+2$ and $X = Y = [-n,n] \times HGr(n,2n)$ and
$\alpha = \beta$ is the restriction of the universal class $\tau_{4k+2}$, then we have the identification
by the construction of $m$.
When we have $2p=2q=4k+2$ but $X$, $Y$, $\alpha$ and $\beta$ are general then by Theorem
\ref{T:universal} there exists an $n$ and morphisms $f_{\alpha} \colon X \to [-n,n] \times HGr(n,2n)$
and $f_{\beta} \colon Y \to [-n,n] \times HGr(n,2n)$ in $\mathcal Sm/S$
such that
$f_{\alpha}^{*}(\tau_{4k+2} |_{[-n,n] \times HGr(n,2n)}) = \alpha$
and
$f_{\beta}^{*}(\tau_{4k+2} |_{[-n,n] \times HGr(n,2n)}) = \beta$. The
identification of the two pairings for $\alpha$ and $\beta$ now follows from that for the restrictions
of the universal classes because both pairings are functorial for morphisms in $\mathcal Sm/S$.
For general $p$ and $q$ pick $4k+2 \geq \max(2p,2q)$. The identification of the two pairings on
$\alpha$ and $\beta$ follows from the identification of the two pairings on
\begin{gather*}
\alpha \times p_{1}(U,\phi)^{\times 2k+1-p} \in KO_{0}^{[4k+2]}(X \times (HP^{1})^{\times 2k+1-p}),
\\
\beta \times p_{1}(U,\phi)^{\times 2k+1-q} \in KO_{0}^{[4k+2]}(Y \times(HP^{1})^{\times 2k+1-q}).
\qedhere
\end{gather*}
\end{proof}
\begin{thm}
\label{monoidBO}
Let $m$ be as in \eqref {E:m}, and
let $e \in \mathbf{BO}^{0,0}(pt_{+})$
be the unit of the partial multiplicative structure on
$(\mathbf{BO}^{*,*},\partial)$
of Definition \ref{partialmultonBO**}.
Then
\( (\mathbf{BO}, m, e)\)
is an almost commutative monoid in $SH(S)$, and the $\times$ and $1$ induced by $m$ and $e$
make $(\mathbf{BO}^{*,*},\partial,\times,1)$ a $\angles {-1}$-commutative bigraded ring cohomology theory on
$\mathbf{M}_{\bullet}^{\text{\textit{small}}}(S)$ and on $\mathcal Sm\mathcal Op/S$.
\end{thm}
\begin{proof}
We prove associativity. The morphisms
\[
\xymatrix @M=5pt @C=60pt {
\mathbf{BO} \wedge \mathbf{BO} \wedge \mathbf{BO}
\ar@<4pt>[r]^-{m \circ (m \wedge id_{\mathbf{BO}} )}
\ar@<-4pt>[r]_-{m \circ (id_{\mathbf{BO}} \wedge m)}
&
\mathbf{BO}
}
\]
define two elements of $\mathbf{BO}^{0,0}(\mathbf{BO} \wedge \mathbf{BO} \wedge \mathbf{BO})$
with images in
\[
\varprojlim \mathbf{BO}^{24k+12,12k+6}(\mathbf{KO}^{[4k+2]} \wedge \mathbf{KO}^{[4k+2]} \wedge
\mathbf{KO}^{[4k+2]}).
\]
This last group is isomorphic to
\[
\varprojlim \mathbf{BO}^{24k+12,12k+6}((\mathbb{Z} \times HGr,(0,x_{0}))^{\wedge 3})
\]
So it is enough to show that the corresponding elements
$(\tau_{4k+2} \boxtimes \tau_{4k+2}) \boxtimes \tau_{4k+2}$ and
$\tau_{4k+2} \boxtimes (\tau_{4k+2} \boxtimes \tau_{4k+2})$
in each group of the inverse system are
equal. However, since we have
\[
\mathbf{BO}^{24k+12,12k+6}((\mathbb{Z} \times HGr,(0,x_{0}))^{\wedge 3})
\cong
\varprojlim \mathbf{BO}^{24k+12,12k+6}(([-n,n] \times HGr(n,2n),(0,x_{0}))^{\wedge 3})
\]
by Theorem \ref{T:A(ZxHGr)} it is enough to show that the restrictions to the
smooth affine schemes
$[-n,n] \times HGr(n,2n)$ are equal. But then by Theorem \ref{T:coincide} the
pairings coincide with the naive products, and those are associative.
The proofs of commutativity and of the unit property are similar.
Thus $(\mathbf{BO},m,e)$ is an almost commutative monoid in $SH(S)$. The rest of the statement
of the theorem follows from that fact by Theorem \ref{T:almost.commutative.product}
except for the value of $\varepsilon$. For that note that $\Sigma_{T}1 \in \mathbf{BO}^{2,1}(T)$ is
the class in $GW^{[1]}(\mathbf{A}^{1},\mathbf{A}^{1}-0)$ of the symmetric complex $(K(\mathcal{O}),\varTheta(\mathcal{O}))$
of \eqref{E:thom.A1}. The pullback of the complex along the endomorphism $x \mapsto -x$
of $\mathbf{A}^{1}$ is isometric to $(K(\mathcal{O}),-\varTheta(\mathcal{O}))$. So we have $\varepsilon = \angles{-1}$.
\end{proof}
\begin{thm}
\label{T:unique.prod}
Suppose $\times$ and $\times'$ are two products on $(\mathbf{BO}^{*,*},\partial)$ on $\mathbf{M}_{\bullet}^{\text{\textit{small}}}(S)$
which associative and $\angles{-1}$-commutative with unit $1 = \angles 1$, and such that
$\alpha \times \Sigma_{\mathbb{G}_{m}}1 = \Sigma_{\mathbb{G}_{m}}\alpha$ and $\alpha \times \Sigma_{S^{1}_{s}}1
= \Sigma_{S^{1}_{s}}\alpha$
for all $\alpha$. If $\times$ and $\times'$
are both compatible
with the products $KO^{[2m]}_{0}(X) \times KO^{[2n]}_{0}(Y) \to KO^{[m+n]}_{0}(X \times Y)$
induced on Grothendieck-Witt groups of smooth schemes by the tensor product,
then we have $\times = \times '$.
\end{thm}
\begin{proof}
Suppose first that we have $\alpha \in \mathbf{BO}^{2i,i}(A)$ and $\beta \in \mathbf{BO}^{2j,j}(B)$ with $A$ and $B$
small pointed motivic spaces. By Theorems \ref{T:varprojlim.a} and \ref{T:small.colim}
there exist $m$ and $n$ such that
$\Sigma_{T}^{4m+2-i}\alpha \in \mathbf{BO}^{8m+4,4m+2}(A \wedge T^{\wedge 2m})$
has a factorisation
\[
A \wedge T^{\wedge 4m+2} \to ([-n,n] \times HGr(n,2n),(0,x_{0})) \hra \mathbb{Z} \times HGr \cong
\mathbf{KO}^{[4m+2]} \to \mathbf{BO}(4m+2)
\]
and such that $\Sigma_{T}^{4m+2-j}\beta$ has a similar factorization. Since $\Sigma_{T}^{4m+2-i}\alpha$
and $\Sigma_{T}^{4m+2-j}\beta$ are thus pullbacks of classes on which $\times$ and $\times'$ agree,
the products agree on $\Sigma_{T}^{4m+2-i}\alpha$
and $\Sigma_{T}^{4m+2-j}\beta$.
The compatibility of the products with the suspension and their $\varepsilon$-commutativity imply that
we also have
$\Sigma_{T}^{8m+4-i-j}(\alpha \times \beta) = \Sigma_{T}^{8m+4-i-j}(\alpha \times' \beta)$.
Since the suspension operation is a bijection on cohomology groups, we have
$\alpha \times \beta = \alpha \times' \beta$.
For $\alpha \in \mathbf{BO}^{p,q}(A)$ and $\beta \in \mathbf{BO}^{p',q'}(B)$ with for instance
$p<2q$ and $p' >2q'$ the products
agree on
$\Sigma_{S^{1}_{s}}^{2q-p}\alpha \in \mathbf{BO}^{2q,q}(A)$ and
$\Sigma_{\mathbb{G}_{m}}^{p'-2q'}\beta \in \mathbf{BO}^{2p'-2q',p'-q'}(B)$ by the previous case. By the same sort of argument they also agree on $\alpha$ and $\beta$. The other cases
are similar.
\end{proof}
\begin{thm}
The assertions of Theorem \ref{uniq2} and \ref{uniq1} hold.
\end{thm}
This follows from Theorems \ref {monoidBO} and \ref{T:unique.prod}.
\section{Spectra of finite and infinite real and quaternionic Grassmannians}
\label{S:finite}
We now wish to switch our sphere from $T$ to the pointed $HP^{1}$. This is possible because
the pointed $HP^{1}$ is isomorphic to $T^{\wedge 2}$ in $H_{\bullet}(S)$. The result is a spectrum
$\mathbf{BO}_{HP^{1}}$ very similar to $\mathbf{BO}$ except that the structural maps are induced by a product
with the Pontryagin class of a symplectic bundle on a pointed scheme rather than a Thom class
on a Thom space. This is important because the universal property of $\mathbb{Z} \times HGr$
deals with Grothendieck-Witt classes of bundles on schemes not of chain complexes on an $X$
acyclic over $U$.
The zigzag \eqref{E:T2=HP1} also gives us an equivalence between the stable homotopy categories
$SH_{T^{\wedge 2}}(S)$ and $SH_{(HP^{1},x_{\infty})}(S)$
\cite[Proposition 2.13]{Jardine:2000aa}.
There is also a Quillen equivalences between $Spt_{T}(S)$ and $Spt_{T^{\wedge 2}}(S)$
given by the forgetful functor and its adjoints.
\begin{thm}
The stable homotopy categories of $T$-spectra and of $(HP^{1},x_{0})$-spectra are
equivalent.
\end{thm}
The class $-p_{1}(\mathcal U_{HP^{1}},\phi_{HP^{1}}) \in
\mathbf{KO}_{0}^{[2]}(HP^{1},x_{\infty})$ corresponds to
$\thom^{\times 2} \in \mathbf{KO}_{0}^{[2]}(\mathbf{A}^{2}/\mathbf{A}^{2}-0)$ under the identifications.
So we may define our new spectrum.
\begin{defn}
\label{D:BO.HP1}
The $HP^{1}$-spectrum $\mathbf{BO}_{HP^{1}}$ corresponding to the $T$-spectrum $\mathbf{BO}$
has spaces
$(\mathbf{KO}^{[0]}, \mathbf{KO}^{[2]}, \mathbf{KO}^{[4]},\dots)$ and bonding maps
adjoint to the maps
\[
{-} \times -p_{1}(\mathcal U_{HP^{1}},\phi_{HP^{1}}) \colon \mathbf{KO}^{[2n]}({-}) \to
\mathbf{KO}^{[2n+2]}({-} \wedge (HP^{1},x_{\infty}))
\]
\end{defn}
The purpose of this section is to prove the following result. For parallelism with the labelling of
real and complex Grassmannians, we write
$HGr'(2r,2n) = HGr(r,n)$ so that $RGr(2r,2n)$ and $HGr'(2r,2n)$ are both open subschemes
of the ordinary Grassmannian $Gr(2r,2n)$ where a certain bilinear form is nondegenerate.
We also write
\[
[-n,n]' = \{ i \in \mathbb{Z} \mid \text{$-n \leq i \leq n$ and $i \equiv n \bmod 2$} \}.
\]
For even $n$ the scheme $[-n,n]' \times RGr(n,2n)$ is pointed in the component corresponding
to $0 \in [-n,n]'$ by the point corresponding to
$\mathsf{H}_{+}^{\oplus n/2} \oplus 0 \subset \mathsf{H}_{+}^{\oplus n}$. For odd $n$
we either do not use a base point or we use a disjoint base point. To compactify our notations
we write
\begin{align}
HGr'_{2n} & = [-n,n] \times HGr'(2n,4n),
\\
RGr_{2n} & = ([-2n,2n]' \times RGr(2n,4n)) \cup ([-2n+1,2n-1]' \times RGr(2n-1,4n-2)).
\end{align}
The first step in the proof of this theorem is the following general construction.
For $(X,x)$ a pointed scheme over $S$, let $(X,x)^{+}$ be the pushout of
\begin{equation}
\label{E:X+}
\xymatrix @M=5pt @C=30pt {
\mathbf{A}^{1}
& pt \ar[r]^-{x} \ar[l]^-{\sim}_-{0}
& X
}
\end{equation}
pointed by $1 \in \mathbf{A}^{1}(pt)$. (This is essentially the $\mathbf{A}^{1}$ mapping cone of the inclusion
$x \colon pt \to X$.)
The natural projection $(X,x)^{+} \to (X,x)$ which is the identity on $X$ and sends $\mathbf{A}^{1} \to x$
is a motivic weak equivalence.
We abbreviate $HP^{1+} = (HP^{1},x_{0})^{+}$. We will actually consider $HP^{1+}$-spectra.
The natural functor
$SH_{(HP^{1},x_{0})}(S) \to SH_{(HP^{1+}}(S)$ is an equivalence, and let $\mathbf{BO}_{HP^{1+}}$
be the $(HP^{1},x_{0})^{+}$-spectrum corresponding to $\mathbf{BO}$.
\begin{thm}
\label{T:finite}
There are $HP^{1+}$-spectra
$\mathbf{BO}^{\text{\textit{fin}}}$ and $\mathbf{BO}^{\text{\textit{geom}}}$ which are isomorphic to
$\mathbf{BO}_{HP^{1+}}$ in $SH_{HP^{1+}}(S)$
with spaces
\begin{gather}
\mathbf{BO}^{\text{\textit{fin}}}_{2i} =
\begin{cases}
RGr_{2\cdot 8^{i}} & \text{for even $i$}, \\
HGr'_{2\cdot 8^{i}} & \text{for odd $i$},
\end{cases}
\\
\mathbf{BO}^{\text{\textit{geom}}}_{2i} =
\begin{cases}
\mathbb{Z} \times RGr & \text{for even $i$}, \\
\mathbb{Z} \times HGr & \text{for odd $i$},
\end{cases}
\end{gather}
which are unions of real and quaternionic Grassmannians.
The bonding maps $\mathbf{BO}^{*}_{2i} \wedge HP^{1+} \to \mathbf{BO}_{2i+2}^{*}$ with $* = \text{\textit{fin}}$ or $\text{\textit{geom}}$
are morphisms of schemes and ind-schemes, respectively, which are constant on
$\mathbf{BO}^{*}_{2i} \vee HP^{1+}$.
\end{thm}
\begin{prop}
\label{P:A1.homotopy}
Let $(X,x)$, $(Y,y)$ and $(Z,z)$ be pointed schemes. There is a natural bijection between
\begin{enumerate}
\item
the set of morphisms of pointed schemes
$g \colon (X,x) \times (Y,y)^{+} \to (Z,z)$ which restrict to the constant map
$\bigl( X \times 1 \bigr) \cup \bigl( x \times (Y,y)^{+} \bigr) \to z$, and
\item
the set of pairs
$(f,h)$ where
\begin{enumerate}
\item
$f \colon (X \times Y, x \times y) \to (Z,z)$ is a morphism of schemes
which restricts to the constant map $x \times Y \to z$ and
\item
$h \colon (X,x) \times \mathbf{A}^{1} \to (Z,z)$ is a pointed $\mathbf{A}^{1}$-homotopy between
$f |_{X \times y}$ and the constant map $X \to z$.
\end{enumerate}
\end{enumerate}
\end{prop}
The idea is that
$(X,x)\times (Y,y)^{+}$ is the union of two subschemes, and the restrictions
of $g$ to these subschemes are
$g |_{X \times Y} = f$ and $g|_{X \times \mathbf{A}^{1}} = h$.
Let $U_{2n}$ and $U$ be the tautological symplectic subbundles on $HGr'(2n,4n)$ and
$HP^{1} = HGr'(2,4)$ resp., and let $V_{16n}$ be the tautological orthogonal subbundle
on $RGr(16n,24n)$. Let $\mathsf{H}$ be the trivial rank $2$ symplectic bundle, and let
$\mathsf{H}_{+}$ be the trivial rank $2$ orthogonal bundle for the split quadratic form
$q(x_{1},x_{2}) = x_{1}x_{2}$.
\begin{lem}
\label{L:HGr.HP1.to.RGr}
There
exist morphisms of pointed schemes
\[
f_{2n} \colon \bigl( [-n,n] \times HGr'(2n,4n) \bigr) \times HP^{1} \to RGr(16n,32n)
\]
such that the Grothendieck-Witt classes satisfy
\begin{equation}
\label{E:GW.formula}
f_{2n}^{*}([V_{16n}]-8n[\mathsf{H}_{+}]) = ([U_{2n}]-(n-i)[\mathsf{H}]) \boxtimes ([U]-[\mathsf{H}])
\end{equation}
\parens{where $i \in [-n,n] \subset \mathbb{Z}$ is the index of the component}
and such that
$f_{2n}|_{pt \times HP^{1}}$
is constant, and
$f_{2n}|_{([-n,n] \times HGr'(2n,4n)) \times pt}$
is pointed $\mathbf{A}^{1}$-homotopic to a constant map.
These maps and homotopies are
compatible with the inclusions $HGr'(2n,4n) \hra HGr'(2(n+1),4(n+1))$
and $RGr(16n,32n) \hra RGr(16(n+1),32(n+1))$.
\end{lem}
\begin{proof}
There are orthogonal direct sums $U_{2n} \oplus U_{2n}^{\perp} \cong \mathsf{H}^{\oplus 2n}$
and $U \oplus U^{\perp} \cong \mathsf{H}^{\oplus 2}$. Consequently we have
\begin{equation*}
([U_{2n}]-(n-i)[\mathsf{H}]) \boxtimes ([U]-[\mathsf{H}])
=
[U_{2n} \boxtimes U]
+ [\mathsf{H}^{\oplus n-i} \boxtimes U^{\perp}]
+ [U_{2n}^{\perp} \boxtimes \mathsf{H}]
- (6n-2i) [\mathsf{H}_{+}].
\end{equation*}
We have inclusions of orthogonal bundles
\begin{align*}
U_{2n} \boxtimes U
& \subset \mathsf{H}^{\oplus 2n} \boxtimes U,
&
U_{2n}^{\perp} \boxtimes \mathsf{H}
& \subset \mathsf{H}^{\oplus 2n} \boxtimes \mathsf{H},
\\
\mathsf{H}^{\oplus n-i} \boxtimes U^{\perp}
& \subset
\mathsf{H}^{\oplus 2n} \boxtimes U^{\perp},
&
\mathsf{H}_{+}^{\oplus 2n+2i}
& \subset
\mathsf{H}_{+}^{\oplus 4n}
\end{align*}
Consequently the orthogonal subbundle
\begin{equation}
\label{E:subbundle}
(U_{2n} \boxtimes U)
\oplus (\mathsf{H}^{\oplus n-i} \boxtimes U^{\perp})
\oplus (U_{2n}^{\perp} \boxtimes \mathsf{H})
\oplus \mathsf{H}_{+}^{\oplus 2n+2i}
\end{equation}
of
\[
\bigl( \mathsf{H}^{\oplus 2n} \boxtimes (U \oplus U^{\perp} \oplus \mathsf{H}) \bigr)
\oplus \mathsf{H}_{+}^{\oplus 4n} = \mathsf{H}_{+}^{\oplus 16n}
\]
is classified by a map $f_{2n} \colon ([-n,n] \times HGr'(2n,4n)) \times HP^{1}
\to RGr(16n,32n)$.
When we restrict to $pt \times HP^{1}$, we have $i=0$, and
the direct sum $U_{2n} \oplus U_{2n}^{\perp}$ becomes
$\mathsf{H}^{\oplus n} \oplus \mathsf{H}^{\oplus n}$.
The orthogonal direct sum
\eqref{E:subbundle} becomes
\[
\bigl( (\mathsf{H}^{\oplus n} \oplus 0)\boxtimes (U \oplus U^{\perp} \oplus 0) \bigr)
\oplus
\bigl( ( 0 \oplus \mathsf{H}^{\oplus n}) \boxtimes (0 \oplus 0 \oplus \mathsf{H}) \bigr)
\oplus
(\mathsf{H}_{+}^{\oplus 2n} \oplus 0).
\]
Since $U \oplus U^{\perp} \oplus 0 = \mathsf{H} \oplus \mathsf{H} \oplus 0$,
this is the same as
the subbundle
\begin{equation}
\label{E:pointing}
\bigl( (\mathsf{H}^{\oplus n} \oplus 0)\boxtimes (\mathsf{H} \oplus \mathsf{H} \oplus 0) \bigr)
\oplus
\bigl( ( 0 \oplus \mathsf{H}^{\oplus n}) \boxtimes (0 \oplus 0 \oplus \mathsf{H}) \bigr)
\oplus
(\mathsf{H}_{+}^{\oplus 2n} \oplus 0).
\end{equation}
corresponding to the pointing of $RGr(16n,32n)$. So
$f_{2n}(pt \times HP^{1}) = pt$.
A similar argument shows that $f_{2n}$ is compatible with the inclusions
of the Grassmannians in higher-dimensional Grassmannians.
When we restrict $f_{2n}$ to $([-n,n] \times HGr'(2n,4n)) \times pt$,
the direct sum over $pt \subset HP^{1}$ becomes $U \oplus U^{\perp} = \mathsf{H} \oplus \mathsf{H}$.
The orthogonal direct sum
\eqref{E:subbundle} becomes
\begin{equation}
\label{E:homotopic.subbundle}
\bigl(U_{2n} \boxtimes (\mathsf{H} \oplus 0 \oplus 0) \bigr)
\oplus \bigl(\mathsf{H}^{\oplus n-i} \boxtimes (0 \oplus \mathsf{H} \oplus 0) \bigr)
\oplus \bigl(U_{2n}^{\perp} \boxtimes(0 \oplus 0 \oplus \mathsf{H}) \bigr)
\oplus \mathsf{H}_{+}^{\oplus 2n+2i}
\end{equation}
This direct sum decomposition of the subbundle of rank $16n$
is compatible with the orthogonal direct sum decomposition
of the bundle of rank $32n$ into two summands
\begin{equation}
\label{E:first.summand}
\mathsf{H}^{\oplus 2n} \boxtimes (\mathsf{H} \oplus 0 \oplus \mathsf{H})
= (U_{2n} \oplus U_{2n}^{\perp}) \boxtimes(\mathsf{H} \oplus 0 \oplus \mathsf{H})
= (\mathsf{H}^{\oplus n} \oplus \mathsf{H}^{\oplus n}) \boxtimes(\mathsf{H} \oplus 0 \oplus \mathsf{H})
\end{equation}
and
\begin{equation}
\label{E:second.summand}
\bigl( \mathsf{H}^{\oplus 2n} \boxtimes (0 \oplus \mathsf{H} \oplus 0) \bigr)
\oplus \mathsf{H}_{+}^{4n} =
\mathsf{H}_{+}^{8n}.
\end{equation}
Because
\[
M_{1} =
\begin{pmatrix}
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & -1 \\
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & -1 \\
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\]
is a product of matrices which are elementary symplectic and elementary orthogonal, there is a morphism $M \colon \mathbf{A}^{1} \to Sp_{4} \cap O_{4}$
with $M(0) = I$ and $M(1) = M_{1}$, namely
\[
M(t) =
\begin{pmatrix}
1-t^{2} & 0 & -t & 0 \\
0 & 1-t^{2} & 0 & -2t+t^{3} \\
2t-t^{3} & 0 & 1-t^{2} & 0 \\
0 & t & 0 & 1-t^{2}
\end{pmatrix}
\]
When we restrict to $pt \times pt$ the direct sum $U \oplus U^{\perp}$ becomes $\mathsf{H} \oplus \mathsf{H}$.
We may see that
\[
\bigl( (1_{\mathsf{H}^{\oplus n} \oplus 0} \boxtimes 1_{\mathsf{H}^{\oplus 2}})
\oplus (1_{0 \oplus \mathsf{H}^{\oplus n}} \boxtimes M )\bigr)
^{-1}
\bigl( (1_{U_{2n}} \boxtimes 1_{\mathsf{H}^{\oplus 2}})
\oplus (1_{U_{2n}^{\perp}} \boxtimes M )\bigr)
\]
is a pointed $\mathbf{A}^{1}$-homotopy between the subbundle
\[
\bigl(U_{2n} \boxtimes (\mathsf{H} \oplus 0 \oplus 0) \bigr)
\oplus \bigl(U_{2n}^{\perp} \boxtimes(0 \oplus 0 \oplus \mathsf{H}) \bigr)
\]
and the subbundle
\[
\bigl( (\mathsf{H}^{\oplus n} \oplus 0)\boxtimes (\mathsf{H} \oplus 0 \oplus 0) \bigr)
\oplus
\bigl( ( 0 \oplus \mathsf{H}^{\oplus n}) \boxtimes (0 \oplus 0 \oplus \mathsf{H}) \bigr)
\]
One may construct a similar pointed $\mathbf{A}^{1}$-homotopy between the subbundles
\[
\bigl( (\mathsf{H}^{\oplus n-i} \oplus 0) \boxtimes ( 0 \oplus \mathsf{H} \oplus 0) \bigr)
\oplus (\mathsf{H}_{+}^{\oplus 2n+2i} \oplus 0)
\]
and
\[
\bigl( (\mathsf{H}^{\oplus n} \oplus 0) \boxtimes ( 0 \oplus \mathsf{H} \oplus 0) \bigr)
\oplus (\mathsf{H}_{+}^{\oplus 2n} \oplus 0)
\]
Combining the two gives a pointed $\mathbf{A}^{1}$-homotopy between \eqref{E:homotopic.subbundle}
and \eqref{E:pointing} and thus between $f_{2n} |_{([-n,n] \times HGr'(2n,4n)) \times pt}$ and the
constant map.
The compatibility of the homotopies with the inclusions of Grassmannians is relatively straightforward.
\end{proof}
The next lemma is proven in the same way as the last one.
\begin{lem}
\label{L:RGr.HP1.to.HGr}
There
exist morphisms of pointed schemes
\[
g_{n} \colon ([-n,n]' \times RGr(n,2n)) \times HP^{1} \to HGr'(8n,16n)
\]
such that the Grothendieck-Witt classes satisfy
\begin{equation}
\label{E:GW.formula.2}
g_{n}^{*}([U_{8n}]-4n[\mathsf{H}]) = ([V_{n}]-\tfrac{1}{2}(n-i)[\mathsf{H}_{+}]) \boxtimes ([U]-[\mathsf{H}])
\end{equation}
\parens{where $i \in [-n,n]' \subset \mathbb{Z}$ is the index of the component}
and such that
$g_{n}|_{pt \times HP^{1}}$ is constant,
and $g_{n}|_{([-n,n]' \times RGr(n,2n)) \times pt}$
is
pointed $\mathbf{A}^{1}$-homotopic to a constant map. These maps and homotopies are
compatible with the inclusions $RGr(n,2n) \hra RGr(n+2,2n+4)$
and $HGr'(8n,16n) \hra HGr'(8(n+2),16(n+2))$.
\end{lem}
\begin{proof}[Proof of Theorem \ref{T:finite}]
By Proposition \ref{P:A1.homotopy} the maps and homotopies of
Lemmas \ref{L:HGr.HP1.to.RGr} and \ref{L:RGr.HP1.to.HGr}
give us maps
\begin{align*}
F_{n} \colon & HGr'_{2n} \wedge HP^{1+} \to RGr_{16n},
\\
G_{n} \colon & RGr_{2n} \wedge HP^{1+} \to HGr'_{16n}.
\end{align*}
We now define $\mathbf{BO}^{\text{\textit{fin}}}$ to be the $HP^{1}$-spectrum with spaces as in the
statement of Theorem \ref{T:finite} and with bonding maps the compositions
\begin{gather*}
RGr_{2 \cdot 8^{2i}} \wedge HP^{1+}
\to
HGr'_{2 \cdot 8^{2i+1}}
\to
HGr'_{2 \cdot 8^{2i+2}}
\\
HGr'_{2 \cdot 8^{2i}} \wedge HP^{1+}
\to
RGr_{2 \cdot 8^{2i+1}}
\to
RGr_{2 \cdot 8^{2i+2}}
\end{gather*}
of the appropriate
$F_{n}$ or $G_{n}$ with the maps induced by the inclusions of Grassmannians.
We define $\mathbf{BO}^{\text{\textit{geom}}}$ to be the $HP^{1}$-spectrum with spaces
\[
\mathbf{BO}^{\text{\textit{geom}}}_{2i} =
\begin{cases}
\operatornamewithlimits{colim}_{n} RGr_{2n} = \mathbb{Z} \times RGr & \text{for even $i$}, \\
\operatornamewithlimits{colim}_{n} HGr'_{2n} = \mathbb{Z} \times HGr & \text{for odd $i$},
\end{cases}
\]
and with bonding maps induced by the $F_{n}$ and $G_{n}$.
We claim that the inclusion map $\mathbf{BO}^{\text{\textit{fin}}} \to \mathbf{BO}^{\text{\textit{geom}}}$
is a stable weak equivalence. To show this we need to show that the maps
$\operatornamewithlimits{colim}_{i} \Omega_{HP^{1}}^{i} (\mathbf{BO}^{\text{\textit{fin}}}_{2i+2j})^{f} \to
\operatornamewithlimits{colim}_{i} \Omega_{HP^{1}}^{i} (\mathbf{BO}^{\text{\textit{geom}}}_{2i+2j})^{f}$ are weak equivalences for all $j$.
The $(-)^{f}$ denotes fibrant replacement.
This is because we have two $\mathbb{N}^{2}$-indexed families of spaces
\begin{align*}
E_{n,i} & =
\begin{cases}
\Omega_{HP^{1}}^{i} (RGr_{2\cdot 8^{i}})^{f} & \text{for even $i$}, \\
\Omega_{HP^{1}}^{i} (HGr'_{2\cdot 8^{i}})^{f} & \text{for odd $i$},
\end{cases}
\\
E'_{n,i} & =
\begin{cases}
\Omega_{HP^{1}}^{i} (HGr'_{2\cdot 8^{i}})^{f} & \text{for even $i$}, \\
\Omega_{HP^{1}}^{i} (RGr_{2\cdot 8^{i}})^{f} & \text{for odd $i$},
\end{cases}
\end{align*}
The inclusions of Grassmannians the $F_{n}$ and $G_{n}$ give us maps $E_{n,i} \to E_{n+1,i}$
and $E_{n,i} \to E_{n+1,i+1}$ which commute, and similarly for the $E'_{n,i}$.
Thus the $E_{n,i}$ and $E'_{n,i}$ are filtered systems of spaces
indexed by a category with set of objects $\mathbb{N}^{2}$ such that there is a unique
arrow $(n,i) \to (n',i')$ if and only if $n \leq n'$ and $i-n \leq i'-n'$. By cofinality we have
isomorphisms $\operatornamewithlimits{colim}_{i} E_{i,2i+2j} = \operatornamewithlimits{colim}_{n,i} E_{n,i} = \operatornamewithlimits{colim}_{i} (\operatornamewithlimits{colim}_{n} E_{n,i})$ for all $j$
and similarly for $E'_{n,i}$. These are the required weak equivalences for even and odd $j$
respectively.
We now construct an isomorphism $\mathbf{BO}^{\text{\textit{geom}}} \cong \mathbf{BO}_{HP^{1+}}$
in $SH_{HP^{1+}}(S)$.
By Theorem \ref{T:varprojlim.b} we have an exact sequence
\begin{equation*}
0 \to \varprojlim\nolimits^{1} \mathbf{BO}_{HP^{1+}}^{4n-1,2n}(\mathbf{BO}^{\text{\textit{geom}}}_{2n})
\to \mathbf{BO}^{0,0}_{HP^{1+}}(\mathbf{BO}^{\text{\textit{geom}}}) \to
\varprojlim \mathbf{BO}^{4n,2n}_{HP^{1+}}(\mathbf{BO}^{\text{\textit{geom}}}_{2n})
\to 0.
\end{equation*}
For every odd $n = 2k+1$ the universal element of Definition \ref{D:tau} gives an isomorphism
$\tau_{4k+2} \colon \mathbb{Z} \times HGr \cong \mathbf{KO}^{[4k+2]}$ in $H_{\bullet}(S)$
and by adjunction a
$\tau_{4k+2}' \in \mathbf{BO}^{8k+4,4k+2}(\mathbf{BO}^{\text{\textit{geom}}}_{4k+2})$. The inverse system
sends $\tau_{4k+2}' \mapsto \tau_{4k-2}'$ because in the diagram
\begin{equation}
\label{E:compatible}
\vcenter{
\xymatrix @M=5pt @C=70pt {
(\mathbb{Z} \times HGr) \wedge HP^{1+} \wedge HP^{1+} \ar[r]
\ar[d]_-{\tau_{4k-2} \wedge 1 \wedge 1}^-{\cong}
& \mathbb{Z} \times HGr \ar[d]_-{\tau_{4k+2}}^-{\cong}
\\
\mathbf{KO}^{[4k-2]} \wedge HP^{1+}\wedge HP^{1+} \ar[r]
& \mathbf{KO}^{[4k+2]}
}}
\end{equation}
both horizontal maps are a $\times$ product with $p_{1}(U)^{\times 2} = ([U]-[\mathsf{H}])^{\times 2}$.
So the $\tau'_{4k+2}$ define an inverse system
$\tau'\in \varprojlim \mathbf{BO}^{8k+4,4k+2}_{HP^{1}}(\mathbf{BO}^{\text{\textit{geom}}}_{4k+2})$. Since
the $\tau'_{4k+2}$ are all isomorphisms, any element of
$\mathbf{BO}^{0,0}_{HP^{1}}(\mathbf{BO}^{\text{\textit{geom}}})$
lifting $\tau'$ is an isomorphism by general facts about homotopy colimits of sequential direct systems
in triangulated categories.
\end{proof}
The inverse system $\tau'$ also lies in
$\varprojlim \mathbf{BO}^{4i,2i}_{HP^{1}}(\mathbf{BO}^{\text{\textit{geom}}}_{2i})$. So it also gives
us maps $\tau_{4k} \colon \mathbb{Z} \times RGr \to \mathbf{KO}^{[4k]}$ in $H_{\bullet}(S)$. Essentially
these are the compositions of $\tau_{4k+2}$ and the maps in the motivic unstable homotopy
category induced by the adjoint bonding maps of the two spectra
\[
\mathbb{Z} \times RGr \to \Omega_{HP^{1+}}(\mathbb{Z} \times HGr)
\xra{\sim} \Omega_{HP^{1+}}\mathbf{KO}^{[4k+2]} \xla{\sim} \mathbf{KO}^{[4k]}.
\]
We do not know whether these are isomorphisms in $H_{\bullet}(S)$. The best we know how
to do is:
\begin{prop}
\label{P:right.inverse}
The morphism $\Omega_{HP^{1}}(\tau_{4k})$ has a right inverse in $H_{\bullet}(S)$.
\end{prop}
This is because in the commutative diagram
\[
\xymatrix @M=5pt {
\mathbb{Z} \times HGr \ar[r] \ar[d]^{\sim}
&
\Omega_{HP^{1+}}(\mathbb{Z} \times RGr) \ar[d]
\\
\mathbf{KO}^{[4k-2]} \ar[r]^-{\sim}
&
\Omega_{HP^{1+}}\mathbf{KO}^{[4k]}
}
\]
the arrows on the left side and bottom of the square are weak equivalences by Theorem \ref{T:MV.symp}
and \ref{OmegaSpectrumBO} respectively.
\section{The commutative monoid structure in \texorpdfstring{$SH(S)$}{SH(S)}}
\label{S:vanishing}
The main technical result of this section is the following theorem. We use it to show that
the almost commutative monoid structure on the $T$-spectrum $\mathbf{BO}$ we constructed in
Theorem \ref {monoidBO} is actually a commutative monoid for $S = \operatorname{Spec} \mathbb{Z}[\frac 12]$.
\begin{thm}
\label{T:lim1}
Let $S$ be a regular noetherian separated $\mathbb{Z}[\frac 12]$-scheme of finite Krull dimension.
Suppose
that $KO_{1}(S)$ and $KSp_{1}(S)$ are finite.
Then for all $m$
the natural map
\[
\mathbf{BO}^{0,0}(\mathbf{BO}^{\wedge m}) \to
\varprojlim\mathbf{BO}^{2mi,mi}((\mathbf{KO}^{[i]})^{\wedge m})
\]
is an isomorphism.
\end{thm}
\begin{proof}
We prove the theorem for the $HP^{1}$-spectrum $\mathbf{BO}_{HP^{1}}$. The theorem then follows
for the $T$-spectrum $\mathbf{BO}$.
By Theorem \ref{T:varprojlim.b} and Theorem \ref{T:finite} there is a commutative diagram with exact rows
\[
\xymatrix @M=5pt @C=12pt @R=18pt {
\raisebox{-18pt}{$\sideset{}{^{1}}\varprojlim\limits_{i}\mathbf{BO}_{HP^{1}}^{4mi-1,2mi}((\mathbf{KO}^{[2i]})^{\wedge m})$}
\ar@{>->}[r] \ar[d]
&
\mathbf{BO}_{HP^{1}}^{0,0}(\mathbf{BO}_{HP^{1}}^{\wedge m})
\ar@{->>}[r] \ar[d]^-{\cong}
&
\raisebox{-18pt}{$\varprojlim\limits_{i}\mathbf{BO}_{HP^{1}}^{4mi,2mi}((\mathbf{KO}^{[2i]})^{\wedge m})$}
\ar[d]^{\text{$\cong$ by cofinality}}
\\
\raisebox{-18pt}{$\sideset{}{^{1}}\varprojlim\limits_{i}\mathbf{BO}_{HP^{1}}^{4mi-1,2mi}((\mathbf{BO}^{\text{\textit{fin}}}_{2i})^{\wedge m})$} \ar@{>->}[r]
&
\mathbf{BO}^{0,0}_{HP^{1}}((\mathbf{BO}^{\text{\textit{fin}}})^{\wedge m}) \ar@{->>}[r]
&
\raisebox{-18pt}{$\varprojlim\limits_{i}\mathbf{BO}^{4mi,2mi}_{HP^{1}}((\mathbf{BO}^{\text{\textit{fin}}}_{2i})^{\wedge m})$}
}
\]
The middle vertical arrow is an isomorphism because
the morphism
$(\mathbf{BO}^{\text{\textit{fin}}}_{2i})^{\wedge m} \to \mathbf{BO}_{HP^{1}}^{\wedge m}$ is a stable
weak equivalence. The righthand vertical arrow is an isomorphism by a cofinality argument
as in the proof of Theorem \ref{T:finite}. It follows that the lefthand vertical arrow is also an
isomorphism.
The $\varprojlim^{1}$ in the lower row is isomorphic to
\[
\sideset{}{^{1}}\varprojlim\limits_{\text{$i$ odd}} KO_{1}(HGr'_{2N_{i}}{}^{\wedge m})
\qquad \text{or} \qquad
\sideset{}{^{1}}\varprojlim\limits_{\text{$i$ odd}} KSp_{1}(HGr'_{2N_{i}}{}^{\wedge m})
\]
for even $m$ and odd $m$, respectively, where $N_{i} = 8^{2i}$. By Theorem \ref {T:Grass}
each group in the system
is a direct sum of a finite number of copies of $KO_{1}(S)$ and of $KSp_{1}(S)$. By the hypothesis
it
follows that each group in this system is finite, and so the $\varprojlim^{1}$ in the lower row
of the diagram vanishes.
Therefore the $\varprojlim^{1}$ in the upper row also vanishes, proving the isomorphism.
\end{proof}
\begin{thm}
\label{T:KSp1.euclidean}
Let $R$ be a Euclidean domain. Then we have $KSp_{1}(R) = 0$.
\end{thm}
This is classical. It is proven essentially
by showing that the action of the group $ESp_{2n}(R)$
on unimodular vectors is transitive.
\begin{thm}
\label{T:pid}
Let $R$ be a Euclidean domain with $\frac 12 \in R$.
Then we have
$KO_{1}(R) \cong \mathbb{Z}/ 2 \mathbb{Z} \times R^{\times}/R^{\times 2}$.
\end{thm}
This must be very well known to the experts. We include the proof for completeness' sake.
\begin{proof}
[Proof of Theorem \ref{T:pid}]
We use the long exact sequences of Karoubi's fundamental theorem
\cite{Karoubi:1980aa,Schlichting:2006aa}
\[
\cdots \to KO_{i}^{[n]}(R) \xrightarrow{F} K_{i}(R) \xrightarrow{H}
KO_{i}^{[n+1]}(R) \xrightarrow{\eta} KO_{i-1}^{[n]}(R) \to \cdots.
\]
with $F$ the forgetful map and $H$ the hyperbolic map. This amounts to four exact sequences including
\begin{gather}
\label{E:fund.3}
\cdots \to
{}_{-1}V \to K_{1} \to KO_{1} \to {}_{1}U \xra{0} K_{0} \hra GW^{+} \to W^{0} \to 0
\\
\label{E:fund.4}
\cdots \to
KSp_{1} \to K_{1} \twoheadrightarrow {}_{-1}V \xra{0} GW^{-} \hra K_{0} \to {}_{1}U \to W^{-1} \to 0
\end{gather}
with the $GW^{+}$ and $GW^{-}$ the Grothendieck-Witt groups of symmetric and skew-symmetric bilinear forms respectively,
the ${}_{-1}V$ and ${}_{1}U$ the groups of \cite[Appendices 2 et 3]{Karoubi:1975aa},
and the $W^{i}$ the Witt groups with cohomological indexing \textit{\`a la} Balmer.
For a principal ideal domain containing $\frac 12$ we have $W^{i} = 0$ for $i \equiv 2$ or $3 \pmod 4$,
while the
the forgetful map $GW^{-} \to K_{0}$ of \eqref{E:fund.4}
is the inclusion $2\mathbb{Z} \subset \mathbb{Z}$, and the hyperbolic map $K_{0} \to GW^{+}$
of \eqref{E:fund.3} is injective. It then follows from the two sequences that we have a short exact sequence
\[
0 \to \coker(K_{1} \twoheadrightarrow {}_{-1}V \to K_{1}) \to KO_{1} \to \mathbb{Z}/2\mathbb{Z} \to 0.
\]
By \cite[\S 4.1]{Karoubi:1980aa} or \cite[\S 4.5]{Barge:2008it} the composition
$K_{1} \twoheadrightarrow {}_{-1}V \to K_{1}$ is
$[x] \mapsto [x]-[{}^{t}\bar{x}^{-1}]$. For a euclidean domain with trivial involution, this
is $[x] \mapsto [x^{2}]$. So we have an exact sequence
$1 \to R^{\times}/R^{\times 2} \to KO_{1} \to \mathbb{Z}/2\mathbb{Z} \to 0$.
The image of $[b] \in R^{\times}/R^{\times 2}$ in $KO_{1}$
is the class of the auto-isometry $\bigl[\begin{smallmatrix} b & 0 \\ 0 & b^{-1} \end{smallmatrix}\bigr]$ of the
hyperbolic quadratic form $q(x_{1},x_{2}) = x_{1}x_{2}$.
The class of the auto-isometry $\bigl[\begin{smallmatrix} 0&1\\1&0 \end{smallmatrix}\bigr]$ is not in that image
because its determinant is $-1$. It provides a splitting of the surjection in the exact sequence.
\end{proof}
For a nice discussion of the group ${}_{1}V$ and a bit of the other two exact sequences of Karoubi's
fundamental theorem see \cite[\S 4.5]{Barge:2008it}.
\begin{thm}
\label{T:commutative.monoid}
Suppose $KO_{1}(S)$ and $KSp_{1}(S)$ are finite, for instance $S = \operatorname{Spec} \mathbb{Z}[\frac 12]$.
Let $m \in Hom_{SH(S)}(\mathbf{BO} \wedge \mathbf{BO} , \mathbf{BO})$ be the morphism of \eqref{E:m}.
Let $e \in Hom_{SH(S)}(pt_{+},\mathbf{BO}) = \mathbf{BO}^{0,0}(pt_{+})$ be the element corresponding
to $\angles{1} \in GW^{+}(pt) = KO_{0}^{[0]}(pt)$.
\parens{a} Then $(\mathbf{BO},m,e)$ is a commutative monoid in $SH(S)$.
\parens{b} The map $m$ is the unique element of $Hom_{SH(S)}(\mathbf{BO} \wedge \mathbf{BO} , \mathbf{BO})$ defining
a pairing which, when restricted to pairing
\[
\mathbf{BO}^{4p,2p}(X_{+}) \times \mathbf{BO}^{4q,2q}(Y_{+}) \to
\mathbf{BO}^{4p+4q,2p+2q}(X_{+} \wedge Y_{+})
\]
with $X,Y \in \mathcal Sm/S$ coincides with the tensor product pairing
\[
KO_{0}^{[2p]}(X) \times KO_{0}^{[2q]}(Y)
\to KO_{0}^{[2p+2q]}(X \times Y)
\]
of Grothendieck-Witt groups.
\end{thm}
\begin{proof}
(a) By Theorem \ref{monoidBO} $(B,m,e)$ is an almost commutative monoid in $SH(S)$.
By Definition \ref{D:almost.monoid} the obstructions to $(B,m,e)$ being a commutative monoid
are three classes in the kernels of the maps of Theorem \ref{T:lim1} for $m=1,2,3$. Those
classes vanish.
(b) The product on the $KO_{0}^{[2p]}(X_{+})$ determined uniquely the
$\bar m$ of \eqref{E:bar.m}. By Theorem \ref{T:lim1} $m$ is the unique element of
$Hom_{SH(S)}(\mathbf{BO} \wedge \mathbf{BO} , \mathbf{BO})$ mapping onto $\bar m$.
\end{proof}
We now wish to use the closed motivic model structure
of \cite[Appendix A]{Panin:2009aa}.
Among its properties are:
\begin{enumerate}
\item
The closed motivic model structure
and the local injective motivic model structure
used in \S\S \ref{S:KO.motivic.spaces}--\ref{S:BO}
have the same weak equivalences, but the closed motivic model structure
has fewer cofibrations
and more fibrations than the local injective motivic model structure.
\item
A pointed smooth $S$-scheme $(X,x_{0})$ is cofibrant in the closed motivic model structure.
More generally, a closed embedding $Z \rightarrowtail X$ in $\mathcal Sm/S$ induces a cofibration
$Z_{+} \rightarrowtail X_{+}$
\cite[Lemma A.10]{Panin:2009aa}.
Hence the pointed scheme $HP^{1+}$ is cofibrant for the closed motivic model structure, so we may
define levelwise and stable closed motivic model structures for $HP^{1+}$-spectra.
\item
For any morphism $u \colon S\to S'$ of noetherian schemes of finite Krull dimension,
the pullback $u^{*}\colon \mathbf{M}_{\bullet}(S') \to \mathbf{M}_{\bullet}(S)$ is a
strict symmetric monoidal left Quillen functor for the closed motivic model structure
\cite[Theorem A.17]{Panin:2009aa}.
Consequently $\mathbf{L}u^{*} \colon SH(S') \to SH(S)$ can be computed by
taking levelwise closed cofibrant replacements and then applying $u^{*}$.
\end{enumerate}
To extend $m$ to other base schemes $S$, we need to discuss base change for morphisms
$u \colon S \to S'$. For any $X \in \mathcal Sm/S'$ there is a duality-preserving pullback functor inducing
morphisms of hermitian $K$-theory spaces
$(1 \times u)^{*} \colon KO^{[n]}(X) \to KO^{[n]}(X \times_{S'}S)$.
This gives us maps $KO^{[n]}_{S'} \to u_{*}KO^{[n]}_{S}$ and adjoint maps $u^{*} KO_{S'}^{[n]} \to
KO_{S}^{[n]}$. These maps are compatible with Thom isomorphisms, inducing maps of spectra.
The maps $u^{*}\mathbf{BO}^{\text{\textit{geom}}}_{S'} \to \mathbf{BO}^{\text{\textit{geom}}}_{S}$ are isomorphisms in
$SH(S)$ because $u^{*}$ acts as base change on the quaternionic and real Grassmannians and
their direct colimits.
The maps $\mathbf{L}u^{*}\mathbf{BO}^{\text{\textit{geom}}}_{S'} \to u^{*}\mathbf{BO}^{\text{\textit{geom}}}_{S'}$
are isomorphisms $SH(S)$ because for the closed motivic model structure
$\mathbf{BO}^{\text{\textit{geom}}}_{S'}$ is levelwise cofibrant
and $u^{*}$ is a levelwise left Quillen functor.
Setting $S' = \operatorname{Spec} \mathbb{Z}[\frac 12]$ with $u \colon S \to \operatorname{Spec} \mathbb{Z}[\frac 12]$
the canonical map, we can now define the monoidal structure on $\mathbf{BO}_{S}$ in $SH(S)$
as in \cite[Definition 3.7]{Panin:2009aa} as the composition
\[
m_{S} \colon
\mathbf{BO}_{S} \wedge \mathbf{BO}_{S} \cong
u^{*}\mathbf{BO}_{\mathbb{Z}[\frac 12]} \wedge u^{*}\mathbf{BO}_{\mathbb{Z}[\frac 12]}
\cong
u^{*}(\mathbf{BO}_{\mathbb{Z}[\frac 12]} \wedge \mathbf{BO}_{\mathbb{Z}[\frac 12]})
\xra{u^{*}m_{\mathbb{Z}[\frac 12]}}
u^{*}\mathbf{BO}_{\mathbb{Z}[\frac 12]}
\cong
\mathbf{BO}_{S}
\]
\begin{thm}
\label{T:unique.2}
The assertions of Theorem \ref{T:unique} hold.
\end{thm}
For $S = \operatorname{Spec} \mathbb{Z}[\frac 12]$ this is part of Theorem \ref{T:commutative.monoid}.
For other $S$ it is deduced by base change from $\operatorname{Spec} \mathbb{Z}[\frac 12]$.
\begin{thm}
The assertions of Theorem \ref{T:SLc.ring} hold.
\end{thm}
Theorem \ref{T:SLc.oriented} shows that hermitian $K$-theory is an $SL^{c}$-oriented cohomology
theory with a partial multiplicative structure. The ring structure is given by Theorem
\ref{T:unique.2}. The compatibility of the two multiplications is Theorem \ref{T:coincide}.
Schlichting's multiplicative structure, which we mentioned when discussing Theorem \ref{T:compatible},
could replace our partial multiplicative structure for
Theorems \ref{T:SLc.ring}, \ref{uniq2}, \ref{uniq1}, etc.
However, as we understand it,
Schlichting's product is defined in unstable homotopy theory.
To get our main Theorem \ref{T:unique}
with the monoid structure for $T$-spectra, we need our argument with the $\varprojlim^{1}$.
\section{\texorpdfstring
{$CP^{1+}$-spectra $\mathbf{BGL}^{\text{\textit{fin}}}$ and $\mathbf{BGL}^{\text{\textit{geom}}}$ for algebraic $K$-theory}
{CP\^{ }1+ spectra BGL\^{ }fin and BGL\^{ } for algebraic K-theory}
}
\label{S:K.theory}
The $HP^{1+}$-spectra
constructed in \S\ref{S:finite} have
an analogue for ordinary algebraic $K$-theory: the $CP^{1}$-spectra $\mathbf{BGL}^{\text{\textit{fin}}}$
and $\mathbf{BGL}^{\text{\textit{geom}}}$.
We sketch their construction. The first can be used to show that the uniqueness results concerning
the algebraic $K$-theory spectrum $\mathbf{BGL}$ and its $\times$ product
of \cite[Remark 2.19 and Theorem 3.6]{Panin:2009aa} hold for any base scheme $S$
which is noetherian of finite Krull dimension with finite $K_{1}(S)$ and not just
for $S = \operatorname{Spec} \mathbb{Z}$.
We use the affine Grassmannians which can be defined as
\[
CGr(m,n) = GL_{n}/(GL_{m} \times GL_{n-m})
\]
or as the open subscheme
\[
CGr(m,n) \subset Gr(m,n) \times Gr(n-m,n)
\]
where the two tautological subbundles of $\mathcal{O}^{\oplus n}$ are supplementary
or as the closed subscheme of the space on $n \times n$ matrices parametrizing projectors of rank $m$.
Each $CGr(m,n)$ is affine over the base scheme and
an $\mathbf{A}^{m(n-m)}$-bundle over the ordinary Grassmannnian
$Gr(m,n)$. Morphisms $V \to CGr(m,n)$ are in bijection with direct sum decompositions
$\mathcal{O}_{V}^{\oplus n} = U'_{m} \oplus U''_{n-m}$ with $U'_{m}$ and $U''_{n-m}$ subbundles
of ranks $m$ and $n-m$ respectively. We let $CGr = \operatornamewithlimits{colim}_{n} CGr(n,2n)$.
In particular
$CP^{1} = CGr(1,2) \cong \mathbf{P}^{1} \times \mathbf{P}^{1} - \Delta$ is an $\mathbf{A}^{1}$-bundle over
$\mathbf P^{1}$. We may point $CGr(1,2)$ by $CGr(0,0)$. Let $CP^{1+}$ then be the pointed scheme
constructed in \eqref{E:X+}.
The motivic stable homotopy categories of $\mathbf{P}^{1}$-spectra,
of $CP^{1}$-spectra and of $CP^{1+}$-spectra are equivalent. In particular there is a $CP^{1+}$-spectrum
$\mathbf{BGL}_{CP^{1+}}$ corresponding to the $\mathbf{P}^{1}$-spectrum $\mathbf{BGL}$
of \cite{Panin:2009aa}. For any smooth $S$-scheme $X$ we write $n = n[\mathcal{O}_{X}] \in K_{0}(X)$.
\begin{lem}
\label{L:Gr.P1.to.Gr}
There
exist morphisms of pointed schemes
\[
h_{n} \colon \bigl( [-n,n] \times CGr(n,2n) \bigr) \times CP^{1} \to CGr(4n,8n)
\]
such that the classes in $K_{0}$ satisfy
\begin{equation}
\label{E:K0.formula}
h_{n}^{*}([U'_{4n}]-4n) = ([U'_{n}]-(n-i)) \boxtimes ([U'_{1}]-1)
\end{equation}
\parens{where $i \in [-n,n] \subset \mathbb{Z}$ is the index of the component}
and such that
$h_{n}|_{pt \times CP^{1}}$ is constant, and
$h_{n}|_{([-n,n] \times CGr(n,2n)) \times pt}$ is
pointed $\mathbf{A}^{1}$-homotopic to a constant map. Moreover, these maps and homotopies are
compatible with the inclusions $CGr(n,2n) \hra CGr(n+1,2(n+1))$
and $CGr(4n,8n) \hra CGr(4(n+1),8(n+1))$.
\end{lem}
This lemma is proven in the same way as Lemma \ref{L:HGr.HP1.to.RGr} using the equality
\[
([U'_{n}]-(n-i)) \boxtimes ([U'_{1}]-1) = [U'_{n} \boxtimes U'_{1}] + [\mathcal{O}^{\oplus n-i}\boxtimes U''_{1}]
+ [U''_{n} \boxtimes \mathcal{O}] - (3n-i)[\mathcal{O} \boxtimes \mathcal{O}]
\]
in $K_{0}([-n,n] \times CGr(n,2n))$
and the direct sum decompositions of vector bundles
\begin{gather*}
(U'_{n} \boxtimes U'_{1}) \oplus (U''_{n} \boxtimes U'_{1})
= \mathcal{O}^{\oplus 2n} \boxtimes U'_{1},
\\
(\mathcal{O}^{\oplus n-i} \boxtimes U''_{1}) \oplus (\mathcal{O}^{\oplus n+i} \boxtimes U''_{1})
=
\mathcal{O}^{\oplus 2n} \boxtimes U''_{1},
\\
(U''_{n} \boxtimes \mathcal{O}) \oplus (U'_{n} \boxtimes \mathcal{O})
= \mathcal{O}^{\oplus 2n} \boxtimes \mathcal{O},
\\
(\mathcal{O}^{\oplus n+i} \boxtimes \mathcal{O}) \oplus (\mathcal{O}^{\oplus n-i} \boxtimes \mathcal{O})
=
\mathcal{O}^{\oplus 2n} \boxtimes \mathcal{O},
\end{gather*}
yielding a decomposition of the trivial bundle of rank $8n$ on $([-n,n] \times CGr(n,2n)) \times CP^{1}$
as the direct sum of two
subbundles of rank $4n$.
\begin{thm}
\label{T:BGL.finite}
There are $CP^{1+}$-spectra
$\mathbf{BGL}^{\text{\textit{fin}}}$ and $\mathbf{BGL}^{\text{\textit{geom}}}$ isomorphic to
$\mathbf{BGL}_{CP^{1}}$ in $SH_{CP^{1}}(S)$
with spaces
\begin{align*}
\mathbf{BGL}^{\text{\textit{fin}}}_{n} & =
[-4^{n},4^{n}] \times CGr(4^{n}, 2 \cdot 4^{n})
&
\mathbf{BGL}^{\text{\textit{geom}}}_{n} & = \mathbb Z \times CGr
\end{align*}
which are unions of affine Grassmannians.
The bonding maps $\mathbf{BGL}^{*}_{n} \wedge CP^{1+} \to \mathbf{BGL}_{n+1}^{*}$
of the two spectra
are morphisms of schemes or ind-schemes which are constant on the wedge
$\mathbf{BGL}^{*}_{n} \vee CP^{1+}$.
\end{thm}
This theorem is proven in the same way as Theorem \ref{T:finite}.
|
1,116,691,499,341 | arxiv | \section{Introduction \label{s1}}\rm
Throughout this paper $\K$ is an arbitrary field, $\Z_+$ is the
set of non-negative integers and $\N$ is the set of positive
integers. For a set $X$, $\K\langle X\rangle$ stands for the free
associative algebra over $\K$ generated by $X$. We deal with
{\it quadratic algebras}, that is, algebras $R$ given as
$\K\langle X\rangle/I$, where $I$ is the ideal in $\K\langle X\rangle$
generated by a collection of homogeneous elements (called relations)
of degree $2$.
Algebras of this class, their growth, their Hilbert series and
nil/nilpotency properties have been extensively studied, see
\cite{popo,smo,ufnar} and references therein. One of the most
challenging questions in the area (see \cite{smo,zelm}) is the Kurosh problem
of whether there is an infinite dimensional nil algebra in this class.
A version of this question dealing with algebras of finite Gelfand--Kirillov
dimension was solved in \cite{lesmo}. The Golod--Shafarevich type lower estimates
for the dimensions of the graded components of an algebra play a crucial role
in the study of quadratic algebras. These estimates have many other applications, for instance, to $p$-groups and class field theory \cite{gosh,zelm1}.
Recall that a $\K$-algebra $R$ defined by the set $X$ of generators and a set of
homogeneous relations inherits the degree grading from the free algebra $\K\langle X\rangle$.
If $X$ is finite, one can consider the Hilbert series of $R$:
\begin{equation*
H_R(t)=\sum_{q=0}^\infty (\dimk R_q)\,t^q,
\end{equation*}
where $R_q$ is the $q^{\rm th}$ homogeneous component of $R$.
The original Golod--Shafarevich theorem provides a lower estimate
for the coefficients of $H_R$. In the case of quadratic algebras the
theorem reads as follows \cite{gosh,popo}. For two power series $a(t)$ and
$b(t)$ with real coefficients we write $a(t)\geq b(t)$ if $a_j\geq b_j$
for any $j\in\Z_+$, while $|a(t)|$ stands for the power series obtained from $a(t)$ by
replacing by zeros all coefficients starting from the first
non-positive one.
\begin{thmgs}Let, $n\in\N$, $0\leq d\leq
n^2$ and $R$ be a quadratic $\K$-algebra with $n$ generators and $d$
relations. Then $H_R(t)\geq |(1-nt+dt^2)^{-1}|$.
\end{thmgs}
In particular, Theorem~GS provides a lower estimate on the order of
nilpotency of $R$.
\begin{definition}\label{kstep} A graded algebra $R$ is called
{\it $k$-step nilpotent} if $R_k=\{0\}$.
\end{definition}
Analysing the series $K(t)=|(1-nt+dt^2)^{-1}|$ in a standard way, one can
easily see that it is a polynomial of degree $<k$ if and only if
\begin{equation}\label{phik}
\frac d{n^2}\geq \phi_k,\ \ \text{where}\ \
\phi_k=\frac{1}{4}\cos^{-2}\Bigl(\frac{\pi}{k+1}\Bigr).
\end{equation}
For the sake of convenience, we outline the argument. If $(1-nt+dt^2)^{-1}=\sum\limits_{m=0}^\infty c_mt^m$
(the Taylor series expansion), then $K(t)$ is not a polynomial of degree $<k$ precisely when $c_m>0$ for $0\leq m\leq k$. Next, if $x^2-nx+d=(x-a)(x-b)$ ($a$ and $b$ are complex numbers in general), then an easy computation yields that $c_m=(m+1)(n/2)^m$ if $a=b$ and $c_m=\frac{a^{m+1}-b^{m+1}}{a-b}$ otherwise for $m\in\Z_+$. It follows that $c_m>0$ for all $m\in\Z_+$ if $a$ and $b$ are real, which happens precisely when $d\leq\frac{n^2}{4}$. If $n^2\geq d>\frac{n^2}{4}$, then $a,b=\sqrt{d}e^{\pm i\alpha}$, where $\alpha=\arccos\frac{n}{\sqrt d}$. Hence $c_m=\frac{a^{m+1}-b^{m+1}}{a-b}=d^{m/2}\frac{\sin(m+1)\alpha}{\sin\alpha}$ for $m\in\Z_+$. Clearly $c_m$ for $0\leq m\leq k$ are positive precisely when $(k+1)\alpha<\pi$. After plugging in $\alpha=\arccos\frac{n}{\sqrt d}$, (\ref{phik}) follows.
Formula (\ref{kstep}) together with Theorem~GS and the obvious fact that
the sequence $\{\phi_k\}$ decreases and converges to $\frac14$
implies the following corollary, which can be found in \cite{popo}.
\begin{corgs}If $R$ is a quadratic $\K$-algebra given by $n$
generators and $d<\phi_k n^2$ relations, then $\dim R_k>0$, where
$\phi_k$ is defined in $(\ref{phik})$. That is, $R$ is not $k$-step
nilpotent. In particular, if $d\leq\frac{n^2}{4}$, then $\dim R_k>0$
for every $k\in\N$ and therefore $R$ is infinite dimensional.
\end{corgs}
Asymptotic optimality of the last statement in Corollary~GS was
proved by Wisliceny \cite{wis}.
\begin{thmw}For every $n\in\N$, there exists a quadratic $\K$-algebra
$R$ given by $n$ generators and $d_n$ relations such that $R$ is
finite dimensional and
$\lim\limits_{n\to\infty}\frac{d_n}{n^2}=\frac14$.
\end{thmw}
More specifically, Wisliceny has constructed a quadratic algebra
given by $n$ generators and $\bigl\lceil
\frac{n^2+2n}{4}\bigr\rceil$ semigroup relations (that is, every
relation is either a degree 2 monomial or a difference of two degree
2 monomials), which is finite dimensional. Note that here and everywhere
below $\lfloor t\rfloor$ is the largest integer $\leq t$,
while $\lceil t\rceil$ is the smallest integer $\geq t$, where $t$
is a real number. The authors \cite{ns}
have improved the last result by showing that the minimal number of
semigroup quadratic relations needed for finite dimensionality of an
algebra with $n$ generators is exactly $\bigl\lceil
\frac{n^2+n}{4}\bigr\rceil$. The number $\bigl\lceil
\frac{n^2+1}{4}\bigr\rceil$ remains a conjectural answer to the same
question in the class of general quadratic (not necessarily
semigroup) algebras.
\subsection{Results}
Note that if $R$ is $k$-step nilpotent, then $R_m=\{0\}$ for $m\geq
k$ and therefore $R$ is finite dimensional provided $|X|<\infty$,
where $X$ is the set of generators of $R$.
Thus $R$ is $k$-step nilpotent if and only if $H_R$ is a polynomial
of degree $<k$.
In this article we show that the first statement in Corollary~GS is
asymptotically optimal for every $k\geq 2$. In order to formulate
the exact statement, we shall introduce the following numbers. For
$n\in\N$ and $k\geq 2$ let
\begin{equation}\label{dnk}
d_{n,k}=\min_{n=a_1+{\dots}+a_{k-1}}\ \max_{1\leq j\leq
k-1}(a_1+{\dots}+a_j)(a_j+{\dots}+a_{k-1}),
\end{equation}
where $a_j$ are assumed to be non-negative integers. It turns out
that the integers $d_{n,k}$ are not too far from $\phi_k n^2$.
\begin{lemma}\label{dnph} For each $n,k\in\N$ with $k\geq 2$,
\begin{equation}\label{estim}
\textstyle
\phi_k n^2\leq d_{n,k}\leq \phi_k n^2+\frac{(1+\phi_k)n}{2}+\frac14.
\end{equation}
In particular, $\lim\limits_{n\to\infty}\frac{d_{n,k}}{\phi_k n^2}=1$ for each
$k\geq 2$.
\end{lemma}
We have defined the numbers $d_{n,k}$ since they feature in the
following theorem.
\begin{theorem}\label{asop}Let $k\geq2$.
Then for every $n\in\N$, there exists a quadratic $\K$-algebra $R$
given by $n$ generators and $d_{n,k}$ relations such that $R$ is
$k$-step nilpotent.
\end{theorem}
Corollary~GS, Theorem~\ref{asop} and Lemma~\ref{dnph} imply that the
first statement in Corollary~GS is asymptotically optimal. Note that
Anick \cite{ani1,ani2} conjectured that for any $n\in\N$ and $0\leq
d\leq n^2$, there is a quadratic $\K$-algebra $R$ with $n$
generators and $d$ relations such that $H_R(t)=|(1-nt+dt^2)^{-1}|$.
The problem whether this conjecture is true remains open.
Theorem~\ref{asop} can be considered as an affirmative solution of
its natural asymptotic version. It is also worth noting that for
$k=2$, the statement of Theorem~\ref{asop} is trivial, while the
case $k=3$ was done by Anick \cite{ani1}. It is also worth
mentioning that the asymptotic optimality of the first statement in
Corollary~GS for $k=4$ and for $k=5$ in the case $|\K|=\infty$ was
earlier obtained by the authors \cite{na}
building upon the ideas set in \cite{cana} and using a completely
different approach. We refer to \cite{asyy} for a result on asymptotic
optimality of Theorem~GS in a completely different sense.
Curiously enough, for some pairs $(n,k)$ the estimate provided by
Theorem~\ref{asop} hits the mark. We illustrate this observation by
the following result dealing with the cases $k=4$ and $k=5$. Note
that $\phi_4=\frac{3-\sqrt 5}{2}$ and $\phi_5=\frac13$. Recall that
Fibonacci numbers are the members of the recurrent sequence defined
by $F_0=F_1=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq2$.
\begin{theorem}\label{k45} The equality
$d_{n,4}=\bigl\lceil\frac{3-\sqrt 5}{2}n^2\bigr\rceil$ holds if and
only if $n$ is a Fibonacci number. The equality
$d_{n,5}=\bigl\lceil\frac{n^2}{3}\bigr\rceil$ holds if and only if
$n\in\{1,2\}$ or $n$ is divisible by $6$.
\end{theorem}
Note that Theorem~\ref{k45}, Theorem~\ref{asop} and Corollary~GS
imply that if $k=4$ and $n$ is a Fibonacci number or if $k=5$ and
$6$ divides $n$, then the minimal number of quadratic relations
needed for the finite dimensionality of an algebra with $n$
generators is exactly $\lceil \phi_k n^2\rceil$. The proof of
Theorem~\ref{asop} is based upon the following general result. We
start by introducing some notation.
\begin{definition}\label{paror}Let $X$ be the union of pairwise
disjoint sets $A_1,\dots,A_k$ and
\begin{equation}\label{M}
M=M(A_1,\dots,A_k)=\bigcup_{1\leq j\leq q\leq n}\!\!A_q\times
A_j\subseteq X\times X.
\end{equation}
We introduce the following {\it partial ordering on $M$, generated
by the partition} $\{A_1,\dots,A_k\}$. Namely, for distinct elements
$(a,b)$ and $(c,d)$ of $M$, we write $(a,b)\prec(c,d)$ if $(a,b)\in
A_l\times A_j$ and $(c,d)\in A_m\times A_r$ with $m\geq r>l\geq j$.
\end{definition}
\begin{definition}\label{supp} For a homogeneous degree $2$
polynomial $g$ in the free algebra $\K\langle X\rangle$, the
$($uniquely determined$)$ finite subset $S$ of $X\times X$ such that
$g=\sum\limits_{(x,y)\in S} c_{x,y}xy$ with
$c_{x,y}\in\K\setminus\{0\}$ is called the {\it support} of $g$ and
is denoted $S=\supp(g)$.
\end{definition}
The next result is one of the main tools in the proof of
Theorem~\ref{asop}.
\begin{theorem}\label{chin} Let $k\in\N$, $\{A_1,\dots,A_k\}$ be
a partition of a set $X$ and $M$ be the set defined in $(\ref{M})$.
Assume also that $\{f_\alpha\}_{\alpha\in \Lambda}$ is a family of
homogeneous degree $2$ elements of the free algebra $\K\langle
X\rangle$ such that
$\bigcup\limits_{\alpha\in\Lambda}\supp(f_\alpha)=M$ and each
$\supp(f_\alpha)$ is a chain in $M$ with respect to the partial
ordering $\prec$ on $M$, generated by the partition
$\{A_1,\dots,A_k\}$ as in Definition~$\ref{paror}$. Then the algebra $R=\K\langle X\rangle/I$ with
$I={\tt Id}\{f_\alpha:\alpha\in\Lambda\}$ is $(k+1)$-step nilpotent.
\end{theorem}
We conclude the introduction by providing a specific example of an
application of Theorem~\ref{chin}.
\begin{example} \label{ex8} Let $X=\{a,b,c,p,q,x,y,z\}$ be an
$8$-element set partitioned into $3$ subsets $A_1=\{a,b,c\}$,
$A_2=\{p,q\}$ and $A_3=\{x,y,z\}$. Let $M$ and the partial
ordering $\prec$ on $M$ be as in Definition~{\rm\ref{paror}}. Consider
the following $25$ quadratic relations:
\begin{align*}
f_1&=xc,&f_2&=xa,&f_3&=xp+ab,&f_4&=yz+qc,\qquad
f_5=pq,
\\
f_6&=yc,&f_7&=ya,&f_8&=yp+bb,&f_9&=yy+qb,
\\
f_{10}&=zc,&f_{11}&=za,&f_{12}&=zp+cb,&f_{13}&=yx+qa,
\\
f_{14}&=xb,&f_{15}&=xq+ac,&f_{16}&=xz+pc,&f_{17}&=zz+qq+ca,
\\
f_{18}&=yb,&f_{19}&=yq+bc,&f_{20}&=xy+pb,&f_{21}&=zy+qp+ba,
\\
f_{22}&=zb,&f_{23}&=zq+cc,&f_{24}&=xx+pa,&f_{25}&=zx+pp+aa.
\end{align*}
It is straightforward to verify that the support of each $f_j$ is a
chain in $(M,\prec)$ and that the union of $\supp(f_j)$ for $1\leq
j\leq 25$ is $M$. Theorem~{\rm\ref{chin}} ensures that the algebra given
by the $8$-element generator set $X$ and the relations $f_j$
with $1\leq j\leq 25$ is $4$-step nilpotent. Incidentally,
$25=\bigl\lceil\phi_4\cdot8^2\bigr\rceil$, which means $($see
Corollary~{\rm GS)} that a quadratic algebra given by $8$ generators and
$\leq24$ relations is never $4$-step nilpotent.
\end{example}
\section{Combinatorial lemmas}
Theorem~\ref{chin} allows us to construct $k$-step nilpotent
quadratic algebras with few relations. In order to do this, we need
an estimate on the number of relations in an algebra featuring in
Theorem~\ref{chin}. Recall that the {\it width} $w(X,<)$ of a
partially ordered set $(X,<)$ is the supremum of the cardinalities
of antichains in $X$.
\begin{lemma}\label{esti} Let $k\in\N$, $\{A_1,\dots,A_k\}$ be
a partition of a finite set $X$ and $M\subseteq X^2$ be the set
defined in $(\ref{M})$ with the partial ordering $\prec$ introduced
in Definition~$\ref{paror}$. For $1\leq q\leq k$, let
$B_q=\bigcup\limits_{j\geq q\geq m}A_j\times A_m$. Then
$w(M,\prec)=\max\{|B_1|,\dots,|B_k|\}$.
\end{lemma}
\begin{proof} It is a straightforward exercise to verify
that each $B_q$ is an antichain in $(M,\prec)$ and that every
antichain is contained in at least one of the sets $B_q$.
\end{proof}
We also need the following observation.
\begin{lemma}\label{alpk} Let $k\geq 2$ and
$\alpha_0,\alpha_1,\dots,\alpha_{k-1}\geq 0$ be defined by the
fromulae $\alpha_0=0$, $\alpha_1=\phi_k$ and
$\alpha_j=\frac{\phi_k}{1-\alpha_{j-1}}$ for $2\leq j\leq k-1$. Then
\begin{align}
&\text{$0=\alpha_0<\alpha_1<{\dots}<\alpha_{k-1}=1$}, \label{alpk00}
\\
&\text{$\alpha_j(1-\alpha_{j-1})=\phi_k$ for $1\leq j\leq k-1$} \label{alpk01}
\\
&\text{and $\max\limits_{1\leq j\leq k-1}(\alpha_j-\alpha_{j-1})=\phi_k$
$($attained for $j=1$ and for $j=k-1)$.}\label{alpk02}
\end{align}
\end{lemma}
\begin{proof} Obviously, (\ref{alpk01}) is a direct consequence of the definition of $\alpha_j$. Next, (\ref{alpk02}) follows easily from (\ref{alpk00}). Indeed, assuming that (\ref{alpk00}) holds, we have $\alpha_{k-1}=1$, which implies $\alpha_{k-2}=1-\phi_k$. Since $\alpha_j-\alpha_{j-1}=\frac{\phi_k}{1-\alpha_{j-1}}-\alpha_{j-1}$ and $0\leq \alpha_{j-1}\leq 1-\phi_k$ for $1\leq j\leq k-1$, (\ref{alpk02}) follows from the elementary fact that the function $\frac{\phi_k}{1-x}-x$ on the interval $[0,1-\phi_k]$ attains its maximal value at the end-points.
Thus it remains to verify (\ref{alpk00}). For $0<t\leq 1$ consider the rational function $f_t(x)=\frac{t}{1-x}$ and for $m\in\Z_+$ let $f_t^{[m]}$ be the $m^{\rm th}$ iterate of $f_t$: $f_t^{[0]}(x)=x$ and $f_t^{[m]}=f_t\circ{\dots}\circ f_t$ $m$ times for $m\in\N$. We start with an elementary observation
\begin{equation}\label{tlqu1}
\begin{array}{l}
\text{if $0\leq t\leq\frac14$, then the sequence $\{f_t^{[m]}(0)\}_{m\in\Z_+}$ is strictly increasing}
\\
\text{and converges to the fixed point $w_t=\frac{1-\sqrt{1-4t}}{2}\in\bigl[0,\frac12\bigr]$ of $f_t$.}
\end{array}
\end{equation}
For instance, to justify (\ref{tlqu1}), one can use induction with respect to $m$ to prove the chain of inequalities $0\leq f_t^{[m]}(0)<f_t^{[m+1]}(0)<w_t$.
Next, it is easy to verify that if $\frac14<t\leq 1$, then $f_t(x)>x$ for $x\in[0,1)$. Hence,
\begin{equation}\label{tlqu2}
\text{$f_t^{[m+1]}(0)>f_t^{[m]}(0)$ provided $0\leq f_t^{[m]}(0)<1$}.
\end{equation}
For each $m\in\Z_+$, we consider the rational function $h_m(t)=f_t^{[m]}(0)$ of the variable $t$. Now we observe that (\ref{alpk02}) follows from the claim
\begin{equation}\label{tlqu3}
\text{for every $m\in\N$, $\phi_{m+1}$ is the smallest solution of the equation $h_m(t)=1$ on
$\textstyle\bigl(\frac14,1\bigr]$.}
\end{equation}
Indeed, assume that (\ref{tlqu3}) holds. By (\ref{tlqu1}), $0<h_m(t)<\frac12$ for every $m\in\N$ and $t\in\bigl(0,\frac14\bigr]$. Since the sequence $\{\phi_m\}$ is decreasing, $h_j(t)<1$ whenever $j\leq m$ and $0\leq t< \phi_{m+1}$. Using (\ref{tlqu3}) with $m=k-1$ and (\ref{tlqu2}), we now have
$$
0=f^{[0]}_{\phi_k}(0)<f^{[1]}_{\phi_k}(0)<{\dots}<f^{[k-1]}_{\phi_k}(0)=h_{k-1}(\phi_k)=1.
$$
On the other hand, by definition of $\alpha_j$, $\alpha_j=f_{\phi_k}^{[j]}(0)$ for $0\leq j\leq k-1$ and (\ref{alpk02}) follows.
Thus it remains to prove (\ref{tlqu3}). Using the obvious recurrent relation $h_{j+1}(t)=\frac{t}{1-h_j(t)}$ together with the initial data $h_0=0$, one can use the induction with respect to $m$ to verify that
$$
\textstyle h_m(t)=t\frac{a^m-\overline{a}^m}{a^{m+1}-\overline{a}^{m+1}}\ \ \text{for $m\in\Z_+$ and $t\in \bigl[\frac14,1\bigr]$, where $a=a(t)=\frac{1+i\sqrt{4t-1}}{2}$}.
$$
Hence for $t\in\bigl[\frac14,1\bigr]$,
\begin{equation}\label{tlqu4}
\textstyle h_m(t)=1\iff (a/\overline{a})^m=(\overline{a}-t)/(a-t)\iff
e^{im\alpha(t)}=e^{i\beta(t)},
\end{equation}
where
$$
\textstyle\alpha(t)=2\arccos\frac1{2\sqrt t} \ \ \text{and}\ \ \beta(t)=2\pi-2\arccos\bigl(\frac1{2t}-1\bigr)
$$
are the arguments of the unimodular complex numbers $a/\overline{a}$ and $(\overline{a}-t)/(a-t)$. The case $m=1$ is trivial. Assuming that $m\geq 2$ and using (\ref{tlqu4}), we see that the smallest $t\in \bigl[\frac14,\frac12\bigr]$ satisfying $h_m(t)=1$ must satisfy $m\alpha(t)=\beta(t)$. Since the function $m\alpha(t)-\beta(t)$ on the interval $\bigl[\frac14,\frac12\bigr]$ is strictly increasing (look at the derivative) and has values of opposite signs at the ends, there is exactly one $t_m\in \bigl[\frac14,\frac12\bigr]$ satisfying $m\alpha(t_m)=\beta(t_m)$. Then $t_m$ is the smallest solution of the equation $h_m(t)=t$ on the interval $\bigl[\frac14,1\bigr]$. Since $\phi_{m+1}\in \bigl[\frac14,\frac12\bigr]$, (\ref{tlqu3}) will follow if we show that $m\alpha(\phi_{m+1})=\beta(\phi_{m+1})$. This is indeed true: plugging in $\phi_{m+1}=\frac1{4\cos^2(\pi/(m+2))}$, we have
\begin{align*}
&\textstyle m\alpha(\phi_{m+1})=2m\arccos\bigl(\cos\bigl(\frac{\pi}{m+2}\bigr)\bigr)=\frac{2\pi m}{m+2};
\\
&\textstyle\beta(\phi_{m+1})=2\pi-2\arccos\bigl(2\cos^2\bigl(\frac{\pi}{m+2}\bigr)-1\bigr)=
2\pi-2\arccos\bigl(\cos\bigl(\frac{2\pi}{m+2}\bigr)\bigr)=2\pi-\frac{4\pi}{m+2}=\frac{2\pi m}{m+2}.
\end{align*}
Hence $m\alpha(\phi_{m+1})=\beta(\phi_{m+1})$, which completes the proof.
\end{proof}
\section{Proof of Theorem~\ref{chin}}
For $k\in\N$, we denote $\N_k=\{1,2,\dots,k\}$.
Assume the contrary. Then the set $\Omega$ of
$j=(j_1,\dots,j_{k+1})\in\N_k^{k+1}$ such that there are $x_1\in
A_{j_1}$, $\dots$, $x_{k+1}\in A_{j_{k+1}}$ for which $x_1\dots
x_{k+1}\notin I$ is non-empty. We endow $\N_k^{k+1}$ with the
lexicographical ordering $<$ counting from the right-hand side. That
is, $j<m$ if and only if there is $l\in\N_{k+1}$ such that $j_l<m_l$
and $j_r=m_r$ for $r>l$. Since $<$ is a total ordering on the finite
set $\N_k^{k+1}$ and $\Omega\subseteq \N_k^{k+1}$ is non-empty,
$\Omega$ has a unique element $j$ minimal with respect to $<$. Since
$j\in\Omega$, there are $x_1\in A_{j_1}$, $\dots$, $x_{k+1}\in
A_{j_{k+1}}$ for which $x_1\dots x_{k+1}\notin I$.
Now we shall construct inductively $m_1,\dots,m_{k+1}\in\N_{k}$ and
monomials $u_1,\dots,u_{k+1}$ in $\K\langle X\rangle$ of degree
$k+1$ such that
\begin{align}
&\text{$m_l>m_{l-1}$ if $l\geq 2$}; \label{ind1}
\\
&u_l\notin I; \label{ind2}
\\
&\text{$u_l=v_lw_lx_{l+1}x_{l+2}\dots x_{k+1}$, where $w_l\in
A_{m_l}$ and $v_l$ is a monomial of degree $l-1$}. \label{ind3}
\end{align}
We start by setting $u_1=x_1\dots x_{k+1}$ and $m_1=j_1$ and
observing that (\ref{ind1}--\ref{ind3}) with $l=1$ are satisfied.
Assume now that $2\leq l\leq k+1$ and that $m_1,\dots,m_{l-1}$ and
$u_1,\dots,u_{l-1}$ satisfying the desired conditions are already
constructed.
If $m_{l-1}<j_l$, then we set $m_l=j_l$, $w_l=x_l$, $u_l=u_{l-1}$
and $v_l=v_{l-1}w_{l-1}$. Using the induction hypothesis, we see
that (\ref{ind1}--\ref{ind3}) are satisfied. It remains to consider
the case $m_{l-1}\geq j_l$. In this case $w_{l-1}x_l\in M$ and
therefore there is $\alpha\in\Lambda$ such that
$(w_{l-1},x_l)\in\supp(f_\alpha)$. Let
$S=\supp(f_\alpha)\setminus\{(w_{l-1},x_l)\}$. Since $f_\alpha\in
I$,
$$
w_{l-1}x_l=\sum_{(a,b)\in S} c_{a,b}ab\ \ (\bmod I)\ \ \ \text{with
$c_{a,b}\in\K$.}
$$
Using (\ref{ind3}) for $l-1$ and the above display, we get
$$
u_{l-1}=\sum_{(a,b)\in S}c_{a,b}v_{l-1}abx_{l+1}\dots x_{k+1}\ \
(\bmod I).
$$
Since $\supp(f_\alpha)$ is a chain in $M$ with respect to $\prec$,
for every $(a,b)\in S$, either $(a,b)\prec (w_{l-1},x_l)$ or
$(w_{l-1},x_l)\prec (a,b)$. If $(a,b)\prec (w_{l-1},x_l)$, $b$ is
contained in $A_q$ with $q<j_l$. Using the definition of $\Omega$
and the minimality of $j$ in $\Omega$, we obtain
$$
v_{l-1}abx_{l+1}\dots x_{k+1}\in I\ \ \text{if $(a,b)\in S$,
$(a,b)\prec (w_{l-1},x_l)$.}
$$
According to the last two displays
$$
u_{l-1}=\sum_{(a,b)\in S\atop (w_{l-1},x_l)\prec
(a,b)}c_{a,b}v_{l-1}abx_{l+1}\dots x_{k+1}\ \ (\bmod I).
$$
By (\ref{ind2}) for $l-1$, $u_{l-1}\notin I$. Thus, using the above
display, we can pick $(a,b)\in S$ such that $(w_{l-1},x_l)\prec
(a,b)$ and $v_{l-1}abx_{l+1}\dots x_{k+1}\notin I$. Now we set
$u_l=v_{l-1}abx_{l+1}\dots x_{k+1}$, $w_l=b$, $v_l=v_{l-1}a$ and
take $m_l$ such that $w_l=b\in A_{m_l}$.
Since $w_{l-1}\in A_{m_{l-1}}$ and $(w_{l-1},x_l)\prec
(a,b)=(a,w_l)$, we have $m_l>m_{l-1}$. Thus (\ref{ind1}--\ref{ind3})
are satisfied. This completes the inductive procedure of
constructing $m_1,\dots,m_{k+1}$ and $u_1,\dots,u_{k+1}$. By
(\ref{ind1}), $m_j$ for $1\leq j\leq k+1$ are $k+1$ pairwise
distinct elements of the $k$-element set $\N_k$. We have arrived to
a contradiction, which proves that $R$ is $(k+1)$-step nilpotent.
\section{Proofs of Theorem~\ref{asop} and Lemma~\ref{dnph}}
Let $k\geq2$, $n\in\N$ and $a_1,\dots,a_{k-1}\in\Z_+$ be such that
$a_1+{\dots}+a_{k-1}=n$. In order to prove Theorem~\ref{asop}, it
suffices to prove that there is a quadratic $\K$-algebra $R$ given
by $n$ generators and
$$
d=\max_{1\leq j\leq k-1}(a_1+{\dots}+a_j)(a_j+{\dots}+a_{k-1})
$$
relations such that $R$ is $k$-step nilpotent.
Let $X$ be an $n$-element set of generators. Since
$a_1+{\dots}+a_{k-1}=n$, we can present $X$ as the union of the
pairwise disjoint sets $A_1,\dots,A_{k-1}$ with $|A_j|=a_j$ for
$1\leq j\leq k-1$. Consider the set $M\subset X^2$ defined in
(\ref{M}) and the partial ordering $\prec$ on $M$ generated by the
partition $\{A_1,\dots,A_{k-1}\}$. For $1\leq j\leq k-1$, let
$B_j=\bigcup\limits_{q\geq j\geq m}A_q\times A_m$. Clearly,
$|B_j|=(a_1+{\dots}+a_j)(a_j+{\dots}+a_{k-1})$. Hence
$d=\max\{|B_1|,\dots,|B_{k-1}|\}$. By Lemma~\ref{esti},
$w(M,\prec)=d$. According to the Dilworth theorem (see \cite{gal}
for a short inductive proof) the width of a finite partially ordered
set $P$ is precisely the minimal number of chains needed to cover
$P$. Hence, we can write $M=\bigcup\limits_{q=1}^d C_q$, where each
$C_q$ is a chain in $M$. Now we consider the homogeneous degree 2
elements of $\K\langle X\rangle$ given by
$$
f_q=\sum_{(a,b)\in C_q}ab\ \ \text{for $1\leq q\leq d$}.
$$
Clearly $\supp(f_q)=C_q$. Thus the union of the supports of $f_q$ is
$M$ and each $\supp(f_q)$ is a chain in $M$. By Theorem~\ref{chin},
the algebra $R$ given by the relations $f_q$ for $1\leq q\leq d$ is
$k$-step nilpotent. This completes the proof of Theorem~\ref{asop}.
Now we shall prove Lemma~\ref{dnph}. By Theorems~GS and~\ref{asop},
$d_{n,k}\geq \phi_k n^2$ for every $k\geq 2$ and $n\in\N$. This
proves the first inequality in (\ref{estim}). It remains to prove
the second one. By Lemma~\ref{alpk}, there are
$\alpha_0,\dots,\alpha_{k-1}\in[0,1]$ such that
$0=\alpha_0<\alpha_1<{\dots}<\alpha_{k-1}=1$ and
$\alpha_j(1-\alpha_{j-1})=\phi_k$ for $1\leq j\leq k-1$. Now for
$0\leq j\leq k-1$ let $b_j=\lceil n\alpha_j-\frac12\rceil$. Clearly
$0=b_0\leq b_1\leq{\dots}\leq b_{k-1}=n$. Now we set
$a_j=b_j-b_{j-1}$ for $1\leq j\leq k-1$. Then $a_j\in\Z_+$ and
$a_1+{\dots}+a_{k-1}=n$. Hence
$$
d_{n,k}\leq\max_{1\leq j\leq
k-1}(a_1+{\dots}+a_j)(a_j+{\dots}+a_{k-1})=\!\!\max_{1\leq j\leq
k-1} b_j(n-b_{j-1})=\!\!\max_{1\leq j\leq k-1}\bigl\lceil
n\alpha_j-{\textstyle \frac12}\bigr\rceil \cdot \bigl\lfloor
n(1-\alpha_{j-1})+{\textstyle \frac12}\bigr\rfloor.
$$
It is easy to see that for every $\alpha,\beta\in[0,1]$,
$$
\textstyle\bigl\lceil n\alpha-\frac12\bigr\rceil \cdot \bigl\lfloor
n\beta+\frac12\bigr\rfloor -\alpha\beta n^2\leq
\frac{\alpha+\beta}{2}n+\frac14.
$$
From the last two displays and the equalities
$\alpha_j(1-\alpha_{j-1})=\phi_k$ it follows that
$$
d_{n,k}\leq \phi_k n^2+\frac n2\max_{1\leq j\leq
k-1}(1+\alpha_j-\alpha_{j-1})+\frac14.
$$
By Lemma~\ref{alpk}, the maximum in the above display equals
$\phi_k$. Thus $d_{n,k}\leq \phi_k n^2+\frac{1+\phi_k}{2}n+\frac14$,
which completes the proof of Lemma~\ref{dnph}.
\section{4-Step nilpotency and the Fibonacci numbers}
First, we derive an explicit formula for $d_{n,4}$.
\begin{lemma}\label{dn4} For every $n\in\N$,
\begin{equation}\label{dn4f}
d_{n,4}=\min\bigl\{\bigl\lceil {\textstyle
\frac{\sqrt5-1}{2}}n\bigr\rceil^2,n\bigl\lceil {\textstyle
\frac{3-\sqrt5}{2}}n\bigr\rceil\bigr\}.
\end{equation}
\end{lemma}
\begin{proof} Using (\ref{dnk}) with $k=4$ and denoting $a=a_1$ and
$b=a_3$, we obtain
$$
d_{n,4}=\min\{\max\{na,nb,(n-a)(n-b)\}:a,b\in\Z_+,\ a+b\leq n\}.
$$
An obvious symmetry consideration yields
$$
d_{n,4}=\min\{\max\{na,nb,(n-a)(n-b)\}:a,b\in\Z_+,\ b\leq a,\
a+b\leq n\}.
$$
Since $nb\leq na$ and $(n-a)(n-b)\geq (n-a)^2$ when $a,b\in\Z_+$
satisfy $b\leq a\leq n$, we have
\begin{equation}\label{dn4eq}
d_{n,4}=\min\{\max\{na,(n-a)^2\}:a\in\Z_+,\ 2a\leq n\}.
\end{equation}
Now, assume that $a\in\Z_+$ satisfies $2a\leq n$. Solving a
quadratic inequality we see that $na\geq (n-a)^2$ holds precisely
when $a\geq \phi_4n$. Hence (\ref{dn4eq}) can be rewritten as
$$
\begin{array}{l}d_{n,4}=\min\{a_n,b_n\},\ \ \text{where}\\
a_n=\min\{na:a\in\Z_+,\ \phi_4n\leq a\leq n/2\}\ \ \text{and}\ \
b_n=\min\{(n-a)^2:a\in\Z_+,\ a\leq \phi_4n\}.
\end{array}
$$
Clearly, the minimum in the definition of $a_n$ is attained for
$a=\lceil\phi_4 n\rceil$ and the minimum in the definition of $b_n$
is attained for $a=\lfloor \phi_4n\rfloor$. Hence
$a_n=n\lceil\phi_4 n\rceil$ and $b_n=\lceil(1-\phi_4)n\rceil^2$.
Using the equalities $\phi_4=\frac{3-\sqrt 5}{2}$ and
$1-\phi_4=\frac{\sqrt 5-1}{2}$, we see that (\ref{dn4f}) follows
from the above display.
\end{proof}
\begin{corollary}\label{ttt} The equality $d_{n,4}=\bigl\lceil\phi_4
n^2\bigr\rceil$ holds if and only if either $\bigl\lceil\phi_4
n^2\bigr\rceil$ is divisible by $n$ or $\bigl\lceil\phi_4
n^2\bigr\rceil$ is a square of a positive integer.
\end{corollary}
\begin{proof} Let $m=\bigl\lceil\phi_4 n^2\bigr\rceil$. From Lemma~\ref{dn4}
it follows that $d_{n,4}$ is always either divisible by $n$ or is a
square. Thus the equality $m=d_{n,4}$ can only hold if either $m$ is
divisible by $n$ or $m$ is a square.
If $m$ is divisible by $n$, we can write $m=nj$ for some $j\in\N$.
Now it is easy to see that $j=\bigl\lceil \frac{3-\sqrt
5}{2}n\bigr\rceil$ and therefore, by Lemma~\ref{dn4}, $d_{n,4}\geq
jn=m$. On the other hand, choosing $a=j$ and using (\ref{dn4eq}), we
get $d_{n,4}\leq \max\{nj,(n-j)^2\}=nj$. Thus $d_{n,4}=nj=m$.
If $m$ is a square, we can write $m=j^2$ for some $j\in\N$. Now it
is easy to see that $j=\bigl\lceil \frac{\sqrt 5-1}{2}n\bigr\rceil$
and therefore, by Lemma~\ref{dn4}, $d_{n,4}\geq j^2=m$. On the other
hand, choosing $a=n-j$ and using (\ref{dn4eq}), we get $d_{n,4}\leq
\max\{n(n-j),j^2\}=j^2$. Thus $d_{n,4}=j^2=m$.
\end{proof}
\begin{proof}[Proof of the first part of Theorem~{\rm\ref{k45}}]
Let $F_0,F_1,\dots$ be the Fibonacci sequence and
$\phi=\frac{\sqrt5+1}2$ be the golden ratio number. Using the
formula $F_n=\frac{\phi^n-(-\phi)^{-n}}{\sqrt 5}$ together with the
equality $\phi_4=\phi^{-2}$, one can easily verify that $\bigl\lceil
\phi_4 F_k^2\bigr\rceil=F_{k-1}^2$ if $k$ is odd and $\bigl\lceil
\phi_4 F_k^2\bigr\rceil=F_kF_{k-2}$ if $k$ is even. Thus if $n$ is a
Fibonacci number, then $\bigl\lceil \phi_4 n^2\bigr\rceil$ is either
divisible by $n$ or is a square.
To show the converse, we use the following criterion of recognizing
the Fibonacci numbers due to M\"obius \cite{mob}. It says that a
positive integer $n$ is a Fibonacci number if and only if the
interval $(\phi n-n^{-1},\phi n+n^{-1})$ contains an integer.
Furthermore, if $m$ is an integer belonging to $(\phi n-n^{-1},\phi
n+n^{-1})$, then $m$ is the next Fibonacci number after $n$.
First, assume that $n\in\N$ and $\bigl\lceil \phi_4 n^2\bigr\rceil$
is divisible by $n$. Then $\phi_4 n^2+\theta=nk$, where $k\in\N$ and
$0<\theta<1$. Since $\phi_4=2-\phi$, it follows that $\phi
n-(2n-k)=\frac{\theta}n$ and therefore $2n-k\in (\phi n-n^{-1},\phi
n+n^{-1})$. By the criterion of M\"obius, $n$ is a Fibonacci number.
Finally, assume that $\bigl\lceil \phi_4 n^2\bigr\rceil$ is a square
number. Since $\phi_4=\phi^{-2}$, this means that
$\frac{n^2}{\phi^2}+\theta=k^2$, where $k\in\N$ and $0<\theta<1$. It
immediately follows that $k=\bigl\lceil \frac{n}{\phi}\bigr\rceil$.
In other words $k=\frac{n}{\phi}+\alpha$ with $0<\alpha<1$. Squaring
the last equality, we get
$k^2=\frac{n^2}{\phi^2}+\theta=\frac{n^2}{\phi^2}+\frac{2n\alpha}{\phi}+\alpha^2$.
In particular, $\frac{2n\alpha}{\phi}<\theta<1$. Hence
$\phi\alpha<\frac{\phi^2}{2n}$. Thus the equality
$k=\frac{n}{\phi}+\alpha$ implies $n=\phi k-\phi\alpha$ and
$$
\phi\alpha<\frac{\phi^2}{2n}=\frac{\phi^2}{2(\phi k-\phi\alpha)}<
\frac{\phi^2}{2(\phi k-\phi^2/2n)}.
$$
Since $n\geq k$, we have
$$
\phi\alpha<\frac{\phi^2}{2(\phi k-\phi^2/2k)}<\frac1k,
$$
where the last inequality is satisfied for $k>2$. Now the above
display and the equality $n=\phi k-\phi\alpha$ imply that $n$
belongs to the interval $(\phi k-k^{-1},\phi k+k^{-1})$. By the
criterion of M\"obius, both $k$ and $n$ are Fibonacci numbers provided $k>2$.
If $k=1$ or $k=2$, a direct computation yields $n=2$ or $n=3$ respectively, which
are Fibonacci numbers as well.
Thus we have proven that $\bigl\lceil \phi_4 n^2\bigr\rceil$ is
either divisible by $n$ or is a square number precisely when $n$ is
a Fibonacci number. By Lemma~\ref{ttt}, $d_{n,4}=\bigl\lceil \phi_4
n^2\bigr\rceil$ if and only if $n$ is a Fibonacci number.
\end{proof}
\section{5-Step nilpotency}
In this section we prove the second part of Theorem~\ref{k45}. As in
the previous section we start by simplification the formula defining
$d_{n,5}$.
\begin{lemma}\label{D5} If $n\in\N$ is even, then $d_{n,5}=\frac n2\bigl\lceil
\frac{2n}{3}\bigr\rceil$. If $n\in\N$ is congruent to $-1$ modulo
$6$, then $d_{n,5}=n\bigl\lceil\frac{n(n+1)}{3n+1}\bigr\rceil$. If
$n\in\N$ is congruent to $1$ or to $3$ modulo $6$, then
$d_{n,5}=\frac{n+1}{2} \bigl\lceil\frac{2n^2}{3n+1}\bigr\rceil$.
\end{lemma}
\begin{proof} Using the symmetry in (\ref{dnk}) with respect to
reversing the order of $a_j$, we have
\begin{equation}\label{dn5}
\begin{array}{l}d_{n,5}=\min\{S(a):a\in\Z_+^4,\ a_1+a_2+a_3+a_4=n,\ a_1\leq
a_4\},\ \ \text{where}\\
S(a)=\max\{na_1,na_4,(a_1+a_2)(a_2+a_3+a_4),(a_1+a_2+a_3)(a_3+a_4)\}.
\end{array}
\end{equation}
It is easy to see that the minimum in (\ref{dn5}) can not be
attained when $a_2=0$ if $n>1$ (the case $n=1$ is trivial anyway).
If $a_1<a_4$ and $a_2>0$, one can easily check that $S(a')\leq
S(a)$, where $a'$ is obtained from $a$ by increasing $a_1$ by $1$
with simultaneous decreasing of $a_2$ by $1$. Similarly, if
$a_1=a_4$ and $|a_2-a_3|>1$, $S(a')\leq S(a)$, where $a'$ is
obtained from $a$ by increasing the smaller of $a_2$ and $a_3$ by
$1$ with simultaneous decreasing of the bigger one by $1$. It
follows that among $a\in\Z_+^4$ for which the minimum in (\ref{dn5})
is attained there must be at least one point satisfying $a_1=a_4$
and $|a_2-a_3|\leq 1$. Thus the minimum in (\ref{dn5}) is attained
at a point $a$ of the shape $a=(\alpha,\beta,\beta,\alpha)$ if $n$
is even and it is attained at a point $a$ of the shape
$a=(\alpha,\beta+1,\beta,\alpha)$ if $n$ is odd. Substituting this
data into (\ref{dn5}), we get
\begin{equation}\label{d5even}
d_{n,5}=\frac n2\min\{\max\{2a,n-a\}:a\in\Z_+,\ a\leq n/2\}\ \
\text{if $n$ is even}
\end{equation}
and
\begin{equation}\label{d5odd}
d_{n,5}=\min\{\max\{na,(n+1)(n-a)/2\}:a\in\Z_+,\ a\leq n/2\}\ \
\text{if $n$ is odd}.
\end{equation}
Since $\max\{2a,n-a\}=n-a$ if $3a\leq n$ and $\max\{2a,n-a\}=2a$ if
$3a\geq n$, (\ref{d5even}) implies that
$d_{n,5}=\min\bigl\{n\bigl\lceil\frac{n}3\bigr\rceil,\frac
n2\bigl\lceil\frac{2n}{3}\bigr\rceil\bigr\}=\frac
n2\bigl\lceil\frac{2n}{3}\bigr\rceil$ if $n$ is even (the two
numbers in the last minimum are equal in all cases except for the
numbers $n$ congruent to $-2$ modulo $6$ in which case the second
one is less by $1$).
Next, $\max\{na,(n+1)(n-a)/2\}=(n+1)(n-a)/2$ if $a\leq
\frac{n(n+1)}{3n+1}$ and $\max\{na,(n+1)(n-a)/2\}=na$ if $a\geq
\frac{n(n+1)}{3n+1}$. Plugging this into (\ref{d5odd}), we get
$d_{n,5}=\min\{n\bigl\lceil\frac{n(n+1)}{3n+1}\bigr\rceil,\frac{n+1}{2}
\bigl\lceil\frac{2n^2}{3n+1}\bigr\rceil\}$. Considering the cases of
$n$ being $1$, $3$ and $-1$ modulo $6$ separately, we see that
$d_{n,5}=n\bigl\lceil\frac{n(n+1)}{3n+1}\bigr\rceil$ if $n$ is
congruent to $-1$ modulo $6$ and $d_n=\frac{n+1}{2}
\bigl\lceil\frac{2n^2}{3n+1}\bigr\rceil$ ff $n\in\N$ is congruent to
$1$ or to $3$ modulo $6$.
\end{proof}
From Lemma~\ref{D5} it immediately follows that
$d_{n,5}=\frac{n^2}{3}=\phi_5 n^2$ if $6$ is a factor of $n$.
Considering the exact formula provided by Lemma~\ref{D5} and
treating the possible remainders for the division of $n$ by $6$ as
separate cases, one easily sees that $d_{n,5}-\frac{n^2}{3}\geq1$
and therefore $d_{n,5}>\lceil \phi_5n^2\rceil$ if $n$ is not
divisible by $6$ and $n\geq 3$. It is easy to verify that the
equality $d_{n,5}=\bigl\lceil\phi_5 n^2\bigr\rceil$ holds for $n=1$
and for $n=2$. This completes the Proof of Theorem~\ref{k45}.
\bigskip
We conclude by reminding that the following particular cases of the
Anick's conjecture \cite{ani1} remain unproved.
\begin{conj}\label{con1} There is a $k$-step nilpotent $\K$-algebra given by $n$
generators and $d$ quadratic relations whenever $d\geq \phi_kn^2$.
\end{conj}
\begin{conj}\label{con2} There is a finite dimensional
$\K$-algebra given by $n$ generators and $d$ quadratic relations
whenever $d>\frac{n^2}{4}$.
\end{conj}
\bigskip
{\bf Acknowledgements}
We are grateful to the Max-Planck-Institute for Mathematics in Bonn and to IHES, where parts of this research have been done, for hospitality, support, and excellent research atmosphere.
This work is funded by the ERC grant 320974, and partially supported by the project PUT9038.
We also would like to thank a referee for valuable suggestions, which helped to improve the presentation.
\small\rm
|
1,116,691,499,342 | arxiv | \section{#1}}
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\begin{document}
\large
\parskip2pt
\parindent20pt
\mathsurround1pt
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\begin{center}
{\tif Surveying points in the complex projective plane}
\vskip15pt
{\auf Lane Hughston and Simon Salamon}
\vskip25pt
\begin{quote}\small
We classify SIC-POVMs of rank one in $\mathbb{CP}^2,$ or equivalently sets of
nine equally-spaced points in $\mathbb{CP}^2,$ without the assumption of group
covariance. If two points are fixed, the remaining seven must lie on a
pinched torus that a standard moment mapping projects to a circle in
$\mathbb{R}^3.$ We use this approach to prove that any SIC set in $\mathbb{CP}^2$ is
isometric to a known solution, given by nine points lying in
triples on the equators of the three 2-spheres each defined by the
vanishing of one homogeneous coordinate. We set up a system of
equations to describe hexagons in $\mathbb{CP}^2$ with the property that any
two vertices are related by a cross ratio (transition probability) of
$1/4.$ We then symmetrize the equations, factor out by the known
solutions, and compute a Gr\"obner basis to show that no SIC sets
remain. We do find new configurations of nine points in which 27 of
the 36 pairs of vertices of the configuration are equally spaced.
\end{quote}
\end{center}
\vskip10pt
\section*{Introduction}
A symmetric, informationally complete, positive-operator valued
measure or SIC-POVM on the Hermitian vector space $\mathbb{C}^n$ is a set
$\{P_j\}$ of $n^2$ rank-one projection operators such that
\begin{equation*}
\frac1n\!\sum_{i=1}^{n^2} P_i = I,
\end{equation*}
and
\begin{equation*}
\mathop{\mathrm{tr}}(P_jP_k)=\frac1{n+1}(n\delta_{jk}+1)
\end{equation*}
for all $j,k.$ Such objects attracted wide attention following
conjectures about their existence made by Zauner \cite{Z} in 1999 and
Renes \emph{et al.\ } \cite{RSC} in 2004, and since then have been investigated
by a large number of authors, along with higher rank versions and the
allied concept of mutually unbiased basis. See, for example,
\cite{App,AFZ,AYZ,DBBA,Durt,Gour,Grl,Hogg,Hugh,SG,Woo,Zhu}, and
references cited therein.
SIC-POVMs arise in the theory of quantum measurement (see Davies
\cite{Dav} and Holevo \cite{Hol} for the significance of general
POVMs), and are of great interest in connection with their potential
applications to quantum tomography. The idea is the following. Suppose
that one has a large number of independent identical copies of a
quantum system (say, a large molecule), the state (or `structure') of
which is unknown and needs to be determined. A SIC-POVM can be thought
of as a kind of symmetrically oriented machine that can be used to
make a single tomographic measurement on each independent copy of the
molecule, with the property that once the results of the various
measurements have been gathered for a sufficiently large number of
molecules, the state of the molecule can be efficiently determined to
a high degree of accuracy. The `symmetric orientation' is not with
respect to ordinary three-dimensional physical space (as in the
classical tomography of medical imaging), but rather with respect to
the space of pure quantum states.
Since each element $P_i$ of a SIC-POVM is a matrix of rank one and
trace unity, it determines a point in complex projective space
$\mathbb{CP}^{n-1}.$ It is well known that a SIC-POVM can then be defined as a
configuration of $n^2$ points in $\mathbb{CP}^{n-1}$ that are mutually
equidistant under the standard K\"ahler metric \cite{LS,Wel}. This is
the definition that we shall adopt in \S\ref{HW}, and the distance is
determined by Lemma~\ref{CSI}. Such a set of points is often called a
`SIC', but we favour the expression `SIC set'.
The existence of such configurations (for example, nine equidistant
points in $\mathbb{CP}^2,$ or sixteen equidistant points in $\mathbb{CP}^3$) is
counterintuitive to our everyday way of thinking in which a regular
simplex in $\mathbb{R}^n$ has $n+1$ vertices (but see \cite{GKMS}). It has
been conjectured that $\mathbb{CP}^{n-1}$ possesses such a configuration for
every $n$ \cite{RSC,Z}. There is evidence for this for $n$ up to at
least $67,$ and various explicit solutions have been found in lower
dimensions. Most of the known SIC sets in higher dimensions are
constructed as orbits of a Heisenberg group $W\times H$ acting on
$\mathbb{CP}^{n-1}$ (see Section~\ref{HW}), and representative vectors occur
as eigenvectors of an isometry that is an outer automorphism of $W\times H.$
In the case $n=5$, the automorphisms of $W\times H$ play a key role in the
construction of the celebrated Horrocks-Mumford bundle over $\mathbb{CP}^4$ in
\cite{HM}, which is an excellent reference for this group theory. In
the case $n=3$ (and more generally, when $n$ is prime) any finite
group of isometries whose orbit is a SIC set must be conjugate to
$W\times H$ \cite{Zhu}, but in this paper we work without the assumption of
group covariance (see Grassl \cite{Grl}).
The space $\mathbb{CP}^1,$ endowed with the Fubini-Study metric, is isometric
to the standard two-sphere, and embedding this in $\mathbb{R}^3$ is a simple
example of the representation of $\mathbb{CP}^{n-1}$ as an adjoint orbit in
the Lie algebra $\mathfrak{su}(n)$ of its isometry group. The existence of a SIC
set can then be interpreted as a statement about the placement of such
orbits. The problem can also be formulated so as to apply to more
general (co-)adjoint orbits in a Lie algebra.
The vertices of any inscribed regular tetrahedron in $S^2$ provide a
SIC set for $\mathbb{CP}^1$ ($n=2$). The situation for the projective plane
$\mathbb{CP}^2$ is already surprisingly intricate, and the case $n=3$ is
characterized by the existence of continuous families of non-congruent
SIC sets. It is easy to begin their study. Using homogeneous
coordinates, any three equally-spaced points on the equator
$\{[0,z_2,z_3]:|z_2|=|z_3|\}$ of the two-sphere $z_1=0$ lie in a SIC
set formed by adding three equally-spaced points from each of the
equators of the two-spheres $z_2=0$ and $z_3=0.$ If the diameter of
$\mathbb{CP}^2$ is chosen to be $\pi,$ all nine points are a distance $2\pi/3$
apart. Moreover, if the three triples match up so as to lie on a total
of twelve projective lines, the nine points are the flexes of a plane
cubic curve \cite{Hugh}.
In this paper, we show that any SIC set in $\mathbb{CP}^2$ is congruent to one
of those just described (see Theorem~\ref{strong}). This result will
not surprise the experts; it has perhaps been verified numerically,
and is apparently a consequence of computer-aided results in
\cite{Sz}. Our proof relies on a computation for its final step but is
predominantly analytical. We use the two-point homogeneity of $\mathbb{CP}^2$
to fix two points of a SIC set; applying the moment mapping relative
to a maximal torus shows that the remaining seven points lie in a
pinched torus above a circle $\mathscr{C}$ in $\mathbb{R}^3$ (illustrated in
Figure~1). We exhibit the known solutions in a different form
(Proposition~\ref{7}) and characterize them by a symmetry condition
(Lemma~\ref{LR}). Adding three more points a distance $2\pi/3$ from
the first two and from each other leads to a polynomial equation that
is symmetric in three variables $x,y,z$ that represent the tangents of
angles measured around $\mathscr{C}$ (Theorem~\ref{fa}). The resulting
geometry is illustrated in Section~\ref{Gi}.
Adding a sixth point allows us to write down four equations in four
variables $t,x,y,z.$ When these are totally symmetrized, we obtain a
system that represents a necessary condition for the six points to
form part of a SIC set. For the known solutions, at least one of the
four points on the pinched torus must project to $\mathscr{C}$ with an angle
equal to $\pm\pi/6.$ This fact enables us to focus attention on the
so-called quotient ideal that parametrizes `extra' solutions, and to
describe it by means of an appropriate Gr\"obner basis. Once one root
$t$ is fixed, the extra solutions form a finite set and the final step
is to determine its size. There are too few extra solutions for these
to arise from an undiscovered SIC set.
This paper had its origins in a number of survey talks aimed at
bringing elements of the SIC-POVM problem in various low dimensions to
the attention of a wider audience, and the title and figures reflect
this. We focus on the case $n=3$ from Section~\ref{mt} onwards, and
Sections~6--10 contain the more specialized material required to
achieve our goal. The Fubini-Study metric on an ambient projective
space plays a central role in the construction or approximation of
K\"ahler-Einstein metrics on algebraic varieties, and it is our hope
that more general theory may shed further light on the discrete
problem outlined above.
\sect{Hermitian preliminaries}
\noindent We begin with a few remarks to fix conventions. The complex
vector space
\begin{equation}
\mathbb{C}^n=\{\mathbf{z}=(z_1,\ldots z_n)^\top:z_i\in\mathbb{C}\}
\end{equation}
of column vectors comes equipped with a Hermitian form
\begin{equation}\left<\mathbf{w},\mathbf{z}\right> = \left<\mathbf{w}|\,\mathbf{z}\right> =
\sum_{i=1}^n\overline w_iz_i\end{equation}
which is anti-linear in the first (bra) position. Each fixed $\mathbf{w}$
defines a linear functional $\mathbf{z}\mapsto\left<\mathbf{w},\mathbf{z}\right>,$ and
\begin{equation}
\mathbf{w}\ \mapsto\ \left<\mathbf{w},\ \hbox{\footnotesize$\bullet$}\ \right>
\end{equation} is an anti-linear
bijective mapping $h\colon V \to V^*,$ equivalently an
\emph{isomorphism} $V\cong\overline V^*$ of complex vector spaces. Complex
projective space is the quotient
\begin{equation}
\mathbb{CP}^{n-1}=\frac{\mathbb{C}^n\setminus 0}{\mathbb{C}^*},
\end{equation}
consisting of one-dimensional subspaces of $\mathbb{C}^n$ or \emph{rays}, and
is a compact topological space. For any non-zero $\mathbf{w}\in\mathbb{C}^n$ the
associated point in $\mathbb{CP}^{n-1}$ will be denoted by $[\mathbf{w}].$ Each such
point determines a conjugate hyperplane $W$ defined by
\begin{equation}
W=\mathbb{P}(\ker h(\mathbf{w}))\cong\mathbb{CP}^{n-2}\subset\mathbb{CP}^{n-1}.
\end{equation} This is
the geometrical content of the map $h.$
Two points $[\mathbf{w}],[\mathbf{z}]$ lie on a unique projective line
$\mathbb{L}\cong\mathbb{CP}^1.$ The associated conjugate hyperplanes $W,Z$ intersect
$\mathbb{L}$ in $[\mathbf{w}'],[\mathbf{z}'],$ where
\begin{equation}
\mathbf{w}' = \left<\mathbf{w},\mathbf{z}\right>\mathbf{w}-\left<\mathbf{w},\mathbf{w}\right>\mathbf{z},
\qquad
\mathbf{z}' = \left<\mathbf{z},\mathbf{z}\right>\mathbf{w}-\left<\mathbf{z},\mathbf{w}\right>\mathbf{z}.
\end{equation}
The resulting four points, taken in the order $[\mathbf{w}],[\mathbf{z}],[\mathbf{z}'],[\mathbf{w}'],$
have inhomogeneous coordinates
\begin{equation} \infty,\quad 0,\quad
-\big<\mathbf{z},\mathbf{z}\big>/\big<\mathbf{z},\mathbf{w}\big>,\quad
-\left<\mathbf{w},\mathbf{z}\right>/\big<\mathbf{w},\mathbf{w}\big>,\end{equation}
and a real cross ratio
\begin{equation}
\kappa([\mathbf{w}],[\mathbf{z}]) =
\frac{\big<\mathbf{w},\mathbf{z}\big>\big<\mathbf{z},\mathbf{w}\big>}{\big<\mathbf{w},\mathbf{w}\big>\big<\mathbf{z},\mathbf{z}\big>}
= \frac{|\big<\mathbf{w},\mathbf{z}\big>|^2}{\|\mathbf{w}\|^2\|\mathbf{z}\|^2}\in[0,1].
\end{equation}
When the points of $\mathbb{CP}^{n-1}$ are interpreted as pure quantum states,
$\kappa$ can be regarded as a transition probability \cite{AshS,
BH,Gib, Hugh2,Hugh3}. The Fubini-Study distance $d$ between the
points $[\mathbf{w}]$ and $[\mathbf{z}]$ is defined by expressing the cross ratio as
$\cos^2(d/2),$ so that
\begin{equation}\label{FS}
d([\mathbf{w}],[\mathbf{z}]) =
2\arccos\left(\frac{|\big<\mathbf{w},\mathbf{z}\big>|}{\|\mathbf{w}\|\|\mathbf{z}\|}\right)\in[0,\pi].
\end{equation}
When $n=2$ we get $\mathbb{CP}^1\cong S^2.$ We shall see in Example~\ref{abc}
that $d$ is the spherical distance
\begin{equation}
\theta=\arccos\big|\big<\mathbf{u},\mathbf{v}\big>\big|,\qquad \mathbf{u},\mathbf{v}\in S^2,
\end{equation}
measuring the arclength of a great circle joining $\mathbf{u}$ and $\mathbf{v}.$ The
$\mathbb{CP}^1$ calculation confirms that $d$ is the usual distance measured
along geodesics of $\mathbb{CP}^{n-1}$ since any two points of the latter lie
on a unique projective line $\mathbb{CP}^1.$ The distance \eqref{FS} satisfies
the triangle inequality
\begin{equation}
d([\mathbf{w}],[\mathbf{z}])\leqslant d([\mathbf{w}],[\mathbf{y}])+d([\mathbf{y}],[\mathbf{z}]).
\end{equation}
This can be verified by working inside the $\mathbb{CP}^2$ that contains
$[\mathbf{w}],$ $[\mathbf{y}],$ $[\mathbf{z}].$
The so-called Fubini-Study metric is the square $ds^2$ of the
infinitesimal distance between $[\mathbf{z}]$ and $[\mathbf{z}+d\mathbf{z}],$ computed using
\begin{equation}\label{taylor}
\begin{array}{rcl}
\kappa([\mathbf{z}],[\mathbf{z}+d\mathbf{z}]) &=& \displaystyle
\frac{\|\mathbf{z}\|^2+2\mathop{\mathrm{Re}}\big<\mathbf{z},d\mathbf{z}\big>+|\big<\mathbf{z},d\mathbf{z}\big>|^2/\|\mathbf{z}\|^2}
{\|\mathbf{z}\|^2+2\mathop{\mathrm{Re}}\big<\mathbf{z},d\mathbf{z}\big>+\|d\mathbf{z}\|^2}\\[20pt]
&=& \displaystyle
1-\frac{\|d\mathbf{z}\|^2}{\|\mathbf{z}\|^2}+\frac{|\big<\mathbf{z},d\mathbf{z}\big>|^2}{\|\mathbf{z}\|^4}
+ O(\|d\mathbf{z}\|^3).
\end{array}
\end{equation}
There are no first-order terms, and we obtain the Riemannian metric
$g=ds^2$ where
\begin{equation}
ds^2\ =\ 4\frac{\|\mathbf{z}\|^2\|d\mathbf{z}\|^2-|\big<\mathbf{z},d\mathbf{z}\big>|^2}{\|\mathbf{z}\|^4}.
\end{equation}
If we set $z_n=1,$ and use the summation convention over the remaining
indices $z_1,\ldots,z_{n-1},$ then in the traditional notation we have
\begin{equation}
g_{\alpha\beta}dz^\alpha d\overline z^\beta\ =\
4\frac{(\overline z_\alpha z^\alpha+1)dz_\beta d\overline z^\beta-
\overline z_\alpha z_\beta dz^\alpha d\overline z^\beta}{(\overline z_\alpha z^\alpha+1)^2}.
\end{equation}
See, for example, Arnold \cite{Arn} and Kobayashi and Nomizu
\cite{KN}. When $n=2,$ we obtain the classical first fundamental form
\begin{equation}
ds^2\ =\ \frac{4\,dz d\overline z}{(1+|z|^2)^2}\ =\
\frac{8(dx^2+dy^2)}{(1+x^2+y^2)^2}
\end{equation}
on the two-sphere $S^2,$ in which $x,y$ are isothermal coordinates.
\sect{The special unitary group}
\noindent The Hermitian form $h$ is invariant under the action of the
unitary group
\begin{equation}
\mathrm{U}(n)=\{X\in\mathbb{C}^{n,n}:\overline X\!X^\top=I\}.
\end{equation}
Its centre consists of scalar multiples $e^{it}I$ that act trivially
on $\mathbb{CP}^{n-1}.$ So we consider the special unitary group
\begin{equation}
\mathrm{SU}(n)=\{X\in\mathrm{U}(n):\det X=1\},
\end{equation}
whose centre is $\mathbb{Z}_n=\big<e^{2\pi i/n}I\big>.$ The next result is due
to Wigner \cite{Wig}; a modern treatment is given in \cite{Fr}.
\begin{theorem}
The isometry group of the Fubini-Study space $\mathbb{CP}^{n-1},$ i.e.\ the
group of bijections preserving the distance $d,$ is generated by
$\mathrm{SU}(n)/\mathbb{Z}_n$ and $[\mathbf{z}]\mapsto[\overline\mathbf{z}].$
\end{theorem}
The Lie algebra $\mathfrak{su}(n)$ can (as a vector space) be defined as the
tangent space $T_I\mathrm{SU}(n)$ at the identity. It consists of tangent
vectors $A=\dot X_0$ to curves $X_t=I+tA+O(t^2)$ in $U(n).$ Thus
\begin{equation}
\mathfrak{su}(n)=\{A\in \mathbb{C}^{n,n}:\overline A+A^\top=0,\ \mathop{\mathrm{tr}} A=0\}.
\end{equation}
A matrix $M\in\mathrm{SU}(n)$ acts on $\mathfrak{su}(n)$ by the adjoint representation
\begin{equation}
A\mapsto M\!AM^{-1}=M\!AM^\top.
\end{equation}
The space $\mathfrak{su}(n)$ carries an invariant inner product
\begin{equation} \big<A,B\big>=-\mathop{\mathrm{tr}}(AB),\end{equation}
and $\mathrm{SU}(n)$ itself carries a bi-invariant Riemannian invariant.
We shall work with the corresponding affine space
\begin{equation}
\mathscr{H}_n=\{A\in \mathbb{C}^{n,n}: \overline A=A^\top,\ \mathop{\mathrm{tr}} A=1\}
\end{equation}
of Hermitian matrices of trace one. There is an obvious bijection
\begin{equation}\label{obv}
\mathscr{H}_n\ \stackrel\cong\longrightarrow\ \mathfrak{su}(n),
\end{equation}
given by $A\mapsto i(A-n^{-1}I).$
\bigbreak
The \emph{canonical embedding} of $\mathbb{CP}^{n-1}$ into $\mathscr{H}_n$ is a
variant of the moment mapping for the adjoint action of $\mathrm{SU}(n).$ To
describe it, assume for convenience that all vectors are
normalized. Thus, we set $\|\mathbf{z}\|=1$ ($\mathbf{z}\in S^{2n-1}$) and there
remains only a phase ambiguity in passing to a point $[\mathbf{z}]=[e^{it}\mathbf{z}]$
of $\mathbb{CP}^{n-1}.$ Map $[\mathbf{z}]$ to
\begin{equation}\label{can}
P_\mathbf{z} = \mathbf{z}\overline\mathbf{z}^\top = \textstyle
\left(\begin{array}{l}
|z_1|^2 \quad z_1\overline z_2 \quad z_1\overline z_3\cdots\\
z_2\overline z_1 \quad |z_z|^2 \quad z_2\overline z_3\cdots\\
z_3\overline z_1\quad \cdots\\
\cdots\quad \cdots\end{array}\right),
\end{equation}
which is a projection operator (meaning $P^2=P$) of rank one.
The injective map
\begin{equation}\label{i}
i\colon\mathbb{CP}^{n-1}\hookrightarrow\mathscr{H}_n
\end{equation}
defined by $[\mathbf{z}]\mapsto P_\mathbf{z}$ is $\mathrm{SU}(n)$-equivariant. We can use it
to measure distances since
\begin{equation}
\kappa([\mathbf{w}],[\mathbf{z}]) = \big|\big<\mathbf{w},\mathbf{z}\big>\big|^2 = \mathop{\mathrm{tr}}(P_\mathbf{w} P_\mathbf{z}),
\end{equation}
assuming $\|\mathbf{z}\|=1=\|\mathbf{w}\|.$ Moreover, the derivative
\begin{equation}
i_*\colon T_x\mathbb{CP}^{n-1}\hookrightarrow T_x\mathscr{H}_n\cong\mathbb{R}^N
\end{equation}
is $\mathrm{U}(n-1)$-equivariant, and \eqref{i} is an isometric embedding.
\begin{exa} [The Bloch sphere] \label{abc} \rm
For $n=2,$ the image of this map consists of
the matrices
\begin{equation} \left(\!\begin{array}{cc}
|z_1|^2 & \overline z_1z_2\\[3pt] z_1\overline z_2 & |z_2|^2\end{array}\!\right) =
\fr12\!\left(\!\begin{array}{cc} 1+a & b+ic\\[2pt] b-ic &
1-a\end{array}\!\right)\end{equation} with $|z_1|^2+|z_2|^2=1$ and
$a^2+b^2+c^2=1.$ This provides the well-known isomorphism $\mathbb{CP}^1\cong
S^2.$ The angle $\theta$ between two unit vectors in $\mathbb{R}^3$ is given by
\begin{equation}
aa'+bb'+cc'=\cos\theta.
\end{equation}
The inner product in $\mathscr{H}_2$ is then
\begin{equation}
\textstyle \fr12(1+aa'+bb'+cc')=\fr12(1+\cos\theta)=\cos^2(\theta/2).
\end{equation}
But $\theta$ is also the standard distance $d$ along the great circle on
the surface of the sphere joining the endpoints of the two unit
vectors.
\end{exa}
In the example above, fix (say) the north pole $p\in S^2,$ and
consider the function $\kappa_p=\sin^2(\theta/2)$ where $\theta$ is now
latitude in radians. Its gradient $\nabla\kappa_p$ is tangent to the
meridians joining $p$ to the south pole $p',$ whereas
$J(\nabla\kappa_p)$ is a vector field that represents rotation about
$pp'.$ This situation is generalized to higher dimensions as follows.
The composition
\begin{equation}
\mathbb{CP}^{n-1}\ \longrightarrow\ \mathscr{H}_n\ \stackrel\cong\longrightarrow\ \mathfrak{su}(n),
\end{equation}
where $[\mathbf{z}]\mapsto i(P_\mathbf{z}-n^{-1}I),$ is a \emph{moment mapping} of the
type determined whenever a Lie group acts on a symplectic
manifold. The image (isomorphic to $\mathbb{CP}^{n-1}$) inside $\mathfrak{su}(n)$ is an
orbit for the action of $\mathrm{SU}(n).$ Any such adjoint orbit carries a
K\"ahler metric by general principles. Fix a point
$p=[\mathbf{z}]\in\mathbb{CP}^{-1},$ and consider the function $\kappa_p$ defined by
\begin{equation}
\kappa_p([\mathbf{w}])=\kappa([\mathbf{z}],[\mathbf{w}])=\mathop{\mathrm{tr}}(P_\mathbf{z} P_\mathbf{w}).
\end{equation}
We have
\begin{prop}
The rotated gradient $J(\nabla\kappa_p)$ is the infinitesimal isometry
(Killing field) associated to $i(P_\mathbf{z}-n^{-1}I).$
\end{prop}
For further details of various aspects of the K\"ahlerian geometry of
the space of pure quantum states, see Anandan and Aharonov \cite{AA},
Ashtekar and Schilling \cite{AshS}, Bengtsson and Zyczkowski
\cite{BZ}, Brody and Hughston \cite{BH}, Gibbons \cite{Gib}, Hughston
\cite{Hugh2,Hugh3}, and Kibble \cite{Kib}.
\sect{Sets of points in projective space}\label{HW}
\noindent We choose to begin with
\begin{defi}
A SIC-POVM or SIC set is a collection $\mathbb{S}$ of $n^2$ points $[\mathbf{z}_i]$
in $\mathbb{CP}^{n-1}$ that are mutually equidistant, so if $\|\mathbf{z}_i\|=1$ then
\begin{equation}
|\big<\mathbf{z}_i,\mathbf{z}_j\big>|^2=\kappa,\qquad i\ne j,
\end{equation}
for some fixed cross ratio $\kappa\in[0,1).$
\end{defi}
We can associate to $[\mathbf{z}_i]$ the point $P_i=P_{[\mathbf{z}_i]}$ in $\mathscr{H}_n.$ A SIC set then
consists of a regular simplex embedded in
\begin{equation} \mathscr{H}_n\cong\mathfrak{su}(n)\cong\mathbb{R}^N,\quad N=n^2\-1\end{equation}
with $n^2$ vertices $\{P_i\}$ that lie in the adjoint orbit
$\mathbb{CP}^{n-1}.$ The latter requirement is the crucial one, since a
regular simplex with $n^2$ vertices in $\mathbb{R}^N$ is readily obtained by
projecting an arbitrary orthonormal basis of $\mathbb{R}^{N+1}.$
Do SIC sets exist?
\begin{exa}\rm
A SIC set in $\mathbb{CP}^1=S^2$ is an inscribed tetrahedron in the
two-sphere. Any two such tetrahedrons are congruent by
$\mathrm{SO}(3)=\mathrm{SU}(2)/\mathbb{Z}_2,$ though that does not stop us seeking the
`neatest' set of vertices to write down. One set is
\begin{equation}\Big\{\ [0,1],\quad [\sqrt2,1],\quad
[\sqrt2,\omega],\quad [\sqrt2,\omega^2]\ \Big\},\end{equation} where
$\omega=e^{2\pi i/3}.$ Another set of vertices, which is perhaps less obvious, is
\begin{equation}\label{tet2}
\Big\{\ [1,\varpi],\quad [\varpi, 1],\quad
[1,-\varpi],\quad [\varpi,-1]\ \Big\},
\end{equation}
where $\varpi=(1+i)/(1+\sqrt3).$ This second set nevertheless plays an
important role, as we shall see.
\end{exa}
If $n\ge3,$ any two SIC sets in $\mathbb{CP}^{n-1}\subset\mathbb{R}^N$ are congruent by
$\mathrm{SO}(N)$ (where $N=n^2-1$), but \emph{not} in general by $\mathrm{SU}(n).$
One can present more SIC sets by generalizing the second tetrahedron
\eqref{tet2}. We define two cyclic groups of order $n.$ Let $W$ be the
group generated by the cyclic permutation
\begin{equation}\label{cyc}
[z_1,z_2,\ldots,z_n]\mapsto[z_n,z_1,\ldots,z_{n-1}];
\end{equation}
let $\omega=e^{2\pi i/n},$ and denote by $H$ the group generated by
\begin{equation}\label{run}
[z_1,z_2,\ldots,z_n]\mapsto[z_1,\omega z_2,\ldots,\omega^{n-1}z_n].
\end{equation}
$W\times H$ acts on $\mathbb{CP}^{n-1}$ as a subgroup of $\mathrm{SU}(n)$ isomorphic to
$\mathbb{Z}_n\times\mathbb{Z}_n.$ This subgroup is sometimes called the
Weyl-Heisenberg group after \cite{Weyl}. It can be regarded as the
projectivization of an extended finite group, namely the Heisenberg
group of three-by-three matrices with coefficients in the ring $\mathbb{Z}_n.$
For this reason, it is legitimate to refer to the action of $W\times H$
simply as that of the \emph{Heisenberg group}. The following two
results can be verified by direct calculation:
\begin{prop}\label{011}
The orbit
\begin{equation}
(W\times H)\cdot[0,1,1]
\end{equation}
is a SIC set consisting of nine points in $\mathbb{CP}^2.$
\end{prop}
\begin{prop}
Let $r=\sqrt2$ and $s=\sqrt{2+\sqrt5}.$ Then
\begin{equation}
(W\times H)\cdot[-s-i(r\+s),\ 1\-r+i,\ s+i(s\-r),\ 1+r+i]
\end{equation}
is a SIC set of sixteen points in $\mathbb{CP}^3.$
\end{prop}
\noindent An element $\mathbf{z}\in\mathbb{C}^n$ such that the orbit $(W\times H) \cdot[\mathbf{z}]$
is a SIC set is called a \emph{fiducial vector} for the action of
$W\times H.$\smallbreak
In his 1999 Vienna PhD thesis \cite{Z}, Zauner made a number of
conjectures that extended the basic
\begin{conj}
$\mathbb{CP}^{n-1}$ possesses a SIC set for all $n.$
\end{conj}
\noindent It is widely believed that such a set can always be realized
as an orbit of $W\times H,$ and that the number of non-congruent solutions
(meaning solutions that are not related to one another by an isometry
or element of $\mathrm{SU}(n)$) increases with $n.$ There are sporadic
constructions of SIC sets using different finite groups (see
Remark~\ref{HP3}).\medbreak
Explicit algebraic solutions are known for
$n=2,3,4,\ldots,15,19,24,35$ and $48,$ from work of Zauner \cite{Z},
Appleby \cite{App}, Renes \emph{et al.\ } \cite{RSC}, Flammia \cite{Fla},
Grassl \cite{Grl}, Zhu \cite{Zhu}, and many other authors (see
\cite{AFZ,DBBA} and references cited therein). All such examples lie
(up to isometry) in solvable extensions of $\mathbb Q$
\cite{AYZ}. Extensive numerical verification has been carried out for
$n\le67$ (Scott and Grassl \cite{SG}).
The next result is well known, but we include it
for completeness. Let $\{[\mathbf{z}_i]\}$ be a SIC set in $\mathbb{CP}^{n-1}$ and
$\{P_i\}$ its image in $\mathscr{H}_n.$ Recall that $\mathop{\mathrm{tr}}(P_iP_j)=\kappa$ if $i\ne
j.$ Thus $\kappa$ is the cross ratio or transition probability between
any two points in the SIC set.
\begin{lem}\label{CSI}
Any SIC set in $\mathbb{CP}^{n-1}$ satisfies
$\kappa=1/(n+1),$ and \begin{equation}\frac1n\sum P_i=I.\end{equation}
\end{lem}
\begin{proof}
Define $Q_j=P_j-\kappa I.$ Then
\begin{equation} \mathop{\mathrm{tr}}(P_iQ_j)=
\left\{\begin{array}{ll} 1-\kappa\quad & i=j\\0& i\ne j\end{array}\right.\end{equation}
So $(P_i)$ is a basis of $i\mathfrak{u}(n)$ (called a \emph{quorum})
and we can set
\begin{equation}\label{I}
I=\sum_{i=1}^{n^2} c_iP_i.
\end{equation}
Applying $\mathop{\mathrm{tr}}(Q_j\,\cdot),$ we get $1-\kappa n=c_j(1-\kappa),$ so all the
$c_i$ are equal. To complete the proof, take the trace of
\eqref{I}. This gives
\begin{equation} n=n^2\frac{1-\kappa n}{1-\kappa},\end{equation}
and $\kappa=(1-n)/(1-n^2)=1/(1+n).$
\end{proof}
It will be convenient in our analysis of SIC sets to introduce the
following.
\begin{defi}\label{cs}
Two points in $\mathbb{CP}^{n-1}$ will be said to be `correctly separated' if
the cross ratio that they define equals $1/(n+1).$
\end{defi}
Suppose that $\mathbb{CP}^{n-1}$ admits a SIC set $\mathbb{S}.$ Then, up to isometry,
two points form part of a SIC set if and only if they are correctly
separated. This follows from the fact that $\mathbb{CP}^{n-1}$ is a
\emph{two-point homogeneous space}, meaning that there exists an
isometry that maps any two points to any other two points the same
distance apart \cite{Wang}. The lemma above is then a key result that
enables one to go some way in attempting to construct a SIC set
without knowing for sure that it exists.
\begin{rem}\label{HP3}\rm
Lemma~\ref{CSI} precludes the existence of four or more points of a
SIC set from lying on a projective line $\mathbb{CP}^1$ whenever $n\ge3,$
since their cross ratio would have to be that for $n=2,$ namely $1/3.$
An application relates to the SIC set in $\mathbb{CP}^7$ constructed by Hoggar
\cite{Hogg}. It consists of a $(\mathbb{Z}_2)^6$ orbit of 64 points that the
Hopf fibration $\pi\colon\mathbb{CP}^7\to\mathbb{HP}^3$ projects down to an equal
number of points in the quaternionic projective space $\mathbb{HP}^3.$ It
would be impossible to find a SIC set in $\mathbb{CP}^7$ with four points in
each fibre of $\pi,$ but we wonder whether there exists a SIC set
arising from 32 points in $\mathbb{HP}^3$ with two points in each fibre. Such
questions are related to work by Armstrong \emph{et al.\ } on twistor lifts
\cite{APS}.
\end{rem}
\sect{The action of a maximal torus}\label{mt}
Starting in this section, we restrict the discussion mainly to the
case $n=3.$ We shall develop the concept of moment mapping, but
restricted to a maximal torus in $\mathrm{SU}(3),$ acting on $\mathbb{CP}^2.$ Fix the
torus
\begin{equation}\label{T}
T = \left\{\left(\!\begin{array}{ccc}
e^{ix_1} & 0 & 0\\
0 & e^{ix_2} & 0\\
0 & 0 & e^{ix_3}
\end{array}\!\right)\ :\ \textstyle
\sum\limits_{i=1}^3 x_i=0\hbox{ mod\,$2\pi$}\right\},
\end{equation}
which is, of course, homeomorphic to $S^1\times S^1.$ The hyperplane
$x_1+x_2+x_3=0$ in $\mathbb{R}^3$ represents the Lie algebra $\mathfrak t$ of
$T,$ which we also identify with $\mathfrak{t}^*$ using the induced
inner product. The moment mapping for $T$ acting on $\mathbb{CP}^2$ is then
the composition
\begin{equation}
\mathbb{CP}^2 \longrightarrow\mathfrak{su}(3)\longrightarrow \mathfrak{t}
\end{equation}
obtained by projecting the adjoint orbit orthogonally to $\mathfrak t.$
When we pass from $\mathfrak{su}(3)$ to $\mathscr{H}_3$ via \eqref{obv}, we can identify
this composition with the mapping $[\mathbf{z}]\mapsto(x_1,x_2,x_3),$ where
\begin{equation}\label{mu}
(x_1,x_2,x_3)=\mu([\mathbf{z}])=\frac1{\|\mathbf{z}\|^2}(|z_1|^2,|z_2|^2,|z_3|^2)
\end{equation}
consists of the diagonal entries in \eqref{can}. Here,
$\|\mathbf{z}\|^2=|z_1|^2+|z_2|^2+|z_3|^2,$ though it is convenient to assume
$\|\mathbf{z}\|=1.$ After the shift from traceless matrices to $\mathscr{H}_3,$ the
image of $\mu$ is the two-simplex $\mathscr{T},$ a filled equilateral triangle
lying in the plane $x_1+x_2+x_3=1,$ illustrated in Figure~1. The
residual three-fold symmetry visible is that of the Weyl group
$W=N(T)/T\cong\mathbb{Z}_3.$
It is well known that $\mathscr{T}$ parametrizes the orbits of $T$ on $\mathbb{CP}^2$
via \eqref{mu}. See, for example, Guillemin and Sternberg \cite{GS}.
The inverse image of an interior point of $\mathscr{T}$ is a two-torus
$T/\mathbb{Z}_3;$ the inverse image of a vertex is a single point in $\mathbb{CP}^2;$
and the inverse image of any other boundary point is a circle $S^1.$
Topologically, this leads to a description of the complex projective
plane as a quotient
\begin{equation}
\mathbb{CP}^2 = \frac{\mathscr{T}\times T^2}{{}\sim{}}.\end{equation}
Here $\sim$ is the equivalence relation that collapses points over the
boundary of $\mathscr{T}$ in accordance with the scheme outlined above.
\begin{figure}[b]
\scalebox{1.4}{\includegraphics{tri.pdf}}
\caption{The image of the moment mapping $\mu\colon\mathbb{CP}^2\to\mathbb{R}^3$ is a
two-simplex $\mathscr{T}$ that takes the form of a filled equilateral
triangle. If $\mathbf{z}=(z_1,z_2,z_3)$ is a unit vector in $\mathbb{C}^3,$ the
point $[\mathbf{z}]$ is mapped to $(x_1,x_2,x_3)=(|z_1|^2,|z_2|^2,|z_3|^2).$
The inscribed circle is the intersection of the plane
$x_1+x_2+x_3=1$ containing the (coloured) image of $\mu$ and the
(invisible) sphere $x_1^2+x_2^2+x_3^2=\frac12.$}
\end{figure}
Let $m_1,m_2,m_3$ be the midpoints of the sides of $\mathscr{T},$ and consider
the circles
\begin{equation} \label{cimi}
C_i=\mu^{-1}(m_i),\qquad i=1,2,3.
\end{equation}
The first circle $C_1$ consists of those points $[0,z_2,z_3]$ of
$\mathbb{CP}^2$ with $|z_2|=|z_3|.$ Any set of three
equidistant points in $C_1$ has the form
\begin{equation}\label{phs}
[0,e^{i\sigma},1],\quad [0,e^{i\sigma},\omega],\quad [0,e^{i\sigma},\omega^2],
\end{equation}
where $\omega=e^{2\pi i/3}.$ The cross ratio defined by any two of these
points is given by
\begin{equation}\textstyle
\left|\frac12(1+\omega)\right|^2 = \frac14,
\end{equation}
so they are indeed correctly separated. Similarly, $C_2$ consists of
points $[z_1,0,z_3]$ with $|z_1|=|z_3|,$ and $C_3$ points
$[z_1,z_2,0]$ with $|z_1|=|z_2|.$ Now choose three equidistant points
in $C_2,$ and three equidistant ones in $C_3.$ It is easy to check
that the resulting nine points constitutes a SIC set. This generalizes
Proposition~\ref{011}.
\begin{defi}
By a midpoint solution, we mean a SIC set in $\mathbb{CP}^2$ consisting of
three points in each of the three circles $C_1,C_2,C_3.$
\end{defi}
\noindent This construction defines a one-parameter family of SIC sets
up to isometry, since the stabilizer in $\mathrm{SU}(3)$ of the points in
$C_1$ is a subgroup $\mathrm{U}(1)$ that can be used to remove the phase
ambiguity in $C_2.$ See the discussion surrounding \eqref{U1}.
Let $\mathscr{C}$ denote the circle passing through the midpoints
$m_1,m_2,m_3,$ illustrated in Figure~1. As a curve in $\mathbb{R}^3,$ it is
the intersection of the plane $x_1+x_2+x_3=1$ with the sphere
$x_1^2+x_2^2+x_3^2=1/2.$ It was actually plotted using the next
result.
\begin{lem}\label{paraC}
In $\mathbb{R}^3,$ the inscribed circle $\mathscr{C}$ is parametrized by
\begin{equation}\fr23\Big(\!\cos^2\theta,\,\cos^2(\theta+\fr23\pi),\,
\cos^2(\theta+\fr43\pi)\Big),\end{equation} for
$\theta\in[-\frac12\pi,\frac12\pi).$
\end{lem}
\begin{proof}
First, consider the effect of $\mu$ on \emph{real} vectors. Suppose
that $\mathbf{z}=(a,b,c)$ is a unit vector with $a,b,c\in\mathbb{R},$ and set
$s_k=a^k+b^k+c^k.$ There is an identity
\begin{equation}\label{Sid}
s_1(-a+b+c)(a-b+c)(a+b-c)=s_2^2-2s_4.
\end{equation}
If $\mu([\mathbf{z}])=(a^2,b^2,c^2)\in\mathscr{C}$ then $s_2=1$ and $s_4=1/2,$ so the
right-hand side of \eqref{Sid} vanishes. It follows $a\pm b\pm c=0$
for some choice of signs. Now set
\begin{equation}\label{abc=}\textstyle
a=\sqrt{\frac23}\cos \theta,\quad
b=\sqrt{\frac23}\cos(\theta+\fr23\pi),\quad
c=\sqrt{\frac23}\cos(\theta+\fr43\pi).
\end{equation}
Trig-expanding $b$ and $c$ shows that $a+b+c=0$ and $a^2+b^2+c^2=1.$
It follows from \eqref{Sid} that $s_4=1/2$ and $(a^2,b^2,c^2)\in\mathscr{C}.$
The midpoints $m_1,m_2,m_3$ are given respectively by
$\theta=\pm\pi/2,-\pi/6,\pi/6.$ This confirms the stated range for $t.$
\end{proof}
\sect{Weyl-Heisenberg orbits}
In this section, we show how the moment mapping \eqref{mu} helps one
to understand the action of the groups $W$ and $H$ defined in
\eqref{cyc} and \eqref{run} with $n=3.$ We shall see that the $\mathscr{C}$
plays a prominent role, and Lemma~\ref{paraC} will be the basis
for the parametrization of elements of a SIC set.
\begin{lem}\label{nicefibres}
The $H$-orbit of a point $[\mathbf{z}]$ in $\mathbb{CP}^2$ consists of three points
that are correctly separated from one another if and only if
$\mu([\mathbf{z}])\in\mathscr{C}.$
\end{lem}
\begin{proof}
Suppose that $\mathbf{z}=\mathbf{z}^{(0)}$ is a unit vector. The orbit $H\cdot[\mathbf{z}]$
consists of the projective classes of the vectors
\begin{equation}\label{Horb}
\mathbf{z}^{(0)}=(z_1,z_2,z_3),\quad
\mathbf{z}^{(1)}=(z_1,\omega z_2,\omega^2 z_3),\quad
\mathbf{z}^{(2)}=(z_1,\omega^2 z_2,\omega z_3)
\end{equation}
generated by \eqref{run}. We can express
\begin{equation}
|\big<\mathbf{z}^{(0)},\mathbf{z}^{(1)}\big>|^2
=(|z_1|^2+ \omega|z_2|^2+\omega^2|z_3|^2)(|z_1|^2+\omega^2|z_2|^2+\omega|z_3^2|)
\end{equation}
in the form $\alpha-\beta,$ where
\begin{equation}
\alpha=|z_1|^4+|z_2|^4+|z_3|^4,\quad
\beta=|z_2|^2|z_3|^2+|z_3|^2|z_1|^2+|z_1|^2|z_2|^2.
\end{equation}
Therefore $\mathbf{z}^{(0)}$ and $\mathbf{z}^{(1)}$ are correctly separated if and only if
$\alpha-\beta=1/4.$ But
\begin{equation}\label{JK}
\alpha+2\beta=(|z_1|^2+|z_2|^2+|z_3|^2)^2=1,
\end{equation}
since $\mathbf{z}=\mathbf{z}^{(0)}$ is normalized, so the condition of correct
separation is $\alpha=1/2.$ Since $x_i=|z_i|^2$ are the Cartesian
coordinates in $\mathbb{R}^3,$ correct separation of $\mathbf{z}^{(0)}$ and $\mathbf{z}^{(1)}$
implies that $\mu([\mathbf{z}])\in\mathscr{C}.$ This condition only depends on
$\mu([\mathbf{z}])$ since $H$ is a subgroup of $T$ and its action commutes
with all the elements of $T.$ Therefore if $\mu([\mathbf{z}])\in\mathscr{C},$ all
three points in \eqref{Horb} will be correctly separated.
\end{proof}
\begin{exa}\rm
Lemma~\ref{nicefibres} is really an assertion about the induced metric
on the fibres $\mu^{-1}(p)$ for $p\in\mathscr{C}.$ This metric will depend
crucially on the position of $p$ in $\mathscr{C},$ since it degenerates as $p$
approaches any one of the midpoints $m_i$ (over which the fibres are
circles rather than 2-tori). This behaviour is illustrated in
Figure~2, which provides a visualization of the fibres $\mu^{-1}(p_i)$
for $i=1,2,$ where
\begin{equation}\label{p12}
p_1=\big(\fr23,\fr16,\fr16\big),\qquad
p_2=\big(\fr18(3+\sqrt5),\fr14,\fr18(3-\sqrt5)\big).
\end{equation}
are two points of $\mathscr{C}$. Note that $p_1$ is the point
diametrically opposite $m_1,$ whereas $p_2$ lies between $p_1$ and
$m_3.$
The coordinates used in Figure~2 are derived from the action of the
maximal torus \eqref{T}, which is represented by translation. Scalar
multiplication by $\omega=e^{2\pi i/3}$ on vectors in $\mathbb{C}^3$ generates
the action of the centre $\mathbb{Z}_3$ of $\mathrm{SU}(3),$ so that $(z_1,z_2,z_3)$
and $(\omega z_1,\omega z_2,\omega z_3)$ appear as distinct points in the
diagrams, although they determine the same point of $\mathbb{CP}^2.$ The
centre is responsible for the evident three-fold symmetry, which is
best represented by the hexagonal fundamental domain on the right-hand
side. Comparing this with the left-hand parallogram and its
translates, one sees that a 2-torus can be formed by identifying the
opposite edges of a hexagon, a fact that is well known (see, for
example, Thurston \cite{Thur}).
\begin{figure}[t]
\scalebox{.95}{\includegraphics{fibres.pdf}}
\caption{A representation of the fibre $\mu^{-1}(p_1)$ (left) and
$\mu^{-1}(p_2)$ (right) in $\mathbb{CP}^2$ for the points \eqref{p12}. The
coloured regions are two different fundamental regions for the
torus, and the red curves are points a distance $2\pi/3$ from the
centre point.} \vskip10pt
\end{figure}
Both diagrams display exactly three distinct points of $\mathbb{CP}^2$ in the
closure of each coloured fundamental domain, and each of these triples
of points forms an equilateral triangle. This can be seen from an
inspection of the curves that are the loci of points a distance
$2\pi/3$ from the centre point. The latter is correctly separated from
each of the other two points, and these two points are correctly
separated from each other because distances are translation
invariant.
\end{exa}
We are now in a position to give a full description of those SIC sets
that are orbits of the group generated by \eqref{cyc} and \eqref{run}.
\begin{theorem}\label{class}
Let $\mathbf{z}=(z_1,z_2,z_3)\in\mathbb{C}^3.$ Then $(W\times H)\cdot[\mathbf{z}]$ is a SIC set if
and only if one of the variables $z_1,z_2,z_3$ vanishes, or
\begin{equation}\label{cos3}
[\mathbf{z}] =
\Big[\!\cos\theta,\,\omega^j\cos(\theta+\fr23\pi),\,\omega^k\cos(\theta+\fr43\pi)\Big],
\end{equation}
for some $\theta\in[-\frac12\pi,\frac12\pi)$ and $j,k\in\{0,1,2\}.$
\end{theorem}
\begin{proof}
Suppose that $\mathbf{z}$ is a unit vector and that $(W\times H)\cdot[\mathbf{z}]$ is a SIC
set.
Let $\mathbf{z}'=(z_3,z_1,z_2).$ In the notation \eqref{Horb}, we have
\begin{equation}\label{RDe}
|\big<\mathbf{z}^{(k)},\mathbf{z}'\big>|^2
= \big|z_1\overline{z}_3+\omega^kz_2\overline{z}_1+\omega^{2k}z_3\overline{z}_2\big|^2
= \beta + 2\,\mathrm{Re}\big[\omega^{2k}\Delta\big],
\end{equation}
where
\begin{equation}\label{De}
\Delta=z_1^2\overline{z}_2\overline{z}_3+z_2^2\overline{z}_3\overline{z}_1+z_3^2\overline{z}_1\overline{z}_2.
\end{equation}
By assumption, \eqref{RDe} equals $1/4$ for all $k=0,1,2.$ From
\eqref{JK} we have $\beta=1/4,$ so the expression in square brackets
above must be purely imaginary. This happens for all $k$ if and only
if $\Delta=0.$
By assumption, $\mu([\mathbf{z}])\in\mathscr{C},$ so $[\mathbf{z}]$ must lie in a $T$-orbit of
$[a,b,c]$ where $a,b,c$ are given by \eqref{abc=} for some $\theta.$
Since $a+b+c=0,$ \eqref{De} and \eqref{RDe} tell us that $\mathbf{z}=(a,b,c)$
is a fiducial vector. Let us look for other fiducials in the same
$T$-orbit by considering
\begin{equation}\label{zzz}
(z_1,z_2,z_3)=(a,\,e^{i\beta}b,\,e^{i\gamma}c),
\end{equation}
having normalized the coefficient of $a.$ Let us assume that
$abc\ne0,$ so $b+c\ne0.$ Since $\Delta=0,$ we have
\begin{equation}
e^{3i\beta}b+e^{3i\gamma}c = b+c.
\end{equation}
Taking the moduli of both sides gives $\cos(3\beta-3\gamma)=1,$ so
$\beta$ equals $\gamma$ mod $2\pi/3.$ It follows that both $\beta$
and $\gamma$ are multiples of $2\pi/3,$ and that $[\mathbf{z}]$ has the
form \eqref{cos3}.
Conversely, the vector \eqref{cos3} satisfies \eqref{RDe} and projects
to $\mathscr{C}.$ Thus its $W\times H$ orbit is a SIC set.
\end{proof}
To summarize, any three equally-spaced points on $\mathscr{C}$ form the `base'
of a group covariant SIC set. If these are the three midpoints $m_i$
of the sides then \emph{any} point in $\mu^{-1}(m_i)$ is a fiducial
vector. But for a generic point $p\in\mathscr{C},$ the choices are restricted
to nine points on the two-torus $\mu^{-1}(p).$ As $p$ approaches a
midpoint, these nine points become three.
\begin{rem}\rm
The methods of this section can be extended to the study of SIC sets
in $\mathbb{CP}^{n-1}$ that arise as orbits of $W\times H$ for $n>3.$ Using the
moment mapping $\mu\colon\mathbb{CP}^{n-1}\to\mathbb{R}^n,$ one can define a subset of
the simplex $\mu(\mathbb{CP}^{n-1})$ consisting of points whose inverse image
contains $H$-orbits of correctly-separated points. For $\mathbb{CP}^3$ the
relevant subset consists of two circular arcs inside a solid
tetrahedron, but is no longer one-dimensional if $n>4,$ as discussed
by Lora Lamia \cite{NLLD}. For applications of the use of
$\mu\colon\mathbb{CP}^3\to\mathbb{R}^4$ in classifying almost-Hermitian structures on
manifolds of real dimension six, see Mihaylov \cite{Mih}.
\end{rem}
The SIC sets in $\mathbb{CP}^2$ described above have been discussed by Renes
\emph{et al.\ } \cite{RSC}, Zhu \cite{Zhu}, and various other authors. In
particular, it is known that any SIC set arising from
Theorem~\ref{class} is isometric to a midpoint solution. This can be
proved by adapting the proof of Proposition~\ref{7} below, but we
shall prove a much stronger result in this paper, namely
\begin{theorem}\label{strong}
Any SIC set $\mathbb{S}$ in $\mathbb{CP}^2$ is congruent modulo $\mathrm{SU}(3)$ to a
midpoint solution.
\end{theorem}
\noindent In the next section, we shall work with yet another
description of the isometry class of a midpoint solution, in which
each circle $C_i$ contains exactly two points of the SIC set.
\sect{Two-point homogeneity}\label{2pH}
Suppose, going forward, that $\mathbb{S}$ is a SIC set in $\mathbb{CP}^2,$ consisting
of nine points $[\mathbf{z}_i],$ $i=1,\ldots,9.$ Up to the action of the
isometry group, we are free to assume that $\mathbb{S}$ contains the two
points of $C_1$ represented by the unit vectors
\begin{equation}\label{zz12}
\textstyle\mathbf{z}_1=\fr1{\sqrt2}(0,1,-\omega),\quad \mathbf{z}_2=\fr1{\sqrt2}(0,1,-\omega^2),
\end{equation}
which are a distance $2\pi/3$ apart. This is on account of the
two-point homogeneity of $\mathbb{CP}^2.$ Lemma~\ref{CSI} tells us that any
other point $[\mathbf{z}]$ of $\mathbb{S}$ must satisfy
\begin{equation}\label{must}
\big|\big<\mathbf{z},\mathbf{z}_j\big>\big|^2 = \fr14\|\mathbf{z}\|^2,\qquad j=1,2.
\end{equation}
Using this equation, we can prove another lemma that emphasizes the
important role played by the incircle $\mathscr{C}.$
\begin{lem}\label{Zst}
The moment map $\mu$ projects any remaining point $[\mathbf{z}]$ of $\mathbb{S}$ to a
point of $\mathscr{C}.$ Indeed, we may take $\mathbf{z}$ to be a unit vector of the form
\begin{equation}\label{zst}
\mathbf{z}(\sigma,\phi) = \sqrt{\!\fr23}\Big(\,e^{i\sigma}\cos \phi,\
\cos(\phi+\fr23\pi),\,\cos(\phi+\fr43\pi)\,\Big)
\end{equation}
for some $\sigma\in(-\pi,\pi]$ and some $\phi\in(-\frac12\pi,\frac12\pi].$
\end{lem}
\noindent To lighten the notation, we shall write $\mathbf{z}[\sigma,\phi]$ as a
shorthand for $[\mathbf{z}(\sigma,\phi)],$ so that square brackets on either side
of `$\mathbf{z}$' indicate a projective class.\smallbreak
Lemma~\ref{paraC} tells us that $\mathbf{z}[\sigma,\phi]$ lies over $\mathscr{C}.$
Observe that $\mu(\mathbf{z}[\sigma,\phi])$ depends only on the \emph{angle}
$\phi$ measured around $\mathscr{C},$ and not on the \emph{phase} $\sigma.$
Moreover, as $\sigma$ and $\phi$ vary, $\mathbf{z}[\sigma,\phi]$ parametrizes a
pinched two-torus, the pinch point being
\begin{equation}\textstyle
\mathbf{z}[\sigma,-\frac12\pi]=\mathbf{z}[\sigma,\frac12\pi]=[0,1,-1],
\end{equation}
which is evidently independent of $\sigma.$ Having chosen $[\mathbf{z}_1],[\mathbf{z}_2]$
on $C_1,$ we can see that any third point of $C_1\cap\mathbb{S}$ must be this
point, which explains the pinching. One should note that
$\mathbf{z}(\sigma,\phi)=-\mathbf{z}(\sigma,\phi+\pi),$ which is why $\phi=-\pi/2$ is
excluded from the non-projective representation \eqref{zst}.\medbreak
\begin{proof}[Proof of Lemma~\ref{Zst}]
Let us suppose that $[\mathbf{z}] = [z_1,z_2,z_3]\in\mathbb{S}.$ If $z_2=0$ then
$[\mathbf{z}]\in C_2$ and \eqref{zst} will be valid for $\phi=-\pi/6.$ We may
therefore take $z_2=1$ and set $z_1=a,$ $z_3=c$ where $a,c\in\mathbb{C}.$ Then
by assumption, we have
\begin{equation}
|1-c\overline\omega|^2 = |1-c\omega|^2,
\end{equation}
which implies that $c$ is real, and
\begin{equation}
[\mathbf{z}]=\left[a,\ 1,\ \pm|c|\right].
\end{equation}
Using \eqref{must}, we see that
\begin{equation}
\fr12(1\pm|c|+|c|^2)=\fr14(|a|^2+1+|c|^2),
\end{equation}
so $|a|^2= (1\pm|c|)^2,$ and
\begin{equation}
|a|^2+1+|c|^2= 2(1 \pm|c|+|c|^2).
\end{equation}
Therefore,
\begin{equation} \begin{array}{rcl}
|a|^4+1+|c|^4
&=& 2\pm4|c|+6|c|^2\pm4|c|^3+2|c|^4\\[4pt]
&=& \fr12(|a|^2+1+|c|^2)^2.
\end{array}\end{equation}
It follows that $\mu$ does indeed map $[\mathbf{z}]$ into $\mathscr{C}.$ In view of
Lemma~\ref{paraC}, we must be able to express $[\mathbf{z}]$ in the stated
form for some $e^{i\sigma}\in\mathrm{U}(1).$
\end{proof}
The points $[\mathbf{z}_1],[\mathbf{z}_2]$ are both fixed by the subgroup $\mathrm{U}(1)$ of \eqref{T}
generated by
\begin{equation}\label{U1}
\left(\!\begin{array}{ccc}
e^{-2ix} & 0 & 0\\
0 & e^{ix} & 0\\
0 & 0 & e^{ix}
\end{array}\!\right),
\end{equation}
so we may assume that a third point of $\mathbb{S}$ is $\mathbf{z}[0,\theta]$. The next
result shows that there \emph{does} exist a SIC set containing this
point for any $\theta.$
\begin{prop}\label{7}
For any $\theta\in[-\frac12\pi,\frac12\pi),$ the six points
\begin{equation}\label{6pts}
\begin{array}{lll}
[0,1,-\omega]=[\mathbf{z}_1],\qquad & [0,1,-\omega^2]=[\mathbf{z}_2] &\in C_1,\\[5pt]
[1,0,-\omega]=\mathbf{z}[-\frac23\pi,\!-\frac16\pi],\quad &
[1,0,-\omega^2]=\mathbf{z}[\frac23\pi,\!-\frac16\pi] &\in C_2,\\[5pt]
[1,-\omega,0]=\mathbf{z}[-\frac23\pi,\frac16\pi], &
[1,-\omega^2,0]=\mathbf{z}[\frac23\pi,\frac16\pi] &\in C_3,
\end{array}\end{equation}
combine with three points
\begin{equation}\label{3pts}\textstyle
\mathbf{z}[0,\theta-\frac13\pi],\qquad \mathbf{z}[0,\theta],
\qquad\mathbf{z}[0,\theta+\frac13\pi]
\end{equation}
to form a SIC set isometric to a midpoint solution.
\end{prop}
\noindent We shall denote this SIC set by $\mathbb{S}_\th;$ it is illustrated in
Figure~3.
\begin{proof}
Consider the matrix
\begin{equation}
M = \frac1{\sqrt3}\!\left(\begin{array}{ccc} \omega^2 & \omega & 1 \\ 1 & \omega &
\omega^2\\ 1 & 1 & 1 \end{array}\right).
\end{equation}
It is easy to check that $M\in\mathrm{U}(3)$ and that $M^3=i\omega^2I.$ A
calculation shows that
\begin{equation}\textstyle
M\kern1pt\mathbf{z}(0,\theta)^{\!\top} =
\fr1{\sqrt2}\!\left(e^{i\theta}\omega^2,\ e^{-i\theta},\ 0\right)^{\!\top},
\end{equation}
so that $M$ maps $\mathbf{z}[0,\theta]$ to the point $[e^{2i\theta},\,\omega,\,0]$
of $C_3.$ Moreover, $M$ maps the array \eqref{6pts} to the array
\begin{equation}\begin{array}{ll}
[0,1,-1], & [1,0,-\omega],\\[5pt]
[1,0,-\omega^2],\qquad &[0,1,-\omega^2],\\[5pt]
[0,1,-\omega], & [1,0,-1] \end{array}\end{equation} of points
in $C_1\sqcup\,C_2.$ It follows that $M$ maps $\mathbb{S}_\th$ onto three
triples of points, each triple belonging to $C_i$ for some $i=1,2,3.$
\end{proof}
The first six points \eqref{6pts} of $\mathbb{S}_\th$ do not depend on $\theta,$
whereas the last triple of points can be rotated at will (by varying
$\theta$) around a circle $C_3'$ covering $\mathscr{C}$. For example, $\mathbf{z}[0,0]$
lies over the point $p_1$ of $\mathscr{C}$ diametrically opposite $m_1$ (see
\eqref{p12}).
\begin{figure}[b]
\scalebox{.72}{\includegraphics{pins2.pdf}}
\caption{The SIC set $\mathbb{S}_\th$ defined by Proposition~\ref{7} contains
two points in each circle $C_i$ (including $[\mathbf{z}_1],[\mathbf{z}_2]$ in $C_1$)
that do not depend on $\theta,$ together with a triple of points
(including $[\mathbf{z}_3]$) that $\mu$ also projects to $\mathscr{C}$ for which
$\theta$ represents the angle around $\mathscr{C}.$}
\end{figure}
\begin{rem}\rm
Nine points in $\mathbb{CP}^2$ are the inflection points of a non-singular
cubic curve if and only if the line determined by any two of them
contains a third. This being the case, there are twelve such lines
altogether, on which the nine points lie by threes, with four of the
twelve lines through each of the nine points, thus forming the
so-called Hesse configuration $\{9_4, 12_3\}$. For the points of
$\mathbb{S}_\th$ to arise in this way, and as described by Hughston \cite{Hugh}
and Dang \emph{et al.\ } \cite{DBBA}, the projective line $\mathbb{L}_1\cong\mathbb{CP}^1$
generated $[\mathbf{z}_1],[\mathbf{z}_2]$ must contain a third point of $\mathbb{S}_\th.$ But
$\mathbb{L}_1$ is the inverse image by $\mu$ of the side of $\mathscr{T}$ containing
$m_1,$ and will only contain another point if $\theta$ assumes one of the
values $\pm\pi/2,\pm\pi/6.$ This occurs when the three red legs (the
ones generated by $[\mathbf{z}_3]$ by rotation by $2\pi/3$) in Figure~3 line
up with the green legs (the ones over the midpoints), and $\mathbb{S}_\th$ is
then itself a special midpoint solution.
\end{rem}
\begin{exa}\label{eigen}\rm
The unitary transformation $M$ maps $C_3'$ to $C_3.$ It permutes the
elements of the SIC set $(W\times H)\cdot[0,1,-1],$ though it fixes none of
them. The matrices
\begin{equation}
A = \left(\!\begin{array}{ccc}0&1&0\\0&0&1\\1&0&0\end{array}\!\right),\quad
B = \left(\!\begin{array}{ccc}1&0&0\\0&\omega&0\\0&0&\omega^2\end{array}\!\right)
\end{equation}
generate $W$ and $H$ respectively, and satisfy
\begin{equation}\label{conj}
M\kern-1pt AM^{-1} = \omega B,\qquad
M\kern-1pt BM^{-1} = \omega^2A^{-1}B^{-1}.
\end{equation}
It follows that $M$ is an element of the so-called Clifford group, the
normalizer of $W\times H$ in $\mathrm{U}(3).$ Modulo phase, this normalizer is
isomorphic to a semidirect product
$\mathrm{SL}(2,\mathbb{Z}_3)\ltimes(\mathbb{Z}_3)^2$ (see Appleby \cite{App} and
Horrocks-Mumford \cite{HM}). Equation \eqref{conj} asserts that $M$
induces the automorphism of $W\times H$ given by \begin{equation}
\left(\!\begin{array}{cc}0&-1\\1&-1\end{array}\!\right)\in\mathrm{SL}(2,\mathbb{Z}_3).
\end{equation}
It is conjectured that a fiducial vector can always be found in an
eigenspace of some element of the Clifford group (see Zauner
\cite{Z}). In the case of $M,$ a computation shows that any one of its
eigenvectors defines a point of $\mathbb{CP}^2$ whose orbit under $W\times H$ is a
configuration of nine points arranged in nine lines. Each of the 27
pairs of points lying on one of the nine lines has a cross ratio
$\kappa=1/3,$ whereas the remaining nine pairs of points have
$\kappa=0.$ Compared to the Hesse configuration above, this means that
three of the twelve triples of points are not collinear, but each of
these three triples forms an orthonormal basis of $\mathbb{C}^3.$
\end{exa}
\begin{rem}\rm
If an isometry is to fix both $[\mathbf{z}_1]$ and $[\mathbf{z}_2],$ there is no
ambiguity remaining in the choice of $\phi\in[\-\pi/2,\pi/2)$ in
Lemma~\ref{Zst}. However, we are at liberty to interchange $[\mathbf{z}_1]$
and $[\mathbf{z}_2]$ by applying either complex conjugation or the unitary
\begin{equation}
\left(\!\begin{array}{ccc} 1 & 0 &
0\\0 & 0 & 1\\0 & 1 & 0\end{array}\!\right).
\end{equation}
The former has the effect of replacing $\sigma$ by $-\sigma,$ and the latter
of replacing $\phi$ by $-\phi$ in \eqref{zst}. In particular, the
congruence class of the unordered set $\mathbb{S}_\th$ uniquely specifies
$|\theta|.$ This fact can also be verified using a triple product that
measures the signed area of the planar geodesic triangle spanned by
three points. See, for example, Brody and Hughston \cite{BH} and
references cited therein.
\end{rem}
\sect{Trigonometry}
From now on, we shall assume that $\mathbb{S}$ is a SIC set in $\mathbb{CP}^2$ that
contains the points $[\mathbf{z}_1],[\mathbf{z}_2]$ defined by
\eqref{zz12}. Lemma~\ref{Zst} tells that any other point of $\mathbb{S}$ has
the form
\begin{equation}\label{ZST}
\mathbf{z}[\sigma,\phi]=\Big[\,e^{i\sigma}\cos\phi,\
\cos(\phi+\fr23\pi),\,\cos(\phi+\fr43\pi)\omega^2\,\Big],
\end{equation}
where $(\sigma,\phi)$ belongs to the rectangle
\begin{equation}
\mathscr{R} = (-\pi,\pi]\times(-\fr12\pi,\fr12\pi].
\end{equation}
The next result, from which many others follow, translates distance
into the new `rectangular' coordinates.
\begin{lem}\label{corr}
Suppose that
$\phi,\psi\in(-\frac12\pi,\frac12\pi)\setminus\{-\frac16\pi,\frac16\pi\}.$ Then the
points $\mathbf{z}[\sigma,\phi]$ and $\mathbf{z}[\tau,\psi]$ are the correct distance
$2\pi/3$ apart if and only if
\begin{equation}\label{CS}
1-\cos(\sigma-\tau) =
\frac{9(1+2\cos(2(\phi-\psi)))\sec\phi\sec\psi}
{16(\cos\phi\cos\psi + 3\sin\phi\sin\psi)}.
\end{equation}
\end{lem}
\begin{proof}
Not only do we have to establish the formula, but we also need to show
that the assumptions imply that the denominator of the fraction is
non-zero. We use the abbreviated notation
\begin{equation}\begin{array}{rcl}
\Gamma_0 &=& \cos\phi\cos \psi,\\[4pt] \Gamma_1 &=&
\cos(\phi+\frac23\pi)\cos(\psi+\frac23\pi),\\[4pt] \Gamma_2 &=&
\cos(\phi+\frac43\pi)\cos(\psi+\frac43\pi).
\end{array}\end{equation}
The condition on the cross ratio for correct separation is that
\begin{equation}
\fr49\big|e^{i(\sigma-\tau)}\Gamma_0+\Gamma_1+\Gamma_2\big|^2=\fr14,
\end{equation}
which gives
\begin{equation}
2\cos(\sigma-\tau)\Gamma_0(\Gamma_1+\Gamma_2)+\Gamma_0^2+(\Gamma_1+\Gamma_2)^2=\fr9{16},
\end{equation}
and therefore
\begin{equation}\label{minus9}
32\Gamma_0(\Gamma_1+\Gamma_2)(1-\cos(\sigma-\tau)) =
32\Gamma_0(\Gamma_1+\Gamma_2)+ 16\Gamma_0^2+16(\Gamma_1+\Gamma_2)^2 - 9.
\end{equation}
A calculation shows that the right-hand side of \eqref{minus9} is equal to
\begin{equation}
9(1+2\cos(2(\phi-\psi))),
\end{equation}
which vanishes when $\cos(\phi -\psi)=\pm1/2.$ By hypothesis,
$\Gamma_0\ne0.$ If
\begin{equation}
\Gamma_1+\Gamma_2 = \fr12[\cos \phi\cos \psi + 3\sin\phi\sin\psi]
\end{equation}
vanishes, then
\begin{equation}
\cos \phi\cos \psi +\sin \phi\sin \psi=\cos(\phi-\psi)
=\pm\fr12,
\end{equation}
and hence
\begin{equation}
\cos \phi\cos \psi=\pm\fr34,\quad\sin \phi\sin \psi=\mp\fr14.
\end{equation}
Now set $x=\tan \phi$ and $y=\tan\psi.$ Then $xy=-1/3$ and it holds
that
\begin{equation}
\pm\sqrt3=\tan(\phi-\psi)=\frac{x-y}{1+xy}=\fr32(x-y).
\end{equation}
We therefore have
\begin{equation} (x+y)^2=(x-y)^2 +4xy=0,
\end{equation}
and $\phi=-\psi=\pm\pi/6,$ values that are excluded. We may
therefore assume that $\Gamma_0(\Gamma_1+\Gamma_2)\ne0,$ and \eqref{CS}
follows.
\end{proof}
\begin{lem}\label{pi2}
If $\mathbb{S}$ contains the pinch point $[0,1,-1]$ as well as $[0,1,-\omega]$
and $[0,1,-\omega^2],$ then $\mathbb{S}$ is a midpoint solution.
\end{lem}
\begin{proof}
By hypothesis, $\mathbb{S}$ contains three points of the circle $C_1.$ If
$\mathbf{z}[\sigma,\phi]$ is a fourth point of $\mathbb{S},$ then \eqref{ZST} is
correctly separated from $[0,1,-1]$ and
\begin{equation}
\cos(\phi+\fr23\pi)-\cos(\phi+\fr43\pi)=\pm\fr{\sqrt3}2.
\end{equation}
This implies that $\sin\phi=\pm1/2,$ and forces $\mathbf{z}[\sigma,\phi]$ to lie
on $C_2\sqcup C_3.$ Therefore $\mathbb{S}$ lies in the disjoint union
$C_1\sqcup C_2\sqcup C_3.$
\end{proof}
One can rewrite \eqref{CS} as
\begin{equation}\label{rew}\begin{array}{rcl}
\cos(\sigma - \tau)
&=& \displaystyle
1-\frac{9(1 + 2 \cos 2(\phi-\psi))}
{4(4\cos^2\phi\cos^2\psi+3\sin 2\phi\sin2\psi)}\\[15pt]
&=& \displaystyle
\frac{-5 + 4\cos2\phi+ 4\cos2\psi- 14\cos2\phi\cos2\psi-6\sin2\phi\sin2\psi}
{4(1 + \cos2\phi+\cos2\psi+\cos2\phi\cos2\psi+ 3\sin2\phi\sin2\psi)}.
\end{array}\end{equation}
We shall convert the right-hand side into a rational function by setting
\begin{equation}\label{tans}
x=\tan\phi,\quad y=\tan\psi.
\end{equation}
In the light of Lemma~\ref{pi2}, we assume from now on that $x,y$ are
finite.
Equation~\eqref{rew} simplifies to
\begin{equation}\label{stxy}
\cos(\sigma - \tau) =
\frac{-11+9x^2+9y^2-27x^2y^2-24xy}{16(1+3xy)}.
\end{equation}
If $1+3xy=0,$ then the numerator on top of it must also vanish, so
$x^2+y^2=2/3$ and $(x+y)^2=0.$ Thus (as in the previous proof)
$x=-y=\pm1/\kern-2pt\sqrt3.$ This means that $\mathbf{z}[\sigma,\phi]$ lies on one of the circles
$C_2,C_3,$ and $\mathbf{z}[\tau,\psi]$ lies on the other, so there are no
restrictions on $\sigma$ and $\tau.$
The main result of this section is the following, which establishes a
criterion for the existence in $\mathbb{CP}^2$ of five points that are
correctly separated from one another.
\begin{theorem}\label{fa}
Suppose that $\mathbf{z}[\sigma,\phi],\mathbf{z}[\tau,\psi],\mathbf{z}[\upsilon,\chi]$ are three
points of $\mathbb{CP}^2,$ a distance $2\pi/3$ away from each other (and from
$[\mathbf{z}_1],[\mathbf{z}_2]$). Set $p=x+y+z,$ $q=yz+zx+xy,$ $r=xyz,$ where
$x=\tan\phi,$ $y=\tan\psi,$ $z=\tan\chi.$ Then $F(p,q,r)=0$ where
\begin{equation}\label{F0}\begin{array}{c}
F(p,q,r)= 9 - 22 p^2 + 9 p^4 + 87 q - 126 p^2 q + 27 p^4 q + 298 q^2 - 226 p^2q^2\\
+ 24 p^4 q^2+ 414 q^3 - 138 p^2 q^3 + 189 q^4 + 27 q^5 - 3 p r - 50 p^3 r - 15 p^5 r\\
+ 88 p q r- 48 p^3 q r + 234 p q^2 r + 18 p^3 q^2 r - 144 p q^3 r + 81 p q^4 r + 189 r^2\\
- 480 p^2 r^2 - 153 p^4 r^2 + 1398 q r^2 - 306 p^2 q r^2 + 2736 q^2 r^2- 486 p^2 q^2 r^2\\
+ 810 q^3 r^2 + 243 q^4 r^2 - 558 p r^3 - 486 p^3 r^3 + 2376 p q r^3 - 810 p q^2 r^3\\
+ 567 r^4 - 162 p^2 r^4 + 6399 q r^4 + 486 q^2 r^4 + 1701 p r^5 + 2187 r^6.
\end{array}\end{equation}
\end{theorem}
\begin{proof}
Although \eqref{F0} is rather complicated, the existence of such an
expression is a consequence of the elementary trigonometric identity
\begin{equation}\label{trig}
A+B+C=0\ \Rightarrow\ \cos^2\!A+\cos^2\!B+\cos^2\!C=1+2\cos A\cos
B\cos C,
\end{equation}
which is tailor made for \eqref{stxy}. The identity itself can be
proved by writing applying more standard ones to the sum $A+(B+C).$
Denote the right-hand side of \eqref{stxy} by the symmetric function
$c(x,y).$ Then
\begin{equation}\label{ccc}
c(x,y)^2+c(y,z)^2+c(z,x)^2 = 1 + 2c(x,y)c(y,z)c(z,x).
\end{equation}
This simplifes into the vanishing of the quotient
\begin{equation}\label{243}
\frac{243\,f(x,y,z)}{2048(1+3xy)^2(1+3yz)^2(1+3zx)^2},
\end{equation}
in which $f$ is a totally symmetric polynomial. We can then use the
\emph{Mathematica} command $\mathtt{SymmetricReduction}$ to express
\begin{equation}
f(x,y,z)=F(p,q,r)
\end{equation}
as a function of the elementary symmetric polynomials, and the result
follows.
\end{proof}
\sect{Graphical interpretation}\label{Gi}
Suppose once again that $\mathbb{S}$ is a SIC set in $\mathbb{CP}^2$ containing
$[\mathbf{z}_1]=[0,1,-\omega]$ and $[\mathbf{z}_2]=[0,1,-\omega^2],$ and (in view of
Lemma~\ref{pi2}) \emph{not} the third point $[0,1,-1]$ of $C_1.$ The
planar parametrization \eqref{CS} of the remaining points of $\mathbb{S}$
enables us to describe graphically the quest for such SIC sets. Before
we do this, we prove two results that help with their classification.
Setting $\phi=-\pi/6$ in \eqref{zst} defines the circle $C_2,$ and
$\phi=\pi/6$ the circle $C_3.$ It will be convenient to consider three
more circles $C_-,C_0,C_+$ given by $\sigma=-2\pi/3$, $0$, $2\pi/3$
respectively. Unlike $C_1,C_2,C_3,$ these three are not disjoint: they
meet in $[0,1,-1].$ The circles $C_2,C_3$ are represented by
horizontal lines in $\mathscr{R},$ and $C_-,C_0,C_+$ by equally-spaced
vertical lines; all five have diameter $\pi.$ The lines representing
$C_-,C_0$ and $C_2$ are visible in Figure~7.
\begin{lem}\label{pi6}
If $\mathbb{S}$ contains a point $\mathbf{z}[\sigma,\phi]$ with $|\phi|=\pi/6$ then
$\mathbb{S}$ is isometric to a midpoint solution.
\end{lem}
\begin{proof}
We can use the isometry \eqref{U1} to shift all points of $\mathbb{S}$ by a
translation parallel to the horizontal axis within our rectangle
$\mathscr{R}.$ We may therefore assume that $\sigma=0.$ Suppose for definiteness
that $\phi=\pi/6,$ so that $x=1/\kern-2pt\sqrt3$ and $\mathbf{z}[\sigma,\phi]\in C_3.$ Suppose
that $\mathbf{z}[\tau,\psi]$ is a fourth point of $\mathbb{S},$ and apply
\eqref{stxy}. The numerator equals
\begin{equation}
-11+9x^2+9y^2-27x^2y^2-24xy=-8(1+\sqrt3y),
\end{equation}
and the right-hand side of \eqref{stxy} becomes $-1/2$ unless
$y=-1/\kern-2pt\sqrt3.$ It follows that $\tau=\pm2\pi/3$ or $\psi=-\pi/6.$ Indeed,
the set of points correctly separated from $[\mathbf{z}_1],$ $[\mathbf{z}_2]$ and
$\mathbf{z}[0,\pi/6]$ is the union $C_-\cup C_2\cup C_+.$ This union must now
contain six points of $\mathbb{S},$ and no circle can contain more than
three.
Now suppose that $\mathbb{S}$ contains distinct points $\mathbf{z}[\frac23\pi,\psi]
\in C_+$ and $\mathbf{z}[\upsilon,-\frac16\pi]\in C_2$. Then \eqref{stxy} tells us
that either $y=1/\kern-2pt\sqrt3$ (and so $\psi=\pi/6$), or else
\begin{equation}\textstyle
\cos(\frac23\pi-\upsilon)=-\frac12
\end{equation}
and $\upsilon=-2\pi/3$ or $\upsilon=0$. So either the first point lies
on $C_3$, or else the second point lies on $C_-\sqcup C_0$. Now
suppose that $\mathbb{S}$ contains $\mathbf{z}[-\frac23\pi,\psi]\in C_-$ and
$\mathbf{z}[\frac23\pi,\chi]\in C_+$ This time, \eqref{stxy} yields
\begin{equation}
(3y^2-1)(3z^2-1)=0,
\end{equation}
and at least one of the two points is one of the last four in
\eqref{6pts}. We may also suppose that $[0,1,-1]\not\in\mathbb{S}$ by
Lemma~\ref{pi2}. It then follows that $\mathbb{S}$ consists of $[\mathbf{z}_1]$ and
$[\mathbf{z}_2]$, the two points in $C_-\cap(C_2\sqcup C_3),$ the two points
in $C_0\cap(C_2\sqcup C_3)$ and three points in $C_+,$ \emph{or} the
same thing with $C_-$ and $C_+$ interchanged. Applying \eqref{U1} with
$x=\pm2\pi/3,$ we obtain exactly the SIC set $\mathbb{S}_\th$ for some $\theta$
(like the one that includes the green points in Figures~6 and 7). Then
the result follows from Proposition~\ref{7}.
\end{proof}
\begin{lem}\label{LR}
Suppose that $\mathbb{S}$ is a SIC set that contains $[\mathbf{z}_1],[\mathbf{z}_2]$ and
$\mathbf{z}[0,\theta].$ Recall that any SIC set has this property up to
isometry. If $\mathbb{S}$ contains distinct points
$\mathbf{z}[\sigma_1,\phi],\mathbf{z}[\sigma_2,\phi]$ with $\sigma_1,\sigma_2\in(-\pi,\pi)$ then
it is isometric to a midpoint solution.
\end{lem}
\begin{proof}
First observe that $\sigma_1+\sigma_2=0;$ this follows by applying
Lemma~\ref{corr} in which we can set $(\tau,\psi)=(0,\theta)$ to obtain
$\cos\sigma_1=\cos\sigma_2.$ So take $\sigma=\sigma_1.$
In view of Lemma~\ref{pi6}, we may suppose that $x=\tan\phi$ is
different from $\pm1/\kern-2pt\sqrt3.$ We can choose a sixth point $\mathbf{z}[\tau,\psi]$
of $\mathbb{S}$ such that $\tau\ne\pi,$ since the circle $\tau=\pi$ can
contain at most three points a distance $2\pi/3$ apart. It follows
from \eqref{stxy} that either $1+3xy=0$ (and we can apply
Lemma~\ref{pi6}) or
\begin{equation}
\cos(\sigma-\tau)=\cos(-\sigma-\tau).
\end{equation}
Since $\sigma=0$ and $\sigma=\pi$ do not yield distinct points, the only
possibility remaining from our assumption is that $\tau=0.$ If
$t=\tan\theta$ and $y=\tan\psi$, \eqref{stxy} implies that
\begin{equation}
t^2+y^2-3t^2y^2-8ty-3=0.
\end{equation}
This gives
\begin{equation}
y=\frac{t\pm\sqrt3}{1\mp\sqrt3\,t},
\end{equation}
$\psi=\theta\pm\pi/3$ modulo $\pi.$ This is the configuration of three
points visible on the central vertical axis in Figure~7. All together,
$\mathbb{S}$ now contains at most seven points including $[\mathbf{z}_1]$ and
$[\mathbf{z}_2],$ which is a contradiction. Using \eqref{stxy}, one can in
fact show that given the sixth point, either $\phi$ or $\psi$ must
equal $\pm\pi/6.$
\end{proof}
\def\vskip10pt{\vskip10pt}
\begin{figure}[b]
\scalebox{1.1}{\includegraphics{oval1.pdf}}
\caption{The black (resp., red) point is correctly separated from all
points on the black (resp., red) curve. The two points cannot belong
to a SIC set because there are only four remaining points correctly
separated from them both.} \vskip10pt
\end{figure}
\begin{figure}[b]
\scalebox{1.1}{\includegraphics{oval11.pdf}}
\caption{The points that are correctly separated from the black point
can form a disconnected set. Here, there appear to be five points
correctly separated from the red and black points, but these five
points do not in fact form part of a SIC set.} \vskip10pt
\end{figure}
We are now in a position to illustrate the problem of finding SIC sets
that contain $[\mathbf{z}_1]$ and $[\mathbf{z}_2].$ We can (and shall) assume that a
third point of $\mathbb{S}$ is $\mathbf{z}[0,\theta]$ for some fixed
$\theta\in(-\pi/2,\pi/2).$ This point corresponds to one on the central
vertical axis of the rectangle $\mathscr{R},$ and will be displayed by a black
dot in the figures. We shall draw some curves to illustrate the
concept of correctly separated points in $\mathscr{R},$ meaning that the
distance between the points they represent in $\mathbb{CP}^2$ equals $2\pi/3.$
A fourth point $\mathbf{z}[\sigma,\phi]$ of $\mathbb{S}$ will be displayed by a red dot.
\begin{figure}[t]
\scalebox{1.1}{\includegraphics{oval2.pdf}}
\caption{Here the fourth point $\mathbf{z}[0,\frac1{16}\pi-\frac13\pi]$ belongs
to $\mathbb{S}_\th$ which is generated by the remaining five points on the
intersection of the black and red curves.} \vskip10pt
\end{figure}
\begin{figure}[t]
\scalebox{1.1}{\includegraphics{oval3.pdf}}
\caption{Here the fourth point $\mathbf{z}[\frac23\pi,\frac16\pi]\in\mathbb{S}_\th$ is
equidistant from all points on the circles $C_-,C_0,C_2$,
represented by straight lines in $\mathscr{R}.$ The segments top and bottom
collapse to the pinch point.} \vskip10pt
\end{figure}
In Figure~4, $\theta=\pi/16$ so that the third point $\mathbf{z}[0,\theta]$ of $\mathbb{S}$
is close to centre of $\mathscr{R}.$ The black curve is the set of points
$\mathbf{z}[\sigma,\phi]$ which are a distance $2\pi/3$ from $\mathbf{z}[0,\theta].$ The
remaining six points of $\mathbb{S}$ must therefore lie on this curve. One
such example is represented by the red dot, which actually has
$\phi=\pi/4.$ Points $\mathbf{z}[\tau,\psi]$ a distance $2\pi/3$ apart from
this red point are those on the red curve (which has two
components). The intersection of the black and red curves consists of
points which are correctly separated from both the third and fourth
points. Since there are only four of these (we require five), the
value $x=\tan\phi=1$ cannot in fact occur when $\theta=\pi/16.$
The nature of the black curve is heavily dependent on the value
chosen of $\theta$ and $t=\tan\theta.$ If $\mathbf{z}[\pi,\phi]$ is correctly
separated from $\mathbf{z}[0,\theta]$ and $x=\tan\phi$ then
\begin{equation}\label{pi}
9(1-3t^2)x^2+24t x+9t^2+5=0.
\end{equation}
Computing the roots of the discriminant as a function of $t ,$
\eqref{pi} has distinct roots if and only if
$|t|>\sqrt{5/27}=0.430\ldots$ In this case, the black curve has two
connected components, and an example is visible in Figure~5 for which
$\theta=\pi/7.$ This time the red point $(\sigma,\phi)$ is chosen (with $x$
approximately $4.75$) so that there are exactly five points correctly
separated from both the (black and red) third and fourth
points. Subsequent analysis will show that these five points are not
correctly separated from each other.
Although the fourth (red) point in Figure~4 is not admissible (nor, in
fact, is that in Figure~5), Proposition~\ref{7} implies that there
\emph{does} exists a SIC set, namely $\mathbb{S}_\th,$ containing the first
three points, so there must be at least six points on the black curve
that \emph{are} admissible. For Figures~6 and 7, we return to the
value $\theta=\pi/16,$ and display these six points in green.
In Figure~6, we have chosen the fourth point $\mathbf{z}[\sigma,\phi]$ to be the
admissible one with $\sigma=0$ and $\phi$ negative. In Figure~7, we have
chosen the fourth point to be one of the points of $\mathbb{S}_\th$ that does
not depend on $\theta.$ Recall that the top and bottom boundary of $\mathscr{R}$
is a single point, and that the horizontal lines $\phi=\pm\pi/6$ are
the circles $C_2,C_3.$ Figure~7 illustrates the fact that any point of
$C_2$ is correctly separated from $[\mathbf{z}_1],$ $[\mathbf{z}_2]$ and a given point
of $C_3,$ as we explained in the proof of Lemma~\ref{pi6}.
\sect{Symmetrization}\label{Sym}
We suppose now that $\mathbb{S}$ is a SIC set containing, in addition to
$[\mathbf{z}_1]=[0,1,-\omega]$ and $[\mathbf{z}_2]=[0,1,-\omega^2],$ four more points
$\mathbf{z}[\sigma_i,\phi_i],$ $i=3,4,5,6,$ with $t=\phi_3,$ $x=\phi_4,$
$y=\phi_5,$ $z=\phi_6.$
In view of Theorem~\ref{fa} and equation \eqref{243}, our task is to
investigate the system of polynomial equations given by
\begin{equation}\label{task}
f(x,y,z)=0,\ \ f(t,y,z)=0,\ \ f(t,x,z)=0,\ \ f(t,x,y)=0.
\end{equation}
Since $f$ is itself symmetric, the whole system is invariant under the
action of the group of permutations of $t,x,y,z.$ There are
refinements of Buchberger's algorithm for dealing with symmetric
ideals, but we shall adopt the technique outlined in
\cite{St}. Namely, we shall convert the system into a system four
equations, each of which involves only the elementary symmetric
polynomials defined by
\begin{equation}\label{esp}\begin{array}{l}
a=t+x+y+z,\\
b=t x+t y+t z+yz+zx+xy,\\
c=xyz+t yz+t xz+t xy,\\
d=t xyz.
\end{array}\end{equation}
To accomplish this, first define
\begin{equation}
F_1=f(x,y,z)+f(t,y,z)+f(t,x,z)+f(t,x,y),
\end{equation}
and consider
\begin{equation}
g(t,x,y,z)=\frac{f(t,x,y)-f(t,x,z)}{y-z}.
\end{equation}
This is a \emph{polynomial} in $t,x,y,z,$ so we can symmetrize it to get
\begin{equation}\begin{array}{r}
F_2=g(t,x,y,z)+g(t,y,z,x)+g(t,z,x,y)\\
+g(x,y,t,z)+g(x,z,t,y)+g(y,z,x,t).
\end{array}\end{equation}
Next, set
\begin{equation}
h(t,x,y,z)=\frac{g(t,x, y, z)-g(t, y, x, z)}{x - y},
\end{equation}
so as to define
\begin{equation}\begin{array}{r}
F_3= h(t,x,y,z) + h(t,y,z,x) + h(t,z,x,y)\\
+h(x,y,t,z) + h(x,z,t,y) + h(y,x,t,z)\\
+h(z,x,y,t) + h(x,y,z,t) + h(y,z,x,t)\\
+ h(z,y,t,x) + h(y,z,t,x) + h(z,x,t,y),\\[10pt]
\displaystyle F_4= \frac{h(t,x,y,z)-h(x,t,y,z)}{t- x}.
\phantom{oooooo}
\end{array}\end{equation}
\medbreak
Each of $F_1,F_2,F_3,F_4$ is a symmetric polynomial in $t,x,y,z,$
and can therefore be expressed as a polynomial in $a,b,c,d.$ The proof
of Theorem~\ref{strong} proceeds by examination of the system
\begin{equation}\label{4F}
F_1(a,b,c,d)=0,\ \ F_2(a,b,c,d)=0,\ \
F_3(a,b,c,d)=0,\ \ F_4(a,b,c,d)=0.
\end{equation}
To determine the polynomials $F_i$ in practice, we used again the
\emph{Mathematica} command $\mathtt{SymmetricReduction}$. For
completeness we list them explicitly:
{\normalsize\begin{equation*}\begin{array}{l} F_1=36 -
66 a^2 + 27 a^4 + 218 b - 288 a^2 b + 54 a^4 b + 614 b^2 - 452 a^2
b^2 + 48 a^4 b^2 + 828 b^3 - 276 a^2 b^3\\[-.7pt] + 378 b^4 + 54 b^5 - 41 a
c + 159 a^3 c - 63 a^5 c - 567 a b c + 270 a^3 b c - 246 a b^2 c +
18 a^3 b^2 c - 279 a b^3 c\\[-.7pt] + 81 a b^4 c + 834 c^2 - 708 a^2 c^2 -
153 a^4 c^2 + 1968 b c^2 - 171 a^2 b c^2 + 2871 b^2 c^2 - 486 a^2
b^2 c^2 + 810 b^3 c^2\\[-.7pt] + 243 b^4 c^2 - 693 a c^3 - 486 a^3 c^3 +
2376 a b c^3 - 810 a b^2 c^3 + 567 c^4 - 162 a^2 c^4 + 6399 b c^4 +
486 b^2 c^4\\[-.7pt] + 1701 a c^5 + 2187 c^6 - 712 d + 687 a^2 d + 414 a^4
d - 2632 b d + 351 a^2 b d + 216 a^4 b d - 4470 b^2 d + 1107 a^2 b^2
d\\[-.7pt] - 5454 b^3 d + 243 a^2 b^3 d - 1782 b^4 d - 486 b^5 d + 453 a c
d + 927 a^3 c d - 3330 a b c d + 2835 a^3 b c d - 7857 a b^2 c d\\[-.7pt] +
1458 a b^3 c d + 666 c^2 d - 1485 a^2 c^2 d - 4968 b c^2 d + 2268
a^2 b c^2 d - 24786 b^2 c^2 d - 1944 b^3 c^2 d - 6075 a c^3 d\\[-.7pt] -
9477 a b c^3 d - 1701 c^4 d - 13122 b c^4 d + 4656 d^2 - 531 a^2 d^2
- 2673 a^4 d^2 + 14436 b d^2 + 9774 a^2 b d^2 + 12042 b^2 d^2\\[-.7pt] -
2349 a^2 b^2 d^2 + 13608 b^3 d^2 + 972 b^4 d^2 + 3861 a c d^2 - 1944
a^3 c d^2 + 37665 a b c d^2 + 13365 a b^2 c d^2 + 6966 c^2 d^2\\[-.7pt] +
8991 a^2 c^2 d^2 + 7776 b c^2 d^2 + 19683 b^2 c^2 d^2 + 13122 a c^3
d^2 - 11448 d^3 - 11907 a^2 d^3 - 35640 b d^3 - 12393 a^2 b d^3\\[-.7pt] -
7290 b^2 d^3 - 4374 b^3 d^3 - 16281 a c d^3 - 26244 a b c d^3 -
13122 c^2 d^3 + 8748 d^4 + 6561 a^2 d^4 + 13122 b d^4.
\end{array}\end{equation*}}\\[-15pt]
{\normalsize\begin{equation*}\begin{array}{l} F_2= 63 a -
243 a^3 + 81 a^5 + 829 a b - 846 a^3 b + 81 a^5 b + 1706 a b^2 - 642
a^3 b^2 + 1092 a b^3 + 18 a^3 b^3 + 45 a b^4\\[-.7pt] + 81 a b^5 - 1086 c +
741 a^2 c - 18 a^4 c - 2348 b c + 123 a^2 b c - 153 a^4 b c + 24 b^2
c - 630 a^2 b^2 c + 2493 b^3 c\\[-.7pt] - 486 a^2 b^3 c + 810 b^4 c + 243
b^5 c + 135 a c^2 + 81 a^3 c^2 - 18 a b c^2 - 486 a^3 b c^2 + 2376 a
b^2 c^2 - 810 a b^3 c^2\\[-.7pt] - 162 c^3 + 567 b c^3 - 162 a^2 b c^3 +
6399 b^2 c^3 + 486 b^3 c^3 + 1701 a b c^4 + 2187 b c^5 - 1512 a d +
2583 a^3 d\\[-.7pt] + 405 a^5 d - 8709 a b d + 2232 a^3 b d - 11583 a b^2 d
+ 1944 a^3 b^2 d - 7479 a b^3 d - 891 a b^4 d + 1926 c d + 1080 a^2
c d\\[-.7pt] + 1701 a^4 c d + 270 b c d - 5859 a^2 b c d - 2862 b^2 c d +
4374 a^2 b^2 c d - 11988 b^3 c d - 972 b^4 c d - 2916 a c^2 d\\[-.7pt] +
486 a^3 c^2 d - 25272 a b c^2 d - 7533 a b^2 c^2 d - 5103 a^2 c^3 d
- 1701 b c^3 d - 8748 b^2 c^3 d - 6561 a c^4 d + 10422 a d^2\\[-.7pt] -
3807 a^3 d^2 + 37071 a b d^2 - 3888 a^3 b d^2 + 29160 a b^2 d^2 +
4617 a b^3 d^2 + 486 c d^2 + 14337 a^2 c d^2 + 41472 b c d^2\\[-.7pt] +
17010 a^2 b c d^2 + 7290 b^2 c d^2 + 6561 b^3 c d^2 + 15309 a c^2
d^2 + 26244 a b c^2 d^2 + 13122 c^3 d^2 - 46656 a d^3\\[-.7pt] - 6561 a^3
d^3 - 28431 a b d^3 - 10935 a b^2 d^3 - 10206 c d^3 - 13122 a^2 c
d^3 - 30618 b c d^3 + 19683 a d^4.\end{array}\end{equation*}}\\[-15pt]
{\normalsize\begin{equation*}\begin{array}{l} F_3=780 - 978 a^2 - 180 a^4 + 27
a^6 + 3468 b - 1320 a^2 b - 396 a^4 b + 4968 b^2 - 81 a^2 b^2 + 54
a^4 b^2 + 2268 b^3\\[-.7pt] - 108 a^2 b^3 + 324 b^4 + 243 a^2 b^4 - 594 a c
- 693 a^3 c - 459 a^5 c + 2250 a b c - 1161 a^3 b c + 6993 a b^2 c -
1458 a^3 b^2 c\\[-.7pt] + 2430 a b^3 c + 729 a b^4 c + 900 c^2 - 1188 a^2
c^2 - 1458 a^4 c^2 + 648 b c^2 + 7128 a^2 b c^2 - 2430 a^2 b^2 c^2 +
1701 a c^3\\[-.7pt] - 486 a^3 c^3 + 19197 a b c^3 + 1458 a b^2 c^3 + 5103
a^2 c^4 + 6561 a c^5 - 5304 d + 4446 a^2 d + 4509 a^4 d - 22644 b
d\\[-.7pt] - 6318 a^2 b d + 3645 a^4 b d - 32076 b^2 d - 10044 a^2 b^2 d -
10692 b^3 d - 486 a^2 b^3 d - 2916 b^4 d + 6642 a c d\\[-.7pt] + 4779 a^3 c
d - 34668 a b c d + 5832 a^3 b c d - 26244 a b^2 c d - 2916 a b^3 c
d - 7776 c^2 d - 16281 a^2 c^2 d - 76788 b c^2 d\\[-.7pt] - 18225 a^2 b c^2
d - 5832 b^2 c^2 d - 25515 a c^3 d - 26244 a b c^3 d - 26244 c^4 d +
19440 d^2 + 18954 a^2 d^2 - 2187 a^4 d^2\\[-.7pt] + 50868 b d^2 + 19440 a^2
b d^2 + 75816 b^2 d^2 + 9477 a^2 b^2 d^2 + 5832 b^3 d^2 + 85050 a c
d^2 + 11664 a^3 c d^2\\[-.7pt] + 65610 a b c d^2 + 19683 a b^2 c d^2 +
20412 c^2 d^2 + 19683 a^2 c^2 d^2 + 78732 b c^2 d^2 - 68040 d^3 -
39366 a^2 d^3\\[-.7pt] - 32076 b d^3 - 13122 a^2 b d^3 - 26244 b^2 d^3 -
65610 a c d^3 + 26244 d^4. \end{array}\end{equation*}}\\[-15pt]
{\normalsize\begin{equation*}\begin{array}{l}
F_4= 70 a - 168 a^3 + 9 a^5 + 464 a b - 228 a^3 b + 525 a b^2 + 18 a^3 b^2 - 36 a b^3 + 81 a b^4 - 26 c - 399 a^2 c - 153 a^4 c\\[-.7pt]
+ 1158 b c - 387 a^2 b c + 2655 b^2 c - 486 a^2 b^2 c + 810 b^3 c + 243 b^4 c - 504 a c^2 - 486 a^3 c^2 + 2376 a b c^2 - 810 a b^2 c^2\\[-.7pt]
+ 567 c^3 - 162 a^2 c^3 + 6399 b c^3 + 486 b^2 c^3 + 1701 a c^4 + 2187 c^5 - 762 a d + 963 a^3 d - 3942 a b d + 1215 a^3 b d\\[-.7pt]
- 4320 a b^2 d - 162 a b^3 d + 486 c d - 675 a^2 c d - 3348 b c d + 1944 a^2 b c d - 11988 b^2 c d - 972 b^3 c d - 6075 a c^2 d\\[-.7pt]
- 6075 a b c^2 d - 1701 c^3 d - 8748 b c^3 d + 4266 a d^2 - 729 a^3 d^2 + 10692 a b d^2 + 3159 a b^2 d^2 + 5994 c d^2\\[-.7pt]
+ 3888 a^2 c d^2 + 4374 b c d^2 + 6561 b^2 c d^2 + 6561 a c^2 d^2 - 4374 a d^3 - 4374 a b d^3 - 4374 c d^3.
\end{array}\end{equation*}}
When $d=0$ (so at least one of $t,x,y,z$ vanishes), the expressions
for the $F_i$ simplify greatly, and explicit solutions to \eqref{4F}
can be computed. Not all of the solutions are valid because both
\eqref{4F} is only a necessary (not a sufficent) condition on the
variables $t,x,y,z.$ The symmetrization process can introduce
solutions that arise when these quantities are not distinct, as in
\eqref{tt} below. Another problem is the ambiguity of sign in the
horizontal coordinate of $\mathscr{R},$ and this will result in our method
capturing solutions like that illustrated in Figure~8.
\sect{Conclusion}
We shall use the theory of Gr\"obner bases to analyse the ideal
\begin{equation}
I=\left<F_1,F_2,F_3,F_4\right>
\end{equation}
of the polynomial ring $\mathbb{R}[a,b,c,d].$ In view of Lemma~\ref{pi6}, we
are not interested in solutions to \eqref{4F} for which $\pm1/\kern-2pt\sqrt3$ is a
root of the polynomial
\begin{equation}\label{qr}
g(x)=x^4-ax^3+bx^2-cx+d.
\end{equation}
Equivalently we want solutions for which
\begin{equation}
\begin{array}{rcl} G(a,b,c,d) &=& 81g(\frac1{\sqrt3})g(-\frac1{\sqrt3})\\[8pt]
&=& 1 - 3a^2 + 6b+ 9b^2 - 18ac- 27c^2 + 18d+ 54bd+ 81d^2
\end{array}\end{equation} is non-zero. Nor are we interested in solutions of
\eqref{4F} that give rise to a repeated root of \eqref{qr}, for these
can be ignored thanks to Lemma~\ref{LR}.
Using the notion of quotient ideal (see Cox \emph{et al.\ } \cite[Chap.~4,
\S4]{CLO}) we compute the quotient $I:\left<G\right>.$ This is done
by finding a Gr\"obner basis for
\begin{equation}
J=\left<uF_1,uF_2,uF_3,uF_4,(1-u)G\right>
\end{equation}
using a lexicographic ordering with the dummy variable $u$ first in
the dictionary. Those basis elements that do not involve $u$ are
necessarily divisible by $G$ and provide a basis for the quotient. The
order of the remaining variables is also important, and we used the
\emph{Mathematica} command
$\mathtt{gb:=GroebnerBasis[J,\{u,a,c,b,d\}]}.$ The first element
$\mathtt{gb[[1]]/G}$ equals
\begin{equation}
-(d-1)^3(3 d-1)^3(3+b+3d)(9d-1)^3(1+3b+9d)^3(19+9b+27d),
\end{equation}
and thus we obtain
\begin{theorem}\label{key5}
Let $\mathbb{S}=\{[\mathbf{z}_i]\}$ be a SIC set satisfying \eqref{zz12}. Let
$t,x,y,z$ be the `vertical tangents' of $[\mathbf{z}_3],[\mathbf{z}_4],[\mathbf{z}_5],[\mathbf{z}_6],$
and define $b,d$ as in \eqref{esp}. If $\mathbb{S}$ is not isometric to a
midpoint solution then one or more of the following equations must
hold:
\begin{equation}
d=1,\ d=\fr13,\ d=\fr19,
\ b=-(3d+3),\ b=-\fr13(9d+1),\ -\fr19(27d+19).
\end{equation}
\end{theorem}
We shall examine each possibility in turn.
\smallbreak\noindent \textbf{Case (i).} $d=1/9.$ If $I'$ denotes the
ideal $I$ with $9d-1$ adjoined (in practice, we can merely set
$d=1/9$), one repeats the procedure to determine a basis of
$I':\left<G\right>.$ The new second element $\mathtt{gb[[2]]/G}$
equals
\[ (12b+8-27c^2)(3b+2)(3b+10)(9b+22).\]
First suppose that $b=(27c^2-8)/12.$ This leads to $a+3c=0$ and the
quartic \eqref{qr} has a pair of double roots
\begin{equation}
x=\fr1{12}(-9c\pm\sqrt{48+81c^2}).
\end{equation}
Expressed more simply, the roots are
\begin{equation}\label{tt}
x=t,\ t,\ -\fr1{3t},\ -\fr1{3t},
\end{equation}
and we can ignore this solution in view of Lemma~\ref{LR}.
If $b=-2/3$ we get $a=c=0$ and all the roots of \eqref{qr}
are $\pm1/\kern-2pt\sqrt3.$
If $b=-10/3$ we get $a=0$ and $c=\pm8/(3\sqrt3);$ one root
of \eqref{qr} is still $\pm1/\kern-2pt\sqrt3.$
If $b=-22/9,$ we have an instance of Case (iv) in which $a=0$ and
\begin{equation}\label{97}
c=\pm\fr8{27}\sqrt{\smash{26\pm2\sqrt{97}}\vphantom{I^I}}.
\end{equation}
Provided we take a minus sign inside the square root, \eqref{qr} has
four distinct roots, and provides the `fake SIC set' discussed below
and illustrated in Figure~8.
\smallbreak\noindent\noindent\textbf{Case (ii).} Setting $1+3b+9d=0$
and re-evaluating the quotient ideal forces $d$ to equal one of
$1,1/3,1/9.$ The first leads to
\[a=0,\ b=-\fr{10}3,\ c=0,\ d=1,\]
giving roots of \eqref{qr} that are repeated and include
$\pm1/\kern-2pt\sqrt3.$ The case $d=1/3$ produces no new solutions.
\smallbreak\noindent\noindent\textbf{Case (iii).} $3+b+3d=0.$ This
leads to the solutions
\[a=\pm\fr8{\sqrt3},\ b=-6,\ c=0,\ d=1\]
and
\[a=\pm\fr8{\sqrt3},\ b=0,\ c=0,\ d=-1.\]
In the former case, $\pm1/\kern-2pt\sqrt3$ is still a root of \eqref{qr}.
In the latter case, the quartic has two non-real roots.
\smallbreak\noindent\noindent\textbf{Case (iv).} $19+9b+27d=0.$ This
is in some sense the generic case. It leads to
\begin{equation}\label{144} \begin{array}{r}
16+9a^2+27ac-144d=0,\hskip200pt\\[4pt] 4194304 - 73728 a^2 - 132192 a^4 +
6561 a^6 - 4866048ac -746496 a^3c\\ + 78732 a^5 c - 8626176 c^2-
699840 a^2 c^2 + 354294 a^4 c^2 + 1679616 a c^3\\ + 708588 a^3 c^3 +
1889568 c^4 +531441 a^2 c^4=0, \end{array}\end{equation} giving rise to a
one-parameter family of solutions to \eqref{4F}. To describe this
family, we fix $t=\tan\theta$ exactly as we did in the
figures of Section~\ref{Sym}. We set
\begin{equation}\label{a2p}
a=t+p,\ \ b=t p+q,\ \ c=t q+r,\ \ d=t r
\end{equation}
(the notation is as in Theorem~\ref{fa}), and compute a Gr\"obner
basis of the ideal $K$ generated by $19+9b+27d$ and the left-hand
sides of \eqref{144} in terms of $t.$ This can be accomplished with
the \emph{Mathematica} command\\
\hphantom{ooo}$\mathtt{GroebnerBasis[K,\{r,q,p\},
CoefficientDomain\to RationalFunctions]}.$\\ Provided $t\ne0$ and
$|t|\ne1/\kern-2pt\sqrt3,$ the leading terms are $p,q,r^6.$ This means that the
non-leading monomials are $1,r,r^2,r^3,r^4,r^5,$ and that there exist
six solutions over $\mathbb{C}$ counting multiplicity \cite{Sturm}.
\begin{proof}[Completion of the proof of Theorem~\ref{strong}]
Let us first summarize the argument so far. The existence of six
correctly-separated points in $\mathbb{CP}^2,$ including the ones
$[\mathbf{z}_1],[\mathbf{z}_2]$ we fixed from the start of Section~\ref{2pH} onwards,
leads to a solution of the system \eqref{4F}. Lemma~\ref{pi6} allows
us to dispense with cases in which one root $t$ of \eqref{qr} (or, one
root of $x^3-px^2+qx-r=0$) equals $\pm1/\kern-2pt\sqrt3;$ such cases give rise to
SIC sets isometric to $\mathbb{S}_\th.$
Theorem~\ref{key5} provides conditions for any extra solutions, and we
are led to focus on Case (iv), which does supply a family of solutions
to \eqref{4F}. We must show that these do not harbour an undetected
SIC set. In accordance with \eqref{U1}, we can assume that a third
point of $\mathbb{S}$ equals $\mathbf{z}[0,\theta]$ and apply Lemma~\ref{LR}. The
remaining six points of a SIC set would give rise to $\binom63=20$
solutions for each fixed $x.$ But Case (iv) provides at most six sets
of roots.
\end{proof}
We can now be certain that the solutions in Case (iv) are not SIC
sets. For any given rational value of $t,$ the solutions are roots of
polynomials whose coefficients are known exactly. Experimentally, the
number of real solutions varies from two to five according to the
following table:
\begin{equation*}
\begin{array}{|c|c|}\hline
\hbox{real solutions} & \hbox{range}\\\hline\hline
2 & \hphantom{0.11<}|t|<0.1898\\\hline
3 & \quad 0.1899<|t|<0.4386\quad\\\hline
5 & 0.4387<|t|<1/\kern-2pt\sqrt3\\\hline
4 & 1/\kern-2pt\sqrt3<|t|<1.1546\\\hline
3 & 1.1547<|t|\hphantom{<0.11}\\\hline
\end{array}
\end{equation*}
\noindent Although not SIC sets, these solutions validate \eqref{4F}
by virtue of `cross-field passes' of the type described below. Their
changing number as $|t|$ increases reflects the transitional nature of
the curves displayed in Figures~4 to 7.
\begin{exa}\label{fake}\rm
Take $(a,b,c,d)=(0,-22/9,c,1/9)$ where $c$ is given by \eqref{97} with
both minus signs. Then \eqref{qr} becomes
\begin{equation}
27x^4-66x^2+8\sqrt{\smash{26-2\sqrt{97}}\vphantom{I^I}}x+3=0,
\end{equation}
and has four real roots, namely $x_3=t=-1.687\ldots$ and
\begin{equation}\label{456}
x=x_4=-0.109\ldots\quad y=x_5=0.442\ldots\quad z=x_6=1.354\ldots
\end{equation}
Let $\phi_i=\arctan x_i.$ Set $\sigma_3=0,$ but for $i=4,5,6$ choose
$\sigma_i>0$ such that $\mathbf{z}[\sigma_i,\phi_i]$ is correctly separated from
$\mathbf{z}[0,\phi_3].$ Then $[\mathbf{z}_1],[\mathbf{z}_2],\mathbf{z}[0,\phi_3]$ are
all a distance $2\pi/3$ from each of the six
points $\mathbf{z}[\pm\sigma_i,\phi_i]$ for $i=4,5,6.$ Moreover, the pairs
\begin{equation}\label{last6}\begin{array}{lll}
\mathbf{z}[\sigma_4,\phi_4],\ \mathbf{z}[\sigma_5,\phi_5]; &
\mathbf{z}[\sigma_4,\phi_4],\ \mathbf{z}[\sigma_6,\phi_6]; &
\mathbf{z}[\sigma_5,\phi_5],\ \mathbf{z}[-\sigma_6,\phi_6];\\[3pt]
\mathbf{z}[-\sigma_4,\phi_4],\mathbf{z}[-\sigma_5,\phi_5];\quad &
\mathbf{z}[-\sigma_4,\phi_4],\mathbf{z}[-\sigma_6,\phi_6];\quad &
\mathbf{z}[-\sigma_5,\phi_5],\ \mathbf{z}[\sigma_6,\phi_6] \end{array}\end{equation} are a distance
$2\pi/3$ apart. This does not contradict Lemma~\ref{LR} because
$\mathbf{z}[-\sigma_i,\phi_i]$ and $\mathbf{z}[\sigma_i,\phi_i]$ are not correctly
separated. All together, we have constructed nine points in $\mathbb{CP}^2$
for which 27 of the $\binom92$ pairs are correctly separated, though
the resulting configuration is less symmetrical than that of
Example~\ref{eigen}. The seven points $\mathbf{z}[\pm\sigma_i,\phi_i]$ are shown
in Figure~8; distinguishing a different root $x_3$ from the list
\eqref{456} would give a different picture of the same phenomenon.
\end{exa}
\begin{figure}[t]
\scalebox{1.3}{\includegraphics{fake2.pdf}}
\caption{A representation of seven points in $\mathbb{CP}^2$ all a distance
$2\pi/3$ from $[\mathbf{z}_1],[\mathbf{z}_2].$ Pairs joined by dashed edges are also
$2\pi/3$ apart.}
\vskip20pt
\end{figure}
\vskip20pt
\normalsize
\section*{Acknowledgments}
LPH acknowledges support from the Fields Institute, Ontario, the
Perimeter Institute, Ontario, and Tulane University, New Orleans, and
wishes to thank S.~Abramsky, D.~Appleby, R.~Blume-Kohout, D.~Brody,
H.~Brown, S.~Flammia, C.~Fuchs, L.~Hardy, and H.~Zhu for stimulating
discussions. SMS acknowledges support arising from visits to the
University of Nijmegen, the University of Sofia and the University of
Turin, and thanks N.~Lora Lamia for helpful comments. The authors are
grateful to J.~Armstrong for suggesting the use of quotient ideals and
their computation, which improved an earlier approach involving a
count of multiplicities of known solutions.
\bibliographystyle{plain}
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1,116,691,499,343 | arxiv | \section{#1}
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\begin{document}
\title[Asymptotics for the perfect conductivity problem ]{Asymptotics for the electric field when $M$-convex inclusions are close to the matrix boundary}
\author[Z.W. Zhao]{Zhiwen Zhao}
\address[Z.W. Zhao]{1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. }
\address{2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands.}
\email{[email protected].}
\date{\today}
\begin{abstract}
In the perfect conductivity problem of composites, the electric field may become arbitrarily large as $\varepsilon$, the distance between the inclusions and the matrix boundary, tends to zero. The main contribution of this paper lies in developing a clear and concise procedure to establish a boundary asymptotic formula of the concentration for perfect conductors with arbitrary shape in all dimensions, which explicitly exhibits the singularities of the blow-up factor $Q[\varphi]$ introduced in \cite{LX2017} by picking the boundary data $\varphi$ of $k$-order growth. In particular, the smoothness of inclusions required for at least $C^{3,1}$ in \cite{LLY2019} is weakened to $C^{2,\alpha}$, $0<\alpha<1$ here.
\end{abstract}
\maketitle
\section{Introduction and main results}
It is well known that field concentrations appear widely in nature and industrial applications. These fields include extreme electric, heat fluxes and mechanical loads. Motivated by the issue of material failure initiation, in this paper we are devoted to the investigation of blow-up phenomena arising from high-contrast fiber-reinforced composites with the densely packed fibers. The key feature of the concentrated fields is that the blow-up comes from the narrow regions between fibers and the thin gaps between fibers and the matrix boundary. It is worth emphasizing that the latter is more interesting due to the interaction from the boundary data. Although there has made great progress in the engineering and mathematical literature since Babu\u{s}ka et al's famous work \cite{BASL1999} over the past two decades, accurate numerical computation of the concentrated field are still very hard for lack of fine characterization to develop an efficient numerical scheme. So, it is significantly important from a practical point of view to precisely describe the singular behavior of such high concentration.
In the context of electrostatics, the field is the gradient of a solution to the Laplace equation and the blow-up rate of the gradient were captured accurately. Denote the distance between two inclusions or between inclusions and the matrix boundary by $\varepsilon$. It has been proved that for the perfect conductivity problem, the blow-up rate of the gradient is $\varepsilon^{-1/2}$ in two dimensions \cite{AKLLL2007,BC1984,BLY2009,AKL2005,Y2007,Y2009,K1993}, while it is $|\varepsilon\ln\varepsilon|^{-1}$ in three dimensions \cite{BLY2009,LY2009,BLY2010,L2012}.
Besides these foregoing estimates of the singularities for the field, there is another direction of investigation to establish the asymptotic formula of $\nabla u$ in the thin gap of electric field concentration. In two dimensions, consider the following conductivity problem
\begin{align}\label{per001}
\begin{cases}
\Delta u=0,&\hbox{in}\;\mathbb{R}^{2}\setminus\overline{D_{1}\cup D_{2}},\\
u=C_{j}, &\hbox{on}\;\partial D_{j},\;j=1,2,\\
u(\mathbf{x})-H(\mathbf{x})=O(|\mathbf{x}|^{-1}),&\mathrm{as}\;|\mathbf{x}|\rightarrow\infty,\\
\int_{\partial D_{j}}\frac{\partial u}{\partial\nu}\big|_{+}=0,&j=1,2,
\end{cases}
\end{align}
where $H$ is a given harmonic function in $\mathbb{R}^{2}$ and
$$\frac{\partial u}{\partial\nu}\Big|_{+}:=\lim_{\tau\rightarrow0}\frac{u(x+\nu\tau)-u(x)}{\tau}.$$
Here and throughtout this paper $\nu$ is the unit outer normal of $D_{j}$ and the subscript $\pm$ shows the limit from outside and inside the domain, respectively. For problem (\ref{per001}), Kang, Lim and Yun \cite{KLY2013} obtained a complete characterization of the singularities of $\nabla u$ with $D_{1}$ and $D_{2}$ being disks as follows
\begin{align}\label{singular}
\nabla u(\mathbf{x})=\frac{2r_{1}r_{2}}{r_{1}+r_{2}}(\mathbf{n}\cdot\nabla H)(\mathbf{p})\nabla h(\mathbf{x})+\nabla g(\mathbf{x}),
\end{align}
where $h(\mathbf{x})=\frac{1}{2\pi}(\ln|\mathbf{x}-\mathbf{p}_{1}|-\ln|\mathbf{x}-\mathbf{p}_{2}|)$ with $\mathbf{p}_{1}\in D_{1}$ and $\mathbf{p}_{2}\in D_{2}$ being the fixed point of $R_{1}R_{2}$ and $R_{2}R_{1}$ respectively, $R_{j}$ is the reflection with respect to $\partial D_{j}$, $\mathbf{n}$ is the unit vector in the direction of $\mathbf{p}_{2}-\mathbf{p}_{1}$, $\mathbf{p}$ is the middle point of the shortest line segment connecting $\partial D_{1}$ and $\partial D_{2}$, and $|\nabla g|$ is bounded independently of $\varepsilon$ on any bounded subset of $\mathbb{R}^{2}\setminus\overline{D_{1}\cup D_{2}}$. Obviously $\nabla h$ characterizes the singular behavior of $\nabla u$ explicitly. Ammari, Ciraolo, Kang, Lee, Yun \cite{ACKLY2013} extended the characterization (\ref{singular}) to the case when inclusions $D_{1}$ and $D_{2}$ are strictly convex domains in $\mathbb{R}^{2}$ by utilizing disks osculating to convex domains. In three dimensions, Kang, Lim and Yun \cite{KLY2014} derived an asymptotic formula of $\nabla u$ for two spherical perfect conductors with the same radii. The asymptotics for perfectly conducting particles with the different radii can be seen in \cite{LWX2019}. Recently, a great work on establishing an asymptotic formula in dimensions two and three for two arbitrarily $2$-convex inclusions has been completed by Li, Li and Yang in \cite{LLY2019}. It is worth mentioning that for high-contrast composites with the matrix described by nonlinear constitutive laws such as $p$-Laplace, Gorb and Novikov \cite{G2012} captured the stress concentration factor. Additionally, the asymptotics of the eigenvalues of the Poincar\'{e} variational problem for two close-to-touching inclusions were obtained by Bonnetier and Triki in \cite{BT2013}. More related work can be seen in \cite{ABTV2015,AKLLZ2009,BCN2013,BLL2015,BLL2017,BT2012,BV2000,DL2019,G2015,KY2019,KLY2015,LLBY2014,LY2015,M1996,MMN2007,BJL2017,LX2017}.
However, to the best of our knowledge, previous investigations on the asymptotics of the field concentration only focused on the narrow region between inclusions. This paper, by contrast, aims at deriving a completely asymptotic characterization for the perfect conductivity problem (\ref{con002}) with $m$-convex inclusions close to the matrix boundary and the boundary data of $k$-order growth in all dimensions. The asymptotic results in this paper also provide an efficient way to compute the electrical field numerically.
To state our main works in a precise manner, we first describe our domain and notations. Let $D\subset\mathbb{R}^{n}\,(n\geq2)$ be a bounded domain with $C^{2,\alpha}~(0<\alpha<1)$ boundary, which has a $C^{2,\alpha}$-subdomain $D_{1}^{\ast}$ touching matrix boundary $\partial D$ only at one point. That is, by a translation and rotation of the coordinates, if necessary,
\begin{align*}
\partial D_{1}^{\ast}\cap\partial D=\{0'\}\subset\mathbb{R}^{n-1}.
\end{align*}
Throughout the paper, we use superscript prime to denote ($n-1$)-dimensional domains and variables, such as $\Sigma'$ and $x'$. After a translation, we set
\begin{align*}
D_{1}^{\varepsilon}:=D_{1}^{\ast}+(0',\varepsilon),
\end{align*}
where $\varepsilon>0$ is a sufficiently small constant. For the sake of simplicity, denote
\begin{align*}
D_{1}:=D_{1}^{\varepsilon},\quad\mathrm{and}\quad\Omega:=D\setminus\overline{D}_{1}.
\end{align*}
The conductivity problem with inclusions close to touching matrix boundary can be modeled by the following scalar equation with piecewise constant coefficients
\begin{align}\label{con001}
\begin{cases}
\mathrm{div}{(a_{k}(x)\nabla u)}=0,&\hbox{in}\;D,\\
u=\varphi, &\hbox{on}\;\partial{D},
\end{cases}
\end{align}
where
\begin{align*}
a_{k}(x)=
\begin{cases}
k\in[0,1)\cup(1,\infty],&\hbox{in}\;D_{1},\\
1,&\hbox{on}\;D\setminus D_{1}.
\end{cases}
\end{align*}
Actually, equation (\ref{con001}) can also be used to describe more physical phenomenon, such as dielectrics, magnetism, thermal conduction, chemical diffusion and flow in porous media.
When the conductivity of $D_{1}$ degenerates to be infinity, problem (\ref{con001}) turns into the perfect conductivity problem as follows
\begin{align}\label{con002}
\begin{cases}
\Delta u=0,&\hbox{in}\;D\setminus D_{1},\\
u=C_{1}, &\hbox{in}\;\overline{D}_{1},\\
\int_{\partial D_{1}}\frac{\partial u}{\partial\nu}\big|_{+}=0,\\
u=\varphi, &\mathrm{on}\;\partial D,
\end{cases}
\end{align}
where the free constant $C_{1}$ is determined later by the third line of (\ref{con002}). There has established the existence, uniqueness and regularity of weak solutions to (\ref{con002}) in \cite{BLY2009} with a minor modification. We further assume that there exists a small constant $R>0$ independent of $\varepsilon$, such that the portions of $\partial D$ and $\partial D_{1}$ near the origin can be written as
\begin{align*}
x_{n}=\varepsilon+h_{1}(x')\quad\mathrm{and}\quad x_{n}=h(x'),\quad\quad x'\in B_{2R}',
\end{align*}
where $h_{1}$ and $h$ satisfy that for $m\geq2$,
\begin{enumerate}
{\it\item[(\bf{\em H1})]
$h_{1}(x')-h(x')=\lambda|x'|^{m}+O(|x'|^{m+1}),$
\item[(\bf{\em H2})]
$|\nabla_{x'}^{i}h_{1}(x')|,\,|\nabla_{x'}^{i}h(x')|\leq \kappa_{1}|x'|^{m-i},\;\,i=1,2,$
\item[(\bf{\em H3})]
$\|h_{1}\|_{C^{2,\alpha}(B'_{2R})}+\|h\|_{C^{2,\alpha}(B'_{2R})}\leq \kappa_{2},$}
\end{enumerate}
where $\lambda$ and $\kappa_{j},j=1,2$, are three positive constants independent of $\varepsilon$.
To explicitly uncover the effect of boundary data $\varphi$ on the singularities of the field, we classify $\varphi\in C^{2}(\partial D)$ according to its parity as follows. Denote the bottom boundary of $\Omega_{R}$ by $\Gamma^{-}_{R}=\{x\in\mathbb{R}^{n}|\,x_{n}=h(x'),\,|x'|<R\}$. Suppose that for $x\in\Gamma^{-}_{R}$,
\begin{itemize}
{\it
\item[({\bf{\em S1}})] $\varphi$ satisfies the $k$-order growth condition, that is,
\begin{align*}
\varphi(x)=\eta\,|x'|^{k};
\end{align*}
\item[({\bf{\em S2}})] $\varphi$ is odd with respect to some $x_{i_{0}}$, $i_{0}\in\{1,\cdots,n-1\}$,}
\end{itemize}
where $\eta>0$ and $k>1$ is a positive integer.
For $z'\in B'_{R},\,0<t\leq2R$, denote
\begin{align*}
\Omega_{t}(z'):=&\left\{x\in \mathbb{R}^{n}~\big|~h(x')<x_{n}<\varepsilon+h_{1}(x'),~|x'-z'|<{t}\right\}.
\end{align*}
We will use the abbreviated notation $\Omega_{t}$ for the domain $\Omega_{t}(0')$. Before stating our main results, we first introduce two scalar auxiliary functions $\bar{u}\in C^{2}(\mathbb{R}^{n})$ and $\bar{u}_{0}\in C^{2}(\mathbb{R}^{n})$ such that $\bar{u}=1$ on $\partial D_{1}$, $\bar{u}=0$ on $\partial D$ and
\begin{align}\label{con009}
\bar{u}(x)=\frac{x_{n}-h(x')}{\varepsilon+h_{1}(x')-h(x')},\;\,\mathrm{in}\;\,\Omega_{2R},\quad\;\|\bar{u}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq C,
\end{align}
and $\bar{u}_{0}=0$ on $\partial D_{1}$, $\bar{u}_{0}=\varphi(x)$ on $\partial D$, and
\begin{align}\label{con016}
\bar{u}_{0}=\varphi(x',h(x'))(1-\bar{u}),\;\,\mathrm{in}\;\Omega_{2R},\quad\;\|\bar{u}_{0}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq C.
\end{align}
To simplify notations used in the following, for $i=0$, and $i=k$, $k$ is the order of growth defined in ({\bf{\em S1}}), we denote
\begin{align*}
\rho_{i}(n,m;\varepsilon)=&
\begin{cases}
\varepsilon^{\frac{n+i-1}{m}-1},&m>n+i-1,\\
|\ln\varepsilon|,&m=n+i-1,\\
1,&m<n+i-1,
\end{cases}
\end{align*}
and
\begin{align*}
\Gamma_{m}^{n+i}=&
\begin{cases}
\Gamma\left(1-\frac{n+i-1}{m}\right)\Gamma\left(\frac{n+i-1}{m}\right),&m>n+i-1,\\
1,&m=n+i-1,
\end{cases}
\end{align*}
where $\Gamma(s)=\int^{+\infty}_{0}t^{s-1}e^{-t}\,dt$, $s>0$ is the Gamma function. Denote by $\omega_{n-1}$ the area of the surface of unit sphere in $(n-1)$-dimension. For $(z',z_{n})\in\Omega_{2R}$, denote
\begin{align}\label{ZZW666}
\delta(z'):=\varepsilon+h_{1}(z')-h(z').
\end{align}
Let $\Omega^{\ast}:=D\setminus\overline{D^{\ast}_{1}}$. We define a linear functional with respect to $\varphi$,
\begin{align}\label{linear001}
Q^{\ast}[\varphi]:=\int_{\partial D_{1}}\frac{\partial v_{0}^{\ast}}{\partial\nu},
\end{align}
where $v_{0}^{\ast}$ is a solution of the following problem:
\begin{align}\label{con003}
\begin{cases}
\Delta v_{0}^{\ast}=0,\quad\quad\;\,&\mathrm{in}\;\Omega^{\ast},\\
v_{0}^{\ast}=0,\quad\quad\;\,&\mathrm{on}\;\partial D_{1}^{\ast},\\
v_{0}^{\ast}=\varphi(x),\quad\;\,&\mathrm{on}\;\partial D.
\end{cases}
\end{align}
Note that the definition of $Q^{\ast}[\varphi]$ is valid under case ({\bf{\em S2}}) but only valid for $m<n+k-1$ under case ({\bf{\em S1}}). For $m<n-1$, define
\begin{align}\label{zz1}
a_{11}^{\ast}:=\int_{\Omega^{\ast}}|\nabla v_{1}^{\ast}|^{2},
\end{align}
where $v_{1}^{\ast}$ satisfies
\begin{equation}\label{con022}
\begin{cases}
\Delta v_{1}^{\ast}=0,\quad\;\,&\mathrm{in}\;\,\Omega^{\ast},\\
v_{1}^{\ast}=1,\quad\;\,&\mathrm{on}\;\,\partial D_{1}^{\ast}\setminus\{0\},\\
v_{1}^{\ast}=0,\quad\;\,&\mathrm{on}\;\,\partial D.
\end{cases}
\end{equation}
Unless otherwise stated, in what following $C$ represents a constant, whose values may vary from line to line, depending only on $\lambda$, $\kappa_{1},\kappa_{2},R$ and an upper bound of the $C^{2,\alpha}$ norms of $\partial D_{1}$ and $\partial D$, but not on $\varepsilon$. We also call a constant having such dependence a $universal$ $constant$. Without loss of generality, we set $\varphi(0)=0$. Otherwise, we substitute $u-\varphi(0)$ for $u$ throughout this paper. For simplicity of discussions, we assume that convexity index $m\geq2$ and growth order index $k>1$ are all positive integers in the following.
\begin{theorem}\label{thm001}
Assume that $D_{1}\subset D\subseteq\mathbb{R}^{n}\,(n\geq2)$ are defined as above, conditions ({\bf{\em H1}})--({\bf{\em H3}}) and ({\bf{\em S1}}) hold. Let $u\in H^{1}(D;\mathbb{R}^{n})\cap C^{1}(\overline{\Omega};\mathbb{R}^{n})$ be the solution of (\ref{con002}). Then for a sufficiently small $\varepsilon>0$ and $x\in\Omega_{R}$,
$(i)$ for $m\geq n+k-1$,
\begin{align*}
\nabla u=\frac{\eta\Gamma^{n+k}_{m}}{\lambda^{\frac{k}{m}}\Gamma^{n}_{m}}(1+O(r_{\varepsilon}))\rho_{k;0}(n,m;\varepsilon)\nabla\bar{u}+\nabla\bar{u}_{0}+O(\mathbf{1})\delta^{1-\frac{2}{m}}\|\varphi\|_{C^{2}(\partial D)};
\end{align*}
$(ii)$ for $n-1\leq m<n+k-1$, if $Q^{\ast}[\varphi]\neq0$,
\begin{align*}
\nabla u=\frac{m\lambda^{\frac{n-1}{m}}Q^{\ast}[\varphi]}{(n-1)\omega_{n-1}\Gamma^{n}_{m}}\frac{1+O(r_{\varepsilon})}{\rho_{0}(n,m;\varepsilon)}\nabla\bar{u}+\nabla\bar{u}_{0}+O(\mathbf{1})\delta^{1-\frac{2}{m}}\|\varphi\|_{C^{2}(\partial D)};
\end{align*}
$(iii)$ for $m<n-1$, if $Q^{\ast}[\varphi]\neq0$,
\begin{align*}
\nabla u=\frac{Q^{\ast}[\varphi]}{a_{11}^{\ast}}(1+O(r_{\varepsilon}))\nabla\bar{u}+\nabla\bar{u}_{0}+O(\mathbf{1})\delta^{1-\frac{2}{m}}\|\varphi\|_{C^{2}(\partial D)},
\end{align*}
where $\rho_{k;0}(n,m;\varepsilon)=\rho_{k}(n,m;\varepsilon)/\rho_{0}(n,m;\varepsilon)$, $\bar{u}$ and $\bar{u}_{0}$ are defined by (\ref{con009}) and (\ref{con016}), respectively, $\delta$ is defined by (\ref{ZZW666}), $a_{11}^{\ast}$ is defined by (\ref{zz1}), and
\begin{align}\label{CCC}
r_{\varepsilon}=&
\begin{cases}
\varepsilon^{\frac{1}{m}},&m>n+k,\\
\varepsilon^{\frac{1}{m}}|\ln\varepsilon|,&m=n+k,\\
|\ln\varepsilon|^{-1},&m=n+k-1,\\
\varepsilon^{\frac{n+k-1-m}{(n+k-1)(m+1)}},&n-1<m<n+k-1,\\
|\ln\varepsilon|^{-1},&m=n-1,\\
\max\{\varepsilon^{\frac{n+k-1-m}{(n+k-1)(m+1)}},\varepsilon^{\frac{1}{6}}\},&m<n-1.
\end{cases}
\end{align}
\end{theorem}
\begin{remark}
There is a great difference between interior asymptotics and boundary asymptotics. Specifically, the blow-up factor $Q_{\varepsilon}[\varphi]$ defined in \cite{LLY2019} is bounded for any boundary data $\varphi$, while $Q[\varphi]$ here can increase the singularities of the field by $\varepsilon^{\frac{n+k-1}{m}-1}$ if $m>n+k-1$ or $|\ln\varepsilon|$ if $m=n+k-1$ for the boundary data $\varphi$ with $k$-order growth. In addition, when $m>2$, the remainder of order $O(\varepsilon^{1-2/m})$ in the shortest line segment between the conductors and the matrix boundary provides a more precise characterization on the asymptotic behavior of the concentration than that of $m=2$. Finally, the concisely main terms $\nabla\bar{u}$ and $\nabla\bar{u}_{0}$ together with their coefficients can completely describe the singular effect of the geometry, which will greatly reduce the complexity of numerical computation for $\nabla u$.
\end{remark}
\begin{remark}
The asymptotics of $\nabla u$ in Theorem \ref{thm001} indicate that
$(1)$ if $m\leq n+k-1$, then its maximum achieves only at $\{x'=0'\}\cap\Omega$;
$(2)$ if $m>n+k-1$, then the maximum achieves at both $\{x'=0'\}\cap\Omega$ and $\{|x'|=\varepsilon^{\frac{1}{m}}\}\cap\Omega$.
\end{remark}
\begin{remark}
In order to further reveal the effect of principal curvatures of the geometry, we take $n=3$ relevant to physical dimension for example. Consider
\begin{align*}
\varphi=\eta_{1}|x_{1}|^{k}+\eta_{2}|x_{2}|^{k},\quad x\in\{\lambda_{1}|x_{1}|^{m}+\lambda_{2}|x_{2}|^{m}<R,\;x_{3}=h(x')\},
\end{align*}
and
\begin{align}\label{Geometry}
h_{1}(x')-h(x')=\lambda_{1}|x_{1}|^{m}+\lambda_{2}|x_{2}|^{m},\quad x'\in\{\lambda_{1}|x_{1}|^{m}+\lambda_{2}|x_{2}|^{m}<R\},
\end{align}
where $\lambda_{i},\eta_{i}$, $i=1,2$, are four positive constant independent of $\varepsilon$. Then by the same method as in Theorem \ref{thm001}, we find that the coefficient of the main term $\nabla\bar{u}$ has an explicit dependence on $\lambda_{i}$ and $\eta_{i}$ in the form of that $\eta_{1}\lambda_{1}^{-\frac{k}{m}}+\eta_{2}\lambda_{2}^{-\frac{k}{m}}$ for $m\geq k+2$ and $\sqrt[m]{\lambda_{1}\lambda_{2}}$ for $m<k+2$.
\end{remark}
\begin{theorem}\label{coro002}
Assume that $D_{1}\subset D\subseteq\mathbb{R}^{n}\,(n\geq2)$ are defined as above, conditions ({\bf{\em H1}})--({\bf{\em H3}}) and ({\bf{\em S2}}) hold, $Q^{\ast}[\varphi]\neq0$. Let $u\in H^{1}(D;\mathbb{R}^{n})\cap C^{1}(\overline{\Omega};\mathbb{R}^{n})$ be the solution of (\ref{con002}). Then for a sufficiently small $\varepsilon>0$,
$(i)$ for $m\geq n-1$,
\begin{align*}
\nabla u=\frac{m\lambda^{\frac{n-1}{m}}Q^{\ast}[\varphi]}{(n-1)\omega_{n-1}\Gamma^{n}_{m}}\frac{1+O(\tilde{r}_{\varepsilon})}{\rho_{0}(n,m;\varepsilon)}\nabla\bar{u}+\nabla\bar{u}_{0}+O(\mathbf{1})\delta^{1-\frac{2}{m}}\|\varphi\|_{C^{2}(\partial D)};
\end{align*}
$(ii)$ for $m<n-1$,
\begin{align*}
\nabla u=\frac{Q^{\ast}[\varphi]}{a_{11}^{\ast}}(1+O(\tilde{r}_{\varepsilon}))\nabla\bar{u}+\nabla\bar{u}_{0}+O(\mathbf{1})\delta^{1-\frac{2}{m}}\|\varphi\|_{C^{2}(\partial D)},
\end{align*}
where $\bar{u}$ and $\bar{u}_{0}$ are defined by (\ref{con009}) and (\ref{con016}), respectively, $\delta$ is defined by (\ref{ZZW666}), $a_{11}^{\ast}$ is defined by (\ref{zz1}), and
\begin{align*}
\tilde{r}_{\varepsilon}=&
\begin{cases}
\varepsilon^{\frac{m+n-2}{(m+1)(2m+n-2)}},&m>n-1,\\
|\ln\varepsilon|^{-1},&m=n-1,\\
\max\{\varepsilon^{\frac{m+n-2}{(m+1)(2m+n-2)}},\varepsilon^{\frac{1}{6}}\}.&m<n-1.
\end{cases}
\end{align*}
\end{theorem}
\begin{remark}
The asymptotics of $\nabla u$ in Theorem \ref{coro002} imply that
$(1)$ if $m<n$, then its maximum attains only at $\{x'=0'\}\cap\Omega$;
$(2)$ if $m=n$, then the maximum attains at $\{x'=0'\}\cap\Omega$ and $\{|x'|=\varepsilon^{\frac{1}{m}}\}\cap\Omega$ simultaneously;
$(3)$ if $m>n$, then the maximum attains at $\{|x'|=\varepsilon^{\frac{1}{m}}\}\cap\Omega$.
\end{remark}
\begin{remark}
If (\ref{Geometry}) holds in Theorem \ref{coro002}, we can obtain that the coefficient of the main term $\nabla\bar{u}$ has an explicit dependence of $\sqrt[m]{\lambda_{1}\lambda_{2}}$.
\end{remark}
The organization of this paper is as follows. In section 2, we carry out a linear decomposition of the solution $u$ to problem (\ref{con002}) as $v_{0}$ and $v_{1}$, defined by (\ref{con005}) and (\ref{con006}) below, and we prove the correspondingly main terms $\bar{u}_{0}$ and $\bar{u}$ constructed by (\ref{con009}) and (\ref{con016}), respectively, in Lemma \ref{lem001} and Theorem \ref{thm002}. Based on the results obtained in section 2, we give the proofs of Theorem \ref{thm001} and Theorem \ref{coro002} consisting of the asymptotics of blow-up factor $Q[\varphi]$ and $a_{11}$ in section 3.
\section{Preliminary}
\subsection{Solution split}
As in \cite{LX2017}, we decompose the solution $u$ of (\ref{con002}) as follows
\begin{align}\label{con0033}
u(x)=C_{1}v_{1}(x)+v_{0}(x),\quad\;\,\mathrm{in}\;D\setminus\overline{D}_{1},
\end{align}
where $v_{i}$, $i=0,1$, verify
\begin{align}\label{con005}
\begin{cases}
\Delta v_{0}=0,\quad\quad\;\,&\mathrm{in}\;\Omega,\\
v_{0}=0,\quad\quad\;\,&\mathrm{on}\;\partial D_{1},\\
v_{0}=\varphi(x),\quad\;\,&\mathrm{on}\;\partial D,
\end{cases}
\end{align}
and
\begin{align}\label{con006}
\begin{cases}
\Delta v_{1}=0,\quad\quad\;\,&\mathrm{in}\;\Omega,\\
v_{1}=1,\quad\quad\;\,&\mathrm{on}\;\partial D_{1},\\
v_{1}=0,\quad\;\,&\mathrm{on}\;\partial D,
\end{cases}
\end{align}
respectively. Similarly as (\ref{linear001}) and (\ref{con003}), we define a linear functional of $\varphi$ as follows
\begin{align}\label{linear002}
Q[\varphi]=\int_{\partial D_{1}}\frac{\partial v_{0}}{\partial\nu},
\end{align}
where $v_{0}$ is defined by (\ref{con005}). Denote
\begin{align*}
a_{11}:=\int_{\Omega}|\nabla v_{1}|^{2}dx.
\end{align*}
Then, it follows from the third line of (\ref{con002}) and the decomposition (\ref{con0033}) that
\begin{align*}
C_{1}\int_{\partial D_{1}}\frac{\partial v_{1}}{\partial\nu}+\int_{\partial D_{1}}\frac{\partial v_{0}}{\partial\nu}=0.
\end{align*}
Recalling the definition of $v_{1}$ and making use of integration by parts, we have
\begin{align}\label{con007}
\nabla u=\frac{Q[\varphi]}{a_{11}}\nabla v_{1}+\nabla v_{0}.
\end{align}
\subsection{A general boundary value problem}
To obtain the asymptotic expansion for $\nabla u$, we first consider the following general boundary value problem:
\begin{equation}\label{con008}
\begin{cases}
\Delta v=0,\quad\;\,&\mathrm{in}\;\,\Omega,\\
v=\psi,&\mathrm{on}\;\,\partial D_{1},\\
v=0,&\mathrm{on}\;\,\partial D,
\end{cases}
\end{equation}
where $\psi\in C^{2}(\partial D_{1})$ is a given scalar function. Note that if $\psi=1$ on $\partial D_{1}$, then $v_{1}=v$. Extend $\psi\in C^{2}(\partial D_{1})$ to $\psi\in C^{2}(\overline{\Omega})$ such that $\|\psi\|_{C^{2}(\overline{\Omega\setminus\Omega_{R}})}\leq C\|\psi\|_{C^{2}(\partial D_{1})}$. Construct a cutoff function $\rho\in C^{2}(\overline{\Omega})$ satisfying $0\leq\rho\leq1$, $|\nabla\rho|\leq C$ on $\overline{\Omega}$, and
\begin{align}\label{con011}
\rho=1\;\,\mathrm{on}\;\,\Omega_{\frac{3}{2}R},\quad\rho=0\;\,\mathrm{on}\;\,\overline{\Omega}\setminus\Omega_{2R}.
\end{align}
For $x\in\Omega$, we define
\begin{align*}
\bar{v}(x)=[\rho(x)\psi(x',\varepsilon+h_{1}(x'))+(1-\rho(x))\psi(x)]\bar{u}(x),
\end{align*}
where $\bar{u}$ is defined by (\ref{con009}). Specially,
\begin{align*}
\bar{v}(x)=\psi(x',\varepsilon+h_{1}(x'))\bar{u}(x),\quad\;\,\mathrm{in}\;\Omega_{R}.
\end{align*}
Due to (\ref{con009}), we have
\begin{align}\label{KK6}
\|\bar{v}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq C\|\psi\|_{C^{2}(\partial D_{1})}.
\end{align}
Similarly as in \cite{LX2017}, we can obtain an asymptotic expansion of the gradient for problem (\ref{con006}).
\begin{theorem}\label{thm002}
Assume as above. Let $v\in H^{1}(\Omega)$ be a weak solution of (\ref{con008}). Then, for a sufficiently small $\varepsilon>0$,
\begin{align}\label{con013}
|\nabla(v-\bar{v})(x)|\leq C\delta^{1-\frac{2}{m}}(|\psi(x',\varepsilon+h_{1}(x'))|+\delta^{\frac{1}{m}}\|\psi\|_{C^{2}(\partial D_{1})}),\quad\mathrm{in}\;\,\Omega_{R}.
\end{align}
Consequently, (\ref{con013}), together with choosing $\psi=1$ on $\partial D_{1}$, yields that
\begin{align}\label{con015}
\nabla v_{1}=\nabla\bar{u}+O(\mathbf{1})\delta^{1-\frac{2}{m}},\quad\;\,\mathrm{in}\;\Omega_{R},
\end{align}
and
\begin{align*}
\|\nabla v\|_{L^{\infty}(\Omega\setminus\Omega_{R})}\leq C\|\psi\|_{C^{2}(\partial D_{1})}.
\end{align*}
where $v_{1}\in H^{1}(\Omega)$ is a weak solution of (\ref{con006})
\end{theorem}
Note that when $m>2$, the remainder of order $O(1)$ in \cite{LX2017} is improved to that of order $O(\varepsilon^{1-2/m})$ for $x\in\{x'=0'\}\cap\Omega_{R}$ here. For readers' convenience, the detailed proof of Theorem \ref{thm002} is left in the Appendix. Similarly, by applying Theorem \ref{thm002}, we can find that the leading term of $\nabla v_{0}$ is $\nabla\bar{u}_{0}$ in the following.
\begin{lemma}\label{lem001}
Assume as above. Let $v_{0}$ be the weak solution of (\ref{con005}). Then, for a sufficiently small $\varepsilon>0$,
\begin{align}\label{con018}
\nabla v_{0}=\nabla\bar{u}_{0}+O(\mathbf{1})\delta^{1-\frac{2}{m}}(|\varphi(x',h(x'))|+\delta^{\frac{1}{m}}\|\varphi\|_{C^{2}(\partial D)}),\quad\;\,\mathrm{in}\;\Omega_{R},
\end{align}
and
\begin{align}\label{con01818}
\|\nabla_{x'}v_{0}\|_{L^{\infty}(\Omega_{R})}\leq C\|\varphi\|_{C^{2}(\partial D)},\;\,\|\nabla v_{0}\|_{L^{\infty}(\Omega\setminus\Omega_{R})}\leq C\|\varphi\|_{C^{2}(\partial D)},
\end{align}
where $\bar{u}_{0}$ is defined by (\ref{con016}).
\end{lemma}
Therefore, recalling the decomposition (\ref{con007}) and in view of (\ref{con015}) and (\ref{con018}), for the purpose of deriving the asymptotic of $\nabla u$, it suffices to establish the following two aspects of expansions:
(i) Expansion of $Q[\varphi]$;
(ii) Expansion of $a_{11}$.
\section{Proofs of Theorem \ref{thm001} and Theorem \ref{coro002}}
\subsection{Expansion of $Q[\varphi]$}
Before proving Theorem \ref{thm001} and Theorem \ref{coro002}, we first give an expansion of $Q[\varphi]$ with respect to $\varepsilon$.
\begin{lemma}\label{lem002}
Assume as above. Then, for a sufficiently small $\varepsilon>0$,
$(a)$ if ({\bf{\em S1}}) holds for $m\geq n+k-1$ in Theorem \ref{thm001},
\begin{align*}
Q[\varphi]=&\frac{(n-1)\omega_{n-1}\eta\Gamma^{n+k}_{m}}{m\lambda^{\frac{n+k-1}{m}}}\rho_{k}(n,m;\varepsilon)
\begin{cases}
1+O(1)\varepsilon^{\frac{1}{m}},&m>n+k,\\
1+O(1)\varepsilon^{\frac{1}{m}}|\ln\varepsilon|,&m=n+k,\\
1+O(1)|\ln\varepsilon|^{-1},&m=n+k-1;
\end{cases}
\end{align*}
$(b)$ if ({\bf{\em S1}}) holds for $m<n+k-1$ in Theorem \ref{thm001},
\begin{align*}
Q[\varphi]=&
Q^{\ast}[\varphi]+O(1)\varepsilon^{\frac{n+k-1-m}{(n+k-1)(m+1)}}.
\end{align*}
$(c)$ if ({\bf{\em S2}}) holds in Theorem \ref{coro002},
\begin{align*}
Q[\varphi]=&
Q^{\ast}[\varphi]+O(1)\varepsilon^{\frac{m+n-2}{(m+1)(2m+n-2)}}.
\end{align*}
\end{lemma}
\begin{proof}
{\bf Step 1.} Proof of $(a)$. Note that the unit outward normal $\nu$ to $\partial D_{1}$ is given by
$$\nu=\frac{(\nabla_{x'}h_{1}(x'),-1)}{\sqrt{1+|\nabla_{x'}h_{1}(x')|^{2}}},\quad\;\,\mathrm{in}\;\Omega_{R}.$$
In light of ({\bf{\em H2}}), we obtain that for $i=1,\cdots,n-1$,
\begin{align}\label{con021}
|\nu_{i}|\leq C|x'|^{m-1},\quad|\nu_{n}|\leq1,\quad\;\,\mathrm{in}\;\Omega_{R}.
\end{align}
Recalling the definition of $Q[\varphi]$, it follows from (\ref{con018})--(\ref{con01818}) and (\ref{con021}) that
\begin{align*}
Q[\varphi]=&\int_{\partial D_{1}}\partial_{n}v_{0}\nu_{n}+\int_{\partial D_{1}}\sum^{n-1}_{i=1}\partial_{i}v_{0}\nu_{i}\\
=&\int_{|x'|<R}\frac{\eta|x'|^{k}}{\varepsilon+\lambda|x'|^{m}}+O(1)\int_{|x'|<R}\frac{\eta|x'|^{k+1}}{\varepsilon+\lambda|x'|^{m}}+O(1)\|\varphi\|_{C^{2}(\partial D)}\\
=&\frac{(n-1)\omega_{n-1}\eta\Gamma^{n+k}_{m}}{m\lambda^{\frac{n+k-1}{m}}}\begin{cases}
\varepsilon^{\frac{n+k-1}{m}-1}+O(1)\varepsilon^{\frac{n+k}{m}-1}\|\varphi\|_{C^{2}(\partial D)},&m>n+k,\\
\varepsilon^{-\frac{1}{m}}+O(1)|\ln\varepsilon|\|\varphi\|_{C^{2}(\partial D)},&m=n+k,\\
|\ln\varepsilon|+O(1)\|\varphi\|_{C^{2}(\partial D)},&m=n+k-1.
\end{cases}
\end{align*}
{\bf Step 2.} Proofs of $(b)$ and $(c)$. In view of the definitions of $Q[\varphi]$ and $Q^{\ast}[\varphi]$, it follows from integration by parts that
\begin{align*}
Q[\varphi]=\int_{\partial D}\frac{\partial v_{1}}{\partial\nu}\varphi(x),\quad\quad Q^{\ast}[\varphi]=\int_{\partial D}\frac{\partial v_{1}^{\ast}}{\partial\nu}\varphi(x),
\end{align*}
where $v_{1}$ and $v_{1}^{\ast}$ are defined by (\ref{con006}) and (\ref{con022}). Thus,
\begin{align*}
Q[\varphi]-Q^{\ast}[\varphi]=\int_{\partial D}\frac{\partial(v_{1}-v_{1}^{\ast})}{\partial\nu}\cdot\varphi(x).
\end{align*}
To estimate $v_{1}-v_{1}^{\ast}$, we first introduce a scar auxiliary functions $\bar{u}^{\ast}$ satisfying $\bar{u}^{\ast}=1$ on $\partial D_{1}^{\ast}\setminus\{0\}$, $\bar{u}^{\ast}=0$ on $\partial D$, and
$$\bar{u}^{\ast}=\frac{x_{n}-h(x')}{h_{1}(x')-h(x')},\quad\mathrm{in}\;\,\Omega_{2R}^{\ast},\quad\;\,\|\bar{u}^{\ast}\|_{C^{2}(\Omega^{\ast}\setminus\Omega_{R}^{\ast})}\leq C,$$
where $\Omega^{\ast}_{r}:=\Omega^{\ast}\cap\{|x'|<r\},$ $0<r\leq2R$. In view of ({\bf{\em H2}}), we obtain that for $x\in\Omega_{R}^{\ast}$,
\begin{align}\label{con023}
|\nabla_{x'}(\bar{u}-\bar{u}^{\ast})|\leq\frac{C}{|x'|},
\end{align}
and
\begin{align}\label{con025}
|\partial_{n}(\bar{u}-\bar{u}^{\ast})|\leq\frac{C\varepsilon}{|x'|^{m}(\varepsilon+|x'|^{m})}.
\end{align}
Applying Theorem \ref{thm002} to (\ref{con022}), it follows that for $x\in\Omega_{R}^{\ast}$,
\begin{align}\label{con026}
|\nabla(v_{1}^{\ast}-\bar{u}^{\ast})|\leq C|x'|^{m-2},
\end{align}
and
\begin{align}\label{con027}
|\nabla_{x'}v_{1}^{\ast}|\leq\frac{C}{|x'|},\quad\;|\partial_{n}v_{1}^{\ast}|\leq\frac{C}{|x'|^{m}}.
\end{align}
For $0<r<R$, denote
\begin{align}\label{con028}
\mathcal{C}_{r}:=\left\{x\in\mathbb{R}^{n}\Big|\;|x'|<r,\,\frac{1}{2}\min_{|x'|\leq r}h(x')\leq x_{n}\leq\varepsilon+2\max_{|x'|\leq r}h_{1}(x')\right\}.
\end{align}
We now divide into two steps to estimate $|Q[\varphi]-Q^{\ast}[\varphi]|$.
{\bf Step 2.1.} Note that $v_{1}-v_{1}^{\ast}$ solves
\begin{align*}
\begin{cases}
\Delta(v_{1}-v_{1}^{\ast})=0,&\mathrm{in}\;\,D\setminus(\overline{D_{1}\cup D_{1}^{\ast}}),\\
v_{1}-v_{1}^{\ast}=1-v_{1}^{\ast},&\mathrm{on}\;\,\partial D_{1}\setminus D_{1}^{\ast},\\
v_{1}-v_{1}^{\ast}=v_{1}-1,&\mathrm{on}\;\,\partial D_{1}^{\ast}\setminus(D_{1}\cup\{0\}),\\
v_{1}-v_{1}^{\ast}=0,&\mathrm{on}\;\,\partial D.
\end{cases}
\end{align*}
We first estimate $|v_{1}-v_{1}^{\ast}|$ on $\partial(D_{1}\cup D_{1}^{\ast})\setminus\mathcal{C}_{\varepsilon^{\gamma}}$, where $0<\gamma<1/2$ to be determined later. In light of the definition of $v_{1}^{\ast}$, we derive that
$$|\partial_{n}v_{1}^{\ast}|\leq C,\quad\;\,\mathrm{in}\;\Omega^{\ast}\setminus\Omega^{\ast}_{R}.$$
Therefore,
\begin{align}\label{con029}
|v_{1}-v_{1}^{\ast}|\leq C\varepsilon,\quad\;\,\mathrm{for}\;\,x\in\partial D_{1}\setminus D_{1}^{\ast}.
\end{align}
It follows from (\ref{con015}) that
\begin{align}\label{con030}
|v_{1}-v_{1}^{\ast}|\leq C\varepsilon^{1-m\gamma},\quad\;\,\mathrm{on}\;\,\partial D_{1}^{\ast}\setminus(D_{1}\cup\mathcal{C}_{\varepsilon^{\gamma}}).
\end{align}
Combining Theorem \ref{thm002} and (\ref{con025})--(\ref{con026}), we obtain that for $x\in\Omega_{R}^{\ast}\cap\{|x'|=\varepsilon^{\gamma}\}$,
\begin{align*}
|\partial_{n}(v_{1}-v_{1}^{\ast})|\leq&|\partial_{n}(v_{1}-\bar{u})|+|\partial_{n}(\bar{u}-\bar{u}^{\ast})|+|\partial_{n}(v_{1}^{\ast}-\bar{u}^{\ast})|\\
\leq&C\left(\frac{1}{\varepsilon^{2m\gamma-1}}+\varepsilon^{(m-2)\gamma}\right),
\end{align*}
which together with $v_{1}-v_{1}^{\ast}=0$ on $\partial D$ yields that
\begin{align}\label{con031}
|(v_{1}-v_{1}^{\ast})(x',x_{n})|=&|(v_{1}-v_{1}^{\ast})(x',x_{n})-(v_{1}-v_{1}^{\ast})(x',h(x'))|\notag\\
\leq&C\big(\varepsilon^{1-m\gamma}+\varepsilon^{2(m-1)\gamma}\big).
\end{align}
Take $\gamma=\frac{1}{m+1}$. Then, it follows from (\ref{con029})--(\ref{con031}) that
$$|v_{1}-v_{1}^{\ast}|\leq C\varepsilon^{\frac{1}{m+1}},\quad\;\,\mathrm{on}\;\,\partial\big(D\setminus\big(\overline{D_{1}\cup D_{1}^{\ast}\cup\mathcal{C}_{\varepsilon^{\frac{1}{m+1}}}}\big)\big).$$
Making use of the maximum principle, we obtain
$$|v_{1}-v_{1}^{\ast}|\leq C\varepsilon^{\frac{1}{m+1}},\quad\;\,\mathrm{in}\;\,D\setminus\big(\overline{D_{1}\cup D_{1}^{\ast}\cup\mathcal{C}_{\varepsilon^{\frac{1}{m+1}}}}\big).$$
This, together with the standard interior and boundary estimates, leads to that, for any $\frac{m-1}{m(m+1)}<\tilde{\gamma}<\frac{1}{m+1}$,
$$|\nabla(v_{1}-v_{1}^{\ast})|\leq C\varepsilon^{m\tilde{\gamma}-\frac{m-1}{m+1}},\quad\;\,\mathrm{in}\;\,D\setminus\big(\overline{D_{1}\cup D_{1}^{\ast}\cup\mathcal{C}_{\varepsilon^{\frac{1}{m+1}-\tilde{\gamma}}}}\big),$$
which implies that
\begin{align}\label{con032}
|\mathcal{A}^{out}|:=\left|\int_{\partial D\setminus\mathcal{C}_{\varepsilon^{\frac{1}{m+1}-\tilde{\gamma}}}}\frac{\partial(v_{1}-v_{1}^{\ast})}{\partial\nu}\cdot\varphi(x)\right|\leq C\|\varphi\|_{L^{\infty}(\partial D)}\varepsilon^{m\tilde{\gamma}-\frac{m-1}{m+1}},
\end{align}
where $\frac{m-1}{m(m+1)}<\tilde{\gamma}<\frac{1}{m+1}$ to be determined later.
{\bf Step 2.2.} We further estimate
\begin{align*}
\mathcal{A}^{in}:=&\int_{\partial D\cap\mathcal{C}_{\varepsilon^{\frac{1}{m+1}-\tilde{\gamma}}}}\frac{\partial(v_{1}-v_{1}^{\ast})}{\partial\nu}\cdot\varphi(x)\\
=&\int_{\partial D\cap\mathcal{C}_{\varepsilon^{\frac{1}{m+1}-\tilde{\gamma}}}}\frac{\partial(w_{1}-w_{1}^{\ast})}{\partial\nu}\cdot\varphi(x)+\int_{\partial D\cap\mathcal{C}_{\varepsilon^{\frac{1}{m+1}-\tilde{\gamma}}}}\frac{\partial(\bar{u}-\bar{u}^{\ast})}{\partial\nu}\cdot\varphi(x)\\
=&:\mathcal{A}_{w}+\mathcal{A}_{u},
\end{align*}
where $w_{1}=v_{1}-\bar{u}$ and $w_{1}^{\ast}=v_{1}^{\ast}-\bar{u}^{\ast}$. To begin with, applying Theorem \ref{thm002}, we obtain that
\begin{align}\label{ZW12}
|\mathcal{A}_{w}|\leq C\eta\int_{\partial D\cap\mathcal{C}_{\varepsilon^{\frac{1}{m+1}-\tilde{\gamma}}}}|x'|^{m+k-2}\leq C\eta\varepsilon^{(\frac{1}{m+1}-\tilde{\gamma})(m+n+k-3)}.
\end{align}
To estimate $\mathcal{A}_{u}$, we split it into two parts as follows.
\begin{align*}
\mathcal{A}_{u}=&\int_{\partial D\cap\mathcal{C}_{\varepsilon^{\frac{1}{m+1}-\tilde{\gamma}}}}\sum^{n-1}_{i=1}\partial_{i}(\bar{u}-\bar{u}^{\ast})\nu_{i}\varphi(x)+\int_{\partial D\cap\mathcal{C}_{\varepsilon^{\frac{1}{m+1}-\tilde{\gamma}}}}\partial_{n}(\bar{u}-\bar{u}^{\ast})\nu_{n}\varphi(x)\\
=&:\mathcal{A}^{1}_{u}+\mathcal{A}^{2}_{u}.
\end{align*}
{\bf Case 1.} If ({\bf{\em S1}}) holds for $m<n+k-1$ in Theorem \ref{thm001}, owing to (\ref{con021}) and (\ref{con023})--(\ref{con025}), we obtain that
\begin{align*}
|\mathcal{A}^{1}_{u}|\leq C\eta\varepsilon^{\left(\frac{1}{m+1}-\tilde{\gamma}\right)(n+k+m-3)},\;\,|\mathcal{A}^{2}_{u}|\leq C\eta\varepsilon^{\left(\frac{1}{m+1}-\tilde{\gamma}\right)(n+k-m-1)}.
\end{align*}
Then
\begin{align*}
|\mathcal{A}_{u}|\leq C\eta\varepsilon^{\left(\frac{1}{m+1}-\tilde{\gamma}\right)(n+k-m-1)}.
\end{align*}
This, together with (\ref{con032})--(\ref{ZW12}) and picking $\tilde{\gamma}=\frac{n+k-2}{(n+k-1)(m+1)}$, yields that
\begin{align*}
|Q[\varphi]-Q^{\ast}[\varphi]|\leq&C(\eta+\|\varphi\|_{L^{\infty}(\partial D)})\varepsilon^{\frac{n+k-m-1}{(n+k-1)(m+1)}}.
\end{align*}
{\bf Case 2.} If ({\bf{\em S2}}) holds in Theorem \ref{coro002}, based on the fact that the integrating domain is symmetric with respect to $x_{i}$, $i=1,\cdots,n-1$, we have
\begin{align*}
|\mathcal{A}^{1}_{u}|\leq C\eta\varepsilon^{\left(\frac{1}{m+1}-\tilde{\gamma}\right)(n+m-2)},\;\,\mathcal{A}^{2}_{u}=0.
\end{align*}
Hence,
\begin{align*}
|\mathcal{A}_{u}|\leq C\eta\varepsilon^{\left(\frac{1}{m+1}-\tilde{\gamma}\right)(n+m-2)}.
\end{align*}
This, together with (\ref{con032})--(\ref{ZW12}) and taking $\tilde{\gamma}=\frac{2m+n-3}{(2m+n-2)(m+1)}$, leads to that
\begin{align*}
|Q[\varphi]-Q^{\ast}[\varphi]|\leq C(\eta+\|\varphi\|_{L^{\infty}(\partial D)})\varepsilon^{\frac{m+n-2}{(m+1)(2m+n-2)}}.
\end{align*}
Consequently, it follows from {\bf Step 1} and {\bf Step 2} that Lemma \ref{lem002} holds.
\end{proof}
\subsection{Expansion of $a_{11}$}
Before stating the asymptotic of $a_{11}$ with respect to $\varepsilon$, we first introduce a notation used in the following. Denote
\begin{align}
A:=&\int_{\Omega^{\ast}\setminus\Omega_{R}^{\ast}}|\nabla v_{1}^{\ast}|^{2}+2\int_{\Omega_{R}^{\ast}}\nabla\bar{u}^{\ast}\cdot\nabla(v_{1}^{\ast}-\bar{u}^{\ast})\notag\\
&+\int_{\Omega^{\ast}_{R}}\big(|\nabla(v_{1}^{\ast}-\bar{u}^{\ast})|^{2}+|\partial_{x'}\bar{u}^{\ast}|^{2}\big).\label{con03333}
\end{align}
\begin{lemma}\label{lem003}
Assume as in Theorem \ref{thm001} and Theorem \ref{coro002}. Then, for a sufficiently small $\varepsilon>0$,
$(i)$ for $m\geq n-1$,
\begin{align*}
a_{11}=&
\frac{(n-1)\omega_{n-1}\Gamma^{n}_{m}}{m\lambda^{\frac{n-1}{m}}}\rho_{0}(n,m;\varepsilon)
\begin{cases}
1+O(1)\varepsilon^{\frac{1}{m}},&m>n,\\
1+O(1)\varepsilon^{\frac{1}{m}}|\ln\varepsilon|,&m=n,\\
1+O(1)|\ln\varepsilon|^{-1},&m=n-1;
\end{cases}
\end{align*}
$(ii)$ for $m<n-1$,
\begin{align*}
a_{11}=&a_{11}^{\ast}+O(1)\varepsilon^{\frac{1}{6}},
\end{align*}
where $a_{11}^{\ast}$ is defined by (\ref{zz1}).
\end{lemma}
\begin{proof}
Fix $\bar{\gamma}=\frac{1}{6m}$. We first split $a_{11}$ into three parts as follows.
\begin{align*}
a_{11}=\int_{\Omega_{\varepsilon^{\bar{\gamma}}}}|\nabla v_{1}|^{2}+\int_{\Omega_{R}\setminus\Omega_{\varepsilon^{\bar{\gamma}}}}|\nabla v_{1}|^{2}+\int_{\Omega\setminus\Omega_{R}}|\nabla v_{1}|^{2}=:\mathrm{I}+\mathrm{II}+\mathrm{III}.
\end{align*}
{\bf Step 1.} As for $\mathrm{I}$, recalling the definition of $\bar{u}$ and using Theorem \ref{thm002}, we obtain that
\begin{align}\label{con03365}
\mathrm{I}=&\int_{\Omega_{\varepsilon^{\bar{\gamma}}}}|\partial_{n}\bar{u}|^{2}+\int_{\Omega_{\varepsilon^{\bar{\gamma}}}}|\partial_{x'}\bar{u}|^{2}+2\int_{\Omega_{\varepsilon^{\bar{\gamma}}}}\nabla\bar{u}\cdot\nabla(v_{1}-\bar{u})+\int_{\Omega_{\varepsilon^{\bar{\gamma}}}}|\nabla(v_{1}-\bar{u})|^{2}\notag\\
=&\int_{|x'|<\varepsilon^{\bar{\gamma}}}\frac{dx'}{\varepsilon+h_{1}(x')-h(x')}+O(1)\varepsilon^{\frac{n+m-3}{6m}}.
\end{align}
For the second term $\mathrm{II}$, we further decompose it into three parts as follows
\begin{align*}
\mathrm{II}_{1}=&\int_{(\Omega_{R}\setminus\Omega_{\varepsilon^{\bar{\gamma}}})\setminus(\Omega^{\ast}_{R}\setminus\Omega^{\ast}_{\varepsilon^{\bar{\gamma}}})}|\nabla v_{1}|^{2},\\
\mathrm{II}_{2}=&\int_{\Omega^{\ast}_{R}\setminus\Omega^{\ast}_{\varepsilon^{\bar{\gamma}}}}|\nabla(v_{1}-v_{1}^{\ast})|^{2}+2\int_{\Omega^{\ast}_{R}\setminus\Omega^{\ast}_{\varepsilon^{\bar{\gamma}}}}\nabla v_{1}^{\ast}\cdot\nabla(v_{1}-v_{1}^{\ast}),\\
\mathrm{II}_{3}=&\int_{\Omega^{\ast}_{R}\setminus\Omega^{\ast}_{\varepsilon^{\bar{\gamma}}}}|\nabla v_{1}^{\ast}|^{2}.
\end{align*}
Due to the fact that the thickness of $(\Omega_{R}\setminus\Omega_{\varepsilon^{\bar{\gamma}}})\setminus(\Omega^{\ast}_{R}\setminus\Omega^{\ast}_{\varepsilon^{\bar{\gamma}}})$ is $\varepsilon$, it follows from (\ref{con015}) that
\begin{align}\label{con0333355}
\mathrm{II}_{1}\leq&C\varepsilon\int_{\varepsilon^{\bar{\gamma}}<|x'|<R}\frac{dx'}{|x'|^{2m}}\leq C
\begin{cases}
\varepsilon^{\frac{4m+n-1}{6m}},&m>\frac{n-1}{2},\\
\varepsilon|\ln\varepsilon|,&m=\frac{n-1}{2},\\
\varepsilon,&m<\frac{n-1}{2}.
\end{cases}
\end{align}
By picking $\gamma=\frac{1}{2m}$ in {\bf Step 2.1} of the proof of Lemma \ref{lem002}, it follows from (\ref{con029})--(\ref{con031}) and the maximum principle that
\begin{align*}
|v_{1}-v_{1}^{\ast}|\leq C\varepsilon^{\frac{1}{2}},\quad\;\,\mathrm{in}\;\,D\setminus\big(\overline{D_{1}\cup D_{1}^{\ast}\cup\mathcal{C}_{\varepsilon^{\frac{1}{2m}}}}\big).
\end{align*}
Similarly as before, utilizing the standard interior and boundary estimates, we derive that
\begin{align}\label{con035}
|\nabla(v_{1}-v_{1}^{\ast})|\leq C\varepsilon^{\frac{1}{6}},\quad\;\,\mathrm{in}\;\,D\setminus\big(\overline{D_{1}\cup D_{1}^{\ast}\cup\mathcal{C}_{\varepsilon^{\frac{1}{3m}}}}\big).
\end{align}
Then combining (\ref{con027}) and (\ref{con035}), we obtain that
\begin{align}\label{con036}
|\mathrm{II}_{2}|\leq&C\varepsilon^{\frac{1}{6}}.
\end{align}
As for $\mathrm{II}_{3}$, it follows from (\ref{con026}) and (\ref{con027}) that
\begin{align*}
\mathrm{II}_{3}=&\int_{\Omega^{\ast}_{R}\setminus\Omega^{\ast}_{\varepsilon^{\bar{\gamma}}}}|\nabla\bar{u}^{\ast}|^{2}+2\int_{\Omega^{\ast}_{R}\setminus\Omega^{\ast}_{\varepsilon^{\bar{\gamma}}}}\nabla\bar{u}^{\ast}\cdot\nabla(v_{1}^{\ast}-\bar{u}^{\ast})+\int_{\Omega^{\ast}_{R}\setminus\Omega^{\ast}_{\varepsilon^{\bar{\gamma}}}}|\nabla(v_{1}^{\ast}-\bar{u}^{\ast})|^{2}\\
=&\int_{\varepsilon^{\bar{\gamma}}<|x'|<R}\frac{dx'}{h_{1}(x')-h(x')}+A-\int_{\Omega^{\ast}\setminus\Omega_{R}^{\ast}}|\nabla v_{1}^{\ast}|^{2}+O(1)\varepsilon^{(n+m-3)\bar{\gamma}},
\end{align*}
where $A$ is defined by (\ref{con03333}). This, together with (\ref{con0333355}) and (\ref{con036}), leads to that
\begin{align}\label{con037}
\mathrm{II}=&\int_{\varepsilon^{\bar{\gamma}}<|x'|<R}\frac{dx'}{h_{1}(x')-h(x')}+A-\int_{\Omega^{\ast}\setminus\Omega_{R}^{\ast}}|\nabla v_{1}^{\ast}|^{2}\notag\\
&+O(1)
\begin{cases}
\varepsilon^{\frac{m-1}{6m}},&n=2,\\
\varepsilon^{\frac{1}{6}},&n\geq3.
\end{cases}
\end{align}
For the last term $\mathrm{III}$, due to the fact that $|\nabla v_{1}|$ is bounded in $D_{1}^{\ast}\setminus(D_{1}\cup\Omega_{R})$ and $D_{1}\setminus D_{1}^{\ast}$ and the fact that the volume of $D_{1}^{\ast}\setminus(D_{1}\cup\Omega_{R})$ and $D_{1}\setminus D_{1}^{\ast}$ is of order $O(\varepsilon)$, it follows from (\ref{con035}) that
\begin{align*}
\mathrm{III}=&\int_{D\setminus(D_{1}\cup D_{1}^{\ast}\cup\Omega_{R})}|\nabla v_{1}|^{2}+O(1)\varepsilon\\
=&\int_{D\setminus(D_{1}\cup D_{1}^{\ast}\cup\Omega_{R})}|\nabla v_{1}^{\ast}|^{2}+2\int_{D\setminus(D_{1}\cup D_{1}^{\ast}\cup\Omega_{R})}\nabla v_{1}^{\ast}\cdot\nabla(v_{1}-v_{1}^{\ast})\\
&+\int_{D\setminus(D_{1}\cup D_{1}^{\ast}\cup\Omega_{R})}|\nabla(v_{1}-v_{1}^{\ast})|^{2}+O(1)\varepsilon\\
=&\int_{\Omega^{\ast}\setminus\Omega^{\ast}_{R}}|\nabla v_{1}^{\ast}|^{2}+O(1)\varepsilon^{\frac{1}{6}}.
\end{align*}
This, together with (\ref{con03365}) and (\ref{con037}), yields that
\begin{align*}
a_{11}=&\int_{\varepsilon^{\bar{\gamma}}<|x'|<R}\frac{dx'}{h_{1}(x')-h(x')}+\int_{|x'|<\varepsilon^{\bar{\gamma}}}\frac{dx'}{\varepsilon+h_{1}(x')-h(x')}\\
&+A+O(1)
\begin{cases}
\varepsilon^{\frac{m-1}{6m}},&n=2,\\
\varepsilon^{\frac{1}{6}},&n\geq3.
\end{cases}
\end{align*}
{\bf Step 2.} Denote
$$\mathbf{Main}:=\int_{\varepsilon^{\bar{\gamma}}<|x'|<R}\frac{dx'}{h_{1}(x')-h(x')}+\int_{|x'|<\varepsilon^{\bar{\gamma}}}\frac{dx'}{\varepsilon+h_{1}(x')-h(x')}.$$
(i) For $m\geq n-1$,
\begin{align*}
\mathbf{Main}=&\int_{|x'|<R}\frac{dx'}{\varepsilon+h_{1}-h}+\int_{\varepsilon^{\bar{\gamma}}<|x'|<R}\frac{\varepsilon\,dx'}{(h_{1}-h)(\varepsilon+h_{1}-h)}\\
=&\int_{|x'|<R}\frac{1}{\varepsilon+\lambda|x'|^{m}}+\int_{|x'|<R}\left(\frac{1}{\varepsilon+h_{1}-h}-\frac{1}{\varepsilon+\lambda|x'|^{m}}\right)+O(1)\varepsilon^{\frac{4m+n-1}{6m}}\\
=&(n-1)\omega_{n-1}\int_{0}^{R}\frac{s^{n-2}}{\varepsilon+\lambda s^{m}}+O(1)\int^{R}_{0}\frac{s^{n-1}}{\varepsilon+\lambda s^{m}}\\
=&
\frac{(n-1)\omega_{n-1}\Gamma^{n}_{m}}{m\lambda^{\frac{n-1}{m}}}
\begin{cases}
\varepsilon^{\frac{n-1}{m}-1}+O(1)\varepsilon^{\frac{n}{m}-1},&m>n,\\
\varepsilon^{-\frac{1}{m}}+O(1)|\ln\varepsilon|,&m=n,\\
|\ln\varepsilon|+O(1),&m=n-1;
\end{cases}
\end{align*}
(ii) For $m<n-1$,
\begin{align*}
\mathbf{Main}=&\int_{|x'|<R}\frac{dx'}{h_{1}-h}-\int_{\varepsilon^{\bar{\gamma}}<|x'|<R}\frac{\varepsilon\,dx'}{(h_{1}-h)(\varepsilon+h_{1}-h)}\\
=&\int_{\Omega_{R}^{\ast}}|\partial_{n}\bar{u}^{\ast}|^{2}+O(1)\varepsilon^{\frac{4m+n-1}{6m}}.
\end{align*}
Therefore, it follows from {\bf Step 1} and {\bf Step 2} that Lemma \ref{lem003} holds.
\subsection{Proof of Theorem \ref{thm001}}
Recalling decomposition (\ref{con007}) and combining the results derived in Theorem \ref{thm002}, Lemma \ref{lem001}, Lemma \ref{lem002} and Lemma \ref{lem003}, we complete the proofs of Theorem \ref{thm001} and Theorem \ref{coro002}.
\end{proof}
\section{Appendix:\,The proof of Theorem \ref{thm002}}
In light of assumptions ({\bf{\em H1}}) and ({\bf{\em H2}}), it follows from a direct calculation that for $i=1,\cdots,n-1$, $x\in\Omega_{2R}$,
\begin{align}
|\partial_{i}\bar{v}|\leq&\frac{C|\psi(x',\varepsilon+h_{1}(x'))|}{\sqrt[m]{\varepsilon+|x'|^{m}}}+C\|\nabla\psi\|_{L^{\infty}(\partial D_{1})},\label{ADE001}\\
|\partial_{n}\bar{v}|=&\frac{|\psi(x',\varepsilon+h_{1}(x'))|}{\delta(x')},\quad\partial_{nn}\bar{v}=0,\label{ADE002}
\end{align}
and
\begin{align}\label{ADE003}
|\Delta\bar{v}|\leq\frac{|\psi(x',\varepsilon+h_{1}(x'))|}{(\varepsilon+|x'|^{m})^{\frac{2}{m}}}+\frac{\|\nabla\psi\|_{L^{\infty}(\partial D_{1})}}{\sqrt[m]{\varepsilon+|x'|^{m}}}+\|\nabla^{2}\psi\|_{L^{\infty}(\partial D_{1})}.
\end{align}
Here and throughout this section, for simplicity of notations, we use $\|\nabla\psi\|_{L^{\infty}}$, $\|\nabla^{2}\psi\|_{L^{\infty}}$ and $\|\psi\|_{C^{2}}$ to denote $\|\nabla\psi\|_{L^{\infty}(\partial{D}_{1})}$, $\|\nabla^{2}\psi\|_{L^{\infty}(\partial{D_1})}$ and $\|\psi\|_{C^{2}(\partial D_{1})}$, respectively.
Define
\begin{equation}\label{def_w}
w:=v-\bar{v}.
\end{equation}
\noindent{\bf STEP 1.}
Let $v\in H^1(\Omega)$ be a weak solution of (\ref{con008}). Then
\begin{align}\label{ADE005}
\int_{\Omega}|\nabla w|^2dx\leq C\|\psi\|_{C^{2}(\partial D_{1})}^{2}.
\end{align}
Invoking (\ref{def_w}), $w$ satisfies
\begin{align}\label{ADE0006}
\begin{cases}
\Delta w=-\Delta\bar{v},&
\hbox{in}\ \Omega, \\
w=0, \quad&\hbox{on} \ \partial\Omega.
\end{cases}
\end{align}
Multiplying the equation in (\ref{ADE0006}) and integrating by parts, it follows from the Poincar\'{e} inequality, Sobolev trace embedding theorem, (\ref{KK6}) and (\ref{ADE001})--(\ref{ADE002}) that
\begin{align*}
\int_{\Omega}|\nabla w|^{2}=&\int_{\Omega_{R}}\omega\Delta\bar{v}+\int_{\Omega\setminus\Omega_{R}}\omega\Delta\bar{v}\\
\leq&\sum^{n-1}_{i=1}\left|\int_{\Omega_{R}}\omega\partial_{ii}\bar{v}\right|+C\|\psi\|_{C^{2}}\int_{\Omega\setminus\Omega_{R}}|w|\\
\leq&C\|\nabla w\|_{L^{2}(\Omega_{R})}\|\nabla_{x'}\bar{v}\|_{L^{2}(\Omega_{R})}+C\|\psi\|_{C^{2}(\partial D_{1})}\|\nabla w\|_{L^{2}(\Omega\setminus\Omega_{R})}\\
\leq&C\|\psi\|_{C^{2}(\partial D_{1})}\|\nabla w\|_{L^{2}(\Omega)}.
\end{align*}
Then (\ref{ADE005}) is proved.
\noindent{\bf STEP 2.}
Proof of
\begin{align}\label{step2}
\int_{\Omega_\delta(z')}|\nabla w|^2dx
&\leq C\delta^{n+2-\frac{4}{m}}\left(|\psi(z',\varepsilon+h_{1}(z'))|^2+\delta^{\frac{2}{m}}\|\psi\|_{C^2(\partial D_1)}^2\right),
\end{align}
where $\delta$ is defined by (\ref{ZZW666}). As seen in \cite{LX2017}, we have the iteration formula as follows:
\begin{align*}
\int_{\Omega_{t}(z')}|\nabla w|^{2}dx\leq\frac{C}{(s-t)^{2}}\int_{\Omega_{s}(z')}|w|^{2}dx+C(s-t)^{2}\int_{\Omega_{s}(z')}|\Delta\bar{v}|^{2}dx.
\end{align*}
We next divide into two cases to prove (\ref{step2}).
{\bf Case 1.} If $|z'|<\varepsilon^{\frac{1}{m}},\,0<s<\varepsilon^{\frac{1}{m}}$, we have $\varepsilon\leq\delta(x')\leq C\varepsilon$ in $\Omega_{\sqrt[m]{\varepsilon}}(z')$. In light of (\ref{ADE003}), we derive
\begin{align}\label{ADE006}
&\int_{\Omega_{s}(z')}|\Delta\bar{v}|^{2}\leq C|\psi(z',\varepsilon+h_{1}(z'))|^{2}\frac{s^{n-1}}{\varepsilon^{\frac{4}{m}-1}}+Cs^{n-1}\varepsilon^{1-\frac{2}{m}}\|\psi\|^{2}_{C^{2}},
\end{align}
while, due to the fact that $w=0$ on $\Gamma^{-}_{R}:=\{x\in\mathbb{R}^{n}|\,x_{n}=h(x'),\,|x'|<R\}$,
\begin{align}\label{ADE007}
\int_{\Omega_{s}(z')}|w|^{2}\leq C\varepsilon^{2}\int_{\Omega_{s}(z')}|\nabla w|^{2}.
\end{align}
Denote
$$F(t):=\int_{\Omega_{t}(z')}|\nabla w_{1}|^{2}.$$
It follows from (\ref{ADE006}) and (\ref{ADE007}) that for $0<t<s<\varepsilon^{\frac{1}{m}}$,
\begin{align}\label{ADE008}
F(t)\leq &\left(\frac{c_1\varepsilon}{s-t}\right)^2F(s)+C(s-t)^2s^{n-1}\bigg(\frac{|\psi(z',\varepsilon+h_{1}(z'))|^2}{\varepsilon^{\frac{4-m}{m}}}+\frac{\|\psi\|_{C^2}}{\varepsilon^{\frac{2-m}{m}}}\bigg),
\end{align}
where $c_{1}$ and $C$ are universal constants.
Pick $k=\left[\frac{1}{4c_{1}\sqrt[m]{\varepsilon}}\right]+1$ and $t_{i}=\delta+2c_{1}i\varepsilon,\;i=0,1,2,\cdots,k$. Then, (\ref{ADE008}), together with $s=t_{i+1}$ and $t=t_{i}$, leads to
$$F(t_{i})\leq\frac{1}{4}F(t_{i+1})+C(i+1)^{n-1}\varepsilon^{n+2-\frac{4}{m}}\left[|\psi(z',\varepsilon+h_{1}(z'))|^{2}+\varepsilon^{\frac{2}{m}}\|\psi\|^{2}_{C^{2}}\right].$$
It follows from $k$ iterations and (\ref{ADE005}) that for a sufficiently small $\varepsilon>0$,
\begin{align}\label{ADE009}
F(t_{0})\leq C\varepsilon^{n+2-\frac{4}{m}}\left(|\psi(z',\varepsilon+h_{1}(z'))|^{2}+\varepsilon^{\frac{2}{m}}\|\psi\|^{2}_{C^{2}}\right).
\end{align}
{\bf Case 2.} If $\varepsilon^{\frac{1}{m}}\leq|z'|\leq R$ and $0<s<\frac{2|z'|}{3}$, we have $\frac{|z'|^{m}}{C}\leq\delta(x')\leq C|z'|^{m}$ in $\Omega_{\frac{2|z'|}{3}}(z')$. Similar to (\ref{ADE006}) and (\ref{ADE007}), we obtain
\begin{align*}
\int_{\Omega_{s}(z')}|\Delta\bar{v}|^{2}\leq&C|\psi(z',\varepsilon+h_{1}(z'))|^{2}\frac{s^{n-1}}{|z'|^{4-m}}+Cs^{n-1}|z'|^{m-2}\|\psi\|^{2}_{C^{2}},
\end{align*}
and
$$\int_{\Omega_{s}(z')}|w|^{2}\leq C|z'|^{2m}\int_{\Omega_{s}(z')}|\nabla w|^{2}.$$
Moreover, for $0<t<s<\frac{2|z'|}{3}$, estimate (\ref{ADE008}) becomes
\begin{align*}
F(t)\leq\left(\frac{c_{2}|z'|^{m}}{s-t}\right)^{2}F(s)+C(s-t)^{2}s^{n-1}\left(\frac{|\psi(z',\varepsilon+h_{1}(z'))|^{2}}{|z'|^{4-m}}+|z'|^{m-2}\|\psi\|_{C^{2}}^{2}\right).
\end{align*}
Similarly as above, pick $k=\left[\frac{1}{4c_{2}|z'|}\right]+1,\,t_{i}=\delta+2c_{2}i|z'|^{m},\,i=0,1,2,\cdots,k$ and take $s=t_{i+1},\;t=t_{i}$. Then, we obtain
$$F(t_{i})\leq\frac{1}{4}F(t_{i+1})+C(i+1)^{n-1}|z'|^{m(n+2)-4}\left(|\psi(z',\varepsilon+h_{1}(z'))|^{2}+|z'|^{2}\|\psi\|^{2}_{C^{2}}\right).$$
Likewise, by using $k$ iterations, we have
\begin{align}\label{ADE010}
F(t_{0})\leq C|z'|^{m(n+2)-4}\left(|\psi(z',\varepsilon+h_{1}(z'))|^{2}+|z'|^{2}\|\psi\|^{2}_{C^{2}}\right).
\end{align}
Consequently, (\ref{ADE010}), together with (\ref{ADE009}), yields that (\ref{step2}) holds.
\noindent{\bf STEP 3.}
Proof of
\begin{align}\label{ADE011}
|\nabla w(x)|\leq C\delta^{1-\frac{2}{m}}(|\psi(x',\varepsilon+h_{1}(x'))|+\delta^{\frac{1}{m}}\|\psi\|_{C^{2}(\partial D_{1})}),\quad\mathrm{in}\;\Omega_{R}.
\end{align}
As in \cite{LX2017}, combining the rescaling argument, Sobolev embedding theorem, $W^{2,p}$ estimate and bootstrap argument, we obtain
\begin{align*}
\|\nabla w\|_{L^{\infty}(\Omega_{\delta/2}(z'))}\leq\frac{C}{\delta}\left(\delta^{1-\frac{n}{2}}\|\nabla w\|_{L^{2}(\Omega_{\delta}(z'))}+\delta^{2}\|\Delta\bar{v}\|_{L^{\infty}(\Omega_{\delta}(z'))}\right).
\end{align*}
In view of (\ref{ADE003}) and (\ref{step2}), we obtain that for $|z'|\leq R$,
$$\delta\|\Delta\bar{v}\|_{L^{\infty}(\Omega_{\delta}(z'))}\leq C\delta^{1-\frac{2}{m}}(|\psi(z',\varepsilon+h_{1}(z'))|+\delta^{\frac{1}{m}}\|\psi\|_{C^{2}}),$$
and
\begin{align*}
\delta^{-\frac{n}{2}}\|\nabla w\|_{L^{2}(\Omega_{\delta}(z'))}\leq C\delta^{1-\frac{2}{m}}(|\psi(z',\varepsilon+h_{1}(z'))|+\delta^{\frac{2}{m}}\|\psi\|_{C^{2}}).
\end{align*}
Consequently, for $h(z')<z_{n}<\varepsilon+h_{1}(z')$,
$$|\nabla w(z',z_{n})|\leq C\delta^{1-\frac{2}{m}}(|\psi(z',\varepsilon+h_{1}(z'))|+\delta^{\frac{1}{m}}\|\psi\|_{C^{2}}).$$
Estimate (\ref{con013}) is established. On the other hand, it follows from the standard interior estimates and boundary estimates for the Laplace equation that
\begin{align*}
\|\nabla v\|_{L^{\infty}(\Omega\setminus\Omega_{R})}\leq C\|\psi\|_{C^{2}(\partial D_{1})}.
\end{align*}
Thus, Theorem \ref{thm002} is proved.
\noindent{\bf{\large Acknowledgements.}} The author is greatly indebted to Professor HaiGang Li for his constant encouragement and very helpful discussions. The author was partially supported by NSFC (11971061) and BJNSF (1202013).
\bibliographystyle{plain}
\def$'${$'$}
|
1,116,691,499,344 | arxiv | \section{Introduction}
The shell model of the nucleus has remained its most credible microscopic description through
more than seven decades now. Testing the model across the nuclear chart and refining the inputs,
towards accomplishing better overlap with data, has been an agenda of nuclear structure
studies through their evolving practice. The exercise is facilitated by developments in
computational resources that help circumvent the dimensional challenges incurred in the application
of shell model, particularly to heavier systems such as those around Pb ($Z = 82$).
It may be noted that the very validity of the shell model for describing level structures
around the proton $Z = 82$ closure
was a subject of early investigations in the region. While the closure at $Z = 82$ was
identified to be sufficiently stable against collective excitations \cite{Rah85}, it was also observed that
light Hg ($Z = 80$) isotopes do exhibit collectivity and there were predictions of similar phenomena
in the proton-rich side of the ($Z = 82$) closure, for the light Po ($Z = 84$) nuclei \cite{Wec85}.
The studies undertaken towards resolving the proposition, however, froze on describing the excitations
of light-Po isotopes, such as $^{199-201}$Po, within the framework of the shell model. This was also
commensurate with the systematically calculated \cite{Wec85} shapes of the Pb isotopes starting from
$^{208}$Pb ($Z = 82, N = 126$) and extending to the lighter ones. The doubly-magic $^{208}$Pb, quite
expectedly, exhibited deep energy minimum for a spherical shape; the minimum became shallower for
lighter systems in the isotopic chain and eventually evolved into a double minima corresponding to
both prolate and oblate deformations for nuclei as light as $^{190}$Pb ($Z = 82, N = 108$).
Such a scenario, however, wasn't established in $^{198}$Pb or $^{202}$Pb that still manifested
near spherical shapes and it was found valid to interpret the excitation schemes of the neighboring
light Po isotopes from the perspectives of the shell model. The merits of such interpretation
notwithstanding, it was largely extracted from the evolution of experimentally
observed level energies and their spacings across the isotopic and/or the isotonic chains. That was
presumably owing to the limited wherewithal then available for computational endeavors but, nevertheless,
could provide insights into the particle excitations underlying the level scheme of the nuclei being
studied. The experimental findings in these studies mostly followed population of the nuclei of interest
in $\alpha$- or heavy-ion induced fusion-evaporation reactions and detection of the $\gamma$-rays using
modest setups of few Ge detectors and, at times, using conversion electron measurements alongwith. \\
The only existing precedence of spectroscopic study of the $^{203}$Po ($Z = 84, N = 119$) nucleus,
following its population in a fusion-evaporation reaction, was by Fant {\it{et al.}} \cite{Fan86}.
The nucleus was populated using $\alpha$-induced reaction on $^{204}$Pb and the de-excitation $\gamma$-rays
were detected using small planar Ge(Li) detectors, large coaxial Ge(Li) detectors and intrinsic Ge detectors.
Conversion electrons were also measured in conjunction. The level scheme of the nucleus was established
upto an excitation energy of $\sim$ 4.4 MeV and spin $\sim$ 18$\hbar$. However, only a selected number of
$\gamma$-ray transitions, presumably the strongest ones, and levels were identified above the 25/2$^+$ state;
the spin-parity assignments were considerably tentative therein. The configurations of the excited states
were largely ascribed to the coupling of an odd neutron hole to the excitations of the even $^{204}$Po-core ($Z = 84, N = 120$).
Two configurations, based on proton excitations outside the closed proton shell of the $^{208}$Pb-core, were
identified in the latter. These were $\pi h_{9/2}^2$ and $\pi h_{9/2}i_{13/2}$ that resulted in maximum spins
8 and 11 respectively. The available single particle orbitals for the odd neutron are $2f_{7/2}, 1h_{9/2}, 1i_{13/2},
3p_{3/2}, 2f_{5/2}, 3p_{1/2}$ and the first $5/2^-, 3/2^-, 1/2^-, 13/2^+$ states, in $^{203}$Po, were identified with
single neutron excitations therein. The 17/2$^+$, 21/2$^+$, and 25/2$^+$ yrast states in odd-A Po isotopes
were attributed to the odd neutron hole $\nu i_{13/2}^{-1}$ coupled to the excitations of the corresponding Pb-core
or of the two valence protons of the Po-core, resulting in states 2$^+$ - 8$^+$. This followed the systematics
of the yrast states in odd-A Pb and Po isotones. It may be noted that the yrast 17/2$^+$ and the 21/2$^+$ states
in isotopes $^{199-205}$Pb had been ascribed to pure neutron excitations, such as $\nu p_{1/2}^{-1}f_{5/2}^{-1}i_{13/2}^{-1}$
and $\nu f_{5/2}^{-2}i_{13/2}^{-1}$. However, such (pure neutron) excitations would result into states of
higher excitation energies than those of the yrast 17/2$^+$ and the 21/2$^+$ levels in odd-A Po isotopes. It was thus
found reasonable to assign the pure neutron excitations to the respective non-yrast states. The 27/2$^+$ and the 29/2$^+$
levels in the odd Po nuclei were identified with three-quasiparticle configurations $\pi h_{9/2}^2\otimes\nu i_{13/2}^{-1}$.
The configurations for the isomeric 25/2$^-$ and 29/2$^-$ were derived from their overlap with the systematics
of these states observed in the Pb isotopes. Accordingly, their configuration in $^{203}$Po was identified
to be similar to that in $^{201}$Pb and the same is $(\pi (h_{9/2}^2)_{0^+}\otimes\nu p_{1/2}^{-2}f_{5/2}^{-3}(i_{13/2}^{-2})_{12^+})_{25/2^{-}29/2^{-}}$. The findings in $^{203}$Po thus upheld the interpretation of
its excitation scheme within the framework
of the single particle excitations, as had been established for the still lighter isotopes
of the nucleus \cite{Wec85}. This was also a
continuing trend from the heavier isotopes such as $^{205,207}$Po \cite{Rah85}. The absence of collectivity was
further corroborated by the absence of enhanced B(E2) in these nuclei \cite{Fan86}. \\
The present paper reports a spectroscopic investigation of the level structure of $^{203}$Po, using
updated experimental facilities as well as contemporary framework for shell model calculations. The objective
was to explore possible features in the excitation scheme of the nucleus, through the use of a
large array of high-resolution gamma-ray detectors in the setup, and to test the reproducibility of the observed
level energies in the calculations of the interacting shell model. The computational exercise is a validation
of the model Hamiltonian used for the purpose as well as of facility in identifying and quantifying the single
particle excitations that contribute to the observed level scheme. \\
\section{Experimental Details and Data Analysis}
Excitations of the $^{203}$Po nucleus were investigated following its population in the $^{194}$Pt($^{13}$C,4n) reaction
at $E_{lab}$ = 74 MeV. The target was 13 mg/cm$^2$ thick self-supporting foil of enriched (99\%) $^{194}$Pt. The
beam was delivered by the 15 UD Pelletron at IUAC, New Delhi and the beam energy was so chosen after an excitation
function measurement at the commencement of the experiment. As per the predictions of the statistical model calculations,
at this beam energy, the aforementioned reaction would be of dominant cross section amongst the possible
compound nucleus fusion-evaporation channels while the fission (exit) channel would amount to
$\sim$ 25\% of the total fusion cross-section. Indeed, the yield of $^{203}$Po was observed to be maximum
when compared to the other fusion-evaporation products, that principally included isotopes of Po ($Z = 84$),
Bi ($Z = 83$) and Pb ($Z = 82$), as illustrated in Fig. 1.
The detection system was the Indian National Gamma Array (INGA) setup
at IUAC \cite{Mur10}
and (then) consisted of eighteen Compton suppressed HPGe clover detectors positioned at 148$^o$ (4 detectors),
123$^o$ (4 detectors), 90$^o$ (6 detectors), 57$^o$ (2 detectors), and 32$^o$ (2 detectors). An assembly
of three absorber sheets of lead, tin, and copper was afixed on the face of the heavymet collimator of the
Anti Compton Shield (ACS) in each detector. The absorbers facilitated in reducing the intensity of the X-rays,
from the thick target, being incident on the detectors (and thus contributing in the event trigger).
Data was principally acquired under the condition that at least two Compton suppressed HPGe clover
detectors needed to fire in coincidence for generating the event trigger. The two- and higher-fold events
acquired was $\sim$ 2$\times$10$^9$.\\
\begin{figure}
\includegraphics[angle=-90,scale=.35,trim=2.0cm 2.0cm 0.0cm 1.0cm,clip=true]{fig1.ps}
\caption{\label{fig1} Part of the $\gamma$-ray spectrum corresponding to the full projection of a
$\gamma$-$\gamma$ symmetric matrix and illustrating the different product nuclei populated
in the present experiment.}
\end{figure}
The data was sorted into spectra, symmetric and asymmetric (angle dependent) $\gamma$-$\gamma$ matrices as
well as $\gamma$-$\gamma$-$\gamma$ cube using SPRINGZ \cite{Das17_2} and INGASORT \cite{Bho01} codes
and subsequently analyzed using the RADWARE \cite{Rad95} package. The methodology and the
objectives of the exercise were identical to that of any regular investigation of nuclear level structure
using $\gamma$-ray spectroscopy. These have been detailed in numerous papers, such as Ref. \cite{Sam18,Sam19},
and are briefly mentioned herein. The coincidence relationships between the observed $\gamma$-ray transitions
were extracted from the symmetric $\gamma$-$\gamma$ matrix and $\gamma$-$\gamma$-$\gamma$ cube. The
coincidences along with the intensity considerations were applied for the placement of the $\gamma$-ray
transitions in the level scheme of the nucleus. The assignment of multipolarities of the
$\gamma$-rays followed determination of their $R_{ADO}$ (Ratio of Angular Distribution from Oriented Nuclei)
values using
\begin{equation}
R_{ADO} = \frac{I_{\gamma1} \ at \ 32^o \ (Gated \ by \ \gamma_2 \ at \ all \ angles)}{I_{\gamma1} \ at \ 123^o \ (Gated \ by \ \gamma_2 \ at \ all \ angles)}
\end{equation}
\noindent{where $I$ is the intensity of the transition (of interest, $\gamma_1$ in the above equation)
in the relevant gated spectrum that is generated
from the appropriate angle dependent matrix. As far as this analysis is concerned, the $R_{ADO}$
value for the stretched dipole ($\Delta$J = 1) transitions is 0.73$\pm$0.01 while for the
stretched quadrupole ($\Delta$J = 2) ones,
it is 1.34$\pm$0.01. These values were derived from $R_{ADO}$s of transitions with previously
established multipolarities and belonging to other Po isotopes populated in the same experiment.
The $R_{ADO}$ values determined for different $\gamma$-ray transitions, observed in this study,
are represented in Fig. 2.} \\
\begin{figure}
\includegraphics[angle=-90,scale=.35,trim=1.0cm 1.0cm 0.0cm 1.0cm,clip=true]{fig2.ps}
\caption{\label{fig2}$R_{ADO}$ values for transitions of $^{203}$Po, as determined
in the current analysis. Those for selected transitions of $^{202,204}$Po are plotted
as reference.}
\end{figure}
\begin{figure}
\includegraphics[angle=-90,scale=.30,trim=1.0cm 1.0cm 0.0cm 1.0cm,clip=true]{fig3a.ps}
\includegraphics[angle=-90,scale=.30,trim=1.0cm 1.0cm 0.0cm 1.0cm,clip=true]{fig3b.ps}
\includegraphics[angle=-90,scale=.30,trim=1.0cm 1.0cm 0.0cm 1.0cm,clip=true]{fig3c.ps}
\caption{\label{fig3}(a) Plot of geometrical asymmetry as a function of $\gamma$-ray energy.
(b) Polarization asymmetry of transitions of $^{203}$Po. (c) Linear
polarization values for transitions of $^{203}$Po along with the corresponding theoretical estimates for
some of them (of pure multipolarity). The $\Delta$ and $P$ values for selected transitions of other isotopes, that were
populated in the same experiment, are included for validation.}
\end{figure}
The electromagnetic nature of the transitions were assigned on the basis of their polarization asymmetry
evaluated using,
\begin{equation}
\Delta = \frac{aN_\perp \ - \ N_\parallel}{aN_\perp \ + \ N_\parallel}
\end{equation}
\noindent{where $N_\perp$ and $N_\parallel$ are respectively the number of photons of the $\gamma$-ray of interest
that are scattered perpendicular to and parallel to the reference plane. The latter is defined by
the beam direction and the direction of emission of the $\gamma$-ray. Each of the four crystals of a HPGe
clover detector operates as scatterer while the two adjacent ones, parallel and perpendicular to the
scatterer, operate as absorbers and facilitate in identifying the scattering events in the respective
directions. The asymmetry between the two scattering possibilities is known to be maximum at 90$^o$.
Thus, the $N_\perp$ ($N_\parallel$) for $\gamma$-rays is extracted from a
matrix that has been constructed with the perpendicular (parallel) scattering events in the detectors
at 90$^o$ on one axis and the coincident detections in detectors at all other angles on the
other axis. The coincidences aid in the unambiguous identification of the $\gamma$-ray transition
being analyzed. The $a$ in Eq. (2) represents the asymmetry that is characteristic to the geometry
of the detection setup. It was determined from the asymmetry between $N_\perp$ and $N_\parallel$
for $\gamma$-rays of (unpolarized) radioactive sources, such as $^{152}$Eu, and using $a = N_\parallel/N_\perp$.
The typical plot of $a$, as a function of $\gamma$-ray energy, for the present setup is
illustrated in Fig. 3(a). The observed asymmetry between $N_\perp$ and $N_\parallel$ for polarized
$\gamma$-rays, such as those emitted by spin oriented ensemble of nuclei produced in
fusion-evaporation reactions, depends on the degree of their polarization ($P$) and the sensitivity ($Q$)
of the measurement setup. These are related through,}
\begin{equation}
P = \frac{\Delta}{Q}
\end{equation}
\noindent{with,}
\begin{equation}
Q(E_\gamma) = Q_0(E_\gamma)(CE_\gamma \ + \ D)
\end{equation}
\noindent{where,}
\begin{equation}
Q_0(E_\gamma) = \frac{\alpha + 1}{\alpha^2 + \alpha + 1}
\end{equation}
\noindent{$\alpha$ being $E_\gamma/m_ec^2$, $m_ec^2$ is the electron rest mass energy. The $C$ and $D$
parameters for the purpose were adopted from those following the work by Palit {\it{et al.}} \cite{Pal00}
and are $C = 0.000099 \ keV^{-1}$ and $D = 0.446$.} \\
As per the regular methodology of nuclear structure studies, using $\gamma$-ray spectroscopy,
the information on coincidence relationships between the $\gamma$-rays along with their
intensities, multipolarities and electromagnetic nature, as resulting from the aforementioned analysis,
were used to identify the excitation scheme of the nucleus and the same is discussed in the
next section.
\section{Results}
\begin{turnpage}
\begin{figure*}
\includegraphics[angle=-90,scale=1.00,trim=4.0cm 0.0cm 6.0cm 1.0cm,clip=true]{fig4.ps}
\caption{\label{fig4} Excitation scheme of $^{203}$Po following the present work. The $\gamma$-ray
transitions that have been newly identified in this study are indicated in red.}
\end{figure*}
\end{turnpage}
\clearpage
The excitation scheme of $^{203}$Po, as established or confirmed in the present investigation, is
illustrated in Fig. 4. Figs. 5 and 6 illustrate the representative gated spectra respectively projected from
$\gamma$-$\gamma$ matrix and $\gamma$-$\gamma$-$\gamma$ cube. The observed coincidences
have been used to identify the placement of transitions in the level scheme.
Twenty new $\gamma$-ray transitions have been placed in the level scheme of the nucleus and the
following modifications have been made in the existing \cite{Fan86,nndc}
assignments therein. The details of the $\gamma$-ray transitions are recorded in Table I.
(The energies of the transitions and the levels
are rounded off to the nearest integer in the discussions herein.)
\begin{figure}
\includegraphics[angle=-90,scale=.35,trim=3.0cm 1.0cm 0.0cm 0.0cm,clip=true]{fig5a.ps}
\includegraphics[angle=-90,scale=.35,trim=3.0cm 1.0cm 0.0cm 0.0cm,clip=true]{fig5b.ps}
\caption{\label{fig5}Representative spectra projected out of $\gamma$-$\gamma$ matrix with gate on transitions of $^{203}$Po, as indicated in the inset of the respective spectrum. The $\gamma$-rays newly identified in the present work are marked with *. Those resulting from overlapping coincidences in other nuclei, populated in the same experiment, are also labeled accordingly.}
\end{figure}
\begin{figure}
\includegraphics[angle=-90,scale=.35,trim=1.0cm 1.0cm 0.0cm 0.0cm,clip=true]{fig6.ps}
\caption{\label{fig6}Representative spectrum projected out of $\gamma$-$\gamma$-$\gamma$ cube with double gate on transitions of $^{203}$Po, as indicated in the inset of the spectrum. The $\gamma$-rays newly identified in the present work are marked with *.}
\end{figure}
\begin{enumerate}
\item{The placement of 397-keV transition has been changed with respect to that assigned in the
literature, as de-exciting a $\sim$ 1527-keV level \cite{Sem87}. The level and the $\gamma$-ray
was not reported by Fant {\it{et al.}} while in the present study the placement of the transition
has been modified to one de-exciting the $\sim$ 3264-keV state. The level has been marked as a new one
in the level scheme (Fig. 4) while the $\gamma$-ray transition is identified to have been observed
previously, albeit with a different placement.}
\item{The 219-keV transition de-exciting the 2274-keV level has been identified as a M1 and the state
has been identified to be of spin-parity 27/2$^+$. There was no spin-parity assignment for this level,
identified as $\sim$ 2277-keV by Fant {\it{et al.}} \cite{nndc}, in the previous studies.}
\item{The 959-keV transition, de-exciting the 3014-keV state, has been assigned a mixed M1+(E2)
nature, following the present measurements. Accordingly, the state has been assigned a spin-parity of
27/2$^+$ that is at variance with the assignment by Fant {\it{et al.}} \cite{nndc}. The latter
had identified the $\gamma$-ray as a pure E2 one and had tentatively assigned the spin-parity of the
state ($\sim$ 3018-keV, as per Fant {\it{et al.}}) as (29/2$^+$).}
\item{The 543-keV $\gamma$-ray, from the 3066-keV state, has been assigned a multipolarity of
E1 in this study. It was tentatively identified as M1, by Fant {\it{et al.}}, and the spin of the
level (at $\sim$ 3070-keV, as per Fant {\it{et al.}}) was accordingly assigned to be 29/2.}
\item{The 585-keV transition de-exciting the 3108-keV state has been established as a pure dipole
in this study. However, the electromagnetic nature of the same could not be unambiguously determined in the
present investigation. The multipolarity of the transition was undetermined in the work by
Fant {\it{et al.}} and consequently there was no spin assignment for the state (at $\sim$ 3112-keV,
as quoted by Fant {\it{et al.}}) therein.}
\item{The spin-parity of the 3236-keV state, de-excited by the 369-keV transition, has been confirmed
to be 33/2$^+$ in this study. The assignment for the level (at $\sim$ 3241-keV, as per Fant {\it{et al.}})
was only tentative in the previous work \cite{nndc}.}
\item{The spin-parity of the 3712-keV level has been assigned as 33/2$^+$ in this work, following the E2 assignment
of the 845-keV $\gamma$-ray that de-excites the state. There was no multipolarity assignment for the
transition or spin-parity assignment for the state (at $\sim$ 3717-keV, as per Fant {\it{et al.}})
in the previous studies \cite{nndc}.}
\item{The 3877-keV state has been assigned spin-parity of 33/2$^-$, in this measurement. This is
following the identification of the 450-keV transition, that de-excites the level,
as an E2 one herein. Previously \cite{nndc}, the transition
was assigned as M1 and the spin-parity of the state (at $\sim$ 3882-keV, as quoted by Fant {\it{et al.}})
as 31/2$^-$. Fig. 7 represents the spectra of the transition corresponding to the perpendicular
and the parallel scattering events and illustrates the dominance of the former that leads to
positive value of polarization asymmetry (Eq. 2) or polarization (Eq. 3).}
\item{The spin-parity of the 4352-keV level has been confirmed to be 35/2$^-$ in the present work. The
assignment was tentative for the state (at $\sim$ 4358-keV, quoted by Fant {\it{et al.}}) in the
previous studies. The 476-keV transition, de-exciting the state, has been identified as M1 in this study and this is
different from the E2 assignment by Fant {\it{et al.}}.}
\end{enumerate}
An additional proposition can be put forth on the multipolarity assignment of the 182-keV transition de-exciting
the 2156-keV state. Since the state is known to be an isomer of $T_{1/2}$ $>$ 200 ns \cite{Fan86}, the multipolarity
of the transition could not be ascertained from its $ADO$ ratio and its polarization asymmetry. These measurements
are valid for transitions emitted by spin oriented ensemble of nuclei, such as produced in fusion-evaporation
reactions, while the aforementioned isomeric lifetime is sufficient to induce dealignment.
If the observed intensity of this 182-keV transition is corrected for electron conversion, using
codes such as BrICC \cite{Kib08}, it is $\sim$ 60\% increased if the transition is an E2 one and $\sim$ 300\%
enhanced if it is a M1. The latter would result in an unbalanced intensity across the 1975-keV state that is
fed by the 182-keV transition and de-excited by the 596-keV one. If the 182-keV transition is thus interpreted
to be of E2 nature, the 2156-keV level can be assigned a spin-parity of 25/2$^+$. However, since there is no
direct experimental evidence for the same, this proposition has not been indicated in the level scheme (Fig. 4) and the
assignment has not been included in the table (Table I).
The sharp decrease in the relative intensity of the $\gamma$-ray transitions across the 2156-keV
state is also noteworthy and can be ascribed to the state being an isomer of T$_{1/2}$ $>$ 200 ns \cite{Fan86}. \\
\begin{figure}
\includegraphics[angle=-90,scale=.35,trim=1.0cm 0.0cm 1.0cm 1.0cm,clip=true]{fig7.ps}
\caption{\label{fig7}Spectra of 450-keV transition peak corresponding to the perpendicular and the parallel scattering events in the HPGe clover detectors at 90$^o$.}
\end{figure}
Previous studies \cite{Fan86,Fan90} on the Po isotopes had reported a number of isomers therein.
Some of these, with half-lives around few ns, have been reexamined in the current study
using the centroid shift method \cite{Mad22,Yad22}.
In the present implementation of the technique, the time difference between the feeding and the decaying
$\gamma$-ray transitions of a state is histogrammed alternately by defining one as the start (stop) and the
other as stop (start). The difference between the centroids of the two distribution is known to be 2$\tau$,
$\tau$ being the average lifetime of the state. Fig. 8 illustrates the time difference spectra between the
(i) 788- and 262-keV transitions that respectively feeds and de-excites the 3387-keV state in $^{204}$Po \cite{Fan90}, also
populated in the present experiment, and (ii) 637- and 304-keV transitions that respectively feeds and de-excites the
2789-keV state in $^{203}$Po. The half-life of the 3387-keV state in $^{204}$Po was determined
by Fant {\it{et al.}} \cite{Fan90} as 9$\pm$3 ns, presumably following an analysis of the time profile
of the decaying transition with respect to the RF of the accelerator. The present analysis has resulted
in T$_{1/2}$ = 6.2$\pm$0.8 ns, that is within the limits of uncertainty on the previous estimate and validates
the present analysis. The latter carried out for the 2789-keV state in $^{203}$Po yields its
T$_{1/2}$ = 7.1$\pm$0.1 ns that is lesser than the previous value, also reported by Fant {\it{et al.}} \cite{Fan86},
of 12$\pm$2 ns. \\
\begin{figure}
\includegraphics[angle=0,scale=.30,trim=1.0cm 1.0cm 0.0cm 1.0cm,clip=true]{fig8a.ps}
\includegraphics[angle=0,scale=.30,trim=1.0cm 1.0cm 0.0cm 1.0cm,clip=true]{fig8b.ps}
\caption{\label{fig8}(Color online) Time difference spectra between transitions indicated in the inset, for determining isomeric lifetimes. The upper panel corresponds to the state at 3387-keV state in $^{204}$Po while the bottom panel is for 2789-keV state in $^{203}$Po. (Please refer to the text for details.)}
\end{figure}
The experimentally observed level scheme of the $^{203}$Po nucleus has been interpreted
through single particle excitations in the framework of the shell model. The same is detailed
in the next section. \\
\LTcapwidth=\textwidth
\begin{longtable*}{ccccccccccc}
\caption{\label{tab1}Details of the levels and the $\gamma$-ray transitions of $^{203}$Po nucleus observed in the present work. The energy of a $\gamma$-ray transition is the weighted average of its value in multiple gates. The relative intensities ($I_\gamma$) of the $\gamma$-ray transitions are normalized with respect to the intensity of 466-keV transition as observed in 612-keV gated spectrum. The ADO ratios($R_{ADO}$), polarization asymmetry ($\Delta_{pol}$), and linear polarization ($P$) of the transitions are determined using the procedure described in Section II. The $N$ and $a$ superscripts indicate the assignments that have been respectively adopted from NNDC \cite{nndc} and/or Fant {\it{et al.}} \cite{Fan86}.} \\
\hline
$E_i (keV) $ & $E_{\gamma} (keV) $ &$I_{\gamma}$ & $J_i^{\pi}$ & $J_f^{\pi}$&$R_{ADO}$ &$\Delta_{pol}$ & P & Multipolarity\\
\hline
\hline
\endfirsthead
\multicolumn{11}{c}%
{{ \tablename \thetable{} -- continued from previous page}} \\
\hline
$E_i (keV) $ & $E_{\gamma} (keV) $ & $I_{\gamma}$ & $J_i^{\pi}$ & $J_f^{\pi}$&$R_{ADO}$ &$\Delta_{pol}$ & P & Multipolarity\\
\hline
\endhead
\hline
\multicolumn{11}{c}{Continued in next page}\\
\hline
\endfoot
\endlastfoot
638.7 $\pm$0.1 & 577.2$\pm$0.1 &11$\pm$1 & 7/2$^{-}$ & (3/2$^{-})$& & & &E2$^N$ \\
& 638.7$\pm$0.1 &48$\pm$1 & 7/2$^{-}$ & $5/2^{-}$ & & & &M1$^N$ \\
641.7 $\pm$0.2 & 641.7$\pm$0.2 & & 13/2$^{+}$ & $5/2^{-}$ & & & &M4$^N$ \\
1055.2 $\pm$0.1 & 416.5$\pm$0.1 &59$\pm$1 & (11/2$^{-})^a$& $7/2^{-}$& & & &M1+E2$^N$ \\
1254.0 $\pm$0.2 & 612.3$\pm$0.1 & & 17/2$^{+}$ & $13/2^{+}$& & & &E2$^N$ \\
1378.8 $\pm$0.3 & 737.1$\pm$0.2 &127$\pm$3 &(17/2$^{+})^a$& $13/2^{+}$& & & &(E2)$^N$ \\
1697.1 $\pm$0.8 & 318.3$\pm$0.7 &23$\pm$6 & &($17/2^{+})^a$& & & & \\
1719.8 $\pm$0.2 & 465.8$\pm$0.1 &1000 & 21/2$^{+}$ & $17/2^{+}$& & & &E2$^N$ \\
1974.5 $\pm$0.3 & 595.7$\pm$0.1 &251$\pm$2 &(21/2$^{+})^a$& $(17/2^{+})$& & & &E2$^N$ \\
2054.7 $\pm$0.2 & 334.9$\pm$0.1 &643$\pm$14 & 25/2$^{+}$ & $21/2^{+}$ & & & &E2$^N$ \\
2077.0 $\pm$0.2 & 356.8$\pm$0.1 &64$\pm$2 & 21/2$^{+}$ & $21/2^{+}$ & & & &M1$^N$ \\
2156.4 $\pm$0.3 & 181.9$\pm$0.1 &156$\pm$1& &$(21/2^{+})^a$& & & & \\
2184.4 $\pm$0.7 & 805.6$\pm$0.6 &4$\pm$1 & &($17/2^{+})^a$& & & & \\
2273.6 $\pm$0.2 & 219.1$\pm$0.1 &58$\pm$1 & 27/2$^{+}$ & $25/2^{+}$&0.75$\pm$0.01 &-0.16$\pm$0.09&-0.43$\pm$0.24&M1 \\
2404.2 $\pm$0.2 & 349.2$\pm$0.1 &36$\pm$1 & 25/2$^{+}$ & $25/2^{+}$& & & &M1$^N$ \\
& 684.1$\pm$0.1 &71$\pm$2 & 25/2$^{+}$ & $21/2^{+}$& & & & \\
2485.8 $\pm$0.2 & 408.6$\pm$0.2 &21$\pm$1 & 23/2$^{+}$ & $21/2^{+}$& & & &M1$^N$ \\
& 765.9$\pm$0.1 &76$\pm$2 & 23/2$^{+}$ & $21/2^{+}$& & & &M1$^N$ \\
2500.2 $\pm$0.2 & 780.1$\pm$0.1 &40$\pm$1 & 23/2$^{+}$ & $21/2^{+}$& & & &(M1)$^N$ \\
2523.0 $\pm$0.2 & 468.3$\pm$0.1 &176$\pm$4 & 27/2$^{+}$ & $25/2^{+}$& & & &M1$^N$ \\
2527.6 $\pm$0.7 &1148.8$\pm$0.6 & & & (17/2$^{+})^a$& & & & \\
2765.1 $\pm$0.4 & 710.4$\pm$0.3 &8$\pm$1 & & 25/2$^{+}$& & & & \\
2789.1 $\pm$0.2 & 289.3$\pm$0.1 &74$\pm$2 & 25/2$^{-}$ & 23/2$^{+}$& & & &(E1)$^N$ \\
& 303.5$\pm$0.1 &102$\pm$2 & 25/2$^{-}$ & 23/2$^{+}$& & & &E1$^N$ \\
& 385.1$\pm$0.1 &54$\pm$2 & 25/2$^{-}$ & 25/2$^{+}$& & & &E1$^N$ \\
2820.9 $\pm$0.2 & 297.7$\pm$0.1 &28$\pm$1 & 29/2$^{-}$ & 27/2$^{+}$& & & &(E1)$^N$ \\
2863.0 $\pm$0.2 & 589.4$\pm$0.1 &55$\pm$2 & 31/2$^{+}$ & 27/2$^{+}$&1.05$\pm$0.05 &0.05$\pm$0.04 &0.21$\pm$0.17 &E2+M3 \\
2867.2 $\pm$0.2 & 812.5$\pm$0.1 &226$\pm$5 & 29/2$^{+}$ & 25/2$^{+}$&1.34$\pm$0.01 &0.06$\pm$0.01 &0.32$\pm$0.05 &E2 \\
2950.0 $\pm$0.4 & 793.6$\pm$0.2 & & & &0.70$\pm$0.03 & & &D \\
2976.7 $\pm$0.3 & 922.2$\pm$0.2 &13$\pm$1 & 27/2$^{+}$ & 25/2$^{+}$&0.82$\pm$0.06 &-0.05$\pm$0.06&-0.31$\pm$0.37 &M1 \\
3013.5 $\pm$0.2 & 959.1$\pm$0.1 &44$\pm$1 & 27/2$^{+}$ & 25/2$^{+}$&0.83$\pm$0.02 &-0.01$\pm$0.03&-0.06$\pm$0.19 &M1+(E2) \\
3066.1 $\pm$0.2 & 542.7$\pm$0.1 &46$\pm$1 & 29/2$^{-}$ & 27/2$^{+}$&0.81$\pm$0.01 &0.04$\pm$0.03 &0.16$\pm$0.12 &E1 \\
3093.2 $\pm$0.4 &1038.5$\pm$0.3 &16$\pm$3 & 29/2$^{ }$ & 25/2$^{+}$&1.24$\pm$0.09 & & &Q \\
3107.5 $\pm$0.2 & 584.5$\pm$0.1 &21$\pm$1 & 29/2$^{ }$ & 27/2$^{+}$&0.66$\pm$0.02 & & &D \\
3108.3 $\pm$0.4 & 704.1$\pm$0.3 &11$\pm$1 & & 25/2$^{+}$& & & & \\
3220.5 $\pm$0.4 &1165.8$\pm$0.3 &6$\pm$1 & & 25/2$^{+}$& & & & \\
3231.8 $\pm$0.2 & 364.5$\pm$0.1 &40$\pm$1 & 31/2$^{+}$ & 29/2$^{+}$&0.79$\pm$0.02 &-0.04$\pm$0.03&-0.13$\pm$0.10 &M1 \\
3236.3 $\pm$0.2 & 368.8$\pm$0.1 &79$\pm$2 & 33/2$^{+}$ & 29/2$^{+}$&1.39$\pm$0.03 &0.03$\pm$0.02 &0.10$\pm$0.06 &E2 \\
3241.9 $\pm$0.2 & 374.7$\pm$0.1 &15$\pm$1 & 33/2$^{+}$ & 29/2$^{+}$&1.41$\pm$0.04 &0.08$\pm$0.10 &0.26$\pm$0.32 &(E2) \\
3254.4 $\pm$0.2 & 390.9$\pm$0.1 &18$\pm$1 & 33/2$^{-}$ & 31/2$^{+}$&0.97$\pm$0.05 &0.08$\pm$0.07 &0.26$\pm$0.23 &E1+M2 \\
3262.9 $\pm$0.2 & 196.6$\pm$0.1 &16$\pm$1 & 33/2$^{ }$ & 29/2$^{-}$&1.28$\pm$0.04 & & &Q \\
3264.0 $\pm$0.2 & 397.0$\pm$0.1 &22$\pm$1 & 33/2$^{+}$ & 29/2$^{+}$&1.20$\pm$0.04 &0.05$\pm$0.06 &0.17$\pm$0.20 &(E2) \\
3292.0 $\pm$0.3 &1237.2$\pm$0.2 &5$\pm$1 & & 25/2$^{+}$& & & & \\
3416.1 $\pm$0.3 & 627.4$\pm$0.2 &17$\pm$1 & & 25/2$^{-}$& & & & \\
3426.5 $\pm$0.2 & 637.3$\pm$0.1 &140$\pm$3 & 29/2$^{-}$ & 25/2$^{-}$&1.39$\pm$0.02 &0.08$\pm$0.01 &0.35$\pm$0.05 &E2 \\
3443.2 $\pm$0.3 & 211.4$\pm$0.2 &29$\pm$1 & 33/2$^{ }$ & 31/2$^{+}$&0.72$\pm$0.02 & & &D \\
3456.2 $\pm$0.2 & 192.3$\pm$0.1 &11$\pm$1 & 35/2$^{ }$ & 33/2$^{+}$&0.68$\pm$0.05 & & &D \\
3488.8 $\pm$0.3 & 234.3$\pm$0.2 &15$\pm$1 & 35/2$^{ }$ & 33/2$^{-}$&0.69$\pm$0.01 & & &D \\
3603.8 $\pm$0.3 &1329.9$\pm$0.2 &9$\pm$1 & & 27/2$^{+}$& & & & \\
3711.9 $\pm$0.2 & 844.7$\pm$0.1 &29$\pm$1 & 33/2$^{+}$ & 29/2$^{+}$&1.25$\pm$0.05 &0.04$\pm$0.02 &0.22$\pm$0.11 &E2 \\
3741.7 $\pm$0.2 & 315.1$\pm$0.1 &34$\pm$1 & 31/2$^{-}$ & 29/2$^{-}$&0.98$\pm$0.04 &-0.08$\pm$0.05&-0.24$\pm$0.15 &M1+E2 \\
3876.6 $\pm$0.2 & 450.3$\pm$0.1 &97$\pm$2 & 33/2$^{-}$ & 29/2$^{-}$&1.33$\pm$0.01 &0.07$\pm$0.02 &0.25$\pm$0.07 &E2 \\
3934.7 $\pm$0.3 & 491.5$\pm$0.2 &21$\pm$1 & 37/2$^{ }$ & 33/2$^{ }$&1.36$\pm$0.07 & & &Q \\
3979.4 $\pm$0.3 & 742.9$\pm$0.2 &23$\pm$1 & 35/2$^{+}$ & 33/2$^{+}$&0.77$\pm$0.03 &-0.04$\pm$0.03&-0.20$\pm$0.15 &M1 \\
4146.2 $\pm$0.2 & 891.8$\pm$0.1 &24$\pm$1 & 35/2$^{+}$ & 33/2$^{-}$&0.82$\pm$0.03 &0.01$\pm$0.03 &0.06$\pm$0.18 &(E1) \\
4352.3 $\pm$0.2 & 475.9$\pm$0.1 &27$\pm$1 & 35/2$^{-}$ & 33/2$^{-}$&0.77$\pm$0.04 &-0.06$\pm$0.02&-0.22$\pm$0.07 &M1 \\
4528.2 $\pm$0.4 & 175.9$\pm$0.3 &13$\pm$1 & & 35/2$^{-}$& & & & \\
4612.0 $\pm$0.3 & 735.4$\pm$0.2 &26$\pm$1 & 35/2$^{-}$ & 33/2$^{-}$&0.90$\pm$0.04 &-0.05$\pm$0.07&-0.25$\pm$0.35 &M1+E2 \\
4623.3 $\pm$0.4 & 746.7$\pm$0.3 &8$\pm$1 & & 33/2$^{-}$& & & & \\
4645.6 $\pm$0.2 &1189.3$\pm$0.1 &24$\pm$1 & 37/2$^{ }$ & 35/2$^{ }$&0.73$\pm$0.04 &-0.03$\pm$0.07&-0.24$\pm$0.56 &(M1) \\
4929.8 $\pm$0.5 &1053.2$\pm$0.4 & & & 33/2$^{-}$& & & & \\
\hline
\bigskip
\end{longtable*}
\section{Discussions}
One of the objectives of this endeavor has been to probe the efficacy of the shell model
in interpreting the excitation scheme of the nuclei in the A $\sim$ 200 region.
There have been similar efforts, in recent times, wherein level structures
of nuclei around the $^{208}$Pb-core are calculated in the shell model framework.
Bothe {\it{et al.}} \cite{Bot22} have reported such calculations for the isomeric states
in $^{203}$Tl ($Z = 81, N = 122$) while Yadav {\it{et al.}} \cite{Yad22} and
Madhu {\it{et al.}} \cite{Mad22} have used them for deciphering the particle excitations
associated with the observed states of $^{215,216}$Fr ($Z = 87, N = 128,129$) nuclei.
These studies have identified a general overlap, between the experimental and the calculated
level energies, of within $\sim$ 250-keV as reasonable. \\
Large basis shell model calculation has been carried out in the present work using KHH7B \cite{Her72} Hamiltonian
in the model space spanning $Z = 58-114$ and $N = 100-164$. The latter includes proton orbitals
$d_{5/2}$, $h_{11/2}$, $d_{3/2}$ and $s_{1/2}$ below $Z = 82$ and the $h_{9/2}$, $f_{7/2}$, and $i_{13/2}$
above; the neutron orbitals are $i_{13/2}$, $p_{3/2}$, $f_{5/2}$, and $p_{1/2}$ below $N = 126$
and the $g_{9/2}$, $i_{11/2}$, and $j_{15/2}$ above. Proton excitations across $Z = 82$ closure and neutron excitations
across the closure at $N = 126$ have not been allowed in the calculations. The matrix
diagonalization has been carried out using the OXBASH \cite{Bro04} code. The comparison between
the calculated and the experimental level energies is illustrated in Figs. 9 and 10. The dominant
particle configurations along with the energy values of the states are recorded in Table II. \\
\begin{figure}
\includegraphics[angle=-90,scale=.40,trim=0.0cm 0.0cm 0.0cm 0.0cm,clip=true]{fig9.ps}
\caption{\label{fig9}Comparison between the calculated and the experimental level energies of the negative parity states in $^{203}$Po.}
\end{figure}
\begin{figure*}
\includegraphics[angle=-90,scale=.70,trim=0.0cm 0.0cm 5.0cm 0.0cm,clip=true]{fig10.ps}
\caption{\label{fig10}Comparison between the calculated and the experimental level energies of the positive parity states in $^{203}$Po.}
\end{figure*}
The calculated energies of the negative parity states with spin $<$ 29/2 are excellent
overlap with their experimental values, even within $\sim$ 100-keV for some of them.
The 25/2$^-$ level is an exception for which the theoretical and the measured level energies
differ by $\sim$ 800 keV. The dominant particle configurations associated with these states
negative parity states
have been calculated to be $\pi(h_{9/2}^2)\otimes\nu(f_{5/2}^{3,2}p_{3/2}^{2,3}i_{13/2}^{14})$.
The negative parity states at higher spins, $\ge$ 31/2, are
poorly represented in the calculations wherein their energies are deviant by as much as
500-keV - 1-MeV with respect to the experimental values. The energy of the calculated yrast
33/2$^-$ state, however, reasonably overlaps with the measured energy within $\sim$ 250-keV.
The most probable particle configurations for the negative parity states at higher spins
correspond to $\pi(h_{9/2}^{1}i_{13/2}^{1})\otimes\nu(f_{5/2}^{2}p_{3/2}^{4}i_{13/2}^{13})$.
However, those of the yrare 33/2$^-$ and the 35/2$^-$ are different
($\pi(h_{9/2}^{2})\otimes\nu(f_{5/2}^{3}p_{3/2}^{4}i_{13/2}^{12})$ but, as indicated by the
widely deviant calculated energies vis-a-vis the experimental ones, these configurations do not appropriately
represent the relevant states, similar to the other high spin levels of odd parity. \\
\begin{longtable*}{cccccccccccccccc}
\caption{\label{tab2}Main partitions of wave functions of the positive and negative parity states in $^{203}$Po for KHH7B interaction} \\
\hline
&&Level Energy& & $J^{\pi}$ & Probability & Proton & Neutron \\
&& & & & & & \\
& EXPT&&SM & & & & & \\
\hline
\hline
\endfirsthead
\multicolumn{16}{c}%
{{ \tablename \thetable{} -- continued from previous page}} \\
\hline
&&Level Energy& & $J^{\pi}$ & Probability &Proton & Neutron \\
&& & & & & & \\
& EXPT&&SM & & & & & \\
\hline
\hline
\endhead
\hline
\multicolumn{16}{c}{Continued in next page}\\
\hline
\endfoot
\endlastfoot
& & & & & & & & \\
& & & & &NEGATIVE PARITY & & & \\
& & & & & & & & \\
& 0 && 0 & $5/2^{-}$ & 29.33 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^3p_{3/2}^2p_{1/2}^0i_{13/2}^{14}$ \\
&& && & & & & \\
& 62 && 181 & $(3/2^{-})$ & 39.21 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^3p_{1/2}^0i_{13/2}^{14}$ \\
&& && & & & & \\
& 639 && 755 & $7/2^{-}$ & 19.96 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^3p_{1/2}^0i_{13/2}^{14}$ \\
&& && & & & & \\
& 1055 && 993 & $(11/2^{-})$& 37.64 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^3p_{1/2}^0i_{13/2}^{14}$ \\
&& && & & & & \\
& 2789 && 1987 & $25/2^{-}$ & 39.78 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^3p_{3/2}^2p_{1/2}^0i_{13/2}^{14}$ \\
&& && & & & & \\
& 2821 && 2715 & $29/2^{-}$ & 70.95 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^3p_{3/2}^2p_{1/2}^0i_{13/2}^{14}$ \\
&& && & & & & \\
& 3066 && 2758 & $29/2^{-}$ & 31.65 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^3p_{3/2}^4p_{1/2}^0i_{13/2}^{12}$ \\
&& && & & & & \\
& 3254 && 2999 & $33/2^{-}$ & 26.06 & $h_{9/2}^1f_{7/2}^0i_{13/2}^1$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && & & & & \\
& 3427 && 2812 & $29/2^{-}$ & 24.93 & $h_{9/2}^1f_{7/2}^0i_{13/2}^1$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && & & & & \\
& 3742 && 2851 & $31/2^{-}$ & 25.53 & $h_{9/2}^1f_{7/2}^0i_{13/2}^1$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && & & & & \\
& 3877 && 3402 & $33/2^{-}$ & 42.34 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^3p_{3/2}^4p_{1/2}^0i_{13/2}^{12}$ \\
&& && & & & & \\
& 4352 && 3465 & $35/2^{-}$ & 24.51 & $h_{9/2}^1f_{7/2}^0i_{13/2}^1$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && & & & & \\
& 4612 && 3671 & $35/2^{-}$ & 29.32 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^3p_{3/2}^4p_{1/2}^0i_{13/2}^{12}$ \\
&& && & & & & \\
& & & & & & \\
& & & & &POSITIVE PARITY & \\
& & & & & & \\
& 642 && 700 & $13/2^{+}$ & 24.13 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 1254 && 1365 & $17/2^{+}$ & 28.43 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 1379 && 1861 & ($17/2^{+}$) & 23.95 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 1720 && 1745 & $21/2^{+}$ & 23.00 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 1975 && 1947 & ($21/2^{+}$) &30.87 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^4p_{3/2}^2p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 2055 && 1981 & $25/2^{+}$ & 32.32 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 2077 && 2023 & $21/2^{+}$ & 27.80 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 2404 && 2347 & $25/2^{+}$ & 35.75 & $h_{9/2}^1f_{7/2}^0i_{13/2}^1$ & $f_{5/2}^3p_{3/2}^2p_{1/2}^0i_{13/2}^{14}$ \\
&& && && & & \\
& 2486 && 2001 & $23/2^{+}$ & 12.82 & $h_{9/2}^0f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^0p_{3/2}^0p_{1/2}^0i_{13/2}^{0}$ \\
&& && && & & \\
& 2500 && 2337 & $23/2^{+}$ & 19.05 & $h_{9/2}^0f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^0p_{3/2}^0p_{1/2}^0i_{13/2}^{0}$ \\
&& && && & & \\
& 2274 && 2318 & $27/2^{+}$ & 22.42 & $h_{9/2}^1f_{7/2}^0i_{13/2}^1$ & $f_{5/2}^1p_{3/2}^4p_{1/2}^0i_{13/2}^{14}$ \\
&& && && & & \\
& 2523 && 2375 & $27/2^{+}$ & 26.22 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 2977 && 2667 & $27/2^{+}$ & 31.61 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^3p_{3/2}^3p_{1/2}^0i_{13/2}^{13}$ \\
&& && & & & \\
& 2867 && 2620 & $29/2^{+}$ & 35.78 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && & & & \\
& 2863 && 2758 & $31/2^{+}$ & 53.28 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 3232 && 2949 & $31/2^{+}$ & 32.75 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^4p_{3/2}^2p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 3236 && 2782 & $33/2^{+}$ & 52.68 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && && & & \\
& 3242 && 3185 & $33/2^{+}$ & 57.96 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^3p_{3/2}^3p_{1/2}^0i_{13/2}^{13}$ \\
&& && & & & \\
& 3264 && 3226 & $33/2^{+}$ & 30.14 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && & & & \\
& 3979 && 3230 & $35/2^{+}$ & 51.41 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^2p_{3/2}^4p_{1/2}^0i_{13/2}^{13}$ \\
&& && & & & \\
& 4146 && 3517 & $35/2^{+}$ & 54.27 & $h_{9/2}^2f_{7/2}^0i_{13/2}^0$ & $f_{5/2}^3p_{3/2}^3p_{1/2}^0i_{13/2}^{13}$ \\
\hline
\hline
\end{longtable*}
The calculated level energies for most of the positive parity states with spin $<$ 27/2 are
in excellent overlap, within or around 100-keV, with their experimental values.
The yrare 17/2$^+$ state is an exception for which the calculated and the experimental energies
differ by $\sim$ 500-keV. However, it is noteworthy that the spin-parity assignment of the
1379-keV state as second 17/2$^+$ was by Fant {\it{et al.}} \cite{Fan86} and is tentative. This could not be
confirmed in the present study. If the parity assignment of the state is changed, it would be the yrast
(and only observed) 17/2$^-$ level with calculated energy of 1214-keV that is in reasonable
overlap with the experimental value. It is noted that, in such a scenario, the 737-keV (17/2$^-$ $\rightarrow$ 13/2$^+$)
and 596-keV (21/2$^+$ $\rightarrow$ 17/2$^-$) transitions would be M2 ones and, according to the
Weisskopf estimate, would translate into lifetimes of $\sim$ few ns for the states they de-excite.
These lifetimes are much less than the $\gamma$-$\gamma$ coincidence window (200 ns) of the experiment
and, thus, will not impact the observed intensity of the transitions.
The yrast and the yrare 23/2$^+$ state respectively exhibit differences of $\sim$ 500-keV
and $\sim$ 200-keV between their theoretical and measured values. While the latter can still be perceived as
a reasonable overlap, a deviation of the calculated energy by $\sim$ 500-keV with respect to the
experimental one indicates an aberrant representation of the state in the framework of the shell model
calculations. It is also noteworthy that the yrast 23/2$^+$ state is calculated to be
of substantially mixed configurations, compared to the other states of the nucleus, and the
numerically dominant partition is only of 13\% probability. As far as the positive parity states of spin
$\gtrsim$ 29/2 are concerned, the overlap of experimental and calculated energies is
of considerable variance. While they excellently agree for the yrast 31/2$^+$, the yrare and the
third 33/2$^+$ levels, within $\sim$ 100-keV, the difference is $\sim$ 250-450 keV for the 29/2$^+$
and the yrast 31/2$^+$ and 33/2$^+$. It is still higher, $\sim$ 700-keV, for 35/2$^+$.
Such deviations, at the highest excitations observed in the nucleus, can be ascribed to the
limitations of the model calculations in representing the associated multiparticle
configurations based on the high-j orbitals (that characterize the relevant model space).
Most of the positive parity states have been calculated to be of dominant configuration
$\pi(h_{9/2}^{2})\otimes\nu(f_{5/2}^{2-4}p_{3/2}^{2-4}i_{13/2}^{13}$. The exceptions
are the yrare 25/2$^+$ state, for which the calculated dominant configuration is
$\pi(h_{9/2}^{1}i_{13/2}^{1})\otimes\nu(f_{5/2}^{3}p_{3/2}^{2}i_{13/2}^{14}$, and the
yrast 27/2$^+$ state, for which the most probable configuration is
$\pi(h_{9/2}^{1}i_{13/2}^{1})\otimes\nu(f_{5/2}^{1}p_{3/2}^{4}i_{13/2}^{14}$. It is
noteworthy that the calculated and the experimental energies of these states agree within $\sim$ 50-keV
that presumably vindicates the interpretation of their underlying excitations. \\
If the 2156-keV state is assigned a spin-parity of 25/2$^+$, as discussed in the previous section,
following an E2 assignment for the 182-keV transition (that de-excites the level), the level is then
the yrare 25/2$^+$ and exhibits a reasonable overlap, within $\sim$ 200 keV,
with the calculated energy (2347-keV). The current yrare 25/2$^+$ at 2404-keV is then
the third 25/2$^+$ state and its energy is in excellent agreement with the theoretical value
of 2383-keV. Once again, since there is no direct experimental evidence to corroborate the
spin-parity assignment of the state at 2156-keV, this has not been included in the table. \\
It may thus be summed up that the observed excitation scheme of the $^{203}$Po nucleus
could be satisfactorily interpreted within the framework of the large basis shell model
calculations. The specific deviations might have resulted from the limitations
of the Hamiltonian that requires further refinements. The latter is expected to be facilitated
by the availability of experimental data through endeavors such as the present study. \\
\section{Conclusion}
The level structure of the $^{203}$Po nucleus has been probed following its population
in $^{194}$Pt($^{13}$C,4n) reaction at E$_{lab}$ = 74 MeV. The excitation scheme of the
nucleus has been established upto $\sim$ 5 MeV and spin $\sim$ 18$\hbar$. Twenty new $\gamma$-ray
transitions have been added in the level scheme of the the nucleus and spin-parity assignments
have been either made or confirmed for a number of states therein. The observed level scheme
has been satisfactorily interpreted within the framework of large basis shell model calculations
wherein the excited states of the nucleus have been ascribed to proton excitations in
$h_{9/2}$ and $i_{13/2}$ orbitals outside the $Z = 82$ closure and neutron excitations in
$f_{5/2}$, $p_{3/2}$ and $i_{13/2}$ orbitals in the $N = 126$ shell.
The overlap between the experimental and the calculated level energies, of $^{203}$Po,
upholds the credibility of the shell model in catering to a microscopic description
of the excitation scheme even for heavy nuclei in the $A \sim 200$ region and in model space
consisting of high-j orbitals. Further refinements in the model calculations are envisaged
to follow the availability of experimental data. \\
\section*{Acknowledgments}
The authors wish to thank the staff associated with the
Pelletron Facility at IUAC, New Delhi, for their help and support during the experiment.
We record our deepest gratitude for Late Prof. Asimananda Goswami and Mr. Pradipta Kumar Das, of
the Saha Institute of Nuclear Physics (SINP), Kolkata, for their guidance, help
and active contribution in the target preparation.
Help and support received from Mr. Kausik Basu (UGC-DAE CSR, KC) during the experiment, is appreciated.
Help of V. Vishnu Jyothi during the experiment is also acknowledged.
P.C.S. acknowledges a research grant from SERB (India), CRG/2019/000556.
B.M. acknowledges the support from Department of Science and Technology, Government of India through
DST/INSPIRE Fellowship (IF200310).
U.G. acknowledges the support from US National Science Foundation through Grant No. PHY1762495.
This work is partially supported by the Department of Science
and Technology, Government of India (No. IR/S2/PF-03/2003-II).
|
1,116,691,499,345 | arxiv | \section{Introduction}
\label{sec:intro}
Gases in astrophysics are commonly multiphase, that is, phases with vastly different temperatures exist co-spatially.
We know, for instance, that the interstellar medium is kept in a stable at three phase state due to thermal feedback processes \citep{McKee1977}. More quiescent, the intracluster medium (ICM) or cirgumgalactic medium (CGM) are found to have two main phases, a $T \sim 10^4\,$K `cold' and a $T \gtrsim 10^6\,$K hot phase \citep[e.g.,][]{Tumlinson2017}.
Modeling these gases proves to be extremely difficult due to the corresponding different spatial scales, and large simulations struggle with convergence of the cold gas properties \citep{Faucher-Giguere2016,VandeVoort2018,Hummels2018}. This is worrisome as this phase corresponds to the fuel for future star-formation and is most commonly compared to observations (e.g., via quasar absorption line studies, \citealp{Crighton2015,Chen2017,Haislmaier2021}, or emission measurements, \citealp{Steidel2011,Hennawi2015,Battaia2018}).
One of the key properties to constrain is therefore a characteristic size of this cold phase where -- hopefully -- one would find convergence in at least the total cold gas mass and other relevant observables.
Several past and current studies suggested `characteristic length scales' of cold gas \citep{Field1965,McCourt2016,Gronke2018}. Most of them focused on fragmentation processes leading to smaller cold gas `droplets' as a result. Here, we want to focus instead on coagulation between cold gas clouds leading to bigger structures.
\citet{Waters2019} have studied this recently, however -- as we will show below -- in a different regime where the coagulation speed is much slower than the one found in this work.
There are several examples where cooling-induced coagulation appears to be important. For instance:
\begin{itemize}
\item{{\it Cloud-Crushing.} In wind-tunnel simulations of an isolated cold cloud subject to a wind, the cloud can initially have a `near death' experience as cloud material is dispersed both streamwise and laterally \citep{Armillotta2017,Gronke2018,Gronnow2018,Li2019a,Kanjilal2020,Farber2021}, particularly for clouds close to the survival radius $r_{\rm crit}$ \citep[cf.][]{Gronke2018,Gronke2020}. As the cloud becomes entrained and shear is reduced, however, cold gas fragments rapidly coagulate back to form a cometary structure. Subsequently, cloud fragments which are peeled off the side of the cloud are refocused back onto the downstream tail.}
\item{{\it Cloud shattering.} In simulations of `cloud-shattering', under-pressured clouds lose sonic contact with their surroundings due to rapid radiative cooling, and are crushed by surrounding hot gas \citep{McCourt2016,Gronke2020}. Since cloud compression overshoots, the cloud subsequently re-expands, and flings small droplets into its surroundings. However, for clouds with a final overdensity (after regaining pressure balance with surroundings) $\chi_{\rm f} \le 300$, the outflowing droplets turn around and coagulate to once again form a monolithic cloud.}
\item{{\it Turbulence.} In simulations of radiatively cooling multi-phase gas in the presence of extrinsic turbulent driving, coagulation of cold gas clumps are frequent, and play a critical role in maintaining a scale-free power-law distribution ${\rm d}n/{\rm d}M \propto M^{-2}$ \citep{Gronke2022}. While this could simply be geometric (i.e., collisions which occur because clumps are entrained in the turbulent velocity field), there are hints of cooling-induced `focussing'. For instance, we see deviations from this power law at low Mach numbers, which will be presented in future work.}
\end{itemize}
In this work, we want to systematically study the effect of cooling induced coagulation.
This short paper is structured as follows: in Sec.~\ref{sec:methods} we describe our (numerical) methods, in Sec.~\ref{sec:results} we present our results, discuss them in Sec.~\ref{sec:discussion} before we conclude in \S~\ref{sec:conclusion}. Videos visualizing our results can be found at \url{https://max.lyman-alpha.com/coagulation}.
\section{Methods}
\label{sec:methods}
For our hydrodynamical simulation, we use \texttt{Athena} 4.0 \citep{Stone2008} and \texttt{Athena++} \citep{Stone2020}.
We use the HLLC Riemann solver, second-order reconstruction with slope limiters in the primitive variables, and the van Leer unsplit integrator \citep{Gardiner2008}. In both codes, we implemented the \citet{Townsend2009} cooling algorithm which allows for fast and accurate computations of the radiative losses. We adopt a solar metallicity cooling curve to which we fitted a power-law.
For this work, we use three different setups:
\begin{itemize}
\item \textit{Isolated cloud.} This three-dimensional setup is similar to the one used in \citep{Gronke2020}, i.e., we placed an isolated cloud of size $\sim r_\mathrm{cl}$\footnote{As the cloud is non-spherical, the effective radius is slightly larger. See \citet{Gronke2020} for details.} with temperature $T_{\rm cl}$ and overdensity $\chi\equiv \rho_\mathrm{cl} / \rho_{\rm h}$ in a hot medium. While the setup is initially in pressure equilibrium, the cloud will (rapidly) cool to $T_{\rm floor}$ leaving $r_\mathrm{cl}$, $\chi$, and $T_{\rm cl}/T_{\rm floor}$ the most important parameters. The purpose of this setup is to systematically study the pulsation induced mass growth discussed in \citep{Gronke2019,Gronke2020,Tan2020}.
\item \textit{Cloud-droplet.} Here, in addition to a cloud as described above, we place a droplet of size $r_{\rm d}$ and temperature $T_{\rm d}$ at a distance $d_0$ away from the cloud. In some cases we also give the droplet an initial velocity $v_{\rm d}$ away from the cloud. The purpose of this setup is to study the coagulation process of the cloud and the droplet. As we need to resolve the droplet sufficiently, we here resort to 2D simulations -- but also carry out 3D ones to study the dimensionality dependence of our results.
\item \textit{Multiple droplets.} We place $N_{\rm d,0}$ droplets with properties as described above randomly within a radius $d$. Again, we perform 2D and 3D simulations with the purpose of studying the coagulation behavior.
\item \textit{Turbulent droplets.} The placement is identical to the 3D `multiple droplets' setup described above but we continuously stir the box in the same manner as in \citet{Gronke2022}, that is, with decaying turbulence as well as continuous driving (to produce a roughly constant kinetic energy) at the scale of the simulation domain with ratio of solenoidal to compressive components of $\sim 1/3$.
\end{itemize}
For all our setups, we strive to resolve the cold gas by at least $\sim 16$ cells to ensure convergent behavior. We do, however, increase the resolution to $\sim 64$ cells to check this explicitly in some cases (see Appendix~\ref{sec:convergence}; also see \citealp{Tan2020} for an extensive discussion on resolution requirements). Furthermore, we employ `outflowing' boundary conditions -- except in the `turbulent droplets' setup where we used periodic ones.
\section{Results}
\label{sec:results}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/mdot_evolution_multiplot01.pdf}
\caption{Evolution of pulsating clouds with various initial conditions. The upper panel shows the cold gas volume normalized by its initial value, and the lower panel shows the mass growth rate normalized by the theoretically expected value.}
\label{fig:mdot_evolution_multiplot}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/mdot_vs_rcl_overview.pdf}
\caption{Mass growth rate of different pulsating clouds. Note that we display simulations that do not fragment, i.e., have either $\chi_{\rm final}\le 300$ or a perturbation of $T_{\rm cl}/T_{\rm floor} < 1.6$. We display simulations with different overdensities $\chi$ (color coded) and resolutions (marker type) and a minimum perturbation of $T_{\rm cl}/T_{\rm floor}>1.1$. The dashed line shows the theoretical expectation.}
\label{fig:mdot_vs_rcl_overview}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/mdot_vs_X.pdf}
\caption{Mass growth versus the initial perturbation $T_{\rm cl} / T_{\rm floor}$. If $r_\mathrm{cl} \gg \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$, $\dot m$ is independent of $T_{\rm cl} / T_{\rm floor}$ for $T_{\rm cl} / T_{\rm floor}\gtrsim 1.5$. The minimum perturbation shown with filled symbols is $T_{\rm cl} / T_{\rm floor}=1.1$. As an unfilled black circle, we also show a simulation with $T_{\rm cl} / T_{\rm floor}=1$ which does not grow.
}
\label{fig:mdot_vs_X}
\end{figure}
\subsection{Pulsations \& mass growth in a static medium}
\label{sec:convergence_mdot}
If a cloud does not fragment, it instead oscillates. These oscillations are accompanied by cold gas mass growth\footnote{Note, that the oscillations are crucial in order to obtain a converged mass growth, as we illustrate in Appendix~\ref{sec:osci_conv}.} -- which analogous to our findings in \citet{Gronke2019} we expect to be
\begin{equation}
\label{eq:mdot}
\dot m \sim v_{\mathrm{mix}} A_\mathrm{cl} \rho_{\mathrm{hot}}
\end{equation}
with a cold gas surface area $A_\mathrm{cl}$, and a surrounding hot gas density $\rho_{\mathrm{hot}}$. The characteristic mixing velocity is given by
\begin{equation}
\label{eq:vmix}
v_{\mathrm{mix}}\sim \alpha c_{\mathrm{s}} \left( \frac{t_{\mathrm{cool}}}{t_{sc}} \right)^{-1/4}\sim \alpha c_{\mathrm{s}} \left( \frac{r_\mathrm{cl}}{\ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi} \right)^{1/4},
\end{equation}
where all the quantities $c_{\mathrm{s}}$, $t_{\mathrm{coo}}$, and $t_{sc}$ are evaluated at the floor, that is, $v_{\mathrm{mix}}$ is of the order of the cold gas sound speed, and $\alpha$ is a dimensionless quantity of order unity we calibrate to simulations. This scaling has been confirmed with high-resolution turbulent mixing layer simulations \citep{Tan2020,Fielding2020}.
Figure~\ref{fig:mdot_evolution_multiplot} shows examples of our simulations with different initial overdensities and temperatures ($\chi$, $T_\mathrm{cl}$, respectively), and different cloud sizes. The upper panel shows the cold gas volume from which we see that the oscillations take place on the order of the final sound crossing time $t_{\rm sc, floor}\sim r_\mathrm{cl} / c_{\rm s,floor}$.
The lower panel of Fig.~\ref{fig:mdot_evolution_multiplot} shows the mass growth rate -- which we normalize by the analytic estimate Eq.~\eqref{eq:mdot}. We see that the for all the simulations, the values oscillate around $\sim 0.5$, implying $\alpha\sim 0.5$. Moreover, mass growth at this rate keeps this value for many $t_{\rm sc,floor}$, which is longer than we naively expect the initial turbulence in the mixing layer between the hot and cold medium to last. Instead, mixing is facilitated by cooling induced pulsations.
On overview of the mass growth rate for a range of simulations is shown in Fig.~\ref{fig:mdot_vs_rcl_overview}. Shown are simulations which did not \textit{shatter}, i.e., we excluded the simulations for which the maximum number of droplets was $>100$ which occurs for $\chi_{\mathrm{final}}= T_\mathrm{cl} / T_{\mathrm{floor}} \chi\gtrsim 300$ \citep{Gronke2020}. Note that for this plot we normalized the radii by the theoretical estimate by using the overdensity, temperature, and cloud radius at the point at which the cloud loses sonic contact, i.e., $\chi^* = \chi (r_\mathrm{cl} / r_\mathrm{cl}^*)^3$ with $r_\mathrm{cl}^*=\sqrt{\gamma T_\mathrm{cl}^*} t_{\rm cool}(T^*_\mathrm{cl}, \chi^* \rho_{\mathrm{hot}})$ \citep{Gronke2020}.
Figure~\ref{fig:mdot_vs_rcl_overview} shows that \textit{(i)} the mass growth follows the scaling relation of Eq.~\eqref{eq:vmix} over $\gtrsim 5$ orders of magnitude in cloud size and $\gtrsim 2$ orders of magnitude in overdensity, \textit{(ii)} for small clouds ($r_{\rm cl}^{*} \lesssim 100\ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$ for $\chi\gtrsim 100$, larger for smaller overdensities) the mass growth is less than expected, and \textit{(iii)} the high-resolution runs (of $l_{\rm cell} / r_\mathrm{cl}=64$, i.e., a factor of $4$ improvement compared to our fiducial resolution) are consistent with these findings.
As stated above, the clouds in the simulations shown in Fig.~\ref{fig:mdot_vs_rcl_overview} were `sufficiently' perturbed to allow mass growth without shattering (i.e., keeping $\chi_{\rm final}\lesssim 300$ or $T_{\rm cl}/T_{\rm floor}\lesssim 2$). The impact of this initial perturbation -- which sheds light on what `sufficiently' exactly means -- is shown in Fig.~\ref{fig:mdot_vs_X}. In this, we can see that \textit{(i)} as seen before Eq.~\eqref{eq:vmix} is valid only for clouds $r_{\rm cl} \gg \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$ which will pulsate and grow\footnote{Note that Fig.~\ref{fig:mdot_vs_X} shows a small overdensity of only $\chi=10$ which we show to be able to explore a range of $T_\mathrm{cl} / T_{\rm floor}$ values without $\chi_{\rm final}\gtrsim \chi_{\rm crit}$ and, thus, shattering.}, \textit{(ii)} if $T_\mathrm{cl} / T_{\mathrm{floor}}\gtrsim 1.5$, the mass growth does not depend on the extent of the perturbation, and \textit{(iii)} for smaller perturbations ($T_{\rm cl}/T_{\rm floor}\lesssim 1.5$), the mass growth does grow with the perturbation but even a value $T_{\rm cl}/T_{\rm floor}\sim 1.01$ (representing our initial random fluctuations, cf.~\S~\ref{sec:methods}) does to lead a significantly larger mass growth than for an unperturbed cloud, where mixing is only due to numerical diffusion.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/2d_smr_chif50_multiplot01.pdf}
\caption{Evolution of $2$D simulations of a droplet located at $d_0/r_\mathrm{cl}$ merging with a cloud of radius $r_\mathrm{cl}$ which cools from $T_\mathrm{cl}$ to $T_{\mathrm{floor}}$. \textit{Top panel:} cold gas mass as a function of time. \textit{Central panel:} Ratio of measured to predicted cold gas mass growth. \textit{Bottom panel:} velocity of the droplet as a function of time. The dashed lines in the upper and lower panel show the curves stemming from solving Eq.~\eqref{eq:eom_mass_growth} with $\alpha=0.45$ which is marked as a black line in the central panel.}
\label{fig:coag2d_multiplot}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/2d_smr_veldrop_multiplot01.pdf}
\caption{Evolution of $2$D simulations of a droplet located at $d_0/r_\mathrm{cl}=1.1$, and an initial velocity $v_{\rm d,0}$ merging with a cloud of radius $r_\mathrm{cl}$ which cools from $T_\mathrm{cl}\approx 2 T_{\rm floor}$. \textit{Top panel:} mass growth rate normalized by the expected value from Eq.~\eqref{eq:mdot}. \textit{Bottom panel:} location of the droplet as a function of time. The dashed lines correspond to Eq.~\eqref{eq:eom_mass_growth} with a velocity dependent $\alpha_{\rm d}\sim (v_{\rm d} / c_{\rm s,floor})^{1/2}$}
\label{fig:coag2d_veldrop_multiplot}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/3d_smr_chif50_multiplot02.pdf}
\caption{Evolution of $3$D simulations of a droplet located at $d_0/r_\mathrm{cl}$ merging with a cloud of radius $r_\mathrm{cl}$ which cools from $T_\mathrm{cl}$ to $T_{\mathrm{floor}}$. The solid, dashed and dotted lines show runs with overdensities of $\chi\sim 50$, $\sim 10^3$ and $\sim 10^4$, respectively. The runs marked with ${}^*$ are the ones were we perturbed the droplet, i.e., $T_{\rm d}=T_{\rm cl}$. \textit{Top panel:} ratio of measured to predicted cold gas mass growth. \textit{Bottom panel:} normalized velocity of the droplet as a function of time.}
\label{fig:coag3d_multiplot}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{plots/multi2d_dens_N50_static_coag.pdf}
\caption{Projections of a $3$D simulations with $N_{\rm d} = 50$ droplets of size $r_{\rm d}\sim 500\ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$ placed randomly in a sphere with radius $15 r_{\rm d}$ (marked as white dashed line). The droplets coagulate on a timescale of $\sim 40 t_{\rm sc,cl}$.}
\label{fig:multi2d_3dcoag}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/2d_dist_vs_time.pdf}
\caption{Evolution of $2$D simulations of a fog of droplets randomly places within $r<100 r_{\rm d}$. Plotted is the distance of the droplet initially furthest away from the origin as a function of time. The dashed lines are our analytical estimate of this scenario using $\alpha=0.1$.}
\label{fig:coag2d_fog}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/3d_dist_vs_time_new.pdf}
\caption{Evolution of $3$D simulations of a fog of droplets randomly places within $r<d_0$. Plotted is the distance of the droplet initially furthest away from the origin as a function of time. The dashed lines are our analytical estimate of this scenario using $\alpha=0.2$. Note how the coagulation is very slow if $r_{\rm d}\lesssim \mathcal{O}(\ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi)$ due to the lack of pulsations.}
\label{fig:coag3d_fog}
\end{figure}
\subsection{Cooling induced coagulation}
\label{sec:result_coag}
As we have seen in the previous section, the mass transfer rate from hot to cold medium depends on the size of the cold gas cloud, and is generally an important prediction to compare to observations. In the circumgalactic medium, for instance, characteristic scales of the cold $\sim 10^4\,$K gas are commonly inferred from absorption line studies \citep[e.g.,][]{Schayelargepopulation2007,Lan2017,Churchill2020} or through emission properties \citep[e.g.,][]{Cantalupo2014,Hennawi2015,Li2021} which indicate the presence of small $\lesssim 100\,$pc clouds. This finding has sparked a range of theoretical studies. As mentioned above, \citet{McCourt2016} suggested droplets of the size of $\ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi\equiv \mathrm{min}(c_{\rm s}t_{\rm cool})$ to be the outcome of a cooling and fragmentation process. Furthermore, a characteristic size of a cloud $r_{\rm cl}\gtrsim r_{\rm cl,crit} =v_{\rm wind} t_{\rm cool,mix}/\sqrt{\chi}$ is also required for it to survive ram pressure acceleration \citep{Gronke2018,Li2019a,Kanjilal2020}. These predictions can be compared to observations; they also set resolution requirements for larger-scale simulations. Using the example of the circumgalactic medium again, current cosmological simulations are not yet numerically converged in cold gas properties, which makes comparisons to observations problematic \citep[e.g.,][]{Faucher-Giguere2016,Hummels2018}.
Fragmentation and mixing are processes lowering the size of the cloud. On the other hand, mass growth through cooling (as discussed in the last section), and coagulation of clouds are processes increasing the characteristic cloud size. Coagulation of cold gas clouds is seen to occur in larger scale simulations \citep{Gronke2022}. Here, we want to study the coagulation process due to cooling in highly idealized setups.
\subsubsection{Static, 2D setup}
Figure~\ref{fig:coag2d_multiplot} shows the outcome of two-dimensional simulations where we placed a single droplet of size $r_{\rm d}\sim 0.1 r_\mathrm{cl}$ at a distance $d_0$. We perturb the cloud and the droplet as in the previous section by initializing their temperature to $T_\mathrm{cl} > T_{\rm floor}$. As seen before, the cold gas mass growth (upper and central panel of Fig.~\ref{fig:coag2d_multiplot}) follows the expected evolution given by Eq.~\eqref{eq:mdot}.
Due to this mass growth, which is dominated by the cloud, the surrounding hot gas streams towards, and entrains the droplet.
In the lower panel of Fig.~\ref{fig:coag2d_multiplot}, we show the droplet velocity as a function of time.
Note that the droplet gets accelerated both via ram pressure and momentum transfer due to cooling of the mixed material which take place on timescales of $t_{\mathrm{drag}}\sim \chi r_{\mathrm{d}} / v_{\mathrm{hot}}$ and $t_{\mathrm{grow}}\equiv m / \dot m$, respectively.
The ratio of these two timescales is
\begin{equation}
\frac{t_{\mathrm{drag}}}{t_{\mathrm{grow}}} \sim \frac{v_{\mathrm{mix}}}{v_{\mathrm{hot}}}\sim \frac{d}{r_\mathrm{cl}}
\end{equation}
where we used $v_{\rm hot}\sim v_{\mathrm{mix}} (r_\mathrm{cl} / d)$, which comes from mass conservation in 2D. This shows that we expect the momentum transfer via mass growth to dominate.
The net force acting on the droplet is $F \sim \dot{p} \sim \dot{m} v + m \dot{v} \sim \dot{m} (v_{\rm hot} + \dot{d})\sim 0$. Hence, the equation of motion of the droplet is
\begin{equation}
\label{eq:eom_mass_growth}
\dot m (v_{\mathrm{mix,cl}} r_\mathrm{cl} / d + \dot d) = -m(t) \ddot d
\end{equation}
with $\dot m\sim 2 \pi v_{\mathrm{mix,d}}r_{\mathrm{d}}\rho_{\mathrm{h}}$ as before.
This can be improved by taking into account the growth of the cloud and the, thus, shrinking distance by using $r_{\mathrm{cl}}^2\sim m_\mathrm{cl} / (\pi \rho_{\mathrm{cl}})$ and solving for the time-dependent cloud mass $m(t)$ by integrating $\dot m\sim 2 \pi v_{\mathrm{mix,cl}}r_{\mathrm{cl}}\rho_{\mathrm{h}}$ as well.
Note that $v_{\rm mix} \propto r^{1/4}$ is a scale dependent quantity, and thus it is distinct for the cloud and the droplet. However, Eq.~\eqref{eq:vmix}
was derived in $3$D (and with larger perturbations), so it is unclear if it holds here.
The dashed lines in Fig.~\ref{fig:coag2d_multiplot} shows the outcome of this analytic model and we see that (using $\alpha\sim 0.45$) it fits the numerical solution reasonably well.
\subsubsection{Static, 3D setup including large $\chi$}
Figure~\ref{fig:coag3d_multiplot} shows the evolution of three dimensional runs of the same setup. Note that as here $v_{\rm hot}\propto d^{-2}$ the coagulation process is much slower compared to the 2D runs described above. Nevertheless, the droplets do move towards the cloud and they do so approximately with the velocity expected.
In Fig.~\ref{fig:coag3d_multiplot}, we also show the results of a run with a much larger overdensity of $\chi\sim 1000$ (with dashed lines). We can note that (i) the mass growth follows the predicted scaling Eq.~\eqref{eq:mdot}, (ii) the droplet's motion is independent of $\chi$. This might seem counter-intuitive since the acceleration (both via drag and mass growth) is to first order proportional to $\chi$. However, since the droplet growth time is short compared to the travel time\footnote{This is consistent with the simulations in which we see the droplet's mass increase by a factor of $\sim 5$ ($\sim 25$) for $\chi\sim 10^3$ ($\sim 50$) in the travel time.} $t_{\rm travel}\sim d/v_{\rm hot}\sim (d/r_{\rm cl})^3 t_{\rm sc,cl}$, we can see them as comoving with the hot gas, and the acceleration period is so short that its $\chi$ dependence is unimportant.
This is no longer the case once the growth time of the droplet is smaller than the travel time, i.e., setting $t_{\rm grow,d}\sim t_{\rm travel}$ yields a critical overdensity of
\begin{equation}
\label{eq:chi_static}
\chi_{\rm stuck} \sim \beta \frac{d^3}{r_{\mathrm{cl}}^2 r_{\rm d}} \left( \frac{r_{\rm d}}{r_{\mathrm{cl}}} \right)^{1/4}
\end{equation}
above which the droplet should not move. Here, $\beta$ is a fudge factor encapsulating the initial deviation of $t_{\rm grow,cl}/t_{\rm grow,d}$ from its steady state (expected) value. Since the cloud is initially perturbed, $\dot m > \dot m_{\rm predicted}$ initially (cf. Fig.~\ref{fig:coag3d_multiplot} upper panel), but for the droplet it takes a bit for the mixing turbulence to develop, hence, the reverse is true. Setting $\beta\sim 0.1$ (consistent with the mass growths from the simulation), we obtain a $\chi_{\rm stuck}\sim 3600$ (for $d\sim 4 r_{\rm cl}$, $r_\mathrm{cl} / r_{\rm d}\sim 10$). Fig.~\ref{fig:coag3d_multiplot} also shows a simulation with $\chi\sim 10^4$ where indeed the velocity of the droplet $v\sim 0$ (dotted red line in the lower panel of Fig.~\ref{fig:coag3d_multiplot}).
\subsubsection{Static, multidroplet setup}
Instead of placing a single droplet next to a large cloud, we placed a large number of droplets randomly within a sphere. We again perturb them using an initial temperature of $T/T_{\rm floor}\sim 2$. Due to their combined growth, these droplets will merge to form a single blob. Fig.~\ref{fig:multi2d_3dcoag} visualizes this evolution via density projections of a $3$D simulation. Fig.~\ref{fig:coag2d_fog} and Fig.~\ref{fig:coag3d_fog} show the distances of the droplets initially furthest away from the center of the sphere to it for two and three-dimensional simulations, respectively.
An increased droplet number density implies more mass growth, and thus faster coagulation. We adopted our cloud-droplet model to this fog of droplets by using a cloud of mass $m_{\rm cl}=N_{\rm drop} m_{\rm drop}$, i.e., considering the combined mass growth.
This simple model (shown as dashed lines in Figs.~\ref{fig:coag2d_fog}, \ref{fig:coag3d_fog}) reproduces the contraction process reasonably well. Discrepancies occur at extremely dense droplet placement when the free-streaming of the hot gas no longer occurs (i.e., shielding becomes important), and for small droplets $r_{\rm d}\lesssim \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$ (thick lines in Fig.~\ref{fig:coag3d_fog}). As shown in \S~\ref{sec:convergence_mdot} for these clouds the mass growth -- and hence, the speed of coagulation -- is significantly slower.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{plots/multi2d_dens_N10_Evar.pdf}
\caption{Time evolution of turbulent multiphase boxes with different Mach numbers and $10$ droplets of size $r_{\rm d}\sim 500\ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$. While the $\mathcal{M}\sim 1$ case shows fast fragmentation, in the $\mathcal{M}\sim 0.1$ case some droplets have coagulated.}
\end{figure*}
\subsubsection{Droplets with initial velocity (2D)}
Using this simple model of cooling induced coagulation, we can add additional complexities. Figure~\ref{fig:coag2d_veldrop_multiplot} shows the evolution of 2D runs in which we impose an initial droplet velocity $v_{\rm d,0}$ away from the cloud. This is akin to the situation for `shattering' clouds when droplets fly away with high ($v_{\rm d,0}\lesssim \text{a few}\times c_{\rm s,cold}$) velocities \citep{Gronke2020}. With the model described above, we can reproduce the droplets trajectory quite accurately but note that we use a velocity dependent fudge factor\footnote{In 3D shearing layers, $v_{\rm mix} \sim (u^{\prime})^{3/4} (r/t_{\rm cool})^{1/4}$ scales with the turbulent velocity $u^{\prime}$ rather than the cold gas sound speed, where $u^{\prime} \propto v^{4/5}$, giving $v_{\rm mix} \propto v^{3/5}$ \citep{Tan2020}. Since we have not investigated this in 2D, and also the modification of droplet surface area by the initial velocity, we merely note that this fudge factor (which is $\sim 2$ or less in our numerical experiments) works well.} for the droplet's mass growth rate of $\alpha_{\rm d}\sim (v_{\rm d}/c_{\rm s,floor})^{1/2}$.
In summary, the coagulation process of cold gas structures embedded within a hotter surrounding is driven by the cold gas mass growth in two ways. First, in order to sustain the global cold mass growth, the hot medium is moving at a velocity $v \propto v_{\rm mix}(r_{\rm cl}/d)^{2}$ in 3D towards the cold gas. And secondly, due to their own mass growth, droplets become rapidly entrained in this velocity field. We developed a simple model describing this system, which reproduces our numerical results reasonably well. Such a static setup does not represent, however, reality for most astrophysical systems. We therefore study cold gas mass growth and coagulation in a turbulent setup next.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/nclumps_vs_mass.pdf}
\caption{Number of clumps versus the cold gas mass for six simulations of turbulent, multiphase media with different Mach numbers and cold gas sizes. The runs with lower Mach number and larger cloud sizes show stronger coagulation in line with Eq.~\eqref{eq:Mach_coag_numerical}. The lines are horizontally slightly offset for better visualization.}
\label{fig:turb_nclumps_vs_mass}
\end{figure}
\subsection{Coagulation in a turbulent medium}
\label{sec:res_coag_turb}
As we saw in the last section, the coagulation velocity is $\sim c_{\rm s,c}$, i.e., rather small. In typical astrophysical systems with turbulent velocity dispersion $\sim c_{\rm s,hot}$ it seems at first sight that coagulation cannot `win' over dispersion. This is in line with simulations of multiphase gas in a turbulent medium which show fragmentation of the cold gas \citep[e.g.,][]{Saury2014,Gronke2022,Mohapatra2022a}. However, since the dispersion is not a directed bulk motion like coagulation but instead more akin to a random walk, it is of interest to study the threshold $v_{\rm coag}\sim v_{\rm disp}$. There are two interesting questions: (i) when does a system of clouds coagulate? (ii) when does an individual cloud fragment in the face of turbulence?
Turbulent dispersion is a large area of research in fluid mechanics \citep[for reviews see, e.g.,][]{Sawford2001,Salazar2009} with a long history. \citet{Batchelor1950} found that initially the mean separation of two particles in a turbulent medium scales as $\langle d^2 \rangle\propto t^2$ whereas for latter times\footnote{Specifically for $t\gg t_{\rm B}\sim d_0^{2/3} \langle \epsilon \rangle^{-1/3}$ where $d_0$ and $\epsilon$ is the initial separation and the turbulent dissipation, respectively.} the particles `forget' their initial separation and $\langle d^2 \rangle\propto t^3$. In both cases, turbulent dispersion is superdiffusive, compared to the customary diffusive expectation $\langle d^2 \rangle \propto t$.
Since we are interested in the dominant process initially -- which governs the further evolution -- we equate the velocity dispersion prior to the ``Batchelor time'' to the coagulation velocity.
In this regime, the mean dispersion velocity is given by
\begin{equation}
\label{eq:vdisp}
\bar v \sim a v_{\rm turb} \left( \frac{d_0}{L} \right)^{1/3}
\end{equation}
where $v_{\rm turb}\sim \mathcal{M}c_{\rm c,hot}$ is the driving velocity on the scale of the box and $a\sim 2$ a numerical prefactor\footnote{Specifically, $a=\sqrt{11/6} C_2$ with $C_2$ being the Kolmogorov constant for the second order velocity structure function. \citet{2013PhRvE..87b3002N} find $C_2\sim 4.02$.}. Note that while Eq.~\eqref{eq:vdisp} follows from the $\langle d^2 \rangle\propto t^2$ scaling described above, it simply represents the Kolmogorov scaling.
If we evaluate turbulence and coagulation at the scale of the cloud $d_0 = r_{\rm cl}$, and require $v_{\rm coag}\sim v_{\rm mix} > \bar{v}$, this yields a critical Mach number:
\begin{align}
\label{eq:Mach_coag}
\mathcal{M}_{\rm coag}\sim& \frac{\alpha (r_\mathrm{cl} / \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi)^{1/4}}{a (r_\mathrm{cl} / L)^{1/3} \chi^{1/2}}\\
\sim & 0.16 \left( \frac{r_\mathrm{cl} / \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi}{500} \right)^{1/4} \left( \frac{L/r_\mathrm{cl}}{40} \right)^{1/3} \left( \frac{\chi}{100} \right)^{-1/2}
\label{eq:Mach_coag_numerical}
\end{align}
below which coagulation is stronger than dispersion and clouds should be more robust to fragmentation.
In Eq.~\eqref{eq:Mach_coag_numerical}, we plugged in typical values and used the fiducial values $a=2$ and $\alpha= 0.2$ as suggested by the result presented in \S~\ref{sec:convergence_mdot} and \citet{Gronke2019}.
In \S\ref{sec:coag_grav}, we also estimate critical Mach numbers below which clouds can coagulate (Eq.~\eqref{eq:Mach_coag_collection2}). The point is that although there is strong inverse square geometric dimming of coagulation forces, the critical Mach number for coagulation is still $\mathcal{M} \sim v_{\rm mix}/c_{\rm s,h} \sim 0.1 (\chi/100)^{-1/2}$ if cold gas covering fractions are high.
Figure~\ref{fig:coag3d_fog} shows snapshots of simulations with multiphase, turbulent media. The boxes were initiated with decaying as well as driven turbulence to ensure approximately constant Mach number, and $10$ cold clumps were placed in them (with overdensity $\chi\sim 100$ and size $r_{\rm d}\sim 500\ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$; the numerical setup is identical to \citealp{Gronke2022}). The $\mathcal{M}\sim 1$ simulation shows the most fragmentation, whereas in the $\mathcal{M}\sim 0.1$ run, coagulation of droplets occurs.
Figure~\ref{fig:turb_nclumps_vs_mass} shows this behavior in a more quantitative manner. As the turbulent, multiphase medium evolves, the cold gas mass grows (if it is initially larger than some critical size; see \citealp{Gronke2022}) -- and fragments. The extent of this fragmentation depends on the competition between coagulation and dispersion. In Fig.~\ref{fig:turb_nclumps_vs_mass} we show the results from six simulations with different cloud sizes and Mach numbers (with $256^3$ cells, $L_{\rm box}/r_{\rm cl}=40$, and $\chi\sim 100$) for in which we analyzed using a clump finding algorithm. In the $r_\mathrm{cl} / \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi\sim 500$ case, the $\mathcal{M}\sim 1$ and $\mathcal{M}\sim 0.3$ simulations fragment into $\gtrsim 100$ clumps while in the runs with $r_\mathrm{cl} / \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi \sim 5000$ this is only true for $\mathcal{M}\sim 1$. Note that in these simulations, we have kept $L/r_{\rm cl} \sim 40$ constant. Our results are in line with the discussion in \S\ref{sec:coag_grav}. Eq.~\eqref{eq:Mach_coag_numerical} which yields a critical mach number of $\mathcal{M}_{\rm coag}\sim 0.16$ and $0.28$ for the smaller and larger cloud, respectively.
Due to numerical constraints, we can only probe small dynamic temporal and spatial range. However, we showed that coagulation does affect the dynamics of turbulent, multiphase media. Naturally, also other potentially observable properties such as the cloud size distribution are also affected. We will study this point in detail in future work.
\section{Discussion}
\label{sec:discussion}
\subsection{Analogies between coagulation and gravity}
\label{sec:coag_grav}
Consider two clouds separated by a distance $d$. Cloud 1 experiences a force due to cloud 2 given by:
\begin{equation}
F_{1,2} \sim \dot{m}_1 v_{\rm coag,2} \sim \rho_{\rm h} v_{\rm mix,1} A_1 v_{\rm mix,2} \frac{A_2}{4 \pi d^{2}}
\label{eq:F_1,2}
\end{equation}
On the other hand, cloud 2 experiences a force due to cloud 1 given by:
\begin{equation}
F_{2,1} \sim \dot{m}_2 v_{\rm coag,1} \sim \rho_{\rm h} v_{\rm mix,2} A_2 v_{\rm mix,1} \frac{A_1}{4 \pi d^{2}}\;.
\label{eq:F_2,1}
\end{equation}
Thus, the two clouds exert equal and opposite attractive forces on one another, with magnitude scaling as the inverse square of their separation $F \propto d^{-2}$. This reminds us of another extremely well-studied force -- gravity -- with the same characteristics, $|F_{1,2}| = |F_{2,1}| \sim G m_1 m_2/d^{2}$. Despite the fact that gravity is relatively `weak'\footnote{For instance, the `gravitational fine-structure constant' $\alpha_{\rm G} \sim G m_p^2/\hbar c \sim 10^{-38}$ is orders of magnitude weaker than the electromagnetic fine-structure constant, $\alpha = e^{2}/\hbar c =1/137$.} and also decays as $F \propto d^{-2}$, it is of course crucial in structuring mass distributions -- despite the simple nature of Newtonian gravity, it gives rise to very rich and complex behavior \citep[e.g.,][]{binney2008}. This is in part because it is a long range attractive force without any shielding --- unlike electromagnetism, there are no negative charges. Similarly, cooling-induced coagulation is a wholly attractive force with no negative charges\footnote{There can be geometric shielding in an optically thick flow (where a cloud blocks hot gas and thus modulates hot gas flow behind it), but we will ignore this complication for now.}. While there are important differences\footnote{For instance, smaller signal speed: coagulational forces propagate at the sound speed of hot gas, and time delay effects can be important.}, the parallels between gravity and coagulation are strong enough to be a useful avenue for thinking about this problem.
From examining equations \ref{eq:F_1,2} and \ref{eq:F_2,1}, we can identify the analog of gravitational mass to be area $ m \rightarrow A$, and the analog of the gravitational constant to be a peculiar form of kinetic energy density\footnote{Note that $v_{\rm mix} \propto r^{1/4}$ is size dependent.
We adopt a value $\langle v_{\rm mix} \rangle$ which is understood to be averaged over the size spectrum of cloudlets in the system.}
$G \rightarrow \rho_{\rm h} v_{\rm mix}^{2}$. Already this tells us about an important difference between gravitational and coagulational dynamics. Mass is conserved under fragmentation and coagulation. However, area is {\it not} conserved: for instance, if one `shatters' a cloud into tiny droplets of radius $r_{\rm d}$, with the number of droplets $N \sim (r_{\rm cl}/r_{\rm d})^{3}$, then the area increases by a factor $r_{\rm cl}^{2}/(N r_{\rm d}^{2}) \sim (r_{\rm cl}/r_{\rm d})$, so that coagulation becomes significantly more important\footnote{As discussed in \S~\ref{sec:convergence_mdot}, this only holds for sizes down to $\sim \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$ after which no pulsations -- and thus no `cooling induced' coagulation -- will occur. However, mixing, cooling (and coagulation) due to external factors such as shear flows can still play a role for these tiny fragments.}. This surface area dependence is key to the strong modulation of coagulation -- `shattering' (which rapidly increases the surface area to mass ratio) boosts the importance of coagulation, while mergers/coagulation itself (which decrease the surface area to mass ratio) reduces the importance of coagulation. In a multi-body system, each cloud is weighted by area, not by mass, and we can follow the motion of an extended distribution by writing an equation for the `center of area' $\langle \mathbf{r}_{\rm CA} \rangle = \int \mathbf{r} \mathrm{d} A/ \int \mathrm{d} A$, rather than the center of mass $\langle \mathbf{r}_{\rm CM} \rangle = \int \mathbf{r} \mathrm{d} M/ \int \mathrm{d} M$. We can also think about the analog of the free fall time, $t_{\rm ff} \sim 1/\sqrt{G\rho}$. Consider the total forces acting on a single cloud of mass $m_{\rm cl}$ and area $A_{\rm cl}$ at distance $d$ to the `center of area' of a collection of clouds with total area $A(<d)$ in a sphere with $r=d$,
\begin{equation}
m \ddot{d} \sim \rho_{\rm h} v_{\rm mix}^{2} \frac{A_{\rm cl} A_{\rm tot}(<d)}{4 \pi d^{2}} \sim \rho_h v_{\rm mix}^{2} f_A A_{\rm cl}
\end{equation}
where $f_A \sim A_{\rm tot}(<d)/4 \pi d^{2}$,
the number of times a random line of sight with impact parameter less than d to the `center of area intersects a surface\footnote{Similar to optical depth, $f_{\rm A} > 1$ is possible, which boosts the importance of coagulation and decreases $t_{\rm coag}$.}, we can obtain the coagulation time for a cloud embedded in a collection of clouds:
\begin{equation}
t_{\rm coag} \sim \left( \frac{\chi}{f_{\rm A}} \right)^{1/2} \frac{(r_{\rm cl} d)^{1/2}}{v_{\rm mix}}.
\label{eq:tcoag}
\end{equation}
Note the appearance of $r_{\rm cl}$ in $t_{\rm coag}$: there will be mass segregation in coagulational collapse, with larger clouds falling to the center more slowly. In gravity, we have the principle of equivalence, due to the equivalence of gravitating and inertial mass: $F=ma = mg$, so $a=g$, independent of mass-- feathers and rocks fall at the same rate in a vacuum. However, for coagulation, $F=ma = m_{\rm coag} g_{\rm coag} \propto A g_{\rm coag}$, so $a \propto A/m \propto 1/r$; larger objects fall more slowly\footnote{Of course, mergers and fragmentation will modulate $t_{\rm coag}$ of a cloud, just as evolving density modulates $t_{\rm ff} \sim 1/\sqrt{G \rho}$.}.
We can compare the coagulation time Eq.~\eqref{eq:tcoag} with the results shown in Fig.~\ref{fig:coag3d_fog}, where $N_\mathrm{cl}=50$ clouds of size $r_{\rm cl}$ are randomly distributed within a sphere of size $d= 15 r_{\rm cl}$. This gives $f_{\rm A} \approx \int \mathrm{d} V \, n_{\rm cl} \pi r_{\rm cl}^{2}/(4 \pi r^{2}) \approx N_\mathrm{cl} (r_{\rm cl}/d)^{2}$, where the cloud number density $n_{\mathrm{cl}} \approx N_\mathrm{cl}/(4 \pi/3 d^{3})$. Inserting into Eq.~\eqref{eq:tcoag} yields:
\begin{equation}
\frac{t_{\rm coag}}{t_{\rm sc,cl}} \sim 80 \left( \frac{\chi}{100} \right)^{1/2} \left( \frac{N_\mathrm{cl}}{50} \right)^{-1/2} \left( \frac{d/r_{\rm cl}}{15} \right)^{3/2}
\end{equation}
which is a factor of 2 larger than the simulation result of ${t_{\rm coag}}/{t_{\rm sc,cl}} \approx 40$. This is good for an order of magnitude estimate, since clouds accelerate as they fall towards the center ($F_{\rm coag} \propto d^{-2}$). Moreover, Fig. \ref{fig:coag3d_fog} shows rough agreement with a $t_{\rm coag} \propto N_\mathrm{cl}^{-1/2}$ as well as the $t_{\rm coag} \propto d^{3/2}$ scaling. \\
In practice, the clouds, or the hot medium itself, will often be endowed with some relative velocities, which can cause the clouds to disperse. It would be nice to have some rule of thumb or intuition as to when the system coagulates or when it flies apart. In self-gravitating systems, we can compare potential energy $U$ with kinetic energy $K$. If $|U| > K$, the system collapse; if $|U| < K$, it is unbound and flies apart. Could a similar criterion be helpful in coagulating systems? Let us first study how to define potential energy $U$. Consider the work done to separate two clouds from $d_1$ to $d_2$:
\begin{equation}
\Delta U = \int_{d_1}^{d_2} F_{\rm coag} \mathrm{d} r = \frac{\rho_h v_{\rm mix}^{2}}{4 \pi} A_1 A_2 \left( \frac{1}{d_1} - \frac{1}{d_2} \right).
\end{equation}
Since $F_{\rm coag}$ is a radial force, it is conservative: $\Delta U$ is independent of the path taken from $d_1$ to $d_2$, and any closed loop (i.e., a path that ends back up at $d_1$ means that no net work\footnote{There {\it will} be work done by other drag forces; we only consider work done by $F_{\rm coag}$.} is done by $F_{\rm coag}$.
Thus, we can meaningfully define a potential energy $U$ where $F_{\rm coag} = - \nabla U$. If we set $U(\infty) = 0$, we can write:
\begin{equation}
U(d) \sim \frac{\rho_{\rm h} v_{\rm mix}^{2} A_1 A_2}{4 \pi d} \sim 3 \rho_h v_{\rm mix}^{2} \Omega_1 \Omega_2 V_d
\end{equation}
where $V_d \sim (4 \pi/3) d^{3}$, and $\Omega_{i} = A_i/(4 \pi d^{2})$ is the solid angle subtended by cloud $i$. The potential energy density is $\rho v_{\rm mix}^{2} \Omega_1 \Omega_2$: the kinetic energy density $\rho_{\rm h} v_{\rm mix}^{2}$ modulated by the area covering fractions $\Omega_1, \Omega_2$. As the covering fractions increase, so does $|U|$. Thus, fragmentation increases $|U|$, and mergers/coagulation decrease $|U|$.
For a collection of clouds, we sum the potential energy contributions from all pairs of clouds. From the analogy to Newtonian gravity, where $U \sim G M_{\rm tot}^{2}/\langle d \rangle$, where $\langle d \rangle$ is a characteristic scale (such as the half mass radius), we can write the total potential energy as:
\begin{equation}
U_{\rm tot} \sim \frac{\rho_{\rm h} v_{\rm mix}^{2}}{4 \pi} \frac{A_{\rm tot}^{2}}{\langle d \rangle} \sim M_{\rm h} v_{\rm mix}^{2} f_A^2
\label{eq:U_pot}
\end{equation}
where $M_{\rm h}\sim \rho_{\rm h} \langle d \rangle^3$ is the hot gas mass, and the area covering fraction/enhancement factor $f_{\rm A} \sim A_{\rm tot}/\langle d \rangle^{2}$ modulates the strength of potential energy. Thus, if $f_{\rm A} > 1$ (and indeed, $f_{\rm A} \gg 1$ is possible in 'fog-like' cloud topology), the potential energy will exceed the naive bound $M_{\rm h} v_{\rm mix}^{2}$, due to the superposition of the flows from multiple small clouds. Of course, at that point a more careful treatment which takes geometric shielding into account is necessary.
What about the kinetic energy? There are at least two classes of problems: (i) the hot gas is initially static and the cloudlets have some initial relative velocity. A prototypical example is cloud shattering. This statement is also approximately true of the cloud growth problem in the frame of the wind, when cloud fragments of different size have undergone differential acceleration. In this case the relevant kinetic energy is $K \sim M_c \sigma_c^2$, where $\sigma_c^2(d)$ is the velocity dispersion of cold gas at scale $d$. (ii) The hot gas velocity field has significant velocity structure (e.g., in the form of shear or turbulence), and can potentially entrain the clouds. In this case, the relevant kinetic energy is $K \sim M_h \sigma_h^2$, where $\sigma_h^2$ is the velocity dispersion of hot gas.
Although energy is not strictly conserved\footnote{In the first case, the hot medium provides an additional drag force which slows dispersal and promotes coagulation. In the second case, the hot medium (if it entrains the clouds) promotes dispersal. Therefore, unlike the self-gravitating case, there are additional dissipative or driving forces acting, besides the conservative force. Thus, there is no energy conservation: in the first case, kinetic energy decays (due to `friction' against the hot gas); in the second case, cloud entrainment transfers kinetic energy from the hot to cold gas. And, as previously noted, fragmentation/mergers modulates potential energy.}, we can use this to estimate whether coagulation is likely to happen. For coagulation to happen, we require that $|U_{\rm tot}| < |K_{\rm tot}|$, or $\sigma_h(d) < v_{\rm mix} f_A$. If we use Kolmogorov scalings for $\sigma_h(d)$, this gives a critical Mach number for coagulation:
\begin{align}
\label{eq:Mach_coag_collection1}
\mathcal{M}_{\rm coag}\sim& \frac{\alpha (r_\mathrm{cl} / \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi)^{1/4} f_A }{(r_\mathrm{cl} / L)^{1/3} \chi^{1/2}}\\
\sim & 0.2 \left( \frac{r_\mathrm{cl} / \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi}{500} \right)^{1/4} \left( \frac{L/d}{5} \right)^{1/3} \left( \frac{\chi}{100} \right)^{-1/2} \left(\frac{f_A}{0.25} \right)
\label{eq:Mach_coag_collection2}
\end{align}
below which clouds will coagulate. While the normalization of this equation needs to be calibrated against simulations (and will likely change), it provides testable scalings for the dependence of equation \ref{eq:Mach_coag} on physical parameters. We defer this to future work.
We can also use this to understand why there is a critical final overdensity $\chi_{\rm crit} \approx 300$ for recollapse and coagulation during `shattering' \citep{Gronke2020}. For an expanding cloud to achieve momentum contact with its surroundings and decelerate, it must sweep up of order its own mass: $\rho_{c} r_{\rm cl}^3 \approx \rho_{\rm h} d^{3}$, which gives $d \approx \chi^{1/3} r_{\rm cl}$. Assuming droplets are launched at a velocity $v \sim c_{\rm s,c}$, we have:
\begin{equation}
\frac{U_{\rm tot}}{K_{\rm tot}} \sim \frac{M_{\rm h}} {M_{\rm c}} \frac{v_{\rm mix}^{2}}{c_{\rm s,c}^{2}} \left(\frac{N r_d^2}{\chi^{2/3} r_{\rm cl}^2} \right)^{2} \propto \frac{N^{2/3}}{\chi^{4/3}}
\end{equation}
where $A_{\rm tot} \sim N r_{\rm d}^{2} \sim r_{\rm cl}^{2} (r_{\rm cl}/r_{\rm d})$, where $N \sim (r_{\rm cl}/r_{\rm d})^{3}$. The first two factors $M_{\rm h}/M_{\rm c}$ and $v_{\rm mix}^{2}/c_{\rm s,c}^{2}$ are order unity. The number of cloudlets $N$ is difficult to model, but it is clear that as overdensity $\chi$ increases, the ratio $U_{\rm tot}/K_{\rm tot}$ decreases, and eventually coagulation is not possible. Overdense gas is lauched out to larger distances $d$ before it is slowed down by the hot gas, and by that time, the covering fraction $f_{\rm A}$ drops sufficiently that coagulation is suppressed.
Of course, more careful study and detailed comparisons to simulations are required to transform these remarks into a quantitative theory, which we defer to future work.
\subsection{The competition of coagulation versus dispersion}
At first blush, the results of this paper might suggest that coagulation should be unimportant. The coagulation velocity $v_{\rm coag} \sim v_{\rm mix} \sim c_{\rm s,c}$ is small and diminishes rapidly with distance, $v_{\rm coag} \propto d^{-2}$. This corresponds to a small Mach number, even a relatively small distance from the cloud:
\begin{equation}
{\mathcal M} \sim \frac{v_{\rm mix}}{c_{\rm s,h}} \left( \frac{r_{\rm cl}}{d}\right)^{2} \sim 10^{-2} \left( \frac{v_{\rm mix}}{c_{\rm s,c}} \right) \left( \frac{\chi}{100} \right)^{-1/2} \left(\frac{d}{3 r_{\rm cl}} \right)^{-2}
\end{equation}
which would appear miniscule compared to other velocities in the system, so that coagulational inflow is a negligibly small perturbation. Yet, there are configurations, such as cloud crushing and cloud shattering, where coagulation is undeniably important. Indeed, a multi-phase mixing layer \citep{Kwak2010,Tan2020} is itself an example of coagulation -- despite the high velocity of shearing hot gas, $v_{\rm shear} \sim {\mathcal M} c_{\rm s,h} \gg v_{\rm mix} \sim c_{\rm s,c}$, cooling gas fragments in the mixing layer advect towards the cold gas layer.
The previous section (\S\ref{sec:coag_grav}) addressed the $v_{\rm coag} \propto d^{-2}$ fall-off. This only holds for a single cloud. If surface area is enhanced (e.g, by fragmentation), so that the area covering fraction $f_{\rm A}$ is large, then the fall-off with distance is supressed. Thus, for instance, $U_{\rm tot} \sim M_{\rm h} v_{\rm mix}^{2}$ when $f_{\rm A} \sim 1$ (Eq.~\eqref{eq:U_pot}); all the hot gas is moving with velocity $v_{\rm mix}$. This is similar to Obler's paradox: if every sightline in an infinite static universe ends on the surface of a star, then the surface brightness of the night sky would be that of a stellar surface, since the reduced solid angle (which increases the number of stars which tile the sky) and inverse square dimming behave in the same way. Similarly, if $f_{\rm A} \sim 1$, then $v_{\rm coag} \sim v_{\rm mix}$, regardless of distance. Alternatively, we can use the analogy between gravity and coagulation to use Gauss's law to find how $v_{\rm coag}$ diminishes with distance. For the cometary tail of a cloud in a wind or a filamentary cold gas structure, $F_{\rm coag} \propto v_{\rm coag} \propto d^{-1}$ (as for the gravitational force of a filament). For a semi-infinite slab of cold gas (as in a mixing layer), $F_{\rm coag} \propto v_{\rm coag} \sim v_{\rm mix}$ is independent of distance (as for the gravitational field above a mass sheet).
Still, that leaves the second question: even if $v_{\rm coag} \sim v_{\rm mix} \sim c_{\rm s,c}$, how can that compete against much larger turbulent velocities $\sigma_{\rm t} \sim \mathcal{M}_{\rm h} c_{\rm s,h}$? Indeed, it cannot in general\footnote{It is true that $\sigma \propto l^{1/3}$ in Kolmogorov turbulence, so that turbulence decreases at small scales, but $\sigma_{\rm l} < c_{\rm s,c}$ is only true for $l < (\chi^{-1/2} \mathcal{M}_{\rm h})^3 L$, where $L$ is the driving scale. For instance, for $\chi \sim 100, M_{\rm h} \sim 0.5$, then $l \lsim 0.01 L$. Such scales are at best only a few grid cells apart in simulations where coagulation is seen, and coagulating cloudlets are generally separated by larger distances.}.
However, it can in laminar bulk flows (where even if the bulk flow velocity is large, the relative velocity between cold gas fragments is small as they entrain in the hot wind), or in quiescent regions of a turbulent medium. For instance, as hot and cold gas mix, the `mass loading' of cold gas into the hot gas results in a new velocity dispersion $\sigma_{\rm mix}$, where $\langle \rho \rangle \sigma_{\rm mix}^2 \approx \rho_{\rm h} \sigma_{\rm t}^{2}$, and $\langle \rho \rangle \sim f_{\rm c} \rho_{\rm c} + (1-f_{\rm c}) \rho_{\rm h} \approx f_{\rm c} \rho_{\rm c}$. This gives $\sigma_{\rm mix} \sim \sigma_{\rm t}/(
\chi f_{\rm c})^{1/2} \sim \mathcal{M}_{\rm h} f_{\rm c}^{-1/2} c_{\rm s,c}$, so that $\sigma_{\rm mix} < c_{\rm s,c}$ if $f_{\rm c} > \mathcal{M}_{\rm h}^2$. Note that here $\sigma_{\rm mix}$ is the velocity dispersion of the multi-phase (hot and cold) gas mixture; it is {\it not} the velocity dispersion of mixed gas at some intermediate temperature. All situations where coagulation is observed to be important (e.g., coagulation onto the cometary tail of a cloud; shattering; turbulent mixing layers) are those where cold gas mass loading $f_{\rm c}$ is fairly large and the gas turbulent velocity {\it does} in fact obey $\sigma_{\rm mix} \lsim v_{\rm mix}$.
The same is not true if $\langle \rho \rangle \ll \rho_{\rm c}$, and thus $\sigma_{\rm mix} > v_{\rm mix}$. For instance, in the driven turbulence multi-phase setup of \citep{Gronke2022} (where $\sigma_{\rm mix} > v_{\rm mix}$; see fig 19 in that paper), coagulation indeed does not outcompete fragmentation by turbulence. Instead of coalescing into a large central cloud, a scale free power law mass distribution of clouds forms.
To summarize: coagulation is efficient, despite the small amplitude $v_{\rm coag} \sim v_{\rm mix} \sim c_{\rm s,c}$ and rapid fall-off $v_{\rm coag} \propto d^{-2}$, in regions where \textit{(i)} the extrinsic dispersion velocity is low, e.g., if the cold gas fraction $f_{\rm c}$ is high, and $\langle \rho \rangle \approx f_{\rm c} \rho_{\rm c}$ (since this mass loading reduces the turbulent velocity to $\sigma_{\rm mix} < c_{\rm s,c}$ if $f_{\rm c} > \mathcal{M}_{\rm h}^2$) and \textit{(ii)} the geometrical dimming can be overcome, for instance, through the geometry of the cold medium or if the cold gas covering fraction $f_{\rm A}$ is high (since the fall-off with distance in $v_{\rm coag}$ goes away as $f_{\rm A} \rightarrow 1$). \\
Next, we discuss some of these cases where coagulation is important in more detail:
\begin{itemize}
\item{{\it Mixing layers, clouds and streams.} Plane parallel Kelvin Helmholtz mixing layers have $f_A \sim 1$, and thus $v_{\rm coag} \sim v_{\rm mix}$ does not decline with distance. Also, regions where cold gas mass loading is substantial have turbulent velocities $u^{\prime} \sim v_{\rm shear}/\sqrt{\chi} \sim c_{\rm c,s}$, thus comparable to $v_{\rm mix}$, as one might expect from the above arguments. Clouds in a hot wind develop an extended cometary tail with a cylindrical structure \citep[e.g.][]{Gronke2019}. Thus, entrained clouds, or cold gas streams \citep[e.g.,][]{mandelkerInstabilitySupersonicCold2020,Bustard2022} correspond to our 2D (Fig \ref{fig:coag2d_veldrop_multiplot}), rather than our 3D (Fig \ref{fig:coag3d_multiplot}) simulations, with $v_{\rm in} \propto d^{-1}$ instead of $v_{\rm in} \propto d^{-2}$. Similar to Fig \ref{fig:coag2d_veldrop_multiplot}, the droplet returns on a timescale $\sim \tilde{\alpha} t_{\rm sc,floor}$ (where $\tilde{\alpha} \sim 5-10$), during which time it travels a distance $\sim v_{\rm w} \tilde{\alpha} t_{\rm sc,floor} \sim \tilde{\alpha} \chi^{1/2} \mathcal{M} r_{\rm cl}$.}
\item {\textit{Expulsion from a central origin.}
Droplets dispersed from a central origin can eventually coagulate back together. In \citet{Gronke2020}, we argued that the competition between dispersion and coagulation sets the threshold of `shattering' which we found to be $\chi_{\rm final}\gtrsim 300 (r_\mathrm{cl} / 10^4 \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi)^{1/6}$ (for $r_\mathrm{cl}\gg \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$ and $\delta P / P \gtrsim 1.5$). In our simulations, clouds straddling this boundary had vastly different outcomes.
In principle, since drag forces cause kinetic energy to decay, coagulation could potentially once again dominate at late times\footnote{Similarly, pulsations could potentially damp in an otherwise static medium. However, we showed that they continue on timescales much longer than $t_{\rm sc}$, and in practice turbulence will always perturb the cloud.}, though in practice turbulence will further separate the fragments and shape the mass distribution.
While the exact mechanism of fragmentation and dispersion in the `shattering' scenario needs revisiting, in broad terms the role of coagulation here is clear.}
\item {\textit{Coagulation in extrinsic turbulence.} In \citet{Gronke2022}, we studied turbulent, multiphase dynamics in more detail and found that the droplets follow a power law mass distribution $\mathrm{d} n/\mathrm{d} m \propto m^{-2}$ (which is also found in larger scale simulations of the intracluster medium; cf. \citealp{LiMODELINGACTIVE2014a}). This simulations were all run at Mach numbers above the critical Mach number (Eq.~\eqref{eq:Mach_coag_numerical}) where we might expect coagulation to play a role. For low Mach numbers, we expect deviations from this power law, which we will analyze in future work.}
\end{itemize}
\subsection{Caveats}
\label{sec:disc_caveats}
Our study does not address a range of topics which we hope to revisit in future work.
\begin{itemize}
\item Magnetic fields. Most plasmas are magnetized, which affects the mixing and thus the mass transfer process \citep{Ryu1995,Ji2018}. Furthermore, $B$-fields imply a non-thermal pressure support which can become large in the cold medium even with initially large plasma $\beta$ due to magnetic compression \citep{Sharma2010,Gronke2019}.
\item Cosmic rays. Similar to magnetic fields, cosmic rays are present in astrophysical plasmas and provide non-thermal pressure support which changes the cooling rates of the gas \citep[e.g.,][]{2016MNRAS.456..582S,Butsky2020}.
\item Simplified setup. The goal of this study was to develop a simple model for coagulation. Thus, we focused on very simplified initial conditions. In future work, we want to apply this model in more realistic scenarios such as multiphase galactic winds.
\item Resolution and dynamic range. As all numerical studies, we suffer from finite resolution and limited dynamic range. We tried to support our findings with resolution studies throughout as well as analytic models matching our numerical findings.
\end{itemize}
\subsection{Comparison to the literature}
\label{sec:comparison_literature}
The oscillations for cooling clouds were previously discussed in the literature, in particular in \citet{Waters2019b} and \citet{Das2020} in 1D and in \citet{Gronke2020} in 3D simulations. Notably, \citet{Waters2019b} and \citet{Das2020} carry out in-depth analyses of linear, non-isobaric thermal instability and
found pulsations for `large clouds'.
\citet{Waters2019b} show that the pulsations decay on a long ($\gtrsim 10 t_{\rm sc}$) timescale, and that the cloud settle eventually ($\gtrsim 50 t_{\rm sc}$) back to the equilibrium state. However, note that gas mixing -- which can fuel cooling and further pulsations -- is not captured in 1D.
The further mass growth associated with these pulsations was not studied in these simplified setups. In \citet{Gronke2018}, similar pulsations were seen in the entrained state of a `windtunnel' simulations. There, the pulsations were, thus, not seeded by the initial cooling but by the shock hitting the cloud. The mass growth rates of the cold gas agree well including the characteristic $\propto t_{\rm cool}^{-1/4}$ scaling. The rates as well as the scalings have been confirmed in turbulent mixing layer simulations
\citep{Fielding2020,Tan2020}\footnote{Note that, more recently, these scalings were extended into the high-$\mathcal{M}$ regime (\citealp{Yang2022}; see also \citealp{Bustard2022}).}, although there gas mixing is driven by shear, rather than pulsations from overstable sound waves.
Similarly, coagulation was observed in previous studies. \citet{ZelDovich1969} computed the cooling rate at the (laminar) boundary of a two-phase medium, and noted that this leads to coagulation of the cold medium. However, they also point out that this velocity of the front is minuscule. Building upon this work, \citet{Elphick1991} constructed a 1D framework to study the coagulation of an ensemble of cold gas fronts -- which they extend to include bulk fluid motions in \citet{Elphick1992}. This work was later extended to more dimensions \citep{Shaviv1994}.
More recently, \citet{Koyama2004} and \citet{Waters2019} also discuss cooling induced coagulation in one- and two-dimensions, respectively. They point out that not only do the fronts move due to growth of the cold gas, but that motion is induced by the cooling (see, e.g., Fig. 2 in \citealp{Koyama2004}). In particular, \citet{Waters2019} analyze the coagulation behavior and note the coalescence timescales.
Our work differs from these previous studies in several aspects. Firstly, we carry out two- and three-dimensional simulations and build an analytic model reproducing our hydrodynamical results. More importantly, however, we focus on the production of intermediate temperature gas by turbulent mixing--which cannot be captured in 1D--as opposed to laminar heat transport due to thermal conduction alone. For this reason, our coagulation velocities are much greater (in the same way as the turbulent diffusion dominates over laminar heat transport; cf. \citealp{Tan2020}). For instance, for their fiducial 2D run in which they placed a $r \sim 8\ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$ and a $r \sim 23 \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$ clouds at a distance of $d\sim 33 \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi$, \citet{Waters2019} found a coagulation velocity of order $v\sim d/t_{\rm coag}\sim 0.02 c_{\rm s}$ (cf. their table 4)
which is much less than the $v_{\rm coag}\sim v_{\rm mix} r/d \sim c_{\rm s}$ we find\footnote{Note that the small cloud sizes they employed would lead to reduced pulsations and thus an actual slower coagulation velocity; see Fig.~\ref{fig:mdot_vs_X}.}.
Coagulation can be observed in many larger scale simulations. As already mentioned, many `cloud crushing' simulations with radiative cooling display signs of cold gas coagulation \citep[e.g.,][]{Schneider2016,Gronke2019,Abruzzo2021}. Also in simulations studying thermal instabilities, coagulation has been observed \citep{Sharma2010,Butsky2020}.
In even larger scale simulations, e.g., of the multiphase dynamics in the CGM \citep[e.g.][]{Hafen2019,Hummels2018}, the ICM \citep{LiMODELINGACTIVE2014}, or in ram pressure stripped tails of galaxies \citep{Tonnesen2010,Farber2022} coagulation should take place and play a role -- however, it is unclear whether current resolutions are sufficient to capture this effect.
\section{Conclusion}
\label{sec:conclusion}
We investigated cooling driven coagulation process of cold gas in a multi-phase
medium, a phenomenon which has been seen in diverse simulations. For instance, it is observed in the `focusing' of cold gas droplets onto the cometary tail of a cold cloud in a hot wind. To gain understanding, we first investigated cooling induced coagulation in a static medium.
Our findings can be summarized as follows:
\begin{enumerate}
\item Perturbed cold gas blobs of size $> \ifmmode{\ell_{\mathrm{shatter}}}\else $\ell_{\mathrm{shatter}}$\fi \equiv \mathrm{min}(c_{\rm s}t_{\rm cool})$ develop overstable soundwaves which lead to continuous pulsations and mass growth of the cold gas.
\item This process leads to a flow of hot gas with velocity $v_{\rm coag} \sim v_{\rm mix} (r_{\rm cl}/d)^2$ in 3D, where $v_{\rm mix}$ (given by Eq.~\eqref{eq:vmix}) is of order the cold gas sound speed. Cold droplets can get rapidly entrained in this hot gas flow (due to their own growth and the associated momentum transfer), eventually merging with other cold gas structures.
\end{enumerate}
We furthermore developed an analytic model describing the mass growth and coagulation process which fits our numerical results reasonably well. Although $v_{\rm coag} \sim v_{\rm mix} (r_{\rm cl}/d)^2$ may appear small, note that: (i) turbulent gas velocities can be small, e.g., if the cold gas mass fraction is high. Also, in bulk flows (as in a wind), the relative velocities between entrained gas fragments becomes small. (ii) The geometric $v_{\rm coag} \propto d^{-2}$ dimming can be much weaker in different geometries (e.g., $v_{\rm coag} \propto d^{-1}$ for a cometary tail), or if the cold gas covering fraction $f_A$ is high (where $v_{\rm coag} \approx v_{\rm mix} \approx$const).
Our finding supports the idea that cooling driven coagulation of adjacent cold gas is possible and we establish a criterion defining the regimes where coagulation or dispersion in transonic turbulence dominates.
Due to the similar $F \propto d^{-2}$ force, we draw analogies to gravity. The monopole term for coagulation is surface area, rather than mass. Thus, fragmentation, which increases area at fixed mass, also increases coagulation.
We have neglected magnetic fields, cosmic rays, and the inclusion of a more realistic (turbulent and stratified) background. We plan to address these issues in future work.
\section*{Acknowledgments}
This research made use of \texttt{yt} \citep{2011ApJS..192....9T}, \texttt{matplotlib} \citep{Hunter:2007}, \texttt{numpy} \citep{van2011numpy}, and \texttt{scipy} \citep{scipy}.
We acknowledge support from NASA grant NNX17AK58G, HST grant HST-AR-15039.003-A, and XSEDE grant TG-AST180036 the Texas Advanced Computing Center (TACC) of the University of Texas at Austin.
MG thanks the Max Planck Society for support through the Max Planck Research Group.
This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.
\bibliographystyle{mnras}
|
1,116,691,499,346 | arxiv | \section{Introduction}
\label{introduction}
Synchronized behavior arising among the constituents of an ensemble is common in
nature. Examples include the synchronized flashing of fireflies, the
blossoming of flowers, cardiac cells giving rise to the pacemaker role of the
sinoatrial node of the heart and the electrical pulses of neurons. This global
behavior can originate from a common response to an external stimulus or might
appear in autonomous, non-forced, systems. The theoretical basis for the
understanding of synchronization in non-forced systems was put forward by
Winfree
(\cite{Winfree67}) who showed that the interaction --i.e. coupling-- between
the constituents is an essential ingredient for the existence of a synchronized
output. The seminal work on coupled oscillators by Kuramoto
(\cite{Kuramoto1975})
offered a model case whose solution confirms the basic hypothesis of Winfree:
while interaction helps towards the achievement of a common behavior, a perfect
order can be achieved only in the absence of diversity --heterogeneity-- among
the components of an ensemble. In the Kuramoto model, diversity manifests when
the oscillators have different natural frequencies, those they display when
uncoupled from each other. While this result holds for systems that can be
described by coupled oscillators, recent results indicate that in other cases
diversity among the constituents might actually have a positive role in the
setting of a resonant behavior with an external signal. This was first
demonstrated in
(\cite{Tessone2006}) and has been since extended to many other systems
(\cite{Gosak2009,Ullner2009,Zanette2009,Tessone2009,Postnova09,Chen09,Wu09,
Zanette2009,
Ullner2009,Perc08,Acebron07}). In the case of non-forced excitable systems, a
unifying treatment of the role of noise and diversity has been developed in
\cite{TSTP:2007}.
Many biological systems, including neurons, display excitability as a response
to an
external perturbation. Excitability is
characterized by the existence of a threshold, a largely independent
response to a suprathreshold input and a refractory time. It is well established
that the dynamical features of neurons can be described by excitable models: when a neuron is
perturbed by a single impulse, the neuron can generate a single spike, and when perturbed by a
continuum signal, a train of spikes with a characteristic firing frequency can appear instead.
Although the creation and propagation of electrical signals has been thoroughly studied by physiologists for at least
a century, the most important landmark in this field is due to Hodgkin and Huxley
(\cite{HH1952d}), who developed the first quantitative model to describe the
evolution of the membrane potential in the squid giant axon. Because of the
central importance of cellular electrical activity in biological systems and
because this model forms the basis for the study of excitable systems, it
remains to this date an important model for analysis. Subsequently a simplified
version of this model, known as the FitzHugh-Nagumo (FHN) model
(\cite{FHN1961}), that captures many of the qualitative features of the
Hodgkin-Huxley model was developed. The FHN model has two variables, one fast
and one slow. The fast variable represents the evolution of the membrane potential and it is
known as the excitation variable. The slow variable accounts for the $K^{+}$
ionic current and is known as the recovery variable. One virtue of this model is that it can be studied
using phase-plane methods, because it is a two variable system. For instance,
the fast variable has a cubic nullcline while the slow variable has a linear
nullcline and the study of both nullclines and their intersections allow to
determine the different dynamical regimes of the model. The FHN model
can then be extended by coupling the individual elements, and provides an
interesting model, for example, of coupled neuron populations.
In the initial studies of the dynamics of the FHN and similar models,
the individual units (e.g. neurons) were treated as identical. However, it is
evident that real populations of neurons
display a large degree of variability, both in morphology and dynamical
activity; that is, there is a diversity in the population of these biological units. Then, it is natural
to ask what role diversity plays in the global dynamical behavior of these systems and a lot
of activity has been developed along these lines in the recent years. In general,
it has been found, as noted above, that diversity can in fact be an important
parameter in controlling the dynamics. In particular, it has been shown
that both excitable and bistable systems can improve their response to an
external stimulus if there is an adequate degree of diversity in the
constituent units (\cite{Tessone2006}).
In this paper we continue our study of the effect of diversity, investigating in
some depth its role in the FHN model of excitable systems. We consider in detail a system of many neurons
coupled either chemically or electrically. We show that in both cases a right
amount of diversity can indeed enhance the response to a periodic external
stimulus and we discuss in detail the difference between the two types of
coupling. The outline of the paper is as follows:
In section \ref{sec:model} we present the FHN model and the coupling schemes
considered. Next, in section \ref{sec:results} we present the main results we
have obtained from numerical simulations. Afterwards, in section \ref{sec:OPE} we develop an approximate theoretical treatment based on an order
parameter expansion which allows us to obtain a quantitative description of the behavior of the model and compare
its predictions with the numerical results. Finally, in section \ref{sec:conclusions} we summarize the conclusions.
\section{FitzHugh-Nagumo model}
\label{sec:model}
\subsection{Dynamical equations}
Let us consider a system of excitable neurons described by the FHN model.
The dynamical equations describing the activity of a single neuron are:
\begin{eqnarray}
\epsilon \frac{d x}{dt}&=&x(1-x)(x-b)-y+d,
\label{FHNvoltage-eq}\\
\frac{d y}{dt}&=&x-cy+a,
\label{FHNrecover-eq}
\end{eqnarray}
where $x(t)$ and $y(t)$ represent, respectively, the fast membrane potential
and the slow potassium gating variable
of a neuron. We assume that the time scales of these variables are well
separated by the small parameter $\epsilon=0.01$. Other parameters are fixed to
$b=0.5$, $c=4.6$ and $d=0.1$ (\cite{Glatt08}), while the value of $a$ will
fluctuate from one neuron to the other, so reflecting the intrinsic diversity in
the neuronal ensemble.
Let us concentrate first on the dynamics of a single neuron as described by Eqs.
(\ref{FHNvoltage-eq})-(\ref{FHNrecover-eq}). This dynamics has a
strong dependence on the parameter $a$. Three different operating
regimes are identify: for $a\lesssim-0.09$ the system has a stable focus in the
right branch of the cubic nullcline leading the system to an excitable regime;
for $-0.09\lesssim a\lesssim0.01$ a limit cycle around an unstable focus
appears (oscillatory regime) and for $0.01\lesssim a$ a stable focus appears
again, now at the left side of the cubic nullcline (excitable regime). Figure
\ref{f0FHN} (a) shows the nullclines $f(x)=x(1-x)(x-b)+d$ and $g(x,a)=(x+a)/c$
of the system in the three operating regimes described above, for $a=-0.1$, $0.0$ and $0.06$, respectively. In
the excitable regime, spikes (also known as pulses), can appear as a result of
an external perturbation of large enough amplitude. A convenient definition is
that a spike appears when the membrane potential
of the neuron exceeds a certain threshold value, e.g. $x\geq 0.5$. In the
oscillatory regime, spikes appear spontaneously with an intrinsic firing
frequency $\nu$ which, as shown in Figure \ref{f0FHN} (b), does not depend much
on the value of $a$.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.495\linewidth,clip]{Figure1a.eps}
\includegraphics[width=0.495\linewidth,clip]{Figure1b.eps}
\caption{(a) Nullclines of the FHN system for three different values of the
parameter $a$.
$f(x)$ (solid line) and $g(x,a)$ for $a=-0.1$ (dotted line); $a=0.0$
(dash-dotted line) and
$a=0.06$ (dashed line). (b)
Dependence of the firing frequency $\nu$ on $a$.}
\label{f0FHN}
\end{figure}\\
To illustrate the dynamical behavior of $x(t)$ and $y(t) $, we show in Figure
\ref{f2} the phase-portrait for three different values of $a=-0.1$, $0.0$ and $0.06$
corresponding to the three nullclines represented in Figure \ref{f0FHN} (a).
\begin{figure}[htb]
\centering
\includegraphics[width=0.99\linewidth,height=4cm,clip]{Figure2.eps}
\caption{Phase-space portrait of the FHN system for different values of $a$.
Gray line represents
the evolution of $\{x,y\}$. $f(x)$ (black line) and $g(x,a)$ for (a)
$a=-0.1$ dotted line (excitable regime); (b) $a=0.0$ dash-dotted line
(oscillatory regime) and
(c) $a=0.06$ dash line (excitable regime).}
\label{f2}
\end{figure}
\subsection{Coupling scenarios: electrical versus chemical interaction}
\label{diffcoup}
We consider now that we have an ensemble of $N$ coupled neurons. Each one is
described by dynamical variables $x_i(t),\, y_i(t)$, $i=1,\dots,N$, obeying the
FitzHugh-Nagumo equations. The neurons are not isolated from each other, but
interact mutually. The full set of equations is now:
\begin{eqnarray}
\epsilon \frac{d x_i}{dt}&=&x_i(1-x_i)(x_i-b)-y_i+d+I^{syn}_i(t),
\label{FHNvoltage-eqi1}\\
\frac{d y_i}{dt}&=&x_i-cy_i+a_i,
\label{FHNrecover-eqi}
\end{eqnarray}
where $I^{syn}_i(t)$ is the coupling term accounting for all the interactions of
neuron $i$ with other neurons.
To take into account the natural diversity of the units, we assume that
the parameter $a_i$, that controls the degree of excitability of the neuron,
varies from neuron to neuron. In particular we assign to $a_i$ random values
following a Gaussian distribution with mean
$\left\langle a_i \right\rangle= a$ and correlations
$\left\langle (a_i- a)(a_j- a) \right\rangle=\delta_{ij}\sigma^2$. We use
$\sigma$ as a measure of the heterogeneity of the system and, in the following,
we use the value of $\sigma$ as an indicator of the degree of diversity. If
$\sigma=0$, all neurons have exactly the same set of parameters, while large
values of $\sigma$ indicate a large dispersion in the dynamical properties of
individual neurons.
The most common way of communication between neurons is via chemical synapses,
where the transmission is carried by an agent called neurotransmitter. In these
synapses, neurons are separated by a synaptic cleft and the neurotransmitter has
to diffuse to reach the post-synaptic receptors. In the chemical coupling case
the synaptic current is modeled as:
\begin{eqnarray}
I^{syn}_i(t)=\frac{K}{N_c}\sum_{j=1}^{N_c} g_{ij}r_j(t)\left(E_i^s-x_i\right).
\label{eqnIsyn}
\end{eqnarray}
In this configuration, we consider that neuron $i$ is connected to $N_c$ neurons
randomly chosen from the set of $N-1$ available neurons. Once a connection
is established between neuron \textit{i} and \textit{j}, we assume that the reciprocal
connection is also created. Then, the connection fraction of each neuron is defined as $f=N_c/(N-1)$.
In Eq. (\ref{eqnIsyn}) K determines the coupling strength and $g_{ij}$ represents the maximum conductance
in the synapse between the neurons \textit{i} and \textit{j}. For simplicity, we
limit our study to the homogeneous coupling configuration, where $g_{ij}=1$ if
neurons $i$ and $j$ are connected and $g_{ij}=0$ otherwise.
The character of the synapse is determined by the synaptic reversal potential of the receptor neuron,
$E_i^s$. An excitatory (resp. inhibitory) synapse is characterized by a value of
$E_i^s$ greater (resp. smaller) than the
membrane resting potential. We consider $E_i^s=0.7$ for the
excitatory synapses and $E_i^s=-2.0$ for the inhibitory ones. We also define the fraction
of excitatory neurons (those that project excitatory synapses) in the system as $f_e=N_e/N$ being $N_e$
the number of excitatory neurons.
Finally, $r_j(t)$ is a time dependent function
representing the fraction of bound receptors and it is given by:
\begin{equation}
r_j(t)=\cases{
1-e^{-\alpha t} & {\rm for}\ $t\leq t_{on}$,\cr
\left(1-e^{-\alpha t_{on}}\right)e^{-\beta \left( t-t_{on}\right)} & {\rm for}\, $t > t_{on}$
}
\end{equation}
where $\alpha=2.5$ and $\beta=3.5$ are the rise and decay time constants,
respectively.
Here $t_{on}=0.1$ represents the time the synaptic connection remains active and $t$
is the time from the spike generation in the presynaptic neuron.
There is another type of synapse where the membranes of the neurons are in close contact and thus the
transmission of the signal is achieved directly (electrical synapses). In this case of
electrically-mediated interactions, also known as diffusive coupling, the total synaptic current is
proportional to the sum of the membrane potential difference between a neuron
and its neighbors, and it is given by:
\begin{eqnarray}
I^{syn}_i(t)=\frac{K}{N_c}\sum_{j=1}^{N_c} \left( x_j-x_i\right).
\end{eqnarray}%
The last ingredient of our model is the presence of an external forcing that
acts upon all neurons simultaneously. Although our results are very general, for
the sake of concreteness we use a periodic forcing of amplitude $A$ and period
$T$. More precisely, the dynamical equations under the presence of this forcing
are modified as:
\begin{eqnarray}
\epsilon \frac{d x_i}{dt}&=&x_i(1-x_i)(x_i-b)-y_i+d+I^{syn}_i(t),
\label{FHNvoltage-eqi}\\
\frac{d y_i}{dt}&=&x_i-cy_i+a_i+A\sin\left(\frac{2\pi}{T}t\right),
\label{FHNrecover-eqi2}
\end{eqnarray}
which is the basis of our subsequent analysis.
\section{Results}
\label{sec:results}
We are interested in analysing the response of the global system to the external
forcing. We will show that its effect can be enhanced under the presence of the
right amount of diversity in the set of parameters $a_i$, i.e. a conveniently
defined response will reach its maximum amplitude at an intermediate value of the root
mean square value $\sigma$.
In order to quantify the global response of the system with respect to the diversity, we use
the spectral amplification factor defined as
\begin{eqnarray}
\eta &=& \frac{4}{A^{2}} \left| \left\langle e^{-i\frac{2\pi}{T}t} X(t)
\right\rangle\right|^2. \label{saf}
\end{eqnarray}
where $X(t)=\frac{1}{N} \sum_{i=1}^N x_i(t)$ is the global, average collective
variable of the system and $\left\langle \cdot \right\rangle$ denotes a time average. We will analyze
separately the cases of electrical and chemical coupling.
\subsubsection{Electrical coupling}
In this subsection we concentrate on the situation in which the neurons are
electrically (diffusively) coupled in a random network, where a neuron \textit{i} is connected
randomly with $N_c=f(N-1)$ other neurons. The mean value of the Gaussian
distribution of the parameters $a_i$ is fixed to
$a =0.06$ and the coupling strength to $K=0.6$. Figure \ref{f1FHN} shows the
spectral amplification factor $\eta$ as a function of the diversity $\sigma$ for
fixed values of the amplitude $A=0.05$ and two different values of the period $T$ of the external
forcing, for an increasing connection fraction $f$. It can be seen from
Figure \ref{f1FHN}
that intermediate values of $\sigma$ give a maximum response in the spectral
amplification factor.
Moreover, the maximum value shifts slightly to smaller values of $\sigma$ as the
fraction of
connected neurons $f$ increases. We have also observed that a period $T$ of the
external forcing close
to the inverse of the intrinsic firing frequency of the neurons
($\nu\approx0.9$, according to Figure \ref{f0FHN}b) yields the largest response.
\begin{figure}[h]
\centering
\includegraphics[width=0.99\linewidth,clip]{Figure3.eps}
\caption{Spectral amplification factor $\eta$ as a function of $\sigma$ for an
increasing fraction of connected neurons $f$ for two different periods of the
external modulation.
(a) $T=1.6$ and (b) $T=1.11$. Other parameters: $a =0.06$, $K=0.6$ and
$A=0.05$.}
\label{f1FHN}
\end{figure}\\
In order to further illustrate the response of the system to the external
sinusoidal
modulation, Figure \ref{f1bFHN} shows the raster plot of the ensemble (lower
panels) and the
time traces of ten randomly chosen neurons (upper panels) for different values
of the diversity
parameter. It can be seen in both the top and bottom panels of this figure that
an intermediate level
of diversity gives a more regular behavior than either smaller or larger values
of $\sigma$. This fact is more
evident in the time traces where the amplitude of the oscillation varies
randomly for large $\sigma$ values.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.95\linewidth,clip]{Figure4.eps}
\caption{Time traces of ten randomly chosen neurons and raster plot (every time a neuron spikes a dot is drawn) of the
fully connected, $f=1$, ensemble in the case of electrical coupling for three
different values of the diversity parameter: (a)
$\sigma=0.1$, (b) $\sigma=0.4$ and (c) $\sigma=0.9$. Other parameters: $a
=0.06$, $K=0.6$, $A=0.05$ and $T=1.6$.}
\label{f1bFHN}
\end{figure}
\subsubsection{Chemical coupling}
We consider in this subsection the situation in which the units are chemically
coupled. Figure
\ref{f2FHN} shows the spectral amplification factor $\eta$ as a function of the
diversity $\sigma$
for fixed values of the amplitude $A=0.05$ and two different periods of the
external
modulation when the fraction of randomly connected neurons $f$ increases.
\begin{figure}[h]
\centering
\includegraphics[width=0.99\linewidth,clip]{Figure5.eps}
\caption{Spectral amplification factor $\eta$ as a function of $\sigma$ for an
increasing fraction of connected neurons $f$. The fraction of excitatory neurons
was fixed to $f_e=0.8$. Two different periods of the external modulation have been
considered: (a) $T=1.6$ and (b) $T=1.11$.}
\label{f2FHN}
\end{figure}
The coupling strength is fixed to $K=1.5$. The fraction of excitatory neurons in the system
is set to $f_e=0.8$.
It can be seen from Figure \ref{f2FHN} that the spectral
amplification factor $\eta$ increases as $f$ increases, reaching the maximum
response at $f \simeq 0.05$. Interestingly, beyond
this value $\eta$ does not change significantly, indicating that the response of
the system does not improve when the percentage of connected
neurons is larger than $5\%$. Or, put in another way, with a $5\%$
connectivity, the system already behaves as being fully
connected as far as the response to the external forcing is concerned. It is also
worth noting that the maximum response is always
at the same value of $\sigma$, independent of $f$.
The effect of changing the ratio of
excitatory/inhibitory synapses in our system is shown in Figure \ref{f3FHN} in the globally coupled case $f=1$. The
spectral amplification factor increases as the fraction of excitatory
connections $f_e$
increases, while the position of the maximum shifts slightly to larger values of
$\sigma \simeq 0.05$.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\linewidth,clip]{Figure6.eps}
\caption{Spectral amplification factor $\eta$ as a function of $\sigma$ for an
increasing fraction of excitatory synapses $f_e$ in the case that neurons are globally coupled, $f=1$. Other parameters: $a=0.06$,
$K=1.5$, $A=0.05$ and $T=1.6$.}
\label{f3FHN}
\end{figure}
Comparing the results from both electrical and chemical coupling schemes, it can
be seen that the
electrical coupling gives a larger value of $\eta$ but requires, at the same
time, a larger diversity.
The electrical coupling also exhibits a
larger range of diversity values for which the system has an optimal response
compared with the chemical
coupling. In contrast, the optimal response in the
chemical coupling scheme occurs for small values of the diversity and does not
significantly change in amplitude
and width when the percentage of connected neurons $f$ is increased above $5\%$.
\section{Order Parameter Expansion}
\label{sec:OPE}
It is possible to perform an approximate analysis of the effect of the diversity
in the case of diffusive (electrical) coupling. The analysis allows to gain
insight into the amplification mechanism by showing how the effective nullclines
of the global variable $X(t)$ are modified when varying $\sigma$. The
theoretical analysis is based upon a modification of the so-called order
parameter expansion developed by Monte and D\textquoteright Ovidio
(\cite{Monte03a,Monte05}) along the lines of \cite{KT:2010}. The approximation
begins by expanding the
dynamical variables around their average values $X(t)=\frac{1}{N}\sum_i x_i$
and
$Y(t)=\frac{1}{N}\sum_i y_i$ as $x_j(t)=X(t)+\delta^x_j(t)$,
$y_j(t)=Y(t)+\delta^y_j(t)$ and the diversity
parameter around its mean value $a_j=a+\delta^a_j$. The validity of this
expansion relies on
the existence of a coherent behavior by which the individual units $x_j$ deviate
in a small amount $\delta^x_j$ from the global behavior characterized by the
global average variable $X(t)$. It also assumes that the deviations $\delta^a_j$
are small. We expand equations (\ref{FHNvoltage-eqi})
for $\frac{dx_i}{dt}$ and (\ref{FHNrecover-eqi2}) for $\frac{dy_i}{dt}$
up to second order in $\delta^x_i$, $\delta^y_i$ and $\delta^a_i$; the resulting
equations are:
\begin{eqnarray}
\epsilon\frac{dx_i}{dt}&=&f\left(X,Y\right)+f_x\left(X,Y\right)\delta^x_i+
f_y\left(X,Y\right)\delta^y_i+\frac{1}{2}f_{xx}\left(X,Y\right)
\left(\delta^x_i\right)^2 , \label{exp1-eq}\\
\frac{dy_i}{dt}&=&g\left(X,Y,a\right)+g_x\left(X,Y,a\right)\delta^x_i+
g_a\left(X,Y,a\right)\delta^a_i , \label{exp2-eq}
\end{eqnarray}
where
\begin{equation}
\begin{array}{rcl}
f(x,y)&=& x(1-x)(x-b)-y+d-Kx , \cr
g(x,y,a)&=&x-cy+a+A\sin\left(\frac{2\pi}{T} t\right),
\end{array}
\end{equation}
and we used the notation $f_x$ to indicate the derivative of $f$ with respect to
$x$ and so forth.
Note that Eq. (\ref{exp2-eq}) is exact since it is linear in all the variables.
If we average Eq. (\ref{exp1-eq})-(\ref{exp2-eq}) using $\langle \cdot
\rangle=\frac{1}{N}\sum_i\cdot$ we obtain:
\begin{eqnarray}
\epsilon\frac{dX}{dt}&=&f\left(X,Y\right)+
\frac{1}{2}f_{xx}\left(X,Y\right)\Omega^x ,\label{exp3-eq}\\
\frac{dY}{dt}&=& g\left(X,Y,a\right) , \label{exp4-eq}
\end{eqnarray}
where we have used $\langle \delta^x_j \rangle=\langle
\delta^y_j \rangle=\langle \delta^a_j \rangle=0$ and defined the
second moment $\Omega^x=\langle(\delta^x_j)^2 \rangle$. We also defined
$\Omega^y=\langle(\delta^y_j)^2 \rangle$,
$\sigma^2=\langle(\delta^a_j)^2 \rangle$, and the shape factors
$\Sigma^{xy}=\langle\delta^x_j\delta^y_j\rangle$,
$\Sigma^{xa}=\langle\delta^x_j\delta^a_j\rangle$ and
$\Sigma^{ya}=\langle\delta^y_j\delta^a_j\rangle$.
The evolution equations for the second moments are found from
the first-order expansion $\dot{\delta}^x_j=\dot{x}_j-\dot{X}$, so
that $\dot{\Omega}^x=2\langle\delta^x_j\dot{\delta}^x_j\rangle$
and
$\dot{\Sigma}^{xy}=\langle\dot{\delta}^x_j\delta^y_j+\delta^x_j\dot{
\delta}^y_j\rangle$, were the dot stands for time derivative.
\begin{eqnarray}
\epsilon\dot{\delta}^x_i&=&f_x\delta^x_i+f_y\delta^y_i+\frac{1}{2}f_{xx}
\left[\left(\delta^x_i\right)^2-
\Omega^x\right] ,\label{exp5-eq} \\
\dot{\delta}^y_i&=& g_x\delta^x_i+g_y\delta^y_i+g_a\delta^a_i ,
\label{exp6-eq}\\
\dot{\Omega}^x&=&\frac{2}{\epsilon}\left[f_x\Omega^x+
f_y\Sigma^{xy}\right] ,\label{exp7-eq}\\
\dot{\Omega}^y&=&2\left[g_x\Sigma^{xy}+g_y\Omega^{y}+g_a\Sigma^{ya}\right] ,
\label{exp8-eq}\\
\dot{\Sigma}^{xy}&=&\frac{1}{\epsilon}\left[f_x\Sigma^{xy}+
f_y\Omega^y\right]+g_x\Omega^x+g_y\Sigma^{xy}+g_a\Sigma^{xa} ,\label{exp9-eq}\\
\dot{\Sigma}^{xa}&=&\frac{1}{\epsilon}\left[f_x\Sigma^{xa}+
f_y\Sigma^{ya}\right] ,\label{exp10-eq}\\
\dot{\Sigma}^{ya}&=&g_x\Sigma^{xa}+g_y\Sigma^{ya}+g_a\sigma^2 .\label{exp11-eq}
\end{eqnarray}
The system of Eq. (\ref{exp3-eq})-(\ref{exp4-eq}) together with Eq.
(\ref{exp7-eq})-(\ref{exp11-eq}) forms a closed set of differential equations
for the average collective variables $X(t)$ and $Y(t)$:
\begin{eqnarray}
\epsilon\dot{X}&=&-X^3+(1+b)X^2-(b+3\Omega^x)X+(1+b)\Omega^x+d-Y ,
\label{exp12-eq}\\
\dot{Y}&=& X-cY+a+A\sin\left(\frac{2\pi}{T} t\right) ,\label{exp13-eq}\\
\epsilon\dot{\Omega}^x &=& 2(-3X^2+2(1+b)X-b-K)\Omega^x-2\Sigma^{xy} ,
\label{exp14-eq}\\
\dot{\Omega}^y&=&2\left[\Sigma^{xy}-c\Omega^y+\Sigma^{ya}\right] ,
\label{exp15-eq}\\
\dot{\Sigma}^{xy}&=&\frac{1}{\epsilon}\left[(-3X^2+2(1+b)X-b-K)\Sigma^{xy}
-\Omega^y\right]{} \nonumber \\ & & {}
+\Omega^x-c\Sigma^{xy}+\Sigma^{xa} ,\label{exp16-eq}\\
\epsilon\dot{\Sigma}^{xa}&=&(-3X^2+2(1+b)X-b-K)\Sigma^{xa}
-\Sigma^{ya} ,\label{exp17-eq}\\
\dot{\Sigma}^{ya}&=&\Sigma^{xa}-c\Sigma^{ya}+\sigma^2 .\label{exp18-eq}
\end{eqnarray}
Numerical integration of this system allows us to obtain $X(t)$, from which
we can compute the spectral amplification factor $\eta$. The value of $\eta$
obtained
from the expansion is later compared with that obtained from the numerical
integration of
the Eqs. (\ref{FHNvoltage-eqi})-(\ref{FHNrecover-eqi}) (see Figure \ref{f1theo}
below).
We can obtain another set of closed equations for $X(t)$ and $Y(t)$ if we
perform an adiabatic
elimination of the fluctuations, i.e.,
$\dot{\Omega}^x=\dot{\Omega}^y=\dot{\Sigma}^{xy}=\dot{\Sigma}^{xa}=\dot{\Sigma}^
{ya}=0$, yielding to:
\begin{equation}
\begin{array}{lll}
\displaystyle \Sigma^{xa}=\frac{\sigma^2}{cH(x)-1} , & \Sigma^{ya}=\frac{H(x)\sigma^2}{cH(x)-1}, &\Sigma^{xy}=\frac{H(x)\sigma^2}{(cH(x)-1)^2} ,\label{adiabatic_3}\cr
\displaystyle \Omega^x=\frac{\sigma^2}{(cH(x)-1)^2} ,& \Omega^y=\frac{H^2(x)\sigma^2}{(cH(x)-1)^2} ,
\end{array}
\end{equation}
with $H(x)=-3x^2+2(1+b)x-b-K$.
Substituting $\Omega^x$ in Eqs. (\ref{exp12-eq})-(\ref{exp13-eq}), we find
a
closed form for the equations describing the evolution of the mean-field
variables $X(t)$
and $Y(t)$:
\begin{eqnarray}
%
\epsilon\dot{X}&=&-X^3+(1+b)X^2-\left[b+\frac{3\sigma^2}{(cH(X)-1)^2}\right]X
+\frac{(1+b)\sigma^2}{(cH(X)-1)^2}+d-Y, \label{MF1}\\
\dot{Y}&=&X-cY+a +A\sin\left(\frac{2\pi}{T} t\right) \label{MF2}
\end{eqnarray}
These equations provide a closed form for the nullclines of
the global
variables $X$ and $Y$ for the non-forcing case $A=0$. They also reflect how diversity influences these
variables.
Figure \ref{fig:f03} shows these nullclines $Y_1(X,\sigma)$ and $Y_2(X,a)$ of
Eqs. (\ref{MF1})-(\ref{MF2}) respectively for
$a=0.06$ and different values of the diversity $\sigma$.
\begin{figure}[h!]
\centering
\includegraphics[width=0.65\linewidth,clip]{Figure7.eps}
\caption{Nullclines of Eq. (\ref{MF1})-(\ref{MF2}) for different values of the
diversity $\sigma$. $Y_1(X,\sigma)$ for
$\sigma$: (a) $0.0$ (red line), (b) $0.5$ (green line), (c) $0.8$ (blue line), (d) $1.0$ (violet line),
(e) $1.2$ (cian line) and (f) $1.4$ (yellow line). The nullcline $Y_2(X,a)$ of Eq. (\ref{MF2})
for $a=0.06$ is represented with a black line.}
\label{fig:f03}
\end{figure}
It can be seen in the figure
that the diversity changes the shape of the cubic nullcline
$Y_1(X,\sigma)$ leading to a loss of stability of the fixed point of the system
that, for a
certain range of $\sigma$, becomes a limit cycle.
To schematize the behavior of the global variables $X$ and $Y$ when the
diversity changes, we show in Figure \ref{fig:f04} the phase-portrait for
different values of $\sigma=0.0$, $0.5$, $0.8$, $1.0$, $1.2$ and $1.4$
(corresponding to the values represented in Figure \ref{fig:f03}). It can be
seen that there is a range of $\sigma$ for which the system exhibits a
collective oscillatory behavior even in the absence of the weak external
modulation. \\
\begin{figure}[h!]
\centering
\includegraphics[width=0.99\linewidth,clip]{Figure8.eps}
\caption{Phase-space portrait for different values of $\sigma$. Grey line
represents the evolution
of $\{X,Y\}$. The black lines represent the nullcline $Y_2(X,a)$ for $a=0.06$
and the cubic nullcline $Y_1(X,\sigma)$ for
$\sigma$: (a) $0.0$, (b) $0.5$, (c) $0.8$, (d) $1.0$, (e) $1.2$ and (f)
$1.4$.}
\label{fig:f04}
\end{figure}
With the collective variable $X(t)$ obtained from the adiabatic elimination we
can estimate the spectral amplification factor $\eta$. Figure \ref{f1theo} shows the results obtained from the numerical integration
of Eqs. (\ref{exp12-eq})-(\ref{exp18-eq}),
together with the numerical simulation of the full system, Eqs.
(\ref{FHNvoltage-eqi})-(\ref{FHNrecover-eqi}) and the adiabatic
approximation obtained from Eqs. (\ref{MF1}) and (\ref{MF2}). It can be seen that our order parameter expansion is in good agreement
with the numerical integration of the full system, even in the case in which the
second
moments are adiabatically eliminated.
\begin{figure}[h!]
\centering
\includegraphics[width=0.99\linewidth,clip]{Figure9.eps}
\caption{Order parameter expansion versus numerical integration of the
full system. An adiabatic approximation is also included (see text). Two
different periods of the
external modulation have been considered: (a) $T=1.6$ and (b) $T=1.11$.
Other parameter as in Figure \ref{f1FHN}.}
\label{f1theo}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
We have studied the effect of the diversity in an ensemble of coupled
neurons described by the FHN model. We have observed that an intermediate
value of diversity can enhance the response of the system to an
external periodic forcing. We have studied both electrical and
chemical coupling schemes finding that the electrical coupling
induces a larger response of the system
to an external weak modulation, as well as existing for a wider range. In
contrast, the chemical coupling scheme exhibits a smaller optimal amplitude and
narrower range of response, however, for smaller values of the diversity. We have also
found that the response of the system in the electrical
coupling scheme strongly depends on the fraction of connected neurons in
the system whereas it does not improve much above a
small fraction of connected neurons in the chemical coupling scheme.
We have also developed an order parameter
expansion whose results are in good agreement with those obtained numerically for
the electrically (diffusively) coupled FHN system. By an adiabatic elimination of the fluctuations
we have found a closed form of the effective nullclines
of the global collective variables of the system obtaining a simple expression
of how the diversity influences the collective variables of the system.
The microscopic mechanism leading the system to a resonant
behavior with the external signal is as follows: in the homogeneous situation,
where all the units are identical, the weak external modulation cannot induce
any spike in the system. When the diversity increases, a fraction of the neurons
enters into the oscillatory regime and, due to the interactions, pull the other
neurons with them. This leads the system to an oscillatory collective behavior
that follows the
external signal. For larger values of the diversity, the fraction of neurons
in the oscillatory regime decreases and the rest of neurons offer some
resistance to being pulled by the oscillatory ones; thus, the system cannot
respond
to the external signal anymore. These
results suggest that the diversity present in biological systems may have an
important role in enhancing the response of the system to the
detection of weak signals.
We acknowledge financial support from the following organizations:
National Science Foundation (Grant DMR- 0702890);
G. Harold and Leila Y. Mathers Foundation;
European Commission Project GABA (FP6-NEST Contract 043309); EU NoE Biosim (LSHB-CT-2004-005137); and
MEC (Spain) and Feder (project FIS2007-60327).
|
1,116,691,499,347 | arxiv | \section{Introduction}
Cuprate thin films are a promising avenue for exploring issues of dimensionality, disorder, and free carrier density in these strongly correlated materials\cite{ahn,Konstantinovic2000859,ahn1999, PhysRevB.51.3257}. Mechanical exfoliation has proven to be an effective way of producing a variety of thin crystals\cite{geim2005,staley:184505,top,Ye:2010fk}, although it has mostly been applied to graphene. Compared to traditional techniques, mechanical exfoliation allows a large variety of deposited materials and substrates. To determine the viability of this approach in the high-temperature superconductors, we have produced thin cuprate crystals on oxidized Si substrates via mechanical exfoliation and characterized them with atomic force microscopy (AFM) and micro-Raman spectroscopy. The measured Raman spectra of these exfoliated samples suggest differences in the magnetic properties versus bulk and provide a useful tool for monitoring sample quality.
\section{Experiment}
\subsection{Exfoliation Methods and Microscopy}
Heavily under-doped Bi$_2$Sr$_2$Ca$_{1-x}$Dy$_x$Cu$_2$O$_{8+\delta}$ (Bi-2212) crystals (x = 0.3,0.4) were grown by the floating zone method and characterized by SQUID magnetometry, resistivity, Hall, and thermal transport measurements.\cite{PhysRevB.77.094515} These samples are highly insulating and exhibit no magnetic or superconducting order down to 2 K. Thin Bi-2212 samples were deposited onto Si substrates capped with a 280 nm thick SiO$_2$ layer via an exfoliation technique similar to those applied to graphene\cite{geim2005}. To minimize sample exposure to adsorbed water and air, the substrates were baked in N$_2$ gas at 400 K prior to deposition. Next the substrates are allowed to cool to room temperature, and then exfoliation is performed in the same N$_2$ environment. Upon deposition, exfoliated crystals are located using an optical microscope. The reflectivity of these samples is sensitive to thickness, allowing the identification of thin crystals. Indeed, we have identified crystals as thin as two unit cells ($\approx 6$ nm) in this manner. Finally, the samples are stored in vacuum when not in use.
Example exfoliated crystals of a variety of thicknesses (and thus colors) are shown in figure \ref{fig:flakes}. After being identified optically, potentially thin crystals were then examined using a Digital Instruments Nanoscope III AFM operating in contact mode to measure their dimensions. An example AFM image is shown in figure \ref{fig:afm2}. These exfoliated crystals can be tens of microns across and as thin as three unit cells. As seen in figure \ref{fig:afm2}, samples tend to be planar (RMS roughness $<$ 1.5 nm comparable to the bare substrate roughness) and have well-defined edges. The small thickness and large cross-sectional area of these samples makes them promising candidates for studies of electrostatic doping and the effects of dimensionality.
\begin{figure}
\includegraphics[scale=1]{f1}
\caption{\label{fig:flakes} Optical microscope images of exfoliated crystals. All scale bars are 20 microns. A) 13 nm thick crystal (dark cyan) B) 17 nm (dark blue) and 20 nm (light blue) thick crystals C) 42 nm thick crystal (bright green region) and 160 nm thick crystal (dark green region) D) Exfoliated crystal of varying height. The yellow and orange regions are both thicker than 100 nm.}%
\end{figure}
Our AFM and optical microscopy studies allow us to explain the observed color variation with height. This behavior can be understood as an interference effect. Specfically, contrast can be defined as the difference in reflected intensity from an exfoliated crystal and the bare substrate normalized to the substrate value. Using bulk optical constants from reference \onlinecite{PhysRevB.60.14917}, a calculation of crystal contrast similar to reference \onlinecite{blake:063124} suggests that the peak contrast should blue-shift with decreasing thickness. The results of this calculation are shown in figure \ref{fig:visibility} where we display the contrast as a function of crystal thickness and wavelength. The contrast for a 20 unit cell crystal peaks in the green. As the crystals become thinner, the peak contrast blue-shifts and decreases.
\begin{figure}
\includegraphics[scale=1]{f2}%
\caption{\label{fig:afm2} AFM image (left panel) and profile (right panel) of a 42 nm exfoliated crystal. The red (gray) line in the image corresponds to the profile displayed in the right panel. An optical image of this same crystal is shown in figure \ref{fig:flakes} C).} %
\end{figure}
\begin{figure}
\includegraphics[scale=1]{f3}%
\caption{\label{fig:visibility}Calculated exfoliated crystal visibility versus wavelength and crystal thickness on a 280 nm SiO$_2$/Si substrate.} %
\end{figure}
\begin{figure}
\includegraphics[scale=1]{f4}%
\caption{\label{fig:grig} CCD visibility measurements for red (top) and blue (bottom) channels. The solid curves represent the visibility expected from the calculation illustrated in figure \ref{fig:visibility} while open circles are the measured crystals visibilities.} %
\end{figure}
To confirm this calculation, visbility measurements were performed using a color CCD camera coupled to an optical microscope. Figure \ref{fig:grig} displays the measured (open circles) and calculated (curves) crystal visibilities for a variety of heights. The calculated visibilities were determined by integrating the calculated contrast over the spectral response of a given CCD channel. These measurements show the results of the contrast calculation to be qualitatively correct. While this data suggests that these exfoliated pieces are indeed Bi-2212, it is necessary to establish whether these exfoliated crystals remain Bi-2212 or form some other phase.
\subsection{Raman Spectroscopy}
Polarized Raman spectroscopy is a well established tool for identifying materials and has been widely applied to the cuprates\cite{weber_merlin,PhysRevB.68.184504,devereaux:175,PhysRevB.46.6505,PhysRevB.37.2353,PhysRevB.42.8760,PhysRevB.43.3009,PhysRevB.42.4842,PhysRevB.61.9752,Weber:89}. A Horiba Jobin Yvon LabRam Raman microscope with a 532 nm excitation source and 100x (0.8 NA) microscope objective was used to measure the Raman spectra of bulk and exfoliated samples. Measurements were performed in the backscattering geometry. The polarizations of the incident and scattered light are defined with respect to the sample crystallographic axes. The notation XY, for instance, refers to incident light polarized along the $a$ axis (along the Cu-O bond) with scattered light polarized along the $b$ axis; while X' and Y' refer to directions 45 degrees from X and Y respectively. Raman spectra contain components corresponding to the different symmetry projections of the material and these components can be distinguished by their different polarization dependences. Dy-doped Bi-2212 is known to be nearly tetragonal and so we take spectra in the XX, X'X', XY, and X'Y' polarization geometries. For tetragonal (D$_{4h}$) crystal symmetry, XX corresponds to B$_{1g}$ + A$_{1g}$, X'X' to B$_{2g}$ + A$_{1g}$, XY to B$_{2g}$ + A$_{2g}$, and X'Y' to B$_{1g}$ + A$_{2g}$.\cite{devereaux:175} At Raman shifts less than 1000 cm$^{-1}$ exfoliated sample spectra are dominated by signal from the silicon substrate, making phonon peaks difficult to identify. Fortunately, bulk Bi-2212 has two features at higher energies: a collection of two-phonon peaks broadened into a single feature around 1250 cm$^{-1}$ and a broad two-magnon feature centered between 2000-3000 cm$^{-1}$ (depending on doping) \cite{weber_merlin,PhysRevB.68.184504,G.Blumberg11211997,PhysRevB.53.8619}. The two-magnon feature has been studied extensively due to the importance of magnetism in cuprates physics\cite{fluc}. Since thin materials are easily damaged by laser radiation\cite{staley:184505}, we tried a variety of exposure times and excitation powers to maximize the signal to noise ratio while minimizing exposure to the Raman excitation source. Limiting the samples exposure to the laser to approximately ten seconds at 0.5 mW produced no observable changes in the Raman response. Finally, the measured spectra have been corrected for interference effects as described elsewhere\cite{yoon:125422}.
\begin{figure}
\includegraphics[scale=1]{f5}%
\caption{\label{fig:raman} Polarized Raman spectra for a) bulk b) 42 nm exfoliated crystal c) degraded 30 nm exfoliated crystal. In each case, XX spectra are shown in solid black, X'X' in dashed red (dark gray), XY in dashed green (light gray), and X'Y' in solid blue (dark gray). XX and X'X' spectra have been offset for clarity. The spectra in a) and b) show a marked polarization dependence compared with c).} %
\end{figure}
The bulk spectra shown in figure \ref{fig:raman}a show both the two-phonon and the two-magnon features seen in previous experiments. Raman intensity units are arbitrary but consistent between plots. In agreement with previous studies, the two-phonon feature near 1300 cm$^{-1}$ appears primarily in the fully symmetric A$_{1g}$ channel (XX and X'X'). In addition, a broad two-magnon feature appears in B$_{1g}$ (XX and X'Y') near 2500 cm$^{-1}$. For the most part, these two features are also visible in the exfoliated samples. Note that only data for x = 0.4 crystals is displayed, although the behavior of x = 0.3 bulk and exfoliated crystals are qualitatively identical. The Raman features observed in typical exfoliated crystals (figure \ref{fig:raman}b) generally retain the selection rules expected from bulk measurements and from the literature for an $ab$-face. This confirms that the cleavage plane is in fact the $ab$-plane, as observed in other experiments\cite{Tanaka:1989uq}. In both exfoliated and bulk samples,we find that the two-magnon feature is principally in the B$_{1g}$ symmetry channel while the two phonon excitation appears predominantly in A$_{1g}$. This suggests that exfoliated samples retain the crystal symmetry of the bulk. The polarization dependence of the two-magnon feature in Bi-2212 also allows the identification of the crystal axes of an exfoliated sample, although it cannot distinguish between the $a$ and $b$ axes.
The polarization dependence of the Raman spectra also reveals information about degradation in the exfoliated crystals. Certain exfoliated sample spectra, particularly from samples overexposed to laser light and/or air, do not exhibit the expected selection rules. As shown in figure \ref{fig:raman}c, spectra from such samples show only a slight polarization dependence and a weak two-magnon peak. This is attributed to sample degradation and is discussed in more detail later. There are, however, important differences between bulk and exfoliated spectra even when the expected selection rules are observed. Figure \ref{fig:disc}a illustrates the evolution of the two-magnon line shape with thickness. Spectra from thin crystals ($<$ 50 nm) show a large enhancement in the amplitude of the two-magnon peak, as well as an apparent blue shift in the peak positions of the two-magnon (from around 2500 cm$^{-1}$ to 2900 cm$^{-1}$). The shift is more clearly seen in figure \ref{fig:disc}b, which shows both bulk and 30 nm spectra. It should be noted that the bulk spectrum has been scaled to match the peak heights of the exfoliated crystal.
\begin{figure}
\includegraphics[scale=1]{f6}%
\caption{\label{fig:disc} a) Variation of XX spectra with thicknesses. b) XX spectra for bulk and a 10 unit cell exfoliated sample showing a shift in the two-magnon peak position. Data has been normalized by the two-magnon peak height. c) Effect of oxide layer variation on interference corrected spectra.} %
\end{figure}
\section{Discussion}
There are several factors that could potentially cause significant change in the Raman spectra of the exfoliated samples. The first is the interference correction mentioned earlier. This correction is sensitive to the thickness of the SiO$_2$ layer, so any variation from the nominal value (280 nm) would produce a distortion of the Raman spectra. The thickness of the thermal oxide layers used in this work can vary by 5 percent, according to the manufacturer and confirmed by ellipsometry measurements. Figure \ref{fig:disc}c shows the effect of this variation on the corrected spectra of a 10 unit cell exfoliated sample. The thickness of the oxide layer can affect the ratio of the two-phonon and two-magnon features as well as influence their lineshapes. However, this effect is insufficient to account for the observed blue shift and enhancement of the two-magnon. The Raman response therefore suggests a significant change in the magnetic interactions of the exfoliated crystals with respect to bulk.
A change in doping level could account for the observed change in the exfoliated crystals' effective exchange constant. One possibility is environmental doping by charged impurities on the SiO$_2$ surface. This effect has previously been observed in graphene\cite{PhysRevLett.99.246803}. Typically holes are added to the graphene layer and we might expect a similar effect in exfoliated Bi-2212. However, the two-magnon feature in hole-based cuprates is known to red-shift, weaken and broaden with doping\cite{weber_merlin,PhysRevB.68.184504}, while our spectra blue-shift and grow (see figures \ref{fig:disc}a and \ref{fig:disc}b). Another possibility is oxygen outdiffusion. This effect has been observed in Bi-2212 microwhiskers\cite{ISI:000229544200081,synchrotron} and in cuprate thin films\cite{ISI:000263564500011}. Oxygen loss would be expected to reduce the number of holes in the exfoliated crystal and so is consistent with the observed change in the two-magnon feature. Indeed, the peak position of the two-magnon feature seen in the exfoliated crystals is consistent with the peak location reported elsewhere for antiferromagnetic and insulating Bi-2212 \cite{PhysRevB.68.184504}. Loss of holes through oxygen outdiffusion is also consistent with the insulating behavior observed in optimally-doped exfoliated samples\cite{geim2005}. Lastly, one might conjecture that a structural change in the exfoliated Bi-2212 sample could be responsible for shifting the two-magnon peak. As the energy and efficiency of the two-magnon process depends on the hopping integral $t$\cite{weber_merlin,PhysRevLett.74.3057,PhysRevB.53.8619}, an increase in this parameter could explain the peak enhancement and shift observed in thin exfoliated crystals.
Finally, Bi-2212 is known to degrade in atmosphere,\cite{0953-2048-6-7-008} which would produce a corresponding change in the Raman spectra. Given the the thickness of these exfoliated samples, even thin surface degradation can be expected to have a pronounced effect on the measured Raman features. Indeed, earlier work done on exfoliated Bi-2212 crystals found that even optimally doped exfoliated crystals were insulating\cite{geim2005} when deposited in air. We find that older samples tend to show less distinct symmetry, as evidenced by the XX and X'X' spectra shown in figure \ref{fig:raman}c. Specifically, the difference between the XX and X'X' geometries is less pronounced and the overall scattered intensity is reduced. The particular sample shown in \ref{fig:raman}c was exposed to air for tens of hours as well as potentially damaged by over exposure to the Raman excitation laser. It should be noted that the loss of selection rules suggests the crystals are becoming amorphous. The two-phonon feature is also absent in \ref{fig:raman}c, consistent with a degradation of the lattice structure. Note that this degradation is distinct from the structural change discussed above as a possible cause of the enhancement and blue-shift of the two-magnon Raman feature. Raman spectroscopy therefore provides a powerful method for identifying degradation in exfoliated samples.
\section{Conclusion}
The work described here suggests that exfoliated cuprate thin crystals can be produced with dimensions suitable for device applications. FET devices would allow the investigation of a number of important issues relating to cuprate physics, principally the disentanglement of disorder and carrier density. Significant differences were observed between the two-magnon Raman spectra of exfoliated and bulk Bi-2212, indicating a change in the effective exchange constant of the exfoliated crystals. However, when properly handled, these samples demonstrate selection rules and energy scales similar to bulk. This suggests that exfoliated crystals remain sufficiently bulk-like to merit study. Changes in the Raman spectra of these materials with time also indicate the importance of minimizing sample exposure to atmosphere and to laser radiation. The two most important directions for further research are transport measurements on exfoliated crystal devices (field effect and transport) as well as depositing Bi-2212 thin crystals on other substrates. In particular, transport measurements could help elucidate the role of disorder\cite{K.McElroy08122005} in the formation of the pseudogap and in normal state transport\cite{PhysRevLett.85.638,PhysRevB.71.014514,PhysRevLett.75.4662}. A Raman-inactive substrate would allow study of the lower energy Bi-2212 phonon modes and thus provide important insight into the structure of these cuprate thin crystals. Nonetheless, we have identified a change in the magnetic exchange constant of the exfoliated crystals with respect to bulk and have demonstrated the utility of AFM and in particular Raman microscopy in establishing the quality of these crystals.
\begin{acknowledgments}
We are grateful for numerous discussions with Y.J. Kim, A. Paramekanti, A.B. Kuzmenko, Z.Q. Li, M.Y. Han, and P. Kim. AFM was performed with the assistance of R. McAloney and M.C. Goh. We would also like to acknowledge the help of the ECTI Open Research Facility. Work at the University of Toronto was supported by NSERC, CFI, ORF and OCE.
\end{acknowledgments}
|
1,116,691,499,348 | arxiv | \section{Introduction}
The study of variational problems with Euclidean symmetry is an old problem, indeed, Euler's 1744 study of elastic beams
is such a case. However, methods to analyse such problems efficiently and effectively, are still of interest.
In this paper, we consider variational problems for curves in 3-space for which the Lagrangian is invariant under
the special Euclidean group $SE(3)=SO(3)\ltimes \mathbb{R}^3$ acting linearly in the standard way, that is,
\begin{equation}\label{SE3act}
\begin{pmatrix} x\\y\\z\end{pmatrix} \mapsto R\begin{pmatrix} x\\y\\z\end{pmatrix} + \begin{pmatrix} a\\b\\c\end{pmatrix},\qquad R\in SO(3).
\end{equation}
The Euler--Lagrange equations satisfied by the extremising curves have $SE(3)$ as a Lie symmetry group, and can be therefore be written
in terms of the differential invariants of the action, and their derivatives with respect to arc-length. Further, the six dimensional space of Noether's laws
are key to analysing the space of extremals.
To date, the Frenet--Serret\ frame has been used to analyse Euclidean invariant variational problems, and this requires that the Lagrangian can be written in terms of the Euclidean curvature and torsion. Because the Frenet--Serret\ frame can be derived using \textit{algebraic} equations (at each point) on the relevant jet bundle,
the powerful symbolic calculus of invariants can be used, to obtain not only the Euler--Lagrange equations directly in terms of the curvature and torsion, but the full set of Noether's laws can also be written down directly using both the invariants and the frame \cite{GonMan2}.
Let us denote the space curve as $s\mapsto P(s)\in\mathbb{R}^3$, where $s$ is arc-length, and the tangent vector to this curve by $P'$, so that
$'={\rm d}/{\rm d}s$. By the definition of arc-length, $|P'|^2=P'\cdot P'=1$. Then provided $P''\ne 0$, the left Frenet--Serret frame is given
by
\begin{equation}\label{FSdef}
\sigma_{FS}^{\ell}=\begin{pmatrix} P'(s) & \frac{P''(s)}{|P''(s)|} & \frac{P'(s) \times P''(s)}{|P''(s)|}\end{pmatrix} \in SO(3).
\end{equation}
From a computational point of view, the Frenet--Serret\ frame is convenient as it can be computed straightforwardly at arbitrary points along the curve. However, it is undefined wherever the curvature is degenerate, such as at inflection points or along straight sections of the curve.
The left Frenet--Serret\ frame is \textit{left equivariant}, that is, if at any point $z=P(s)$ on the curve, since
$R\in SO(3)$ acts linearly in the standard way on the tangent space $T_z\mathbb{R}^3$, then it is readily seen that
\[ \sigma_{FS}^{\ell} \mapsto \begin{pmatrix} R P'(s) & \frac{R P''(s)}{| R P''(s)|} & \frac{R P'(s) \times R P''(s)}{| R P''(s)|}\end{pmatrix}=R \sigma^{\ell}_{FS}.\]
The Euclidean curvature $\kappa$ and the torsion $\tau$ at the point $P(s)$ are then the nonzero components of the invariant so-called curvature matrix, specifically,
\begin{equation}\label{FScurvMx} \left(\sigma^{\ell}_{FS}\right)^{-1} \left(\sigma^{\ell}_{FS}\right)' = \begin{pmatrix}0 & -\kappa &0\\ \kappa &0&-\tau\\ 0&\tau&0\end{pmatrix}.\end{equation}
In contrast to this frame, \textit{relatively parallel} frames were described by \cite{Bishop} who detailed what is now known variously as the Normal, Parallel, Bishop or Rotation Minimizing\ frame.
The Rotation Minimizing\ frame has many advantages over the Frenet--Serret\ frame. First of all, unlike the Frenet--Serret\ frame, the Rotation Minimizing\ frame is
defined at all points of a smooth curve.
The Rotation Minimizing\ frame may be used to study a larger class of variational problems, because
while the generating invariants for the symbolic invariant calculus given by the Frenet--Serret\ frame, curvature and torsion, are of order
2 and 3 respectively, those given by the Rotation Minimizing\ frame are both of order only 2. Finally,
the Rotation Minimizing\ frame, its computation, approximation and its applications, have been extensively used and studied in the Computer Aided Design literature,
\cite{BR,F1,F2,F3,G,H,K,PFL,PW,SW,WJZL}.
One reason is that the sweep surfaces they generate are, in general, superior, \cite{WangJoe}; as illustrated in Figure \ref{sweep1}, sweep surfaces
generated from the Frenet--Serret\ frame can exhibit strong twisting at inflection points.
Bishop, \cite{Bishop}, defines a normal vector field along a curve to be \textit{relatively parallel} if its derivative is proportional to the tangent vector.
The equation used in the Computer Aided Design literature for the relatively parallel normal
vector $V=V(s)$ to the curve $s\mapsto P(s)$ is \cite{WangJoe},
\begin{equation}\label{Vdef} V' = -(P''\cdot V) P'.\end{equation}
The function of proportionality between $V'$ and $P'$ is chosen to guarantee that, without loss of generality,
we may suppose that $|V|\equiv 1$ and $P'\cdot V\equiv 0$, see Proposition \ref{propVprops}.
Then the left Rotation Minimizing\ frame is
\begin{equation}\label{RMdef}
\sigma_{RM}^{\ell} = \begin{pmatrix} P' & V & P'\times V\end{pmatrix}.
\end{equation}
We have that $\sigma_{RM}^{\ell}$ is left equivariant and, as shown by Bishop, the invariant curvature matrix $\left(\sigma^{\ell}_{RM}\right)^{-1}\left(\sigma^{\ell}_{RM}\right)'$ takes the form
\begin{equation}\label{RMcurvMx} \left(\sigma^{\ell}_{RM}\right)^{-1}\left(\sigma^{\ell}_{RM}\right)'= \begin{pmatrix} 0 & -\kappa_1 & -\kappa_2\\\kappa_1 & 0 &0\\ \kappa_2 &0&0\end{pmatrix},
\end{equation}
that is, where the $(2,3)$-component is guaranteed to be zero.
\begin{figure}
\caption{Given a curve in space, we compare the Frenet--Serret\ frame with the Rotation Minimizing\ frame along it.
\label{sweep1}}
\centering
\[
\begin{array}{cc}
\includegraphics[scale=0.32]{introNF} &
\includegraphics[scale=0.3]{introSF}\\[9pt]
\text{Surface sweeping given by} \ V & \text{Surface sweeping given by} \ P''\\
\text{using the Rotation Minimizing\ frame} & \text{using the Frenet--Serret\ frame}
\end{array}
\]
\end{figure}
Since both the Rotation Minimizing\ and the Frenet--Serret\ frames share the same first column, we have for some angle $\theta=\theta(s)$, (see Figure \ref{gaugeFSRM}),
\begin{equation}\label{sigmarm}
\sigma^{\ell}_{RM} = \sigma^{\ell}_{FS} \left( \begin{array}{ccc} 1 & 0 & 0\\ 0 & \cos{\theta} & \sin{\theta} \\ 0 & -\sin{\theta} & \cos{\theta} \end{array}\right).
\end{equation}
\begin{figure}
\caption{\label{gaugeFSRM}Diagram of a Rotation Minimizing\ frame and a Frenet--Serret\ frame of a curve $P(s)$ in $\mathbb{R}^3$. Note that $P'(s)$ is common in both frames.}
\centering
\includegraphics[width=0.8\textwidth]{NFtikz1a.pdf}
\end{figure}
Calculating $\left(\sigma^{\ell}_{RM}\right)^{-1} \left(\sigma^{\ell}_{RM}\right)'$, using \eqref{FScurvMx}, and \eqref{sigmarm}, and comparing the result to \eqref{RMcurvMx} leads to
the well known relations,
\begin{equation}\label{k1k2} \kappa_1=\kappa\cos\theta,\qquad \kappa_2 = \kappa\sin\theta, \qquad \theta_s = \tau.\end{equation}
Treating the Rotation Minimizing\ frame as a gauge transformation of the Frenet--Serret\ frame, together with
\[
\theta(t) - \theta_0 = \int^t_{t_0} \tau(t) |P'(t)| \ {\rm d}t
\]
has been proven to lack numerical robustness for a general space curve, (see \cite{G}).
This makes the use of the Rotation Minimizing\ frame defined in terms of the normal vector $V$, as in \eqref{RMdef},
to be a better choice in the application literature, and is our choice here.
The theory and applications of Lie group based moving frames are now well established, and provides an invariant calculus to study differential systems that are either invariant or equivariant under the action of a Lie group, (see the graduate text \cite{Mansfield:2010aa} and references therein).
Beginning with \cite{FO2,FO1} who found the recurrence formulae for the invariant differentiation of a computable set of generating invariants,
given algebraic equations for the frame,
there is now a rigorous and constructive symbolic invariant calculus,
\cite{hubertAA,hubertAD,hubertAC} and \cite{hubertB,hubertA}. This calculus has been applied to study problems
in the calculus of variations where the Lagrangian has a Lie group symmetry \cite{kogan,GonMan, GonMan2, GonMan3}.
The case of Euclidean invariance using the Frenet--Serret\ frame was studied by \cite{GonMan2}.
The formulae for the recurrence relations in the symbolic invariant calculus require the equations defining the frame to be algebraic at each point in the domain of the frame, and indeed,
the equations defining the Frenet--Serret\ frame,
despite involving the components of $P(s)$, $P'(s)$ and $P''(s)$, are algebraic at each point of the relevant jet bundle.
However, the recurrence formulae for the invariant derivatives defined using the Rotation Minimizing\ frame need to be derived in another way, because the equations defining the frame are not algebraic
in the jet variables. Indeed, considering \eqref{k1k2}, it would seem that the Rotation Minimizing\ frame is defined by a relation on the
invariants, $\tau$ and $\theta_s$, or, a differential equation on an extended space, one which includes either $\theta$, or $V$.
Our approach is to extend the manifold on which the group acts, to include the vector $V$ and its derivatives, in such a way that
the differential equation defining $V$ is a simple constraint for our variational problem. Because the group acts linearly on $P'$, $V$ and their derivatives,
it turns out to be straightforward to write down a set of generating invariants, the recurrence formulae for their invariant differentiation and their differential
syzygies. With these to hand, the methods used by \cite{GonMan2} can be adapted to obtain Euler--Lagrange equations directly in terms of the invariants
and to write down the six Noether conservation laws.
In \S \ref{MFsection}, we introduce the notions of a Lie-group based moving frame, the frame-defined invariants and the curvature matrices.
The symbolic invariantized form of the curvature matrices for the Rotation Minimizing\ frame are found, and we derive the recurrence formulae for the symbolic differential invariants and the syzygy operator we will need in the sequel. In \S \ref{ELsection}, we obtain the Euler--Lagrange equations
and Noether's laws for a Lagrangian with a Euclidean symmetry, using the results of \S \ref{MFsection}. In \S \ref{IPsection}, the use of Noether's laws to ease the integration problem is carried out. In \S \ref{EXsection}, some examples and applications are presented.
The final section, \S \ref{CCsection} is devoted to our conclusions and the implication of our results to extending the range of applicability of the symbolic invariant calculus to frames not defined algebraically.
\section{Moving frames}\label{MFsection}
\begin{defn}\label{leftact}
Given a Lie group $G$ and a manifold $M$, a \emph{left} action of $G$ on $M$ is defined to be a smooth map
\[G\times M\rightarrow M, \qquad (g, z) \mapsto g\cdot z \]
such that
\[ g\cdot (h\cdot z) = (gh)\cdot z.\]
\end{defn}
\begin{defn}[Moving Frame]
Given a smooth left Lie group action $G\times M\rightarrow M$, a moving frame on the domain $\mathcal{U}\subset M$ is an equivariant map $\rho: \mathcal{U}\rightarrow G$, that is
\[\begin{array}{ll}\rho(g\cdot z) = g\rho(z) &\qquad \mbox{\emph{left equivariance}}\\
\rho(g\cdot z)=\rho(z)g^{-1} &\qquad \mbox{\emph{right equivariance}}\end{array}\]
\end{defn}
The frame is called left or right accordingly. Given a left frame, its (group) inverse is a right frame, and vice versa. In practice, the ease of calculation can differ considerably depending on the choice of parity.
Moving frames exist when the action is free and regular on its domain
$\mathcal{U}\subset M$. This means, the orbits foliate $\mathcal{U}$, and for any cross section $\mathcal{K}\subset M$ which is transverse to the orbits $\mathcal{O}(z)$, the set $\mathcal{K}\cap \mathcal{O}(z)$ has just one element, the projection of $z$ onto $\mathcal{K}$, (see \cite{Mansfield:2010aa} for full details).
The standard method to calculate a moving frame for the group action on a neighbourhood $\mathcal{U}\subset M$ of $z$ is as follows.
Using a cross-section $\mathcal{K}$, given by a system of equations $\psi_r(z)=0$, for $r=1,\ldots,R,$ where $R$ is the dimension of the group $G$, one then solves the so-called normalization equations,
\begin{equation}\label{frameEqA}
\psi_r(g\cdot z) = 0, \qquad r=1,\ldots, R,
\end{equation}
for $g$ as a function of $z$.
The solution is the group element $g=\rho(z)$ that maps $z$ to its projection on $\mathcal{K}$.
The conditions on the action above are those for the Implicit Function Theorem to hold (see \cite{Hirsch}), so the solution $\rho$ is unique. A consequence of uniqueness is that
$$
\rho(g\cdot z)=\rho(z) g^{-1},
$$
that is, the frame is \textit{right equivariant}, as both $\rho(g\cdot z) $ and $ \rho(z) g^{-1}$ solve the equation $\psi_r\left(\rho(g\cdot z)\cdot \left(g\cdot z\right)\right)=0$.
The equivariance of the frame enables one to obtain invariants of the group action.
\begin{lem}[Normalized Invariants]
Given a left or right action $G\times M \rightarrow M$ and a \textit{right} frame $\rho$, then
$\iota(z)=\rho(z)\cdot z$, for $z$ in the domain of the frame $\rho$, is invariant under the group action.
\end{lem}
\begin{defn}\label{normInvsnames} The normalized, or frame-defined, invariants are the components of $\iota(z)$.
\end{defn}
When the frame is not known explicitly, the normalized invariants are said to be known symbolically. The power of the symbolic invariant calculus
derives from the fact that these symbolic invariants can be used effectively in a wide range of calculations.
We now state the Replacement Rule, from which it follows that the normalized invariants provide a set of generators for the algebra of invariants.
\begin{theorem}[Replacement Rule]\label{reprule}
Given a right moving frame on $M$ for the action $G\times M\rightarrow M$,
and an invariant $F(z)$ of this action, then $F(z)=F(\iota(z))$.
\end{theorem}
\begin{defn}[Invariantization]\label{invOpDef} Given a right moving frame $\rho$, the map $z\mapsto \iota(z)=\rho(z)\cdot z$ is called the \emph{invariantization} of the point $z$, and the map
$F(z)\mapsto F(\iota(z))$,
is called the \emph{invariantization} of $F$.\end{defn}
In this paper, we will consider derivatives with respect to arc-length $s$ of our curve $s\mapsto P(s)$, where we note that arc-length is a Euclidean invariant, and we will also consider the
evolution of this curve with respect to a `time' parameter $t$, which we declare to be invariant under our $SE(3)$ action. In general, if the independent variables of our curves and surfaces, with respect to which we differentiate, are \textit{all} invariant, we may make the following definition.
\begin{defn}[Curvature matrices]
The curvature matrix with respect to the independent variable $x$ is defined to be, for a right frame $\rho$,
\[
Q^x=\left(\frac{{\rm d}}{{\rm d} x} \rho\right) \rho^{-1}.
\]
\end{defn}
The non-constant components of the curvature matrices are differential invariants of the action. These are
referred to as the \textit{curvature invariants} or the Maurer--Cartan invariants.
\subsection{The extended right Rotation Minimizing\ frame}
Since the symbolic invariant calculus is standardly carried out for a right frame, we consider
a right Rotation Minimizing\ frame, $\rho_{RM}$, which we need for our application to include the translation component of the
Special Euclidean group $SE(3)$. We consider
the Lie group $SE(3)$ to act on an enlarged manifold (jet bundle) having local coordinates to be the components of
\[ P, P', P'', \dots , P^{(n)}=\frac{{\rm d}^n}{{\rm d}s^n} P, \dots, V, V', V'',\dots V^{(n)}=\frac{{\rm d}^n}{{\rm d}s^n} V, \dots\]
where the left action is, for $g=(R, \mathbf{a})\in SE(3)=SO(3)\ltimes \mathbb{R}^3$,
\[ P\mapsto R P + \mathbf{a},\qquad P^{(n)}\mapsto R P^{(n)}, n>0, \qquad V^{(n)}\mapsto R V^{(n)}, n\ge 0.\]
In the standard representation of $SE(3)$ in $GL(4,\mathbb{R})$,
\[ g=(R, \mathbf{a})\mapsto \begin{pmatrix} R & \mathbf{a}\\ 0 & 1\end{pmatrix},\]
our \textit{extended right Rotation Minimizing\ frame} for this action is defined to be,
\begin{equation}\label{rhoRMdef} \rho_{RM}=\begin{pmatrix} \sigma_{RM} & -\sigma_{RM} P \\ 0 & 1\end{pmatrix}\end{equation}
where
\begin{equation}\label{defRMright}
\sigma_{RM} = \left(\sigma_{RM}^{\ell}\right)^T\in SO(3).
\end{equation}
The curvature matrix is, by direct calculation and noting that $\sigma_{RM}P'=\begin{pmatrix} 1&0&0\end{pmatrix}^T$,
\begin{equation}\label{ExCurvMx} Q^s=\rho_{RM}'\rho_{RM}^{-1} = \begin{pmatrix} \sigma_{RM}'\sigma_{RM}^{-1} & \begin{array}{r} -1\\ 0\\ 0\end{array}\\ 0 & \phantom{-}0\end{pmatrix}.\end{equation}
To obtain the complete set of normalized invariants and the (reduced) curvature matrix $\sigma_{RM}'\sigma_{RM}^{-1}$,
we first consider solutions of the defining equation for $V$.
\begin{prop}\label{propVprops}
Given a curve $s\mapsto P(s)\in \mathbb{R}^3$ such that $P'\cdot P'=|P'|^2=1$, and suppose that $V=V(s)$ satisfies equation \eqref{Vdef},
which for convenience we give again here,
\begin{equation}\label{NVdef}
V' = -(P''\cdot V) P'
\end{equation}
together with the initial conditions $V(s_0)=1$, $V(s_0)\cdot P'(s_0)=0$. Then
\begin{enumerate}
\item $V\cdot P' \equiv 0$
\item $ V\cdot V\equiv 1$
\item For any constant $\psi\in\mathbb{R}$,
\[ W=\cos\psi\, V + \sin\psi\, P'\times V\]
also solves equation \eqref{NVdef} with $|W|\equiv1$ and $W\cdot P'\equiv 0$
\end{enumerate}
\end{prop}
\begin{proof}
\begin{enumerate}
\item By direct calculation,
the scalar product $V\cdot P'$ is constant with respect to $s$. The result follows from the assumption on the initial data.
\item Equation \eqref{NVdef} implies $V'\cdot V = -(P''\cdot V)(P'\cdot V)=0$ by 1. above. Hence $V\cdot V$ is constant
with respect to $s$. The result follows from the assumption on the initial data.
\item Since \eqref{NVdef} is linear, it suffices to prove that $W=P'\times V$ also solves Equation \eqref{NVdef}.
We have by the orthogonality of both $V$ and $P''$ to $V$ that $V= b(s) P'' + c(s) P'\times P''$ for some coefficients $b(s)$, $c(s)$.
Then $P'\times V= b(s) P'\times P'' - c(s)P''$ and
\[\begin{array}{rcl}
(P'\times V)' &=& P''\times V + P'\times V'\\
&=& P''\times V\\ &=& c(s)(P''\cdot P'') P'.
\end{array}\]
But $P''\cdot (P'\times V)= -c(s) P''\cdot P''$ and hence
\[ W' = -(P'' \cdot W)\cdot P'\]
as required.
\end{enumerate}
\end{proof}
The proposition shows that if $V$ solves \eqref{NVdef} and for some $s_0$, $V(s_0)$ has unit length and is orthogonal to $P(s_0)$,
then $\sigma_{RM}\in SO(3)$ for all $s$, and this we now assume.
In the applications, it is necessary to ensure the initial data for $V$ holds when integrating for the frame. The proposition shows further that in fact there is a one-parameter family
of Rotation Minimizing\ frames, determined by the initial data for $V$.
Let $\mathfrak{so}(3)$ denote the set of $3\times 3$ skew-symmetric matrices, the Lie algebra of $SO(3)$.
We have by direct calculation that
\begin{equation}\label{CurvMat} \sigma_{RM}'\sigma_{RM}^{-1}=\left(\begin{array}{ccc}
0 & P''\cdot V & P''\cdot (P'\times V) \\
-P''\cdot V & 0 & 0\\
-P''\cdot (P'\times V) & 0 &0\end{array}\right)\in \mathfrak{so}(3)\end{equation}
We now write down the symbolic normalized invariants, and obtain $\sigma_{RM}'\sigma_{RM}^{-1}$ in terms of them.
We denote the components of $P(s)$ as $P(s)=(X(s), Y(s), Z(s))$ and that
of the $n$-th derivative with respect to $s$ as
$P^{(n)}=(X^{(n)}, Y^{(n)}, Z^{(n)})$.
By construction,
\[ \rho_{RM}\cdot P=0\]
and by definition of the action,
\[\rho_{RM}\cdot P^{(n)}=\sigma_{RM}P^{(n)},\quad n>0.\]
We now recall the standard symbolic names of these normalized invariants (see Definition \ref{normInvsnames}), as
\begin{equation}\label{DefIXn} \sigma_{RM} P^{(n)}=( \iota(X^{(n)}), \iota(Y^{(n)}), \iota(Z^{(n)}) )^T.\end{equation}
Since
\[
((\iota(X'), \iota(Y'), \iota(Z')){^{T}}= \sigma_{RM} P' = (P'\cdot P', V\cdot P', (P'\times V)\cdot P'){^{T}}= (1, 0, 0)^T,
\]
we make the following definition.
\begin{defn}[arc-length constraint]\label{arclengCon}
The equation $ \iota(X')=1$ is denoted as the \emph{arc-length constraint}.
\end{defn}
Differentiating (\ref{DefIXn}) with respect to $s$, yields
\begin{equation}\label{RecDiff}
\frac{{\rm d}}{{\rm d} s} \begin{pmatrix} \iota(X^{(n)})\\ \iota(Y^{(n)})\\ \iota(Z^{(n)}) \end{pmatrix}
= \frac{{\rm d}}{{\rm d} s}({\sigma_{RM}})\sigma_{RM}^{-1} \begin{pmatrix} \iota(X^{(n)})\\ \iota(Y^{(n)})\\ \iota(Z^{(n)}) \end{pmatrix}
+ \begin{pmatrix} \iota(X^{(n+1)})\\ \iota(Y^{(n+1)})\\ \iota(Z^{(n+1)}) \end{pmatrix}.
\end{equation}
Setting $n=1$
and recalling $ \sigma_{RM} P' = (1, 0, 0)^T$, we have from (\ref{CurvMat}) and (\ref{RecDiff}) that
\[ \left(\begin{array}{c} 0\\ 0\\0\end{array}\right)
= \left(\begin{array}{c} \iota(X'')\\ \iota(Y'')\\\iota(Z'')\end{array}\right) + \left(\begin{array}{c} 0\\ -P''\cdot V\\-P''\cdot (P'\times V) \end{array}\right).\]
Therefore
we can write down $\frac{{\rm d}}{{\rm d} s}({\sigma_{RM}})\sigma_{RM}^{-1}$ in terms of the normalized invariants, specifically,
\begin{equation}\label{QsigRMNormInvP}
\frac{{\rm d}}{{\rm d} s}({\sigma_{RM}})\sigma_{RM}^{-1}=\begin{pmatrix} 0& \iota(Y'') & \iota(Z'')\\ -\iota(Y'') &0&0\\-\iota(Z'') &0&0\end{pmatrix}.
\end{equation}
Inserting this into Equation \eqref{RecDiff} yields the all important recurrence formulae for the symbolic invariant differentiation of the normalized
invariants of the $P^{(n)}$.
We next consider the normalized invariants of the $V^{(n)}$, which are
\begin{equation}\label{SSSDI}
\sigma_{RM} V^{(n)}=(\iota(V_1^{(n)}), \iota(V_2^{(n)}), \iota(V_3^{(n)}))^T, \quad n\ge 0.
\end{equation}
Differentiating both sides of \eqref{SSSDI} with respect to $s$ yields the recurrence formula for the invariant differentiation
of the symbolic normalized invariants of the components of $V^{(n)}$,
\begin{equation}\label{QsigRMNormInvV}
\frac{{\rm d}}{{\rm d} s} \begin{pmatrix}\iota(V_1^{(n)})\\ \iota(V_2^{(n)})\\ \iota(V_3^{(n)})\end{pmatrix} =
\frac{{\rm d}}{{\rm d} s}({\sigma_{RM}})\sigma_{RM}^{-1} \begin{pmatrix}\iota(V_1^{(n)})\\ \iota(V_2^{(n)})\\ \iota(V_3^{(n)})\end{pmatrix}
+ \begin{pmatrix}\iota(V_1^{(n+1)})\\ \iota(V_2^{(n+1)})\\ \iota(V_3^{(n+1)})\end{pmatrix}. \end{equation}
Setting $n=0$ into this, and since $\sigma_{RM} V = (0, 1, 0)^T$ we have that
\begin{equation}\label{VSymbInvs}
\left(\begin{array}{c} 0\\ 0\\0\end{array}\right)
= \left(\begin{array}{c} \iota(V_1')\\ \iota(V_2')\\ \iota(V_3')\end{array}\right) + \left(\begin{array}{c} \iota(Y'')\\ 0\\0 \end{array}\right).\end{equation}
Finally, we note that if we take a right orthonormal frame $\sigma_{RM}=\begin{pmatrix} P' & V & P'\times V\end{pmatrix}^T$,
where we have momentarily relaxed the differential equation condition on $V$, calculate $\sigma_{RM}'\sigma_{RM}^{-1}$ and write the components in terms
of the normalized invariants using the Replacement Rule, Theorem \ref{reprule}, we obtain
\begin{equation}\label{ResQsV3NotZero} \sigma_{RM}'\sigma_{RM}^{-1} = \begin{pmatrix} 0& \iota(Y'') & \iota(Z'')\\ -\iota(Y'')&0&\iota(V_3')\\ -\iota(Z'') & -\iota(V_3') & 0\end{pmatrix}.\end{equation}
We thus see that $(2,3)$-component of $\sigma_{RM}'\sigma_{RM}^{-1}$ being zero,
which is what makes $\sigma_{RM}$ a Rotation Minimizing\ frame, yields a constraint on the symbolic invariant $\iota(V_3') $.
The invariantization of the differential equation for $V$ yields
\[\begin{pmatrix}\iota(V_1') \\ \iota(V_2')\\ \iota(V_3')\end{pmatrix} = -\iota(Y'')\begin{pmatrix}1 \\ 0\\ 0\end{pmatrix}.\]
Using calculations similar to those above, it can be seen that the first two components of this equation relate to the orthonormality of $V$
with respect to $P'$. We thus make the following definition.
\begin{defn}[Rotation Minimizing\ frame constraint]\label{NFproxy}
The equation $\iota(V_3')=0$ is denoted as the \emph{Rotation Minimizing\ frame constraint}.
\end{defn}
When deriving the differential syzygy needed in the sequel, we will write the (reduced) curvature matrix with respect to
$s$ for the Rotation Minimizing\ frame as
\begin{equation}\label{QsNnoCon}
\frac{{\rm d}}{{\rm d} s}({\sigma_{RM}})\sigma_{RM}^{-1} =\left(\begin{array}{ccc} 0 & \iota(Y'') & \iota(Z'')\\ -\iota(Y'') & 0 &\iota(V_3')\\ -\iota(Z'') & -\iota(V_3')&0\end{array}\right),\qquad \iota(V_3')=0.
\end{equation}
This is because we need to calculate the evolution of $\iota(V_3')$ with respect to time, for our application.
\subsection{The time evolution of the frame}
We now suppose that our curve $s\mapsto P(s)$ evolves in time. The time derivatives of our variables are denoted as
\[ \frac{{\rm d}}{{\rm d}t} P^{(n)} = P_t^{(n)}, \qquad \frac{{\rm d}}{{\rm d}t} V^{(n)} = V_t^{(n)}\]
and the action is, for
$g=(R,\mathbf{a})\in SO(3)\ltimes \mathbb{R}$, and all $n\ge 0$,
\[ \begin{array}{rcl} P_t^{(n)}\mapsto g\cdot P_t^{(n)} &=&R P_t^{(n)} \\
V_t^{(n)}\mapsto g\cdot V_t^{(n)}&=&R V_t^{(n)}.
\end{array}\]
The normalized differential invariants are the components of
\[ \iota(P^{(n)}_t) = \sigma_{RM} P^{(n)}_t,\qquad \iota(V^{(n)}_t) = \sigma_{RM} V^{(n)}_t,\quad n=0,1,2,\dots \]
The curvature matrix for the extended Rotation Minimizing\ frame, with respect to time, is
\begin{equation}\label{ExCurvMxTime}
\begin{array}{rcl}
\frac{{\rm d}}{{\rm d} t}\rho_{RM} \rho_{RM}^{-1}&=&\begin{pmatrix} \frac{{\rm d}}{{\rm d} t}\sigma_{RM}\, \sigma_{RM}^{-1} & -\sigma_{RM} P_t\\
0&0\end{pmatrix}\\[15pt] &=&
\begin{pmatrix}\frac{{\rm d}}{{\rm d} t}\sigma_{RM}\, \sigma_{RM}^{-1} &
\begin{matrix} -\iota(X_t)\\ -\iota(Y_t)\\-\iota(Z_t)\end{matrix}\\0&0\end{pmatrix}.
\end{array}\end{equation}
Calculating the invariant matrix $\frac{{\rm d}}{{\rm d} t}({\sigma_{RM}})\sigma_{RM}^{-1} \in \mathfrak{so}(3)$ yields
\[\begin{array}{rcl}
\frac{{\rm d}}{{\rm d} t}({\sigma_{RM}})\sigma_{RM}^{-1} &=&\left( \begin{array}{ccc} 0& P'_t\cdot V & P'_t\cdot (P'\times V)\\
-P'_t V & 0 & V_t\cdot (P'\times V)\\
-P'_t\cdot (P'\times V) & -V_t\cdot (P'\times V)&0\end{array}\right)\\[25pt]
&=&\left( \begin{array}{ccc} 0& \iota(Y'_t) & \iota(Z'_t)\\
-\iota(Y'_t) & 0 & \iota(V'_{3,t})\\
-\iota(Z'_t)& - \iota(V'_{3,t})&0\end{array}\right)\end{array}
\]
where we have used the Replacement Rule, Theorem \ref{reprule}, recalling $\sigma_{RM}P'=\begin{pmatrix} 1&0&0\end{pmatrix}^T$ and
$\sigma_{RM}V=\begin{pmatrix} 0&1&0\end{pmatrix}^T$.
Differentiating both sides of $\sigma_{RM} P' = (1,0,0)^T$ with respect to $t$ yields
\[\sigma_{RM} P'_t + \left(\frac{{\rm d}}{{\rm d} t}({\sigma_{RM}})\sigma_{RM}^{-1} \right) \left(\sigma_{RM} P'\right) =\begin{pmatrix}0\\0\\0\end{pmatrix}\]
so that indeed,
\[ \begin{pmatrix}\iota(X'_t)\\ \iota(Y'_t)\\ \iota(Z'_t)\end{pmatrix}= \begin{pmatrix}0\\ P'_t\cdot V\\ P'_t\cdot (P'\times V)\end{pmatrix}.\]
Further, differentiating both sides of $\sigma_{RM} V = (0,1,0)^T$ with respect to $t$ yields
\[ \sigma_{RM} V_t +\left(\frac{{\rm d}}{{\rm d} t}({\sigma_{RM}})\sigma_{RM}^{-1} \right) \left(\sigma_{RM} V\right) =\begin{pmatrix}0\\0\\0\end{pmatrix}\] so that
\[ \begin{pmatrix}\iota(V_{1,t})\\ \iota(V_{2,t})\\ \iota(V_{3,t})\end{pmatrix}= \begin{pmatrix}-P'_t\cdot V \\0\\ V_t\cdot (P'\times V)\end{pmatrix}.\]
\subsection{The syzygy operator $\mathcal{H}$}
Recall the extended Rotation Minimizing\ frame, $\rho_{RM}$, and the curvature matrices, $Q^s=\rho_{RM}'\rho_{RM}^{-1}$,
$Q^t=\frac{{\rm d}}{{\rm d} t}\rho_{RM}\rho_{RM}^{-1}$, Equations (\ref{rhoRMdef}), (\ref{ExCurvMx}), (\ref{ExCurvMxTime}) and
repeated here for convenience,
\begin{equation}\label{rhoRMdefN} \rho_{RM}=\begin{pmatrix} \sigma_{RM} & -\sigma_{RM} P \\ 0 & 1\end{pmatrix},\end{equation}
\begin{equation}\label{ExCurvMxN} Q^s=\rho_{RM}'\rho_{RM}^{-1} = \begin{pmatrix} \sigma_{RM}'\sigma_{RM}^{-1} & \begin{matrix} -\iota(X')\\ \phantom{-}0\\ \phantom{-} 0\end{matrix}\\ 0 & \phantom{-}0\end{pmatrix}\end{equation}
where we have not yet imposed the arc length constraint $\iota(X')=1$ since we need its time evolution, and
\begin{equation}\label{ExCurvMxTimeN}
Q^t=\frac{{\rm d}}{{\rm d} t}\rho_{RM} \rho_{RM}^{-1}=
\begin{pmatrix}\frac{{\rm d}}{{\rm d} t}\sigma_{RM}\, \sigma_{RM}^{-1} &
\begin{matrix} -\iota(X_t)\\ -\iota(Y_t)\\-\iota(Z_t)\end{matrix}\\0&0\end{pmatrix}.
\end{equation}
The non-constant components of $Q^s$ are the generating invariants of the algebra of invariants of the form
$F=F(P,P',P'',\dots, V,V',V'',\dots)$; every invariant of this form can be written as a function of $\iota(Y'')$, $\iota(Z'')$ and their
derivatives with respect to $s$.
The syzygy operator $\mathcal{H}$ that we need for our calculations in the Calculus of Variations,
relates the time derivatives
of these generating invariants to the $s$ derivatives of the components of $\iota(P_t)$ and $\iota(V_t)$, occurring in $Q^t$.
In our case here, the syzygy operator $\mathcal{H}$ can be calculated by examining the components of the compatibility condition
of the curvature matrices $ Q^s$ and $Q^t$,
\begin{equation}\label{CCrhoRM} \frac{{\rm d}}{{\rm d} t} Q^s - \frac{{\rm d}}{{\rm d} s}Q^t = \left[ Q^t, Q^s\right] \end{equation}
which follows from the fact the derivatives with respect to $t$ and $s$ commute (see \cite{Mansfield:2010aa}, \S 5.2).
We use $\sigma_{RM}'\sigma_{RM}^{-1} $ in the form of Equation (\ref{ResQsV3NotZero}), that is, with the Rotation Minimizing\ constraint
not yet imposed, as we will need its variation with respect to time in the sequel.
Calculating the components of Equation (\ref{CCrhoRM}) yields,
\begin{equation}\label{DiffSyz}
\begin{array}{rcl}
\frac{{\rm d}}{{\rm d} t} \iota(X') &=&\frac{{\rm d}}{{\rm d} s} \iota(X_t)-\iota(Y'')\iota(Y_t)-\iota(Z'')\iota(Z_t),\\
\frac{{\rm d}}{{\rm d} t} \iota(Y'') &=& \frac{{\rm d}^2}{{\rm d} s^2} \iota(Y_t)+\frac{{\rm d}}{{\rm d} s}\left(\iota(Y'')\iota(X_t)\right)+\iota(V_{3,t})\iota(Z''),\\
\frac{{\rm d}}{{\rm d} t} \iota(Z'') &=& \frac{{\rm d}^2}{{\rm d} s^2} \iota(Z_t)+\frac{{\rm d}}{{\rm d} s}\left(\iota(Z'')\iota(X_t)\right)-\iota(V_{3,t})\iota(Y''),\\
\frac{{\rm d}}{{\rm d} t} \iota(V_3')&=&\frac{{\rm d}}{{\rm d} s}\iota(V_{3,t}) +\iota(Y'')\frac{{\rm d}}{{\rm d} s}\iota(Z_t)-\iota(Z'')\frac{{\rm d}}{{\rm d} s}\iota(Y_t)
\end{array}\end{equation}
or in the form we require,
\[ \frac{{\rm d}}{{\rm d} t}\left(\begin{array}{c} \iota(X') \\ \iota(Y'') \\ \iota(Z'') \\
\iota(V_3') \end{array}\right) = \mathcal{H}
\left(\begin{array}{c}\iota(X_t) \\ \iota(Y_t) \\\iota(Z_t)\\ \iota(V_{3,t})\end{array}\right)\]
where
\begin{equation}\label{Hdef}
\mathcal{H}=\left(\begin{array}{cccc}
\frac{{\rm d}}{{\rm d} s} & -\iota(Y'') & \iota(Z'') & 0 \\
\iota(Y'')\frac{{\rm d}}{{\rm d} s}+\frac{{\rm d}}{{\rm d} s}\iota(Y'') & \frac{{\rm d}^2}{{\rm d} s^2} & 0 & \iota(Z'')\\
\iota(Z'')\frac{{\rm d}}{{\rm d} s}+\frac{{\rm d}}{{\rm d} s}\iota(Z'') & 0 & \frac{{\rm d}^2}{{\rm d} s^2} & -\iota(Y'')\\
0 & -\iota(Z'')\frac{{\rm d}}{{\rm d} s} & \iota(Y'')\frac{{\rm d}}{{\rm d} s} &\frac{{\rm d}}{{\rm d} s}
\end{array}\right).
\end{equation}
We note that $\mathcal{H}$ is an invariant, linear, matrix differential operator.
\section{Invariant calculus of variations}\label{ELsection}
We consider an $SE(3)$ invariant Lagrangian of the form
\[ \mathcal{L}[X',Y',Z',X'',Y'',Z'',...] = \int L(\kappa_1, \kappa_2,\kappa_{1,s},\kappa_{2,s},...) + \mu \zeta + \lambda(\eta -1)\, {\rm d}s\]
where we have set $\zeta=\iota(V_3')$, $\eta = \iota(X')$, $\kappa_1 = \iota(Y'')$ and $\kappa_2=\iota(Z'')$, and
where $\mu$ and $\lambda$ are Lagrange multipliers for the Rotation Minimizing\ frame constraint (Definition \ref{NFproxy}) and the arc-length constraint (Definition \ref{arclengCon}) respectively.
Recall the Euler operator with respect to a dependent variable $u$ is defined by
\[ \mathrm{E}^u(L) = \sum_n (-1)^n \frac{{\rm d}^n}{{\rm d}s^n} \frac{\partial L}{\partial u^{(n)}} \]
where $u^{(n)} = \frac{{\rm d}^n}{{\rm d}s^n} u$.
We apply the invariantized version of the calculation of the Euler--Lagrange equations (see \cite{Mansfield:2010aa}, \S 7.3, also \cite{GonMan2}),
to obtain
\[ 0=\begin{pmatrix} \mathrm{E}^X \\ \mathrm{E}^Y \\ \mathrm{E}^Z \\ \mathrm{E}^{V_3} \end{pmatrix}=\mathcal{H}^* \begin{pmatrix} \mathrm{E}^{\eta}\\ \mathrm{E}^{\kappa_1} \\ \mathrm{E}^{\kappa_2} \\ \mathrm{E}^{\zeta}\end{pmatrix}\]
that is,
\begin{align}\label{eulerlagrangeequations}
0=\mathrm{E}^X\,&=-\kappa_1\frac{{\rm d}}{{\rm d} s} \mathrm{E}^{\kappa_1} -\kappa_2\frac{{\rm d}}{{\rm d} s} \mathrm{E}^{\kappa_2} - \lambda_s, \\
0=\mathrm{E}^{Y}\,&=\frac{{\rm d}^2}{{\rm d} s^2} \mathrm{E}^{\kappa_1} +\frac{{\rm d}}{{\rm d} s}\left( \kappa_2 \mu \right)- \kappa_1 \lambda ,\\
0=\mathrm{E}^{Z}\,&=\frac{{\rm d}^2}{{\rm d} s^2} \mathrm{E}^{\kappa_2} -\frac{{\rm d}}{{\rm d} s} \left( \kappa_1 \mu \right) - \kappa_2 \lambda ,\\
0=\mathrm{E}^{V_3}&=\mathrm{E}^{\kappa_1} \kappa_2- \mathrm{E}^{\kappa_2} \kappa_1 - \mu_s.
\end{align}
\begin{rem}
Note that
\[
-\kappa_1\frac{{\rm d}}{{\rm d} s}\mathrm{E}^{\kappa_1} -\kappa_2\frac{{\rm d}}{{\rm d} s}\mathrm{E}^{\kappa_2} = -\frac{{\rm d}}{{\rm d} s}\left( \kappa_1 \mathrm{E}^{\kappa_1} + \kappa_2 \mathrm{E}^{\kappa_2}\right) +\kappa_{1,s} \mathrm{E}^{\kappa_1}+\kappa_{2,s} \mathrm{E}^{\kappa_2}.
\]
Also, by arguments similar to that of equation $(7.17)$ in \cite{Mansfield:2010aa} we have
\begin{align*}
& \kappa_{1,s} \mathrm{E}^{\kappa_1} +\kappa_{2,s} \mathrm{E}^{\kappa_1} \\
&\qquad= \frac{{\rm d}}{{\rm d} s} \left(L - \sum_{m=1} \sum^{m-1}_{k=0} {\left(-1\right)}^k \left(\left(\frac{{\rm d}^k}{{\rm d} s^k} \frac{\partial L}{\partial \kappa_{1,m}}\right)\kappa_{1,m-k}
+ \left(\frac{{\rm d}^k}{{\rm d} s^k} \frac{\partial L}{\partial \kappa_{2,m}}\right)\kappa_{2,m-k}\right)\right).
\end{align*}
Therefore, $\lambda_s$ is a total derivative and we obtain
\begin{equation}\label{lambda}
\lambda = - \kappa_1 \mathrm{E}^{\kappa_1} - \kappa_2 \mathrm{E}^{\kappa_2} + L - \sum_{m=1} \sum^{m-1}_{k=0} {\left(-1\right)}^k \left(\left(\frac{{\rm d}^k}{{\rm d} s^k} \frac{\partial L}{\partial \kappa_{1,m}}\right)\kappa_{1,m-k}
- \left(\frac{{\rm d}^k}{{\rm d} s^k} \frac{\partial L}{\partial \kappa_{2,m}}\right)\kappa_{2,m-k}\right)
\end{equation}
where the constant of integration has been absorbed into $\lambda$ by Remark $7.1.9$ of \cite{Mansfield:2010aa}. This result for $\lambda$ relates
to the invariance of the Lagrangian under translation in $s$, that is, we have invariance under $s\mapsto s+\epsilon$ and hence a
corresponding Noether law.
\end{rem}
To apply the results of \cite{GonMan2} to obtain the Noether conservation laws, we need to calculate the infinitesimals of our group action,
its associated \textit{matrix of infinitesimals}, and the right Adjoint action of the Lie group $SE(3)$ on the infinitesimal vector fields. Here we give the basic definitions, for completeness. For the Lie group $SE(3)$ and the left linear action, the precise calculations appear in \cite{GonMan2} with the end results needed for our case here recorded in the proof of the following Theorem.
Elements in the Lie group $SE(3)$ are, in a neighbourhood of the identity element, described by six parameters, three translation parameters,
$a$, $b$ and $c$, and
three rotation parameters, $\theta_{xy}$, $\theta_{yz}$ and $\theta_{xz}$ where $\theta_{xy}$ is the (anticlockwise) rotation in the $(x,y)$-plane,
and similarly for $\theta_{yz}$ and $\theta_{xz}$. For a point with coordinates $(X,Y,Z, X',Y', Z',V_1,V_2,V_3,\dots)=(z_1, z_2, \dots)$, we define the infinitesimal vector field with respect to the group parameter $a_i$ to be
\[ \mathbf{v}_{a_i} = \sum_j \frac{\partial g\cdot z_j}{\partial a_i}\Big\vert_{g=e}\, \partial_{z_j}\]
where $e$ is the identity element of the group. The \textit{matrix of infinitesimals} is then the matrix
\[ \Phi=(\phi_{ij}),\qquad \phi_{ij} = \frac{\partial g\cdot z_j}{\partial a_i}\Big\vert_{g=e}\]
and the invariantized matrix of infinitesimals is
\[ \Phi(I)=(\iota(\phi_{ij})).\]
Finally, given an infinitesimal vector field $\mathbf{v}=\sum_j \xi^j(z)\partial_{z_j}$, the right Adjoint action of $G$ on $\mathbf{v}$ is given by
\[ \sum_j \xi^j(z)\partial_{z_j}\mapsto \sum_j \xi^j(g\cdot z)\partial_{g\cdot z_j}.\]
This determines a linear map, $\mathbf{v}_{a_i}\mapsto \sum_{k}(\mathcal{A}d(g) )_{ik}\mathbf{v}_{a_k}$ called the right Adjoint action of
$G$ on its Lie algebra of vector fields. (See \cite{Mansfield:2010aa} for further details.)
Continuing to apply the results of \cite{GonMan2}, we obtain that Noether's laws are as given in the following theorem,
\begin{theorem} The conservation laws are of the form
\begin{equation}\label{NCL}
\left(\begin{array}{cccccc} \sigma_{RM}^T & 0 \\ D {\bf X} \sigma_{RM}^T & D \sigma_{RM}^T D \end{array}\right)\left( \begin{array}{c} \lambda \\ -\frac{{\rm d}}{{\rm d} s} \mathrm{E}^{\kappa_1} - \mu \kappa_2 \\ -\frac{{\rm d}}{{\rm d} s} \mathrm{E}^{\kappa_2} + \mu \kappa_1 \\ \mu \\ \mathrm{E}^{\kappa_2} \\ \mathrm{E}^{\kappa_1} \end{array}\right) = \left( \begin{array}{c} c_1 \\ c_2 \\ c_3 \\ c_4 \\ c_5 \\ c_6 \end{array}\right)
\end{equation}
where
\[
{\bf X}=\left(\begin{array}{ccc} 0 & -Z & Y\\Z&0&-X\\ -Y& X &0\end{array}\right)
\]
$D=\textrm{diag}(1,-1,1)$, and the $c_i$ are constants.
\end{theorem}
\begin{proof}
In order to compute the conservation laws, we need the boundary terms $\mathcal{A}_{\mathcal{H}}$, the (right) Adjoint representation
of the frame $\rho_{RM}$ and the invariantized matrix of infinitesimals, which we defined above. We now consider these in turn.
Let $\mathrm{E}(L)=\begin{pmatrix} \mathrm{E}^{\eta} & \mathrm{E}^{\kappa_1} & \mathrm{E}^{\kappa_2} & \mathrm{E}^{\zeta} \end{pmatrix}$ and let
$\phi^t=\begin{pmatrix} \iota(X_t) & \iota(Y_t) & \iota(Z_t) & \iota(V_{3,t})\end{pmatrix}^T$. Then
the boundary terms $\mathcal{A}_{\mathcal{H}} $ are defined by
\[ \frac{{\rm d}}{{\rm d} s} \mathcal{A}_{\mathcal{H}} = \mathrm{E}(L) \mathcal{H}\phi^t - \mathcal{H}^* \mathrm{E}(L) \phi^t.\]
By direct calculation, we obtain
\begin{align*}
\mathcal{A}_{\mathcal{H}}&= \lambda \iota(X_t) + \left( -\frac{{\rm d}}{{\rm d} s} \mathrm{E}^{\kappa_1} - \mu \kappa_2\right) \iota(Y_t) + \left( -\frac{{\rm d}}{{\rm d} s} \mathrm{E}^{\kappa_2} + \mu \kappa_1\right) \iota(Z_t) \\ & \qquad + \mathrm{E}^{\kappa_1} \iota(Y'_t) + \mathrm{E}^{\kappa_2} \iota(Z'_t) + \mu \iota(V_{3,t})\\
&= \mathcal{C}^X \iota(X_t) + \mathcal{C}^Y \iota(Y_t) + \mathcal{C}^Z \iota(Z_t) + \mathcal{C}^{Y'}\iota(Y'_t)+ \mathcal{C}^{Z'}\iota(Z'_t)
+\mathcal{C}^{V_{3,t}} \iota(V_{3,t})
\end{align*}
where this defines the coefficients $\mathcal{C}$ and where we have used the syzygies
\begin{align*}
\iota(Y'_t) & = \frac{{\rm d}}{{\rm d} s} \iota(Y_t) + \kappa_1 \iota(X_t),\\
\iota(Z'_t) & = \frac{{\rm d}}{{\rm d} s} \iota(Z_t) + \kappa_2 \iota(X_t)
\end{align*}
to eliminate derivatives of $\iota(Y_t)$ and $\iota(Z_t)$ in the boundary terms.
In \cite{GonMan2}, the authors show the (right) Adjoint representation of ${\rm S}E(3)$ with respect to the generating infinitesimal vector fields of the action,
\begin{equation}\label{VOI}
\begin{array}{lll}
{\bf v}_{a} = \partial_X, & {\bf v}_{b} = \partial_Y, & {\bf v}_{c} = \partial_Z, \\ {\bf v}_{YZ} = Y \partial_Z - Z \partial_Y, & {\bf v}_{XZ} = X \partial_Z - Z \partial_X, & {\bf v}_{XY} = X \partial_Y -Y \partial_X\end{array}
\end{equation}
is of the form, for $g=(R,\mathbf{a})$,
\[
\mathcal{A}d(g)=\left(\begin{array}{cc} R & 0 \\ DAR & DRD \end{array}\right)
\]
where $R \in {\rm S}O(3)$, $D$ is the diagonal matrix $D=\mbox{diag}(1,-1,1)$ and $A$ is the matrix
\[
A=\left(\begin{array}{ccc} 0 & -c & b\\c&0&-a\\ -b& a &0\end{array}\right)
\]
where $\mathbf{a}={(a,b,c)}^{T}$ is the translation vector component of $g$.
Hence
\[
{\mathcal{A}d(\rho_{RM})}^{-1}=\left(\begin{array}{cccccc} \sigma_{RM}^{T} & 0 \\ D {\bf X} \sigma_{RM}^{T} & D \sigma_{RM}^{T} D \end{array}\right)
\]
where
\[
{\bf X}=\left(\begin{array}{ccc} 0 & -Z & Y\\Z&0&-X\\ -Y& X &0\end{array}\right).
\]
The invariantized matrix of infinitesimals with respect to the basis \eqref{VOI} is
\[ \Phi(I) =\bordermatrix{ & X & Y & Z & Y' & Z' & V_3
\cr a & 1 & 0 & 0 & 0 & 0 & 0
\cr b &0&1&0& 0 & 0 & 0
\cr c & 0&0&1& 0 & 0 & 0
\cr\theta_{yz} & 0 & 0 & 0& 0 & 0 & 1
\cr \theta_{xz} &0&0&0& 0 & 1 & 0
\cr\theta_{xy} & 0&0&0& 1 & 0 & 0
\cr}.\]
Finally, the conservation laws obtained via Noether's theorem for the unidimensional case are, see \cite{GonMan2},
\begin{equation}\label{conservatiolaws}
{\mathcal{A}d(\rho)}^{-1} v(I)={\bf c}
\end{equation}
where
\begin{equation}\label{infinitesimals}
v(I)=\sum_{\alpha} \Phi^{\alpha}(I) \mathcal{C}^{\alpha}={\left( \begin{array}{cccccc} \lambda & -\frac{{\rm d}}{{\rm d} s} \mathrm{E}^{\kappa_1} - \mu \kappa_2 & -\frac{{\rm d}}{{\rm d} s} \mathrm{E}^{\kappa_2} + \mu \kappa_1 & \mu & \mathrm{E}^{\kappa_2} & \mathrm{E}^{\kappa_1} \end{array}\right)}^T
\end{equation}
as required.
\end{proof}
\begin{rem} A quick check on this result is obtained by noting the following.
Differentiating \eqref{NCL} with respect to $s$ and multiplying by $\mathcal{A}d(\rho_{RM})$, we get
\[
\frac{{\rm d}}{{\rm d} s}v(I)=\frac{{\rm d}}{{\rm d} s}\left(\mathcal{A}d(\rho_{RM}) \right){\mathcal{A}d(\rho)}^{-1} v(I)
\]
i.e,
\begin{equation}\label{diffv}
\frac{{\rm d}}{{\rm d} s} v(I) = \left( \begin{array}{cccccc} 0 & \kappa_1 & \kappa_2 & 0 & 0 & 0 \\ -\kappa_1 & 0 & 0 & 0 & 0 & 0 \\ -\kappa_2 & 0 & 0 & 0 &0 & 0\\ 0 & 0 & 0 & 0 & -\kappa_1 & \kappa_2 \\ 0 & 0 & -1 & \kappa_1 & 0 & 0 \\ 0 & -1 & 0 & -\kappa_2 & 0 & 0 \end{array}\right) v(I).
\end{equation}
We observe that the first four rows are equivalent to the Euler-Lagrange equations while last two rows are identically 0, as expected.
\end{rem}
\section{Solution of the integration problem}\label{IPsection}
The conservation laws \eqref{NCL} can reduce the integration problem. We write these in the form,
\begin{equation}\label{NCLinv}
\left(\begin{array}{cccccc} \sigma^T_{RM} & 0 \\ D {\bf X} \sigma^T_{RM} & D \sigma^T_{RM} D \end{array}\right)\left(\begin{array}{c} \textbf{w}_1(I)\\ \textbf{w}_2(I)\end{array}\right)=\left(\begin{array}{c} \textbf{c}_1\\ \textbf{c}_2\end{array}\right)
\end{equation}
where $v(I)=( \textbf{w}_1(I), \textbf{w}_2(I))^T$, $\textbf{c}=(\textbf{c}_1,\textbf{c}_2)^T$ and recalling
\[
\mathbf{X}=\left(\begin{array}{ccc} 0 & -Z & Y\\Z&0&-X\\ -Y& X &0\end{array}\right).
\]
Since $\sigma_{RM}\in {\rm S}O(3)$ we have from
\begin{equation}\label{CLeqnCompOne} \sigma_{RM} \textbf{c}_1=\textbf{w}_1(I)\end{equation}
that
\begin{equation}\label{modcweqn}
|\textbf{c}_1|=|\textbf{w}_1(I)|.
\end{equation}
Further, multiplying the second component of Equation (\ref{NCLinv}) on the left by $ \textbf{c}_1(I)^T D$, since $D^2=I$, we obtain
\begin{equation}\label{SecondIntELeqn}
\textbf{w}_1^TD\textbf{w}_2 = \textbf{c}_1^TD\textbf{c}_2.
\end{equation}
In order to solve Equation (\ref{CLeqnCompOne}), as far as we can, for the components of $\sigma_{RM}$ in terms of the components of
$\textbf{c}_1$ and $\textbf{w}_1(I)$, we use the Cayley representation $\Phi$ of elements of $SO(3)$. We define
\[ \Phi(x_1,x_2,x_3,x_4)=\left(\begin{array}{ccc}
x_1^2+x_2^2-x_3^3-x_4^2 & -2(x_1x_4-x_2x_3) & 2(x_1x_3+x_2x_4)\\
2(x_1x_4+x_2x_3) & x_1^2-x_2^2+x_3^3-x_4^2 & -2(x_1x_2-x_3x_4)\\
-2(x_1x_3-x_2x_4)& 2(x_1x_2+x_3x_4) & x_1^2-x_2^2-x_3^3+x_4^2\end{array}\right).
\]
Then provided $x_1^2+x_2^2+x_3^3+x_4^2=1$, $\Phi(x_1,x_2,x_3,x_3)\in SO(3)$, has an axis of rotation $(x_2, x_3, x_4)^T$ and
the angle of rotation $\psi$ satisfies $2 x_1^2-1 = \cos\psi$. Hence we may define, for an angle $\psi$ and axis of rotation $\textbf{a}=(a_1,a_2, a_3)^T\ne 0$,
\[ R(\psi, \textbf{a}) = \Phi\left(\cos\left(\frac{\psi}2\right), \sin\left(\frac{\psi}2\right) \frac{a_1}{|\mathbf{a}|},\sin\left(\frac{\psi}2\right) \frac{a_2}{|\mathbf{a}|},
\sin\left(\frac{\psi}2\right) \frac{a_3}{|\mathbf{a}|}\right) \in {\rm S}O(3).\]
There are two cases.
\noindent \textbf{Case 1}.\quad If $\mathbf{w}_1 + \mathbf{c}_1$ is bounded away from zero,
we note that $\sigma_{RM}$ may be taken to be a product of a rotation about $\mathbf{c}_1 + {(0,0,|\mathbf{c}_1|)}^{T}$ with angle $\pi$ followed by a rotation about ${(0,0,|\mathbf{c}_1|)}^{T}$ with any angle $\psi$ and a rotation about $\mathbf{w}_1 + {(0,0,|\mathbf{c}_1|)}^{T}$ with angle $\pi$, that is,
\[ \sigma_{RM}=R(\pi, \mathbf{w}_1 + {(0,0,|\mathbf{c}_1|)}^{T})R(\psi(s), {(0,0,|\mathbf{c}_1|)}^{T})R(\pi, \mathbf{c}_1 + {(0,0,|\mathbf{c}_1|)}^{T} ).\]
This solves for $\sigma_{RM}$ up to the angle $\psi$.
If we differentiate this with respect to $s$, right multiply by $\sigma_{RM}^{-1}$
\[ \sigma^{-1}_{RM}=R(\pi, \mathbf{c}_1 + {(0,0,|\mathbf{c}_1|)}^{T})R(-\psi(s), {(0,0,|\mathbf{c}_1|)}^{T})R(\pi, \mathbf{w}_1 + {(0,0,|\mathbf{c}_1|)}^{T} )\]
using \eqref{diffv} and taking into account that
\[
\frac{{\rm d}}{{\rm d} s}(\sigma_{RM})\sigma_{RM}^{-1}=\left(\begin{array}{ccc} 0 & \kappa_1 & \kappa_2 \\ -\kappa_1 & 0 & 0 \\ -\kappa_2 & 0 & 0 \end{array}\right)
\] we obtain a remarkable equation for
$\psi$, specifically,
\begin{equation}\label{PsiEqnCaseOne}
\psi_s = - \kappa_1 +
\frac{v_2(I)}{|\mathbf{c}_1| + v_3(I)}\, \kappa_2
\end{equation}
where recall $v_2(I)$ and $v_3(I)$ are the second and third components of the vector of invariants, $v(I)$, an also, be definition,
the second and third components of $\mathbf{w}_1$.
\noindent \textbf{Case 2}.\quad If $\mathbf{w}_1 - \mathbf{c}_1$ is bounded away from zero, we note that $\sigma_{RM}$ may be taken to be a product of a rotation about $\mathbf{c}_1 + {(0,0,-|\mathbf{c}_1|)}^{T}$ with angle $\pi$ followed by a rotation about ${(0,0,-|\mathbf{c}_1|)}^{T}$ with any angle $\psi$ and a rotation about $\mathbf{w}_1 + {(0,0,-|\mathbf{c}_1|)}^{T}$ with angle $\pi$, that is,
\[ \sigma_{RM}=R(\pi, \mathbf{w}_1 + {(0,0,-|\mathbf{c}_1|)}^{T})R(\psi(s), {(0,0,-|\mathbf{c}_1|)}^{T})R(\pi, \mathbf{c}_1 + {(0,0,-|\mathbf{c}_1|)}^{T} ).\]
Since the matrix on the right and the matrix on the left are constant, we obtain the same equation for $\psi$ as above, but with the signs of $\mathbf{c}_1$ reversed.
Hence in this case,
\begin{equation}\label{PsiEqnCaseTwo}
\psi_s = \kappa_1 +
\frac{v_2(I)}{|\mathbf{c}_1| - v_3(I)}\, \kappa_2.
\end{equation}
In either case, we obtain $\sigma_{RM}$ up to a quadrature. There is a significant overlap in the domains of the two cases, and matching one to the other, as needed, is not a problem.
Next, we seek $P$. We note the first row of $\sigma_{RM}$ is $P'$,
and so we can always obtain $P$ by quadrature. However, we note that only one component needs to be calculated this way, as the
second component of Equation (\ref{NCLinv}) provides algebraic equations for two of the components of $P$, i.e,
\begin{align*}
X &=\frac{1}{v_3(I)} (v_4(I) + Zv_2(I) - {(\sigma D {\bf c_2})}_1), \\
Y &= \frac{1}{v_3(I)} (v_5(I) + Zv_1(I) + {(\sigma D {\bf c_2})}_2) \\
\end{align*}
where $Z$ has been solved previously by quadrature.
We conclude by noting that the conservation laws provide two first integrals of the Euler--Lagrange equations.
They may be used to solve for $P$ in terms of two quadratures, and they also solve for the normal vector $V$ in terms of
one quadrature, that of $\psi$.
Finally, we note that it is easy to obtain the Frenet--Serret\ frame from our calculations, since it is defined in terms of $P'$ and $P''$.
\section{Examples and applications}\label{EXsection}
We examine a Lagrangian which is not possible to study in the Frenet--Serret\ framework. Secondly, we study functionals used to model some biological structures, invariant under ${\rm S}E(3)$ and depending on the curvature, torsion and their derivatives, but using our results for the Rotation Minimizing\ frame.
We first show that every Lagrangian which can be written in terms of the Euclidean curvature $\kappa$ and torsion $\tau$ can be written
in terms of the invariants, $\kappa_1$ and $\kappa_2$.
From \eqref{k1k2} we have that
\[
\kappa_1 = \kappa\cos{\theta}, \qquad \kappa_2 = \kappa \sin{\theta}
\]
and therefore, using $\tan\theta = \kappa_2/\kappa_1$ and $\theta_s=\tau$ we have,
\begin{equation}\label{RMtoFS}
\kappa=\sqrt{\kappa_1^2 + \kappa_2^2}, \qquad \tau = \frac{\kappa_1 \kappa_{2,s} - \kappa_{1,s}\kappa_2}{\kappa_1^2 + \kappa_2^2}.
\end{equation}
But the converse is not true. Lagrangians which depend only on $\kappa_2/\kappa_1$ cannot be written in terms of $\kappa$ and $\tau$.
Our first example is the simplest such Lagrangian, which we study simply because we can.
\subsection{Invariant Lagrangians involving only $\kappa_2/\kappa_1$}\label{tantheta}
Let us consider the Lagrangian
\begin{align*}
\mathcal{L}[\kappa_2/\kappa_1]
&= \int \frac{1}{2} {\left( \frac{\kappa_2}{\kappa_1}\right)}^2 + \lambda \left(\eta - 1 \right) + \mu \zeta \ {\rm d}s\\
&=\int \tan{\theta}^2 + \lambda \left(\eta - 1 \right) + \mu \zeta \ {\rm d}s
\end{align*}
where recall $\eta=1$ is the arc-length constraint and $\zeta=0$ is the Rotation Minimizing\ constraint.
Using the results of the previous section,
we obtain the Euler--Lagrange equations
\begin{equation}\label{tanEL1}
\begin{aligned}
&\left( -\frac{12 {\frac{{\rm d}}{{\rm d} s}\kappa_1}^2}{\kappa_1^5} +\frac{3 \frac{{\rm d}^2}{{\rm d} s^2}\kappa_1}{\kappa_1^4} - \frac{1}{2\kappa_1} \right) \kappa_2^2 +
\left( \frac{12 {\frac{{\rm d}}{{\rm d} s}\kappa_1}{\frac{{\rm d}}{{\rm d} s}\kappa_2}}{\kappa_1^4} -\frac{2 \frac{{\rm d}^2}{{\rm d} s^2}\kappa_2}{\kappa_1^3} - \mu_s \right) \kappa_2 \\&
\qquad \qquad -\frac{2\frac{{\rm d}}{{\rm d} s}\kappa_2^2}{\kappa_1^3}+\mu \frac{{\rm d}}{{\rm d} s} \kappa_2 = 0,\\
\end{aligned}
\end{equation}
\begin{align}
& \label{tanEL2}-\frac{\kappa_2^3}{2\kappa_1^2} + \left( \frac{6 \frac{{\rm d}}{{\rm d} s}\kappa_1^2}{\kappa_1^4} - \frac{2\frac{{\rm d}^2}{{\rm d} s^2}\kappa_1}{\kappa_1^3} \right)\kappa_2 \frac{\frac{{\rm d}^2}{{\rm d} s^2} \kappa_2}{\kappa_1^2} -\frac{4\frac{{\rm d}}{{\rm d} s}\kappa_1\frac{{\rm d}}{{\rm d} s}\kappa_2}{\kappa_1^3}-\mu_s \kappa_1 -\mu \frac{{\rm d}}{{\rm d} s}\kappa_1 = 0,\\
&\mu_s + \frac{\kappa_2^3}{\kappa_1^3} + \frac{\kappa_2}{\kappa_1} =0
\end{align}
where $\lambda= \frac{1}{2}{\left(\frac{\kappa_2}{\kappa_1}\right)}^2$ has been solved using \eqref{lambda}.
Further, the vector of invariants $v(I)$ needed for the conservation laws is
\[
v(I)=\left( \begin{array}{c} \frac{1}{2}{\left(\frac{\kappa_2}{\kappa_1}\right)}^2 \\
-\frac{\kappa_2}{\kappa_1^4}(\kappa_1^4\mu-2\kappa_1\frac{{\rm d}}{{\rm d} s}\kappa_2 + 3\kappa_2 \frac{{\rm d}}{{\rm d} s}\kappa_1)\\
-\frac{\frac{{\rm d}}{{\rm d} s}\kappa_2}{\kappa_1^2} + \frac{2 \kappa_2\frac{{\rm d}}{{\rm d} s}\kappa_1}{\kappa_1^3} + \mu\kappa_1\\
\mu \\
\frac{\kappa_2}{\kappa_1^2}\\
-\frac{\kappa_2^2}{\kappa_1^3}
\end{array}\right).
\]
Solving \eqref{tanEL1}, \eqref{tanEL2} along with \eqref{SecondIntELeqn}, \eqref{PsiEqnCaseOne} and \eqref{PsiEqnCaseTwo} for $\kappa_1,\kappa_2,\mu$ and $\psi$ with initial conditions
\[
\begin{array}{llll}
\kappa_1(0) = 1, &\kappa_2(0) = \frac{1}{2}, &\frac{{\rm d}}{{\rm d} s}\kappa_1(0) = 1,& \frac{{\rm d}}{{\rm d} s}\kappa_2(0) = 1,\\[10pt]
\lambda(0) = 1, &\mu(0) = 1, &Z(0)=1, &\psi(0)=0
\end{array}
\]
we obtain the following solutions, see Figures \ref{App0a}, \ref{App0b}, \ref{App0c}.
\begin{figure}
\caption{Solutions for the invariants $\kappa_1$, $\kappa_2$, $\theta$ and $\kappa$\label{App0a}}
\vspace{5mm}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{kappa1vskappa2.pdf} &
\includegraphics[width=0.3\textwidth]{thetas.pdf} &
\includegraphics[width=0.3\textwidth]{svskappa.pdf}\\
$\kappa_1$ vs $\kappa_2$ & $s$ vs $\theta$ & $s$ vs $\kappa$ \\
\end{tabular}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\caption{Plots of the first integrals\label{App0b}}
\vspace{5mm}
\begin{tabular}{cc}
\includegraphics[width=0.3\textwidth]{firstintegral} &
\includegraphics[width=0.3\textwidth]{secondintegral} \\
$v_1^2+v_2^2+v_3^2=c_1^2+c_2^2+c_3^3$ & $v_1v_4-v_2v_5+v_3v_6=c_1c_4-c_2c_5+c_3c_6$ \\
\end{tabular}
\end{center}
\end{figure}
\begin{figure}
\caption{Sweep surfaces using the Rotation Minimizing\ frame and the Frenet--Serret\ frame along the extremal curve\label{App0c}}
\vspace{5mm}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\textwidth]{SNF1} &
\includegraphics[width=0.4\textwidth]{SFS1} \\
Plot of $V$ along the extremal curve & Plot of $P''$ along the extremal curve \\
using the Rotation Minimizing\ frame & using the Frenet--Serret frame \\
\end{tabular}
\end{figure}
\pagebreak
\subsection{Further examples}
In order to model strands of proteins, nucleid acids and polymers, some authors have made use of the classic calculus of variations and studied the Euler--Lagrange equations of an energy functional depending on the curvature, torsion and their first derivatives.
In \cite{McCoy} and \cite{TMH} the authors consider protein backbones and polymers as a smooth curve in $\mathbb{R}^3$ and use the Frenet--Serret equations in order to compute a variation to the curve. The Euler--Lagrange equations are obtained for these type of functionals. In \cite{FNS} the same method is used to obtain the Euler--Lagrange equations for functionals which are linear in the curvature.
In this section we study two examples from the families of functionals studied, but in terms of the invariants $\kappa_1$ and $\kappa_2$.
The conversion of a functional given in terms of Euclidean curvature and torsion to one given in terms of $\kappa_1$ and $\kappa_2$ is given in Equation \eqref{RMtoFS}.
\subsubsection{The Lagrangian $\int \kappa^2 \tau \ {\rm d}s = \int \kappa_1 \kappa_{2,s}-\kappa_{1,s}\kappa_2 \ {\rm d}s$}
For the Lagrangian
\[
\int \kappa_1 \kappa_{2,s} - \kappa_{1,s}\kappa_2 \ {\rm d}s
\]
the Euler--Lagrange equations are
\begin{align}\label{eulerlagrangelag2a}
2 \kappa_{2,sss}+3\kappa_{2,s}\kappa^2&=0,\\\label{eulerlagrangelag2b}
-2 \kappa_{1,sss}-3\kappa_{1,s}\kappa^2&=0.
\end{align}
The conservation laws are of the form \eqref{conservatiolaws} where
\[
v(I)=(2(\kappa_{1,s}\kappa_2-\kappa_1\kappa_{2,s}) \quad -2\kappa_{2,ss} - \kappa_2\kappa^2 \quad -2\kappa_{1,ss} + \kappa_1\kappa^2 \quad \kappa^2 \quad -2\kappa_{1,s} \quad 2\kappa_{2,s} )^T.
\]
Solving \eqref{eulerlagrangelag2a}, \eqref{eulerlagrangelag2b} along with \eqref{PsiEqnCaseOne} and \eqref{PsiEqnCaseTwo} for
$\kappa_1,\kappa_2$ and $\psi$ with initial conditions
\[
\begin{array}{llll}
\kappa_1(0) = 1, &\kappa_2(0) = \frac{1}{2}, &\frac{{\rm d}}{{\rm d} s}\kappa_1(0) = 1,& \frac{{\rm d}}{{\rm d} s}\kappa_2(0) = 1,\\[10pt]
\frac{{\rm d}^2}{{\rm d} s^2}\kappa_1(0) = 1,& \frac{{\rm d}^2}{{\rm d} s^2}\kappa_2(0) = 1, &\psi(0)=0 &
\end{array}
\]
and integrating to obtain the extremizing curve and its Rotation Minimizing\ frame, we obtain the following solutions, see Figures \ref{App1a}, \ref{App1b}, \ref{App1c}.
\begin{figure}
\caption{Solutions for the invariants $\kappa_1$, $\kappa_2$, $\theta$ and $\kappa$\label{App1a}}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{k1k2lag2} &
\includegraphics[width=0.3\textwidth]{thetalag2} &
\includegraphics[width=0.3\textwidth]{kappalag2}\\
$\kappa_1$ vs $\kappa_2$ & $s$ vs $\theta$ & $s$ vs $\kappa$ \\
\end{tabular}
\begin{center}
\caption{Plots of the conservation laws\label{App1b}}
\begin{tabular}{cc}
\includegraphics[width=0.3\textwidth]{cl1llag1} &
\includegraphics[width=0.3\textwidth]{cl2lag2} \\
$v_1^2+v_2^2+v_3^2=c_1^2+c_2^2+c_3^3$ & $v_1v_4-v_2v_5+v_3v_6=c_1c_4-c_2c_5+c_3c_6$ \\
\end{tabular}
\end{center}
\caption{Sweep surface using $V$ from the Rotation Minimizing\ frame along the extremal curve\label{App1c}}
\vspace{5mm}
\centering
\includegraphics[width=0.6\textwidth]{plotlag2c}
\end{figure}
\subsubsection{The Lagrangian $\int \kappa_{1,s} \kappa_{2,ss} - \kappa_{1,ss}\kappa_{2,s} \ {\rm d}s$}
We now consider
\[
\int \kappa_{1,s} \kappa_{2,ss} - \kappa_{1,ss}\kappa_{2,s} \ {\rm d}s
\]
In terms of the Euclidean curvature and torsion, this Lagrangian can be written as
$\int \left(\kappa^2\tau^3 + \tau\left(2\kappa_s^2-\kappa\kappa_{ss}\right)+\kappa\kappa_s\tau_s\right)\,{\rm d}s$.
The Euler--Lagrange equations are
\begin{align}\label{eulerlagrangelag3a}
-2\kappa_{2,ssss}+\frac{{\rm d}}{{\rm d} s}(\kappa_2 \mu)-\kappa_1\lambda&=0,\\\label{eulerlagrangelag3b}
2\kappa_{1,ssss}-\frac{{\rm d}}{{\rm d} s}(\kappa_1 \mu)-\kappa_2\lambda&=0
\end{align}
where
\[
\lambda = 2\kappa_{2,sss}\kappa_1 - 2\kappa_{1,s}\kappa_{2,ss} + 2\kappa_{2,s}\kappa_{1,ss}-2\kappa_2\kappa_{1,sss}
\]
and
\[
\mu=\kappa_{1,s}^2+\kappa_{2,s}^2-2(\kappa_1\kappa_{1,ss}+\kappa_2\kappa_{2,ss}).
\]
The conservation laws are of the form \eqref{conservatiolaws} where
\[
v(I)=(\lambda \quad 2\kappa_{2,ssss} - \mu\kappa_2 \quad -2\kappa_{1,ssss} + \mu\kappa_1 \quad \mu \quad 2\kappa_{1,sss} \quad -2\kappa_{2,sss} ).
\]
Solving \eqref{eulerlagrangelag3a}, \eqref{eulerlagrangelag3b} along with \eqref{PsiEqnCaseOne} and \eqref{PsiEqnCaseTwo} for $\kappa_1,\kappa_2$ and $\psi$ with initial conditions
\[
\begin{array}{lllll}
\kappa_1(0) = 1, &\kappa_2(0) = \frac{1}{2}, &\frac{{\rm d}}{{\rm d} s}\kappa_1(0) = 1,& \frac{{\rm d}}{{\rm d} s}\kappa_2(0) = 1,&\\[10pt]
\frac{{\rm d}^2}{{\rm d} s^2}\kappa_1(0) = 1,& \frac{{\rm d}^2}{{\rm d} s^2}\kappa_2(0) = 1, & \frac{{\rm d}^3}{{\rm d} s^3}\kappa_1(0) = 1,& \frac{{\rm d}^3}{{\rm d} s^3}\kappa_2(0) = 1, &\psi(0)=0
\end{array}
\]
and integrating to obtain the extremizing curve and its Rotation Minimizing\ frame, we obtain the following solutions,
see Figures \ref{App2a}, \ref{App2b}, \ref{App2c}.
\begin{figure}
\caption{Solutions for the invariants $\kappa_1$,$\kappa_2$,$\theta$ and $\kappa$\label{App2a}}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{k1k2lag3} &
\includegraphics[width=0.3\textwidth]{thetalag3} &
\includegraphics[width=0.3\textwidth]{kappalag3}\\
$\kappa_1$ vs $\kappa_2$ & $s$ vs $\theta$ & $s$ vs $\kappa$ \\
\end{tabular}
\begin{center}
\caption{Plots of the first integrals\label{App2b}}
\begin{tabular}{cc}
\includegraphics[width=0.3\textwidth]{cl1lag3} &
\includegraphics[width=0.3\textwidth]{cl2lag3} \\
$v_1^2+v_2^2+v_3^2=c_1^2+c_2^2+c_3^3$ & $v_1v_4-v_2v_5+v_3v_6=c_1c_4-c_2c_5+c_3c_6$ \\
\end{tabular}
\end{center}
\caption{Sweep surface using $V$ from the Rotation Minimizing\ frame along the extremal curve.\label{App2c}}
\vspace{5mm}
\centering
\includegraphics[width=0.5\textwidth]{plotlag3c}
\end{figure}
\section{Conclusions}\label{CCsection}
In this paper we have developed the Calculus of Variations for invariant Lagrangians under the Euclidean action of rotations and translations on curves in 3-space, using the Rotation Minimizing\ frame. We obtain the Euler-Lagrange equations in their invariant form and their corresponding conservation laws. These results yield an easier form than those obtained in \cite{GonMan3}. We also show how to ease the integration problem using the conservation laws and to recover the extremals in the original variables. We show how to minimize the angle between the normal and binormal vector and give an application in the study of biological problems.
It is clear that our results can be generalized to obtain a symbolic calculus of invariants for a broad class of problems in which the frame is not
defined in terms of algebraic equations, in the coordinates of the manifold on which the Lie group actions. This is a topic for further study.
Future work would include the construction of a discrete Rotation Minimizing\ frame and obtaining the invariant Euler-Lagrange equations and conservation laws using the discrete invariant calculus of variations developed in \cite{newpaper}. The investigation of the minimization of functionals that are invariant under higher dimensional Euclidean actions is also of interest well as the study of joint invariants in problems where two helices appear and interact with each other.
\section*{Acknowledgements}
The authors would like to thank Evelyne Hubert for pointing out the use of the Rotation Minimizing\ frame in the Computer Aided Design literature, and the Maplesoft Customer Support for help with the plots of the sweep surfaces, which were performed using Maple 2018. The second author
would like to thank the SMSAS department at the University of Kent and the EPSRC (grant EP/M506540/1) for generously funding this research.
|
1,116,691,499,349 | arxiv | \section{Introduction}
Control of magnetic states by optical excitations in
magnetically ordered materials
has attracted considerable attention
since the demonstration of ultrafast demagnetization
in Ni within 1 ps, explored by time-resolved
magneto-optical Kerr effect studies \cite{Beaurepaire1996}.
Ultrafast demagnetization was also observed
for other elementary ferromagnetic transition metals
such as Co and Fe, and intermetallic alloys
\cite{Stohr,Kirilyuk2010a}.
Several mechanisms have been proposed
to understand the ultrafast demagnetization.
Beaurepaire {\it et al.} proposed a phenomenological
``three-temperature model''
in order to understand the ultrafast demagnetization of Ni, which
considers three interacting reservoirs of electrons, spins, and lattice,
and suggested the importance of direct electron-spin interactions.
Since the electron, spin, and lattice systems are quite tightly
coupled to each other in strongly correlated $3d$ transition metal
oxides, it is interesting to investigate the photoinduced
dynamics with respect to the electronic states and magnetism
\cite{Miyano1997,Kise2000,Cavalleri2005,Ogasawara2005,
Zhang2006,Muller2009,Koopmans2010,Okamoto2011,
Radu2011,DeJong2013,Bergeard2014,Beaud2014}.
For this study, we chose
fully oxidized single crystalline BaFeO$_{3}$ thin films,
which show unusual behaviors of
ferromagnetic and insulating properties
with saturation magnetization and Curie temperature
of 3.2 $\mu_B/\mbox{formula unit}$ and 115 K,
respectively \cite{Chakraverty2013}.
The large magnetic moment of BaFeO$_{3}$
thin films results in quite large peak intensity of
Fe $2p$ x-ray magnetic circular dichroism (XMCD), $\sim$ 18 \% of
the x-ray absorption peak intensity \cite{Tsuyama2015}.
Thus, BaFeO$ _{3} $ thin films are appropriate samples
to carry out time-resolved magnetic circular dichroism experiments.
Furthermore, the investigation of
the demagnetization dynamics of insulators allows one
to relate electronic structure to magnetic dynamics.
In order to investigate the magnetic dynamics of ferromagnetic
insulating BaFeO$_{3}$ thin films, we performed time-resolved
reflectivity studies at the Femtospex slicing facility
at the synchrotron radiation source
BESSY II \cite{holl}, using circularly polarized x-ray pulses.
Our experimental method has the advantage that,
in one reflectivity experiment, we can probe
electronic structure as well as magnetism.
BaFeO$ _{2.5} $ thin films were grown on SrTiO$_{3}$ (001)
substrate with the film thickness of 50 nm, using pulsed
laser deposition \cite{Chakraverty2013}.
After the deposition, BaFeO$ _{2.5} $ thin films were annealed
at 200 $ ^{\circ} $C under ozone atmosphere to obtain BaFeO$_{3}$
thin films \cite{Chakraverty2013}.
The quality of the thin-film samples were confirmed by x-ray
diffraction, Fe 2$p$ x-ray absorption spectroscopy,
and Fe $2p$ core-level hard x-ray photoemission spectroscopy
measurements by comparing cluster-model calculations, which
found that the formal valence of Fe was $4+$
\cite{Chakraverty2013,Tsuyama2015}.
The experimental geometry is shown schematically
in Fig.~\ref{Reflectivity} (a). We used fixed circular polarization
and created magnetic contrast by switching the direction of the
magnetic field ($H$), which was oriented along the sample surface
([010] direction). We recorded specular reflectivity data
for two magnetization directions, $R^+$ and $R^-$.
The average reflectivity $R=(R^++R^-)/2$ is a measure
of the electronic and structural properties, while the magnetic circular
dichroism in reflectivity (MCDR) signal
$DR=(R^+-R^-)/2$ is a measure of the sample
magnetization \cite{mertins}.
A Ti-Sapphire laser ($\lambda=$ 800 nm,
$h\nu=$ 1.55 eV) with the pulse width of $\sim 50$ fs
was employed as a pump laser with $\pi$ polarization.
The spot size of the pump laser was
$\sim$ 0.40 mm (horizontal) $\times$ 0.25 mm (vertical),
and that of the probe x-ray was
$\sim$ 0.1 mm $\times$ 0.1 mm.
The repetition rate of the time-resolved measurement
was 3 kHz, limited by the frequencies of the pump laser.
The pumped and unpumped signals were obtained alternatively.
The time resolution was 70 ps, corresponding
to the pulse length of the probe x-ray.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7cm]{Geo2.eps}
\includegraphics[width=8cm]{MCD.eps}
\caption{(a) Geometry of the measurements.
Panel (b) shows the reflectivity (top) and MCDR (bottom)
intensity from BaFeO$_3$ thin films
as a function of $2\theta$. Panel (c) shows the photon energy
dependence of reflectivity (top) and MCDR (bottom).}
\label{Reflectivity}
\end{center}
\end{figure}
The angular dependences of reflectivity and MCDR are
shown in Fig.~\ref{Reflectivity} (b)
for a photon energy of 703 eV.
The angular dependence of reflectivity show oscillating structures,
attributed to the interference between x-rays reflected from
the surface and interface of the thin-film sample.
It enables us to estimate the film thickness of the sample to be
(48 $\pm$ 1) nm, in good agreement with the evaluation of
$\sim$ (50 $\pm$ 1) nm by reflection high-energy electron diffraction.
The oscillating structures in the angular dependence of reflectivity
also imply a probing depth larger than the film thickness.
MCDR also shows an angular dependence with sign reversal
as shown in Fig.~\ref{Reflectivity} (b).
We fixed 2$\theta$ to be $\sim$ 15$^{\circ}$, which maximizes the MCDR
to about 12 \% of the average reflectivity.
Figure \ref{Reflectivity} (c) shows
the photon-energy dependence of reflectivity and MCDR.
Although the photon-energy dependence of MCDR
is different from the circular dichroism obtained
from x-ray absorption spectra \cite{Tsuyama2015},
it is a suited measure of the sample magnetization \cite{mertins}.
We fixed the photon energy to be $h\nu$ = 714 eV, and the time evolution
of the intensities of reflectivity as well as XMCD at this photon energy are
traced in order to investigate the electronic and magnetic dynamics
of BaFeO$_{3}$ thin films.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=8cm]{Layout1.eps}
\caption{Time evolution of (a) XMCD intensity and (b) reflectivity
of BaFeO$_{3}$ thin films for various pump laser fluence.
All the curves, except for the case of 10 mJ/cm$^2$,
are shifted upward for clarity.
(c) The reflectivity spectra with and without the pump effect (top).
The difference of the spectra (bottom).}
\label{TrXMCD}
\end{center}
\end{figure}
Figure~\ref{TrXMCD} (a) shows the time evolution of the MCDR intensities
for different pump fluence.
The vertical axis shows the excited MCDR intensities normalized
by the unpumped signal
($i.e.$ $\it{\Delta}R_{Pump}$/$\it{\Delta}R_{No Pump}$).
Here, the subscript of Pump and No Pump denote the signals with and
without the laser excitations, respectively.
The MCDR intensities decrease after the incidence
of the pump laser at $t=0$.
The time evolution of the MCDR intensity show different behaviors
with the change of the pump fluence ($F$).
When $F$ is smaller than 5.0 mJ/cm$^{2}$, the demagnetization
time is relatively slow and
magnetization recovery sets in after about 400 ps.
When the pump $F$ is larger than 6.6 mJ/cm$^{2}$, on the other hand,
the demagnetization time is quite fast and no recovery of the
magnetization can be observed within the first 800 ps.
We assign the different behavior of the demagnetization dynamics
to a laser-induced insulator-to-metal transitions
for $F\geq$ 6.6 mJ/cm$^{2}$, as discussed in the following.
We show the time evolution of the intensity of the average reflectivity
in Fig.~\ref{TrXMCD} (b), which allows us
to investigate the electronic dynamics.
The vertical axis shows the excited reflectivity intensities normalized
by those without excitation ($i.e.$ $R_{Pump}$/$R_{No Pump}$).
No pump effects were observed for $F\leq$ 5.0 mJ/cm$^{2}$
with our time resolution of 70 ps.
Pump effects, on the other hand, were clearly observed
for $F\geq$ 6.6 mJ/cm$^{2}$.
They occur via a transfer of spectral weight from the maximum of
the Fe $2p_{3/2}$ resonance (710 eV) towards lower excitation energies
as evidenced by comparing the spectra with and without laser
in Fig.~\ref{TrXMCD} (c). The transfer to lower energies
is particularly evident in the difference plot in the lower frame.
A similar behavior with spectral weight transfers above a fluence
threshold was observed in time-resolved x-ray absorption
spectroscopy from VO$_2$ \cite{Cavalleri2005}
and interpreted as a laser induced insulator-to-metal transition.
Since BaFeO$_3$ thin films are near the phase boundary
of a metal-insulator transition, we assign the spectral weight transfer
in this material to a similar mechanism.
In order to discuss the results quantitatively,
we fitted them using exponential functions for decay and
recovery (solid lines in Fig.~\ref{TrXMCD} (a) and (b));
$\tau_{decay}$ and $\tau_{recovery}$ denote
the corresponding time constants.
To take temporal resolution into account,
the fitting function was convoluted by a Gaussian
with a full width at half maximum of $\tau_{reso}$ = 70 ps.
The parameters extracted from the experimental delay scans
are summarized in TABLE~\ref{table1}.
\begin{table}[ht]
\begin{flushleft}
\caption{Fitting parameters of
the time evolution of
MCDR and reflectivity intensity in Fig.~\ref{TrXMCD}.
We fixed $\tau_{reso}$ = 70 ps.}
\begin{center}
\begin{tabular}{c|cc|cc}
\hline\hline
$F$ & $\tau_{decay,MCDR}$ & $\tau_{recovery,MCDR}$ &
$\tau_{decay,ref}$ & $\tau_{recovery,ref}$ \\
$[\mbox{mJ/cm}^2]$ & [ps] & [ps] & [ps] & [ps] \\
\hline
$3.3$ & $140$ & $1000$ & $-$ & $-$ \\
$5.0$ & $150$ & $1200$ & $-$ & $-$ \\
$6.6$ & $29$ & $-$ & $16$ & $1200$ \\
$10$ & $23$ & $-$ & $9.0$ & $1000$ \\
\hline\hline
\end{tabular}
\label{table1}
\end{center}
\end{flushleft}
\end{table}
The quantitative analysis confirms the existence of two different kinds
of dynamics for high and low fluences respectively.
We show the demagnetization time, $\tau_{decay}$ determined
by the fitting functions in the top panel of Fig.~\ref{Summary}.
The demagnetization time is estimated to
be $\tau_{decay}$ $\sim$ 150 ps for the weaker excitations
($F\leq $ 5.0 mJ/cm$^{2}$), and demagnetization faster
than our temporal resolution was
observed for the strong pump fluence ($F\geq $ 6.6 mJ/cm$^{2}$).
In the bottom panel of Fig.~\ref{Summary}, we show the
amplitude of the reflectivity change, which is a measure of
the spectral weight transfer indicative
of the insulator-to-metal transition,
as a function of laser fluence.
From the clear threshold behavior of this quantity
we deduce that a sufficiently high density of
excited carriers reduces
electron-electron and electron-phonon interactions,
which are the origins of the insulating properties
in BaFeO$_{3}$ thin films \cite{Chakraverty2013, Tsuyama2015}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7cm]{fig3.eps}
\caption{The demagnetization time $\tau_{decay}$ (top),
determined from Fig.~\ref{TrXMCD}, and the magnitude
of decay of reflectivity due to the pump (bottom) are shown.}
\label{Summary}
\end{center}
\end{figure}
We then conclude that the drastic changes of in the demagnetization
dynamics with laser fluence are due to the photoinduced
insulator-to-metal transition as shown by the time-resolved
reflectivity measurement, since the electronic structure
significantly influences the demagnetization
dynamics \cite{Kise2000,Zhang2006,Muller2009,Kirilyuk2010a}.
Half metals and ferromagnetic insulators usually show relatively
slow demagnetization in the range of from 100 ps to 1000 ps,
while itinerant ferromagnets (with low spin polarization)
can show ultrafast demagnetization
\cite{Kise2000,Zhang2006,Muller2009,Kirilyuk2010a}.
Within common models for ultrafast magnetic dynamics, these differences
in demagnetization dynamics are explained
by the differences of strength of coupling between
electronic and spin reservoirs.
Ferromagnetic metals with low spin polarization,
such as Ni and Fe, have strong coupling between
electron and spin reservoirs because the electronic structure
of ferromagnetic metals can accommodate spin-flip excitations with
quasiparticle scatterings, resulting in the ultrafast
demagnetization due to the rapid increase of spin temperature
via the electron reservoir.
Half metals and ferromagnetic insulators, on the other hand,
do not have strong coupling between the reservoirs.
This is because the spin-flip excitations are significantly
suppressed after the rapid increase of electron temperature
in their electronic structure, resulting in slow increase
of spin temperature via heating of the lattice.
Since undisturbed BaFeO$_{3}$ thin films are ferromagnetic insulators,
the spin scattering after the electronic excitations should be suppressed.
The slow demagnetization time of $\sim$ 150 ps in BaFeO$_{3}$ thin
films with small pump fluence ($F\leq$ 5.0 mJ/cm$^{2}$) can be explained
by heating via the lattice.
The large pump fluence ($F\geq$ 6.6 mJ/cm$^{2}$), however, induces
an insulator-to-metal transition in BaFeO$_{3}$ thin films quite rapidly,
which results in the unusually fast demagnetization in BaFeO$_{3}$
thin films for a ferromagnetic insulator.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7cm]{fig4.eps}
\caption{Mechanism of insulator-to-metal transition
induced by the strong laser excitation.}
\label{IM}
\end{center}
\end{figure}
When the pump fluence is weaker than 5.0 mJ/cm$^{2}$,
magnetizations in BaFeO$ _{3} $ thin films recover with the time
constant of $\tau_{recovery}$ $\sim$ 1000 ps.
The time scale of $\sim$ 1000 ps can be assigned to heat diffusion
needed to cool the sample below the magnetic ordering temperature
after electron, lattice, and spin systems have reached
thermal equilibrium.
Remarkably, the time-resolved reflectivity change
for strong pump fluence also shows a recovery
on this time scale of $\tau_{recovery}$ $\sim$ 1000 ps,
indicating that also here heat diffusion is the relevant mechanism.
This latter observation is quite notable,
because equilibrium between the electron and lattice temperature
should be reached within 1 ps. The slow reopening of the band-gap
on time scales of $\sim$ 1000 ps shows that for high excitation
densities we drive the system into a metastable state.
As a mechanism for the long life time of the metallic state,
we consider that hot carriers, generated by the quasiparticle
scattering and closing the band gap, prevent it from opening again
by reducing electron-electron and
electron-phonon interactions \cite{Okamoto2011}.
This process is schematically shown in Fig.~\ref{IM}.
In conclusion, we investigated the electronic and magnetic
dynamics by time-resolved reflectivity and MCDR measurement
on BaFeO$_{3}$ thin films.
When the pump laser fluence is smaller than 5.0 mJ/cm$^{2}$,
relatively slow demagnetization of $\tau_{decay}$ $\sim$ 150 ps was
observed, due to the insulating properties of the ground state
in BaFeO$_{3}$ thin films without any changes in Fe $2p$ x-ray
reflectivity.
When the pump laser fluence is stronger than 6.6 mJ/cm$^{2}$, on the
other hand, rapid changes in Fe $2p$ x-ray reflectivity are
observed, which is attributed to a transition into a metallic state,
resulting in an unusually fast demagnetization
with $\tau_{decay} < 70$ ps.
Since BaFeO$_{3}$ thin films are near the phase boundary
of a metal-insulator transition, the insulating phase is
quite sensitive to carrier density. Thus, the origin of the
insulator-to-metal transition is a photoinduced Mott transition
into a metastable state stabilized by screened electron-electron
and electron-phonon interactions.
Our findings indicate a mechanism for tuning magnetic dynamics
in correlated materials, which resembles heat-assisted
magnetic switching in metallic magnets. By creating
a sufficiently high excitation density, spin flip scattering channels
open up which increase the spin systems susceptibility
to external manipulation.
This research was supported by the Japan Society
the Promotion of Science (JSPS) through the Funding Program
for World-Learning Innovative R\&D on Science and Technology
(FIRST program), JSPS Giant-in-Aid for Scientific
Research, and by Grant for Basic Science Research Projects
from the Sumitomo Foundation. This work was also partially
supported by the Ministry of Education, Culture, Sports,
Science and Technology of Japan (X-ray Free Electron Laser
Priority Strategy Program).
H. Y. H. acknowledges support by the Department of Energy,
Basic Energy Sciences, Materials Science and Engineering
Division, under Contract No. DE-AC02- 76SF00515.
|
1,116,691,499,350 | arxiv |
\chapter{Background}
\section{Scalability in general-purpose processors}
Assuming continued demand for computers where \emph{new functions can
be defined by the end-user by writing and using their own software}
(cf. \cref{sn:gpdemand}), the question remains of how to make
general-purpose processors that are both fast (operations/second) and
efficient (operations/watt). However, conventional approaches on
silicon seem to have reached ``walls'' on both fronts around year
2000~\cite{ronen.01.ieee}.
Since technology progress still delivers increasingly more transistors
per chip (Moore's law), the trend has become to glue individual
processors together on the same chip, i.e. design ``multi-cores.''
The issue with this is that software is mostly written using
sequential algorithms: introducing hardware parallelism (multiple
processors) immediately raises the question of how to introduce
explicit concurrency in software. Software concurrency is hard and
both hardware architects and programming language designers have been
making only baby steps since 2000.
\begin{sidenote}
\caption{On the continued demand for general-purpose
computers.}\label{sn:gpdemand} One needs to accept the premise that
general-purpose computers are highly desirable and that the future of
computing hangs on their continued development to fully value the
remainder of this report. I have explained my own reasons to accept
this premise in \cite[sect.~1.2--1.4]{poss.12}; in short, I propose
that general-purpose computers are, like “stem cells,” necessary to
the continuation of computer science. I also propose they are
essential to the democratic freedom of any citizen to create their own
tools (in software) in this numeric age.
Meanwhile, I also acknowledge that it is not the role of mere
academicians to decide “what people really want.” If market, fashion
and politics determine that science should research instead all sorts of
maximally efficient special-purpose computing devices, the deconstruction
phase of this report would even be easier: there would be simply
no place at all for D-RISC and Microgrids.
\end{sidenote}
\section{D-RISC and Microgrids: what has been done}
The \emph{Microgrid} many-core architecture is a research project at
the University of Amsterdam, which investigates whether concurrency
management (thread scheduling, synchronization, and inter-thread
communication) traditionally under control of software operating
systems can be accelerated in hardware to obtain higher efficiency and
performance. Microgrids are clusters of a simple RISC core design
called D-RISC~\cite{bolychevsky.96.ieee}; each D-RISC core supports
hardware multi-threading (HMT) using a dataflow scheduler, and is also equipped with a hardware Thread
Management Unit (TMU) which can coordinate with neighbouring TMUs for
automatic thread and data distribution (cf. \cref{sn:tmu}). In short:
\begin{center}
\bfseries
D-RISC = simple RISC + dataflow HMT scheduler + TMU $-$ interrupt management
Microgrid = n$\times$D-RISC + TMU-to-TMU NoC + custom cache/memory protocol
\end{center}
\begin{sidenote}
\caption{Details about D-RISC/Microgrids architecture}\label{sn:tmu}
The rest of the text assumes passing familiarity with the D-RISC
and Microgrids architecture, as presented in chapters 3 and 4 of~\cite{poss.12}.
\end{sidenote}
Prior to 2007, research on D-RISC and the Microgrid was focused on
programmability issues and carried out with high-level simulators:
both using traditional software multithreading and an API to emulate
the TMU services~\cite{tol.09.jsa}, and using a custom functional ISA
emulator~\cite{bousias.06.cj}. As the initial phases of the D-RISC and
Microgrid design were encouraging~\cite{bell.06.jpp,bousias.09.jsa},
the EU-funded project Apple-CORE (2008-2011) was started to study its
implementability in a system, including a full vertical tooling stack
from an FPGA implementation up to benchmarks in higher-level
programming languages.
The outcome of the Apple-CORE project is summarized
in~\cite{poss.12.dsd,poss.12}: the D-RISC core was implemented on FPGA
as UTLEON3~\cite{danek.12}, a model of Microgrids was implemented in
MGSim, software tooling was delivered to program
Microgrids~\cite{saougkos.11,poss.12.sl,grelck.09.cpc}, and D-RISC and
Microgrids were confirmed using both UTLEON3 and MGSim to deliver
higher performance and efficiency for \emph{some of the selected
benchmarks}.
\section{D-RISC and Microgrids: what is going on}
At the time of this writing, research in this area continues on two
fronts. An industry-backed project has funded more effort towards
tailoring D-RISC for real-time embedded systems, by adding
priority-based scheduling and fault tolerance. Next to this, four
doctoral candidates are planning to defend their thesis on extensions
and improvements to D-RISC and Microgrids, and simulations thereof.
The Microgrid-related technology produced so far is also used for
graduate and undergraduate education in computer architecture and
compiler construction.
\section*{Disclaimer}
{\itshape
The arguments presented hereafter are my own, and thus may not be shared
by my colleagues or work partners. To my knowledge, at the time of
this writing there is no acknowledgement or consensus around the
D-RISC/Microgrids enterprise, other than my own experience and
impressions, that give credit to the perspective presented here.
}
\section*{Purpose and rationale}
This report lays flat my personal views on D-RISC and Microgrids as of
March 2013. It reflects the opinions and insights that I have gained
from working on this project during the period 2008-2013.
The origin of this report is a case of cognitive dissonance. On the
one hand, using critical thought against my “achievements” of the past
few years is causing a growing discomfort, discontent and
disappointment at the way the design and implementation of
D-RISC/Microgrids have been carried out so far, both by myself and my
colleagues. On the other hand, my optimism combined with an unusual
combination of curiosity and fascination for theoretical computer
science is sustaining a belief that despite its flaws, the project has
produced an interesting conceptual framework which deserves further
investigation at least by academics and teachers. Only by resolving
this cognitive dissonance can I satisfy myself that my continued work
in this area is compatible with my aspirations as a scientific
researcher. By writing this report, I hope I can resolve this
dissonance by externalizing both sides and construct rationally their
resolution.
\clearpage
\section*{Executive summary}
\paragraph{Shortcomings in the research results so far.}
The D-RISC/Microgrids project was purportedly intended to solve major issues in
micro-architecture research, related to scalability in performance and
efficiency in general-purpose microprocessors.
The strategy to solve these issues was to implement a
combination of dataflow scheduling with hardware support for thread
concurrency management within and across cores on chip.
Implementation
was carried out, but the results are inconclusive. On the one hand, the proposed
hardware does indeed provide higher performance and efficiency in
regular, data-parallel computation kernels. On the other hand, no evidence has yet
been produced that the proposed hardware benefits larger applications
with more irregular workloads.
Power efficiency and intelligent resource
management was regularly advertised but not actively researched.
Effort has been invested into widening the scope of the technology
towards applications and industrial relevance, but these applications
have not yet materialized.
\paragraph{Shortcomings in methodology.}
Research on D-RISC/Microgrids is not following the scientific method.
It is instead currently carried out as an engineering enterprise, but without clear
technology ouputs and without identifying
its potential applications. Its relevance in a university research group is thus questionable.
\paragraph{Obstacles to further progress.}
The D-RISC/Microgrids project has the ambitious aim to produce a
general-purpose processor chip able to disrupt the current
state-of-the-art. However, the limited human resources dedicated to
the project are insufficient to reach this aim in isolation.
The expansion of the research group to a community of users and and research
partners is blocked by a fundamental lack of compatibility with existing operating
systems and application software. This lack of
compatibility is not properly justified, neither by practical nor
theoretical reasons.
Meanwhile, the scientific effort to test the hypotheses that underly
the D-RISC/Microgrids project is poorly directed, and not
enough attention has been given to negative results that
invalidate these hypotheses.
Finally, the multi-core research field is nowadays much
more crowded than it was ten years ago, yet the research on D-RISC/Microgrids
does not acknowledge its competition nor attempts to differentiate
its contributions from the state of the art.
\paragraph{Actual contributions.}
The research has produced interesting discussions
that challenge some tacit assumptions of the research community,
experimental results that can be reused by future work, improvements
to partner technologies and new simulation techniques.
Most of the software designed and implemented during the research
can be reused by third parties, and not only for research
directly related to D-RISC/Microgrids.
The intellectual framework educates practitioners to think about
two general separations of concerns, namely concurrency vs.~ parallelism and
using memory for storage vs.~ synchronization.
\paragraph{Individual architectural features.}
The D-RISC core combines features found in other processors, such as a RISC pipeline
and hardware multithreading, with custom features (e.g.~ its TMU) and optimizations
to the conventional features (e.g.~ switch annotations for the HMT scheduler).
Some architectural optimizations found in D-RISC/Microgrids could be reused
with other processors, for example switch annotations and bulk coherency in the memory network.
The key feature of D-RISC/Microgrids, namely its TMU and inter-TMU control NoC,
does not depend on the other features specific to D-RISC and could be potentially reused with other processors.
\paragraph{Follow-up strategies.}
I can see three follow-up strategies for new investments around
D-RISC/Microgrids: exploitation, i.e.~ apply the technology produced
so far to other uses than research; salvaging and opening the
technology, i.e.~ extracting individual features from the
D-RISC/Microgrids design and evaluating them as extensions of
existing processors; and distillation of the main ideas in the realm
of fundamental computer science.
Ongoing research towards doctoral theses should be careful
to rephrase research questions in the light of our recent shared
understanding of the project's issues.
\chapter{Conclusion}
In traditional academic research projects, the abstract and general
questions receive most attention, and technology and engineering
``happen'' as a by-product. In contrast, the D-RISC/Microgrids project
was primarily a technology and engineering enterprise, with some
occasional and incidental scientific output.
My opinion is that further work in this direction faces two fundamental problems.
Firstly, a continued focus on engineering makes the project increasingly
difficult to host in an academic institution and impedes the growth
of an academic network.
The product of the current and past effort is made of chip
blueprints, simulation software, ancillary programming tools,
education materials and demonstration tools. Unfortunately, the
metrics used in academia to reward scientific effort are
peer-reviewed academic publications, conference attendance, invited
talks and lectures, successfully defended doctoral theses, etc. This
mismatch implies that the work has become extraordinarily difficult to
defend in academic communities. Moreover, any team member expecting
to receive an academic training from this project risks facing a
strong sense of disconnect between expectations and reality that may drive
them away. This is detrimental to the growth of a network of
supporting researchers around the project.
Secondly, the lack of connections with related work, especially
a continued disregard for software compatibility, constitutes a serious
management issue that threatens the project.
This disconnect has not always been an issue. In general, at
the start of a new line of research in computer architecture, compatibility
can be readily sacrificed to simplify the research environment
and quickly obtain preliminary evaluation results using simple, ad-hoc
experiments. Moreover, ten years ago when the research strategy
was being shaped, there did not yet exist any pervasive software culture for
multi-core programming and software interfaces to concurrency management.
In this context, a new, immature approach was simply competing with a host of other equally
new, immature approaches. But this context has thus evolved, and
the research risks facing irrelevance if the circumstancial changes in context and expectations are not addressed soon.
The question then remains: what to do now? For this, I have detailed
in \cref{sec:new} three possible strategies for new investments which
I know are viable from the current status of the research and would
address the two problems identified above.
One is to \emph{exploit}: take the shortest practical route to
maximize visibility of the current results and apply the technology. I
am currently driving exploitation towards academia, using the produced
tools for education in chip architecture and code generation. I am
seeking support from undergraduate students to design a minimal but
working form of exception handling and system-level compatibility with
existing software. I am also keeping ready to partner with industry to
work on exploitation projects that do not require further design.
Another is to \emph{salvage and open}: bring the technology apart and
offer its most salient bits and pieces as reusable components, able
to ground partnerships for follow-up joint research projects using
existing platforms and processor core designs. I may be interested
to support work in this direction, but not to drive the work myself.
The third is \emph{distill and reincarnate}: extract the underlying
fundamental research questions that are still relevant in this day and
age, and create a new research direction to explore. I have started
some preliminary work in this direction myself already.
An incidental, more personal but more fundamental question in the
bigger picture is whether any of these strategies is favorable to the
development of a researcher's career in the current academic
institution where the project is currently hosted. According to my
hierarchy, the answer is currently: ``not likely.'' I may try to
convince them otherwise by sublimating the work somehow, but personal
circumstances may prevent my long-term dedication to D-RISC/Microgrids
in favor of more aligned research topics instead.
\chapter{Actual contributions}
\section{Concrete scientific contributions}
The scientific output of the project is positive on at least four angles.
First, some published discussion-oriented articles have
articulated interesting challenges to the tacit assumptions of the
architecture community about the exploitation of concurrency in processors,
e.g.~~\cite{bell.06.jpp,jesshope.08.samos,tol.11.apc,jesshope.09.parco,bernard.10.ppl,bernard.10,poss.10.amp,poss.12,vantol.13}.
Second, all results-oriented articles accepted by peer review
for publication are based on real experimental results using
``honest and best effort'' implementations of the proposed ideas,
e.g.~~\cite{luo.02,jesshope.09.arcs,hasasneh.07.jsa,vu.08.icamst,bousias.06.cj,bousias.09.jsa,poss.12.dsd}.
Regardless of the conclusions drawn from them, these results
constitute a sound database of prior work to all future
researchers working on related areas.
Third, the research group has indirectly contributed to other projects
via its few partnerships. For example, the close work relationship
with the designers of Single-Assignment C and S-NET have yielded both
joint scientific
outputs~\cite{jesshope.08.apcsac,a-c-d44,poss.12.interact,grelck.09.cpc}
and technology improvements, directly or indirectly inspired by the
work on D-RISC/Microgrids.
Fourth, it the project has enabled ancillary research in novel techniques for system
simulation when cores are hardware multi-threaded~\cite{tol.09.jsa,tol.11.dutc,mirfan.11,poss.12.rapido,mirfan.12,lankamp.13.mgsim}, whose results are
scientific contributions on their own regardless of the specific merits of D-RISC/Microgrids.
\section{Technology products}
The research efforts have produced the following components and tools:
\begin{itemize}
\item svp-ptl and d-utc~\cite{tol.09.jsa,tol.11.dutc}, a library of TMU-like services implemented in software over POSIX threads,
ready to implement concurrent software over multi-cores and distributed memory systems;
\item MGSim~\cite{poss.12.rapido,lankamp.13.mgsim}, a combination of:
\begin{itemize}
\item a general discrete-event, component-based simulation framework in C++, and
\item a library of component models that can be used to simulated D-RISC/Microgrid-based architectures;
\end{itemize}
\item HLSim~\cite{mirfan.11,mirfan.12}, a discrete-event, thread-based simulation of multi-scale systems using the TMU protocol
and the API from svp-ptl;
\item the ``SL core'' package~\cite{poss.12.sl}, a combination of:
\begin{itemize}
\item a code translator from SL to D-RISC code, using any of its 3 possible ISAs,
\item a code translator from SL to the API of HLSim and svp-ptl,
\item a code translator from SL to ``vanilla'', sequential ISO C,
\item an incomplete port of a standard C library suitable for use in the simulated D-RISC/Microgrid environments, and
\item operating system components for resource management and interfacing with I/O services on D-RISC cores;
\end{itemize}
\item a set of micro-benchmarks using the SL language extensions that exercise the architecture and demonstrate the features
of the simulation frameworks;
\item via research partners, the UTLEON3 core design in VHDL~\cite{danek.12} which implements one D-RISC core with a partial TMU
for use on FPGA.
\end{itemize}
Some of these tools are specific to the D-RISC/Microgrid architecture and are not applicable outside of this project, whereas
others could be reused by researchers that have never been exposed to D-RISC/Microgrids. On a different scale,
some of these tools have been explicitly packaged for reuse and tested for portability by 3rd party users, whereas
others are not readily reusable due to dependencies on the local research environment. I summarize how the tools map to
these two scales in \cref{fig:tech}.
\begin{figure}
\centering
\includegraphics[scale=.6]{tech}
\caption{Technology products as of 2012.}\label{fig:tech}
\end{figure}
\section{Conceptual separation of concerns}
The research around D-RISC/Microgrids has strongly promoted two
intellectual exercises and shaped a generation of researchers able to
converse fluently about the following two issue separations.
The first is the separation between concurrency and parallelism, i.e.~
the distinction between \emph{opportunity for parallelism} that can be
encoded in software by relaxing synchronization constraints, and the
\emph{actual simultaneity of execution at run-time} which depends on
resource duplication over space.
This first separation is promoted/enabled by D-RISC/Microgrids by
promoting a TMU protocol which does not guarantee the availability of
parallel resources at run-time. Software that wishes to use the TMU
can only relax synchronization, i.e.~ introduce concurrency, whereas
actual parallelism is introduced later, by the TMU at run-time,
depending on resource availability. Although this separation of
concerns can be promoted by other means, any researcher working on
D-RISC/Microgrids \emph{cannot avoid} acquiring a sharp consciousness
of these issues.
The second separation is between ``memory as storage'' and ``memory as
a synchronization mechanism.'' In commodity architectures, the only
interface between the individual core and its environment in a
multi-core chip is its memory interface. This implies that the same
memory interface is used for both loading and storing values to main
memory within individual threads, and for coordination of work between
cores. The latter, in particular, has historically mandated the
extension of memory systems with transactional mechanisms (bus
locking, compare-and-swap, test-and-set) which would otherwise be
unneeded.
As explained in~\cite[Chap.~7]{poss.12}, the proposed
D-RISC/Microgrids architecture separates\footnotemark{} the memory
network for data storage from a ``control network-on-chip'' in charge
of synchronizing and coordinating work between TMUs. This forces the
researchers writing software for the platform to realize that the data
structures for synchronization traditionally implemented in memory,
such as producer-consumer FIFOs, mutexes or semaphores, are really specific instances of
\emph{abstract synchronization services} whose behavior can be
obtained in other ways, possibly more cheaply and efficiently.
\footnotetext{The aim of this separation was to test whether a custom
NoC can achieve cheaper and more efficient synchronization than a
memory system, and to avoid the research overhead of developing a
complex cache coherency protocol that also supports transactions.
This aim was reached, insofar that a custom NoC is indeed cheaper
than a comprehensive memory protocol, but it requires special
support in software, cf.~~\cite[Chap.~7]{poss.12}.}
These separation of concerns are not yet widely understood and commonly accepted in
the research community. The increased intellectual acuity
of the researchers ``educated'' by working on D-RISC/Microgrids
forms an advantage that can thus be considered a contribution of the enterprise.
\begin{summary}
\begin{itemize}
\item The research has produced interesting discussions
that challenge some tacit assumptions of the research community,
experimental results that can be reused by future work, improvements
to partner technologies and new simulation techniques.
\item Most of the software designed and implemented during the research
can be reused by third parties, and not only for research
directly related to D-RISC/Microgrids.
\item The intellectual framework educates practitioners to think about
two general separations of concerns, namely concurrency vs.~ parallelism and
using memory for storage vs.~ synchronization.
\end{itemize}
\end{summary}
\chapter{Methodology issues}
What the body of published and unpublished materials reveal is a
large, loosely scoped enterprise to define a multi-core processor chip
with the ambitious aim to solve the most significant problems of
architecture research in the period 2010-2020.
\section{Actual methodology}
Although never explicit, a strategy guides the effort:
\begin{enumerate}
\item accumulate technology around the simple ideas of \emph{dataflow
scheduling} and \emph{partial hardware support for concurrency
management}, so as to define an execution platform able to run
parallel benchmark programs;
\item ``try it out'' and measure how it behaves;
\item if the measurements are unsatisfactory, return to step \#1;
otherwise publish results and claim success.
\end{enumerate}
\begin{figure}
\centering
\includegraphics[scale=.4]{process}
\caption{High-level overview of research dynamics around D-RISC/Microgrids.}\label{fig:mgproc}
\end{figure}
An overview of this process is given in \cref{fig:mgproc}. From an
outsider's perspective, this research activity is independent, as its
only input is human effort and financial investment. Its overall
output is scientific articles on measured results, and a regular stream
of educated practitioners.
Over the year, two patterns have emerged. The first is that the
research group often stalls, busy looping from unsatisfactory results
back to implementing more features without questioning the overall
strategy (thick blue arrow in the figure). The consequence is an irregular, unfocused publication
throughput and doctoral candidates abandoning their research from lack
of focus. The second pattern is that only positive, ``competitive''
results are retained as candidates for publication. The consequence is
a lack of visibility on the research process, methodologies and
shortcomings, although these could also be useful and valuable to the
scientific community.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{science-activities}
\caption{Activities related to science.}\label{fig:science}
\end{figure}
\section{Relationship with other scientific activities}
Compare the process above with \cref{fig:science}, explained as follows in \cite[Chap.~1]{poss.12}:
\begin{quote}
\itshape
The traditional purpose of the fundamental sciences is the acquisition
of new knowledge pertaining to observed phenomena, in an
attempt to describe ``what is.'' In parallel to the discovery of
new knowledge through scientific inquiry, philosophers, or
theoreticians, derive ideas of ``what could be.'' Via
formalisms, they construct structures of thought to validate
these ideas and derive iteratively new ideas from them.
We can focus for a moment on the human dynamics around these
activities. On the one hand, the intellectual pleasure that
internally motivates the human scientists is mostly to be found
in the acquisition of knowledge and ideas. For natural scientists, the
focus is on accuracy relative to the observed phenomena, whereas for
philosophers the focus is on consistency. On the other hand, the
external motivation for all fields of science, which materially
sustains their activities, is the need of humans for either
discovery or material benefits to their physical
existence. From this position, the outcome of
scientific inquiry and philosophical thought, namely knowledge and
ideas, is not directly what human audiences are interested in. The
``missing link'' between scientific insight and its practical benefits
is \emph{innovation}, an \emph{engineering} process in two steps.
The first step of innovation is \emph{foundational engineering}: the
creative, nearly artistic process where humans find a new way to
assemble parts into a more complex artifact, following the inspiration
and foreshadowing of their past study of knowledge and ideas, and
guided by human-centered concerns. Foundational engineering, as
an activity, consumes refined matter from the
physical world and produces new more complex things, usually tools and
machines, whose function and behavior are intricate, emergent
composition of their parts. The novelty factor is key: the outcome
must have characteristics yet unseen to qualify as foundational;
merely reproducing the object would just qualify as manufacturing.
The characteristic human factor in this foundational step
is \emph{creativity}, which corresponds to the serendipitously
successful, mostly irrationally motivated selection of ideas,
knowledge and material components in a way that only reveals itself as
useful, and thus can only be justified, \textsl{a posteriori}.
The other step is \emph{applicative engineering}, where humans
assemble artifacts previously engineerd into complex systems that
satisfy the needs of fellow humans. In contrast to foundational
engineering, the characteristic human factor here is meticulousness in
the realization and scrupulousness in recognizing and following an
audience's expectations---if not fabricating them on the spot.
The entire system of activities around science is driven by
a \emph{demand for applications}: the need of mankind to improve its
condition creates demand for man-made systems that solve its problems,
which in turn creates demand for new sorts of devices and artifacts to
construct these systems, which in turn creates demand for basic
materials as input, on the one hand, and intellectual diversity and
background in the form of knowledge and ideas. We illustrate this general view
in \cref{fig:science} [...]. The role of education, in turn, is
to act as a glue, ensuring that the output of the various activities
are duly and faithfully communicated to the interested parties.
\end{quote}
In this context, the activity around D-RISC/Microgrids can be
recognized to actually constitute \emph{foundational engineering}: the process
of invention that produces tools and artefacts that can subsequently
solve ``real-world'' problems.
This immediately highlights two major issues:
\begin{itemize}
\item the output of foundational engineering is measured by the tools
and artefacts it produces, not merely their description in the form
of academic publications. For the effort on D-RISC/Microgrids to be
recognized and valued as innovation, it must be accompanied by the
marketing of its \emph{technology}, including its flaws and
limitations, and ultimately exploitation to real-world applications.
\item it is not the primary purpose of the academic institutions of
science to fund and support foundational engineering. Although it
is not uncommon to see foundational engineering occur in academic
environments, it is usually only accepted as a by-product
exploitation of the other activities of science, namely natural and
fundamental sciences. To justify continued effort on D-RISC/Microgrids
in a university research group, the fundamental principles of the technology
must be extracted, abstracted, studied formally and generalized
in a relevant, technology-independent fashion. This work has not been performed so far
other than via the three isolated publications \cite{tdvu.07.icfem,vu.08.icamst} and \cite[Chap.~7]{poss.12}.
\end{itemize}
\begin{summary}
\begin{itemize}
\item Research on D-RISC/Microgrids is not following the scientific method.
It is instead currently carried out as an engineering enterprise, but without clear
technology ouputs and without identifying
its potential applications.
\item
Its relevance in a university research group is thus questionable.
\end{itemize}
\end{summary}
\chapter{Obstacles to further progress}
In \cref{chap:outcomes} I have presented the status of the research so
far, and identified the areas where it did not deliver on its own
self-set expectations. In this chapter, I follow up by reviewing the
likely causes of these shortcomings.
\section{Large project scope}
The project's aim is to eventually deliver a full general-purpose processor. However, the design
of a new processor architecture requires large research and implementation investment on \emph{all} the following fronts:
\begin{enumerate}
\item micro-processor logic (per core); \label{e:mp}
\item NoC interconnect: flow control, routing, failure management; \label{e:net}
\item cache management and inter-cache coherency; \label{e:caches}
\item core-NoC interfaces; \label{e:if}
\item external memory interfaces;
\item external I/O interfaces;
\item hardware/software interface design: ISA, but also I/O address space layout, MMU control, access to performance counters, etc; \label{e:isa}
\item ISA code generation in low-level compilers; \label{e:cgen}
\item architecture-specific optimizations in compilers; \label{e:opt}
\item architecture-specific support in software operating systems and programming language libraries; \label{e:os}
\item individual component validation, both from formal analysis and unit testing; \label{e:val}
\item component-level, then system-level modeling and simulation; \label{e:sim}
\item design of the integration strategy in a larger system (exploring system parameters and how system-level interconnects will impact intra-chip behavior);
\item circuit synthesis and prototype production (either ASIC or FPGA); \label{e:impl}
\item show casing and marketing via extensive comparisons with competition products; \label{e:cmp}
\item seeking and partnering with industry to define marketable products. \label{e:ind}
\end{enumerate}
Each of these areas would require multiple man-years worth of
investment before evidence of success in reaching the aim ``delivering a
full general-purpose processor'' can be strongly claimed.
To this date, the research effort around D-RISC/Microgrids was focused
on \cref{e:mp,e:caches,e:sim,e:isa}, although for \cref{e:isa} no clear
picture of the interface to virtual memory and I/O has yet emerged. A
lot of effort was also spent towards \cref{e:cgen,e:os} to enable
benchmarking, albeit half-heartedly because the research group lacked
expertise in these areas until recently. Some effort was spent on
\cref{e:impl} with a research partner~\cite{danek.12}, although the resulting FPGA model only
implements 1 core connected to a bus, i.e.~ it only shows limited
benefits of latency tolerance and does not implement the TMU's
multi-core coordination features. The other items have not been
investigated yet. Even if the scope of the research was reduced to
``design a Microgrid component that can be integrated as accelerator
in a larger processor chip,'' as advertised in \cite{poss.12.dsd},
effort should still be invested on
\cref{e:net,e:if,e:opt,e:val,e:ind,e:cmp}. This is out of the reach of
an isolated university research group expected to deliver strong
results in periods of five years with a bandwidth of 1 to 5
contributors per year.
\section{``Here Is My Chip, You Figure It Out''}
Following the original aim thus requires partnerships with other
organizations to carry out the work together. However, in the
scientific community, partnerships only come into existence based on
mutually beneficial arrangements: a peer researcher or instution may
be willing to contribute effort and technology towards the betterment
of D-RISC/Microgrids, only if they get something in return.
However, most of the interactions with potential partners so far were
carried out thus: ``here is our technology, we think it is good for
such-and-such use cases, what about you try to use it and tell us what
you do with it?'' For the reasons discussed extensively
in~\cite[Sect.~1.5, 15.1 \& Chap.~16]{poss.12}, this approach is
subject to the ``Here Is My Chip, You Figure It Out'' (HIMCYFIO)
hazard: faced with alien, unrecognizable technology, a potential
partner or user will be reluctant to invest the effort necessary to
cross the \emph{comprehension} threshold, even before they start to
think about potential shared endeavours. The producer of the
technology must cross this threshold preemptively to avoid the
HIMCYFIO pitfall.
Although my thesis \cite{poss.12} made one step in that direction,
further work is needed by this research group to bring the
D-RISC/Microgrid technology ``to the level'' of its potential
partners. The participants must identify what challenges potential
partners are facing, and preemptively shape the technology into a
palatable solution to the partners' problems.
To this day, the issues faced by the scientific peers in the same
research domain have not been investigated thoroughly by the research
group. The main reason for this is not lack of intent; rather a
crucial technical practical/historical obstacle: validation and peer
recognition in the micro-architecture community heavily relies on the
ability to exchange hardware platforms \emph{without modifying the
software}, so as to enable sound comparisons between
solutions. Although source-level compatibility is sufficient
(recompiling code towards a new platform has become acceptable in the
community), it is also necessary: given the large effort necessary to
design and deliver hardware, little effort can be spent
rewriting/adapting benchmark code, which has often accumulated
man-years of design, towards new platforms. However, for the reasons
described below and as of this writing, D-RISC/Microgrid's
\emph{cannot be made source-compatible} with most existing benchmarks.
This limitation is a high obstacle to publication and thus visibility,
and cuts short most opening discussions with potential partners.
\section{Lack of features needed for compatibility}\label{sec:incompat}
In my opinion, the main obstacle to the usability of D-RISC/Microgrids
by third parties is the lack of the following features:
\begin{itemize}\bfseries
\item \emph{process virtualization}, including per-process virtual memory address spaces and virtual I/O channels;
\item interrupt-like \emph{mechanisms to handle faults and unexpected external events};
\item the \emph{ability to stop a process and inspect it} externally, e.g.~ using a debugger;
\item the ability to \emph{preempt a running program and reclaim its resources}.
\end{itemize}
Programmers, in particular operating system and language implementers,
have been accustomed in the last 50 years to expect these features from any
general-purpose computer; the corollary is that all the existing
\emph{operating software} underlying existing application software
makes pervasive use of these services from the hardware platform.
The standpoint of the research group was that these features
are an ``historical artefact'' that were motivated at a time when
on-chip resources (cores, memory) were limited, and must thus be reconsidered at an
age where there are thousands of cores on chip and 64-bit addresses to
memory. As suggested in~\cite{tol.06,jesshope.08.samos,tol.11.apc,vantol.13}
and \cite[Sect.~3.3.2 \& Chap.~14--15]{poss.12}, this research group's public answer to
queries about these features goes as follows:
\begin{itemize}
\item ``preemptive event handlers should not be needed when any two
system-level tasks waiting on events could be active at the same
time in different hardware threads or cores (of which there are
thousands available);''
\item ``separate virtual address spaces should not be needed when a single virtual 64-bit
address space can be partitioned a thousand-fold while still
providing petabytes addressable to each process;''
\item ``once processes are allocated over space and not over time,
process boundaries are congruent to areas on chip and resource reclamation can be implemented simply by fully resetting the
corresponding hardware resources;''
\item ``issues of debugging and investigation do not need special supports as long
as simulators and emulators are available: debugging can be performed
from the simulator/emulator's host.''
\end{itemize}
As an insider, I can also report the second half of the answer: it is
possible to add support for these features, but the fear is that doing
so would make the research more difficult because a larger set of
issues would need to be considered. An ungrounded assumption is that
introducing support for virtualization would introduce overheads in
the hardware and make D-RISC and Microgrids less competitive against
other processors, including for the workloads where it currently
shines. The assumption is ungrounded because the consequences of
extending D-RISC in that direction have simply not been investigated
yet.
{\bfseries
In short, the position of the research group is ``these are
complicated engineering issues, but we think they are only superficial
usability concerns so they do not deserve our attention yet.''
}
The net effect is that existing operating software cannot be reused
with the proposed platform. Even assuming that new, custom-built
operating software \emph{could potentially show} that these
traditional mechanisms for virtualization can be avoided, the very
lack of software compatibility caused by the current situation may
well form the unsurmountable barrier preventing the group from forming
the partnerships needed for further developments, for the reasons
outlined in the previous section.
(For the sake of clarity, according to my own analysis these features
are both necessary and sufficient to immediately enable porting and
reusing operating software and existing major programming framework on
the proposed platform. To my knowledge, these features constitute
exactly the remaining obstacle to compatibility.)
\section{Weakly grounded and tested hypotheses}
The foundation for the ``D-RISC enterprise'' is the observation
that the static and dynamic costs of OoOE in GP processors is largely caused
by the logic necessary to discover instruction-level parallelism at run-time.
The hypotheses of the D-RISC/Microgrids research are then articulated as follows:
\begin{enumerate}\itshape
\item By shifting the responsibility to
discover concurrency, from the run-time to design-time (or compile-time),
these costs can be avoided. And instead of encoding concurrency the
VLIW way, which is weak when faced with the unpredictable variance of on-chip access
latencies in large chips, the concurrency can be encoded via threads
instead.
\item If thread creation is not more expensive than simple branches,
many sequential patterns including function calls and loops can be
transparently replaced by threads, and this
transformation can be performed casually in code generators for any
language. As a result, just using the platform's compilation tools
can introduce concurrency automatically in any sequential software and
solve the general ``programmability'' challenge of parallel hardware.
\item If concurrency management is encoded in the ISA with lightweight
instructions, the same binary code can be run under any amount of
resources, starting with a single thread on 1 core where it can be
as fast as an equivalent branch-based sequential code.
\item If concurrency management is encoded without explicit reference to parallel
hardware resources (``resource-agnostic''), the execution platform
can adapt the code at run-time to maximize performance and efficiency
to the resources effectively available.
\end{enumerate}
The first hypothesis has been largely confirmed to hold for large
classes of applications, but then not by D-RISC specifically: Intel's
HyperThreaded cores, then Sun/Oracle's Niagara cores, have been endorsing
the benefits of hardware multithreading for general-purpose computing
(especially in the datacenter domain) for a few years already.
The other hypotheses are more problematic.
The 2nd hypothesis heavily relies on the assumption that the
sequential composition of activation frames (for function calls and
loop iterations) can be cheaply and automatically replaced by parallel
composition. As I reveal partially in~\cite[Chap.~9]{poss.12}, this
research group had ``forgotten'' to consider that
\emph{general-purpose computations may use arbitrarily large amounts
of storage} at any step. This implies a local, ``private'' memory
area \emph{of varying size} for each activation frame \emph{while it
is running}.
With sequential execution, this is possible to implement cheaply with
a stack because only the most recently called activation frame is
running and can use the entire remaining memory. With parallel
execution, multiple activation frames are running and compete for the
available memory. Without much care (and thus ``intelligence'' that
must in turn be implemented in the TMU, raising its cost), the management of all these ``memory clients''
that all grow and shrink their local memory dynamically becomes a
bottleneck to parallel scalability.
Note that this problem is avoided in most highly-parallel
``accelerators'' by simply stating they do not support general-purpose
computations~\cite[Sect.~1.4\&12.6]{poss.12}, and in supercomputers by
providing large amounts of local RAM next to each processor. The
``supercomputer approach,'' applied to D-RISC/Microgrids, would imply
embedding large SRAM modules next to each core on chip or DRAM using
3D stacking, a direction not yet envisioned by this research group.
The 3rd hypothesis about automatic sequentialization was partly shown to hold with
D-RISC for threads that mostly perform local computations on
non-shared data and resources. As soon as shared or non-local
resources are used, sequentialization must choose a computation order
that maximizes locality of access and reuse. In sequential code,
typically the specified order of operations carries domain knowledge
from the programmer or compiler about locality; with threaded code,
this knowledge has disappeared from the code and must be reintroduced
at run-time while sequentializing. So far, the results produced show
that the automatic sequentialization is doing a poor
job~\cite[Sect.~10.4]{poss.12} but no research is being performed to
improve this state of affairs.
The 4th hypothesis about performance portability has been only
confirmed with D-RISC for data-parallel code with regular
data access patterns. For other types of concurrent code,
experimentation has shown that performance and efficiency are largely
dependent on the congruence between the data access patterns and
the physical topology of the interconnect between cores and
memory~\cite[Chap.~13]{poss.12}. However, to this date the concurrency
management interface of D-RISC's TMU does not allow to specify data
access patterns, so the TMU cannot make ``intelligent'' decisions
about placement. No further research is being performed in this
direction either.
All in all, these hypotheses seem \emph{attractive}: our peers have
reviewed these hypotheses through our publications and confirmed
implicitly, by accepting publication and funding further research,
that the hypotheses have merit and that our research efforts to test
them are scientifically worthwhile. However, I can also practically
observe that the work is not organized around strategic experiments
that would provide clear answers on these hypotheses. Meanwhile, I
have observed negative results that tend to invalidate the
hypotheses as they now stand, and I have also observed that these negative
results are not publicly exposed; instead, they are casually treated
as ``bugs'' and addressed by ad-hoc workarounds in the architecture
design.
\section{Competition and lack of differentiation}
Research on D-RISC/Microgrids was initiated in the late 1990's: a time
where multi-core chips were not yet widely used, and were there was
still a lot of uncertainty about concurrent programming. Back then,
D-RISC's approach was not only fresh, it was also spearheading
research in this area. There was thus not much to consider in terms of
``competition'' and ``related work.''
This has now changed. Since 2005, we have observed an explosion of
hardware multithreaded cores, multi-core chips and concurrent
programming frameworks. There now even exists architectures where
concurrency management is partly implemented in hardware: NVidia's
Fermi, Kalray's MPPA, Tilera's TILE are examples. Parallel and
multi-core programming is now being studied by students as a basic
course, and a wealth of software frameworks have evolved to manage
large number of ``micro threads'' efficiently on today's multi-core
chips: qthreads, codelets, green threads, Erlang's run-time system,
etc. Moreover, many high-level constructs to expose concurrency in
programming languages have been designed, e.g. in C/C++ (in the new
2011 standards), Scala, Haskell, etc. These have gradually introduced
\emph{expectations} in programmers' sub-cultures about what features
programming environments should and should not provide. Finally, some
architectural features have become widely accepted as fundamental to
the continued relevance of multi-cores, e.g. transactional memory and
heterogeneity, which have yet not been analyzed nor picked up by the
D-RISC/Microgrids enterprise.
In short, the research field has become crowded. To attract attention
and thus gather momentum, it is essential to \emph{acknowledge the
competition}, \emph{stay competitive} by keeping up and integrating
the good ideas from other projects, and simultaneously
\emph{differentiate} the new technology by pitting it against its
competition systematically. To this date, the research effort around
D-RISC/Microgrids has not focused on studying and integrating the
growing state-of-the-art, and differentiation is not expressed in
publications.
\begin{summary}
\begin{itemize}
\item
The D-RISC/Microgrids project has the ambitious aim to produce a
general-purpose processor chip able to disrupt the current
state-of-the-art. However, the limited human resources dedicated to
the project are insufficient to reach this aim in isolation.
\item
The expansion of the research group to a community of users and and research
partners is blocked by a fundamental lack of compatibility with existing operating
systems and application software.
\item This lack of
compatibility is not properly justified, neither by practical nor
theoretical reasons.
\item Meanwhile, the scientific effort to test the hypotheses that underly
the D-RISC/Microgrids project is poorly directed, and not
enough attention has been given to negative results that
invalidate these hypotheses.
\item Finally, the multi-core research field is nowadays much
more crowded than it was ten years ago, yet the research on D-RISC/Microgrids
does not acknowledge its competition nor attempts to differentiate
its contributions from the state of the art.
\end{itemize}
\end{summary}
\chapter{Outcomes vs.~ original intents: a retrospective}
\label{chap:outcomes}
The research on these topics was funded based on multiple research
proposals and statements of intent over time. In this section, I
explore \emph{published} motivations, where the scientific community has
agreed via the publication approval process that the motivations and
justifications were worthwhile.
Although most scientific projects are expected to diverge from their
original intents, published objectives reveal the general motivation
that drives the effort. Note that I do not focus here on
``result-oriented'' papers that report factually on research outcomes;
only on those publications that made statements of intent and
motivation before the corresponding work was carried out.
\section{I. Bell, N. Hasasneh \& C. Jesshope, JPP 2006}
Here we explore the contributions originally advertised in~\cite{bell.06.jpp}.
\subsection{Summary of the article}
Abstract:
\begin{quote}
\itshape Chip multiprocessors (CMPs) hold great promise for achieving
scalability in future systems. Microthreaded CMPs add a means of
exploiting legacy code in such systems. Using this model, compilers
generate parametric concurrency from sequential source code, which
can be used to optimise a range of operational parameters such as
power and performance over many orders of magnitude, given a scalable
implementation. This paper shows scalability in performance, power
and most importantly, in silicon implementation, the main contribution
of this paper. The microthread model requires dynamic register
allocation and a hardware scheduler, which must support hundreds of
microthreads per processor. The scheduler must support thread
creation, context switching and thread rescheduling on every machine
cycle to fully support this model, which is a significant
challenge. Scalable implementations of such support structures are
given and the feasibility of large-scale CMPs is investigated by
giving detailed area estimate of these structures.
\end{quote}
Problem statement:
\begin{quote}\itshape
In general, there are only a few requirements for the design of
efficient and powerful general-purpose CMPs, these are: scalability of
performance, area and power with issue width, and programmability from
legacy sequential code. Issue width is defined here as the number of
instructions issued on chip simultaneously, whether in a single
processor or in multiple processors and no distinction is made
here. To meet these requirements a number of problems must be solved,
including the extraction of ILP from legacy code, managing locality,
minimising global communication, latency tolerance, power-efficient
instruction execution strategies (i.e. avoiding speculation),
effective power management, workload balancing, and finally, the
decoupling of remote and local activity to allow for an asynchronous
composition of synchronous processors. Most CMPs address only some of
these issues as they attempt to reuse elements of existing processor
designs, ignoring the fact that these are suitable only for chips with
relatively few cores.
\end{quote}
Proposed main contribution:
\begin{quote}
\bfseries\itshape
In this paper a CMP is evaluated, that is based on microthreading, which addresses either directly or indirectly, all of the above issues and, theoretically, provides the ability to scale systems to very large number of processors.
\end{quote}
\subsection{Analysis}
The ``CMP'' in the paper's text refers to the D-RISC/Microgrid technology. Does this paper, and all the research
since then (8 years) support the claim that it solves ``all of the above issues''?
We list here how the technology addresses the issues in decreasing order of success.
\paragraph{Extraction of ILP from legacy code:} this was successful. ILP is extracted implicitly by D-RISC's dataflow scheduler, although
the ILP width is limited by the number of active threads and the number of registers per thread, because the reordering
information is stored in registers.
\paragraph{Decoupling of remote and local activity:} this is mostly successful, insofar the D-RISC's TMU control protocol
has different primitives to spawn work locally and remotely.
\paragraph{Scalability of performance, area and power with issue width:} each D-RISC core uses a single-issue pipeline,
so this claim states that performance, area and power scales with the number of cores. The research has indeed
shown this to be true for large, regular, data-parallel computation kernels, but the picture is not so clear for
small or more heterogeneous workloads because of the inter-core latencies in the concurrency management protocol itself
and on-chip network contention due to cache coherency protocols.
\paragraph{Power-efficient instruction execution strategies:} this is only partly successful; with one thread
the execution is not too power efficient compared to traditional
single-issue, in-order designs, and efficiency only increases with the number of threads
active. The problem is that operand availability is only tested at the
read stage of the pipeline. When there are multiple threads,
annotations at the fetch stage prevent ``potentially suspending''
instructions from entering the pipeline, but if only one thread is
active, the instructions with missing operands will still enter the
pipeline, create a bubble after the read stage, and subsequently need
to be rescheduled. This is a form of speculation, thus inherently less
power-efficient than the traditional approach to stall the pipeline during
issue.
\paragraph{Latency tolerance:} this is only partly successful. The latency of intra-core operations is tolerated by intra-thread ILP, but then the same is possible with conventional barrel processors or out-of-order execution\footnote{Albeit possibly at a larger area and power budget. However, the actual area and power requirement of D-RISC are yet to be evaluated.}. Longer latencies can be tolerated as long as there are sufficient threads active on the core to interleave with a waiting thread. If most active threads are busy communicating, then latency tolerance is highly dependent on a full end-to-end support for split-phase transactions, i.e.~ the rest of the system must support
a large number of in-flight transactions. In practice, the caches and
external memory interfaces become a bottleneck, and to this day no clear solution has emerged on this front.
\paragraph{Minimising global communication:} this is only mildly successful. For synchronization and concurrency management, global communication is limited by constraining the TMU's automatic workload distribution to adjacent cores only. The responsibility of choosing an area of the chip where to start the distribution is left to an hypothetical resource manager, not yet researched/implemented. Next to this, one should also consider memory communication. Here all results so far use memory protocols that incur global communication for coherency. No results show yet that global memory communication has been minimized.
\paragraph{Workload balancing:} this is only very mildly successful. D-RISC's TMU can automatically spread a batch of threads to multiple cores using an even N/P distribution, but this is the only form of distribution supported. Due to bulk reuse and synchronization, this simple distribution causes irrecoverable imbalances as soon as the batch is heterogeneous.
\paragraph{Managing locality:} this is not achieved, insofar that the data used by instructions is invisible to D-RISC's ``intelligence'' (its hardware TMU), and the memory and core networks are not topologically congruent. The software has to negociate locality of code and data explicitly with knowledge of the chip's layout.
\paragraph{Programmability from legacy sequential code:} to this date, most existing sequential code cannot
be reused as-is with D-RISC/Microgrids, because of incomplete support
for operating system services, cf.~ also \cref{sec:incompat}.
\paragraph{Effective power management:} to this date, power management has not been explored.
\section{NWO Microgrids: 2006-2010}
\subsection{Research question}
The project NWO Microgrids was funded by the Dutch government based on
the following research question:
\begin{quote} \itshape
Is it possible, through the introduction of simple and explicit concurrency controls, to develop a systematic approach to:
\begin{enumerate}
\item incrementally designing new processor architectures (i.e. based on an existing ISA and infrastructure);
\item dynamically managing and optimising the available resources for a variety of goals such as performance, power and reliability (i.e. resulting in autonomous and self-adaptive microgrids);
\item formally defining the architectures' execution properties;
\item incrementally developing the architectures' infrastructure (i.e. simulators, compilers, binary-to-binary translators and even silicon intellectual property);
\end{enumerate}
all within the context of ten to fifteen years of silicon-technology scaling (i.e. maintaining scalability over a thousand fold increase in chip density)?
\end{quote}
Note that this is a yes/no question.
To answer ``yes,'' it suffices to propose at least one set of ``simple
and explicit concurrency controls'' and a corresponding ``systematic
approach'' that delivers on the four other points. Alternatively,
``yes'' can also be given, perhaps less satisfactorily, if a
theoretical analysis indirectly merely proves the systematic approach
exists (``it is possible to develop it'') without actually developing
it. To answer ``no,'' in contrast, it is necessary to demonstrate
that there cannot exist any set of ``simple and explicit concurrency
controls'' which makes a ``systematic approach'' possible.
Strategically, this question suggests its own answer:
\begin{itemize}
\item a ``no'' answer would be a formidable theoretical endeavour, likely very
difficult to obtain (possibly impossible within the proposed 10-15
years time frame);
\item a ``yes'' answer based on theoretical proof of existence would be equally difficult;
\item therefore, the question suggests a ``yes'' is expected, hinged
on the ability of the researchers to use the features of an existing
architecture as their candidate ``simple and explicit concurrency
controls,'' and consequently show that a corresponding ``systematic
approach'' delivers on the 4 other points.
\end{itemize}
In short, NWO Microgrids's ``declaration of scientific intent'' can be reformulated as follows:
\begin{quote} \itshape
We will show that D-RISC/Microgrids make it possible to develop a systematic approach to:
\begin{enumerate}
\item incrementally designing new processor architectures (i.e. based on an existing ISA and infrastructure);
\item dynamically managing and optimising the available resources for a variety of goals such as performance, power and reliability (i.e. resulting in autonomous and self-adaptive microgrids);
\item formally defining the architectures' execution properties;
\item incrementally developing the architectures' infrastructure (i.e. simulators, compilers, binary-to-binary translators and even silicon intellectual property);
\end{enumerate}
all within the context of ten to fifteen years of silicon-technology scaling.
\end{quote}
\subsection{Outcome vs.~ expectations}
Did NWO Microgrids deliver on its self-set expectations?
\paragraph{Did NWO Microgrids deliver a ``systematic approach'' with the desired properties?} No, the design of D-RISC/Microgrids
was instead carried out in an ad-hoc fashion, with multiple phases of
trial-and-error and backtracking. I highlight an ethical issue here,
because on the one hand the Dutch NWO funded a project on the
assumption that its outcome would be a \emph{systematic approach
(method)} that could be reused with different architectures, and on
the other hand the research team knew well in
advance that systematization would not be studied.
For the sake of deconstruction, let us however stretch the word
``systematic approach'' to encompass ``the process of designing and
building D-RISC/Microgrids.'' Does this extended definition match the
other required properties?
\paragraph{Did the process of designing and building D-RISC/Microgrids incrementally designed a processor architecture, based on an existing ISA and infrastructure?} Here the answer is only partially ``yes'': the design was indeed incremental (starting from a simple, known-to-work RISC pipeline) and used an existing ISA (Alpha), but it did not reuse an existing infrastructure. Instead, all the infrastructure for the project was built from scratch.
\paragraph{Did the process of designing and building D-RISC/Microgrids enabled the dynamic management and optimisation of available resources for a variety of goals such as performance, power and reliability?} The jury is still out on this one; no answer was given yet after many years of research. Even a published doctoral thesis on the topic~\cite{vantol.13}, which merely touched performance-driven resource management, did not yield definite answers. As of this writing, another doctoral candidate is working on the reliability issue, but issues of power are still left untouched.
\paragraph{Did the process of designing and building D-RISC/Microgrids include a formal definition of the architecture's execution properties?} Yes, namely in~\cite{tdvu.07.icfem} and~\cite[Chap.~7]{poss.12}.
\paragraph{Did the process of designing and building D-RISC/Microgrids include an incremental development of the architecture's infrastructure? (i.e. simulators, compilers, binary-to-binary translators and even silicon intellectual property)} Here the answer
is only partially ``yes.'' Simulators and compilers were developed,
and parts of silicon IP (cf.~\cite{danek.12}), however there were no
binary-to-binary translators produced to establish a compatibility
path with existing code, as initially envisioned.
\subsection{Restrospective on the research question}
Since the process of designing D-RISC/Microgrids did not exhibit all
the expected properties, it cannot be used to answer ``yes'' firmly to
that project's research question. However, ``no'' cannot be
confidently given either. In other words, the question is still mostly
left unanswered.
Instead, I can only summarize the situation by proposing that the NWO
simply funded some additional \emph{development} of D-RISC/Microgrids,
and the project's description only merely \emph{guided the development
process} without pressuring it into delivering scientific output.
Moreover, it is clear to me that the original research question
\emph{was so ill-phrased that it cannot be answered scientifically}: I
cannot see any experimental path that would yield a definite answer
within a reasonable time frame.
Consequently, any rephrasing would be equally improductive, for example
determine whether there is any \emph{other} set of concurrency
controls that yield a ``yes'' answer on the same research question, or
whether D-RISC/Microgrids can be fixed/enhanced to this aim.
Therefore, in my opinion, the original phrasing for the project NWO
Microgrids cannot be used to motivate further work in this area. The
corollary is that researchers should not exploit the past attention
given by NWO's to this question as justification to spend more effort
in this area. If justification is needed, it must be found somewhere
else.
\section{C. Jesshope, APC 2008}
This whitepaper/article~\cite{jesshope.08.apc} made a statement of
intent about the applicability and the aims of D-RISC/Microgrids. It
introduces the ``SVP model,'' an intellectual construction used from
2008 to 2011. SVP intended to abstract the specific inner workings of
D-RISC/Microgrids, keeping only the high-level semantics of its TMU
concurrency management protocol visible to programmers.
\begin{table}
\begin{tabular}{p{.48\textwidth}p{.48\textwidth}}
Problem & Proposed answer \\
\hline \hline
How to effectively program distributed multiprocessor systems & Use SVP's simple concurrency control primitives \\
\hline
How to make architectures that are both efficient and can tolerate a large latency in responding to external events &
Use a combination of native support for dataflow scheduling and split-phase transactions throughout the system, such as found
in SVP implementations \\
\hline
How to design programming model that is both deterministic and free from deadlock under concurrent composition & Use SVP's strictly
hierarchical concurrency and forward-only communication patterns \\
\hline
How to ensure binary compatibility across a range of implementations from a single processor to the highest level of concurrency a particular application can support & Use SVP's granularity-independent abstraction of concurrency resources \\
\end{tabular}
\caption{Problems purportedly solved by SVP.}\label{tab:mgprob}
\end{table}
According to this article, D-RISC/Microgrids as abstracted in SVP
should have solved the problems listed in \cref{tab:mgprob}.
Note that this article defined SVP and its benefits \emph{before D-RISC's TMU, and thus Microgrids were fully defined.}
As it happened, the advertised features of SVP
ended up \emph{not being implementable in D-RISC/Microgrids}. Specifically:
\begin{itemize}
\item end-to-end asynchrony stops at the chip boundary, both at
the memory and I/O interfaces, and these latencies cannot be fully
tolerated;
\item the lack of hardware mechanisms to virtualize resources prevented the
proper implementation of deadlock-free composition.
\end{itemize}
Moreover, although binary compatibility is possible across chip
technologies, the execution performance of the code ended up not being
portable between different number of cores and interconnect
topologies.
Since 2011, when D-RISC's TMU was well-enough defined that it was
both obviously \emph{different from and more powerful than SVP's abstractions},
the SVP model has been downplayed and is not a central component of
publications any more.
\section{EU Apple-CORE: 2008-2011}
The project Apple-CORE was funded by the European Union based on a
statement of intent via an \emph{abstract}, and via a list of explicit
\emph{objectives}. There was no ``research question'' per se, as
the goal of the project was to build infrastructure and show
that the objectives were reached as a consequence.
\subsection{Summary of outcomes}
To an outsider,
the EU seemed to have funded research to develop a new general-purpose processor,
that would extend and possibly even replace the technology currently
in use in commodity hardware. Had that objective been reached, the
project would have been disruptive indeed. This \emph{potential} to
both disrupt the state of the art and advance technology in a way
largely beneficial to society was sufficient, to the proposal's
initial reviewers, to justify the investment.
However, there is an ethical issue at hand. First, the Apple-CORE
proposal stated that D-RISC and Microgrids were already
designed and a code generator for $\mu$TC was available prior
to the start of the project, whereas it was known to the authors of the proposal that
this was not true.
Only after the first year, after reviewers had been induced to believe
the issues were minor, did it become clear that the EU was also
funding this prerequisite technology. Also, at that point there was no
evidence that this ``initial'' technology would be sufficient to
research all the project's original objectives in a timely fashion.
Indeed, what happened is that the overhead of producing these
prerequisites prevented the consortium from exploring all the issues.
In short, Apple-CORE \emph{did not actually have the potential to
disrupt the state of the art and advance technology in the way
announced using the budget requested}. Besides, the details of scientific
outcomes (cf.~ next section) do not reveal any strong evidence that the
Apple-CORE technology can replace existing processors (cf.~ also \cref{sec:incompat}).
\paragraph{Disclaimer}
{\itshape The results and circumstances described below have been
brought to the attention of the project's reviewers while
the project was ongoing, and the project was judged successful
by both the reviewers and the project officer \emph{despite} these
issues. As I have learned since then, most
large, publicly-funded projects suffer from more serious issues and
the issues described here pale in comparison.}
\subsection{Outcomes vs.~ project objectives}
I present below how the project's outcome, as can be observed at the
end of 2012, relates to each stated objective in the project's
description of work. I list the objectives in decreasing order of
success, and mark with ``$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$'' those
points that most diverge from the overall initial goal of the project.
\paragraph{Apple-CORE will investigate the support structures in
implementing the SVP model in the LEON 3 processor and develop an SVP soft-core prototype.}
This was achieved, and was quite successful~\cite{danek.10.ddecs,sykora.11.lncs,danek.12}.
\paragraph{Apple-CORE will investigate the integration of instruction-set extensions to support custom accelerators based on both microthreads and families of microthreads.}
This was achieved, and was quite successful~\cite{danek.12}.
\paragraph{Apple-CORE will explore the gains of SVP in the context of data-parallel
programming, investigate the implications of functional concurrency and explore the possible design space.}
This was achieved, and was quite successful~\cite{a-c-d44}.
\paragraph{Apple-CORE will support many-core processors by
capturing concurrency systematically using
instructions in the processors’ ISA and by dynamically mapping
and scheduling that concurrency in the processors’ implementation (the SVP model).}
This was achieved~\cite{poss.12.dsd}.
\paragraph{Apple-CORE will derive a set of loop transformations to transform iterative computations into a combination of independent and dependent families of threads respecting the communication restrictions in the SVP model.}
This was achieved~\cite{saougkos.09.cpc,saougkos.11}.
\paragraph{Apple-CORE will extract task, loop and, implicitly in the SVP model, instruction level concurrency in the parallelising C compiler.}
This was only partially achieved: only loop and ILP was extracted. Task concurrency wasn't.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will provide binary-code compatibility across generations of multi-cores from few- to many-cores.}
This was achieved, although Apple-CORE also showed that code that performs well on
a small number of cores typically does not scale (performance- and
efficiency-wise) to large number of cores. Conversely, code that runs
with interesting speedups on large number of cores do not run
efficiently on small number of cores. The binary compatibility is thus
merely functional: it is possible to run the same code and obtain the
same results, but the performance is not portable. One can easily
argue that this form of compatibility is not really what was desired.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will implement and evaluate memory models and coherency protocols for many- core systems.}
This was achieved, only to conclude that the proposed models and
protocols were cumbersome to use, inefficient and otherwise
detrimental to performance for any configuration larger than 30-60
cores.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will study the resource management issues that are exposed in exploiting massive concurrency as it arises from data-parallel or functional program specifications.}
This was achieved: the management issues were indeed studied, but only
to conclude that the Apple-CORE strategy did not significantly
simplify the problem, which is otherwise shared by all research projects
in this field.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will provide high-level programming environments that
improve the programming productivity and automate the generation of
concurrency, or at least separate the concerns of concurrent
programming from its implementation, i.e. automate all scheduling and
synchronisation.} This was only partly achieved. By funding extra
development on Single-Assignment C, Apple-CORE did indeed ``improve
the programming productivity and automate the generation of
concurrency.'' However this effort was not directly related to
D-RISC/Microgrids: the improvements on SaC are portable to any
parallel hardware supported by the SaC compiler. Furthermore
Apple-CORE did not ``separate the concerns of concurrent programming
from its implementation,'' and neither did it ``automate all
synchronization.''
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will investigate and implement memory protection and security issues for many- core systems.}
This was investigated but not implemented.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will implement a port of a micro-kernel operating system onto one or more of the processor platforms (emulation and/or soft core).}
This was not investigated and not achieved, cf.~ also \cref{sec:incompat}.
\paragraph{
Apple-CORE will promote the $\mu$TC language as a standard front-end
to the gcc compiler and will use it as a target for all user-level
compiler development.} This did not occur, and $\mu$TC is not being
used any more, for the reasons presented in~\cite[App.~G]{poss.12}. Instead,
the project used another front-end to D-RISC/Microgrids called SL~\cite{poss.12.sl}, which
is riddled with practical limitations and has yet to gain credentials in the scientific community.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will investigate and evaluate programming productivity issues for the tools developed.}
This was not investigated nor evaluated.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will select a range of benchmark applications of interest to potential users of the SVP model within the European computer industry.}
This was only partly achieved: the industrial participation in the project was low, therefore the relevance of the resulting benchmark selection
cannot be confidently ascertained.
\paragraph{Apple-CORE will build an infrastructure of tools that will enable the SVP model to be evaluated and adopted by the European Computer Industry.}
This objective is untestable. Although an infrastructure of tools was produced, no adoption by the European Computer Industry has yet occurred.
\subsection{Outcomes vs.~ intents in the project's abstract}
The project's abstract also declared research intents not
covered otherwise in the objectives. I review them here:
\paragraph{The benefits are large, [...] as compilers need only capture
concurrency in a virtual way rather than capturing, mapping and
scheduling it.}
These benefits were not observed. Instead, Apple-CORE taught us is not sufficient to capture concurrency;
some semantics that brings an intuition of the machine back to the programmer
was necessary after all.
\paragraph{This separates the
concerns of programming and concurrency engineering and opens the door
for successful parallelising compilers.} There was no breakthrough in
programming and concurrency engineering during Apple-CORE, and no
``successful parallelising compiler'' has been produced as a
result. Technically, there were parallelising compilers produced, but
they are not yet ``successful'' insofar they have not yet gathered any
user base other than their own developers.
\paragraph{Particular benefits can be expected for data-parallel
and functional programming languages as they expose their concurrency
in a way that can be easily captured by a compiler.}
This was indeed shown, although this can be equally shown using
most parallel platforms in this day and age.
\paragraph{Another advantage
of this approach is the binary compatibility the new processor has
with the modified ISA. [...] Once code is compiled with the new
tools, binary-code is executable on an arbitrary numbers of processors
and hence provides future binary-code compatibility.} See above: although
the code is binary-compatible, the performance is not portable.
\paragraph{The concurrency controls also allow for management of
partial failure, which together with the binary-code compatibility
provide the necessary support for reliable systems.} This was
not shown in practice by Apple-CORE, although Apple-CORE's
infrastructure does simplify a research project on this topic,
started later on.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Finally, this
approach exposes information about the work to be executed on each
processor and how much can be executed at any given time.
This
information can provide powerful mechanisms for the management of
power by load balancing processors based on clock/frequency
scaling. }
This was not researched in Apple-CORE.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
In particular, the binary compatibility provides a unique
opportunity to make an impact on commodity processors in Europe.}
This was not achieved, as there is no binary compatibility with
existing processors. Actually, the argument of ``binary
compatibility'' throughout the project proposal diverges from usual
expectations. ``Binary compatibility'' is usually understood to mean
``backward compatibility with existing binary code,'' meaning that
existing software from other platforms can be reused on the new
platform. However, Apple-CORE instead promotes ``binary compatibility
between multiple instances of the Apple-CORE technology,'' i.e.~ no
binary backward compatibility with other platforms.
\section{C. Jesshope et al., ParCo 2009}
This article~\cite{jesshope.09.parco} rephrased the motivation
behind the D-RISC/Microgrid work, two years in the Apple-CORE project:
\begin{quote}
\itshape In a more general market [than embedded and special-purpose
accelerators], the labour-intensive approach of hand mapping an
application is not feasible, as the effort required is large and
compounded by the many different applications. {\bfseries A more automated
approach from the tool chain is necessary. This investment in the tool
chain, in turn, demands an abstract target to avoid these
compatibility issues. That target or concurrency model then needs to
be implemented on a variety of platforms to give portability, whatever
the granularity of that platform. Our experience suggests that an
abstract target should adopt concurrent rather than sequential
composition, but admit a well-defined sequential schedule. It must
capture locality without specifying explicit communication. Ideally,
it should support asynchrony using data-driven scheduling to allow for
high latency operations. However, above all, it must provide safe
program composition, i.e. guaranteed freedom from deadlock when two
concurrent programs are combined. Our SVP model is designed to meet
all of these requirements.} Whether it is implemented in the ISA of a
conventional core, as described here or encapsulated as a software API
will only effect the parameters described above, which in turn will
determine at what level of granularity one moves from parallel to
sequential execution of the same code.
\end{quote}
A few years afterwards, is the D-RISC/Microgrids management protocol
matching the claims? I review them here in decreasing order of
success.
\paragraph{The model should support asynchrony using data-driven scheduling to
allow for high-latency operations.} This was achieved (primary feature of D-RISC).
\paragraph{The model should adopt concurrent rather than sequential composition.}
This was achieved, although \emph{both} concurrent and sequential composition are equally promoted.
\paragraph{The model must admit a well-defined sequential schedule.}
This was mostly achieved~\cite[Chap.~10]{poss.12}. A sequential schedule is not properly defined
as soon as a program manipulates on-chip resources (in particular cores) explicitly.
\paragraph{The model needs to be implemented on a variety of platforms.} This was not achieved.
A software emulation of D-RISC's \emph{envisioned} TMU was implemented early on~\cite{tol.09.jsa}, but
the actual D-RISC TMU ended up with different semantics which have not yet been implemented elsewhere.
\paragraph{The model must capture locality without specifying explicit communication.}
This was not achieved: explicit communication is required between different threads.
\paragraph{Above all, it must provide safe program composition, i.e. guaranteed freedom from deadlock
when two concurrent programs are combined.}
This was not achieved: the ability of code to manipulate resources explicitly, combined with the lack
of full resource virtualization, may cause compositions to deadlock from resource starvation.
\section{ASCI 5-year research plan, 2010}
This document was submitted to a consortium of Dutch universities
at the start of 2010, to define the overall research plan of the consortium
over the period 2010-2014.
Over D-RISC/Microgrids this report states:
\begin{quote}
\itshape
Our work on fine-grain threaded architectures with data-driven
scheduling using the SVP concurrency model will continue but we will
also explore software implementations of SVP on other emerging
multi-core architectures such as Niagara and Intel's SCC. This will
allow us to explore multi-grain architecture and develop an
infrastructure to support such an approach. One of the major
directions in this work will be the development of a coherent set of
operating system services that support space sharing in these
heterogeneous environments and yet provide a secure operating
environment that can be scaled from chip-level micro-grids to globally
distributed Grids. One of the major challenges, especially in
mainstream computing, will be in making these systems programmable
without specialized concurrency knowledge and we have designed
programming language support to express parallel computations and
systems at a very high level of abstraction and developed compilation
technologies that effectively map the abstract descriptions to
concurrent computing environments. We have international
collaborations developing the functional, data parallel language SAC
(Single Assignment C) and the asynchronous co-ordination language
S-Net. Our long-term vision is in the direction of a compilation
infrastructure that automatically adapts running programs derived from
high-level specifications to a heterogeneous and dynamically varying
execution environment based on continuous reflection of execution
parameters.
\end{quote}
To this date, SVP was not implemented on other architectures. No
operating systems services have yet developed that support space
sharing in heterogeneous environments and provide a secure exeuction
environment. The D-RISC/Microgrids language tools are not yet able to
map the abstraction of concurrent resources to maximize performance
and efficiency. ``Automatic adaption of running programs towards
heterogeneous and dynamically varying execution parameters based
on continuous reflection'' was not achieved either yet.
\begin{summary}
\begin{itemize}
\item
The D-RISC/Microgrids project was purportedly intended to solve major issues in
micro-architecture research, related to scalability in performance and
efficiency in general-purpose microprocessors.
\item
The strategy to solve these issues was to implement a
combination of dataflow scheduling with hardware support for thread
concurrency management within and across cores on chip.
\item
Implementation
was carried out, but the results are inconclusive. On the one hand, the proposed
hardware does indeed provide higher performance and efficiency in
regular, data-parallel workloads, but these are also the ``boring''
applications which benefit equally well from vector units or accelerators
in conventional processors. On the other hand, no evidence has yet
been produced that the proposed hardware benefits larger applications
with more irregular workloads.
\item
Power efficiency and intelligent resource
management was regularly advertised but not actively researched.
\item
Effort has been invested into widening the scope of the technology
towards applications and industrial relevance, but these applications
have not yet materialized.
\end{itemize}
\end{summary}
\section*{Chapter summary}%
}{\end{snugshade}
\end{minipage}
}
\pagestyle{headings}
\begin{document}
\author{Raphael ‘kena’ Poss\\University of Amsterdam, The Netherlands\\\url{[email protected]}}
\title{On whether and how \\ D-RISC and Microgrids can be kept relevant \\
(self-assessment report)}
\maketitle
\input{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\clearpage
\input{begin}
\input{background}
\part{Deconstruction}\label{part:decons}
\input{outcomes}
\input{methodology}
\input{obstacles}
\part{Reconstruction}
\input{contrib}
\input{parts}
\input{strategy}
\input{conclusion}
\newcommand{\etalchar}[1]{#1}
\addcontentsline{toc}{chapter}{References}
\bibliographystyle{is-alphaurl}
\chapter{Individual architectural features}
Publications, posters and talks usually present the D-RISC core and
Microgrid clusters thereof as a single coherent technology made from
inter-dependent features. In reality, a gradual composition is
possible, as well as adding TMU-like features to other processor cores
than D-RISC.
I shortly present here my understanding of this
composition, which I have started to recognize while
writing~\cite[Chap.~3]{poss.12}.
\section{Overview}
\begin{figure}[b!]
\centering
\includegraphics[width=.6\textwidth]{features2}
\caption{Overview of the characteristic features of D-RISC/Microgrids.}\label{fig:features2}
\end{figure}
To start with, I summarize
the characteristic features in \cref{fig:features2}; in this
diagram I denote with a double edge the features not found
in other processors, and with a striped red edge those
features found in other processors but not on D-RISC/Microgrids. This diagram
exposes the composition of features in the design, as follows:
\begin{itemize}
\item the D-RISC core itself is a composition of the following features:
\begin{itemize}
\item a pretty conventional in-order, single-issue RISC pipeline,
\item a dataflow instruction scheduler that can execution instructions out-of-order while respecting data dependencies,
\item multiple hardware threads (separate program counters and registers),
\item a memory-mapped interface to an I/O subsystem,
\item a custom hardware Thread Management Unit (TMU) that manages logical tasks and maps and schedules them over hardware threads;
\end{itemize}
\item a Microgrid is a cluster of D-RISC cores together with:
\begin{itemize}
\item asynchronous functional units (FUs) between cores for seldom-used instructions;
\item a shared MMU that provides a single virtual address space to all cores;
\item a custom control network-on-chip to coordinate concurrency management between TMUs;
\item optionally, a custom memory interconnect.
\end{itemize}
\end{itemize}
Of these features, we can distinguish those that provide ``added
value'' compared to other processor architectures, namely the TMU,
dataflow scheduler and control NoC, from those that are ``unique'' and
make the design fundamentally incompatible with conventional wisdom, namely
the lack of support for preemption and the lack of per-core (and
per-hardware thread) MMU, which were discussed in \cref{sec:incompat}.
Remarkably, this overview alone reveals that the combination of
features found in D-RISC could be obtained by starting from an existing RISC
core design. The Tera MTA, for example, already features hardware
multithreading, a dataflow scheduler and memory-mapped I/O, and could
be \emph{extended} with a TMU. Similarly, Sun/Oracle's Niagara T4
cores already feature hardware multithreading, memory-mapped I/O and
out-of-order instruction execution (via reservation stations, which is
really a form of dataflow scheduling), and could thus be extended with
a TMU as well.
\section{Details}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{features}
\caption{Overview of the characteristic features of D-RISC/Microgrids (expanded).}\label{fig:features}
\end{figure}
To understand how D-RISC and Microgrids additionally benefit from re-implementing
its own ``version'' of features already found in existing processors,
it is useful to dig one level deeper, as illustrated in
\cref{fig:features}.
(Again, I denote with a double edge the features
not found in other processors, and with a striped red edge those
features found in other processors but not on D-RISC/Microgrids. Features
with dotted borders are experimental, not yet ready for production. The
red triangles highlight the missing features that fundamentally hurt
software reuse.)
Besides highlighting the wealth of features of D-RISC's TMU and the inter-TMU NoC, this diagram draws
attention to the following points.
\subsection{Dataflow scheduling from the register file}
All dataflow schedulers rely on a \emph{matching store}, which retains
information about ``what to do'' when an operation has completed,
while the operation is ongoing. In conventional architectures,
matching stores are implemented:
\begin{itemize}
\item as part of the memory
sub-system, starting at the data cache (e.g. Tera MTA, Niagara T4+), to
determine ``what to do when a memory operation completes,''
\item via reservation stations next to functional units in out-of-order
execution (e.g. PowerPC), to determine ``what to do when a local operation completes.''
\end{itemize}
Like MTA and T4, D-RISC uses the L1 D-cache as matching store; but
contrary to conventional OoOE techniques it uses the main register
file instead of reservation stations for out-of-order execution of
instructions.
This choice \emph{simplifies the circuits} that connects the pipeline,
functional units and the register file together. The trade-off is that
the number of in-flight instructions \emph{per thread} is limited to
the ISA register space, typically 31, whereas it can grow arbitrarily
with reservation stations.
This way, we can recognize that \emph{in the large design domain where
out-of-order instruction execution is desirable, D-RISC is exploring
the sub-domain where per-thread ILP can be traded off with simpler
circuits.}
\subsection{Optimizations to hardware multithreading}
\Cref{fig:features} also highlight two features that optimize
hardware multithreading, regardless of the presence of the TMU
and of a dataflow scheduler.
The first is a tight integration of the schedule queue and the
I-cache. As detailed in~\cite{lankamp.08} and
\cite[Sect.~3.2]{poss.12}, a a thread is not considered by the fetch
stage unless its code is already in the I-cache, so as to prevent
pipeline bubbles due to I-cache misses. This technique decouples code
fetching and thread scheduling and \emph{yields higher utilization of
the pipeline overall}. I do not know whether this feature
is used by other hardware multithreaded core designs, but
I would be surprised if it were not.
The other feature is the ability to \emph{hint} the fetch stage of the
pipeline to switch preemptively to another thread after fetching an
instruction, when that instruction is \emph{likely to cause a pipeline
bubble at a later stage}. This can be used for any instruction that
reads the result of a previously issued long-latency instruction. For
example, in ``ld r1 $\leftarrow$ [x]; add r2 $\leftarrow$ r1+r1'' the ``add''
instruction would be annotated, so that the fetch stage preemptively
switches to another thread: if the ``add'' suspends due to a missing
input (e.g.~ due to a cache miss on the previous ``ld'' instruction), no
other instruction from the same thread needs to be flushed from the
pipeline. This feature also \emph{yields higher utilization of the
pipeline overall}, to my knowledge is not present in other
processors, but could be possibly be added to them.
To this date, switch hints are implemented in D-RISC by
\emph{interleaving} hint bits with the instruction stream, which in
turns requires a custom assembler to generate the binary program
code. However, other implementations are possible, cf.~
\cite[Sect.~4.4]{poss.12}.
\subsection{Bulk coherency in the shared memory network}
As soon as the designer of a chip architecture combines multiple cores
with memory-level parallelism and still wishes
to expose a ``shared memory'' interface to programmers,
a \emph{coherency protocol} must be designed to ensure
that the memory updates performed by one core on one part
of the memory become eventually visible to other cores
connected to other parts of the memory.
(Note that this problem disappears if all cores are physically
connected to the same memory or shared cache with a bus.)
\emph{Necessarily}, a coherency protocol implies communication across
the memory network to propagate the data ``placed into the memory'' by
store requests, to the point where it may be needed by subsequent load
requests. Also, since the memory network cannot ``predict the
future,'' it must make a decision to effect this propagation
preemptively at some granularity: either after each individual store,
or while evicting cache lines, or by allowing the program code to
``grab exclusivity'' for a range of addresses over the entire system (write-invalidate).
In the Microgrid design, knowledge from the TMU about the program's
structure is used to implement an optimization to coherency: when a
bulk of tasks are created together (a feature of the TMU), the TMU
informs the cache network that it can wait to propagate the stores performed by
that bulk until all tasks in it have terminated, or until when a task creates
a sub-task. This ``makes sense,'' i.e.~ it is valid, because D-RISC's programming
model specifies that stores are only visible to other tasks after a
task terminates or to the sub-tasks it subsequently creates. The
benefit of this optimization is a \emph{reduced number of
coherency-related communication events in the memory network} for some workloads.
To my knowledge, this optimization opportunity is also exploited in
SIMD/SPMD accelerators, and in general it could be readily
considered in any design where high-level information about the
clustering of software operations is visible to the hardware.
\section{TMU reusability}
The purpose and consequence of decomposing the D-RISC design is to
isolate its TMU and recognize that the TMU is really a hardware
accelerator for system management functions that would be otherwise
realized in software. It is actually possible to describe the TMU as
an extension to any generic RISC core, even a core that does not
offer the other features of D-RISC:
\begin{itemize}
\item without hardware multithreading, the TMU would be constrained to
schedule logical tasks over a single hardware thread. This would
restrict the amount of instruction-level parallelism (because the
maximum number of in-flight instructions is restricted by ISA
register window), but would still save up the cost of branches and
increments to implement repetition, and thus accelerate loops;
\item without out-of-order execution (either dataflow scheduling or
via other means), instructions that control the TMU would cause the
processor to wait until the TMU operation has completed. This may
imply large waiting times for ``complex'' operations, for example
allocating a group of cores on another part of the chip. It would
also mandate the use of interrupts to signal asynchronous
completion, for example the termination of a task, but would still
save up the overhead of doing the thread management entirely in a
software operating system.
\end{itemize}
\begin{summary}
\begin{itemize}
\item The D-RISC core combines features found in other processors, such as a RISC pipeline
and hardware multithreading, with custom features (e.g.~ its TMU) and optimizations
to the conventional features (e.g.~ switch annotations for the HMT scheduler).
\item Some architectural optimizations found in D-RISC/Microgrids could be reused
with other processors, for example switch annotations and bulk coherency in the memory network.
\item The key feature of D-RISC/Microgrids, namely its TMU and inter-TMU control NoC,
does not depend on the other features specific to D-RISC and could be potentially reused with other processors.
\end{itemize}
\end{summary}
\chapter{Follow-up strategies}
\Cref{part:decons} has highlighted shortcomings in the methodology and
obstacles to further progress. This analysis raised the question of
how to move forward from there \emph{differently}, so as to
avoid these shortcomings and obstacles. This chapter presents my view on this question, articulated in two directions: first,
what would constitute ``sane approaches'' for new projects (\cref{sec:new}); then
how to ``fix'' or ``improve'' ongoing projects/research (\cref{sec:ongoing}).
\section{Possible strategies for new investments}\label{sec:new}
\paragraph{Exploit.}
Apply the technology produced
so far to other uses than research.
A successful application so far has been education: with only
a minor but regular maintenance effort, the simulation tools
can provide support in architecture and compiler courses
for the coming 5-10 years.
However, given the processor is unable to support any C code that requires a ``hosted''
environment (or other
languages whose RTS is written using hosted C), applications in industry will be limited to small
embedded systems.
Possibly, with only a minor effort investment, an ad-hoc form of preemption
and per-core MMU can be added to a simulation model and obtain a
limited compatibility with software frameworks. Without
significant research, this would yield
sub-efficient (non-competitive) performance, but the gained
compatibility might be sufficient to activate further
external interest in the work.
\paragraph{Salvage and open.}
Extract individual features from the D-RISC/Microgrids
design and evaluate them as extensions
of existing processors.
Small ``first steps'' in this direction can be made
by starting with the switch annotations and the coupling
of the fetch stage with the I-cache. These features
seem readily applicable to the Niagara architecture
and the latest ARM multithreaded cores.
A more significant project would be to extract
the TMU and offer it as a reusable accelerator
component where the processor designer
can choose which TMU feature are activated. For example,
the features related to bulk creation/synchronization
or multi-core resource management may not be always relevant,
and a designer should not need to pay the price of
their integration if they end up not being used.
Conversely, the D-RISC core stripped of hardware multithreading and
its TMU could be offered as a SoC building block, marketing its dataflow
scheduler as a lightweight implementation of out-of-order execution. For
this block to be moderately competitive, a branch predictor
may be proposed as an option.
\paragraph{Distill and reincarnate.}
From the perspective of theoretical computer science, the
D-RISC/Microgrids enterprise has raised two questions that may warrant
a wealth of further fundamental research.
The first was opened on purpose: \emph{how does the cost intuition of
programmers evolve when complex operating system services are
available at nearly the cost of basic arithmetic?} This is one of the key
questions that the TMU was designed to answer. The \emph{desired} answer was
originally: ``once programmers are comfortable about the costs of
concurrency, they would use concurrency everywhere and obtain parallel
speedups at every level.'' This particular answer was not obtained by
the research so far, but it may well be that other interesting answers can be
obtained instead.
Further effort in this direction could be bootstrapped as follows.
First, get acquainted with a software community already comfortable
using concurrency without too much assumptions about hardware.
Haskell and Erlang programmers are interesting candidates. Then,
observe and inventory which specific patterns of concurrency they
already use, and those they are striving to implement. Then re-design
a custom TMU that accelerates their favorite language run-time
system. Then, demonstrate the net effect on existing programs, and
document how the programmers modify their software over time to take
advantage of this accelerator.
I discovered the other fundamental question while dismantling
the ``SVP model'' proposed in~\cite{jesshope.08.apc} and used
subsequently in the period 2008-2011. SVP has captured, using its
``places,'' the notion that a group of processors should be considered
as a \emph{single}, \emph{fungible resource} that can be allocated
dynamically from the computing environment and sub-partitioned
dynamically using abstract operators such as those implemented by
D-RISC's TMU. The designers of SVP then claimed that ``places are the
fundamental currency of computing'' and that their abstract operators
were ``general-purpose,'' i.e.~ sufficiently general to carry out any computation.
As I discuss in~\cite[Chap.~9\&12]{poss.12}, I believe this particular claim is invalid,
because a ``general'' model of computing resource should offer and
define memory and means for I/O as well, which SVP
places do not. However, studying SVP raises the complement question:
is it possible to \emph{extend a general model of computation with a
cost model that uses entire virtual parallel computers with multiple
cores and multiple memories as the basic resource unit}? A strategy
for exploring this question would
probably benefit from starting with an inherently concurrent model, which
Turing and queue machines are not. The Actor model~\cite{agha.85} and
Milner's $\pi$-calculus~\cite{milner.92,milner.92.2} may be more suitable candidates, as their intuitive
implementations have well-understood operational semantics already.
Further effort in this direction could be bootstrapped as follows. First,
select a technology which already uses virtualizations of entire
parallel resources as a basic building block. Modern Unix systems
and VM hypervisors are candidates. Then formalize its
basic concurrency operations (e.g.~ fork, wait in Unix) in the conceptual
terminology of a general model. Then, based on this formalization and
expert knowledge of the actual behavior of the technology on parallel hardware,
design a cost algebra that is reasonably predictive. Then implement
a framework that visualizes and predicts cost for existing applications
using that technology. Use the interest gained in this way to attract
funding on the fundamental question.
\section{Possible strategies for ongoing projects}\label{sec:ongoing}
\subsection{Partnership with industry: 150k€ at stake}
An industry partner has recently funded some initial research effort to add
priority scheduling to D-RISC's thread scheduler and to explore fault
detection and recovery. Initial results suggest an opportunity to fund
further development effort in that direction, with the understanding
that the partner can use the benefits of the technology in their embedded
aeronautics controllers, which already use space-hardened custom SPARC
cores, in a 1-core or 2-core configuration.
Here the two strategies ``exploit'' and ``salvage and open'' described
above are applicable.
For the ``exploit'' strategy, the partner would need to fund
simultaneously a rewrite of the D-RISC specification in a language
suitable for both simulation and synthesis, so as to avoid maintaining
two source bases over time, and an extension of the current D-RISC
design to support preemption and resource reclamation, as much as
required by the partner's software.
For the ``salvage and open'' strategy, the partner could simply fund
a rewrite of D-RISC's HMT scheduler and the subset
of D-RISC's TMU that is sufficient for the partner's software as an extension
of the partner's favorite/desired existing processor core.
\subsection{Ongoing PhD theses: 400k€ at stake}
Both my peers who already defended a doctoral thesis founded on D-RISC and
Microgrids~\cite{bernard.10,vantol.13}, and myself, have been assailed during
our defenses with variation of the following:
\begin{itemize}
\item ``why did you choose this platform?''
\item ``what makes this platform especially attractive?''
\item ``why is your evaluation by software applications so poor?''
\end{itemize}
As answers, all three of us formulated variations of ``I was told this
platform was general enough and/or had great potential when I started,
and only later I recognized some of the obstacles, but I did my
part nonetheless. And look, by the way, I found some nice answers to
side research questions of my own, not initially phrased in the
D-RISC/Microgrids enterprise!''
Meanwhile, our unspoken thought was: ``I trusted my supervisor this
was the right place to start my PhD study and obtain the scientific
merit needed to graduate successfully, and as a beginner scientist I
did not have yet the critical acuity to recognize our shared
methodological shortcomings. But everyone can make mistakes, and
should be forgiven for them. After all, my PhD defense committee finds
me worthy of a doctorate, so it couldn't be as bad as it looks.''
In principle, the currently ongoing PhD research projects could
be concluded on the same note, and numerous new projects started
with the understanding they will conclude similarly.
In practice however, as I am sitting next to them and entertain close
social contact, I feel dishonest letting my peers employ this strategy: given
I now understand the shortcomings, is it fair to let my peers struggle
with the large friction to academic publication and peer acceptance
caused by our communal continued use of a flawed approach? The risk
is great also that they recognize this friction but feel the obstacle
is insurmountable, or worse, that this realization engenders distrust
against the potentials of further research in the area.
Here, unfortunately, I do not have the experience sufficient to
guarantee better outcomes with an alternate strategy with any
confidence. The essence of any sane approach, to me, would be to
retrospectively \emph{reverse-engineer properly formulated research
questions} that happen to be suitably answered by the work effectively performed,
independently from the initial initiative. This question should then
be phrased as generally as possible so that it does not hinge on the
specifics of D-RISC/Microgrids. Only in a second phase, subsequently
propose the current implementation of D-RISC/Microgrids as a case
study. To illustrate, I list some possible rephrasings in
\cref{tab:foo}. In nearly all cases, I think it would be useful to
acknowledge early on that the restrictions described in
\cref{sec:incompat} are arbitrary, and seek actively means to overcome
them to gain access to more software benchmarks. This may even imply partial uses of the ``salvage and open''
strategy described earlier.
\begin{table}
\begin{tabular}{p{.35\textwidth}cp{.55\textwidth}}
Initial impulse & & Reverse-engineered research questions \\
\hline
how to build a D-RISC TMU? & $\rightarrow$ & what are the costs/benefits of
accelerating OS functions for thread management with a hardware unit? \\
& $\rightarrow$ & what insights about how the hw/sw interface
influences programming language semantics, are gained while building a TMU? \\
how to build a D-RISC/Microgrids simulator? & $\rightarrow$ & what simulation
framework would be suitable for research in micro-architecture design while
keeping simulation performance high enough to run significant multi-core workloads? \\
& $\rightarrow$ & to which level of accuracy can a model in this framework simulate
the behavior of a hardware implementation? \\
how to improve D-RISC's memory performance? & $\rightarrow$ & what are the costs/benefits
of modifying memory interfaces and protocols to increase the latency tolerance abilities of cores that use HMT and/or dataflow schedulers? \\
& $\rightarrow$ & what are the quantitative benefits of exploiting the concurrency awareness available
in hardware in memory protocols? \\
how to implement priority scheduling in D-RISC? & $\rightarrow$ & what are the cost/benefits
of extending a HMT scheduler with priorities? \\
how to implement fault tolerance in D-RISC/Microgrids? & $\rightarrow$ & what are the costs/benefits
of exploiting the concurrency awareness available in hardware in fault tolerance protocols? \\
& $\rightarrow$ & is it possible to abstract fault tolerance to a general computing model equipped
with a resource/cost model?
\end{tabular}
\caption{Example reverse-engineering of research questions.}\label{tab:foo}
\end{table}
\begin{summary}
\begin{itemize}
\item I can see three follow-up strategies for new investments around
D-RISC/Microgrids: exploitation, i.e.~ apply the technology produced
so far to other uses than research; salvaging and opening the
technology, i.e.~ extracting individual features from the
D-RISC/Microgrids design and evaluating them as extensions of
existing processors; and distillation of the main ideas in the realm
of fundamental computer science.
\item Ongoing research towards doctoral theses should be careful
to rephrase research questions in the light of our recent shared
understanding of the project's issues.
\end{itemize}
\end{summary}
|
1,116,691,499,351 | arxiv | \section{Introduction}
Imagine you are driving a car on a highway and suddenly an object appears in your visual field, crossing the road. Your initial reaction is to slam on the brakes even before recognizing the object. This highlights a core difference between human vision and current machine learning strategies for object recognition. In machine learning, object recognition is often viewed as a feedforward, bottom up process, where image features are extracted from local to global in a hierarchical manner; whereas in human vision, we can capture the gist of an image at a glance without processing the details of it, a crucial ability for animals to survive in competitive natural environments. This strategic difference has been demonstrated by a large volume of experimental data. For example, Sugase et al. found that neurons in the inferior temporal cortex (IT) of macaque monkeys convey the global information of an object much faster than the fine information of it~\citep{Sugase1999Nature}; FMRI and MEG studies on humans showed that the activation of orbitofrontal cortex (OFC) precedes that of the temporal cortex when a blurred object was shown to the subject~\citep{Bar2006PNAS}.
Indeed, the Reverse Hierarchy Theory
for visual perception has proposed that although the representation of image features in the ventral pathway goes from local to global, our perception of an image goes inversely from global to local~\citep{Hochstein2002Neuron}.
How does this happen in the brain? Experimental studies show that there are two separable signal pathways for visual information processing in the brain
(see Fig.\ref{fig-demo}). One is called the parvocellular pathway (P-pathway), which starts from midget retina ganglion cells (MRGCs), projects to layers 3-6 in the lateral geniculate nucleus (LGN), and then
primarily goes downstream along the ventral pathway. The other is called the magnocellular pathway (M-pathway), which starts from parasol retina ganglion cells (PRGCs), projects to layers 1-2 of LGN or the superior colliculus (SC), and then goes downstream to higher cortical areas. The two pathways have different neural response characteristics and complementary computational roles. Specifically, the P-pathway is sensitive to color and responds primarily to visual inputs of high spatial frequency; whereas the M-pathway is color blind and responds primarily to visual inputs of low spatial frequency~\citep{Derrington1984JP}. It has been suggested that the M-pathway serves as a short-cut to extract global information of images rapidly; while the P-pathway extracts fine features of images slowly; the interplay between two pathways endows the neural system with the capacity of processing visual information rapidly, adaptively, and robustly~\citep{Bar2003JCN,Wang2020bioRxiv, Bullier2001BRR}.
\begin{figure}[tbp]
\begin{center}
\centerline{\includegraphics[width=0.7\linewidth]{figs/demo2.eps}}
\caption{Two signal pathways for visual information processing in the brain. The P-pathway starts from midget retina ganglion cells (MRGCs) and goes through the ventral stream to Fusiform and other cortical regions. The M-pathway starts from parasol retina ganglion cells (PRGCs) and goes through superior colliculus (SC) or the dorsal stream to prefrontal cortex (PFC) and other cortical
regions. The P-pathway receives detailed visual inputs and extracts fine features of images. The M-pathway receives low-pass filtered visual inputs and extracts coarse features of images. The two pathways are associated in higher cortical areas to
accomplish visual recognition.
For instance, the M-pathway may generate candidate objects based on the coarse information of an image, which serves as a cognitive bias guiding the fine recognition of the image mediated by the P-pathway.}
\label{fig-demo}
\end{center}
\end{figure}
Although the P- and M- pathways are well known in the field, exactly how their interplay facilitates object recognition remains poorly understood.
Conventionally, machine learning learns from neuroscience to develop new models, e.g., convolution neural networks (CNNs) constructed by learning from the hierarchical, feedforward architecture of the visual pathway~\citep{Fukushima1980BC, Lecun1998IEEE} and new machine learning models for scene recognition inspired by the central and peripheral vision~\cite{Wang2017JOV, Wu2018CISS}.
Conversely, recent state-of-the-art machine learning models, such as deep CNNs, were applied to interpret neural data and obtained encouraging results. For example, by using deep CNNs to model the ventral pathway, the response properties of neurons and their computational roles were well interpreted~\citep{Yamins2013nips, Kriegeskorte2015AV}.
In this study, based on the biological relevance and expressive power of CNNs, we build up a two-pathway model to elucidate the computational advantages associated with the interplay between P- and M- pathways. The outcome of this study will naturally give us insight into developing new object recognition architectures.
Our model consists of three parts, two parallel CNNs and an associative memory network that integrates them (Fig.~\ref{fig-model}). One CNN mimics the P-pathway, which is relatively deep, has small-size kernels, and receives detailed visual inputs. It aims to extract fine features of images, referred to as FineNet hereafter. The other CNN mimics the M-pathway, which is relatively shallow, has large-size kernels, and receives low-pass filtered or binarized visual inputs. It aims to extract coarse features of images, referred to as CoarseNet hereafter. The two CNNs are associated with each other via a Restrict Boltzmann Machine (RBM). Based on this model, we unveil a number of appealing properties associated with the interplay between two pathways. First, since CoarseNet is shallow and unable to learn an object recognition task well, FineNet can teach CoarseNet through imitation to improve its performance considerably. Second, since CoarseNet has large convolution kernels and receives coarse inputs, its performance is robust to noise corruptions, which in return can enhance noise robustness of FineNet via association. Third, since CoarseNet is faster (mimicking the M-pathway),
its output can serve as a cognitive bias to leverage the performance of FineNet.
\begin{figure}[tbp]
\begin{center}
\centerline{\includegraphics[width=0.7\linewidth]{figs/model.eps}}
\caption{The two-pathway model consisting of two CNNs (FineNet and CoarseNet) and an RBM. (A) The imitation learning phase. CoarseNet learns from FineNet through imitation. (B) The association phase. FineNet and CoarseNet are associated with each other through the RBM.
(C) The interplay phase in which CoarseNet improves the noise robustness of FineNet. The example illustrates how the result of CoarseNet helps FineNet recognize a
noise corrupted image of eagle via iterating the state of the RBM. LPF: low-pass filtered.}
\label{fig-model}
\end{center}
\end{figure}
\section{The Two-pathway Model}
The structure of our two-pathway model, together with its learning and functioning processes, are illustrated in Fig.~\ref{fig-model}.
Since the detailed structures and functions of the P- and M- pathways are far from clear, we adopt two CNNs of different complexities to capture their essential computational properties
(CNNs have been shown to be effective for modeling visual information processing~\citep{Yamins2013nips, Kriegeskorte2015AV}),
with a focus on their different characteristics in feature extraction. Specifically,
FineNet is deeper than CoarseNet, reflecting that the P-pathway goes through more feature analyzing relays
(V1-V2-V4-IT in the primate brain) than the M-pathway.
It also has smaller convolution kernels and receives much more detailed visual inputs, i.e., raw visual images with RGB channels, reflecting that PRGCs are color selective and have much smaller receptive fields than MRGCs
anatomically.
Due to the large receptive field sizes of MRGCs and the electrical couplings between them (which leads to long-range coherent activity in the retina that is believed to be crucial for global object perception~\citep{Roy2017PNAS}), we set CoarseNet to have relatively large-size
convolution kernels and process low-pass filtered visual inputs.
Moreover, since MRGCs are color blind, we consider that CoarseNet processes grayed images.
Biologically, information processing along the M-pathway is much faster than that along the P-pathway, but this is not reflected in our model, as CNNs do not include the dynamics of neurons.
Experimental data indicates that multiple brain regions are likely
involved in the association between the two pathways, such as the medial temporal lobe, hippocampus~\citep{Eichenbaum2000, Ranganath2004}, and the parahippocampal cortex~\citep{Aminoff2007}. Here, we simplify this as an associative memory process mediated by an RBM, whereby the outputs of two CNNs can interact with each other.
\subsection{The imitation learning phase}
Given an input image $\bm{x}$,
denote the output of FineNet to be $\bm{p}^F(\bm{x})=\bm{f}^F\left(\bm{g}^F\left(\bm{x},\bm{\theta}^F\right),\bm{w}^F\right)$, where $\bm{g}^F(\cdot,\bm{\theta}^F)$ represents
the mapping function from the input to the penultimate layer, $\bm{f}^F(\cdot,\bm{w}^F)$ the readout function, and
$\{\bm{\theta}^F,\bm{w}^F\}$ the trainable parameters. $\bm{p}^F(\bm{x})$ is a K-dimensional vector,
with $K$ the number of image classes.
Similarly, the output of CoarseNet is written as
$\bm{p}^C(\hat{\bm{x}})=\bm{f}^C\left(\bm{g}^C\left(\hat{\bm{x}},\bm{\theta}^C\right),\bm{w}^C\right)$, where $\hat{\bm{x}}$ represents the coarse input with respect to the image $\bm{x}$.
$\bm{g}^C(\cdot,\bm{\theta}^C)$ and $\bm{f}^C(\cdot,\bm{w}^C)$ represent, respectively,
the mapping functions from the input to the penultimate layer and
the readout function, and
$\{\bm{\theta}^C,\bm{w}^C\}$ the trainable parameters.
The coarse input to CoarseNet is obtained by either low-pass filtering
a grayed image using a 2D Gaussian filter or binarizing an image (see examples in Fig.~\ref{fig-data}A).
First, to get a model of the P-pathway,
we optimize FineNet through minimizing a cross-entropy loss, which is written as
\begin{equation}
L_F=-\frac{1}{N}\sum^N_{i=1}\sum^K_{j=1}y_{i,j}
\ln{p^F_j(\bm{x}_i)},
\end{equation}
where $p^F_j$ is the $j$th element of $\bm{p}^F$, i.e., the likelihood of
the $j$th class, and $y_{i,j}$ is the $j$th element of the
one-hot label $\bm{y}_i$ for the image $\bm{x}_i$, which
is $1$ for the correct class and $0$ otherwise.
The summation runs over all images $N$ and all classes $K$.
Second, to get a model of the M-pathway,
we optimize CoarseNet via imitation learning from the optimized FineNet, and the corresponding loss function is given by
\begin{equation}
L_{C}=\frac{1}{N}\sum^N_{i=1}
\left[-\alpha
\sum^K_{j=1}y_{i,j}\ln p^C_j(\hat{\bm{x}_i})+\frac{1-\alpha}{2}\|
\bm{g}^C(\hat{\bm{x}}_i)-\bm{g}^F(\bm{x}_i)\|^2
\right],
\label{eq-imitation}
\end{equation}
where
$\bm{g}^F(\bm{x}_i)$ is the imitation target for feature representations, i.e., the neural activity at the penultimate layer of FineNet. The symbol $\|\cdot \|$ denotes $L_{2}$ normal, and $\alpha$ is a hyper-parameter controlling the balance between the cross-entropy and imitation losses. Through imitating the features representations of FineNet,
CoarseNet learns to classify images based on the coarse inputs.
\subsection{The association and interplay phases}
\label{asso-interlay}
After training two networks, we perform association learning to establish their correlation via an RBM.
The RBM is a simplified version of the Boltzmann Machine (BM), with the latter being an extension of the Hopfield model including stochastic dynamics~\citep{Hinton2006NC}. Both the BM~\citep{Ackley1985CS} and the Hopfield model~\citep{Hopfield1982PNAS} aim to
capture how memory patterns are stored as stationary states of neural circuits via recurrent connections between neurons. An RBM consists of a visible and a hidden layers with no within-layer connections.
It is unclear yet how two visual
pathways interact with each other biologically, which is complex and task-dependent. Hence we consider that the associated information via the RBM is slightly different for the two tasks investigated in the present study.
Specifically, for the task of using the output of CoarseNet to improve the robustness of FineNet to noise, referred to as the robustness task (see Sec.~\ref{sec-robustness}), we
concatenate the neural activities in the penultimate layers of CoarseNet and FineNet in response to the same images, i.e,
$\left\{\bm{g}^C(\hat{\bm{x}}), \bm{g}^F(\bm{x})\right\}$,
as the inputs to the visible layer of the RBM.
For the task of using the output of CoarseNet as a cognitive bias to improve the performance of FineNet, referred to as the cognitive-bias task (see Sec.~\ref{sec-bias}), we concatenate
the neural activity in the penultimate layer of CoarseNet $\bm{g}^C(\hat{\bm{x}})$ and the corresponding context vector $\bm{c}(\hat{\bm{x}})$ as the inputs to the visible layer of the RBM.
Once the inputs to the visible layer (i.e., the data pairs to be stored)
are specified, we optimize the connections between the visible and hidden layers, such that the data pairs are stored as local minimums of the energy function of the RBM.
After training, since the data pairs are stored as memory patterns in the RBM, it is expected that when one component of a data pair is presented as a cue, the RBM will retrieve the other component automatically. The retrieved information can then be used for the recognition task.
Denote the updating dynamics of the RBM to be $[\bm{v}_{t+1}^C,\bm{v}_{t+1}^F]=\bm{A}([\bm{v}_{t}^C,\bm{v}_{t}^F])$, with
$\bm{A}(\cdot)$ the mapping function implemented by the RBM.
In the noise robustness task (Sec.~\ref{sec-robustness}),
we clamp $\bm{v}_t^C=\bm{g}^C(\hat{\bm{x}})$ unchanged over time,
and set the initial state $\bm{v}_{0}^F=\bm{g}^F(\bm{x})$. After iterating the state of the RBM by a number of steps $T$, the output $\bm{v}_{T}^F$ gives the associated feature representation of FineNet.
In the cognitive-bias task (Sec.~\ref{sec-bias}),
we clamp $\bm{v}_t^C=\bm{g}^C(\hat{\bm{x}})$ unchanged over time, and set the initial state $\bm{v}_{0}^F=0$. After iterating the state of the RBM by a number of steps $T$, the output $\bm{v}_{T}^F$ gives the retrieved context vector $\bm{c}(\hat{\bm{x}})$ for the image $\bm{x}$. This context vector is then combined with the feature representation of FineNet,
denoted as $\bm{g}^F(\bm{x},\bm{c}(\hat{\bm{x}}),\bm{\theta}^F)$,
to carry out the recognition.
\section{Interplay between Two Pathways}\label{exp}
\subsection{Implementation details}\label{imple-detail}
Based on the proposed model, we carry out simulation experiments to explore the computational properties associated with the interplay between two pathways. Three datasets, CIFAR-10, Pascalvoc-mask and CIFAR-100 are used.
CIFAR-10 is for demonstrating the effect of imitation learning (Fig.~\ref{fig-result1}A-C) and the robustness of the model to noise (Fig.~\ref{fig-result2}).
Pascalvoc-mask (binarized images) is used as another form of
coarse
inputs to evaluate the performance of CoarseNet (Fig.~\ref{fig-result1}D-E).
CIFAR-100 is for demonstrating the effect of cognitive bias
(Fig.~\ref{fig-result3}). An example of visual inputs used in the experiments is displayed in Fig.~\ref{fig-data}.
\begin{figure}[tbp]
\begin{center}
\centerline{\includegraphics[width=0.7\linewidth]{figs/dataset.eps}}
\caption{Examples of visual inputs used in the experiments. (A) Examples of visual inputs used for training FineNet and CoarseNet.
From left to right, a raw image to FineNet, a low-pass filtered image (LPF data) to CoarseNet, and a binarized image (mask data) to CoarseNet. (B-D) Visual inputs corrupted with different kinds of noise for evaluating the model. (B) Examples with uniform noise sampled from the range of $[-U,U]$. From left to right, the width $U$ is $0.1$, $0.5$, or $0.8$, respectively. (C) Examples with salt-and-pepper noise. From left to right, the proportion of image pixels replaced by white and black ones is $0.1$, $0.5$, or $0.8$, respectively. (D) Adversarial noise.
Left: the adversarial noise of the example image in (A)-left, obtained by the Fast Gradient Sign Method~\citep{Goodfellow2015}; Middle and Right: the adversarial examples with noise level of $0.1$ and $0.5$, respectively. }
\label{fig-data}
\end{center}
\end{figure}
FineNet used in this work consists of three stacked layers, each of which comprises a 128-filter $3\times3$ convolution, followed by a batch normalization, a ReLU nonlinearity, and $2\times2$ max-pooling. CoarseNet has two stacked layers with the same composition as in FineNet, except that it comprises 64-filter $11\times11$ convolution in the first layer and 128-filter $9\times9$ convolution in the second layer.
The parameter
$\alpha=0.4$ is used when training CoarseNet. Both FineNet and CoarseNet have a fully-connected layer of $1000$ units before the readout layer. Except for normalizing with the channel-wise mean and standard deviation of the whole dataset, no other pre-processing strategies are adopted.
During the training of FineNet and CoarseNet, the total number of epochs is $150$. SGD with momentum $0.9$, batch size $64$, and an initial learning rate $0.1$ is used. The learning rate is multiplied with $0.1$ after $100$ and $125$ epochs. During the training of the RBM, the total number of epochs is $2000$. We also use SGD to optimize the RBM with an initial learning rate of $0.1$, which is multiplied with $0.1$ after $500$ and $1000$ epochs. The visible and hidden layers of the RBM have $2000$ units and $400$ units, respectively.
\begin{figure}[htbp]
\begin{center}
\centerline{\includegraphics[width=0.7\linewidth]{figs/result1.eps}}
\caption{Imitation learning from FineNet improves the performance of CoarseNet. (A-C): the performances of CoarseNet trained on the low-pass filtered images from CIFAR10 with or without imitation. (A) Performance vs. the number of convolution channels. (B) Performance vs. the size of convolution kernel. (C) Performance vs. STD of the Gaussian filter. (D-E): the performances of CoarseNet training on the binarized images from Pascalvoc-mask. (D) Performance vs. the number of convolution channels. (E) Performance vs. the size of convolution kernel. Parameters: STD of the Gaussian filter in (A-B) is $2.0$. Other parameters are the same as described in Sec.~\ref{imple-detail}.}
\label{fig-result1}
\end{center}
\end{figure}
\subsection{CoarseNet learning from FineNet via imitation}
The first computational issue we address is about improving the
performance of CoarseNet. Because it is shallow, has large convolution kernels, and receives coarse inputs, CoarseNet is
normally unable to learn an object recognition task well. Therefore, we explore whether CoarseNet can imitate FineNet to improve its performance (Eq.\ref{eq-imitation}). The implication of the result is discussed in Sec.~\ref{discussion}.
Fig.~\ref{fig-result1} shows that with imitation, the classification accuracy of CoarseNet is improved considerably, compared to that without imitation over a wide range of parameters.
Specifically, with respect to the number of convolution kernels in CoarseNet, the improvement is significant when the number of kernels
is large (Fig.~\ref{fig-result1}A for low-pass filtered inputs;
Fig.~\ref{fig-result1}D for binarized inputs);
with respect to the size of kernels in CoarseNet,
the improvement is also significant
(Fig.~\ref{fig-result1}B for low-pass filtered inputs; Fig.~\ref{fig-result1}E for binarized inputs);
with respect to variation of the low-pass filter bandwidth
(quantified by the standard deviation (STD) of the Gaussian filter),
the performance is consistently improved
(Fig.~\ref{fig-result1}C).
The fact that the effect of imitation learning also depends on
the network parameters (Fig.~\ref{fig-result1}A-B) indicates
that in reality there is a trade-off between
having a simple structure for
the M-pathway and the
capability of the M-pathway imitating the P-pathway.
\subsection{Improved robustness via CoarseNet}
\label{sec-robustness}
The second computational issue we address concerns the robustness of our model to noise.
FineNet, as a deep CNN trained for image classification, is known to overly rely on local textures rather than the global shape of objects, and it is sensitive to unseen noise.
Since CoarseNet processes low-pass filtered (or binarized) visual inputs, whereby the local texture information is no longer the main cue supporting the classification, we expect that the performance of CoarseNet is robust to noise corruptions. Furthermore, through association, we expect that the robustness of FineNet is also leveraged. We carry out simulations to test this hypothesis.
The results are presented in Fig.~\ref{fig-result2}.
We first confirm that CoarseNet is indeed
robust to various forms of noise corruptions,
including uniform noise (Fig.~\ref{fig-result2}A), salt-and-pepper noise (Fig.~\ref{fig-result2}B), and
adversarial noise (Fig.~\ref{fig-result2}C)
Notably, the robustness of CoarseNet increases when the filter bandwidth
decreases (the red vs. the orange lines). This is understandable, since the filter
bandwidth
controls the spatial precision of visual inputs (the extent of local texture information) to be extracted by CoarseNet. Combining this observation with that in Fig.~\ref{fig-result1} (the accuracy of CoarseNet decreases with the filter bandwidth), it indicates a trade-off between the robustness and accuracy of the network. Remarkably,we observe that through interplay with CoarseNet, the robustness of FineNet is also improved significantly with respect to
uniform noise (Fig.~\ref{fig-result2}D), salt-and-pepper noise (Fig.~\ref{fig-result2}E), and adversarial noise (Fig.~\ref{fig-result2}F).
The implication of this result is discussed in Sec.~\ref{discussion}.
\begin{figure}[htbp]
\begin{center}
\centerline{\includegraphics[width=0.7\linewidth]{figs/result2.eps}}
\caption{Robustness of the model to noise. (A-C) CoarseNet is much more robust to noise disruption than FineNet without association.
STD of the Gaussian filter is $2.0$ for CoarseNet-LPF2.0 (red line) and $0.2$ for CoarseNet-LPF0.2 (orange line).
(D-F) After association, the robustness of FineNet is improved significantly. Interplay steps refers to the number of iteration of the state of the RBM
(A,D) Performance vs. uniform noise. (B,E) Performance vs. salt-and-pepper noise. (C,F) Performance vs. adversarial noise. STD of the Gaussian filter is $2.0$ in (D-E). Other parameters are the same as described in Sec.~\ref{imple-detail}.}
\label{fig-result2}
\end{center}
\end{figure}
\subsection{Cognitive bias via the interplay between two networks}
\label{sec-bias}
The third computational issue we address concerns the effect of cognitive bias in visual information process. Cognitive biases, such as the context information, can narrow down object candidates and facilitate recognition significantly. It has been suggested that the output of the M-pathway can serve as a cognitive bias to facilitate the performance of the P-pathway~\citep{Bar2007}. We test this property in our model. Specifically, we consider that the super-class
of an object (e.g., animal) serves as a cognitive bias to help the recognition of the sub-class of the object (e.g., cat). Each super-class is corresponding to a vector which can modulate the outputs of FineNet and are called context vectors here. Images forming
$5$ super-classes and $25$ sub-classes from CIFAR100 are randomly sampled for training
CoarseNet and FineNet are trained to recognize the super- and sub- class of images, respectively. As described in Sec.~\ref{asso-interlay}, we train the RBM to retrieve
the context vector conveying the super-class information of an image, when the output of CoarseNet is available. This super-class information is then used to boost the classification of FineNet
The results are presented in Fig.~\ref{fig-result3}, which show that the output of CoarseNet indeed can serve as a cogntive bias to improve the accuracy of FineNet on classifying images based on the sub-class information significantly.
\begin{figure}[htbp]
\begin{center}
\centerline{\includegraphics[width=0.7\linewidth]{figs/result3.eps}}
\caption{The output of CoarseNet serving as a cogntive bias which improves the performance of FineNet significantly. The blue line represents the performance of FineNet without a cognitive bias, and the other line
the performance of FineNet with a cognitive bias generated by CoarseNet. (A) Performance vs. uniform noise. (B) Performance vs. salt-and-pepper noise. STD of the Gaussian filter is $1.4$.
Other parameters are the same as described in Sec.~\ref{imple-detail} }
\label{fig-result3}
\end{center}
\end{figure}
\section{Conclusion and Discussion}\label{discussion}
In the present study, we have proposed a two-pathway model to mimic the P- and M- signal pathways for visual information processing in the brain. The model is composed of two CNNs, FineNet and CoarseNet, with the former being deeper and having smaller convolution kernels than the latter. The former receives detailed visual inputs and extracts fine features of images, while the latter receives low-passed filtered or binarized visual inputs and extracts coarse
features of images. The two networks interact with each other through an association process mediated by an RBM. We demonstrate that the interplay between the two networks leads to a number of appealing properties. 1) Through imitation from FineNet, the performance of CoarseNet is improved considerably compared to the case of no imitation.
2) CoarseNet is robust to noise due to its simple structure; and interestingly, through association between two networks, the robustness of FineNet is also improved significantly.
3) the result of CoarseNet can serve as a cognitive bias to leverage the performance of FineNet significantly.
Although we have used very simple CNNs to mimic the very complicated visual pathways, our model reveals some general advantages
associated with the interplay between two networks of different complexities, which may have some far-reaching implications about our understanding of
visual information processing in the brain.
Firstly, an evolutionary drive for the brain to have the M-pathway is for
rapid reaction when facing danger. This speed requirement means that the M-pathway needs to be shallow and process coarse visual inputs in order to save time, but meanwhile it should efficiently generate approximated, if not accurate, recognition of the object, which can serve as a cognitive bias for further improved processing. However, it is a well-known fact that a shallow neural network alone is unable to achieve object recognition well (this has actually motivated the development of deep neural networks). So, how does the brain resolve this dilemma? Here, we argue that the imitation learning strategy proposed in machine learning~\citep{Hinton2015CS} provides a natural solution to this challenge,
that is, the M-pathway can learn from the P-pathway through imitation to improve its performance. Imitation learning has been widely observed in human behaviors, although previous studies mainly focus on imitation between individual persons, without exploring whether it can occurs between neural circuits in the brain. We presume that the brain has resources to implement imitation learning across cortical regions, e.g., the widely observed synchronized oscillations between cortical regions may mediate this~\citep{Buzsaki2004Science}. It will be interesting and exciting to investigate this issue in neurophysiological experiments.
Secondly, a large volume of comparative studies has shown that human vision is much more robust to noise than machine learning models, with the classical example of adversarial noise~\citep{Goodfellow2015}. Here, our model suggests the potential underlying neural mechanism, that is, the brain exploits two separate pathways of different complexities to process visual inputs at different granularities, such that noise that is harmful to one pathway is not harmful to the other; and through interplay, both pathways eventually become robust to the noise. Previous studies also demonstrated that human vision is robust to binarized images, whereas deep CNNs are not~\citep{Baker2018Plos}. The M-pathway naturally accounts for this phenomenon as demonstrated by CoarseNet in our model.
Thirdly, experimental findings have shown that early activation in the PFC (orbitofrontal cortex in particular) is elicited by the low-pass filtered information from the M-pathway, which serves as a rapid detector or predictor of potential content based on the coarse information of the input (i.e., gist)~\citep{Bar2003JCN}. This rapid predictor may modulate neural activation in the inferior temporal cortex through feedback, and facilitates recognition by biasing the bottom-up process to concentrate on a small set of the most likely object representations. We partly demonstrate this cognitive bias effect using the two-pathway model. This is different from
the previous attentional neural network~\citep{Wang2014NIPS}, which has adopted the label information related to an object as feedback to modulate features, but without specifying how the label information is generated. Our model suggests a more biologically plausible way to implement the cognitive feedback.
Our proposed two-pathway model is also inspirational to machine learning. Deep neural networks (DNNs), which mainly mimic the P-pathway, have proved to be very effective in many applications~\citep{Krizhevsky2012NIPS, He2016CVPR}, yet they still suffer from a lot of shortcomings, including sensitivity to unseen noise. As demonstrated in this study, the two-pathway model provides a potential new architecture to solve this noise sensitivity problem. Conventionally, DNNs focus on extracting local features of images. Recently, there are also efforts aiming to extend DNNs to process ``global information" of images. The methods proposed include, for instance, augmenting training data to have various variations in local features of objects, such that the network is forced to learn information about the global-shape of objects in order to accomplish the recognition task~\citep{Geirhos2019ICLR}; or inducing recurrent connections between neurons in the same layer, so that the network is able to extract texture information over a wide range~\citep{Montobbio2019ARXIV, Spoerer2017FP}. Nevertheless, these methods are very different from ours. Specifically, our model considers having an extra pathway with a coarse structure to extract the global information of images. Furthermore, our model holds advanced cognitive capabilities not shared by a single pathway model. For instance, as partly demonstrated in this work, CoarseNet can quickly capture the gist of an image, which can subsequently serve as a cognitive bias to guide FineNet to perform fine recognition of the image. We expect that the two-pathway model will inspire us to develop new network architectures for implementing more human-like object recognition behaviors.
\clearpage
|
1,116,691,499,352 | arxiv |
\section{Introduction}
\label{sec:intro}
Mobility-on-Demand (MoD) systems have become a fixture in urban transportation networks, with the popularity of ride-hailing services operated by transport network companies (TNCs) such as Uber and Lyft. The popularity of such services has also stimulated the deployment of ridepooling \parencite{alonso2017demand} and on-demand microtransit \parencite{shaheen2020sharing}, and the study of the potential for shared automated vehicle systems \parencite{fagnant2014travel}.
The rapid growth of MoD systems, driven by the ubiquity of smartphones, electronic payments, and advances in vehicle dispatch algorithms, has changed the landscape for urban mobility and has the potential to reduce auto ownership. While such services in the form of ride-hailing (non-shared rides) are not likely to reduce the negative externalities associated with auto travel (e.g., pollution, congestion), and in-fact might even increase them~\parencite{FEHR}, high-capacity MoD applications (e.g., ridepooling, microtransit) have the potential to do so by improving the ratio between passenger miles traveled (PMT) and vehicle miles traveled (VMT)~\parencite{santi2014quantifying,alonso2017demand}.
These studies provide insights into the potential benefits of high-capacity door-to-door mobility service as a more affordable, efficient, and sustainable alternative to the more prevalent ride-hailing services.
However, many questions remain regarding the availability and adoption of such services based on the ability of such systems to generate profits (relative to ride-hailing) and user preferences (e.g., the price convenience trade-off).
Customers who appreciate convenience or privacy will seek exclusive services (i.e., not sharing rides with others).
In contrast, others may prefer to use low-cost services as long as delays due to detours are acceptable.
In short, the proportion of customers opting for the shared option in an MoD platform depends on (a)~the perception of quality of service (QoS) from exclusive and shared rides and (b)~their prices.
Therefore, MoD platforms need to optimize their services to maximize gross profits subject to operational constraints and commuter choice---requiring the development optimal vehicle dispatching and trip pricing mechanisms across the services that they provide.
For example, in the case of an operator that provides ride-hailing and ridepooling options, the goal is to optimize both vehicle dispatch and pricing for both services, which leads to a complex, coupled optimization problem.
\smallskip
\noindent \textbf{Joint vehicle dispatching and pricing problem (JVDPP).} \quad
Approaches for optimal vehicle dispatching and trip pricing have been extensively investigated in the literature.
Vehicle dispatching involves matching available vehicles with pending requests and directing empty vehicles towards locations likely to have future requests.
Efficient dispatching algorithms can reduce wait times and increase the overall throughput \parencite{alonso2017demand}.
Pricing is an equally important instrument in the MoD platform because it controls the demand for services and profits. In MoD systems, pricing is used to balance supply and demand temporarily \parencite{ke2020pricing}, via what is commonly known as surge pricing, as well as in the long run \parencite{bimpikis2019spatial}, via spatial pricing.
For example, a pricing policy can simultaneously control the demand side by adjusting trip fares and control the supply side by redistributing fare splits \parencite{he2018pricing, zha2018geometric}.
In our context, where an operator provides multiple service types, the platform must optimize vehicle dispatching and pricing jointly for the mixed service types and trip requests, which we refer to as the JVDPP.
This problem is non-trivial because the induced demand depends on both of these operational policies.
A menu of exclusive and shared services with corresponding prices and estimated journey times is offered to each customer accessing the platform with a ride request.
The rider is then assumed to choose the utility-maximizing option, while the MoD platform is expected to ensure that all customers who have ride options are served.
The discrete trip assignment decision and continuous pricing decision are coupled in JVDPP, leading to an optimization problem that is challenging to solve.
\subsection{Preliminaries and Related Work}
This section summarizes the growing literature on vehicle dispatching and pricing, and highlights how this work integrates these two streams into a cohesive joint optimization framework for MoD platforms.
Table \ref{tab:1} summarizes the methods and findings in recent studies that focus on either topic or the joint problem.
We refer the reader to \citet{wang2019ridesourcing}
for a comprehensive review of related studies.
\begin{table}[!htb]
\centering
\caption{Summary of related work} \label{tab:1}
\begin{tabular}{l|l|l|l}
\toprule
Topic
& Setting & Methods & Reference \\ \hline
\multirow{4}{*}{
\begin{tabular}[c]{@{}l@{}}
Dispatching, \\
assignment\\
and rebalancing
\end{tabular} }
& \multirow{2}{*}{Single-ride } & Sequential dispatch & \begin{tabular}[c]{@{}l@{}}
\citet{zhang2017taxi} \\
\citet{lei2020efficient}
\end{tabular} \\ \cline{3-4}
& & Batched dispatch &
\begin{tabular}[c]{@{}l@{}}
\citet{jintao2020learning} \\
\citet{qin2020optimal} \end{tabular}
\\ \cline{2-4}
& \multirow{2}{*}{Ridepooling } & \begin{tabular}[c]{@{}l@{}} Dynamic \\ programming \end{tabular} & \begin{tabular}[c]{@{}l@{}}
\citet{yu2019integrated} \\
\citet{shah2020neural} \\
\end{tabular} \\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}} Combinatorial/ \\
heuristic \end{tabular} & \begin{tabular}[c]{@{}l@{}} \cite{alonso2017demand} \\
\citet{liu2018framework} \\
\citet{luo2021efficient} \\
\citet{simonetto2019real} \\
\citet{lowalekar2021zone}
\end{tabular}
\\ \cline{2-4} \hline
\multicolumn{1}{l|}{\multirow{3}{*}{Pricing}} & \multicolumn{1}{l|}{\multirow{2}{*}{Single-ride }} & \multicolumn{1}{l|}{ Aggregate model } & \multicolumn{1}{l}{ \begin{tabular}[c]{@{}l@{}}
\citet{castillo2017surge} \\
\citet{he2018pricing} \\
\citet{zha2018geometric} \\
\citet{guda2019your}
\end{tabular} } \\ \cline{3-4}
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{ \begin{tabular}[c]{@{}l@{}} Dynamic \\ programming \end{tabular} } & \multicolumn{1}{l}{ \begin{tabular}[c]{@{}l@{}}
\citet{sayarshad2018scalable}
\end{tabular} } \\ \cline{2-4}
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{Ridepooling } & \multicolumn{1}{l|}{ Network modeling } & \multicolumn{1}{l}{
\begin{tabular}[c]{@{}l@{}}
\citet{zhang2019pool} \\
\citet{ke2020pricing}
\end{tabular}
} \\ \hline
\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}} Joint vehicle dispatch \\ and pricing \end{tabular}}
& Single-ride & \begin{tabular}[c]{@{}l@{}} Dynamic \\ programming \end{tabular} & \begin{tabular}[c]{@{}l@{}}
\citet{chen2020dynamic} \\
\citet{banerjee2017pricing} \\
\cite{ozkan2020dynamic} \end{tabular} \\ \cline{2-4}
& \begin{tabular}[c]{@{}l@{}} Mixed pooling\\ fleet \end{tabular} & \begin{tabular}[c]{@{}l@{}} Combinatorial/ \\
heuristic \end{tabular} &
\begin{tabular}[c]{@{}l@{}}
\citet{yan2020dynamic} \\
\textbf{This work} \end{tabular}
\\
\bottomrule
\end{tabular}
\end{table}
\noindent \textbf{Vehicle dispatch methods.} \quad
Two standard vehicle dispatch methods are sequential (continuous/instant) dispatch and batched dispatch.
The sequential dispatch method assigns each request to nearby available vehicles upon their arrival \parencite{zhang2017taxi, lei2020efficient}.
This method is
simple to implement and has modest performance \parencite{zha2018geometric}.
A batched dispatch method collects requests during a short interval and solves a global assignment problem at the end of each interval.
Putting trip requests in batches can potentially find a better matching with available vehicles or shareable rides compared to the sequential method \parencite{yan2020dynamic, jintao2020learning, qin2020optimal}.
The new operational challenge of vehicle dispatch for ridepooling services is how to combine multiple trip requests going in a similar direction into a single ride by solving the \emph{trip-vehicle assignment} problem.
Most models are limited to solve ridepooling with two customers per vehicle \parencite{santi2014quantifying, sundt2021heuristics}.
The computational challenge of solving the trip-vehicle assignment increases substantially when the vehicle capacity exceeds two.
However, state of the art algorithms for ridepooling can now solve large-scale problem instances with larger vehicle capacities in real-time. In one of the first such works, \citet{alonso2017demand} proposed an anytime-optimal algorithm that dynamically assigns multiple requests to vehicles by extending the idea of the shareability graph from \citet{santi2014quantifying} and exploiting the sparsity of the number of feasible trip configurations in practical problem instances. There is also a growing literature on proactive rebalancing/routing of idle/partially occupied vehicles to meet estimated future demand using historical data \parencite{bent2007waiting, wen2017rebalancing, spieser2016shared, liu2020proactive, fielbaum2021demand} and improving the computational efficiency \parencite{simonetto2019real, luo2021efficient}.
All of these approaches are myopic in the sense that they do not consider future demands. The non-myopic problem formulations are significantly harder as they are high-dimensional stochastic optimization problems in a setting where the myopic problem is already very challenging. \citet{yu2019integrated} proposed an approximate dynamic programming approach with a spatial-and-temporal decomposition heuristic for improving the computational efficiency, but even this approach has limited scalability. To counter this challenge, \citet{shah2020neural} utilize a neural network based approach to learn the value function, which allows for solving larger problem instances.
Deep reinforcement learning has also recently been adopted in industry \parencite{tang2021value,qin2020ride}.
\noindent \textbf{Pricing methods.} \quad Pricing can improve the revenue management of MoD systems by smoothing out the supply and demand imbalance temporarily through surge prices or permanently through optimal prices at steady states. There is a large and growing literature on this topic. \citet{banerjee2017pricing} modeled the MoD system as a continuous-time Markov chain where the state is the number of vehicles at each vertex and solved the optimal prices for each Origin-Destination (O-D) pair. In perhaps the most well known (currently) analysis of this pricing strategies for ride-hailing systems,
\citet{castillo2017surge} proposed an aggregate model for relieving the so-called Wild Goose Chase (WGC) phenomenon in ride-hailing systems, using data-driven optimization.
\citet{qiu2018dynamic} studied the pricing problem as a dynamic program that focuses on the temporal discrepancy of the problem.
Other studies addressed problems such as demand-supply imbalance, cost split, and driver incentives \parencite{he2018pricing,guda2019your,zhang2019pool,ke2020pricing, bimpikis2019spatial, ozkan2020dynamic}.
\noindent \textbf{Research opportunities.} \quad
The above works study the vehicle dispatch and pricing problems separately, and ignore the following operational issues which have not been fully addressed in the literature.
In many of the pricing models, trip fares are computed from aggregate models of operations, and therefore may not be consistent with what happens in practice.
Furthermore, even though platforms typically provide shared and exclusive services simultaneously, prior studies typically optimize the systems independently. They do not determine the prices and vehicles dispatch strategies for both options in tandem, even though both options are available to each potential customer and compete with each other.
The network effect, which reflects the connection between the elasticity of supply and demand, has not been well-addressed in the literature, especially regarding strategic drivers' relocation decisions and customers' choices between different service options.
In this respedt, the work of \citet{yan2020dynamic} is the closest to ours.
They studied the JVDPP in the conventional ride-hailing setting (i.e., exclusive services) or ridepooling under a simple time-window-based policy.
A queueing-based heuristic determined the surge prices and matching time intervals at the market equilibrium.
In contrast, we consider an MoD platform with mixed fleets where pricing and dispatching decisions are made at the trip level, and develop MIP formulation that can compute the optimal vehicle dispatching and prices in real-time.
\subsection{Overview and Main Contributions}
We consider a transportation system with three travel modes that coexist---exclusive MoD service (e.g., UberX), shared MoD service (e.g., Uberpool), and an outside alternative (e.g., taxi service).
An MoD platform operates a mixed fleet of two services to satisfy different consumer groups.
\emph{Endogenous demand} is considered throughout our analysis.
Particularly, customers will make the mode choice based on the prices and QoS, which follows a multinomial logit (MNL) model \parencite{liu2018framework, qiu2018dynamic}.
The main goals of this work are to develop the following:
\begin{enumerate}
\item A sequential pricing and dispatch (SPD) framework that assigns each trip request to an available vehicle and simultaneously determines the optimal prices.
\item A batched pricing and dispatching (BPD) framework that determines globally optimal (for all requests in the batch) trip assignment policies and prices over fixed intervals
\end{enumerate}
\noindent The corresponding technical contributions include:
\begin{enumerate}
\item Developing an exact approach that determines the optimal prices and vehicles to dispatch in real-time for exclusive and shared MoD services by harnessing the power of multi-product pricing.
\item
Proving that the objective of the JVDPP problem is jointly concave for the special case that at most two customers can share rides, and deriving analytical forms for optimal prices.
\item Establishing a non-myopic mixed integer programming (MIP) framework by introducing a separable cost structure consisting of instantaneous costs and forward-looking control-dependent costs (termed ``retrospective costs'').
\end{enumerate}
The introduction of retrospective costs is a simple and robust approach for decoupling the endogenous supply and demand constraints---by considering the duration a vehicle is blocked from accepting future requests after being assigned.
A dispatching policy that considers retrospective costs allows for more tractably optimizing supply configurations based on future demands.
Since each customer may choose between two service types, we develop an \textit{overbooking} policy in BPD to ensure that vehicles are available for both options over the planning horizon.
These techniques can substantially reduce the computational challenge in other formulations (e.g., the dynamic programming formulation in \citet{yu2019integrated}).
The remainder of the article is organized as follows:
Section~\ref{c4sec: problem} describes the background of the JVDPP and motivates a flexible and computationally efficient optimization framework to solve it.
Section~\ref{c4sec: experiments} discusses the experimental results using the taxi data in Manhattan, New York City (NYC).
The conclusions are drawn in Section~\ref{sec:conclusion}.
The notation used in this work is summarized in Table \ref{table:app1} in Appendix \ref{sec:app1}.
\section{Problem Description and Model Formulation}\label{c4sec: problem}
\subsection{JVDPP on MoD Platforms with Mixed Fleets}
We consider an
MoD platform providing two types of services: exclusive service (indicated by subscript ``$e$'') and shared service (indicated by subscript ``$s$'').
Exclusive service is the conventional ride-hailing service where each customer takes a private ride.
Shared service corresponds to a ridepooling option where customers agree to share rides with others on a similar route and pay reduced fares.
As the exclusive service is more desirable (e.g., is faster, more private, and more reliable) and costly to operate, it is usually priced higher than the shared option. The MoD platform offers both premium and budget options to cater to various types of potential customers and these two options provide commuters service-specific prices and expected wait times.
\subsubsection{Overview of SPD and BPD. }
SPD and BPD are two common operational frameworks adopted by TNCs.
SPD assigns trip requests to available vehicles upon arrival.
BPD allows new trip requests and vehicles in each region to wait for a fixed time interval to be matched so the platform can determine the dispatch plans and prices in batches.
The sequence of interactions between the arriving customers and the MoD platform is described in Figure \ref{fig:1}.
\begin{figure}[!htb]
\centering
\includegraphics[width = 0.95\textwidth]{figures/fig1.png}
\caption{Road-map for implementing the BPD policy}
\label{fig:1}
\end{figure}
Both frameworks have their respective merits and the overall performance depends on the market conditions (i.e., the supply and demand in the transportation network) \parencite{qin2020optimal}.
Since the prices and the QoS of exclusive and shared services affect how each customer chooses between them (possibly with outside alternatives also present), the pricing and dispatching problems are tightly coupled.
This research will simplify the analysis under the following assumptions: (a)~customers make irrevocable decisions after being assigned to rides or choosing alternative outside options, and (b)~drivers specify their preferred service type before the assignment.
These assumptions are commonly imposed in the literature \parencite{qiu2018dynamic}.
Considering more sophisticated driver and customer behavior (e.g., ride cancellation) is possible in our general framework.
\subsubsection{Preliminaries for trip-vehicle assignment (TVA).}
\label{sec212}
Vehicle dispatching in SPD can be easily addressed by rule-based assignment (e.g., customers are matched with nearest available vehicles in \citet{zhang2019pool}) or optimal assignment based on an objective (e.g., minimizing total system travel time or passenger inconvenience in \citet{ota2016stars}).
Since each assignment is made instantaneously, there is little room for improvement from considering more sophisticated online matching algorithms \parencite{ashlagi2018maximum}.
However, the problem is more computationally challenging in BPD, especially for high-capacity vehicle settings.
One of the most popular approaches for high-capacity ridepooling is based on the work of \citet{alonso2017demand}, which develops an efficient TVA formulation for vehicle dispatching in BPD.
Our work builds on this general formulation, which we briefly introduce here for completeness.
TVA describes the matchable relationship between the system's \textit{supply} (available vehicles) and \textit{demand} (pending trip requests) on a so-called shareablity graph \parencite{santi2014quantifying} (See Figure \ref{fig2}).
\citet{alonso2017demand} proposed a multi-step procedure from constructing this graph to solving the optimal assignment---leading to a computationally efficient approach for solving vehicle dispatch problem in high-capacity MoD settings.
Whenever a new request $r$ arrives in the system, a set of criteria is applied to check whether each vehicle $v$ can be matched with the request and whether any existing request $r' \in \mathcal{R}$ can be matched with $r$. This allows the creation of a request and vehicle compatibility graph known as the shareability graph.
Requests are assignable to a vehicle if the following conditions are satisfied:
\begin{enumerate}
\item The wait time of each customer must be below the maximum wait time $\Omega$.
\item The total delay~\footnote{The total delay is the difference between the trip's actual arrival time and the earliest possible arrival time.} must not exceed the maximum delay $\Delta$.
\item The number of customers in the vehicle has to be smaller than or equal to the capacity at any time.
\end{enumerate}
The shareability graph is then use to compute all feasible vehicle request pairings are determined at a predefined batching window (e.g., 30 seconds), utilizing the fact that any feasible grouping (trip) must form a clique in the shareability graph (a necessary but not sufficient condition for feasibility). One the feasible trips are computed, an ILP is solved to determine the optimal trip-vehicle assignment.
This procedure also rebalances the vehicle fleet by routing empty vehicles to locations with potential for future demand at the end of each batch.
Since the main contribution of our work is to integrate pricing with the TVA formulation in \citet{alonso2017demand}, we focus on the problem of pricing at the request level in the remainder of this paper.
\subsubsection{Profit-maximizing pricing for mixed fleets. }
Trip fares for the exclusive service $p_e$ and shared service $p_s$ are presented to each customer entering the platform.
Let $P_e$, $P_s$, and $P_o$ denote the probability of choosing the exclusive services, the shared service, and some alternative outside options, such that $P_e + P_s + P_o = 1$.
A set of trip requests $\mathcal{R}$ are revealed one by one in SPD or in batches in BPD.
The expected profit obtained by serving these demands $\mathcal{R}$ under a fixed pricing policy $\pi_p$ and an assignment policy $\pi_{a}$ is given by:
\begin{align} \label{eq1}
\mathbb{E}_{\pi_p, \pi_{a} } \Big[ \sum_{r \in \mathcal{R} } \Phi_r \Big] = \sum_{r \in \mathcal{R}_e} P_e (p_e^r - c_e^r ) + \sum_{r \in \mathcal{R}_s } P_s (p_s^r - c_s^r),
\end{align}
where $\Phi_r$ is the profit from request $r\in \mathcal{R}$.
The MoD platform's goal is to search for optimal policies that maximize the expected profit.
More specifically, policy $\pi_p$ determines prices for exclusive service $p_e$ and shared service $p_s$ for each request $r\in \mathcal{R}$; policy $\pi_{a}$ determines the vehicle assignment process, either in a sequential or batched manner, for matching requests and available vehicles.
Costs for the exclusive option are $c_e$ and for the shared option are $c_s$.
Since the prices and assignments are determined at the request level for each $r \in \mathcal{R}$, the trip request set $\mathcal{R}$ is split into demand served by exclusive service $\mathcal{R}_e$ and by shared service $\mathcal{R}_s$.
Due to the existence of outside alternatives, $\mathcal{R}_e \cup \mathcal{R}_s \subset \mathcal{R}$.
Throughout, we omit the superscript $r$ whenever it is clear from the context.
Direct calculation of the expected profit is not practical in JVDPP due to the challenge of estimating the continuation value, which is the expected future profit under the current pricing and dispatching policies.
For example, a dynamic program formulation models all evolutionary dynamics of supply and demand and seeks non-myopic policies, which is computationally intractable due to the ``curse of dimensionality''.
This work seeks to develop a structured model that \emph{approximates} the system dynamics and customers' behavior under various pricing strategies and finds near-optimal policies in a more tractable manner.
The proposed method assumes a centralized dispatch system where drivers follow the MoD platform's guidance strictly, and the derived method is appropriate for applications such as shared automated vehicles \parencite{fagnant2014travel}, high-capacity on-demand transit \parencite{alonso2017demand}, and ride-hailing platforms with permanent employees \parencite{dong2020optimal}.
The following sections discuss how to efficiently seek the optimal SPD and BPD policies with mild assumptions.
\subsection{Pricing in SPD Framework} \label{sec: seq}
This section described how to set prices under the SPD framework.
SPD is simple to implement and short matching times can strengthen customers' adherence to the current MoD platform in the face of competition.
The main challenge is the estimation of the expected profit obtained from serving requests in Equation~\eqref{eq1}, which includes the option of rejecting them to keep the vehicle open for future demand (e.g., if the request is sending the vehicle to a low demand region at a high demand time period).
Denote feasible vehicles for a request by $\mathcal{V}_e$ for the exclusive service and $\mathcal{V}_s$ for the shared service, respectively.
The SPD framework handles an arbitrarily arriving request $r$ as follows:
\begin{enumerate}
\item For vehicles for exclusive service $v_e \in \mathcal{V}_e$ and those for shared service $v_s \in \mathcal{V}_s$, the platform computes the optimal prices $p_e^*$ and $p_s^*$
and picks a vehicle from each type as
$v_e \in \arg \min_{v_e' \in \mathcal{V}_e} \{ c_e(v_e', r) \}$ and $v_s \in \arg \min_{v_s' \in \mathcal{V}_s}\{ c_s(v_s', r)\}$, respectively. The cost $c_e(v_e, r)$ (respectively, $c_s(v_s, r)$) measures the operational and retrospective costs associated with assigning request $r$ to vehicle $v_e$ (respectively, $v_s$).
\item The platform presents a menu of the best prices of each mode along with an outside alternative.
\item If the customer declines the MoD service, the procedure terminates; otherwise, the request is assigned to the corresponding vehicle.
\end{enumerate}
The key step in obtaining an explicit form for the profit function \eqref{eq1} is separating each cost into the \emph{operational} cost and the \emph{retrospective} cost.
Operational costs consider all costs incurred in picking up customers and searching for rides (e.g., fuel, personnel).
Retrospective costs include penalties for lost demands.
The remainder of this section focuses on finding all feasible vehicles (Step 1) and computing optimal prices $p_e^*, p_s^*$ for each trip request (Step 2).
Given that both types of vehicles are available, the pricing policy $\pi_p$ aims to compute two prices $p_e$ and $p_s$ to maximize the expected profit in Equation~\eqref{eq1}:
\begin{equation}\label{c4equ: seqobj}
(p_e^*, p_s^*)= \arg \max \mathbb{E}_{\pi_p, \pi_{a}} [\Phi_r ] = \arg \max \{ P_e (p_{e} - c_e ) + P_s (p_{s} - c_s ) \}.
\end{equation}
As the pricing decision is \emph{non-myopic}, the key challenge is to estimate the value function (i.e., the continuation value of making a pricing and assignment decision for $r$).
Since Equation~\eqref{c4equ: seqobj} addresses that the mode choice parameters $P_e, P_s$ and the costs $c_e, c_s$ are both dependent on the pricing policy $\pi_p$, we can find analytical forms for $\Phi_r$ and derive a heuristic-based pricing policy detailed below.
\subsubsection{MNL model for service type choice.}
The expected profit per request depends on each customer's mode choice parametrized by $P_e$ and $P_s$.
The MNL model is used for customers' choice between two MoD service types and an outside alternative.
This model has been widely used in the revenue management and transportation literature \parencite{li2011pricing, he2018pricing, qiu2018dynamic}.
Customers choose from a menu of modes and their corresponding prices, an option that maximizes their expected utilities.
Let $U_e$, $U_s$, and $U_o$ denote the respective utilities associated with three travel modes (exclusive MoD service, shared MoD service, and outside alternative).
Their utilities for each travel mode $m \in \{e, s, o\}$ are given by:
\begin{equation}\label{equ:u}
U_{m} = \beta_{p} \, p_{m} + \beta_{w} \, w_{m} + \beta_{t} \, t_{m},
\end{equation}
where $p_{m}$, $w_{m}$, and $t_{m}$ are the trip price, estimated wait time, and estimated travel time for choosing mode $m$.
All these terms are computed after an arbitrary trip request $r$ is revealed.
$\beta_{p}$, $\beta_{w}$, and $\beta_{t}$
are the corresponding coefficients estimated from historical data \parencite{liu2018framework}.
Note that $\pi_p$ affects the first term in Equation~\eqref{equ:u} and $\pi_{a}$ affects the other terms.
The current setting considers homogeneous customers throughout the analysis for simplicity, but this assumption can be relaxed if there are additional data sources.
The predicted probabilities of choosing travel mode $m$ are:
\begin{equation}
P_m = \frac{e^{U_{m}}}{e^{U_{e}} + e^{U_{s}} + e^{U_{o}}}\quad\quad \forall m \in \{e, s, o\},
\end{equation}
where the utilities of choosing modes $m \in \{e,s,o\}$ can be rewritten as:
\begin{align*}
U_m = U_m^{\pi_p} + U_m^{\pi_{a}} = \beta_{p} \, p_m + U_m^{\pi_{a}}.
\end{align*}
The following analysis focuses on how to compute prices that approximately optimize Equation~\eqref{c4equ: seqobj}
in the SPD framework, which assumes that
profits earned from serving trip requests in $\mathcal{R}$ are drawn from a known distribution (e.g., the empirical distribution).
\subsubsection{Operational and retrospective costs.}
The costs $c_e$ and $c_s$ depend on the pricing policy at the request level, and consists of two parts: the operational cost $c_{ \text{operational} } $ and the retrospective cost $c_{ \text{retrospective} }$.
The operational costs include fuel, maintenance, and fixed tiers that may vary for different modes.
For exclusive service, the fixed operational cost $c_{e, \text{operational} }$ is deterministic.
In contrast, the operational cost $c_{s, \text{operational}}$ for shared service is an expectation conditional on whether the system will match this ride with others in the future and how the cost is split, and is computed as follows:
\begin{equation}\label{equ: expec_cost}
c_{s, \text{operational} } = \left(1 - \sum\limits_{r' \in \mathcal{R}_{m}} \alpha_{r'}\right) c_0 + \sum\limits_{r' \in \mathcal{R}_{m}} \alpha_{r'} \left(c_{r', r} - p_s^{r'} \right),
\end{equation}
where $c_0$ is the operational cost if the request is not matched with any requests in the future; $\mathcal{R}_m$ is the set of all possible future requests for mode $m$ that might be matched to the request $r$; $\alpha_{r'}$ is the probability that request $r'$ is matched to $r$ conditioned on the customer submitting $r$ choosing the shared option; $c_{r', r}$ is the additional cost of adding request $r'$ to the vehicle matched to $r$, which is the difference between the cost of serving both requests and the cost of serving only the first request; and $p_s^{r'} $ is the price of request $r'$.
Although the distribution for each $r' \in \mathcal{R}_m$ cannot be computed in exact form, expected operational costs can be estimated empirically.
The retrospective cost $c_{ \text{retrospective} } $ penalizes a suboptimal assignment (i.e., one in which the vehicle could have been matched to a better request in the future).
When a vehicle $v$ is assigned to serve a request $r$ (or multiple requests if $v \in \mathcal{V}_s$), it is blocked from future operations within a given duration.
The retrospective cost considers the spatio-temporal distribution of requests that occur during this time period, because the supply levels at both origin and destination areas are affected by a suboptimal assignment.
The actual retrospective cost is assumed to follow a function $g_{\pi_p, \pi_{a}}(\mathcal{R}_m)$.
A surrogate function $\hat{g}(\cdot)$ is used as a proxy for the retrospective cost because the exact spatio-temporal demand distributions are not available.
Any advanced estimators for the continuation value of future requests $\mathcal{R}_m$ can be easily integrated into this pricing framework.
For example, TNCs have used recurrent neural networks to predict the profit and estimated time-of-arrival in real-time \parencite{li2019efficient,shah2020neural}.
This work proposes a straightforward estimator $\hat{g}(\cdot)$ that requires minimal access to contextual demand and supply data.
We divide a fixed horizon into $M$ time intervals and the road network into $N$ clusters.
For each time interval and cluster, we compute the average profit made per unit time and per vehicle in the cluster based on the steady-state pricing model \parencite{castillo2017surge} and extend it to the ridepooling case.
The MoD system follows supply flow balance equations:
\begin{align}
& \quad L = L_e + L_s, \label{eq6} \\
& \quad Y = Y_e + Y_s, \nonumber
\end{align}
where $L$ is the total number of vehicles in the cluster at the current time interval, and $Y_e$ and $Y_s$ are the steady-state throughput of exclusive vehicles and sharing vehicles, respectively.
The total number of vehicles per type can be calculated as:
\begin{align*}
\begin{cases}
L_e = O_e + \eta_e Y_e + T_e Y_e \\
L_s = O_s + \frac{\eta_s }{\zeta_s } Y_s + \frac{ T_s }{\zeta_s} Y_s
\end{cases}, \nonumber
\end{align*}
where $O_e$ and $O_s$ are the number of available vehicles of each group, and $\eta_e$ and $\eta_s$ are the average waiting times for the customers choosing exclusive and shared services. The expected waiting time is assumed to be equal in each group because the assignment policy $\pi_{a}$ is fixed.
$T_e$ and $T_s$ represent the average trip duration of each group at steady-state.
According to Little's law, $\eta Y$ represents the number of vehicles in pick-up trips and $T \cdot Y$ represents the number of en-route vehicles.
$\zeta_s$ is the average utilization of sharing vehicles in ridepooling over time. We explain how to calculate the average utilization in Appendix~\ref{sec:app2}.
The waiting times $\eta_e$ and $\eta_s$ are determined by the density of available vehicles near the origin location \parencite{zha2018geometric}.
Without loss of generality, we assume $\eta_e = F_e(O_e)$ and $\eta_s = F_s(O_s)$, and obtain the approximate trip throughput as:
\begin{align}
&Y_e(\eta_e, O_e) \triangleq \frac{L_e - O_e}{F_e(O_e) + T_e}, \nonumber \\
& Y_s(\eta_s, O_s) \triangleq \frac{ \zeta_s ( L_s - O_s)}{F_s(O_s) + T_s}.
\end{align}
For a request $r$ traveling from the origin region $o$ to the destination region $d$, we can compute the average profit obtained in each region by combining the throughput $Y_e, Y_s$ and listed prices $p_e, p_s$. For each region $k$, $\bar{\epsilon}^k_e$ and $\bar{\epsilon}^k_s$ are the respective average profits.
Given the trip duration $t_r$ for an exclusive service vehicle, the retrospective costs $c^{\pi_p}_{m, \text{retrospective}}$ for mode $m\in \{e,s\}$ are as follows:
\begin{align} \label{equ: potential}
c_{m, \text{retrospective}} & = \hat{g}(t_r, t_r') = \bar{\epsilon}^{o}_m t_r + ( \bar{\epsilon}^{o}_m - \bar{\epsilon}^{d}_m ) t_r^\prime,
\end{align}
where $\bar{\epsilon^{o}_m } \cdot t_r$ is the type $m$ vehicle's average profit collected from staying in the origin $o$, and
$(\bar{\epsilon}_m^{o} - \bar{\epsilon}_m^{d}) \cdot t_r^\prime$ is the average profit difference that the vehicle makes between the two clusters in time $t_r^\prime$.
The computation of retrospective cost connects the supply and demand information across the MoD network.
A similar approach has been used in modeling taxi and ride-hailing vehicles' movement \parencite{yang2002demand, bimpikis2019spatial}; the derivation of these approximate forms are based on a steady-state analysis of the MoD system dynamics.
\subsubsection{Multi-product pricing (MPP) problem for mixed fleets.}
The expected profit of the MPP problem under the SPD framework is:
\begin{equation}\label{equ: obj}
\mathbb{E}_{\pi_p, \pi_{a} } [\Phi_r] = \frac{ e^{ \beta_{p} p_{e} + U_e^{\pi_{a}} } (p_{e} - c_e) + e^{ \beta_{p} p_{s} + U_s^{\pi_{a}} } (p_{s} - c_s) }{ e^{ \beta_{p} p_{e} + U_e^{\pi_{a}} } + e^{ \beta_{p} p_{s} + U_s^{\pi_{a}} } + e^{ \beta_{p} p_{o} + U_o^{\pi_{a}} } }.
\end{equation}
In general, the objective function \eqref{equ: obj} is not convex in prices \parencite{hanson1996optimizing}.
A common convexification technique is to project the pricing to market shares and transform the objective function \parencite{li2011pricing} as follows.
For the special case where each vehicle serves at most two rides per trip, we show a closed-form expression of unique optimal prices.
\begin{proposition}\label{prop: seq}
The objective function has a unique critical point given by
\begin{equation}
p_e^* = \frac{1}{\beta_p} \left\{ \log \Bigg[ \dfrac{ e^{U_0} \cdot W\left( \frac{ (1 + e^{U^{\pi_{a} }_{s} - U^{\pi_{a} }_{e} + \beta_p c_s - \beta_p c_e }) \cdot e^{U^{\pi_{a} }_{e} + \beta_p c_e - 1} }{ e^{U_0} } \right) }{1 + e^{U^{\pi_{a} }_{s} - U^{\pi_{a} }_{e} + \beta_p c_s - \beta_p c_e }} \Bigg] - U^{\pi_{a} }_{e} \right\},
\end{equation}
where $W(\cdot)$ is the Lambert W function, and
\begin{equation*}
p_{e}^* - p_{s}^* = c_e - c_s.
\end{equation*}
\end{proposition}
\begin{remark}
The difference between the price for exclusive service and the price for shared service at the critical point is the expected cost savings from choosing the shared service.
\end{remark}
Since the Lambert W function is positive and increasing when the input is nonnegative, the objective function~\eqref{equ: obj} has a unique solution.
Hence, the optimal price for two services corresponding to an arbitrary origin-destination pair can be computed by checking all the boundary points and the critical point giving the maximum profit.
The proof of Proposition~\ref{prop: seq} can be found in Appendix~\ref{proof:prop:seq}.
In summary, the MoD platform can present the prices computed by the SPD framework to each arriving customer so that she will choose the mode that maximizes her utility (or opts out for an outside alternative).
The SPD framework can be implemented in real time as each service's optimal price has a closed-form solution.
\subsection{Pricing in BPD Framework}
This section presents how to set prices under the BPD framework.
Since BPD holds upcoming trip requests in batches and optimizes each batch's operations, it is more advantageous for a platform seeking to improve the matching quality, reduce the overall operational costs, and optimize routes for multiple pickups and dropoffs \parencite{ke2020ride}.
This section focuses on a BPD framework with a vehicular capacity of two, which is later extended to higher capacity cases.\footnote{Most TNCs currently operate a low-capacity services due to the capacity of their vehicle fleet and other factors \parencite{ke2021data}.}
The main challenge is to jointly determine the trip-vehicle assignment and prices for each batch.
\subsubsection{Vehicle dispatch with overbooking.}
The construction of the shareability graph in \citet{santi2014quantifying}, which describes the matchable relationship between available vehicles and pending requests, needs to be expanded to consider mixed service types.
However,
extending the formulation in \citet{alonso2017demand} to consider endogenous future demand is non-trivial.
The main research question is how to incorporate the value of reserving supply or demand in the exact optimal assignment to maximize the cumulative profit in batches.
At a high level, we first construct a Request-Vehicle (RV) graph and convert it to a Request-Trip-Vehicle (RTV) graph as in \citet{alonso2017demand}.
Finally, we can create a so-called Exclusive-Sharing-Vehicle (ESV) graph that accounts for both exclusive and shared services.
These graphs describe the feasible assignment between vehicles and pending trip requests.
Subsequently, we solve a global integer optimization to find the optimal prices and trip assignment for maximizing expected profit.
More concretely, given a batch of supply (available vehicles $\mathcal{V}$) and demand (trip requests $\mathcal{R}$), the procedure is as follows:
\begin{enumerate}
\item Construct a RV graph (Figure \ref{fig2-1}).
The RV graph gives all possible pairwise matchings between vehicles and requests.
In this graph, two requests $r_i$ and $r_j$ or a request $r$ and a vehicle $v$ are connected if the conditions discussed in Section \ref{sec212} are satisfied.
\item Construct a RTV graph (Figure \ref{fig2-2}) using the matchings in the RV graph.
A trip $T$ is a clique of requests that can be served by a vehicle without violating any feasibility constraints.
\item Compute an ESV graph (Figure \ref{fig2-3}) graph using both the RV graph and the RTV graph.
Each ESV matching contains either one request or two requests:
\begin{itemize}
\item If the ESV matching contains one request, the clique contains a combination of the following vehicles based on the RV graph:
\begin{enumerate}
\item An exclusive service vehicle $v_e$ available for this request.
\item A shared service vehicle $v_s$ available for this request.
\end{enumerate}
\item If the ESV matching contains two requests, the clique contains a combination of the following vehicles based on both the RV graph and the RTV graph:
\begin{enumerate}
\item A shared service vehicle $v_s$ that is able to serve these two requests.
\item An exclusive service vehicle $v_e$ available for the first request.
\item An exclusive service vehicle $v_e'$ available for the second request.
\end{enumerate}
\end{itemize}
\end{enumerate}
\noindent
\begin{figure}[!htb]
\hspace{-0.3in}
\subfloat[RV graph \label{fig2-1} ]{ \includegraphics[width = 0.33\textwidth]{figures/fig2-1.png} }
\subfloat[RTV graph \label{fig2-2} ]{ \includegraphics[width = 0.33\textwidth]{figures/fig2-2.png} }
\subfloat[ESV graph \label{fig2-3} ]{ \includegraphics[width = 0.33\textwidth]{figures/fig2-3.png} }
\caption{Constructing a shareability graph in batched pricing; $e_1$ is an exclusive vehicle and $s_1$ is a sharing vehicle; We show two sample ESV matchings containing one and two requests, respectively. } \label{fig2}
\end{figure}
Compared to the standard TVA formulation in \parencite{alonso2017demand}, we need to relax the packing constraints in the MIP to include the customers' mode choice behavior, as follows:
\begin{subequations}\label{eq:BDP}
\begin{align}
\text{maximize: } & \displaystyle\sum\limits_{i \in \mathcal{M}} u_{i} y_{i} - \sum\limits_{j \in \mathcal{V}} w_j \tag{\ref{eq:BDP} } \\
\text{subject to: }& \displaystyle \sum\limits_{i \in \mathcal{I}_{R = k}^{\mathcal{M}}} y_i \leq 1, & \forall k \in \mathcal{R} \label{eq:BDPa} \\
&w_{j} \geq (c_p + \bar{ \epsilon}_j) \cdot (\sum\limits_{i \in \mathcal{I}_{j }^{\mathcal{M}}} y_i \gamma_{i, j} - 1), & \forall j\in \mathcal{V} \label{eq:BDPb} \\
&w_{j} \geq 0, & \forall j\in \mathcal{V} \label{eq:BDPc} \\
& y_{i} \in \{0,1\}, & \forall i\in \mathcal{M} \label{eq:BDPd}
\end{align}
\end{subequations}
where $u_{i}$ is the expected profit of the ESV matching $i$, and $\mathcal{M}$ is the set of ESV matchings.
The binary decision variables $y_i$ represents whether the system choose an ESV matching $i \in \mathcal{M}$.
In the packing constraints Eq.~\eqref{eq:BDPa}, we split the potential ESV matchings into two sets: (a) a set of matchings that is connected to vehicle $j$ as $\mathcal{I}_{j}^{\mathcal{M}}$,
and (b) a set of matchings that contains request $k$ as $\mathcal{I}_{k }^\mathcal{M}$.
This packing constraint ensures that each request is assigned to at most one vehicle.
Since the mismatch of vehicle and requested service type is possible in the BPD framework, we adopt an overbooking strategy to enhance the overall performance in Eq.~\eqref{eq:BDPb}.
The platform with mixed fleets allows each vehicle to be assigned to multiple ESV matchings.
$\gamma_{i, j}$ is the probability that, as the ESV matching $i$ containing the vehicle $j$ is selected, vehicle $j$ is used at the end.
If vehicle $j$ is designated for the exclusive service, $\gamma_{i, j}$ is the predicted probability of the corresponding customer accepting the trip.
If vehicle $j$ is designated for the shared service, $\gamma_{i, j}$ is the probability that at least one of the assigned customers accepts the trip.
If more than one customer has accepted the assignment, then at least one request is either reassigned or rejected when there is no available vehicle nearby.
We impose a fixed penalty $c_p$ for each lost demand.
Let $\bar{\epsilon}_j$ be the mean expected profit earned by vehicle $j$ among all feasible matchings.
Variables $x_j$ are the overbooking penalty for vehicle $j \in \mathcal{V}$.
However, the system needs to evaluate an exponential number of scenarios to calculate the exact number of lost demands.
Therefore, we use $\sum_{ i \in \mathcal{I}_{j }^{\mathcal{M}} } y_i \cdot \gamma_{i, j} - 1$ as an approximation and simplify this computation.
The BPD prices are computed using the approach in Section \ref{sec: batchpricing} and the maximum waiting time and delay for each request is updated for each batch.
\subsubsection{Price mechanism for ESV matchings.}\label{sec: batchpricing}
In this section, we show how to find the optimal prices for each trip request in ESV matchings.
If the matching contains only one request, the optimization procedure is the same as shown in the SPD framework.
Therefore, we focus on the case where the ESV matching consists of two requests.
Suppose that $r_1$ and $r_2$ are the two requests in an ESV matching. The expected profit function of MPP is given by:
\begin{equation}\label{equ: batchprofit}
\begin{split}
\mathbb{E}_{\pi_p} [ \Phi_{r_1, r_2} ] &= P_{1, s} (p_{1, s} - c_{1, s}) + P_{2, s} (p_{2, s} - c_{2, s}) + P_{1, e} (p_{1, e} - c_{1, e}) \\
&+ P_{2, e} (p_{2, e} - c_{2, e}) + P_{1, s} P_{2, s} (c_{1, s} + c_{2, s} - c_{s, s}),
\end{split}
\end{equation}
where $p_{1, s}$ and $p_{1, e}$ are the shared service price and exclusive service price for $r_1$, respectively.
$p_{2, s}$ and $p_{2, e}$ are prices for $r_2$.
$c_{1, e}$ and $c_{2, e}$ are the cost incurred in serving $r_1$ and $r_2$ exclusively.
$c_{1, s}$ and $c_{2, s}$ are the expected cost of serving $r_1$ and $r_2$ if they do not form a sharing trip.
$c_{s, s}$ is the cost of serving $r_1$ and $r_2$ in a single sharing trip.
The cost is an expectation that depends on whether these requests are shared with other requests and whom to be shared with.
We have the following property of the objective function in regard with the probabilities $Pr_{1, e}$, $Pr_{1, s}$, $Pr_{2, e}$, and $Pr_{2, s}$:
\begin{proposition}\label{prop: bat}
The objective function Eq.~\eqref{equ: batchprofit} is jointly concave in the predicted probabilities if
\begin{equation}
\min\{c_{1, e}, c_{2, e}\} \leq -\frac{1}{\beta_p}.
\end{equation}
\end{proposition}
The proof is provided in Appendix \ref{proof: prop: bat}.
If this condition is not satisfied, we use a brute-force method to find the optimal solution for the nonconvex profit function.
Specifically, we can enumerate $P_{1, s}$ in the range $[0, 1]$ with a predefined step length.
Given the value of $P_{1, s}$, we can prove that the problem to optimize $P_{1, e}$, $Pr_{2, s}$ and $P_{2, e}$ is jointly concave in the variables using similar methods.
After enumerating all $P_{1, s}$, we can return the prices with the highest expected profit.
\section{Numerical Experiments}\label{c4sec: experiments}
\subsection{Overview of NYC case study}
The data used to estimate the model were obtained from a stated-preference study in NYC.
We use the publicly available Taxi and Limousine Commission (TLC) trip record data in Manhattan, NYC as a proxy for the real O-D demands, which is an established procedure in literature \parencite{santi2014quantifying, alonso2017demand}.
In this market, an MoD platform operates mixed fleets and competes for customers who use the existing taxicab service.
This numerical case study uses taxicab trip data as the requests $\mathcal{R}$ for both MoD and outside services.
We run the simulation using multiple consecutive weekdays' demand data (April 1, 2013 to April 9, 2013) in Manhattan, NYC.
To serve the travel demand in the area of study, the platform operates a fleet of 2000 vehicles with capacity one as $V_e$ and 1000 vehicles with capacity two as $V_s$.
The road network in Manhattan, NYC consists of 4092 nodes and 9453 edges.
The edge travel time is estimated by the daily mean travel time using the method in \citet{santi2014quantifying}.
The pickup time in the taxicab trip data is considered as the starting time of each trip request.
The reader interested in more details on the MoD simulator can refer to the previous work \parencite{alonso2017demand}.
All aforementioned methods are implemented using Python $3.5$, and all experiments are conducted on an Intel Core i7 computer (3.4 GHz, 16 GB RAM).
In this section, we present a series of numerical experiments to show:
\begin{enumerate}
\item The profit improvement by considering retrospective costs.
\item The performance comparison between SPD and BPD.
\item The profit increase of our framework compared with static pricing methods (discussed in Section~\ref{sec: calibration}).
\end{enumerate}
\subsection{Data Description and Parameter Estimation}
We estimate the parameters in the implementation of SPD and BPD frameworks in three main steps.
First, most parameters in the pricing policies can be estimated from historical data.
Second, we check main modeling assumptions a) the convexity of the objective and b) the convergence of mean costs to the derived analytical forms.
Finally, we calibrate the mode choice model.
\subsubsection{Estimation of parameters in pricing policies.}
The average trip duration, trip costs, and the customers' arrival rate at each time interval are computed from historical data.
Currently, TNCs mainly hire freelancer drivers who receive a fixed proportion of fares as their commission fees.
However, since our operational setting assumes a centrally controlled vehicle fleet (e.g., automated vehicles) that follows dispatching and re-positioning guidance from the platform, driver compensation are not considered throughout this experiment.
Next, we calibrate the functions $F_e(\cdot)$ and $F_s(\cdot)$.
Since the system's supply and demand are equal at the equilibrium, we can obtain the prices for both services given the number of open vehicles and compute the expected total profit.
In summary, given the total number of vehicles for the exclusive service and the shared service $L_e$ and $L_s$, we can enumerate $O_e$ and $O_s$ and compute the corresponding prices $p_e$ and $p_s$, respectively.
Finally, we calculated the prices offered to customers by the developed heuristic because it is difficult to determine $t_r^\prime$ at the request level.
The prices are calculated as maximizers to the average profit per period for each vehicle providing services $\bar{\epsilon}_e$ and $\bar{\epsilon}_s$.
The period is set it to be 20 minutes throughout the experiment.
\subsubsection{Test of modeling assumptions. }
Two critical assumptions are tested in this numerical experiment. First, whether the objective function is jointly concave; Second, if cost functions converge to their analytical forms.
\noindent \textbf{Test of concavity in Proposition \ref{prop: seq}.} \quad
According to our estimated mode choice model, $-1/\beta_p \approx 13.5$ in the numerical experiment.
In other words, if the average operational cost in each ESV matching is smaller than \$13.5, the problem is jointly concave in the predicted probabilities.
The area of interest in our experiment, Manhattan borough, has an area of 22.7 square miles (13.4 miles long and 2.3 miles wide at its widest).
If the MoD carriers are automated vehicles, the driving cost is about \$0.1458 per mile including the cost of fuel, maintenance and tiers \parencite{liu2018framework}.
Therefore, if the shorter trip in each ESV matching, $r_1$ and $r_2$, is less than 92.6 miles, the problem is jointly concave in the predicted probabilities and we can compute the unique optimal prices for each request.
\noindent \textbf{The convergence of expected costs.} \quad
As mentioned in Eq.~\eqref{equ: expec_cost}, if a request is the first being assigned to a sharing trip, the operational cost is an expectation that depends on whether and who this request will be matched with.
Since the expected cost does not have an explicit form, we run simulations to obtain an empirical estimator.
Specifically, we test the convergence by the following procedure:
\begin{enumerate}
\item As the demand is sparsely distributed in the road network, we use the K-means clustering algorithm to cluster the nodes according to their geo-coordinates spatially using the method in \citet{liu2020proactive}.
Assuming that each customer has a walking range of $\alpha$ miles, we choose the number of clusters $K$ such that these clusters cover the entire area approximately.
\item The simulation runs for a long period (e.g., a week) and aggregates these trips into O-D pairs between these clusters.
\item Operational costs for each trip are recorded assuming that travel demand is exogenous and unaffected by the implemented policies.
\end{enumerate}
The expected cost for each O-D cluster is initialized to be 0 at the beginning.
We compute the average realized cost for each pair of clusters at the end of simulation, and use it as the expected operational cost for each O-D pair that belongs to these clusters.
After running each day's simulation, we update the average cost for each O-D cluster based on completed results, and use it as the starting value for the next day.
Although the expected cost and the mode choice decisions in the simulation are coupled, the expected cost between each cluster can converge to a relatively stable throughout the simulation.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.65\textwidth]{figures/fig3-difference.png}
\caption{\label{fig: MAD_cost} The mean absolute difference of the expected cost among O-D clusters.}
\end{figure}
Figure~\ref{fig: MAD_cost} shows the mean absolute difference of the expected cost among all O-D clusters between each day and the previous day.
The curve has a decreasing trend and fluctuates around a relatively low value, which indicates the numerical convergence of the expected cost. \par
\subsubsection{Mode choice model calibration.} \label{sec: calibration}
We use a \emph{static} price adopted by UberX service in the past \parencite{Uber} as a benchmark throughout this experiment.
The price is computed by:
\begin{equation}\label{equ: trip_cost}
p_r = \max (f_{min}, f_{base} + f_t t_r + f_d d_r)
\end{equation}
where $f_{min}$ and $f_{base}$ are the minimum fare and base fare for the ride-hailing service, $t_r$ and $d_r$ are the travel time and travel distance for $r$, $f_t$ is the time rate per second for the ride-hailing service, and $f_d$ is the distance rate per mile for the ride-hailing service.
We use the same coefficients from a previous NYC Uber empirical study \parencite{Uberprice}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.65\textwidth]{figures/fig4-price.png}
\caption{\label{fig: utility_scale} The price comparison between the static price and the price of our framework under different coefficient multipliers.}
\end{figure}
The choice model that reflects the customers' preference between MoD and conventional taxi service is calibrated as in the prior work \parencite{liu2018framework}.
Customers choosing between MoD services and public transit mainly consider the QoS, such as total travel time and variations of waiting time.
In comparison, customers are more price-sensitive as MoD and taxi service are similar in this dense urban network.
Practitioners may conduct extra surveys to estimate the mode choice model's coefficients more accurately.
We circumvent the issue with parameter errors by incorporating a multiplier greater than 1 to scale the price coefficients in the MNL mode choice model.
We simulate the pricing-and-assignment process using an arbitrary day's data (April 10, 2013) using different multipliers $\{1.2, 1.4, 1.6, 1.8, 2.0 \}$.
The goal is to find a multiplier such that the average price determined by our framework is similar to the average static price computed by Eq.~\eqref{equ: trip_cost}.
Figure \ref{fig: utility_scale} shows the average price between applying the proposed SPD/BPD framework and that of the static price benchmark.
Since the multiplier of 1.8 gives the same average price as the average static price, we fix it in the remainder of the experiments and rerun the above experiments to estimate O-D clusters' expected operational costs.
\subsection{The Effects of Retrospective Cost Function}
Retrospective costs represent the expected sunk profit of assigning an available vehicle to future requests.
In this section, we use the simulation of demand data on April 16, 2013, to illustrate the effect of including the retrospective cost in Eq.~\eqref{equ: potential}.
The overall profit decreases after employing retrospective costs directly.
Possible reasons include:
\begin{enumerate}
\item Eq.~\eqref{equ: potential} is based on a steady-state analysis \parencite{castillo2017surge,yan2020dynamic}.
The actual demand and supply processes are nonstationary, and hence $\hat{g}(\cdot)$ is not a precise estimator for $g(\cdot)$.
\item The retrospective cost represents the predicted profit a vehicle could make during the travel time.
This estimation may not be accurate in a clustered network as the travel time may vary significantly from individual requests.
\end{enumerate}
\begin{figure}[htb]
\centering
\includegraphics[width=0.65\textwidth]{figures/fig5-profit.png}
\caption{\label{fig: opportunity} The profit of both the batching pricing framework and the sequential pricing framework under different retrospective cost multiplier. }
\end{figure}
As mentioned in Section \ref{sec: batchpricing}, finding the exact form of $\hat{g}(\cdot)$ is not the focus of this work.
We conduct a sensitivity analysis using a multiplier to scale the retrospective cost and demonstrate the effect of including retrospective costs.
The multiplier ranges from 0.0 to 1.0 with a step of 0.1 in the following experiment.
We run the experiments separately for each instance and use multipliers that give maximum total profits.
Note that optimal multiplier may differ for the BPD framework and the SPD framework.
Figure~\ref{fig: opportunity} shows profits of both frameworks under different retrospective cost multipliers.
There exists an optimal multiplier larger than $0$ that maximizes the expected profit (0.2 and 0.3 for the batching pricing framework and the sequential pricing framework, respectively).
The total profits increase by 2.7\% and 9.2\% for the BPD framework and the SPD framework, respectively.
The introduction of retrospective costs has less impact on the BPD framework because the setup of batched requests has already considered the partial effect of delayed reward.
\subsection{Performance Comparison}
In this section, we compare the performance of our BPD and SPD frameworks and the results are presented in Table~\ref{c4: model_compare}.
The simulation uses the demand data on April 17, 2013, in Manhattan, NYC.
As this work intends to improve the performance of high-capacity MoD dispatch algorithms in \citet{alonso2017demand}, we use two benchmark models that employ a static pricing mechanism (i.e., fixed rules of splitting trip fares in ridepooling) in the TVA formulation.
In the static pricing method, the fractions of customers choosing shared services are calculated by the probability of the customer sharing with other requests in the future.
We assume that the shared service's fares are split based on the corresponding exclusive service prices.
Let $\kappa$ be the discounter of the shared service and $\theta$ be the probability of the request $r$ sharing with other requests in the future, which is computed empirically using simulation on historical data for each O-D cluster.
Then, given the exclusive service price $p_e$, we compute the shared service price $p_s$ as $ p_s = (1 - 0.3 \cdot \theta + 0.2) \cdot p_e$ \parencite{Uberprice}.
This simple price structure can be easily calibrated with access to real-world ride-hailing trip data.
Table~\ref{c4: model_compare} lists the profit, the market share, the mean price, and the mean waiting time of customers.
The overall profits of the BPD framework and SPD framework increase significantly and decrease the mean waiting time compared to the corresponding static pricing method (23.6\% increase and 39.1\% increase).
However, the market shares decrease by 8.0\% with the BPD framework and 6.6\% with the SPD framework, respectively.
One explanation is that our pricing frameworks set the prices higher than the status-quo static pricing mechanism.
Although the market share decreases, the profit per trip grows significantly, and total profits increase with either framework.
\begin{table}[!htb]
\caption{Model comparison}
\label{c4: model_compare}
\resizebox{\textwidth}{!}{%
\begin{tabular}{lcccc}
\toprule
Model & Profit (million) & Market share (\%) & Mean price & Mean waiting time (s) \\
\midrule
BPD & 1.99 & 34.54 & 14.37 & 135.60 \\
Batched-Static & 1.61 & 42.49 & 9.73 & 144.52 \\
SPD & 1.92 & 32.58 & 14.72 & 117.88 \\
Sequential-Static & 1.38 & 39.15 & 9.44 & 119.56 \\
\bottomrule
\end{tabular}
}
\smallskip
{\small Batched-Static: Matching the requests in batches and set prices using the static price in Eq.~\eqref{equ: trip_cost}. \\
\noindent Sequential-Static: Matching the requests sequentially and set prices using the static price in Eq.~\eqref{equ: trip_cost}.
}
\end{table}
Compared to the SPD framework, the BPD frameworks can increase profits due to more efficient matching and pricing in batches.
Specifically, the BPD framework increases the profit by about 3.6\% compared to the SPD framework, and the profit of the \textit{Batched-Static} framework is 16.7\% higher than the \textit{Sequential-Static} framework.
The profit increase of BPD is at the expense of an increase in the mean waiting time because the batched supply and demand uses a 30-second matching window.
In practice, the MoD platform should find the optimal trade-off between the overall service level and profits.
\section{Conclusion} \label{sec:conclusion}
This study proposes a computationally efficient approach for solving the JVDPP on an MoD platform.
The platform providing exclusive and shared services must continuously balance each option's QoS and prices to accommodate various demands and maximize the total profit.
An important step towards operating such a platform is to design a scalable dispatching and pricing method with modest performance guarantees.
Leveraging analytical MPP and spatial pricing results, we propose a sequential framework and a batched framework that simultaneously compute prices and dispatch ridepooling trips at the request level.
We show a closed-form solution to the optimal prices in the SPD framework and prove the convexity of the BPD's objective with regard to mode choice probabilities under mild technical assumptions on the cost structures.
Based on these results, we propose a scalable MIP formulation with overbooking to overcome the myopicity in the BPD framework. The notion of retrospective costs, a spatiotemporal proxy to the suboptimal assignment, can increase the overall profit by about 3\% to 9\% in numerical experiments.
The SPD/BPD framework yields significant profit increases (24\% to 39\%) compared to the benchmark static pricing method.
There are multiple avenues for future research.
First, the tractability of this model is based on MPP results for up to two requests.
It is worth investigating how to improve the brute-force search procedure for high-capacity MoD fleets with mixed fleets.
Second, considering more realistic cost structures with advanced data-driven approaches than the steady-state analysis and reevaluating SPD/BPD frameworks' effectiveness is a promising direction.
\section*{Acknowledgement}
This research is supported by NSF DMS-1839346 and SCC-1952011.
\clearpage
\ECSwitch
\ECHead{Proofs and supplementary materials}
\begin{appendices}
\section{Summary of notation} \label{sec:app1}
We summarize the notation used throughout this work in the following table:
\begin{center}
\begin{longtable}{c|l}
\caption{Summary of notation} \label{table:app1} \\
\toprule
\multicolumn{1}{c|}{Notation} & \multicolumn{1}{c}{Definition} \\ \midrule
\multicolumn{2}{l}{Terminology} \\ \midrule
MoD & Mobility-on-Demand \\ \hline
TNC & Transportation network company \\ \hline
JVDPP & Joint vehicle dispatching and pricing problem \\ \hline
SPD & Sequential pricing and dispatch \\ \hline
BPD & Batched pricing and dispatch \\ \hline
MIP & Mixed integer programming \\ \hline
MNL & Multinomial logit model \\ \hline
O-D pair & Origin-destination pair \\ \hline
QoS & Quality of service \\ \hline
TVA & Trip-vehicle assignment formulation \\ \hline
RV graph & Request-vehicle graph \\ \hline
RTV graph & Request-trip-vehicle graph \\ \hline
ESV graph & Exclusive-sharing-vehicle graph \\ \midrule
\multicolumn{2}{l}{Decision variables} \\ \midrule
$\pi_{a}$ & Trip-vehicle assignment policy \\ \hline
$\pi_p$ & Pricing policy \\ \hline
$p_e^r$ & MoD's price for request $r$ to use the exclusive service \\ \hline
$p_s^r$ & MoD's price for request $r$ to use the shared service \\ \hline
$y_i$ & Binary variable for choosing matching $i\in \mathcal{M}$ \\ \hline
$x_j$& Overbooking penalty for vehicle $j \in \mathcal{V}$ \\ \midrule
\multicolumn{2}{l}{Mixed MoD fleet parameters} \\ \midrule
$\mathcal{R}$ & Set of trip requests \\ \hline
$\Phi_r$ & Profit collected from request $r$ \\ \hline
$Pr_e$ & Probability of choosing the exclusive MoD service \\ \hline
$Pr_s$ & Probability of choosing the shared MoD service \\ \hline
$Pr_o$ & Probability of choosing the outside option \\ \hline
$c_e$ & Cost of offering exclusive MoD service \\ \hline
$c_s$ & Cost of offering shared MoD service \\ \hline
$c_{m, \text{operational} } $ & Operational cost for mode $m\in \{e,s\}$ \\ \hline
$c_{m, \text{potential} } $ & retrospective cost for mode $m\in \{e,s\}$ \\ \hline
$\Omega$ & Maximum waiting time for matching \\ \hline
$\Delta $ & Maximum delay \\ \hline
$U_m$ & Customer's utility from choosing the mode $m \in \{e,s,o \}$ \\ \hline
$w_m$ & Waiting time of mode $m$, $m\in \{e,s\}$ \\ \hline
$t_m$ & Travel time of mode $m$, $m\in \{e,s\}$ \\ \hline
$\beta_{p}$ & Coefficient of price in utility function \\ \hline
$\beta_{w}$ & Coefficient of waiting time in utility function \\ \hline
$\beta_{t}$ & Coefficient of travel time in utility function \\ \hline
$\mathcal{V}_e$ & Set of exclusive MoD vehicles \\ \hline
$\mathcal{V}_s$ & Set of shared MoD vehicles \\ \hline
$c_0$ & Operational cost if a request is not matched with any other request \\ \hline
$\alpha_{r'}$ & The probability request $r'$ is matched with vehicle carrying $r$ \\ \hline
$L$ & Total number of MoD vehicles in the cluster \\ \hline
$L_e$ & Total number of exclusive vehicles \\ \hline
$L_s$ & Total number of sharing vehicles \\ \hline
$Y$ & The steady-state throughput of MoD vehicles \\ \hline
$Y_e$ & The steady-state throughput of exclusive service vehicles \\ \hline
$Y_s$ & The steady-state throughput of shared service vehicles \\ \hline
$O_e$ & The number of available exclusive service vehicles \\ \hline
$O_s$ & The number of available shared service vehicles \\ \hline
$T_e$ & Average trip duration of exclusive services \\ \hline
$T_s$ & Average trip duration of shared services \\ \hline
$\eta_e$ & Average waiting time for exclusive service \\ \hline
$\eta_s$ & Average waiting time for shared service \\ \hline
$\zeta_s$ & Average utilization of sharing service \\ \hline
$\bar{\epsilon}_o$ & Average profit if vehicle stays in the origin \\ \hline
$\bar{\epsilon}_d$ & Average profit if vehicle travels to destination \\ \hline
$\mathcal{M}$ & Set of ESV matching \\ \hline
$\mathcal{V}$ & Set of vehicles in the batch \\ \hline
$\mathcal{I}_{j }^\mathcal{M}$ & A set of matchings contains vehicle $j$ for given ESV matchings $\mathcal{M}$ \\ \hline
$\mathcal{I}_{k }^\mathcal{M}$ & A set of matchings contains requests $k$ for given ESV matchings $\mathcal{M}$ \\ \hline
$u_i$ & Expected profit of the ESV matching $i \in \mathcal{M}$ \\ \hline
$\gamma_{i,j}$ & Probability that vehicle $j$ is used if the ESV matching $i$ is selected \\ \hline
$\Phi_{r_1, r_2}$ & Expected profit of serving two requests $r_1, r_2$ in an ESV matching \\ \bottomrule
\end{longtable}
\end{center}
\section{Utilization of ridepooling vehicles in sequential pricing model} \label{sec:app2}
We use an aggregate mean utilization $\zeta_s$ in the fleet conservation model.
\begin{align}
& Y_s(\eta, O) \triangleq \frac{ \zeta_s ( L_s - O_s)}{F_s(O_s) + T_s}.
\end{align}
We can derive this parameter from a Markov chain model of the ridepooling process.
The following formulation is for the special case that each ride takes at most two requests, which can be easily extended to higher capacity cases.
The number of customers on vehicle has three states $\{0,1,2\}$. The transition probabilities are $P_{ij}$ for $i,j \in \{0,1,2\}$. We let the parameters of empty vehicle be $O_s, Y_s$ and vehicle with one customer be $O_s', Y_s'$. It
At the steady state, the number of vehicles in each state denoted by $N_i$ are:
\begin{align*}
\begin{cases}
N_0 = O_s + \eta_s Y_s \\
N_1 = O_s' + \eta_s' Y_s' + T_s Y_s \\
N_2 = T_s' Y_s'
\end{cases}.
\end{align*}
The detailed balance equations are:
\begin{align*}
& N_0 P_{01} = N_1 P_{10} \\
& N_1 P_{12} = N_ 2 P_{21}.
\end{align*}
We can use these equations to calibrate the throughput $Y_s, Y_s'$. For example, if we assume the that vehicles with zero and one customer are equal, we have $P_{01} = P_{12 }$. The average utilization is thus $\zeta_s = ( N_1 +2 N_2) / L_s $. Note these two probabilities are roughly the ratio of time
\begin{align*}
\zeta_s = \frac{ O_s' + \eta_s' Y_s' + T_s Y_s + 2 T_s' Y_s' }{ L_s }
\end{align*}
\section{Proof of proposition~\ref{prop: seq}}\label{proof:prop:seq}
\begin{proof}
The partial derivative for $p_s$ is:
\begin{equation}
\frac{\partial \mathop{\mathbb{E}_{\pi_p}} [\Phi_r]}{\partial p_{s}} \\= \frac{\left(\beta \cdot e^{U_{s}} \cdot (p_{s} - m) + e^{U_{s}} \right) \cdot (e^{U_{s}} + e^{U_{e}} + D) - \beta \cdot e^{U_{s}} \cdot \left(e^{U_{s}}\cdot (p_{s} - m)+ e^{U_{e}} \cdot (p_{e} - c)\right)}{(e^{U_{s}} + e^{U_{e}} + D)^2}
\end{equation}
The partial derivatives are well defined everywhere.
Since critical points are either points such that the partial derivatives do not exist or the partial derivatives are 0.
For simplicity, let $e^{U_{s}}$ and $e^{U_{e}}$ be $x$ and $y$. Then, we can infer that $p_{s} = \frac{\log(x) - d_{s}}{\beta}$ and that $p_{e} = \frac{\log(y) - d_{e}}{\beta}$. A critical point will satisfy the following equations if it exists:
\begin{align}
&x + y + y \cdot \log(\frac{x}{y}) - (d_{s} + \beta \cdot m - d_{e} - \beta \cdot c) \cdot y + D \cdot \log(x) + D \cdot (1 - d_{s} - \beta \cdot m) = 0 \label{equ: 1}, \\
&x + y - x \cdot \log(\frac{x}{y}) + (d_{s} + \beta \cdot m - d_{e} - \beta \cdot c) \cdot x + D \cdot \log(y) + D \cdot (1 - d_{e} - \beta \cdot c) = 0 \label{equ: 2}
\end{align}
Let Equation~\eqref{equ: 1} subtract Equation~\eqref{equ: 2}, and we can find the following solution:
\begin{equation}
x = e^{d_{s} + \beta \cdot m - d_{e} - \beta \cdot c} \cdot y
\end{equation}
We can further infer that a critical point satisfies the following condition:
\begin{equation}
p_{e} - p_{s} = c - m
\end{equation}
Employing this relationship in Eq.~\eqref{equ: 1}, we can get the critical point:
\begin{equation}
p_e = \dfrac{\log\left(\dfrac{W\left((1 + e^{d_{s} - d_{e} + \beta \cdot m - \beta \cdot c}) \cdot e^{d_{e} + \beta \cdot c - 1}/ D\right) \cdot D)}{1 + e^{d_{s} - d_{e} + \beta \cdot m - \beta \cdot c}}\right) - d_{e}}{\beta}
\end{equation}
\end{proof}
\section{Proof of proposition~\ref{prop: bat}}\label{proof: prop: bat}
\begin{proof}
We first replace the price variables in the function by the following equations:
\begin{equation}
p_{1, s} = \dfrac{\log{\frac{D_1 P_{1, s}}{1 - P_{1, s} - P_{1, e}}} - d_{1,s}}{\beta}
\end{equation}
where $D_1$ is exponential to the power of the utility of $r_1$ taking the taxi service. Similarly, we can also use the predicted probabilities to represent the other price variables. Since the price vector and the predicted vector has a one-to-one matching~\parencite{li2011pricing}, we can obtain the price variables if we can find the optimal predicted probabilities. Therefore, minimizing the following function will be equivalent to maximizing the expected profit shown in Eq.~\eqref{equ: batchprofit}. \par
\begin{align}\label{equ: replaceobj}
\beta \mathbb{E}_{\pi_p} [\Phi_r] &= P_{1, s} \left(\log (\frac{D_1 \cdot P_{1, s}}{1 - P_{1, s} - P_{1, e}}) - \beta c_{1, s} \right)
+ P_{1, e} \left(\log(\frac{D_1 P_{1, e}}{1 - P_{1, s} - P_{1, e}}) - \beta c_{1, e}\right) \nonumber \\
&+ P_{2, s} \left(\log (\frac{D_2 P_{2, s}}{1 - P_{2, s} - P_{2, e}}) - \beta c_{2, s}\right)
+ P_{2, e} \left(\log(\frac{D_2 P_{2, e}}{1 - P_{2, s} - P_{2, e}}) - \beta c_{2, e}\right) \nonumber \\
&+ P_{1, s} P_{2, s} \beta (c_{1, s} + c_{2, s} - c_{s, s})
\end{align}
Let $\phi_1 := 1 - P_{1, s} - P_{1, e}$ and $\phi_2 := 1 - P_{2, s} - P_{2, e} $.
The Hessian matrix of the above equation $H$ is as follows, where the order of each row and column is $P_{1, s}$, $P_{1, e}$, $P_{2, s}$ and $P_{2, e}$:
\begin{align*}
\begin{bmatrix}
\dfrac{1}{P_{1, s}} + \dfrac{1}{\phi_1} + \dfrac{1}{\phi_1^2} & \dfrac{1}{\phi_1} + \dfrac{1}{\phi_1^2} & \beta (c_{1, s} + c_{2, s} - c_{s, s}) & 0 \\
\dfrac{1}{\phi_1} + \dfrac{1}{\phi_1^2} & \dfrac{1}{P_{1, e}} + \dfrac{1}{\phi_1} + \dfrac{1}{\phi_1^2} & 0 & 0 \\
\beta (c_{1, s} + c_{2, s} - c_{s, s}) & 0 & \dfrac{1}{P_{2, s}} + \dfrac{1}{\phi_2} + \dfrac{1}{\phi_2^2} & \dfrac{1}{\phi_2} + \dfrac{1}{\phi_2^2} \\
0 & 0 & \dfrac{1}{\phi_2} + \dfrac{1}{\phi_2^2} & \dfrac{1}{P_{2, e}} + \dfrac{1}{\phi_2} + \dfrac{1}{\phi_2^2}
\end{bmatrix}
\end{align*}
If Eq.~\eqref{equ: replaceobj} is jointly convex in the predicted probabilities, we know that the problem has at most one critical point.
We can find the optimal solution by finding the point satisfying the first-order derivative condition and compare its objective function value with the boundary points or using the second-order derivative test.
If the above Hessian matrix is positive definite, we conclude that the convexity condition is satisfied. \par
Let $\Vec{y} = \{y_1, y_2, y_3, y_4\}$ be a non-zero vector of four any real numbers. We can derive the following: \par
\begin{equation}\label{equ: secondorder}
\begin{split}
\Vec{y} \cdot H \cdot \Vec{y}^{T} &= (\frac{1}{1 - P_{1, s} - P_{1, e}} + \frac{1}{(1 - P_{1, s} - P_{1, e})^2}) (y_1 + y_2)^2 \\
&+ (\frac{1}{1 - P_{2, s} - P_{2, e}} + \frac{1}{(1 - P_{2, s} - P_{2, e})^2}) (y_3 + y_4)^2 \\
&+ \frac{y_1^2}{P_{1, s}} + \frac{y_2^2}{P_{1, e}} + \frac{y_3^2}{P_{2, s}} + \frac{y_4^2}{P_{2, e}} + 2 y_1 y_3 \beta (c_{1, s} + c_{2, s} - c_{s, s})
\end{split}
\end{equation}
If $\frac{y_1^2}{P_{1, s}} + \frac{y_3^2}{P_{2, s}} + 2 y_1 y_3 \beta (c_{1, s} + c_{2, s} - c_{s, s}) \geq 0$, $\Vec{y} \cdot H \cdot \Vec{y}^{T} \geq 0$ because all the other parts in the equation are non-negative. Let $C = c_{1, s} + c_{2, s} - c_{s, s}$ for readability, we can derive the following:
\begin{equation*}
\frac{y_1^2}{P_{1, s}} + \frac{y_3^2}{P_{2, s}} + 2 y_1 y_3 \beta C = (\sqrt{-\beta C} (y_1 - y_3))^2 + (\frac{1}{P_{1, s}} + \beta C) y_1^2 + (\frac{1}{P_{2, s}} + \beta C) y_3^2
\end{equation*}
Therefore, the condition to prove the problem is jointly concave in the predicted probabilities is the following:
\begin{equation}\label{equ: oricondition}
C \leq \min\{-\frac{1}{\beta P_{1, s}}, -\frac{1}{\beta P_{2, s}}\}
\end{equation}
Since $\beta < 0$, $P_{1, s} \leq 1$ and $P_{2, s} \leq 1$, the following condition is a sufficient condition to satisfy Eq.~\eqref{equ: oricondition}:
\begin{equation}\label{equ: sufcondition}
C \leq -\frac{1}{\beta}
\end{equation}
where $C = c_{1, s} + c_{2, s} - c_{s, s}$ is the operational cost savings if $r_1$ and $r_2$ can share a ride compared to the sum of the expected cost for either of them choosing the shared service.
The cost savings should always be larger than 0.
Otherwise, the service provider should not match $r_1$ and $r_2$ as a sharing trip.
We know that $c_{s, s} \geq \max\{c_{1, e}, c_{2, e}\}$, $c_{1, s} \leq c_{1, e}$ and $c_{2, s} \leq c_{2, e}$, and thus Eq.~\eqref{equ: sufcondition} is satisfied if the following equation is satisfied:
\begin{equation*}
\min\{c_{1, e}, c_{2, e}\} \leq -\frac{1}{\beta}
\end{equation*}
\end{proof}
\end{appendices}
\printbibliography
\end{document}
|
1,116,691,499,353 | arxiv | \section{Introduction}
We consider finite, undirected graphs without loops and multiple edges. For a graph $G$, $V(G)$ and $E(G)$ denotes the vertex-set and the edge-set respectively. A $(p,q)$ graph $G$ is a graph such that $|V(G)|=p$ and $|E(G)|=q$. We refer the reader to \cite{WA} and \cite{WE} for all other terms and notation not provided in this paper.
A labeling of a graph $G$ is any mapping that sends some set of graph elements to a set of non-negative integers. If the domain is the vertex-set or the edge-set, the labelings are called \emph{vertex labeling or edge labeling} respectively. Moreover, if the domain is $V(G)\cup E(G)$ then the labeling is called \emph{total labeling}.
Let $f$ be a vertex labeling of a graph $G$, we define the edge-weight of $uv\in E(G)$ to be $wt_f(uv)=f(u)+f(v)$. If $f$ is a total labeling, then the edge-weight of $uv$ is $wt_f(uv)=f(u)+f(v)+f(uv)$. The vertex-weight of a vertex $v$, $v\in E(G)$ is defined by $$wt_f(v)=\sum_{u\in N(v)}f(uv)+f(v)$$
where $N(v)$ is the set of the neighbors of $V$. If the vertices are labeled with the smallest posssible numbers i.e. $f(V(G))=\{1,2,3,...,p\}$, then the total labeling is called \emph{super}.
A labeling $f$ is called \emph{edge-antimagic total(vertex-antimagic total)}, for short EAT(VAT), if all edge-weights (vertex-weights) are pairwise distinct. A graph that admits EAT (VAT) labeling is called an EAT (VAT) graph. If the edge-weights (vertex-weights) are all the same, then the total labeling is called \emph{edge-magic total (vertex-magic total)}. For an edge labeling, a \emph{vertex-antimagic edge} (VAE) labeling is a labeling whereby a vertex-weight is the sum of the labels of all edges incident with the vertex.
In 1990, Harsfield and Ringel \cite{HR} introduced the concept of an antimagic labeling of graphs whereby they conjectured that every tree except $P_2$ has a VAE labeling. This conjecture was proved to be true for all graphs having minimum degree $log|V(G)|$ by Alon \emph{et al} \cite{AK}.
If a VAE labeling satisfies the condition that the set of all the vertex-weights is $\{a,a+d,...,a+(p-1)d\}$ where $a>0$ and $d\geq 1$ are two fixed integers, then the labeling is called an \emph{(a,d)-VAE labeling}. For further results on graph labeling see \cite{BAM}, \cite{GA} and \cite{MD}.
In \cite{MPJ}, Miller \emph{et al} proved that all graphs are (super) EAT. They also proved that all graphs are (super )VAT.
If the labeling is simultaneously vertex-antimagic total and edge-antimagic total, then it is referred to as \emph{totally-antimagic total} (TAT) labeling and a graph that admits such labeling is a \emph{totally-antimagic total} (TAT) graph. The definition of totally antimagic total labeling is a natural extension of the concept of totally magic labeling. In \cite{BE1}, Baca \emph{et al} deals with totally antimagic total graphs. They found totally-antimagic total labeling of some classes of graphs and proved that paths, cycles, stars, double-stars and wheels are totally antimagic total. Moreover, they showed that a union of regular totally antimagic total graphs is a totally antimagic total graph. Also, in \cite{AJ}, Akwu and Ajayi showed that complete bipartite graphs are totally antimagic total graphs.
In this paper, we deal with totally antimagic total labeling of ladders, prisms and generalised Petersen graphs. We also show that the chain graphs obtained by concatenation of totally antimagic total graphs are totally antimagic total graphs.
First we provide some definitions which are related to the present work.
\begin{definition}
\emph{Ladder} is a graph obtained by the cartesian product of path $P_n$ and path $P_2$ denoted by $L_n$, i.e. $L_n \simeq P_n\times P_2$ where $V(L_n)=\{u_iv_i:1\leq i\leq n\}$ and $E(L_n)=\{u_iu_{i+1},v_iv_{i+1}:1\leq i\leq n-1\}\cup \{u_iv_i:1\leq i\leq n\}$.
\end{definition}
\begin{definition}
A labeling $g$ is ordered (sharp ordered) if $wt_g(u)\leq wt_g(v)$ $(wt_g(u)<wt_g(v))$ holds for every pair of vertices $u,v\in G$ such that $g(u)<g(v)$. A graph that admits a sharp ordered labeling is called a \emph{(sharp) ordered } graph. Also, if the vertex set can be partitioned into $n$ sets such that each set is sharp ordered, then the graph is referred to as \emph{weak ordered} graph.
\end{definition}
\begin{definition}
The \emph{prism} graph can be defined as the cartesian product $C_n\times P_2$ of a cycle on $n$ vertices with a path of length $2$. The vertex set is $V(C_n\times P_2)=\{u_iv_i:1\leq i\leq n\}$ and the edge set is
$E(C_n\times P_2)=\{u_iu_{i+1},v_iv_{i+1}:1\leq i\leq n\}\cup \{u_iv_i:1\leq i\leq n\}$
where $i$ is calculated modulo $n$. The orders of the vertex set and the edge set are $2n$ and $3n$ respectively.
\end{definition}
\begin{definition}
The \emph{generalised Pertersen} graph $P(n,m), \ n\geq 3$ and $1\leq m\leq \lfloor \frac{n-1}{2} \rfloor$ consists of an outer $n$-cycle $u_i,u_2,...,u_n$, a set of $n$ spokes $u_iv_i, 1\leq i\leq n$ and $n$ edges $v_iv_{i+m}$, $1\leq i\leq n$ with indices taken modulo $n$.
\end{definition}
\section{Totally antimagic total graphs}
In this section, we prove that ladders, prism graphs and generalised Petersen graphs are totally antimagic total graphs.
\begin{thm}
The ladder graph $L_n$, $n\geq 2$ is a weak ordered super TAT.
\end{thm}
\begin{proof}
We denote the vertices of $L_n$ by the following symbols $V(L_n)=\{u_iv_i:1\leq i\leq n\}$ such that $E(L_n)=\{u_iu_{i+1},v_iv_{i+1}:1\leq i\leq n-1\}\cup \{u_iv_i:1\leq i\leq n\}$. Consider the labeling $g$ of $L_n$ as follows:
$$g(u_i)=\left\{\begin{array}{ll}
2i-1, & 1\leq i\leq \lceil\frac{n}{2}\rceil\\ \ \\
2(n-i+1), & \lceil\frac{n}{2} \rceil+1\leq i\leq n
\end{array}\right.$$
and
$$g(v_i)=\left\{\begin{array}{ll}
n+2i-1, & 1\leq i\leq \lceil\frac{n}{2}\rceil\\ \ \\
3n+2(1-i), & \lceil\frac{n}{2} \rceil+1\leq i\leq n
\end{array}\right.$$
Also, the labeling of the edges is as follows:
$$g(u_iv_i)=\left\{\begin{array}{ll}
2(n+i)-1, & 1\leq i\leq \lceil\frac{n}{2} \rceil\\ \ \\
2(2n-i+1), & \lceil\frac{n}{2} \rceil+1\leq i\leq n-1
\end{array}\right.$$
$$g(u_iu_{i+1})=\left\{\begin{array}{ll}
3n+2i-1, & 1\leq i\leq \lfloor\frac{n}{2} \rfloor\\ \ \\
5n-2i, & \lfloor\frac{n}{2} \rfloor+1\leq i\leq n-1
\end{array}\right.$$
$$g(v_iv_{i+1})=\left\{\begin{array}{ll}
2(2n+i-1), & 1\leq i\leq \lfloor\frac{n}{2} \rfloor\\ \ \\
2(3n-i)-1, & \lfloor\frac{n}{2} \rfloor+1\leq i\leq n-1
\end{array}\right.$$
The above labeling is super. Next, we show that the vertex-weights are pairwise distinct.\\
The vertex-weights of $V(L_n)$ are
$$wt_g(u_i)=g(u_i)+\sum_{u\in N(u)}g(u_iu)$$ where vertex $u$ is any vertex adjacent to vertex $u_i$.
For $i=1$ and $i=n$, we have the weights $$wt_g(u_1)= 3+5n$$ and $$wt_g(u_n)=6+5n.$$
Also, for $i=2,...,n-1$, we have the weight $$wt_g(u_i)=\left\{\begin{array}{lll} \ \\
2(4i+4n-3), & 2\leq i\leq \lfloor\frac{n}{2} \rfloor\\ \ \\
12n-3, & i=\lfloor\frac{n}{2} \rfloor+1\\ \ \\
2(4(2n-i)+3), & \lfloor \frac{n}{2} \rfloor +2 \leq i\leq n-1
\end{array}\right.$$
Furthermore,
$$wt_g(v_i)=\left\{\begin{array}{ll}
W_1, \\ \ \\
W_2
\end{array}\right.$$
where $$W_1=2(2i-1)+k(2n+i-1)-(t+3n)$$ with the following conditions:
for $i=1, k=1$ and $t=0$,
for $2\leq i\leq \lfloor \frac{n}{2} \rfloor, k=t=2$.\\
While $$W_2=2(2(1-i)+k(3n-i))+(t-k)+7n$$ with the following conditions:
$k=t=2$ whenever $\lfloor \frac{n}{2}\rfloor +2\leq i\leq n-1$,
$k=1$ and $t=2$ for $i=n$,
for $n$ even and $i= \frac{n}{2} +1$, $k=2$ and $t=1$,\\
for $n$ odd and $i=\lfloor \frac{n}{2}\rfloor +1$, $k=2$ and $t=-3$.
In view of the above labeling, the weights of all the vertices are different, that is the labeling is vertex-antimagic total.
Now, we show that the edge-weights are pairwise distinct. The edge-weight of the edges under labeling $g$ is as follows:
$$wt_g(u_iu_{i+1})=\left\{\begin{array}{ll}
W_3, \\ \ \\
W_4
\end{array}\right.$$
where $$W_3=3(2i+n-1)+t$$ with the following conditions:
$t=2$ for $1\leq i\leq \lceil \frac{n}{2}\rceil -1$,
$t=1$ for $n$ even and $i=\frac{n}{2}$,
$t=-2$ for $n$ odd and $i=\lceil \frac{n}{2}\rceil$,
while $$W_4=3(3n-2i)+2$$ for $\lceil \frac{n}{2}\rceil +1\leq i\leq n-1$.
Also, $$wt_g(v_iv_{i+1})=\left\{\begin{array}{ll}
W_5, \\ \ \\
W_6
\end{array}\right.$$
where $$W_5=2(3(n+i)-2)+t$$ with the following conditions:
$t=2$ whenever $1\leq i\leq \lceil \frac{n}{2}\rceil -1$,
$t=1$ whenever $n$ is even and $i=\frac{n}{2}$,
$t=-2$ whenever $i=\lceil \frac{n}{2} \rceil $ and $n$ is odd.
Also,
$$W_6=3(2(2n-i)+1)-2, \lceil \frac{n}{2} \rceil +1\leq i\leq n-1$$
In view of the above labeling, the edge-weights are pair-wise distinct, which implies that the labeling is edge-antimagic labeling, i.e. ladders are edge-antimagic total graphs. \ \\ \ \\
If we partition the vertex set into two, i.e. $u_i$ and $v_i$, $1\leq i\leq n$, each vertex set is sharp ordered which implies that the graph is a weak ordered graph. Therefore ladders are super weak ordered TAT graphs since they are both edge-antimagic total graphs and vertex-antimagic total graphs.
\end{proof}
The following theorem shows that prism graphs are TAT graphs.
\begin{thm}
The prism graph $C_n\times P_2$ is a super TAT graph for every $n>2$.
\end{thm}
\begin{proof}
Denote the vertices of prism graph $C_n\times P_2$ by $\{u_iv_i:1\leq i\leq n\}$ and the edges by $\{u_iu_{i+1},v_iv_{i+1}:1\leq i\leq n\}\cup \{u_iv_i:1\leq i\leq n\}$. Let $g$ be the labeling on $C_n\times P_2$. Define the labeling $g$ on the vertices as follows:
Whenever $i=1$, $g(u_1)=1$ and $g(v_1)=n+1$.
Also,
$$g(u_i)=\left\{\begin{array}{ll}
2(i-1), & 2\leq i\leq \lfloor\frac{n}{2} \rfloor +1\\ \ \\
2(n-i)+3, & \lfloor\frac{n}{2} \rfloor+2\leq i\leq n
\end{array}\right.$$
$$g(v_i)=\left\{\begin{array}{ll}
n+2(i-1), & 2\leq i\leq \lfloor\frac{n}{2} \rfloor +1\\ \ \\
3(n+1)-2i, & \lfloor\frac{n}{2} \rfloor+2\leq i\leq n
\end{array}\right.$$
From the labeling above, we have the labeling $g$ to be super.\\
Moreover, define the labeling $g$ on the edge set as follows:
$$g(u_iu_{i+1})=\left\{\begin{array}{ll}
2(n+i)-1, & 1\leq i\leq \lceil\frac{n}{2} \rceil\\ \ \\
2(2n+1-i, & \lceil\frac{n}{2} \rceil+1\leq i\leq n
\end{array}\right.$$
$$g(v_iv_{i+1})=\left\{\begin{array}{ll}
2(2n+i)-1, & 1\leq i\leq \lceil\frac{n}{2} \rceil\\ \ \\
2(3n-i+1), & \lceil\frac{n}{2} \rceil+1\leq i\leq n
\end{array}\right.$$
$$g(u_iv_i)=\left\{\begin{array}{ll}
2(2n-1)+3, & 2\leq i\leq \lfloor\frac{n}{2} \rfloor +1\\ \ \\
2(n-1+i), & \lfloor\frac{n}{2} \rfloor+2\leq i\leq n
\end{array}\right.$$
For $i=1$, we have $g(u_1v_1)=4n$.
Next, we consider the vertex-weights and show that they are pairwise distinct. The vertex-weights of $C_n\times P_2$ is as follows:
$$wt_g(u_i)=\left\{\begin{array}{ll}
W_7\\ \ \\
W_8
\end{array}\right.$$
where $$W_ 7=4(2n+i)+(t-1)$$ with the following conditions:
whenever $i=1$, $t=1$,
whenever $2\leq i\leq \lceil \frac{n}{2}\rceil$, $t=-2$.
and $$W_8=4(3n-i)+(t+5)$$ with the following conditions:
when $i=\frac{n}{2}+1, n$ even, $t=-1$,
when $i=\lceil \frac{n}{2}\rceil +1, n$ odd, $t=1$,
when $\lceil \frac{n}{2} \rceil +2\leq i\leq n$, $t=2$.
Also, $$wt_g(v_i)=\left\{\begin{array}{ll}
W_9\\ \ \\
W_{10}
\end{array}\right.$$
where $$W_9=13n+4i+(t-1)$$ with the following conditions:
$t=1$ whenever $i=1$ and $t=-2$ whenever $2\leq i\leq \lceil \frac{n}{2} \rceil$.\\
Also, $$W_{10}= 17n-4i+(t+5)$$ with the following conditions:
$t=-1$ whenever $i=\frac{n}{2}+1$ and $n$ even,
$t=1$ for $i=\lceil \frac{n}{2}\rceil +1$ and $n$ odd,
$t=2$ whenever $\lceil \frac{n}{2}\rceil +2 \leq i\leq n$.\\
In view of the above, the set $\{W_i\}_{i=7}^{10}$ are pairwise distinct which shows that the labeling $g$ is a vertex-antimagic total.\\
For the edge-weights under the labeling $g$, we have the following:\\
$$wt_g(u_iu_{i+1})=\left\{\begin{array}{ll}
W_{11}\\ \ \\
W_{12}
\end{array}\right.$$
where $$W_{11}=2(n+3i)+(t-5)$$ satisfying the following:
$t=3$ for $i=1$,
$t=2$ for $2\leq i\leq \lfloor\frac{n}{2}\rfloor$,
$t=1$ for $i=\lceil \frac{n}{2} \rceil$, $n$ odd.\\
Also, $$W_{12}=2(4(n+1)-3i)+t$$ satisfying the following conditions:
$t=-3$ whenever $i=\frac{n}{2}+1, n$ even,
$t=-2$ whenever $\lfloor \frac{n}{2}\rfloor +2\leq i\leq n$.
$$wt_g(v_iv_{i+1})=\left\{\begin{array}{ll}
W_{13}\\ \ \\
W_{14}\end{array}\right.$$
Where $$W_{13}=6(n+i)+(t-5)$$ with the following conditions:
for $i=1$, $t=3$,
for $2\leq i\leq \lfloor \frac{n}{2}\rfloor$, $t=3$,
for $i=\lceil \frac{n}{2}\rceil$ and $n$ odd, $t=1$.
Furthermore, $$W_{14}=2(3(2n-1)+4)+t$$ satisfying the followings:
$t=-3$ whenever $i=\frac{n}{2}+1$ and $n$ even,
$t=-2$ whenever $\lfloor \frac{n}{2}\rfloor +2\leq i\leq n$.\\
Moreover, $$wt_g(u_iv_i)=\left\{\begin{array}{ll}
5n+2i+t, & t=0 \ \ for\ \ i=1\ \ and\ \ t=-1 \ \ for\ \ 2\leq i\leq \lfloor \frac{n}{2} \rfloor+1\\ \ \\
7n+2(2-i), & \lfloor\frac{n}{2} \rfloor+2\leq i\leq n
\end{array}\right.$$
The edge-weights of the edges in $C_n\times P_2$ under the labeling $g$ are all different which implies that the labeling is edge-antimagic total.
Therefore the labeling $g$ is TAT labeling since it is both vertex-antimagic total and edge-antimagic total. Thus the prism graphs $C_n\times P_2$ is a totally antimagic total graph.
\end{proof}
Next, we give the TAT labeling of generalised Petersen graph.
\begin{thm}
The generalised Petersen graph $P(n,m), n\geq 3$ and $1\leq m\leq \lfloor \frac{n-1}{2}\rfloor$ is a super TAT graph.
\end{thm}
\begin{proof}
Denotes the vertices of the graph $P(n,m)$ by the symbols $u_iv_i, 1\leq i\leq n$. Let $g$ be a labeling on the graph $P(n,m)$ defined in the following way:
$$g(u_i)=i, 1\leq i\leq n$$
$$g(v_i)=\left\{\begin{array}{ll}
n+1, & 1\leq i\leq n-1\\ \ \\
2(3n-i+1), & j= n
\end{array}\right.$$
Also, define labeling $g$ on the edges as follows:
$$g(u_iu_{i+1})=3n-(i-1), 1\leq i\leq n$$
where $i$ is calculated modulo $n$.
$$g(u_iv_i)=\left\{\begin{array}{ll}
3n+1, & i=1\\ \ \\
4n-(i-2), & 2\leq i\leq n
\end{array}\right.$$
$$g(v_iv_{i+m})=5n-(i-1), 1\leq i\leq n$$
with indices $i+m$ taken modulo $n$.\\
It is easy to see that the labeling $g$ is super. Also, the vertex-weights under the labeling $g$ is as follows:
$$wt_g(u_i)=\left\{\begin{array}{ll}
10n+2-i, & i=1\\ \ \\
5(2n+1)-2i, & 2\leq i\leq n
\end{array}\right.$$
$$g(v_iv_{i+1})=\left\{\begin{array}{lll}
14n+2-m, & i=1\\ \ \\
15n-2i+5-t, &t=m+1 \ \ for \ \ 2\leq i\leq \lfloor\frac{n}{2} \rfloor \ \\
& and \ \ t=-m\ \ for \ \ \lfloor \frac{n}{2}\rfloor+1\leq i\leq n-1\\ \ \\
12n+m+5, & i=n
\end{array}\right.$$
In view of the above labeling, the vertex-weights are pairwise distinct which implies that the labeling $g$ is super vertex-antimagic total.\\
Next we consider the edge-weights of the graph $P(n,m)$ as follows:
$$wt_g(u_iu_{i+1})=\left\{\begin{array}{ll}
3n+i+2, & 1\leq i\leq n-1\\ \ \\
3n+2, & i=n
\end{array}\right.$$
$$wt_g(v_iv_{i+m})=7n+3-t$$ with the following conditions:
$t=-(k+j)$ whenever $1\leq i\leq \lfloor \frac{n}{2}\rfloor$,
$t=k-j+1$ whenever $\lfloor \frac{n}{2}\rfloor +1\leq i\leq n-1$,
$t=k+1$ whenever $i=n$.
Also, $$wt_g(u_iv_i)=\left\{\begin{array}{lll}
4(n+1), & i=1\\ \ \\
5n+3+i, & 2\leq i\leq n-1\\ \ \\
5n+3, & i=n
\end{array}\right.$$
This means that all the edges in $P(n,m)$ have different edge-weights which implies that the graph $P(n,m)$ is an edge-antimagic total graph. Therefore the generalised Petersen graph $P(n,m)$ is a super totally antimagic total graph since it admits both vertex-antimagic total labeling and edge-antimagic total labeling.
\end{proof}
\section{Totally antimagic total labeling of chain graphs}
In this section, we prove that the chain graphs of totally antimagic total graphs is totally antimagic total graph. Suppose now that the graphs $B_1,B_2,...,B_m$ are blocks and that for any $i\in \{1,2,...,m\}$, $B_i$ and $B_{i+1}$ have a vertex in common in such a way that the block-cut point is a path. The graph $G$ obtained by the concatenation will be called a \emph{chain} graph.
\begin{thm}\label{RT}
The chain graph $G$ obtained by concatenation of totally antimagic total graphs is a TAT graph.
\end{thm}
\begin{proof}
Let $G$ denotes that chain graph with blocks $G_i, 1\leq i\leq m$. Let $g_i,1\leq i\leq m$ be a TAT labeling of $G_i$. Also, let $p=|V(G_i)|$ and $$g_i:V(G_i)\cup E(G_i)\rightarrow \{1,2,...,|V(G_i)|+|E(G_i)|\}$$ such that $wt_{g_i}(v)\neq wt_{g_i}(u)$ for all $u,v\in V(G_i),u\neq v$ and $wt_{g_i}(e)\neq wt_{g_i}(h)$ for all $e,h\in E(G_i), e\neq h$.\\
Consider the cut-vertex between $G_i$ and $G_{i+1}$ as the concatenation of the vertex labeled with $1\in V(G_i)$ and vertex labeled with $p\in V(G_{i+1})$ denoted by $r_i$. Without loss of generality we may assume that $r_{i+1}>r_i$. Define a labeling for $G$ such that
$$f(x)=\left\{\begin{array}{ll}
g_1(x), & x\in V(G_1)\\ \ \\
g_i(x)+\disp{\sum_{j=1}^{i-1}|V(G_j)|+\sum_{j=1}^{i-1}|E(G_j)|-(i-2)}, & x\in V(G_i),i=1,2,...,m
\end{array}\right.$$
and $$f(e)=\left\{\begin{array}{ll}
g_1(e), & e\in E(G_1)\\ \ \\
g_i(e)+\disp {\sum_{j=1}^{i=1}|V(G_j)|+\sum_{j=1}^{i-1}|E(G_j)|-(i-1)}, & e\in E(G_i),i=1,2,...,m
\end{array}\right.$$
It is easy to see that $f$ is a total antimagic total labeling of $G$. For the edge-weights under the labeling $f$, we obtain
$$wt_f(e)=\left\{\begin{array}{ll}
wt_{g_1}(e), & e\in E(G_1)\\ \ \\
wt_{g_i}(e)+3(\disp {\sum_{j=1}^{i-1}|V(G_j)|+\sum_{j=1}^{i-1}|E(G_j)|})-1, & e\in E(G_i),i=1,2,...,m
\end{array}\right.$$
Also, the edge-weights for the edges in $G_{i+1}$ incidents with the cut-vertex $r_i$, under the labeling $f$ is as follows:
$$wt_f(e)=\left\{\begin{array}{ll}
wt_{g_1}(e), & e\in E(G_1)\\ \ \\
wt{g_i}(e)+2(\disp {\sum_{j=1}^{i-1}|V(G_j)|+\sum_{j=1}^{i-1}|E(G_j)|})-p+r_i-r_1, & e\in E(G_i),i=1,2,...,m
\end{array}\right.$$
As $g_i,i=1,2,...,m$ is edge-antimagic labeling, the edge-weights of all edges in $G$ under the labeling $f$ are pairwise distinct.\\
The maximum edge-weight of an edge $e\in E(G_i),1\leq i\leq m$ is
$$wt_f^{max}(e)\leq 3(\disp {\sum_{j=1}^{i}|V(G_j)|+\sum_{j=1}^{i-1}|E(G_j)|})-|V(G_1)|+|E(G_1)|, e\in E(G_i),i=1,2,...,m.$$
Thus $f$ is an edge-antimagic labeling of $G$.\\
For the vertex-weight under labeling $f$, we get
$$wt_f(v)=\left\{\begin{array}{ll}
wt_{g_1}(v), & v\in V(G_1)\\ \ \\
wt{g_i}(v)+(deg(v)_+1)(\disp {\sum_{j=1}^{i-1}|V(G_j)|+\sum_{j=1}^{i-1}|E(G_j)|}-1)+1, & e\in E(G_i),i=1,2,...,m
\end{array}\right.$$
For the cut-vertices $r_i$, the weights under labeling $f$, for cycle-like structure, we get
$$wt_f(r_i)=wt_{g_1}(v_p)+wt_{g_1}(v_1)+deg(v)(\disp {\sum_{j=1}^{i-1}|V(G_j)|+|E(G_j)|})+$$
$$(deg(v)+1)(\disp{\sum_{j=1}^{i-2}|V(G_j)+E(G_j)|})-deg(r_i)-p.$$
where $v_p$ and $v_1$ are the vertices in $G_1$ labeled with $p$ and $1$ respectively.\\
Also, the cut-vertices weights under labeling $f$ for path-like structure is as follows:
$$wt_f(r_i)=wt_{g_1}(v_p)+wt_{g_1}(v_1)+(deg(v)+1)(2(\sum_{j=1}^{i-1}|V(G_j)|+|E(G_j)|)$$$$+|V(G_{j-1})|+|E(G_{j-1})|)-deg(r_i)-p$$
In view of the above labeling, the chain graph $G$ is a TAT graph.
\end{proof}
\begin{cor}
The tree graph formed from the concatenation of paths is a TAT graph.
\end{cor}
\begin{proof}
This follows from directly from theorem \ref{RT}.
\end{proof}
\emph{Conjecture}: All trees are TAT graph.
|
1,116,691,499,354 | arxiv | \section{Introduction}
Power electronic loads have found a wider application in power system networks especially after their advancement in the late 1900s. Many loads require essential power electronic converters for stage conversion. Some common examples of power electronic loads include uninterrupted power supply (UPS) devices, personal computers, laptops, electric vehicle chargers, etc. These non-linear loads contribute to non-linear sinusoidal currents. The non-sinusoidal currents, when passing through network impedance, create a non-sinusoidal voltage drop \cite{grady2012understanding}. The non-sinusoidal voltage and current components are integer multiples of the fundamental component called \textit{``harmonics"}. The deterioration of the supply voltage creates stress on the electrical equipment and can potentially damage it, resulting in increased operating costs and downtime \cite{Fuchs2008}.
Increased voltage and current harmonics are found to have a direct relationship to premature aging and degradation of transformers. Initial transformer designs were made considering conventional load models, i.e., Constant Impedance (Z), Constant Current (I), and Constant Power (P) or \textit{``ZIP"} models, that would operate at fundamental 60Hz or 50Hz frequency \cite{McLorn2017,bokhari2013experimental}. Under the increased penetration of non-linear loads, the design of power transformers needs to be reassessed to ensure proper and safe operation. Increased non-linear loads increase the transformer losses due to overheating of the core, creating a larger derating factor \cite{Lavers1999,Masoum2008}.
The problem of harmonics is more evident with customers on the low-voltage end. The common household equipment includes but not limited to desktops, laptops, LED lamps, variable speed drives, solar panels, etc. To compound the challenges, as more and more electric vehicles come to the market, they rely majorly on at-home charging that produces a large fraction of non-linear voltage and current. Certain power electronic devices like VFD's contribute more $3^{rd}$ harmonics, if they are not properly compensated it would lead to additional losses and loss-of-life for the transformer. The addition of harmonics has an effect on transformer protection as well. Addition of the $5^{th}$ harmonic needs to be compensated below a certain threshold before the protection relays can be engaged. The distorted harmonic waveforms results in loss of essential information for protection, this might result in the protection devices operating slower \cite{jain2011harmonics}.
Currently, there is a gap in high-fidelity load models that can capture the typical characteristics of non-linear models, i.e., the cross-coupling effect of voltages and current. A harmonic-rich current/voltage dataset is essential to understand the effect on transformer losses, heating, etc. \textit{``ZIP"} based harmonic load models suffer from a lack of enhanced harmonic spectrum that can be observed through the operation of various non-linear devices \cite{collin_component-based_2010}. Real-world field data is not publicly available to perform such analysis. Methods relying on laboratory setups to develop such datasets fail to capture the effect on other nonlinear load currents.
Therefore, in this paper, (i) we perform an analysis of residential transformer heating and losses encountered due to the presence of non-linear power electronic residential loads using PSCAD/EMTdc. Detailed power electronic models are developed to create harmonic rich datasets to entail their effect on transformer operation; (ii) Different loading scenarios on the transformer are assessed with increasing PV penetration to understand its effect on THD(\%), eddy current losses, and the subsequent impact on transformer derating.
The rest of the paper has been organized in the following way. Section \ref{Modelling} discusses the modeling approach for the study. Section \ref{sec:trans} discusses the calculation of eddy current losses due to nonlinear harmonic current in transformers. Section \ref{Results} discusses the results, and Section \ref{Conclusion} concludes the paper with major findings and proposed future enhancements.
\section{Modeling} \label{Modelling}
\subsection{System Description}
Usually, the residential customers are supplied through a single split-phase connection in the USA. In this work, we are modeling 5 houses that have been connected to a 7.2 kV distribution transformer, as shown in Fig. \ref{fig:house_combo}.
Each home is comprised of four power electronic load combinations shown in Table \ref{tab:load_models} and discussed in detail in \cite{Ankit2022}.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{house_combo.png}
\caption{Schematic representation of the simulation setup in PSCAD.}
\label{fig:house_combo}
\end{figure}
\begin{table}[H]
\renewcommand{\arraystretch}{1.2}
\caption{Power electronics-based load models representing house appliances }
\label{tab:load_models}
\centering
\begin{tabular}{l l}
\hline
Load model & House appliances \\
\hline
Rectifier + Buck DC-DC converter & Desktop, home entertainment\\
Rectifier + Flyback DC--DC converter & Laptop charger \\
VFD + Induction motor & HVAC, washer, dryer \\
Boost converter + inverter & PV system, EV charger\\
\hline
\end{tabular}
\vspace{-3mm}
\end{table}
\subsection{Data Generation}
The power electronic load combinations were modeled using PSCAD/EMTdc. The steady-state values of current were recorded at the secondary of the distribution transformer as shown in Fig. \ref{fig:house_combo}. A similar process was repeated for several load combinations, and a few scenarios were selected that draw significantly harmonic-rich current for further analysis as discussed in Section \ref{Results}.
\section{Transformer Degradation Analysis due to Harmonics} \label{sec:trans}
Most power electronic loads are fitted with a single or distributed capacitor at the terminal that helps to maintain a constant dc voltage along with the parasitic inductors. Since there is a periodic change in the load impedance, the current waveform varies from the supplied voltage waveform. The non-sinusoidal current can be represented as a sum of the fundamental and integer multiple of the fundamental \textit{``harmonics"}.
\subsection{Fast Fourier Transform (FFT)} \label{FFT}
One can transform a given sequence in time into its respective frequency components using Discrete Fourier Transform (DFT) \cite{He2016}. FFT is useful for performing the DFT of a sequence. FFT performs the computation of the DFT matrix as a product of sparse factors. The DFT for such a sequence can be given as (\ref{Eqn:DFT}),
\begin{equation}
X[k] = \sum_{n= 0}^{N-1}x[n]e^{-j2\pi kn /N}
\label{Eqn:DFT}
\end{equation}
\noindent where \emph{N} is the length of the signal. Since the sampling frequency of the signal is $20kHz$, the maximum represented frequencies are half of the sampling frequency. We try to capture all the representative frequencies in that range. The harmonic components in the measured current are a function of the fundamental $60Hz$ frequency. A frequency scan is performed to identify the magnitude of the current harmonics $I_h$.
Since the measured signal is not an integer multiple, the endpoints of the frequency spectrum are discontinuous. FFT produces a smeared spectral version of the original signal where the energy of one frequency leaks into adjacent frequencies. This phenomenon is known as spectral leakage. To get the best estimate of the current harmonics, we perform a scan of the frequencies adjacent to the integer harmonic frequency. Practices like windowing are utilized to reduce the effect of the non-integer frequencies, but this was not considered as a part of our work.
\subsection{Eddy Current Losses} \label{sec:loss_eddy}
In a residential set-up, most of the losses are due to heating $I^2R$ losses. A residential home consists of both linear and non-linear loads; in this study, we majorly focus on the increase in non-linear residential loads that create harmonics that inadvertently contribute to more losses. The effect is more tremendous at the grid-edge locations where a lot of power electronic loads are connected, for example, the distribution service transformers.
The losses occurring in a transformer can be divided into two categories (\ref{eqn:ploss}), (i) \textit{No-Load losses} and (ii) \textit{Load losses}. In this paper, we study the losses due to non-linear loads. When current flows through a conductor, it generates heat which is either utilized or lost in the surrounding environment.
\begin{equation}
Tr_{Loss} = Tr_{NL} + Tr_{LL} \label{eqn:ploss}
\end{equation}
The load losses ($Tr_{LL}$) can be further subdivided into a summation of eddy current losses ($Tr_{EC}$) and structural stray losses ($Tr_{ST})$.
The eddy current component can be written as (\ref{eqn:eddy}),
\begin{equation}
Tr_{EC} = \sum_{h=1}^{h_{max}}I_h^2R_{h} \label{eqn:eddy}
\end{equation}
\noindent The winding losses increase as a square of the harmonic current component ($I_h$), $R_h$ is the effective resistance comprising of the non-frequency dc component and the resistance that varies with harmonic content.
When harmonic current flows through the conductive materials of the transformer, it leads to a variation in temperature.
From Newton's law of cooling (\ref{eqn:Newton}),
\begin{equation}
P\delta t = mc\delta\theta + \alpha \theta A \delta t \label{eqn:Newton}
\end{equation}
\noindent where $P$ is the $I^2R$ losses for material, $m$ is the mass of the material, $c$ is the specific heat capacity of the material, $\delta \theta$ rise of temperature above ambient temperature for time $\delta t$, $A$ is the surface area of the material and $\alpha$ is the emissivity factor.
The change in temperature can be written as (\ref{eqn:temp}) \cite{Tony2012},
\begin{equation}
\theta (t) = \theta_{final}[1-e^{-t/\tau}] \label{eqn:temp}
\end{equation}
In steady state (\ref{eqn:Newton}), $mc\delta\theta = 0$ and (\ref{eqn:Newton}) can be re-written as (\ref{eqn:new_eqn}),
\begin{equation}
\begin{array}{l}
P \delta t = \alpha \theta A \delta t, \\
\theta = \frac{P}{\alpha A}
\end{array} \label{eqn:new_eqn}
\end{equation}
\noindent From (\ref{eqn:new_eqn}), we can say that $\theta \propto P \propto I^2R_h$, as more non-linear current passes through the transformer, the ambient temperature of the material changes, resulting in more losses.
\begin{equation}
R_h = R_{DC}+h^2P_{EC-R}
\end{equation}
\noindent where $R_{DC}$ is the dc-winding resistance at $h^{th}$ harmonic and $P_{EC-R}$ is the winding eddy current loss factor, that ranges between 0.01 in low voltage transformers to 0.10 for substation transformers. For our study, we consider $P_{EC-R}$ as 0.05.
By replacing $R_h$ in (\ref{eqn:eddy}), we get
\begin{equation}
Tr_{EC} = I_1^2R_{DC}+\sum_{h=3,5,7,...}^{h_{max}}I_h^2h^2P_{EC-R} \label{eqn:eddy_loss}
\end{equation}
\noindent The first term of (\ref{eqn:eddy_loss}) $I_1^2R_{DC}$ is the non-frequency dependant part, and $I_h^2h^2P_{EC-R}$ reflects the frequency dependant part of the transformer eddy current losses. Thus, the harmonic driven transformer eddy current losses can be summarized as (\ref{eqn:eddy_final}),
\begin{equation}
Tr_{EC} = P_{EC-R}\sum_{h=1}^{h_{max}}I^2h^2 \label{eqn:eddy_final}
\end{equation}
\subsection{Harmonic Loss Factor \& Transformer Derating} \label{k-factor}
Harmonic loss factor ($F_{HL}$) is defined as the ratio of the total loss due to eddy current due to harmonics and the winding current losses in the absence of harmonics \cite{IEEEC57}. It is expressed as (\ref{eqn:f-hl}),
\begin{equation}
F_{HL} = \frac{\sum_{h=1}^{h_{max}}I_h^2 h^2}{\sum_{h=1}^{h_{max}}I_h^2}
\label{eqn:f-hl}
\end{equation}
To reduce the loss-of-life of a transformer due to an increased non-linear current, they are usually derated (i.e., reduced transformer loading). The derating \% helps to understand the transformer operational capability below its rating to prolong its duration.
\section{Results and Discussion} \label{Results}
\subsection{Simulation Setup}
To understand the effect of different loading scenarios on transformers, the simulation setup described in Fig. \ref{fig:house_combo} is used. A total of 5 scenarios are constructed with different power electronic load combinations to analyze the impact on the transformer, as shown in Table \ref{tab:scenario}. All these 5 scenarios are assumed to represent the peak loading condition for a given transformer with increasing solar PV units. Scenarios 1, 2, and 3 are evening peaking cases where solar generation. Whereas scenarios 4 and 5 have solar generation high enough to cause a reverse power peak during daytime when the load is low. It is assumed that 1 PV unit generates 3.5 kW and 1.5 kW during the daytime and evening, respectively. The exact loading condition (mix of VFD, laptop, desktop, and PV) of each of the 5 houses connected to a service transformer in each scenario is shown in Fig. \ref{fig:cases}.
\begin{table}[]
\renewcommand{\arraystretch}{1.1}
\caption{Scenarios of different load combinations with increasing PV penetration}
\label{tab:scenario}
\begin{tabular}{cccccc}
\hline
\multirow{2}{*}{Scenarios} & \multirow{2}{*}{PV units} & Peak Load & Total PV & Total other & \multirow{2}{*}{Net load} \\
& & Time & generation & load & \\ \hline
1 & 0 & evening & 0 kW & 9.5 kW & 9.5 kW \\
2 & 1 & evening & 1.5 kW & 9.5 kW & 8 kW \\
3 & 2 & evening & 3 kW & 9.5 kW & 6.5 kW \\
4 & 3 & day & 10.5 kW & 2.5 kW & -8 kW \\
5 & 4 & day & 14 kW & 2.5 kW & -11.5 kW \\ \hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{cases.jpg}
\caption{Load combination for 5 houses corresponding to a different scenario, each scenario represents the peak load condition with a given PV penetration.}
\label{fig:cases}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{trans_current.png}
\caption{Equivalent transformer secondary current for different scenarios.}
\label{fig:trans_curr}
\end{figure}
\subsection{Impact of increasing PV on Transformer Current Harmonics and Eddy Losses}
The current waveforms drawn by the transformer secondary in all 5 scenarios are shown in Fig. \ref{fig:trans_curr}. The reverse current can be noticed in scenarios 3 and 4 due to high solar PV generation. These waveforms were analyzed to understand the harmonic contribution in each scenario. The harmonic contents of the measured current signals were extracted using FFT as described in Section \ref{FFT}. The THD(\%) for each of the scenarios was calculated using (\ref{eqn:thd}),
\begin{equation}
THD = \frac{\sqrt{\sum_{h=2}^{h_{max}}I_h^2}}{I_1} \label{eqn:thd}
\end{equation}
Where $I_h$ denotes the $h^{th}$ harmonic current magnitude. Eddy current losses calculations are based on the discussion in Section \ref{sec:loss_eddy}. THD(\%) and corresponding eddy losses for each scenario are shown in Fig. \ref{fig:eddy_loss}. It is observed in scenarios 1, 2, and 3 that increasing solar PV units cause more harmonic distortion in transformer currents. On the contrary, in scenarios 4 and 5, it is observed that the addition of PV generation decreases THD. To understand this pattern, we need to look at the frequency spread of $1^{st}$, $3^{rd}$, and $5^{th}$ harmonic of the current as shown in Fig. \ref{fig:freq}. Observing scenarios 1, 2, and 3, we find that the increasing PV generation reduces the net fundamental current magnitude as expected. However, the $3^{rd}$ harmonic current magnitude remains the same in scenarios 1, 2, and 3, as shown in Fig. \ref{fig:freq}. It can be inferred that the PV inverter is primarily compensating for the fundamental component of the load currents, not the $3^{rd}$ harmonic. This leads to an increased \% of $3^{rd}$ harmonic in scenario 3, as shown in Fig. \ref{fig:freq_norm}, resulting in high THD(\%).
On the other hand, scenarios 4 and 5 have very high PV generation but a much lower amount of other power electronics load compared to scenarios 1-3. Therefore, the net current is mainly composed of PV current. Since PV units are mandated to maintain less than 5\% THD, they come equipped with harmonic filters by the vendors, as discussed in \cite{Ankit2022}. Therefore, in scenarios 4 and 5, harmonic distortions are the lowest.
Transformer eddy losses tend to follow the current THD \% and are shown in Fig. \ref{fig:eddy_loss}. It can be seen that scenario 3 has the highest eddy losses, close to 30\%.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{eddy_loss_4scenario.png}
\caption{Variation of THD(\%) along with eddy current losses for different load combinations and PV penetration.}
\label{fig:eddy_loss}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{harmonic_contribution.png}
\caption{The harmonic frequency spectrum of the secondary transformer under various load combinations and PV penetrations.}
\label{fig:freq}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{harmonic_contribution_norm.png}
\caption{Variation of normalized frequency spectrum of $3^{rd}$ \& $5^{th}$ harmonic with respect to fundamental current magnitude with different load and PV penetration scenarios.}
\label{fig:freq_norm}
\end{figure}
\subsection{Impact on Transformer Derating}
All 5 scenarios represent the peak loading situation for the given PV penetration. It is important for the transformer derating analysis as it is usually performed in a full-loading condition. Note that the up to 2 PV unit penetration (scenario 1-3) peak loading is assumed to occur during the evening when other loads are high. Whereas, for higher penetration (scenarios 4-5), solar PV generation can create its own reverse power peak during the daytime.
The harmonic loss factor helps us to understand the \% at which the transformers should be operated to prolong its life. The derating \% for the different scenarios is shown in the last column of the Table \ref{tab:scenario}. The worst derating is observed in scenario 3, where the transformer operates at 75.88\% of its rated capacity resulting in significant loss of life. If the penetration of power electronic loads are further increased the transformers would need to be further derated for their operation.
\begin{table}[]
\centering
\caption {The overall impact of increasing PV penetration on transformer degradation in terms of THD, eddy current losses, and transformer derating}
\label{tab:results}
\begin{tabular}{ccccc}
\hline
Scenarios & THD (\%) & $Tr_{EC}$ (\%) & Derating (\%) \\ \hline
1 & 18.30 & 6.89 & 85.59 \\
2 & 26.51 & 8.66 & 78.59 \\
3 & 29.05 & 9.41 & 75.88 \\
4 & 5.55 & 5.22 & 98.01 \\
5 & 3.52 & 5.11 & 98.95 \\ \hline
\end{tabular}
\vspace{-4mm}
\end{table}
Total impact on the transformer in terms of THD(\%), eddy losses, and derating are listed in Table \ref{tab:results}. Overall, if PV units come equipped with a filter as mandated by the standards, their individual effect on the transformer loading is positive. Therefore, in the presence of low power-electronic loads (scenarios 4 \& 5), increasing PV penetration has a positive impact on transformer degradation. However, in the presence of high power-electronic loads (scenarios 1-3), increasing PV penetration may have adverse impacts on transformer degradation since it is not able to compensate for the $3^{rd}$ harmonic consumed by the loads such as VFDs, thus increasing $3^{rd}$ harmonic contribution.
\section{Conclusion} \label{Conclusion}
The proposed work represents the challenges faced by a low-voltage distribution transformer due to the high penetration of a power electronic-dominated residential infrastructure via EMTP simulations. By operating principle, transformers are linear devices, and the addition of non-linear power electronic loads makes their operation non-linear. The harmonic load currents increase the losses in a transformer. The non-linear current causes a rise in temperature that affects the effective resistance of the transformer, as discussed in the Section \ref{sec:trans}. The eddy current losses depend on the magnitude of the harmonic current, and higher THD(\%) contributes to more eddy current losses. Integration of PV resources to compensate for the transformer loading had an adverse effect with increased levels of $3^{rd}$ harmonic. Such scenario's are particularly visible for scenario 2 \& 3 where the load of the network is mostly compensated by PV generation. Although, under low power electronic loading conditions, the addition of PV helped to reduce THD(\%) in the transformer.
To have a better estimate of the transformer performance, detailed power electronic models are necessary. However, the scalability of inverter models in different simulation tools beyond a certain number is infeasible. Thus developing mathematical models of harmonic loads via frequency coupled matrix (FCM) will be considered in future work to generate high-fidelity time-series data for further analysis.
\bibliographystyle{IEEEtran}
|
1,116,691,499,355 | arxiv | \section{Introduction}
The study of topological defects has wide applicability in many areas of
physics. In the cosmological arena, defects have been put forward as a possible
mechanism for structure formation~\cite{VS}, and while recent work on global
defects~\cite{ABR,PST} indicates that these were probably not responsible for
structure formation, the intriguing discovery of a non-gaussian signature in
the microwave background~\cite{FMG} leaves open the possibility that
defects were around at some point in the evolution of the universe.
Of all the topological defects, domain walls are the most deceptively simple to
study. They correspond to solitons in 1+1-dimensions, which are extended in two
spatial directions to form a wall structure. Because they depend on only one
spatial coordinate, the distance from the wall, the solutions in the absence of
gravity can often be written in closed analytic form, and for certain
potentials the models are completely integrable and the defects have the full
interpretation of solitary travelling waves. In the presence of gravity
however, the situation changes, gravity destroys the integrability of the
theory, and also, except in a perturbative sense, the analytic nature of the
solutions. Fortunately, since the walls correspond approximately to
hypersurfaces in spacetime, there is a well defined way of analysing their
gravity using Israel's thin wall formalism \cite{Israel}, in which the wall is
approximated by an infinitesimally thin hypersurface, and most of the
literature concerning domain wall gravity uses this method.
The domain wall is a rather interesting object gravitationally: unlike almost
all of the other topological defects (the exception being the global
string~\cite{R}), its metric is not static but time-dependent~\cite{Vil,IS},
having a de Sitter-like expansion in the plane of the wall. Observers
experience a repulsion from the domain wall, and there is an horizon at finite
proper distance from the defect's core. This horizon can be interpreted as a
facet of the choice of coordinates, which usually use the flat space wall
solution as a starting point, and impose planar symmetry on the domain wall
spacetime. However, it is possible to use a different set of coordinates
\cite{IS,GWG}\ in which the wall has the appearance of a bubble which contracts
in from infinite radius to some minimum radius, then re-expands, undergoing
uniform acceleration from the origin. The `horizon' is then simply the
lightcone of the origin in these coordinates, and is somewhat similar to the
horizon of Rindler spacetime.
The crucial physical difference of a domain wall spacetime, as opposed to that
of the local cosmic string or monopole, is the presence of this cosmological
horizon, which introduces a second length scale into the system. Ordinarily, a
defect possesses one length scale, $w$, which is a measure of its thickness.
However, the distance to the event horizon of the domain wall gives another
length scale,
$z_{\rm h}$, which can be compared to $w$. Since these lengths are given in
terms of the coupling constants of the theory, taking a thin wall limit turns
out to be a very artificial construction in terms of these underlying
parameters, and the issue of the self-gravity of thick walls becomes more
pertinent. After the original work by Vilenkin, Ipser and
Sikivie~\cite{Vil,IS}, attempts focussed on trying to find a perturbative
expansion in the wall thickness~\cite{RD,larryw} both for the purpose of
discovering the motion of the wall, as well as verifying that the hypersurface
formalism was a good approximation to the true gravitational field. These
results backed up the thin wall approximation, the main difference being the
presence of a sub-dominant tension transverse to the wall.
With the suggestion of Hill, Schramm and Fry~\cite{HSF} of a late time phase
transition with thick domain walls, there was some effort at finding exact
thick solutions~\cite{Goetz,Mukh}, see also \cite{Tom,TI}. However, such
defects were supposed to be thick by virtue of the low temperature of the phase
transition responsible for defect formation. The suggestion that Planck scale
topological defects could be responsible for inflation \cite{Linde,Vil2} then
reopened the issue of thick domain walls. `Thick' then means thick compared to
their natural de Sitter horizon, and therefore this appears to be a very strong
gravity situation. The issue of whether a domain wall can survive in a
Friedmann-Robertson-Walker (FRW) universe with horizon size comparable to wall
thickness has been analysed in some detail for the case of no gravitational
back reaction of the wall on the FRW background~\cite{BV1,BBGH}, and in the
case of Euclidean instantons on a de Sitter background including self-gravity
\cite{BV2,BGV};
however, to our knowledge, a systematic analysis of the strong self-gravity
of thick domain walls with an arbitrary field theory potential has
not been carried out.
In this paper we perform a detailed analysis of the strong gravity of thick
domain walls. With a combination of analytical and numerical results, we map
out the parameter space in which a wall-like solution exists. Although we focus
on two main examples of the sine-Gordon and $\lambda\Phi^4$ wall, we also
illustrate to what extent, and for what potentials, the results will also hold
in general. We begin in the next section by setting out the general formalism
for a thick domain wall, deriving the metric and Einstein equations, as well as
defining what we mean by a `wall' solution. We also show how to generalise a
coordinate transformation which takes a planar domain wall spacetime with an
horizon into a flat spacetime with an accelerating wall to the case of walls
with finite thickness. The solution outside the wall horizon is shown to be
related to an FRW cosmology with a slowly rolling scalar field.
In the following section we derive analytic results for the thick wall. It
turns out that there are two possibilities for the scalar field from which the
`wall' is constituted: it can either be a planar wall or a de Sitter solution.
Since the de Sitter solution is always possible, but not necessarily stable, we
derive analytic bounds on the gravitational coupling strength of the wall for
when the de Sitter solution is the only possible solution, and when it becomes
unstable to decay into a wall-like solution. Note that the de Sitter solution
\emph{always} contains an instability to the field rolling coherently down the
potential well~--- we will not be interested in these instabilities, only in
those which would lead to the formation of a domain wall. We derive these
bounds for both our chosen field theory models, as well as for a general
potential. We then provide, in the case of weak gravitational coupling, some
perturbative solutions for the wall and its gravitational fields. These can be
readily compared to existing results in the literature. We then present
numerical results backing up and extending the analytic work. In the
penultimate section we consider the domain wall in the presence of a
cosmological constant, and conclude in the final section.
\section{Plane-symmetric spacetimes}\label{sec2}
We start by setting up the general framework for a domain wall coupled to
gravity. We initially consider a general matter Lagrangian
\begin{equation}
{\cal L}_\mscr{M} = ( \nabla_a \Phi )^2 - U \left ( \Phi \right ),
\label{wallag}
\end{equation}
where $\Phi$ is a real Higgs field and the symmetry breaking potential
$U(\Phi)$, has a discrete set of degenerate minima. We assume that the spacing
of these minima is proportional to the (dimensionful) parameter $\eta$, which
sets the symmetry breaking scale, and that $U(\Phi)$ is characterised by a
scale $V_\mscr{F} = U(\Phi_\mscr{F})$, where $\Phi_\mscr{F}$ is a local false
vacuum situated between successive minima ($\Phi_\mscr{F} = 0$ is a
conventional choice). For example, in the usual `kink' model, $U(\Phi) =
\lambda \left ( \Phi^2 - \eta^2 \right )^2$, and we see $\eta$ directly, with
$V_\mscr{F} = \lambda \eta^4$. For convenience, we scale out dimensionful
parameters via
\begin{equation} \label{epsilon}
X = \Phi/\eta, \qquad \epsilon = 8\pi G\eta^2.
\end{equation}
The dimensionless parameter $\epsilon$, which we call gravitational strength
parameter, characterises the gravitational interaction of the Higgs field.
Then, defining $V(X) = U(\eta X)/V_\mscr{F}$,\footnote{Note that $V(X_{\rm
F})=1$ by definition.}
\begin{equation}
8\pi G{\cal L}_\mscr{M} = {\epsilon \over w^2} \left [
w^2 (\nabla_a X)^2 - V(X) \right ],
\label{rescaledlag}
\end{equation}
where $w = \sqrt{\epsilon \over 8\pi G V_\mscr{F}}$ represents the inverse mass
of the scalar after symmetry breaking, and of course will also characterise the
width of the wall defect within the theory. The equations of motion following
from (\ref{rescaledlag}) are simply
\begin{equation}
\Box X + {1 \over 2w^2} {\partial V \over \partial X} = 0.
\label{walleqs}
\end{equation}
Without loss of generality we can set $w=1$ (which amounts to choosing `wall'
units rather than Planck units) and, looking for a static solution in flat
space, we see that (\ref{walleqs}) can be integrated directly to give
\begin{equation}
X'^2 = V(X),
\label{flatnlde}
\end{equation}
which has an implicit solution
\begin{equation}
\int^{X}_{X_\mscr{F}} {dX\over\sqrt{V(X)}} = z - z_0,
\label{flatx}
\end{equation}
where $X_\mscr{F} = X(z_0)$ is the false vacuum. For example, in the
$\lambda\Phi^4$ model above, $z-z_0 = \int_0^X dX/(1-X^2) = \tanh^{-1} X$, and
we get the usual kink solution centered on $z_0$: $X = \tanh(z-z_0)$. Another
model that we will be exploring in detail is the sine-Gordon model, with
$V(\Phi) = V_\mscr{F} [1 + \cos (\Phi/\eta) ]/2$, in which case $z - z_0 =
2\ln\tan (\pi/4 + X/4)$.
We now look for a plane-symmetric gravitating domain wall solution, since this
represents the most obvious intuitive generalization of the flat space domain
wall. We will consider coordinate transformations of this solution at the end
of the section. The metric therefore will have planar symmetry (i.e. Killing
vectors $\partial_x, \partial_y, x\partial_y - y\partial_x$), and in addition
will display reflection symmetry around a surface, $z=0$ say, which represents
the location of the wall (defined by $X = X_{\rm F}$). If we choose $z$ to be
the proper distance from the wall, then the metric may be written as
\begin{equation}
\label{met1}
ds^2 = A^2(z) dt^2 - B^2(z,t) \left( dx^2+dy^2 \right) - dz^2,
\end{equation}
with the associated Einstein equations derived from the
Lagrangian~(\ref{wallag}) coupled to gravity through the usual Hilbert term,
${\cal L}_\mscr{G} = - R/ 2\epsilon$, being
\begin{equation}
R_{ab} = 2\epsilon X_{,a} X_{,b} - \epsilon \, V(X) \, g_{ab}.
\label{eeqs}
\end{equation}
Before writing these equations explicitly, we will first examine what we mean
by a domain wall solution, since this will require some rather specific
boundary conditions at $z=0$, and for large $z$. We will define a \emph{wall
solution} to be a function $X(z)$ of the proper distance from the wall which
at $z=0$ is at a local maximum of $V(X)$, $X_\mscr{F}$, and which falls towards
distinct minima on either side of the wall. Assuming that $V(X)$ is locally
symmetric around the maximum, then $(X(z) - X_{\rm F})$ will be an odd function
of $z$. This restriction embodies the idea of a defect, in which the field
falls to distinct vacua on either side of its core, and settles to a
topological, rather than radiative, configuration. It is possible that the only
nonsingular solution satisfying these criteria is $X \equiv X_\mscr{F}$, in
which case the spacetime will be de Sitter; indeed, this is always a possible
solution to the equations of motion satisfying the above criteria, though it
will not necessarily be stable. Clearly, there is always an instability
corresponding to a coherent roll of the field towards one of its vacuum values,
however, since we are interested in spacetimes with the interpretation of a
domain wall, we will ignore this mode, and say that de Sitter spacetime is
`stable' if there is no odd perturbation of $(X - X_{\rm F})$ which is growing
in time. As we will see, the scalar field does not fall all the way to its
minimum value within the range of validity of the coordinates in (\ref{met1}),
which is why we do not place any specific boundary conditions on $X$ for large
$z$, however, it will turn out that we require $X'(z_{h}) = 0$ for a
nonsingular solution. Finally, we can choose $t$ to set $A(0) = 1$, and
reflection symmetry requires $A'(0)=0$.
Turning to the Einstein equations (\ref{eeqs}), we see that
\begin{equation}
R_{zt} = {{\dot B} A'\over BA} - {{\dot B}' \over B} = 0
\end{equation}
which implies $B = b(t) A(z)$. Then the relation
\begin{equation}
R^t_t - R^x_x = A^2 \left ( {{\dot b}^2 \over b^2} - {{\ddot b}\over b}
\right ) = 0
\end{equation}
yields
\begin{equation}
b(t) = {\rm e}^{kt},
\end{equation}
so that the equations of motion for the gravitating wall finally reduce simply
to
\begin{mathletters}\begin{eqnarray}
X'' + 3 {A'\over A} X' &=& {1\over2} {\partial V\over \partial X}
\label{gweqb} \\
{A''\over A} &=& -{\epsilon\over3} \left [ 2 X^{\prime 2} + V(X) \right ]
\label{gweqa}\\
\left ( {A'\over A} \right )^2 &=& {k^2\over A^2} + {\epsilon\over3}
\left [ X^{\prime 2} - V(X) \right ]. \label{gweqc}
\end{eqnarray}\label{gweqs}\end{mathletters}
Note that since $A''\leq 0$, once $A'$ becomes negative it will always be
bounded away from zero, therefore there is some finite $z_{\rm h}$ for which
$A(z_{\rm h})=0$, and we have either a physical or coordinate singularity.
Thus we see immediately that there is no nonsingular solution with $k^{2}=0$.
It is also easy to see that the false vacuum de Sitter solution is given by $X
= X_\mscr{F}$, $A=\cos kz$, $k^2 = \epsilon/3$.\footnote{This somewhat less
familiar form is merely one of the many coordinate transformations of de
Sitter, and can be reduced to the more familiar form $ds^2 = d\tau^2 -
{\rm e}^{2k\tau} d{\bf x}^2$ via the transformation ${\rm e}^{k\tau} = {\rm e}^{kt}\cos kz$,
$\zeta = \tan(kz) {\rm e}^{-kt} /k$.}
Determining therefore whether or not a wall solution exists reduces to
investigating the system of equations (\ref{gweqs}). Clearly, for small
$\epsilon$ we might expect a wall solution to the above equations to exist, and
to be given by a perturbative expansion around flat space. For large
$\epsilon$, since $A' = O(\epsilon)$, the $A'X'/A$ term in (\ref{gweqb}) will
drive the solution to a singularity for nonzero $(X-X_\mscr{F})$, hence
for large $\epsilon$ we expect only the de Sitter solution. For intermediate
values of $\epsilon$, except for some very special cases, an
analytic solution does not exist and we have to numerically integrate the
equations.
Before proceeding with the details of this analysis, we conclude this section
by commenting on a coordinate transformation of the plane symmetric metric
(\ref{met1}), which transforms the defect into an accelerating bubble wall.
Defining
\begin{mathletters}\label{starred}\begin{eqnarray}
x^* &=& A(z) {\rm e}^{kt} x \\
y^* &=& A(z) {\rm e}^{kt} y \\
t^*-z^* &=& -{1\over k} A(z) {\rm e}^{kt} \\
t^*+z^* &=& \phantom{-}{1\over k} A(z) {\rm e}^{-kt} - k(x^2+y^2) A(z) {\rm e}^{kt}
\end{eqnarray}\end{mathletters}
gives an alternate form of the line element:
\begin{equation}
\label{starmet}
ds^2 = dt^{*2} - dx^{*2} - dy^{*2} - dz^{*2} + \left ( {A^{\prime2}\over k^2}
-1 \right ) dz^2,
\end{equation}
where $z$ is given implicitly by
\begin{equation}
k^2 (t^{*2} - {\bf x}^{*2}) = - A^2(z).
\end{equation}
The wall (i.e.\ the zero of $X-X_{\rm F}$) is located at ${\bf x}^{*2} - t^{*2}
= 1/k^2$, and as we will see, for small values of the gravitational coupling
$\epsilon$, $A'$ rapidly approaches $k = O(\epsilon)$ outside the core of the
wall. Therefore, we see that the spacetime in these coordinates is
approximately flat, with the wall being located at a spacelike hyperboloid at a
distance $1/k$ from the origin. This corresponds to a wall undergoing uniform
acceleration, contracting in from infinity, reaching a minimum radius, and
re-expanding outwards. The horizon ($A(z_{\rm h})=0$ in the old coordinates) is
now the lightcone centered on the origin, and the region exterior to the
horizon is the causal future and past of the origin. Thus the coordinate
transformation (\ref{starred}) generalises the thin wall transformation given
in \cite{IS}, and discussed in detail in section three of \cite{GWG}.
For larger values of $\epsilon$ the spacetime will not be flat, and $X$
will not be at its true vacuum value near the horizon. In particular, setting
$t^* = \tau\cosh\psi$, $x^* = \tau \sinh \psi \sin \theta \cos\phi$, etc., then
gives a rather familiar form for the metric exterior to the horizon, i.e.\
inside the lightcone of the origin:
\begin{equation}
ds^2 = C^2(\tau) d\tau^2 - \tau^2 \left [ d\psi^2 + \sinh^2\psi (
d\theta^2 + \sin^2\theta d\phi^2 )\right ],
\end{equation}
the metric of an open FRW universe. Thus if the scalar field is not at its
vacuum value at the horizon, its evolution outside the horizon is given by the
rolling of a scalar towards its minimum in an open FRW model, a well studied
problem!
\section{self-gravitating domain walls}\label{sgwall}
We now want to find solutions to~(\ref{gweqs}) representing an isolated domain
wall for various values of the parameter $\epsilon$. We start by considering a
general potential, $V(X)$, finding an analytic perturbative solution for small
$\epsilon$, and showing that if $\epsilon$ is sufficiently large there is no
wall solution. We also demonstrate an instability of the false vacuum de Sitter
solution to wall formation for $\epsilon \leq \epsilon_{\rm max}$, where
$\epsilon_{\rm max}$ depends on the second derivative of the potential at the
origin. We then explore the particular cases of the $\lambda \Phi^4$ kink and
the sine-Gordon model in some detail, first making reference to the analytic
work, then describing the solutions for intermediate values of $\epsilon$ with
the help of numerical work. Both analytic and numerical work show the presence
of a phase transition in the behaviour of the solutions between wall existence
and nonexistence.
It is clear that when the gravitational strength parameter is set to zero, one
gets the flat space solution $A=1$, with $X$ being given by (\ref{flatx}). Let
us now consider small values of $\epsilon$, typically $\epsilon\ll 1$. Then we
can expand the fields $X$ and $A$ in powers of $\epsilon$
\begin{mathletters}\begin{eqnarray}
X &=& X_0 + \epsilon X_1 + O\left(\epsilon^2\right)\\
A &=& A_0 + \epsilon A_1 + O\left(\epsilon^2\right),
\end{eqnarray}\end{mathletters}
where $X_n$, $A_n$ are now independent of $\epsilon$ and $A_0, X_0$ are the
flat space solutions.
The field equations (\ref{gweqs}) to lowest order in $\epsilon$ give
\begin{mathletters}\begin{eqnarray}
A_1'' &=& -{1\over3} \left [ 2 X_0^{\prime2} + V(X_0) \right ]
=-X_0^{\prime2} \label{onea}\\
X_1'' &=& - 3 A_1' X_0' + {1\over 2} X_1 {\partial ^2 V \over \partial X^2 }
\bigg | _{X_0(z)}. \label{oneb}
\end{eqnarray}\end{mathletters}
The boundary conditions of $A$ and $X$ give
\begin{equation}
A_1(0) = A_1'(0) = 0, \qquad X_1(0) =0, \qquad X_{1} \to 0 \mbox{ for large
}z; \label{fobcs}
\end{equation}
(\ref{onea}) and~(\ref{oneb}) can be integrated to give
\begin{equation}
A_1 = - \int \!\!\!\! \int V(X_{0}) dz = -
\int {dX\over \sqrt{V}}\int\sqrt{V}dX
\label{aone}
\end{equation}
and
\begin{equation}
X_1 = - {3\over2} X_0' \int {dz\over X_0^{\prime2}} \left ( A_1'^{2}-
{k^{2}\over \epsilon^{2}} \right ) dz,
\label{xone}
\end{equation}
which can also be expressed as an implicit integral in terms of $V(X)$ and $X$.
Finally, noting that
\begin{equation}
X_0'X_1' - X_1 X_0'' = {3\over2} A_1^{\prime2} + X_0'(0) X_1'(0),
\end{equation}
(\ref{gweqc}) implies
\begin{equation}
k^2 = \epsilon^2 \left [ A_1^{\prime2} + {2\over3} (X_1X_0'' - X_0'X_1')
\right ] = - {2\epsilon^2\over3} X_0'(0) X_1'(0).
\label{kone}
\end{equation}
Before exploring these solutions for specific models for $V(X)$, we can make
some general statements about the self-gravitating wall. First of all, since
$A_1''$ is strictly negative, $A$ will asymptote $1 - kz$, where $k$ is
O($\epsilon$) from the above relation. This means that $A_1$ will cease to be
small at a distance of order $\epsilon^{-1}$ from the wall, and our expansion
procedure strictly speaking breaks down. However, since $A_1'$ is not growing,
it is clear that continuing the expansion to higher orders in $\epsilon$ will
merely produce minor corrections to $A$, and will not alter the qualitative
behaviour, namely, that $A(z)$ has a zero at a distance of order
$\epsilon^{-1}$ from the wall (see figure~\ref{fig:comp}). Since $g_{tt}=A^2$
this is simply the event horizon which is familiar from the thin wall
approximation.
As $\epsilon$ increases, the effect of the scalar field's energy-momentum on
the geometry increases, and we expect the horizon to move closer to the wall,
roughly until $\epsilon = {\rm O}(1)$. For large $\epsilon$, the expected
horizon would be well inside the core of the wall and two possibilities now
emerge. First of all, the scalar field could simply ignore the geometry, and
fall away minutely from its false vacuum, causing an horizon at some small
value of $z$. Since the spatial gradient of such a solution would be
relatively small, the energy-momentum would be vacuum dominated, and we would
expect the spacetime to be very close to the de Sitter solution. Outside the
horizon, the scalar field would roll to its vacuum value as described in the
last section.
The alternative
is that there is a phase transition in the behaviour of $X$, that is, that $X$
either has some nontrivial odd form, approaching reasonably close to its vacuum
value at the horizon, or $X \equiv X_\mscr{F}$. In other words, $X$ must either
roll significantly away from $X_\mscr{F}$, or not at all. There are two reasons
why we expect the latter scenario to hold. The first is that Basu and Vilenkin
\cite{BV2} observed just such a phase transition in studying the problem of
wall instantons.
The other reason for suspecting a phase transition lies in the behaviour of
field theory solutions on compact surfaces. For example, two of the authors
have studied a cosmic string interacting with the event horizon of an extremal
black hole~\cite{BEG}. There, there are nontrivial solutions with the string
piercing the horizon while the string fields can fall reasonably close to their
vacuum values around the horizon; however, there is a transition when the
string becomes sufficiently thick relative to the black hole: the event horizon
can no longer support a nontrivial solution and the flux of the string is
expelled~--- the only horizon solution is the trivial one.
First of all, let us show that for small $\epsilon$ the false vacuum de Sitter
solution is unstable. Recall that the de Sitter solution is
\begin{equation}
\label{sitter}
ds^2 = \cos^2(kz) dt^2 - {\rm e}^{2kt} \cos^2(kz) (dx^2+dy^2) - dz^2
\end{equation}
with $X \equiv X_\mscr{F}$ and $k^2 = \epsilon/3$. This solution will be
`stable' (i.e., stable to wall formation) if there is no perturbation of $X -
X_{\rm F}$ which is an odd function of $z$ and growing in time. Setting $X =
X_\mscr{F} + \xi$, and noting that corrections to the geometry are O($\xi^2$),
we see that an instability must satisfy the time-dependent linearized
perturbation equation
\begin{equation}
\label{stab2}
\xi'' - 3k\tan(kz) \xi' - \sec^2(kz) [ \ddot{\xi} + 2k \dot{\xi} ] -
\frac12 \xi {\partial^2 V\over \partial X^2} \bigg | _{X_\mscr{F}} = 0
\end{equation}
(where a dot indicates differentiation with respect to $t$). This equation does
indeed have unstable solutions for $k^2 = \epsilon/3 <|V''(X_{\rm F})|/8$, the
dominant instability being given by
\begin{equation}
\xi = {\rm e}^{k\nu t} \sin kz (\cos kz) ^\nu
\end{equation}
with
\begin{equation}
\nu = -{5\over2} + {1\over2} \sqrt{ 9 - 2V''(X_\mscr{F})/k^2}.
\label{nudef}
\end{equation}
Thus for $\epsilon$ smaller than
\begin{equation}\label{epsmax}
\epsilon_{\rm max} = \frac38 \; \left|V''(X_\mscr{F})\right|,
\end{equation}
the de Sitter solution is unstable to wall formation.
Now let us examine whether a wall solution can exist for large $\epsilon$. For
a nontrivial wall solution, we require $X'(0)>0$, and for a nonsingular
solution, we require $X'(z_{\rm h})=0$. Now taking the derivative of the field
equation (\ref{gweqb}) we get
\begin{equation}
X'''= -3{A' \over A}X''+X'\left[-3{A'' \over A}+3\left({A' \over A}\right)^2
+ \frac12 {\partial^2V\over\partial X^2} \right]
= -3{A' \over A}X''+X' F(z)
\label{nowall1}
\end{equation}
where $F(z)$ may be rewritten using~(\ref{gweqs}) as
\begin{equation}
F(z) = \left[ 3\epsilon X^{\prime2} + 3{k^2\over A^2}
+ \frac12 {\partial^2V\over\partial X^2} \right].
\end{equation}
Now, at $z=0$ (\ref{gweqc}) gives $3k^2 = \epsilon[1-X'(0)^2]$, hence
\begin{equation}
X'''(0) = X'(0) \left [ 3\epsilon - 6 k^2 + \frac12 V''(X_\mscr{F}) \right]
> X'(0) \left [ \epsilon + \frac12 V''(X_\mscr{F}) \right]
\end{equation}
Therefore, if $\epsilon > |V''(X_\mscr{F})|/2$, $X'''(0)>0$, and $X'$ is
increasing away from $z=0$. Moreover, if $|V''(X)|$ is maximized at
$X_\mscr{F}$, then $F(z)$ is strictly increasing away from $z=0$ and $X'$ can
never be zero at the horizon. Thus with a minimal assumption on the nature of
the potential, we have shown that there is no wall solution possible if
\begin{equation}
\epsilon > \frac12 \, |V''(X_\mscr{F})|.
\label{esup}
\end{equation}
This argument does not make any assumptions as to the behaviour of the
geometry, it simply relies on general properties of the potential.
To reiterate, we have shown that for $\epsilon \leq \epsilon_{\rm max}$ there
are two solutions: a wall spacetime and a de Sitter solution, the latter
being unstable to wall formation. For $\epsilon > \epsilon_{\rm max}$,
the de Sitter solution is `stable', and for $\epsilon > 4\epsilon_{\rm max}/3$,
the domain wall solution can analytically be shown not to exist. To
determine whether de Sitter is the only solution for $\epsilon_{\rm max} <
\epsilon < 4\epsilon_{\rm max}/3$, the problem must be examined numerically.
We now do this, and obtain explicitly the perturbative solution for the
$\lambda\Phi^4$ and sine-Gordon potentials.
\subsection{The $\lambda\Phi^4$ kink}
In this case the potential $V(X)$ is given by $ V(X)=(X^2-1)^2 $, and in flat
spacetime (O$\left(\epsilon^0\right)$) we have the usual flat domain wall
solution plotted in figure~\ref{fig:flatwall},
\begin{equation}
X_0 = \tanh(z).
\end{equation}
\begin{figure}[hbtp]
\centerline{\epsfig{file=flatwall.eps,width=6.8cm}}
\vspace*{0mm}
\caption{The flat spacetime solution. The solid line is $X = \tanh(z)$, the
solution for the Goldstone model, and the broken line is $X = 4
\arctan \left( {\rm e}^{\frac z2} \right) - \pi$, the solution for the
sine-Gordon case of section~\ref{sec:sG}.}
\label{fig:flatwall}
\end{figure}
Integrating (\ref{aone}), (\ref{xone}) and calculating (\ref{kone}) one gets,
\begin{mathletters}\begin{eqnarray}
X_1 &=& -{1\over2} {\rm sech}^2 z \left [ z + {1\over 3} \tanh z \right ] \\
A_1 &=& -{2\over 3} \log\cosh z - {1\over6}\tanh^2 z \\
k &=& \phantom{-}{2 \over 3}\epsilon+{\rm O}\left(\epsilon^2\right),
\end{eqnarray}\end{mathletters}
which represent the first order gravitational corrections to $X$ and $A$. Note
also that $X_1$ does indeed satisfy the boundary conditions (\ref{fobcs}). The
distance to the event horizon is given by $z_{\rm h} \simeq 3 / 2 \epsilon$.
Putting together our results we get to order O$(\epsilon)$
\begin{mathletters}
\label{eq:wallseries1}
\begin{eqnarray}
X &=& \tanh z - \frac \epsilon 2 {\rm sech}^2 z \left[ z + \frac13 \tanh z
\right] \\
A &=& 1 - \frac \epsilon 3 \left[ 2\ln \cosh z + \frac 12 \tanh^2 z
\right].
\end{eqnarray}
\end{mathletters}
This solution is compared on figure~\ref{fig:comp} with the one found
numerically, for $\epsilon = 0.1$.
\begin{figure}[htbp]
\centerline{\epsfig{file=golcomp.eps,width=6.8cm}}
\caption{Comparison between the solution obtained numerically (solid line)
for $\epsilon = 0.1$ and the series to order O($\epsilon$)
[equation~(\protect\ref{eq:wallseries1})]. (The two solutions for
$X$ appear identical at this scale.)}
\label{fig:comp}
\end{figure}
For larger values of $\epsilon$, we must resort to numerical methods to find
solutions of~(\ref{gweqs}). Here, we have used the routine \textsc{solvde}
from~\cite{NumRec}. The wall solutions that we obtain are qualitatively the
same as the one shown on figure~\ref{fig:soln}. Note that as mentioned
previously, $X$ does not go to its asymptotic value at $z_{\rm h}$ (and
consequently that the energy density does not tend to zero at the horizon). In
fact, we do not solve~(\ref{gweqs}) as written in section~\ref{sec2};
instead, we rewrite equation~(\ref{gweqa}) as
\begin{equation}
\left( \frac{A'}A \right)' + \left( \frac{A'}A \right)^2 + \frac \epsilon 3
\left[ 2X'{}^2 + V(X) \right] = 0.
\end{equation}
The system~(\ref{gweqb},~\ref{gweqa}) can now be written as three coupled first
order ordinary differential equations (ODE's),
\begin{mathletters}\begin{eqnarray}
X' &=& Y \\
Y' &=& - 3 Y Z + \frac12 \frac{\partial V}{\partial X} \\
Z' &=& - \frac \epsilon 3 \left[ 2Y{}^2 + V(X) \right] - Z^2,
\end{eqnarray}\end{mathletters}
where $Z = A'/A$. These equations were solved for the boundary conditions
$X(0) = Z(0) = Y(z_\mscr{h}) = 0$.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=solhig.eps,width=6.8cm} \qquad
\epsfig{file=soltaa.eps,width=6.8cm} \\
\epsfig{file=sola.eps,width=6.8cm} \qquad
\epsfig{file=solgtt.eps,width=6.8cm}
\end{center}
\caption{Numerical solution of the equations~(\protect\ref{gweqs}) for the
$\lambda \Phi^4$ model. This solution was obtained for $\epsilon =
0.9$ (in which case the horizon was situated at a proper distance
$z_\mscr{h} = 2.789$). The figure shows~({\it a\/}) the Higgs field,
({\it b\/}) the energy momentum tensor $T^x{}_x = T^y{}_y = T^t{}_t$
and $T^z{}_z$, ({\it c\/}) the function $A(z)$ and~({\it d\/}) the
metric component $g_{tt}(z) = A^2(z)$.}
\label{fig:soln}
\end{figure}
To determine whether there is indeed a phase transition in the behaviour
of the solutions, we examine the evolution of the value of the Higgs field at
the horizon, $X|_{z_{\rm h}}$, as a function of $\epsilon$. For a wall
solution, this value represents the maximum of the function $X$ (see for
instance figure~\ref{fig:soln}{\it a\/}), and a value $X|_{z_{\rm h}} = X_{\rm
F} = 0$ corresponds to the the false vacuum de Sitter solution. We expect
therefore that $X|_{z_{\rm h}}$ will drop from $1$ at $\epsilon=0$ to $0$ for
some $\epsilon$ in the range $[\epsilon_{\rm max}, {4\over3}\epsilon_{\rm
max}]$. According to our previous discussion, it is the solutions in an
intermediate range of $\epsilon$ which interest us most. We find
(figure~\ref{fig:phtrans}{\it a\/}) that the scalar field undergoes a phase
transition at the value $\epsilon = \epsilon_{\rm max} = 3/2$, in perfect
agreement with the prediction~(\ref{epsmax}), at which point wall solutions
cease to exist, and only the de Sitter configuration remains.
Figure~\ref{fig:phtrans}{\it b\/} shows the evolution of the proper distance to
the horizon as a function of $\epsilon$; for small $\epsilon$ this approaches
the first order prediction $z_{\rm h} \simeq 3 / 2\epsilon$ (dashed line), but
higher order corrections rapidly spoil the agreement. The proper distance to
the horizon at the phase transition can be predicted by the condition $\cos
(kz_{\rm h}) = 0$, which~--- with $k^2 = \epsilon_{\rm max} / 3 = 1/2$~---
implies $z_{\rm h} = \pi/\sqrt{2} \approx 2.221$.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=phtrans.eps,width=6.8cm}
\epsfig{file=loghor.eps,width=6.8cm}
\end{center}
\caption{({\it a\/}) The evolution of $X|_{z_{\rm h}}$ as a function of
$\epsilon$. ({\it b\/}) Log-log plot of the proper distance to the
horizon as a function of $\epsilon$ (solid line) compared with the
first order prediction of $z_{\rm h} = 3 / 2 \epsilon$ (dashed
line). The dash-dotted line indicates the phase transition at
$\epsilon = 3/2, X|_{z_{\rm h}} \approx 2.221$.}
\label{fig:phtrans}
\end{figure}
Thus we see that the numerical work confirms the general analytic
derivations given earlier in the section, and indicates that at
$\epsilon_{\rm max}=3/2$, the domain wall solution disappears entirely.
\subsection{The Sine-Gordon Potential}
\label{sec:sG}
Consider now the periodic sine-Gordon potential, $ V(X) = \frac12 (1 + \cos
X) = \cos^{2} X/2$. As before to zeroth order in $\epsilon$, one gets
the usual sine-Gordon soliton
\begin{equation}
\label{sg1}
X_0 = 4 \arctan \left({\rm e}^{{z\over2}}\right) - \pi,
\end{equation}
as shown in figure~\ref{fig:flatwall}.
Making use of (\ref{aone}, \ref{xone}), we obtain the gravitational
back reaction to order O$(\epsilon)$,
\begin{mathletters}
\label{sg2}
\begin{eqnarray}
A &=& 1 -4 \epsilon \; \ln \cosh {z\over2}\\
X &=& 4 \arctan \left({\rm e}^{{z\over2}}\right) - \pi
-6\epsilon\, z \; {\rm sech} {z\over2}
\end{eqnarray}
\end{mathletters}
and $k=2\epsilon+{\rm O}\left(\epsilon^2\right)$, which fixes the horizon
distance to $z_{\rm h} \simeq 1/2\epsilon$.
Note the agreement with equation (3.19) of Widrow's paper \cite{larryw}, who
also considered a sine-Gordon domain wall, although he did not compute
the correction to the scalar field.
Again, we must turn to numerical methods to find solutions for higher values of
$\epsilon$. The results we find are qualitatively very similar to those
obtained for $\lambda \Phi^4$. In particular we observe again the phase
transition predicted in the previous section for a general potential. This
time, however, the analytic results predict $\epsilon_{\rm max} = 3/16$
and $z_{\rm h} = \pi/2k
= 2\pi$; again, this is in excellent agreement with the numerical results.
\section{Domain walls with a cosmological constant}
In this section we consider the previous theories in a universe with a non-zero
cosmological constant $\Lambda$, which would correspond to gravitating domain
walls in an inflating universe. The effect of this constant can be readily
taken into account by modifying equations~(\ref{gweqs}) as follows:
\begin{mathletters}\begin{eqnarray}
X'' + 3 {A'\over A} X' &=& {1\over2} {\partial V\over \partial X}
\label{gweqbL} \\
{A''\over A} &=& -{\epsilon\over3} \left [ 2 X^{\prime 2} + V(X) \right ] -
\frac13 \Lambda \label{gweqaL} \\
\left ( {A'\over A} \right )^2 &=& {k^2\over A^2} + {\epsilon\over3}
\left [ X^{\prime 2} - V(X) \right ] - \frac13 \Lambda. \label{gweqcL}
\end{eqnarray}\end{mathletters}
There are two qualitatively distinct cases: if $\Lambda > 0$, the wall is
embedded in a de Sitter background, whereas if $\Lambda < 0$ it is in an
anti-de Sitter background.\footnote{Strictly speaking, for an anti-de Sitter
background, in order to have the reflection symmetry around $z=0$ we need to
have $k^2<0$, which for a real metric would require $b(t) = \cos kt$. This in
turn requires the $\{x,y\}$ sections to be hyperbolic (see for example
\cite{CGS}); however, since this does not affect the
equations of motion for $A(z)$, we will not discuss it further, and instead
refer the reader to \cite{CS}\ (and references therein)
for a detailed review of anti-de Sitter domain
walls.} The latter is of particular interest because the effect of the
cosmological constant should counteract the (effective) cosmological
constant created by the wall's back reaction.
First let us review how the analytic arguments of the previous section are
affected by a cosmological constant. Note that a false vacuum solution $X =
X_{\rm F}$ will now have an effective cosmological constant $\Lambda_{\rm eff}=
\Lambda + \epsilon$, hence the metric will be of the form (\ref{sitter}) with
$k^{2} = \Lambda_{\rm eff}/3$. Therefore, the previous arguments go through
essentially unchanged, but with $\Lambda_{\rm eff}$ instead of $\epsilon$.
Therefore
\begin{equation}
\epsilon_{\rm max} = {3\over 8} |V''(X_{\rm F})| - \Lambda
\end{equation}
Obviously, the range of the instability is increased for negative $\Lambda$,
and decreased for positive $\Lambda$. If $\Lambda > {3\over 8} |V''(X_{\rm
F})|$, then the false vacuum solution is `stable' (and indeed the only one).
This is illustrated in figures~\ref{fig:hig} and~\ref{fig:hor}, which show the
evolution of $X|_{z_{\rm h}}$ and $z_{\rm h}$ with $\epsilon$ for both $\lambda
\Phi^4$ and the sine-Gordon models as well as for several values of the
cosmological constant, $\Lambda = -0.3, -0.2, \ldots, 0.3$. In partiular, note
that for sine-Gordon (figure~\ref{fig:hig}{\it b\/}) the formula above tells us
that for $\Lambda > 3/16 = 0.1875$ the only solution is $X \equiv X_{\rm F}$;
this is why we do not see the curves for $\Lambda = 0.2$ and $0.3$.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=golhig.eps,width=6.8cm} \qquad
\epsfig{file=sGhig.eps,width=6.8cm}
\end{center}
\caption{Evolution of $X|_{z_{\rm h}}$ in function of $\epsilon$ and
$\Lambda$ (from right to left, $\Lambda = -0.3, -0.2, \ldots
0.2, 0.3$). ({\it a\/}) shows the $\lambda \Phi^4$ case, and ({\it
b\/}) shows the sine-Gordon case. In ({\it b\/}) we have actually
divided $X|_{z_{\rm h}}$ by $\pi$ to help the comparison with case
({\it a\/}).}
\label{fig:hig}
\end{figure}
Note as well that the value of $z_{\rm h}$ at which the phase transition occurs
($\pi / \sqrt{2}$ for $\lambda \Phi^4$, and $2\pi$ for sine-Gordon) remains
unaltered by the inclusion of the cosmological constant, as expected from the
discussion above. In fact, all the de Sitter solutions remain identical if
$\epsilon$ and $\Lambda$ are allowed to vary but $\Lambda_{\rm eff}$ remains
constant. (This is obviously not the case for the wall solutions, as $\epsilon$
then multiplies terms containing the Higgs field.)
\begin{figure}[htbp]
\begin{center}
\epsfig{file=golhor.eps,width=6.8cm} \qquad
\epsfig{file=sGhor.eps,width=6.8cm}
\end{center}
\caption{Distance from the wall to the horizon, as a function of $\epsilon$
and for the same values of $\Lambda$ as in figure~\ref{fig:hig}.
({\it a\/}) was obtained for $\lambda \Phi^4$ and ({\it b\/})
for sine-Gordon. Again, the broken lines show the values of
$z_\mscr{h}$ at the phase transition.}
\label{fig:hor}
\end{figure}
Now let us turn to the solution in the anti-de Sitter case, $\Lambda < 0$. We
now find \emph{three} qualitatively distinct solutions. For very small
$\epsilon$, the wall's self-gravitation cannot compete with the anti-de Sitter
expansion and $A'/A$ is strictly positive; in fact, it is easy to check that
the solution plotted on figure~\ref{fig:seq}{\it a\/} is $A(z) = \cosh(\sqrt{
|\Lambda|/3}\, z)$. As one increases $\epsilon$, the potential is observed to
decrease close to the wall's core, whereas the Higgs profile is slightly
smoothed~(figure \ref{fig:seq}{\it b\/}). In fact, this is the beginning of a
complete change in the metric function $A(z)$: as the wall's gravitational
interaction is switched on, $A$ assumes the shape of a ``double well,'' with a
local maximum at the imposed boundary value $A(0) = 1$ and two local minima
symmetrically situated at $A(\pm z_{\rm m})$ for some $z_{\rm m}$. As
$\epsilon$ increases, this double well becomes deeper, whereas $z_{\rm m}$
moves away from the wall. Notice that so far the function $A(z)$ is strictly
positive, and therefore none of these solutions exhibit an event horizon.
Eventually, however, for some critical value, $\epsilon_{\rm c}$, of
$\epsilon$, the two minima of $A(z)$ vanish as $z_{\rm m} \to \infty$
(figure~\ref{fig:seq}{\it c\/}). This would appear to be a thick wall version
of the type II extreme domain wall spacetime of Cvetic and Griffies \cite{CG},
and is therefore presumably supersymmetrizable. For $\epsilon > \epsilon_{\rm
c}$, the metric becomes negative at a finite distance $z_{\rm h}$, thus
giving rise to the wall's horizon. The Goetz solution \cite{Goetz}\ lies in
this range.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=seq1.eps,width=6.8cm} \qquad
\epsfig{file=seq2.eps,width=6.8cm} \\
\epsfig{file=seq3.eps,width=6.8cm} \qquad
\epsfig{file=seq4.eps,width=6.8cm}
\end{center}
\caption{Solutions $X(z)$ (solid lines) and $A(z)$ to the sine-Gordon
equations for $\Lambda = -0.3$ and $\epsilon = $ 0 ({\it a\/}); 0.2
({\it b\/}); 0.367\ldots ({\it c\/}) and 0.4 ({\it d\/}).}
\label{fig:seq}
\end{figure}
Figure~\ref{fig:par} shows the parameter space $(\Lambda, \epsilon)$, and the
different kinds of solution that we find. It is interesting to note that the
two lines separating the three phases seem to run parallel to each other in
both cases, indicating that a phenomenon similar to the triple point observed
in the phase diagram of water never occurs. This is to be expected, since as
long as the wall does not have an event horizon it is constrained to take its
asymptotic value at infinity. Of course, this topological constraint does not
imply that the lines are parallel, merely that they cannot meet in the physical
range $\epsilon > 0$; figure~\ref{fig:par} then shows that the range of the
parameter $\epsilon$ over which the value of the Higgs field at the horizon is
allowed to drop from 1 to 0 is fixed.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=golpar.eps,width=6.8cm} \qquad
\epsfig{file=sGpar.eps,width=6.8cm}
\end{center}
\caption{Parameter space $(\epsilon, \Lambda)$ and the types of solutions
found.}
\label{fig:par}
\end{figure}
\section{Discussion}
To summarize, we have investigated the spacetimes of thick, gravitating domain
walls in detail, using both analytical arguments and numerical integration.
Both methods demonstrate the existence of a phase transition in the nature of
the `wall' solution, from being wall-like to a pure false vacuum de Sitter
spacetime. We find that for walls much thinner than their cosmological horizon,
the Vilenkin thin wall solution is a good description of the spacetime. For
thicker walls the spacetime differs more markedly until finally there is only
the de Sitter solution. The transition occurs abruptly, when the gravitational
strength of the wall is order of magnitude unity. For walls less strongly
gravitating than this critical value, the spacetime has the appearance of a
gravitating domain wall, possibly lying within a background de Sitter or
anti-de Sitter universe. For very strongly gravitating domain walls however,
the de Sitter universe is the only solution with the symmetries we have
imposed, the cosmological constant being provided by the false vacuum
energy of the wall. Of
course this does not prove that the de Sitter solution is the {\it only}
solution to the coupled Einstein-scalar field equations; by transforming to
a global coordinate system for de Sitter, one can find a complete set of
solutions for the tachyonic wave equation, hence we can find an instability of
the required parity which is given in the notation of section~\ref{sgwall}\
by
\begin{equation}
\xi = \sin (kz)
[v(x,y,z,t)]^{-(\nu+5)/2}
~_2{\rm F}_1 \left [ {\nu+1\over2},{\nu+5\over2},\nu +{7\over2} ; v\right]
\end{equation}
where $\nu$ is given by (\ref{nudef}), and $v$ by
\begin{equation}
v(x,y,z,t) = \left [ 1 + \cos^2kz \left ( \sinh kt + {\textstyle{1\over2}} k^2 (x^2+y^2)
e^{kt} \right )^2 \right ] ^{-1}
\end{equation}
and F is a hypergeometric function. However, since this instability
depends on the $\{x,y\}$ coordinates, it does not correspond to decay to
a solution of the type considered in this paper,
and presumably corresponds to decay to a time dependent spherical wall
of the type found numerically by Sakai et.\ al.\ \cite{SSTM} which are
of relevance to topological inflation \cite{Linde,Vil2}. These
solutions are not solitons in the sense of having a fixed profile in
time, and therefore are outside the consideration of this paper\footnote{
We would like to thank Alex Vilenkin for discussions on this point.}.
This phase transition behaviour of the scalar is reminiscent of the flux
expulsion of a vortex core by the horizon of an extremal black hole
\cite{BEG,CCES}. There, it was the fact that in the extremal limit the black
hole horizon $\cal H$ decoupled from the exterior spacetime that allowed a
partially analytic analysis of the vortex equations on the $S^2$ surface that
was the event horizon. The findings in that case, \cite{BEG}, parallel our
results here very closely. There was always a solution corresponding to the
fields taking their false vacuum values on $\cal H$, but for sufficiently thin
vortices (relative to the horizon radius) there was an additional solution
corresponding to a vortex anti-vortex pair at opposite poles (in the case of a
string threading the black hole). In that case however, the false vacuum
solution for thin strings could be shown not to extend to a full solution in
the exterior spacetime. Here, we always have a false vacuum de Sitter solution,
which is unstable to wall formation, as well as the defect solution for low
gravitational coupling. It is easy to see the common feature in these two
problems~--- the compact nature of the spatial section upon which the defect
must live. Defects on compact spaces have been analysed; for example, Avis and
Isham~\cite{AI} explored some years ago the $\lambda \Phi^4$ solutions on a
circle. They found exact solutions for the scalar field in terms of elliptic
functions, and no solution other than false vacuum if the radius of the circle
was too small. However, the crucial difference of our work to~\cite{AI}\
and~\cite{BEG} is that we are not looking at defects on a fixed background,
but looking for self-gravitating wall solutions without specifying their
topology \emph{ab initio}.
The topology of a black hole event horizon is obviously compact, however, it
turns out that in fact the topology of a domain wall spacetime is also
compact~\cite{GWG}. To see that not only de Sitter spacetime but also the
domain wall spacetime is topologically $S^3\times$I\negthinspace R, consider the
coordinate transformation (\ref{starred}). If we define a fifth coordinate,
$w^*$, by
\begin{equation}
\label{wstar}
w^* = \int_0^z{\sqrt{1 - {A^{\prime2}\over k^2}} \; dz}
\end{equation}
then the wall metric becomes a slice of a five dimensional flat metric, and we
can view our spacetime as a four-dimensional hypersurface embedded in
five-dimensional flat spacetime in an analogous fashion to the de Sitter
hyperboloid. From (\ref{starmet}) the equation for this hypersurface is
\begin{equation}
t^{*2} - {\bf x}^{*2} = - {A^2(w^*)\over k^2}.
\label{surface}
\end{equation}
For example, the de Sitter solution is $A = \cos kz$, therefore $w^* = {1\over
k} \sin kz$ from (\ref{wstar}), and (\ref{surface}) reduces to $t^{*2} - {\bf
x}^{*2} - w^{*2} = -1/k^2$, the de Sitter hyperboloid. For small $\epsilon$
on the other hand, the sine-Gordon wall from (\ref{sg2}) gives
\begin{equation}
w^* = \int_0^z {\rm sech}{z\over2} dz = 4 \arctan {\rm e}^{z\over2} - \pi
=X_0(z), \label{wsg}
\end{equation}
hence the hypersurface is given by a hyperboloid which has been deformed by
squashing in the $w^*$ direction,
\begin{equation}
\label{oonesurf}
t^{*2}-{\bf x}^{*2} = -{1\over 4\epsilon^2} \left (
1 + 4\epsilon \log \cos {w^*\over2}\right )^2.
\end{equation}
The spatial section is depicted in figure~\ref{fig:discus} which shows the
$t^*=z^*=0$ slice to O($\epsilon$) for $\epsilon = 1/30$.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=discus.eps,width=6.8cm}
\end{center}
\caption{The spatial section of the weakly gravitating sine-Gordon
domain wall. The surface shown is the $t^* = z^*=0$ surface
for $\epsilon = 1/30$.}
\label{fig:discus}
\end{figure}
The spatial geometry of the domain wall is therefore topologically $S^3$, and
is similar to a discus, although the upper and lower surfaces are flat almost
to the edge of the discus. This corresponds exactly to our intuitive idea of
the spacetime exterior to the wall being flat, with a highly localised region
of curvature generated by the wall itself located at the rim of the discus. It
fleshes out the thin wall description of the spacetime, smoothing over the
distributional singularity of the thin wall hypersurface. The two length
scales of the domain wall spacetime correspond to the two radii of the discus,
and as $\epsilon$ increases, the discus radius shrinks, with its height
remaining much the same, until the geometry is almost spherical, at which point
the radius becomes too small to support a defect solution and we make the
transition to a false vacuum de Sitter hyperboloid, with an exactly spherical
spatial geometry.
What is interesting in comparison with the Avis and Isham scenario is that it
is the self gravity of the domain wall which produces a compactification of
spacetime at its own characteristic scale, which ultimately becomes too small
to support the defect itself.
Finally, in analogy with \cite{AI}, we can make a plot of
the normalized action for the domain wall solution versus the de Sitter
solution
\begin{equation}
S = \int {\cal L}_{\rm G} + {\cal L}_{\rm M} = \eta^2 \int V(X) \, {\rm e}^{2kt}
A^3 d^4x = N \int_0^{z_{\rm h}} A^3 V(X) dz,
\end{equation}
where $N$ is a normalization factor, and there are no boundary terms from the
gravitational part of the action. For the false vacuum de Sitter solution
$\bar{S} = S/N = 2/3k = 2/\sqrt{3\epsilon}$, and for the weakly gravitating
$\lambda\Phi^4$ model, ${\bar S} = 2/3$. Figure~\ref{fig:bifurc} shows a plot
of the action of the $\lambda \Phi^4$ wall against the false vacuum de Sitter
solution, which indicates clearly the instability of the latter solution to
wall formation for $\epsilon < 3/2$. (This can be compared to the
instanton action plot obtained in \cite{BV2}.)
\begin{figure}
\centerline{\epsfig{file=bifurc.eps,width=6.8cm}}
\caption{Bifurcation diagram for the gravitating domain wall. We plot here
the normalized action $\bar{S}$ as a function of $\epsilon$.
Solid lines indicate the stable solutions found numerically, and the
dashed line represents the unstable de Sitter solution.}
\label{fig:bifurc}
\end{figure}
\section*{Acknowledgements}
We would like to thank Alex Vilenkin and Richard Ward for helpful
discussions.
F.B. is supported by an ORS award and a Durham University award.
C.C. is supported by EPSRC and by the `R\'egion Centre.'
R.G. is supported by the Royal Society.
\def\apj#1 #2 #3.{{\it Astrophys.\ J.\ \bf#1} #2 (#3).}
\def\cmp#1 #2 #3.{{\it Commun.\ Math.\ Phys.\ \bf#1} #2 (#3).}
\def\comnpp#1 #2 #3.{{\it Comm.\ Nucl.\ Part.\ Phys.\ \bf#1} #2 (#3).}
\def\cqg#1 #2 #3.{{\it Class.\ Quant.\ Grav.\ \bf#1} #2 (#3).}
\def\jmp#1 #2 #3.{{\it J.\ Math.\ Phys.\ \bf#1} #2 (#3).}
\def\mpla#1 #2 #3.{{\it Mod.\ Phys.\ Lett.\ \rm A\bf#1} #2 (#3).}
\def\ncim#1 #2 #3.{{\it Nuovo Cim.\ \bf#1\/} #2 (#3).}
\def\npb#1 #2 #3.{{\it Nucl.\ Phys.\ \rm B\bf#1} #2 (#3).}
\def\phrep#1 #2 #3.{{\it Phys.\ Rep.\ \bf#1\/} #2 (#3).}
\def\plb#1 #2 #3.{{\it Phys.\ Lett.\ \bf#1\/}B #2 (#3).}
\def\pr#1 #2 #3.{{\it Phys.\ Rev.\ \bf#1} #2 (#3).}
\def\prd#1 #2 #3.{{\it Phys.\ Rev.\ \rm D\bf#1} #2 (#3).}
\def\prl#1 #2 #3.{{\it Phys.\ Rev.\ Lett.\ \bf#1} #2 (#3).}
\def\prs#1 #2 #3.{{\it Proc.\ Roy.\ Soc.\ Lond.\ A.\ \bf#1} #2 (#3).}
|
1,116,691,499,356 | arxiv | \section*{Abstract}
This manuscript aims to develop and describe gain scheduling control concept for a gas turbine engine which drives a variable pitch propeller. An architecture for gain-scheduling control is developed that controls the turboshaft engine for large thrust commands in stable fashion with good performance. Fuel flow and propeller pitch angle are the two control inputs of the system. New stability proof has been developed for gain scheduling control of gas turbine engines using global linearization and LMI techniques. This approach guarantees absolute stability of the closed loop gas turbine engines with gain-scheduling controllers.
\section*{Nomenclature}
$N_1$: Non-dimensional Fan Spool Speed\\
$N_2$: Non-dimensional Core Spool Speed\\
$T$: hrust (N)\\
TSFC: Thrust Specific Fuel Consumption\\
$\alpha$: Scheduling Parameter\\
$\sigma$: Singular Value\\
$\lambda$: Eigenvalue\\
\section{Introduction}
The gain-scheduling approach is perhaps one of the most popular nonlinear control design approaches which has
been widely and successfully applied in fields ranging from aerospace to process control \cite{research-rugh-2000, surveyGS-leith-2000}. Gain-scheduling, specifically has been used for gas turbine engine control, some of these works are \cite{lpv-balas-2002, lpv-gilbert-2010, lpv-bruzelius-2002, lpv-Shuqing-2010, approximate-zhao-2011, approximate-yu-2011}. In general, stability and control of gas turbine engines have been of interest to researchers and engineers from a variety of perspectives. Stability of axial flow fans operating in parallel has been investigated in \cite{fanStability-simon-1985}. An application of robust stability analysis tools for uncertain turbine engine systems is presented in \cite{robustAeroengine-arriffin-1997}. Application of the Linear-Quadratic-Gaussian with Loop-Transfer-Recovery methodology to design of a control system for a simplified turbofan engine model is considered in \cite{lqg-garg-1989}. A unified robust multivariable approach to propulsion control design has been developed in \cite{turbofanControl-fredrick-2000}. A simplified scheme for scheduling multivariable controllers for robust performance over a wide range of turbofan engine operating points is presented in \cite{turbofanSched-garg-1997}.
In the previous work by authors \cite{distributed-pakmehr-2009, decentralized-pakmehr-2010} controllers developed for single spool and twin spool turboshaft system. Those controllers were designed for small transients, and small throttle commands. In this work we develop a gain-scheduling control structure for JetCat SPT5 turboshaft engine using the method presented in \cite{gainsched-shamma-1988, research-rugh-2000, gain-shamma-2006, overview-shamma-2012}. the controller is designed to be used for entire flight envelope of the twin spool turboshaft engine.
In this manuscript, first a linear representation of the turbofan system dynamics is developed. Then control theoretic concepts for gain-scheduling control of this model is presented. The developed controller can be used for the entire flight envelope of the engine with guaranteed stability. Finally the simulation results for gain scheduling control of a physics-based nonlinear model of the JetCat SPT5 turboshaft engine are presented.
\section{Gain Scheduling Control Design}
Consider the nonlinear dynamical system
\begin{equation}\label{eqn_gs1}
\begin{array}{c}
\dot{x}^p(t)= f^p(x^p(t),u(t)),\\[5pt]
y(t)=g^p(x^p(t),u(t)),
\end{array}
\end{equation}
where $x^p \in \Re^n$ is the state vector, $u\in \Re^m$ is the control input vector, $y\in \Re^k$ is the output vector, $f^p(.)$ is an $n$-dimensional, and $g^p(.)$ is an $k$-dimensional differentiable nonlinear vector functions.
We want to design a feedback control such that $y(t) \rightarrow r(t)$ as $t \rightarrow \infty$, where $r(t) \in D_r \subset \Re^k$ is the output reference signals vector.
Assume that for each $r(t) \in D_r$, there is a unique pair $(x^p_e, u_e)$ that depends continuously on $r$ and satisfies the equations:
\begin{equation}\label{eqn_gs2}
\begin{array}{c}
0= f^p(x^p_e,u_e),\\[5pt]
r=g^p(x^p_e,u_e),
\end{array}
\end{equation}
in case of a constant $r$. $x^p_e$ is the desired equilibrium point and $u_e$ is the steady-state control that is needed to maintain equilibrium at $x^p_e$.
\newtheorem{deff}{Definition}
\begin{deff} \label{def1}
The functions $x^p_e(\alpha), u_e(\alpha)$, and $r_e(\alpha)$ define an equilibrium family for the plant (\ref{eqn_gs1}) on the set $\Omega$ if
\begin{equation}\label{eqn_gs222}
\begin{array}{l}
f^p(x^p_e(\alpha),u_e(\alpha), r_e(\alpha))=0, \\[5pt]
g^p(x^p_e(\alpha),u_e(\alpha))=r_e(\alpha), ~\alpha \in \Omega.
\end{array}
\end{equation}
\end{deff}
Let $\Omega \subset \Re^{m+n}$ be the region of interest for all possible system state and control vector $(x^p,u)$ during the system operation, and denote $x^{p*}_i$ and $u^*_i$, $i\in I = {1, 2, . . . , l}$, as a set of (constant) operating points located at some representative (and properly separated) points inside $\Omega$. Introduce a set of $l$ regions $\Omega_i$ centered at the chosen operating points $(x^{p*}_i, u^*_i)$, and denote their interiors as $\Omega_{i0}$, such that $\Omega_{j0} \bigcap \Omega_{k0}={\oslash}$ for all $j \neq k$, and $\bigcup_{i=1}^{l} \Omega_i=\Omega$. The linear model around each equilibrium point is
\begin{equation}\label{eqn_gs3}
\begin{array}{l}
\dot{x}^p = A^p_i (x^p-x_i^{p*}) + B^p_i (u-u_i^*),\\
y = C^p_i (x^-x_i^{p*}) + D^p_i (u-u_i^*) + y_i^* ,
\end{array}
\end{equation}
where the matrices are obtained as follows
\begin{equation}\label{eqn_gs4}
\begin{array}{l}
\displaystyle A^p_i=\frac{\partial f^p}{\partial x^p}|_{(x^{p*}_i, u^*_i)}, ~~ \forall (x^p,u) \in \Omega_i, \\[5pt]
\displaystyle B^p_i=\frac{\partial f^p}{\partial u}|_{(x^{p*}_i, u^*_i)}, ~~ \forall (x^p,u) \in \Omega_i, \\[5pt]
\displaystyle C^p_i=\frac{\partial g^p}{\partial x^p}|_{(x^{p*}_i, u^*_i)}, ~~ \forall (x^p,u) \in \Omega_i, \\[5pt]
\displaystyle D^p_i=\frac{\partial g^p}{\partial u}|_{(x^{p*}_i, u^*_i)}, ~~ \forall (x^p,u) \in \Omega_i.
\end{array}
\end{equation}
Here we assume that the common boundary of two regions $\Omega_j$ and $\Omega_z$ belongs to only one of $\Omega_j$ and $\Omega_z$. Note that at each moment, $(x^p,u)$ belongs to only one $\Omega_i$.
Performing linearizations at a series of trim points gives a linearization family described by
\begin{equation}\label{eqn_gs5}
\begin{array}{l}
\delta \dot{x}^p = A^p(\alpha) \delta x^p + B^p(\alpha) \delta u,\\[5pt]
\delta y = C^p(\alpha) \delta x^p + D^p(\alpha) \delta u.
\end{array}
\end{equation}
where
\begin{equation}\label{eqn_gs6}
\begin{array}{l}
\delta x^p = x^p-x^p_e(\alpha) \\[5pt]
\delta y = y-y_e(\alpha),\\[5pt]
\delta u = u-u_e(\alpha), ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
Gain scheduled controller for plant (\ref{eqn_gs5}), is designed as follows. First, a set of parameter values $\alpha_i$ are selected, which represent the range of the plant's dynamics, and a linear time-invariant controller for each is designed. Then, in between operating points, the controller gains are linearly interpolated such that for all frozen values of the parameters, the closed loop system has excellent properties, such as nominal stability and robust performance. To guarantee that the closed loop system will retain the feedback properties of the frozen-time designs, the scheduling variables should vary slowly withe respect to the system dynamics \cite{gainsched-shamma-1988}.
The parameter $\alpha$ called the scheduling variable in gain scheduling control. $A^p(\alpha), B^p(\alpha), C^p(\alpha)$, and $D^p(\alpha)$ are the parameterized linearization system matrices and $x^p_e(\alpha), u_e(\alpha)$, and $y_e(\alpha)$ are the parameterized steady-state system variables, which form the equilibrium manifold of system (\ref{eqn_gs1}). The subscript $'e'$ stands for steady-state throughout this paper.
Let the controller have the following structure
\begin{equation}\label{eqn_gs7}
\begin{array}{l}
\dot{x}^c = A^c(\alpha) \delta x^c + B^c(\alpha) [\delta y-\delta r],\\[5pt]
\delta u = C^c(\alpha) \delta x^c + D^c(\alpha) [\delta y-\delta r], ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
where
\begin{equation}\label{eqn_gs8}
\begin{array}{l}
\delta x^c = x^c-x^c_e(\alpha) \\[5pt]
\delta r = r-r_e(\alpha), ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
A standard realization of the parameterized controller can be written in the following form with the reference signal explicitly displayed
\begin{equation} \label{eqn_gs9}
\begin{array}{l}
\left[
\begin{array}{c}
\dot{x}^c \\
\delta u
\end{array}
\right] =
\left[
\begin{array}{ccc}
A^c(\alpha) & B^c(\alpha) & -B^c(\alpha) \\
C^c(\alpha) & D^c(\alpha) & -D^c(\alpha)
\end{array}
\right] ~
\left[
\begin{array}{c}
\delta x^c \\
\delta y \\
\delta r
\end{array}
\right], ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
Controller has the general form
\begin{equation}\label{eqn_gs10}
\begin{array}{c}
\dot{x}^c(t)= f^c(x^c(t),y(t), r(t)),\\[5pt]
u(t)=g^c(x^c(t),y(t), r(t)),
\end{array}
\end{equation}
with the input and output signals corresponding to the nonlinear plant (\ref{eqn_gs1}).
The objective in linearization scheduling is that the equilibrium family of the controller (\ref{eqn_gs10}) match the plant equilibrium family, so that the closed loop system maintains suitable trim values, and second the linearization family of the controller is the designed family of linear controllers \cite{research-rugh-2000}.
For the equilibrium conditions of plant (\ref{eqn_gs1}) and controller (\ref{eqn_gs10}) to match, there must exist a function $x^c_e(\alpha)$ such that
\begin{equation}\label{eqn_gs11}
\begin{array}{l}
0= f^c(x^c_e(\alpha),y_e(\alpha),r_e(\alpha)),\\[5pt]
u_e(\alpha)=g^c(x^c_e(\alpha),y_e(\alpha), r_e(\alpha)), ~~~ \forall \alpha \in \Omega,
\end{array}
\end{equation}
where
\begin{equation}\label{eqn_gs12}
\begin{array}{l}
\displaystyle A^c(\alpha)=\frac{\partial f^c}{\partial x^c}|_{(x^c_e(\alpha),y_e(\alpha),r_e(\alpha))}, \\[5pt]
\displaystyle B^c(\alpha)=\frac{\partial f^c}{\partial y}|_{(x^c_e(\alpha),y_e(\alpha),r_e(\alpha))}, \\[5pt]
\displaystyle C^c(\alpha)=\frac{\partial g^c}{\partial x^c}|_{(x^c_e(\alpha),y_e(\alpha),r_e(\alpha))}, \\[5pt]
\displaystyle D^c(\alpha)=\frac{\partial g^c}{\partial y}|_{(x^c_e(\alpha),y_e(\alpha),r_e(\alpha))}, ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
So the controller family has the form
\begin{equation}\label{eqn_gs13}
\begin{array}{l}
\dot{x}^c = A^c(\alpha) [x^c-x^c_e(\alpha)] + B^c(\alpha) [y-r],\\[5pt]
u = C^c(\alpha) [x^c-x^c_e(\alpha)]+ D^c(\alpha) [y-r]+ u_e(\alpha), ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
Note that $r_e(\alpha)=y_e(\alpha)$, as a result $\delta y - \delta r=y-r$. The scheduling parameter $\alpha$ is treated as a parameter throughout the design process, and then it becomes a time-varying input signal to the gain-scheduled controller implementation through the dependence $\alpha(t)=p(y(t))$. Thus the gain-scheduled controller becomes
\begin{equation}\label{eqn_gs14}
\begin{array}{l}
\dot{x}^c = A^c(p(y)) [x^c-x^c_e(p(y))] + B^c(p(y)) [y-r],\\[5pt]
u = C^c(p(y)) [x^c-x^c_e(p(y))]+ D^c(p(y)) [y-r]+ u_e(p(y)).
\end{array}
\end{equation}
Linearization of (\ref{eqn_gs14}) about an equilibrium specified by $\alpha$ gives
\begin{equation}\label{eqn_gs15}
\begin{array}{l}
\delta \dot{x}^c = A^c(\alpha) \delta x^c + B^c(\alpha) [y-r] \displaystyle -[A^c(\alpha) \frac{\partial x^c_e (\alpha)}{\partial \alpha}] \times [\frac{\partial p }{\partial y} (y_e(\alpha)) (y-r)], \\[5pt]
\displaystyle \delta u = C^c(\alpha) \delta x^c + D^c(\alpha) [y-r] \displaystyle +[\frac{\partial u_e (\alpha)}{\partial \alpha} - C^c(\alpha) \frac{\partial x^c_e (\alpha)}{\partial \alpha}] \displaystyle \times [\frac{\partial p }{\partial y} (y_e(\alpha)) (y-r)].
\end{array}
\end{equation}
Comparing this to (\ref{eqn_gs9}), we see there are additional terms, which we refer to them as hidden coupling terms \cite{research-rugh-2000}.
In order to get rid of these terms we have to design the controller such that
\begin{equation}\label{eqn_gs16}
\begin{array}{l}
\displaystyle A^c(\alpha) \frac{\partial x^c_e (\alpha)}{\partial \alpha} = 0, \\[5pt]
\displaystyle \frac{\partial u_e (\alpha)}{\partial \alpha} - C^c(\alpha) \frac{\partial x^c_e (\alpha)}{\partial \alpha} =0.
\end{array}
\end{equation}
It is not always easy to come up with solutions to satisfy this condition. In order to make the design process easier, we can augment integrators at the plant input, so there is no need for equilibrium control value. By augmenting integrators at the plant (\ref{eqn_gs1}) input we obtain
\begin{equation} \label{eqn_gs61}
\begin{array}{l}
\left[
\begin{array}{c}
\dot{x}^p(t) \\
\dot{u}(t)
\end{array}
\right] =
\left(
\begin{array}{c}
f^p(x^p(t),u(t)) \\
-\eta_c u(t)
\end{array}
\right) +
\left[
\begin{array}{c}
0 \\
\eta_c \times I
\end{array}
\right] v(t).
\end{array}
\end{equation}
The new controller has the general form
\begin{equation}\label{eqn_gs110}
\begin{array}{c}
\dot{x}^c(t)= f^c(x^c(t),y(t), r(t)),\\[5pt]
v(t)=g^c(x^c(t),y(t), r(t)),
\end{array}
\end{equation}
with the input and output signals corresponding to the nonlinear plant (\ref{eqn_gs1}).
Combining (\ref{eqn_gs61}) and (\ref{eqn_gs110})leads to
\begin{equation} \label{eqn_gs111}
\begin{array}{l}
\underbrace{\left[
\begin{array}{c}
\dot{x}^p(t) \\
\dot{u}(t) \\
\dot{x}^c(t)
\end{array}
\right]}_{\dot{x}} =
\underbrace{\left(
\begin{array}{c}
f^p(x^p(t),u(t)) \\
-\eta_c u(t) \\
f^c(x^c(t),g^p(x^p(t),u(t)) ,r(t))
\end{array}
\right)}_{f(x,r)} +
\underbrace{\left[
\begin{array}{c}
0 \\
\eta_c \times I \\
0
\end{array}
\right]}_{B} v(t). \\[5pt]
v(t)=\underbrace{g^c(x^c(t),g^p(x^p(t),u(t)), r(t))}_{g(x,r)},
\end{array}
\end{equation}
Then the closed loop nonlinear system is
\begin{equation}\label{eqn_gs112}
\begin{array}{l}
\dot{x}(t)= f(x(t),r(t))+B g(x(t),r(t )),\\[5pt]
~~~~~~ =F(x(t),r(t))
\end{array}
\end{equation}
The augmented linear family of systems for (\ref{eqn_gs61}) becomes
\begin{equation}\label{eqn_gs62}
\begin{array}{l}
\underbrace{\left[
\begin{array}{c}
\dot{x}^p(t) \\
\dot{u}(t)
\end{array}
\right]}_{\dot{x}_{aug}} =
\underbrace{\left[
\begin{array}{cc}
A^p(\alpha) & B^p(\alpha)\\
0 & -\eta_c \times I
\end{array}
\right]}_{A_{aug}(\alpha)}
\underbrace{\left[
\begin{array}{c}
\delta x^p \\
\delta u
\end{array}
\right]}_{\delta x_{aug}}+
\underbrace{\left[
\begin{array}{c}
0 \\
\eta_c \times I
\end{array}
\right]}_{B_{aug}} v(t) ,\\[5pt]
\delta y = \underbrace{[C^p(\alpha), D^p(\alpha)]}_{C_{aug}(\alpha)}
\underbrace{\left[
\begin{array}{c}
\delta x^p \\
\delta u
\end{array}
\right]}_{\delta x_{aug}}.
\end{array}
\end{equation}
Now, the control realization for this system is
\begin{equation}\label{eqn_gs63}
\begin{array}{l}
\dot{x}^c = A^c(\alpha) x^c + B^c(\alpha) [y-r],\\[5pt]
v = C^c(\alpha) x^c+ D^c(\alpha) [y-r] ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
One of the options for control design is to set controller matrices as follows
\begin{equation}\label{eqn_gs17}
\begin{array}{ll}
A^c(\alpha)= A^c= -\epsilon_c I, & ~B^c(\alpha)=B^c=I, \\[5pt]
C^c(\alpha) = -K_i(\alpha), & ~D^c(\alpha)=-K_p(\alpha).
\end{array}
\end{equation}
which is a kind of PI control, where $K_i(\alpha)$ is the integral gain matrix, and $K_p(\alpha)$ is the proportional gain matrix.
Hence the control for the augmented system has the final form
\begin{equation} \label{eqn_gs64}
\begin{array}{l}
\left[
\begin{array}{c}
\dot{x}^c \\
v
\end{array}
\right] =
\left[
\begin{array}{ccc}
-\epsilon_c I & I & -I\\
K_i(\alpha) & K_p(\alpha) & -K_p(\alpha)
\end{array}
\right] ~
\left[
\begin{array}{c}
x^c \\
y \\
r
\end{array}
\right], ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
With these choices for control matrices, the control input is
\begin{equation}\label{eqn_gs19}
\begin{array}{l}
x^c = \displaystyle \int \! \left(-\epsilon_c x^c +(y-r) \right) \, \mathrm{d}\tau, \\[5pt]
v = - K_i(\alpha) x^c - K_p(\alpha) (y-r), ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
Figure~\ref{fig:GainSched_Cont_Struc}, shows schematically how the gain scheduling controller works.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.5\textwidth]{figs/GainSched_Cont_Struc02}
\caption{Gain scheduling controller diagram}
\label{fig:GainSched_Cont_Struc}
\end{figure}
The linearized closed loop system (\ref{eqn_gs62}) with controller (\ref{eqn_gs63}) becomes
\begin{equation} \label{eqn_gs21}
\begin{array}{l}
\underbrace{\left[
\begin{array}{c}
\delta \dot{x}^p \\
\delta \dot{u} \\
\dot{x}^c
\end{array}
\right]}_{\dot{x}} =
\underbrace{\left[
\begin{array}{ccc}
A^p(\alpha) &~~~ B^p(\alpha) &~~~ 0 \\
\eta_c D^c(\alpha) C^p(\alpha) &~~~ -\eta_c I+ D^c(\alpha) D^p(\alpha) & ~~~ \eta_c C^c(\alpha) \\
B^c(\alpha) C^p(\alpha) &~~~ B^c(\alpha)D^p(\alpha) &~~~ A^c(\alpha)
\end{array}
\right]}_{A_{cl}(\alpha)}
\underbrace{\left[
\begin{array}{c}
\delta x^p \\
\delta u \\
x^c
\end{array}
\right]}_{x} +
\underbrace{\left[
\begin{array}{c}
0 \\
-\eta_c D^c(\alpha) \\
- B^c(\alpha)
\end{array}
\right]}_{B_{cl}(\alpha)} \delta r, ~~~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
For the case where we have simplified output $\delta y = \delta x^p$, (i.e. $C^p(\alpha)=I, D^p(\alpha)=0$) the linearized closed loop system (\ref{eqn_gs62}) with controller (\ref{eqn_gs64}) becomes
\begin{equation} \label{eqn_gs22}
\begin{array}{l}
\underbrace{\left[
\begin{array}{c}
\delta \dot{x}^p \\
\delta \dot{u} \\
\dot{x}^c
\end{array}
\right]}_{\dot{x}} =
\underbrace{\left[
\begin{array}{ccc}
A^p(\alpha) &~~~ B^p(\alpha) &~~~ 0 \\
-\eta_c K_p(\alpha) &~~~ -\eta_c I & ~~~ -\eta_c K_i(\alpha) \\
I &~~~ 0 &~~~ -\epsilon_c I
\end{array}
\right]}_{A_{cl}(\alpha)}
\underbrace{\left[
\begin{array}{c}
\delta x^p \\
\delta u \\
x^c
\end{array}
\right]}_{x} +\underbrace{\left[
\begin{array}{c}
0 \\
\eta_c K_p(\alpha) \\
- I
\end{array}
\right]}_{B_{cl}(\alpha)} \delta r, ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
\subsection{Stability Analysis}
In closed loop system (\ref{eqn_gs22}), let $\delta r =0$, and consider the unforced linear time varying system
\begin{equation}\label{eqn_gs50}
\begin{array}{l}
\dot{x}=A_{cl}(\alpha) x, ~~ x(0)= x_0, ~~ \forall \alpha \in \Omega, \\[5pt]
\delta y= \delta x^p.
\end{array}
\end{equation}
\newtheorem{ass}{Assumption}
\begin{ass} \label{ass1}
The matrix $A_{cl}$ is bounded and Lipschitz continuous as follows
\begin{equation}\label{eqn_gs51}
\begin{array}{l}
||A_{cl}(t)|| \leq k_A, ~~ \forall t>0, \\[5pt]
||A_{cl}(t)-A_{cl}(\tau)|| \leq L_A ||t- \tau||, ~~ \forall t , \tau >0,
\end{array}
\end{equation}
\end{ass}
\begin{ass} \label{ass2}
The constant eigenvalues of matrix $A_{cl}(y)$ are uniformly bounded away from the closed complex right-half plane for all constant $y$.
\end{ass}
\newtheorem{thm}{Theorem}
\begin{thm} \label{thm1}
Consider system (\ref{eqn_gs50}), under assumptions \ref{ass1} and \ref{ass2}, then there exists constants $m$, $\lambda$, and $\epsilon > 0$ such that if
\begin{equation}\label{eqn_gs52}
||\dot{y}(t)|| \leq \epsilon_y, ~~ \forall t \in [0,T],
\end{equation}
then
\begin{equation}\label{eqn_gs53}
||x(t)|| \leq me^{-\lambda t} ||x_0||, ~~\forall t \in [0,T].
\end{equation}
\end{thm}
To analyse the stability of the nonlinear closed-loop system, we use a technique known as "global linearization" developed in \cite{lmi-boyd-1994}.
\begin{thm}\label{thm2}
Consider nonlinear system (\ref{eqn_gs112}), and assume there are a family of equilibrium points $(x_e,r_e)$ such that $F(x_e,r_e)=0$. Then $A^{nl}_{cl} = \frac{\partial F}{\partial x} \in S, ~\forall x$, where $S$ is a polytope, and it is described by a list of its vertices, i.e. in the form
\begin{equation}\label{eqn_gs113}
S:= \textbf{Co}\{ A^{nl}_{cl_1}, ..., A^{nl}_{cl_L} \},
\end{equation}
where $A^{nl}_{cl_i}$s are obtained by linearizing nonlinear system (\ref{eqn_gs112}) near equilibrium points (steady state condition), and also non-equilibrium points (transient condition). Now, assume their exist a common symmetric positive definite matrix $P=P^\mathsf{T} > 0$ such that: \\
\begin{equation}\label{eqn_gs54}
P A^{nl}_{cl_i}+A^{nl \mathsf{T}}_{cl_i} P < 0, ~~~ \forall i \in \{ 1,2,..., L \}.
\end{equation}
then system (\ref{eqn_gs112}) is absolutely stable. Since by design $A_{cl}(\alpha) \in S, ~\forall \alpha$, then system (\ref{eqn_gs50}) is also stable.
\end{thm}
If there are no hidden coupling terms involving $\delta y$, then the design of a stabilizing linear controller family can be assumed to guarantee stability of the linearized closed-loop system in a neighborhood of every $\alpha \in \Omega$. The closed-loop system is not restricted to remain in a neighborhood of any single equilibrium, but is assumed to be \emph{slowly-varying} and to have initial state sufficiently close to some equilibrium in S. Then the conclusion is that the closed-loop system remains in a neighborhood of the equilibrium manifold \cite{research-rugh-2000}. Using results developed in \cite{gainsched-shamma-1988}, we can figure out if a system is slowly-varying or not. Here we rewrite theorem Theorem 12 from \cite{research-rugh-2000}:
\begin{thm} \label{thm3}
For plant (\ref{eqn_gs1}), suppose the gain-scheduled controller (\ref{eqn_gs14}) is such that there are no hidden coupling terms and the eigenvalues of the linearized closed-loop system satisfy $Re[\lambda] \leq - \epsilon < 0$ for every $\alpha \in \Omega$. Then given $\rho > 0$ there exist positive constants $\mu$ and $\gamma$ such that the response of the nonlinear closed-loop system satisfies the following property. If the exogenous signal $|| \dot{r}(t)|| < \mu$, for $t \geq 0$, and if for some $\alpha \in \Omega$,
\begin{equation} \label{eqn_gs23}
\begin{array}{l}
\left|\left|
\left[
\begin{array}{c}
x^p(0) \\
u(0) \\
x^c(0)
\end{array}
\right] -
\left[
\begin{array}{c}
x^p_e(\alpha) \\
u_e(\alpha) \\
0
\end{array}
\right]
\right|\right| < \gamma,
\end{array}
\end{equation}
then
\begin{equation} \label{eqn_gs24}
\begin{array}{l}
\left|\left|
\left[
\begin{array}{c}
x^p(t) \\
u(t) \\
x^c(t)
\end{array}
\right] -
\left[
\begin{array}{c}
x^p_e(p(y(t))) \\
u_e(p(y(t))) \\
0
\end{array}
\right]
\right|\right| < \rho, ~~\forall t \geq 0.
\end{array}
\end{equation}
\end{thm}
\subsection{Integration Anti-Windup}
It is desirable to have integral action in the controller since the presence of an integral term eliminates steady state error in the controlled variable. Since the allowable values for control inputs are limited, if any controller reaches its limit, and error is produced form the difference of the control signal and the actual limited signal applied to the plant. This phenomenon is known as integral wind-up. Because of this, Integral Wind-Up Protection (IWUP) is used to reduce the effect of the integral term of the controller.
An approach to IWUP from \cite{development-martin-2008, control-csnak-2010} was adopted for our control problem. The main idea with this approach is to decrease the error seen by the integrators. This allows the integrator to increase to an appropriate value and decrease the size of the instantaneous change in magnitude when the controller becomes saturated. First, the generated control signal to the fuel-metering valve is subtracted from the saturation value. The resulting difference is then amplified by an integral feedback gain (IFB) and subtracted from the input to the integrator. The IFB is empirically tuned to provide adequate performance. The IFB is not gain scheduled, a constant value is sufficient for good performance.
\section{Turboshaft Engine Example}
We apply the developed gain-scheduling controller to a physics-based model of a turboshaft engine driving a variable pitch propeller developed in \cite{fitzgerald-model-2012, pakmehr-decentmodel-2011}. For a standard day at sea level condition we found five equilibrium points for linearizing the dynamics near them. The linearization matrices for these five equilibrium points and steady state values of the engine variables and control parameters are:
\begin{itemize}
\item Equilibrium Point 1 (Full Thrust):\\
$u_1^*=1.0,~ u_2^*=16 ~(\text{deg}),~ x_1^*=1.0,~ x_2^*=0.9524,~ T^*=255.8685 ~(N),~ \alpha^*=1.3810,$ and the matrices are
\begin{eqnarray} \label{eqn_gs70}
\begin{array}{c}
A_1=
\left[
\begin{array}{cc}
-5 & 0 \\
3.5 & -2.3
\end{array}
\right],~ B_1=
\left[
\begin{array}{cc}
1.4 & 0 \\
0.63 & -0.085
\end{array}
\right], ~ C_1=I, \\[10pt]
Ki_1=
\left[
\begin{array}{cc}
0.7 & 0.7 \\
0.7 & 0.6
\end{array}
\right],~ Kp_1=
\left[
\begin{array}{cc}
1.2 & 1.2 \\
1.2 & 1.2
\end{array}
\right].
\end{array}
\end{eqnarray}
\item Equilibrium Point 2:\\
$u_1^*=0.7,~ u_2^*=16 ~(\text{deg}),~ x_1^*=0.8826,~ x_2^*=0.6263,~ T^*=181.9711 ~(N),~ \alpha^*=1.0822,$ and the matrices are
\begin{eqnarray} \label{eqn_gs71}
\begin{array}{c}
A_2=
\left[
\begin{array}{cc}
-2.83 & -0.0008 \\
1.20 & -2.10
\end{array}
\right],~ B_2=
\left[
\begin{array}{cc}
1.14 & 0 \\
0.78 & -0.054
\end{array}
\right], ~ C_2=I, \\[10pt]
Ki_2=
\left[
\begin{array}{cc}
0.6 & 0.6 \\
0.6 & 0.5
\end{array}
\right],~ Kp_2=
\left[
\begin{array}{cc}
1.1 & 1.1 \\
1.1 & 1.1
\end{array}
\right].
\end{array}
\end{eqnarray}
\item Equilibrium Point 3 (Cruise):\\
$u_1^*=0.4685,~ u_2^*=16 ~(\text{deg}),~ x_1^*=0.7264,~ x_2^*=0.5,~ T^*=70.5125 ~(N),~ \alpha^*=0.8818,$ and the matrices are
\begin{eqnarray} \label{eqn_gs72}
\begin{array}{c}
A_3=
\left[
\begin{array}{cc}
-1.9 & 0.061 \\
0.45 & -1.1
\end{array}
\right],~ B_3=
\left[
\begin{array}{cc}
1.57 & 0 \\
0.3 & -0.023
\end{array}
\right], ~ C_3=I, \\[10pt]
Ki_3=
\left[
\begin{array}{cc}
0.5 & 0.5 \\
0.5 & 0.4
\end{array}
\right],~ Kp_3=
\left[
\begin{array}{cc}
1 & 1 \\
1 & 1
\end{array}
\right].
\end{array}
\end{eqnarray}
\item Equilibrium Point 4:\\
$u_1^*=0.3,~ u_2^*=16 ~(\text{deg}),~ x_1^*=0.5327,~ x_2^*=0.3678,~ T^*=38.155 ~(N),~ \alpha^*=0.6473,$ and the matrices are
\begin{eqnarray} \label{eqn_gs73}
\begin{array}{c}
A_4=
\left[
\begin{array}{cc}
-0.85 & 0.032 \\
0.32 & -0.64
\end{array}
\right],~ B_4=
\left[
\begin{array}{cc}
1.1 & 0 \\
0.17 & -0.011
\end{array}
\right], ~ C_4=I, \\[10pt]
Ki_4=
\left[
\begin{array}{cc}
0.4 & 0.4 \\
0.4 & 0.3
\end{array}
\right],~ Kp_4=
\left[
\begin{array}{cc}
0.8 & 0.8 \\
0.8 & 0.8
\end{array}
\right].
\end{array}
\end{eqnarray}
\item Equilibrium Point 5 (Idle):\\
$u_1^*=0.145,~ u_2^*=16 ~(\text{deg}),~ x_1^*=0.295,~ x_2^*=0.161,~ T^*=7.317 ~(N),~ \alpha^*=0.3361,$ and the matrices are
\begin{eqnarray} \label{eqn_gs74}
\begin{array}{c}
A_5=
\left[
\begin{array}{cc}
-0.38 & -0.0008 \\
0.26 & -0.34
\end{array}
\right], ~ B_5=
\left[
\begin{array}{cc}
0.7 & 0 \\
0.1 & -0.0024
\end{array}
\right], ~C_5= I, \\[10pt]
Ki_5=
\left[
\begin{array}{cc}
0.3 & 0.3 \\
0.3 & 0.2
\end{array}
\right],~ Kp_5=
\left[
\begin{array}{cc}
0.6 & 0.6 \\
0.6 & 0.6
\end{array}
\right].
\end{array}
\end{eqnarray}
\end{itemize}
Other controller parameters are set to
\begin{equation}\label{eqn_gs75}
\epsilon_c=1, ~~ \eta_c=3, ~~ Q= 3 \times I.
\end{equation}
To show the stability of the closed loop system, 20 different (10 equilibrium, and 10 non-equilibrium) linearizations have been used, to solve inequality (\ref{eqn_gs54}), in Matlab with the aid of YALMIP \cite{YALMIP-lofberg-2004} and SeDuMi \cite{sedumi-Sturm-2001} packages. The numerical value for the common matrix $P$ is:
\begin{eqnarray} \label{eqn_gs76}
P = \left[
\begin{array}{cccccc}
0.639 & 0.035 & 0.121 & -0.015 & -0.073 & -0.036 \\
0.034 & 0.391 & 0.036 & -0.002 & -0.103 & -0.029 \\
0.121 & 0.036 & 0.184 & -0.048 & -0.029 & -0.017 \\
-0.015 & -0.002 & -0.048 & 0.130 & 0.028 & 0.022 \\
-0.073 & -0.103 & -0.029 & 0.028 & 0.322 & 0.028 \\
-0.036 & -0.029 & -0.017 & 0.022 & 0.028 & 0.298
\end{array}
\right]
\end{eqnarray}
To show that the designed gain scheduled controller works properly on JetCat engine we used it to control the engine from idle to cruise condition and then again back to idle condition in a stable manner and with good performance. Simulation results are shown in figures \ref{fig_cg_01} to \ref{fig_cg_23}.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_01}}
\caption{Norm of closed-loop system matrix ($||A_{cl}(t)||$), and its rate of change ($||\dot{A}_{cl}(t)||$) }\label{fig_cg_01}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_02}}
\caption{Closed-loop system eigenvalues ($\lambda[A_{cl}(\alpha)]$)}\label{fig_cg_02}
\end{minipage}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_03}}
\caption{Scheduling Parameter ($\alpha(t)=||x(t)||$) and its rate of change ($\dot{\alpha}(t)=\frac{x^T \dot{x}}{||x(t)||}$)}\label{fig_cg_03}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_04}}
\caption{Norm of measured output of the system ($||y(t)||$), and its rate of change ($||\dot{y}(t)||$) }\label{fig_cg_04}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_01}, shows the history of the norm of closed-loop system matrix $||A_{cl}(t)||$, and its rate $||\dot{A}_{cl}(t)||$. As it can be seen the figure shows the boundedness of these two variables in accordance with Assumption \ref{ass1} where $k_A=7.4539$, and $L_A=1.0106$. Figure \ref{fig_cg_02}, shows the history of the closed-loop system matrix eigenvalues $\lambda \{ A_{cl} \}$. As it is apparent, all the six eigenvalues remain negative with the time change of the scheduling parameter $\alpha$, and hence satisfies assumption \ref{ass2} of the stability theorem.
Figure \ref{fig_cg_03}, shows the history of the scheduling parameter which is $\alpha=p(y)=||y||=||x||$. It is also shows the history of the switching function, which is defined based on the norm of the spool speed equilibrium values vector. As it is apparent from the plot, engine operated in the vicinity of at least three equilibrium points to be able to accelerate from idle to cruise condition. The norm of the scheduling parameter rate $\dot{\alpha}(t)=\frac{x^T \dot{x}}{||x(t)||}$, also has been plotted. Figure \ref{fig_cg_04}, shows the history the norms of the output vector and its rate. This satisfies the condition of Theorem \ref{thm1} with $\epsilon_y=0.0025$. Using formulas from \cite{gainsched-shamma-1988}, we can compute $m=496.7476$, and $\lambda=0.5271$.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_08}}
\caption{History of the unforced closed-loop system Lyapunov function $V(t)$, and its rate of change $\dot{V}(t)$ }\label{fig_cg_08}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_09}}
\caption{Rate of change of reference signals ($\dot{r}$)}\label{fig_cg_09}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_08}, shows the history of the quadratic time varying Lyapunov function of the unforced closed loop system (\ref{eqn_gs50}). As it is apparent, $V(t)=\delta X^T P(t)\delta X$, is decrescent and bounded from above and below. The history of $\dot{V}$, shows that it is non-positive for all $t>0$, so the exponential stability of the slowly varying system (\ref{eqn_gs50}) with a gains-scheduling controller is guaranteed. Figure \ref{fig_cg_09}, shows the rate of change of the reference signals for the outputs of the system. The outputs in this simulation are core and spool speed. $||\dot{r}|| < 0.15$, which corresponds to the assumption of the $||\dot{r}||$ boundedness in the theorem \ref{thm3}.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_10}}
\caption{Plant states: core and fan spool speeds} \label{fig_cg_10}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_11}}
\caption{Controller states}\label{fig_cg_11}
\end{minipage}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_12}}
\caption{Core and fan spool speeds vs. core and fan spool accelerations}\label{fig_cg_12}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_13}}
\caption{Core and fan spools accelerations}\label{fig_cg_13}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_10}, shows the history of the plant states which are core and fan spool speeds. Figure \ref{fig_cg_11}, shows the time histories of the controller states. Figure \ref{fig_cg_12}, shows the phase plot for core and fan spool dynamics. Figure \ref{fig_cg_13}, shows the time history of the fan and core spool accelerations, i.e. $\dot{N}_1$ and $\dot{N}_2$.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_14}}
\caption{Output: core spool speed and its reference signal}\label{fig_cg_14}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_15}}
\caption{Output: fan spool speed and its reference signal}\label{fig_cg_15}
\end{minipage}
\end{figure}
Figures \ref{fig_cg_14} and \ref{fig_cg_15}, show the core and fan spool speeds tracking their reference signals. Figure \ref{fig_cg_16}, shows the history of thrust and it is following its reference command from idle to cruise condition and then back to the idle for standard day, sea level condition. Figure \ref{fig_cg_17}, shows the control inputs to the augmented system, $v(t)=[v_1(t), v_2(t)]^T$, each element corresponding to one of the control inputs to the original system. To keep the engine dynamics within the limits of the operation for the available engine model, some limits have been defined on the augmented system control input, $-0.18 \leq v(t) \leq 0.18$. This limits will help to keep the fuel control input non-negative and also limits the rate of the control, $\dot{u}(t)$.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_16}}
\caption{Thrust and its reference signal}\label{fig_cg_16}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_17}}
\caption{Control inputs to the augmented system ($v(t)$)}\label{fig_cg_17}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_18}, shows time rate of fuel and prop pitch angle inputs. Figure \ref{fig_cg_19}, shows fuel flow and propeller pitch angle histories as control inputs. For better performance and also to keep the engine in the safe range of operation limits has been defined for the augmented control inputs. Figures \ref{fig_cg_20} and \ref{fig_cg_21}, show the baseline fuel controller integral ($K_i(\alpha)$) and proportional ($K_p(\alpha)$) gain matrices histories. These gains have been obtained by interpolation using the previously deigned fixed-gain controllers, each one corresponding to a equilibrium point of the engine. The numerical values of these gains are mentioned in equations (\ref{eqn_gs71}) to (\ref{eqn_gs73}).
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_18}}
\caption{Rate of change for fuel and prop pitch angle control inputs ($\dot{u}(t)$)}\label{fig_cg_18}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_19}}
\caption{Fuel and prop pitch angle control inputs ($u(t)$)}\label{fig_cg_19}
\end{minipage}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_20}}
\caption{Controllers integral gain matrix ($K_i(\alpha)$) parameters histories}\label{fig_cg_20}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_21}}
\caption{Controllers proportional gain matrix ($K_p(\alpha)$) parameters histories }\label{fig_cg_21}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_22}, shows JetCat SPT5 turboshaft engine compressor map. In this map the approximate stall line and also the operating line for this simulation has been shown. The engine operates in a safe region with a big stall margin during its acceleration from idle to cruise and again decelerating back to idle condition. Figure \ref{fig_cg_23}, shows the histories of turbine temperature, thrust specific fuel consumption (TSFC), compressor pressure ratio and corrected air flow rate.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_22}}
\caption{JetCat SPT5 engine compressor map with operating line}\label{fig_cg_22}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_23}}
\caption{Turbine temperature, TSFC, compressor overall pressure ratio and air flow rate histories}\label{fig_cg_23}
\end{minipage}
\end{figure}
Using gain-scheduling control technique, this simulation shows the possibility of controlling the linear parameter varying model of the turboshaft engine for a large transient throttle. This case study, simulates the engine accelerating from idle thrust to the cruise condition and then decelerates back to the idle condition in the standard day sea level condition. To tune the controller we need to trade off between the settling time and overshoot percentage. For large throttle transients, the existing controller regulates the controlled variable, additional limiters, will be developed to protect critical engine variables from exceeding physical bounds and to ensure safer operation.
\section{Conclusions}
We developed a gain-scheduling controller with stability guarantees for nonlinear gas turbine engine systems. Using global linearization and LMI techniques, we showed guaranteed absolute stability for closed loop gas turbine engine systems with gain-scheduling controllers. Simulation results presented to show the applicability of the proposed controller to the nonlinear physics-based JetCat SPT5 turboshaft engine model for large transients from idle to cruise condition and vice versa.
\section*{Acknowledgment}
This material is based upon the work supported by the Air Force Research Laboratory (AFRL).
\bibliographystyle{plain}
\section*{Abstract}
This manuscript aims to develop and describe gain scheduling control concept for a gas turbine engine which drives a variable pitch propeller. An architecture for gain-scheduling control is developed that controls the turboshaft engine for large thrust commands in stable fashion with good performance. Fuel flow and propeller pitch angle are the two control inputs of the system. New stability proof has been developed for gain scheduling control of gas turbine engines using global linearization and LMI techniques. This approach guarantees absolute stability of the closed loop gas turbine engines with gain-scheduling controllers.
\section*{Nomenclature}
$N_1$: Non-dimensional Fan Spool Speed\\
$N_2$: Non-dimensional Core Spool Speed\\
$T$: hrust (N)\\
TSFC: Thrust Specific Fuel Consumption\\
$\alpha$: Scheduling Parameter\\
$\sigma$: Singular Value\\
$\lambda$: Eigenvalue\\
\section{Introduction}
The gain-scheduling approach is perhaps one of the most popular nonlinear control design approaches which has
been widely and successfully applied in fields ranging from aerospace to process control \cite{research-rugh-2000, surveyGS-leith-2000}. Gain-scheduling, specifically has been used for gas turbine engine control, some of these works are \cite{lpv-balas-2002, lpv-gilbert-2010, lpv-bruzelius-2002, lpv-Shuqing-2010, approximate-zhao-2011, approximate-yu-2011}. In general, stability and control of gas turbine engines have been of interest to researchers and engineers from a variety of perspectives. Stability of axial flow fans operating in parallel has been investigated in \cite{fanStability-simon-1985}. An application of robust stability analysis tools for uncertain turbine engine systems is presented in \cite{robustAeroengine-arriffin-1997}. Application of the Linear-Quadratic-Gaussian with Loop-Transfer-Recovery methodology to design of a control system for a simplified turbofan engine model is considered in \cite{lqg-garg-1989}. A unified robust multivariable approach to propulsion control design has been developed in \cite{turbofanControl-fredrick-2000}. A simplified scheme for scheduling multivariable controllers for robust performance over a wide range of turbofan engine operating points is presented in \cite{turbofanSched-garg-1997}.
In the previous work by authors \cite{distributed-pakmehr-2009, decentralized-pakmehr-2010} controllers developed for single spool and twin spool turboshaft system. Those controllers were designed for small transients, and small throttle commands. In this work we develop a gain-scheduling control structure for JetCat SPT5 turboshaft engine using the method presented in \cite{gainsched-shamma-1988, research-rugh-2000, gain-shamma-2006, overview-shamma-2012}. the controller is designed to be used for entire flight envelope of the twin spool turboshaft engine.
In this manuscript, first a linear representation of the turbofan system dynamics is developed. Then control theoretic concepts for gain-scheduling control of this model is presented. The developed controller can be used for the entire flight envelope of the engine with guaranteed stability. Finally the simulation results for gain scheduling control of a physics-based nonlinear model of the JetCat SPT5 turboshaft engine are presented.
\section{Gain Scheduling Control Design}
Consider the nonlinear dynamical system
\begin{equation}\label{eqn_gs1}
\begin{array}{c}
\dot{x}^p(t)= f^p(x^p(t),u(t)),\\[5pt]
y(t)=g^p(x^p(t),u(t)),
\end{array}
\end{equation}
where $x^p \in \Re^n$ is the state vector, $u\in \Re^m$ is the control input vector, $y\in \Re^k$ is the output vector, $f^p(.)$ is an $n$-dimensional, and $g^p(.)$ is an $k$-dimensional differentiable nonlinear vector functions.
We want to design a feedback control such that $y(t) \rightarrow r(t)$ as $t \rightarrow \infty$, where $r(t) \in D_r \subset \Re^k$ is the output reference signals vector.
Assume that for each $r(t) \in D_r$, there is a unique pair $(x^p_e, u_e)$ that depends continuously on $r$ and satisfies the equations:
\begin{equation}\label{eqn_gs2}
\begin{array}{c}
0= f^p(x^p_e,u_e),\\[5pt]
r=g^p(x^p_e,u_e),
\end{array}
\end{equation}
in case of a constant $r$. $x^p_e$ is the desired equilibrium point and $u_e$ is the steady-state control that is needed to maintain equilibrium at $x^p_e$.
\newtheorem{deff}{Definition}
\begin{deff} \label{def1}
The functions $x^p_e(\alpha), u_e(\alpha)$, and $r_e(\alpha)$ define an equilibrium family for the plant (\ref{eqn_gs1}) on the set $\Omega$ if
\begin{equation}\label{eqn_gs222}
\begin{array}{l}
f^p(x^p_e(\alpha),u_e(\alpha), r_e(\alpha))=0, \\[5pt]
g^p(x^p_e(\alpha),u_e(\alpha))=r_e(\alpha), ~\alpha \in \Omega.
\end{array}
\end{equation}
\end{deff}
Let $\Omega \subset \Re^{m+n}$ be the region of interest for all possible system state and control vector $(x^p,u)$ during the system operation, and denote $x^{p*}_i$ and $u^*_i$, $i\in I = {1, 2, . . . , l}$, as a set of (constant) operating points located at some representative (and properly separated) points inside $\Omega$. Introduce a set of $l$ regions $\Omega_i$ centered at the chosen operating points $(x^{p*}_i, u^*_i)$, and denote their interiors as $\Omega_{i0}$, such that $\Omega_{j0} \bigcap \Omega_{k0}={\oslash}$ for all $j \neq k$, and $\bigcup_{i=1}^{l} \Omega_i=\Omega$. The linear model around each equilibrium point is
\begin{equation}\label{eqn_gs3}
\begin{array}{l}
\dot{x}^p = A^p_i (x^p-x_i^{p*}) + B^p_i (u-u_i^*),\\
y = C^p_i (x^-x_i^{p*}) + D^p_i (u-u_i^*) + y_i^* ,
\end{array}
\end{equation}
where the matrices are obtained as follows
\begin{equation}\label{eqn_gs4}
\begin{array}{l}
\displaystyle A^p_i=\frac{\partial f^p}{\partial x^p}|_{(x^{p*}_i, u^*_i)}, ~~ \forall (x^p,u) \in \Omega_i, \\[5pt]
\displaystyle B^p_i=\frac{\partial f^p}{\partial u}|_{(x^{p*}_i, u^*_i)}, ~~ \forall (x^p,u) \in \Omega_i, \\[5pt]
\displaystyle C^p_i=\frac{\partial g^p}{\partial x^p}|_{(x^{p*}_i, u^*_i)}, ~~ \forall (x^p,u) \in \Omega_i, \\[5pt]
\displaystyle D^p_i=\frac{\partial g^p}{\partial u}|_{(x^{p*}_i, u^*_i)}, ~~ \forall (x^p,u) \in \Omega_i.
\end{array}
\end{equation}
Here we assume that the common boundary of two regions $\Omega_j$ and $\Omega_z$ belongs to only one of $\Omega_j$ and $\Omega_z$. Note that at each moment, $(x^p,u)$ belongs to only one $\Omega_i$.
Performing linearizations at a series of trim points gives a linearization family described by
\begin{equation}\label{eqn_gs5}
\begin{array}{l}
\delta \dot{x}^p = A^p(\alpha) \delta x^p + B^p(\alpha) \delta u,\\[5pt]
\delta y = C^p(\alpha) \delta x^p + D^p(\alpha) \delta u.
\end{array}
\end{equation}
where
\begin{equation}\label{eqn_gs6}
\begin{array}{l}
\delta x^p = x^p-x^p_e(\alpha) \\[5pt]
\delta y = y-y_e(\alpha),\\[5pt]
\delta u = u-u_e(\alpha), ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
Gain scheduled controller for plant (\ref{eqn_gs5}), is designed as follows. First, a set of parameter values $\alpha_i$ are selected, which represent the range of the plant's dynamics, and a linear time-invariant controller for each is designed. Then, in between operating points, the controller gains are linearly interpolated such that for all frozen values of the parameters, the closed loop system has excellent properties, such as nominal stability and robust performance. To guarantee that the closed loop system will retain the feedback properties of the frozen-time designs, the scheduling variables should vary slowly withe respect to the system dynamics \cite{gainsched-shamma-1988}.
The parameter $\alpha$ called the scheduling variable in gain scheduling control. $A^p(\alpha), B^p(\alpha), C^p(\alpha)$, and $D^p(\alpha)$ are the parameterized linearization system matrices and $x^p_e(\alpha), u_e(\alpha)$, and $y_e(\alpha)$ are the parameterized steady-state system variables, which form the equilibrium manifold of system (\ref{eqn_gs1}). The subscript $'e'$ stands for steady-state throughout this paper.
Let the controller have the following structure
\begin{equation}\label{eqn_gs7}
\begin{array}{l}
\dot{x}^c = A^c(\alpha) \delta x^c + B^c(\alpha) [\delta y-\delta r],\\[5pt]
\delta u = C^c(\alpha) \delta x^c + D^c(\alpha) [\delta y-\delta r], ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
where
\begin{equation}\label{eqn_gs8}
\begin{array}{l}
\delta x^c = x^c-x^c_e(\alpha) \\[5pt]
\delta r = r-r_e(\alpha), ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
A standard realization of the parameterized controller can be written in the following form with the reference signal explicitly displayed
\begin{equation} \label{eqn_gs9}
\begin{array}{l}
\left[
\begin{array}{c}
\dot{x}^c \\
\delta u
\end{array}
\right] =
\left[
\begin{array}{ccc}
A^c(\alpha) & B^c(\alpha) & -B^c(\alpha) \\
C^c(\alpha) & D^c(\alpha) & -D^c(\alpha)
\end{array}
\right] ~
\left[
\begin{array}{c}
\delta x^c \\
\delta y \\
\delta r
\end{array}
\right], ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
Controller has the general form
\begin{equation}\label{eqn_gs10}
\begin{array}{c}
\dot{x}^c(t)= f^c(x^c(t),y(t), r(t)),\\[5pt]
u(t)=g^c(x^c(t),y(t), r(t)),
\end{array}
\end{equation}
with the input and output signals corresponding to the nonlinear plant (\ref{eqn_gs1}).
The objective in linearization scheduling is that the equilibrium family of the controller (\ref{eqn_gs10}) match the plant equilibrium family, so that the closed loop system maintains suitable trim values, and second the linearization family of the controller is the designed family of linear controllers \cite{research-rugh-2000}.
For the equilibrium conditions of plant (\ref{eqn_gs1}) and controller (\ref{eqn_gs10}) to match, there must exist a function $x^c_e(\alpha)$ such that
\begin{equation}\label{eqn_gs11}
\begin{array}{l}
0= f^c(x^c_e(\alpha),y_e(\alpha),r_e(\alpha)),\\[5pt]
u_e(\alpha)=g^c(x^c_e(\alpha),y_e(\alpha), r_e(\alpha)), ~~~ \forall \alpha \in \Omega,
\end{array}
\end{equation}
where
\begin{equation}\label{eqn_gs12}
\begin{array}{l}
\displaystyle A^c(\alpha)=\frac{\partial f^c}{\partial x^c}|_{(x^c_e(\alpha),y_e(\alpha),r_e(\alpha))}, \\[5pt]
\displaystyle B^c(\alpha)=\frac{\partial f^c}{\partial y}|_{(x^c_e(\alpha),y_e(\alpha),r_e(\alpha))}, \\[5pt]
\displaystyle C^c(\alpha)=\frac{\partial g^c}{\partial x^c}|_{(x^c_e(\alpha),y_e(\alpha),r_e(\alpha))}, \\[5pt]
\displaystyle D^c(\alpha)=\frac{\partial g^c}{\partial y}|_{(x^c_e(\alpha),y_e(\alpha),r_e(\alpha))}, ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
So the controller family has the form
\begin{equation}\label{eqn_gs13}
\begin{array}{l}
\dot{x}^c = A^c(\alpha) [x^c-x^c_e(\alpha)] + B^c(\alpha) [y-r],\\[5pt]
u = C^c(\alpha) [x^c-x^c_e(\alpha)]+ D^c(\alpha) [y-r]+ u_e(\alpha), ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
Note that $r_e(\alpha)=y_e(\alpha)$, as a result $\delta y - \delta r=y-r$. The scheduling parameter $\alpha$ is treated as a parameter throughout the design process, and then it becomes a time-varying input signal to the gain-scheduled controller implementation through the dependence $\alpha(t)=p(y(t))$. Thus the gain-scheduled controller becomes
\begin{equation}\label{eqn_gs14}
\begin{array}{l}
\dot{x}^c = A^c(p(y)) [x^c-x^c_e(p(y))] + B^c(p(y)) [y-r],\\[5pt]
u = C^c(p(y)) [x^c-x^c_e(p(y))]+ D^c(p(y)) [y-r]+ u_e(p(y)).
\end{array}
\end{equation}
Linearization of (\ref{eqn_gs14}) about an equilibrium specified by $\alpha$ gives
\begin{equation}\label{eqn_gs15}
\begin{array}{l}
\delta \dot{x}^c = A^c(\alpha) \delta x^c + B^c(\alpha) [y-r] \displaystyle -[A^c(\alpha) \frac{\partial x^c_e (\alpha)}{\partial \alpha}] \times [\frac{\partial p }{\partial y} (y_e(\alpha)) (y-r)], \\[5pt]
\displaystyle \delta u = C^c(\alpha) \delta x^c + D^c(\alpha) [y-r] \displaystyle +[\frac{\partial u_e (\alpha)}{\partial \alpha} - C^c(\alpha) \frac{\partial x^c_e (\alpha)}{\partial \alpha}] \displaystyle \times [\frac{\partial p }{\partial y} (y_e(\alpha)) (y-r)].
\end{array}
\end{equation}
Comparing this to (\ref{eqn_gs9}), we see there are additional terms, which we refer to them as hidden coupling terms \cite{research-rugh-2000}.
In order to get rid of these terms we have to design the controller such that
\begin{equation}\label{eqn_gs16}
\begin{array}{l}
\displaystyle A^c(\alpha) \frac{\partial x^c_e (\alpha)}{\partial \alpha} = 0, \\[5pt]
\displaystyle \frac{\partial u_e (\alpha)}{\partial \alpha} - C^c(\alpha) \frac{\partial x^c_e (\alpha)}{\partial \alpha} =0.
\end{array}
\end{equation}
It is not always easy to come up with solutions to satisfy this condition. In order to make the design process easier, we can augment integrators at the plant input, so there is no need for equilibrium control value. By augmenting integrators at the plant (\ref{eqn_gs1}) input we obtain
\begin{equation} \label{eqn_gs61}
\begin{array}{l}
\left[
\begin{array}{c}
\dot{x}^p(t) \\
\dot{u}(t)
\end{array}
\right] =
\left(
\begin{array}{c}
f^p(x^p(t),u(t)) \\
-\eta_c u(t)
\end{array}
\right) +
\left[
\begin{array}{c}
0 \\
\eta_c \times I
\end{array}
\right] v(t).
\end{array}
\end{equation}
The new controller has the general form
\begin{equation}\label{eqn_gs110}
\begin{array}{c}
\dot{x}^c(t)= f^c(x^c(t),y(t), r(t)),\\[5pt]
v(t)=g^c(x^c(t),y(t), r(t)),
\end{array}
\end{equation}
with the input and output signals corresponding to the nonlinear plant (\ref{eqn_gs1}).
Combining (\ref{eqn_gs61}) and (\ref{eqn_gs110})leads to
\begin{equation} \label{eqn_gs111}
\begin{array}{l}
\underbrace{\left[
\begin{array}{c}
\dot{x}^p(t) \\
\dot{u}(t) \\
\dot{x}^c(t)
\end{array}
\right]}_{\dot{x}} =
\underbrace{\left(
\begin{array}{c}
f^p(x^p(t),u(t)) \\
-\eta_c u(t) \\
f^c(x^c(t),g^p(x^p(t),u(t)) ,r(t))
\end{array}
\right)}_{f(x,r)} +
\underbrace{\left[
\begin{array}{c}
0 \\
\eta_c \times I \\
0
\end{array}
\right]}_{B} v(t). \\[5pt]
v(t)=\underbrace{g^c(x^c(t),g^p(x^p(t),u(t)), r(t))}_{g(x,r)},
\end{array}
\end{equation}
Then the closed loop nonlinear system is
\begin{equation}\label{eqn_gs112}
\begin{array}{l}
\dot{x}(t)= f(x(t),r(t))+B g(x(t),r(t )),\\[5pt]
~~~~~~ =F(x(t),r(t))
\end{array}
\end{equation}
The augmented linear family of systems for (\ref{eqn_gs61}) becomes
\begin{equation}\label{eqn_gs62}
\begin{array}{l}
\underbrace{\left[
\begin{array}{c}
\dot{x}^p(t) \\
\dot{u}(t)
\end{array}
\right]}_{\dot{x}_{aug}} =
\underbrace{\left[
\begin{array}{cc}
A^p(\alpha) & B^p(\alpha)\\
0 & -\eta_c \times I
\end{array}
\right]}_{A_{aug}(\alpha)}
\underbrace{\left[
\begin{array}{c}
\delta x^p \\
\delta u
\end{array}
\right]}_{\delta x_{aug}}+
\underbrace{\left[
\begin{array}{c}
0 \\
\eta_c \times I
\end{array}
\right]}_{B_{aug}} v(t) ,\\[5pt]
\delta y = \underbrace{[C^p(\alpha), D^p(\alpha)]}_{C_{aug}(\alpha)}
\underbrace{\left[
\begin{array}{c}
\delta x^p \\
\delta u
\end{array}
\right]}_{\delta x_{aug}}.
\end{array}
\end{equation}
Now, the control realization for this system is
\begin{equation}\label{eqn_gs63}
\begin{array}{l}
\dot{x}^c = A^c(\alpha) x^c + B^c(\alpha) [y-r],\\[5pt]
v = C^c(\alpha) x^c+ D^c(\alpha) [y-r] ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
One of the options for control design is to set controller matrices as follows
\begin{equation}\label{eqn_gs17}
\begin{array}{ll}
A^c(\alpha)= A^c= -\epsilon_c I, & ~B^c(\alpha)=B^c=I, \\[5pt]
C^c(\alpha) = -K_i(\alpha), & ~D^c(\alpha)=-K_p(\alpha).
\end{array}
\end{equation}
which is a kind of PI control, where $K_i(\alpha)$ is the integral gain matrix, and $K_p(\alpha)$ is the proportional gain matrix.
Hence the control for the augmented system has the final form
\begin{equation} \label{eqn_gs64}
\begin{array}{l}
\left[
\begin{array}{c}
\dot{x}^c \\
v
\end{array}
\right] =
\left[
\begin{array}{ccc}
-\epsilon_c I & I & -I\\
K_i(\alpha) & K_p(\alpha) & -K_p(\alpha)
\end{array}
\right] ~
\left[
\begin{array}{c}
x^c \\
y \\
r
\end{array}
\right], ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
With these choices for control matrices, the control input is
\begin{equation}\label{eqn_gs19}
\begin{array}{l}
x^c = \displaystyle \int \! \left(-\epsilon_c x^c +(y-r) \right) \, \mathrm{d}\tau, \\[5pt]
v = - K_i(\alpha) x^c - K_p(\alpha) (y-r), ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
Figure~\ref{fig:GainSched_Cont_Struc}, shows schematically how the gain scheduling controller works.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.5\textwidth]{figs/GainSched_Cont_Struc02}
\caption{Gain scheduling controller diagram}
\label{fig:GainSched_Cont_Struc}
\end{figure}
The linearized closed loop system (\ref{eqn_gs62}) with controller (\ref{eqn_gs63}) becomes
\begin{equation} \label{eqn_gs21}
\begin{array}{l}
\underbrace{\left[
\begin{array}{c}
\delta \dot{x}^p \\
\delta \dot{u} \\
\dot{x}^c
\end{array}
\right]}_{\dot{x}} =
\underbrace{\left[
\begin{array}{ccc}
A^p(\alpha) &~~~ B^p(\alpha) &~~~ 0 \\
\eta_c D^c(\alpha) C^p(\alpha) &~~~ -\eta_c I+ D^c(\alpha) D^p(\alpha) & ~~~ \eta_c C^c(\alpha) \\
B^c(\alpha) C^p(\alpha) &~~~ B^c(\alpha)D^p(\alpha) &~~~ A^c(\alpha)
\end{array}
\right]}_{A_{cl}(\alpha)}
\underbrace{\left[
\begin{array}{c}
\delta x^p \\
\delta u \\
x^c
\end{array}
\right]}_{x} +
\underbrace{\left[
\begin{array}{c}
0 \\
-\eta_c D^c(\alpha) \\
- B^c(\alpha)
\end{array}
\right]}_{B_{cl}(\alpha)} \delta r, ~~~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
For the case where we have simplified output $\delta y = \delta x^p$, (i.e. $C^p(\alpha)=I, D^p(\alpha)=0$) the linearized closed loop system (\ref{eqn_gs62}) with controller (\ref{eqn_gs64}) becomes
\begin{equation} \label{eqn_gs22}
\begin{array}{l}
\underbrace{\left[
\begin{array}{c}
\delta \dot{x}^p \\
\delta \dot{u} \\
\dot{x}^c
\end{array}
\right]}_{\dot{x}} =
\underbrace{\left[
\begin{array}{ccc}
A^p(\alpha) &~~~ B^p(\alpha) &~~~ 0 \\
-\eta_c K_p(\alpha) &~~~ -\eta_c I & ~~~ -\eta_c K_i(\alpha) \\
I &~~~ 0 &~~~ -\epsilon_c I
\end{array}
\right]}_{A_{cl}(\alpha)}
\underbrace{\left[
\begin{array}{c}
\delta x^p \\
\delta u \\
x^c
\end{array}
\right]}_{x} +\underbrace{\left[
\begin{array}{c}
0 \\
\eta_c K_p(\alpha) \\
- I
\end{array}
\right]}_{B_{cl}(\alpha)} \delta r, ~~~ \forall \alpha \in \Omega.
\end{array}
\end{equation}
\subsection{Stability Analysis}
In closed loop system (\ref{eqn_gs22}), let $\delta r =0$, and consider the unforced linear time varying system
\begin{equation}\label{eqn_gs50}
\begin{array}{l}
\dot{x}=A_{cl}(\alpha) x, ~~ x(0)= x_0, ~~ \forall \alpha \in \Omega, \\[5pt]
\delta y= \delta x^p.
\end{array}
\end{equation}
\newtheorem{ass}{Assumption}
\begin{ass} \label{ass1}
The matrix $A_{cl}$ is bounded and Lipschitz continuous as follows
\begin{equation}\label{eqn_gs51}
\begin{array}{l}
||A_{cl}(t)|| \leq k_A, ~~ \forall t>0, \\[5pt]
||A_{cl}(t)-A_{cl}(\tau)|| \leq L_A ||t- \tau||, ~~ \forall t , \tau >0,
\end{array}
\end{equation}
\end{ass}
\begin{ass} \label{ass2}
The constant eigenvalues of matrix $A_{cl}(y)$ are uniformly bounded away from the closed complex right-half plane for all constant $y$.
\end{ass}
\newtheorem{thm}{Theorem}
\begin{thm} \label{thm1}
Consider system (\ref{eqn_gs50}), under assumptions \ref{ass1} and \ref{ass2}, then there exists constants $m$, $\lambda$, and $\epsilon > 0$ such that if
\begin{equation}\label{eqn_gs52}
||\dot{y}(t)|| \leq \epsilon_y, ~~ \forall t \in [0,T],
\end{equation}
then
\begin{equation}\label{eqn_gs53}
||x(t)|| \leq me^{-\lambda t} ||x_0||, ~~\forall t \in [0,T].
\end{equation}
\end{thm}
To analyse the stability of the nonlinear closed-loop system, we use a technique known as "global linearization" developed in \cite{lmi-boyd-1994}.
\begin{thm}\label{thm2}
Consider nonlinear system (\ref{eqn_gs112}), and assume there are a family of equilibrium points $(x_e,r_e)$ such that $F(x_e,r_e)=0$. Then $A^{nl}_{cl} = \frac{\partial F}{\partial x} \in S, ~\forall x$, where $S$ is a polytope, and it is described by a list of its vertices, i.e. in the form
\begin{equation}\label{eqn_gs113}
S:= \textbf{Co}\{ A^{nl}_{cl_1}, ..., A^{nl}_{cl_L} \},
\end{equation}
where $A^{nl}_{cl_i}$s are obtained by linearizing nonlinear system (\ref{eqn_gs112}) near equilibrium points (steady state condition), and also non-equilibrium points (transient condition). Now, assume their exist a common symmetric positive definite matrix $P=P^\mathsf{T} > 0$ such that: \\
\begin{equation}\label{eqn_gs54}
P A^{nl}_{cl_i}+A^{nl \mathsf{T}}_{cl_i} P < 0, ~~~ \forall i \in \{ 1,2,..., L \}.
\end{equation}
then system (\ref{eqn_gs112}) is absolutely stable. Since by design $A_{cl}(\alpha) \in S, ~\forall \alpha$, then system (\ref{eqn_gs50}) is also stable.
\end{thm}
If there are no hidden coupling terms involving $\delta y$, then the design of a stabilizing linear controller family can be assumed to guarantee stability of the linearized closed-loop system in a neighborhood of every $\alpha \in \Omega$. The closed-loop system is not restricted to remain in a neighborhood of any single equilibrium, but is assumed to be \emph{slowly-varying} and to have initial state sufficiently close to some equilibrium in S. Then the conclusion is that the closed-loop system remains in a neighborhood of the equilibrium manifold \cite{research-rugh-2000}. Using results developed in \cite{gainsched-shamma-1988}, we can figure out if a system is slowly-varying or not. Here we rewrite theorem Theorem 12 from \cite{research-rugh-2000}:
\begin{thm} \label{thm3}
For plant (\ref{eqn_gs1}), suppose the gain-scheduled controller (\ref{eqn_gs14}) is such that there are no hidden coupling terms and the eigenvalues of the linearized closed-loop system satisfy $Re[\lambda] \leq - \epsilon < 0$ for every $\alpha \in \Omega$. Then given $\rho > 0$ there exist positive constants $\mu$ and $\gamma$ such that the response of the nonlinear closed-loop system satisfies the following property. If the exogenous signal $|| \dot{r}(t)|| < \mu$, for $t \geq 0$, and if for some $\alpha \in \Omega$,
\begin{equation} \label{eqn_gs23}
\begin{array}{l}
\left|\left|
\left[
\begin{array}{c}
x^p(0) \\
u(0) \\
x^c(0)
\end{array}
\right] -
\left[
\begin{array}{c}
x^p_e(\alpha) \\
u_e(\alpha) \\
0
\end{array}
\right]
\right|\right| < \gamma,
\end{array}
\end{equation}
then
\begin{equation} \label{eqn_gs24}
\begin{array}{l}
\left|\left|
\left[
\begin{array}{c}
x^p(t) \\
u(t) \\
x^c(t)
\end{array}
\right] -
\left[
\begin{array}{c}
x^p_e(p(y(t))) \\
u_e(p(y(t))) \\
0
\end{array}
\right]
\right|\right| < \rho, ~~\forall t \geq 0.
\end{array}
\end{equation}
\end{thm}
\subsection{Integration Anti-Windup}
It is desirable to have integral action in the controller since the presence of an integral term eliminates steady state error in the controlled variable. Since the allowable values for control inputs are limited, if any controller reaches its limit, and error is produced form the difference of the control signal and the actual limited signal applied to the plant. This phenomenon is known as integral wind-up. Because of this, Integral Wind-Up Protection (IWUP) is used to reduce the effect of the integral term of the controller.
An approach to IWUP from \cite{development-martin-2008, control-csnak-2010} was adopted for our control problem. The main idea with this approach is to decrease the error seen by the integrators. This allows the integrator to increase to an appropriate value and decrease the size of the instantaneous change in magnitude when the controller becomes saturated. First, the generated control signal to the fuel-metering valve is subtracted from the saturation value. The resulting difference is then amplified by an integral feedback gain (IFB) and subtracted from the input to the integrator. The IFB is empirically tuned to provide adequate performance. The IFB is not gain scheduled, a constant value is sufficient for good performance.
\section{Turboshaft Engine Example}
We apply the developed gain-scheduling controller to a physics-based model of a turboshaft engine driving a variable pitch propeller developed in \cite{fitzgerald-model-2012, pakmehr-decentmodel-2011}. For a standard day at sea level condition we found five equilibrium points for linearizing the dynamics near them. The linearization matrices for these five equilibrium points and steady state values of the engine variables and control parameters are:
\begin{itemize}
\item Equilibrium Point 1 (Full Thrust):\\
$u_1^*=1.0,~ u_2^*=16 ~(\text{deg}),~ x_1^*=1.0,~ x_2^*=0.9524,~ T^*=255.8685 ~(N),~ \alpha^*=1.3810,$ and the matrices are
\begin{eqnarray} \label{eqn_gs70}
\begin{array}{c}
A_1=
\left[
\begin{array}{cc}
-5 & 0 \\
3.5 & -2.3
\end{array}
\right],~ B_1=
\left[
\begin{array}{cc}
1.4 & 0 \\
0.63 & -0.085
\end{array}
\right], ~ C_1=I, \\[10pt]
Ki_1=
\left[
\begin{array}{cc}
0.7 & 0.7 \\
0.7 & 0.6
\end{array}
\right],~ Kp_1=
\left[
\begin{array}{cc}
1.2 & 1.2 \\
1.2 & 1.2
\end{array}
\right].
\end{array}
\end{eqnarray}
\item Equilibrium Point 2:\\
$u_1^*=0.7,~ u_2^*=16 ~(\text{deg}),~ x_1^*=0.8826,~ x_2^*=0.6263,~ T^*=181.9711 ~(N),~ \alpha^*=1.0822,$ and the matrices are
\begin{eqnarray} \label{eqn_gs71}
\begin{array}{c}
A_2=
\left[
\begin{array}{cc}
-2.83 & -0.0008 \\
1.20 & -2.10
\end{array}
\right],~ B_2=
\left[
\begin{array}{cc}
1.14 & 0 \\
0.78 & -0.054
\end{array}
\right], ~ C_2=I, \\[10pt]
Ki_2=
\left[
\begin{array}{cc}
0.6 & 0.6 \\
0.6 & 0.5
\end{array}
\right],~ Kp_2=
\left[
\begin{array}{cc}
1.1 & 1.1 \\
1.1 & 1.1
\end{array}
\right].
\end{array}
\end{eqnarray}
\item Equilibrium Point 3 (Cruise):\\
$u_1^*=0.4685,~ u_2^*=16 ~(\text{deg}),~ x_1^*=0.7264,~ x_2^*=0.5,~ T^*=70.5125 ~(N),~ \alpha^*=0.8818,$ and the matrices are
\begin{eqnarray} \label{eqn_gs72}
\begin{array}{c}
A_3=
\left[
\begin{array}{cc}
-1.9 & 0.061 \\
0.45 & -1.1
\end{array}
\right],~ B_3=
\left[
\begin{array}{cc}
1.57 & 0 \\
0.3 & -0.023
\end{array}
\right], ~ C_3=I, \\[10pt]
Ki_3=
\left[
\begin{array}{cc}
0.5 & 0.5 \\
0.5 & 0.4
\end{array}
\right],~ Kp_3=
\left[
\begin{array}{cc}
1 & 1 \\
1 & 1
\end{array}
\right].
\end{array}
\end{eqnarray}
\item Equilibrium Point 4:\\
$u_1^*=0.3,~ u_2^*=16 ~(\text{deg}),~ x_1^*=0.5327,~ x_2^*=0.3678,~ T^*=38.155 ~(N),~ \alpha^*=0.6473,$ and the matrices are
\begin{eqnarray} \label{eqn_gs73}
\begin{array}{c}
A_4=
\left[
\begin{array}{cc}
-0.85 & 0.032 \\
0.32 & -0.64
\end{array}
\right],~ B_4=
\left[
\begin{array}{cc}
1.1 & 0 \\
0.17 & -0.011
\end{array}
\right], ~ C_4=I, \\[10pt]
Ki_4=
\left[
\begin{array}{cc}
0.4 & 0.4 \\
0.4 & 0.3
\end{array}
\right],~ Kp_4=
\left[
\begin{array}{cc}
0.8 & 0.8 \\
0.8 & 0.8
\end{array}
\right].
\end{array}
\end{eqnarray}
\item Equilibrium Point 5 (Idle):\\
$u_1^*=0.145,~ u_2^*=16 ~(\text{deg}),~ x_1^*=0.295,~ x_2^*=0.161,~ T^*=7.317 ~(N),~ \alpha^*=0.3361,$ and the matrices are
\begin{eqnarray} \label{eqn_gs74}
\begin{array}{c}
A_5=
\left[
\begin{array}{cc}
-0.38 & -0.0008 \\
0.26 & -0.34
\end{array}
\right], ~ B_5=
\left[
\begin{array}{cc}
0.7 & 0 \\
0.1 & -0.0024
\end{array}
\right], ~C_5= I, \\[10pt]
Ki_5=
\left[
\begin{array}{cc}
0.3 & 0.3 \\
0.3 & 0.2
\end{array}
\right],~ Kp_5=
\left[
\begin{array}{cc}
0.6 & 0.6 \\
0.6 & 0.6
\end{array}
\right].
\end{array}
\end{eqnarray}
\end{itemize}
Other controller parameters are set to
\begin{equation}\label{eqn_gs75}
\epsilon_c=1, ~~ \eta_c=3, ~~ Q= 3 \times I.
\end{equation}
To show the stability of the closed loop system, 20 different (10 equilibrium, and 10 non-equilibrium) linearizations have been used, to solve inequality (\ref{eqn_gs54}), in Matlab with the aid of YALMIP \cite{YALMIP-lofberg-2004} and SeDuMi \cite{sedumi-Sturm-2001} packages. The numerical value for the common matrix $P$ is:
\begin{eqnarray} \label{eqn_gs76}
P = \left[
\begin{array}{cccccc}
0.639 & 0.035 & 0.121 & -0.015 & -0.073 & -0.036 \\
0.034 & 0.391 & 0.036 & -0.002 & -0.103 & -0.029 \\
0.121 & 0.036 & 0.184 & -0.048 & -0.029 & -0.017 \\
-0.015 & -0.002 & -0.048 & 0.130 & 0.028 & 0.022 \\
-0.073 & -0.103 & -0.029 & 0.028 & 0.322 & 0.028 \\
-0.036 & -0.029 & -0.017 & 0.022 & 0.028 & 0.298
\end{array}
\right]
\end{eqnarray}
To show that the designed gain scheduled controller works properly on JetCat engine we used it to control the engine from idle to cruise condition and then again back to idle condition in a stable manner and with good performance. Simulation results are shown in figures \ref{fig_cg_01} to \ref{fig_cg_23}.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_01}}
\caption{Norm of closed-loop system matrix ($||A_{cl}(t)||$), and its rate of change ($||\dot{A}_{cl}(t)||$) }\label{fig_cg_01}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_02}}
\caption{Closed-loop system eigenvalues ($\lambda[A_{cl}(\alpha)]$)}\label{fig_cg_02}
\end{minipage}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_03}}
\caption{Scheduling Parameter ($\alpha(t)=||x(t)||$) and its rate of change ($\dot{\alpha}(t)=\frac{x^T \dot{x}}{||x(t)||}$)}\label{fig_cg_03}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_04}}
\caption{Norm of measured output of the system ($||y(t)||$), and its rate of change ($||\dot{y}(t)||$) }\label{fig_cg_04}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_01}, shows the history of the norm of closed-loop system matrix $||A_{cl}(t)||$, and its rate $||\dot{A}_{cl}(t)||$. As it can be seen the figure shows the boundedness of these two variables in accordance with Assumption \ref{ass1} where $k_A=7.4539$, and $L_A=1.0106$. Figure \ref{fig_cg_02}, shows the history of the closed-loop system matrix eigenvalues $\lambda \{ A_{cl} \}$. As it is apparent, all the six eigenvalues remain negative with the time change of the scheduling parameter $\alpha$, and hence satisfies assumption \ref{ass2} of the stability theorem.
Figure \ref{fig_cg_03}, shows the history of the scheduling parameter which is $\alpha=p(y)=||y||=||x||$. It is also shows the history of the switching function, which is defined based on the norm of the spool speed equilibrium values vector. As it is apparent from the plot, engine operated in the vicinity of at least three equilibrium points to be able to accelerate from idle to cruise condition. The norm of the scheduling parameter rate $\dot{\alpha}(t)=\frac{x^T \dot{x}}{||x(t)||}$, also has been plotted. Figure \ref{fig_cg_04}, shows the history the norms of the output vector and its rate. This satisfies the condition of Theorem \ref{thm1} with $\epsilon_y=0.0025$. Using formulas from \cite{gainsched-shamma-1988}, we can compute $m=496.7476$, and $\lambda=0.5271$.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_08}}
\caption{History of the unforced closed-loop system Lyapunov function $V(t)$, and its rate of change $\dot{V}(t)$ }\label{fig_cg_08}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_09}}
\caption{Rate of change of reference signals ($\dot{r}$)}\label{fig_cg_09}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_08}, shows the history of the quadratic time varying Lyapunov function of the unforced closed loop system (\ref{eqn_gs50}). As it is apparent, $V(t)=\delta X^T P(t)\delta X$, is decrescent and bounded from above and below. The history of $\dot{V}$, shows that it is non-positive for all $t>0$, so the exponential stability of the slowly varying system (\ref{eqn_gs50}) with a gains-scheduling controller is guaranteed. Figure \ref{fig_cg_09}, shows the rate of change of the reference signals for the outputs of the system. The outputs in this simulation are core and spool speed. $||\dot{r}|| < 0.15$, which corresponds to the assumption of the $||\dot{r}||$ boundedness in the theorem \ref{thm3}.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_10}}
\caption{Plant states: core and fan spool speeds} \label{fig_cg_10}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_11}}
\caption{Controller states}\label{fig_cg_11}
\end{minipage}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_12}}
\caption{Core and fan spool speeds vs. core and fan spool accelerations}\label{fig_cg_12}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_13}}
\caption{Core and fan spools accelerations}\label{fig_cg_13}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_10}, shows the history of the plant states which are core and fan spool speeds. Figure \ref{fig_cg_11}, shows the time histories of the controller states. Figure \ref{fig_cg_12}, shows the phase plot for core and fan spool dynamics. Figure \ref{fig_cg_13}, shows the time history of the fan and core spool accelerations, i.e. $\dot{N}_1$ and $\dot{N}_2$.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_14}}
\caption{Output: core spool speed and its reference signal}\label{fig_cg_14}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_15}}
\caption{Output: fan spool speed and its reference signal}\label{fig_cg_15}
\end{minipage}
\end{figure}
Figures \ref{fig_cg_14} and \ref{fig_cg_15}, show the core and fan spool speeds tracking their reference signals. Figure \ref{fig_cg_16}, shows the history of thrust and it is following its reference command from idle to cruise condition and then back to the idle for standard day, sea level condition. Figure \ref{fig_cg_17}, shows the control inputs to the augmented system, $v(t)=[v_1(t), v_2(t)]^T$, each element corresponding to one of the control inputs to the original system. To keep the engine dynamics within the limits of the operation for the available engine model, some limits have been defined on the augmented system control input, $-0.18 \leq v(t) \leq 0.18$. This limits will help to keep the fuel control input non-negative and also limits the rate of the control, $\dot{u}(t)$.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_16}}
\caption{Thrust and its reference signal}\label{fig_cg_16}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_17}}
\caption{Control inputs to the augmented system ($v(t)$)}\label{fig_cg_17}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_18}, shows time rate of fuel and prop pitch angle inputs. Figure \ref{fig_cg_19}, shows fuel flow and propeller pitch angle histories as control inputs. For better performance and also to keep the engine in the safe range of operation limits has been defined for the augmented control inputs. Figures \ref{fig_cg_20} and \ref{fig_cg_21}, show the baseline fuel controller integral ($K_i(\alpha)$) and proportional ($K_p(\alpha)$) gain matrices histories. These gains have been obtained by interpolation using the previously deigned fixed-gain controllers, each one corresponding to a equilibrium point of the engine. The numerical values of these gains are mentioned in equations (\ref{eqn_gs71}) to (\ref{eqn_gs73}).
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_18}}
\caption{Rate of change for fuel and prop pitch angle control inputs ($\dot{u}(t)$)}\label{fig_cg_18}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_19}}
\caption{Fuel and prop pitch angle control inputs ($u(t)$)}\label{fig_cg_19}
\end{minipage}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_20}}
\caption{Controllers integral gain matrix ($K_i(\alpha)$) parameters histories}\label{fig_cg_20}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_21}}
\caption{Controllers proportional gain matrix ($K_p(\alpha)$) parameters histories }\label{fig_cg_21}
\end{minipage}
\end{figure}
Figure \ref{fig_cg_22}, shows JetCat SPT5 turboshaft engine compressor map. In this map the approximate stall line and also the operating line for this simulation has been shown. The engine operates in a safe region with a big stall margin during its acceleration from idle to cruise and again decelerating back to idle condition. Figure \ref{fig_cg_23}, shows the histories of turbine temperature, thrust specific fuel consumption (TSFC), compressor pressure ratio and corrected air flow rate.
\begin{figure}[!ht]
\centering
\begin{minipage}[l]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_22}}
\caption{JetCat SPT5 engine compressor map with operating line}\label{fig_cg_22}
\end{minipage}
\begin{minipage}[r]{3.2in}
\centering
\resizebox{3.2in}{!}{\includegraphics{./figs/CentLin_23}}
\caption{Turbine temperature, TSFC, compressor overall pressure ratio and air flow rate histories}\label{fig_cg_23}
\end{minipage}
\end{figure}
Using gain-scheduling control technique, this simulation shows the possibility of controlling the linear parameter varying model of the turboshaft engine for a large transient throttle. This case study, simulates the engine accelerating from idle thrust to the cruise condition and then decelerates back to the idle condition in the standard day sea level condition. To tune the controller we need to trade off between the settling time and overshoot percentage. For large throttle transients, the existing controller regulates the controlled variable, additional limiters, will be developed to protect critical engine variables from exceeding physical bounds and to ensure safer operation.
\section{Conclusions}
We developed a gain-scheduling controller with stability guarantees for nonlinear gas turbine engine systems. Using global linearization and LMI techniques, we showed guaranteed absolute stability for closed loop gas turbine engine systems with gain-scheduling controllers. Simulation results presented to show the applicability of the proposed controller to the nonlinear physics-based JetCat SPT5 turboshaft engine model for large transients from idle to cruise condition and vice versa.
\section*{Acknowledgment}
This material is based upon the work supported by the Air Force Research Laboratory (AFRL).
\bibliographystyle{plain}
|
1,116,691,499,357 | arxiv |
\section{Introduction}
Conversational Recommender Systems (CRS) aim to make recommendations by learning users' preferences through interactive conversations~\cite{cikm18-saur,wsdm20-ear,nips18-redial}. CRS has become one of the trending research topics for recommender systems and is gaining increasing attention, due to its natural advantage of explicitly acquiring users' real-time preferences and providing a user-engaged recommendation procedure.
Based on different scenarios, various CRS have been proposed, either from the perspective of recommender systems, being an enhanced interactive recommender system~\cite{sigir18-crm,kdd18-q&r,wsdm20-ear}, or from the perspective of dialogue systems, being a variation of goal-oriented conversational systems~\cite{nips18-redial,emnlp19-goredial,lei2018sequicity}.
Most of these CRS assume that users always know what they want and the system passively and solely targets at making the successful recommendation on users' desired items.
Recently, several efforts have been made on developing multi-goal conversational recommender systems (MG-CRS)~\cite{durecdial,tgredial} that can proactively discover users' interests and naturally lead user-engaged dialogues with multiple conversation goals or topics, not just recommendations.
As the examples illustrated in Fig.~\ref{example}, MG-CRS is expected to dynamically adjust the conversational goals (\textit{e.g.}, from QA, Chit-chat, to Recommendation in Fig.~\ref{example}(a)) and adaptively lead the dialogue topics (\textit{e.g.}, from Alien, Hero, to Animation in Fig.~\ref{example}(b)).
Accordingly, the problem of MG-CRS can be decomposed into four tasks:
\begin{itemize}
\item \textit{Goal Planning} aims to plan the goal sequence to guide the conversation to the final recommendation.
\item \textit{Topic Prediction} predicts the appropriate topics\footnote{According to different applications, the topics can be specific topic classes (\textit{e.g.}, TG-ReDial), topical knowledge entities (\textit{e.g.}, DuRecDial), etc. } for the current conversational goal.
\item \textit{Item Recommendation} provides suitable items that meet the user's need.
\item \textit{Response Generation} produces appropriate natural language responses to users.
\end{itemize}
\begin{figure*}
\centering
\includegraphics[width=0.7\textwidth]{fig/example.pdf}
\caption{Two illustrative examples of Multi-goal CRS from DuRecDial~\cite{durecdial} and TG-ReDial~\cite{tgredial} datasets.}
\label{example}
\end{figure*}
Early works typically adopt modularized frameworks that address different tasks in MG-CRS with independent models. For example, \citet{durecdial} propose a CNN-based goal planning model to predict the next goal and a multi-GRU-based response model to jointly select topical knowledge and generate responses with the guidance of the predicted goal.
\citet{tgredial} propose three separated modules based on pre-trained language models for topic prediction, item recommendation, and response generation, while the conversational goals (Chit-chat or Recommend) at each turn are pre-defined. Due to the complexity of the whole problem of MG-CRS, some recent studies tend to simplify the MG-CRS problem by either (i) assuming some information (\textit{e.g.}, the goal sequence) is priorly known~\cite{aaai21-gokc,KERS} or (ii) only performing joint learning on some of the tasks (\textit{e.g.,} topic prediction and response generation)~\cite{topicrefine}, instead of solving the whole problem of MG-CRS.
Despite their effectiveness, there remain some limitations in the existing systems:
(i) The four tasks in MG-CRS are supposed to be closely related, while existing systems often learn each task individually without considering inter-task interactions for mutual enhancement.
(ii) In reality, it is impossible to always have the pre-defined goal sequences or topic threads for guiding the conversation, which is also the challenge that characterizes MG-CRS from other CRS. Therefore, all tasks are expected to be learned and inferred by the system automatically.
(iii) There are some substantial differences among the four tasks, including various task paradigms (from multi-label classification, ranking, to text generation) and diverse required information (\textit{e.g.,} dialogue context, user profile, knowledge base, etc.). It would be time-consuming and labor-intensive to design and train several independent models for these distinct tasks.
To tackle the aforementioned challenges, we propose a novel Unified MultI-goal conversational recommeNDer system, namely \textbf{UniMIND}, which unifies four tasks in MG-CRS into the same sequence-to-sequence (Seq2Seq) paradigm and utilizes prompt-based learning strategies to endow the model with the capability of multi-task learning.
In specific, motivated by recent successes of paradigm shifts~\cite{paradigm} on many NLP tasks, we reformulate each task in MG-CRS as a Seq2Seq problem. Seq2Seq is a general and flexible paradigm that can handle any task whose input and output can be recast as a sequence of tokens, and better leverage the semantic relationships between input and output.
For example, topic prediction is a multi-label classification problem, where the Seq2Seq paradigm can fully utilize the label semantics~\cite{coling18-sgm}.
Item recommendation requires to rank a list of candidate items, where we expand the original vocabulary of the Seq2Seq model with an extra item vocabulary to capture the relationship between context words and candidate items~\cite{kbrd}.
Furthermore, pre-trained language models (PLMs), \textit{e.g.,} GPT2~\cite{gpt2}, T5~\cite{t5}, have become the de-facto methods for dialogue generation tasks.
In order to adapt PLMs to each task of MG-CRS, we investigate prompt-based learning strategies~\cite{prompt} to manipulate the model behavior so that the PLM itself can be used to predict the desired output and facilitate multi-task learning.
The contributions are summarized as follows
\begin{itemize}
\item We propose a novel method, namely UniMIND, that tackle all tasks in MG-CRS with a unified model. To the best of our knowledge, it is the first attempt towards a unified framework for MG-CRS.
\item We reformulate all tasks in MG-CRS with diverse modeling paradigms into the same Seq2Seq paradigm to seamlessly unify the complex MG-CRS problem.
\item We investigate prompt-based learning strategies to enable the multi-task learning of all tasks in MG-CRS, and develop a special token prompting strategy that bridges the relationships within each type of information.
\item Experimental results on two benchmark MG-CRS datasets show that UniMIND achieves state-of-the-art performance on all tasks. Extensive analyses provide some new insights of the features in different types of dialogues, and some takeaways for future MG-CRS studies.
\end{itemize}
\section{Related Work}
\subsection{Conversational Recommendation}
Conversational recommender system (CRS)~\cite{crs-survey1,crs-survey2} generally consists of two main components: a dialogue component to interact with the user and a recommender component to select items for recommendations based on user preference. According to the form of conversation, existing mainstream CRS can be divided into two groups: attribute-based CRS and open-ended CRS~\cite{rid}.
Attribute-based CRS~\cite{cikm18-saur,sigir21-hoops,sigir21-compar,cikm21-crs,sigir21-learn2ask} asks clarification questions about the item attributes to acquire user preferences for making better recommendation.
For these CRS, the system usually asks questions about the user’s preferences or makes recommendations multiple times, with the goal of achieving engaging and successful recommendations with fewer turns of conversations~\cite{kdd20-scpr,sigir21-crs,dasfaa21-crs}.
Open-ended CRS~\cite{nips18-redial,emnlp19-goredial,kdd20-redial-kg,kbrd} focuses on how to understand users' preferences and intentions from their utterances and interacts with user through natural language conversations.
All aforementioned studies on CRS typically target at a single goal, \textit{i.e.}, making successful recommendations.
Some latest studies~\cite{tgredial,durecdial,inspired} aim at solving the problem of multi-goal conversational recommender systems (MG-CRS), which involves a sequence of goals to lead a user-engaged conversation, such as recommendation, chit-chat, QA, topic-focused dialogue, etc.
Existing studies mainly focus on some of the tasks in MG-CRS.
For example, \citet{aaai21-gokc} and \citet{KERS} assume that the complete goal sequence is given and study the task as a knowledge-grounded response selection problem.
\citet{topicrefine} jointly predict the topics and generate responses without the consideration of item recommendation.
In order to tackle the entire problem of MG-CRS, we investigate a unified framework for all the tasks.
\subsection{Pre-trained Seq2Seq Models for Dialogue}\label{sec:related_plm}
Conventional dialogue systems~\cite{dialogue-survey} can be generally categorized into chitchat-based dialogue systems, which aim at conversing with users on open-domain topics, and task-oriented dialogue systems, which target at assisting users to accomplish certain goals.
Recently, both kinds of dialogue systems benefit from the advances in pre-trained Seq2Seq models, \textit{e.g.}, GPT-2~\cite{gpt2} and T5~\cite{t5}. For example, DialoGPT~\cite{dialogpt} and Plato~\cite{plato} extend GPT-2~\cite{gpt2} and BERT~\cite{bert}, respectively, to pre-train on open-domain conversational data for dialogue generation.
In task-oriented dialogue systems, several attempts have been made on leveraging pre-trained Seq2Seq models for generative dialogue state tracking~\cite{simpletod,emnlp21-dst-prompt}, or further applying multi-task dialogue pre-training over external dialogue corpora~\cite{soloist,pptod}.
In the scope of CRS, pre-trained Seq2Seq models are typically adopted as the response generation module~\cite{tgredial,topicrefine}.
In this work, we aim to maximize the utility of pre-trained Seq2Seq models on all tasks in MG-CRS.
\subsection{Paradigm Shift \& Prompt-based Learning}
In the past years, modeling for most NLP tasks have converged to
several mainstream paradigms~\cite{paradigm}, including Classification, Matching/Ranking, Sequence Labeling, Seq2Seq, etc.
Recent work has shown that models under some paradigms also generalize well on tasks with other paradigms.
For example, multi-label or multi-task classification may be challenging for conventional classification modeling, where \citet{coling18-sgm} adopt the Seq2Seq paradigm to better capture interactions between the labels and \citet{icml20-class-rank} adopt the Matching paradigm to predict whether the text is matched with the label with descriptions.
In order to better utilize powerful pre-trained language models (PLMs) with diverse pre-training paradigms, prompt-based learning~\cite{prompt} has been widely studied for shifting the target task to adaptive modeling paradigms with the PLM by using appropriate prompts.
Text classification tasks can also be solved by PLMs with the Masked Language Modeling (MLM) paradigm~\cite{acl21-class-mlm,eacl21-class-mlm,naacl21-class-mlm} or the Seq2Seq paradigm~\cite{t5}. In addition, several attempts have been made on adapting pre-trained Seq2Seq models to the task of document ranking~\cite{emnlp20-ranking-generation,emnlp20findings-ranking-generation}.
Some latest studies investigate paradigm shift or prompt-based learning on more diverse tasks, such as information extraction~\cite{acl22-infoext}, sentiment analysis~\cite{acl21-gabsa,emnlp21-quad}, etc.
In the field of dialogue systems, such paradigm shift techniques also bring inspiring progresses on task-oriented dialogue systems.
As mentioned in Section~\ref{sec:related_plm}, recent state-of-the-art performance~\cite{simpletod,emnlp21-dst-prompt} on dialogue state tracking is achieved by the reformulation of such a structure prediction task as a text generation task.
In addition, \citet{tickettalk} investigate the unified Seq2Seq paradigm for action prediction in transaction-based dialogue systems.
However, there are more complicated subtasks with diverse paradigms in MG-CRS, varying from multi-label classification, ranking/recommendation, to text generation.
In this work, we propose to unify all the subtasks with different paradigms in MG-CRS into Seq2Seq paradigm, and design prompt-based learning approaches for utilizing pre-trained Seq2Seq models for MG-CRS.
\section{Problem Definition}
Let $\mathcal{C}_t=\{c_1,...,c_{t-1}\}$ denote the dialogue context at the current conversation turn $t$. Correspondingly, let $\mathcal{G}_t=\{g_1,...,g_{t-1}\}$ and $\mathcal{K}_t=\{k_1,...,k_{t-1}\}$ denote the historical goal sequence and topic thread, respectively.
The CRS maintains a pre-defined set of goals $\mathbb{G}$, topics $\mathbb{K}$ to be predicted, and a large set of items $\mathbb{V}$ to be recommended during the conversation.
In some applications, there also exist the user profiles $\mathcal{P}_u$ for each user $u$, which can be historical item interaction data or certain personal knowledge.
Overall, MG-CRS aims to (1) plan the next goal $g_t\in \mathbb{G}$, (2) predict the next topic $k_t\in \mathbb{K}$, (3) recommend appropriate items $v_t\in\mathbb{V}$, and (4) produce a proper response $c_t$ for the current turn $t$.
Specifically, the problem of MG-CRS can be decomposed into the following four tasks:
\begin{itemize}
\item \textit{\textbf{Goal Planning}}. At each turn $t$, given the dialogue context $\mathcal{C}_t$ and the goal history $\mathcal{G}_t$, MG-CRS first selects the appropriate goal $g_t\in \mathbb{G}$ to determine where the conversation goes.
\item \textit{\textbf{Topic Prediction}}. The second task is to predict the next conversational topics $k_t\in \mathbb{K}$ for completing the planned goal $g_t$, with respect to the dialogue context $\mathcal{C}_t$, the historical topic thread $\mathcal{K}_t$, and the user profile $\mathcal{P}_u$ (if exists).
\item \textit{\textbf{Item Recommendation}}. If the selected goal $g_t$ is to make recommendations, the CRS should recommend an item $v_t\in\mathbb{V}$, based on the dialogue context $\mathcal{C}_t$ and the user profile $\mathcal{P}_u$ (if exists). In general, the recommended item $v_t$ is supposed to be related to the predicted topics $k_t$.
\item \textit{\textbf{Response Generation}}. The end task is to generate a proper response $c_t$ concerning the predicted topics $k_t$ for completing the selected goal $g_t$. When the goal is to make recommendation, the generated response is also expected to provide persuasive reasons for the recommended item $v_t$.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{fig/method.pdf}
\caption{Overview and examples of the input and output sequences for UniMIND.}
\label{overview}
\end{figure}
\section{Method}\label{sec:method}
In this section, we first describe the paradigm shift that reformulates each task in MG-CRS into the unified Seq2Seq paradigm, and then introduce the prompt-based learning strategies for the multi-task learning of all tasks.
The overview and examples of the input and output sequences for UniMIND are illustrated in Fig.~\ref{overview}.
Overall, the learning and inference procedure consists of three stages, including multi-task learning, prompt-based learning, and inference.
\subsection{Unified Paradigm Shifts}\label{sec:paradigm}
\subsubsection{\textbf{Goal Planning}}
To dynamically adjust the conversation goal in the multi-goal conversational recommendation systems, \citet{durecdial} divide the task of goal planning into two sub-tasks, goal completion estimation and goal prediction. Goal completion estimation aims at estimating the probability of goal completion by performing a \textit{binary classification}.
Goal prediction aims to predict the next goal when the previous goal is completed by performing \textit{multi-task classification}:
\begin{equation}
\bm{y}_\text{GP} = \mathbf{CLS}(\mathbf{Enc}(\mathcal{C}_t,\mathcal{G}_t)) \in \{0,1\}^{|\mathbb{G}|},
\end{equation}
where $\mathbf{Enc}(\cdot)$ is the encoders that encode the dialogue context and the goal history, and $\mathbf{CLS}(\cdot)$ is a multi-task classifier. $|\mathbb{G}|$ is the total number of all possible goal classes.
We unify these two sub-tasks as one Seq2Seq task, which aims at directly generating the label of the next goal as natural language with a text generation model, \textit{i.e.,} $\mathbf{UniMIND}(\cdot)$:
\begin{gather}\label{eq:goal}
g_t = \mathbf{UniMIND}(\mathcal{C}_t,\mathcal{G}_t).
\end{gather}
The objective of the Goal Planning task $G$ is to maximize the sequential log-likelihood:
\begin{gather}
\mathcal{L}_G = \log p(g_t|\mathcal{C}_t,\mathcal{G}_t) = \sum^{L_g}_{l=1}\log p_\theta(g_{t,l}|g_{t,<l};\mathcal{C}_t,\mathcal{G}_t),
\end{gather}
where $L_g$ denotes the target length of the generated goal label. Such a paradigm shift can alleviate the error propagation in the two-stage classification method.
\subsubsection{\textbf{Topic Prediction}}
According to different applications, the topic can be knowledge entities or specific topic classes, and there are also different corresponding solutions.
For instance, \citet{durecdial} implicitly select the relevant knowledge triples by assigning attention weights to all candidate knowledge and then fusing them into a single vector.
\citet{tgredial} predict the next topics by performing a \textit{text matching} task:
\begin{equation}
{y_\text{KS}}_i = \mathbf{CLS}(\mathbf{Enc}(k_i), \mathbf{Enc}(\mathcal{C}_t, \mathcal{K}_t, \mathcal{P}_u)),
\end{equation}
where the predicted topic is ranked with the highest relevance score, \textit{i.e.}, $\arg\max_i {y_\text{KS}}_i$.
However, these approaches may fail to handle those scenarios with a large set of topics to be predicted. Besides, the fixed number of predicted topics may also affect the cases where there is no topic needed or multiple topics are involved.
We regard the topic prediction as an multi-label classification problem and reformulate it into the Seq2Seq paradigm, where the predicted labels are concatenated into a single sequence as the target sequence to be generated. A special token, \textit{e.g.}, ``\texttt{</k>}'', is used to separate each individual label.
\begin{align}\label{eq:topic}
k_t &= \mathbf{UniMIND}(\mathcal{C}_t, \mathcal{K}_t, \mathcal{P}_u, g_t),\\
\begin{split}
\mathcal{L}_K &= \log p(k_t|\mathcal{C}_t,\mathcal{K}_t, \mathcal{P}_u, g_t) \\
&= \sum^{L_k}_{l=1}\log p_\theta(k_{t,l}|k_{t,<l};\mathcal{C}_t, \mathcal{K}_t, \mathcal{P}_u, g_t),
\end{split}
\end{align}
where $L_k$ denotes the target length of generated sequence of topic labels, and $\mathcal{L}_K$ is the objective function of the Topic Prediction task $T$. The final prediction results can be recovered by splitting the sequence using ``\texttt{</k>}''.
By doing so, the MG-CRS will be more scalable and flexible to different situations, even when the number of possible topics is extremely large and variable.
In addition, it can be observed from Fig.~\ref{example} that the next topic is highly related to the dialogue context. And the Seq2Seq paradigm can make full use of the semantic relationships between the dialogue context and the predicted topic labels as well as among different topic labels.
\subsubsection{\textbf{Item Recommendation}}
There are two main-stream solutions for item recommendation in CRS: (i) Following traditional recommendation systems, the recommendation module~\cite{tgredial,kdd20-redial-kg} aims to rank all the items by computing the probability that recommend an item $v_i$ to the user $u$:
\begin{equation}
{y_\text{Rec}}_i = \mathbf{CLS}(e_i,\mathbf{Enc}(\mathcal{C}_t, \mathcal{P}_u)),
\end{equation}
where $e_i$ is the trainable item embedding, and the recommended item is ranked with the highest recommendation probability, \textit{i.e.}, $\arg\max_i {y_\text{Rec}}_i$.
(ii) Some studies~\cite{kbrd,rid} perform the item recommendation within an end-to-end CRS, by regarding the item set as an additional vocabulary for generation, \textit{i.e.}, each item index as a word in the expanded vocabulary.
Motivated by the second type of approach, we extend the idea of vocabulary expansion with the candidate item set to further take into account the semantic information of the item.
Specifically, the target sequence of the item recommendation task is composed of the item expressions in both the original vocabulary and the expanded item vocabulary.
For example, if the target item in a sample is ``\texttt{The Witness}'' with the item index as 100, then the target sequence for this sample will be ``\texttt{\_100\_ The Witness}''.
By doing so, not only can the model capture the relationship between context words and candidate items, but also exploit the semantic information of items.
\begin{align}
v_t &= \mathbf{UniMIND}(\mathcal{C}_t, \mathcal{P}_u, g_t, k_t),\\
\begin{split}
\mathcal{L}_R &= \log p(v_t|\mathcal{C}_t, \mathcal{P}_u, g_t, k_t) \\
&= \sum^{L_r}_{l=1}\log p_\theta(v_{t,l}|v_{t,<l};\mathcal{C}_t, \mathcal{P}_u, g_t, k_t),
\end{split}
\end{align}
where $L_r$ denotes the target length, and $\mathcal{L}_R$ is the objective function of the Item Recommendation task $R$.
In the inference, we just conduct decoding for one step and rank the largest probability of the item in the expanded item vocabulary to recommend:
\begin{equation}\label{eq:item}
v_t = \arg\max_{v^{(i)}\in\mathbb{V}} p_\theta(v^{(i)}|\texttt{[sos]};\mathcal{C}_t, \mathcal{P}_u, g_t, k_t),
\end{equation}
where \texttt{[sos]} denote the start-of-sentence token.
\subsubsection{\textbf{Response Generation}}
Since response generation is a standard text generation problem in the Seq2Seq paradigm, it does not require any paradigm shift.
\begin{align}\label{eq:resp}
c_t &= \mathbf{UniMIND}(\mathcal{C}_t, g_t, k_t, v_t),\\
\begin{split}
\mathcal{L}_D &= \log p(c_t|\mathcal{C}_t, g_t, k_t, v_t) \\
&= \sum^{L_d}_{l=1}\log p_\theta(c_{t,l}|c_{t,<l};\mathcal{C}_t, g_t, k_t, v_t),
\end{split}
\end{align}
where $L_d$ denotes the target length of generated responses, and $\mathcal{L}_D$ is the objective function of the Response Generation task $D$.
\subsection{Prompt-based Learning}
After unifying all tasks in MG-CRS into the Seq2Seq paradigm, each input and output sequence pair in each task forms an annotated data instance for training a unified encoder-decoder model.
Inspired by prompt-based learning~\cite{prompt,t5}, we add a task-specific prompt to the original input sequence to specify which task the model should perform as well as enrich the input with task-specific information.
As discussed in Section~\ref{sec:paradigm}, the combination of required information in the input sequence varies in different tasks.
In order to indicate each information segment with a specific type, all sub-sequences are concatenated with special segment tokens, such as \texttt{[user]}, \texttt{[system]}, \texttt{[goal]}, \texttt{[topic]}, \texttt{[item]}, \texttt{[profile]}, etc.
These special tokens are regarded as additional vocabulary with randomly initialized embeddings to be learned.
For instance, the original input $X$ can be represented as:
\begin{align*}
X_G=&\texttt{[goal]}g_1\texttt{[user]}c_1\texttt{[goal]}g_2\texttt{[system]}...\texttt{[user]}c_{t-1};\\
X_T=&\texttt{[profile]}\mathcal{P}_u\texttt{[topic]}k_1\texttt{[user]}c_1\texttt{[topic]}k_2\texttt{[system]}...\texttt{[user]}c_{t-1}\texttt{[goal]}g_t;\\
X_R=&\texttt{[profile]}\mathcal{P}_u\texttt{[user]}c_1\texttt{[system]}...\texttt{[user]}c_{t-1}\texttt{[goal]}g_t\texttt{[topic]}k_t;\\
X_D=&\texttt{[user]}c_1\texttt{[system]}...\texttt{[user]}c_{t-1}\texttt{[goal]}g_t\texttt{[topic]}k_t\texttt{[item]}v_t.
\end{align*}
Specific examples are presented in Fig.~\ref{overview}(b).
The input for prompt-based learning is composed of the original input sequence $X$ and a task-specific prompt $Z$. We investigate two types of task-specific prompts, namely \textit{Natural Language Prompt} and \textit{Special Token Prompt}.
\textbf{Natural Language Prompt (UniMIND$_\text{N}$).} Similar to T5~\cite{t5}, the natural language prompt employs a guidance sentence to indicate each task as follows:
\begin{align*}
Z_G&=``\texttt{Plan the next goal:}";\\ Z_T&=``\texttt{Predict the next topic:}";\\
Z_R&=``\texttt{Recommend an item:}";\\
Z_D&=``\texttt{Generate the response:}".
\end{align*}
\textbf{Special Token Prompt (UniMIND$_\text{S}$).} Another realization of the task-specific prompt is based on the special segment tokens as follows:
\begin{align*}
Z_G&=\texttt{[goal]};\quad Z_T=\texttt{[topic]};\\ Z_R&=\texttt{[item]};\quad Z_D=\texttt{[system]},
\end{align*}
which can be categorized into the family of \textit{continuous prompts}~\cite{prompt}.
Originally, the special segment tokens are designed to indicate the beginning of each type of information in the input sequence. Therefore, the learned representations are supposed to preserve the knowledge of the different forms or patterns of each type of information. For example, \texttt{[goal]} and \texttt{[topic]} can separate two groups of classification labels, while \texttt{[system]} can also distinguish the speaker features of the system responses from the user utterances.
\begin{algorithm}[t]
\caption{Learning and Inference Procedure}
\label{algo}
\KwIn{Train-set $\mathcal{D}_1=\{\mathcal{D}_G,\mathcal{D}_T,\mathcal{D}_R,\mathcal{D}_D\}=\{(X,Y,Z)_i\}_{i=1}^{|\mathcal{D}_1|}$;
Test-set $\mathcal{D}_2=\{(\mathcal{C},\mathcal{G},\mathcal{K},\mathcal{P}_u)_i\}_{i=1}^{|\mathcal{D}_2|}$; Pre-trained parameters $\theta$; Training epoch number $e_1$; Fine-tuning epoch number $e_2$; Loss function $\mathcal{L}_\theta$;}
// Multi-task Training\;
\For{$\mathit{epoch} = 1, 2, \ldots , e_1$}{
\For{Batch $B$ in Shuffle($\mathcal{D}_1$)}{
Use $B=\{(X,Y,Z)_i\}_{i=1}^{|B|}$ to optimize $\mathcal{L}_\theta$ in Eq.~(\ref{eq:loss})\;
}
}
// Prompt-based Learning\;
\For{Task $t$ in $[G,T,R,D]$}{
$\theta_t = \theta$\;
\For{$\mathit{epoch} = 1, 2, \ldots , e_2$}{
\For{Batch $B$ in Shuffle($\mathcal{D}_t$)}{
Use $B=\{(X,Y,Z)_i\}_{i=1}^{|B|}$ to optimize $\mathcal{L}_{\theta_t}$ in Eq.~(\ref{eq:loss})\;
}
}
}
// Inference\;
\For{Sample $d_i$ in $\mathcal{D}_2$}{
$d_i = (\mathcal{C},\mathcal{G},\mathcal{K},\mathcal{P}_u)_i$\;
Obtain $g_i$, $k_i$, $v_i$, $c_i$ in order via Eq.~(\ref{eq:goal}, \ref{eq:topic}, \ref{eq:item}, \ref{eq:resp}) using tuned model $\theta_G$, $\theta_T$, $\theta_R$, $\theta_D$, respectively\;
}
\KwOut{Models $\theta_G$, $\theta_T$, $\theta_R$, $\theta_D$; Prediction results $\{g,k,v,c\}_i^{|\mathcal{D}_2|}$.}
\end{algorithm}
\subsection{Multi-task Training and Inference}
The proposed UniMIND model is initialized with weights from a pre-trained LM in an encoder-decoder fashion, \textit{e.g.}, BART~\cite{bart} or T5~\cite{t5}.
The overall learning and inference procedure of UniMIND is presented in Algorithm~\ref{algo}, which consists of three stages, including multi-task training, prompt-based learning, and inference.
In the multi-task training stage, the model is trained to perform all tasks in MG-CRS with all training data.
Given the training sample $(X,Y,Z)$, the objective $\mathcal{L}_\theta$ is to maximize the log-likelihood:
\begin{equation}\label{eq:loss}
\mathcal{L}_\theta = \sum_{l=1}^L \log p_\theta(y_l|y_{<l};X,Z),
\end{equation}
where $L$ denotes the length of the target sequence. When applying the trained UniMIND to a specific task, we first use the same objective function as in the multi-task training stage, \textit{i.e.}, Eq.~(\ref{eq:loss}), to perform task-specific prompt-based learning.
In the inference stage, given a data sample $d = (\mathcal{C},\mathcal{G},\mathcal{K},\mathcal{P}_u)$, we perform the four tasks in order using corresponding tuned models.
Finally, the predicted output contains the next goal $g$, the next topic $k$, the recommended item $v$, and the generated response $c$.
\section{Experimental Setups}\label{sec:exp_setup}
\subsection{Research Questions}
The empirical analysis targets the following research questions:
\begin{itemize}
\item \textbf{RQ1}: How is the performance of the proposed method on the end task of MG-CRS, \textit{i.e.}, Response Generation, compared to existing methods?
\item \textbf{RQ2}: How is the performance of the proposed method on each sub-task of MG-CRS, including Goal Planning, Topic Prediction, and Item Recommendation, compared to existing methods?
\item \textbf{RQ3}: How does the three-stage training and inference procedure as well as each sub-task contribute to the overall performance?
\item \textbf{RQ4}: What is the difference between multi-goal conversational recommender systems and traditional conversational recommender systems?
\end{itemize}
\begin{table}
\caption{Satistics of datasets.}
\centering
\begin{tabular}{lrr}
\toprule
Dataset & TG-ReDial & DuRecDial \\
\midrule
\#Dialogues&10,000&10,190\\
train/dev/test&8,495/757/748&6,618/946/2,626\\
\#Utterances&129,392&155,477\\
\#Goals&8&21\\
\#Topics/Entities&2,571&701\\
\#items&33,834&11,162\\
\bottomrule
\end{tabular}
\label{dataset}
\end{table}
\subsection{Datasets}
We conduct the experiments on two multi-goal conversational recommendation datasets, namely DuRecDial\footnote{\url{https://github.com/PaddlePaddle/Research/tree/48408392e152ffb2a09ce0a52334453e9c08b082/NLP/ACL2020-DuRecDial}}~\cite{durecdial} and TG-ReDial\footnote{\url{https://github.com/RUCAIBox/TG-ReDial}}~\cite{tgredial}. The dataset statistics are presented in Table~\ref{dataset}. We adopt the same train/dev/test split in the original datasets.
\begin{itemize}
\item \textbf{DuRecDial} is a goal-oriented knowledge-driven conversational recommendation dataset, which contains dialogues across multiple domains, including movie, music, restaurant, etc. The conversational goals and knowledge entities at each conversation turn are given. There are 21 types of conversational goals. We treat the knowledge entities as conversational topics, as the example in Fig.~\ref{example}(a). The dataset provides the user profiles with historical item interactions and the knowledge base related to each dialogue.
\item \textbf{TG-ReDial} is a topic-guided conversational recommendation dataset in the movie domain. We regard the labeled actions in the TG-ReDial dataset as the conversational goals at each turn. There are 8 types of conversational goals in TG-ReDial. An example is shown in Fig.~\ref{example}(b). The user profiles with historical item interactions are also given.
\end{itemize}
\subsection{Baselines \& Evaluation Metrics}
\subsubsection{Response Generation}
Since the original settings of TG-ReDial and DuRecDial have some differences, we compare to different groups of response generation baselines. For TG-ReDial, we consider the following baselines for comparisons:
\begin{itemize}
\item \textbf{ReDial}~\cite{nips18-redial} and \textbf{KBRD}~\cite{kbrd} apply the hierarchical RNN and the Transformer architecture, respectively, for response generation in conversational recommendation.
\item \textbf{Trans.}~\cite{transformer} and \textbf{GPT-2}~\cite{gpt2} are two general text generation baselines, which are also adopted for the evaluation on the DuRecDial dateset.
\item \textbf{Union}~\cite{tgredial} and \textbf{TopRef.}~\cite{topicrefine} employ GPT-2 to generate the response conditioned on the predicted topic or the recommended item.
\end{itemize}
For DuRecDial, we compare to the following baselines:
\begin{itemize}
\item \textbf{seq2seq}~\cite{seq2seq} and \textbf{PGN}~\cite{pgn} are two classic text generation methods. For seq2seq and Transformer, we also report the knowledge-grounded performance, \textit{i.e.}, seq2seq+kg and Trans.+kg.
\item \textbf{PostKS}~\cite{postks} is a knowledge-grounded response generation method.
\item \textbf{MGCG}\footnote{Following previous studies~\cite{aaai21-gokc,KERS}, we only compare our model with MGCG\_G, since MGCG\_R is a retrieval-based dialogue model.}~\cite{durecdial} adopts multi-type GRUs to encode the dialogue context, the goal sequence, and the topical knowledge and uses another GRU to generate responses.
\item \textbf{GOKC}~\cite{aaai21-gokc} and \textbf{KERS}~\cite{KERS} both assume that the goal sequence is given and focus on the knowledge-grounded response generation problem. Therefore, we also report their performance without the given goal sequence (``- w/o goal'').
\end{itemize}
For both datasets, we also adopt the vanilla \textbf{BART}~\cite{bart} as a baseline.
Following previous studies~\cite{durecdial,tgredial,aaai21-gokc,KERS}, we adopt word-level F1, BLEU, Distinct scores (Dist), and Perplexity (PPL) as automatic evaluation metrics.
\subsubsection{Goal Planning}
We compare to three text classification baselines:
\begin{itemize}
\item \textbf{MGCG}~\cite{durecdial} employs two CNN-based classifiers to perform two sub-tasks, goal completion estimation and goal prediction.
\item \textbf{BERT} uses the dialogue context as the input for performing a classification task to predict the next goal.
\item \textbf{BERT+CNN} combines the obtained representations from MGCG and BERT to predict the next goal.
\end{itemize}
We adopt Macro-averaged Precision (P), Recall (R), and F1 as the evaluation metrics.
\subsubsection{Topic Prediction}
We compare to five methods:
\begin{itemize}
\item \textbf{MGCG}~\cite{durecdial} adopts multi-type GRUs to encode the dialogue context, the historical topic sequence, and the user profile for performing a text matching task to rank the candidate topics.
\item \textbf{Conv/Topic/Profile/-BERT}~\cite{tgredial} utilizes BERT to encode historical utterances/topics/user profiles to rank the candidate topics, respectively.
\item \textbf{Union}~\cite{tgredial} combines the obtained representations from Conv/Topic/Profile-BERT.
\end{itemize}
Following \citet{tgredial}, we adopt Hit@$k$ as the evaluation metrics for ranking all the possible topics. Besides, we also report Micro-average P, R, and F1 of all the test instances. Since there are some responses that have no topic, if the prediction on a response with no gold labels is also empty, it means the model performs well and we set the P, R, F1 to 1, otherwise 0.
\subsubsection{Item Recommendation}
We compare to five recommendation baselines:
\begin{itemize}
\item \textbf{GRU4Rec}~\cite{gru4rec} and \textbf{SASRec}~\cite{sasrec} apply GRU and Transformer, respectively, to encode user interaction history without using conversation data.
\item \textbf{TextCNN}~\cite{textcnn} and \textbf{BERT}~\cite{bert} adopt a CNN-based model and a BERT-based model, respectively, to encode the dialogue context without using historical user interaction data.
\item \textbf{Union}~\cite{tgredial} combines the learned representations from SASRec and BERT to make recommendations.
\end{itemize}
Following \citet{tgredial}, we adopt NDCG@$k$ and MRR@$k$ as the evaluation metrics.
\subsection{Implementation Details}
Most results of the baselines are reported in previous works. For reproducing some additional results, we implement those baselines with the open-source CRS toolkit, CRSLab\footnote{\url{https://github.com/RUCAIBox/CRSLab}}~\cite{crslab}. In order to make a fair comparison with other baselines, we choose BART$_\text{base}$ as the pre-trained Seq2Seq model for UniMIND, which shares a similar number of model parameters with BERT$_\text{base}$ and GPT-2.
The pre-trained weights of BART$_\text{base}$ are initialized using the Chinese BART\footnote{\url{https://huggingface.co/fnlp/bart-base-chinese}}~\cite{bart-chinese}.
We use the same hyper-parameter settings for these two datasets. The learning rate and the weight decay rate are set to be 5e-5 and 0.01, respectively. The max source sequence length and the max target sequence length are 512 and 100, respectively.
For DuRecDial dataset, we extract the knowledge triples concerning the predicted topical entities from the given knowledge base, and regard these knowledge triples as the complete topic context for the response generation task, whose maximum length is set to be 256.
We train all the baselines up to 20 epochs. For the proposed method, UniMIND, we conduct multi-task training for 15 epochs and prompt-based learning for 5 epochs.\footnote{The code will be publicly released via \url{https://github.com/dengyang17/UniMIND}.}
\section{Experimental Results}\label{sec:exp}
We evaluate the proposed method on both the end-to-end response generation and each sub-task of MG-CRS.
\subsection{Evaluation on Response Generation (RQ1)}
We conduct both automatic and human evaluation on the response generation task.
\subsubsection{\textbf{Automatic Evaluation}}
\begin{table}
\caption{End-to-end Evaluation of Response Generation on TG-ReDial. The mark $\bigcirc$ denotes that the ground-truth labels are given on both training and testing, $\checkmark$ denotes that the ground-truth labels are given for training only, and $\times$ for none. $^\dagger$ indicates statistically significant improvement ($p$<0.05) over \underline{the best baseline}.}
\centering
\begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{Model}&\multicolumn{2}{c}{Ground-truth}&\multicolumn{4}{c}{Evaluation Metrics}\\
\cmidrule(lr){2-3}\cmidrule(lr){4-7}
&Goal&Topic& F1 & BLEU-1/2 & Dist-2 & PPL \\
\midrule
ReDial$^*$~\cite{nips18-redial}&$\times$&$\times$&-&0.177/0.028&0.025&81.61\\
KBRD$^*$~\cite{kbrd}&$\times$&$\times$&-&0.221/0.028&0.025&28.02\\
Trans.$^*$~\cite{transformer}&$\times$&$\times$&-&0.287/0.071&0.083&32.86\\
GPT-2$^*$~\cite{gpt2}&$\times$&$\times$&-&0.279/0.066&0.094&13.38\\
Union~\cite{tgredial}&$\bigcirc$&$\checkmark$&-&0.280/0.065&0.094&\underline{7.22}\\
TopicRef.~\cite{topicrefine}&$\bigcirc$&$\checkmark$&-&\underline{0.294/0.086}&-&-\\
BART~\cite{bart}&$\times$&$\times$&\underline{32.80}&0.291/0.070&\underline{0.097}&7.59\\
\midrule
\textbf{UniMIND}$_\text{N}$&$\checkmark$&$\checkmark$&35.40$^\dagger$&0.310/0.089$^\dagger$&\textbf{0.200}$^\dagger$&6.81$^\dagger$\\
\textbf{UniMIND}$_\text{S}$&$\checkmark$&$\checkmark$&\textbf{35.62}$^\dagger$&\textbf{0.314/0.090}$^\dagger$&0.198$^\dagger$&\textbf{5.22}$^\dagger$\\
\bottomrule
\multicolumn{5}{l}{$^*$ Results reported from \citet{tgredial}.}
\end{tabular}
\label{exp:resp_tg}
\end{table}
\begin{table}
\caption{End-to-end Evaluation of Response Generation on DuRecDial. The mark $\bigcirc$ denotes that the ground-truth labels are given on both training and testing, $\checkmark$ denotes that the ground-truth labels are given for training only, and $\times$ for none. $^\dagger$ indicates statistically significant improvement ($p$<0.05) over \underline{the best baseline}.}
\centering
\begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{Model}&\multicolumn{2}{c}{Ground-truth}&\multicolumn{4}{c}{Evaluation Metrics}\\
\cmidrule(lr){2-3}\cmidrule(lr){4-7}
&Goal&Topic& F1 & BLEU-1/2 & Dist-2 & PPL \\
\midrule
seq2seq$^*$~\cite{seq2seq}&$\times$&$\times$&26.08&0.188/0.102&0.013&22.82\\
seq2seq+kg$^{**}$~\cite{seq2seq}&$\times$&$\checkmark$&24.52&0.165/0.079&0.013&24.75\\
PGN$^*$~\cite{pgn}&$\times$&$\times$&33.95 &0.243/0.161& 0.039& 24.28\\
PostKS$^*$~\cite{postks}&$\times$&$\checkmark$&39.87& 0.343/0.244& 0.056& 15.32\\
MGCG~\cite{durecdial}&$\checkmark$&$\checkmark$&42.04& 0.362/0.252& 0.081 &14.89\\
Trans.$^{**}$~\cite{transformer}&$\times$&$\times$&41.79& 0.393/0.288&0.050&9.78\\
Trans.+kg$^{**}$~\cite{transformer}&$\times$&$\checkmark$&44.73& 0.419/0.318&0.055&9.40\\
GPT-2~\cite{gpt2}&$\times$&$\times$&47.01& 0.392/0.295& 0.165& 15.56\\
GOKC~\cite{aaai21-gokc}&$\bigcirc$&$\checkmark$&47.28& 0.413/0.318& \underline{0.084}& 11.38\\
- w/o goal&$\times$&$\checkmark$&45.59&0.401/0.303& 0.081& 12.45\\
KERS~\cite{KERS}&$\bigcirc$&$\checkmark$&\underline{50.47}&\underline{0.463/0.362}&0.079&\underline{8.34}\\
- w/o goal&$\times$&$\checkmark$&48.95&0.450/0.351& 0.082& 8.76\\
BART~\cite{bart}&$\times$&$\times$&48.41&0.418/0.328&0.049&8.72\\
\midrule
\textbf{UniMIND}$_\text{N}$&$\checkmark$&$\checkmark$&\textbf{52.19}$^\dagger$&\textbf{0.479/0.398}$^\dagger$&0.079&\textbf{6.63}$^\dagger$\\
\textbf{UniMIND}$_\text{S}$&$\checkmark$&$\checkmark$&51.87$^\dagger$&0.477/0.397$^\dagger$&\textbf{0.086}&6.69$^\dagger$\\
\bottomrule
\multicolumn{7}{l}{$^*$ Results reported from \citet{aaai21-gokc}. $^{**}$ Results reported from \citet{KERS}.}
\end{tabular}
\label{exp:resp_du}
\end{table}
Table~\ref{exp:resp_tg} and Table~\ref{exp:resp_du} summarize the experimental results on the end task of MG-CRS, \textit{i.e.}, Response Generation, with different conversational recommender systems.
Most of the baseline systems simplify the whole MG-CRS problem by assuming the conversational goals are pre-defined ($\bigcirc$) at each turn or ignoring some tasks ($\times$), including the current state-of-the-art methods on both datasets, \textit{i.e.,} TopicRef. and KERS.
Given pre-defined goals, the systems perform much better than their original counterparts, \textit{e.g.}, GOKC and KERS, indicating the importance of the conversational goals in MG-CRS.
Besides, simply adapting pre-trained language models to response generation in MG-CRS fails to achieve a promising performance improvement, \textit{e.g.}, GPT-2 and BART.
Finally, UniMIND not only achieves the state-of-the-art performance on the content preservation metrics (F1, BLEU) but also has a promising performance on diversity (Dist) and fluency (PPL) on both datasets.
Overall, the experimental results provide the answer to \textbf{RQ1}: \textit{UniMIND substantially and consistently outperforms existing strong baselines, including those baselines with pre-defined goals, with a noticeable margin on Response Generation, which is the end task of MG-CRS.}
\subsubsection{\textbf{Human Evaluation}}
\begin{table}
\caption{Human Evaluation of Response Generation.}
\centering
\begin{tabular}{lcccc}
\toprule
Model & Fluency & Informativeness & Appropriateness & Proactivity \\
\midrule
GPT-2&1.36&1.39&1.25&1.71\\
Union&1.31&1.24&1.58&1.80\\
BART&1.81&1.31&1.40&1.77\\
\textbf{UniMind}$_\text{N}$&1.93&1.52&1.70&\textbf{1.96}\\
\textbf{UniMind}$_\text{S}$&\textbf{1.94}&\textbf{1.62}&\textbf{1.72}&\textbf{1.96}\\
\midrule
Human&1.98&1.90&1.99&1.98\\
\bottomrule
\end{tabular}
\label{exp:human}
\end{table}
We conduct human evaluation to evaluate the generated response from four aspects:
\begin{itemize}
\item \textbf{Fluency}: how fluent and coherent the generated response is?
\item \textbf{Informativeness}: how rich is the generated response in information?
\item \textbf{Appropriateness}: is the generated response appropriate for the current topic?
\item \textbf{Proactivity}: how well does the generated response proactively complete the current goal?
\end{itemize}
We randomly sample 100 dialogues from TG-ReDial and compare their responses produced by four methods (GPT-2, Union, BART, and UniMIND)\footnote{The generated responses of GPT-2 and Union on the TG-ReDial dataset are provided by \url{https://github.com/RUCAIBox/TG_CRS_Code}.}.
Three annotators are asked to score each generated response with \{0: bad, 1: ok, 2: good\}. These annotators are all well-educated research assistants. As the results presented in Table~\ref{exp:human}, UniMIND consistently outperforms these strong baselines from different aspects of human evaluation.
It is noteworthy that the scores on \textit{Appropriateness} and \textit{Proactivity} are substantially improved by UniMIND, which demonstrates that UniMIND can effectively lead a proactive conversation with appropriate content.
However, compared with the reference responses (Human), there is still much room for improvement on \textit{Informativeness} and \textit{Appropriateness} of the responses generated by UniMIND.
The human judgements further support the above answer to the \textbf{RQ1}: \textit{The responses generated by the proposed method preserve a higher degree of fluency, informativeness as well as explicitly reflect the target conversational topics and lead a proactive conversation, which contributes to a higher overall quality. }
\subsection{Evaluation on Each Task (RQ2)}
\subsubsection{\textbf{Evaluation on Goal Planning}}
\begin{table}
\caption{Evaluation of Goal Planning. $^\dagger$ indicates statistically significant improvement ($p$<0.05) over \underline{the best baseline}.}
\centering
\begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{Model} & \multicolumn{3}{c}{TG-ReDial} & \multicolumn{3}{c}{DuRecDial} \\
\cmidrule(lr){2-4} \cmidrule(lr){5-7}
& P & R & F1 & P & R & F1\\
\midrule
MGCG&0.7546&0.8093& 0.7794&0.5739& 0.6324& 0.5787\\
BERT&0.8742&0.9004& 0.8858&0.9174& 0.9337& 0.9187\\
BERT+CNN&\underline{0.8777}& \underline{0.9182}& \underline{0.8971}& \underline{0.9248}& \underline{0.9357}& \underline{0.9229}\\
\midrule
\textbf{UniMIND}$_\text{N}$&0.8879$^\dagger$& 0.9403$^\dagger$& 0.9122$^\dagger$&\textbf{0.9327}$^\dagger$& \textbf{0.9466}$^\dagger$& \textbf{0.9357}$^\dagger$\\
\textbf{UniMIND}$_\text{S}$&\textbf{0.8887}$^\dagger$& \textbf{0.9425}$^\dagger$& \textbf{0.9137}$^\dagger$&0.9326$^\dagger$& 0.9369$^\dagger$& 0.9335$^\dagger$\\
\bottomrule
\end{tabular}
\label{exp:goal}
\end{table}
Table~\ref{exp:goal} presents the experimental results on the Goal Planning task.
Among the baselines, BERT and BERT+CNN perform much better than MGCG by fine-tuning a BERT-based model to encode the contextual information from the dialogue history, which attaches great importance in determining where the conversation should go at the next turn.
Two variants of UniMIND achieve similar performance and both of them significantly outperform these baselines on two datasets.
\subsubsection{\textbf{Evaluation on Topic Prediction}}
\begin{table*}
\caption{Evaluation of Topic Prediction on TG-ReDial. We report P@1, R@1, and F1@1 scores for the matching-based baselines, which achieve the highest F1@$k$ scores. We regard the first generated topic as the top-ranked topic to compute the Hit@1 score for UniMIND. $^\dagger$ indicates statistically significant improvement ($p$<0.05) over \underline{the best baseline}.}
\centering
\begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{Model} & \multicolumn{6}{c}{TG-ReDial} \\
\cmidrule(lr){2-7}
& Hit@1 & Hit@3 & Hit@5 & P & R &F1 \\
\midrule
MGCG&0.3635& 0.5173& 0.6009& 0.5211&0.5013& 0.5079\\
Topic-BERT&0.4381& 0.5823& 0.6246&0.6221& 0.5548& 0.5772\\
Profile-BERT&0.0907& 0.1597& 0.2248& 0.3945&0.3740& 0.3808\\
Conv-BERT&0.4348& 0.5873& 0.6329&0.6264& \underline{0.5589}& 0.5814\\
Union&\underline{0.4420}&\underline{0.5923}&\underline{0.6374}&\underline{0.6301}& 0.5586& \underline{0.5824}\\
\midrule
\textbf{UniMIND}$_\text{N}$&0.7319$^\dagger$&-&-&0.6876$^\dagger$& 0.6915$^\dagger$& 0.6889$^\dagger$\\
\textbf{UniMIND}$_\text{S}$&\textbf{0.7351}$^\dagger$&-&-&\textbf{0.6912}$^\dagger$& \textbf{0.6951}$^\dagger$& \textbf{0.6925}$^\dagger$\\
\bottomrule
\end{tabular}
\label{exp:know_tg}
\end{table*}
\begin{table*}
\caption{Evaluation of Topic Prediction on DuRecDial. We report P@1, R@1, and F1@1 scores for the matching-based baselines, which achieve the highest F1@$k$ scores. We regard the first generated topic as the top-ranked topic to compute the Hit@1 score for UniMIND. $^\dagger$ indicates statistically significant improvement ($p$<0.05) over \underline{the best baseline}.}
\centering
\begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{Model} & \multicolumn{6}{c}{DuRecDial} \\
\cmidrule(lr){2-7}
& Hit@1 & Hit@3 & Hit@5 & P & R &F1 \\
\midrule
MGCG&0.6639& 0.7510& 0.7810&0.6697&0.5419& 0.5880\\
Topic-BERT&0.6337& 0.7460& 0.7785&0.6957& 0.6015& 0.6289\\
Profile-BERT&0.4908& 0.6844& 0.7698&0.4576& 0.4036& 0.4181\\
Conv-BERT&0.7791& 0.8226& 0.8425&0.8122& 0.7065& 0.7377\\
Union&\underline{0.7877}& \underline{0.8462}& \underline{0.8696}&\underline{0.8327}& \underline{0.7270}& \underline{0.7582}\\
\midrule
\textbf{UniMIND}$_\text{N}$&\textbf{0.9056}$^\dagger$&-&-&\textbf{0.8981}$^\dagger$& \textbf{0.8994}$^\dagger$& \textbf{0.8978}$^\dagger$\\
\textbf{UniMIND}$_\text{S}$&0.9023$^\dagger$&-&-&0.8957$^\dagger$& 0.8964$^\dagger$& 0.8952$^\dagger$\\
\bottomrule
\end{tabular}
\label{exp:know_du}
\end{table*}
Table~\ref{exp:know_tg} and Table~\ref{exp:know_du} present the experimental results on the Topic Prediction task.
Note that the results on TG-ReDial reported in \cite{tgredial} are based on the assumption that the target recommendation topic is given, so that all the baselines achieve a similar result, due to the strong supervision signal of the target topic.
In fact, without the guidance of the target topic, the task of topic prediction becomes more difficult, since the system is required to predict the next topic based on the coherency and relevancy to the dialogue context, the historical topic thread, and the user profile.
Therefore, we can observe that Profile-BERT barely works on the topic prediction task.
Overall, UniMIND significantly outperforms these strong baselines on both Hit scores and F1 scores, where the Hit@1 scores of UniMIND are even higher than the Hit@5 scores of Union.
Since it is difficult to determine the number of topic classes with these matching-based baselines, UniMIND owns remarkable flexibility and scalability to this multi-label classification problem, which is proven by the F1 scores.
\subsubsection{\textbf{Evaluation on Item Recommendation}}
\begin{table*}
\caption{Evaluation of Item Recommendation on TG-ReDial. $^\dagger$ indicates statistically significant improvement ($p$<0.05) over \underline{the best baseline}.}
\centering
\begin{tabular}{lcccc}
\toprule
\multirow{2}{*}{Model} & \multicolumn{4}{c}{TG-ReDial} \\
\cmidrule(lr){2-5}
& NDCG@10 & NDCG@50 & MRR@10 & MRR@50 \\
\midrule
GRU4Rec&0.0028&0.0062&0.0014&0.0020\\
SASRec&0.0092&0.0179&0.0050&0.0068\\
TextCNN&0.0144&0.0215&0.0119&0.0133\\
BERT&0.0246&0.0439&0.0182&0.0211\\
Union&\underline{0.0348}&\underline{0.0527}&\underline{0.0240}&\underline{0.0277}\\
\midrule
\textbf{UniMIND}$_\text{N}$&0.0306& 0.0499& 0.0236& 0.0277\\
\textbf{UniMIND}$_\text{S}$&\textbf{0.0386}$^\dagger$& \textbf{0.0638}$^\dagger$& \textbf{0.0283}$^\dagger$& \textbf{0.0319}\\
\bottomrule
\end{tabular}
\label{exp:rec_tg}
\end{table*}
\begin{table*}
\caption{Evaluation of Item Recommendation on DuRecDial. $^\dagger$ indicates statistically significant improvement ($p$<0.05) over \underline{the best baseline}.}
\centering
\begin{tabular}{lcccc}
\toprule
\multirow{2}{*}{Model} & \multicolumn{4}{c}{DuRecDial} \\
\cmidrule(lr){2-5}
& NDCG@10 & NDCG@50 & MRR@10 & MRR@50 \\
\midrule
GRU4Rec&0.2188& 0.2734& 0.1713& 0.1833\\
SASRec&0.3686& 0.4130& 0.3071& 0.3174\\
TextCNN&0.5049& 0.5344& 0.4516& 0.4584\\
BERT&0.5455& 0.5719& 0.4983& 0.5043\\
Union&\underline{0.5568}& \underline{0.5831}& \underline{0.5101}& \underline{0.5159}\\
\midrule
\textbf{UniMIND}$_\text{N}$&0.5986$^\dagger$& 0.6099$^\dagger$& 0.5922$^\dagger$& 0.5944$^\dagger$\\
\textbf{UniMIND}$_\text{S}$&\textbf{0.6343}$^\dagger$& \textbf{0.6471}$^\dagger$& \textbf{0.6291}$^\dagger$& \textbf{0.6318}$^\dagger$\\
\bottomrule
\end{tabular}
\label{exp:rec_du}
\end{table*}
Table~\ref{exp:rec_tg} and Table~\ref{exp:rec_du} summarize the experimental results on the Item Recommendation task.
Among these baselines, due to the sparsity of the historical user-item interaction data, traditional recommendation methods (\textit{e.g.}, GRU4Rec and SASRec) fall short of handling the item recommendation task in MG-CRS, while text-based recommendation methods (\textit{e.g.}, TextCNN and BERT) show more promising performance. Union further improves the performance by combining the advantages of BERT and SASRec.
Without using the historical interaction data, UniMIND achieves competitive performance with Union.
In specific, UniMIND$_\text{S}$ significantly and consistently outperforms Union on both datasets and UniMIND$_\text{N}$ outperforms Union on DuRecDial.
Different from the observations on other tasks, UniMIND$_\text{S}$ performs much better than UniMIND$_\text{N}$ on item recommendation, showing that the natural language prompts can not fully utilize the relationships with the expanded item vocabulary.
Overall, the experimental results provide the answer to \textbf{RQ2}: \textit{UniMIND significantly outperforms existing strong baselines on each sub-task of MG-CRS. The strong performance can not only contribute to the final quality of the generated responses, but also provide useful and convenient adaptation for different sub-task applications.}
\section{Detailed Analyses \& Discussions}
In this section, we provide a variety of detailed analyses and discussions to look deeper into the proposed method. Note that the following analyses are all conducted with UniMIND$_\text{S}$, since this variant has generally better performance on both datasets according to Section~\ref{sec:exp}.
\subsection{Ablation Study (RQ3)}
\begin{table}
\caption{Ablation Study (Results on Response Generation).}
\centering
\begin{tabular}{lcccc}
\toprule
\multirow{2}{*}{Model} & \multicolumn{2}{c}{TG-ReDial} & \multicolumn{2}{c}{DuRecDial} \\
\cmidrule(lr){2-3} \cmidrule(lr){4-5}
& F1 & BLEU-1/2 & F1 & BLEU-1/2 \\
\midrule
\textbf{UniMIND}$_\text{S}$&35.62&0.314/0.090&51.87&0.477/0.397\\
- w/o MTL&34.73&0.305/0.086&50.98&0.465/0.381\\
- w/o PL &33.94&0.299/0.084&50.67&0.454/0.373\\%
\midrule
\multicolumn{5}{l}{\textit{Response Generation}}\\
- OracleGen&39.43&0.350/0.104&55.07&0.525/0.444\\
- DirectGen&32.80&0.291/0.070&48.41&0.418/0.328\\
\midrule
\multicolumn{5}{l}{\textit{Goal Planning}}\\
- Oracle&36.78&0.328/0.097&53.28&0.498/0.417\\
- BERT+CNN&35.51&0.314/0.090&51.67&0.475/0.396\\
- w/o goal&35.13&0.309/0.086&50.97&0.467/0.380\\
\midrule
\multicolumn{5}{l}{\textit{Topic Prediction}}\\
- Oracle&38.22&0.336/0.100&52.68&0.492/0.413\\
- Union&34.32&0.301/0.084&51.09&0.469/0.383\\
- w/o topic&35.01&0.307/0.085&48.73&0.426/0.340\\
\midrule
\multicolumn{5}{l}{\textit{Item Recommendation}}\\
- Oracle&35.72&0.315/0.091&52.42&0.488/0.409\\
- Union&35.60&0.314/0.090&51.25&0.471/0.394\\
- w/o item&35.73&0.315/0.090&51.10&0.469/0.385\\
\bottomrule
\end{tabular}
\label{exp:ablation}
\end{table}
In order to investigate the effect of the proposed training procedure and each task, the results of the ablation study are presented in Table~\ref{exp:ablation}.
\subsubsection{Effect of Training Procedure}
We first evaluate the effectiveness of the multi-task learning and the prompt-based learning strategies.
``- w/o MTL'' denotes that we train four independent Seq2Seq models for each task.
``- w/o PL'' denotes that we only train one unified multi-task learning model for all tasks without task-specific prompt-based learning.
\textit{The results show that the best performance can only be achieved by combining multi-task learning and prompt-based learning}, which can answer the first part of \textbf{RQ3}.
\subsubsection{Effect of Each Sub-task}
Besides, we also present the performance of ``- OracleGen'', which denotes that the input sequence for response generation is composed of the ground-truth goals, topics, and items, while ``- DirectGen'' denotes the input sequence only contains the dialogue history.
The results show that the performance is improved by 15\%-25\% (DirectGen $\rightarrow$ OracleGen), which demonstrates the importance of these kinds of information in MG-CRS.
In addition, we further investigate the effect of each task, by replacing them with the ground-truth labels (``- Oracle'') or predicted results from other strong baselines (``- BERT+CNN'' and ``- Union''), or discarding the information (``- w/o'').
The results show that Topic Prediction and Goal Planning largely affect the final performance, where precisely predicting topics can bring the most prominent improvement on the final response generation.
However, Item Recommendation has the least effect on the final response generation for both datasets.
Overall, for the answer to the second part of \textbf{RQ3}, \textit{the experimental results show that all sub-tasks attach more or less importance to the final Response Generation task, among which Topic Prediction and Goal Planning are more influential than Item Recommendation.}
\subsection{Performance w.r.t. Goal Type (RQ4)}
\begin{table}
\caption{Performance w.r.t. Goal Type.}
\centering
\setlength{\tabcolsep}{1.1mm}{
\begin{tabular}{lcccccc}
\toprule
\multirow{2}{*}{Goal Type}&\multirow{2}{*}{\%}&Goal&Topic&\multicolumn{3}{c}{Response Gen.}\\
\cmidrule(lr){3-3}\cmidrule(lr){4-4}\cmidrule(lr){5-7}
&&F1&F1&F1&BLEU-1/2&Dist-2\\
\midrule
\multicolumn{7}{c}{TG-ReDial}\\
\midrule
Recommend.&54.4&\textbf{0.9629}&\textbf{0.8864}&37.6&0.337/0.072&0.218\\
Chit-chat&39.0&0.9428&0.3886&30.5&0.254/0.071&\textbf{0.327}\\
Rec. Request&31.9&0.8352&0.6926&\textbf{45.4}&\textbf{0.404/0.167}&0.251\\
\midrule
\multicolumn{7}{c}{DuRecDial}\\
\midrule
Recommend.&37.2&0.9235&0.7933&45.9&0.455/0.376&0.101\\
Chit-chat&15.5&0.8734&0.9787&41.7&0.396/0.309&\textbf{0.132}\\
QA&16.7&0.9298&0.9278&62.5&0.587/0.505&0.122\\
Task&11.3&\textbf{0.9456}&\textbf{0.9963}&\textbf{68.5}&\textbf{0.701/0.637}&0.114\\
\bottomrule
\end{tabular}}
\label{exp:goal_type}
\end{table}
In order to further analyze the characteristics of MG-CRS against traditional CRS, we present the performance with respect to different goal types in Table~\ref{exp:goal_type}.
In order to provide a comprehensive scope, we aggregate the results of several dominant goal types for each dataset.
For the TG-ReDial dataset that contains responses with multiple goals, the type-wise scores are averaged from all the samples that contain the goal type.
For the DuRecDial dataset that contains multi-domain dialogues, the type-wise scores are averaged from all the samples that contain the goal type across all domains.
On both datasets, ``\textit{Recommendation}'' is still the most important goal type with the largest number of samples in MG-CRS.
There are several notable observations as follows:
\begin{itemize}
\item As for Goal Planning, ``\textit{Recommendation}'' is the easiest goal type to plan for TG-ReDial, since this type always comes after ``\textit{Recommendation Request}''. However, the timing for ``\textit{Recommendation Request}'' would be more difficult to determine.
\item As for Topic Prediction, the topic labels in TG-ReDial are the abstractive conversational topics, while those in DuRecDial are the knowledge entities discussed in the current conversation.
The difference in annotations leads to the different observations in the two datasets.
For ``\textit{Chit-chat}'' dialogues, the abstractive topics are hard to predict in TG-ReDial, while the discussed entities can be effectively predicted in DuRecDial.
Conversely, for ``\textit{Recommendation}'' dialogues, the abstractive topics are often the genre of the recommended movies in TG-ReDial, while there might be multiple knowledge entities related to the recommended item in DuRecDial.
\item As for Response Generation, ``\textit{Chit-chat}'' dialogues reach the lowest scores on the content preservation metrics (\textit{i.e.}, F1 and BLEU), while achieving the highest scores on the diversity metrics (\textit{i.e.}, Dist) in both datasets.
This phenomenon is prevalent in chit-chat dialogue studies.
``\textit{Recommendation}'' dialogues reach the lowest scores on Dist, due to the similar expressions when making recommendations.
\end{itemize}
Therefore, we can derive the answer to \textbf{RQ4} from this analysis: \textit{The performance w.r.t. different goal types demonstrate the difficulties of MG-CRS, since there are some great differences among different types of dialogues, not just making recommendations in traditional conversational recommender systems.}
\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{fig/case.pdf}
\caption{Case Study.}
\label{case}
\end{figure}
\subsection{Case Study}
In order to intuitively differentiate UniMIND from other baselines, Fig.~\ref{case} presents a specific dialogue from TG-ReDial.
At the $t$-$1$-th turn, the ground-truth conversational goal and topics are ``\textit{Chit-chat}'' and ``\textit{Love}'', which means that the system is expected to provide a causal response talking about love.
However, Union predicts the topic to be ``\textit{Starry Sky}'' in advance, due to the strong supervision signals of the target topics of the movie to be recommended.
Moreover, without the goal planning task, Union and BART tend to make recommendations frequently, since the majority of the conversational goals in MG-CRS is ``\textit{Recommendation}'' as shown in Table~\ref{exp:goal_type}.
This is likely to degrade the user experience.
UniMIND generates an appropriate response to discuss about love with the user, by making good goal planning and topic prediction.
At the $t$-th turn, the ground-truth conversational goal and topics are ``\textit{Movie Recommendation}'' and ``\textit{Love/Starry Sky}'', which means that the system is expected to recommend a movie about love and starry sky.
UniMIND can better capture the multiple topics and provide a more coherent response to the dialogue context.
\subsection{Error Analysis and Limitations}\label{sec:error}
Despite the effectiveness of the proposed UniMIND framework for MG-CRS, we would like to better understand the failure modes of UniMIND for further improvement in future studies.
After analyzing those cases with low human evaluation scores, we identify the following limitations and discuss the potential solutions:
\begin{itemize}
\item \textbf{Low Recommendation Success Rate}. All the baselines and UniMIND fail to reach a promising recommendation performance on TG-ReDial as shown in Table~\ref{exp:rec_tg}, due to the sparsity of the user-item interactions. Since the historical interaction data is not utilized in UniMIND, one possible direction is to study how to incorporate this kind of data into the Seq2Seq framework for improving the recommendation performance.
\item \textbf{Informativeness}. As shown in Table~\ref{exp:human}, there is still a gap between the generated and the ground-truth response on \textbf{Informativeness}. In order to diversify and enrich the information in dialogue systems, a common practice is to leverage open-domain dialogue corpus to post-train the generation model~\cite{dialogpt}, which can also be easily applied to our unified Seq2Seq framework.
\item \textbf{Error Propagation}. This is a typical issue of solving multiple tasks in sequential order. Table~\ref{error} presents the Exact Match (EM) scores between the generated input sequence and the oracle input sequence for each task, which inevitably go down along with the sequential completion of each task. There are some techniques studied to alleviate this issue in cascaded generation methods, such as introducing contrastive objectives~\cite{soloist} or noisy channel models~\cite{tacl21-noisy}.
\end{itemize}
\begin{table}
\caption{EM between generated and oracle input sequences.}
\centering
\begin{tabular}{lccc}
\toprule
Dataset & Topic Pred. & Item Rec. & Response Gen. \\
\midrule
TG-ReDial&87.20\%&80.17\%&15.44\%\\
DuRecDial&92.32\%&89.16\%&84.74\%\\
\bottomrule
\end{tabular}
\label{error}
\end{table}
\section{Conclusions}
In this work, we propose a novel unified multi-task learning framework for multi-goal conversational recommender systems, namely UniMIND.
Specifically, we unify four tasks in MG-CRS into the same sequence-to-sequence (Seq2Seq) paradigm and utilize prompt-based learning strategies to endow the model with the capability of multi-task learning.
Experimental results on two MG-CRS datasets show that the proposed method achieves state-of-the-art performance on each task with a unified model.
Extensive analyses demonstrate the importance of each task and the difficulties of handling different types of dialogues in MG-CRS.
This work is the first attempt towards a unified multi-task learning framework for MG-CRS. There are some limitations and room for further improvement.
As discussed in Section~\ref{sec:error}, the error analyses and limitation discussions shed some potential directions for future studies. For example, it can be beneficial to incorporate historical user-item interaction data into the unified framework for making better recommendations or leverage open-domain dialogue corpus to post-train the generation model for generating more informative and diverse responses. It would be also worth investigating approaches to alleviate the error propagation issue in the training procedure.
In addition, similar to other prompt-based learning studies, the proposed method can be extended to handle the low-resource scenarios or few-shot learning settings in MG-CRS, which will be more practical in real-world applications.
|
1,116,691,499,358 | arxiv | \section{Introduction} %
\label{introduction}
Type III solar radio bursts are caused by semi-relativistic electrons
streaming through and perturbing the ambient coronal or interplanetary plasma.
A recent review is given by \citet{Reid14}.
The dominant theory, proposed by \citet{Ginzburg58}, invokes a two-step
process beginning with the stimulation of Langmuir waves (plasma oscillations)
in the background plasma by an electron beam.
A small fraction of the Langmuir wave energy is then converted into electromagnetic
radiation at either the local electron plasma frequency ($f_p$) or its harmonic
(2$f_p$; see reviews by \citealt{Robinson00,Melrose09}).
The emission frequency depends mainly on the ambient electron density ($n_e$) because
$f_p \propto \sqrt{n_e}$.
This relationship produces the defining feature of type III bursts, a rapid drift from high to low frequencies
as the exciter beam travels away from the Sun through decreasing densities \citep{Wild50}.
The rate at which the emission frequency drifts ($df/dt$) is therefore related to the
electron beam speed, which can be obtained in the radial direction by assuming a density model $n_e(r)$.
Many authors have employed this technique for various events with various models, generally
finding modest fractions of light speed (0.1--0.4 $c$; e.g. \citealt{Alvarez73,Aschwanden95,Mann99,Melendez99,Krupar15,Kishore17}).
Alternatively, the coronal and/or interplanetary density gradient can be inferred by instead assuming a
beam speed (e.g. \citealt{Fainberg71,Leblanc98}) or by simply assuming that the beam speed is constant \citep{Cairns09}.
While these methods can yield robust estimates for the density gradient, they cannot be converted into an explicit
density structure $n_e(r)$ without normalizing the gradient to a specific value at a specific heliocentric distance.
This normalization has typically been done using estimates from white light polarized brightness data close to the Sun,
\textit{in situ} data in the interplanetary medium, or the observed height of type III burst sources at various
frequencies.
Densities inferred from type III source heights, particularly at lower frequencies,
have frequently conflicted with those obtained from other methods.
The earliest spatial measurements found larger source heights than would be expected from
fundamental plasma emission, implying density enhancements of an order of magnitude or more \citep{Wild59}.
This finding was confirmed by subsequent investigations (e.g. \citealt{Morimoto64,Malitson66}), and along
with other arguments, led many authors to
two conclusions:
First, that harmonic (2$f_p$) emission likely dominates
(e.g. \citealt{Fainberg71,Mercier74,Stewart76}).
This brings the corresponding densities down by a factor of 4, then implying only a moderate enhancement
over densities inferred from white light data.
(Counterarguments for the prevalence of fundamental emission will be referenced in Section~\ref{density}.)
Second, that the electron beams preferentially traverse overdense flux tubes
(e.g. \citealt{Bougeret84}), a conclusion bolstered by
spatial correlations between several type III bursts and white light streamers
(e.g. \citealt{Trottet82,Kundu84,Gopalswamy87,Mugundhan18}).
The overdense hypothesis has been challenged by evidence that the large
source heights can instead be explained by propagation effects.
If type III emission is produced in thin, high-density structures, then it can escape relatively
unperturbed through its comparatively rarefied surroundings.
However, if the emission is produced in an environment near the associated plasma level
(i.e. with an average $n_e$ corresponding to the radio waves' equivalent $f_p$), then refraction and scattering
by density inhomogeneities may substantially shift an observed source from its
true origin (e.g. \citealt{Leblanc73,Riddle74,Bougeret77}).
\citet{Duncan79} introduced the term \textit{ducting} in this context,
\edit{which refers to emission being guided to larger heights within a low-density structure though
successive reflections against the high-density ``walls" of the duct.}
This concept was generalized for a more realistic corona by \citet{Robinson83}, who showed that random
scattering of radio waves by thin, overdense fibers has the \edit{same} net effect of elevating an observed source
radially above its emission site.
\edit{Additional details on this topic, along with coronal refraction, will be given in Section~\ref{propagation}.}
Many authors came to favor propagation effects instead of the overdense structure interpretation
for a few reasons.
Despite the aforementioned case studies, type IIIs did not appear to be statistically
associated with regions of high average density in the corona \citep{Leblanc74,Leblanc77} or
in the solar wind \citep{Steinberg84}.
Interplanetary (kHz-range) type III source regions are also so large as to demand angular broadening
by propagation effects (e.g. \citealt{Steinberg85,Lecacheux89}).
Invoking propagation effects can also
be used to explain apparent spatial differences between fundamental and harmonic sources (e.g. \citealt{Stewart72,Kontar17})\edit{,
along with large offsets between radio sources on the disk and their likely electron acceleration sites (e.g. \citealt{Bisoi18}).
These arguments are reviewed by \citet{Dulk00}, and further discussion with additional recent references will be presented
in Sections~\ref{propagation} and \ref{discussion}.}
\edit{Both the interpretation of electron beams moving along overdense structures and of radio propagation effects
elevating burst sources} rely on the presence of thin, high-density fibers.
Either the electron beams are traveling within these structures or the type III emission is being scattered
by them. In this paper, we will suggest that propagation effects are important but cannot entirely
explain the density enhancements for our events.
Section~\ref{observations} describes our observations: \ref{mwa} outlines our data reduction,
\ref{events} details our event selection criteria, and \ref{context} describes the multi-wavelength
context for the selected type III bursts.
Section~\ref{analysis} describes our analysis and results: \ref{density} infers densities from type III
source heights, \ref{speed} estimates electron beam speeds from imaging data, and
\ref{propagation} examines propagation effects by comparing the extent of the quiescent
corona to model predictions.
In Section~\ref{discussion}, we discuss the implications of our results, along with other recent
developments, on the debate between the overdense and propagation effects hypotheses.
Finally, our conclusions are summarized in Section~\ref{conclusion}.
\section{Observations} %
\label{observations} %
\subsection{Murchison Widefield Array (MWA)}
\label{mwa}
The MWA is a low-frequency radio interferometer in Western Australia
with an instantaneous bandwidth of 30.72 MHz that can be flexibly distributed
from 80 to 300 MHz \citep{Tingay13}.
Our data were recorded with a 0.5 s time cadence and a 40 kHz spectral
resolution, which we average over 12 separate 2.56 MHz bandwidths
centered at
80, 89, 98, 108, 120, 132, 145, 161, 179, 196, 217, and 240 MHz.
We use the same data processing scheme as \citet{McCauley17}, and
what follows is a brief summary thereof.
Visibilities were generated with the standard MWA correlator \citep{Ord15} and the
\texttt{cotter} software \citep{Offringa12,Offringa15}.
Observations of bright and well-modelled calibrator sources were used
to obtain solutions for the complex antenna gains \citep{Hurley14}, which
were improved by imaging the calibrator and iteratively self-calibrating from there \citep{Hurley17}.
\texttt{WSClean} \citep{Offringa14} was used to perform the imaging
with a Briggs -2 weighting \citep{Briggs95} to maximize spatial resolution and minimize point spread function (PSF) sidelobes.
The primary beam model of \citet{Sutinjo15} was used to produce Stokes I
images from the instrumental polarizations,
and the SolarSoftWare (SSW\footnote{SSW: \url{https://www.lmsal.com/solarsoft/}}, \citealt{Freeland98})
routine \texttt{mwa\_prep} \citep{McCauley17} was used to translate the images onto solar coordinates.
Flux calibration was achieved by comparison with thermal bremsstrahlung and gyroresonance emission predictions from
FORWARD\footnote{FORWARD: \url{https://www2.hao.ucar.edu/modeling/FORWARD-home}} \citep{Gibson16}
based on the Magnetohydrodynamic Algorithm outside a Sphere model
(MAS\footnote{MAS: \url{http://www.predsci.com/hmi/data\_access.php}}; \citealt{Lionello09}).
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{masked_max_centroid_3panel_sub_figure.eps}}
\caption{MWA type III burst contours at 50\% of the peak intensity
for each channel overlaid on 240 MHz images of the quiescent corona.
The solid circle represents the optical disk, and dotted lines bound the region included
in the dynamic spectra (Figs.~\ref{fig:ds_comp} \& \ref{fig:spectra}).
Colored ellipses in the lower-right corners show the synthesized beam sizes for each channel.}
\label{fig:centroids}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{20141014_010112_1097290888_hera_sc3_0_spectra_comparison.eps}}
\caption{Dynamic spectra constructed from image intensities
for the type III burst near 03:05:20 UT on 2014-10-14.
Panel A includes the full FOV, while B includes only
the segment bounded by the dotted lines in Fig.~\ref{fig:centroids}.
Dotted horizontal lines show the locations of the 12 channels, each having a spectral width of 2.56 MHz.
Intensities have been divided by the background level and plotted a logarithmic scale.
}
\label{fig:ds_comp}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{all_spectra_figure_set2.eps}}
\caption{Dynamic spectra constructed from partial image intensities, including only the FOV segment bounded
by the dotted lines in Fig.~\ref{fig:centroids}.
The left column shows the full 5-min observation intervals, while the right column shows 20-sec
periods surrounding the selected type III bursts.
Circles and crosses denote the onset and peak burst times for each channel.
}
\label{fig:spectra}
\end{figure}
\subsection{Event Selection}
\label{events}
These data are part of an imaging survey of many
type III bursts observed by the MWA during 45 separate observing
periods in 2014 and 2015.
\citet{McCauley17} performed a case study of an event that exhibits
unusual source motion, and future work will present statistical analyses.
Burst periods during MWA observing runs were identified using the daily
National Oceanic and Atmospheric Administration (NOAA) solar event
reports\footnote{NOAA event reports: \url{http://www.swpc.noaa.gov/products/solar-and-geophysical-event-reports}}
based on observations from the \textit{Learmonth} \citep{Guidice81,Kennewell03}
and \textit{Culgoora} \citep{Prestage94} solar radio spectrographs, which overlap with
the MWA's frequency range at the low and high ends, respectively.
Three events were selected from the full sample based on the following criteria.
First, the burst sites needed to be located at the radio limb with roughly
radial progressions across frequency channels.
Limb events minimize projection effects, allowing us to reasonably approximate the
projected distance from Sun-center as the actual radial height.
Second, to eliminate potential confusion between multiple events and to maximize spectral coverage,
the bursts needed to be sufficiently isolated in time and frequency, with a
coherent drift from high to low frequencies across the full MWA bandwidth.
Third, the source regions needed to be relatively uncomplicated ellipses with
little-to-no intrinsic motion of the sort described by \citet{McCauley17}.
This again minimizes projection effects and ensures that we follow a single
beam trajectory for each event.
Figure~\ref{fig:centroids} shows the burst contours for each channel overlaid on
quiescent background images at 240 MHz.
Each of the three events occurred on a different day, and we refer to them by the
UTC date on which they occurred.
Figure~\ref{fig:ds_comp} shows dynamic spectra for the 2014-10-14 event, with the left panel
covering the full Sun and the right panel including only the region demarcated by the dotted
lines in Figure~\ref{fig:centroids}.
The partial Sun spectrum excludes a neighboring region that is active over the
same period, allowing the type III frequency structure to be more easily followed.
This approach is similar to that of \citet{Mohan17}, who discuss the utility of spatially resolved dynamic spectra.
Figure~\ref{fig:spectra} shows the masked spectra for all three events.
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{lasco_overlay_figure.eps}}
\caption{Overlays of the 50\% burst contours onto
AIA 171 \AA{} and LASCO C2 images.
Contour colors are for spectral channels from 80--240 MHz as in Fig.~\ref{fig:centroids}.
\edit{UTC observation times are shown for LASCO in the left panel, for AIA in the upper-left
of the right panel, and for MWA in the middle of the right panel.
The MWA times reflect the average peak time across frequency channels (see Fig.~\ref{fig:spectra}).
The AIA images are 10-min (50-image) averages processed with a radial filter to accentuate off-limb features;
times reflect the middle of these 10-min windows, which begin at the burst onsets and cover the subsequent periods
over which associated EUV signatures would be expected.}
Images are rotated in the right column such that the burst progression
is roughly horizontal, which helps illustrate the extent to which each event
progresses radially.
Cyan arrows point to the EUV structures that exhibit activity
during or just after the radio bursts.
\edit{The black arrow in the lower-left panel points to a CME that originated behind the limb and
passed the C2 occulting disk around 20-min prior to the type III burst.}
}
\label{fig:overlay}
\end{figure}
\subsection{Context}
\label{context}
In this section, we briefly describe the context for each of the radio bursts with
respect to observations at other wavelengths and associated phenomena.
Figure~\ref{fig:overlay} overlays the burst contours from Figure~\ref{fig:centroids} onto
contemporaneous extreme ultraviolet (EUV) and white light data.
The white light images were produced by the
Large Angle and Spectrometric C2 Coronagraph (LASCO C2; \citealt{Brueckner95})
onboard the Solar and Heliospheric Observatory (SOHO; \citealt{Domingo95}).
\edit{C2 has an observing cadence of 20 min, and Figure~\ref{fig:overlay}
includes the nearest images in time to our radio bursts.}
The EUV data come from the
Atmospheric Imaging Assembly (AIA; \citealt{Lemen12})
onboard the Solar Dynamics Observatory (SDO; \citealt{Pesnell12}).
We use the 171 \AA{} AIA channel, which is dominated by Fe IX
emission produced by plasma at around 0.63 MK, because it most
clearly delineates the fine magnetic structures along which type III beams
are expected to travel.
To further accentuate off-limb features, we apply a radial filter
using the SSW routine \texttt{aia\_rfilter} \citep{Masson14}.
Note that the apparent brightness of a given pixel in a radial filter image corresponds to
its true intensity relative only to pixels of the same radial height (i.e. equally bright structures
at different heights do not have the same physical intensity).
\edit{AIA has an observing cadence of 12 s, and Figure~\ref{fig:overlay} uses 10-min (50-image)
averages that cover the periods during and immediately after the radio bursts.
This time window is used because a potential EUV signature associated with a
type III burst will propagate at a much lower speed than the burst-driving electron beam and
will likely be most apparent in the minutes following the burst (e.g. \citealt{McCauley17,Cairns17}).}
In all cases, the radio bursts appear to be aligned with dense structures visible to AIA at lower
heights and to LASCO C2 at larger heights.
The latter case is obvious, with each set of burst contours situated just below bright
white light streamers.
Cyan arrows in the right panels of Figure~\ref{fig:overlay} identify the associated EUV structures, each
of which exhibits a mild brightening and/or outflow during or immediately after the corresponding radio burst.
This activity may be indicative of weak EUV jets, which are frequently associated
with type III bursts (e.g. \citealt{Chen13,Innes16,McCauley17,Cairns17}), but robust outflows are not observed here.
The alignment between the EUV and radio burst structure is particularly
striking for the 2015-09-23 event in that both appear to follow roughly the same non-radial arc.
A correspondence between EUV rays and type III bursts was previously reported by \citet{Pick09}.
Type III bursts are commonly, but not always, associated with X-ray flares (e.g. \citealt{Benz05,Benz07,Cairns17}) and occasionally with
Coronal Mass Ejections (CMEs; e.g. \citealt{Cane02,Cliver09}).
Our 2014-10-14 event is not associated with either, but the other two are.
On 2015-09-23, a weak B-class flare occurred just to the north of
our radio sources from active region 12415.
The flare peaked around 3:11 UT, which corresponds to a period of relatively
intense coherent radio emission that precedes the weaker burst of interest here (see Figure~\ref{fig:spectra}).
Given the radio source positions and associated EUV structure, we do not believe the flare site
to be the source of accelerated electrons for our event, though the flare may have been responsible
for stimulating further reconnection to the south.
\edit{On 2015-10-27, a CME was ongoing at the time of the radio burst, and its leading edge,
indicated by the black arrow in the lower left panel of Figure~\ref{fig:overlay},
can be seen just above the C2 occulting disk.
Inspection of images from the Extreme Ultraviolet Imager (EUVI; \citealt{Howard08}) onboard the
STEREO-A spacecraft shows that the CME originated from a large active region close to the east limb
but occulted by the disk from AIA's perspective.
The CME was launched well before our type III burst, but the region that produced it was very active
over this period and is likely connected to the activity visible to AIA immediately after the radio burst
along the structure indicated by the cyan arrows in the lower right panel of Figure~\ref{fig:overlay}.
So while we do not think the CME was directly involved in triggering the radio burst, it may have impacted
the medium through which the type III electron beam would later propagate, which is relevant
to a hypothesis proposed by \citet{Morosan14} that will be discussed in Section~\ref{discussion}.}
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{baseline_plot.eps}}
\caption{Light curves for the 2015-09-23 observation, shown to illustrate the background level determination.
Backgrounds (dotted lines) are obtained by taking the median intensity,
excluding points 2 standard deviations above that,
and iterating until no more points are excluded.
The dashed lines mark the burst period from the right column of Fig.~\ref{fig:spectra}.}
\label{fig:baseline}
\end{figure}
\section{Analysis and Results}
\label{analysis}
\subsection{Density Profiles}
\label{density}
Standard plasma emission theory expects type III radiation
at either the ambient electron plasma frequency ($f_{\rm p}$) or its harmonic
($2f_{\rm p}$).
The emission frequency $f$ is related to electron density ($n_e$) in the following way \edit{(in cgs units)}:
\edit{
\begin{equation} \label{eq:fp}
f = \textrm{N}f_{\rm p} = \textrm{N}\sqrt{\frac{e^2n_e}{\pi{}m_e}}
~~~\Rightarrow{}~~~
n_e = \pi{}m_{e}\left(\frac{f}{\textrm{N}e}\right)^2,
\end{equation}
}
\noindent where
$e$ is the electron charge,
$m_e$ is the electron mass,
and \edit{N} is either 1 (fundamental) or 2 (harmonic).
For frequencies in \edit{Hz} and densities in cm\tsp{-3}, $n_e \approx 1.24\e{-8}f^2$ for fundamental
and $3.10\e{-9}f^2$ for harmonic emission.
Density can thus be easily extracted given the emission mode and location.
Unfortunately, neither property is entirely straightforward.
Harmonic emission is often favored in the corona because being produced above the ambient
$f_p$ makes it less likely to absorbed \citep{Bastian98} and because type IIIs tend to be
more weakly circularly polarized than expected for fundamental emission \citep{Dulk80}.
Harmonic emission also implies lower densities by a factor of 4, which are easier to
reconcile with the large heights often observed (see Section \ref{introduction}).
However, fundamental--harmonic pairs can be observed near our frequency range (e.g. \citealt{Kontar17}),
fundamental emission is expected to contribute significantly to interplanetary type III burst spectra (e.g. \citealt{Robinson98}),
and fundamental emission is thought to be the more efficient process from a theoretical perspective (e.g. \citealt{Li13b,Li14}).
As described in Section~\ref{introduction}, a source's apparent height may also be
augmented by \edit{propagation effects}, which we will consider in Section~\ref{propagation}.
We measure source heights at the onset of burst emission, which we define as
when the total intensity reaches 1.3$\times$ the background level.
Background levels are determined for each frequency by taking the median intensity, excluding
points 2 standard deviations above that, and iterating until no more points are excluded.
Figure~\ref{fig:baseline} shows the result of this baseline procedure for three frequencies from the 2015-09-23 event,
which is shown because it exhibits the most complicated dynamic spectrum.
Onset times are represented by circles
in Figure~\ref{fig:spectra}, and centroids are obtained at these
times from 2-dimensional (2D) Gaussian fits.
As mentioned in Section \ref{events}, these events were chosen because they appear at the radio limb
and thus the 2D plane-of-sky positions can reasonably approximate the
physical altitude.
Geometrically, these heights are lower limits to the true radial height, but
propagation effects that increase apparent height
are likely to be more important than the projection angle (see Section \ref{propagation}).
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{type3_densities_plot_data_3model_harmonic.eps}}
\caption{Densities inferred from the type III source positions assuming fundamental ($f_p$; dashed)
or harmonic ($2f_p$; solid) emission.
Background coronal models based on white light data near solar minimum \citep{Saito77} and
maximum \citep{Newkirk61} are shown for comparison, along with a recent streamer model
based on EUV data \citep{Goryaev14}.
Only the average uncertainties are shown for clarity; the dark gray bars represent the 1$\sigma$ centroid
variability over the full burst, and the light gray bars represent the major axes of the synthesized beams.
}
\label{fig:dens1}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{type3_densities_morosan_comparison_harmonic.eps}}
\caption{Densities assuming harmonic emission compared to recent type III results at higher \citep{Chen13}
and lower \citep{Morosan14} frequencies.
The dotted lines apply the $n_e(r) = C(r - 1)^{-2}$ profile detailed by \citet{Cairns09}, where
the constant $C$ has been normalized to the density implied by our 240 MHz source positions.
}
\label{fig:dens2}
\end{figure}
Figure~\ref{fig:dens1} plots height versus density for both the fundamental and harmonic assumptions.
Two sets of height uncertainties are shown for the average density profiles.
The smaller, dark gray error bars reflect the 1$\sigma$ position variability over the full burst durations, and the larger,
light gray bars reflect the full width at half maximum (FWHM) of the synethesized beam major axes.
Note that if the source is dominated by a single compact component, which would be a reasonable assumption here,
then the FWHM resolution uncertainty can be reduced by a factor inversely proportional to the signal-to-noise ratio (SNR) \citep{Lonsdale18,Reid88}.
\edit{Given our high SNRs, which average 217$\sigma$ at the burst onsets,} this ``spot mapping" approach \edit{typically} results in sub-arcsecond position uncertainties on the apparent source location.
However, spatial shifts may be introduced by changes in the ionosphere between the solar and calibration observation
times, and more importantly, an apparent source may differ significantly from its actual emission site due to propagation effects (i.e. refraction and scattering).
For these reasons, we opt to show the more conservative uncertainties outlined above.
For comparison, Figure~\ref{fig:dens1} includes radial density models from \citet{Saito77}, \citet{Newkirk61}, and \citet{Goryaev14}.
The \citeauthor{Saito77} profile refers to the equatorial background near solar minimum based on white
light polarized brightness data, while the \citeauthor{Newkirk61} curve is based on similar data near solar maximum
and implies the largest densities among ``standard" background models.
The \citeauthor{Goryaev14} model instead refers to a dense streamer and is based on a
novel technique using widefield EUV imaging.
This profile is somewhat elevated above streamer densities
inferred from contemporary white light (e.g. \citealt{Gibson99})
and spectroscopic (e.g. \citealt{Parenti00,Spadaro07}) measurements at similar heights, though
some earlier white light studies found comparably large streamer densities (e.g. \citealt{Saito67}).
For additional coronal density profiles, see also \citet{Allen47,Koutchmy94,Guhathakurta96,Mann99,Mercier15,Wang17}
and references therein.
From Figure~\ref{fig:dens1}, we see that the type III densities assuming fundamental emission are an average
of 3--4$\times$ higher than the EUV streamer model.
These values may be unreasonably large, meaning either that the fundamental emission hypothesis is
not viable here or that fundamental emission \edit{originating} from a lower altitude
\edit{was observed a larger height due to propagation effects} (see Section \ref{propagation}).
Assuming harmonic emission, the 2014-10-14 burst implies electron densities of 1.8\e{8} cm\tsp{-3} (240 MHz)
at \rsolar{1.40} down to
0.20\e{8} cm\tsp{-3} (80 MHz) at \rsolar{2.10}.
This represents a moderate ($\sim1.4\times$) enhancement over the \citeauthor{Goryaev14} streamer model
or a significant ($\sim4.1\times$) enhancement over the \citeauthor{Newkirk61} background.
The other two events fall between the EUV streamer and solar maximum background models, with the
2015-10-27 source heights implying densities of 1.8\e{8} cm\tsp{-3} (240 MHz) at \rsolar{1.25} down to
0.20\e{8} cm\tsp{-3} (80 MHz) at \rsolar{1.68}.
Note that the 2015-09-23 burst implies an unusually steep density gradient that is not consistent with standard radial
density models, perhaps because that event
deviates significantly from the radial direction (see Figure~\ref{fig:overlay}).
Figure~\ref{fig:dens2} shows how our results compare to densities inferred from recent type III imaging at
higher and lower frequencies, all assuming harmonic emission.
The high-frequency (1.0--1.5 GHz) results come from \citet{Chen13}, who used the Very Large Array (VLA)
to find densities around an order of magnitude above the background.
The low-frequency (30--60 MHz) points were obtained using the
Low Frequency Array (LOFAR) by \citet{Morosan14}, who also found large enhancements.
We plot data from their ``Burst 2" (see Figures 3 \& 4) because it began beyond
our average radio limb height at 80 MHz.
Their other two events exhibit 60 MHz emission near the optical limb, which may indicate
that the 2D plane-of-sky positions significantly underestimate the true altitudes
(i.e. those electron beams may have been inclined toward the observer).
Figure~\ref{fig:dens2} also includes density curves of the form $n_e(r) = C(r - 1)^{-2}$, where
$r$ is in solar radii and $C$ is normalized to match the densities implied by our 240 MHz source heights.
This model was introduced by \citet{Cairns09} based on type III frequency drift rates over 40--180 MHz
and was subsequently validated over a larger frequency range by \citet{Lobzin10}.
The \citeauthor{Cairns09} model is somewhat steeper than others over the MWA's height range
($\sim$ 1.25--2.10\rsolar{}) but becomes more gently-sloping at larger heights,
effectively bridging the corona to solar wind transition.
From Figure~\ref{fig:dens2}, we see that this model is a good fit to the 2014-10-14 and 2015-10-27 data.
The 2015-09-23 event is not well-fit by this or any other standard model,
which may be attributed to its aforementioned non-radial structure.
Extending these gradients to larger heights matches the LOFAR data fairly well
and likewise with the VLA data at lower heights, which come from higher frequencies than have been
examined with this model previously.
\subsection{Electron Beam Kinematics}
\label{speed}
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{speeds_plot.eps}}
\caption{Exciter speed estimates from the time- and frequency-varying source positions.
The dashed orange line includes the high time outlier (orange asterisk).
The uncertainties shown in the lower right are the same for a given frequency and reflect the time and spatial resolutions.
The black bar represents the smallest synthesized beam size at 240 MHz (corresponding to the lower-left points),
and the gray bar represents the largest beam size at 80 MHz (corresponding to the upper-right points).
}
\label{fig:speeds}
\end{figure}
Type III beams speeds are known primarily from frequency drift rates ($df/dt$) observed in dynamic spectra.
Assuming either fundamental or harmonic emission, a given burst frequency can be straightforwardly
converted into a radial height given a density model $n_e(r)$, and $df/dt$ then becomes $dr/dt$.
The literature includes a wide range of values using this technique, reflecting the variability
among models as well as any intrinsic variability in electron speed.
Modest fractions of light speed are typically inferred from drift rates of coronal bursts ($\sim$0.1--0.4 c;
\citealt{Alvarez73,Aschwanden95,Melendez99,Kishore17}), though speeds larger
than 0.5 c have been reported by some studies \citep{Poquerusse94,Klassen03,Carley16}.
Our imaging observations allow us to measure the exciter speed without assuming $n_e(r)$ by following the
apparent height progression of type III sources at different frequencies.
As in the previous section, we obtain radial heights from centroid positions at the
onset of burst emission for each frequency.
These data are plotted in Figure~\ref{fig:speeds} along with linear least-squares fits to the speed using
the time and spatial resolutions as uncertainties.
The 2014-10-14 event exhibits an anomalously \edit{late} onset time at 80 MHz
\edit{(see the circles in Figure~\ref{fig:spectra}a and the orange asterisk in Figure~\ref{fig:speeds}).
This is likely due to the diminished intensity at that frequency, which precludes an appropriate comparison
to the onset times at higher frequencies where the burst is much more intense.
Figure~\ref{fig:speeds} shows fits both including (0.29 c) and excluding (0.60 c) the 80 MHz point for
the 2014-10-14 event, and the latter value is used in the discussion to follow because of the better overall fit.
Note that while the onset of 80 MHz emission is at a later time than expected given the prior frequency progression,
the source location is consistent with the other channels and thus
its inclusion not does not impact the inferred density profile from Figures~\ref{fig:dens1} and \ref{fig:dens2}.}
We find an average speed across events of 0.39 c, which is consistent with results from other imaging observations.
The same strategy was recently employed at lower frequencies by \citet{Morosan14}, who found an
average of 0.45 c.
\citet{McCauley17} indirectly inferred a beam speed of 0.2 c from MWA imaging.
\citet{Chen13} also tracked centroid positions at higher frequencies, though
in projection across the disk, finding 0.3 c.
\edit{\citet{Mann18} recently examined the apparent speeds of three temporally adjacent type III bursts imaged by
LOFAR. They find that the sources do not propagate with uniform speed, with each burst exhibiting an
acceleration in apparent height, and they conclude that the exciting electron beams must have broad velocity distributions.
From Figure~\ref{fig:speeds}, we observe an apparent acceleration only for one event (2015-10-27), with the other two events
exhibiting the opposite trend to some extent.
However, our data are consistent with \citet{Mann18} in that a uniform speed is not a particularly good fit for any of our events, but
the MWA's 0.5 s temporal resolution limits our ability to characterize the source speeds in great detail.}
\begin{table}
\caption{Imaging Beam Speeds vs $df/dt$ Model Predictions}
\label{tab:speeds}
\begin{tabular}{l|c|ccc|c}
\hline
& \multicolumn{5}{c}{Beam Speed (c)} \\
& & \multicolumn{3}{c|}{Assuming $f_p$ -- $2f_p$ emission} & \\
Event & Imaging & Goryaev \textit{et al.} & Newkirk & Saito \textit{et al.}\tabnote{$f_p$ case not viable because model does not include densities above $f_p \approx$ 116 MHz.} & Cairns \textit{et al.}\tabnote{Model normalized to match the densities implied by our 240 MHz heights.} \\
\hline
2014-10-14\tabnote{Excludes the 80 MHz outlier (orange asterisk in Fig. 10).} & 0.60 $\pm$ 0.13 & 0.38 -- 0.45 & 0.22 -- 0.31 & *** -- 0.30 & 0.58 \\
2015-09-23 & 0.24 $\pm$ 0.10 & 0.34 -- 0.40 & 0.20 -- 0.28 & *** -- 0.27 & 0.50 \\
2015-10-27 & 0.32 $\pm$ 0.12 & 0.44 -- 0.55 & 0.26 -- 0.36 & *** -- 0.48 & 0.40 \\
\hline
\end{tabular}
\end{table}
Taken together, we see that speeds measured from imaging observations tend to produce values
at the higher end of what is typical for $df/dt$ inferences.
We compare the two approaches for the same events in Table~\ref{tab:speeds} using the same
models shown in Figure~\ref{fig:dens1}.
We also include speeds derived using the \citet{Cairns09} model, normalized to the densities
implied by our 240 MHz source heights.
These values are separated from the others in Table~\ref{tab:speeds} because the normalization
precludes direct comparisons to the other models.
The $df/dt$-inferred speeds are consistently smaller than the imaging results for the 2014-10-14 event, which
was also true for the bursts studied by \citet{Morosan14}, but there is no major difference between the
two approaches for our other events given the range of values.
Note that this comparison is arguably a less direct version of the
height versus density comparison from the previous section in that
the extent to which the imaging and model-dependent $df/dt$ speeds agree unsurprisingly mirrors the extent
to which the density profiles themselves agree.
The 2014-10-14 speeds are closest to those derived using $n_e(r)$ from \citeauthor{Goryaev14}, and
the 2015-10-27 result is closest to the \citeauthor{Newkirk61}-derived speed, both assuming
harmonic emission, because those density profiles are most closely matched in Figure~\ref{fig:dens1}.
Likewise, the speeds from those events agree well with $df/dt$ speeds obtained using the
normalized \citeauthor{Cairns09} curves because a $C(r - 1)^{-2}$ gradient fits those data nicely.
The 2015-09-23 speed is between the two values derived using the \citeauthor{Newkirk61} model assuming either
fundamental and harmonic emission, but this may be coincidence given that the modeled and observed
density profiles are widely discrepant.
That event's non-radial
profile may also prevent meaningful agreement with any simple $n_e(r)$ model (see Figure~\ref{fig:overlay}).
\subsection{Propagation Effects}
\label{propagation}
As described in Section \ref{introduction}, a number of authors have argued that radio propagation
effects, namely refraction and scattering, can explain the large source heights frequently exhibited by type III bursts.
\citet{Bougeret77} introduced the idea of scattering by overdense fibers in the
context of radio burst morphologies, and \citet{Stewart74} suggested that type III
emission may be produced in underdense flux tubes as a way of explaining observed
harmonic--fundamental ratios.
These two concepts were combined by \citet{Duncan79}, who introduced the
term \textit{ducting} to refer to radiation that is produced in an underdense environment and subsequently
guided to a larger height by reflections against a surrounding ``wall" of much
higher-density material, which eventually becomes transparent with sufficient altitude.
\edit{While plausible, this concept generalizes poorly in that electron beams are not
expected to be found preferentially within coherent sets of low-density structures that would be conducive to ducting.}
\edit{\citet{Robinson83} addressed this by showing}
that random reflections against overdense fibers can
have the same effect of elevating an observed burst site above
its true origin, but without requiring any peculiarities of the emission site (i.e. low-density).
\edit{Because the high density fibers known to permeate the corona are not randomly
arranged and are generally radial, random scattering against them does not randomly modulate
the aggregate ray path--scattering instead tends to guide the emission outward to larger heights in a manner
that is analogous to the classic ducting scenario.
For this reason, other authors (e.g. \citealt{Poquerusse88}) have chosen to retain \textit{ducting}
to refer to the similar but more general impact of scattering, without implying that the emission is
guided within a particular density structure as originally proposed by \citet{Duncan79}.
Here, we will simply refer to \textit{scattering} to avoid potential confusion between the two concepts.}
\edit{After being scattered for the last time upon reaching a height with sufficiently-low densities, a radio wave will
then be refracted through the corona before reaching an observer, further shifting the source location.
As the coronal density gradient generally decreases radially, radio waves will tend to refract toward
to the radial direction such that a source originating at the limb will appear at a somewhat lower height than its origin, which
could be either the actual emission site (e.g. \citealt{Stewart76}) or, more likely, the point of last scatter (e.g. \citealt{Mann18}).
Accounting for the refractive shift, which becomes larger with decreasing frequency, therefore requires that the emission be generated at
or scattered to an even larger height than is implied by the observed source location.
Recent results on this topic from \citet{Mann18} will be discussed in the next section.}
Propagation effects are also thought to be important to the observed structure of the quiescent corona,
where the dominant emission mechanism is thermal
bremsstrahlung (free-free) radiation at MWA frequencies.
Outside of coronal holes, this emission is expected to be in or close to the optically thick regime (e.g. \citealt{Kundu82,Gibson16}), which means that the observed brightness temperature
should be the same as the coronal temperature.
However, well-calibrated 2D measurements have generally found lower brightness
temperatures than expected from temperatures derived at other wavelengths (see review by \citealt{Lantos99}).
Additionally, the size of the corona appears to be larger than expected at low frequencies
(e.g. \citealt{Aubier71,Thejappa92,Sastry94,Subramanian04,Ramesh06}).
The prevailing explanation for these effects is also scattering by density inhomogeneities
(e.g. \citealt{Melrose88,Alissandrakis94,Thejappa08})\edit{, though the refractive effect described in the
previous paragraph is also important \citep{Thejappa08}.}
Thus, the \edit{scattering} process that may act to elevate type III sources also
affects quiescent emission, increasing the apparent size of the corona.
We will take advantage of this by using the difference in extent between
observed and modeled quiescent emission
as a proxy for the net effect of propagation effects on our type III source heights.
\edit{This approach is limited in that, although both are related to scattering, the extent
to which the magnitudes of these two phenomena are related is unclear.
In particular, previous studies on the broadening of the radio Sun by scattering
have considered random density inhomogeneities as opposed to the more realistic
case of high density fibers capable of producing the ducting-like effect for type III sources.}
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{20141014_010112_1097290888_hera_sc3_0_forward_comparson_beamfix2.eps}}
\caption{MWA background images for the 2014-10-14 event (top) and corresponding MAS--FORWARD synthetic
images convolved with the MWA beam (bottom).
Beam ellipses are shown in the lower-left corners,
and the cyan curves are the 50\% burst contours from Figs.~\ref{fig:centroids} \& \ref{fig:overlay}.
This day is shown because thermal emission is only barely distinguishable at 132 MHz, precluding
the Fig.~\ref{fig:offset} analysis at that frequency, which was not the case for any other event-channel combination.
}
\label{fig:forward}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{20141014_010112_1097290888_hera_sc3_0_scatter_offset_plot_beamfix2_median.eps}}
\caption{\textit{Left}: Average intensity versus radial distance obtained from the Fig.~\ref{fig:forward} images
and normalized by the median value below \rsolar{1}. $\Delta{h}$ refers to the height offset between the observed and
modeled intensity profiles at the apparent type III burst height at 80 MHz.
\textit{Right}: $\Delta{h}$ for each frequency and event.
An orange asterisk marks the one instance where data was available but a measurement could not be made
because the thermal background emission was not well-detected (see Fig.~\ref{fig:forward}), so an average of the
adjacent points is used.
The uncertainties reflect the sensitivity of $\Delta{h}$ to the normalization choice in the left panel (see Sec.~\ref{propagation}).
}
\label{fig:offset}
\end{figure}
Figure~\ref{fig:forward} shows the observed background emission versus synthetic images
based on a global MHD model.
The MWA images are obtained by averaging every frame with a total intensity less than
2$\sigma$ above the background level, which is determined via the procedure shown in Figure~\ref{fig:baseline}.
Synthetic images are obtained using
FORWARD \citep{Gibson16}, a
software suite that calculates the expected bremsstrahlung and gyroresonance emission
given a model atmosphere.
We use the Magnetohydrodynamic Algorithm outside a Sphere
(MAS; \citealt{Lionello09}) medium resolution (\texttt{hmi\_mast\_mas\_std\_0201}) model, which extrapolates
the coronal magnetic field from photospheric magnetograms (e.g. \citealt{Miki99})
and applies a heating model adapted from \citet{Schrijver04} to compute density and temperature.
\citet{McCauley17} established the use of these model images for flux calibration and included a qualitative
comparison to MWA observations.
As in the aforementioned literature,
the radial extent of the corona is somewhat larger in the observations than
in the beam-convolved model images.
To quantify this difference, we divide both image sets into concentric rings about Sun-center.
The average intensity within each ring is plotted against its radial distance in
the left panel of Figure~\ref{fig:offset}, where the intensities have been normalized by the
median value below \rsolar{1}.
We then measure the offset $\Delta{h}$ between the observed and modeled profiles at the
heights obtained from the type III positions.
$\Delta{h}$ is sensitive to how the intensity curves are normalized, and we quantify
this uncertainty by repeating the procedure for 10 different normalization factors that
reflect the median intensities within radial bins of width \rsolar{0.1} from 0 to \rsolar{1}.
The right panel of Figure~\ref{fig:offset} plots the $\Delta{h}$ results for each event,
which have 1$\sigma$ uncertainties of less than $\pm$\rsolar{0.025}.
The offset appears to depend roughly
linearly on frequency, with larger offsets at lower frequencies.
Fitting a line through all of the points, we find that:
\begin{equation}
\label{eq:dh}
\Delta{h} \approx{} -1.5\e{-3}f + 0.41; ~80\leq{} f \leq{240}~\rm{MHz}
\end{equation}
\noindent where $\Delta{h}$ is in solar radii and $f$ is in MHz.
This yields \rsolar{0.30} at 80 MHz and \rsolar{0.06} at 240 MHz.
We do not expect this expression to be relevant much outside of the prescribed frequency range,
but extrapolating slightly, we obtain \rsolar{0.32} at 60 MHz. This value is a bit more than
half of the $<$ \rsolar{0.56} limit found by \citet{Poquerusse88}.
\citeauthor{Poquerusse88}, and others who have quantified the scattering effect (e.g. \citealt{Robinson83}),
obtained their results by computing ray trajectories through a model corona.
That approach allows a fuller understanding of the propagation physics,
but the result is dependent on the
assumed concentration and distribution of high density fibers, which are not well constrained.
Our critical assumption is that emission produced at significantly lower heights would be absorbed, as
would be the case in our optically-thick model corona.
However, low coronal brightness temperatures could also be
explained by lower opacities (e.g. \citealt{Mercier09}) or a low filling factor, which would allow burst emission
to escape from lower heights and lead us to underestimate the potential impact of \edit{propagation effects}.
Figure~\ref{fig:dens3} shows how the Figure~\ref{fig:dens1} density results change after subtracting the
height offsets from Figure~\ref{fig:offset}.
The 2014-10-14 harmonic ($2f_p$) profile remains reasonable with the offsets, lying just below the
\citet{Goryaev14} model instead of just above it, while the
fundamental emission densities for that event would still be quite large.
Given that the \citeauthor{Goryaev14} model is among the highest-density streamer models
in the literature, we conclude that harmonic emission from a beam traveling
along an overdense structure is consistent with 2014-10-14 data.
Our assessment for this event is that
propagation effects may contribute to some but not all the apparent density enhancement.
The other two events exhibit unusually steep density profiles once the offsets are subtracted.
That was true also for the original 2015-09-23 results, which we attributed to its non-radial
trajectory in Section~\ref{density}.
However, the original 2015-10-27 densities were well-matched to the
\citet{Cairns09} $C(r - 1)^{-2}$ gradient but become too steep to match any standard density gradient
with the inferred offsets.
This may simply reflect the intrinsic density gradient of the particular structure.
Alternatively, it is possible that we have over- or underestimated the impact of \edit{propagation effects} at the low or high end of our
frequency range, respectively.
However, the frequency dependence of scattering \edit{and refraction} means that any treatment will
steepen the density gradient.
Aside from their slopes, the offsets bring the densities implied by both bursts to generally within the
normal background range assuming harmonic emission or to a moderately enhanced level
assuming fundamental emission.
\begin{figure}
\centerline{\includegraphics[width=\textwidth,clip=]{type3_densities_style4_plot_beamfix2_harmonic_offset.eps}}
\caption{Imaging density profiles after applying the $\Delta{h}$ offsets from Fig.~\ref{fig:offset} and
assuming harmonic emission.
Model curves are as in Figs.~\ref{fig:dens1} \& \ref{fig:dens2}.
}
\label{fig:dens3}
\end{figure}
\section{Discussion} %
\label{discussion} %
The previous section suggests that propagation effects can partially explain
the apparent density enhancements implied by our type III source heights.
Assuming harmonic emission, our estimates for the potential magnitude of \edit{propagation effects} bring
the densities to within normal background levels for two events, while one
event remains enhanced at a level consistent with a dense streamer.
In Section~\ref{density}, we showed that our original density profiles are
consistent with those found from recent type III burst studies at lower
and higher frequencies, which together are well-fit by the
\citet{Cairns09} $C(r - 1)^{-2}$ gradient.
Both the low- and high-frequency studies conclude that their large densities
imply electron beams traveling along overdense structures,
but neither consider the impact of propagation effects.
\citet{Morosan14} propose a variation of the overdense hypothesis based
on their 30--60 MHz LOFAR observations. They suggest that
the passage of a CME just prior to an electron beam's arrival may compress streamer plasma
enough to facilitate type III emission at unusually large heights.
This was consistent with their events being associated with a CME and could be relevant to
our 2015-09-23 event, \edit{which was also preceded by a CME} (see~Section \ref{context}).
While this interpretation is plausible,
we do not think such special conditions need to be invoked given that the densities inferred from the
\citeauthor{Morosan14} results are consistent with ours (Figure~\ref{fig:dens2}) and are broadly consistent with the large
type III source heights found using the previous generations of low frequency instruments (e.g. \citealt{Wild59,Morimoto64,Malitson66,Stewart76,Kundu84}).
\edit{Propagation effects} seem particularly likely to have contributed (at least partially) to their inferred density enhancement, as the
effects become stronger with decreasing frequency.
\edit{Recently, \citet{Mann18} also examined the heights of type III sources observed at the limb by LOFAR.
After accounting for the refractive effect described in Section~\ref{propagation}, and relying on scattering to direct
emission toward the observer at large heights prior to being refracted,
their results imply a density enhancement of around 3.3$\times$ over the \citet{Newkirk61} density
model, assuming fundamental emission.
Incorporating our offsets from Section~\ref{propagation} gives us an average enhancement of 4.6$\times$
over the same model across our three events, also assuming fundamental emission.
Our results are therefore consistent with those of \citet{Mann18}, though our attempts quantify the impact
of propagation effects are quite different.}
\citet{Chen13} also found large densities using VLA data at higher frequencies (1.0--1.5 GHz).
Scattering is also thought to be important at high frequencies given the apparent lack
of small-scale structure \citep{Bastian94}, but the
extent to which \edit{scattering may also elevate radio sources} in that regime has
not been addressed to our knowledge.
\citeauthor{Chen13} observed an on-disk event, from which they obtain source heights by comparing
their projected positions to stereoscopic observations of an associated EUV jet, which is assumed
to have the same inclination as the type III electron beam.
This method would also be impacted by any source shifting caused by \edit{scattering}.
Although these shifts would be much smaller than at low frequencies, the background
gradient is much steeper, so a reasonably small shift may still strongly influence the
inferred density relative to the background.
If we accept the densities obtained at higher frequencies, albeit from just one example,
then their consistency with low frequency observations in general is striking.
As we describe in Section~\ref{introduction}, the community largely came to favor propagation effects
over the overdense hypothesis in the 1980s, and
the topic has not had much consideration since.
If new observations at low heights (high frequencies) also suggest beams moving
preferentially along dense structures, then it begs the question of whether or not that interpretation is again
viable at larger heights (lower frequencies).
In that case, this would need to be reconciled with the
fact that electron beams have not been found to be preferentially associated
with particularly high density regions in \textit{in situ} solar wind measurements \citep{Steinberg84}, along with
the evidence for other impacts of scattering such as angular broadening
(e.g. \citealt{Steinberg85,Bastian94,Ingale15}).
Selection effects may be relevant, as radiation produced well above the ambient plasma frequency is
less likely to be absorbed before reaching the observer.
Thus, coronal type III bursts may imply high densities because beams traveling along dense structures are
likelier to be observed.
Type III bursts also have a range of starting frequencies,
which has been interpreted in terms of a range in acceleration (i.e. reconnection) heights that
are often larger than those inferred from X-ray observations \citep{Reid14b}.
Alternatively, a beam may be accelerated at a smaller height than is implied by
the resultant type III starting frequency due to
unfavorable radiation escape conditions (absorption) below the apparent starting height.
Simulations also suggest that electron beams may travel a significant distance before producing
observable emission \citep{Li13b,Li14} and that they may be radio loud at
some frequencies but not at others due to variations in the ambient
density \citep{Li12,Loi14} and/or temperature \citep{Li11b,Li11}.
The magnetic structures along which electron beams travel also evolve with distance from
the Sun.
A popular open flux tube model is an expanding funnel that is thin at the base of the corona
and increasingly less so into the solar wind (e.g. \citealt{Byhring08,He08,Pucci10}).
Such structures may allow a dense flux tube to become less dense relative
to the background as it expands with height.
Moreover, a beam following a particular magnetic field line from the corona into the high-$\beta$ solar wind may not
necessarily encounter a coherent density structure throughout the heliosphere.
Turbulent mixing, corotation interaction regions, CMEs, and other effects influence
solar wind density such that it is not obvious that an electron beam traversing an overdense structure near the Sun
should also be moving in an overdense structure at large heliocentric distances.
We also note that one of the main conclusions from many of the type III studies referenced here is unchanged
in either the overdense or \edit{propagation effects} scenarios.
Both cases require a very fibrous corona that can supply dense structures
along which beams may travel and/or dense structures capable of \edit{scattering} radio emission \edit{to larger heights}.
This consistent with the fine structure known from eclipse observations (e.g. \citealt{Woo07}) that has more recently
been evidenced by EUV observations.
For instance, analyses of a sungrazing comet \citep{Raymond14} and of EUV spectra \citep{Hahn16}
independently suggest large density contrasts ($\gtrsim$ 3--10) between neighboring flux tubes in
regions where the structures themselves are undetected.
As our understanding of such fine structure improves, better constraints can be placed on them
for the purpose of modeling the impact of \edit{propagation effects} on radio sources.
\section{Conclusion} %
\label{conclusion} %
We presented imaging of three isolated type III bursts observed at the
limb on different days using the MWA.
Each event is associated with a white light streamer
and plausibly associated with EUV rays that exhibit activity around the
time of the radio bursts.
Assuming harmonic plasma emission, density profiles derived from
the source heights imply enhancements of
$\sim$2.4--5.4$\times$ over background levels.
This corresponds to electron densities
of 1.8\e{8} cm\tsp{-3} (240 MHz)
down to 0.20\e{8} cm\tsp{-3} (80 MHz) at average heights of 1.3 to \rsolar{1.9}.
These values are consistent with the highest streamer densities inferred from other
wavelengths and with the large radio source heights found using older instruments.
The densities are also consistent with recent type III results at higher and lower frequencies,
which combined are well-fit by a $C(r - 1)^{-2}$ gradient.
By comparing the extent of the radio limb to model predictions, we estimated that
\edit{radio propagation effects, principally the ducting-like effect of random scattering by high density fibers,}
may be responsible for 0.06--0.30\rsolar{}
of our apparent source heights.
This shift brings the results from 2 of our 3 events to within a standard range of background densities.
We therefore conclude that propagation effects can partially explain the apparent density enhancements but
that beams moving along overdense structures cannot be ruled out.
We also used the imaging data to estimate electron beam speeds of 0.24--0.60 c.
\begin{acks}
Support for this work was provided by the Australian Government through an Endeavour Postgraduate Scholarship.
We thank Stephen White and Don Melrose for helpful discussions
\edit{and the anonymous referee for their constructive comments}.
This scientific work makes use of the Murchison Radio-astronomy Observatory (MRO), operated by
the Commonwealth Scientific and Industrial Research Organisation (CSIRO).
We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site.
Support for the operation of the MWA is provided by the Australian Government's
National Collaborative Research Infrastructure Strategy (NCRIS),
under a contract to Curtin University administered by Astronomy Australia Limited.
We acknowledge the Pawsey Supercomputing Centre, which is supported by the
Western Australian and Australian Governments.
The SDO is a National Aeronautics and Space Administration (NASA) satellite, and
we acknowledge the AIA science team for providing open
access to data and software.
The SOHO/LASCO data used here are produced by a consortium of the Naval Research Laboratory (USA),
Max-Planck-Institut fuer Aeronomie (Germany), Laboratoire d'Astronomie (France),
and the University of Birmingham (UK).
SOHO is a project of international cooperation between ESA and NASA.
This research has also made use of NASA's Astrophysics Data System (ADS),
along with JHelioviewer \citep{Muller17} and the Virtual Solar Observatory (VSO, \citealt{Hill09}).
\end{acks}
\bibliographystyle{spr-mp-sola}
\tracingmacros=2
|
1,116,691,499,359 | arxiv | \section{Introduction}
The central engine of $\gamma$-ray bursts (GRBs, see, \citealp[e.g.][]{kumarandzhang2015} for a review) is
hidden to direct observation. However, its workings may be imprinted in observational signals
of this phenomenon. \cite{bromberg2012,bromberg2013} proposed that the prompt $\gamma$-ray duration distribution
of collapsar GRBs exhibits a plateau, indicating the fact that GRB jets launched at the core of a
collapsing star must drill their way out of the star and break out from its surface before producing the observed
$\gamma$-rays (for the collapsar model, see,
\citealp[e.g.][]{paczynski1998, macfadyen1999}). This claim can be
extended to understand low-luminosity GRBs as jets that barely failed to
break out \citep{bromberg2011}. Recently, these arguments
have been used to suggest that failed jets may operate in all Type Ib/c supernovae (SNe) \citep{sobacchi2017}.
Previous works have focused on the $\gamma$-ray duration distribution and assumed a single breakout time
for all collapsar GRBs \citep[e.g.,][]{sobacchi2017}. However, differences in the properties of long-duration GRB engines
should yield different breakout times. In particular, one expects that more powerful jets will propagate more easily
through the star and will break out from it much quicker than weaker jets \citep[e.g.][]{zhang2003, morsony2007, mizuta2009, lazzatietal2012}.
Both analytical estimates \citep{bromberg2011, bromberg2011b} and
numerical simulations \citep[see][and references
therein]{lazzatietal2012} suggest that the jet's breakout time
depends upon the engine's isotropic luminosity as $L_{\rm e}^{-\chi}$; the
power-law index lies between 1/3 and 1/2 depending on properties of
the stellar envelope (e.g., density profile, radius and mass) and/or
the properties of the jet (e.g., jet composition).
In this paper, we consider the luminosity dependence of the breakout time in a scenario where the jet needs to
drill through the collapsing star before it can power a GRB. An extended distribution in engine luminosities is motivated by the GRB luminosity function itself, which extends several orders of magnitude in isotropic luminosity \citep[e.g.,][]{wanderman_piran2010}. We show that a broken power-law GRB luminosity function is an expected outcome of the jet-envelope interaction for central engines having a single power-law luminosity distribution. After matching the parameters of the model-predicted GRB luminosity function to the observed one, we derive a {\sl mono-parametric} $\gamma$-ray duration distribution. The maximum breakout time (i.e., its single tunable parameter) can be inferred by comparison to the observed GRB collapsar duration distribution. This framework is quite powerful as it connects observable quantities with
the properties of the GRB central engine and of the collapsing star. It also makes a tantalizing connection between GRBs and jet-driven
core-collapse supernovae (e.g., \citealp{soderberg2010, lazzatietal2012, marguttietal2014, sobacchi2017, piranetal2017, bear2017}, for a review, see, e.g., \citealp{soker2016} and references therein).
\section{Model setup}
We consider a scenario similar to that presented in \cite{bromberg2012}. A long-duration GRB central engine that launches a jet\footnote{Hence, we use the terms central engine and (injected) jet interchangeably.} for
a duration $t_{\rm e}$ must be active for a duration longer than the time $t_{\rm b}$ it takes the jet to break out of the stellar
envelope in order to produce a $\gamma$-ray signal. The duration of the prompt $\gamma$-ray emission is thus
$t_{\gamma} = t_{\rm e} - t_{\rm b}$. For $t_{\rm e} < t_{\rm b}$ the jet is unable to break out from the star and ``fails'', but still injects its energy to
the stellar envelope, possibly powering the supernova explosion (see section 4). Contrary to previous studies
\citep{bromberg2012, sobacchi2017}, we consider a distribution of engine luminosities which, in turn, translates
into (i) a distribution of observed GRB luminosities and (ii) a distribution of breakout times.
In this section we describe our assumptions regarding the distribution of engine luminosities and their duration.
We consider a normalized power-law distribution of isotropic engine luminosities $L_{\rm e}$ as
\begin{eqnarray}
\label{eq:pL}
p(L_{\rm e})\equiv \frac{{\rm d}N}{{\rm d}L_{\rm e}}=\frac{a-1}{L_{\rm e, \min}}\left( \frac{L_{\rm e}}{L_{\rm e, \min}}\right)^{-a}, \ a> 1,
\end{eqnarray}
where $L_{\rm e,min}$ is the engine's minimum isotropic luminosity. Whether or not the
jet breaks out from the star depends on the breakout time $t_{\rm b}$, which is related to $L_{\rm e}$ as:
\begin{eqnarray}
\label{eq:tb}
t_{\rm b}=t_0 \left(\frac{L_{\rm e}}{L_{\rm e,0}} \right)^{-\chi}.
\end{eqnarray}
where $L_{\rm e,0}=10^{51}$~erg~s$^{-1}$. Henceforth, we consider a narrow
range of possible power-law indices that are motivated by relativistic hydrodynamic simulations of jet propagation in collapsars,
namely $1/3 \lesssim \chi \lesssim 1/2$ \citep[e.g.][]{bromberg2011, lazzatietal2012, nakar2015}. The normalization $t_0$ encodes information about the jet's collimation and the properties of the stellar envelope. In our analysis, we will treat $t_0$ as a free parameter and assume it is the same among all GRB collapsars.
The distribution of breakout times, $p_{\rm b}(t_{\rm b})$, can be determined by the condition $p_{\rm b}(t_{\rm b}) {\rm d}t_{\rm b} = p(L_{\rm e}) {\rm d} L_{\rm e}$ using equations (\ref{eq:pL}) and (\ref{eq:tb}). The resulting distribution can be written as:
\begin{eqnarray}
\label{eq:pb}
p_{\rm b}(t_{\rm b}) & = & A_{\rm b} \left( \frac{t_{\rm b}}{t_0} \right)^{s}H[t_{\rm b, max}-t_{\rm b}]
\end{eqnarray}
where $H[x]$ is the Heaviside step function, $s=(a-1-\chi)/\chi$, and
\begin{eqnarray}
A_{\rm b} & = & \frac{a-1}{\chi t_0}\left(\frac{L_{\rm e,0}}{L_{\rm e,min}}\right)^{-a+1}.
\end{eqnarray}
The maximum breakout time $t_{\rm b, max}$ corresponds to an engine with the minimum luminosity and is given by:
\begin{eqnarray}
t_{\rm b, max} & = & t_0\left(\frac{L_{\rm e,min}}{L_{\rm e,0}}\right)^{-\chi}.
\label{eq:tbmax}
\end{eqnarray}
The distribution of engine durations $t_{\rm e}$ is assumed to be unrelated to the distribution of engine luminosities $L_{\rm e}$ and to follow a power law, similar to previous studies \citep[e.g.,][]{bromberg2012, sobacchi2017}:
\begin{eqnarray}
\label{eq:pe}
p_{\rm e}(t_{\rm e}) = A_{\rm e} \left( \frac{t_{\rm e}}{t_{\rm e, min}}\right)^{-\beta}H[t_{\rm e} - t_{\rm e, min}], \ \beta > 1,
\end{eqnarray}
where $t_{\rm e} =t_{\gamma}+t_{\rm b}$ and $t_{\rm e, min}$ is the minimum duration of the central engine. The normalization is
\begin{eqnarray}
A_{\rm e}=\frac{\beta-1}{t_{\rm e, min}}
\end{eqnarray}
and ensures that $\int_0^{\infty}{\rm d}t_{\rm e} p_{\rm e}(t_{\rm e})=1$.
\section{Results}
In this section we connect the engine distributions described previously with the observed distributions of luminosities and durations
of long (collapsar) GRBs.
Of all possible central engines, only those with $t_{\rm e} > t_{\rm b}$ can lead to ``successful'' GRBs, that is,
jets that can break out from the star and produce a $\gamma$-ray signal while the engine is still active.
The fraction of such successful jets is given by:
\begin{eqnarray}
f(t_{\rm b}) = \int_{t_{\rm b}}^{\infty} {\rm d}t_{\rm e} p_{\rm e}(t_{\rm e})
\end{eqnarray}
and can be written as:
\begin{eqnarray}
\label{eq:fout}
f(t_{\rm b}; L_{\rm e}) = \left(\frac{t_{\rm b}}{t_{\rm e, min}} \right)^{-\beta+1}H[t_{\rm b} - t_{\rm e, min}] + H[t_{\rm e, min} - t_{\rm b}],
\end{eqnarray}
where the dependence upon $L_{\rm e}$ comes through $t_{\rm b}$. The last term in
the right hand-side of the equation shows that
all engines with breakout times shorter than the minimum duration $t_{\rm e, min}$ will be successful in producing GRBs.
\subsection{Intrinsic GRB luminosity function}
We argue that the (normalized) isotropic GRB luminosity function is
the product of the fraction $f$ of successful jets and the engine luminosity function $p(L_{\rm e})$:
\begin{eqnarray}
\frac{{\rm d}N_{\rm GRB}}{{\rm d}L} = \frac{f(t_{\rm b};L_{\rm e})}{\eta}\frac{{\rm d}N}{{\rm d}L_{\rm e}},
\end{eqnarray}
where we assumed that the isotropic GRB (radiated) luminosity $L$ is a constant fraction of the
isotropic engine luminosity (i.e., $L=\eta L_{\rm e}$), as motivated by the narrow distribution of the $\gamma$-ray efficiency of GRBs \citep[e.g.][]{fan_2006, beniamini_2016}. Depending on whether $L$ denotes the peak or average burst luminosity, the numerical value of $\eta$ will differ by a factor of $\sim 3$.
Using equations (\ref{eq:pL}), (\ref{eq:tb}), and (\ref{eq:fout}) we find:
\begin{eqnarray}
\label{eq:pGRB}
\frac{{\rm d}N_{\rm GRB}}{{\rm d}L} =N_0\left(\frac{L}{L_{\min}}\right)^{-a}
\left\{ \begin{array}{l}
\left(\frac{L}{L_{\rm br}}\right)^{-w}, L \le L_{\rm br} \\ \\
1, \ L > L_{\rm br},
\end{array}
\right.
\end{eqnarray}
where $N_0\equiv (a-1)/L_{\min}$, $w=\chi(-\beta+1)<0$ and the break luminosity is defined as:
\begin{eqnarray}
\label{eq:Lbr}
L_{\rm br} \equiv L_0 \left(\frac{t_{\rm e, min}}{t_0} \right)^{-1/\chi}.
\end{eqnarray}
Therefore, a power-law distribution of the central engine luminosity
results in a broken power-law distribution of the GRB luminosity
function. The break at $L_{\rm br}$ is due to the fact that an
increasing fraction of engines with lower luminosities are not able
to produce successful GRBs. In contrast, the GRB luminosity function reflects the luminosity function of the central engine for $L > L_{\rm br}$.
Observationally, the GRB luminosity function is, indeed, best described by a broken power law \citep[e.g.][]{wanderman_piran2010, salvaterra2012, pescalli2016}:
\begin{eqnarray}
\label{eq:pgrb_intrinsic}
\phi(L) \equiv \frac{{\rm d}N_{\rm GRB}}{{\rm d}\log L} = \left \{
\begin{array}{ll}
\left(\frac{L}{L_*}\right)^{-\alpha_L} & L \le L_* \\
\left(\frac{L}{L_*}\right)^{-\beta_L} & L > L_*.
\end{array}
\right.
\end{eqnarray}
Comparison of the GRB luminosity function with the predicted one, equation (\ref{eq:pGRB}), allows us to uniquely determine the critical parameters of our model that describe the engine luminosity distribution and its duration:
\begin{eqnarray}
L_{\rm br} & = & L_* \label{eq:Lstar} \\
a & = & \beta_L +1 \label{eq:a} \\
\beta & = & \frac{\beta_L-\alpha_L}{\chi}+1 \label{eq:beta} \\
t_{\rm e, min} & = & t_0 \left(\frac{10 L_*}{\eta_{-1} L_{\rm e,0}} \right)^{-\chi} \label{eq:temin},
\label{eq:param}
\end{eqnarray}
where we use the standard $Q_{\rm x} = 10^{-x} Q$ notation. We present a graphical view of our comparison in Fig.~\ref{fig:fig1} and summarize our results for the two indicative $\chi$ values in Table~\ref{tab:tab1}. Interestingly, the minimum engine activity timescale cannot be arbitrarily short for a given $t_0$.
\begin{figure}
\centering
\includegraphics[trim={0cm 0cm 0cm 0.8cm},clip,width=0.49\textwidth]{Luminosity_function.eps}
\caption{The distribution of engine luminosities, ${\rm d}N/{\rm d}L = (1/\eta){\rm d}N/{\rm d}L_{\rm e}$, assumed to be a power-law (blue dashed line), experiences a break as we consider only jets that can break out of the star to produce a distribution of successful GRBs, ${\rm d}N_{\rm GRB}/{\rm d}L$ (red solid line). We match this distribution to that found by \protect\cite{wanderman_piran2010}. A similar matching can be done for other luminosity functions. The inset shows the fraction of successful jets as a function of luminosity, see equation (\ref{eq:pGRB}).}
\label{fig:fig1}
\end{figure}
\begin{table}
\centering
\caption{Model parameters for jet propagation through the stellar envelope as determined by the GRB
luminosity function and the collapsar GRB duration distribution. Results for two theoretically motivated $\chi$
values are shown. The power-law indices are defined in equations (\ref{eq:pL}) and (\ref{eq:pe})
$t_{\rm b, max}$ is obtained from the observed $\gamma$-ray duration
distribution, see Fig. \ref{fig:fig2}. The parameter values of the GRB luminosity function are fixed to the best-fit values of \citet{wanderman_piran2010}.
}
\begin{tabular}{ccccc}
\hline
$\chi$ & $a$ & $\beta$ & $t_{\rm e, min}$ & $t_{\rm b, max}$ \\
\hline
1/3& 2.4 & 4.6 & $t_0/7$ & 69~s \\
1/2& 2.4 & 3.5 & $t_0/18$& 47~s \\
\hline
\end{tabular}
\label{tab:tab1}
\end{table}
\subsection{Distribution of $\gamma$-ray durations}
The distribution of rest-frame $\gamma$-ray durations of GRBs $t_{\gamma} = t_{\rm e} - t_{\rm b}$ can be calculated using the procedure outlined
in \cite{sobacchi2017}. However, we relax their assumption of a fixed
$t_{\rm b}$ value by considering a distribution of breakout times as determined by
the jet luminosity. In this case, $p_{\gamma}$ is calculated as:
\begin{eqnarray}
\label{eq:pg-def}
p_{\gamma}(t_{\gamma}) = \int_0^{\infty} {\rm d}t_{\rm b} p_{\rm b}(t_{\rm b}) \frac{p_{\rm e}(t_{\gamma}+t_{\rm b})}{f(t_{\rm b}; L_e)}.
\end{eqnarray}
The above equation can be recast in the form:
\begin{eqnarray}
\label{eq:pg}
p_{\gamma}(t_{\gamma})& = & A_{\rm b}A_{\rm e}t_0^{-s}\left(\frac{t_{\gamma}}{t_{\rm e, min}}\right)^{-\beta}\left[I_1(t_{\gamma}) + I_2(t_{\gamma}) \right] \\
I_1(t_{\gamma})& = & \int_{t_{\rm e, min}-t_{\gamma}}^{t_{\rm e, min}} {\rm d}t_{\rm b} t_{\rm b}^s \left(1+\frac{t_{\rm b}}{t_{\gamma}}\right)^{-\beta} \\
I_2(t_{\gamma})& = & t_{\rm e, min}^{1-\beta}\int_{t_{\rm e, min}}^{t_{\rm b, max}} {\rm d}t_{\rm b} t_{\rm b}^{s+\beta-1} \left(1+\frac{t_{\rm b}}{t_{\gamma}}\right)^{-\beta}
\end{eqnarray}
where $t_{\rm b, max}$ corresponds to the breakout time of the minimum isotropic engine luminosity $L_{\rm e, min}$,
see equation (\ref{eq:tbmax}). The distribution of $\gamma$-ray durations can be analytically derived in two limiting regimes \citep[see also][]{bromberg2012}:
\begin{enumerate}
\item $t_{\gamma} \ll t_{\rm b}$, where $t_{\rm e} \approx t_{\rm b}$. Using also $t_{\rm b, max} \gg t_{\rm e, min}$, equation (\ref{eq:pg}) can be cast in the form:
\begin{eqnarray}
\label{eq:pgshort}
p_{\gamma}(t_{\gamma}) \simeq \frac{\beta_L(\beta_L-\alpha_L)}{\chi(\beta_L-\chi)}t_{\rm b, max}^{-1},
\end{eqnarray}
which is independent of $t_{\gamma}$. For short enough
$\gamma$-ray durations, $p_{\gamma}$ is set by the maximum breakout
timescale and the power-law indices of the GRB luminosity function.
\item $t_{\gamma} \gg t_{\rm b}$, where $t_{\rm e} \approx t_{\gamma}$. Equation (\ref{eq:pg-def}) then results in:
\begin{eqnarray}
\label{eq:pglong}
p_{\gamma}(t_{\gamma}) \simeq \frac{\beta_L(\beta_L-\alpha_L)}{\chi (2\beta_L-\alpha_L) t_{\rm b, max}}\left( \frac{t_{\gamma}}{t_{\rm b, max}}\right)^{-1-\frac{\beta_L-\alpha_L}{\chi}}.
\end{eqnarray}
In this regime, the distribution of durations reflects the
distribution of engine timescales ($p_{\gamma} \propto t_{\gamma}^{-\beta}$), in agreement with previous works
\citep{bromberg2012, sobacchi2017}. However, in our study, the
power-law slope of the engine time distribution $\beta$ is {\it not} a free
parameter to be determined by comparison to the GRB duration distribution. It is instead
uniquely determined by the power-law slopes of
the GRB luminosity function (i.e. $\alpha_L, \beta_L$) and $\chi$
and can be used as a test of the model against the observed duration distribution of GRBs.
\end{enumerate}
For a specific jet propagation theory ($\chi$ value) and GRB luminosity function,
the only free parameter which enters the calculation of $p_{\gamma}(t_{\gamma})$ is the maximum breakout time. This affects both the ``plateau'' and the turnover of the distribution $p_{\gamma}$, see equations (\ref{eq:pgshort}) and (\ref{eq:pglong}). Thus, $t_{\rm b, max}$ can be determined by fitting the model, equation (\ref{eq:pg}), to the duration distribution of collapsar GRBs.
To derive the observed GRB duration distribution, we used the $T_{90}$\footnote{This is a measure of the GRB duration and is defined as the time interval during which 90 per cent of the photon counts have been detected \citep{kouveliotou1993}.} value of 1066 GRBs detected by the {\it Swift} satellite during the period of 2004 -- 2017\footnote{\url{https://swift.gsfc.nasa.gov/archive/grb_table/}}. Redshift information is available only for $\sim 1/3$ of the sample, with a median redshift of $z\simeq 1.7$. This was adopted as the typical redshift of the entire {\it Swift} sample in order to construct the histogram of rest-frame $\gamma$-ray durations as $t_{90}=T_{90}/(1+z)$.
The normalized histogram of $t_{90}$ of all {\it Swift} GRBs is plotted in Fig.~\ref{fig:fig2} with a black line. The corresponding histogram of {\it Swift} GRBs with measured redshifts is also shown for comparison (orange line). Overall, the two histograms are qualitatively similar with the main differences appearing at short durations. Regardless, GRBs with $t_{90} \lesssim 0.3$~s (i.e., short GRBs) do not have a collapsar origin and are neglected in this study. Overplotted as coloured curves are our model
predictions with $t_{\rm b, max}\simeq 69$~s for $\chi=1/3$ and $t_{\rm b, max} \simeq 47$~s for $\chi=1/2$ (see also Table~\ref{tab:tab1}). Both provide
very good quantitative descriptions to the collapsar GRB duration distribution. In fact, the plateau of the duration distribution is expected to be similar for models with the same product of $\chi$ and $t_{\rm b, max}$, as long as $\beta_{\rm L} \gg \chi$. More specifically, $p_{\gamma} \propto (\chi t_{\rm b, max})^{-1}$ at short enough durations, see equation (\ref{eq:pgshort}). From the ``fit'' to the duration distributions with $\chi=1/3$ and $\chi=1/2$ we find $\chi t_{\rm b, max} \approx T_{\max}$ where $T_{\max}\simeq 23$~s.
A timescale of $\sim 60$~s -- beyond which the duration distribution turns into a steep power law -- is also found by \cite{sobacchi2017}. The authors associate this timescale with a {\it single} breakout time assumed for all
GRBs, whereas in our framework it is clearly related to the maximum breakout time. In addition,
\cite{sobacchi2017} fit their data in order to derive $\beta$, whereas in our model $\beta$ is {\it uniquely} determined
by the GRB luminosity function and $\chi$ -- see equation (\ref{eq:pglong}). A fit to the data is not guaranteed in our model. For instance, the observed $p_{\gamma}$ distribution
cannot be explained by luminosity functions with $\beta_L-\alpha_L < 1$ for $\chi$ in the range $1/3-1/2$, as motivated by theory
\citep{bromberg2011, lazzatietal2012}.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth, trim=20 100 20 80]{t90_v4.eps}
\caption{Histogram (normalized) of rest-frame $\gamma$-ray durations of all {\it Swift} GRBs (black line)
assuming a typical redshift of $\sim 1.7$.
The corresponding histogram of the 319 {\it Swift} GRBs with measured redshift is also shown (orange line).
Short GRBs ($t_{90} \lesssim 0.3$~s) do not have a collapsar origin and are therefore neglected in this study.
The model predictions for $\chi=1/3$ and $\chi=1/2$ are overplotted
for comparison. The model achieves a good quantitative description of the data by adjusting its single parameter: the maximum
breakout time $t_{\rm b, max}$ (shown with dashed vertical lines). The results are obtained for the GRB luminosity function of \citet{wanderman_piran2010}.
}
\label{fig:fig2}
\end{figure}
The maximum breakout time inferred from the observed GRB duration distribution can also be translated to a constraint between $t_0$ and a minimum luminosity for a given $\chi$ value. Using $\chi t_{\rm b, max}\approx T_{\max}$ together with equation (\ref{eq:tbmax}), we find:
\begin{eqnarray}
\label{eq:constraint-1}
T_{\rm max} \approx \chi t_{\rm b, max} = \chi t_0 \left(\frac{L_{\min}}{\eta L_{\rm e,0}} \right)^{-\chi}.
\end{eqnarray}
\section{Discussion}
The shape of the $\gamma$-ray duration distribution of collapsar GRBs (i.e., plateau followed by a power-law)
is generic and the result of: (i) the relation between the three
timescales, i.e. $t_{\gamma} = t_{\rm e} - t_{\rm b}$, which stems from the jet-stellar
envelope interaction, (ii) {\it at least} one power-law distribution
(either of $t_{\rm b}$ or $t_{\rm e}$), and (iii) a characteristic
timescale set by either $t_{\rm b}$ or $t_{\rm e}$. The plateau in $p_{\gamma}$ at short
durations is an even more general result of the jet-envelope
interaction model, as it only requires that $p_{\rm e}$ is a smooth function of $t_{\rm e}$ \citep{bromberg2012}.
\subsection{Completeness of the GRB sample}
In our analysis we assumed that the {\it Swift} GRB sample is complete with respect to
the observed burst duration. In other words, for a fixed luminosity the detection of the burst does not depend on its duration.
This is a simplifying assumption, for the minimum detectable flux of a GRB depends upon the exposure time $T$ as $T^{-1/2}$ \citep{baumgartner_2013, lien_2016}. For GRBs, the maximum exposure time is set by their duration. For simplicity, let us identify
$T$ as the observed duration $T_{90}$. This implies that the flux limit for detection is higher for GRBs with shorter duration. Indeed,
there is lack of luminous and short bursts (i.e., $L_{\rm iso} > 10^{52}$ erg s$^{-1}$ and $T_{90} < 2$~s) in the third
{\it Swift}/BAT GRB catalog (see Fig.~25 in \citealp{lien_2016}). This can be the result of a genuine lack of GRBs with these properties
or of unavailable redshift measurements. Regardless, these results suggest that the {\it Swift}/BAT sample is complete with respect to burst duration for long GRBs. Therefore, we do not expect our main conclusions to depend strongly on this assumption.
\subsection{Progenitors of collapsar GRBs}
For a distribution of breakout times we showed that the only free parameter that controls $p_{\gamma}(t_{\gamma})$
(both the plateau and position of the break) is the maximum breakout time. This, in turn, depends upon
$t_0$ and $L_{\min}$, see equations (\ref{eq:tbmax}) and (\ref{eq:constraint-1}).
Here, $L_{\min}$ should not be interpreted as the intrinsic GRB minimum luminosity. In fact,
the GRB sample becomes progressively less complete for lower isotropic GRB luminosities \citep[e.g.,][]{wanderman_piran2010}. Thus,
the duration distribution probes GRBs with isotropic luminosities down to an ``effective'' minimum luminosity $L_{\rm eff}$.
Substitution of $L_{\rm eff}\simeq 10^{51}$ erg s$^{-1}$
to equation (\ref{eq:constraint-1}) leads to a conservative constraint:
\begin{eqnarray}
\label{eq:constraint}
\frac{t_0}{T_{\max}} \approx \frac{10^{\chi}}{\chi}\left(\frac{L_{\rm eff, 51}}{\eta_{-1}} \right)^{\chi},
\end{eqnarray}
which translates to $t_0\sim 150$~s for all $\chi$ values in the range $1/3-1/2$ and $T_{\max}\sim 23$~s. Values of $t_0 \gg 20$~s cannot be easily reconciled with the scenario of jet propagation through a compact progenitor, because of the weak dependence that $t_0$ has on stellar properties \citep[see][]{bromberg2011, bromberg2012}. Our analysis suggests the presence of an extended low-mass envelope surrounding the GRB progenitor, as concluded independently by \cite{sobacchi2017}. The weak dependence of the breakout time on $L_{\rm eff}$ makes this conclusion quite robust.
Assuming a common $t_0 \sim 150$~s for all GRB collapsars, we can further predict that
the duration distribution will extend towards longer durations (i.e., larger $t_{\rm b, max}$ and $T_{\max}$) as the
sample becomes more complete at lower luminosities thanks to more sensitive future missions. Falsification of this prediction would provide indirect evidence for a distribution of $t_0$ among GRB progenitors, which lies beyond the scope of this paper.
\subsection{Energetics of central engines}
We next examine the energetics of central engines and their potential in powering a supernova explosion. All engines with
$L_{\rm e} > L_{\rm e, *}\equiv \eta^{-1} L_*$ produce successful GRBs. The associated energy deposited to the
stellar envelope by successful engines is
\begin{eqnarray}
\label{eq:Ees}
E_{\rm e, s} \approx f_{\rm b} L_{\rm e} t_{\rm b}(L_{\rm e}) \approx E_{\rm e,*} \left(\frac{L_{\rm e}}{L_{\rm e, *}}\right)^{1-\chi},
\end{eqnarray}
where $f_{\rm b} = 0.01 \ f_{\rm b, -2}$ is the jet beaming fraction and
\begin{eqnarray}
E_{\rm e, *}\equiv f_{\rm b} L_{\rm e,*} t_{\rm e, min}.
\end{eqnarray}
The minimum engine timescale $t_{\rm e, min}$ is given by equation (\ref{eq:temin}). Failed jets can also inject a considerable amount of energy to the stellar envelope. For $L_{\rm e, min} \le L_{\rm e} \le L_{\rm e,*}$ the engine activity timescale $t_{\rm e}$ lies between $t_{\rm b, max}$ and $t_{\rm e, min}$. The latter is still a good proxy for the average $t_{\rm e}$ in the range of $[t_{\rm e, min}, t_{\rm b, max}]$ because of the steepness of $p_{\rm e}(t_{\rm e})$. The energy of a failed GRB engine is then given by
\begin{eqnarray}
\label{eq:Eef}
E_{\rm e,f} \approx E_{\rm e,*} \frac{L_{\rm e}}{L_{\rm e,*}}.
\end{eqnarray}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth, trim=15 15 0 0]{Ee_Ne_v1.eps}
\caption{Energy deposited to the stellar envelope by successful and failed GRB engines for $\chi=1/3-1/2$ and $t_0=150$~s as obtained by our analysis (shaded region). Also shown are the number of successful (magenta dotted line) and failed (magenta dash-dotted line) engines as a function of engine luminosity. The total number of engines is overplotted (blue dashed line); a normalization of one engine with $L_{\rm e, *}= \eta_{-1}10^{53.5}$~erg~s$^{-1}$ was used.}
\label{fig:fig3}
\end{figure}
Figure~\ref{fig:fig3} presents the beaming-corrected energy, $E_{\rm e}$, of successful and failed engines (see equations (\ref{eq:Ees}) and (\ref{eq:Eef}), respectively) as a function of $L_{\rm e}$ for $\chi=1/3-1/2$ and $t_0=150$~s. For a typical GRB engine (i.e., $L_{\rm e} = L_{\rm e,*}$), we find
\begin{eqnarray}
\label{eq:Ees-num}
E_{\rm e,*} \approx (2-7) \times 10^{52} f_{\rm b,-2} \left(\frac{L_{*, 52.5}}{\eta_{-1}}\right)^{1-\chi} \, {\rm erg}
\end{eqnarray}
for $\chi = 1/3 - 1/2$. This is comparable to the kinetic energy inferred in SNe associated to
long GRBs \cite[e.g.,][]{hjorthandbloom2012} and provides further support that GRB-SNe are jet-driven \cite[e.g.,][]{hjorth2013}.
Failed jets that inject $\sim 10^{51}$ erg (1 foe) to the stellar envelope -- an energy comparable to the
binding energy of the stellar envelope and the typical kinetic energy of core-collapse SNe -- could facilitate the
explosion mechanism in these SNe \citep{sobacchi2017, piranetal2017}, but they are not expected to drive mildly relativistic SNe ejecta \citep[see e.g.][]{soderberg2010}. The isotropic engine luminosity in this case is $L_{\rm e, foe}=10^{52}\ E_{\rm e, 51} f_{\rm b, -2}^{-1} t_{\rm e, min, 1}^{-1}$~erg~s$^{-1}$.
Jets can deposit a significant energy into the stellar envelope on timescales of tens of seconds (i.e., $t \gtrsim t_{\rm e, min}$), but it is unlikely that they can produce the necessary mass of radioactive $^{56}$Ni to partially power the optical light curve of core-collapse SNe \citep[i.e., $\sim 0.02\,M_{\sun}-0.2\,M_{\sun}$,][]{hamuy_2003, mueller_2017}. Being a by-product of the explosive silicon burning process, $^{56}$Ni can be synthesized under specific temperature and density conditions \citep[e.g.][]{woosley_1973, woosley_1995, jerkstrand_2015}, which cannot be easily met once the ejecta has expanded far from the central engine.
Although failed jets cannot be the sole mechanism that drives the SN explosion, a picture emerges where they may be an important component of this process \citep[e.g.][]{lazzatietal2012, sobacchi2017, piranetal2017}. The limitation on jet-driven explosions stems from the energetic requirement of injecting $\sim 10^{51}$~erg and not from a lack of engines at these lower luminosities (see dash-dotted line in Fig.~\ref{fig:fig3}).
The number of failed GRB engines with $L_{\rm e} < L_{\rm e,*}$ can be written as
$N_{\rm f}(L_{\rm e})\approx \left(L_{\rm e}/L_{\rm e,*}\right)^{-\beta_L}$, where a normalization of one engine at luminosity $L_{\rm e,*}$ was assumed. The true rate of failed engines capable of injecting $\sim 10^{51}$ erg at typical $\hat{z}=2$ can be then estimated as:
\begin{eqnarray}
R_{\rm f}(\hat{z}) \!\! & \approx & \!\! N_{\rm f}(L_{\rm e, foe}) R_{\rm GRB}(\hat{z}) f_{\rm b}^{-1}\\
\!\! & \approx & \!\! 10^5 \, f_{\rm b,-2}^{0.4} \left(\frac{\eta_{-1} E_{\rm e, 51}}{t_{\rm e, min,1} L_{*, 52.5}}\right)^{-1.4}\!\!\!\!{\rm Gpc}^{-3} {\rm yr}^{-1}
\end{eqnarray}
where $R_{\rm GRB}(\hat{z})\approx 10$~Gpc$^{-3}$~yr$^{-1}$ is the observed (i.e., uncorrected for beaming) collapsar GRB rate \citep[][]{wanderman_piran2010, sobacchi2017}. We also used $\beta_L=1.4$ and assumed a redshift independent engine luminosity function. Interestingly, this order-of-magnitude prediction is consistent with inferred rates of Type Ib/c SNe at $\hat{z}\sim 2$ \citep[e.g.][]{dahlen2004, dahlen2012, cappellaro2015, strolger2015} suggesting an active role of the central engine in all Ib/c SNe \citep[see also][]{sobacchi2017, bear2017}.
Our results, although consistent with those by \cite{sobacchi2017}, are derived with a very different method. The number
and energetics of our failed jets, as well as the minimum engine timescale are naturally obtained by matching
the distribution of successful engines with the GRB luminosity function, instead of relying on an extrapolation
of the central engine duration distribution to an (almost) arbitrary value.
Our analysis was performed independently of the nature of the GRB central engine, which can be either a newly born black hole or a magnetar (i.e., a rapidly spinning highly magnetized neutron star), see, e.g., \cite{kumarandzhang2015} for a review on GRB central engines. Our predictions for the energetics of central engines are summarized in Fig.~\ref{fig:fig3} and can be used to constrain their nature. We showed that the beaming-corrected energy of a typical GRB engine that is active for $t_{\rm e, min} \sim 10$~s is a few $\times 10^{52}$~erg, see also equation (\ref{eq:Ees-num}). This might be already in tension with the magnetar scenario, although recent studies cast doubt on previous estimates on the maximum energy extractable in the magnetar model \citep{brianetal2015}. In addition, the distributions of the engine duration and luminosity are expected to differ between models of the central engine. In the magnetar scenario, for example, where the luminosity and duration of the engine depend both on the magnetic field and spin period of the magnetar (e.g., \citealp{usov1992}), the distributions of $L_{\rm e}$ and $t_{\rm e}$ will be correlated. A careful comparison of our model predictions against several observational constraints, such as the GRB distribution on the $L-T_{90}$ plane, will be the subject of a follow-up study.
\section{Conclusions}
The propagation of the jet through the envelope of a collapsing star, which acts as a filter of less powerful jets, may explain
the observed GRB luminosity function and the GRB duration distribution. Our work helps to better understand the properties of long GRB central engines, while it provides interesting hints for the jet-driven SN scenario.
\section*{Acknowledgements}
We thank the anonymous referee for a constructive report. We also thank P. Beniamini, B. Metzger, and E. Nakar for useful comments on the manuscript. We thank Georgios Vasilopoulos for useful discussions. We acknowledge support from NASA through the grants NNX16AB32G and NNX17AG21G issued through the Astrophysics Theory Program. We also acknowledge support from the Research Corporation for Science Advancement’s Scialog program.
\bibliographystyle{mnras}
|
1,116,691,499,360 | arxiv | \section{Introduction}
There has been much interest in high-temperature (unconventional) superconductivity (SC) over more than last two decades.
Moreover, the phase separations phenomenon involving superconducting states (SS) is a very current topic after it has been evidenced in a broad range of intensely investigated materials including iron pnictides, cuprates and organic conductors (discussed below, also see for examples in Refs.~[\onlinecite{ksenofontov.wartmann.11,johnston.10,stewart.11,park.inosov.09,ricci.poccia.11,goko.aczel.09,simonelli.mizokawa.14,xu.tan.08,
udby.andersen.09,savici.fudamoto.02,mohottala.wells.06,pan.oneal.01,kornilov.pudalov.04,colin.salameh.08,taylor.carrington.08,fernandes.schmalian.10}] and references therein).
The phase separation is a coexistence of two (homogeneous) phases. In such a state coexisting phases form domains, which can differ from each other by, for example, electron concentration or order parameter.
It is worth to note that the phase separations involving superconducting states have been evidenced in a broad range of currently intensely investigated materials including iron pnictides, cuprates and organic conductors.
In particular, there are experimental evidences of phase separation between superconducting and (anti-)ferromagnetic~\cite{ksenofontov.wartmann.11,fernandes.schmalian.10} or magnetic and non-magnetic (superconducting) order~\cite{park.inosov.09,ricci.poccia.11,goko.aczel.09,simonelli.mizokawa.14} in iron-based superconductors.
Moreover, the coexistence of superconductivity and magnetic order~\cite{xu.tan.08,savici.fudamoto.02,udby.andersen.09} as well as charge-order~\cite{pan.oneal.01} has been reported in cuprates.
Organic compounds also exhibit the superconductor-insulator phase separations as a~result of the external pressure (e.g. quasi-one dimensional (TMTSF)$_2$PF$_6$~\cite{kornilov.pudalov.04} and (TMTSF)$_2$ReO$_4$~\cite{colin.salameh.08}) and fast cooling rate through the glass-like structure transition (e.~g. $\kappa$-(ET)$_2$Cu[N(CN)$_2$]Br~\cite{taylor.carrington.08}).
The Penson-Kolb-Hubbard (PKH) model is one of the conceptually simplest phenomenological models for studying correlations and for description of superconductivity in narrow-band systems with short-range, almost unretarded pairing~\cite{micnas.rannniger.90,hui.doniach.93,japardze.muller.97,japaridze.kampf.01,robaszkiewicz.bulka.99,czart.robaszkiewicz.01,czart.robaszkiewicz.04,czart.robaszkiewicz.06,czart.robaszkiewicz.15a,czart.robaszkiewicz.15b,dolcini.montorsi.00,robaszkiewicz.czart.01}. The PKH model includes also a nonlocal pairing mechanism (the intersite pair hopping term $J$) that is distinct from the on-site interaction $U$ in the attractive Hubbard model and that is the driving force of pair formation and also of their condensation~\cite{micnas.rannniger.90,robaszkiewicz.bulka.99}. Notice that in the absence of the on-site $U$ term the PKH model reduces to the Penson-Kolb model~\cite{penson.kolb.86,kolb.penson.86}, whereas in the absence of the intersite $J$ term it reduces to the standard Hubbard model~\cite{micnas.rannniger.90,hubbard.63}.
In this paper we determine the phase diagrams (for $T=0$ as well as $T>0$) of the PKH model for two dimensional (2D) square lattice within Hartree-Fock mean-field (HF--MF) theory focusing on a behavior of superconducting phases with changing model parameters and on a possibility of the occurrence of the state with phase separation. We obtain that the phase separation state, in which two different superconducting phases ($s$-wave and $\eta$-wave) can coexists, occurs in a definite range of electron concentration. In addition, increasing temperature can change the symmetry of the superconducting order parameter (from $\eta$-wave into $s$-wave).
The paper is organized as follows. In Section~\ref{sec:model} we present the derivation of HF--MF grand canonical potential and introduce some basic concept of phase separation. Next, Section~\ref{sec:results} is devoted for presentation of the results of numerical computations for the ground state ($T=0$, Section~\ref{sec:groundstate}) and finite temperatures ($T>0$, Section~\ref{sec:finitetemp}). Section~\ref{sec:summary} reports the important conclusions and provides supplementary discussion.
\section{Model and methods}\label{sec:model}
The purpose of the present work is the analysis of phase diagrams, in particular including phase separation, of the extended Hubbard model with pair hopping interaction, i.e. the so-called PKH model, which Hamiltonian in the real space has the following form
\begin{eqnarray}
\label{eq.ham} \mathcal{H} & = & \mathcal{H}_{0} + \mathcal{H}_{int}, \\
\label{eq.ham.zero} \mathcal{H}_{0} &=& \sum_{ \langle i,j \rangle \sigma } \left( - t - \mu \delta_{ij} \right) c_{i\sigma}^{\dagger} c_{j\sigma} , \\
\label{eq.ham.int} \mathcal{H}_{int} &=& U \sum_{i} c_{i\uparrow}^{\dagger} c_{i\uparrow} c_{i\downarrow}^{\dagger} c_{i\downarrow} + J \sum_{ \langle i,j \rangle } c_{i\uparrow}^{\dagger} c_{i\downarrow}^{\dagger} c_{j\downarrow} c_{j\uparrow},
\end{eqnarray}
where $c^{\dagger}_{i\sigma}$ ($c_{i\sigma}$) denotes the creation (annihilation) operator of an electron with spin $\sigma$ at site $i$.
$t$ is the single electron hopping integral between nearest-neigbors (NN), $\mu$ is the chemical potential, $U$ is the on-site density-density interaction, and $J$ is the intersite charge-exchange (pair hopping) interaction between NN, respectively.
The electron hopping amplitude ($t$) will be taken as a scale of energy in the system.
The pair-hopping term ($J$) was first proposed in Refs.~[\onlinecite{penson.kolb.86,kolb.penson.86}] by Penson and Kolb in 1986 and it can be derived from a general microscopic tight-binding Hamiltonian, where the Coulomb repulsion may lead to the pair hopping interaction $J = \langle ii | e^{2} / r | jj \rangle$ \cite{micnas.rannniger.90,czart.robaszkiewicz.01,czart.robaszkiewicz.04,czart.robaszkiewicz.06,czart.robaszkiewicz.15a,czart.robaszkiewicz.15b,robaszkiewicz.bulka.99,kapcia.robaszkiewicz.12,kapcia.robaszkiewicz.13,kapcia.14a,kapcia.14b,kapcia.15a,kapcia.15b,mierzejewski.maska.04}. In such a case $J$ is positive (\emph{repulsive} model $J > 0$, favoring $\eta$-wave SC), but in this case the magnitude of $J$ is very small. However, the effective attractive form ($J<0$, favoring $s$-wave SC) is also possible (as well as an enhancement of the magnitude of $J>0$) and it can originate from the coupling of electrons with intersite (intermolecular) vibrations via modulation of the hopping integral or from the on-site hybridization term in the general periodic Anderson model (cf. e.g. Ref.~[\onlinecite{micnas.rannniger.90,czart.robaszkiewicz.01,czart.robaszkiewicz.04,czart.robaszkiewicz.06,czart.robaszkiewicz.15a,czart.robaszkiewicz.15b,robaszkiewicz.bulka.99,kapcia.robaszkiewicz.12,kapcia.robaszkiewicz.13,kapcia.14a,kapcia.14b,kapcia.15b,kapcia.15a,mierzejewski.maska.04}] and references therein). It can also be included in the effective models for Fermi gas in an optical lattice in the strong interaction limit~\cite{rosch.rasch.08}. The role of $J$ interaction in a multiorbital model is of a particular interest because of its presence in the iron pnictides~\cite{ptok.crivelli.15}. It should be stressed that the Hubbard model on bipartite lattice has been rigorously proved to have $\eta$-wave SS states as eigenstates~\cite{yang.1989}. Moreover, $\eta$-wave pairing has been found as a mechanism of superconductivity in a large class of models of strongly correlated electron system (extended Hubbard models)~\cite{boer.korepin.1995}. It has been found that SC originating from the $J>0$ interaction is unique in that it is robust against the orbital (diamagnetic) pair breaking mechanism~\cite{mierzejewski.maska.04}.
The recent studies show that pair-hopping term can also play an important role in nano-space layered structures~\cite{kusakabe.12} and nano-devices~\cite{azema.dare.13}.
Moreover, its effects in various fermionic systems~\cite{matsuura.miyake.12,nishiguchi.kuroki.13,ding.zhang.14,kraus.dalmonte.13,ptok.crivelli.15} have been studied, in particular on the charge-Kondo effect~\cite{matsuura.miyake.12} and Majorana edge states~\cite{kraus.dalmonte.13} as well as in cuprates~\cite{nishiguchi.kuroki.13} and iron-pnictides~\cite{ptok.crivelli.15}. The pair-hopping interactions have been also intensively studied in Bose systems (e.g. Ref.~[\onlinecite{sarker.lovorn.12,zhang.yin.13}]).
In order to analyze superconducting phases we perform the mean-field factorization of interaction Hamiltonian~(\ref{eq.ham.int}):
\begin{eqnarray}
\label{eq.ham_space_mf}
\mathcal{H}^{HF}_{int} &=& U \sum_{i} \left( \Delta_{i}^{\ast} c_{i\downarrow} c_{i\uparrow} + h.c.
\right) - U \sum_{i} | \Delta_{i} |^{2} \\
\nonumber
&+& J \sum_{ \langle i,j \rangle } \left( \Delta^{\ast}_{j} c_{i\downarrow} c_{i\uparrow} + h.c.
\right) - J \sum_{ \langle i,j \rangle } \Delta^{\ast}_{i} \Delta_{j},
\end{eqnarray}
where we define local superconducting order parameter (SOP) as $\Delta_{i} = \langle c_{i\downarrow}c_{i\uparrow} \rangle$.
In general case, the local SOP is given as $\Delta_{i} = \Delta_{0} \exp ( i {\bm Q} \cdot {\bm r}_{i} )$, where $\Delta_0$ is the amplitude of SOP, ${\bm Q} = ( 0, 0)$ for $s$-wave SS and ${\bm Q} = ( \pi , \pi)$ for $\eta$-wave SS.
The mean-field Hamiltonian $\mathcal{H}^{HF} = \mathcal{H}_{0} + \mathcal{H}^{HF}_{int}$ in the momentum space takes the form~\cite{ptok.mierzejewski.08,ptok.maska.09}:
\begin{eqnarray}
\label{eq.ham_mom_mf} \mathcal{H}^{HF} &=& \sum_{{\bm k} \sigma } \mathcal{E}_{{\bm k}\sigma} c_{{\bm k}\sigma}^{\dagger} c_{{\bm k}\sigma} \\
\nonumber &+& U_{\bm Q}^{eff} \sum_{\bm k} \left( \Delta_{0}^{\ast} c_{-{\bm k}+{\bm Q} \downarrow} c_{{\bm k}\uparrow} + H.c. \right) - U_{\bm Q}^{eff} N | \Delta_{0} |^{2} ,
\end{eqnarray}
where $\mathcal{E}_{{\bm k}\sigma} = - t \gamma_{\bm k} - \mu$ is a dispersion relation of free electrons (independent of spin $\sigma$ in the absence of external magnetic field, with lattice constants $a=1$ and $b=1$), $\gamma_{\bm k} = 2 \left ( \cos ( k_{x} ) + \cos ( k_{y} ) \right)$ (for 2D square lattice), $U_{\bm k}^{eff} = U + J \gamma_{\bm k}$ is a effective pairing interaction in the momentum space, and $N=N_{x} \times N_{y}$ is number of sites in the lattice.
Notice that HF-MF Hamiltonians (\ref{eq.ham_space_mf}) and~(\ref{eq.ham_mom_mf}) exhibit the particle-hole symmetry (with respect to half-filling ($n=1$, $\mu=0$), so the phase diagrams will be presented as a function of $|\mu|$ and $|1-n|$~\cite{micnas.rannniger.90,kapcia.robaszkiewicz.12}.
In particular, the phase diagrams are symmetric with respect to $\mu = 0$ or $n=1$.
Let us stress that Hamiltonian (\ref{eq.ham}) also exhibits this symmetry, but at half-filling the corresponding value of chemical potential is $\mu=U/2$, which is exact result for Hamiltonian (\ref{eq.ham}).
\subsection{The Bogoliubov transformation}\label{sec:bogoliubov}
\begin{figure
\includegraphics[scale=1.2]{fig1}
\caption{Ground state $J/t$ vs. $U/t$ phase diagram for $n=1$ ($\mu=0$, obtained for $k_{B} T = 10^{-5} t$).
The numbers: $1$, $2$, and $3$ (in circles) denote the regions of occurrence of particular phases: $\eta$-wave SS, $s$-wave SS, and NO, respectively. The transition between regions $1$ and $2$ is discontinuous, whereas the transitions between regions $1$ and $3$ as well as between $2$ and $3$ are continuous.
The colour intensity indicates a value of the amplitude $\Delta_0$ of SOP (compare also with Ref.~[\onlinecite{ptok.kapcia.15}]).
}
\label{fig:fig1}
\end{figure}
In the Nambu notation Hamiltonian~(\ref{eq.ham_mom_mf}) takes the following form:
\begin{eqnarray}
\mathcal{H}^{HF} &=& \sum_{\bm k} \Psi_{\bm k}^{\dagger} H_{\bm k} \Psi_{\bm k} + \mbox{const}
\end{eqnarray}
with
\begin{eqnarray}
\quad H_{\bm k} &=& \left( \begin{array}{cc}
\mathcal{E}_{{\bm k}\uparrow} & U_{\bm Q}^{eff} \Delta_{0} \\
U_{\bm Q}^{eff} \Delta_{0}^{\ast} & -\mathcal{E}_{-{\bm k}+{\bm Q} \downarrow}
\end{array} \right) ,
\end{eqnarray}
where $\Psi_{\bm k}^{\dagger} = ( c_{{\bm k}\uparrow}^{\dagger} , c_{-{\bm k}+{\bm Q} \downarrow} )$ are Nambu's spinors.
Eigenvalues of matrix $H_{\bm k}$ are given by:
\begin{eqnarray}
\lambda_{{\bm k},\pm} = \zeta_{\bm k} \pm \vartheta_{\bm k},
\end{eqnarray}
where plus (minus) sign corresponds to the particle (hole) excitations.
Next, we assume that
\begin{eqnarray}
\vartheta_{\bm k} = \sqrt{ ( \eta_{\bm k} - \mu )^{2} + | U_{\bm Q}^{eff} \Delta_{0} |^{2} } , \\
\zeta_{\bm k} = - t \frac{ \gamma_{\bm k} - \gamma_{-{\bm k}+{\bm Q}} }{2} , \quad \eta_{\bm k} = - t \frac{ \gamma_{\bm k} + \gamma_{-{\bm k}+{\bm Q}} }{2} .
\end{eqnarray}
Matrix $H_{\bm k}$ can be diagonalized using unitary transformation $\mathcal{U}_{\bm k}$, which has the form:
\begin{eqnarray}
\mathcal{U}_{\bm k} &=& \frac{1}{2} \left( \begin{array}{cc}
u_{\bm k} & v_{\bm k} \\
- v_{\bm k} & u_{\bm k}
\end{array} \right),
\end{eqnarray}
where
\begin{eqnarray}
u_{\bm k} &=& \frac{1}{2} \sqrt{ 1 + ( \eta_{\bm k} - \mu ) / \vartheta_{\bm k} } , \\
v_{\bm k} &=& \frac{1}{2} \sqrt{ 1 - ( \eta_{\bm k} - \mu ) / \vartheta_{\bm k} } .
\end{eqnarray}
Then $H_{\bm k} = \mathcal{U}_{\bm k}^{\dagger} \cdot \mbox{diag} ( \lambda_{{\bm k},+} , \lambda_{{\bm k},-} ) \cdot \mathcal{U}_{\bm k}$, and the grand canonical potential takes the form:
\begin{eqnarray}
\nonumber
\Omega &\equiv& -k_{B} T \ln \left\{ {\rm Tr} \left[ \exp(-\beta \mathcal{H}^{HF})\right] \right\} \\
\label{eq.free_ene}
&=& - k_{B} T \sum_{{\bm k},\tau=\pm} \ln \left( 1 + \exp ( - \beta \lambda_{{\bm k}\tau} ) \right) \\
\nonumber &+& \sum_{\bm k} \left( \mathcal{E}_{{\bm k},\downarrow} - U_{{\bm Q}_{SC}}^{eff} | \Delta_{0} |^{2} \right),
\end{eqnarray}
where $\beta = 1 / k_{B} T$ and $T$ is absolute temperature. The electron concentration $n$ in the system is determined from the condition:
\begin{eqnarray}\label{eq.conc}
n & \equiv & \langle n \rangle \equiv - \frac{1}{N} \frac{d\Omega}{d\mu} \\
\nonumber &=& 1 + \frac{1}{N_{x} N_{y}} \sum_{\bm k} \frac{ \eta_{\bm k} - \mu }{\vartheta_{\bm k}} \left[ f ( \lambda_{{\bm k},+} ) - f ( \lambda_{{\bm k},-} ) \right ]
\end{eqnarray}
where $f ( \omega ) = 1 / ( 1 + \exp ( \beta \omega ) )$ is the Fermi-Dirac distribution.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=1.3]{fig2}
\end{center}
\caption{
Ground state $J/t$ vs. $U/t$ phase diagram for (a) $\mu=0.0t$, (b) $\mu=\pm 0.25t$, (c) $\mu=\pm 0.5t$ ($k_{B} T = 10^{-5} t$).
Denotations as in Fig.~\ref{fig:fig1}.
For $U\rightarrow-\infty$ the discontinuous transition between two SS phases occurs at $J=0$ (independently of $\mu$).
}
\label{fig.dfuj_lim}
\end{figure}
\begin{figure*
\includegraphics[scale=1]{fig3}
\caption{Ground state ($ k_{B} T = 10^{-5} t$) phase diagrams $J/t$ vs. $\mu/t$ (first and second column) and $J/t$ vs. $|n-1|$ (third and fourth column) for fixed values of $U/t$ (fixed in all panels in the row), $U/t=-2.0,-2.5,-5.0$, respectively.
The numbers: $1$, $2$, and $3$ (in circles) denote the regions of occurrence of particular phases: $\eta$-wave SS, $s$-wave SS, and NO.
The transition between regions $1$ and $2$ is discontinuous (for fixed $\mu$, only first and second column), whereas the transitions between regions $1$ and $3$ as well as between $2$ and $3$ are continuous.
Between regions $1$ and $2$ there are the regions of the PS state occurrence (for fixed $n$, only third and fourth column). They are not denoted, because these regions are very narrow (narrower than thickness of the curves in the figure)
The colour intensity indicates a value of the amplitude $\Delta_0$ of SOP (first and third column), the electron concentration $|1-n|$ (second column) and the chemical potential $|\mu|/t$ (fourth column).
}
\label{fig:fig2}
\end{figure*}
\subsection{The state with phase separation}\label{sec:ps}
In this subsection we would like to introduce the concept of phase separation and introduce the basics of its theory \change{(also cf. e.g. Refs.~[\onlinecite{arrigoni.strinati.91,bak.04,fernandes.schmalian.10,kapcia.robaszkiewicz.12,kapcia.robaszkiewicz.13}]).}
Phase separation (PS) is a state in which two domains with different electron concentration: $n_+$ and $n_-$ exist in the system
(coexistence of two homogeneous phases). The free energies of the PS states are calculated from the expression:
\begin{eqnarray}
\label{row:freeenergyPS}
f_{PS}(n_{+},n_{-}) = m f_{+}(n_{+}) + (1-m) f_{-}(n_{-}),
\end{eqnarray}
where $f_{\pm}(n_{\pm})$ are values of a free energy of two separating phases at $n_{\pm}$ corresponding to the
lowest homogeneous solution for a~given phase ($f=\Omega/N+\mu n$, calculated using~(\ref{eq.free_ene}) and~(\ref{eq.conc})),
$m$ is a fraction of the system with electron concentration $n_{+}$, $1-m$ is a~fraction with electron concentration $n_-$ ($n_{+}>n_{-}$) and
\begin{eqnarray}
\label{eq.PSn}
mn_+ +(1-m)n_-=n,
\end{eqnarray}
where $n$ is fixed.
The minimization of~(\ref{row:freeenergyPS}) with respect to $n_+$ and $n_-$ ($n$ fixed) yields the equality between the chemical potentials in both domains:
\begin{eqnarray}
\label{row:PS1}
\mu_+(n_+)=\mu_-(n_-)
\end{eqnarray}
(chemical equilibrium)
and the following equation (so-called Maxwell's construction):
\begin{eqnarray}
\label{row:PS2}
\mu_+(n_+)=\frac{f_{+}(n_{+})-f_{-}(n_{-})}{n_{+}-n_{-}},
\end{eqnarray}
which is equivalent with equality of grand potentials per site in domains: $\omega_+(\mu_+)=\omega_-(\mu_-)$. It implies that the transitions with a~discontinuous change of $n$ from $n_-$ to $n_+$ in the system considered for fixed $\mu$ can lead to occurrence of the regions of phase separation in the concentration range $n_-<n<n_+$ on the diagrams obtained as a function of $n$. In these regions the homogeneous phases can be metastable as well as unstable, depending on the $n$-dependence of $\mu$. In the PS states the chemical potential $\mu=\mu_+(n_+)=\mu_-(n_-)$ is independent of the electron concentration, i.e. $\partial \mu/\partial n=0$.
\change{%
On the other hand, there is another more intuitive approach. In such an approach the grand canonical potential $\omega = f -\mu n$ is used and chemical potential is independent variable instead of $n$ for free energy $f$, as it was in previous case. In both separating phases chemical potential has the same value. As a consequence we obtain the following simplified procedure:
at first step we solve the equations for homogeneous phases and next we determined the transition point between both phases (for the condition: $\omega_+(\mu)=\omega_-(\mu)$). Usually, the electron concentrations in these both phases are different ($n_+>n_-$). In such a case, for electron densities between $n_-<n<n_+$ the PS separation state can occur, in which considered phases coexist.}
\change{The PS instability is specific to the short-range nature of the interactions in the model. In the presence of (unscreened)
long-range Coulomb interactions, only a frustrated PS can occur (mesoscale, nanoscale) with the formation of various possible textures and the large-scale (macroscopic) PS can be prevented \cite{coleman.yukalova.95,yukalov.yukalova.04,yukalov.yukalova.14}.}
\section{Numerical results}\label{sec:results}
All calculations have been performed on graphic processor units using NVIDIA CUDA parallel computing technology, in momentum space on a square lattice grid $N_x \times N_y = 1000 \times 1000$, using the algorithm described in Ref.~[\onlinecite{januszewski.ptok.14}].
All phase transition boundaries, necessary to construct the complete phase diagram for fixed $\mu$, have been obtained numerically by comparing the grand potential~(\ref{eq.free_ene}) for the solutions found. The transition boundaries for fixed $n$ have been determined by comparing the free energies $f=\Omega/N+\mu n$ for homogeneous phase (it is calculated by using~(\ref{eq.free_ene}), the concentration $n$ is determined by~(\ref{eq.conc})) and phase separated states (determined by~(\ref{row:freeenergyPS})). It has been also checked that these results are consistent with the boundaries obtained from the results for fixed $\mu$ (discussed above) by determining the values of electron concentration (equation~(\ref{eq.conc})) on the both sides of transition boundaries derived at fixed $\mu$, which is thermodynamically conjugate to $n$.
\subsection{The ground state ($T=0$)}\label{sec:groundstate}
In this section we discuss the phase diagrams for model~(\ref{eq.ham}) at the ground state ($T=0$).
For $T=0$ on the phase diagrams as a function of the chemical potential $\mu$ the following three homogeneous phases occurs: $s$-wave SS, $\eta$-wave SS and normal (non-ordered, NO) phase.
The diagrams (Figs.~\ref{fig:fig1}--\ref{fig:fig2}, as a function of $\mu$) are nonsymmetric with respect to $J=0$ and consists of three regions in which phases mentioned above occur.
The $\eta$-wave SS phase can occur only for $J>0$, whereas $s$-wave SS phase can be stable for $J<0$ as well as for $J>0$ (in restricted ranges). The transition between both SS phases is discontinuous with a discontinuous change of global SOP defined as $\Delta_{\bm Q} = \tfrac{1}{N} \sum_i \Delta_i \exp(i {\bm Q} {\bm R}_i) $, where ${\bm Q} = (0,0)$ for the $s$-wave SS phase and ${\bm Q} = (\pi,\pi)$ for the $\eta$-wave SS phase ($\Delta_{(0,0)} = \Delta_0$ or $\Delta_{(\pi,\pi)} = \Delta_0$ in the $s$- or $\eta$-wave SS phase, respectively). The transitions between the SS phases ($s$-wave or $\eta$-wave, $\Delta_0 \neq 0$) and the NO phase ($\Delta_0 = 0$) are continuous ones.
The diagram for the half-filling ($n=1$, $\mu=0$) is shown in Fig.~\ref{fig:fig1} (also cf. Ref.~[\onlinecite{ptok.kapcia.15}]).
In the range presented in Fig.~\ref{fig:fig1} the boundary between two SS phases is decreasing function of $U/t$.
Notice that a necessary condition for the SS phases occurrence is $U^{eff}_{i} \leq0$ (or $U^{eff}_{\bm k} \leq 0$), thus the regions of the SS phases must be restricted at least by lines $U \pm 4 J = 0$, which are also the boundaries of the phases occurrence determined by minimization of $\Omega$ for $\mu=0$. However, for a general case of $\mu\neq0$ the SS ($s$- or $\eta$-wave) phases are stable only if $|U^{eff}_i|$ is higher than some critical value and the boundary of stability of the particular phases determined by minimization of $\Omega$ are moved towards lower values of $U/t$ (cf. Fig.~\ref{fig.dfuj_lim}).
In the limit $U/t \rightarrow - \infty$ the discontinuous boundary between both SS phases is located at $J/t = 0$ (independently of $\mu$). In such a limit there is a full symmetry between $s$-wave SS and $\eta$-wave SS phases~\cite{kapcia.robaszkiewicz.12,kapcia.14a,kapcia.robaszkiewicz.13,kapcia.14b,kapcia.15a,kapcia.15b}.
Notice also that in this limit model~(\ref{eq.ham}) is equivalent with the hard-core boson model on the lattice~\cite{micnas.rannniger.90,kapcia.robaszkiewicz.12}.
In Fig.~\ref{fig:fig2} we presents $J/t$ vs. $|\mu|/t$ diagrams (first and second column) as well as $J/t$ vs. $|1-n|$ diagrams (third and fourth column) for the fixed values of $U<0$. The transition between both SS phases for fixed $\mu$ is associated with a discontinuous change of electron concentration $n$ (it is visible especially in Fig.~\ref{fig:muvsn}).
Thus on the phase diagrams as a function of $n$, between the regions of the homogeneous $s$-wave SS and $\eta$-wave SS phases, there are regions of the PS state occurrence, where both SS phases coexist. In Fig.~\ref{fig:fig2} they are not denoted, because these regions are very narrow (narrower than thickness of the curves in the figure). In general, the regions of the SS phases occurrence extend with increasing on-site attractive interaction $|U|$, whereas the regions of the NO phases are reduced by increasing $|U|$.
In Fig.~\ref{fig:muvsn} we present the electron concentration $n$ as a function of the chemical potential $\mu$ for fixed model parameters. It is clearly visible that for $U/t=-5$ there is a discontinuity of $n$ (from $n_-$ to $n_+$) at the transition between two SS phases. The discontinuous transitions between the $\eta$-wave SS and $s$-wave SS phases are indicated by arrows. For $U/t=-2.0,-2.5$ the discontinuity is much smaller, but it still occurs.
\begin{figure
\includegraphics[scale=1]{fig4}
\caption{The $\mu/t$-dependence of electron concentration $|n-1|$ for $U/t=-0.5$ and various values of $J/t=0.4,0.5,0.6,0.7$ (as labelled) at the ground state ($k_BT=10^{-5}t$). In the inset the $\mu/t$-dependence of electron concentration $|n-1|$ for $J/t=0.4$ and $U/t=-2.0,-2.5$ (as labelled). The discontinuous transitions between two SS phases are indicated by arrows.}
\label{fig:muvsn}
\end{figure}
\begin{figure*
\includegraphics[scale=1]{fig5}
\caption{(a) $k_BT/t$ vs. $\mu/t$ and (b), (c) $k_BT/t$ vs. $|n-1|$ phase diagrams for $U/t = -5.0$ and $J/t =0.4$. The colour intensity indicates a value of the electron concentration $|1-n|$ (a), the amplitude $\Delta_0$ of SOP (b), and the chemical potential $\mu/t$ (c).
The numbers: $1$, $2$, $3$, and $4$ (in circles) denote the regions of occurrence of particular phases and states: $\eta$-wave SS, $s$-wave SS, NO, and PS respectively.
The transition between regions $1$ and $2$ is discontinuous (only in panel (a)), whereas the transitions between regions $1$ and $3$ as well as between $2$ and $3$ are continuous.
}
\label{fig:fig5}
\end{figure*}
\subsection{Finite temperatures ($T>0$)}\label{sec:finitetemp}
In this section we discuss the evolution of the phase diagram of the model considered with increasing temperature $T$ and chemical potential $\mu$ (or electron concentration $n$).
As an example, the finite temperature phase diagrams for $U/t=-5.0$ and $J/t=0.4$ are shown in Fig.~\ref{fig:fig5} as a function of $\mu$ and $n$.
On the phase diagrams three homogeneous phases (s-wave SS, $\eta$-wave SS and NO) occur.
The transition between the SS phases and the NO phase are continuous one (second order) and they are decreasing functions of $|\mu|/t$ and $|1-n|$.
The highest transition temperature is for the half-filling ($n=1$, $\mu=0$, the transition from $\eta$-wave SS phase into the NO phase).
The transition from the $s$-wave SS phase to the $\eta$-wave SS phase with increasing temperature is discontinuous (first order) for fixed $\mu$ (Fig.~\ref{fig:fig5}(a)) and its temperature increases with $|\mu|$.
All transition lines (two of second order and one of first order) merge in the bicritical point.
On the phase diagram for fixed $n$ (Figs.~\ref{fig:fig5}(b),(c)), the two SS phases can coexist in the state with (macroscopic) phase separation.
The temperatures of the transitions between the PS state and the homogeneous SS phases increases with $|1-n|$.
In particular, at the ground state ($T=0$) the electron concentrations in the domains are: $n_{-} = 0.2788$ (the $\eta$-wave SS domain) and $n_+ = 0.3125$ (the $s$-wave SS domain).
\begin{figure
\includegraphics[scale=1.2]{fig6}
\caption{The $k_BT/t$ vs. $J/t$ phase diagrams at half-filling ($n=0$, $\mu=0$) for (a) $U/t = -2.0$, (b) $U/t = -3.0$, and (c) $U/t = -5.0$.
Denotations as in Fig.~\ref{fig:fig1}.}
\label{fig:fig6}
\end{figure}
Notice that in the PS state values of $\Delta_0$ (or $\Delta_{\bm Q}$) are undetermined (there are two different order parameters at every domain) and the chemical potential is constant (not dependent on $n$). The different order parameters and electron concentrations in both domains are not dependent on the concentration $n$ of the electrons in the whole system. The general discussion of the PS states properties can be found in e.g. Ref.~[\onlinecite{bak.04,kapcia.robaszkiewicz.12}].
In addition, the phase diagrams for fixed $U/t$ and half-filling ($n=1$, $\mu=0$) are presented in Fig.~\ref{fig:fig6}.
The structure of the diagrams is not dependent on a value of attractive $U$ interaction.
Temperatures of the continuous transition between the $s$-wave SS phase and the NO phase and the discontinuous transition between both SS phase decrease with increasing $J/t$, whereas boundary between the $\eta$-wave SS phase and the NO phase (continuous transition) increase with increasing $J/t$ (for fixed $U/t$).
The regions of the SS phases occurrence extends with increasing $|U|/t$ and $|J|/t$.
These two interactions induce superconductivity in the system and also stabilize both SS phases.
Let us stress that it is possible (for fixed $\mu$ as well for fixed $n$) to change the type of superconductivity occurring in the system with increasing temperature (the homogeneous $\eta$-wave SS phase can exist in higher temperatures that the homogeneous $s$-wave SS phase, but not contrariwise).
\section{Conclusion and supplementary discussion}\label{sec:summary}
In this paper we studied the superconducting states of the PKH model focusing on the states with phase separation states between two different superconducting phases. We derived phase diagrams of the model and found that two superconducting phases with different symmetry of order parameter can coexist in a state with phase separation. Moreover, the results predict the change of a symmetry of superconducting order (from $\eta$-wave to $s$-wave) with increasing temperature (for fixed $\mu$ as well as fixed $n$).
Notice that one of the results of this paper that the temperature can change the symmetry of superconductivity pairing is consistent with other works~\cite{aperis.kotetes.11,thomale.platt.11a,thomale.platt.11b,maiti.korshunov.11,fernandes.millis.13,livanas.aperis.15}. One of the real materials, which exhibit a pronounced fragility of the gap symmetry, are the iron-based SCs (for review see e.g. Ref.~[\onlinecite{johnston.10,stewart.11,livanas.aperis.15}]).
In fact, recent works have demonstrated the possibility of gap symmetry transitions~\cite{aperis.kotetes.11,thomale.platt.11a,thomale.platt.11b,maiti.korshunov.11,fernandes.millis.13,livanas.aperis.15}, independently of the pairing mechanism. The small-$\vec{q}$ electron-phonon interaction~\cite{aperis.kotetes.11} and the spin-fluctuations scenario~\cite{thomale.platt.11a,thomale.platt.11b,maiti.korshunov.11,fernandes.millis.13} are both compatible, but gap symmetry transitions constitute a characteristic feature of the first mechanism, which leads to a loss of rigidity of the gap function in momentum space (momentum decoupling)~\cite{oppeneer.varelogiannis.03}.
One should also notice that it was also derived that increasing magnetic field can change symmetry of SOP from $s$-wave (or $d$-wave) into $\eta$-wave~\cite{ptok.maska.09}.
Let us stress that it has been reported that superconductivity can coexist with charge-ordered and magnetically ordered phases in states with phase separations~\cite{kapcia.12,kapcia.13,kapcia.15b,kapcia.15a,kapcia.robaszkiewicz.12}.
In present work we do not consider the charge and magnetic orderings. In the PKH model, they can both occur for $U > 0$ (charge ordering can be also present for $U < 0$)~\cite{micnas.rannniger.90,robaszkiewicz.bulka.99,japardze.muller.97,japaridze.kampf.01,japaridze.sarkar.02,japaridze.sarkar.02}.
Although the existence of $\eta$-wave SS has not been confirmed experimentally, the recent theoretical results (e.g. Ref.~[\onlinecite{mierzejewski.maska.04,ptok.kapcia.15,czart.robaszkiewicz.15a,czart.robaszkiewicz.15b,czart.robaszkiewicz.15b}]) give the explicit suggestions how it can be distinguished from the $s$-wave SS. The main issue in the differentiate these two phases explicitly is that only absolute value of the order parameter (energy gap) can be measured experimentally (e.g. by STM spectroscopy) and only the behaviour of thermodynamical properties~\cite{mierzejewski.maska.04,czart.robaszkiewicz.15a,czart.robaszkiewicz.15b,czart.robaszkiewicz.15b} or the influence of impurities on superconducting properties~\cite{ptok.kapcia.15} can give information about the SS symmetry.
\begin{acknowledgments}
The authors thank Stanis\l{}aw Robaszkiewicz for very fruitful discussions and comments.
K.J.K. is supported by National Science Centre (NCN, Poland) -- the grant No. DEC-2013/08/T/ST3/00012 in years 2013--2015.
\end{acknowledgments}
|
1,116,691,499,361 | arxiv | \section*{Proof}
Define $f\colon \mathbb{R}^2 \mapsto \mathbb{R}^k$ and $K: \mathbb{R}^2 \mapsto \mathbb{R}^{k\times l}$, where $k$ is the number of the input channels and $l$ is the number of the output channel. The convolution can be expressed by
$(K\ast f)(\vec{v})= \sum_{\vec{w}}f(\vec{v}+\vec{w})K(\vec{w})$, where $\vec{v}=(i,j)\in\mathbb{R}^2$.
\begin{Claim}
$(T_g^0(K\ast f))(\vec{v})= (T_g^0 K) \ast (T_g^0 f)(\vec{v})$
\end{Claim}
\begin{proof}\\
$ RHS&=\sum_{\vec{w}} f(g^{-1}(\vec{v}+\vec{w}))K(g^{-1}\vec{w})$; \\ \\
Let $\vec{y}=g^{-1}\vec{w}$, $LHS&= (K \ast f)(g^{-1}\vec{v})= \sum_{\vec{y}} f(g^{-1}\vec{v} +\vec{y})K(\vec{y}) = \sum_{\vec{w}}f((g^{-1}\vec{v} +g^{-1}\vec{w})K(g^{-1}\vec{w})$\\ \\
$RHS \Leftrightarrow LHS$
\end{proof}
\begin{Claim}
$f\colon \mathbb{R}^2 \mapsto \mathbb{R}^k$ and $K_{i=1\colon n} \colon \mathbb{R}^2 \mapsto \mathbb{R}^{1\times1}$.
$\tilde{K}=\mathrm{Diag}(K_1,\ldots,K_n) \colon \mathbb{R}^2 \mapsto \mathbb{R}^{n\times n}$. $\tilde{f} &= Dup(f) \colon \mathbb{R}^2 \mapsto \mathbb{R}^n$, where $\tilde{f}(\vec{x})= (f(\vec{x}),\ldots,f(\vec{x}))$. So $(\tilde{K}\ast \tilde{f})_i &= K_i \ast f$. For $g\in C_n$, $(\rho_{\mathrm{reg}}(g)\tilde{K}) \ast \tilde{f} &= \rho_{\mathrm{reg}}(g)(\tilde{K}\ast \tilde{f})$.
\end{Claim}
\begin{proof}
Assume $n$ kernels $K_{i&=1\colon n}$, then we convolve each kernel with the input image $h_i=(K_i\ast f)$. Clearly permuting the kernels $K_i$ also permutes $h_i$, so $\rho_{\mathrm{reg}}(g)h = (\rho_{\mathrm{reg}}(g)K)\ast f$
\end{proof}
\begin{Claim}
$\mathcal{R}_n(T_g^0 f) = \rho_{\mathrm{reg}}(-g)\mathcal{R}_n(f) $
\end{Claim}
\begin{proof}
$\mathcal{R}_n(f)(\vec{x})=( f(\vec{x}), f(g^{-1}\vec{x}), f(g^{-2}\vec{x}),\ldots ,f(g^{-(n-1)}\vec{x}))$;\\ \\
$RHS &= LHS = (f(g^{-1}\vec{x}), f(g^{-2}\vec{x}),\ldots,f(g^{-(n-1)}\vec{x}), f(\vec{x}))$
\end{proof}
\subsection{Proof of proposition 1}
The equivariance of Transporter Net under rotations of the picked object:
\begin{equation}
\psi(\mathcal{R}_n(T^0_{g}c)) \star \phi(o_t) = \rho_{\mathrm{reg}}(-g)( \psi(\mathcal{R}_n(c)) \star) \phi(o_t)
\end{equation}
\begin{proof}
Since $\psi$ is applied independently to each of the rotated channels in $\mathcal{R}_n(c)$, we can define $\psi_n((f_1,\ldots,f_n))=(\psi(f_1),\ldots,(\psi(f_n))$.
\begin{align*}
LHS &= \psi_n(\rho_{\mathrm{reg}}(g)\mathcal{R}_n(c)) \star \phi(o_t)\\
&= (\rho_{\mathrm{reg}}(-g) \psi_n(\mathcal{R}_n(c)) \star \phi(o_t) \:\: {\text{claim 0.3}}\\
&= \rho_{\mathrm{reg}}(-g) (\psi_n(\mathcal{R}_n(c) \star \phi(o_t)) \:\: {\text{claim 0.2}}\\
& = RHS
\end{align*}
\end{proof}
\subsection{Proof of proposition 2}
To prove the equivariance of Equivariant Transporter Net under rotations of the picked object and the placement:
\begin{equation}
\Psi(T^0_{g_1}(c)) \star \phi (T^0_{g_2} (o_t\setminus c)) =\\ \rho_{reg}(g_2-g_1)(T^0_{g_2}[\Psi(c) \star \phi ( o_t\setminus c)])
\label{equ:proof1}
\end{equation}
We need first prove the equivariance under rotations of the placement:
\begin{equation}
\Psi(c) \star \phi (T^0_g(o_t\setminus c)) = T_g^{\mathrm{reg}}( \Psi(c) \star \phi (o_t\setminus c)).\\
\end{equation}
\begin{proof}
\begin{align*}
LHS &= \Psi(c) \star T^0_g \phi (o_t) \:\:\text{the equivariance of $\phi$}\\
&= T^0_g T^0_{g^-1}\Psi(c) \star T^0_g \phi (o_t)\\
&= T^0_g (T^0_{g^-1}\Psi(c) \star \phi (o_t))\:\: \text{claim 0.1}\\
&= T^0_g (T^0_{g^-1}\mathcal{R}_n(\psi (c)) \star \phi (o_t))\\
&= T^0_g ((\rho_{\mathrm{reg}}(g)\Psi(c) \star \phi (o_t)) \:\:\text{claim 0.3}\\
&= T^0_g \rho_{\mathrm{reg}}(g)(\Psi(c) \star \phi (o_t)) \:\:\text{commute since one acts on channel space and one acts on base space}\\
&= T_g^{\mathrm{reg}}( \Psi(c) \star \phi (o_t))\\
\end{align*}
With the equivariance of $\psi$, proposition 1 could be reformulated as
\begin{equation}
\Psi(T^0_{g}c) \star \phi(o_t\setminus c) = \rho_{\mathrm{reg}}(-g)( \Psi(c) \star) \phi(o_t \setminus c)
\label{equ:proof2}
\end{equation}
Note we use $o_t\setminus c$ to emphasize the target placement of $o_t$ since the object and the placement are nonoverlapping. Combining euq1 and equ2 realizes the proposition 2.
\end{proof}
\section*{Special Equivariance of Proposition 2:}
In this subsection, we make a summary of some important properties inside our placing networks and also provide a intuitive explanation for each one. \\\\
\textbf{Equivariance property:}
When $g_1=0$ or $g_2=0$, we can get
\begin{equation*}
\Psi(T^0_g(c)) \star \phi (o_t \backslash c) = \rho_{reg}(-g) (\Psi(c) \star \phi (o_t\backslash c))
\label{equi_original_our}
\end{equation*}
\begin{equation*}
\Psi(c) \star \phi (T^0_g(o_t \backslash c)) = T^{reg}_g(\Psi(c) \star \phi (o_t\backslash c))
\label{equi_2_simple}
\end{equation*}
Both of and show the equivariance of our networks under the rotation $g\in C_n$ of either the object or the placement.
\\\\
\textbf{Invariance property:} when $g_1=g_2$,
\begin{equation}
\Psi(T^0_g(c)) \star \phi (T^0_g( o_t\backslash c)) = T^0_g (\Psi(c) \star \phi (o_t\backslash c))
\end{equation}
The equation above demonstrates that a rotation $g$ on the whole observation $o_t$ doesn't change the placing angle but rotates the placing location by $g$. Although data augmentation could help non-equivairant models learn this property, our networks obtain it by nature.
\\\\
\textbf{Relativity property:}\\
\begin{equation}
\begin{aligned}
\Psi(T^0_g(c)) \star \phi (o_t\backslash c) = \rho_{reg}(-g)(T^0_g[ (\Psi(c) \star \phi ((T^0_{-g}( o_t\backslash c))])
\end{aligned}
\label{relativity}
\end{equation}
Equation~\eqref{relativity} explores the relationship between a rotation on $c$ by $g$ and a inverse rotation $-g$ on $o_t\backslash c$. They are equivariant under some transformation. Intuitively, $c$ could be considered as the block and $o_t\backslash c$ can be regarded as the slot.
\newpage
\section*{Task descriptions of Ravens-10:}
Here we provide a short description of Ravens-10 Environment, we refer readers to~\cite{zeng2020transporter} for details.
The poses of objects and placements in each task are randomly sampled in the workspace without collision. Performance on each task is evaluated in one of two ways: 1) pose: translation and rotation error relative to target pose is less than a threshold $ \tau=1\mathrm{cm}$ and $\omega=\frac{\pi}{12}$ respectively. Tasks: block-insertion, towers-of-hanoi, place-red-in-green, align-box-corner, stack-block-pyramid, assembling-kits. Partial scores are assigned to multiple-action tasks. 2) Zone: Ravens-10 discretizes the 3D bounding box of each object into $2cm^3$ voxels. The Total reward is calculated by $\frac{\text{\# of voxels in target zone}}{\text{total \# of voxels}}$. Tasks: palletizing-boxes, packing-boxes, manipulating-cables, sweeping-piles.
Note that pushing objects could also be parameterized with $a_{\mathrm{pick}}$ and $a_\mathrm{{place}}$ that correspond to the starting pose and the ending pose of the end effector.
\begin{enumerate}
\item \textbf{block-insertion:} pick up an L-shape block and place it into an L-shaped fixture.
\item \textbf{place-red-in-green:} picking up red cubes and place them into green bowls. There could be multiple bowls and cubes with different colors.
\item \textbf{towers-of-hanoi:} sequentially picking up disks and placing them into pegs such that all 3 disks initialized on one peg are moved to another, and that only smaller disks can be on top of larger ones.
\item \textbf{align-box-corner:} picking up a randomly sized box and place it to align one of its
corners to a green L-shaped marker labeled on the tabletop.
\item \textbf{stack-block-pyramid:} sequentially picking up 6 blocks and stacking them into a pyramid of 3-2-1.
\item \textbf{palletizing-boxes:} picking up 18 boxes and stacking them on top of a pallet.
\item \textbf{assembling-kits:} picking 5 shaped objects (randomly sampled with replacement from a set of 20) and fitting them to corresponding silhouettes of the objects on a board.
\item \textbf{packing-boxes:} picking and placing randomly sized boxes tightly into a randomly sized container.
\item \textbf{manipulating-rope:} manipulating a deformable rope such that it connects the two endpoints of an incomplete 3-sided square (colored in green).
\item \textbf{sweeping-piles:} pushing piles of small objects (randomly initialized) into a desired target goal zone on the tabletop marked with green boundaries. The task is implemented with a pad-shaped end effector.
\end{enumerate}
\section*{Training settings:}
We train Equivariant Transporter Network with Adam~\cite{} optimizer with a fixed learning rate of $10^{-4}$. It takes about 0.8 seconds to complete one SGD~\cite{} step with a batch size of 1 on a NVIDIA Tesla V100 SXM2 GPU. We evaluate the performance every 10k steps on 100 unseen tests for each task and best performances on most tasks can be achieved before 10K steps.
\section*{Ablation study:}
\end{document}
\section{Introduction}
Many challenging robotic manipulation problems can be viewed through the lens of a single pick and place operation. This is the approach taken in the Transporter Network framework~\cite{zeng2020transporter} where the model first detects a task-appropriate pick position and then detects a task-appropriate place position and orientation. Since the choices of pick and place pose are conditioned on the current manipulation scene, this model can be used to express multi-step pick-place policies that solve complex tasks. An important part of the Transporter Net model is the cross convolutional layer that matches an image patch around the picked object with an appropriate place position. By performing the cross convolution between an encoding of the scene and an encoding of a stack of differently rotated image patches around the pick, this model detects the task-appropriate place pose.
\begin{figure}[b]
\centering
\includegraphics[clip,width=0.33\textwidth]{figs/cn.png}
\caption{If Transporter Network~\cite{zeng2020transporter} learns to pick and place an object when it is presented in one orientation, the model is immediately able to generalize to new object orientations. We view this as $C_n$-equivariace of the model.}
\label{fig:transporter_symmetry}
\end{figure}
As a result of this design, Transporter Net is equivariant with respect to pick orientation. That is, if the model can pick and place an object correctly when the object is presented in one orientation, it is automatically able to pick and place the same object when it is presented in a different orientation. This is illustrated in Figure~\ref{fig:transporter_symmetry}. The left side of Figure~\ref{fig:transporter_symmetry} shows a pick/place problem where the robot must pick the pink object and place it inside the green outline. Because the model is equivariant, the ability to solve the pick/place task on the left side of Figure~\ref{fig:transporter_symmetry} immediately implies an ability to solve the task on the right side of Figure~\ref{fig:transporter_symmetry} where the object to be picked has been rotated. This symmetry over object orientation enables Transporter Net to generalize well and it is fundamentally linked to the sample efficiency of the model. Assuming that pick orientation is discretized into $n$ possible gripper rotations, we will refer to this as a $C_n$ pick symmetry, where $C_n$ is the finite cyclic subgroup of $\mathrm{SO}(2)$ that denotes a set of $n$ rotations.
\begin{figure}[b]
\centering
\includegraphics[clip,width=0.33\textwidth]{figs/cn_by_cn.png}
\caption{Our proposed Equivariant Transporter Network is able to generalize over both pick and place orientation. We view this as $C_n \times C_n$-equivariace of the model.}
\label{fig:place_symmetry}
\end{figure}
Although Transporter Net is $C_n$-equivariant with regard to pick, the model does not have a similar equivariance with regard to place. That is, if the model learns how to place an object in one orientation, that knowledge does not generalize immediately to different place orientations. This paper seeks to add this type of equivariance to the Transporter Network model by incorporating $C_n$-equivarant convolutional layers into both the pick and place models. Our resulting model is equivariant both to changes in pick object orientation and changes in place orientation. This symmetry is illustrated in Figure~\ref{fig:place_symmetry} and can be viewed as a direct product of two cyclic groups, $C_n \times C_n$. Our specific contributions are as follows. 1) We propose a novel version of Transporter Net that is equivariant over $C_n \times C_n$ rather than just $C_n$ and evaluate it on the Raven-10 benchmark proposed in~\cite{zeng2020transporter}. 2) We augment our Equivariant Transporter Net model with the ability to grasp using a gripper rather than just a suction cup and demonstrate it on gripper-augmented versions of five of the Ravens-10 tasks. 3) We demonstrate the approach on real-robot versions of three of the gripper-augmented tasks. Our results indicate that our approach is more sample efficient than the baseline version of Transporter Net and therefore learns better policies from a small number of demonstrations.
\section{Related Work}
\textbf{Pick and Place.} Pick and place is an important topic in manipulation due to its value in industry. Traditional assembly methods in factories require customized workstations so that fixed pick and place actions can be manually predefined. Recently, considerable research has focused on vision-based manipulation. Some work~\cite{narayanan2016discriminatively,chen2019grip,gualtieri2021robotic} assumes that object mesh models are available in order to run ICP~\cite{besl1992method} and align the object model with segmented observations or completions~\cite{yuan2018pcn,huang2021gascn}. Other work learns a category-level pose estimator~\cite{yoon2003real,deng2020self} or key-point detector~\cite{nagabandi2020deep,liu2020keypose} from training on a large dataset. However, these methods often require expensive object-specific labels, making them difficult to use widely. Although end-to-end models~\cite{zakka2020form2fit,khansari2020action,devin2020self,berscheid2020self} that directly map input observations to actions can learn quickly and generalize well, these methods still need to be trained on large datasets. For example, \citet{khansari2020action} collects a dataset with 7.2 million samples. \citet{devin2020self} collects $40K$ grasps and places per task. \citet{zakka2020form2fit} collects 500 disassembly sequences for each kit.
\textbf{Equivariance Learning in Manipulation.}
Fully Convolutional Networks (FCN) are translationally equivariant, and have been shown to improve learning efficiency in many manipulation tasks~\cite{zeng2018robotic,morrison2018closing}. The idea of encoding SE(2) symmetries in the structure of neural networks is first introduced in G-Convolution~\cite{cohen2016group}. The extension work proposes an alternative architecture, Steerable CNN~\cite{cohen2016steerable}.~\citet{weiler2019general} proposes a general framework for implementing E(2)-Steerable CNNs. In the context of robotics learning, ~\citet{wang2022equivariant} uses SE(2) equivariance in Q learning to solve multi-step sequential manipulation tasks; \citet{anonymous2022mathrmsoequivariant} extends it to $\mathrm{SO}(2)$-equivairant reinforcement learning. Our work tackles manipulation rearrangement tasks by extracting inherent SE(2) equivariance through the imitation learning approach~\cite{hussein2017imitation,hester2018deep,vecerik2017leveraging}.
\section{Background on Symmetry Groups}
\label{sect:background}
\subsection{The Groups $\mathrm{SO}(2)$ and $C_n$}
We are primarily interested in rotations expressed by the group $\mathrm{SO}(2)$ and its cyclic subgroup $C_n \leq \mathrm{SO}(2)$. $\mathrm{SO}(2)$ contains the continuous planar rotations $\{ \mathrm{Rot}_{\theta}: 0\leq \theta < 2\pi\}$. The discrete subgroup $C_n = \{ \mathrm{Rot}_{\theta}: \theta \in \{\frac{2\pi i}{n}| 0\leq i < n\} \}$ contains only rotations by angles which are multiples of $2\pi/n$. The special Euclidean group $\mathrm{SE}(2) = \mathrm{SO}(2) \times \mathbb{R}^2$ describes all translations and rotations of $\mathbb{R}^2$.
\subsection{Representation of a Group}
A $d$-dimensional \emph{representation} $\rho \colon G \to \mathrm{GL}_d$ of a group $G$ assigns to each element $g \in G$ an invertible $d\!\times\! d$-matrix $\rho(g)$. Different representations of $\mathrm{SO}(2)$ or $C_n$ help to describe how different signals are transformed under rotations. For example, the trivial representation $\rho_0 \colon \mathrm{SO}(2) \to \mathrm{GL}_1$ assigns $\rho_0(g) = 1$ for all $g \in G$, i.e. no transformation under rotation. The standard representation
\[\rho_1(\mathrm{Rot}_\theta) = \begin{pmatrix}
\cos{\theta} & -\sin{\theta} \\
\sin{\theta} & \cos{\theta}
\end{pmatrix}
\]
represents each group element by its standard rotation matrix. Notice that $\rho_0$ and $\rho_1$ can be used to represent elements from either $\mathrm{SO}(2)$ or $C_n$. The regular representation $\rho_{\mathrm{reg}}$ of $C_n$ acts on a vector in $\mathbb{R}^{n}$ by cyclically permuting its coordinates $\rho_{\mathrm{reg}}(\mathrm{Rot}_{2\pi /n})(x_0,x_1,...,x_{n-2},x_{n-1})=(x_{n-1} ,x_0,x_1,...,x_{n-2})$. We can rotate by multiples of $2\pi/n$ by $\rho_{\mathrm{reg}}(\mathrm{Rot}_{2\pi i /n}) = \rho_{\mathrm{reg}}(\mathrm{Rot}_{2\pi /n})^i$. The regular representation for elements of the quotient group is denoted $\rho_{\mathrm{quot}}^{C_n/C_k}$ and acts on $\mathbb{R}^{n/k}$ by permuting $|C_n|/|C_k|$ channels. This gives a quotient representation of $C_n$ defined $\rho_{\mathrm{quot}}^{C_n/C_k}(\mathrm{Rot}_{2\pi i /n})(\mathbf{x})_j = (\mathbf{x})_{j + i\ \mathrm{mod} (n/k)}$, which implies features that are invariant under the action of $C_k$. For more details, we refer the reader to~\citet{serre1977linear,weiler2019general}.
\subsection{Feature Map Transformations}
\begin{wrapfigure}{r}{0.15\textwidth}
\begin{center}
\includegraphics[width=0.15\textwidth]{figs/rho_n.png}
\end{center}
\caption{Illustration of the action of $T_g^{\mathrm{reg}}$ on a $2 \times 2$ image.}
\label{fig:rotation_illustration}
\end{wrapfigure}
We formalize images and feature maps as feature vector fields, i.e. functions $f \colon \mathbb{R}^2 \rightarrow \mathbb{R}^c$, which assign a feature vector $f(\mathbf{x}) \in \mathbb{R}^c$ to each position $\mathbf{x} \in \mathbb{R}^2$. While in practice we discretize and truncate the domain of $f$ $\lbrace (i,j) : 1 \leq i \leq W, 1 \leq j \leq W \rbrace$, here we will consider it to be continuous for the purpose of analysis. The action of a rotation $g \in \mathrm{SO}(2)$ on $f$ is a combination of a rotation in the domain of $f$ via $\rho_1$ (this rotates the pixel positions) and a rotation in the channel space $\mathbb{R}^c$ by $\rho \in \{\rho_0, \rho_{\mathrm{reg}}\}$. If $\rho = \rho_{\mathrm{reg}}$, then the channels cyclically permute according to the rotation. If $\rho = \rho_0$, the channels do not change. We denote this action (the action of $g$ on $f$ via $\rho$) by $T^{\rho}_g(f)$:
\begin{equation}
\label{eqn:rotation_illustration}
[T^{\rho}_g(f)](\mathbf{x}) = \rho(g) \cdot f( \rho_1(g)^{-1} \mathbf{x}).
\end{equation}
For example, the action of $T^{\rho_{\mathrm{reg}}}_g(f)$ is illustrated in Figure~\ref{fig:rotation_illustration} for a rotation of $g = \pi/2$ on a $2 \times 2$ image $f$ that uses $\rho_{\mathrm{reg}}$. The expression $\rho_1(g)^{-1} \mathbf{x}$ rotates the pixels via the standard representation. Multiplication by $\rho(g) = \rho_{\mathrm{reg}}(g)$ permutes the channels. For brevity, we will denote $T_g^{\mathrm{reg}} = T_g^{\rho_{\mathrm{reg}}}$ and $T_g^{0} = T_g^{\rho_{0}}$.
\subsection{Equivariant Mappings}
A function $F$ is equivariant if it commutes with the action of the group,
\begin{equation}
\label{eqn:equivariance}
T_g^{\mathrm{out}}[F(f)] = F(T_g^{\mathrm{in}}[f])
\end{equation}
where $T_g^{\mathrm{in}}$ transforms the input to $F$ by the group element $g$ while $T_g^{\mathrm{out}}$ transforms the output of $F$ by $g$. For example, if $f$ is an image, then $\mathrm{SO}(2)$-equivariance of $F$ implies that it acts on $f$ in the same way regardless of the orientation in which $f$ is presented. That is, if $F$ takes an image $f$ rotated by $g$ (RHS of Equation~\ref{eqn:equivariance}), then it is possible to recover the same output by evaluating $F$ for the un-rotated image $f$ and rotating its output (LHS of Equation~\ref{eqn:equivariance}).
\section{Transporter Network}
Before describing our variation on Transporter Net, we summarize the the pick and place problem and the original Transporter Net architecture described in~\cite{zeng2020transporter}.
\subsection{Problem Statement}
We define the \emph{Planar Pick and Place} problem as follows. Given a visual observation $o_t$, the problem is to learn a probability distribution $p(a_{\mathrm{pick}} | o_t)$ over picking actions $a_{\mathrm{pick}} \in \mathrm{SE}(2)$ and a distribution $p(a_{\mathrm{place}}|o_t,a_{\mathrm{pick}})$ over placing actions $a_{\mathrm{place}} \in \mathrm{SE}(2)$ conditioned on $a_{pick}$ that accomplishes some task of interest.
The visual observation $o_t$ is typically a projection of the scene (e.g., top-down RGB-D images) and the pose of the end effector is expressed as $(u,v,\theta)$ where $u,v$ denote the pixel coordinates of the gripper position and $\theta$ denotes gripper orientation. (Since~\cite{zeng2020transporter} uses suction cups to pick, that work ignores pick orientation.)
\begin{figure}[htp]
\centering
\includegraphics[width=0.38\textwidth]{figs/transporter_architecture.png}
\caption[Transporter Network]{The Architecture of Transporter Net.}
\label{fig:transporter_architecture}
\end{figure}
\subsection{Description of Transporter Net}
\label{sect:transporter_desc}
Transporter Network~\cite{zeng2020transporter} solves the planar pick and place problem
using the architecture shown in Figure~\ref{fig:transporter_architecture}. The pick network $f_{\mathrm{pick}} \colon o_t \mapsto p(u,v)$ maps $o_t$ onto a probability distribution $p(u,v)$ over pick position $(u,v) \in \mathbb{R}^2$. The output pick position $a_{\mathrm{pick}}^*$ is calculated by maximizing $f_{\mathrm{pick}}(o_t)$ over $(u,v)$. The place position and orientation is calculated as follows. First, an image patch $c$ centered on $a_{\mathrm{pick}}^*$ is cropped from $o_t$ to represent the pick action as well as the object. Then, the crop $c$ is rotated $n$ times to produce a stack of $n$ rotated crops. Using the notation of Section~\ref{sect:background}, we will denote this stack of crops as
\begin{equation}
\mathcal{R}_n(c) = (T^0_{2\pi i/n}(c))_{i=0}^{n-1},
\end{equation}
where we refer to $\mathcal{R}_n$ as the ``lifting'' operator. Then, $\mathcal{R}_n(c)$ is encoded using a neural network $\psi$.
The original image, $o_t$, is encoded by a separate neural network $\phi$. The distribution over place location is evaluated by taking the cross correlation between $\psi$ and $\phi$,
\begin{equation}
\label{eqn:transporter1}
f_{\mathrm{place}}(o_t,c) = \psi(\mathcal{R}_n(c)) \star \phi(o_t),
\end{equation}
where $\psi$ is applied independently to each of the rotated channels in $\mathcal{R}_n(c)$. Place position and orientation is calculated by maximizing $f_{\mathrm{place}}$ over the pixel position (for position) and the orientation channel (for orientation).
\begin{figure}[tp]
\centering
\includegraphics[width=0.35\textwidth]{figs/transporter_1.png}
\caption{Illustration of the main part of the proof of Proposition~\ref{prop:equivtransporter}. Rotating the crop $c$ induces a cyclic shift in the channels of the output $\psi(\mathcal{R}_n(T_g^0 c)) = \rho_\mathrm{reg}(-g)\psi(\mathcal{R}_n(c)).$
}
\label{fig:euqi_transporter}
\end{figure}
\subsection{Equivariance of Transporter Net}
The model architecture described above gives Transporter Network the following equivariance property.
\begin{proposition}
\label{prop:equivtransporter}
The Transporter Net place network $f_{\mathrm{place}}$ is $C_n$-equivariant. That is, given $g \in C_n$, object image crop $c$ and scene image $o_t$,
\begin{equation}
\label{eqn:transporter3}
f_{\mathrm{place}}(o_t,T^0_g(c)) = \rho_{\mathrm{reg}}(-g) f_{\mathrm{place}}(o_t,c).
\end{equation}
\end{proposition}
Proposition~\ref{prop:equivtransporter} expresses the following intuition. A rotation of $g$ applied to the orientation of the object to be picked results in a $-g$ change in the placing angle, which is represented by a permutation along the channel axis of the placing feature maps. This is a symmetry over the cyclic group $C_n \leq \mathrm{SO}(2)$ which is encoded directly into the model. It enables it to immediately generalize over different orientations of the object to be picked and thereby improves sample efficiency.
The main idea of the proof is shown in Figure \ref{fig:euqi_transporter}. Namely $\psi(\mathcal{R}_n(\cdot))$ is equivariant in the sense that rotating the crop $c$ induces a cyclic shift in the channels of the output. Formally, $\psi(\mathcal{R}_n(T_g^0 c)) = \rho_\mathrm{reg}(-g)\psi(\mathcal{R}_n(c)).$ Noting that a permutation of the filters $K$ in the convolution $K \star \phi(o_t)$ induces the same permutation in the output feature maps completes the proof. The full proof is given in Appendix~\ref{proof_proposition}. Note that here $\psi$ is a simple CNN with no rotational equivariance. The equivariance results from the lifting $\mathcal{R}_n$.
\section{Equivariant Transporter}
\subsection{Equivariant Pick}
Our approach to the pick network is similar to that in Transporter Net~\cite{zeng2020transporter} except that: 1) we explicitly encode equivariance constraints into the pick networks, thereby making pick learning more sample efficient; 2) we infer pick orientation so that we can use parallel jaw grippers rather than just suction grippers.
\begin{wrapfigure}[17]{r}{0.25\textwidth}
\vspace{-0.5cm}
\begin{center}
\includegraphics[width=0.25\textwidth]{figs/attention1.png}
\end{center}
\caption{Equivariant Transporter Pick model. First, we find the pick position $a^*_{pick}$ by evaluating the argmax over $f_p(o_t)$. Then, we evaluate $f_\theta$ for the image patch centered on $a^*_{pick}$.}
\label{fig:pick_archi}
\end{wrapfigure}
\subsubsection{Model}
We propose a $C_n$-equivariant model for detecting the planar pose for the pick operation. First, we decompose the learning process of $a_{\mathrm{pick}} \in \mathrm{SE}(2)$ into two parts,
\begin{equation}
p(a_{\mathrm{pick}}) = p(u,v) p(\theta|(u,v)),
\end{equation}
where $p(u,v)$ denotes the probability that a pick exists at pixel coordinates $u,v$ and $p(\theta|(u,v))$ is the probability that the pick at $u,v$ should be executed with a gripper orientation of $\theta$. The distributions $p(u,v)$ and $p(\theta|(u,v))$ are modeled as two neural networks:
\begin{align}
\label{eqn:pick1}
f_{p}(o_t) &\mapsto p(u,v), \\
\label{eqn:pick2}
f_{\theta}(o_t,(u,v)) &\mapsto p(\theta|(u,v)).
\end{align}
Given this factorization, we can query the maximum of $p(a_{\mathrm{pick}})$ by evaluating $(\hat{u},\hat{v}) = \argmax_{(u,v)}[p(u,v)]$ and then $\hat{\theta} = \argmax_{\theta}[p(\theta|\hat{u},\hat{v})]$.
This is illustrated in \figref{fig:pick_archi}. The left side of \figref{fig:pick_archi} shows the maximization of $f_p$ at $a^*_{pick}$. The right side shows evaluation of $f_\theta$ for the image patch centered at $a^*_{pick}$.
\subsubsection{Equivariance Relationships}
There are two equivariance relationships that we would expect to be satisfied for planar picking:
\begin{align}
\label{eqn:equi1}
f_p(T^0_g(o_t)) &= T^0_g (f_p(o_t))\\
\label{eqn:equi2}
f_\theta(T^0_g(o_t),T^0_g(u,v)) &= \rho_{reg}(g)(f_{\theta}(o_t,(u,v))).
\end{align}
Equation~\ref{eqn:equi1} states that the grasp points found in an image rotated by $g \in \mathrm{SO}(2)$, (LHS of Equation~\ref{eqn:equi1}), should correspond to the grasp points found in the original image subsequently rotated by $g$, (RHS of Equation~\ref{eqn:equi1}). Equation~\ref{eqn:equi2} says that the grasp orientation at the rotated grasp point $T^0_g(u,v)$ in the rotated image $T^0_g(o_t)$ (LHS of Equation~\ref{eqn:equi2}) should be shifted by $g = 2\pi i$ relative to the grasp orientation at the original grasp points in the original image (RHS of Equation~\ref{eqn:equi2}). We encode both $f_p$ and $f_\theta$ using equivariant convolutional layers~\cite{weiler2019general} which constrain the models to represent only those functions which satisfy Equations~\ref{eqn:equi1} and~\ref{eqn:equi2}.
\subsubsection{Gripper Orientation Using the Quotient Group}
A key observation in planar picking is that, for many robots, the gripper is bilaterally symmetric, i.e. grasp outcome is invariant when the gripper is rotated by $\pi$. We can encode this additional symmetry to reduce redundancy and save computational cost using the regular representation of the quotient group $C_n / C_2$ which identifies orientations that are $\pi$ apart. When using this quotient group for gripper orientation, $\rho_{\mathrm{reg}}$ in Equation~\ref{eqn:equi2} is replaced with $\rho_{\mathrm{reg}}^{C_n/C_2}$.
\subsection{Equivariant Place}
Given the picked object represented by the image patch c centered on $a_{\mathrm{pick}}$, the place network models the distribution of $a_{\mathrm{place}}=(u_{\mathrm{place}},v_{\mathrm{place}},\theta_{\mathrm{place}})$ by:
\begin{equation}
f_{\mathrm{place}}(o_t,c) \mapsto p(a_{\mathrm{place}}|o_t,a_{\mathrm{pick}}),
\end{equation}
where $p(a_{\mathrm{place}}|o_t,a_{\mathrm{pick}})$ denotes the probability that the object at $a_{\mathrm{pick}}$ in scene $o_t$ should be placed at $a_{\mathrm{place}}$.
Our place model architecture closely follows that of Transporter Net~\cite{zeng2020transporter}. The main difference is that we explicitly encode equivariance constraints on both $\phi$ and $\psi$ networks. As a result of this change: 1) we are able to simplify the model by transposing the lifting operation $\mathcal{R}_n$ and the processing by $\phi$; 2) our new model is equivariant with respect to a larger symmetry group $C_n \times C_n$, compared to Transporter Net which is only equivariant over $C_n$.
\subsubsection{Equivariant $\phi$ and $\psi$}
\label{sect:transporter_desc1}
We explicitly encode both $\phi$ and $\psi$ as $C_n$-equivariant models that satisfy the following constraints:
\begin{align}
\label{eqn:psi}
\psi(T^0_g(c)) &= T^0_g (\psi(c)) \\
\label{eqn:phi}
\phi(T^0_g(o_t)) &= T^0_g(\phi(o_t)),
\end{align}
for $g \in \mathrm{SO}(2)$.
The equivariance constraint of Equation~\ref{eqn:phi} says that when the input image rotates, we would expect the place location to rotate correspondingly. This constraint helps the model generalize across place orientations. The constraint of Equation~\ref{eqn:psi} says that when the picked object rotates (represented by the image patch $c$), then the place orientation should correspondingly rotate.
\subsubsection{Place Model}
When the equivariance constraint of Equation~\ref{eqn:psi} is satisfied, we can exchange $\mathcal{R}_n$ (the lifting operation) with $\psi$: $\psi(\mathcal{R}_n(c)) = \mathcal{R}_n(\psi(c))$. This equality is useful because it means that we only need to evaluate $\psi$ for one image patch rather than the stack of image patches $\mathcal{R}_n(c)$ -- something that is computationally cheaper. The resulting place model is then:
\begin{eqnarray}
\label{eqn:place_equiv_pre}
f'_{\mathrm{place}}(o_t,c) & = & \mathcal{R}_n(\psi(c)) \star \phi(o_t) \\
\label{eqn:place_equiv}
& = & \Psi(c) \star \phi(o_t),
\end{eqnarray}
where Equation~\ref{eqn:place_equiv} substitutes $\Psi(c) = \mathcal{R}_n[\psi (c)]$ to simplify the expression. Here, we use $f'_{\mathrm{place}}$ to denote Equivariant Transporter Net defined using equivariant $\phi$ and $\psi$ in contrast to the baseline Transporter Net $f_{\mathrm{place}}$ of Equation~\ref{eqn:transporter1}. Note that both $f_{\mathrm{place}}$ and $f'_{\mathrm{place}}$ satisfy Proposition~\ref{prop:equivtransporter}. However, $f_{\mathrm{place}}$ accomplishes this by symmetrizing a non-equivariant network (i.e. evaluating $\psi(\mathcal{R}_n(c))$) whereas our model $f'_{\mathrm{place}}$ encodes the symmetry directly into $\psi$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.42\textwidth]{figs/placing_network_7.png}
\caption[Place Network Architecture]{Equivariance of our placing network under the rotation of the object and the placement. A $90\degree$ rotation on $c$ and a $-90\degree$ rotation on $o_t\backslash c$ are equivariant to: i), a $-90\degree$ rotation on the placing location, and ii), the shift on the channel of placing rotation angle from $270\degree$ (the last channel) to $90\degree$ (the second channel).}
\label{fig:place_archi}
\end{figure}
\subsection{Equivariance Properties of the Placing Network}
As Proposition~\ref{prop:equivtransporter} demonstrates, the baseline Transporter Net model~\cite{zeng2020transporter} encodes the symmetry that rotations of the object to be picked (represented by $c$) should result in corresponding rotations of the place orientation for that object. However, pick and place problems have a second symmetry that is not encoded in Transporter Net: that rotations of the scene image (represented by $o_t$) should also result in corresponding rotations of the place orientation. In fact, as we demonstrate in Proposition~\ref{proposition_ours} below, we encode this second type of symmetry by enforcing the constraints of Equations~\ref{eqn:psi} and~\ref{eqn:phi}. Essentially, we go from a $C_n$-symmetric model to a $C_n \times C_n$-symmetric model.
\begin{proposition}
\label{proposition_ours}
Equivariant Transporter Net $f_{\mathrm{place}}'$ is $C_n \times C_n$-equivariant. That is, given rotations $g_1 \in C_n$ of the picked object and $g_2 \in C_n$ of the scene, we have that:
\begin{equation}
f'_\mathrm{place}(T^0_{g_1}(c), T^0_{g_2} (o_t)) = \rho_{\mathrm{reg}}(g_2-g_1)T^0_{g_2} f'_\mathrm{place}(c, o_t).
\label{eqn:prop2}
\end{equation}
\end{proposition}
Proposition~\ref{proposition_ours} is proven in Appendix~\ref{proof_proposition} and illustrated in Figure~\ref{fig:place_archi}. The top of Figure~\ref{fig:place_archi} going left to right shows the rotation of both the object by $g_1$ (in orange) and the place pose by $g_2$ (in green). The LHS of Equation~\ref{eqn:prop2} evaluates $f'_{\mathrm{place}}$ for these two rotated images. The lower left of Figure~\ref{fig:place_archi} shows $f'_{\mathrm{place}}(c,o_t)$. Going left to right at the bottom of Figure~\ref{fig:place_archi} shows the pixel-rotation by $T^0_{g_2}$ and the channel permutation by $g_2 - g_1$ (RHS of Equation~\ref{eqn:prop2}).
Note that in additional to the two rotational symmetries enforced by our model, it also has translational symmetry. Since the rotational symmetry is realized by additional restrictions to the weights of kernels of convolutional networks, the rotational symmetry is in addition to the underlying shift equivariance of the convolutional network. Thus, the full symmetry group enforced is the group generated by $C_n \times C_n \times (\mathbb{R}^2,+)$.
Enforcing equivariance with respect to an even larger symmetry group than Transporter Net leads to even greater sample efficiency. Equivariant neural networks learn effectively on a lower dimensional space, the equivalence classes of samples under the group action. Thus a larger group results in an even smaller dimensional sample space and thus better coverage by the training data.
\subsection{Model Architecture Details}
\subsubsection{Pick model $f_p$ (Equation~\ref{eqn:pick1})}
The input to $f_p$ is a $4$-channel RGB-D image $o_t \in \mathbb{R}^{4\times H \times W}$. The output is a feature map $p(u,v) \in \mathbb{R}^{H\times W}$ which encodes a distribution over pick location. $f_p$ is implemented as an 18-layer euqivariant residual network with a U-Net~\cite{ronneberger2015u} as the main block. The U-net has 8 residual blocks (each block contains 2 equivariant convolution layers~\cite{weiler2019general} and one skip connection): 4 residual blocks~\cite{he2016deep} are used for the encoder and the other 4 residual blocks are used for the decoder. The encoding process trades spatial dimensions for channels with max-pooling in each block; the decoding process upsamples the feature embedding with bilinear-upsampling operations. The first layer maps the trivial representation of $o_t$ to regular representation and the last equivariant layer transforms the regular representation back to the trivial representation, followed by image-wide softmax. ReLU activations~\cite{nair2010rectified} are interleaved inside the network.
\subsubsection{Pick model $f_\theta$ (Equation~\ref{eqn:pick2})}
Given the picking location $(u^*,v^*)$, the pick angle network $f_\theta$ takes as input a crop $c\in \mathbb{R}^{4\times H_1 \times W_1}$ centered on $(u^*,v^*)$ and outputs the distribution $p(\theta|u,v) \in \mathbb{R}^{n/2}$, where $n$ is the size of the rotation group(i.e. $n=| C_n |$). The first layer maps the trivial representation of $c$ to a quotient regular representation followed by 3 residual blocks containing max-pooling operators. This goes to two equivariant convolution layers and then to an average pooling layer.
\subsubsection{Place models $\phi$ and $\psi$}
Our place model has two equivariant convolution networks, $\phi$ and $\psi$, and both have similar architectures to $f_p$. The network $\phi$ takes as input a zero-padded version of $o_t$, $\mathrm{pad}(o_t)\in \mathbb{R}^{4\times (H+d) \times (W+d)}$, and generates a dense feature map, $\phi(\mathrm{pad}(o_t))\in\mathbb{R}^{(H+d) \times (W+d)}$, where $d$ is the padding size. The network $\psi$ takes as input the image patch $c \in \mathbb{R}^{4\times H_2 \times W_2}$ and outputs $\psi(c)\in \mathbb{R}^{H_2 \times W_2}$. After applying rotations of $C_n$ to $\psi(c)$, the transformed dense embeddings $\Psi(c)\in \mathbb{R}^{n\times H_2 \times W_2}$ are cross-correlated with $\phi(\mathrm{pad}(o_t))$ to generate the placing action distribution $p(a_{\mathrm{place}}|o_t,a_{\mathrm{pick}}) \in \mathbb{R}^{n\times H \times W}$, where the channel axis $n$ corresponds to placing angles, $\frac{2\pi i}{n}$ for $0\leq i < n$.
\subsubsection{Group Types and Sizes}
The networks $f_p$, $\psi$, and $\phi$ are all defined using $C_6$ regular representations. The network $f_\theta$ is defined using the regular quotient representation $C_{36}/C_2$, which corresponds to the number of allowed place orientations.
\subsubsection{Training Details:}
\label{traing_settings}
We train Equivariant Transporter Network with the Adam~\cite{kingma2014adam} optimizer with a fixed learning rate of $10^{-4}$. It takes about 0.8 seconds to complete one SGD step with a batch size of 1 on a NVIDIA Tesla V100 SXM2 GPU. For each task, we evaluate performance every 10k steps on 100 unseen tests. On most tasks, the best performance is achieved in less than 10k SGD steps. Our model converges in a few hours on all tasks.
\section{Experiments}
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{figs/gripper.png}
\caption{ Simulated environment for parallel-jaw gripper tasks. From left to right: (a) inserting blocks into fixtures, (b) placing red boxes into green bowls, (c) align box corners to green lines, (d) stacking a pyramid of blocks, (e) palletizing boxes.}
\label{fig:gripper_env}
\end{figure*}
We evaluate Equivariant Transporter using the Ravens-10 Benchmark~\cite{zeng2020transporter} and variations thereof.
\subsection{Tasks}
\subsubsection{Ravens-10 Tasks}
Ravens-10 is a behaviour cloning simulation environment for manipulation, where each task owns an oracle that can sample expert demonstrations from the distribution of successful picking and placing actions with the access to the ground-truth pose of each object. The 10 tasks of Ravens can be classified into 3 categories: \textit{Single-object manipulation tasks} (block-insertion, align-box-corner); \textit{Multiple-object manipulation tasks} (place-red-in-green, towers-of-hanoi, stack-block-pyramid, palletizing-boxes, assembling-kits, packing-boxes); \textit{Deformable-object manipulation task} (manipulating-rope, sweeping-piles). Detailed explanations of the tasks can be found in Appendix~\ref{raven_10_details}
\subsubsection{Ravens-10 Tasks Modified for the Parallel Jaw Gripper}
\label{parallel env}
We selected 5 tasks (block-insertion, align-box-corner, place-red-in-green, tack-block-pyramid, palletizing-boxes) from Ravens-10 and replaced the suction cup with the Franka Emika gripper. \figref{fig:gripper_env} illustrates the initial state and completion state for each of these five tasks. For each of these five tasks, we defined an expert trajectory generator. Since the Transporter Net framework assumes that the object does not move during picking, we defined these expert generators such that this was the case.
\subfile{tables/table1}
\subfile{tables/table2}
\subsection{Training and Evaluation}
\subsubsection{Training}
For each task, we produce a dataset of $n$ expert demonstrations, where each demonstration contains a sequence of one or more observation-action pairs $(o_t,\Bar{a}_t)$. Each action $\Bar{a}_t$ contains an expert picking action $\Bar{a}_{\mathrm{pick}}$ and an expert placing action $\Bar{a}_{\mathrm{place}}$. We use $\Bar{a}_t$ to generate one-hot pixel maps as the ground-truth labels for our picking model and placing model. The models are trained using a cross-entropy loss.
\subsubsection{Metrics}
We measure performance the same way as it was measured in ~\cite{zeng2020transporter} -- using a metric in the range of 0 (failure) to 100 (success). Partial scores are assigned to multiple-action tasks. For example, in the block-stacking task where the agent needs to construct a 6-block pyramid, each successful rearrangement is credited with a score of 16.667. We report highest validation performance during training, averaged over 100 unseen tests for each task.
\subsubsection{Baselines}
We compare our method against Transporter Net~\cite{zeng2020transporter} as well as the following baselines previously used in the Transporter Net paper~\cite{zeng2020transporter}. \textit{Form2Fit}~\cite{zakka2020form2fit} introduces a matching module with the measurement of $L_2$ distance of high-dimension descriptors of picking and placing locations. \textit{Conv-MLP} is a common end-to-end model~\cite{levine2016end} which outputs $a_{\mathrm{pick}}$ and $a_{\mathrm{place}}$ using convolution layers and MLPs (multi-layer perceptrons). \textit{GT-State MLP} is a regression model composed of an MLP that accepts the ground-truth poses and 3D bounding boxes of objects in the environment. \textit{GT-State MLP 2-step} outputs the actions sequentially with two MLP networks and feeds $a_{\mathrm{pick}}$ to the second step. All regression baselines learn mixture densities~\cite{bishop1994mixture} with log likelihood loss.
\subsubsection{Adaptation of Transporter Net for Picking Using a Parallel Jaw Gripper}
In order to compare our method against Transporter Net for the five parallel jaw gripper tasks, we must modify Transporter to handle the gripper. We accomplish this by~\cite{zeng2018learning} lifting the input scene image over $C_n$, producing a stack of differently oriented input images which is provided as input to the pick network $f_{\mathrm{pick}}$. The results are counter-rotated at the output of $f_{\mathrm{pick}}$.
\subsection{Results for the Ravens-10 Benchmark Tasks}
\subsubsection{Task Success Rates}
\begin{figure}
\centering
\subfigure[]{ \includegraphics[clip,width=0.23\textwidth]{figs/fast.png}}
\subfigure[]{ \includegraphics[clip,width=0.232\textwidth]{figs/fast2.png}}
\caption[faster learning]{Equivariant Transporter Network converges faster than Transporter Network. Left: Block-insertion task. Right: sweeping-piles task. On the block insertion task, Equivariant Transporter can hit greater than $90\%$ success rate after 10 training steps and achieve $100\%$ succeess rate with less than 100 training steps.}
\label{fig:fast_converge}
\end{figure}
Table~\ref{table:sample-efficiency1} shows the performance of our model on the Raven-10 tasks for different numbers of demonstrations used during training. All tests are evaluated on unseen configurations, i.e., random poses of objects, and three tasks (align-box-corner, assembling-kits, packing-box) use unseen objects.
Our proposed Equivariant Transporter Net outperforms all the other baselines in nearly all cases. The amount by which our method outperforms others is largest when the number of demonstrations is smallest, i.e. with only 1 or 10 demonstrations. With just 10 demonstrations per task, our method can achieve $\geq 95\%$ success rate on 7/10 tasks.
\subsubsection{Training Efficiency}
Another interesting consequence of our more structured model is that training is much faster. Figure~\ref{fig:fast_converge} shows task success rates as a function of the number or SGD steps for two tasks (Block Insertion and Sweeping Piles). Our equivariant model converges much faster in both cases.
\subsection{Results for Parallel Jaw Gripper Tasks}
\subsubsection{Task Success Rates}
Table~\ref{table:sample-efficiency2} compares the performance of Equivariant Transporter with the baseline Transporter Net for the Parallel Jaw Gripper tasks. Again, our method outperforms the baseline in nearly all cases.
\subsubsection{Comparison with Ravens-10}
One interesting observation that can be made by comparing Tables~\ref{table:sample-efficiency1} and~\ref{table:sample-efficiency2} is that both Equivariant Transporter and baseline Transporter do better on the gripper versions of the task compared to the original Ravens-10 versions. This is likely caused by the fact that the expert demonstrations we developed for the gripper version on the task have less stochastic gripper placement during pick than is the case in the original Ravens-10 benchmark.
\subsection{Ablation Study}
\label{ablation_study}
\subsubsection{Ablations}
We performed an ablation study to evaluate the relative importance of the equivariant models in pick ($f_p$ and $f_\theta$) and place ($\psi$ and $\phi$). We compare four versions of model with various levels of equivariance: non-equivariant pick and non-equivariant place (baseline Transporter), equivariant pick and non-equivariant place,
non-equivariant pick and equivariant place, and
equivariant pick and equivariant place (Equivariant Transporter).
\begin{figure}
\centering
\subfigure[]{ \includegraphics[width=0.23\textwidth]{figs/ablation2.png}}
\subfigure[]{ \includegraphics[width=0.232\textwidth]{figs/ablation1.png}}
\vfill
\subfigure[]{ \includegraphics[width=0.23\textwidth]{figs/ablation3.png}}
\subfigure[]{ \includegraphics[width=0.232\textwidth]{figs/ablation4.png}}
\caption[Ablation Study]{Ablation study. Performance is evaluated on 100 unseen tests of each task.}
\label{fig:ablation}
\end{figure}
\subsubsection{Results}
\figref{fig:ablation} shows the performance of all four ablations as a function of the number of SGD steps for the scenario where the agent is given 10 or 100 expert demonstrations. The results indicate that place equivariance (i.e. equivariance of $\psi$ and $\phi$) is namely responsible for the gains in performance of Equivariant Transporter versus baseline Transporter. This finding is consistent with the argument that it is the larger $C_n \times C_n$ symmetry group (only present with equivariant place) that is responsible for our performance gains. Though the non-equivariant and equivariant pick networks result in comparable performance, the equivariant network is far more computationally efficient, taking a single image as input versus 36 for the non-equivariant network.
\begin{table}[]
\centering
\begin{tabular}{c|c|c|c}
\hline
Task & \# demos & \# completions / \# trials & success rate\\
\hline
stack-block-pyramid & 10 & 17/20 & 95.8\%\\
\hline
place-box-in-bowl & 10 & 20/20 & 100\%\\
\hline
block-insertion & 10 & 20/20 & 100\%\\
\hline
\end{tabular}
\caption{Task success rates for physical robot evaluation tasks.}
\label{tab:real_test_performance}
\end{table}
\begin{figure}
\centering
\centering
\subfigure[]{ \includegraphics[clip,width=0.23\textwidth]{figs/bi.png}}
\subfigure[]{ \includegraphics[clip,width=0.226\textwidth]{figs/pig.png}}
\caption[depth image]{Real robot experiment: initial observation $o_t\in\mathbb{R}^{200\times200}$ from the depth sensor. The left figure shows the block insertion task; the right figure shows the task of placing boxes in bowls. The depth value ($\mathrm{meter}$) is illustrated in the color bar.}
\label{fig:depth}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figs/robot1.png}
\caption[depth image]{Stack-block-pyramid task on the real robot. The left figure shows the initial state; the right figure shows the completion state.}
\label{fig:robot}
\end{figure}
\subsection{Experiments on a Physical Robot}
We evaluated Equivariant Transporter on a physical robot in our lab. There was no use of the simulator in this experiment -- all demonstrations were performed on the real robot.
\subsubsection{Setup}
We used a UR5 robot with a Robotiq-85 end effector. The workspace was a $40cm \times 40cm$ region on a table beneath the robot (see Figure~\ref{fig:robot}). The observations $o$ were $200 \times 200$ depth images obtained using a Occipital Structure Sensor that was mounted pointing directly down at the table (see Figure~\ref{fig:depth}).
\subsubsection{Tasks}
We evaluated Equivariant Transporter on three of the Ravens-10 gripper-modified tasks: block insertion, placing boxes in bowls, and stacking a pyramid of blocks. Since our sensor only measures depth (and not RGB), we modified the box-in-bowls task such that box color was irrelevant to success, i.e. the task is simply to put any box into a bowl.
\subsubsection{Demonstrations}
We obtained 10 human demonstrations of each task. These demonstrations were obtained by releasing the UR5 brakes and pushing the arm physically so that the harmonic actuators were back-driven.
\subsection{Training and Evaluation}
For each task, our model was trained for 10k SGD steps. During testing, objects were randomly placed on the table. A task was considered to have failed when a single incorrect pick or place occurred. We evaluated on 20 unseen configurations of each task.
\subsubsection{Results}
Table\:\ref{tab:real_test_performance} shows results from 20 runs of each of the three tasks. Notice that the success rates here are higher than they were for the corresponding tasks performed in simulation (Table~\ref{table:sample-efficiency2}). This is likely caused by the fact that the criteria for task success in simulation (less than 1 cm translational error and less than $\frac{\pi}{12}$ rotation error were more conservative than is actually the case in the real world. Videos can be found in supplementary materials.
\section{Conclusion and Limitations}
\label{sec:conclusion}
This paper explores the symmetries present in the pick and place problem and finds that they can be described by the direct product group $C_n \times C_n$, where $C_n$ denotes the cyclic group of discrete orientations. This corresponds to the group of different pick and place orientations. We evaluate the Transporter Network model proposed in~\cite{zeng2020transporter} and find that it encodes one of these symmetries (the pick symmetry), but not the other (the place symmetry). We propose a novel version of Transporter Net, Equivariant Transporter Net, which we show encodes both types of symmetries. We evaluate our model on the Ravens-10 Benchmark and evaluate against multiple strong baselines. Finally, we demonstrate that the method can effectively be used to learn manipulation policies on a physical robot. One limitation of our framework as it is presented in this paper is that it relies entirely on behavior cloning. A clear direction for future work is to integrate more on-policy learning which we believe would enable us to handle more complex tasks.
\bibliographystyle{plainnat}
\section{Introduction}
This demo file is intended to serve as a ``starter file" for the
Robotics: Science and Systems conference papers produced under \LaTeX\
using IEEEtran.cls version 1.7a and later.
\section{Section}
Section text here.
\subsection{Subsection Heading Here}
Subsection text here.
\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{RSS citations}
Please make sure to include \verb!natbib.sty! and to use the
\verb!plainnat.bst! bibliography style. \verb!natbib! provides additional
citation commands, most usefully \verb!\citet!. For example, rather than the
awkward construction
{\small
\begin{verbatim}
\cite{kalman1960new} demonstrated...
\end{verbatim}
}
\noindent
rendered as ``\cite{kalman1960new} demonstrated...,''
or the
inconvenient
{\small
\begin{verbatim}
Kalman \cite{kalman1960new}
demonstrated...
\end{verbatim}
}
\noindent
rendered as
``Kalman \cite{kalman1960new} demonstrated...'',
one can
write
{\small
\begin{verbatim}
\citet{kalman1960new} demonstrated...
\end{verbatim}
}
\noindent
which renders as ``\citet{kalman1960new} demonstrated...'' and is
both easy to write and much easier to read.
\subsection{RSS Hyperlinks}
This year, we would like to use the ability of PDF viewers to interpret
hyperlinks, specifically to allow each reference in the bibliography to be a
link to an online version of the reference.
As an example, if you were to cite ``Passive Dynamic Walking''
\cite{McGeer01041990}, the entry in the bibtex would read:
{\small
\begin{verbatim}
@article{McGeer01041990,
author = {McGeer, Tad},
title = {\href{http://ijr.sagepub.com/content/9/2/62.abstract}{Passive Dynamic Walking}},
volume = {9},
number = {2},
pages = {62-82},
year = {1990},
doi = {10.1177/027836499000900206},
URL = {http://ijr.sagepub.com/content/9/2/62.abstract},
eprint = {http://ijr.sagepub.com/content/9/2/62.full.pdf+html},
journal = {The International Journal of Robotics Research}
}
\end{verbatim}
}
\noindent
and the entry in the compiled PDF would look like:
\def\tmplabel#1{[#1]}
\begin{enumerate}
\item[\tmplabel{1}] Tad McGeer. \href{http://ijr.sagepub.com/content/9/2/62.abstract}{Passive Dynamic
Walking}. {\em The International Journal of Robotics Research}, 9(2):62--82,
1990.
\end{enumerate}
where the title of the article is a link that takes you to the article on IJRR's website.
Linking cited articles will not always be possible, especially for
older articles. There are also often several versions of papers
online: authors are free to decide what to use as the link destination
yet we strongly encourage to link to archival or publisher sites
(such as IEEE Xplore or Sage Journals). We encourage all authors to use this feature to
the extent possible.
\section{Conclusion}
\label{sec:conclusion}
The conclusion goes here.
\section*{Acknowledgments}
\bibliographystyle{plainnat}
|
1,116,691,499,362 | arxiv | \section{\large{S\MakeLowercase{upplemental} M\MakeLowercase{aterials for:} \\
K\MakeLowercase{ondo} C\MakeLowercase{orrelation} I\MakeLowercase{nduced} L\MakeLowercase{ow-}F\MakeLowercase{ield} M\MakeLowercase{agnetoresistance} A\MakeLowercase{nomalies in} I\MakeLowercase{n}S\MakeLowercase{b} N\MakeLowercase{anowire} J\MakeLowercase{osephson} Q\MakeLowercase{uantum} D\MakeLowercase{ot} D\MakeLowercase{evices}
}}
\vspace{25mm}
\large{The supplemental materials include figures showing measurement results for Device~3 and figures showing supporting datasets for Device~1 and Device~2. Figure descriptions and associated discussions are all presented in the captions.
\vspace{25mm}
\begin{itemize}
\item \hspace{1mm} Figure 1: Device~3 and its characterization measurements
\vspace{5mm}
\item \hspace{1mm} Figure 2: Magnetic field dependent measurements of the conductance spectra and the supercurrent in Device~3.
\vspace{5mm}
\item \hspace{1mm} Figure 3: Observation of the low field NMR in Device~2 in a ``$\pi$"-junction.
\vspace{5mm}
\item \hspace{1mm} Figure 4: Observation of the coexistence of the anomalous NMR and field-induced recovered Kondo Peak in Devices~1 and 2 close to Coulomb resonances.
\vspace{5mm}
\item \hspace{1mm} Figure 5: Measurements of the conductance spectra for Device~2 in the open regime.
\vspace{5mm}
\item \hspace{1mm} Figure 6: Observation of weak NMR features at the finite-bias conductance peaks in all the three studied devices.
\end{itemize}
}
\begin{figure}[h]
\centering
\includegraphics[width=17 cm]{SI-1}
\caption{\label{SI-1} \normalsize{\textbf{Device~3 and its characterization measurements.} (a) Scanning electron microscope image of Device~3 taken at a 30 degrees tilted angle from the substrate normal direction. The device is fabricated in the same manner as for Devices~1 and 2, but without side gates. The width and orientation of the Ti/Al contact electrodes are different from that in Devices~1 and 2 and, thus, the metal-semiconductor contact areas in Device~3 are smaller. The scale bar is 500~nm. (b) Charge stability diagram measured for the device at perpendicularly applied magnetic field $B=200$~mT, i.e., at the the normal state of the device. Here a Coulomb diamond structure marked with ``odd" for the device with an odd-number electron occupation in the dot and its two neighboring Coulomb diamond structures marked with ``even" for the device with even-number electron occupations in the dot are shown. The charging energy of the dot extracted from the measurements is about 6~meV. A split Kondo ridge can be seen in the odd-number electron occupation region. (c) Differential conductance measured as a function of $V_{sd}$ at $V_{\rm bg}$=-585 mV as indicated by the dashed line in (b) and at magnetic fields of $B$= 70 mT to 700 mT. Traces are sequentially offset by 0.005 $e^2/h$. An effective g-factor $|g^*|\approx42$ can be extracted from the Kondo ridge splittings. (d) Low energy conductance spectrum measured at $B=0$ for the device in a back gate voltage region similar to that in panel (b), (e) Line-cut view for the measurements shown in panel (d). Traces are sequentially offset by 0.12 $e^2/h$. From (d) and (e), we can clearly see conductance peaks at finite bias voltages, which originate from MARs and ABSs. Zero-bias peak originated from the Kondo enhanced superconductivity is also observed in the odd-number occupation region. (f) Differential resistance measured in a two-terminal current-bias setup for the device in a similar back gate region as in (d). A 100~$\Omega$ constant circuit resistance is subtracted from the measured values. The two lower insets A and B are close-up views of the the measurements shown in the dashed rectangular regions A and B, respectively. In the Kondo regime, Josephson current (dark color) can be identified although the critical current is small. (g) Voltage drop across the device measured as a function of the applied source-drain current at a few selected back gate voltages from the region marked by the dashed rectangle A in panel (e). (h) Zero-bias conductance measured in the superconducting states $G_s$ (black solid line) and in the normal state $G_n$ (black dashed line) as a function of $V_{\rm bg}$. Measured supercurrent (blue circles) in the same back gate region is also plotted. The presence of the Kondo resonance greatly enhances Josephson current. Note that the smeared switching from dissipationless branch to dissipative causes large errors and fluctuations in the supercurrent measurements. }}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=17 cm]{SI-2}
\caption{\label{SI-2} \normalsize{\textbf{Magnetic field dependent measurements of the conductance spectra and the supercurrent in Device~3.} (a) Magnetic field evolution of the conductance spectrum measured for Device 3 at $V_{\rm bg}=-585$ mV, i.e., along the dashed line in Fig.~\ref{SI-1}(d) of the Supplemental Materials. The critical magnetic field in this device is about $B=60$~mT and is higher than that in Devices~1 and 2. This is presumably because the contact electrodes have a smaller width in Device~3. At magnetic fields above the critical field, two differential conductance peaks at finite bias voltages are observable, see a line-cut plot (red solid line) in the upper part of the figure. The two peaks show linear magnetic field dependencies as indicated by two straight dashed lines in the figure and can be traced to the Zeeman splitting of the normal state Kondo conductance ridge. (b) Close-up view of the zero-bias peak in (a). Here, we can see that the zero-bias peak is also associated with a pronounced NMR at low magnetic fields ($|B|<30$~mT).} (c) Magnetic field dependence of the critical current measured at $V_{\rm bg}=-150$~mV where the device is in the open regime. Here no field-enhanced critical current can be seen.}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=17 cm]{SI-3}
\caption{\label{SI-3} \normalsize{\textbf{Observation of the low field NMR in Device~2 in a ``$\pi$"-junction.} (a) Low energy conductance spectrum of Device~2 as a function of $V_{\rm bg}$ measured at $B=0$, $V_{\rm g1}=-2.4$~V and $V_{\rm g2}=1.2$~V. From the profiles of the two finite-bias conductance peaks (with one of them indicated by the black dashed line), the device in this odd-number occupation region forms a ``$\pi$"-junction. (b) and (c) Magnetic field evolutions of the conductance spectra measured for Device 2 at the back gate voltages indicated by white lines A and B in (a), respectively. Here, we can see that the anomalous NMRs are present in both panels. (d) Zero-bias conductance as a function of magnetic field measured for Device 2 at selected $V_{\rm bg}$ values. \st{Note that the curves are successively offset by ????.} The lower and upper curves are the measurements at the the values of $V_{\rm bg}$ marked by the white lines A and B in (a), respectively. The other curves are for the values of $V_{\rm bg}$ taken to be between the two. It is seen that all the curves show a NMR conductance valley. The widths of the valleys are around 6~mT in all curves, but the depths of the valleys are dependent on $V_{\rm bg}$. Since the Kondo temperature strongly depends on the position of the Fermi level with respect to the level position of the dot and, in general, has its minimum in the middle of the deep Coulomb blockade region while reaches its maximum near the Coulomb resonance, we can conclude that the NMR valley width is nearly independent of $T_{\rm K}$ while the valley depth does show a $T_{\rm K}$ dependence. The measurements here can exclude the magnetic field induced quantum phase transition as a possible mechanism for the observed anomalous NMR.}}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=17 cm]{SI-4}
\caption{\label{SI-4} \normalsize{\textbf{Observation of the coexistence of the anomalous NMR and field-induced recovered Kondo Peak in Devices~1 and 2 close to Coulomb resonances.} (a) Low energy conductance spectrum measured for Device~1 at $B=0$, $V_{\rm g1}=-300$~mV and $V_{\rm g2}=-600$~mV. (b) Magnetic field evolution of the spectrum measured at the back gate $V_{\rm bg}$=-235 mV, i.e., along the black dashed line in (a). (c) Close-up view of the zero-bias peak region in (b). (d) Line-cut at $V_{\rm sd}= 0$ taken from (c), i.e., along the white dashed line in (c). Indicated by the blue arrow in (d), the zero-bias conductance peak shows a NMR at low magnetic fields. There is an enhanced zero-bias conductance peak occurring around $B=18$~mT (red arrow), which can be attributed to the recovered Kondo screening as the superconducting gap shrinks to a much smaller value at high fields than the Kondo temperature [see, e.g., Lee \emph{et.al.}, Phys. Rev. Lett. \textbf{109}, 186802 (2012)]. (e)-(h) The same as in (a)-(d) but for the measurements of Device~2 at $V_{\rm g1}=2.2$~V and $V_{\rm g2}=-3$~V. The coexistence of the anomalous NMR and field-recovered Kondo Peak is also observed here. }}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=17 cm]{SI-5}
\caption{\label{SI-5} \normalsize{\textbf{Measurements of the conductance spectra for Device~2 in the open regime}. (a) Differential conductance as a function of $V_{sd}$ and $V_{\rm bg}$ measured for Device~2 in the open regime at $B=30$~mT and $V_{\rm g1}=V_{\rm g2}=0$. A Fabry-P\'{e}rot interference conductance chess-board pattern can be identified, which shows that the device is in the open ballistic transport regime. (b) The same a (a) but the measurements at $B=0$. The Fabry-P\'{e}rot pattern remains but is greatly modulated by the superconductivity in the low bias voltage region [see Li \emph{et.al.}, Sci. Rep. \textbf{6}, 24822 (2016)]. (c) Measured differential conductance as a function of $V_{\rm sd}$ and $B$ at $V_{\rm bg}$= 1.88 V, i.e., along the dashed line in (b). Supercurrent induced zero-bias peak and finite-bias MAR peaks are observed to gradually vanishes as magnetic field increases. (d) Line-cut at $V_{\rm sd}=0$ taken along the dashed line in (c). Here, the zero-bias conductance is seen to monotonically decrease with increasing magnetic field and no NMR is observed. It shows that no anomalous NMR could appear in the the Josephson junction device without the Kondo correlations.}}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=17 cm]{SI-6}
\caption{\label{SI-6} \normalsize{\textbf{Observation of weak NMR features at the finite-bias conductance peaks in all the three studied devices.} (a) Conductance spectrum of Device~1 taken from Fig.~2(b) in the main article. Here, the colormap is rescaled in order to make the weak NMR feature associated with the finite-bias conductance peak visible. (b) Contour plot of (a). Here the NMR feature at the finite-bias conductance peak can be more clearly identified. (c)-(h) Line cut plots of magnetic field evolution of the conductance taken at finite-bias conductance peaks from all the three studied devices. Red lines are the line cuts taken at the positive bias voltage peaks and blue lines are the line cuts taken at the negative bias voltage peaks. The NMRs at the finite-bias conductance peaks are seen to appear in similar magnetic field ranges as their zero-bias counterparts. It is also seen that no NMR is observable when $T_{\rm K}$ is much larger than the superconductor gap.}}
\end{figure}
\end{document} |
1,116,691,499,363 | arxiv | \section{Introduction}
\label{intro}
Consider an undirected graph G=(V, E) with $v=\{1, 2,... , n\}$, where specific vertex 1 is root vertex and edge set is $E=\{(i,j): i,j\in V,i<j\}$, that each edge has an associated cost $c_{ij}\geq 0$. Also, we define a positive integer $H$ to show the maximum allowance distance of edge from the root. Let $T$ be a tree in $G$, and let $V(T)$ be the set of vertices belonging to $T$. Our goal is finding a tree $T$, rooted at vertex 1, subject to constraints and the number of edges $h_i$ between $i\in V (T)$ and the root vertex limited to the maximum value H.
Since HCST is a generalization of Steiner tree problem, it is Np-hard problem\cite{109}. We consider hop constraints in graph because of reasons such as reliability or transmission delay in networks. The extensive uses of hop constraints in various settings have been proposed in the literature \cite{103,102,101,107,108}.
Intensive researches on the Minimum Spanning Tree problem with hop constraints (HCMST), which is a special case of the HCST problem where all vertices in the graph are terminals exist.
Many surveys regarding of HCMST problem can be found in \cite{5,111,110}. Though, there aren't much attention to Steiner tree problem with hop constraints. In \cite{7}, Gouveia has mentioned HCST problem with developing a strengthened version of a multi-commodity flow model for the minimum spanning tree problem.
%
The LP lower bounds of this model are equal to the ones from a Lagrangian relaxation approach of a weaker MIP model introduced by Gouveia in \cite{8}. Voss presents MIP formulations based on Miller-Tucker-Zemlin subtour elimination constraints \cite{14}. The formulation is then strengthened by disaggregation of variables indicating used edges.
%
The author develops a simple heuristic to find starting solutions and improves them with an exchange procedure based on tabu search. Gouveia Also gives a survey of hop-indexed tree and flow formulations for the hop constrained spanning and Steiner tree problem in \cite{9}.
%
Costa et. al. give a comparison of three methods for a generalization of the HCSTP which is called Steiner tree problem with revenues, budget and hop constraints (STPRBH) in \cite{1}. The considered methods comprise greedy algorithm, destroy-and-repair method and tabu search approaches.
%
Computational results are reported for instances with up to 500 vertices and 12500 edges. Costa et al. in \cite{2} introduce two new models for he STPRBH. Both models are based on the generalized sub-tour elimination constraints and a set of exponential size hop constraints. The authors provide a theoretical and computational comparison with two models based on Miller-Tucker-Zemlin constraints presented in Gouveia \cite{10} and Voss \cite{14}. Theoretical and computational comparisons of flow-based vs. path-based mixed integer programming models for HCST problem are presented by Gouveia et al.. They propose formulations to solve the problem with promising optimality and implement branch-and-price algorithms for all of the formulations\cite{11}.
%
Boeck et al. used layered graphs for hop constrained problems to build extended formulations by techniques presented to reduce the size of the layered graphs\cite{65}. They also presented variation of this problem arising in the context of multicast transmission in telecommunications.
Dokeroglu et al. recently proposed novel self-adaptive and stagnation-aware breakout local search algorithm for the solution of Steiner tree problem with revenue, budget and hop constraints with parallel algorithms \cite{15}.
In this paper, we focus on efficient optimization iterative greedy algorithms to find efficient solution for HCST. The new algorithms find a feasible solution for HCST are based on Voss's approach in \cite{14}. The main feature of our last proposed algorithm is that it uses the idea of Kruskal algorithm in the problem with hop constraint for the first time and enhances the results of this NP-hard problem in polynomial time.
The rest of this paper is organized as following. In Section 2 we formulate a model for HCST problem. Section 3 and 4 provide two complement greedy algorithms to solve HCST. In Section 5 we propose Non Root Based Insertion Algorithm based on the idea of Kruskal algorithm\cite{3}. In Section 6 we report extensive set of computational experiments with our iterative greedy algorithms on well known benchmark, consisting variant kinds of graphs. Finally, Section 7 presents concluding remarks .
\section{Mathematical formulation}
\label{MathModel}
A non-directed graph $G = (V, E)$ is given with edge cost $c_{ij}\geq 0$, $(i, j)\in E$. Let consider the set $Q\subseteq V$ represent basic vertices. To obtain a subgraph with minimum cost, other vertices could be involved. Such vertices are called Steiner vertices $(S=V-Q)$. Consider $x^p_{ij}$ as an edge $(i, j) \in E$ that shows the $p^{th}$ position from the root and $\delta (S)$ as a set of edges with one vertex from $S$. The HCST model can be written as follow:
\begin{equation}
min \sum_{p=1}^{H}\sum_{(i,j)\in E}c_{ij}x_{ij}^{p}
\end{equation}
\begin{equation}
\sum_{p=1}^{H}\sum_{(i,j)\in \delta (S)}x_{ij}^p\geq 1 \hspace{1cm}\forall S \subseteq V, S\cap Q\neq \emptyset , (V\backslash S)\cap Q\neq \emptyset
\end{equation}
\begin{equation}
x_{ij}^p\in \{0, 1\} \hspace{1cm}\forall (i, j)\in E
\end{equation}
The goal is to find a subgraph with minimum cost due to the hop constraints; Constraints in (2) present the connectivity constraints of basic vertices attaching to at least one edge of the vertices from $S$ and index $p$ is used to avoid creation of any cycle.
\section{Minimum-Hop Iterative Greedy (MinHIG) Algorithm}
\label{}
In this section, we present an iterative greedy algorithm for HCST problem. The basic idea is from \cite{14}. The first three steps of the algorithm use the generalization of prime algorithm \cite{12} considering the path with number of hop instead of a direct edge between two vertices (Algorithm 1).
We start with a partial solution $G=({root},\emptyset)$ that just contains the root. As we mentioned before, $Q$ is the set of all basic vertices. At each step the set $T$ is equal to $Q\backslash G$ and $H$ is the maximum allowed number of hops that the root can connects other vertices in the tree.
Suppose $V_G$ denote the vertex set of $G$ and $V_T$ denote the vertex set of $T$. Cost of path $p(u,v)$ between vertices $u$ and $v$ is presented by $d_{uv}$. Also, for every vertex $v$, we define $U_v$ equal to the number of needed hops to reach $v$ from the root. For initialization, we set $U_{root}$ to zero. Phase 1 in MinHIG algorithm finds $U_v$ for all vertices $v$.
\begin{algorithm}[H]
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{An undirected graph.}
\Output{Find U values.}
{\bf Step 1}. {\bf Initialization} $G=(\{root\},\emptyset)$;
\While{$Q\not\subseteq G$}{
{\bf Step 2}. {\bf Find} vertices $u^*\in V_G$, $v^*\in V_T$ and path $P(u^*,v^*)$, where
$U_v^* = U_u^* +|P(u^*,v^*)|\leq H$,
$d_{u^* v^*}=min\{d_{uv} |u\leq V_G ,v^*\in V_T\}$;
{\bf Step 3}. {\bf Add} the vertices and edges of path $P(u^*,v^*)$ to G,
{\bf update} U values of the vertices of path $P(u^*,v^*)$.
}
\caption{MinHIG algorithm (Phase 1)}
\end{algorithm}
\begin{algorithm}[H]
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{U values.}
\Output{An efficient feasible solution.}
{\bf Step 1}. {\bf Initialization} $Tree=(\{root\},\emptyset)$, h=1;
\While{$h\neq H$}{
{\bf Step 2}. {\bf Find} vertices $u^*\in Q$, $v^*\in V_{Tree}$, and path $P(u^*,v^*)$, where
$U_u^*=h$,
$d_{u^* v^*}=min\{d_{u^*v} |v\in V_{Tree}, U_{u^*v^*} = U_u^* +|P(u^*,v^*)|\leq H\}$;
{\bf Step 3}. {\bf Add} the vertices and edges of path $P(u^*,v^*)$ to Tree.
$h\leftarrow h+1$
}
\caption{MinHIG algorithm (Phase 2)}
\end{algorithm}
Note that in MinHIG algorithm, we suppose that in the efficient optimal solution, vertices with high $U$ values are connected to vertices with low $U$ values.
In phase 2 in MinHIG, first we add the root vertex to the final tree $Tree$.
Then, we start with $h=1$ and and add it until reach maximum hop constraint $h=H$. Vertices with fewer number of hops have been added to the tree earlier. Therefor, with constraint $U_u^* +|P(u^*,v^*)|\leq H$, we find $P(u^*,v^*)$, which is the shortest path among all paths of vertex $u^*$ to the all previous added vertices to $Tree$. Thus, all edges and vertices of this path will be added to the $Tree$. We repeat this procedure until $Tree$ contains all basic vertices.
Figure 1 shows an example of instances where MinHIG gets a solution while Voss's algorithm \cite{14} doesn't work.
\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{01.png}
\caption{An illustrative example to build HCST with 3 mandatory vertices and 3 Steiner vertices(Circles).}
\label{figNNfdsds}
\end{figure}
For Figure 1 consider maximum allowed $H$ is 3. With Voss's algorithm \cite{14}, at first, path $P(r\rightarrow 2\rightarrow 4\rightarrow 5)$ with cost 7 is added to the tree. Then path $P(r\rightarrow 1\rightarrow 3)$ with cost 8 will be added. The final Steiner tree is built has cost 15, although, it is obvious that the optimal cost of the Steiner tree is 10. In Step 3 in phase 2 of Algorithm 1 we have tree reconstruction which gives us the cost of 10 for this example.
Suppose that vertices with higher amount of $U$ should be connected to the vertices with lower $U$ values. Therefore, in Step 3 in phase 2, without considering set G, vertices with less U values are added to the final tree. In fact, when we add each vertex, we are sure that other vertices with lower $U$ have been added to the tree earlier and if the current vertex should be connected to one of the basic vertices, we are sure that all possible vertices have been already added to the tree.
With implementation of MinHIG on this specific example, after first phase values of $U_r=0, U_1=1, U_3= 2, U_2 =1, U_4= 2,$ and $U_5 =3$ have been obtained. After the root is added, basic vertex 3 with the lowest $U$ value among basic vertices with shortest path $P(r\rightarrow 1\rightarrow 3)$ and cost 8 will be added to the tree. Then, the next basic vertex which has not been added to the $Tree$ yet and has minimum value $U$ is vertex 5 with shortest path $P(3\rightarrow 5)$ and cost 2. At the end, the cost of $Tree$ would be 10.
\section{Maximum Hop Iterative Greedy (MaxHIG) Algorithm}
\label{}
In MinHIG algorithm, it was assumed that in the Steiner tree, the vertices with higher $U$ are connected to vertices with lower $U$ ,but this approach doesn't cover all groups of graphs. There are other kinds of graphs in which the vertices by lower hops are connected to the path of the other vertices with more hops.
For example, consider Figure 2 where maximum allowed number of $H$ is 4.
\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{02}
\caption{An illustrative example to build HCST with 4 mandatory vertices and 6 Steiner vertices}
\label{figNNfdsds}
\end{figure}
By MinHIG algorithm, first we add path $P(r\rightarrow 2\rightarrow 6)$ with cost 5 to the $Tree$. Then, path $P(r\rightarrow 1\rightarrow 4\rightarrow 8)$ with cost 6 and path $P(r\rightarrow 3\rightarrow 5\rightarrow 7\rightarrow 9)$ with cost 8 will be added to the $Tree$ respectively. Therefor, the result cost of the achieved Steiner tree by this algorithm would be 19, while the optimal cost is 10.
\bigskip
\begin{algorithm}[H]
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{An undirected graph.}
\Output{An efficient feasible solution.}
{\bf Run Algorithm 1}
{\bf Step 1}. {\bf Initialization} $Tree=(\{root\},\emptyset)$, h=H;
\While{$h\neq 1$}{
{\bf Step 2}. {\bf Find} vertices $u^*\in Q$, $v^*\in V_{Tree}$, and path $P(u^*,v^*)$, where
$U_u^*=h$,
$d_{u^* v^*}=min\{d_{u^*v} |v\in V_{Tree}, U_{u^*v^*} = U_u^* +|P(u^*,v^*)|\leq H\}$.
{\bf Step 3}. {\bf Add} the vertices and edges of path $P(u^*,v^*)$ to Tree.
}
\caption{MaxHIG algorithm}
\end{algorithm}
\bigskip
For these types of graphs we substitute algorithm MinHIG phase 2 (Algorithm 2) with MaxHIG algorithm (Algorithm3). The following is implementation of algorithm MaxHIG on the example of Figure 2:
First we start with vertex 9 which has the maximum $U$, $U_9=4$, and add path $P(3\rightarrow 5\rightarrow 7\rightarrow 9)$ with cost 8 to the tree $Tree$. In the next step, vertex 8 with $U_8=3$ and cost 1 will be added to the tree. Then, the path $P(3\rightarrow 6)$ with cost 1 is added to the tree. The final result cost of the $Tree $ would be 10.
%
In fact, these two greedy algorithms are helpful for two different types of trees with different features. The first category that is solved with MinHIG are graphs, which in their optimal Steiner tree the basic vertices with high amount hops, $U$, are connected to the basic vertices with lower $U$. The latter that can be solved with MaxHIG are graphs that in their optimal Steiner tree, some of the basic vertices are connected to the path of other basic vertices. To obtain Steiner tree in a given arbitrary graph, we run both algorithms and the best answer is considered as the final answer (we will see in last section that in most cases the result by combining of these two algorithms is better than the algorithm by Voss in \cite{14}).
The complexity analysis of presented algorithms is as follows:\\
-$O(QEH)$ is needed to find the shortest path between basic vertices. \\
-$O(ELogE)$ is need for Prim algorithm implementation.\\
-$O(QV)$ is need for adding a basic vertex at each time and the comparison to all vertices inside $Tree$.\\
\section{Non Root Based Insertion (NRBI) Algorithm}
\label{}
In this part, first we give an informal overview of NRBI algorithm and then we present the analysis in details.
The algorithm uses generalized idea of both Prime and Kruskal algorithm. We use new variable $itr_v$ (Algorithm 4) to every basic vertex $v$ as time of entrance to the set $G$. Again here We assume that $itr_{root}$ is zero at first and then the first basic vertex $v$ entered to set $G$ after root has $itr_v=1$.
In phase 2 of NRBI algorithm, basic vertices are sorted in descending order due to their $iter$ amount. Then, they would be added to the $Tree$ similar to idea of Kruskal algorithm. In the Voss's algorithm, the final tree is a subset of set $G$ constructed in the first phase. In MinHIG and MaxHIG algorithms, set $G$ is completely neglected and the final tree is created only based on $U$ values. In this algorithm, as we will see further, set $G$ is used as an auxiliary set. When vertex $v$ is added to the tree, there will be two cases:
1- Connecting vertex $v$ to vertices with less $itr$.
2- Connecting vertex $v$ to vertices, which will be added to $G$ after vertex $v$.
In the former one, the result is clear and vertex $v$ should be connected to same vertices that it was connected to in $G$. If the latter, result will not be clear. First we have to find all vertices with more $itr$ than $v$ to find the needed Steiner vertices and add them to the $Tree$ and then we add vertex $v$.
Suppose all vertices except basic vertex $v$ have been added in the optimal tree and now we have to add the last remaining vertex $v$ to the Steiner tree. For similar conditions without loss of generality, we assume that the tree is two pieces.
First one contains all vertices with $itr$ less than $k$ (this sub-tree may be a sub-forest and consists of many disconnected sub-trees) and second one is optimal sub-tree containing all vertices with $itr$ value more than $k$. Our goal is to connect $v$ to the first or second set in the best way. Each time, vertex $v$ is connected to one of the two pieces until we get the whole tree of all vertices especially vertex $v$. The optimal sub-tree of the first set of the vertices with $itr$ less than $k$ is the optimal sub-tree generated by the set $S$ containing all vertices added before $v$ to $S$.
\bigskip
\begin{algorithm}[H]
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{An undirected graph.}
\Output{U and itr values.}
{\bf Step 1}. {\bf Initialization} $G=(\{root\},\emptyset)$,
\While{$Q\not\subseteq S$}{
{\bf Step 2}. {\bf Find} vertices $u^*\in V_G$ and $v^*\in V_T$ and path P(u*,v*), where
$U_v^* = U_u^* +|P(u^*,v^*)|\leq H$,
$d_{u^* v^*}=min\{d_{uv} |u\leq V_G ,v^*\in V_T\}$;
{\bf Step 3}. {\bf Add} the vertices and edges of path $P(u^*,v^*)$ to G,
{\bf update} U values of the vertices of path $P(u^*,v^*)$,
{\bf Save} $itr[Basic Vertex\hspace{0.2cm} v \in P(u^*,v^* )]$.
}
\caption{NRBI (Phase 1)}
\end{algorithm}
\bigskip
\begin{algorithm}[H]
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{$U_i$ and itr. }
\Output{An efficient feasible solution.}
{\bf Initialization} $Tree=\{\emptyset\}$
\For{Maximum $Itr_v$ to 1}{
{\bf Find} $P(v,u^* )$ where
$u^*\in Tree$,
$U_u^*+|P(u^*, v )|\leq U_v$;
\If {$Cost(P(v,u^* ))<Cost(P(v,u^* )\in G)$}
{{\bf Add} all vertices and edges of $P(v,u^* )$ to Tree.}
\Else{ {\bf Add} all vertices and edges of $P(v,u^* )\in G$ to Tree.}
}
\caption{NRBI (Phase 2)}
\end{algorithm}
\bigskip
Now we want to find the most optimal connection of the vertex $v$ to the tree. So, in the second step, we try to make the best connections for optimal subtree of $v$. Since having $U$ values generated from the first phase of the generalized of prim algorithm, we can guarantee that the hop constraint is not violated. In the second part of the algorithm, by inspiring the idea of Kruskal algorithm, we allow to connect each vertex to other vertices without violating $U$ (Algorithm 5).
\paragraph{The analysis of NRBI algorithm}
Here we analyze our algorithm in details. We show that the output is tree and the constructed set doesn't include any cycle or cross. All basic vertices in $Q$ are connected together with $Q-1$ paths by idea of Kruskal algorithm. Because all $U$ values generated in the phase 1 are connected by this precondition that each vertex can only be connected to the vertex with less $U$, so no cycle would be created.
Since, in this problem we consider adding path instead of edge due to of the idea of Kruskal algorithm, we should show that no cross is gonna happened either.
In NRBI algorithm, we connect vertices based on their $itr$ in descending order. The cross happens when some vertices of some paths are connected to each other. Since we know that paths that are constructed by vertices with more $itr$. These vertices are the only ones capable of changing $U$ values of vertices that are produced before in path by lower $itr$ and reverse of it is not possible. Thus cross wont happen.
In the second phase of the algorithm, when we add vertices with more $itr$ to the forest respectively, we assure that during choosing a shortest path for each basic vertex, no cross will be created (No changes can happen in previous paths of the forest).
%
To find the closest vertex to the current $itr$, distances from all basic and Steiner vertices that have been added to the $Tree$ before are calculated. Thus, there wont be any possibility of creating the cross. If we assume that cross is created, so the vertex at the intersection has been added to the $Tree$ before, then the vertex with current $itr$ was closer to the vertex of the cross and this is in contrast to our assumption that we have connected the current vertex to the closest vertex with respect to hop constraint. Therefore, there is no cross in graph and the result is tree.
Time complexity of this algorithm in the first part is similar to the previous presented algorithms, equal to $O(ELogE)$. In the second part in tree construction, all vertices are connected to each other in the forest. Since for every basic vertex when we add it, it will be compared to all the Steiner and basic vertices in the tree, thus the running time is $O(QV)$. Before, in the generalized Bellman-Ford, all paths from vertices to each other were calculated in time $O(QEH)$ and stored in the table. The only cost that we nee to calculate to find the path is the time of adding selected path from the table for every vertex in $Q$ which is at most $H$. Therefor, the time complexity would be $O(QH)$ and the total time of last part of the algorithm is $O(QV$).
\begin{figure}[h!]
\centering
\includegraphics[scale=0.25]{03}
\caption{An illustrative example to build HCST with 4 mendatory vertices and 4 Steiner vertices}
\label{A}
\end{figure}
Figure 3 presents an example that if we apply all previous algorithms on that we obtain 31 as the optimal Steiner tree. Though here is the implementation of the NRBI algorithm on the same graph:
{\bf First Step:}
First, paths $P(r\rightarrow 1\rightarrow 2)$ with cost 8 and basic vertex 2, $P(2\rightarrow 4\rightarrow 6\rightarrow 7)$ with cost of 6 and basic vertex 7, and then path $P(r\rightarrow 3\rightarrow 5)$ with cost 17 are added to set $G$ respectively. Now we see that set $G$ contains all basic vertices.
The updated variables $itr$ and $U$ are as follows,
$itr_7=2, itr_5=3, itr_2=1,$ and $itr_0= 0$,
$U_5=2, U_2=2, U_7=5,$ and $U_0 = 0$
{\bf Second Step:}
So, we added basic vertices 0,2,5,7 to the tree.
Starting with the basic vertex with maximum $itr$, we select vertex 5. The shortest path between vertex 5 and existed vertices $u^*$ in set $Tree$ with $U_u^*+|P(u^*,5)|\leq U_5$ is $P(r\rightarrow 3\rightarrow 5)$ . All edges and vertices of this path are added to the tree (This path is the same path of vertex 5 in set $G$).
The next basic vertex is the one with itr = 2, which is vertex 7.
The best way to connect vertex 7 to the one of the vertices in the set $Tree$ is path $P(2\rightarrow 4\rightarrow 6\rightarrow 7)$ with cost 6. For next vertex with $itr=1$, vertex 2, the best way to connect it to one of the vertices of the tree is $P(2\rightarrow 3)$ with cost 7. Although the best path for 2 in $G$ is $P(r\rightarrow 1\rightarrow 2)$ with cost 8, we add vertex 2 with cost 7. The optimal Steiner tree with cost 30 is obtained.
Note, the tree at the beginning of the second phase is not connected same as idea of Kruskal algorithm when we add edges to find the shortest path, but with adding $Q-1$ paths, at the end the Steiner tree would be a connected tree.
\section{Computation}
\label{}
In order to assess the performance of all proposed algorithms, we used instances $\{c, d\}$n, $n\in \{5,10,15,20\}$ randomly chosen from the OR-Library\footnote{
http://people.brunel.ac.uk/mastjjb/jeb/orlib/steininfo.html}[6].
The size of these instances for ST problem have been defined between 500 and 1000 nodes from 625 up to 25000 edges (see \cite{105} for more details about these instances).
In general, ST instances generate HCST input graphs in this way that ST files provide edge-costs and Steiner vertices so that we select 200 and 300 basic vertices due to random vectors. All algorithms have been implemented in C++ \footnote{https://github.com/Farzaneh9696/HC-Steiner} and run on all instances, sparse graphs $\{c5,c10,d5,d10\}$ and dense graphs $\{c15,c20,d15,d20\}$.
For the sake of quality of all algorithms for every sample library, we generate number of vectors equal to 100 times of the hop limitation $H$ to specify different group of basic vertices. For example, to implement the algorithms on instances for $H=3$, we generate 300 vectors to create different groups of basic vertices.
All algorithms have been implemented on every vector on every instances. Therefore, the number of implementation for every algorithm on specified hop is equal to $100 H \times 16$.
We compare all of our algorithms with algorithm in \cite{14} (Named in the table Voss). In the tables, MinHIG algorithm, MaxHIG algorithm, and NRBI algorithm are represented as MinH, MaxH, and NRBI respectively. We also combine first two algorithms, MinHIG and MaxHIG, and show the results of this combination as MM in our tables.
Tables 1 to 8 compare results of algorithms as follows:
First algoritm vs second algorithm: {\bf FOS,\hspace{0.2cm} SFOS,\hspace{0.2cm} SOF, and\hspace{0.2cm} SSOF}
{ \bf FOS} = represents number of times that results of first algorithm is better than the second one.
{ \bf SFOS} = sum of the cost amount that the first algorithm is less than the second one.
{\bf SOF}= represents the number of times that results of the second algorithm is better than the first one.
{\bf SSOF}= sum of the cost amount that the second algorithm is less than the first one.
In the following, we show that how we compare results of every two algorithms in every row of the table:
For example: Voss vs MinH: 6\hspace{0.2cm} 20 \hspace{0.2cm} 224\hspace{0.2cm} 2739 means that in this comparison the Voss's algorithm is 6 times better than MinH and the sum of the difference of their cost is 20. MinH is 224 times better than Voss and the sum of the difference of their cost is 2739.
Tests are based on the benchmarks c and d with 200 and 300 basic vertices for 3, 5, 7, and 10 hop. Tables 1 to 4 present our computational results on sparse sets $\{c5,d5\}$ and $\{c10,d10\}$ with 200 and 300 basic vertices, respectively. We focus on NRBI and MM to show that they are the best ones among all five algorithms, but all algorithms are compared two each other one by one and results are provided in tables 1 to 8.
For $H=3$, in 8.54$\%$ of 300 vectors, NRBI is better than Voss algorithm. The cost difference is 69.12 in average. In the rest percent of the vectors, the cost calculation for both Voss and NRBI are the same, and in no cases Voss could get better than NRBI. For this hop limitation, We also saw MM is better than Voss with cost difference 49.37 in 5.79$\%$ of 300 vectors and in 2.5$\%$ vectors, Voss is better with cost difference 21.62 . In the remaining ones, two algorithms found the same cost.
For $H=5$, NRBI in 67.37$\%$ of 500 vectors is better compared to the Voss's algorithm, with cost difference 12940.11 in average. In the remaining percent of the vectors the cost calculations for both Voss and NRBI are equal, and in no cases Voss was seen better than NRBI. In this hop, we also found that in 52.27$\%$ of 500 vectors, MM is better than Voss with cost difference 5879. In 14.2$\%$ vectors Voss is better with cost difference 638 and in the remaining ones two algorithms found the result with similar cost.
For $H=7$, NRBI in 99.71$\%$ of 700 vectors is better than Voss with cost difference 34029.62 in average and for the rest of the vectors, the minimum costs found by Voss and NRBI are equal. In no cases Voss is better than NRBI. In this hop, in 63.14$\%$ of 700 vectors MM is better than Voss with cost difference 9040.12 and in 34.30$\%$ with cost 3604.37 vector Voss is better. The remaining vectors, two algorithms calculated same cost.
\begin{table}[H]
\begin{center}
\begin{adjustbox}{width=\textwidth,totalheight=\textheight,keepaspectratio}
\begin{tabular}{cc c c c|c c c c c}
\caption{Steiner results on c5 and d5 instances with 200 basic vertices}
\\
Methods Comparison&\multicolumn{4}{c}{c5} & \multicolumn{4}{|c}{d5}&Number of Hop \\
\hline
&FOS &SFOS &SOF &SSOF &FOS &SFOS &SOF &SSOF& \\
\cline{2-9}
Voss vs MinH: &0 &0 &0 &0 &0 &0 &0 &0 & \\
Voss vs MaxH: &0 &0 &0 &0 &0 &0 &0 &0 & \\
Voss vs NRBI: &0 &0 &0 &0 &0 &0 &0 &0 &\\
Voss vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MinH vs MaxH: &0 &0 &0 &0 &0 &0 &0 &0 &H=3\\
MinH vs NRBI: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
NRBI vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &96 &638 &6 &20 &224 &2739 &\\
Voss vs MaxH: &0 &0 &96 &638 &22 &118 &219 &2655 &\\
Voss vs NRBI: &0 &0 &96 &638 &0 &0 &231 &2805 &\\
Voss vs MM: &0 &0 &96 &638 &4 &16 &224 &2741 &\\
MinH vs MaxH: &0 &0 &0 &0 &33 &188 &3 &6 &H=5\\
MinH vs NRBI: &0 &0 &0 &0 &0 &0 &21 &86 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &3 &6 &\\
MaxH vs NRBI: &0 &0 &0 &0 &0 &0 &49 &268 &\\
MaxH vs MM: &0 &0 &0 &0 &0 &0 &33 &188 &\\
NRBI vs MM: &0 &0 &0 &0 &18 &80 &0 &0 &\\
\hline
Voss vs MinH: &222 &1689 &448 &5801 &281 &3297 &399 &5753 &\\
Voss vs MaxH: &382 &4682 &302 &3422 &441 &7373 &240 &2814 &\\
Voss vs NRBI: &0 &0 &691 &14835 &0 &0 &697 &19983 &\\
Voss vs MM: &201 &1379 &470 &6515 &248 &2714 &428 &6258 &\\
MinH vs MaxH: &524 &6396 &155 &1024 &512 &8103 &115 &1088 &H=7\\
MinH vs NRBI: &24 &68 &662 &10791 &11 &26 &683 &17553 &\\
MinH vs MM: &0 &0 &155 &1024 &0 &0 &115 &1088 &\\
MaxH vs NRBI: &9 &48 &684 &16143 &0 &0 &698 &24542 &\\
MaxH vs MM: &0 &0 &524 &6396 &0 &0 &512 &8103 &\\
NRBI vs MM: &651 &9811 &31 &112 &682 &16465 &11 &26 &\\
\hline
Voss vs MinH: &296 &5671 &697 &18648 &563 &13046 &422 &8136 &\\
Voss vs MaxH: &762 &27232 &231 &4129 &921 &42692 &73 &1065 &\\
Voss vs NRBI: &0 &0 &1000 &49505 &0 &0 &1000 &46771 &\\
Voss vs MM: &279 &4712 &715 &19223 &553 &12259 &432 &8400 &\\
MinH vs MaxH: &894 &37614 &101 &1534 &902 &37768 &92 &1051 &H=10\\
MinH vs NRBI: &13 &75 &986 &36603 &1 &6 &999 &51687 &\\
MinH vs MM: &0 &0 &101 &1534 &0 &0 &92 &1051 &\\
MaxH vs NRBI: &1 &7 &999 &72615 &0 &0 &1000 &88398 &\\
MaxH vs MM: &0 &0 &894 &37614 &0 &0 &902 &37768 &\\
NRBI vs MM: &985 &35076 &14 &82 &999 &50636 &1 &6 &\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\end{table}
For $H=10$, in 99.75$\%$ of 1000 vectors, NRBI could find a better solution than Voss with cost difference 9683 in average. In the remaining percent of the vectors Voss and NRBI are equal. In 39.95$\%$ of 1000 vectors, MM is better than Voss with cost difference 38545.75. Also, Voss is better in 58.67$\%$ vectors with cost difference 1789.75.Remaining had same results.
\begin{table}[H]
\begin{center}
\begin{adjustbox}{width=\textwidth,totalheight=\textheight,keepaspectratio}
\begin{tabular}{cc c c c|c c c c c}
\\
Methods Comparison&\multicolumn{4}{c}{c5} & \multicolumn{4}{|c}{d5}&Number of Hop \\
\hline
&FOS &SFOS &SOF &SSOF &FOS &SFOS &SOF &SSOF& \\
\cline{2-9}
Voss vs MinH: &0 &0 &0 &0 &0 &0 &0 &0 &\\
Voss vs MaxH: &0 &0 &0 &0 &0 &0 &0 &0 &\\
Voss vs NRBI: &0 &0 &0 &0 &0 &0 &0 &0 &\\
Voss vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MinH vs MaxH: &0 &0 &0 &0 &0 &0 &0 &0 &H=3\\
MinH vs NRBI: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
NRBI vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
\hline
Voss vs MinH: &94 &518 &46 &264 &7 &22 &52 &459 &\\
Voss vs MaxH: &160 &670 &48 &269 &64 &230 &250 &2929 &\\
Voss vs NRBI: &0 &0 &116 &540 &0 &0 &253 &3107 &\\
Voss vs MM: &74 &365 &54 &304 &1 &3 &252 &2950 &\\
MinH vs MaxH: &109 &340 &39 &193 &67 &248 &220 &2510 &H=5\\
MinH vs NRBI: &6 &36 &152 &830 &0 &0 &222 &2670 &\\
MinH vs MM: &0 &0 &39 &193 &0 &0 &220 &2510 &\\
MaxH vs NRBI: &7 &34 &217 &975 &0 &0 &112 &408 &\\
MaxH vs MM: &0 &0 &109 &340 &0 &0 &67 &248 &\\
NRBI vs MM: &130 &642 &7 &41 &46 &160 &0 &0 &\\
\hline
Voss vs MinH: &291 &2909 &387 &5108 &171 &1623 &513 &8615 &\\
Voss vs MaxH: &358 &5280 &317 &4029 &350 &5619 &340 &5118 &\\
Voss vs NRBI: &0 &0 &698 &17979 &0 &0 &698 &19602 &\\
Voss vs MM: &221 &1922 &453 &6487 &134 &1080 &548 &9751 &\\
MinH vs MaxH: &424 &5816 &247 &2366 &515 &9172 &170 &1679 &\\
MinH vs NRBI: &5 &13 &693 &15793 &19 &80 &673 &12690 &H=7\\
MinH vs MM: &0 &0 &247 &2366 &0 &0 &170 &1679 &\\
MaxH vs NRBRI: &5 &13 &693 &19243 &5 &26 &693 &20129 &\\
MaxH vs MM: &0 &0 &424 &5816 &0 &0 &515 &9172 &\\
NRBI vs MM: &687 &13436 &9 &22 &666 &11037 &24 &106 &\\
\hline
Voss vs MinH: &277 &5560 &718 &25206 &380 &7035 &598 &14452 &\\
Voss vs MaxH: &711 &27168 &279 &7472 &784 &31241 &207 &3655 &\\
Voss vs NRBI: &0 &0 &1000 &64750 &0 &0 &1000 &62950 &\\
Voss vs MM: &270 &5223 &725 &25720 &352 &6131 &628 &15403 &\\
MinH vs MaxH: &925 &40193 &71 &851 &866 &36858 &126 &1855 &H=10\\
MinH vs NRBI: &14 &92 &984 &45196 &0 &0 &999 &55533 &\\
MinH vs MM: &0 &0 &71 &851 &0 &0 &126 &1855 &\\
MaxH vs NRBI: &1 &1 &999 &84447 &0 &0 &1000 &90536 &\\
MaxH vs MM: &0 &0 &925 &40193 &0 &0 &866 &36858 &\\
NRBI vs MM: &983 &44346 &15 &93 &999 &53678 &0 &0 &\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\caption{Steiner results on c5 and d5 instances with 300 basic vertices}
\end{table}
\begin{table}[H]
\begin{center}
\begin{adjustbox}{width=\textwidth,totalheight=\textheight,keepaspectratio}
\begin{tabular}{cc c c c|c c c c c}
\\
Methods Comparison&\multicolumn{4}{c}{c10} & \multicolumn{4}{|c}{d10}&Number of Hop \\
\hline
&FOS &SFOS &SOF &SSOF &FOS &SFOS &SOF &SSOF& \\
\cline{2-9}
Voss vs MinH: &21 &70 &18 &64 &0 &0 &0 &0 &\\
Voss vs MaxH: &29 &109 &32 &105 &0 &0 &0 &0 &\\
Voss vs NRBI: &0 &0 &60 &166 &0 &0 &0 &0 &\\
Voss vs MM: &16 &46 &37 &120 &0 &0 &0 &0 &\\
MinH vs MaxH: &18 &78 &24 &80 &0 &0 &0 &0 &H=3\\
MinH vs NRBI: &0 &0 &63 &172 &0 &0 &0 &0 &\\
MinH vs MM: &0 &0 &24 &80 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &57 &170 &0 &0 &0 &0 &\\
MaxH vs MM: &0 &0 &18 &78 &0 &0 &0 &0 &\\
NRBI vs MM: &39 &92 &0 &0 &0 &0 &0 &0 &\\
\hline
Voss vs MinH: &39 &327 &456 &14096 &264 &3296 &222 &2386 &\\
Voss vs MaxH: &175 &2562 &317 &6787 &233 &2722 &257 &2533 &\\
Voss vs NRBI: &0 &0 &500 &28924 &0 &0 &500 &12442 &\\
Voss vs MM: &28 &200 &468 &14740 &196 &2019 &293 &3284 &\\
MinH vs MaxH: &416 &10315 &76 &771 &215 &1454 &247 &2175 &H=5\\
MinH vs NRBI: &1 &3 &498 &15158 &0 &0 &498 &13352 &\\
MinH vs MM: &0 &0 &76 &771 &0 &0 &247 &2175 &\\
MaxH vs NRBI: &0 &0 &499 &24699 &1 &1 &498 &12632 &\\
MaxH vs MM: &0 &0 &416 &10315 &0 &0 &215 &1454 &\\
NRBI vs MM: &497 &14387 &1 &3 &496 &11178 &1 &1 &\\
\hline
Voss vs MinH: &375 &6439 &309 &4436 &497 &17432 &195 &5241 &\\
Voss vs MaxH: &680 &34162 &19 &150 &347 &9000 &344 &9944 &\\
Voss vs NRBI: &0 &0 &700 &23774 &0 &0 &700 &52782 &\\
Voss vs MM: &374 &6287 &310 &4450 &311 &7199 &378 &11278 &\\
MinH vs MaxH: &682 &32175 &17 &166 &178 &3135 &517 &16270 &H=7\\
MinH vs NRBI: &1 &3 &699 &25780 &0 &0 &700 &64973 &\\
MinH vs MM: &0 &0 &17 &166 &0 &0 &517 &16270 &\\
MaxH vs NRBI: &0 &0 &700 &57786 &0 &0 &700 &51838 &\\
MaxH vs MM: &0 &0 &682 &32175 &0 &0 &178 &3135 &\\
NRBI vs MM: &699 &25614 &1 &3 &700 &48703 &0 &0 &\\
\hline
Voss vs MinH: &834 &14577 &149 &1182 &986 &56132 &12 &150 &\\
Voss vs MaxH: &1000 &97700 &0 &0 &999 &101492 &1 &23 &\\
Voss vs NRBI: &0 &0 &982 &13866 &0 &0 &1000 &39112 &\\
Voss vs MM: &834 &14577 &149 &1182 &985 &54496 &13 &173 &\\
MinH vs MaxH: &999 &84305 &0 &0 &903 &47146 &93 &1659 &H=10\\
MinH vs NRBI: &0 &0 &997 &27261 &0 &0 &1000 &95094 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &93 &1659 &\\
MaxH vs NRBI: &0 &0 &1000 &111566 &0 &0 &1000 &140581 &\\
MaxH vs MM: &0 &0 &999 &84305 &0 &0 &903 &47146 &\\
NRBI vs MM: &997 &27261 &0 &0 &1000 &93435 &0 &0 &\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\caption{Steiner results on c10 and d10 instances with 200 basic vertices}
\end{table}
Tables 5 to 8 present our computational results on dense sets $\{c15,d15\}$ and $\{c20,d20\}$ with 200 and 300 basic vertices, respectively. When $H=3$, NRBI is better than Voss in 100$\%$ of 300 vectors. The cost difference is 9683 in average. In the remaining part of the vectors Voss and NRBI both have equal result, and in no cases Voss is better than NRBI. In this hop, in 58.83$\%$ of 300 vectors, MM is better than Voss with cost difference 2407.12 and in 37.16$\%$ of them Voss gets better with cost 797. For rest of the vectors they calculate same cost.
\begin{table}[H]
\begin{center}
\begin{adjustbox}{width=\textwidth,totalheight=\textheight,keepaspectratio}
\begin{tabular}{cc c c c|c c c c c}
\\
Method Comparison&\multicolumn{4}{c}{c10} & \multicolumn{4}{|c}{d10}&Number of Hop \\
\hline
&FOS &SFOS &SOF &SSOF &FOS &SFOS &SOF &SSOF& \\
\cline{2-9}
Voss vs MinH: &68 &266 &51 &79 &0 &0 &0 &0 &\\
Voss vs MaxH: &87 &311 &91 &247 &0 &0 &0 &0 &\\
Voss vs NRBI: &0 &0 &145 &387 &0 &0 &0 &0 &\\
Voss vs MM: &44 &127 &102 &275 &0 &0 &0 &0 &\\
MinH vs MaxH: &59 &212 &93 &335 &0 &0 &0 &0 &H=3\\
MinH vs NRBI: &0 &0 &154 &574 &0 &0 &0 &0 &\\
MinH vs MM: &0 &0 &93 &335 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &116 &451 &0 &0 &0 &0 &\\
MaxH vs MM: &0 &0 &59 &212 &0 &0 &0 &0 &\\
NRBI vs MM: &81 &239 &0 &0 &0 &0 &0 &0 &\\
\hline
Voss vs MinH: &20 &125 &473 &20236 &278 &3889 &210 &2631 &\\
Voss vs MaxH: &181 &3499 &314 &7960 &281 &3901 &207 &2101 &\\
Voss vs NRBI: &0 &0 &500 &36101 &0 &0 &499 &18964 &\\
Voss vs MM: &16 &101 &479 &20437 &218 &2686 &272 &3302 &\\
MinH vs MaxH: &472 &15875 &24 &225 &253 &2416 &218 &1874 &H=5\\
MinH vs NRBI: &1 &1 &497 &15991 &0 &0 &500 &20222 &\\
MinH vs MM: &0 &0 &24 &225 &0 &0 &218 &1874 &\\
MaxH vs NRBI: &0 &0 &500 &31640 &0 &0 &500 &20764 &\\
MaxH vs MM: &0 &0 &472 &15875 &0 &0 &253 &2416 &\\
NRBI vs MM: &497 &15766 &1 &1 &500 &18348 &0 &0 &\\
\hline
Voss vs MinH: &194 &2423 &491 &10725 &260 &6758 &435 &15972 &\\
Voss vs MaxH: &694 &48742 &3 &69 &520 &22808 &176 &4618 &\\
Voss vs NRBI: &0 &0 &700 &31022 &0 &0 &700 &92260 &\\
Voss vs MM: &194 &2421 &491 &10730 &238 &5833 &458 &16852 &\\
MinH vs MaxH: &698 &56982 &2 &7 &591 &29209 &105 &1805 &H=7\\
MinH vs NRBI: &1 &5 &699 &22725 &0 &0 &700 &83046 &\\
MinH vs MM: &0 &0 &2 &7 &0 &0 &105 &1805 &\\
MaxH vs NRBI: &0 &0 &700 &79695 &0 &0 &700 &110450 &\\
MaxH vs MM: &0 &0 &698 &56982 &0 &0 &591 &29209 &\\
NRBI vs MM: &699 &22718 &1 &5 &700 &81241 &0 &0 &\\
\hline
Voss vs MinH : &560 &6496 &397 &3949 &861 &40240 &137 &2817 &\\
Voss vs MaxH : &1000 &127553 &0 &0 &996 &138991 &4 &119 &\\
Voss vs NRBI : &0 &0 &998 &20109 &0 &0 &1000 &68787 &\\
Voss vs MM : &560 &6496 &397 &3949 &861 &40024 &137 &2832 &\\
MinH vs MaxH : &1000 &125006 &0 &0 &989 &101680 &10 &231 &H=10\\
MinH vs NRBI: &1 &3 &998 &22659 &0 &0 &1000 &106210 &\\
MinH vs MM : &0 &0 &0 &0 &0 &0 &10 &231 &\\
MaxH vs NRBI: &0 &0 &1000 &147662 &0 &0 &1000 &207659 &\\
MaxH vs MM : &0 &0 &1000 &125006 &0 &0 &989 &101680 &\\
NRBI vs MM : &998 &22659 &1 &3 &1000 &105979 &0 &0 &\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\caption{Steiner results on c10 and d10 instances with 300 basic vertices}
\end{table}
\begin{table}[H]
\begin{center}
\begin{adjustbox}{width=\textwidth,totalheight=\textheight,keepaspectratio}
\begin{tabular}{cc c c c|c c c c c}
\\
Methods Comparison&\multicolumn{4}{c}{c15} & \multicolumn{4}{|c}{d15}&Number of Hop \\
\hline
&FOS &SFOS &SOF &SSOF &FOS &SFOS &SOF &SSOF& \\
\cline{2-9}
Voss vs MinH: &67 &677 &227 &4224 &39 &182 &250 &2380 &\\
Voss vs MaxH: &151 &2010 &137 &1738 &93 &578 &190 &1439 &\\
Voss vs NRBI: &0 &0 &300 &15814 &0 &0 &299 &5292 &\\
Voss vs MM: &61 &583 &231 &4389 &35 &169 &254 &2451 &\\
MinH vs MaxH: &255 &4078 &42 &259 &238 &1421 &20 &84 &H=3\\
MinH vs NRBI: &0 &0 &299 &12267 &1 &1 &293 &3095 &\\
MinH vs MM: &0 &0 &42 &259 &0 &0 &20 &84 &\\
MaxH vs NRBI: &0 &0 &300 &16086 &1 &3 &298 &4434 &\\
MaxH vs MM: &0 &0 &255 &4078 &0 &0 &238 &1421 &\\
NRBI vs MM: &299 &12008 &0 &0 &292 &3014 &2 &4 &\\
\hline
Voss vs MinH: &13 &141 &486 &20420 &5 &22 &495 &22888 &\\
Voss vs MaxH: &304 &5007 &187 &2488 &37 &221 &459 &11779 &\\
Voss vs NRBI: &0 &0 &500 &21862 &0 &0 &500 &21389 &\\
Voss vs MM: &306 &5019 &185 &2434 &41 &241 &455 &11297 &\\
MinH vs MaxH: &488 &22864 &8 &66 &440 &11810 &56 &502 &H=5\\
MinH vs NRBI: &0 &0 &500 &19343 &0 &0 &500 &32947 &\\
MinH vs MM: &0 &0 &8 &66 &0 &0 &56 &502 &\\
MaxH vs NRBI: &0 &0 &500 &42141 &0 &0 &500 &44255 &\\
MaxH vs MM: &0 &0 &488 &22864 &0 &0 &440 &11810 &\\
NRBI vs MM: &500 &19277 &0 &0 &500 &32445 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &700 &64212 &1 &3 &699 &44764 &\\
Voss vs MaxH: &97 &649 &593 &10549 &6 &20 &694 &21112 &\\
Voss vs NRBI: &0 &0 &699 &12872 &0 &0 &700 &11019 &\\
Voss vs MM: &97 &649 &593 &10549 &6 &20 &694 &21016 &\\
MinH vs MaxH: &700 &54312 &0 &0 &680 &23765 &18 &96 &H=7\\
MinH vs NRBI: &0 &0 &700 &22772 &0 &0 &700 &32111 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &18 &96 &\\
MaxH vs NRBI: &0 &0 &700 &77084 &0 &0 &700 &55780 &\\
MaxH vs MM: &0 &0 &700 &54312 &0 &0 &680 &23765 &\\
NRBI vs MM: &700 &22772 &0 &0 &700 &32015 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &1000 &123718 &0 &0 &1000 &83140 &\\
Voss vs MaxH: &39 &171 &947 &18998 &0 &0 &1000 &34330 &\\
Voss vs NRBI: &0 &0 &980 &7150 &0 &0 &956 &5709 &\\
Voss vs MM: &39 &171 &947 &18998 &0 &0 &1000 &34322 &\\
MinH vs MaxH: &1000 &104891 &0 &0 &997 &48818 &3 &8 &H=10\\
MinH vs NRBI: &0 &0 &1000 &25977 &0 &0 &1000 &40039 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &3 &8 &\\
MaxH vs NRBI: &0 &0 &1000 &130868 &0 &0 &1000 &88849 &\\
MaxH vs MM: &0 &0 &1000 &104891 &0 &0 &997 &48818 &\\
NRBI vs MM: &1000 &25977 &0 &0 &1000 &40031 &0 &0 &\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\caption{Steiner results on c15 and d15 instances with 200 basic vertices}
\end{table}
For $H=5$, in 92.70$\%$ of 500 vectors NRBI result is better than what Voss got with cost difference 13012.25 in average. In the rest percent of the vectors, the min cost calculated by Voss and
NRBI are same. In no cases Voss is better than NRBI. For this hop, also in 77.12$\%$ of 500 vectors MM is better than Voss. The cost difference of them is 4101. In 20.8$\%$ of vectors Voss is better with cost difference 1657.62. For rest of the vectors two algorithms calculate same cost.
\begin{table}[H]
\begin{center}
\begin{adjustbox}{width=\textwidth,totalheight=\textheight,keepaspectratio}
\begin{tabular}{cc c c c|c c c c c}
\\
Methods Comparison&\multicolumn{4}{c}{c15} & \multicolumn{4}{|c}{d15}&Number of Hop \\
\hline
&FOS &SFOS &SOF &SSOF &FOS &SFOS &SOF &SSOF& \\
\cline{2-9}
Voss vs MinH: &29 &233 &268 &7410 &116 &690 &179 &1768 &\\
Voss vs MaxH: &131 &2032 &162 &3205 &168 &1560 &119 &1071 &\\
Voss vs NRBI: &0 &0 &300 &22042 &0 &0 &300 &7822 &\\
Voss vs MM: &27 &219 &271 &7594 &112 &648 &181 &1856 &\\
MinH vs MaxH: &270 &6202 &25 &198 &238 &1697 &45 &130 &H=3\\
MinH vs NRBI: &0 &0 &300 &14865 &0 &0 &300 &6744 &\\
MinH vs MM: &0 &0 &25 &198 &0 &0 &45 &130 &\\
MaxH vs NRBI: &0 &0 &300 &20869 &0 &0 &300 &8311 &\\
MaxH vs MM: &0 &0 &270 &6202 &0 &0 &238 &1697 &\\
NRBI vs MM: &300 &14667 &0 &0 &300 &6614 &0 &0 &\\
\hline
Voss vs MinH: &9 &158 &490 &34526 &2 &24 &498 &34875 &\\
Voss vs MaxH: &367 &7289 &120 &1084 &63 &592 &434 &12094 &\\
Voss vs NRBI: &0 &0 &500 &25233 &0 &0 &500 &27998 &\\
Voss vs MM: &367 &7289 &120 &1067 &64 &600 &433 &11977 &\\
MinH vs MaxH: &499 &40590 &1 &17 &481 &23474 &17 &125 &H=5\\
MinH vs NRBI: &0 &0 &500 &19028 &0 &0 &500 &39500 &\\
MinH vs MM: &0 &0 &1 &17 &0 &0 &17 &125 &\\
MaxH vs NRBI: &0 &0 &500 &59601 &0 &0 &500 &62849 &\\
MaxH vs MM: &0 &0 &499 &40590 &0 &0 &481 &23474 &\\
NRBI vs MM: &500 &19011 &0 &0 &500 &39375 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &700 &78845 &0 &0 &700 &59576 &\\
Voss vs MaxH: &250 &2070 &421 &4763 &17 &124 &677 &19948 &\\
Voss vs NRBI: &0 &0 &700 &15325 &0 &0 &700 &18595 &\\
Voss vs MM: &250 &2070 &421 &4763 &17 &124 &677 &19940 &\\
MinH vs MaxH: &700 &76152 &0 &0 &697 &39760 &1 &8 &H=7\\
MinH vs NRBI: &0 &0 &700 &18018 &0 &0 &700 &38419 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &1 &8 &\\
MaxH vs NRBI: &0 &0 &700 &94170 &0 &0 &700 &78171 &\\
MaxH vs MM: &0 &0 &700 &76152 &0 &0 &697 &39760 &\\
NRBI vs MM: &700 &18018 &0 &0 &700 &38411 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &1000 &145523 &0 &0 &1000 &114865 &\\
Voss vs MaxH: &90 &472 &892 &11882 &2 &4 &998 &36087 &\\
Voss vs NRBI: &0 &0 &992 &8653 &0 &0 &994 &9344 &\\
Voss vs MM: &90 &472 &892 &11882 &2 &4 &998 &36087 &\\
MinH vs MaxH: &1000 &134113 &0 &0 &1000 &78782 &0 &0 &H=10\\
MinH vs NRBI: &0 &0 &1000 &20063 &0 &0 &1000 &45427 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &1000 &154176 &0 &0 &1000 &124209 &\\
MaxH vs MM: &0 &0 &1000 &134113 &0 &0 &1000 &78782 &\\
NRBI vs MM: &1000 &20063 &0 &0 &1000 &45427 &0 &0 &\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\caption{Steiner results on c15 and d15 instances with 300 basic vertices}
\end{table}
\begin{table}[H]
\begin{center}
\begin{adjustbox}{width=\textwidth,totalheight=\textheight,keepaspectratio}
\begin{tabular}{cc c c c|c c c c c}
\\
Methods Comparison&\multicolumn{4}{c}{c20} & \multicolumn{4}{|c}{d20}&Number of Hop \\
\hline
&FOS &SFOS &SOF &SSOF &FOS &SFOS &SOF &SSOF& \\
\cline{2-9}
Voss vs MinH: &166 &706 &105 &412 &229 &1819 &57 &288 &\\
Voss vs MaxH: &286 &3298 &9 &30 &293 &4831 &6 &29 &\\
Voss vs NRBI: &0 &0 &300 &3484 &0 &0 &300 &6621 &\\
Voss vs MM: &164 &696 &107 &425 &229 &1808 &57 &293 &\\
MinH vs MaxH: &289 &2997 &9 &23 &289 &3287 &8 &16 &H=3\\
MinH vs NRBI: &0 &0 &299 &3778 &0 &0 &300 &8152 &\\
MinH vs MM: &0 &0 &9 &23 &0 &0 &8 &16 &\\
MaxH vs NRBI: &0 &0 &300 &6752 &0 &0 &300 &11423 &\\
MaxH vs MM: &0 &0 &289 &2997 &0 &0 &289 &3287 &\\
NRBI vs MM: &299 &3755 &0 &0 &300 &8136 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &500 &20868 &0 &0 &500 &21531 &\\
Voss vs MaxH: &14 &26 &475 &3163 &5 &17 &493 &5853 &\\
Voss vs NRBI: &0 &0 &380 &987 &0 &0 &493 &2560 &\\
Voss vs MM: &14 &26 &475 &3163 &5 &17 &493 &5852 &\\
MinH vs MaxH: &500 &17731 &0 &0 &499 &15696 &1 &1 &H=5\\
MinH vs NRBI: &0 &0 &499 &4124 &0 &0 &500 &8396 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &1 &1 &\\
MaxH vs NRBI: &0 &0 &500 &21855 &0 &0 &500 &24091 &\\
MaxH vs MM: &0 &0 &500 &17731 &0 &0 &499 &15696 &\\
NRBI vs MM: &499 &4124 &0 &0 &500 &8395 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &700 &34709 &0 &0 &700 &39599 &\\
Voss vs MaxH: &0 &0 &694 &5152 &1 &2 &699 &8067 &\\
Voss vs NRBI: &0 &0 &231 &275 &0 &0 &596 &1834 &\\
Voss vs MM: &0 &0 &694 &5152 &1 &2 &699 &8067 &\\
MinH vs MaxH: &700 &29557 &0 &0 &700 &31534 &0 &0 &H=7\\
MinH vs NRBI: &0 &0 &696 &5427 &0 &0 &700 &9899 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &700 &34984 &0 &0 &700 &41433 &\\
MaxH vs MM: &0 &0 &700 &29557 &0 &0 &700 &31534 &\\
NRBI vs MM: &696 &5427 &0 &0 &700 &9899 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &1000 &49957 &0 &0 &1000 &64645 &\\
Voss vs MaxH: &1 &1 &990 &7125 &0 &0 &1000 &13254 &\\
Voss vs NRBI: &0 &0 &263 &307 &0 &0 &325 &465 &\\
Voss vs MM: &1 &1 &990 &7125 &0 &0 &1000 &13254 &\\
MinH vs MaxH: &1000 &42833 &0 &0 &1000 &51391 &0 &0 &H=10\\
MinH vs NRBI: &0 &0 &994 &7431 &0 &0 &1000 &13719 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &1000 &50264 &0 &0 &1000 &65110 &\\
MaxH vs MM: &0 &0 &1000 &42833 &0 &0 &1000 &51391 &\\
NRBI vs MM: &994 &7431 &0 &0 &1000 &13719 &0 &0 &\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\caption{Steiner results on c20 and d20 instances with 200 basic vertices}
\end{table}
For $H=7$, NRBI in 75.50$\%$ of 700 vectors is better than Voss with cost difference 7608.3 in average. In the remained percent of the vectors the minimum costs calculated by both Voss and NRBI are equal. In no cases Voss is better than NRBI. For this hop, in 91.6$\%$ of 700 vectors MM is better than Voss with cost difference 10210.5 and in 5.32$\%$ vectors Voss is better with cost difference 1090.37.
\begin{table}[H]
\begin{center}
\begin{adjustbox}{width=\textwidth,totalheight=\textheight,keepaspectratio}
\begin{tabular}{cc c c c|c c c c c}
\\
Methods Comparison&\multicolumn{4}{c}{c20} & \multicolumn{4}{|c}{d20}&Number of Hop \\
\hline
&FOS &SFOS &SOF &SSOF &FOS &SFOS &SOF &SSOF& \\
\cline{2-9}
Voss vs MinH: &59 &299 &228 &1763 &206 &1875 &83 &481 &\\
Voss vs MaxH: &288 &4347 &8 &27 &290 &5431 &8 &51 &\\
Voss vs NRBI: &0 &0 &300 &5951 &0 &0 &300 &10438 &\\
Voss vs MM: &59 &293 &228 &1766 &205 &1860 &83 &483 &\\
MinH vs MaxH: &296 &5793 &3 &9 &284 &4003 &6 &17 &H=3\\
MinH vs NRBI: &1 &1 &299 &4488 &0 &0 &300 &11832 &\\
MinH vs MM: &0 &0 &3 &9 &0 &0 &6 &17 &\\
MaxH vs NRBI: &0 &0 &300 &10271 &0 &0 &300 &15818 &\\
MaxH vs MM : &0 &0 &296 &5793 &0 &0 &284 &4003 &\\
NRBI vs MM: &299 &4479 &1 &1 &300 &11815 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &500 &29162 &0 &0 &500 &27343 &\\
Voss vs MaxH: &23 &31 &440 &1950 &12 &38 &484 &5848 &\\
Voss vs NRBI: &0 &0 &339 &684 &0 &0 &496 &3386 &\\
Voss vs MM: &23 &31 &440 &1950 &12 &38 &484 &5848 &\\
MinH vs MaxH: &500 &27243 &0 &0 &500 &21533 &0 &0 &H=5\\
MinH vs NRBI: &0 &0 &488 &2603 &0 &0 &500 &9196 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &500 &29846 &0 &0 &500 &30729 &\\
MaxH vs MM: &0 &0 &500 &27243 &0 &0 &500 &21533 &\\
NRBI vs MM: &488 &2603 &0 &0 &500 &9196 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &700 &45658 &0 &0 &700 &54858 &\\
Voss vs MaxH: &6 &6 &654 &2886 &1 &2 &699 &9311 &\\
Voss vs NRBI: &0 &0 &216 &253 &0 &0 &391 &694 &\\
Voss vs MM: &6 &6 &654 &2886 &1 &2 &699 &9311 &\\
MinH vs MaxH: &700 &42778 &0 &0 &700 &45549 &0 &0 &H=7\\
MinH vs NRBI: &1 &1 &672 &3134 &0 &0 &700 &10003 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &700 &45911 &0 &0 &700 &55552 &\\
MaxH vs MM: &0 &0 &700 &42778 &0 &0 &700 &45549 &\\
NRBI vs MM: &672 &3134 &1 &1 &700 &10003 &0 &0 &\\
\hline
Voss vs MinH: &0 &0 &1000 &66914 &0 &0 &1000 &81809 &\\
Voss vs MaxH: &9 &9 &935 &3952 &0 &0 &1000 &13594 &\\
Voss vs NRBI: &0 &0 &257 &309 &0 &0 &393 &491 &\\
Voss vs MM: &9 &9 &935 &3952 &0 &0 &1000 &13594 &\\
MinH vs MaxH: &1000 &62971 &0 &0 &1000 &68215 &0 &0 &H=10\\
MinH vs NRBI: &0 &0 &959 &4252 &0 &0 &1000 &14085 &\\
MinH vs MM: &0 &0 &0 &0 &0 &0 &0 &0 &\\
MaxH vs NRBI: &0 &0 &1000 &67223 &0 &0 &1000 &82300 &\\
MaxH vs MM: &0 &0 &1000 &62971 &0 &0 &1000 &68215 &\\
NRBI vs MM: &959 &4252 &0 &0 &1000 &14085 &0 &0 &\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\caption{Steiner results on c20 and d20 instances with 300 basic vertices}
\end{table}
For $H=10$, NRBI gets better min cost than Voss in 64.48$\%$ of 1000 vectors. The cost difference of them is 4053.5 in average. In the remaining percent of the vectors the min costs returned by both Voss and NRBI are equal. Voss on no cases is better than NRBI. For this hop, we also saw that in 97.02$\%$ of 1000 vectors, MM is better than Voss with cost difference 17401.75 and in 1.76$\%$ vectors Voss is better with cost difference 82.12. On the remaining of vectors, both algorithms found same cost.
\begin{table}[H]
\begin{center}
\begin{adjustbox}{width=\textwidth,totalheight=\textheight,keepaspectratio}
\begin{tabular}{cc c c c c c c c c c}\\
\hline
& &&\multicolumn{2}{c}{MinH} &\multicolumn{2}{c}{MaxH} &\multicolumn{2}{c}{MM} &\multicolumn{2}{c}{NRBI} \\
\cline{4-11}
Inctances&Hop &Voss &MinCost &Imp$\%$ &MinCost &Imp$\%$ &MinCost &Imp$\%$ &MinCost &Imp$\%$ \\
\hline
&3& 92.91 &90.24 &2.87 &90.18 &2.93 &90.12 &3 &89.80 &3.34\\
$\{c5,d5\}$& 5 &517.63 &508.81 &1.70 &514.58 &0.58 &494.26 &4.51 &492.47 &4.86\\
$\{c10,d10\}$& 7 &889.99 &867.31 &2.54 &870.79 &2.15 &831.67 &6.55 &825.64 &7.23\\
& 10 &1044.39 &972.10 &6.92 &981.38 &6.03 &944.72 &9.54 &928.45 &11.10\\
\hline
& 3 &446.06 &438.25 &1.75 &443.35 &0.6 &415.46 &6.86 &412.39 &7.54\\
$\{c15,d15\}$& 5 &420.25 &367.43 &12.56 &375.19 &10.72 &348.80 &17.00 &345.09 &17.88\\
$\{c20,d20\}$& 7 &378.12 &304.00 &19.60 &302.72 &19.94 &296.90 &21.40 &292.11 &22.70\\
& 10 &361.05 &265.70 &26.40 &356.03 &1.39 &24.08 &33.20 &235.907 &34.60\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\caption{Performances of proposed algorithms in comparison to Voss}
\end{table}
Now we report optimality of algorithms on two groups of graphs, sparse and dense. The analysis in Table 9 indicates that NRBI greedy algorithm performs quite well and consistent. The column $"Imp"$ indicates the amount of improvement over the minimum objective function of all proposed algorithms to Voss. As the density of a graph and number of hop increases, its performance improves. The average of improvements of NRBI to Voss on sparse graph instances are presented for all hops and it is in the range of 3.34 to 11.10.
The highest improvement on these graphs is when the hop is 10. Also, For dense graphs the improvement is between 7.54 to 34.60 when the hop is 10. This amount for MM which is combination of MinHIG and MaxHIG is between 3 to 9.54 on sparse graphs and 6.86 to 33.20 on dense graphs. Table 9 also shows all improvements of MinHIG and MaxHIG algorithms on both sparse and dense instances with hops 3, 5, 7, and 10.
Figure 4 shows an average decrease of cost of all algorithms on all different hops, where one can observe that the average amount of improvement on all hops for MM and NRBI algorithm to Voss.
\begin{figure}[H]
\centering
\includegraphics[scale=0.45]{chart.png}
\caption{Average of Minimum cost found with algorithms on H=3, 5, 7, 10.}
\label{figNNfdsds}
\end{figure}
\section{Conclusion}
\label{}
In this paper we have proposed three greedy algorithms to solve Steiner tree problem with hop constraint, which is a important class of network design. The basic ideas of first two greedy algorithms, MinHIG and MaxHIG, were limited to prim algorithm. They were good on some graphs, but we could not arrive to near optimal solution on all kind of graphs. In fact, the structure of graphs has effect on these algorithms' solutions, since the algorithms are root based.
Furthermore, we proposed an algorithm ,NRBI, which expand the idea of Kruskal algorithm. The algorithm seemed to be quite robust even when more generalized problems are considered, such as having different hop constraints for all basic vertices. In addition, some comprehensive analysis of all proposed algorithms and the comparison to the Voss's algorithm has been shown that NRBI algorithm arrived to best solution almost in all cases. The improvement is significantly 34.60$\%$ in the best case when hop is 10 on dense graphs and in the worst case 3.34$\%$ for hop 3 on sparse graphs on these problems. An interesting future work could be use of heuristic algorithms to try to improve the feasible solutions of the proposed greedy algorithms.
|
1,116,691,499,364 | arxiv | \section{Introduction}\label{sec:1}
We are concerned with numerical methods for Zakai equations,
linear stochastic partial differential equations of the form
\begin{equation}
\label{eq:1.1}
\begin{split}
du(t,x) = L_0u(t,x)dt + \sum_{k=1}^mL_ku(s,x)dW_k(t),
\quad 0\le t\le T,
\end{split}
\end{equation}
with initial condition $u(0,x)=u_0(x)$,
where the process $\{W(t)=(W_1(t),\ldots,W_m(t))\}_{0\le t\le T}$ is an $m$-dimensional
standard Wiener process on a complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$.
Here, for each $k=0,1,\ldots, m$, the partial differential operator $L_k$ is given by
\begin{align*}
L_0f(x)&=\frac{1}{2}\sum_{i,j=1}^d\frac{\partial^2}{\partial x_i\partial x_j}
(a_{ij}(x)f(x))
+ \sum_{i=1}^d\frac{\partial}{\partial x_i}(b_i(x)f(x)), \\
L_{k}f(x) &= \beta_{k}(x)f(x) +\sum_{i=1}^d\frac{\partial}{\partial x_i}
(\gamma_{ik}(x)f(x)), \quad k=1,\ldots, m,
\end{align*}
where $a=(a_{ij})$ is $\mathbb{R}^{d\times d}$-valued,
$b=(b_i)$ is $\mathbb{R}^d$-valued,
$\beta=(\beta_k)$ is $\mathbb{R}^{m}$-valued,
$\gamma=(\gamma_{ik})$ is $\mathbb{R}^{d\times m}$-valued,
and $u_0$ is $\mathbb{R}$-valued,
all of which are defined on $\mathbb{R}^d$.
The conditions for these functions are described in Section \ref{sec:2} below.
It is well known that solving Zakai equations is amount to computing the optimal filter for
diffusion processes.
We refer to Rozovskii \cite{roz:1990}, Kunita \cite{kun:1990},
Liptser and Shiryaev \cite{lip-shi:2001}, Bensoussan \cite{ben:1992},
Bain and Crisan \cite{bai-cri:2009}, and the references therein
for Zakai equations and their relation with nonlinear filtering.
It is also well known that for linear diffusion processes the optimal filters allow for
finite dimensional realizations, i.e., they can be represented by
some stochastic and deterministic differential equations in finite dimensions.
For nonlinear diffusion processes, it is difficult to obtain such realizations except
some special cases (see Bene{\v{s}} \cite{ben:1981} and \cite{ben:1992}).
Thus one may be led to numerical approach to Zakai equations for computing the optimal filter.
Several efforts have been made to obtain approximation methods for the
equations during the past several decades. For example, the finite difference method
(see Yoo \cite{yoo:1999}, Gy{\"o}ngy \cite{gyo:2014} and the references therein),
the particle method (see Crisan et al.~\cite{cri-etal:1998}),
a series expansion approach (Lototsky et al.~\cite{lot-etal:1997}),
Galerkin type approximation (Ahmed and Radaideh \cite{ahm-rad:1997} and
Frey et al.~\cite{fre-etal:2013})
and the splitting up method (Bensoussan et al.~\cite{ben-etal:1990}).
In the present paper, we examine the approximation of $u(t,x)$ by a collocation method with
kernel-based interpolation. Given a points set
$\Gamma=\{x_1,\ldots,x_N\}\subset\mathbb{R}^d$ and a positive definite function
$\Phi:\mathbb{R}^d\to\mathbb{R}$, the function
\[
I(f)(x):=\sum_{j=1}^N(A^{-1}f|_{\Gamma})_j\Phi(x-x_j), \quad x\in\mathbb{R}^d,
\]
interpolates $f$ on $\Gamma$. Here, $A=\{\Phi(x_j-x_{\ell})\}_{j,\ell=1,\ldots,N}$,
$f|_{\Gamma}$ is the column vector composed of $f(x_j)$,
$j=1,\ldots,N$, and
$(A^{-1}z)_j$ denotes the $j$-th component of $A^{-1}z$ for $z\in\mathbb{R}^N$.
Thus, with time grid $\{t_0,\ldots,t_n\}$, the function $u^h$ recursively defined by
\begin{align*}
u^h(t_i,x) &= u^h(t_{i-1},x)+L_0I(u^h(t_{i-1},\cdot)(x)(t_i-t_{i-1}) \\
&\quad + \sum_{k=1}^mL_kI(u^h(t_{i-1},\cdot))(x)(W_k(t_i)-W_k(t_{i-1})),
\quad i=0,\ldots,n,\;\; x\in \mathbb{R}^d,
\end{align*}
is a good candidate for an approximate solution of \eqref{eq:1.1}.
The approximation above can be seen as a kernel-based (or meshfree) collocation method for
stochastic partial differential equations.
The meshfree collocation method is proposed by Kansa \cite{kan:1990b},
where deterministic partial differential equations are concerned.
Since then many studies on numerical experiments and practical applications
for this method are generated. As for rigorous convergence,
Schaback \cite{sch:2010} and Nakano \cite{nak:2017} study the case of deterministic linear
operator equations and fully nonlinear parabolic equations, respectively.
However, at least for parabolic equations, there is little known about
explicit examples of the grid structure and kernel functions that ensure rigorous convergence.
An exception is Hon et.al \cite{hon-etal:2014}, where
an error bound is obtained for a special heat equation in one dimension.
A main difficulty lies in handling the process of the iterative kernel-based interpolation.
A straightforward estimates for $|I(f)(x)|$ involves the condition number of the matrix $A$,
which in general rapidly diverges to infinity (see Wendland \cite{wen:2010}).
Thus we need to take a different route.
Our main idea is to introduce a condition on the decay of
the inverse of the interpolation matrix and to choose an appropriate
approximation domain whose radius goes to infinity such that the interpolation is still effective.
From this together with standard argument in error estimation we find that
the approximation error is bounded by the order of the square root of the time step and
the error that comes from a single step interpolation.
See Lemma \ref{lem:3.7} and Theorem \ref{thm:3.2} below.
The structure of this paper is as follows:
Section \ref{sec:2} introduces some notation, and
describes the basic results for Zakai equations and
the kernel-based interpolation, which are used in this paper.
We derive an approximation method for Zakai equations and
prove its convergence in
Section \ref{sec:3}. Numerical experiments are performed in Section \ref{sec:4}.
\section{Preliminaries}\label{sec:2}
\subsection{Notation}\label{sec:2.1}
Throughout this paper,
we denote by $a^{\mathsf{T}}$ the transpose of a vector or matrix $a$.
For $a=(a_i)\in\mathbb{R}^{\ell}$
we set $|a|=(\sum_{i=1}^{\ell}(a_{i})^2)^{1/2}$.
For a multiindex $\alpha=(\alpha_1,\ldots,\alpha_d)$ of nonnegative integers,
the differential operator $D^{\alpha}$ is defined as usual by
\begin{equation*}
D^{\alpha}=\frac{\partial^{|\alpha|_1}}
{\partial x_1^{\alpha_1}\cdots \partial x_d^{\alpha_d}}
\end{equation*}
with $|\alpha|_1=\alpha_1+\cdots +\alpha_d$.
For an open set $\mathcal{O}\subset\mathbb{R}^d$, we denote
by $C^{\kappa}(\mathcal{O})$ the space of continuous real-valued functions
on $\mathcal{O}$ with continuous derivatives up to the order $\kappa\in\mathbb{N}$,
with the norm
\[
\|f\|_{C^{\kappa}(\mathcal{O})}=\max_{|\alpha|_1\le\kappa}\sup_{x\in\mathcal{O}}
|D^{\alpha}f(x)|.
\]
Further, we denote by $C^{\infty}_0(\mathbb{R}^d)$
the space of infinitely differentiable functions on $\mathbb{R}^d$ with compact supports.
For any $p\in [1,\infty)$ and any open set $\mathcal{O}\subset\mathbb{R}^d$,
we denote by $L^p(\mathcal{O})$ the space of all measurable functions
$f:\mathcal{O}\to \mathbb{R}$ such that
\[
\|f\|_{L^p(\mathcal{O})}:=\left\{\int_{\mathcal{O}}|f(x)|^pdx\right\}^{1/p}<\infty.
\]
For $\kappa\in\mathbb{N}$,
we write $H^{\kappa}(\mathcal{O})$ for the space of all measurable functions
$f$ on $\mathcal{O}$ such that
the generalized derivatives $D^{\alpha}f$ exist for all $|\alpha|_1\le \kappa$ and
that
\begin{equation*}
\|f\|_{H^{\kappa}(\mathcal{O})}^2
:= \sum_{|\alpha|_1\le\kappa}\|D^{\alpha}f\|^2_{L^2(\mathcal{O})}<\infty.
\end{equation*}
In addition, for $0< r<1$,
we write $H^{\kappa+r}(\mathcal{O})$ for the space of all measurable functions
$f$ on $\mathcal{O}$ such that
the generalized derivatives $D^{\alpha}u$ exist for all $|\alpha|_1\le \kappa$ and
that $\|f\|_{H^{\kappa+r}(\mathcal{O})}^2
:= \|f\|_{H^{\kappa}(\mathcal{O})}^2 + |f|_{H^{\kappa+r}(\mathcal{O})}^2<\infty$ with
\begin{equation*}
|f|_{H^{\kappa +r}(\mathcal{O})}^2
=\sum_{|\alpha|_1=\kappa}
\int_{\mathcal{O}}\int_{\mathcal{O}}\frac{|D^{\alpha}f(x)-D^{\alpha}f(y)|^2}
{|x-y|^{d+2r}}dxdy.
\end{equation*}
For $x\in\mathbb{R}$ we use the notation $\lfloor x\rfloor=\max\{n\in\mathbb{Z}: n\le x\}$.
By $C$ we denote positive constants that may vary from line to line
and that are independent of $h$ introduced below.
\subsection{Zakai equations}\label{sec:2.2}
We impose the following conditions on the coefficients of the equation \eqref{eq:1.1}:
\begin{assum}
\label{assum:2.1}
\begin{enumerate}[\rm (i)]
\item All components of the functions $a$, $b$, $\beta$, $\gamma$,
and $u_0$ are infinitely differentiable with bounded continuous derivatives of any order.
\item For any $x\in\mathbb{R}^d$,
\begin{equation*}
\xi^{\mathsf{T}}(a(x)-\gamma(x)\gamma(x)^{\mathsf{T}})\xi\ge 0, \quad \xi\in\mathbb{R}^d.
\end{equation*}
\end{enumerate}
\end{assum}
It follows from Assumption \ref{assum:2.1} and
Gerencs{\'e}r et.al~\cite[Theorem 2.1]{ger-etal:2015} that
there exists a unique predictable process
$\{u(t)\}_{0\le t\le T}$ such that the following are satisfied:
\begin{enumerate}[\rm (i)]
\item $u(t,\cdot,\omega)\in H^{\nu}(\mathbb{R}^d)$ for
any $(t,\omega)\in [0,T]\times\Omega_0$, where $\Omega_0\in\mathcal{F}$ with
$\mathbb{P}(\Omega_0)=1$ and for any $\nu\in\mathbb{N}$;
\item for $\varphi\in C^{\infty}_0(\mathbb{R}^d)$,
\begin{equation}
\label{eq:2.1}
\begin{split}
(u(t),\varphi) = (u_0,\varphi) + \int_0^t(u(s,\cdot),L_0^*\varphi)ds
+ \sum_{k=1}^m\int_0^t(u(s,\cdot),L_k^*\varphi)dW_k(s),
\quad 0\le t\le T, \;\;\text{a.s.}
\end{split}
\end{equation}
\end{enumerate}
Here, $(\cdot,\cdot)$ denotes the inner product in $L^2(\mathbb{R}^d)$, and
for each $k=0,1,\ldots, m$, the partial differential operator $L_{k}^*$ is the
formal adjoint of $L_k$.
Moreover, $u(t,x)$ satisfies
\begin{equation*}
\mathbb{E}\left[\sup_{0\le t\le T}\|u(t,\cdot)\|^2_{H^{\nu}(\mathbb{R}^d)}\right]
\le C\|u_0\|^2_{H^{\nu}(\mathbb{R}^d)}, \quad \nu\in\mathbb{N}.
\end{equation*}
Further, as in \cite[Proposition 3, Section 1.3, Chapter 4]{roz:1990},
there exists a version $\tilde{u}$, with
respect to $x$, of $u$ such that $\tilde{u}(t,x,\omega)\in C^{\infty}(\mathbb{R}^d)$ for
$(t,\omega)\in [0,T]\times \Omega$ and that for any $\kappa\in\mathbb{N}$ and
$|\alpha|_1\le\kappa$,
\begin{equation}
\label{eq:2.2}
D^{\alpha}\tilde{u}(t,x)=D^{\alpha}u_0(x)+\int_0^tD^{\alpha}L_0\tilde{u}(s,x)dx
+ \sum_{k=1}^m\int_0^t
D^{\alpha}L_k\tilde{u}(s,x)dW_k(s), \quad\text{a.s.}, \;\; (t,x)\in [0,T]\times\mathbb{R}^d.
\end{equation}
In particular, $\tilde{u}$ is a solution to the Zakai equation in the strong sense, i.e.,
$\tilde{u}$ satisfies
\begin{equation*}
\tilde{u}(t,x)=\tilde{u}_0(x)+\int_0^tL_0\tilde{u}(s,x)dx + \sum_{k=1}^m\int_0^t
L_k\tilde{u}(s,x)dW_k(s), \quad\text{a.s.}, \;\; (t,x)\in [0,T]\times\mathbb{R}^d.
\end{equation*}
We remark that in (\ref{eq:2.2}) the stochastic integral is taken to be a continuous version
with respect to $(t,x)$. With this version, \eqref{eq:2.2} holds with
probability one uniformly on $[0,T]\times\mathbb{R}^d$.
\subsection{Kernel-based interpolation}\label{sec:2.3}
In this subsection, we recall the basis of the interpolation theory with
positive definite functions.
We refer to \cite{wen:2010} for a complete account.
Let $\Phi: \mathbb{R}^d\to \mathbb{R}$ be a radial and positive definite function,
i.e., $\Phi(\cdot)=\phi(|\cdot|)$ for some $\phi:[0,\infty)\to\mathbb{R}$ and
for every $\ell\in\mathbb{N}$, for all pairwise distinct
$y_1,\ldots, y_{\ell}\in\mathbb{R}^{d}$ and for all
$\alpha=(\alpha_i)\in\mathbb{R}^{\ell}\setminus\{0\}$, we have
\begin{equation*}
\sum_{i,j=1}^{\ell}\alpha_i\alpha_j\Phi(y_i-y_j)>0.
\end{equation*}
Let $\Gamma=\{x_1,\cdots,x_N\}$ be a finite subset of $\mathbb{R}^d$ and
put $A=\{\Phi(x_i-x_j)\}_{1\le i,j\le N}$.
Then $A$ is invertible and thus for any $g:\mathbb{R}^d\to\mathbb{R}$ the function
\begin{equation*}
I(g)(x) = \sum_{j=1}^N(A^{-1}(g|_{\Gamma}))_j\Phi(x-x_j), \quad x\in\mathbb{R}^d,
\end{equation*}
interpolates $g$ on $\Gamma$.
If $\Phi\in C(\mathbb{R}^d)\cap L^1(\mathbb{R}^d)$, then
\begin{equation*}
\mathcal{N}_{\Phi}(\mathbb{R}^d):=\left\{f\in C(\mathbb{R}^d)\cap L^1(\mathbb{R}^d):
\widehat{f}/\sqrt{\widehat{\Phi}}\in L^2(\mathbb{R}^d)\right\},
\end{equation*}
called the native space, is a real Hilbert space with the inner product
\begin{equation*}
(f,g)_{\mathcal{N}_{\Phi}(\mathbb{R}^d)}=(2\pi)^{-d/2}\int_{\mathbb{R}^d}
\frac{\widehat{f}(\xi)\overline{\widehat{g}(\xi)}}{\widehat{\Phi}(\xi)}d\xi
\end{equation*}
and the norm
$\|f\|_{\mathcal{N}_{\Phi}(\mathbb{R}^d)}^2:=(f,f)_{\mathcal{N}_{\Phi}(\mathbb{R}^d)}$.
Here, for $f\in L^1(\mathbb{R}^d)$, the function $\widehat{f}$ is the Fourier transform
of $f$, defined as usual by
\begin{equation*}
\widehat{f}(\xi)=(2\pi)^{-d/2}\int_{\mathbb{R}^d}f(x)e^{-\sqrt{-1}x^{\mathsf{T}}\xi} dx,
\quad \xi\in\mathbb{R}^d.
\end{equation*}
Moreover, $\mathbb{R}^d\times\mathbb{R}^d\ni (x,y)\mapsto \Phi(x-y)$
is a reproducing kernel for $\mathcal{N}_{\Phi}(\mathbb{R}^d)$.
If $\widehat{\Phi}$ satisfies
\begin{equation}
\label{eq:2.3}
c_1(1+|\xi|^2)^{-\kappa}\le \widehat{\Phi}(\xi)\le c_2(1+|\xi|^2)^{-\kappa},
\quad \xi\in\mathbb{R}^d,
\end{equation}
for some constants $c_1,c_2>0$ and $\kappa>d/2$,
then we have from Corollary 10.13 in \cite{wen:2010} that
$H^{\kappa}(\mathbb{R}^d)= \mathcal{N}_{\Phi}(\mathbb{R}^d)$ and
\begin{equation}
\label{eq:2.4}
c_1\|f\|_{H^{\kappa}(\mathbb{R}^d)}\le \|f\|_{\mathcal{N}_{\Phi}(\mathbb{R}^d)}
\le c_2\|f\|_{H^{\kappa}(\mathbb{R}^d)}, \quad f\in H^{\kappa}(\mathbb{R}^d).
\end{equation}
Namely, the native space $\mathcal{N}_{\Phi}(\mathbb{R}^d)$
coincides with the Sobolev space $H^{\kappa}(\mathbb{R}^d)$ with equivalent norm.
Further, we mention that \eqref{eq:2.4} and Corollary 10.25 in \cite{wen:2010} implies
\begin{equation}
\label{eq:2.5}
\|I(g)\|_{H^{\kappa}(\mathbb{R}^d)}\le C\|g\|_{H^{\kappa}(\mathbb{R}^d)}, \quad
\|g-I(g)\|_{H^{\kappa}(\mathbb{R}^d)}\le C\|g\|_{H^{\kappa}(\mathbb{R}^d)},
\quad g\in H^{\kappa}(\mathbb{R}^d).
\end{equation}
The so-called Wendland kernel is a typical example of $\Phi$ satisfying
\eqref{eq:2.3}--\eqref{eq:2.5}, which is defined as follows: for a given $\tau\in\mathbb{N}$,
set the function $\Phi_{d,\tau}$ satisfying
$\Phi_{d,\tau}(x)=\phi_{d,\tau}(|x|)$, $x\in\mathbb{R}^d$, where
\begin{equation*}
\phi_{d,\tau}(r)=\int_r^{\infty}r_{\tau}\int_{r_{\tau}}^{\infty}r_{\tau-1}\int_{r_{\tau-1}}^{\infty}
\cdots\: r_2\int_{r_2}^{\infty}r_1\max\{1-r_1,0\}^{\nu}dr_1dr_2\cdots dr_{\tau}, \quad
r\ge 0
\end{equation*}
with $\nu =\lfloor d/2\rfloor +\tau+1$.
For example,
\begin{align*}
\phi_{1,2}(r)&\doteq \max\{1-r,0\}^5(8r^2+5r+1), \\
\phi_{1,3}(r)&\doteq \max\{1-r,0\}^7(21r^3 + 19r^2 + 7r + 1), \\
\phi_{1,4}(r) &\doteq \max\{1-r,0\}^9(384r^4 + 453r^3 + 237r^2 + 63r + 7), \\
\phi_{2,4}(r) &\doteq \max\{1-r,0\}^{10}(429r^4+450r^3+210r^2+50r+5), \\
\phi_{2,5}(r) &\doteq
\max\{1-r,0\}^{12}(2048r^5 + 2697r^4 + 1644r^3 + 566r^2 + 108r + 9),
\end{align*}
where $\doteq$ denotes equality up to a positive constant factor.
Then, $\Phi_{d,\tau}\in C^{2\tau}(\mathbb{R}^d)$ and
$\mathcal{N}_{\Phi_{d,\tau}}(\mathbb{R}^d)=H^{\tau+(d+1)/2}(\mathbb{R}^d)$.
Furthermore, $\Phi_{d,\tau}$ satisfies \eqref{eq:2.3}--\eqref{eq:2.5} with
$\kappa=\tau+(d+1)/2$.
\section{Collocation method for Zakai equations}\label{sec:3}
Let us describe the collocation methods for (\ref{eq:2.1}).
In what follows, we always consider the version of $u$, and thus by abuse of notation,
we write $u$ for $\tilde{u}$.
Moreover, we restrict ourselves to the class of Wendland kernels
$\Phi=\Phi_{d,\tau}$ described in Section \ref{sec:2.2}.
Suppose that the open rectangle $(-R,R)^d$ for some $R>0$ is the set of points at which
the approximate solution is to be computed.
We choose a set $\Gamma=\{x_1,\ldots,x_N\}$ consisting of
pairwise distinct points such that
\begin{equation*}
\Gamma=\{x_1,\ldots,x_N\}\subset (-R,R)^d.
\end{equation*}
To construct an approximate solution of Zakai equation,
we first take a set $\{t_0,\ldots,t_n\}$ of time discretized points such that
$0=t_0<t_1<\cdots<t_n=T$.
The solution $u$ of the Zakai equation approximately satisfies
\begin{equation*}
u(t_i,x)\simeq u(t_{i-1},x) + \sum_{k=0}^mL_ku(t_{i-1},x)\Delta W_k(t_i),
\end{equation*}
where $W_0(t)=t$ and $\Delta W_k(t_i)=W_k(t_i)-W_k(t_{i-1})$.
Since
$L_ku(t_{i-1},x)\simeq L_kI(u(t_{i-1},\cdot))(x)$, we see
\begin{equation*}
u(t_i,x)\simeq u(t_{i-1},x) + \sum_{k=0}^m L_kI(u(t_{i-1},\cdot))(x)\Delta W_k(t_i).
\end{equation*}
Thus, we define the function $u^h$, a candidate of an approximate solution
parametrized with a parameter $h>0$, by
\begin{equation}
\label{eq:3.1}
\begin{aligned}
u^{h}(t_0,x)&=u_0(x), \quad x\in\mathbb{R}^d, \\
u^{h}(t_i,x)&=u^{h}(t_{i-1},x)+\sum_{k=0}^mL_k
I(u^h(t_{i-1},\cdot))(x)
\Delta W_k(t_i), \quad x\in\mathbb{R}^d, \;\; i=1,\ldots,n.
\end{aligned}
\end{equation}
With this definition, the $N$-dimensional vector
$u_i^{h}=(u_{i,1}^{h},\ldots, u_{i,N}^{h})^{\mathsf{T}}$ of the collocation points
satisfies
\begin{equation*}
\begin{aligned}
u_0^{h}&=(u_0(x_1),\ldots,u_0(x_N))^{\mathsf{T}}, \\
u_i^{h} &= u_{i-1}^{h} + \sum_{k=0}^m(A_kA^{-1}u_{i-1}^{h}) \Delta W_k(t_i),
\quad i=1,\ldots,n.
\end{aligned}
\end{equation*}
Here, we have set $A_k=(A_{k,j\ell})_{1\le j,\ell\le N}$ with
$A_{k,j\ell}=L_k\Phi(x-x_{\ell})|_{x=x_j}$.
This follows from
\begin{equation*}
L_ku^{h}(t_i,x_j)=\sum_{\ell=1}^N(A^{-1}u_i^{h})_{\ell}L_k\Phi(x-x_{\ell})|_{x=x_j}
=(A_kA^{-1}u_i^{h})_j.
\end{equation*}
To discuss the error of the approximation above, set
$\Delta t=\max_{1\le i\le n}(t_i-t_{i-1})$ and consider
the Hausdorff distance $\Delta_1 x$ between $\Gamma$ and $(-R,R)^d$,
and the separation distance
$\Delta_2 x$ defined respectively by
\begin{equation*}
\Delta_1 x = \sup_{x\in (-R,R)^d}\min_{j=1,\ldots,N}|x-x_j|, \quad
\Delta_2 x = \frac{1}{2}\min_{i\neq j}|x_i-x_j|.
\end{equation*}
Then suppose that
$\Delta t$, $R$, $N$, $\Delta_1 x$ and $\Delta_2 x$ are functions of $h$.
In what follows, $\#\mathcal{K}$ denotes the cardinality of a finite set $\mathcal{K}$.
\begin{assum}
\label{assum:3.1}
\begin{enumerate}[\rm (i)]
\item The parameters $\Delta t$, $R$, $N$, and $\Delta_1x$ satisfy
$\Delta t\to 0$, $R\to \infty$, $N\to\infty$, and $\Delta_1x\to 0$ as $h\searrow 0$.
\item There exist $c_1,c_2,c_3,c_4$ and $\lambda$, positive constants independent of $h$,
such that for any $i=1,\ldots,N$,
\[
\#\left\{j\in\{1,\ldots,N\}: |(A^{-1})_{ij}|> c_1\frac{(\Delta_2x)^d}{N}\right\}
\le c_2(\Delta_2x)^{-\lambda d},
\]
and that
\begin{equation*}
c_3(\Delta_2x)^{-(1+\lambda)d}\le R^{1/2}\le c_4(\Delta_1 x)^{-(\tau-3/2)/d}.
\end{equation*}
\end{enumerate}
\end{assum}
\begin{rem}
\label{rem:3.1.5}
It is known that $\Delta_2x\le \Delta_1x$ holds (see Chapter 14 in \cite{wen:2010}).
Thus the condition $\Delta_1x\to 0$ implies $\Delta_2x\to 0$ as $h\searrow 0$.
\end{rem}
\begin{rem}
\label{rem:3.2}
Suppose that $\Gamma$ is quasi-uniform in the sense that
\[
c_5RN^{-1/d}\le \Delta_2 x\le c_6RN^{-1/d},
\quad c_5^{\prime}RN^{-1/d}\le \Delta_1x\le c_6^{\prime}RN^{-1/d}
\]
hold for some positive constants $c_5,c_6,c_5^{\prime},c_5^{\prime}$.
In this case,
a sufficient condition for which the latter part of Assumption \ref{assum:3.1} (ii) holds is
\begin{equation*}
c_7N^{(1-1/(1+2d(1+\lambda))\frac{1}{d}}\le R\le c_8 N^{(1-d/(d+2\tau-3))\frac{1}{d}}
\end{equation*}
with $\tau\ge 3/2 + (1+\lambda)d^2$, for some positive constants $c_7$ and $c_8$.
\end{rem}
The approximation error for the Zakai equation is
estimated as follows:
\begin{thm}
\label{thm:3.2}
Suppose that Assumptions $\ref{assum:2.1}$ and $\ref{assum:3.1}$ hold.
Suppose moreover that $\tau\ge 3$.
Then, there exists $h_0>0$ such that
\begin{equation*}
\max_{i=1,\ldots,n}\sup_{x\in (-R,R)^d}\mathbb{E}\left[|u(t_i,x)- u^{h}(t_i,x)|^2
\right] \le C\left(\Delta t+ (\Delta_1x)^{2\tau-3}\right), \quad h\le h_0.
\end{equation*}
\end{thm}
The rest of this section is devoted to the proof of Theorem \ref{thm:3.2}.
To this end, for every $x\in (-R,R)^d$, put
\begin{equation*}
\mathcal{I}(x)=\{i\in\{1,\ldots,N\}: |x-x_i|\le 1\}.
\end{equation*}
\begin{lem}
\label{lem:3.4}
Suppose that $\Delta_2x\le 1$. Then
there exists $\nu\in\mathbb{N}$ such that
\begin{equation*}
\#\mathcal{I}(x)\le \nu\lfloor {(\Delta_2x)}^{-d}\rfloor, \quad x\in (-R,R)^d.
\end{equation*}
\end{lem}
\begin{proof}
Fix $x\in (-R,R)^d$. Put $q=\Delta_2x$ for notational simplicity.
It follows from the definition of $\mathcal{I}(x)$ that
\begin{equation*}
\bigcup_{i\in\mathcal{I}(x)}B_{q/2}(x_i)\subset B_{3/2}(x),
\end{equation*}
where $B_r(y)$ denotes the closed ball centered at $y\in\mathbb{R}^d$ with radius $r\ge 0$.
Further, $\{B_{q/2}(x_i)\}_{i\in\mathcal{I}(x)}$ is disjoint.
Indeed, otherwise, there exist $y\in\mathbb{R}^d$, $i,j\in\mathcal{I}(x)$ such that
$|y-x_i|\le q/2$ and $|y-x_j|\le q/2$. This implies
$|x_i-x_j|\le q=(1/2)\min_{i^{'}\neq j^{'}}|x_{i^{'}}-x_{j^{'}}|$, and so
$x_i=x_j$. Since we have assumed that $x_i$'s are pairwise distinct, we have $i=j$.
Denote by $\mathrm{Leb}(A^{\prime})$ for the Lebesgue measure for $A^{\prime}$.
Then we have $C\#\mathcal{I}(x)q^d
= \mathrm{Leb}(\cup_{i\in\mathcal{I}(x)}B_{q/2}(x_i))\le
\mathrm{Leb}(B_{3/2}(x))$. Thus, $\#\mathcal{I}(x)\le \tilde{C}q^{-d}$ for some
$\tilde{C}>0$ that is independent of $x$.
\end{proof}
\begin{lem}
\label{lem:3.5}
Suppose that Assumption {\rm \ref{assum:3.1} (i)} and $\tau\ge 3$ hold. Then,
there exists $h_0>0$ such that
for any multi-index $\alpha$ with $|\alpha|_1\le 2$ and $f\in H^{\tau+(d+1)/2}(\mathbb{R}^d)$,
we have
\begin{equation*}
\|D^{\alpha}f-D^{\alpha}I(f)\|_{L^{\infty}(-R,R)^d}\le
C(\Delta_1x)^{\tau+1/2-|\alpha|_1}\|f\|_{H^{\tau+(d+1)/2}(\mathbb{R}^d)},
\quad h\le h_0.
\end{equation*}
\end{lem}
\begin{proof}
This result is reported in \cite[Corollary 11.33]{wen:2010} for more general domains.
However, a simple application of that result leads to an ambiguity of the dependence of
the constant $C$ on $R$.
Here we will confirm that we can take $C$ to be independent of $R$.
Let $f\in H^{\tau+(d+1)/2}(\mathbb{R}^d)$ with $f|_{\Gamma}=0$.
Set $\tilde{\Gamma}=\{x_1/R,\ldots,x_N/R\}$ and
$\tilde{f}(z)=f(Rz)$, $z\in (-1,1)^d$.
Then, $\tilde{f}|_{\tilde{\Gamma}}=f|_{\Gamma}=0$ and
\begin{equation*}
\sup_{z\in (-1,1)^d}\min_{\xi\in\tilde{\Gamma}}|\xi-y|
=\sup_{y\in (-R,R)^d}\min_{j=1,\ldots,N}\left|\frac{x_j}{R}-\frac{y}{R}\right|
=\frac{\Delta_1x}{R}.
\end{equation*}
Since $\Delta_1x/R\to 0$ as $h\searrow 0$ and $\tau\ge 3$,
we can apply \cite[Theorem 11.32]{wen:2010} to
$\tilde{f}$ to obtain
\begin{equation}
\label{eq:3.2}
|D^{\alpha}\tilde{f}(z)|\le C(\Delta_1x/R)^{\tau+1/2-|\alpha|_1}
|\tilde{f}|_{H^{\tau+(d+1)/2}((-1,1)^d)}, \quad h\le h_0
\end{equation}
for some $h_0>0$.
It is straightforward to see that
\[
D^{\alpha}\tilde{f}(z)=R^{|\alpha|_1}(D^{\alpha}f)(Rz), \quad
|\tilde{f}|_{H^{\tau+(d+1)/2}((-1,1)^d)}=R^{\tau+1/2}|f|_{H^{\tau+(d+1)/2}((-R,R)^d)}.
\]
Substituting these relations into \eqref{eq:3.2}, we have
\begin{equation}
\label{eq:3.3}
|D^{\alpha}f(y)|\le C(\Delta_1x)^{\tau+1/2-|\alpha|_1}|f|_{H^{\tau+(d+1)/2}((-R,R)^d)},
\quad y\in (-R,R)^d.
\end{equation}
This and \eqref{eq:2.5} yield
\begin{align*}
\|D^{\alpha}f-D^{\alpha}I(f)\|_{L^{\infty}((-R,R)^d)}
&\le C(\Delta_1x)^{\tau+1/2-|\alpha|_1}|f-I(f)|_{H^{\tau+(d+1)/2}((-R,R)^d)} \\
&\le C(\Delta_1x)^{\tau+1/2-|\alpha|_1}\|f\|_{H^{\tau+(d+1)/2}(\mathbb{R}^d)}.
\end{align*}
Thus the lemma follows.
\end{proof}
Observe that for any $f:\mathbb{R}^d\to\mathbb{R}$,
\begin{equation*}
I(f)(x)=\sum_{j=1}^N(A^{-1}f|_{\Gamma})_j\Phi(x-x_j)
= \sum_{j=1}^Nf(x_j)Q_j(x), \quad x\in\mathbb{R}^d,
\end{equation*}
where
\begin{equation*}
Q_j(x)=\sum_{i=1}^N(A^{-1})_{ij}\Phi(x-x_i), \quad j=1,\ldots,N.
\end{equation*}
The following result tells us that the process of iterative kernel-based interpolation is stable,
which is a key to our convergence analysis.
\begin{lem}
\label{lem:3.7}
Suppose that Assumption $\ref{assum:3.1}$ and $\tau\ge 3$ hold. Then,
there exists $h_0>0$ such that
\begin{equation*}
\sup_{0<h\le h_0}\sup_{x\in (-R,R)^d}\sum_{j=1}^N|D^{\alpha}Q_j(x)|<\infty, \quad |\alpha|_1\le 2.
\end{equation*}
\end{lem}
\begin{proof}
Fix $\tilde{x}\in (-R,R)^d$ and $|\alpha|_1\le 2$.
Set $q=\Delta_2x$ for simplicity.
First consider the set
\begin{equation*}
\mathcal{J}(\tilde{x}):=\left\{j\in\{1,\ldots,N\}: |(A^{-1})_{ij}|\le c_1\frac{q^d}{N}, \;\;
i\in\mathcal{I}(\tilde{x})\right\}.
\end{equation*}
Then we have
\begin{equation*}
|(A^{-1})_{ij}|\le c_1\frac{q^d}{N}, \quad
j\in\mathcal{J}(\tilde{x}), \;\; i\in\mathcal{I}(\tilde{x}).
\end{equation*}
This together with Lemma \ref{lem:3.4} leads to
\begin{equation}
\label{eq:3.4}
\begin{split}
\sum_{j\in\mathcal{J}(\tilde{x})}|D^{\alpha}Q_j(\tilde{x})|
&=\sum_{j\in\mathcal{J}(\tilde{x})}|\sum_{i\in\mathcal{I}(\tilde{x})}(A^{-1})_{ij}D^{\alpha}
\Phi(\tilde{x}-x_i)|\le C\sum_{j\in\mathcal{J}(\tilde{x})}
\sum_{i\in\mathcal{I}(\tilde{x})}|(A^{-1})_{ij}| \\
&\le C\sum_{j\in\mathcal{J}(\tilde{x})}\sum_{i\in\mathcal{I}(\tilde{x})}\frac{q^d}{N}
\le CNq^{-d}\frac{q^d}{N}\le C.
\end{split}
\end{equation}
Now by Assumption \ref{assum:3.1} (ii) and Lemma \ref{lem:3.4},
there exists $\tilde{\nu}\in\mathbb{N}$ such that
\begin{equation*}
\#\{j: j\notin\mathcal{J}(\tilde{x})\}\le \tilde{\nu}\lfloor R^{1/2}\rfloor.
\end{equation*}
Then, by Kergin interpolation (see Kergin \cite{ker:1980})
there exists a polynomial $p$ on $\mathbb{R}^d$ with degree at most
$\tilde{\nu}\lfloor R^{1/2}\rfloor$ that interpolates
$\mathrm{sgn}(D^{\alpha}Q_j(\tilde{x}))$ at $x_j$ for all $j\notin\mathcal{J}(\tilde{x})$.
This leads to
\begin{equation}
\label{eq:3.5}
\sum_{j\notin\mathcal{J}(\tilde{x})}|D^{\alpha}Q_j(\tilde{x})|
=\sum_{j\notin\mathcal{J}(\tilde{x})}\mathrm{sgn}(D^{\alpha}Q_j(\tilde{x}))
D^{\alpha}Q_j(\tilde{x})
= \sum_{j\notin\mathcal{J}(\tilde{x})}p(x_j)D^{\alpha}Q_j(\tilde{x}).
\end{equation}
Bernstein inequality (see
Proposition 11.6 in \cite{wen:2010}) and Assumption \ref{assum:3.1} (ii) implies that
\[
\max_{|\alpha|_1\le 1}\sup_{x\in (-R,R)^d}|D^{\alpha}p(x)|\le C\sup_{x\in (-R,R)^d}|p(x)|.
\]
Thus, for $x\in (-R,R)^d$, take a nearest $x_j$ to observe
\[
|p(x)|\le |p(x_j)| + |p(x)-p(x_j)|
\le 1 + C\Delta_1 x\sup_{y\in (-R,R)^d}|p(y)|,
\]
from which and $\Delta_1x\to 0$ this polynomial $p$ satisfies
$|p(x)|\le 2$ for $x\in (-R,R)^d$.
Further, we have
$\max_{|\alpha|_1\le 1}\sup_{x\in (-R,R)^d}|D^{\alpha}p(x)|\le C_0$
for some $C_0>0$ that is independent of $R$ and $\tilde{x}$.
In particular, $p$ is Lipschitz continuous on $(-R,R)^d$
with Lipschitz coefficient $C_0$. Hence the function
\[
\tilde{p}(x):=\inf_{y\in (-R,R)^d}\{p(y)+C_0|x-y|\}, \quad x\in\mathbb{R}^d,
\]
is Lipschitz continuous on $\mathbb{R}^d$ with the same Lipschitz coefficient and
satisfies $\tilde{p}=p$ on $(-R,R)^d$.
Further, for $\varepsilon>0$ to be specified later,
define the function $\bar{p}$ on $\mathbb{R}^d$ by
\[
\bar{p}(x)= \varepsilon^{-d}\int_{\mathbb{R}^d}\tilde{p}(y)
\zeta\left(\frac{x-y}{\varepsilon}\right)dy,
\]
where
$\zeta$ is a $C^{\infty}$-function such that $0\le \zeta(x)\le 1$ for $x\in\mathbb{R}^d$,
$\zeta(x)=1$ for $|x|\le 1$, and $\zeta(x)=0$ for $|x|>1+\tilde{c}$ for some $\tilde{c}>0$.
It is straightforward to verify that this function satisfies
\begin{equation}
\label{eq:3.5.1}
|\bar{p}(x)-\tilde{p}(x)|\le C_1\varepsilon, \quad
|D^{\alpha}\bar{p}(x)|\le C_1\varepsilon^{-|\alpha|_1}, \quad |\alpha|_1\le\nu_1,
\;\; x\in\mathbb{R}^d
\end{equation}
for some $C_1>0$ that is independent of $R$ and $\varepsilon$.
Here $\nu_1=\tau+\min\{\kappa\in\mathbb{Z}: \kappa\ge (d+1)/2\}$.
Then consider the function $\hat{p}\in H^{\nu_1}(\mathbb{R}^d)$ defined by
$\hat{p}(x)=\bar{p}(x)\zeta(x/R)$, $x\in\mathbb{R}^d$.
With these modifications and in view of \eqref{eq:3.4}--\eqref{eq:3.5.1}, we obtain
\begin{align*}
\sum_{j=1}^N|D^{\alpha}Q_j(\tilde{x})|
&\le \sum_{j\notin\mathcal{J}(\tilde{x})}p(x_j)D^{\alpha}Q_j(\tilde{x}) + C
= \sum_{j=1}^Np(x_j)D^{\alpha}Q_j(\tilde{x}) -\sum_{j\in\mathcal{J}(\tilde{x})}
p(x_j)D^{\alpha}Q_j(\tilde{x}) + C \\
&\le \sum_{j=1}^Np(x_j)D^{\alpha}Q_j(\tilde{x}) +2\sum_{j\in\mathcal{J}(\tilde{x})}
|D^{\alpha}Q_j(\tilde{x})| + C \\
&\le \sum_{j=1}^N\tilde{p}(x_j)D^{\alpha}Q_j(\tilde{x}) + C
\le \sum_{j=1}^N\hat{p}(x_j)D^{\alpha}Q_j(\tilde{x})
+ C_1\varepsilon\sum_{j=1}^N|D^{\alpha}Q_j(\tilde{x})| + C.
\end{align*}
Setting $\varepsilon$ to satisfy $C_1\varepsilon<1$, we obtain
\begin{equation*}
\sum_{j=1}^N|D^{\alpha}Q_j(\tilde{x})|
\le \frac{1}{1-C_1\varepsilon}|D^{\alpha}I(\hat{p})(\tilde{x})|
+ \frac{C}{1-C_1\varepsilon}.
\end{equation*}
Moreover, by Lemma \ref{lem:3.5} and \eqref{eq:3.5.1},
\begin{align*}
|D^{\alpha}I(\hat{p})(\tilde{x})|&\le |D^{\alpha}I(\hat{p})(\tilde{x})-D^{\alpha}\hat{p}(\tilde{x})|
+|D^{\alpha}\bar{p}(\tilde{x})| \\
&\le C(\Delta_1x)^{\tau-3/2}\|\hat{p}\|_{H^{\tau+(d+1)/2}(\mathbb{R}^d)}
+ C \\
&\le C(\Delta_1x)^{\tau-3/2}\|\bar{p}\|_{H^{\nu_1}((-(1+\tilde{c})R,(1+\tilde{c})R)^d)} + C \\
&\le C\|\bar{p}\|_{C^{\nu_1}(\mathbb{R}^d)}R^{d/2}(\Delta_1x)^{\tau-3/2} + C.
\end{align*}
Assumption \ref{assum:3.1} (ii)
and the boundedness of $\|\bar{p}\|_{C^{\nu_1}(\mathbb{R}^d)}$ now
lead to the conclusion of the lemma.
\end{proof}
\begin{rem}
One might ask if Assumption \ref{assum:3.1} (ii) can be simplified in some sense.
This problem, however, seems to be nontrivial.
For example, the classical result by Demko et.el \cite{dem-etal:1984} tells us that
if a matrix $A$ is $m$-banded, symmetric and positive definite then we have
\[
|(A^{-1})_{ij}|\le \frac{2}{\lambda_{min}}\left(1-\frac{2}{\sqrt{r}+1}\right)^{\frac{2}{m}|i-j|}.
\]
Here, $\lambda_{min}$ and $r$ are the minimum eigenvalue and the condition number of $A$,
respectively. Our interpolation matrix satisfies $\lambda_{min}\ge Cq^{2\tau+1}$ and
$r\le Cq^{-2\tau-d-1}$, where $q=\Delta_2 x$.
Moreover, if the matrix is banded, then it is necessarily $Cq^{-d}$-banded.
So a sufficient condition for which
$|(A^{-1})_{ij}|\le Cq^d/N$ holds is
\[
q^{-2\tau-1}\exp(-Cq^{\tau+(3d+1)/2}|i-j|)\le Cq^d/N.
\]
This is equivalent to
$|i-j|\ge Cq^{-\tau-(3d+1)/2}\log(Nq^{-2\tau-d-1})$.
The arguments in the proof of Lemma 3.7 then leads to the condition
\[
Cq^{-\tau-(5d+1)/2}\log(Nq^{-2\tau-d-1})\le \sqrt{R}\le
C(\Delta_1x)^{-(\tau-3/2)/d},
\]
which is similar to the latter part of Assumption \ref{assum:3.1} (ii).
If this condition holds, then we must have
$(\tau-3/2)/d\ge \tau+(5d+1)/2$ and this is of course impossible.
\end{rem}
\begin{proof}[Proof of Theorem \ref{thm:3.2}]
First, for $i=0,\ldots,n-1$ and $x\in\mathbb{R}^d$, we have
\begin{align*}
(u(t_{i+1},x)-u^{h}(t_{i+1},x))^2&=(u(t_i,x)-u^{h}(t_i,x))^2+
(S_{i+1}(x))^2 +(\Theta_{i+1}(x))^2 \\
&\quad + 2(u(t_i,x)-u^{h}(t_i,x))S_{i+1}(x)
+2\Theta_{i+1}(x)S_{i+1}(x) \\
&\quad + 2(u(t_i,x)-u^{h}(t_i,x))\Theta_{i+1}(x).
\end{align*}
Here, for $i=0,\ldots,n-1$ and $x\in\mathbb{R}^d$,
\begin{align*}
S_{i+1}(x) &= \sum_{k=0}^mL_kI(u(t_i)-u^{h}(t_i))(x)\Delta W_{t_{i+1}}^k, \\
\Theta_{i+1}(x)&=\sum_{k=0}^m\int_{t_i}^{t_{i+1}}(L_ku(s,x)-L_kI(u(t_i))(x))dW_s^k.
\end{align*}
It is straightforward to see that
\begin{align*}
&\mathbb{E}(S_{i+1}(x))^2 \\
&=\mathbb{E}|L_0I(u(t_i)-u^h(t_i))(x)|^2(t_{i+1}-t_i)^2
+\sum_{k=1}^m\mathbb{E}|L_kI(u(t_i)-u^h(t_i))(x)|^2(t_{i+1}-t_i) \\
&\le C\sum_{|\alpha|_1\le 2}\mathbb{E}|D^{\alpha}I(u(t_i)-u^h(t_i))|^2\Delta t.
\end{align*}
By Cauchy-Schwartz inequality and Lemma \ref{lem:3.7},
\begin{align*}
&\mathbb{E}|D^{\alpha}I(u(t_i)-u^h(t_i))(x)|^2
= \mathbb{E}\left|\sum_{j=1}^N(u(t_i,x_j)-u^h(t_i,x_j))D^{\alpha}Q_j(x)\right|^2 \\
&=\sum_{j,\ell=1}^N\mathbb{E}[(u(t_i,x_j)-u^h(t_i,x_j))
(u(t_i,x_{\ell})-u^h(t_i,x_{\ell}))]D^{\alpha}Q_j(x)D^{\alpha}Q_{\ell}(x) \\
&\le \sum_{j,\ell=1}^N(\mathbb{E}|u(t_i,x_j)-u^h(t_i,x_j)|^2)^{1/2}
(\mathbb{E}|u(t_i,x_{\ell})-u^h(t_i,x_{\ell})|^2)^{1/2}
|D^{\alpha}Q_j(x)||D^{\alpha}Q_{\ell}(x)| \\
&\le \sup_{y\in (-R,R)^d}\mathbb{E}|u(t_i,y)-u^h(t_i,y)|^2
\left(\sum_{j=1}^N|D^{\alpha}Q_j(x)|\right)^2 \\
&\le C\sup_{y\in (-R,R)^d}\mathbb{E}|u(t_i,y)-u^h(t_i,y)|^2.
\end{align*}
Hence,
\begin{equation}
\label{eq:3.6}
\mathbb{E}(S_{i+1}(x))^2\le C\sup_{y\in (-R,R)^d}\mathbb{E}|u(t_i,y)-u^h(t_i,y)|^2
\Delta t.
\end{equation}
Next, it follows from It{\^o} isometry that
\begin{align*}
&\mathbb{E}|\Theta_{i+1}(x)|^2 \\
&\le 2\mathbb{E}\left|\int_{t_i}^{t_{i+1}}
(L_0u(s,x)-L_0I(u(t_i))(x))ds\right|^2
+ 2\mathbb{E}\left|\sum_{k=1}^m\int_{t_i}^{t_{i+1}}
(L_ku(s,x)-L_kI(u(t_i))(x))dW_s^k\right|^2 \\
&\le 2\Delta t\mathbb{E}\int_{t_i}^{t_{i+1}}|L_0u(s,x)-L_0I(u(t_i))(x)|^2ds
+ 2\sum_{k=1}^m\mathbb{E}\int_{t_i}^{t_{i+1}}|L_ku(s,x)-L_kI(u(t_i))(x)|^2ds \\
&\le C\sum_{|\alpha|_1\le 2}\mathbb{E}\int_{t_i}^{t_{i+1}}
|D^{\alpha}u(s,x)-D^{\alpha}I(u(t_i))(x)|^2ds.
\end{align*}
Again by It{\^o} isometry,
\begin{align*}
\mathbb{E}|D^{\alpha}u(s,x)-D^{\alpha}u(t_i,x)|^2
&=\mathbb{E}\left|\sum_{k=0}^m\int_{t_i}^sD^{\alpha}u(r,x)dW_r^k\right|^2
\le C\sum_{k=0}^m\mathbb{E}\int_{t_i}^s|D^{\alpha}L_ku(r,x)|^2dr \\
&\le C\mathbb{E}\sup_{0\le t\le T}\|u(t)\|_{C^4(\mathbb{R}^d)}^2(s-t_i).
\end{align*}
Further, Lemma \ref{lem:3.5} means
\begin{align*}
\mathbb{E}|D^{\alpha}u(t_i,x)-D^{\alpha}I(u(t_i))(x)|^2
\le C(\Delta_1x)^{2\tau-3}\mathbb{E}\|u\|^2_{H^{\tau+(d+1)/2}(\mathbb{R}^d)}.
\end{align*}
Thus,
\begin{equation}
\label{eq:3.7}
\mathbb{E}|\Theta_{i+1}(x)|^2\le C(\Delta t)^2 + C(\Delta_1x)^{2\tau-3}\Delta t.
\end{equation}
The arguments used in the estimations above yield
\begin{equation}
\label{eq:3.8}
\begin{aligned}
&\mathbb{E}(u(t_i,x)-u^h(t_i,x))S_{i+1}(x) \\
&=\mathbb{E}(u(t_i,x)-u^h(t_i,x))L_0I(u(t_i)-u^h(t_i))(x)(t_{i+1}-t_i) \\
&\le (\mathbb{E}|u(t_i,x)-u^h(t_i,x)|^2)^{1/2}
(\mathbb{E}|L_0I(u(t_i)-u^h(t_i))(x)|^2)^{1/2}\Delta t \\
&\le C\sup_{y\in (-R,R)^d}\mathbb{E}|u(t_i,y)-u^h(t_i,y)|^2\Delta t.
\end{aligned}
\end{equation}
Here we have again used the boundedness of the coefficients of $L_0$
and Lemma \ref{lem:3.7} to derive the last inequality.
Furthermore, we obtain
\begin{equation*}
\begin{aligned}
&\mathbb{E}(u(t_i,x)-u^h(t_i,x))\Theta_{i+1}(x) \\
&=\mathbb{E}(u(t_i,x)-u^h(t_i,x))
\int_{t_i}^{t_{i+1}}(L_0u(s,x)-L_0I(u(t_i))(x))ds \\
&\le (\mathbb{E}|u(t_i,x)-u^h(t_i,x)|^2)^{1/2}
\left(\mathbb{E}\left|\int_{t_i}^{t_{i+1}}(L_0u(s,x)
-L_0I(u(t_i))(x))ds\right|^2\right)^{1/2} \\
&\le (\mathbb{E}|u(t_i,x)-u^h(t_i,x)|^2)^{1/2}
\left(\Delta t\mathbb{E}\int_{t_i}^{t_{i+1}}\left|L_0u(s,x)
-L_0I(u(t_i))(x)\right|^2ds\right)^{1/2} \\
&=(\mathbb{E}|u(t_i,x)-u^h(t_i,x)|^2\Delta t)^{1/2}
\left(\mathbb{E}\int_{t_i}^{t_{i+1}}\left|L_0u(s,x)
-L_0I(u(t_i))(x)\right|^2ds\right)^{1/2} \\
&\le 2\mathbb{E}|u(t_i,x)-u^h(t_i,x)|^2\Delta t
+ 2\mathbb{E}\int_{t_i}^{t_{i+1}}\left|L_0u(s,x)
-L_0I(u(t_i))(x)\right|^2ds.
\end{aligned}
\end{equation*}
and so
\begin{equation}
\label{eq:3.9}
\begin{aligned}
&\mathbb{E}(u(t_i,x)-u^h(t_i,x))\Theta_{i+1}(x) \\
&\le 2\sup_{y\in (-R,R)^d}\mathbb{E}|u(t_i,y)-u^h(t_i,y)|^2)\Delta t
+C(\Delta t)^2 + C(\Delta_1x)^{2\tau-3}\Delta t.
\end{aligned}
\end{equation}
Then, from \eqref{eq:3.6}--\eqref{eq:3.9} we have, for $i=0,\ldots,n-1$,
\begin{align*}
&\sup_{x\in(-R,R)^d}\mathbb{E}|u(t_{i+1},x)-u^{h}(t_{i+1},x)|^2 \\
&\le (1+C\Delta t)\sup_{x\in (-R,R)^d}\mathbb{E}|u(t_{i},x)-u^{h}(t_{i},x)|^2
+ C(\Delta t)^2 + C(\Delta_1x)^{2\tau-3}\Delta t.
\end{align*}
A simple application of the discrete Gronwall lemma now leads to what we aim to prove.
\end{proof}
\section{Numerical experiments}\label{sec:4}
In this section, we apply our collocation method to the one-dimentional Zakai equation
\begin{equation}
\label{eq:4.1}
\left\{
\begin{aligned}
du(t,x)&=\left(\frac{1}{2}\frac{\partial^2}{\partial x^2} u(t,x)
- \frac{\partial}{\partial x}(\tanh(x)u(t,x))\right)dt
+ u(t,x)dW(t), \quad 0\le t\le 1, \\
u(0,x)&=\frac{1}{\sqrt{2\pi}}\cosh (x)e^{-|x|^2/2}.
\end{aligned}
\right.
\end{equation}
The unique solution $u(t,x)$ to \eqref{eq:4.1} is given by
\begin{equation*}
u(t,x)=\frac{1}{\sqrt{2\pi}}\cosh(x)
\exp\left(W(t)-\frac{3t}{2} - \frac{|x|^2}{2(1+t)}\right).
\end{equation*}
We use the Wendland kernel $\phi_{1,4}$ scaled by some positive constant for
the performance test.
We choose the time grid as a uniform one in $[0,1]$, and
as suggested in Remark \ref{rem:3.2}, we take the uniform spatial grid points
on $[-R+2R/(N+1),R-2R/(N+1)]$ where
$R = (1/5)N^{(1-1/(2\tau-2))}$.
To check the validity of Assumption \ref{assum:3.1} (ii), we plot
\[
\iota(N) = \max_i\#\{j: |(A^{-1})_{ij}|> \Delta_2x/N\}
\]
in Figure \ref{fig:4.1}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\columnwidth, bb=0 0 512 384]
{band_v2_plot_d1tau4_v3.png}
\caption{Plotting $\iota(N)$ and $5/\Delta_2x$ for $N=1,2,\ldots,1000$.}
\label{fig:4.1}
\end{figure}
We can see that $\iota(N)<5/\Delta_2x$ for all $N\le 1000$.
Thus, Assumption \ref{assum:3.1} (ii) seems to be satisfied with $c_1=1$, $c_2=5$, and
$\lambda=1$ for the sequence of the tuning parameters defined by $N$ from $1$
at least to $1000$.
This is consistent with the condition
$\tau\ge 3/2 + (1+\lambda)d^2$ in Remark \ref{rem:3.2}.
\begin{figure}[htbp]
\centering
\subfigure{\includegraphics[width=0.425\columnwidth, bb=0 0 512 384]
{d1tau4N32n256x-1v4.png}}~
\subfigure{\includegraphics[width=0.425\columnwidth, bb=0 0 512 384]
{d1tau4N32n256x-05v4.png}} \\
\subfigure{\includegraphics[width=0.425\columnwidth, bb=0 0 512 384]
{d1tau4N32n256x05v4.png}}
\subfigure{\includegraphics[width=0.425\columnwidth, bb=0 0 512 384]
{d1tau4N32n256x1v4.png}} \\
\caption{Comparing the exact solution (solid line) and
the approximated one (dashed line) at $x$ =
$-1, -1/2, 1/2, 1$, in the case of $N=2^5$ and $n=2^{8}$.}
\label{fig:4.2}
\end{figure}
\begin{figure}[htbp]
\centering
\subfigure{\includegraphics[width=0.425\columnwidth, bb=0 0 512 384]
{d1tau4N32n256t2-7v4.png}}~
\subfigure{\includegraphics[width=0.425\columnwidth, bb=0 0 512 384]
{d1tau4N32n256t2-5v4.png}} \\
\subfigure{\includegraphics[width=0.425\columnwidth, bb=0 0 512 384]
{d1tau4N32n256t2-3v4.png}}
\subfigure{\includegraphics[width=0.425\columnwidth, bb=0 0 512 384]
{d1tau4N32n256t2-1v4.png}} \\
\caption{Comparing the exact solution (solid line) and
the approximated one (dashed line)
at time = $2^1\Delta t, 2^3\Delta t, 2^5\Delta t, 2^7\Delta t$, in the case of
$N=2^5$ and $n=2^{8}$.}
\label{fig:4.3}
\end{figure}
Figure \ref{fig:4.2} plots sample paths of $u(\cdot,x)$ and
$u^h(\cdot,x)$ for several spatial points.
Figure \ref{fig:4.3} plots snapshots of the time evolutions
of $u(t,\cdot)$ and $u^h(t,\cdot)$.
The both show that our collocation method yields a good approximation as well as
the accumulation of the error near the time maturity cannot be negligible.
To compare an averaged performance, we compute the root mean squared errors
averaged over $10000$ samples, defined by
\begin{align*}
\text{RMSE} &:= \sqrt{\frac{1}{10000\times 41(n+1)}\sum_{i=0}^n\sum_{j=1}^{41}
\sum_{\ell=1}^{10000}
|u_{\ell}(t_i,\xi_j)-u^{h}_{\ell}(t_i,\xi_j)|^2},
\end{align*}
for several values of $N$ and $n$.
Here, $u_{\ell}$ and $u^{h}_{\ell}$ are
the exact solution and approximate solution at $\ell$-th trial, respectively,
and $\{\xi_1,\ldots,\xi_{41}\}$ is the set of evaluation points consisting of
the equi-spaced grid points on $[-2,2]$.
As another comparison, we compute numerical solutions by
the implicit Euler finite difference method for the test equation, which is described as follows:
\[
\left\{
\begin{aligned}
\tilde{u}(0,x) &= u(0,x), \quad x\in\tilde{\Gamma}, \\
\tilde{u}(t_i,\pm R) &= 0, \quad i=0,\ldots,n, \\
\tilde{u}(t_{i+1},x)&=\tilde{u}(t_i,x) + \tilde{L}\tilde{u}(t_{i+1},x) + \tilde{u}(t_i,x)\Delta W(t_{i+1}),
\quad i=0,\ldots,n-1, \;\; x\in\tilde{\Gamma}.
\end{aligned}
\right.
\]
Here, $\tilde{\Gamma}=\{-R+j\Delta x: j=1,\ldots,N\}$ with $\Delta x=2R/(N+1)$ and
$\tilde{L}$ denotes the corresponding finite difference operator.
Then $\tilde{u}$ converges to $u$ as $R\to\infty$, $\Delta t\to 0$, and $\Delta x\to 0$.
See Gerencs{\'e}r and Gy{\"o}ngy \cite{ger-gyo:2017}.
Table \ref{table:4.1} shows that the resulting RMSE's
are sufficiently small for all pairs $(N,n)$ although its decrease is nonmonotonic.
Here $\text{RMSE}_{\text{fd}}$ denotes the corresponding the root mean squared errors
for the finite difference method, where the set of evaluation points is taken to be
$\tilde{\Gamma}$ itself.
We can conclude that Theorem \ref{thm:3.2} is well consistent with the results of
our experiments, and that the kernel-based collocation method outperforms the implicit Euler
finite difference method when the length $R$'s are set to be the ones that guarantee
the convergence of the former.
\begin{table}[htb]
\centering
\begin{tabular}[t]{ccccccc}
\toprule
$N$ & $R$ & $(\Delta_1x)^{\tau-3/2}$ & $n$ & $\sqrt{\Delta t}$ & $\text{RMSE}$ &
$\text{RMSE}_{\text{fd}}$ \\ \midrule
$2^4$ & 2.0159 & 0.0274 & $2^{6}$ & 0.1250 & 0.0714 & 0.2163 \\
& & & $2^{8}$ & 0.0625 & 0.0726 & 0.2171 \\
& & & $2^{10}$ & 0.0312 & 0.0745 & 0.2267 \\ \midrule
$2^5$ & 3.5919 & 0.0221 & $2^{6}$ & 0.1250 & 0.0323 & 0.1981 \\
& & & $2^{8}$ & 0.0625 & 0.0252 & 0.2017 \\
& & & $2^{10}$ & 0.0312 & 0.0234 & 0.1991 \\ \midrule
$2^6$ & 6.4000 & 0.0172 & $2^{6}$ & 0.1250 & 0.0333 & 0.2020\\
& & & $2^{8}$ & 0.0625 & 0.0261 & 0.2088 \\
& & & $2^{10}$ & 0.0312 & 0.0241 & 0.2067 \\ \bottomrule
\end{tabular}
\caption{The resulting root mean squared errors
for several pairs $(N,n)$.}
\label{table:4.1}
\end{table}
\subsection*{Acknowledgements}
This study is partially supported by JSPS KAKENHI Grant Number JP17K05359.
\bibliographystyle{plain}
|
1,116,691,499,365 | arxiv | \section{Introduction}\label{Intro}
We recall that, in the \textit{Riemannian case}, \textit{harmonic maps} are the critical points of the {\em energy functional}
\begin{equation}\label{energia}
E(\varphi)=\frac{1}{2}\int_{M}\,\|d\varphi\|^2\,dV \, ,
\end{equation}
where $\varphi:M\to N$ is a smooth map between two Riemannian
manifolds $(M^m,g)$ and $(N^n,h)$. In particular, $\varphi$ is harmonic if and only if it is a solution of the Euler-Lagrange system of equations associated to \eqref{energia}, i.e.,
\begin{equation}\label{harmonicityequation}
- d^* d \varphi = {\trace} \, \nabla d \varphi =0 \, .
\end{equation}
The left member of \eqref{harmonicityequation} is a vector field along the map $\varphi$ or, equivalently, a section of the pull-back bundle $\varphi^{-1} TN$: it is called {\em tension field} and denoted $\tau (\varphi)$. In addition, we recall that, if $\varphi$ is an \textit{isometric immersion}, then $\varphi$ is a harmonic map if and only if the immersion $\varphi$ defines a minimal submanifold of $N$ (see \cite{EL83, EL1} for background).
Next, in order to define the notion of an $r$-harmonic map, we consider the following family of functionals which represent a version of order $r$ of the classical energy \eqref{energia}. If $r=2s$, $s \geq 1$:
\begin{eqnarray}\label{2s-energia}
E_{2s}(\varphi)&=& \frac{1}{2} \int_M \, \langle \, \underbrace{(d^* d) \ldots (d^* d)}_{s\, {\rm times}}\varphi, \,\underbrace{(d^* d) \ldots (d^* d)}_{s\, {\rm times}}\varphi \, \rangle_{_N}\, \,dV \nonumber\\
&=& \frac{1}{2} \int_M \, \langle \,\overline{\Delta}^{s-1}\tau(\varphi), \,\overline{\Delta}^{s-1}\tau(\varphi)\,\rangle_{_N} \, \,dV\,.
\end{eqnarray}
In the case that $r=2s+1$:
\begin{eqnarray}\label{2s+1-energia}
E_{2s+1}(\varphi)&=& \frac{1}{2} \int_M \, \langle\,d\underbrace{(d^* d) \ldots (d^* d)}_{s\, {\rm times}}\varphi, \,d\underbrace{(d^* d) \ldots (d^* d)}_{s\, {\rm times}}\varphi\,\rangle_{_N}\, \,dV\nonumber \\
&=& \frac{1}{2} \int_M \,\sum_{j=1}^m \langle\,\nabla^\varphi_{e_j}\, \overline{\Delta}^{s-1}\tau(\varphi), \,\nabla^\varphi_{e_j}\,\overline{\Delta}^{s-1}\tau(\varphi)\, \rangle_{_N} \, \,dV \,.
\end{eqnarray}
Here, \(\overline{\Delta}=d^* d\) represents the Laplacian on the pull-back bundle $\varphi^{-1} TN$.
Then a map $\varphi:(M^m,g)\to(N^n,h)$ is \textit{$r$-harmonic} if, for all variations $\varphi_t$,
$$
\left .\frac{d}{dt} \, E_{r}(\varphi_t) \, \right |_{t=0}\,=\,0 \,\,.
$$
In the case that $r=2$, the functional \eqref{2s-energia} is called \textit{bienergy} and its critical points are the so-called \textit{biharmonic maps}. A very ample literature on biharmonic maps is available and we refer to \cite{Chen, Jiang, SMCO, Ou} for an introduction to this topic. More generally, the \textit{$r$-energy functionals} $E_r(\varphi)$ defined in \eqref{2s-energia}, \eqref{2s+1-energia} have been intensively studied (see \cite{Volker, Volker2, BMOR1, Maeta1, Maeta3, Maeta2, MOR-space forms, Mont-Ratto4, Na-Ura, Wang, Wang2}, for instance).
We say that an $r$-harmonic map is {\it proper} if it is not harmonic (similarly, an $r$-harmonic submanifold, i.e., an $r$-harmonic isometric immersion, is {\it proper} if it is not minimal). We point out that, as observed by Maeta in his series of papers {\cite{Maeta1, Maeta3, Maeta2}}, in general $r$-harmonic does not imply $r'$-harmonic for $r'>r$ unless the target manifold is flat.
In our recent work \cite{MOR-space forms} we proved some general results for $r$-harmonic hypersurfaces into space forms and deduced that the value of the integer $r$ plays a crucial role to generate geometric phenomena which differ substantially from the classical situation corresponding to the biharmonic and triharmonic cases. For instance, if $\ell \geq 3$, there exists no isoparametric hypersurface of ${\mathbb S}^{m+1}$ of degree $\ell$ which is proper biharmonic or triharmonic. By contrast, when $r \geq 5$, there are several examples of such hypersurfaces which are proper $r$-harmonic (see \cite{MOR-space forms}). From the point of view of the differential geometry of submanifolds, the difficulties which one encounters in studying the equations which define a general $r$-harmonic submanifold are huge. Therefore, a reasonable starting point is to focus on the case that the ambient is a space form $N^{m+1}(c)$ (here and below, $c$ denotes the sectional curvature) and study CMC hypersurfaces with constant squared norm $\|A\|^2$ of the shape operator. In this order of ideas, in the Riemannian case we obtained the following general result:
\begin{theorem}\label{Th-existence-hypersurfaces-c>0-e-c<0}\cite{MOR-space forms} Let $M^m$ be a non-minimal CMC hypersurface in a Riemannian space form $N^{m+1}(c)$ and assume that $\|A\|^2$ is constant. Then
$M^m$ is proper $r$-harmonic ($r \geq 3$) if and only if
\begin{equation*}\label{r-harmonicity-condition-in-general}
\|A\|^4-m\,c\,\|A\|^2-(r-2)m^2 \,c\, \alpha^2=0 \,,
\end{equation*}
where the constant $\alpha$ denotes the mean curvature of $M^m$.
\end{theorem}
\begin{remark} In the biharmonic case, a similar result is available under a less restrictive hypothesis. Indeed,
\begin{theorem}\label{Th-bihar-hypersurfaces-spheres}
(See \cite{CMO02, CMO03, Jiang}) Let $M^m$ be a non-minimal CMC hypersurface in $N^{m+1}(c)$. Then $M^m$ is proper biharmonic if and only if $\|A\|^2=cm$. In particular, if $c \leq 0$, then no such $M^m$ can exist.
\end{theorem}
\end{remark}
The first goal of this paper is to establish a version of Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0} when the ambient is a pseudo-Riemannian space form. Next, we shall describe several new examples and some geometric applications. In order to state our results, it is convenient to introduce first some basic notions concerning pseudo-Riemannian manifolds and geometry. Therefore, our paper is organised as follows.
In Section~\ref{Sec-preliminaries} we shall review some basic aspects of the theory of pseudo-Riemannian space forms and pseudo-Riemannian geometry. Moreover, we shall describe how to generalise the notion of $r$-harmonicity in this context.
In Section~\ref{Sec-results} we shall state our main results and their geometric applications.
Finally, in Section~\ref{Sec-proofs}, we shall provide all the proofs.
\vspace{2mm}
For the sake of completeness, we mention in this introduction that another possible, interesting definition of an $r$-order version of the energy functional, which was proposed by Eells-Sampson and Eells-Lemaire (see \cite{EL83, ES}), is
\begin{equation}\label{r-energy-Eells-Lemaire}
E_{r}^{ES}(\varphi)= \frac{1}{2} \int_M \, \left\|(d+d^*){^r}\varphi \right \|^2 \,dV \,.
\end{equation}
As for a detailed discussion and comparison between definitions \eqref{2s-energia}, \eqref{2s+1-energia} and \eqref{r-energy-Eells-Lemaire}, we refer to \cite{BMOR1}. We believe that most of the techniques used in \cite{BMOR1} could also be applied in the pseudo-Riemannian context, but we shall not pursue this option in this paper.\\
{\bf Acknowledgements}. The authors would like to thank Professor Miguel Dom\'{i}nguez V\'{a}zquez for very useful correspondence on Lorentzian isoparametric hypersurfaces.
\section{Pseudo-Riemannian geometry, pseudo-Riemannian space forms and $r$-harmonicity}\label{Sec-preliminaries}
A basic reference for pseudo-Riemannian geometry is the classical book of O'Neill (see \cite{Neill}), but for the specific topics treated in this section we also refer to \cite{Abe, Beem, Dong, Liu, Sasahara, Zhang}.
Let $(M^m_t,g)$ be a pseudo-Riemannian manifold of dimension
$m$ with a \textit{nondegenerate} metric of \textit{index} $t$ ($0 \leq t \leq m$). In order to clarify the notion of index $t$, let us first recall that nondegeneracy means that the only vector $X \in T_pM$ satisfying $g_p(X, Y ) = 0$ for all $Y \in T_pM$
is $X = 0$, for any $p \in M $. A local \textit{orthonormal} frame field
of $(M^m_t,g)$ is a set of local vector fields $\{e_i\}_{i=1}^m$ such that $g(e_i, e_j) = \varepsilon_i \delta_{ij}$, with
$\varepsilon_1=\ldots \varepsilon_t=-1$, $\varepsilon_{t+1}=\ldots \varepsilon_m=1$.
Next, let us fix terminology and notations concerning pseudo-Riemannian space forms.
The $m$-dimensional pseudo-Euclidean space with index $t$ is denoted by ${\mathbb R}^m_t=({\mathbb R}^m,\langle,\rangle)$, where
\[
\langle x,y \rangle= -\sum_{i=1}^t x_i y_i +\sum_{i=t+1}^m x_i y_i \,.
\]
The $m$-dimensional pseudo-Riemannian sphere, denoted by ${\mathbb S}^m_t (c)$ is defined as follows:
\begin{equation}\label{eq-def-pseudosfere}
{\mathbb S}^m_t (c)= \left \{x \in {\mathbb R}^{m+1}_t \colon \langle x,x \rangle = \frac{1}{c} \right \} \quad \quad (c>0)\,.
\end{equation}
${\mathbb S}^m_t (c)$, with the induced metric from ${\mathbb R}^{m+1}_t$, is a complete pseudo-Riemannian manifold with index $t$ and constant positive sectional curvature $c$.
The $m$-dimensional pseudo-Riemannian hyperbolic space, denoted by ${\mathbb H}^m_t (c)$ is defined as follows:
\begin{equation}\label{eq-def-pseudohyperbolic}
{\mathbb H}^m_t (c)= \left \{x \in {\mathbb R}^{m+1}_{t+1} \colon \langle x,x \rangle =\frac{1}{c} \right \} \quad \quad (c<0)\,.
\end{equation}
${\mathbb H}^m_t (c)$, with the induced metric from ${\mathbb R}^{m+1}_{t+1}$, is a complete pseudo-Riemannian manifold with index $t$ and constant negative sectional curvature $c$.
A pseudo-Riemannian space form refers to one of the three spaces ${\mathbb R}^m_t$, ${\mathbb S}^m_t (c),{\mathbb H}^m_t (c)$. We shall write ${\mathbb S}^m_t $ and ${\mathbb H}^m_t $ for ${\mathbb S}^m_t (1)$ and ${\mathbb H}^m_t (-1)$ respectively. Sometimes, to provide a unified treatment, we also use $N^m_t(c)$ to denote a pseudo-Riemannian space form of sectional curvature $c$.
The flat ($c=0$) pseudo-Riemannian space ${\mathbb R}^m_t$ is called \textit{Minkowski space}, while ${\mathbb S}^m_t (c)$ and ${\mathbb H}^m_t (c)$ are known as \textit{de Sitter space} and \textit{anti-de Sitter space} respectively. When the index is $t=1$, these spaces are also referred to as \textit{Lorentz space forms}. We also point out that ${\mathbb S}^m_t (c)$ is diffeomorphic to ${\mathbb R}^t \times {\mathbb S}^{m-t}$, while ${\mathbb H}^m_t (c)$ is diffeomorphic to ${\mathbb S}^{t} \times {\mathbb R}^{m-t}$. In particular, ${\mathbb S}^m_{m-1} (c)$ and ${\mathbb H}^m_1 (c)$ are not simply connected.
Finally, we point out that in this paper we restrict our study to \textit{connected} manifolds. Therefore, even if in some definitions formally one has two connected components (for instance, according to \eqref{eq-def-pseudosfere}, \eqref{eq-def-pseudohyperbolic}, this happens to ${\mathbb S}^m_m (c)$ and ${\mathbb H}^m_0 (c)$), we shall implicitly assume that we just work with one connected component.
Next, we recall for future use that the sectional curvature tensor field of $N^m_t(c)$ is described by the following simple expression:
\begin{equation}\label{tensor-curvature-N(c)}
R^{N(c)}(X,Y)Z=c\, \big(\langle Y,Z \rangle X-\langle X,Z \rangle Y \big) \quad \quad \forall \,X,Y,Z \in C(TN(c)) \,.
\end{equation}
Pseudo-Riemannian space forms have important applications in the theory of general relativity and ${\mathbb R}^3_1$, ${\mathbb S}^3_1 (c)$ and ${\mathbb H}^3_1 (c)$ are model spaces for Minkowski, de Sitter and anti-de Sitter space-time respectively.
\vspace{3mm}
Now, in order to make this paper as self-contained as possible, we follow \cite{Dong} and recall here how the basic operators of Riemannian geometry extend to the pseudo-Riemannian context. In a chart of $(M^m_t,g)$, for a local orthonormal frame field $\{e_i\}_{i=1}^m$ we have the following basic identities, which hold for any $X \in {C(TM)}$, $f \in {C^\infty(M)}$ and bilinear form $b \in C( \odot^2 TM)$:
\begin{eqnarray}\label{key-formulas-pseudo}\nonumber
X &=&\sum_{i=1}^m \varepsilon_i g(X,e_i) e_i \,;\\\nonumber
\grad f &=&\sum_{i=1}^m \varepsilon_i df(e_i) e_i \,;\\
{\rm div} X &=&\sum_{i=1}^m \varepsilon_i g(\nabla_{e_i}X,e_i) \,;\\\nonumber
\Delta f &=&-\sum_{i=1}^m \varepsilon_i \left [e_i(e_if)-\big (\nabla_{e_i}e_i\big)f\right ] \,;\\\nonumber
\trace b& =&\sum_{i=1}^m \varepsilon_i b(e_i,e_i)\,.
\end{eqnarray}
In this paper we shall focus on the study of \textit{pseudo-Riemannian} (or, \textit{nondegenerate}) hypersurfaces of a pseudo-Riemannian space form. We can describe such hypersurfaces by means of a smooth
$\varphi:M^m_{t'} \to N^{m+1}_t$ and the hypothesis that $M^m_{t'}$ is pseudo-Riemannian simply means that the metric induced by $\varphi$ is nondegenerate. Therefore we can assume that, locally, there always exists an orthonormal frame field $\{e_i \}_{i=1}^m$ on $M^m_{t'}$. Moreover, denoting $\eta$ a unit normal vector field, $\{e_i ,\eta\} $ is a local orthonormal frame field on $N^{m+1}_t$ and we have two possibilities: either $\langle \eta, \eta \rangle=\varepsilon=1$ and $t'=t$, or $\varepsilon=-1$ and $t'=t-1$.
In particular, when the ambient space is Lorentzian and $\varepsilon=-1$, then the hypersurface is Riemannian and the standard terminology is to say that it is \textit{space-like}. Let us denote $B$ the second fundamental form and $A_\eta=A$ the associated shape operator. The classical Gauss and Weingarten formulas hold as in the Riemannian case, i.e.,
\begin{eqnarray}\label{Gauss-Weingarten-eq}
\nabla^N_X Y&=&\nabla^M_X Y +B(X,Y) {\,;}\\\nonumber
\nabla^N_X \eta&=& - A(X)
\end{eqnarray}
for all tangent vector fields to $M^m_{t'}$. We observe that it is easy to deduce from \eqref{Gauss-Weingarten-eq} that
\begin{equation}\label{eq-legame-A-B}
B(X,Y)= \varepsilon \left < A(X),Y\right > \eta\,.
\end{equation}
The mean curvature vector field of $M^m_{t'}$, denoted ${\mathbf H}$, is defined by
\[
{\mathbf H}= \frac{1}{m}\,\trace B = f\,\eta
\]
where, using \eqref{eq-legame-A-B}, we deduce that the mean curvature function $f$ is given by
\begin{equation}\label{mean-curv-fuction-pseudo}
f= \frac{1}{m} \varepsilon \sum_{i=1}^m \varepsilon_i \langle A(e_i),e_i \rangle \,.
\end{equation}
In this paper, we shall always say that a hypersurface of a pseudo-Riemannian space is \textit{minimal} if $f$ vanishes identically. However, we mention that, when the ambient space is Lorentzian and the hypersurface is space-like, the term \textit{maximal} instead of minimal is also used in the literature, because in this case small regions are local maximizers of the volume functional.
For future use, we recall that
\begin{equation}\label{norma-A}
\trace A^2=\,\sum_{i=1}^m \varepsilon_i \langle A(A(e_i)),e_i \rangle=\sum_{i=1}^m \varepsilon_i \langle A(e_i),A(e_i)\rangle\,.
\end{equation}
We point that in \cite{Dong, Liu} the right hand side of \eqref{norma-A} is denoted by $\|A\|^2$ instead of $\trace A^2$.
\vspace{3mm}
In the last equality of \eqref{norma-A} we used the fact that the shape operator $A$ is self-adjoint. We point out that, in general, $A$ is \textit{not} diagonalizable. Indeed, since $\langle \,,\rangle$ is not positive definite, it is well-known that complex eigenvalues may appear or, in some cases, $A$ may not even be diagonalizable over ${\mathbb C}$ (see \cite{Hahn}). We shall study these situations in detail in Section~\ref{Sec-results}.
Next, we introduce an important family of pseudo-Riemannian hypersurfaces of $N={\mathbb S}^{m+1}_t$ ($m \geq2$, $1 \leq t \leq m$) and give, in Table~\ref{Hypersurfaces-pseudo-sphere}, the expression for $A$ and $\varepsilon$ (see \cite{Abe}). The definition of the hypersurfaces in Table~\ref{Hypersurfaces-pseudo-sphere} are given in Table~\ref{Hypersurfaces-pseudo-sphere-notations1} and Table~\ref{Hypersurfaces-pseudo-sphere-notations2}.
\begin{table}[h!]
\begin{tabular}{ |p{3.5cm}|p{6cm}|p{1.5cm}| }
\hline
\multicolumn{3}{|c|}{Pseudo-Riemannian hypersurfaces of ${\mathbb S}^{m+1}_t=N\subset {\mathbb R}^{m+2}_t $} \\
\hline
Hypersurface & Shape operator $A$ & $\varepsilon$\\
\hline\vspace{.5mm}
${\mathbb S}^m_t(c) $ &\vspace{.5mm} $\pm \sqrt{c-1}\,I$ &\vspace{.5mm} 1\vspace{.5mm}\\
${\mathbb S}^m_{t-1}(c)$ & $\pm \sqrt {1-c} \,I$ & $-1$ \vspace{1mm}\\
${\mathbb R}^m_{t-1}$ &$\pm \,I $ & $-1$\vspace{1mm}\\
${\mathbb H}^m_{t-1}(c)$ &$\pm \sqrt {1-c}\, I$ & $-1$\vspace{1mm}\\
${\mathbb S}^{k}_{\ell}(c) \times {\mathbb S}^{m-k}_{t-\ell}\left (\frac{c}{c-1}\right )$ & $\pm(\sqrt{c-1}\, I_k \oplus - \sqrt {1/(c-1)}\, I_{m-k})$ & $1$\vspace{1mm}\\
${\mathbb S}^{k}_{\ell}(c) \times {\mathbb H}^{m-k}_{t-\ell-1}\left (\frac{c}{c-1} \right )$ & $\pm(\sqrt{1-c}\, I_k \oplus \sqrt {1/(1-c)}\, I_{m-k}) $ & $-1$ \\
\hline
\end{tabular}
\vspace{1mm}
\caption{}
\label{Hypersurfaces-pseudo-sphere}
\end{table}
\begin{table}[h!]
\begin{tabular}{ |p{1.cm}p{.2cm}p{8.1cm}|p{1.6cm}| }
\hline
${\mathbb S}^m_t(c)$ & =&$\big\{ x=(x_1,\ldots,x_{m+2}) \in N \colon x_{m+2}=\sqrt{1-(1/c)}\big\}$& $1 \leq c$\vspace{1mm}
\\
${\mathbb S}^m_{t-1}(c)$&=&$\big \{ x\in N \colon x_1=\sqrt{(1/c)-1} \big \}$ & $0<c\leq 1$\vspace{1mm}\\
${\mathbb R}^m_{t-1}$&=&$\big \{ x\in N \colon x_{1}=x_{m+2}+a\big \}$ & $a>0$\vspace{1mm}\\
${\mathbb H}^m_{t-1}(c)$&=&$\big\{ x\in N \colon x_{m+2}=\sqrt{1-(1/c)}\big \}$ & $ c<0$\\
\hline
\end{tabular}
\vspace{1mm}
\caption{}
\label{Hypersurfaces-pseudo-sphere-notations1}
\end{table}
{\small
\begin{table}[h!]
\begin{tabular}{ |p{3.1cm}p{.1cm}p{5.6cm}|p{2.4cm}| }
\hline
$\displaystyle{{\mathbb S}^{k}_{\ell}(c) \times {\mathbb S}^{m-k}_{t-\ell} (\frac{c}{c-1} )}$ &=&$\displaystyle{ \Big\{ x\in N \colon -\sum_{i=1}^{\ell}x_i^2+\sum_{i=t +1}^{t+k-\ell+1}x_i^2= \frac{1}{c}},$ &$c>1$ \\
&&$ \displaystyle{-\sum_{i=\ell+1}^{t}x_i^2+\sum_{i=t+k-\ell +2}^{m+2}x_i^2= \frac{c-1}{c} \Big\}}$ &$1\leq k \leq m-1$\\
&&&$0\leq \ell \leq k$\\
&&&$0\leq \ell \leq t$\\
&&&$t-\ell \leq m-k$\\
\hline
$\displaystyle{{\mathbb S}^{k}_{\ell}(c) \times {\mathbb H}^{m-k}_{t-\ell-1} (\frac{c}{c-1} )}$ &=&$\displaystyle{ \Big\{ x\in N \colon -\sum_{i=1}^{\ell}x_i^2+\sum_{i=t +1}^{t+k-\ell+1}x_i^2= \frac{1}{c}},$ &$1>c>0$ \\
&&$ \displaystyle{-\sum_{i=\ell+1}^{t}x_i^2+\sum_{i=t+k-\ell +2}^{m+2}x_i^2= \frac{c-1}{c} \Big\}}$ &$1\leq k \leq m-1$\\
&&&$0\leq \ell \leq k$\\
&&&$0\leq \ell \leq t-1$\\
&&&$t-\ell-1 \leq m-k$\\
\hline
\end{tabular}
\vspace{1mm}
\caption{}
\label{Hypersurfaces-pseudo-sphere-notations2}
\end{table}
}
The importance of the families of hypersurfaces described in Table~\ref{Hypersurfaces-pseudo-sphere} lays in the fact {that} essentially any pseudo-Riemannian hypersurface of ${\mathbb S}^{m+1}_t$ with diagonalizable shape operator having at most two distinct constant principal curvatures is locally congruent to one of these (see Theorem 5.1 of \cite{Abe}, and also Theorem~\ref{Th-nostro-rigidity-2-curv}, for more details).
\begin{remark}\label{Rem-basta-c>0}
A similar family of hypersurfaces of pseudo-hyperbolic spaces is available, but we omit its description because all the properties of $r$-harmonic submanifolds in ${\mathbb H}^{m+1}_{t^*}$ can be deduced from those of $r$-harmonic submanifolds in ${\mathbb S}^{m+1}_{t}$, with $t=m+1-t^*$. This is a consequence of the fact that, up to a multiplicative constant factor -1 in the metric (that determines a {non-isometric} transformation), irrelevant for the $r$-harmonic equation, we can identify ${\mathbb S}^{m+1}_t$ with ${\mathbb H}^{m+1}_{m+1-t}$. For this reason, without loss of generality, in this paper all the existence and classification results for $r$-harmonic submanifolds that do not depend on a given signature of the metric will be stated only for the cases that the curvature of the ambient space is either $c=1$ or $c=0$.
\end{remark}
Next, to prepare the ground for the study of $r$-harmonicity, we consider a general smooth map $\varphi:(M^m_{t'},g) \to (N^n_t,h)$ between two pseudo-Riemannian manifolds. Let us denote $\nabla^{\varphi}$ the induced connection on the pull-back bundle $\varphi ^{-1}TN$. In the pseudo-Riemannian context, the operator corresponding to the classical Riemannian \textit{rough Laplacian} on sections of $\varphi^{-1} TN$, which will still be denoted $\overline{\Delta}$, becomes
\begin{equation} \label{roughlaplacian-pseudo}
\overline{\Delta}=d^* d =-\sum_{i=1}^m\varepsilon_i\left(\nabla^{\varphi}_{e_i}
\nabla^{\varphi}_{e_i}-\nabla^{\varphi}_
{\nabla^M_{e_i}e_i}\right)\,,
\end{equation}
where again $\{e_i\}_{i=1}^m$ is a local orthonormal frame field tangent to $M^m_{t'}$.
We are now in the right position to summarise the key points which enable us to describe the generalisation of the notions of harmonicity and, more generally, $r$-harmonicity to the pseudo-Riemannian context.
A smooth map $\varphi:(M^m_{t'},g) \to (N^n_t,h)$ between two pseudo-Riemannian manifolds is \textit{harmonic} if its tension field vanishes identically, i.e.,
\begin{equation}\label{tension-field-pseudo}
\tau(\varphi)=\trace \nabla d \varphi= \sum_{i=1}^m \varepsilon_i \left [ \nabla^{\varphi}_{e_i}d\varphi(e_i)-d \varphi \left (\nabla^M_{e_i} e_i \right )\right ] =0\,,
\end{equation}
where $\{e_i \}_{i=1}^m$ is a local orthonormal frame field on $M^m_{t'}$ as above. As for papers and examples in this context, we cite \cite{Konderak, Ratto}.
Similarly, taking into account \eqref{roughlaplacian-pseudo}, we can define the $r$-energy for a map between two pseudo-Riemannian manifolds precisely as in \eqref{2s-energia}, \eqref{2s+1-energia}. In the Riemannian case, the explicit expression for the $r$-tension field associated to $E_r(\varphi)$ was obtained by Maeta and Wang (see \cite{Maeta1, Wang}). Essentially, in the pseudo-Riemannian context, the only relevant difference appears when one has to take a trace (for instance, see the last equation in \eqref{key-formulas-pseudo}). More specifically, we have the following expression for the $r$-tension field, where $\overline{\Delta}$ and $\tau$ are given in \eqref{roughlaplacian-pseudo} and \eqref{tension-field-pseudo} respectively:
\begin{eqnarray}\label{2s-tension}
\tau_{2s}(\varphi)&=&\overline{\Delta}^{2s-1}\tau(\varphi)-\varepsilon_i R^N \left(\overline{\Delta}^{2s-2} \tau(\varphi), d \varphi (e_i)\right ) d \varphi (e_i) \nonumber\\
&& -\, \sum_{\ell=1}^{s-1}\, \left \{\varepsilon_i R^N \left( \nabla^\varphi_{e_i}\,\overline{\Delta}^{s+\ell-2} \tau(\varphi), \overline{\Delta}^{s-\ell-1} \tau(\varphi)\right ) d \varphi (e_i) \right .\\ \nonumber
&& \qquad \qquad -\, \left . \varepsilon_i R^N \left( \overline{\Delta}^{s+\ell-2} \tau(\varphi),\nabla^\varphi_{e_i}\, \overline{\Delta}^{s-\ell-1} \tau(\varphi)\right ) d \varphi (e_i) \right \} \,\, ,
\end{eqnarray}
where $\overline{\Delta}^{-1}=0$ and $\{e_i\}_{i=1}^m$ is a local orthonormal frame field tangent to $M^m_{t'}$ (the sum over $i$ is not written but understood). Similarly,
\begin{eqnarray}\label{2s+1-tension}
\tau_{2s+1}(\varphi)&=&\overline{\Delta}^{2s}\tau(\varphi)-\varepsilon_i R^N \left(\overline{\Delta}^{2s-1} \tau(\varphi), d \varphi (e_i)\right ) d \varphi (e_i)\nonumber \\
&& -\, \sum_{\ell=1}^{s-1}\, \left \{\varepsilon_i R^N \left( \nabla^\varphi_{e_i}\,\overline{\Delta}^{s+\ell-1} \tau(\varphi), \overline{\Delta}^{s-\ell-1} \tau(\varphi)\right ) d \varphi (e_i) \right .\\ \nonumber
&& \qquad \qquad -\, \left . \varepsilon_i R^N \left( \overline{\Delta}^{s+\ell-1} \tau(\varphi),\nabla^\varphi_{e_i}\, \overline{\Delta}^{s-\ell-1} \tau(\varphi)\right ) d \varphi (e_i) \right \} \\ \nonumber
&& \,-\,\varepsilon_i R^N \Big( \nabla^\varphi_{e_i}\,\overline{\Delta}^{s-1} \tau(\varphi), \overline{\Delta}^{s-1} \tau(\varphi)\Big ) d \varphi (e_i)\,\,.
\end{eqnarray}
From the analytic point of view, one of the major differences with respect to the Riemannian case is the fact the PDE's system $\tau_r(\varphi)=0$ is \textit{not elliptic} when $1 \leq t' \leq m-1$.
\section{Statement of the results}\label{Sec-results}
In the Riemannian case, when the ambient space form has nonpositive sectional curvature there are several results which assert that, under suitable conditions, an $r$-harmonic submanifold is minimal (see \cite{Chen}, \cite{Maeta1}, \cite{Maeta4} and \cite{Na-Ura}, for instance). Things drastically change when the ambient is the Euclidean sphere ${\mathbb S}^{m+1}$. Indeed, in this case several examples of proper $r$-harmonic hypersurfaces have been constructed and studied (see \cite{BMOR1, Maeta2, MOR-space forms, Mont-Ratto4} and references therein).
In the pseudo-Riemannian setting there are in the literature several interesting results, but most of them are limited to the biharmonic case. One of the instances which has attracted more attention is the study of space-like biharmonic hypersurfaces in a Lorentzian space form. In this case, it seems that the sign of the curvature of the ambient produces phenomena which are, in some sense, dual with respect to the Riemannian case. In other words, positive curvature increases the rigidity of space-like biharmonic hypersurfaces. For example, the following interesting result was proved by Ouyang:
\begin{theorem}\label{Th-Ouyang} \cite{Ouyang} Let $M^m$ be a CMC space-like biharmonic hypersurface in either ${\mathbb R}^m_1$ or ${\mathbb S}^{m+1}_1$. Then $M^m$ is minimal.
\end{theorem}
In this order of ideas, we also have:
\begin{theorem}\label{Th-Zhang} \cite{Zhang} Let $M$ be a complete, space-like biharmonic surface in ${\mathbb R}^3_1$ or ${\mathbb S}^{3}_1$. Then $M$ must be totally geodesic, i.e., ${\mathbb R}^2$ or ${\mathbb S}^2$.
\end{theorem}
This trend is confirmed in the $r$-harmonic case ($r \geq 3$), as we shall show in Corollary~\ref{Cor-no-space-like}.
By contrast, when the index of the hypersurface is positive, some results in the biharmonic case are available but their interpretation is less evident. For the purpose of comparison with the results of this paper, we report here, using our notations, the following interesting result of Liu and Du (see Theorem\link1.2 of \cite{Liu}, where the cases $c\neq 0,1$ are also dealt with explicitly):
\begin{theorem}\label{Th-Liu-Dong} \cite{Liu} Let $M^m_{t'}$ be a pseudo-Riemannian proper biharmonic hypersurface in $N^{m+1}_t(c)$. If $M^m_{t'}$ has diagonalizable shape operator with at most two distinct principal curvatures, then $c\neq 0$. Furthermore, when $c =1$, then $t'=t$ and $M^m_{t'}$ is congruent to either ${\mathbb S}^m_t(2)$ or ${\mathbb S}^{m_1}_{t_1}(2) \times {\mathbb S}^{m-m_1}_{t-t_1}( 2 )$ with $m_1 \neq m-m_1$.
\end{theorem}
\begin{remark} When $c=1$, our statement of Theorem~\ref{Th-Liu-Dong} is equivalent to Theorem\link1.2 of \cite{Liu}, but our formulation makes it easier the comparison with the classical results for biharmonic hypersurfaces in the Riemannian case.
\end{remark}
The results of \cite{Liu} in the biharmonic case were recently refined in \cite{Dong}. Finally, for the sake of completeness, we also have to cite the interesting paper \cite{Sasahara} by Sasahara, where the author classifies proper biharmonic curves and surfaces in de Sitter 3-space and anti-de Sitter 3-space. In Theorem~\ref{Th-Sasah-r>2} below, we shall extend this classification for surfaces in $N^3_1(c)$ to the case $r\geq3$.
Our investigation of this type of problems in the pseudo-Riemannian context starts with the following general result, which is a pseudo-Riemannian version of Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0}. To the purpose of a quick comparison between Theorems~\ref{Th-existence-hypersurfaces-c>0-e-c<0} and \ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo}, we point out that, if the induced metric on the hypersurface is Riemannian, then $\trace A^2=||A||^2$.
\begin{theorem}\label{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} Assume that $m \geq2$ and $1 \leq t \leq m$. Let $M^m_{t'}$ be a non-minimal CMC pseudo-Riemannian hypersurface in a pseudo-Riemannian space form $N^{m+1}_t(c)$ and assume that $\trace A^2$ is constant. Then
$M^m_{t'}$ is proper biharmonic if and only if
\begin{equation}\label{2-harmonicity-condition-in-general-pseudo}
\varepsilon \trace A^2-m\,c=0 \,.
\end{equation}
If $r \geq 3$, then
$M^m_{t'}$ is proper $r$-harmonic if and only if either
\begin{equation}\label{eq-Tr-A^2=0}
\trace A^2=0
\end{equation}
or
\begin{equation}\label{r-harmonicity-condition-in-general-pseudo}
\varepsilon \left (\trace A^2\right )^2-m\,c\,\left (\trace A^2\right )-(r-2)m^2 \,c\, \alpha^2=0 \,,
\end{equation}
where $\alpha$ denotes the mean curvature of $M^m_{t'}$ and $\varepsilon=\langle \eta,\eta\rangle$.
\end{theorem}
\begin{remark} The special case $r=2$ in Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} is part of the result proved in \cite{Liu}, where the condition of biharmonicity for a general hypersurface $M^m_{t'} \hookrightarrow N^{m+1}_t(c)$ was computed. In the case of surfaces \eqref{2-harmonicity-condition-in-general-pseudo} was first obtained in \cite{Sasahara}.
\end{remark}
\begin{remark} It is important to point out that in Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} we do not require that the shape operator $A$ be diagonalizable. By way of example, in Theorem~\ref{Th-Xiao-c=-1} we shall exhibit a new family of $r$-harmonic surfaces in ${\mathbb H}^3_1$ whose shape operator is \textit{not} diagonalizable.
\end{remark}
A first, immediate consequence of Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} is the following
\begin{corollary}\label{Cor-no-space-like} Assume $r \geq 3$ and $m\geq2$. Let $M^m$ be a space-like, $r$-harmonic CMC hypersurface in a Lorentzian space form $N^{m+1}_1(c)$. If $\trace A^2$ is constant and $c \geq 0$, then $M^m$ is minimal.
\end{corollary}
In the same spirit:
\begin{corollary}\label{Cor-no-space-like-bis} Assume that $r \geq 3$, $m \geq2$ and $1 \leq t \leq m$. Let $M^m_{t'}$ be a pseudo-Riemannian $r$-harmonic CMC hypersurface in $N^{m+1}_t(c)$. If $\trace A^2$ is a positive constant and $\varepsilon c < 0$, then $M^m_{t'}$ is minimal.
\end{corollary}
Next, we shall use Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} to construct new examples of $r$-harmonic hypersurfaces. Indeed,
\begin{corollary}\label{Cor-r-harmonic-pseudo-hyperspheres} Assume that $r \geq 3$, $m \geq2$ and $1 \leq t \leq m$. Let ${\mathbb S}^m_{t}(c)$ be a small pseudo-hypersphere in ${\mathbb S}^{m+1}_t$. Then ${\mathbb S}^m_{t}(c)$ is proper $r$-harmonic if and only if $c=r$.
\end{corollary}
With the notation of Tables~\eqref{Hypersurfaces-pseudo-sphere}-\eqref{Hypersurfaces-pseudo-sphere-notations1}-\eqref{Hypersurfaces-pseudo-sphere-notations2}:
\begin{theorem}\label{Th-r-harmonic-pseudo-Clifford} Assume that $r \geq 3$, $m \geq2$ and $1 \leq t \leq m$. Let $M^m_t={\mathbb S}^{k}_{\ell}(c) \times {\mathbb S}^{m-k}_{t-\ell}\left (\frac{c}{c-1}\right )$ $(c>1)$ be a generalised pseudo-Clifford torus in ${\mathbb S}^{m+1}_t$. Then $M^m_t$ is proper $r$-harmonic if and only if $c \neq \frac{m}{k}$ and
\begin{equation}\label{condizione-pseudo-tori-Clifford}
P_3(c)=k c^3 - k (r+2) c^2 + [m (r-1) + k (r+2)] c - m r=0\,.
\end{equation}
\end{theorem}
\begin{remark} The third order polynomial $P_3(c)$ in \eqref{condizione-pseudo-tori-Clifford} is equivalent to the one which appears in the Riemannian case. To see this, it is enough to set
\[
k=p \,\,; \quad m-k=q \,\,; \quad c=\frac{1}{x} \,.
\]
Then, up to a constant factor $-1/x^3$, $P_3(c)$ becomes the third order polynomial obtained in Theorem\link1.2 of \cite{Mont-Ratto4}. We point out that, in the special case that $k=m-k$, we have
\[
P_3(c)=(-2 + c) k (c^2 + r - c r) \,.
\]
Therefore, when $k=m-k$, the pseudo-Clifford torus ${\mathbb S}^{k}_{\ell}(c) \times {\mathbb S}^{m-k}_{t-\ell}\left (\frac{c}{c-1}\right )$ is proper $r$-harmonic in ${\mathbb S}^{m+1}_t$ if and only if
\begin{equation}\label{eq-k=m-k}
c=\frac{r \pm \sqrt{r^2-4r}}{2} \qquad \,\, (r \geq 5) \,.
\end{equation}
For a more detailed discussion on the existence and qualitative behaviour of admissible roots of $P_3(c)$ in the general case we refer to \cite{Mont-Ratto4}.
\end{remark}
Next, we show that the examples given in Corollary~\ref{Cor-r-harmonic-pseudo-hyperspheres} and Theorem~\ref{Th-r-harmonic-pseudo-Clifford} are the only possible ones within a certain class of hypersurfaces. More precisely, we prove the following result:
\begin{theorem}\label{Th-nostro-rigidity-2-curv} Assume that $r \geq 3$, $m \geq2$ and $1 \leq t \leq m$. Let $M^{m}_{t'}$ be a pseudo-Riemannian hypersurface with diagonalizable shape operator in a pseudo-Riemannian space form $N^{m+1}_t (c)$, $c=0$ or $c=1$. Assume that there exist at most two distinct principal curvatures and that they are constant on $M^{m}_{t'}$. If $M^{m}_{t'}$ is proper $r$-harmonic, then $M^{m}_{t'}$ is one of the examples given in Corollary~\ref{Cor-r-harmonic-pseudo-hyperspheres} or Theorem~\ref{Th-r-harmonic-pseudo-Clifford}.
\end{theorem}
\begin{remark} Of course, it is straightforward to state explicitly the version of Corollary~\ref{Cor-r-harmonic-pseudo-hyperspheres}, Theorem~\ref{Th-r-harmonic-pseudo-Clifford} and Theorem~\ref{Th-nostro-rigidity-2-curv} in the case that $c=-1$. However, for the reasons explained in Remark~\ref{Rem-basta-c>0}, we omit the details.
\end{remark}
Next, we obtain some geometric results for triharmonic surfaces which provide a version in the pseudo-Riemannian case of some facts which we proved in \cite{MOR-space forms} in the Riemannian case.
\begin{theorem}\label{Th-non-existence-surfaces-c<=0}
Assume $r \geq 3$. Let $M^2_{t'}$ be a pseudo-Riemannian triharmonic CMC surface in $N^{3}_t(c)$ and assume that its shape operator $A$ is diagonalizable. If $\varepsilon c \leq 0$, then $M^2_{t'}$ is minimal.
\end{theorem}
Next, we focus on the case that the ambient space is ${\mathbb S}^{3}_t$. Our result is:
\begin{theorem}\label{Th-structure-surfaces-c>0}
Let $M^2_{t'}$ be a CMC proper triharmonic surface in ${\mathbb S}^3_t$ and assume that its shape operator $A$ is diagonalizable. Then $M^2_{t'}$ is an open part of the small pseudo-hypersphere ${\mathbb S}^2_t(3)$.
\end{theorem}
A widely studied family of hypersurfaces in the pseudo-Riemannian setting is that of isoparametric Lorentzian hypersurfaces of a Lorentzian space-form. For this specific topic, we refer to \cite{Magid, Xiao} and, for further background, to \cite{Hahn, Nomizu-pseudo}.
We recall that a Lorentzian hypersurface $M^m_1$ in a Lorentzian space form $N^{m+1}_1(c)$ is said to be \textit{isoparametric} if the minimal polynomial of the shape operator $A$ is constant on $M^m_1$.
We know by \cite[Proposition~2.1]{Hahn} that $M^m_1$ has constant principal curvatures with constant algebraic multiplicities.
Moreover, according to \cite[Chapter~9]{Neill} there exist bases where the shape operator $A$ assumes one of the following Jordan canonical forms:
\\
$
{\rm I}\quad \begin{bmatrix}
a_1&\cdots&0\\
\vdots& \ddots&\vdots\\
0&\cdots& a_m
\end{bmatrix}
\quad\hfill
{\rm II}\quad \begin{bmatrix}
a_0&0&&&\\
1&a_0&&&&\\
&&a_1&&\\
&&&\ddots&\\
&&&&a_{m-2}
\end{bmatrix}
$
$
{\rm III}\quad \begin{bmatrix}
a_0&0&0&&&\\
0&a_0&1&&&\\
-1&0&a_0&&&\\
&&&a_1&&\\
&&&&\ddots&\\
&&&&&a_{m-3}
\end{bmatrix}
\quad\hfill
{\rm IV}\quad\begin{bmatrix}
a_0&b_0&&&\\
-b_0&a_0&&&&\\
&&a_1&&\\
&&&\ddots&\\
&&&&a_{m-2}
\end{bmatrix}
$
\\
Here $b_0$ is assumed to be non-zero. In cases I, II and III the eigenvalues are
real, while $a_0 \pm i b_0$ are complex eigenvalues in case IV. A
Lorentzian isoparametric hypersurface in $N^{m+1}_1(c)$ is called of type I, II, III or IV according to the form of its shape operator $A$.
We observe that a Lorentzian isoparametric hypersurface in $N^{m+1}_1(c)$ is a CMC hypersurface with $\trace A^2$ constant. Therefore, Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} applies to this type of hypersurfaces.
Now, a direct computation shows that for a Lorentzian isoparametric hypersurface of type
I, II and III, with $\trace A\neq 0$, we have $\trace A^2>0$. Differently, for hypersurfaces of type IV it is possible to have $\trace A\neq 0$ and $\trace A^2\leq0$. If the curvature of the ambient space is $c=0$, we easily deduce from Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} that a non-minimal Lorentzian isoparametric hypersurface is $r$-harmonic ($r\geq 2$) if and only if $\trace A^2= 0$.
These observations prove our first result in this context, that is
\begin{proposition}\label{Th-magid} Assume $r \geq 2$ and $m\geq2$. Let $M^m_1$ be an isoparametric Lorentzian $r$-harmonic hypersurface in the flat Lorentzian space form ${\mathbb R}^{m+1}_1$. Then either $M^m_1$ is minimal or its shape operator is of type IV.
\end{proposition}
In \cite[Theorem~4.10]{Magid} Magid gave a proof that there exist no isoparametric Lorentzian hypersurfaces of type IV in the flat Lorentzian space form ${\mathbb R}^{m+1}_1$. However, it was pointed out in \cite{RamosVasquezLopez} that there could be some gaps in the arguments of \cite{Magid}.
By contrast, in the case that the curvature of the ambient space is $c=-1$, we shall construct some new examples of proper $r$-harmonic Lorentzian isoparametric surfaces in ${\mathbb H}^{3}_1$ with non-diagonalizable shape operator of type IV.
More precisely, first, as above, we observe that Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} implies that a Lorentzian isoparametric hypersurface in ${\mathbb H}^{m+1}_1$ can be $r$-harmonic only if its shape operator is non-diagonalizable and of type IV.
The following result proves that {\em complex circles} (see \cite{Magid2}) provide a geometrically significant example of such hypersurfaces:
\begin{theorem}\label{Th-Xiao-c=-1} Let $M^2_1$ be a complex circle in ${\mathbb H}^{3}_1\subset {\mathbb R}^4_2$ parametrized by
\[
\begin{split}
x(s,t)= &\big\{b \cos (s) \cosh (t)-a \sin (s)
\sinh (t),\\
&a \cos (s) \sinh (t)+b
\sin (s) \cosh (t),\\
&a \cos (s)
\cosh (t)+b \sin (s) \sinh (t),\\
&b
\cos (s) \sinh (t)-a \sin (s)
\cosh (t)\big\}\,,
\end{split}
\]
where $a$ and $b$ are real numbers such that $b^2-a^2=1$ and $ab\neq 0$.
Then $M^2_1$ is proper $r$-harmonic provided that either
\begin{enumerate}
\item $r>2$ and
\[
{a^2=\frac{\sqrt{2}}{2}-\frac{1}{2}\,,\quad
b^2=\frac{\sqrt{2}}{2}+\frac{1}{2}}
\]
or
\item $r=3$ and
\[
a^2=\frac{\sqrt{3}}{3}-\frac{1}{2}\,, \quad b^2=\frac{\sqrt{3}}{3}+\frac{1}{2}\,.
\]
\end{enumerate}
\end{theorem}
\begin{remark} The shape operator $A$ of the $r$-harmonic complex circles obtained in Theorem~\ref{Th-Xiao-c=-1} is non-diagonalizable and of type IV. As we shall see in the proof of Theorem~\ref{Th-Xiao-c=-1}, the instances of Case $(1)$ have $\trace A^2=0$, while those of Case $(2)$ have $\trace A^2<0$. In accordance with the result of Sasahara (see \cite[Theorem~5.4]{Sasahara}), the family of isoparametric surfaces studied in Theorem~\ref{Th-Xiao-c=-1} does not contain any proper biharmonic immersion.
\end{remark}
Next, we recall that a \textit{null curve} $\gamma(s)$ in $N^3_1(c) \subset {\mathbb R}^4_t$ is a smooth curve such that $\langle \gamma'(s), \gamma'(s)\rangle \equiv 0$. A {\em $B$-scroll} over a null curve $\gamma(s)$ is a surface of index $1$ in $N^3_1(c)$ parametrized by
\begin{equation}\label{eq-B-Scroll}
x(s,u)=\gamma(s)+u\,B(s) \,,
\end{equation}
where $\{A,B,C \}$ is a pseudo-orthonormal frame field, or a \textit{Cartan frame field}, along $\gamma(s)$, i.e.,
\begin{equation}\label{eq-Cartan-frame}
\begin{split}
\langle A,A\rangle &=\langle B,B\rangle =0\,, \qquad \langle A,B\rangle=-1 \,, \\
\langle A,C\rangle &=\langle B,C\rangle =0\,, \qquad \langle C,C\rangle=1\\
&{\rm and} \\
\gamma'(s)&=A(s) \,,\\
C'(s)&=- \lambda \, A(s)- k(s)\,B(s) \,,
\end{split}
\end{equation}
where $\lambda$ is a real constant and $k(s) \neq 0$ (see \cite{Alias, Hahn, Sasahara} for details).
We obtain a version of \cite[Theorem~5.4]{Sasahara} in the case that $r \geq3$. More precisely, we shall prove:
\begin{theorem}\label{Th-Sasah-r>2} Assume that $r \geq3$. Let $M^2_{t'}$ be an isoparametric pseudo-Riemannian surface in a $3$-dimensional Lorentz space form $N^3_1(c)$, where $c \in \{-1,1\}$.
Then $M^2_{t'}$ is proper $r$-harmonic if and only if it is congruent to an open subset of one of the following:
\begin{itemize}
\item[(1)] ${\mathbb S}^2_1(r) \subset {\mathbb S}^3_1$;
\item[(2)] ${\mathbb H}^2(-r) \subset {\mathbb H}^3_1$;
\item[(3)] ${\mathbb S}^{1}_{\ell}(c) \times {\mathbb S}^{1}_{1-\ell}\left (\frac{c}{c-1}\right ) \subset {\mathbb S}^3_1$, where $0 \leq \ell \leq 1$ and
\begin{equation*}\label{eq-condiz-c}
c=\frac{r \pm \sqrt{r^2-4r}}{2} \qquad \,\, {(r \geq 5)} \,;
\end{equation*}
\item[(4)] ${\mathbb H}^{1}_{1-\ell}(-c) \times {\mathbb H}^{1}_{\ell}\left (-\,\frac{c}{c-1}\right ) \subset {\mathbb H}^3_1$, where $0 \leq \ell \leq 1$ and $c$ is as in Case~{\rm (3)};
\item[(5)] a $B$-scroll over a null curve in ${\mathbb S}^3_1$ whose Gauss curvature $K$ is constant and equal to $r$;
\item[(6)] an $r$-harmonic complex circle in ${\mathbb H}^3_1$ of the type described in Theorem~\ref{Th-Xiao-c=-1}.
\end{itemize}
\end{theorem}
\begin{remark} Comparison of \cite[Theorem~5.4]{Sasahara} with our Theorem~\ref{Th-Sasah-r>2} shows that the relevant differences between the biharmonic case and the case $r \geq 3$ is the appearance in Theorem~\ref{Th-Sasah-r>2} of the family of solutions of type $(6)$ for all $r \geq3$, and of type $(3)$ and $(4)$ for $r \geq 5$. These facts, together with Theorem~\ref{Th-Xiao-c=-1}, support the following general idea: we can find geometrically interesting situations where there is no biharmonic instance, but there exist examples of $r$-harmonic immersions, $r \geq 3$. Moreover, comparing the result of the present paper with those of \cite{MOR-space forms}, we see that the notion of $r$-harmonicity is more flexible in the pseudo-Riemannian setup compared to the Riemannian case as it allows for a larger class of solutions. In this order of ideas, we also cite the phenomenon illustrated in Remark~\ref{Remark-appendix} below.
\end{remark}
\begin{example}\label{Example-B-scroll} First, we point out that the family of $r$-harmonic surfaces in ${\mathbb S}^3_1$ obtained in Case (5) of Theorem~\ref{Th-Sasah-r>2} is very ample. This is a consequence of the fact that, given any real constant $\lambda$ and smooth function $k(s)$, it is always possible to determine (at least locally) a null curve $\gamma$ such that its associated $B$-scroll \eqref{eq-B-Scroll} verifies \eqref{eq-Cartan-frame}. As proved in \cite{Kim}, the existence of such a null curve $\gamma$ can be deduced by solving a suitable Cauchy problem for a first order linear system of ordinary differential equations. In the proof of Theorem~\ref{Th-Sasah-r>2} we shall show that $K=\lambda^2+1$ and any such surface is proper $r$-harmonic provided that $\lambda^2=r-1\, (r \geq 2)$.
As a special case, following the procedure described in the Appendix of \cite{Kim}, here we give the explicit expression of the null curve $\gamma(s)$ and of its associated vector field $B(s)$ assuming that $k(s) \equiv 1$. Note that when, as in our example, the function $k(s)$ is bounded, $\gamma(s)$ is defined on the whole ${\mathbb R}$. To this end, let
{\footnotesize
\[
\begin{split}
\gamma(s)=&\Big\{
\frac{\sqrt{c} (c+d-2) \sin(\sqrt{d} s)+2 c \cos
(\sqrt{d} s)+\sqrt{d} (c+d+2) \sinh (\sqrt{c}
s)+2 d \cosh(\sqrt{c} s)}{2 (c+d)},\\
&
\frac{
\sqrt{c} (c+d) \sin(\sqrt{d} s)+2 c \cos(\sqrt{d}
s)+ \sqrt{d} (c+d) \sinh(\sqrt{c} s)+2d
\cosh(\sqrt{c} s)}{2(c+d)},\\
&
\frac{-\sqrt{c} \sin(\sqrt{d} s)+c \cos(\sqrt{d} s)+\sqrt{d} \sinh(\sqrt{c} s)+d \cosh(\sqrt{c}
s)}{c+d},\\
&
\frac{\cos(\sqrt{d} s)-\cosh(\sqrt{c}
s)}{c+d}\Big\}
\end{split}
\]
}
where
\[
c=\sqrt{1+\lambda^2}+\lambda\,,\quad d=\sqrt{1+\lambda^2}-\lambda\,.
\]
Then $\gamma(s)$ is a null curve in ${\mathbb S}^3_1$ with associated vector field
{\footnotesize
\[
\begin{split}
B(s)=&\frac{1}{4}\Big\{2 \sqrt{c} \sin(\sqrt{d} s)+(2-c-d)
\cos(\sqrt{d} s)+2 \sqrt{d} \sinh(\sqrt{c}
s)+(2+c+d) \cosh(\sqrt{c} s) ,\\
&
2 \sqrt{c} \sin(\sqrt{d} s)+2 \sqrt{d} \sinh(\sqrt{c} s)+(c+d) \left(\cosh(\sqrt{c} s)-\cos(\sqrt{d} s)\right),\\
&
2\sqrt{c} \sin(\sqrt{d} s)+2\sqrt{d} \sinh(\sqrt{c} s)+2\cosh(\sqrt{c} s)+2\cos(\sqrt{d}
s),\\
&
2\sqrt{d} \sin(\sqrt{d}
s)-2\sqrt{c} \sinh(\sqrt{c} s)\Big\}
\end{split}
\]
}
and the parametrized surface
\[
x(s,u)=\gamma(s)+u\,B(s)
\]
defines a $B$-scroll with Gauss curvature $K=1+\lambda^2$ and $k(s)\equiv1$. Choosing $\lambda=\pm\sqrt{r-1}$ we obtain the desired $r$-harmonic surface in ${\mathbb S}^3_1$.
For the sake of completeness, the method of computation used to determine the curve $\gamma(s)$ and the vector field $B(s)$ will be given at the end of this paper, in Appendix~\ref{Appendix}.
\end{example}
\section{Proofs}\label{Sec-proofs}
As a preliminary step, in the following lemma we state without proof some standard facts which we shall use in this section.
\begin{lemma}\label{Lemma-tecnico-1} Let $\varphi:M^m_{t'} \to N^{m+1}_t(c)$ be a pseudo-Riemannian hypersurface. Let $A$ denote the shape operator and $f=(1/m)\varepsilon \trace A$ the mean curvature function. Then
\begin{itemize}
\item[{\rm (a)}] $(\nabla A) (\cdot,\cdot)$ is symmetric;
\item[{\rm (b)}] $\langle (\nabla A) (\cdot,\cdot), \cdot \rangle$ is totally symmetric;
\item[{\rm (c)}] $\trace (\nabla A) (\cdot,\cdot)= m \,\varepsilon\,\grad f$.
\end{itemize}
\end{lemma}
Next, we perform our first computation:
\begin{lemma}\label{Lemma-Delta-H} Let $\varphi:M^m_{t'} \to N^{m+1}_t(c)$ be a pseudo-Riemannian hypersurface and denote by $\eta$ the unit normal vector field. Then
\begin{equation}\label{Delta-H-formula}
\overline{\Delta} {\mathbf H}= (\Delta f + f \varepsilon \trace A^2) \eta + 2 A(\grad f ) + m f \varepsilon \grad f \,.
\end{equation}
\end{lemma}
\begin{proof} We work with a geodesic frame field $\left \{ X_i \right \}_{i=1}^m$ around an arbitrarily fixed point $p \in M^m_{t'}$. Also, we simplify the notation writing $\nabla$ for $\nabla^{M}$. Since ${\mathbf H}=f \eta$, around $p$ we have:
\[
\nabla_{X_i}^{\varphi} {\mathbf H}=\nabla_{X_i}^{\perp} {\mathbf H}-A_{{\mathbf H}}(X_i)=\left ( X_i f \right )\eta -f A \left (X_i \right )\,.
\]
Then at $p$ we have:
\begin{eqnarray*}
\nabla_{X_i}^{\varphi}\nabla_{X_i}^{\varphi} {\mathbf H}&=&\left(X_i X_i f \right )\eta- \left( X_i f \right )A \left (X_i \right )\\
&&- \left( X_i f \right )A \left (X_i \right )-f \big( \nabla_{X_i} A \left (X_i \right )+B\left( X_i, A \left (X_i \right )\right )\big )\nonumber\\
&=&\left(X_i X_i f \right )\eta- 2\left( X_i f \right )A \left (X_i \right )-f (\nabla A)(X_i,X_i)\\
&&-f\varepsilon \langle A(X_i), A(X_i)\rangle \eta \,,
\label{eq:nabla2H}
\end{eqnarray*}
where, for the second equality, we also used \eqref{eq-legame-A-B}.
Now we take the sum over $i$ as in \eqref{roughlaplacian-pseudo} and, using \eqref{key-formulas-pseudo} and Lemma~\ref{Lemma-tecnico-1}, we obtain \eqref{Delta-H-formula} (note that the sign convention for $\Delta$ and $\overline{\Delta}$ is given in \eqref{key-formulas-pseudo}, \eqref{roughlaplacian-pseudo}).
\end{proof}
Next, we assume that the mean curvature function $f$ is constant and we obtain
\begin{lemma}\label{Lemma-Delta2-H} Let $\varphi:M^m_{t'} \to N^{m+1}_t(c)$ be a pseudo-Riemannian hypersurface and assume that its mean curvature function $f$ is equal to a constant $\alpha$. Then
\begin{equation}\label{Delta-2-H-formula}
\overline{\Delta}^2 {\mathbf H}= \alpha \left (\varepsilon\Delta \trace A^2+ (\trace A^2)^2 \right ) \eta + 2 \alpha \varepsilon A \big(\grad \trace A^2 \big ) \,.
\end{equation}
\end{lemma}
\begin{proof} Since $f$ is constant, according to Lemma~\ref{Lemma-Delta-H} we have $\overline{\Delta}{\mathbf H}=\alpha \varepsilon \trace A^2 \eta$ and so
\[
\overline{\Delta}^2{\mathbf H}=\alpha \varepsilon\overline{\Delta}\big ( \trace A^2 \eta \big)\,.
\]
Now, around $p$:
\[
\nabla_{X_i}^{\varphi} \big ( \trace A^2 \eta\big )=\big ( X_i \trace A^2 \big )\eta -\trace A^2 A\left (X_i \right )\,.
\]
At $p$:
\begin{eqnarray*}
\nabla_{X_i}^{\varphi}\nabla_{X_i}^{\varphi} \big ( \trace A^2 \eta\big )&=&\left(X_i X_i \trace A^2 \right )\eta- \left( X_i \trace A^2 \right )A \left (X_i \right )\\
&&- \left( X_i \trace A^2 \right )A \left (X_i \right )-\trace A^2 \big( \nabla_{X_i} A \left (X_i \right )+B\left( X_i, A \left (X_i \right )\right )\big )\,.\\
\end{eqnarray*}
Next, computing as in Lemma~\ref{Lemma-Delta-H}, we find
\[
\overline{\Delta}\big ( \trace A^2 \eta \big)=\left (\Delta \trace A^2 \right ) \eta+ 2 A \big(\grad \trace A^2\big ) +m \trace A^2 \varepsilon\grad f + \varepsilon (\trace A^2)^2 \eta
\]
and, since $f$ is constant, the proof ends immediately.
\end{proof}
We are now in the right position to prove our first theorem.
\begin{proof}[Proof of Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo}] As $\trace A^2$ is constant, it follows from Lemma~\ref{Lemma-Delta2-H} that
\begin{equation}\label{Delta-H-A2-constant}
\overline{\Delta}^2 {\mathbf H}= \alpha (\trace A^2)^2 \eta\,.
\end{equation}
Now, from the computations above, we observe that
\begin{equation}\label{Delta-eta}
\overline{\Delta}\eta=\varepsilon \trace A^2 \eta \,.
\end{equation}
Next, putting together \eqref{Delta-H-formula}, \eqref{Delta-H-A2-constant} and \eqref{Delta-eta}, we easily deduce that
\begin{equation}\label{Delta-H-potenza-p}
\overline{\Delta}^p {\mathbf H}= \alpha \,\varepsilon ^p \left(\trace A^2\right)^{p} \eta \quad \quad \forall p \in {\mathbb N}^* \,.
\end{equation}
Now we are in a good position to perform the explicit calculation of the $r$-tension field $\tau_{r}(\varphi)$ described in \eqref{2s-tension}, \eqref{2s+1-tension}. We begin with $\tau_{2s}(\varphi)$, $s \geq2$. Using \eqref{Delta-H-potenza-p}, \eqref{tensor-curvature-N(c)} and computing we obtain (as in \eqref{2s-tension}, we omit to write the sum over $i$):
\begin{eqnarray*}
\frac{1}{m} \tau_{2s}(\varphi)&=&\alpha \varepsilon\left(\trace A^2\right)^{2s-1} \eta-c \,\Big \{ \varepsilon_i\langle d\varphi(X_i),d\varphi(X_i) \rangle \alpha \left(\trace A^2\right)^{2s-2} \eta \\
&& - \varepsilon_i \langle d\varphi(X_i), \alpha \left(\trace A^2\right)^{2s-2} \eta \rangle d\varphi(X_i) \Big \}\\
&& -c\, m\varepsilon \sum_{\ell=1}^{s-1} \Big \{ \varepsilon_i\langle d\varphi(X_i), \alpha \left(\trace A^2\right)^{s-\ell-1} \eta \rangle \big ( -\alpha \left(\trace A^2\right)^{s+\ell-2} A(X_i) \big ) \\
&& - \varepsilon_i\langle d\varphi(X_i), -\alpha \left(\trace A^2\right)^{s+\ell-2} A(X_i) \rangle \alpha \left(\trace A^2\right)^{s-\ell-1} \eta \Big \}\\
&&+c\, m \varepsilon\sum_{\ell=1}^{s-1} \Big \{\varepsilon_i \langle d\varphi(X_i),- \alpha \left(\trace A^2\right)^{s-\ell-1} A(X_i) \rangle \alpha \left(\trace A^2\right)^{s+\ell-2} \eta \\
&& - \varepsilon_i\langle d\varphi(X_i), \alpha \left(\trace A^2\right)^{s+\ell-2} \eta \rangle \big ( -\alpha \left(\trace A^2\right)^{s-\ell-1} A(X_i) \big ) \Big \}\\
&=&\alpha\varepsilon \left(\trace A^2\right)^{2s-1} \eta-c \, m \alpha \left(\trace A^2\right)^{2s-2} \eta \\
&& -c\, \varepsilon m \Big \{ \sum_{\ell=1}^{s-1} \big [\varepsilon m \alpha^3 \left(\trace A^2\right)^{2s-3} \eta \big ] + \sum_{\ell=1}^{s-1} \big [ \varepsilon m \alpha^3 \left(\trace A^2\right)^{2s-3} \eta \big ] \Big \}\\
&=&\alpha \left(\trace A^2\right)^{2s-3} \Big \{\varepsilon \left(\trace A^2\right)^2-m \,c\, \trace A^2- (2s-2)m^2\,c\,\alpha^2 \Big \}\eta\,.
\end{eqnarray*}
This completes the proof in the case $r=2s$. The case $r=2s+1$ is similar and so we omit the details.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{Cor-no-space-like}] $\,$
Since a space-like hypersurface is Riemannian, $\trace A^2=||A||^2$. Then, either the hypersurface is totally geodesic, or it follows from \eqref{r-harmonicity-condition-in-general-pseudo} with $\varepsilon=-1$ that $M^m$ cannot be proper $r$-harmonic, thus the only possibility is that $M^m$ is minimal, that is $\alpha=0$.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{Cor-no-space-like-bis}] Since, by assumption, $\trace A^2$ is a positive constant, condition \eqref{eq-Tr-A^2=0} does not hold. Similarly, since $\varepsilon$ and $c$ have opposite sign, equation \eqref{r-harmonicity-condition-in-general-pseudo} cannot be verified. Therefore, the only possibility is that the hypersurface is minimal.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{Cor-r-harmonic-pseudo-hyperspheres}] We use \eqref{mean-curv-fuction-pseudo}, \eqref{norma-A} and Table~\ref{Hypersurfaces-pseudo-sphere} to compute $\trace A^2=m \,(c-1)$ and $\alpha=\sqrt{c-1}$. Then the thesis follows by direct substitution in \eqref{r-harmonicity-condition-in-general-pseudo}, where the curvature of the ambient space is $1$ and also $\varepsilon=1$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Th-r-harmonic-pseudo-Clifford}] The hypotheses of Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} are verified and so the condition for $r$-harmonicity is equation \eqref{r-harmonicity-condition-in-general-pseudo} where, as in the proof of Corollary~\ref{Cor-r-harmonic-pseudo-hyperspheres}, the curvature of the ambient space is $1$ and $\varepsilon=1$. Then we compute using the explicit expression for the shape operator given in Table~\ref{Hypersurfaces-pseudo-sphere}:
\begin{eqnarray*}
\trace A^2&=& k(c-1)+\frac{(m-k)}{(c-1)}\,; \\
\alpha^2&=&\frac{1}{m^2}\left (k \sqrt{c-1}-\frac{(m-k)}{ \sqrt{c-1}} \right )^2\\
&=&\frac{(c k - m)^2}{m^2 (c-1)} \,.
\end{eqnarray*}
Next, after direct substitution and simplification, we find that equation \eqref{r-harmonicity-condition-in-general-pseudo} is equivalent to
\[
(ck-m) P_3(c)=0 \,,
\]
where $P_3(c)$ is the third order polynomial defined in \eqref{condizione-pseudo-tori-Clifford}. Now, since $ ck-m=0$ corresponds to $\alpha=0$, the conclusion of the theorem follows immediately.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Th-nostro-rigidity-2-curv}] When the curvature of the ambient is $c=1$, according to Theorem 5.1 of \cite{Abe} $M^m_{t'}$ is one of the hypersurfaces listed in Table~\ref{Hypersurfaces-pseudo-sphere}. Then a case by case direct inspection of \eqref{r-harmonicity-condition-in-general-pseudo}, using again \eqref{mean-curv-fuction-pseudo}, \eqref{norma-A} and Table~\ref{Hypersurfaces-pseudo-sphere} to compute $\trace A^2$ and $\alpha$, shows that the only $r$-harmonic hypersurfaces in this family are those given in Corollary~\ref{Cor-r-harmonic-pseudo-hyperspheres} and Theorem~\ref{Th-r-harmonic-pseudo-Clifford}. In the case that $c=0$, Theorem 5.1 of \cite{Abe} says that $M^m_{t'}$ is one of the hypersurfaces listed in (R-1)--(R-6), p. 131 of \cite{Abe}. Then, again, the thesis follows easily by direct inspection.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Th-non-existence-surfaces-c<=0}] The $3$-tension field is described by \eqref{2s+1-tension} with $s=1$. In the first part of the proof, for future reference, we do not make any assumption on the dimension $m$ and the curvature $c$. We observe that $\tau(\varphi)=m{\mathbf H}$ and use Lemma~\ref{Lemma-Delta-H} with $f$ constant and \eqref{tensor-curvature-N(c)}. We have:
\begin{eqnarray}\label{Curv-tensor-expr-1}\nonumber
\sum_{i=1}^m \varepsilon_i R^{N(c)} \left( \overline{\Delta}\tau(\varphi),d \varphi(X_i)\right ) d \varphi (X_i)&=&c\, m \sum_{i=1}^m \varepsilon_i \Big \{ \langle d \varphi (X_i), d \varphi (X_i) \rangle \overline{\Delta}{\mathbf H}\\\nonumber
&&\qquad \quad - \langle d \varphi (X_i), \overline{\Delta}{\mathbf H} \rangle d \varphi (X_i) \Big \}\\
&=&c\, m \big \{ m \alpha \varepsilon\,\trace A^2\, \eta - 0 \big \}\\\nonumber
&=&c\, m^2 \alpha \varepsilon\,\trace A^2 \,\eta \,.
\end{eqnarray}
Similarly, we compute
\begin{equation}\label{Curv-tensor-expr-2}
\sum_{i=1}^m \varepsilon_i R^{N(c)} \left( \nabla^{\varphi}_{X_i} \tau(\varphi),\tau(\varphi) \right ) d \varphi (X_i)=c\, \varepsilon m^3 \alpha^3 \eta \,.
\end{equation}
Using \eqref{Delta-2-H-formula}, \eqref{Curv-tensor-expr-1} and \eqref{Curv-tensor-expr-2} into \eqref{2s+1-tension} and Lemma~\ref{Lemma-Delta2-H} we obtain the explicit expression of the $3$-tension field:
\begin{eqnarray*}\label{tri-tension-field-explicit}
\tau_3(\varphi)&=&m \alpha \big[\varepsilon\Delta \trace A^2 + \left (\trace A^2\right)^2 -m\, c \,\varepsilon\,\trace A^2 -m^2\, c\, \varepsilon \, \alpha^2 \big ] \eta \\\nonumber
&&+ 2m\alpha \varepsilon\, A \left ( \grad \trace A^2 \right )
\end{eqnarray*}
Therefore, we conclude that $M^m_{t'}$ is a triharmonic hypersurface in $N^{m+1}_t(c)$ if and only if either it is minimal or
\begin{equation}\label{tri-harmonicity-system-explicit}
\left \{
\begin{array}{ll}
{\rm (i)}\quad &\Delta \trace A^2 + \varepsilon\, \left ( \trace A^2\right)^2 -m\, c \,\trace A^2 -m^2\, c\, \alpha^2=0 \\
\\
{\rm (ii)}\quad & A \left ( \grad \trace A^2\right ) =0 \,.
\end{array}
\right .
\end{equation}
From now on, we assume that $M^m_t$ is not minimal and we use the hypothesis $\varepsilon c \leq 0$. First, we analyse the case $c=0$. If $\trace A^2$ is constant, then it follows immediately from \eqref{tri-harmonicity-system-explicit} that $ \left ( \trace A^2\right)^2=0$ and so, since $A$ is diagonalizable, the hypersurface is totally geodesic, a contradiction. If $\trace A^2$ is not a constant, then there exists an open set $U$ of the surface such that $\grad \trace A^2\neq 0$ on $U$. We deduce from \eqref{tri-harmonicity-system-explicit}(ii) that $0$ is an eigenvalue of $A$ on $U$. Now we use the assumption that $m=2$. Since $M^2_{t'}$ is CMC, it is easy to conclude that necessarily $\trace A^2$ is again a constant on $U$, and this is a contradiction. Next, if $c\neq0$, by the assumption $\varepsilon c \leq 0$ we deduce that $c$ and $\varepsilon$ have opposite sign. Then, using this observation in \eqref{tri-harmonicity-system-explicit} (i), the proof follows essentially the same argument as in the case $c=0$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Th-structure-surfaces-c>0}] First, from an argument similar to the proof of Theorem~\ref{Th-non-existence-surfaces-c<=0}, we deduce that $\trace A^2$ must be a constant on $M^2_{t'}$ and also the two principal curvatures are constant.
Then, according to Theorem~\ref{Th-nostro-rigidity-2-curv}, we deduce that the only possibility is that $M^2_{t'}$ is an open part of a small pseudo-hypersphere ${\mathbb S}^2_t(3)$ because, according to Theorem~\ref{Th-r-harmonic-pseudo-Clifford}, in these dimensions there exists no generalised pseudo-Clifford torus which is proper triharmonic. Indeed, using $k=1, r=3$ and $m=2$, \eqref{condizione-pseudo-tori-Clifford} becomes
\[
P_3(c)=c^3-5 c^2+9 c-6\,.
\]
Now, the only real root of $P_3(c)$ is $c=2$ and it corresponds to the minimal pseudo-Clifford torus.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Th-Xiao-c=-1}]
We have to find under what conditions on $a,b$ the complex circle $x(s,t)$ is $r$-harmonic. By using standard techniques of the theory of surfaces in ${\mathbb H}^{3}_1\subset {\mathbb R}^4_2$, we can compute the shape operator of the complex circle $x(s,t)$ and obtain
\[
A=\frac{1}{a^2+b^2}\begin{bmatrix}
2 ab & 1 \\
&\\
-1& 2 ab \\
\end{bmatrix}\,.
\]
The condition $ab\neq 0$ ensures that $\trace A\neq 0$, that is $M^2_1$ is not minimal.
The case $r=2$. From Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo} we deduce that $M^2_1$ is biharmonic if and only if $\trace A^2+2=0$. Now, a direct computation gives
\[
\trace A^2+2= \frac{16 b^2 (b^2-1)}{(a^2+b^2)^2}\,,
\]
from which, since $b^2-1=a^2$, we deduce that $M^2_1$ cannot be proper biharmonic.
The case $r>2$. In this case, using Theorem~\ref{Th-existence-hypersurfaces-c>0-e-c<0-pseudo}, we have two possibilities, that is:
(i) either
\[
0=\trace A^2=2 \,\frac{4a^2b^2-1}{(a^2+b^2)^2}\,;
\]
(ii) or $4a^2b^2-1\neq 0$ and, according to \eqref{r-harmonicity-condition-in-general-pseudo},
\[
\begin{split}
0&=\left(\trace A^2\right )^2+2\,\left (\trace A^2\right )+(r-2) (\trace A)^2\\
&=\frac{16 b^2 \left(b^2-1\right)
\left(\left(a^2+b^2\right)^2
r-4\right)}{\left(a^2+b^2\right)^4
}\,.
\end{split}
\]
Case (i), together with the hypothesis $b^2-a^2=1$, give exactly point (1) of the theorem. As for Case (ii), we have a proper solution if and only if
\[
1<a^2+b^2=\frac{2}{\sqrt{r}}
\]
from which we deduce that the only possible value is $r=3$. Taking into account that, by assumption, $b^2-a^2=1$, it is easy to conclude Case (2) of the theorem.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Th-Sasah-r>2}] Case (i). Here we assume that the shape operator $A$ is diagonalizable. Since the hypersurface has dimension $m=2$ and constant principal curvatures, we can apply Theorem~\ref{Th-nostro-rigidity-2-curv}. Moreover, taking into account Remark~\ref{Rem-basta-c>0}, the proper $r$-harmonic surfaces in ${\mathbb H}^3_1$ can be deduced from those of ${\mathbb S}^3_2$. Putting these facts together and using \eqref{eq-k=m-k} we obtain Cases {$(1)-(4)$} in the statement of the theorem.
Case (ii). Now, we assume that the shape operator $A$ is non-diagonalizable. In this case, $t'=1$ and so $\varepsilon =1$. If $A$ has a double, real eigenvalue $\lambda$, then the argument of \cite[Proposition\link4.1]{Alias} enables us to conclude that $M^2_1$ is a $B$-scroll in $N^3_1(c)$ over a null curve $\gamma$, and its shape operator $A$, with respect to the coordinate frame field $\left \{ \partial \slash \partial s,\,\partial \slash \partial u \right \}$ associated to \eqref{eq-B-Scroll}, is given by
\[
A= \left [
\begin{array}{cc}
\lambda &0 \\
k(s)&\lambda
\end{array}
\right ]
\]
with $k(s) \neq 0$. Therefore, $\trace A^2= 2 \lambda^2$ and $\alpha= \lambda$.
Next, we assume that the $B$-scroll is not minimal, i.e., $\lambda \neq 0$, and we apply \eqref{r-harmonicity-condition-in-general-pseudo}: if $c=-1,\,m=2, \,\varepsilon=1$ then there are no solutions; if $c=1,\,m=2, \,\varepsilon=1$, then it is easy to deduce that the $B$-scroll is proper $r$-harmonic if and only if
\[
\lambda^2 =r-1 \,.
\]
Finally, observing that the Gauss curvature $K$ of the $B$-scroll is given by $K=\det (A)+c$, with $c=1$, we conclude Case (5).
It remains to analyse the case that the shape operator $A$ has two complex eigenvalues. This will lead us to Case (6). Indeed, from \cite[p.~453]{Alias}, we conclude that $M^2_1$ is a flat Lorentzian surface in ${\mathbb H}^3_1$ with parallel second fundamental form in ${\mathbb R}^4_2$. Then, according to \cite{Magid2}, $M^2_1$ is locally congruent to a complex circle and therefore the conclusion follows from Theorem~\ref{Th-Xiao-c=-1}.
\end{proof}
|
1,116,691,499,366 | arxiv | \section{Introduction and Related Work}
\label{Sec:Intro}
Dictionary learning is a useful procedure by which dependencies among input features can be represented in terms of suitable bases\cite{aharon2006ksvd, chi2013petrels,DictionaryLearningSurvey,elad2006image,mairal2010online,zou2006sparse,shen2008sparse,lee2010biclustering,mairal2008supervised,Kasiviswanathan12}. It has found applications in many machine learning and inference tasks including image denoising\cite{elad2006image,mairal2010online}, dimensionality-reduction \cite{zou2006sparse,shen2008sparse}, bi-clustering \cite{lee2010biclustering}, feature-extraction and classification \cite{mairal2008supervised}, and novel document detection \cite{Kasiviswanathan12}. Dictionary learning usually alternates between two steps: (i) an inference (sparse coding) step and (ii) a dictionary update step. The first step finds a sparse representation for the input data using the existing dictionary by solving, for example, a regularized regression problem, while the second step usually employs a gradient descent iteration to update the dictionary entries.
With the increasing complexity of various learning tasks, it is not uncommon for the size of the learning dictionaries to be demanding in terms of memory and computing requirements. It is therefore important to study scenarios where the dictionary is not necessarily available in a single central location but its components are possibly spread out over multiple locations. This is particularly true in Big Data scenarios where large dictionary components may already be available at separate locations and it is not feasible to aggregate all dictionaries in one location due to communication and privacy considerations. This observation motivates us to examine how to learn a dictionary model that is stored over a network of agents, where each agent is in charge of only a portion of the dictionary elements. Compared with other works, the problem we solve in this article is how to learn a distributed dictionary model, which is, for example, different from the useful work in \cite{Chainais2013learning} where it is assumed instead that each agent maintains the \emph{entire} dictionary model.
In this paper, we first formulate a general dictionary learning problem, where the residual error function and the regularization function can assume different forms in different applications. As we shall explain, this form turns out not to be directly amenable to distributed implementations. However, when the regularization is strongly convex, we will show that the problem has a dual function that can be solved in a distributed manner using diffusion strategies \cite{Cattivelli10,sayed2013diffusion,chen2012AllertonLimit,chen2013JSTSPpareto}. In this solution, the agents will not need to share their (private) dictionary elements but only the dual variable. Useful consensus strategies \cite{kar2008distributed,lee2013distributed,bertsekas1997parallel,tsitsiklis1986distributed} can also be used for the same purpose. However, since it has been shown that diffusion strategies have enhanced stability and learning abilities over consensus strategies \cite{sayed2014proc,sayed2014adaptation,tu2012diffusion}, we will continue our presentation by focusing on diffusion strategies.
We will test our proposed algorithm on two important applications of dictionary learning: (i) novel document detection\cite{Kasiviswanathan12,Aiello2013sensing,Takahashi2014Jan}, and (ii) bi-clustering on microarray data \cite{lee2010biclustering}. A third application related to image denoising is considered in \cite{chen2014icasspdictionary}. In the novel document detection problem \cite{Kasiviswanathan12,Aiello2013sensing,Takahashi2014Jan}, each learner receives documents associated with certain topics, and wishes to determine if an incoming document is associated with a topic that has already been observed in previous data. This application is useful, for example, in finance when a company wishes to mine news streams for factors that may impact stock prices. Another example is the mining of social media streams for topics that may be unfavorable to a company. In these applications, our algorithm is able to perform distributed non-negative matrix factorization tasks, with the residual metric chosen as the Huber loss function \cite{huber1964robust}, and is able to achieve a high area under the receiver operating characteristic (ROC) curve. In the bi-clustering experiment, our algorithm is used to learn relations between genes and types of cancer. From the learned dictionary, the patients are subsequently clustered into groups corresponding to different manifestations of cancer. We show that our algorithm can obtain similar clustering results to those in \cite{lee2010biclustering}, which relies instead on a batched (centralized) implementation.
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.7}
\caption{Examples of tasks solved by the general formulation \eqref{Equ:ProbForm:DictLearn_Objective}--\eqref{Equ:ProbForm:DictLearn_Constraint}. The loss functions $f(u)$ are illustrated in Fig. \ref{Fig:ScalarLossFunctions}. }
\label{Tab:Task}
\centering
\begin{threeparttable}
\begin{tabular}{c||c|c|c|c}
\hline \hline
\rowcolor[gray]{0.9} \rule[-1ex]{0pt}{4ex} \textbf{Tasks} & $f(u)$ & $h_y(y)$ & $h_W(W)$ & $\mathcal{W}_k$\\
\hline
\rule[-1ex]{0pt}{4ex} \textbf{Sparse SVD} & $\frac{1}{2} \|u\|_2^2$ & $\gamma \|y\|_1 + \frac{\delta}{2} \|y\|_2^2$ & 0 & $\left\{W_k: \| [ W_k ]_{:,q} \|_2 \le 1\right\}$ \\
\hline
\rule[-1ex]{0pt}{4ex} \textbf{Bi-Clustering} & $\frac{1}{2}\|u\|_2^2$ & $\gamma \|y\|_1 + \frac{\delta}{2} \|y\|_2^2$ & $\beta \cdot \vertiii{W}_1$ \tnote{a} & $\left\{W_k: \| [ W_k ]_{:,q} \|_2 \le 1\right\}$ \\
\hline
\multirow{1}{*}{\textbf{Nonnegative Matrix}} & $\frac{1}{2}\|u\|_2^2$ & $\gamma \|y\|_{1,+} + \frac{\delta}{2} \|y\|_2^2$ \tnote{b} & 0 & $\left\{W_k: \| [ W_k ]_{:,q} \|_2 \le 1, \; W_k \succeq 0\right\}$ \\ \cline{2-5}
\multirow{1}{*}{\textbf{Factorization}} & $\displaystyle\sum_{m=1}^M L(u_m)$ \tnote{c} & $\gamma \|y\|_{1,+} + \frac{\delta}{2} \|y\|_2^2$ & 0 & $\left\{W_k: \| [ W_k ]_{:,q} \|_2 \le 1, \; W_k \succeq 0\right\}$ \\
\hline \hline
\end{tabular}
\begin{tablenotes}
\vspace{0.5em}
\item[a]
The notation $\vertiii{W}_1$ is used to denote the sum of all absolute entries in the matrix $W$: $\vertiii{W}_1 = \sum_{m=1}^M \sum_{q=1}^K |W_{mq}|$, which is different from the conventional matrix $1-$norm defined as the maximum absolute column sum: $\|W\|_1 = \max_{1 \le q \le K} \sum_{m=1}^M |W_{mq}|$.
\vspace{0.5em}
\item[b]
The notation $\|y\|_{1,+}$ is defined as $\|y\|_{1,+} = \|y\|_1$ if $y \succeq 0$ and $\|y\|_{1,+} = +\infty$ otherwise. It imposes infinite penalty on any negative entry appearing in the vector $y$. Since negative entries are already penalized in $\|y\|_{1,+}$, there is no need to penalize it again in the $\frac{\delta}{2}\|y\|_{2}^2$ term.
\item[c]
The scalar Huber loss function is defined as $L(u_m) \triangleq \begin{cases}
\frac{1}{2\eta} u_m^2, & |u_m| < \eta \\
|u_m| - \frac{\eta}{2}, & \textrm{otherwise}
\end{cases}$,
where $\eta$ is a positive parameter.
\end{tablenotes}
\end{threeparttable}
\end{table*}
The paper is organized as follows. In Section \ref{Sec:ProbForm}, we introduce the dictionary learning problem over distributed models. In Section \ref{Sec:DictLearn}, using the concepts of conjugate function and dual decomposition, we transform the original dictionary learning problem into a form that is amenable to distributed optimization. In Section \ref{Sec:Experiments}, we test our proposed algorithm on two applications. In Section \ref{Sec:Conclusion} we conclude the exposition.
\section{Problem Formulation}
\label{Sec:ProbForm}
\subsection{General Dictionary Learning Problem}
We seek to solve the following general form of a \emph{global} dictionary learning problem over a network of $N$ agents connected by a topology:
\begin{align}
\min_{W} \quad& \mathbb{E}
\Big[
f( \bm{x}_t - W \bm{y}_{t}^o )
+
h_y( \bm{y}_t^o)
\Big]
+
h_{W}(W)
\label{Equ:ProbForm:DictLearn_Objective}
\\
\mathrm{s.t.}
\quad&
W \in \mc{W}
\label{Equ:ProbForm:DictLearn_Constraint}
\end{align}
where $\mathbb{E}[\cdot]$ denotes the expectation operator, $\bm{x}_t$ is the $M \times 1$ input data vector at time $t$ (we use boldface letters to represent random quantities), $\bm{y}_t^o$ is a $K\times 1$ coding vector defined further ahead as the solution to \eqref{Equ:ProbForm:InferenceProblem}, and $W$ is an $M \times K$ dictionary matrix. Moreover, the $q$-th column of $W$, denoted by $[W]_{:,q}$, is called the $q$-th dictionary element (or \emph{atom}), $f(u)$ in \eqref{Equ:ProbForm:DictLearn_Objective} denotes a differentiable convex loss function for the residual error, $h_y(y)$ and $h_W(W)$ are convex (but not necessarily differentiable) regularization terms on $y$ and $W$, respectively, and $\mc{W}$ denotes the convex constraint set on $W$. Depending on the application problem of interest, there are different choices for $f(u)$, $h_y(y)$, $h_W(W)$ and $\mc{W}$. Table \ref{Tab:Task} lists some typical tasks and the corresponding choices for these functions. In regular dictionary learning \cite{mairal2010online}, the constraint set $\mc{W}$ is
\begin{align}
\mc{W} = \left\{
W: \;
\| [ W ]_{:,q} \|_2 \le 1, \; \forall q
\right\}
\label{Equ:ProbForm:W_subUnitNormConstraint}
\end{align}
and in applications of nonnegative matrix factorization \cite{mairal2010online} and novel document detection (topic modeling) \cite{Kasiviswanathan12}, it is
\begin{align}
\mc{W} = \left\{
W: \;
\| [ W ]_{:,q} \|_2 \le 1, \; W \succeq 0, \; \forall q
\right\}
\label{Equ:ProbForm:W_subUnitNormNonnegConstraint}
\end{align}
where the notation $W \succeq 0$ means each entry of the matrix $W$ is nonnegative. We note that if there is a constraint on $y$, it can be absorbed into the regularization factor $h_y(y)$, by including an indicator function of the constraint into this regularization term. For example, if $y$ is required to satisfy $ y \in \mathcal{Y} = \{y: 0 \preceq y \preceq \mathds{1} \}$, where $\mathds{1}$ denotes the all-one vector, we can modify the original regularization $h_y(y)$ by adding an additional indicator function:
\begin{align}
h_y(y) \leftarrow h_y(y) + I_{\mathcal{Y}}(y)
\end{align}
where the indicator function $I_{\mathcal{Y}}(y)$ for $\mathcal{Y}$ is defined as
\begin{align}
I_{\mathcal{Y}}(y) \triangleq
\begin{cases}
0, & \mathrm{if~} 0 \preceq y \preceq \mathds{1}
\\
+\infty, & \mathrm{otherwise}
\end{cases}
\label{Equ:ProbForm:IndicatorFun_def}
\end{align}
The vector $\bm{y}_t^o$ in \eqref{Equ:ProbForm:DictLearn_Objective} is the solution to the following general inference problem for each input data sample $x_t$ at time $t$ for a given $W$ (the regular font for $x_t$ and $y_t^o$ denotes realizations for the random quantities $\bm{x}_t$ and $\bm{y}_t^o$):
\begin{align}
y_t^o \triangleq \arg\min_{y}
\left[
f( x_t - W y ) + h_y(y)
\right]
\label{Equ:ProbForm:InferenceProblem}
\end{align}
Note that dictionary learning consists of two steps: the inference step (sparse coding) for $x_t$ at each time $t$ in \eqref{Equ:ProbForm:InferenceProblem}, and the dictionary update step (learning) in \eqref{Equ:ProbForm:DictLearn_Objective}--\eqref{Equ:ProbForm:DictLearn_Constraint}.
\subsection{Dictionary Learning over Networked Agents}
\label{Sec:ProbForm:DictLearnNetwork}
\begin{figure}
\centering
\includegraphics[width=0.3\textwidth]{Fig_DistributedModel_v5}
\caption{The data sample $x_t$ at time $t$
is available to a subset $\mc{N}_I$ of agents in the network (e.g., agents
$3$ and $6$ in the figure),
and each agent $k$ is in charge of one sub-dictionary, $W_k$, and the corresponding optimal
sub-vector of coefficients estimated at time $t$, $y_{k,t}^o$. Each agent
$k$ can only exchange information with its immediate neighbors (e.g.,
agents $5$, $2$ and $6$ in the figure and $k$ itself). We use $\mc{N}_k$ to denote the
set of neighbors of agent $k$. }
\label{Fig:DistributedModel}
\end{figure}
Let the matrix $W$ and the vector $y$ be partitioned in the following block forms:
\begin{align}
W &= \begin{bmatrix}
W_1 & \cdots & W_N
\end{bmatrix},
\quad
y = \mathrm{col}\{ y_1, \; \ldots, \; y_N\}
\label{Equ:ProbForm:W_y_def}
\end{align}
where $W_k$ is an $M \times N_k$ \emph{sub-dictionary} matrix and $y_k$ is an $N_k \times 1$ sub-vector.
{ Note that the sizes of the sub-dictionaries add up to the total size of the dictionary, $K$, i.e.,
\begin{align}
N_1+\cdots+N_N = K
\end{align}
}%
Furthermore, we assume the regularization terms $h_y(y)$ and $h_W(W)$ admit the following decompositions:
\begin{align}
h_y(y) &= \sum_{k=1}^N
h_{y_k}(y_k),
\quad
h_W(W) = \sum_{k=1}^N
h_{W_k}(W_k)
\label{Equ:ProbForm:hW_factorization}
\end{align}
Then, the objective function of the inference step \eqref{Equ:ProbForm:InferenceProblem} can be written as
\begin{align}
Q(W, y; x_t)
\triangleq
f\Big(
x_t - \sum_{k=1}^N W_k y_k
\Big)
+
\sum_{k=1}^N
h_{y_k}(y_k)
\label{Equ:ProbForm:InferenceProblem_FactoredForm}
\end{align}
We observe from \eqref{Equ:ProbForm:InferenceProblem_FactoredForm} that the sub-dictionary matrices $\{W_k\}$ are linearly combined to represent the input data $x_t$. By minimizing $Q(W,y;x_t)$ over $y$, the first term in \eqref{Equ:ProbForm:InferenceProblem_FactoredForm} helps ensure that the representation error for $x_t$ is small. The second term in \eqref{Equ:ProbForm:InferenceProblem_FactoredForm}, which usually involves a combination of $\ell_1$ and $\ell_2$ measures, as indicated in Table~\ref{Tab:Task}, helps ensure that each of the resulting combination coefficients $\{y_k\}$ is sparse and small. We will make the following assumption regarding $h_{y_k}(y_k)$ throughout the paper
\begin{assumption}[Strongly convex regularization]
\label{Assumption:StronglyConvexRegularization_hyk}
The regularization terms $h_{y_k}(y_k)$ are assumed to be strongly convex for $k=1,\ldots,N$.
\hfill \qed
\end{assumption}
\noindent This assumption will allow us to develop a fully distributed strategy that enables the sub-dictionaries $\{W_k\}$ and the corresponding coefficients $\{y_k\}$ to be stored and learned in a distributed manner over the network; each agent $k$ will infer its own $y_k$ and update its own sub-dictionary $W_k$ with limited interaction with its neighboring agents. Requiring $\{h_{y_k}(y_k)\}$ to be strongly convex is not restrictive since we can always add a small $\ell_2$ regularization term to make it strongly convex. For example, in Table \ref{Tab:Task}, we add an $\ell_2$ term to $\ell_1$ regularization so that the resulting $h_{y_k}(y_k)$ ends up amounting to elastic net regularization, in the manner advanced in \cite{zou2006sparse}.
Figure \ref{Fig:DistributedModel} shows the assumed configuration of the knowledge and data distribution over the network. The sub-dictionaries $\{W_k\}$ can be interpreted as the ``wisdom'' that is distributed over the network, and which we wish to combine in a distributed manner to form a greater ``intelligence'' for interpreting the data $\bm{x}_t$. Observe that we are allowing $\bm{x}_t$ to be observed by only a subset, $\mc{N}_I$, of the agents. By having the dictionary distributed over the agents, we would then like to develop a procedure that enables these networked agents to find the \emph{global} solutions to both the inference problem \eqref{Equ:ProbForm:InferenceProblem} and the learning problem \eqref{Equ:ProbForm:DictLearn_Objective}--\eqref{Equ:ProbForm:DictLearn_Constraint} with interactions that are limited to their neighborhoods.
\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{Fig_LossFunctions_v5}
\caption{Illustration of the loss functions, and the elastic net regularization.}
\label{Fig:ScalarLossFunctions}
\end{figure}
\subsection{Relation to Prior Work}
\label{Sec:ProbForm:RelatedWork}
\subsubsection{Model Distributed vs. Data Distributed}
The problem we are solving in this paper is different from the useful work \cite{chainais2013distributed,Chainais2013learning} on distributed dictionary learning and from the traditional distributed learning setting \cite{Cattivelli10,chen2013JSTSPpareto,sayed2013diffusion,chouvardas2011adaptive}, where it is assumed that the \emph{entire} dictionary $W$ is maintained by each agent or that individual data samples generated by the same distribution, denoted by $\bm{x}_{k,t}$, are observed by the agents at each time $t$. That is, these previous works study \emph{data distributed} formulations. What we are studying in this paper is to find a distributed solution where each agent is only in charge of a \emph{portion} of the dictionary ($W_k$ for each agent $k$) and where the incoming data, $\bm{x}_t$, is observed by only a subset of the agents. This scenario corresponds to a \emph{model distributed} (or dictionary-distributed) formulation. A different formulation is also considered in \cite{dean2013large} in the context of distributed deep neural network (DNN) models over computer networks. In these models, each computer is in charge of a portion of neurons in the DNN, and the computing nodes exchange their private activation signals. As we will see further ahead, our distributed model requires exchanging neither the private combination coefficients $\{y_k\}$ nor the sub-dictionaries $\{W_k\}$.
The distributed-model setting we are studying is important in practice because agents tend to be limited in their memory and computing power and they may not be able to store large dictionaries locally. Even if the agents were powerful enough, different agents may still have access to different databases and different sources of information. Rather than aggregate the information in the form of large dictionaries at every single location, it is often more advantageous to keep the information distributed due to costs in exchanging large dataset and dictionary models, and also due to privacy considerations where agents may not be in favor of sharing their private information.
\subsubsection{Distributed Basis Pursuit}
\label{Sec:RelatedWork:DBP}
{
Other useful related works appear in the studies \cite{mota2012distributedBP,mota2013DADMM,yuan2013convergence} on distributed basis pursuit, which also rely on dual decomposition arguments. However, there are some key differences in problem formulation, generality, and technique, as explained in \cite{towfic2014dictionary}. For example, the works \cite{mota2012distributedBP,mota2013DADMM,yuan2013convergence} do not deal with dictionary learning problems and focus instead on the solution of special cases of the inference problem \eqref{Equ:ProbForm:InferenceProblem}. Specifically, the problem formulations in \cite{mota2012distributedBP,mota2013DADMM,yuan2013convergence} focus on determining sparse solutions to (underdetermined) linear systems of equations, which can be interpreted as corresponding to scenarios where the dictionaries are \emph{static} and not learned from data. In comparison, in this article, we show how the inference {\em and}
{
learning problems \eqref{Equ:ProbForm:InferenceProblem}
and \eqref{Equ:ProbForm:DictLearn_Objective}--\eqref{Equ:ProbForm:DictLearn_Constraint}
}%
can be \emph{jointly} integrated into a common framework. Furthermore, our proposed distributed dictionary learning strategy is an \emph{online} algorithm, which updates the dictionaries sequentially in response to streaming data. We also only require the data sample $x_t$ to be available to a subset of the agents (e.g., one agent) while it is assumed in \cite{mota2012distributedBP,mota2013DADMM,yuan2013convergence} that all agents have access to the same data $x_t$.
For instance, one of the problems studied in \cite{mota2012distributedBP} is the following inference problem (compare with \eqref{Equ:ProbForm:InferenceProblem}):
\begin{subequations}
\label{Equ:ProbForm:motaBP_noisefree}
\begin{align}
y_t^o \triangleq \underset{y}{\arg\min} \quad&
\sum_{k=1}^N \left[\gamma\|y_k\|_1 + \frac{\delta}{2} \|y_k\|_2^2\right]
\label{Equ:ProbForm:mota1_cost}
\\
\mathrm{s.t.}\quad &\sum_{k=1}^N W_k y_k = x_t
\label{Equ:ProbForm:mota1}
\end{align}
\end{subequations}
This formulation can be recast as a special case of \eqref{Equ:ProbForm:InferenceProblem} by selecting:
\begin{subequations}
\begin{align}
h_{y_k}(y_k) &= \gamma \|y_k\|_1 + \frac{\delta}{2} \|y_k\|_2^2
\label{eq:mota1_regularizer}\\
f(x_t-Wy) &= I_{\mathcal{B}}\Big(x_t-\sum_{k=1}^N W_k y_k\Big)
\label{eq:mota1_residual}
\end{align}
\end{subequations}
where $I_{\mathcal{B}}(\cdot)$ is the indicator function defined by:
\begin{align}
I_{\mathcal{B}}(u) = \begin{cases}
0, & u \in \mathcal{B}\\
\infty, & u \notin \mathcal{B}
\end{cases} \label{eq:indicator_function}
\end{align}
where $\mathcal{B} \triangleq \left\{0_M\right\}$ is a set consisting of the zero vector in $\mathbb{R}^M$. Equality constraints of the form \eqref{Equ:ProbForm:mota1}, or a residual function of the form \eqref{eq:mota1_residual}, are generally problematic for problems that require {\em both} learning and inference since modeling and measurement errors usually seep into the data and the $\{W_k\}$ may not be able to represent the $x_t$ accurately with a precise equality as in \eqref{Equ:ProbForm:mota1}. To handle the modeling error, the work \cite{mota2013DADMM} considered instead:
\begin{subequations}
\label{Equ:ProbForm:motaBP_noisy}
\begin{align}
y_t^o \triangleq \underset{y}{\arg\min} \quad&
\sum_{k=1}^N \left[\gamma\|y_k\|_1 + \frac{\delta}{2} \|y_k\|_2^2\right] \label{Equ:ProbForm:mota2_lasso_obj}\\
\mathrm{s.t.}\quad &\Big\|\sum_{k=1}^N W_k y_k - x_t\Big\|_2 \leq \sigma
\label{Equ:ProbForm:mota2_lasso}
\end{align}
\end{subequations}
for some $\sigma \geq 0$, which again can be viewed as a special case of problem \eqref{Equ:ProbForm:InferenceProblem} for the same $h_{y_k}(\cdot)$ from \eqref{eq:mota1_regularizer} and with the indicator function in \eqref{eq:mota1_residual} replaced by $I_{\mathcal{C}}(u)$ relative to the set
\begin{align}
\mathcal{C} \triangleq \left\{u \in \mathbb{R}^{M\times 1} : \|u\|_2 \leq \sigma \right\}
\label{Equ:ProbForm:C_set}
\end{align}
An alternative problem formulation that removes the indicator functions is considered in \cite{mateos2010distributed,mota2013DADMM}, namely,
\begin{align}
y_t^o \triangleq \underset{y}{\arg\min}
\left[
\frac{1}{2} \| x_t - W y \|^2 + \gamma \|y\|_1
\right]
\label{Equ:ProbForm:InferenceProblem_mota2_BPDN}
\end{align}
Here, we now have $h_y(y)=\gamma \|y\|_1$ and $f(u)=\frac{1}{2}\|u\|^2$. However, for problem \eqref{Equ:ProbForm:InferenceProblem_mota2_BPDN}, the dictionary elements as well as the entries of $x_t$, were partitioned in \cite{mateos2010distributed,mota2013DADMM} by {\em rows} across the network as opposed to our column-wise partitioning in \eqref{Equ:ProbForm:W_y_def}:
\begin{align}
W = [U_1^T, \ldots, U_N^T]^T
\end{align}
In this case, it is straightforward to rewrite problem \eqref{Equ:ProbForm:InferenceProblem_mota2_BPDN} in the form
\begin{align}
y_t^o \triangleq \underset{y}{\arg\min} \sum_{k=1}^N
\left[
\frac{1}{2} \| x_{k,t} - U_k y \|^2 + \frac{\gamma}{N} \|y\|_1
\right]
\end{align}
which is naturally in a ``sum-of-costs'' form; such forms are directly amenable to distributed optimization and do not require transformations --- see \eqref{Equ:DictLearnDist:J_glob_sumOFcosts} further ahead. However, the more challenging problem where the matrix $W$ is partitioned column-wise as in \eqref{Equ:ProbForm:W_y_def}, which leads to the ``cost-of-sum'' form showed earlier in \eqref{Equ:ProbForm:InferenceProblem_FactoredForm}, was not examined in \cite{mota2013DADMM,mateos2010distributed}.
In summary, we will solve the more challenging problem of \emph{joint} inference and dictionary learning (instead of inference alone under static dictionaries) under the \emph{column-wise} partitioning of $W$ (rather than row-wise partitioning) and general penalty functions $f(\cdot)$ and $\{h_{y_k}(\cdot)\}$ (instead of the special indicator choices in \eqref{eq:indicator_function} and \eqref{Equ:ProbForm:C_set}).
}
\section{Learning over Distributed Models}
\label{Sec:DictLearn}
\subsection{``Cost-of-Sum'' vs. ``Sum-of-Costs''}
\label{Sec:DictLearn:CostSum}
We thus start by observing that the cost function \eqref{Equ:ProbForm:InferenceProblem_FactoredForm} is a regularized ``\emph{cost-of-sum}''; it consists of two terms: the first term has a sum of quantities associated with different agents appearing as an argument for the function $f(\cdot)$ and the second term is a collection of separable regularization terms $\{h_{y_k}(y_k)\}$. This formulation is different from the classical ``\emph{sum-of-costs}'' problem, which usually seeks to minimize a global cost function, $J^{\mathrm{glob}}(w)$, that is expressed as the aggregate sum of individual costs $\{J_k(w)\}$, say, as:
\begin{align}
J^{\mathrm{glob}}(w) = \sum_{k=1}^N J_k(w)
\label{Equ:DictLearnDist:J_glob_sumOFcosts}
\end{align}
The ``sum-of-costs'' problem \eqref{Equ:DictLearnDist:J_glob_sumOFcosts} is amenable to distributed implementations\cite{Cattivelli10,sayed2013diffusion,chen2012AllertonLimit,chen2013JSTSPpareto,kar2008distributed,lee2013distributed,bertsekas1997parallel,tsitsiklis1986distributed,sayed2014proc}. In comparison, minimizing the regularized ``cost-of-sum'' problem in \eqref{Equ:ProbForm:InferenceProblem_FactoredForm} directly would require knowledge of all sub-dictionaries $\{W_k\}$ and coefficients $\{y_k\}$. Therefore, this formulation is not directly amenable to the distributed techniques from \cite{Cattivelli10,sayed2013diffusion,chen2012AllertonLimit,chen2013JSTSPpareto,kar2008distributed,lee2013distributed,bertsekas1997parallel,tsitsiklis1986distributed,sayed2014proc}. In \cite{chang2013distributed}, the authors proposed a useful consensus-based primal-dual perturbation method to solve a similar constrained ``cost-of-sum'' problem for smart grid control. In their method, an averaging consensus step was used to compute the sum inside the cost. We follow a different route and arrive at a more efficient distributed strategy by transforming the original optimization problem into a dual problem that has the same form as \eqref{Equ:DictLearnDist:J_glob_sumOFcosts} --- see \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm}--\eqref{Equ:DictLearnDist:DualProblem_Constraint_newForm} further ahead, and which can then be solved efficiently by means of diffusion strategies. There will be no need to exchange any information among the agents beyond the dual variable, or to employ a separate consensus step to evaluate the sum inside the cost in order to update their own sub-dictionaries.
\subsection{Inference over Distributed Models: A Dual Formulation}
\label{Sec:DictLearn:Inference_Dual}
To begin with, we first transform the minimization of \eqref{Equ:ProbForm:InferenceProblem_FactoredForm} into the following equivalent optimization problem { by introducing a splitting variable $z$}:
\begin{subequations}
\begin{align}
\min_{ \{y_k\}, z } \quad& f( x_t - z ) + \sum_{k=1}^N h_{y_k}(y_k)
\label{Equ:DictLearnDist:NewInference_Objective}
\\
\mathrm{s.t.} \quad& z = \sum_{k=1}^N W_k y_k
\label{Equ:DictLearnDist:NewInference_Constraint}
\end{align}
\end{subequations}
Note that the above problem is convex over both $\{y_k\}$ and $z$ since the objective is convex and the equality constraint is linear. Problem \eqref{Equ:DictLearnDist:NewInference_Objective}--\eqref{Equ:DictLearnDist:NewInference_Constraint} is a convex optimization problem with linear constraints so that strong duality holds\cite[p.514]{bertsekas1999nonlinear}, meaning that the optimal solution to \eqref{Equ:DictLearnDist:NewInference_Objective}--\eqref{Equ:DictLearnDist:NewInference_Constraint} can be found by solving its corresponding dual problem (see \eqref{Equ:DictLearnDist:DualProblem} below) and then recovering the optimal primal variables $\{y_k\}$ and $z$ (to be discussed in Sec. \ref{Sec:DictLearn:RecoveryPrimalVar}):
\begin{align}
\max_{\nu} g(\nu; x_t)
\label{Equ:DictLearnDist:DualProblem}
\end{align}
where $g(\nu; x_t)$ is the dual function associated with the optimization problem \eqref{Equ:DictLearnDist:NewInference_Objective}--\eqref{Equ:DictLearnDist:NewInference_Constraint}, and is defined as follows. First, the Lagrangian $L(\{y_k\}, z, \nu; x_t)$ over the primal variables $\{y_k\}$ and $z$ is given by
\begin{align}
L&( \{y_k\}, z, \nu; x_t )
\nonumber\\
&=
f( x_t - z ) + \nu^T z
+
\sum_{k=1}^N
\Big[
h_{y_k}(y_k)
-
\nu^T W_k y_k
\Big]
\label{Equ:DictLearnDist:Lagrangian}
\end{align}
Then, the dual function $g(\nu; x_t)$ can be expressed as:
\begin{align}
g&(\nu;x_t) \nonumber\\ &\triangleq
\inf_{\{y_k\}, z}
L(\{y_k\}, z, \nu; x_t)
\nonumber\\
&=
\inf_{ z } \!
\left[
f( x_t \!-\! z ) \!+\! \nu^T z
\right]
\!+\!\!
\sum_{k=1}^N
\inf_{y_k} \!
\Big[
h_{y_k}(y_k)
\!-\!
\nu^T W_k y_k
\Big]
\label{Equ:DictLearnDist:DualFunction_def}
\\
&\overset{(a)}{=}
\inf_{u}
\left[
f(u) \!-\! \nu^T u \!+\! \nu^T x_t
\right]
\!+\!
\sum_{k=1}^N
\inf_{y_k} \!
\Big[
h_{y_k}(y_k)
\!-\!
\nu^T W_k y_k
\Big]
\nonumber\\
&=
-\!
\sup_{u}
\left[
\nu^T u \!-\! f(u)
\right]
\!+\!
\nu^T x_t
\!-\!\!
\sum_{k=1}^N \!
\sup_{y_k}
\left[
\nu^T W_k y_k \!-\! h_{y_k}\!(y_k)
\right]
\nonumber\\
&=
- f^{\star}(\nu) + \nu^T x_t
-
\sum_{k=1}^N h_{y_k}^{\star}(W_k^T \nu)
\label{Equ:DictLearnDist:DualFunction_expr}
\\
&
\qquad\qquad
\nu \in \mathcal{V}_f \cap \mathcal{V}_{h_{y_1}} \cap \cdots \cap \mathcal{V}_{h_{y_N}}
\nonumber
\end{align}
where in step (a) we introduced $u \triangleq x_t - z$, and $f^{\star}(\cdot)$ and $h_{y_k}^{\star}(\cdot)$ are the conjugate functions of $f(\cdot)$ and $h_{y_k}(\cdot)$, respectively, with the corresponding domains denoted by $\mathcal{V}_f$ and $\mathcal{V}_{h_{y_k}}$, respectively. We note that the conjugate function { (or \emph{Legendre-Fenchel} transform\cite[p.37]{urruty1993convex2})}, $r^{\star}(\nu)$, for a function $r(x)$ is defined as \cite[pp.90-95]{boyd2004convex}:
\begin{align}
r^{\star}(\nu) \triangleq \sup_{x}
\left[
\nu^T x - r(x)
\right]
,
\quad
\nu \in \mathcal{V}_r
\label{Equ:DictLearnDist:r_conj_def}
\end{align}
where the domain $\mathcal{V}_r$ is defined as the set of $\nu$ where the above supremum is finite. { The conjugate function $r^{\star}(\nu)$ and its domain $\mathcal{V}_r$ are convex regardless of whether $r(x)$ is convex or not \cite[p.530]{bertsekas1999nonlinear}\cite[p.91]{boyd2004convex}.} In particular, it holds that $\mathcal{V}_r = \mathbb{R}^M$ if $r(x)$ is strongly convex \cite[p.82]{urruty1993convex2}. Now since $h_{y_k}(\cdot)$ is assumed in Assumption \ref{Assumption:StronglyConvexRegularization_hyk} to be strongly convex, its domain $\mathcal{V}_{h_{y_k}}$ is the entire $\mathbb{R}^M$. If $f(u)$ happens to be strongly convex (rather than only convex, e.g., if $f(u)=\frac{1}{2}\|u\|_2^2$), then $\mathcal{V}_{f}$ would also be $\mathbb{R}^M$, otherwise it is a convex subset of $\mathbb{R}^M$. Therefore, the dual function in \eqref{Equ:DictLearnDist:DualFunction_expr} becomes
\begin{align}
g(\nu; x_t) &= - f^{\star}(\nu) + \nu^T x_t
-
\sum_{k=1}^N h_{y_k}^{\star}(W_k^T \nu),
\; \nu \in \mathcal{V}_{f}
\label{Equ:DictLearn:g_expression_final}
\end{align}
Now, maximizing $g(\nu; x_t)$ is equivalent to minimizing $-g(\nu; x_t)$ so that the dual problem \eqref{Equ:DictLearnDist:DualProblem} is equivalent to
\begin{subequations}
\label{Equ:DictLearn:Inference_Dual}
\begin{align}
\min_{\nu} \quad& -g(\nu; x_t)
=
f^{\star}(\nu) - \nu^T x_t
+
\sum_{k=1}^N h_{y_k}^{\star}(W_k^T \nu)
\label{Equ:DictLearn:Inference_Dual_Objective}
\\
\mathrm{s.t.} \quad& \nu
\in
\mathcal{V}_{f}
\label{Equ:DictLearn:Inference_Dual_Constraint}
\end{align}
\end{subequations}
Note that the objective function in the above optimization problem is an aggregation of (i) individual costs associated with sub-dictionaries at different agents (last term in \eqref{Equ:DictLearn:Inference_Dual_Objective}), (ii) a term associated with the data sample $x_t$ (second term in \eqref{Equ:DictLearn:Inference_Dual_Objective}), and (iii) a term that is the conjugate function of the residual cost (first term in \eqref{Equ:DictLearn:Inference_Dual_Objective}). In contrast to \eqref{Equ:ProbForm:InferenceProblem_FactoredForm}, the cost function in \eqref{Equ:DictLearn:Inference_Dual_Objective} is now in a form that is amenable to distributed processing. In particular, diffusion strategies \cite{chen2011TSPdiffopt,sayed2013diffusion,sayed2014proc}, consensus strategies \cite{kar2008distributed,lee2013distributed,bertsekas1997parallel,tsitsiklis1986distributed}, or ADMM strategies \cite{mota2012distributedBP,mota2013DADMM,schizas2008consensus1,zhu2009distributed,ling2012multi,towfic2014dictionary} can now be applied
to obtain the optimal dual variable $\nu_t^o$ in a distributed manner at the various agents.
To arrive at the distributed solution, we proceed as follows. We denote the set of agents that observe the data sample $x_t$ by $\mc{N}_I$. Motivated by \eqref{Equ:DictLearn:Inference_Dual_Objective}, with each agent $k$, we associate the local cost function:
\begin{align}
J_k(\nu; x_t)
&\triangleq \!
\begin{cases}
- \frac{ \nu^T x_t }{|\mc{N}_I|}
\!+\!
\frac{1}{N} f^{\star}(\nu)
\!+\!
h_{y_k}^{\star}(W_k^T \nu),
& k \in \mc{N}_I \\
\frac{1}{N} f^{\star}(\nu)
\!+\!
h_{y_k}^{\star}(W_k^T \nu),
& k \notin \mc{N}_I
\end{cases} \label{Equ:DictLearn:Split_Nodes} \!\!
\end{align}
where $| \mc{N}_I |$ denotes the cardinality of $\mc{N}_I$. Then, the optimization problem \eqref{Equ:DictLearn:Inference_Dual_Objective}--\eqref{Equ:DictLearn:Inference_Dual_Constraint} can be rewritten as
\begin{subequations}
\begin{align}
\min_{\nu} \quad& \sum_{k=1}^N J_k(\nu; x_t)
\label{Equ:DictLearnDist:DualProblem_Objective_newForm}
\\
\mathrm{s.t.} \quad& \nu \in \mathcal{V}_{f}
\label{Equ:DictLearnDist:DualProblem_Constraint_newForm}
\end{align}
\end{subequations}
In Sections \ref{Sec:DictLearn:Inference_Diffusion} and \ref{Sec:DictLearn:Inference_ADMM}, we will first discuss the solution of \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm}--\eqref{Equ:DictLearnDist:DualProblem_Constraint_newForm} for the optimal dual variable, $\nu_t^o$, in a distributed manner. And then in Sec. \ref{Sec:DictLearn:RecoveryPrimalVar}, we will reveal how to recover the optimal primal variables $y_{k,t}^o$ and $z_t^o$ from $\nu_t^o$.
\subsection{Inference over Distributed Models: Diffusion Strategies}
\label{Sec:DictLearn:Inference_Diffusion}
Note that the new equivalent form \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm} is an aggregation of individual costs associated with different agents; each cost $J_k(\nu; x_t)$ only requires knowledge of $W_k$. Consider first the case in which $f(u)$ is strongly convex. Then, it holds that $\mathcal{V}_f = \mathbb{R}^M$ and problem \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm}--\eqref{Equ:DictLearnDist:DualProblem_Constraint_newForm} becomes an unconstrained optimization problem of the same general form as problems studied in \cite{chen2013JSTSPpareto,chen2012AllertonLimit}. Therefore, we can directly apply the diffusion strategies developed in these works to solve \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm}--\eqref{Equ:DictLearnDist:DualProblem_Constraint_newForm} in a fully distributed manner. The adapt-then-combine (ATC) implementation of the diffusion algorithm then takes the following form:
\begin{subequations}
\begin{align}
\psi_{k,i} &= \nu_{k,i-1} - \mu \cdot \nabla_{\nu} J_k( \nu_{k,i-1}; x_t )
\label{Equ:DictLearnDist:ATC_adapt}
\\
\nu_{k,i} &= \sum_{ \ell \in \mc{N}_k } a_{\ell k} \psi_{\ell,i}
\label{Equ:DictLearnDist:ATC_combine}
\end{align}
\end{subequations}
where $\nu_{k,i}$ denotes the estimate of the optimal $\nu_t^o$ at agent $k$ at iteration $i$ (we will use $i$ to denote the $i$-th iteration of the inference, and use $t$ to denote the $t$-th data sample), $\psi_{k,i}$ is an intermediate variable, $\mc{N}_k$ denotes the neighborhood of agent $k$, $\mu$ is the step-size parameter chosen to be a small positive number, and $a_{\ell k}$ is the combination coefficient that agent $k$ assigns to the information received from agent $\ell$ and it satisfies
\begin{align}
\sum_{\ell \in \mc{N}_k} a_{\ell k} = 1,
\;\;
a_{\ell k} > 0 \mathrm{~if~} \ell \in \mc{N}_k,\;\;
a_{\ell k} = 0 \mathrm{~if~} \ell \notin \mc{N}_k
\label{Equ:DictLearnDist:A_condition_general}
\end{align}
Let $A$ denote the $N \times N$ matrix that collects $a_{\ell k}$ as its $(\ell,k)$-th entry.
Then, it is shown in \cite{chen2013JSTSPpareto} that as long as the matrix $A$ is doubly-stochastic (i.e., satisfies $A\mathds{1}=A^T\mathds{1}=\mathds{1}$) and $\mu$ is selected such that
\begin{align}
0 < \mu < \min_{1 \le k \le N} \frac{1}{\sigma_{k}}
\label{Equ:DictLearnDist:StepSizeCondition}
\end{align}
where $\sigma_k$ is the Lipschitz constant\footnote{{
If $J_k(\nu; x_t)$ is twice-differentiable, then the Lipschitz gradient condition \eqref{Equ:DictLearnDist:LipschitzGradient} is equivalent to requiring an upper bound on the Hessian of $J_k(\nu; x_t)$, i.e.,
$0 \le \nabla_{\nu}^2 J_{k}(\nu; x_t) \le \sigma_{k} I_M$.
}} of the gradient of $J_k(\nu; x_t)$:
\begin{align}
\| \nabla_{\nu} J_{k}(\nu_1; x_t) - \nabla_{\nu} J_{k}(\nu_2; x_t) \|
\le \sigma_{k} \cdot \| \nu_1 - \nu_2 \|
\label{Equ:DictLearnDist:LipschitzGradient}
\end{align}
then algorithm \eqref{Equ:DictLearnDist:ATC_adapt}--\eqref{Equ:DictLearnDist:ATC_combine} converges to a fixed point that is $O(\mu^2)$ away from the optimal solution of \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm} in squared Euclidean distance. We remark that a doubly-stochastic matrix is one that satisfies $A\mathds{1}=A^T\mathds{1}=\mathds{1}$.
\begin{table*}[!ht]
\caption{Conjugate functions used in this paper for different tasks}
\label{Tab:ConjProx}
\centering
\renewcommand{\arraystretch}{2.0}
\begin{threeparttable}
\begin{tabular}{c||c|c|c|c|c|c|c|c}
\hline \hline
\rowcolor[gray]{0.9} \rule[-1ex]{0pt}{4ex} \textbf{Tasks} & $f(u)$ & $f^{\star}(\nu)$ & $\mathcal{V}_f$ & $ z_t^o$ & $h_{y_k}(y_k)$ & $h_{y_k}^{\star}(W_k^T \nu)$ & $\mathcal{V}_{h_{y_k}}$ & $y_{k,t}^o$\\
\hline
\rule[-1ex]{0pt}{4ex} \textbf{Sparse SVD} & $\frac{1}{2} \|u\|_2^2$ & $\frac{1}{2}\|\nu\|_2^2$ & $\mathbb{R}^M$ & $x_t - \nu_t^o$ & $\gamma \|y_k\|_{1} + \frac{\delta}{2} \|y_k\|_2^2$ & $\mc{S}_{\frac{\gamma}{\delta}}\left(\frac{W_k^T \nu}{\delta}\right)$ \tnote{b} & $\mathbb{R}^M$ & $\mc{T}_{\frac{\gamma}{\delta}}\left(\frac{W_k^T \nu_t^o}{\delta}\right)$\tnote{a} \\
\hline
\rule[-1ex]{0pt}{4ex} \textbf{Bi-Clustering} & $\frac{1}{2}\|u\|_2^2$ & $\frac{1}{2}\|\nu\|_2^2$ & $\mathbb{R}^M$ & $x_t - \nu_t^o$ & $\gamma \|y_k\|_{1} + \frac{\delta}{2} \|y_k\|_2^2$ & $\mc{S}_{\frac{\gamma}{\delta}}\left(\frac{W_k^T \nu}{\delta}\right)$ & $\mathbb{R}^M$ & $\mc{T}_{\frac{\gamma}{\delta}}\left(\frac{W_k^T \nu_t^o}{\delta}\right)$ \\
\hline
\rule[0ex]{0pt}{4ex}\multirow{1}{*}{\textbf{Nonnegative Matrix}} & $\frac{1}{2}\|u\|_2^2$ & $\frac{1}{2}\|\nu\|_2^2$ & $\mathbb{R}^M$ & $x_t - \nu_t^o$ & $\gamma \|y_k\|_{1,+} + \frac{\delta}{2} \|y_k\|_2^2$ & $\mc{S}_{\frac{\gamma}{\delta}}^{+}\left(\frac{W_k^T \nu}{\delta}\right)$ \tnote{d} & $\mathbb{R}^M$ & $\mc{T}_{\frac{\gamma}{\delta}}^{+}\left(\frac{W_k^T \nu_t^o}{\delta}\right)$\tnote{c} \\
\cline{2-9}
\multirow{1}{*}{\textbf{Factorization}} & $\displaystyle\sum_{m=1}^M L(u_m)$ & $\frac{\eta}{2}\|\nu\|_2^2$ & $\{\nu: \|\nu\|_{\infty} \le 1\}$ & \cancel{\phantom{Nothing}} & $\gamma \|y_k\|_{1,+} + \frac{\delta}{2} \|y_k\|_2^2$ & $\mc{S}_{\frac{\gamma}{\delta}}^{+}\left(\frac{W_k^T \nu}{\delta}\right)$ & $\mathbb{R}^M$ & $\mc{T}_{\frac{\gamma}{\delta}}^{+}\left(\frac{W_k^T \nu_t^o}{\delta}\right)$ \\
\hline \hline
\end{tabular}
\begin{tablenotes}
\vspace{0.5em}
\item[a]
$\mc{T}_{\lambda}(x)$ denotes the entry-wise soft-thresholding operator on
the vector $x$: $[\mc{T}_{\lambda}(x)]_n \triangleq
(| [x]_n |-\lambda)_{+} \mathrm{sgn}([x]_n)$, where
$(x)_{+} = \max( x, 0 )$.
\vspace{0.5em}
\item[b]
$\mc{S}_{\frac{\gamma}{\delta}}(x)$ is the function defined by
$\mc{S}_{\frac{\gamma}{\delta}}(x) \triangleq
-\frac{\delta}{2}
\cdot
\big\| \mc{T}_{\frac{\gamma}{\delta}}
(x)
\big\|_2^2
-
\gamma
\cdot
\big\|\mc{T}_{\frac{\gamma}{\delta}}
(x)
\big\|_1
+
\delta \cdot x^T
\mc{T}_{\frac{\gamma}{\delta}}
\left(
x
\right)$
for $x \in \mathbb{R}^M$.
\vspace{0.5em}
\item[c]
$\mc{T}_{\lambda}^{+}(x)$ denotes the entry-wise one-side
soft-thresholding operator on the vector $x$:
$[\mc{T}_{\lambda}^{+}(x)]_n \triangleq ([x]_n - \lambda)_{+}$.
\vspace{0.5em}
\item[d]
$\mc{S}_{\frac{\gamma}{\delta}}^{+}(x)$ is defined by
$\mc{S}_{\frac{\gamma}{\delta}}^{+}(x) \triangleq
-\frac{\delta}{2}
\cdot
\big\| \mc{T}_{\frac{\gamma}{\delta}}^{+}(x)
\big\|_2^2
-
\gamma
\cdot
\big\|\mc{T}_{\frac{\gamma}{\delta}}^{+}
(x)
\big\|_1
+
\delta \cdot x^T
\mc{T}_{\frac{\gamma}{\delta}}^{+}
\left(
x
\right)$
for $x \in \mathbb{R}^M$.
\vspace{0.5em}
\item[e]
The functions $\mc{T}_{\lambda}(x)$, $\mc{T}_{\lambda}^{+}(x)$,
$\mc{S}_{\frac{\gamma}{\delta}}(x)$, and
$\mc{S}_{\frac{\gamma}{\delta}}^{+}(x)$ for the case of a scalar argument $x$ are illustrated in Fig.
\ref{Fig:soft_thres_S_func}.
\end{tablenotes}
\end{threeparttable}
\end{table*}
Consider now the case in which the constraint set $\mathcal{V}_f$ is \emph{not} equal to $\mathbb{R}^{M}$ but is still known to all agents. This is a reasonable requirement. In general, we need to solve the supremum in \eqref{Equ:DictLearnDist:r_conj_def} with $r(x)=f(x)$ to derive the expression for $f^{\star}(\nu)$ and determine the set $\mathcal{V}_f$ that makes the supremum in \eqref{Equ:DictLearnDist:r_conj_def} finite. Fortunately, this step can be pursued in closed-form for many typical choices of $f(u)$. We list in Table \ref{Tab:ConjProx} the results that will be used in Sec. \ref{Sec:Experiments}; part of these results are derived in Appendix \ref{Appendix:DerivationConjugateFun} and the rest is from \cite[pp.90-95]{boyd2004convex}. Usually, $\mathcal{V}_f$ for these typical choices of $f(u)$ are simple sets whose projection operators\footnote{The projection operator onto the set $\mathcal{V}_f$ is defined as $\displaystyle \Pi_{\mathcal{V}_f}(\nu) \triangleq \arg\min_{x \in \mathcal{V}_f} \| x - \nu \|_2$.} can be found in closed-form --- see also \cite{parikh2013proximal}.
For example, the projection operator onto the set
\begin{align}
\mathcal{V}_f = \{\nu: \| \nu \|_{\infty} \le 1\}
= \{ \nu: - \mathds{1} \preceq \nu \preceq \mathds{1} \}
\end{align}
that is listed in the third row of Table \ref{Tab:ConjProx} is given by
\begin{align}
[\Pi_{ \mathcal{V}_f }( \nu)]_m
= \begin{cases}
1 & \mathrm{if~} \nu_m > 1 \\
\nu_m & \mathrm{if~} -1 \le \nu_m \le 1 \\
-1 & \mathrm{if~} \nu_m < -1
\end{cases} \label{Equ:DictLearnDist:Projection_Inf_Norm}
\end{align}
where $[x]_m$ denotes the $m$-th entry of the vector $x$ and $\nu_m$ denotes the $m$-th entry of the vector $\nu$. Once the constraint set $\mathcal{V}_f$ is found, it can be enforced either by incorporating local projections onto $\mathcal{V}_f$ into the combination step \eqref{Equ:DictLearnDist:ATC_combine} at each agent \cite{theodoridis2011adaptive} or by using the penalized diffusion method \cite{towfic2013adaptive2}. For example, the projection-based strategy replaces \eqref{Equ:DictLearnDist:ATC_adapt}--\eqref{Equ:DictLearnDist:ATC_combine} by:
\begin{subequations}
\begin{align}
\psi_{k,i} &= \nu_{k,i-1} - \mu \cdot \nabla_{\nu} J_k( \nu_{k,i-1}; x_t )
\label{Equ:DictLearnDist:ATC_adapt_projection}
\\
\nu_{k,i} &= \Pi_{\mathcal{V}_f}
\left[
\sum_{ \ell \in \mc{N}_k } a_{\ell k} \psi_{\ell,i}
\right]
\label{Equ:DictLearnDist:ATC_combine_projection}
\end{align}
\end{subequations}
where $\Pi_{\mathcal{V}_f}[\cdot]$ is the projection operator onto $\mathcal{V}_f$.
{
\subsection{Inference over Distributed Models: ADMM Strategies}
\label{Sec:DictLearn:Inference_ADMM}
An alternative approach to solving the dual inference problem \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm}--\eqref{Equ:DictLearnDist:DualProblem_Constraint_newForm} is the distributed alternating direction multiplier method (ADMM) \cite{mota2012distributedBP,mota2013DADMM,Barbarossa2014distributed,schizas2008consensus1,zhu2009distributed}. Depending on the configuration of the network, there are different variations of distributed ADMM strategies. For example, the method proposed in \cite{schizas2008consensus1} relies on a set of bridge nodes for the distributed interactions among agents, and the method in \cite{mota2012distributedBP,mota2013DADMM} uses a graph coloring approach to partition the agents in the network into different groups, and lets the optimization process alternate between different groups with one group of agents engaged at a time. In \cite{zhu2009distributed} and \cite{Barbarossa2014distributed}, the authors developed ADMM strategies that adopt Jacobian style updates with all agents engaged in the computation concurrently. Below, we describe the Jacobian-ADMM strategies from\cite[p.356]{Barbarossa2014distributed} and briefly compare them with the diffusion strategies.
The Jacobian-ADMM strategy solves \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm}--\eqref{Equ:DictLearnDist:DualProblem_Constraint_newForm} by first transforming it into the following equivalent optimization problem:
\begin{subequations}
\begin{align}
\min_{\nu} \quad& \sum_{k=1}^N \big[ J_k(\nu_k; x_t) + I_{\mathcal{V}_f}(\nu_k) \big]
\label{Equ:DictLearnDist:DualProblem_Objective_newForm_ADMM}
\\
\mathrm{s.t.} \quad& \nu_k = \nu_{\ell}, \quad \ell \in \mathcal{N}_k\backslash\{k\},
\;\; k=1,\ldots,N
\label{Equ:DictLearnDist:DualProblem_Constraint_newForm_ADMM}
\end{align}
\end{subequations}
where the cost function is decoupled among different $\{\nu_k\}$ and the constraints are coupled through neighborhoods. Then, the following recursion is used to solve \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm_ADMM}--\eqref{Equ:DictLearnDist:DualProblem_Constraint_newForm_ADMM}:
\begin{subequations}
\begin{align}
\nu_{k,i} &= \arg\min_{\nu_k}
\sum_{k=1}^N
\bigg\{
\big[ J_k(\nu_k; x_t) + I_{\mathcal{V}_f}(\nu_k) \big]
\nonumber\\
&\quad+ \!
\sum_{\ell=1}^N
\! b_{k \ell}
\Big[ \!
\lambda_{k \ell,i\!-\!1} ^T
( \nu_{\ell,i \!-\! 1} \!-\! \nu_{k} )
\!+\!
\| \nu_{\ell,i-1} \!-\! \nu_{k} \|_2^2
\Big] \!
\bigg\}
\label{Equ:DictLearnDist:ADMM_primal}
\\
\lambda_{k\ell,i} \! &= \! \lambda_{k\ell,i-1}
+
\mu \; b_{k \ell} \cdot
\left(
\nu_{k,i} - \nu_{\ell,i}
\right)
\label{Equ:DictLearnDist:ADMM_dual}
\end{align}
\end{subequations}
where $b_{k\ell}$ is the $(k, \ell)$-th entry of the adjacency matrix $B = [b_{k\ell}]$ of the network, which is defined as:
\begin{align}
b_{k\ell} = 1 \mathrm{~if~} \ell \in \mathcal{N}_k \backslash \{k\},
\; b_{k \ell}=0 \mathrm{~otherwise}
\end{align}
\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{Fig_ThresholdingFunctions_v2}
\caption{Illustration of the functions $\mc{T}_{\lambda}(x)$, $\mc{T}_{\lambda}^+(x)$, $\mc{S}_{\lambda}(x)$, and $\mc{S}_{\lambda}^+(x)$.}
\label{Fig:soft_thres_S_func}
\end{figure}%
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{Fig_ADMMvsDiffusion}
\caption{Comparison between the ADMM strategy and the diffusion strategy. The diffusion strategy
has two time scales and the ADMM strategy may have three time scales. The first time scale is
the dictionary update over the data stream (see Sec. \ref{Sec:DictLearn:DictUpdate}),
the second time scale is the iterative algorithm for solving the inference problem for each data
sample $x_t$, and the third time scale in ADMM is to solve the ``argmin'' in
\eqref{Equ:DictLearnDist:ADMM_primal}.}
\label{Fig:ADMMvsDiffusion}
\end{figure}%
From recursion \eqref{Equ:DictLearnDist:ADMM_primal}--\eqref{Equ:DictLearnDist:ADMM_dual}, we observe that ADMM requires solving a separate optimization problem ($\arg\min$) for each ADMM step. This optimization problem generally requires an iterative algorithm to solve when it cannot be solved in closed-form, which adds a third time scale to the algorithm, as explained in\cite{towfic2014dictionary} in the context of dictionary learning. This situation is illustrated in Fig. \ref{Fig:ADMMvsDiffusion}. The need for a third time-scale usually translates into requiring faster processing at the agents between data arrivals, which can be a hindrance for adaptation in real-time.
}
\subsection{Recovery of the Primal Variables}
\label{Sec:DictLearn:RecoveryPrimalVar}
Returning to the diffusion solution \eqref{Equ:DictLearnDist:ATC_adapt}--\eqref{Equ:DictLearnDist:ATC_combine} or \eqref{Equ:DictLearnDist:ATC_adapt_projection}--\eqref{Equ:DictLearnDist:ATC_adapt_projection}, once the optimal dual variable $\nu_t^o$ has been estimated by the various agents, the optimal primal variables $y_{k,t}^o$ and $z_t^o$ can now be recovered uniquely if $f(u)$ and $\{h_{y_k}(y_k)\}$ are strongly convex. In this case, the infimums in \eqref{Equ:DictLearnDist:DualFunction_def} can be attained and become minima. As a result, optimal primal variables can be recovered via
\begin{align}
z_t^o &= \arg\min_{ z }
\left\{
f( x_t - z ) + (\nu_t^o)^T z
\right\}
\nonumber\\
&\overset{(a)}{=}
x_t - \arg\max_{u} \big[ (\nu_t^o)^T u - f(u) \big]
\label{Equ:DictLearnDist:z_optimal_primal}
\\
y_{k,t}^o &=
\arg\min_{y_k}
\Big\{
h_{y_k}(y_k)
\!-\!
(\nu_t^o)^T W_k y_k
\Big\}
\nonumber\\
&=
\arg\max_{y_k}
\big[
(W_k^T \nu_t^o)^T y_k - h_{y_k}(y_k)
\big]
\label{Equ:DictLearnDist:yk_optimal_primal}
\end{align}
where step (a) performs the variable substitution $u=x_t-z$.
By \eqref{Equ:DictLearnDist:z_optimal_primal}--\eqref{Equ:DictLearnDist:yk_optimal_primal}, we obtain the optimal solutions of \eqref{Equ:DictLearnDist:NewInference_Objective}--\eqref{Equ:DictLearnDist:NewInference_Constraint} (and also of the original inference problem \eqref{Equ:ProbForm:InferenceProblem}) after first solving the dual problem \eqref{Equ:DictLearnDist:DualProblem}. For many typical choices of $f(\cdot)$ and $h_{y_k}(\cdot)$, the solutions of \eqref{Equ:DictLearnDist:z_optimal_primal}--\eqref{Equ:DictLearnDist:yk_optimal_primal} can be expressed in closed form in terms of $\nu_t^o$. In Table \ref{Tab:ConjProx}, we list the results that will be used later in Sec. \ref{Sec:Experiments} with the derivation given in Appendix \ref{Appendix:DerivationConjugateFun}.
The strong convexity of $f(u)$ and $\{h_{y_k}(y_k)\}$ is needed if we want to uniquely recover $z_t^o$ and $\{y_{k,t}^o\}$ from the dual problem \eqref{Equ:DictLearnDist:DualProblem}. As we will show further ahead in \eqref{Equ:DictLearnDist:DictionaryUpdate_final}, the quantities $\{y_{k,t}^o\}$ are always needed in the dictionary update. For this reason, we assumed in Assumption \ref{Assumption:StronglyConvexRegularization_hyk} that the $\{h_{y_k}(y_k)\}$ are strongly convex, which can always be satisfied by means of elastic net regularization as explained earlier. On the other hand, depending on the application, the recovery of $z_t^o$ is not always needed and neither is the strong convexity of $f(u)$ (in these cases, it is sufficient to assume that $f(u)$) is convex). For example, as explained in \cite{chen2014icasspdictionary}, the image denoising application requires recovery of $z_t^o$ as the final reconstructed image. On the other hand, the novel document detection application discussed further ahead does not require recovery of $z_t^o$ but the maximum value of the dual function, $g(\nu; x_t)$, which, by strong duality, is equal to the minimum value of the cost function \eqref{Equ:DictLearnDist:NewInference_Objective} and that of \eqref{Equ:ProbForm:InferenceProblem}.
\subsection{Choice of Residual and Regularization Functions}
\label{Sec:DictLearn:ChoiceResReg}
In Tables \ref{Tab:Task}--\ref{Tab:ConjProx}, we list several typical choices for the residual function, $f(u)$, and the regularization functions, $\{h_{y_k}(y_k)\}$. In general, a careful choice of $f(u)$ and $\{h_{y_k}(y_k)\}$ can make the dual cost \eqref{Equ:DictLearn:Inference_Dual_Objective} better conditioned than in the primal cost \eqref{Equ:DictLearnDist:NewInference_Objective}. Recall that the primal cost \eqref{Equ:DictLearnDist:NewInference_Objective} may not be differentiable due to the choice of $h_{y_k}(y_k)$ (e.g., the elastic net). However, if $f(u)$ is chosen to be strictly convex with Lipschitz gradients and the $\{h_{y_k}(y_k)\}$ are chosen to be strongly convex (not necessarily differentiable), then the conjugate function $f^{\star}(\cdot)$ will be a differentiable strongly convex function with Lipschitz gradient and the $\{h_{y_k}^{\star}(\cdot)\}$ will be differentiable convex functions with Lipschitz gradients \cite[pp.79--84]{urruty1993convex2}. Adding $f^{\star}(\cdot)$ and $\{h_{y_k}^{\star}(\cdot)\}$ together in \eqref{Equ:DictLearn:Inference_Dual_Objective} essentially transforms a non-differentiable primal cost \eqref{Equ:DictLearnDist:NewInference_Objective} into a differentiable strongly convex dual cost \eqref{Equ:DictLearn:Inference_Dual_Objective} with Lipschitz gradients. As a result, the algorithms that optimize the dual problem \eqref{Equ:DictLearn:Inference_Dual_Objective}--\eqref{Equ:DictLearn:Inference_Dual_Constraint} can generally enjoy a fast (geometric) convergence rate \cite{poliak1987introduction,chen2013JSTSPpareto,sayed2014adaptation}.
\subsection{Distributed Dictionary Updates}
\label{Sec:DictLearn:DictUpdate}
Now that we have shown how the inference task \eqref{Equ:ProbForm:InferenceProblem} can be solved in a distributed manner, we move on to explain how the local sub-dictionaries $W_k$ can be updated through the solution of the stochastic optimization problem \eqref{Equ:ProbForm:DictLearn_Objective}--\eqref{Equ:ProbForm:DictLearn_Constraint}, which is rewritten as:
\begin{subequations}
\label{Equ:DictLearnDist:DictUpdate_total}
\begin{align}
\min_{W} \quad& \mathbb{E}
Q(W, \bm{y}_t^o; \bm{x}_t)
+
\sum_{k=1}^N
h_{W_k}(W_k)
\label{Equ:DictLearnDist:DictUpdate_Objective}
\\
\mathrm{s.t.}
\quad&
W_k \in \mc{W}_k,
\quad
k = 1,\ldots, N
\label{Equ:DictLearnDist:DictUpdate_Constraint}
\end{align}
\end{subequations}
where the loss function $Q(W, \bm{y}_t^o; \bm{x}_t)$ is given in \eqref{Equ:ProbForm:InferenceProblem_FactoredForm}, $\bm{y}_t^o \triangleq \mathrm{col}\{ \bm{y}_{1,t}^o,\ldots,\bm{y}_{N,t}^o \}$, the decomposition for $h_W(W)$ from \eqref{Equ:ProbForm:hW_factorization} is used, and we assume the constraint set $\mc{W}$ can be decomposed into a set of constraints $\{\mc{W}_k\}$ on the individual sub-dictionaries $W_k$; this condition usually holds for typical dictionary learning applications --- see Table \ref{Tab:Task}.
{
Problem \eqref{Equ:DictLearnDist:DictUpdate_Objective}--\eqref{Equ:DictLearnDist:DictUpdate_Constraint} can also be written as the following unconstrained optimization problem by introducing indicator functions for the sets $\{\mathcal{W}_k\}$:
\begin{align}
\min_{W} \quad& \mathbb{E}
Q(W, \bm{y}_t^o; \bm{x}_t)
\!+\!
\sum_{k=1}^N
\Big[
h_{W_k}(W_k)
+
I_{\mathcal{W}_k}(W_k)
\Big]
\label{Equ:DictLearnDist:DictUpdate_Objective_Unconstrained}
\end{align}
}%
Note that the cost function in \eqref{Equ:DictLearnDist:DictUpdate_Objective_Unconstrained} consists of two parts, where the first term is differentiable\footnote{Note from \eqref{Equ:ProbForm:InferenceProblem_FactoredForm} that $Q(\cdot)$ depends on $W$ via $f(\cdot)$, which is assumed to be differentiable.} with respect to $W$ while the second term, if it exists, is non-differentiable but usually consists of simple components --- see Table \ref{Tab:Task}. A typical approach to optimizing cost functions of this type is the \emph{proximal gradient} method\cite{figueiredo2005bound,figueiredo2007majorization,beck2009fast,parikh2013proximal}, which applies gradient descent to the first differentiable part followed by a proximal operator to the second non-differentiable part. This method is known to converge faster than applying the subgradient descent method to both parts.
{ However, the proximal gradient methods in\cite{figueiredo2005bound,figueiredo2007majorization,beck2009fast,parikh2013proximal} are developed for deterministic optimization, where the exact form of the objective function is known. In constrast, our objective function in \eqref{Equ:DictLearnDist:DictUpdate_Objective_Unconstrained} assumes a stochastic form and is unknown beforehand because the statistical distribution of the data $\{\bm{x}_t\}$ is not known.
}%
Therefore, our strategy is to apply the proximal gradient method to the cost function in \eqref{Equ:DictLearnDist:DictUpdate_Objective_Unconstrained} and remove the expectation operator to obtain an instantaneous approximation to the true gradient; this is the approach typically used in adaptation \cite{Sayed08,sayed2014adaptation,sayed2014proc} and stochastic approximation\cite{kushner2003stochastic}:
\begin{align}
W_{k,t} \!=\! \mathrm{prox}_{\mu_w \cdot (h_{W_k}\!+\!I_{\mathcal{W}_k})}
\Big\{
W_{k,t-1} \!-\! \mu_w \nabla_{W_k} Q(W_{t-1}, y_t^o; x_t)
\Big\}
\label{Equ:DictLearn:DictUpdate_interm1_alternative}
\end{align}
Recursion \eqref{Equ:DictLearn:DictUpdate_interm1_alternative} is effective as long as the proximal operator of $h_{W_k}(W_k)+I_{\mathcal{W}_k}(W_k)$ can be solved easily in closed-form. When this is not possible but the proximal operators of $h_{W_k}(\cdot)$ and $I_{\mathcal{W}_k}(\cdot)$ are simple, it is preferable to apply a stochastic gradient descent step, followed by the proximal operator of $h_{W_k}(\cdot)$, and then the proximal operator of $I_{\mathcal{W}_k}(\cdot)$ (equivalent to $\Pi_{\mathcal{W}_k}(\cdot)$\cite{parikh2013proximal}, which is the projection onto $\mathcal{W}_k$) in an \emph{incremental} manner \cite{bertsekas2010incremental}, thus leading to the following recursion:
\begin{align}
W_{k,t} \!=\! \Pi_{\mc{W}_k}\!\!
\left\{\!
\mathrm{prox}_{\mu_w \!\cdot h_{W_k}}
\!\!
\big(
W_{k,t-1} \!-\! \mu_w \nabla_{W_k} Q(W_{t-1}, y_t^o; x_t)
\big)\!
\right\}
\label{Equ:DictLearn:DictUpdate_interm1}
\end{align}
where $W_{t-1} \triangleq [ W_{1,t-1}, \cdots, W_{N,t-1} ]$, and $\mathrm{prox}_{\mu_w \cdot h_{W_k}}(\cdot)$ denotes the proximal operator of $\mu_w \cdot h_{W_k}(W_k)$. The expression for the gradient $\mu_w \nabla_{W_k} Q(W_{t-1}, y_t^o; x_t)$ will be given further ahead in \eqref{Equ:DictLearn:DictUpdate_gradWk_interm1}--\eqref{Equ:DictLearnDist:DictionaryUpdate_final}. We recall that the proximal operator of a vector function $h(u)$ is defined as \cite[p.6]{parikh2013proximal}:
\begin{align}
\mathrm{prox}_{h}(x)
\triangleq
\arg\min_{u}
\left(
h(u) + \frac{1}{2} \| u - x \|_2^2
\right)
\label{Equ:DictLearnDist:Prox_def}
\end{align}
For a matrix function $h(U)$, the proximal operator assumes the same form as \eqref{Equ:DictLearnDist:Prox_def} except that the Euclidean norm in \eqref{Equ:DictLearnDist:Prox_def} is replaced by the Frobenius norm. The proximal operator for $\mu_w \cdot h_{W_k}(W_k) = \mu_w\beta \cdot \vertiii{W_k}_1$ used in the bi-clustering task in Table \ref{Tab:Task} is the entry-wise soft-thresholding function \cite[p.191]{parikh2013proximal}:
\begin{align}
\mathrm{prox}_{\mu_w \cdot h_{W_k}}(\cdot)
=
\mathrm{prox}_{\mu_w \beta \cdot \vertiii{W_k}}(\cdot)
=
\mathcal{T}_{\mu_w \cdot \beta}(\cdot)
\label{Equ:DictLearn:DictUpdate_ProxW_SoftThresh}
\end{align}
and the proximal operator for $h_{W_k}(W_k) = 0$ for other cases in Table \ref{Tab:Task} is the identity mapping: $\mathrm{prox}_{0}(x) =x$. With regards to the projection operator used in \eqref{Equ:DictLearn:DictUpdate_interm1}, we provide some examples of interest for the current work. If the constraint set $\mc{W}_k$ is of the form:
\begin{align}
\mc{W}_k = \{ W_k: \; \|[W_k]_{:,q}\|_2 \le 1\}
\end{align}
then the projection operator $\Pi_{\mc{W}_k}(\cdot)$ is given by\cite{theodoridis2011adaptive,parikh2013proximal}:
\begin{align}
[\Pi_{\mc{W}_k}(X)]_{:,n}
= \begin{cases}
[X]_{:,n}, & \|[X]_{:,n}\|_2 \le 1 \\
\frac{[X]_{:,n}}{\|[X]_{:,n}\|_2}, & \|[X]_{:,n}\|_2 > 1
\end{cases}
\label{Equ:DictLearn:DictUpdate_Projection_normball}
\end{align}
On the other hand, if the constraint set $\mc{W}_k$ is of the form:
\begin{align}
\mc{W}_k = \{ W_k: \; \|[W_k]_{:,q}\|_2 \le 1, \; \mc{W} \succeq 0\}
\end{align}
then the projection operator $\Pi_{\mc{W}_k}(\cdot)$ becomes
\begin{align}
[\Pi_{\mc{W}_k}(X)]_{:,n}
= \begin{cases}
\big([X]_{:,n}\big)_{+},
& \|\big([X]_{:,n}\big)_{+}\|_2 \le 1
\\
\displaystyle
\frac{\big([X]_{:,n}\big)_{+}}
{\|\big([X]_{:,n}\big)_{+}\|_2},
& \|\big([X]_{:,n}\big)_{+}\|_2 > 1
\end{cases}
\label{Equ:DictLearn:DictUpdate_Projection_nonnegativenormball}
\end{align}
where $(x)_{+} = \max(x,0)$, i.e., it replaces all the negative entries of a vector $x$ with zeros.
\noindent\begin{algorithm}[t]
\caption{{\small Model-distributed diffusion strategy for dictionary learning (Main algorithm)}}
\label{alg:DiffusionSparseCoding}
{\small{
\begin{algorithmic}
{
\STATE {\bf Initialization:} The sub-dictionaries $\{W_k\}$ are randomly initialized and then projected onto either the constraint \eqref{Equ:ProbForm:W_subUnitNormConstraint} or \eqref{Equ:ProbForm:W_subUnitNormNonnegConstraint}, depending on the task in Tab. \ref{Tab:Task}.}
\FOR{each input data sample $x_t$}
\STATE Compute $\nu_{t}^o$ by iterating \eqref{Equ:DictLearnDist:ATC_adapt}-\eqref{Equ:DictLearnDist:ATC_combine} until convergence: $\nu_t^o\approx \nu_{k,i}$. That is:
\begin{align*}
\begin{cases}
\psi_{k,i} = \nu_{k,i-1}
-
\mu \cdot \nabla_{\nu} J_k( \nu_{k,i-1}; x_t )
\\
\nu_{k,i} = \displaystyle
\Pi_{\mathcal{V}_f}\Big\{\sum_{ \ell \in \mc{N}_k } a_{\ell k} \psi_{\ell,i}\Big\}
\end{cases}
\end{align*}
{ with initialization $\{\nu_{k,0} = 0, \; k=1,\ldots,N\}$.}
\FOR{each agent $k$}
\STATE Compute coefficient $y_{k,t}^o$ using Table \ref{Tab:ConjProx} or
\eqref{Equ:DictLearnDist:yk_optimal_primal}:
\begin{align*}
y_{k,t}^o = \arg\max_{y_k}
\big[
(W_k^T \nu_t^o)^T y_k - h_{y_k}(y_k)
\big]
\end{align*}
\vspace{-0.8em}
\STATE Adjust dictionary element $W_{k,t}$ using \eqref{Equ:DictLearnDist:DictionaryUpdate_final}:
\begin{align*}
W_{k,t} = \Pi_{\mc{W}_k}
\left\{
\mathrm{prox}_{\mu_w \cdot h_{W_k}}
\big(
W_{k,t-1} + \mu_w \nu_t^o (y_{k,t}^o)^T
\big)
\right\}
\end{align*}
\ENDFOR
\ENDFOR
\end{algorithmic}}}
\end{algorithm}
Now, we return to derive the expression for the gradient $\nabla_{W_k} Q(W_{t-1}, y_t^o; x_t)$ in \eqref{Equ:DictLearn:DictUpdate_interm1}. By \eqref{Equ:ProbForm:InferenceProblem_FactoredForm}, we have
\begin{align}
\nabla_{W_k} \! Q(W_{t-1}, y_t^o; x_t)
\!&= \!-\!
f_{u}'
\!
\Big(\!
x_t \!-\! \sum_{k=1}^N W_{k,t-1} y_{k,t}^o
\!
\Big)
(y_{k,t}^o)^T \!\!
\label{Equ:DictLearn:DictUpdate_gradWk_interm1}
\end{align}
where $f_{u}'(u)$ denotes the gradient of $f(u)$ with respect to the residual $u$. On the face of it, expression \eqref{Equ:DictLearn:DictUpdate_gradWk_interm1} requires global knowledge by agent $k$ of all sub-dictionaries $\{W_k\}$ across the network, which goes against the desired objective of arriving at a distributed implementation. However, we can develop a distributed algorithm by exploiting the structure of the problem as follows. Note from \eqref{Equ:DictLearnDist:Lagrangian} that the optimal inference result should satisfy:
\begin{align}
\begin{cases}
0 = \nabla_z L ( \{y_{k,t}^o\}, z_t^o, \nu_t^o; x_t )
\\
0 = \nabla_\nu L ( \{y_{k,t}^o\}, z_t^o, \nu_t^o; x_t )
\end{cases}
\!\! \Leftrightarrow \;\;
\begin{cases} \!
0 = - \! f_{u}'( x_t \!-\! z_t^o ) \!+\! \nu_t^o
\\
\displaystyle\!
z_t^o = \sum_{k=1}^N
W_{k,t-1} y_{k,t}^o
\end{cases}\!\!\!\!\!\!\!\!\!
\label{Equ:DictLearn:DictUpdate_OptimalInferenceCondition}
\end{align}
which leads to
\begin{align}
&0 = - f_{u}'\Big(
x_t - \sum_{k=1}^N W_{k,t-1} y_{k,t}^o
\Big)
+
\nu_t^o
\nonumber\\
&\quad\Leftrightarrow\quad
\nu_t^o = f_{u}'\Big(
x_t - \sum_{k=1}^N W_{k,t-1} y_{k,t}^o
\Big)
\label{Equ:Dictlearn:DictUpdate_grad_nu_relation}
\end{align}
In other words, we find that the optimal dual variable $\nu_t^o$ is equal to the desired gradient vector. Substituting \eqref{Equ:Dictlearn:DictUpdate_grad_nu_relation} into \eqref{Equ:DictLearn:DictUpdate_gradWk_interm1}, the dictionary learning update \eqref{Equ:DictLearn:DictUpdate_interm1} becomes
\begin{align}
W_{k,t} = \Pi_{\mc{W}_k}
\left\{
\mathrm{prox}_{\mu_w \cdot h_{W_k}}
\big(
W_{k,t-1} + \mu_w \nu_t^o (y_{k,t}^o)^T
\big)
\right\}
\label{Equ:DictLearnDist:DictionaryUpdate_final}
\end{align}
which is now in a fully-distributed form. At each agent $k$, the above $\nu_t^o$ can be replaced by the estimate $\nu_{k,i}$ after a sufficient number of inference iterations (large enough $i$). We note that the dictionary learning update \eqref{Equ:DictLearnDist:DictionaryUpdate_final} has the following important interpretation. Let
\begin{align}
u_t^o \triangleq
x_t - \sum_{k=1}^N W_{k,t-1} y_{k,t}^o
\end{align}
which is the optimal prediction residual error using the entire existing dictionary set $\{W_{k,t-1}\}_{k=1}^N$. Observe from \eqref{Equ:Dictlearn:DictUpdate_grad_nu_relation} that $\nu_t^o$ is the gradient of the residual function $f(u)$ at the optimal $u_t^o$. The update term for dictionary element $k$ in \eqref{Equ:DictLearnDist:DictionaryUpdate_final} is effectively the correlation between $\nu_t^o$, the gradient of the residual function $f(u_t^o)$, and the coefficient $y_{k,t}^o$ (the activation) at agent $k$. In the special case of $f(u) = \frac{1}{2} \|u\|_2^2$, expression \eqref{Equ:Dictlearn:DictUpdate_grad_nu_relation} implies that
\begin{align}
\nu_t^o = u_t^o
=
x_t - \sum_{k=1}^N W_{k,t-1} y_{k,t}^o
\end{align}
In this case, $\nu_t^o$ has the interpretation of being equal to the optimal prediction residual error, $u_t^o$, using the entire existing dictionary set $\{W_{k,t-1}\}_{k=1}^N$. Then, the update term for dictionary element $k$ in \eqref{Equ:DictLearnDist:DictionaryUpdate_final} becomes the correlation between the optimal prediction error $\nu_t^o = u_t^o$ and the coefficient $y_{k,t}^o$ at agent $k$. Furthermore, recursion \eqref{Equ:DictLearnDist:DictionaryUpdate_final} reveals that, for each input data sample $x_t$, after the dual variable $\nu_t^o$ is obtained at each agent, there is no need to exchange any additional information among agents in order to update their own sub-dictionaries; the dual variable $\nu_t^o$ already provides sufficient information to carry out the update. The fully distributed algorithm for dictionary learning is listed in Algorithm \ref{alg:DiffusionSparseCoding} and is also illustrated in Fig. \ref{Fig:Fig_DictLearn}.
\section{Important Special Cases and Experiments}
\label{Sec:Experiments}
In this section, we apply the dictionary learning algorithm to two problems involving novel document/topic detection and bi-clustering. A third application to image denoising is considered in \cite{chen2014icasspdictionary,towfic2014dictionary}.\footnote{The software code for the experiments in this manuscript is available online at \url{http://www.ee.ucla.edu/asl}} In our experiments below, we will use the diffusion strategy \eqref{Equ:DictLearnDist:ATC_adapt}--\eqref{Equ:DictLearnDist:ATC_combine} or \eqref{Equ:DictLearnDist:ATC_adapt_projection}--\eqref{Equ:DictLearnDist:ATC_combine_projection} to solve the dual inference problem \eqref{Equ:DictLearn:Inference_Dual_Objective}--\eqref{Equ:DictLearn:Inference_Dual_Constraint}.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{Fig_DictLearn_v4}
\caption{The distributed inference step and the dictionary update step over distributed models.
In the inference step, after each data sample $x_t$ arrives a subset of the agents in the network,
all the agents find the corresponding optimal dual variable $\nu_t^o$ by exchanging the estimates of
$\nu_t^o$ with neighbors. In the dictionary update step, agents update their sub-dictionaries locally
on their own using a step of proximal stochastic gradient descent as
\eqref{Equ:DictLearnDist:DictionaryUpdate_final}.}
\label{Fig:Fig_DictLearn}
\end{figure}
\subsection{Tuning of the parameters}
\label{Sec:Discussion}
In the following experiments, it is necessary to select properly the step-size $\mu$ for the diffusion algorithm \eqref{Equ:DictLearnDist:ATC_adapt}--\eqref{Equ:DictLearnDist:ATC_combine} to ensure that the estimate for $\nu_t^o$ converges sufficiently close to it after a reasonable number of iterations. { Table \ref{Tab:StepSizeCond} lists the step-size conditions that guarantee the convergence of the diffusion algorithm for different applications, which are derived from the general condition \eqref{Equ:DictLearnDist:StepSizeCondition}. Note that as long as the agents know the regularization parameter $\delta$ and the maximum number, $N_{\max}$, of dictionary atoms that are allowed at each agent, the agents can select the step-size in a distributed manner.}
{ For the convenience of the experiments in this section and only to get an idea about how many iterations are typically needed for the inference step, we choose a data sample $x$ from the training dataset, and use a non-distributed optimization package such as CVX \cite{cvx} to compute the optimal solution $y^o \triangleq \mathrm{col}\{y_{1}^o,\ldots, y_N^o\}$ and its respective dual variable $\nu^o$ as the ground truth for the inference problem \eqref{Equ:DictLearnDist:NewInference_Objective}--\eqref{Equ:DictLearnDist:NewInference_Constraint}. We plot the signal-to-noise measures $\|y^o\|^2/\|y_i-y^o\|^2$ and $\|\nu^o\|^2/\|\nu_{k,i}-\nu^o\|^2$ against the iteration number $i$ in Fig.~\ref{Fig:Discussion:LearningCurve}. The value $\nu_{k,i}$ is obtained from the distributed algorithm (see \eqref{Equ:DictLearnDist:ATC_combine} or \eqref{Equ:DictLearnDist:ATC_combine_projection}) at each iteration $i$ and $y_i \triangleq \mathrm{col}\{y_{1,i},\ldots,y_{N,i}\}$ is calculated at each iteration according to:
\begin{align}
y_{k,i} = \arg\max_{y_k}
\big[
(W_k^T \nu_{k,i})^T y_k - h_{y_k}(y_k)
\big]
\label{Equ:DictLearnDist:yk_optimal_primal_tuning_section}
\end{align}
Observe from Fig. \ref{Fig:Discussion:LearningCurve} that in order to achieve satisfactory SNR values (e.g., $40$-$50$dB) for both $y$ and $\nu$, the required number of diffusion iterations is about $500$. Also note that the primal variable $y$ generally reaches a high SNR value before the dual variable $\nu$, but both are required to be found with reasonable accuracy for the dictionary update step (see \eqref{Equ:DictLearnDist:DictionaryUpdate_final}). Furthermore, although the number of iterations by diffusion seems to be large for solving the inference problem, the actual wall-clock time it takes is short because of the relatively low complexity per step.
We further note that there was no restriction imposed on the size of the network. In our experiments, network consists of $N=196$ nodes in the image denoising example \cite{chen2014icasspdictionary} and employs from $N=10$ to $N=80$ nodes in the novel document detection example. In the bi-clustering example, the network size, $N$, is three because of the application setup and the nature of the data from \cite{lee2010biclustering}, where the rank of the data matrix is low so that three dictionary atoms are sufficient to represent the data.
}
\begin{table}[!t]
\caption{Conditions of the step-size parameter $\mu$ for inference step.}
\label{Tab:StepSizeCond}
\centering
\renewcommand{\arraystretch}{2.0}
\begin{threeparttable}
\begin{tabular}{c|c||c}
\hline \hline
\rowcolor[gray]{0.9} \rule[-1ex]{0pt}{4ex} $f(u)$ & $h_{y_k}(y_k)$ & Step-size condition\\
\hline
\rule[-1ex]{0pt}{4ex} $\frac{1}{2}\|u\|_2^2$ & $\gamma \|y_k\|_{1} + \frac{\delta}{2} \|y_k\|_2^2$ & $0< \mu < \frac{1}{1 + {N_{\max}}/{\delta}}$ \\
\hline
\rule[0ex]{0pt}{4ex} $\frac{1}{2}\|u\|_2^2$ & $\gamma \|y_k\|_{1,+} + \frac{\delta}{2} \|y_k\|_2^2$ & $0< \mu < \frac{1}{1 + {N_{\max}}/{\delta}}$ \\
\hline
Huber loss & $\gamma \|y_k\|_{1,+} + \frac{\delta}{2} \|y_k\|_2^2$ & $0< \mu < \frac{1}{\eta + {N_{\max}}/{\delta}}$ \\
\hline \hline
\end{tabular}
\begin{tablenotes}
\vspace{0.5em}
\item[a]
$N_{\max} \triangleq \max_{1 \le k \le N} N_k$ is the maximum number of the dictionary atoms
that are allowed at each agent.
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{LearningCurve_mu_0_5_v2}
\caption{Learning curve for the Huber document detection example described by Alg.~\ref{alg:DocumentDetection:Huber} with $\mu = 0.5$.}
\label{Fig:Discussion:LearningCurve}
\end{figure}
\subsection{Novel Document Detection via Dictionary Learning}
\label{Sec:Experiment:NovelDocumentDetection}
In the novel document detection application \cite{Kasiviswanathan12,Aiello2013sensing,Takahashi2014Jan}, a stream of documents arrives in blocks at the network, and the task is to detect which of the documents in the incoming batch are associated with topics that have not been observed previously, and to incorporate the new block of data into the knowledge database to detect new topics/documents in future incoming batches. We refer to each such step as a ``time-step'' and we use $x_t^s$ to denote the $t$th data sample in the $s$th time-step, where $1 \le t \le T_s$ with $T_s$ being the number of samples in the $s$th time-step ($T_s=1000$ for all $s$ in this example), and $1\leq s \leq 8$ since our dataset only contains enough data for eight time-steps. We simulate our dictionary learning algorithm using the Huber cost function as the residual metric. We compare our algorithm performance to that proposed in \cite{Kasiviswanathan12} under the same setup proposed there. The data is from the TDT2 dataset, which contains news documents associated with their dominant topics collected over the first $27$ weeks of $1998$. The data is compiled into a term frequency-inverse document frequency (TF-IDF) matrix $X \in \mathbb{R}^{M \times T}$, where $M = 19527$ and $T = 9394$. The documents have been processed so that only the most frequent $30$ topics (and documents associated with them) are preserved. In this experiment, we allow all agents in the network to observe the incoming data. The key observation is that if a document belongs to a topic that has been observed previously, then it is expected that the objective value of the optimization problem \eqref{Equ:DictLearnDist:NewInference_Objective}--\eqref{Equ:DictLearnDist:NewInference_Constraint} will be ``small'' since the document should be well modeled by the available dictionary. On the other hand, when the objective value is ``large,'' then this is an indication that the document is not well modeled by the available dictionary.
In this application, we let $f(u) = \sum_{m=1}^M L(u_m)$, where $L(u_m)$ is chosen to be the scalar Huber function defined in Table \ref{Tab:Task}. We choose Huber loss for the following reasons. {The work \cite{Kasiviswanathan12} points out that some of the coefficients of the representation error $u = x_t-W y$ in text documents contain large, impulsive values. For this reason, the work \cite{Kasiviswanathan12} adopts the $\ell_1$ loss $f(u) = \|u\|_1$ because this loss grows only linearly for large $u$ and is less sensitive to large outliers. However, $\ell_1$ loss is not differentiable and has a conjugate function of zero with domain $\mathcal{V}_f = \{ \nu: \|\nu\|_{\infty} \le 1\}$. In comparison, the Huber loss, while preserving the linear growth for large $u$, is smooth and has Lipschitz gradients, which gives a quadratic conjugate function (see Tab. \ref{Tab:ConjProx} and Sec. \ref{Sec:DictLearn:ChoiceResReg}) that naturally regularizes the dual cost \eqref{Equ:DictLearn:Inference_Dual_Objective} to make it strongly convex. In this way, we end up with a better conditioned optimization problem, which allows first-order methods (e.g., diffusion) to achieve relatively fast convergence and satisfactory performance on the dual inference problem \eqref{Equ:DictLearn:Inference_Dual_Objective}--\eqref{Equ:DictLearn:Inference_Dual_Constraint}.
} The setup is the same as in \cite{Kasiviswanathan12},\footnote{We would like to thank S. P. Kasiviswanathan for kindly sharing his MATLAB code through e-mail communication in order to reproduce the simulation in \cite{Kasiviswanathan12}, including the ordered data.} except that we start with only ten dictionary atoms, and add ten additional atoms after each time-step. We simulate the last line of the non-negative matrix factorization setup in Table~\ref{Tab:Task}. We compare our algorithm to the one proposed in \cite{Kasiviswanathan12}, which simulates the setup where $f(u) = \|u\|_1$, $h_y(y)=\|y\|_1$, and $\mathcal{W}_k = \{w: \|w\|_1 \leq 1\}$. { Therefore, the choice of the penalty function $f(u)$ is also slightly different, as we use Huber loss while \cite{Kasiviswanathan12} uses $\ell_1$ loss.}
For the simulation of the diffusion algorithm, the data are normalized so that $\|x_t^s\|_2 = 1$. In contrast, when testing on the centralized ADMM-based algorithm from \cite{Kasiviswanathan12}, the data are normalized so that $\|x_t^s\|_1 = 1$ in keeping with the proposed simulation setup there. The constraint set for $W$ for the diffusion-based algorithm is $\left\{W: \| [ W ]_{:,q} \|_2 \le 1, \; W \succeq 0\right\}$, while the constraint set for the ADMM-based algorithm from \cite{Kasiviswanathan12} is $\left\{W: \| [ W ]_{:,q} \|_1 \le 1, \; W \succeq 0\right\}$. We choose $\gamma = 0.05$ and $\delta = 0.1$. For the initialization of the dictionary for the ADMM algorithm from \cite{Kasiviswanathan12}, we let the algorithm iterate between the sparse coding step and the dictionary learning step $35$ times. The diffusion algorithm runs through the data once. We choose $\eta = 0.2$ for the connection point between the quadratic part and the linear part of the Huber loss function. Both the fully connected and distributed algorithms utilize a learning step-size of $\mu_w(s) = 1/s$, where $s$ is the current time-step for learning of the dictionary. For the inference, the fully connected algorithm utilizes $\mu^\textrm{FC} = 0.5$, while the distributed algorithm uses $\mu = 0.05$. The fully connected algorithm performs $100$ iterations for the inference, while the distributed algorithm utilizes $1000$ iterations for the inference. Samples $1$-$1000$ are used for the initialization of the dictionary. Novel documents are only introduced at the first (samples $1001$-$2000$), second ($2001$-$3000$), fifth ($5001$-$6000$), sixth ($6001$-$7000$), and eighth ($8001$-$9000$) time-steps. For this reason, we only execute the novel document detection part of the algorithm at those time-steps, and present the ROC curves for those time-steps. We run our algorithm using the fully connected case, where $A = \frac{1}{N} \mathds{1} \mathds{1}^T$ and the distributed case where the probability that two nodes are connected is $0.5$, and the combination matrix is the Metropolis rule.
To obtain the distributed algorithm, we note from \eqref{Equ:DictLearn:Split_Nodes} that
\begin{align}
J_k(\nu; x_t^s) \triangleq
\displaystyle
\frac{1}{N} (f^{\star}(\nu) - \nu^T x_t^s)
\!+\!
h_{y_k}^{\star}(w_k^T \nu)
\label{Equ:DictLearn:Experiments:NovelDocument:HuberResidual:J_k}
\end{align}
where we are using $w_k$ instead of $W_k$ because each agent $k$ is in charge of one atom of the dictionary (i.e., the $k$-th column of $W$).
Since we now use $f(u) = \sum_{m=1}^M L(u_m)$ and $h_{y_k}(y_k) = \gamma \|y\|_{1,+} + \frac{\delta}{2} \|y\|_2^2$ (according to the last row of Table~\ref{Tab:Task}), we obtain that $f^\star(\nu) = \frac{\eta}{2} \|\nu\|_2^2$, $\mathcal{V}_f = \{\nu: \|\nu\|_\infty \leq 1\}$, and $h_{y_k}^{\star}(w_k^T \nu) = \mathcal{S}_{\frac{\gamma}{\delta}}^+\left(\frac{w_k^T \nu}{\delta}\right)$ according to Table~\ref{Tab:ConjProx}. A straightforward calculation then shows that
\begin{align}
\nabla_\nu f^\star(\nu) &= \eta\cdot \nu,
\quad
\nabla_\nu h_{y_k}^{\star}(w_k^T \nu) = \frac{1}{\delta} \mathcal{T}_\gamma^+ (w_k^T \nu) w_k \label{Equ:DictLearn:Experiments:NovelDocument:HuberResidual:nabla_h_star}
\end{align}
Substituting \eqref{Equ:DictLearn:Experiments:NovelDocument:HuberResidual:nabla_h_star} into the gradient of \eqref{Equ:DictLearn:Experiments:NovelDocument:HuberResidual:J_k}, we obtain:
\begin{align}
\nabla_\nu J_k(\nu;x_t) &= \frac{1}{N} (\eta\cdot \nu - x_t)
\!+\!
\frac{1}{\delta} \mathcal{T}_\gamma^+(w_k^T \nu) w_k \label{Equ:DictLearn:Experiments:NovelDocument:HuberResidual:nabla_J_k}
\end{align}
where we let $\mathcal{N}_I = \mathcal{N}$ and all agents in the network have access to $x_t^s$. By substituting \eqref{Equ:DictLearn:Experiments:NovelDocument:HuberResidual:nabla_J_k} into the inference part of Alg.~\ref{alg:DiffusionSparseCoding}, we immediately obtain the inference part of Alg.~\ref{alg:DocumentDetection:Huber}. For the learning portion of the algorithm, we need to compute $y_{k,t}^o$ at node $k$ once $\nu_{t}^o$ has been estimated. With our choices of $f(u)$ and $h(y_k)$, we observe from Table \ref{Tab:ConjProx} that $y_{k,t}^o$ may be obtained as $y^o_{k,t} = \mathcal{T}_{\frac{\gamma}{\delta}}^+\left(\frac{w_k^T \nu_t^o}{\delta}\right) = \frac{1}{\delta} \mathcal{T}_{\gamma}^+\left(w_k^T \nu_t^o\right)$ (as listed in Alg.~\ref{alg:DocumentDetection:Huber}). Now, using the fact that $h_{w_k}(w_k) = 0$ (see Table~\ref{Tab:Task}), we have that the update rule for $w_k$ from Alg.~\ref{alg:DiffusionSparseCoding} becomes
\begin{align}
w_{k,t} = \Pi_{\mathcal{W}_k}\left\{w_{k,t-1} \!+\! \mu_w \nu_t^o y_{k,t}^{o}\right\}
\label{Equ:DictLearn:Experiments:NovelDocument:DictUpdate}
\end{align}
where $\mathcal{W}_k = \{w: \|w\|_2 \leq 1, w \succeq 0\}$ (see Table~\ref{Tab:Task}).
{ When recursion \eqref{Equ:DictLearn:Experiments:NovelDocument:DictUpdate} finishes going through the data samples in the $s$-th time-step, the most up-to-date dictionary is denoted by
$W^s = [w_1^s \cdots w_N^s]$.}
In this example, we do not need to recover $z_t^o$ in \eqref{Equ:DictLearnDist:z_optimal_primal}, but we only need to recover the cost value for representing a test data sample $\xi_t$ using dictionary $W^s$ learned up to the $s$-th time-step:
\begin{align}
\min_{\{y_k\}}
\left[
f\Big(
\xi_t - \sum_{k=1}^N w_{k}^s y_k
\Big)
+
\sum_{k=1}^N
h_{y_k}(y_k)
\right]
\label{Equ:DictLearn:Experiment:NovelDocument:PrimalCostValue}
\end{align}
where we use $\xi_t$ to differentiate it from the training data sample $x_t^s$.
Interestingly, since strong duality holds for this example, based on the argument from \eqref{Equ:DictLearnDist:NewInference_Objective} to \eqref{Equ:DictLearn:Inference_Dual_Objective}, the above minimum primal cost \eqref{Equ:DictLearn:Experiment:NovelDocument:PrimalCostValue} is equal to the maximum value of its associated dual cost:
\begin{align}
\max_{\nu} g(\nu, \xi_t) = g(\nu_t^o,\xi_t) = -\sum_{k=1}^N J_k(\nu_t^o,\xi_t)
\label{Equ:DictLearn:Experiment:NovelDocument:DualCostValue}
\end{align}
where the first equality follows from the fact that $\nu_t^o$ is the optimizer of the dual problem.
Therefore, we can obtain the minimum primal cost \eqref{Equ:DictLearn:Experiment:NovelDocument:PrimalCostValue} by computing the maximum dual cost \eqref{Equ:DictLearn:Experiment:NovelDocument:DualCostValue}, which can be done in many ways with one of them being the diffusion strategy. In order to obtain a scaled multiple of \eqref{Equ:DictLearn:Experiment:NovelDocument:DualCostValue}, we setup the following scalar optimization problem:
\begin{align}
\min_g\ \sum_{k=1}^N V_k(g)
\label{Equ:DictLearn:Experiments:NovelDocument:SquareResidual:consensus}
\end{align}
where
\begin{align}
V_k(g) \triangleq \frac{1}{2} \left(J_k(\nu_t^o,\xi_t)+g\right)^2
\end{align}
from which we can obtain the following scalar diffusion algorithm \cite{chen2013JSTSPpareto}:
\begin{align}
\begin{cases}
\phi_{k}(i) = g_{k}(i-1) - \mu_g (J_k(\nu_t^o,\xi_t)+g_{k}(i-1))\\
\displaystyle g_{k}(i) = \sum_{\ell \in \mathcal{N}_k} a_{\ell k} \phi_{\ell}(i)
\end{cases}
\label{Equ:DictLearn:Experiments:NovelDocument:SquareResidual:ScalarDiffusion}
\end{align}
After sufficient iterations, recursion \eqref{Equ:DictLearn:Experiments:NovelDocument:SquareResidual:ScalarDiffusion} approximates the minimizer of \eqref{Equ:DictLearn:Experiments:NovelDocument:SquareResidual:consensus}, which is $g^o_t = -\frac{1}{N} \sum_{k=1}^N J_k(\nu_t^o,\xi_t)$. { Comparing $g^o_t$ to \eqref{Equ:DictLearn:Experiment:NovelDocument:DualCostValue}, we note that there is an additional positive scaling factor, $1/N$, in $g^o_t $. However, it does not affect the result since it can be absorbed into the threshold parameter:
\begin{align}
-\sum_{k=1}^N J_k(\nu_t^o,\xi_t)
\underset{H_0}{\overset{H_1}{\gtrless}} \chi'
\quad\Leftrightarrow\quad
&g^o_t \underset{H_0}{\overset{H_1}{\gtrless}} \chi \triangleq \frac{\chi'}{N}
\end{align}
where $H_1$ and $H_0$ denote the hypotheses of ``the document is novel'' and ``the document is not novel'', respectively. In other words, using a threshold $\chi'$ for the original cost \eqref{Equ:DictLearn:Experiment:NovelDocument:DualCostValue}, is equivalent to using the threshold $\chi = \chi'/N$ for $g_t^o$.
}
The final algorithm is listed in Alg.~\ref{alg:DocumentDetection:Huber}. Each node in the network is responsible for a single dictionary atom. The sparse coding stages of the centralized ADMM-based algorithm from \cite{Kasiviswanathan12} utilize $35$ iterations, and the number of iterations of the dictionary update steps are capped at $10$ for all iterations other than the initialization step, which are the default setup in the code of \cite{Kasiviswanathan12}. We observe that the performance of the centralized ADMM-based algorithm reproduced in this manuscript is competitive with that in \cite{Kasiviswanathan12}, even though the initial dictionary size is chosen to be ten, as opposed to $200$ atoms (as was done in the experiment in \cite{Kasiviswanathan12}). Furthermore, for our algorithm, since we are simulating a network of $N$-agents on a single machine, we expect the computation time to be $N$ times as much as that in \cite{Kasiviswanathan12} in order to have a fair comparison. This is because the gradient descent steps and the combination steps in \eqref{Equ:DictLearnDist:ATC_adapt}--\eqref{Equ:DictLearnDist:ATC_combine} should be finished concurrently in an actual $N$-agent network, while our single-machine simulation can only perform them sequentially. For this reason, we choose the setup for our algorithm (such as the number of inference iterations) to be about $N$ times of that in \cite{Kasiviswanathan12} to ensure a fair comparison.\footnote{When applying the centralized gradient descent to the dual inference problem \eqref{Equ:DictLearnDist:DualProblem_Objective_newForm}--\eqref{Equ:DictLearnDist:DualProblem_Constraint_newForm} with $1000$ iterations at a single machine, we found that the entire learning time over one time-step ($1000$ samples) is approximately the same as that of the ADMM-based method from \cite{Kasiviswanathan12} using the same MATLAB implementation for the time benchmark.}
\noindent\begin{algorithm}[t]
\caption{{\small{Model-distributed diffusion strategy for distributed novel document detection (Huber Loss Residual).}}}
\label{alg:DocumentDetection:Huber}
{\small{
\begin{algorithmic}
{
\STATE {\bf Initialization:} The sub-dictionaries $\{W_k\}$ are randomly initialized and then projected onto \eqref{Equ:ProbForm:W_subUnitNormNonnegConstraint} using \eqref{Equ:DictLearn:DictUpdate_Projection_nonnegativenormball}.}
\FOR{each time step $s = 1, 2, \ldots, 8$}
\STATE \underline{\textbf{\emph{Dictionary Learning:}}}
\FOR{each training sample $x_t^s$ from time-step $s$, ($t=1,\ldots, T_s$)}
\STATE Each node $k$ repeats until convergence:
\!\!\!\!{\footnotesize{\begin{equation*}
\begin{cases}
\!\psi_{k,i} \!\!=\! \nu_{k,i-1} \!-\!\! \frac{\mu}{N} (\!\eta \nu_{k,i-1} \!-\! x_t^s\!) \!-\! \frac{\mu}{\delta} \mathcal{T}_\gamma^+(\!w_{k,t-1}^T \nu_{k,i-1}\!)w_{k,t-1}\\
\!\nu_{k,i} \!=\! \Pi_{\nu \in [-1,1]}\left\{\sum_{\ell \in \mathcal{N}_k} a_{\ell k} \psi_{\ell,i}\right\}
\end{cases}
\end{equation*}}}%
{with initialization $\{\nu_{k,0} = 0, \; k=1,\ldots,N\}$.}
where the above projection is carried out according to \eqref{Equ:DictLearnDist:Projection_Inf_Norm}.
\STATE Set $\nu_t^o = \nu_{k,i}$. Compute $y_{k,t}^o = \frac{1}{\delta}\mathcal{T}_\gamma^+(w_{k,t-1}^T \nu_t^o)$.
\STATE Update the dictionary using:
\begin{equation*}
w_{k,t} = \Pi_{\|w\|_2 \leq 1} \left\{\Pi_{w \succeq 0}\left\{w_{k,t-1} \!+\! \mu_w(s) \nu_t^o y_{k,t}^{o }\right\}\right\}
\end{equation*}
\ENDFOR
\STATE Let $w_{k}^s$ denote the most up-to-date sub-dictionary at agent $k$.
\STATE \underline{\textbf{\emph{Novel Document Detection:}}}
\FOR{each test data sample $\xi_t$, each node $k$}
\STATE Repeat until convergence:
\!\!\!\!{\footnotesize{\begin{equation*}
\begin{cases}
\!\psi_{k,i} \!\!=\! \nu_{k,i-1} \!\!-\!\! \frac{\mu}{N} (\!\eta \nu_{k,i-1} \!-\! \xi_t\!) \!-\! \frac{\mu}{\delta} \mathcal{T}_\gamma^+\big(\!(w_{k}^s)^T \nu_{k,i-1}\!\big)w_{k}^s\\
\!\nu_{k,i} = \Pi_{\nu \in [-1,1]}\left\{\sum_{\ell \in \mathcal{N}_k} a_{\ell k} \psi_{\ell,i}\right\}
\end{cases}
\end{equation*}}}
\STATE Set $\nu_t^o = \nu_{k,i}$.
\STATE Perform diffusion strategy to optimize \eqref{Equ:DictLearn:Experiments:NovelDocument:SquareResidual:consensus} until convergence:
\begin{align*}
\begin{cases}
\phi_{k}(i) = g_{k}(i-1) - \mu_g (J_k(\nu_t^o,\xi_t)+g_{k}(i-1))\\
g_{k}(i) = \sum_{\ell \in \mathcal{N}_k} a_{\ell k} \phi_{\ell}(i)
\end{cases}
\end{align*}
where $J_k(\nu,\cdot)$ is defined in \eqref{Equ:DictLearn:Experiments:NovelDocument:HuberResidual:J_k}.
\STATE Set $g^o_t = g_{k,i}$.
\IF{$g^o_t > \chi$}
\STATE declare document as novel.
\ELSE
\STATE declare document as not novel.
\ENDIF
\ENDFOR
\STATE Add nodes to network (expand the dictionary)
\ENDFOR
\end{algorithmic}}}
\end{algorithm}
\begin{figure}[th!]
\centering
\includegraphics[width=0.5\textwidth]{Fig_DocumentDetection_HuberLabel_v5}
\caption{Application of dictionary learning to novel document/topic detection. At each time step, the algorithms receive $1000$ documents. The task is to determine which documents are associated with topics that have already been observed, and which documents are associated with topics that have not yet been observed. These curves represent the ROC curve associated with each time step against a changing test set. The $x$-axis represents probability of false alarm while the $y$-axis represents probability of detection. The area under each cuve is listed in Table~\ref{Tab:DocumentDetection:L1:AUC}.}
\label{Fig:DocumentDetection:L1}
\end{figure}
\begin{table}[t!]
\caption{Area under ROC curve for the three tested algorithms. Novel documents not presented in time-steps $3$, $5$, $7$.}
\label{Tab:DocumentDetection:L1:AUC}
\begin{center}
\begin{tabular}{c||c|c|c}
\hline \hline
\rowcolor[gray]{0.9} Time Step & ADMM \cite{Kasiviswanathan12} & Diffusion (Fully Connected) & Diffusion \\
\hline
$1$ & $0.69$ & $\textbf{0.79}$ & $\textbf{0.79}$ \\
\hline
$2$ & $0.65$ & $\textbf{0.94}$ & $0.93$ \\
\hline
$5$ & $0.70$ & $0.94$ & $\textbf{0.95}$ \\
\hline
$6$ & $0.77$ & $\textbf{0.96}$ & $0.95$ \\
\hline
$8$ & $0.76$ & $0.93$ & $\textbf{0.94}$ \\
\hline \hline
\end{tabular}
\end{center}\vspace{-1\baselineskip}
\end{table}
The performance of the algorithms is illustrated in Fig.~\ref{Fig:DocumentDetection:L1}. We observe that the Huber loss function improves performance relative to the $\ell_1$ function. The area under each ROC curve is listed in Table~\ref{Tab:DocumentDetection:L1:AUC}. Since the different algorithms were initialized with different dictionaries, it may be possible for the sparsely-connected diffusion strategy to slightly outperform the fully-connected diffusion strategy. We observe this effect in Table~\ref{Tab:DocumentDetection:L1:AUC}, where the sparsely-connected network outperforms the fully-connected network by $0.01$ (area under ROC curve).
\subsection{Biclustering via Sparse Singular-Value-Decomposition}
\label{Sec:Experiment:BiClustering}
Consider next the cancer data matrix $X \in \mathbb{R}^{M\times T}$ from \cite{lee2010biclustering}, where $M=56$ and $T=12,625$. Each row of $X$ contains the genetic information for each of $56$ patients. Each patient belongs to one of four cancer categories: Normal, Carcinoid, Colon, and SmallCell. The algorithm is unaware of the true category (label) of any patient, but wants to cluster patients into groups with different cancer types using the genetic information. The problem was formulated in \cite{lee2013distributed} as a bi-clustering task (see also Tables \ref{Tab:Task}--\ref{Tab:ConjProx}) that factorizes $X$ as
\begin{align}
X \approx \sum_{k=1}^N w_{k} y_{k}^T
\label{eq:SVD}
\end{align}
with both $w_k \in \mathbb{R}^{M \times 1}$ and $y_k \in \mathbb{R}^{T \times 1}$ being sparse.
\noindent\begin{algorithm}
\caption{\small Simplified algorithm from \cite{lee2010biclustering} for biclustering.}
\label{alg:lee}
\begin{algorithmic}
{\small
\FOR{each $k$}
\STATE Apply standard SVD to $X = w_\textrm{old} s_\textrm{old} y_\textrm{old}^T$. Repeat until convergence:
\begin{enumerate}
\STATE Set $\tilde{y} = \mathcal{T}_{\lambda}(X^T w_\textrm{old})$,
and $y_\textrm{new} = \tilde{y}/\|\tilde{y}\|_2$.
\STATE Set $\tilde{w} = \mathcal{T}_{\beta}(X y_\textrm{new})$,
and $w_\textrm{new} = \tilde{w}/\|\tilde{y}\|_2$.
\STATE Set $w_\textrm{old} = w_\textrm{new}$.
\end{enumerate}
\STATE Set $w_k = w_\textrm{new}$, $s_k=w_\textrm{new}^T X y_\textrm{new}$, and $y_k = s_k y_\textrm{new}$.
\STATE Set $X = X - w_k y_k^T$.
\ENDFOR
}
\end{algorithmic}
\end{algorithm}
\noindent\begin{algorithm}
\caption{\small Model-distributed diffusion strategy for { online} biclustering.}
\label{alg:biclustering}
\begin{algorithmic}
{\small
\STATE {\bf Initialization:} The sub-dictionaries $\{W_k\}$ are randomly initialized and then projected onto \eqref{Equ:ProbForm:W_subUnitNormConstraint} using \eqref{Equ:DictLearn:DictUpdate_Projection_normball}.
\FOR{each input data sample $x_t$, each node $k$}
\STATE Repeat until convergence:
\!\!\!\!\begin{equation*}
\begin{cases}
\!\psi_{k,i} \!\!=\! \nu_{k,i-1} \!-\! \mu_\nu \frac{1}{N} (\nu_{k,i-1} - x_t) -\\
\!\!\quad\quad\quad\quad\quad\quad \frac{\mu_\nu}{\delta} \mathcal{T}_\gamma(w_{k,t-1}^T \nu_{k,i-1})w_{k,t-1}\\
\!\nu_{k,i} = \sum_{\ell \in \mathcal{N}_k} a_{\ell k} \psi_{\ell,i}
\end{cases}
\end{equation*}
{with initialization $\{\nu_{k,0}=0, \;k=1,\ldots,N\}$.}
\STATE Set $\nu_k^o = \nu_{k,i}$. Compute $y_k^o = \frac{1}{\delta}\mathcal{T}_\gamma(w_{k,t-1}^T \nu^o)$.
\STATE Update the dictionary using:
\begin{equation*}
w_{k,t} = \Pi_{\|w\| \leq 1}\left\{\mathcal{T}_\beta\left(w_{k,t-1} \!+\! \mu_w \nu_k^o y_k^{o T}\right)\right\}
\end{equation*}
\ENDFOR
}
\end{algorithmic}
\end{algorithm}
\begin{figure}[th!]
\centerline{
\subfigure[Data clusters obtained by Alg.~\ref{alg:lee}.]{
\includegraphics[width=0.45\textwidth]{Fig_Biclustering_Lee_v2}
}
}\vspace{0.5em}
\centerline{
\subfigure[Data clusters obtained by Alg.~\ref{alg:biclustering}.]{
\includegraphics[width=0.45\textwidth]{Fig_Biclustering_DiffFullyConnected_v3}
}
}
\caption{Application of microarray biclustering. Each marker represents one patient, and $[w_k]_m$ denotes the $m$-th entry of the dictionary atom $w_k$, where $m=1,\ldots, 56$ is the index of the patients and $k=1,2,3$ is the index of the dictionary atoms. The algorithm is unaware of the ground truth of the cancer categories of each patient. { After the bi-clustering is done, we add colors to different markers according to the ground truth (label) to visualize the success of the bi-clustering task.}}
\label{Fig:biclustering_clusters}
\end{figure}
In Alg.~\ref{alg:lee}, we list the algorithm from \cite{lee2010biclustering}, which alternates between two sparse coding steps to obtain $y$ and $w$, respectively. Observe that the algorithm is a batch algorithm, in that it utilizes the entire data set at each iteration. In addition, the algorithm works by computing the best sparse rank-$1$ approximation for the matrix $X \approx w_1 y_1^T$, then computes the best rank-$1$ approximation for $X-w_1 y_1^T \approx w_2 y_2^T$, and so on. In contrast, our proposed Alg. \ref{alg:DiffusionSparseCoding}, when specialized to the bi-clustering application (see Alg.~\ref{alg:biclustering}), runs through the data in an online manner and obtains the $\{w_k\}$ simultaneously.
We choose $N=3$ to be consistent with the setup in \cite{lee2010biclustering}, where each node is responsible for a single dictionary atom. We set $\gamma = 0.5$ and $\beta = 0.01$. Since the number of nodes is small, we simulate the fully connected case where the combination matrix $A = \frac{1}{N} \mathds{1}_N \mathds{1}_N^T$ (i.e., each node is effectively averaging the estimate of $\psi_{k,i}$). We run Alg.~\ref{alg:lee} until $\|w_\textrm{new}-w_\textrm{old}\|_\infty < 1\times 10^{-10}$. We run our algorithm's sparse coding for a total of $2000$ iterations. We choose $\mu_\nu = 0.01$, $\mu_w = 5\times 10^{-3}$, and $\delta = 0.01$. In Fig.~\ref{Fig:biclustering_clusters}, we plot, in the same manner as\cite{lee2010biclustering}, the clustering results of Algorithms \ref{alg:lee}--\ref{alg:biclustering}. This is an unsupervised learning task, meaning that, during the learning process, the algorithms are unaware of the ground truth of the cancer categories of each patient. Still, the algorithms are required to cluster patients into different groups according to their underlying genetic information, hoping that patients of similar genetic information will be clustered together. After the clustering is done, we add colors to different markers according to the ground truth (label) to visualize and evaluate the result of the clustering. The clustering will be more successful if (i) markers of the same color are clustered together, and (ii) markers of different colors are well separated. We observe that both algorithms, without the use of the cancer labels, can successfully cluster the data into 4 distinct clusters according to the genetic information, with each cluster corresponding to a different type of cancer. The advantage of the diffusion strategy is that it only requires each node to observe each data sample (each column of $X$) once (not batched) and obtain $\{w_1,w_2,w_3\}$ simultaneously. Note that in this example an additional data collection process is required to gather all the $w_1$, $w_2$, and $w_3$ to generate the final bi-clustering plots in Fig. \ref{Fig:biclustering_clusters}. This is because we need to use $([w_1]_m, [w_2]_m, [w_3]_m)$ to represent the genetic profile of each patient $m$. This step is usually less demanding than learning the $\{w_k\}$, especially in large-scale genetic data analysis. The agents may choose to report the obtained results periodically. Nevertheless, the computation-intensive learning process in bi-clustering is still distributed over the network, where the agents learn different $\{w_k\}$ in an online and simultaneous manner.
\section{Conclusion}
\label{Sec:Conclusion}
In this paper, we studied the online dictionary learning problem over distributed models, where each agent is in charge of a portion of the dictionary atoms and the agents collaborate to represent the data. Using the concepts of conjugate function and dual decomposition, we transform the original learning problem into a form that is amenable to distributed optimization, which is then solved by means of a diffusion strategy. The collaborative inference step generates dual variables that are used by the agents to update their dictionary atoms without the need to share their dictionaries or even the coefficient models for the training data. The proposed algorithm is tested over two typical tasks of dictionary learning, namely, novel document detection and bi-clustering. The results demonstrate that our proposed algorithm can solve the dictionary learning tasks effectively in a distributed and online manner.
In relation to the convergence behavior, we remark that the general learning problem \eqref{Equ:ProbForm:DictLearn_Objective}--\eqref{Equ:ProbForm:DictLearn_Constraint} is not jointly convex with respect to both $W$ and $y$. This fact explains why convergence guarantees towards a global minimum, when it exists, are generally not available in the literature. A common technique for solving such coupled optimization problems is to alternate between the minimization over one variable while keeping the other variable fixed. In this article, we followed a similar construction albeit one that operates in an online and distributed manner. For the inference problem \eqref{Equ:ProbForm:InferenceProblem}, we applied the diffusion strategy, which has already been shown in prior studies \cite{chen2013JSTSPpareto} to converge within $O(\mu^2)$ to the optimal inference solution. For the dictionary update step, we used a proximal projection step. Simulation results in this article and by other authors have indicated that such alternating optimization solutions tend to perform well in practice.
\appendices
\section{Derivation of Some Typical Conjugate Functions}
\label{Appendix:DerivationConjugateFun}
In this appendix, we derive the conjugate functions listed in Table \ref{Tab:ConjProx}. The conjugate functions for $\frac{1}{2}\| u \|_2^2$, and their corresponding domains can be found in \cite[pp.90-94]{boyd2004convex}. The conjugate function for the scalar Huber loss $L(u_m)$ can be found in \cite{zach2010practical} as $L^{\star}(\nu_m) = \frac{1}{2} \nu_m^2$ with $| \nu_m | \le 1$.
Therefore, by the ``sums of independent functions'' property\footnote{\label{FN:ConjProperty_SumOfIndependentFunc}If $f(x_1,\ldots,x_N)=f_1(x_1) + \cdots f_N(x_N)$, then the conjugate function for $f(x_1,\ldots,x_N)$ is given by $f^{\star}(\nu_1,\ldots,\nu_N) = f_1^{\star}(\nu_1) + \cdots + f_N^{\star}(\nu_N)$, where $f_1^{\star}(\nu_1),\ldots, f_N^{\star}(\nu_N)$ are the conjugate functions for $f_1(x_1), \ldots, f_N(x_N)$, respectively.} in \cite[p.95]{boyd2004convex}, the conjugate function of $\sum_{m=1}^M L(u_m)$ is:
\begin{align}
\sum_{m=1}^M L^{\star}(\nu_m) = \sum_{m=1}^M \frac{1}{2} \nu_m^2
= \frac{1}{2}\| \nu \|_2^2,
\end{align}
where the domain is given by
\begin{align}
| \nu_m | \le 1, \quad m =1, \ldots, M
\quad \Leftrightarrow \quad
\| \nu \|_{\infty} \le 1
\end{align}
Next, we derive the conjugate functions for the elastic net regularization term $ h_{y_k}(y_k) = \gamma \| y_k \|_{1} + \frac{\delta}{2} \| y_k \|_2^2$.
By the definition of conjugate functions in \eqref{Equ:DictLearnDist:r_conj_def}, we have
\begin{align}
h_{y_k}^{\star}(W_k^T\nu) &=
\sup_{y_k}
\left[
(W_k^T\nu)^T y_k - h_{y_k}(y_k)
\right]
\nonumber\\
&=
-\inf_{y_k}
\left[
h_{y_k}(y_k) - (W_k^T\nu)^T y_k
\right]
\nonumber\\
&=
-\inf_{y_k}
\left[
\gamma \| y_k \|_{1}
\!+\!
\frac{\delta}{2} \| y_k \|_2^2
\!-\!
(W_k^T\nu)^T y_k
\right]
\label{Equ:Appendix:hyk_conj_interm0}
\\
&=
- \delta
\!\cdot\!
\inf_{y_k}
\left[
\frac{\gamma}{\delta}
\| y_k \|_{1}
\!+\!
\frac{1}{2}
\Big\| y_k \!-\! \frac{1}{\delta} W_k^T\nu \Big\|_2^2
\right]
\nonumber\\
&\quad
+
\frac{1}{2\delta}
\| W_k^T \nu \|_2^2
\label{Equ:Appendix:hyk_conj_interm1}
\end{align}
where the last step completes the square. Note from \eqref{Equ:DictLearnDist:Prox_def} that the optimal $y_k$ that minimizes the term inside the bracket of \eqref{Equ:Appendix:hyk_conj_interm1} can be expressed as the proximal operator of $(\gamma/\delta)\|y_k\|_1$, which is known to be given by the entry-wise soft-thresholding operator\cite[p.188]{parikh2013proximal} \cite{donoho1995noising}:
\begin{align}
y_{k,t}^o
&=
\arg\min_{y_k}
\left[
\frac{\gamma}{\delta}
\| y_k \|_{1}
+
\frac{1}{2}
\Big\| y_k \!-\! \frac{1}{\delta} W_k^T\nu \Big\|_2^2
\right]
\nonumber\\
&=
\mathrm{prox}_{\frac{\gamma}{\delta}\|\cdot\|_1}
\left(
\frac{W_k^T\nu}{\delta}
\right)
=
\mc{T}_{\frac{\gamma}{\delta}}
\left(
\frac{W_k^T\nu}{\delta}
\right)
\label{Equ:Appendix:hyk_conj_interm2}
\end{align}
where $[\mc{T}_{\lambda}(x)]_n \triangleq (| [x]_n |-\lambda)_{+} \mathrm{sgn}([x]_n)$
and $(x)_{+} = \max( x, 0 )$. Substituting \eqref{Equ:Appendix:hyk_conj_interm2} into \eqref{Equ:Appendix:hyk_conj_interm0}, we obtain
\begin{align}
h_{y_k}^{\star}(W_k^T\nu)
&=
\mc{S}_{\frac{\gamma}{\delta}}
\left(
\frac{W_k^T\nu}{\delta}
\right)
\end{align}
where
\begin{align}
\mc{S}_{\frac{\gamma}{\delta}}
\left(
x
\right)
&\triangleq
-\gamma
\!\cdot\!
\big\|
\mc{T}_{\frac{\gamma}{\delta}}
\left(
x
\right)
\!
\big\|_1
\!-\!
\frac{\delta}{2}
\big\|
\mc{T}_{\frac{\gamma}{\delta}}
\left(
x
\right)
\!
\big\|_2^2
\!+\!
\delta
\cdot
x^T
\mc{T}_{\frac{\gamma}{\delta}}
\left(
x
\right)
\end{align}
Finally, we derive the conjugate function for the nonnegative elastic net regularization function $h_{y_k}(y_k) = \gamma \| y_k \|_{1, + } + \frac{\delta}{2} \|y_k\|_2^2$. Following the same line of argument from \eqref{Equ:Appendix:hyk_conj_interm0}--\eqref{Equ:Appendix:hyk_conj_interm1}, we get
\begin{subequations}
\begin{align}
h_{y_k}^{\star}(W_k^T\nu)
&=
-\inf_{y_k}
\left[
\gamma \| y_k \|_{1,+}
\!+\!
\frac{\delta}{2} \| y_k \|_2^2
\!-\!
(W_k^T\nu)^T
\!
y_k
\!
\right] \!\!
\label{Equ:Appendix:hyk_nonnegative_conj_interm0}
\\
&=
- \delta
\!\cdot\!
\inf_{y_k}
\left[
\frac{\gamma}{\delta}
\| y_k \|_{1,+}
\!+\!
\frac{1}{2}
\Big\| y_k \!-\! \frac{1}{\delta} W_k^T\nu \Big\|_2^2
\right]
\nonumber\\
&\quad
+
\frac{1}{2\delta}
\| W_k^T \nu \|_2^2
\label{Equ:Appendix:hyk_nonnegative_conj_interm1}
\end{align}
\end{subequations}
By \eqref{Equ:DictLearnDist:Prox_def}, the optimal $y_{k,t}^o$ that minimizes the term inside the bracket of \eqref{Equ:Appendix:hyk_nonnegative_conj_interm1} is given by
\begin{align}
y_{k,t}^o &= \arg\min_{y_k}
\left[
\frac{\gamma}{\delta}
\| y_k \|_{1,+}
+
\frac{1}{2}
\Big\| y_k \!-\! \frac{1}{\delta} W_k^T\nu \Big\|_2^2
\right]
\label{Equ:Appendix:ykto_argmin}
\end{align}
Applying an argument similar to the one used in \cite{beck2009fast},
we can express the optimal $y_{k,t}^o$ in \eqref{Equ:Appendix:ykto_argmin} as
\begin{align}
y_{k,t}^o = \mc{T}_{\frac{\gamma}{\delta}}^{+}
\left(
\frac{W_k^T \nu}{\delta}
\right)
\label{Equ:Appendix:ykto_Tplus_expr}
\end{align}
where $[\mc{T}_{\lambda}^{+}(x)]_n \triangleq ([x]_n - \lambda)_{+}$.
Substituting \eqref{Equ:Appendix:ykto_Tplus_expr} into \eqref{Equ:Appendix:hyk_nonnegative_conj_interm0}:
\begin{align}
h_{y_k}^{\star}(W_k^T\nu)
&=
\mc{S}_{\frac{\gamma}{\delta}}^{+}
\left(
\frac{W_k^T\nu}{\delta}
\right)
\end{align}
where
\begin{align}
\mc{S}_{\frac{\gamma}{\delta}}^{+}
\left(
x
\right)
&\triangleq
-\gamma
\!\cdot\!
\big\|
\mc{T}_{\frac{\gamma}{\delta}}^{+}
\left(
x
\right)
\!
\big\|_{1,+}
\!-\!
\frac{\delta}{2}
\big\|
\mc{T}_{\frac{\gamma}{\delta}}^{+}
\left(
x
\right)
\!
\big\|_2^2
\!+\!
\delta
\cdot
x^T
\mc{T}_{\frac{\gamma}{\delta}}^{+}
\left(
x
\right)
\nonumber\\
&=
-\gamma
\!\cdot\!
\big\|
\mc{T}_{\frac{\gamma}{\delta}}^{+}
\left(
x
\right)
\!
\big\|_{1}
\!-\!
\frac{\delta}{2}
\big\|
\mc{T}_{\frac{\gamma}{\delta}}^{+}
\left(
x
\right)
\!
\big\|_2^2
\!+\!
\delta\!
\cdot\!
x^T
\mc{T}_{\frac{\gamma}{\delta}}^{+}
(
x
)
\end{align}
where the last step uses the fact that the output of $\mc{T}_{\gamma}^{+}(\cdot)$ is always nonnegative so that $\|\mc{T}_{\frac{\gamma}{\delta}}^{+} \left(x\right)\|_{1,+} = \|\mc{T}_{\frac{\gamma}{\delta}}^{+} \left(x\right)\|_{1}$.
\bibliographystyle{IEEEbib}
|
1,116,691,499,367 | arxiv | \section{Introduction}
The continual gravitational collapse of a massive
matter cloud within the framework of general relativity was
investigated for the first time by the
classic works of Oppenheimer and Snyder, and Datt (OSD)
\cite{OSD}.
Such a treatment of dynamical collapse would be
essential to determine the final fate of a massive collapsing
star which shrinks catastrophically under the force
of its own gravity when its internal
nuclear fuel is exhausted.
The outcome in the above case is seen to be a
black hole developing in the spacetime. As the gravitational
collapse progresses, an event horizon forms within
the collapsing cloud and from the region
within the horizon no material particles or light rays can escape,
thus forming a black hole. The continually collapsing
star enters the horizon and finally ends up forming a
spacetime singularity, which is hidden inside the black
hole and which is unseen to all the outside observers
in the universe. The matter and energy densities, spacetime
curvatures, and all physical quantities blow up and
take extreme values in the limit of approach to
such a spacetime singularity.
This classic picture became the foundation of
an extensive theory and astrophysical applications
of modern day black hole physics, further to the
suggestion that {\it all} realistic massive stars undergoing a
continual gravitational collapse would have the same
qualitative behaviour. This means that, while the general
theory of relativity necessarily implies the formation
of a spacetime singularity as the endstate for a massive
collapsing star, such a singularity will always be
necessarily hidden within a black hole. Such an assumption
is known as the cosmic censorship hypothesis
\cite{Penrose},
and taking
it to be valid, the theory and applications
of black hole physics have developed extensively in
past many decades.
The cosmic censorship has, however, remained an
unproved conjecture as yet in gravitation theory, despite
numerous attempts to establish the same.
Therefore, in past many years, much effort
has also been devoted towards understanding and analyzing
the final fate of a physically realistic dynamical gravitational
collapse scenario. The current status is, despite much work in
studying the censorship and its implications, the
issue of final fate of a complete gravitational collapse
of a massive star remains far from being fully resolved.
In particular, we need to formulate in a precise
manner the conditions in gravitational collapse that
would lead to the formation of black holes
necessarily. We now know that under
a wide variety of physically realistic situations, the
collapse ends in a black hole or a naked singularity, depending
on the initial conditions from which the collapse develops
and the dynamical evolutions as allowed by the
Einstein equations
(see e.g. \cite{Ref, Joshibook} and references therein).
It is now clear that naked
singularities are to be considered as a general feature
of general relativistic physics and that they may
develop as the end-state of collapse in a broad
variety of physical collapse situations.
It follows that a careful and extensive study of gravitational
collapse phenomena in general relativity is the key to put the theory
of black holes and their astrophysical implications on a firm footing.
From such a perspective, we investigated here
the effect of introducing small stress perturbations in the collapse
dynamics of the classic Oppenheimer-Snyder-Datt gravitational
collapse, an idealized model assuming zero
pressure, which terminates in a black hole final fate.
Our key purpose here is to study the stability
of the OSD black hole under introduction of small tangential
pressures. Clearly, stresses within a massive collapsing star are very
important physical forces to be taken into account while
considering its dynamical evolution and the final fate
of collapse (see for example
\cite{Press}).
We show here explicitly the existence of classes of
stress perturbations such that the introduction of a smallest
tangential pressure within the collapsing OSD cloud changes the
endstate of collapse to formation of a naked singularity,
rather than a black hole. It follows that the OSD
black hole is not stable under small stress perturbations
within the collapsing cloud. As we point out below,
this can also be viewed as perturbing the spacetime metric
of the cloud in a small way. Our work thus clarifies the role played
by tangential stresses in a well known gravitational
collapse scenario. The class of stress perturbations considered
here, although specific, is physically reasonable and generic
enough so as to provide a good insight into the stability
of the OSD black hole. Clearly, such a result provides an
important insight into the structure of the censorship principle
which as yet remains to be properly understood. This has
also implications towards the physical consequences
of final outcomes of a continual collapse, some of which are
indicated in the concluding remarks.
\vspace{\baselineskip}
The general spherically symmetric line element
describing the collapsing matter cloud can be written
as,
\begin{equation}\label{metric}
ds^2=-e^{2\nu(t, r)}dt^2+e^{2\psi(t, r)}dr^2+R(t, r)^2d\Omega^2,
\end{equation}
with the stress-energy tensor for a generic matter source given by,
$T_t^t=-\rho, \; T_r^r=p_r, \; T_\theta^\theta=T_\phi^\phi=p_\theta$.
The above is a general scenario, in that it involves no assumptions
on the form of the matter or the equation of state.
In order to decide on the stability or otherwise of the
OSD model under the injection of small stress perturbations, we need
to consider the dynamical development of the collapsing cloud,
as governed by the Einstein equations.
The visibility or otherwise of the final singularity is determined
by the behaviour of apparent horizon in the spacetime,
which is the boundary of the trapped surface region
that develops as the collapse progresses. First, we define a
scaling function $v(r,t)$ by the relation $R=rv$
\cite{GJ4}.
The Einstein equations for the above spacetime geometry
can then be written as,
\begin{eqnarray}\label{p2}
p_r&=&-\frac{\dot{F}}{R^2\dot{R}}, \; \rho = \frac{F'}{R^2R'} \; ,\\ \label{nu2}
\nu'&=&2\frac{p_\theta-p_r}{\rho+p_r}\frac{R'}{R}-\frac{p_r'}{\rho+p_r} \; ,\\ \label{G2}
2\dot{R}'&=&R'\frac{\dot{G}}{G}+\dot{R}\frac{H'}{H} \; ,\\
\label{F2}
F&=&R(1-G+H) \; ,
\end{eqnarray}
where the functions $F$ and $G$ are defined as,
$H =e^{-2\nu(r, v)}\dot{R}^2 , \; G=e^{-2\psi(r, v)}R'^2$.
The above are five equations in
seven unknowns, namely $\rho,\; p_r, \; p_{\theta}, \; R,\; F,\; G,\; H$.
Here $\rho$ is the mass-energy density, $p_r$ and $p_\theta$ are
the radial and tangential stresses respectively, $R$ is the physical
radius for the matter cloud, and $F$ is the Misner-Sharp mass
function.
With the above definitions of $v, H$ and $G$, we can
substitute the unknowns $R, H$ with $v, \nu$. Without loss of
generality, the scaling function $v$ can be set $v(t_i, r)=1$ at the
initial time $t_i=0$ when the collapse commences.
It then goes to zero at the spacetime singularity $t_s$, which
corresponds to $R=0$, {\it i.e.} we have $v(t_s, r)=0$.
The above amounts to the scaling $R=r$ at the initial
epoch, which is an allowed freedom. The collapse condition
here is $\dot R<0$ throughout the evolution,
which is equivalent to $\dot{v}<0$.
We can integrate \eqref{G2} by defining a suitably regular
function $A(r, v)$ by $\nu'\equiv A_{,v}(r,v)R'$ (the function $A$ is
defined in full generality here, while often the restriction to
the class $\nu=\nu(R)$, implying $A(R)=\nu(R)$, is made, see e.g.
\cite{JGM}).
This gives $G(r,t)=b(r)e^{2rA(r,v)}$.
The arbitrary function of integration $b(r)$ can be interpreted
following the analogy with dust collapse models, where pressures vanish.
It turns out to be related to the velocity of the collapsing shells, and
once we write it as $b(r)=1+r^2b_0(r)$,
we can see that values $b_0=const.$ in the dust limit correspond
to the open ($b_0<0$), closed ($b_0>0$) or flat ($b_0=0$) Friedmann-
Robertson-Walker models. The radial stress $p_r$ and the energy
density $\rho$ are obtained from equations \eqref{p2},
once a specific choice for the mass function $F(r,t)$ is made. The function
$\nu$ can be taken as the second free function for the system so
that once a particular form of $\nu$ is specified, equation \eqref{nu2}
provides the tangential stress profile $p_{\theta}$. Finally, from the
equation of motion \eqref{F2}, we can integrate to obtain $v(r, t)$,
thus solving the system of Einstein equations.
We can also invert the function $v(r,t)$,
which is monotonically decreasing in $t$, to obtain the
time needed by the matter shell at any radial value $r$ to
reach the event with a particular value $v$. We write
the function $t(r, v)$ from equation \eqref{F2} as,
\begin{equation}
t(r, v)= \int^1_{v}\frac{e^{-\nu}}{\sqrt{\frac{F}{r^3\tilde{v}}+\frac{be^{2rA}-1}{r^2}}}d\tilde{v} \; .
\end{equation}
The time taken by the shell at $r$ to reach the spacetime singularity
at $v=0$ is then $t_s(r)=t(r, 0)$.
Since $t(r, v)$ is in general at least $C^2$ everywhere in the spacetime
(because of the regularity of the functions involved), and is continuous at the
center, we can write it as,
\begin{equation}\label{t}
t(r, v)= t(0, v)+r\chi(v)+O(r^2) \;
\end{equation}
When $t(r,v)$ is differentiable, we can make a Taylor expansion
near the center $r=0$.
Here, $t(0,v)$ is the above integral evaluated at $r=0$
and $\chi(v)=\left.\frac{dt}{dr}\right|_{r=0}$. As we point out below, the
quantity $\chi(0)$ plays an important role towards determining the
nature of the final singularity of collapse.
We consider collapse from a regular initial data, and
so the Einstein equation (5) implies that the Misner-Sharp
mass $F(r, v)$
must go as $r^3$ near the center $r=0$ in order for the density to be regular
at the center, and also to have $t(0, v)$ well defined.
Therefore, in general, $F$ must have the form,
$F(r,v) = r^3 M(r,v)$,
where $M$ is a suitably regular function.
Then, by continuity, the time for the shell located at any $r$ close to
center to reach the singularity is given as,
$t_s(r)= t_s(0)+r\chi(0)+O(r^2)$.
Basically, this means that the singularity curve should have a well-defined
tangent at the center. Regularity at the center also implies that
the metric function $\nu$ cannot have constant or linear terms in
$r$ in a close neighborhood of $r=0$, and it must go as $\nu\sim r^2$
near the center. Therefore the most general choice of the free function
$\nu$ is,
\begin{equation}
\nu(r,v)=r^2g(r,v) \;
\end{equation}
Since $g(r, v)$ is a regular function (at least $C^2$), it can be
written near $r=0$ as,
\begin{equation}\label{expand-g}
g(r, v)=g_0(v)+g_1(v)r+g_2(v)r^2+...
\end{equation}
We would now like to investigate how the OSD gravitational
collapse scenario, which is a homogeneous pressureless dust cloud collapsing
to give rise to a black hole, gets altered when small stress perturbations are
introduced in the dynamical evolution of collapse.
To that end, we first note that the dust scenario is obtained if
$p_r=p_{\theta}=0$ in the above. In that case, from equation \eqref{nu2} it
follows that $\nu'=0$ and that together with the condition $\nu(0)=0$ gives
$\nu=0$ identically.
These models have been widely studied in the literature, and it is seen
that for generic dust collapse the final outcome can be either a black hole
or a naked singularity, depending on the nature of the initial density
and velocity profiles of the collapsing matter shells
\cite{dust}.
In the OSD collapse to a black hole, the trapped surfaces
or the apparent horizon in the spacetime develop much earlier before
the formation of the final singularity of collapse. On the other hand,
when inhomogeneities are allowed in the initial density profile, such as a
higher density at the center of the star, then the trapped surface formation
is delayed in a natural manner within the collapsing cloud and the final
singularity becomes visible to faraway observers in the universe
\cite{JDM}.
The OSD case is obtained from above when
we further assume that the collapsing dust is necessarily homogeneous at all
epochs of collapse. This is of course an idealized scenario because realistic
stars would have typically higher densities at the center, which slowly falls
off with increasing radius, and they also would have non-zero internal stresses.
Specifically, the conditions that must be imposed to obtain the OSD case
from the above are given by,
\begin{itemize}
\item[(a)] $M=M_0$\; ,
\item[(b)] $v=v(t)$\; ,
\item[(c)] $b_0(r)=k$\; .
\end{itemize}
Then we have $F'=3M_0r^2$, $R'=v$, and the energy density is homogeneous
throughout the collapse with $\rho=\rho (t)= {3M_0}/{v^3}$.
The spacetime geometry then becomes the Oppenheimer-Snyder metric,
\begin{equation}
ds^2=-dt^2+\frac{v^2}{1+kr^2}dr^2+r^2v^2d\Omega^2 \; ,
\end{equation}
where the function $v(t)$ is solution of the equation of motion,
$ \frac{dv}{dt}=\sqrt{(M_0/v)+k}$,
obtained from Einstein equation \eqref{F2}. In this case we get
$\chi(0)=0$ identically. All the matter shells then collapse into a
simultaneous singularity (due to condition (b)), which
is necessarily covered by the event horizon that developed in the
spacetime at an earlier time, thus giving rise to a black hole.
To examine the effect of introducing stress perturbations
in the above scenario and to study the models thus obtained which are close
to the Oppenheimer-Snyder in this sense, we need to relax
and perturb one or more of the above conditions (a), (b) or (c).
If the collapse outcome would not to be a black hole, the final
singularity of collapse cannot be simultaneous. We are thus led to relax
condition (b) above, allowing $v = v(t,r)$, rather than $v=v(t)$ only.
At the same time, in order not to depart too much from the OSD model,
we keep (a) and (c) unchanged. This also brings out more clearly the role
played by the stress perturbations in the model.
In terms of the spacetime metric \eqref{metric},
while the metric function $\nu(t,r)$ must be identically vanishing for the
dust case, the above amounts to allowing for small perturbations in $\nu$,
and allowing it to be non-zero now. This is equivalent to introducing
small stress perturbations in the model, and we show below how that
affects the apparent horizon developing in the collapsing cloud.
We note immediately that taking $M=M_0$ leads to $ F=r^3M_0$.
We have $R'=v+rv'\rightarrow v$
for $r\rightarrow 0$ and therefore we get
$ A_{,v}= {\nu'}/{v}$.
With the expansion near $r=0$ for
both $A$ and $g$ we get the relation between the coefficients of
the expansion of $g$ and those
for the expansion of $A$. Integrating \eqref{G2} in the small $r$ limit we thus
obtain $G(r,t)=b(r)e^{2\nu(r, v)}$.
The radial stress $p_r$ vanishes in this case as $\dot F=0$, while
the tangential pressure, obtained from equation \eqref{nu2}, has the form,
$p_\theta= p_1 r^2 + p_2 r^3 +...$, where $p_1, p_2$ are naturally
evaluated in terms of coefficients of $m, g$, and $R$ and its derivatives,
\begin{equation}\label{pt}
p_\theta=3\frac{M_0g_0}{vR'^2}r^2+\frac{9}{2}\frac{M_0g_1}{vR'^2}r^3+...
\end{equation}
Here the choice of sign of the functions $g_0$
and $g_1$ is enough to ensure positivity or negativity of $p_\theta$.
We note that scenarios with vanishing radial stresses but
non-vanishing tangential stresses have been considered in past,
with the most physically significant model (though not the only relevant one)
being the so called `Einstein cluster' (see
\cite{cluster}),
which describes a cloud of collapsing counter rotating particles.
Naked singularities and black holes are found to arise as the endstate of
such models, depending on the initial density, velocity and stress configurations \cite{GJ5}.
The first order coefficient $\chi$ in
equation of the time curve of the singularity $t_s(r)$ is now obtained as
\begin{equation}\label{chi}
\chi(0)=-\int^1_0\frac{v^{\frac{3}{2}}g_1(v)}{(M_0+vk+2vg_0(v))^{\frac{3}{2}}}dv \; .
\end{equation}
As mentioned above, it is $\chi(0)$ that governs the
nature of the singularity curve, and whether it is increasing or decreasing
away from the center. Clearly, it is the matter initial data in terms of
density and stress profiles, the velocity of the collapsing shells, and
the allowed dynamical evolutions that govern and fix
the value of $\chi(0)$.
The quantity $\chi(0)$ also governs the behaviour of
apparent horizon and trapped surface formation, as we show
below, which in turn governs the nakedness or otherwise of
the singularity. The equation for the apparent horizon is
given by ${F}/{R}=1$. It is analogous to that of the dust case
since ${F}/{R}={rM}/{v}$ in both cases
\cite{JDM}.
So the apparent horizon curve $r_{ah}(t)$ is given by
$ r_{ah}^2=\frac{v_{ah}}{M_0}$,
with $v_{ah}=v(r_{ah}(t), t)$, which can also be inverted as a
time curve $t_{ah}(r)$.
The visibility of the singularity at the center of the collapsing cloud
to faraway observers is determined by the
nature of this apparent horizon curve which is given by,
\begin{equation}\label{t-ah}
t_{ah}(r)=t_s(r)-\int_0^{v_{ah}}\frac{e^{-\nu}}{\sqrt{\frac{M_0}{v}+\frac{be^{2\nu}-1}{r^2}}}dv
\end{equation}
where $t_s(r)$ is the singularity time curve, whose initial point is
$t_0=t_s(0)$. Near $r=0$ the equation \eqref{t-ah} becomes,
\begin{equation}
t_{ah}(r) =t_0+\chi(0)r+o(r^2) \; .
\end{equation}
From the above, it is now easy to see how the stress perturbation
affects the time of formation of the apparent horizon, and therefore the
formation of a black hole or naked singularity.
A naked singularity typically occurs as collapse endstate when a
co-moving observer at fixed $r$ does not encounter any trapped surfaces
till the time of singularity formation. For a black hole to form, trapped
surfaces develop before the singularity, so it is needed that,
\begin{equation}
t_{ah}(r) \le t_0 ~~\mbox{for}~~ r>0, ~~\mbox{near}~~ r=0 \; .
\end{equation}
It is clear that for all functions $g_1(v)$ for which $\chi(0)$ is
positive, this
condition is violated and the apparent horizon is forced to appear
after the formation of the central singularity. The apparent horizon curve
then initiates at the central singularity $r=0$ at $t=t_0$ and increases
with increasing $r$, moving to the future, {\it i.e.} $t_{ah} > t_0$ for
$r > 0 $ near the center. The behaviour of outgoing families of
null geodesics has been analyzed in detail in such a case
when $\chi(0)>0$ and we know
that geodesics terminate at the singularity in the past.
Thus timelike and null geodesics come out from the singularity,
making it visible to external observers
\cite{Geo}.
It follows that $g_1$ is the term in the stresses $p_\theta$
which decides the black hole or naked singularity final fate.
We can choose it to be arbitrarily small, and we now see how introducing
a generic tangential stress perturbation in the model would change
drastically the final outcome of collapse. For all non-vanishing
tangential stresses with $g_0=0$ and $g_1<0$, even
the slightest perturbation of the Oppenheimer-Snyder-Datt
scenario, injecting a small tangential stress would result in a naked
singularity. The space of all functions $g_1$ that make $\chi(0)$
positive, which includes all the strictly negative functions $g_1$,
causes the collapse to end in a naked singularity. We note that
while this is an explicit example, by no means this is the only
class.
The remarkable feature of this class is that it
corresponds to a collapse model for a simple and straightforward
perturbation of the Oppenheimer-Snyder-Datt spacetime metric,
where the geometry near the center can be written as,
\begin{equation}\label{pert}
ds^2=-(1-2g_1 r^3)dt^2+\frac{(v+rv')^2}{1+kr^2-2g_1 r^3}dr^2+r^2v^2d\Omega^2 \; ,
\end{equation}
The metric above satisfies Einstein equations in the neighborhood
of the center of the cloud when the function $g_1(v)$ is small and bounded.
We could take for example, $0<|g_1(v)|<\epsilon$, so that the smaller we
take the parameter $\epsilon$ the bigger will be the radius where the
approximation is valid.
The function $v(r,t)$ above is governed by the equation of motion
\eqref{F2} which in the small $r$ limit becomes,
$ {dv}/{dt}=(1-g_1(v) r^3) ({\frac{M_0}{v}+k-2g_1(v) r})^{1/2}.$
Finally, $\chi(0)$ in this case is given by equation \eqref{chi} with $g_0=0$,
and in certain cases can also be integrated.
We note that any realistic matter model must satisfy some
energy conditions ensuring the positivity of mass and energy density.
In general, the weak energy condition implies restrictions on the density
and pressure profiles. The energy density as given by the second of equations
\eqref{p2} must be positive. Since $R$ is positive, to ensure positivity
of $\rho$ we require $F>0$ and $R'>0$. The choice of
positive $M(r)$ (which obviously holds for $M_0>0$ and is physically reasonable)
ensures positivity of the mass function. Then $R'>0$ is a sufficient condition for the
avoidance of shell crossing singularities. The tangential stress can be written
from \eqref{nu2} where $p_r=0$, and is given by $p_\theta=\frac{1}{2}\frac{R}{R'}\rho\nu'$.
So the sign of the function $\nu'$ determines the sign of $p_\theta$.
Positivity of $\rho+p_\theta$ is then ensured for small values of $r$
throughout collapse for any form of $p_\theta$. In fact, regardless
of the values taken by $M$ and $g$, there will always be a neighborhood of $r=0$
for which $|p_\theta|<\rho$ and therefore $\rho+p_\theta\geq0$.
The black hole and naked singularity outcomes of gravitational
collapse are very different from each other physically, and would have
quite different observational signatures. In the naked singularity case we have
the possibility to observe the physical effects happening in the vicinity of
the ultra dense regions that form in the very final stages of collapse.
However, in a black hole scenario, such regions are necessarily hidden
within the event horizon. The fact that a slightest stress perturbation
of the OSD collapse could change the outcome drastically, taking it from a
black hole to naked singularity formation, means that the naked singularity
final state for a collapsing star must be studied carefully to deduce its
physical consequences which are not well understood so far.
The existence of subspaces of collapse solutions as we have
shown here, that go to a naked singularity final state rather than a black
hole, in the arbitrary vicinity of the OSD black hole, presents an intriguing
scenario. It gives an idea of the richness of the structure present in
gravitation theory and the complex solution space of Einstein equations
which are a complicated set of highly non-linear partial differential equations.
What we see here is there are classes of stress perturbations such
that an arbitrarily small change from the OSD model is a solution going
to naked singularity. In this sense, this manifests an instability in the
black hole formation process in gravitational collapse. This also provides
an intriguing insight into the nature of cosmic censorship, namely that
the collapse must be properly fine-tuned necessarily if it is to produce
a black hole only as the final endstate.
Traditionally it was believed that the presence of
stresses or pressures in the collapsing matter cloud would increase the chance
of black hole formation, thereby ruling out dust models
that were found to lead to a naked singularity as collapse endstate.
That is no longer the case.
The model described here not only provides a new class of collapses
ending in a naked singularity, but more importantly, shows how the bifurcation
line that separates the phase space of `black hole formation' from that of
the `naked singularity formation' runs directly over the simplest and most
studied of black hole scenarios such as the OSD model, thus making
it unstable under perturbations.
|
1,116,691,499,368 | arxiv | \@startsection{subsubsection}{3}{10pt}{-1.25ex plus -1ex minus -.1ex}{0ex plus 0ex}{\normalsize\bf}{\@startsection{subsubsection}{3}{10pt}{-1.25ex plus -1ex minus -.1ex}{0ex plus 0ex}{\normalsize\bf}}
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\noindent\LARGE{\textbf{Comment on ``A single level tunneling model for molecular junctions: evaluating the simulation methods''
}}
\vspace{0.6cm}
\noindent\large{\textbf{Ioan B\^aldea
\textit{$^{a \ast}$}
}}\vspace{0.5cm}
\vspace{0.6cm}
\noindent
\normalsize{Abstract:\\
The present Comment demonstrates important flaws of the paper Phys. Chem. Chem. Phys. 2022, 24, 11958 by Opodi~\emph{et al.}
Their crown result (``applicability map'') aims at indicating parameter ranges wherein two approximate methods (called method 2 and 3)
apply. My calculations reveal that the applicability map is a factual error.
Deviations of $I_2$ from the exact current $I_1$ do not exceed 3\%
for model parameters where Opodi~\emph{et al.} claimed that method 2 is inapplicable.
As for method 3, the parameter range of the applicability map is beyond its scope,
as stated in papers cited by Opodi~\emph{et al.}~themselves.
}
$ $ \\
{{\bf Keywords}:
molecular electronics, nanojunctions, single level model}
\vspace{0.5cm}
\footnotetext{\textit{$^{a}$~Theoretical Chemistry, Heidelberg University, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany}}
\footnotetext{$^\ast$~E-mail: [email protected]
}
Comparing currents $I_1$, $I_2$, and $I_3$
through tunneling molecular junctions computed via three single level models (see below),
Opodi \emph{et al.}\cite{Opodi:22} claimed, \emph{e.g.},~that:\\[0.4ex]
\indent (i) The applicability of the method based on $I_3$,\cite{Baldea:2012a} which was previously validated against experiments on
benchmark molecular junctions (\emph{e.g.}, ref~\citenum{Baldea:2015b,Baldea:2019d,Baldea:2019h})
is ``quite limited''.
\\[0.4ex]
\indent (ii) The ``applicability map'' (Fig.~5 of ref~\citenum{Opodi:22}) should be used in practice
as guidance for the applicability of methods 2 and 3 (\emph{i.e.}, based on $I_2$ and $I_3$) because
(ii$_1$) not only method 3 (ii$_2$) but also method 2 is drastically limited.\\[0.4ex]
\indent (iii) Model parameters for molecular junctions previously extracted from
experimental $I$-$V$-data need revision.
Before demonstrating that these claims are incorrect,
let me briefly summarize
the relevant information available prior to ref~\citenum{Opodi:22}.
Unless otherwise noted
(\emph{e.g.}, the difference between $\Gamma$ and $\tilde{\Gamma}$ expressed by eqn~(\ref{eq-factor-Gamma})),
I use the same notations as ref~\citenum{Opodi:22}.
\begin{equation}
\label{eq-I1}
I_1 =
\frac{2 e}{h} \left( N \Gamma_{g}^2 \right) \int_{-\infty}^{\infty}
\frac{f\left(\varepsilon - e V/2\right) - f\left(\varepsilon + e V/2 \right)}{\left(\varepsilon - \varepsilon_0\right)^2 + \Gamma_{a}^2} d\!\varepsilon
\end{equation}
\begin{equation}
\label{eq-I2}
I_2 = \frac{2 e}{h \Gamma_a} \left( N\Gamma_{g}^2\right)
\left(
\tan^{-1}\frac{\varepsilon_0 + eV/2}{\Gamma_{a}} -
\tan^{-1}\frac{\varepsilon_0 - eV/2}{\Gamma_{a}}
\right)
\end{equation}
\begin{equation}
\label{eq-I3}
I_3 = \frac{2 e}{h} \left( N \Gamma_{g}^2\right) \frac{e V}{\varepsilon_0^2 - (e V/2)^2}
= \underbrace{G_0 \frac{\left( N \Gamma_{g}^2\right)}{\varepsilon_0^2}}_{G_3} \frac{\varepsilon_0^2 V}{\varepsilon_0^2 - (e V/2)^2}
\end{equation}
(a) As a particular case of a formula \cite{Caroli:71a},
eqn~(\ref{eq-I1}) expresses the exact current in the coherent tunneling regime
through a junction consisting of $N$ molecules (set to $N=1$ unless otherwise specified)
mediated by a single level whose energy offset relative to the electrodes' unbiased
Fermi energy is $\varepsilon_0$, coupled to wide, flat band electrodes (hence Lorentzian transmission).
The effective level coupling to electrodes $\Gamma_{g} \equiv \sqrt{\Gamma_s \Gamma_t}$
is expressed in terms of energy independent quantities $\Gamma_{s,t}$,
representing the level couplings to the two electrodes--- substrate (label $s$) and tip (label $t$)---,
which also contribute to the finite level width $\Gamma_{a} = \left(\Gamma_s + \Gamma_t\right)/2$.
In the symmetric case assumed following Opodi \emph{et al.}
\begin{equation}
\label{eq-sym}
\tilde{\Gamma} = \Gamma_s = \Gamma_t = \Gamma_g = \Gamma_a
\end{equation}
and $\varepsilon_0$ is independent of bias.
In this Comment, I use the symbol $\tilde{\Gamma}$--- a quantity denoted by $\Gamma$
in ref~\citenum{Baldea:2012a} and in all studies on
junctions fabricated with the conducting probe atomic force microscopy (CP-AFM) platform
cited in ref~\citenum{Opodi:22}---
in order to distinguish it from the quantity denoted by $\Gamma$
by Opodi \emph{et al.}\cite{Opodi:22}
Comparison of the present eqn~(\ref{eq-I2}) and (\ref{eq-I3})--- in ref~\citenum{Baldea:2012a} these are
eqn~(3) and (4), respectively--- with eqn~(2) and (3) of ref~\citenum{Opodi:22} makes it clear why:
$\tilde{\Gamma}$ is one half of the quantity denoted by $\Gamma = \Gamma_L + \Gamma_R = 2 \Gamma_L = 2 \Gamma_R $
by Opodi \emph{et al.}\cite{Opodi:22}
\begin{equation}
\label{eq-factor-Gamma}
\tilde{\Gamma} = \Gamma / 2
\end{equation}
(b) Eqn~(\ref{eq-I2}) follows as an exact result from eqn~(\ref{eq-I1}) in the zero temperature limit ($T\to 0$),
when the Fermi distribution
$f(\varepsilon) \equiv 1/\left[1 + \exp\left(\varepsilon / k_B T\right)\right]$
reduces to the step function.
This low temperature limit
(expressed by eqn~(\ref{eq-e0-T}) and (\ref{eq-Gamma-T}) below)
assumes a negligible variation of
the transmission function (which is controlled by $\varepsilon_0$ and $\tilde{\Gamma}$)
within energy ranges of widths $\sim k_B T$ around the electrodes' Fermi level
wherein electron states switch between full ($f \approx 1$) and
empty ($f\approx 0$) occupancies.
Away from resonance, $\tilde{\Gamma}$ (usually a small value,
$\tilde{\Gamma} \ll \left\vert\varepsilon_0 \right\vert$,
see eqn~(\ref{eq-Gamma-vs-e0}))
plays a negligible role and
\begin{subequations}
\label{eq-low-T}
\begin{equation}
\label{eq-e0-T}
\left\vert\varepsilon_0 \pm eV/2\right\vert \gg k_B T
\end{equation}
is sufficient for the low temperature limit to apply.\cite{Lambert:11}
However, closer to resonance the aforementioned weakly energy-dependent transmission also implies
a sufficiently large $\tilde{\Gamma}$. This implies a relationship between the transmission width ($\sim\tilde{\Gamma}$)
and the width ($\sim k_B T$) of the range ($\sim \left(\varepsilon \pm e V/2 - k_B T, \varepsilon \pm e V/2 + k_B T\right)$, \emph{cf.}~\ref{eq-I1}))
wherein the electrode Fermi functions rapidly vary. In practice, very close to resonance,
loosely speaking, this means \cite{Baldea:2017d,Baldea:2022c,Baldea:2022j}
\begin{equation}
\label{eq-Gamma-T}
\tilde{\Gamma} \sim k_B T
\end{equation}
\end{subequations}
(c) Eqn~(\ref{eq-I3}) was \emph{analytically} derived \cite{Baldea:2012a}
from eqn~(\ref{eq-I2}) for sufficiently large arguments of the inverse trigonometric functions
\begin{equation}
\label{eq-arctan}
\tan^{-1}(x) \simeq
\frac{\pi}{2} - \frac{1}{x} \mbox{ (holds within 1\% for } x > x_0 = 2.929)
\end{equation}
The bias range wherein eqn~(\ref{eq-I3}) holds within the above accuracy can be expressed as follows
\begin{equation}
\label{eq-tmp}
e \vert V \vert < 2 \left(\left\vert \varepsilon_0\right\vert - x_0 \tilde{\Gamma}\right)
\simeq 2 \left\vert \varepsilon_0\right\vert \left(1 - x_0 \sqrt{g}\right)
\end{equation}
where
\begin{equation}
\label{eq-g}
g = \frac{1}{N}\frac{G}{G_0} = \frac{\tilde{\Gamma}^2}{\varepsilon_0^2 + \tilde{\Gamma}^2}
= \underbrace{\frac{\tilde{\Gamma}^2}{\varepsilon_0^2}}_{g_3}
\left[1 + \mathcal{O}\left(\frac{\tilde{\Gamma}^2}{\varepsilon_0^2}\right)\right]
\end{equation}
is the zero-temperature low bias conductance \emph{per molecule} ($G/N$)
in units of the universal conductance quantum $G_0 = 2 e^2/h = 77.48\,\mu$S.
Strict on-resonance ($\varepsilon_0 \equiv 0$) single-channel transport is characterized by $g \equiv 1$.
In the vast majority of molecular junctions fabricated so far, tunneling transport
is off-resonant ($g \ll 1$), and $ g < g_{max} = 0.01$ safely holds
in all experimental situations of which I am aware, including all CP-AFM junctions considered in ref~\citenum{Opodi:22}.
Imposing
\begin{equation}
\label{eq-g-max}
g < 0.01
\end{equation}
in eqn~(\ref{eq-g}) yields
\begin{equation}
\label{eq-Gamma-vs-e0}
\tilde{\Gamma} < 0.1005 \left \vert \varepsilon_0\right\vert \simeq \left\vert \varepsilon_0\right\vert / 10
\end{equation}
and \emph{via} eqn~(\ref{eq-tmp})\cite{Baldea:2015b,Baldea:2017g}
\begin{subequations}
\label{eq-low-V}
\begin{eqnarray}
\label{eq-low-bias}
\vert e V \vert & < & 1.4\,\left\vert \varepsilon_0\right\vert \mbox{ or, equivalently}\\
\label{eq-low-bias-vt}
\vert e V \vert & < & 1.25\, e V_t \mbox{ (\emph{cf.}~eqn~(\ref{eq-vt0}))}
\end{eqnarray}
\end{subequations}
Along with the low-$T$ limit assumed by eqn~(\ref{eq-I2}),
eqn~(\ref{eq-Gamma-vs-e0}) and (\ref{eq-low-bias}) are necessary conditions for eqn~(\ref{eq-I3}) to apply.
Aiming at aiding experimentalists interested in $I$-$V$ data processing, who do not know
$\varepsilon_0$ \emph{a priori}, in ref~\citenum{Baldea:2012a} I rephrased eqn~(\ref{eq-arctan}) by saying that
eqn~(\ref{eq-I3}) holds for biases not much larger than the transition voltage $V_t$ (eqn~(\ref{eq-low-bias-vt})).
$V_t$ is a quantity that can be directly extracted from experiment without any theoretical
assumption from the maximum of $V^2/\vert I\vert$ plotted \emph{vs} $V$.\cite{Beebe:06,Baldea:2015b}
The fact that eqn~(\ref{eq-I3}) should be applied \emph{only}
for biases compatible with eqn~(\ref{eq-low-bias}) has been steadily emphasized (\emph{e.g.},
ref~\citenum{Baldea:2012b} and discussion related
to Fig.~2 and 3, and eqn~4 of ref~\citenum{Baldea:2017g}).
(d) Eqn~(\ref{eq-I3}) is particularly useful because it allows expression of the transition voltage $V_t$
in terms of the level offset \cite{Baldea:2012a}
\begin{equation}
\label{eq-vt0}
e V_t =
2 \left\vert \varepsilon_0 \right\vert/\sqrt{3}
\end{equation}
which can thus be easily estimated.
$\varepsilon_0$ is a key quantity in discussing the structure-function relationship in molecular electronics.
Thermal corrections to eqn~(\ref{eq-vt0}), which are significant for small offsets ($\left\vert \varepsilon_0 \right\vert \raisebox{-0.3ex}{$\stackrel{<}{\sim}$} 0.4$\,eV)
even at the room temperature ($k_B T = 25$\,meV) assumed by Opodi \emph{et al.}~were
also quantitatively analyzed (\emph{e.g.}, Fig.~4 in ref~\citenum{Baldea:2017g}).
\emph{To sum up, method 2 applies to situations compatible with
eqn~(\ref{eq-low-T}), and method 3 applies in situations compatible with eqn~(\ref{eq-low-T}) and (\ref{eq-low-V}).
This is a conclusion of a general theoretical analysis that needs no additional confirmation from numerical calculations like those
of ref~\citenum{Opodi:22}.}
Switching to the above claims,
the following should be said:
\underline{To claim (i)}:
Fig.~2 of ref~\citenum{Opodi:22} shows (along with $\vert I_{1,2} \vert$ also) currents $\vert I_{3} \vert$ computed
for biases $-1.5\,\mbox{V} < V < +1.5\,\mbox{V}$ at couplings $\Gamma = 2 \tilde{\Gamma} = \{1; 5; 10; 100\}$\,meV
and offsets $\varepsilon_0 = \{0.1; 0.5; 1\}$\,eV.
The lower cusps visible there depict currents
vanishing ($I \to 0, \log\vert I\vert \to -\infty$) at $V = 0$.
The upper symmetric cusps ($\log\vert I\vert \to +\infty$ at $V \to \pm 2 \varepsilon_0 $)
depicted by the blue lines were obtained by \emph{mathematical} application of eqn~(\ref{eq-I3})
beyond the \emph{physically meaningful} bias range of eqn~(\ref{eq-low-bias}), for which eqn~(\ref{eq-I3})
was theoretically deduced.
To make clear this point, I corrected in the present Fig.~\ref{fig:fig2}
Fig.~2 of ref~\citenum{Opodi:22}. The blue lines in the present Fig.~\ref{fig:fig2} depict
$I_3$ in the bias range $e\vert V\vert < 1.4\,\left\vert\varepsilon_0\right\vert $ for which eqn~(\ref{eq-I3})
was theoretically deduced and for which it makes physical sense.
The dashed orange lines are merely mathematical curves for $I_3$ computed using
eqn~(\ref{eq-I3}) at
$e\vert V\vert > 1.4\,\left\vert\varepsilon_0\right\vert $, where they have no physical meaning.
``By definition'', these orange curves are beyond the scope of method 3.
The presentation adopted in Fig.~2 by Opodi \emph{et al.}~also masks
{the fallacy}
of applying eqn~(\ref{eq-I3}) at biases $\vert e V \vert > 2 \left\vert \varepsilon_0\right\vert$.
There, the denominator in the RHS becomes negative and
bias and current have opposite directions (\emph{i.e.}, $I>0$ for $V<0$ and $I<0$ for $V>0$).
Visible at $\vert e V \vert = 2\left\vert \varepsilon_0 \right\vert$ are the nonphysical cusps of the blue ($I_3$)
curves in Fig.~2 by ref.~\citenum{Opodi:22},
same as the cusps of the orange curves of the present Fig.~\ref{fig:fig2}.
With this correction, the present Fig.~\ref{fig:fig2}
reveals what it should. Namely that, as long as the off-resonance condition of eqn~(\ref{eq-Gamma-vs-e0})
is satisfied ---
\emph{i.e.}, excepting for Fig.~\ref{fig:fig2}A---,
in all other panels
the blue and green curves ($I_3$ and $I_2$, respectively) practically coincide.
Significant differences between the exact red curve ($I_1$) and the approximate green and blue ($I_2$ and $I_3$, respectively)
are only visible in situations violating the low temperature condition (eqn~(\ref{eq-low-T})):
in panels D, E, G, and H violating eqn~(\ref{eq-Gamma-T}),
and at biases incompatible with eqn~(\ref{eq-e0-T}).
\begin{figure*}[htb]
\centerline{
\includegraphics[width=0.3\textwidth,angle=0]{Fig2a}
\includegraphics[width=0.3\textwidth,angle=0]{Fig2b}
\includegraphics[width=0.3\textwidth,angle=0]{Fig2c}
}
\centerline{
\includegraphics[width=0.3\textwidth,angle=0]{Fig2d}
\includegraphics[width=0.3\textwidth,angle=0]{Fig2e}
\includegraphics[width=0.3\textwidth,angle=0]{Fig2f}
}
\centerline{
\includegraphics[width=0.3\textwidth,angle=0]{Fig2g}
\includegraphics[width=0.3\textwidth,angle=0]{Fig2h}
\includegraphics[width=0.3\textwidth,angle=0]{Fig2i}
}
\centerline{
\includegraphics[width=0.3\textwidth,angle=0]{Fig2j}
\includegraphics[width=0.3\textwidth,angle=0]{Fig2k}
\includegraphics[width=0.3\textwidth,angle=0]{Fig2l}
}
\caption{Currents $I_1$, $I_2$, and $I_3$ computed using the parameters of Fig.~2 of Opodi \emph{et al.}.
Redrawing their figure emphasizes the difference between the current $I_3$ in the bias range
for which eqn~(\ref{eq-I3}) was theoretically deduced \cite{Baldea:2012a} (blue curves) and $I_3$ computed
outside of bias range (orange dashed curves), wherein eqn~(\ref{eq-I3}) is merely a mathematical formula without any physical sense.
Notice that, throughout, the green ($I_2$) and blue ($I_3$) curves excellently agree with the exact red curves ($I_1$)
precisely in the parameter ranges predicted by theory,
\emph{i.e.}~eqn~(\ref{eq-low-T}) and eqn~(\ref{eq-low-bias}).
The tick symbols at $V=1.5$\,V depicted in all panels emphasize that method 2 is very accurate,
invalidating thereby the ``applicability map'' shown by Opodi \emph{et al.} in their Fig.~5 (also reproduced in the present Fig.~\ref{fig:errors}a).}
\label{fig:fig2}
\end{figure*}
\underline{To claim (ii)}:
Refuting \underline{claim (ii$_1$)} is {straightforward}.
Based on their Fig.~5, Opodi \emph{et al.}~cannot make {a} statement on method 3: they consider
parameters $\varepsilon_0 < 1$\,eV at the bias $V = 1.5$\,V($>1.4\,\left\vert \varepsilon_0\right\vert $),
{which is} incompatible with eqn~(\ref{eq-low-bias}).
Noteworthily, the condition expressed by eqn~(\ref{eq-low-bias}) defied by ref~\citenum{Opodi:22}~was clearly
stated in references that Opodi \emph{et al.}~have cited.
To reject \underline{claim (ii$_2$)},
I show in Fig.~\ref{fig:errors}b, c, and d deviations of the current $I_2$ from the exact value $I_1$
in snapshots taken
horizontally (\emph{i.e.}, constant $\Gamma$) and vertically (\emph{i.e.}, constant $\varepsilon_0$)
across the ``applicability map'' (\emph{cf.}~Fig.~\ref{fig:errors}a).
As visible in Fig.~\ref{fig:errors}b and c,
in all regions where Opodi \emph{et~al.} claimed that method 2 is invalid
the contrary is true; the largest relative deviation $ I_2 / I_1 - 1$ does not exceed 3\%.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.28\textwidth,angle=0]{map_emphasis_incorrect}}
\centerline{\includegraphics[width=0.3\textwidth,angle=0]{fig_error_alongGamma_all}}
\centerline{\includegraphics[width=0.3\textwidth,angle=0]{fig_error_alongE0_all}}
\caption{
(a) Tick symbols depicting excellent agreement between $I_2$ and $I_1$ in all panels of Fig.~\ref{fig:fig2}
overimposed on the ``applicability map'' adapted after Fig.~5 (courtesy Xi Yu) of ref.~\citenum{Opodi:22}
contradict the claim of Opodi \emph{et al.} on the inaplicability of method 2.
(b,c) Deviations of $I_2$ from the exact current $I_1$
reveal that method 2 excellently works in situations where Opodi \emph{et al}~claimed the contrary.
Importantly, showing parameter values $\left\vert\varepsilon_0\right\vert < 1$\,eV at bias $V=1.5$\,V, panel a is beyond the scope of method 3
(\emph{cf.}~eqn~(\ref{eq-low-bias})).}
\label{fig:errors}
\end{figure}
What the physical quantity is underlying the color code depicted in their Fig.~5 (reproduced here in Fig.~\ref{fig:errors}a)
is not {explained by Opodi \emph{et~al.}}
Anyway, the conclusion of Opodi \emph{et~al.}~summarized in their Fig.~5 contradicts their results shown in their Fig.~2;
all panels of that figure reveal excellent agreement between the green ($I_2$) and red (exact $I_1$)
curves at $V = 1.5$\,V.
For the reader's convenience, the thick symbols at $V = 1.5$\,V in the present Fig.~\ref{fig:fig2} overlapped on the
``applicability map''of ref~\citenum{Opodi:22} emphasize this aspect.
Inspection of these symbols (indicating that method 2 is excellent) overimposed on Fig.~\ref{fig:errors}a
reveals that they (also) lie in regions where
Opodi \emph{et~al.}~claimed that method 2 fails.
Once more, their ``applicability map'' is factually incorrect.
\underline{To claim (iii)}:
In their Fig.~3, 4A, 4B, S1 to S4, and S8 as well as in Table~2
Opodi \emph{et~al.}~made unsuitable comparisons: the values for the CP-AFM junctions
taken from their ref~38, 39, 44, and 57
(present ref~\citenum{Baldea:2018a,Baldea:2019d,Baldea:2019h,Frisbie:21a})
are values of $\tilde{\Gamma} = \Gamma/2$, while those estimated by themselves are values of $\Gamma = 2 \tilde{\Gamma}$.
Confusing $\tilde{\Gamma}$ and $\Gamma$, no wonder that they needed re-fitting of the original $I$-$V$ data.
If they had correctly re-fitted the CP-AFM data using $I_3$, with all the values of
$N$ given in the original works (namely, their ref~38, 39, 44, and 57,
the present ref~\citenum{Baldea:2018a,Baldea:2019d,Baldea:2019h,Frisbie:21a}), up to minor
inaccuracies inherently arising from digitizing the experimental $I$-$V$-curves, they would have
reconfirmed the values of $\varepsilon_0$ reported in the original publications, and
would have obtained values of $\Gamma = 2\tilde{\Gamma}$
two times larger than those originally reported for $\tilde{\Gamma}$ (\emph{cf.}~eqn~(\ref{eq-factor-Gamma})).
{I still have to emphasize} a difference of paramount importance between $I$-$V$ data fitting based on
eqn~(\ref{eq-I1}) and (\ref{eq-I2}) on one side, and eqn~(\ref{eq-I3}) on the other side.
Eqn~(\ref{eq-I1}) and (\ref{eq-I2}) have three independent fitting parameters
$\left(\varepsilon_0, N \Gamma_g^2, \Gamma_a\right) \to
\left(\left\vert\varepsilon_0\right\vert, N {\Gamma}^2, {\Gamma}\right)$
while eqn~(\ref{eq-I3}) has only two independent fitting parameters
$\left(\left\vert\varepsilon_0\right\vert, N \Gamma_g^2\right) \to \left(\left\vert\varepsilon_0\right\vert, N {\Gamma}^2\right)$.
All the narrative on the $N$-$\Gamma$-entanglement
and wording on ``twin sisters'' used in Sec.~3.4 of the original article clearly reveal
that ref.~\citenum{Opodi:22} overlooked that, when using $I_3$, $N$ and $\Gamma$
are two parameters whose values are \emph{impossible} to separate; they
build a unique fitting parameter $N {\Gamma}^2 \equiv 4 N \tilde{\Gamma}^2$.
Data fitting using $I_3$ and three fitting parameters $\left(\varepsilon_0, \Gamma, N\right)$
has an infinity number of solutions for $\Gamma$ and $N$ but they all yield a unique value of $N {\Gamma}^2$.
Were method 3 ``quite limited'' and the deviations of $I_3$ from $I_1$ or $I_2$ significant,
Opodi \emph{et~al.}~would have been able to determine three best fit parameters
$\left(\left\vert \varepsilon_0\right\vert, \Gamma, N\right)$; at least for the ``most problematic'' junctions
where they claimed important departures of $I_3$-based estimates from those based on $I_1$ and $I_2$.
If this is indeed the case, the value of $N$ can be determined from data fitting \cite{Baldea:2022j}.
Their MATLAB code \ib{(additionally relevant details in the ESI)} clearly reveals how they \ib{arrived at}
showing such differences for
real junctions considered. In that code, they keep $N$ fixed and adjust $ \varepsilon_0 $ and $ \Gamma$.
As long it is reasonably realistic, an arbitrarily chosen value of $N$ has no impact on
directly measurable properties. It changes the value of $\Gamma$ but neither $N \Gamma^2 \propto G_3 \approx G$
nor the level offset $ \varepsilon_0 $ changes, because method 3 performs well in almost all real cases.
However, defying available values of $N$ for the CP-AFM junctions to which they referred,
Opodi \emph{et al.}~spoke of values up to $N\sim 10^5$. Employing such artificially large $N$'s
nonphysically reduces $\Gamma$ ($N \Gamma^2 \approx \mbox{constant}$, $\Gamma \propto 1/\sqrt{N}$)
down to values incompatible with eqn~(\ref{eq-Gamma-T}),
arriving thereby at the idea that eqn~(\ref{eq-I3}) no longer applies.
In spot checks, I also interrogated curves shown by Opodi \emph{et al.}~for single-molecule mechanically controllable
break junctions. I arrived so at the junction of 4,4'-bisnitrotolane (BNT),\cite{Zotti:10} the real junction for
which they claimed the most severe failure of method 3.
If Fig.~S7D to S7F and 4D of ref.~\citenum{Opodi:22} were correct, both $\varepsilon_0 $ and $\Gamma$ based on $I_3$ would be in error by a factor of two.
To reject this claim, in Fig.~\ref{fig:mcbj} I show curves for $I_1$, $I_2$, and $I_3$ computed with the values of $ \varepsilon_0 $ and $\Gamma$
indicated by Opodi \emph{et al.} in their Fig.~S7D, E, and F, respectively.
They should coincide with the black curves of Fig.~S7D, E, and F, respectively if the latter were correct.
According to Opodi \emph{et al.}, all these
would represent fitting curves of the \emph{same} experimental curve (red points in Fig.~S7D, E, and F).
\begin{figure}[htb]
\centerline{\includegraphics[width=0.28\textwidth,angle=0]{FigS7_iv_BNT}}
\centerline{\includegraphics[width=0.28\textwidth,angle=0]{FigS7_iv_BNT_ext}}
\caption{(a)
$I$-$V$ curves for single-molecule junctions \cite{Zotti:10} obtained using $N=1$ and
parameters taken from the figures of ref.~\citenum{Opodi:22}
indicated in the legends.
To convince himself or herself that the present curves for $I_2$ and $I_3$ are correct and
different from those of Fig.~S7E and S7E of ref.~\citenum{Opodi:22},
the reader can easily generate the present green and blue curves by using the GNUPLOT script of the ESI$^\dagger$.
(b) The curves for $I_1$, $I_2$, and $I_3$ computed with the parameter values (indicated in the inset)
taken from Fig.~S7D of ref.~\citenum{Opodi:22} do not support the failure of method 3 claimed by {Opodi \emph{et al.}}
}
\label{fig:mcbj}
\end{figure}
\ib{Provided that MATLAB is available, the reader
can run the code ``generateIVfitIV.m'' included in the ESI$^\dag$ to convince himself or herself}
that the red curve of Fig.~\ref{fig:mcbj}a and not the black curve of Fig.~S7D
represents the exact current $I_1$ computed using eqn~(\ref{eq-I1}).
\ib{Otherwise,} running the GNUPLOT script also put in the ESI$^\dag$
will at least convince \ib{the} reader
that the green and the blue lines of this figure and not the black curves in Fig.~S7E and F, respectively
do represent the currents $I_2$ and $I_3$ computed using eqn~(\ref{eq-I2}) and (\ref{eq-I3})
for the parameters and the bias range indicated.
\ib{The} reader will realize that the three curves shown in Fig.~\ref{fig:mcbj} cannot represent best fits of the \emph{same}
experimental $I$-$V$-curve (red points in Fig.~S7D, E, and F of ref~\citenum{Opodi:22}).
If Opodi \emph{et~al.}~had calculated $I_2$ and $I_3$ using the parameters indicated in their Fig.~S7D, E, and F
(same as in the present Fig.~\ref{fig:mcbj}a), they would not have obtained the black curves of their Fig.~S7D, E, and F
but the red, green, and blue curves of Fig.~\ref{fig:mcbj}a. The values $\varepsilon_0 = 0.57$\,eV and $\Gamma = 147$\,meV
of Fig.~S7F (so much different from $\varepsilon_0 = 0.27$\,eV and $\Gamma = 71$\,meV of Fig.~S7D and $\varepsilon_0 = 0.28$\,eV and $\Gamma = 78.6$\,meV
of Fig.~S7E) can by no means be substantiated from these calculations.
All aforementioned values of $\varepsilon_0$ and $\Gamma$ of Fig.~S7D, E, and F
are exactly the same as the values shown in Table~2 and Fig.~4D of ref~\citenum{Opodi:22}, and used as argument against method 3.
As additional support, I also show (Fig.~\ref{fig:mcbj}b) the curves for $I_1$, $I_2$, and $I_3$, all computed with the same
parameters, namely those of Fig.~S7D of ref~\citenum{Opodi:22} ($\varepsilon_0 = 0.27$\,eV and $\Gamma = 71$\,meV).
In accord with the general theoretical considerations presented under (b) and (c),
the differences between $I_2$ and $I_3$ (blue and green curves) are negligible,
while deviations of $I_2$ from $I_1$ are notable only for biases close $\pm 2\varepsilon_0$
that invalidate eqn~(\ref{eq-e0-T}).
To sum up, the claim of Opodi \emph{et~al.}~on the failure of method 3 for the specific case considered above is incorrect because
is based on values incompatible with calculations.
As made clear under (c) above, eqn~(\ref{eq-low-bias}) is a condition deduced analytically.
If it holds, eqn~(\ref{eq-I3}) is within $\sim 1\%$ as good as eqn~(\ref{eq-I2}).
It makes little sense to check numerically
a general condition deduced analytically,
or even worse (as done in ref.~\citenum{Opodi:22}) to claim that it does not apply for biases incompatible with eqn~(\ref{eq-low-bias}).
The interested scholar needs not the ``applicability map''
(Fig.~5 of ref~\citenum{Opodi:22}, \ib{to be corrected elsewhere (I.~B\^aldea, to be submitted)}).
In the whole parameter ranges where Opodi \emph{et al.}~claimed the opposite, method 2 turned out to be extremely accurate
(\emph{cf.}~Fig.~\ref{fig:errors}b and c).
Likewise, ``by definition'' (cf.~eqn~(\ref{eq-low-bias})), method 3
should not be applied at biases above $ e V > 2 \left\vert\varepsilon_0\right\vert $
shown there, which makes the ``applicability map'' irrelevant for method 3.
Theory should clearly indicate the parameter ranges where an analytic formula is valid.
This is a task accomplished in case of eqn~(\ref{eq-I3}).
In publications also cited by Opodi \emph{et~al.} \cite{Baldea:2015b,Baldea:2017g},
particular attention has been drawn on not to apply eqn~(\ref{eq-I3}) at biases
violating eqn~(\ref{eq-low-bias})\cite{Baldea:2015b} and/or for energy offsets
($\left\vert\varepsilon_0\right\vert \raisebox{-0.3ex}{$\stackrel{<}{\sim}$} 0.5$\,eV) where thermal effects ($k_B T \simeq 25\,\mbox{meV} \neq 0$)
matter \cite{Baldea:2017g}.
It is experimentalists' responsibility not to apply it
under conditions that defy the boundaries under which it was theoretically deduced.
Financial support from the German Research Foundation
(DFG Grant No. BA 1799/3-2) in the initial stage of this work and computational support by the
state of Baden-W\"urttemberg through bwHPC and the German Research Foundation through
Grant No.~INST 40/575-1 FUGG
(bwUniCluster 2.0, bwForCluster/MLS\&WISO 2.0/HELIX, and JUSTUS 2.0 cluster) are gratefully acknowledged.
\renewcommand\refname{Notes and references}
\footnotesize{
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\newpage
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\caption{ }
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|
1,116,691,499,369 | arxiv | \section{Introduction}
This paper is a continuation of a collection of works about applications of
the theory of quadratic differentials in quantum theory. In fact, quadratic
differentials have provided an important tool in the asymptotic study of some
solutions of algebraic equations. In quantum theory, trajectories of some
quadratic differentials have crucial role in the WKB analysis.
We consider the eigenvalue problem
\begin{equation}
-y^{\prime\prime}+V_{s,m}(x)y=\lambda y,\text{ } \label{sch eq}%
\end{equation}
where the potential $V_{s,J}(x)$ is given by :%
\[
V_{s,m}(x)=\frac{(4s-1)(4s-3)}{x^{2}}+(x^{6}-(4s+4m-2)x^{2}).
\]
This problem was studied by A.Turbiner \cite{ref01},\cite{ref3}. For
$s\in\left] \frac{1}{4},\frac{3}{4}\right[ $, there is an attractive
centrifugal term. For $s\in\left] -\infty,\frac{1}{4}\right[ \cup$ $\left]
\frac{3}{4},+\infty\right[ ,$ the centrifugal term is repulsive. For
$s=\frac{1}{4}$ or $\frac{3}{4},$ the centrifugal core term disappears leaving
a sextic oscillator
\[
V_{p,m}(x)=\text{ }x^{6}-(4m+p)x^{2};\text{ }p\in\left\{ -1,1\right\}
\]
We will study the case $s=\frac{1}{4}$ (so $p=-1$), with a perturbed
potential
\[
V_{-1,m+1}(x)=V_{m}(x)=\text{ }x^{6}+\gamma m^{1/2}x^{4}+(\frac{\gamma^{2}%
m}{4}-(4m+3))x^{2},\text{ }%
\]
with boundray condition $y(\pm\infty)=0.$This problem is exactly solvable:
that means that for every $\gamma$ there exist $m+1$ eigenfunctions of the
form
\[
\phi(x)=q_{m}(x)e^{-\frac{x^{4}}{4}-\frac{\gamma m^{1/2}}{4}x^{2}},
\]
where $q_{m}$ is an even-parity polynomial of degree $2m$, corresponding to
$m+1$ eigenvalues $\lambda_{m};$ see \cite{ref2}. Note that in the case $p=1$
we have the same form of eigenfunctions with an odd-parity polynomials; for
more details see again \cite{ref2}.
Substitiuting in the Schr\"{o}dinger equation (\ref{sch eq}), we find that
\[
-q_{m}^{\prime\prime}(x)+(2x^{3}+\gamma m^{\frac{1}{2}}x)q_{m}^{\prime
}(x)-(4mx^{2}-\frac{\gamma m^{1/2}}{2})q_{m}(x)=\lambda_{m}q_{m}(x).
\]
The differential operator%
\[
K=-\frac{d^{2}}{dx^{2}}+(2x^{3}+\gamma m^{1/2}x)\frac{d}{dx}-(4mx^{2}%
-\frac{\gamma m^{1/2}}{2})
\]
preserves the $(m+1)$-dimentional linear space of all even polynomials of
degree $\leq2m$. Using $z=x^{2},$ we find that
\begin{equation}
-4zq_{m}^{\prime\prime}(z)+(4z^{2}+2\gamma m^{1/2}z-2)q_{m}^{\prime
}(z)-(4mz-\frac{\gamma m^{1/2}}{2})q_{m}(z)=\lambda_{m}q_{m}(z). \label{eq2}%
\end{equation}
Our main goal is the study of the asymptotic root-counting measure
$\vartheta_{m}$ of an appropriate re-scaled polynomial sequence $\left\{
Q_{m}(z)=q_{m}(m^{\varepsilon}z)\right\} .$ Let $\nu_{m}$ be the normalized
root-counting measure of the sequence $\left( q_{m}\right) $. The Cauchy
transform $C_{\nu_{m}}$ and $C_{\vartheta_{m}}$ of respectively $\nu_{m}$ and
$\vartheta_{m}$ satisfies the following equations :
\[
4zmC_{\nu_{m}}^{2}(z)-(4z^{2}+2\gamma m^{1/2}z-2)C_{\nu_{m}}(z)+\frac
{(4mz-\frac{\gamma m^{1/2}}{2}+\lambda_{m})}{m}+4zC_{\nu_{m}}^{\prime}(z)=0.
\]%
\[
4zm^{1-\varepsilon}C_{\vartheta_{m}}^{2}(z)-(4m^{\varepsilon}z^{2}+2\gamma
m^{1/2}z-2m^{-\varepsilon})C_{\vartheta_{m}}(z)+\frac{(4m^{1+\varepsilon
}z-\frac{\gamma m^{1/2}}{2}+\lambda_{m})}{m}+4m^{-\varepsilon}zC_{\vartheta
_{m}}^{\prime}(z)=0.
\]
For $\varepsilon=1/2,$ we get
\begin{equation}
-4zC_{\vartheta_{m}}^{2}(z)+(4z^{2}+2\gamma z-\frac{2}{m})C_{\vartheta_{m}%
}(z)-(4z-\frac{\gamma}{2m}+\frac{\lambda m}{m^{3/2}})-4z\frac{C_{\vartheta
_{m}}^{\prime}(z)}{m}=0. \label{eqdiff1}%
\end{equation}
It was shown in \cite{ref4}, that the sequence $\left( \lambda_{m}%
/m^{4/3}\right) $ is bounded. By the Helly selection Theorem, we may assume
that
\[
\underset{m\rightarrow\infty}{\lim}\left( \lambda_{m}/m^{4/3}\right)
=\delta,
\]
and then, there exists a compactly-supported positive measure $\nu$ such that
\[
\underset{m\rightarrow\infty}{\lim}\vartheta_{m}=\nu,\underset{m\rightarrow
\infty}{\lim}C_{\vartheta_{m}}\mathcal{=C}_{\nu}=\mathcal{C}.
\]
Finally, taking the limits in (\ref{eqdiff1}), we obtain the algebraic
equation :%
\begin{equation}
z\mathcal{C}^{2}(z)-(z^{2}+\frac{\gamma}{2}z)\mathcal{C}(z)+(z+\frac{\delta
}{4})=0. \label{algeq}%
\end{equation}
\bigskip In this paper, we discuss the existence of solutions of equation
(\ref{algeq}) as Cauchy transform of compactly-supported signed measure.
Section \ref{connection alg}, we make the connection between this a algebraic
equation and a particular quadratic differential. In section \ref{qudar dif},
we describe the critical graph of the related quadratic differential in the
Riemann sphere $\overline{%
\mathbb{C}
},$ more precisely, we discuss the number of its finite critical trajectories.
\section{\bigskip A quadratic differential \label{qudar dif}}
\bigskip Below, we describe the critical graphs of the the family of quadratic
differentials
\begin{equation}
\varpi_{q}=-\frac{q\left( z\right) }{z}dz^{2}, \label{qd}%
\end{equation}
where $q$ is a monic polynomial of degree $3.$ We begin our investigation by
some immediate observations from the theory of quadratic differentials. For
more details, we refer the reader to \cite{Strebel},\cite{jenkins},.
Recall that \emph{finite critical points }of a given meromorphic quadratic
differential $-Q\left( z\right) dz^{2}$ on the Riemann sphere $\overline{%
\mathbb{C}
}$ are its zeros and simple poles; poles of order $2$ or greater then 1 called
\emph{infinite critical points. }All other points of $\overline{%
\mathbb{C}
}$ are called \emph{regular points}.
\emph{Horizontal trajectories} (or just trajectories) of the quadratic
differential are the zero loci of the equation%
\[
-Q\left( z\right) dz^{2}>0,
\]
or equivalently%
\begin{equation}
\mathcal{\Re}\int^{z}\sqrt{Q\left( t\right) }\,dt=\text{\emph{const}}.
\label{eq traj}%
\end{equation}
If $z\left( t\right) ,t\in%
\mathbb{R}
$ is a horizontal trajectory, then the function
\[
t\longmapsto\Im\int^{t}\sqrt{Q\left( z\left( u\right) \right) }z^{\prime
}\left( u\right) du
\]
is monotone.
The \emph{vertical} (or, \emph{orthogonal}) trajectories are obtained by
replacing $\Im$ by $\Re$ in equation (\ref{eq traj}). The horizontal and
vertical trajectories produce two pairwise orthogonal foliations of the
Riemann sphere $\overline{%
\mathbb{C}
}$.
A trajectory passing through a critical point is called \emph{critical
trajectory}. In particular, if it starts and ends at a finite critical point,
it is called \emph{finite critical trajectory }or\emph{\ short trajectory},
otherwise, we call it an \emph{infinite critical trajectory}. A short
trajectory is called \emph{unbroken }if it does not pass through any finite
critical points except its two endpoints. The closure the set of finite and
infinite critical trajectories is called the \emph{critical graph}.
A necessary condition for the existence of a short trajectory connecting
finite critical points is the existence of a Jordan arc $\gamma$ connecting
them, such that
\begin{equation}
\Re\int_{\gamma}\sqrt{Q\left( t\right) }dt=0. \label{cond necess}%
\end{equation}
However, this condition is sufficient in general; see counter-example in
\cite{F.Thabet}.
The local structure of the trajectories is as follow :
\begin{itemize}
\item At any regular point, horizontal (resp. vertical) trajectories look
locally as simple analytic arcs passing through this point, and through every
regular point passes a uniquely determined horizontal (resp. vertical)
trajectory; these horizontal and vertical trajectories are locally orthogonal
at this point.
\item From each zero of multiplicity $r$, there emanate $r+2$ critical
trajectories spacing under equal angle $2\pi/\left( r+2\right) $.
\item At a simple pole, there emanates exactly one horizontal trajectory.
\item At the pole of order $r>2$, there are $r-2$ asymptotic directions
(called \emph{critical directions}) spacing under equal angle $2\pi/\left(
r-2\right) $ and a neighborhood $\mathcal{U}$, such that each trajectory
entering $\mathcal{U}$ stays in $\mathcal{U}$ and tends to the pole in one of
the critical directions. See Figure \ref{FIG1}.
\end{itemize}
\begin{figure}[tbh]
\begin{minipage}[b]{0.28\linewidth}
\centering\includegraphics[scale=0.25]{1.pdf}
\end{minipage}\hfill
\begin{minipage}[b]{0.28\linewidth} \includegraphics[scale=0.25]{2.pdf}
\end{minipage}
\hfill\begin{minipage}[b]{0.28\linewidth} \includegraphics[scale=0.3]{3.pdf}
\end{minipage}
\caption{Structure of the trajectories near a simple zero (left), a simple
pole (center), and a pole of order 6 (right).}%
\label{FIG1}%
\end{figure}
\bigskip A very helpful tool that will be used in our investigation is the
Teichm\"{u}ller lemma (see \cite[Theorem 14.1]{Strebel}).
\begin{definition}
\bigskip A domain in $\overline{%
\mathbb{C}
}$ bounded only by segments of horizontal and/or vertical trajectories of
$\varpi_{q}$ (and their endpoints) is called $\varpi_{q}$-polygon.
\end{definition}
\begin{lemma}
[Teichm\H{u}ller]\label{teich lemma} Let $\Omega$ be a $\varpi_{q}$-polygon,
and let $z_{j}$ be the critical points on the boundary $\partial\Omega$ of
$\Omega,$ and let $\theta_{j}$ be the corresponding interior angles with
vertices at $z_{j},$ respectively. Then%
\begin{equation}
\sum\left( 1-\dfrac{\left( n_{j}+2\right) \theta_{j}}{2\pi}\right) =2+\sum
m_{i}, \label{Teich equality}%
\end{equation}
where $n_{j}$ are the multiplicities of $z_{j}=1,$ and $m_{i}$ the
multiplicities of critical points inside $\Omega.$\bigskip
\end{lemma}
\bigskip We will focus on the case where
\[
q\left( z\right) =q_{a}\left( z\right) =\left( z-1\right) \left(
z-a\right) \left( z-\overline{a}\right) ,
\]
with
\[
a\in%
\mathbb{C}
^{+}=\left\{ a\in%
\mathbb{C}
\mid\Im\left( a\right) >0\right\} .
\]
We have the following immediate observations :
\begin{itemize}
\item The finite critical points of $\varpi_{q}$ are $1,a,\overline{a}$ as
simple zeros, and $-1$ as a simple pole.
\item With the parametrization $u=1/z$, we get
\[
\varpi_{q}\left( u\right) =\left( -\frac{1}{u^{6}}+\mathcal{O}\left(
\frac{1}{u^{5}}\right) \right) du^{2},\text{ }u\longrightarrow0,
\]
thus, infinity is an infinite critical point of $\varpi_{q},$ as a pole of
order $6.$
\item Since $\infty$ is the only infinite critical point of $\varpi_{q},$ any
critical trajectory which is not finite diverges to $\infty$ following one of
the 4 directions :
\[
D_{k}=\left\{ z\in%
\mathbb{C}
\mid\arg\left( z\right) =\left( 2k+1\right) \frac{\pi}{4}\right\}
,k=0,1,2,3.
\]
The same thing happen to the orthogonal trajectories at $\infty$, but the
critical directions are :%
\[
D_{k}^{\perp}=\left\{ z\in%
\mathbb{C}
:\arg\left( z\right) =\frac{k\pi}{2}\right\} ;k=0,1,2,3.
\]
Observe that if two trajectories diverge to $\infty$ in a same direction
$D_{k},$ then there exists a neighborhood $\mathcal{V}$ of $\infty$, such that
any orthogonal trajectory traversing $D_{k}$ in $\mathcal{V},$ must traverse
these two trajectories.
\item \bigskip Since the quadratic differential $\varpi_{q}$ has two poles,
Jenkins Three-pole Theorem (see \cite[Theorem 15.2]{Strebel}) asserts that the
situation of the so-called recurrent trajectory (whose closure might be dense
in some domain in $%
\mathbb{C}
$) cannot happen.
\end{itemize}
\bigskip
\begin{lemma}
\label{at infinity}Two critical trajectories of $\varpi_{q}$ emanating from
the same zero cannot diverge to $\infty$ in the same direction.
\end{lemma}
\begin{lemma}
\label{residue}For any Jordan arc $\gamma$ connecting $a$ and $\overline{a}$
in $%
\mathbb{C}
\setminus\left[ 0,1\right] $ we have%
\[
\Re\int_{\gamma}\sqrt{\frac{\left( z-1\right) \left( z-a\right) \left(
z-\overline{a}\right) }{z}}dz=0,
\]
and then, condition (\ref{cond necess}) is fulfilled.
\end{lemma}
We consider the set
\[
\Sigma=\left\{ z\in%
\mathbb{C}
:\Re\int_{0}^{z}\sqrt{\frac{\left( t-1\right) \left( t-z\right) \left(
t-\overline{z}\right) }{t}}dt=0\right\}
\]
\begin{lemma}
\label{curve}The set $\Sigma$ is symmetric with respect to the real axis, and
it is formed by $3$ Jordan arcs :
\begin{itemize}
\item the segment $\left[ 0,1\right] ;$
\item two curves $\Sigma^{\pm}$ emerging from $z=1,$ and diverging
respectively to infinity in $%
\mathbb{C}
_{\pm}.$
\end{itemize}
\end{lemma}
\begin{figure}[th]
\centering\includegraphics[height=2in,width=3in]{4.pdf}\caption{Approximate
plot of the curve $\Sigma$}%
\label{FIG2}%
\end{figure}
We give here the behavior of $\Sigma$ at $z=1$ and at the infinity:
\begin{lemma}
\label{asympt of the curve}The following results hold :
\begin{align*}
\lim\limits_{z\rightarrow\infty,z\in\Sigma\cap%
\mathbb{C}
^{+}}\arg\left( z\right) & =\frac{\pi}{2},\\
\lim\limits_{z\rightarrow1,z\in\Sigma\cap%
\mathbb{C}
^{+}}\arg\left( z\right) & =\frac{\pi}{3}.
\end{align*}
\end{lemma}
From Lemma \ref{curve}, $\Sigma$ splits $%
\mathbb{C}
$ into 2 connected domains :
\begin{itemize}
\item $\Omega_{1}$ limited by $\Sigma^{\pm}$ and containing $z=2;$
\item $\Omega_{2}=%
\mathbb{C}
\setminus$ $\left( \Omega_{1}\cup\Sigma^{\pm}\cup\left[ 0,1\right] \right)
.$ See Figure \ref{FIG2}.
\end{itemize}
\begin{proposition}
\label{main}For any complex number $a,$ the quadratic differential $\varpi
_{q}$ has :
\begin{itemize}
\item two short trajectories if $a\in\Omega_{1}$: the segment $\left[
0,1\right] $ and another one connecting $a$ and $\overline{a}$ in $\Omega
_{1};$see Figure \ref{FIG3};
\item three short trajectories if $a\in\Sigma^{\pm}$ : the segment $\left[
0,1\right] $ and two others connecting $z=1$ with $a$ and $\overline{a};$ see
Figure \ref{FIG4};
\item two short trajectories if $a\in\Omega_{2}$ : the segment $\left[
0,1\right] $ and another one connecting $a$ and $\overline{a}$ in $\Omega
_{2};$see Figure \ref{FIG5}.
\end{itemize}
\end{proposition}
\begin{figure}[tbh]
\begin{minipage}[b]{0.48\linewidth}
\centering\includegraphics[scale=0.5]{5.pdf}
\end{minipage}\hfill
\begin{minipage}[b]{0.48\linewidth} \includegraphics[scale=0.5]{6.pdf}
\end{minipage}
\caption{Critical graphs when $a\in\Omega_{1},$ here $a=1.6+2i$ (left) and
$a=1.8+2i$ (right).}%
\label{FIG3}%
\end{figure}
\begin{figure}[th]
\centering\includegraphics[height=2in,width=3in]{7.pdf}\caption{Critical graph
when $a$ $\in\Sigma,$ here $a=1.55+2i.$}%
\label{FIG4}%
\end{figure}
\begin{figure}[tbh]
\begin{minipage}[b]{0.48\linewidth}
\centering\includegraphics[scale=0.5]{8.pdf}
\end{minipage}\hfill
\begin{minipage}[b]{0.48\linewidth} \includegraphics[scale=0.5]{9.pdf}
\end{minipage}
\caption{Critical graphs when $a\in\Omega_{2},$ here $a=0.5+2i$ (left) and
$a=2i$ (right).}%
\label{FIG5}%
\end{figure}
\begin{remark}
The case when $q=\prod_{i=1}^{3}\left( z-a_{i}\right) \in%
\mathbb{R}
\left[ X\right] ,$ with real zeros is quite simple; if the zeros are simple
: $a_{1}<a_{2}<a_{3},$ then the segments $\left[ a_{1},a_{2}\right] $ and
$\left[ a_{3},a_{4}\right] $ are two short trajectories of $\varpi_{q}$. See
Figure \ref{FIG6}.
\end{remark}
\begin{figure}[tbh]
\begin{minipage}[b]{0.48\linewidth}
\centering\includegraphics[scale=0.5]{10.pdf}
\end{minipage}\hfill
\begin{minipage}[b]{0.48\linewidth} \includegraphics[scale=0.5]{11.pdf}
\end{minipage}
\caption{Critical graphs for $\varpi_{q}$ when $q=\left( z-1\right) \left(
z-2\right) \left( z-3\right) $ (left) and $q=\left( z-1\right) \left(
z-2\right) ^{2}$ (right).}%
\label{FIG6}%
\end{figure}\bigskip\bigskip
\section{Connection with the algebraic equation \label{connection alg}}
\bigskip The Cauchy transform $\mathcal{C}_{\nu}$ of a compactly supported
Borelian complex measure $\nu$ is defined in $%
\mathbb{C}
\setminus$\emph{supp}$\left( \nu\right) $ by :
\[
\mathcal{C}_{\nu}\left( z\right) =\int_{%
\mathbb{C}
}\frac{d\nu\left( t\right) }{z-t}.
\]
It satisfies
\[
\mathcal{C}_{\nu}\left( z\right) =\frac{\nu\left(
\mathbb{C}
\right) }{z}+\mathcal{\allowbreak O}\left( z^{-2}\right) ,z\rightarrow
\infty,
\]
and the inversion formula (which should be understood in the distributions
sense) :%
\[
\nu=\frac{1}{\pi}\frac{\partial\mathcal{C}_{\nu}}{\partial\overline{z}}.\text{
}%
\]
In particular, the normalized root-counting measure $\nu_{n}=\nu(P_{n})$ of a
complex polynomial $P_{n}$ of a complex polynomial $P_{n}$ of degree $n$ is
defined in $%
\mathbb{C}
$ by :
\[
\nu_{n}=\frac{_{1}}{n}\sum\limits_{P_{n}\left( a\right) =0}\delta_{a},\text{
(each zero is counted with its multiplicity);}%
\]
the Cauchy transform of $\nu_{n}$ is :
\[
\mathcal{C}_{\nu_{n}}(z)=\int_{%
\mathbb{C}
}\frac{d\nu_{n}\left( t\right) }{z-t}=\frac{P_{n}^{^{\prime}}(z)}{nP_{n}%
(z)};P_{n}(z)\neq0.
\]
Let us come back to the algebraic equation (\ref{algeq}). We are seeking a
compactly-supported signed measure $\nu$ such that, its Cauchy transform
$\mathcal{C}_{\nu}$ satisfies almost everywhere in $%
\mathbb{C}
$ equation (\ref{algeq}).
With the choice of the square root of the discriminant%
\[
\Delta\left( z\right) =\frac{z}{4}\left( 4z^{3}+4\gamma z^{2}+\left(
\gamma^{2}-16\right) z-16\delta\right)
\]
of the quadratic equation (\ref{algeq}) (as a quadratic equation) with
condition
\[
\sqrt{\Delta\left( z\right) }\sim z^{2},z\rightarrow\infty,
\]
it is easy to check that independently of the complex numbers $\gamma$ and
$\delta$, we have :
\[
\mathcal{C}(z)=\frac{2z^{2}+\gamma z-2\sqrt{\Delta\left( z\right) }}%
{4z}=\frac{1}{z}+\mathcal{\allowbreak O}\left( z^{-2}\right) ,z\rightarrow
\infty,
\]
which let us be hopeful for the existence of the measure $\nu.$
The following Lemma gives a sufficient condition on a solution of
(\ref{algeq}) to be the Cauchy transform of some compactly supported measure
in $%
\mathbb{C}
$ :
\begin{lemma}
[{\cite[comp. Th. 1.2, Ch. II,]{garnet}}]Suppose $f\in L_{loc}^{1}\left(
\mathbb{C}
\right) $ and that $f(z)\rightarrow0$ as $z\rightarrow\infty$ and let $\mu$
be a compactly-supported measure in $%
\mathbb{C}
$ such that%
\[
\mu=\frac{1}{\pi}\frac{\partial f}{\partial\overline{z}}%
\]
in the sense of distributions. Then $f(z)=\mathcal{C}_{\mu}\left( z\right) $
almost everywhere in $%
\mathbb{C}
.$
\end{lemma}
The following Proposition gives a necessary condition the existence of
measures $\nu$ :
\begin{proposition}
\label{connection}Let us consider the quadratic differential
\begin{equation}
-\frac{\Delta\left( z\right) }{z^{2}}dz^{2}. \label{qd2}%
\end{equation}
If the signed measure $\nu$ exists, then, the quadratic differential
(\ref{qd2}) has two short trajectories, and, the support of $\nu$ coincides
with these short trajectories. In particular, if $\Delta\left( z\right) $ is
a real polynomial, then the problem of finding the measure $\nu$ is solved.
\end{proposition}
\section{Proofs}
\begin{proof}
[Proof of Lemma \ref{at infinity}]Suppose that $\gamma_{1}$ and $\gamma_{2}$
are two such trajectories emanating from the zero $a$ or $1$, spacing with
angle $\theta\in\left\{ 2\pi/3,4\pi/3\right\} .$ Consider the $\varpi_{q}%
$-polygon with edges $\gamma_{1}$ and $\gamma_{2},$ and vertices $z_{j},$ and
infinity. The right side of (\ref{Teich equality}) can take only the values
$0$ or $-1,$ while the left side is at list $2;$ a contradiction. (Observe the
$\varpi_{qp}$-polygon cannot contain the pole $z=0,$ otherwise it contains
$z=1$ and, again we get a contradiction with (\ref{Teich equality}%
).\bigskip\bigskip
\end{proof}
\begin{proof}
[Proof of Lemma \ref{residue}]Since $\frac{q\left( t\right) }{t}$ is a real
rational fraction, then
\begin{equation}
\overline{\sqrt{\frac{q\left( t\right) }{t}}}=\sqrt{\frac{q\left(
\overline{t}\right) }{\overline{t}}},t\neq0, \label{ration symm}%
\end{equation}
and we get, after the change of variable $u=\overline{t}$ in second integral
:
\begin{align*}
\Re\left( \int_{\overline{z}}^{z}\sqrt{\frac{q\left( t\right) }{t}%
}dt\right) & =\Re\left( \int_{1}^{z}\sqrt{\frac{q\left( t\right) }{t}%
}dt-\int_{1}^{\overline{z}}\sqrt{\frac{q\left( t\right) }{t}}dt\right) \\
& =\Re\left( \int_{1}^{z}\sqrt{\frac{q\left( t\right) }{t}}dt-\overline
{\int_{1}^{z}\sqrt{\frac{q\left( t\right) }{t}}dt}\right) \\
& =\Re\left( 2i\Im\left( \int_{1}^{z}\sqrt{\frac{q\left( t\right) }{t}%
}dt\right) \right) \\
& =0.
\end{align*}
Let us give a necessary condition to get two short trajectories joining two
different pairs of finite critical points of $\varpi_{q}$ in the general case
:
\[
\frac{q\left( z\right) }{z}=\frac{z^{3}+\alpha z^{2}+\beta z+\gamma}%
{z}=\frac{\left( z-a\right) \left( z-b\right) \left( z-c\right) }%
{z},a,b,c\in%
\mathbb{C}
.
\]
Consider two disjoint oriented Jordan arcs $\gamma_{1}$ and $\gamma_{2}$
connecting two distinct pairs of zeros. We define the single-valued function
$\sqrt{\frac{q\left( z\right) }{z}}$ in $%
\mathbb{C}
\setminus\left( \gamma_{1}\cup\gamma_{2}\right) $ with condition
$\sqrt{\frac{q\left( z\right) }{z}}\sim z,z\rightarrow\infty.$ For
$s\in\gamma_{1}\cup\gamma_{2},$ we denote by $\left( \sqrt{p\left( s\right)
}\right) _{+}$ and $\left( \sqrt{p\left( s\right) }\right) _{-}$the
limits from the $+$ and $-$ sides, respectively. (As usual, the $+$ side of an
oriented curve lies to the left and the $-$ side lies to the right, if one
traverses the curve according to its orientation.)
From the Laurent expansion at $\infty$ of $\sqrt{q\left( z\right) }:$%
\[
\sqrt{\frac{q\left( z\right) }{z}}=\allowbreak z+\frac{\alpha}{2}-\left(
\frac{\alpha^{2}-4\beta}{8z}\right) +\allowbreak\mathcal{O}\left(
z^{-2}\right) ,
\]
we deduce the residue%
\[
res_{\infty}\left( \sqrt{\frac{q\left( z\right) }{z}}\right) =\frac{1}%
{8}\allowbreak\left( \alpha^{2}-4\beta\right) .
\]
Let%
\[
I=\int_{\gamma_{1}}\left( \sqrt{\frac{q\left( s\right) }{s}}\right)
_{+}ds+\int_{\gamma_{2}}\left( \sqrt{\frac{q\left( s\right) }{s}}\right)
_{+}ds.
\]
Since
\[
\left( \sqrt{\frac{q\left( s\right) }{s}}\right) _{+}=-\left( \sqrt
{\frac{q\left( s\right) }{s}}\right) _{-},s\in\gamma_{1}\cup\gamma_{2},
\]
we have
\[
2I=\int_{\gamma_{1}\cup\gamma_{2}}\left[ \left( \sqrt{\frac{q\left(
s\right) }{s}}\right) _{+}-\left( \sqrt{\frac{q\left( s\right) }{s}%
}\right) _{-}\right] ds=\oint_{\Gamma_{q}}\sqrt{\frac{q\left( z\right)
}{z}}dz,
\]
where $\Gamma_{q}$ is a closed contours encircling the curves $\gamma_{1}$ and
$\gamma_{2}$. After the contour deformation, we pick up the residue at
$z=\infty,$ and we get
\begin{align*}
I & =\frac{1}{2}\oint_{\Gamma_{q}}\sqrt{\frac{q\left( z\right) }{z}}dz=\pm
i\pi res_{\infty}\left( \sqrt{p\left( z\right) }\right) \\
& =\pm\frac{\pi i}{8}\allowbreak\allowbreak\left( \alpha^{2}-4\beta\right)
\end{align*}
and the necessary condition is
\[
\Im\left( \allowbreak\alpha^{2}-4\beta\right) =0,
\]
which is satisfied for the case when $q$ is real.
\end{proof}
\begin{proof}
[Proof of Lemma \ref{curve}]It is clear that $\Sigma\cap%
\mathbb{R}
=\left[ 0,1\right] .$ The fact that $\Sigma$ is symmetric with respect to
the real axis follows from the observation (\ref{ration symm}). In order to
prove that $\Sigma$ is a curve, we consider the real functions $F$ and $G$
defined for $\left( x,y\right) $ in $%
\mathbb{C}
_{+}$ by:
\begin{align*}
F\left( x,y\right) & =\Re\left( \int_{0}^{x}\sqrt{\frac{\left( u-\left(
x+iy\right) \right) \left( u-\left( x-iy\right) \right) \left(
u-1\right) }{u}}du\right) \\
& =\Re\left( \int_{0}^{x}\sqrt{\frac{\left( \left( u-x\right) ^{2}%
+y^{2}\right) \left( u-1\right) }{u}}du\right) ;\\
G\left( x,y\right) & =\Re\left( \int_{x}^{x+iy}\sqrt{\frac{\left(
u-\left( x+iy\right) \right) \left( u-\left( x-iy\right) \right)
\left( u-1\right) }{u}}du\right) \\
& =-\int_{0}^{1}y^{2}\sqrt{1-t^{2}}\Im\sqrt{1-\frac{1}{x+ity}}dt.
\end{align*}
Observe that
\[
\Sigma=\left\{ \left( x,y\right) \in%
\mathbb{R}
^{2}\mid\left( F+G\right) \left( x,y\right) =0\right\} .
\]
We prove first that $\Sigma\setminus\left[ 0,1\right] \subset\left\{ z\in%
\mathbb{C}
\mid\Re z>1\right\} .$ If $x\leq1$ and $y>0,$ then, it is obvious that,
$F\left( x,y\right) =0.$ By the other hand, we have for $0<t\leq1$ :%
\begin{align}
0 & <\arg\left( x+ity\right) <\arg\left( x-1+ity\right) <\pi
\label{eq arg}\\
& \Longrightarrow0<\arg\left( 1-\frac{1}{x+ity}\right) <\pi\Longrightarrow
\arg\sqrt{1-\frac{1}{x+ity}}\in\left] 0,\frac{\pi}{2}\right[ \nonumber\\
& \Longrightarrow\Im\sqrt{1-\frac{1}{x+ity}}>0\Longrightarrow G\left(
x,y\right) <0.\nonumber
\end{align}
Hence, $\left( F+G\right) \left( x,y\right) <0$ which proves that $\left(
x,y\right) \notin\Sigma.$
Let us prove now that $\Sigma$ is a curve in the set
\[
\Pi=\left\{ \left( x,y\right) ;x>1,y>0\right\} .
\]
We have
\[
\frac{\partial F}{\partial x}\left( x,y\right) =\sqrt{\frac{y^{2}\left(
x-1\right) }{x}}+\int_{1}^{x}\frac{\left( x-u\right) \left( u-1\right)
}{\sqrt{\left( \left( u-x\right) ^{2}+y^{2}\right) \left( u-1\right) u}%
}dt>0.
\]
By the other hand, with $u_{t}=x+ity,$ $t\in\left[ 0,1\right] ,$ we get
\begin{align*}
\frac{\partial G}{\partial x}\left( x,y\right) & =\frac{\partial}{\partial
x}\left[ \Re\left( \int_{0}^{1}iy^{2}\sqrt{1-t^{2}}\sqrt{1-\frac{1}{u_{t}}%
}dt\right) \right] \\
& =-\int_{0}^{1}\frac{y^{2}\sqrt{1-t^{2}}}{2}\Im\left( \frac{1}{u_{t}%
^{2}\sqrt{1-\frac{1}{u_{t}}}}\right) dt
\end{align*}
It is sufficient to prove that,
\[
\forall t\in\left[ 0,1\right] ,\Im\left( \frac{1}{u_{t}^{2}\sqrt{1-\frac
{1}{u_{t}}}}\right) \leq0,
\]
which is equivalent to prove that,%
\[
\forall t\in\left[ 0,1\right] ,\arg\left( \frac{1}{u_{t}^{2}\sqrt
{1-\frac{1}{u_{t}}}}\right) \in\left[ \pi,2\pi\right[ ,
\]
where the argument is taken in $\left[ 0,2\pi\right[ $. It follows from
(\ref{eq arg}) that for any $t\in\left[ 0,1\right] $ :%
\[
\arg\left( \frac{1}{u_{t}^{2}\sqrt{1-\frac{1}{u_{t}}}}\right) =2\pi-\left(
\frac{3}{2}\arg\left( u_{t}\right) +\frac{1}{2}\arg\left( u_{t}-1\right)
\right) \in\left] \pi,2\pi\right[ .
\]
We deduce that for any $0\leq t\leq1,$ $\Im\left( \frac{1}{u_{t}^{2}%
\sqrt{1-\frac{1}{u_{t}}}}\right) \leq0,$ and then $\frac{\partial G}{\partial
x}\left( x,y\right) \geq0.$ Finally, we just proved that
\[
\frac{\partial\left( F+G\right) }{\partial x}\left( x,y\right)
\neq0,\left( x,y\right) \in\Sigma\cap\Pi.
\]
We conclude that the set $\Sigma$ is a curve in $%
\mathbb{C}
$ by $\allowbreak$applying the Implicit Function Theorem to the function $F+G$.
\end{proof}
\begin{proof}
[Proof of Lemma \ref{asympt of the curve}]Let us put $z=re^{ix}\in
\Sigma,r>1,x\in\left[ 0,\frac{\pi}{2}\right] .$ With the change of variable
$t=sre^{ix}$, we get%
\[
\Re\left( e^{2ix}\int_{0}^{1}\sqrt{\frac{\left( s-\frac{1}{r}e^{-ix}\right)
\left( s-1\right) \left( s-e^{-2ix}\right) }{s}}ds\right) =0.
\]
Taking the limits when $r\rightarrow\infty,$ we get%
\begin{equation}
0=\Re\int_{0}^{1}e^{2ix}\sqrt{\left( s-1\right) \left( s-e^{-2ix}\right)
}. \label{0}%
\end{equation}
We see that $x\neq0;$ suppose that $x\neq\frac{\pi}{2}.$ With the change of
variable $t=\alpha u+\beta,$ where%
\[
\beta=\frac{1+e^{-2ix}}{2},\alpha=i\frac{1-e^{-2ix}}{2},
\]
(\ref{0}) gives%
\begin{align*}
0 & =\Re\left( \int_{\cot x}^{i}\sqrt{u^{2}+1}du\right) =\Re\left(
\int_{\cot x}^{0}\sqrt{u^{2}+1}du+\int_{0}^{i}\sqrt{u^{2}+1}du\right) \\
& =\Re\left( \int_{\cot x}^{0}\sqrt{u^{2}+1}du\right) >0;
\end{align*}
a contradiction.
The Laurent serie of $\sqrt{\frac{\left( t-1\right) \left( t-z\right)
\left( t-\overline{z}\right) }{t}}$ when $t\rightarrow1$ is :%
\[
\sqrt{\frac{\left( t-1\right) \left( t-z\right) \left( t-\overline
{z}\right) }{t}}=\left\vert z-1\right\vert \allowbreak\sqrt{t-1}+o\left(
\left( t-1\right) ^{\frac{1}{2}}\right) .
\]
We conclude that
\[
0=\lim\limits_{z\rightarrow1,z\in\Sigma^{+}}\Re\int_{1}^{z}\sqrt{\frac{\left(
t-1\right) \left( t-z\right) \left( t-\overline{z}\right) }{t}}%
dt=\frac{2}{3}\left\vert z-1\right\vert \Re\left( z-1\right) ^{\frac{3}{2}%
},
\]
and then%
\[
\arg\left( z-1\right) ^{\frac{3}{2}}\equiv\frac{\pi}{2}\operatorname{mod}%
\left( \pi\right) ,
\]
which finishes the proof.
\end{proof}
\begin{proof}
[Proof of Proposition \ref{main}]It is clear that the segment $\left[
0,1\right] $ is always a short trajectory of $\varpi_{q}$. If $a\notin%
\Sigma,$ then, from (\ref{cond necess})there is no short trajectory connecting
$a$ to $0$ or $1.$ From Lemma \ref{at infinity}, at most two critical
trajectories emanating from $a$ can diverge to $\infty$ in the upper
half-plane $%
\mathbb{C}
^{+}.$ By consideration of symmetry with respect to the real axis, at list one
critical trajectory emanating from $a$ meets a critical trajectory emanating
from $\overline{a},$ at some point $b\in%
\mathbb{R}
\setminus\left[ 0,1\right] .$ Since $b$ cannot be a zero of the quadratic
differential $\varpi_{q},$ we conclude that these two critical trajectories
form a short one.
If $a\in\Sigma$, and no short trajectory connecting $a$ to $z=1,$ then, there
exist two critical trajectories $\gamma_{a}$ and $\gamma_{1}$ emanating
respectively from $a$ and $1$ and diverging to infinity in a same direction
$D_{k}$. \bigskip From the behaviour of orthogonal trajectories at $\infty,$
we can take an orthogonal trajectory $\sigma$ that hits $\gamma_{1}$ and
$\gamma_{a}$ respectively in two points $b$ and $c$ (there are infinitely many
such orthogonal trajectories $\sigma$ ). We consider a path $\gamma$
connecting $z=1$ and $a$ formed by the part of $\gamma_{1}$ from $z=1$ to $b,$
the part of $\sigma$ from $b$ to $c,$ and the part of $\gamma_{a}$ from $c$ to
$a.$ Then
\begin{align*}
\Re\int_{\gamma}\sqrt{p\left( t\right) }dt & =\Re\int_{1}^{b}%
\sqrt{p\left( t\right) }dt+\Re\int_{b}^{c}\sqrt{p\left( t\right) }%
dt+\Re\int_{c}^{a}\sqrt{p\left( t\right) }dt\\
& =\Re\int_{b}^{c}\sqrt{p\left( t\right) }dt\neq0,
\end{align*}
which contradicts the fact $a\in\Sigma.$
\end{proof}
\begin{proof}
[Proof of Proposition \ref{connection}]The fact that the support of $\nu$ is
formed by horizontal trajectories of the quadratic differential (\ref{qd2}) is
classic and it is based on the so-called Plemelj-Sokhotsky Formula. For more
details, we refer the reader to \cite{amf rakh1},\cite{pritsker}%
,\cite{Shapiro},\cite{bullgard}...
Since the Cauchy transform is a single-valued function in $%
\mathbb{C}
\setminus$\emph{supp}$\left( \nu\right) ,$ then, the support of $\nu$ should
include all the singular points (finite critical points) of the quadratic
differential (\ref{qd2}). But, $\allowbreak$the horizontal trajectories that
contain all finite critical points are exactly short trajectories. The measure
$\nu$ is absolutely continuous with respect to the linear Lebesgue measure,
and it is given on its support (with an adequate orientation) by the
expression :
\[
d\nu\left( t\right) =\frac{1}{8i\pi}\left( \sqrt{\frac{\Delta\left(
t\right) }{t}}\right) _{+}dt.
\]
It is easy to check that the Cauchy transform of $\nu$ satisfies
(\ref{algeq}), indeed :
\begin{align*}
C_{\nu}\left( z\right) & =\frac{1}{2i\pi}\int\frac{\left( \sqrt
{\frac{\Delta\left( t\right) }{t}}\right) _{+}}{4\left( z-t\right)
}dt=\frac{1}{4i\pi}\oint\frac{\sqrt{\frac{\Delta\left( t\right) }{t}}%
}{4\left( z-t\right) }dt\\
& =\frac{1}{4}\left( res_{z}\frac{\sqrt{\frac{\Delta\left( t\right) }{t}}%
}{z-t}-res_{\infty}\frac{\sqrt{\frac{\Delta\left( t\right) }{t}}}%
{z-t}\right) \\
& =\frac{-\sqrt{\frac{\Delta\left( z\right) }{z}}+\gamma+2z}{4}%
=\frac{-\sqrt{z\Delta\left( z\right) }+\gamma z+2z^{2}}{4z},
\end{align*}
where the path of integration in the first integral is formed by the two short
trajectories, and, in the second integral is a closed contour including the
two short trajectories and far away from $z.$
\end{proof}
\begin{acknowledgement}
\bigskip\ This work was partially supported by the laboratory research
"Mathematics and Applications", Faculty of sciences of Gab\`{e}s. Tunisia.
\end{acknowledgement}
\section{Introduction}
This paper is a continuation of a collection of works about applications of
the theory of quadratic differentials in quantum theory. In fact, quadratic
differentials have provided an important tool in the asymptotic study of some
solutions of algebraic equations. In quantum theory, trajectories of some
quadratic differentials have crucial role in the WKB analysis.
We consider the eigenvalue problem
\begin{equation}
-y^{\prime\prime}+V_{s,m}(x)y=\lambda y,\text{ } \label{sch eq}%
\end{equation}
where the potential $V_{s,J}(x)$ is given by :%
\[
V_{s,m}(x)=\frac{(4s-1)(4s-3)}{x^{2}}+(x^{6}-(4s+4m-2)x^{2}).
\]
This problem was studied by A.Turbiner \cite{ref01},\cite{ref3}. For
$s\in\left] \frac{1}{4},\frac{3}{4}\right[ $, there is an attractive
centrifugal term. For $s\in\left] -\infty,\frac{1}{4}\right[ \cup$ $\left]
\frac{3}{4},+\infty\right[ ,$ the centrifugal term is repulsive. For
$s=\frac{1}{4}$ or $\frac{3}{4},$ the centrifugal core term disappears leaving
a sextic oscillator
\[
V_{p,m}(x)=\text{ }x^{6}-(4m+p)x^{2};\text{ }p\in\left\{ -1,1\right\}
\]
We will study the case $s=\frac{1}{4}$ (so $p=-1$), with a perturbed
potential
\[
V_{-1,m+1}(x)=V_{m}(x)=\text{ }x^{6}+\gamma m^{1/2}x^{4}+(\frac{\gamma^{2}%
m}{4}-(4m+3))x^{2},\text{ }%
\]
with boundray condition $y(\pm\infty)=0.$This problem is exactly solvable:
that means that for every $\gamma$ there exist $m+1$ eigenfunctions of the
form
\[
\phi(x)=q_{m}(x)e^{-\frac{x^{4}}{4}-\frac{\gamma m^{1/2}}{4}x^{2}},
\]
where $q_{m}$ is an even-parity polynomial of degree $2m$, corresponding to
$m+1$ eigenvalues $\lambda_{m};$ see \cite{ref2}. Note that in the case $p=1$
we have the same form of eigenfunctions with an odd-parity polynomials; for
more details see again \cite{ref2}.
Substitiuting in the Schr\"{o}dinger equation (\ref{sch eq}), we find that
\[
-q_{m}^{\prime\prime}(x)+(2x^{3}+\gamma m^{\frac{1}{2}}x)q_{m}^{\prime
}(x)-(4mx^{2}-\frac{\gamma m^{1/2}}{2})q_{m}(x)=\lambda_{m}q_{m}(x).
\]
The differential operator%
\[
K=-\frac{d^{2}}{dx^{2}}+(2x^{3}+\gamma m^{1/2}x)\frac{d}{dx}-(4mx^{2}%
-\frac{\gamma m^{1/2}}{2})
\]
preserves the $(m+1)$-dimentional linear space of all even polynomials of
degree $\leq2m$. Using $z=x^{2},$ we find that
\begin{equation}
-4zq_{m}^{\prime\prime}(z)+(4z^{2}+2\gamma m^{1/2}z-2)q_{m}^{\prime
}(z)-(4mz-\frac{\gamma m^{1/2}}{2})q_{m}(z)=\lambda_{m}q_{m}(z). \label{eq2}%
\end{equation}
Our main goal is the study of the asymptotic root-counting measure
$\vartheta_{m}$ of an appropriate re-scaled polynomial sequence $\left\{
Q_{m}(z)=q_{m}(m^{\varepsilon}z)\right\} .$ Let $\nu_{m}$ be the normalized
root-counting measure of the sequence $\left( q_{m}\right) $. The Cauchy
transform $C_{\nu_{m}}$ and $C_{\vartheta_{m}}$ of respectively $\nu_{m}$ and
$\vartheta_{m}$ satisfies the following equations :
\[
4zmC_{\nu_{m}}^{2}(z)-(4z^{2}+2\gamma m^{1/2}z-2)C_{\nu_{m}}(z)+\frac
{(4mz-\frac{\gamma m^{1/2}}{2}+\lambda_{m})}{m}+4zC_{\nu_{m}}^{\prime}(z)=0.
\]%
\[
4zm^{1-\varepsilon}C_{\vartheta_{m}}^{2}(z)-(4m^{\varepsilon}z^{2}+2\gamma
m^{1/2}z-2m^{-\varepsilon})C_{\vartheta_{m}}(z)+\frac{(4m^{1+\varepsilon
}z-\frac{\gamma m^{1/2}}{2}+\lambda_{m})}{m}+4m^{-\varepsilon}zC_{\vartheta
_{m}}^{\prime}(z)=0.
\]
For $\varepsilon=1/2,$ we get
\begin{equation}
-4zC_{\vartheta_{m}}^{2}(z)+(4z^{2}+2\gamma z-\frac{2}{m})C_{\vartheta_{m}%
}(z)-(4z-\frac{\gamma}{2m}+\frac{\lambda m}{m^{3/2}})-4z\frac{C_{\vartheta
_{m}}^{\prime}(z)}{m}=0. \label{eqdiff1}%
\end{equation}
It was shown in \cite{ref4}, that the sequence $\left( \lambda_{m}%
/m^{4/3}\right) $ is bounded. By the Helly selection Theorem, we may assume
that
\[
\underset{m\rightarrow\infty}{\lim}\left( \lambda_{m}/m^{4/3}\right)
=\delta,
\]
and then, there exists a compactly-supported positive measure $\nu$ such that
\[
\underset{m\rightarrow\infty}{\lim}\vartheta_{m}=\nu,\underset{m\rightarrow
\infty}{\lim}C_{\vartheta_{m}}\mathcal{=C}_{\nu}=\mathcal{C}.
\]
Finally, taking the limits in (\ref{eqdiff1}), we obtain the algebraic
equation :%
\begin{equation}
z\mathcal{C}^{2}(z)-(z^{2}+\frac{\gamma}{2}z)\mathcal{C}(z)+(z+\frac{\delta
}{4})=0. \label{algeq}%
\end{equation}
\bigskip In this paper, we discuss the existence of solutions of equation
(\ref{algeq}) as Cauchy transform of compactly-supported signed measure.
Section \ref{connection alg}, we make the connection between this a algebraic
equation and a particular quadratic differential. In section \ref{qudar dif},
we describe the critical graph of the related quadratic differential in the
Riemann sphere $\overline{%
\mathbb{C}
},$ more precisely, we discuss the number of its finite critical trajectories.
\section{\bigskip A quadratic differential \label{qudar dif}}
\bigskip Below, we describe the critical graphs of the the family of quadratic
differentials
\begin{equation}
\varpi_{q}=-\frac{q\left( z\right) }{z}dz^{2}, \label{qd}%
\end{equation}
where $q$ is a monic polynomial of degree $3.$ We begin our investigation by
some immediate observations from the theory of quadratic differentials. For
more details, we refer the reader to \cite{Strebel},\cite{jenkins},.
Recall that \emph{finite critical points }of a given meromorphic quadratic
differential $-Q\left( z\right) dz^{2}$ on the Riemann sphere $\overline{%
\mathbb{C}
}$ are its zeros and simple poles; poles of order $2$ or greater then 1 called
\emph{infinite critical points. }All other points of $\overline{%
\mathbb{C}
}$ are called \emph{regular points}.
\emph{Horizontal trajectories} (or just trajectories) of the quadratic
differential are the zero loci of the equation%
\[
-Q\left( z\right) dz^{2}>0,
\]
or equivalently%
\begin{equation}
\mathcal{\Re}\int^{z}\sqrt{Q\left( t\right) }\,dt=\text{\emph{const}}.
\label{eq traj}%
\end{equation}
If $z\left( t\right) ,t\in%
\mathbb{R}
$ is a horizontal trajectory, then the function
\[
t\longmapsto\Im\int^{t}\sqrt{Q\left( z\left( u\right) \right) }z^{\prime
}\left( u\right) du
\]
is monotone.
The \emph{vertical} (or, \emph{orthogonal}) trajectories are obtained by
replacing $\Im$ by $\Re$ in equation (\ref{eq traj}). The horizontal and
vertical trajectories produce two pairwise orthogonal foliations of the
Riemann sphere $\overline{%
\mathbb{C}
}$.
A trajectory passing through a critical point is called \emph{critical
trajectory}. In particular, if it starts and ends at a finite critical point,
it is called \emph{finite critical trajectory }or\emph{\ short trajectory},
otherwise, we call it an \emph{infinite critical trajectory}. A short
trajectory is called \emph{unbroken }if it does not pass through any finite
critical points except its two endpoints. The closure the set of finite and
infinite critical trajectories is called the \emph{critical graph}.
A necessary condition for the existence of a short trajectory connecting
finite critical points is the existence of a Jordan arc $\gamma$ connecting
them, such that
\begin{equation}
\Re\int_{\gamma}\sqrt{Q\left( t\right) }dt=0. \label{cond necess}%
\end{equation}
However, this condition is sufficient in general; see counter-example in
\cite{F.Thabet}.
The local structure of the trajectories is as follow :
\begin{itemize}
\item At any regular point, horizontal (resp. vertical) trajectories look
locally as simple analytic arcs passing through this point, and through every
regular point passes a uniquely determined horizontal (resp. vertical)
trajectory; these horizontal and vertical trajectories are locally orthogonal
at this point.
\item From each zero of multiplicity $r$, there emanate $r+2$ critical
trajectories spacing under equal angle $2\pi/\left( r+2\right) $.
\item At a simple pole, there emanates exactly one horizontal trajectory.
\item At the pole of order $r>2$, there are $r-2$ asymptotic directions
(called \emph{critical directions}) spacing under equal angle $2\pi/\left(
r-2\right) $ and a neighborhood $\mathcal{U}$, such that each trajectory
entering $\mathcal{U}$ stays in $\mathcal{U}$ and tends to the pole in one of
the critical directions. See Figure \ref{FIG1}.
\end{itemize}
\begin{figure}[tbh]
\begin{minipage}[b]{0.28\linewidth}
\centering\includegraphics[scale=0.25]{1.pdf}
\end{minipage}\hfill
\begin{minipage}[b]{0.28\linewidth} \includegraphics[scale=0.25]{2.pdf}
\end{minipage}
\hfill\begin{minipage}[b]{0.28\linewidth} \includegraphics[scale=0.3]{3.pdf}
\end{minipage}
\caption{Structure of the trajectories near a simple zero (left), a simple
pole (center), and a pole of order 6 (right).}%
\label{FIG1}%
\end{figure}
\bigskip A very helpful tool that will be used in our investigation is the
Teichm\"{u}ller lemma (see \cite[Theorem 14.1]{Strebel}).
\begin{definition}
\bigskip A domain in $\overline{%
\mathbb{C}
}$ bounded only by segments of horizontal and/or vertical trajectories of
$\varpi_{q}$ (and their endpoints) is called $\varpi_{q}$-polygon.
\end{definition}
\begin{lemma}
[Teichm\H{u}ller]\label{teich lemma} Let $\Omega$ be a $\varpi_{q}$-polygon,
and let $z_{j}$ be the critical points on the boundary $\partial\Omega$ of
$\Omega,$ and let $\theta_{j}$ be the corresponding interior angles with
vertices at $z_{j},$ respectively. Then%
\begin{equation}
\sum\left( 1-\dfrac{\left( n_{j}+2\right) \theta_{j}}{2\pi}\right) =2+\sum
m_{i}, \label{Teich equality}%
\end{equation}
where $n_{j}$ are the multiplicities of $z_{j}=1,$ and $m_{i}$ the
multiplicities of critical points inside $\Omega.$\bigskip
\end{lemma}
\bigskip We will focus on the case where
\[
q\left( z\right) =q_{a}\left( z\right) =\left( z-1\right) \left(
z-a\right) \left( z-\overline{a}\right) ,
\]
with
\[
a\in%
\mathbb{C}
^{+}=\left\{ a\in%
\mathbb{C}
\mid\Im\left( a\right) >0\right\} .
\]
We have the following immediate observations :
\begin{itemize}
\item The finite critical points of $\varpi_{q}$ are $1,a,\overline{a}$ as
simple zeros, and $-1$ as a simple pole.
\item With the parametrization $u=1/z$, we get
\[
\varpi_{q}\left( u\right) =\left( -\frac{1}{u^{6}}+\mathcal{O}\left(
\frac{1}{u^{5}}\right) \right) du^{2},\text{ }u\longrightarrow0,
\]
thus, infinity is an infinite critical point of $\varpi_{q},$ as a pole of
order $6.$
\item Since $\infty$ is the only infinite critical point of $\varpi_{q},$ any
critical trajectory which is not finite diverges to $\infty$ following one of
the 4 directions :
\[
D_{k}=\left\{ z\in%
\mathbb{C}
\mid\arg\left( z\right) =\left( 2k+1\right) \frac{\pi}{4}\right\}
,k=0,1,2,3.
\]
The same thing happen to the orthogonal trajectories at $\infty$, but the
critical directions are :%
\[
D_{k}^{\perp}=\left\{ z\in%
\mathbb{C}
:\arg\left( z\right) =\frac{k\pi}{2}\right\} ;k=0,1,2,3.
\]
Observe that if two trajectories diverge to $\infty$ in a same direction
$D_{k},$ then there exists a neighborhood $\mathcal{V}$ of $\infty$, such that
any orthogonal trajectory traversing $D_{k}$ in $\mathcal{V},$ must traverse
these two trajectories.
\item \bigskip Since the quadratic differential $\varpi_{q}$ has two poles,
Jenkins Three-pole Theorem (see \cite[Theorem 15.2]{Strebel}) asserts that the
situation of the so-called recurrent trajectory (whose closure might be dense
in some domain in $%
\mathbb{C}
$) cannot happen.
\end{itemize}
\bigskip
\begin{lemma}
\label{at infinity}Two critical trajectories of $\varpi_{q}$ emanating from
the same zero cannot diverge to $\infty$ in the same direction.
\end{lemma}
\begin{lemma}
\label{residue}For any Jordan arc $\gamma$ connecting $a$ and $\overline{a}$
in $%
\mathbb{C}
\setminus\left[ 0,1\right] $ we have%
\[
\Re\int_{\gamma}\sqrt{\frac{\left( z-1\right) \left( z-a\right) \left(
z-\overline{a}\right) }{z}}dz=0,
\]
and then, condition (\ref{cond necess}) is fulfilled.
\end{lemma}
We consider the set
\[
\Sigma=\left\{ z\in%
\mathbb{C}
:\Re\int_{0}^{z}\sqrt{\frac{\left( t-1\right) \left( t-z\right) \left(
t-\overline{z}\right) }{t}}dt=0\right\}
\]
\begin{lemma}
\label{curve}The set $\Sigma$ is symmetric with respect to the real axis, and
it is formed by $3$ Jordan arcs :
\begin{itemize}
\item the segment $\left[ 0,1\right] ;$
\item two curves $\Sigma^{\pm}$ emerging from $z=1,$ and diverging
respectively to infinity in $%
\mathbb{C}
_{\pm}.$
\end{itemize}
\end{lemma}
\begin{figure}[th]
\centering\includegraphics[height=2in,width=3in]{4.pdf}\caption{Approximate
plot of the curve $\Sigma$}%
\label{FIG2}%
\end{figure}
We give here the behavior of $\Sigma$ at $z=1$ and at the infinity:
\begin{lemma}
\label{asympt of the curve}The following results hold :
\begin{align*}
\lim\limits_{z\rightarrow\infty,z\in\Sigma\cap%
\mathbb{C}
^{+}}\arg\left( z\right) & =\frac{\pi}{2},\\
\lim\limits_{z\rightarrow1,z\in\Sigma\cap%
\mathbb{C}
^{+}}\arg\left( z\right) & =\frac{\pi}{3}.
\end{align*}
\end{lemma}
From Lemma \ref{curve}, $\Sigma$ splits $%
\mathbb{C}
$ into 2 connected domains :
\begin{itemize}
\item $\Omega_{1}$ limited by $\Sigma^{\pm}$ and containing $z=2;$
\item $\Omega_{2}=%
\mathbb{C}
\setminus$ $\left( \Omega_{1}\cup\Sigma^{\pm}\cup\left[ 0,1\right] \right)
.$ See Figure \ref{FIG2}.
\end{itemize}
\begin{proposition}
\label{main}For any complex number $a,$ the quadratic differential $\varpi
_{q}$ has :
\begin{itemize}
\item two short trajectories if $a\in\Omega_{1}$: the segment $\left[
0,1\right] $ and another one connecting $a$ and $\overline{a}$ in $\Omega
_{1};$see Figure \ref{FIG3};
\item three short trajectories if $a\in\Sigma^{\pm}$ : the segment $\left[
0,1\right] $ and two others connecting $z=1$ with $a$ and $\overline{a};$ see
Figure \ref{FIG4};
\item two short trajectories if $a\in\Omega_{2}$ : the segment $\left[
0,1\right] $ and another one connecting $a$ and $\overline{a}$ in $\Omega
_{2};$see Figure \ref{FIG5}.
\end{itemize}
\end{proposition}
\begin{figure}[tbh]
\begin{minipage}[b]{0.48\linewidth}
\centering\includegraphics[scale=0.5]{5.pdf}
\end{minipage}\hfill
\begin{minipage}[b]{0.48\linewidth} \includegraphics[scale=0.5]{6.pdf}
\end{minipage}
\caption{Critical graphs when $a\in\Omega_{1},$ here $a=1.6+2i$ (left) and
$a=1.8+2i$ (right).}%
\label{FIG3}%
\end{figure}
\begin{figure}[th]
\centering\includegraphics[height=2in,width=3in]{7.pdf}\caption{Critical graph
when $a$ $\in\Sigma,$ here $a=1.55+2i.$}%
\label{FIG4}%
\end{figure}
\begin{figure}[tbh]
\begin{minipage}[b]{0.48\linewidth}
\centering\includegraphics[scale=0.5]{8.pdf}
\end{minipage}\hfill
\begin{minipage}[b]{0.48\linewidth} \includegraphics[scale=0.5]{9.pdf}
\end{minipage}
\caption{Critical graphs when $a\in\Omega_{2},$ here $a=0.5+2i$ (left) and
$a=2i$ (right).}%
\label{FIG5}%
\end{figure}
\begin{remark}
The case when $q=\prod_{i=1}^{3}\left( z-a_{i}\right) \in%
\mathbb{R}
\left[ X\right] ,$ with real zeros is quite simple; if the zeros are simple
: $a_{1}<a_{2}<a_{3},$ then the segments $\left[ a_{1},a_{2}\right] $ and
$\left[ a_{3},a_{4}\right] $ are two short trajectories of $\varpi_{q}$. See
Figure \ref{FIG6}.
\end{remark}
\begin{figure}[tbh]
\begin{minipage}[b]{0.48\linewidth}
\centering\includegraphics[scale=0.5]{10.pdf}
\end{minipage}\hfill
\begin{minipage}[b]{0.48\linewidth} \includegraphics[scale=0.5]{11.pdf}
\end{minipage}
\caption{Critical graphs for $\varpi_{q}$ when $q=\left( z-1\right) \left(
z-2\right) \left( z-3\right) $ (left) and $q=\left( z-1\right) \left(
z-2\right) ^{2}$ (right).}%
\label{FIG6}%
\end{figure}\bigskip\bigskip
\section{Connection with the algebraic equation \label{connection alg}}
\bigskip The Cauchy transform $\mathcal{C}_{\nu}$ of a compactly supported
Borelian complex measure $\nu$ is defined in $%
\mathbb{C}
\setminus$\emph{supp}$\left( \nu\right) $ by :
\[
\mathcal{C}_{\nu}\left( z\right) =\int_{%
\mathbb{C}
}\frac{d\nu\left( t\right) }{z-t}.
\]
It satisfies
\[
\mathcal{C}_{\nu}\left( z\right) =\frac{\nu\left(
\mathbb{C}
\right) }{z}+\mathcal{\allowbreak O}\left( z^{-2}\right) ,z\rightarrow
\infty,
\]
and the inversion formula (which should be understood in the distributions
sense) :%
\[
\nu=\frac{1}{\pi}\frac{\partial\mathcal{C}_{\nu}}{\partial\overline{z}}.\text{
}%
\]
In particular, the normalized root-counting measure $\nu_{n}=\nu(P_{n})$ of a
complex polynomial $P_{n}$ of a complex polynomial $P_{n}$ of degree $n$ is
defined in $%
\mathbb{C}
$ by :
\[
\nu_{n}=\frac{_{1}}{n}\sum\limits_{P_{n}\left( a\right) =0}\delta_{a},\text{
(each zero is counted with its multiplicity);}%
\]
the Cauchy transform of $\nu_{n}$ is :
\[
\mathcal{C}_{\nu_{n}}(z)=\int_{%
\mathbb{C}
}\frac{d\nu_{n}\left( t\right) }{z-t}=\frac{P_{n}^{^{\prime}}(z)}{nP_{n}%
(z)};P_{n}(z)\neq0.
\]
Let us come back to the algebraic equation (\ref{algeq}). We are seeking a
compactly-supported signed measure $\nu$ such that, its Cauchy transform
$\mathcal{C}_{\nu}$ satisfies almost everywhere in $%
\mathbb{C}
$ equation (\ref{algeq}).
With the choice of the square root of the discriminant%
\[
\Delta\left( z\right) =\frac{z}{4}\left( 4z^{3}+4\gamma z^{2}+\left(
\gamma^{2}-16\right) z-16\delta\right)
\]
of the quadratic equation (\ref{algeq}) (as a quadratic equation) with
condition
\[
\sqrt{\Delta\left( z\right) }\sim z^{2},z\rightarrow\infty,
\]
it is easy to check that independently of the complex numbers $\gamma$ and
$\delta$, we have :
\[
\mathcal{C}(z)=\frac{2z^{2}+\gamma z-2\sqrt{\Delta\left( z\right) }}%
{4z}=\frac{1}{z}+\mathcal{\allowbreak O}\left( z^{-2}\right) ,z\rightarrow
\infty,
\]
which let us be hopeful for the existence of the measure $\nu.$
The following Lemma gives a sufficient condition on a solution of
(\ref{algeq}) to be the Cauchy transform of some compactly supported measure
in $%
\mathbb{C}
$ :
\begin{lemma}
[{\cite[comp. Th. 1.2, Ch. II,]{garnet}}]Suppose $f\in L_{loc}^{1}\left(
\mathbb{C}
\right) $ and that $f(z)\rightarrow0$ as $z\rightarrow\infty$ and let $\mu$
be a compactly-supported measure in $%
\mathbb{C}
$ such that%
\[
\mu=\frac{1}{\pi}\frac{\partial f}{\partial\overline{z}}%
\]
in the sense of distributions. Then $f(z)=\mathcal{C}_{\mu}\left( z\right) $
almost everywhere in $%
\mathbb{C}
.$
\end{lemma}
The following Proposition gives a necessary condition the existence of
measures $\nu$ :
\begin{proposition}
\label{connection}Let us consider the quadratic differential
\begin{equation}
-\frac{\Delta\left( z\right) }{z^{2}}dz^{2}. \label{qd2}%
\end{equation}
If the signed measure $\nu$ exists, then, the quadratic differential
(\ref{qd2}) has two short trajectories, and, the support of $\nu$ coincides
with these short trajectories. In particular, if $\Delta\left( z\right) $ is
a real polynomial, then the problem of finding the measure $\nu$ is solved.
\end{proposition}
\section{Proofs}
\begin{proof}
[Proof of Lemma \ref{at infinity}]Suppose that $\gamma_{1}$ and $\gamma_{2}$
are two such trajectories emanating from the zero $a$ or $1$, spacing with
angle $\theta\in\left\{ 2\pi/3,4\pi/3\right\} .$ Consider the $\varpi_{q}%
$-polygon with edges $\gamma_{1}$ and $\gamma_{2},$ and vertices $z_{j},$ and
infinity. The right side of (\ref{Teich equality}) can take only the values
$0$ or $-1,$ while the left side is at list $2;$ a contradiction. (Observe the
$\varpi_{qp}$-polygon cannot contain the pole $z=0,$ otherwise it contains
$z=1$ and, again we get a contradiction with (\ref{Teich equality}%
).\bigskip\bigskip
\end{proof}
\begin{proof}
[Proof of Lemma \ref{residue}]Since $\frac{q\left( t\right) }{t}$ is a real
rational fraction, then
\begin{equation}
\overline{\sqrt{\frac{q\left( t\right) }{t}}}=\sqrt{\frac{q\left(
\overline{t}\right) }{\overline{t}}},t\neq0, \label{ration symm}%
\end{equation}
and we get, after the change of variable $u=\overline{t}$ in second integral
:
\begin{align*}
\Re\left( \int_{\overline{z}}^{z}\sqrt{\frac{q\left( t\right) }{t}%
}dt\right) & =\Re\left( \int_{1}^{z}\sqrt{\frac{q\left( t\right) }{t}%
}dt-\int_{1}^{\overline{z}}\sqrt{\frac{q\left( t\right) }{t}}dt\right) \\
& =\Re\left( \int_{1}^{z}\sqrt{\frac{q\left( t\right) }{t}}dt-\overline
{\int_{1}^{z}\sqrt{\frac{q\left( t\right) }{t}}dt}\right) \\
& =\Re\left( 2i\Im\left( \int_{1}^{z}\sqrt{\frac{q\left( t\right) }{t}%
}dt\right) \right) \\
& =0.
\end{align*}
Let us give a necessary condition to get two short trajectories joining two
different pairs of finite critical points of $\varpi_{q}$ in the general case
:
\[
\frac{q\left( z\right) }{z}=\frac{z^{3}+\alpha z^{2}+\beta z+\gamma}%
{z}=\frac{\left( z-a\right) \left( z-b\right) \left( z-c\right) }%
{z},a,b,c\in%
\mathbb{C}
.
\]
Consider two disjoint oriented Jordan arcs $\gamma_{1}$ and $\gamma_{2}$
connecting two distinct pairs of zeros. We define the single-valued function
$\sqrt{\frac{q\left( z\right) }{z}}$ in $%
\mathbb{C}
\setminus\left( \gamma_{1}\cup\gamma_{2}\right) $ with condition
$\sqrt{\frac{q\left( z\right) }{z}}\sim z,z\rightarrow\infty.$ For
$s\in\gamma_{1}\cup\gamma_{2},$ we denote by $\left( \sqrt{p\left( s\right)
}\right) _{+}$ and $\left( \sqrt{p\left( s\right) }\right) _{-}$the
limits from the $+$ and $-$ sides, respectively. (As usual, the $+$ side of an
oriented curve lies to the left and the $-$ side lies to the right, if one
traverses the curve according to its orientation.)
From the Laurent expansion at $\infty$ of $\sqrt{q\left( z\right) }:$%
\[
\sqrt{\frac{q\left( z\right) }{z}}=\allowbreak z+\frac{\alpha}{2}-\left(
\frac{\alpha^{2}-4\beta}{8z}\right) +\allowbreak\mathcal{O}\left(
z^{-2}\right) ,
\]
we deduce the residue%
\[
res_{\infty}\left( \sqrt{\frac{q\left( z\right) }{z}}\right) =\frac{1}%
{8}\allowbreak\left( \alpha^{2}-4\beta\right) .
\]
Let%
\[
I=\int_{\gamma_{1}}\left( \sqrt{\frac{q\left( s\right) }{s}}\right)
_{+}ds+\int_{\gamma_{2}}\left( \sqrt{\frac{q\left( s\right) }{s}}\right)
_{+}ds.
\]
Since
\[
\left( \sqrt{\frac{q\left( s\right) }{s}}\right) _{+}=-\left( \sqrt
{\frac{q\left( s\right) }{s}}\right) _{-},s\in\gamma_{1}\cup\gamma_{2},
\]
we have
\[
2I=\int_{\gamma_{1}\cup\gamma_{2}}\left[ \left( \sqrt{\frac{q\left(
s\right) }{s}}\right) _{+}-\left( \sqrt{\frac{q\left( s\right) }{s}%
}\right) _{-}\right] ds=\oint_{\Gamma_{q}}\sqrt{\frac{q\left( z\right)
}{z}}dz,
\]
where $\Gamma_{q}$ is a closed contours encircling the curves $\gamma_{1}$ and
$\gamma_{2}$. After the contour deformation, we pick up the residue at
$z=\infty,$ and we get
\begin{align*}
I & =\frac{1}{2}\oint_{\Gamma_{q}}\sqrt{\frac{q\left( z\right) }{z}}dz=\pm
i\pi res_{\infty}\left( \sqrt{p\left( z\right) }\right) \\
& =\pm\frac{\pi i}{8}\allowbreak\allowbreak\left( \alpha^{2}-4\beta\right)
\end{align*}
and the necessary condition is
\[
\Im\left( \allowbreak\alpha^{2}-4\beta\right) =0,
\]
which is satisfied for the case when $q$ is real.
\end{proof}
\begin{proof}
[Proof of Lemma \ref{curve}]It is clear that $\Sigma\cap%
\mathbb{R}
=\left[ 0,1\right] .$ The fact that $\Sigma$ is symmetric with respect to
the real axis follows from the observation (\ref{ration symm}). In order to
prove that $\Sigma$ is a curve, we consider the real functions $F$ and $G$
defined for $\left( x,y\right) $ in $%
\mathbb{C}
_{+}$ by:
\begin{align*}
F\left( x,y\right) & =\Re\left( \int_{0}^{x}\sqrt{\frac{\left( u-\left(
x+iy\right) \right) \left( u-\left( x-iy\right) \right) \left(
u-1\right) }{u}}du\right) \\
& =\Re\left( \int_{0}^{x}\sqrt{\frac{\left( \left( u-x\right) ^{2}%
+y^{2}\right) \left( u-1\right) }{u}}du\right) ;\\
G\left( x,y\right) & =\Re\left( \int_{x}^{x+iy}\sqrt{\frac{\left(
u-\left( x+iy\right) \right) \left( u-\left( x-iy\right) \right)
\left( u-1\right) }{u}}du\right) \\
& =-\int_{0}^{1}y^{2}\sqrt{1-t^{2}}\Im\sqrt{1-\frac{1}{x+ity}}dt.
\end{align*}
Observe that
\[
\Sigma=\left\{ \left( x,y\right) \in%
\mathbb{R}
^{2}\mid\left( F+G\right) \left( x,y\right) =0\right\} .
\]
We prove first that $\Sigma\setminus\left[ 0,1\right] \subset\left\{ z\in%
\mathbb{C}
\mid\Re z>1\right\} .$ If $x\leq1$ and $y>0,$ then, it is obvious that,
$F\left( x,y\right) =0.$ By the other hand, we have for $0<t\leq1$ :%
\begin{align}
0 & <\arg\left( x+ity\right) <\arg\left( x-1+ity\right) <\pi
\label{eq arg}\\
& \Longrightarrow0<\arg\left( 1-\frac{1}{x+ity}\right) <\pi\Longrightarrow
\arg\sqrt{1-\frac{1}{x+ity}}\in\left] 0,\frac{\pi}{2}\right[ \nonumber\\
& \Longrightarrow\Im\sqrt{1-\frac{1}{x+ity}}>0\Longrightarrow G\left(
x,y\right) <0.\nonumber
\end{align}
Hence, $\left( F+G\right) \left( x,y\right) <0$ which proves that $\left(
x,y\right) \notin\Sigma.$
Let us prove now that $\Sigma$ is a curve in the set
\[
\Pi=\left\{ \left( x,y\right) ;x>1,y>0\right\} .
\]
We have
\[
\frac{\partial F}{\partial x}\left( x,y\right) =\sqrt{\frac{y^{2}\left(
x-1\right) }{x}}+\int_{1}^{x}\frac{\left( x-u\right) \left( u-1\right)
}{\sqrt{\left( \left( u-x\right) ^{2}+y^{2}\right) \left( u-1\right) u}%
}dt>0.
\]
By the other hand, with $u_{t}=x+ity,$ $t\in\left[ 0,1\right] ,$ we get
\begin{align*}
\frac{\partial G}{\partial x}\left( x,y\right) & =\frac{\partial}{\partial
x}\left[ \Re\left( \int_{0}^{1}iy^{2}\sqrt{1-t^{2}}\sqrt{1-\frac{1}{u_{t}}%
}dt\right) \right] \\
& =-\int_{0}^{1}\frac{y^{2}\sqrt{1-t^{2}}}{2}\Im\left( \frac{1}{u_{t}%
^{2}\sqrt{1-\frac{1}{u_{t}}}}\right) dt
\end{align*}
It is sufficient to prove that,
\[
\forall t\in\left[ 0,1\right] ,\Im\left( \frac{1}{u_{t}^{2}\sqrt{1-\frac
{1}{u_{t}}}}\right) \leq0,
\]
which is equivalent to prove that,%
\[
\forall t\in\left[ 0,1\right] ,\arg\left( \frac{1}{u_{t}^{2}\sqrt
{1-\frac{1}{u_{t}}}}\right) \in\left[ \pi,2\pi\right[ ,
\]
where the argument is taken in $\left[ 0,2\pi\right[ $. It follows from
(\ref{eq arg}) that for any $t\in\left[ 0,1\right] $ :%
\[
\arg\left( \frac{1}{u_{t}^{2}\sqrt{1-\frac{1}{u_{t}}}}\right) =2\pi-\left(
\frac{3}{2}\arg\left( u_{t}\right) +\frac{1}{2}\arg\left( u_{t}-1\right)
\right) \in\left] \pi,2\pi\right[ .
\]
We deduce that for any $0\leq t\leq1,$ $\Im\left( \frac{1}{u_{t}^{2}%
\sqrt{1-\frac{1}{u_{t}}}}\right) \leq0,$ and then $\frac{\partial G}{\partial
x}\left( x,y\right) \geq0.$ Finally, we just proved that
\[
\frac{\partial\left( F+G\right) }{\partial x}\left( x,y\right)
\neq0,\left( x,y\right) \in\Sigma\cap\Pi.
\]
We conclude that the set $\Sigma$ is a curve in $%
\mathbb{C}
$ by $\allowbreak$applying the Implicit Function Theorem to the function $F+G$.
\end{proof}
\begin{proof}
[Proof of Lemma \ref{asympt of the curve}]Let us put $z=re^{ix}\in
\Sigma,r>1,x\in\left[ 0,\frac{\pi}{2}\right] .$ With the change of variable
$t=sre^{ix}$, we get%
\[
\Re\left( e^{2ix}\int_{0}^{1}\sqrt{\frac{\left( s-\frac{1}{r}e^{-ix}\right)
\left( s-1\right) \left( s-e^{-2ix}\right) }{s}}ds\right) =0.
\]
Taking the limits when $r\rightarrow\infty,$ we get%
\begin{equation}
0=\Re\int_{0}^{1}e^{2ix}\sqrt{\left( s-1\right) \left( s-e^{-2ix}\right)
}. \label{0}%
\end{equation}
We see that $x\neq0;$ suppose that $x\neq\frac{\pi}{2}.$ With the change of
variable $t=\alpha u+\beta,$ where%
\[
\beta=\frac{1+e^{-2ix}}{2},\alpha=i\frac{1-e^{-2ix}}{2},
\]
(\ref{0}) gives%
\begin{align*}
0 & =\Re\left( \int_{\cot x}^{i}\sqrt{u^{2}+1}du\right) =\Re\left(
\int_{\cot x}^{0}\sqrt{u^{2}+1}du+\int_{0}^{i}\sqrt{u^{2}+1}du\right) \\
& =\Re\left( \int_{\cot x}^{0}\sqrt{u^{2}+1}du\right) >0;
\end{align*}
a contradiction.
The Laurent serie of $\sqrt{\frac{\left( t-1\right) \left( t-z\right)
\left( t-\overline{z}\right) }{t}}$ when $t\rightarrow1$ is :%
\[
\sqrt{\frac{\left( t-1\right) \left( t-z\right) \left( t-\overline
{z}\right) }{t}}=\left\vert z-1\right\vert \allowbreak\sqrt{t-1}+o\left(
\left( t-1\right) ^{\frac{1}{2}}\right) .
\]
We conclude that
\[
0=\lim\limits_{z\rightarrow1,z\in\Sigma^{+}}\Re\int_{1}^{z}\sqrt{\frac{\left(
t-1\right) \left( t-z\right) \left( t-\overline{z}\right) }{t}}%
dt=\frac{2}{3}\left\vert z-1\right\vert \Re\left( z-1\right) ^{\frac{3}{2}%
},
\]
and then%
\[
\arg\left( z-1\right) ^{\frac{3}{2}}\equiv\frac{\pi}{2}\operatorname{mod}%
\left( \pi\right) ,
\]
which finishes the proof.
\end{proof}
\begin{proof}
[Proof of Proposition \ref{main}]It is clear that the segment $\left[
0,1\right] $ is always a short trajectory of $\varpi_{q}$. If $a\notin%
\Sigma,$ then, from (\ref{cond necess})there is no short trajectory connecting
$a$ to $0$ or $1.$ From Lemma \ref{at infinity}, at most two critical
trajectories emanating from $a$ can diverge to $\infty$ in the upper
half-plane $%
\mathbb{C}
^{+}.$ By consideration of symmetry with respect to the real axis, at list one
critical trajectory emanating from $a$ meets a critical trajectory emanating
from $\overline{a},$ at some point $b\in%
\mathbb{R}
\setminus\left[ 0,1\right] .$ Since $b$ cannot be a zero of the quadratic
differential $\varpi_{q},$ we conclude that these two critical trajectories
form a short one.
If $a\in\Sigma$, and no short trajectory connecting $a$ to $z=1,$ then, there
exist two critical trajectories $\gamma_{a}$ and $\gamma_{1}$ emanating
respectively from $a$ and $1$ and diverging to infinity in a same direction
$D_{k}$. \bigskip From the behaviour of orthogonal trajectories at $\infty,$
we can take an orthogonal trajectory $\sigma$ that hits $\gamma_{1}$ and
$\gamma_{a}$ respectively in two points $b$ and $c$ (there are infinitely many
such orthogonal trajectories $\sigma$ ). We consider a path $\gamma$
connecting $z=1$ and $a$ formed by the part of $\gamma_{1}$ from $z=1$ to $b,$
the part of $\sigma$ from $b$ to $c,$ and the part of $\gamma_{a}$ from $c$ to
$a.$ Then
\begin{align*}
\Re\int_{\gamma}\sqrt{p\left( t\right) }dt & =\Re\int_{1}^{b}%
\sqrt{p\left( t\right) }dt+\Re\int_{b}^{c}\sqrt{p\left( t\right) }%
dt+\Re\int_{c}^{a}\sqrt{p\left( t\right) }dt\\
& =\Re\int_{b}^{c}\sqrt{p\left( t\right) }dt\neq0,
\end{align*}
which contradicts the fact $a\in\Sigma.$
\end{proof}
\begin{proof}
[Proof of Proposition \ref{connection}]The fact that the support of $\nu$ is
formed by horizontal trajectories of the quadratic differential (\ref{qd2}) is
classic and it is based on the so-called Plemelj-Sokhotsky Formula. For more
details, we refer the reader to \cite{amf rakh1},\cite{pritsker}%
,\cite{Shapiro},\cite{bullgard}...
Since the Cauchy transform is a single-valued function in $%
\mathbb{C}
\setminus$\emph{supp}$\left( \nu\right) ,$ then, the support of $\nu$ should
include all the singular points (finite critical points) of the quadratic
differential (\ref{qd2}). But, $\allowbreak$the horizontal trajectories that
contain all finite critical points are exactly short trajectories. The measure
$\nu$ is absolutely continuous with respect to the linear Lebesgue measure,
and it is given on its support (with an adequate orientation) by the
expression :
\[
d\nu\left( t\right) =\frac{1}{8i\pi}\left( \sqrt{\frac{\Delta\left(
t\right) }{t}}\right) _{+}dt.
\]
It is easy to check that the Cauchy transform of $\nu$ satisfies
(\ref{algeq}), indeed :
\begin{align*}
C_{\nu}\left( z\right) & =\frac{1}{2i\pi}\int\frac{\left( \sqrt
{\frac{\Delta\left( t\right) }{t}}\right) _{+}}{4\left( z-t\right)
}dt=\frac{1}{4i\pi}\oint\frac{\sqrt{\frac{\Delta\left( t\right) }{t}}%
}{4\left( z-t\right) }dt\\
& =\frac{1}{4}\left( res_{z}\frac{\sqrt{\frac{\Delta\left( t\right) }{t}}%
}{z-t}-res_{\infty}\frac{\sqrt{\frac{\Delta\left( t\right) }{t}}}%
{z-t}\right) \\
& =\frac{-\sqrt{\frac{\Delta\left( z\right) }{z}}+\gamma+2z}{4}%
=\frac{-\sqrt{z\Delta\left( z\right) }+\gamma z+2z^{2}}{4z},
\end{align*}
where the path of integration in the first integral is formed by the two short
trajectories, and, in the second integral is a closed contour including the
two short trajectories and far away from $z.$
\end{proof}
\begin{acknowledgement}
\bigskip\ This work was partially supported by the laboratory research
"Mathematics and Applications", Faculty of sciences of Gab\`{e}s. Tunisia.
\end{acknowledgement}
|
1,116,691,499,370 | arxiv | \section{Background and Overview}\label{sec:bo}
The arc of this paper begins and ends with a discussion of the Prisoner's Dilemma, but it passes through a new result in provability logic. Thus, it will hopefully be of interest to game theorists and logicians alike.
\subsection{Open-source Prisoner's Dilemma}
Consider the Prisoner's Dilemma, a game with two possible actions C (Cooperate) and D (Defect), with the following payoff matrix:
\begin{table}[ht]
\centering
\setlength{\extrarowheight}{2pt}
\begin{tabular}{cc|c|c|}
& \multicolumn{1}{c}{} & \multicolumn{2}{c}{Player $2$}\\
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{$C$} & \multicolumn{1}{c}{$D$} \\\cline{3-4}
\multirow{2}*{Player $1$} & $C$ & $(2,2)$ & $(0,3)$ \\\cline{3-4}
& $D$ & $(3,0)$ & $(1,1)$ \\\cline{3-4}
\end{tabular}
\end{table}
\noindent In other words, by choosing $D$ over $C$, each player can destroy 2 units of its opponent's utility to gain 1 unit of its own. As long as the payoffs are truly represented in the matrix---for example, there are no reputational costs of choosing $D$ that are not already imputed in the payoffs---then $(D,D)$ is the only Nash equilibrium, and the only correlated equilibrium. In fact, irrespective of the opponent's move, it is better to defect. It is therefore broadly believed that $(D,D)$ is an inevitable outcome between ``rational'' agents in a truly represented (non-iterated) Prisoner's Dilemma.
But consider a version of the game---as first studied by \citet{Tennenholtz:2004:Program}---wherein each player is an algorithm which can read its opponent's source code, as well as its own, before the game. Is $D$ still the obvious correct strategy? As a warm-up, one can imagine designing various algorithmic ``agents'' to compete in such games. For example, an agent who always cooperates:
\begin{Verbatim}[frame=single]
def CooperateBot(Opponent) :
return C
\end{Verbatim}
\citet{Tennenholtz:2004:Program} considers a simple agent which cooperates if and only if the opponent is identitically equal to itself:
\begin{Verbatim}[frame=single]
def IsMeBot(Opponent) :
if Opponent=IsMeBot
return C
else
return D
\end{Verbatim}
When playing against IsMeBot, the opponent is incentivized to ``be IsMeBot'', and in particular, cooperate. To capture this intuition, Tenenholtz defines a {\em \bf program equilibrium} to be a pair of agents (programs) competing in a game, with access to one another's source code, such that replacing either agent by a different agent would decrease its expected payoff. Thus, a program equilibrium is a Nash equilibrium of the `meta-game' of choosing which program to play.
Agents in a program equilibrium can return outputs that do not constitute a Nash equilibrium (of the object-level game), even in a one-shot game, as can be seen here: (IsMeBot,IsMeBot) is a program equilibrium, returning outputs (C,C) and payoffs (2,2). This program equilibrium of IsMeBot is highly fragile, however: if we let IsMeBot' be the same program but with a tiny irrelevant change to its code---a comment perhaps---then IsMeBot will defect against it. Agents studied by \citet{Fortnow:2009:Program} are similarly fragile.
To the end of someday designing real-world cooperative agents, it is therefore interesting to design a more ``robust'' cooperative agent, whose behavior does not depend too heavily on the details of the implementation of its opponent, but which nonetheless incentivizes its opponent to cooperate. For this, consider:
\begin{Verbatim}[frame=single]
def FairBot_k(Opponent) :
search for a proof of length k that
Opponent(FairBot_k) = C
if found,
return C
else
return D
\end{Verbatim}
\noindent Here a `proof of length k' means a mathematical proof---say, in some implementation of Peano Arithmetic---using fewer than $k$ characters (symbols) to write out as a text file. To begin thinking about these agents, observe that
\begin{itemize}
\item $\mathrm{CooperateBot}(\mathrm{FairBot}_k) = C$, because $\mathrm{CooperateBot}$ always returns $C$;
\item $\mathrm{FairBot}_k(\mathrm{CooperateBot}) = C$ when $k$ is large enough to complete the shortest proof that $\mathrm{CooperateBot}(\mathrm{FairBot}_k)=C$ (which, given the simplicity of $\mathrm{CooperateBot}$, will be very short), and $D$ when $k$ is too small to complete the proof.
\end{itemize}
\subsection{The Example of FairBot vs FairBot}
\noindent The first interesting question that arises is then:
\begin{center}
What is $\mathrm{FairBot}_k(\mathrm{FairBot}_k)$?
\end{center}
\noindent It is not so hard to see that if $k$ is too small to complete any proofs, each FairBot returns $D$. But suppose $k$ is extremely large, for example, $10^{100}$. Does $\mathrm{FairBot}_k(\mathrm{FairBot}_k)$ find a proof that $\mathrm{FairBot}_k(\mathrm{FairBot}_k)=C$ and therefore return $C$, validating the proof? Or does it continue searching for a proof that $\mathrm{FairBot}_k(\mathrm{FairBot}_k)=C$ until the proof bound is reached, and having found no such proof, return $D$, consistent with the failed proof search?
It is worth pausing a moment to reflect on this question, since, when given no hints, 100\% of the dozens of mathematicians and computer scientists I've seen asked it have answered incorrectly at first (myself included).
Consider that each instance of $\mathrm{FairBot}_k$ is waiting for a proof that the other $\mathrm{FairBot}_k$ will return $C$ before it will return $C$ itself, and since neither algorithm has a clause in its code to take a ``leap of faith'' in such a situation, it seems that neither algorithm will ``make the first move'', so their proof searches must simply keep searching until they reach their limit $k$ and return $D$.
However, this reasoning turns out to be incorrect, because of a version of L\"{o}b's Theorem that is the main result of this paper, proven in Section \ref{sec:blob}. It implies that $\mathrm{FairBot}_k(\mathrm{FairBot}_k) = C$ for large $k$. Aside from being surprising, this result opens up a whole class of behaviors that can outperform the classical $(D,D)$ equilibrium in a truly formulated, non-iterated Prisoner's Dilemma. Moreover, this performance can be made more robust, into a statement about any two agents willing to cooperate based on a proof of their opponents' cooperation.
Such interesting ``L\"{o}bian" behavior first seemed plausible from the work of \citet{Barasz:2014:RobustCooperation} and \citet{LaVictoire:2014:PrisDilemmaLob}, who illustrated something like program equilibria among certain non-computable logical entities they called ``modal agents'', including an analog of FairBot in that context.
\subsection{Robust Cooperative Program Equilibria}
The main application of this paper is to establish robust cooperative program equilibria for computationally bounded agents. In particular, it is possible to write algorithms which are unexploitable in a Prisoner's Dilemma---that is, they never receive the undesirable outcome $(C,D)$ as Player 1---and which achieve the outcome $(C,C)$ against a variety of opponents, such that there is no incentive for their opponents to deviate from cooperation, even though there is no iteration or reputation to be earned in the game. This is what we mean by ``robust cooperation''.
To summarize the result, we write $\bx{k}p$ for the statement ``$p$ can be proven using $k$ or fewer written symbols''.
Given a nonnegative increasing function $G$, we say that an agent $A_k$ taking a parameter $k \in \mathbb{N}$ is {\bf G-fair} if
$$\proves{} \bx{k+G(\textrm{LengthOf}(Opp))}[Opp(A_k) = Cooperate] \implies A_k(Opp) = Cooperate$$
In other words, if $A_k$ finding a proof that its opponent cooperates is sufficient for $A_k$ to cooperate, we say it is $G$-fair, provided the proof lengths in the search did not exceed $k+G(\textrm{LengthOf}(Opp))$. Then we have, in terms to be made precise later,
\begin{theorem*}[{\bf Robust cooperation of bounded agents}] If certain bounds are satisfied by the G\"{o}del encoding of our proof system, and the function $G$ exceeds a certain asymptotic lower bound, then for any $G$-fair agents $A_k$ and $B_k$, we have for all sufficiently large $m,n$,
$$A_m(B_n) = B_n(A_m) = Cooperate$$
\end{theorem*}
This result depends crucially on a new version of L\"{ob}'s Theorem.
\subsection{L\"{o}b's Theorem}
L\"{o}b's Theorem states that, if $\bx{}p$ denotes the provability of statement $p$ in Peano Arithmetic (or any extension of it), then
$$\bx{}(\bx{}p \implies p) \implies \bx{}p$$
If the reader has never encountered this result, consider the case where $p$ is the Riemann Hypothesis, $RH$. Suppose that the Riemann Hypothesis is, unbeknownst to us, false. Without yet knowing whether $RH$ is true, it is tempting for us to claim at least that {\em if RH is provable, then RH is true}, i.e. $\bx{}RH \implies RH$. However, if that claim were itself provable, i.e. if $\bx{}(\bx{}RH \implies RH)$, then L\"{o}b's Theorem tells us that $\bx{}RH$---the Riemann Hypothesis is provable---which is very bad news for the soundness of our proof system if the Riemann Hypothesis is actually false!
Thus, L\"{o}b's Theorem defies the intuition that we might soundly prove the ``self-trust'' statement that {\em if we prove $p$, then $p$ is true}. This counterintuitiveness is in fact the same phenomenon as the surprising outcome that $\mathrm{FairBot}_k(\mathrm{FairBot}_k)=C$ from earlier, except that the FairBots---being algorithms which halt---only concern proofs up to a certain bounded length, $k$. Hence the motivation of this paper: to establish a version of L\"{o}b's theorem for proofs bounded in length by a parameter,~$k$. In rough terms, we prove:
\begin{theorem*}[Parametric Bounded L\"{o}b]
Suppose $p(-)$ is a logical formula with a single unquantified variable, and that $f:\mathbb{N} \to \mathbb{N}$ is computable and exceeds a certain asymptotic lower bound. Then $\exists\hat k$ :
\begin{align*}
&\proves{} \forall k,\; \bx{f(k)}p(k) \implies p(k)\\
\;\;\Rightarrow\;\; &\proves{} \forall k>\hat k, \; p(k)
\end{align*}
\end{theorem*}
\subsection{Comparison to previous work}
As mentioned, \citet{Tennenholtz:2004:Program} first defined program equilibria and studied various `non-robust' examples similar to IsMeBot above, which depend on program equality. In particular, if one agent is written in C++ while the other is written in Python, they will defect against each other. Examples studied by \citet{Fortnow:2009:Program} are similarly fragile.
Later, \citet{Peters:2012:Definable} consider agents encoded as a first-order formulas over the integers which can reference the G\"{o}del-numbering of the formula for the other player as well as its own, but these agents are non-computable in a way similar to those of \citet{Barasz:2014:RobustCooperation} and \citet{LaVictoire:2014:PrisDilemmaLob}.
By comparison, the program equilibria exhibited here are both computable and {\em robust}, in that they do not depend on tests for program equality, and generally exist between many pairs of agents provided they both follow a certain principle of fairness, in which a new bounded L\"{o}b's Theorem plays a crucial role.
\subsection{Long-Term Relevance}
As automated reasoning and decision-making systems improve, it is plausible that some such systems might exhibit a capacity to reason in generality about their own design principles, and those of other systems. As an illustrative example, such a system can be designed expressly today: a theorem-prover can be handed a copy of its own source code and queried to write proofs about it. Less contrivedly, there might be economic value in creating systems that can reason about themselves and others, such as for collaboration or negotiation. For example, a human can reason that he is mentally outclassed in the middle of a competitive game of Go against a new player, and therefore resign to hedge his losses. Such reasoning invokes a theory of the reasoning capacity of one's opponent, and of oneself: algorithms reasoning about algorithms.
It therefore seems prudent to explore what game-theoretic dynamics emerge from algorithms reasoning about each other, beginning with the simplest cases we can currently state and examine, similar in spirit to the way RAND Corporation's Thomas Schelling began his understanding of nuclear deterrence~\citep{Schelling:1958:ConflictProspectus,Schelling:1966:Arms}, by analyzing simple examples of non-zero-sum games \citep{Schelling:1958:Reinterpretation}.
In this paper, we find that classical game theory---and more generally, causal decision theory \citep{Gibbard:1978}---is not an adequate framework for describing the competitive interactions of algorithms that reason about the source codes of their opponent algorithms and themselves. When given read access to one another's source code---an extreme scenario for two humans, but trivial for computer systems---competing algorithms can exhibit counterintuitive ``L\"{o}bian'' behaviors which, among other things, can robustly achieve cooperative outcomes that outperform classical Nash equilibria and correlated equilibria. Moreover, the time at which each algorithm outputs its cooperative decision occurs later in time than the causal pathway by which it benefits from the decision (namely, the pathway wherein its opponent predicts its behavior using its source code; see Section \ref{sec:cdt}).
Thus, without further investigation, our more classical intuitions about what group-level behaviors will emerge from such algorithms may miss the mark entirely.
\section{Fundamentals}
Here we begin building up the main technical result of the paper. The algorithms examined here will make use of provability logic as a way of ``reasoning about reasoning'', and the main resource bounds on the algorithms, for simplicity, will be the lengths of the proofs they may discover.
\subsection{Proof Length and Notation}
If the first line of a three-line proof is so long that it would not fit on any physical computer system, saying the proof is ``only three lines long'' is not very descriptive. Therefore, we will measure proof length in {\em characters} instead of lines, the way one might measure the size of a text file on a computer. An extensive analysis of proof lengths measured in characters is covered by \citet{Pudlak:1998}.
We will fix a proof system $S$ (e.g. an extension of Peano Arithmetic) throughout, and write
$$S \proves{n} \phi, \quad \text{or simply} \quad \proves{n} \phi$$
to mean that there exists an $S$-proof of $\phi$ using $n$ or fewer characters. After a choice of G\"{o}del encoding for $S$, it is customary to write $\bx{}\phi$ for $\exists n : Bew(n,\qquote{\phi})$, i.e., there exists a number $n$ encoding a proof of $\phi$. This allows $S$ to indirectly talk about the existence of proofs in $S$. We will extend this definition to talk about proof lengths:
$$\bx{n}\phi \quad \text{means} \quad \exists m : Bew(m,\qquote{\phi}) {\textrm{ and }} \mathrm{ProofLength}(m)<n$$
where $\mathrm{ProofLength}(m)$ denotes the length, in characters, of the proof encoded by $m$. In other words, $\bx{n}\phi$ is the $S$-encoded statement that $\phi$ can be proven in $S$ with $n$ or fewer characters.
\subsection{Proof System}\label{sec:system}
We let $S$ be any first-order proof system that
\begin{itemize}
\item[1)] can represent computable functions in the sense of Section \ref{sec:rep},
\item[2)] can write any number $k\in\mathbb{N}$ using $\mathcal{O}\lg k$ symbols, and
\item[3)] allows the definition and expansion of abbreviations during proofs.
\end{itemize}
For example, we could take Peano Arithmetic, where each proof line is either
\begin{itemize}
\item an axiom, or
\item an application of Modus Ponens from lines above it,
\end{itemize}
and additionally allow ourselves to write numbers in a binary format, and allow proof lines which are
\begin{itemize}
\item the definition of an abbreviation that may be used in subsequent lines, or
\item an expansion of an abbreviation used in a previous line.
\end{itemize}
\noindent We have chosen to allow abbreviations in our proof system for two reasons. The first is that real-world automated proof systems will tend to use abbreviations because of memory constraints. The second is that abbreviations make the lengths of the shortest proofs in this system slightly easier to analyze: for example, if a number $N$ with a very large number of digits occurs in the shortest proof of a proposition, it will not occur multiple times; instead, it will occur only once, in the definition of an abbreviation for it. Then, we don't have to carefully count the number of times the numeral occurs in the proof to determine its contribution to the proof length; its contribution will simply be linear in its length, or $\lg N$.
We write
\begin{itemize}
\item[] $\mathrm{Lang}(S)$ for the language of $S$,
\item[] $\mathrm{Lang}_r(S)$ for the formulas in $\mathrm{Lang}(S)$ with $r$ free variables, and
\item[] $\mathrm{Const}(S)$ for the set of constants in $S$ (e.g. $0$, $\mathcal{S} 0$, etc.).
\end{itemize}
\subsection{G\"{o}del Encoding}
We fix throughout a G\"{o}del numbering
$$\#(-) : \mathrm{Lang}(S) \to \mathbb{N}$$
and a ``numeral'' mapping
$${}^\circ(-) : \mathbb{N} \to \mathrm{Const}(S)\subseteq \mathrm{Lang}(S)$$
for expressing naturals as constants in $S$. Note that in traditional $\mathcal{P}\!\mathcal{A}$, for example, ${}^\circ 5 = \mathcal{S}\Ss\mathcal{S}\Ss\Ss0$. However, to be more realistic we have assumed that $S$ uses a binary encoding to be more efficient, so e.g.,
$${}^\circ 5 = 101.$$
The maps $\#(-)$ and ${}^\circ(-)$ combine to form a G\"{o}del encoding
$$\qquote{(-)} : \mathrm{Lang}(S)\to\mathrm{Const}(S)$$
$$\qquote{\phi}:={}^\circ \# \phi$$
which allows $S$ to write proofs about itself.
\subsection{Convention for Representing Computable Functions}\label{sec:rep}
The astute reader will notice that throughout, although $\mathcal{P}\!\mathcal{A}$ and related first-order theories typically have no symbols for functions, we will often write objectionable expressions like
$$\proves{} \ldots \text{something about $f(x)$} \ldots$$
where $f:\mathbb{N}\to\mathbb{N}$ is some computable function.
However, there is a convention for interpreting such statements. It is known \citep[see, e.g. Theorem 6.8 of][Part II]{Cori:2001} that for any computable function $f:\mathbb{N}\to\mathbb{N}$, there exists a ``graph'' predicate $\Gamma_f(-,-)\in\mathrm{Lang}_2(\mathcal{P}\!\mathcal{A})$ such that
$$\forall x\in\mathbb{N}, \; \mathcal{P}\!\mathcal{A} \proves{} \forall y, \; \Gamma_f({}^\circ x,y) \iff y = {}^\circ f(x)$$
\noindent We have assumed that $S$ is capable of representing computable functions in this way (e.g., by being an extension of $\mathcal{P}\!\mathcal{A}$).
The two-place predicates $\Gamma_f$ are cumbersome in writing because each usage introduces a quantifier. For example, if we we have functions $f$, $g$ and $h$ and we want to say that $S$ proves that for any $x$ value, $f(x) < g(x) + h(x)$, technically we should write
$$\proves{} \forall x \forall y_1 \forall y_1 \forall y_3,\; \Gamma_f(x,y_1) {\textrm{ and }} \Gamma_g(x,y_2){\textrm{ and }} \Gamma_h(x,y_3) \implies y_1 < y_2 + y_3$$
However, for easier reading, in such cases we will abuse notation and write
$$\proves{} \forall x, \; f(x) < g(x) + h(x),$$
leaving the expansion in terms of $\Gamma$'s and $\forall y$'s as an exercise to any willing reader.
\subsection{Asymptotic Notation}
We use the convention that $f \prec g$ means that for any $M\in \mathbb{N}$, there exists an $N\in\mathbb{N}$ such that $\forall n>N, Mf(n) < g(n)$. We write $\mathcal{O} g$ for the set of functions $f\preceq g$, and for a specific function $\mathcal{E}$ we will sometimes write write $\mathcal{E} \mathcal{O} g$ for the set of functions of the form $\mathcal{E} \circ f$ where $f\in \mathcal{O} g$.
\section{A Parametric Diagonal Lemma}
L\"{o}b's Theorem can be proven via the classical Diagonal Lemma \citep{Carnap:1934}, which states that for any formula $F(-)\in\mathrm{Lang}_1(S)$ (having one free variable), there exists a sentence $\psi\in\mathrm{Lang}_0(S)$ (with no free variables) such that
$$\proves{} \psi \iff F(\qquote\psi).$$
However, to reason about computer systems with certain as-yet unset parameters, we will need a generalization of the Diagonal Lemma for formulas with free variables to represent those parameters in a way that avoids writing a separate proof for every instance of the parameters:
\begin{proposition}[Parametric Diagonal Lemma]Suppose $S$ is a first-order theory capable of representing all computable functions, as in Section \ref{sec:rep}. Then for any predicate $G\in\mathrm{Lang}_{r+1}(S)$, there exists a predicate $\psi\in\mathrm{Lang}_r(S)$ such that
$$\proves{} \forall \bar k = (k_1,\ldots,k_r), \; \psi(\bar k) \iff G(\qquote\psi,\bar k)$$
\end{proposition}
\begin{proof}
We define a ``partial self-evaluation function'' $e:\mathbb{N}\to\mathbb{N}$ as follows:
$$e(n) = \begin{cases}
\#\left[\theta(\qquote\theta,-,\ldots,-)\right] &\mbox{if } n = \#\theta \text{ for some }\theta\in\mathrm{Lang}_{r+1}(S) \\
0 & \mbox{otherwise }
\end{cases}
$$
Now, $e$ is computable, and therefore representable in $\mathrm{Lang}(S)$, so we can define $\beta\in\mathrm{Lang}_{r+1}(S)$ by
$$\beta(n,\bar k) := G(e(n),\bar k)$$
(using the notational convention of Section \ref{sec:rep} to avoid writing extra quantifiers and $\Gamma_e$'s). Then, $\forall \theta\in\mathrm{Lang}_{r+1}(S)$,
$$\proves{} \forall \bar k\; \beta(\qquote{\theta},\bar k) \iff G(\qquote{\theta(\qquote{\theta},-,\ldots,-)},\bar k)$$
Now let $\theta = \beta$, so we have
$$\proves{} \forall \bar k\; \beta(\qquote(\beta),\bar k) \iff G(\qquote{\beta(\qquote{\beta},-,\ldots,-)},\bar k)$$
Finally, taking $\psi(\bar k) = \beta(\qquote{\beta},\bar k)$ yields the desired result
$$\proves{} \forall \bar k\; \psi(\bar k) \iff G(\qquote{\psi},\bar k)$$
\end{proof}
\section{A Bounded Provability Predicate, \texorpdfstring{$\bx{k}$}{box k}}
\subsection{Defining \texorpdfstring{$\bx{k}$}{box k}}
Given a choice of G\"{o}del encoding for Peano Arithmetic, it is classical that a predicate $Bew(-,-) \in \mathrm{Lang}_2(S)$ exists such that $Bew(m,n)$ means, in natural language, that the number $m$ encodes a proof in $\mathcal{P}\!\mathcal{A}$, and that the number $n$ encodes the statement it proves. So, the standard provability operator $\bx{}:\mathrm{Lang}(\mathcal{P}\!\mathcal{A})\to\mathrm{Lang}(\mathcal{P}\!\mathcal{A})$ can be defined as
$$\bx{}\phi := \exists m : Bew(m,\qquote\phi).$$
We take for granted that $Bew$ exists for $S$ and can be extended to a three-place predicate $Bew(-,-,-) \in \mathrm{Lang}_2(S)$ such that $Bew(m,n,k)$ means that
\begin{itemize}
\item $m$ encodes a proof in $S$,
\item $n$ encodes the statement it proves, and
\item the proof encoded by $m$ uses at most $k$ characters when written in the language of $S$ ({\em not} when written using the encoding.)
\end{itemize}
Then we can define a ``bounded'' box operator:
$$\bx{k}\phi = \exists m : Bew(m,\qquote{\phi},k).$$
We also take for granted a computable ``single variable evaluation'' function, $Eval_1:\mathbb{N}\to\mathbb{N}$, such that for any $\phi(-)\in\mathrm{Lang}_1(S)$,
$$Eval_1(\qquote\phi,k) = \qquote{\phi({}^\circ k)}$$
Since $Eval_1$ is computable, it can be represented in $\mathrm{Lang}(S)$ as in Section \ref{sec:rep}.
This allows us to extend the $\bx{k}$ operator to act on sentences $\phi(-)$ with an unbound variable:
$$(\bx{k}\phi)(\ell) := \exists m : Bew(m,Eval_1(\qquote\phi,\ell),k)$$
In words, ``There is a proof using $k$ or fewer characters of the formula $\phi(\ell)$''.
\subsection{Basic Properties of \texorpdfstring{$\bx{k}$}{box k}}
Each of the following properties will be needed multiple times during the proof of Parametric Bounded L\"{o}b. Since the proof is already highly symbolic, we give these properties English names to recall them.
\begin{property}[Implication Distribution]
There is a constant $c\in \mathrm{Const}(S)$ such that for any $p,q\in\mathrm{Lang}(S)$,
$$\proves{} \forall a\forall b,\; \bx{a}(p\implies q) \implies (\bx{b}p \implies \bx{a+b+c} q).$$
\end{property}
\begin{proof}[Proof sketch]
The fact that one can combine a proof of an implication with the proof of its antecedent to obtain a proof of its consequent can be proven in general, with quantified variables in place of the G\"{o}del numbers of the particular statements involved. Let us suppose this general proof has length $c_0$. Then, we need only instantiate the statements in it to $p$ and $q$. However, if $p$ and $q$ are long expressions, they can have been abbreviated in the earlier proofs without lengthening them, so they can be written in abbreviated form again during this step. Hence, the total cost of combining the two proofs is around $c=2c_0$, which is constant with respect to $p$ and $q$.
\end{proof}
\begin{property}[Quantifier Distribution]
There is a constant $C\in \mathrm{Const}(S)$ such that for any $\phi(-) \in \mathrm{Lang}_1(S)$,
\begin{align*}
&\proves{}\bx{N}\left(\forall k \phi(k)\right)\\
\;\;\Rightarrow\;\; &\proves{} \forall k \; \bx{C+2N+\lg k} \phi(k)\text{, which in turn}\\
\;\;\Rightarrow\;\; &\proves{} \forall k \; \bx{\mathcal{O}\lg k} \phi(k)
\end{align*}
\end{property}
\begin{proof}
An encoded proof of $\phi({}^\circ K)$ for a specific $K$ can be obtained by specializing the conclusion of an $N$-character encoded proof of $\forall k \phi(k)$ and appending the specialization with ${}^\circ K$ in place of $k$ at the end. To avoid repeating ${}^\circ K$ numerous times in the final line (in case it is large), we will use an abbreviation for $\phi$. Thus the appended lines can say:
\begin{itemize}
\item[(1)] let $\Phi$ stand for $\qquote{\phi}$
\item[(2)] $\Phi({}^\circ K)$
\end{itemize}
Let us analyze how many characters are needed to write such lines. First, we need a string $\Phi$ to use as an abbreviation for $\phi$. Since no string of length $\frac{N}{2}$ has yet been used as an abbreviation in the earlier proof (otherwise we can shorten the proof by not defining and using the abbreviation), we can surely have $\mathrm{Length}(\Phi)<\frac{N}{2}$. We also need some constant $c$ number of characters to write out the system's equivalent of ``let'', ``stand for'', ``('', and ``)''. Finally, we need $\lg K$ characters to write ${}^\circ K$. Altogether, the proof was extended by $C+N+\lg(k)$ characters, for a total length of $2N+c+\lg k$.
\end{proof}
\section{Parametric Bounded L\"{o}b}\label{sec:blob}
\begin{definition}[Proof expansion function]\label{def:E}We choose a computable function $\mathcal{E}:\mathbb{N}\to\mathbb{N}$ to bound the expansion of proof lengths when we G\"{o}del-encode them. Its definition is that it must be large enough to satisfy the following two properties:
\end{definition}
\begin{property}[Bounded Necessitation]
$\forall \phi \in \mathrm{Lang}(S)$,
\begin{align}
&\proves{k} \phi \\
\;\;\Rightarrow\;\; &\proves{\mathcal{E} k} \bx{k} \phi
\end{align}
\end{property}
\begin{property}[Bounded Inner Necessitation]
For any $\phi \in \mathrm{Lang}(S)$,
$$\proves{} \bx{k}\phi \implies \bx{\mathcal{E} k}\bx{k}\phi.$$
\end{property}
\noindent {\bf Estimating $\mathcal{E}$.} How large must $\mathcal{E}$ be in practice? G\"{o}del numberings for sequences of integers can be achieved in $\mathcal{O} n$ space \citep{Tsai:2002}, as can G\"{o}del numberings of term algebras \citep{Tarau:2013}. To check that one line is an application of Modus Ponens from previous lines, if the proof encoding indexes the implication to which MP is applied, is a test for string equality that is linear in the length of the lines. Finally, to check that an abbreviation has been applied or expanded, if the proof encoding indexes where the abbreviation occurs, is also a linear time test for string equality.
Thus, it seems reasonable to expect $\mathcal{E} \in \mathcal{O} k$ for real-world theorem-provers. But however large it may be, in any case we have:
\begin{theorem}[Parametric Bounded L\"{o}b]\label{thm:pblob}
Suppose $p(-)\in\mathrm{Lang}_1(S)$ is a formula with a single unquantified variable, and that $f:\mathbb{N} \to \mathbb{N}$ is computable and satisfies $f(k) \succ \mathcal{E}\mathcal{O}\lg k$. Then $\exists\hat k$ :
\begin{align*}
&\proves{} \forall k,\; \bx{f(k)}p(k) \implies p(k)\\
\;\;\Rightarrow\;\; &\proves{} \forall k>\hat k, \; p(k)
\end{align*}
\end{theorem}
\noindent{\em Note:} In fact a weaker statement
$$\proves{} \forall k>k_1,\; \bx{f(k)}p(k) \implies p(k)$$
is sufficient to derive the consequent, since we could just redefine $f(k)$ to be $0$ for $k\leq k_1$ and then $\bx{f(k)}p(k) \implies p(k)$ is vacuously true and provable for $k \leq k_1$ as well.
\begin{proof}
{\em (In this proof, each centered equation will follow directly from the one above it unless otherwise noted.)}
We begin by choosing some function $g(k)$ such that $\lg k \prec g(k)$ and $\mathcal{E} g(k) \prec f(k)$. For example, we could take $g(k) = \floor{\sqrt{(\lg k)(\mathcal{E}\- f(k))}}$. Define a predicate $G(-,-)\in\mathrm{Lang}_2(S)$ by
$$G(n,k) := \left(\exists m : Bew(m,Eval_1(n,k),g(k))\right) \implies p(k)$$
so that for any $\phi(-)\in\mathrm{Lang}_1(S)$,
$$G(\qquote\phi,k) = \bx{g(k)}\phi(k) \implies p(k).$$
Now, by the Parametric Diagonal Lemma, $\exists \psi(-)\in\mathrm{Lang}_1(S)$ such that in some number of characters $n$,
\eqn{\label{eqn1}
\proves{n} \forall k \; \psi(k) \iff G(\qquote\psi,k)
}
By Bounded Necessitation,
$$\proves{} \bx{n} \left(\forall k \; \psi(k) \iff G(\qquote\psi,k)\right)$$
By Quantifier Distribution, since $n$ is constant with respect to $k$,
$$\proves{} \forall k \; \bx{\mathcal{O}\lg k} \left(\psi(k) \iff G(\qquote\psi,k)\right),$$
in which we can specialize to the forward implication,
$$\proves{} \forall k \; \bx{\mathcal{O}\lg k} \left(\psi(k) \implies G(\qquote\psi,k)\right)$$
By Implication Distribution of $\bx{\mathcal{O}\lg k}$,
$$\proves{} \forall k \forall a \; \bx{a}\psi(k) \implies \bx{a+\mathcal{O}\lg k}G(\qquote\psi,k)$$
By Implication Distribution again, this time of $\bx{a+\mathcal{O}\lg k}$ over the implication $G(\qquote\psi,k) = \bx{g(k)}\phi(k) \implies p(k)$, we obtain
$$\proves{} \forall k \forall a \forall b \; \bx{a}\psi(k) \implies
\left(\bx{b}\bx{g(k)}\psi(k) \implies \bx{a+b+\mathcal{O}\lg k}p(k)\right)$$
Now we specialize this equation to $a=g(k)$ and $b=h(k)$, where $h:\mathbb{N}\to\mathbb{N}$ is a computable function satisfying $\mathcal{E} g(k)\prec h(k)\prec f(k)$, for example $h(k) = \floor{\sqrt{f(k)\mathcal{E} g(k)}}$:
$$\proves{} \forall k \; \bx{g(k)} \psi(k) \implies
\left(\bx{h(k)}\bx{g(k)}\psi(k) \implies \bx{g(k)+h(k)+\mathcal{O}\lg k} p(k)\right)$$
Then since $g(k)+h(k)+\mathcal{O}\lg k < f(k)$ after some bound $k > k_1$, we have
$$\proves{} \forall k>k_1, \; \bx{g(k)}\psi(k) \implies \left(\bx{h(k)}\bx{g(k)}\psi(k) \implies \bx{f(k)} p(k)\right)$$
Now, by hypothesis, $\proves{} \forall k\; \bx{f(k)}p(k)\implies p(k)$, thus
\eqn{\label{eqn9}
\proves{} \forall k>k_1, \; \bx{g(k)}\psi(k) \implies \left(\bx{h(k)}\bx{g(k)}\psi(k) \implies p(k)\right)
}
Also, without any of the above, from Bounded Inner Necessitation we can write
$$\proves{} \forall k \forall a \; \bx{a}\psi(k)\implies \bx{\mathcal{E} a}\bx{a}\psi(k)$$
From this, with $a=g(k)$, we have
$$\proves{} \forall k \; \bx{g(k)}\psi(k)\implies \bx{\mathcal{E} g(k)}\bx{g(k)}\psi(k)$$
Now, since $\mathcal{E} g(k) < h(k)$ after some bound $k>k_2$, we have
\eqn{\label{eqn12}
\proves{} \forall k > k_2 \; \bx{g(k)}\psi(k)\implies \bx{h(k)}\bx{g(k)}\psi(k)
}
Next, from Equations \ref{eqn9} and \ref{eqn12}, assuming we chose $k_2\geq k_1$ for convenience, we have
\eqn{\label{eqn13}
\proves{} \forall k>k_2,\; \bx{g(k)}\psi(k)\implies p(k)
}
But from Equation \ref{eqn1}, the implication here is equivalent to $\psi(k)$, so we have
$$ \proves{N} \forall k>k_2,\; \psi(k),$$
where $N$ is the number of characters needed for the proof above. From this, by Bounded Necessitation, we have
$$\proves{} \bx{N} [\forall k>k_2,\; \psi(k)].$$
By Quantifier Distribution of $\bx{N}$,
$$\proves{} \forall k>k_2,\; \bx{\mathcal{O}\lg k} \psi(k)$$
and since $\mathcal{O}\lg k < g(k)$ after some bound $k>\hat k$, taking $\hat k \geq k_2$ for convenience, we have
\eqn{\label{eqn17}
\proves{}\forall k>\hat k,\; \bx{g(k)}\psi(k).
}
Finally, from Equations \ref{eqn13} and \ref{eqn17} we have
$$\proves{} \forall k>\hat k,\; p(k),$$
as required.
\end{proof}
\section{Robust Cooperation of Bounded Agents in the Prisoner's Dilemma}
\citet{Barasz:2014:RobustCooperation}, \citet{LaVictoire:2014:PrisDilemmaLob}, and others have exhibited various proof-based agents who robustly cooperate in the Prisoner's Dilemma by basing their decisions on proofs about each other's cooperation. However, their agents are purely logical entities which can discover proofs of unbounded length, and so are impossible to run on a physical computer. This leaves open the question of whether such behavior is achievable by agents with bounded computational resources.
So, consider the following bounded agent, where $G$ is some increasing, non-negative function to be determined later, and $G=0$ recovers the definition of FairBot from Section 1:
\begin{Verbatim}[frame=single]
def FairBot_k(Opponent) :
let B = k + G(LengthOf(Opponent))
search for proof of length at most B that
Opponent(FairBot_k) = Cooperate
if found,
return Cooperate
else
return Defect
\end{Verbatim}
Question: What is $\mathrm{FairBot}_k(\mathrm{FairBot}_k)$? It seems intuitive that each FairBot is waiting for the other to provably cooperate, in a bottomless regression that will exhaust the proof bound B. Thus, they will find no proof of cooperation, and hence defect.
However, this turns out not to be the case, as a consequence of Parametric Bounded L\"{o}b. We let $$p(k) := [\mathrm{FairBot}_k(\mathrm{FairBot}_k) = Cooperate].$$
Since $G\geq 0$, $k \leq B$ in the definition of FairBot, so we have
$$\proves{} \bx{k}p(k) \implies \bx{B}p(k).$$
Now since $\bx{B}p(k)$ is FairBot's criterion for cooperation, we also have
$$\proves{} \bx{B}p(k) \implies p(k), \textrm{ \ so}$$
$$\proves{} \forall k, \; \bx{k} p(k) \implies p(k),$$
whence for sufficiently large $\hat k$, by Parametric Bounded L\"{o}b,
$$\proves{} \forall k>\hat k,\; p(k).$$
In other words, $FairBot_k$ cooperates with $FairBot_k$ for large $k$.
This result is interesting for three reasons:
\begin{itemize}
\item[1.] It is {\em surprising}. 100\% of the dozens of mathematicians and computer scientists that I've asked to guess the output of $\mathrm{FairBot}_k(\mathrm{FairBot}_k)$ have guessed incorrectly (expecting the proof searches to enter an infinite regress and thus reach their bounds), or have given an invalid argument for cooperation (such as ``it would be better to cooperate, so they will").
\item[2.] It is {\em advantageous}. FairBot outperforms the classical Nash/correlated equilibrium solution (Defect, Defect) to the Prisoner's Dilemma, in a one-shot game with no iteration or future reputation. Moreover, it does so {\em while being unexploitable}: if an opponent will defect against FairBot, FairBot will find no proof of the opponent's cooperation, so it will also defect.
\item[3.] It is {\em robust}. Previous examples of cooperative program equilibria studied by \citet{Tennenholtz:2004:Program} and \citet{Fortnow:2009:Program} all involved cooperation based on {\em equality of programs}, a very fragile condition. For example, the agent IsMeBot from the introduction will mutually defect against an identical opponent written in a different programming language, or even in a slightly different style. Such fragility is not desirable if we wish to build real-world cooperative systems.
\end{itemize}
Taking this robustness further, we next demonstrate mutual cooperative program equilibria among a wide variety of (unequal) agents, provided only that they employ a certain ``principle of fairness". Given a non-negative increasing function $G$, we say that an agent $A_k$ taking a parameter $k \in \mathbb{N}$ is {\bf G-fair} if
$$\proves{} \bx{k+G(\textrm{LengthOf}(Opp))}[Opp(A_k) = C] \implies A_k(Opp) = C$$
In other words, if $A_k$ finding a proof that its opponent cooperates is sufficient for $A_k$ to cooperate, we say it is $G$-fair, provided the proofs in the search did not exceed length $k+G(\textrm{LengthOf}(Opp))$. The agents $\mathrm{FairBot}_k$ defined above are $G$-fair, and the reader is encouraged to keep these examples in mind for the following result:
\begin{theorem}[{\bf Robust cooperation of bounded agents}]Suppose that
\begin{itemize}
\item the proof expansion function $\mathcal{E}$ (defined in Section \ref{sec:blob}) of our proof system satisfies $\mathcal{E} \mathcal{O} \lg k \prec k$,
\item $f$ is any function satisfying $\mathcal{E}\mathcal{O}\lg k \prec f(k) \prec k$, and
\item $G$ is any increasing function satisfying $G(\ell) > 6f(2^\ell)$.
\end{itemize}
Then, for any $G$-fair agents $A_k$ and $B_k$, we can choose a threshold $r$ such that for all $m,n>r$,
$$A_m(B_n) = B_n(A_m) = Cooperate$$
\end{theorem}
\noindent {\bf Feasibility of bounds.} Before proceeding, recall from Section \ref{sec:blob} that we can achieve $\mathcal{E} \in \mathcal{O} k$ for automatic proof systems that are designed for easy verifiability, in which case $\mathcal{E} \mathcal{O} \lg k = \mathcal{O} \lg k$, well below the $\prec k$ requirement.
\begin{proof}
For brevity, we let
\begin{align}
a(k) &:= G(\mathrm{LengthOf}(A_k)), \\
b(k) &:= G(\mathrm{LengthOf}(B_k)), \\
\alpha(m,n) &:= [A_m(B_n) = Cooperate], \text{ and }\\
\beta(n,m) &:= [B_n(A_m) = Cooperate]
\end{align}
so we can write the $G$-fairness conditions more compactly as
\begin{align}
\label{eqnC1} \proves{} &\bx{m+b(n)} \beta(n,m) \implies \alpha(m,n) \text{ and}\\
\nonumber \proves{} &\bx{n+a(m)} \alpha(m,n) \implies \beta(n,m).
\end{align}
Now, $\mathrm{LengthOf}(A_k) > \lg k$ and $\mathrm{LengthOf}(B_k) > \lg k$ since they must reference the parameter $k$ in their code. Applying $G$ to both sides yields
\eqn{\label{eqnC2} a(k),b(k) > G(\lg k) > 6f(k).}
Define an ``eventual cooperation" predicate:
$$p(k) := \forall m> k,\; \forall n> k, \; \alpha(m,n) {\textrm{ and }} \beta(n,m).$$
Using Quantifier Distribution once on the definition of $p(k)$,
$$\proves{} \forall k [\bx{f(k)}p(k) \implies \forall m>k, \; \bx{C+2f(k)+\lg m } [\forall n > k, \; \alpha(m,n) {\textrm{ and }} \beta(n,m)]]$$
Applying Quantifier Distribution again,
\eqn{\label{eqnC2a}\proves{} \forall k [\bx{f(k)}p(k) \implies \forall m>k, \forall n > k, \; \bx{3C+4f(k)+2\lg m +\lg n }[\alpha(m,n) {\textrm{ and }} \beta(n,m)]]}
Now, for $m,n$ large and $>k$, we have
\begin{align*}
3C + \lg n &< n &&\textrm{ \quad and by (\ref{eqnC2}),}\\
4f(k) + 2\lg m < 6f(m) &< a(m). &&
\end{align*}
Adding these inequalities yields
$$3C+4f(k)+2\lg m +\lg n < n+a(m),$$
so for some $k_1$, from (\ref{eqnC2a}) we derive
$$\proves{} \forall k>k_1,\; [\bx{f(k)}p(k) \implies \forall m>k, \forall n > k, \; \bx{n+a(m)}\alpha(m,n)].$$
Similarly, we also have
\begin{align*}
3C + 2\lg m &< m &&\textrm{ \quad and }\\
4f(k) + \lg n < 5f(n) &< b(n), &&\text{ \quad so for some $k_2\geq k_1$,}
\end{align*}
$$\proves{} \forall k>k_2 \; [\bx{f(k)}p(k) \implies \forall m>k, \forall n > k, \; \bx{n+a(m)}\alpha(m,n) {\textrm{ and }} \bx{m+b(n)}\beta(n,m)]$$
Thus by (\ref{eqnC1}),
$$\proves{} \forall k > k_2 [\bx{f(k)}p(k) \implies \forall m>k, \forall n > k, \; c(n,m) {\textrm{ and }} c(m,n)]\text{, i.e.}$$
$$\proves{} \forall k > k_2, \; \bx{f(k)}p(k) \implies p(k)$$
Therefore, by Parametric Bounded L\"{o}b (and the note following it), for some $\hat k$ we have
$$\proves{} \forall k > \hat k, \; p(k).$$
In other words, for all $m,n>\hat k + 1$,
$$A_m(B_n)=B_n(A_m)=Cooperate.$$
\end{proof}
\subsection{Ramifications for Causal Decision Theory}\label{sec:cdt}
Causal Decision Theory \citep{Gibbard:1978} is a framework for evaluating the desirability of an action by assessing the causal consequences of the action itself. The interaction of $\mathrm{FairBot}_m$ and $\mathrm{FairBot}_n$ present a challenge to Causal Decision Theory, in a way similar to Newcomb's Problem \citep{Nozick:1969}, a classic scenario wherein one agent is able to predict the actions of another.
Concretely, imagine $\mathrm{FairBot}_m$ and $\mathrm{FairBot}_n$ are played against each other while being run on separate computers in separate rooms, and that they will print their final responses, $C$ or $D$, at the same time. When $\mathrm{FairBot}_m$ decides to cooperate with $\mathrm{FairBot}_n$, it does so {\em after} computing a proof that $\mathrm{FairBot}_n(\mathrm{FairBot}_m)=C$, but {\em before} its opponent $\mathrm{FairBot}_n$ actually prints its response. There is therefore no causal effect transmitted from the value that $\mathrm{FairBot}_n$ prints to its screen to the value that $\mathrm{FairBot}_m$ prints to its screen. So from a purely causal perspective, there is an ``incentive'' for $\mathrm{FairBot}_n$ to print $D$ instead of $C$, since that would have ``no effect'' on its opponent, and would counterfactually yield the better outcome $(D,C)$ in place of $(C,C)$. Thus one might argue that $\mathrm{FairBot}_n$ is acting sub-optimally in this scenario: its response could be changed to obtain a better outcome, $(D,C)$.
However, such reasoning is misplaced from a strategic standpoint. $\mathrm{FairBot}_n$ cannot output $D$ while its opponent $\mathrm{FairBot}_m$ outputs $C$, for that outcome would be logically incoherent. Although the instance of $\mathrm{FairBot}_n$ running as Player 2 has no causal effect on the $\mathrm{FairBot}_m$ running as Player 1, it cannot treat its decision as independent: the outcome $(C,D)$ is simply not attainable by any agent under any circumstances when Player 1 is $\mathrm{FairBot}_m$.
This prompts a re-thinking of what it means to make an optimal decision as an algorithm whose source code is transparent. Such questions, and some of their long-term relevance, have already been considered at length in \citet{Soares:2015:TowardIDT}.
\section{Summary}
We have discovered a version of L\"{o}b's Theorem which can be applied to algorithms with bounded computational resources. This result, in turn, can be used by algorithmic agents that have access to one another's source codes to achieve cooperative outcomes (among other things) that out-perform classical Nash equilibria and correlated equilibria, via conditions that are much more robust than previously known examples depending on program equality. Moreover, the causal pathway by which each agent benefits from its own decision to cooperate happens {\em before} the agent actually computes its decision, which prompts a re-thinking of the causal analysis of optimal decision-making known as Causal Decision Theory in a setting where decision-making agents are algorithms with transparent source-codes.
In light of these findings, classical game theoretic results and the intuitions we derive from them may be quite far from describing what we should actually expect from systems of agents capable of reasoning about each other's design. In order to ensure robust and beneficial long-term deployment of advanced AI technologies in the future, as described in \citet{Russell:2015:ResearchPriorities} and supported by over 100 researchers in the Future of Life Institute's Open Letter \citep{Tegmark:2015:FLIOpenLetter}, it seems prudent to investigate these dynamics ahead of time, so as to be prepared for the sorts of game-theoretic scenarios that might arise between algorithmic agents in the future.
As a direction for potential future investigation, it seems inevitable that other agents described in the purely logical (non-computable) setting of \citet{Barasz:2014:RobustCooperation} and \citet{LaVictoire:2014:PrisDilemmaLob} will likely have bounded, algorithmic analogs, and that many more general consequences of L\"{o}b's Theorem---perhaps all the theorems of G\"{o}del--L\"{o}b provability logic---will have resource-bounded analogs as well.
\section*{Acknowledgements}
My decision to search for a result in this area was strongly influenced by Paul Christiano's belief that some such result should exist. As well, conversations with Patrick LaVictoire, Jessica Taylor, Sam Eisenstat, and Jacob Tsimerman were helpful in sanity-checking my ideas and maintaining my interest in the problem.
This research was supported as part of the Future of Life Institute (futureoflife.org) FLI-RFP-AI1 program, grant~\#2015-144576.
\printbibliography
\end{document} |
1,116,691,499,371 | arxiv | \section*{Introduction}
In their paper \cite{bps} M.A. Barja, R. Pardini and L. Stoppino introduce and study the \emph{continuous rank function} associated to a line bundle $M$ on a variety $X$ equipped with a morphism $X\buildrel f\over\rightarrow A$ to a polarized abelian variety.
Motivated by their work, we consider more generally \emph{cohomological rank functions} -- defined in a similar way -- of a bounded complex $\mathcal{F}$ of coherent sheaves on a polarized abelian variety $(A,{\underline l})$ defined over an algebraically closed field of characteristic zero. As it turns out, these functions often encode interesting
geometric information. The purpose of this paper is to establish some general structure results about them and show some examples of application.
Let $L$ be an ample line bundle on an abelian variety $A$, let ${\underline l}=c_1(L) $ and let $\varphi_{\underline l}:A\rightarrow \widehat A$ be the corresponding isogeny. The cohomological rank functions of $\mathcal{F}\in \mathrm{D}^b(A)$ with respect to the polarization ${\underline l}$ are initially defined (see Definition \ref{def} below) as certain continuous rational-valued functions
\[h^i_{\mathcal{F},{\underline l}}:\mathbb Q\rightarrow\mathbb Q^{\ge 0}. \>\footnote{In the present context the above-mentioned continuous rank function of Barja-Pardini-Stoppino is recovered as $h^0_{f_*M,\, {\underline l}}$ (see the notation above).} \]
The definition of these functions is peculiar to abelian varieties (and more generally to irregular varieties), as it uses the isogenies $\mu_b:A\rightarrow A$, \ $z\mapsto bz$. For $x\in\mathbb Z$, \ $h^i_{\mathcal{F},{\underline l}}(x):=h^i_\mathcal{F}(x{\underline l})$ coincides with the generic value of $h^i(A,\mathcal{F}\otimes L^x)$, for $L$ varying among all line bundles representing ${\underline l}$. This is extended to all $x\in\mathbb Q$ using the isogenies $\mu_b$. In fact the rational numbers $h^i_\mathcal{F}(x{\underline l})$ can be interpreted as generic cohomology ranks of the $\mathbb Q$-twisted coherent sheaf (or, more generally, $\mathbb Q$-twisted complex of coherent sheaves) $\mathcal{F}\langle x{\underline l}\rangle$ (in the sense of Lazarsfeld \cite[\S6.2A]{laz2}).
The above functions are closely related to the Fourier-Mukai transform $\Phi_{\mathcal P}:\mathrm{D}^b(A)\rightarrow \mathrm{D}^b(\widehat A)$ associated to the Poincar\'e line bundle and our first point consists in exploiting systematically such relation. We prove the following transformation formula (Proposition \ref{inversion-a} below)
\begin{align}\label{align}
h^i_{\mathcal{F}}(x{\underline l})\> = &
\> \frac{(-x)^g}{\chi({\underline l})}h^i_{\varphi_{\underline l}^*\Phi_{\mathcal P}(\mathcal{F})}(-\frac{1}{x}{\underline l})&\quad\hbox{for $x\in \mathbb Q^-$}\\
h^i_{\mathcal{F}}(x{\underline l}) \> = &
\> \frac{x^g}{\chi({\underline l})}h^{g-i}_{\varphi_{\underline l}^*\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)}(\frac{1}{x}{\underline l})&\quad\hbox{for $x\in \mathbb Q^+$}
\end{align}
This has several consequences, summarized in the following theorem. The proof and discussion of the various items are found in Sections 2,3 and 4.
\begin{theoremalpha}\label{summary} Let $\mathcal{F}\in \mathrm{D}^b(A)$ and $i\in\mathbb Z$. Let $g=\dim A$.\\
\emph{(1) (Corollaries \ref{inversion}, \ref{inversion-Q}, \ref{continuity}.) } For each $x_0\in\mathbb Q$ there are $\epsilon^-,\epsilon^+>0$ and two \emph{(explicit, see below)} polynomials $P_{i,\, \mathcal{F},\,x_0}^+,P_{i,\, \mathcal{F},\,x_0}^-\in\mathbb Q[x]$ of degree $\le g$ such that $P_{i,\, \mathcal{F},\,x_0}^+(x_0)=P_{i,\, \mathcal{F},\,x_0}^-(x_0)$ and
\begin{eqnarray*}h^i_{\mathcal{F}}(x{\underline l})=&P_{i,\,\mathcal{F},\,x_0}^-(x)&\quad\hbox{for $x\in (x_0-\epsilon^-,x_0]\cap\mathbb Q$} \\
h^i_{\mathcal{F}}(x{\underline l})=&P^+_{i,\, \mathcal{F},\,x_0}(x)&\quad\hbox{for $x\in [x_0,x_0+\epsilon^+)\cap\mathbb Q$}
\end{eqnarray*}
\noindent \emph{(2) (Proposition \ref{derivatives})} Let $k<g$ and $x_0\in\mathbb Q$. If the function $h^i_{\mathcal{F},{\underline l}}$ is strictly of class ${\mathcal C}^k$ at $x_0$ then the jump locus $J^{i+}(\mathcal{F}\langle x_0\rangle)$ \emph{(see \S4 for the definition)} has codimension $\le k+1$.
\noindent \emph{(3) (Theorem \ref{c0})} The function $h^i_{\mathcal{F},{\underline l}}$ extends to a continuous function $h^i_{\mathcal{F},{\underline l}}:\mathbb R\rightarrow \mathbb R^{\ge 0}$.
\footnote{ This theorem provides partial answers to some questions raised, in the specific case of the above mentioned continuous rank functions $h^0_{f_*M,\, {\underline l}}$, in \cite{bps}, e.g. Question 8.11. We also point out that for such functions item (3) of the present Theorem, as well as some additional properties, were already proved in \emph{loc. cit.} via different methods.}
\end{theoremalpha}
It follows from (1) that for $x_0\in\mathbb Q$ the function $h^i_{\mathcal{F},{\underline l}}$ is smooth at $x_0$ if and only if the two polynomials $P^-_{i,\,\mathcal{F},\, x_0}$ and $P^+_{i,\,\mathcal{F},\, x_0}$ coincide. If this is not the case $x_0$ is called a \emph{critical point}.
It turns out (Corollary \ref{inversion}) that for $x_0\in\mathbb Z$ the two polynomials $P^-_{i,\,\mathcal{F}\,\,x_0}(x)$ and $P^+_{i,\,\mathcal{F},\,x_0}(x)$ are obtained from the Hilbert polynomials (with respect to the polarization ${\underline l}$) of the two coherent sheaves $\mathcal{G}^{i,+}_{x_0}:=\varphi_{\underline l}^*R^i\Phi_{\mathcal P}(\mathcal{F}\otimes L^{x_0})$ and $\mathcal{G}^{i,-}_{x_0}:=\varphi_{\underline l}^*R^{g-i}\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee\otimes L^{-x_0})$ in the following way:
\begin{align}\label{align2}P^-_{i,\,x_0,\mathcal{F}}(x)\> = &
\> \frac{(-x)^g}{\chi({\underline l})}\chi_{\mathcal{G}^{i,+}_{x_0}}(-\frac{1}{x}{\underline l})\\
P^+_{i,\,x_0,\mathcal{F}}(x)\>=&\> \frac{x^g}{\chi({\underline l})}\chi_{\mathcal{G}^{i,-}_{x_0}}(\frac{1}{x}{\underline l})\>.
\end{align}
For non-integer $x_0\in \mathbb Q$ the two polynomials $P^-_{i,\,\mathcal{F}\,\,x_0}(x)$ and $P^+_{i,\,\mathcal{F},\,x_0}(x)$ have a similar description after reducing to the integer case (Corollary \ref{inversion-Q}).
Thus item (2) of Theorem \ref{summary} tells that, for $x_0\in\mathbb Q$, the first $k$ coefficients
of the
polynomials $P^-_{i,\,x_0,\mathcal{F}}(x)$ and $P^+_{i,\,x_0,\mathcal{F}}(x)$
coincide as soon as the rank function \ ${\rm Pic}^0 A\rightarrow \mathbb Z^{\ge 0}$ defined by $\alpha\mapsto h^i(\mathcal{F}\otimes L^{x_0}\otimes P_\alpha)$ has jump locus of codimension $\ge k+1$. In this last formulation we are implicitly assuming that $x_0$ is integer but for rational $x_0$ the situation is completely similar. However there might be irrational critical points (see e.g. Example \ref{irrational}), and at present we lack any similar interpretation for them.
In Section 5 we relate cohomological rank functions with the notions of GV, M-regular and IT(0)-sheaves, which are extended here to the $\mathbb Q$-twisted setting. We provide formulations of Hacon's results (\cite{hac}), and some related ones, which are simpler and more convenient even for usual sheaves. Finally, in Section 6 we point out some integral properties of cohomological rank functions.
It seems that the critical points of the function
and the polynomials $P^-_{i,\> \mathcal{F},\, x_0}$ and $P^+_{i,\mathcal{F},x_0}$ are interesting and sometimes novel invariants in many concrete geometric situations. We exemplify this in the following two applications.
\noindent{\bf Application to GV-subschemes. } Our first example concerns GV-subschemes of principally polarized abelian varieties (here will assume that the ground field is $\mathbb C$). This notion (we refer to Section 7 below for the definition and basic properties) was introduced in \cite{minimal} in the attempt of providing a Fourier-Mukai approach to the minimal class conjecture (\cite{debarre}), predicting that the only effective algebraic cycles representing the minimal classes $\frac{\underline\theta^{g-d}}{(g-d)!}\in H^{2(g-d)}(A,\mathbb Z)$ are (translates of) the subvarieties $\pm W_d(C)$ of Jacobians $J(C)$, and $\pm F$, the Fano surface in the intermediate Jacobian of a cubic threefold.
It is known that the subvarieties $W_d(C)$ of Jacobians, as well as the Fano surface
(\cite{ho1}) are GV-subschemes and that, on the other hand, geometrically non-degenerate GV-subschemes have minimal classes (\cite{minimal}). Therefore it was conjectured in \emph{loc. cit.} that geometrically non-degenerate GV-subschemes are either (translates of) $\pm W_d(C)$ or $\pm F$ as above. Denoting $g$ the dimension of the p.p.a.v. and $d$ the dimension of the subscheme, this is known only in a few cases: (i) for $d=1$ and $d=g-2$ (\emph{loc. cit.});
(ii) for $g=5$, settled in the recent work \cite{cps}, (iii) for Jacobians and intermediate Jacobians of generic cubic threefolds, as consequences of the main results of respectively \cite{debarre} and \cite{ho2}. In the recent work \cite{s} it is proved that geometrically non-degenerate GV-subschemes are reduced and irreducible and that the geometric genus of their desingularizations is the expected one, namely $g\choose d$.
As an application of cohomological rank functions we prove that the Hilbert polynomial as well as all $h^i({\mathcal O}_X)$'s are the expected ones:
\begin{theoremalpha}\label{introGV} Let $X$ be geometrically non-degenerate GV-subvariety of dimension $d$ of a principally polarized complex abelian variety $(A,\theta)$. Then:
\noindent (1) (Theorem \ref{gv1}) \ $\chi_{{\mathcal O}_X}(x\underline\theta)=\sum_{i=0}^d{g\choose i}(x-1)^i$.
\noindent (2) (Theorem \ref{hodgenumber}) $h^i({\mathcal O}_X)={g\choose i}$ for all $i=1,\dots, d$.
\end{theoremalpha}
The proof of (1) is based on the study of the function $h^0_{{\mathcal O}_X}(x\underline\theta)$ at the highest critical point (which turns out to be $x=1$). (2) follows from (1) via another argument involving the Fourier-Mukai transform.
As a corollary of $(2)$, combining with the results of \cite{s} and \cite{cps}, we have
\begin{propositionalpha} \emph{(Corollary \ref{rational-singularity}).} A 2-dimensional geometrically non-degenerate GV-subscheme is normal with rational singularities.
\end{propositionalpha}
\noindent{\bf Application to multiplication maps of global sections of line bundles and normal generation of abelian varieties. } Finally we illustrate the interest of cohomological rank functions in another example: the ideal sheaf of one (closed) point $p\in A$. The functions
$h^i_{\mathcal{I}_p}(x{\underline l})$
seem to be highly interesting ones, especially in the perspective of basepoint-freeness criteria for primitive line bundles on abelian varieties. While we defer this to a subsequent paper, here we content ourselves to point out an elementary -- but surprising -- relation with multiplication maps of global sections of powers of line bundles.
We consider the critical point
\[\beta({{\underline l}})
=\inf \{x\in\mathbb Q\>|\> h^1_{\mathcal{I}_p}(x{\underline l})=0\}\]
(as the notation suggests, such notion does not depend on $p\in A$).
A standard argument shows that in any case $\beta({\underline l})\le 1$ and $\beta({\underline l})=1$ if and only if the polarization ${\underline l}$ has base points, i.e. a line bundle $L$ representing ${\underline l}$ (or, equivalently, all of them) has base points. Therefore, given a rational number $x=\frac{a}{b}$, it is suggestive to think that the inequality $\beta({\underline l})<x$ holds if and only if ``the rational polarization $x{\underline l}$ is basepoint-free". Explicitly, this means the following: let $\mu_b:A\rightarrow A$ be the multiplication-by-$b$ isogeny. Then, as it follows from the definition of cohomological rank functions, $\beta({\underline l})<\frac{a}{b}$ means that the finite scheme $\mu_b^{-1}(p)$ imposes independent conditions to all translates of a given line bundle $L^{ab}$ with $c_1(L)={\underline l}$.
In turn $\frac{a}{b}=\beta({\underline l})$ means that $\mu_b^{-1}(p)$ imposes dependent conditions to a proper closed subset of translates of the line bundle $L^{ab}$ as above.\footnote{Writing $1=\frac{b}{b}$ one recovers the usual notions of basepoint-freeness and base locus.} At present we don't know how to compute, or at least bound efficiently, the invariant $\beta({\underline l})$ of a \emph{primitive} polarization ${\underline l}$ (except for principal polarizations of course).
Here is one of the reasons why one is lead to consider the number $\beta({\underline l})$. Let $\underline n$ be another polarization on $A$. We assume that $\underline n$ is basepoint-free. Let $N$ be a line bundle representing $\underline n$ and let $M_N$ be the kernel of the evaluation map of global sections of $N$.
We consider the critical point
\[s({\underline n})=
\inf\{x\in\mathbb Q\>|\> h^1_{M_N}(x\underline n)=0\}\]
(again this invariant does not depend on the line bundle $N$ representing $\underline n$). Well known facts about the vector bundles $M_N$ yield that,
given $x\in\mathbb Z^+$, $s(\underline n)\le x$ if and only if the multiplication maps of global sections
\begin{equation}\label{mult} H^0(N)\otimes H^0(N^x\otimes P_\alpha)\rightarrow H^0(N^{x+1}\otimes P_\alpha)
\end{equation} are surjective for general $\alpha\in \widehat A$ and, furthermore, $s({\underline n})< x$ if and only if the surjectivity holds for all $\alpha\in \widehat A$. Now the cohomological rank function leads to consider a "fractional" version of the maps (\ref{mult}). Writing $x=\frac{a}{b}$, these are the multiplication maps of global sections
\begin{equation}\label{mult-frac1}H^0(N)\otimes H^0(N^{ab}\otimes P_\alpha)\rightarrow H^0(\mu_b^*(N)\otimes N^{ab}\otimes P_\alpha)
\end{equation}
obtained by composing with the natural inclusion $H^0(N)\hookrightarrow H^0(\mu_b^*N)$. It follows
that $s(\underline n)\le \frac{a}{b}$ if and only if the maps (\ref{mult-frac1}) are surjective for general $\alpha\in\widehat A$.
The strict inequality holds if the surjectivity holds for all $\alpha\in \widehat A$. \footnote{Again a simple computation shows that when $x$ is an integer, writing $x=\frac{xb}{b}$ one recovers the usual notions of surjectivity of the maps (\ref{mult}) for every (resp. for general) $\alpha\in\widehat A$.}
As a simple consequence of the formulas (\ref{align}) applied to ${\underline n}=h{\underline l}$ we have
\begin{theoremalpha}\label{b-s}Let $h$ be an integer such that the polarization $h{\underline l}$ is basepoint-free \emph{(hence $h\ge 1$ if ${\underline l}$ is basepoint-free, $h\ge 2$ otherwise)}. Then
\begin{equation}\label{mult-frac}s(h {{\underline l}})= \frac{\beta({{\underline l}})}{h-\beta({{\underline l}})}\>.
\end{equation}
\end{theoremalpha}
Since $\beta({\underline l})\le 1$ it follows that
\[s(h {{\underline l}})\le \frac{1}{h-1}\]
and equality holds if and only if $\beta({\underline l})=1$, i.e. ${\underline l}$ has base points.
Surprisingly, this apparently unexpressive result summarizes, generalizes and improves what is known about the surjectivity of multiplication maps of global sections and projective normality of line bundles on abelian varieties. For example, the case $h=2$ alone tells that $s(2{\underline l})\le 1$, with equality if and only if $\beta({\underline l})=1$, i.e. ${\underline l}$ has base points. In view of the above, this means that the multiplication maps (\ref{mult}) for a second power $N=L^2$ and $x=1$ are in any case surjective for general $\alpha\in\widehat A$, and in fact for all $\alpha\in\widehat A$ as soon as ${\underline l}$ is basepoint free. This is a classical result which implies all classical results on projective normality of abelian varieties proved via theta-groups by Mumford, Koizumi, Sekiguchi, Kempf, Ohbuchi and others (see \cite{kempf} \S6.1-2, \cite{birke-lange} \S7.1-2 and references theiren, see also \cite{sav} and \cite{pp1} for a theta-group-free treatment). We refer to Section 8 below for more on this.
Finally if ${\underline l}$ is basepoint-free and $h=1$ the above Theorem tells that $\beta({\underline l})<\frac{1}{2}$ if and only if the multiplication maps (\ref{mult}) for $N=L$ and $x=1$ are surjective for all $\alpha\in \widehat A$. Using a well known argument, this implies
\begin{corollaryalpha} Assume that ${\underline l}$ is basepoint-free and $\beta({\underline l})<\frac{1}{2}$. Then ${\underline l}$ is projectively normal \emph{(this means that all line bundles $L\otimes P_\alpha$ are projective normal)}.
\end{corollaryalpha}
This is at the same time an explanation and a generalization of Ohbuchi's theorem (\cite{oh}) asserting that, given a polarization ${\underline n}$, $2{\underline n}$ is projectively normal
as soon as ${\underline n}$ is basepoint-free.
Finally, we remark that, although the applications presented in this paper concern abelian varieties and their subvarieties, the study of cohomological rank functions can be applied to the wider context of \emph{irregular varieties},
namely varieties having non-constant morphisms to an abelian varieties, say $f:X\rightarrow A$ (as mentioned above this is indeed the point of view of the paper \cite{bps}). Given
an element $\mathcal{F}\in \mathrm{D}^b(X)$, this can be done by considering the cohomological rank functions of the complex $Rf_*\mathcal{F}$.
\subsection*{Acknowledgements}
We thank Federico Caucci, Rob Lazarsfeld, Luigi Lombardi and Stefan Schreieder for useful comments and suggestions. We are especially grateful to Schreieder for pointing out a gap in Section 7 of a previous version of this paper.
\section{Notation and background material}
We work on an algebraically closed ground field of characteristic zero.
\noindent A polarization \emph{${\underline l}$} on an abelian variety is the class of an ample line bundle $L$ in $\mathrm{Pic} A/{\rm Pic}^0 A$. The corresponding isogeny is denoted
\[\varphi_{\underline l}:A\rightarrow \widehat A\]
where $\widehat A:={\rm Pic}^0 A$. For $b\in\mathbb Z$
\[\mu_b:A\rightarrow A\qquad z\mapsto bz\]
denotes the multiplication-by-$b$ homomorphism.
Let $A$ be a $g$-dimensional abelian variety.
We denote ${\mathcal P}$ the Poincar\'e line bundle on $A\times \widehat A$. For $\alpha\in\widehat A$ the corresponding line bundle in $A$ is denoted by $P_\alpha$, i.e. $P_\alpha={\mathcal P}_{|A\times\{ \alpha\}}$. We always denote $\hat e$ the origin of $\hat{A}$.
Let $\mathrm{D}^b(A)$ be the bounded derived category of coherent sheaves on $A$ and denote by
\[\Phi^{A\rightarrow \widehat A}_{\mathcal P}: \mathrm{D}^b(A)\rightarrow \mathrm{D}^b(\widehat{A})\]
the Fourier-Mukai functor associated to ${\mathcal P}$. It is an equivalence (\cite{mukai}), whose quasi-inverse is
\begin{equation}\label{mukai-0}\Phi^{\widehat A\rightarrow A}_{{\mathcal P}^\vee[g]}: \mathrm{D}^b(\widehat A)\rightarrow \mathrm{D}^b(A)
\end{equation}
When possible we will suppress the direction of the functor from the notation, writing simply $\Phi_{\mathcal P}$. Since ${\mathcal P}^\vee=(-1_A,1_{\widehat A})^*{\mathcal P}=(1_A,-1_{\widehat A})^*\mathcal P$ it follows that $\Phi_{{\mathcal P}^\vee}=(-1)^*\Phi_{\mathcal P}$.
Finally, we will denote $R^i\Phi_{\mathcal P}$ the induced $i$-th cohomology functors.
For the reader's convenience we list some useful facts, in use throughout the paper, concerning the above Fourier-Mukai equivalence.
\noindent - \emph{Exchange of direct and inverse image of isogenies \emph{(\cite{mukai} (3.4))}. } Let $\varphi:A\rightarrow B$ be an isogeny of abelian varieties and let $\hat\varphi: \widehat B\rightarrow \widehat A$ be the dual isogeny. Then
\begin{equation}\label{mukai-1}\hat\varphi^*\Phi_{{\mathcal P}_A}(\mathcal{F})=\Phi_{{\mathcal P}_B}\varphi_*(\mathcal{F}), \qquad \hat\varphi_*\Phi_{{\mathcal P}_B}(\mathcal{G})=\Phi_{{\mathcal P}_A}\varphi^*(\mathcal{G})
\end{equation}
\noindent - \emph{Exchange of derived tensor product and derived Pontryagin product \emph{(\cite{mukai} (3.7))}. }
\begin{equation}\label{mukai-2}
\Phi_{\mathcal P}(\mathcal{F}*\mathcal{G})=(\Phi_{\mathcal P}\mathcal{F})\otimes (\Phi_{\mathcal P}\mathcal{G})\qquad \Phi_{\mathcal P}(\mathcal{F}\otimes \mathcal{G})=(\Phi_{\mathcal P}\mathcal{F})* (\Phi_{\mathcal P}\mathcal{G})[g]
\end{equation}
\noindent - \emph{Serre-Grothendieck duality \emph{(\cite{mukai} (3.8). See also \cite{pp2} Lemma 2.2)}. }
As customary, for a given projective variety $X$ (in what follows $X$ will be either $A$ or $\widehat A$) and $\mathcal{F}\in \mathrm{D}^b(X)$, we denote $\mathcal{F}^{\vee}:={\mathcal R} Hom(\mathcal{F}, {\mathcal O}_X)\in \mathrm{D}^b(X)$. Then
\begin{equation}\label{mukai-3}
(\Phi_{\mathcal P}\mathcal{F})^\vee=\Phi_{{\mathcal P}^\vee}(\mathcal{F}^\vee)[g]
\end{equation}
\noindent - \emph{The transform of a non-degenerate line bundle \emph{(\cite{mukai} Prop. 3.11(1))}. } Given an ample line bundle on $A$, the Fourier-Mukai transform $\Phi_\mathcal P(L)$ is a locally free sheaf (concentrated in degree $0$) on $\widehat A$, denoted by $\widehat L$, of rank equal to $h^0(L)$. Moreover
\begin{equation}\label{mukai-4}
\varphi_{{\underline l}}^*\widehat L\simeq H^0(L)\otimes L^{-1}=(L^{-1})^{\oplus h^0(L)}
\end{equation}
\noindent - \emph{The Pontryagin product with a non-degenerate line bundle \emph{(\cite{mukai} (3.10))}. } Given a non-degenerate line bundle $N$ on $A$, we denote $\underline n=c_1(N)$. Let $\mathcal{F}\in \mathrm{D}^b(A)$. Then
\begin{equation}\label{mukai-5} \mathcal{F}*N=N\otimes\varphi_{\underline n}^*\bigl(\Phi_{\mathcal P}((-1)^*\mathcal{F})\otimes N)\bigr)
\end{equation}
\noindent - \emph{(Hyper)cohomology and derived tensor product \emph{(\cite {pp2} Lemma 2.1)}. } Let $\mathcal{F}\in \mathrm{D}^b(A)$ and $\mathcal{G}\in\mathrm{D}^b(\widehat A)$.
\begin{equation}\label{exchange} H^i(A,\mathcal{F}\otimes \Phi^{\widehat A\rightarrow A}_{\mathcal P}(\mathcal{G}))=H^i(\widehat A, \Phi^{A\rightarrow \widehat A}_{\mathcal P}(\mathcal{F})\otimes \mathcal{G} )
\end{equation}
\section{Cohomological rank functions on abelian varieties}
In this section we define a certain non-negative rational number as the rank of the cohomology of a coherent sheaf (or, more generally, of the hypercohomology of a complex of coherent sheaves) twisted with a rational power of a polarization. This definition is already found in \cite{barja} and, somewhat implicitly, a notion like that was already in use in \cite{kollar} (proof of Thm 17.12) and \cite{pp3} (proof of Thm 4.1). This provides rational cohomological rank functions satisfying certain transformation formulas under Fourier-Mukai transform (Prop. \ref{inversion} below). It follows that these functions are polynomial almost everywhere and extend to continuous functions on an open neighborhood of $\mathbb Q$ in $\mathbb R$ (Corollaries \ref{inversion-Q} and \ref{continuity}).
\begin{definition}\label{def}
(1) Given $\mathcal{F}\in \mathrm{D}^b(A)$ and $i\in \mathbb Z$, define
\[h^i_{gen}(A, \mathcal{F})\]
as the dimension of hypercohomology $H^i(A,\mathcal{F}\otimes P_\alpha)$, for $\alpha$ general in $\widehat A$. \footnote{It is well known that hypercohomology groups as the above satisfy the usual base-change and semicontinuity properties, see e.g. \cite{pp2} proof of Lemma 3.6 and \cite{g} 7.7.4 and Remarque 7.7.12(ii).}
\noindent (2) Given $\mathcal{F}\in \mathrm{D}^b(A)$, a polarization ${\underline l}$ on $A$ and $x={\frac{a} {b}}\in\mathbb Q$, $b>0$, we define
\[h^i_{\mathcal{F}}(x{\underline l})=b^{-2g}\, h^i_{gen}(A,\mu_b^*(\mathcal{F})\otimes L^{ab})\]
\end{definition}
The definition is dictated from the fact that the degree of $\mu_b:A\rightarrow A$ is $b^{2g}$ (see the previous section for the notation) and $\mu_b^*({\underline l})=b^2{\underline l}$. Therefore the pullback via $\mu_b$ of the class $\frac{a}{b}{\underline l}$ is $ab{\underline l}$. It is easy to check that the definition does not depend on the representation $x=\frac{a}{b}$. For example, if $n\in\mathbb Z$, writing $n=\frac{nb}{b}$ one gets
\[b^{-2g}h^i_{gen}((\mu_b^*\mathcal{F})\otimes L^{b^2n})=b^{-2g}h^i_{gen}(\mu_b^*(\mathcal{F}\otimes L^n))=b^{-2g}\sum_{\alpha\in \hat\mu_b^{-1}(\hat e)}h^i_{gen}(\mathcal{F}\otimes L^n\otimes P_\alpha)=h^i_{gen}(\mathcal{F}\otimes L^n)\]
where $\hat e$ is the identity point of $\widehat A$ and $\hat\mu_b:\widehat A\rightarrow \widehat A$ is the dual isogeny.
\begin{remark}\label{Q-twisted}[Coherent sheaves $\mathbb Q$-twisted by a polarization] Let ${\underline l}$ be a polarization on our abelian variety $A$. Following Lazarsfeld (\cite{laz2}), but somewhat more restrictively, we will define \emph{ coherent sheaves $\mathbb Q$-twisted by ${\underline l}$} as equivalence classes of pairs $(\mathcal{F},x{\underline l})$ where $\mathcal{F}$ is a coherent sheaf on $A$ and $x\in\mathbb Q$, with respect to the equivalence relation generated by $(\mathcal{F}\otimes L^h,x{\underline l})\sim (\mathcal{F},(h+x){\underline l})$, for $L$ a line bundle representing ${\underline l}$ and $h\in\mathbb Z$. Such thing is denoted $\mathcal{F}\langle x{\underline l}\rangle$ (note that\ $\mathcal{F}\otimes P_\alpha\langle x{\underline l}\rangle=\mathcal{F}\langle x{\underline l}\rangle$\ for $\alpha\in \widehat A$). Similarly, one can define complexes of coherent sheaves $\mathbb Q$-twisted by the polarization ${\underline l}$.
Now the quantity $h^i_\mathcal{F}(x{\underline l})$ depends only on the $\mathbb Q$-twisted complex $\mathcal{F}\langle x{\underline l}\rangle$ and one may think of it as the (generic) cohomology rank
$h^i(A,\mathcal{F}\langle x{\underline l}\rangle)$.
\end{remark}
Some immediate basic properties of generic cohomology ranks defined above are:
\noindent (a) \ $\chi_\mathcal{F}(x{\underline l})=\sum_i (-1)^ih^i_{\mathcal{F}}(x{\underline l})$, \
where $\chi_\mathcal{F}(x{\underline l})$ is the Hilbert polynomial, i.e. the Euler characteristic.
\noindent (b) Serre duality: $h^i_{\mathcal{F}}(x{\underline l})=h^{g-i}_{\mathcal{F}^{\vee}}(-x{\underline l})$.
\noindent (c) Serre vanishing: \emph{given a coherent sheaf $\mathcal{F}$ there is a $x_0\in \mathbb Q$ such that $h^i_\mathcal{F}(x{\underline l})=0$ for all $i>0$ and for all rational $x\ge x_0$. }
\noindent \emph{Proof of} (c). It is well known that there is $n_0\in \mathbb Z$ such that $h^i(A,\mathcal{F}\otimes L^{n_0}\otimes P_\alpha)=0$ for all $i>0$ and for all $\alpha\in{\rm Pic}^0 X$. Following the terminology of Mukai, this condition is referred to as follows: \emph{$\mathcal{F}\otimes L^{n}$ satisfies IT(0)} (the Index Theorem with index 0, see also \S5 below). Therefore, for all $b\in \mathbb Z^+$, $ \mu_b^*(\mathcal{F})\otimes L^{b^2n_0}$ satisfies IT(0). The tensor product of a coherent IT(0) sheaf with a locally free IT(0) sheaf is IT(0) (see e.g. \cite[Prop. 3.1]{pp3} for a stronger result). Therefore $\mu_b^*(\mathcal{F})\otimes L^m$ satisfies IT(0) for all $b\in \mathbb Z^+$ and $m\ge b^2n_0$. This is more than enough to ensure that $h^i_{\mathcal{F}}(x{\underline l})=0$ for all rational numbers $x\ge n_0$.\footnote{More precisely this proves, in the terminology of Section 5 below, that the $\mathbb Q$-twisted coherent sheaves $\mathcal{F}\langle x{\underline l}\rangle$ satisfy IT(0) for all $x\in\mathbb Q^{\ge n_0}$.} \endproof
The following Proposition describes the behavior of the generic cohomology ranks with respect to the Fourier-Mukai transform.
\begin{proposition}\label{inversion-a} Let $\mathcal{F}\in \mathrm{D}^b(A)$ and let ${\underline l}$ be a polarization on $A$. Then, for $x\in\mathbb Q^+$
\[h^i_{\mathcal{F}}(x{\underline l}) = \frac{x^g}{\chi({\underline l})}h^{g-i}_{\varphi_{\underline l}^*\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)}(\frac{1}{x}{\underline l})\]
and, for $x\in\mathbb Q^-$,
\[h^i_{\mathcal{F}}(x{\underline l}) =
\frac{ (-x)^g}{\chi({\underline l})}h^i_{\varphi_{\underline l}^*\Phi_{\mathcal P}(\mathcal{F})}(-\frac{1}{x}{\underline l})\]
\end{proposition}
\begin{proof}
Let us start with the case $x=\frac{a}{b}\in\mathbb Q^+$. Then,
\begin{eqnarray*}
h^i_{\mathcal{F}}(x{\underline l})=\frac{1}{b^{2g}}h^i_{gen}(A, \mu_b^*\mathcal{F}\otimes L^{ab})&=&\frac{1}{b^{2g}}\dim \mathrm{Ext}^i_A(\mu_b^*\mathcal{F}^{\vee}, L^{ab}_{\alpha})\\&=&\frac{1}{b^{2g}}\dim \mathrm{Ext}^i_{\hat{A}}(\Phi_{\mathcal P}(\mu_b^*\mathcal{F}^{\vee}),\Phi_{\mathcal P}( L^{ab}_{\alpha})),
\end{eqnarray*}
where $\alpha\in\hat{A}$ is general, $L^{ab}_{\alpha}:=L^{ab}\otimes P_{\alpha}$, and the last equality holds by Mukai's equivalence \cite{mukai}.
Note that, by (\ref{mukai-1}), $\Phi_{\mathcal P}(\mu_b^*\mathcal{F}^{\vee})\simeq \hat{\mu}_{b*}\Phi_{\mathcal P}(\mathcal{F}^{\vee})$
where $\hat{\mu}_{b}: \widehat{A}\rightarrow \widehat{A}$ is the multiplication by $b$ on $\widehat{A}$. By (\ref{mukai-4}) $R\Phi_{\mathcal P}( L^{ab}_{\alpha}):=\widehat{L^{ab}_{\alpha}}$ is a vector bundle on $\widehat{A}$ and
\begin{equation}\label{to-be-inserted}\mu_{ab}^*\varphi_{\underline l}^*\widehat{L^{ab}_{\alpha}}=\varphi_{ab{\underline l}}^*\widehat{L^{ab}_{\alpha}}
\simeq ((L^{ab}_{\alpha})^{-1})^{\oplus h^0(L^{ab})}.
\end{equation}
Hence, for general $\alpha\in\widehat A$,
\begin{eqnarray*}
h^i_{\mathcal{F}}(x{\underline l})&=&\frac{1}{b^{2g}}\dim \mathrm{Ext}^i_{\widehat{A}}(\hat{\mu}_{b*}\Phi_{\mathcal P}(\mathcal{F}^{\vee}), \widehat{L^{ab}_{\alpha}})=\frac{1}{b^{2g}}\dim \mathrm{Ext}^i_{\widehat{A}}(\Phi_{\mathcal P}(\mathcal{F}^{\vee}), \hat{\mu}_{b}^*\widehat{L^{ab}_{\alpha}})\\
&=&\frac{1}{b^{2g}}\dim \mathrm{Ext}^{g-i}_{\widehat{A}}(\hat{\mu}_{b}^*\widehat{L^{ab}_{\alpha}},\Phi_{\mathcal P}(\mathcal{F}^{\vee}) )\\&=&\frac{1}{\deg \hat{\mu}_a\deg \varphi_{\underline l}}\frac{1}{b^{2g}}\dim \mathrm{Ext}^{g-i}_{A}(\varphi_
{\underline l}^*\hat{\mu}_a^*\hat{\mu}_{b}^*\widehat{L^{ab}_{\alpha}},\varphi_{\underline l}^*\hat{\mu}_a^*\Phi_{\mathcal P}(\mathcal{F}^{\vee}) )\\&=&\frac{1}{\chi({\underline l})^2}\frac{1}{a^{2g}b^{2g}}\dim \mathrm{Ext}^{g-i}_{A}(\varphi_{ab{\underline l}}^*\widehat{L^{ab}_{\alpha}},\mu_a^*\varphi_{\underline l}^*\Phi_{\mathcal P}(\mathcal{F}^{\vee}) )\\&\buildrel{(\ref{to-be-inserted})}\over =&\frac{1}{\chi({\underline l})}\frac{1}{a^{g}b^{g}}h^{g-i}(A,\mu_a^*\varphi_{\underline l}^*\Phi_{\mathcal P}(\mathcal{F}^{\vee}) \otimes L^{ab}_{\alpha}).
\end{eqnarray*}
We also note that $(-1)_{\widehat{A}}^*\Phi_{\mathcal P}(\mathcal{F}^{\vee})=\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^{\vee})$ and $(-1)_{{A}}^*{\underline l}={\underline l}$. Hence, applying $(-1)_A^*$, we get
\begin{eqnarray*}h^i_{\mathcal{F}}(x{\underline l})\> =\> \frac{1}{\chi({\underline l})}\frac{1}{a^{g}b^{g}}h^{g-i}_{gen}(\mu_a^*\varphi_{\underline l}^*\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^{\vee}) \otimes L^{ab})\> =\>
\frac{1}{\chi({\underline l})}\frac{a^g}{b^g}h^{g-i}_{\varphi_{\underline l}^*\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^{\vee})}(\frac{b}{a}{\underline l})).\end{eqnarray*}
By similar argument (or by Serre duality) we get the equalities when $x\in\mathbb Q^-$.
\end{proof}
\begin{corollary}\label{inversion} Under the same hypothesis and notation of the previous Proposition, for each $i\in\mathbb Z$ there are $\epsilon^-,\epsilon^+ >0$ and two polynomials $P^-_{i,\mathcal{F}},P^+_{i,\mathcal{F}}\in\mathbb Q [x]$ of degree $\le \dim A$ such that, for $x\in (-\epsilon^-,0)\cap \mathbb Q$,
\[h^i_\mathcal{F}(x{\underline l})\> =\> P^-_{i,\mathcal{F}}(x) \]
and, for $x\in (0,\epsilon^+)\cap \mathbb Q$
\[h^i_\mathcal{F}(x{\underline l})\> =\> P^+_{i,\mathcal{F}}(x)\]
More precisely
\[h^i_{\mathcal{F}}(x{\underline l}) =
\frac{(-x)^g}{\chi({\underline l})}\chi_{\varphi_{\underline l}^*R^i\Phi_{\mathcal P}(\mathcal{F})}(-\frac{1}{x}{\underline l}) \quad\hbox{\emph{for} $x\in (-\epsilon^-,0)\cap \mathbb Q$}\]
\[h^i_{\mathcal{F}}(x{\underline l})=
\frac{x^g}{\chi({\underline l})}\chi_{\varphi_{\underline l}^*R^{g-i}\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)}(\frac{1}{x}{\underline l})\quad\hbox{\emph{for} $x\in(0,\epsilon^+)\cap \mathbb Q$}\]
\end{corollary}
\proof The statement follows from Proposition \ref{inversion} via Serre vanishing (see (c) above in this section). Indeed for a sufficiently small $x\in\mathbb Q^+$ we have that $h^k_{\varphi_{\underline l}^*R^{j}\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)}(\frac 1 {x} {\underline l}))=0$ for all $k\ne 0$ and all $j$. Therefore the hypercohomology spectral sequence computing $ h^{g-i}_{\varphi_{\underline l}^*\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)}(\frac 1 {x} {\underline l}))$ \footnote{this means the spectral sequence computing the hypercohomogy groups $H^{g-i}(A,\mu_a^*(\varphi_{\underline l}^*\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee))\otimes L^{ab}\otimes P_\alpha)$ for $\frac{ b}{a}=\frac{1}{x}$ and $\alpha\in{\rm Pic}^0 X$ general} collapses so that
\[ h^{g-i}_{\varphi_{\underline l}^*\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)}(\frac 1 {x}{\underline l}))=h^0_{\varphi_{\underline l}^*R^{g-i}\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)}(\frac 1 {x} {\underline l}))=\chi_{\varphi_{\underline l}^*R^{g-i}\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)}(\frac 1 {x}{\underline l}))\>.\]
This proves the statement for $x>0$. The proof for the case $x<0$ is the same.
\endproof
\begin{remark}\label{eff-serre} It follows from the proof that one can take as $\epsilon^-$ the minimum, for all $i$, of $\frac{1}{x_i}$, where $x_i$ is a bound ensuring Serre vanishing for twists with powers of $L$ of the sheaf $\varphi_{\underline l}^*R^i\Phi_{\mathcal P}(\mathcal{F})$. Similarly for $\epsilon^+$.
\end{remark}
The next Corollary shows that the statement of the previous Corollary holds more generally in $\mathbb Q$-twisted setting.
\begin{corollary}\label{inversion-Q} Same hypothesis and notation of the previous Proposition. Let $x_0\in\mathbb Q$. For each $i\in\mathbb Z$ there are $\epsilon^-,\epsilon^+ >0$ and two polynomials $P^-_{i,\mathcal{F},x_0},P^+_{i,\mathcal{F},x_0}\in\mathbb Q [x]$ of degree $\le \dim A$ such that, for $x\in (x_0-\epsilon^-,x_0)\cap \mathbb Q$,
\[h^i_\mathcal{F}(x{\underline l})\> =\> P^-_{i,\mathcal{F},x_0}(x) \]
and, for $x\in (x_0,x_0+\epsilon^+)\cap \mathbb Q$
\[h^i_\mathcal{F}(x{\underline l})\> =\> P^+_{i,\mathcal{F},x_0}(x)\]
\end{corollary}
\proof This follows by reducing to the previous Corollary via the formula
\[h^i_\mathcal{F}((x_0+y){\underline l})=b^{-2g}h^i_{\mu_b^*(\mathcal{F})\otimes L^{ab}}(b^2y{\underline l})\]
for $x_0=\frac{a}{b}$, $b>0$.
\endproof
As a consequence we have
\begin{corollary}\label{continuity}
The functions $h^i_{\mathcal{F},{\underline l}}:\mathbb Q\rightarrow \mathbb Q^{\ge 0}$ extend to a continuous functions $h^i_{\mathcal{F},L}: U\rightarrow \mathbb R$, where $U$ is an open subset of $\mathbb R$ containing $\mathbb Q$, satisfying the condition of Corollary \ref{inversion-Q} above, namely for each $x_0\in U$ there exist $\epsilon^-,\epsilon^+>0$ and two polynomials $P_{i,\mathcal{F},x_0}^-,P_{i,\mathcal{F},x_0}^+\in\mathbb Q[x]$ of degree $\le \dim A$, having the same value at $x_0$, such that
\[h^i_{\mathcal{F}}(x_{\underline l})=\begin{cases}P^-_{i,\mathcal{F},x_0}(x)&\hbox{for $x\in (x_0-\epsilon^-,x_0]$}\\
P^+_{i,\mathcal{F},x_0}(x)&\hbox{for $x\in [x_0,x_0+\epsilon^+)\>.$}
\end{cases}\]
\end{corollary}
\proof Let $x_0\in \mathbb Q$.
The assertion follows from Corollary \ref{inversion-Q} because, for $x_0\in\mathbb Q$, $P^-_{i,\mathcal{F},x_0}(x_0)=P^+_{i,\mathcal{F},x_0}(x_0)=h^i_{\mathcal{F}}(x_0{\underline l})$.
To prove this we can assume, using Corollary \ref{inversion-Q}, that $x_0=0$. By Corollary \ref{inversion} we have that $P^-_{i,\mathcal{F},x_0}(0)$ (resp. $P^+_{i,\mathcal{F},x_0}(0)$) coincide, up the the same multiplicative constant, with the coefficients of degree $g$ of the Hilbert polynomial of the sheaves $\varphi_{\underline l}^*R^i\Phi_{\mathcal P}(\mathcal{F})$, resp. $\varphi_{\underline l}^*R^{g-i}\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)$. Hence they coincide, up to the same multiplicative constant, with the generic ranks of the above sheaves. By cohomology and base change, and Serre duality, such generic ranks coincide with $h^i_{\mathcal{F},{\underline l}}(0)$.
\endproof
\begin{remark}\label{acc} It seems likely that $U=\mathbb R$, hence the cohomological rank functions would be piecewise-polynomial (compare \cite[Question 8.11]{bps}). This would follow from the absence of accumulation points in $\mathbb R\smallsetminus U$, but at present we don't know how to prove that. In any case, in the next section we prove that the cohomological rank functions extend to continuous function on the whole $\mathbb R$.
\end{remark}
Given two objects $\mathcal{F}$ and $\mathcal{G}$ in $\mathrm{D}^b(A)$ and $f\in\text{Hom}_{\mathrm{D}^b(A)}(\mathcal{F},\mathcal{G})$ one can define similarly the $i$-th cohomological rank, nullity and corank of the maps
twisted with a rational multiple of a polarization ${\underline l}$ as the generic rank, nullity and corank of the maps
\[H^i(A, \mu_b^*(\mathcal{F})\otimes L^{ab}\otimes P_\alpha)\rightarrow H^i(A,\mu_b^*(\mathcal{G})\otimes L^{ab}\otimes P_\alpha).\]
This gives rise to functions \ $\mathbb Q\rightarrow \mathbb Q^\ge 0$ \ satisfying the same properties.
Let us consider, for example, the rank, (the kernel and the corank have completely similar description) and let us denote it $rk\bigl(h^i_f(x{\underline l})\bigr)$.
\begin{proposition}\label{functions}
Let $x_0\in\mathbb Q$. For each $i\in\mathbb Z$ there are $\epsilon^-,\epsilon^+ >0$ and two polynomials $P^-_{i,f,x_0},P^+_{i,f,x_0}\in\mathbb Q [x]$ of degree $\le \dim A$ such that, for $x\in (x_0-\epsilon^-,x_0)\cap \mathbb Q$,
\[rk\bigl(h^i_f(x{\underline l})\bigr)\> =\> P^-_{i,f, x_0}(x) \]
and, for $x\in (x_0,x_0+\epsilon^+)\cap \mathbb Q$
\[rk\bigl(h^i_f(x{\underline l})\bigr)\> =\> P^+_{i,f,x_0}(x).\]
\end{proposition}
\begin{proof} As above, we can assume that $x_0=0$. By Corollary \ref{inversion} and its proof there is a $\epsilon^->0$ such that for $x=\frac{a}{b}\in (-\epsilon^-,0)$ (with $a<0$ and $b>0$), $rk\bigl(h^i_f(x{\underline l})\bigr)$ coincides with
$\frac{(-x)^g}{\chi({\underline l} )} rk (F_{-\frac{1}{x}})$
where $F_{-\frac{1}{x}}$ is the natural map
\[ F_{-\frac{1}{x}}: H^0\bigl(\mu_{-a}^*(\varphi_{\underline l}^*R^i\Phi_{\mathcal P}\mathcal{F})\otimes L^{-ab}\bigr)\rightarrow H^0\bigl(\mu_{-a}^*(\varphi_{\underline l}^*R^i\Phi_{\mathcal P}\mathcal{G})\otimes L^{-ab}\bigr) \]
By an easy calculation with Serre vanishing (see (c) in this Section), up to taking a smaller $\epsilon^-$ the image of the map $F_{-\frac{1}{x}}$ is $H^0$ of the image of the map coherent sheaves
\[\mu_{-a}^*(\varphi_{\underline l}^*R^i\Phi_{\mathcal P}\mathcal{F})\otimes L^{-ab} \rightarrow \mu_{-a}^*(\varphi_{\underline l}^*R^i\Phi_{\mathcal P}\mathcal{G})\otimes L^{-ab}\]
and its dimension is
\[\chi_{Im(\varphi_{\underline l}^*R^i\Phi_{\mathcal P}(f))}(-\frac{1}{x}{\underline l})\]
In conclusion, for $x\in (-\epsilon^-,0)$
\[rk\bigl(h^i_f(x{\underline l})\bigr)=\frac{(-x)^g}{\chi({\underline l})}\chi_{Im(\varphi_{\underline l}^*R^i\Phi_{\mathcal P}(f))}(-\frac{1}{x}{\underline l}):=P^-_{i,f,0}(x)\]
Similarly, for $x\in (0,\epsilon^+)$
\[rk\bigl(h^i_f(x{\underline l})\bigr)=\frac{(x)^g}{\chi({\underline l})}\chi_{Im(\varphi_{\underline l}^*R^{g-i}\Phi_{\mathcal P^\vee}(f))}(\frac{1}{x}{\underline l}):=P^+_{i,f,0}(x).\]
\end{proof}
\section{Continuity as real functions}
The aim of this section is to prove Theorem \ref{c0} below, asserting that the cohomological rank functions extend to continuous functions on the whole $\mathbb R$ (see Remark \ref{acc}). We
start with a version of Serre's vanishing needed in the proof.
\begin{lemma}\label{effectivebound} Let $A$ be an abelian variety and let $L$ be a very ample line bundle on $A$. Let $\mathcal{F}$ be a coherent sheaf of dimension $n$ on $A$. There exist two integers $M^-$ and $M^+$ such that for all integers $m\in \mathbb Z^+$, for all $k=0,\dots n$ and all sufficiently general complete intersections $Z_k=D_1\cap D_2\cap\cdots \cap D_k$ of $k$ divisors $D_i\in |m^2\rho_iL|$ with $0<\rho_i<1$ rational with $m^2\rho_i\in\mathbb Z$ \emph{(here we understand $Z_0=A$)}, and for all $\alpha\in\widehat A$, the following conditions hold:
\[\begin{cases}h^i(\mu_m^*\mathcal{F}\mid_{Z_k}\otimes L^{m^2t}\otimes P_\alpha)=0&\hbox{for all \ \ $i\geq 1$ and $t\in \mathbb Z^{\ge M^+}$}\\
h^i(\mu_m^*\mathcal{F}\mid_{Z_k}\otimes L^{m^2t}\otimes P_\alpha)=\chi(\mathcal{E}xt^{g-i}(\mu_m^*\mathcal{F}\mid_{Z_k}, \mathcal O_A)\otimes L^{-m^2t})&\hbox{ for all $i\leq n-1-s$ and $t\in \mathbb Z^{\leq M^-}$}
\end{cases}\]
The pair $(M^-,M^+)$ will be referred to as \emph{an effective cohomological bound for $\mathcal{F}$}.
\end{lemma}
\begin{proof} Note that the statement makes sense since $L$ is assumed to be very ample and $\tau_i:=m^2\rho_i\in \mathbb Z^+$.
Since $Z_k$ is a general complete intersection, the Koszul resolution of ${\mathcal O}_{Z_k}$, tensored with $\mu_m^*\mathcal{F}$
\begin{equation}\label{koszul}0\rightarrow \mu_m^*F\otimes L^{-\sum_i\tau_i} \rightarrow \cdots\rightarrow \mu_m^*\mathcal{F}\otimes(\oplus_i L^{-\tau_i})\rightarrow \mu_m^*\mathcal{F}\rightarrow \mu_m^*F\mid_{Z_k}\rightarrow 0
\end{equation}
is exact. Therefore the bound of the upper line is a variant of Serre vanishing, in the version of the previous section, via a standard diagram-chase.
Concerning the lower line, we first prove it for $k=0$. By Serre duality
\begin{eqnarray*}H^i(\mu_m^*\mathcal{F}\otimes L^{m^2t}\otimes P_\alpha)=H^{g-i}((\mu_m^*\mathcal{F})^\vee\otimes L^{-m^2t}\otimes P_\alpha^{\vee} )
= H^{g-i}(\mu_m^*(\mathcal{F}^\vee)\otimes L^{-m^2t}\otimes P_\alpha^{\vee} ).\end{eqnarray*}
Since $\mathcal{F}$ is a coherent sheaf of dimension $n$, $\mathcal{E}xt^j(F, \mathcal{O}_A)$ vanishes for $j<g-n$ while it has
codimension $\geq j$ with support contained in the support of $\mathcal{F}$ for $j\ge g-n$ (see for instance \cite[Proposition 1.1.6]{hl}). We apply Serre vanishing to find an integer $N$ such that
such that
\[H^j(\mu_m^*\mathcal{E}xt^{i}(\mathcal{F}, \mathcal{O}_A)\otimes L^{-m^2t}\otimes P_\alpha^{\vee})=0\quad\hbox{ for all $t\in \mathbb Z$ such that $-t\ge N$ and $j\geq 1$.}\]
The statement of the bottom line for $k=0$ follows via the spectral sequence
\[H^{h}(\mu_m^*{\mathcal Ext}^{g-i-h}(\mathcal{F},{\mathcal O}_A)\otimes L^{-m^2t}\otimes P_\alpha^\vee)\Rightarrow H^{g-i}((\mu_m^*\mathcal{F})^\vee\otimes L^{-m^2t}\otimes P_\alpha^{\vee} ).\]
At this point the statement of the lower line for all $k\le g-n$ follows as above from the case $k=0$ and the fact that for a general choice of a very ample divisor $D$ we have a short exact sequence for all $j\geq 0$ $$0\rightarrow\mu_m^* \mathcal Ext^j(\mathcal{F}, \mathcal O_A)\rightarrow\mu_m^* \mathcal Ext^j(\mathcal{F}, \mathcal O_A)\otimes \mathcal{O}_A(D)\rightarrow \mathcal Ext^{j+1}(\mu_m^*\mathcal{F}\mid_D, \mathcal{O}_A)\rightarrow 0.$$
(see \cite[Lemma 1.1.13]{hl}).
\end{proof}
\begin{theorem}\label{c0} Let $A$ be an abelian variety, let ${\underline l}$ be a polarization on $A$ and $\mathcal{F}\in \mathrm{D}^b(A)$. The functions $x\mapsto h_{\mathcal{F}}^i(x{\underline l})$ extend to continuous functions on $\mathbb R$.
Such functions are bounded above by a polynomial function of degree at most $n=\dim \mathcal{F}$, whose coefficients involve only the intersection numbers of the support of $\mathcal{F}$ with powers of $L$, the ranks of the cohomology sheaves of $\mathcal{F}$ on the generic points of their support, and an effective bound $(N^-, N^+)$ of generic cohomology of $\mathcal{F}$.
\end{theorem}
\begin{proof} We can assume that ${\underline l}$ is very ample. The proof will be in some steps. To begin with, we prove the statement under the assumption that $\mathcal{F}$ is a pure sheaf.
Let $V_1,\ldots, V_s$ be the irreducible components of the support of $\mathcal{F}$ with reduced scheme structures. Hence each $V_j$ is an integral variety. Let $t_j$ be the length of $\mathcal{F}$ at the generic point of $V_j$ and define
$$u(\mathcal{F}):=\sum_jt_j(V_j\cdot L^n)_A.$$ We have seen in the previous lemma that $h_{\mathcal{F}}^i(x{\underline l})$ are natural polynomial functions for $x\leq M^-$ and $x\geq M^+$. We now deal with the case when $M^-\leq x\leq M^+$. More precisely we will prove, by induction on $n=\dim \mathcal{F}$, the following statements
\begin{itemize}
\item[(a)] $h^i_{\mathcal{F}, {\underline l}}$ extends to a continuous function on $\mathbb R$;
\item[(b)] $h^0_{\mathcal{F}, {\underline l}}(x)\leq \frac{u(\mathcal{F})}{n!}(x-M^-)^n$, for $x\geq M^-$; \\$h^i_{\mathcal{F}, {\underline l}}(x)\leq 2^{n-1}u(\mathcal{F})(M^+-M^-)^{n-1}$, for $M^-\leq x\leq M^+$,\\ and $h^n_{\mathcal{F}, {\underline l}}(x)\leq \frac{u(\mathcal{F})}{n!}(M^+-x)^n$, for $x\leq M^+$.
\end{itemize}
These assertions are clear if $\dim\mathcal{F}=0$.
Assume that they hold for all pure sheaves of dimension $\leq n-1$. We will prove that they imply the following assertions:
\noindent\emph{ For all pure sheaves $\mathcal{F}$ with $\dim\mathcal{F}=n$ and for all rational numbers $x$ and $0<\epsilon<1$}
\begin{eqnarray}\label{derivativebound}
\label{1}&& h_{\mathcal{F}}^0((x+\epsilon){\underline l})-h_{\mathcal{F}}^0(x{\underline l})\leq \epsilon\frac{u(\mathcal{F})}{(n-1)!}(x+\epsilon-M^-)^{n-1}\; \mathrm{for}\; x\geq M^-;\\
\label{2}&&| h_{\mathcal{F}}^i((x+\epsilon){\underline l})-h_{\mathcal{F}}^i(x{\underline l})|\leq \epsilon 2^{n-1}u(\mathcal{F})(M^+-M^-)^{n-1}\; \mathrm{for}\; M^-\leq x<x+\epsilon\leq M^+;\\
\label{3}&& h_{\mathcal{F}}^{n}(x{\underline l} )-h_{\mathcal{F}}^{n}((x+\epsilon){\underline l})\leq \epsilon\frac{u(\mathcal{F})}{(n-1)!}(M^+-x-\epsilon)^{n-1}\; \mathrm{for}\; x+\epsilon\leq M^+.
\end{eqnarray}
Take $M$ sufficiently large and divisible such that $Mx$ and $M\epsilon$ are integers, take a general divisor $D\in |M^2\epsilon L|$ and consider the short exact sequence:
$$0\rightarrow \mu_M^*\mathcal F\otimes L^{M^2x} \xrightarrow{\cdot D} \mu_M^*\mathcal F\otimes L^{M^2(x+\epsilon)}\rightarrow \mu_M^*\mathcal F\otimes L^{M^2(x+\epsilon)}\mid_D\rightarrow 0.$$
Taking the long exact sequence of cohomology of the above sequence tensored with a general $P_\alpha\in \widehat A$, we see that
\begin{equation}
h_{\mathcal{F}}^0((x+\epsilon){\underline l})-h_{\mathcal{F}}^0(x{\underline l})\leq \frac{1}{M^{2g}}h^{0}_{gen}( \mu_M^*\mathcal F\otimes L^{M^2(x+\epsilon)}\mid_D)
=\frac{1}{M^{2g}}h^0_{\mu_M^*\mathcal{F}\mid_D}(M^2(x+\epsilon){\underline l}) .
\end{equation}
Note that $\mu_M^*\mathcal{F}\mid_D$ is a pure sheaf on $A$ of dimension $n-1$. It is also easy to see that an effective cohomological bound of $\mu_M^*\mathcal{F}\mid_D$ is $(M^2M^-, M^2M^+)$. Hence condition $\text{(b)}_{n-1}$ above yields that
\[h_{\mu_M^*\mathcal{F}\mid_D}^0(M^2(x+\epsilon){\underline l})\leq \frac{u(\mu_M^*\mathcal{F}\mid_D)}{(n-1)!}M^{2n-2}(x+\epsilon-M^-)^{n-1}.\]
The components of the support of $\mu_M^*\mathcal{F}\mid_D$ are $\mu^{-1}V_1\cap D,\ldots, \mu^{-1}V_s\cap D$. Hence
\[u(\mu_M^*\mathcal{F}\mid_D)=\sum_jt_j(\mu_M^{-1}V_j\cdot D\cdot L^{n-1})_A=\frac{\epsilon}{M^{2n-2}}\sum_jt_j(\mu_M^{*}V_j\cdot \mu_M^*L^{n})_A\\= \epsilon M^{2g-2n+2}r(\mathcal{F}).\]
It follows that
\[h_{\mathcal{F}}^0((x+\epsilon){\underline l})-h_{\mathcal{F}}^0(x{\underline l})\leq \epsilon\frac{u(\mathcal{F})}{(n-1)!}(x+\epsilon-M_0)^{n-1}\]
i.e. $(\ref{1})_n$.
The estimate $(\ref{3})_n$ is proved exactly in the same way as the $h_{\mathcal{F}, L}^0$ case. Concerning $(\ref{2})_n$ note that for $M^-\leq x<x+\epsilon\leq M^+$,
\begin{eqnarray*} &&|h_\mathcal{F}^i((x+\epsilon){\underline l})-h_{\mathcal{F}}^i(x{\underline l})|\\ &\leq &\frac{1}{M^{2g}} \big(h_{gen}^{i-1}(\mu_M^*\mathcal{F}\mid_{D}\otimes L^{M^2(x+\epsilon)})+h_{gen}^{i}(\mu_M^*\mathcal F\mid_{D}\otimes L^{M^2(x+\epsilon)})\big)\\
&\leq & \epsilon 2^{n-1}u(\mathcal{F})(M^+-M^-)^{n-1}.
\end{eqnarray*} This concludes the proof of the estimates (\ref{1}) (\ref{2}) (\ref{3}) under the assumption that $(b)_{n-1}$ holds.
Turning to $(a)_n$ and $(b)_n$, note that the functions $h_{\mathcal{F}}^i(x{\underline l})$ satisfy the statement of Corollaries \ref{inversion-Q} and \ref{continuity}. Therefore
the left derivative $D^-h_{\mathcal{F}}^i(x{\underline l})$ and the right derivative $D^+h_{\mathcal{F}}^i(x{\underline l})$ exist on all $x\in\mathbb Q$ (in fact on all $x\in U$ of Cor. \ref{continuity}), and they coincide away of a discrete subset. The inequalities $(\ref{1})_n$, $(\ref{2})_n$ and $(\ref{3})_n$ show that both derivatives are bounded above by the corresponding polynomials of degree $n-1$. Note that by Lemma \ref{effectivebound} and the assumption that $\mathcal{F}$ is pure, we have $h^0_{\mathcal{F}}(M^-{\underline l})=0$ and $h^i_{\mathcal{F}}(M^+{\underline l})=0$ for $i\geq 1$, and hence by integration, the above bounds for derivatives imply $(a)_n$ and $(b)_n$ for all $x\in U$ as above and hence, by continuity, for all $x\in\mathbb R$.
This concludes the proof of the Theorem for pure sheaves.
Next, we prove the Theorem for all coherent sheaves $\mathcal{F}$ on $A$. Assume that $\dim \mathcal{F}=n$.
We consider the torsion filtration of $\mathcal F$:
\[T_0(\mathcal F)\subset T_1(\mathcal F)\subset\cdots\subset T_{n-1}(\mathcal F)\subset T_n(\mathcal F)= \mathcal F,\]
where $ T_i(\mathcal F)$ is the maximal subsheaf of $\mathcal F$ of dimension $i$ and hence $\mathcal Q_i:=T_i(\mathcal F)/T_{i-1}(\mathcal F)$ is a pure sheaf of dimension $i$.
We see that $h^0_{\mathcal{F}}(x{\underline l})\leq h^0_{T_{n-1}(\mathcal{F})}(x{\underline l})+h^0_{\mathcal Q_n}(x{\underline l})$ and we also have, adopting the previous notation,
\begin{eqnarray*}h^0_{\mathcal{F}}((x+\epsilon){\underline l})-h^0_{\mathcal{F}}(x{\underline l})&\leq &\frac{1}{M^{2g}}h^0_{\mu_M^*\mathcal{F}\mid_D}(M^2(x+\epsilon){\underline l})
\\ &\leq &\frac{1}{M^{2g}}\big(h^0_{\mu_M^*T_{n-1}(\mathcal{F})\mid_D}(M^2(x+\epsilon){\underline l})+h^0_{\mu_M^*\mathcal{Q}_n\mid_D}(M^2(x+\epsilon){\underline l})\big).
\end{eqnarray*}
We then proceed by induction on $\dim \mathcal{F}$ and the results on the pure sheaf case to prove the continuity of the function $h^0_{\mathcal{F}}(x{\underline l})$ and its boundedness. The proof of continuity for other cohomology rank functions of $\mathcal{F}$ is similar.
Similarly the proof of the statement of the Theorem for objects of the bounded derived category follows the same lines, using the functorial hypercohomology spectral sequences
\[E_2^{h,k}(\mu^*_m(\mathcal{F})\otimes L^r\otimes P_\alpha):=H^h(A, \mathcal H^k(\mu_m^*\mathcal{F}\otimes L^r\otimes P_\alpha))\Rightarrow H^{h+k}(A, \mu_m^*\mathcal{F}\otimes L^r\otimes P_\alpha)\]
This time, for $x\in\mathbb Q$, $x=\frac{a}{b}$ with $b>0$ one defines the cohomological rank functions for the groups appearing at each page: $e_{r,\mathcal{F}}^{h,k}(x{\underline l}):=b^{-2g}
\dim E_r^{h,k}(\mu^*_b(\mathcal{F})\otimes L^a\otimes P_\alpha)$ for general $\alpha\in \widehat A$. Using Proposition \ref{functions} these functions are already defined in $\mathbb R$ minus a discrete set satisfying the property stated in Corollary \ref{inversion-Q}. By induction on $r$ and on the dimension of the cohomology sheaves one proves that these functions can be extended to continuous functions satisfying the same property. They are bounded as above. From this and the convergence one gets the same statements for the functions $h^i_\mathcal{F}(x{\underline l})$. We leave the details to the reader.
\end{proof}
\section{Critical points and jump loci}
A \emph{critical point for the function $x\mapsto h^i_\mathcal{F}(x{\underline l})$} is a $x_0\in\mathbb R$ where the function is not smooth. We denote $S_{\mathcal{F},{\underline l}}^i$ the set of critical points of $h_{\mathcal{F}}^i(x{\underline l})$ and let $S_{\mathcal{F}, {\underline l}}=\cup_i S_{\mathcal{F},{\underline l}}^i$. This is the subject of this section. In all examples we know, the critical points of a cohomological rank function are finitely many, and satisfy the conclusion of Corollary \ref{continuity}. We expect this to be true in general.
It follows from the results of Section 2 that for $x_0\in\mathbb Q$, or more generally for $x_0$ in the open set $U$ of Corollary \ref{continuity}, $x_0$ is a critical point if and only if the polynomials $P^-_{i,\mathcal{F},x_0}$ and $P^+_{i,\mathcal{F},x_0}$ do not coincide. As we will see below it is easy to produce examples of rational critical points. However they can be irrational -- even for line bundles on abelian varieties -- as shown by the following example.
\begin{example}\label{irrational} Let $(A,{\underline l})$ be a polarized abelian variety and let $M$ be a non-degenerate line bundle on $A$.
Consider the polynomial $P(x)=\chi_M(x{\underline l})$. By Mumford, all roots of $P(x)$ are real numbers. Denote them by $\lambda_1>\lambda_2>\cdots>\lambda_k$, let $m_i$ be the multiplicity of $\lambda_i$ and finally denote by $\lambda_0=\infty$, $m_0=0$, and $\lambda_{k+1}=-\infty$. We know by \cite[Page 155]{mumford} that for any rational number $x=\frac{a}{b}\in (\lambda_{i+1}, \lambda_i)$ with $b>0$, the line bundle $M^{b}\otimes L^a$ is non-degenerate and has index $a_i:=\sum_{0\leq k\leq i}m_k$. Hence
\[h_{M}^{k}(x{\underline l})=\begin{cases}(-1)^{k}P(x)&\hbox{if $k=a_i$ for some $0\leq i\leq k$, and $x\in (\lambda_{i+1}, \lambda_i) $}\\
0&\hbox{ otherwise}\end{cases}\]
Hence the set of critical points of $M$ is $\{\lambda_1,\ldots,\lambda_k\}$.
If $A$ is a simple abelian variety and $M$ and $L$ are linearly independent in $\mathrm{NS}(A)$ (such abelian varieties exist and they are called Shimura-Hilbert-Blumenthal varieties, see for instance \cite{dl}), then all roots of $P(x)$ are irrational. Actually, if some root $\lambda_i=\frac{a}{b}$ is rational, then $M^b\otimes L^a$ is a non-trivial degenerate line bundle and hence its kernel is a non-trivial abelian subvariety of $A$, which is a contradiction.
\end{example}
A critical point $x_0\in\mathbb R$ is said to be \emph{of index $k$} if the function $h^i_\mathcal{F}(x{\underline l})$ is of class $\mathcal C^{k}$ but not $\mathcal C^{k+1}$ at $x_0$. By Corollaries \ref{inversion-Q} and \ref{continuity} if $x_0\in\mathbb Q$ (or, more generally, $x_0\in U$ as above) this can be equivalently stated as follows
\[P^+_{i,\mathcal{F},x_0}-P^-_{i,\mathcal{F},x_0}=(x-x_0)^{k+1}Q(x)\quad\hbox{with $Q(x)\in\mathbb Q[x]$ of degree $\le g-k-1$ such that $Q(x_0)\ne 0$}\]
(in particular, it follows that the index is at most $g-1$).The main result of this section is Proposition \ref{derivatives}, relating the index of a rational critical point with the dimension of the jump locus.
It is not difficult to exhibit cohomological rank functions with critical points even of index zero. i.e. the function is non-differentiable at such a point. The simple examples below serve also as illustration of the method of calculation provided by the results of Section 2.
\begin{example}\label{counterexample} Let $A=B\times E$, a principally polarized product of a principally polarized $(g-1)$-dimensional abelian variety $B$ and an elliptic curve $E$. Let $\Theta_B$ be a principal polarization on $B$ and $p$ a closed point of $E$. Let $\mathcal{F}=\mathcal{O}_B(\Theta_B)\boxtimes\mathcal{O}_E$. It is well known that:\\
(a) the FM transform -- on $E$ -- of the sheaf ${\mathcal O}_E$ is $k(\hat e)[-1]$, the one-dimensional skyscraper sheaf at the origin, in cohomological degree $1$.\\
\noindent (b) The FM transform -- on B -- of the sheaf ${\mathcal O}_B(-\Theta_B)$ is equal to ${\mathcal O}_{\widehat B}(\Theta_{\widehat B})[-(g-1)]$.
\noindent By K\"unneth formula it follows from (a) that $R^0\Phi_{\mathcal P}(\mathcal{F})=0$, hence $h_{\mathcal{F}}^0(x{\underline l})=0$ for $x<0$ (of course this was obvious from the beginning).
On the other hand, again from K\"unneth formula together with (a) and (b) it follows that
\[R\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)=R^g\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)[-g]=i_{\widehat B*}({\mathcal O}_{\widehat B}(\Theta_{\widehat B}))[-g],\]
where $i_{\widehat B}:\widehat B\rightarrow \widehat A$ is the natural inclusion $\hat b\mapsto (\hat b,\hat e)$.
Hence, for $x>0$
\[h^0_{\mathcal{F}}(x{\underline l})=(x)^gh^0_{R^g\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)}(\frac{1}{x})= x^g(1+\frac{1}{x})^{g-1}=x(1+x)^{g-1}\]
In conclusion
\[h^0_{\mathcal{F}}(x{\underline l})=\begin{cases}0&\hbox{for $x\leq 0$}\\
x(1+x)^{g-1}&\hbox{for $x\geq 0$}\end{cases}\]
(Of course the same calculation could have been worked out in a completely elementary way).
Hence $x_0=0$ is critical point of index zero.
\end{example}
\begin{example}\label{AJ-0} Let $A$ be the Jacobian of a smooth curve of genus $g$, equipped with the natural principal polarization and let $i: C\hookrightarrow A$ be an Abel-Jacobi embedding. Let $p\in C$ and let $\mathcal{F}=i_*{\mathcal O}_C((g-1)p)$. We claim that $x_0=0$ is a critical point of index zero for the function $h^0_\mathcal{F}(x{\underline l})$.
Notice that $\mathcal{F}^{\vee}=i_*\omega_C(-(g-1)p)[1-g]$ and $\deg_C(\omega_C(-(g-1)p))=g-1$. Hence $R^0\Phi_{\mathcal P}(\mathcal{F})=0$, while
\[R\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)=R^{g}\Phi_{\mathcal P^\vee}(\mathcal{F}^\vee)[-g]=R^1\Phi_{\mathcal P^\vee}\bigl(i_*\omega_C(-(g-1)p)\bigr)[-g]:=\mathcal H [-g]\]
is a torsion sheaf in cohomological degree $g$ (supported at a translate of a theta-divisor, where it is of generic rank equal to $1$).
From Proposition \ref{inversion-a} it follows that $h_{\mathcal{F}}^0(x{\underline l})=0$ for $x\le 0$ and $h_{\mathcal{F}}^1(x{\underline l})=0$ for $x\ge 0$. Hence, by (a) of \S1,
\[h_{\mathcal{F}}^0(x{\underline l})=\begin{cases}0&\hbox{for $x\leq 0$}\\
\chi_\mathcal{F}(x{\underline l})=gx&\hbox{for $x\geq 0$}
\end{cases}\]
This proves what claimed.\\
One can show that $x_0=0$ is a critical point of index $g-d-1$ of the $h^0$-function of the sheaf $i_*{\mathcal O}_C(dp)$, with $0\le d\le g-1$.
\end{example}
As it will be clear in the sequel, the previous examples are explained by the presence of a jump locus of codimension one.
\noindent\textbf{Jump loci. } We introduce some terminology. Let $(A,{\underline l})$ be a polarized abelian variety. Let $\mathcal{F}\in\mathrm{D}^b(A)$ and $x_0\in\mathbb Q$. The \emph{jump locus} of the $i$-the cohomology of $\mathcal{F}$ at $x_0=\frac{a}{b}$ is the closed subscheme of $\widehat A$ consisting of the points $\alpha$ such that $h^i(A, (\mu_b^*\mathcal{F})\otimes L^{ab}\otimes P_\alpha))$ is strictly greater than the generic value, where $L$ is a line bundle representing ${\underline l}$. A different choice of the line bundle $L$ changes the jump locus in a translate of it while a different fractional representation of $x_0$, say $x_0=\frac{ah}{bh}$ changes the jump locus in its inverse image via the isogeny $ \mu_h:\widehat A\rightarrow \widehat A$. Therefore, strictly speaking, for us the jump locus at $x_0$ of a cohomological rank function $h^i_\mathcal{F}(x{\underline l})$ will be an equivalence class of (reduced) subschemes with respect to the equivalence relation generated by translations
and multiplication isogenies. In this paper we will be only concerned with the dimension of these loci. We will denote it by $\dim J^{i+}(\mathcal{F}\langle x_0{\underline l}\rangle)$.
\begin{proposition}\label{derivatives} Let $\mathcal{F}\in\mathrm{D}^b(A)$. If $ x_0\in \mathbb Q$ is a critical point of index $k$ for $h^i_{\mathcal{F},{\underline l}}$, then
$\mathrm{codim}_{\widehat A}\, J^{i+}(\mathcal{F}\langle x_0\rangle)\le k+1$.
\end{proposition}
\begin{proof}
We may assume that $ x_0=0$.
By Corollary \ref{inversion} we know that in a left neighborhood of $0$, $h_{\mathcal F}^i(x{\underline l})=\frac{(-x)^g}{\chi({\underline l})} \chi_{\varphi_{{\underline l}}^*R^i\Phi_{\mathcal P}(\mathcal{F})}(-\frac{1}{x}{\underline l})$ and in a right neighborhood of $0$, $h_{\mathcal F}^i(x{\underline l})=\frac{x^g}{\chi({\underline l})} \chi_{\varphi_{\underline l}^*R^{g-i}\Phi_{\mathcal P^\vee}(\mathcal{F}^{\vee})}(\frac{1}{x}{\underline l})$.
We denote
\[P_1(x):=\chi_{\varphi_{\underline l}^*R^i\Phi_{\mathcal P}(\mathcal{F})}(x{\underline l})=a_gx^g+a_{g-1}x^{g-1}+\cdots+a_{1}x+a_0\]
and
\[P_2(x):=\chi_{\varphi_{\underline l}^*R^{g-i}\Phi_{\mathcal P^\vee}(\mathcal{F}^{\vee})}(x{\underline l})=b_gx^g+b_{g-1}x^{g-1}+\cdots+b_{1}x+b_0.\]
It follows $h_{\mathcal{F}}^i(x{\underline l}) $ is strictly of class $\mathcal C^k$ at $0$ if and only if
\begin{equation}\label{change-sign}(-1)^{j}a_{g-j}=b_{g-j}\quad\hbox{for \ $j=0,\>\cdots\>, k$}\quad \mathrm{and} \quad (-1)^{k+1}a_{g-k-1}\neq b_{g-k-1}
\end{equation}
We also note that for a coherent sheaf $\mathcal Q$, $$\chi_{\mathcal Q}(x{\underline l})=\int_{A}\mathrm{ch}(\mathcal Q)e^{x{\underline l}}=\sum_{j\geq 0}\frac{1}{(g-j)!}(\mathrm{ch}_j({\mathcal Q})\cdot {\underline l}^{g-j})_Ax^{g-j}.$$
On the other hand, by Grothendieck duality (\ref{mukai-3}) we have $\Phi_{\mathcal P}(\mathcal F)^\vee=\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^{\vee})[g]$. Thus we have a natural homomorphism $R^{g-i}\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^{\vee})\rightarrow \mathcal{H}om(R^i\Phi_{\mathcal P}(\mathcal{F}), \mathcal{O}_{\widehat A}):=\mathcal H$ by the Grothendieck spectral sequence. By base change and Serre duality such homomorphism is an isomorphism of vector bundles on the open set $V$ whose closed points are the $\alpha\in\widehat A$ such that $h^i(\mathcal{F}\otimes P_{\alpha})$ takes the generic value. Now assume that the complement of $V$, i.e. a representative of $J^{i+}(\mathcal{F})$, has codimension $>k+1$. Hence $\mathrm{ch}(R^{g-i}\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^{\vee}))-\mathrm{ch}(\mathcal H)\in \mathrm{CH}^{> k+1}(\widehat{A}).$ Since $R^i\Phi_{\mathcal P}(\mathcal F)$ is a vector bundle on $V$, thus $\mathrm{ch}_j(\mathcal{H})=(-1)^j\mathrm{ch}_j(R^i\Phi_{\mathcal P})(\mathcal F)$ for $j\leq k+1$. This implies that
$$(-1)^ja_{g-j}=\frac{(-1)^j}{(g-j)!}(\varphi_{\underline l}^*\mathrm{ch}_j\big(R^i\Phi_{\mathcal P})(\mathcal F)\cdot {\underline l}^{g-j}\big)_A=\frac{1}{(g-j)!}(\varphi_{\underline l}^*\mathrm{ch}_j\big(R^{g-i}\Phi_{\mathcal P^{\vee}})(\mathcal F^{\vee})\cdot {\underline l}^{g-j}\big)_A=b_{g-j},$$
for $j=0,\ldots, k+1$, which contradicts (\ref{change-sign}). We conclude the proof.
\end{proof}
\section{Generic vanishing, M-regularity and IT(0) of $\mathbb Q$-twisted sheaves on abelian varieties}
The notions of GV, M-regular and IT(0)-sheaves (and other related ones) are useful in the study of the geometry of abelian and irregular varieties via the Fourier-Mukai transform associated to the Poincar\'e line bundle. In this section we extend such notions
to the $\mathbb Q$-twisted setting. In doing that we don't claim any originality, as this point of view was already taken, at least implicitly, in the work \cite{pp3}, Proof of Thm 4.1, and goes back to work of Hacon (\cite{hac}). It turns out that the $\mathbb Q$-twisted formulation of Hacon's criterion for being GV, and related results, is simpler and more expressive even for usual (non-$\mathbb Q$-twisted) coherent sheaves or, more generally, objects of $\mathrm{D}^b(A)$. In the last part of the section we go back to cohomological rank functions. First we show how they can be used to provide a characterization of M-regularity and related notions. Finally we show the maximal critical points relates to the notion of $\mathbb Q$-twisted GV sheaves.
As for jump loci (see the previous Section), one can define the \emph{cohomological support locus} of the $i$-th cohomology of the $\mathbb Q$-twisted object of $\mathrm{D}^b(A)$, say $\mathcal{F}\langle x_0{\underline l} \rangle$, as the equivalence class (with respect to the equivalence relation generated by translations and inverse images by multiplication-isogenies) of the loci
\[\{\alpha\in\widehat A\>|\> h^i(A, (\mu_b^*\mathcal{F})\otimes L^{ab}\otimes P_\alpha))>0\}.\]
If $h^i_\mathcal{F}(x_0{\underline l})=0$ it coincides with the jump locus while it is simply $\widehat A$ if $h^i_\mathcal{F}(x{\underline l})>0$. Its dimension is well-defined, and we will denote it $\dim V^i(\mathcal{F}\langle x_0{\underline l}\rangle)$.
The $\mathbb Q$-twisted object $\mathcal{F}\langle x_0{\underline l}\rangle $ is said to be \emph{GV} if $\text{codim}_{\widehat A}V^i(\mathcal{F}\langle x_0{\underline l}\rangle)\ge i$ for all $i>0$. It is said to be \emph{a M-regular sheaf} if $\text{codim}_{\widehat A}V^i(\mathcal{F}\langle x_0{\underline l}\rangle)> i$ for all $i>0$.
It is said to \emph{satisfy the index theorem with index 0, IT(0)} for short, if $V^i(\mathcal{F}\langle x_0{\underline l}\rangle)$ is empty for all $i\ne 0$.
If $\mathcal{F}\langle x_0\rangle $ is $\mathbb Q$-twisted coherent sheaf or, more generally, a $\mathbb Q$-twisted object of $\mathrm{D}^b(A)$ such that $V^i(\mathcal{F}\langle x_0\rangle)$ is empty for $i<0$, such conditions can be equivalently stated described as follows:
\begin{theorem}\label{A} (a) $\mathcal{F}\langle x_0{\underline l}\rangle $ is GV if and only if, for one (hence for all) representation $x_0=\frac{a}{b}$
\[\Phi_{\mathcal P^\vee}(\mu_b^*\mathcal{F}^\vee\otimes L^{-ab})=R^g\Phi_{\mathcal P^\vee}(\mu_b^*\mathcal{F}^\vee\otimes L^{-ab})[-g]\>.\>\footnote{This condition is usually expressed by saying that $\mu_b^*\mathcal{F}^\vee\otimes L^{-ab}$ satisfies the Weak Index Theorem with index $g$.}\]
If this is the case
\[R^i\Phi_{\mathcal P}(\mu_b^*(\mathcal{F})\otimes L^{ab})=\mathcal{E}xt^i_{{\mathcal O}_{\widehat A}}(R^g\Phi_{\mathcal P^\vee}(\mu_b^*\mathcal{F}^\vee\otimes L^{-ab}),{\mathcal O}_{\widehat A})\]
\noindent (b) Assume that $\mathcal{F}\langle x_0{\underline l}\rangle $ is GV. Then it is M-regular if and only if the sheaf $R^g\Phi_{\mathcal P^\vee}(\mu_b^*\mathcal{F}^\vee\otimes L^{-ab})$ is torsion-free.
\noindent (c) Assume that $\mathcal{F}\langle x_0{\underline l}\rangle $ is GV, Then it is IT(0) if the sheaf $R^g\Phi_{\mathcal P^\vee}(\mu_b^*\mathcal{F}^\vee\otimes L^{-ab})$ is locally free. Equivalently
\begin{equation}\label{it0}\Phi_{\mathcal P}(\mu_b^*(\mathcal{F})\otimes L^{ab})=R^0\Phi_{\mathcal P}(\mu_b^*(\mathcal{F})\otimes L^{ab})\>.
\end{equation}
\end{theorem}
These results follow immediately from the same statements for coherent sheaves or objects in $\mathrm{D}^b(A)$, see e.g. the survey \cite[\S1]{msri}, or \cite[\S3]{pp2}, where the subject is treated in much greater generality.
In this language well known criteria of Hacon (\cite{hac}) can be stated as follows:
\begin{theorem}\label{B} \noindent (a) $\mathcal{F}\langle x_0{\underline l}\rangle$ is GV if and only if $\mathcal{F}\langle (x_0+x){\underline l}\rangle$ is IT(0) for sufficiently small $x\in \mathbb Q^+$. Equivalently
$\mathcal{F}\langle x_0{\underline l}\rangle$ is GV if and only if $\mathcal{F}\langle (x_0+x){\underline l}\rangle$ is IT(0) for all $x\in \mathbb Q^+$.
\noindent (b) If $\mathcal{F}\langle x_0{\underline l}\rangle$ is GV but not IT(0) then $\mathcal{F}\langle (x_0-x){\underline l}\rangle$ is not GV for all $x\in\mathbb Q^+$.
\noindent (c) $\mathcal{F}\langle x_0{\underline l}\rangle$ is IT(0) if and only if $\mathcal{F}\langle (x_0-x){\underline l}\rangle $ is IT(0) for sufficiently small $x\in\mathbb Q^+$.
\end{theorem}
\begin{proof} (a) Let $x_0=\frac{a}{b}$. We have that $\mathcal{F}\langle x_0{\underline l}\rangle$ is GV if and only if $\mu_b^*(\mathcal{F})\otimes L^{ab}$ is GV. Hacon's criterion (see \cite[Thm A]{pp2}) states that this is the case if and only if
\begin{equation}\label{haconbis}H^i(\mu_b^*(\mathcal{F})\otimes L^{ab}\otimes \Phi^{\widehat A\rightarrow A}_{\mathcal P}(N^{-k})[g])=0
\end{equation}
for all $i\ne 0$ and for all sufficiently big $k\in \mathbb Z$, where $N$ is an ample line bundle on $\widehat A$. Equivalently (up to taking a higher lower bound for $k$),
\begin{equation}\label{hacon}
\mu_b^*(\mathcal{F})\otimes L^{ab}\otimes \Phi^{\widehat A\rightarrow A}_{\mathcal P}(N^{-k})[g]\quad\hbox{is}\quad IT(0)
\end{equation}
for sufficiently big $k$.
We take as $N=L_\delta$ a line bundle representing the polarization ${\underline l}_\delta$ dual to ${\underline l}$ (\cite{birke-lange} \S14.4). By Prop 14.4.1 of \emph{loc. cit.} we have that
\begin{equation}\label{np-2}
\varphi_{{\underline l}}^*{\underline l}_\delta=d_1d_g{\underline l}
\end{equation}
and
\begin{equation}\label{np-3}
\varphi_{{\underline l}_\delta}\circ\varphi_{{\underline l}}=\mu_{d_1d_g}
\end{equation}
where $(d_1,\dots ,d_g)$ is the type of ${\underline l}$.
Combining with (\ref{mukai-4}) we get
\[\mu_{d_1d_gk}^*\Phi^{\widehat A\rightarrow A}_{\mathcal P}(L_\delta^{k})= \varphi_{{\underline l}}^*\varphi_{{\underline l}_\delta}^*\mu_k^*\Phi^{\widehat A\rightarrow A}_{\mathcal P}(L_\delta^{k})=(\varphi_{\underline l}^*(L_\delta^{-k}))^{\oplus k^g\chi({\underline l}_\delta)}=
(L^{-d_1d_gk})^{\oplus k^g\chi({\underline l}_\delta)}\]
Loosely speaking, we can think of the vector bundle $\Phi^{\widehat A\rightarrow A}_{\mathcal P}(L_\delta^{k})$ as representative of $(-\frac{1}{d_1d_gk}{\underline l})^{\oplus k^g\chi({\underline l}_\delta)}$. It follows, after a little calculation, that (\ref{hacon}), hence the fact that $\mathcal{F}\langle x_0{\underline l}\rangle$ is GV, is equivalent to the fact that $\mathcal{F}\langle (x_0+\frac{1}{d_1d_gk}){\underline l}\rangle$ is IT(0) for sufficiently big $k$. This is in turn equivalent to the fact that $\mathcal{F}\langle (x_0+x){\underline l}\rangle$ is IT(0) for sufficiently small $x\in \mathbb Q^+$ because the tensor product of an IT(0) (or, more generally, GV) sheaf and a locally free IT(0) sheaf is IT(0) (\cite[Prop. 3.1]{pp3} ). This proves the first statement of (a).
The second statement follows again from \emph{loc.cit.}\\
(b) follows from (a).\\
(c) is proved as (a) using a similar Hacon's criterion telling that (\ref{it0}) is equivalent to the fact that
$\mu_b^*(\mathcal{F})\otimes L^{ab}\otimes \Phi^{\widehat A\rightarrow A}_{\mathcal P}(N^{k})$ is IT(0) for sufficiently big $k$.
\end{proof}
Using the cohomological rank functions on the left neighborhood of a rational point, we have the following characterization of GV-sheaves and M-regular sheaves.
\begin{proposition}\label{gv&M}
(a) \ $\mathcal{F}\langle x_0{\underline l}\rangle$ is GV, if and only if $h_{\mathcal{F}}^i((x_0-x){\underline l})=O(x^i)$ for sufficiently small $x\in\mathbb Q^+$, for all $i\geq 1$.
\noindent (b) \ $\mathcal{F}\langle x_0{\underline l}\rangle$ is M-regular, if and only if $h_{\mathcal{F}}^i((x_0-x){\underline l})=O(x^{i+1})$ for sufficiently small $x\in\mathbb Q^+$, for all $i\geq 1$.
\end{proposition}
\begin{proof}
We may suppose that $x_0=0$. Then $\mathcal{F}$ is GV (resp. M-regular) is equivalent to say that $\mathrm{codim}\; R^i\Phi_{\mathcal{P}}(\mathcal{F})\geq i$ (resp. $\mathrm{codim}\;R^i\Phi_{\mathcal{P}}(\mathcal{F})> i$) for all $i\geq 1$ (see \cite{pp2} Lemma 3.6). Then we conclude by Corollary \ref{inversion}.
\end{proof}
It turns out that, more generally, the notion of \emph{gv}-index (\cite{duke} Def. 3.1) can be extended to the $\mathbb Q$-twisted setting and described via cohomological rank functions as in Proposition \ref{gv&M}. We leave this to the reader.
It is likely that a sort converse of Proposition \ref{derivatives} holds, namely the rational critical points arise only in presence of non-empty jump loci, although not necessarily for the same cohomological index. A partial result in this direction is the following
\begin{proposition}\label{maxthreshold} Let $ x_0\in \mathbb Q$. If the $\mathbb Q$-twisted sheaf $\mathcal{F}\langle x_0{\underline l}\rangle$ is GV but not IT(0) then $x_0\in S_{\mathcal{F},{\underline l}}$. In fact it is the maximal element of $S_{\mathcal{F},{\underline l}}$.
\end{proposition}
However notice that, given a coherent sheaf $\mathcal{F}$, in general there is no reason to expect that there is an $x_0\in\mathbb Q$ such that the hypothesis of the Proposition holds. In other words, the maximal critical point might be irrational.
\begin{proof}
As before, we may assume that $ x_0=0$ and we need to compare the coefficients of the two polynomials $P_1(x)=\chi_{\varphi_{{\underline l}}^*R^0\Phi_{\mathcal P}(\mathcal{F})}(x)$ and $P_2(x)=\chi_{\varphi_{{\underline l}}^*R^{g}\Phi_{{\mathcal P}^{\vee}}(\mathcal{F}^{\vee})}(x)$. By assumption, $\mathcal{F}$ is a GV sheaf, hence by Theorem \ref{A}(a) $R\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^\vee)=R^g\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^{\vee})[-g]$ and $R^i\Phi_{\mathcal P}(\mathcal{F})=\mathcal{E}xt^i(R^g\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^{\vee}), \mathcal{O}_{\widehat{A}})$. Moreover the condition that $\mathcal{F}$ is GV but not IT(0) implies that $R^g\Phi_{\mathcal{P}^{\vee}}(\mathcal{F}^{\vee})$ is not locally free. Hence for some $i>0$, $R^i\Phi_{\mathcal P}(\mathcal{F})$ is nonzero. Thus, $h_{\mathcal{F}}^i(x{\underline l})$ is nonzero for $x$ in a left neighborhood of $0$ and obviously $h_{\mathcal{F}}^i(x{\underline l})=0$ for $x$ positive. Hence $x_0\in S_{\mathcal{F},{\underline l}}$.
\end{proof}
\begin{remark}\label{maximal-example}
Under the above assumption, it is in general not true that $x_0$ is a critical point of $h_{\mathcal{F}}^0(x{\underline l})$ as shown by the following example.
Let $(A, \underline \theta)$ be a principally polarized abelian variety and let $\Theta$ be a theta-divisor. Let $\mathcal{F}=\mathcal O_A\oplus \mathcal O_{\Theta}(\Theta)$. Then $\mathcal{F}$ is GV and not IT(0). It is easy to see that \[h_{\mathcal{F}}^0(x{\underline l})=\begin{cases}(1+x)^g&\hbox{for $x\geq -1$}\\
0&\hbox{for $x<-1$}\end{cases}\] However notice that $h^{g-1}_{\mathcal{F}}(x{\underline l})=h^g_{\mathcal{F}}(x{\underline l})=(-x)^g$
for $-1\leq x\leq 0$.
\end{remark}
\section{Some integral properties of the coefficients}
In this section we point out some interesting integrality property of the polynomials involved in cohomological rank functions. We will use the results of \S2 throughout.
\begin{lemma}Let $\mathcal{F}\in \mathrm{D}^b(A)$. Assume that $h^i_{\mathcal{F}}(x{\underline l})=P(x)$ is a polynomial function for $x$ in an interval $U_1\subseteq \mathbb R$. Then all coefficients of $P(x)$ belong to $\frac{1}{g!}\mathbb Z$.
\end{lemma}
\begin{proof}
We already know that $P(x)\in \mathbb{Q}[x]$ is a polynomial of degree at most $g$.
We may choose $p$ sufficiently large such that there exists $q\in \mathbb Z$ such that the numbers $\frac{q}{p},\ldots, \frac{q+g}{p}$ and $\frac{q}{p+1},\ldots, \frac{q+g}{p+1}$ belong to $U_1$. By definition, $a_i:=P(\frac{q+i}{p})=h_{\mathcal{F}}^i(\frac{q+i}{p})=\frac{1}{p^{2g}}h_{gen}^i(\mathcal{F}\otimes L^{p(q+i)})\in \frac{1}{p^{2g}}\mathbb Z$. Then we know that
$$P(x)=\sum_{i=0}^g\frac{a_i}{\prod_{j\neq i}(\frac{i-j}{p})}\prod_{j\neq i}(x-\frac{q+j}{p})=\sum_{i=0}^g\frac{a_i}{\prod_{j\neq i}(i-j)}\prod_{j\neq i}(px-q-j).$$
Hence all coefficients of $P(x)$ belong to $\frac{1}{g!}\frac{1}{p^{2g}}\mathbb Z$. Apply the same argument to $P(\frac{q}{p+1}),\ldots, P(\frac{q+g}{p+1})$, we see that all coefficients of $P(x)$ belong to $\frac{1}{g!}\frac{1}{(p+1)^{2g}}\mathbb Z$. Hence they belong to $\frac{1}{g!}\mathbb Z$.
\end{proof}
\begin{remark}By a slightly different argument, we can say something more. Let $\frac{a}{b}\in U_1$. Then by Corollary \ref{inversion}, we know that, for $x>0$ small enough, \begin{eqnarray}Q(x):=P(\frac{a}{b}-x)&=&x^g\frac{1}{\chi({\underline l})}\chi_{\varphi_{{\underline l}}^*R^i\Phi_{\mathcal P}(\mu_b^*\mathcal{F}\otimes L^{ab})}(\frac{1}{b^2}x{\underline l})\\\nonumber
&=& \frac{1}{\chi({\underline l})}\sum_{k=0}^g\big(ch_k(\varphi_{{\underline l}}^*R^i\Phi_{\mathcal P}(\mu_b^*\mathcal{F}\otimes L^{ab}))\cdot {\underline l}^{g-k}\big)_A\>\frac{1}{b^{2g-2k}}x^k.\end{eqnarray}
Hence the coefficient $b_k$ of $x^k$ of $Q(x)$ is $\frac{1}{\chi({\underline l})}\big(ch_k(\varphi_{{\underline l}}^*R^i\Phi_{\mathcal P}(\mu_b^*\mathcal{F}\otimes L^{ab}))\cdot {\underline l}^{g-k}\big)_A\>\frac{1}{b^{2g-2k}}$. Note that \begin{eqnarray*}\big(ch_k(\varphi_{{\underline l}}^*R^i\Phi_{\mathcal P}(\mu_b^*\mathcal{F}\otimes L^{ab}))\cdot {\underline l}^{g-k}\big)_A=\big(\varphi_{{\underline l}}^*ch_k(R^i\Phi_{\mathcal P}(\mu_b^*\mathcal{F}\otimes L^{ab}))\cdot {\underline l}^{g-k}\big)_A\\=\frac{\deg\varphi_{{\underline l}}}{(d_1d_g)^{g-k}}\big( ch_k(R^i\Phi_{\mathcal P}(\mu_b^*\mathcal{F}\otimes L^{ab}))\cdot {\underline l}_{\delta}^{g-k}\big)_{\widehat{A}},\end{eqnarray*} where ${\underline l}_\delta$ is the dual polarization (\cite{birke-lange} \S14.4) and the last equality holds because of (\ref{np-2}). Moreover, since
\begin{equation}
\chi({\underline l}_\delta)=\frac{(d_1d_g)^g}{\chi({\underline l})}
\end{equation}
we note that the class $[{\underline l}_{\delta}]^{g-k}$ belongs to $(g-k)!\frac{(d_1d_g)^{g-k}}{d_g\cdots d_{k
+1}}H^{2g-2k}(\widehat{A}, \mathbb Z)$. On the other hand, it is clear that $[ch_k(R^i\Phi_{\mathcal P}(\mu_b^*\mathcal{F}\otimes L^{ab}))]\in \frac{1}{k!}H^{2k}(\widehat{A}, \mathbb Z)$. Thus $b_k\in (d_1\cdots d_k)\frac{(g-k)!}{k!}\frac{1}{b^{2g-2k}}\mathbb Z$. From this computation, we see easily that the coefficient $a_k$ of $x^k$ in $P(x)$ belongs to $ (d_1\cdots d_k)\frac{(g-k)!}{k!}\mathbb Z$.
\end{remark}
We have the following strange corollary.
\begin{corollary} Let $\mathcal{F}\in \mathrm{D}^b(A)$. Then $b^g\mid h_{gen}^i(\mu_b^*\mathcal{F}\otimes L^a)$ for all $b$ such that $(b, g!)=1$ and $a\in \mathbb Z$.
\end{corollary}
\begin{proof}Since $h_{\mathcal{F}}^i(x{\underline l})$ is a polynomial of degree at most $g$ whose coefficients belong to $\frac{1}{g!}\mathbb Z$, we have that $g!b^gh_{\mathcal{F}}^i(\frac{a}{b}{\underline l})\in \mathbb Z$. As $(b, g!)=1$, we conclude that $ b^g\mid h_{gen}^i(\mu_b^*\mathcal{F}\otimes L^a)$.
\end{proof}
\section{GV-subschemes of principally polarized abelian varieties}
Let $(A,\underline\theta)$ be a $g$-dimensional principally polarized abelian variety.
A subscheme $X$ of $A$ is called a \emph{GV-subscheme} if its twisted ideal sheaf
$\mathcal{I}_X(\Theta)$ is GV. This technical definition is motivated by the fact that the subvarieties $\pm W_d$ of Jacobians and $\pm F$, the Fano surface of lines of intermediate jacobians of cubic threefolds, are the only known examples of (non-degenerate) GV-subschemes.
We summarize some basic results on the subject in use in the sequel. One considers the ``theta-dual" of $X$, namely the cohomological support locus
\[V(X):=V^0(\mathcal{I}_X(\Theta))=\{\alpha\in \widehat A\>|\> h^0(\mathcal{I}_X(\Theta)\otimes P_\alpha)>0\}\]
equipped with its natural scheme structure (\cite{minimal} \S4). Let $X$ be a geometrically non-degenerate GV-subscheme of pure dimension $d$. Then
\noindent (a) \cite[Theorem 2(1)]{s} \emph{$X$ and $V(X)$ are reduced and irreducible.}
\noindent (b) (\cite{minimal}) \emph{$V(X)$ is a geometrically non-degenerate GV-scheme of pure dimension $g-d-1$ \emph{(the maximal dimension)}. Moreover $V(V(X))=X$ and both $X$ and $V(X)$ are Cohen-Macaulay.}
\noindent (c) (\emph{loc. cit.}) \ \emph{$\Phi_{\mathcal P}({\mathcal O}_X(\Theta))=\bigl(\mathcal{I}_{V(X)}(\Theta)\bigr)^\vee$. Equivalently, by Grothendieck duality \emph{(see (\ref{mukai-3}))}, $\Phi_{\mathcal P^\vee}(\omega_X(-\Theta))=\mathcal{I}_{V(X)}(\Theta)[-d]$. }
\noindent (d) (\emph{loc. cit.}) \ \emph{$X$ has minimal class $[X]=\frac{\underline\theta^{g-d}}{(g-d)!}$. }
In \emph{loc. cit.} it is conjectured that the converse of (d) holds. According to the conjecture of Debarre, this would imply that that the only geometrically non-degenerate GV-subschemes are the subvarieties $\pm W_i$ and $\pm F$ as above. We refer to the Introduction for what is known in this direction.
\noindent\textbf{Generalities on GV-subschemes. } We start with some general results on GV-subschemes, possibly of independent interest. The first Proposition does not follow from Green-Lazarsfeld's Generic Vanishing Theorem because a GV-subscheme can be singular. It does follow, via Lemma \ref{normality} below, from the Generic Vanishing Theorem of \cite{pareschi} which works for \emph{normal} Cohen-Macaulay subschemes of abelian varieties. However the following ad-hoc proof is much simpler.
\begin{proposition}\label{GL} Let $X$ be a non-degenerate reduced GV-subscheme. Then its dualizing sheaf $\omega_X$ is a GV-sheaf.
\end{proposition}
\begin{proof} By Hacon's criterion (\ref{haconbis}), it is enough to show that
\[H^i(\omega_X\otimes \Phi_{\mathcal P}({\mathcal O}_A(-k\Theta))[g])=0\quad\hbox{ for $k$ sufficiently big}\]
We have that $\Phi_{\mathcal P}({\mathcal O}_A(-k\Theta))[g]$ is a vector bundle (in degree $0$) which will be denoted, as usual, $\widehat{{\mathcal O}_A(-k\Theta)}$.
We can write
\[H^i(\omega_X\otimes \widehat{{\mathcal O}_A(-k\Theta)})=H^i(\omega_X(-\Theta)\otimes\widehat{{\mathcal O}_A(-k\Theta)}(\Theta))\]
Applying the inverse of $\Phi_{\mathcal P}$, namely $\Phi_{{\mathcal P}^\vee[g]}$ (see (\ref{mukai-0})) to the last formula of (c) of this section we get that $\omega_X(-\Theta)=\Phi_{\mathcal P}(\mathcal{I}_{V(X)}(\Theta))[g-d]$. Therefore, by (\ref{exchange})
\[H^i((\omega_X(-\Theta))\otimes \widehat{{\mathcal O}_A(-k\Theta)}(\Theta)))
= H^{i+g-d}\Bigl(\mathcal{I}_{V(X)}(\Theta)\otimes \Phi_{\mathcal P}\bigl(\widehat{{\mathcal O}_A(-k\Theta)}(\Theta)\bigr)\Bigr)\]
Clearly $\widehat{{\mathcal O}_A(-k\Theta)} $ is an IT(0) sheaf for $k\ge 1$ ((\ref{mukai-4})). Therefore $\Phi_{\mathcal P}\bigl(\widehat{{\mathcal O}_A(-k\Theta)}(\Theta)\bigr)$ is a sheaf (locally free)
in cohomological degree $0$. Hence the above cohomology groups vanish for $i>0$ because $\dim V(X)=g-d-1$.
\end{proof}
\begin{proposition} Let $X$ be a non-degenerate reduced GV-subscheme. Then
\[\Phi_{\mathcal P^\vee}(\omega_X)=\bigl(\Phi_{\mathcal P}(\mathcal I_{V(X)})\bigr)(-\Theta)[g-d]\]
\end{proposition}
\begin{proof}
\begin{eqnarray*}\Phi_{\mathcal P}(\omega_X)&=&
\Phi_{\mathcal P}\bigl(\omega_X(-\Theta)\otimes\mathcal{O}_A(\Theta)\bigr)\\
&\buildrel{(\ref{mukai-2})}\over =&\Phi_{\mathcal P}(\omega_X(-\Theta))*\Phi_{\mathcal P}({\mathcal O}_A(\Theta)[g]\\
&\buildrel{(c),(\ref{mukai-4})}\over =&
=(-1)^*\mathcal I_{V(X)}(\Theta)[-d]*{\mathcal O}_A(-\Theta)[g]\\
&\buildrel{(\ref{mukai-5})}\over =& (\Phi_{\mathcal P}(\mathcal I_{-V(X)}))(-\Theta)[g-d].
\end{eqnarray*}
\end{proof}
\begin{corollary}\label{gv-strong} Let $X$ be a non-degenerate reduced GV-subscheme. Then
\begin{equation}\label{strong}R^i\Phi_{\mathcal P^\vee}(\omega_X)=0 \quad\hbox{for $i\ne 0,d$}\quad\hbox{and}\quad R^d\Phi_{\mathcal P^\vee}(\omega_X)=k(\hat e).\footnote{The last assertion was proved under very general assumptions in \cite[Prop. 6.1]{catalans}.}
\end{equation}
\end{corollary}
\begin{proof} This follows from the fact that also $V(X)$ is reduced Cohen-Macaulay ((b) of this section). Therefore, by Proposition \ref{GL}, combined by the duality characterization of Theorem \ref{haconbis}(a)
\[\Phi_{\mathcal P}({\mathcal O}_{V(X)})=R^{d-g-1}\Phi_{\mathcal P}({\mathcal O}_{V(X)})[-(g-d-1)]\] Hence it follows from the standard
exact sequence $0\rightarrow \mathcal{I}_{V(X)}\rightarrow {\mathcal O}_A\rightarrow {\mathcal O}_{V(X)}\rightarrow 0$ that $R^i\Phi_{\mathcal P}(\mathcal{I}_{V(X)})=0$ for $i\ne g-d,g$ and $R^g\Phi_{\mathcal P}(\mathcal{I}_{V(X)})=k(\hat e)[-g]$.
Therefore the assertion follows from the previous Proposition.
\end{proof}
Corollary \ref{gv-strong} is quite strong. Its implications hold in a quite general context but, for sake of brevity, here we will stick to an ad-hoc treatment of the case of GV-subschemes.
\begin{lemma}\label{cup} Let $X$ be a subscheme of an abelian variety $A$. If $X$ satisfies (\ref{strong}) then the natural maps $\Lambda^iH^1({\mathcal O}_A)\rightarrow H^i({\mathcal O}_X)$ are isomorphisms for all $i<d$ and injective for $i=d$.
\end{lemma}
\begin{proof} It the first place we note that (as it is well known) for all $\mathcal{F}\in \mathrm{D}^b(A)$ we have a natural isomorphism
\begin{equation}\label{ext} H^i(A,\mathcal{F})\cong \text{Ext}^{g+i}(k(\hat e),\Phi_{\mathcal P}(\mathcal{F}))
\end{equation}
where $\hat e$ denotes the origin of $\widehat A$. Indeed, by the Fourier-Mukai equivalence,
\[H^i(A,\mathcal{F})=\text{Hom}_{D(A)}({\mathcal O}_A,\mathcal{F}[i])\cong
\text{Hom}_{D(\widehat A)}(k(\hat e)[-g],\Phi_{\mathcal P}(\mathcal{F}))\cong \text{Ext}^{g+i}(k(\hat e),\Phi_{\mathcal P}(\mathcal{F})) \]
Applying (\ref{ext}) to $\mathcal{F}=\omega_X$ we are reduced to compute the hypercohomology spectral sequence
\[\text{Ext}^{g+i-k}(k(\hat e),R^k\Phi_{\mathcal P}(\omega_X))\Rightarrow \text{Ext}^{g+i}(k(\hat e),\Phi_{\mathcal P}(\omega_X))\cong H^i(\omega_X) \]
We recall that $\text{Ext}^j_{\widehat A}(k(\hat e),k(\hat e))=\Lambda^jH^1({\mathcal O}_A)$. Condition (\ref{strong}) makes the above spectral sequence very easy. In fact we get the maps
\begin{eqnarray*}H^i(\omega_X)\cong \text{Ext}^{g+i}(k(\hat e), \Phi_{\mathcal P}(\omega_X))&\rightarrow &\text{Ext}^{g+i-d}(k(\hat e), R^d\Phi_{\mathcal P}(\omega_X)[d])\\
&=&\text{Ext}^{g+i-d}(k(\hat e), k(\hat e))\\
&=&\Lambda^{g+i-d}H^1({\mathcal O}_A)\\
&=&\Lambda^{d-i}H^1({\mathcal O}_A)^\vee
\end{eqnarray*}
which are isomorphisms for $i>0$ and surjective for $i=0$. The Lemma follows by duality. \end{proof}
\noindent\textbf{The Poincar\'e polynomial of a GV-subscheme. } As an application of cohomological rank functions we prove that the Hilbert polynomial of geometrically non-degenerate GV-subschemes is the conjectured one.
\begin{theorem}\label{gv1} Let $X$ be a geometrically non-degenerate GV-subscheme of dimension $d$ of a principally polarized abelian variety $(A,
\underline\theta)$. Then
\[\chi_{{\mathcal O}_X}(x\underline\theta)=\sum_{i=0}^d{g\choose i}(x-1)^i\]
\end{theorem}
\begin{proof} We compute the functions $h^i_{{\mathcal O}_X}(x\underline\theta)$ in a neighborhood of $x_0=1$. First we compute it in an interval $(1-\epsilon^-,1]$ as in Corollary \ref{inversion}. By (b) and (c) of this section we have that $R^0\Phi_{\mathcal P}({\mathcal O}_X(\Theta))={\mathcal O}_A(-\Theta)$, $R^d\Phi_{\mathcal P}({\mathcal O}_X(\Theta))=\omega_{V(X)}(-\Theta)$ and $R^i\Phi_{\mathcal P}({\mathcal O}_X(\Theta))=0$ for $i\ne 0,d$ (see e.g. \cite[Prop. 5.1(b)]{minimal}). Therefore, in a small interval $[-\epsilon^-,0]$
\[h^i_{{\mathcal O}_X(\Theta)}(y\underline\theta)=\begin{cases}(-y)^g\chi_{{\mathcal O}_A(-\Theta)}(-\frac{1}{y})=y^g(1+\frac{1}{y})^g=(1+y)^g&\text{for }\>i=0\\
(-y)^gQ(-\frac{1}{y})&\text{for } \> i=d\\
0&\text{for } \> i\ne 0,d
\end{cases}\]
where $Q$ is the Hilbert polynomial of the sheaf $\omega_{V(X)}(-\Theta)$, hence a polynomial of degree $g-d-1$. It follows that $(-y)^gQ(-\frac{1}{y})=y^{d+1}T(y)$, where $T$ is a polynomial of degree $g-d-1$ such that $T(0)\ne 0$. Setting $x=1+y$ we get that in the interval $(1-\epsilon^-,1]$ \footnote{actually in Proposition \ref{gv2} below it will be shown that $\epsilon^-=1$, but this is not necessary for the present Theorem}
\[h^i_{{\mathcal O}_X}(x\underline\theta)=\begin{cases}x^g&\text{for }\>i=0\\
(x-1)^{d+1}T(x-1)&\text{for } \> i=d, \>\text {where } \> \deg T=g-d-1 \> \> \text{and } \> T(1)\ne 0 \>\> \\
0&\text{for } \> i\ne 0,d
\end{cases}\]
Writing $x^g$ as its Taylor expansion centered at $x_0=1$ this yields the equality of polynomials
\[\chi_{{\mathcal O}_X}(x\underline\theta)= \sum_{i=0}^g{g\choose i}(x-1)^i +(-1)^{d}(x-1)^{d+1}T(x-1).\]
It follows that $\chi_{{\mathcal O}_X}(x\underline\theta)$, which is a polynomial of degree $d$, is the Taylor expansion of $x^g$ at order $d$. \end{proof}
In the following proposition we compute the cohomological rank functions (with respect to the polarization $\underline\theta$) of the structure sheaf of a GV-scheme. In particular, this answers to Question 8.10 of \cite{bps}, asking, in the present terminology, for the cohomological rank functions of the structure sheaf of a curve in its Jacobian.
\begin{proposition}\label{gv2} In the same hypotheses of the previous Theorem
\[h^0_{{\mathcal O}_X}(x\underline\theta)=\begin{cases}0&\text{for }\> x\le 0\\
x^g&\text{for }\> x\in[0,1]\\
\chi_{{\mathcal O}_X}(x\underline\theta)=\sum_{i=0}^d{g\choose i}(x-1)^i&\text{for } \> x\ge 1\\
\end{cases}\]
\end{proposition}
\begin{proof} The assertion for $x\le 0$ is obvious. The assertion for $x\ge 1$ follows from the fact that ${\mathcal O}_X(\Theta)$ is a GV-sheaf (in fact M-regular), this last assertion being well known, as it follows at once from the definition of GV-subscheme and the exact sequence
\[0\rightarrow \mathcal{I}_X(\Theta)\rightarrow {\mathcal O}_A(\Theta)\rightarrow {\mathcal O}_X(\Theta)\rightarrow 0.\]
Therefore, by Theorem \ref{B}(a), ${\mathcal O}_X\langle (1+x)\underline\theta
\rangle$ is IT(0) for $x>0$. In the proof of the previous Theorem, we computed the function $h^0_{{\mathcal O}_X}(x\underline\theta)=x^g$ for $x$ in an interval $(1-\epsilon^-,1]$. Therefore, to conclude the proof, we need to show that we can take $\epsilon^-=1$. By Proposition \ref{GL}, the dualizing sheaf $\omega_{V(X)}$ is a GV-sheaf. Hence, again by Theorem \ref{B}(a), $\omega_{V(X)}\langle x\underline\theta\rangle$ is IT(0) for $x>0$. Therefore both $R^0\Phi_{\mathcal P}({\mathcal O}_X(\Theta))\langle n\underline\theta\rangle={\mathcal O}_A(-\Theta)\langle n\underline\theta\rangle$ and $R^d\Phi_{\mathcal P}({\mathcal O}_X(\Theta))\langle n\underline\theta\rangle=\omega_{V(X)}(-\Theta)\langle n\underline\theta\rangle$ are IT(0) for $n>1$. Therefore one can take $\epsilon^-=1$ (see Remark \ref{eff-serre}).
\end{proof}
A consequence of Corollary \ref{gv-strong} and Theorem \ref{gv1} we get
\begin{theorem}\label{hodgenumber} Let $X$ be a geometrically non-degenerate $d$-dimensional $GV$-subscheme of a $g$-dimensional p.p.a.v. $A$. Then
\[h^i({\mathcal O}_X)={g\choose i}\quad\hbox{for all $i\le d$}\]
\end{theorem}
\begin{proof} Lemma \ref{cup} implies that the natural map $\Lambda^iH^1({\mathcal O}_A)\rightarrow H^i({\mathcal O}_X)$ is an isomorphism for $i<d$ and injective for $i=d$. By Theorem \ref{gv1} the coefficient of degree $0$ of the Poincar\'e polynomial, namely $\chi({\mathcal O}_X)$, is equal to $\sum_{i=0}^d(-1)^i{g\choose i}$. The result follows.
\end{proof}
If one believes to the conjectures mentioned at the beginning of this section, geometrically non-degenerate GV-subschemes should be normal with rational singularities. On a somewhat different note, we take the opportunity to prove some partial results in this direction, using the results of \cite{s} and \cite{cps}.
\begin{lemma}\label{normality}
Let $(A, \underline\theta)$ be an indecomposable PPAV. Assume that $X$ is a geometrically non-degenerate GV-subscheme. Then $X$ is normal.
\end{lemma}
\begin{proof}
We know that $X$ is reduced and irreducible by (a) of this section. Since an irreducible theta divisor is smooth in codimension $1$, by the argument in \cite[Theorem 4.1]{cps}, GV-subschemes of $A$ are smooth in codimension $1$. Since $X$ is Cohen-Macaulay, we conclude that $X$ is normal by Serre's criterion.
\end{proof}
\begin{corollary}\label{rational-singularity}
Let $(A, \underline\theta)$ be an indecomposable PPAV. Assume that $X$ is a geometrically non-degenerate GV-subscheme of dimension $2$. Then $X$ has rational singularities.
\end{corollary}
\begin{proof}
Fix $\mu: X'\rightarrow X$ a resolution of singularities. By Lemma \ref{normality}, we only need to prove that $\mu_*\omega_{X'}=\omega_X$ to conclude that $X$ has rational singularities.
We first claim that $h^1(\mathcal O_{X'})=h^0(\Omega_{X'}^1)=g$. Assume the contrary. Then the Albanese variety $ A_{X'}$ has dimension $h^1(\mathcal O_{X'})>g$. We consider the commutative diagram
\begin{eqnarray*}
\xymatrix{
X'\ar[dr]_{\tau}\ar[d]^{\mu}\ar[r]^{a_{X'}} & A_{X'}\ar[d]\\
X \ar@{^{(}->}[r]& A.}
\end{eqnarray*}
Since $X$ is normal, $\mu_*\mathcal{O}_{X'}=\mathcal{O}_X$. Hence $h^1(\mathcal O_{X'})=h^1(\mathcal{O}_X)+h^0(X, R^1\mu_*\mathcal{O}_{X'})>g$. Thus $h^0(X, R^1\mu_*\mathcal{O}_{X'})>0$ and hence there exists an irreducible curve $C$ on $X'$ which is contracted by $\mu$ and $a_{X'}\mid_C$ is generically finite onto its image. In particular, there exists a holomorphic $1$-form $\omega_0\in H^0(X', \Omega_{X'}^1)$ such that $(\omega_0)_{|C}$ is non-zero. Pick $p\in C$ a general point
and consider the following local calculation around $p$. Let $x, y$ be local analytic coordinates of $X'$ around $p$ and assume that $C$ is defined by $y=0$. We may assume that $\tau(p)$ is the origin of $A$ and $\tau(x, y)=(f_1(x, y), \ldots, f_g(x,y))$ in an analytic neighborhood of $p$. Let $m$ be the multiplicity of $C$ in the fiber $\mu^*(0)$. Then the holomorphic functions $f_i$ can be written as $y^mg_i(x,y)$ with $g_i$ holomorphic around $p$. For each holomorphic $1$-form $\omega\in H^0(A, \Omega_A^1)$, we write $\tau^*\omega=h_1dx+h_2dy$ in a neighborhood of $p$. Then $y^m\mid h_1$ and $y^{m-1}\mid h_2$. Thus for any $s\in \tau^*H^0(A, \Omega_A^2)\subset H^0(X', K_{X'}) $, the corresponding divisor $D(s)$ has multiplicity of $C$ $\geq 2m-1$. On the other hand, since $\omega_0\mid_C\neq 0$, writing locally $\omega_0=g_1dx+g_2dy$ around $p$, we have that $(g_1)_{|_C}$ is non-zero. Hence there exists $t\in \omega_0\wedge \tau^*H^0(A, \Omega_A^1)\subset H^0(X', K_{X'})$ such that the corresponding divisor $D(t)$ whose multiplicity of $C$ is $m-1$. But then have a contradiction, since, by a result of Schreieder mentioned in the Introduction, the natural map $H^0(X', K_{X'})\simeq \tau^*H^0(A, \Omega_A^2)$ is an isomorphism (\cite[Theorem 2]{s}). This proves what claimed. It follows that the natural map $H^1({\mathcal O}_A)\rightarrow H^1({\mathcal O}_{X^\prime})$ is an isomorphism.
Consider the short exact sequence $$0\rightarrow \mu_*\omega_{X'}\rightarrow \omega_X\rightarrow \tau\rightarrow 0.$$ By Theorem \ref{hodgenumber} and the result of Schreieder it follows that $H^0(\mu_*\omega_{X'})\rightarrow H^0(\omega_X)$ is an isomorphism. By Theorem \ref{hodgenumber} and the above claim it follows that the map $H^1(\mu_*\omega_{X'})\rightarrow H^1(\omega_X)$ is an isomorphism. Thus $h^0(\tau)=0$ hence the cohomological support locus $V^0(\tau)$ is strictly contained in $\widehat A$. On the other hand, we know that $\omega_X$ is M-regular by Corollary \ref{gv-strong}. This, together with the fact that $\mu_*\omega_{X'}$ is GV (\cite{hac}) yields that $\tau$ is M-regular. Since $V^0(\tau)$ is strictly contained in $\widehat A$, this implies (as it is well known, see e.g. \cite[Lemma 1.12(b)]{msri}) that $\tau=0$.
\end{proof}
\begin{corollary}\label{l-t} In the hypothesis of the previous Corollary, let $X^\prime$ be any desingularization of $X$. Then the induced morphism $\tau:X^\prime \rightarrow A$ is the Albanese morphism of $X^\prime$.
\end{corollary}
\begin{proof} In view of the fact that $h^1({\mathcal O}_{X^\prime})=g=\dim A$, it is enough to prove that $\tau$ does not factor through any non-trivial isogeny. This means that if $\alpha\in \widehat A$ is such that $\tau^*P_\alpha$ is trivial
then $\alpha=\hat e$. But this follows from the last part of Lemma \ref{gv-strong}, which implies by base change that the cohomological support locus $V^d(\omega_X)=\{\hat e\}$ and Lemma \ref{rational-singularity}.
Alternatively, one can use the results of \cite{lt}.
\end{proof}
\section{Cohomological rank functions of the ideal of one point, multiplication maps of global sections, and normal generation of abelian varieties}
We refer to the Introduction for a general presentation of the contents of this section. Let $A$ be an abelian variety and ${\underline l}$ and ${\underline n}$ be polarizations on $A$ (in our applications ${\underline n}$ will be a multiple of ${\underline l}$). Assume moreover that ${\underline n}$ is basepoint free. Let $N$ be an ample and basepoint free line bundle representing ${\underline n}$. We consider the evaluation bundle of $N$, defined by the exact sequence
\begin{equation}\label{evaluation}0\rightarrow M_N\rightarrow H^0(N)\otimes{\mathcal O}_A\rightarrow N\rightarrow 0
\end{equation}
Finally, let $p\in A$. We consider the cohomological rank functions $h^i_{\mathcal{I}_p}(x{\underline l})$ and $h^i_{M_N}(x{\underline n})$.\footnote{Note that they don't depend respectively on $p$ and on $N$.} In both cases for $x\ge 0$ the functions are zero for $i\ge 2$ and for $x\le 0$ they are zero for $i\ne 1, g$. We consider their maximal critical points, namely
\begin{eqnarray*}\beta({{\underline l}})
&=&\inf \{x\in\mathbb Q\>|\> h^1_{\mathcal{I}_p}(x{\underline l})=0\}\\
s({\underline n})&=&
\inf\{x\in\mathbb Q\>|\> h^1_{M_N}(x\underline n)=0\}
\end{eqnarray*}
As it is easy to see, the problem illustrated by Remark \ref{maximal-example} about Proposition \ref{maxthreshold} does not occur for these two sheaves. Hence if $x_0\in\mathbb Q$ is such that $\mathcal{I}_p\langle x_0{\underline l}\rangle$ (resp. $M_N\langle x_0{\underline n}\rangle$) is GV but not IT(0) then $\beta({\underline l})$ (resp. $s({\underline n})$) is a critical point of $h^i_{\mathcal{I}_p}(x{\underline l})$ (resp. $h^i_{M_N}(x{\underline n})$) for $i=0,1$, in fact the maximal one. In any case, for $x\in \mathbb Q$, the fact that $x>\beta({\underline l})$ (resp. $y> s({\underline n})$) is equivalent to the fact that $\mathcal{I}_p\langle x{\underline l}\rangle$ (resp. $M_N\langle y{\underline n}\rangle$) is IT(0).
Let us spell what the IT(0) (resp. GV) condition mean for the above $\mathbb Q$-twisted sheaves. For $x=\frac{a}{b}\in \mathbb Q^{>0}$, the fact that $\mathcal{I}_p\langle x{\underline l}\rangle$ is IT(0) means that
\[h^1(\mu_b^*(\mathcal{I}_p)\otimes L_\alpha^{ab})=0\]
for all line $\alpha\in \widehat A$, where as usual $L_\alpha^{ab}$ denotes $L^{ab}\otimes P_\alpha$. This means that the finite scheme $p+\mu_b^{-1}(0)$ imposes independent conditions to the line bundles $L^{ab}_\alpha$ for all $\alpha\in\widehat A$. Since the $L^{ab}_\alpha$'s are all translates of the same line bundle this means that \emph{for all} $p\in A$ the finite scheme $p+\mu_b^{-1}(0)$ impose independent conditions to the global sections of the line bundle $L^{ab}$ (hence the same happens for \emph{all} line bundles $L_\alpha^{ab}$). This condition can be interpreted as basepoint-freeness for the fractional polarization $x{\underline l}$. Note that if $x\in \mathbb Z$, writing $x=\frac{xb}{b}$ one finds back the usual basepoint-freeness. In turn the fact that $\mathcal{I}_p\langle x{\underline l}\rangle$ is GV but not IT(0) means that for all $\alpha$ in a proper closed subset of $ \widehat A$ the finite scheme $p+\mu_b^{-1}(0)$ does not impose independent conditions to the global sections of the line bundle $L^{ab}_\alpha$. As above this means that for $p$ in a proper subset of $ A$ the finite scheme $p+\mu_b^{-1}(0)$ does not impose independent conditions to the global sections of $L^{ab}$ (hence the same property holds for all line bundles $L^{ab}_\alpha$). Again, for $x\in \mathbb Z$ one finds back the usual notion of base points and base locus. It follows that $\mathcal{I}_p(L)$ is any case GV and it is IT(0) if and only if ${\underline l}$ is basepoint free. In other words: $\beta({\underline l})\le 1$ and equality holds if and only if ${\underline l}$ has base points.
Similarly, for $y=\frac{a}{b}$, the fact that $M_N\langle y{\underline n}\rangle$ is IT(0) (resp. GV) means that
\[h^1(\mu_b^*(M_N)\otimes N^{ab}_\alpha)=0\]
for all (resp. for general) $\alpha\in \widehat A$. Pulling back the exact sequence (\ref{evaluation}) via $\mu_b$ and tensoring with $L^{ab}_\alpha$ this has the meaning mentioned in the introduction, namely that the multiplication maps obtained by composing with the natural inclusion $H^0(N)\hookrightarrow H^0(\mu_b^*N)$
\begin{equation}\label{mult-frac-new}H^0(N)\otimes H^0(N^{ab}_\alpha)\rightarrow H^0(\mu_b^*(N)\otimes N^{ab}_\alpha)
\end{equation} are surjective for all (resp. for general) $\alpha\in \widehat A$. The above maps (\ref{mult-frac-new}) can be thought as the multiplication maps of global sections of $N$ and of a representative of the rational power $N^{\frac{a}{b}}$ (twisted by $P_\alpha$)
\begin{proposition} Let $(A,{\underline n})$ be a polarized abelian variety and assume that $\underline n$ is basepoint free. Let $p\in A$. For $i=0,1$ and $y<1 $
\[h^i_{\mathcal{I}_p}(y\underline n)=\frac{(1-y)^g}{\chi({\underline n})}h^i_{M_N}((-1+\frac{1}{1-y})\underline n)\]
Consequently
\begin{equation}\label{s-r}s({\underline n})=-1+\frac{1}{1-\beta({\underline n})}=\frac{\beta({\underline n})}{1-\beta({\underline n})}
\end{equation}
\end{proposition}
\begin{proof}
We can assume that $p=e$ (the origin of $A$). The essential point of the proof is that
\begin{equation}\label{f}\varphi_{\underline n}^*(R^0\Phi_{\mathcal P}(\mathcal{I}_e(N)))=M_N\otimes N^{-1}
\end{equation}
Indeed, by the exact sequence $0\rightarrow \mathcal{I}_e(N)\rightarrow N\rightarrow N\otimes k(e)\rightarrow 0$ it follows that $R^0\Phi_{\mathcal P}(\mathcal{I}_e(N))$ is the kernel of the map
\[R^0\Phi_{\mathcal P}(N)=\widehat N\buildrel f \over \rightarrow R^0\Phi_{\mathcal P}(N\otimes k(e))={\mathcal O}_{\widehat A}.\]
By (\ref{mukai-4}) the map $\varphi_{\underline n}^*(f)$ is identified to a map
$H^0(N)\otimes N^{-1}\rightarrow {\mathcal O}_A$ which is easily seen to be the evaluation map tensored with $N^{-1}$.
Next, we notice that, since the polarization ${\underline n}$ is assumed to be basepoint free, we have that
\begin{equation}\label{R0}
\Phi_{\mathcal P}(\mathcal{I}_e(N))=R^0\Phi_{\mathcal P}(\mathcal{I}_e(N))
\end{equation}
To prove this, we first notice that $H^i(\mathcal{I}_e\otimes N_\alpha)=0$ for all $\alpha\in \widehat A$ and $i>1$. By base change this implies that the support $R^i\Phi_{\mathcal P}(\mathcal{I}_e(N))$ is equal to $V^1(\mathcal{I}_e(L))=\{\alpha\in\widehat A\>|\> h^1(\mathcal{I}_e\otimes N\otimes P_\alpha)>0\}$ which is non-empty if and only if ${\underline n}$ has base points. This proves (\ref{R0}).
Therefore, by Proposition \ref{inversion-a} and degeneration of the spectral sequence computing the hypercohomology, we have that for $i=0,1$ and $t< 0$
\[h^i_{\mathcal{I}_e(N)}(t{\underline n})=\frac{(-t)^g}{\chi({\underline n})}h^i_{\varphi_{\underline n}^*R^0\Phi_{\mathcal P}(\mathcal{I}_0(N))}(-\frac{1}{t}\underline n)\buildrel{(\ref{f})}\over =\frac{(-t)^g}{\chi({\underline n})}h^i_{M_N}((-1-\frac{1}{t})\underline n)\]
The first statement of the Proposition follows setting $y=1+t$. The second statement follows from the first one.
\end{proof}
Applying the previous proposition to divisible polarizations ${\underline n}=h{\underline l}$ we get Theorem \ref{b-s} of the Introduction, namely
\begin{corollary} Let $(A,{\underline l})$ be a polarized abelian variety and let $h$ be an integer such that $h{\underline l}$ is basepoint free \emph{(hence $h\ge 2$, and $h\ge 1$ if ${\underline l}$ is basepoint-free)}. Then
\begin{equation}\label{s-beta}
s(h {\underline l})= \frac{\beta({\underline l})}{h-\beta({\underline l})}
\end{equation}
Consequently:
\noindent (a)
\[s(h {\underline l})\le \frac{1}{h-1}\]
and equality holds if and only if ${\underline l}$ has base points.
\noindent (b) Assume that ${\underline l}$ is base point free. Then $s({\underline l})< 1$ if and only if $\beta({\underline l})<\frac{1}{2}$. In particular, if $\beta({\underline l})<\frac{1}{2}$ then ${\underline l}$ is normally generated.
\end{corollary}
\begin{proof} (a) By definition, $h^i_{\mathcal{I}_e}(x(h{\underline l}))=h^i_{\mathcal{I}_e}((xh){\underline l})$. Hence $\beta(h{\underline l})=\frac{1}{h}\beta({\underline l})$.
By the previous Proposition,
\[s(h{\underline l})
=\frac{\beta(h{\underline l})}{1-\beta(h{\underline l})}=\frac{\beta({\underline l})}{h-\beta({\underline l})}\]
The last statement follows from the fact that $\beta({\underline l})\le 1$ and equality holds if and only if ${\underline l}$ has base points.
\noindent (b) The first assertion follows immediately from (\ref{s-beta}). Concerning the last assertion, we have that $s({\underline l})<1$ if and only if the multiplication maps
\[H^0(L)\otimes H^0(L_\alpha)\rightarrow H^0(L^2_\alpha)\]
are surjective for all $\alpha\in\widehat A$. A well known argument (e.g. \cite{kempf}, proof of Thm 6.8(c) and Cor. 6.9, or \cite{birke-lange}, proof of Theorem 7.3.1) proves that this implies that $L_\alpha$ is normally generated for all $\alpha\in\widehat A$.
\end{proof}
Item (b) as well as the case $h=2$ of item (a) of the above Corollary have been already commented in the Introduction. Here we note that the Proposition implies that, for an integer $h\ge 2$ and $\frac{a}{b}\ge \frac{1}{h-1}$ the ``fractional" multiplication maps of global sections
\[H^0(L^h)\otimes H^0(L^{hab}_\alpha)\rightarrow H^0(\mu_b^*(L^h)\otimes L^{hab}_\alpha)\]
are surjective for general $\alpha\in \widehat A$ and in fact for all $\alpha \in \widehat A$ as soon as $\frac{a}{b}> \frac{1}{h-1}$ or ${\underline l}$ is basepoint free. This is much stronger than the known results on the subject. For example, for $h=3$ Koizumi's theorem on projective normality, in a slightly stronger version (\cite{kempf} Cor.6.9, \cite{birke-lange} Th. 7.3.1) tells that the above maps are surjective for all $\alpha\in\widehat A$ for $b=1$ and $a=\frac{2}{3}$
while the Corollary asserts that the same happens for $\frac{a}{b}>\frac{1}{2}$. Moreover, for the critical value $\frac{a}{b}=\frac{1}{2}$, the Corollary tells that the maps
\[H^0(L^3)\otimes H^0(L^{6}_\alpha)\rightarrow H^0(\mu_2^*(L^3)\otimes L^{6}_\alpha)\]
are surjective for general $\alpha\in \widehat A$ and in fact for all $\alpha\in\widehat A$ as soon as ${\underline l}$ is basepoint free. For arbitrary $h$ the same happens for the ``critical" maps
\[H^0(L^h)\otimes H^0(L^{h(h-1)}_\alpha)\rightarrow H^0(\mu_{h-1}^*(L^h)\otimes L^{h(h-1)}_\alpha)\]
Note that, when ${\underline l}$ is a principal polarization, the dimension of the source of the above maps is equal to the dimension of the target, namely $(h^2(h-1))^g$.
\providecommand{\bysame}{\leavevmode\hbox
to3em{\hrulefill}\thinspace}
|
1,116,691,499,372 | arxiv | \section{Introduction and summary}
\label{sec:intro}
\input{sections/intro}
\section{Double holography for Interface CFTs}
\label{sec:double_holo}
\input{sections/double}
\section{Correlation functions of heavy operators in ICFT}
\label{sec:corr_heavy}
\input{sections/corr_heavy}
\section{Entanglement entropy and the island formula}
\label{sec:Ent_entropy}
\input{sections/Ent_entropy}
\section{Seafaring from AdS/ICFT to AdS/BCFT}
\label{sec:BCFT}
\input{sections/BCFT}
\section{Conclusions}
\label{sec:conc}
\input{sections/conc}
\subsection*{Acknowledgements}
We would like to thank Costas Bachas, Ivano Basile, Lorenzo Bianchi, Nikolay Bobev, Shira Chapman, Ben Craps, Stefano De Angelis, Oleksandr Gamayun, Raghu Mahajan, Vassilis Papadopoulos, Edgar Shaghoulian and Manus Visser. This work has been supported in part by the Fonds National Suisse de la Recherche Scientifique (Schweizerischer Nationalfonds zur F\"orderung der wissenschaftlichen Forschung) through Project Grants 200020\_ 182513, the NCCR 51NF40-141869 The Mathematics of Physics (SwissMAP), and by the DFG Collaborative Research Center (CRC) 183 Project No. 277101999 - project A03. T.A. is supported by the Delta ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). M.M is supported by the Swiss National Science Foundation through the Ambizione grant number 193472.
\subsection{Geodesics and entanglement entropy in BCFT from ICFT}
\label{subsec:TakaProof}
To obtain the physics of BCFT from ICFT, we must tune the AdS lengths such that one becomes much smaller than the other, i.e.
\begin{equation}
\nu \equiv \frac{L_{\rm I}}{L_{\rm II}} \to 0 \ .
\end{equation}
This ensures that $c_{\rm II}\gg c_{\rm I}$, meaning that the interface becomes impermeable to the transmission of excitations, as the CFT$_{\rm I}$ has comparatively too few degrees of freedom per unit volume to acommodate a generic excitation coming in from the right.
How do we extract the vacuum geometry in this limit? First recall that the tension $T$ is constrained to lie between $T_{\rm min}$ and $T_{\rm max}$ as defined in \eqref{T_constraints}. We can parametrize the range of possible $T$ via
\begin{equation}\label{eq:etadef}
T^2 = \frac{1}{L_{\rm I}^2} + \frac{1}{L_{\rm II}^2} + \frac{2 \eta}{L_{\rm I}L_{\rm II}} \ ,
\end{equation}
where the constraint is now translated into $-1 < \eta < 1$. Let us rewrite \eqref{tanh_rhos}, in terms of of these parameters:
\begin{equation}
\sin(\psi_{\rm I}) = \frac{1 + \eta\, \nu }{\sqrt{ 1 + 2 \eta\, \nu+\nu^2}} \ , \qquad\qquad \sin(\psi_{\rm II}) = \frac{\eta + \nu }{\sqrt{ 1 + 2 \eta\, \nu+\nu^2}} \ .
\end{equation}
In the limit $\nu \to 0$, this means
\begin{equation}
\sin(\psi_{\rm I}) \to 1~, \qquad\qquad \sin(\psi_{\rm II}) = \eta ~.
\label{sinPsiToEta}
\end{equation}
Meaning that $\psi_{\rm I} \to \pi/2$ and the geometry becomes that of figure \ref{Takayanagi_prescription}, where we have also displayed a geodesic between two boundary points across the interface, as studied in section \ref{sec:intcrossinggeods}. Eq. \eqref{sinPsiToEta} can be used to relate $\eta$ to the tension of the EOW brane.
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}
\shade[top color=white, bottom color=ForestGreen, opacity = 0.25] (0,0) -- (5,0) -- ({5},{5*sin(60)}) -- ({-5*cos(60)},{5*sin(60)}) -- cycle;
\scoped[transform canvas={rotate around={120:(0,0)}}]
\shade[top color=white, bottom color=Cerulean, opacity = 0.25] (-3,0) -- ({5},{0}) -- ({5},{3.5}) -- ({-3},{3.5}) -- cycle;
\draw[dashed] (0,0) -- (0,5);
\draw (0, 1.5) arc (90:120:1.5) node [midway, anchor=south] {\small $\psi_{\rm II}$};
\draw (0, 1.45) arc (90:120:1.45);
\draw[very thick, ForestGreen, opacity = 0.8, name path=L1] (0,0) -- (5,0);
\draw[very thick, Cerulean, opacity = 0.8] (0,0) -- ({3*cos(60)},{-3*sin(60)});
\draw[very thick, NavyBlue, opacity = 0.8, name path=I] (0,0) -- ({-5*cos(60)},{5*sin(60)});
\draw[very thick, Cerulean, opacity = 0, name path=L2] ({5*cos(45)},{5*sin(45)}) -- ({4.5*cos(60)},{-4.5*sin(60)});
\draw[very thick, blue] ({2*cos(60)},{-2*sin(60)}) arc (300:120:2.8);
\draw[very thick, blue] (3.6,0) arc (0:120:3.6);
\fill[NavyBlue] (0,0) circle (2pt);
\draw (1.9, 0) node [anchor=north] {\small $\sigma_{\rm II}$};
\draw (0.95, -0.8) node [anchor=north] {\small $\sigma_{\rm I}$};
\fill[red] ({2*cos(60)},{-2*sin(60)}) circle (2pt);
\fill[red] (3.6,0) circle (2pt);
\end{tikzpicture}
\end{center}
\caption{The geodesic for the two point function in the BCFT limit proposed in (\ref{BCFT_lambda_0}). In such regime the angle $\psi_{\rm I}$ goes to $\pi/2$, the length doesn't depend on $\sigma_{\rm I}$ anymore and the geodesic approaches the brane at a right angle, as proposed by Takayanagi in \cite{Takayanagi:2011zk}.}
\label{Takayanagi_prescription}
\end{figure}
Let us return to the types of geodesics shown in figure \ref{Takayanagi_prescription}. We see clearly that in the limit $\nu\to 0$, the angle $\varphi$ as shown in figure \ref{Generic_sigma_2} is tending to $\pi$, meaning that, according to \eqref{theta_phi},
\begin{equation}
\theta\to \frac{\pi}{2}+\psi_{\rm II}~.
\end{equation}
For this configuration, $r$ and $R$ given in \eqref{sol_r1} and \eqref{sol_r2} tend to $\frac{\sigma_{\rm I}+\sigma_{\rm II}}{2}$ and $\sigma_{\rm II}$ respectively. We can obtain the length of the geodesic connecting these two points by taking $L_{\rm I}= \nu L_{\rm II}$ in \eqref{length_geodesic}, yielding
\begin{equation}
d(\sigma_{\rm I}, \sigma_{\rm II})\underset{\nu\to0}{=} L_{\rm II} \log \left[ \frac{2\sigma_{\rm II}}{\varepsilon_{\rm II}}\tan\left(\frac{\psi_{\rm II}}{2}+\frac{\pi}{4}\right) \right]=\rho^*_{\rm II} + L_{\rm II} \log \left[ \frac{2\sigma_{\rm II}}{\varepsilon_{\rm II}} \right]~, \label{eq:bcftlength}
\end{equation}
where the dependence on $\sigma_{\rm I}$ has dropped out entirely, and we have thus reproduced the result in \cite{Takayanagi:2011zk}.
Some interpretation is in order. A geodesic connecting two boundary points is usually associated with a two-point function. However, in sending $\nu\to 0$ we are sending the ratio of dimensions $\Delta_{\rm I}/\Delta_{\rm II}$ to zero as well, effectively producing the correlation function of $\mathcal{O}_{\rm II}$ with the identity. Thus in the $\nu\to 0$ limit, the ICFT two point function effectively becomes a BCFT one point function, whose universal form is
\begin{equation}
\langle \mathcal O_{\rm II}(\sigma_{\rm II}) \rangle = \frac{a_{\mathcal O}}{(2 \sigma_{\rm II})^{\Delta_{\rm II}}} \ ,
\end{equation}
as can be seen by exponentiating \eqref{eq:bcftlength} using the geodesic approximation.
In \cite{Takayanagi:2011zk}, this result was obtained by minimizing over the length of all curves ending on the EOW brane. That procedure yields a geodesic which arrives on the brane at a right angle. In figure \ref{Takayanagi_prescription}, we see how this condition arises in the $\nu\to0$ limit of the smooth geodesics considered in the ICFT construction: the geodesic in the AdS$_{\rm I}$ region becomes a complete semi-circle.
What about entanglement entropies? RT surfaces that compute EEs are codimension-2, so in the specific case of AdS$_3$/CFT$_2$, they would correspond to geodesics, but not so in higher dimensions. Furthermore, the emergence of minimal surfaces in both three and higher dimensions does not stem from the worldline path integral of a massive particle, so we need an alternative argument to convince us that we should expect minimal surfaces to compute entanglement entropies in ICFT, and that their BCFT limits give rise to Takayanagi's prescription. Luckily the reasoning of \cite{Lewkowycz:2013nqa} still holds in our holographic interface setup (even generalized to higher dimensions) since the crux of their argument relied on the $\mathbb{Z}_n$ symmetry of the $n-$replicated entangling region at the locally-asymptotically-AdS boundary---which one must consider when calculating the entanglement entropy using the standard replica trick. In this construction, the minimality of the RT surface stems from the leading order effects of the codimension-2 locus in the bulk left invariant under the action of this $\mathbb{Z}_n$, even as we take $n\to 1$. At the asymptotic boundary in ICFT we will still specify boundary conditions at the entangling surface, and will therefore have a $\mathbb{Z}_n$ symmetry in the replicated geometry, even if the entangling surface crosses the interface. Since the bulk metric is continuous due to the Israel-Lanczos conditions, there will again be a $\mathbb{Z}_n$-invariant codimension-2 surface that will ultimately be responsible for the minimal RT surfaces in ICFT whose effect will survive the $n\to 1$ limit. Having established that minimal RT surfaces compute EEs in AdS geometries with a thin brane, the remaining step is to take the BCFT limit.
\subsection{Quantum Extremal Surfaces for BCFTs}
As in section \ref{subsec:EEvacuum}, we can now try to interpret the BCFT limit of the three-dimensional geodesic as a QES in the ``intermediate'' 2d braneworld picture in the large-$T$ limit. This section mirrors \ref{subsec:EEvacuum} very closely, so we will be brief. We will also take
\begin{equation}
c_{\rm II}=c~, \qquad\qquad\sigma_{\rm II}=\sigma~, \qquad\qquad \epsilon_{\rm II}=\varepsilon
\end{equation}
for notational convenience. As before, we view the effective action on the brane as resulting from integrating out bulk degrees of freedom up to the brane \cite{Henningson:1998ey} (similar ideas have been discussed in \cite{Verheijden:2021yrb, Chen:2020uac, Chen:2020hmv, Hernandez:2020nem, grimaldi2022quantum}), and the generalized entropy is the entropy of this weakly gravitating two-dimensional conformal theory, where we take the UV cutoff along the brane insertion to be $y$ dependent $\varepsilon_{\rm brane}({y})= y\cos\psi_{\rm II}$ given the angle the brane makes with the boundary at $z=0$. We find
\begin{equation}
S_{\text{gen}} ( y) = \frac{c}{6} \log \left[ \frac{(\sigma + y)^2}{ y \varepsilon} \frac{1}{\cos(\psi_{\rm II})}\right] \ .
\label{BCFT_gen_entropy2}
\end{equation}
The effect of gravity on the brane is accounted for by requiring that the quantum extremal surface be an extremum over all possible island locations $y$ in (\ref{BCFT_gen_entropy2}). Thus
\begin{equation}
\partial_{ y} S_{\text{gen}} = 0 \qquad \Rightarrow \qquad y^* = \sigma \ ,
\end{equation}
Notice that the position of the QES matches exactly the point where Takayanagi's geodesic meets the EOW brane \cite{Takayanagi:2011zk, Fujita:2011fp}. The entropy of the island saddle in the intermediate picture then reads
\begin{equation}
S_{\text{island}} = \frac{c}{6} \log\left[\frac{2\sigma}{\varepsilon} \frac{2}{\cos(\psi_{\rm II})}\right] \ .
\end{equation}
We can compare this result with the one obtained using Takayanagi's prescription in three dimensions,
\begin{equation}
S^{\text{3D}}_{\text{RT}} = \frac{c}{6} \log\left[\frac{2 \sigma}{\varepsilon} \tan\left(\frac{\psi_{\rm II}}{2}+\frac{\pi}{4}\right)\right] \ .
\end{equation}
To see that these match in the large-tension limit, recall that $T_{\rm max}$ is reached by taking $\eta\to 1$ in \eqref{eq:etadef}. Parametrizing $\eta=1-\delta^2/2$, this translates to $\psi_{\rm II}=\frac{\pi}{2}-\delta+\mathcal{O}(\delta^3)$ as $\delta\to 0$. Plugging this into the above formulas, we find
\begin{align}
S_{\text{island}}&=-\frac{c}{6}\log(\delta)+\frac{c}{6}\log\left[\frac{4\sigma}{\varepsilon}\right]+\mathcal{O}\left(\delta^2\right)\\
S^{\text{3D}}_{\text{RT}}&=-\frac{c}{6}\log(\delta)+\frac{c}{6}\log\left[\frac{4\sigma}{\varepsilon}\right]+\mathcal{O}\left(\delta^2\right)
\end{align}
where we find a mismatch between these formulas only at $\mathcal{O}\left(\delta^2\right)$. The match between the QES formula and the RT calculation is somewhat cleaner in the BCFT limit as compared to ICFT, since the location of the island agrees for any value of the tension, although the entropies still only match in the limit of $T\to T_{\rm max}$.
It is curious to remark that a different choice of position dependent cutoff, namely
\begin{equation}
\varepsilon_{\rm brane}(y)= \frac{2 y}{\tan\left(\frac{\psi_{\rm II}}{2}+\frac{\pi}{4}\right)}
\end{equation}
would have resulted in an \emph{exact} match between the island prescription and the three-dimensional RT formula, valid for all values of the tension $T$. While this would be appealing, we have no geometric interpretation of such a formula and moreover do not expect the theory on the brane to be locally conformal away from the large-$T$ limit. We thus cannot justify using the CFT entropy formula when computing the generalized entropy for arbitrary $T$.
\subsection{BCFT two-point functions} \label{S_p_a_2pt}
Let us now analyze the BCFT limit of geodesics connecting two points on the same CFT (figure \ref{2pt_same_side_1}), where we take the insertions on the side whose central charge we are keeping finite. We readily see that our limit gives the structure of geodesics in AdS/BCFT studied in \cite{Kastikainen:2021ybu}.
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}
\shade[top color=white, bottom color=ForestGreen, opacity = 0.25] (0,0) -- (5,0) -- ({5},{5*sin(60)}) -- ({-5*cos(60)},{5*sin(60)}) -- cycle;
\scoped[transform canvas={rotate around={120:(0,0)}}]
\shade[top color=white, bottom color=Cerulean, opacity = 0.25] (-1,0) -- ({5},{0}) -- ({5},{3.5}) -- ({-1},{3.5}) -- cycle;
\draw[very thick, ForestGreen, opacity = 0.8, name path=L1] (0,0) -- (5,0);
\draw[very thick, Cerulean, opacity = 0.8] (0,0) -- ({1*cos(60)},{-1*sin(60)});
\draw[very thick, NavyBlue, opacity = 0.8, name path=I] (0,0) -- ({-5*cos(60)},{5*sin(60)});
\draw[very thick, Cerulean, opacity = 0, name path=L2] ({5*cos(45)},{5*sin(45)}) -- ({2.7*cos(60)},{-2.7*sin(60)});
\draw[very thick, Violet] (3.6,0) arc (0:120:3.6)node [midway, anchor=south west] {\small $2$};
\draw[very thick, Violet] (1.2,0) arc (0:120:1.2);
\draw[very thick, Violet] ({-3.6*cos(60)},{3.6*sin(60)}) arc (120:300:1.2);
\draw[very thick, Magenta] (3.6,0) arc (0:180:1.2) node [midway, anchor=south] {\small $1$};
\draw[very thick, blue] (1.2,0) arc (0:103:1.84);
\draw[very thick, blue] (3.6,0) arc (0:137:2.67) node [midway, anchor=north east] {\small $3$};
\fill[NavyBlue] (0,0) circle (2pt);
\fill[red] (1.2,0) circle (2pt);
\fill[red] (3.6,0) circle (2pt);
\fill[blue] (-1.039,1.8) circle (1pt);
\end{tikzpicture}
\end{center}
\caption{The various geodesic saddles in BCFT, recovered from the $\nu\to 0$ limit of ICFT.}
\label{BCFT_saddle_pts}
\end{figure}
We label the possible RT geodesics by (1), (2) and (3) in figure \ref{BCFT_saddle_pts}, where (1) is the standard ``connected'' RT surface whereas (2) and (3) are ``disconnected'' surfaces that can end on the EOW brane. Based on our ICFT setup, we expect (2) and (3) to be related to the two solution branches labeled $s_1^*$ and $s_2^*$ in section \ref{sec:same_side}.
What happens as $\nu \to0 $ in the ICFT setup is that surfaces of type (3) probe the region behind the brane less as $\nu$ is decreased, and ultimately reflect off the EOW brane. This is precisely the structure described in \cite{Kastikainen:2021ybu}. Let us return to \eqref{equation_for_s} in the BCFT limit. Now we must solve:
\begin{equation}\label{eq:bcftsequation}
\Theta=\frac{\sigma_{\rm II}^P}{\sigma_{\rm II}^Q}=\frac{s(s+2)+2\sin(\psi_{\rm II})\left(\sin(\psi_{\rm II})+\sqrt{s(s+2)+\sin^2(\psi_{\rm II})}\right)}{s^2}
\end{equation}
The solutions that describe geodesics of type (2) are given by $s_2^*\to \infty$. Indeed in the $\nu\to0$ limit, the red curve of figure \ref{Two_saddles} is pushed upwards towards infinity. The solutions that describe curves of type (3) on the other hand are given by
\begin{equation}
s^*_{1} = \frac{2}{\Theta -1}\left( 1+ \sin(\psi_{\rm II}) \sqrt{\Theta}\right) \ ,
\label{s3_psi_BCFT}
\end{equation}
however this only satisfies \eqref{eq:bcftsequation} for
\begin{equation}
\sin(\psi_{\rm II})>-\frac{2\sqrt{\Theta}}{1+\Theta}~.
\end{equation}
We also remind the reader that
\begin{equation}
s> \begin{cases} 0~, &\psi_{\rm II} \in [0,\pi/2]\\ -1+\cos(\psi_{\rm II})~,&\psi_{\rm II} \in [-\pi/2,0]\end{cases}~.
\end{equation}
We see that, for negative tension, \emph{i.e.} $\psi_{\rm II}<0$, when $\Theta$ is greater than some critical value set by $\psi_{\rm II}$, reflecting geodesics of type (3) cease to exist. In this case geodesics of type (1) also do not exist and we are left with the ``disconnected'' geodesics of type (2). Thus, in a rather intuitive way, starting from ICFT, we have recovered some of the results in \cite{Kastikainen:2021ybu}.
\subsection{Geodesic approximation}
Primary operators $\mathcal O$ with $\Delta \gg 1$, are dual to scalar fields of mass $m L \approx \Delta$, meaning they are \emph{heavy} in AdS units. The two point correlation function can thus be estimated using the geodesic approximation. In the holographic dual of ICFT described above (see figure \ref{double_holo}), some geodesics will inevitably intersect the brane, especially if we are considering correlation functions such as $\langle\mathcal{O}_{\rm I} (x)\mathcal{O}_{\rm II}(y)\rangle$. We therefore need to understand the geodesic approximation applied to this case.
Let us begin by briefly reviewing the geodesic approximation in the absence of a brane. Consider a free scalar field $\phi$ of mass $m$, with Euclidean action
\begin{equation}
S[\phi] = \int \text{d}^3 x \, \sqrt{g} \left( \frac{1}{2} g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi + \frac{1}{2} \, m^2 \phi^2 \right) \ ,
\end{equation}
where $g_{\mu\nu}$ is the metric on (Euclidean) AdS$_{3}$. The two point function is simply the propagator,
\begin{equation}
\langle \phi(x_1) \phi(x_2) \rangle = \int \mathcal D \phi(x) \, \phi(x_1) \phi(x_2) \ e^{-S[\phi]} \ ,
\end{equation}
which can be expressed in the worldline formalism as
\begin{equation}
\langle \phi(x_1) \phi(x_2) \rangle = \underset{u(0) \, = \, x_1}{\underset{u(1) \, = \, x_2}{\int}} \frac{\mathcal D u(\tau) \mathcal D e(\tau)}{{\rm Vol}({\rm Gauge})} \ e^{- S[u(\tau),e(\tau)]}\equiv \langle x_2| x_1\rangle ~,
\label{2pointWorldLineForm}
\end{equation}
(see e.g. \cite{Strass1992,corradini2012quantum}).
In the equation above
\begin{equation}
S[u(\tau),e(\tau)] = \int_0^1 \text{d} \tau \left( \frac{\dot u^2}{2e} + \frac{m^2e}{2} \right)
\end{equation}
is the worldline action,
\begin{equation}
\dot u^{2} = g_{\mu\nu}(u)\dot u^{\mu} \dot u^{\nu}
\end{equation}
and $e(\tau)$ is an \textit{einbein} along the wordline. One way to see that this functional integral is designed to give us the propagator is to look at the constraint equation that arises from varying $S$ with respect to $e(\tau)$:\footnote{The extra factor of $i$ in the definition of $p_\mu$ stems from our choice of a Euclidean target space. }
\begin{equation}\label{eq:hclassical}
H\equiv g^{\mu\nu}(u)p_\mu p_\nu+m^2=0~, \qquad\qquad p_\mu\equiv i\frac{\partial L}{\partial \dot u^\mu}~.
\end{equation}
Upon canonically quantizing the theory, $H$ will be promoted to an operator, and the above constraint tells us that
\begin{equation}\label{eq:hilbertspaceconstraint}
\langle x_2|H| x_1\rangle=\hat{H}\langle x_2|x_1\rangle=0~,
\end{equation}
where $\hat{H}$ is a differential operator. Determining the form of $\hat{H}$ by canonically quantizing the classical expression \eqref{eq:hclassical} will generally suffer from ordering ambiguities due to the coordinate dependence in the background metric $g^{\mu\nu}$. Luckily, braver souls have attempted this before us \cite{dewitt2003global,bastianelli2006path}. Moreover we expect by general covariance that:
\begin{equation}
\hat H = - \Box_{x_{1,2}}+m^2~,
\end{equation}
where $\Box$ is the Laplacian for the metric $g_{\mu\nu}$ and the coordinate it acts on depends on if it is taken to act to the right or to the left in \eqref{eq:hilbertspaceconstraint}. Taken together this means our worldline path integral is a generally covariant expression for a propagator from $x_1$ to $x_2$ on the background $g_{\mu\nu}$.\footnote{The correct delta function at coincident points is also accounted for by eq. \eqref{2pointWorldLineForm}, as can be seen for instance by going to a locally inertial frame when the two insertions are close, and comparing to the flat space version of the world-line path integral. The latter explicitly gives the expected $(p^2+m^2)^{-1}$. }
Alternatively, we can integrate out the einbein entirely, leaving us with the standard Nambu-Goto action along the worldline, meaning our two-point function can be expressed as:
\begin{equation}
\langle \phi(x_1) \phi(x_2) \rangle = \underset{u(0) \, = \, x_1}{\underset{u(1) \, = \, x_2}{\int}} \mathcal D u(\tau)\ e^{ - m \int_0^1 \text{d} \tau \sqrt{\dot u^2}}\ .
\end{equation}
This latter expression admits a saddle point approximation in the limit of large mass:
\begin{equation}
\langle \phi(x_1) \phi(x_2) \rangle \sim \sum_{\mathcal P} e^{- m d_{\mathcal P}(x_1,x_2)} \ ,
\end{equation}
where $\mathcal P$ is the set of geodesic paths connecting $x_1$ and $x_2$ and $d_{\mathcal P}(x_1,x_2)$ is the length of the trajectory. To connect this calculation to that of a CFT two-point function for a primary operator of dimension $\Delta$, we identify
\begin{equation}
m^2 L^2 = \Delta (\Delta - 2) \approx \Delta^2
\end{equation}
and
\begin{equation}
\langle \mathcal O(x_1) \mathcal O (x_2) \rangle = \lim_{z_1, z_2 \to 0} \frac{1}{z_1^{\Delta} z_2^{\Delta}} \langle \phi(x_1) \phi(x_2) \rangle \ ,
\end{equation}
where we have used the notation $x_1 \equiv (z_1, \vec x_1)$ and similarly for $x_2$ in the Poincar\'{e} coordinates of \eqref{eq:poincarecoordinates}.
\subsection{Scalar field on a thin-brane background}
The case we are interested in deals with a scalar field on a background that has a thin brane between $\mathcal{M}_{\rm I}$ and $\mathcal{M}_{\rm II}$. Let us denote the scalar field as $\phi_{\rm I} (x)$ if $x \in \mathcal M_{\rm I}$ and $\phi_{\rm II} (x)$ if $x \in \mathcal M_{\rm II}$. The Euclidean action for the scalar field can now be written as:
\begin{equation}
S[\phi] = \int_{\mathcal M_{\rm I}} \text{d}^3 x \, \sqrt{g} \left( \frac{1}{2} g^{\mu\nu} \partial_{\mu} \phi_{\rm I} \partial_{\nu} \phi_{\rm I} + \frac{1}{2} \, m^2 \phi_{\rm I}^2 \right) + \int_{\mathcal M_{\rm II}} \text{d}^3 x \, \sqrt{g} \left( \frac{1}{2} g^{\mu\nu} \partial_{\mu} \phi_{\rm II} \partial_{\nu} \phi_{\rm II} + \frac{1}{2} \, m^2 \phi_{\rm II}^2 \right) \ .
\end{equation}
Varying the above action produces the following boundary term at the brane:
\begin{equation}
\delta S[\phi] = \int \text{d}^2 y \, \sqrt{h} \left( \delta \phi_{\rm I} \partial_{n} \phi_{\rm I} - \delta \phi_{\rm II} \partial_{n} \phi_{\rm II} \right) \ ,
\label{variation}
\end{equation}
where $y$ parameterizes the coordinates {along} the brane, $h_{\mu\nu}$ is the induced metric {on} the brane, and $n^\mu$ a unit normal {to} the brane (such that $\partial_n\equiv n^\mu\partial_{y^\mu})$. This contribution vanishes if we set, for example, decoupled Dirichlet or Neumann boundary conditions. However, we want to identify the scalar field across the brane, meaning we impose that the field configuration is continuous across the surface, thus we cannot independently vary $\phi_{\rm I}$ and $\phi_{\rm II}$ along the brane. Vanishing of (\ref{variation}) therefore requires that the normal derivatives also be equal. In formulas:
\begin{equation}
\phi_{\rm I}(y) = \phi_{\rm II}(y) \ , \qquad \partial_{n}\phi_{\rm I}(y) = \partial_n \phi_{\rm II}(y) \ .
\label{bdy_scalar_across_interface}
\end{equation}
With these boundary conditions we can reconsider the discussion above for the scalar field in AdS$_3$. Since the dynamics of a scalar field in each space is given by a second order (partial) differential equation, (\ref{bdy_scalar_across_interface}) implies that $\phi_{\rm I}(x)$ and $\phi_{\rm II}(x)$ can naturally be extended to a global scalar field $\phi(x)$ defined on the spacetime $\mathcal{M}_{\rm I}\cup\mathcal{M}_{\rm II}$, with:
\begin{equation}
\phi (x) \Big|_{x \in \text{AdS}_{\rm I}} = \phi_{\rm I} (x) \ , \qquad \phi (x) \Big|_{x \in \text{AdS}_{\rm II}} = \phi_{\rm II} (x) \ .
\end{equation}
Moreover, based on the general arguments above, the propagator of this global field $\phi(x)$ can again be computed in the worldline formalism:
\begin{equation}
\langle \phi(x_1) \phi(x_2) \rangle \sim \sum_{\mathcal P'} e^{- m d_{\mathcal P'}(x_1,x_2)} \ ,
\label{geodApproxICFT}
\end{equation}
where $\mathcal P'$ is a path in $\mathcal{M}_{\rm I}\cup\mathcal{M}_{\rm II}$ that can, in principle, cross the brane many times. Indeed, away from the brane, the same argument as in the previous subsection ensures that eq. \eqref{geodApproxICFT} solves the Klein-Gordon equation. At the location of the brane, we only have to check the boundary conditions \eqref{bdy_scalar_across_interface}: we will show in the next subsection that geodesics across the interface precisely obey them. However, since the AdS scale jumps across the brane, we deduce that this scalar field is dual to operators of \emph{different} conformal dimensions in the CFT$_{\rm I,II}$, namely:
\begin{equation}
m^2 L_{\rm I, II}^2 = \Delta_{\rm I, II} (\Delta_{\rm I, II} - 2) \ .
\end{equation}
Taking this into account, the correlation function of, for instance, operators placed on either side of the boundary can be read off from the bulk formula using:
\begin{equation}
\langle \mathcal O_{\rm I}(x_1) \mathcal O_{\rm II} (x_2) \rangle = \lim_{z_1, z_2 \to 0} \frac{1}{z_1^{\Delta_{\rm I}} z_2^{\Delta_{\rm II}}} \langle \phi_{\rm I}(x_1) \phi_{\rm II}(x_2) \rangle \ .
\label{extrapolateDic}
\end{equation}
Analogously to the rules familiar from the usual AdS/CFT dictionary, the setup can be generalised beyond the simple gluing condition \eqref{bdy_scalar_across_interface}. We could have considered adding scalar field self-interactions localized on the brane, as was considered in \cite{Kastikainen:2021ybu}. For instance, we could add to the effective action for the field $\phi$ a polynomial term of the form
\begin{equation}
\int_{\mathcal S} \text{d}^2 y \, \sqrt{h} \, V\big(\phi,\nabla \phi\big) \subset S_{\text{brane}} \ ,
\end{equation}
then the interactions present in $V(\phi)$ will affect the correlation functions. For example, a tadpole in $V(\phi)$ would generate non–trivial correlation functions between $\phi(x)$ on the conformal boundary and on the brane. Or similarly, if $V(\phi)$ includes a $\phi^n$ interaction, then one would need to include the appropriate $n$-point vertices on the brane and fix their positions by minimizing the length of the incident geodesics. We do not analyze these cases in this paper, especially because we will be ultimately interested in the application of the formalism to the computation of entanglement entropy, where the need for computing geodesics arises from the Ryu-Takayanagi prescription, rather than from a specific form of the scalar potential.
\subsection{Continuity and smoothness of the geodesic crossing the brane}
Let us try and understand the implications these considerations have when the geodesic approximation is valid. We will now show that the Israel--Lanczos conditions imply that the geodesics are continuous and once differentiable across the brane (and hence in the class $C^1$). As discussed in the previous subsection, these are the only saddles contributing in the worldine formalism, as long as eq. (\ref{bdy_scalar_across_interface}) holds, and in the absence of localized couplings along the brane.
Taking inspiration from \cite{poisson_2004}, let us introduce a system of coordinates in a neighborhood of the interface $\mathcal S$ such that the local topology of spacetime is $\mathbb R \times \mathcal S$ in $\mathcal{M}_{\rm I}\cup\mathcal{M}_{\rm II}$. Specifically, this parametrization can be constructed using the set of geodesics labeled by a function $\lambda(x^\mu)$ such that the locus $\lambda = 0$ lies on the surface $\mathcal S$. Moreover, these geodesics will be constructed and such that their first derivative is normal to $\mathcal S$. Thus the function $\lambda (x^\mu)$, appropriately normalized, defines the proper distance (with sign) of the point $x^\mu$ from the surface $\mathcal S$. The unit normal to the brane, up to a normalization, is then
\begin{equation}
n_{\alpha} = \partial_{\alpha} \lambda \Big |_{\lambda=0} \ .
\end{equation}
Using the Heaviside $\theta$–function, the metric on $\mathcal{M}_{\rm I}\cup\mathcal{M}_{\rm II}$ in the vicinity of $\mathcal{S}$ can then be conveniently written as
\begin{equation}
g_{\alpha \beta} = \theta(\lambda) g_{\alpha \beta}^{\rm I} + \theta(-\lambda) g_{\alpha \beta}^{\rm II} \ ,
\end{equation}
where we remind the reader that $g_{\alpha \beta}^{\rm{I,II}}$ are the metrics on either side of the brane. The derivative of the metric is then
\begin{align}
\partial_\gamma g_{\alpha \beta} \; = & \; \theta(\lambda) \partial_\gamma g_{\alpha \beta}^{\rm I} + \partial_\gamma \theta(\lambda) g_{\alpha \beta}^{\rm I} + \theta(-\lambda) \partial_\gamma g_{\alpha \beta}^{\rm II} + \partial_\gamma \theta(-\lambda) g_{\alpha \beta}^{\rm II} \nonumber \\
= & \; \theta(\lambda) \partial_\gamma g_{\alpha \beta}^{\rm I} + \theta(-\lambda) \partial_\gamma g_{\alpha \beta}^{\rm II} + \delta(\lambda) (g_{\alpha \beta}^{\rm I} - g_{\alpha \beta}^{\rm II}) n_\gamma \ .
\label{derivative_g}
\end{align}
The first Israel–Lanczos condition,
\begin{equation}
g_{\alpha \beta}^{\rm I} - g_{\alpha \beta}^{\rm II} \Big |_{\lambda=0} = 0 \ ,
\end{equation}
removes the $\delta$–discontinuity from (\ref{derivative_g}), leaving only step–like ones. Consequently, the connection
\begin{equation}
{\Gamma^{\mu}}_{\nu\rho} = \frac{1}{2} g^{\mu \alpha} (\partial_{\nu}g_{\alpha \rho} + \partial_{\rho}g_{\alpha \nu} - \partial_{\alpha}g_{\nu \rho})
\end{equation}
is at most step–wise discontinuous, while the Riemann curvature
\begin{equation}
{R^\mu}_{\nu \rho \sigma} = \partial_{\rho} {\Gamma^{\mu}}_{\nu\sigma} - \partial_{\sigma} {\Gamma^{\mu}}_{\nu\rho} + {\Gamma^{\mu}}_{\rho\alpha} {\Gamma^{\alpha}}_{\nu\sigma} - {\Gamma^{\mu}}_{\sigma\alpha} {\Gamma^{\alpha}}_{\nu\rho}
\end{equation}
has a $\delta$–like discontinuity, required to satisfy the Einstein equations (since the stress--energy tensor has itself a $\delta$–discontinuity at the brane). However, we can deduce from the the geodesic equation,
\begin{equation}
\frac{\text{d}^2 x^{\mu}}{\text{d}\lambda^2} + {\Gamma^{\mu}}_{\nu\rho} \frac{\text{d} x^{\nu}}{\text{d}\lambda}\frac{\text{d} x^{\rho}}{\text{d}\lambda} = 0 \ ,
\end{equation}
that the absence of a $\delta$–like discontinuity in the connection implies that both the geodesic and its derivative must be continuous. The upshot of this discussion, as we will show, is that since boundary-anchored geodesics are continuous and smooth, they satisfy very simple geometric conditions. In the next section we will show how to solve the simple geometrical problems associated with finding various different boundary-anchored geodesics.
\subsection{A panoply of geodesics}\label{sec:panoply}
Let us begin with a few facts about geodesics in locally AdS spacetimes. In Poincar\'{e} coordinates, geodesics minimize the action functional
\begin{equation}
\int \text{d} s \frac{L}{z(s)}\sqrt{\left(\frac{d\tau}{ds}\right)^2+\left(\frac{dx}{ds}\right)^2+\left(\frac{dz}{ds}\right)^2}~,
\end{equation}
and it is straightforward to show that constant-$\tau$ geodesics trace out circular arcs:
\begin{equation}
\left(x-\frac{\sigma_1+\sigma_2}{2}\right)^2+z^2=\left(\frac{\sigma_1-\sigma_2}{2}\right)^2
\end{equation}
parametrized by their endpoints on the cutoff surface boundary at $x=\sigma_{1,2}$ and $z=0$. Importantly these circles are centered along the conformal boundary at $z=0$.
\subsubsection{Brane-crossing geodesics at equal-time }\label{sec:intcrossinggeods}
We now have everything in place to compute the geodesic distance between points on either side of the defect. For simplicity we will first take them to lie at equal time $\tau=\tau' = 0$, but will generalize to arbitrary times later. The distance of the two points from the interface on the conformal boundary is denoted as $\sigma_{\rm I}$ and $\sigma_{\rm II}$ respectively. A pictorial representation of such a geodesic is shown in figure \ref{Generic_sigma_2}. This case was considered before, see \cite{Czech:2016nxc,Chapman:2018bqj}.
Our task is to find a continuous and smooth geodesic that lands on the boundary at particular marked points. But we have just argued that geodesics in AdS$_3$ are circles centered along the conformal boundary. Therefore, finding the length of the geodesic in figure \ref{Generic_sigma_2} is equivalent to finding two circular arcs that smoothly connect the boundary endpoints through the brane.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale=1.3]
\shade[top color=white, bottom color=ForestGreen, opacity = 0.25] (0,0) -- (5,0) -- ({5},{5*sin(60)}) -- ({-5*cos(60)},{5*sin(60)}) -- cycle;
\scoped[transform canvas={rotate around={45:(0,0)}}]
\shade[top color=white, bottom color=Cerulean, opacity = 0.25] (0,0) -- ({-3},{0}) -- ({-3},{5*cos(15))}) -- ({5*sin(15)},{5*cos(15)}) -- cycle;
\draw[very thick, ForestGreen, opacity = 0.8, name path=L1] (0,0) -- (5,0);
\draw[very thick, Cerulean, dashed, opacity = 0.8] (0,0) -- ({2*cos(45)},{2*sin(45)});
\draw[very thick, NavyBlue, opacity = 0.8, name path=I] (0,0) -- ({-5*cos(60)},{5*sin(60)});
\draw[very thick, Cerulean, opacity = 0.8] (0,0) -- ({-3*cos(45)},{-3*sin(45)});
\draw[very thick, Cerulean, opacity = 0, name path=L2] ({-3*cos(45)},{-3*sin(45)}) -- ({2*cos(45)},{2*sin(45)});
\draw[opacity = 0, name path=L] (-2, 2.8) -- (2.5, -1);
\draw[dashed] (0,0) -- (0,5);
\draw[dashed] (0,0) -- ({-5*cos(45)},{5*sin(45)});
\draw (0, 3.6) arc (90:120:3.6) node [midway, anchor=south] {\small $\psi_{\rm II}$};
\draw (0, 3.55) arc (90:120:3.55);
\draw ({3.5*cos(120)}, {3.5*sin(120)}) arc (120:135:3.5) node [midway, anchor=south east] {\small $\psi_{\rm I}$};
\draw ({3.45*cos(120)}, {3.45*sin(120)}) arc (120:135:3.45);
\path [name intersections={of=L and I, by=S}];
\path [name intersections={of=L and L1, by=A}];
\path [name intersections={of=L and L2, by=B}];
\draw (-1.7, 2.5458) -- (1.9079, -0.5);
\fill[NavyBlue] (0,0) circle (2pt);
\tikzmath{coordinate \C;
\C = (S)-(A);
\R = sqrt((\Cx)^2+(\Cy)^2);
}
\tikzmath{coordinate \D;
\D = (S)-(B);
\r = sqrt((\Dx)^2+(\Dy)^2);
}
\centerarc[very thick, blue, name path=c1](A)(0:140:\R pt);
\centerarc[very thick, blue, name path=c2](B)(140:225:\r pt);
\path [name intersections={of=c1 and L1, by=a}];
\path [name intersections={of=c2 and L2, by=b}];
\fill[red] (a) circle (2pt) node[below=0.9] {$P$};
\fill[red] (b) circle (2pt)node[anchor=north west] {$Q$};
\centerarc[](A)(0:140: 0.3);
\centerarc[](A)(0:140: 0.25);
\centerarc[](B)(140:225:0.3);
\centerarc[](B)(140:225:0.25);
\node at (1.55, 0.45) {$\theta$};
\node at (0.15, 0.6) {$\varphi$};
\fill (S) circle (1pt);
\fill (B) circle (1pt);
\draw (0.6,0.85) node {$B$};
\fill (A) circle (1pt) node[below=0.9] {$A$};
\draw[] (-0.95, 2.2) node{$S$};
\fill[NavyBlue] (0,0) circle (2pt);
\node[NavyBlue] (0,0) [below=0.9] {$O$};
\draw[<->] (0.3,-0.6) -- ($(a)-(0.2,0.6)$);
\draw (2.2,-0.8) node[font=\small] {$\sigma_{\rm II}$};
\draw[<->] ({0.2*cos(135)-(-0.6)*sin(135)},{-0.2*sin(135)-(-0.6)*cos(135)}) -- ($(b) + ({-0.2*cos(135)-(-0.6)*sin(135)},{0.2*sin(135)-(-0.6)*cos(135)})$);
\draw (-0.05,-1.2) node[font=\small] {$\sigma_{\rm I}$};
\end{tikzpicture}
\end{center}
\caption{The smooth geodesic through the brane that connects two points on different CFTs which are respectively $\sigma_{\rm I}$ and $\sigma_{\rm II}$ away from the defect.}
\label{Generic_sigma_2}
\end{figure}
Let us denote by $A$ and $B$ the centers of the two circular arcs along their respective boundaries, as shown in figure \ref{Generic_sigma_2}. We will work this out specifically for the case where $\sigma_{\rm I}\leq \sigma_{\rm II}$, but the result \eqref{length_geodesic} is independent of this choice, as required by conformal invariance. Recall that these boundaries intersect at an angle along their common interface. The smoothness condition imposes that these circular arcs meet at a point $S$ on the brane that lies along a line that goes through both $A$ and $B$. To compute the length of the arcs we then need to compute the angles $\theta$ and $\varphi$ which subtend these arcs, and the radii of the circles $\overline{AP}$ and $\overline{BQ}$. First notice that
\begin{equation}
\widehat{AOB} = \psi_{\rm I} + \psi_{\rm II} \ .
\end{equation}
Then, considering the angles of the triangle $\triangle AOB$ we get
\begin{equation}
\varphi = \pi + \psi_{\rm I} + \psi_{\rm II} - \theta \ .
\label{theta_phi}
\end{equation}
The law of cosines applied to this triangle gives
\begin{equation}
\overline{OB}^2 = \overline{OA}^2 + \overline{AB}^2 + 2 \cdot \overline{OA} \cdot \overline{AB} \cdot \cos(\theta) \ , \label{eq:lawofcosines1}
\end{equation}
while the law of sines leads to:
\begin{equation}
\overline{OB} \cdot \sin(\psi_{\rm I} + \psi_{\rm II}) = \overline{AB} \cdot \sin( \theta) \ . \label{eq:lawofsines1}
\end{equation}
On the other hand, looking at the triangle $\triangle OBS$, the law of sines yields
\begin{equation}
\overline{OS}\cos(\varphi-\psi_{\rm I})=\overline{OB}\sin (\varphi)~,
\label{eq:lawofsines2}
\end{equation}
while the law of cosines applied to this triangle gives:
\begin{equation}
\overline{OS}^2 = \overline{OB}^2 + \overline{BS}^2 - 2 \cdot \overline{OB} \cdot \overline{BS} \cdot \cos(\varphi) \ . \label{eq:lawofcosines2}
\end{equation}
We denote by $r$ the radius of the arc in AdS$_{\rm I}$ (blue region) and by $R$ the radius of the arc in the green AdS$_{\rm II}$ (green region). Thus:
\begin{equation}
\overline{AP} = \overline{AS} = R~,\qquad \text{and} \qquad \overline{BQ} = \overline{BS} = r \ .
\end{equation}
We also note the relations
\begin{equation}
\overline{OA} = \sigma_{\rm II} - R \ , \qquad\qquad
\overline{OB} = r - \sigma_{\rm I} \ , \qquad\qquad
\overline{AB} = R - r \ .
\end{equation}
Recall that in our notation $\sigma_{\rm I,II}$ are positive quantities denoting distances along the boundary from the interface at $O$. Using \eqref{theta_phi} along with the two relations \eqref{eq:lawofsines1}-\eqref{eq:lawofsines2}, we can write
\begin{equation}
R=r+(r-\sigma_{\rm I})\sin(\psi_{\rm I}+\psi_{\rm II})\csc(\theta)~, \qquad \overline{OS}=(r-\sigma_{\rm I})\sin(\psi_{\rm I}+\psi_{\rm II}-\theta)\sec(\theta-\psi_{\rm II})~.
\end{equation}
Plugging these into the remaining two equations gives
\begin{multline}
(r-\sigma_{\rm I})\left[(r-\sigma_{\rm I})\sin(\psi_{\rm I}+\psi_{\rm II})+(r-\sigma_{\rm II})\sin(\psi_{\rm I}+\psi_{\rm II})\right]\sin(\psi_{\rm I}+\psi_{\rm II})\\=(\sigma_{\rm I}+\sigma_{\rm II}-2r)(\sigma_{\rm I}-\sigma_{\rm II})\cos^2\left(\frac{\theta}{2}\right)~,\label{eq:req1}
\end{multline}
and
\begin{equation}
r^2+(r-\sigma_{\rm I})\left[(r-\sigma_{\rm I})\cos(\psi_{\rm I})\cos(2\theta-\psi_{\rm I}-2\psi_{\rm II})\sec^2(\theta-\psi_{\rm II})+2 r \cos(\theta-\psi_{\rm I}-\psi_{\rm II})\right]=0~.\label{eq:req2}
\end{equation}
We now have two equations in the remaining two unknowns $(r,\theta)$. Being quadratic equations in $r$, it is important that we keep certain limits in mind when selecting the correct solution branch. Namely, when $\psi_{\rm I}=\psi_{\rm II}=0 $, we expect:
\begin{equation}
r=R=\frac{\sigma_{\rm I}+\sigma_{\rm II}}{2}~, \qquad \cos(\theta)=\frac{\sigma_{\rm I}-\sigma_{\rm II}}{\sigma_{\rm I}+\sigma_{\rm II}}~,\qquad \psi_{\rm I},\psi_{\rm II}\rightarrow0~. \label{eq:correctlimitpsi}
\end{equation}
The solutions to \eqref{eq:req1} and \eqref{eq:req2} that satisfy \eqref{eq:correctlimitpsi} are:
\begin{align}
r &= \frac{1}{2}\csc\left(\frac{\varphi}{2}\right)\sec\left(\frac{\psi_{\rm I}+\psi_{\rm II}}{2}\right)\,\Bigg[\sigma_{\rm II} \cos \left( \frac{\theta}{2} \right)- \sigma_{\rm I}\,\cos \left( \frac{\theta}{2}+\varphi \right)\Bigg]~,\label{sol_r1}\\
R&= \frac{1}{2}\csc\left(\frac{\theta}{2}\right)\sec\left(\frac{\psi_{\rm I}+\psi_{\rm II}}{2}\right)\Bigg[\,\sigma_{\rm I}~\cos\left(\frac{\varphi}{2}\right)-\sigma_{\rm II}\cos\left(\frac{\varphi}{2}+\theta\right)\Bigg]~.\label{sol_r2}
\end{align}
Setting \eqref{sol_r1} equal to \eqref{sol_r2} (recalling that $\varphi = \pi + \psi_{\rm I} + \psi_{\rm II} - \theta $) along with a healthy amount of massaging gives us the following equation for $\theta$:
\begin{equation}
\sigma_{\rm I}\cos\left(\theta-\frac{\psi_{\rm I}+3\psi_{\rm II}}{2}\right)+\sigma_{\rm II}\cos\left(\theta+\frac{\psi_{\rm I}-\psi_{\rm II}}{2}\right) =(\sigma_{\rm I}-\sigma_{\rm II})\cos\left(\frac{\psi_{\rm I}-\psi_{\rm II}}{2}\right)~.
\end{equation}
Taking $\sigma_{\rm I}=\sigma_{\rm II}$, one sees that the solution is $\theta=\psi_{\rm II}+\frac{\pi}{2}$. More generally,
the above equation can be rewritten as a quadratic equation in $\cos(\theta)$, with solution:
\begin{align}
&\cos (\theta)=\frac{\cos\left(\frac{\psi_{\rm I}-\psi_{\rm II}}{2}\right)}{\sigma_{\rm I}^2+\sigma_{\rm II}^2+2 \sigma_{\rm I}\sigma_{\rm II}\cos(\psi_{\rm I}+\psi_{\rm II})}\, \times\nonumber\\
&\left\lbrace - \, \sigma_{\rm II}^2\cos\left(\frac{\psi_{\rm I}-\psi_{\rm II}}{2}\right) + \sigma_{\rm I}^2\cos\left(\frac{\psi_{\rm I}+3\psi_{\rm II}}{2}\right) + 2\sigma_{\rm I}\sigma_{\rm II}\sin(\psi_{\rm II})\sin\left(\frac{{\psi_{\rm I}+\psi_{\rm II}}}{2}\right)\right.\nonumber\\
&\left.- \left[\sigma_{\rm I}\sin\left(\frac{\psi_{\rm I}+3\psi_{\rm II}}{2}\right)-\sigma_{\rm II}\sin\left(\frac{\psi_{\rm I}-\psi_{\rm II}}{2}\right)\right]\sqrt{\left[\frac{(\sigma_{\rm I}+\sigma_{\rm II})^2-(\sigma_{\rm I}-\sigma_{\rm II})^2\cos(\psi_{\rm I}-\psi_{\rm II})+4\sigma_{\rm I}\sigma_{\rm II}\cos(\psi_{\rm I}+\psi_{\rm II})}{2\cos^2\left(\frac{\psi_{\rm I}-\psi_{\rm II}}{2}\right)}\right]}\right\rbrace\label{eq:cossol}
\end{align}
and the branch of the square root in \eqref{eq:cossol} is selected such that $\cos\theta=-\sin\psi_{\rm II}$ in the limit $\sigma_{\rm I}=\sigma_{\rm II}$.
To convert this data into a geodesic length, recall that, as reviewed in section \ref{sec:embeddingcoordinates}, any two points $x^\mu$ and $x'^\mu$ in AdS$_3$ can be labeled by their embedding coordinates on the hyperboloid: $X^A$ and $ X'^A$ where ${A=0,\dots3}$. The geodesic distance $d$ between any two points is therefore determined by
\begin{equation}
-G_{\mu\nu} X^\mu X'^\nu = L^2 \cosh \left(\frac{d}{L}\right)~.
\end{equation}
In the Poincar\'{e} coordinates of \eqref{eq:poincarecoordinates}, we thus have
\begin{equation}
{d}=L\, \cosh^{-1}\left(\frac{(\tau-\tau')^2+(x-x')^2+z^2+z'^2}{2 z\, z'}\right)~.
\end{equation}
For the section of the geodesic in AdS$_{\rm I}$, we can treat the point $B$ as the origin of our coordinates. Thus we want to compute the distance betwen a point at $(x,z)=(-r,\varepsilon_{\rm I})$ and the point $(x',z')=(-r\cos\varphi,r\sin\varphi)$, while in the AdS$_{\rm II}$ we want to compute the distance betwen a point at $(x,z)=(R,\varepsilon_{\rm II})$ and the point $(x',z')=(R\cos\theta,R\sin\theta)$, both with $\tau=\tau'$. In this expression we are allowing for the distinct possibility that the CFT duals have different UV cutoffs. The leading order result as $\varepsilon_{\rm I,II}\rightarrow 0$ is:
\begin{equation}
d(\sigma_{\rm I}, \sigma_{\rm II}) = L_{\rm I} \log \left[ \frac{2r}{\varepsilon_{\rm I}}\tan\left(\frac{\varphi}{2}\right) \right] + L_{\rm II} \log \left[ \frac{2R}{\varepsilon_{\rm II}} \tan\left(\frac{\theta}{2}\right) \right] \ ,
\label{length_geodesic}
\end{equation}
where we remind the reader of the relation (\ref{theta_phi}) between $\theta$ and $\varphi$. Notice that when $\sigma_{\rm II} = \sigma_{\rm I} = \sigma$, the geodesic length simplifies to
\begin{equation}
R = r = \sigma, \qquad\qquad \varphi=\psi_{\rm I}+\frac{\pi}{2}~,\qquad\qquad\theta=\psi_{\rm II}+\frac{\pi}{2}~,
\end{equation}
so that
\begin{equation}
d(\sigma, \sigma) = L_{\rm I} \log\left[ \frac{2\sigma}{\varepsilon_{\rm I}} \tan\left(\frac{\psi_{\rm I}}{2}+\frac{\pi}{4}\right) \right] + L_{\rm II} \log \left[ \frac{2\sigma}{\varepsilon_{\rm II}} \tan\left(\frac{\psi_{\rm II}}{2}+\frac{\pi}{4}\right) \right] \ .
\label{eq:distanceequal}
\end{equation}
This is expected, since in this case $A$ and $B$ meet at the origin $O$ in figure \ref{Generic_sigma_2}. Using \eqref{angles}, we can rewrite this as:
\begin{equation}
d(\sigma, \sigma) = \rho^*_{\rm I}+\rho^*_{\rm II}+L_{\rm I} \log\left[ \frac{2\sigma}{\varepsilon_{\rm I}}\right] + L_{\rm II} \log \left[ \frac{2\sigma}{\varepsilon_{\rm II}} \right] .
\label{dEqualrhoStar}
\end{equation}
The dramatic simplification of this formula is easy to understand from the point of view of the dual field theory. Recall that the exponential of the distance computes a correlation function in the ICFT. When the points are in mirroring positions with respect of the interface, like in eq. \eqref{dEqualrhoStar}, the conformal group acts on the correlator in the same way as on a one-point function in a BCFT. This can be easily seen, for instance, via the folding trick---see \emph{e.g.} \cite{Bachas:2001vj}. In turn, the dependence on the coordinates of a one-point function is completely fixed by symmetry, and this yields eq. \eqref{dEqualrhoStar}.
\subsubsection{Points on different sides at generic positions and conformal properties}
The result of the previous section allows us also to compute the geodesic length between two points with $\tau_1 \neq \tau_2$. Indeed, we can take advantage of the invariance of the geodesic distance under the isometries. Our main interest lies in the dual CFT, so let us think about the points on the conformal boundary. It is easy to show \cite{McAvity:1993ue,McAvity:1995zd} that knowledge of a two-point function on the line $\tau_1 = \tau_2$ is sufficient to reconstruct the correlator everywhere. Indeed, there is only one cross ratio for two points with a flat boundary:
\begin{equation}
\xi=\frac{(\sigma_{\rm I}-\sigma_{\rm II})^2+(\tau_{\rm I}-\tau_{\rm II})^2}{4\sigma_{\rm I}\sigma_{\rm II}}~.
\label{xiCross}
\end{equation}
$\xi$ is positive and vanishes when the operators are in the mirroring position discussed above, and diverges as one point is brought close to the interface. Specifically, the two-point function of our scalar primary must take the form
\begin{equation}
\langle\mathcal{O}(x_{\rm I})\mathcal{O}(x_{\rm II})\rangle
= \frac{1}{\sigma_{\rm I}^{\Delta_{\rm I}}\sigma_{\rm II}^{\Delta_{\rm II}}} \tilde{g}(\xi)~,
\label{corrOfXi}
\end{equation}
with $\tilde{g}$ a function which is not fixed by symmetry
Comparing this equation with eq. \eqref{geodApproxICFT} and with the extrapolate dictionary \eqref{extrapolateDic}, we see that the geodesic distance \eqref{length_geodesic} must take the form
\begin{equation}
d(\sigma_{\rm I},\sigma_{\rm II})=L_1 \log\frac{\sigma_{\rm I}}{\varepsilon_{\rm I}} +
L_2 \log\frac{\sigma_{\rm II}}{\varepsilon_{\rm II}}+g(\xi)~,
\label{distofXi}
\end{equation}
where
\begin{equation}
g(\xi) = L_{\rm I} \log \left[ \frac{2r}{\sigma_{\rm I}}\tan\left(\frac{\varphi}{2}\right) \right] + L_{\rm II} \log \left[ \frac{2R}{\sigma_{\rm II}} \tan\left(\frac{\theta}{2}\right) \right]~.
\label{gxiGeodesic}
\end{equation}
Consistently, $g(\xi)$ only depends on the ratio $\sigma_{\rm I}/\sigma_{\rm II}$. Eq. \eqref{distofXi} encodes the dependence of the distance, and therefore of the correlator, from generic positions of the endpoints.\footnote{We are keeping the cutoffs $\epsilon_{\rm I}$ and $\epsilon_{\rm II}$ fixed. This is the correct procedure to obtain physical correlators in the CFT. Of course, if one was to apply an AdS isometry to the endpoints of the geodesic, including their Poincaré coordinate distance from the conformal boundary, the length would stay the same.} Indeed, one can invert eq. \eqref{xiCross} in the $\tau_{\rm I}=\tau_{\rm II}$ case, and obtain
\begin{equation}
\frac{\sigma_{\rm I}}{\sigma_{\rm II}} =
1+2\xi \pm 2 \sqrt{\xi (\xi+1)}~.
\label{ratioofXi}
\end{equation}
Plugging eq. \eqref{ratioofXi} into eq. \eqref{gxiGeodesic}, one gets the explicit expression as a function of the cross ratio. It's a nice check of eq. \eqref{length_geodesic} that the result does not depend on the branch chosen for the square root. Indeed, swapping branches corresponds to sending $\sigma_{\rm I}/\sigma_{\rm II} \to \sigma_{\rm II}/\sigma_{\rm I}$. This is a conformal transformation (for instance, an inversion), and is the only invariance which is not explicit in eq. \eqref{length_geodesic}. We checked this fact numerically: it would be nice to simplify eq. \eqref{length_geodesic} further to write it as a simple function of $\xi$. Finally, the correlator \eqref{corrOfXi} can be computed in any configuration by simply evaluating $\xi$ in eq. \eqref{xiCross} at the desired position.
For completeness, we report the explicit form of a conformal Killing vector which produces the generic configuration from the $\tau_{\rm I}=\tau_{\rm II}$ case. The subgroup of the isometries of AdS which preserves the position of the brane is simply the one which fixes $\rho$ in the parametrization \eqref{eq:AdS2slices}. This is the isometry group of AdS$_2$, which acts on the complex coordinate $w=\tau+i y$ via the $sl(2,\mathbb{R})$ transformation
\begin{equation}
w \to \frac{a w+b}{c w+d}~, \quad ad-bc \neq 0~, \quad a,\,b,\,c,\,d \in \mathbb{R}~.
\label{sl2R}
\end{equation}
If $c \neq 0$, the transformation includes a special conformal transformation on each slice (including the conformal boundary). It is easy to check that this isometry will do the job. For instance, the choice $a=d=1$, $b=-c=\lambda$ generates circular orbits in the $(\tau,y)$ plane, as a function of $\lambda$, which leave invariant the point $(0,1)$, as well as the the boundary $y=0$.\footnote{The action of this conformal Killing vector is just a rotation of the half sphere obtained as a stereographic projection of the upper half plane} Clearly, placing one operator at $\sigma_{\rm I}=1$ and varying the position of the other on the $\tau=0$ line, we obtain any configuration, up to a translation and a dilatation. One can easily map the transformation \eqref{sl2R} to Poincaré coordinates, if needed, via eq. \eqref{zxToyChi}. For instance, the special conformal $a=d=1$, $b=0$ $c \in \mathbb{R}$ becomes
\begin{equation}
x \to \frac{x}{1 +2 c \tau + c^2(\tau^2+x^2+z^2)} \ , \quad z \to \frac{z}{1 +2 c \tau + c^2(\tau^2+x^2+z^2)} \ , \quad \tau \to \frac{\tau +c\,(\tau^2+x^2+z^2)}{1 +2 c \tau + c^2(\tau^2+x^2+z^2)}~,
\end{equation}
which is nothing but a special conformal transformation for the full AdS$_3$ with parameter along the $\tau$ direction.
\subsubsection{Points on the same side at $\tau = 0$} \label{sec:same_side}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=1.5]
\shade[top color=white, bottom color=ForestGreen, opacity = 0.25] (0,0) -- (5.5,0) -- ({5.5},{5*sin(75)}) -- ({-5*cos(75)},{5*sin(75)}) -- cycle;
\scoped[transform canvas={rotate around={85:(0,0)}}]
\shade[top color=white, bottom color=Cerulean, opacity = 0.25] (0,0) -- ({-1},{0}) -- ({-1},{5*cos(70))}) -- ({5*sin(70)},{5*cos(70)}) -- cycle;
\draw[very thick, ForestGreen, opacity = 0.8, name path=C1] (0,0) -- (5.5,0);
\draw[very thick, Cerulean, dashed, opacity = 0.8, name path=L2] (0,0) -- ({2*cos(85)},{2*sin(85)});
\draw[very thick, NavyBlue, opacity = 0.8, name path=I] (0,0) -- ({-5*cos(75)},{5*sin(75)});
\draw[very thick, ForestGreen, dashed, opacity = 0.8, name path=L1] (0,0) -- (-2,0);
\draw[very thick, Cerulean, opacity = 0.8] (0,0) -- ({-1*cos(85)},{-1*sin(85)});
\draw[very thick, Cerulean, opacity = 0, name path=L2] ({-1*cos(85)},{-1*sin(85)}) -- ({2*cos(85)},{2*sin(85)});
\draw[name path=L] (-0.6, -0.7059) -- (0.4, 2.2349);
\path [name intersections={of=L and I, by=S}];
\path [name intersections={of=L and L1, by=A}];
\path [name intersections={of=L and L2, by=B}];
\fill[NavyBlue] (0,0) circle (2pt);
\draw[NavyBlue] (0.1,0) node[anchor = north]{\small $O$};
\fill (A) circle (1pt);
\draw (-0.28, 0) node[anchor=north]{\small $C$};
\fill (B) circle (1pt);
\draw (0.35,1.47) node{\small $B$};
\draw (-0.8,1.95) node{\small $S$};
\draw (-0.35,0.45) node{\small $V$};
\tikzmath{coordinate \C;
\C = (S)-(A);
\R = sqrt((\Cx)^2+(\Cy)^2);
}
\tikzmath{coordinate \D;
\D = (S)-(B);
\r = sqrt((\Dx)^2+(\Dy)^2);
}
\centerarc[very thick, blue, name path=c1](A)(0:74:\R pt);
\centerarc[very thick, blue, name path=c2](B)(138:250:\r pt);
\path [name intersections={of=c1 and L1, by=a}];
\path [name intersections={of=c2 and L2, by=b}];
\draw[name path=L3] (-1.4,2.763) -- (2.45,-0.621);
\path [name intersections={of=L3 and C1, by=c}];
\path [name intersections={of=c2 and L3, by=e}];
\fill (c) circle (1pt) node[anchor=north east]{\small $A$};
\centerarc[](c)(0:140:0.2);
\centerarc[](c)(0:140:0.25);
\centerarc[](A)(180:250:0.2);
\centerarc[](A)(180:250:0.25);
\draw (2.15,0.55) node[anchor=north east]{\small $\gamma$};
\draw (-0.55,-0.15) node[anchor=north east]{\small $\alpha$};
\tikzmath{coordinate \E;
\E = (e)-(c);
\p = sqrt((\Ex)^2+(\Ey)^2);
}
\centerarc[very thick, blue, name path=c3](c)(139:0:\p pt);
\path [name intersections={of=c3 and C1, by=f}];
\path [name intersections={of=c1 and C1, by=g}];
\fill[red] (g) circle (2pt) node[anchor=south west]{\small $Q$};
\fill[red] (4.77,0) circle (2pt) node[anchor=north east]{\small $P$};
\draw[<->] (0,-0.7) -- (0.25,-0.7) node [midway, below] {\small $\sigma_{\rm II}^Q$};
\draw[<->] (0,-1.2) -- (4.77,-1.2) node [midway, below] {\small $\sigma_{\rm II}^P$};
\fill (S) circle (1pt);
\fill ({-2.07*cos(75)},{2.07*sin(75)}) circle (1pt);
\fill[NavyBlue] (0,0) circle (2pt);
\end{tikzpicture}
\end{center}
\caption{Geometric construction to find geodesics that explore the space behind the brane. The point $O$ refers to the origin where the defects lies, not one of the boundary points of the geodesic.}
\label{2pt_same_side_1}
\end{figure}
In this section we will be interested in CFT two-point functions with operator insertions restricted to one side of the interface. Recall that we have chosen to parametrize our CFTs such that $c_{\rm I}< c_{\rm II}$, without loss of generality. This has implications for correlation functions of operators placed in the CFT$_{\rm II}$ region, such as $\langle\mathcal{O}_{\rm II}(x_1)\mathcal{O}_{\rm II}(x_2)\rangle$. This is because, when $L_{\rm I}<L_{\rm II}$, there exist boundary-anchored geodesics such as the one depicted in figure \ref{2pt_same_side_1}, that probe the geometry behind the brane. However no such geodesic exists for e.g. $\langle\mathcal{O}_{\rm I}(x_1)\mathcal{O}_{\rm I}(x_2)\rangle$ when $L_{\rm I}<L_{\rm II}$.
Proving the existence of such geodesics follows the same logic as before: given points $P$ and $Q$, we want to show that there exist continuous and smooth geodesics that can be built piecewise out of circular arcs centered on the AdS boundary (or its continuation through the brane). We will instead ask a related question: for each (allowable) choice of the point $C$ and the angle $\alpha$ defined in figure \ref{2pt_same_side_1}, how many give geodesics that end on the pair of points $P$ and $Q$?
Considering figure \ref{2pt_same_side_1}, by simple geometric reasoning we note that (recall that $\widehat{AOB}=\psi_{\rm I}+\psi_{\rm II}$)
\begin{equation}
\widehat{BSO} =\gamma-\psi_{\rm II}-\frac{\pi}{2}
\end{equation}
while
\begin{equation}
\widehat{BVS} = \frac{\pi}{2}-\alpha+\psi_{\rm II} \ .
\label{BVS}
\end{equation}
Since the points $S$ and $V$ lie along a circular arc in the AdS$_{\rm I}$ region, the triangle $\triangle BSV$ is isosceles and therefore $\widehat{BSO}=\widehat{BVS}$, implying that the angles $\alpha$ and $\gamma$ are related:
\begin{equation}
\gamma = \pi + 2 \psi_{\rm II} - \alpha \ .
\label{gamma_alpha}
\end{equation}
Moreover, defining:
\begin{equation}
\overline{OQ} = \sigma^Q_{\rm II} \ , \qquad \overline{OP} = \sigma^P_{\rm II} \ , \qquad \overline{OC} = \lambda \ ,
\end{equation}
we note that the point $A$ is entirely determined in terms of $\lambda$ and $\alpha$:
\begin{equation}
\overline{OA} = \frac{\lambda \sin (\alpha ) \sin (\alpha + \psi_{\rm I} - \psi_{\rm II}) }{\sin (\alpha - 2 \psi_{\rm II}) \sin (\psi_{\rm I} + \psi_{\rm II} - \alpha )} ~ \ ,
\label{OA}
\end{equation}
and, using elementary geometry, we readily compute the location of the endpoints $P$ and $Q$ along the boundary in terms of $\lambda$ and $\alpha$ as well:
\begin{equation}
\sigma^Q_{\rm II} = \lambda \left(\frac{ \cos (\psi_{\rm II})}{\cos (\alpha - \psi_{\rm II})} - 1 \right)~,\qquad \sigma^P_{\rm II} = \lambda \frac{\sin (\alpha ) \sin (\alpha + \psi_{\rm I} - \psi_{\rm II}) [\cos (\psi_{\rm II}) + \cos (\alpha - \psi_{\rm II})]}{ \sin (\alpha - 2 \psi_{\rm II}) \sin (\psi_{\rm I} + \psi_{\rm II} - \alpha) \cos (\alpha - \psi_{\rm II})} \ .
\label{sigma_12}
\end{equation}
Given the external data $\{\sigma_{\rm II}^Q,\sigma_{\rm II}^P\}$, we would like to determine the allowed set of $\lambda$ and $\alpha$ that satsify \eqref{sigma_12}.
Going forward, it will be convenient to parametrize $\alpha$ as :
\begin{equation}
\alpha=\psi_{\rm II} +\cos^{-1}\left[\frac{\cos(\psi_{\rm II})}{1+s}\right]
\end{equation}
for $s>-1+\cos(\psi_{\rm II})$. The geometric interpretation of $s$ is clear plugging this parametrization into the first equation in \eqref{sigma_12}, which gives $\lambda=\sigma^Q_{\rm II}/s$. Notice that $\lambda>0$ must hold for $\psi_{\rm II} \in [0,\pi/2]$, thus our parameter $s$ is valued in:
\begin{equation}
s> \begin{cases} 0~, &\psi_{\rm II} \in [0,\pi/2]\\ -1+\cos(\psi_{\rm II})~,&\psi_{\rm II} \in [-\pi/2,0]\end{cases}~.
\end{equation}
Substituting this into the second equation in \eqref{sigma_12} we obtain
\begin{multline}
\Theta\equiv\frac{\sigma^P_{\rm II}}{\sigma^Q_{\rm II}}=\\
-\frac{(s+2)}{s} \frac{(1+2 s (s+2)) \cos (\psi_{\rm I})-\cos (\psi_{\rm I} + 2 \psi_{\rm II}) + 2\sin (\psi_{\rm I}+\psi_{\rm II}) \sqrt{(1+s)^2-\cos^2(\psi_{\rm II})}}{(1+2 s (s+2)) \cos (\psi_{\rm I})-\cos (\psi_{\rm I} + 2 \psi_{\rm II}) - 2\sin (\psi_{\rm I}+\psi_{\rm II}) \sqrt{(1+s)^2-\cos^2(\psi_{\rm II})} } \ .
\label{equation_for_s}
\end{multline}
In figure \ref{Two_saddles}, we plot a contour in the $\Theta-s$ plane that satisfies \eqref{equation_for_s} for given $\psi_{\rm I,II}$. We note that for $\Theta$ sufficiently large, there are two branches of solutions, which we denote by by $s_1^*$ and $s_2^*$, meaning that there exists either two geodesics of the type depicted in figure \ref{2pt_same_side_1} for a given set of endpoints, or none.
\begin{figure}[t]
\centering
\includegraphics[width=0.55\linewidth]{figures/Saddle_point_analysis.pdf}
\caption{The solutions $s_{1,2}^*$ for $\psi_{\rm I} = 1.5$ and $\psi_{\rm II} = 0.5$. Recall that $\Theta\equiv\frac{\sigma^P_{\rm II}}{\sigma^Q_{\rm II}} \geq 1$. For $\Theta$ large enough there are always two geodesics that connects them of the kind of figure \ref{2pt_same_side_1}. }
\label{Two_saddles}
\end{figure}
Since we have taken $L_{\rm I}< L_{\rm II}$ we expect paths of the type in figure \ref{2pt_same_side_1} to exist by virtue of trying to take a shortcut through the spacetime that is more highly curved. Interestingly, no solutions to \eqref{equation_for_s} exist if we take $\psi_{\rm II}>\psi_{\rm I}$, meaning that the above analysis above actually shows that no such geodesics exist for $L_{\rm I} > L_{\rm II} $.
The geodesics found in this section also have a natural interpretation in the BCFT limit of our setup, which we explore in Section \ref{S_p_a_2pt}. Recall that this limit consists of taking $L_{\rm I}/L_{\rm II}\rightarrow 0$, thus these saddles should, in principle, exist. Indeed, in holographic BCFT setups one often finds disconnected geodesics (as well as reflecting ones) which contribute to correlation functions, as has been noted in \cite{Kastikainen:2021ybu}. In Section \ref{S_p_a_2pt} we reinterpret such connected and disconnected contributions as a limit of the ICFT geodesics that cross the brane. The reader may consult Appendix \ref{app:same_side} for a computation of the length of these geodesics.
\paragraph{Saddles with more brane crossings:}
We have only touched on the possibility of geodesics that cross the brane at most twice. One may also consider paths that cross the brane more times. However, their existence is not general, and they may appear only for specific regions of the parameter space. Moreover, the length of these geodesics is always bigger than the saddles found above. We can prove this formally. Let $I$ be the set of points in which the path $\mathcal P$ crosses the brane. If the cardinality $\operatorname{card}(I) \geq 3$, there is always a pair of points in $I$ that are connected by an arc that lies in the AdS with smaller curvature. However, there always is another geodesic stretching between the same two points and lying in the other AdS. The new path $\mathcal P'$, obtained by performing this replacement, is shorter.\footnote{Although in general $\mathcal P'$ is not an extremizer because it's not $C^1$.} This means that such paths do not contribute to the computation of entanglement entropies or correlation functions at leading order.
\subsection{Review of ICFTs}
Physical systems at criticality are the realm of conformal field theory. We can probe these systems by measuring their response to local excitations. Alternatively, we can turn on couplings along extended submanifolds, and measure their effect on observables. A natural possibility is to couple two different critical systems along a mutual boundary. If the latter is conformal invariant, the combined system is known as an \emph{interface conformal field theory} (ICFT), and has been the subject of many papers over the years, both in condensed matter \cite{Cardy:1984bb,Kane:1992zza,Ghoshal:1993tm,Oshikawa:1996dj,LeClair:1997gz,Saleur:1998hq,Saleur:2000gp,Quella:2006de,eisler2012entanglement,Gaiotto:2012,Gliozzi:2015qsa,Meineri:2019ycm}, and in holography \cite{Karch:2000gx,Bachas:2001vj,DeWolfe:2001pq,Bhaseen:2013ypa,Erdmenger:2014xya,Sonner:2017jcf,Simidzija:2020ukv,Bachas:2020yxv,Bachas:2021tnp,Bachas:2021fqo}. Semi-transparent interfaces can be engineered in various ways. An obvious possibility is to first pick conformal boundary conditions for two $d$-dimensional CFTs---always denoted as CFT$_{\rm I}$ and CFT$_{\rm II}$ in the following. A conformal boundary condition, or BCFT, is defined by the property that observables are constrained by the subgroup of the conformal symmetry which preserves the (flat) boundary. If there are two boundary operators $\bhat{O}_{\rm I}$ and $\bhat{O}_{\rm II}$ respectively, such that their scaling dimensions obey $\wh{\Delta}_{\rm I}+\wh{\Delta}_{\rm II}<d-1$, then one can turn on the coupling
\begin{equation}
\lambda \int_\textup{boundary} \bhat{O}_{\rm I}\, \bhat{O}_{\rm II}~.
\label{intCouplingOO}
\end{equation}
Generically, we expect observables at large distances to be again constrained by conformal symmetry. However, rather than a tensor product of boundary conditions, the interface might be permeable and show non-vanishing correlations between the two sides.
The simple example \eqref{intCouplingOO} could be complicated in various ways: one can couple CFT$_{\rm I}$ and CFT$_{\rm II}$ indirectly, via lower dimensional matter localized on the interface, or consider multi-parameter flows. If the two CFTs coincide, one can construct defects by integrating bulk local operators on a codimension one surface \cite{Oshikawa:1996dj}. It is also possible to tune a marginal coupling to different values on two half-spaces, or flow via a relevant deformation on half of the space, thus constructing Janus \cite{Bachas:2001vj,Bak:2003jk} and renormalization group (RG) \cite{Gaiotto:2012} interfaces.
In this paper, we shall mostly consider the case of two dimensional CFTs, although the qualitative picture easily generalizes, and we expect many specific results to be the same. The CFTs on either side of the interface are characterized by central charges $c_{\rm I}$ and $c_{\rm II}$. We will be interested in the case where both CFTs are holographic, so in particular $c_{\rm I/II}\gg1$ and both theories have a sparse spectrum. The holographic dual of this setup will consist of two asymptotically AdS$_3$ spacetimes with the AdS radii determined by the central charges of the two CFTs:
\begin{equation}
c_{\rm I , II}= \frac{3L_{\rm I , II}}{2 G_{(3)}}~.
\label{cBrownHenneaux}
\end{equation}
The full bulk spacetime interpolates between these two AdS$_3$'s as we move along a spatial direction, and we will work in the approximation that the region connecting these geometries is simply a thin brane, as in \cite{Azeyanagi:2007qj,Takayanagi:2011zk,Simidzija:2020ukv,Bachas:2021fqo, Bachas:2021tnp, Erdmenger:2014xya}. This setup readily generalizes the BCFT setup of \cite{Takayanagi:2011zk,Fujita:2011fp}. The precise geometry will be reviewed in subsection \ref{subsec:bottomup}.
When the ratio $c_{\rm I}/c_{\rm II}$ is generic, the brane allows the trasmission of energy across the interface \cite{Quella:2006de,Meineri:2019ycm,Bachas:2020yxv}. However, in the limit $c_{\rm I}\ll c_{\rm II}$, it becomes impossible for generic waves built from the CFT$_{\rm II}$ degrees of freedom to scatter into the CFT$_{\rm I}$. Thus, by this simple reasoning we obtain the physics of BCFT as a limit of ICFT. This will guide our story.
\subsection{Changing faces of two-faced geometries}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}
\path[-stealth, thick, draw=black, snake it] (0.1,0) -- (0.8,0);
\path[-stealth, thick, draw=black, snake it] (0.9,0) -- (1.6,0);
\path[-stealth, thick, draw=black, snake it] (1.7,0) -- (2.4,0);
\path[-stealth, thick, draw=black, snake it] (-0.1,0) -- (-0.8,0);
\path[-stealth, thick, draw=black, snake it] (-0.9,0) -- (-1.6,0);
\path[-stealth, thick, draw=black, snake it] (-1.7,0) -- (-2.4,0);
\filldraw[NavyBlue] (0,0) circle (5pt);
\draw[NavyBlue] (0,-0.6) node{Quantum};
\draw[NavyBlue] (0,-1) node{dot};
\fill[Cerulean, opacity=0.2, very thick] (-2.5,-0.75) rectangle (-4.5,0.75);
\fill[ForestGreen, opacity=0.2, very thick] (2.5,-0.75) rectangle (4.5,0.75);
\draw[Cerulean, very thick] (-2.5,-0.75) rectangle (-4.5,0.75);
\draw[ForestGreen, very thick] (2.5,-0.75) rectangle (4.5,0.75);
\draw[ForestGreen] (3.5, 0) node{Bath$_{\rm II}$};
\draw[Cerulean] (-3.5, 0) node{Bath$_{\rm I}$};
\draw[Cerulean, very thick] (0,-2.5) -- (-3.5, -2.5);
\draw[Cerulean] (-2.8, -3) node{CFT$_{\rm I}$};
\draw[ForestGreen, very thick] (0,-2.5) -- (3.5, -2.5);
\draw[ForestGreen] (2.8, -3) node{CFT$_{\rm II}$};
\filldraw[NavyBlue] (0,-2.5) circle (5pt);
\end{tikzpicture}
\end{center}
\caption{A possible physical interpretation of the system: a quantum dot coupled to two baths. The quantum dot has finite entropy, the boundary entropy of the defect, and if we put the system in the thermofield double state the dot can be seen as an eternal black hole coupled to two reservoirs.}
\label{it}
\end{figure}
As announced, our setup consists of two AdS regions glued together at the location of a brane, as shown in the first panel of figure \ref{double_holo}. The three interpretations of the system were referred to in the introduction and are explained in the caption of the same figure. This doubly holographic interface allows us to explore a variety of physical questions\cite{Bachas:2020yxv, Bachas:2021fqo, Bachas:2021tnp, Erdmenger:2014xya}, some of which we shall consider in detail in the rest of the paper.
One of the main reasons of interest is the same highlighted in the recent literature \cite{Almheiri:2019hni, Rozali:2019day, Chen:2020uac, Chen:2020hmv, grimaldi2022quantum}. The quantum dot which separates the two CFTs has finite, albeit possibly large, entropy. If the system is put in a generic time dependent state, energy and entropy will be exchanged with the baths, until equilibrium is reached, see figure \ref{it}. The time evolution of the entanglement between the dot and the baths must be compatible with the finiteness of the Hilbert space of the defect, a fact which is the cornerstone of the Page curve \cite{Page:1993df,Page:1993wv}. While this does not imply a paradox yet, one can choose a state---the thermofield double---and a Hamiltonian---the difference of the global time translations on the two sides---such that stationary observers see a horizon on the brane, in the intermediate description of figure \ref{double_holo}. What is more tantalizing, the system is under analytic control in the large $N$ limit, and the Page curve can be computed via the Ryu-Takayanagi formula. In section \ref{sec:Ent_entropy} we will expand on this topic.
Of course, ICFTs are interesting in their own right, and the possibility of exact computations at strong coupling extends beyond the thermofield double, where the black-hole arises, and also beyond entanglement entropy altogether. General two-point functions of single-trace heavy operators can be computed via the geodesic approximation. They are not fixed by symmetry, and they do not decouple from the interface. In the conformal bootstrap language, they exchange an infinite number of conformal blocks, despite being fixed by a single geodesic stretching between two boundary points. In section \ref{sec:corr_heavy}, we describe in detail how to compute these geodesics, and illustrate a variety of scenarios depending on the tension of the parameters of the ICFT and the position of the local operators. The resulting structure is quite rich. In particular, when the operators are thought of as twist fields, the results express the entanglement of subregions of the two CFTs. Via entanglement wedge reconstruction, they could shed light on the way a local bulk is encoded in the boundary, when the state is strongly perturbed away from the vacuum.
\subsection{Bottom-up model}
\label{subsec:bottomup}
Our focus in this paper will be on interface CFTs dual to semiclassical gravity in AdS$_3$, particularly a geometry described by a thin brane separating two locally AdS$_3$ geometries with respective curvatures scales $L_{\rm I/II}$ coupled through a permeable membrane. The Euclidean gravity action describing our system on $\mathcal{M}_{\rm I}$ and $\mathcal{M}_{\rm II}$ is:
\begin{multline}
S_{\rm EH}=-\frac{1}{16\pi G_{(3)}}\Bigg[\int_{\mathcal{M}_{\rm I}}\text{d}^3 x \sqrt{g_{\rm I}} \left( R_{\rm I} + \frac{2}{L_{\rm I}^2} \right)+ \int_{\mathcal M_{\rm II}} \text{d}^3 x \sqrt{g_{\rm II}} \left( R_{\rm II} + \frac{2}{L_{\rm II}^2} \right) \\
+2\int_{\mathcal{S}}\text{d}^2 y\sqrt{h}\left(K_{\rm I}-K_{\rm II}\right)-2T\int_{\mathcal{S}}\text{d}^2 y\sqrt{h} \Bigg] +\text{corner and counterterms}
\label{Action_bottom_up}
\end{multline}
where $T$ represents the tension of the brane, $h_{ab}$ is the induced metric along it, and the extrinsic curvatures $K_{\rm I,II}$ are computed with outward normal pointing from I$\rightarrow$II in both cases. For details on the corner term and counterterms see for instance appendix A of \cite{Bachas:2021fqo}.
The location of the brane between the two spacetimes $\mathcal{S}\equiv\partial \mathcal M_{\rm I} \cap \partial \mathcal M_{\rm II}$ is determined by the two Israel--Lanczos matching conditions \cite{Israel:1966rt, Lanczos:1924}. These matching conditions follow from the equations of motion of the action (\ref{Action_bottom_up}), and the saddle point approximation instructs us that they represent a reliable approximation of the position of the brane in the limit
\begin{equation}
\frac{T L_{\rm I, II}^2}{G_{(3)}} \gg 1 \ .
\label{weaklyCoupledBrane}
\end{equation}
This is satisfied in the usual regime $c_{\rm I , II} \sim L_{\rm I , II}/ G_{(3)} \gg 1$, and assuming $T L_{\rm I , II}$ an order one number. The solution of the equations of motion will suggest that the latter is generally respected, since it will enforce the constraint (\ref{T_constraints}).
If $x_{\rm I}( y)$ is the embedding of the brane in $\mathcal M_{\rm I}$ and $x_{\rm II}( y)$ is its embedding in $\mathcal M_{\rm II}$, the first matching condition imposes equality of $h_{ab}$, as viewed from either spacetime:
\begin{equation}
h_{ab} \equiv \frac{\partial x_{\rm I}^{\mu}}{\partial y^a} \frac{\partial x_{\rm I}^{\nu}}{\partial y^b} g^{ \rm{ I}}_{\mu\nu} = \frac{\partial x_{\rm{II}}^{\mu}}{\partial y^a} \frac{\partial x_{\rm II}^{\nu}}{\partial y^b} g^{\rm{ II}}_{\mu\nu} \ .
\label{first_matching_cond}
\end{equation}
The second matching condition involves the two extrinsic curvatures $K_{i,ab}$, and requires that their discontinuity across the brane be proportional to the tension $T$. In terms of
\begin{equation}
\Delta K_{ab} \equiv K_{{\rm{I}},ab} - K_{{\rm{II}},ab} \ ,
\end{equation}
the second matching condition is
\begin{equation}
\Delta K_{ab} - h_{ab} \Delta K = - \, T \, h_{ab} \ .
\label{second}
\end{equation}
Using the trace of (\ref{second}), we can simplify the above expression in three dimensions\footnote{The brane is a codimension one surface, so in three dimensions $h_{ab} h^{ab} = 2$.} to:
\begin{equation}
\Delta K_{ab} = T h_{ab} \ .
\label{second_matching_cond}
\end{equation}
\subsection{Bulk geometry and coordinates}\label{sec:embeddingcoordinates}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}
\draw[gray!60!white, name path=A] plot [smooth, tension=2] coordinates { ({2.5*cos(60)},{2.5*sin(60)}) (1.75, 2.05) ({3.5 - 2.5*cos(60)},{2.5*sin(60)})};
\draw[gray!60!white, name path=B] plot [smooth, tension=2] coordinates { ({2.2*cos(60)},{2.2*sin(60)}) (1.75, 2) ({3.5 - 2.2*cos(60)},{2.2*sin(60)})};
\draw[gray!60!white, name path=C] plot [smooth, tension=2] coordinates { ({1.5*cos(60)},{1.5*sin(60)}) (1.7, 1.43) ({3.5 - 2*cos(60)},{2*sin(60)})};
\draw[gray!60!white, name path=D] plot [smooth, tension=1] coordinates { ({1.1*cos(60)},{1.1*sin(60)}) (1.75, 1.4) ({3.5 - 1.8*cos(60)},{1.8*sin(60)})};
\draw[gray!60!white, name path = E] plot [smooth, tension=2] coordinates { ({0.8*cos(60)},{0.8*sin(60)}) (1.7, 0.55) ({3.5 - 0.9*cos(60)},{0.9*sin(60)})};
\draw[gray!60!white, name path = F] plot [smooth, tension=1] coordinates { ({0.5*cos(60)},{0.5*sin(60)}) (1.7, 0.5) ({3.5 - 0.4*cos(60)},{0.4*sin(60)})};
\draw[gray!60!white, name path = G] plot [smooth, tension=0.5] coordinates { ({3.5 - 0.75*cos(60)},{0.75*sin(60)}) ({3.5 - 0.55*cos(60)-0.9},{0.55*sin(60)+0.04}) ({3.5 - 0.5*cos(60)},{0.5*sin(60)}) };
\filldraw[rotate=30, rounded corners=1, Red!50!Yellow] (1.25,-1.5) -- (1.75,-1.5) -- (1.75, -1.2) -- (1.25, -1.2) -- cycle;
\filldraw[rotate=30, rounded corners=1, Red!50!Yellow] (1.65, -1.2) -- (1.35, -1.2) -- (1.47,-0.9) -- (1.53,-0.9) -- cycle;
\draw[rotate=30, rounded corners=5, Gray!50!black] (0.9,-1.5) -- (2.1,-1.5) -- (2.1, -3.2) -- (0.9, -3.2) -- cycle ;
\draw[rotate=30, Bittersweet, very thin] (1.65, -1.19) -- (1.35, -1.19) ;
\draw[rotate=30, thick, rounded corners=0.5, blue!50!NavyBlue, fill = white] (1,-2) -- (2,-2) -- (2, -2.7) -- (1, -2.7) -- cycle ;
\node[rotate=30, blue!50!NavyBlue] at (2.38, -1.15) {\tiny \fontfamily{augie}\selectfont Israel};
\node[rotate=30, blue!50!NavyBlue] at (2.51, -1.35) {\tiny \fontfamily{augie}\selectfont Lanczos};
\shade[top color=white, bottom color=Cerulean, opacity = 0.25] (-3,0) -- (0,0) -- (1.5,3*0.866) -- (-3,3*0.866) -- cycle;
\shade[top color=white, bottom color=ForestGreen, opacity = 0.25] (3.5,0) -- (6.5,0) -- (6.5,3*0.866) -- (2,3*0.866) -- cycle;
\draw[very thick, Cerulean] (-3,0) -- (0,0);
\draw (-1.5,0) node[below, font=\small] {\textcolor{Cerulean}{CFT$_{\rm I}$}};
\draw[very thick, NavyBlue] (0,0) -- (1.5,{3*sin(60)});
\draw[very thick, ForestGreen] (3.5,0) -- (6.5,0);
\draw (5,0) node[below, font=\small] {\textcolor{ForestGreen}{CFT$_{\rm II}$}};
\draw[very thick, NavyBlue] (3.5,0) -- (2,{3*sin(60)});
\draw[] (6, 2.2) node[font=\small] {$\mathcal M_{\rm II}$};
\draw[] (-2.5, 2.2) node[font=\small] {$\mathcal M_{\rm I}$};
\filldraw[NavyBlue] (0,0) circle (2pt);
\filldraw[NavyBlue] (3.5,0) circle (2pt);
\end{tikzpicture}
\end{center}
\caption{The bulk geometry for a linear defect in AdS$_3$/CFT$_2$. The two dark blue lines represent the brane, and the two spaces are glued with the Israel--Lanczos matching conditions. }
\label{bulk_vacuum}
\end{figure}
On either side of the brane, the spacetime will be locally AdS$_3$. Everything about the geometry of (Euclidean-)AdS$_3$ can be surmised by thinking of it as a hyperboloid
\begin{equation}\label{embed:hyp}
-\left(X^0\right)^2+\left(X^3\right)^2+\sum_{i=1}^2\left(X^i\right)^2=-L^2\,,
\end{equation}
embedded in four-dimensional flat Minkowski spacetime:
\begin{equation}
G_{\mu\nu} \text{d} X^\mu \text{d} X^\nu=-\left(\text{d} X^0\right)^2+\left(\text{d} X^3\right)^2+\sum_{i=1}^2\left(\text{d} X^i\right)^2~. \label{eq:twotimesmetric}
\end{equation}
The metric of AdS$_3$ can be obtained by finding a set of coordinates that `solve' \eqref{embed:hyp}. One such set are the \emph{global coordinates},
\begin{align}\label{eq:embglobal}
&X^0=L\sqrt{1+\frac{r_g^2}{L^2}}\cosh \left(\frac{\tau_g}{L}\right)~, &X^1=r_g\cos \theta~,\nonumber\\
&X^3=L\sqrt{1+\frac{r_g^2}{L^2}}\sinh \left(\frac{\tau_g}{L}\right)~, &X^2=r_g\sin \theta~.
\end{align}
In this coordinates we have the local form of the metric
\begin{equation}\label{eq:globalmet}
\text{d} s^2 = \left(1+\frac{r_g^2}{L^2}\right) \text{d} \tau_g^2 +\frac{\text{d} r_g^2}{\left(1+\frac{r_g^2}{L^2}\right) } +r_g^2 \text{d} \theta^2 ~.
\end{equation}
The spatial boundary of AdS$_3$ is the location where $\left(X^1\right)^2+\left(X^2\right)^2\rightarrow \infty$, and in these coordinates:
\begin{equation}
\left(X^1\right)^2+\left(X^2\right)^2=r_g^2~.
\end{equation}
Thus we conclude that the spatial boundary coincides with $r_g\rightarrow \infty$, as expected.
An alternative set of coordinates, one that is particularly useful set for solving the junction conditions described above, is the AdS$_2$ slicing of AdS$_3$:
\begin{align}
&X^0=\frac{L^2+\tau^2+y^2}{2y}\cosh\left(\frac{\rho}{L}\right)~, &&X^1=L\sinh\left(\frac{\rho}{L}\right)~, \nonumber\\
&X^3=\frac{L \tau}{y}\cosh\left(\frac{\rho}{L}\right)~, &&X^2=\frac{L^2-\tau^2-y^2}{2y}\cosh\left(\frac{\rho}{L}\right)~,
\end{align}
where $-\infty<\rho<\infty$ and $y\geq 0$~. This gives rise to the local form of the metric
\begin{equation}\label{eq:AdS2slices}
\text{d} s^2 = \text{d} \rho^2 + L^2 \cosh^2 \left(\frac{\rho}{L} \right) \left( \frac{\text{d} \tau^2+\text{d} y^2 }{y^2} \right) \ ,
\end{equation}
which is related to the standard Poincar\'{e} slicing of AdS$_3$ through the following coordinate transformation
\begin{equation}
z=\frac{y}{\cosh\left(\frac{\rho}{L}\right)} ~, \qquad x= y\tanh \left(\frac{\rho}{L}\right)~,
\end{equation}
with metric
\begin{equation}\label{eq:poincarecoordinates}
\text{d} s^2 = L^2 \, \frac{\text{d} \tau^2+\text{d} x^2+ \text{d} z^2}{z^2}~.
\end{equation}
Either by looking at the embedding space coordinates $\left(X^1\right)^2+\left(X^2\right)^2$ or the Poincar\'{e} coordinates, we note that the boundary of AdS$_3$ at $z=0$ can be reached either by taking $|\rho|\rightarrow\infty$ or $y\rightarrow 0$. Although not immediatly obvious, $\rho$ actually parametrizes an angular coordinate. Defining:
\begin{equation}
\tanh\left(\frac{\rho}{L}\right)\equiv \sin\chi~,
\end{equation}
then the Poincar\'{e} coordinates are simply:
\begin{equation}
z= y\cos\chi~, \qquad x= y \sin\chi,
\label{zxToyChi}
\end{equation}
thus $y$ is a radial coordinate in the $xz$-plane. The asymptotic boundary can then be thought of as the locus $\chi=\pm \pi/2$ and $y$ then becomes the coordinate along the boundary.
Finally, AdS$_3$ is special because there exists a parametrization that induces a black hole horizon on the hyperboloid. These coordinates are
\begin{align}\label{eq:bhemb}
&X^0=L\frac{ r_b}{r_H}\cosh \left(\frac{r_H}{L}\theta\right)~, &&X^1=L\sqrt{\frac{r_b^2}{r_H^2}-1}\cos \left(\frac{r_H \tau_b}{L^2}\right)~,\nonumber\\
&X^3=L\sqrt{\frac{r_b^2}{r_H^2}-1}\sin \left(\frac{r_H \tau_b}{L^2}\right)~, &&X^2=L\frac{r_b}{r_H}\sinh \left(\frac{r_H}{L}\theta\right)~.
\end{align}
leading to the following metric for the BTZ black hole:
\begin{equation}\label{eq:bhmet}
\text{d} s^2 = \left(\frac{r_b^2-r_H^2}{L^2}\right) \text{d} \tau_b^2 +\left(\frac{r_b^2-r_H^2}{L^2}\right)^{-1}\text{d} r_b^2 +r_b^2 \text{d} \theta^2 ~.
\end{equation}
It is evident from \eqref{eq:bhemb} that regularity of the Euclidean-BTZ metric will require $\tau_b$ to be periodic with periodicity $\tau_b \sim \tau_b + \frac{2\pi L^2}{r_H}$. Moreover, choosing $r_H=i L$ gives us back a double analytic continuation of \eqref{eq:embglobal}, with the roles of $X^0$ and $X^1$ swapped.
It is possible to go from Poincar\'{e} slicing \eqref{eq:AdS2slices} to global coordinates \eqref{eq:globalmet} via the coordinate transformation:
\begin{equation}
\rho= L \sinh^{-1}\left(\frac{r_g}{L}\cos\theta\right)~,\quad \tau=\frac{L \sinh\left(\frac{\tau_g}{L}\right)\sqrt{1+\frac{r_g^2}{L^2}}}{\cosh\left(\frac{\tau_g}{L}\right)\sqrt{1+\frac{r_g^2}{L^2}}-\frac{r_g}{L}\sin\theta} ~,\quad y= \frac{L \sqrt{1+\frac{r_g^2}{L^2}\cos^2\theta}}{\cosh\left(\frac{\tau_g}{L}\right)\sqrt{1+\frac{r_g^2}{L^2}}-\frac{r_g}{L}\sin\theta}~,
\end{equation}
and similarly, we can obtain the black hole metric \eqref{eq:bhmet} from Poincar\'{e} slicing using:
\begin{align}
&\rho= L \sinh^{-1}\left[\cos\left(\frac{r_H \tau_b}{L^2}\right)\sqrt{\frac{r_b^2}{r_H^2}-1}\right]~, \nonumber \\
&\tau=e^{\frac{r_H}{L}\theta}\sin\left(\frac{r_H \tau_b}{L^2}\right)\sqrt{1-\frac{r_H^2}{r_b^2}}~, \quad y=e^{\frac{r_H}{L}\theta}\sqrt{1-\left(1-\frac{r_H^2}{r_b^2}\right)\sin^2\left(\frac{r_H \tau_b}{L^2}\right)}~.
\end{align}
Note that this crucially means that a solution to the junction conditions in one set of coordinates can be brought into a solution in another set of coordinates via a coordinate transformation. Having set out some geometric basics and defined a number of convenient systems of coordinates we will now move on to actually solving the junction conditions.
\subsection{Solving the junction conditions}
The simplest way to solve the junction conditions is to work with the coordinates \eqref{eq:AdS2slices}, that is we will take the coordinates on each side of the brane to be
\begin{equation}
\text{d} s^2_{\mathcal M_{i}} = \text{d} \rho^2_{i} + L_{i}^2 \cosh^2 \left(\frac{\rho_{i}}{L_{i}} \right) \left( \frac{\text{d} y_i^2 + \text{d} \tau_i^2}{y_i^2} \right)
\label{M1}
\end{equation}
for $i= \{ {\rm I,II} \}$. From here on, we work in Euclidean signature, although there is no obstruction to continuing back to Lorentzian signature. We would like to consider a static junction in these coordinates, meaning that our brane is located at $\rho_i=\rho_i^*$ in each spacetime patch. In each patch of spacetime $\mathcal{M}_i$ the coordinates $\rho_i$ will range from $-\infty<\rho<\rho_i^*$. The induced metric on the brane is therefore
\begin{equation}
\text{d}^2 \hat s \equiv h_{ab} \text{d} \hat x^a \text{d} \hat x^b = L_{\rm I}^2 \cosh^2\left(\frac{\rho^*_{\rm I}}{L_{\rm I}} \right) \left( \frac{\text{d} y_{\rm I}^2 + \text{d} \tau_{\rm I}^2}{y_{\rm I}^2} \right) = L_{\rm II}^2 \cosh^2\left(\frac{\rho^*_{\rm II}}{L_{\rm II}} \right) \left( \frac{\text{d} y_{\rm II}^2 + \text{d} \tau_{\rm II}^2}{y_{\rm II}^2} \right) \ ,
\label{brane_metric}
\end{equation}
and the first junction condition enforces the relation
\begin{equation}
y_{\rm I}=y_{\rm II}~, \qquad \tau_{\rm I}=\tau_{\rm II}~, \qquad L_{\rm I} \cosh \left(\frac{\rho^*_{\rm I}}{L_{\rm I}} \right) = L_{\rm II} \cosh \left(\frac{\rho^*_{\rm II}}{L_{\rm II}} \right)~.
\label{linear_1}
\end{equation}
The extrinsic curvatures can be readily computed
\begin{align}
K_{{\rm I},ab} \, =& \, \left. \frac{1}{2} \frac{\partial h_{ab}}{\partial \rho_{\rm I}} \right|_{\rho_{\rm I} = \rho_{\rm I}^*} = \frac{1}{L_{\rm I}} \tanh\left(\frac{\rho^*_{\rm I}}{L_{\rm I}} \right) h_{ab} \ , \nonumber \\
K_{{\rm II},ab} \, =& \, \left. -\frac{1}{2} \frac{\partial h_{ab}}{\partial \rho_{\rm II}}\right|_{\rho_{\rm II} = \rho_{\rm II}^*} = - \frac{1}{L_{\rm II}} \tanh\left(\frac{\rho^*_{\rm II}}{L_{\rm II}} \right) h_{ab}~,
\end{align}
where the sign difference comes from the fact that the outward normals on either side point in opposite directions. The second junction condition therefore takes the form
\begin{equation}
\frac{1}{L_{\rm I}} \tanh \left(\frac{\rho^*_{\rm I}}{L_{\rm I}} \right) + \frac{1}{L_{\rm II}} \tanh \left(\frac{\rho^*_{\rm II}}{L_{\rm II}} \right) = T \ .
\label{linear_2}
\end{equation}
These two conditions are solved by
\begin{equation}
\tanh \left(\frac{\rho^*_{\rm I}}{L_{\rm I}} \right) = \frac{L_{\rm I}}{2T} \left( T^2 + \frac{1}{L_{\rm I}^2} - \frac{1}{L_{\rm II}^2} \right)~,\qquad \tanh \left(\frac{\rho^*_{\rm II}}{L_{\rm II}} \right) = \frac{L_{\rm II}}{2T} \left( T^2 + \frac{1}{L_{\rm II}^2} - \frac{1}{L_{\rm I}^2} \right)~.
\label{tanh_rhos}
\end{equation}
Since $-1 < \tanh(x) < 1$, one obtains a consistency constraint on the tension, namely
\begin{equation}
T_{\text{min}} = \left| \frac{1}{L_{\rm I}} - \frac{1}{L_{\rm II}} \right| < T < \frac{1}{L_{\rm I}} + \frac{1}{L_{\rm II}} = T_{\text{max}} \ .
\label{T_constraints}
\end{equation}
In the following it will be useful to parametrize the location of the brane by its angle with respect to the perpendicular to the conformal boundaries in each patch.
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}
\draw[gray!60!white, name path=A] plot [smooth, tension=2] coordinates { ({2.5*cos(60)},{2.5*sin(60)}) (1.75, 2.05) ({3.5 - 2.5*cos(60)},{2.5*sin(60)})};
\draw[gray!60!white, name path=B] plot [smooth, tension=2] coordinates { ({2.2*cos(60)},{2.2*sin(60)}) (1.75, 2) ({3.5 - 2.2*cos(60)},{2.2*sin(60)})};
\draw[gray!60!white, name path=C] plot [smooth, tension=2] coordinates { ({1.5*cos(60)},{1.5*sin(60)}) (1.7, 1.43) ({3.5 - 2*cos(60)},{2*sin(60)})};
\draw[gray!60!white, name path=D] plot [smooth, tension=1] coordinates { ({1.1*cos(60)},{1.1*sin(60)}) (1.75, 1.4) ({3.5 - 1.8*cos(60)},{1.8*sin(60)})};
\draw[gray!60!white, name path = E] plot [smooth, tension=2] coordinates { ({0.8*cos(60)},{0.8*sin(60)}) (1.7, 0.55) ({3.5 - 0.9*cos(60)},{0.9*sin(60)})};
\draw[gray!60!white, name path = F] plot [smooth, tension=1] coordinates { ({0.5*cos(60)},{0.5*sin(60)}) (1.7, 0.5) ({3.5 - 0.4*cos(60)},{0.4*sin(60)})};
\draw[gray!60!white, name path = G] plot [smooth, tension=0.5] coordinates { ({3.5 - 0.75*cos(60)},{0.75*sin(60)}) ({3.5 - 0.55*cos(60)-0.9},{0.55*sin(60)+0.04}) ({3.5 - 0.5*cos(60)},{0.5*sin(60)}) };
\shade[top color=white, bottom color=Cerulean, opacity = 0.25] (-3,0) -- (0,0) -- (1.5,3*0.866) -- (-3,3*0.866) -- cycle;
\shade[top color=white, bottom color=ForestGreen, opacity = 0.25] (3.5,0) -- (6.5,0) -- (6.5,3*0.866) -- (2,3*0.866) -- cycle;
\draw[very thick, Cerulean] (-3,0) -- (0,0);
\draw (-2.5,0) node[below, font=\small] {\textcolor{Cerulean}{CFT$_{\rm I}$}};
\draw (-1,0) node[below, font=\small] {\textcolor{Cerulean}{$\rho_{\rm I} \to - \infty$}};
\draw (-0.9,-0.25) node[below, font=\small] {\textcolor{Cerulean}{$\chi_{\rm I} \to - \pi/2$}};
\draw[very thick, NavyBlue] (0,0) -- (1.5,3*0.866);
\draw[very thick, ForestGreen] (3.5,0) -- (6.5,0);
\draw (6.2,0) node[below, font=\small] {\textcolor{ForestGreen}{CFT$_{\rm II}$}};
\draw (4.5,0) node[below, font=\small] {\textcolor{ForestGreen}{$\rho_{\rm II} \to - \infty$}};
\draw (4.6,-0.25) node[below, font=\small] {\textcolor{ForestGreen}{$\chi_{\rm II} \to -\pi/2$}};
\draw (2.73,2) node[font=\small] {\textcolor{NavyBlue}{$\rho^*_{\rm II}$}};
\draw (0.85,2) node[font=\small] {\textcolor{NavyBlue}{$\rho^*_{\rm I}$}};
\draw[very thick, NavyBlue] (3.5,0) -- (2,3*0.866);
\draw[] (6, 2.2) node[font=\small] {$\mathcal M_{\rm II}$};
\draw[] (-2.5, 2.2) node[font=\small] {$\mathcal M_{\rm I}$};
\draw[] (3.5,1) arc (90:120:1);
\draw[] (3.5,1.05) arc (90:120:1.05);
\draw (3.2,1.3) node[font=\small] {$\psi_{\rm II}$};
\draw[] (0,1) arc (90:60:1);
\draw[] (0,1.05) arc (90:60:1.05);
\draw (0.35,1.3) node[font=\small] {$\psi_{\rm I}$};
\draw[dashed] (0, -0.3) -- (0, 2.8);
\draw[dashed] (3.5, -0.3) -- (3.5, 2.8);
\filldraw[NavyBlue] (0,0) circle (2pt);
\filldraw[NavyBlue] (3.5,0) circle (2pt);
\end{tikzpicture}
\end{center}
\caption{The definition of the coordinates $\psi_{\rm I}$ and $\psi_{\rm II}$ as the geometric angles between the brane and the normal direction of the conformal boundaries. }
\label{psi_coord}
\end{figure}
These angles are defined as follows:
\begin{equation}
\sin(\psi_{\rm I,II}) = \tanh \left(\frac{\rho^*_{\rm I,II}}{L_{\rm I,II}} \right) \ , \qquad\text{or} \,\qquad \psi_{\rm I, II}= \chi^*_{\rm I, II}~,
\label{angles}
\end{equation}
and shown in Figure \ref{psi_coord}.
In later sections we will consider the \textit{large tension limit}, meaning
\begin{equation}
T \to T_{\text{max}} = \frac{1}{L_{\rm I}} + \frac{1}{L_{\rm II}} \ .
\label{largeT}
\end{equation}
To understand the geometry in this regime it is useful write
\begin{equation}
T^2 = \frac{1}{L_{\rm I}^2} + \frac{1}{L_{\rm II}^2} +\frac{2 - \delta^2}{L_{\rm I}L_{\rm II}}
\label{Ttodelta}
\end{equation}
and expand for $\delta \to 0$. In this limit
\begin{align}
\sin(\psi_{\rm I}) = \tanh \left(\frac{\rho^*_{\rm I}}{L_{\rm I}} \right) =& \; 1 - \frac{L_{\rm I}^2}{2(L_{\rm I}+L_{\rm II})^2} \delta^2 + \mathcal O(\delta^{4})\ , \nonumber \\
\sin(\psi_{\rm II}) = \tanh \left(\frac{\rho^*_{\rm II}}{L_{\rm II}} \right) =& \; 1 - \frac{L_{\rm II}^2}{2(L_{\rm I}+L_{\rm II})^2} \delta^2 + \mathcal O(\delta^{4})\ .
\end{align}
Geometrically, this means that
\begin{equation}
\psi_{\rm I} = \frac{\pi}{2} - \frac{L_{\rm I}}{L_{\rm I}+L_{\rm II}} \delta + \mathcal O(\delta^2) \ , \qquad \qquad \psi_{\rm II} = \frac{\pi}{2} - \frac{L_{\rm II}}{L_{\rm I}+L_{\rm II}} \delta + \mathcal O(\delta^2) \ ,
\label{psiofdelta}
\end{equation}
and thus in the large tension limit both AdS$^{\rm I}_3$ and AdS$^{\rm II}_3$ are reconstructed, since the brane approaches the conformal boundary. This fact will be important in section \ref{sec:Ent_entropy}, in order to derive the island formula from the RT prescription.
\subsection{The BCFT limit}\label{sec.BCFTlimit}
It is sometimes useful to consider the limiting situation in which one of the two radii, let's say $L_{\rm II}$ for concreteness, is much larger than the other. To gain intuition about this scenario, let us think of the limit $L_{\rm I} \to 0$ first\cite{Brown:2011gt}, although the latter lies well outside the semi-classical gravitational regime.
If we just apply the Brown-Henneaux relation for AdS$_3$/CFT$_2$---eq. \eqref{cBrownHenneaux}---we find that the central charge of one of the two CFTs vanishes in this limit.
Given that the central charge measures the number of degrees of freedom of a conformal field theory, the limit of $c_{I} \to 0$ eliminates all degrees of freedom of the CFT$_{\rm{I}}$. No excitation can be transmitted from CFT$_{\rm{II}}$ through the interface, which therefore acts as a boundary. In other words, the ICFT reduces to a BCFT. This argument can be made precise by using unitarity and its consequences on Cardy gluing conditions \cite{Billo:2016cpy,Meineri:2019ycm}: the vanishing of the transmission coefficient is a necessary and sufficient condition for a conformal interface to be factorizing. We can now go back to the limit
\begin{equation}
\frac{L_{\rm I}}{L_{\rm II}} \to 0 \ ,
\label{BCFT_lambda_0}
\end{equation}
which allow us to keep both $L_{\rm I}$ and $L_{\rm II}$ large in Planck units. It is intuitive, and also rigorously true due to unitarity \cite{Billo:2016cpy,Meineri:2019ycm}, that the transmission coefficient vanishes in the strict limit. Hence, eq. \eqref{BCFT_lambda_0} corresponds to a BCFT limit which we can take without abandoning the classical regime.
Accordingly, one also expects the bulk brane to effectively become an EOW brane. We shall confirm this expectation in section \ref{sec:BCFT}, where the limit \eqref{BCFT_lambda_0} will be performed on results found in previous sections for AdS/ICFT, and the BCFT expectations met. From this point of view, we would like to regard the AdS/BCFT framework as a subset of AdS/ICFT.
The interest around this idea is that the bulk dual to an ICFT has the same topology as the asymptotically AdS spaces familiar from AdS/CFT. This allows us to easily extend some of the entries of the usual AdS/CFT dictionary. Taking the BCFT limit at the end, one derives the corresponding rules for AdS/BCFT, which are then justified and do not need to be separately conjectured. An example is the famous Takayanagi prescription to compute entanglement entropies in AdS/BCFT \cite{Takayanagi:2011zk, Fujita:2011fp}, where the RT geodesics are allowed also to end on the EOW brane. Such a rule arises naturally from the interpolation CFT~$\to$~ICFT~$\to$~BCFT, as we will show in section \ref{subsec:TakaProof}.
It is interesting to note that, while the BCFT limit works perfectly in classical gravity, accounting for $1/c$ corrections might be subtle and require modifications. Indeed, since $L_{\rm I}/L_{\rm II}=c_{\rm I}/c_{\rm II}$, we cannot disregard transmission effects across the thin brane while keeping $1/c_{\rm II}$ contributions.
\subsection{Semi-infinite intervals in the vacuum}
\label{subsec:EEvacuum}
Let us start by asking the following question. What is the von Neumann entropy of the density matrix obtained from the vacuum state of the system in figure \ref{it} by tracing over the conformal interface and a part of the two baths? Specifically, we choose the entangling surface to be composed of two points at distances $\sigma_{\rm I}$ and $\sigma_{\rm II}$ from the defect respectively, as depicted on the left panel of figure \ref{Island_figure}.
The answer in the large $c$ limit is given by the RT formula \cite{Ryu:2006bv}:
\begin{equation}
S_{\sigma_{\rm I},\sigma_{\rm II}} = \frac{d(\sigma_{\rm I}, \sigma_{\rm II})}{4G_{(3)}}~,
\end{equation}
where $d(\sigma_{\rm I}, \sigma_{\rm II})$ is the length of the geodesic depicted in figure \ref{Island_figure}. Using eq. \eqref{length_geodesic}, one finds
\begin{equation}
S_{\sigma_{\rm I},\sigma_{\rm II}} = \frac{c_{\rm I}}{6} \log \left[ \frac{2r}{\varepsilon} \tan \left( \frac{\varphi}{2} \right) \right] + \frac{c_{\rm II}}{6} \log \left[ \frac{2R}{\varepsilon} \tan \left( \frac{\theta}{2} \right) \right]\ ,
\label{S_bulk}
\end{equation}
where the dependence on $\sigma_{\rm I}$ and $\sigma_{\rm II}$ is through $r,\,R,\,\theta$ and $\varphi$. The explicit expressions can be found in eqs. (\ref{theta_phi},\ref{sol_r1},\ref{sol_r2},\ref{eq:cossol}), while the geometric meanings of these parameters are clear from figure \ref{Generic_sigma_2}, where $r=BQ$ and $R=AP$.
Eq. \eqref{S_bulk} is complicated in general, but it becomes transparent if we take $\sigma_{\rm I}=\sigma_{\rm II}=\sigma$. In this case, as remarked in subsection \ref{sec:panoply}, conformal symmetry is more constraining, and forces the $\sigma$ dependence to simplify---see eq. \eqref{eq:distanceequal}:
\begin{equation}
S_{\sigma} = \frac{c_{\rm I}}{6} \log \frac{2\sigma}{\varepsilon}+\frac{c_{\rm II}}{6} \log \frac{2\sigma}{\varepsilon}+ \frac{\rho^*_{\rm I}+\rho^*_{\rm II}}{4 G_{(3)}} \ .
\label{EEequalSigma}
\end{equation}
Subtracting the bulk contribution off, we read the interface entropy
\begin{equation}
S_\textup{interface} = \frac{\rho^*_{\rm I}+\rho^*_{\rm II}}{4 G_{(3)}}~.
\end{equation}
Let us now reinterpret eq. \eqref{S_bulk} from the point of view of the `brane+baths' system. As advocated in the introduction, we should expect a simple picture to emerge in the large tension limit of eq. \eqref{largeT}:
\begin{equation}
T \to T_{\text{max}} ~.
\end{equation}
The brane is pushed towards the boundary---see eq. \eqref{psiofdelta}---and the isometries of AdS$_3$ act as conformal transformations on points of the brane. In other words, the theory on the brane is now a CFT coupled to a (weakly) fluctuating metric, deformed by a UV cutoff which behaves locally like the standard cutoff of Poincaré AdS. The nature of the CFT on the brane also follows from the geometry. Integrating out the bulk on either side, we end up, by symmetry, with two copies of the Polyakov action with central charges $c_{\rm I}$ and $c_{\rm II}$. All in all, we have two holographic CFTs, each defined on the whole real line. On half of the line, the CFTs are decoupled and live on a frozen conformally flat metric. On the other half, the CFTs still interact through the common metric, as it is clear from the fact that the transmission coefficient does not vanish even when the tension is the maximal one \cite{Bachas:2020yxv}. It would be interesting to compute energy transmission and reflection in the `brane+baths' picture.
\begin{figure}[t]
\begin{flushright}
\begin{tikzpicture}[scale=0.9]
\shade[top color=white, bottom color=black, opacity = 0.1] (2,1) -- (6,1) -- ({6},{4}) -- ({5*cos(175)+2},{4}) -- ({5*cos(175)+2},{5*sin(175)+1}) -- cycle;
\scoped[transform canvas={rotate around={170:(1.8,0.9)}}]
\shade[top color=white, bottom color=black, opacity = 0.1] (2,1) -- (-2,1) -- ({-2},{4}) -- ({-5*cos(175)+2},{4}) -- ({-5*cos(175)+2},{5*sin(175)+1}) -- cycle;
\draw[very thick, ForestGreen, opacity = 0.8, name path=L1] (2,1) -- (6,1);
\draw[very thick, Cerulean, opacity = 0.8] (2,1) -- ({4*cos(350)+2},{4*sin(350)+1});
\draw[very thick, NavyBlue, opacity = 0.8, name path=I] (2,1) -- ({5*cos(175)+2},{5*sin(175)+1});
\centerarc[very thick, black!50, opacity = 0.5, name path=c1](2.5,1)(0:177:42.7pt);
\centerarc[very thick, black!50, opacity = 0.5, name path=c1]({1*cos(350)+2},{1*sin(350)+1})(350:174:57pt);
\fill[red] (4,1) circle (2pt);
\fill[NavyBlue] (2,1) circle (2pt);
\fill[red] ({3*cos(350)+2},{3*sin(350)+1}) circle (2pt);
\fill[blue] ({1*cos(175)+2},{1*sin(175)+1}) circle (2pt);
\fill[blue] ({1*cos(175) -0.35+2},{1*sin(175)-0.25+1}) node[]{\small $y^*$};
\draw [blue, opacity = 0.3, line width = 3pt] ({1*cos(175)+2},{1*sin(175)+1}) -- ({5*cos(175)+2},{5*sin(175)+1});
\draw [blue] ({3*cos(175)+2},{3*sin(175)+0.3+1}) node [rotate = -5] {\small Island} ;
\fill[color = NavyBlue, opacity = 0.15] (8, 4) -- (8,-2) -- (10.5,-2) -- (10.5, 4) -- cycle;
\draw[thick, blue] (9, 1) -- (8,1);
\draw[NavyBlue] (8.65, -1.6) node[font=\small] {AdS$_2$};
\draw[thick, NavyBlue, opacity = 0.8] (8, 4) -- (8,-2);
\draw[thick, NavyBlue, opacity = 0.8] (10.5, 4) -- (10.5,-2);
\draw[dashed] (8, 1) -- (10.5,3.5);
\filldraw[blue] (9, 1) circle (1.5pt) node[font=\small, anchor = north] {$y^*$};
\draw[dashed] (8, 1) -- (10.5,-1.5);
\draw[thick, color = Cerulean, opacity = 0.8] (10.5,3.5) -- (13,2) -- (10.5, -1.5) -- cycle;
\filldraw[red] (12, 1.7) circle (1.5pt) node[font=\small, anchor = south] {$\sigma_{\rm I}$};
\draw[thick, red] (12, 1.7) -- (13,2);
\fill[color = Cerulean, opacity = 0.2] (10.5,3.5) -- (13,2) -- (10.5, -1.5) -- cycle;
\draw[thick, color = ForestGreen, opacity = 0.8] (10.5,3.5) -- (13,0) -- (10.5, -1.5) -- cycle;
\fill[color = ForestGreen, opacity = 0.2] (10.5,3.5) -- (13,0) -- (10.5, -1.5) -- cycle;
\draw[Cerulean] (12.2, 3.1) node[font=\small] {CFT$_{\rm I}$};
\draw[ForestGreen] (12.2, -1.2) node[font=\small] {CFT$_{\rm II}$};
\filldraw[red] (12, 0.3) circle (1.5pt) node[font=\small, anchor = north] {$\sigma_{\rm II}$};
\draw[thick, red] (12, 0.3) -- (13,0);
\draw[] (14,0);
\end{tikzpicture}
\end{flushright}
\caption{In the spirit of double holography we can interpret the geodesic of Figure \ref{Generic_sigma_2} in a a two dimensional perspective. In particular we can integrate out the bulk degrees of freedom and consider the induced effective action on the brane. Then, the bulk geodesic can be interpreted as the island in the intermediate two dimensional picture. In particular, for large values of the tension, one can prove that the point where the geodesic of Figure \ref{Generic_sigma_2} crosses the brane is the position where the island appears minimizing the 2$d$ generalized entropy.}
\label{Island_figure}
\end{figure}
With this in mind, it is easy to realize what the geodesic in the left panel of figure \ref{Island_figure} computes. The arc on either side is an RT surface for CFT$_{\rm I}$ and CFT$_{\rm II}$ respectively. It computes the entanglement entropy of two intervals stretching between $\sigma_{\rm I/II}$ respectively and $y^*$. Also, recall that the value $y^*$ can be obtained by minimizing the length of all the curves which join $\sigma_{\rm I}$ to $\sigma_{\rm II}$, are piecewise geodesics, and pass through a point $y$ on the brane. These entropies can be computed via the universal CFT formula \cite{Cardy2004Calabrese}. However, it pays off to be careful with the role of the cutoff. At the points $\sigma_{\rm I/II}$, we should of course maintain the uniform cutoff $\varepsilon$ used so far. On the brane, the cutoff, chosen again as the coordinate distance from the boundary of AdS in Poincaré coordinates, is point dependent, and easily seen to be $\varepsilon_\text{brane}(y)=y \cos \psi$.\footnote{Alternatively, one can think of the theory on the brane to live in AdS$_2$, and keep into account the appropriate Weyl factor, but, for consistency, no additional cutoff.} Explicitly, we expect to find
\begin{equation}
S_{\sigma_{\rm I}\sigma_{\rm II}} \sim \min_y \left[
\frac{c_{\rm I}}{6} \log \left( \frac{(y + \sigma_{\rm I})^2}{y \, \varepsilon} \frac{1}{\cos(\psi_{\rm I})}\right) + \frac{c_{\rm II}}{6} \log \left( \frac{(y + \sigma_{\rm II})^2}{y \, \varepsilon} \frac{1}{\cos(\psi_{\rm II})}\right)
\right] \equiv S_{\text{island}}~, \quad T \to T_\textup{max}~.
\label{island_S_int}
\end{equation}
This is, remarkably, the island formula \cite{Penington:2019npb, Almheiri:2019hni, Almheiri:2019yqk, Almheiri:2019qdq, Almheiri:2020cfm}. Indeed, the quantity to minimize is a special case of the generalized entropy:
\begin{equation}
S_{\text{gen}} = \frac{A(\partial I)}{4 G_N} + S_{\rm vN} (I \cup R) \ , \qquad S_{\text{island}}=\min_y S_{\text{gen}}~,
\label{S_gen}
\end{equation}
where $I$ is the island and $R$ the region of the bath whose entanglement entropy we want to compute. In the present case, the island is the region highlighted in blue in figure \ref{Island_figure}, and the area term is missing because there is no Hilbert-Einstein term, nor a dilaton, on the brane.
Eq. \eqref{island_S_int} can be verified explicitly, also making precise how many terms in an expansion in $(T \to T_\textup{max})$ can be matched. The equation
\begin{equation}
\partial_{y} S_{\text{gen}} = 0 \qquad \Rightarrow \qquad \frac{2 c_{\rm I}}{y + \sigma_{\rm I}} - \frac{c_{\rm I}}{y} + \frac{2 c_{\rm II}}{y + \sigma_{\rm II}} - \frac{c_{\rm II}}{y} = 0
\label{minimization_S_gen}
\end{equation}
implies that the minimum, \emph{i.e.} the position of the quantum extremal surface (QES), is at
\begin{equation}
y^* = \frac{(c_{\rm I} - c_{\rm II})(\sigma_{\rm I}- \sigma_{\rm II})+ \sqrt{(c_{\rm I}-c_{\rm II})^2 (\sigma_{\rm I}-\sigma_{\rm II})^2 + 4 \sigma_{\rm I} \sigma_{\rm II} (c_{\rm II}+c_{\rm II})^2}}{2 (c_{\rm I}+c_{\rm II})} \ .
\label{island_position}
\end{equation}
Notice that the second solution of the quadratic related to (\ref{minimization_S_gen}) is always negative and $y^*$ must be positive, so we disregard it. In the limit $T \to T_\textup{max},$ it can be checked that the position of the island $y^*$ approaches the point where the geodesic in the bulk crosses the brane. Moreover, the equality \eqref{island_S_int} is verified. Ideed, writing the tension of the brane as in eq. \eqref{Ttodelta}:
\begin{equation}
T^2 = \frac{1}{L_{\rm I}^2} + \frac{1}{L_{\rm II}^2} + \frac{2 - \delta^2}{L_{\rm I} L_{\rm II}} \ ,
\end{equation}
in the $\delta \to 0$ limit one finds
\begin{equation}
\cos(\psi_{\rm I}) = \frac{L_{\rm I}}{L_{\rm I} + L_{\rm II}} \, \delta + \mathcal O(\delta^2) \ , \qquad \qquad \cos(\psi_{\rm II}) = \frac{L_{\rm II}}{L_{\rm I} + L_{\rm II}} \, \delta + \mathcal O(\delta^2) \ ,
\end{equation}
and the minimum of the generalized entropy can be expanded as
\begin{multline}
S_{\text{island}} = -\frac{c_{\rm I} + c_{\rm II}}{6} \log(\delta) + \frac{c_{\rm I}}{6} \log \left[ \frac{(y^* + \sigma_{\rm I})^2}{y^* \, \varepsilon} \left(\frac{L_{\rm I} + L_{\rm II}}{L_{\rm I}} \right)\right] \\ + \frac{c_{\rm II}}{6} \log \left[ \frac{(y^* + \sigma_{\rm II})^2}{y^* \, \varepsilon} \left(\frac{L_{\rm I} + L_{\rm II}}{L_{\rm II}} \right) \right] + \mathcal O(\delta) \ .
\end{multline}
The 3d bulk computation (\ref{S_bulk}) agrees with this result up to terms which vanish as $\delta \to 0$. This agreement is quite remarkable---we have matched a nontrivial formula (not fixed by symmetry) in ICFT using the QES prescription in the 2d braneworld, giving strong evidence of the validity of this formula in the setting of 2d gravity.
\subsection{Black hole evaporation in double holography and AdS/ICFT}
\label{subsec:thermofield}
The same ideas can be applied in the context of the black hole information paradox \cite{Penington:2019npb, Almheiri:2019hni, Almheiri:2019yqk, Almheiri:2019qdq, Almheiri:2020cfm}. The simplest setup consists of the thermofield double state, which is connected to the vacuum by a conformal transformation.
This approach was originally presented in \cite{Rozali:2019day} in the case of AdS/BCFT, using the RT prescription. Here we will extend this computation to the case of AdS/ICFT,\footnote{In the recent paper \cite{Grimaldi:2022suv} a related construction is used. In that case, the compact spatial direction in the two copies of the bath forces the two CFTs to be the same.} and we will complement it with the two dimensional computation in the intermediate picture. Again, this will allow us to derive the island formula from the RT prescription. The agreement between the two perspectives, in this case, is all the more relevant, because it implies that a UV finite quantity, the Page time, can be matched exactly.
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}
\fill[red] (1, 1) circle (2pt);
\fill[red] (-2, 1) circle (2pt);
\draw[red] (1, 1) node[anchor=north west]{\small $q^2_{\rm II}$};
\draw[red] (-2, 1) node[anchor=north east]{\small $q^2_{\rm I}$};
\draw[very thick, ForestGreen, name path=I, dashed] (3.5,2) arc (90:270:0.5 and 1);
\draw[very thick, ForestGreen, name path=J] (3.5,0) arc (-90:90:0.5 and 1);
\draw[very thick, ForestGreen, name path=A] (0,0) -- (4,0);
\draw[very thick, ForestGreen, name path=B] (0,2) -- (4,2);
\draw[very thick, ForestGreen] (4.2,0) -- (4.4,0);
\draw[very thick, ForestGreen] (4.2,2) -- (4.4,2);
\draw[very thick, Cerulean, name path=D] (0,0) -- (-4,0);
\draw[very thick, Cerulean, name path=E] (0,2) -- (-4,2);
\draw[very thick, Cerulean, name path=G] (-3.5,2) arc (90:270:0.5 and 1);
\draw[very thick, Cerulean, name path=H] (-3.5,0) arc (-90:90:0.5 and 1);
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\tikzfillbetween[of=C and I]{ForestGreen, opacity=0.25};
\draw[very thick, NavyBlue, dashed] (0,2) arc (90:270:0.5 and 1);
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\tikzfillbetween[of=F and J]{ForestGreen, opacity=0.25};
\tikzfillbetween[of=C and F]{NavyBlue, opacity=0.25};
\draw[very thick, NavyBlue] (0,0) arc (-90:90:0.5 and 1);
\draw[very thick, Cerulean] (-4.2,0) -- (-4.4,0);
\draw[very thick, Cerulean] (-4.2,2) -- (-4.4,2);
\draw[thick] (1.5,0) arc (-90:90:0.5 and 1);
\draw[thick, dashed] (1.5,2) arc (90:270:0.5 and 1);
\draw[thick] (-1.5,0) arc (-90:90:0.5 and 1);
\draw[thick, dashed] (-1.5,2) arc (90:270:0.5 and 1);
\draw[<->] (0.05,-0.3) -- (1.45, -0.3);
\draw[<->] (-0.05,-0.3) -- (-1.45, -0.3);
\draw (0.75, -0.5) node[font=\small] {$u_0$};
\draw (-0.75, -0.5) node[font=\small] {$- \, u_0$};
\fill[red] (2, 1) circle (2pt);
\fill[red] (-1, 1) circle (2pt);
\draw[red] (2, 1) node[anchor=north west]{\small $q^1_{\rm II}$};
\draw[red] (-1, 1) node[anchor=north east]{\small $q^1_{\rm I}$};
\draw[-stealth, thick, domain=0:330, smooth,variable=\t] plot ({0.5 * sin(\t) - 5}, {cos(\t) +1 });
\draw[-stealth, domain=0:300, smooth,variable=\t, opacity=0] plot ({0.4 * sin(\t) + 5}, {0.8*cos(\t) +1 });
\draw[] ({-5.8}, {1}) node{\small $\beta$};
\end{tikzpicture}
\end{center}
\caption{Thermofield double state of the system considered. The red dots are the twist operators. To obtain the Lorentzian evolution, one has to cut the cylinder at $v_E = 0$ and $v_E = \frac{i \beta}{2}$, and then perform the Wick rotation $v_E = iv$.}
\label{fig:cylinder}
\end{figure}
Let us review the basic idea. On the conformal boundary, the thermofield double state is prepared by the Euclidean path integral on half of the infinite cylinder, with the defect running along the circle---see figure \ref{fig:cylinder}. The initial condition for the Lorentzian evolution consists therefore of $\emph{two}$ copies of the system in figure \ref{it}, whose state is entangled. Tracing over one of the copies, we get a thermal state for the other. The holographic dual to this state contains an eternal black string in AdS$_3$ \cite{Maldacena:2001kr}, which crosses the brane and induces a horizon on it as well. The Lorentzian geometry is sketched in figure \ref{Island_TFD}.
It is important to emphasize that Lorentzian time evolution on the two boundaries is obtained by Wick rotating the generator of the rotation on the circle, as appropriate for a finite temperature system. In figure \ref{Island_TFD}, the corresponding Killing vector moves time `upward' in the left asymptotic region and `downward' in the right one.
The simplest quantum-information theoretic quantity to compute in this system is the entanglement entropy of the density matrix obtained tracing over both copies of the quantum dot. In the `brane+bath' perspective, this is the entropy of the Hawking radiation deposited in the baths by the AdS$_2$ black hole. In this case we have four twist operators and not two, since we have doubled the system: they are marked with red dots in figures \ref{fig:cylinder}, \ref{Island_TFD} and \ref{Page_curve}. For simplicity, we choose them to lie all at the same distance from the quantum dots.\footnote{The symmetries of the problem are then the same as in the BCFT case treated in \cite{Rozali:2019day}, and the $3d$ computation is expected to match. Following the logic presented here, however, one can straightforwardly extend the result to the asymmetric case, using formulas from section \ref{sec:corr_heavy}.} The time evolution we are interested in evolves `upward' in both copies of the system. This is not a symmetry, and the entanglement entropy is time dependent. In particular, since the quantum dot has finite thermodynamic entropy, the radiation cannot increase the entropy in the baths indefinitely.
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale = 0.75]
\draw[thick, orange, opacity = 0.5] plot [smooth, tension=1] coordinates { (-15, 1.7) (-19, 3) (-23, 1.7)};
\draw[thick, blue, opacity = 0.5] plot [smooth, tension=1.5] coordinates { (-15, 1.7) (-16.6, 2.17) (-17.5, 2)};
\draw[thick, blue, opacity = 0.5] plot [smooth, tension=1.5] coordinates { (-20.5, 2) (-21.4, 2.17) (-23, 1.7)};
\fill[color = NavyBlue, opacity = 0.2] (-16.5, 4.5) -- (-16.5,-2.5) -- (-21.5,-2.5) -- (-21.5, 4.5) -- cycle;
\draw[thick, Gray!50!Black] plot [smooth, tension=1] coordinates { (-17.5, 2) (-19, 2.2) (-20.5,2)};
\draw[dashed] (-16.5, -1.5) -- (-21.5,3.5);
\draw[dashed] (-16.5, 3.5) -- (-21.5,-1.5);
\filldraw[Gray!50!Black] (-20.5, 2) circle (1.5pt);
\filldraw[Gray!50!Black] (-17.5, 2) circle (1.5pt);
\draw[thick, color = Cerulean, opacity = 0.8] (-21.5,3.5) -- (-24,2) -- (-21.5, -1.5) -- cycle;
\filldraw[red] (-23, 1.7) circle (1.5pt);
\fill[color = Cerulean, opacity = 0.2] (-21.5,3.5) -- (-24,2) -- (-21.5, -1.5) -- cycle;
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\draw[thick, color = Cerulean, opacity = 0.8] (-16.5,3.5) -- (-14,2) -- (-16.5, -1.5) -- cycle;
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\fill[color = Cerulean, opacity = 0.2] (-16.5,3.5) -- (-14,2) -- (-16.5, -1.5) -- cycle;
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\filldraw[red] (-15, 0.3) circle (1.5pt);
\draw[thick, blue] plot [smooth, tension=1.5] coordinates { (-15, 0.3) (-16.5, 1.05) (-17.5, 2)};
\draw[thick, blue] plot [smooth, tension=1.5] coordinates { (-20.5, 2) (-21.5, 1.05) (-23, 0.3)};
\draw[thick, orange] plot [smooth, tension=1] coordinates { (-15, 0.3) (-19, -1) (-23, 0.3)};
\path[thick, draw=black, snake it] (-16.5, -1.5) -- (-21.5,-1.5);
\path[thick, draw=black, snake it] (-16.5, 3.5) -- (-21.5,3.5);
\fill[color = NavyBlue, opacity = 0.2] (-5.5, 4.5) -- (-5.5,-2.5) -- (-10.5,-2.5) -- (-10.5, 4.5) -- cycle;
\draw[thick, blue] plot [smooth, tension=1] coordinates { (-6.5, 2) (-8, 2.2) (-9.5,2)};
\draw[dashed] (-5.5, -1.5) -- (-10.5,3.5);
\draw[dashed] (-5.5, 3.5) -- (-10.5,-1.5);
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\filldraw[blue] (-6.5, 2) circle (1.5pt);
\draw[thick, color = Cerulean, opacity = 0.8] (-10.5,3.5) -- (-13,2) -- (-10.5, -1.5) -- cycle;
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\draw[thick, red] (-12, 0.3) -- (-13,0);
\draw[thick, color = Cerulean, opacity = 0.8] (-5.5,3.5) -- (-3,2) -- (-5.5, -1.5) -- cycle;
\filldraw[red] (-4, 1.7) circle (1.5pt);
\draw[thick, red] (-4, 1.7) -- (-3,2);
\fill[color = Cerulean, opacity = 0.2] (-5.5,3.5) -- (-3,2) -- (-5.5, -1.5) -- cycle;
\draw[thick, color = ForestGreen, opacity = 0.8] (-5.5,3.5) -- (-3,0) -- (-5.5, -1.5) -- cycle;
\fill[color = ForestGreen, opacity = 0.2] (-5.5,3.5) -- (-3,0) -- (-5.5, -1.5) -- cycle;
\filldraw[red] (-4, 0.3) circle (1.5pt);
\draw[thick, red] (-4, 0.3) -- (-3,0);
\path[thick, draw=black, snake it] (-5.5, -1.5) -- (-10.5,-1.5);
\path[thick, draw=black, snake it] (-5.5, 3.5) -- (-10.5,3.5);
\end{tikzpicture}
\end{center}
\caption{The information paradox for the thermofield double state. {\it Left}: the computation of the entanglement entropy of the radiation region using RT surfaces. The orange lines are the RT saddles that dominate at early times, in which the entropy increases linearly in time. The blue lines are the RT saddles that cross the brane, and are constant in time, thus saturate the entanglement entropy. The gray line is the island. {\it Right}: the same computation in the two dimensional perspective using the generalized entropy formula. The blue dots represent the QES, and the blue line is the island.}
\label{Island_TFD}
\end{figure}
As promised, we will compute such entropy in two ways: both using RT surfaces in three dimensions and using the island prescription in two dimensions, as it is sketched in figure \ref{Island_TFD}. As in the previous section, we expect the two entropies to match in the limit of large tension, and that an island in two dimensions forms when the two RT saddles exchange dominance (the Page time $t_P$). Moreover, the points where the blue geodesics in figure \ref{Island_TFD} cross the brane are the boundaries of the island in two dimensions.
It is also useful to conformally map the CFTs from the cylinder to flat space. The Euclidean path integral now is done with the defect placed on a circle of length $l$, which is an arbitrary scale. Later it will only appear in a dimensionless combination with the UV cutoff $\varepsilon$, and cancel out from universal quantities like the Page time. The transformation is
\begin{equation}
p = l \, e^{\frac{2 \pi}{\beta} q} \ ,
\label{plane_cylinder}
\end{equation}
where $p = \tilde x + i \tilde \tau$ is the complex coordinate on the plane and $q = u + i v_E$ is the complex coordinate on the cylinder, such that $v_E \simeq v_E + \beta$, as depicted in figure \ref{fig:cylinder}. Later we will consider only the Lorentzian evolution obtained by the Wick rotation $v_E = iv$. Correlation functions in the planar geometry can be computed by conformally mapping the circle to a straight line and using the results of section \ref{sec:corr_heavy}. The analytic continuation to real time described above now turns the defect into a hyperbola, and stationary observers in the thermofield double state lie on hyperbolic trajectories on the plane. This coordinate system is depicted in the left panel of figure \ref{Page_curve}.
In details, we can connect the geometry of the previous section (coordinates $x$ and $t$) with the one of this section (coordinates $\tilde x$, $\tilde t$ in Lorentzian) with the AdS isometry which acts on the boundary by turning the flat interface into a circle. Explicitly,
\begin{align}
\tilde x_i \; = \; \frac{\displaystyle x_i - \frac{x_i^2+z_i^2-t_i^2}{2l}}{\displaystyle 1 - \frac{x_i}{l} + \frac{x_i^2+z_i^2-t_i^2}{4 l^2}} \, & + \, l \ , \qquad \qquad \tilde z_i \; = \; \frac{z_i}{\displaystyle 1 - \frac{x_i}{l} + \frac{x_i^2+z_i^2-t_i^2}{4 l^2}} \ , \nonumber \\
\tilde t_i \; =& \; \frac{t_i}{\displaystyle 1 - \frac{x_i}{l} + \frac{x_i^2+z_i^2-t_i^2}{4 l^2}} \ .
\label{Lorentz_conformal}
\end{align}
Notice that we are already in Lorentzian signature. In these coordinates the brane is the locus of points
\begin{equation}
\tilde x_{\rm I}^2 - \tilde t_{\rm I}^2 + \left( \tilde z_{\rm I} - l \tan(\psi_{\rm I}) \right)^2 = \frac{l^2}{\cos^2(\psi_{\rm I})} \ ,
\label{conf_euc_brane_1}
\end{equation}
on side ${\rm I}$ of the brane, and
\begin{equation}
\tilde x_{\rm II}^2 - \tilde t_{\rm II}^2 + \left( \tilde z_{\rm II} + l \tan(\psi_{\rm II}) \right)^2 = \frac{l^2}{\cos^2(\psi_{\rm II})} \ ,
\label{conf_euc_brane_2}
\end{equation}
on side ${\rm II}$. The brane ends on the conformal boundary along the hyperbola depicted in figure \ref{Page_curve}. As announced above, we take the twist operators at the same distance $-u^{1,2}_{\rm I} = u^{1,2}_{\rm II} = u_0$ in the cylinder geometry of figure \ref{fig:cylinder}. Their Lorentzian trajectories in the planar geometry are
\begin{align}
\tilde x_{\rm I}(q^1_{\rm I}) \;&=\; l e^{- \frac{2 \pi}{\beta} u_0} \cosh\Big(\frac{2 \pi}{\beta} v\Big) \ , &\tilde t_{\rm I}(q^1_{\rm I}) \; &=\; l e^{- \frac{2 \pi}{\beta} u_0} \sinh\Big(\frac{2 \pi}{\beta} v\Big) \ , \\
\tilde x_{\rm II}(q^1_{\rm II}) \;&= \; l e^{\frac{2 \pi}{\beta} u_0} \cosh\Big(\frac{2 \pi}{\beta} v\Big) \ , &\tilde t_{\rm II}(q^1_{\rm II}) \;& =\; l e^{\frac{2 \pi}{\beta} u_0} \sinh\Big(\frac{2 \pi}{\beta} v\Big) \ ,\\
\tilde x_{\rm I}(q^2_{\rm I}) \;&=\; - l e^{- \frac{2 \pi}{\beta} u_0} \cosh\Big(\frac{2 \pi}{\beta} v\Big) \ , &\tilde t_{\rm I}(q^2_{\rm I}) \; &=\; l e^{- \frac{2 \pi}{\beta} u_0} \sinh\Big(\frac{2 \pi}{\beta} v\Big) \ ,\\
\tilde x_{\rm II}(q^2_{\rm II}) \;&=\; - l e^{\frac{2 \pi}{\beta} u_0} \cosh\Big(\frac{2 \pi}{\beta} v\Big) \ , &\tilde t_{\rm II}(q^2_{\rm II}) \; &=\; l e^{\frac{2 \pi}{\beta} u_0} \sinh\Big(\frac{2 \pi}{\beta} v\Big) \ .
\end{align}
The length of the orange geodesics in figure \ref{Island_TFD} and \ref{Page_curve} grows with $v$ as
\begin{equation}
S_{\text{early}}^{\text{plane}} = \frac{c_{\rm I}}{3} \log \left[ \frac{2l}{\varepsilon} e^{- \frac{2 \pi}{\beta} u_0} \cosh\Big(\frac{2 \pi}{\beta} v\Big) \right] + \frac{c_{\rm II}}{3} \log \left[ \frac{2l}{\varepsilon} e^{\frac{2 \pi}{\beta} u_0} \cosh \Big(\frac{2 \pi}{\beta} v\Big) \right] \sim \frac{c_{\rm I} + c_{\rm II}}{3} \frac{2 \pi}{\beta} \, v \ ,
\end{equation}
thus linearly in time after a short transient. To express this result in the coordinates of the cylinder, we have to take into account that the transformation (\ref{plane_cylinder}) generates a Weyl factor between the metric on the plane and on the cylinder, namely
\begin{equation}
\text{d} s_{\text{plane}}^2 = \text{d} p \, \text{d} \bar p = l^2 \left(\frac{2 \pi}{\beta}\right)^2 e^{\frac{2 \pi}{\beta} (q + \bar q)} \, \text{d} q \, \text{d} \bar q = \Omega^2(q, \bar q) \, \text{d} s_{\text{cyl}}^2\ .
\label{Weyl_transf}
\end{equation}
Taking this into account, the entropy in the cylinder geometry reads
\begin{equation}
S_{\text{early}}^{\text{cyl}} = \frac{c_{\rm I} + c_{\rm II}}{3} \log \left[\frac{\beta}{\pi \varepsilon} \cosh\left(\frac{2 \pi}{\beta} v\right) \right] \sim \frac{c_{\rm I} + c_{\rm II}}{3} \frac{2 \pi}{\beta} \, v \ ,
\label{Searly}
\end{equation}
This saddle dominates at early time, and does not cross the brane: eq. \eqref{Searly} is independent of the tension $T$. In terms of the conformal block decomposition of the correlator of twist operators, it is obtained by only keeping the identity operator in the fusion of each pair on either side of the interface. Notice moreover that the dependence on the position of the twist operators (namely $u_0$) drops from the final form. This is due to the fact that this saddle is insensitive to the presence of the defect, which is the only source of breaking of the translational symmetry of the cylinder.
On the other hand, the island saddle can be obtained by appropriately transforming the one obtained in the vacuum in the previous subsection. Applying the transformation (\ref{Lorentz_conformal}) on eq. \eqref{EEequalSigma}, accounting for the second pair of twist operators and appropriately transforming the cutoffs, we get for the entropy of the blue geodesics
\begin{equation}
S_{\text{late}}^{\text{plane}} = \frac{c_{\rm I}}{3} \log\left[ \frac{l}{\varepsilon} \frac{e^{\frac{4 \pi}{\beta} u_0}-1}{e^{\frac{4 \pi}{\beta} u_0}} \right] + \frac{c_{\rm II}}{3} \log \left[\frac{l}{\varepsilon} \left( e^{\frac{4 \pi}{\beta} u_0}-1 \right) \right] + \frac{\rho^*_{\rm I}+\rho^*_{\rm II}}{2 G_{(3)}} \ .
\end{equation}
Taking into account the same Weyl transformation (\ref{Weyl_transf}), we get
\begin{equation}
S_{\text{late}}^{\text{cyl}} = \frac{c_{\rm I} + c_{\rm II}}{3} \log\left[ \frac{\beta}{\pi \varepsilon} \sinh\left( \frac{2 \pi}{\beta} u_0 \right) \right] + \frac{\rho^*_{\rm I}+\rho^*_{\rm II}}{2 G_{(3)}} \ .
\label{Slate}
\end{equation}
This saddle is constant in $v$ and dominates at late times. The result depends on the boundary entropy and on the position of the twist operators, as expected.
Moreover, since we are considering a doubled system, the saturation is twice the interface entropy.
The Page curve is shown in Figure \ref{Page_curve} (right), in which we can distinguish an early rising phase and a saturation phase. The exchange of dominances is at the Page time, defined as
\begin{equation}
S_{\text{early}}^{\text{cyl}} (v_{P}) = S_{\text{late}}^{\text{cyl}} \ ,
\end{equation}
and this is a renormalization scheme independent quantity, as it can be checked by the fact that the cutoff $\varepsilon$ drops out when equating eqs. \eqref{Searly} and \eqref{Slate}. As emphasized above, the saturation value for the entropy is around twice the value of the defect entropy, i.e. the entropy of the black hole, provided we take the twist operators sufficiently close to the defect.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\begin{tikzpicture}[scale = 0.7]
\draw[thin, -stealth] (-5.5,0) -- (5.5,0) node[anchor = north east]{\small $\tilde x$};
\draw[thin, -stealth] (0,0) -- (0,4.5) node[anchor = north east]{\small $\tilde t$};
\begin{scope}
\clip(-6.2,-2)rectangle(6.2,5.5);
\clip plot[domain=0:1.8, smooth,variable=\t]({2*cosh(\t)}, {1.5*sinh(\t)}) -- ({2*cosh(1.8)}, -2) -- ({-2*cosh(1.8)}, -2) -- ({-2*cosh(1.8)}, {1.5*sinh(1.8)}) -- plot[domain=-1.8:0, smooth,variable=\t]({-2*cosh(\t)}, {-1.5*sinh(\t)}) -- plot[domain=0:180, smooth, variable=\t]({-2*cos(\t)}, {-1.5*sin(\t)});
\fill[ForestGreen,opacity=.2] (-7,-2)rectangle(7,5.5);
\end{scope}
\begin{scope}
\clip(-6.2,-1.5)rectangle(6.2,5.5);
\clip plot[domain=0:1.8, smooth, variable=\t]({2*cosh(\t)}, {1.5*sinh(\t)}) -- (6.2,5.5) -- (-6.2,5.5) -- plot[domain=-1.8:0, smooth, variable=\t]({-2*cosh(\t)}, {-1.5*sinh(\t)}) -- plot[domain=0:180, smooth, variable=\t]({-2*cos(\t)}, {-1.5*sin(\t)});
\fill[Cerulean,opacity=.2] (-7,-1.5)rectangle(7,5.5);
\end{scope}
\draw[dashed] (0,0) -- (6, 4.5);
\draw[dashed] (-6, 4.5) -- (0,0);
\fill[red, opacity = 0.5] ({2*0.77*cosh(0.9)}, {2*0.77*(3/4)*sinh(0.9)}) circle (2pt);
\fill[red, opacity = 0.5] ({-2*0.77*cosh(0.9)}, {2*0.77*(3/4)*sinh(0.9)}) circle (2pt);
\draw[very thick, orange, opacity = 0.5, domain=0:180, smooth,variable=\t] plot ({2*0.77*cosh(0.9)*cos(\t)}, {2*0.77*cosh(0.9)*(3/4)*sin(\t) + 2*0.77*(3/4)*sinh(0.9)});
\draw[thick, blue, opacity = 0.5] plot [smooth, tension=1] coordinates {({2*cosh(0.9)*cos(20)}, {1.5*sinh(0.9) + 1.5*cosh(0.9)*sin(20)}) (2.27, 1.8) ({2*0.77*cosh(0.9)}, {2*0.77*(3/4)*sinh(0.9)})};
\draw[thick, blue, opacity = 0.5] plot [smooth, tension=1] coordinates {({-2*cosh(0.9)*cos(20)}, {1.5*sinh(0.9) + 1.5*cosh(0.9)*sin(20)}) (-2.27, 1.8) ({-2*0.77*cosh(0.9)}, {2*0.77*(3/4)*sinh(0.9)})};
\begin{scope}
\clip(-6.2,-1.5)rectangle(6.2,5.5);
\clip plot[domain=0:1.8, smooth, variable=\t]({2*cosh(\t)}, {1.5*sinh(\t)}) -- (6.2,5.5) -- (-6.2,5.5) -- plot[domain=-1.8:0, smooth, variable=\t]({-2*cosh(\t)}, {-1.5*sinh(\t)}) -- plot[domain=0:180, smooth, variable=\t]({-2*cos(\t)}, {-1.5*sin(\t)});
\fill[NavyBlue,opacity=.2] (-7,-1.5)rectangle(7,5.5);
\end{scope}
\draw[thick, NavyBlue, domain=0:1.8, smooth,variable=\t] plot ({2*cosh(\t)}, {1.5*sinh(\t)});
\draw[thick, NavyBlue, domain=0:1.8, smooth,variable=\t] plot ({-2*cosh(\t)}, {1.5*sinh(\t)});
\draw[thick, NavyBlue, domain=0:180, smooth,variable=\t] plot ({-2*cos(\t)}, {-1.5*sin(\t)});
\fill[red] ({2*1.3*cosh(0.9)}, {2*1.3*(3/4)*sinh(0.9)}) circle (2pt);
\fill[red] ({-2*1.3*cosh(0.9)}, {2*1.3*(3/4)*sinh(0.9)}) circle (2pt);
\draw[very thick, orange, domain=0:180, smooth,variable=\t] plot ({2*1.3*cosh(0.9)*cos(\t)}, {2*1.3*cosh(0.9)*(3/4)*sin(\t) + 2*1.3*(3/4)*sinh(0.9)});
\fill[Gray!50!Black] ({2*cosh(0.9)*cos(20)}, {1.5*sinh(0.9) + 1.5*cosh(0.9)*sin(20)}) circle (2pt);
\fill[Gray!50!Black] ({-2*cosh(0.9)*cos(20)}, {1.5*sinh(0.9) + 1.5*cosh(0.9)*sin(20)}) circle (2pt);
\draw[thick, blue] plot [smooth, tension=1] coordinates { ({2*1.3*cosh(0.9)}, {2*1.3*(3/4)*sinh(0.9)}) (3.25, 2.35) ({2*cosh(0.9)*cos(20)}, {1.5*sinh(0.9) + 1.5*cosh(0.9)*sin(20)}) };
\draw[thick, blue] plot [smooth, tension=1] coordinates { ({-2*1.3*cosh(0.9)}, {2*1.3*(3/4)*sinh(0.9)}) (-3.25, 2.35) ({-2*cosh(0.9)*cos(20)}, {1.5*sinh(0.9) + 1.5*cosh(0.9)*sin(20)}) };
\draw[<->] (14.5, -1.2) -- (14.5, 2.1) node[anchor = west, midway]{$\sim 2 S_{\text{BH}}$};
\draw[] (-4.5, 0) node[anchor=south]{\small Lorentzian};
\draw[] (-4.5, 0) node[anchor=north]{\small Euclidean};
\end{tikzpicture}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.43\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Page_curve.pdf}
\end{subfigure}
\hfill
\caption{{\it Left}: the system in the planar thermofield double state (in Lorentzian signature). The brane (dark blue) becomes an hyperboloid, and the red dots during the dynamics follow hyperbolic trajectories. The orange RT surfaces are dominant at early times, and the entropy increases linearly. The blue RT surfaces are dominant at late times, and the entropy is constant. Such geodesics are the corresponding island saddle in the two dimensional perspective. {\it Right}: the dynamics of the entanglement entropies in time corresponding to the different RT surfaces (orange and blue), in the cylinder coordinates. The total entanglement entropy is always the smaller between (\ref{Searly}) and (\ref{Slate}), thus it increases (approximately) linearly until it saturates at (approximately) twice of the boundary entropy. In this plot we have chosen $c_{\rm I} = 6$, $c_{\rm II} = 12$, $\beta = 2\pi$, $u_0 = 1$, $\varepsilon = 1$ and $\frac{\rho^*_{\rm I}+\rho^*_{\rm II}}{2 G_{(3)}} = 50$.}
\label{Page_curve}
\end{figure}
The same computation can be done in the two dimensional perspective---see the right panel of figure \ref{Island_TFD}. The reasoning is exactly the same as in subsection \ref{subsec:EEvacuum}. In particular, since the early time contribution is only sensitive to the degrees of freedom in the baths, it automatically matches in the two perspectives, so we don't have to compute it again. On the other hand, for the island contribution we first notice that in the static coordinate system at $t = 0$, the twist operators $q^1_{\rm I, II}$ are at position
\begin{equation}
x_{\rm I}(q^1_{\rm I}) = - \, 2 l \tanh \left( \frac{\pi u_0}{\beta} \right) \ , \qquad \qquad x_{\rm II}(q^1_{\rm II}) = 2 l \tanh \left( \frac{\pi u_0}{\beta} \right) \ .
\end{equation}
Then, the generalized entropy we must minimize in the static coordinate system is (for only one pair of twist)
\begin{equation}
S_{\text{gen}}^{\text{static}} = \frac{c_{\rm I}}{6} \log \left[ \frac{\left (\tilde y + 2 l \tanh ( \pi u_0/\beta ) \right)^2}{\tilde y \, \varepsilon} \frac{1}{\cos(\psi_{\rm I})} \right] + \frac{c_{\rm II}}{6} \log \left[ \frac{\left (\tilde y + 2 l \tanh ( \pi u_0/\beta ) \right)^2}{\tilde y \, \varepsilon} \frac{1}{\cos(\psi_{\rm II})} \right]
\end{equation}
In the planar $p$-coordinates, accounting for both pairs and transforming as usual the cutoffs, this becomes
\begin{equation}
S_{\text{gen}}^{\text{plane}} = \frac{c_{\rm I}}{3} \log \left[ \frac{\left (\tilde y + 2 l \tanh ( \pi u_0/\beta ) \right)^2}{\tilde y \, \varepsilon} \frac{\left(e^{\frac{2 \pi}{\beta} u_0} + 1\right)^2}{4 e^{\frac{4 \pi}{\beta} u_0} \cos(\psi_{\rm I})} \right] + \frac{c_{\rm II}}{3} \log \left[ \frac{(\tilde y + 2 l \tanh ( \pi u_0/\beta ))^2}{\tilde y \, \varepsilon} \frac{\left(e^{\frac{2 \pi}{\beta} u_0} + 1\right)^2}{4 \cos(\psi_{\rm II})} \right] \ ,
\end{equation}
and finally on the cylinder
\begin{equation}
S_{\text{gen}}^{\text{cyl}} = \frac{c_{\rm I} + c_{\rm II}}{3} \log \left[ \left( \frac{\beta}{2 \pi l} \right)\frac{\left(\tilde y + 2 l \tanh ( \pi u_0/\beta ) \right)^2}{\tilde y \, \varepsilon}\frac{\left(e^{\frac{2 \pi}{\beta} u_0} + 1\right)^2}{4 e^{\frac{2 \pi}{\beta} u_0} \cos(\psi_{\rm I})} \right] \ .
\label{Sgen_TFD}
\end{equation}
The minimum of (\ref{Sgen_TFD}) is at
\begin{equation}
\tilde y^* = 2 l \tanh \left( \frac{\pi u_0}{\beta} \right) \ ,
\end{equation}
and plugging this into (\ref{Sgen_TFD}), we obtain
\begin{equation}
S_{\text{gen}}^{\text{cyl}} = \frac{c_{\rm I} + c_{\rm II}}{3} \log \left[ \frac{\beta}{\pi \varepsilon} \sinh\left( \frac{2 \pi}{\beta} u_0 \right) \right] + \frac{c_{\rm I}}{3} \log \left[ \frac{2}{\cos(\psi_{\rm I})}\right] + \frac{c_{\rm II}}{3} \log \left[ \frac{2}{\cos(\psi_{\rm II})}\right] \ .
\label{S_isl_2D}
\end{equation}
It is not hard to see that, in the large tension limit, the two-dimensional entropy (\ref{S_isl_2D}) agrees, in the large tension limit, with its three-dimensional counterpart (\ref{Slate}), recalling the relation (\ref{angles}). While this result is only one conformal transformation away from the computation of subsection \ref{subsec:EEvacuum}, it has a different relevance in this context. Indeed, the agreement obtained both at early and late times implies that the Page time matches in the two description. This is a well-defined quantity, free from any subtleties related to the choice of UV cutoffs. In the CFT, it can be expressed in terms of CFT data: the central charges and the defect entropy. It is remarkable that the island formula reproduces it exactly.
|
1,116,691,499,373 | arxiv | \section{Introduction}
Submillimetre (submm) observations probe the Rayleigh-Jeans side of the blackbody emission of dust in galaxies.
In that regime, the dimming of the submm flux density of a galaxy due to its cosmological distance is counterbalanced by the redshifting of its spectral energy distribution (SED).
Consequently, submm observations can trace galaxies with the same infrared luminosities over a broad range of redshifts, and are thus a very powerful tool for studying the cosmic star-formation history \citep{blain_1996a}.
Unfortunately, most current deep submm surveys have spatial resolutions on the order of ten arcseconds.
This large beam size, combined with the steep submm number counts \citep[e.g., ][]{coppin_2006}, leads to a high level of confusion, which ultimately limits the sensitivity of submm observations.
Submm surveys are therefore limited to the brightest sources and submm-selected galaxies\footnote{Note that here we use the term SMGs to refer to sources selected by ground-based facilities in the 850-1200$\,\mu$m window.} \citep[SMGs;][for a review]{smail_1997,barger_1998,hughes_1998,blain_2002} have thus been primarily used for probing the most luminous tail of the high-redshift star-forming galaxy population.\\
\indent{
Substantial efforts have been invested in high-resolution multi-wavelength identifications of SMGs using (sub)mm, radio, mid- or near-infrared observations \citep[e.g.,][]{downes_1999,dannerbauer_2002,ivison_2002,pope_2005, bertoldi_2007, biggs_2011}.
It has been found that SMGs lie at high-redshift, $z\thicksim2$ \citep{hughes_1998,carilli_1999,barger_2000,smail_2000,chapman_2005,pope_2006,wardlow_2011}, and are massive systems \citep[M$_{\ast}\thicksim10^{10}$-$10^{11}\,M_{\odot}$,][]{swinbank_2004,tacconi_2006,tacconi_2008,hainline_2011}.
Extrapolation of their infrared luminosities ($L_{{\rm IR}}$) from submm, radio or mid-infrared observations, have shown that SMGs are extremely luminous \citep[$L_{{\rm IR}}(8-1000\,\mu{\rm m})>10^{12}\,{\rm L_{\odot}}$; e.g.,][]{chapman_2005,pope_2006,pope_2008,kovacs_2006,kovacs_2010}.
Their infrared luminosities are mainly powered by star-formation rather than by active galactic nucleus (AGN) activity \citep{alexander_2005,lutz_2005,valiante_2007,menendez-delmestre_2007,pope_2008,menendez-delmestre_2009,laird_2010}, and correspond to star-formation rates (SFRs) of a few 100s to few 1000s of M$_{\odot}\,$yr$^{-1}$.
The most luminous SMGs are therefore peculiar galaxies because their SFRs are higher than that of typical galaxies of similar mass at similar redshift \citep{daddi_2007a}.
Interferometric observations of their CO molecular gas suggest that the most luminous $z\thicksim2$ SMGs (flux density at 850 $\mu$m, $S_{850}> 5$ mJy) are major mergers in various stages, characterised by compact or very disturbed CO kinematics/morphologies \citep{tacconi_2006,tacconi_2008,engel_2010,bothwell_2010}.
The gas to total baryonic mass fraction of SMGs is comparable to that of typical galaxies at the same redshift \citep[$30-60\%$;][]{tacconi_2008,tacconi_2010}, implying that SMGs have higher star-formation efficiencies \citep{daddi_2008,daddi_2010,genzel_2010}.
Finally, although the comoving volume density of SMGs with $S_{850}> 5$ mJy is low \citep[$\thicksim10^{-5}\,$Mpc$^{-3}$;][]{chapman_2005}, their contribution to the SFR density of the Universe at $z\thicksim2$ is $\thicksim10\%$ \citep{chapman_2005}.
\\ \\}
\indent{
Based on these derived properties, a picture of the nature of the most luminous SMGs has emerged.
SMGs with $S_{850}>5\,$mJy are thought to exhibit very intense short-lived star-formation bursts, triggered by mergers, and to be the high-redshift progenitors of local massive early-type galaxies \citep{lilly_1999,swinbank_2006,daddi_2007,daddi_2007a,tacconi_2008,cimatti_2008}.
In that picture, SMGs belong to a class of galaxies offset from the so-called ``main sequence of star-formation'' which links the SFRs and stellar masses of normal star-forming galaxies (SFGs) over a broad range of redshifts \citep{noeske_2007a,daddi_2007a,elbaz_2007,rodighiero_2010b,rodighiero_2011}.
The existence of this main sequence of star-formation is usually interpreted as evidence that the bulk of the SFG population is forming stars gradually with a long duty cycle, likely sustained by the accretion of cold gas from the intergalactic medium (IGM) and along the cosmic web \citep{dekel_2009,dave_2010}.
Occasional major merger events create extreme systems with intense short-lived starbursts, like SMGs, which are offset from the main sequence of star-formation and which likely evolve into ``red and dead'' galaxies.
\\}
\indent{
The picture of SMGs as a homogeneous population of major mergers has now been weakened by new observational constraints.
The (sub)mm selection method does not correspond to a perfect bolometric selection but rather selects galaxies in the $T_{{\rm dust}}-L_{{\rm IR}}$ parameter space favouring, at low infrared luminosities, galaxies with colder dust temperature \citep[][]{chapman_2005,magnelli_2010}.
Thus, current SMG samples can contain a significant fraction of relatively low luminosity galaxies with cold dust temperature, i.e., galaxies with lower SFRs in the main sequence regime.
The diversity of the SMG population is also supported by high-resolution observations.
Some submm sources are actually composed of two galaxies (with normal ongoing star-formation) which are soon to merge and are observed as one submm source because of the large submm beam \citep[][]{younger_2009, kovacs_2010,wang_2011}.
Finally, constraints from simulations also support this diversity.
While simulations of major mergers are able to reproduce the extreme SFRs of bright SMGs \citep{chakrabarti_2008b,narayanan_2010,hayward_2011}, there might be issues (depending on the exact merger condition needed to create these properties) to match the comoving volume density of SMGs using the high-redshift major merger rates \citep{dave_2010}.
Thus, \citet{dave_2010} have tried to reproduce the properties of SMGs using hydrodynamic simulations in a cosmological context.
Their simulations cannot simultaneously reproduce the measured SFRs and comoving densities of SMGs, because the bulk of their simulated SMGs evolve secularly and exhibit lower SFRs than those inferred from observations (by a factor $\thicksim2-3$).
These results are also consistent with those of semi-analytic models which have great difficulties accounting simultaneously for the measured luminosities/SFRs and number counts of SMGs \citep{baugh_2005,swinbank_2008}.
\\}
\indent{
Due to all these difficulties some questions remain: How homogenous is the SMG population? Have SMG luminosities been overestimated? What triggers their SFRs?
\\}
\indent{
One of the ingredients needed to shed light on the nature of SMGs is direct and robust measurements of their infrared luminosities and SEDs.
Indeed, while SMGs have been studied at all wavelengths, in most cases their infrared luminosities are still based on large extrapolations from radio, submm or mid-infrared observations.
Using 350$\,\mu$m SHARC-2 observations, \citet{kovacs_2006,kovacs_2010} provided more robust estimates of the infrared luminosity of a handful of SMGs.
However, these studies still lacked rest-frame far-infrared observations on both sides of the peak of the SEDs.
Using observations by the 1.8-m Balloon-borne Large Aperture Submillimetre Telescope (BLAST) at 250, 350, 500 $\mu$m, \citet{chapin_2011} studied the far-infrared SED of SMGs at its peak and thus robustly constrained their dust temperatures.
Nevertheless, this study was limited to a relatively small SMG sample (23 sources with spectroscopic redshift estimates) and suffered from observations with large beam size (i.e., $\thicksim19\arcsec$ at 250$\,\mu$m).
Now, thanks to the advent of the \textit{Herschel} Space Observatory \citep{pilbratt_2010}, we can go further in the analysis of the far-infrared SED of SMGs.
Using deep observations at 100 and 160$\,\mu$m by the Photodetector Array Camera and Spectrometer \citep[PACS;][]{poglitsch_2010} onboard the \textit{Herschel} Space Observatory, \citet{magnelli_2010} estimated the infrared luminosities and dust temperatures of a small sample of SMGs (17 sources).
Soon after, \citet{chapman_2010} provided similar estimates using deep observations at 250, 350 and 500~$\mu$m using the Spectral and Photometric Imaging REceiver \citep[SPIRE;][]{griffin_2010} also on \textit{Herschel}.
Both studies revealed the diversity of the SMG population and its bias, with respect to a bolometric selection, towards galaxies with cold dust temperature.
Some galaxies exhibit extreme infrared luminosities of $\thicksim10^{13}\,{\rm L_{\odot}}$ and relatively warmer dust components, while others have much lower luminosities (i.e., a few $10^{12}\,{\rm L_{\odot}}$) and colder dust components.
\\}
\indent{
After more than two years of operation, \textit{Herschel} has now produced deep observations of the most widely studied blank and lensed extragalactic fields.
These combined new PACS and SPIRE data provide for the first time a wide coverage of the far-infrared SEDs of a large sample of SMGs, allowing us to go further in our understanding of their properties.
Our results unambiguously reveal the true infrared luminosity of SMGs and can be used to test the quality of pre-\textit{Herschel} estimates based on monochromatic extrapolations.
These infrared luminosities and dust temperatures also shed light on the diversity of this population and can be used to test the different modes of star formation that could power their luminosities.
Finally using the large wavelength coverage provided by the \textit{Herschel} observations, we can constrain the dust emissivity spectral index, $\beta$, of SMGs.
\\}
\indent{
Here, we use PACS and SPIRE data for a sample of 61 SMGs with known spectroscopic redshifts to provide an insight into the properties and nature of the SMG population.
A comprehensive analysis of the complete SMG samples in the fields studied here will be the subject of other papers.
\\}
\indent{
The paper is structured as follows.
In section \ref{sec: observations} we present the \textit{Herschel} data used in our study.
Section \ref{ref:sample} presents our \textit{Herschel}-detected SMG sample with known spectroscopic redshifts and discusses the selection function of this sample.
Section \ref{sec:SED} is dedicated to SED analysis, describing how we have derived dust temperatures and infrared luminosities using a single-temperature modified blackbody model and a power-law temperature distribution model.
We consistently refer to temperatures as $T_{{\rm dust}}$ if based on a $\beta=1.5$ modified blackbody, and $T_{{\rm c}}$ for the minimum temperature in the power-law distribution model.
Scientific conclusions drawn from these estimates are discussed in Section \ref{sec:discussion} and in section \ref{sec:nature} we discuss the nature of SMGs.
Finally, we summarize our findings in Section \ref{sec:summary}.
Throughout the paper we use a cosmology with $H_{0}=71 \rm{km\,s^{-1}\,Mpc^{-1}}$, $\Omega_{\Lambda}=0.73$ and $\Omega_{\rm M}=0.27$.
A \citet{chabrier_2003} initial mass function (IMF) is always assumed.
}
\section{Observations\label{sec: observations}}
In this study, we used deep PACS 70, 100 and 160$\,\mu$m and SPIRE 250, 350 and 500$\,\mu$m observations provided by the \textit{Herschel} Space Observatory.
PACS observations were taken as part of the PACS Evolutionary Probe \citep[PEP\footnote{http://www.mpe.mpg.de/ir/Research/PEP};][]{lutz_2011} guaranteed time key programme, while the SPIRE observations were taken as part of the \textit{Herschel} Multi-tiered Extragalactic Survey (HerMES\footnote{http://hermes.sussex.ac.uk}; Oliver et al. 2011, MNRAS submitted.).
These two large key programmes are structured as ``wedding cakes" (i.e., with large area wide surveys and smaller pencil beam deep surveys) and include many widely studied blank and lensed extragalactic fields.
Many of these fields being common to both programmes, their combination provides an unique and powerful tool to study the SED of galaxies over a broad range of wavelength.
The PEP and HerMES surveys and data reduction methods are described in \citet{lutz_2011} and Oliver et al. (2011, MNRAS submitted) and references therein, respectively.
Here, we only summarise the properties relevant for our study.\\ \\
\indent{
From the PEP and HerMES programmes, we used the observations of the Great Observatories Origins Deep Survey-North (GOODS-N) and -South (GOODS-S) fields, the Lockman Hole (LH) field, the Cosmological evolution survey (COSMOS) field and the lensed fields Abell 2218, Abell 1835, Abell 2219, Abell 2390, Abell 370, Abell 1689, MS1054, CL0024 and MS045.
Table \ref{tab:field} summarises the main properties of these fields.
\textit{Herschel} flux densities were derived with a point-spread-function-fitting analysis guided using the position of sources detected in deep 24~$\mu$m observations from the Multiband Imaging Photometer \citep[MIPS;][]{rieke_2004} onboard the \textit{Spitzer} Space Observatory.
This method has the advantage that it deals with a large part of the blending issues encountered in dense fields and providing a straightforward association between MIPS, PACS and SPIRE sources.
This MIPS-24$\,\mu$m-guided extraction is also very reliable for the purpose of this study, because here we focus on a subsample of SMGs which already have, for the most part, a MIPS-24$\,\mu$m identification \citep[e.g.,][]{hainline_2009}.
\\}
\indent{
In PEP, prior source extraction was performed using the method presented in \citet{magnelli_2009}, while in HerMES it was performed using the method presented in \citet{roseboom_2010}, both consortia using consistent MIPS-24$\,\mu$m catalogues.
In GOODS-N and -S, we used the GOODS MIPS-24$\,\mu$m catalogue presented in \citet{magnelli_2009,magnelli_2011a} reaching a 3$\sigma$ limit of $20\,\mu$Jy.
In the LH, we used the MIPS-24$\,\mu$m catalogue provided by a \textit{Spitzer} legacy programme (PI: E. Egami), reaching a 3$\sigma$ limit of $30\,\mu$Jy (Egami et al., in prep.).
In COSMOS, we used the latest MIPS-24$\,\mu$m catalogue available, reaching a 3$\sigma$ limit of $45\,\mu$Jy \citep{lefloch_2009}.
In the lensed fields, we used the public MIPS-24$\,\mu$m observations (PI: G. Rieke).
The data processing and catalogue extraction follow the standard MIPS processing with some improvements, this is described in more detail in Valtchanov et al. (in prep.).
In the central region these MIPS-24$\,\mu$m data reaches a 1$\sigma$ limit of $\thicksim20$-$100\,\mu$Jy depending on the cirrus contamination \citep[e.g.,][]{marcillac_2007,bai_2007}.
Using all these MIPS-24$\,\mu$m source positions as prior, we created our PACS and SPIRE catalogues.
The reliability, completeness and contamination of our PACS and SPIRE catalogues were tested via Monte-Carlo simulations (see Lutz et al., \citeyear{lutz_2011} and Oliver et al. 2011, MNRAS submitted for details).
All these properties are given in \citet{berta_2011} and \citet{roseboom_2010}.
Table \ref{tab:field} only summarises the depth of all these catalogs.
\\}
\indent{
We note that the SPIRE prior catalogues reach a 3$\sigma$ limit of $\thicksim10\,$mJy, $\thicksim12\,$mJy and $\thicksim15\,$mJy at 250, 350 and 500$\,\mu$m, respectively, while the formal 3$\sigma$ extragalactic confusion limits at these wavelengths are $14.4\,$mJy, $16.5\,$mJy and $18.3\,$mJy \citep{nguyen_2010}.
Sources detected below these formal 3$\sigma$ confusion limits should thus be treated with caution.
In our \textit{specific} SMG sample (with robust spectroscopic redshift estimates), only a small fraction of galaxies has SPIRE measurements below these formal confusion limits (less than $10$\%).
For these sources, we follow the prescription of \citet{elbaz_2010}, i.e., we take advantage of the higher spatial resolution of the MIPS-24$\,\mu$m observations to flag some galaxies as more ``isolated'' than others and for which SPIRE flux densities can potentially be more robust.
Using this diagnostic, we conclude that in our final SMG sample only three sources (i.e., 5\% of our sample) have SPIRE measurements potentially affected by confusion.
While useful, we note that this diagnostic might not be fully reliable in fields where only shallow MIPS-24$\,\mu$m observations are available.
In our case, only the COSMOS field can significantly be affected by this limitation and in this field none of our SMGs only relies on SPIRE flux densities below the formal 3$\sigma$ confusion limit.
}
\section{Galaxy sample\label{ref:sample}}
In order to infer dust temperatures, infrared luminosities and more generally dust properties, we have to rely on SMGs with robust redshift estimates obtained through secure multi-wavelength identifications.
In this section, we present the construction of such a sample and discuss its selection function.\\
\indent{
In every field the construction of our sample follows three steps.
(i) First, we search in the literature for samples of SMGs, i.e., galaxies selected by ground-based facilities in the 850-1200$\,\mu$m window, with robust multi-wavelength identifications and spectroscopic redshift estimates.
In some of our fields, more than one such SMG sample were available.
For example in GOODS-N, multi-wavelength identification of Submillimetre Common User Bolometer Array \citep[SCUBA;][]{holland_1999} and AzTEC \citep[][]{wilson_2008} sources have been separately published.
In that case, we cross-match these samples using a matching radius of 9$\arcsec$ (i.e., about the half-width at half maximum, HWHM, of the submm observations\footnote{This radius also corresponds to the 3$\sigma_{{\rm pos}}$ positional error of submm observations ($\sigma_{{\rm pos}}\thicksim$FWHM$/(2\times{\rm SN})$), assuming that the bulk of our submm detections has a signal to noise ratio (SN) of $\thicksim3$.}) and keep, for sources presented in more than one sample, the more secure multi-wavelength identifications \citep[i.e., the one with the lowest probability, $P$, of chance association,][]{downes_1986}.
(ii) We complement the far-infrared SED coverage of the SMGs defined in step (i) by searching for their submm/mm counterparts in all blind catalogues available (i.e., catalogues with no multi-wavelength identifications).
In this step we again use a matching radius of 9$\arcsec$.
(iii) Finally, we cross-match the SMG sample defined in step (i) (and which SED coverage has been complemented in step (ii)) with our MIPS-PACS-SPIRE catalogues.
In this step we use the optical, MIPS or radio positions of the SMGs, the MIPS-24$\,\mu$m positions from our MIPS-PACS-SPIRE catalogues and a matching radius of 3$\arcsec$ (i.e., corresponding to the MIPS-24$\,\mu$m HWHM).
\\}
\indent{
Some of our SMGs with robust spectroscopic redshift estimates might correspond to a PACS/SPIRE detection missed by our source extraction method because of a lack of a MIPS-24$\,\mu$m prior.
For that reason, we visually check in our PACS/SPIRE images that the absence of a PACS/SPIRE detection was not due to a lack of a MIPS-24$\,\mu$m prior.
We find no such cases.
}
\subsection{GOODS-N}
In GOODS-N, we use the multi-wavelength identification of SCUBA-850$\,\mu$m sources made by \citet{pope_2006,pope_2008}\footnote{For GN05 we use the spectroscopic redshift revised in \citet{pope_2008}; for GN20 and GN20.2 we use the spectroscopic redshifts revised in \citet{daddi_2009b,daddi_2009a}; and finally for GN07 we use the redshift from \citet{chapman_2005}} using data and redshift informations mainly from \citet{borys_2003} and \citet{chapman_2005}.
We also use the multi-wavelength identification of AzTEC-$1.1\,$mm sources made by \citet{chapin_2009}.
From the Pope et al. sample we only use the SMGs with spectroscopic redshift estimates.\\
\indent{
From the AzTEC sample of Chapin et al., we only keep the two sources with robust spectroscopic redshifts that are not detected by SCUBA (i.e., not already included in the Pope et al. sample).
We complement the Pope et al. sample with AzTEC flux densities when available.
\\}
\indent{
\citet{greve_2008} present the Max Planck Millimeter Bolometer (MAMBO, at $1.2\,$mm) observations of the GOODS-N field.
Some of these MAMBO sources have robust radio identifications in this paper but the corresponding radio positions are not provided.
Consequently we only consider MAMBO counterparts of our SCUBA and AzTEC sources.
\\ \\}
\indent{
This sample of 25 SMGs with robust redshift estimates is cross-matched with our MIPS-PACS-SPIRE multi-wavelength catalogue.
Fourteen SMGs are detected in at least one of the PACS-SPIRE bands.
Among those 14 sources, 10 are detected by both PACS and SPIRE, 3 are only detected with SPIRE and 1 only detected with PACS.
The final sample of 14 SMGs in GOODS-N is presented in Tables \ref{tab:GOODSN} and \ref{tab:GOODSN sup}.
\\}
\indent{
Four SMGs are detected only in the SPIRE-250$\,\mu$m band with flux density below the formal 3$\sigma$ confusion limit, namely, GN5, GN15, GN20 and GN20.2.
For these four sources we compute their ``cleanness'' index as defined in \citet{elbaz_2010}, i.e., sources are defined as ``isolated'' if they have at most one MIPS-24$\,\mu$m neighbour within 20$\arcsec$ with $S_{24}> 50\%$ of the central MIPS-24$\,\mu$m source.
Among those four sources, one is found to be ``isolated'' and hence with robust SPIRE measurements (GN15).
Therefore, results derived for GN5, GN20 and GN20.2 have to be treated with caution.
}
\subsection{GOODS-S}
In GOODS-S we use the multi-wavelength identification of sources observed by the Large APEX Bolometer Camera (LABOCA) ECDFS Submm Survey at 870$\,\mu$m \citep[LESS;][]{weiss_2009b}, as presented by \citet{biggs_2011}.
This sample contains 75 SMGs robustly associated to MIPS, radio and optical counterparts but only 15 are situated in the deep GOODS-S field observed by \textit{Herschel}\footnote{PEP and HerMES have both observed the Extended Chandra Deep Field South. These observations are shallower than those of GOODS-S and are not used in this analysis.}.
Redshift information is taken from zLESS (Danielson et al., in prep.) which provides spectroscopic follow-up of the Wei\ss$\ $et al. sources.\\
\indent{
\citet{scott_2010} presented the AzTEC observations of the GOODS-S field, but no multi-wavelength identifications of these sources are available.
\\ \\}
\indent{
This yielded seven SMGs with robust spectroscopic redshift estimates.
This sample is then cross-matched with our MIPS-PACS-SPIRE multi-wavelength catalogue.
These seven SMGs are all detected in at least one PACS/SPIRE band.
Multi-wavelength properties of these seven SMGs are presented in Tables \ref{tab:ECDFS} and \ref{tab:ECDFS sup}.
}
\subsection{Lockman Hole (LH)}
In LH, we start from the multi-wavelength identifications of 44 SCUBA HAlf Degree Extragalactic Survey \citep[SHADES; ][]{coppin_2006} sources made by \citet{ivison_2007}.
Eleven have a spectroscopic redshift in \citet{chapman_2005}.
These SCUBA sources were associated in \citet{ivison_2007} with their MAMBO counterparts \citep{greve_2004}.
We also used the AzTEC counterparts of these sources provided in \citet{austermann_2010}.\\
\indent{
\citet{chapman_2005} provide redshift information for two additional SCUBA SMGs that are not in the Ivison et al. sample (SMMJ105225.79+571906.4 and SMMJ105238.19+571651.1).
The absence of these two SMGs in this sample could be explained by their low S/N submm detections.
We decided to include those two galaxies in our sample of SMGs with robust redshift estimates.
\\}
\indent{
Recently, \citet{coppin_2010} derived the spectroscopic redshifts of six SMGs using the PAH signatures observed in the \textit{Spitzer}-IRS spectra.
This study added one SHADES source (LOCK850.15) and four AzTEC sources (AzTEC.01, AzTEC.05, AzTEC.10 and AzTEC.62) to our SMG sample.
This study also revised the redshift of LOCK850.01 from $z=2.148$ to $z=3.38$.
We adopt this new redshift because previous estimates were based on the spectroscopic follow-up of a galaxy $\thicksim3\arcsec$ away from the radio counterpart of this submm source.
\\ \\}
\indent{
The resulting sample of 18 SMGs with robust redshift estimates was cross-matched with our MIPS-PACS-SPIRE multi-wavelength catalogue.
Fifteen are detected in at least one of the PACS/SPIRE bands.
Tables \ref{tab:LH} and \ref{tab:LH sup} present the multi-wavelength properties of this subsample.
}
\subsection{COSMOS}
In the COSMOS field we use the multi-wavelength identification of LABOCA and MAMBO sources carried out by Aravena et al. (in prep.) and \citet{bertoldi_2007}, respectively.
From the Aravena et al. sample we only keep sources with radio identifications.
This limits our sample to 46 SMGs out of the 163 LABOCA sources.
In the Bertoldi et al. sample there are 27 MAMBO sources with robust radio identifications.
Among those sources, nine are already included in the Aravena et al. sample.
For those sources we keep the radio identification obtained by Aravena et al. because it is based on the latest version of the deep COSMOS radio catalogue.\\
\indent{
We cross-match this sample of 64 SMGs with the AzTEC catalogue of \citet{scott_2008}, which has no multi-wavelength identifications.
AzTEC sources with no LABOCA or MAMBO counterparts but with Submillimeter Array (SMA) follow-up \citep{younger_2007,younger_2009} are included in our sample (i.e., 5 sources).
\\}
\indent{
Capak et al. (in prep.) provide redshift follow-up for some of these 69 SMGs with robust multi-wavelength identifications.
So far this spectroscopic follow-up programme has obtained redshift estimates for 15 of these SMGs.
\\ \\}
\indent{
These 15 SMGs with robust redshift estimates are cross-matched with our MIPS-PACS-SPIRE multi-wavelength catalogue yielding 11 SMGs detected in at least one of the PACS/SPIRE bands.
Tables \ref{tab:COSMOS} and \ref{tab:COSMOS sup} present the multi-wavelength properties of this subsample.
}
\subsection{Cluster fields}
\indent{
We gather from the literature a sample of well-known lensed SMGs with \textit{both} spectroscopic redshifts and lensing magnification estimates.
In the A2218 field, our SMG sample is assembled from \citet{kneib_2004} and \citet{knudsen_2006, knudsen_2008} and contains six lensed sources.
Among these six lensed sources, three correspond to the same lensed galaxy \citep[SMMJ16359+6612; ][]{kneib_2004}.
In A1835, submm observations are taken from \citet{ivison_2000}.
The redshift of SMMJ14011+0252 is also taken from \citet{ivison_2000}, while the redshift estimate of SMMJ14009+0252 is from \citet{weiss_2009}.
In MS0451 and A2219, submm observations are taken from \citet{chapman_2002}.
Each field contains only one lensed SMG with both spectroscopic redshifts and lensing magnification estimates, namely, SMMJ16403+4644 and SMMJ04554+0301 \citep[][respectively]{rigby_2008,borys_2004}.
In MS1054, we use submm observations and redshift information provided in \citet[][SMMJ10570-0336]{knudsen_2008}.
For A1689, we use submm observations and lensing magnification estimates from \citet[][SMMJ13115-1208]{knudsen_2008} while redshift informations are from \citet{rigby_2008}.
Finally in CL0024, A2390 and A370 submm observations are taken from \citet{smail_2002}.
The redshift of SMMJ00266+1708 comes from \citet{valiante_2007}, the redshift of SMMJ02399-0136 comes from \citet{ivison_1998} \citep[see also][]{lutz_2005} and the redshift of SMMJ02399-0134 comes from \citet{smail_2002}.
For SMMJ21536+1742 we use \citet{barger_1999} \citep[K3 counterpart;][]{frayer_2004}.
\\}
\indent{
All but one of these sixteen lensed SMGs have been detected in at least one of the PACS/SPIRE bands.
Because these galaxies are magnified, their mid-to-far infrared fluxes are de-magnified prior to further analysis using magnification factors from the literature.
Tables \ref{tab:CLUSTERS} and \ref{tab:CLUSTERS sup} present our lensed SMG sample.
\\ \\}
\indent{
The infrared luminosities of our lensed SMGs strongly depend on their magnification factors.
These factors are estimated from complex lens models, constrained by the many lensed features seen in these clusters.
We adopt a characteristic error of $20\%$ on their luminosities to account for uncertainties in the lens models.
}
\subsection{SMGs with multiple counterparts}
Our SMG sample contains 62 sources detected by PACS/SPIRE and with secure spectroscopic redshift estimates.
Among these 62 SMGs, eleven have multiple optical/radio/MIPS counterparts.
Six of them (GN04, GN07, GN19, GN39, AzTECJ$100008$+$024008$ and MAMBO11) are treated as one single system because they are assumed to be interacting galaxies.
The optical counterparts of GN19 and GN39 are spectroscopically confirmed to lie at the same redshift \citep{chapman_2005,swinbank_2004} and the optical counterparts of GN04 and GN07 exhibit IRAC photometry consistent with both optical sources being at the same redshift.
The optical counterpart of MAMBO11 without any spectroscopic redshift estimate (MAMBO11W) has a photometric redshift supporting the assumption of an interacting system \citep{bertoldi_2007}.
AzTECJ$100008$+$024008$ has two SMA counterparts within the submm beam with consistent redshifts \citep{younger_2009}.
Because these multiple counterparts are thought to be part of an interacting system, to derive the dust properties of these galaxies we sum the mid-infrared, far-infrared and radio flux densities of their optical/radio/MIPS counterparts.\\
\indent{
For four SMGs we have a spectroscopic follow up for only one of their multiple MIPS/radio counterparts, LOCK850.03, LOCK850.04, LOCK850.15 and LESS10, namely.
Thus we cannot assess whether these galaxies are interacting systems.
We assume that only the source with a redshift estimate significantly contributes to the submm and far-infrared flux-densities.
This assumption is supported by the fact that the MIPS-24$\,\mu$m and radio flux densities of these sources agree with the infrared luminosities derived from their far-infrared/submm flux densities.
The inclusion or exclusion of these four sources would not change the conclusions of our study.
\\}
\indent{
LOCK850.41 has two robust radio counterparts coinciding with two MIPS-24$\,\mu$m sources.
Spectroscopic follow-up of these counterparts shows that they do not correspond to an interacting system, one galaxy is situated at $z=0.689$ \citep{menendez-delmestre_2009} and the other at $z=0.974$ \citep{coppin_2010}.
IRS observations show that while the low redshift galaxy exhibits strong PAH signatures, the galaxy situated at $z=0.974$ has a continuum-dominated mid-infrared spectrum with no visible PAH features, consistent with an AGN classification.
This suggests that the high-redshift galaxy has very low ongoing star-formation, incompatible with bright far-infrared and submm emission.
However, because this assumption is still highly uncertain, we decide to remove this source from our final sample.
}
\subsection{Stellar mass estimates\label{subsec:stellar masses}}
Due to the significant obscuration at rest-frame optical wavelengths, and to the possible presence of a rest-frame near-IR continuum excess in numerous SMGs \citep{hainline_2011}, the determination of the stellar mass of SMGs is still highly debated.
For example, different assumptions about the star-formation history or about the contribution of an AGN to the rest-frame near-IR continuum excess could lead to systematic variations in the median stellar mass estimates of SMGs of more than a factor 2 \citep[see][]{hainline_2011,michalowski_2010,michalowski_2011}.
Due to all these different methods and assumptions, it was impossible to find stellar masses homogeneously derived for all our SMGs in the literature.
Therefore, we decided to infer the stellar masses of our SMGs using a single method.
We would like to stress that resolving the problem of the stellar mass estimates of SMGs is beyond the scope of this paper.
The absolute values of our estimates might not be fully reliable, but the fact that we are using a homogeneous method and assumptions over our sample should provide a good tool to study relative variations in stellar mass.
Lensed SMGs are not considered in that study because of the difficulty to obtain coherent optical-to-near infrared data for these galaxies, making any stellar mass estimates very uncertain.\\
\indent{
Optical-to-near-infrared photometry was obtained using the radio or optical positions of our SMGs.
In GOODS-N and COSMOS, we used the multi-wavelength catalogue built by the PEP consortium and presented in \citet{berta_2010,berta_2011}.
In GOODS-S, we used the MUSIC catalogue \citep{santini_2009} and the optical-to-near-infrared photometry of SMGs presented in \citet{wardlow_2011}.
In the LH field, we used the optical-to-near-infrared photometry of SMGs presented in \citet{dye_2008} and \citet{coppin_2010}.
Stellar masses were then calculated by fitting the multi-wavelength photometry to \citet{bruzual_2003} templates through a $\chi^2$ minimization, using the method described in \citet{fontana_2004} and updated as in \citet{santini_2009}.
We looked at all fits and rejected those sources with problematic fits.
Among the 46 SMGs considered in this study (all our blank field SMGs), 39 SMGs have good optical-to-near-infrared SED fits.
The stellar masses of these 39 SMGs are provided in Table \ref{tab:temperature}.
In the LH field, we find that our stellar mass estimates are in perfect agreement with results from \citet{hainline_2011}.
The agreement between our findings is encouraging because CO observations and dynamic mass arguments \citep{engel_2010} favour these lower stellar mass estimates, more consistent with the findings of \citet{hainline_2011} than those of \citet{michalowski_2010}.
The median log$(M_{\ast})$ of $10.86$ for our sample is also fully consistent with log$(M_{\ast})$$\,\thicksim\,$$11.0$ for SMGs estimated from the SMG halo mass of \citet{hickox_2012}, using the conversion to stellar mass by \citet{moster_2010}.
}
\subsection{Final sample and selection biases\label{subsec: bias}}
Our final SMG sample contains 61 sources detected by PACS/SPIRE and with secure spectroscopic redshift estimates.
Because this sample requires MIPS detections, PACS or SPIRE detections and robust redshift estimates, it is affected by several selection biases.
Previous studies have already discussed the biases introduced by (sub)mm observations and/or SPIRE-like (i.e., BLAST) observations \citep[e.g.,][]{casey_2009a,chapin_2011,symeonidis_2011} but none of them have examined our peculiar selection function.
In this section we list all our selection biases and try to estimate how representative our sample is of the SMG population and more generally of the high-redshift star-forming galaxy population.
Here, we only focus on the blank field SMG population because lensed SMGs are affected by more complex selection function depending on their positions with respect to the foreground lenses.\\ \\
\indent{
Because (sub)mm and far-infrared surveys observe the thermal emission of dust they are limited, at a given redshift, in the range of infrared luminosities and dust temperatures probed.
In order to quantify these selection biases we studied the $T_{{\rm dust}}-L_{{\rm IR}}$ parameter space reachable with our far-infrared, submm and radio observations.
For that purpose we took a model describing the far-infrared SED of SMGs (a power-law temperature distribution parameterized with $T_{{\rm c}}$, i.e., the temperature of the coldest dust component of the model, see section \ref{subsec:distri-T}) and estimated for each point of the $T_{{\rm c}}-L_{{\rm IR}}$ parameter space its detectability by the PACS (100$\,\mu$m or 160$\,\mu$m but mainly by the 160$\,\mu$m band), SPIRE (250~$\mu$m, 350~$\mu$m or 500~$\mu$m but mainly by the 250$\,\mu$m band) and SCUBA (850~$\mu$m) instruments.
Then, in order to compare these estimates with the local $T_{{\rm dust}}-L_{{\rm IR}}$ relation derived by \citet{chapman_2003} using a single temperature optically thin modified blackbody model, we simply converted $T_{{\rm c}}$ into $T_{{\rm dust}}$ with $T_{{\rm c}}=0.6\times T_{{\rm dust}}+3\,$K (see section \ref{subsubsec:distri-T PEP} and Fig. \ref{fig: lir kovacs single}).
This study cannot be directly performed using a single temperature optically thin modified blackbody function because that model cannot reproduce the PACS 100$\,\mu$m measurements sometimes dominated by warmer or transiently heated dust components (see Section \ref{subsec:single-T}).
For the radio detectability we used the local far-infrared/radio correlation\footnote{In Section \ref{subsec:lir lir} we find that the parameterization of the far-infrared/radio correlation, $\langle q\rangle$, is slightly lower in our SMG sample than in the local universe, $\langle q\rangle=2.0$ versus $\langle q\rangle=2.34$. However, here, we prefer to use the local value of $\langle q\rangle$ because our sample cannot be used to fully constrain this parameter. This is a conservative approach because using a lower value of $\langle q\rangle$ one would decrease the selection bias introduced by radio observations, i.e., radio observations could reach lower infrared luminosities at a given redshift.} \citep{helou_1988,yun_2001} and for the MIPS-24$\,\mu$m detectability we used the \citet{chary_2001} templates\footnote{In Section \ref{subsec:lir lir} we find that the infrared luminosities estimated from the MIPS-24$\,\mu$m fluxes densities and the \citet{chary_2001} library are overestimated. Therefore, in this exercise, the use of the \citet{chary_2001} library is a conservative approach because at high-redshift the MIPS-24$\,\mu$m observations could reach even lower infrared luminosities.}.
In this exercise we used the typical 3$\sigma$ limits of GOODS-N observations, i.e., $20\,\mu$Jy, $3\,$mJy, $6\,$mJy, $10\,$mJy, 12$\,$mJy, 12$\,$mJy, 3$\,$mJy and 15$\,\mu$Jy at 24$\,\mu$m, 100$\,\mu$m, 160$\,\mu$m, 250$\,\mu$m, 350$\,\mu$m, 500$\,\mu$m, 850$\,\mu$m and $1.4\,$GHz, respectively.
The left panel of Fig. \ref{fig: biais} shows the selection limits observed in the GOODS-N field.
To obtain the selection functions of the other fields, one would simply shift the lines of Fig.$\,$\ref{fig: biais} towards higher infrared luminosities according to the depth of the observations with respect to the GOODS-N field (see Table \ref{tab:field ancillary}).
\\}
\begin{figure*}
\centering
\includegraphics[width=9.cm]{18312fg1a.eps}
\includegraphics[width=9.cm]{18312fg1b.eps}
\caption{\label{fig: biais}\small{
(\textit{Left}) Selection limits introduced in the $T_{{\rm dust}}-L_{{\rm IR}}$ parameter space by single-wavelength detection techniques.
Continuous, dashed, dotted-dashed, triple-dotted-dashed and dotted lines show the lower limits on $L_{{\rm IR}}$ introduced by the submm, PACS, SPIRE, MIPS-24$\,\mu$m and radio observations, respectively, at $z=1.5$ (thin blue lines) and at $z=2.5$ (thick red lines).
The parameter space reachable by a given single-wavelength detection technique corresponds to the area situated to the right of the lines.
As an example, the red arrows show the parameter space probed at $z\thicksim2.5$ by our GOODS-N SMG sample.
The shaded area shows the local $T_{{\rm dust}}-L_{{\rm IR}}$ relation found by \citet{chapman_2003}, linearly extrapolated to $10^{13}\,{\rm L_{\odot}}$.
The striped area presents results for SMGs extrapolated by \citet{chapman_2005} from radio and submm data.
(\textit{Right}) The hatched histogram shows the redshift distribution of our PACS/SPIRE detected SMG sample.
The empty histogram shows the redshift distribution of its parent sample, i.e., SMGs with robust redshift estimates obtained through secure multi-wavelength identifications.
}}
\end{figure*}
\indent{
The first selection bias introduced in our SMG sample come from the (sub)mm detections.
This selection bias is almost redshift independent, but selects, at a given infrared luminosity, only galaxies with cooler dust.
The bias decreases at high infrared luminosities where submm observations probe a large range in dust temperature.
In fields where (sub)mm observations are shallower than in GOODS-N\footnote{One can convert the MAMBO or AzTEC flux density limits into its corresponding SCUBA-850$\,\mu$m flux density limit using the Raleigh-Jeans approximation.}, these selection functions shift towards higher infrared luminosities.
Nevertheless, shallow (sub)mm observations would still probe, at high infrared luminosities, a large dynamic range in dust temperature.
Therefore, assuming that the local $T_{{\rm dust}}-L_{{\rm IR}}$ relation holds at high redshift \citep[e.g.,][]{hwang_2010, chapin_2011,marsden_2011}, and extrapolating it towards higher infrared luminosities, we can assume that \textit{at high luminosities} ($L_{{\rm IR}}\gtrsim10^{12.5}\,{\rm L_{\odot}}$), \textit{SMGs are a representative sample of the underlying star-forming galaxy population}.
\\}
\indent{
The second selection bias affecting our sample comes from the necessity of having robust redshift estimates.
This requirement translates into accurate positions and multi-wavelength identifications mainly obtained via radio observations (among the 69 SMGs with redshift estimates in our blank fields, 59 have been identified using radio observations while only 5 have been identified using MIPS-24$\,\mu$m observations and 5 using SMA observations).
Radio observations probe the synchrotron emission of galaxies and suffer from positive \textit{k}-corrections, independent of the dust temperature.
This biases our sample towards higher infrared luminosities as the redshift increases (see dotted lines in the left panel of Fig.$\,$\ref{fig: biais}).
The redshift estimates of these radio sources, obtained mainly through optical spectroscopy, introduce additional selection biases.
For example, just for feasibility of the optical spectroscopy in a reasonable amount of time and/or success of detection, SMGs with spectroscopic redshifts might be biased towards optically-bright SMGs \citep[see e.g.,][]{chapman_2005} and are also likely to have a higher incidence of strong emission lines than typical SMGs.
In addition, spectroscopic follow up of SMGs might also miss some objects at $1.2<z<1.8$ (namely the ``redshift desert''), due to the lack of strong emission lines in the rest-frame wavelength range observed by ground-based spectroscopic instruments \citep[see e.g.,][]{chapman_2005}.
All these selection biases are very difficult to quantify because they depend on the follow-up strategy used.
Here, using a Kolmogorov-Smirnov (KS) analysis, we simply verify that the radio and submm flux density distribution of SMGs with spectroscopic redshift is consistent with that of its parent sample, i.e., SMGs with radio counterparts.
This suggests that the spectroscopic follow-up of radio-identified SMGs does not introduce strong biases towards any particular infrared luminosity or dust temperature.
On the contrary, we find that the distribution of submm to radio flux ratio of the SMGs with spectroscopic redshift is different than that of its parents sample (only 30$\%$ of chance for being drawn from the same distribution).
Because the submm to radio flux ratio has been used as a redshift indicator by many early works \citep[e.g.,][]{carilli_1999,chapman_2005}, we conclude that spectroscopic follow-up of SMGs might be slightly biased towards low redshift galaxies.
However, in terms of luminosities and dust temperatures, we assume that \textit{at high infrared luminosities ($L_{{\rm IR}}\gtrsim10^{12.5}\,{\rm L_{\odot}}$), SMGs with robust spectroscopic redshift estimates are still a good representation of the underlying SMG population and therefore of the entire high luminosity star-forming galaxy population}.
At low infrared luminosities, however, SMGs with redshift estimates represent a subsample of SMGs biased towards lower redshift galaxies, essentially because of the need for a radio-based identification.
\\}
\indent{
Our final SMG sample is also affected by the MIPS-PACS-SPIRE detection requirement.
The MIPS-24$\,\mu$m requirement should not significantly influence our sample because it corresponds, up to $z\thicksim3\,$-$\,4$ and in all our fields, to selection limits several times lower in term of infrared luminosities than those introduced by radio observations (see triple-dotted-dashed line in the left panel of Fig. \ref{fig: biais}).
On the contrary, the PACS/SPIRE requirement affect our sample and is redshift dependent.
PACS observations, which suffer from positive \textit{k}-corrections, are slightly biased towards galaxies with hotter dust while SPIRE observations are biased towards cooler dust.
The SPIRE selection bias is also redshift dependent because SPIRE detections are mainly obtained in the 250$\,\mu$m band which suffers from positive \textit{k}-corrections as it reaches the peak of the far-infrared SED of galaxies at $z$$\,\thicksim\,$$1.5$.
In GOODS-N, the selection bias due to the PACS/SPIRE observations is almost equivalent to that introduced by the combination of submm and radio observations.
In other fields, the PACS/SPIRE requirement is even less constraining because the SPIRE observations are as deep as in GOODS-N while radio and (sub)mm observations are shallower.
This is reflected by the fact that the PACS/SPIRE detection rate of SMGs with robust spectroscopic redshift estimates is very high, and much higher than that observed by \citet{dannerbauer_2010} for the entire SMG population, i.e., 73\% versus 39\%.
\\ \\}
\indent{
In summary, our final SMG sample should provide \textit{a good representation of the high infrared luminosity ($L_{{\rm IR}}\gtrsim10^{12.5}\,{\rm L_{\odot}}$) SMG population and more generally, of the entire high infrared luminosity galaxy population}.
On the other hand, as we go to lower infrared luminosities ($L_{{\rm IR}}\lesssim10^{12.5}\,{\rm L_{\odot}}$), our final SMG sample is biased towards low redshift galaxies with cold dust.
Most of these biases are not inherent to our PACS/SPIRE SMG subsample but are intrinsic to any SMG sample requiring robust spectroscopic follow-up aided by secure radio/MIPS multi-wavelength identifications.
\\ \\}
\indent{
The right panel of Fig.$\,$\ref{fig: biais} presents the redshift distribution of our PACS/SPIRE-detected SMG sample.
This redshift distribution is consistent with that of the entire SMG sample with robust redshift estimates.
The median redshift of our PACS-SPIRE detected SMG sample is $z=2.4$ and is consistent with the median redshift of the entire SMG population, i.e., $z\thicksim2.3$ \citep[][]{chapman_2005}.
}
\section{SED analysis\label{sec:SED}}
In this section we describe the models used to infer the dust properties of the SMGs.
Scientific conclusions drawn from these properties are discussed in Section \ref{sec:discussion}.\\
\subsection{Single modified blackbody model\label{subsec:single-T}}
In order to infer the dust temperatures and infrared luminosities of our galaxies we fitted their far-infrared and (sub)mm flux densities with a single temperature modified blackbody model.
This model provides a very simple description of the far-infrared SED of a galaxy, because it assumes that the emission-weighted sum of all the dust components could be reasonably well fitted by only one blackbody function at a given temperature.
Despite its simplicity and the fact that it is known that this model cannot fully reproduce the Wien side of the far-infrared SED of galaxies \citep[e.g.,][]{blain_2003,magnelli_2010,hwang_2010}, we adopted this model for two reasons:
(i) studies of the \textit{Infrared Astronomical Satellite} (\textit{IRAS}) galaxies have demonstrated that it still provides an accurate diagnostic of the typical heating conditions in the interstellar medium of big grains in thermal equilibrium \citep{desert_1990}; and (ii) it allows direct comparison with most of the pre-\textit{Herschel} studies.
The far-infrared flux densities of our galaxies were thus fitted, in the optically thin approximation, with a single modified blackbody function : \\
\begin{equation}
S_{\nu}\propto\frac{\nu^{3+\beta}}{{\rm exp}(h\nu/kT_{{\rm dust}})-1},
\end{equation}
where $S_{\nu}$ is the flux density, $\beta$ is the dust emissivity spectral index and $T_{{\rm dust}}$ is the dust temperature.
This single temperature modified blackbody model cannot reproduce the full rest-frame 8$-$to$-$1000$\,\mu$m SED over which the total infrared luminosities ($L_{{\rm IR}}$[8$-$1000\,$\mu$m]) are classically defined.
A significant amount of energy emitted at relatively short rest-frame wavelengths (i.e., where the backbody function drop sharply) would thus be missed by a simple integration of the blackbody function over the rest-frame 8$-$to$-$1000$\,\mu$m wavelengths.
Therefore, the total infrared luminosities of our galaxies were inferred using the far-infrared luminosity definition ($L_{{\rm FIR}}[$40$-$120$\,\mu$m]) given by Helou et al. (\citeyear{helou_1988}) and a bolometric-correction term.
This bolometric-correction is equal to $1.91$ \citep[][$L_{{\rm IR}}=1.91\times L_{{\rm FIR}}$]{dale_2001} but introduces uncertainties in our estimates because it varies ($\pm30\%$) with the intrinsic shape of the galaxy SED \citep{dale_2001}.
\\
\subsubsection{Constraints on $\beta$\label{subsubsec:single}}
The exact value of the dust emissivity spectral index $\beta$ is still debated.
Laboratory experiments as well as observations in diverse Galactic environments suggest a broad range of values for $\beta$ \citep[][and references therein]{dunne_2001, dupac_2003}.
The value of $\beta$ seems to depend on the chemical composition, the temperature and the size of the dust grains.
Despite its variability on Galactic scales, extragalactic constraints on $\beta$ converge to a narrow range of values ($1.5<\beta<2.0$).
In particular, \citet{dunne_2001} found a constant dust emissivity spectral index $\beta$ of $\thicksim2$ using a sample of galaxies probing a broad range of infrared luminosities.
Based on this latter conclusion, we assume that $\beta$ could be considered as universal over the SMG population. \\
\indent{
Assuming $\beta$ to be universal, we can constrain its value globally using our sample of 61 SMGs.
To perform this global fit we gridded the $\beta$ parameter space $[0.1$$-$$3.0]$ with steps of $0.05$.
Then, for each value of $\beta$, we performed a $\chi^{2}$ minimization for each galaxy, varying $T_{{\rm dust}}$ and the blackbody normalization.
The $\chi^{2}$ value at a given $\beta$ is then defined as the sum of the $\chi^{2}$ value of all galaxies (i.e., $\chi^{2}_{{\rm \beta_{i}}}=\sum\,\chi^{2}_{{\rm gal}}$).
Our $\chi^{2}$ minimization was done using a standard Levenberg-Marquardt method.
\\}
\indent{
We apply this global fit to three different wavelength coverages.
First, we fit the full wavelength coverage provided by the \textit{Herschel} and (sub)mm observations (i.e., from the PACS~70$\,\mu$m to the (sub)mm wavelength);
second, we exclude from the fits the PACS~70 and 100$\,\mu$m data points;
and third, we exclude from the fits the PACS~70, 100 and 160$\,\mu$m data points.
For these three different wavelength coverages the best fit is obtained at $\beta=0.6\pm0.2$, $\ $$\beta=1.2\pm0.2$ and $\beta=1.7\pm0.3$, respectively (using the $95\%$ confidence level, i.e., $\Delta\chi^2$$=$$\,\chi^2_{{\rm min}}+3.8$; note that these errors stand for the mean values, rather than for the standard deviation of the population).
Fits of the full wavelength coverage systematically lead to significantly larger $\chi^{2}_{{\rm gal}}$ values than for the other cases (i.e., $\chi^{2}_{{\rm gal}}$$\thicksim$$18$ for N$_{{\rm dof}}$$\thicksim$$4$).
On the contrary, fits excluding the PACS~70 and 100$\,\mu$m data points or the PACS~70, 100$\,\mu$m and 160 $\,\mu$m data points lead in both cases to low $\chi^{2}_{{\rm gal}}$ values, i.e., with $\chi^{2}_{{\rm gal}}\thicksim6$ for N$_{{\rm dof}}\thicksim3$ and $\chi^{2}_{{\rm gal}}\thicksim4$ for N$_{{\rm dof}}\thicksim2$, respectively.
\\}
\indent{
The large $\chi^{2}_{{\rm gal}}$ values observed when we try to reproduce the full wavelength coverage provided by the \textit{Herschel} and (sub)mm observations perfectly illustrate the limits of a single temperature model.
Such a simple model cannot fully describe the Wien side of the far-infrared SED of galaxies \citep[e.g.,][]{blain_2003,magnelli_2010,hwang_2010}.
The PACS 70 and 100~$\mu$m flux densities are likely dominated by a warmer or transiently heated dust component.
Consequently, the PACS 70 and 100$\,\mu$m data points have to be excluded from the fitting procedure.
A precise description of the far-infrared SEDs of galaxies requires a more complex model which includes multiple dust components (see Section \ref{subsec:distri-T}).
\\}
\indent{
The increase of $\beta$ when excluding short-wavelength measurements from the fits agrees with the conclusions of \citet{shetty_2009} studying galactic dense cores$\,$: constraints on $\beta$ are highly sensitive to the wavelength coverage used in the fits as well as to the noise properties of the observations.
Although interesting, our constraints on $\beta$ should thus be used with caution.
}
\subsubsection{Fitting the full SMG sample\label{subsubsec:single full}}
In the following, we decide to fix the dust emissivity spectral index $\beta$ to its standard value of 1.5.
This choice is driven by two reasons.
First, this value is fully compatible with our findings (i.e., $1.2<\beta<1.7$) and second, it allows direct comparison with all pre-\textit{Herschel} studies.
We also decide to exclude from our fits the PACS~70 and 100$\,\mu$m data points because they are likely dominated by a warmer or transiently heated dust component.
The PACS~160$\,\mu$m data points are kept because their exclusion does not significantly improve our fits while their inclusion allows better constraints of the dust temperature estimates (see Fig. \ref{fig: PACS SPIRE}).\\
\indent{
Figure \ref{fig: fit single T} presents results of this fitting procedure to each individual SED, while Table \ref{tab:temperature} gives the inferred dust temperatures and infrared luminosities.
Uncertainties are estimated using the distribution of $T_{{\rm dust}}$ and $L_{\rm IR}$ values that correspond to models with $\chi^{2}< \chi^{2}_{\rm min}+1$.
\\}
\begin{figure*}
\centering
\includegraphics[width=7.5cm]{18312fg2a.eps}
\includegraphics[width=7.5cm]{18312fg2b.eps}
\caption{\label{fig: PACS SPIRE}\small{
(\textit{Left}) Dust temperatures inferred from the combination of PACS only (or SPIRE only) together with submm observations, compared with the reference values inferred using PACS, SPIRE and submm observations.
These comparisons are for a single dust temperature modified blackbody model.
Blue squares represent SMGs situated in blank fields while green diamonds represent lensed-SMGs.
(\textit{Right}) Same comparison but for the inferred infrared luminosities.
The dust temperatures and infrared luminosities of galaxies can be reasonably inferred from their PACS+submm or their SPIRE+submm observations alone using a single temperature modified blackbody model.
}}
\end{figure*}
\indent{
We observe in Fig. \ref{fig: fit single T} that a single dust temperature model provides a reasonable fit to the 160$\,\mu$m$-$to$-$mm data points (with $\chi^{2}\thicksim7$ for N$_{{\rm dof}}\thicksim3$).
Figure \ref{fig: fit single T} also shows the limits of this model at short wavelengths and why we excluded from our fits the PACS 70$\,\mu$m and 100$\,\mu$m data points.
The modified blackbody functions drop quickly at short wavelengths and cannot reproduce the PACS 70$\,\mu$m and 100$\,\mu$m data points of most of our SMGs.
\\ \\}
\indent{
For some of the SMGs we do not have both PACS and SPIRE detections.
For those galaxies, we can expect the inferred dust temperatures and infrared luminosities to be more uncertain, and potentially biased because PACS and SPIRE measurements probe different parts of the blackbody emission of the dust (Wien and Rayleigh-Jeans side, respectively).
To assess this issue, we compared the dust temperatures and infrared luminosities inferred using the combination of PACS and submm observations, or SPIRE and submm observations, to the reference values inferred using the continuous wavelength coverage provided by the combination of PACS, SPIRE and submm observations.
This analysis is based on 50 SMGs detected by both PACS and SPIRE.
Results are shown in Fig. \ref{fig: PACS SPIRE}.
\\}
\indent{
For most of our sources the dust temperatures and infrared luminosities estimated from the combination of PACS (or SPIRE) and submm observations are in good agreement with our reference values, i.e., $\sigma[T_{{\rm dust}}^{{\rm Ref}}-T_{{\rm dust}}^{{\rm PACS}}]=2.5\,$K ($\,\sigma[T_{{\rm dust}}^{{\rm Ref}}-T_{{\rm dust}}^{{\rm SPIRE}}]=3.1\,$K$\,$) and $\sigma[L_{{\rm IR}}^{{\rm Ref}}/L_{{\rm IR}}^{{\rm PACS}}]=0.08\,$dex ($\,\sigma[L_{{\rm IR}}^{{\rm Ref}}/L_{{\rm IR}}^{{\rm SPIRE}}]=0.1\,$dex).
However, the dust temperatures inferred using SPIRE and submm observations are slightly underestimated at high dust temperature ($T_{{\rm dust}}>35\,$K).
At these temperatures, the SPIRE observations start to be affected by the shift of the far-infrared SED peak towards rest-frame wavelengths barely probed by the SPIRE 250$\,\mu$m passband.
This effect slightly biases these estimates.
\\ \\}
\indent{
There are only a few sources with large uncertainties (i.e., $\Delta T$$>$$8\,$K or $\Delta {\rm log}(L_{{\rm IR}})$$>$$0.3\,$, COSLA127R1I, AzTECJ100019$+$0232, SMMJ105238$+$5716, GN26, and SMMJ163541$+$6611).
Examining the SED fits of these galaxies, we find that all of them exhibit large $\chi^{2}$ (i.e., $\gtrsim15$ for N$_{{\rm dof}}$$\thicksim$$3$) when combining their PACS, SPIRE and submm observations.
These large $\chi^2$ values seem to be explained by one or two inconsistent flux densities in their SEDs.
These inconsistent data points do not correspond to a specific rest-frame wavelength but randomly affect the PACS, SPIRE or the ground based data points.
Thus they are unlikely due to strong emission lines (like the [C II] emission line, Smail et al. \citeyear{smail_2011}) which are not included in our simple modified blackbody model.
We conclude that the observed discrepancies are not directly due to our simple modified blackbody model but to some outlying flux densities, as expected when working close to the non-Gaussian confusion limit which can create significant outliers.
\\ \\}
\indent{
Finally, one can expect the accuracy of the estimates inferred from the combination of PACS (or SPIRE) and submm observations to vary as function of the redshift: high(low) redshift galaxies with PACS (SPIRE) only measurements could have inaccurate dust temperature estimates because their far-infrared SED peak shifts outside the PACS (SPIRE) bands.
However, we find no significant evolution of $\Delta T$ or $\Delta {\rm log}(L_{{\rm IR}})$ with the redshift.
At low redshift, the shift of the far-infrared SED peak towards shorter wavelengths is counterbalanced by the fact that at these redshifts, galaxies exhibit relatively low infrared luminosities and dust temperatures, shifting back their far-infrared SED peak towards the SPIRE bands.
Likewise, at high redshift, SMGs exhibit higher infrared luminosities and dust temperatures, shifting back their far-infrared SED peak towards the PACS bands.
\\ \\}
\indent{
We conclude that the dust temperatures and infrared luminosities of galaxies can be reasonably inferred from their PACS+submm or their SPIRE+submm observations alone using a single temperature modified blackbody model.
This may be important for survey regions covered at sufficient depth with one of these instruments only.
}
\subsection{Power-law temperature distribution\label{subsec:distri-T}}
Although a single-temperature model gives a good description of the far-infrared peak and Rayleigh-Jeans side of the SED of SMGs, it fails to reproduce short wavelength observations (e.g., the PACS 70 and 100$\,\mu$m passbands) which are affected by warmer or transiently heated dust components.
Consequently, the total infrared luminosity of SMGs (i.e., $L_{{\rm IR}}[8$$-$$1000\,\mu$m]) has to be extrapolated from $L_{{\rm FIR}}[40$$-$$120\,\mu$m] and short wavelength observations have to be excluded from the fit.
In order to reproduce these short wavelength observations we need to use a more complex model, taking into account warmer dust components.\\
\indent{
To describe the dust emission of galaxies, \citet{dale_2001} and \citet{dale_2002} assumed that they are the superposition of regions heated by different radiation fields.
In that framework, they assumed that the dust mass submitted to a radiation field $U$ is given by $dM_{{\rm dust}}/dU\propto U^{-\alpha}$.
Then using simple assumptions they showed that $\alpha\sim2.5$ is appropriate for a diffuse medium while $\alpha\sim1$ describes a dense medium.
Following the same idea, \citet{kovacs_2010} described the SEDs of galaxies by a power-law distribution of temperature components ($dM_{{\rm dust}}/dT\propto T^{-\gamma}$) with a low-temperature cutoff $T_{{\rm c}}$.
Under the assumption that the dust is only heated by radiation (and not by non-radiative processes like shocks), the main parameters of this model and that of Dale \& Helou are linked by $\gamma\approx4+\alpha+\beta_{{\rm eff}} $ (where $\beta_{{\rm eff}}$ is the dust emissivity spectral index observed near the peak of the far-infrared emission).
This model can accurately describe the mid-to-far-infrared SEDs of local starbursts \citep{kovacs_2010} and is convenient for our purposes as it is parameterized in dust temperature rather than radiation field.
Consequently, while other models could have been used \citep[e.g.,][]{dale_2002,draine_2007}, we adopted this prescription as a natural extension of our single dust temperature model.
\\ \\}
\indent{
The parameterization of this power-law temperature distribution model is fully described in \citet{kovacs_2010}, and briefly summarized here.
In particular we do not give the analytical derivation of the infrared luminosity because here we derive this quantity using a simple discrete numerical integration.
\\}
\indent{
Expressed in observable parameter space, the emission from a single modified blackbody emission, not in the optically thin approximation, is given by
\begin{equation}
S_{\nu_{{\rm obs}}}(T_{{\rm obs}})=m\,d\Omega\,(1-e^{-\tau})\,B_{\nu_{{\rm obs}}}(T_{{\rm obs}}),
\end{equation}
where $B_{\nu}$ is the Planck function, $T_{{\rm obs}}$ is the observed-frame temperature (i.e., $T_{{\rm obs}}=T/(1+z)$), $\tau$ is the optical depth, $d\Omega$ is the solid angle subtended by the galaxy and $m$ is a magnification correction for lensed galaxies ($=1$ in all other cases).
In the model proposed by \citet{kovacs_2010}, the optical depth is expressed as a function of the dust mass ($M_{{\rm dust}}$) and the projected source diameter ($R$), together with the usual power-law frequency dependence for the emissivity of dust,
\begin{equation}
\tau({\nu_{r}})=\kappa_{0}\left(\frac{\nu_{{\rm r}}}{\nu_{0}}\right)^\beta\frac{M_{{\rm dust}}}{\pi R^2},
\end{equation}
where $\tau$ is expressed in the rest-frame ($\nu_{{\rm r}}=\nu_{{\rm obs}}(1+z)$) and $\kappa_{0}$ is the photon cross-section to mass ratio of particles at the reference frequency $\nu_{0}$.
To allow direct comparison with \citet{kovacs_2010}, we adopted $\kappa_{850}=0.15\,{\rm m^2\,kg^{-1}}$ at $\nu_{0}={c/850\,\mu{\rm m}}$ \citep{dunne_2003}, even though the exact value of this parameter is still under active discussion \citep[e.g., ][]{hildebrand_1983,krugel_1990,sodroski_1997,james_2002}.
Using this formalism a power-law temperature distribution model can be expressed as,
\begin{equation}
S_{\nu_{{\rm obs}}}^{{\rm tot}}(T_{{\rm c}})=(\gamma-1)T^{\gamma-1}_{{\rm c}}\int_{T_{{\rm c}}}^{\infty}S_{\nu_{{\rm obs}}}(T_{{\rm obs}})T^{-\gamma}dT,
\end{equation}
where $T_{{\rm c}}$ is the low-temperature cutoff of the model.
}
\begin{figure*}
\centering
\includegraphics[width=8.cm]{18312fg3a.eps}
\includegraphics[width=8.cm]{18312fg3b.eps}
\includegraphics[width=8.cm]{18312fg3c.eps}
\caption{\label{fig: fit beta gamma A}\small{
Constraints on $\gamma$, $\beta$ and $R$ obtained from a $\chi^2$ minimization analysis using 20 SMGs with PACS, SPIRE, submm and mm observations.
These constraints correspond to our power-law temperature distribution model.
Isocontours show the 99\%, 95\% and 68\% confidence level.
}}
\end{figure*}
\subsubsection{Constraints on $\beta$, $\gamma$ and $R$\label{subsubsec:distri-T GOODSN}}
The power-law temperature distribution model has five free parameters, $T_{{\rm c}}$, $M_{{\rm dust}}$, $\beta$, $\gamma$ and $R$.
It can only be constrained from observations that probe the full far-infrared SEDs of galaxies, i.e., probing the Wien-side, the peak and Rayleigh-Jeans-side of these SEDs.
Such broad spectral coverage can only be obtained through the combination of PACS, SPIRE, submm and millimeter observations and thus can only be applied to a small fraction of our SMG sample.
Therefore here we investigate the possibility that some of those parameters are universal over the full SMG population. \\ \\
\indent{
As already mentioned in Section \ref{subsubsec:single}, considering that the exact value of the dust emissivity spectral index $\beta$ is still debated, one can assume this value to be universal over the SMG population.
\\}
\indent{
\citet{kovacs_2010} found little variation of $\gamma$ in the local star-forming galaxy population.
Based on this finding they assumed a constant value of $\gamma$ for high-redshift luminous starbursts and obtained a good fit to their SEDs.
Therefore, in the following, we consider $\gamma$ as universal over the SMG population.
\\}
\indent{
Finally, we considered the projected radius of the emitting region, $R$, as universal over the SMG population.
This consideration is perhaps questionable because in high-redshift star-forming galaxies, the diameter of the region forming stars spans a wide range of values from 1 to 10 kpc \citep{chapman_2004,muxlow_2005,tacconi_2006,tacconi_2008,biggs_2008,casey_2009a,iono_2009,lehnert_2009,carilli_2010,swinbank_2010,tacconi_2010,younger_2010}.
However, in the power-law temperature distribution model the variation of $R$ does not strongly affect the estimates of $L_{{\rm IR}}$ ($<5\%$) but only affects the physical interpretation that one can draw from the absolute value of $T_{{\rm c}}$: smaller values of $R$ imply higher values for $T_{{\rm c}}$.
In any case, the study of the relative variation of $T_{{\rm c}}$ from one galaxy to the other is not qualitatively affected by the exact value of $R$.
\\}
\indent{
Assuming these three parameters to be universal, we constrained them globally using a subsample of 19 SMGs detected in all PACS and SPIRE passbands and with at least one detection longward of 1 mm (needed to obtain good constraints on the dust emissivity $\beta$).
To perform this global fit we first gridded the $\beta$, $\gamma$ and $R$ parameter space using ranges of $[1.0$$-$$2.5]$, $[6.5$$-$$9.0]$ and $[0.5\,{\rm kpc}$$-$$9.0\,{\rm kpc}]$ and steps of $0.05$, $0.1$ and $0.25$, respectively; then, for each node of this grid, we performed a $\chi^{2}$ minimization for each galaxy, varying $T_{{\rm c}}$ and $M_{{\rm dust}}$.
The $\chi^{2}$ value of the node is then defined as the sum of the $\chi^{2}$ value of all galaxies (i.e., $\chi^{2}_{{\rm node}}=\sum\,\chi^{2}_{{\rm gal}}$).
Our $\chi^{2}$ minimization was done using a standard Levenberg-Marquardt method.
Figure \ref{fig: fit beta gamma A} presents the confidence levels obtained for $\beta$, $\gamma$ and $R$.
Confidence levels are computed using $\Delta\chi^2$$=$$\,\chi^2_{{\rm min}}+[2.3, 6.0, 11.6]$ for the 68\%, 95\% and 99\% confidence level, respectively.
The best fit is obtained at $\beta=2.0\pm0.2$, $\gamma=7.3\pm0.3$ and $R=3\pm1\,$kpc (using the $95\%$ confidence level; note that these errors stand for the mean values, rather than for the standard deviation of the population), and corresponds to $\chi^{2}_{{\rm gal}}$$\thicksim$$8$ for N$_{{\rm dof}}$$\thicksim$$5$.
These $\chi^{2}_{{\rm gal}}$ values confirm that our model provides a good description of the far-infrared SEDs of SMGs even if three parameters are considered common to all galaxies.
\\ \\}
\indent{
In Fig. \ref{fig: fit beta gamma A}, we observe only small degeneracies between $\beta$, $\gamma$ and $R$, e.g., an increase of $\beta$ could be compensated, in terms of $\chi^{2}$ minimization, by an increase of $R$.
The wide wavelength coverage provided by our data allows us to reasonably constrain our model.
Constraints on $\beta$, $\gamma$ and $R$ are also in line with the physical expectations and with independent estimates.
A dust emissivity spectral index $\beta$ of $2.0\pm0.2$ is in agreement with conclusions based on local LIRG/ULIRG \citep{dunne_2001,chakrabarti_2008}.
The dust emissivity spectral index found using our power-law temperature distribution model is different than that used in our single temperature model, i.e., $\beta=2.0$ instead of 1.5.
However, this difference is expected, because, as already noticed in \citet{dunne_2001}, single temperature models require lower values of $\beta$ than multi-component models.
\\}
\indent{
Constraints on $\gamma$ found in our study are in very good agreement with estimates made by \citet{kovacs_2010} on local starbursts, i.e., $\gamma=7.22\pm0.09$.
However, using a sample of high-redshift starbursts, \citet{kovacs_2010} found a lower value of $\gamma$, i.e., $\gamma=6.71\pm0.11$.
This discrepancy might arise from the fact that to infer this value, \citet{kovacs_2010} could only rely on uncertain MIPS-24$\,\mu$m continuum estimates, extrapolated from broadband observations contaminated by PAH emission.
\\}
\indent{
We find an average emission diameter of $6\pm2$ kpc (i.e., $R=3$ kpc), which is consistent with estimates from various studies using various high-resolution observations that have inferred diameters of order 1$-$10 kpc for SMGs \citep{chapman_2004,muxlow_2005,tacconi_2006,tacconi_2008,biggs_2008,casey_2009a,iono_2009,lehnert_2009,carilli_2010,swinbank_2010,tacconi_2010,younger_2010}.
\citet{kovacs_2010} found an emission diameter of $\thicksim$$\,2\,$kpc for their high-redshift star-forming galaxies.
As already mentioned, this discrepancy might arise from the fact that \citet{kovacs_2010} relied on extrapolated MIPS-24$\,\mu$m continuum measurements to make these estimates.
We would like to stress that while our constraints on $R$ are in line with previous estimates, its exact value should still be treated with caution.
Indeed, robust constraints on the size of the emitting region would require the use of a complex radiative transfer model, taking into account the geometry of the star-forming regions.
For example, \citet{chakrabarti_2008}, using a self-consistent radiative transfer model and assuming a spherical geometry, found $R_{{\rm c}}$$\thicksim$$10\,$kpc.
The agreement, within a factor 2$-$3, between our findings is encouraging in view of the approximations of our simple model.
\\ \\}
\indent{
Based on these results, we conclude that $\beta$, $\gamma$ and $R$ can be considered as universal for these 19 SMGs.
Nevertheless, how representative are these 19 SMGs of the full 61 SMG sample$\,$?
Using a KS analysis, we find that the redshift distribution of these two samples are fully compatible but that their infrared luminosity distributions are slightly different (only $40\%$ of chance of being drawn from the same distribution).
The sample of 19 SMGs exhibits slightly higher infrared luminosities than the full SMG sample, a median $L_{{\rm IR}}$ of $6\times10^{12}\,{\rm L_{\odot}}$ versus $4\times10^{12}\,{\rm L_{\odot}}$.
These 19 SMGs are therefore not a perfect subsample of our full SMG sample.
However, because these two samples are also far from being incompatible, we consider that the inferred values of $\beta$, $\gamma$ and $R$ are universal for our 61 SMGs.
This assumption is further supported by the fact that these parameters provide a good description of the far-infrared SED of the rest of our SMG sample (see Section \ref{subsubsec:distri-T PEP}).
}
\begin{figure*}
\centering
\includegraphics[width=7.cm]{18312fg4a.eps}
\includegraphics[width=7.cm]{18312fg4b.eps}
\caption{\label{fig: kovacs pacs spire}\small{
(\textit{Left}) Dust temperatures inferred from the combination of PACS only (or SPIRE only) together with submm observations, compared with the reference values inferred using PACS, SPIRE and submm observations.
These comparisons are for our power-law temperature distribution model.
Symbols are the same as in Fig. \ref{fig: PACS SPIRE}.
(\textit{Right}) Same comparison but for the inferred infrared luminosities.
The dust temperatures and infrared luminosities of galaxies can be reasonably inferred from their PACS+submm or their SPIRE+submm observations alone using a temperature distribution model.
}}
\end{figure*}
\subsubsection{Fitting the full SMG sample\label{subsubsec:distri-T PEP}}
\begin{figure*}
\centering
\includegraphics[width=9.cm]{18312fg5a.eps}
\includegraphics[width=9.cm]{18312fg5b.eps}
\caption{\label{fig: lir kovacs single}\small{
(\textit{Left}) Comparison of the infrared luminosities inferred using a power-law temperature distribution model with those inferred using a single dust temperature model.
Symbols are the same as in Fig. \ref{fig: PACS SPIRE}.
The black solid line shows the one-to-one relation.
The green dotted-dashed line shows the bias introduced in our single dust temperature model by the use of a constant bolometric-correction term of 1.91 to convert $L_{{\rm IR}}[40-120\,\mu{\rm m}]$ into $L_{{\rm IR}}[8-1000\,\mu{\rm m}]$.
To compute this line we measure $L_{{\rm IR}}[40-120\,\mu{\rm m}]$ and $L_{{\rm IR}}[8-1000\,\mu{\rm m}]$ on a power-law temperature template library normalized to reproduce the $T_{{\rm c}}-L_{{\rm IR}}$ correlation (see the red dashed line in the right panel of Figure \ref{fig: T vs LIR}).
We then plot on the \textit{x}-axis $1.91\times L_{{\rm IR}}[40-120\,\mu{\rm m}]$ and on the \textit{y}-axis $L_{{\rm IR}}[8-1000\,\mu{\rm m}]$.
(\textit{Right}) Comparison of the dust temperatures inferred using a power-law temperature distribution model ($T_{{\rm c}}$) with those inferred using a single dust temperature model ($T_{{\rm dust}}$).
The red dashed line show a linear fit to the $T_{{\rm c}}$-$T_{{\rm dust}}$ relation, $T_{{\rm c}}=0.6\times T_{{\rm dust}}+3\,$K.
Symbols are the same as in the left panel.
Note that $T_{{\rm c}}$ indicates the temperature of the coldest dust component of the multi-component model while $T_{{\rm dust}}$ measures an effective dust temperature.
}}
\end{figure*}
We now fit the full SMG sample (including their PACS 70$\,\mu$m and PACS 100$\,\mu$m detections) leaving $T_{{\rm c}}$ and $M_{{\rm dust}}$ as the only free parameters of the model.
Results of these fits are shown in Fig. \ref{fig: fit single T}.
Table \ref{tab:temperature} summarizes the results inferred from these fits.
Uncertainties are estimated using the distribution of $T_{{\rm c}}$, $M_{{\rm dust}}$ and $L_{\rm IR}$ values that correspond to models with $\chi^{2}< \chi^{2}_{\rm min}+1$.
For most of our SMGs this model provides (even with fixed $\beta$, $\gamma$ and $R$ parameters) a very good fit to our data points (i.e., $\chi^{2}_{{\rm gal}}$$\thicksim$$7$ for N$_{{\rm qof}}$$\thicksim$$3$).
Almost all the highest $\chi^{2}_{{\rm gal}}$ values ($>25$) correspond to the lensed-SMGs with relatively low infrared luminosities and high dust temperatures.
This might suggest that for these galaxies $\beta$, $\gamma$ and $R$ are slightly different.
Consequently, the infrared luminosities and dust temperatures inferred for these galaxies using our power-law temperature distribution model might be biased.
These possible biases are discussed later on in this section.
\\ \\
\indent{
As for the single temperature model, we would like to verify that fits of SMGs with only PACS and submm observations, or only SPIRE and submm observations, are not biased compared to fits of SMGs with PACS, SPIRE and submm observations.
Therefore, we compare the dust temperatures and infrared luminosities that one would infer using only the PACS (or SPIRE) and submm observations and our power-law temperature distribution model (with $\beta=2.0$, $\gamma=7.3$ and $R=3\,$kpc), to that inferred using the combination of PACS, SPIRE and submm observations.
This analysis is based on 50 SMGs detected by both PACS and SPIRE and results are shown in Fig. \ref{fig: kovacs pacs spire}.
We find that the dust temperatures and infrared luminosities inferred using the combination of PACS (or SPIRE) and submm observations are in very good agreement with those inferred using the combination of PACS, SPIRE and submm observation:
$\sigma[T_{{\rm dust}}^{{\rm Ref}}-T_{{\rm dust}}^{{\rm PACS}}]=1.2\,$K ($\,\sigma[T_{{\rm c}}^{{\rm Ref}}-T_{{\rm c}}^{{\rm SPIRE}}]=2.3\,$K$\,$) and $\sigma[L_{{\rm IR}}^{{\rm Ref}}/L_{{\rm IR}}^{{\rm PACS}}]=0.10\,$dex ($\,\sigma[L_{{\rm IR}}^{{\rm Ref}}/L_{{\rm IR}}^{{\rm SPIRE}}]=0.09\,$dex)
This agreement is even better than that obtained in the case of our single temperature model.
Consequently, estimates made on SMGs with only PACS or only SPIRE observations can be used with confidence.
\\ \\}
\indent{
One of the main advantages of this power-law temperature distribution model is that it provides robust estimates of the total infrared luminosity ($L_{{\rm IR}}[8-1000\,\mu{\rm m}]$) of galaxies.
The left panel of Fig. \ref{fig: lir kovacs single} shows the difference between the infrared luminosity extrapolated from a single temperature modified blackbody model and that inferred from our power-law temperature distribution model.
We find a very tight correlation between those two estimates, ${\rm log}(L_{{\rm IR}}^{{\rm Multi-T}})=0.84(\pm0.02)\times{\rm log}(L_{{\rm IR}}^{{\rm single-T}})+2.0(\pm0.2)$.
However, we observe that the single dust temperature model systematically overestimates the luminosity of galaxies at high infrared luminosities and underestimates the luminosity of galaxies at low infrared luminosities.
These discrepancies can be explained by the fact that in our single temperature model we were using a constant bolometric-correction term to convert $L_{{\rm IR}}[40$$-$$120\,\mu{\rm m}]$ into $L_{{\rm IR}}[8$$-$$1000\,\mu{\rm m}]$, while its value changes with dust temperature (as with infrared luminosity, because there is a broad $T_{{\rm c}}-L_{{\rm IR}}$ correlation; see the right panel of Fig. \ref{fig: T vs LIR}).
For example, at high infrared luminosity (i.e., $L_{{\rm IR}}\gtrsim3\times10^{12}\,{\rm L_{\odot}}$) all galaxies have $T_{{\rm c}}>25\,$K.
At these temperatures, the bolometric-correction term is, in our power-law temperature distribution model, of the order of $\thicksim1.5$.
The difference between our constant bolometric-correction term of 1.91 and this one, fully explains the observed discrepancies.
This bias is illustrated by the \textit{green dotted-dashed} line in the left panel of Fig. \ref{fig: lir kovacs single}.
\\}
\indent{
The right panel of Fig. \ref{fig: lir kovacs single} shows the comparison between the dust temperature inferred using a single-temperature modified blackbody and that inferred using our power-law temperature distribution model.
There is a tight correlation between these estimates and a very small dispersion.
However, we can observe significant differences between these two estimates.
$T_{{\rm c}}$ indicates the temperature of the coldest dust component while $T_{{\rm dust}}$ measures an effective dust temperature, therefore it is not surprising that $T_{{\rm dust}}$ yields values warmer than $T_{{\rm c}}$.
Some lensed-SMGs significantly deviate from this $T_{{\rm dust}}$-$T_{{\rm c}}$ relation.
These galaxies correspond to the ones with the largest $\chi^{2}_{{\rm gal}}$ values, suggesting that in these systems $\beta$, $\gamma$ and $R$ might be slightly different.
These dust temperatures are systematically shifted towards lower values while the corresponding infrared luminosities are not affected (see the left panel of Fig. \ref{fig: lir kovacs single}).
Consequently, when studying the $T_{{\rm c}}-L_{{\rm IR}}$ plane, one has to keep in mind these slight shifts, or refer to the $T_{{\rm dust}}-L_{{\rm IR}}$ plane which is not affected by this effect.
\\ \\}
\indent{
In the rest of the paper we use the infrared luminosities derived using the power-law temperature distribution model, unless stated otherwise.
}
\section{Results and Discussion\label{sec:discussion}}
\subsection{The Infrared luminosity of SMGs\label{subsec:lir lir}}
The nature of SMGs has been greatly debated for more than a decade and in particular the reliability of their measured extreme SFRs.
Indeed, while simulations of major mergers are able to reproduce such extreme SFRs, simulations in a cosmological context have had great difficulties accounting for the estimated SFRs and number counts \citep{baugh_2005,dave_2010}.
Thus, the question remains: are the infrared luminosities of SMGs overestimated$\,$?
Thanks to \textit{Herschel} observations we can now assess this question by measuring the true infrared luminosity of SMGs, studying their evolution as function of the redshift and testing the quality of pre-\textit{Herschel} estimates based on monochromatic extrapolations. \\ \\
\begin{figure*}
\centering
\includegraphics[width=16.cm]{18312fg6.eps}
\caption{\label{fig: lir s850 z}\small{
(\textit{Left}) Infrared luminosities as function of the submm flux density.
Blue squares represent SMGs situated in blank fields while green diamonds represent lensed-SMGs.
OFRGs from \citet{magnelli_2010} are presented with left red arrows.
The solid and dashed lines show the linear fit to the $L_{{\rm IR}}-S_{850}$ relation and the 1$\sigma$ envelope ($L_{{\rm IR}}[{\rm L_{\odot}}]=10^{11.33\pm0.29}\times S_{850}^{1.59}\,$[mJy]).
Dotted lines show the $L_{{\rm IR}}-S_{850}$ relation followed by single modified ($\beta=1.5$) blackbody functions at 20, 35 and 50$\,$K.
(\textit{Right}) Infrared luminosities as function of the redshift.
The symbols are same as in the left panel but OFRGs are represented by red filled circles.
Blue dotted, red dashed and green dotted-dashed lines present the lower limit of the parameter space reachable using our deep radio (i.e., 20 $\mu$Jy), PACS 160 $\mu$m (i.e., 3 mJy) and MIPS-24$\,\mu$m (i.e., 20$\,\mu$Jy) observations of the GOODS-N field, respectively.
Note that in these figures galaxies with high $\chi^{2}$ value do not lie in a particular region of these plots but are rather randomly distributed.
\vspace{1.cm}
}}
\end{figure*}
\indent{
Figure \ref{fig: lir s850 z} shows the infrared luminosities of SMGs as a function of their 850 $\mu$m flux densities\footnote{for sources with no 850 $\mu$m observations we used extrapolations assuming $\beta=2.0$, i.e., $S_{850}^{{\rm extrapolated}}=S_{\lambda_{{\rm submm}}}\times\,\left(\lambda_{\rm submm}/850\right)^{4}$ where $\lambda_{{\rm submm}}$ is the (sub)mm wavelength at which the SMG has been detected.} and their redshifts.
Our results unambiguously confirm the remarkably large infrared luminosities of SMGs.
The vast majority exhibit infrared luminosity larger than $10^{12}\,{\rm L_{\odot}}$, and some even have $L_{{\rm IR}}>10^{13}\,{\rm L_{\odot}}$.
The first, second and third quartiles of our sample are $10^{12.0}\,{\rm L_{\odot}}$, $10^{12.6}\,{\rm L_{\odot}}$ and $10^{12.8}\,{\rm L_{\odot}}$, respectively.
These infrared luminosities correspond to SFRs of $100\,$M$_{\odot}\,$yr$^{-1}$, $400\,$M$_{\odot}\,$yr$^{-1}$ and $630\,$M$_{\odot}\,$yr$^{-1}$, respectively (using SFR~$[{\rm M}_{\odot}~ {\rm yr}^{-1}] = 1\times 10^{-10} L_{\rm IR}~[{\rm L}_{\odot}]$, assuming a Chabrier IMF and no significant AGN contribution to the far-infrared luminosity).
The existence of this large sample of star-forming galaxies with extreme infrared luminosities illustrates the strong evolution with redshift of the infrared galaxy population: in the local Universe such luminous infrared galaxies are very rare but their comoving space density increases by a factor $\thicksim$$400$ between $z\,$$\thicksim$$\,0$ and $z\,$$\thicksim$$\,2$ \citep{magnelli_2011a,chapman_2005}.
Consequently, the characterization of the mechanisms triggering their starbursts becomes crucial in order to obtain a good census of the star formation history of the Universe.
\\ \\}
\indent{
We observe a weak trend between $S_{850}$ and $L_{{\rm IR}}$ (left panel of Fig. \ref{fig: lir s850 z}).
However, this correlation is likely driven by selection effects.
Indeed, since submm observations at low luminosity are biased towards cold dust temperatures (see section \ref{subsec: bias}), they miss the bulk of the star-forming galaxy population at low and intermediate infrared luminosities.
This missing population should have warm dust components and therefore relatively faint 850 $\mu$m flux densities \citep[see also][]{chapman_2004,casey_2009a,magnelli_2010,chapman_2010,magdis_2010}.
This hypothesis is strengthened by the position of some of the lensed SMGs, which give us a glimpse into the bulk of the population of galaxies with low infrared luminosities.
The underlying $S_{850}-L_{{\rm IR}}$ relation cannot be probed using a submm-selected sample.
\\}
\indent{
Submm observations have the great advantage of being subject to negative \textit{k}-correction which makes an galaxy equally detectable in the submm over a very wide range of redshift.
Therefore, one can expect the redshift distribution of submm galaxies to be relatively uniform if there were no strong evolution of the underlying galaxy population.
Instead, we observe a strong correlation between the infrared luminosities of galaxies and their redshifts (right panel of Fig. \ref{fig: lir s850 z}).
This trend can be explained by an evolution of the underlying galaxy population and by selection effects.
The increase with redshift of the number of very luminous SMGs is due to the evolution of the infrared galaxy population and a volume effect: at high redshift, the comoving space density of luminous infrared galaxies is larger \citep{magnelli_2011a,chapman_2005} and the comoving volume probed by our survey increases.
On the other hand, the lack of low luminosity galaxies at high redshift is quite surprising.
Indeed, simply due to a volume effect, we would expect to see many more low luminosity galaxies at high redshift than at low redshift.
We argue that this trend can be easily understood as a pure selection effect.
Indeed, as illustrated in the right panel of Fig. \ref{fig: lir s850 z}, the depth of the deepest radio observations used to provide robust multi-wavelength counterparts creates the low boundary in the $L_{{\rm IR}}-z$ plane.
\citet{pope_2006} and \citet{banerji_2011} argue instead that this lack of low-luminosity galaxies at high redshift could be due to an evolution of their SEDs.
To be missed by submm observations, those galaxies should exhibit hotter dust temperatures than low redshift galaxies of the same luminosity.
This seems to be incompatible with the modest evolution with redshift of the $T_{{\rm dust}}-L_{{\rm IR}}$ relation observed up to $z\thicksim2$ \citep{hwang_2010,chapin_2011,marsden_2011}.
\\ \\}
\indent{
Using our reference infrared luminosities (i.e., inferred from the power-law temperature distribution model) we can now test the quality of pre-\textit{Herschel} estimates.
One of the most common pre-\textit{Herschel} monochromatic extrapolations was based on the MIPS-24$\,\mu$m flux densities and the \citet[][hereafter CE01]{chary_2001} SED library.
We applied these extrapolations to our SMG sample and compared those estimates (hereafter $L_{{\rm IR}}^{24}$) to our reference infrared luminosities (left panel of Fig. \ref{fig: LIR vs LIR}).
\\}
\indent{
Our results reveal that the use of the MIPS-24$\,\mu$m emission and of the CE01 SED library yields inaccurate estimates of the infrared luminosities, characterized by a large scatter ($\sigma[{\rm log}(L_{{\rm IR}}^{24}/L_{{\rm IR}}^{{\rm ref}})]$$\thicksim$$0.47\,$dex) and a systematic overestimate for the most luminous galaxies.
These results are in line with conclusions of \citet{hainline_2009} studying SMGs and of \citet{papovich_2007}, \citet{murphy_2009}, \citet{nordon_2010,nordon_2012} and \citet{elbaz_2010,elbaz_2011} studying bolometrically selected high-redshift galaxies.
Our study also agrees with the fact that the overestimate of the infrared luminosity by the MIPS-24$\,\mu$m flux density and the CE01 SED library occurs at $z>1.5$, i.e., when the MIPS-24$\,\mu$m passband starts probing rest-frame wavelengths dominated by PAH emission \citep{nordon_2010,nordon_2012,elbaz_2010,elbaz_2011}.
Indeed, SMGs with infrared luminosities below $10^{12}\,{\rm L_{\odot}}$ are all at $z<1.5$ and exhibit better agreement between $L_{{\rm IR}}^{24}$ and $L_{{\rm IR}}^{{\rm ref}}$.
\\}
\begin{figure*}
\centering
\includegraphics[width=16.cm]{18312fg7.eps}
\caption{\label{fig: LIR vs LIR}\small{
Infrared luminosities for submm sources detected at 24 $\mu$m and 1.4 GHz.
The \textit{x}-axis shows the infrared luminosities extrapolated from the MIPS-24$\,\mu$m (left) or the radio (right) flux density, using the CE01 library or the FIR/radio correlation (with $q=2.34$), respectively.
The \textit{y}-axis shows the ratio of the infrared luminosities extrapolated from the MIPS-24$\,\mu$m or radio flux density and the reference infrared luminosities inferred from our power-law temperature distribution model.
The symbols are same as in Fig \ref{fig: lir s850 z}.
\vspace{1.cm}
}}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=16.cm]{18312fg8.eps}
\caption{\label{fig: Q vs LIR}\small{
Evolution of $\langle q\rangle$ as function of the infrared luminosity (left panel) and the redshift (right panel).
On the left panel, solid and dashed lines show the local relation and its 1$\,\sigma$ dispersion as found by \citet{yun_2001}.
On the right panel, solid and dashed lines show the redshift evolution of $\langle q\rangle\propto(1+z)^{-0.15\pm0.03}$ from its local value as inferred in \citet{ivison_2010a}.
The symbols are same as in Fig \ref{fig: lir s850 z}.
Note that here, we did not attempt to correct for any incompleteness, e.g., using a Kaplan-Meier estimator, and biases introduced in our sample.
So these results should be taken with caution because they only apply to our specific selection function, i.e., SMG with spectroscopic redshift estimates mainly obtained through robust radio identifications.
}}
\end{figure*}
\indent{
All these studies show that the SEDs of star-forming galaxies strongly evolve with redshift.
This evolution might be interpreted as a modification of the physical conditions prevailing in their star-forming regions.
\citet{elbaz_2011} and \citet{nordon_2012} found that the SEDs of these high-redshift galaxies with extreme star-formation could be described using local SEDs of less luminous galaxies \citep[see also][]{papovich_2007,magnelli_2011a}.
This SED evolution is thus likely due to an increase of the PAH emission strength: the star-forming regions in those extreme high-redshift starbursts might be less compact than in their local analogues (i.e., ULIRGs), resulting in stronger PAH emission \citep{menendez-delmestre_2009}.
This hypothesis is supported by the observations in SMGs of larger star-forming regions than in local ULIRGs \citep{tacconi_2006,tacconi_2008,tacconi_2010} and by the observations of strong PAH signatures in their IRS spectra \citep{lutz_2005,valiante_2007,pope_2008,menendez-delmestre_2007,menendez-delmestre_2009}.
\\ \\}
\begin{figure*}
\centering
\includegraphics[width=9.1cm]{18312fg9a.eps}
\includegraphics[width=9.1cm]{18312fg9b.eps}
\caption{\label{fig: T vs LIR}\small{
(\textit{Left}) Dust temperature-luminosity relation inferred from our single temperature model.
The symbols are same as in Fig \ref{fig: lir s850 z}.
Red circles present the OFRG sample of \citet{magnelli_2010}.
The striped area presents results for SMGs extrapolated by \citet{chapman_2005} from radio and submm data.
The \citet{chapman_2003} derivation of the median and interquartile range of the $T_{{\rm dust}}$-$L_{{\rm IR}}$ relation observed at $z\thicksim0$ is shown by solid and dashed-dotted lines, linearly extrapolated to $10^{13}\,{\rm L_{\odot}}$.
The dashed line represent the dust temperature-luminosity relation derived in \citet{roseboom_2011} for mm-selected sample observed with SPIRE and assuming a single modified blackbody model.
(\textit{Right}) Dust temperature-luminosity relation inferred from our power-law temperature distribution model.
Symbols are the same as in the left panel.
The red dashed line presents the $T_{{\rm c}}$-$L_{{\rm IR}}$ relation inferred from a least-square second degree polynomial fit.
}}
\end{figure*}
\indent{
Another popular pre-\textit{Herschel} monochromatic extrapolation was to use radio flux densities and the local FIR/radio correlation \citep{helou_1988,yun_2001},
\begin{equation}
q={\rm log}\left(\frac{L_{{\rm FIR}}[{\rm W}]}{3.75\times10^{12}\times L_{1.4\,{\rm GHz}}[{\rm W Hz^{-1}}]}\right),
\end{equation}
where $L_{{\rm FIR}}$ is the infrared luminosity from rest frame 40 $\mu$m to 120 $\mu$m and $L_{1.4\,{\rm GHz}}$ is the \textit{k}-corrected radio luminosity density \citep[here we assume a standard radio slope $\alpha=0.8$; ][]{ibar_2010}.
In the following, we derived the infrared luminosities of our galaxies using this FIR/radio correlation and $\langle q\rangle=2.34$, as observed in the local Universe by \citet{yun_2001}.
Those estimates are compared to our reference values in the right panel of Fig. \ref{fig: LIR vs LIR}.
\\}
\indent{
We find a tighter correlation between our reference infrared luminosities and those inferred using radio flux densities and the local FIR/radio correlation ($\sigma[{\rm log}(L_{{\rm IR}}^{{\rm Radio}}/L_{{\rm IR}}^{{\rm ref}})]$$\thicksim$$0.29\,$dex).
The accuracy of these extrapolations is also supported by the good agreement found between those estimates in our lensed SMG sample.
Nevertheless, we also observe a trend with the infrared luminosity: at high luminosities, the FIR/radio correlation systematically overestimates the luminosity.
Since there is a tight correlation between $L_{{\rm IR}}$ and $z$, one can suspect this trend to be driven by an evolution of $\langle q\rangle$ with redshift.
As illustrated in the right panel of Fig. \ref{fig: Q vs LIR}, this trend is indeed in very good agreement with the evolution of $\langle q\rangle$ proportional to $(1+z)^{-0.15\pm0.03}$ found in \citet{ivison_2010a}.
Nevertheless, one has to keep in mind that our sample \textit{cannot be used to probe the evolution of $\langle q\rangle$ with redshift}, since it is, by construction via the radio identifications, biased towards galaxies with high radio flux densities.
Therefore, because here we did not attempt to correct for any of these incompleteness, e.g., using a Kaplan-Meier estimator, any of our results on $\langle q\rangle$ should be taken with caution.
The evolution of $\langle q\rangle$ could only been studied through carefully selected samples and using radio stacking.
So far, no clear conclusion on the evolution of $\langle q\rangle$ with redshift has been made (see Ivison et al. \citeyear{ivison_2010a}, \citeyear{ivison_2010b}, Roseboom et al. \citeyear{roseboom_2011}).
}
\subsection{The $T_{{\rm dust}}-L_{{\rm IR}}$ plane\label{subsec: t vs lir}}
The left panel of Fig. \ref{fig: T vs LIR} shows the $T_{{\rm dust}}-L_{{\rm IR}}$ plane inferred from our single temperature model.
The use of this simple model provides a comparison to other studies.
Compared to previous \textit{Herschel}-based results \citep{magnelli_2010,chapman_2010}, our large SMG sample populates the low (i.e., $L_{{\rm IR}}<10^{11.5}\,{\rm L_{\odot}}$) and high (i.e., $10^{13}\,{\rm L_{\odot}}>L_{{\rm IR}}$) luminosity regions of the $T_{{\rm dust}}-L_{{\rm IR}}$ diagram.
This large dynamic range allows a clear characterization of the $T_{{\rm dust}}-L_{{\rm IR}}$ correlation.\\
\indent{
The left panel of Fig. \ref{fig: T vs LIR} clearly confirms the selection bias introduced by submm observations:
At low luminosities SMGs are biased towards cold dust temperatures.
The upper envelope of the SMG $T_{{\rm dust}}-L_{{\rm IR}}$ distribution only depends on the depth of the submm observations (see Section \ref{subsec: bias}).
The existence of a population of dusty star-forming galaxies missed by submm observations is corroborated by the presence, in the upper part of the $T_{{\rm dust}}-L_{{\rm IR}}$ diagram, of some of the lensed SMGs and the optically faint radio galaxies \citep[OFRGs, ][]{magnelli_2010,casey_2009a}.
\\}
\indent{
Our SMG sample, together with our lensed SMG sample and the OFRG sample of \citet{magnelli_2010}, suggests that high-redshift dusty star-forming galaxies exhibit a wide range of dust temperatures \citep[see also][]{casey_2009a,magdis_2010}.
This might indicate that the $T_{{\rm dust}}-L_{{\rm IR}}$ relation at high redshift has a higher scatter than locally.
However, this conclusion can be driven by selection effects, because a significant fraction of the galaxies with intermediate dust properties are missed by our current sample.
This missing population will probably reconcile our finding with those of \citet{hwang_2010}, who found modest changes in the $T_{{\rm dust}}$-$L_{{\rm IR}}$ relation as function of the redshift using an $L_{{\rm IR}}$-selected sample of galaxies observed with \textit{Herschel}.
This conclusion is also strengthened by the fact that at high luminosities (i.e., few times $10^{12}\,{\rm L_{\odot}}$, where SMGs are a representative sample of the entire high luminosity galaxy population) SMGs exhibit dust temperatures that are in line with the $T_{{\rm dust}}-L_{{\rm IR}}$ relation extrapolated from local observations of \citet{chapman_2003} \citep[see also][]{clements_2010b,planck_2011}.
\\}
\indent{
As illustrated by the striped region in the left panel of Fig. \ref{fig: T vs LIR}, our dust temperatures and infrared luminosities largely agree with those extrapolated by pre-\textit{Herschel} studies using the local FIR/radio correlation.
This agreement of course reflects the broad consistency found between the local value of $\langle q\rangle$ and that observed in our sample (see Section \ref{subsec:lir lir}).
Our results also agree with those found by \citet{roseboom_2011} on a mm-selected sample observed with SPIRE and assuming a single modified blackbody model (see the dashed line in the left panel of Fig. \ref{fig: T vs LIR}).
\\ \\}
\indent{
From the wide range of dust temperatures, we can conclude that although submm observations are very useful to select extreme star-forming galaxies at high redshift, they cannot be used to obtain a complete census of the dusty star-forming galaxy population with relatively low infrared luminosities ($L_{{\rm IR}}\lesssim10^{12.5}\,{\rm L_{\odot}}$).
This census is now possible using bolometric selections provided by deep \textit{Herschel} observations \citep[e.g.,][]{magdis_2010} but still limited to relatively low redshift galaxies ($z<2.5$) due to the positive \textit{k}-correction affecting \textit{Herschel} data.
In the near future, very deep mm observations provided by the Atacama Large Millimeter Array (ALMA) might help to obtain this census even at high redshift.
\\ \\}
\indent{
The right panel of Fig. \ref{fig: T vs LIR} shows the $T_{{\rm c}}-L_{{\rm IR}}$ plane inferred from our power-law temperature distribution model.
This plane cannot be compared to any pre-\textit{Herschel} studies.
$T_{{\rm c}}$ is the temperature of the coldest dust component while $T_{{\rm dust}}$ gives an average dust temperature.
Thus, $T_{{\rm c}}$ is systematically lower than $T_{{\rm dust}}$, but their relative variations are tightly correlated (see also Fig. \ref{fig: lir kovacs single}).
Conclusions that one can draw from the $T_{{\rm c}}-L_{{\rm IR}}$ plane are the same as those drawn from the $T_{{\rm dust}}-L_{{\rm IR}}$ plane.
\\}
\indent{
We fitted the $T_{{\rm c}}-{\rm log}(L_{{\rm IR}})$ and $T_{{\rm dust}}-{\rm log}(L_{{\rm IR}})$ relation with a second order polynomial function and studied the scatter around these fits.
We find $\sigma_{T_{{\rm c}}}=1.9\,$K and $\sigma_{T_{{\rm dust}}}=3.8\,$K.
The decrease of the scatter (by a factor $2$) is in line with expectations from the relation between $T_{{\rm c}}$ and $T_{{\rm dust}}$, i.e., a factor $1.7$ because $T_{{\rm c}}=0.6\times T_{{\rm dust}}+3\,$K.
We note that our single-temperature model is also sensitive to the rest-frame wavelengths used in the fits; even if all galaxies at a given infrared luminosity have had the same dust temperature, our single-temperature model would still be affected by their redshift distribution, i.e., by the rest-frame wavelength probed by the PACS 160$\,\mu$m data point.
This redshift distribution would then introduce an artificial $T_{{\rm dust}}$ scatter.
In contrast, our power-law temperature distribution model is less affected by this effect, since it is constructed to reproduce cold and warm dust components.
}
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{18312fg10.eps}}
\caption{\label{fig: sed lir}\small{
Mean rest-frame SED of SMGs for four infrared luminosity bins, from bottom to top: $L_{{\rm IR}}<10^{12}\,{\rm L_{\odot}}$; $10^{12}\,{\rm L_{\odot}}<L_{{\rm IR}}<10^{12.7}\,{\rm L_{\odot}}$; $10^{12.7}\,{\rm L_{\odot}}<L_{{\rm IR}}<10^{13}\,{\rm L_{\odot}}$ and $10^{13}\,{\rm L_{\odot}}<L_{{\rm IR}}$.
The solid lines show the power-law temperature distribution SED corresponding to the mean dust mass and dust temperature of the bin.
The photometry of each of the sources was slightly renormalized to match these SED templates at submm wavelengths.
Dashed lines represent the CE01 template corresponding to the mean infrared luminosity of the bin, i.e., these templates were not fitted to the photometry of our SMGs.
}}
\end{figure}
\subsection{The spectral energy distribution of SMGs}
As discussed in section \ref{subsec: t vs lir}, SMGs can not be treated as a homogenous galaxy population, because they probe wide ranges in infrared luminosity and dust temperature.
Moreover, while at high infrared luminosity SMGs are a representative sample of the underlying high luminosity galaxy population, at lower infrared luminosities, SMGs are only a subsample of the entire infrared galaxy population, and are biased towards cold dust temperatures.
Thus, the SEDs of SMGs have to be analysed as a function of their infrared luminosities.
Figure \ref{fig: sed lir} presents the photometry of our SMGs split into four different infrared luminosity bins, i.e., $L_{{\rm IR}}<10^{12}\,{\rm L_{\odot}}$, $10^{12}\,{\rm L_{\odot}}<L_{{\rm IR}}<10^{12.7}\,{\rm L_{\odot}}$, $10^{12.7}\,{\rm L_{\odot}}<L_{{\rm IR}}<10^{13}\,{\rm L_{\odot}}$ and $10^{13}\,{\rm L_{\odot}}<L_{{\rm IR}}$.
In these panels, we show the mean SED inferred from our power-law temperature distribution model.
These SEDs correspond to $\beta=2.0$, $\gamma=7.3$, $R=3$ kpc and to the mean $M_{{\rm dust}}$ and $T_{{\rm c}}$ inferred for the galaxies of the bin.
In these panels, we also show the CE01 SED corresponding to the mean infrared luminosity inferred for the galaxies of the bin, i.e., the CE01 SEDs are not fitted to the photometry of the individual galaxies here.\\ \\
\indent{
At high infrared luminosities, the peak of the CE01 SED template is in agreement with that of our power-law temperature distribution model.
This indicates that in this range of luminosities, the local $T_{{\rm dust}}-L_{{\rm IR}}$ relation used in the CE01 library does not significantly evolve with redshift.
In contrast, the MIPS-24$\,\mu$m observations are systematically above predictions from the CE01 SED template.
As already mentioned, these discrepancies are likely due to an increase of the PAH emission strength in these galaxies and produce inaccurate infrared luminosity extrapolations from the MIPS-24$\,\mu$m flux density using the CE01 library \citep{elbaz_2011,nordon_2012}.
\\}
\indent{
As we go to lower infrared luminosities we observe larger discrepancies between the peak of the CE01 SED and that of our power-law temperature distribution model.
While the CE01 SED templates follow the local $T_{{\rm dust}}-L_{{\rm IR}}$ relation, our SMG sample is more and more biased towards cold dust temperatures.
At such low infrared luminosities, the SMG population represents the low-temperature-end of the real $T_{{\rm dust}}-L_{{\rm IR}}$ distribution (see Fig. \ref{fig: T vs LIR}).
In this low luminosity range, we note the better agreement than at high luminosities between observed and predicted MIPS-24$\,\mu$m.
}
\section{Toward a better understanding of the nature of SMGs$\,$?\label{sec:nature}}
Our results unambiguously reveal the diversity of the SMG population.
Some of these galaxies exhibit extreme infrared luminosities, with no local analogues ($L_{{\rm IR}}\gtrsim10^{13}\,{\rm L_{\odot}}$), while others have relatively low infrared luminosities ($10^{12}\,{\rm L_{\odot}}\lesssim L_{{\rm IR}}\lesssim10^{13}\,{\rm L_{\odot}}$).
Is this diversity reflecting differences in the mechanisms triggering their SFRs? \\ \\
\indent{
Recent hydrodynamic simulations, coupled with radiative transfer calculations, have found that while SMGs with relatively low infrared luminosities can be created by different scenarios (two gas rich galaxies soon to merge and observed as one submm source, or an isolated star-forming galaxy with large gas fraction), SMGs with the most extreme infrared luminosities/SFRs (i.e., $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}$, equivalently $\thicksim500\,$M$_{\odot}\,$yr$^{-1}$) can only be induced by strong starbursts at the coalescence of major mergers \citep{hayward_2011}.
These results are consistent with those of \citet{dave_2010} who found that SFRs induced by a secular mode of star formation reach at most, at $z\thicksim2$, a value of $\thicksim$$\,500\,$M$_{\odot}\,$yr$^{-1}$ (i.e., $L_{{\rm IR}}\thicksim10^{12.7}\,{\rm L_{\odot}}$).
This value of $\thicksim$$\,500\,$M$_{\odot}\,$yr$^{-1}$ can thus be considered as the ``maximum non-merger SFR'' (hereafter SFR$_{{\rm max}}^{{\rm secular}}$) and be used to separate, at $z\thicksim2$, merger-induced starbursts from galaxies with a secular mode of star formation.
Moreover, in a steady-state between SFR and gas accretion, one could expect SFR$_{{\rm max}}^{{\rm secular}}$ and the gas fraction of galaxies to be strongly related \citep{bouche_2010,dave_2011b}.
Therefore, SFR$_{{\rm max}}^{{\rm secular}}$ should decrease at low redshift with the gas fraction of galaxies \citep{tacconi_2010,geach_2011}.
Qualitatively this assumption is supported by observations of local ULIRGs, which are mostly associated with major mergers but which exhibit SFRs lower than $\thicksim$$\,500\,$M$_{\odot}\,$yr$^{-1}$, likely because they have relatively low gas fraction ($\thicksim10\%$; see Fig. 9 of Saintonge et al. \citeyear{saintonge_2011b}).
Therefore, in the redshift range $z=0$$-$$2$, we can separate merger-induced starbursts from non major-merging ones using a threshold of $500\,\times(1+z)^{2.2}_{z=2}\,$M$_{\odot}\,$yr$^{-1}$, while at $z>2$, we can use a threshold of $500\,$M$_{\odot}\,$yr$^{-1}$.
Here, the redshift dependence, $(1+z)^{2.2}_{z=2}\equiv((1+z)/3)^{2.2}$, comes from the evolution of the gas fraction found in \citet{geach_2011}.
Using a Chabrier IMF, these SFR thresholds correspond to the most luminous SMGs of our sample, i.e., $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}\times(1+z)^{2.2}_{z=2}$ at $0<z<2$ and $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}$ at $z>2$.
\\}
\indent{
A correlation between the SFR and the stellar mass of star-forming galaxies has been observed over the last $10\,$Gyr of lookback time \citep[SFR$\,\propto\,M_{\ast}^\alpha$ or SSFR$\,=\,$SFR$/M_{\ast}$$\,\propto\,$$M_{\ast}^{\alpha-1}$ with $0.5<\alpha<1.0$; ][]{noeske_2007a,elbaz_2007,daddi_2007a,rodighiero_2010b,oliver_2010,karim_2011,mancini_2011}.
The existence of this ``main sequence of star formation'' (MS) is usually interpreted as a piece of evidence that the bulk of the star-forming galaxy population is forming stars gradually with long duty cycles.
Galaxies situated on the main sequence would be consistent with a secular mode of star formation, likely sustained by a continuous gas accretion from the IGM and along the cosmic web \citep{dekel_2009,dave_2010}, while star-forming galaxies located far above the main sequence would be consistent with strong starbursts with short duty-cycles, mainly triggered by major mergers.
In that picture, to separate galaxies triggered by major mergers from those with secular mode of star formation, one should use the offset of a galaxy with respect to the MS, rather than simply using its infrared luminosity \citep[][Magnelli et al., in prep.]{wuyts_2011,elbaz_2011,nordon_2012,rodighiero_2011}.
\\}
\indent{
There are thus two ways to identify major-merger induced starbursts.
In the following, we apply these two criteria to our SMG sample, compare their results, and more importantly test their ability to effectively select major-merger induced starbursts.
For the criterion using the offset of a galaxy with respect to the main sequence (i.e., ${\rm \Delta log(SSFR)_{MS}=log[SSFR(galaxy)/SSFR_{MS}}(M_{\ast},z)]$), we use the stellar masses of 39 blank field SMGs derived in Section \ref{subsec:stellar masses} and the definition of the MS given by \citet{rodighiero_2010b}, i.e., ${\rm log(SSFR)_{MS}=\alpha\,log}(M_{\ast})$$\,+\,$$\beta$ where $(\alpha\,,\,\beta)$$\,=\,$$(-0.27,2.6)$, $(-0.51,5.3)$ and $(-0.49,5.2)$ at $0.5<z<1.0$, $1.0<z<1.5$ and $z>1.5$, respectively.
We adopt the definition of \citet{rodighiero_2010b} for consistent use of the FIR as a star-formation indicator.
None of our results strongly depend on this specific definition.
\\ \\}
\indent{
In the left panel of Fig. \ref{fig: SSFR vs z} we observe that our SMGs are systematically above the main sequence of star-formation, consistently with previous findings \citep[e.g.,][]{daddi_2007a,hainline_2011,wardlow_2011}.
Nevertheless, while low luminosity SMGs are within 2$\sigma$ from the MS, SMGs above our merger-induced starburst separation (i.e., SFR$_{{\rm max}}^{{\rm secular}}$) are at least 2$\sigma$ above it.
This segregation shows that for the relatively narrow range of stellar masses probed by our SMG sample ($1\times10^{10}\,$M$_{\ast}\,$$-$$\,5\times10^{11}\,$M$_{\ast}$), a simple cut in SFR allows us to accurately select the galaxies lying above the main sequence.
The fact that these two independent criteria (one is based on hydrodynamic simulations while the other is empirically derived using duty cycle arguments) select the same sample of galaxies strengthen their accuracy and therefore supports the assumption of a major-merger induced scenario.
We note that the SFR/luminosity criterion selects galaxies located $\thicksim$1$\,$dex above the main sequence.
This is consistent with values used in studies selecting merger-induced starbursts based on their location with respect to the main sequence \citep[][]{elbaz_2011,nordon_2012,rodighiero_2011}.
We conclude that for our specific SMGs sample these two criteria are equivalent.
\\ \\}
\indent{
Half of the galaxies in our sample (29 SMGs) have SFRs above our merger-induced starburst separation (i.e., SFR$_{{\rm max}}^{{\rm secular}}$; hereafter we call these galaxies luminous-SMGs, because they have $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}\times(1+z)^{2.2}_{z=2}$ at $0<z<2$ and $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}$ at $z>2$).
Their median infrared luminosity is $6.4\times10^{12}\,{\rm L_{\odot}}$, their median $T_{{\rm c}}$ is 27~K and they are at least 2$\sigma$ above the MS of star formation.
The high dust temperatures of these luminous-SMGs agree with those observed in local ULIRGs (see the agreement between the mean SED of these SMGs and the CE01 template in the top panel of Fig. \ref{fig: sed lir}).
This agreement suggests similar physical conditions prevailing in the star-forming regions of local ULIRGs and those of luminous-SMGs.
Since local ULIRGs are triggered by major mergers, this suggest that luminous-SMGs might also be produced by major mergers.
\\}
\indent{
The relatively high dust temperatures of the luminous-SMG subsample (compared to the rest of the SMG population, see the right panel of Fig. \ref{fig: SSFR vs z}) also agrees, qualitatively, with the large increase of the dust temperature predicted by \citet{hayward_2011} at the coalescence of their major merger simulations.
To quantitatively confirm this agreement we compare our dust temperatures with those of the hydrodynamic simulations of \citet{hayward_2011}.
First, we redshifted their simulated SEDs to match our SMG redshift distribution, second, we convolved these SEDs with the PACS, SPIRE, submm and mm filters and, third, we applied cuts in flux densities to match the properties of the GOODS-N field (i.e., a field with deep submm and \textit{Herschel} observations, probing a large dynamic range in the $T_{{\rm c}}-L_{{\rm IR}}$ plane).
Then, we fitted our power-law temperature distribution model with $\beta=2.0$, $\gamma=7.3$ and $R=3$ kpc to this set of simulated SEDs, leaving $M_{{\rm dust}}$ and $T_{{\rm c}}$ as the only free parameters of the model\footnote{If we constrained $\beta$, $\gamma$ and $R$ on the simulated SEDs, we find $\beta=1.6\pm0.2$, $\gamma=8.7\pm0.7$ and $R=2\pm1\,$kpc.
These values are different that those obtained on our SMGs and lead, systematically, to higher dust temperatures ($\Delta T_{{\rm c}}\thicksim7\,$K).
Nevertheless, we believe that using these constraints will not provide a fair dust temperature comparison with our SMGs.
First, while the exact values of $\beta$, $\gamma$ and $R$ strongly affect the inferred $T_{{\rm c}}$, the location of the FIR peak of the simulated SEDs stays unchanged.
Simulated SEDs of major mergers peak at shorter wavelength than those of isolated starburst, and the localization of these peaks are consistent with those of our SMGs.
Second, if the constraints on $\beta$, $\gamma$ and $R$ from our SMGs do not provide the optimal fit to the simulated SEDs, they still provide a fairly good fit to them.
Third, the simulated SEDs cannot be used to constrain $\beta$, $\gamma$ and $R$ because they do not yet include stochastically heated very small grains (Hayward et al., in prep).
}.
As for our data, the power-law temperature distribution model provides a good fit to the simulated SEDs, characterised by reasonably low $\chi^{2}$ values (i.e., $\thicksim8$ for N$_{{\rm dof}}\thicksim3$).
We find that simulated galaxies populate the same region of the $T_{{\rm c}}-L_{{\rm IR}}$ plane as our SMG sample.
Extreme infrared luminosities (i.e., $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}$) are indeed only observed in simulations of strong starbursts at the coalescence of major mergers \citep{hayward_2011}.
Simulations of two gas rich galaxies soon to merge (i.e., at an epoch where tidal effects have not yet caused strong starbursts) and observed as one submm source, always have lower infrared luminosities (i.e., $L_{{\rm IR}}\lesssim10^{12.7}\,{\rm L_{\odot}}$).
In the simulations, strong starbursts at the coalescence of major mergers exhibit higher dust temperatures ($\overline T_{{\rm c}}\thicksim28\,$K) than isolated starbursts ($\overline T_{{\rm c}}\thicksim22\,$K).
The agreement between the dust temperatures of simulated major mergers and that of our luminous-SMGs (i.e., $\overline T_{{\rm c}}$ is $27\,$K) supports the assumption that these luminous-SMGs are observed at a late-stage of a major merger.
\\ \\}
\indent{
We conclude that the most luminous SMGs exhibit properties, including their extreme infrared luminosity ($\overline L_{{\rm IR}}$$\,=$$\,6.4\times10^{12}\,{\rm L_{\odot}}$), their hot dust temperature ($\overline T_{{\rm c}}$$\,=\,$$27\,$K) and their location with respect to the main sequence ($>2\sigma$), which favour the scenario in which they correspond to intense starbursts with short duty-cycles, mainly triggered by major mergers.
On the other hand, SMGs with low infrared luminosities exhibit properties, including their relatively cold dust temperatures ($\overline T_{{\rm c}}=20\,$K) and their location with respect to the main sequence (within $\thicksim$$\,2\sigma$), which favour the scenario of isolated star-forming galaxies or pairs about to merge, i.e., at a time where tidal effects have not yet caused strong starbursts.
The distinction between these two modes (i.e., isolated star-forming galaxy or early-stage major merger) cannot be assessed using our data.
However, the existence in some SMGs of two galaxies about to merge and being contained in the same submm beam is confirmed by some high resolution CO line observations \citep[][HDF242 aka GN19]{tacconi_2008} and submm continuum observations \citep{younger_2009, kovacs_2010,wang_2011}.
\\}
\begin{figure*}
\centering
\includegraphics[width=9.cm]{18312fg11a.eps}
\includegraphics[width=9.cm]{18312fg11b.eps}
\caption{\label{fig: SSFR vs z}\small{
(\textit{Left}) Distribution of ``distance'' with respect to the main sequence observed in our SMG sample having accurate stellar masses estimates (empty histogram).
The hatched histogram shows the distribution observed in a subsample of luminous SMGs with SFR$>$SFR$_{{\rm max}}^{{\rm secular}}$, i.e., $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}\times(1+z)^{2.2}_{z=2}$ at $0<z<2$ and $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}$ at $z>2$.
(\textit{Right}) Dust temperature of SMGs as function of their distance with respect to the main sequence of star-formation.
Blue squares show luminous SMGs with $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}\times(1+z)^{2.2}_{z=2}$ at $0<z<2$ and $L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}$ at $z>2$.
Green squares show SMGs with infrared luminosities below these thresholds.
Red points represent the OFRGs, i.e., galaxies with relatively low infrared luminosities below our treshold.
In both plots the location of the main sequence as function of the redshift is taken from \citet{rodighiero_2010b}.
The 1$\,\sigma$ scatter around this main sequence is illustrated by the shaded area.
}}
\end{figure*}
\indent{
We stress that the conclusions drawn for the low luminosity SMG population should not be extrapolated to the entire low luminosity galaxy population.
Indeed, at these luminosities SMGs do not constitute a representative sample of the underlying population (see Section \ref{subsec: bias}).
In particular, galaxies with relatively low infrared luminosities but warm dust (e.g., the OFRGs) might be triggered by major mergers \citep{casey_2011}.
\\ \\}
\indent{
In the right panel of Fig. \ref{fig: SSFR vs z} we observe a clear correlation between the location of a galaxy with respect to the main sequence (i.e., ${\rm \Delta log(SSFR)_{MS}}$) and its dust temperature $T_{{\rm c}}$.
Nevertheless, because our sample is affected by strong selection biases (in term of luminosity and dust temperature as well as in term of being preferentially optically bright), this $T_{{\rm c}}$$\,-\,$${\rm \Delta log(SSFR)_{MS}}$ relation has to be treated with caution.
At low infrared luminosities our sample is biased towards galaxies with cooler dust temperatures.
Thus, we can expect ``main sequence'' galaxies to exhibit a broader range of dust temperatures than our current SMGs sample, i.e., weakening the $T_{{\rm c}}$$\,-\,$${\rm \Delta log(SSFR)_{MS}}$ relation.
On the other hand, the location of the OFRGs in this figure seems to qualitatively confirm the existence of a $T_{{\rm c}}$$\,-\,$${\rm \Delta log(SSFR)_{MS}}$ relation.
Indeed, even at relatively low infrared luminosities ($10^{12}\,{\rm L_{\odot}}\lesssim L_{{\rm IR}}\lesssim10^{13}\,{\rm L_{\odot}}$) and over the same range of redshift (i.e., $1.0<z<2.5$), galaxies with hotter dust temperatures (i.e., the OFRGs) are more offset from the MS than galaxies of the same infrared luminosities but with cooler dust temperature (i.e., the SMGs represented with green squares in the right panel of Fig. \ref{fig: SSFR vs z}).
The existence of a $T_{{\rm c}}$$\,-\,$${\rm \Delta log(SSFR)_{MS}}$ relation is also consistent with the fact that \citet{elbaz_2011} and \citet{nordon_2012} find a correlation between ${\rm \Delta log(SSFR)_{MS}}$ and the SED properties of star-forming galaxies.
The PAH-to-$L_{{\rm IR}}$ ratio of main sequence galaxies is constant, but decreases with increasing offset above the main sequence.
Finally, \citet{elbaz_2011} and \citet{wuyts_2011} also find that ${\rm \Delta log(SSFR)_{MS}}$ correlates with the compactness of the star-forming region; the SFR density of main sequence galaxies is roughly constant while it increases with increasing offset above the main sequence.
All these correlations are strong observational support of the physical interpretation given to the main sequence of star-formation.
Galaxies offset from the main sequence, likely triggered by major mergers, have compact star-forming regions resulting in warmer dust temperatures and weaker PAH emission.
}
\section{Summary\label{sec:summary}}
Using the \textit{Herschel} PACS and SPIRE observations of several deep cosmologcial fields, we study in detail the far-infrared properties of a sample of 61 SMGs which have secure redshift estimates.
We find that at high infrared luminosities this sample provides a good representation of the entire SMG population and more generally of the entire high luminosities star-forming galaxy population.
At low infrared luminosities, our sample is less representative, because it is biased towards low redshift galaxies with cooler dust.
Dust properties of these SMGs are inferred using two different approaches.
First, we use a single dust temperature modified blackbody model which provides a very simple description of the dust emission of galaxies and allows comparisons with all pre-\textit{Herschel} estimates.
Then, in order to obtain a better description of the Wien side of the dust emission, we use a power-law temperature distribution model.
This model provides an accurate description of the rest frame far-infrared SEDs of SMGs.
From this model we can constrain the dust emissivity spectral index, the characteristic emission diameter, the temperature index, the dust temperatures and the infrared luminosities of SMGs.
These properties are analysed and put into perspective with the more general question of the formation and evolution of star-forming galaxies.
Our main conclusions are:
\begin{enumerate}
\item We find that a single dust temperature model provides a good description of the far-infrared peak and Rayleigh-Jeans part of SED of SMGs, but fails to reproduce its Wien-side.
The dust temperatures and infrared luminosities inferred using the combination of only PACS (or only SPIRE) and submm observations are in very good agreement with the reference estimates based on PACS+SPIRE+submm data.
\item Using a power-law temperature distribution model we obtain a good description of the far-infrared SED of SMGs at its peak, on the Rayleigh-Jeans side and on the Wien side.
Using this model and the combination of PACS, SPIRE and submm observations, we obtain constraints on the dust emissivity spectral index of SMGs $\beta=2.0\pm0.2$ and the temperature index $\gamma=7.3\pm0.3$.
The dust emissivity spectral index found in our sample is in line with estimates by Dunne \& Eales (2001).
\item We find that luminosity extrapolations based on the radio emission are considerably more reliable than those based on the mid-infrared emission and the \citet{chary_2001} library.
For our sample, the FIR/radio correlation is parameterized with $\langle q\rangle=2.0\pm0.3$.
However, this value could not be applied to the full high-redshift star-forming galaxy populations because our sample is not well-suited to study the evolution of $\langle q\rangle$ with redshift.
\item Our study unambiguously reveals the diversity of the SMG population, which probes large ranges in infrared luminosity (from $L_{{\rm IR}}$$\,\thicksim$$\,2\times10^{11}\,{\rm L_{\odot}}$ to $\thicksim$$\,3\times10^{13}\,{\rm L_{\odot}}$) and dust temperature (from $T_{{\rm c}}=14\,$K to $T_{{\rm c}}=36\,$K) and is strongly biased towards galaxies with cold dust.
This bias decreases at high luminosities, and at $L_{{\rm IR}}\gtrsim10^{12.5}\,{\rm L_{\odot}}$, SMGs are a representative sample of the entire high infrared luminosity star-forming galaxies population.
At lower infrared luminosities, a complete census on the high-redshift star-forming galaxy population requires the use of the bolometric selection provided by deep \textit{Herschel} observations.
\item Our study clearly reveals that some SMGs exhibit extreme infrared luminosities ($L_{{\rm IR}}\gtrsim10^{12.7}\,{\rm L_{\odot}}$) which correspond to SFRs of $>500\,$M$_{\odot}$yr$^{-1}$.
We also observe that these luminous-SMGs exhibit warm dust temperatures ($\overline T_{{\rm c}}$$=$$\,27\,$K) and are outliers of the main sequence of star-formation ($\thicksim$2$\sigma$ above it).
The extreme SFRs of these luminous-SMGs are difficult to reconcile with a secular mode of star formation \citep[e.g., ][]{dave_2010} and could correspond to a merger-driven stage in the evolution of these galaxies.
This hypothesis is supported by the fact that these SMGs exhibit warm dust temperatures consistent with estimates from hydrodynamic simulations of major mergers coupled with radiative transfer calculation \citep{hayward_2011}, and that as outliers of the main sequence they are commonly assumed to be intense starbursts with short duty-cycles, likely triggered by major mergers.
\item At low infrared luminosities, the dust temperatures and the infrared luminosities of SMGs are consistent with a secular mode of star formation.
This hypothesis is also supported by the fact that those galaxies are situated close the main sequence of star-formation and hence are assumed to have large duty-cycles of star formation.
\end{enumerate}
\acknowledgement{
We thank the anonymous referee for suggestions which greatly enhanced this work.
We thank C. Hayward for providing us with his simulated SEDs.
PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain).
SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including University of Lethbridge (Canada), NAOC (China), CEA, LAM (France), IFSI, University of Padua (Italy), IAC (Spain), Stockholm Observatory (Sweden), Imperial College London, RAL, UCL-MSSL, UKATC, University of Sussex (UK), Caltech, JPL, NHSC, University of Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK) and NASA (USA).
The SPIRE data presented in this paper will be released through the {\em Herschel} Database in Marseille HeDaM ({hedam.oamp.fr/HerMES}).We acknowledge support from the Science and Technology Facilities Council [grant number ST/F002858/1] and [grant number
ST/I000976/1].
This study is based on observations made with ESO telescopes at the Paranal and Atacama Observatories under programme numbers: 171.A-3045, 168.A-0485, 082.A-0890 and 183.A-0666.
}
\bibliographystyle{aa}
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1,116,691,499,374 | arxiv | \section{Introduction}%
The analysis of transient and time-dependent queueing models is of great relevance for numerous applications, such as call centres \cite{brown2005statistical} and outpatient wards of hospitals where the server operates only over a finite amount of time \cite{ kim2014callcenter, kim2014choosing}. Besides their practical relevance, these systems provide a substantial mathematical challenge because the standard tools of renewal theory and ergodic theory are unsuited for their study. In other words, the steady-state distribution provides, if any, a poor approximation for the performance measures of transient queueing systems.
Here we focus on a particular class of transient queues, in which a finite (but large) number $n$ of customers can potentially join. As time passes, fewer customers can join the queue, so that eventually the queue length process will be identically zero and only its time-dependent behavior is of interest. We exploit ideas from the heavy-traffic approximation literature to prove that the queue length process can be approximated by a diffusion consisting of a Brownian motion with parabolic drift, reflected at zero.
The heavy-traffic approximation approach has been pioneered by Iglehart and Whitt \cite{iglehart1970multipleI} and has since been extended to a wide variety of settings where the time-dependent behavior is of interest, see \cite{glynn1990diffusions} for an excellent overview. Indeed, our result should be contrasted with \cite{iglehart1970multipleI}, where the queue length process is shown to converge to a reflected Brownian motion. The additional parabolic drift captures the effect of the diminishing pool of customers.
Transient queueing models have been studied lately by Honnappa et al.~\cite{honnappa2015delta, honnappa2014transitory}. However, interest in non-ergodic queues dates back to the pioneering work of Newell on the so-called $M_t/M_t/1$ queue \cite{newell1968queuesI, newell1968queuesII, newell1968queuesIII}. Later, Keller \cite{keller1982time} rederived Newell's heuristic results by methods of asymptotic expansion of the transition probabilities. Massey \cite{massey1985asymptotic} expanded and formalized these earlier results by using operator techniques.
More recently, Honnappa, Jain and Ward \cite{honnappa2015delta} introduced the $\DG$ queue as a model for systems in which a finite number of customers can join and/or which operate only over a finite time window. In \cite{honnappa2015delta} the authors prove a Functional Law of Large Numbers (FLLN) and a Functional Central Limit Theorem (FCLT) for the $\DG$ queue under very mild assumptions. In \cite{bet2014heavy}, by exploiting a general \emph{martingale} FCLT from \cite{MarkovProcesses}, it is shown that, when the arrival times are exponentially distributed and under the additional assumption that the queue satisfies a certain heavy-traffic condition, the rescaled queue length process converges in distribution to a reflected Brownian motion with parabolic drift.
The martingale FCLT is a convenient and powerful tool, but comes at a high cost in terms of computations to verify technical conditions. On the other hand, both the pre-limit and the limit queue length processes are easily characterized through explicit formulas. This suggests that it should possible to prove the convergence result in \cite{bet2014heavy} by using the ``straightforward'' approach to stochastic-process convergence, as detailed e.g.~in \cite{billingsley1999convergence}. As an example, assume a sequence of processes $(\mathcal S_n(\cdot))_{n\geq1}$ and a candidate limit $\mathcal S(\cdot)$ are given. The ``straightforward'' approach consists in proving separately the tightness of the family $(\mathcal S_n(\cdot))_{n\geq1}$, seen as measures on a certain function space, and the convergence of the finite-dimensional distributions, that is, as $n\rightarrow\infty$,
\begin{equation}\label{eq:finite_dimensional_convergence}%
\mathbb P(S_n(t_1)\in A_1,\ldots, S_n(t_k)\in A_k)\rightarrow\mathbb P(S(t_1)\in A_1,\ldots, S(t_k)\in A_k),
\end{equation}%
for each $k\geq1$ and $t_1,\ldots,t_k$. Note that condition \eqref{eq:finite_dimensional_convergence} characterizes the limit process uniquely.
By exploiting this method, we prove that the queue length process of the $\DG$ queue converges in distribution to a Brownian motion with negative quadratic drift, reflected at zero. In particular, the proof we give is substantially simpler than the one in \cite{bet2014heavy}, requiring only the standard notions of stochastic-process convergence theory \cite{billingsley1999convergence}. This approach has two advantages. First, we impose mild assumptions on the arrival time distribution, thus generalizing \cite{bet2014heavy}, where the arrival times were assumed to be exponentially distributed. Second, as a consequence of our main theorem, several results relating quantities of interest other than the queue length can be deduced. As an example of this, we prove a sample path Little's Law.
The rest of the paper is organized as follows. In Section \ref{sec:model} we describe the $\DG$ model, our assumptions, the processes of interest and state the main result. In Section \ref{sec:proof} we prove the main theorem, by separately proving convergence of the terms appearing in the expression for the queue length process. In Section \ref{sec:sample_path_little_law} we prove a transient version of Little's Law by building on the techniques and results of Section \ref{sec:proof}. In Section \ref{sec:conclusions} we summarize our result and sketch some interesting future research directions.
\section{The model and the main result}\label{sec:model}
We consider a population of $n$ customers. Each customer is assigned a clock $T_i$, with $i=1,\ldots, n$. We assume $(T_i)_{i=1}^{\infty}$ to be a sequence of positive i.i.d.~random variables with common density function $\fT(\cdot)$ and distribution function $\FT(\cdot)$. In particular, $\FT(\cdot)$ is continuous. Customers arrive at a single server with an infinite buffer and are served on a First-Come-First-Served basis.
The number of arrivals in $[0,t]$ is then given by
\begin{equation}%
A^n(t) := \sum_{i=1}^n \mathds 1_{\{T_i\leq t\}}.
\end{equation}%
Note that $A^n(t)/n$ is the empirical cumulative distribution function associated with $(T_i)_{i=1}^n$. The service times $(S_i)_{i=1}^{\infty}$ are i.i.d.~random variables such that $\sigma^2 := \Var(S) <\infty$. The corresponding (rescaled) renewal process is defined as
\begin{equation}\label{eq:service_renewal_process}%
S^n(t) : = \sup \Big\{m\geq1 \mid \sum_{i=1}^m S_i \leq nt\Big\}.
\end{equation}%
We further assume that at time zero the system obeys the \emph{heavy-traffic condition}
\begin{equation}%
\fT(0)= \sup_{t\geq0}\fT(t),
\end{equation}%
and that
\begin{equation}\label{eq:heavy_traffic_condition}%
\E[S]\fT(0) = 1,
\end{equation}%
which can be interpreted as follows. The number of arrivals in the interval $[0,\mathrm d t]$ is approximately $n(\FT(0+\mathrm dt) - \FT(0)) \approx n\fT(0)\mathrm d t$. Consequently, $\lambda_n = n\fT(0)$ represents the \emph{instantaneous} arrival rate in zero. On the other hand, because of the time scaling in \eqref{eq:service_renewal_process}, the service rate is $\mu_n = n/\E[S]$. The heavy-traffic condition is then equivalent to assuming that
\begin{equation}\label{eq:heavy_traffic_condition_interpretation}%
\frac{\lambda_n}{\mu_n} = 1.
\end{equation}%
More generally, condition \eqref{eq:heavy_traffic_condition_interpretation} could be replaced by $\frac{\lambda_n}{\mu_n} = 1 + \varepsilon_n$, for some $\varepsilon_n\rightarrow0$, but we refrain from doing it here. For a detailed explanation of the condition \eqref{eq:heavy_traffic_condition}, see \cite{bet2014heavy}.
Our main object of interest is the queue length process, defined as
\begin{equation}\label{eq:def_queue_length_process}%
Q^n(t) = A^n(t) - S^n(B^n(t)).
\end{equation}%
Here $B^n(t)$ is a continuous process that increases at rate $1$ if the server is working, and is constant otherwise.
Note that $A^n(t)$ and $S^n(t)$ are independent as they only depend respectively on $(T_i)_{i\geq1}$ and $(S_i)_{i\geq1}$. They interact through the time-change $t\mapsto B^n(t)$, which depends on both $(T_i)_{i\geq1}$ and $(S_i)_{i\geq1}$.
The \emph{diffusion-scaled heavy-traffic} queue length process is defined as
\begin{equation}\label{eq:diffusion_scaled_queue_length}%
\hat Q ^n(t) := \frac{{Q}^n(tn^{-1/3})}{n^{1/3}}.
\end{equation}%
Recall that the Skorokhod reflection map is the functional
defined by
\begin{align}%
\psi(f)(t) &= - \inf_{ s\leq t} (f(s))^-,\\
\phi(f)(t) &= f(t) + \psi(f)(t).
\end{align}%
We are now able to state our main theorem.
\begin{theorem}[Scaling limit of the queue length process]\label{th:main_theorem}
As $n\rightarrow\infty$,
\begin{equation}\label{eq:main_theorem_reflected_convergence}%
\hat Q ^n(t) \stackrel{\mathrm d}{\rightarrow} \phi(\hat X)(t),\qquad\mathrm{in}~(\mathcal D,J_1),
\end{equation}%
where
\begin{equation}\label{eq:limiting_free_process}
\hat X(t) = B_1(f_{\sss T}(0) t) - \frac{\sigma}{\E[S]^{3/2}}B_2(t) - \frac{f'_{\sss T}(0)}{2} t^2,
\end{equation}%
and $B_1(\cdot), B_2(\cdot)$ are two independent standard Brownian motions.
\end{theorem}
\textbf{Notation.}~Here $\mathcal D(\mathbb R) = \mathcal D$ denotes the space of c\`adl\`ag functions with values in $\mathbb R$, that is of functions $f(\cdot):\mathbb R^+ \rightarrow\mathbb R$ which are continuous from the right at every point and such that $\lim_{s\rightarrow t^-} f(s)$ exists for all $t>0$. $\mathcal D$ is endowed with the usual Skorokhod $J_1$ topology. For a sequence of stochastic processes $(X_n)_{n\geq1}$, $X_n \sr{\mathrm d}{\rightarrow} X$ in $(\mathcal D, J_1)$ means that $(X_n)_{n\geq1}$, seen as a sequence of random variables on $\mathcal D$, converges to $X$ in distribution, when $\mathcal D$ is endowed with the $J_1$ topology. Analogously, $X_n \sr{\mathrm d}{\rightarrow} X$ in $(\mathcal D, U)$ means that $(X_n)_{n\geq1}$ converges to $X$ in distribution, uniformly over compact subsets. Recall that, for a sequence $(x_n)_{n\geq1}\subset \mathcal D$, if $x_n\rightarrow x$ in $(\mathcal D, J_1)$ as $n\rightarrow\infty$, and $x$ is continuous, then $x_n\rightarrow x$ in $(\mathcal D, U)$, see \cite[p.~124]{billingsley1999convergence}. When dealing with vectors of functions we make use of the \emph{weak} $J_1$ topology $JW_1$. This coincides with the product topology on $\mathcal D\times \mathcal D\times\cdots \times \mathcal D = \mathcal D^k$. Given two (possibly random) functions, either on the real numbers or on the integers, $f, g$ the notation $f\sim g$ means $\lim_{x\rightarrow\infty} f(x)/g(x) = 1$, where $x\in\mathbb R$ or $x\in\mathbb N$. The notation $f(x) = \oP(g(x))$ means that $f(x)/g(x)\sr{\mathbb{P}}{\rightarrow}0$ as $x\rightarrow\infty$. The notation $f(x) = \Theta(g(x))$ means $f(x) = O(g(x)) $ and $g(x) = O(f(x))$. Finally, $f(x)^+=\max\{0,f(x)\}$ and $f(x)^-=\max\{0,-f(x)\}$ denote the positive and negative part of a function $f(\cdot)$ respectively.
\textbf{The cumulative busy time process.}\qquad We now give an explicit analytical characterization of $B^n(\cdot)$. To this end, we need to introduce several auxiliary processes.
The \emph{cumulative input} process is defined as
\begin{equation}%
C^n(t) := \sum_{i=1}^{A^n(t)}\frac{S_i}{n}.
\end{equation}%
$C^n(t)$ can be seen as the (rescaled) total amount of work that has entered the queue by time $t$.
Assuming that the server works at speed one, the \emph{net-put process} $N^n(t)$ is defined as
\begin{equation}%
N^n(t) := C^n(t) - t.
\end{equation}%
The \emph{workload} process is then defined as
\begin{equation}%
L^n(t) := \phi(N^n)(t) = N^n(t) - \inf_{s\leq t} (N^n(s))^-.
\end{equation}%
Note that $L^n(t)$ is positive if and only if
\begin{align}\label{eq:workload_positive_condition}%
C^n(t) &\geq t + \inf_{s\leq t} (N^n(s))^-\nnl
&= t -\psi(N^n)(t).
\end{align}%
By construction, $\psi(N^n)(t)$ increases (linearly) if and only if the server is idling, and is constant otherwise. In other words, $I^n(t) := \psi(N^n)(t)$ can be interpreted as the cumulative idle time proess. Consequently the term on the right-hand side of \eqref{eq:workload_positive_condition} can be interpreted as the \emph{cumulative busy time} process, and we define it as
\begin{equation}\label{eq:def_cumulative_busy_time}%
B^n(t) := t -\psi(N^n)(t).
\end{equation}%
Note that $B^n(t)$ increases only if the server is working, and is constant otherwise. With this definition, \eqref{eq:workload_positive_condition} reads
\begin{equation}%
C^n(t) \geq B^n(t),
\end{equation}%
so that the workload is positive if and only if the cumulative input up to time $t$ is larger than the total time the server has spent processing jobs, and in that case it decreases linearly in time.
\textbf{The queue length process.}\qquad It is more convenient to express $Q^n(t)$ as a reflection of a simpler process $X^n(t)$. We will refer to $X^n(t)$ as the \emph{free process}. To do so, we rewrite \eqref{eq:def_queue_length_process} as
\begin{align}%
Q^n(t) &= \Big(A^n(t) - S^n(B^n(t)) +\frac{B^n(t)}{\E[S]} -\fT(0)t\Big)- \Big(\frac{B^n(t)}{\E[S]} - \fT(0)t\Big)\nnl
&= \Big(A^n(t) - S^n(B^n(t)) +\frac{B^n(t)}{\E[S]} -\fT(0)t\Big) + \fT(0) I^n(t),
\end{align}%
where we used \eqref{eq:heavy_traffic_condition} in the second equality. We define
\begin{equation}%
X^n(t) = A^n(t) - S^n(B^n(t)) +\frac{B^n(t)}{\E[S]} -\fT(0)t.
\end{equation}%
We recall that, for a given process $X^n(t)$, the \emph{Skorokhod problem} associated with $X^n(t)$ consists in finding two processes $P(t)$ and $R(t)$ such that $P(t) = X^n(t) + R(t)\geq 0$, $R(t)$ is increasing, and $\int_0^{\infty} X^n(t) \mathrm d R(t) =0$.
Note that $I^n(\cdot)$ is increasing and, by definition of $Q^n(t)$ and $I^n(t)$,
\begin{equation}%
\int_0^{\infty} Q^n(t)\mathrm d I^n(t) = 0.
\end{equation}%
Then $Q^n(t)$ and $I^n(t)$ are a solution to the Skorokhod problem associated with $X^n(t)$ and, by applying \cite[Proposition 2.2, p.251]{asmussen2003applied} we have the representation
\begin{equation}%
Q^n(t) = X^n + \psi (X^n)(t) = \phi (X^n) (t),
\end{equation}%
where
\begin{equation}\label{eq:idle_time_representation}%
\psi (X^n)(t) = -\Big( \frac{B^n(t)}{\E[S]}-\fT(0) t\Big).
\end{equation}%
\textbf{The fluid and diffusive scaling regimes.}\qquad The \emph{fluid-scaled heavy-traffic} queue length process is defined as
\begin{equation}%
\bar{Q}^n (t) := \frac{Q^n(tn^{-1/3})}{n^{2/3}} = n^{1/3}\Big(\frac{A^n(tn^{-1/3})}{n} - \frac{S^n(B^n(tn^{-1/3}))}{n}\Big).
\end{equation}%
Correspondingly, $\bar X ^n(t)$ is defined as
\begin{align}%
\bar{X}^n (t) &:= n^{1/3}\Big(\frac{A^n(tn^{-1/3})}{n} - \frac{S^n(B^n(tn^{-1/3}))}{n}\Big) + n^{1/3}\frac{B^n(tn^{-1/3})}{\E[S]} -\fT(0)t \nnl
&= n^{1/3}\Big(\frac{A^n(n^{-1/3}t)}{n} -\FT(tn^{-1/3})\Big) - n^{1/3}\Big(\frac{S^n(B^n(tn^{-1/3}))}{n} - \frac{B^n(tn^{-1/3})}{\E[S]}\Big) \nnl
&\quad+ (n^{1/3}\FT(tn^{-1/3}) - \fT(0)t).
\end{align}%
where in the second equality we have added and subtracted $\FT(t)$ in order to rewrite $\bar X^n(t)$. It can be shown through an application of the functional Law of Large Numbers that, as $n\rightarrow\infty$, the fluid-scaled process $\bar Q^n (\cdot)$ converges to a deterministic process $\bar Q(\cdot)$. However, under our heavy-traffic assumption the process $\bar Q(\cdot)$ is identically zero. Because of this, the diffusion-scaled queue length process can be rewritten as
\begin{equation
\hat Q ^n(t) = n^{1/3}\bar{Q}^n(t) = n^{1/3} (\bar{Q}^n(t) - \bar Q(t) ).
\end{equation}%
Accordingly, $\hat X^n(t)$ is defined as
\begin{align}\label{eq:diffusion_scaled_free_process}%
\hat X^n (t) &:= n^{1/3} \bar{X}^n(t)\nnl
&= n^{2/3}\Big(\frac{A^n(tn^{-1/3})}{n} - \FT(tn^{-1/3})\Big) - n^{2/3}\Big(\frac{S^n(B^n(tn^{-1/3}))}{n} - \frac{B^n(tn^{-1/3})}{\mathbb{E}[S]}\Big) \nnl
&\quad+n^{2/3}(\FT(tn^{-1/3}) - f_{\sss T}(0) tn^{-1/3}).
\end{align}%
In order to prove Theorem \ref{th:main_theorem} we will rely on an analogous result for $\hat X^n(\cdot)$. In fact, Theorem \ref{th:main_theorem} is a straightforward consequence of the following:
\begin{theorem}[Scaling limit of the free process]
As $n\rightarrow\infty$,
\begin{equation}\label{eq:main_theorem}%
\hat X^n (t) \stackrel{\mathrm d}{\rightarrow} \hat X(t),\qquad\mathrm{in}~(\mathcal D,J_1),
\end{equation}%
where $\hat X(\cdot)$ is given by
\begin{equation}
\hat X(t) = B_1(f_{\sss T}(0) t) - \frac{\sigma}{\E[S]^{3/2}}B_2(t) - \frac{f'_{\sss T}(0)}{2} t^2,
\end{equation}%
and $B_1(\cdot), B_2(\cdot)$ are two independent standard Brownian motions.
\end{theorem}
\textbf{The scaling exponents.}\qquad Let us now give an heuristic motivation for the scaling exponents in \eqref{eq:diffusion_scaled_free_process}. Define the general time scaling exponent as $-\alpha$ and the spatial scaling exponent as $\beta$, for some $\alpha, \beta>0$ to be determined, so that $\difX$ is given by
\begin{align}\label{eq:arrival_process_diffusion_scaled_generic_exponents}%
\difX &= n^{\beta}\Big(\frac{A^n(tn^{-\alpha})}{n}-\FT (tn^{-\alpha}) \Big) + n^{\beta}\Big(\frac{S^n(B^n(tn^{-\alpha}))}{n} - \frac{B^n(tn^{-\alpha})}{\mathbb{E}[S]}\Big)\\
&\quad+ n^{\beta}(F(tn^{-\alpha}) - f_{\sss T}(0) tn^{-\alpha}).\notag
\end{align}%
For the deterministic drift to converge to a non-trivial limit it is necessary that $\alpha, \beta$ be such that $2\alpha = \beta$. Indeed, replacing $\FT(t n^{-\alpha})$ with its Taylor expansion up to the second term, we get
\begin{equation}%
n^{\beta}(\FT(t n^{-\alpha}) - \fT(0) t n^{-\alpha}) =n^{\beta}\Big(\frac{\fT'(0)}{2} t^2 n^{-2\alpha} + o (n^{-2\alpha})\Big).
\end{equation}%
Moreover, a necessary condition for $\hat A^n (\cdot)$ in \eqref{eq:arrival_process_diffusion_scaled_generic_exponents} to converge to a non-trivial random process is that, for fixed time $t>0$, the variance of $\difA$ be $\Theta(1)$. This is given by
\begin{align}%
\Var(\difA) &= \frac{n^{2\beta}}{n}\Var (\mathds 1_{\{T\leq tn^{-\alpha}\}})\nnl
&= \frac{n^{2\beta}}{n} \mathbb P(T\leq tn^{-\alpha})(1 - \mathbb P(T\leq tn^{-\alpha}))\nnl
&= \frac{n^{2\beta}}{n}(\fT(0)tn^{-\alpha} + o(n^{-\alpha})).
\end{align}%
Then, $\alpha$ and $\beta$ should be such that
\begin{equation}%
\frac{n^{2\beta-\alpha}}{n} = O(1),
\end{equation}%
which, together with $\beta = 2\alpha$, imply that $\alpha = 1/3$ and $\beta = 2/3$.
\textbf{Comparison with known results.}\qquad We conclude by drawing a connection between Theorem \ref{th:main_theorem} and the analogous result in \cite{bet2014heavy}. There, the queue length process is shown to converge to $\phi(X)(t)$, where $X(t) = \sigma B(t) -t^2/2$, where $\sigma^2 = \E[S^2]/\E[S]^3$ and $B(t)$ is a standard Brownian motion. The random process consisting of the sum of two Brownian motions in \eqref{eq:limiting_free_process} is equivalent in distribution to a single Brownian motion with variance equal to
\begin{equation}%
\fT(0)+\frac{\E[S^2]-\E[S]^2}{\E[S]^3}.
\end{equation}%
By the heavy-traffic condition \eqref{eq:heavy_traffic_condition} this can be simplified to
\begin{equation}%
\frac{\E[S]^2+\E[S^2]-\E[S]^2}{\E[S]^3} = \frac{\E[S^2]}{\E[S]^3}.
\end{equation}%
Therefore, the two limits are equal in distribution, as expected.
\section{Proof of Theorem \ref{th:main_theorem}}\label{sec:proof}
\subsection{Overview of the proof}
The proof of Theorem \ref{th:main_theorem} proceeds in several steps. These consist in proving convergence of the three terms in \eqref{eq:diffusion_scaled_queue_length} to the respective terms in \eqref{eq:limiting_free_process} separately. The first term in \eqref{eq:diffusion_scaled_queue_length} is the centred and rescaled empirical distribution function of the sequence $(T_i)_{i\geq1}$. Therefore, its convergence to $B_1(\fT(0)t)$ can be seen as a `local Donsker's Theorem', in which the limiting Brownian Bridge is replaced by a Brownian motion. The second term in \eqref{eq:diffusion_scaled_queue_length} is a time-changed, centred and rescaled renewal process and thus converges by a random time-change theorem and the FCLT for renewal processes. The third term also converges trivially to the limiting quadratic drift. Then,
the convergence \eqref{eq:main_theorem_reflected_convergence} follows immediately from \eqref{eq:main_theorem} by the continuity of the Skorokhod reflection $\phi(x)$ in all $x\in\mathcal C$, the space of real-valued continuous functions, see \cite[Theorem 13.5.1]{StochasticProcess}.
\subsection{A local Donsker's Theorem}
For sake of simplicity, let us define
\begin{equation}\label{eq:arrival_process_diffusion_scaled}%
\difA := n^{2/3}\Big(\frac{A^n(t n^{-1/3})}{n}-\FT (t n^{-1/3}) \Big)
\end{equation}%
and
\begin{equation}%
\hat A(t) := B_1(\fT(0)t).
\end{equation}%
The goal of this section is to prove the following:
\begin{lemma}[Convergence of the arrival process]\label{lem:local_donsker_theorem}
As $n\rightarrow\infty$,
\begin{equation}\label{eq:claim_local_donsker_theorem}%
\difA\stackrel{\mathrm d}{\rightarrow} \hat A(t),\qquad \mathrm{in}~(\mathcal D, J_1).
\end{equation}%
\end{lemma}
\begin{proof}
The proof proceeds in two steps. First, we prove convergence of the finite-dimensional distributions. This characterizes the limit uniquely. Second, we prove tightness of the family $(\difA)_{n\geq1}$, seen as elements of $\mathcal P ( \mathcal D)$, the space of measures on the Polish space $\mathcal D$ of c\`adl\`ag functions.
By definition, we say that the finite-dimensional distributions of $\hat A^n (\cdot)$ converge to the finite-dimensional distributions of $\hat A(\cdot)$ if, for every $n\in\mathbb N$ and for each choice of $(t_i)_{i=1}^n$ such that $0 < t_1 < t_2 < \ldots < t_n < \infty$ it holds that, as $n\rightarrow\infty$,
\begin{equation}\label{eq:finite_dimensional_convergence_definition}%
(\hat A^n(t_1), ,\ldots,\hat A^n(t_n)) \dconv (\hat A(t_1), \ldots, \hat A(t_n)).
\end{equation}%
For simplicity we shall prove \eqref{eq:finite_dimensional_convergence_definition} for $t_1<t_2$, the generalization to an arbitrary choice of $(t_i)_{i=1}^n$ being straightforward. We then aim to show that, as $n\rightarrow\infty$,
\begin{equation}%
(\hat A^n (t_1), \hat A^n(t_2) )\dconv (\hat A (t_1), \hat A(t_2) ).
\end{equation}%
Let $\mathcal N (m, v)$ denote a normally distributed random variable with mean $m$ and covariance matrix $v$. Then $(\hat A (t_1), \hat A(t_2))\sim \mathcal N(m, V_{t_1,t_2})$, with mean $m=(0,0)$ and covariance matrix $V_{t_1,t_2}$ given by
\begin{equation}
V_{t_1,t_2} = \fT(0)\left(
\begin{array}{cc}
t_1 & t_1\wedge t_2\\
t_1\wedge t_2 & t_2
\end{array}\right),
\end{equation}%
where $a\wedge b = \min\{a,b\}$. To show joint convergence, we apply the Cram\'er-Wold device. Given an arbitrary vector $\gamma = (\gamma_1,\gamma_2)\in\mathbb R^2$, we aim to show that, as $n\rightarrow\infty$,
\begin{equation}\label{eq:cramer_wold_device}%
\gamma_1\hat A^n (t_1)+\gamma_2\hat A^n(t_2)\dconv \gamma_1 \hat A(t_1) + \gamma_2 \hat A(t_2).
\end{equation}%
This is done through the following straightforward generalization of the Lindeberg-Feller CLT.
\begin{theorem}[Lindeberg-Feller CLT \cite{klenke2008probability}]\label{th:lindeberg_CLT}%
Let $(X_{n,l})_{l=1}^n$ be an array of random variables such that $\E[X_{n,l}] = 0$ for all $n\geq 1$ and $l\leq n$ and $\sum_{l=1}^n \Var (X_{n,l})\rightarrow1$. Define
\begin{equation}%
S_n := X_{n,1} + \ldots + X_{n,n}.
\end{equation}%
Assume that the \emph{Lindeberg condition} holds, i.e. for $\varepsilon>0$,
\begin{equation}%
\frac{1}{\Var(S_n)}\sum_{l=1}^n \E[X_{n,l}^2\mathds 1_{\{X_{n,l}^2> \varepsilon^2\Var(S_n)\}}]\rightarrow 0,\qquad n\rightarrow\infty.
\end{equation}%
Then $(S_n)_{n\geq1}$ converges in distribution to a standard normal random variable.
\end{theorem}%
We remark that in the usual formulation of the Lindeberg-Feller CLT it is assumed that $\sum_{l=1}^n \Var (X_{n,l}) = 1$. The proof of the theorem, as presented e.g.~in \cite{klenke2008probability} can be directly generalized to accommodate for the assumption that $\sum_{l=1}^n \Var (X_{n,l})\rightarrow1$. We now take $X_{n,l}$ to be
\begin{equation}%
X_{n,l} = \gamma_1\frac{ \mathds 1_{\{T_l \leq t_1n^{-1/3}\}} - \FT(t_1n^{-1/3})}{n^{1/3}v_{t_1,t_2}} + \gamma_2\frac{ \mathds 1_{\{T_l \leq t_2n^{-1/3}\}} - \FT(t_2n^{-1/3})}{n^{1/3}v_{t_1,t_2}},
\end{equation}%
where $v_{t_1,t_2}$ is a normalizing constant and is given by
\begin{equation}%
v_{t_1,t_2} = \sqrt{\fT(0) (\gamma_1^2 t_1 + \gamma_2^2 t_2 + 2 \gamma_1\gamma_2t_1)}.
\end{equation}%
Recall that $t_1<t_2$ by assumption. In order to deduce the desired convergence \eqref{eq:cramer_wold_device} we are left to check the conditions of Theorem \ref{th:lindeberg_CLT}. Trivially, $ \E[X_{n,l}] = 0$. Moreover, it is possible to explicitly compute $\Var (X_{n,l})$ as follows:
\begin{align}%
\Var (X_{n,l}) &= \frac{\gamma_1 ^2}{n^{2/3}v_{t_1,t_2}^2} (\FT(t_1n^{-1/3}) - \FT(t_1n^{-1/3})^2)\nnl
&\quad+\frac{\gamma_2 ^2}{n^{2/3}v_{t_1,t_2}^2} (\FT(t_2n^{-1/3}) - \FT(t_2n^{-1/3})^2) \nnl
&\quad+\frac{2\gamma_1\gamma_2}{n^{2/3}v_{t_1,t_2}^2}(\FT(t_1n^{-1/3}) - \FT(t_1n^{-1/3})\FT(t_2n^{-1/3}))\nnl
&= \frac{\fT(0)}{v_{t_1,t_2}^2} \Big(\frac{\gamma_1^2}{n} t_1 + \frac{\gamma_2^2}{n} t_2 + 2 \frac{\gamma_1\gamma_2}{n}t_1\Big) + O(n^{-4/3}),
\end{align}%
where in the second equality the distribution function $\FT(\cdot)$ was Taylor expanded. In particular,
\begin{equation}
\sum_{l=1}^n \Var(X_{n,l}) = 1 + O(n^{-1/3}).
\end{equation}
The \emph{Lindeberg condition} is also satisfied, since
\begin{align}%
\sum_{l=1}^n &\frac{1}{n^{2/3}v_{t_1,t_2}}\E[(\mathds 1_{\{T_i\leq t_1n^{-1/3}\}}-\FT(t_1 n^{-1/3}))^2\mathds 1_{\{(\mathds 1_{\{T_i\leq t_1n^{-1/3}\}}-\FT(t_1 n^{-1/3}))\geq \varepsilon n^{1/3}\}}]\nnl
&=\frac{n^{1/3}}{v_{t_1,t_2}}\E[(\mathds 1_{\{T_1\leq t_1n^{-1/3}\}}-\FT(t_1 n^{-1/3}))^2\mathds 1_{\{(\mathds 1_{\{T_1\leq t_1n^{-1/3}\}}-\FT(t_1 n^{-1/3}))\geq \varepsilon n^{1/3}\}}]\nnl
&\leq \frac{n^{1/3}}{v_{t_1,t_2}}\sqrt{\E[(\mathds 1_{\{T_1\leq t_1n^{-1/3}\}}-\FT(t_1 n^{-1/3}))^4]}\sqrt{ \mathbb P (\mathds 1_{\{T_1\leq t_1n^{-1/3}\}}-\FT(t_1 n^{-1/3})\geq \varepsilon n^{1/3})},
\end{align}%
by the Cauchy-Schwartz inequality. The first term is of the order $O(n^{-1/3})$, while the second is identically zero for $n$ large enough.
By Theorem \ref{th:lindeberg_CLT},
\begin{equation}\label{eq:cramer_wold_device_applied}%
\frac{1}{v_{t_1,t_2}}(\gamma_1,\gamma_2)\cdot(\hat{A}^n(t_1),\hat{A}^n(t_2))^{\mathrm t}\dconv \mathcal N(0,1),
\end{equation}%
where $\cdot$ denotes the usual scalar product and $q^{\mathrm t}$ denotes the transpose of a vector $q$. However, since
\begin{equation}%
(\gamma_1,\gamma_2)\cdot V_{t_1,t_2}\cdot (\gamma_1,\gamma_2)^{\mathrm t} = v_{t_1,t_2}^2,
\end{equation}%
then
\begin{equation}%
\mathcal N(0,1)\sr{\mathrm d}{=} \frac{1}{v_{t_1,t_2}}(\gamma_1,\gamma_2)\cdot\mathcal N ((0,0), V_{t_1,t_2}),
\end{equation}%
and this together with \eqref{eq:cramer_wold_device_applied} implies \eqref{eq:cramer_wold_device}. By an application of the Cram\'er-Wold device, joint convergence follows.
The last step of the proof is to show that $(\hat{A}^n(\cdot))_{n=1}^{\infty}$ is a tight family of random variables on $\mathcal D$. By \cite[Theorem 13.5]{billingsley1999convergence}, in particular equation (13.14), it is enough for $(\hat{A}^n(\cdot))_{n=1}^{\infty}$ to satisfy the following condition. For every $T>0$,
\begin{equation}\label{eq:tightness_criterion}%
\E[\vert \difA - \hat A^n(t_1)\vert^2\vert \hat A^n(t_2) - \difA\vert^2] \leq (f_{\textup{inc}}(t_2) - f_{\textup{inc}}(t_1))^2,
\end{equation}%
for $0\leq t_1\leq t\leq t_2\leq T$ and $f_{\textup{inc}}(\cdot)$ is a non-decreasing function. Checking \eqref{eq:tightness_criterion} amounts to computing the mean appearing on the left side of the equation. Define
\begin{align}%
p_1 &:= \FT(tn^{-1/3}) - \FT (t_1n^{-1/3}), \nnl
p_2 &:= \FT(t_2n^{-1/3}) - \FT(tn^{-1/3}).
\end{align}%
Define also
\begin{equation}%
\alpha_i:=\left\{ \begin{array}{ll}
1-p_1, & \mathrm{if}~\frac{T_i}{n^{1/3}}\in(t_1,t],\\
-p_1, & \mathrm{if}~\frac{T_i}{n^{1/3}}\nin(t_1,t],
\end{array}\right.
\end{equation}%
and
\begin{equation}%
\beta_i:=\left\{ \begin{array}{ll}
1-p_2, & \mathrm{if}~\frac{T_i}{n^{1/3}}\in(t,t_2],\\
-p_2, & \mathrm{if}~\frac{T_i}{n^{1/3}}\nin(t,t_2],
\end{array}\right.
\end{equation}%
where we have omitted dependencies on $n$ to avoid cumbersome notation. Note that $\E[\alpha_1] = \E[\beta_1] = 0$. With the help of these definitions, \eqref{eq:tightness_criterion} can be rewritten in the following form:
\begin{equation}\label{eq:tightness_criterion_rewritten}%
\E\Big[\big(\sum_{i=1}^n\alpha_i\big)^2\big(\sum_{i=1}^n\beta_i\big)^2\Big]\leq n^{4/3}(f_{\textup{inc}}(t_2) - f_{\textup{inc}}(t_1))^2.
\end{equation}%
We will take $f_{\mathrm {inc}} (t) = \sqrt{K} t$ for a certain constant $K>0$. By definition $\alpha_i$ (resp. $\beta_i$) is independent from $\alpha_j$ and $\beta_j$ for $j\neq i$, so that the left side of \eqref{eq:tightness_criterion_rewritten} can be simplified as
\begin{align}%
n\E[\alpha_1^2\beta_1^2] + n(n-1) \E[\alpha_1^2]\E[\beta_2^2] + 2n(n-1)\E[\alpha_1\beta_1]\E[\alpha_2\beta_2].
\end{align}%
The first term $n\E[\alpha_1^2\beta_1^2]$ is of lower order, so we focus on the remaining two. A simple computation gives
\begin{align}%
\E[\alpha_1^2] &= p_1(1-p_1)\leq p_1,\nnl
\E[\beta_1^2] &= p_2(1-p_2)\leq p_2,\nnl
\E[\alpha_1\beta_2] &= -p_1p_2,
\end{align}%
so that, since $p_1\leq(p_1 + p_2)$ and $p_2\leq (p_1 + p_2)$,
\begin{align}%
\E\Big[\big(\sum_{i=1}^n\alpha_i\big)^2\big(\sum_{i=1}^n\beta_i\big)^2\Big] &\leq C_0 n^2 p_1p_2 \leq C_0 n^2 (p_2 + p_1)^2\nnl
&= C_0 n^2 (\FT(t_2 n^{-1/3}) - \FT(t_1 n^{-1/3}))^2 \nnl
&\leq C_1 n^{4/3} \fT(0) (t_2 - t_1)^2,
\end{align}%
for a sufficiently large $C_1>0$. Therefore, we have verified \eqref{eq:tightness_criterion_rewritten} with $f_{\mathrm {inc}} (t) = \sqrt{C_1 \fT(0)}t$.
\end{proof}
\subsection{A functional CLT for renewal processes}
We define
\begin{equation}\label{eq:service_process_diffusion_scaled}%
\hat S^n(t) := n^{2/3}\Big(\frac{S^n(tn^{-1/3})}{n} - \frac{1}{\E[S]}tn^{-1/3}\Big)
\end{equation}%
and
\begin{equation}%
\hat S(t) := \frac{\sigma}{\E[S]^{3/2}}B_2(t),
\end{equation}%
where $\sigma^2$ is the variance of $S$.
The goal of this section is then to prove the following:
\begin{lemma}[Convergence of the service process]\label{lem:clt_renewal_processes}
As $n\rightarrow\infty$,
\begin{equation}\label{eq:claim_clt_renewal_processes}%
\hat S^n(t)\stackrel{\mathrm d}{\rightarrow} \hat S(t), \qquad\mathrm{in}~(\mathcal D, J_1).
\end{equation}%
\end{lemma}
\begin{proof}
Note that $S^n(tn^{-1/3}) = S^{n^{2/3}}(t)$. Moreover,
\begin{equation}%
n^{2/3}\Big(\frac{S^n(tn^{-1/3})}{n} - \frac{1}{\E[S]}tn^{-1/3}\Big) =
\frac{S^{n^{2/3}}(t)- \E[S]^{-1}tn^{2/3}}{n^{1/3}}.
\end{equation}%
Therefore, the claim \eqref{eq:claim_clt_renewal_processes} can be proven by directly applying a FCLT for renewal processes, see e.g.~\cite[Theorem 14.6]{billingsley1999convergence}.
\end{proof}
\subsection{Convergence of the cumulative busy time}
In this section we exploit Lemma \ref{lem:clt_renewal_processes} and the random time change theorem to prove that the rescaled service process in \eqref{eq:diffusion_scaled_queue_length} converges.
First, we prove some scaling limits for the arrival process. Define the fluid-scaled arrival process as
\begin{equation}%
\bar A^n(t) := \frac{A^n(tn^{-1/3})}{n^{2/3}}.
\end{equation}%
The following straightforward generalization of the Chebyshev inequality is useful when proving the strong Law of Large Numbers:
\begin{lemma}[Generalized Chebyshev inequality]\label{lem:generalized_chebyshev_inequality}
For any $p=1,2,\ldots$, and any random variable $X$ such that $\E[\vert X\vert^p]<\infty$,
\begin{equation}%
\mathbb P(\vert X\vert \geq \varepsilon) \leq \frac{\E[\vert X\vert ^p]}{\varepsilon^p}.
\end{equation}%
\end{lemma}%
By using Lemma \ref{lem:generalized_chebyshev_inequality} together with the Borel-Cantelli lemma, we can prove the following:
\begin{lemma}[LLN for the arrival process]\label{lem:LLN_arrival_process}%
As $n\rightarrow\infty$,
\begin{equation}\label{eq:arrival_process_pointwise_local_LLN}%
\Big\vert \bar A^n(t) - \fT(0)t \Big\vert\asconv0,
\end{equation}%
for fixed $t\geq0$.
\end{lemma}%
\begin{proof}
First, we rewrite
\begin{equation}%
\bar A ^n(t) -\fT(0)t = \frac{1}{n}\sum_{i=1}^n (n^{1/3}\mathds 1_{\{T_i\leq tn^{-1/3}\}} - n^{1/3}\FT(tn^{-1/3})) =: \frac{1}{n}\sum_{i=1}^n Y_i.
\end{equation}%
In order to apply the Borel-Cantelli lemma, we compute
\begin{align}%
\mathbb P\Big(\vert\sum_{i=1}^n Y_i\vert \geq \varepsilon n\Big) & \leq \frac{\E[\vert \sum_{i=1}^n Y_i\vert^4]}{n^4\varepsilon^4} \nnl
&= \frac{n \E[\vert Y_1 \vert ^4] + 3n(n-1)\E[\vert Y_1\vert^2]^2}{n^4\varepsilon^4}.
\end{align}%
It is immediate to see that the leading orders of the expectation values are
\begin{align}%
\E[\vert Y_1\vert^4] &= O(n^{4/3}\mathbb P(T_i\leq tn^{-1/3}))= O(tn),\nnl
\E[\vert Y_1\vert^2] &= O(n^{2/3}\mathbb P(T_i\leq tn^{-1/3})) = O(tn^{1/3}).
\end{align}%
We conclude that, for a large constant $C_1>0$,
\begin{equation}%
\mathbb P\Big(\vert\sum_{i=1}^n Y_i\vert \geq \varepsilon n\Big) \leq C_1 \frac{tn^2 + 3tn^{8/3}}{n^4\varepsilon^4}.
\end{equation}%
Define the event $\mathcal A := \{\vert\sum_{i=1}^n Y_i\vert \geq \varepsilon n ~\mathrm{for~infinitely~many~}n\}$. Since
\begin{equation}%
\sum_{n=1}^{\infty} \mathbb P\Big(\vert\sum_{i=1}^n Y_i\vert \geq \varepsilon n\Big) \leq C_1 \sum_{n=1}^{\infty} \frac{tn^2 + 3tn^{8/3}}{n^4\varepsilon^4} \leq C_2 \sum_{n=1}^{\infty} \frac{1}{n^{4/3}\varepsilon^4}<\infty,
\end{equation}%
for some large constant $C_2>0$, by the Borel-Cantelli lemma,
\begin{equation}%
\mathbb P(\mathcal A)=0.
\end{equation}%
Since $\varepsilon>0$ is arbitrary, this concludes the proof of \eqref{eq:arrival_process_pointwise_local_LLN}.
\end{proof}%
We are now interested in obtaining a Glivenko-Cantelli-type theorem which extends the convergence \eqref{eq:arrival_process_pointwise_local_LLN} to uniform convergence over compact subsets of the positive half-line. This is summarized in the following lemma.
\begin{lemma}[Glivenko-Cantelli Theorem for the arrival process]\label{th:glivenko-cantelli_arrival_process}%
As $n\rightarrow\infty$,
\begin{equation}\label{eq:glivenko-cantelli_arrival_process}%
\bar A^n(t) \asconv \fT(0)t,\qquad \mathrm{in}~(\mathcal D,U).
\end{equation}%
Consequently, as $n\rightarrow\infty$,
\begin{equation}\label{eq:fCLT_cumulative_input_process}%
n^{1/3}C^n(t n^{-1/3}) \asconv t \qquad\mathrm{in}~(\mathcal D, U).
\end{equation}%
\end{lemma}%
\begin{proof}%
Let $T>0$ be arbitrary. The claim \eqref{eq:glivenko-cantelli_arrival_process} is then equivalent to
\begin{equation}%
\lim_{n\rightarrow\infty}\sup_{t\leq T}\Big\vert \bar A^n(t) - \fT(0) t\Big\vert = 0,\qquad\mathrm{a.s.}
\end{equation}%
Let $N$ be a large but arbitrary natural number and define
\begin{equation}%
t_j := \frac{1}{\fT(0)}\frac{j}{N}T,\qquad j=1,\ldots, N,
\end{equation}%
so that $\fT(0)t_j = \frac{j}{N}T$. The idea is that both $A^n(t)$ and $\fT(0)t$ are increasing, so for $t\in(t_{j-1},t_j)$ the difference of the two can be bounded by their values in $t_{j-1}$ and $t_j$. Then, we have convergence because of Lemma \ref{lem:LLN_arrival_process} and because $N$ is fixed. Formally, define the error as
\begin{equation}%
E_{n,N}:= \max_{j=1,\ldots,N}(\vert A^n(t_jn^{-1/3})/n^{2/3} - \fT(0)t_j\vert + \vert A^n(t_j^-n^{-1/3})/n^{2/3} - \fT(0)t_j^-\vert).
\end{equation}%
For $t\in(t_{j-1},t_j)$ we upper bound $\bar A^n(t)$ as follows
\begin{equation}%
\bar A^n(t)\leq \bar A^n(t_j^-)\leq \fT(0) t_j^- + E_{n,N} \leq \fT(0)t + E_{n,N} + \frac{T}{N},
\end{equation}%
where in the last inequality we used the bound $\vert \fT(0)t_{j}-\fT(0)t_{j-1}\vert \leq \frac{T}{N}$. Analogously, for the lower bound
\begin{equation}%
\bar A^n(t) \geq \bar A^n(t_{j-1})\geq \fT(0) t_{j-1} - E_{n,N} \geq \fT(0)t - E_{n,N} - \frac{T}{N}.
\end{equation}%
Summarizing the two bounds, since $E_{n,N}$ and $T/N$ do not depend on the choice of the sequence $(t_j)_{j=1}^N$,
\begin{equation}%
\sup_{t\leq T}\vert \bar A^n(t) - \fT(0) t \vert \leq E_{n,N} + \frac{T}{N}.
\end{equation}%
Since $N$ is fixed, almost surely
\begin{equation}%
\lim_{n\rightarrow \infty}E_{n,N} = 0,
\end{equation}%
by Lemma \ref{lem:LLN_arrival_process}. Letting $N\rightarrow\infty$, we obtain \eqref{eq:glivenko-cantelli_arrival_process}.
The convergence \eqref{eq:fCLT_cumulative_input_process} follows from \eqref{eq:glivenko-cantelli_arrival_process}. Indeed, by the functional strong Law of Large Numbers \cite[Theorem 5.10]{chen2001fundamentals} we have that
\begin{equation}%
\sum_{i=1}^{tn^{2/3}} \frac{S_i}{n^{2/3}}\asconv \E[S]t \qquad\mathrm{in}~(\mathcal D, U).
\end{equation}%
Since $\bar A^n(t)$ converges to a deterministic limit, we also have the joint convergence
\begin{equation}%
\Big(\sum_{i=1}^{tn^{2/3}} \frac{S_i}{n^{2/3}},\bar A^n(t)\Big)\asconv (\mathbb E[S]t, \fT(0)t),\qquad \mathrm{in}~(\mathcal D^2, WJ_1).
\end{equation}%
Note that $A^n(t)$ is non-decreasing. Then, by a time-change theorem \cite[Lemma p.151]{billingsley1999convergence},
\begin{equation}%
\sum_{i=1}^{A^n(tn^{-1/3})}\mkern-9mu \frac{S_i}{n^{2/3}} \asconv\E[S]\fT(0) t\qquad\mathrm{in}~(\mathcal D,U).
\end{equation}%
Recall that convergence in $(\mathcal D, J_1)$ to a continuous function implies convergence in $(\mathcal D, U)$. Moreover, $\E[S]\fT(0) = 1$ by the heavy-traffic condition \eqref{eq:heavy_traffic_condition}, and this concludes the proof of \eqref{eq:fCLT_cumulative_input_process}.
\end{proof}%
Since $t\mapsto \fT(0) t$ is not a proper distribution function, Theorem \ref{th:glivenko-cantelli_arrival_process} should also be interpreted as a \emph{local} version of the usual Glivenko-Cantelli Theorem.
Let us now define the fluid-scaled cumulative busy time process as
\begin{equation}%
\bar B^n (t) := n^{1/3}B^n(tn^{-1/3}).
\end{equation}%
We are able to prove the following lemma:
\begin{lemma}[Convergence of the time-changed service process]
With assumptions as above, as $n\rightarrow\infty$,
\begin{equation}%
\bar B^n(t)\asconv t,\qquad \mathrm{in}~(\mathcal D, U),
\end{equation}%
\end{lemma}
\begin{proof}
Recall that $B^n$ can be rewritten as
\begin{equation}%
B^n(t) = t + \Psi(N^n)(t)= t +\inf_{s\leq t} (C^n(s) - s)^-.
\end{equation}%
By Lemma \ref{th:glivenko-cantelli_arrival_process}, $n^{1/3}(C^n(t n^{-1/3}) - t n^{-1/3})\asconv 0$ in $(\mathcal D, U)$. Moreover, the null function is a continuity point of $\Psi(\cdot)$ with probability one \cite[Lemma 13.4.1]{StochasticProcess}. The claim then follows from the Continuous Mapping Theorem \cite[Theorem 3.4.3]{StochasticProcess}.
\end{proof}
\subsection{Proof of Theorem \ref{th:main_theorem}}
Since $\bar B^n(\cdot)$ converges to a deterministic limit, we have
\begin{equation}%
(\hat A^n(t), \hat S^n(t), \bar B^n(t))\dconv (\hat A(t), \hat S(t), t),\qquad\mathrm{in}~(\mathcal D^3, WJ_1).
\end{equation}%
Note also that $\hat A^n (\cdot)$ and $\hat S^n(\cdot)$ are independent processes, so that $\hat A(\cdot)$ and $\hat S(\cdot)$ are also independent. Applying the random time-change theorem \cite[Lemma p.151]{billingsley1999convergence}, we get
\begin{equation}%
(\hat A^n(t), \hat S^n(\bar B^n(t)) )\dconv (\hat A(t), \hat S(t)),\qquad\mathrm{in}~(\mathcal D^2, WJ_1).
\end{equation}%
Since the limit points are continuous, by \cite[Theorem 4.1]{whitt1980some} addition is also continuous, so that
\begin{equation}%
\hat A^n(t) - \hat S^n(t) + n^{2/3}(\FT(tn^{-1/3}) - \fT(0)tn^{-1/3})\dconv \hat A(t) - \hat S(t) -\frac{\fT'(0)}{2}t^2,\quad\mathrm{in}~(\mathcal{D}, J_1),
\end{equation}%
which is the first claim \eqref{eq:main_theorem}. By \cite[Theorem 13.5.1]{StochasticProcess}, the reflection map $\phi(\cdot)$ is continuous when $\mathcal D$ is endowed with the $J_1$ topology, from which the second claim \eqref{eq:main_theorem_reflected_convergence} follows.
\qed
\section{Sample path Little's Law}\label{sec:sample_path_little_law}
In this section we apply the ideas and results from the previous sections to derive a sample path version of Little's Law. The standard formulation of Little's Law relates the expected waiting time $\E[W]$, to the expected queue length $\mathbb E[L_q]$ as $\E[L_q] = \lambda \E[W]$, where $\lambda$ is the rate at which customers arrive. We will work instead with the \emph{virtual} waiting time $W^n(t)$, defined as
\begin{equation}\label{eq:virtual_waiting_time}
W^n(t) := C^n(t) - B^n(t).
\end{equation}%
Accordingly, we define the diffusion-scaled virtual waiting time as
\begin{equation}%
\hat W^n(t) := n^{2/3}\Big(C^n(tn^{-1/3}) - B^n(tn^{-1/3})\Big) = n^{1/3}\Big(\sum_{i=1}^{A^n(tn^{-1/3})}\frac{S_i}{n^{2/3}} - \bar B^n(t)\Big).
\end{equation}%
First, we rewrite the expression for $\hat W^n(t)$ as
\begin{align}\label{eq:virtual_waiting_time_diffusion_scaled}%
\hat W^n(t) &= n^{1/3}\Big(\sum_{i=1}^{\bar A^n(t)n^{2/3}}\frac{S_i}{n^{2/3}} - \E[S] \bar A^n(t)\Big)+ n^{1/3}\E[S]( \bar A^n(t) - n^{1/3} \FT(tn^{-1/3})) \nnl
&\quad + n^{1/3}\E[S](\FT(tn^{-1/3})-\fT(0)t) + n^{1/3}\E[S](\fT(0)t-\bar B^n(t)/\E[S]).
\end{align}%
By \eqref{eq:idle_time_representation}, $n^{1/3}(\fT(0)t- \bar B^n(t) /\E[S]) = \psi (\hat X^n)(t)$, so that \eqref{eq:virtual_waiting_time_diffusion_scaled} can be further simplified as
\begin{align}\label{eq:virtual_waiting_time_diffusion_scaled_rewritten}%
\hat W^n (t) &= \E[S]\hat Q^n(t)+ n^{1/3}\Big(\sum_{i=1}^{\bar A^n(t)n^{2/3}}\frac{S_i}{n^{2/3}} - \E[S]\bar A^n(t)\Big) + \E[S]\hat S^n(\bar B^n(t)).
\end{align}%
We now focus on the second and third terms in \eqref{eq:virtual_waiting_time_diffusion_scaled_rewritten}. Let us ignore the time change $t\mapsto \bar A^n(t)$ and $t\mapsto\bar B^n(t)$ for the moment. Then, the second and third terms in \eqref{eq:virtual_waiting_time_diffusion_scaled_rewritten} represent the difference between the diffusion-scaled partial sums and the (negative) diffusion-scaled counting process associated with the sequence of random variables $(S_i)_{i\geq1}$. These converge to the same limiting Brownian motion, so that their contribution to $\hat W^n(t)$ vanishes in the limit. We now aim to make this reasoning rigorous.
\begin{theorem}[Diffusion sample path Little's Law]\label{th:diffusion_sample_path_little_law}
As $n\rightarrow\infty$,
\begin{equation}%
\hat W^n(t) \dconv \hat W(t),\qquad\mathrm{in}~(\mathcal{D}, J_1),
\end{equation}%
where
\begin{equation}%
\hat W(t) := \E[S]\hat Q(t).
\end{equation}%
\end{theorem}%
\begin{proof}
Define the diffusion-scaled partial sum process as
\begin{equation}%
\hat P^n(t) = n^{1/3} \Big(\sum_{i=1}^{tn^{2/3}}\frac{S_i}{n^{2/3}}-\E[S]\bar A^n(t)\Big).
\end{equation}%
By \cite[Theorem 7.3.2]{StochasticProcess}, $\hat P^n(\cdot)$ and $\hat S^n(\cdot)$ jointly converge as follows:
\begin{equation}%
(\hat P^n(t), \hat S^n(t)) \dconv (- \E[S]\hat S(\E[S]t), \hat S(t)),\qquad\mathrm{in}~(\mathcal D^2, WJ_1),
\end{equation}%
where $\hat S^n(t)$ is the same as in \eqref{eq:claim_clt_renewal_processes}. Since $\hat A^n(t)$ is independent from $\hat P^n(t)$ and $\hat S^n(t)$,
\begin{equation}%
(\hat A^n(t), \hat P^n(t), \hat S^n(t)) \dconv (\hat A(t), - \E[S]\hat S(\E[S]t), \hat S(t)),\qquad\mathrm{in}~(\mathcal D^3, WJ_1).
\end{equation}%
Moreover, since $\bar A^n(\cdot)$ and $\bar B^n(\cdot)$ converge to deterministic limits, by \cite[Theorem 11.4.5]{StochasticProcess} the above convergence can be strengthened to
\begin{equation}%
(\hat A^n(t),\hat P^n(t), \hat S^n(t),\bar A^n(t),\bar B^n(t)) \dconv (\hat A(t), - \E[S]\hat S(\E[S]t), \hat S(t), \fT(0)t, t),\quad\mathrm{in}~(\mathcal D^4, WJ_1).
\end{equation}%
It follows that
\begin{equation}\label{eq:little_law_proof_joint_convergence}%
(\hat A^n(t),\hat P^n(\bar A^n(t)), \E[S]\hat S^n(\bar B^n(t))) \dconv (\hat A(t),- \E[S]\hat S(t),\E[S]\hat S(t)),\qquad\mathrm{in}~(\mathcal D^3, WJ_1),
\end{equation}%
by the heavy-traffic assumption \eqref{eq:heavy_traffic_condition}. The limit functions are continuous with probability one, and thus their sums converge to the sums of the limits. This observation, together with the Continuous Mapping Theorem and \eqref{eq:little_law_proof_joint_convergence} imply that
\begin{equation}%
\E[S]\hat Q^n(t)+ \hat P^n(\bar A^n(t)) + \E[S]\hat S^n(\bar B^n(t))\dconv \hat Q(t),\qquad\mathrm{in}~(\mathcal D, J_1),
\end{equation}%
as $n\rightarrow\infty$, as desired.
\end{proof}%
By our assumptions, $\E[S] = 1/\fT(0)$ so that we retrieve the usual form of Little's Law as
\begin{equation}%
\hat Q(t) = \fT(0)\hat W(t).
\end{equation}%
Note that $\fT(0) = \lambda$ when $T$ is exponentially distributed with mean $1/\lambda$.
Theorem \ref{th:diffusion_sample_path_little_law} should be contrasted with the analogous result in \cite[Proposition 4]{honnappa2015delta}. There, an extra diffusion term appears. This term is a function of the fluid limit of the queue length process. However, in our setting, this limit is the zero process, as can be seen in \eqref{eq:diffusion_scaled_queue_length}, where no centering is needed and thus the term disappears.
\section{Conclusions}\label{sec:conclusions}
While the $\DG$ queue originated as a simple model for the study of general time-inhomogeneous queueing systems, it has very recently gained much attention since it represents the standard model for queues in which only a finite number of customers request for service \cite{honnappa2014transitory}. In this paper we have shown how techniques from the theory of stochastic-process limits, and more specifically heavy-traffic diffusion approximations, can be successfully employed to prove convergence results for the $\DG$ queue. In particular, we have proven that when suitably rescaled (according to a non-standard scaling) the queue length process converges in distribution to a reflected Brownian motion with downwards parabolic drift. Our result is a generalization of \cite{bet2014heavy}, where the arrival times are assumed to follow an exponential distribution. There, more demanding embedding and martingale techniques were used. Therefore, the techniques we have introduced offer a significant computational and conceptual advantage, allowing one to easily study other quantities of interest of the $\DG$ model other than the queue length process. As an example of the strength of our approach, we have proven a sample path diffusion Little's Law, which relates the virtual waiting time and the queue length processes.
\section*{Acknowledgments}%
The author is very grateful to Debankur Mukherjee and Jori Selen for suggesting many improvements to the manuscript and numerous helpful discussions.
\bibliographystyle{abbrv}
|
1,116,691,499,375 | arxiv | \section{Introduction}
Statisticians are interested in inferring properties about a population based on independently sampled data. In the parametric regime, the inference problem boils down to constructing point estimates and confidence intervals for a finite number of unknown parameters. When the data-generation process is well-specified by the parametric family, an elegant asymptotic theory --- credited to Ronald Fisher in the 1920s --- has been established for maximum likelihood estimation (MLE). This asymptotic theory is readily generalizable to the model mis-specification setting, for a properly chosen risk function $\ell(\theta, z)$ and the corresponding empirical risk minimizer (ERM)
\begin{align*}
\widehat{\theta}_{\rm ERM} & \triangleq \argmin_{\theta} ~ \frac{1}{N} \sum_{i=1}^{N} \ell(\theta, z_i) &\text{empirical risk minimizer}, \\
\theta_* &\triangleq \argmin_{\theta} \operatorname{\mathbb{E}}_{\mathbf{z} \sim P} \ell(\theta, \mathbf{z}) &\text{population minimizer},
\end{align*}
with
\begin{align*}
\sqrt{N} \left( \widehat{\theta}_{\rm ERM} - \theta_* \right) \xrightarrow[]{\mathcal{L}} \mathcal{N}\left(0, \mathbf{H}(\theta_*)^{-1} \mathbf{\Sigma}(\theta_*) \mathbf{H}(\theta_*)^{-1}\right).
\end{align*}
Here $\theta$ is the parameter of the model, $z_i$'s are i.i.d draws from an unknown distribution $P$, Hessian $\mathbf{H}(\theta) \triangleq \operatorname{\mathbb{E}} \left[\nabla^2_\theta \ell(\theta, \mathbf{z}) \right]$, and $\mathbf{\Sigma}(\theta) \triangleq \operatorname{\mathbb{E}} \left[ \nabla_\theta \ell(\theta, \mathbf{z}) \otimes \nabla_\theta \ell(\theta, \mathbf{z}) \right]$.
Define the \textit{population landscape} $L(\theta)$ as\footnote{It is also called loss function in the statistical learning literature. In generalized methods of moment, $\mathbb{E}_{\mathbf{z} \sim P} \nabla_\theta \ell(\theta, \mathbf{z}) = 0$ is also called moment condition. The MLE can be also viewed as a special case with $\ell(\theta, \mathbf{z}) = -\log p_{\theta}(\mathbf{z})$ and the data-generation process being $P = P_{\theta_*}$.}
\begin{align}
L(\theta) \triangleq \operatorname{\mathbb{E}}_{\mathbf{z} \sim P} \ell(\theta, \mathbf{z}).
\end{align}
One should notice that the elegant statistical theory for inference holds under rather mild regularity conditions, without requiring a convex $L(\theta)$. However, it overlooks one important aspect: the optimization difficulty of the landscape on $\theta$.
Optimization techniques are required to solve for the above estimator $\widehat{\theta}$, as they rarely take closed form. Global convergence and computational complexity is only well-understood when the sample analog $\frac{1}{N} \sum_{i=1}^N \ell(\theta, z_i)$ is convex. The optimization is done iteratively
\begin{align}
\label{eq:updates}
\theta_{t+1} = \theta_t - \eta \mathbf{h}(\theta_t),
\end{align}
where the vector field $\mathbf{h}$ is based on the first- and/or second-order information, $\eta$ is step-size.
For the non-convex case, the convergence becomes less clear, but in practice people still employ these iterative methods. Nevertheless, in either case, the available convergence results fall short of the statistical goal: after a certain number of iterations, one is interested in knowing the sampling distribution of $\theta_t$, for uncertainty quantification of the optimization algorithm.
The goal of the present work is to combine the strength of the two worlds in inference and optimization: to characterize the statistical distribution of the iterative methods, with good optimization guarantee. Specifically,
we study particular stochastic optimization methods for the (possibly non-convex) population landscape $L(\theta)$ in the fixed dimension regime, and at the same time characterize the sampling distribution at each step, through establishing a non-asymptotic theory. We allow for model mis-specification, and require only mild moment conditions on the data-generating process.
\subsection{Motivation}
Observe the simple fact that what one actually wishes to optimize is the population objective $L(\theta) = \operatorname{\mathbb{E}}_{\mathbf{z} \sim P} \ell(\theta, \mathbf{z})$, not the sample version. Therefore, stochastic approximation pioneered by \cite{robbins1951stochastic, kiefer1952stochastic} stands out as a natural optimization approach for the statistical inference problem.
In modern practice, \textit{Stochastic Gradient Descent} (SGD) with mini-batches of size $n$ is widely used,
\begin{align}
\theta_{t+1} = \theta_{t} - \eta \widehat{\operatorname{\mathbb{E}}}_n \nabla_\theta \ell(\theta_t, \mathbf{z}),
\end{align}
where $\widehat{\operatorname{\mathbb{E}}}_n$ is the empirical expectation over $n$ independently sampled mini-batch data.
Our first observation follows from the intuition that Gaussian approximation holds for each step when $n$ is not too small, which we will make rigorous in a moment. Define
\begin{align}
\mathbf{b}(\theta) &= \operatorname{\mathbb{E}}_{\mathbf{z} \sim P} \nabla_{\theta} \ell(\theta, \mathbf{z}), \label{eq:grad} \\
\mathbf{V}(\theta) &= \left\{ \Cov[\nabla_{\theta} \ell(\theta, \mathbf{z}) ] \right\}^{1/2}, \label{eq:cov}
\end{align}
then observe the following approximation for \eqref{eq:heuristic} via Central Limit Theorem (CLT)
\begin{align}
\theta_{t+1} &= \theta_{t} - \eta \widehat{\operatorname{\mathbb{E}}}_n \nabla_\theta \ell(\theta_t, \mathbf{z}) \nonumber \\
&= \theta_{t} - \eta \operatorname{\mathbb{E}} \nabla_\theta \ell(\theta_t, \mathbf{z}) + \eta \left[ \operatorname{\mathbb{E}} \nabla_\theta \ell(\theta_t, \mathbf{z}) - \widehat{\operatorname{\mathbb{E}}}_n \nabla_\theta \ell(\theta_t, \mathbf{z}) \right] \nonumber \\
& \approx \theta_{t} - \eta \mathbf{b}(\theta_t) + \sqrt{2\beta^{-1} \eta} \mathbf{V}(\theta_t) \mathbf{g}_t \label{eq:heuristic}, \quad \text{with}~\beta \triangleq \frac{2n}{\eta},
\end{align}
where $\mathbf{g}_t, t\geq 0$ are independent isotropic Gaussian vectors\footnote{CLT states that for $X_i, i\in [n]$ i.i.d sampled, asymptotically the following convergence in distribution holds $\sqrt{n} \left[ \frac{1}{n} \sum_{i=1}^n X_i - \operatorname{\mathbb{E}} X \right] \xrightarrow[]{\mathcal{L}} \mathcal{N}(0, \Cov(X)).$
If we substitute $X_i = \mathbf{V}(\theta_t)^{-1} \nabla_\theta \ell(\theta_t, Z_i)$, condition on $\theta_t$, one can see where the isotropic Gaussian emerges.}.
The combination of $n,\eta$ provides a stronger approximation guarantee at each iteration
for large $n$, in contrast to the asymptotic normal approximation for the average trajectory in \cite{polyak1992acceleration} as $t\rightarrow \infty$.
The $\beta^{-1}$ quantifies the ``variance'' injected in each step (due to sampled mini-batches), or the ``temperature'' parameter: the larger the $\beta$ is, the closer the distribution is concentrated near the deterministic steepest gradient descent updates. The scaling of the step-size $\eta$ relates to Cauchy discretization of the It\^{o} diffusion process (as $\eta \rightarrow 0$) $$d \theta_t = - \mathbf{b}(\theta_t) dt + \sqrt{2\beta^{-1}} \mathbf{V}(\theta_t) dB_t.$$
Our second observation comes from a classic ``standardization'' idea in statistics --- we want to adjust the stochastic gradient vector at step $t$ by $\mathbf{V}(\theta_t)$ so that the conditional noise (conditioned on $\theta_t$) for each coordinate is independent and homogenous,
\begin{align}
\theta_{t+1} &= \theta_{t} - \eta \mathbf{V}(\theta_t)^{-1} \widehat{\operatorname{\mathbb{E}}}_n \nabla_\theta \ell(\theta_t, \mathbf{z}) \nonumber \\
& \approx \theta_{t} - \eta \mathbf{V}(\theta_t)^{-1} \mathbf{b}(\theta_t) + \sqrt{2\beta^{-1} \eta} \mathbf{g}_t \label{eq:heuristic-masg}.
\end{align}
Namely, noisier gradient information is weighted less.
This standardization trick in statistics is similar to the Newton/quasi-Newton method in second-order optimization, though with notable difference. The similarity lies in the fact the noisy gradient information is weighted according to some local version of ``curvature.'' However, the former uses root of second moment matrix, while the latter uses Hessian (second-order derivatives).
To answer the inference question about $L(\theta)$ using the ``moment-adjusted'' iterative method proposed in \eqref{eq:heuristic-masg}, one needs to know the sampling distribution of $\theta_t$ for a fixed $t$. One hopes to directly describe the distribution in a non-asymptotic fashion, instead of characterizing this distribution either through the asymptotic normal limit \citep{polyak1992acceleration} (passing over data one at a time) in the convex senario, or through the invariant distribution which could in theory take exponential time to converge for general non-convex $L(\theta)$ \citep{bovier2004metastability,raginsky2017non}. One thing to notice is that, at a fixed time $t$, the distribution is distinct from Gaussian, for general $\mathbf{b}$ and $\mathbf{V}$.
From an optimization angle, one would like the iterative algorithm to converge (to a local optima) quickly. This is also important for the purpose of inference: given the distribution can be approximately characterized at each step, one hopes that the distribution will concentrate near a local minimum of the population landscape $L(\theta)$ within a reasonable time budget, before the error accumulates in the stochastic process and invalidates the approximation.
\paragraph{Notations}
For a vector $v$, $\| v \| = \sqrt{v^T v}$ denotes the $\ell_2$ norm, and $v \otimes v = vv^T$ denotes the outer-product. We use $\| M \|$ to denote the operator norm for a matrix $M$. For a positive semi-definite matrix $M$, $\langle v, w \rangle_{M} = v^T M w$. We use $t \in [T]$ to denote indices $0\leq t\leq T$, and ``$\xrightarrow[]{\mathcal{L}}$'' for convergence in distribution. For two matrices $A$ and $B$, we use $A \otimes_K B$ to represent the Kronecker product. Moreover, $O, o$ are the Bachmann-Landau notations and $O_{\bf p}$ denotes stochastic boundedness. In the discussion, we use $O_{\epsilon, \delta}(\cdot)$ to denote the order of magnitude for parameters $\epsilon, \delta$ only, treating others as constants. For two probability measures $\mu, \nu$, we use $D_{\rm KL}(\mu, \nu)$ and $D_{\rm TV}(\mu, \nu)$ to denote the Kullback-Leibler and total variation distance respectively. Throughout, we denote the population gradient $\b \in \mathbb{R}^p$, moment matrix $\mathbf{V}, \mathbf{\Sigma} \in \mathbb{R}^{p \times p}$ using the boldface notation, with the hope of emphasizing their role in the paper.
\subsection{Contributions and Organization}
We propose the \textit{\textbf{M}oment-\textbf{a}djusted \textbf{s}tochastic \textbf{Grad}ient descent} (MasGrad), an iterative optimization method that infers the stationary points of the population landscape $L(\theta)$, namely $\{ \theta \in \mathbb{R}^p: \|\nabla L(\theta) \| = 0 \}$. The MasGrad is a simple variant of SGD that adjusts the descent direction using $\mathbf{V}(\theta_t)^{-1}$ (defined in \eqref{eq:cov}, the square root of the inverse covariance matrix) at the current location,
\begin{align*}
\text{MasGrad}: \quad \theta_{t+1} &= \theta_{t} - \eta \mathbf{V}(\theta_{t})^{-1} \widehat{\operatorname{\mathbb{E}}}_n \nabla_\theta \ell(\theta_t, \mathbf{z}).
\end{align*}
We summarize our main contributions in two perspectives. Extensions including estimation and computation of the moment-adjusted gradients will be discussed later in Section~\ref{sec:est-comp-mat-root}.
\smallskip
\noindent \textbf{Inference.} \quad The distribution of MasGrad updates $\theta_{t} \in \mathbb{R}^p$, with $n$ independently sampled mini-batch data at each step, can be characterized in a non-asymptotic fashion. Informally, for any data-generating distribution $\mathbf{z} \sim P$ under mild conditions, the distribution of $\theta_t$ --- denoted as $\mu(\theta_{t})$ --- satisfies,
\begin{align*}
D_{\rm TV} (\mu(\theta_t), \nu_{t, \eta}) \leq O_{t, n} \left( \sqrt{\frac{t}{n}} \right) \quad \Rightarrow \quad \mu(\theta_t) \xrightarrow[]{\mathcal{L}} \nu_{t, \eta}, ~\text{converge in distribution as $n \rightarrow \infty$}.
\end{align*}
Here $\nu_{t, \eta}$ is the distribution of $\xi_t$ that follows the update initialized with $\xi_0 = \theta_0$
\begin{align}
\label{eq:xi}
\xi_{t+1} = \xi_t - \eta \mathbf{V}(\xi_t)^{-1} \mathbf{b}(\xi_t) + \sqrt{2\beta^{-1} \eta} \mathbf{g}_t, ~ \mathbf{g}_t \sim \mathcal{N}(0, I_p)~\text{and}~\beta = \frac{2n}{\eta}.
\end{align}
Remark that $\nu_{t, \eta}$ only depends on $t, \eta$, and the first and second moments $\mathbf{b},\mathbf{V}$ of $\nabla \ell(\theta, \mathbf{z})$, regardless of the specific data-generating distribution $\mathbf{z} \sim P$.
The rigorous statement is deferred to Thm.~\ref{thm:couple.p}, and further extensions to the continuous time analog are discussed in Appendix~\ref{sec:continuous-langevin}.
\smallskip
\noindent \textbf{Optimization.} \quad Interestingly, in the strongly convex case such as in generalized linear models (GLMs), the ``standardization'' idea achieves the Nesterov acceleration \citep{nesterov1983method, nesterov2013introductory}. Informally, the number of iterations for an $\epsilon$-minimizer for gradient descent requires
\begin{align*}
T_{\rm GD} = O_{\epsilon, \kappa}\left( \kappa \log \frac{1}{\epsilon} \right), \quad \text{for some $\kappa > 1$}.
\end{align*}
We show that for GLMs under mild conditions, MasGrad reduces the number of iterations to
\begin{align*}
T_{\rm MasGrad} = O_{\epsilon, \kappa}\left( \sqrt{\kappa} \log \frac{1}{\epsilon} \right),
\end{align*}
which matches Nesterov's acceleration in the strongly convex case. The formal statement is deferred to Section~\ref{sec:acceleration}, where extensions including proximal updates are discussed.
\smallskip
Combining the inference and optimization theory together, we present informally the results for both the \textit{convex} and \textit{non-convex} cases. Recall that $\theta \in \mathbb{R}^p$.
\smallskip
\noindent \textbf{Convex.} \quad In the strongly convex case, MasGrad with a properly chosen step-size and the following choice of parameters
\begin{align*}
T = O_{\epsilon}\left(\log \frac{1}{\epsilon}\right) ~~\text{and}~~ n = O_{\epsilon,p}\left( \frac{p}{\epsilon} \right),
\end{align*}
satisfies
\begin{align*}
&\text{inference}: \quad D_{\rm TV}\left( \mu(\theta_T), \mu(\xi_T) \right) \leq O_{\epsilon}\left( \sqrt{\epsilon \log 1/\epsilon} \right),\\
&\text{optimization}: \quad \operatorname{\mathbb{E}} L(\theta_T) - \min_{\theta} L(\theta) \leq \epsilon, ~\operatorname{\mathbb{E}} L(\xi_T) - \min_{\theta} L(\theta) \leq \epsilon,~~\text{where $\xi_T \sim \nu_{T, \eta}$,}
\end{align*}
where the evolution of $\xi_t$ is defined in \eqref{eq:xi}.
Here the total number of samples needed is $nT = O_\epsilon(\epsilon^{-1} \log 1/\epsilon)$. The formal result is stated in Thm.~\ref{thm:converge}.
\smallskip
\noindent \textbf{Non-convex.} \quad Under mild smoothness conditions, MasGrad with a proper step-size and the following choice of parameters
\begin{align*}
T = O_{\epsilon,\delta, p}\left( \frac{1 \vee p\delta^2}{\epsilon^2}\right) ~~\text{and}~~ n = O_{\epsilon,\delta, p}\left( \frac{\delta^{-2} \vee p}{\epsilon^2}\right),
\end{align*}
satisfies
\begin{align*}
&\text{inference}: \quad D_{\rm TV}\left( \mu(\theta_t, t\in [T]), \mu(\xi_t, t\in [T]) \right) \leq O_{\delta}(\delta),\\
&\text{optimization}: \quad \operatorname{\mathbb{E}} \min_{t \leq T} \| \nabla L(\theta_t) \| \leq \epsilon, ~\operatorname{\mathbb{E}} \min_{t \leq T} \| \nabla L(\xi_t) \| \leq \epsilon,~~\text{where $\xi_t \sim \nu_{t,\eta}$, for $t \in [T]$.}
\end{align*}
Here the total number of samples needed is $nT = O_{\epsilon,\delta}(\epsilon^{-4} \delta^{-2})$. The formal result is deferred to Thm.~\ref{thm:non-convex}.
\section{Relations to the Literature}
\label{sec:literature}
In the case of a differentiable convex $L(\theta)$, finding a minimum is equivalent to solving $\nabla L(\theta) = 0$. This simple equivalence reveals that the vanilla SGD, which takes the form\footnote{Recognize that $\nabla_\theta \ell(\theta_t, z_t)$ is an unbiased estimate of the population gradient as $\nabla_\theta L(\theta_t) = \operatorname{\mathbb{E}}_{\mathbf{z} \sim P}[\nabla_\theta \ell(\theta_t, \mathbf{z})]$.}
\begin{align}
\label{eq:sto.approx}
\theta_{t+1} = \theta_{t} - \eta_t \nabla_\theta \ell(\theta_t, z_t),
\end{align}
is an instance of stochastic first-order approximation methods. This class of methods are iterative algorithms that attempt to solve fixed-point equations (for example, $\nabla L(\theta) = 0$) provided noisy observations (for example, $\nabla_\theta \ell(\theta_t, z_t)$) \citep{robbins1951stochastic,kiefer1952stochastic,toulis2017asymptotic,chen2016statistical,li2017statistical}. Using slowly diminishing step-sizes $\eta_t = O(1/t^{\alpha})$ ($\alpha<1$), \citet{ruppert1988} and \citet{polyak1990} showed that acceleration using the average over trajectories of this recursive stochastic approximation algorithm attains optimal convergence rate for a strongly convex $L$ (see \cite{polyak1992acceleration} for more details). Recently, the running time of stochastic first-order methods are considerably improved using combinations of variance-reduction techniques \citep{roux2012stochastic,johnson2013accelerating} and Nesterov's acceleration \citep{ghadimi2016accelerated,cotter2011better,jofre2017variance,ghadimi2012optimal,arjevani2016oracle}.
Despite the celebrated success of stochastic first-order methods in modern machine learning tasks, researchers have kept improving the per-iteration complexity of second-order methods such as Newton or quasi-Newton methods, due to their faster convergence.
A fruitful line of research has focused on how to improve asymptotic convergence rate as $t \rightarrow \infty$ through pre-conditioning, a technique that involves approximating the unknown Hessian $\mathbf{H}(\theta) = \nabla^2_\theta L(\theta)$ (see, for instance, \citet{bordes2009sgd} and references therein).
Utilizing the curvature information reflected by various efficient approximations of the Hessian matrix, stochastic quasi-Newton methods \citep{moritz2016linearly,byrd2016stochastic,wang2017stochastic,schraudolph2007stochastic,mokhtari2015global,becker2012quasi}, Newton sketching or subsampled Newton \citep{pilanci2015newton,xu2016sub,berahas2017investigation,bollapragada2016exact}, and stochastic approximation of the inverse Hessian via Taylor expansion \citep{agarwal2017second} have been proposed to strike balance between convergence rate and per-iteration complexity.
In the information geometry literature, one closely related method is the natural gradient \citep{amari1998natural,amari2012differential}. When the parameter space enjoys a certain structure, it has been shown that natural gradient outperforms the classic gradient descent both theoretically and empirically. To adapt the natural gradient to our setting, we relate the loss function to a generative model $\ell(\theta, z) = - \log p_{\theta}(z)$. The Riemannian structure of the parameter space (manifold) of the statistical model is defined by the Fisher information
\begin{align*}
\mathbf{I}(\theta) = \operatorname{\mathbb{E}}_{\mathbf{z} \sim P} \left[ \nabla_{\theta} \ell(\theta, \mathbf{z}) \otimes \nabla_{\theta} \ell(\theta, \mathbf{z}) \right].
\end{align*}
The natural gradient can be viewed as the steepest descent induced by the Riemannian metric
\begin{align*}
\theta_{t+1} &= \argmin_{\theta} \left[ L(\theta_t) + \langle \nabla_{\theta} L(\theta_t), \theta - \theta_t \rangle + \frac{1}{2\eta_t} \| \theta - \theta_t \|_{\mathbf{I}(\theta_t)}^2 \right] \\
&= \theta_t - \eta_t \mathbf{I}(\theta_t)^{-1} \nabla_{\theta} L(\theta_t).
\end{align*}
Note the intimate connection between natural gradient descent and approximate second-order optimization method, as the Fisher information can be heuristically viewed as an approximation of the Hessian \citep{schraudolph2002fast, martens2014new}.
Another popular and closely related example as such is AdaGrad \citep{duchi2011adaptive}, which is a variant of SGD that adaptively determines learning rates for different coordinates by incorporating the geometric information of past iterates. In its simplest form, AdaGrad records previous gradient information through
\[
G_t = \sum_{i=1}^t \nabla \ell(\theta_i, z_i) \otimes \nabla \ell(\theta_i, z_i),
\]
and this procedure then updates iterates according to
\[
\theta_{t+1} = \theta_t - \gamma G^{-\frac12}_t \nabla \ell(\theta_t, z_t),
\]
where $\gamma > 0$ is fixed. In large-scale learning tasks, evaluating $G_t^{-\frac12}$ is computationally prohibitive and thus is often suggested to use $\text{diag}(G_t)^{-\frac12}$ instead. It should be noted, however, that the theoretical derivation of regret bound for AdaGrad considers $G_t^{-\frac12}$. AdaGrad is a flexible improvement on SGD and can easily extend to non-smooth optimization and non-Euclidean optimization such as mirror descent. With the geometric structure $G_t$ learned from past gradients, AdaGrad assigns different learning rates to different components of the parameter, allowing infrequent features to take relatively larger learning rates. This adjustment is shown to speed up convergence dramatically in a wide range of empirical problems \citep{pennington2014glove}.
Stochastic Gradient Langevin Dynamics (SGLD) has been an active research field in sampling and optimization in recent years \citep{welling2011bayesian, dalalyan2017theoretical,
bubeck2015sampling, raginsky2017non, mandt2017stochastic, brosse2017sampling, tzen2018local, durmus2018efficient}. SGLD injects an additional $\sqrt{2\beta^{-1}\eta}$ level isotropic Gaussian noise to each step of SGD with step-size $\eta$, where $\beta$ is the inverse temperature parameter. Besides similar optimization benefits as SGD such as convergence and chances of escaping stationary points, the injected randomness of SGLD provides an efficient way of sampling from the targeted invariant distribution of the continuous-time diffusion process, which has been shown to be useful statistically in Bayesian sampling \citep{welling2011bayesian, mandt2017stochastic, durmus2018efficient}.
In the current paper, we take a distinct approach: we motivate and analyze a variant of SGD through the lens of Langevin dynamics, from a frequentist point of view, and then present the optimization benefits as a by-product of the statistical motivation. The approximation in Eqn.~\eqref{eq:heuristic} relates the density evolution of $\theta_s$ to a discretized version of It\^{o} diffusion process (as $\eta \rightarrow 0$)
\begin{align*}
d \theta_s = - \mathbf{b}(\theta_s) ds + \sqrt{2\beta^{-1}} \mathbf{V}(\theta_s) dB_s.
\end{align*}
The invariant distribution $\pi(\theta)$ satisfies the following Fokker--Planck equation
\begin{align*}
\beta^{-1} \sum_{i,j} \frac{\partial^2}{\partial x_i x_j} (\pi \mathbf{a}_{ij}) + \sum_{i} \frac{\partial}{\partial x_i} (\pi \mathbf{b}_{i}) = 0
\end{align*}
where $\mathbf{a}_{ij}(x) = (\mathbf{V}(x) \mathbf{V}(x)')_{ij}$.
In general, the stationary distribution is hard to characterize unless both $\mathbf{V}$ and $\mathbf{b}$ take special simple forms.
For example, when $\mathbf{b}(x)$ is linear and $\mathbf{V}(x)$ is independent of $x$ as in \citep{mandt2017stochastic}, the diffusion process reduces to Ornstein-Uhlenbeck process with multivariate Gaussian as the invariant distribution. Another simple case is when $\mathbf{V}(x) = \mathbf{I}$, the diffusion process is also referred to as Langevin dynamics, with the Gibbs measure $\pi(\theta) \propto \exp(-\beta L(\theta))$ as the unique invariant distribution \citep{welling2011bayesian, dalalyan2017theoretical, raginsky2017non}.
\section{Statistical Inference via Langevin Diffusion}
\label{sec:stat}
In this section we will explain why \textit{\textbf{M}oment-\textbf{a}djusted \textbf{s}tochastic \textbf{Grad}ient descent} (MasGrad) produces recursive updates whose statistical distribution can be characterized. We would like to mention that MasGrad at the same time achieves significant acceleration in optimization in the strongly convex case (detailed in Section~\ref{sec:acceleration}). For the general non-convex case, we provide non-asymptotic theory for inference and optimization in Section~\ref{sec:non-convex}. We first present the simplest version of the algorithm, assuming that $\mathbf{V}(\theta)^{-1}$ can be evaluated at any given $\theta$. Statistical estimation and efficient direct computation of $\mathbf{V}(\theta)^{-1}$ will be discussed in Section~\ref{sec:est-comp-mat-root}.
Recall the MasGrad we introduced, which adjusts the gradient direction using the root of the inverse covariance matrix at the current location,
\begin{align}
\label{eq: MasGrad}
\text{MasGrad}: \quad \theta_{t+1} &= \theta_{t} - \eta \mathbf{V}(\theta_{t})^{-1} \widehat{\operatorname{\mathbb{E}}}_n \nabla_\theta \ell(\theta_t, \mathbf{z}).
\end{align}
As we have heuristically outlined in Eqn.~\eqref{eq:heuristic}, the MasGrad can be approximated by the following discretized Langevin diffusion,
\begin{align}
\label{eq:discrete.diff}
\text{Discretized diffusion}: \quad \xi_{t+1} &= \xi_{t} - \eta \mathbf{V}(\xi_{t} )^{-1} \mathbf{b}(\xi_{t}) + \sqrt{2\beta^{-1} \eta} \mathbf{g}_t.
\end{align}
In this section, we establish non-asymptotic bounds on the distance between the distribution of MasGrad process $\mathcal{L}(\theta_t, t \in [T])$ and discretized diffusion process $\mathcal{L}(\xi_t, t \in [T])$.
The proof is based on the entropic Central Limit Theorem (entropic CLT) \citep{barron1986entropy, bobkov2013, bobkov2014berry}. The classic CLT based on convergence in distribution is too weak for our purpose: we need to translate the non-asymptotic bounds at each step to the whole stochastic process. It turns out that the entropic CLT couples naturally with the chain-rule property of relative entropy, which together provides non-asymptotic characterization on closeness of the distributions for the stochastic processes.
Let's first state the standard assumptions for entropic CLT. These assumptions can be found in \citep{bobkov2013}. Remark that we are focusing on fixed dimension setting.
\begin{enumerate}[label={\bf (A.\arabic*)}]
\item Absolute continuity to Gaussian: assume random vector $X \in \mathbb{R}^p$ has bounded entropic distance to the Gaussian distribution, for some constant $D_1$
\begin{align*}
D_{\rm KL}\left( \mu(X) || \mu(\mathbf{g}) \right) < D_1, \quad \text{where $\mathbf{g} \sim \mathcal{N}(0,I_p)$.}
\end{align*}
\item Finite $(4+\delta)$-th moments: assume that there exists constant $D_2$
\begin{align*}
\mathbb{E} \| X \|^{4+\delta} < D_2, \quad \text{for some small $\delta > 0$.}
\end{align*}
\end{enumerate}
Define $\forall i$, the stochastic component of the adjusted gradient direction
\begin{align}
\label{eq:asmp}
X_i(\theta) = \mathbf{V}(\theta)^{-1} \left[ \nabla_\theta \ell(\theta, z_i) - \operatorname{\mathbb{E}}_{\mathbf{z} \sim P} \nabla_\theta \ell(\theta, \mathbf{z}) \right].
\end{align}
It is clear that $X_i$'s are i.i.d. with $\operatorname{\mathbb{E}} X_i(\theta) = 0$ and $\Cov [X_i(\theta)] = I_p$. Here $X_i(\theta)$ is defined on the same $\sigma$-field as $z_i$ drawn from $P$.
\begin{thm}[Non-asymptotic bound for inference]
\label{thm:couple.p}
Let $\mu(\theta_t, t \in [T])$ denote $\mathcal{L}(\theta_t, t \in [T])$, the joint distribution of MasGrad process, and $\mu( \xi_t, t \in [T] )$ be the joint distribution of the discretized diffusion process in \eqref{eq:discrete.diff}. Consider the same initialization $\theta_0 = \xi_0$.
Assume that uniformly for any $\theta$, $X(\theta)$ defined in \eqref{eq:asmp} satisfies {\bf(A.1)} and {\bf(A.2)} with constants $D_1, D_2$ that only depends on $p$. Then the following bound holds,
\begin{align}
D_{\rm TV}\left( \mu(\theta_t, t \in [T]), \mu( \xi_t, t \in [T] ) \right) \leq C \sqrt{\frac{T}{n} + o\left( \frac{T(\log n)^{\frac{p - (4+\delta)}{2}}}{n^{1+\frac{\delta}{2}}} \right) },
\end{align}
where $C$ is some constant that depends on the $D_1$ and $D_2$ only.
\end{thm}
\begin{remark}
\rm
The above theorem characterizes the sampling distribution of MasGrad -- $\theta_t$, using a measure that only depends on the first and second moments of $\nabla \ell(\theta, \mathbf{z})$, namely $\mathbf{V}(\theta)^{-1}\mathbf{b}(\theta)$, regardless of the specific the data-generating distribution $\mathbf{z} \sim P$. Observe that the distribution closeness is established in a strong total variation distance sense, for the two stochastic processes $\{\theta_t, t\in [T]\}$ and $\{ \xi_t, t \in [T]\}$. If we dig in to the proof, one can easily obtain the following marginal result
\begin{align*}
D_{\rm TV}\left( \mu(\theta_T), \mu(\xi_T) \right) \leq \sqrt{2 D_{\rm KL} \left( \mu(\theta_T) || \mu(\xi_T) \right) } \leq \sqrt{2 D_{\rm KL} \left( \mu(\theta_t, t\in [T]) || \mu(\xi_t, t\in [T]) \right) },
\end{align*}
where the last inequality follows from the chain-rule of relative entropy.
Therefore, one can as well prove for the last step distribution
$$
D_{\rm TV}\left( \mu(\theta_T), \mu(\xi_T) \right) \leq C \sqrt{\frac{T}{n} }.
$$
\end{remark}
\begin{remark}
\rm
One important fact about Thm.~\ref{thm:couple.p} is that it holds for any step-size $\eta$, which provides us additional freedom of choosing the optimal step-size for the optimization purpose. This theorem is stated in the fixed dimensional setting when $p$ does not change with $n$. Remark in addition that the Gaussian approximation at each step still holds with high probability, in the moderate dimensional setting when $p = o(\frac{\log n}{\log \log n})$, as shown in the non-asymptotic bound in the above Thm.~\ref{thm:couple.p}. We would like to emphasize that the current paper only considers the fixed dimension setting, while considering the mini-batch sample size $n$ and running time $T$ varying. Assumptions (A.1) and (A.2) are standard assumptions in entropic CLT: (A.1) states that the distribution for each stochastic gradient is non-lattice with bounded relative entropy to Gaussian; (A.2) is the standard weak moment condition. Note here that the constants $D_1$ and $D_2$ depend on the dimension implicitly.
For the purpose of statistical inference, one can always approximately characterize the distribution of MasGrad using Thm.~\ref{thm:couple.p}. As an additional benefit, the result naturally provides us an algorithmic way of sampling this target universal distribution $\mu(\xi_t)$.
For some particular tasks, it remains of theoretical interest to analytically characterize the distribution of MasGrad using the continuous time Langevin diffusion and its invariant distribution. We defer the analysis of the discrepancy between the discretized diffusion to the continuous time analog to Appendix~\ref{sec:continuous-langevin}.
\end{remark}
\section{Convexity and Acceleration}
\label{sec:acceleration}
In this section, we will demonstrate that the ``moment-adjusting'' idea motivated from standardizing the error from an inference perspective achieves similar effect as acceleration in convex optimization. We will investigate \textit{Generalized Linear Models} (GLMs) as the main example. Later, we will also discuss the case with non-smooth regularization. It should be noted that using first-order information to achieve acceleration was first established in the seminal work by \cite{nesterov1983method, nesterov2013introductory} based on the ingenious notion of estimating sequence. Before diving into the technical analysis, we would like to point out that in MasGrad the moment-adjusting matrix $\mathbf{V}(\theta)$ can be estimated using only first-order information, however, as one will see, MasGrad achieves acceleration for GLMs in a way resembles the approximate second-order method such as quasi-Newton.
\subsection{Inference and optimization for optima}
Now we are ready to state the theory for inference and optimization using MasGrad in the strongly convex case. Let $L(w): \mathbb{R}^p \rightarrow \mathbb{R}$ be a smooth convex function. Recall $\mathbf{b}(w) = \nabla L(w)$, $\mathbf{H}(w) = \nabla^2 L(w)$ and $\mathbf{V}(w) \in \mathbb{R}^{p \times p}$ are positive definite matrices.
Define
\begin{align}
\label{eq:conv.glm}
\alpha &\triangleq \min_{v, w}~ \lambda_{\min} \left( \mathbf{V}(w)^{-1/2} \mathbf{H}(v) \mathbf{V}(w)^{-1/2} \right) >0, \nonumber \\
\gamma &\triangleq \max_{v, w}~ \lambda_{\max} \left( \mathbf{V}(w)^{-1/2} \mathbf{H}(v) \mathbf{V}(w)^{-1/2} \right) >0.
\end{align}
\begin{thm}[MasGrad: strongly convex]
\label{thm:converge}
Let $\alpha, \gamma$ be defined as in \eqref{eq:conv.glm}.
Consider the MasGrad updates $\theta_t$ in \eqref{eq: MasGrad} with step-size $\eta = 1/\gamma$, and the corresponding discretized diffusion $\xi_t$,
\begin{align*}
\xi_{t+1} = \xi_t - \eta \mathbf{V}(\xi_t)^{-1} \mathbf{b}(\xi_t)+ \sqrt{2\beta^{-1} \eta} \mathbf{g}_t,\quad \text{where $\beta = \frac{2n}{\eta}$}.
\end{align*}
Then for any precision $\epsilon >0$, one can choose
\begin{align}
\label{eq:conv-T-n}
T = \frac{\gamma}{\alpha} \log \frac{2(L(\theta_0) - \min_{\theta} L(\theta))}{\epsilon} ~~\text{and}~~ n = \frac{4p \max_{\theta} \| \mathbf{V}(\theta) \|}{\alpha \epsilon},
\end{align}
such that
\begin{align*}
&(1)\quad D_{\rm TV}\left( \mu(\theta_t, t \in [T]), \mu( \xi_t, t \in [T] ) \right) \leq O_{\epsilon}\left( \sqrt{\epsilon \log (1/\epsilon)} \right),\\
&(2)\quad \operatorname{\mathbb{E}} L(\theta_t) - \min_{\theta} L(\theta) \leq \epsilon, ~ \operatorname{\mathbb{E}} L(\xi_t) - \min_{\theta} L(\theta) \leq \epsilon,
\end{align*}
with in total $O_{\epsilon}(\epsilon^{-1} \log 1/\epsilon)$ independent data samples.
\end{thm}
\begin{remark}
\rm
In plain language, discretized diffusion process $\xi_t, t \in [T]$, whose distribution only depends on the adjusted moments $\mathbf{V}^{-1}\b$, approximates the sampling distribution of MasGrad $\theta_t, t \in [T]$ in a strong sense, i.e., the distribution of paths are close in TV distance. In addition, as a stochastic optimization method, MasGrad's optimization guarantee depends on the ``modified'' condition number defined in \eqref{eq:conv.glm}. Let's sketch the proof. Using Lemma~\ref{lem:conv.glm} in Appendix~\ref{sec:proof}, for all $t>0$, one can prove
\begin{align*}
\operatorname{\mathbb{E}} L(\xi_{t}) - \min_{\theta} L(\theta) \leq \left(1 - \frac{\alpha}{\gamma}\right)^t (L(\theta_0) - \min_{\theta} L(\theta)) + \max_{\theta} \| \mathbf{V}(\theta) \| \cdot \frac{\gamma}{\alpha} \beta^{-1} p.
\end{align*}
Therefore we can define the condition number of MasGrad as
\begin{align}
\label{eq:cond.num}
\kappa_{\rm MasGrad} = \frac{\max_{w, v} \lambda_{\max} \left( [\mathbf{V}(w)]^{-1/2} \mathbf{H}(v) [\mathbf{V}(w)]^{-1/2} \right)}{\min_{w, v} \lambda_{\min} \left( [\mathbf{V}(w)]^{-1/2} \mathbf{H}(v) [\mathbf{V}(w)]^{-1/2} \right) },~~ \kappa_{\rm GD} = \frac{\max_{v} \lambda_{\max} \left( \mathbf{H}(v) \right)}{\min_{v} \lambda_{\min} \left( \mathbf{H}(v) \right)},
\end{align}
in contrast to the condition number in gradient descent.
If $\beta = \frac{2n}{\eta}$ and $T, n$ are chosen as in \eqref{eq:conv-T-n},
we know that $\operatorname{\mathbb{E}} L(\xi_{T}) - L(\theta_*) \leq \epsilon$. Recall the result we establish in Thm.~\ref{thm:couple.p}, the total variation distance between MasGrad and the discretize diffusion in this case is bounded by
$\sqrt{T/n} = O_{\epsilon} \left( \sqrt{\epsilon \log (1/\epsilon) } \right),$
and the total number of samples used is of the order $nT = O_{\epsilon, p}(p/\epsilon \log (1/\epsilon))$. This result can be contrasted with the classical asymptotic normality for MLE or ERM: to achieve an $\epsilon$-minimizer,
\begin{align*}
\epsilon \geq L(\widehat{\theta}_N) - L(\theta_*) \asymp \| \widehat{\theta}_N - \theta_* \|^2 \asymp \frac{p}{N} \Leftrightarrow N = O_{\epsilon,p}(p/\epsilon),
\end{align*}
the asymptotic sample complexity scales $O_{\epsilon, p}(p/\epsilon)$. Similar calculations also hold with the Ruppert--Polyak average on stochastic approximation with a carefully chosen decreasing step-size. As we can see, our result holds non-asymptotically, and it achieves both the optimization and inference goal, with an additional logarithmic factor.
\end{remark}
\subsection{Acceleration for GLMs}
Now let's take GLMs as an example to articulate the effect of acceleration. We will first use an illustrating toy example to show the intuition in an informal way, and then present the rigorous acceleration result for GLMs.
\smallskip
\paragraph{Toy example (informal).} Consider $y_i = \langle x_i, \theta_\ast \rangle + \epsilon_i$, $\epsilon_i \sim \mathcal{N}(0, \sigma^2)$ i.i.d. for $i \in [N]$. Let's focus on the fixed design case (where the expectation is only over $\mathbf{y}$), the loss $\ell(\theta, (x, y)) = \frac{1}{2}(\langle x, \theta \rangle - y)^2$. Denote $X \in \mathbb{R}^{N \times p}$, then we have
\begin{align*}
\mathbf{b}(\theta) &= \mathbb{E} \left[ \frac{1}{N} \sum_{i=1}^N ( x_i^T \theta - y_i) x_i \right] = \frac{1}{N} \sum_{i=1}^N x_i x_i^T (\theta - \theta_*) = \frac{1}{N} X^T X (\theta - \theta_*), \\
\mathbf{V}(w) &= \left[ \frac{1}{N} \sum_{i=1}^N x_i x_i^T \sigma^2 \right]^{1/2} = \sigma \left[\frac{1}{N} X^T X\right]^{1/2},
\end{align*}
and the Hessian is $\mathbf{H}(w) = X^T X/N.$
Therefore, in this case, we have $$\kappa_{\rm MasGrad} = \sqrt{\kappa_{\rm GD}}.$$
By applying Lemma~\ref{lem:conv.glm} in Appendix~\ref{sec:proof}, one achieves the same effect as Nesterov's acceleration in the strongly convex case \citep{nesterov2013introductory}. Remark that the above analysis is to demonstrate the intuition, and is not rigorous --- as MasGrad is sensible with the random design.
\paragraph{Generalized linear models, random design, mis-specified model.}
Now let's provide a rigorous and unified treatment for the generalized linear models.
Consider the generalized linear model \citep{mccullagh1984generalized} where the response random variable $\mathbf{y}$ follows from the exponential family parametrize by $(\theta, \phi)$,
\begin{align*}
f(y; \theta,\phi) = b(y, \phi) e^{\frac{y\theta - c(\theta)}{d(\phi)}}
\end{align*}
where $\mu = \mathbb{E} [\mathbf{y} | \mathbf{x} = x] = c'(\theta)$, $c''(\theta)>0$, and the natural parameter satisfies the linear relationship $\theta = \theta(\mu) = x^Tw$. In this case, we choose the loss function according to the negative log-likelihood
\begin{align*}
\ell(w, (x, y)) = - y_i x_i^T w + c(x_i^T w).
\end{align*}
Special cases include,
\begin{itemize}
\item Bernoulli model (Logistic regression): $c(\theta) = \log (1 + e^\theta), ~\text{where}~x_i^Tw = \theta =\log \frac{\mu}{1 - \mu}$;
\item Poisson model (Poisson regression): $c(\theta) = e^\theta, ~\text{where}~x_i^Tw = \theta= \log \mu$;
\item Gaussian model (linear regression): $c(\theta) = \frac{1}{2}\theta^2, ~\text{where}~x_i^Tw = \theta = \mu$.
\end{itemize}
We are interested in inference even when the model can be \textit{mis-specified}.
Consider the statistical learning setting where $z_i = (x_i, y_i) \sim P = P_{\mathbf{x}} \times P_{\mathbf{y}|\mathbf{x}}, i\in [N]$ i.i.d. from some unknown joint distribution $P$. We are trying to infer the parameters $w$ by fitting the data using a parametric exponential family, however, we allow the flexibility that the exponential family model for $P(\mathbf{y}|\mathbf{x}=x)$ can be mis-specified. Specifically, the true regression function $m_*(x) = \mathbb{E}(\mathbf{y}|\mathbf{x} = x)$ may not be $c'(x^T w)$ for all $w$, namely, may not be realized by any model in the exponential family model class.
We have the population landscape
\begin{align}
\label{eq:glm.landscape}
L(w) = \operatorname{\mathbb{E}}_{(\mathbf{x}, \mathbf{y})\sim P} \left[ - \mathbf{y} \mathbf{x}^T w + c(\mathbf{x}^T w) \right].
\end{align}
Define the conditional variance $\xi(x) = \operatorname{\mathbb{V}\textnormal{ar}}(\mathbf{y}|\mathbf{x}=x) \in \mathbb{R}$ and the bias $\beta(\mathbf{x}, w) \triangleq c'(\mathbf{x}^T w) - m_*(\mathbf{x}) \in \mathbb{R}$, we have the following acceleration result for GLMs.
\begin{thm}[Acceleration]
\label{thm:acceleration}
Consider the condition number defined in ~\eqref{eq:cond.num} for MasGrad and GD, and assume that there exists constant $C>1$ such that for any $x, w, v$,
\begin{align*}
0 < \max \left\{ \frac{\xi(x)^2 + \beta(x, w)^2}{c''(x^T v)}, \frac{c''(x^T v)}{\xi(x)^2} \right\} < C^{1/3}.
\end{align*}
Then for the optimization problem associated with GLMs defined in \eqref{eq:glm.landscape}, the following holds
$$
\kappa_{\rm MasGrad} < C \sqrt{\kappa_{\rm GD}}.
$$
\end{thm}
\begin{remark}
\rm
The above theorem together with Lemma~\ref{lem:conv.glm} in Appendix~\ref{sec:proof} states that in the noiseless setting, the time complexity for MasGrad is
$O\left( \sqrt{\kappa_{\rm GD}} \log 1/\epsilon \right)$
in contrast to the complexity of GD --
$O\left( \kappa_{\rm GD} \log 1/\epsilon \right)$, which is crucial when the condition number is large. The proof is based on matrix inequalities and the following analytic expressions,
\begin{align*}
&\mathbf{b}(w) = \mathbb{E}\left[ -\mathbf{y} \mathbf{x} + c'(\mathbf{x}^T w) \mathbf{x} \right] = \mathbb{E}\left[ (c'(\mathbf{x}^T w) - m_*(\mathbf{x})) \mathbf{x} \right], \\
&\mathbf{V}(w) = \left( \mathbb{E}[ \xi(\mathbf{x})^2 \mathbf{x} \mathbf{x}^T] + \Cov[\beta(\mathbf{x}, w) \mathbf{x} ] \right)^{1/2}, \quad \mathbf{H}(w) = \operatorname{\mathbb{E}}\left[ c''(\mathbf{x}^T w) \mathbf{x} \mathbf{x}^T \right].
\end{align*}
\end{remark}
\subsection{Non-smooth regularization}
In this section, we extend the acceleration result to problems with non-smooth regularization.
The main results are based on a simple modification called \textit{\textbf{M}oment-\textbf{ad}justed \textbf{Prox}imal Gradient descent} (MadProx).
Consider the population loss function that can be decomposed into
\begin{align}
\label{eq:non-smooth-decom}
L(w) = g(w) + h(w)
\end{align}
where $g(w)$ is a smooth and convex function in $w$, and $h(w)$ is a non-smooth regularizer that is convex. Special cases include,
\begin{itemize}
\item sparse regression with $\ell(w, (x_i, y_i)) = \frac{1}{2} (x_i^T w - y_i)^2 + \lambda \| w \|_1$ and
\begin{align*}
L(w) = \operatorname{\mathbb{E}}_{(\mathbf{x}, \mathbf{y}) \sim P} \left[\frac{1}{2} (\mathbf{x}^T w - \mathbf{y})^2 \right] + \lambda \| w \|_1 := g(w) + h(w);
\end{align*}
\item low rank matrix trace regression with $\ell(W, (X_i, y_i)) = \frac{1}{2} (\langle X_i, W \rangle - y_i)^2 + \lambda \| W \|_*$
\begin{align*}
L(W) = \operatorname{\mathbb{E}}_{(\mathbf{X}, \mathbf{y}) \sim P} \left[\frac{1}{2} (\langle \mathbf{X}, W \rangle - \mathbf{y})^2 \right] + \lambda \| W \|_* := g(W) + h(W).
\end{align*}
\end{itemize}
Now we will show the role of moment matrix $\mathbf{V}$ in ``speeding up'' the convergence of proximal gradient descent in the following proposition. Here we focus on an easier case when $\mathbf{V}(w)$ does not depend on $w$\footnote{As is in the linear regression fixed design case, where $\mathbf{V}(w) = \left( \mathbb{E}[ \xi(\mathbf{x})^2 \mathbf{x} \mathbf{x}^T] \right)^{1/2}$ does not depend on $w$.}.
Define the moment-adjusted proximal function and MadProx
\begin{align}
\prox_{\eta, \mathbf{V}}(w) = \argmin_{u} \left[ \frac{1}{2\eta}\| u - w \|_{\mathbf{V}}^2 + h(u) \right], \\
\label{eq:mad-proximal}
\text{MadProx:}\quad w_{t+1} = \prox_{\eta, \mathbf{V}}(w_t - \eta \mathbf{V}^{-1} \nabla g(w_t)).
\end{align}
\begin{proposition}[Moment-adjusted proximal]
\label{thm:prox}
Consider $L(w) = g(w) + h(w)$ as in \eqref{eq:non-smooth-decom}.
Denote $\mathbf{H}$ as the Hessian of $g$, and define
\begin{align*}
\alpha \triangleq \min_{v}~ \lambda_{\min} \left( \mathbf{V}^{-1/2} \mathbf{H}(v) \mathbf{V}^{-1/2} \right) >0 , \quad \gamma \triangleq \max_{v}~ \lambda_{\max} \left( \mathbf{V}^{-1/2} \mathbf{H}(v) \mathbf{V}^{-1/2} \right) >0.
\end{align*}
Consider the MadProx updates defined in \eqref{eq:mad-proximal} with step-size $\eta = 1/\gamma$ and adjusting matrix $\mathbf{V}$. If
$$
T \geq \frac{\gamma}{\alpha} \log \left( \frac{\alpha}{2\epsilon} \|w_0 - w_* \|_{\mathbf{V}}^2 + 1 \right),
$$
we have $L(w_T) - \min_w L(w) \leq \epsilon.$
\end{proposition}
\begin{remark}
One can see that MadProx
implements moment-adjusted gradient (using implicit updates)
because $w_{t+1}$ satisfies the implicit equation
\begin{align*}
w_{t+1} = w_{t} - \eta \mathbf{V}^{-1} (\nabla g(w_t) + \partial h(w_{t+1})),
\end{align*}
in comparison to the sub-gradient step (explicit updates)
\begin{align*}
w_{t+1} = w_{t} - \eta \mathbf{V}^{-1} (\nabla g(w_t) + \partial h(w_{t})).
\end{align*}
Remark that as in the GLMs case, the moment-adjusted idea speed up the computation as the number of proximal steps scales with adjusted condition number
$\kappa_{\rm MadProx} \approx \sqrt{\kappa_{\rm GD}}$.
However, to be fair, it can be computationally hard to implement each proximal step for a non-diagonal $\mathbf{V}$. Motivated from the diagonalizing idea in AdaGrad \citep{duchi2011adaptive}, one can substitute $\mathbf{V}$ by ${\rm diag}(\mathbf{V})$ to save the per-iteration computation.
\end{remark}
\section{Non-Convex Inference}
\label{sec:non-convex}
In this section, we study the non-asymptotic inference and optimization for stationary points of a smooth non-convex population landscape $L(\theta)$, via our proposed MasGrad.
\subsection{Inference and optimization for stationary points}
First we state a theorem that quantifies how well our proposed MasGrad achieves both the inference and optimization goal.
\begin{thm}[MasGrad: non-convex]
\label{thm:non-convex}
Let $L(w): \mathbb{R}^p \rightarrow \mathbb{R}$ be a smooth function. Recall $\mathbf{b}(w) = \nabla L(w)$, and $\mathbf{H}(w)$ being the Hessian matrix of $L$. $\mathbf{V}(w) \in \mathbb{R}^{p \times p}$ is a positive definite matrix.
Assume
\begin{align*}
\gamma &\triangleq \max_{v, w}~ \lambda_{\max} \left( \mathbf{V}(w)^{-1/2} \mathbf{H}(v) \mathbf{V}(w)^{-1/2} \right) >0.
\end{align*}
Consider the MasGrad updates $\theta_t$ in \eqref{eq: MasGrad} with step-size $\eta = 1/\gamma$, and the corresponding discretized diffusion $\xi_t$,
\begin{align*}
\xi_{t+1} = \xi_t - \eta \mathbf{V}(\xi_t)^{-1} \mathbf{b}(\xi_t)+ \sqrt{2\beta^{-1} \eta} \mathbf{g}_t,\quad \text{where $\beta = \frac{2n}{\eta}$}.
\end{align*}
Then for any precision $\epsilon, \delta > 0$, one can choose
\begin{align}
T = \frac{2\gamma(L(\theta_0) - \min_\theta L(\theta)) + p\delta^2 }{\epsilon^2} \cdot (\max_\theta \|\mathbf{V}(\theta)\| \vee 1),~ \text{and}~~ n = \frac{T}{\delta^2},
\end{align}
such that
\begin{align*}
&(1)\quad D_{\rm TV}\left( \mu(\theta_t, t \in [T]), \mu( \xi_t, t \in [T] ) \right) \leq O_{\delta}(\delta), \\
&(2)\quad \operatorname{\mathbb{E}} \min_{t \leq T} \| \nabla L(\theta_t) \|\leq \epsilon, ~\operatorname{\mathbb{E}} \min_{t \leq T} \| \nabla L(\xi_t) \| \leq \epsilon,
\end{align*}
with in total $O_{\epsilon,\delta}(\epsilon^{-4} \delta^{-2})$ independent data samples.
\end{thm}
\begin{remark}
\rm
We would like to contrast the optimization part of the above theorem with the sample complexity result of classic SGD. To obtain an $\epsilon$-stationary point $w$ such that in expectation $\| \nabla L(w) \| \leq \epsilon$, SGD needs $O_\epsilon(\epsilon^{-4})$ iterations for non-convex smooth functions (with step size $\eta_t = \min\{ 1/\gamma, 1/\sqrt{t}\}$). Here we show that one can achieve this accuracy with the same dependence on $\epsilon$ with MasGrad, while being able to make statistical inference at the same time. And the additional price we pay for $\delta$-closeness in distribution for statistical inference is a factor of $\delta^{-2}$.
The result can also be compared to Thm.~\ref{thm:converge} (the strongly convex case). In both cases, statistically, we have shown that the discretized diffusion $\xi_t$ tracks the non-asymptotic distribution of MasGrad $\theta_t$, as long as the data-generating process satisfies conditions like weak moment and bounded entropic distance to Gaussian. The distribution of $\xi_t$ is universal regardless of the specific data-generating distribution, and only depends on the moments $\mathbf{V}(\theta)^{-1}\b(\theta)$. In terms of optimization, to obtain an $\epsilon$-minimizer, the discretized diffusion approximation to MasGrad --- with the proper step-size $\eta$, and inverse temperature $\beta = 2n/\eta$ --- achieves the acceleration in the strongly convex case, and enjoys the same dependence on $\epsilon$ as SGD in the non-convex case in terms of sample complexity.
\end{remark}
\subsection{Why local inference}
For a general non-convex landscape, let us discuss why we focus on inference about local optima, or more precisely stationary points. Our Thm.~\ref{thm:non-convex} can be read as, within reasonable number of steps, the MasGrad converges to a population stationary point, and the distribution is well-described by the discretized Langevin diffusion. One can argue that the random perturbation introduced by the isotropic Gaussian noise in Langevin diffusion makes the process hard to converge to a typical saddle point. Therefore, intuitively, the MasGrad will converge to a distribution that is well concentrated near a certain local optima (depending on the initialization) as the temperature parameter $\beta^{-1} = \eta/2n$ is small. In this asymptotic low temperature regime, the Eyring-Kramer Law states that the transiting time from one local optimum to another local optimum, or the exiting time from a certain local optimum, is very long --- roughly $e^{\beta h}$ where $h$ is the depth of the basin of the local optimum \citep{bovier2004metastability, tzen2018local}. Therefore, a reasonable and tangible goal is to establish statistical inference for population local optima, for a particular initialization.
\section{Estimation and Computation of MasGrad Direction}
\label{sec:est-comp-mat-root}
We address in this section how to estimate and efficiently approximate the MasGrad direction $\mathbf{V}(\theta)^{-1} \b(\theta)$ at a current parameter location $\theta$. The estimation part undertakes a plug-in approach relying on the theory of self-normalized processes \citep{pena2008self}. For efficient computation of the pre-conditioning matrix, we devise a fast iterative algorithm to directly approximate the root of the inverse covariance matrix, which in a way resembles the advantage of quasi-Newton methods \citep{wright1999numerical}, however, with noticeable differences. The quasi-Newton methods approximate Hessian with first-order information, while MasGrad uses stochastic gradient information to approximate the root of the inverse covariance matrix as pre-conditioning.
In this section we deliberately state all propositions working with general sample covariance matrix $\widehat{\Sigma}$ with dimension $d$, to emphasize that the results extend beyond the discussions for MasGrad.
\subsection{Statistical estimation and self-normalized processes}
\label{sec:est-mat-root}
Recall that $\mathbf{V}(\theta)$ is the matrix root of the covariance. We estimate the moment-adjusted gradient direction $\mathbf{V}(\theta)^{-1} \mathbf{b}(\theta)$ at current location $\theta$, base on a mini-batch of size $n$. This section concerns this estimation part, borrowing tools from self-normalized processes. Define the sample estimates based on i.i.d data $z_i$ as
\begin{align*}
\widehat{\b}(\theta) &\triangleq \frac{1}{n} \sum_{i=1}^n \nabla_{\theta} \ell(\theta, z_i) \\
\widehat{\mathbf{\Sigma}}(\theta) &\triangleq \frac{1}{n-1} \sum_{i=1}^n [\nabla_{\theta}\ell(\theta, z_i) - \widehat{\b}(\theta)] \otimes [\nabla_{\theta}\ell(\theta, z_i) - \widehat{\b}(\theta)]
\end{align*}
and $\widehat{\mathbf{V}}(\theta)$ satisfies
$\widehat{\mathbf{V}}(\theta) \widehat{\mathbf{V}}(\theta)^T = \widehat{\mathbf{\Sigma}}(\theta),$
we will show that the plug-in approach $\widehat{\mathbf{V}}(\theta)^{-1} \widehat{\b}(\theta)$ estimates the population moment-adjusted gradient direction $\mathbf{V}(\theta)^{-1} \mathbf{b}(\theta)$ consistently at a parametric rate, in the fixed dimension setting.
\begin{proposition}[Connection to self-normalized processes]\label{lem:self-norm-process}
Consider $\{x_i \in \mathbb{R}^d, 1\leq i \leq n\}$ i.i.d with mean $\mu$, $\bar{x}$ and $\widehat{\Sigma}$ to be sample mean vector and sample covariance. Consider $d \ll n$ and $\widehat{\Sigma}$ is invertible. Denote the centered moments
\begin{align*}
S_n \triangleq \sum_{i=1}^n (x_i - \mu), \quad V_n^2 \triangleq \sum_{i=1}^n (x_i - \mu) \otimes (x_i - \mu)
\end{align*}
and the multivariate self-normalized process
\begin{align*}
M_n \triangleq V_n^{-1} S_n \in \mathbb{R}^d.
\end{align*}
Then there exists $\widehat{V}$, which satisfies
$
\widehat{V} \widehat{V}^T = \widehat{\Sigma}
$
such that
\begin{align*}
\sqrt{n} \widehat{V}^{-1} (\bar{x} - \mu) = M_n \cdot \sqrt{\frac{n-1}{n - \| M_n \|^2}}.
\end{align*}
\end{proposition}
\begin{remark}
\rm
In the case of $d=1$, the above proposition reduces to a standard result in \cite{pena2008self}. In our matrix version,
the proof relies on Sherman-Morrison-Woodbury matrix identity, together with a rank-one update formula for matrix root we derived in Lemma~\ref{lem:matrix-root} in Appendix~\ref{sec:proof}. Recall the Law of the Iterated Logarithm (LIL) on the norm of self-normalized process $\|M_n\|^2 \sim \log \log n$ (Theorem 14.11 in \citep{pena2008self}, in the case when dimension is fixed), a direct application of the above formula implies
\begin{align*}
\widehat{\mathbf{V}}(\theta)^{-1} \left( \widehat{\b}(\theta) - \mathbf{b}(\theta) \right) = \frac{1}{\sqrt{n}} M_n \cdot \sqrt{\frac{n-1}{n - \| M_n \|^2}} = \frac{1 + O_{\bf p}(\log\log n/n)}{\sqrt{n}} M_n,
\end{align*}
where $M_n$ is a self-normalized process with asymptotic distribution being $\mathcal{N}(0, I_p)$. By Lemma~\ref{lem:consistency} in Appendix~\ref{sec:proof}, when $p\ll n$, the following approximation holds
\begin{align*}
\widehat{\mathbf{V}}(\theta)^{-1} \widehat{\b}(\theta) - \mathbf{V}(\theta)^{-1} \mathbf{b}(\theta) = \overbrace{\widehat{\mathbf{V}}(\theta)^{-1} \left( \widehat{\b}(\theta) - \mathbf{b}(\theta) \right)}^{\text{self-normalized processes}} + O_{\bf p} \left(\sqrt{\frac{p \log n}{n}} \right),
\end{align*}
where the approximation is with respect to $\ell_2$ norm.
All together, the above implies that one can estimate $\mathbf{V}(\theta)^{-1} \mathbf{b}(\theta)$ consistently in the fixed dimension $p$ and large $n$ setting.
\end{remark}
\subsection{Efficient computation via direct rank-one updates}
\label{sec:comp-mat-root}
In this section we devise a fast iterative formula for calculating $\widehat{\mathbf{V}}(\theta)^{-1}$ directly via rank-one updates.
Recall the brute-force approach of calculating $\widehat{\mathbf{\Sigma}}(\theta)$ first
then solving for the inverse root $\widehat{\mathbf{V}}(\theta)^{-1}$ involves $O(n p^2 + p^3)$ complexity in the computation. Instead, we will provide an algorithm that approximates $\widehat{\mathbf{V}}(\theta)^{-1}$ directly through iterative rank-one updates, that is only $O(np^2)$ in complexity, utilizing the fact that the sample covariance is a finite sum of rank-one matrices. To the best of our knowledge, this direct approach of calculating root of inverse covariance matrix is new.
\begin{proposition}[Iterative rank-one updates of matrix inverse root]
\label{lem:rank-one-mat-root-inv}
Initialize $H_0 = I_d$, and define the recursive rank-one updates for matrix inverse root, for $v_i \in \mathbb{R}^d$
\begin{align}
\label{eq:rank-one-update}
H_{i+1} = H_{i} - \frac{1}{\alpha_i} H_i v_{i+1} v_{i+1}^T H_i^T H_i
\end{align}
with $\alpha_i \triangleq (1 + \sqrt{1+v_{i+1}^T H_i^T H_i v_{i+1}})\sqrt{1+v_{i+1}^T H_i^T H_i v_{i+1}} \in \mathbb{R}$.
Then for all $n$, $H_n$ is the matrix inverse root of $I_d + \sum_{i=1}^{n} v_i \otimes v_i$.
In other words, define $V_n \triangleq H_n^{-1},$
then $V_n V_n^T = I_d + \sum_{i=1}^{n} v_i \otimes v_i.$
\end{proposition}
\begin{remark}
\rm
One can directly apply the above result to evaluate $\widehat{\mathbf{V}}(\theta)^{-1}$ efficiently.
Define $v_i = \nabla_{\theta}\ell(\theta, \mathbf{z}_i) - \widehat{\b}(\theta)$, one can use \eqref{eq:rank-one-update} in the above proposition for fast iterative calculations, and that
$$
\left( \sqrt{n-1} H_{n} \right)^{-1} \left( \sqrt{n-1} H_{n}^T \right)^{-1} = \frac{1}{n-1} I_d + \widehat{\mathbf{\Sigma}} \approx \widehat{\mathbf{\Sigma}}.
$$
Therefore $\widehat{\mathbf{V}}(\theta)^{-1}$ is approximated by $\sqrt{n-1} H_{n}$. We remark that the quality of the approximation depends on the spectral decay of the true covariance $\Sigma$.
For each iteration, the computation complexity for \eqref{eq:rank-one-update} is $4d^2$, with some careful design in calculation: it takes $d^2$ operations to calculate $H_i v_{i+1} \in \mathbb{R}^d$, then an additional $d^2$ to calculate $(H_i v_{i+1})^T H_i \in \mathbb{R}^d$, another $d^2$ operations for multiplication of rank-one vectors $H_i v_{i+1} \times (H_i v_{i+1})^T H_i$, and finally $d^2$ operations for matrix addition. Hence, the total complexity is $O(nd^2)$ (for MasGrad, simply substitute $d = p$).
\end{remark}
\subsection{Optimal updates for online least-squares}
\label{sec:optim-updat-online}
In the case of a least-squares loss $\ell(\theta, z) = \frac12 (y - x^T \theta)^2$, we offer a simple and efficient online rule for estimating $\mathbf{V}(\theta)$ without any accuracy loss compared with offline counterparts. This is based on the fact that the data points $z_i$ and the parameter $\theta$ can be ``decoupled'' in least-squares. To show this, first write the covariance as
\[
\begin{aligned}
\mathbf{V}(\theta)^2 &= \Cov[(\mathbf{y} - \mathbf{x}^T \theta) \mathbf{x}]\\
&= \Cov(\mathbf{x}\mathbf{x}^T \theta) + \Cov(\mathbf{y}\mathbf{x}) - 2 \Cov(\mathbf{x}\mathbf{x}^T \theta, \mathbf{x}\mathbf{y}).
\end{aligned}
\]
To efficiently estimate $\Cov(\mathbf{x}\mathbf{x}^T \theta)$ in an online fashion, we observe that
\begin{equation}\label{eq:cov_two}
\Cov(\mathbf{x}\mathbf{x}^T \theta) = \operatorname{\mathbb{E}}_{\mathbf{z} \sim P} (\mathbf{x}\mathbf{x}^T \theta \theta^T \mathbf{x}\mathbf{x}^T) - \left[\operatorname{\mathbb{E}}_{\mathbf{z} \sim P} (\mathbf{x}\mathbf{x}^T \theta) \right] \left[\operatorname{\mathbb{E}}_{\mathbf{z} \sim P} (\mathbf{x}\mathbf{x}^T \theta) \right]^T.
\end{equation}
Recalling $\otimes_{\text{K}}$ denotes the Kronecker product and letting $\vecs(X)$ be the vector that is formed by stacking the columns of $X$ into a single column, we express $\mathbf{x}\mathbf{x}^T \theta \theta^T \mathbf{x}\mathbf{x}^T$ as
\[
\vecs (\mathbf{x}\mathbf{x}^T \theta \theta^T \mathbf{x}\mathbf{x}^T) = \left[(\mathbf{x} \mathbf{x}^T) \otimes_{\text{K}} (\mathbf{x}\mathbf{x}^T) \right] (\theta \otimes_{\text{K}} \theta).
\]
This expression shows that
\begin{equation}\nonumbe
\vecs\left[\operatorname{\mathbb{E}}_{\mathbf{z} \sim P} (\mathbf{x}\mathbf{x}^T \theta \theta^T \mathbf{x}\mathbf{x}^T) \right] = \operatorname{\mathbb{E}}_{\mathbf{z} \sim P}\left[(\mathbf{x} \mathbf{x}^T) \otimes_{\text{K}} (\mathbf{x}\mathbf{x}^T) \right] (\theta \otimes_{\text{K}} \theta).
\end{equation}
Accordingly, one can simply keep track of $\sum_{i=1}^t (\mathbf{x}_i \mathbf{x}_i^T) \otimes_{\text{K}} (\mathbf{x}_i\mathbf{x}_i^T)$ in the online setting and estimate $\operatorname{\mathbb{E}}_{\mathbf{z} \sim P} (\mathbf{x}\mathbf{x}^T \theta \theta^T \mathbf{x}\mathbf{x}^T)$ through mapping the vector
\[
\left[ \frac{\sum_{i=1}^t (\mathbf{x}_i \mathbf{x}_i^T) \otimes_{\text{K}} (\mathbf{x}_i\mathbf{x}_i^T)}{t} \right] (\theta \otimes_{\text{K}} \theta).
\]
to its associated matrix. It remains to estimate $\operatorname{\mathbb{E}}_{\mathbf{z} \sim P} (\mathbf{x}\mathbf{x}^T \theta)$ in \eqref{eq:cov_two}. Recognizing $\operatorname{\mathbb{E}}_{\mathbf{z} \sim P} (\mathbf{x}\mathbf{x}^T \theta) = \left[ \operatorname{\mathbb{E}}_{\mathbf{z} \sim P} (\mathbf{x}\mathbf{x}^T) \right] \theta$, this can be done by simply recording the sum $\mathbf{x}_1\mathbf{x}_1^T + \cdots + \mathbf{x}_t\mathbf{x}_t^T$ in an online manner and replacing $\operatorname{\mathbb{E}}_{\mathbf{z} \sim P} (\mathbf{x}\mathbf{x}^T)$ by the average $(\mathbf{x}_1\mathbf{x}_1^T + \cdots + \mathbf{x}_t\mathbf{x}_t^T)/t$. Likewise, $\Cov(\mathbf{y}\mathbf{x})$ and $\Cov(\mathbf{x}\mathbf{x}^T \theta, \mathbf{x}\mathbf{y})$ can be estimated in the online setting regardless of a varying $\theta$. We omit this part in the interest of space.
\section{Numerical Experiments}
In this section we present results for numerical experiments. Full details of the experiments are deferred to Appendix~\ref{sec:exp-details}.
\smallskip
\noindent \textbf{Linear models.} \quad
The first numerical example is the simple linear regression, as in Fig.~\ref{fig:linear}. Here we generate two plots as a proof of concept. The top one summarizes the trajectory of several methods for inference --- our proposed \textit{MasGrad}, the discretized diffusion approximation \textit{diff\_MasGrad}, as well as the classical \textit{SGD}, and the diffusion approximation \textit{diff\_SGD} --- with the confidence intervals (95\% coverage) at each time step $t$. In this convex setting, we can solve for the global optimum, which is labeled as the \textit{truth}. Here the mini-batch size is $n=50$. We run $100$ independent chains to calculate the confidence intervals at each step. We look at the low dimensional case $p = 4$, and the four subfigures (on top) each corresponds to one
coordinate of the parameter $w_i, i \in [p]$. The $x$-axis is $t$, the time of the evolution, and $y$-axis is the value of the parameter $w$. We remark that \textit{MasGrad} and \textit{diff\_MasGrad} are path-wise close in terms of distribution, which verifies our statistical theory in Thm.~\ref{thm:couple.p}. This also holds for \textit{GD} and \textit{diff\_GD}.
Remark that in this simulation, the condition number of the empirical Gram matrix is $30.98$, and the first and third coordinates have very small population eigenvalues, which explains why in those coordinates \textit{MasGrad} has significant acceleration compared to \textit{SGD} as shown in the figure. To be fair, at each time step, both MasGrad and SGD sample the same amount of data, and the step-size is chosen as in Thm.~\ref{thm:acceleration}. All four chains start with the same random initialization.
To examine the optimization side of the story, we plot the logarithm of the $\ell_2$-error according to time $t$, for \textit{diff\_MasGrad} and \textit{diff\_SGD}, in the bottom plot. Remark that the error bar quantifies the confidence interval for the log error. In theory, we should expect that the slope of MasGrad is twice that of the slope of SGD. In simulation, it seems that the acceleration is slightly better than what the theory predicts.
We would like to remark that compared to GD, in which different coordinates make uneven progress (fast progress in the second and fourth coordinates, but slow on the others), MasGrad adaptively adjusts the relative step-size on each coordinate for synchronized progress. This effect has also been observed in AdaGrad and natural gradient descent.
\begin{figure}[pht]
\centering
\includegraphics[width = 0.9\textwidth]{./linear-inf.pdf}
\includegraphics[width = 0.9\textwidth]{./linear-opt.pdf}
\caption{Linear regression}
\label{fig:linear}
\end{figure}
\noindent \textbf{Logistic model.} \quad
Fig.~\ref{fig:logistic} illustrates the acceleration for inference in logistics regression. The figure should be read the same way as in the linear case. In this case, we sample a much larger number of samples ($N = 500$) and then use the GLMs package in R to fit the global optimum. For MasGrad and SGD, we generate bootstrap subsamples ($n = 25$) to evaluate stochastic descents at each iteration. Again, we run $100$ independent chains to calculate the confidence interval at each step.
In this case, there is no theoretically optimal way of choosing the step-size, so we choose the same step-size ($\eta = 0.2$) for both MasGrad and SGD.
Statistically, the \textit{MasGrad} and \textit{diff\_MasGrad} are close in distribution when $t < 100$, and they both reach a stationary distribution after around $50$ steps, simultaneously for all $p=4$ coordinates. Then the distribution fluctuates around stationarity. However, \textit{GD} and \textit{diff\_GD} make much slower progress, and they fail to reach the global optimum in $100$ steps.
For optimization, empirically the acceleration in the log error plot seems to be better than what the theoretical results predict. Remark that the confidence intervals are on the scale of log error, therefore, it is negative-skewed.
\begin{figure}[pht]
\centering
\includegraphics[width = 0.9\textwidth]{./logistic-inf.pdf}
\includegraphics[width = 0.9\textwidth]{./logistic-opt.pdf}
\caption{Logistic regression}
\label{fig:logistic}
\end{figure}
\noindent \textbf{Gaussian mixture.} \quad
Here we showcase inference via MasGrad for non-convex case, using the Gaussian mixture model. We will consider a simple setting: the data $z_i \in \mathbb{R}^n, 1\leq i \leq [N]$ generated from a mixture of $p$ Gaussians, with mean $[\theta_1, \theta_2,\ldots, \theta_p]\triangleq \theta$ respectively, and variance $\sigma^2$. The goal is to infer the unknown mean vector $\theta \in \mathbb{R}^p$. The problem is non-convex due to the mixture nature: the maximum likelihood is multimodal, as we can shuffle the coordinates of $\theta$ to obtain the equivalent class of local optima.
Fig.~\ref{fig:mixture} illustrates the acceleration for inference in the Gaussian mixture model. Here we run two simulations, according to the difficulty (or separability) of the problem defined as the signal-to-noise ratio ${\rm SNR} \triangleq \min_{i\neq j} |\theta_{i} - \theta_{j}|/\sigma$. The top one is for the easy case with ${\rm SNR} = 3.3$ and the bottom one for the hard case with ${\rm SNR} = 1$. In both simulations, $\theta = (1,2,3) \in \mathbb{R}^3$, and we choose a random initial point to start the chains. The plot is presented as before. At each iteration, we subsample $n = 20$ data points to calculate the decent direction, and the step-size is fixed to be $\eta = 0.05$. Remark that there are many population local optima (at least $3! = 6$), and both \textit{MasGrad} and \textit{diff\_MasGrad} seem to be able to find a good local optimum relatively quickly (which concentrates near a permutation of $1,2,3$ for each coordinate), compared to \textit{SGD} and \textit{diff\_SGD}. The acceleration effect in both cases seems to be apparent. Again, we want to emphasize that the convergence for each coordinate in MasGrad seems to happen around the same number of iterations, which is not true for SGD.
\begin{figure}[pht]
\centering
\includegraphics[width =0.9\textwidth]{./mixture-inf.pdf}
\includegraphics[width =0.9\textwidth]{./mixture-inf-2.pdf}
\caption{Gaussian mixture}
\label{fig:mixture}
\end{figure}
\noindent \textbf{Shallow neural networks.} \quad
We also run MasGrad on a two-layer ReLU neural network, as a proof of concept for non-convex models. Define the ReLU activation $\sigma(x) = \max(x, 0)$, a two-layer neural network (with $k$ hidden units) represents a function
\begin{align*}
f_w(x) = \sigma(W_2 \sigma(W_1 x)), \quad \text{where $x \in \mathbb{R}^d$, $w = \{ W_1 \in \mathbb{R}^{k \times d}, W_2 \in \mathbb{R}^{1 \times k} \}$}.
\end{align*}
In our experiment, we work with the square loss
$\ell(w, (x,y)) = \frac{1}{2}(y - f_w(x))^2.$
The gradients can be calculated through back-propagation.
In this case, it is harder to calculate the global optimum; instead, in order to compare the \textit{diff\_MasGrad} and \textit{SGD}, we run $50$ experiments with random initializations to explore the population landscape.
For each experiment (as illustrated in the top figure in Fig.~\ref{fig:neural}), we randomly initialize the weights using standard Gaussians. As usual, we run 100 independent chains with the same initial points for \textit{diff\_MasGrad} and \textit{SGD} to calculate the confidence interval. As anticipated, the distribution is rather non-Gaussian (for instance, in coordinate $2$ and $6$). We run the chain for 100 steps, and then evaluate the population loss function for the two methods. Out of the $50$ experiments, $45/50= 90\%$ of the time the population loss returned by \textit{diff\_MasGrad} is much smaller than that of the \textit{SGD}. The bottom figure in Fig.~\ref{fig:neural} plots the histogram (dotplot using ggplot2 \citep{Wickham:2009aa}) of the population error (test accuracy). Empirically, the $\textit{diff\_MasGrad}$ seems to converge to ``better'' local optima most of the time. There could be several explanations: first, MasGrad uses better local geometry (similar to natural gradient) so that it induces better implicit regularization; second, MasGrad as an optimization method accelerates the chain so that it mixes to a local optima faster, compared to SGD which may not yet converge within a certain time budget.
\begin{figure}[pht]
\includegraphics[width = 0.9\textwidth]{./neural_nets.pdf}
\includegraphics[width = 0.9\textwidth]{./neural_nets_error.pdf}
\centering
\caption{Shallow neural nets}
\label{fig:neural}
\end{figure}
\section{Further Discussions}
Let us continue to discuss more about $\mathbf{V}(\theta_t)$. Note that in the fixed-dimension setting, one can estimate the covariance matrix of the gradient $\nabla \ell(\theta, \mathbf{z})$ using the empirical version with $N$ independent samples, when $N$ is large. Let us be more careful in this statement: (1) When the population landscape is convex, then the global optimum of $\widehat{L}_N(\theta)$ and $L(\theta)$ are within $1/\sqrt{N}$. We can always treat $\widehat{L}_N(\theta)$ as the population version and at each step we bootstrap subsamples of size $n$ to evaluate the stochastic gradients, adjusted using the empirical covariance $\widehat{\mathbf{V}}_N$ calculated using $N$ data points. Intuitively, when $\eta < O(n/N)$ (so that $\beta > N$), we know the MasGrad will concentrate near the optimum of $\widehat{L}_N(\theta)$ with better accuracy than $1/\sqrt{N}$. (2) In the non-convex case, things become unclear. However, under stronger conditions such as strongly Morse \citep{mei2016landscape}, i.e., when there is nice one-to-one correspondence between the stationary points of $\widehat{L}_N(\theta)$ and $L(\theta)$, one may still use the bootstrap idea above with $\widehat{\mathbf{V}}_N$. (3) Computation of $\widehat{\mathbf{V}}_N$ and its inverse could be burdensome, thus one may want to use the efficient rank-one updates designed in Section~\ref{sec:optim-updat-online}, or to calculate a diagonalized version of $\widehat{\mathbf{V}}_N$ as done in AdaGrad \citep{duchi2011adaptive}. (4) To have fully rigorous non-asymptotic theory in the case where $\mathbf{V}$ is known, one may require involved tools from self-normalized processes \citep{pena2008self} to establish a similar version of entropic CLT for multivariate self-normalized processes, where we standardize $\widehat{\operatorname{\mathbb{E}}}_n[\nabla \ell(\theta, \mathbf{z})]$ by the empirical covariance matrix $\widehat{\mathbf{V}}_n$ calculated based on the same samples. To the best of our knowledge, this is an ambitious and challenging goal that is beyond the scope and focus of the current paper.
We would like to conclude this section by discussing the connections between pre-conditioning methods and our moment-adjusting method.
Pre-conditioning considers performing a linear transformation $\xi = A^{-1} \theta$ on the original parameter space on $\theta$. In other words, consider $\tilde{L}(\xi) \triangleq L(A\xi)$, and perform the updates on $\xi$ yields
\begin{align*}
\xi_{t+1} = \xi_{t} - \eta \nabla_\xi \tilde{L}(\xi)= \xi_{t} - \eta A \mathbf{b} (A \xi_t) ~\Rightarrow~ \theta_{t+1} = \theta_{t} - \eta A^2 \mathbf{b}(\theta_t),
\end{align*}
Therefore, in the noiseless case, the moment-adjusting method is equivalent to pre-conditioning when the moment matrix $\mathbf{V}(\theta)$ is a constant matrix w.r.t. $\theta$. However, in Langevin diffusion when the isotropic Gaussian noise is presented, the connection becomes more subtle --- as $\mathbf{V}^{-1} (\theta) \mathbf{b}(\theta)$ may not be the gradient vector field for any function. The moment-adjusting idea motivated from standardizing noise in statistics is different from the pre-conditioning idea in optimization. We would also like to point out that a nice idea using Hessian information to speed up the Langevin diffusion for sampling from log-concave distribution has been considered in \cite{dalalyan2017theoretical}. Remark that we use the moment matrix at the current
point $\theta_t$ (time varying) instead of the optimal point $\theta_*$ (which is unknown). We also use the matrix root instead of the covariance matrix itself. In the case when the model is well-specified and the loss function chosen to be the negative log-likelihood, the $V(\theta_*)$ is the root of the Fisher information matrix.
\section*{Supplemental Materials}
Due to space constraints, we have
relegated further discussion of Langevin diffusion to Appendix~\ref{sec:continuous-langevin}, the detailed proofs to Appendix~\ref{sec:proof}, and remaining details about experiments to Appendix~\ref{sec:exp-details} in the Supplement to ``Statistical Inference for the Population Landscape via Moment-Adjusted Stochastic Gradients."
\section*{Acknowledgement}
The authors would like to thank the Associate Editor and the anonymous referees for the constructive feedback that significantly improves the content and presentation of the paper.
\bibliographystyle{abbrvnat}
|
1,116,691,499,376 | arxiv | \section{Introduction}
One of the most important parameters of QCD is the total number of dynamical quarks. Asymptotic freedom is lost at the one-loop level for $N_f > 33/2$, while chiral symmetry is broken spontaneously for two or more flavors until QCD becomes conformal for a number of flavors that can be determined by means of lattice QCD simulations (see \cite{Kuti:2013} for a review). In this paper we consider one-flavor QCD, where chiral symmetry is broken explicitly by the anomaly rather than by spontaneous symmetry breaking. As a consequence, the sign of the chiral condensate does not change when the sign of the quark mass is reversed. On the other hand, the Banks-Casher relation \cite{Banks:1979yr} predicts that the chiral condensate does change sign when the quark mass changes sign. The resolution of this apparent contradiction is well-known \cite{Leutwyler:1992yt, Kanazawa:2011tt}: In the derivation of the Banks-Casher relation it is assumed that the fermionic measure (or equivalently the spectral density of the Dirac operator) is positive definite, but this assumption is violated for one-flavor QCD when the quark mass is negative. In this paper we will show that in this case the relation between the Dirac spectrum and the chiral condensate may be determined by an alternative mechanism \cite{Osborn:2005ss}, which was first observed for QCD at nonzero baryon number chemical potential. In the latter case the spectral density is not positive definite due to the phase of the fermion determinant, and the discontinuity of the chiral condensate when the quark mass crosses the imaginary axis does not arise from a dense spectrum of eigenvalues on the imaginary axis but rather from an oscillating spectral density in the complex plane with an amplitude that increases exponentially with the volume and a period that is inversely proportional to the volume. An analogous mechanism is at work in other physical situations, e.g., in one-dimensional one-flavor U(1) gauge theory at nonzero chemical potential, where the Dirac spectrum is an ellipse in the complex plane while the chiral condensate only has a discontinuity across the imaginary axis \cite{Ravagli:2007rw}, in two-color QCD at nonzero chemical potential \cite{Akemann:2010tv}, or in QCD at large isospin density with mismatched quark chemical potentials \cite{Kanazawa:2014lga}.
In this paper we obtain simple analytical expressions for the one-flavor microscopic spectral density of the Dirac operator at zero $\theta$-angle. These expressions allow us to explicitly apply the above-mentioned alternative mechanism in the case of one-flavor QCD. Analytical results for the spectral density at fixed topological charge $\nu$ are well-known \cite{Verbaarschot:1993pm,Damgaard:1997ye,Wilke:1997gf,Damgaard:1998xy} (see \cite{Verbaarschot:2000dy,Verbaarschot:2009jz} for reviews), but simple expressions for the spectral density at fixed $\theta$-angle have not yet appeared in the literature. In \cite{Damgaard:1999ij,Kanazawa:2011tt}, the spectral density at fixed $\theta$-angle was studied numerically, and several analytical results were obtained as well. It was also realized that an analytical expression for the spectral density could be derived by combining the expression for the spectral density in terms of microscopic partition functions \cite{Akemann:1998ta} with expressions for the partition function at fixed $\theta$-angle. At the time the paper \cite{Damgaard:1999ij} was published these expressions were only known for two flavors \cite{Leutwyler:1992yt}, while an expression for the three-flavor partition function is required to obtain the one-flavor spectral density. General results for more flavors were derived in \cite{Lenaghan:2001ur}, and they were used to obtain the spectral density at fixed $\theta$ in \cite{Kanazawa:2011tt}. However, simpler analytical results can be obtained using identities for sums of products of Bessel functions which we derive in this paper, and as far as we know, some of these identities are not known in the literature.
The physics of one-flavor QCD was recently reviewed in \cite{Creutz:2006ts}, where some of the questions that are raised in the present paper were also addressed. However, the relation between the chiral condensate and the spectral density of the Dirac operator turns out to be much more intricate than anticipated in \cite{Creutz:2006ts}. We show that for negative quark mass the chiral condensate results from large cancellations between the contributions of the zero and nonzero modes \cite{Kanazawa:2011tt} and that the oscillations in the spectral density are essential for the continuity of the chiral condensate.
This paper is organized as follows. In Section~\ref{sec:relation} we discuss the relation between the spectrum of the Dirac operator and the chiral condensate for one-flavor QCD. Also, we briefly comment on the case of several flavors. Analytical results for the spectral density are derived in Section~\ref{sec:density}, and the condensate is evaluated in Section~\ref{sec:condensate}. Concluding remarks are made in Section~\ref{sec:conclusions}. In Appendix~\ref{app:uniform} we investigate under what conditions the thermodynamic limit and the sum over topological sectors can be interchanged. In Appendix~\ref{app:sums} we derive addition theorems for products of Bessel functions. Asymptotic results for the spectral density are worked out in Appendix~\ref{app:asympt}. In Appendix~\ref{app:integrals} we give integrals that are used to compute the chiral condensate. Asymptotic results for the chiral condensate are worked out in Appendix~\ref{app:condensate}. The mass independence of the chiral condensate is shown in Appendix~\ref{appd}.
\section{Chiral condensate and spectral density of one-flavor QCD}
\label{sec:relation}
For $m \ll 1/\Lambda_\text{QCD} \sqrt V$ the one-flavor partition
function at fixed $\theta$-angle is given by \cite{Leutwyler:1992yt}
\begin{align}
Z(m,\theta) = e^{mV \Sigma\cos\theta}\,,
\label{znf1}
\end{align}
where $V$ is the volume of space-time, $m$ is a quark mass which we take to be real, and $\Sigma$ is the absolute value of the chiral condensate in the limit $m=0$ and $\theta=0$. To avoid unnecessary minus signs we define the quantity $\Sigma(m,\theta)$ (which we will also refer to as the chiral condensate) by
\begin{align}
\Sigma(m,\theta) = -\langle \bar q q\rangle =\frac 1V \frac d{dm} \log Z(m,\theta)
= \Sigma \cos\theta \,.
\label{con-sig}
\end{align}
Since $\Sigma(m,\theta)$ is independent of $m$ it does not change when $m$ crosses the imaginary axis on which the eigenvalues of the Dirac operator are located. In terms of the Dirac eigenvalues $i\lambda_k$ (with real $\lambda_k$) $\Sigma(m)$ is given by
\begin{align}
\Sigma(m)
&=\frac1V\Big\langle\sum_k\frac1{i\lambda_k+m}\Big\rangle\notag\\
&=\frac 1V \int_{-\infty}^\infty d\lambda\,
\frac{\rho(\lambda,m)}{i\lambda+m}\,,
\label{eq:BC}
\end{align}
where we have not displayed the dependence on $\theta$ explicitly
because Eq.~\eqref{eq:BC} is valid not only for fixed $\theta$ but
also in sectors of fixed topological charge, i.e., as a relation
between $\Sigma_\nu(m)$ and $\rho_\nu(\lambda)$ defined below. In the
first line of Eq.~\eqref{eq:BC}, the average is over the gauge fields
weighted by the fermion determinant. In the second line,
$\rho(\lambda,m)$ is the spectral density of the Dirac operator in the
one-flavor theory, which is symmetric with respect to $\lambda=0$ and
also includes the contributions from exact zero modes. For a non-negative spectral density, $\Sigma(m)$ in
Eq.~\eqref{eq:BC} changes sign when the quark mass changes
sign. However, for negative quark mass the fermion determinant and
hence the spectral density is not positive definite, and we will see
below that this allows for a constant chiral condensate.
The partition function at fixed $\theta$-angle can be decomposed into
partition functions at fixed topological charge,
\begin{align}
Z(m,\theta) = \sum_\nu e^{i\nu\theta} Z_\nu(m)\,,
\end{align}
resulting in the decomposition
\begin{align}
\label{consig}
\Sigma(m,\theta) = \frac 1{Z(m,\theta)}\sum_\nu
e^{i\nu\theta} Z_\nu(m) \Sigma_\nu(m)
\end{align}
of the chiral condensate with
\begin{align}
\Sigma_\nu(m) = \frac1V\frac d{dm} \log Z_\nu(m)\,.
\end{align}
Since $Z_\nu(m)/Z(m,\theta)$ is the probability of finding a gauge-field
configuration with topological charge $\nu$, the spectral density can
be decomposed as \cite{Damgaard:1999ij}
\begin{align}
\label{eq:rho_theta}
\rho(\lambda,m,\theta) = \frac 1{Z(m,\theta)}\sum_\nu e^{i\nu\theta}
Z_\nu(m) \rho_\nu(\lambda,m)\,,
\end{align}
where $\rho_\nu(\lambda,m)$ is the spectral density for gauge-field
configurations with fixed topological charge $\nu$.
The one-flavor partition function in the sector of topological charge
$\nu$ is given by
\begin{align}
Z_\nu(m) = \frac 1{2\pi} \int_{-\pi}^\pi d\theta\, e^{-i\nu \theta }
Z(m,\theta) = I_\nu(mV \Sigma) \,,
\end{align}
resulting in a chiral condensate at fixed $\nu$ equal to
\begin{align}
\Sigma_\nu(m) = \Sigma \frac {I'_\nu(mV\Sigma)}{I_\nu(mV\Sigma)}\,.
\end{align}
In the thermodynamic limit $V\to\infty$ we thus have
\begin{align}
\lim_{V\to\infty}\Sigma_\nu(m) = \sign(m)\Sigma\,.
\end{align}
If the thermodynamic limit and the sum over $\nu$ in Eq.~\eqref{consig} could be interchanged, the chiral condensate would be given by
\begin{align}
\label{eq:qm1}
\lim_{V\to\infty}\Sigma(m,\theta) \overset?= \sign(m)\Sigma\,.
\end{align}
Here and below, the question mark indicates that the result holds only under certain conditions. In fact, \eqref{eq:qm1} contradicts Eq.~\eqref{con-sig} unless $m>0$ and $\theta=0$ or $m<0$ and $\theta=\pi$, which implies that the thermodynamic limit and the sum over $\nu$ can only be interchanged in these two cases. Why this is so is explained in detail in Appendix~\ref{app:uniform}.
Let us now try to understand Eq.~\eqref{eq:qm1} in terms of the eigenvalue density. The rescaled spectral density at fixed $\nu$ is defined by
\begin{align}
\label{eq:rescaled}
\hat\rho_\nu(\hat x,\hat m) \equiv \lim_{V\to \infty}\frac 1{V\Sigma}\,\rho_\nu\left
(\frac{\hat x}{V\Sigma},\frac{\hat m}{V\Sigma} \right ) ,
\end{align}
where we have introduced the dimensionless variables $\hat x=\lambda V\Sigma$ and $\hat m=mV\Sigma$. If the thermodynamic limit in \eqref{eq:rescaled} is taken for \emph{fixed} $\lambda=\hat x/V\Sigma$ we obtain $\hat\rho_\nu^\nz(\hat x,\hat m)=1/\pi$ \cite{Verbaarschot:1993pm}, where the superscript nz (``nonzero'') indicates that we have momentarily ignored the contribution from the zero modes (which will be reinstated below). Assuming that the thermodynamic limit and the sum over $\nu$ in Eq.~\eqref{eq:rho_theta} can be interchanged, we thus obtain
\begin{align}
\hat\rho^\nz(\hat x,\hat m,\theta)\overset?=\frac1\pi\,.
\label{rhoav}
\end{align}
At fixed $\theta$-angle the spectral density \eqref{rhoav} then results, via Eq.~\eqref{eq:BC}, in a chiral condensate given by Eq.~\eqref{eq:qm1}. Again, Eq.~\eqref{rhoav} and the resulting Eq.~\eqref{eq:qm1} are only correct for $m>0$ and $\theta=0$ or $m<0$ and $\theta=\pi$, i.e., if the thermodynamic limit and the sum in Eq.~\eqref{eq:rho_theta} can be interchanged (which can be shown in analogy to the arguments for Eq.~\eqref{consig} in Appendix~\ref{app:uniform}). Note that in these cases the zero modes do not contribute to the chiral condensate in the thermodynamic limit, see Eq.~\eqref{eq:zm} below (which is valid for $m>0$ and $\theta=0$).
We now set $\theta=0$ for simplicity to study only the dependence on the sign of $m$ and decompose the spectral density as
\begin{align}
\hat\rho(\hat x,\hat m,\theta=0) = \frac1\pi + \Delta\hat\rho(\hat x,\hat m)\,,
\end{align}
where the additional part also includes the contribution from the zero modes, given explicitly in Eq.~\eqref{eq:rho_zm} below. To obtain a condensate that is constant as a function of $\hat m$, the contribution from the additional part of the spectral density must be equal to
\begin{align}
\int_{-\infty}^\infty d\hat x\, \frac{\Delta \hat\rho(\hat x,\hat m)}{i\hat x+\hat m} = 2\Theta(-\hat m)\,,
\label{eq:heavi}
\end{align}
where $\Theta$ denotes the Heaviside function. Combined with the contribution $\sign(m) \Sigma$ from the $1/\pi$ term in the spectral density this gives the correct result $\Sigma(m,\theta=0)=\Sigma$.
The solution of Eq.~\eqref{eq:heavi} for $\Delta\hat\rho$ is not unique. Before discussing the case of one-flavor QCD, let us look at a simpler example. From the Fourier decomposition of the Heaviside function, we consider the spectral density
\begin{align}
\label{eq:heavi_ex}
\Delta \rho(\lambda, m) = -\frac{V\Sigma}\pi \left(e^{i\lambda V\Sigma -m V\Sigma}+e^{-i\lambda V\Sigma -m V\Sigma}\right),
\end{align}
which is symmetric in $\lambda$ as in QCD. After integration according to Eq.~\eqref{eq:heavi}, the second term gives the desired $\Theta$-function and thus the mass independence of the condensate, while the first term gives a result proportional to $\Theta(m)e^{-2mV\Sigma}$ that vanishes in the thermodynamic limit. A similar mechanism is at work in one-flavor QCD, but there are some differences. In the next section we derive an explicit expression for $\Delta\hat\rho$ for this case. In Section~\ref{sec:condensate} we show that this expression indeed satisfies Eq.~\eqref{eq:heavi}, for any value of $\hat m$ and not only in the thermodynamic limit. We will see that for $m<0$ the contributions to $\Delta\hat\rho$ are strongly oscillating. After integration according to Eq.~\eqref{eq:heavi} we get contributions to the condensate that diverge exponentially in the thermodynamic limit, i.e., for $\hat m\to-\infty$. The desired $\Theta$-function discontinuity is obtained through cancellations of the different contributions. We will also see that the contribution of the zero modes plays an essential role for $m<0$.
Let us briefly comment on the case of several flavors. In that case chiral symmetry is broken spontaneously, resulting in a partition function that is dominated by Nambu-Goldstone (NG) bosons in the chiral limit. In the thermodynamic limit, the NG fields $U$ align themselves with the mass term. After diagonalizing the NG fields the mean-field action (for degenerate quark masses and with $\theta=0$) can be written as \cite{Smilga:1998dh}
\begin{align}
S_\text{mf}&=-\frac12mV\Sigma\tr(U+U^\dagger)\notag\\
&=-mV\Sigma\Bigg(\sum_{i=1}^{N_f-1}\cos\phi_i+\cos\sum_{i=1}^{N_f-1}\phi_i\Bigg)\,,
\end{align}
which is to be minimized as a function of the $\phi_i$. For even $N_f$ the minimum is obtained for $\phi_i=0$ if $m>0$ and for $\phi_i=\pi$ if $m<0$. This results in a mass dependence of the partition function in the thermodynamic limit given by
\begin{align}
\label{eq:Nfeven}
Z(m) = e^{N_f |m|V \Sigma}\quad(N_f\text{ even})\,,
\end{align}
in contrast to the one-flavor partition function \eqref{znf1}. As a consequence, the chiral condensate is discontinuous at $m=0$. For odd $N_f$ the minimum is obtained for $\phi_i=0$ if $m>0$ and for $\phi_i=\pi(1-1/N_f)$ if $m<0$. Hence in this case the mass dependence of the partition function is given by
\begin{align}
Z(m) =
\begin{cases}
e^{N_f mV \Sigma}\,, & m>0\,,\\
e^{N_fmV\Sigma\cos\pi(1-1/N_f)}\,, & m<0\,,
\end{cases}
\quad(N_f\text{ odd})
\end{align}
which agrees with \eqref{znf1} for $N_f=1$ and approaches \eqref{eq:Nfeven} for large $N_f$. For odd $N_f>1$ the chiral condensate is also discontinuous at $m=0$.
\section{\boldmath Spectral Density at $\theta = 0$}
\label{sec:density}
Let us split up the spectral density at fixed $\nu$ into the zero-mode
contribution, the quenched nonzero-mode contribution, and the
nonzero-mode contribution due to dynamical quarks,
\begin{align}
\label{eq:decomp}
\rho_\nu(\lambda) = \rho^\zm_\nu(\lambda) + \rho^\q_\nu(\lambda) +
\rho^\d_\nu(\lambda) \,,
\end{align}
where we have suppressed the dependence on $m$ for simplicity. In the
microscopic limit of one-flavor QCD these contributions are given by
\cite{Verbaarschot:1993pm,Damgaard:1997ye,Wilke:1997gf,Damgaard:1998xy}
\begin{align}
\label{eq:zm_nu}
\hat\rho^\zm_\nu(\hat x) &= {|\nu|}\delta(\hat x)\,, \\
\hat\rho^\q_\nu(\hat x)&=\frac{|\hat x|}2\big[J_{\nu}^2(\hat x)
-J_{\nu+1}(\hat x)J_{\nu-1}(\hat x)\big]\,, \\
\hat\rho^\d_\nu(\hat x,\hat m) &=\frac {-|\hat x|}{\hat x^2+\hat m^2}
\Big [\hat x J_\nu(\hat x) J_{\nu+1}(\hat x)\!+\!\hat m \frac
{I_{\nu+1}(\hat m)}{I_\nu(\hat m)}J_\nu^2(\hat x)\Big].
\end{align}
\begin{figure}
\centering
\includegraphics{rhoq.pdf}\\[5mm]
\includegraphics{rhod.pdf}
\caption{Three-dimensional plot of the quenched (top) and dynamical
(bottom) part of the rescaled spectral density as a function of
$\hat x$ and $\hat m$. The normalization is chosen such that the
asymptotic value of the rescaled spectral density is equal to
$1/\pi$.}
\label{fig:3d}
\end{figure}
For the spectral density at fixed $\theta=0$ we use a decomposition
analogous to Eq.~\eqref{eq:decomp}, only with the subscript $\nu$
omitted. For the zero-mode part of the spectral density the sum over
$\nu$ can be evaluated explicitly
\cite{Leutwyler:1992yt,Damgaard:1999ij}, resulting in
\begin{align}
\hat\rho^\zm(\hat x,\hat m) &=e^{-\hat m} \sum_\nu I_\nu(\hat m)|\nu| \delta(\hat x)
\notag\\
&=e^{-\hat m} \hat m \big[I_0(\hat m)+I_1(\hat m)\big] \delta(\hat x)\,.
\label{eq:rho_zm}
\end{align}
The quenched part follows from the identities \eqref{eq:sum000} and
\eqref{eq:sum01-1} derived in Appendix~\ref{app:sums},
\begin{align}
\hat\rho^\q(\hat x,\hat m) &= e^{-\hat m} \sum_\nu I_\nu(\hat m)
\frac{|\hat x|}2\big[J_{\nu}^2(\hat x) -J_{\nu+1}(\hat x) J_{\nu-1}(\hat x)\big]\notag \\
&=\frac 1\pi \int_0^1 \frac {dt}{t\sqrt{1-t^2}}\, e^{-2\hat m t^2} J_1(2|\hat x| t)\,,
\label{eq:rhoq}
\end{align}
and using the identities \eqref{eq:sum001} and \eqref{eq:sum100}
the dynamical part of the spectral density is given by
\begin{align}
\hat\rho^\d(\hat x,\hat m) &= e^{-\hat m} \sum_\nu I_\nu(\hat m) \rho_\nu^\d(\hat x,\hat m)
\notag\\ &= -\frac 2\pi \frac{|\hat x|}{\hat x^2+\hat m^2}\int_0^1
\frac {dt}{\sqrt{1-t^2}}\,e^{-2\hat m t^2} \notag\\
&\qquad \times\big[ \hat x t J_1(2\hat x t) +\hat m(1-2t^2)J_0(2\hat x t)\big]\,.
\label{eq:rhod}
\end{align}
These formulas are valid for both positive and negative quark mass.
In Fig.~\ref{fig:3d} we show three-dimensional plots of the quenched
(top) and the dynamical (bottom) part of the spectral density. The
quenched part oscillates about the asymptotic value of $1/\pi$, while
the dynamical part oscillates about zero. For negative mass the
amplitude of the oscillations increases exponentially with the volume
(i.e., with the rescaled quark mass $\hat m=mV\Sigma$), while the period
in terms of $\lambda=\hat x/V\Sigma$ is of order $1/V$. In
Fig.~\ref{fig:rhoqd} we plot $\hat\rho^\q(\hat x,\hat m)+ \hat\rho^\d(\hat x,\hat m) $. This
figure shows that the exponentially large oscillations do not cancel,
which also follows from the asymptotic results given below.
\begin{figure}
\centerline{\includegraphics{rhoqd.pdf}}
\caption{Three-dimensional plot of $\hat\rho^\q(\hat x,\hat m)+\hat\rho^\d(\hat x,\hat m)$.}
\label{fig:rhoqd}
\end{figure}
Let us consider the large-$|\hat m|$ limit of these results. For the
zero-mode part we find
\begin{align}
\hat\rho^\zm(\hat x,\hat m)\sim
\begin{cases}
\displaystyle \sqrt{\frac{2\hat m}\pi}\delta(\hat x) \,,
& \hat m\to\infty\,,\\
\displaystyle -\frac{e^{2|\hat m|}}{\sqrt{8\pi|\hat m|}}\delta(\hat x) \,,
& \hat m\to-\infty\,.
\end{cases}
\end{align}
For the nonzero-mode parts the integrals over $t$ can be evaluated in
saddle-point approximation. For $\hat m\to\infty$ a universal scaling
function is obtained by taking the limit (see Appendix~\ref{appc})
\begin{align}
\lim_{\hat m \to \infty}\hat\rho^\q(u \sqrt{\hat m},\hat m)
&= \frac {|u| e^{-\frac{u^2}4}} {\sqrt{8 \pi}}
\big[I_0(u^2/4)\!+\!I_1(u^2/4)\big],\label{as-plus}\\
\lim_{\hat m \to \infty}\hat\rho^\d(u\sqrt{\hat m},\hat m)
&= -\frac {|u| e^{-\frac{u^2}4}}{\hat m\sqrt{ 2\pi}} I_0(u^2/4)\,. \end{align}
This shows that the dynamical part of the spectral density is
suppressed by $1/\hat m$ so that we recover the quenched result in the
large-$\hat m$ limit (or, equivalently, the thermodynamic limit). In
Fig.~\ref{fig:dens_pos} we compare the $\hat m\to \infty $ limit of the
quenched part of the rescaled spectral density with the exact result
for $\hat m=40$.
\begin{figure}
\centerline{\includegraphics{rhoq-asym-min-2.pdf}}
\caption{Comparison of the exact result \eqref{eq:rhoq} for
the quenched part of the spectral density for $\hat m = 40$ (solid
red curve) with the asymptotic result \eqref{as-plus} (dashed
black curve).}
\label{fig:dens_pos}
\end{figure}
For large negative mass, the spectral density factorizes into
functions that only depend on $\hat x$ or $\hat m$. A leading-order saddle
point-approximation results in (see Appendix~\ref{appa})
\begin{alignat}{2}
\rho^\q(\hat x,\hat m) &\sim
\frac{e^{2|\hat m|}}{\sqrt{8\pi|\hat m|}}J_1(2|\hat x|)\,,
&& \hat m\to-\infty\,, \label{as-min}\\
\rho^\d(\hat x,\hat m) &\sim
\frac{e^{2|\hat m|}}{\sqrt{2\pi|\hat m|^3}}\, |\hat x| J_0(2\hat x)\,,\quad
&& \hat m\to-\infty\,.
\end{alignat}
In agreement with our naive expectation, also in this case the
dynamical contribution to the spectral density is suppressed by
$1/\hat m$.
The result for large negative $\hat m$ increases exponentially with the
volume. In Fig.~\ref{fig:dens_neg} we show the asymptotic rescaled
result of the quenched part together with the exact result for $\hat m =
-40$.
\begin{figure}
\centerline{\includegraphics{rhoq-asym-2.pdf}}
\caption{Comparison of the rescaled exact result \eqref{eq:rhoq} for
the quenched part of the spectral density for $\hat m =-40$ (solid
red curve) with the asymptotic result \eqref{as-min} (black dashed
curve).}
\label{fig:dens_neg}
\end{figure}
In the remainder of this section we briefly discuss an alternative
approach to compute the spectral density at fixed $\theta$-angle,
based on the expression for the one-flavor spectral density at fixed
$\nu$ derived in \cite{Akemann:1998ta},
\begin{align}
\hat\rho_\nu(\hat x,\hat m) =
\frac{(-1)^\nu}2 |\hat x|(\hat x^2+\hat m^2)
\frac{\hat Z^{N_f=3}_\nu(\hat m,i\hat x,i\hat x)}{\hat Z_\nu^{N_f=1}(\hat m)}\,,
\end{align}
where we have defined $\hat Z(\hat m)=Z(\hat m/V\Sigma)$. Using
Eq.~\eqref{eq:rho_theta} this gives \cite{Damgaard:1999ij,Kanazawa:2011tt}
\begin{align}
\hspace*{-2mm}
\hat\rho(\hat x,\hat m,\theta) &= \frac 1{\hat Z(\hat m,\theta)} \sum_\nu e^{i\nu\theta}
\hat Z_\nu(\hat m)\hat\rho_\nu(\hat x,\hat m)\notag \\
&= \frac{|\hat x|(\hat x^2+\hat m^2)}{2\hat Z(\hat m,\theta)}\sum_\nu e^{i\nu\theta}(-1)^\nu
\hat Z^{N_f=3}_\nu(\hat m,i\hat x,i\hat x)\notag \\
&= \frac{|\hat x|}2(\hat x^2+\hat m^2)\frac {\hat Z^{N_f=3}(\hat m,i\hat x,i\hat x,\theta+\pi)}
{\hat Z^{N_f=1}(\hat m,\theta)} \,.
\end{align}
An integral representation of the microscopic partition function at
fixed $\theta$ was worked out in Ref.~\cite{Lenaghan:2001ur},
which gives us an explicit, though quite involved, analytical
expression for the spectral density at fixed $\theta$ \cite{Kanazawa:2011tt}. The approach
we followed above appears to be simpler. For the quenched
nonzero-mode part of the spectral density at $\theta=0$ it is easy to
see that the two methods lead to the same final expression. This part
can be shown to be
\begin{align}
\hat\rho^\q(\hat x,\hat m) =
\frac{|\hat x|}2 e^{-\hat m } \int_{-\pi}^\pi \frac {d\phi}{2\pi}\,
e^{\hat m \cos \phi} \hat Z^{N_f=2}(i\hat x,i\hat x,\phi+\pi)\,,
\end{align}
and using the expression for the two-flavor partition function derived
in \cite{Leutwyler:1992yt} we reproduce Eq.~\eqref{eq:rhoq}.
\section{\boldmath Chiral condensate at $\theta=0$}
\label{sec:condensate}
In this section we answer the question raised in the introduction, namely in what way a non-vanishing eigenvalue density can result in a chiral condensate that remains constant when the quark mass becomes negative. Because of the Banks-Casher relation this is not possible for a positive definite eigenvalue density. In essence, the discontinuity \eqref{eq:qm1} predicted by the sign-quenched theory must be canceled by another discontinuity \eqref{eq:heavi} due to the oscillating part of the spectral density. Here we show that to obtain this discontinuity a similar mechanism is at work in one-flavor QCD as in the other cases discussed in the introduction.
We restrict ourselves to $\theta=0$, although our results can in principle be extended to nonzero $\theta$ using Eq.~\eqref{eq:S}. The chiral condensate is related to the spectral density via Eq.~\eqref{eq:BC}. Using the same decomposition as for the spectral density the chiral condensate can be decomposed as
\begin{align}
\Sigma(m) = \Sigma^\zm(m) + \Sigma^\q(m) + \Sigma^\d(m)\,.
\end{align}
We define $\hat\Sigma(\hat m)=\Sigma(\hat m/V\Sigma)/\Sigma$ to simplify the
notation in the microscopic domain. The contribution from the zero
modes in this domain is well known \cite{Leutwyler:1992yt},
\begin{align}
\hat\Sigma^\zm(\hat m) = e^{-\hat m}\big[I_0(\hat m) + I_1(\hat m)\big]\,.
\label{cond-zero}
\end{align}
The contributions of $\rho^\q(\hat x,\hat m) $ and $\rho^\d(\hat x,\hat m)$ to the
chiral condensate can be obtained using Eqs.~\eqref{eq:rhoq} and
\eqref{eq:rhod} and performing the integral in Eq.~\eqref{eq:BC},
which in our notation and using the symmetry of the density becomes
\begin{align}
\hat\Sigma^{\q,\d}(\hat m)=2\hat m \int_0^\infty d\hat x\,
\frac{\hat\rho^{\q,\d}(\hat x,\hat m)}{\hat x^2+\hat m^2}\,.
\end{align}
These integrals are known analytically (see
Appendix~\ref{app:integrals}), resulting in
\begin{align}
\label{cond-anq}
\hat\Sigma^\q(\hat m) &=\frac 1{\pi\hat m} \int_0^1
\frac {dt\, e^{-2\hat m t^2}}{t^2\sqrt{1-t^2}}
\left[ 1 - 2t|\hat m| K_1(2t|\hat m|)\right],\\
\label{cond-an}
\hat\Sigma^\d(\hat m) &= -\frac {4}\pi \int_0^1 \frac {dt\,t\,e^{-2\hat m t^2}}
{\sqrt{1-t^2}} \\
&\quad\times
\big[t\hat m K_0(2t|\hat m|)+(1-2t^2)|\hat m|K_1(2t|\hat m|)\big]\,.\notag
\end{align}
For $|\hat m|\to0$ we have $\hat\Sigma^\zm(\hat m)\to1$, i.e., in this limit the
chiral condensate is entirely due to the zero modes. It is
straightforward to show that both $\hat\Sigma^\q(\hat m)$ and $\hat\Sigma^\d(\hat m)$
vanish for $|\hat m|\to0$.
Before giving the exact result for $\hat\Sigma(\hat m)$, let us look at the asymptotic behavior for $|\hat m| \gg 1$. For the zero-mode contribution we find (see Appendix~\ref{app:condensate})
\begin{align}
\hat\Sigma^\zm(\hat m)\sim
\begin{cases}
\displaystyle \sqrt{\frac2{\pi\hat m}}\,, & \hat m\to\infty\,,\\[3mm]
\displaystyle \frac{e^{2|\hat m |}}{\sqrt{8\pi |\hat m|^3}}\,, & \hat m\to-\infty\,,
\end{cases}
\label{eq:zm}
\end{align}
i.e., while the contribution of the zero modes is suppressed as
$\hat m\to\infty$, it grows exponentially as $ \hat m\to-\infty$. As was
already observed in \cite{Kanazawa:2011tt}, in order to get a
mass-independent chiral condensate, this exponential growth must be
canceled by the contributions of the nonzero modes. The large-$|\hat m|$
behavior of these contributions can be analyzed using a similar
approach as in Appendix~\ref{app:asympt}, and we obtain
(see Appendix~\ref{app:condensate})
\begin{align}
\label{eq:q}
\hat\Sigma^\q(\hat m) & \sim
\begin{cases}
\displaystyle 1-\sqrt{\frac2{\pi\hat m}}+\frac 1{2\hat m}\,,
&\hat m \to \infty\,, \\[3mm]
\displaystyle -\frac {e^{2|\hat m|}}{\sqrt{8\pi |\hat m|^3}}\,,
& \hat m \to -\infty\,,
\end{cases}\\
\hat\Sigma^\d(\hat m) & \sim
\begin{cases}
\displaystyle -\frac 1{2\hat m}\,, & \hat m \to \infty\,, \\[3mm]
\displaystyle 2 -\frac {1}{|\hat m|}\,, & \hat m \to -\infty\,.
\end{cases}
\label{asymd}
\end{align}
We see that the dynamical part is finite, while the quenched part
diverges in the thermodynamic limit for negative mass. As already
observed in \cite{Kanazawa:2011tt}, the leading divergence in
Eq.~\eqref{eq:q} exactly cancels the leading divergence of the
zero-mode part in Eq.~\eqref{eq:zm}. To extract a finite result for
the chiral condensate, the cancellation has to be implemented
analytically for arbitrary $\hat m$. This can be achieved by observing
that the zero-mode contribution can be rewritten as
\begin{align}
\hat\Sigma^\zm(\hat m) &= e^{-\hat m}\big[I_0(\hat m) + I_1(\hat m)\big] \notag\\
&= \frac 1{\pi\hat m} \int_0^1 \frac{dt}{t^2\sqrt{1-t^2}}
\big(1-e^{-2\hat m t^2}\big)\,,
\end{align}
which can be checked by Mathematica \cite{mathematica}. Adding this result to the quenched part of the chiral condensate we obtain
\begin{align}
&\hat\Sigma^\q(\hat m)+\hat\Sigma^\zm(\hat m) = \frac 1{\pi\hat m} \int_0^1
\frac{dt}{t^2\sqrt{1-t^2}} \notag\\
&\qquad\qquad\qquad\quad \times
\left [ 1 - e^{-2\hat m t^2} 2t|\hat m| K_1(2t|\hat m|) \right].
\label{cond-zmq}
\end{align}
The asymptotic behavior of this result is given by (see
Appendix~\ref{app:condensate})
\begin{align}
\hat\Sigma^\q(\hat m)+\hat\Sigma^\zm(\hat m) \sim
\begin{cases}
\displaystyle 1+\frac 1{2\hat m}\,, & \hat m \to \infty\,,\\[3mm]
\displaystyle -1+\frac 1{|\hat m|}\,, & \hat m \to -\infty\,.
\end{cases}
\label{asymq}
\end{align}
We observe that the asymptotic forms of $\hat\Sigma^\q(\hat m)+\hat\Sigma^\zm(\hat m)$ and
$\hat\Sigma^\d(\hat m)$ add up to one. In fact, the relation
\begin{align}
\hat\Sigma^\zm(\hat m)+\hat\Sigma^\q(\hat m)+\hat\Sigma^\d(\hat m) = 1
\end{align}
holds for all $\hat m$, which shows explicitly that the chiral condensate
is continuous when the quark mass crosses the imaginary axis. We
prove this relation in Appendix~\ref{appd} by showing that the second
derivative with respect to $\hat m$ can be expressed as an integral over
the total derivative of a function $f(t,\hat m)$ that vanishes at the
endpoints of the integration domain.
In Fig.~\ref{fig:cond} we show $\hat\Sigma^\q(\hat m)+\hat\Sigma^\zm(\hat m)$ (red),
$\hat\Sigma^\d(\hat m)$ (blue), and the sum of the two contributions (black),
which indeed equals one.
\begin{figure}[t!]
\centerline{\includegraphics{sigtot.pdf}}
\caption{The rescaled chiral condensate (black solid curve) for
one-flavor QCD as a function of $\hat m$ is the sum of a quenched
contribution, which includes the contribution of the zero modes,
(red curve) and a contribution due to the dynamical quarks (blue
curve). In the thermodynamic limit both the red and the blue curve
develop a discontinuity at $m=\hat m/V\Sigma=0$.}
\label{fig:cond}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
We have obtained simple analytical expressions for the microscopic spectral density of the Dirac operator for one-flavor QCD at zero $\theta$-angle. These results enabled us to clarify the relation between the spectral density and the chiral condensate and to explain the puzzle that in the thermodynamic limit the chiral condensate develops a discontinuity in sectors of fixed topological charge, while after summing over all sectors the discontinuity disappears. The underlying reason is that for negative mass the spectral density is no longer positive definite, which invalidates the Banks-Casher relation. A different mechanism, first discovered within the context of QCD at nonzero chemical potential, takes over. The essence of this mechanism is that an oscillating part of the spectral density with a period inversely proportional to the volume and an amplitude that diverges exponentially with the volume can give rise to a contribution to the chiral condensate that is discontinuous in the thermodynamics limit at a location where there is no dense line of eigenvalues. For QCD at nonzero chemical potential this mechanism creates a discontinuity of the chiral condensate across the imaginary axis, while the eigenvalues of the Dirac operator are scattered in a two-dimensional area around the imaginary axis. For one-flavor QCD the discontinuity due to the oscillating part of the spectral density cancels the discontinuity of the chiral condensate of the sign-quenched theory so that the chiral condensate remains constant when the mass crosses the imaginary axis. An important difference to QCD at nonzero chemical potential is the role played by the zero modes, which cancel a divergent contribution due to the nonzero modes. The Silver Blaze property \cite{Cohen:2003kd} of the chiral condensate could be shown by rewriting the contributions to the chiral condensate in terms of a total derivative. Remarkably, cancellations in the baryon number for QCD at nonzero chemical potential could also be explained in terms of total derivatives \cite{Greensite:2013vza}. Whether this is a coincidence or a generic feature of the Silver Blaze problem will be deferred to future work. Our results can be generalized to arbitrary $\theta$-angle and more flavors, which we also hope to address in a future publication.
\begin{acknowledgments}
We acknowledge support by the Alexander-von-Hum\-boldt Foundation (JV), U.S.\ DOE Grant No.\ DE-FG-88ER40388 (JV), and DFG Grant SFB/TRR-55 (TW). We also thank Jacques Bloch, Poul Damgaard, Takuya Kanazawa, Mario Kieburg, and Kim Splittorff for useful discussions.
\end{acknowledgments}
|
1,116,691,499,377 | arxiv | \section{Introduction}
The abundance profiles of chemical elements constitute one of the key properties of galactic disks. They depend on the past history of the disk and the various physical effects that affected it: star formation, infall and outflows, radial flows of gas, radial motions of stars and tidal interactions or mergers with other galaxies. Most semi-analytical studies of abundance profiles were performed in the framework of the so-called "independent-ring" model, where the galactic disk is simulated as an ensemble of independently evolving annuli
(e.g. \citet{Guesten1982,Matteucci1989,Ferrini1994,PranAub1995,Chiappini1997,BP99,Hou2000,Prantzos2000})
and concerned the MW disk, for which a large number of other constraints,
both local and global are available. Those studies focused mainly on the interplay between the local star formation and infall rates, or the impact of variable stellar
IMF. They also revealed the key issue of the evolution of the abundance profile, some studies supporting a flattening of it with time (e.g.
\citet{Ferrini1994,PranAub1995,Hou2000} while others concluded the opposite (e.g. \citet{Tosi1988,Chiappini1997}).
Pioneering work of \cite{Tinsley1978} and \cite{Mayor1981} noticed the potential importance of radial gaseous flows for the chemical evolution of galactic disks. \cite{Lacey1985} presented a systematic investigation of the causes of such flows, and explored with parametrized calculations the impact of such effects on the chemical evolution of the Galaxy. Further parametrized investigations with simple 1D models of disk evolution are made in e.g. \cite{Tosi1988,Clarke1989,Sommer1990,Goetz1992,Chamcham1994,Edmunds1995,Portinari2000} and more recently in \cite{Spitoni2011,Bilitewski2012,Mott2013,Cavichia2014}. They have various motivations (mostly to fit the abundance profiles, but also gas and star profiles) and they are generally applied to the study of the MW disk. As expected, results are not conclusive, because they depend not only on the parametrization of the unknown inflow velocity patterns, but also on the other unknown (and parameterized) ingredients of the models, especially the adopted SFR and infall profiles as functions of time.
Among the alleged causes of radial inflows, the impact of a galactic bar is well established, both from simulations and from observations. Numerical simulations \citep{Athanassoula1992,Friedli1993,Shlosman1993} showed that the presence of a non-axisymmetric potential from a bar can drive important amounts of gas inwards of corotation (CR) fuelling star formation in the galactic nucleus, while at the same time gas is pushed outwards outside corotation. In a disk galaxy, this radial flow mixes gas of metal-poor regions into metal-rich ones (and vice-versa) and may flatten the abundance profile (e.g. \citet{Friedli1994,Zaritsky1994,Martin1994,Dutil1999}), although \cite{Sanchez2012} find little difference in that respect between barred and non-barred disks. The study of \cite{KPA2013} suggests that bars may
be changing the chemical abundance profile inside the corotation
radius but they have only a small impact outside it, while \cite{Martel2013} find a rather complex situation of continuous exchange of gas and metals between the bar and the central region of the disk.
The investigation of radial motions of stars - due to inhomogeneities of the galactic gravitational potential - on the chemical evolution of disks, has a more recent history. The role of the bar has been studied to some extent with N-body+SPH codes by \cite{Friedli1993} and \cite{Friedli1994}. Observations in the 90ies revealed that the MW does have a bar \citep{Blitz1991}, but its size and age are not well known yet. \cite{SellwoodBinney2002} showed that, in the presence of recurring transient spirals, stars in a galactic disk could
undergo important radial displacements: stars
found at corotation with a spiral arm may be scattered to different galactocentric radii (inwards or outwards), a process which preserves
overall angular momentum distribution and does not contribute to the radial heating of the stellar disk.
Using a simple model, they showed how this process can increase the dispersion in the local metallicity vs age relation, well above the amount due to the epicyclic motion.
\cite{minchevfamaey2010} suggested that resonance overlap of the bar and spiral structure \citep{Sygnet1988} produces a more efficient redistribution of angular momentum in the disk. This bar-spiral coupling was studied in detail with N-body simulations by \cite{Brunetti2011} who
found that radial migration can be assimilated to a diffusion process, albeit with time- and position-dependent diffusion coefficients. That idea was confirmed by the analysis of N-body+SPH simulations of a disk galaxy by \cite{KPA2013} who showed that radial migration moves around not only "passive" tracers of chemical evolution (i.e. long-lived stars, keeping on the surfaces the chemical composition of the gas at the time and place of their birth), but also "active" agents of chemical evolution, i.e. long-lived nucleosynthesis sources (mainly SNIa producing Fe and $\sim$1.5 M$_{\odot}$ \ stars producing s-process elements).
The implications of radial migration for the chemical evolution of MW-type disks
were studied with N-body codes by \cite{Roskar2008} , who found that the stellar abundance profiles flatten with stellar age, even if the gaseous abundance profiles were steeper in the past.
\cite{SB2009} introduced a parametrised prescription of
radial migration (distinguishing epicyclic motions
from migration due to transient spirals) in a semi-analytical chemical evolution code. They suggested that radial mixing could explain not only local observables (e.g. the dispersion in the age-metallicity relation) but also the formation of the Galaxy's thick disk, by bringing to the solar neighborhood a kinematically "hot" stellar population from the inner disk.
That possibility was subsequently investigated
with N-body models, but controversial results are obtained up to now:
while \cite{Loebman2011} find that secular processes (i.e. radial migration) are sufficient to explain the kinematic properties of the local thick disk, \cite{Minchev2012b} find this mechanism insufficient and suggest that an external agent (e.g. early mergers) is required for that.
Following the pioneering work of \cite{SB2009}, the properties of the MW disk were studied in detail with semi-analytical models accounting for radial migration by
\cite{Minchev2013} and \cite{Kubryk2014}. The three models differ in several ways:
\cite{SB2009} use a toy-model of star transfer between adjacent radial zones (with coefficients tuned to reproduce properties of the local disk), whereas the other two are inspired by the results of N-body simulations (but they adopt different implementation techniques of those results). Radial gaseous flows are included in \cite{SB2009} and \cite{Kubryk2014}, but not in \cite{Minchev2013}; however, in \cite{SB2009} the radial flows concern mainly the outer disk, where in \cite{Kubryk2014} they concern the inner disk, since that work simulates the action of a bar. The dimension vertical to the galactic plane is considered in \cite{SB2009} and \cite{Minchev2013}, but not in \cite{Kubryk2014}. Finally, the star formation and radial infall laws are different in the three works. All models consider explicitly Fe production by SNIa (albeit with different prescriptions for the SNIa rate) and the finite lifetimes of stars.
Despite those differences, all three models find good agreement with the main observables of the MW, both locally (dispersion in age-metallicity relation, metallicity distribution, the characteristic "two-branch" behaviour between thick and thin disk in the O/Fe vs Fe/H plane) and globally (stellar and abundance profiles). This agreement suggests that, despite their sophistication, such models still involve too many parameters and suffer from degeneracy problems. We note
here the difference in the final abundance gradient of Fe/H between \cite{SB2009} and \cite{Minchev2014}, who find slopes of the corresponding exponential profiles of -0.1 dex kpc$^{-1}$ and -0.06 dex kpc$^{-1}$, respectively. Recent observations of statistically significant samples of Cepheids are consistent with the latter value, as we shall discuss in Sec. \ref{sec:abundance-evol}.
In this work, we study the evolution of abundance profiles of all elements from H to Ni in the MW, using the model presented in Kubryk et al. (2014, hereafter KPA2014).
The plan of the paper is as follows: The main ingredients of the model are briefly presented in Sec. \ref{sec:Model}, where we also discuss some of the results concerning the impact of radial migration on the disk properties. We illustrate that impact by comparing a model with radial migration to one without it. In Sec. \ref{sec:abundance-evol} we discuss in some detail the profiles of the most important metals, namely O (Sec. \ref{sub:O-profiles}) and Fe
(Sec. \ref{sub:Fe-profiles}) and we compare them to a large number of recent observations from various metallicity tracers. The impact of radial migration on
the evolution of the abundance profiles is discussed in Sec. \ref{subsec:Evolution_profiles}, where we compare our results to those of a similar study \citep{Minchev2014} and to a compilation of extragalactic observations by \cite{Jones2013}. In Sec. \ref{sub:OFe} we adress the issue of the O/Fe ratio; in view of the small dispersion displayed by that ratio as a function of time, it can be used as a robust proxy for stellar age in studies of the evolution of the abundance profiles in the disk. In Sec. \ref{sub:Other} we present our results for all elements from H to Ni and we compare them to several sets of observational data. We find good overall agreement with observations, but a systematically larger (in absolute value) slope of the abundance profiles for the Fe-peak elements compared to observations. We also reveal - and we draw attention to - interesting differences between the results obtained with different sets of stellar yields, as well as a
manifestation of the "odd-even" effect of nucleosynthesis, which does not appear, however, in the observational data.
A summary of the results is presented in Sec. \ref{sec:Summary}.
\section{The model}
\label{sec:Model}
The model presented in KPA14 for the evolution of the MW disk, involves radial motions of both gas and stars. The MW disk is built gradually by infall of primordial gas \footnote{The composition of the infall is equally important when it comes to discuss the evolution of abundances and abundance ratios in the MW disk. Observations are of little help at present: they generally find low metallicities for gas clouds {\it presently} falling to the MW disk ($\sim$0.1 Z$_{\odot}$, e.g. \citealt{Wakker1999}), but they provide no information on the past metallicity of such clouds or on their abundance ratios. Here we adopt the simplest possible assumption, namely that the infalling gas has always primordial composition. This assumption hardly affects the results for the chemical evolution of the disk, but it allows for the existence of disk stars with metallicities lower than [Fe/H]=-1 (see \citet{Bensby2013a} and references therein).} in the potential well of a "typical" dark matter halo of final mass 10$^{12}$ M$_{\odot}$, the evolution of which is extracted from numerical simulations (from
\citet{Li2007}). The infall rate is a parametrized function, its timescale increasing monotonically with galactocentric radius and ranging from 1 Gyr at 1 kpc to 7 Gyr at 7 kpc and slightly increasing further outwards (see Fig. \ref{Fig:ModelGen1}). Star formation depends on the local surface density of molecular gas, which is calculated by the semi-empirical prescriptions of \cite{BlitzRos06}: it depends on a combination of the stellar surface density profile (steeply decreasing with radius) and the gas surface density profile (essentially flat today). This allows us to use the final profiles of atomic and molecular gas as supplementary constraints to the model (see Fig. 6 in KPA14). The adopted prescription produces a steep profile of H$_{\rm 2}$ \ (as observed in the Galaxy) and, thereoff, a steep SFR profile in the inner disk during most of the Galactic evolution; this impacts directly on the corresponding abundance profiles, as we discuss below. For the radial flows of gas, we consider only the case of a MW-like bar operating for the last 6 Gyr and driving gas inwards and outwards of corotation. The radial velocity profile of the gas flow induced by the bar is similar to the one adopted in \cite{Portinari2000} (their Case B for the bar), but not exactly the same: \cite{Portinari2000} consider additional radial flows inwards, acting all over the disk, while we limit ourselves to the case of the bar alone. The adopted radial velocity profile is given in Fig. \ref{Fig:ModelGen1}.
We considered separately the epicyclic motion of stars (blurring) from the true variation of their guiding radius (churning), as in \cite{SB2009}.
For the former, we developed an analytic formalism
based on the epicyclic approximation; for the latter, we adopted a parametrised description, using time- and radius- dependent diffusion coefficients, extracted from the N-body+SPH simulation of KPA13, which concerns a disk galaxy with a strong bar; as discussed in KPA2014, we adapted those transfer coefficients taking into account the smaller size of the MW bar. For the chemical evolution, we adopted recent sets of metallicity-dependent yields from \cite{Nomoto2013}, providing a homogeneous and fine grid of data, well adapted to the case of the MW disk. For comparison purposes, we also used older yields from \cite{WW95} and \cite{CL2004}. We adopted the stellar IMF of \cite{Kroupa02} with a slope of X=1.7 (the Scalo slope) for the high masses. For the rate of SNIa, we adopted the empirical law of $R_{SNIa} \ \propto t^{-1.1}$, after the observed delayed time distribution of those objects in external galaxies (see e.g. \cite{Maoz2012} and references therein). In contrast with usual practice in studies of galactic chemical evolution, we adopted the formalism of Single Particle Population (SSP), which is the only one applicable to the case of radial migration, since it allows one to consider the radial displacements of nucleosynthesis sources and in particular of SNIa (see KPA13 for a discussion of that effect). Finally, we adopted a large and diverse set of recent observational data to constrain our model.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{F_mod1.eps}
\caption[]{{\it Left } (from top to bottom): Model profiles of stars, gas, SFR and molecular gas fraction $f_{Mol}$ at 4, 8 ({\it thin} curves) and 12 Gyr ({\it thick} curve). The curve at 12 Gyr is compared to observational data for the present-day profiles of the correspoding quantities ({\it shaded aereas} for the stars, gas and $f_{Mol}$ and points with error bars for the SFR); data sources are provided in KPA14. {\it Right} (from top to bottom): Infall timescales, infall rates, velocity profiles of radial inflow (positive values towards the Galactic center), ratio of SNIa/CCSN ; in all panels but the top one, the three curves correspond to 4, 8 and 12 Gyr (thickest curve).
}
\label{Fig:ModelGen1}
\end{center}
\end{figure}
Our model reproduces well the present day values of most of the main global observables of the MW bulge (assumed to correspond to radii $r<$2 kpc) and disk ($r>$2 kpc): present-day masses of stars, atomic and molecular gas, star formation rates as well as core collapse supernova (CCSN) and SNIa rates (see Fig. 2 in KPA14). The corresponding radial profiles of all those quantities (azimuthally averaged) are also reproduced in a satisfactory way (Fig. 6 in KPA14). The azimuthally averaged radial velocity of gas inflow in the bar region is constrained to be less than a few tenths of km/s in the framework of that model. The local properties of the MW disk, i.e. metallicity distribution and age-metallicity relation, are also well reproduced. In particular, following \cite{SellwoodBinney2002}, we showed how radial migration can be constrained by the observed dispersion in the age-metallicity relation. . We emphasize, however, that the observational samples that we used - from \cite{Bensby2014} for the age-metallicity relation and from \cite{Adibekyan2011} and \cite{Bensby2014} for the metallicity distribution - have various selection biases (kinematic, limited by magnitude or volume), which have not been applied to our results: the model predictions for the solar neighborhood concern the "solar cylinder", of diameter 0.5 kpc (the size of our radial bin), centered on the Sun.
{\it Assuming } that the thick disk is the oldest ($>$9 Gyr) part of the disk, we found that the adopted radial migration scheme can reproduce quantitatively the main local properties of the thin and thick disk: metallicity-distributions, the characteristic "two-branch" behaviour of the local O/Fe vs Fe/H relation, local surface densities of stars (10 M$_{\odot}$/pc$^2$ \ and 28 M$_{\odot}$/pc$^2$ \ for the thick and thin disk, respectively). The thick disk extends up to $\sim$11 kpc and has a scale length of 1.8 kpc; this is consistent with recent evaluations (e.g. \citealt{Bovy2013} and references therein) and it is considerably shorter than the one of the thin disk, consistent with the inside-out formation scheme.
Some of the main results of the model relevant for this work appear in Fig. \ref{Fig:ModelGen1}. The inside-out formation of the disk is evidenced by the evolution of both the infall rate profile and the stellar profile. The gaseous profile, mostly flat today, with a local surface density of $\sim$12 M$_{\odot}$/pc$^2$, is well reproduced.
The profile of the molecular fraction $f_{Mol} = \frac{\rm H_2}{\rm (HI+H_2)}$ is well reproduced also, after the prescriptions of \cite{BlitzRos06}. As already mentioned,
this profile plays an important role in our model, because it determines the molecular profile and, thereoff, the profile of the star formation rate $\Psi (r) = f_{Mol}(r) \Sigma_{Gas}(r)$.
The profile of the SNIa/CCSN ratio (bottom right panel) becomes steeper as one moves from the outer to the inner Galaxy, because the SNIa rate - being a mixture of old and young population objects - follows a combination of the stellar and gas profiles. The steep stellar profile increases substantially the SNIa/CCSN ratio in the inner disk; the outer disk, populated mostly by gas and young stars, has essentially SNIa belonging to the young stellar population, as the CCSN: as a result, the SNIa/CCSN ratio in the outer disk is practically constant. Notice that similar results for the SNIa/CCSN ratio are obtained in the independent-ring model for the MW disk by
\cite{Boissier2009}, both with the numerical prescription of \cite{Greggio05} and with an analytical prescription for the SNIa rate (their Fig. 11). In the present study, a fraction of SNIa - those belonging to the old stellar population - is affected by radial migration, mostly in the region 3-12 kpc (see next paragraph). All these features affect directly the resulting O and Fe profiles, as well as those of all other elements (see discussion in next section).
Some aspects of the stellar radial migration of the model appear in Fig. \ref{Fig:ModelGen2}. The top panel displays the fraction of stars born in radius $r$ (in a radial bin of width $\Delta r=\pm$0.25 kpc) and found in the end in all radii. It can be seen that the action of the bar brings a large fraction of the stars of the inner disk in the outer regions: some stars born in $r$=3 kpc are found int he solar neighborhood in the end of the simulation. As we show in KPA14, these are the most metallic stars presently found in the solar neighborhood, with metallicities [Fe/H]$\sim$0.4 and they are 3-5 Gyr old. In contrast, a negligible fraction of the stars born in $r>$12 kpc reaches the solar vicinity\footnote{ For each "birth radius" there are approximately as many stars migrating outwards as inwards. However, there are MORE STARS in the inner regions than in the outer ones, because of the exponentially declining outwards surface density profile. As a result, the outer regions receive more stars from the inner ones than what they send to them: the radial profile of the fraction of stars that each region receives from the others is not symmetric but biased towards the inner regions, i.e. each disk annulus receives more stars from the inner regions than from the outer ones.}
The middle panel of Fig. \ref{Fig:ModelGen2} displays the original vs final guiding radii of stars. Stars
found in the radial range 3-14 kpc have been formed, on average, inwards of their present position; the effect is most pronounced for stars in the region 6-9 kpc, where the average outwards displacement reaches $\Delta r\sim$1.5 kpc, and it is reduced to negligible values inside 4 kpc and outside 14 kpc. Also, the dispersion around those average values is large in the 6-9 kpc range and decreases outside it. Clearly, however, radial migration affects to some extent regions at all galactocentric distances. We stress that the extent of radial migration depends strongly on the adopted (time- and radius-dependent) diffusion coefficients and on the morphology of the disk galaxy: a larger amount of radial migration is expected in the case of barred disks, through the coupling of the bar to the spiral arms \citep{minchevfamaey2010}.
Finally, the bottom panel of Fig. \ref{Fig:ModelGen2} displays the average age of the stellar populations as function of Galactocentric radius, both for all the stars and for the stars formed {\it in situ}. In the latter case, the average age varies little with radius outside $r$=6 kpc, because the adopted profile of the infall timescale (Fig. \ref{Fig:ModelGen1}) varies little in that region. However, radial migration brings older (on average) stars from the inner disk in intermediate radii: as a result, a clear age gradient is developed throughout the disk (solid curve in Fig. \ref{Fig:ModelGen2} bottom).
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{F_mod2_2.eps}
\caption[]{{\it Top:} Fractions of stars born in radii $r_{Origin}$=3, 8 and 13 kpc and found in radius $r_{Final}$ at $t$=12 Gyr.
{\it Second row:} Numbers of stars born in annuli of radius $r_{Origin}$=3, 8 and 13 kpc and width $\Delta r$=0.5 kpc and found in radius $r_{Final}$ at $t$=12 Gyr.
{\it Third row:} Original vs final guiding radii of stars ({\it solid curve}); the shaded aerea includes $\pm$1 $\sigma$ values (i.e. from 16\% to 84\% of the stars) and the dotted diagonal line indicates the stellar guiding radii in the absence of radial migration.
{\it Bottom:} Average age of stars vs Galactocentric radius ({\it solid curve}); the shaded aerea includes $\pm$1 $\sigma$ values and the {\it dotted} curve indicates the average age of stars formed {\it in-situ}.
}
\label{Fig:ModelGen2}
\end{center}
\end{figure}
As emphasized in KPA14, the impact of radial migration on the properties of the disk is not intuitively straightforward, because migrating stars may return their gas (metal-rich, if originating from the inner disk or metal-poor if originating from the outer disk) in places far away from their home radius; this gas may affect the local metallicity and also fuel star formation (depending on the local star formation efficiency). The situation becomes even more complex in the case of a bar: the bar drives inwards gas - also fuelling star formation in the inner disk - which is more metal poor, in general, than the local gas and thus the metallicity in the inner disk decreases; the overall result depends, however, also on the ratio of the local infall rate to the SFR and on the previous history of the disk (which determines the metallicity at a given time).
Oxygen is affected differently than Fe, because the main source of the latter, namely SNIa, is affected by radial migration, while the source of O (massive stars) is not.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{F_MigrVsNoMigr.eps}
\caption[]{{\it Top:} Comparison of various properties of our model, expressed as the ratio of a given quantity in the case of radial migration to the same quantity when stars do not migrate. From top to bottom, {\it left}: star surface density, gas surface density, gas fraction, O/H profile; {\it right}: SFR surface density, SNIa rate, ratio of SNIa/CCSN, Fe/H profile. In all panels, {\it thick curves} represent results at 12 Gyr and thin curves at 8 and 4 Gyr, respectively (as in Fig. \ref{Fig:ModelGen1}).
}
\label{Fig:ModelGen3}
\end{center}
\end{figure}
We attempt an illustration of this complex behaviour
in Fig. \ref{Fig:ModelGen3}, where we plot the ratio of several quantities of the model (with radial migration and radial inflow) to those same quantities obtained by an identical model (same boundary conditions, same SFR and infall rates) without radial migration or radial inflow.
One can easily see that the quantities affected mostly by radial migration
are the long-lived stars and SNIa: their radial profiles are affected over most of the disk. The radial profiles of SNIa are affected to smaller extent than those of stars, because a substantial fraction of SNIa results from a young population, unaffected by radial migration ($\sim$40\% of them explode within 1 Gyr after the formation of their progenitor system, see Fig. C1 in Appendix C of KPA2013); in contrast, most stars are low-mass and long-lived ($\sim$90\% by number for a normal IMF) and their population is affected by radial migration.
Gas is affected mainly not by radial migration, but by the radial inflow induced by the bar. Its surface density is depleted in the 2-4 kpc region and slightly increased outside it; notice that the latter increase is also due, to a small extent, to the gas returned to the ISM by the migrating, dying stars.
The evolution of the gas profile is reflected in the one of the SFR profile,
which is also affected by the radially dependent fraction $f_{H_2}$ of molecular gas: there is less SFR in the 2-4 kpc region than without
radial migration and gas inflow. That region is also the one corresponding to the peak of the infall rate during the late evolution of the disk (see Fig. \ref{Fig:ModelGen1}), and shows a higher infall/SFR ratio than the model with no radial migration. For those reasons (smaller SFR and higher dilution of the metallicity through the primordial infall), this region is found to have a lower O/H ratio than in the model with no radial migration. The decrease in the Fe/H ratio is smaller, because some Fe is contributed in those zones by SNIa migrating inwards. But the largest impact on the chemical evolution concerns the SNIa migrating outwards: they increase the Fe content of the region between 5 and 8 kpc by $\sim$30\% after 4 Gyr and by $\sim$10\% in the end of the simulation.
As a result, the final Fe/H radial profile is somewhat steeper than in the model without radial migration.
Overall, the effects of radial migration on the profiles of stars, SNIa, SNIa/CCSN ratio and Fe/H appear to become less important at late times. This
result appears counter-intuitive, at first sight, because more radial migration occurs at longer timescales (everything else kept equal). Our counter-intuitive result is due to the inside-out formation of the disk.
At early times, there are few stars (and SNIa) in the outer Galaxyl: any radial transfer from the inner regions (where a large stellar population has been formed) to the outer ones, increases the surface density of the latter by a large amount. At late times, the stellar population is in place all over the disk: the impact of radial migration from the inner to the outer disk (where a lot of stars are formed {\it in situ}) is proportionally smaller then.
KPA13 performed a similar exploration of the effects of radial migration for the case of a N-body+SPH simulation concerning a barred disk galaxy, evolving without gaseous infall (as a closed box). They found that, in that case, the strong bar induced
a much larger amount of radial migration all over the disk, affecting particularly its outmost regions.
As discussed in KPA14, we adapted the description of the radial migration of that model to the one of the MW, by taking into account the size of the bar in the two cases. The smaller bar of the MW implies smaller extent of radial migration than in the barred disk of KPA13.
\section{Abundance evolution}
\label{sec:abundance-evol}
Our model includes the detailed evolution of 83 isotopes, from H to Zn. The abundances of the corresponding 32 elements are obtained at each time step by summing over the isotopic ones.
IN KPA14 we have found that the adopted parameters of the model (SFR efficiency, infall timescale, SNIa rate, IMF, etc.) allow us to reproduce quite well the solar abundance of O and Fe for the {\it average 4.5 Gyr old star} in the solar neighborhood\footnote{We calculate the average metallicity $<[Z/H]>=\frac{\Sigma N_i[Z/H]_i}{\Sigma N_i}$ of stars of age $A$ found in zone $r$, where $N_i$ is the number of stars with metallicity $[Z/H]_i$ in that zone.} The average birth radius of those stars is found to be at a Galactocentric distance of $\sim$6.5 kpc, i.e. $\sim$1.5 kpc inwards of the present day position of the Sun. This implies that the Sun is an average star of 4.5 Gyr in the Solar neighborhood as far as its chemistry is concerned, but such stars are not born {\it in situ}, they have migrated here from the inner disk.
Regarding the other elements and isotopes, we find that the agreement with the solar abundances is quite good for most of them (to better than a factor of 2, see Fig. 17 in KPA14). In several cases, however, and in particular in the region of Sc to Mn, a clear underabundance is obtained with the adopted yields of \cite{Nomoto2013}. The reason of that disagreement is obviously a deficiency of the yields. A similar deficiency is obtained when using those yields to study the evolution of the halo(see Fig. 10 in \cite{Nomoto2013}). To cure for that, we normalised the model results as to have the average abundances of the 4.5 Gyr old stars currently present in the solar neighborhood equal to their solar values, i.e. we corrected the stellar yields (integrated over the IMF) to various degrees (by 2\% for O, up to a factor of 4 for V). In that way, we were able to compare the corresponding evolution of all the abundance ratios X/Fe vs Fe/H to observational data of two recent surveys \citep{Adibekyan2011,Bensby2014} concerning the local thin and thick disks separately.
We showed how such detailed comparisons in the future will provide valuable constraints to both stellar nucleosynthesis and chemical evolution models.
Here we extent the investigation of the abundance evolution to the whole disk for all the elements of our model;
we leave the isotopic evolution for a future paper.
We use abundance data from various sources and different classes of objects: Cepheids, B-stars, HII regions, planetary nebulae (PN) and open clusters. The first three concern young objects: Cepheids have masses $>$3 M$_{\odot}$ \ and, depending on their metallicity, they are younger than 300-400 Myr, while B-stars and HII regions are less than a few 10$^7$ Myr old; on the other hand, PN and open clusters correspond to objects with ages up to several Gyr. The sources of the adopted data are presented in Table 1.
\begin{table}
\caption{\label{Tab:Data}{References for adopted data on Cepheids, B-stars, HII-regions, planetary nebulae (PN) and Open clusters (OC).}
}
\begin{tabular}{lccccc}
\hline \hline
Element & Cepheids & B-stars & H-II & PN & OC\\
\hline
C & 1 & 4,5 & & & \\
N & 1 & 4,5 & 6 & 7 & \\
O & 1 & 4,5 & 6 & 7,8 &\\
Ne & & & & 7 & \\
Na & 1 & & & & \\
Mg & 1 & 4,5 & & & \\
Al & 1 & 4,5 & & & \\
Si & 1 & 4,5 & & & \\
S & & 4,5 & 6 & 7 & \\
Ar & & & & 7 & \\
Ca & 1 & & & & \\
Sc & 1 & & & & \\
Ti & 1 & & & & \\
V & 1 & & & & \\
Mn & 1 & & & & \\
Cr & 1 & & & & \\
Fe & 1,2,3 & & & & 9,10,11 \\
Co & 1 & & & & \\
Ni & 1 & & & & \\
\hline
\end{tabular}
1: \cite{Luck2011}, 2: \cite{Lemasle2013}, 3: \cite{Genovali2014}, 4:\cite{Gummersbach1998}, 5: \cite{Daflon2004}, 6: \cite{Rudolph2006}, 7: \cite{Henry2004}, 8:\cite{Henry2010}, 9: \cite{Magrini2009},
10: \cite{Yong2012}, 11: \cite{Frinchaboy2013}
\end{table}
\subsection{Oxygen profiles}
\label{sub:O-profiles}
Oxygen is a major product of the nucleosynthesis of massive stars (M$>$10 M$_{\odot}$). Being short-lived (lifetime $<$20 Myr), such stars have no time to migrate away from their birth sites (less than a 100 pc, as suggested by the fact that all CCSN localised up to now in external galaxies are within spiral arms and/or regions of active star formation). As a result, the radial O profile is not affected by radial migration. It is strongly affected, however, by gas radial inflows, as found in numerous studies, with semi-analytical and N-body+SPH models, e.g. \cite{Mayor1981,Lacey1985,Tosi1988,Friedli1994,Portinari2000,Spitoni2011,Bilitewski2012,Cavichia2014}, etc. The results presented here depend directly on the adopted treatment of radial inflow, which corresponds to the action of the galactic bar, as described in Sec. \ref{sec:Model}. Among the aforementioned studies, only \cite{Portinari2000} and \cite{Cavichia2014} considered explicitly radial flows induced by the Galactic bar.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{F_O_profiles.eps}
\caption[]{Oxygen abundance profiles of Cepheids, B-stars, HII regions, and planetary nebulae (PN); data sources are in Table 1. The adopted solar value
of log(O/H)+12=8.73 is from \cite{Asplund2009}. Model curves (from bottom to top, in all panels) represent the gaseous abundance profile at time 4, 8 and 12 Gyr ({\it thick curve}), respectively.
}
\label{Fig:O-profiles}
\end{center}
\end{figure}
In Fig. \ref{Fig:O-profiles} we display the evolution of the gaseous profile of oxygen in our model at three different times (4, 8 and 12 Gyr, respectively). The O profile results from the joint action of three different factors:
- the inside-out formation of the disk, affected by both the adopted infall profile (shorter time-scale in the inner disk, Fig. \ref{Fig:ModelGen1}, top right) and the larger efficiency of star formation in the inner disk (because of the larger molecular fraction there, Fig. \ref{Fig:ModelGen1}, bottom left).
- the radial inflow, which affects the gaseous profile and all the abundance profiles in the inner galaxy ($r<$6 kpc).
In the 2-4 kpc region, the combination of the action of the bar (which pushes gas partly towards the center and partly towards the outer disk) to the metal-poor infall leads to a local depression of O/H with respect to adjacent regions. This is due to the fact that the O rich gas pushed inwards and outwards from that region is replenished by the metal poor infall, the rate of which happens to be maximum in that region (Fig. \ref{Fig:ModelGen1}). In adjacent regions the effect is smaller, because the late infall rate is less intense there.
The disk beyond radius $r\sim$6 kpc is not affected by radial inflows. The resulting O/H profile is smoothly decreasing outwards, but it cannot be described by a single exponential over the whole radial range: its slope is steeper at small radii and flatter at larger ones. If it is fitted with a single exponential, then the slope depends on the radial range considered.
In this work, we shall consider as baseline values those in the range 5-14 kpc, where most of the observational data are available.
There are several shortcomings and uncertainties in the analysis of the observed O abundances of different types of objects across the Galactic disk, which are presented in the recent monograph of \cite{Stasinska2012}. The various surveys lead to widely different results for the O/H abundance gradient
$dlog(O/H)/dr$ (in dex kpc$^{-1}$), ranging from
small values (-0.023$\pm$0.06 in the PN sample of \citet{Stanghellini2010} in the 2-17 kpc range) to high ones (-0.056$\pm$0.013 in the Cepheid sample
of \cite{Luck2011} in the 5-16 kpc range).
In Fig. \ref{Fig:O-profiles} we compare our results to the
data of some recent, representative surveys, of the aforementioned tracers.
We do not plot all the data on the same figure, since this would create confusion and increase artificially the scatter, because of systematic uncertainties between different analysis techniques (see \citet{Stasinska2012}. It can be seen that our results are in overall agreement with the various observations.
There are practically no data corresponding to the inner disk
($r<$4 kpc), where our model predicts lower values than in adjacent reasons, as discussed above. We notice, however, that in their study of PN in the direction of the Galactic bulge, \cite{Chiappini2009b} identified a subsample of 44 objects
which actually belong to the inner disk population (a few kpc from the Galactic center) and have an average value of log(O/H)+12=8.52$\pm$0.23, i.e. less than expected from the extrapolation of the \cite{Henry2010} data for PN in that region. It is not yet clear whether that difference is due to systematic uncertainties between the two studies - \citet{Chiappini2009} collected
line intensities from the literature, unlike \citet{Henry2010} - or to a genuine decline of the oxygen profile in the inner disk, in qualitative agreement with our results.
\subsection{Iron profiles}
\label{sub:Fe-profiles}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{F_Fe_profiles.eps}
\caption[]{Iron abundance profiles of Cepheids (top) and open clusters (bottom). Data sources are provided in Table 1. Symbols in the upper panel correspond to Ref. 3 and in the lower panel to Refs. 9 (blue filled squares), 10 (brown asterisks) and 11 (green open squares), respectively. Curves corespond to model results at time 4, 8 and 12 Gyr (thick curve), respectively). The thick red curve in the {\it top} panel corresponds to the average metallicity of a young stellar population of age 0.2$\pm$0.2 Gyr (Cepheids) and the shaded aerea represents the corresponding $\pm$1-$\sigma$ dispersion.
}
\label{Fig:Fe-profiles}
\end{center}
\end{figure}
While O is exclusively the product of massive stars, Fe has two sources:
massive stars and thermonuclear supernovae.
From the nucleosynthesis and chemical evolution points of view, Fe is then
far more complicated to deal with than O. The reasons are
- The Fe yield of massive stars, exploding as CCSN, are difficult to calculate from first principles, since CCSN explosions are not well understood yet.
Observations suggest that Fe yields depend on the energy of the explosion
(e.g. \cite{Hamuy2003}) but this parameter is not systematically taken into account in yield calculations.
- Most of solar Fe appears to come not from CCSN but from SNIa (on the basis of the observed decline of O/Fe in disk stars), but is is difficult to relate in a unique way the rate of SNIa to that of CCSN (see, however, Appendix C in KPA2014).
Radial migration introduces one more layer of complexity in the story of Fe.
As shown in \cite{KPA2013}, a fraction of SNIa - mainly those resulting
from the oldest stellar populations - may be affected by radial migration as single stars are. The effect is quite important in the disk of the simulation of \cite{KPA2013}, which displays a long and strong bar,
its semi-major axis reaching in the last evolutionary stages between 6 and 8 kpc.
In this work, the effect of radial migration appears to be rather small for SNIa and Fe production (see right panels in Fig. \ref{Fig:ModelGen3}). The reason
is that star formation proceeds at a quasi-constant rate over most of the disk, creating a large number of SNIa {\it at late times}; in those conditions, the migration of some old SNIa progenitors from the inner disk, modifies little the situation (and, in any case not beyond a galactocentric radius of $r\sim$12 kpc). In contrast, in the simulation of \cite{KPA2013}, there is very little star formation in the whole disk after the first couple of Gyr, due to the lack of accreting gas; as a result, the radial migration of SNIa progenitors from the inner disk during the subsequent 8 Gyr of evolution (under the action of the strong bar), increases considerably the
SNIa population and the concomitant Fe production in the outer disk.
Our results for the Fe/H profile are displayed in Fig. \ref{Fig:Fe-profiles}, at three different times: 4, 8 ad 12 Gyr, respectively. In the latter case, we display in the upper panel the average metallicity of stars aged between 0 and 0.4 Gyr, (i.e. covering the range of Cepheid ages), along with the corresponding range of $\pm$1-$\sigma$ values. Being young objects, Cepheids have no time to migrate away from their birth places and theirs radial profile after migration (displayed here) is practically the same as the one they have at their birth.
The upper panel shows clearly that radial migration introduces very little dispersion in [Fe/H] for such young objects. If the observed dispersion in the sample of \cite{Genovali2014} is real and not due to measurement errorss, then its origin should be due to other factors (e.g. uncertainties in radial distance estimates, azimuthal variation of Fe/H, etc.)
A few other features of the Fe/H profile in Fig. \ref{Fig:Fe-profiles} are worth noticing:
- The profile flattens off in the 3-5 kpc region, instead of presenting there a decrease, as the O profile does. The reason is that in this region, at late times there is a population of old progenitors of SNIa (formed early on) which produces some Fe and compensates for the deficiency of CCSN there; this is not the case for O, as we discussed in the previous section. This contribution of old SNIa turns out to be sufficient to smooth the Fe/H profile in that region.
- Outside that region, the Fe/H profile decreases rather steeply, more steeply in any case than the corresponding O profile. The reason is that the ratio of SNIa/CCSN is always higher in the inner disk than in the outer disk (see right bottom panel of Fig. \ref{Fig:ModelGen1}), because the former has both an old and a young population of progenitors, while the latter has only a young population. As a result, Fe production is more important proportionally to the one of O in the inner disk, and the resulting Fe profile is steeper.
- In the outer disk, the Fe profile is less steeply decreasing, for the same reasons as the O profile (see previous section), namely the star formation efficiency of the adopted prescription for the SFR. Again, the profile cannot be described by a single exponential.
Comparison to observations is reasonably good, given the dispersion in the data.
In the case of the open clusters, dispersion appears to be even larger than in the case of Cepheids; here, however, the (poorly determined) age of the clusters, which covers a range of several Gyr, certainly contributes to this effect.
Although our Fe profile flattens in the outer disk, we never obtain a quasi-constant Fe/H abundance beyond $r\sim$15 kpc, in contrast to the observational findings of \cite{Yong2012} and \cite{Heiter2014} for open clusters.
Compared to other models in the literature, our results are closer to those of \cite{Naab2006}, as far as the overall Fe profile is concerned, which is also
steeper in the inner disk and progressively flattens outwards. Manifestly, this is due to the similar dependence of the SFR on radius in the two cases, steeper in the inner disk and flatter in the outer disk. In our case, this dependence results from the adopted SFR proportional to the molecular gas (Fig. 1, left panels), while in the case of \cite{Naab2006} it results from the adopted law $SFR \propto\Sigma_{GAS}/\tau_{DYN}$, with the dynamical timescale $\tau_{DYN}\propto r$ for a flat rotation curve: the factor $1/\tau_{DYN}$ varies considerably in the inner disk and much less outside 10 kpc.
On the other hand, the model of \cite{Magrini2009} does produce a nearly flat Fe profile in the outer disk -
even for intermediate ages - presumably through some appropriate combination of SFR and infall rate there
\subsection{Evolution of abundance profiles}
\label{subsec:Evolution_profiles}
The abundance profiles of gas and stars depend on the interplay between star formation, infall, radial inflow and radial migration of stars.
Observations of the final profiles alone can hardly shed light on this complex interplay. The history of the abundance profiles, if observed through some tracer of well determined age, could help in that respect: indeed, some semi-analytical models predict
gradients steeper in the past (e.g. \citet{Hou2000}) while others predict that gradients are flatter for older objects (e.g. \cite{Chiappini01}). As discussed in \cite{Pilkington2012}, who surveyed 25 models (both semi-analytical and with N-body+SPH codes), models may also differ widely as to the rate of change of the gradients with time (see \cite{Gibson2013} for an update).
Observations of planetary nebulae of different age classes suggested that O
gradients were steeper in the past \citep{Maciel2009}. However, the systematic uncertainties affecting age and distance estimates of those objects make it difficult to use them as tracers of the past gradient evolution
at present. Even worse, radial migration modifies considerably the radial profiles of stellar populations, as found in \cite{Roskar2008}, by mixing metal-rich stars from the inner regions in the outer disk. In those conditions, it becomes difficult to use abundance profiles of old objects to infer directly the
chemical evolution history of a galactic disk. Still, such observations, combined to other data (e.g. photometry profiles, stellar gradients as a function of distance from the galactic plane, etc.) and to appropriate models - taking properly into account the observational biases - may provide valuable information on the history of the Galaxy.
In Fig. \ref{Fig:FeMigvsNoMig} we display the evolution of the Fe profiles of our model for all the stars ever born (right bottom panel) and for stars of different age ranges. We show the average metallicity for stars formed {\it in situ} (dotted curves)\footnote{This is not the same thing as the average gas metallicity during the corresponding time interval: the average stellar metallicity is weighted with the star formation rate during that period, whereas the average gas metallicity is not.} and for all stars found in radius $r$ at the end of the simulation (solid curves); the latter population has been affected by radial migration. It can be seen that for the younger stars (up to 4 Gyr old) the differences between the corresponding profiles is small, for two reasons: i) radial migration does not have time to shuffle stars away from their birth places, and (most importantly) ii) at late times, the abundance profile is flatter than in earlier period, so that even an efficient radial migration cannot produce a large effect, because the abundance differences between different radii are small in any case. Still, radial migration increases steadily the dispersion in metallicity with age at all radii.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{F_Model_MigsVsInSitu.eps}
\caption[]{{\it Top} Evolution of the Fe profile for stars of different ages, born {\it in situ} ({\it dotted} curves) and presently found at radius $R_{fin}$ ({\it solid}); the {\it shaded} aerea indicates the $\pm$1 $\sigma$ range of values.
}
\label{Fig:FeMigvsNoMig}
\end{center}
\end{figure}
For stars older than the Sun, the effect of radial migration on the abundance profiles becomes more and more important, as it makes the profiles appear today flatter than they were at the time of the stellar birth, and flatter than the ones of younger stellar populations (despite the fact that the corresponding gaseous profile was steeper in the past). In particular, the oldest stars (presumably belonging to the thick disk) have a quasi-flat profile in the inner region, extending up to 6 kpc; beyond 9-10 kpc, however, the corresponding metallicity drops rapidly to values characteristic of halo stars.
The impact of radial migration on the past abundance profiles of a galactic disk was first identified by \cite{Roskar2008}: in their Fig. 2 they show how the older stars of their simulation ($>$5 Gyr) have a quasi-flat metallicity profile throughout the disk.
Although it is difficult to compare directly our results with other models of similar scope, because of the many different assumptions involved (see KPA2014 for a brief description of the differences between the models of \cite{SB2009}, \cite{Minchev2013} and KPA2014), we attempt here such a comparison to the results
of \cite{Minchev2014}. In their Fig. 9 (top left), they provide results for the stellar abundances of practically all stars of their model (found in the end of the simulation within a distance of 3 kpc from the galactic plane), as function of galactocentric radius and stellar age. There are differences and similarities with our results, but it is not clear whether the latter are due to similarities in the models or to different boundary conditions. In particular,
they also find that the older stars have a flatter abundance profile
than the younger ones; however, this is probably due not to radial migration, but to the fact that in their case the abundance profile of gas is also flatter in early time than lately, a characteristic feature of the model of \cite{Chiappini01}. In their case, dispersion in metallicity is more important for older stars than for young ones, as in our case and probably for the same reason, i.e. more time for radial migration being available to older objects and/or larger epicyclic motions; however, this dispersion appears to extend further outwards for the younger stellar population, whereas the opposite is obtained in our case.
Finally, the older stars in their simulation display a quasi-flat abundance profile all over the disk, whereas our corresponding metallicity profiles plummet beyond 9-10 kpc. This difference is simply due to the fact that we start our simulation with gas of primordial composition, whereas they adopt an initial metallicity of 0.1 Z$_{\odot}$. Such differences may have negligible impact in some cases (i.e. for almost any observable concerning the solar neighborhood), but turn out to be crucial in others.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{F_Evol_grad.eps}
\caption[]{{\it Top}: Evolution of the gaseous O and Fe abundance gradients. {\it Middle}: The evolution of the gaseous Fe gradient ({\it dotted, same as in top panel}) is compared to the evolution of the stellar Fe gradient, as it appears today after stellar migration ({\it thick dashed}). The squares, connected by solid segments represent the results of \cite{Minchev2014} (their Table 2) for all stars found within distance $Z<$3 kpc from the galactic plane. {\it Bottom}: The evolution of the O abundance gradient in the gas (same as in top panel) is compared to the extragalactic data compiled from \citet{Jones2013}; notice that the three {\it open symbols} correspond to an evaluation of MW data, which is superseded by the recent one of \cite{Maciel2013}, who find no clear indications for an evolution of the abundance gradient. .
}
\label{Fig:Evol_grad}
\end{center}
\end{figure}
We show the evolution of the O and Fe abundance gradients in
Fig. \ref{Fig:Evol_grad}. In the top panel we display the evolution of the
gradients in the gaseous phase. As already discussed, gaseous gradients decrease in absolute value with time, i.e. the abundance profiles become flatter with time (at least in the framework of this type of models). \cite{Hou2000} performed the first comparison between the evolution of the O and Fe abundance gradients and found that Fe gradients are steeper than those of O - by $\sim$0.1 dex - because of the role of SNIa: the ratio of SNIa/CCSN is larger in the inner disk than in the outer one (see Fig. \ref{Fig:ModelGen1}). We confirm this result here,
but we obtain a larger difference between the two gradients - $\sim$0.25 dex - because of the more efficient star formation in the inner disk and the role of radial migration: the latter increases by $\sim$10\% the abundance of Fe, but not the one of O, in the region outside 6 kpc (see bottom panels in Fig. \ref{Fig:ModelGen3}).
As discussed in the previous paragraphs, radial migration modifies the {\it presently observed} evolution of stellar profiles. This is illustrated in the middle panel of Fig. \ref{Fig:Evol_grad}, where the evolution of the Fe gradient
in the gas is compared to the Fe gradient of stellar population as a function of their age. The gradients are the same for the last $\sim$2 Gyr
(see also Fig. \ref{Fig:FeMigvsNoMig}) but beyond that age the two curves start deviating: the one corresponding to the stellar population becomes flatter with age. For the oldest stars, the gradient is close to zero, i.e. the abundance profile is practically flat. Our results are qualitatively similar to those of \cite{Minchev2014}, also displayed in Fig. \ref{Fig:Evol_grad}, although the evolution is milder in the their case.
It is difficult to compare directly the "age effect" of radial migration on the abundance profile to observations, because of the uncertainties in stellar age estimates. However, there
is an indirect way, through the fact that older stars are, on average located further away from the plane of the disk than younger ones, because of the increase in the vertical velocity dispersion with stellar age. Thus,
analysing a sample of old, main sequence stars belonging to the thin and thick disks from the SEGUE survey, \cite{Cheng2012a} find that the Fe gradient in the
region 6$< r$(kpc)$<$16 increase from -0.065 dex kpc$^{-1}$ at
vertical distance from the plane $Z$=0.2 kpc to a positive value at $Z>$1 kpc;
this is a clear signature of older stellar populations having flatter abundance profiles, as found in \cite{Roskar2008,Minchev2014} and in this work.
However, our model lacks the vertical dimension to the galactic plane and thus we cannot compare directly to the data of \cite{Cheng2012a}: at every distance from the plane there is a mixture of stellar populations, the contributions of older stars increasing with the distance. Our results presented in Fig. \ref{Fig:Evol_grad} (middle panel) include all stars found today between galactocentric radii of 4 and 11 kpc. Qualitatively, they are in agreement with the observations, since they suggest a gradient close to nul for the oldest stars and close to -0.07 dex/kpc for the youngest ones. A detailed comparison to the observations would require a model including the $z$ dimension (vertical to the plane), including the observational biases, i.e. slices at appropriate distances from the plane, as in \cite{Minchev2014}.
Alternatively, such a comparison would be possible if a volume limited sample with accurate stellar ages were available.
The bottom panel of Fig. \ref{Fig:Evol_grad} illustrates another way of comparing model results to observations of abundance gradient evolution. The results concern the evolution of the oxygen abundance gradient in the gas (as in top panel). The data are from observations of oxygen in high redshift lensed disk galaxies, from the recent compilation of \cite{Jones2013}; they find that the metallicity gradients flatten with time, by a factor of 2.6$\pm$0.9, on average, between redshifts 2.2 and 0, although they acknowledge that the discrepancy with the MASSIV data - the highest data point in the bottom panel of Fig. \ref{Fig:Evol_grad} - warrants further investigation. Barring that puzzling discrepancy, we find a rather fair agreement of the high redshift data with our results. It should be stressed, however, that the comparison may not be meaningful after all, because its not clear whether those isolated high redshift systems are progenitors of MW-like disks.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{F_OvsFe.eps}
\caption[]{{\it Top} Evolution of the O/Fe profile. Data are for Cepheids ({\it filed circles}, from\cite{Luck2011}) and for open clusters (\cite{Yong2012}, {\it asterisks} and \cite{Frinchaboy2013}, {\it squares}). {\it Dotted} curves correspond to model stars of average ages 11, 8, 4 and 0.2 Gyr (from top to bottom) formed {\it in situ} and {\it solid} curves to stellar populations of the same age found today at radius $r$.
{\it Middle:} Mass weighted stellar [O/Fe] profile vs radius. The {\it solid}
red curve indicates our results and the {\it shaded aerea} the corrsponding $\pm$1-$\sigma$ range. The {\it dashed} curves are from Fig. 2 of \cite{Gibson2013} and indicate results of \cite{SB2009} and \cite{Gibson2013}, the latter obtained with two different models (see text).
{\it Bottom}: The stellar Fe/H gradient of stars found today in the region 5-11 kpc is plotted vs. the average [O/Fe]
ratio of those stars (see text).
}
\label{Fig:F_OvsFe_evol}
\end{center}
\end{figure}
\subsection{O/Fe profile}
\label{sub:OFe}
The variation of the O/Fe ratio provides important information on the
evolutionary status of a galaxian system: high O/Fe values (typically $\sim$3 times solar) indicate a chemically young system, enriched only by the ejecta
of CCSN, while $\sim$solar values indicate systems several Gyr old, enriched also by SNIa. The transition from high O/Fe (and, more generally, high $\alpha$/Fe)
to low O/Fe values constitutes one of the key tracers of the chemical evolution
of the local Galaxy (the halo to disk transition) and of nearby dwarf galaxies as well.
In the case of the MW disk,the O/Fe ratio is expected to vary, from high values in the "young" outer disk, to lower ones in the older inner disk, in the framework of the inside-out formation scheme. In Fig. \ref{Fig:F_OvsFe_evol} (top panel), we plot the O/Fe radial profile for stellar populations of ages 11, 8, 4 and 0 Gyr (from top to bottom), for all the stars found in a given region in the end of the simulation (solid curves) and for stars formed {in situ} (dotted curves). The decrease (with time) of O/Fe occurs first in the inner galaxy and
progressively moves outwards. The youngest objects have [O/Fe]$\sim$0.1 in the outer disk and $\sim$-0.25 in the innermost regions, whereas for the oldest objects the ratio varies from 0.5 to 0.4. As in the case of the Fe/H profile, radial migration modifies the O/Fe profiles by bringing evolved stellar populations (of lower O/Fe) into outer regions; the effect is more important for the oldest stars and affects the region between 4 and 12 kpc.
Similar results, at least qualitatively, appear in Fig. 9 (bottom left panel) of \cite{Minchev2014}, where the [Mg/Fe] profile is plotted for stars of different ages. Mg being an $\alpha$ element, a comparison to the O/Fe profile is meaningful \footnote{Notice, however, that halo stars appear to have, in general higher [O/Fe] than [Mg/Fe] ratios at a given metallicity.}. They obtain a variation of [Mg/Fe] for their youngest stars ranging from $\sim$-0.16 at $r$=6 kpc to $\sim$0.1 at $r$=16 kpc, as well as a flat Mg/Fe profile for the oldest stars; both results are in fair agreement with ours.
Fig. \ref {Fig:F_OvsFe_evol} (top panel) displays observational data from Cepheids and open clusters of various ages. As with the corresponding Fe/H profile of Fig. \ref{Fig:Fe-profiles}, the dispersion in the O/Fe ratio at every galactocentric radius is quite large and cannot be explained with our models; radial migration can play only a marginal role in that respect, for so young objects. No clear trend with radius appear in the case of Cepheids, while the data for open clusters are marginally consistent with such a trend, as discussed in \cite{Yong2012}.
In the middle panel of Fig. \ref{Fig:F_OvsFe_evol}, we plot the results for the mass weighted average of all stars at the end of our simulation.
We obtain a rather flat profile in the inner disk. The average [O/Fe]$\sim$0.1 in that region corresponds to stars older than 8 Gyr, as inferred through a comparison to the upper panel, which have been mixed throughout the inner disk by radial migration. In the outer disk, less affected by radial migration, the average [O/Fe] ratio increases slowly but steadily, up to a value of 0.2. The overall 1-$\sigma$ dispersion is much larger in the inner disk than in the outer one, as indicated by the shaded aerea. We compare our results to the corresponding ones reported in
Fig. 2 of \cite{Gibson2013} and reproduced in the middle panel of our Fig. \ref{Fig:F_OvsFe_evol}. The semi-analytical model of \cite{SB2009} displays
a much steeper slope of [O/Fe] vs. radius, which may result from less radial mixing than in our case or from a much larger gradient of stars formed {\it in situ}. We think that both reasons contribute to the difference with our results, taking into account that \cite{SB2009} obtain quite large Fe gradients (dlog(Fe/H)/dr$\sim$-0.1 dex kpc$^{-1}$) and that they consider radial mixing induced only by the transient spiral mechanism of \cite{SellwoodBinney2002} and not by the more efficient bar-spiral interaction of our model. On the other hand, both models of \cite{Gibson2013} display a very flat profile of [O/Fe] over the whole disk,
which is rather difficult to understand, in view of the enhanced SNIa/CCSN ratio in the inner disk expected from inside-out formation schemes (as discussed in Sec. 2), unless if a very efficient radial mixing occurs for the stars over the whole disk. It is clear, however, that for different reasons, not necessarily well analysed yet, different models make different predictions for the profiles of metallicity and of various abundance ratios and only observations will help clarifying the situation.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{F_Other_other.eps}
\caption[]{Abundance profiles from C to Ar and comparison to data from PN (magenta {\it triangles}), B-stars (green {\it squares}) and HII-regions (brown {\it dots}); data sources are provided in Table 1. Model curves correspond to gaseous profiles at time 4, 8 and 12 Gyr ({\it thick}), respectively.
}
\label{Fig:F_Other_other}
\end{center}
\end{figure}
Finally, the bottom panel of Fig. \ref{Fig:F_OvsFe_evol} illustrates another use of the O/Fe ratio to probe the evolution of the Galactic disk. As found in KPA2014, the O/Fe ratio declines monotonically with time and displays very little dispersion from radial mixing at any age. It constitutes thus a natural "chronometer" as argued in \cite{Bovy2012} and it can be used in cases where stellar ages are not known or accurately measured.
\cite{Toyouchi2014} have analysed 18500 disk stars from the SDSS and HARPS surveys and plotted the Fe/H gradients as a function of the [$\alpha$/Fe]
values of the corresponding stellar populations. They found that, starting
with youngest stars (lowest [$\alpha$/Fe] values), the gradient first decreases, i.e. it becomes more negative and then increases, reaching positive values. However, they found large systematic differences between the samples of the two surveys (concerning the absolute values of the gradients and the turning points in [$\alpha$/Fe]). We display our results in the
bottom panel of Fig. \ref{Fig:F_OvsFe_evol}, showing a qualitative agreement with the findings of \cite{Toyouchi2014}: starting with the youngest stars, the Fe/H gradient shows first a small decline, as slightly older stellar populations are probed (with a steeper Fe gradient because too young to be affected by radial migration); then, older populations are probed, more and more affected by radial migration and displaying flatter Fe/H profiles (as also indicated in the middle panel of our Fig. \ref{Fig:Evol_grad}). The oldest stars have a nearly flat Fe/H profile, but we never find a positive gradient, as \cite{Toyouchi2014} do. We stress again that a meaningful comparison to observations should involve models properly accounting for observational biases.
\subsection{Other elements}
\label{sub:Other}
Oxygen and iron are the most frequently observed elements in the solar neighborhood and the MW disk. Their abundances (as a function of time and/or space) constitute important constraints on the chemical evolution of the Galaxy. The other elements play a marginal role in that respect: their observations mainly serve to support conclusions obtained through the observations of O and Fe or to constrain stellar nucleosynthesis models.
In Fig. \ref{Fig:F_Other_other} we present our results for eight more elements, with abundance profiles derived from observations of B-stars, HII-regions and planetary nebulae.
In most cases, available data are for the region 4-12 kpc, with the exceptions of N and S (and, of course, O) where observations of H-II regions and PN extend up to 17 kpc. The inclusion of PN, presumably covering a wide range of ages, increases considerably the dispersion at every radius.
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,width=0.99\textwidth]{F_Other_Cepheids.eps}
\caption[]{Abundance profiles from C to Zn and comparison to Cepheid data (from \cite{Luck2011}, except for Fe data, provided in \cite{Genovali2014}). Model curves correspond to the gaseous profiles after 4, 8 and 12 ({\it thick curve}) Gyr, respectively.
}
\label{Fig:F_Other_Cepheids}
\end{center}
\end{figure*}
Among the 9 elements of Fig. \ref{Fig:F_Other_other}, those heavier than N are almost exclusively products of massive stars; Si and heavier elements receive a small, but non negligible contribution from SNIa, that we take into account through the adopted yields of SNIa from \cite{Iwamoto99}. All those elements are produced as "primaries", i.e. their stellar yields depend little on the initial metallicity of the stars. C and N have a complex nucleosynthetic origin, since they are produced both by massive and and intermediate mass stars.
Their yields from massive stars are sensitive to yet poorly understood (and metallicity-dependent) stellar properties, as mass loss and rotation. In lower mass stars, C is produced in the shell He-burning of the AGB phase and ejected in the ISM through the 3d dredge-up, while N may be produced in the bottom of the convective envelope of the AGB star ("Hot-bottom burning") at the expense of C. N is, in principle, a "secondary" element (being synthesized from the initial C and O during the CNO cycle, its yield depends on the initial metallicity); yet, it may be produced essentially as primary in the "hot-bottom burning" of intermediate mass AGB stars and in fast rotating massive stars.
The yields from \cite{Nomoto2013} that we adopted in this work, include yields from low-mass stars from \cite{Karakas2010}, but the ones for massive star concern non-rotatings stars. Given the complexity of the nucleosynthesis of C and N, we consider that a dedicated study for the evolution of those two elements would be necessary and we do not attempt it here.
The results displayed in Fig. \ref{Fig:F_Other_other} present similar features to those already discussed for O, namely a flattening of the abundance profiles with time (due to the inside-out formation) and a slightly hollow profile in the bar region at late times (due to the combined action of the bar and of metal-poor infall).
Available data, however, display either too large differences (e.g. between B-stars and PN for S) or too large dispersion (in the case of PN data) or they are too scarce (e.g. for C) to allow for any serious constrains, either on the model or on the yields.
(Fig. \ref{Fig:Evol_grad}, middle panel)
In Fig. \ref{Fig:F_Other_Cepheids} we compare our results for all elements between C and Ni to a homogeneous data set for Cepheids \citep{Luck2011}, large enough for a statistically meaningful comparison with models. Still, Cepheids are relatively massive ($>$3 M$_{\odot}$) and evolved stars, having gone through the first dredge-up. This implies that they are expected to exhibit large amounts of N at their surface, formed {\it in situ} at the expense
of C, and perhaps of O; Na is also possibly affected by H-burning in those stars. In consequence, none of those four elements observed in Cepheids can be used as tracer of the chemical evolution of the Galaxy (but they may certainly be used as probes of the internal nucleosynthesis of Cepheids).
\begin{figure*}
\begin{center}
\includegraphics[angle=-90, width=\textwidth, bb=243 20 593 780, clip]{F_Final_gradients_b.eps}
\caption[]{Present day abundance gradients from H to Ni: models vs. observations. Observations are from H-II regions \citep{Rudolph2006}, B-stars \citep{Daflon2004}, Planetary nebulae \citep{Henry2004} and Cepheids \citep{Luck2011,Luck2011b,Lemasle2013}. Model results
are obtained with the model described here and three different sets of yields, from \cite{Nomoto2013} \cite{WW95} and \cite{CL2004}, as indicated by the open symbols and discussed in the text.
}
\label{Fig:F_FinalGradients}
\end{center}
\end{figure*}
Concerning the other elements, one sees that observed abundances are systematically higher than solar in the solar neighborhood (except for Mn and Ni), in rough agreement with theoretical expectations.
Our final abundance profiles are globally in agreement with the data, although the obtained slope is, in general, slightly larger than the observationally inferred one, as we discuss below. Finally, the Cepheid data do
not extend into the bar region, as to allow us to probe the predictions of the model there.
In Fig. \ref{Fig:F_FinalGradients}, we present a quantitative comparison of the gradients of all elements between H and Ni to the data set of Cepheids from \cite{Luck2011}, complemented with a few other data presented here for completeness. We notice, however, that the abundance profiles are not necessarily perfect exponentials to be fit by straight lines of a single slope in logarithmic space. In our case they are not, and the slope depends on the radial range considered: here we take the range 4-17 kpc, to compare directly with
the slopes provided by \cite{Luck2011} for that same range. The other data sets correspond, however, to different radial ranges, making a direct comparison difficult.
In order to give an idea of the theoretical uncertainties, we provide results for 3 different runs of the same model using 3 different sets of yields. The first one is from \cite{Nomoto2013} (mass range for massive stars: 13-40 M$_{\odot}$) using the low-mass yields of \cite{Karakas2010} (1-6 M$_{\odot}$), as discussed throughout this work. The second set adopts the massive star yields of \cite{WW95} (12-40 M$_{\odot}$) and the ones of \cite{VandenHoek1997} (0.9-8 M$_{\odot}$) for intermediate mass stars. The third one adopts the massive star yields of \cite{CL2004} (13-35 M$_{\odot}$) and those of \cite{VandenHoek1997} for intermediate mas stars.
All those yields are metallicity-dependent, but they cover different ranges of masses and metallicities. The limited cover of the mass grid by all the data sets implies the need for interpolation between the low-mass and massive star ranges, and extrapolation between the most massive star of the calculated yields and the most massive star of the adopted IMF, here taken to be 100 M$_{\odot}$; these operations introduces numerical biases in the results. Regarding the metallicities, the yields of \cite{Nomoto2013} are provided for a finer grid and extend to supersolar metallicities, both features being essential for a
consistent study of the evolution of the MW disk. The calculations of massive stars involve different ingredients,
e.g. no mass loss in \cite{WW95} and \cite{CL2004} vs mass loss in \cite{Nomoto2013}, different prescriptions for some key nuclear reaction rates or the various mixing mechanisms and the description of the final supernova explosion. Similar differences characterise the physics underlying the yields of intermediate mass stars, e.g. the treatment of hot-bottom burning. Those differences, as well as other important ingredients (not considered in those calculation, like e.g. rotation) make any attempt for a systematic comparison between sets of yields rather futile. We provide such a comparison only for illustrative purposes, being fully aware that the overall theoretical uncertainties should be larger than found here. We simply notice that, in all cases we keep the same IMF and the same prescription for the rate and yields
of SNIa.
An inspection of Fig. \ref{Fig:F_FinalGradients} shows that the observed gradients of all elements between C and Ni lie in the narrow range of dlogX/dr $\sim$-0.04 to -0.06 dex kpc$^{-1}$, with the exception of C and (curiously) S. Taking into account the larger error bars, the data for other tracers are in good agreement with those of Cepheids.
Theoretical results present some features common to all sets of adopted yields.
i) quasi-identical slopes for the Fe-peak elements, dominated by SNIa;
ii) quasi-identical slopes for all $\alpha$-elements beyond Ne;
iii) a distinctive difference in the slopes of even vs odd elements, the former been smaller in absolute value than the latter. To our knowledge, it is the first time that the "odd-even" effect of nucleosynthesis, known to affect the behaviour of the abundance ratios in low metallicity stars (e.g. \citet{Goswami2000}, is put in evidence in the case of the Galactic abundance gradients.
There are also some noticeable differences between the various sets of yields.
- The yields of \cite{Nomoto2013} produce flatter profiles for C,N and O than those of \cite{WW95} or \cite{CL2004}; we find that this is due mainly to the impact of the yields of low-mass stars (from \cite{Karakas2010} in the former case), which
have a more pronounced "hot-bottom burning" than the yields of \cite{VandenHoek1997}
adopted in the latter two cases.
- the "odd-even" effect appears to be more pronounced with the yields of \cite{WW95} than with those of \cite{Nomoto2013}, and even more so with the yields of \cite{CL2004}.
- F is a particular case, since \cite{WW95} and \cite{Nomoto2013} include the effect of $\nu$-induced nucleosynthesis during the final SN explosion, while \cite{CL2004} do not.
Comparing model results to observations, one sees a significant offset of the former - by $\sim$0.02 dex downwards - for all elements beyond Cl {\it except Fe}.
For lighter elements, there is satisfactory agreement (within error bars) for the even elements, but significant discrepancy for the odd ones.
Some conclusions may be drawn from the comparison between model and observations on the one hand, and between different sets of yields on the other. First, if the systematic discrepancy of $\sim$0.01-0.02 dex between model results and observations is confirmed, some key ingredients of the model should be revised: this could be the case, for instance, of the timescales of the infall rate, which should be lower in the outer disk than adopted here (as to favour a more rapid evolution and larger final abundances in that region); alternatively, a metal-enriched composition of the infalling gas could be adopted, instead of the primordial one adopted here. We find, however, that in the framework of this model the required infall metallicity is $\sim$0.4 Z$_{\odot}$, too large for the infalling gas (acceptable metallicities, up to 0.2 Z$_{\odot}$, increase the slope by only 0.01 dex/kpc). Second, it is interesting to see whether the abundance gradients present any systematic trends with the atomic number of the element, i.e. either steeper profiles for Fe-peak elements or the "odd-even" effect. If this turns out not to be the case, then
the yields of nucleosynthesis should be revised and the role of the IMF scrutinized (because more massive stars produce larger ratios of $\alpha$/Fe). In any case, the abundance profiles of a large number of elements should be accurately established, on a radial basis as large as possible. Then, the abundance profiles could be used to probe stellar nucleosynthesis, a role played already by abundance ratios in low metallicity halo stars.
We notice that the study of \cite{Romano2010}, concerning the impact of different sets of yields on the chemical evolution of the Milky Way,
reaches conclusions which present similarities to our own, but also some differences. Their model has no radial migration or radial inflows and uses different prescriptions for the SFR and infall rate than ours; thus, they obtain different abundance profiles, flatter than ours by $\sim$0.02 dex/kpc for C and O (their Fig. 17 left). Still, they adopt the same IMF, while their models 1 and 15 adopt similar sets of yields with our study: the former adopts \cite{WW95} for massive stars and \cite{vandenHoek97} for low and intermediate mass stars (LIMS),
while the latter uses the yields of \cite{Nomoto2013} which include those of \cite{Karakas2010} for LIMS. Thus, some comparison with our results becomes possible. \cite{Romano2010} find that "... for some elements, different choices for the yields provide also different slopes for the gradients; this is the case for carbon and oxygen, while a negligible effect is seen for other elements, including those heavier than Si". We confirm the case for C and O, but we attribute it to the differences in the corresponding yields of LIMS, while \cite{Romano2010} attribute it to the yields of rotating, massive stars with mass loss (we did not study that case here). On the other hand, we confirm that the impact on the slope of the abundance profiles is negligible for other {\it even} elements, but we find a significant effect for the {\it odd} elements, as we emphasized in the previous paragraphs.
\section{Summary}
\label{sec:Summary}
In this work we study the abundance profiles of elements between H and Ni in the MW disk, using a semi-analytical model involving radial motions of gas and stars.
We adopt parametrised descriptions of those radial motions, which are based on N-body simulations for the case of stars and on a simple analytical prescription for the gas radial velocity profile and are inspired by the presence of a bar in the Milky Way.
Other key ingredients of the model is the assumption of a SFR dependent on the molecular gas and the use of a fine grid of recent stellar yields from \cite{Nomoto2013}, which include up-dated yields of low-mass stars from \cite{Karakas2010} and cover a large range of initial metallicities.
The model reproduces successfully a large number of observations concerning the solar neighborhood and the disk of the MW, as discussed in KPA14; they include the local age-metallicity relation and metallicity distribution, the $\alpha$/Fe vs Fe/H
relation and the surface density profiles of the thin and thick disks, as well as the profiles of stars, SFR, H${\rm I}$ \ and H$_{\rm 2}$, and the total amounts of gas, stars, SFR, CCSN and SNIa rates in the disk and the bulge of the MW.
In Sec. \ref{sec:Model} we present the key model ingredients and we show how the radial motions affect the profiles of stars, gas, SFR, SNIa and Fe. We find that the effect concerns mainly the inner disk, because of the key role played by the Galactic bar in radial migration according to our assumptions; its effect on the disk is found to be negligible beyond 12 kpc, with the adopted prescriptions.
The effect on the Fe profile in the inner disk is rather small (of the order of 10\% and it is due to the role of SNIa from long-lived progenitors, which have time enough to migrate away from their birth place.
In Sec. \ref{sec:abundance-evol} we present our results and we compare them to a large number of observational data from various metallicity tracers (H-II regions, B-stars, PN, Cepheids and open clusters). We notice that the data base is not homogeneous and does not cover uniformly the radial extent of the MW disk.
Our abundance profiles cannot be characterised by a unique slope, since they flatten progressively towards the outer disk (as a result of the adopted SFR prescription)
and towards the inner disk (as a result of the radial flow induced by the bar).
We find that the abundance profiles flatten with time, as a result of the inside-out formation of the disk. But the observational confirmation of this effect in the MW
becomes impossible because of the effect of radial migration, which cancels and even inverts it.
We confirm (Sec. \ref{subsec:Evolution_profiles}) the main effect of radial migration on the abundance profiles found by \cite{Roskar2008}, namely the flattening of the past abundance profiles of stars, which becomes more pronounced for the older stellar populations. We compare quantitatively our results (Fig. \ref{Fig:Evol_grad}, middle panel) to those of \cite{Minchev2014}, who find similar, albeit somewhat flatter, abundance profiles.
The evolution of our gaseous abundance profiles is in fair agreement with the extragalactic, high redshift, data compiled by \cite{Jones2013}, which are too uncertain at present, however, to draw firm conclusions (Fig. \ref{Fig:Evol_grad}, bottom panel). In Sec. \ref{sub:OFe} we present the evolution of our [O/Fe] profiles. The evolution of both [Fe/H] and [O/Fe] profiles, modified by radial migration, is encoded in the stellar populations currently present in the local disk and is revealed by preliminary observations of those quantities in stars at different distances from the plane of the disk. Our 1D model lacks the dimension vertical to the plane that would allow us to perform a meaningful comparison to such observations, but the results discussed in Sec. \ref{sub:OFe} are in qualitative agreement with the data (the profiles are expected to flatten with distance from the plane).
In Sec. \ref{sub:Other} we present our results for all elements between C and Ni.
From the theory point of view, we stress the systematic differences in the final abundance profiles due to the different sets of stellar yields (the physics of both low and high mass stars still suffering by several uncertainties). We find a rather good agreement with observationally derived slopes of abundance profiles (assuming they can be described by a single exponential) but also some systematic differences
(see Fig. \ref{Fig:F_FinalGradients}: in particular, we obtain slopes systematically 0.01-0.02 dex larger (in absolute value) than the observed ones. We argue that this difference, if definitively established, could be cured by some revision of basic ingredients of the model, namely the need for smaller infall timescales in the outer disk or a non-primordial composition for the infalling gas. We also find an interesting "odd-even" effect of larger slopes for odd elements. This metallicity-dependent effect, already discussed in the context of abundance ratios X/Fe in halo stars, is found here for the first time and it is generic, i.e. it concerns all sets of stellar yields. However, it does not appear in the observational data; if observations are confirmed, then some of the stellar nucleosynthesis results should be revised.
\medskip
\noindent
{\it Acknowledgments}: We are grateful to an anonymous referee for her/his constructive report.
EA acknowledges financial support to the DAGAL network from the People
Programme (Marie Curie Actions) of the European Union's Seventh
Framework Programme FP7/2007-2013/ under REA grant agreement number
PITN-GA-2011-289313 and from the CNES (Centre National d'Etudes
Spatiales - France). We also acknowledge partial support from the PNCG
(Programme National Cosmologie et Galaxies - France).
\bibliographystyle{aa}
|
1,116,691,499,378 | arxiv | \section{Introduction}
\IEEEPARstart{B}{LIND} face restoration (BFR) is a typically ill-posed inverse problem and aims at reproducing realistic and reasonable high-quality (HQ) face images from unknown degraded inputs. Low-quality (LQ) face images are often affected by the combination of multiple unknown degradation factors in the wild, such as low resolution, noises, blur~\cite{shen2018blur}, and compression artifacts~\cite{dong2015compression}, etc.. Therefore, this may cause serious loss of high-frequency details in the image and affect the overall image quality. Furthermore, due to the complexity and diversity of facial postures, it is still a challenge to restore face images with natural and high-quality results for BFR tasks. The human face has a highly complex structure and has special properties different from other objects. Previous works based on Convolutional Neural Networks (CNN) and GAN use various facial semantic priors (including facial landmarks~\cite{jeong2017driver,jin2021facial_landmark,sharma20213Dface_landmark}, facial parsing maps~\cite{shen2018blur,chen2018fsrnet,richardson2021psp,chen2021psfr-gan}, facial heatmaps ~\cite{bulat2018super}, and facial component dictionaries~\cite{li2020DFD}) to guide the networks to recover face shape and details. However, when these facial priors are adopted to restore severely degraded images, there is still much room for improvement in the restoration results due to the limited prior information.
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{figure/first_page_figure.pdf}
\caption{Restoration results from the old photos of the Solvay conference, 1927. Only some face images are displayed. Our method can restore the details of the original images to a large extent and avoid excessive fantasy. \textbf{Please zoom in for the best view}.}
\label{fig0:show results}
\end{figure}
As a generative facial model with excellent performance, StyleGAN~\cite{karras2019stylegan,karras2020stylegan2,karras2021stylegan3} is capable of synthesizing face images with rich textures, changeable styles, and highly realistic vision, and providing rich and diverse priors, such as facial contours and textures of all areas (including hair, eyes, and mouth). So far, StyleGAN2~\cite{karras2020stylegan2} has been widely
applied to face restoration tasks, such as PULSE~\cite{menon2020pulse}, pSp~\cite{richardson2021psp}, PSFR-GAN~\cite{chen2021psfr-gan}, GFPGAN~\cite{wang2021GFPGAN}, GPEN~\cite{yang2021gpen}, GCFSR~\cite{he2022gcfsr}, and Panini-Net~\cite{wang2022panini}. However, although these methods have taken full advantage of the StyleGAN2 network or the pre-trained StyleGAN2 model, there is still limited resilience due to the lack of appropriate and efficient training strategies.
All in all, when used to restore the seriously degraded pose-varied face image, existing methods often have many problems in the output results, such as excessive fantasy, visual artifacts, blurry and unreasonable textures. To address this challenging problem, we propose a novel blind face restoration network with the GAN prior~\cite{karras2020stylegan2}, composed of an asymmetric codec and a StyleGAN2 prior network. Firstly. Since there are few weak texture features in degraded images, we propose a mixed multi-path residual block (MMRB), which mainly adopts a two-branch sparse structure to extract the features of different scales and can achieve spatial interaction and aggregation of shared features through skip connections. In this work, we apply MMRB layers to gradually extract weak texture features of input images from shallow to deep.
Secondly. To better restore natural and high-quality face images, as shown in Fig.~\ref{fig0:show results}, we adopt the StyleGAN2 model with generative facial prior as the primary generator network and jointly fine-tune the generator with the codec. Using facial landmark points of training data, the position coordinates of the local areas are obtained, such as the eyes and mouth. We adopt then local facial losses to promote the authenticity of the output facial results during training. Finally, we introduce a training strategy of freezing a pre-trained discriminator (FreezeD), which helps our network recover more reasonable and high-fidelity results in complex degraded scenes. In particular, this training strategy helps our model to remain stable during training and speed up the fitting of the generated results to the target distribution.
Our main contributions are summarized as follows:
\begin{itemize}
\item We propose a well-designed blind face restoration network with generative facial prior, which can promote the quality of face images with complex facial posture and serious degradation.
\item For the StyleGAN2 prior model, we specially design a novel self-supervised training strategy that freezes a pre-trained discriminator (FreezeD) and jointly fine-tunes the generator with the codec. It helps our model recover high-quality face images more realistically and reasonably and maintain stability during training.
\item To better extract few and weak texture features in low-quality images, we propose a MMRB layer, which adopts a two-branch sparse structure to extract the features of different scales and can achieve spatial interaction and aggregation of shared features through skip connections.
\item Our method achieves state-of-the-art results on multiple datasets. It can observe that our method can tackle seriously degraded face images in diverse poses and expressions and can restore more realistic and natural facial textures.
\end{itemize}
\section{Related Work}
\subsection{Face Restoration}
For blind face restoration in the real world, the restoration problem becomes more complex due to the particularity of the face image itself and the influence of many unknown degradation factors, such as compression artifacts, low resolution, blur, and noise. There are three main research trends~\cite{jiang2021faceSR_survey} for this task: basic CNN-based methods, GAN-based methods, and Prior-guided methods.
\textit{\textbf{Basic CNN-based methods.}} IPFH~\cite{cheng2019identity} utilized deep reinforcement learning and facial illuminator to restore high-quality textures of human face. DPDFN~\cite{jiang2020dual} proposed a simple dual-path deep fusion network for face super-resolution without additional face priors, which constructed two individual branches for learning global facial contours and local face details and then fused the results of the two branches. Chen et al.~\cite{chen2019modeling} adopted a nearest neighbor network to model the neighbor relationship in HR to produce high-quality and high-resolution outputs.
\textit{\textbf{GAN-based methods}}. HiFaceGAN~\cite{yang2020hifacegan} proposed a collaborative suppression and replenishment framework to tackle unconstrained face restoration problems. TDAE~\cite{yu2017hallucinating} presented a discriminative generative network, which is not affected by posture and facial expression and can ultra-resolve low-resolution face images. Super-FAN~\cite{bulat2018super} first proposed an end-to-end model that addresses face super-resolution and alignment via integrating a sub-network for face alignment through heat map regression and optimizing a heatmap loss. In order to improve the accuracy of face recognition, MDFR~\cite{tu2021joint} proposed an multi-degradation face restoration model which can restore a given low-quality face to the high-quality one in arbitrary poses. PSFR-GAN~\cite{chen2021psfr-gan} based on the StyleGAN2 network proposed a semantic-aware style transfer approach to modulate the features of different scales and recover HQ face details progressively. GCFSR~\cite{he2022gcfsr} proposed a generative and controllable face super-resolution model without reliance on any additional priors, which can use to reconstruct faithful images with promising identity information. However, face restoration methods based on CNN and GAN have limited resilience due to lacking generative facial prior.
\begin{figure*}[!t]
\centering
\includegraphics[width=1.0\textwidth]{figure/model_figure.pdf}
\caption{The framework of our proposed method. It is mainly composed of an asymmetric codec and a StyleGAN2 prior network. The MMRB layer is adopted to gradually extract weak texture features of input images in the encoder. Furthermore, the inputs of the GAN prior network include latent codes $\boldsymbol{W}$, a learned constant feature tensor $\boldsymbol{C}$ and noise branches. This prior network can then apply style blocks to gradually restore high-quality face images from coarse to fine.}
\label{fig1:network}
\end{figure*}
\textit{\textbf{GAN Prior-guided methods}}. The pre-trained model based on the StyleGAN2~\cite{karras2020stylegan2} is often used to solve the GAN inversion problem, such as face restoration and face super-resolution. PULSE~\cite{menon2020pulse} adopted a self-supervised training method to iteratively optimize the latent codes of StyleGAN2 model until the distance between outputs and inputs was below a threshold to restore high-quality facial images. pSp~\cite{richardson2021psp} utilized the pre-trained StyleGAN2 model and used a standard feature pyramid as an encoder network to solve image-to-image translation tasks. GFPGAN~\cite{wang2021GFPGAN} also used the pre-trained StyleGAN2 model as a generative facial prior and then proposed channel-split spatial feature transform layers to perform spatial modulation on a split of features to improve the quality of face images. GPEN~\cite{yang2021gpen} directly embedded GAN prior into the codec structure as the decoder and jointly fine-tuned the GAN prior network with the deep neural network. Panini-Net~\cite{wang2022panini} proposed a learnable mask to dynamically fuse the encoder's features with the features generated by GAN blocks in the pre-trained StyleGAN2 model and adopted a contrastive learning strategy~\cite{liu2021divco} to extract discriminative degradation representations for degraded images. Although HQ face images can be restored using the GAN prior, their restoration results are not realistic enough for the face images with multiple poses and serious degradation.
\subsection{Transfering GAN Priors}
Transfering GAN prior model is also regarded as a domain adaptation, which mainly adjusts the data distribution of the pre-trained model to a domain suitable for other tasks~\cite{zhang2021deep_domian_adaptation,zhang2020supervised,zhou2021domain_generalization_survey,karras2020training_generative,zhuang2021towards}. For the pre-trained model, common strategies are to fine-tune or fix the parameters of the pre-trained model~\cite{huang2021unsupervised,menon2020pulse,jiang2021faceSR_survey}. However, directly using GAN priors for training cannot reduce the difference between the two data distributions and improve the restored image quality~\cite{back2021fine-tuning_stylegans}. Therefore, it needs to adopt appropriate training strategies, especially for seriously degraded image restoration tasks. Most relevant to our work, domain adaptation methods (~\cite{zhu2021mind,tov2021designing,song2021agilegan}) built upon the StyleGAN~\cite{karras2019stylegan}~\cite{karras2020stylegan2} demonstrate impressive visual quality and semantic interpretability in the target domain. StyleGAN-ADA~\cite{karras2020training_generative} proposed an adaptive discriminator augmentation method to train the StyleGAN network on limited data samples. Mo et al.~\cite{mo2020FreezedD} froze lower layers of the discriminator to achieve domain adaptation and demonstrated the effectiveness of this simple baseline using various architectures and datasets. Huang et al.~\cite{huang2021unsupervised} proposed an Image-to-Image translation method that generates a new model in the target domain via a series of model transformations on the pre-trained StyleGAN2 model in the source domain.
\subsection{Feature Extraction Block}
So far, various feature extraction blocks have been proposed to improve the learning ability of convolution layers in a network. Multiple versions of inception blocks~\cite{szegedy2015going}~\cite{szegedy2017inception} adopted parallel multi-scale convolution in the network to replace the dense structure, and fused all convolution results, and used the bottleneck layer for dimensionality reduction to reduce the amount of network computing. Kim et al.~\cite{he2016deep} added a short connection to the previous convolution layer and put forward the idea of residual learning, which can effectively alleviate the disappearance of gradients during training and reduce the number of parameters. After that, Zhang et al.~\cite{zhang2018residual} mainly increased the width of convolution by using the information compensation of other feature channels, which can reduce the vanishing gradient, realize the efficient utilization of feature information. Li et al.~\cite{li2018multi-scale} motivated by the inception module proposed a multi-scale residual block to exploit the features of images at different scales. Inspired by the above methods, we propose a mixed multi-path residual block (MMRB) to extract image features of different paths and realize mixed interaction between features to provide better modeling capabilities for face restoration.
\section{Our Proposed Methods}
\subsection{Global Architecture}
In this section, we will describe the well-designed network framework in Fig.~\ref{fig1:network}, which can be used to solve a serious ill-posed inverse problem in the wild. The framework is inspired by StyleGAN~\cite{karras2019stylegan}, GFPGAN~\cite{wang2021GFPGAN}, and GPEN~\cite{yang2021gpen} models, consisting of an asymmetric codec and a StyleGAN2 prior network. Firstly, We input a low-quality image $\boldsymbol{x}$ into the codec, in which the encoder has one more MMRB layer in each scale than the decoder. That is because the encoder needs to provide more natural and reliable face latent features for the generator. The MMRB layer proposed by us can better extract the weak and few texture features in the degraded images. The input image $\boldsymbol{x}$ is then mapped to the closest latent codes $\boldsymbol{Z}$ in the StyleGAN2 by the encoder. It can be seen from the StyleGAN~\cite{karras2019stylegan} and StyleGAN2~\cite{karras2020stylegan2} that images can be generated only by latent codes without noise branches, but the generated results lack rich and real texture information. Aiming to balance the fidelity and authenticity of the restoration results, we use the decoder and skip connections to reconstruct the multi-scale downsampled features, and directly output the reconstructed features at each level to the noise branches in the StyleGAN2. This process also benefits from the different practices of GFPGAN~\cite{wang2021GFPGAN} and GPEN~\cite{yang2021gpen} networks. We have also verified the function of this process during training and have found that the reconstructed image has richer and more reasonable facial textures. Finally, to reduce the influence of degradation factors, we use L1 restoration loss similar to~\cite{wang2021GFPGAN} for each resolution scale of the reconstructed image in the decoder. Finally, the process of codec is formulated by $:$
\begin{align}
\boldsymbol{\hat{x}}, \boldsymbol{Z} = \operatorname{Codec} \left( \boldsymbol{x} \right),
\end{align}
Secondly, latent codes $\boldsymbol{W}$, a learned constant feature tensor $\boldsymbol{C}$ and noise features $\hat{x}$ are input into the pre-trained StyleGAN2, where $\boldsymbol{W}$ are latent codes decoupled by the latent codes $\boldsymbol{Z}$ through a mapping network of several multi-layer perceptron layers (MLP). Decoupling feature space is formulated by$:$
\begin{align}
\boldsymbol{W} = \operatorname{MLP} \left( \boldsymbol{Z} \right),
\end{align}
The $\boldsymbol{W}$ ($\boldsymbol{w} \in \boldsymbol{W}$) are then broadcasted to each style block, in which the Mod and Demod indicate the modulation and demodulation operation of latent codes $\boldsymbol{W}$, respectively. In particular, the input noises $\boldsymbol{\hat{x}}$ and the output $\boldsymbol{\hat{y}}$ of the style block after feature modulation are fused by concatenating rather than directly adding to convolutions in the StyleGAN2 model, which can take advantage of the features introduced by noise branches to flexibly restore the texture of local areas.
\begin{align}
\boldsymbol{y}_{i+1} = \operatorname{Style} \left( \boldsymbol{w}, \boldsymbol{y}_{i}, \operatorname{Concat} \left[ \boldsymbol{\hat{x}}_{i}, \boldsymbol{\hat{y}}_{i} \right] \right).
\end{align}
The output feature maps $\boldsymbol{y}$ will be input to the next style block. On the basis of the fine-tuned training skills, we can apply style blocks to gradually restore high-quality face images from coarse to fine.
\subsection{Mixed Multi-path Residual block}
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{figure/Mixed_Multi-path_Residual_block.pdf}
\caption{Overview of the mixed multi-path residual block (MMRB).}
\label{fig2:MMRB}
\end{figure}
To extract slight and weak high-frequency textures in the degraded image, we propose a mixed multi-path residual block (MMRB). Here we will clearly describe its structure in Fig.~\ref{fig2:MMRB}. We design a two-branch and interactive feature extraction network, unlike previous works~\cite{szegedy2015going,szegedy2017inception,zhang2018residual,li2018multi-scale}. We first equally split the input multi-channel feature maps $\boldsymbol{X}_{I}$ rather than reduce dimensionality using a $ 1 \times 1 $ bottleneck layer, which can reduce the parameters of convolution calculation for each branch.
\begin{align}
\boldsymbol{X}_{1}, \boldsymbol{X}_{2} = \operatorname{Split} \left( \boldsymbol{X}_{I} \right),
\end{align}
Where $\operatorname{Split} \left( \cdot \right)$ represents the separation operation of the feature maps in the channel dimension. $\boldsymbol{X}_{I} \in {\mathbb{R}}^{H \times W \times C}$. $\boldsymbol{X}_{1},\boldsymbol{X}_{2} \in {\mathbb{R}}^{H \times W \times C {/} 2}$.
We adopt then the two-branch sparse structure to extract the features of different scales. In order to improve the utilization of different scale features, skip connections are applied to share extracted features and achieve spatial interaction and aggregation of different features. The detailed definitions are as follows:
\begin{align}
\boldsymbol{P}_{1} = \boldsymbol{F}_{1} \left( \boldsymbol{X}_1, \boldsymbol{W}_1 \right) = \sigma \left( \boldsymbol{W}_1 \left( \boldsymbol{X}_1 \right) \right),
\end{align}
\begin{align}
\boldsymbol{P}_{2} = \boldsymbol{F}_{2} \left( \boldsymbol{X}_2, \boldsymbol{W}_2 \right) = \sigma \left( \boldsymbol{W}_2 \left( \boldsymbol{X}_2 \right) \right),
\end{align}
\begin{align}
\boldsymbol{P}^{'}_{1} = \boldsymbol{F}_{1} \left( \boldsymbol{C}_1, \boldsymbol{W}_1 \right) = \sigma \left( \boldsymbol{W}_1 \left( \operatorname{Concat} \left[ \boldsymbol{P}_1, \boldsymbol{P}_2 \right] \right) \right),
\end{align}
\begin{align}
\boldsymbol{P}^{'}_{2} = \boldsymbol{F}_{2} \left( \boldsymbol{C}_1, \boldsymbol{W}_2 \right) = \sigma \left( \boldsymbol{W}_2 \left( \operatorname{Concat} \left[ \boldsymbol{P}_1, \boldsymbol{P}_2 \right] \right) \right),
\end{align}
\begin{align}
\boldsymbol{P} = \boldsymbol{F}_{3} \left( \boldsymbol{C}_2, \boldsymbol{W}_3 \right) = \sigma \left( \boldsymbol{W}_3 \left( \operatorname{Concat} \left[ \boldsymbol{P}^{'}_1, \boldsymbol{P}^{'}_2 \right] \right) \right),
\end{align}
Where $\boldsymbol{F} \left( \cdot , \cdot \right)$ represents convolution mapping functions. $\sigma \left( \cdot \right)$ stands for the PReLU nonlinear activation function~\cite{he2015delving}. $ \boldsymbol{W}_{1}, \boldsymbol{W}_{2}, \boldsymbol{W}_{3}$ indicate that the convolution kernel size used in the convolution layer are 3, 5 and 1, respectively. $ \operatorname{Concat} \left[\cdot , \cdot \right]$ denotes the concatenation operation. Finally, through the dimensionality reduction of $ 1 \times 1 $ bottleneck layer, the output feature parameters are significantly reduced.
In order to make the block more efficient and practical, we adopt residual learning similar to in ResNet~\cite{zhang2018residual} to each MMRB. Formally, we describe the MMRB layer as:
\begin{align}
\boldsymbol{X}_O = \boldsymbol{X}_I + \boldsymbol{P}.
\end{align}
Where $\boldsymbol{X}_O $ represents the MMRB's output. $\boldsymbol{P} \in \mathbb{R}^{H \times W \times C}$ and is the output of multi-path feature maps fusion. In practice, The MMRB layer is applied to the encoder level by level from shallow layer to deep layer to improve our network's performance. What is more, it introduces fewer parameters and can be plug-and-play to enhance the ability of feature extraction in other networks.
\subsection{Freeze Discriminator}
Thanks to the powerful generation ability of the StyleGAN2~\cite{karras2020stylegan2} prior model, the generated face image is genuinely diversified in structure, texture, and color information. We use the StyleGAN2 model as a generative facial prior and adopt a jointly fine-tuning training strategy for the pre-trained StyleGAN2 and codec. Fine-tuning the pre-trained StyleGAN2 model can better fit the complex degraded multi-pose face data into the target distribution. While the source domain and target domain are the same in our work. To make the distribution generated results better consistent with the target distribution, we introduce a Freeze Discriminator (FreezeD) training skill~\cite{mo2020FreezedD} for the StyleGAN2 to better learn the target distribution by freezing the higher-resolution layers of the discriminator during the training process. We find that simply freezing the lower layers of the discriminator and only fine-tuning the upper layers performs surprisingly well, and the generator is more stable during training.
Where the pre-trained discriminator model comes from the training results published by~\cite{yang2021gpen}, that is because it has a prior knowledge to distinguish high-quality images, and its target distribution for training is the same as ours. What's more, using the discriminator model and generator for joint fine-tuning can improve the quality of the overall image and speed up the training process.
\subsection{Model Objectives}
To jointly fine-tune our restoration model, we use the following loss functions: the adversarial loss, the reconstruction loss, and the identity preserving loss. The adversarial loss is composed of a global adversarial loss and multiple local adversarial losses, in which the global adversarial loss function is defined as
\begin{equation}
L_{adv-g} = \min_{G} \max_{D} \mathbb{E}_{(X)} \log \left(1 + \exp \left(- D \left(G \left( \boldsymbol{X}^{'} \right)\right)\right)\right)
\end{equation}
Where $\boldsymbol{X}$ and $\boldsymbol{X}^{'}$ denote the ground-truth image and the generated one, and G is the pre-trained StyleGAN2 for fine-tuning. D is the pre-trained discriminator model from~\cite{yang2021gpen} and adopt the FreezeD strategy during training.
We believe the StyleGAN2~\cite{karras2019stylegan} model incorporates geometric face priors for BFR tasks. Therefore, we introduce local facial losses to focus generated results on local patch distributions, such as eyes and mouth. By using these local losses, we can effectively distinguish the high-frequency features of local regions to generate more natural and vivid local contents. In particular, it can make our model better to solve the unreal problem of multi-pose face restoration in the wild.
We employ local discriminators for the left eye, right eye, and mouth. The facial regions of interest (ROI) first need to be cropped and aligned based on the 68 key points solved by the facial keypoints detection model~\cite{jin2021facial_landmark} and Mask R-CNN~\cite{he2017mask}. For each region, we build upon the same discriminator network and employ different local losses for adversarial training. The local facial losses are defined as follows:
\begin{equation}
L_{adv-l} = \sum_{ROI} \mathbb{E}_{(X^{'}_{ROI})} \left[ \log \left(1 - D_{ROI} \left( \boldsymbol{X}^{'}_{ROI} \right)\right)\right]
\end{equation}
\begin{equation}
L_{adv} = \lambda_{g} L_{adv-g} + \lambda_{l} L_{adv-l}
\end{equation}
Where the cross-entropy loss function is used and $D_{ROI}$ is the local discriminator for each region. $ \lambda_{g}$ and $ \lambda_{l}$ represent the loss weights of global adversarial loss and local adversarial losses, respectively. We differentiate the weight of the discriminators to force the generator to focus more on local regions rather than the whole image during training. Besides, we adopt the L1-norm loss and feature matching loss as content losses $L_{C}$:
\begin{equation}
L_{C} = \lambda_{l1} \left\lVert \boldsymbol{X} - \boldsymbol{X}^{'} \right\rVert_1 + \lambda_{FM} \mathbb{E} \left[ \sum^N_{i=1} \left\Vert \psi_{i} \left( \boldsymbol{X} \right) - \psi_{i} \left( \boldsymbol{X}^{'} \right) \right\rVert_2 \right],
\end{equation}
$ \lambda_{l1}$ and $ \lambda_{FM}$ represent the loss weights of the L1-norm loss and feature matching loss~\cite{wang2018high}. Inspired by~\cite{zhao2016loss_functions}, we give $ \lambda_{FM}$ a very small value of the weight parameter to avoid generating smooth results and checkerboard artifacts. $\psi_{i}$ indicates the $i$-th convolution layer of the pre-trained VGG network~\cite{simonyan2014very}. $N$ is the total number of intermediate layers used for feature extraction. In order to enforce the restored face to have a small distance with the ground truth in the deep feature space, we introduce a face preserving loss~\cite{huang2017beyond}:
\begin{equation}
L_{FP} = \lambda_{FP} \left\Vert \phi \left( \boldsymbol{X} \right) - \phi \left( \boldsymbol{X}^{'} \right) \right\rVert_1.
\end{equation}
Where $\lambda_{FP}$ denotes the loss weight. $\phi$ represents the face feature extractor that adopts the pre-trained ArcFace~\cite{jiankang2019retinaface} model for face recognition.
The overall loss optimization function used by our model is defined as
\begin{equation}
L_{total} = L_{adv} + L_{C} + L_{FP}
\end{equation}
The loss hyper-parameters are set as follows: $\lambda_g = 0.5$, $\lambda_l = 3$, $\lambda_{l1} = 8$, $\lambda_{FM} = 0.02$, $\lambda_{FP} = 10$.
\section{Experiments}
\subsection{Datasets}
\textit{\textbf{ Training Datasets}}. We train on the 70k high-quality face images from the FFHQ dataset, which is synthesized by~\cite{karras2020stylegan2}. We resize the resolution of all images to 512$\times$512 during training. To make the training data more in line with the degradation of the real scenes, we follow the practice in~\cite{wang2021real-esrgan}~\cite{wang2021GFPGAN}~\cite{yang2021gpen} and adopt the following degradation model with all possible degradation factors in the wild to synthesize LQ training data:
\begin{equation}
\boldsymbol{x} = \mathbb{D} (\boldsymbol{y}) = \left[ \left(\boldsymbol{y} \otimes \boldsymbol{k} \right) \downarrow_{\boldsymbol{s}} + \boldsymbol{n}_\delta \right]_{JPEG_q}
\end{equation}
$\mathbb{D}$ denotes the degradation process for HQ images. The HQ face image $\boldsymbol{y}$ is first convolved $\otimes$ with blur kernel $\boldsymbol{k}$. Then, a downsampling operation with scale factor $\boldsymbol{s}$ is performed. Meanwhile, we continue to superimpose adding noise $\boldsymbol{n}$ on the high-quality image. Finally, the LQ image $\boldsymbol{x}$ is obtained by JPEG compression. To simulate the seriously degraded scenes, we randomly sample $\boldsymbol{k}$, $\boldsymbol{s}$, $\boldsymbol{\delta}$ and $\boldsymbol{q}$ from $41$, $[0.4 : 8]$, $[0 : 25]$, $[5 : 50]$ for each training pair in our experiments, respectively. In particular, we use Gaussian blur and motion blur to deal with the possible blur of the image in the natural scenes.
\textit{\textbf{ Testing Datasets}}. To better verify the performance of our model in complex degraded scenes, we use various types of low-quality datasets for testing, and they do not overlap with the training data. We make a brief description of these datasets.
\textit{1) CelebA Data} is a synthetic dataset with 30000 high-quality face images~\cite{liu2015deep}. We select 2556 face images with complex facial posture and degrade the selected images using the above degradation model and parameters.
\textit{2) LFW Data} comes from~\cite{huang2008LFW} and is composed of more than 13k face pictures of famous people all over the world with different orientations, expressions, and lighting environments. They are real low-quality images with only a single face. The image size is 250 pixels $\times$ 250 pixels. We select 1610 images with relatively low quality for testing.
\textit{3) FDDB Data} comes from~\cite{fddbData} and contains 5171 human faces in 2845 images taken from different natural scenes. The number of faces in the image is uncertain, not a single number. Moreover, the width and height of images are not equal, and the difference is noticeable. We select 1629 images with relatively low quality for testing.
\textit{4) WebFace Data} is collected face images from the Internet by~\cite{yi2014Webface}, including tens of thousands of images with a size of 250 pixels $\times$ 250 pixels. They are real low-quality images with only a single face and have many old black-and-white photos. We select 804 images with relatively low quality for testing.
\subsection{Implementation Details and Evaluation Metrics}
We adopt the FreezeD strategy~\cite{mo2020FreezedD} to the global discriminator from~\cite{yang2021gpen}, and freeze the parameters of the first five convolution layers and fine-tune the remaining deep features. We then jointly fine-tune the Stylegan2 model retrained by~\cite{yang2021gpen} with the codec. The input image dimension mapped to the nearest latent codes is consistent with the pre-trained Stylegan2 model. At the same time, the resolution and dimension for the output feature maps of every level in the reconstruction process are consistent with the noise branch of the pre-trained model. For each MMRB layer, we adopt it layer by layer to extract weak texture features in low-quality images.
\begin{figure*}[!t]
\centering
\includegraphics[width=1.0\textwidth]{figure/BFR_Celeba_comparison_figure.pdf}
\caption{Qualitative comparisons with several state-of-the-art face restoration methods on the CelebA Data. To intuitively feel the performance difference of each method, we enlarge and display local areas. \textbf{Zoom in for best view}.}
\label{fig: BFR celeba results}
\end{figure*}
During the experiment, we train our model with Adam optimizer~\cite{kingma2014adam} and perform a total of 700k iterations with a batch size of 4. The learning rate (LR) varies for different parts, including the codec, the global discriminator, the local discriminator, the pre-trained Stylegan2 model, and noise branches. They are set to $2\times10^{-3}$, $2\times10^{-5}$, $2\times10^{-3}$, $2\times10^{-4}$ and $2\times10^{-3}$, respectively. The local discriminator includes three facial components: left eye, right eye, and mouth. They use the same learning rate and discriminator model. Furthermore, a piece-wise attenuation strategy is adopted. We implement our model with the PyTorch framework and train it using two NVIDIA V100 GPUs.
For the evaluation, we employ a widely-used non-reference perceptual metric: NIQE~\cite{mittal2012NIQE}. Moreover, we also adopt pixel-wise metrics (PSNR and SSIM) and the reference perceptual metrics (LPIPS~\cite{zhang2018LPIPS} and FID~\cite{2017FID}) for the CelebA Data with Ground Truth (GT).
\subsection{Experiments on Synthetic Images}
To better show the practicability and generality of our model, we verify the performance of face restoration and face super-resolution tasks through synthetic CelebA Data.
\begin{table}[]
\vspace{-0.3cm}
\small
\centering
\caption{Quantitative comparisons of various BFR methods on \textbf{CelebA Data}. Bold \textcolor{red}{\bf RED} indicates the best performance, and bold \textcolor{blue}{\bf BLUE} indicates the second. Reference evaluation metrics are adopted, such as PSNR, SSIM, LPIPS, and FID, and non-reference perceptual metric (NIQE) is also adopted.}
\label{tab:celeba_BFR}
\tabcolsep=0.1cm
\scalebox{1.0}{
\hspace{-0.5cm}
\begin{tabular}{c|cc|ccc}
\hline
Methods & PSNR$\uparrow$ & SSIM$\uparrow$ & LPIPS$\downarrow$ & FID$\downarrow$ &NIQE $\downarrow$ \\ \hline
HiFaceGAN~\cite{yang2020hifacegan} & 24.506 & 0.612 & 0.196 & 31.545 & 4.427 \\
DFDNet~\cite{li2020DFD} & 22.470 & 0.663 & 0.232 & 54.358 & 5.916 \\
PSFRGAN~\cite{chen2021psfr-gan} & 24.884 & 0.663 & 0.192 & 34.695 & 4.765 \\
Panini~\cite{wang2022panini} & 24.679 & 0.611 & 0.194 & 44.696 & \textcolor{red}{\bf 3.837} \\
GPEN~\cite{yang2021gpen} & \textcolor{blue}{25.190} & 0.680 & \textcolor{blue}{0.169} & 21.852 & 4.712 \\
GFPGAN~\cite{wang2021GFPGAN} & 24.895 & \textcolor{blue}{0.688} & 0.172 & \textcolor{blue}{21.160} & 4.746 \\\hline
\textbf{Our}& \textcolor{red}{\bf 26.034} & \textcolor{red}{\bf 0.702} & \textcolor{red}{\bf 0.162} & \textcolor{red}{\bf 16.080} & \textcolor{blue}{\bf 4.285} \\ \hline
GT & $\infty$ & 1 & 0 & 2.478 & 4.188 \\ \hline
\end{tabular}}
\vspace{-0.3cm}
\end{table}
\textit{\textbf{Face Restoration.}} To verify the superiority of the method mentioned in this paper, We make qualitative and quantitative comparisons with several of the latest blind face restoration methods, respectively, including DFDNet~\cite{li2020DFD}, PSFRGAN~\cite{chen2021psfr-gan}, Panini~\cite{wang2022panini}, HiFaceGAN~\cite{yang2020hifacegan}, GFPGAN~\cite{wang2021GFPGAN}, and GPEN~\cite{yang2021gpen}. We also introduce metric values corresponding to the GT to understand better the difference between the test results of different methods and the GT in quantitative analyses.
The quantitative and qualitative results of different methods are shown in Tab.~\ref{tab:celeba_BFR} and Fig.~\ref{fig: BFR celeba results}, respectively. Our method is optimal in all indicators and has the best performance according to the quantitative results. Although the NIQE metric is slightly lower than the best, it does not affect the priority of our method. That is because this perceptual metric can only roughly reflect the local difference between tested images in deep space and the learned perceptual features and can not better show the effectiveness of the recovery model~\cite{blau20182018}. What's more, thanks to the fine-tuning training strategy, the visual qualitative comparison results show that our restoration images are more realistic and natural and have the best visual sensory effect, especially in the eyes and mouth areas. And our method can restore the complex degraded multi-pose face image more flexibly. In particular, the difference can be seen more clearly and intuitively by zooming in the local details. In addition, it is also seen that our method can improve the image quality of non-facial regions.
\begin{figure*}[!t]
\centering
\includegraphics[width=1.0\textwidth]{figure/SR_comparison_figure.pdf}
\caption{Qualitative comparisons of 4x face super-resolution by different methods. We do not apply local magnification displays for the qualitative comparison results, which intends to view the differences between them in the form of the whole picture. \textbf{Zoom in for best view}.}
\label{fig:FSR CelebA results}
\end{figure*}
\textit{\textbf{Face Super-resolution.}} We use the first 1500 images of CelebA Data and perform 4x downsampling of the input images to verify the super-resolution performance of different methods in the wild. Furthermore, we also make qualitative and quantitative comparisons with several state-of-the-art face super-resolution methods, including SuperFAN~\cite{bulat2018super}, pSp~\cite{richardson2021psp}, Panini~\cite{wang2022panini}, HiFaceGAN~\cite{yang2020hifacegan}, DFDNet~\cite{2017FID}, GFPGAN~\cite{wang2021GFPGAN}, and GPEN~\cite{yang2021gpen}. Our method needs to upsample tested images to the original scale and then input them into the network for testing.
\begin{table}[]
\vspace{-0.5cm}
\small
\centering
\caption{Quantitative comparisons of various FSR methods on \textbf{CelebA Data}. We adopt the Bicubic interpolation for 4x down-sampling of the input data to verify the super-resolution performance of different methods. Bold \textcolor{red}{\bf RED} indicates the best performance, and bold \textcolor{blue}{\bf BLUE} indicates second one.}
\label{tab:celeba_FSR}
\tabcolsep=0.1cm
\scalebox{1.0}{
\hspace{-0.5cm}
\begin{tabular}{c|cc|ccc}
\hline
Methods & PSNR$\uparrow$ & SSIM$\uparrow$ & LPIPS$\downarrow$ & FID$\downarrow$ &NIQE $\downarrow$ \\ \hline
Bicubic & 23.757 & 0.641 & 0.297 & 148.468 & 10.460 \\
SuperFAN~\cite{bulat2018super} & 23.193 & 0.617 & 0.290 & 152.188 & 7.857 \\
pSp~\cite{richardson2021psp} & 18.493 & 0.580 & 0.242 & 67.596 & 5.647 \\
HiFaceGAN~\cite{yang2020hifacegan} & 19.946 & 0.470 & 0.326 & 230.756 & 10.480 \\
DFDNet~\cite{2017FID} & 21.772 & 0.638 & 0.260 & 93.107 & 6.416 \\
Panini~\cite{wang2022panini} & 23.417 & 0.620 & 0.278 & 165.834 & 6.898 \\
GPEN~\cite{yang2021gpen} & \textcolor{blue}{24.248} & \textcolor{blue}{0.658} & 0.217 & 50.348 & 5.886 \\
GFPGAN~\cite{wang2021GFPGAN} & 23.780 & 0.656 & \textcolor{blue}{0.190} &\textcolor{blue}{32.894} & \textcolor{blue}{4.955} \\\hline
\textbf{Our} & \textcolor{red}{\bf 24.797} & \textcolor{red}{\bf 0.674} & \textcolor{red}{\bf 0.189} & \textcolor{red}{\bf 27.403} & \textcolor{red}{\bf 4.449} \\ \hline
GT & $\infty$ & 1 & 0 & 2.413 & 4.036 \\ \hline
\end{tabular}}
\vspace{-0.3cm}
\end{table}
The quantitative and qualitative results of different methods are shown in Tab.~\ref{tab:celeba_FSR} and Fig.~\ref{fig:FSR CelebA results}, respectively. As can be seen from the quantitative results, Our method can perform super-resolution on LQ images, and the output results are significantly better than other face super-resolution methods in all indicators. In particular, the output results of our model can be closer to the GT by using the MMRB layer, both in qualitative and quantitative results. We do not adopt local magnification displays for the qualitative comparison results, which intends to view the differences between them in the form of the whole picture. By visual comparison results, our results have fewer or less noticeable artifacts and texture blur, whether in facial texture, hair, or background. Although the pSp method~\cite{richardson2021psp} also shows promising results in the visualization effect, it is quite different from the GT due to its design idea.
\subsection{Experiments on Images in the Wild}
\begin{table}[]
\vspace{-0.3cm}
\small
\centering
\caption{Quantitative comparisons of various FSR methods in the wild, such as \textbf{FDDB Data}, \textbf{LFW Data}, \textbf{WebFace Data}. Only the non-reference perceptual metric (NIQE) is adopted. “\textbf{-}” means that the result is unavailable.}
\label{tab:BFR_wild}
\tabcolsep=0.1cm
\scalebox{1.0}{
\hspace{-0.5cm}
\begin{tabular}{c|cc|cc|cc}
\hline
Dataset & \multicolumn{2}{c|}{\textbf{FDDB Data}} & \multicolumn{2}{c|}{\textbf{LFW Data}} & \multicolumn{2}{c}{\textbf{WebFace Data}} \\
Methods & \multicolumn{2}{c|}{NIQE $\downarrow$} & \multicolumn{2}{c|}{NIQE $\downarrow$} &\multicolumn{2}{c}{NIQE $\downarrow$} \\ \hline
Input & \multicolumn{2}{c|}{4.262} & \multicolumn{2}{c|}{5.454} & \multicolumn{2}{c}{4.875} \\
HiFaceGAN~\cite{yang2020hifacegan} & \multicolumn{2}{c|}{\textcolor{blue}{\bf 3.956}} & \multicolumn{2}{c|}{\textcolor{red}{\bf4.548}} & \multicolumn{2}{c}{4.969} \\
DFDNet~\cite{li2020DFD} & \multicolumn{2}{c|}{4.153} & \multicolumn{2}{c|}{4.990} & \multicolumn{2}{c}{5.449} \\
PSFRGAN~\cite{chen2021psfr-gan} & \multicolumn{2}{c|}{4.334} & \multicolumn{2}{c|}{5.461} & \multicolumn{2}{c}{5.830} \\
Panini~\cite{wang2022panini} & \multicolumn{2}{c|}{$-$} & \multicolumn{2}{c|}{4.694} & \multicolumn{2}{c}{\textcolor{red}{\bf 4.605}} \\
GPEN~\cite{yang2021gpen} & \multicolumn{2}{c|}{4.110} & \multicolumn{2}{c|}{5.418} & \multicolumn{2}{c}{5.793} \\
GFPGAN~\cite{wang2021GFPGAN} & \multicolumn{2}{c|}{4.009} & \multicolumn{2}{c|}{4.982} & \multicolumn{2}{c}{5.313} \\ \hline
\textbf{Ours} & \multicolumn{2}{c|}{\textcolor{red}{\bf 3.578}} & \multicolumn{2}{c|}{\textcolor{red}{\bf 4.548}} & \multicolumn{2}{c}{\textcolor{blue}{\bf 4.875}} \\ \hline
\end{tabular}}
\vspace{-0.5cm}
\end{table}
\begin{figure*}[!t]
\centering
\includegraphics[width=1.0\textwidth]{figure/wild_restoration_1.pdf}
\includegraphics[width=1.0\textwidth]{figure/wild_restoration_2.pdf}
\caption{Qualitative comparisons on the \textbf{FDDB Data}, \textbf{LFW Data}, and \textbf{WebFace Data}. \textbf{Zoom in for best view}.}
\label{fig:FSR wild results}
\end{figure*}
We evaluate our method on the FDDB Data, LFW Data, and WedFace Data, which are face images with complex facial poses in real scenes and suffer from multiple complex unknown degradations. To test the generalization ability, we make qualitative and quantitative comparisons with several latest blind face restoration methods, such as DFDNet~\cite{li2020DFD}, PSFRGAN~\cite{chen2021psfr-gan}, Panini~\cite{wang2022panini}, HiFaceGAN~\cite{yang2020hifacegan}, GFPGAN~\cite{wang2021GFPGAN}, and GPEN~\cite{yang2021gpen}. Since there is no GT in the real degraded image, we adopt the non-reference perceptual metric (NIQE) to test the performance of different methods. For the FDDB data, in order to facilitate qualitative comparison, we select some images and utilize the pre-trained RetinaFace face detection model~\cite{deng2020retinaface} to cut out a single face with consistent width and height for testing.
The quantitative and qualitative results are presented in Tab.~\ref{tab:BFR_wild} and Fig.~\ref{fig:FSR wild results}. It can be seen that our method still has prominent leadership in quantitative indicators and visualization results. Thanks to our careful design model and the excellent training strategy, the restored images have more apparent facial texture, more comfortable color information, and higher visual quality. In particular, the local components in the face are more realistic and reasonable. Although the numerical value is not the best on WebFace Data, it is evident from the visual effect that our method still has excellent performance. In the quantitative comparison, Panini method~\cite{wang2022panini} lacks tested results on the FDDB Data, mainly because the images in the dataset have the problem of unequal width and height. What's more, this method strictly requires equal width and height of the input images.
\subsection{Ablation Studies}
To better understand the roles of different components and training strategies in our method, we train them separately and apply qualitative and quantitative comparisons to test the performance. The involved relevant variables are as follows: without MMRB layer (w/o MMRB), without Local Discriminator (w/o LocalD), and without the fine-tuning and Freeze Discriminator strategy (w/o ft+FreezeD). As can be seen from the Fig.~\ref{fig:ablated study} and Tab.~\ref{tab:ablation}, we can find that:
\begin{figure}[!t]
\centering
\includegraphics[width=0.485\textwidth]{figure/ablated_study_figure.pdf}
\caption{Comparative results of different variables in our model. \textbf{Zoom in for best view}.}
\label{fig:ablated study}
\end{figure}
\begin{table}[!t]
\vspace{-0.2cm}
\small
\centering
\caption{Quantitative comparisons of different variants on \textbf{CelebA Data}. Bold \textbf{black} indicates the best performance.}
\vspace{-0.2cm}
\label{tab:ablation}
\tabcolsep=0.1cm
\scalebox{1.0}{
\hspace{-0.5cm}
\begin{tabular}{l|cc|ccc}
\hline
Components & PSNR$\uparrow$ & SSIM$\uparrow$ & LPIPS$\downarrow$ & FID$\downarrow$ & NIQE $\downarrow$ \\ \hline
a) w/o MMRB & 25.977 & 0.699 & 0.163 & 17.027 & 4.682 \\
b) w/o LocalD & 26.010 & 0.697 & 0.163 & 16.908 & 4.614 \\
c) w/o ft+FreezeD & 25.704 & 0.692 & 0.171 & 19.543 & 4.546 \\ \hline
\textbf{All (Our)} & \textbf{26.034} & \textbf{0.702} & \textbf{0.162} & \textbf{16.080} & \textbf{4.285} \\ \hline
\end{tabular}}
\vspace{-0.3cm}
\end{table}
\textit{\textbf{MMRB layer.}} If the MMRB layer in the encoder is adopted, it can better maintain the original features of the input image and avoid the excessive illusion of local contents. However, when we remove the MMRB layers, it can find that the textures of the restored results are not complete, and the authenticity is weakened, especially in local areas.
\textit{\textbf{Local Discriminator.}} if the network lacks local discriminators and the constraint of local facial losses during training, the restoration results are not natural enough and complete, and the visual sense is poor. That is mainly because the eyes and mouth significantly impact the overall quality.
\textit{\textbf{Fine-tuning and FreezeD strategy.}} This training strategy significantly impacts the restoration results of our model, whether whether it is the qualitative or quantitative comparisons. It can also greatly reduce the distribution difference between the restoration results and the target domain and fit the real high fidelity face images better. However, if the training strategy is not adopted, we can find that it has the worst value of all metrics, and the visualization results are also the worst.
\section{Conclusion}
Aiming at the problem that complex degraded pose-varied and multi-expression face images are challenging to restore, we meticulously designed a restoration network with a generative facial prior for BFR tasks. To better preserve the original facial features and avoid excessive fantasy, we exploited MMRB layers to gradually extract weak texture features in the input images. Furthermore, we fine-tuned the pre-trained StyleGAN2 model and adopted the FreezeD strategy for the global discriminator model, which can better fit the distribution of diverse face images suitable for the natural scenes and improve the quality of the overall images. And especially, because eyes and mouth regions are difficult to recover, we adopted different local adversarial losses to constrain our model for these regions to realistically restore face images. Extensive experiments on synthetic and multiple real-world datasets demonstrated that our model can restore complex degraded pose-varied face images and outperform the latest optimal BFR methods. The restoration results have richer textures, more natural details, and better visual sense. Most importantly, our method can also promote the image quality of non-facial regions and can restore old photos, and film and television works. And it can also be applied to other tasks such as face super-resolution.
\bibliographystyle{IEEEtran}
|
1,116,691,499,379 | arxiv | \section{Introduction}
\setcounter{equation}{0}
The study of localized tachyon condensation \cite{aps,vafa,hkmm,dv,many} has been
considered with many interesting developments.
The basic picture is that
tachyon condensation induces cascade of decays of the orbifolds to less singular ones until the spacetime
supersymmetry is restored. Therefore the localized tachyon condensation has a geometric description as the resolution
of the spacetime singularities.
Following the line of Vafa's reformulation of the problem in terms of Mirror Landau-Ginzburg theory, we worked out the detailed analysis on the fate of spectrum and the background geometry under the tachyon condensation as well as the question of what is the analogue of c-theorem with the GSO-projection
in a series of papers\cite{sin,sinlee,gso}. In all these works, super-conformal invariance was used very heavily so that we were working at the string theory on-shell level.
On the other hand, recently, Dabholkar and Vafa\cite{dv} proposed that the tachyon potential is given by the maximal charge.
Strictly speaking, the $U(1)$ charge in the question is defined only on the conformal point not on the off-shell.
On the other hand, the decay process considered as a renormalization group flow is an off-shell process and one needs to extend the concept of charge in order to define the tachyon potential. The way to extend the charge is to consider it as a semi-index which is contributed by BPS objects only \cite{cvi}. These are not just topological since it has anti-holomorphic dependence as well as holomorphic dependence on the deformation parameters. The way to calculate this quantity is to show that it satisfies $tt^*$ equations\cite{cv} and solve it if possible. In \cite{dv}, the process where ${\mathbb{C}}/{\mathbb{Z}}_3$ decays to ${\mathbb{C}}^1$ was discussed.
In this paper we extend the result of \cite{dv} to ${\mathbb{C}}^2/{\mathbb{Z}}_n$ for $n=3,4,5$. Interestingly, the tachyon potential for $n=3$ and 4 is still given in terms of the solutions of Painleve III type
that appeared in the study of ${\mathbb{C}}^1/{\mathbb{Z}}_3$ with different boundary conditions. For ${\mathbb{C}}^2/{\mathbb{Z}}_5$ the governing equation is that of generalized Toda type. The potential is monotonically decreasing function of RG flow, therefore expected so as function of real time as well. For LG model associated to ${\mathbb{C}}^2/{\mathbb{Z}}_5$ without orbifold projection, $tt^*$ equation contains the Bullough-Dodd equation, whose solution is known\cite{kit}.
In this letter, we do not attempt to give all necessary background. For the set up of tachyon condensation in terms of mirror symmetry, see \cite{vafa,sin}. For the application of $tt^*$ to the tachyon condensation, see \cite{dv}.
\section{${\mathbb{C}}^2/{\mathbb{Z}}_3$}
\setcounter{equation}{0}
In this section we study the tachyon condensation in ${\mathbb{C}}^2/{\mathbb{Z}}_n$ with Landau-Ginzburg (LG) description by calculating and solving the corresponding $tt^*$ equations.
First we consider the simplest of all ${\mathbb{C}}^2/{\mathbb{Z}}_n$, namely $n=3$.
The mirror of this is a LG whose superpotential is given by
\begin{equation}
W={x^3 \over 3} +{y^3 \over 3} - t x y,
\end{equation}
with an imposition of an orbifold constraint.
Before we consider the orbifolded LG theory, we work out the $tt^*$ equations for LG
theory without orbifolding for later interests. The fundamental variables are
not $x,y$ but $\log x, \log y$ \cite{dv,hori} so that the chiral ring consists of
\begin{equation}
\{ x y, x^2y,xy^2,x^2y^2 \}.
\end{equation}
from which charges of the elements can be read off to give $NS$-charges
\begin{equation}
\{{2 \over 3},1,1,{4\over 3} \}.\end{equation}
The R-charges in the topological strings are related to those in $NS$ by the spectral flow
\begin{equation}
q_R=q_{NS}-{n\over 2}.
\end{equation}
For ${\mathbb{C}}^2/{\mathbb{Z}}_3$, we have $n=2$ so we get
\begin{equation}
\{-{1\over 3},0,0,{1\over 3} \} .\label{charge1}\end{equation}
This superpotential has symmetries
\begin{eqnarray}
&&x\to \omega x,~~y\to \omega^2 y ; \nonumber \\
{\rm and } &&x\to \omega^2 x,~~y\to \omega y,
\end{eqnarray}
where $\omega=e^{{2\over 3}\pi i}$, which constrains the metric
$g_{ {\bar j}i}:=\langle \phi_{\bar j}\phi_i \rangle (\phi_is~ {\rm are~ chiral
fields})$
to be of the form
\begin{equation}
g=\pmatrix { a_{11} & 0 & 0 & \bar{b} \cr
0 & a_{21}& 0 & 0 \cr
0 & 0 & a_{12} & 0 \cr
b & 0 & 0 & a_{22}
}.
\end{equation}
The topological metric $\eta_{ij}$ is given by residue paring
$$
\eta_{ij}=\langle\phi_i\phi_j\rangle={1\over {(2\pi i)}^n}\int_\Gamma
{ {\phi_i(X)\phi_j(X)dX^1\wedge\cdots\wedge dX^n} \over
{\partial_1 W \partial_2W\cdots \partial_nW}
}~~~~~~
$$
with superpotential $W$, which in this case is calculated to be
\begin{equation}
\eta=\pmatrix{ 0 & 0 & 0 & 1 \cr
0 & 0 & 1 & 0 \cr
0 & 1 & 0 & 0 \cr
1 & 0 & 0 & t^2
}.
\end{equation}
From the reality constraint
\begin{equation}
\eta^{-1} g (\eta^{-1} g)^* =I,
\end{equation}
we have
\begin{equation}
b={t^2 \over 2}a_{11}, ~~a_{12}={1\over a_{21}}, ~~a_{22}={ 1\over a_{11}} +{|t|^2 \over
4}a_{11}. \label{exch1}
\end{equation}
The chiral ring coefficient are defined by $\phi_i \phi_j=C^k_{ij}\phi_k$ and if we denote the matrix for the multiplication by $xy$ by $C_t$, then
\begin{equation}
C_t=\pmatrix { 0 & 0 & 0 & 1 \cr
0 & t^2 & 0 & 0 \cr
0 & 0 & t^2 & 0 \cr
0 & 0 & 0 & t^2
}.
\end{equation}
Putting all these into $tt^*$ equation
\begin{equation}
\partial_{\bar{t}} (g \partial_{t} g^{-1})
=[C_t,g(C_t)^\dagger g^{-1}],
\end{equation}
we get
\begin{eqnarray}
&& \partial_{\bar{t}} \partial_t \log a_{11} =-{1\over a_{11}^2} +
{ |t|^8 \over 16} a_{11}^2, \nonumber \\
&&\partial_{\bar{t}} \partial_t \log a_{12} =0 .\label{ttbar1}
\end{eqnarray}
The exchange symmetry
$x \rightleftarrows y$ due to the special form of perturbation,
together with the reality condition eq.(\ref{exch1}), determines
\begin{equation}
a_{12}=a_{21}=1,
\end{equation}
which is consistent with the second equation of eq.(\ref{ttbar1}).
By changes of variables
\begin{equation}
y=a_{11}^{-1}, ~~ \zeta={1\over 3}t^{3},~~{\rm and}~~ y={1\over
2}|t|^{2}Y, \label{defY}
\end{equation}
the first equation of eq.(\ref{ttbar1}) can be transformed into
\begin{equation}
\partial_{\bar{\zeta}} \partial_\zeta \log Y =\frac{1}{4}\( Y^{2} - Y^{-2}\) .
\label{formY} \end{equation}
In terms of $Y$ and $z=|\zeta|$ the last equation can be written as
\begin{equation}
Y''= {(Y')^2 \over Y} -{Y'\over z} +Y^3 -{1\over Y},
\end{equation}
which is already known as Painleve III equation.
We can further rewrite the eq.(\ref{formY}) as the
sinh-Gordon equation
\begin{equation}
\partial_{\bar{\zeta}} \partial_\zeta u = \sinh u , \label{sinh}
\end{equation}
by introducing $u$ by $u=2 \log y$.
Now, since we are interested in the scaling behavior of the solutions,
we look at the dependence on $z=|\zeta|$ of the eq.(\ref{sinh}),
\begin{equation}
u''+u'/z = 4 \sinh u , \label{sinhz}\end{equation}
where the prime denote the derivative in $z$.
The solution to this is well known\cite{its}.
The real solutions are classified by their asymptotic behavior in $z\to 0$
\begin{equation}
u(z) \simeq r \log z + O(z^{2-|r|}) ~for ~|r|<2.
\end{equation}
In our case, the regularity of $a_{11}$ requires
\begin{equation} u(z) \sim -{4\over3} \log z. \label{asymp0}
\end{equation}
Therefore this fixes $r=-4/3$. For $z\to \infty$
\begin{equation}
u(z) \sim \sqrt{3\over \pi} { e^{-2z} \over \sqrt{z} } ,~~z\to \infty .\label{asympinfty}
\end{equation}
The precise form of regular solution can be written in terms of a convergent expansion
\begin{equation}
u(z;r)=-4\sum^\infty_{n=0} {[2\cos((2-r)\pi/4)]^{2n+1}] \over
{2n+1}} \int^\infty_{-\infty} \prod^{2n+1}_{i=1}{d\theta_i \over {4\pi}}
{e^{-2z\cosh\theta_i }\over {\cosh[(\theta_i-\theta_{i+1})/2] } } \label{exactu}
\end{equation}
with $\theta_{2n+1}\equiv \theta_1$ and $r=-4/3$.
In order to define the charge matrix, we need to look at the scaling behavior of the system.
The scaling
$z \to \lambda z $ induces
$
\int d^2z d^2 \theta \to \lambda \int d^2z d^2 \theta $,
which is equivalent to the field redefinition and coupling change such that $ W \to \lambda W$.
For the given superpotential
\begin{equation}
W={x^3 \over 3} +{y^3 \over 3} -t xy,
\end{equation}
we can identify the necessary field redefinitions and coupling change as
\begin{equation}
x= \lambda^{1\over
3}\tilde{x},~
y= \lambda^{1\over 3}\tilde{y}~~ {\rm and }~~
t = \lambda^{1\over 3}t_0, \label{scalet}
\end{equation}
by which $W$ can be written as
\begin{equation}
W= \lambda({\tilde{x}^3 \over 3} + {\tilde{y}^3 \over 3} -t_0 \tilde{x}\tilde{y}).
\end{equation}
In this way changing $t$ is equivalent to the changing the scale. The metric components
in old and new basis are related by
\begin{equation}
a_{ij}=\langle x^{\bar i}y^{\bar j}|x^iy^j \rangle =|\lambda|^{(i+j)\over n} \langle
\tilde{x}^{\bar i}\tilde{y}^{\bar j}|\tilde{x}^i\tilde{y}^j \rangle :=|\lambda|^{(i+j)\over n} b_{ij} ,\label{anb}
\end{equation}
with $n=3$.
The off-shell `charge' of the system was defined \cite{cv} as
\begin{equation}
Q=g\partial_\tau g^{-1} -{{\hat c}\over 2} \label{Q}
\end{equation}
where $\tau=\log \lambda$ and the metric components in use are in $\tilde{x},\tilde{y}$ basis,
namely $b_{ij}$'s.
From the regularity condition of
the metric in $x,y$ basis at $t=0,$ one can verify that the charge at the
starting point $t=\lambda=0$ is encoded correctly to give the result listed
in eq.(\ref{charge1}).
{\it Orbifolded LG model: $\{W=x^3+y^3-txy\}//{\mathbb{Z}}_3 $}\\
In this case, the chiral ring generated by $xy$: $\{ xy, x^2y^2 \}$ and $NS$-charges are
$\{{2 \over 3},{4\over 3} \}$.
The metric and charges can be obtained from the previous subsection by simply discarding
$a_{12}$ and $a_{21}$.
Following \cite{dv}, the tachyon potential is proposed to be
\begin{equation}
V(t,{\bar t})=2 {\rm max}|Q(t,{\bar t})|= -\frac{1}{2}z\frac{du}{dz},
\end{equation}
where we identified $\zeta=\lambda$ by eqs.(\ref{defY}),(\ref{scalet}) after setting
$t_0=3^{1/3}$.
The factor 2 in the above equation is from
the asymptotic form of $u$ is given by eqs. (\ref{asympinfty}),(\ref{asymp0}).
Its exact form can be written
in terms of $u$ given in eq.(\ref{exactu}) with $r=-4/3$.
The potential vanishes exponentially as $t\to\infty$.
As a consequence of the tachyon condensation, the fate of the ${\mathbb{C}}^2/{\mathbb{Z}}_3$ is just ${\mathbb{C}}^2$
as expected.
\section{${\mathbb{C}}^2/{\mathbb{Z}}_4$}
Let us now consider $n=4$.
Working in the basis with definite ordering the chiral ring consists of
\begin{equation}
\{xy,x^2y,x^3y,xy^2,x^2y^2,x^3y^2,xy^3,x^2y^3,x^3y^3 \},
\end{equation}
The topological metric and metric from sec.3.1 are given by
\begin{equation}
\eta_{x^{i_1}y^{j_1},x^{i_2}y^{j_2}}=1,~{\rm if}~ i_1+i_2=j_1+j_2=4,
~~~\eta_{x^3y^3,x^3y^3}=t
\end{equation}
and
\begin{equation}
g_{i\bar{j}}= \pmatrix{ a_{11} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \bar{b}
\cr 0 & a_{21} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & a_{31} & 0 & 0 &
0 & 0 & 0 & 0 \cr 0 & 0 & 0 & a_{12} & 0 & 0 & 0 & 0 & 0 \cr 0 & 0
& 0 & 0 & a_{22} & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & a_{32} & 0 & 0
& 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & a_{13} & 0 & 0 \cr 0 & 0 & 0 & 0 & 0
& 0 & 0 & a_{23} & 0 \cr b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{33} }.
\end{equation}
Then, the reality condition gives
\begin{equation}
a_{22}=1,~a_{32}=1/a_{12},~a_{13}=1/a_{31},~a_{23}=1/a_{21},~a_{33}=1/a_{11} +|t|^4a_{11}/4,~b=t^2 a_{11}/4.
\end{equation}
$C_t$ can also be calculated easily and given by
\begin{equation}
C_t=\pmatrix{ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0
& 0 & 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & t & 0 & 0 \cr 0 &
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
1 \cr 0 & t^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & t & 0 & 0 &
0 & 0 & 0 & 0 \cr 0 & 0 & 0 & t^2 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 &
0 & 0 & t^2 & 0 & 0 & 0 & 0 }.
\end{equation}
Then the $tt^*$ equation becomes
\begin{eqnarray}
&&\label{1} ~~-\partial_{\bar{t}}\partial_t\log a_{11}=1/a_{11}
-|t|^4a_{11}/4, \\
&&\label{2} ~~-\partial_{\bar{t}}\partial_t\log
a_{21}=1/(a_{21}a_{12})-|t|^4a_{21}a_{12}, \\
&&\label{3} ~~-\partial_{\bar{t}}\partial_t\log a_{31}=|t|^2/a^2_{31}
-|t|^2a^2_{31}, \\
&&\label{4} ~~-\partial_{\bar{t}}\partial_t \log
a_{12}=1/(a_{21}a_{12})-|t|^4a_{21}a_{12}. \end{eqnarray}
Most of the equations are of the form
\begin{equation}
\partial_{\bar{t}}\partial_t
\log y =y^2 -{|t|^{2k} \over m^2}y^{-2}. \label{standard}
\end{equation}
By introducing change of variables
\begin{eqnarray}
\zeta &=&(1/(1+k/2))(16/m^2)^{1/(2k+4)}t^{1+k/2},\cr
y&=&\sqrt{1/m}|t|^{k/2}e^{u/2}, \end{eqnarray}
above equation lead us to the sinh-Gordon equation eq.(\ref{sinh}).
The value of the $r$ in the solution of sinh-Gordon equation
can be obtained from the regularity of the metric component near $t=0$.
Since $y\sim z^{k/k+2}e^{u/2}$, we have
\begin{equation}
u\sim -\frac{2k}{k+2}\log z + \cdots, \label{u0}
\end{equation}
which determines the value $r=-\frac{2k}{k+2}$.
Now from eq.(\ref{Q}) the components of charge matrix is given by \begin{equation}
q_{ij}=b_{ij}\partial_\tau b^{-1}_{ij}-1.
\end{equation}
Using $a_{ij}=|\lambda|^{(i+j)/2}b_{ij}$, ~~ $a^{-1}_{ij}\sim y^l$ for some $l$ and $y\sim z^{k/(k+2)}e^{u/2}$ with identification $z=|\lambda|$,
\begin{equation}
q_{ij}(z)=\frac{l}{4}z\frac{du(z)}{dz}+\frac{lk}{2(k+2)} +\frac{i+j}{n}-1.
\end{equation}
Notice that $(ij)$ is not the matrix index but the vector index.
Using eq.(\ref{u0}), the value of the charge at $t=0$ is
\begin{equation}
q_{ij}(0)= \frac{i+j}{n}-1,
\end{equation}
which confirms that we are in the right track.
Now we apply this result to our system.
For $a_{11}$, by change of variables
$a_{11}^{-1}=2y^2 $ the eq.(\ref{1}) reduces to the standard form eq.(\ref{standard}) with $k=2,~ m=4$. Therefore $r=-1$.
This equation further can be reduced to sinh-Gordon equation
$\partial_{\bar{\zeta}}\partial_\zeta u=\sinh u$ by setting $\zeta=t^2/2$ and $y=|t|e^{u/2}/{2}$. Since $l=2$ in this case, charge is given by
$q_{11}(z)=\frac{1}{2}z\frac{du(z)}{dz}$.
For $a_{12}$ and $a_{21}$, first we show they are equal. From eq.(\ref{2}) and eq.(\ref{4}) $a_{12}=|F(t)|^2a_{21}$ for some holomorphic function $F(t)$. Since $a_{12}=a_{21}$ at $t=0$ as well as at $t=\infty$, the only non-singular holomorphic function with such boundary conditions is a constant function $F(t)=1$, i.e, $a_{12}=a_{21}$. This supports the exchange symmetry $a_{ij}=a_{ji}$.
With this,
the eq.(\ref{2},\ref{4}) are the case of $m=1, k=2$; by setting
$a_{12}^{-1}=y=|t|e^{u/2}$ and $\zeta=t^2/2$, we get sinh-Gordon
equation. $l=1$ lead us to
$q_{12}(z)=q_{21}(z)=\frac{1}{4}z\frac{du(z)}{dz}$.
For $a_{31}$, by $z=t^2$ and $y=a_3^{-1}$
we get sinh-Gordon and the solution is $y=e^{u/2}$. It is easy to see $q_{31}(z)=\frac{1}{4}z\frac{du(z)}{dz}$. Notice that $q_{31}(0)=q_{13}(0)=0$. The monotonicity of the charge in $t$ suggests that $q_{31}(z)=0$. In fact the exchange symmetry $a_{31}=a_{31}$ and the reality condition $a_{31}=1/a_{31}$ sets $a_{31}=1$.
{\it Mirror of ${\mathbb{C}}^2/{\mathbb{Z}}_4$: }
So far, we have been considering the LG model without orbifolding action. To consider the mirror of ${\mathbb{C}}^2/{\mathbb{Z}}_4$ with generator $xy$, we need to consider $a_{ii}$ $i=1,2,3$. Since $a_{22}=1$ and $a_{33}$ is given by $a_{11}$, we only need to consider the equation for $a_{11}$, which is given by eq.(\ref{1}).
\section{${\mathbb{C}}^2/{\mathbb{Z}}_5$}\setcounter{equation}{0}
Here again, we first analyze the general LG model and at the end we comments on the orbifolded case.
The superpotential is given by
\begin{equation}
W={x^5 \over 5} + {y^5 \over 5} -txy.
\end{equation}
The chiral ring is give by $x^4-ty=0$ and $y^4-tx=0$. We order the
basis by dictionary order in charge pair $(i,j)$:
\begin{eqnarray}
&&{\cal R} =\{ xy, x^2y, x^3y, x^4y, xy^2, x^2y^2, x^3y^2, x^4 y^2,
xy^3, \nonumber \\
&&~~~~~~~~ x^2 y^3, x^3 y^3, x^4 y^3, xy^4, x^2 y^4, x^3 y^4, x^4 y^4
\}.
\end{eqnarray}
The $\eta_{ij}$ can be readily written and we avoid to writing it down.
The metric $g_{i\bar{j}}$ has 16 diagonal components which we denote
by $a_{ij}=\langle{x^{\bar i}y^{\bar j}}|{x^iy^j}\rangle, ~i,j=1,2,3,4$,
and two non-vanishing off diagonal elements $b, \bar b$ as before.
The coupling matrix $C_t$ in this basis is given by
\begin{equation}
C_t=\pmatrix{ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &
0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 &
0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & t & 0 & 0 & 0 & 0 &
0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 &
0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 &
0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 &
0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & t &
0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
1 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
0 & 1 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
0 & 0 & 1 \cr 0 & t^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
& 0 & 0 & 0 \cr 0 & 0 & t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
& 0 & 0 & 0 \cr 0 & 0 & 0 & t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
& 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & t^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &
0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & t^2 & 0 & 0 & 0 & 0 & 0 & 0
& 0 & 0 & 0 & 0
}.
\end{equation}
By use of the reality condition, we have 8 independent variables
out of 16+2 real variables and the rests can be written in terms of them:
\begin{eqnarray}
a_{13}&=&1/a_{42},~a_{23}=1/a_{32},~a_{33}=1/a_{22},~a_{43}=1/a_{12},
~a_{14}=1/a_{41},
\nonumber \\
a_{24}&=&1/a_{31},~a_{34}=1/a_{21},~a_{44}=1/a_{11}+|t|^4a_{11}/4,\nonumber \\
b&=&\frac{1}{2} t^2a_{11}.
\end{eqnarray}
Then the $tt^*$ equation becomes
\begin{eqnarray}
&&-\partial_{\bar{t}}\partial_t \log a_{11} = a_{11}^{-1}a_{22} -{1\over
4}|t|^4 a_{11} a_{22}, \nonumber \\
&&-\partial_{\bar{t}}\partial_t \log a_{22}= a_{22}^{-2}
-a_{11}^{-1}a_{22}
-{1\over 4} |t|^4 a_{11} a_{22} \label{a1122},\\
&&-\partial_{\bar{t}}\partial_t \log a_{21}=
-|t|^4 a_{21} a_{12} + a_{21}^{-1}a_{32}, \nonumber \\
&&-\partial_{\bar{t}}\partial_t \log
a_{12}=-|t|^4a_{21} a_{12} + a_{12}^{-1}a_{32}^{-1}, \nonumber \\
&&-\partial_{\bar{t}}\partial_t \log
a_{32}= a_{12}^{-1}a_{32}^{-1} -a_{21}^{-1}a_{32}, \nonumber \\
&&-\partial_{\bar{t}}\partial_t
\log a_{31} = -|t|^2 a_{31}a_{41} + a_{31}^{-1}a_{42}, \nonumber \\
&&-\partial_{\bar{t}}\partial_t \log a_{41}= -|t|^2a_{31} a_{41} +|t|^2
a_{41}^{-1}a_{42}^{-1}, \nonumber \\
&&-\partial_{\bar{t}}\partial_t \log a_{42}= |t|^2
a_{41}^{-1}a_{42}^{-1}-a_{31}^{-1}a_{42}.
\end{eqnarray}
The reality condition together with exchange
symmetry gives us only 4 independent equations
\begin{eqnarray}
&&-\partial_{\bar{t}}\partial_t\log a_{11}=a_{22}/a_{11}-a_{11} a_{22}
|t|^4/4, \label{11} \\
&&-\partial_{\bar{t}}\partial_t \log a_{22} =a_{22}^{-2}-a_{22}/a_{11}
-|t|^4
a_{11}a_{22}/4, \label{22}\\
&&-\partial_{\bar{t}}\partial_t \log a_{21} =-|t|^4 a_{21}^2 +
a_{21}^{-1}, \label{21}\\
&&-\partial_{\bar{t}}\partial_t \log a_{31} =-|t|^2 a_{31} + a_{31}^{-2},
\label{31}
\end{eqnarray}
as well as the predetermined values of some of them.
\begin{equation}
a_{32}=a_{23}=1, ~~ a_{41}=a_{14}=1.\end{equation}
Notice that all monomials
$x^2y, xy^2, x^3, y^3$ have NS charge 1 and these are the marginal operators. Above results show that the marginal operators do not evolve under the condensation of tachyon represented by $xy$.
To eliminate $t$ from above equations, let
\begin{equation}
a_{ij}:=|t|^{c_{ij}} b_{ij}, ~~~\zeta:=a t^b.\end{equation}
Then by using $\partial_{\bar{t}}\partial_t=
(ab)^2|t|^{2b-2}\partial_{\bar{\zeta}}\partial_\zeta$,
and by requiring that the eqs. (\ref{11}), (\ref{22}) are homogenous in $t$,
\begin{eqnarray}
2b-2&=&c_{22}-c_{11}=4+c_{22}+c_{11}, \nonumber \\ &=&-2c_{22}=-c_{11}+c_{22}, \end{eqnarray}
which give $c_{22}=-2/3$, $c_{11}=-2$ and $b=5/3$ with $ab=1$.
Based on this, we introduce $q_{11}$, $q_{22}$ by
\begin{equation}
a_{11}=2(3/5)^{2}|\zeta|^{-6/5}e^{-q_{11}},
~~~a_{22}=2^{1/3}(3/5)^{2/3}|\zeta|^{-2/5}e^{-q_{22}}, \label{a0011}
\end{equation}
and re-scale by $\zeta\to {2}^{1/3}\zeta$ to get
\begin{eqnarray}
&&\partial_{\bar{\zeta}}\partial_\zeta
q_{11}=e^{q_{11}-q_{22}}-e^{-(q_{11}+q_{22})}, \nonumber \\
&&\partial_{\bar{\zeta}}\partial_\zeta
q_{22}=e^{2q_{22}}-e^{q_{11}-q_{22}}-e^{-(q_{11}+q_{22})}. \label{q1122}
\end{eqnarray}
For eqs.(\ref{21}) and (\ref{31}),
\begin{equation}
4+2c_{21}=-c_{21}=2b-2=2+c_{31}=-2c_{31}.\end{equation} Then we have
$c_{31}=-2/3$, $c_{21}=-4/3$ and $b=5/3$.
We introduce $\tau$ and $Y(\tau), Z(\tau)$ by
\begin{equation}
\tau=|\zeta|^2,~~~ a_{21}=\({5}/{3}\)^{-4/5} \tau^{-2/5} e^{-Y} ~~{\rm
and}~
a_{31}=\({5}/{3}\)^{-2/5} \tau^{-1/5} e^Z .\label{a1020}\end{equation}
Then, both eq.(\ref{21}) eq.(\ref{31}) are reduced to
\begin{equation}
\partial_\tau(\tau \partial_\tau Y)=e^Y-e^{-2Y},
\end{equation}
\begin{equation}
\partial_\tau(\tau \partial_\tau Z)=e^Z-e^{-2Z},
\end{equation}
which are known as Bullough-Dodd equation which is a degenerate
Painleve III, which also appears in the case ${\mathbb{C}}^1/{\mathbb{Z}}_4 \to {\mathbb{C}}^1$
transition with $W=x^4-tx$.
The Bullough-Dodd equation
\begin{equation}
\partial_\tau(\tau\partial_\tau u) =e^u -e^{-2u}
\end{equation}
has been studied extensively and
the properties of the asymptotically regular solutions
were given in \cite{kit}. The solutions are parametrized
by four complex numbers $g_1$, $g_2$, $g_3$, and $s$ satisfying
\begin{equation}
g_1+g_2(1-s)+g_3=1,~~~g_2^2-g_1g_3=g_2.
\end{equation}
The asymptotic forms are given by
\begin{eqnarray}
&&\tau\to \infty;~~e^u \sim 1+\sqrt{ {3\over \pi} } {s\over 2}
(3\tau)^{-{1 \over 4}} e^{-2\sqrt{3\tau} }, ~~~g_1=g_2=0,~g_3=1, \nonumber \\
&&\tau \to 0;~~e^u =-{ \mu^2 \over { 2\tau \sin^2 \{ {i\over 2} [\mu
\ln\tau +\ln ( r_1{C_2\over C_0} ) ] \} } } \sim 2
\mu^2 r_1 {C_2\over C_0} \tau^{\mu-1}, \nonumber \\
&&~~~~~~~~s=1+ \cos[{2\pi\mu \over 3}],~~~r_1=g_3-g_1
+(1+\omega)(g_1-g_2),~~~\omega^\mp=e^{\mp{2\pi i \mu \over 3} },\nonumber \\
&&~~~~~~~~{C_2 \over C_0}=3^{-2\mu} {\Gamma(1-{\mu\over
3})\Gamma(1-{2\mu\over 3}) \over
{\Gamma(1+{\mu\over 3})\Gamma(1+{2\mu\over 3})} }.
\end{eqnarray}
Regularity of the metric as $\tau \to 0$ can fix $s$.
First let us consider $a_{21}\sim \tau^{-2/5}e^{-y}$.
From the regularity of $a_{21}$, we have
\begin{equation}
e^u=e^y\sim \tau^{-2/5},
\end{equation}
which gives
\begin{eqnarray} \mu=3/5,
&&s=1+2\cos(2\pi/5) \simeq 1.618, ~~~r_1=1,\nonumber \\
&&{C_2\over C_0}=3^{-6/5}(25/2){\Gamma(4/5)\Gamma(3/5) \over {
\Gamma(1/5)\Gamma(2/5)} }.
\end{eqnarray}
Similarly, for $a_{31}\sim \tau^{-1/5}e^Z$, we have
\begin{equation}
e^u=e^{-Z}\sim \tau^{-1/5},
\end{equation}
from which we get
\begin{eqnarray}
\mu =4/5, &&s=1+\cos(8\pi/15)\simeq 0.791,~~~r_1=1, \nonumber \\
&&{C_2\over C_0} =3^{-8/5}((15)^2/32)
{ \Gamma(11/15)\Gamma(7/15) \over {\Gamma(4/5)\Gamma(8/15) } }.
\end{eqnarray}
These completely fixes behaviors of solutions at both ends.
{\it Mirror of ${\mathbb{C}}^2/{\mathbb{Z}}_5$ :}
In this case, we only need to consider $a_{11},a_{22}$ since $a_{33}$ and $a_{44}$ are determined in terms of $a_{22}$ and $a_{11}$ respectively by the reality condition.
The $tt*$ equations for $a_{11},a_{22}$ are given by eqs.(\ref{a1122}) or eqs.(\ref{q1122}).
They can be identified as the ${\tilde B}_2=D^T(SO(5))$ Toda system. We will investigate the relation between the $tt*$ equations in the orbifold geometry and various Toda systems elsewhere.
The charge matrix $Q=g\partial_\tau g^{-1}-1$ with $\tau=\log\lambda$ can be calculated to be given by
\begin{equation} Q=\left(%
\begin{array}{cccc}
-\frac{3}{5}+a_{11}\partial_\tau a_{11}^{-1} & 0 & 0 & 0 \\
0& -\frac{1}{5}+a_{22}\partial_\tau a_{22}^{-1} & 0 & 0 \\
0 & 0 & \frac{1}{5}-a_{22}\partial_\tau a_{22}^{-1} & 0 \\
t^2 a_{11}\partial_\tau a_{11}^{-1} & 0 & 0 & \frac{3}{5}-a_{11}\partial_\tau a_{11}^{-1} \\
\end{array}%
\right).
\end{equation}
Notice that in terms of $q_{ij}$ and $\lambda(=\zeta)$,
and if we look at the $|\lambda|$ dependence only, the tachyon potential can be identified as
\begin{equation}
V=2Q_{max}= - \zeta\partial_{\zeta}q_{11}(\zeta).
\end{equation}
We expect that this is monotonically decreasing from the value $5/3$ at $t=0$ to $0$ at $t\to \infty$. So far, the mathematical literature on the solution to the equation is not available and the qualitative behavior we suggested above is from physical intuition that in the final stage of tachyon condensation there is no nontrivial chiral primaries with charge other than 0.
\section{Discussion}
In this paper, we calculated $ tt^*$ equations for ${\mathbb{C}}^2/{\mathbb{Z}}_n$ with n=3,4,5. In $n=3,4$ cases, they reduce to a Painleve III equation with different boundary conditions.
In $n=5$ case, they reduce to a simple Toda system whose explicit solutions are not known yet.
Non-orbifolded LG model associated to $n=5$ case involves a Bullough-Dodd equation.
As a limitation of this paper, we mention that we considered the string theories without GSO projection only. Since GSO projection does not provide a supersymmetry immediately in the orbifold background, there is not much point on restricting ourselves to GSO projected theory.
According to the rule given in \cite{gso}, $xy$ term considered in this paper is projected out for type II, and we need to consider the deformation by higher operator,
which result in highly non-trivial equations due to the algebraic complexity of reality condition.
Another very interesting case is the one where the daughter theory is also an orbifold. This also results in a highly non-trivial equations even for ${\mathbb{C}}^1/{\mathbb{Z}}_n$ background. We wish to report on these issues in later publications.
\vskip .5cm
\noindent {\bf \large Acknowledgement} \\
We want to thank J.Raeymaekers H.Yee for discussions and A. Sen for useful suggestions. This work is supported by the Korea Research Foundation Grant (KRF-2004-015-C00098).
|
1,116,691,499,380 | arxiv | \section{introduction}
During the recent years, tremendous progress in B physics has been
made through the fruitful interplay between theory and experiment.
The precise measurements of the $B$-meson decays can provide an
insight into very high energy scales via the indirect loop effects
of new physics beyond the standard model~(SM), which makes the study
of exclusive non-leptonic $B$-meson decays of great interest.
In the SM, the phenomenon of CP violation can be accommodated in an
efficient way through a complex phase entering the quark-mixing
matrix, which governs the strength of the charged-current
interactions of the quarks. This Kobayashi-Maskawa~(KM)~\cite{km}
mechanism of CP violation is the subject of detailed investigation
in these few decades. However, its origin remains unknown as it is
put into the standard model through the complex Yukawa couplings.
Moreover, the baryon asymmetry of the universe requires new sources
of CP violation. Many possible extensions of the SM in the Higgs
sector have been proposed~\cite{HS}, and it was suggested that CP
symmetry may break down spontaneously~\cite{Lee}. A consistent and
simple model, which provides a spontaneous CP violation mechanism,
has been constructed completely in a general two-Higgs-doublet
model~(2HDM)~\cite{YLW1,YLW2} without imposing the \textit{ad hoc}
discrete symmetry, which is now commonly called as type III 2HDM.
The type III 2HDM, which allows flavor-changing neutral
currents~(FCNCs) at tree level but suppressed by approximate $U(1)$
flavor symmetry, has attracted much more interests. It is known that
FCNCs are suppressed in low-energy experiments, especially for the
lighter two generation quarks. Thus, the type III 2HDM can be
parameterized in a way to satisfy the current experimental
constraints. On the other hand, constrains on the general 2HDM from
the neutral mesons mixing~($K^0-\bar{K}^0$, $D^0-\bar{D}^0$, and
$B^0-\bar{B}^0$)~\cite{Wolfenstein:1994jw, wumixings} and from the
radiative decays of bottom quark~\cite{b2gamma} have also been
studied in detail.
In recent years, there are many works about the $B$-meson decays
within the two-Higgs-doublet model. In Refs.~\cite{Xiao:2002mr,
B2PV}, the authors have studied the $B\to PP, PV$ decays~(with $P$
and $V$ denoting the pseudoscalar and vector mesons, respectively)
within the type III 2HDM. Since through the measurements of
magnitudes and phases of various helicity amplitudes, the charmless
hadronic $B\to VV$ decay modes can reveal more dynamics of exclusive
$B$ decays than $B\to PP$ and $B\to PV$ decays, in the present work
we are going to make a detailed study for $B\to VV$ decays within
the type III 2HDM by emphasizing on the new physics contributions.
It will be seen that this specific new physics has remarkable
effects on CP asymmetries, especially on the parameter $S_f$ for the
penguin-dominated decay modes. On the other hand, the new physics is
found to have very small contributions to the branching ratios and
the transverse polarizations. Furthermore, the polarization anomaly
observed in $B\to \rho K^*$ and $B\to \phi K^*$ modes can not be
improved in our current considered parameter spaces.
The paper is organized as follows: In section II, we first describe
the theoretical framework, including a brief introduction for the
two-Higgs-doublet model with spontaneous CP violation, the effective
Hamiltonian, as well as the decay amplitudes and CP violation
formulas, which are the basic tools to estimate the branching ratios
and CP asymmetries of $B$-meson decays. In section III, we list the
Wilson coefficients and the other relevant input parameters. Our
numerical predictions for the branching ratios, CP asymmetries and
longitudinal polarization fractions are presented in Section IV. Our
conclusions are presented in the last section.
\section{Theoretical Framework}
\subsection{Outline of the Two-Higgs-doublet Model}
Motivated solely from the origin of CP violation, a general
two-Higgs-doublet model with spontaneous CP violation~(type III
2HDM) has been shown to provide one of the simplest and attractive
models in understanding the origin and mechanism of CP violation at
the weak scale. In such a model, there exists more physical neutral
and charged Higgs bosons and rich CP violating sources from a single
CP phase of the vacuum. These new sources of CP violation can lead
to some significant phenomenological effects, which are promising to
be tested by the future $B$ factory and the LHCb experiments. In
this paper, we shall focus on the phenomenological applications of
the type III 2HDM on the two-body charmless hadronic $B\to VV$
decays.
The two complex Higgs doublets in the general 2HDM are generally
expressed as~\cite{YLW1,YLW2,LO,Kiers:1998ry}
\begin{equation}
\Phi_1=\left(\begin{array}{c} \phi_1^+\\ \phi_1^0
\end{array}\right), \, \, \, \Phi_2=\left(\begin{array}{c} \phi_2^+\\
\phi_2^0\end{array}\right).
\end{equation}
The corresponding Yukawa Lagrangian is given as
\begin{eqnarray}
\mathcal{L}_{Y}
&=&\eta_{ija}\bar{\psi}_{i,L}\tilde{\Phi}_aU_{j,R}+\xi_{ija}
\bar{\psi}_{i,L}\Phi_aD_{j,R}+h.c.,
\end{eqnarray}
where the parameters $\eta_{ija}$ and $\xi_{ija}$ are real, so that
the lagrangian is CP invariant. After the symmetry is spontaneously
broken down
\begin{equation}
\langle\phi_1^0\rangle=v_1 e^{i \alpha_1}, \, \, \,
\langle\phi_2^0\rangle=v_2 e^{\alpha_2},
\end{equation}
and the Goldstone particles have been eaten, the physical Higgs
bosons are
\begin{equation}
H_1=\frac{1}{\sqrt{2}}\left( \begin{array}{c} 0\\ v+\phi_1^0
\end{array}\right), \, \, \, H_2=\frac{1}{\sqrt{2}}\left(
\begin{array}{c} H ^+\\ \phi_2^0+i \phi_3^0 \end{array}\right).
\end{equation}
where $H^{\pm}$ are the charged scalar mass eigenstates, ($\phi_1^0,
\phi_2^0, \phi_3^0$) are generally not the mass eigenstates but can
be expressed as linear combinations of the mass eigenstates ($H, h,
A$).
Then the Yukawa part of the Lagrangian for physical particles can be
written as
\begin{equation}
\mathcal{L}_{Y}
=\eta_{ij}^U\bar{\psi}_{i,L}\tilde{H}_1U_{j,R}+\eta_{ij}^D
\bar{\psi}_{i,L}H_1D_{j,R}+\xi_{ij}^U\bar{\psi}_{i,L}\tilde{H}_2
U_{j,R}+\xi_{ij}^D\bar{\psi}_{iL}H_2D_{j,R}+h.c.,
\end{equation}
where
\begin{eqnarray}
\eta_{ij}^U&= & \eta_{ij1}\cos\beta+\eta_{ij2}e^{-\delta}\sin\beta,
\nonumber\\
\xi_{ij}^U&= &-\eta_{ij1}e^{-\delta}\sin\beta+\eta_{ij2}\cos\beta,
\nonumber \\
\eta_{ij}^D&= & \xi_{ij1}\cos\beta+\xi_{ij2}e^{-\delta}\sin\beta,
\nonumber\\
\xi_{ij}^D&= &-\xi_{ij1}e^{-\delta}\sin\beta+\xi_{ij2}\cos\beta,
\end{eqnarray}
and these couplings $\eta^U, \eta^D, \xi^U, \xi^D$ are generally
complex, which means CP violation. According to the CKM mechanism,
after diagonalizing the fermion terms' couplings $\eta^U$ and
$\eta^D$, the other couplings become
\begin{eqnarray}
\label{Yukawa}
\mathcal{L}_{Y}&=&\bar{U}_{i}\frac{m^U}{v}U_{R}(v+
\phi_1^0)+\bar{D}_{L}\frac{m^D}{v}D_{R}(v+\phi_1^0)\nonumber\\
&&+\bar{U}_{L}\tilde{\xi}^U U_{R}(\phi_2^0+i \phi_3^0)+\bar{D}_{L}
\hat{\xi}^U U_{R}H^-\nonumber\\
&&+\bar{U}_{L}\hat{\xi}^D D_{R}H^++\bar{D}_{L}\tilde{\xi}^DD_{R}
(\phi_2^0+i \phi_3^0)+h.c,
\end{eqnarray}
with
\begin{eqnarray}
\tilde{\xi}^{U,D}&=&(V_L^{U,D})^{-1} \xi^{U,D}V_R^{U,D},\nonumber\\
\hat{\xi}^{U}&=&\tilde{\xi}^{U} V_{\mbox{CKM}},\nonumber\\
\hat{\xi}^D&=&V_{\mbox{CKM}}\tilde{\xi}^{D}.
\end{eqnarray}
The Yukawa couplings may be parameterized as following
\begin{equation}\label{yukawa}
\tilde{\xi}_{ij}=\lambda_{ij}\frac{\sqrt{m_i m_j}}{v}.
\end{equation}
with $v$ the vacuum expectation value $v=246$ GeV.
\subsection{Effective Hamiltonian and decay amplitudes of $B\to VV$ decays}
Using the operator product expansion and the renormalization group
equation, the low energy effective Hamiltonian for charmless
hadronic $B$-meson decays with $\Delta B=1$ can be written as
\begin{eqnarray}\label{eff}
{\cal{H}}_{eff}= \frac{G_F}{\sqrt2}
\sum_{p=u,c} \!
V_{pb} V^*_{pq} \Big( C_1\,Q_1^p + C_2\,Q_2^p
+ \!\sum_{i=3,\dots, 16}\!\big[ C_i\,Q_i+ C_i^\prime\,Q_i^\prime \big]
\Big) + \mbox{h.c.} \, ,
\end{eqnarray}
where $C_i(\mu)~(i=1,\dots, 16)$ are the Wilson coefficients that
can be calculated by perturbative theory, and $Q_i$ are the quark
and gluon effective operators, with $Q_{1-10}$ and $Q_{11-16}$
coming from the SM and from the type III 2HDM, respectively. Their
explicit forms are defined as follows~(taking $b\to sq\bar q$
transition as an example)~\cite{huang}
\begin{eqnarray}\label{operators}
&&Q_1 = (\bar{s} u)_{V-A} (\bar{u} b)_{V-A},\nonumber\\
&&Q_2 = (\bar{s}_i u_j)_{V-A} (\bar{u}_j b_i)_{V-A},\nonumber\\
&&Q_{3(5)} = (\bar{s} b)_{V-A}\sum_{q} (\bar{q} q)_{V-(+)A},\nonumber\\
&&Q_{4(6)} = (\bar{s}_i b_j)_{V-A}\sum_{q}(\bar{q}_j q_i)_{V-(+)A},
\nonumber\\
&&Q_{7(9)} =\frac{3}{2}(\bar{s} b)_{V-A}\sum_{q} e_{q}
(\bar{q} q)_{V+(-)A},\nonumber\\
&&Q_{8(10)} =\frac{3}{2}(\bar{s}_i b_j)_{V-A}\sum_{q}e_{q}
(\bar{q}_j q_i)_{V+(-)A},\nonumber\\
&&Q_{11(13)} = (\bar{s}b)_{S+P}\sum_q\frac{m_q\lambda_{qq}^{*}
(\lambda_{qq})}{m_b} (\bar{q} q)_{S-(+)P},\nonumber\\
&&Q_{12(14)} = (\bar{s}_i b_j)_{S+P}
\sum_q\,\frac{m_q\lambda_{qq}^{*}
(\lambda_{qq})}{m_b}(\bar q_j \,q_i)_{S-(+)P},\nonumber\\
&&Q_{15} = \bar s \,\sigma^{\mu\nu}(1+\gamma_5) b
\sum_q\,\frac{m_q\lambda_{qq}}{m_b}
\bar q \sigma_{\mu\nu}(1+\gamma_5)\,q,\nonumber\\
&&Q_{16} = \bar s_i \,\sigma^{\mu\nu}(1+\gamma_5) \,b_j \sum_q\,
\frac{m_q\lambda_{qq}}{m_b} \bar q_j\, \sigma_{\mu\nu}(1+\gamma_5)
\,q_i,
\end{eqnarray}
where $(\bar q_1 q_2)_{V\pm A}=\bar q_1\gamma_\mu(1\pm\gamma_5)q_2$
and $(\bar q_1 q_2)_{S\pm P}=\bar q_1(1\pm\gamma_5)q_2$, with $q
u,d,s,c,b$, and $e_{q}$ is the electric charge number of $q$ quark.
The operators $Q^\prime_i$ in Eq. (\ref{operators}) are obtained
from $Q_i$ via exchanging $L \leftrightarrow R$, and we shall
neglect their effects in our calculations for they are suppressed by
a factor $m_s/m_b$ in model III 2HDM. The Wilson coefficients
$C_i~(i=1,\dots,10)$ have been calculated at leading
order~(LO)~\cite{buchalla, paschos} and at next-to-leading
order~(NLO)~\cite{sm:nlo} in the SM and also at LO in
2HDM~\cite{LO}, while $C_i~(i=11,\dots,16)$ at LO can be found in
Refs.~\cite{huang, Dai}.
Having defined the effective Hamiltonian $H_{eff}$ in terms of the
four-quark operators $Q_i$, we can then proceed to calculate the
hadronic matrix elements with the generalized factorization
assumption~\cite{hyc1,hyc2,ali,lucd} based on the naive
factorization approach.
For two-body charmless hadronic $B\to VV$ decays, the decay
amplitude of the local four fermion operators is defined as
\begin{equation}
A_h\equiv \frac{G_F}{\sqrt{2}}\langle V_1(h_1)V_2(h_2)|(\bar
q_2q_3)_{V\pm A}(\bar b q_1)_{V-A} |B\rangle,
\end{equation}
where $h_1$ and $h_2$ are the helicities of the final-state vector
mesons $V_1$ and $V_2$ with four-momentum $p_1$ and $p_2$,
respectively. Since the $B$ meson has spin zero, in the rest frame
of $B$-meson system, the two vector mesons have the same helicity
due to helicity conservation. Therefore three polarization states
are possible in $B\to VV$ decays with one longitudinal~($L$) and two
transverse, corresponding to helicities $h=0$ and $h=\pm$~(here
$h_1=h_2=h$), respectively. We define the three helicity amplitudes
as follows
\begin{eqnarray}
A_0=A(B\to V_1(p_1,\epsilon_1^0) V_2(p_2, \epsilon_2^0)),\nonumber\\
A_{\pm}=A(B\to V_1(p_1,\epsilon_1^{\pm}) V_2(p_2,
\epsilon_2^{\pm})).
\end{eqnarray}
We choose the momentum $\vec{p}_2$ to be directed in the positive
$z$-direction in the $B$-meson rest frame, and the polarization
four-vectors of the light vector mesons such that in a frame where
both light mesons have large momentum along the $z$-axis. They are
given by
\begin{eqnarray}
\epsilon_1^{\pm \mu}&=&\epsilon_2^{\mp\mu}=(0,\pm1,i,0)/\sqrt{2} ,
\nonumber\\
\epsilon_{1,2}^{0\mu}&=&p_{1,2}^\mu/m_{1,2},
\end{eqnarray}
where $m_1$ and $m_2$ are the masses of $V_1$ and $V_2$ mesons,
respectively. Using the definitions for decay constants and form
factors~\cite{Beneke:2000wa}, the tree-level hadronic matrix
elements of the effective operators $Q_i$ can be decomposed as the
following two amplitudes
\begin{equation}\label{amplitude}
A_h=\mathcal{V}_h+\mathcal{T}_h,
\end{equation}
with
\begin{eqnarray}
\mathcal{V}_h &\equiv& \langle V_1(p_1,\epsilon_1^h)|V-A|B\rangle
\langle V_2(p_2,\epsilon_2^h)|V-A|0\rangle,\nonumber\\
\mathcal{T}_h &\equiv& \langle
V_1(p_1,\epsilon_1^h)|\sigma^{\mu\nu}(1+\gamma^5)|B\rangle \langle
V_2(p_2,\epsilon_2^h)|\sigma_{\mu\nu} (1+\gamma^5)|0\rangle.
\end{eqnarray}
Here, for simplicity, we have omitted the quark spinors in the
corresponding current operators in the above definitions. The three
polarization amplitudes for $\mathcal {V}^h$ and $\mathcal {T}^h$
can be further written as
\begin{eqnarray}
\label{factorization_fumula1}
&&\mathcal{V}_0=i f_{V_2}(m_B^2-m_1^2-m_2^2) A^{V_1}_0,\nonumber\\
\label{factorization_fumula2} &&\mathcal{V}_{\pm}=i f_{V_2} m_2
\left[A^{V_1}_1(m_1+m_B) \mp V^{V_1} \frac{2m_B |p_c|}
{m_B+m_1}\right],\nonumber\\ \label{factorization_fumula3}
&&\mathcal{T}_0=0,\nonumber\\ \label{factorization_fumula4}
&&\mathcal{T}_{\pm}=2 i f_{V_2}^{\perp}\bigg[ 2 T^{V_1}_1 m_B |p_c|
\mp T^{V_1}_2 (m_B^2-m_1^2)\bigg].
\end{eqnarray}
From the amplitude given by Eq.~(\ref{amplitude}), the branching
ratio for $B\to VV$ decays then reads
\begin{equation}
Br(B\to VV)=\frac{\tau_B|p_c|}{8\pi
m^2_B}\left(|A_0|^2+|A_+|^2+|A_-|^2\right),
\end{equation}
where $\tau_B$ is the lifetime of the $B$ meson, and $p_c$ is the
center of mass momentum of either final-state meson with
\begin{equation}
|p_c|=\frac{\sqrt{\left[m_B^2-(m_1+m_2)^2\right]
\left[m_B^2-(m_1-m_2)^2\right]}}{2 m_B}.
\end{equation}
In order to compare the relative size of the three different
helicity amplitudes, we can define the longitudinal polarization
fraction as
\begin{equation}
f_L=\frac{|A_0|^2}{|A_0|^2+|A_+|^2+|A_-|^2},
\end{equation}
which measures the relative strength of the longitudinally
polarization amplitude in a given decay mode.
\subsection{CP-violating asymmetries in $B\to VV$ decays}
Since there are abundant CP violation sources in the
two-Higgs-doublet model, it is also necessary and interesting for us
to discuss CP asymmetries in $B\to VV$ decays.
Firstly, for charged $B^{\pm}$-meson decays, there is only one
simple type of CP violating asymmetry, which detects direct CP
violation
\begin{eqnarray}
\mathcal{A}_{\mathcal{CP}}\equiv\frac{\Gamma(B^+\to
f^+)-\Gamma(B^-\to f^-)}{\Gamma(B^+\to f^+)+\Gamma(B^-\to f^-)}
\end{eqnarray}
For neutral $B$-meson decays, there is another type of CP violation
coming from the mixing between $B_q^0-\overline B_q^0$~(here $q=d$
or $s$)
\begin{eqnarray}
|B_q^0(t)\rangle&=&g_+(t)|B_q^0\rangle+\frac{q}{p}g_-(t)|\overline
B^0_q\rangle, \nonumber\\
|\overline B^0_q(t)\rangle&=&\frac{p}{q}g_-(t)|B_q^0\rangle+g_+
|\overline B^0_q\rangle.
\end{eqnarray}
In this case, there are in general four amplitudes which can be
expressed as~\cite{PheCP,WuCP,Yao:2006px}
\begin{eqnarray}\label{generala}
A_f=\langle f|H_{eff}|B_q^0\rangle &, & \overline A_f=\langle
f|H_{eff}|\overline B_q^0\rangle, \nonumber\\
\overline A_{\bar f}=\langle \bar f|H_{eff}|\overline
B_q^0\rangle&,& A_{\bar f} =\langle\bar f |H_{eff}|B_q^0\rangle.
\end{eqnarray}
For the $B_d-\overline B_d$ and $B_s-\overline B_s$ systems, the
following approximations can be made
\begin{eqnarray}\label{approx}
\mbox{both $B_d$ and $B_s$ systems}: \Big|\frac{q}{p}\Big|\sim1;
\qquad \mbox{only $B_d$ system}: \Delta\Gamma\sim 0.
\end{eqnarray}
Using the decay amplitudes and the approximations listed in
Eqs.~(\ref{generala}) and (\ref{approx}), the time-dependent decay
probabilities for $B_d$ system can then be written as
\begin{eqnarray}
\Gamma(B_d^0(t)\to f)=\frac{|A_f|^2(1+|\lambda_f|^2)}{2}e^{-\Gamma
t}\left\{1+C_f \cos(\Delta mt)-S_f\sin(\Delta m t)\right\},
\nonumber\\
\Gamma(\overline B_d^0(t)\to
f)=\frac{|A_f|^2(1+|\lambda_f|^2)}{2}e^{-\Gamma t}\left\{1-C_f
\cos(\Delta mt)+S_f\sin(\Delta m t)\right\},\label{mixcpbd}
\end{eqnarray}
while for $B_s$ system, we have
\begin{eqnarray}
\Gamma(B_s^0(t)\to f)&=&\frac{|A_f|^2(1+|\lambda_f|^2)}{2}e^{-\Gamma
t} \Big[\cosh\Big(\frac{\Delta\Gamma t}{2}\Big)+D_f \sinh\Big(\frac
{\Delta\Gamma t}{2}\Big)\nonumber\\
&& \hspace{3.5cm} +C_f \cos(\Delta m
t)-S_f\sin(\Delta m t)\Big],\nonumber\\
\Gamma({\overline B_s^0}(t)\to f)&=&\frac{|{A}_f|^2(1+|{\lambda_
f|^2)}}{2}e^{-\Gamma t}\Big[\cosh\Big(\frac{\Delta\Gamma
t}{2}\Big)+D_f \sinh\Big(\frac{\Delta\Gamma t}{2}\Big)\nonumber\\
&& \hspace{3.5cm} -C_f \cos(\Delta{m} t)+S_f\sin(\Delta{m} t)\Big],\label{mixcpbs}
\end{eqnarray}
where $\Gamma$ is the average decay width, $\Delta\Gamma$ and
$\Delta m$ are the width and mass difference, respectively. The
other quantities are defined as
\begin{eqnarray}
\lambda_f\equiv\frac{q}{p}\frac{\bar{A}_f}{A_f},\hspace{1cm}
D_f\equiv\frac{2\mbox{Re}(\lambda_f)}{1+|\lambda_f|^2},\nonumber\\
C_f\equiv\frac{1-|\lambda_f|^2}{1+|\lambda_f|^2},\hspace{1cm}
S_f\equiv\frac{2\mbox{Im}(\lambda_f)}{1+|\lambda_f|^2}.
\end{eqnarray}
From Eqs.(\ref{mixcpbd}) and (\ref{mixcpbs}), we can get:
\begin{eqnarray}
\mathcal{A_{CP}}(B_d\to f)&=&-C_f \cos \Delta m t+S_f \sin\Delta m t,\nonumber\\
\mathcal{A_{CP}}(B_s\to f)&=&\frac{-C_f\cos \Delta mt+S_f\sin \Delta mt}{\cosh\left(\frac{\Delta \Gamma t}{2}\right)+D_f\sinh\left(\frac{\Delta \Gamma t}{2}\right)}.
\end{eqnarray}
\section{Input parameters}
The theoretical predictions in our calculations depend on many input
parameters, such as the Wilson coefficients, the CKM matrix
elements, the hadronic parameters, and so on. Here we present all
the relevant input parameters as follows.
It has been shown from $B_{d,s}^0-\overline B_{d,s}^0$ mixings that
the parameters $|\lambda_{cc}|$ and $|\lambda_{ss}|$ in
Eq.~(\ref{operators}) can reach to be around 100~\cite{mphya20},
while their phases are not well constrained. In our present work we
simply fix the phases to be $\pi/4$, and this choice will not cause
any trouble in our numerical results. For the parameters
$\lambda_{tt}$ and $\lambda_{bb}$, the constraints come mainly from
the experiments for $B-\bar{B}$ mixing, $\Gamma(b\to s \gamma)$,
$\Gamma(b\to c\tau\bar{\nu}_\tau$), $\rho_0$, $R_b$, $B\to P V$, and
the electric dipole moments~(EDMS) of the electron and
neutron~\cite{Atwood,huang,LO,Dai,B2PV}. Based on the above
analyses, we choose the following three typical parameter spaces
which are allowed by the present experiments and have been adopted
for the $B\to PV $ decays\cite{B2PV}
\begin{eqnarray*}
\mbox{Case\quad A}:\quad |\lambda_{tt}|&=&0.15; \quad |\lambda_{bb}|=50,\\
\mbox{Case\quad B}:\quad |\lambda_{tt}|&=&0.3;\ \quad |\lambda_{bb}|=30,\\
\mbox{Case\quad C}:\quad |\lambda_{tt}|&=&0.03; \quad
|\lambda_{bb}|=100,
\end{eqnarray*}
and $\theta_{tt}+\theta_{bb}=\pi/2$. For the Higgs masses and the
Wilson coefficients of $C_{1,\dots,10}$ corresponding to the SM, we
use the results listed in the paper~\cite{B2PV}, while for the
Wilson coefficients in the type III 2HDM, we redefine them as
$\tilde{C}_{11,\dots,16}= \frac{m_s\lambda^{(*)}_{ss}}{m_b}
C_{11,\dots,16}$ in order to compare the contributions from those
operators in SM, here the factor $\frac{m_s\lambda^{(*)}_{ss}}{m_b}$
is associated with the operators in 2HDM, the numerical values for
$\tilde{C}_{11,\dots,16}$ are listed in Table \ref{wilsonc}.
\begin{table}[ht]
\begin{center}
\caption{The Wilson coefficients $\tilde{C}_{11,\dots,16}=
\frac{m_s\lambda^{(*)}_{ss}}{m_b} C_{11,\dots,16}$ in $b \to s$
transition at $\mu =m_b=4.2~\rm{GeV}$ in 2HDM.} \label{wilsonc}
\doublerulesep 0.8pt \tabcolsep 0.15in\vspace{0.2cm}
\begin{tabular} {c c c c c }
\hline\hline
& Case A & Case B & Case C \\
\hlin
$\tilde{C}_{11} $&$-0.0085+0.012 i$&$-0.0085+0.018 i$&$-0.010+0.012 i$\\
$\tilde{C}_{12} $ &$0$&$0$&$0$\\
$\tilde{C}_{13} $ &$-0.0030-0.0049 i$&$-0.0052-0.0069 i$&$-0.0029-0.0052 i$\\
$\tilde{C}_{14} $ &$-0.000060-0.00010 i$&$-0.00011-0.00014 i$&$-0.000059-0.00010 i$\\
$\tilde{C}_{15} $ &$0.000033+0.000055 i$&$0.000058+0.000078 i$&$0.000032+0.000059 i$\\
$\tilde{C}_{16} $&$-0.00010-0.00017 i$&$-0.00018-0.00024 i$&$-0.0001-0.00018 i$\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
As for the CKM matrix elements, we shall use the Wolfenstein
parametrization~\cite{ckm} with the values~\cite{Yao:2006px}:
$A=0.8533\pm0.0512$, $\lambda=0.2200\pm0.0026$, $\bar
\rho=0.20\pm0.09$, and $\bar \eta=0.33\pm0.05$.
For the hadronic parameters, the decay constants, and the form
factors, we list them in Tables.~\ref{input} and \ref{formfactors},
respectively.
\begin{table}[ht]
\begin{center}
\caption{The hadronic input parameters~\cite{Yao:2006px} and the
decay constants taken from the QCD sum rules~\cite{ball3} and
Lattice theory~\cite{lattice}.} \label{input} \doublerulesep 0.8pt
\tabcolsep 0.10in \vspace{0.2cm}
\begin{tabular}{cccccc} \hline\hline
$\tau_{B^\pm}$&$\tau_{B_d}$ & $\tau_{B_s}$&$M_{B_d}$ &$M_{B_s}$
&$m_b$ \\ \hline
$1.638$ps &$1.528$ps&$1.472$ps &$5.28$GeV &$5.37$GeV &$4.2$GeV\\
$m_t$&$m_u$&$m_d$&$m_c$&$m_s$&$m_{\rho^0}$\\
$174$GeV&$3.2$MeV&$6.4$MeV&$1.1$GeV&$0.105$GeV&$0.77$GeV\\
$m_{\rho^{\pm}}$&$m_{\omega}$&$m_{\phi}$&$m_{K^{*\pm}}$&$m_{K^{*0}}$
&$\Lambda_{QCD}$\\
$0.77$GeV&$0.782$GeV&$1.02$GeV&$0.892$GeV&$0.896$GeV&$225$MeV\\
$f_{\rho}$&$f_{\omega}$&$f_{K^*}$&$f_{\phi}$&$f_{\rho}^T$
&$f_{\omega}^T$\\
$0.205$GeV&$0.195$GeV&$0.217$GeV&$0.231$GeV&$0.147$GeV&$0.133$GeV\\
$f_{K^*}^T$&$f_{\phi}^T$& & \\
$0.156$GeV&$0.183$GeV\\ \hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
\begin{center}
\caption{The relevant $B\to V$ transition form factors at $q^2=0$
taken from the light-cone sum
rules~(LCSR)~\cite{prd71014029,Wu:2006rd}.}\label{formfactors}
\doublerulesep 0.8pt \tabcolsep 0.15in \vspace{0.2cm}
\begin{tabular}{c c c c c c c }\hline\hline
decay channel& $V$& $A_0$ & $A_1$ & $A_2$ & $T_1$ &$T_3$
\\\hline
$B \to \rho$ & 0.323 &0.303 & 0.242 &0.221 &0.267 &0.176 \\
$B \to \omega$ &0.293 &0.281 & 0.219 &0.198 &0.242 &0.155 \\
$B\to K^*$ &0.411 &0.374 &0.292 &0.259 &0.333 &0.202 \\
$B_s \to \bar{K}^*$ & 0.311 & 0.363 & 0.233 &0.181 &0.26 & 0.136\\
$B_s \to \phi$&0.434 & 0.474 & 0.311 &0.234 & 0.349 &0.175 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\section{Numerical results and discussions}
In this section, we shall classify the $28$ channels of $B^+$, $B^0$
and $B_s$ decays into two light vector mesons according to the
reliability of the calculation for various observables, which is
motivated by the dominated contributing operators. We shall give our
predictions for the branching ratios, the CP asymmetries, and the
longitudinal polarization fractions both in the SM and in the 2HDM.
Comparisons with the current experiment data, if possible, are also
made.
Before moving to the detailed discussions, some general observations
of new physics effects on $B\to VV$ decays should be made. As can be
seen from Eqs.~(\ref{yukawa}) and (\ref{operators}), the
contributions of new physics operators $O_{11,\dots,16}$ are always
proportional to the factor ${m_q}/{v}$. Thus, they are severely
suppressed for the first generation quarks. In this case, for $B\to
\rho K^*, \omega K^*, \rho \rho, \omega \rho, \omega \omega$ and
$B_s\to \rho K^*, \omega K^*, K^* K^*, \rho \phi , \omega \phi$
decay channels, we can safely ignore the contributions from those
new operators. Note that the new physics still has effects on the
Wilson Coefficients $C_{1-10}$. On the other hand, for $B\to \phi
K^*, \phi \rho, \phi \omega$ and $B_s \to \phi K^*, \phi \phi$ decay
channels, since these are all induced by $b\to q s\bar s$~($q=d, s$)
transitions, we could not ignore the new operators' contributions
any more in this case. In the general factorization approach, it is
impossible to produce a vector meson via the scalar and/or
pseudoscalar currents from the vacuum state, and hence the new
operators $Q_{11}$ and $Q_{13}$ have no contributions to $B\to VV$
decays. Moreover, from the results listed in Table \ref{wilsonc}, it
can be seen that all the contributing new operators $Q_{12, 14, 15,
16}$ have only very small~(even zero) Wilson coefficients. It is
therefore expected that the new physics will have very small effects
on the branching ratios and transverse amplitudes~(hence on the
transverse polarization fractions) of $B\to VV$ decays.
\subsection{CP-averaged branching ratios and direct CP violation}
According to different decay modes, we shall give our predictions
for the branching ratios and direct CP violations one by one.
(i), color-allowed tree-dominated decays. Our predictions for the
CP-averaged branching ratios and the direct CP asymmetries are
presented in Table~\ref{tree}. From the numerical results, we can
see that the branching ratios are all at $10^{-5}$ order, and the
direct CP asymmetries are all very small since the penguin amplitude
contributions are much smaller than the ones from the tree diagrams.
Most predictions within the SM are consistent with the current
experiment data, and the new physics has very small effects on these
types of decays.
\begin{table}[htbp]
\caption{The CP-averaged branching ratios~(in unit of $10^{-6}$)
~(first line) and the direct CP violations~(second line) for the
color-allowed tree-dominant processes both in the SM and in the type
III 2HDM. Case A-C stand for the three different parameter spaces
listed in Section III.}\label{tree}
\begin{center}
\doublerulesep 0.8pt \tabcolsep 0.15in
\begin{tabular}{lcccccc}\hline \hline
Decay modes & Case A &Case B &Case C &SM &Exp. \\
\hline\hline
$B^+ \to \rho^+ \rho^{0}$&14.59&14.59&14.59&15.53&18.2$\pm$3.0\\
&-0.004&-0.004&-0.004&-0.002&-0.08$\pm$0.13\\
$B^0 \to \rho^+ \rho^-$&26.33&25.93&26.73&27.49&$24.2^{+3.1}_{-3.2}$\\
&-0.043&-0.043&-0.042&-0.035&\\
$B^+ \to \rho^+\omega$&12.66&12.47&12.85&13.97&$10.6^{+2.6}_{-2.3}$\\
&-0.042&-0.043&-0.042&-0.034&0.04$\pm$0.18\\
$B_s \to \rho^+ K^{*-}$ &36.88 &36.32 &37.44 &38.50 &\\
&-0.043 & -0.043 & -0.042 &-0.035 &\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
(ii), color-suppressed tree-dominated decays. The numerical results
are given in Table~\ref{ctree}, it is interesting to note that the
branching ratios will generally become smaller after including the
new physics contributions except for the $B\to \rho^0\rho^0$ mode.
Furthermore, there are big direct CP violations in these decay
processes except for the $B^+\to \rho^0\omega$ mode, and the new
physics has more effects on the direct CP asymmetries than on the
branching ratios through the Wilson coefficient functions, although
there are no new operator contributions to the hadronic matrix
elements in this type decays within our approximations. Compared to
Case A and Case C, Case B has the biggest corrections to the CP
asymmetries of the SM.
\begin{table}[htbp]
\caption{The same as Table~\ref{tree} but for color-suppressed
tree-dominant processes.}\label{ctree}
\begin{center}
\doublerulesep 0.8pt \tabcolsep 0.15in
\begin{tabular}{lcccccc}\hline \hline
Decay modes & Case A &Case B &Case C &SM &Exp. \\
\hline\hline
$B^0 \to \rho^0 \rho^0$&0.0814&0.0897&0.0754&0.065&0.86$\pm$0.28\\
&0.176&0.218&0.119&0.153&\\
$B^+ \to \omega \omega$&0.112&0.110&0.115&0.160&$<$4.0\\
&-0.117&-0.088&-0.144&-0.207&\\
$B_s \to \rho^0 \bar{K}^{*0}$&0.081 &0.090 &0.073 &0.092 &$<7.67\times 10^{-4}$\\
&0.176 & 0.218 & 0.119 & 0.153 &\\
$B_s \to \omega \bar{K}^{*0}$&0.183 &0.180 &0.187 &0.262 &\\
&-0.167 & -0.088 & -0.144 & -0.207 &\\
$B^+ \to \rho^0 \omega$&0.024&0.024&0.024&0.076&$<$1.5\\
&-0.063&-0.063&-0.063&-0.035&\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
(iii), penguin-dominated decays. We may divide such decays into two
types: $\Delta S=1$ and $\Delta D=1$ decay modes. They are
corresponding to the upper and the lower parts in
Table~\ref{penguin}, respectively. From the numerical results, we
can see that all the eleven $\Delta S=1$ decay modes have branching
ratios up to $10^{-6}$ or even to $10^{-5}$ order, since they
involve the relative large CKM matrix elements $V^*_{ts}$, while the
$\Delta D=1$ ones have much smaller branching ratios of order of
$10^{-7}$ due to the smaller CKM matrix elements $V^*_{td}$. For
$B\to \omega K^*$ and $B_s \to \phi\phi$ decay modes, our
predictions for the branching ratios with including the new operator
contributions have similar results as the ones within the SM, which,
however, are not quite consistent with the current experimental
data; the numerical results for $B\to \omega K^{\ast}$ modes are
larger than the current experiment limit, and the prediction for
$B_s\to \phi\phi$ is about two times larger than the present data.
For the other decay modes, our predictions for the branching ratios
are in general agreement with the data. As for the direct CP
asymmetries, there are big CP violations in some decay modes, and
the new physics can lead to remarkable effects. Our predictions are
consistent to the data in all these decay modes.
\begin{table}[htbp]
\caption{The same as Table~\ref{tree} but for the penguin-dominated
decay modes. The upper and the lower parts correspond to $\Delta
S=1$ and $\Delta D=1$ processes, respectively.}\label{penguin}
\begin{center}
\doublerulesep 0.8pt \tabcolsep 0.15in
\begin{tabular}{l cccccc}\hline \hline
Decay modes & Case A &Case B &Case C &SM &Exp. \\
\hline\hline
$B^+ \to \rho^+ K^{*0}$&7.169&7.409&7.027&7.287&9.2$\pm$1.5\\
&0.084&0.117&0.049&0.018&-0.01$\pm$0.16\\
$B^+ \to \rho^0 K^{*+}$&5.853&6.229&5.526&5.575&$<$6.1\\
&0.184&0.196&0.169&0.122&$0.20^{+0.32}_{-0.29}$\\
$B^0 \to \rho^0 K^{*0}$&6.396&6.513&6.324&6.245&5.6$\pm$1.6\\
&0.054&0.073&0.033&0.018&0.09$\pm$0.19\\
$B^0 \to \rho^- K^{*+}$&6.046&6.738&5.445&5.571&$<$12\\
&0.295&0.301&0.283&0.199&\\
$B^0 \to \omega K^{*0}$&3.412&3.513&3.351&3.498&$<$2.7\\
&0.078&0.107&0.048&0.024&\\
$B^+ \to \omega K^{*+}$&3.247&3.5697&2.965&3.123&$<$3.4\\
&0.265&0.274&0.251&0.176&\\
$B^0 \to \phi K^{*0}$&9.276&9.704&9.221&9.318&9.5$\pm$0.8\\
&0.045&0.081&-0.002&0.020&-0.01$\pm0.06$\\
$B^+ \to \phi K^{*+}$&9.867&10.32&9.775&9.979&10.0$\pm$1.1\\
&0.039&0.074&-0.013&0.020&-0.01$\pm$0.08\\
$B_s \to \phi \phi$&28.99 &30.34 &28.64 &28.85 &$14^{+8}_{-7}\times 10^{-6}$\\
&0.054 & 0.089 &0.006 & 0.020 &\\
$B_s \to \bar{K}^{*0} K^{*0}$&9.303 &9.614 &9.118 &9.456 &$<1.681\times 10^{-3}$\\
&0.084 & 0.117 & 0.049 & 0.018 &\\
$B_s \to K^{*+} K^{*-}$ &8.404 &9.366 &7.569 &7.744 &\\
&0.295 & 0.302 & 0.283 & 0.199 &\\
\hline
$B^0 \to \bar{K}^{*0} K^{*0}$&0.410&0.420&0.413&0.408&$0.49^{+0.17}_{-0.14}$\\
&-0.092&-0.061&-0.133&-0.145&\\
$B^+ \to K^{*+} K^{*0}$&0.439&0.450&0.443&0.437&$<2.2$\\
&-0.092&-0.061&-0.133&-0.145&\\
$B_s \to \phi \bar{K}^{*0}$&0.517 &0.532 &0.521 &0.526 &$<1.013\times 10^{-3}$\\
& -0.094 & -0.056 & -0.145 & -0.161 &\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
(iv), electroweak penguin or QCD flavor singlet dominated decays. As
can be seen from Table~\ref{ewpenguin}, this type of decays are
expected to have smaller branching ratios due to the large
cancelations among the different Wilson coefficients. Although there
are new operator contributions in $B\to \rho \phi$ and $\omega \phi$
decay modes, the predicted branching ratios are still small. The
direct CP asymmetries for these decays are all small, and the new
physics effects on these observables are not prominent. Due to the
lack of accurate experimental data, we couldn't compare our
predictions with the data yet.
\begin{table}[htbp]
\caption{The same as Talbe~\ref{tree} but for the electroweak
penguin or QCD flavor singlet dominated decays.}\label{ewpenguin}
\begin{center}
\doublerulesep 0.8pt \tabcolsep 0.15in
\begin{tabular}{lcccccc}\hline \hline
Decay modes & Case A &Case B &Case C &SM &Exp. \\
\hline\hline
$B^+ \to \rho^+ \phi$&0.0054&0.0054&0.0054&0.0043&$<16$\\
&-0.011&-0.011&-0.011&-0.014&\\
$B^0 \to \rho^0 \phi$&0.0025&0.0025&0.0025&0.0020&$<$13\\
&-0.011&-0.011&-0.011&-0.014&\\
$B^0 \to \omega \phi$&0.0022&0.0022&0.0022&0.0017&$<1.2$\\
&-0.011&-0.011&-0.011&-0.014&\\
$B_s \to \rho^0 \phi$&0.796 &0.796 &0.796 &0.687 &$<6.17\times 10^{-4}$\\
&0.0048 & 0.0048 &0.0048 & 0.0039 &\\
$B_s \to \phi \omega$ &0.038 &0.038 &0.038 &0.045 &\\
&0.020 & 0.020 &0.020 & 0.018 &\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
(v), the pure annihilation decays. Only six decays belong to this
class, namely $B^0\to K^{*+}K^{*-}$, $B^0\to\phi\phi$,
$B_s\to\rho^+\rho^-$, $B_s\to \rho^0\rho^0$, $B_s\to \rho^0\omega$,
and $B_s\to \omega\omega$. Due to the lack of the information for
the $V_1\to V_2$ transition form factor at large momentum transfers,
we shall not consider them in details in this paper.
\subsection{Time-dependent CP violating parameters $C_f$, $S_f$ and $D_f$}
Since there are abundant CP violating sources in type III 2HDM, it
is expected that there are relatively large CP violations in 2HDM
than in the SM. Using the relevant formulas given in section II, we
can predict the time-dependent CP asymmetries in neutral $B_d$ and
$B_s$ decays, with the numerical results given in Tables~\ref{Bd}
and \ref{Bs}, respectively.
From these two tables, it is seen that, for $B^0\to \rho^+\rho^-,
\rho^0 \phi$ and $\omega \phi$ decay modes, the new physics has
hardly any effects on the parameters $C_f$ and $S_f$, even though
there are new operators contributions in $B^0\to \rho^0 \phi$ and
$\omega \phi$ decay modes. On the other hand, the new physics has
remarkable effects on the other decay modes, especially on $B^0\to
\omega \omega$ one~(for this mode the new physics can even change
the sign of the parameter $S_f$). Furthermore, different parameter
spaces also have remarkable effects on these CP violation
parameters.
For $B_s$ system, there are new operator contributions only in
$B_s\to \phi\phi$ mode. As is expected, the new physics has
remarkable influence on the parameters $C_f$, $S_f$, and $D_f$. For
the other four decay modes, although there are no new operator
contributions, the new physics still has big effects on the
parameter $S_f$, but small effects on $C_f$ and $D_f$.
\begin{table}[htbp]
\caption{The time-dependent CP asymmetry parameters $C_f$~(first
line) and $S_f$~(second line) for $B_d$ decays both in the SM and in
the type III 2HDM. Case A-C stand for the three different parameter
spaces listed in Section III.}\label{Bd}
\begin{center}
\doublerulesep 0.8pt \tabcolsep 0.15in
\begin{tabular}{lccccc}\hline \hline
Decay modes & Case A &Case B &Case C &SM \\
\hline\hline
$B^0 \to \rho^+ \rho^-$&0.043&0.043&0.042&0.035\\
&-0.95&-0.95&-0.95&-0.95\\
$B^0 \to \rho^0 \rho^0$&-0.18&-0.22&-0.12&-0.15\\
&0.97&0.92&0.99&0.89\\
$B^0 \to \omega \rho^0$&0.063&0.063&0.063&0.029\\
&-0.61&-0.61&-0.62&-0.97\\
$B^0 \to \phi \rho^0$&0.011&0.011&0.011&0.014\\
&0.70&0.70&0.70&0.70\\
$B^0 \to \omega \phi$&0.011&0.011&0.011&0.014\\
&0.70&0.70&0.70&0.70\\
$B^0 \to \omega \omega$&0.12&0.09&0.14&0.21\\
&0.53&0.65&0.40&-0.18\\
$B^0 \to K^{*0} \bar{K}^{*0}$&0.092&0.061&0.13&0.15\\
&0.85&0.92&0.75&0.57\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
\caption{The time-dependent CP asymmetry parameters $C_f$~(first
line), $S_f$~(second line), and $D_f$~(third line) for $B_s$ decays
both in the SM and in the type III 2HDM }\label{Bs}
\begin{center}
\doublerulesep 0.8pt \tabcolsep 0.15in
\begin{tabular}{lccccc}\hline \hline
Decay modes & Case A &Case B &Case C &SM \\
\hline\hline
$B_s \to \phi \rho^0$&-0.005&-0.005&-0.005&-0.004\\
&0.052&0.052&0.052&0.14\\
&0.99&0.99&0.99&0.99\\
$B_s \to \phi \omega$&-0.020&-0.020&-0.020&-0.018\\
&0.23&0.23&0.23&0.49\\
&0.97&0.97&0.97&0.87\\
$B_s \to \phi \phi$&-0.054&-0.090&-0.060&-0.020\\
&0.33&0.49&0.14&-0.004\\
&0.94&0.87&0.99&1.0\\
$B_s \to K^{*+} K^{*-}$&-0.30&-0.30&-0.28&-0.20\\
&0.92&0.95&0.88&0.79\\
&0.25&0.12&0.39&0.57\\
$B_s \to \bar{K}^{*0} K^{*0}$&-0.085&-0.12&-0.049&-0.018\\
&0.31&0.45&0.15&-0.003\\
&0.95&0.88&0.99&1.0\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\subsection{The polarization in $B\to \rho K^*$ and $\phi K^*$ decays}
Motivated by the polarization anomaly observed by the
BarBar~\cite{babar}, Belle~\cite{belle} and CDF~\cite{cdf}
experiments, we shall study the polarization in $B\to VV$ decays,
especially in $B\to \rho K^*$ and $\phi K^*$ decays in this section.
One important point that should be noted is that the predictions for
the branching ratios of $B\to \rho K^*$ and $\phi K^*$ modes are
well consistent with the experiment data, which means that if we
want to solve the observed polarization anomaly, we need to find
some way to reduce the longitudinal amplitude and enhance transverse
ones simultaneously. Many studies have been made to try to provide
possible resolutions to the anomaly both within the
SM~\cite{Kagan:2004uw,Li:2004ti,Colangelo:2004rd,Beneke:2006hg} and
in various new physics
models~\cite{Yang:2004pm,Chen:2005mka,Chang:2006dh}. Here we only
concentrate on the longitudinal polarization fraction and the main
results are listed in Table~\ref{polarization}.
It is noted that the polarization anomaly could be well resolved by
introducing the tensor operators
$O_{T1}=\bar{s}\sigma^{\mu\nu}(1+\gamma^5) b \, \bar{s}
\sigma_{\mu\nu}(1+\gamma_5) s$ and
$O_{T8}=\bar{s}_i\sigma^{\mu\nu}(1+\gamma^5) b_j \, \bar{s}_j
\sigma_{\mu\nu}(1+\gamma_5) s_i$ in Ref.~\cite{Chang:2006dh}. It is
interesting to see that these two operators have similar forms as
$Q_{15}$ and $Q_{16}$ in Eq.~(\ref{operators}). However, from the
numerical results given by Table~\ref{polarization}, we can see that
the predicted longitudinal polarization fraction $f_L$ for these
decay modes in the type III 2HDM is almost the same as the one
within the SM. Although there are new operator contributions in
$B\to \phi K^*$ modes, we still can not resolve the polarization
anomaly observed in this decay mode. This is due to the fact that
the strength of new operators in 2HDM is severely suppressed by the
factor $m_q\lambda_{qq}/m_b$. Moreover, as has already been
mentioned in the beginning of this section, the Wilson coefficients
of these new operators are very small, which also result in the
small effects on the transverse amplitudes.
\begin{table}[htpb]
\caption{The longitudinal polarization fractions $f_L$ for $B\to
\rho K^*$ and $\phi K^*$ decay modes. Case A-C stand for the three
different parameter spaces in the type III
2HDM.}\label{polarization}
\begin{center}
\doublerulesep 0.8pt \tabcolsep 0.15in
\begin{tabular}{lccccc}\hline \hline
Decay modes &SM & Case A &Case B &Case C &Exp. \\
\hline\hline
$B^+ \to \rho^+ K^{*0}$&0.91&0.91&0.91&0.91&$0.48\pm0.08$\\
$B^0 \to \rho K^{*0}$&0.95&0.95&0.93&0.93&$0.57\pm0.12$\\
$B^+ \to \phi K^*$&0.89&0.89&0.89&0.89&$0.50\pm0.05$\\
$B^0 \to \phi K^{*0}$&0.89&0.89&0.89&0.89&$0.491\pm0.032$\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
For the other $B\to VV$ decay modes, the predictions for
longitudinal polarization fractions are always about $0.90\sim0.95$.
For simplify, we shall not list the results in details anymore.
In conclusion, adopting the current parameter spaces and with the
general factorization method, we could not resolve the polarization
anomaly observed in $B\to \rho K^*$ and $\phi K^*$ modes within the
SM and 2HDM.
\section{conclusions}
Using the general factorization approach, we have studied all the
$B\to VV$ decay modes except for pure annihilation decay channels
both within the SM and in the two-Higgs-doublet model. From the
numerical results given in the previous section, we can see that:
for the branching ratios, our predictions are generally well
consistent with the current experimental data expect for the $B_s\to
\phi\phi$ decay mode, and the new physics has margin or even
negligible effects on this observable. However, the new physics can
give remarkable contributions to the CP asymmetry parameters $C_f$
and $S_f$, especially to $S_f$ in the penguin-dominated decay modes.
Unfortunately, our predictions for the longitudinal polarization
fractions of $B\to \rho K^*$ and $\phi K^*$ decay modes in 2HDM are
still as large as the ones in the SM, which are much larger than the
experimental data. Some new mechanisms may be needed to improve
those discrepancies.
For simplicity, in this paper we have neglected the contributions
from annihilation and exchange diagrams, although they may play a
significant rule in some decay channels. In our numerical
calculations, we have only considered three possible parameter
spaces for the type III 2HDM. Also we have totally neglected the
first generation Yukawa couplings and the off-diagonal matrix
elements of the Yukawa coupling matrix, in order to eliminate the
FCNC at tree level. However, it is possible that the FCNC involving
the third generation quarks still exists at tree level, making the
constraints less stronger. In a word, we do not exclude the
possibility to improve the predictions by using the other
factorization methods with the annihilation and exchange diagram
contributions included, by choosing other parameters spaces, or even
by introducing additional fourth-generation quarks~\cite{Wu:2004kr}.
In conclusion, we have shown that the new Higgs bosons in the type
III 2HDM with spontaneous CP violation can have significant effects
on some charmless $B\to VV$ decays, especially for the
penguin-dominated decay modes, which can be used as good signals to
test the SM and to explore new physics from more precise
measurements in the future $B$-factory experiments.
\begin{acknowledgments}
This work was supported in part by the National Science Foundation
of China (NSFC) under the grant 10475105, 10491306, and the Project
of Knowledge Innovation Program (PKIP) of Chinese Academy of
Sciences.
\end{acknowledgments}
|
1,116,691,499,381 | arxiv | \section{Introduction}
Black holes can be described by two observable parameters, mass and
spin. To date there are over twenty stellar mass black holes with
dynamically constrained mass (for a review see McClintock \& Remillard
2006); however for just a handfull of these systems do we have any
measurements of their spin.
Shafee et al. (2006) and McClintock et al. (2006) have reported values
of the dimensionless spin parameter, {\it a} for 4U 1543-47 of
0.7-0.85, GRO J1655-40 of 0.65-0.75, and GRS 1915+105 of 0.98-1. More
recently Liu et al. (2008) reported a value of $0.77\pm{\thinspace
0.05}$ for M33{\thinspace X-7}. Their approach relies on modelling
the thermal continuum seen in the thermal X-ray spectra and requires
sources to be selected in the high-soft state (for a review on
spectral states see McClintock \& Remillard 2006. Any powerlaw
emission is then minimal and the spectrum resembles a quasi-blackbody
continuum. Furthermore, precise measurements of the mass and distance
to the black hole, as well as the inclination of the system, are
essential.
We have recently reported precise measurements of the spin parameter
in \hbox{\rm GX 339-4}\ (Reis et al. 2008; Miller et al. 2008a) using the reflection
signatures in the spectrum. These reflection features arise due to
reprocessing of hard X-ray by the cooler accretion disk (Ross \&
Fabian 1993), and consist of fluorescent and recombination emission
lines as well as absorption features. In the inner regions of an
accretion disk the various reflection features become highly distorted
due to relativistic effects and Doppler shifts. The shape of the
prominent Fe-$K\alpha$\ fluorescent line, and more importantly the extent of
its red wing, can give a direct indication of the radius of the
reflecting material from the black hole (Fabian et al. 1989, 2000;
Laor 1991). The stable circular orbit around a black hole extends down
to the radius of marginal stability, ${\it r}_{\rm ms}$, which depends on the spin
parameter (e.g Bardeen et al. 1972). A major advantage of using
reflection features to obtain the spin of the black hole is that these
features are completely independent of black hole mass and distance
and can thus be used for any system where both or either of these
parameters are unknown (for a review see Miller 2007). For \hbox{\rm GX 339-4}\ we
found that for both the very-high and the low-hard state the accretion
disk extends to the innermost stable circular orbit, ${\it r}_{\rm ms}$\ at a radius
of $\approx 2.05$${\it r}_{\rm g}$, where ${\it r}_{\rm g}$$ = GM / c^{2}$ (Reis et al. 2008;
Miller et al. 2008a). This implies a spin of $\approx 0.935$ for \hbox{\rm GX 339-4}.
In this paper we use the method adopted by Reis et al. (2008) which
uses a specially developed spectral grid, \rm{\sc REFHIDEN}\ (Ross \& Fabian
2007), to obtain the spin parameter of {\rm SWIFT} \hbox{\rm J1753.5-0127}\ and {\rm
GRO}{\thinspace\hbox{\rm J1655-40}}. \hbox{\rm J1753.5-0127}\ was first detected in hard X-rays by
the Burst Alert Telescope (BAT) on the {\it SWIFT}\
satellite on 2005 May 30 (Palmer et al. 2005). Using {\it RXTE}\ and {\it XMM-Newton}\
data, Miller, Homan \& Miniutti (2006, hereafter M06) showed the
presence of a cool ($kT \approx 0.2 \hbox{$\rm\thinspace keV$}$) accretion disk extending
close to the ${\it r}_{\rm ms}$\ in the low-hard state (LHS) of the system. The
presence of this cool accretion disk was later confirmed by Ramadevi
\& Seetha (2007) and more recently by Soleri et al. (2008) using
multiwavelength observations of the source.
GRO{\thinspace}\hbox{\rm J1655-40}\ was discovered by the Burst and Transient Source
Experiment (BATSE) on-board of the Compton Gamma Ray Observatory
(CGRO) on 1994 July 27 (Zhang et al. 1994). The mass of the compact
object has been dynamically constrained to $>6.0\hbox{$\rm\thinspace M_{\odot}$}$ (Orosz \&
Bailyn 1997). {\it ASCA}\ observation of \hbox{\rm J1655-40}\ from 1994 through 1996
provided the first clear detection of absorption-line features in the
source (Ueda et al. 1998; see also Miller et al. 2008b). Using
archival {\it ASCA}\ data, Miller et al. (2004) showed evidence of highly
skewed, relativistic lines, and suggested an inner-radius of
reflection of $\approx 1.4$${\it r}_{\rm g}$\ which would indicate a rapidly
spinning black hole. This was later confirmed by Diaz-Trigo et
al. (2007, hereafter DT07) using simultaneous {\it XMM-Newton}\ and {\it INTEGRAL}\
observations of \hbox{\rm J1655-40}\ during the 2005 outburst. The high spin suggested
by these authors is in contrast with the spin parameter reported by
Shafee et al. (2006) of 0.65--0.75 using the thermal thermal continuum
method. In the following section we detail the observation, analyses
procedure and results.
\section{Observation and Data reduction}
{\it SWIFT}\ \hbox{\rm J1753.5-0127}\ was observed in its low-hard state by {\it XMM-Newton}\ for 42\hbox{$\rm\thinspace ks$},
starting on 2006 March 24 16:00:31 UT and simultaneously by {\it RXTE}\ for
2.3\hbox{$\rm\thinspace ks$}\ starting at 17:26:06 UT (M06). The {\it EPIC}{\rm-pn}\ camera (Struder et
al. 2001) was operated in ``timing'' mode with a ``medium'' optical
blocking filter. For GRO \hbox{\rm J1655-40}, observations were made by {\it XMM-Newton}\ for
23.9\hbox{$\rm\thinspace ks$}\ on 2005 March 18 15:47:13 (hereafter Obs 1) and again on 2005
March 27 08:43:59 (hereafter Obs 2) for 22.3\hbox{$\rm\thinspace ks$}\ (DT07). The source
was found to be in the high-soft state and was observed with the
{\it EPIC}{\rm-pn}\ camera in ``burst'' mode with a ``thin'' optical blocking
filter. Starting with the unscreened level 1 data files for all
aforementioned observation we extracted spectral data using the latest
{\it XMM-Newton Science analyses System v} 7.1.0 (SAS). For \hbox{\rm J1753.5-0127}\
events were extracted in a stripe in RAWX (20-56) and the full RAWY
range. RAWX 30--43 and RAWY 5--160 was used for \hbox{\rm J1655-40}. For both sources,
bad pixels and events too close to chip edges were ignored by
requiring ``FLAG = 0'' and ``PATTERN$\le4$''. The energy channels were
grouped by a factor of five to create a spectrum. Background spectra
were extracted for both sources from an adjacent region of similar
RAWY and RAWX. In the case of \hbox{\rm J1655-40}\ the background is negligible due to
the high source flux and it was not used for the analyses that
follows. In both cases, individual response files were created using
the SAS tools {\sc rmfgen} and {\sc arfgen}. The {\it EPIC}{\rm-pn}\ resulted in
a total good-exposure time of 40.1\hbox{$\rm\thinspace ks$}\ for \hbox{\rm J1753.5-0127}. Due to the low
duty-cycle of the ``burst'' mode (3\%) the total good exposure time
for \hbox{\rm J1655-40}\ is $\approx 0.7$ and 0.6\hbox{$\rm\thinspace ks$}\ for Obs 1 and 2
respectively. {\it RXTE}\ data for \hbox{\rm J1753.5-0127}\ were reduced in the standard way
using the {\it HEASOFT v 6.0} software package. We used the ``Standard
2 mode'' data from PCU-2 only. Standard screening resulted in a net
Proportional Counter Array (PCA) and High-Energy X-Ray Timing
Experiment exposures of 2.3 and 0.8\hbox{$\rm\thinspace ks$}\ respectively. To account for
residual uncertainties in the calibration of PCU-2, we added 0.6 per
cent systematic error to all energy channels. The {\it HEXTE-B}
cluster was operated in the ``standard archive mode''. The {\it FTOOL}
{\sc grppha} was used to give at least 20 counts per spectral bin.
We restrict our spectral analyses of the {\it XMM-Newton}\ {\it EPIC}{\rm-pn}\ data for \hbox{\rm J1753.5-0127}\
to the 0.7--10.0\hbox{$\rm\thinspace keV$}\ band. For \hbox{\rm J1655-40}\ we use 0.7--9.0\hbox{$\rm\thinspace keV$}\ due to the
uncertain calibration above 9.0\hbox{$\rm\thinspace keV$}\ for the {\it EPIC}{\rm-pn}\ burst mode. The
PCU-2 and HEXTE spectrum are restricted to 5.0--25.0 and
20.0--100.0\hbox{$\rm\thinspace keV$}\ band respectively. A Gaussian line at 2.2--2.3\hbox{$\rm\thinspace keV$}\
is introduced when fitting the {\it EPIC}{\rm-pn}\ spectrum due to the presence
of an instrumental feature in this energy range that resembles an emission
line. This feature is likely to be caused by Au M-shell edges and Si
features in the detector. All parameters in fits involving different
instruments were tied and a normalisation constant was
introduced. {\sc XSPEC}{\rm\thinspace v\thinspace 12.4.0}\ (Arnaud 1996) was used to analyse all
spectra. The quoted errors on the derived model parameters correspond
to a 90 per cent confidence level for one parameter of interest
($\Delta\chi^{2}=2.71$ criterion).
\section{analysis and results}
Fig. 1 shows the data/model ratio for \hbox{\rm J1753.5-0127}\ fitted with a simple
absorbed powerlaw (\rm{\sc PHABS}\ model in \hbox{\sc XSPEC}\ with ${\it N}_{\rm H}$\ fixed at
$1.72\times10^{21} \hbox{$\rm\thinspace cm$}^{-2}$) in the energy range 2.5--5.0 and
8.0--10.0\hbox{$\rm\thinspace keV$}\ and then extended to the full energy range. As first
noted by Miller, Homan \& Miniutti (2006) and more recently confirmed
by Soleri et al. (2008), \hbox{\rm J1753.5-0127}\ shows clear indications of the presence
of a cool accretion disk with a temperature of $\approx0.2$\hbox{$\rm\thinspace keV$}\ in
its low-hard state. The presence of an excess at $\sim 6.9\hbox{$\rm\thinspace keV$}$ is
indicative of Fe-$K\alpha$\ emission. Similarly for \hbox{\rm J1655-40}, we fit the 0.7-4.0
and 7.0-9.0\hbox{$\rm\thinspace keV$}\ energy range with a power law and a further
multi-color disk blackbody (MCD; Mitsuda et al. 1984) modified by
absorption in the interstellar medium. The value of ${\it N}_{\rm H}$\ was kept
constant for the two observations. The data/model ratio for \hbox{\rm J1655-40}\
extended to the full energy range is shown in Fig. 2. The presence of
a broad Fe-$K\alpha$\ line emission extending to just over 3.0\hbox{$\rm\thinspace keV$}\ as well
as absorption features are clearly seen. Fitting the low/hard state of
\hbox{\rm J1753.5-0127}\ with a simple absorbed powerlaw results in an unacceptable fit
(Table 1) with various residuals in both the soft and the Fe-$K\alpha$\ energy
range.
The presence of a soft disk excess and broad Fe-$K\alpha$\ emission line is
usually modelled phenomenologically with a combination of a
multi-color disk component such as \rm{\sc DISKBB}\ (Mitsuda et al. 1984) and
a relativistic line such as the \rm{\sc LAOR}\ line profile (Laor 1991). This
combination of components can give robust spin measurements when the
presence of the Fe-$K\alpha$\ emission line is significantly above the
continuum. However it should be noted that it is only an approximation
since the identification of ${\it r}_{\rm in}$, as determined from \rm{\sc LAOR}\ assumes a
hard wired spin parameter of $a=0.998$. Furthermore, the \rm{\sc LAOR}\ line
profile is a phenomenological model for a relativistic {\it emission
line}. The effects that extreme gravity have on the reflection
signatures are not limited to the Fe-$K\alpha$\ line profile and thus a more
thorough method to constrain the spin would involve modelling all of
the reflection signatures present in the spectra. In what follows, we
use the reflection model developed by Ross \& Fabian (2007, \rm{\sc REFHIDEN})
to model all the reflection signatures as well as the disk
blackbody emission in a self-consistent manner. This approach was
detailed in Reis et al. (2008) for the galactic black hole \hbox{\rm GX 339-4}.
\begin{figure}
\rotatebox{270}{
\resizebox{!}{8.cm}
{\includegraphics{ratio_1.ps}}
}
\caption{Data/model ratio for \hbox{\rm J1753.5-0127}\ obtained by fitting the energy
range 2.5--5.0 and 8.0--10.0\hbox{$\rm\thinspace keV$}\ with an absorbed powerlaw. It is
clear that a semi-blackbody component as well as an iron reflection
signature is present in the low-hard state of \hbox{\rm J1753.5-0127}. The data have
been rebinned for plotting purposes only.}
\end{figure}
\begin{figure}
\rotatebox{270}{
\resizebox{!}{8cm}
{\includegraphics{ratio_2.ps}}
}
\caption{Data/model ratio for \hbox{\rm J1655-40}\ using a simple absorbed power law
and disk blackbody fitted in the energy range 0.7--4.0 and 7.0--9.0
\hbox{$\rm\thinspace keV$}. Obs 1 and 2 are shown in black and red respectively. The
presence of a broad Fe-$K\alpha$\ line and complex absorption is clearly
seen. The data have been rebinned for plotting purposes only.}
\end{figure}
The reflection features produced by the illuminated surface layer of
an accretion disk are largely dependent on the ionisation state of the
disk, and thus an important quantity is the ionisation parameter
$\xi=4\pi F_h/H_{den}$, where $F_h$ is the hard X-ray flux
illuminating a disk with a hydrogen density $H_{den}$ (Matt, Fabian \&
Ross 1993). The \rm{\sc REFHIDEN}\ reflection model incorporates the
importance of the thermal emission from the disk mid-plane in
determining the ionisation state and thus reflection features. Whereas
previous reflection models such as the Constant Density Ionised Disk
(CDID; Ballantyne, Ross \& Fabian 2001) and \rm{\sc REFLIONX}\ (Ross \& Fabian
2005) vary over the ionisation parameter $\xi$ and does not account
for thermal emission, in \rm{\sc REFHIDEN}\ both the number density of
hydrogen, ${\it H}_{\rm den}$, and the ratio of the the total flux
illuminating the disk to the total blackbody flux emitted by the disk
($R_{illum/BB}$) are implicit parameters. The value of {\it kT} for
the blackbody entering the surface layer from below and the power-law
photon index are further parameters in the model. The disk reflection
spectra is convolved with relativistic blurring kernel \rm{\sc KDBLUR}, which
is derived from the code by Laor (1991). The parameters for the
blurring kernel are the emissivity index $q_{in}$, where the
emissivity ($\epsilon_{r}$) is described by a powerlaw-profile such
that $\epsilon_{r}=r^{-q_{in}}$, disk inclination $i$, the inner disk
radius ${\it r}_{\rm in}$, and the outer disk radius which was fixed at 400${\it r}_{\rm g}$. The
power law index of \rm{\sc REFHIDEN}\ is the same as that of the hard
component.
\begin{table}
\begin{center}
\caption{Results of fits to {\it XMM-Newton}\ {\it EPIC}{\rm-pn}\ data for \hbox{\rm J1753.5-0127}\ with both a
simple absorbed powerlaw and the self-consistent reflection model
\rm{\sc REFHIDEN}. }
\begin{tabular}{lcccccccccc}
\hline
\hline
Parameter & Simple & \rm{\sc REFHIDEN} \\
\hline
${\it N}_{\rm H}$\ $(10^{21} \hbox{$\rm\thinspace cm$}^{-2})$ & $1.54\pm0.13$ & $1.52^{+0.02}_{-0.03}$ \\
$\Gamma$ & $1.631\pm0.003$ & $1.54^{+0.01}_{-0.04}$ \\
$N_{\rm PL}$ & $0.0591\pm0.0002$ & $0.050^{+0.003}_{-0.004}$ \\
{\it kT} (\hbox{$\rm\thinspace keV$})& & $0.193^{+0.003}_{-0.002}$ \\
$H_{\rm den}(\times10^{20}{\thinspace {\rm H}\hbox{$\rm\thinspace cm$}^{-3})} $& & $1.40^{+0.13}_{-0.27}$ \\
$R_{Illum/BB}$ && $9.6^{+0.4}_{-0.6}$ \\
$N_{\rm{\sc REFHIDEN}}$ && $0.0013^{+0.0004}_{-0.0006}$ \\
$q_{\rm in}$ && $4^{+6}_{-1}$ \\
${\it r}_{\rm in}$ (${\it r}_{\rm g}$) && $3.1^{+0.7}_{-0.6}$ \\
{\it i} (\rm deg) & &$55^{+2}_{-7}$ \\
$\chi^{2}/\nu$ & $2191 .9/1859$ & $1839.5/1852$ \\
\hline
\hline
\end{tabular}
\end{center}
\small Notes.- The self-consistent model is described in \hbox{\sc XSPEC}\ as PHABS$\times$KDBLUR$\times$(PL+REFHIDEN). The normalisation of each component is referred to as {\it N}. All errors refer to the 90\% confidence range for a single parameter.
\end{table}
\subsection{Self-consistent reflection and disk emission }
$\bullet$ {\bf SWIFT J1753.5-0127}: The model provides an excellent
fit to the data for \hbox{\rm J1753.5-0127}\ with $\chi^{2}/\nu =1839.5/1852$. Table 1
details the parameter values with all errors corresponding to the 90
per cent confidence range. The data/model spectrum is shown in Fig. 3a
with the fit extended to 100.0\hbox{$\rm\thinspace keV$}\ using {\it RXTE}\ data shown in the
inset. A normalisation constant was added to account for flux mismatch
between the instruments. The best fit model prior to gravitational
blurring is shown in Fig. 3b. The value of the inner radius obtained
from the gravitational blurring of the reflection features is
constrained to be ${\it r}_{\rm in}$$ = 3.1^{+0.7}_{-0.6}$${\it r}_{\rm g}$. The strong constraint
on the value of ${\it r}_{\rm in}$\ can be better appreciated in Fig. 4a, where the
90 per cent confidence level is shown in the $\chi^{2}$ plot obtained
with the ``steppar'' command in \hbox{\sc XSPEC}. Fig. 4b shows a similar
constraint obtained for the inner accretion disk inclination of \hbox{\rm J1753.5-0127}.
\begin{figure*}
\centering
\subfigure[]
{
\rotatebox{270}{
\resizebox{!}{7.cm}
{\includegraphics{figure_finalspec.ps}
}}}
\hspace{1cm}
\subfigure[]
{
\rotatebox{270}{
\resizebox{!}{6.cm}
{\includegraphics{model.ps}}
}}
\caption{{\rm (a)}: Data/model ratio for \hbox{\rm J1753.5-0127}\ in its low-hard
state. The model assumes a powerlaw emissivity profile and
constitutes of a powerlaw and the disk reflection model
\rm{\sc REFHIDEN}. The inset shows the model extended to 100.0\hbox{$\rm\thinspace keV$}\ using
{\it RXTE}\ PCA (red) and HEXTE (green). Data have been rebinned for
plotting purposes only. {\rm (b)}: Best-fit model prior to
gravitational blurring showing the reflection features. The
total model, powerlaw and reflection components are shown in black,
blue and red respectively.}
\end{figure*}
\begin{figure*}
\centering
\subfigure[]
{
\rotatebox{270}{
\resizebox{!}{5.cm}
{\includegraphics{steppar_rin_zoomedout.ps}
}}}
\hspace{1cm}
\subfigure[]
{
\rotatebox{270}{
\resizebox{!}{5.cm}
{\includegraphics{steppar_inclinationzoomedout.ps}}
}}
\caption{{\rm (a)}: $\chi^{2}$ vs ${\it r}_{\rm in}$\ plot for \hbox{\rm J1753.5-0127}. A value of
${\it r}_{\rm in}$$=3.1^{+0.7}_{-0.6}$${\it r}_{\rm g}$\ is found at the 90 per cent confidence
level for one parameter of interest ($\Delta\chi^{2} = 2.71$
criterion) shown by the solid horizontal line. {\rm (b)}: Similar plot
for the disk inclination. A value for the disk inclination of
$55^{+2}_{-7}$ is found at the 90 per cent confidence level for one
parameter of interest. }
\end{figure*}
\begin{figure*}
\centering
\subfigure[]
{
\rotatebox{270}{
\resizebox{!}{5.2cm}
{\includegraphics{contour_q_rin.ps}
}}}
\hspace{1cm}
\subfigure[]
{
\rotatebox{270}{
\resizebox{!}{5.2cm}
{\includegraphics{contour_inc_rin.ps}}
}}
\hspace{1cm}
\subfigure[]
{
\rotatebox{270}{
\resizebox{!}{5.2cm}
{\includegraphics{contour_hden_rin.ps}}
}}
\label{fig:sub}
\caption{{\rm (a)}: Emissivity versus inner radius contour plot for \hbox{\rm J1753.5-0127}. The
68, 90 and 95 per cent confidence range for two parameters of
interest are shown in black, red and green respectively. {\rm (b)}:
Similar plot for the inclination versus inner radius and {\rm (c)}:
Hydrogen density versus inner radius. It can be seen that for the
full range of the emissivity, inclination and hydrogen density, the
inner radius is constrained between approximately 2--4${\it r}_{\rm g}$\ at the 90
per cent confidence level for two parameters. }
\end{figure*}
As can be seen from Table 1, the value of the emissivity index,
$q_{in}$ has been poorly constrained. A value of 3 is expected for a
standard accretion disk (Reynolds \& Nowak 2003) with steeper values
usually being interpreted as emission from a small, compact, centrally
located X-ray source, such as expected from the base of a jet
(Miniutti \& Fabian 2004). In order to investigate any degeneracy
between the value of the inner radius and the unconstrained emissivity
we explored their parameter space using the ``contour'' command in
\hbox{\sc XSPEC}. All parameters except for $\Gamma$ and $R_{Illum/BB}$ were
free to vary. Fig. 5a shows the 68, 90 and 95 per cent contour for two
parameters of interest. Although the emissivity index is poorly
constrained, it can be seem from Fig. 5a that the value of the inner
radius is not strongly affected by this uncertainty. In what follows
we will thus freeze the value of the emissivity at the best fit value
shown in Table 1. Fig. 5b and 5c shows similar contour plots for both
inclination and $H_{den}$ versus inner radius respectively. For a
large range of inclination and $H_{den}$ the value of ${\it r}_{\rm in}$\ remains
approximately between 2--4${\it r}_{\rm g}$\ with a best fit value of approximately
$3$${\it r}_{\rm g}$, in accordance to that shown in Fig. 4a. Assuming that ${\it r}_{\rm in}$$ =
3.1^{+0.7}_{-0.6}$${\it r}_{\rm g}$\ (Fig. 4a) is the same as the radius of marginal
stability ${\it r}_{\rm ms}$, we obtain a dimensionless spin parameter of $
0.76^{+0.11}_{-0.15}$ for \hbox{\rm J1753.5-0127}\ in its low-hard state (Fig. 6).
\begin{figure}
\centering
{
{\includegraphics[height=5.5cm, width=7.5cm]{radius_spin_graph1.ps}}
}
\vspace{0.cm}
\caption {Plot of the innermost stable circular orbit vs dimensionless
spin parameter (solid curve). The constraints imposed by the
innermost radius obtained for \hbox{\rm J1753.5-0127}\ are shown by the intersection of
the solid line with the vertical region. The solid region shows
similar constrains for the lower limit of the spin parameter in
\hbox{\rm J1655-40}. Based on this analysis, a dimensionless spin parameters of
$0.76^{+0.11}_{-0.15}$ is found for \hbox{\rm J1753.5-0127}\ and a lower limit of
$\approx0.90$ is found for \hbox{\rm J1655-40}.}
\end{figure}
\begin{table*}
\begin{center}
\caption{Results of fits to {\it XMM-Newton}\ {\it EPIC}{\rm-pn}\ data for \hbox{\rm J1655-40}\ with self-consistent reflection model \rm{\sc REFHIDEN}. }
\begin{tabular}{lccccccccc}
\hline
\hline
Parameter & Model 1 & & Model 2 & & Model 3 & & Model 4 & \\
& Obs 1 & Obs 2 & Obs 1& Obs 2& Obs 1& Obs 2 & Obs 1& Obs 2\\
${\it N}_{\rm H}$ & $0.76^{+0.01}_{-0.02}$&...& $0.759\pm0.001$ &...&$0.765\pm0.001$&...& $0.742^{+0.004}_{-0.002}$&... \\
$\Gamma$ & $2.59^{+0.03}_{-0.01}$&$2.88^{+0.03}_{-0.02}$ & $2.72\pm0.08$ & $2.91^{+0.04}_{-0.053}$& $2.54^{+0.02}_{-0.01}$ & $2.99^{+0.01}_{-0.02}$& $2.54^{+0.06}_{-0.02}$ & $2.69^{+0.03}_{-0.02}$ \\
$N_{\rm PL}$ & $10.3^{+0.2}_{-0.4}$ & $10.4^{+0.3}_{-0.1}$& $9.0^{+0.1}_{-0.2}$ & $11\pm3$& $10.4^{+0.1}_{-0.2}$ & $9.46^{+0.23}_{-0.02}$& $8.18^{+0.16}_{-0.12}$ & $9.35^{+0.14}_{-0.06}$ \\
{\it kT} (\hbox{$\rm\thinspace keV$}) & $0.846^{+0.002}_{-0.007}$ & $0.84\pm0.01$ & $0.778\pm0.009$&$0.848^{+0.003}_{-0.002}$& $0.776^{+0.002}_{-0.008}$&$0.780^{+0.009}_{-0.001}$& $0.804\pm0.004$&$0.816^{+0.007}_{-0.006}$ \\
$H_{\rm den}$& $9.97^{+0.03}_{-0.56}$&$10^{+0.0}_{-0.2}$& $9.0^{+0.3}_{-0.7}$&$8^{+1}_{-2}$& $7.4^{+0.3}_{-0.1}$&$10^{+0}_{-1}$& $10.0^{+0.0}_{-0.6}$&$9.0^{+0.7}_{-0.6}$ \\
$R_{Illum/BB}$ & $0.93^{+0.02}_{-0.04}$&$0.20^{+0.03}_{-0.01}$ & $0.39^{+0.05}_{-0.10}$&$0.6\pm0.2$& $0.11^{+0.01}_{-0.03}$&$0.01^{+0.20}$ &$1.0^{+0.03}_{-0.02}$&$0.169^{+0.074}_{-0.007}$ \\
$N_{ref}$ &$3.3^{+0.4}_{-0.2}$&$0.29^{+0.06}_{-0.09}$ &$1.4^{+0.1}_{-0.3}$&$0.7\pm 0.2$ & $0.46^{+0.31}_{-0.02}$&$0.02^{+0.10}$ & $3.42^{+0.01}_{-0.04}$&$0.251^{+0.002}_{-0.007}$ \\
$q_{\rm in}$ & $10.0_{-0.3}$&$4.5\pm0.1$ & $10.0^{+0.0}_{-0.3}$&$10.0^{+0.0}_{-0.5}$ & $10.0^{+0.0}_{-0.4}$&$7.75^{+0.06}_{-0.20}$ & $3.87\pm0.08$&$2.75^{+0.03}_{-0.02}$ \\
${\it r}_{\rm in}$ (${\it r}_{\rm g}$) & $1.86^{+0.20}_{-0.02}$& ... & $1.31\pm0.01$& ... & $1.38\pm0.01$& ...&$2.17^{+0.15}_{-0.17}$ &...\\
{\it i} & $50\pm1$&... & 70(f)&... &70(f)&... &$30^{+5}_{-10}$&... \\
$E_{Gabs1}$ (\hbox{$\rm\thinspace keV$})& ...&...& ...&...&6.7(f)&6.7(f)&6.7(f)&6.7(f)\\
$\sigma$(\hbox{$\rm\thinspace keV$}) &...&...& ...&...&$0.08\pm0.02$&$0.003\pm0.001$& $0.09\pm0.03$&$0.004\pm0.001$ \\
$\tau$ &...&...& ...&...&$0.030^{+0.003}_{-0.005}$&$0.20\pm0.06$ & $0.031^{+0.006}_{-0.005}$ &$0.22^{+0.08}_{-0.07}$ \\
$E_{Edge1}$ (\hbox{$\rm\thinspace keV$})&...&...& ...&...&8.8(f)&8.8(f)&8.8(f) &8.8(f)\\
$E_{Gabs2}$ (\hbox{$\rm\thinspace keV$})&...&...& ...&...&6.97(f)&6.97(f)&6.97(f)&6.97(f)\\
$\sigma$(\hbox{$\rm\thinspace keV$})&...&...& ...&...&$0.06\pm0.02$&$<0.8$& $0.056^{+0.019}_{-0.020}$& $<0.05$ \\
$\tau$ & ...&...& ...&...&$0.041^{+0.004}_{-0.005}$&$0.014^{+0.016}_{-0.008}$ &$0.039^{+0.005}_{-0.002}$&$0.013^{+0.023}_{-0.005}$\\
$E_{Edge2}$ (\hbox{$\rm\thinspace keV$})&...&...& ...&...&9.3(f)&9.3(f)&9.3(f)&9.3(f) \\
$\chi^{2}/\nu$&4967.3/3296&...&3230.0/2264&....&4425.3/3289&...& 4370.1/3288 \\
\hline
\hline
\end{tabular}
\end{center}
\small Notes.-Model 1 is described in \hbox{\sc XSPEC}\ as PHABS$\times$KDBLUR$\times$(PL+REFHIDEN). The value of ${\it N}_{\rm H}$, inclination and ${\it r}_{\rm in}$\ were tied between the two observations. An instrumental line at $1.876$\hbox{$\rm\thinspace keV$}\ was added to each model. The normalisation of each component is referred to as {\it N}. Frozen values are followed by (f). $H_{den}$ is given in units of $10^{21}{\thinspace {\rm H}\hbox{$\rm\thinspace cm$}^{-3}}$. Model 2 is similar to the previous model and only fitted to the data below 6.6\hbox{$\rm\thinspace keV$}. Model 3 and 4 includes Fe XXV and Fe XXVI absorptions ({\rm GABS} model in \hbox{\sc XSPEC}) at fixed energies of 6.7 and 6.97\hbox{$\rm\thinspace keV$}\ respectively. Their respective absorption edges is modelled with the model {\rm EDGE} in \hbox{\sc XSPEC}\ with energies frozen at 8.8 and 9.3\hbox{$\rm\thinspace keV$}\ respectively. The optical depth $\tau$ of the absorptions and edges are linked for consistency.
\end{table*}
\vspace{1cm}
$\bullet$ {\bf GRO J1655-40}: Strong absorption features are clearly
present in the spectra of \hbox{\rm J1655-40}\ (see Fig. 2), and thus a fit with
\rm{\sc REFHIDEN}\ should not immediately give an acceptable result. Fig. 7
(Top) shows the best-fit data/model ratio using \rm{\sc REFHIDEN}. The various
parameters for this model are described in Table 2 (Model 1). It can
be seen that, whereas in Fig. 2 there is evidence of both a broad line
and various absorption features, this time the presence of the broad
line has diminished (Fig. 7 Top). This fit constrains the inner radius
to ${\it r}_{\rm in}$$=1.86^{+0.20}_{-0.02}$ ${\it r}_{\rm g}$\ with $\chi^{2}/\nu
=4967.3/3296$. The best fit using Model 1 seems to imply a relatively
low inclination of 50\deg$\pm1$\deg, considerably less than the value
of the binary inclination of $70.2$\deg$\pm1.9$\deg\ (Greene, Bailyn
\& Orosz 2001). However, this value is still in agreement with that
presented by Diaz-Trigo et al. (2007, see their Table 1). The low
inclination seen here could be due to the various absorption features
masking themselves as the blue wing of an iron-$K\alpha$\ line
profile{\footnote {The blue wing of the iron-$K\alpha$\ line profile can be
used to determine the inclination of the source (see e.g. Reynolds
\& Nowak 2003; Fabian \& Miniutti 2005)}}. Although various
features contribute to the determination of the inner radius of
emission and thus spin of the black hole, the most important of those
features for the purpose of this work is the extent of the red wing in
the Fe-$K\alpha$\ fluorescent line. In order to explore whether the
absorption features are affecting the value of the inner radius, we
froze the inclination at 70 degrees and modelled the spectra below
6.4\hbox{$\rm\thinspace keV$}. The results of this fit is detailed in Model 2 (Table 2) and
shown in Figure 7 (Bottom). By restricting the inclination to the
known value of the binary inclination ($\approx70$ degrees) and
fitting the spectra below 6.4\hbox{$\rm\thinspace keV$}\ the innermost radius approaches
that of a maximally rotating black hole with ${\it r}_{\rm in}$$=1.31\pm 0.01$${\it r}_{\rm g}$\
and $\chi^{2}/\nu =3230.0/2264$. The majority of the contribution to
$\chi^{2}$ is now coming from residuals between 0.9--1.1\hbox{$\rm\thinspace keV$}\ possibly
due to the photo-ionisation edge of O VIII at $\approx 0.9$\hbox{$\rm\thinspace keV$}. In
order to extend this fit to the full range we modelled the various
absorption features in a phenomenological manner using two negative
Gaussian absorptions (GABS model in \hbox{\sc XSPEC}) fixed at energies of 6.7
and 6.97\hbox{$\rm\thinspace keV$}\ corresponding to absorption of Helium-like Fe-$\rm XXV$
and Hydrogen-like Fe-$\rm XXVI$ respectively as well as absorption
edges fixed at 8.8 and 9.3\hbox{$\rm\thinspace keV$}. The optical depths $\tau$ of the
absorptions and edges were linked for self-consistency. The various
parameters for this model covering the full energy range are detailed
in Table 2 (Model 3). The value of the inner radius remains low
($<1.4$${\it r}_{\rm g}$) similarly to that of Model 2. Allowing the inclination to
vary over its full range improved the statistics with
$\Delta\chi^{2}=-55.2$ for one less degree of freedom (Model 4) and
resulted in an inner radius of ${\it r}_{\rm in}$$=2.17^{+0.15}_{-0.17}$${\it r}_{\rm g}$\
(Fig. 8a) and inclination of $30$\deg$^{+5}_{-10}$ (Fig. 8b). The best
fit spectra for \hbox{\rm J1655-40}\ including the absorption lines and edges is shown
in Figure 9. Note that as we remove the restriction on the
inclination, the emissivity profile of the two observations become
less steep, with the indices constrained to $q_{in}=3.87\pm0.08$ and
$2.75^{+0.03}_{-0.02}$ for obs 1 and 2 respectively. Adding a possible
nickel emission line at $7.48$\hbox{$\rm\thinspace keV$}\ with equivalent width of 30 and
17\hbox{$\rm\thinspace eV$}\ for obs 1 and 2 respectively resulted in an improved fit with
$\Delta\chi^2=-72.2$ for 4 extra degrees of freedom. As was the case
for Model 2, most of the contribution to chi-squared are now coming
from residuals in the soft energy range. By restricting this fit to
above 1.5\hbox{$\rm\thinspace keV$}\ we obtain $\chi^{2}/\nu =3496.0/2957$.
\begin{figure}
\rotatebox{270}{
{\includegraphics[width=35mm,height=80mm]{ratioplot_simple.ps}}
}
\rotatebox{270}{
{\includegraphics[width=35mm,height=80mm]{ratioplot_below64.ps}}
}
\caption {{\it Top:} Best-fit data/model ratio for \hbox{\rm J1655-40}\ using Model 1
(Table 2). The various absorption features are clearly seen. {\it
Bottom:} Similar as above for Model 2 covering the energy range
0.7--6.4\hbox{$\rm\thinspace keV$}. The model clearly results in a good fit for the
red-wing of the iron-$K\alpha$\ fluorescent line. Obs 1 and 2 are shown in
black and red respectively. The data have been rebinned for plotting
purposes only. }
\end{figure}
\begin{figure}
\centering
\subfigure[]
{
\rotatebox{270}{
\resizebox{!}{5.5cm}
{\includegraphics{steppar_rin_j1655.ps}
}}}
\hspace{1cm}
\subfigure[]
{
\rotatebox{270}{
\resizebox{!}{5.5cm}
{\includegraphics{steppar_inclination_j1655.ps}}
}}
\caption{{\rm (a):} $\chi^{2}$ vs ${\it r}_{\rm in}$\ plot for \hbox{\rm J1655-40}. Using Model 4
(Table 2) with the inclination allowed to vary over its full range
we obtain a value of $2.17^{+0.15}_{-0.17}$${\it r}_{\rm g}$\ for the inner
radius at the 90 per cent confidence level for one parameter of
interest ($\Delta\chi^{2} = 2.71$ criterion) shown by the solid
horizontal line.{\rm (b):} Similar plot for the inclination of \hbox{\rm J1655-40}. }
\end{figure}
\begin{figure}
\rotatebox{270}{
\resizebox{!}{8cm}
{\includegraphics{fig_spc_1655.ps}}
}
\caption {Best-fit spectra for \hbox{\rm J1655-40}\ using Model 4. Obs 1 and 2 are
shown in black and red respectively. The complex absorptions are
phenomenologically modelled with two Gaussian absorption (see
text). The data have been rebinned for plotting purposes only.}
\end{figure}
Similarly to \hbox{\rm J1753.5-0127}\ we investigate any degeneracy between the
inclination and inner radius in Model 4 by using the ``contour''
command in \hbox{\sc XSPEC}. All parameters except for $\Gamma$ and
$R_{Illum/BB}$ were free to vary. Fig. 10 shows the 68, 90 and 95 per
cent contour for the two parameters of interest. It can be seem from
Fig. 10 that for a large range of inclination the value of ${\it r}_{\rm in}$\
remains below 2.35${\it r}_{\rm g}$\ with a best fit value of approximately
$2.15$${\it r}_{\rm g}$, in agreement with the result presented in Fig. 8a. It is
clear that the unknown inclination of the system plays an important
role in the determination of the inner radius (and thus spin) of \hbox{\rm J1655-40},
with values as low as 1.3${\it r}_{\rm g}$\ (spin $>0.99$) obtained with an
inclination of 70 degrees (Model 2) rising to 2.32${\it r}_{\rm g}$\ (spin $>0.9$)
with the inclination allowed to vary (Model 4). However, it must be
noted that using the reflection model both with and without accounting
for the absorption features as well as both with and without
constraints on the inclination, gives results with an inner-radius
consistently below 2.32${\it r}_{\rm g}$\ (Table 2). Taking this value as an upper
limit to the innermost radius of emission implies a dimensionless spin
parameter greater than 0.9 for \hbox{\rm J1655-40}\ (Fig. 6).
\begin{figure}
\rotatebox{270}{
\resizebox{!}{8cm}
{\includegraphics{contour_J1655_inc_rin.ps}}
}
\caption {Inclination-inner radius contour plot for \hbox{\rm J1655-40}. The 68, 90
and 95 per cent confidence range for two parameters of interest are
shown in black, red and green respectively. It can be seen that for
the inner radius is constrained to be below approximately 2.5${\it r}_{\rm g}$ for
a large range of inclination at the 90 per cent confidence level for
two parameters.}
\end{figure}
\section{Discussion}
The X-ray spectra of galactic black hole binaries can provide
important information on both the geometry of the system as well as
intrinsic physical parameters such as the spin of the central black
hole. Using the reflection features embedded in the spectra of both
\hbox{\rm J1753.5-0127}\ and \hbox{\rm J1655-40}\ we found clear indications that the accretion disk
extends close to the radius of marginal stability in both cases. For
\hbox{\rm J1753.5-0127}\ we have shown that the innermost emitting region extends down to
${\it r}_{\rm in}$$= 3.1^{+0.7}_{-0.6}$${\it r}_{\rm g}$\ (Fig. 4a) with an innermost inclination
of $55^{+2}_{-7}$ degrees (Fig. 4b). Based on the normalisation of the
disk blackbody component, Miller, Homan \&\ Miniutti (2006) found an
inner radius of ${\it r}_{\rm in}$$=2.0(3)(M/10\hbox{$\rm\thinspace M_{\odot}$})(d/8.5\hbox{$\rm\thinspace kpc$})(cos
i)^{-1/2}$${\it r}_{\rm g}$, similar to ours for a large range of parameters. Our
result is also consistent with that found by Soleri et al. (2008) of
2.6--6.0${\it r}_{\rm g}$\ for a 10\hbox{$\rm\thinspace M_{\odot}$}\ black hole. In both these cases ${\it r}_{\rm in}$\ was
estimated using the normalisation of the MCD component and thus
requires knowledge of the mass, distance and inclination of the
sources. Uncertainties in these values can significantly affect the
value of the inner radius and thus spin parameter.
Estimating the spin of a black hole based on the extent of the
emitting region relies on the assumption that the accretion disk
extends to the innermost stable circular orbit (ISCO) and that
emission within this radius is negligible. Reynolds \& Fabian (2008)
addressed the robustness of this assumption and found that reflection
within the ISCO becomes significantly less as one considers more
rapidly rotating black holes. The way the dimensionless spin parameter
depends on the position of the ISCO is shown in Fig. 6 (solid curve). Assuming that the accretion disk extend to the ISCO for \hbox{\rm J1753.5-0127}\ we
obtain a spin parameters of $0.76^{+0.11}_{-0.15}$. Note that this
estimate is independent of the unknown distance to the source.
The system in \hbox{\rm J1655-40}\ is known to have an orbital-plane inclination of
approximately 70 degrees (Greene, Bailyn \& Orosz 2001). Constraining
the inclination to this value resulted in an inner radius of emission
consistent with that of a maximally rotating black hole (Model 3,
Table 2). Allowing the inclination of the blurring function to be a
free parameter in the spectral fit of \hbox{\rm J1655-40}\ we obtain an improved fit
with an inner radius of $=2.17^{+0.15}_{-0.17}$${\it r}_{\rm g}$\ (Fig. 8a) and an
inner-disk inclination of $30^{+5}_{-10}$ degrees (Fig. 8b). This
value for the inner radius implies a {\it lower limit} for the spin
parameter of 0.9 (Fig. 6a). The possibility of a misalignment between
the innermost region of the accretion disk and the orbital plane
inclination in galactic microquasars have been suggested by various
authors (see e.g. Maccarone 2002) with clear examples seen in both
\hbox{\rm J1655-40}\ (Martin, Tout \& Pringle 2008) and V4641 Sgr (Martin, Reis \&
Pringle 2008). It was shown in Martin, Tout \& Pringle (2008) that the
alignment time scale in microquasar such as \hbox{\rm J1655-40}\ is usually a
significant fraction of the lifetime of the system. Thus if the black
hole in such a system were formed with misaligned angular momentum, as
expected from supernova-induced kicks, then it would be likely that
the system would remain misaligned for most of its lifetime. This
assumption is supported by the apparent misalignment found in the
present work.
The lower limit for the spin parameter of \hbox{\rm J1655-40}\ found here ($>$0.9;
Fig. 6) is not consistent with that reported by Shafee et al. (2006)
of 0.65--0.75. As was mentioned above their method requires prior
knowledge of several other factors including the inner disk
inclination and distance to the source. Pszota \& Cui (2007) have
recently shown that neither disk continuum models {\it KERRBB} (Li et
al. 2005) nor {\it BHSPEC} (Davis \& Hubeny 2006) were able to
successfully model the ultrasoft spectra of \hbox{\rm GX 339-4}, nonetheless their
best fit parameters in conjunction with the best estimates for the
physical parameters of the source, suggested a moderate spin of
0.5--0.6 in comparison with our estimate of $\approx 0.935$ (Reis et
al. 2008; Miller et al. 2008) based on the various reflection features
in the spectra of \hbox{\rm GX 339-4}. Our suggestion that \hbox{\rm J1655-40}\ contains a rapidly
rotating black hole is in agreement with results based on
Quasi-Periodic Oscillations where a value of $>0.91$ is usually
derived (Zhang et al. 1997; Cui et al. 1998; Wagoner et al. 2001;
Rezzolla et al. 2003). Furthermore, in their derivation of the spin
parameter of \hbox{\rm J1655-40}, Shafee et al. (2006) assumed both a lack of
misalignment between the central disk region and the orbital plane and
more importantly they used a distance to the source of
$3.2\pm0.2$\hbox{$\rm\thinspace kpc$}. Foellmi (2008) has recently shown that the distance
to \hbox{\rm J1655-40}\ is most likely to be less than 2\hbox{$\rm\thinspace kpc$}. When this distance is
used in place of 3.2\hbox{$\rm\thinspace kpc$}\ the spin parameter of \hbox{\rm J1655-40}\ derived by the
thermal continuum method becomes greater than 0.91 in agreement with
our results.
\section{Conclusions}
We have studied the {\it XMM-Newton}\ spectra of both SWIFT \hbox{\rm J1753.5-0127}\ and GRO \hbox{\rm J1655-40}. By
estimating the innermost radius of emission in these systems we
constrain their spin parameters by assuming that the accretion disk
extends down to the radius of marginal stability. For \hbox{\rm J1753.5-0127}\ the spin
is found to be $0.76^{+0.11}_{-0.15}$ at 90 per cent confidence. The
innermost disk inclination in \hbox{\rm J1753.5-0127}\ is estimated at $55^{+2}_{-7}$
degrees. In the case of \hbox{\rm J1655-40}\ we find that the best fit requires a disk
which is significantly misaligned to the orbital plane. An inclination
of $30^{+5}_{-10}$ degrees and a dimensionless spin parameter greater
than 0.9 is found at the 90 per cent confidence level. Our method
involves spectral modelling of both the intrinsic disk thermal
emission as well as the reprocessed hard radiation which manifest
itself as various reflection ``signatures''. These reflection
signatures are independent of both black hole mass and distance and
are thus a very useful tool for such measurements.
\section*{Acknowledgements}
RCR acknowledges STFC for financial support. ACF and RRR thanks
the Royal Society and the College of the Holy Cross respectively.
\vspace{0.5cm}
\noindent {\bf Note added after submission:} In a recent arXiv
posting, Hiemstra et al. (2009) published an analysis made on the same
{\it XMM-Newton}\ and {\it RXTE}\ data for \hbox{\rm J1753.5-0127}\ used here. They confirm the existence
of a broad iron-$K\alpha$\ line and show that various continuum models with
the addition of a \rm{\sc LAOR}\ line can successfully fit the data. More
importantly, they claim to successfully fit the X-ray spectrum of \hbox{\rm J1753.5-0127}\
with a continuum model that does not require emission from a disk
extending down to the ISCO. We have investigated this solution and
find that in their case the broad line in a consequence of Compton
broadening of the emitted photons by the hot surface of the accretion
disk (Ross, Fabian \& Young 1999; Ross \& Fabian 2007). Their solution
requires a highly ionised disk ($\xi \sim 5000$ ${\rm erg}\hbox{$\rm\thinspace cm$}\hbox{$\rm\thinspace s$}^{-1})$
truncated at $\sim255$ ${\it r}_{\rm g}$\ with an inclination angle consistent with
zero. Using the same reflection model (\rm{\sc REFLION}; Ross \& Fabian 2005)
convolved with \rm{\sc KDBLUR}\ and including a disk component, we find a
further solution where the disk extends down to the ISCO (6${\it r}_{\rm g}$) and
has a much lower ionisation parameter of $\xi \sim 500$ ${\rm
erg}\hbox{$\rm\thinspace cm$}\hbox{$\rm\thinspace s$}^{-1}$. This second interpretation yields an inclination of
$\sim 65$ degrees, and an improvement of -7.7 in $\chi^2$ for 2 extra
degrees of freedom. The solution presented in our paper reflects that
of the lower ionisation reported above. We are concerned that in a
solution with such high ionisation parameter and a truncation radius
at $\sim 45$ times the ISCO would result in a disk which is
approximately $2\times 10^4$ times less dense that that of a similar
disk extending down to the ISCO. With an estimated mass of
approximately 10\hbox{$\rm\thinspace M_{\odot}$}\ (Cadolle Bel et al. 2007), a radius of 255${\it r}_{\rm g}$\
is the equivalent of a disk truncated at $\approx 3.8\times 10^8$\hbox{$\rm\thinspace cm$}\
from the central black hole. Using the ionisation parameter and
unabsorbed flux presented in Hiemstra et al. (2009) we can estimate
the hydrogen number density to be approximately $2.3\times 10^{15}{\rm
cm}^{-3}$. A modest surface layer of Thomson depth $\tau_T \sim 3$
will thus results in a disk with a half-thickness, $t$ greater than
$2\times 10^9 {\rm cm}$; at least 5 times larger than the truncation
radius. For a disk with inner radius $r>>$${\it r}_{\rm g}$\ we can write the
ionisation parameters as $\xi \approx 1.13\times10^5 \eta\ t/r^2 { \rm
erg}\hbox{$\rm\thinspace cm$}\hbox{$\rm\thinspace s$}^{-1}$, where $\eta$ is the X-ray efficiency (Reynolds \&
Begelman 1997). Fig. 11 shows the ratio $t/r$ as a function of $r$ for
a system with $\eta=0.06$ and an ionisation parameter of 5000 and 500
${\rm erg}\hbox{$\rm\thinspace cm$}\hbox{$\rm\thinspace s$}^{-1}$. For a typical
thin accretion disk $t/r$ should be well below unity and as can be
seen in Fig. 11 there are no solutions in this range for a disk with
$\xi \sim 5000$ ${\rm erg}\hbox{$\rm\thinspace cm$}\hbox{$\rm\thinspace s$}^{-1}$. We conclude that the high
ionisation solution found by Hiemstra et al (2009) is physically
inconsistent; the disc must extend in to small radii.
\begin{figure}
\rotatebox{270}{
\resizebox{!}{8cm}
{\includegraphics{l_r_versus_r_plot.ps}}
}
\caption { Ratio of half-thickness $t$ over inner radius $r$ as a
function of $r$ for $\xi=5000$ (dotted line) and $\xi= 500$ ${\rm
erg}\hbox{$\rm\thinspace cm$}\hbox{$\rm\thinspace s$}^{-1}$ (solid line). For a typical thin accretion disk
this ratio should be well below 1 (horizontal line). }
\end{figure}
|
1,116,691,499,382 | arxiv | \section{Introduction}
\label{sec:Introduction}
Spectrum analyses of pre-main sequence stars (PMS) require special techniques, notably for dealing with peculiarities of cool, low-mass members of young clusters.
Optical spectra of such stars may include the presence of veiling, large broadening due to fast rotation, emission lines due to accretion and/or chromospheric activity, and molecular bands.
The subtraction of inhomogeneous and variable nebular emission may also be problematic and some residual features can remain in spectra of some young clusters' members after the sky-background removal.
One of the main objectives of the Gaia-ESO Survey is to provide radial velocities (RV) with a precision $\approx$ 0.2 - 0.25 km\,s$^{-1}$\ for stars in young open clusters, to complement Gaia proper motions with comparable accuracy for a statistically significant sample \citep{2012Msngr.147...25G,2013Msngr.154...47R} reaching also fainter targets.
This survey also complements Gaia by deriving metallicity and detailed abundances for several elements, including lithium, which is particularly relevant in the studies of the evolution of low-mass stars and in the determination of clusters' age.
This requires a derivation of all fundamental parameters (effective temperatures $T_{\rm eff}$, metallicity [Fe/H], surface gravity $\log g$, and projected rotational velocity $v\sin i$) independently of the Gaia results.
The ${\rm H}\alpha$\ profile of such young low-mass stars bears information on their chromospheric activity, accretion rate, and mass loss.
Because of their common origin, strong accretion is expected to be correlated with veiling; this can be used for both checking our results, as no correlation would be indicative of large uncertainties, and exploring the extent and details of such a correlation.
Chromospheric activity is known to depend on stellar rotation and both evolve in time; the Gaia-ESO is also going to provide the possibility of exploring the activity-rotation relation and their evolution on a large sample of young stars.
The Gaia-ESO target selection aims at producing unbiased catalogues of stars in open clusters.
Selection criteria based mainly on photometry, supported, when possible, by kinematic memberships, have been adopted for this purpose, although this implies that a large number of non-members are also observed, which are identified {\it a posteriori} from the radial velocity measurements (Bragaglia et al., in prep.).
In our case, the GIRAFFE targets are late-type (F to early-M) stars in the magnitude range 12$\le$V$\le$19 mag, in the PMS or main sequence (MS) phase.
Based on available information, the selection of UVES targets tries to include only slowly rotating ($v\sin i$$<$ 15 km\,s$^{-1}$) single G--K stars in the magnitude range 9$<$V$<$15 without or with weak accretion ($\dot{M}_{\rm acc}$$<$ 10$^{-10}$ M$_{\odot}$ yr$^{-1}$).
To optimise the throughput of the survey, observations of cool stars in the fields of young open clusters are only carried out in the GIRAFFE/HR15N setup (R=17\,000, $\lambda$ from 6470 to 6790\,\AA) and the Red 580 UVES setup (R=47\,000 centred at $\lambda=5800$\,\AA\ with a spectral band of 2000\,\AA).
The Medusa mode of the fibre fed system is used throughout the survey,
allowing the simultaneous allocation of 132 and 8 fibres feeding GIRAFFE and UVES, respectively,
with about 20 (GIRAFFE) and 1 (UVES) fibres used to observe the sky background spectrum.
The GIRAFFE/HR15N setup covers both H$\alpha$ and Li\,(6707.84\,\AA)\ lines, and it is therefore particularly useful for the study of young stars.
However, $T_{\rm eff}$, $\log g$, and [Fe/H]\ diagnostics in this wavelength range are poorer than in other settings and still not satisfactorily reproduced by theoretical models.
For example, the paucity of \element{Fe} lines in the HR15N spectral range makes it difficult to derive both $\log g$\ and [Fe/H]\ in G-type stars from the analysis of the equivalent widths of \ion{Fe}{I} and \ion{Fe}{II} lines.
This paper presents the analysis of the Gaia-ESO spectra in the fields of young open clusters (age $<$ 100 Myr) and is one of a series presenting a description of the Gaia-ESO survey in preparation of its first release of advanced data products.
The Gaia-ESO scientific goals, observations strategies, team organisation, target selection strategy, data release schedule, data reduction, analysis of OBA-type and FGK-type stars not in the fields of young open clusters, non-standard objects and outliers, external calibration, and the survey-wide homogenisation process are discussed in other papers of this series.
The paper is organised as follows.
In Sect.\,\ref{sec:data} the data analysed in the first two Gaia-ESO internal data releases are presented.
In Sect.\,\ref{sec:GeneralStrategy} the principles and general strategies of the Gaia-ESO PMS analysis are outlined.
Methods and validation for the initial raw measurements, fundamental parameters ($T_{\rm eff}, \log g$, [Fe/H], micro-turbulence velocity, veiling, and $v\sin i$), and derived parameters (chromospheric activity, accretion rate, and elemental abundances) are presented in Sects.\,\ref{sec:RawMeasurements}, \ref{sec:FundamentalParameters}, and \ref{sec:DerivedParameters}.
The conclusions are in Sect.\,\ref{sec:Conclusions}.
\section{Data}
\label{sec:data}
\begin{table*}[ht]
\centering
\caption{
Young open clusters (age $<$ 100 Myr) observed by the Gaia-ESO survey in the first 18 months of observations, whose analysis is discussed in this paper.
The cluster NGC6705 (M11) has also been included for validation and comparison across the survey.
}
\begin{tabular}{rccrrrrr}
\hline
Cluster & Approximate Age & Distance & \multicolumn{2}{c}{GIRAFFE} & \multicolumn{2}{c}{UVES} \\
& (Myr) & (pc) & All & WTTS/CTTS & All & WTTS/CTTS & iDR \\
\hline
$\rho$ Oph & 1 & 120 & 200 & 30 & 23 & 5 & 2 \\
Cha\,I & 2 & 160 & 674 & 93 & 49 & 14 & 1, 2\\
NGC2264 & 3 & 760 & 1706 & 446 & 118 & 23 & 2\\
$\gamma$ Vel & $\sim$ 5--10 & 350 & 1242 & 200 & 80 & 2 & 1, 2\\
NGC2547 & 35 & 361 & 450 & 44 & 26 & 1 & 2\\
NGC6705 & 250 & 1877 & 1028 & 0 & 49 & 0 & 2\\
\hline
\end{tabular}
\label{tab:targets}
\end{table*}
The survey's analysis is performed in cycles, following the data reduction of newly observed spectra.
Each new analysis cycle improves upon the last one with updated input data (e.g. atomic and molecular data), improved analysis methods, and improved criteria to define the final recommended parameters.
At the end of each cycle an internal data release (iDR) is produced and made available within the Gaia-ESO consortium for scientific validation.
The description of the methods and recommended parameters criteria given in this paper applies to the analysis of the first 2 years of observations, which will form the basis of the first release of advanced data products to ESO.
The validation procedures presented in this paper consider the first 18 months of observations (iDR1 and iDR2).
The young open clusters observed in the first 18 months of observations are listed in Table\,\ref{tab:targets}, along with the total number of observed stars for each cluster and the number of stars identified as T\,Tauri from the properties of ${\rm H}\alpha$\ emission, spectral type, and Li\ absorption (see Sect.\,\ref{sec:halpha}).
A total of 813 and 45 T\,Tauri stars have been identified in the GIRAFFE and UVES spectra, respectively.
The memberships of these young clusters, including stars not clearly showing the T\,Tauri distinctive features, will be discussed in other Gaia-ESO science verification papers \citep[e.g.,][]{2014A&A...563A..94J}.
The 200 Myr cluster NGC6705 (M11) has been observed in different setups \citep[see e.g.][]{2014A&A...569A..17C} to allow inter-comparison and validation of the analysis methods across the survey and, for this purpose, is included in our analysis.
\section{General analysis strategy}
\label{sec:GeneralStrategy}
The Gaia-ESO consortium is structured in several working groups (WGs).
The analysis of PMS stars is carried out by the WG12, to which six nodes contributed:
INAF--Osservatorio Astrofisico di Arcetri, Centro de Astrofisica de Universidade do Porto (CAUP), Universit\`a di Catania and INAF--Osservatorio Astrofisico di Catania (OACT), INAF--Osservatorio Astronomico di Palermo (OAPA), Universidad Complutense de Madrid (UCM), and Eidgen\"ossische Technische Hochschule Z\"urich (ETH).
The main input to the Gaia-ESO PMS spectrum analysis consists of UVES and GIRAFFE spectra of cool stars in the field of young open clusters.
The preliminary selection criteria are briefly outlined in Sect.\,\ref{sec:Introduction} and will be detailed in one of the papers of this series (Bragaglia et al., in prep.).
The data reduction is performed as described in \cite{Sacco_etal:2014} for the UVES spectra, and Lewis et al. (in prep.) for the GIRAFFE spectra.
These are put on a wavelength scale and shifted to a barycentric reference frame.
Sky-background subtraction, as well as a normalisation to the continuum, is also performed in the data reduction process.
Multi-epoch spectra of the same source are combined in the {\it co-added} spectrum.
Quality information is provided including variance spectra, S/N, non-usable pixels, etc.
Additional inputs are the radial and rotational velocities, as described in Gilmore et al. (in prep.)
and \cite{Sacco_etal:2014}, and photometric data.
Clusters' distance and reddening are also considered as input to the spectrum analysis validation.
Double-lined spectroscopic binaries (SB2) and multiple systems are identified by looking at the shape of the cross-correlation function.
These stars are excluded from the current analysis and a multiplicity flag is reported in the final database.
To ensure the highest homogeneity as possible in the quantities derived, all the different Gaia-ESO spectrum analysis methods adopt the same atomic and molecular data (Heiter et al., in prep.), as well as the same set of model atmospheres \citep[MARCS, ][]{2008A&A...486..951G}.
The output parameters of the Gaia-ESO PMS spectrum analysis are listed in Table\,\ref{tab:PMS-parameters}.
To apply a detailed quality control on the output parameters and optimise the analysis according to the star's characteristics, these are divided into three groups: {\em raw}, {\em fundamental}, and {\em derived}.
Raw parameters are the ${\rm H}\alpha$\ emission and Li\ equivalent widths ($W({\rm H}\alpha$)\ and $W({\rm Li})$), and the H$\alpha$ width at 10\% of the line peak \citep[${\rm H}\alpha\,10\%$, see, e.g.,][]{2004A&A...424..603N}.
These are directly measured on the input spectra and do not require any prior information.
Their measurement is carried out before any other procedure to identify PMS stars and their values are used for optimising the evaluation of the {\it fundamental} parameters in one of the methods used (see Sect.\,\ref{sec:FundamentalParameters}).
Besides $T_{\rm eff}$, $\log g$, and [Fe/H]\footnote{The solar Fe abundance of \cite{2007SSRv..130..105G}, $\log\epsilon({\rm Fe})_{\odot} = 7.45$, is adopted.}, the fundamental parameters derived include also micro-turbulence velocity ($\xi$), projected rotational velocity ($v\sin i$), veiling \citep[$r$, see, e.g., ][]{1988BAAS...20R1092H}, and a gravity-sensitive spectral index \citep[$\gamma$, see][]{Damiani_etal:2014}.
Finally, the {\it derived} parameters are those whose derivation requires prior knowledge of the {\it fundamental} parameters, i.e. elemental abundances ($\log\epsilon({\rm X})$\footnote{$\log\epsilon({\rm X})=\log[N({\rm X})/N({\rm H})]+12$, i.e. a logarithmic abundance by number on a scale where the number of hydrogen atoms is 10$^{12}$.}), mass accretion rate ($\dot{M}_{\rm acc}$), chromospheric activity indices ($\Delta W({\rm H}\alpha$)$_{\rm chr}$\ and $\Delta W({\rm H}\beta$)$_{\rm chr}$), and chromospheric line fluxes ($F({\rm H}\alpha$)$_{\rm chr}$\ and $F({\rm H}\beta$)$_{\rm chr}$).
\begin{table*}[ht]
\centering
\caption{Gaia-ESO PMS analysis output parameters. Columns 2--13 list the number of stars in each cluster for which the parameter was derived from GIRAFFE (G) and UVES (U) spectra separately in iDR2. Lithium parameters and $v\sin i$\ counts include upper limit estimates. Accretion and chromospheric activity parameter counts include only non negligible values. For the elemental abundances, the maximum number of derived values for each star/element is reported. See text for an explanation of the notation used.}
\label{tab:PMS-parameters}
\begin{tabular}{lrrrrrrrrrrrr}
\hline
Parameter & \multicolumn{2}{c}{$\rho$ Oph} & \multicolumn{2}{c}{Cha\,I} & \multicolumn{2}{c}{NGC2264} & \multicolumn{2}{c}{$\gamma$ Vel} & \multicolumn{2}{c}{NGC2547} & \multicolumn{2}{c}{NGC6705} \\
\hline
& G & U & G & U & G & U & G & U & G & U & G & U\\
\hline
\hline
\multicolumn{13}{c}{raw} \\
\hline
$W({\rm H}\alpha$) & 25&5 & 87&14 & 387&24 & 203&2 & 106&1 & 0&0 \\
$W({\rm Li})$ & 189&23 & 633&47 & 1610&114 & 1186&75 & 404&25 & 708&48 \\
${\rm H}\alpha\,10\%$ & 33&5 & 103&14 & 807&23 & 264&2 & 239&1 & 0&0 \\
\hline
\multicolumn{13}{c}{fundamental} \\
\hline
$T_{\rm eff}$ & 170&21 & 572&39 & 1324&70 & 1104&51 & 361&24 & 394&32 \\
$\log g$ & 170&21 & 156&39 & 226&70 & 350&51 & 106&24 & 150&32 \\
$\gamma$ & 156& \dots & 508& \dots & 1199& \dots & 1043& \dots & 337& \dots & 0&\dots \\
[Fe/H] & 170&21 & 515&39 & 1203&70 & 1018&51 & 311&24 & 360&32 \\
$\xi$ & \dots & 14 & \dots&23 & \dots&42 & \dots&46 & \dots&15 & \dots&30 \\
$v\sin i$ & 154&23 & 521&42 & 1192&83 & 1004&75 & 332&25 & 107&33 \\
$r$ & 4&3 & 20&7 & 77&6 & 5&0 & 5&0 & 0&0 \\
\hline
\multicolumn{13}{c}{derived} \\
\hline
${\log \epsilon({\rm Li})}$ & 154&23 & 514&40 & 1203&80 & 1017&57 & 311&25 & 356&31 \\
$\log\epsilon(X)$ & \dots &15 & \dots &28 & \dots &39 & \dots &46 & \dots &14 & \dots &31 \\
$\dot{M}_{\rm acc}$ & 14&4 & 56&7 & 212&11 & 40&1 & 21&0 & 0&0 \\
$\Delta W({\rm H}\alpha$)$_{\rm chr}$ & 21&12 & 69&29 & 267&50 & 205&18 & 115&16 & 61&0 \\
$\Delta W({\rm H}\beta$)$_{\rm chr}$ & \dots &10 & \dots &18 & \dots &42 & \dots &14 & \dots &12 & \dots &0 \\
$F({\rm H}\alpha$)$_{\rm chr}$ & 21&12 & 65&28 & 265&47 & 199&17 & 105&16 & 47&0 \\
$F({\rm H}\beta$)$_{\rm chr}$ & \dots &10 & \dots &17 & \dots &41 & \dots &14 & \dots &12 & \dots &0 \\
\hline
\end{tabular}
\end{table*}
Most parameters listed in Table\,\ref{tab:PMS-parameters} are derived from both UVES and GIRAFFE spectra, with the exception of
$\xi$, $\Delta W({\rm H}\beta$)$_{\rm chr}$, $F({\rm H}\beta$)$_{\rm chr}$, and ${\log \epsilon({\rm X})}$, that are derived from UVES spectra only, and the gravity-sensitive spectral index $\gamma$, which is derived from the GIRAFFE spectra only (see Sect.\,\ref{sec:FundamentalParameters}).
In general, whenever possible, the same parameter is derived by different methods; this allows a thorough check of the derived parameters by inter-comparing the results and flagging discrepant results, which are then used to outline possible weaknesses of the methods and discard unreliable results.
In the absence of significant biases, the results from different methods are combined taking a $\sigma$-clipped average to obtain the {\em recommended} parameters.
If significant biases are present, all results obtained with a method that can give rise to inaccurate or unreliable results in some ranges of parameters are rejected before combining the results as above.
These general criteria, whose application is discussed in details in
Sects.\,\ref{sec:RawMeasurements}--\ref{sec:DerivedParameters},
are firstly applied on the {\em raw} parameters, then on the {\em fundamental} parameters, and finally on the {\it derived} parameters.
The {\em fundamental} parameters are also validated by comparing the results of the analysis methods applied to our spectra against fundamental parameters from angular diameter and parallax measurements (Sect.\,\ref{sec:benchmarks}).
A comparison with $T_{\rm eff}$\ derived from photometry for objects that are not affected by photometric excesses is reported in Appendix\,\ref{sec:photometry}.
The {\em recommended raw} and {\em fundamental} parameters are then used to produce the {\em recommended derived} parameters.
When satisfactory comparisons cannot be achieved, recommended parameters are not provided and only results from individual nodes are made available.
Recommended parameter uncertainties are estimated as both node-to-node dispersion and as average of individual node's uncertainties.
The final results minimise -- as much as possible -- biases that can affect individual methods and the associated uncertainties take differences that may arise from the use of different methods and algorithms into account.
Final results are further validated by a general analysis of the output $\log g$-$T_{\rm eff}$\ diagram, consistency of the parameters, and overview of the results based on the comparison of different clusters.
\section{Raw measurements}
\label{sec:RawMeasurements}
Measuring the raw parameters before carrying out any other analysis allows us to:
(a) identify stars with strong accretion whose spectra may be affected by veiling;
(b) perform a quality control on the raw parameters before they are used in the subsequent analysis; and
(c) apply the appropriate masks to the spectra for the determination of fundamental parameters.
To derive raw parameters from a large dataset of spectra it is convenient to use procedures that are as automatic as possible.
However, in the case of PMS sources extending to M spectral type, such procedures must also be capable of dealing with large rotational broadening and the presence of molecular bands.
Here different methods are used, with different levels of automatism, which allows to examine the presence of biases, eliminate systematic discrepancies, and combine the results with a $\sigma$-clipping to disregard casual mistakes and outliers.
In the following we briefly describe the methods used to derive the raw parameters.
\subsection{${\rm H}\alpha$\ equivalent width and ${\rm H}\alpha$\ width at 10\% of the line peak}
\label{sec:halpha}
Spectra with ${\rm H}\alpha$\ in emission are examined to identify stars with strong accretion, and therefore likely to be affected by veiling, using their $W({\rm H}\alpha$)\ and ${\rm H}\alpha\,10\%$\ measurements.
The Arcetri node measures $W({\rm H}\alpha$)\ and ${\rm H}\alpha\,10\%$\ on the continuum-normalised co-added
spectra of all stars that clearly show ${\rm H}\alpha$\ emission, using a semi-automatic procedure.
After manually defining the wavelength range and level of continuum, $W({\rm H}\alpha$)\ is calculated by a direct integration of the flux above the continuum, while ${\rm H}\alpha\,10\%$\ is derived by considering the level corresponding to 10\% of the maximum flux above the continuum in the selected wavelength range.
All measurements are visually checked and repeated in case of miscalculation (e.g. due to the presence of multiple peaks).
Uncertainties are estimated using multi-epoch observations of stars belonging to the first two young clusters that have been observed (i.e. $\gamma$ Vel and Cha\,I).
Specifically, $W({\rm H}\alpha$)\ and ${\rm H}\alpha\,10\%$\ are first measured on each spectrum before co-adding, then the relative uncertainty for each star is estimated as $\Delta W = 2\abs{W_{1} - W_{2}} / (W_{1} + W_{2})$, where $W_{1}$ and $W_{2}$ are two measurements for the same star from spectra observed at different epochs.
A similar formula is used for $\Delta$${\rm H}\alpha\,10\%$.
Finally, the median of $\Delta$$W({\rm H}\alpha$)\ and $\Delta$${\rm H}\alpha\,10\%$\ are assumed as the relative uncertainties for all stars\footnote{This may also be linked to ${\rm H}\alpha$\ variability.}
The CAUP node makes use of an automatic IDL\footnote{IDL\textregistered (Interactive Data Language) is a registered trademark of Exelis Visual Information Solutions.} procedure to first select stars with ${\rm H}\alpha$\ in emission and then measure ${\rm H}\alpha\,10\%$\ and $W({\rm H}\alpha$)\ on the normalised spectra.
Measurement uncertainty is evaluated from the spectrum S/N.
The OACT node pre-selects spectra with ${\rm H}\alpha$\ in emission by visual inspection.
Then, $W({\rm H}\alpha$)\ and ${\rm H}\alpha\,10\%$\ are measured using an IDL procedure.
$W({\rm H}\alpha$)\ is measured by direct integration of the ${\rm H}\alpha$\ emission profile and its uncertainty evaluated by multiplying the integration range by the mean error in two spectra regions close to the ${\rm H}\alpha$\ line.
${\rm H}\alpha\,10\%$\ uncertainty is evaluated by assuming an error of 10\% in the position of the continuum level.
The OAPA node employed two methods, one based on DAOSPEC \citep{2008PASP..120.1332S} and an IDL procedure, the other on a combination of IRAF and IDL tools.
In the first method, DAOSPEC is used to perform a continuum fit of the spectral region around ${\rm H}\alpha$.
The ${\rm H}\alpha$\ profile in the unnormalised input spectrum is masked by giving as input to DAOSPEC a variable FWHM that takes the rotational and instrumental profiles into account.
The fitted continuum is then used to normalise the input spectrum.
Such a continuum normalised spectrum is used to measure ${\rm H}\alpha\,10\%$\ by an automatic IDL procedure.
Since the uncertainties are assumed to be dominated by the fitting of the continuum, this is repeated four times using different orders (10, 15, 20 and 25) of the polynomial fitting in DAOSPEC.
The resulting ${\rm H}\alpha\,10\%$\ values are then averaged to produce the final result.
In the second method the normalisation is performed through IRAF with three different orders of the polynomial fitting (2, 5, and 10), then $W({\rm H}\alpha$)\ and ${\rm H}\alpha\,10\%$\ are measured with an automatic IDL routine and uncertainties derived as above.
A final visual inspection is performed to check the results and identify broad emission and P~Cygni-like profiles.
For the first data release both methods where used, while in the second data release only the second method was used.
The final spectra of NGC2264 are affected by some residual nebular emission, and a good subtraction of this contribution to the ${\rm H}\alpha$\ emission line cannot be achieved as the nebular emission is concentrated in the region near the ${\rm H}\alpha$\ line peak and is spatially variable \citep[see a detailed description of this topic for the analogous case of the cluster NGC6611 in][]{2013A&A...556A.108B}.
In this case, additional visual inspection of the spectra was necessary to ensure that the narrow nebular emission does not affect significantly the measurements.
\begin{figure}[ht]
\centering
\includegraphics[width=90mm]{giraffe_Ha10_EWHaAcc_pms.eps}
\caption{$W({\rm H}\alpha$)\ vs. ${\rm H}\alpha\,10\%$\ for all young open clusters observed in the first 18 months of observations. Filled symbols are used for stars classified as CTTS, open symbols for stars classified as WTTS.}
\label{fig:giraffe_Ha10_EWHaAcc_pms}
\end{figure}
In the node-to-node comparison of the $W({\rm H}\alpha$)\ results, average differences and dispersions $\sim 5$ \AA\ were found in the analysis of both UVES and GIRAFFE spectra, with only a few outliers.
Average differences in ${\rm H}\alpha\,10\%$\ in the node-to-node comparison was $\sim$10 km\,s$^{-1}$, with a dispersion $\sim$50 km\,s$^{-1}$.
Only a $1\sigma$-clipping was therefore applied before computing the average $W({\rm H}\alpha$)\ and ${\rm H}\alpha\,10\%$\ as recommended values.
The recommended uncertainty was given, conservatively, as the largest amongst the average of individual uncertainties and the standard deviation of the mean.
The recommended ${\rm H}\alpha\,10\%$\ is used, together with the recommended $W({\rm Li})$\ (Sect.\,\ref{sec:WLi}), in our WTTS/CTTS classification.
If the ${\rm H}\alpha$\ is in emission and $W({\rm Li})$$>$100\,m\AA, the star is identified as a T\,Tauri.
Following \cite{2003ApJ...582.1109W}, the T\,Tauri star is then classified as CTTS if ${\rm H}\alpha\,10\%$$\ge$270 km\,s$^{-1}$.
A comparison of $W({\rm H}\alpha$)\ vs. ${\rm H}\alpha\,10\%$\ for all young open clusters observed in the first 18 months of observations is shown in Fig.\,\ref{fig:giraffe_Ha10_EWHaAcc_pms}.
Note that the correlation of the two parameters is as expected from other works \citep[e.g.,][]{2003ApJ...582.1109W} and the fraction of CTTS consistently decreases with the age of the cluster.
\subsection{Li\ equivalent width}
\label{sec:WLi}
\begin{figure*}[ht]
\centering
\includegraphics[width=8.0cm]{GIRAFFE_EWLi_OAPA_Arcetri_gamma2vel.eps}
\includegraphics[width=8.0cm]{UVES_EWLi_Arcetri_OACt_gamma2vel.eps}
\caption{Illustrative node-to-node $W({\rm Li})$\ comparison for $\gamma$~Vel. Left panel: Comparison between OAPA (DAOSPEC, iDR1) and Arcetri (iDR2) for GIRAFFE spectra. Right panel: Comparison between Arcetri code and OACT (IRAF) for UVES spectra. Arrows indicate upper limits.}
\label{fig:ewlicomparison_gammaVel}
\end{figure*}
At young ages, the Li\ doublet line is often in the saturation regime and the rotational broadening frequently dominates.
As a consequence, in general, a direct profile integration of the Li\ line is to be preferred to a Gaussian or a Voigt profile fitting in deriving $W({\rm Li})$.
Furthermore, due to rotational broadening, the integration wavelength interval is very different from one spectrum to another.
The Li\ doublet is also superimposed to molecular bands in spectra of M-type stars, which makes the placement of the continuum difficult, particularly when using automatic procedures.
In such cases, despite being slow and prone to human error and subjective choices, interactive procedures, like those available in \texttt{IRAF}, remain one of the best options for measuring $W({\rm Li})$, at least for comparison purposes.
Weak Li\ lines in slow-rotating stars, on the other hand, can reliably be fitted with a Gaussian or a Voigt profile and integrated analytically, a method that can be easily implemented in automatic procedures and is more accurate than low-order numerical integration at low S/N.
The Gaia-ESO PMS analysis makes use of three independent methods for deriving $W({\rm Li})$\ from the GIRAFFE spectra: the direct profile integration available in the \texttt{IRAF-splot} procedure (OACT node); DAOSPEC \citep[][OAPA node]{2008PASP..120.1332S}; and a semi-automatic IDL procedure specifically developed for the Gaia-ESO by the Arcetri node.
The \texttt{IRAF-splot} task was applied by the OACT node to the unnormalised spectra to make use of the built-in Poisson statistics model of the data.
Such measurements are only performed in those cases where the Li\ line and the nearby continuum are clearly identifiable, which implies that, in general, small $W({\rm Li})$\ ($\apprle 10$ m\AA), low S/N ($\apprle 20$), and spectra with very high $v\sin i$\ ($\apprge 200$ km\,s$^{-1}$) are not considered.
The DAOSPEC (OAPA) measurements were applied to all iDR1 spectra with S/N$>20$, and spectra with S/N$<20$ showing a strong lithium line.
The spectra have been re-normalised prior to the equivalent width determination using high order Legendre polynomial fitting, which allows to follow the shape of molecular bands in M-type stars still maintaining a good agreement with the continuum of earlier type stars.
The typical width of absorption lines in each spectrum has been estimated by convolving the instrumental and rotational profile using $v\sin i$\ from the data reduction pipeline.
Relative internal uncertainties are always better than 5\% for large equivalent widths ($>200$\,m\AA) and degrade up to $\sim$ 50\% for very low equivalent widths ($\sim 10$\,m\AA).
The semi-automatic IDL procedure developed by the Arcetri node performs a spline fitting of the continuum over a region of $\pm 20$~\AA\ around the Li line using an iterative $\sigma$-clipping, and masking
both the Li line and the nearby \ion{Ca}{I} line at 6717.7\,\AA.
When the automatic continuum fitting is not satisfactory (generally for poor S/N spectra or M-type stars), the fit is repeated by setting manually the continuum level.
The $W({\rm Li})$\ is then computed by direct integration of the line within a given interval, which depends on the stellar rotation and was determined by measuring the line widths on a series of rotationally broadened synthetic spectra.
Errors are derived using the \cite{1988IAUS..132..345C} formula; when no Li line (including blends) is visible, the upper limit is set as three times the error.
The contribution of lines blended with Li\ in the GIRAFFE spectra was estimated, after the determination of the fundamental parameters (Sect.\,\ref{sec:FundamentalParameters}), by a spectral synthesis using Spectroscopy Made Easy \citep[SME,][]{1996A&AS..118..595V} with MARCS model atmospheres as input, taking the star's $T_{\rm eff}$, $\log g$, and [Fe/H]\ into account.
For solar metallicity dwarfs above 4000\,K the estimated blends are in agreement with the \cite{Soderblom_etal:1993} relation.
Four nodes (Arcetri, CAUP, OACT, and UCM) calculated $W({\rm Li})$\ in the UVES spectra.
At the UVES resolution, when the star is slow rotating ($v\sin i$$\apprle 25$ km\,s$^{-1}$) and the S/N is sufficiently high (S/N$\apprge 60$), it is possible to de-blend the Li\ line from the nearby features.
Both the CAUP and UCM nodes employed the \texttt{splot} task in IRAF on the unnormalised UVES spectra.
When the Li\ line and the nearby blends, mainly with \element{Fe} lines, are distinguishable, these are de-blended, in which case a Gaussian fitting to the line profile is adopted.
On the contrary, when the lines are indistinguishable, the blends contribution is estimated using the \texttt{ewfind} driver within \texttt{MOOG} code \citep{sneden}, and a direct integration of the line is adopted.
The Arcetri node adopted the same method used for GIRAFFE (see above), except in those cases where it was possible to de-blend the line using IRAF as done by the CAUP and UCM nodes.
When this was not possible, the blends were estimated using SME.
For iDR1, the OACT node employed IRAF as for the GIRAFFE spectra, using SME to estimate the blends.
For iDR2, $W({\rm Li})$\ was derived by spectra subtraction with the template having the closest fundamental parameters but no (or negligible) Li absorption.
In this latter case the blends are removed by the spectra subtraction itself.
It is worth stressing that the PMS analysis output includes both blends-corrected and -uncorrected $W({\rm Li})$.
When a node does not provide blends-corrected $W({\rm Li})$, this is estimated using SME and the node's fundamental parameters if available.
Note that, in the analysis of GIRAFFE spectra, blends are estimated using SME in all cases; the recommended blends-corrected $W({\rm Li})$\ are calculated from the recommended blends-uncorrected $W({\rm Li})$\ using the recommended fundamental parameters.
Conversely, in the analysis of UVES spectra the recommended blends-corrected $W({\rm Li})$\ are derived by averaging the node values, as discussed below.
The blends-uncorrected $W({\rm Li})$\ obtained by the three different methods from GIRAFFE spectra were first compared to check for systematic differences before combining them to produce the final results (see Fig.\,\ref{fig:ewlicomparison_gammaVel} for an illustrative comparison).
After discarding results of one node not consistent with the other two, no significant bias remained and the relative standard deviation of the $W$ difference was at 20\% level.
Also, no trend of the node-to-node differences with respect to S/N nor $v\sin i$\ was present in the selected results.
As a conservative uncertainty estimate on the {\it recommended} $W({\rm Li})$\ we adopted the larger of the standard deviation and the mean of the individual method uncertainties.
In iDR1, 90\% of the $W({\rm Li})$\ measurements have relative uncertainties better than 14\% and 28\% in $\gamma$ Vel and Cha\,I, respectively, the differences being due to the higher fraction of stars of low $T_{\rm eff}$\ and spectra with lower S/N in Cha\,I with respect to $\gamma$ Vel.
About 90\% of all the iDR2-GIRAFFE $W({\rm Li})$\ measurements have uncertainty better than 16\,m\AA.
In the $W({\rm Li})$\ UVES measurements no
systematic deviation nor trends of the node-to-node differences with S/N nor $v\sin i$\ were found from the node-to-node comparison (see Fig.\,\ref{fig:ewlicomparison_gammaVel} for an illustrative comparison) and the recommended values were derived by taking the mean with a 1$\sigma$-clipping.
In iDR1
the median uncertainties are 3\,m\AA\ (4\%) and 10\,m\AA\ (3\%) for $\gamma$ Vel and Cha\,I, respectively\footnote{Note that, given the small number of measurements in UVES compared to GIRAFFE, the median uncertainty is chosen to characterise the internal precision achieved rather than the distribution.}.
About 90\% of all the iDR2-UVES $W({\rm Li})$\ measurements have uncertainty better than 22\,m\AA.
When all $W({\rm Li})$\ measurements for a given star are flagged as upper limits, the recommended $W({\rm Li})$\ is also flagged as an upper limit and the lowest measurement is adopted.
Conversely, when at least one $W({\rm Li})$\ measurement for a given star is not flagged as upper limit, all upper limit estimates for that star are disregarded and the recommended value is derived as above.
\begin{figure}[ht]
\centering
\includegraphics[width=8.0cm]{giraffe_Teff_EWcLi_pms.eps}
\caption{Blends-corrected $W({\rm Li})$\ vs. $T_{\rm eff}$\ for all young open clusters observed in the first 18 months of observations. For comparison, the lower and upper envelope in the Pleiades are shown as solid lines, while dashes and dotted lines are the upper envelopes of IC2602 and the Hyades, respectively. Dots are used for non-members. Upper limits are not included for clarity.}
\label{fig:giraffe_Teff_EWcLi_pms}
\end{figure}
{\em Recommended} blends-corrected $W({\rm Li})$\ vs. $T_{\rm eff}$\ are shown in Fig.\,\ref{fig:giraffe_Teff_EWcLi_pms}, compared with the Pleiades upper and lower envelope and the upper envelopes of IC2602 (30 Myr) and the Hyades \citep[][and references therein]{2005A&A...442..615S}.
Note the increasing Li depletion with age at $T_{\rm eff}$$\gtrsim$ 3500\,K and the lack of depletion at lower $T_{\rm eff}$\ as expected.
A comparison with theoretical models can be found in \cite{2014A&A...563A..94J} for $\gamma$ Vel.
The NGC2547 Li depletion pattern is found in remarkable agreement with \cite{2005MNRAS.358...13J}.
\section{Fundamental parameters}
\label{sec:FundamentalParameters}
Two nodes (OACT and OAPA) provide fundamental parameters from the analysis of GIRAFFE spectra, and four nodes (Arcetri, CAUP, OACT, and UCM) from the analysis of UVES spectra.
With the exception of OACT, the nodes analysing UVES spectra use similar, classical procedures, i.e. measurement of equivalent widths and MOOG \citep{sneden}, enforcing the usual equilibrium relations.
However, different strategies are adopted for the selection of the lines to be used and in the automatisation of the procedures, as described in Sects.\,\ref{sec:FunparsArcetri}--\ref{sec:FunparsUCM}.
As anticipated in Sect.\,\ref{sec:GeneralStrategy}, the validation of fundamental parameters is carried out internally by a node-to-node comparison (Sect.\,\ref{sec:internal}), and externally by comparisons with $T_{\rm eff}$\ and $\log g$\ derived from angular diameter measurements on a sample of stars taken as benchmark (Sect.\,\ref{sec:benchmarks})
\subsection{OACT}
A code that has been extensively used for determining fundamental parameters in PMS stars is \texttt{ROTFIT} \citep[e.g.,][]{2006A&A...454..301F}, which compares the target spectrum with a set of template spectra from ELODIE observations of slowly-rotating non-active stars \citep{2001A&A...369.1048P} artificially rotation-broadened and {\it veiled} at varying $v\sin i$\ and~$r$.
In the following we report a brief summary of the method implemented by \texttt{ROTFIT}, together with some adaptations to the case at hand.
In \texttt{ROTFIT}, the template spectra that most closely reproduce the target spectrum when broadened and {\it veiled} are selected, and their weighted average $T_{\rm eff}$, $\log g$, and [Fe/H]\ assigned to the target star.
As a figure of merit the $\chi^2$ calculated on the target spectrum and the rotational-broadened and veiled template spectrum is adopted.
The weight used in the average is proportional to $\chi^{-2}$.
A discussion on the \texttt{ROTFIT} templates, together with the homogenisation with the Gaia-ESO spectrum analysis, is reported in Appendix\,\ref{sec:ROTFIT_templates}.
The \texttt{ROTFIT} analysis requires different wavelength masks for different type of objects.
The masking criteria for the application to the Gaia-ESO survey are reported in Appendix\,\ref{sec:ROTFIT_masks}.
For the GIRAFFE/HR15N spectra the whole spectral range from 6445 to 6680\,\AA\ and the ten best templates (i.e. with the lowest $\chi^2$) are considered.
The analysis of the UVES 580 spectra is independently performed on segments of 100\,\AA\ each (excluding the parts containing strong telluric lines and the core of Balmer lines), by considering only the best five templates for each segment.
The final parameters $T_{\rm eff}$, $\log g$, [Fe/H], and $v\sin i$\ are obtained by taking the weighted averages of the mean values for each segment, with the weight being proportional to $\chi^{-2}$ and to the amount of information contained in the segment, quantified by the total line absorption $f_i=\int(F_{\lambda}/F_{\rm C}-1)d\lambda$
(where $F_{\lambda}$/$F_{\rm C}$ is the continuum normalised flux).
The $T_{\rm eff}$, $\log g$, [Fe/H], and $v\sin i$\ uncertainties are given as the standard errors of the weighted means, to which the average uncertainties of the templates' stellar parameters are added quadratically.
These are estimated to be $\pm$\,50\,K, $\pm$\,0.1\,dex, $\pm$\,0.1\,dex, and $\pm$\,0.5\,km\,s$^{-1}$\ for $T_{\rm eff}$, $\log g$, [Fe/H], and $v\sin i$, respectively.
The target's spectral type corresponds to that of the best template.
The code also provides an estimate of the veiling by searching for the $r$ value that gives the lowest $\chi^2$.
The determination of the fundamental parameters for a star with veiling is more uncertain than in the non-veiled case because:
(a) lines' and molecular bands' depth are smaller in veiled spectra; and
(b) the determination of the veiling parameter $r$ implies the introduction of an additional degree of freedom in the parameters' fitting, degrading the overall accuracy with respect to the non-veiled stars.
However, a preliminary identification of stars with mass accretion, whose spectra are expected to be affected by veiling, can be done based on the values of ${\rm H}\alpha\,10\%$\ or $W({\rm H}\alpha$)\ \citep{2003ApJ...582.1109W,2004A&A...424..603N}.
It is therefore possible to restrict the veiling calculation to likely accreting stars only, thus preserving the accuracy for stars for which no veiling is expected.
Following \cite{2003ApJ...582.1109W}, we assume that stars with ${\rm H}\alpha\,10\%$$>$270 km\,s$^{-1}$\ can be optically veiled, with an extra margin to take uncertainties into account (see Sect.\,\ref{sec:halpha}).
Within the Gaia-ESO analysis, $v\sin i$\ is also provided by WG8 for GIRAFFE spectra (Koposov et al., in prep.),
together with radial velocities and a first estimate of fundamental parameters, using a conceptually similar approach but with a different fitting strategy and different templates.
The comparison between the results of these two methods turned out to be useful in identifying WG8 unsuccessful fitting for some stars with strong veiling and emission lines.
An illustrative comparison of the results of these two methods for $\gamma$ Vel can be found in \cite{Frasca_etal:2014}, who found a mean difference of $\approx$ 2.88 km\,s$^{-1}$\ and $\sigma \approx 6.27$ for stars in the field of $\gamma$ Vel.
An investigation on the lower limit imposed by the resolution of the instruments by means of Monte-Carlo simulations is also reported in \cite{Frasca_etal:2014}, according to which the lower $v\sin i$\ limit has been set to 7 km\,s$^{-1}$\ in GIRAFFE spectra and 3 km\,s$^{-1}$\ in UVES spectra.
\subsection{OAPA}
An alternative approach for GIRAFFE/HR15N spectra, proposed by \cite{Damiani_etal:2014}, is based on spectral indices in different wavelength ranges of the spectrum.
The derived spectral indices are calibrated against stars with known parameters,
yielding quantitative estimates of $T_{\rm eff}$, $\log g$, and [Fe/H].
This type of approach is usually applied to spectra with lower resolution than GIRAFFE.
These narrow-band indices are affected by fast rotation: $T_{\rm eff}$\ becomes unreliable for $v\sin i$$>$90 km\,s$^{-1}$, [Fe/H]\ above 70 km\,s$^{-1}$, and $\log g$\ above 30 km\,s$^{-1}$.
Therefore, depending on the $v\sin i$\ of the star, not all parameters can be provided.
Using an appropriate combination of flux ratios, this method is also capable of producing an independent estimate of the veiling parameter $r$ \citep[see,][for details]{Damiani_etal:2014}.
\subsection{Arcetri}
\label{sec:FunparsArcetri}
The Arcetri node adopted \texttt{DOOp} \citep[DAOSPEC Option Optimiser pipeline,][]{2014A&A...562A..10C} for measuring line equivalent widths and \texttt{FAMA} \citep{2013A&A...558A..38M} for determining the fundamental parameters.
Line equivalent widths are measured using Gaussian fitting after a re-normalisation of the continuum;
$W$ in the range between 20--120 m\AA, for the \ion{Fe}{i} and \ion{Fe}{ii} lines, and in the range between 5--120 m\AA, for the other elements, were used. The \texttt{FAMA} code makes use of the \ion{Fe}{i} and \ion{Fe}{ii} equivalent widths to derive the fundamental stellar parameters.
The stellar parameters are obtained by searching iteratively for the three equilibria (excitation, ionisation, and trend between $\log n$(\ion{Fe}{i}) and $\log (W/\lambda)$), i.e. with a series of recursive steps starting from a set of initial atmospheric parameters and arriving at a final set of atmospheric parameters which fulfils the three equilibrium conditions.
The convergence criterion is set using the information on the quality of the $W$ measurements, i.e. the minimum reachable slopes are linked to the quality of the spectra, as expressed by the dispersion around the average $<\log n$(\ion{Fe}{i})$>$.
This is correct in the approximation that the main contribution to the dispersion is due to the error in the $W$ measurement rather than to inaccuracy in atomic parameters, as e.g. the oscillator strengths ($\log gf$).
\subsection{CAUP}
The fundamental parameters are automatically determined with a method used in previous works \citep[e.g.][]{2008A&A...487..373S,2011A&A...526A..99S} now adapted to the Gaia-ESO survey.
The method is based on the excitation and ionisation balance of iron lines using [Fe/H]\ as a proxy for the metallicity.
The iron lines constraining the parameters were selected from the Gaia-ESO line list using a new procedure detailed in \cite{2014A&A...561A..21S}.
The equivalent widths are automatically measured using the \texttt{ARES}\footnote{\texttt{ARES} is available for download at http://www.astro.up.pt/~sousasag/ares/} code \citep{2007A&A...469..783S} following the approach of \cite{2008A&A...487..373S,2011A&A...526A..99S} that takes the S/N of each spectrum into account.
The stellar parameters are computed assuming LTE using the 2002 version of \texttt{MOOG} \citep{sneden} and the MARCS grid of atmospheric models.
For this purpose, the interpolation code provided with the MARCS grid was modified to produce an output model readable by \texttt{MOOG}.
Moreover, a wrapper program was implemented to the interpolation code to automatise the method.
The atmospheric parameters are inferred from the previously selected \ion{Fe}{i}-\ion{Fe}{ii} line list.
The downhill simplex \citep{1992nrca.book.....P} minimisation algorithm is used to find the best parameters. In order to identify outliers caused by incorrect $W$ values, a 3$\sigma$-clipping of the \ion{Fe}{i} and \ion{Fe}{ii} lines is performed after a first determination of the stellar parameters.
After this clipping, the procedure is repeated without the rejected lines.
The uncertainties in the stellar parameters are determined as in previous works \citep{2008A&A...487..373S,2011A&A...526A..99S}.
\subsection{UCM}
\label{sec:FunparsUCM}
The UCM node employed the \texttt{StePar} code \citep{2012A&A...547A..13T,2013hsa7.conf..673T}, modified to operate with the spherical and non-spherical MARCS models.
For iDR1 the $W$ measurements were carried out with the \texttt{ARES} code \citep{2007A&A...469..783S}\footnote{The approach of \cite{2008A&A...487..373S} to adjust the \texttt{ARES} parameters according to the S/N of each spectrum was followed.}.
For iDR2 the UCM node adopted the \texttt{TAME} code \citep{2012MNRAS.425.3162K}\footnote{\texttt{TAME} is a tool that can be run in automated or manual mode.}
The manual mode has an interface that allows the user control over the $W$ measurements to verify problematic spectra when needed.
The \cite{2012MNRAS.425.3162K} approach to adjust the \texttt{TAME} continuum $\sigma$ rejection parameter according to the S/N of each spectrum was followed.
The \texttt{StePar} code computes the stellar atmospheric parameters using \texttt{MOOG} \citep[][]{sneden}.
The 2002 and 2013 versions of \texttt{MOOG} were used in iDR1 and iDR2, respectively.
Five line lists were built-up for different regimes: metal rich dwarfs, metal poor dwarfs, metal rich giants, metal poor giants and extremely metal poor stars.
The code iterates until it reaches the excitation and ionisation equilibrium and minimises trends of abundance vs. $\log (W/\lambda)$.
The downhill simplex method \citep{1992nrca.book.....P} was employed to minimise a quadratic form composed of the excitation and ionisation equilibrium conditions.
The code performs a new simplex optimisation until the metallicity of the model and the iron abundance are the same.
Uncertainties for the stellar parameters are derived as described in \cite{2012A&A...547A..13T,2013hsa7.conf..673T}.
In addition, a 3$\sigma$ rejection of the \ion{Fe}{i} and \ion{Fe}{ii} lines is performed after a first determination of the stellar parameters;
\texttt{StePar} is then re-run without the rejected lines.
\subsection{Comparison with benchmark stars}
\label{sec:benchmarks}
\begin{figure*}[htp]
\centering
\includegraphics[width=80mm]{benchmarks_giraffe_teff.eps}
\hspace{-0.5cm}
\includegraphics[width=80mm]{benchmarks_giraffe_logg.eps}
\hspace{-0.5cm}
\includegraphics[width=80mm]{benchmarks_giraffe_feh.eps}
\caption{GIRAFFE benchmarks comparison. Black for [Fe/H] $<$ 0.1, blue for [Fe/H] $>0.1$. Size is inversely proportional to $\log g$\ in the $T_{\rm eff}$\ plot, proportional to $T_{\rm eff}$\ in the $\log g$\ and [Fe/H]\ plot.}
\label{fig:GIRAFFE_benchmarks_comparison}
\end{figure*}
\begin{table*}[ht]
\centering
\caption{Average differences from benchmark reference values (see Fig.\,\ref{fig:GIRAFFE_benchmarks_comparison} and \ref{fig:UVES_benchmarks_comparison}). The Arcetri's [Fe/H]\ results are offset by 0.09 dex before computing $\Delta$[Fe/H]\ and $\sigma$[Fe/H]\ (see text for details).}
\begin{tabular}{rrrrrrrrr}
\hline
& $\langle\Delta T_{\rm eff}\rangle$ & $\langle\sigma (T_{\rm eff})\rangle$ & $\langle\Delta \log g\rangle$ & $\langle\sigma (\log g)\rangle$ & $\langle\Delta$ [Fe/H]$\rangle$ & $\langle\sigma($[Fe/H]$)\rangle$ & $\langle\Delta \xi\rangle$ & $\langle\sigma ({\xi})\rangle$ \\
\hline
\multicolumn{9}{c}{GIRAFFE} \\
\hline
OACT & 50. & 124. & 0.19 & 0.29 & $-$0.04 & 0.14 & \dots & \dots \\
OAPA & 18. & 120. & $-$0.15 & 0.28 & $-$0.03 & 0.16 & \dots & \dots \\
\hline
\multicolumn{9}{c}{UVES} \\
\hline
OACT & 38. & 124. & 0.15 & 0.26 & 0.05 & 0.18 & \dots & \dots \\
Arcetri & 55. & 95. & 0.14 & 0.19 & 0.10 & 0.12 & 0.00 & 0.31 \\
CAUP & 34. & 96. & $-$0.02 & 0.28 & $-$0.03 & 0.08 & 0.04 & 0.33 \\
UCM & 56. & 90. & 0.09 & 0.25 & $-$0.01 & 0.08 & 0.04 & 0.38 \\
\hline
\end{tabular}
\label{tab:benchmarks}
\end{table*}
The precision of the fundamental parameters can be assessed by comparison with results from accurate independent methods like interferometric angular diameter measurements \citep[e.g.][]{2012ApJ...746..101B,2012ApJ...757..112B} which, in combination with the \textsc{Hipparcos} parallax and measurements of the star's bolometric flux, allow the computation of absolute luminosities, linear radii, and effective temperatures.
As part of the Gaia-ESO activities, and also in support of the Gaia mission, a list of stars with accurate fundamental parameters derived from such independent methods is being compiled (\citealt{2014A&A...564A.133J}; Heiter et al., in prep.) and included in the Gaia-ESO target list.
For the range of parameters of interest to the PMS analysis, however, only very few benchmark stars spectra are available in iDR1 and iDR2.
A comparison of the iDR2-GIRAFFE fundamental parameters of benchmark stars with those compiled from the literature is shown in Fig.\,\ref{fig:GIRAFFE_benchmarks_comparison} and Table\,\ref{tab:benchmarks}, in the range of interest.
In this case the $T_{\rm eff}$\ deviations are mostly within $\approx200$\,K.
There is a systematic large deviation of OACT values above 6000\,K.
At lower temperatures, deviations larger than $\approx200$\,K are found in OAPA results for \object{HD\,10700}.
Therefore, although the sample analysed is limited, good results are found for both nodes, except for OACT above 6000\,K.
Excluding the OACT values above this limit, the standard deviation is $\approx$ 120\,K for both datasets.
The OACT $T_{\rm eff}$\ upper limit for the GIRAFFE analysis was further lowered to 5500\,K based on comparison with $T_{\rm eff}$\ from photometry (Appendix\,\ref{sec:photometry}; see also Sect.\,\ref{sec:internal}).
Deviations as large as almost 0.7 dex in $\log g$\ are found in the comparison with the benchmarks, with standard deviation $\approx$ 0.3 dex for both datasets.
From the comparison with benchmarks alone it is not possible to identify a range in which one method performs better than the other.
Indeed the node-to-node comparisons for each cluster outlined a rather complex situation that leads to the parameters selection described in Sect.\,\ref{sec:internal}.
In the parameters range of interest
(i.e. excluding very metal-poor stars),
[Fe/H]\ is approximately reproduced with a maximum deviation of 0.3 dex and a standard deviation of $\approx$ 0.15 dex.
\begin{figure*}[htp]
\centering
\includegraphics[width=80mm]{benchmarks_uves_teff.eps}
\includegraphics[width=80mm]{benchmarks_uves_logg.eps}
\includegraphics[width=80mm]{benchmarks_uves_feh.eps}
\includegraphics[width=80mm]{benchmarks_uves_xi.eps}
\caption{UVES benchmarks comparison. Black for [Fe/H] $<$ 0.1, blue for [Fe/H] $>0.1$. Size inversely proportional to $\log g$\ in the $T_{\rm eff}$\ and $\xi$ plot, proportional to $T_{\rm eff}$\ in the $\log g$\ and [Fe/H]\ plot. Reference $\xi$ from Bergemann et al. (2014, in prep.) \citep[see also][]{Smiljanic_etal:2014}.}
\label{fig:UVES_benchmarks_comparison}
\end{figure*}
The comparison of the UVES fundamental parameters of benchmarks with those compiled from the literature is shown in Fig.\,\ref{fig:UVES_benchmarks_comparison} and Table\,\ref{tab:benchmarks}.
The results for the solar spectrum are outlined in Table\,\ref{tab:UVES_solar}.
In general $T_{\rm eff}$\ deviations from benchmarks are all within 300\,K (maximum) with a few outliers.
Amongst these, UCM $T_{\rm eff}$\ for \object{61\,Cyg\,A} differs by about 800\,K,
but this large deviation does not point to particular problems in some parameters' range as verified through the node-to-node comparison.
The OACT systematic discrepancies in $T_{\rm eff}$\ above 6500\,K, on the other hand, indicate a $T_{\rm eff}$\ upper limit also for the validity of \texttt{ROTFIT} UVES analysis.
This discrepancy is seen also in the node-to-node comparison, on which we estimate an upper limit of 6200\,K for the validity of the OACT results.
Excluding such outliers, the standard deviation is $\approx$ 100\,K for the Arcetri, CAUP, and UCM nodes, and $\approx$ 120\,K for the OACT node, with average difference of 34\,K for CAUP, 38\,K for OACT, and 55\,K for Arcetri and UCM.
Within the UVES dataset of young open clusters, very few sources have $T_{\rm eff}$\ $<$4000\,K.
In this range recommended data are based on OACT results only,
as the presence of molecular bands prevents to carry out analysis based on MOOG.
\begin{table*}[ht]
\centering
\caption{UVES results on the solar spectrum.}
\begin{tabular}{rrrrrrrrr}
\hline
& $\Delta T_{\rm eff}$ & $\sigma T_{\rm eff}$ & $\Delta \log g$ & $\sigma \log g$ & $\Delta$ [Fe/H] & $\sigma $ [Fe/H] & $\Delta \xi$ & $\sigma_{\xi}$ \\
\hline
OACT & $-$1. & 67. & $-$0.14 & 0.11 & 0.06 & 0.10 & \dots & \dots \\
Arcetri & 49. & 150. & 0.09 & 0.20 & 0.09 & 0.07 & 1.00 & 0.15 \\
CAUP & $-$48. & 59. & $-$0.18 & 0.12 & 0.00 & 0.07 & 0.87 & 0.08 \\
UCM & 9. & 48. & 0.00 & 0.11 & 0.03 & 0.04 & 0.75 & 0.08 \\
\hline
\end{tabular}
\label{tab:UVES_solar}
\end{table*}
The agreement in $\log g$\ is approximately at the same level for all nodes.
Benchmarks' $\log g$\ is generally reproduced within a maximum deviation of $\approx0.7$ dex and a standard deviation of $\approx0.3$, only one Arcetri value deviating more than that.
[Fe/H]\ is generally reproduced within a maximum deviation of $\approx0.3$ dex , except one and two measurements by the Arcetri and OACT nodes, respectively, with deviations of $\approx0.5$ dex.
The standard deviation is $\lesssim 0.1$ dex for the CAUP and UCM nodes, $\approx0.2$ dex for the Arcetri node, and $\approx0.3$ for the OACT node.
The OACT node tends to overestimate (underestimate) the metallicity below (above) [Fe/H]=0.
However, this does not lead to significant systematic differences in individual clusters and the OACT results are therefore maintained.
The node-to-node comparisons for individual clusters show that the Arcetri node systematically overestimates [Fe/H], which is not evident in the comparison with the benchmarks possibly because of the large and coarse parameters' distribution of this latter.
To overcome this systematic behaviour, in iDR2 the value obtained by the Arcetri node for the solar spectrum ([Fe/H]=0.09, see Table\,\ref{tab:UVES_solar}) is subtracted in all measurements before computing the recommended [Fe/H].
The recommended [Fe/H]\ agrees with the benchmarks within 0.15\,dex r.m.s.
Solutions with large uncertainties or large $\xi$ ($\gtrsim 2$ km\,s$^{-1}$) are disregarded by the nodes.
Differences in $\xi$ with respect to the values tabulated for the benchmarks are generally below 1 km\,s$^{-1}$.
The recommended fundamental parameters are therefore computed taking the average of the nodes' results with a 1$\sigma$-clipping when at least 3 values are provided.
As discussed above, below 4000\,K only the OACT values are given as recommended values.
In iDR2 we disregarded the OACT UVES values for $T_{\rm eff}$$>$6200\,K.
Note that, despite the large difference in resolution and spectral range, the comparison with benchmarks shows that the UVES $T_{\rm eff}$\ accuracy is only marginally better than GIRAFFE's, while $\log g$\ and [Fe/H]\ results from the two setups are of comparable accuracy.
Our recommended values include $T_{\rm eff}$\ and [Fe/H]\ for 11 stars and $\log g$\ for 3 stars (see Sect.\,\ref{sec:internal}) from both the UVES and GIRAFFE setups.
The comparison of our results for the same stars in the two setups shows that the $T_{\rm eff}$\ ratio (GIRAFFE/UVES) has a mean of 0.99 and a median of 1.00.
The differences in [Fe/H]\ (GIRAFFE-UVES) have a mean of 0.13\,dex and a median of 0.16\,dex.
Among the 3 benchmark stars for which we give recommended $\log g$\ from both GIRAFFE and UVES setups according to the criteria described in Sect.\,\ref{sec:internal}, two are in the range of interest ($\log g$ $\approx$ 4.0) and the maximum difference with the benchmark value is -0.09\,dex.
\subsection{Internal comparison}
\label{sec:internal}
The node-to-node comparison for the UVES individual cluster results before data selection and calibration (see Sect.\,\ref{sec:benchmarks}) gives systematic differences in the ranges 80--160\,K in $T_{\rm eff}$, 0.1--0.3\,dex in $\log g$, and 0.06--0.17\,dex in [Fe/H], while dispersions are in the ranges 160--260\,K in $T_{\rm eff}$, 0.1--0.3\,dex in $\log g$, 0.13--0.45\,dex in [Fe/H].
The application of the data selection and calibration discussed in Sect.\,\ref{sec:benchmarks} reduces systematic differences below 100\,K in $T_{\rm eff}$, and below 0.15\,dex in [Fe/H].
The final node-to-node mean dispersion in the recommended data is 110\,K in $T_{\rm eff}$, 0.21\,dex in $\log g$, and 0.10\,dex in [Fe/H].
These values are very close to the median dispersion: 106\,K in $T_{\rm eff}$, 0.17\,dex in $\log g$, and 0.11\,dex in [Fe/H].
Biases in the recommended data are therefore successfully reduced.
For the GIRAFFE results, systematic differences before data selection are in the ranges 110--200\,K in $T_{\rm eff}$, 0.4--0.8\,dex in $\log g$, and 0.01--0.03 in [Fe/H], while dispersions are in the ranges 210--330\,K in $T_{\rm eff}$, 0.65--1.00\,dex in $\log g$, and 0.17--0.26\,dex in [Fe/H].
The situation here is more complex than in the UVES case. The problems to address are:
\begin{enumerate}
\item[(1)]{The OACT (\texttt{ROTFIT}) $\log g$\ for PMS stars tend to be too high, clustering essentially on the MS\footnote{This is due to the basic criteria for defining the templates, identified as slow rotators, inactive stars and with no significant Li-absorption, which imply that no PMS star can be taken as template.};}
\item[(2)]{The OAPA $\log g$\ for PMS stars tends to be too low, often lower than suggested by models\footnote{An absolute calibration of the gravity-sensitive spectral index in the PMS is very difficult (or impossible with currently available data) because of the lack of suitable PMS calibrators.};}
\item[(3)]{The PMS domain is contaminated by non-members with spurious $\log g$\ in both $\log g$-$T_{\rm eff}$\ diagrams.}
\item[(4)]{The RGB in the OACT $\log g$-$T_{\rm eff}$\ diagram follows the calibrated relation taken from \cite{Cox:2000}, while in the OAPA diagram it doesn't.}
\item[(5)]{The OAPA $\log g$-$T_{\rm eff}$\ diagram outside the MS, PMS, and RGB domains is sparsely populated, with both some very low and very high values, which are indicative of possible presence of some large errors.}
\item[(6)]{The OAPA $T_{\rm eff}$, $\log g$, and [Fe/H]\ are valid for $v\sin i$ $<$ 90, 30, and 70 km\,s$^{-1}$, respectively.}
\item[(7)]{Because of a continuum normalisation problem on the ${\rm H}\alpha$\ wings, in iDR1 and iDR2 the OACT parameters need to be discarded for $T_{\rm eff}$ $>$ 5500\,K.}
\end{enumerate}
In order to reduce biases as much as possible and provide reliable recommended results we adopt the following solution:
\begin{enumerate}
\item{The OAPA $T_{\rm eff}$\ are considered only for $v\sin i$\ $<$ 90 km\,s$^{-1}$. The OACT $T_{\rm eff}$\ are considered only below 5500\,K. In cases where both the OACT and OAPA $T_{\rm eff}$\ are available these are averaged. In all other cases the remaining value, if any, is adopted as recommended $T_{\rm eff}$.}
\item{The OAPA [Fe/H]\ are considered only for $v\sin i$\ $<$ 70 km\,s$^{-1}$. The OACT [Fe/H]\ are considered only below 5500\,K. In cases where both the OACT and OAPA [Fe/H]\ are available these are averaged. In all other cases the remaining value, if any, is adopted as recommended [Fe/H].}
\item{The OAPA $\log g$\ are considered only for $v\sin i$\ $<$ 30 km\,s$^{-1}$. The OACT $\log g$\ are considered only for $T_{\rm eff}$$<$5500\,K. In cases where both the OACT and OAPA $\log g$\ are available these are averaged if they differ by less than 0.3 dex. When they differ by more than 0.3 dex, if the OACT $\log g$\ $>$ 4.2 and the OAPA $\log g$\ $>$ 5, the OACT $\log g$\ is given as recommended value. In all other cases we do not give recommended $\log g$.}
\item{The OAPA gravity--sensitive $\gamma$ index \citep{Damiani_etal:2014} is given as a recommended parameter for $v\sin i$\ $<$ 30 km\,s$^{-1}$.}
\end{enumerate}
The application of such criteria leads to a final node-to-node mean dispersion in the recommended data of 98\,K in $T_{\rm eff}$, 0.23\,dex in $\log g$, and 0.14\,dex in [Fe/H].
These values are very close to the median dispersion: 95\,K in $T_{\rm eff}$, 0.22\,dex in $\log g$, and 0.14\,dex in [Fe/H].
Biases in the recommended data are therefore successfully reduced in the GIRAFFE case too.
When a recommended $\log g$\ is not given, it may be still possible to identify an approximate evolutionary status based on the OACT and OAPA results. Those stars for which a trustworthy $\log g$\ cannot be recommended are therefore flagged, when possible, as PMS, MS or post-MS stars according to the criteria listed in Table\,\ref{tab:evolutionary-status}.
\begin{table}[ht]
\centering
\caption{Criteria for the evolutionary status.}
\label{tab:evolutionary-status}
\begin{tabular}{cccl}
\hline\hline
\noalign{\smallskip}
$T_{\rm eff}$ & $\log g$$_{\rm OACT}$ & $\log g$$_{\rm OAPA}$ & Status\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$< 5500$\,K & $> 3$ & $3$--$4.2$ & PMS\\
$< 5500$\,K & $3$--$4.2$ & \dots & PMS\\
\dots & \dots & $> 4.2$ & MS\\
\dots & $< 3$ & $< 3$ & post-MS\\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=90mm]{giraffe_gamma_teff_pms.eps}
\caption{The OAPA gravity-sensitive $\gamma$ index vs. $T_{\rm eff}$\ for all clusters analysed in iDR1 and iDR2. The group of younger clusters ($\rho$ Oph, Cha\,I, and NGC2264) are clearly distinguishable from the group of older clusters ($\gamma$ Vel and NGC2547).}
\label{fig:giraffe_gamma_teff_pms}
\end{figure}
The gravity-sensitive spectral index $\gamma$ obtained by the \cite{Damiani_etal:2014} approach can provide a rank order in age of the clusters.
This can be seen, for the clusters analysed to date, in Fig.\,\ref{fig:giraffe_gamma_teff_pms}, where values for the younger clusters group ($\rho$ Oph, Cha\,I, and NGC2264) are clearly separated from those of older clusters group ($\gamma$ Vel and NGC2547).
However, both the scatter in $\gamma$ and the small age differences between clusters in the younger or the older group still prevent a clear separation in age.
\subsection{Overview in the $\log g$--$T_{\rm eff}$\ plane}
\label{sec:LoggTeff}
As a final check on our recommended fundamental parameters, we examine the $\log g$--$T_{\rm eff}$\ diagram obtained with our data (Fig.\,\ref{fig:giraffe_hr_diagram_pms}) and compare it with the calibration of MK spectral classes reported in \cite{Cox:2000} and the theoretical PMS isochrones from \cite{Allard_etal:2011}.
We note the clustering of field stars on the MS and the RGB, as expected, while for the PMS clusters' members
a residual bias towards the MS and the RGB remains.
In $\lesssim 1/2$ of the cases, $\log g$\ values for PMS stars are located approximately where predicted by the models, although with large uncertainties.
\begin{figure}[htp]
\centering
\includegraphics[width=90mm]{giraffe_hr_diagram_pms_grey.eps}
\caption{$\log g$--$T_{\rm eff}$\ diagram for all targets. Grey filled circles are used for clusters' non-members and stars not classified as CTTS nor WTTS. The red dashed lines are the dwarfs and giants sequences from \cite{Cox:2000}. The blue dot-dashed lines are the isochrones at 1, 5, 10, and 20 Myr from \cite{Allard_etal:2011}.}
\label{fig:giraffe_hr_diagram_pms}
\end{figure}
\subsection{Comparison between fundamental parameters derived from GIRAFFE and UVES}
A number of stars in the $\gamma$ Vel field have been observed with both UVES and GIRAFFE.
For iDR2, our analysis produced $T_{\rm eff}$\ and [Fe/H]\ values for 31 stars and $\log g$\ for 16 stars in this common sample.
Note that the lower number of $\log g$\ values is due to the application of the criteria described in Sect.\,\ref{sec:internal}, which were applied to iDR2 but not to iDR1.
The comparison of the recommended values for this sample is satisfactory (see Fig.\,\ref{fig:best_giraffe_uves} for iDR2) and support the validity of our approach both in the parameters determination and in the derivation of the recommended values.
A similar comparison is reported in \cite{2014A&A...567A..55S} for iDR1.
Indeed the reproducibility of the parameters obtained with the higher resolution and larger wavelength coverage from UVES using a much smaller wavelength range and a lower resolution as in GIRAFFE is a remarkable achievement and increases our confidence in our parameters determination from the much larger GIRAFFE sample.
\begin{figure*}[htp]
\centering
\includegraphics[width=63mm]{Teff_best_GIRAFFE_vs_UVES_gamma2vel.eps}
\hspace{-0.5cm}
\includegraphics[width=63mm]{logg_best_GIRAFFE_vs_UVES_gamma2vel.eps}
\hspace{-0.5cm}
\includegraphics[width=63mm]{feh_best_GIRAFFE_vs_UVES_gamma2vel.eps}
\caption{Comparison between fundamental parameters derived from GIRAFFE and UVES spectra in the $\gamma$ Vel field (iDR2).}
\label{fig:best_giraffe_uves}
\end{figure*}
\subsection{Veiling vs. ${\rm H}\alpha$\ emission}
For iDR1, the \texttt{ROTFIT} veiling parameter was adopted as the recommended one.
In iDR2, however, it was recognised that some residual nebular emission remained after sky-subtraction, particularly in NGC2264, which were not sufficiently masked in the \texttt{ROTFIT} calculations.
As a consequence, the \texttt{ROTFIT} veiling parameter for NGC2264 was clearly overestimated and the OAPA solution was adopted as recommended in iDR2.
Note that this does not invalidate the results of iDR1 as $\gamma$ Vel and Cha\,I spectra are not affected by residual sky emission in the reduced spectra.
\cite{Frasca_etal:2014} found a positive correlation between ${\rm H}\alpha$\ flux and $r$ in the iDR1 data for Cha\,I objects with $r \ge 0.25$, for which the Spearman's rank analysis yielded a coefficient $\rho=0.58$ with a significance of $\sigma=0.003$.
The same analysis for all clusters in iDR2 gives a coefficient $\rho=0.39$ with a significance of $\sigma=0.004$.
However, a correlation between $r$ and ${\rm H}\alpha\,10\%$\ or $W({\rm H}\alpha$)\ is not evident in the iDR2 data (see Fig.\,\ref{fig:veiling_vs_ha}), where we do see an increase of the upper envelope with either ${\rm H}\alpha\,10\%$\ or $W({\rm H}\alpha$), but the large scatter makes the correlation not significant.
This is at variance with what expected from previous work \citep[e.g.,][]{2003ApJ...582.1109W} and therefore it outlines possible limitations in our veiling determination.
Further validation based on comparison with different methods is deferred to future work.
\begin{figure}[t]
\centering
\includegraphics[width=80mm]{iDR2_veil_Ha10.eps}
\includegraphics[width=80mm]{iDR2_veil_HaAcc.eps}
\caption{Veiling parameter $r$ vs. ${\rm H}\alpha\,10\%$\ (top panel) and vs. $W({\rm H}\alpha$)\ (bottom panel) for iDR2. }
\label{fig:veiling_vs_ha}
\end{figure}
\section{Derived parameters}
\label{sec:DerivedParameters}
\subsection{Li abundance}
\label{sec:Li_abundance_methods}
In the whole GIRAFFE analysis, Li abundances, ${\log \epsilon({\rm Li})}$, were computed from the fundamental parameters (Sect.\,\ref{sec:FundamentalParameters}) and the $W({\rm Li})$\ measurements (Sect.\,\ref{sec:WLi}) using the curve of growth (COG) from \cite{Soderblom_etal:1993} and \cite{2007ApJ...659L..41P} above and below 4000\,K, respectively, with a linear interpolation between the tabulated values.
The recommended ${\log \epsilon({\rm Li})}$\ is derived using the recommended fundamental parameters and recommended $W({\rm Li})$\ as input.
Uncertainties were obtained by propagating the $T_{\rm eff}$\ and $W({\rm Li})$\ uncertainties.
The approach adopted in the GIRAFFE case has the advantage of allowing us to focus on the accuracy of the fundamental parameters and $W({\rm Li})$, relying then on the best COG available to derive node-specific and recommended ${\log \epsilon({\rm Li})}$.
Note that the two COGs adopted do not join smoothly at 4000\,K, but the interpolation scheme ensures a smooth transition between the two regimes.
A derivation of a self-consistent COG in the whole $T_{\rm eff}$\ range is planned as a future improvement.
The node-to-node dispersion in the GIRAFFE case (see Fig.\,\ref{fig:sigma_Li1} for the whole iDR2) then propagates only from the $W({\rm Li})$\ measurements and shows a fairly random distribution wit a median of 0.17\,dex.
In the UVES analysis, the OACT and Arcetri ${\log \epsilon({\rm Li})}$\ were derived as in the GIRAFFE case.
The CAUP and UCM nodes, on the other hand, derived ${\log \epsilon({\rm Li})}$\ by a standard LTE analysis using the driver \texttt{abfind} in the revised version of the spectral synthesis code \texttt{MOOG} \citep{sneden} (see also Sects.\, \ref{sec:FundamentalParameters} and \ref{sec:elemental_abundances_methods}).
CAUP used the 2010 version of \texttt{MOOG}, while UCM used the 2002 and 2013 versions for iDR1 and iDR2, respectively.
Uncertainties were estimated by varying each atmospheric parameter within its uncertainty range to derive the propagated uncertainty in ${\log \epsilon({\rm Li})}$.
The propagated uncertainties where then combined quadratically.
In this case the recommended value is given as the average of all nodes' estimates available with a $\sigma$-clipping when at least three measurements are available.
Figure\,\ref{fig:sigma_Li1} shows that also in this case the node-to-node dispersion have a fairly random distribution, with a median uncertainty of 0.12 dex.
Possible $^{6}$Li contribution was neglected in all cases.
\begin{figure}[htp]
\centering
\includegraphics[width=80mm]{histo_li1_uves_idr2.eps}
\includegraphics[width=80mm]{histo_li1_giraffe_idr2.eps}
\caption{Li abundance uncertainty histogram for all sources in iDR2. For the GIRAFFE spectra the ${\log \epsilon({\rm Li})}$\ uncertainty is propagated from the uncertainty in $T_{\rm eff}$\ and $W({\rm Li})$. For the UVES spectra the node-to-node dispersion is considered. A solid line is used for the cumulative probability (right ordinate axis). See text for details.}
\label{fig:sigma_Li1}
\end{figure}
\subsection{Other elemental abundances}
\label{sec:elemental_abundances_methods}
Elemental abundances were computed by three nodes (Arcetri, CAUP, and UCM) when good quality UVES spectra were available in stars not affected by veiling and/or large $v\sin i$.
The Arcetri node computed abundances using \texttt{FAMA}. We refer the reader to \citet{2013A&A...558A..38M} for a description of the method and the way in which lines are selected for the abundance analysis.
The CAUP node derived individual abundances using the driver \texttt{abfind} in the 2010 version
of \texttt{MOOG} \citep[see][for details]{2009A&A...497..563N,2012A&A...545A..32A}
and equivalent widths measured with the \texttt{ARES} code.
The line list for elements other than Fe (with atomic number $A\leq$28) was selected through the cross-matching between the line list used by \cite{2012A&A...545A..32A} and the line list provided by Gaia-ESO.
For elements with $A>$28, lines that were suitable for $W$ measurements (as tested by the Gaia-ESO line-list working group) were first selected and from these the ones that \texttt{ARES} was able to measure were used.
The atomic data from the Gaia-ESO Survey was adopted.
CAUP considered hyperfine splitting in the analysis of Cu, Ba, Nd,
Sm and Eu abundances, i.e. for all the elements affected with
A $>$ 28 (using the driver \texttt{blends} in \texttt{MOOG}).
The errors of the abundances is given as the line-to-line scatter (when more than one line is measured).
The UCM node adopted an approach similar to CAUP.
For iDR1, two line-lists were prepared: one for dwarfs ($\log g \ge 4.0$) and one for giants ($\log g \le 4.0$).
For iDR2 five line lists were used as done for the stellar parameters (see Sect.\,\ref{sec:FundamentalParameters}).
A total of 13 elements were analysed: Fe, the $\alpha$-elements (Mg, Si, Ca, and Ti), the Fe-peak elements
(Cr, Mn, Co, and Ni), and the odd-Z elements (Na, Al, Sc, and V).
To obtain individual abundances, the equivalent widths are fed into \texttt{MOOG} and then a 3$\sigma$-clipping for each chemical element was applied.
The elements for which at least two nodes derived abundances for at least one star and that were considered in the recommended results are:
\element{Na},
\element{Mg},
\element{Al},
\element{Si},
\element{Ca},
\element{Sc},
\element{Ti},
\element{V},
\element{Cr},
\element{Mn},
\element{Fe},
\element{Co},
\element{Ni},
\element{Zn},
\element{Zr},
\element{Mo},
\element{Ce}.
Only one node results were considered for:
\element{Cu},
\element{Y},
\element{Ba},
\element{La},
\element{Pr},
\element{Nd},
\element{Sm},
\element{Eu}.
Abundances are from the neutral species except for
\element{Ba},
\element{La},
\element{Ce},
\element{Pr},
\element{Nd},
\element{Sm},
\element{Eu},
for which they are from the ionised species.
The node-to-node dispersions of elemental abundances in iDR2 is shown in Fig.\,\ref{fig:UVES_abund_dispersion}.
In general, $\approx$ 90\% of the results for each elements have dispersions below $\approx0.2$ dex.
However the tail of the distributions extends to higher values in more difficult cases for which differences that arise from the different $W$ measurements \citep[see][]{Smiljanic_etal:2014} and line selection strategies play a role.
The dispersion tends to be higher also for abundances of ions like \ion{Ti}{II} and \ion{Cr}{II}.
Poor agreement is found for \ion{Zn}{I} and \ion{Zr}{II}.
Note that abundances for elements which require hyper-fine splitting were provided by the CAUP node only.
The internal precision is comparable with that of the UVES spectra of FGK-type analysis \citep[excluding stars in the field of young open cluster;][]{Smiljanic_etal:2014}.
Note that \cite{Smiljanic_etal:2014} make use of the {\it median of the absolute deviations from the median of the data} (MAD) to quantify the node-to-node dispersion, but this cannot be used here because of the small number of nodes providing abundances.
The dispersion from the mean used here should overestimate the node-to-node dispersion with respect to the MAD, although this is mitigated by the $\sigma$-clipping applied.
Overall, all this indicates that our internal precision for elemental abundances is roughly at the same level of the \cite{Smiljanic_etal:2014} one.
A survey inter-comparison with \cite{Smiljanic_etal:2014} results on the common calibration open cluster \object{NGC6705} was carried out for all elements except \element{Ce}, \element{La}, \element{Pr}, \element{Sm}, for which results did not pass the \cite{Smiljanic_etal:2014} quality control criteria.
The inter-comparison was satisfactory and confirmed the comparable precision with the \cite{Smiljanic_etal:2014} results.
\begin{figure*}[htp]
\centering
\includegraphics[width=40mm]{histo_na1_idr2.eps}
\includegraphics[width=40mm]{histo_mg1_idr2.eps}
\includegraphics[width=40mm]{histo_al1_idr2.eps}
\includegraphics[width=40mm]{histo_si1_idr2.eps}
\includegraphics[width=40mm]{histo_ca1_idr2.eps}
\includegraphics[width=40mm]{histo_sc1_idr2.eps}
\includegraphics[width=40mm]{histo_sc2_idr2.eps}
\includegraphics[width=40mm]{histo_ti1_idr2.eps}
\includegraphics[width=40mm]{histo_ti2_idr2.eps}
\includegraphics[width=40mm]{histo_v1_idr2.eps}
\includegraphics[width=40mm]{histo_cr1_idr2.eps}
\includegraphics[width=40mm]{histo_cr2_idr2.eps}
\includegraphics[width=40mm]{histo_fe1_idr2.eps}
\includegraphics[width=40mm]{histo_fe2_idr2.eps}
\includegraphics[width=40mm]{histo_co1_idr2.eps}
\includegraphics[width=40mm]{histo_ni1_idr2.eps}
\includegraphics[width=40mm]{histo_zn1_idr2.eps}
\includegraphics[width=40mm]{histo_y2_idr2.eps}
\includegraphics[width=40mm]{histo_zr2_idr2.eps}
\caption{iDR2 node-to-node dispersion of elemental abundances.}
\label{fig:UVES_abund_dispersion}
\end{figure*}
\subsection{Mass accretion rate}
\begin{figure}[htp]
\centering
\includegraphics[width=90mm]{mdot_m.eps}
\caption{Mass accretion rates vs. mass for all clusters in iDR2. Symbols and colours as in Fig.\,\ref{fig:giraffe_Ha10_EWHaAcc_pms}. The dashed line represents the $\dot{M} \propto M^2$ relationship. }
\label{fig:mdot_m}
\end{figure}
Mass accretion rates are estimated from the ${\rm H}\alpha\,10\%$\ using the \citet[][Eq.\,(1)]{2004A&A...424..603N} formula.
The use of alternative methods, i.e. making use of the $W({\rm H}\alpha$), is discussed in
\cite{Frasca_etal:2014} and will be implemented in the Gaia-ESO PMS analysis in future data releases.
The use of the \cite{2004A&A...424..603N} relationship has, undoubtedly, the advantage of allowing a simple estimate of $\dot{M}_{\rm acc}$\ from just the ${\rm H}\alpha\,10\%$.
The accuracy and validity of this empirical relationship has, however, been questioned \citep[see, e.g.,][and references therein]{2012MNRAS.427.1344C}, especially in cases when only single epoch observations are available.
Recently, \cite{2014A&A...561A...2A} have computed the accretion rate by modelling the excess emission from the UV to the near-IR and provided empirical relationships between accretion luminosity and the luminosity of 39 emission lines from X-Shooter spectra.
In particular, they have shown that the comparison between $\dot{M}_{\rm acc}$\ derived through primary diagnostics (like the UV-excess) and that obtained with the \cite{2004A&A...424..603N} relationship has a large scatter, with this latter tending to underestimate $\dot{M}_{\rm acc}$\ for ${\rm H}\alpha\,10\%$$<400$ km\,s$^{-1}$\ and to overestimate $\dot{M}_{\rm acc}$\ for ${\rm H}\alpha\,10\%$$>400$ km\,s$^{-1}$.
A comparison of the iDR1 $\dot{M}_{\rm acc}$\ of $\gamma$ Vel and Cha\,I with mass accretion rates derived from line luminosity and the \cite{1998apsf.book.....H} relationship has been presented in \cite{Frasca_etal:2014}.
They found discrepancies of $\sim 0.8$ dex for Cha I and $\sim 0.7$ dex for $\gamma$ Vel on average.
\cite{Frasca_etal:2014} also compared the results obtained for Cha\,I with literature values, finding a fair agreement, with differences that can be ascribed to variability, different methodologies and the use of different evolutionary models.
Mass accretion rates derived for all clusters in the first 18 months of observations vs. stellar mass are shown in Fig.\,\ref{fig:mdot_m}.
Stellar mass is estimated from the recommended $T_{\rm eff}$\ and the age of the cluster using the \cite{Baraffe_etal:1998} models\footnote{The results shown here are for a mixing length parameter $\alpha=1.5$; in this analysis, however, the choice of $\alpha$ is uninfluential.}.
The expectations are that $\dot{M} \propto M^\alpha$ with $\alpha \sim 2$ \citep[e.g.,][]{2005ApJ...625..906M,2008ApJ...681..594H,2014A&A...561A...2A}.
As for the $\gamma$ Vel and Cha\,I cases discussed in \cite{Frasca_etal:2014}, however, the large scatter in $\dot{M}$ prevents us to make a meaningful comparison with such a relationship.
The Spearman's rank correlation analysis for Cha\,I gives, in the iDR2 case, $\rho = 0.43$ and $\sigma = 0.005$, i.e. a higher significance than found by \cite{Frasca_etal:2014} in the iDR1 case ($\rho = 0.26, \sigma = 0.16$ for $\dot{M}$ derived from ${\rm H}\alpha\,10\%$), which indicates a better accuracy of our recommended iDR2 ${\rm H}\alpha\,10\%$\ parameter.
Amongst the younger clusters in our sample, we find $\rho = 0.47$ and $\sigma=0.14$ for $\rho$ Oph, while the correlation is rather poor for NGC2264 ($\rho = 0.19$, $\sigma=0.022$) possibly because of the larger uncertainties due to the residual nebular emission in the spectra of this cluster.
Note that the scatter in Fig.\,\ref{fig:mdot_m} is dominated by NGC2264. Ignoring this cluster, the scatter is consistent with what found by \cite{2014A&A...561A...2A} in their validation of the \cite{2004A&A...424..603N} relationship.
Interestingly, for the older clusters in our sample we find a not significant correlation in $\gamma$ Vel ($\rho = 0.29$, $\sigma=0.247$) but a well defined correlation in NGC2547 ($\rho = 0.89$, $\sigma=0.018$).
In both such cases, two kinematically distinct populations with different ages have been discovered \citep[][]{2014A&A...563A..94J,2015arXiv150101330S}, whose possible consequences in the $\dot{M}$ vs. $M$ relationship still need to be explored.
\subsection{Chromospheric ${\rm H}\alpha$\ and ${\rm H}\beta$\ flux}
After the \texttt{ROTFIT} determination of the fundamental parameters, a best matching template within the library of slowly-rotating inactive stars is identified.
The chromospheric excesses $\Delta W({\rm H}\alpha$)$_{\rm chr}$\ and $\Delta W({\rm H}\beta$)$_{\rm chr}$\ are derived using a spectral subtraction method \citep[see, e.g.,][and references therein]{1985ApJ...295..162B,1994A&A...284..883F,1995A&AS..114..287M} that has been extensively used in the past.
The photospheric flux is removed by subtraction of the spectrum of an {\it inactive} template star with very close fundamental parameters, rotationally broadened at the target $v\sin i$, over the line wavelength range.
Such chromospheric $W$ excesses, $\Delta W({\rm H}\alpha$)$_{\rm chr}$\ and $\Delta W({\rm H}\beta$)$_{\rm chr}$, are then converted to flux, $F({\rm H}\alpha$)$_{\rm chr}$\ and $F({\rm H}\beta$)$_{\rm chr}$, by multiplying it by the theoretical continuum flux at the line's wavelength \citep[see, e.g.,][and references therein]{Frasca_etal:2014}.
It may be argued that even the templates may have some chromospheric {\it basal} flux \citep[see, e.g.,][and references therein]{1998ApJ...494..828J}, also variable in time following the stellar cycles \citep[see, e.g.,][]{2012A&A...540A.130S} which a detailed semi-empirical NLTE chromospheric modelling \citep[e.g.,][]{1990A&A...231..459H,1995A&A...302..839L} could take into account.
This latter is, however, unpractical for applications to large datasets like the Gaia-ESO one.
Furthermore, the chromospheric flux in young stars is much larger than the basal flux, so that this latter can be safely neglected.
\begin{figure}[htp]
\centering
\includegraphics[width=90mm]{fluxhachr_teff_wg12_pms.eps}
\caption{Chromospheric ${\rm H}\alpha$\ flux vs. $T_{\rm eff}$\ for all young clusters observed in the first 18 months of observations. Symbols and colours as in Fig.\,\ref{fig:giraffe_Ha10_EWHaAcc_pms}, with filled (open) symbols used for CTTS (WTTS). The dashed line represents the chromospheric activity -- accretion dividing line of \cite{Frasca_etal:2014}.}
\label{fig:fluxhachr_teff_wg12_pms}
\end{figure}
Results for $\gamma$ Vel and Cha\,I (iDR1) are discussed in \cite{Frasca_etal:2014}, who were able to discriminate between chromospheric-dominated and accretion-dominated ${\rm H}\alpha$\ flux.
$\Delta W({\rm H}\alpha$)$_{\rm chr}$\ vs. $T_{\rm eff}$\ for all clusters observed in the first 18 months of observations (iDR2) is shown in Fig.\,\ref{fig:fluxhachr_teff_wg12_pms}.
We note that the chromospheric activity -- accretion dividing line proposed by \cite{Frasca_etal:2014} ($\log F_{H\alpha} = 6.35 + 0.00049 (T_{\rm eff}-3000)$) delimits quite neatly the two regimes in this larger sample as well, with some larger uncertainties in the case of NGC2264 likely due to residual nebular emission.
This dividing line was also found by \cite{Frasca_etal:2014} to be in remarkable agreement with the saturation limit adopted by \cite{2003AJ....126.2997B} to separate CTTS and WTTS.
\begin{figure}[htp]
\centering
\includegraphics[width=90mm]{giraffe_EWHaChr_vsini_teff_pms_rotfit.eps}
\caption{Chromospheric ${\rm H}\alpha$\ equivalent width excess vs. $v\sin i$\ for all young clusters observed in the first 18 months of observations. Colour coding is used for $T_{\rm eff}$.}
\label{fig:giraffe_EWHaChr_vsini_teff_pms_rotfit}
\end{figure}
Finally, in Fig.\,\ref{fig:giraffe_EWHaChr_vsini_teff_pms_rotfit} we show $\Delta W({\rm H}\alpha$)$_{\rm chr}$\ vs. $v\sin i$\ for all young clusters observed in the first 18 months of observations.
While a full discussion on the activity--rotation relationship is deferred to future work, we note that our data display a $T_{\rm eff}$--dependent activity--rotation correlation regime at low $v\sin i$, followed by a $T_{\rm eff}$--dependent saturation regime at high $v\sin i$, as expected.
The behaviour at different $T_{\rm eff}$\ is quite neatly distinguishable, which further confirm the overall consistency of our results.
\section{Summary and conclusions}
\label{sec:Conclusions}
The Gaia-ESO PMS spectrum analysis provides an extensive list of stellar parameters from spectra acquired in the FLAMES/GIRAFFE/HR15N and FLAMES/UVES/580 setups in the field of young open clusters.
These include raw parameters that are directly measured on the input spectra ($W({\rm H}\alpha$), ${\rm H}\alpha\,10\%$, and $W({\rm Li})$), fundamental parameters ($T_{\rm eff}$, $\log g$, [Fe/H], $\xi$, $v\sin i$, and $r$), and derived parameters (${\log \epsilon({\rm Li})}$, ${\log \epsilon({\rm X})}$), $\dot{M}_{\rm acc}$, $\Delta W({\rm H}\alpha$)$_{\rm chr}$, $\Delta W({\rm H}\beta$)$_{\rm chr}$, $F({\rm H}\alpha$)$_{\rm chr}$, and $F({\rm H}\beta$)$_{\rm chr}$) which require prior knowledge of the former.
Our analysis strategy is devised to deal with peculiarities of PMS stars and young stars in general such as veiling, large broadening due to fast-rotation, emission lines due to accretion and/or chromospheric activity, and molecular bands.
The analysis is also made robust against residual sky-background or foreground features that cannot be completely removed as in the case of inhomogeneous nebular emission.
The availability of different methods for deriving stellar parameters increases the confidence on the output of our analysis.
It allows us to efficiently identify and discard outliers, like those deriving from failed fits or problems in the input spectra, as well as deriving realistic uncertainties from the internal dispersion of the data.
For $T_{\rm eff}$\ and $\log g$\ the external precision is estimated by comparison with results from interferometric angular diameter measurements.
These are estimated to be $\approx$ 120\,K r.m.s. in $T_{\rm eff}$\ and $\approx$0.3 dex r.m.s. in $\log g$\ for both the UVES and GIRAFFE setups.
The comparison with $T_{\rm eff}$\ derived from photometry for a selected group of stars in $\gamma$ Vel with the same foreground extinction and free from accretion signatures gives an agreement of $\approx260$\,K r.m.s.
Our recommended [Fe/H]\ results agree with assessed literature values for such a set of {\it benchmark} stars within $\approx$0.15 dex r.m.s.
A comparison with previous [Fe/H]\ determination for Cha\,I is discussed in \cite{2014A&A...568A...2S}.
Weakness or limitations of the methods used were identified by the node-to-node comparisons and by comparison with benchmark stars.
The observation strategy poses significant challenges to the analysis, since, for optimising the observation time, most of the relevant observations are carried out in just the FLAMES/GIRAFFE/HR15N setup.
For our purposes, the wavelength range of this setup is the best available in the optical, as it contains very important diagnostics for young stars like the ${\rm H}\alpha$\ and Li\ line.
At the same time, surface gravity diagnostics in the HR15N setup are poorer than in other wavelength ranges and still not modelled with sufficient accuracy.
$T_{\rm eff}$\ determination for spectral types earlier than early-G is also challenging since it is based mostly on the ${\rm H}\alpha$\ wings.
For such a wavelength range, two methods based on the comparison with spectra or spectral indices of template stars have proved effective in providing fundamental parameters.
A satisfactory self-consistency of the results have been achieved, at the expense of discarding $\log g$\ values when a sufficient agreement between the two methods cannot be reached.
In such cases, however, it is still possible to provide an {\it evolutionary flag}, as it can be established with confidence whether the star is in a PMS, a MS or a post-MS stage.
An uncalibrated gravity-sensitive spectral index is also provided, useful for a rank order in age.
The reproducibility of the parameters obtained with the higher resolution and larger wavelength coverage from UVES using a much smaller wavelength range and a lower resolution as in the GIRAFFE/HR15N setup, together with the comparable accuracy and precision achieved in the two setups, is a remarkable achievement of this work.
This allows us to provide with confidence parameters for the much larger GIRAFFE sample.
The Gaia-ESO is an ongoing project and this paper describes the PMS spectrum analysis carried out on the first two data releases.
Work is ongoing to improve further our analysis for the next releases.
The tables with the public release results will be available through the ESO data archive\footnote{\url{http://archive.eso.org/wdb/wdb/adp/phase3_spectral/form?phase3_collection=GaiaESO}} and through the Gaia-ESO Survey science archive\footnote{\url{http://ges.roe.ac.uk/index.html}} hosted by the Wide Field Astronomy Unit (WFAU) of the Institute for Astronomy, Royal Observatory, Edinburgh, UK
\begin{acknowledgements}
Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 188.B-3002.
This work was partly supported by the European Union FP7 programme through ERC grant number 320360 and by the Leverhulme Trust through grant RPG-2012-541.
We acknowledge the support from INAF and Ministero dell' Istruzione, dell' Universit\`a e della Ricerca (MIUR) in the form of the grant "Premiale VLT 2012" and the grant ``The Chemical and Dynamical Evolution of the Milky
Way and Local Group Galaxies'' (prot. 2010LY5N2T).
The results presented here benefit from discussions held during the Gaia-ESO workshops and conferences supported by the ESF (European Science Foundation) through the GREAT Research Network Programme.
H.M.T. acknowledges the financial support from BES-2009-012182 and the ESF and GREAT for an exchange grant 4158.
H.M.T. and D.M. acknowledges the financial support from the Spanish Ministerio de Econom\'ia y Competitividad (MINECO) under grant AYA2011-30147-C03-02.
J.I.G.H. acknowledges financial support from the MINECO under grants AYA2011-29060, and 2011 Severo Ochoa Program SEV-2011-0187.
S.G.S, EDM, and V.Zh.A. acknowledge support from the Funda\c{c}\~ao para a Ci\^encia e Tecnologia (Portugal) in the form of grants SFRH/BPD/47611/2008, SFRH/BPD/76606/2011, SFRH/BPD/70574/2010, respectively.
T.B. was funded by grant No. 621-2009-3911 from The Swedish Research Council.
\end{acknowledgements}
\bibliographystyle{aa}
|
1,116,691,499,383 | arxiv | \section{Introduction}
In this paper, we characterize the locus of all points $P$ with an isogonal conjugate in a given quadrilateral $ABCD$. This turns out to be a cubic plane curve, which we will call the \textit{isogonal cubic} of $ABCD$. The isogonal cubic is a well-established figure in geometry. However, its properties are often considered with respect to the base quadrilateral $ABCD$, without considering the isogonal cubic as an individual curve, and often neglecting degenerate cases of $ABCD$.
The first half of the paper is dedicated to discovering geometric properties of isogonal cubics, and also providing constructions on the isogonal cubic with straightedge and compass. This sets up the second half, which characterizes all possible non-degenerate cubics $\mathcal C \in \mathbb{RP}^2$ such that there exist $A, B, C, D \in \mathbb R^2$ for which $\mathcal C$ is the isogonal cubic of $ABCD$. We also establish the notion of the spiral center and isogonal conjugation purely with respect to a valid cubic $\mathcal C$. This formalizes constructions on the isogonal cubic with straightedge and compass, only requiring the cubic's unique spiral center and asymptote, without the base quadrilateral $ABCD$. In particular, these constructions do not require intersecting lines with cubic curves. This makes them compatible with software such as Geogebra, where intersecting lines with cubics is not always supported.
The following is the main result we prove in this paper, which underlies these constructions:
\begin{theorem}
Let $\mathcal C$ be a non-degenerate cubic in $\mathbb R^2$, and let $\mathcal C_0$ denote its embedding in $\mathbb{CP}^2$. Then the following two conditions are equivalent:
\begin{enumerate}
\item[(1)] There exist distinct $A, B, C, D \in \mathcal C$ such that $\mathcal C$ is the isogonal cubic of $ABCD$.
\item[(2)] The circular points at infinity (\cite{ref:Circle}) $I, J$ lie on $\mathcal C_0$, and the tangents at $I, J$ meet on $\mathcal C_0$.
\end{enumerate}
\end{theorem}
In particular, the assertion that (2) is a \textit{sufficient} condition requires special care in $\mathbb R^2$ and $\mathbb{CP}^2$.
\subsection{Definitions and Conventions}
\begin{definition}[Isogonality in $\mathbb{RP}^2$]
For points $P, A, B, C, D \in \mathbb R^2$, pairs of lines $(PA, PC), (PB, PD)$ are called \textit{isogonal} if they share the same pair of angle bisectors. If $P$ is a real point at infinity while $A, B, C, D$ remain in $\mathbb R^2$, we slightly modify our definition of isogonality to mean that for any line $\ell$ intersecting $PA, PB, PC, PD$ at points $E, F, G, H \in \mathbb R^2$, directed lengths $EF$ and $HG$ will be equal.
\end{definition}
The definition for points at infinity is equivalent to the midline of parallel lines $PA, PC$ being the same as the midline of parallel lines $PB, PD$, which complies with the idea of angles as a conceptual measure of ``distance" between two lines.
\begin{definition}[Quadrilateral Conventions and Isogonal Conjugates]
We use the term ``quadrilateral" throughout this paper to refer to possibly self-intersecting quadrilaterals, whose vertices are distinct but possibly collinear. Points $P, Q$ are called \textit{isogonal conjugates} in a quadrilateral $ABCD$ if and only if $(AP, AQ), (AB, AD)$ are isogonal and the analogous holds for the other vertices.
\end{definition}
We will exclusively work in directed angles. For points $X, Y$, the notation $XY$ denotes the line $XY$ if $X$ and $Y$ are distinct, while $XY$ denotes the tangent at $X$ if $X \equiv Y$ and the context of the curve containing $X$ is clear (usually the isogonal cubic). The notation $(XYZ)$ denotes the circumcircle of $XYZ$ provided $X, Y, Z$ are distinct.
\begin{definition}[Notation for Intersection]
We will let $\mathcal S \cap \mathcal T$ denote the intersection of sets of points $\mathcal S$ and $\mathcal T$, which is unique when $\mathcal S$ and $\mathcal T$ are distinct lines. When $\mathcal S$ is a cubic and $\mathcal T$ is a line $XY$ such that $X, Y \in \mathcal S$, we will use the notation $XY \cap \mathcal S$ to denote
\begin{itemize}
\item If either $X$ or $Y$ is a singular point, whichever one of $X, Y$ is singular
\item If $X, Y$ are distinct and $XY$ is not tangent to $\mathcal S$, the third intersection of $XY$ with $\mathcal S$
\item If $X, Y$ are distinct and $XY$ is tangent to $\mathcal S$, the tangency point of $XY$ with $\mathcal S$
\item If $X, Y$ are not distinct and $X$ is not an inflection point of $\mathcal S$, the intersection of the tangent to $\mathcal S$ at $X$ with $\mathcal S$
\item If $X, Y$ are not distinct and $X$ is an inflection point, the point $X$
\end{itemize}
\end{definition}
These are essentially equivalent to the third intersection of $XY$ with $\mathcal C$ counting multiplicity.
We begin with this well-known characterization of all points with isogonal conjugates:
\begin{theorem}
For fixed distinct points $A, B, C, D \in \mathbb R^2$ not all collinear, a point $P \in \mathbb{RP}^2$ is called \textit{excellent} if $(PA, PC)$ and $(PB, PD)$ are isogonal. Then $P$ is excellent if and only if it has an isogonal conjugate in $ABCD$.
\end{theorem}
Most proofs for this fact do not address the case when three of $A, B, C, D$ are collinear, so we will provide the full proof of this lemma for the sake of rigor.
\begin{proof}
The first case is when, without loss of generality, $B, C, D$ are collinear. In this case, we need to prove the following: For triangle $ABC$ and $D \in BC$ and point $P$, the isogonal conjugate $Q$ of $P$ satisfies that $BC$ is a bisector of angle $\angle PDQ$ if and only if $(PA, PD)$, $(PB, PC)$ are isogonal.
This is, in turn, equivalent to the following: For isogonal conjugates $P, Q$ in $ABC$, if $Q_A'$ be the reflection of $Q$ over $BC$, then $(PA, PQ_A')$, $(PB, PC)$ are collinear. To prove this, let $P, Q$ have pedal triangles $P_AP_BP_C$, $Q_AQ_BQ_C$ respectively; by \cite{ref:Three}, these share the same circumcircle $\omega$ centered at the midpoint of $PQ$. Let $PP_A$ meet $\omega$ at $R_A \ne P_A$; $PR_AQ_AQ_A'$ is a parallelogram, so
\begin{align*}
\angle Q_A'PC &= \angle Q_A'PP_A + \angle P_APC \\
&= \angle Q_AR_AP_A + 90^\circ - \angle PCB \\
&= \angle Q_AP_CP_A + 90^\circ - \angle PCB \\
&= \angle Q_AP_CP_B + \angle BP_CP_A + 90^\circ - \angle PCB \\
&= \angle Q_AQ_BQ_C + \angle BPP_A + 90^\circ - \angle PCB \\
&= \angle Q_AQ_BQ + \angle QQ_BQ_C + 90^\circ - \angle CBP + 90^\circ - \angle PCB \\
&= \angle BCQ + \angle QAB - \angle CBP - \angle PCB \\
&= \angle PCA + \angle CAP + \angle BPC \\
&= \angle BPC + \angle CPA \\
&= \angle BPA
\end{align*}
as desired.
\begin{figure}[!htbp]\centering
\includegraphics[width=200pt]{0-1.png}
\caption{The Origin Lemma}
\label{fig:0.1}
\end{figure}
Next, we prove this fact when no three of $A, B, C, D$ are collinear. While the proof for the general case is well-known, we will provide it for the sake of completion.
\newline\newline
\textbf{Lemma:} For quadrilateral $ABCD$ and point $P$, let the projections of $P$ onto $AB, BC, CD, DA$ be $E, F, G, H$. Prove that $EFGH$ is cyclic if and only if $\angle APB = \angle DPC$.
\begin{subproof}
With the various cyclic quadrilaterals,
\begin{align*}
\angle EFG + \angle GHE &= \angle EFP + \angle PFG + \angle GHP + \angle PHE \\
&= \angle EBP + \angle PCG + \angle GDP + \angle PAE \\
&= \angle APB + \angle CPD,
\end{align*}
which directly implies the desired statement.
\end{subproof}
Back to the main problem, drop perpendiculars $E, F, G, H$ from $P$ to $AB, BC, CD, DA$. First, we prove that if isogonal conjugate then $\angle APB = \angle DPC$. Let $P$ have isogonal conjugate $P'$ in $ABCD$. Then $P'$ is the isogonal conjugate of $P$ in both $YAB$ and $XAD$. Drop from $P'$ perpendiculars $E', F', G', H'$; by \cite{ref:Three} on $YAB$, $EFHE'F'H'$ is cyclic, and on $XAD$ we get $EGHE'G'H'$ is cyclic. In other words, $F, F', G, G'$ all lie on $(EE'HH')$, implying that $EFGH$ is cyclic, hence $\angle APB = \angle DPC$ as desired.
Now, we prove that if $\angle APB = \angle DPC$ then it has an isogonal conjugate $P'$. Then $EFGH$ is cyclic; let its circumcircle meet $AB, BC, CD, DA$ at $E', F', G', H'$. By \cite{ref:Three}, the perpendiculars to $AB, BC, CD$ at $E', F', G'$ concur at a single point $P'$, the isogonal conjugate of $P$ in $XBC$. Analogously, the perpendiculars to $AB, CD, DA$ at $E', G', H'$ concur at the isogonal conjugate of $P$ in $XAD$. In other words, $P'H' \perp DA$ and is the isogonal conjugate of $P$ in both $XAD$ and $XBC$, implying that $P'$ is the desired isogonal conjugate of $P$ in $ABCD$.
\end{proof}
\subsection{Degenerate Cases}
One case where the locus of isogonal conjugates becomes degenerate is when $A, B, C, D$ are collinear on a line $\ell$. In this case, the locus becomes the line $\ell$ along with the circle centered on $\ell$ whose inversion swaps $A$ with $C$ and $B$ with $D$, if this circle exists.
Another case is when $ABCD$ is a parallelogram, where we have the following characterization:
\begin{theorem}
If $ABCD$ is a parallelogram, the locus of excellent points is the line of infinity, along with the circumhyperbola passing through the points of infinity on the two angle bisectors of $\angle ABC$.
\end{theorem}
\begin{proof}
By our extension of isogonality to $\mathbb{RP}^2$, the line of infinity is part of this locus. Then for all points $P \in \mathbb R^2$, by the Dual of Desargues' involution theorem (\cite{ref:Elementary}, 133), $(PA, PC)$, $(PB, PD)$ are isogonal if and only if angle $APC$ has the same angle bisectors as the pair of lines through $P$ parallel to $AB$ and $AD$. Thus if $P_1$ and $P_2$ are the points of infinity along with these angle bisectors of $\angle BAD$, we essentially need to find the locus $P \in \mathbb R^2$ for which the angle bisectors of $APC$ are parallel to $\ell_1$ and $\ell_2$.
We claim that this locus is the hyperbola $\mathcal H$ centered at the midpoint $M$ of $AC$ passing through $P_1, P_2, A, C$. For any point $P \in \mathcal H$, $\mathcal H$ becomes the circumrectangular hyperbola of triangle $PAC$ centered at $M$, which is the isogonal conjugate of the perpendicular bisector of $AC$ wrt $PAC$. The isogonal conjugates of $P_1, P_2$ in $PAC$ then become the two arc midpoints of $AC$ in $(PAC)$, so $P_1, P_2$ are indeed the points of infinity along the angle bisectors of $PAC$.
For the other direction, take any point $P$ such that $\angle APC$ has angle bisectors passing through $P_1, P_2$. Then the isogonal conjugate $\mathcal H'$ of the perpendicular bisector of $AC$ wrt $PAC$ will also pass through $P_1, P_2$, implying that $\mathcal H \equiv \mathcal H'$, so $P \in \mathcal H$, as desired. It is now clear that $B, D \in \mathcal H$, which completes the proof.
\end{proof}
For the rest of the paper, we will assume quadrilateral $ABCD$ does not fall under either of these cases. In particular, $A, B, C, D$ are not all collinear, and the midpoints of $AC, BD$ are distinct.
\section{Preliminary Lemmas}
Up until Section 6, we will work in $\mathbb{RP}^2$. All angles are directed mod $180^\circ$.
The following provides another well-known characterization of isogonal conjugates.
\begin{definition}
Let $P$ be the spiral center of $ABCD$. For any point $Y$, call the unique point $Y'$ for which $P$ is the spiral center of $AYCY'$ the \textit{Spiral Inverse} of $Y$.
\end{definition}
\begin{theorem}
The spiral inverse $X'$ of a excellent point $X$ is also the isogonal conjugate of $X$.
\end{theorem}
\begin{proof}
Let $E = AD \cap BC, F = AB \cap CD$. If no three of $A, B, C, D$ are collinear, we have no problems, and otherwise we will assume without loss of generality that $A, B, D$ are collinear. Either way, the following relation is true:
\[\angle DX'C = \angle DX'P + \angle PX'C = \angle XBP + \angle PAX = \angle APB + \angle BXA = \angle DEC + \angle CXD.\]
Note that if $A, B, D$ are collinear, then we would have $B \equiv E$ and $D \equiv F$, but the above angle chase would still hold. Similarly, $\angle EX'C = \angle EDC + \angle CXE$, implying that $X, X'$ are isogonal conjugates in $CDE$.
If no three of $A, B, C, D$ are collinear, analogously, $X, X'$ are isogonal conjugates in $BCF$, implying $X, X'$ are isogonal conjugates in $ABCD$, so we are done. Otherwise, under our WLOG that $A, B, D$ are collinear, $X, X'$ will be isogonal conjugates in $BCD$, and since $X$ is excellent, this implies $X, X'$ are isogonal conjugates in $ABCD$, as desired.
\end{proof}
\begin{definition}
Denote by $\mathcal C$ the cubic which is the locus of all excellent points $X$.
\end{definition}
We will sometimes call $\mathcal C$ the ``isogonal cubic" throughout this paper.
\begin{proof}
Proving that the locus is a cubic amounts to examining the equation
\[\frac{\frac{d-x}{a-x}}{\frac{\overline{d-x}}{\overline{a-x}}} = \frac{\frac{c-x}{b-x}}{\frac{\overline{c-x}}{\overline{b-x}}}\]
in the complex plane (\cite{ref:EGMO}, 6.1). Expanding this gives the desired third-degree equation in $x$. Note that the coefficients of 3rd degree coefficients $x^2\ol x, x\ol x^2$ in the expansion are both zero if and only if $a+c = b+d$, which confirms that parallelograms produce degenerate loci.
\end{proof}
For the rest of this paper, we will assume that $\mathcal C$ is non-degenerate.
Now, we may also recall the following well-known fact.
\begin{theorem}[Isogonal Conjugate at Infinity]\label{infinity}
Let $M, N$ be the midpoints of $AC, BD$. Then the isogonal conjugate of $P$ is the point of infinity along $MN$.
\end{theorem}
\begin{proof}
The parabola $\mathcal P$ tangent to the sides of $ABCD$ has focus $P$, and its directrix is the Gauss-Bodenmiller line (\cite{ref:Inconic}), which is perpendicular to $MN$ (\cite{ref:Gauss}).
It is well known (\cite{ref:Three}) that for any conic with foci $X_1, X_2$ and any point $X$ for which tangents from $X$ exist, $XX_1$ and $XX_2$ are isogonal in the angle formed by the tangents from $X$ to the conic. Applying this to $X \equiv A$ and conic $\mathcal P$, we conclude that $AP$ and the perpendicular from $A$ to the directrix are isogonal in $\angle BAD$. Similar relations with $B, C, D$ imply the desired result.
\end{proof}
\begin{figure}[!htbp]\centering
\includegraphics[width=350pt]{1-1.png}
\caption{Main Configuration}
\label{fig:1.1}
\end{figure}
\begin{theorem}\label{reversal}
Consider two pairs $(X, X'), (Y, Y')$ of isogonal conjugates. Then $A, C$ are isogonal conjugates in $XYX'Y'$.
\end{theorem}
\begin{proof}
Since $(AX, AX'), (AY, AY')$ are isogonal, $A, B, C, D$ are excellent in $XYX'Y'$. Since $A, C$ are spiral inverses in $XYX'Y'$, they are isogonal conjugates, as desired.
\end{proof}
The following corollary immediately follows.
\begin{corollary}\label{swallow-reversal}
For isogonal conjugates $X, X'$ and excellent point $Y$, $(YX, YX'), (YA, YC)$ are isogonal.
\end{corollary}
This directly implies the following critical characterization:
\begin{corollary}[Generalization of Isogonal Cubic]\label{general}
Consider two pairs $(X, X'), (Y, Y')$ of isogonal conjugates. Then the isogonal cubic of $ABCD$ is the isogonal cubic of $XYX'Y'$. Furthermore, any pair of isogonal conjugates $(K, L)$ in $ABCD$ are also isogonal conjugates in $XYX'Y'$.
\end{corollary}
\begin{proof}
By \Cref{swallow-reversal}, for any point $Z$, if pairs of lines $(ZA, ZC), (ZB, ZD)$ are isogonal, then lines $(ZX, ZX'), (ZY, ZY')$ are also isogonal, so $ABCD$ and $XYX'Y'$ indeed share the same isogonal cubic. The second part then directly follows from \Cref{swallow-reversal}.
\end{proof}
\begin{figure}[!htbp]\centering
\includegraphics[width=350pt]{1-2.png}
\caption{Quadrilateral Completeness}
\label{fig:1.2}
\end{figure}
Thus the following is true by the Dual of Desargues' Involution Theorem on $XYX'Y'$:
\begin{corollary}[Quadrilateral Completeness]\label{completeness}
For two pairs $(X, X'), (Y, Y')$ of isogonal conjugates, $XY \cap X'Y'$ and $XY' \cap X'Y$ lie on $\mathcal C$.
\end{corollary}
We now illustrate the relationship between isogonality and inconics.
\begin{theorem}
Let $ABCD$ have inconic $\omega$ and isogonal conjugates $X, X'$. Then the tangents to $\omega$ from $X$ and $X'$ intersect at two pairs of isogonal conjugates.
\end{theorem}
\begin{proof}
Call $IJKL$ the quadrilateral formed by these two pairs of tangents such that $IJ, JK, KL, LI$ are tangent to $\omega$. Since $\omega$ is an inconic of $XIX'K$ and the tangents from $A$ to $\omega$ ($AB$ and $AD$) are isogonal in $\angle XAX'$, by the Dual of Desargues' Involution on $XIX'K$ from $A$, $(AI, AK)$ are also isogonal in $\angle BAD$. Similar arguments imply $(I, K), (J, L)$ are isogonal conjugates as desired.
\end{proof}
\begin{corollary}\label{tangency-exists}
For excellent point $X$ and isogonal conjugates $Y, Y'$, the line $XY'$ is tangent to the inconic of lines $AB, BC, CD, DA, XY$.
\end{corollary}
\section{Relationship of Inconics with Excellent Points}
Consider any excellent point $X$. Let $\omega$ be the inconic of $AB, BC, CD, DA, PX$; by \Cref{tangency-exists}, the line $\ell_1$ through $X$ parallel to $MN$ is also tangent to $\omega$. Reflect $\ell_1$ over the center of $\omega$ to get the second tangent $\ell_2$ from the point of infinity $\infty_{MN}$ along $MN$ to $\omega$; let $\ell_2$ meet the second tangent from $P$ to $\omega$ (other than $PX)$ at $X'$. By \Cref{tangency-exists}, $X$ and $X'$ are isogonal conjugates. Let $PX$ meet $\ell_2$ at $Y$, and let $PX'$ meet $\ell_1$ at $Y'$; then $Y, Y'$ are isogonal conjugates as well. We immediately get the following two corollaries:
\begin{figure}[!htbp]\centering
\includegraphics[width=400pt]{2-1.png}
\caption{Tangents to an Inconic}
\label{fig:2.1}
\end{figure}
\begin{theorem}\label{isogonal-MN}
The midpoint of any two isogonal conjugates lies on $MN$.
\end{theorem}
\begin{theorem}\label{isogonal-parallel}
For $X \in \mathcal C$ and $P_\infty$ the point of infinity along $MN$, let $Y = PX \cap \mathcal C$ and $X' = P_\infty Y \cap \mathcal C$. Then $X, X'$ are isogonal conjugates.
\end{theorem}
We may also note that the midpoint of $XY$ lies on $MN$. Since $\ell_1 \parallel \ell_2$, the bisectors of $\angle PX\infty_{MN}, \angle PY\infty_{MN}$ are parallel (perpendicular) to each other, with the perpendicular pairs of bisectors intersecting on $MN$. Thus, the following is a direct result.
\begin{theorem}[Parallel Bisectors]\label{rho-aias}
If $X, Y$ lie on $\mathcal C$ with $XY$ passing through $P$, then the midpoint of $XY$ lies on $MN$, and the bisectors of $\angle AXC$ and $\angle AYC$ are parallel to each other.
\end{theorem}
\begin{figure}[!htbp]\centering
\includegraphics[width=300pt]{2-2.png}
\caption{Angles with Parallel Bisectors}
\label{fig:2.2}
\end{figure}
This produces a neat construction: if we are given an excellent point $X$, we may construct an excellent point $Y$ such that the bisectors of $\angle AXC$ and $\angle AYC$ are parallel to each other. By \Cref{rho-aias}, this is done by letting lines $PX$ and $MN$ intersect at a point $O$ and setting $Y$ to be the reflection of $X$ over $O$.
\section{Singular Points and Constructing Elements of the Isogonal Cubic}
We begin by noting that since $\mathcal C$ has real coefficients, for any two non-singular points $X$ and $Y$ on $\mathcal C$ in $\mathbb{RP}^2$, line $XY$ is either tangent to $\mathcal C$ at $X$ or $Y$, or $XY$ intersects $\mathcal C$ at a third point in $\mathbb{RP}^2$.
We begin with a well-known general lemma about cubics.
\begin{lemma}\label{four}
From any point $X$ on general non-degenerate cubic $\mathcal C$, there are at most $4$ points $Y \in \mathcal C$ other than $X$ for which $XY$ intersects $\mathcal C$ at $Y$ with multiplicity 2.
\end{lemma}
\begin{proof}
Note that if $X$ is singular, there are no such points $Y$, or else line $XY$ would intersect $\mathcal C$ at both $X$ and $Y$ with multiplicity 2, yielding $2+2 = 4$ total intersections. Assuming that $X$ is non-singular now, consider the embedding of $\mathcal C$ in $\mathbb{CP}^2$ with equation $F(x, y, z) = 0$. For any point $Y = (p: q: r)$ on $\mathcal C$, $Y$ is either a singular point, or the equation of the tangent at $Y$ is given by
\[\frac{\partial F}{\partial x}(p, q, r) \cdot x + \frac{\partial F}{\partial y}(p, q, r) \cdot y + \frac{\partial F}{\partial z}(p, q, r) \cdot z = 0\]
We want to $X = (x_0: y_0: z_0)$ to satisfy the above equation, so fixing $X$ gives us an equation in $p, q, r$ with degree $3-1 = 2$. Let $g(x, y, z)$ denote the expression
\[\frac{\partial F}{\partial x}(x, y, z) \cdot x_0 + \frac{\partial F}{\partial y}(x, y, z) \cdot y_0 + \frac{\partial F}{\partial z}(x, y, z) \cdot z_0\]
Regardless of whether $Y$ is a singular point of $\mathcal C$ or $XY$ is tangent to $\mathcal C$ at $Y$, all such points $Y$ will be solutions to the cubic $F(x, y, z) = 0$ and the conic $g(x, y, z) = 0$, which by Bezout's Theorem (\cite{ref:Fulton}, Section 5.3) gives at most $3(3-1)$ total solutions.
Note that $X$ itself also satisfies both equations; we now claim that $X$ is actually a solution with multiplicity at least 2. Let $\ell$ be the tangent to $\mathcal C$ at $X$; then $\ell$ has equation
\[\frac{\partial F}{\partial x}(x_0, y_0, z_0) \cdot x + \frac{\partial F}{\partial y}(x_0, y_0, z_0) \cdot y + \frac{\partial F}{\partial z}(x_0, y_0, z_0) \cdot z = 0\]
To prove that $X$ is a solution to both $F$ and $g$ with multiplicity 2, we use the fact that $I_X(F, g) \ge m_X(F)m_X(g)$, where $I_X$ denotes the multiplicity of the intersection of curves $F$ and $g$ at $X$, and $m_X(F), m_X(g)$ denoting the multiplicity of point $P$ on curves $F, g$ (\cite{ref:Fulton}, Section 3.3).
If $X$ is a singular point of $F$, then $I_X \ge 2$ as desired. Otherwise, equality holds iff the tangents at $X$ to $F$ and $g$ are distinct. Thus it suffices to show that $\ell$ is tangent to the conic $\mathcal H$ formed by $g$. To prove this, the tangent to $\mathcal H$ at $X$ is given by equation $Ax + By + Cz = 0$, where
\[A = \frac{\partial^2 F}{\partial x^2}(x_0, y_0, z_0) \cdot x_0 + \frac{\partial^2 F}{\partial y^2}(x_0, y_0, z_0) \cdot y_0 + \frac{\partial^2 F}{\partial z^2}(x_0, y_0, z_0) \cdot z_0\]
and $B, C$ are defined similarly. By Euler's Homogeneous Function Theorem (\cite{ref:Euler}),
\[2 \cdot \frac{\partial F}{\partial x}(x_0, y_0, z_0) = \frac{\partial^2 F}{\partial x^2}(x_0, y_0, z_0) \cdot x_0 + \frac{\partial^2 F}{\partial y^2}(x_0, y_0, z_0) \cdot y_0 + \frac{\partial^2 F}{\partial z^2}(x_0, y_0, z_0) \cdot z_0\]
which implies that
\[A = 2 \cdot \frac{\partial F}{\partial x}(x_0, y_0, z_0), \quad\quad B = 2 \cdot \frac{\partial F}{\partial y}(x_0, y_0, z_0), \quad\quad C = 2 \cdot \frac{\partial F}{\partial z}(x_0, y_0, z_0)\]
so the tangent to $\mathcal H$ at $X$ indeed has the same equation as $\ell$, as desired.
Thus $X$ is a solution to $\mathcal C$ and $\mathcal H$ with multiplicity at least 2, so there are at most $3(3-1) - 2 = 4$ such points $Y$, as desired.
\end{proof}
The following lemma also better characterizes $\mathcal C$.
\begin{lemma}\label{truth}
In $\mathbb{RP}^2$, $\mathcal C$ contains exactly one point at infinity.
\end{lemma}
\begin{proof}
The embedding of $\mathcal C$ in $\mathbb{CP}^2$ will contain the circular points at infinity $I, J$ by virtue of being isogonal conjugates, so back in $\mathbb{RP}^2$ there can only be one real point at infinity. On the other hand, given $ABCD$ the point of infinity along the Newton-Gauss line will lie on $\mathcal C$, so there is exactly one.
\end{proof}
To better establish tangencies in $\mathcal C$, we first need to examine singular points.
\begin{theorem}[Singular Points on the Isogonal Cubic]\label{singular}
A point $I \in \mathcal C$ is a singular point if and only if the isogonal conjugate of $I$ is itself.
\end{theorem}
\begin{proof}
We remind our readers of our assumption in Section 1 that $\mathcal C$ is not degenerate.
First, we prove that if $I$ is its own isogonal conjugate, then it is a singular point. Assume the contrary, that $I$ is not singular; then there are at most five non-singular points $X \in \mathcal C$ such that $XI$ is tangent to $\mathcal C$ at either $X$ or $I$. For all points $X$ such that $XI$ is \textit{not} tangent to $\mathcal C$, line $XI$ will intersect $\mathcal C$ at a point $Y \ne I, X$. By \Cref{swallow-reversal} this means that the line through $I, X, Y$ bisects angles $\angle AXC$, $\angle AYC$.
In particular, this means line $CX$ is the reflection of line $AX$ over line $XY$, and line $CY$ is the reflection of line $AY$ over line $XY$, which implies that $C$ is the reflection of $A$ over $XY$. Thus $XI$ is the perpendicular bisector of $AC$. But line $XI$ rotates around $I$ as we vary $X$ along the cubic, contradicting the uniqueness of the perpendicular bisector of $AC$, the desired contradiction.
Next, we prove that if $I$ is a singular point, then the isogonal conjugate of $I$ is itself. Assume the contrary; then let $J \ne I$ be the isogonal conjugate of $I$. Choose any isogonal conjugates $K, L$ distinct from $I, J$ (though we can set $K \equiv A$ etc). By \Cref{completeness}, $X = KI \cap LJ$ and $Y = KJ \cap LI$ will lie on $\mathcal C$. Since $I$ is a singular point, $KI$ will not intersect $\mathcal C$ at a point other than $K$ or $I$, so $X$ is either the same as $K$ or $I$.
If $X \equiv I$, then $I, L, J$ are collinear. But since $I$ is singular line $ILJ$ intersects $\mathcal C$ at $I$ with multiplicity 2, so our assumption that $L \ne I, J$ implies that $I \equiv J$, the desired contradiction.
Thus we must have $X \equiv K$, so $K, L, J$ are collinear. Thus we conclude $J$ lies on line $KL$ for any isogonal conjugates $K, L$. Choosing another pair $(R, S)$ of isogonal conjugates such that no three of $K, L, R, S$ are collinear, let $T = KR \cap LS$ and $U = KS \cap LR$; by \Cref{completeness}, $T$ and $U$ are isogonal conjugates in $\mathcal C$, so $KL, RS, TU$ concur at a single point $J$.
Consider a conic $\mathcal H$ passing through $K, L, R, S$ but not tangent to line $TU$. Then the pole of line $TU$ in $\mathcal H$ is the intersection of $KL$ and $RS$, which is precisely $J$, which lies on line $TU$. For the pole of $TU$ in $\mathcal H$ to lie on $TU$ itself, $TU$ must be tangent to $\mathcal H$ at $J$, the desired contradiction.
\end{proof}
Now, we will construct the tangent to $\mathcal C$ at any non-singular point $X$ as follows.
\begin{theorem}[Tangent to Isogonal Cubic]\label{tangent}
For isogonal conjugates $X, X'$, the isogonal $\ell$ to $XX'$ in $\angle AXC$ is tangent to $\mathcal C$ at $X$.
\end{theorem}
\begin{proof}
As we move a point $Y \in \mathcal C$ with isogonal conjugate $Y'$, $XY$ and $XY'$ are isogonal in $\angle AXC$. Therefore, as $Y$ approaches $X'$, $Y'$ will approach $X$, eventually letting $XY'$ intersect $\mathcal C$ with multiplicity $2$ at $X$.
\end{proof}
In particular, the tangent to $\mathcal C$ at the point of infinity along $MN$ is given by the unique (by \Cref{truth}) asymptote of $\mathcal C$.
\begin{theorem}\label{tangents-isogonal}
Let the tangents to $\mathcal C$ at isogonal conjugates $X, X'$ meet at $Y$, and let $XX'$ meet $\mathcal C$ at $Z \ne X, X'$. Then $Y, Z$ are isogonal conjugates.
\end{theorem}
\begin{proof}
Let $Z^*$ be the isogonal conjugate of $Z$. By \Cref{reversal} $(XZ, XZ^*)$ are isogonal in $\angle AXC$, and $(X'Z, X'Z^*)$ are isogonal in $\angle AX'C$, so $Z^* \equiv Y$, as desired.
\end{proof}
\begin{lemma}
The bisectors of $\angle AZC$ are perpendicular and parallel to $XX'$.
\end{lemma}
\begin{figure}[!htbp]\centering
\includegraphics[width=350pt]{3-1.png}
\caption{Tangents to the Cubic}
\label{fig:3.1}
\end{figure}
\begin{proof}
Examining isogonal conjugates $(A, C), (X, X')$, this follows from \Cref{swallow-reversal}.
\end{proof}
\begin{theorem}
Let $PZ$ meet $\mathcal C$ at $W \ne Z$. Then $WX = WX'$.
\end{theorem}
\begin{proof}
By \Cref{rho-aias}, the bisectors of $XWX'$ are perpendicular and parallel to $XX'$, which gives the desired result.
\end{proof}
\begin{corollary}
If we denote $P$ by $0$ on the conic, then $W = X + X'$ under cubic addition (\cite{ref:Fulton}, Proposition 5.6.4). Thus, the cubic sum of any two isogonal conjugates $X, X'$ is equidistant from $X, X'$.
\end{corollary}
We thus obtain the following construction, if we desire to find all pairs of isogonal conjugates $(Y, Y')$ such that $YY'$ passes through a given excellent point $X$.
\begin{theorem}
For $X \in \mathcal C$, let distinct $Y, Z \ne X$ lie on $\mathcal C$ such that $X, Y, Z$ are collinear and $XY$ bisects $\angle AXC$. Then $Y, Z$ are isogonal conjugates.
\end{theorem}
\begin{proof}
Let $Y'$ be the isogonal conjugate of $Y$; then by \Cref{swallow-reversal}, $(XY, XY')$ and $(XA, XC)$ are isogonal. Since $XY$ bisects $\angle AXC$, $XY'$ and $XY$ must be the same line, implying that either $Y \equiv Y'$ or $Z \equiv Y'$.
In the latter case we are done. In the former case, $Y$ must be an incenter or excenter of $ABCD$, so by \Cref{singular} $Y$ is a singular point. But $X, Y, Z$ are collinear and distinct despite line $XYZ$ intersecting $Y$ with multiplicity 2, the desired contradiction.
\end{proof}
\begin{figure}[!htbp]\centering
\includegraphics[width=450pt]{3-2.png}
\caption{Isogonal Conjugates Collinear with a Given Point}
\label{fig:3.2}
\end{figure}
Note that this also gives us a construction of points $Y$ on $\mathcal C$ such that $ZY$ is tangent to $\mathcal C$ at $Y$, where $Z$ is a fixed point on $\mathcal C$. This is done by letting $X$ be the isogonal conjugate of $Z$ and intersecting the angle bisectors of $\angle AXC$ with $\mathcal C$. By the above, there will be up to four such intersections $X_1, X_2, X_3, X_4$ on $\mathcal C$ for which $ZX_1, ZX_2, ZX_3, ZX_4$ are tangent to $\mathcal C$ at $X_1, X_2, X_3, X_4$.
\section{Constructing Intersections with Lines and Circles}
\begin{theorem}[Line Intersection]
Consider excellent points $X, Y$. Denote by $Z$ the intersection of the reflections of $XY$ over the bisectors of $\angle AXC$ and $\angle AYC$. Then the intersection of $XY$ with $\mathcal C$ other than $X, Y$ is also the isogonal conjugate of $Z$. Furthermore, $PXYZ$ is cyclic.
\end{theorem}
\begin{proof}
Let $W = XY \cap \mathcal C$, and let $W$ have isogonal conjugate $W'$. By \Cref{swallow-reversal}, $XW$ and $XW'$ are isogonal in $\angle AXC$, so line $XW'$ is the reflection of $XY$ over the bisectors of $\angle AXC$, implying that $W' \equiv Z$, proving that $XY \cap \mathcal C$ is indeed the isogonal conjugate of $Z$.
To prove $PXYZ$ is cyclic, let $X', Y'$ be the isogonal conjugates of $ABC$. By \Cref{completeness}, $XY \cap \mathcal C$ lies on $X'Y'$, hence $W \in \mathcal X'Y'$. Then under spiral inversion, line $X'Y'W$ is mapped to the circumcircle of $XYZ$, which must pass through $P$, as desired.
\end{proof}
As a direct corollary, we have the following well-known theorem:
\begin{corollary}[Spiral Center of Isogonal Conjugates Lies on Circumcircle]
For isogonal conjugates $(A, C)$, $(B, D)$ in $\triangle XYZ$, the spiral center of $ABCD$ lies on $(XYZ)$.
\end{corollary}
We may remark that this provides a construction of the intersection of $\mathcal C$ with any line $XY$, provided that $X$ and $Y$ lie on $\mathcal C$ themselves. Next, we characterize intersections of $\mathcal C$ with circles.
\begin{theorem}[Circle Intersection]
Consider excellent points $E, F, G$ with isogonal conjugates $E', F', G'$. Then $(EFG)$ meets $\mathcal C$ at one other point which lies on $(EF'G'), (E'FG'), (E'F'G)$.
\end{theorem}
\begin{proof}
There are two parts to this. First, we prove that if $H$ is a point on $\mathcal C$ such that $EFGH$ is cyclic, then $H$ lies on $(E'F'G)$ (which would imply that it lies on $(EF'G'), (E'FG')$ by symmetry). To prove this, by \Cref{reversal} with $EGE'G'$, since $H \in \mathcal C$, $\angle F'GE' = \angle EGF = \angle EHF = \angle F'HE'$.
\begin{figure}[!htbp]\centering
\includegraphics[width=400pt]{4-1.png}
\caption{Intersecting with Circles}
\label{fig:4.1}
\end{figure}
Next, we prove that if $H \equiv (EFG) \cap (E'F'G)$, then $\angle F'HE' = \angle F'GE' = \angle EGF = \angle EHF$, which implies that $E \in \mathcal C$ as desired.
\end{proof}
We may remark that this provides a construction for all points on $(EFG)$ lying on $\mathcal C$, provided $E, F, G$ lie on $\mathcal C$ themselves. The following is in fact true.
\begin{theorem}
All circles intersect $\mathcal C$ in the real plane at at most 4 points.
\end{theorem}
\begin{proof}
The circular points at infinity lie on $\mathcal C$ by virtue of being isogonal conjugates. The result follows from Bezout's Theorem, where curves of degree 2 and 3 meet for at most six points in $\mathbb{CP}^2$.
\end{proof}
\section{Characterizing the Isogonal Cubic}
For this section, we will work in $\mathbb{CP}^2$ and let $I, J$ denote the circular points at infinity. We must first extend the definition of isogonality to $\mathbb{CP}^2$ as follows:
\begin{definition}
For distinct points $P, A, B, C, D \in \mathbb{CP}^2$, we call the two pairs of lines $(PA, PB)$ and $(PC, PD)$ \textit{isogonal} if and only if the three pairs of lines
\[(PA, PB), \quad\quad (PC, PD), \quad\quad (PI, PJ)\]
comprise a single involution, where $I, J$ are the circular points at infinity.
\end{definition}
One can check this complies with the angular definition of isogonality if $P, A, B, C, D \in \mathbb{RP}^2$.
\begin{corollary}
For distinct points $A, B, C, D$ such that neither of $I, J$ lie on any of the lines $AB, BC, CD, DA$, the locus of points $X$ for which $(XA, XB), (XC, XD)$ are isogonal is a cubic (or curve of lesser degree) through $A, B, C, D, I, J$ in $\mathbb{CP}^2$.
\end{corollary}
\begin{proof}
For any four points $A, B, C, D, E, F$, the locus of points $X$ for which $(XA, XB)$, $(XC, XD)$, $(XE, XF)$ comprise a single involution is a cubic through $A, B, C, D, E, F$. Setting $E, F$ as the circular points at infinity gives the desired result.
\end{proof}
Hence we will call a non-degenerate cubic $\mathcal C$ the "isogonal cubic" of quadrilateral $ABCD$ if it is the locus of all points $X$ for which $(XA, XC), (XB, XD)$ are isogonal (using the new definition).
\begin{corollary}[Loci of Isogonality]\label{loci}
If the locus of points $X$ for which $(XA, XC), (XB, XD)$ are isogonal is a non-degenerate cubic, then neither $I$ nor $J$ cannot lie on any of the lines $AB, BC, CD, DA$.
\end{corollary}
\begin{proof}
Assume the contrary, that WLOG $I \in AB$. Then for any point $P$ on line $AB$, pairs $(XA, XC)$, $(XB, XD)$, $(XI, XJ)$ are part of a single degenerate involution. Thus the locus of points $X$ for which $(XA, XC)$, $(XB, XD)$ are isogonal includes line $AB$, contradicting the proposition that the locus is a non-degenerate cubic.
\end{proof}
In other words, if $ABCD$ has a non-degenerate isogonal cubic $\mathcal C$, then $I$ and $J$ will not lie on $AB, BC, CD, DA$.
Now, the main result of this paper is the following:
\begin{theorem}[Characterization of all Isogonal Cubics]
Let $\mathcal C$ be a non-degenerate cubic in $\mathbb{CP}^2$ containing circular points at infinity $I$ and $J$ at non-singular points. Then the following two conditions are equivalent:
\begin{enumerate}
\item[(1)] There exist non-singular $A, B, C, D \in \mathcal C$ such that $\mathcal C$ is the isogonal cubic of $ABCD$.
\item[(2)] The tangents to $\mathcal C$ at $I, J$ intersect each other on $\mathcal C$.
\end{enumerate}
\end{theorem}
We begin with the following direct result of Cayley-Bacharach (\cite{ref:Cayley}).
\begin{lemma}[Cubics Containing Complete Quadrilateral]\label{complete-quad}
For $P, Q$ on non-degenerate cubic $\mathcal C$, consider $T \in \mathcal C$ and let $U = PT \cap \mathcal C$, $V = QT \cap \mathcal C$ such that $P, Q, T, U, V$ are non-singular. Then $PV \cap QU \in \mathcal C$ iff $PP \cap QQ \in \mathcal C$.
\end{lemma}
\begin{proof}
Let $X = PP \cap QQ, Y = PV \cap QU$. Cayley-Bacharach on triples of lines $(XPP, QTV, QUY)$, $(XQQ, PTU, PVY)$ completes both directions.
\end{proof}
\begin{lemma}[Locus of Involution]\label{involution-locus}
Consider distinct points $A, B, C, D, E, F$ in general position and $G = AC \cap BD, H = AD \cap BC, I = AE \cap BF, J = AF \cap BE$, such that none of the ten points are the circular points at infinity. Then there is a unique cubic $\mathcal C$ through these ten points. Furthermore, for every $P \in \mathcal C$, we have
\[(PA, PB), \quad (PC, PD), \quad (PE, PF), \quad (PG, PH), \quad (PI, PJ)\]
are part of a fixed involution.
\end{lemma}
\begin{figure}[!htbp]\centering
\includegraphics[width=200pt]{5-1.png}
\caption{Two Complete Quadrilaterals}
\label{fig:5.1}
\end{figure}
\begin{proof}
By the Dual of Desargues' Involution Theorem, the locus $\mathcal C$ of all points $P$ for which $(PA, PB)$, $(PC, PD)$, $(PE, PF)$ are part of a single involution is a cubic through
\[A, B, C, D, E, F, G, H, I, J.\]
Thus, there exists a cubic through these $10$ points. Since $A, B, C, D, E, F$ are in general position, no four of the $10$ constructed points are collinear. Since $\mathcal C$ passes through these $10$ fixed points, the cubic through these $10$ points must be unique, as desired.
\end{proof}
We are finally set up to prove the main result.
\begin{theorem}[Characterization in $\mathbb{CP}^2$, Condition (2) $\implies$ (1)]\label{class-forward}
Let $\mathcal C$ be a non-degenerate cubic through $I, J$ such that $I$ and $J$ are non-singular, and $II$ intersects $JJ$ at a point $X$ on $\mathcal C$. Then there exist non-singular points $A, B, C, D \in \mathcal C$ apart from $I, J$ such that $\mathcal C$ is the isogonal cubic of $ABCD$.
\end{theorem}
\begin{proof}
Choose any point $A \in \mathcal C$. Let $B = IA \cap \mathcal C, D = JA \cap \mathcal C$; by \Cref{complete-quad}, $ID$ and $JB$ meet a point $C$ on $\mathcal C$. Construct four points $A', B', C', D' \in \mathcal C$ distinct from $A, B, C, D$ analogously, where $I = A'B' \cap C'D'$ and $J = A'D' \cap B'C'$. We may select $A, A'$ such that none of $A, A', B, B', C, C', D, D'$ are singular.
By \Cref{involution-locus}, $\mathcal C$ is the locus of points $P$ for which $(PI, PJ), (PA, PC), (PA', PC')$ are part of a single involution. But since this involution concerns the circular points at infinity, it follows that $\mathcal C$ is the locus for which $(PA, PC), (PA', PC')$ are isogonal. We are done by taking quadrilateral $AA'CC'$.
\end{proof}
\begin{theorem}[Characterization in $\mathbb{CP}^2$, Condition (1) $\implies$ (2)]\label{class-reverse}
Let $\mathcal C$ be the non-degenerate isogonal cubic of $ABCD$ where $A, B, C, D$ are non-singular. Suppose that $I, J$ lie on $\mathcal C$ at non-singular points distinct from $A, B, C, D$. Then $II \cap JJ \in \mathcal C$.
\end{theorem}
\begin{proof}
Let $X = AI \cap CJ$ and $Y = AJ \cap CI$; then $X, Y$ are non-singular. Note that $(XA, XC)$ and $(XI, XJ)$ are the same pair of lines, so $(XA, XC)$, $(XB, XD)$, $(XI, XJ)$ form an involution. (If $X$ is the same point as either $A, C, I,$ or $J$, we instead use the tangent to $\mathcal C$ at $X$ when necessary.)
So by definition, $X \in \mathcal C$; similarly, $Y \in \mathcal C$. By \Cref{complete-quad}, this implies $II \cap JJ \in \mathcal C$ as desired.
\end{proof}
Going back to $\mathbb R^2$, we derive the complete characterization of all non-degenerate isogonal cubics:
\begin{theorem}[Characterization of All Isogonal Cubics in $\mathbb R^2$]\label{class-real}
Let $\mathcal C$ be a non-degenerate cubic in $\mathbb R^2$, and let $\mathcal C_0$ denote its embedding in $\mathbb{CP}^2$. Then the following two conditions are equivalent:
\begin{enumerate}
\item[(1)] There exist distinct $A, B, C, D \in \mathcal C$ such that $\mathcal C$ is the isogonal cubic of $ABCD$.
\item[(2)] The circular points at infinity $I, J$ lie on $\mathcal C_0$, and the tangents to $\mathcal C_0$ at $I, J$ intersect each other on $\mathcal C_0$.
\end{enumerate}
\end{theorem}
\begin{proof}
The only aspects of the proof we need to modify for this new wording are to prove that:
\begin{enumerate}
\item Under the conditions of (1), if $\mathcal C$ is the isogonal cubic of $ABCD$ where $A, B, C, D$ are distinct, then $A, B, C, D$ cannot be singular points of $\mathcal C_0$.
\item Under the conditions of (2), if any cubic $\mathcal C$ in $\mathbb R^2$ satisfies that its embedding $\mathcal C_0$ in $\mathbb{CP}^2$ passes through $I$ and $J$, then $I$ and $J$ are not singular.
\item Under the conditions of (2), for $A \in \mathcal C$ such that $K = AI \cap \mathcal C_0, L = AJ \cap \mathcal C_0, A' = IL \cap JK$ all lie on distinct points of $\mathcal C_0$, then the point $A'$ will be contained in $\mathcal C$ as well.
\end{enumerate}
Let $\mathcal C$ have Cartesian equation $ax^3+bx^2y+cxy^2+dy^3 + G(x, y) = 0$, where $G$ is a second-degree polynomial in $x, y$. Then $a, b, c, d$ must be real, and since $\mathcal C$ is not degenerate, they cannot all be zero. Thus $\mathcal C_0$ has equation $F(x, y, z) = 0$ where $F(x, y, z) = ax^3 + bx^2y + cxy^2 + dy^3 + zP(x, y, z)$, where $P(x, y, z)$ is a second-degree homogeneous polynomial in $x, y, z$.
\begin{subproof}
For (a), assume that $\mathcal C$ is the isogonal cubic of $ABCD$. Note that $A$ is a singular point in $\mathcal C$ iff it is a singular point in $\mathcal C_0$, because both are equivalent to $\frac{\partial F}{\partial x}(A) = \frac{\partial F}{\partial y}(A) = \frac{\partial F}{\partial z}(A) = 0$, the same equation in both $\mathbb{RP}^2$ and $\mathbb{CP}^2$. We just need to show that $A$ is not a singular point in $\mathbb R^2$.
By \Cref{singular}, $A$ is singular if and only if $A$ is the isogonal conjugate of itself in $ABCD$. But the isogonal conjugate of $A$ is $C$, and since $A$ and $C$ are distinct, this cannot happen. Therefore, $A$ and similarly $B, C, D$ are not singular points of $\mathcal C_0$, as desired. This proves part (a).
\end{subproof}
\begin{subproof}
For (b), we consider general cubic $\mathcal C$ which contains $I, J$. Plugging in $I = (1: i: 0)$ yields equation $a+bi-c-di = 0$, which implies that $a = c$ and $b = d$ because $a, b, c, d$ are all real. Thus $\mathcal C$ has equation $(x^2+y^2)(ax+by) + zP(x, y, z)$. We get $\frac{\partial F}{\partial x}(1: i: 0) = 3ax^2 + 2xby + ay^2$.
Assume, for the sake of contradiction, that $I$ is a singular point. We require $\frac{\partial F}{\partial x} = 0$ for $(1: i: 0)$, which rearranges to $2a + 2bi = 0$. Since $a, b$ are real, this implies that $a = b = 0$, so $a, b, c, d$ are all zero - the desired contradiction. This proves part (b).
\end{subproof}
\begin{subproof}
For (c), it suffices to show that $A' \in \mathbb R^2$. Note that $I, J, A, K, L, A'$ all lie on $\mathcal C_0$, which has all real coefficients. Now, $K, L$ do not lie on the line of infinity, or else $A$ would lie on the line of infinity, which would imply $\mathcal C_0$ containing four points on a line and thus be degenerate, a contradiction. Thus $K, L$ they are contained in $\mathbb C^2$ and thus can be expressed in Cartesian coordinates $(k_x, k_y)$ and $(l_x, l_y)$ respectively. Since $A \in \mathbb R^2$, note that $K$ and $L$ cannot lie in $\mathbb R^2$ - or else the entire lines $AK$ and $AL$ will be contained in $\mathbb{RP}^2$ and never intersect the line of infinity at complex points $I$ and $J$.
From part (b), $\mathcal C$ must have equation of the form $ax^3+bx^2y+axy^2+by^3 + G(x, y) = 0$ where $a, b$ and the coefficients of $G$ are real numbers. For $K$ to satisfy this equation, the point $K'$ whose Cartesian coordinates are the complex conjugates of $K$ - that is, $K' = \left(\ol{k_x}, \ol{k_y}\right)$ in Cartesian coordinates - must also satisfy this equation, and thus lie on $\mathcal C_0$. Having started with $A, I, K$ collinear, we now claim that $A, J, K'$ are collinear. It suffices to show that
\[\begin{vmatrix} 1 & -i & 0 \\ a_x & a_y & 1 \\ \ol{k_x} & \ol{k_y} & 1 \end{vmatrix} = 0 \quad\quad \text{given that} \quad\quad \begin{vmatrix} 1 & i & 0 \\ a_x & a_y & 1 \\ k_x & k_y & 1 \end{vmatrix} = 0\]
Letting $k_x = p+qi$ and $k_y = r+si$ for $p, q, r, s \in \mathbb R$, the second determinant equation gives us
\[0 = \begin{vmatrix} 1 & i & 0 \\ a_x & a_y & 1 \\ p+qi & r+si & 1 \end{vmatrix} = -r-si-q+pi+a_y-ia_x\]
where $(a_x, a_y)$ are the Cartesian coordinates of $A$. Equating the real and imaginary parts yields $a_y = q+r, a_x = p-s$. Similarly, the first determinant equation gives us
\[0 = \begin{vmatrix} 1 & -i & 0 \\ a_x & a_y & 1 \\ p-qi & r-si & 1 \end{vmatrix} = -r+si-q-pi+a_y+ia_x\]
and equating the real and imaginary parts yields $a_y = q+r, a_x = p-s$ - the exact same conditions. Therefore, given that $A, I, K$ are collinear, we indeed conclude that $A, J, K'$ are collinear.
In other words, $K'$ is the unique intersection of $\mathcal C_0$ with $AJ$, hence $K' \equiv L$. Thus $A' = IK' \cap JK$, and $A'$ will not be a point at infinity (otherwise $K$ will also be a point at infinity). Hence in quadrilateral $AKA'K' \in \mathbb C^2$, we have $AK$ meets $A'K'$ at a point of infinity, and $AK'$ meets $A'K$ at a point of infinity - so complex segments $AA'$ and $KK'$ share the same midpoint $M$. Letting $A$ have Cartesian coordinates $(m, n)$ in $\mathbb C^2$, this means that $M$ has Cartesian coordinates
\[\left(\frac{a_x+m}{2}, \frac{a_y+n}{2}\right) \quad = \quad \left(\frac{k_x + \ol{k_x}}{2}, \frac{k_y + \ol{k_y}}{2}\right)\]
But $\frac{k_x+\ol{k_x}}{2}$ is just the real part of $k_x$, so the coordinates of $M$ are real as well. Hence $m$ and $n$ are real, so $A' = (m, n)$ lies in $\mathbb R^2$, proving part (c).
\end{subproof}
With (a), (b), (c) proven, for the sake of completion we will show how this fully finishes our characterization. For the direction (1) $\implies$ (2), we start with $ABCD$, and by (a) none of $A, B, C, D$ are singular points. Then the result directly follows from \Cref{class-reverse}.
For the direction (2) $\implies$ (1), we start with circular points at infinity $I, J$ lying on $\mathcal C_0$, which by (b) implies that $I, J$ are not singular points. Assuming that the tangents to $\mathcal C_0$ at $I$ and $J$ intersect each other on $\mathcal C$, we can choose any point $A \in \mathcal C$ and letting $K = AI \cap \mathcal C_0$, $L = AJ \cap \mathcal C_0$, and $A' = IL \cap JK$ where $A' \in \mathcal C_0$, and then choose another point $B \in \mathcal C$ and define $B' \in \mathcal C_0$ the same way, such that all points formed by these intersections are distinct. By \Cref{class-forward}, $\mathcal C_0$ will be the isogonal cubic of $ABA'B'$. In addition, (c) implies that $A'$ and $B'$ will in fact lie in $\mathbb R^2$ as well. Therefore, $ABA'B'$ is fully contained in $\mathbb R^2$, so $\mathcal C$ is indeed the isogonal cubic of $ABA'B'$. This completes the solution.
\end{proof}
\section{Uniqueness in the Isogonal Cubic}
With this algebraic characterization of all isogonal cubics in $\mathbb R^2$ in mind, in this section, we prove that given an isogonal cubic $\mathcal C \in \mathbb{RP}^2$, there is only one possible spiral center $P$, and for any $X \in \mathcal C$, there is only one possible point that could be the isogonal conjugate of $X$.
\begin{theorem}[Uniqueness of the Spiral Center]
Consider non-degenerate $\mathcal C \in \mathbb{RP}^2$ such that there exist $A, B, C, D \in \mathbb R^2$ for which $\mathcal C$ is the isogonal cubic of $ABCD$. Let $ABCD$ have spiral center $P$. Let $\mathcal C_0$ denote the embedding of $\mathcal C$ in $\mathbb{CP}^2$. Then $PI$ and $PJ$ are respectively tangent to $\mathcal C_0$ at $I$ and $J$.
\end{theorem}
\begin{proof}
Assume, for the sake of contradiction, that $PI$ is not tangent to $\mathcal C_0$ at $I$; then $PJ$ cannot be tangent to $\mathcal C_0$ at $J$ either, so by part (c) of \Cref{class-real}, $PI$ and $PJ$ intersect $\mathcal C_0$ at $K, L \in \mathbb C^2$ respectively, distinct from $I, J, P$, and $IL$ and $JK$ intersect $\mathcal C_0$ at $Q \in \mathbb R^2$. Then by \Cref{class-forward}, $\mathcal C$ is the non-degenerate isogonal cubic of the three quadrilaterals $ABCD$, $APCQ$, $BPDQ$.
By \Cref{truth}, there is one point of infinity $P_\infty \in \mathcal C$, which is the point of infinity along the Newton-Gauss lines of $ABCD$, $APCQ$, $BPDQ$. Let $AP$ meet $\mathcal C$ at $E$; then by \Cref{isogonal-parallel}, $C, E, P_\infty$ are collinear. Since $\mathcal C$ is the isogonal cubic of $APCQ$, it follows that $AP \cap CQ \in \mathcal C$, so in fact $Q = CE \cap \mathcal C$. Since $C, E$ lie in $\mathbb R^2$ they are distinct from $P_\infty$.
If $C, E, P_\infty$ are distinct, then $Q \equiv P_\infty$, contradicting $Q \in \mathbb R^2$, as desired. So line $CE$ intersects $\mathcal C$ with multiplicity 2. We assumed that $ABCD \in \mathbb R^2$, so we cannot have $C \equiv P_\infty$, hence either $C \equiv E$ or $E \equiv P_\infty$. In either case, we cannot have $Q \equiv P_\infty$ else $Q \in \mathbb R^2$ is contradicted; thus $Q \equiv C$. But considering $A, P$ are the respective isogonal conjugates of $C, Q$ in $APCQ$, so this implies $A \equiv P$. Now, the isogonal conjugates of $A, P$ in $ABCD$ are $C, P_\infty$, which implies that $C \equiv P_\infty$ - the desired contradiction.
\end{proof}
In other words, a given non-generate isogonal cubic can only have one possible spiral center - we may now call this \textit{the} spiral center of a given isogonal cubic $\mathcal C$. This leads to the following result, allowing us to define isogonal conjugation on any given isogonal cubic without having to construct a base quadrilateral $ABCD$:
\begin{theorem}[Uniqueness of the Isogonal Conjugate]
Consider non-degenerate $\mathcal C \in \mathbb {RP}^2$ such that there exist $A, B, C, D \in \mathbb R^2$ for which $\mathcal C$ is the isogonal cubic of $ABCD$. Then for any point $X \in \mathcal C$, there is only one possible point $X' \in \mathcal C$ which could be the isogonal conjugate of $X$ in $ABCD$.
\end{theorem}
\begin{proof}
Let $P$ be the spiral center of $\mathcal C$, and let $P_\infty$ be the point of infinity of $\mathcal C$. Consider any $X \in \mathcal C$. If $X \equiv P$, its isogonal conjugate is $P_\infty$, and vice versa. If $X$ is neither $P$ nor $P_\infty$, let $Y = PX \cap \mathcal C$ and $X' = P_\infty Y \cap \mathcal C$. Then by \Cref{isogonal-parallel}, $X'$ is the isogonal conjugate of $X$ in $ABCD$ no matter which $ABCD$ we choose. Since $P$ is fixed, $X'$ depends only on $X$, as desired.
\end{proof}
\begin{figure}[!htbp]\centering
\includegraphics[width=300pt]{6-1.png}
\caption{Construction of the Isogonal Conjugatate in a Cubic}
\label{fig:6.1}
\end{figure}
Therefore, given any non-degenerate isogonal cubic $\mathcal C \in \mathbb R^2$ and any point $X \in \mathcal C$, the spiral center $P$ and the isogonal conjugate of $X$ with respect to $\mathcal C$ are well-defined. Thus, we may now revisit our constructions of intersections and tangents, this time with a general isogonal cubic.
\begin{theorem}[Tangents to the Isogonal Cubic]
For non-singular $X \in \mathcal C$, let $X'$ be its isogonal conjugate. Let $\ell$ be the isogonal of $XX'$ wrt lines $XP, XP_\infty$. Then $\ell$ is tangent to $\mathcal C$ at $X$.
\end{theorem}
\begin{theorem}[Line Intersections in the Isogonal Cubic]
For distinct $X, Y \in \mathcal C$, let $\ell_X$ be the isogonal of $XY$ wrt lines $XP, XP_\infty$; define $\ell_Y$ analogously. Then $XY \cap \mathcal C$ is the isogonal conjugate of $\ell_X \cap \ell_Y$.
\end{theorem}
\section{Algebraic Characterization in the Cartesian Plane}
To conclude the paper, we present a purely algebraic characterization of all possible isogonal cubics in $\mathbb R^2$ for the sake of completion.
\begin{theorem}
A non-degenerate cubic $\mathcal C \in \mathbb R^2$ is an isogonal cubic of some quadrilateral $ABCD$ if and only if it has the form $f(x, y) = f(p, q)$, where
\[f(x, y) = Ax^3 + Bx^2y + Axy^2 + By^3 + Cx^2 + Dxy + Ey^2 + Fx + Gy\]
such that all coefficients are real and $(A, B) \ne (0, 0)$, and
\[p = \frac{AE-AC-BD}{2(A^2+B^2)}, \quad\quad q = \frac{BC-AD-BE}{2(A^2+B^2)}.\]
Furthermore, the spiral center of $\mathcal C$ is $(p, q)$, and the unique real asymptote of $\mathcal C$ is given by
\[(A^3+AB^2)x + (A^2B+B^3)y + (A^2E-ABD+B^2C) = 0.\]
\end{theorem}
\begin{proof}
Let the embedding $\mathcal C_0$ of $\mathcal C$ in $\mathbb{CP}^2$ have equation $g(x, y, z) = 0$, where
\[g(x, y, z) = Ax^3 + Bx^2y + Axy^2 + By^3 + Cx^2z + Dxyz + Ey^2z + Fxz^2 + Gyz^2 + Hz^3\]
where the equality of the coefficients of $x^3$ with $xy^2$ and $x^2y$ with $y^3$ is given by part (b) of \Cref{class-real}. Let $g$ denote the left-hand side of the above equation. We compute
\[\frac{\partial g}{\partial x} = 3Ax^2 + 2Bxy + 2Fxz + Ay^2 + Dyz + Fz^2\]
\[\frac{\partial g}{\partial y} = 3By^2 + 2Axy + 2Eyz + Bx^2 + Dxz + Gz^2\]
\[\frac{\partial g}{\partial z} = 3Hz^2 + 2Fxz + 2Gyz + Cx^2 + Dxy + Ey^2\]
Plugging in the partial derivatives for $(1: i: 0)$, the tangent to $\mathcal C_0$ at $(1: i: 0)$ has equation
\[(2A+2Bi)x + (-2B+2Ai)y + (C+Di-E)z = 0\]
and similarly the tangent to $\mathcal C_0$ at $(1: -i: 0)$ has equation
\[(2A-2Bi)x + (-2B-2Ai)y + (C-Di-E)z = 0\]
The spiral center $P$ of $\mathcal C$ is then given by the solution to these two equations. Solving yields
\[(x: y: z) \quad = \quad \left(AE-AC-BD: BC-AD-BE: 2(A^2+B^2)\right)\]
Since $A$ and $B$ are not both $0$, converting back to Cartesian coordinates implies that $P$ indeed has coordinates given by $(p, q)$. In Cartesian coordinates, the value
\[Ax^3 + Bx^2y + Axy^2 + By^3 + Cx^2 + Dxy + Ey^2 + Fx + Gy\]
must be a constant, particularly $-H$. Plugging in $(p, q)$ immediately gives the equation for $\mathcal C$ to be $f(x, y) = f(p, q)$ as desired.
To determine the asymptote, we find that points of infinity on $\mathcal C_0$ are given by
\[0 = Ax^3 + Bx^2y + Axy^2 + By^3 = (x^2+y^2)(Ax+By)\]
so the real point of infinity is given by $P_\infty = (B: -A: 0)$. Plugging this into the equations for the partial derivatives yields that the tangent to $\mathcal C_0$ at $P_\infty$, and by extension the unique real asymptote of $\mathcal C$, indeed takes the above equation. This completes the proof.
\end{proof}
\section{Acknowledgements}
The author would like to acknowledge and thank to Anant Mudgal for providing some of the necessary theory in complex projective geometry, helping to formalize the analytic characterizations, and checking over the solutions. The author would also like to give special thanks to Michael Diao for checking the paper, suggesting additions to the content, and helping to format the document and diagrams.
|
1,116,691,499,384 | arxiv | \section{Introduction}
\label{sec:Introduction}
Cherenkov radiation is induced in the atmosphere when a cascade of secondary particles propagate in the air from an initial point where an ultra-high energy (UHE) cosmic ray (CR, in other words, astroparticle) enters the atmosphere. The necessary condition for a charged particle to produce this radiation is to move at a speed greater than $c/n$, where $c$ is the speed of light in vacuum and $n$ is the index of refraction in air~\cite{Tamm}. There are plenty of particles moving at such speeds in an extensive air shower (EAS) of cosmic rays, so Cherenkov radiation can be easily measured on a moonless night with simple light detectors working in coincidence of signals.
Cherenkov radiation is used to infer information about the energy, composition, and direction of arrival of the primary astroparticle that initiated the shower. Since the first observation by Cherenkov \cite{Cherenkov} in the laboratory, and Galbraith and Jelley~\cite{Jelley} in the atmosphere, the systematic measurement of the properties of air Cherenkov radiation were performed in the Pamir experiment~\cite{Chudakov}, and then with a number of EAS arrays. Particularly, the Yakutsk array experiment applies these detectors to estimate the energy and mass composition of the primaries~\cite{Dyak,JETP2007}.
Generally, in previous measurements, analog signal readout systems were used with a narrow bandwidth, restricting the possibility of the reconstruction of the waveform of the Cherenkov radiation signal detected in EAS.
Alternatively, detectors were designed for the measurement of the integral signal \cite{Dyak,Turver,BLANCA}.
However, a digital data acquisition system (DAQ) was recently implemented in the Tunka-133 Cherenkov array, consisting of a set of photomultiplier tubes (PMT)~\cite{Tunka}.
In our papers~\cite{Tmprl,Dcnvlv} a method was described for reconstructing the temporal characteristics of the Cherenkov radiation from the signal of a wide field-of-view (WFOV) Cherenkov telescope (hereinafter `telescope') measured in EAS detected with the Yakutsk array.
These characteristics were used to estimate the parameters of the development of the shower, specifically, to set an upper limit to the dimensions of the area along the EAS axis where the Cherenkov radiation intensity is above its half-peak amplitude.
As a development from these efforts, we have analyzed the extended dataset of the telescope measurements including the observational period 2012 to 2015, when coincident detection of EAS events with the telescope was possible.
In the present paper, reconstructed Cherenkov radiation signals are used to estimate in a different way the position of the maximum of the shower particle number in the atmosphere, $X_{max}^{Ne}$, employing the results of Monte Carlo simulations of EAS development.
This article is structured as follows. In Section~\ref{sec:Experiment}, the Yakutsk array experiment and data acquisition and selection for analysis are briefly described, including the telescope. The details of the digital signal processing are given in Section~\ref{sec:SignalProcessing}. In Sections \ref{sec:EAS params} and \ref{sec:Nerling} the connection between the Cherenkov radiation signal and the EAS parameters is studied, and applied to estimate $X_{max}^{Ne}$.
\section{The Yakutsk array. \\Data acquisition and selection for analysis}
\label{sec:Experiment}
The Yakutsk array is located at a site with geographical coordinates ($61.7^{\circ}N, 129.4^{\circ}E$), 100 m above sea level~\cite{MSU,Zenith}. A schematic view of the layout of the surface stations of the array in the relevant observational period is given in Fig.~\ref{Fig:Array}. Forty-nine stations are distributed within a triangular grid of total area 8.2 km$^2$.
The shower events are selected based on coincidence signals from $n\geq 3$ stations, which in turn have been triggered by the two scintillation counters in each station. Complementary triggers at lower energies are produced by the central cluster consisting of 20 Cherenkov radiation detectors~\cite{KnurCERN,KnurFlor,KnurWeihai}.
The main components of the EAS are detected using scintillators, four muon detectors, 48 air Cherenkov light detectors, and six radio detectors. In this paper, we focus exclusively on the pulse shape of the Cherenkov radiation signal from EAS.
Residual aspects concerning other components of the phenomenon are covered in previous papers of the Yakutsk array group~\cite{JETP2007,Tokyo,EMcomponent,MinWidth,Knur2020}.
All detectors/controllers and data processing units of the array are connected by a fiber-optic network.
An array modernization program aims to achieve a LAN channel capacity of 1 Gbps, synchronization accuracy of detectors, and a time resolution accuracy of $10$ ns. The planned energy range for EAS detection is $(10^{15}, 10^{19})$ eV~\cite{ASTRA,MainResults}.
\begin{figure}[t]
\includegraphics[width=0.98\columnwidth]{Stations}
\caption{The arrangement of the detectors of the Yakutsk array.
Charged particle detectors are shown by open circles, Cherenkov radiation detectors subset by filled triangles.
The position of the telescope is indicated by the cross.}
\label{Fig:Array}\end{figure}
\subsection{The wide field-of-view telescope detecting the waveform of Cherenkov signals in EAS observed by the Yakutsk array detectors}
\label{sec:Telescope}
The constituent parts of the telescope are a) the spherical mirror (${\o}260$ mm, $f=113$ mm) mounted at the bottom of a metal tube; b) a position-sensitive PMT (Hamamatsu R2486; ${\o}50$ mm) at the focus for which the anode is formed by $16\times16$ crossed wires; c) a voltage-divider circuit and mechanical support attached to the bearing plate; and d) 32 operational amplifiers mounted onto the tube.
The telescope is mounted vertically near an array station (Fig.~\ref{Fig:Telescope}).
A comprehensive description of the telescope can be found in
\cite{ASTRA,Tlscp,Tmprl}.
The data acquisition system of the telescope consists of 32 operational amplifiers that have 300-MHz bandwidth AD8055 chips connected by long (12 m) coaxial cables to 8-bit LA-n4USB ADC digitizers with 4-ns time slicing. All of the ADC output signals from the 32 channels are continuously stored in PC memory. A trigger signal from the EAS array terminates the process and signals in a 32 $\mu$s interval preceding a trigger are dumped. In Fig.~\ref{Fig:Fout}, an example is given of the output signals of the DAQ recorded in coincidence with the Yakutsk array detectors in a particular CR shower. EAS parameters are estimated using the data from all the appropriate array detectors. Nineteen wires of the telescope PMT exhibit significant Cherenkov radiation signals; the other thirteen wires show no signal above the noise level in an event.
\begin{figure}[t]
\includegraphics[width=0.79\columnwidth]{Telescope}
\caption{Wide field-of-view Cherenkov telescope.
A spherical mirror and a multi-anode PMT with voltage divider and holders are visible.
Pre-amplifiers are mounted on the outside of the tube.}
\label{Fig:Telescope}\end{figure}
In this paper, we use the data accumulated during the period from October 2012 to March 2015 (total number of EAS events is 300173) for which EAS events were detected simultaneously by the surface detectors and the telescope (733 events).
Data selection cuts are applied to exclude showers with cores out of the array area and with zenith angles $\theta>60^{\circ}$.
In the present analysis, we do not use the angular dependence of the telescope signals in an individual EAS event: the angular and arrival time differences of signals are ignored. Saturated signals are thrown out. The number of EAS events surviving after these cuts is 386.
The average zenith angle of the showers in a sample is $18^0\pm11^0$, and the energy is $(2\pm0.3)\times10^{17}$ eV.
\begin{figure*}[t]
\includegraphics[width=\textwidth]{Fout}
\caption{The Cherenkov signal from EAS detected with crossed wires of PMT anode of the telescope.
Left panel: 16 X wires; right panel: 16 Y wires.}
\label{Fig:Fout}\end{figure*}
\section{Analysis of Cherenkov radiation signal}
\label{sec:SignalProcessing}
\subsection{EAS simulation results concerning temporal characteristics of the Cherenkov signal}
\label{sec:CORSIKA}
The physical description of the Cherenkov radiation of relativistic charged particles in a medium originated with the paper of Frank and Tamm \cite{Tamm}. The characteristics of the radiation induced by a cascade of particles in the atmosphere used to be exhaustively modeled by the numerical solution of the cascade equations, or widespread Monte Carlo codes, such as CORSIKA \cite{CORSIKA}. The most known applications of these model simulations are: estimation of the energy of the primary astroparticle using the total flux of the Cherenkov radiation in the EAS \cite{Spectrum_vs_Cher,Spectrum_vs_Gmodel,TotalFlux}; special `Hillas' parametrization of the Cherenkov images of showers in IACTs, resulting in an unprecedented separation of the very high energy photons, initiating the EAS, from the nuclear background \cite{Hillas}.
The main results of simulations concerning the temporal structure of the Cherenkov signal in EAS are the finding of a near-spherical shower front and that the duration of the signal increases with the shower core distance. To elucidate these features, it is convenient to apply a toy model using a vertical EAS for simplicity. A detector is placed at a distance of $R_i$ far away from the core, so that `the shining point' approximation \cite{ShinePoint} is applicable, namely, the light emitter with normal angular distribution $f_{cher}(\nu)$, where $\nu$ is the angle between the direction to the detector and the shower axis, is moving along the shower axis with the speed of light; the light intensity is proportional to the cascade curve, i.e., the total number of electrons, $N_e(h)$.
The photon arrival time to detector is defined by
\begin{equation}
ct=\sqrt{h^2+R_i^2}-h,
\label{Eq:ShinePoint}
\end{equation}
where $h$ is the emission height; $t=0$ when the shining point arrives at the array plane. Integrating $N_e(h)f_{cher}(\nu)/(h^2+R_i^2)$ one can estimate the total signal of the detector.
The spherical shower front of the photons is evidence that most of the Cherenkov radiation is bounded within a small area around some height $h_{max}^{cher}$. A deviation from sphericity is connected to the width of the cascade curve.
The duration of the signal, as a function of the core distance, is produced by a plain geometrical effect which can be demonstrated using Eq.~\ref{Eq:ShinePoint}, specifically, with the cascade curve of $\prod$-form, equal to a constant $\neq 0$ between $h_1$ and $h_2$ (Fig.~\ref{Fig:ToyDur}).
\begin{figure}[b]
\includegraphics[width=\columnwidth]{ToyDur}
\caption{Signal duration as a function of the shower core distance, $R_i$, in a toy model with rectangular cascade curve.}
\label{Fig:ToyDur}\end{figure}
\begin{table}[b]
\begin{center}
\caption{Sum of squared residuals of fitted distributions and the waveform functions.
The number of bins are different, so the sums should be compared within columns only.}
\begin{tabular*}{0.48\textwidth}{@{\extracolsep{\fill}}lrr}
\hline\hline
Approximation & Modeled $f_{in}$ & Measured $f_{in}$\\\hline
Normal & 11.68 & - \\
LogNormal & 0.12 & 24.79 \\
Gamma & 0.78 & 26.68 \\
\hline\hline
\end{tabular*}
\end{center}
\end{table}
Chitnis and Bhat found \cite{LogNorm} that the waveform of the Cherenkov signal in the detector is represented by a lognormal distribution function fairly accurately at core distances up to 280 m, employing Monte Carlo simulation studies of showers with CORSIKA v.560 and EGS4 codes. Battistoni et al. fitted lognormal and gamma distributions to the delay distributions of secondary photons and electrons for different EAS primaries \cite{LogNorm2}. They conclude that the lognormal distribution fits the data better mainly because of the long tails at large delays ($\sim 200$ ns). We have demonstrated recently \cite{Dcnvlv} that the Cherenkov radiation signal from EAS can be approximated by the gamma distribution, using digital signal processing of the output data from the detector.
\begin{figure*}[t]
\includegraphics[width=0.45\textwidth]{Fit3}
\includegraphics[width=0.45\textwidth]{ReFit_Fin}
\caption{Fitting the waveform of the Cherenkov signal in the detector with an appropriate pdf.
Left: $f_{in}(t)$ calculated using Eq.~\ref{Eq:Nerling}.
Right: deconvolution of the telescope signal in EAS detected $23^h48^m00^s$ UTC on 21 October 2012.}
\label{Fig:Fit3}\end{figure*}
In order to prove these results we have chosen a method of calculation after Nerling et al.~\cite{Nerling}, from a multitude of Monte Carlo simulations of EAS development, because of the analytical description of the results concerning Cherenkov radiation in the shower.
The investigation uses an approximation for the energy of the electron and its angular distribution in the high-energy domain based on the universality of both distributions. A similar approach was employed in \cite{Universality,deSouza}. The universality of the calculated electron distributions means their independence from different primary energies, particle types, and zenith angles of EAS to a good approximation for the range of electron energy from 1 MeV to a few GeV, covering the range most important for Cherenkov light emission.
The number of Cherenkov photons arriving at a detector of area $S_d$ at the shower core distance $R_i\gg$ core radius is given by~\cite{ShinePoint,JETP2007}
\begin{eqnarray}
Q_{Sd}\propto\int_0^\infty dh\tau(h)\frac{f_{cher}(\nu)S_dLcos\theta}{L_d^3}\nonumber\\
\times\int_{E_{th}}^{E_0}dE\frac{dN(h,E,E_0}{dE}\varsigma(1-\frac{E_{th}^2}{E^2}),
\label{Eq:Nerling}\end{eqnarray}
\noindent where $\tau(h)$ is the light transmission coefficient; $L$ is the distance along the shower axis from the shining point to the array plane; $L_d$ is the distance from the shining point to the detector; $f_{cher}(\nu)$ is the angular distribution of the photons; $dN/dE$ is the electron differential spectrum; $\varsigma(1-\frac{E_{th}^2}{E^2})$ is the number of photons emitted by an electron along 1 g/cm$^2$; and the threshold energy for an electron to emit Cherenkov radiation is $E_{th}=\frac{mn}{\sqrt{(n-1)(n+1)}}$.
In this approximation, the photons are assumed to be produced at the shower axis.
The parametrization of the electron energy spectrum derived by Nerling et al., $a_0E/(E+a_1)/(E+a_2)$ with constants for fixed shower age $s=3/(1+2X_{max}/x)$ (in the Appendix of~\cite{Nerling}) is used in our calculations. The angular distribution of the Cherenkov photons is approximated by
$$
f_{cher}(\nu,h,s)=a_s(s)\frac{exp(-\nu/\nu_c(h))}{\nu_c(h)}+b_s(s)\frac{exp(-\nu/\nu_{cc}(h))}{\nu_{cc}(h)}
$$
and the parameters are given in the Appendix of~\cite{Nerling}. The angular distribution of photons is a direct consequence of the universal electron angular distribution. The total number of particles as a function of depth is approximated by the gamma distribution (the ``Gaisser--Hillas curve'' used by the PAO collaboration~\cite{GH}) with a depth of the maximum $X_{max}=650$ g/cm$^2$.
The resultant waveform of the Cherenkov signal in a detector placed at the core distance $R_i$ is approximated by normal, gamma and lognormal distributions applying the code ``amoeba,'' which implements the downhill simplex method \cite{Amoeba}, to find the least squares deviation from $f_{in}$. For completeness, the experimentally measured waveform is approximated, too (Fig.~\ref{Fig:Fit3}, right panel). In this case, the input signal to the telescope is reconstructed by applying the Wiener deconvolution algorithm \cite{Dcnvlv}.
It seems that the lognormal and gamma distributions fit the waveform almost equally well, particularly in comparison with the variance of the real signal in the experiment, over the whole range of distances far from the shower core. The sums of the squared residuals are listed in Table 1. Of the three, the lognormal distribution has the minimum deviation, so we have chosen it as the best approximation to the waveform of the Cherenkov signal.
\subsection{Deconvolution of the signal measured with the telescope}
\label{sec:Dcnvlv}
The method of deconvolution of the signal observed by the telescope is described in detail in our previous papers~\cite{Dcnvlv,Dcnvlv2}.
In short, an input Cherenkov signal can be reconstructed by the Fourier transform applied to
\begin{equation}
f_{out}(t)=\int_{-\infty}^{\infty} f_{in}(\tau)g(t-\tau)d\tau =(f_{in}*g),
\label{Eq:Cnvltn}
\end{equation}
where $f_{in},f_{out}$ are the input and output signals of the DAQ; and $g(t)$ is a system transfer function \cite{DSP}.
The last is estimated using the dark current impulse of the PMT; an example is given in Fig.~\ref{Fig:Response}.
It is convenient to reconstruct the input Cherenkov radiation signal induced by EAS applying the approximation by the lognormal distribution function $f_{in}^{lognorm}$, discussed above. In this case the deconvolution procedure can be simplified. Namely, the method now consists of adjustment of the time window to $f_{out}$ and fitting the free parameters of the trial function $f_{in}^{lognorm}$ so that the forward convolution result is congruent to the measured output signal. The convolution theorem ensures that the derived lognormal distribution is the only solution.
To evaluate the free parameters of $f_{in}^{lognorm}$, the nonlinear least squares approach is used. The aim is to minimize the sum of the squared differences between the observed signal and the convolution result in the time window. The optimal values of the parameters are found here by applying a downhill simplex method~\cite{Amoeba}.
\begin{figure}[t]
\includegraphics[width=0.95\columnwidth]{Response}
\caption{Impulse response of the data acquisition system to a short input signal.}
\label{Fig:Response}\end{figure}
To decrease the influence of noise on the analyzed signals, we selected DAQ output signals with amplitudes above the threshold 0.075 V.
For instance, in the event no.~906 shown in Fig.~\ref{Fig:Fout}, only eight channels have amplitudes of the signal above this threshold.
The optimized convolution result $(f_{in}^{lognorm}*g)$ in comparison with the observed output signal $f_{out}$ is illustrated in Fig.~\ref{Fig:Fit}.
The optimization of the lognormal distribution means in our case the fitting of two parameters $av,\sigma$ in order to minimize the sum of squared residuals of the output distributions:
\begin{equation}
f_{in}^{lognorm}(t)=\frac{1}{t\sigma\sqrt{2\pi}}exp(-\frac{(\ln(t)-av)^2}{2\sigma^2}),
\label{Eq:GammaParams}
\end{equation}
where $av$ is the mean of $\ln(t)$ and $\sigma$ is the rms deviation.
If one has the measured moments of the $t$-distribution: $\bar{t},D_t$, then $\sigma^2=\ln(1+D_t/\bar{t}^2)$; $av=\ln(\bar{t})-0.5\sigma^2$.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Fit_906}
\caption{The convolution of the impulse response with trial lognormal distribution versus output signal of DAQ.}
\label{Fig:Fit}\end{figure}
\section{Connection of the duration of the Cherenkov radiation signal with the EAS parameters}
\label{sec:EAS params}
As the main parameter of the Cherenkov radiation observed by the telescope, we treat the duration of the signal, i.e., the full width at half maximum, FWHM, of the lognormal distribution recovered from $f_{out}$. It is shown to have a clear dependence on the shower core distance, which can be used to connect it with the development of the EAS in the atmosphere~\cite{Klmkv,Tunka,KnurWeihai,Tmprl}.
Coincident EAS events detected simultaneously with surface detectors of the Yakutsk array and the telescope were selected for analysis.
The shower parameters were estimated based on the data of the surface detectors; signals with amplitudes above the threshold were used from the telescope DAQ channels to infer the average duration of the Cherenkov signals. A bee line from the telescope to the shower axis is used as the core distance of the detector
\begin{equation}
R_i=R_{AP}\sqrt{\sin^2\psi+\cos^2\psi\cos^2\theta},
\label{Eq:Rp}\end{equation}
where $R_{AP}$ is the distance to the core in the array plane, and $\psi$ is the angle between $R_{AP}$ and the projection of the shower axis.
In spite of the additivity of the variance of the signal, we preferred the FWHM because of its ease of use in experiment. Furthermore, it inherits additivity within certain limits. The 32 crossed wires of the anode with private DAQ channels provide at least several independent measurements of a Cherenkov signal above the threshold in an individual EAS event. At another step, showers are selected in the intervals of core distances where the durations of the reconstructed signals are averaged.
The resultant FWHM of the Cherenkov signal measured with the telescope in coincidence with the surface detectors of the Yakutsk array as a function of $R_i$ is shown in Fig.~\ref{Fig:FWHM} in comparison with previous measurements (Haverah Park \cite{Turver}, Tunka \cite{Tunka}, Yakutsk-1975 \cite{Klmkv}). Our own previous efforts to measure signal durations yielded the results given in~\cite{Tmprl,Dcnvlv}.
Since then, the number of measured EAS events has increased, and the reconstruction algorithm has been improved, so the results have become somewhat more enhanced.
\begin{figure}[t]
\includegraphics[width=0.9\columnwidth]{FWHM}
\caption{Full width of half maximum of the input Cherenkov signal from EAS as a function of the shower core distance.
Vertical bars are statistical errors, horizontal bars are intervals of the radial distance.
Solid curve in the interval $R_i\in(200,1000)$ m is the result of the model simulation that will be described in the next section.}
\label{Fig:FWHM}\end{figure}
Sampling EAS arrival angles, we have found the function $FWHM(R_i)$ to be independent of the azimuth and zenith angles within instrumental errors. While the independence from the azimuth is not surprising, the zenith angle dependence may be revealed through the distance to $X_{max}^{Ne}$ rising with $\theta$. A possible reason is the insufficiently large aperture of the telescope to reveal a faint zenith angle effect~\cite{Tlscp}.
In the same way we looked for an energy dependence of the duration of the Cherenkov signal. Similarly, there are no significant differences over the core distance intervals, except in the utmost interval, where there may be some systematic change of duration with energy, although less than two sigma (Fig~\ref{Fig:DurVsE}).
The signal duration is almost constant at core distances below $100$ m due to the radius of the shining area in EAS core, and is rising with the radius at $R_i\gg100$ m because of the greater length of the shining area along the axis and the position of the shower maximum in the atmosphere, as was explained in Section \ref{sec:CORSIKA}.
\section{Application of EAS simulation results to the analysis of Cherenkov radiation signal}
\label{sec:Nerling}
The parameters of the Cherenkov signal, such as the duration of the signal rising with EAS core distance, $\tau(R_i)$, have been used as objects of investigation in a number of experiments. Namely, the SINP MSU group noted that $\tau(R_i)$ at far distances from the shower core is connected with $X_{max}^{Ne}$ calculated in CKP and HMM model simulations~\cite{Klmkv}. Another method of estimating $X_{max}^{Ne}$ was proposed making use of the lateral distribution slope of Cherenkov radiation measured with the Yakutsk array detectors~\cite{MSU}.
The CASA-BLANCA array studied CRs in the energy range 0.3--30 PeV. To find the transformation from the characteristics of the Cherenkov radiation as measured with BLANCA to the depth of shower maximum, the same method as in the Yakutsk array group was used, validated using the CORSIKA simulations with different hadronic interaction models (QGSJET, VENUS, SIBYLL, and HDPM)~\cite{BLANCA}.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{DurVsE}
\caption{The energy dependence of the duration of the Cherenkov signal in the far core distance interval.}
\label{Fig:DurVsE}\end{figure}
In the Tunka experiment, they used two methods of estimating $X_{max}^{Ne}$: the first is based on the shape of the lateral distribution of the intensity of the Cherenkov radiation, just as was the method used in the previous cases; the second uses the sensitivity of the pulse width at the fixed core distance (400 m) to the position of the EAS maximum~\cite{TunkaStatus}.
Using the measured correlation of the duration of the Cherenkov signal with the distance to the shower core, we have estimated an upper limit to the dimensions of the area along the EAS axis where the Cherenkov radiation intensity is above the half-peak amplitude~\cite{Dcnvlv}.
The length of the shining area is found to be less than 1500 m, and the diameter is less than 200 m in EAS with the primary energy $E_0 = 2.5\times10^{17}$ eV and zenith angle $\theta = 20^0$.
Due to the monotonic relation of the shining point height with the photon arrival time to detector, Eq.~\ref{Eq:ShinePoint}, it is straightforward to estimate $h_{max}^{cher}$ using the time of signal maximum in detector, $t_{max}$, in a model-independent way \cite{Tmprl}:
\begin{equation}
h_{max}^{cher}\sec\theta=\frac{R_i^2-(ct_{max})^2}{ct_{max}}+R_{AP}\sin\theta\cos\psi,
\label{Eq:Pythagor}
\end{equation}
where $h_{max}^{cher}$ is the height where the Cherenkov radiation is emitted, which forms the maximum of the signal in detector at $R_i$ from the shower core.
Unfortunately, the Yakutsk array in its present configuration is not able to measure the reference arrival time of the shining point to the array plane with sufficient accuracy \cite{Tmprl}. Therefore, an implementation of this promising method should be postponed until the completion of the array modernization program.
One of the features of the Cherenkov signal in EAS is that its maximum is different from that of the total number of particles, i.e., the position in the atmosphere of the maximum intensity of the Cherenkov radiation, $h_{max}^{cher}$, is higher than $h_{max}^{Ne}$.
Fig.~\ref{Fig:XmCher} illustrates this property caused by the angular distribution of relativistic electrons emitting Cherenkov photons.
Indeed, evaluation with a toy model indicates that the flat $f_{cher}(\nu)$ has a weak effect on the position of the maximum, while narrowing the beam leads the visible radiation maximum to drift higher in the atmosphere. A linear fit $C_h(R_i)$ can be used to estimate $h_{max}^{Ne}$ based on the measured $h_{max}^{cher}$.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{hmCherVsNe}
\caption{The ratio of maximum heights of the Cherenkov radiation intensity and of the number of shower particles as a function of the core distance of detector.}
\label{Fig:XmCher}\end{figure}
For the purpose of applying the EAS simulation results to the analysis of the Cherenkov radiation signal, namely, to estimate $X_{max}^{Ne}$ based on the temporal characteristics of the Cherenkov signal measured at large shower core distances, it is convenient to employ an analytical description of Cherenkov light emission in EAS, i.e., the results of \cite{Nerling}, as was discussed in Section \ref{sec:CORSIKA}.
Nerling et al.~parametrized the results of the CORSIKA simulations with the QGSJET01 model \cite{QGS01}, which describe showers independently of the primary energy, particle type, and zenith angle, with a high accuracy of a few percent (within shower-to-shower fluctuations).
Actually, this approach allows one to make use of a toy model with the implemented parametrizations of the CORSIKA simulations.
Adjusting the main unmeasurable parameters of EAS, e.g., $X_{max}^{Ne}$ and the width of the angular distribution of the photons, $\sigma_{\nu}$, inherent in showers initiated by different nuclei, to measured Cherenkov radiation characteristics, one can find the best fitting values satisfying the conditions of the model.
In general, due to the universality of the electron distributions in EAS, the angular and lateral distributions of the Cherenkov photons emitted by a shower path element depend only on the age of the shower and its height in the atmosphere \cite{UniverCher,UniverAge}.
Consequently, the distributions of photons measured with Cherenkov radiation detectors can be equivalently described by different models having the same $X_{max}^{Ne}$ and $\sigma_{\nu}$. We do not mention the energy spectrum of electrons, bearing in mind that it determines the total number of electrons emitting Cherenkov radiation, and is parametrized by $X_{max}^{Ne}$.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Nerling_GH_min}
\caption{Fitting the maximum depth of electrons, $X_{max}^{Ne}$, and angular distribution width of Cherenkov photons, $\sigma_{\nu}$, in a simulated shower to our observed signal duration as a function of the core distance within $(200,1000)$ m, illustrated in Fig.~\ref{Fig:FWHM}.}
\label{Fig:DurVsXmax}\end{figure}
The difference between this algorithm and that of SINP MSU and Tunka's second approach is in the fixed core distance of $\tau(R_i)$ in their case, and on the contrary, a variety of distances (within the interval $200-1000$ m) in our case, to determine $X_{max}^{Ne}$.
In the latter, the amount of empirical information is definitely greater.
Fig.~\ref{Fig:DurVsXmax} presents a fit to our measurements of calculations with different maximum depths incorporated into the approximations of Nerling et al. by CORSIKA simulations in $R_i\in(200,1000)$ m. Namely, the sum of squared differences between the observed and simulated durations of Cherenkov signals in the $R_i$ interval is minimized. The width of the angular distribution of Cherenkov photons in EAS is a function of the age of the shower and the height of the shining point; we have approximated it by the value at $s=1,h=h_{max}^{cher}$ in order to demonstrate a fit. Variation of the width, $\sigma_{\nu}(s,h)$, is carried out by a scaling factor applied to the angle between the direction to the detector and the shower axis.
It turns out that $X_{max}^{Ne}=670\pm20$ g/cm$^2$ provides the best fit to the experimental values of the durations of the Cherenkov signals in EAS with energy 0.2 EeV and zenith angle $18^0$. A confidence interval of fitted $X_{max}$ is estimated at the $95\%$ level assuming a small sample of equiprobable depths. A comparison of the results of measurements of Cherenkov radiation with the simulations in a toy model employing the fitted parameters is given in Fig.~\ref{Fig:FWHM}.
We have chosen here the depth of the shower maximum, $X_{max}^{Ne}$, as a conventional parameter useful for comparison with other experiments.
At the site of the Yakutsk array, for the Cherenkov radiation measurements in winter nights, when the atmosphere temperature profile is close to isothermal, a plain exponential equation $X=1020\exp(-h/6900)$ g/cm$^2$ can be applied for estimations.
A resultant average depth of shower maximum in the number of EAS particles is compared with previous measurements in Fig.~\ref{Fig:XmaxData} borrowed from \cite{LOFAR}. It is in reasonable accord with a set of experiments: HiRes \cite{HiRes}, PAO \cite{PAO}, TALE \cite{TALE}, Tunka \cite{TunkaStatus}, Yakutsk 2019 \cite{KnurXmax}, LOFAR \cite{LOFAR} within the interval $(1.7-2.3)\times10^{17}$ eV where the depth dispersion is confined to $\sim(640,680)$ g/cm$^2$.
The estimated mean value of $X_{max}^{Ne}$ at $\overline{E}=0.2$ EeV can be used to infer the proton component fraction in the primary beam, within the two-component (H and Fe nuclei) mass composition assumption. Taking into account the shower maximum depths derived in the QGSJetII-04, EPOS-LHC, and Sibyll-2.3d models, one concludes that the proton fraction is $79\pm21\%$, $62\pm19\%$, and $56\pm18\%$, and the mean mass $\overline{lnA}$ is 0.85, 1.53, 1.76 at $E=0.2$ EeV, in the corresponding model. These values are close to the results of PAO \cite{PAO}.
The divergence between the estimated values of $X_{max}^{Ne}$ in experiments can be considered to be caused by the model uncertainties and instrumental errors due to the variety of detectors used, from fluorescent and Cherenkov light detectors to radio wave receivers.
A straightforward way to reduce the uncertainties would be the application of model-independent methods of measurement. Regarding the planned Cherenkov radiation measurement in the EAS investigation, the triangulation method employing the shower front curvature, e.g., Eq.~\ref{Eq:Pythagor}, seems to be the best choice.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Xmax_E}
\caption{World data on $X_{max}^{Ne}$ estimations in EAS collected in \cite{LOFAR} with the present result added.
The lines indicate the simulation results with QGSJetII-04 (solid), EPOS-LHC (dashed) and Sibyll-2.3d (dotted) models for iron and proton primaries.}
\label{Fig:XmaxData}\end{figure}
\section{Conclusions}
The addition of a wide field-of-view telescope to the multitude of the Yakutsk array detectors has expanded its possibilities for EAS investigation for the measurement of the temporal characteristics of the Cherenkov radiation emitted by shower particles.
In the present paper, the results of an enhanced analysis of the temporal features of this radiation detected in coincidence of signals by the telescope and surface detectors have been given.
The input signal of the telescope's DAQ is reconstructed applying a lognormal approximation of the Cherenkov radiation signal from EAS: both measured and simulated by a model. The experimental data are deconvolved from the telescope output signal using an independent method.
The resultant Cherenkov signal reconstruction algorithm is simple and fast, allowing on-the-fly analysis of measured signals.
The main measurable temporal characteristic of Cherenkov radiation induced by EAS is the signal duration. We have enhanced previous measurements of the signal duration and confirmed explicitly that it rises with the shower core distance at $R_i>200$ m. This rise is related to the development of the shower in the atmosphere, and further, we have demonstrated that the behaviour of the signal duration in the interval $R_i\in(200,1000)$ m can be used to estimate $X_{max}^{Ne}$.
An essential requirement for this is the application of EAS modeling under certain assumptions concerning interactions of the particles.
We implemented Monte Carlo simulation results after Nerling et al.~\cite{Nerling} in our toy model calculations. The resultant estimation of the shower maximum depth $X_{max}^{Ne}=670\pm20$ g/cm$^2$ at $E=(2\pm0.3)\times10^{17}$ eV, $\theta=18^0\pm11^0$ is in reasonable agreement with previous results obtained using different experimental techniques. The connected estimation of the proton fraction and of the mean mass of the primary astroparticles under the two-component hypothesis is close to the results of the PAO collaboration.
\acknowledgements
We would like to thank the Yakutsk array group for data acquisition and analysis. This work was supported by the Ministry of Science and Higher Education of the Russian Federation (program ``Unique Scientific Installations,'' no. 73611).
|
1,116,691,499,385 | arxiv | \section*{References}
\begin{description}
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G.Ponzano and T.Regge, {\it Semiclassical limit of Racah coefficients},
in: Spectroscopic and Group Theoretical Methods in Physics, ed. F. Block
{\it et al.} ( North Holland, Amsterdam, 1968) pp. 1-58.
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T.Regge, Nuovo Cimento {\bf 19} (1961) 558-571.
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B. Hasslacher and M.J. Perry, Phys. Lett. {\bf B 103} (1981) 21-24.
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S.M. Lewis, Phys. Lett. {\bf B 122} (1983) 265-267.
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V.G. Turaev and O.Y. Viro, {\it State sum invariants of 3-manifolds and
quantum 6j-symbols}, LOMI Preprint (1990).
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H. Ooguri and N. Sasakura, Mod. Phys. Lett. {\bf A 6} (1991)
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F. Archer and R.M. Williams, Phys. Lett. {\bf B 273} (1991) 438-444.
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\end{description}
\vfill\eject
{\bf Figure captions}
\vskip2cm
\noindent {\bf fig. 1}\par
The diagram of the $6j$-symbol is the $3$-simplex $T$ with
boundary $\partial T \equiv \tau$ embedded in ${\bf R}^3$.
$( T,\tau )$ is homeomorphic to $(D^3, S^2)$, where $D^3$
is the euclidean $3$-disk and $S^2$ is the $2$-sphere.
\vskip1cm
\noindent {\bf fig. 2a}\par
The diagram of the reduced $12j$-symbol is the $3d$-combinatorial
manifold obtained by joining the two tetrahedra $T$ and $T'$.
Notice that the three edges of $T$ $(J_4, J_2, J_6)$
form a face which is glued to the face $(J'_5, J'_1, J'_6)$ of
$T'$.
\vskip1cm
\noindent {\bf fig. 2b}\par
The $4$-simplex $\sigma$ can be represented in ${\bf R}^3$ as the
combinatorial manifold of fig.2a with an additional edge of
lenght $L$ connecting two vertices as indicated ( the heavy
line $L$ should be thought as lying in the fourth dimension).
This drawing gives however the correct vertex-edge-face
scheme of $\sigma$ ($\sigma$ has 5 vertices, 10 edges and 10
triangular faces). $\sigma$ is topologically the $4$-disk in
${\bf R}^4$ with boundary the $3$-sphere.
\end{document}
|
1,116,691,499,386 | arxiv | \section{Introduction}
In this paper we find a new, and perhaps unexpected, connection between
polyanalytic functions and time-frequency analysis, and use it to obtain a
complete characterization of all lattice sampling and interpolating
sequences in the Bargmann-Fock space of polyanalytic functions or,
equivalently, of all lattice vector-valued Gabor frames and vector-valued
Gabor Riesz sequences for $L^{2}(%
\mathbb{R}
,%
\mathbb{C}
^{n})$.
\subsection{Overview}
The Bargmann-Fock space of polyanalytic functions, $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$, consists of all functions satisfying the equation
\begin{equation}
\left( \frac{d}{d\overline{z}}\right) ^{n}F(z)=0\text{,}
\label{defpolyanalytic}
\end{equation}%
and such that%
\begin{equation}
\int_{%
\mathbb{C}
^{d}}\left\vert F(z)\right\vert ^{2}e^{-\pi \left\vert z\right\vert
^{2}}dz<\infty \text{.} \label{growth}
\end{equation}%
\emph{\ }Functions satisfying (\ref{defpolyanalytic}) are known as \emph{%
polyanalytic functions} of order $n$. Since (\ref{defpolyanalytic})
generalizes the Cauchy-Riemann equation
\begin{equation*}
\frac{d}{d\overline{z}}F(z)=0\text{,}
\end{equation*}%
then the space $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ is a generalization of the Bargmann-Fock space of analytic functions,
$\mathcal{F}(%
\mathbb{C}
^{d})=\mathbf{F}^{1}(%
\mathbb{C}
^{d})$. In the case of $\mathcal{F}(%
\mathbb{C}
)$, a complete description of the sets of sampling and interpolation is
known \cite{Ly},\cite{S1},\cite{SW}.
Polyanalytic functions inherit some of the properties of analytic functions,
often in a nontrivial form. However, as in the theory of several complex
variables, many of the properties break down once we leave the analytic
setting. An obvious difference lies in the structure of the zeros. For
instance, while nonzero entire functions do not have sets of zeros with an
accumulation point, polyanalytic functions can vanish along closed curves:
just take $F(z)=\overline{z}z-1=\left\vert z\right\vert ^{2}-1$, a
polyanalytic function of order 2. Polyanalytic functions have been
investigated thoroughly, notably by Balk and his students \cite{Balk}. They
are naturally related to polyharmonic functions, which have an intriguing
structure of zero sets \cite{HayKor}, \cite{BalkMazalov}, \cite{Render}.
We will study the spaces $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ using time-frequency analysis, and think about the polyanalytic
functions in a different way from the classical classical approach.
The link to time-frequency analysis has impressive consequences: by endowing
the spaces $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ with the structure inherent to translations and modulations, one can
use tools that were unavailable with complex variables. This will provide $%
\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ with properties reminiscent of the classical analytic Fock space.
In order to state our main results, we briefly recall some definitions. See
sections 4 and 5 for more details. We associate the sequence $\Lambda
=\{(x,w)\}$ with the sequence of complex numbers $\Gamma =\{(x+iw)\}$. Then $%
\Gamma $ is \emph{an interpolating sequence} for $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ if, for every sequence $\{\alpha _{i,j}\}\in l^{2}$, there exists $%
F\in \mathbf{F}^{n}(%
\mathbb{C}
^{d})$ such that
\begin{equation*}
e^{i\pi x\omega -\frac{\pi }{2}\left\vert z\right\vert ^{2}}F(z)=\alpha
_{i,j},
\end{equation*}%
for every $z\in \Gamma .$ We say that $\Gamma $ is \emph{a} \emph{sampling
sequence} for $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$\ if there exist $A,B>0$ such that, for every $F\in \mathbf{F}^{n}(%
\mathbb{C}
^{d})$,%
\begin{equation*}
A\left\Vert F\right\Vert _{\mathbf{F}^{n}(%
\mathbb{C}
^{d})}^{2}\leq \sum_{z\in \Gamma }\left\vert F(z)\right\vert ^{2}e^{-\pi
\left\vert z\right\vert ^{2}}\leq B\left\Vert F\right\Vert _{\mathbf{F}^{n}(%
\mathbb{C}
^{d})}^{2}.
\end{equation*}
The concept of interpolating sequences has its roots in deep problems in
complex analysis, and sampling sequences are a major issue in signal
processing, since they correspond to the case where stable numerical
reconstructions of a function \ from its samples is possible. Monograph \cite%
{Seipmonograph} is a good introduction to sampling and interpolation and its
interconnections with other branches of pure and applied mathematics.
Our main results, Theorem 4 and Theorem 6 below, use the concept of Beurling
density, which, in the lattice case, is given by $D(\Gamma )=D(\Lambda
)=\left\vert \det A\right\vert ^{-1}$, where $\Lambda =A%
\mathbb{Z}
^{2}$.%
\begin{equation*}
\end{equation*}%
\textbf{Theorem 4. }The lattice $\Gamma $ is a sampling sequence for $%
\mathbf{F}^{n}(%
\mathbb{C}
)$ if and only if:
\begin{equation*}
D(\Gamma )>n.
\end{equation*}%
\textbf{Theorem 6. }The lattice $\Gamma $ is an interpolating sequence for $%
\mathbf{F}^{n}(%
\mathbb{C}
)$ if and only if:
\begin{equation*}
D(\Gamma )<n.
\end{equation*}
These results follow by establishing one duality between sampling in $%
\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ and multiple interpolation in $\mathcal{F}(%
\mathbb{C}
^{d})$ and a second duality between interpolation in $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ and multiple sampling in $\mathcal{F}(%
\mathbb{C}
^{d})$. When $d=1$, these properties allow us to directly apply the results
in \cite{Brekkeseip}. Taking $n=1$ we recover the well known duality between
sampling and interpolation in $\mathcal{F}(%
\mathbb{C}
^{d})$.
Theorems exhibiting a "Nyquist rate" phenomenon tend to be hard to prove.
They have been studied, for general sequences, in spaces of analytic
functions, first in the Paley-Wiener space \cite{Beurling},\cite{Landauacta},%
\cite{LandauIEE} and then in Bargmann-Fock \cite{Ly},\cite{S1},\cite{SW} and
Bergman \cite{Seip2} spaces of analytic functions. There are two reasons for
us to believe that Beurling type methods do not work here. First, they use
complex variables tools that are not available in the polyanalytic
situation. Second, in all of these spaces, there were known "doubly
orthogonal systems" \cite{Seipdoubly}, which provided the eigenfunctions for
the fundamental equation involving the "concentration operator". We are not
aware of a system with this double orthogonality property in the
polyanalytic situation.
Therefore, we introduce new tools.
With a view to relating our problem to one concerning the density of vector
valued Gabor systems, we extend Bargmann%
\'{}%
s work \cite{Bar} to the setting of polyanalytic functions. Once we do this,
our argument, which is conceptual in nature, follows smoothly.
It is also worth noting that the density Theorem in Gabor analysis has
itself a very rich story, beginning with fundamental but imprecise
statements by John Von Neumann and Dennis Gabor, which caught the attention
of mathematicians after conjectures by Daubechies and Grossman \cite%
{DauGross}. See the survey article \cite{Heil}, \cite{CHO} and the important
special windows studied in \cite{JanStr}.
To give a context to our approach, recall the connection between the
classical (analytic) Bargmann-Fock space and time-frequency analysis.
It is well known that, up to a certain weight, the Gabor transform with a
gaussian window belongs to the Fock space of analytic functions. Moreover,
it has been shown that this is the only choice leading to spaces of analytic
functions \cite{AscensiBruna}.
However, a nice picture emerges when we take Hermite functions as windows.
The analytic situation generated by the gaussian window then becomes the tip
of the iceberg of a larger structure involving spaces of polyanalytic
functions. Indeed, the Gabor transform with the $nth$ Hermite function is,
up to a certain weight (the same as in the analytic case), a polyanalytic
function of order $n+1$.
To fully understand the situation, we will need the spaces constituted by
the functions satisfying (\ref{growth}), which are polyanalytic of order $n$%
, but are \emph{not} polyanalytic of any lower order (in particular they
have no analytic functions). These are the \emph{true} polyanalytic Fock
spaces $\mathcal{F}^{n}(%
\mathbb{C}
^{d})$. The polyanalytic Fock and true polyanalytic Fock spaces are related
by the following orthogonal decomposition (see Corollary 1 in section 3):%
\begin{equation*}
\mathbf{F}^{n}(%
\mathbb{C}
^{d})=\mathcal{F}^{0}(%
\mathbb{C}
^{d})\oplus ...\oplus \mathcal{F}^{n-1}(%
\mathbb{C}
^{d})\text{.}
\end{equation*}
Then, each space $\mathcal{F}^{n}(%
\mathbb{C}
^{d})$ is associated with Gabor transforms with the $nth$ Hermite window.
Such occurrence, which seems to have been hitherto unnoticed, will be
fundamental in our discussion. This observation is related to some recent
developments in Gabor analysis with Hermite functions \cite{CharlyYura},\cite%
{CharlyYurasuper},\cite{Fuhr}, to Janssen%
\'{}%
s approach to the density Theorem \cite{Janssen},\cite{J3} and also to the
techniques used in \cite{Hutnikcr},\cite{Hutnikm},\cite{VasiBergman}, which
suggest that wavelet spaces and polyanalytic functions share intriguing
patterns.
Fock spaces of polyanalytic functions are briefly mentioned in Balk%
\'{}%
s monograph \cite{Balk} and they are implicit in quantum mechanics, in
connection with the Landau levels of the Schr\"{o}dinger operator with
magnetic field \cite{Shigekawa},\cite{LuefGosson} and displaced Fock states
\cite{Wunsche}. However, we were not able to find any reference to
polyanalytic functions in the mathematical physics literature, apart from
\cite{VasiFock}, where creation and annihilation operators are used.
\subsection{The results of Gr\"{o}chenig-Lyubarskii and of
Balan-Casazza-Landau}
Our results are connected to a very recent result of Gr\"{o}chenig and
Lyubarskii, which deserves more specific comment in this introduction.
Denote by $G(\mathbf{h}_{n},\Lambda )$ the set of the translations and
modulations indexed by the lattice $\Lambda $ and acting coordinate-wise on
the vector $\mathbf{h}_{n}.$ Then, the following result holds.
\textbf{Theorem }\cite{CharlyYurasuper}: \emph{Let }$\mathbf{h}%
_{n}=(h_{0},...,h_{n-1})$\emph{\ be the vector of the first }$n$\emph{\
Hermite functions. Then }$G(\mathbf{h}_{n},\Lambda )$\emph{\ is a frame for }%
$L^{2}(%
\mathbb{R}
,%
\mathbb{C}
^{n})$\emph{\ if and only if }%
\begin{equation*}
D(\Lambda )>n.
\end{equation*}%
One may wonder if the equivalence of this condition to the one in Theorem 4
reflects causality or casuality. As we will see, \emph{causality} is the
answer. Actually, one of the key steps in our approach consists of showing
that Theorem 4 is equivalent to the above Theorem.
The original proof of this Theorem in \cite{CharlyYurasuper} combines the
use of the so-called Wexler-Raz biorthogonality relations with complex
analysis techniques based on properties of the Weierstrass sigma function.
We will give an alternative proof, which is considerably shorter (at the
cost of using deeper results from the literature) and has the advantage of
also characterizing the vector valued Gabor Riesz sequences with Hermite
windows in the following statement, which is equivalent to Theorem 6:
\textbf{Theorem 7: }$\mathcal{G}(\mathbf{h}_{n},\Lambda )$ \emph{is a Riesz
sequence for }$L^{2}(%
\mathbb{R}
,%
\mathbb{C}
^{n})$\emph{\ if and only if }%
\begin{equation*}
D(\Gamma )<n\emph{.}
\end{equation*}
One should also remark that the necessity of the condition $D(\Lambda )\geq
n $ for vector valued frames, follows from a result of Balan \cite%
{BalanDensity}. Moreover, this condition holds for general sets, since a
Ramanathan-Steger \cite{RS} \ type argument is used.
Now let us look closer at the problem of deciding when the translations and
modulations of a \emph{single Hermite }function constitute a frame.
It can be easily seen as a corollary of the above Theorem that $D(\Lambda
)>n $ is sufficient for the system $G(h_{n-1},\Lambda )$ to be a frame. It
has been observed by Gr\"{o}chenig and Lyubarskii that there are some
examples which support the intriguing conjecture that such a result might be
sharp.
A recent result of Balan, Casazza and Landau shows that this conjecture
cannot hold for general sets. Let $S_{0}$ stand for the \emph{Feichtinger
algebra }\cite{Fei}.
\textbf{Theorem \cite{BCL} }\emph{Assume }$G(g,\Lambda )$\emph{\ is a Gabor
frame for }$L^{2}\left(
\mathbb{R}
^{d}\right) $\emph{\ with }$g\in S_{0}$\emph{. Then, for every }$\epsilon >0$%
\emph{, there exists a subset }$J_{\epsilon }\subset \Lambda $\emph{\ so
that }$G(g,J_{\epsilon })$\emph{\ is a Gabor frame for }$L^{2}\left(
\mathbb{R}
^{d}\right) $\emph{\ and its upper Beurling density satisfies }$%
D^{+}(J_{\epsilon })\leq 1+\epsilon $\emph{.}
This Theorem implies that Gr\"{o}chenig-Lyubarskii conjecture can only be
true for lattices. Indeed, for every $n$, the $S_{0}$ norm of the Hermite
functions (see \cite{JanssenHermite}) is%
\begin{equation*}
\left\Vert h_{n}\right\Vert _{S_{0}}=2^{\frac{n}{2}+1}\frac{\Gamma (\frac{1}{%
2}n+1)}{\sqrt{n!}},
\end{equation*}%
and therefore,
\begin{equation*}
h_{n}\in S_{0}.
\end{equation*}%
Thus, the Balan-Casazza-Landau Theorem tells us that we can always find, for
every $n$, a set $J$ with Beurling density arbitrarily close to one and such
that $G(h_{n},\Lambda )$ is a frame. However, it is still possible that the
conjecture is true for lattices, since a subset of a lattice may not be a
lattice.
\begin{conjecture}
Let $\Lambda $ be a lattice in $%
\mathbb{R}
^{2}$. If $G(h_{n},\Lambda )$ is s Gabor frame for $L^{2}\left(
\mathbb{R}
^{d}\right) $, then%
\begin{equation*}
D(\Lambda )>n.
\end{equation*}
\end{conjecture}
\subsection{New concepts}
We will introduce some new concepts troughout this paper, in order to extend
Bargmann%
\'{}%
s theory \cite{Bar} and its connection to Gabor analysis in the polyanalytic
setting.
\begin{itemize}
\item We first introduce what we call the \emph{true-polyanalytic Bargmann
transform}:
\begin{equation*}
(\mathcal{B}^{n}f)(z)=(\pi ^{\left\vert n\right\vert }n!)^{-\frac{1}{2}%
}e^{\pi \left\vert z\right\vert ^{2}}\frac{d^{n}}{dz^{n}}\left[ e^{-\pi
\left\vert z\right\vert ^{2}}F(z)\right] \text{.}
\end{equation*}%
Here $F$ stands for the Bargmann transform of $f$. As we will see, this is a
unitary mapping from $L^{2}(%
\mathbb{R}
^{d})$ to $\mathcal{F}^{n}(%
\mathbb{C}
^{d})$. This mapping relates to Gabor transforms with Hermite windows $\Phi
_{n}$ in the following way:%
\begin{equation*}
V_{\Phi _{n}}f(x,\omega )=e^{i\pi x\omega -\pi \frac{\left\vert z\right\vert
^{2}}{2}}(\mathcal{B}^{n}f)(z)\text{,}
\end{equation*}%
and we will provide its basic theory starting from\ this relation, following
as much as possible the presentation of the Bargmann transform given in Gr%
\"{o}chenig%
\'{}%
s book \cite[section 3.3]{Charly}.
\item For vector-valued functions $\mathbf{f}=(f_{0},...,f_{n-1})$, we
define the \emph{polyanalytic Bargmann} \emph{transform},
\begin{equation*}
(\mathbf{B}^{n}\mathbf{f})=\sum_{0\leq k\leq n-1}(\mathcal{B}^{k}f_{k}),
\end{equation*}%
which will be unitary between $L^{2}(%
\mathbb{R}
^{d},%
\mathbb{C}
^{n})$ and $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$.
\item In the last section we will see that the polyanalytic Bargmann
transform is a special case of a vector-valued version of the Gabor
transform. Although this transform plays no direct role in the proofs of the
main Theorems, it must be in the picture for completeness, since it is the
natural time-frequency transformation of which the polyanalytic Bargmann
transform is a special case:
\begin{equation*}
\mathbf{V}_{\mathbf{g}}\mathbf{f}(x,\omega
)=\sum_{k=0}^{n-1}V_{g_{k}}f_{k}(x,\omega ).
\end{equation*}%
In the case where $\{g_{k}\}_{k=0,...n-1}$ constitutes an orthonormal
sequence, it provides an isometry between $L^{2}(%
\mathbb{R}
^{d},%
\mathbb{C}
^{n})$ and between $L^{2}(%
\mathbb{R}
^{2d})$. This transform is the continuous counterpart of the \emph{%
superframes} discussed in \cite{DL}, \cite{Balan}, \cite[Theorem 2.7]%
{CharlyYurasuper}.
\end{itemize}
\subsection{\textbf{Technical summary} \textbf{of proofs }}
With the tools described above at hand, our main argument will depend on two
profound results. More specifically, we will combine variations on the
Janssen-Ron-Shen duality principle \cite{RonShen} with the characterization
of multiple sampling and interpolation sequences in the Fock space \cite%
{Brekkeseip}. The duality principles reflect all the rich inner structure of
Gabor frames. The second result uses a deep elaboration on Beurling%
\'{}%
s balayage technique \cite{Beurling} developed by Seip in \cite{Seip2}.
We will proceed as follows.
First, using an orthogonal basis for the polyanalytic Fock spaces, we prove
the unitarity of $\mathcal{B}^{n}$ and $\mathbf{B}^{n}$. Then we study
sampling in $\mathbf{F}^{n}(%
\mathbb{C}
)$. Using the unitary mapping $\mathbf{B}^{n}$ we show that the problem is
equivalent to the study of vector valued frames with Hermite windows, also
known as superframes \cite{Balan},\cite{CharlyYurasuper}. This problem has
been recently studied in \cite{CharlyYurasuper}, but we provide an
alternative proof, which is more natural in the context of sampling and
interpolation: applying a vector valued version of the Janssen-Ron-Shen
duality we translate the statement into a problem concerning unions of Riesz
sequences. After noticing that the latter is equivalent to a multiple
interpolation problem in Fock spaces of analytic functions, we apply the
interpolation result in \cite{Brekkeseip}. We then study interpolation in $%
\mathbf{F}^{n}(%
\mathbb{C}
)$. In order to do this, we "dualize" the arguments that we have used in the
sampling part, once again using the Janssen-Ron-Shen duality, this time
between vector-valued Riesz sequences and multi-frames with Hermite
functions. This translates our interpolation problem into one of multiple
sampling. Noticing that this problem is equivalent to multiple sampling in
Fock spaces, we apply the sampling result from \cite{Brekkeseip}.
\subsection{\textbf{Organization of the paper}}
The next section contains the classical tools that we are going to use. We
list the basic properties of the Gabor transform, the Bargmann transform and
the Hermite functions.
In the third section, we introduce the true polyanalytic Bargmann and the
polyanalytic Bargmann transforms. By making a connection with the Gabor
transform, we study their basic properties, find an orthogonal basis for the
polyanalytic Fock spaces and prove the unitarity properties.
\ Our main results are in the fourth and fifth sections, where we derive the
duality principles and study sampling and interpolation for $\mathbf{F}^{n}(%
\mathbb{C}
)$.
In our last section we introduce the super Gabor transform and make some
remarks of a more informal character concerning applications and open
problems.
\section{Background}
\subsection{The Gabor transform}
Fix a function $g\neq 0$. Then the Gabor (short-time) Fourier transform of a
function $f$ with respect to the \textquotedblright
window\textquotedblright\ $g$ is defined, for every $x,\omega \in
\mathbb{R}
^{d}$, as
\begin{equation}
V_{g}f(x,\omega )=\int_{%
\mathbb{R}
^{d}}f(t)\overline{g(t-x)}e^{-2\pi it\omega }dt. \label{Gabor}
\end{equation}%
The following relations are usually called \emph{the orthogonal relations
for the short-time Fourier transform}. Let $f_{1},f_{2},g_{1},g_{2}\in L^{2}(%
\mathbb{R}
^{d})$. Then $V_{g_{1}}f_{1},V_{g_{2}}f_{2}\in L^{2}(%
\mathbb{R}
^{2d})$ and
\begin{equation}
\left\langle V_{g_{1}}f_{1},V_{g_{2}}f_{2}\right\rangle _{L^{2}(%
\mathbb{R}
^{2d})}=\left\langle f_{1},f_{2}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}\overline{\left\langle g_{1},g_{2}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}}\text{.} \label{ortogonalityrelations}
\end{equation}
The Gabor transform provides an isometry
\begin{equation*}
V_{g}:L^{2}(%
\mathbb{R}
^{d})\rightarrow L^{2}(%
\mathbb{R}
^{2d})\text{,}
\end{equation*}%
that is, if $f,g\in L^{2}(%
\mathbb{R}
^{d})$, then
\begin{equation}
\left\Vert V_{g}f\right\Vert _{L^{2}(%
\mathbb{R}
^{2d})}=\left\Vert f\right\Vert _{L^{2}(%
\mathbb{R}
^{d})}\left\Vert g\right\Vert _{L^{2}(%
\mathbb{R}
^{d})}\text{.} \label{Gabor isometry}
\end{equation}%
For every $x,\omega \in
\mathbb{R}
^{d}$ define the operators translation by $x$ and modulation by $\omega $ as
\begin{eqnarray*}
T_{x}f(t) &=&f(t-x), \\
M_{\omega }f(t) &=&e^{2\pi i\omega t}f(t).
\end{eqnarray*}%
Using these operators we can write (\ref{Gabor}) as%
\begin{equation*}
V_{g}f(x,\omega )=\left\langle f,M_{\omega }T_{x}g\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}\text{.}
\end{equation*}
\subsection{The Bargmann transform}
Here we will use multi-index notation: $z=(z_{1},...z_{d})$, $%
n=(n_{1},...,n_{d})$ and $\left\vert n\right\vert =n_{1}+...+n_{d}$. The
Bargmann transform, defined by
\begin{equation*}
(\mathcal{B}f)(z)=\int_{%
\mathbb{R}
^{d}}f(t)e^{2\pi tz-\pi z^{2}-\frac{\pi }{2}t^{2}}dt\text{,}
\end{equation*}%
is an isomorphism
\begin{equation*}
\mathcal{B}:L^{2}(%
\mathbb{R}
^{d})\rightarrow \mathcal{F}(%
\mathbb{C}
^{d}),
\end{equation*}%
where $\mathcal{F}(%
\mathbb{C}
^{d})$ stands for the Bargmann-Fock space of analytic functions in $%
\mathbb{C}
^{d}$ with the norm
\begin{equation}
\left\Vert F\right\Vert _{\mathcal{F}(%
\mathbb{C}
^{d})}^{2}=\int_{%
\mathbb{C}
^{d}}\left\vert F(z)\right\vert ^{2}e^{-\pi \left\vert z\right\vert ^{2}}dz.
\label{norm}
\end{equation}%
The collection of the monomials of the form
\begin{equation}
e_{n}(z)=\left( \frac{\pi ^{\left\vert n\right\vert }}{n!}\right) ^{\frac{1}{%
2}}z^{n}=\prod_{j=1}^{d}\frac{\pi ^{\frac{n_{j}}{2}}}{\sqrt{n_{j}!}}z^{n_{j}}%
\text{,} \label{ort}
\end{equation}%
where $n=(n_{1,...}n_{d})$, with $n_{i}\geq 0$, constitutes an orthonormal
basis of $\mathcal{F}(%
\mathbb{C}
^{d})$. The reproducing kernel of $\mathcal{F}(%
\mathbb{C}
^{d})$ is the function $e^{\pi \overline{z}w}$. This means that, for every $%
F\in \mathcal{F}(%
\mathbb{C}
^{d})$,%
\begin{equation*}
\left\langle F(w),e^{\pi \overline{z}w}\right\rangle _{\mathcal{F}(%
\mathbb{C}
^{d})}=F(z)\text{.}
\end{equation*}%
Differentiating $n-k$ times the corresponding reproducing equation, we obtain%
\begin{equation}
\left\langle F(w),w^{n-k}e^{\pi \overline{z}w}\right\rangle _{\mathcal{F}(%
\mathbb{C}
^{d})}=\pi ^{k-n}F^{(n-k)}(z), \label{repder}
\end{equation}%
where for $d>1$ we are using the multi-index derivative%
\begin{equation*}
\frac{d}{dz}f=\frac{df}{dz_{1}...dz_{d}}.
\end{equation*}
A simple calculation shows that the Bargmann transform is related to the
Gabor transform with the Gaussian window $\varphi (t)=2^{\frac{d}{4}}e^{-\pi
t^{2}}$ by the formula
\begin{equation}
V_{\varphi }f(x,-\omega )=e^{i\pi x\omega -\pi \frac{\left\vert z\right\vert
^{2}}{2}}(\mathcal{B}f)(z)\text{,} \label{formulaBar}
\end{equation}%
where $z=x+i\omega $.
We will need one more operator. Define a "translation" $\beta _{\zeta }$ on $%
\mathcal{F}(%
\mathbb{C}
^{d})$ by
\begin{equation}
\beta _{z}F(\zeta )=e^{i\pi x\omega -\pi \frac{\left\vert z\right\vert ^{2}}{%
2}}e^{\pi \overline{z}\zeta }F(\zeta -z)\text{. } \label{shift}
\end{equation}%
The operator $\beta _{z}$ satisfies the intertwining property
\begin{equation}
\beta _{z}\mathcal{B}=\mathcal{B}M_{\omega }T_{x}\text{, \ \ \ \ }%
z=x+i\omega . \label{intertwining}
\end{equation}
\subsection{The Hermite functions}
The \emph{Hermite functions }can be defined via the so called Rodrigues
Formula
\begin{equation*}
h_{n}(t)=c_{n}e^{\pi t^{2}}\left( \frac{d}{dt}\right) ^{n}\left( e^{-2\pi
t^{2}}\right) .
\end{equation*}%
where $c_{n}$ is chosen in such a way that they can provide an orthonormal
basis of $L^{2}(%
\mathbb{R}
)$. Now let $n=(n_{1},...,n_{d})$ and $x\in
\mathbb{R}
^{d}$. The $d$\emph{-dimensional Hermite functions} are%
\begin{equation*}
\Phi _{n}(x)=\prod_{j=1}^{d}h_{n_{j}}(x_{j})\text{.}
\end{equation*}%
They form a complete orthonormal system of $L^{2}(%
\mathbb{R}
^{d})$.
A very important property of the Hermite functions (see for instance \cite%
{JanssenHermite}) is that they are mapped onto a basis of the Bargmann-Fock
space via the Bargmann transform.
\begin{equation}
(\mathcal{B}\Phi _{n})(z)=e_{n}(z). \label{BargHermite}
\end{equation}
\section{Polyanalytic Fock spaces and polyanalytic Bargmann transforms}
\subsection{Definitions}
In this section we use multi-index notation in such a way that there will be
no difference between the one and the $d$-dimensional case. Only at the end
of the last two sections is it necessary to specialize $d=1$.
It is well known \cite{Balk} that every polyanalytic function of order $n$
can be uniquely expressed in the form
\begin{equation}
F(z)=\sum_{0\leq k\leq n-1}\overline{z}^{k}\varphi _{k}(z),
\label{polyexpression}
\end{equation}%
where $\{\varphi _{p}(z)\}_{p=0}^{n-1}$ are analytic functions, each of them
with a power series expansion,%
\begin{equation*}
\varphi _{p}(z)=\sum_{j\geq 0}c_{j,p}z^{j}\text{,}
\end{equation*}%
As a result, there is also a power series expansion for the polyanalytic
function $F$:
\begin{equation}
F(z)=\sum_{0\leq p\leq n-1}\overline{z}^{p}\sum_{j\geq 0}c_{j,p}z^{j}\text{.}
\label{seriespoly}
\end{equation}%
We will often use the inner product in the polyanalytic Fock space, given by%
\begin{equation*}
\left\langle F,G\right\rangle _{\mathbf{F}^{n}(%
\mathbb{C}
^{d})}=\int_{%
\mathbb{C}
^{d}}F(z)\overline{G(z)}e^{-\pi \left\vert z\right\vert ^{2}}dz.
\end{equation*}%
Observe also that this implies%
\begin{equation*}
\left\langle F,G\right\rangle _{\mathbf{F}^{n}(%
\mathbb{C}
^{d})}=\left\langle e^{-\pi \frac{\left\vert z\right\vert ^{2}}{2}}F,e^{-\pi
\frac{\left\vert z\right\vert ^{2}}{2}}G\right\rangle _{L^{2}(%
\mathbb{R}
^{2d})}\text{.}
\end{equation*}
\subsection{The true polyanalytic Bargmann transform}
\begin{definition}
The \emph{true polyanalytic Bargmann transform of order n}, of a function on
$%
\mathbb{R}
^{d}$, is defined by the formula
\begin{equation}
(\mathcal{B}^{n}f)(z)=(\pi ^{\left\vert n\right\vert }n!)^{-\frac{1}{2}%
}e^{\pi \left\vert z\right\vert ^{2}}\frac{d^{n}}{dz^{n}}\left[ e^{-\pi
\left\vert z\right\vert ^{2}}F(z)\right] , \label{polyBargman}
\end{equation}%
where $F(z)=(\mathcal{B}f)(z)$.
\end{definition}
Clearly $\mathcal{B}^{0}f=\mathcal{B}f$ and $\mathcal{B}^{n}$ is a
generalization of the Bargmann transform. We now provide the fundamental
properties of $\mathcal{B}^{n}$. We try to stay as close as possible to the
presentation of section 3.4 in \cite{Charly}. The next proposition is the
departing point of our study.
\begin{proposition}
If $f$ is a function on $%
\mathbb{R}
^{d}$ with polynomial growth, then its true polyanalytic Bargmann transform $%
\mathcal{B}^{n}f$ is a polyanalytic function of order $n+1$ on $%
\mathbb{C}
^{d}.$ If we write $z=x+i\omega $, then this transform is related to the
Gabor transform with Hermite windows in the following way:
\begin{equation}
V_{\Phi _{n}}f(x,\omega )=e^{i\pi x\omega -\pi \frac{\left\vert z\right\vert
^{2}}{2}}(\mathcal{B}^{n}f)(z)\text{.} \label{rel}
\end{equation}%
Moreover, if $f\in L^{2}(%
\mathbb{R}
)$, then
\begin{equation}
\left\Vert \mathcal{B}^{n}f\right\Vert _{L^{2}(%
\mathbb{C}
^{d},e^{-\pi \left\vert z\right\vert ^{2}})}=\left\Vert f\right\Vert _{L^{2}(%
\mathbb{R}
^{d})}. \label{polybfisometry}
\end{equation}
\end{proposition}
\begin{proof}
Let $F=\mathcal{B}f$. The following calculation is from Proposition 3.2 in
\cite{CharlyYura}, where (\ref{repder}) is used:
\begin{eqnarray*}
V_{\Phi _{n}}f(x,\omega ) &=&\left\langle f,M_{x}T_{\omega }\Phi
_{n}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})} \\
&=&\left\langle F,\beta _{z}\mathcal{B}\Phi _{n}\right\rangle _{\mathcal{F}(%
\mathbb{C}
^{d})} \\
&=&(\pi ^{\left\vert n\right\vert }n!)^{-\frac{1}{2}}e^{i\pi x\omega -\frac{%
\pi }{2}\left\vert z\right\vert ^{2}}\left\langle F(w),e^{\pi \overline{z}%
w}(w-z)^{n}\right\rangle _{\mathcal{F}(%
\mathbb{C}
^{d})} \\
&=&(\pi ^{\left\vert n\right\vert }n!)^{-\frac{1}{2}}e^{i\pi x\omega -\frac{%
\pi }{2}\left\vert z\right\vert ^{2}}\sum_{0\leq k\leq n}\binom{n}{k}(-\pi
\overline{z})^{k}F^{(n-k)}(z)\text{.}
\end{eqnarray*}%
Now, since the Bargmann transform of a function in $%
\mathbb{R}
^{d}$ is an entire function \cite[Proposition 3.4.1]{Charly}, the functions $%
F^{(n-k)}(z)$ are also entire, and from (\ref{polyexpression}) we recognize
the sum as a polyanalytic function of order $n+1$. To prove (\ref{rel})
observe that the last expression can be written as
\begin{equation*}
e^{i\pi x\omega -\pi \frac{\left\vert z\right\vert ^{2}}{2}}(\pi
^{\left\vert n\right\vert }n!)^{-\frac{1}{2}}e^{\pi \left\vert z\right\vert
^{2}}\frac{d^{n}}{dz^{n}}\left[ e^{-\pi \left\vert z\right\vert ^{2}}F(z)%
\right] =e^{i\pi x\omega -\pi \frac{\left\vert z\right\vert ^{2}}{2}}(%
\mathcal{B}^{n}f)(z)\text{.}
\end{equation*}%
The isometric property (\ref{polybfisometry}) is an immediate consequence of
(\ref{rel}) and (\ref{Gabor isometry}).
\end{proof}
\subsection{Orthogonal decomposition}
\begin{definition}
For $k,m\in
\mathbb{N}
_{0}^{d}$, consider the functions $e_{k,m}$ defined as
\begin{equation}
e_{k,m}(z)=(\pi ^{\left\vert k\right\vert }k!)^{-\frac{1}{2}}e^{\pi
\left\vert z\right\vert ^{2}}\left( \frac{d}{dz}\right) ^{k}\left[ e^{-\pi
\left\vert z\right\vert ^{2}}e_{m}(z)\right] \text{.} \label{basis}
\end{equation}
\end{definition}
From (\ref{BargHermite}) one can easily see that
\begin{equation}
e_{k,m}(z)=(\mathcal{B}^{k}\Phi _{m})(z)\text{.} \label{Barbasis}
\end{equation}
\begin{proposition}
The set $\left\{ e_{k,m}\right\} _{0\leq k\leq n-1;\text{...}m\geq 0}$is an
orthogonal basis of $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$.
\end{proposition}
\begin{proof}
The orthogonality follows from (\ref{Barbasis}), (\ref{rel}) and (\ref%
{ortogonalityrelations}), since
\begin{eqnarray*}
\left\langle e_{k,m},e_{l,j}\right\rangle _{L^{2}(%
\mathbb{R}
^{2d})} &=&\left\langle \mathcal{B}^{k}\Phi _{m},\mathcal{B}^{l}\Phi
_{j}\right\rangle _{\mathcal{F}(%
\mathbb{C}
^{d})} \\
&=&\left\langle e^{\pi \frac{\left\vert z\right\vert ^{2}}{2}-i\pi x\omega
}V_{\Phi _{k}}\Phi _{m},e^{\pi \frac{\left\vert z\right\vert ^{2}}{2}-i\pi
x\omega }V_{\Phi _{l}}\Phi _{j}\right\rangle _{\mathcal{F}(%
\mathbb{C}
^{d})} \\
&=&\left\langle V_{\Phi _{k}}\Phi _{m},V_{\Phi _{l}}\Phi _{j}\right\rangle
_{L^{2}(%
\mathbb{R}
^{2d})} \\
&=&\left\langle \Phi _{m},\Phi _{j}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}\overline{\left\langle \Phi _{k},\Phi _{l}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}} \\
&=&\delta _{m,j}\delta _{k,l}.
\end{eqnarray*}%
To prove completeness of $\{e_{k,m}\}$ in $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$, suppose that $F\in \mathbf{F}^{n}(%
\mathbb{C}
^{d})$ is such that
\begin{equation*}
\left\langle F,e_{k,m}\right\rangle _{\mathcal{F}(%
\mathbb{C}
^{d})}=0\text{, \ \ \ \ \ }0\leq k\leq n-1;\text{ \ \ \ \ }m\geq 0\text{.}
\end{equation*}%
For $k=0$, we can use the representation of $F$ in power series (\ref%
{seriespoly}). Interchanging the sums with the integrals and using the
orthogonality of the functions (\ref{ort}), the result is
\begin{equation}
\left\langle F,e_{0,m}\right\rangle _{\mathcal{F}(%
\mathbb{C}
^{d})}=\sum_{0\leq p\leq n-1}c_{p+m,p}\frac{(p+m)!}{\sqrt{m!}\pi
^{\left\vert 2p+m\right\vert }}=0\text{, \ \ \ }m\geq 0. \label{kzero}
\end{equation}%
For $k\geq 1$, a calculation using integration by parts gives:
\begin{eqnarray*}
\left\langle F,e_{k,m}\right\rangle _{\mathcal{F}(%
\mathbb{C}
^{d})} &=&\int_{%
\mathbb{C}
^{d}}F(z)\overline{e_{k,m}(z)}e^{-\pi \left\vert z\right\vert ^{2}}dz \\
&=&\int_{%
\mathbb{C}
^{d}}e^{-\pi \left\vert z\right\vert ^{2}}\overline{e_{m}(z)}%
p...(p-k+1)\sum_{k\leq p\leq n-1}\overline{z}^{p-k}\sum_{j\geq
0}c_{j,p}z^{j}dz \\
&=&\sum_{k\leq p\leq n-1}\sum_{j\geq 0}c_{j,p}\frac{p...(p-k+1)\pi
^{\left\vert m\right\vert }}{\sqrt{m!}}\int_{%
\mathbb{C}
^{d}}z^{j}\overline{z}^{m+p-k}e^{-\pi \left\vert z\right\vert ^{2}}dz.
\end{eqnarray*}%
As a result,
\begin{equation*}
\sum_{k\leq p\leq n-1}\frac{p...(p-k+1)(p+m-k)!}{\pi ^{\left\vert
m+2p-2k\right\vert }\sqrt{m!}}c_{m+p-k,p}=0\text{, \ \ }m\geq 0\text{, }%
0\leq k\leq n-1\text{,}
\end{equation*}%
resulting in a triangular system for each $m$. Solving this system we obtain
$c_{j,p}=0$ for $k\leq p\leq n-1$ and $j\geq 0$. Therefore, $F=0$.
\end{proof}
\begin{remark}
Clearly the orthogonality in Proposition 2 can be obtained directly by
integration by parts and moving to polar coordinates. For $k=0$ this has the
advantage of showing that the functions are also orthogonal in the polydisk,
providing the useful "double orthogonality" property as in the proof of \cite%
[Theorem 3.4.2.]{Charly}. However, for $k\geq 0$, the boundary behavior
required in the integration by parts eliminates this advantage, making the
functions $e_{k,m}$ less likely to possess such a property.
\end{remark}
\begin{remark}
It is clear that these functions are reminiscent of the so-called special
Hermite functions, which are the Wigner transforms of two Hermite functions
\cite{Thang}. They also appear in the study of Landau levels in \cite%
{LuefGosson}.
\end{remark}
\begin{definition}
The \emph{true} polyanalytic Fock space of order\emph{\ }$n$ is defined as
\begin{equation}
\mathcal{F}^{n}(%
\mathbb{C}
^{d})=Span\left[ \left\{ e_{n,m}(z)\right\} _{m\geq 0.}\right] \text{.}
\label{true}
\end{equation}
\end{definition}
\begin{remark}
Observe that%
\begin{equation*}
\left( \frac{d}{dz}\right) ^{n}\left[ e^{-\pi \left\vert z\right\vert
^{2}}z^{m}\right] =\frac{d^{m+n}}{dz^{n}d\overline{z}^{m}}\left[ e^{-\pi
\left\vert z\right\vert ^{2}}\right] .
\end{equation*}%
Therefore, our functions $e_{n,m}$ are essentially the complex Hermitian
functions introduced in \cite[pag. 126]{Shigekawa} and, as a result,
according to Theorem 7.1 in \cite{Shigekawa}, the true polyanalytic Fock
spaces are the eigenspaces of the Schr\"{o}dinger operator with magnetic
field in $%
\mathbb{R}
^{2}$, associated with the eigenvalue $n+\frac{1}{2}$. Also, observe that
the basis used in \cite{Ramazanov} approaches this one by a formal limit
procedure.
\end{remark}
The orthogonal basis property has the following consequence:
\begin{corollary}
The polyanalytic Fock space, $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$, admits the following decomposition in terms of true polyanalytic
Fock spaces $\mathcal{F}^{k}(%
\mathbb{C}
^{d})$.
\begin{equation}
\mathbf{F}^{n}(%
\mathbb{C}
^{d})=\mathcal{F}^{0}(%
\mathbb{C}
^{d})\oplus ...\oplus \mathcal{F}^{n-1}(%
\mathbb{C}
^{d})\text{.} \label{decomposition}
\end{equation}
\end{corollary}
This results in a definition equivalent to the one in \cite{VasiBergman},
where the spaces were defined using the decomposition. Observe that $\mathbf{%
F}^{1}(%
\mathbb{C}
^{d})=\mathcal{F}^{0}(%
\mathbb{C}
^{d})=\mathcal{F}(%
\mathbb{C}
^{d})$ and that functions in $\mathcal{F}^{n}(%
\mathbb{C}
^{d})$ are polyanalytic of order $n+1$.
\subsection{Unitarity of $\mathcal{B}^{n}$}
The true polyanalytic Bargmann transform keeps the unitarity property
\begin{theorem}
The true polyanalytic Bargmann transform is an isometric isomorphism
\begin{equation*}
\mathcal{B}^{n}:L^{2}(%
\mathbb{R}
^{d})\rightarrow \mathcal{F}^{n}(%
\mathbb{C}
^{d})\text{.}
\end{equation*}
\end{theorem}
\begin{proof}
Since we know from (\ref{polybfisometry}) that $\mathcal{B}^{n}$ is
isometric, we only need to show that $\mathcal{B}^{n}[L^{2}(%
\mathbb{R}
^{d})]$ is dense in $\mathcal{F}^{n}(%
\mathbb{C}
^{d})$. This is now easy, since the Hermite functions constitute a basis of $%
L^{2}(%
\mathbb{R}
^{d})$ and, by (\ref{Barbasis}), they are mapped onto the basis $\left\{
e_{n,m}(z)\right\} $ of $\mathcal{F}^{n}(%
\mathbb{C}
)$. Since $\mathcal{B}^{n}[L^{2}(%
\mathbb{R}
^{d})]$ contains a basis of $\ \mathcal{F}^{n}(%
\mathbb{C}
^{d})$, then it must be dense.
\end{proof}
\subsection{The polyanalytic Bargmann transform}
Now consider the Hilbert space $\mathcal{H}=L^{2}(%
\mathbb{R}
^{d},%
\mathbb{C}
^{n})$ consisting of vector-valued functions $\mathbf{f}=(f_{0},...,f_{n-1})$
with the inner product
\begin{equation}
\left\langle \mathbf{f,g}\right\rangle _{\mathcal{H}}=\sum_{0\leq k\leq
n-1}\left\langle f_{k},g_{k}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}\text{.} \label{innersuper}
\end{equation}
The \emph{polyanalytic Bargmann transform }of a function $\mathbf{f}%
=(f_{0},...,f_{n-1})$ is defined as
\begin{equation}
(\mathbf{B}^{n}\mathbf{f})(z)=\sum_{0\leq k\leq n-1}(\mathcal{B}^{k}f_{k})(z)%
\text{.} \label{polybargmann}
\end{equation}
The next Theorem, which may have independent interest as a generalization of
Bargmann%
\'{}%
s unitary transform, will be the cornerstone in the proof of our main
results regarding sampling and interpolation.
\begin{theorem}
The polyanalytic Bargmann transform is an isometric isomorphism
\begin{equation*}
\mathbf{B}^{n}:\mathcal{H}\rightarrow \mathbf{F}^{n}(%
\mathbb{C}
^{d})\text{.}
\end{equation*}
\end{theorem}
\begin{proof}
For the isometry, first observe that, from (\ref{ortogonalityrelations}) and
(\ref{rel}),
\begin{equation*}
\left\langle \mathcal{B}^{k}f_{k},\mathcal{B}^{j}f_{j}\right\rangle _{%
\mathcal{F}^{n}(%
\mathbb{C}
^{d})}=\delta _{k,j}.
\end{equation*}%
Then, using the isometric property of $\mathcal{B}^{n}$,
\begin{eqnarray*}
\left\Vert \mathbf{B}^{n}\mathbf{f}\right\Vert _{\mathbf{F}^{n}(%
\mathbb{C}
^{d})}^{2} &=&\sum_{0\leq k\leq n-1}\left\Vert \mathcal{B}%
^{k}f_{k}\right\Vert _{\mathcal{F}^{n}(%
\mathbb{C}
^{d})}^{2} \\
&=&\sum_{0\leq k\leq n-1}\left\Vert f_{k}\right\Vert _{L^{2}(%
\mathbb{R}
^{d})}^{2}=\left\Vert f\right\Vert _{\mathcal{H}}^{2}\text{.}
\end{eqnarray*}%
Moreover, $\mathbf{B}^{n}[L^{2}(%
\mathbb{R}
^{d})]$ is dense in $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$, since, by the decomposition (\ref{decomposition}), every element $%
F\in \mathbf{F}^{n}(%
\mathbb{C}
^{d})$ can be written as
\begin{equation*}
F=\sum_{0\leq k\leq n-1}F_{k}\text{,}
\end{equation*}%
with $F_{k}\in \mathcal{F}^{k}(%
\mathbb{C}
^{d})$, $0\leq k\leq n-1$. Since $\mathcal{B}^{k}$ is unitary, there exists $%
f_{k}\in L^{2}(%
\mathbb{R}
^{d})$ such that $F_{k}=\mathcal{B}^{k}f_{k}$, for every $0\leq k\leq n-1$.
It follows that $F=\mathbf{B}^{n}\mathbf{f}$, with $\mathbf{f=}%
(f_{0},...,f_{n-1})$.
\end{proof}
\section{Sampling in $\mathbf{F}^{n}(%
\mathbb{C}
)$}
\subsection{Definitions}
We will work with lattices. A lattice is a discrete subgroup in $%
\mathbb{R}
^{2d}$ of the form $\Lambda =A%
\mathbb{Z}
^{2d}$, where $A$ is an invertible $2d\times 2d$ matrix. We will define the
\emph{density} of the lattice by%
\begin{equation}
D(\Lambda )=\frac{1}{\left\vert \det A\right\vert }\text{.}
\label{densitylattice}
\end{equation}%
If we write $z=x+i\omega $ and $\pi _{z}g=M_{\omega }T_{x}g$, the adjoint
lattice $\Lambda ^{0}$ is defined by the commuting property as%
\begin{equation*}
\Lambda ^{0}=\{\mu \in
\mathbb{R}
^{2d}:\pi _{z}\pi _{\mu }=\pi _{\mu }\pi _{z}\text{, for all }z\in \Lambda \}%
\text{.}
\end{equation*}%
If $\Lambda =\alpha
\mathbb{Z}
\times \beta
\mathbb{Z}
$, then $\Lambda ^{0}=\beta ^{-1}%
\mathbb{Z}
\times \alpha ^{-1}%
\mathbb{Z}
$. In general,
\begin{equation*}
\Lambda ^{0}=\mathcal{J}(A^{T})^{-1}%
\mathbb{Z}
^{d}\text{,}
\end{equation*}%
where $A^{T}$ is the transpose of $A$ and $\mathcal{J=}\left[
\begin{array}{cc}
0 & I \\
-I & 0%
\end{array}%
\right] $ (consisting of $d\times d$ blocks) is the matrix defining the
standard sympletic form (see \cite{CharlyYurasuper} and \cite{FK}).
Therefore,
\begin{equation}
D(\Lambda ^{0})=\frac{1}{D(\Lambda )}\text{.} \label{reldensities}
\end{equation}%
We will use the notation $\Gamma =\{z=x+i\omega \}$ to indicate the complex
sequence associated with the sequence $\Lambda =(x,\omega )$. The density of
$\Gamma $ will be the density of the associated lattice, that is $D(\Gamma
)=D(\Lambda )$.
\begin{definition}
$\Gamma $ is a sampling sequence for the space $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$\ if there exist positive constants $A$ and $B$ such that, for every $%
F\in \mathbf{F}^{n}(%
\mathbb{C}
^{d})$,
\begin{equation}
A\left\Vert F\right\Vert _{\mathbf{F}^{n}(%
\mathbb{C}
^{d})}^{2}\leq \sum_{z\in \Gamma }\left\vert F(z)\right\vert ^{2}e^{-\pi
\left\vert z\right\vert ^{2}}\leq B\left\Vert F\right\Vert _{\mathbf{F}^{n}(%
\mathbb{C}
^{d})}^{2}. \label{sampling}
\end{equation}
\end{definition}
The definition of sampling in the spaces $\mathcal{F}^{k}(%
\mathbb{C}
^{d})$ is exactly the same.
Now we take the following definition, obtained from \cite[page 114]%
{Brekkeseip}, by making a small simplification (in the notation of \cite[%
page 114]{Brekkeseip} we set $\nu (z)=n$ ) and rewriting it in our context
(observe that the weight $e^{i\pi x\omega }$ makes no diference).
\begin{definition}
A sequence $\Gamma _{n}$, consisting of $n$ copies of $\Gamma $ is a
multiple interpolating sequence in the Fock space $\mathcal{F}(%
\mathbb{C}
^{d})$ if, for every sequence $\{\alpha _{i,j}^{(k)}\}_{k=0,...n-1}$ such
that $\{\alpha _{i,j}^{(k)}\}_{k=0,...n-1}\in l^{2}$, there exists $F\in
\mathcal{F}(%
\mathbb{C}
^{d})$ such that $\left\langle F,\beta _{z}e_{k}\right\rangle =\alpha
_{i,j}^{(k)},$ for all $0\leq k\leq n-1$ and every $z\in \Gamma $.
\end{definition}
Consider again the Hilbert space $\mathcal{H}=L^{2}(%
\mathbb{R}
^{d},%
\mathbb{C}
^{n})$ consisting of vector-valued functions $\mathbf{f}=(f_{0},...,f_{n-1})$
with the inner product (\ref{innersuper}). The time-frequency shifts act
coordinate-wise in an obvious way.
\begin{definition}
The vector valued system $\mathcal{G}(\mathbf{g},\Lambda )=\{M_{\omega }T_{x}%
\mathbf{g}\}_{(x,w)\in \Lambda }$ is a $\emph{Ga}$\emph{b}$\emph{or}$ \emph{%
superframe} for $\mathcal{H}$ if there exist constants $A$ and $B$ such
that, for every $\mathbf{f}\in \mathcal{H}$,
\begin{equation}
A\left\Vert \mathbf{f}\right\Vert _{\mathcal{H}}^{2}\leq \sum_{(x,w)\in
\Lambda }\left\vert \left\langle \mathbf{f},M_{\omega }T_{x}\mathbf{g}%
\right\rangle _{\mathcal{H}}\right\vert ^{2}\leq B\left\Vert \mathbf{f}%
\right\Vert _{\mathcal{H}}^{2}. \label{superframe}
\end{equation}
\end{definition}
Superframes were introduced in a more abstract form in \cite{DL} and in the
context of "multiplexing" in \cite{Balan}. We will need a fundamental
structure Theorem from time-frequency analysis, namely the following version
of the Janssen-Ron-Shen duality principle \cite[Theorem 2.7]{CharlyYurasuper}%
.%
\begin{equation*}
\end{equation*}
\textbf{Theorem A. }Let $\mathbf{g}=(g_{0},...,g_{n-1})$.\ The vector valued
system $\mathcal{G}(\mathbf{g},\Lambda )$ is a $\emph{Ga}$\emph{b}$\emph{or}$
\emph{superframe} for $\mathcal{H}$ if and only if the union of Gabor
systems $\cup _{k=0}^{n-1}\mathcal{G}(g_{k},\Lambda ^{0})$ is a Riesz
sequence for $L^{2}(%
\mathbb{R}
^{d})$.
\subsection{Duality principle}
In this section we will obtain the following duality principle.
\begin{theorem}
$\Gamma $ is a sampling sequence for $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ if and only if the adjoint sequence $\Gamma _{n}^{0}$ is a multiple
interpolating sequence in the Fock space $\mathcal{F}(%
\mathbb{C}
^{d})$.
\end{theorem}
We first prove two Lemmas. Combining them with Theorem A gives Theorem 3.
\begin{lemma}
The union of Gabor systems $\cup _{k=0}^{n-1}\mathcal{G}(h_{k},\Lambda )$ is
a Riesz sequence for $L^{2}(%
\mathbb{R}
^{d})$ if and only if $\Gamma _{n}$ is a multiple interpolating sequence in
the Fock space $\mathcal{F}(%
\mathbb{C}
^{d})$.
\end{lemma}
\begin{proof}
The union of Gabor systems $\cup _{k=0}^{n-1}\mathcal{G}(h_{k},\Lambda )$ is
a Riesz sequence for $L^{2}(%
\mathbb{R}
^{d})$ if, for every sequence $\{\alpha _{i,j}^{(k)}\}_{k=0,...n-1}\in l^{2}$%
, there exists a $f\in L^{2}(%
\mathbb{R}
^{d})$ such that $\left\langle f,M_{\omega }T_{x}h_{k}\right\rangle =\alpha
_{i,j}^{(k)}$, for all $k=0,...n-1$ and every $(x,\omega )\in \Lambda $.
Using the unitarity of $\mathcal{B}$ and the intertwining property (\ref%
{intertwining}) gives
\begin{equation*}
\left\langle f,M_{\omega }T_{x}h_{k}\right\rangle =\left\langle \mathcal{B}%
f,\beta _{z}e_{k}\right\rangle \text{,}
\end{equation*}%
and setting $F=\mathcal{B}f$ shows that $\Gamma _{n}$ is a multiple
interpolating sequence in the Fock space $\mathcal{F}(%
\mathbb{C}
^{d})$.
\end{proof}
The next Lemma is a key step in our argument and it is at this point that
the unitarity of the polyanalytic Bargmann transform is essential.
\begin{lemma}
Let $\mathbf{h}_{n}=(h_{0},...,h_{n-1})$. Then the set $\mathcal{G}(\mathbf{h%
}_{n},\Lambda )$ is a Gabor superframe for $\mathcal{H}=L^{2}(%
\mathbb{R}
^{d},%
\mathbb{C}
^{n})$ if and only if the associated complex sequence $\Gamma $ is a
sampling sequence for $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$.
\end{lemma}
\begin{proof}
Using the definition of the inner product (\ref{innersuper}), identity (\ref%
{rel}) and the definition of the polyanalytic Bargmann transform, it is
clear that
\begin{eqnarray}
\left\langle \mathbf{f},M_{\omega }T_{x}\mathbf{h}_{n}\right\rangle _{%
\mathcal{H}} &=&\sum_{0\leq k\leq n-1}\left\langle f_{k},M_{\omega
}T_{x}h_{k}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})} \label{1} \\
&=&\sum_{0\leq k\leq n-1}e^{i\pi x\omega -\frac{\pi }{2}\left\vert
z\right\vert ^{2}}(\mathcal{B}^{k}f_{k})(z) \notag \\
&=&e^{i\pi x\omega -\frac{\pi }{2}\left\vert z\right\vert ^{2}}(\mathbf{B}%
^{n}\mathbf{f})(z)\text{.} \label{3}
\end{eqnarray}%
Therefore, setting $F=\mathcal{B}^{n}\mathbf{f}$, the unitarity of $\mathbf{B%
}^{n}$ shows that\ the inequalities (\ref{superframe}) are equivalent to (%
\ref{sampling}).
\end{proof}
\subsection{Main result}
We will need the concept of \emph{Beurling density} of a sequence.
Let $I$ be a compact set of measure $1$ in the complex plane and let $%
n^{-}(r)$ denote the smallest (and $n^{+}(r)$ the biggest) number of points
from $\Gamma $ to be found in a translate of $rI$. We define the \emph{lower}
and the \emph{upper} Beurling density of $\Gamma $ to be%
\begin{equation*}
D^{-}(\Gamma )=\lim_{r\rightarrow \infty }\sup \frac{n^{-}(r)}{r^{2}}\text{
and }D^{+}(\Gamma )=\lim_{r\rightarrow \infty }\sup \frac{n^{+}(r)}{r^{2}}%
\text{,}
\end{equation*}%
respectively. When $\Gamma $ is associated with the lattice $\Lambda $, $%
D^{-}(\Gamma )=D^{+}(\Gamma )=D(\Gamma )=D(\Lambda )$.
We will now use the following result, which is Theorem 2.2 in \cite%
{Brekkeseip}. Observe that we can remove the uniformly discrete condition
from the statement in \cite{Brekkeseip} since we are dealing only with
lattices.%
\begin{equation*}
\end{equation*}%
\textbf{Theorem B. }The sequence\textbf{\ }$\Gamma _{n}$ is a multiple
interpolating lattice sequence in the Fock space $\mathcal{F}(%
\mathbb{C}
)$ if and only if $D(\Gamma _{n})<1$.%
\begin{equation*}
\end{equation*}
From this we obtain the characterization of sampling lattices in $\mathbf{F}%
^{n}(%
\mathbb{C}
)$.
\begin{theorem}
The lattice $\Gamma $ is a sampling sequence for $\mathbf{F}^{n}(%
\mathbb{C}
)$ if and only if $D(\Gamma )>n$.
\end{theorem}
\begin{proof}
We know by the duality principle that $\Gamma $ is a sampling sequence for $%
\mathbf{F}^{n}(%
\mathbb{C}
)$ if and only if the adjoint sequence $\Gamma _{n}^{0}$ is a multiple
interpolating sequence in the Fock space $\mathcal{F}(%
\mathbb{C}
)$. By definition of Beurling density, it is obvious that%
\begin{equation*}
D(\Gamma _{n}^{0})=nD(\Gamma ^{0})\text{.}
\end{equation*}%
Therefore, Theorem B states that $\Gamma ^{0}$ is a multiple interpolating
sequence in the Fock space $\mathcal{F}(%
\mathbb{C}
)$ if and only if $D(\Gamma ^{0})<\frac{1}{n}.$ Using (\ref{reldensities}),
we conclude that $\Gamma $ is a sampling sequence for $\mathbf{F}^{n}(%
\mathbb{C}
)$ if and only if $D(\Gamma )>n$.
\end{proof}
Using Lemma 1, we recover Theorem 1.1 of \cite{CharlyYurasuper}.
\begin{corollary}
Let $\mathbf{h}_{n}=(h_{0},...,h_{n-1})$. Then the set $\mathcal{G}(\mathbf{h%
}_{n},\Lambda )$ is a Gabor super frame for $\mathcal{H}=L^{2}(%
\mathbb{R}
,%
\mathbb{C}
^{n})$ if and only if $D(\Gamma )>n$.
\end{corollary}
\section{Interpolation in $\mathbf{F}^{n}(%
\mathbb{C}
)$}
\subsection{Definitions}
\begin{definition}
The sequence $\Gamma $ is an interpolating sequence for $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ if, for every sequence $\{\alpha _{i,j}\}\in l^{2}$, there exists $%
F\in \mathbf{F}^{n}(%
\mathbb{C}
^{d})$ such that $e^{i\pi x\omega -\frac{\pi }{2}\left\vert z\right\vert
^{2}}F(z)=\alpha _{i,j},$ for every $z\in \Gamma $.
\end{definition}
\begin{definition}
The sequence $\Gamma _{n}$, consisting of $n$ copies of $\Gamma $ is is said
to be a multiple sampling sequence for $\mathcal{F}(%
\mathbb{C}
^{d})$ if there exist numbers $A$ and $B$ such that%
\begin{equation}
A\left\Vert F\right\Vert _{\mathcal{F}(%
\mathbb{C}
^{d})}^{2}\leq \sum_{z\in \Gamma }\sum_{0\leq k\leq n-1}\left\vert
\left\langle F,\beta _{z}e_{k}\right\rangle \right\vert ^{2}\leq B\left\Vert
F\right\Vert _{\mathcal{F}(%
\mathbb{C}
^{d})}^{2}. \label{multisampling}
\end{equation}
\end{definition}
\begin{definition}
The set $\cup _{k=0}^{n-1}\mathcal{G}(g_{k},\Lambda )$ is said to generate a
\emph{Gabor multi-frame} in $L^{2}(%
\mathbb{R}
^{d})$\ if there exist constants $A$ and $B$ such that, for every $f\in
L^{2}(%
\mathbb{R}
^{d})$,%
\begin{equation}
A\left\Vert f\right\Vert _{L^{2}(%
\mathbb{R}
^{d})}^{2}\leq \sum_{(x,\omega )\in \Lambda }\sum_{0\leq k\leq
n-1}\left\vert \left\langle f,M_{\omega }T_{x}g_{k}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}\right\vert ^{2}\leq B\left\Vert f\right\Vert _{L^{2}(%
\mathbb{R}
^{d})}^{2}. \label{multiframe}
\end{equation}
\end{definition}
The dual of the duality principle contained in Theorem A is now required. As
stated at the end of \cite{CharlyIMRN}, it reads as follows:%
\begin{equation*}
\end{equation*}%
\textbf{Theorem C. }The set $\mathcal{G}(\mathbf{g},\Lambda )$ is a Riesz
sequence for $L^{2}(%
\mathbb{R}
^{d})$ if and only if $\cup _{k=0}^{n-1}\mathcal{G}(g_{k},\Lambda ^{0})$ is
a Gabor multi-frame in $L^{2}(%
\mathbb{R}
^{d})$.
\subsection{Duality principle}
Now we prove the following duality.
\begin{theorem}
The sequence $\Gamma $ is an interpolating sequence for $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ if and only if $\Gamma _{n}^{0}$ is a multiple sampling sequence for $%
\mathcal{F}(%
\mathbb{C}
^{d})$.
\end{theorem}
As in the sampling section, we first prove two Lemmas which, combined with
Theorem C, give the result. The next Lemma requires only the unitarity of
the Bargmann transform.
\begin{lemma}
The set $\cup _{k=0}^{n-1}\mathcal{G}(h_{k},\Lambda )$ is a Gabor
multi-frame in $L^{2}(%
\mathbb{R}
^{d})$\ if and only if $\Gamma _{n}$ is a multiple sampling sequence for $%
\mathcal{F}(%
\mathbb{C}
^{d})$.
\begin{proof}
Similar to Lemma 1: using the unitarity of $\mathcal{B}$ and the
intertwining property (\ref{intertwining}) gives $\left\langle f,M_{\omega
}T_{x}h_{k}\right\rangle =\left\langle \mathcal{B}f,\beta
_{z}e_{k}\right\rangle $; setting $F=\mathcal{B}f$ it follows from the
unitarity of the Bargmann transform that (\ref{multisampling}) and (\ref%
{multiframe}) are equivalent.
\end{proof}
\end{lemma}
Again, we make the key connection in the next step, where the unitarity of
the polyanalytic Bargmann transform is required.
\begin{lemma}
The sequence $\Gamma $ is an interpolating sequence for $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ if and only if $\mathcal{G}(\mathbf{h}_{n},\Lambda )$ is a Riesz
sequence for $\mathcal{H}$.
\end{lemma}
\begin{proof}
The sequence $\Gamma $ is an interpolating sequence for $\mathbf{F}^{n}(%
\mathbb{C}
^{d})$ if, for every sequence $\{\alpha _{i,j}\}\in l^{2}$, there exists $%
F\in \mathbf{F}^{n}(%
\mathbb{C}
^{d})$ such that $e^{i\pi x\omega -\frac{\pi }{2}\left\vert z\right\vert
^{2}}F(z)=\alpha _{i,j},$ for every $z\in \Gamma $. Using the unitarity of $%
\mathbf{B}^{n}$, we find, for every $F\in \mathbf{F}^{n}(%
\mathbb{C}
^{d})$, a vector valued function $\mathbf{f}\in \mathcal{H}\ $such that $%
\mathbf{B}^{n}\mathbf{f}=F$ or, by (\ref{1})-(\ref{3}),$\ \left\langle
\mathbf{f},M_{\omega }T_{x}\mathbf{h}_{n}\right\rangle _{\mathcal{H}%
}=e^{i\pi x\omega -\frac{\pi }{2}\left\vert z\right\vert ^{2}}F$. Therefore,
the first assertion is equivalent to say that, for every sequence $\{\alpha
_{i,j}\}\in l^{2}$, there exists a $\mathbf{f}\in \mathcal{H}$ such that $%
\left\langle \mathbf{f},M_{\omega }T_{x}\mathbf{h}_{n}\right\rangle _{%
\mathcal{H}}=\alpha _{i,j}$, for every $z\in \Gamma $. This says that $%
\mathcal{G}(\mathbf{h}_{n},\Lambda )$ is a Riesz sequence for $\mathcal{H}$.
\end{proof}
\subsection{Main result}
We will need the following result, which is contained in Theorem 2.1 in \cite%
{Brekkeseip}:%
\begin{equation*}
\end{equation*}
\textbf{Theorem D. }The sequence\textbf{\ }$\Gamma _{n}$ is a multiple
interpolating sequence in the Fock space $\mathcal{F}(%
\mathbb{C}
)$ if and only if $D(\Gamma _{n})<1$.%
\begin{equation*}
\end{equation*}
As before, we can obtain our main result from this one.
\begin{theorem}
The lattice $\Gamma $ is an interpolating sequence for $\mathbf{F}^{n}(%
\mathbb{C}
)$ if and only if $D(\Gamma )<n$.
\end{theorem}
\begin{proof}
We know from the duality principle that $\Gamma $ is an interpolating
sequence for $\mathbf{F}^{n}(%
\mathbb{C}
)$ if and only if $\Gamma _{n}$ is a multiple sampling sequence for $%
\mathcal{F}(%
\mathbb{C}
)$. Once again we have $D(\Gamma _{n}^{0})=nD(\Gamma ^{0})$. Therefore,
Theorem D states that $\Gamma ^{0}$ is a multiple interpolating sequence in
the Fock space $\mathcal{F}(%
\mathbb{C}
)$ if and only if $D(\Gamma ^{0})>\frac{1}{n}$. As in Theorem 5 it follows
that $\Gamma $ is an interpolating sequence for $\mathbf{F}^{n}(%
\mathbb{C}
)$ if and only if $D(\Gamma )<n$.
\end{proof}
From this and Lemma 4 we obtain a new result characterizing all the lattices
which generate vector valued Gabor Riesz sequences with Hermite functions$.$
This reveals, at least for lattices, the existence of a critical density for
vector-valued Gabor systems with Hermite functions.
\begin{theorem}
$\mathcal{G}(\mathbf{h}_{n},\Lambda )$ is a Riesz sequence for $\mathcal{H}$
if and only if $D(\Gamma )<n$.
\end{theorem}
\begin{remark}
We should remark that the reason we did not care about the Bessel condition
in the equivalence of the Riesz sequence and interpolating property, used
several times in the previous section, is that the Hermite functions belong
to Feichtinger%
\'{}%
s algebra $S_{0}$ (see \cite{FeiZim},\cite{Fei}):%
\begin{equation*}
\left\Vert h_{n}\right\Vert _{S_{0}}=\int_{%
\mathbb{R}
}\left\vert \left\langle h_{n}\text{,}M_{\omega }T_{x}\varphi \right\rangle
\right\vert dz<\infty \text{,}
\end{equation*}%
where $\varphi $ is the $L^{2}$-normalized Gaussian. As a result they
satisfy the Bessel condition \cite[theorem 12]{Heil}.
\end{remark}
\section{Generalizations, applications and open problems}
\subsection{The super Gabor transform}
The polyanalytic Bargmann transform is an instance of a more general
transform. Although it plays no role in the proofs of our main results, we
briefly describe it here for completeness of the picture.
The \emph{super Gabor transform} of a function $\mathbf{f}$ with respect to
the \textquotedblright window\textquotedblright\ $\mathbf{g=(}%
g_{0},...g_{n-1}\mathbf{)}$ is defined, for every $x,\omega \in
\mathbb{R}
^{d}$, as
\begin{equation}
\mathbf{V}_{\mathbf{g}}\mathbf{f}(x,\omega )=\left\langle \mathbf{f,}M%
\mathbf{_{\omega }}T_{x}\mathbf{g}\right\rangle _{\mathcal{H}}=\sum_{0\leq
k\leq n-1}\left\langle f_{k},M_{\omega }T_{x}g_{k}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}. \label{supergabor}
\end{equation}
That is to say,%
\begin{equation*}
\mathbf{V}_{\mathbf{g}}\mathbf{f}(x,\omega )=\sum_{0\leq k\leq
n-1}V_{g_{k}}f_{k}(x,\omega )\text{.}
\end{equation*}%
This defines a map%
\begin{equation*}
\mathbf{V}_{\mathbf{g}}\mathbf{f:}\mathcal{H\rightarrow }L^{2}(%
\mathbb{R}
^{2d})\text{.}
\end{equation*}
In the case when the vector $\mathbf{g}$ is extracted from an orthogonal
sequence $\{g_{k}\}_{k\geq 0}$, the essential properties of the Gabor
transform are kept. As an example of how the results concerning Gabor
analysis can be generalized to this setting, we obtain the isometric
properties and the orthogonality relations (the latter valid under the
slightly weaker condition of biorthogonality).
\begin{proposition}
If
\begin{equation}
\left\langle g_{i},g_{j}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}=\delta _{i,j}, \label{ortg}
\end{equation}%
then $\mathbf{V}_{\mathbf{g}}\mathbf{f}$ is an isometry between Hilbert
spaces, that is%
\begin{equation*}
\left\Vert \mathbf{V}_{g}\mathbf{f}\right\Vert _{L^{2}(%
\mathbb{R}
^{2d})}=\left\Vert \mathbf{f}\right\Vert _{\mathcal{H}}.
\end{equation*}
\end{proposition}
\begin{proof}
Using (\ref{ortg}) and (\ref{Gabor isometry}) gives%
\begin{eqnarray*}
\left\Vert \mathbf{V}_{g}\mathbf{f}\right\Vert _{L^{2}(%
\mathbb{R}
^{2d})}^{2} &=&\sum_{0\leq k\leq n-1}\left\langle V_{g_{k}}f_{k}\mathbf{,}%
V_{g_{k}}f_{k}\right\rangle _{L^{2}(%
\mathbb{R}
^{2d})} \\
&=&\sum_{0\leq k\leq n-1}\left\Vert V_{g_{k}}f_{k}\right\Vert _{L^{2}(%
\mathbb{R}
^{2d})}^{2} \\
&=&\sum_{0\leq k\leq n-1}\left\Vert f_{k}\right\Vert _{L^{2}(%
\mathbb{R}
^{d})}^{2} \\
&=&\left\Vert \mathbf{f}\right\Vert _{\mathcal{H}}^{2}.
\end{eqnarray*}
\end{proof}
The orthogonality relations are valid under the slightly weaker condition of
biorthogonality.
\begin{proposition}
If%
\begin{equation}
\left\langle g_{1,i},g_{2,j}\right\rangle _{L^{2}(%
\mathbb{R}
^{d})}=\delta _{i,j}, \label{biort}
\end{equation}%
then $\mathbf{V}_{g}\mathbf{f}$ satisfies%
\begin{equation}
\left\langle \mathbf{V}_{g_{1,i}}\mathbf{f}_{1},\mathbf{V}_{g_{2,j}}\mathbf{f%
}_{2}\right\rangle _{L^{2}(%
\mathbb{R}
^{2d})}=\left\langle \mathbf{f}_{1}\mathbf{,f}_{2}\right\rangle _{\mathcal{H}%
}\text{.} \label{vectorort}
\end{equation}
\end{proposition}
\begin{proof}
Using (\ref{biort}) and (\ref{Gabor isometry}) gives%
\begin{equation*}
\left\langle \mathbf{V}_{g_{1,i}}\mathbf{f}_{1},\mathbf{V}_{g_{2,j}}\mathbf{f%
}_{2}\right\rangle _{L^{2}(%
\mathbb{R}
^{2d})}=\sum_{0\leq k\leq n-1}\left\langle \mathbf{f}_{1,k},\mathbf{f}%
_{2,k}\right\rangle _{L^{2}(%
\mathbb{R}
^{2d})}=\left\langle \mathbf{f}_{1}\mathbf{,f}_{2}\right\rangle _{\mathcal{H}%
}\text{.}
\end{equation*}
\end{proof}
Clearly, when we take $\mathbf{g=(}\Phi _{0},...\Phi _{m-1}\mathbf{)}$, we
have the following relation with the polyanalytic Bargmann transform:%
\begin{equation*}
(\mathbf{B}^{n}\mathbf{f})(z)=e^{i\pi x\omega -\pi \frac{\left\vert
z\right\vert ^{2}}{2}}\mathbf{V}_{\mathbf{g}}\mathbf{f}(x,\omega )\text{.}
\end{equation*}%
This observation removes some of the mystery from the previous sections. Now
we are in a position to say that the polyanalytic Bargmann-Fock spaces play
exactly the same role as the Bargmann-Fock space in the scalar case. We have
thus all the basic ingredients to build a theory of vector valued (or super)
Gabor analysis:
\begin{itemize}
\item A vector valued Gabor transform.
\item A discrete theory for $L^{2}(%
\mathbb{R}
^{d})$ frames and Riesz basis.
\item A special vector of windows providing the connection with complex
analysis, where, in the case $d=1$, a Nyquist rate phenomenon can be
observed.
\end{itemize}
The analyzing vector can be extracted from orthogonal systems other than the
Hermite functions, though they probably do not lead to very structured
situations. As a first example we may think of the Haar basis. Other
wavelets may also be used, but we will not pursue this question further in
this paper.
\subsection{Applications}
Although we are here dealing mostly with questions of a conceptual nature,
there are potential applications of these results in multiplexing, an
important method in telecommunications, computer networks and digital video,
as indicated in \cite{Balan} and \cite{CharlyYurasuper}. The idea of
multiplexing is to encode $n$ independent signals $f_{k}\in L^{2}(%
\mathbb{R}
^{d})$ as a single sequence $\mathbf{f}$ that captures the time-frequency
information of each one. With suitable windows\ $\mathbf{g=(}g_{0},...g_{n-1}%
\mathbf{)}$, the time-frequency content of each signal can be measured by
the associated super-Gabor systems. The super Gabor transform $\mathbf{V}_{%
\mathbf{g}}\mathbf{f}(x,\omega )$, gives the precise value one wants to
approximate via the discrete systems.
\subsection{Further questions}
As in the scalar-valued case, when the connection to complex analysis is
missing the lattices generating super-frames should be hard to describe as,
for instance, the case of a "rectangular" window. What we mean by
rectangular window is one obtained from the Haar basis, which is the natural
generalization of the characteristic function of an interval. We wonder how
Janssen%
\'{}%
s tie \cite{J2} would generalize to this situation.
It is still unclear if there is a relation between the Bargmann-Fock space
of polyanalytic functions and vector valued coherent states, as considered
in \cite{Ali} and \cite{AliEngGaz}.
A rather mysterious topic is sampling and interpolation in the true\emph{\ }%
polyanalytic space $\mathcal{F}^{n-1}(%
\mathbb{C}
)$. Partial results follow from ours. Using decomposition (\ref%
{decomposition}), it is easy to see that we obtain two propositions
concerning $\mathcal{F}^{n-1}(%
\mathbb{C}
)$: one, a sufficient condition for sampling, the other a necessary
condition for interpolation.
\begin{proposition}
If $D(\Lambda )>n$ then $\Gamma $ is a sampling sequence for $\mathcal{F}%
^{n-1}(%
\mathbb{C}
)$.
\end{proposition}
\begin{proof}
This is obvious from decomposition (\ref{decomposition}) because, if $%
D(\Lambda )>n$, then inequality (\ref{sampling}) holds for every $F\in
\mathcal{F}^{n}(%
\mathbb{C}
)$. In particular it also holds for every $F\in \mathbf{F}^{n}(%
\mathbb{C}
)$.
\end{proof}
\begin{proposition}
If $\Gamma $ is an interpolating sequence for $\mathcal{F}^{n-1}(%
\mathbb{C}
)$, then
\begin{equation}
D(\Lambda )<n. \label{estd}
\end{equation}
\end{proposition}
\begin{proof}
If $D(\Lambda )>n$, then $\Gamma $ is not an interpolating sequence for $%
\mathbf{F}^{n}(%
\mathbb{C}
)$. As a result, there exists $\{\alpha _{i,j}\}\in l^{2}$, such that it is
impossible to find $F\in \mathbf{F}^{n}(%
\mathbb{C}
)$, verifying $e^{i\pi x\omega -\frac{\pi }{2}\left\vert z\right\vert
^{2}}F(z)=\alpha _{i,j}$. Again, from the decomposition (\ref{decomposition}%
), one sees that in particular it is impossible to find $F\in \mathcal{F}%
^{n}(%
\mathbb{C}
)$ verifying $e^{i\pi x\omega -\frac{\pi }{2}\left\vert z\right\vert
^{2}}F(z)=\alpha _{i,j}$.
\end{proof}
We may wonder whether these conditions are sharp.
In the case of Proposition 6 the answer is a definite "no". This is because,
if $\Gamma $ is an interpolating sequence for $\mathcal{F}^{n-1}(%
\mathbb{C}
)$, then $\{\left\langle f,M_{\omega }T_{x}h_{n-1}\right\rangle _{L^{2}(%
\mathbb{R}
)}\}_{(x,w)\in \Lambda }$ is a Riesz basis for $L^{2}(%
\mathbb{R}
)$. As a result, $\{\left\langle f,M_{\omega }T_{x}h_{n-1}\right\rangle
_{L^{2}(%
\mathbb{R}
)}\}_{(x,w)\in \Lambda ^{0}}$ is a frame for $L^{2}(%
\mathbb{R}
)$ and consequently, using Rieffel-Ramanathan-Steger Theorem \cite{Rief},
\cite{RS}, we must have $D(\Lambda ^{0})>1$ and, by scalar Ron-Shen duality,
this implies $D(\Lambda )<1$. Therefore, estimate (\ref{estd}) is far from
being sharp.
Let us look closer at Proposition 5. Here things become rather intriguing.
It is easy to see that the set $\mathcal{G}(h_{n},\Lambda )$ is a Gabor
frame if and only if the associated sequence $\Gamma $ is a sampling
sequence for $\mathcal{F}^{n}(%
\mathbb{C}
)$: since $\left\langle f,M_{\omega }T_{x}h_{n-1}\right\rangle _{L^{2}(%
\mathbb{R}
)}=V_{h_{n}}f(x,w)$, then (\ref{rel}) can be written as
\begin{equation}
\left\langle f,M_{\omega }T_{x}h_{n-1}\right\rangle _{L^{2}(%
\mathbb{R}
)}=e^{i\pi x\omega -\frac{\pi }{2}\left\vert z\right\vert ^{2}}\mathcal{B}%
^{n}f(z). \label{relsamp}
\end{equation}
By setting $F=\mathcal{B}^{n}f$, the isometry of $\mathcal{B}^{n}:L^{2}(%
\mathbb{R}
)\rightarrow \mathcal{F}^{n}(%
\mathbb{C}
)$ and the relation (\ref{relsamp}) show that definition (\ref{sampling}) is
equivalent to the fact that $\{\left\langle f,M_{\omega
}T_{x}h_{n-1}\right\rangle _{L^{2}(%
\mathbb{R}
)}\}_{(x,w)\in \Lambda }$ is a frame.
Therefore, Proposition 5 is equivalent to the sufficient condition obtained
in \cite{CharlyYura}, where the authors prove that, if $D(\Lambda )>n$, then
$\{\left\langle f,M_{\omega }T_{x}h_{n}\right\rangle _{L^{2}(%
\mathbb{R}
)}\}_{(x,w)\in \Lambda }$ is a frame and give some evidence to support their
conjecture that the result is sharp. If true, this would be quite a
surprising statement, in face of decomposition (\ref{decomposition}).
Moreover, by duality, it would bring estimate (\ref{estd}) down to $%
D(\Lambda )<1/n$.
\textbf{Acknowledgement. }I would like to thank Hans Feichtinger for his
postdoctoral guidance and the NuHAG group at the University of Vienna, where
I first had contact with some of the ideas developed in this paper.
Moreover, specific acknowledgement is due to Karlheinz Gr\"{o}chenig, who
gave generous local seminars with fresh ideas on Gabor frames with Hermite
windows and on the vectorial duality of Gabor frames, and to Yurii
Lyubarskii, who read the first version of the manuscript, providing comments
and corrections which are incorporated in this version. The Referees and the
Editor helped to improve the readability of the paper and they have drawn my
attention to the recent preprint \cite{BCL}. Finally, Radu Balan explained
me the results in \cite{BCL} and gave insightful remarks on the whole topic
while I was preparing the revised version of the manuscript.
|
1,116,691,499,387 | arxiv | \section{Introduction}
Recently, cosmological theories and observations have provided interesting information about the universe. Data and measurements from the cosmic microwave background (CMB) show that the matter and energy fluctuations are always unstable on a large scale \cite{2,3}. The leading cause of these cosmic fluctuations is still unknown, and there is no specific explanation. But, like all other phenomena, cosmologists use different scenarios to explain the reasons for these fluctuations, including the inflationary world \cite{6,7,8}. Hence, cosmic inflation is a model for the production of perturbations related to the initial density of the universe, which somehow involves the structure formation. As can be seen from the inflation patterns, the universe has gone through an early period of accelerated expansion to solve the problems in cosmology, such as the horizon, flatness, and monopole problems \cite{10}. This accelerated expansion also led to these quantum fluctuations, and over time, these perturbations intensified under gravity, creating the structure of galaxies and everything in the universe on a large scale \cite{10,12,14}. Researchers have studied different inflation models that the simplest model is as inflation by a slow-roll scalar field \cite{B1, B2}. However, many reasons show that inflationary models may be practical for more than one field. First, in many theories, such as string theory or supersymmetry, and many other areas, we are practically dealing with several fields. Second, using two or more scalar fields, may offer desirable features and have many implications in cosmology. For example, hybrid models that include two scalar fields achieve both inflationary ranges and the area of density fluctuations. These are consistent with observable data and they occur at sub-Planckian scales\cite{17,18,19,20}. However, due to the advantages of studying multi-field inflation models, their analysis is complicated and has particular complexities due to the observable data. When we are faced with two or more scalar fields, perturbations in the relative contribution to the energy density are also possible, along with perturbations in total energy density \cite{22}. These isotropic perturbations may be source of curvature perturbations, and their evolution at the super-horizon scale, which confronts calculations with certain complexities, such as the density power spectrum \cite{26,27,28}. As a result, these fields are associated with several initial conditions that affect the power spectrum \cite{29,31}. The important thing is that the complexity of these multi-field inflation models will show against the observable data. Therefore, we should always consider a complete framework for these models and test multi-field models \cite{31,32,33,38,39,41,42,43}. One of these cases is the use of two-field inflation models. In this article, we want to analyze a new perspective of these two-field inflation models according to specific conditions. For two-field inflation, a number of specific models have been considered already \cite{46,47,48,49,52,53,54,55,56,57,59,60,61,62,63,64,65,66,67}. In general, two-field inflation has been used in many cases and has several implications in cosmology. It is including approximate solutions to the metric perturbations in slow-roll approximation. Also, this scenario include the evolution equation for adiabatic and entropy perturbations for certain models with kinetic corrections. There are also models with specific non-canonical corrections, or some unconventional kinetic corrections \cite{68,70,71,72,73}.\\
Given all the concepts mentioned above, we now want to consider a two-field inflation model concerning swampland conjecture \cite{80}. A conjecture, called weak gravity, has recently been introduced \cite{76,77,78,79,q,w}. According to this conjecture, gravity introduced as the weakest force at high energy limit of theories coupled to the gravity \cite{81,83,84}. There is an area that is consistent with quantum gravity called landscape, but at low energy, the landscape is surrounded by a larger area called swampland that contradicts quantum gravity \cite{85,86,86b}.\\
For inflation models to be compatible with quantum gravity, they must meet two criteria. Some inflation models consistent with these criteria, and many inflation models were inconsistent with them. In general, these two conditions are called swampland distance conjecture, which provides an upper limit for $\Delta\phi$ (variation of scalar fields), and swampland de Sitter (dS) conjecture, which provides a limit for potential slope. In this article, we use the swampland dS conjecture. In general, based on Planck mass $M_{pl}$, these conditions are expressed in the following forms \cite{88,89,90,91,e},
\begin{equation}\label{1}
\frac{\Delta\phi}{M_{pl}}<\mathcal{O}(1),
\end{equation}
and
\begin{equation}\label{2}
M_{pl}\frac{V'}{V}>c,
\end{equation}
where $V$ is a scalar field potential, and $c$ is a positive constant.\\
Given the above concepts, in this paper, we present a two-field inflation model method based on the swampland dS conjecture. Then, we evaluate the compatibility or incompatibility of this model with respect to observable data. This paper is organized as follows. In section 2, we study the two-field inflation model and introduce different types of cosmological parameters. In section 3, according to the swampland dS conjecture and the concepts expressed in this paper, we reobtain all of the parameters in section 2 with new points of view. Then, we investigate the compatibility or incompatibility of this inflation model according to the observable data by plotting some figures related to each of these cosmological parameters. We determine the range of these parameters in the final section before conclusion.
\section{Two-field inflation model}
In this section, we first briefly introduce a two-field inflation model, then evaluate these expressed models with the swampland dS conjecture. In order to examine the inflation models, we first consider the corresponding action and metrics. So, for two scalar fields, the action has following form,
\begin{equation}\label{3}
S=\int d^{4}x\sqrt{-g}\left(\frac{M_{pl}^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi-\frac{1}{2}\partial_{\mu}\chi \partial^{\mu}\chi-V(\phi,\chi)\right),
\end{equation}
where $R$ and ($\phi,\chi$) are Ricci scalar and two scalar fields, respectively, and $V(\phi,\chi)$ is corresponding potential. The Friedmann-Robertson-Walker (FRW) space-time introduced as following \cite{91b},
\begin{equation}\label{4}
ds^{2}=-dt^{2}+\alpha^{2}(t)\delta_{ij}dx^{i}dx^{j}.
\end{equation}
Concerning the above equation and scale factor $\alpha(t)$, the Friedmann equations for the evolution of $\alpha(t)$ are as follows \cite{92},
\begin{equation}\label{5}
H^{2}=\frac{1}{3M_{pl}^{2}}(\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\dot{\chi}^{2}+V(\phi,\chi)),
\end{equation}
and
\begin{equation}\label{6}
-2\dot{H}=\frac{1}{M_{pl}}^{2}(\dot{\phi}^{2}+\dot{\chi}^{2})
\end{equation}
The slow-roll parameters such as $\epsilon$ and $\eta$ are given by,
\begin{equation}\label{7}
\epsilon=\frac{3(\dot{\phi}^{2}+\dot{\chi}^{2})}{\dot{\phi}^{2}+\dot{\chi}^{2}+2V},
\end{equation}
and
\begin{equation}\label{8}
\eta=-\frac{2(\dot{\phi}\ddot{\phi}+\dot{\chi}\ddot{\chi})}{H(\dot{\phi}^{2}+\dot{\chi}^{2})}.
\end{equation}
According to the above concepts and using the Hamilton-Jacobi equation, and also the description of inflation dynamics, we consider the following form of Hubble parameter \cite{92},
\begin{equation}\label{9}
H=H_{0}+H_{1}\phi+H_{2}\chi.
\end{equation}
Also, equations of motion with respect to the mentioned method is as follows,
\begin{equation}\label{10}
\dot{\phi}=-2M_{pl}^{2}\frac{dH}{d\phi},
\end{equation}
and
\begin{equation}\label{11}
\dot{\chi}=-2M_{pl}^{2}\frac{dH}{d\chi}
\end{equation}
Solving differential equations (\ref{10}) and (\ref{11}) one can obtain,
\begin{equation}\label{12}
\phi(t)=-2M_{pl}^{2}H_{1}t+\phi_{0}
\end{equation}
and
\begin{equation}\label{13}
\chi(t)=-2M_{pl}^{2}H_{2}t+\chi_{0}
\end{equation}
Also, the corresponding potential of two scalar fields calculated as following,
\begin{eqnarray}\label{14}
V(\phi,\chi)&=&(3H_{0}^{2}-2M_{pl}^{2}H_{1}^{2}-2M_{pl}^{2}H_{2}^{2})+6H_{1}H_{2}\phi\chi\nonumber\\
&+&6H_{0}H_{1}\phi+6H_{0}H_{2}\chi+3H_{1}^{2}\phi^{2}+3H_{2}^{2}\chi^{2}.
\end{eqnarray}
In order to calculate the scalar spectral index $(n_{s})$ and tensor-to scalar ratio \cite{EPJP}, we can use the power spectrum with respect to $C_{s}k=\alpha H$ with $C_{s}^{2}=1$ and $W_{s}\equiv\frac{(\dot{\phi}^{2}+\dot{\chi}^{2})}{2M_{pl}^{2}H^{2}}$, which yields,
\begin{equation}\label{15}
A_{s}=\frac{H^{2}}{8\pi^{2}W_{s}C{s}^{2}}.
\end{equation}
Therefore, we can obtain,
\begin{equation}\label{16}
n_{s}-1=-2\epsilon-\frac{1}{H}\frac{d}{dt}\ln\epsilon,
\end{equation}
and
\begin{equation}\label{17}
r=16\epsilon.
\end{equation}
We have introduced a two-field inflation model, and we expressed different values for cosmological parameters. In addition to the above, other cosmological parameters such as the number of enfolds, running spectrum index, etc., can be obtained. In next section, we reproduce all these values related to the above concepts. We investigate the two-fields inflation modes with respect to swampland dS conjecture. Finally, we plot some figures to determine the ranges of each of these parameters.
\section{The de Sitter conjecture}
According to all mentioned motivations, and by using a series of direct calculations, we will recalculate the potential and other cosmological parameters. We also specify the range associated with each of the parameters using the swampland dS conjecture. Then, we plot some figures, and determine these ranges. Hence, concerning equations of previous section, the potential can be written as following,
\begin{equation}\label{18}
V=3(H_{0}-2M_{pl}^{2}(H_{1}^{2}+H_{2}^{2})t)^{2}-2M_{pl}^{2}(H_{1}^{2}+H_{2}^{2}).
\end{equation}
Now, according to equation (\ref{2}) and above equation, we obtain,
\begin{equation}\label{19}
\frac{6H_{1}H_{2}}{3(H_{0}-2M_{pl}^{2}(H_{1}^{2}+H_{2}^{2})t)^{2}-2M_{pl}^{2}(H_{1}^{2}+H_{2}^{2})}>c.
\end{equation}
The above equation is the dS conjecture concerning mentioned equations. Also, using the equation (\ref{5}) we can write,
\begin{equation}\label{20}
\epsilon=\frac{4M_{pl}^{2}(H_{1}^{2}+H_{2}^{2})}{2M_{pl}^{2}(H_{0}-2M_{pl}^{2}(H_{1}^{2}+H_{2}^{2})t)^{2}}.
\end{equation}
The scalar spectral index $n_{s}$ and tensor-to-scalar ratio ($r$) regarding the equations (\ref{16}) and (\ref{17}), one can obtain,
\begin{equation}\label{21}
n_{s}-1=-\frac{8M_{pl}^{2}(H_{1}^{2}+H_{2}^{2})}{(H_{0}-2M_{pl}^{2}(H_{1}^{2}+H_{2}^{2})t)^{2}},
\end{equation}
and
\begin{equation}\label{22}
r=16\frac{4M_{pl}^{2}(H_{1}^{2}+H_{2}^{2})}{2M_{pl}^{2}(H_{0}-2M_{pl}^{2}(H_{1}^{2}+H_{2}^{2})t)^{2}}.
\end{equation}
After calculating the above-mentioned values related to cosmological parameters, we will examine the relations proportional to the swampland dS conjecture. It is noteworthy that by inverting equation (\ref{21}), two values are obtained according to the scalar spectrum index, and we will have two relations by placing them in the equation (\ref{19}). Therefore, according to the equation (\ref{21}), by inverting the function and obtaining the relation according to the scalar spectral index $(n_{s})$, and replacing it in the equation (\ref{19}), the equations related to swampland conjecture convert to the following forms,
\begin{equation}\label{23}
-\frac{6M_{pl}^{2}(-1+n_{s})^{2}t^{2}H_{1}H_{2}}{(11+n_{s})(M_{pl}^{2}(-1+n_{s})tH_{0}-2(M_{pl}^{2}+\sqrt{M_{pl}^{4}(1-(-1+n_{s})tH_{0})}))}>c,
\end{equation}
and
\begin{equation}\label{24}
-\frac{6M_{pl}^{2}(-1+n_{s})^{2}t^{2}H_{1}H_{2}}{(11+n_{s})(M_{pl}^{2}(-2+(-1+n_{s})tH_{0})+2\sqrt{M_{pl}^{4}(1+(t-n_{s}t)H_{0})})}>c.
\end{equation}
Now, we perform the same procedure for another parameter, i.e., the tensor-to-scalar ratio. So, according to the equation (\ref{22}), by inverting the function and obtaining the relation in terms of tensor-to-scalar ratio ($r$) and replacement in the equation (\ref{19}), the equations related to swampland conjecture are given as follows. In this part, precisely like the previous part, two different values are obtained,
\begin{equation}\label{25}
-\frac{6(4\sqrt{M_{pl}^{4}(4+rtH_{0})}+M_{pl}^{2}(8+rtH_{0}))H_{1}H_{2}}{M_{pl}^{2}(-48+r)H_{0}^{2}}>c,
\end{equation}
and
\begin{equation}\label{26}
-\frac{6M_{pl}^{2}r^{2}t^{2}H_{1}H_{2}}{(-48+r)(4\sqrt{M_{pl}^{4}(4+rtH_{0})}+M_{pl}^{2}(8+rtH_{0}))}>c.
\end{equation}
After the above calculations, we obtain the range appropriate to each cosmological parameters. Therefore, we determine the range of each of these cosmological parameters by plotting some figures. Of course, with respect to the values of (\ref{16}) and (\ref{17}), the relation between the two parameters of cosmology can be well obtained. Now, according to the above concepts, we plot some figures related to each cosmological parameter, so as you can see in the figures, we have plotted the range of each of the cosmological parameters, such as the scalar spectrum index and the tensor-to-scalar ratio, according to the swampland conditions. The area of each parameter has been compared according to the observable data.\\
Assuming $M_{pl}=1$, we can obtain behavior of the swampland dS conjecture using the equation (\ref{23}) which is represented by Fig. \ref{1}. For $n_{s}<1$, it is decreasing function, while for $n_{s}>1$ it is increasing function, it vanishes for $n_{s}=1$. Also, we show that increasing Hubble parameter increases value of $c$ parameter. In Fig. \ref{1} (a) we vary $H_{2}$, while in Fig. \ref{1} (b) vary $H_{1}$ to see similar result. In Fig. \ref{1} (c) we vary $H_{0}$ and see that is not so important parameter.
\begin{figure}[h!]
\begin{center}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig1.eps}
\label{1a}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig2.eps}
\label{1b}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig3.eps}
\label{1c}}
\caption{\small{Behavior of the swampland dS conjecture ($c$) in term of $n_{s}$ in units of constant parameters, consistent to the Eq. (\ref{23}).}}
\label{1}
\end{center}
\end{figure}
Similarly, we obtained behavior of the swampland dS conjecture in term of $n_{s}$ consistent to the equation (\ref{24}) and see that is decreasing function of $n_{s}$ with slow variation (see Fig. \ref{2}). Results are symmetric by variation of $H_{2}$ (Fig. \ref{2} (a)) or $H_{1}$ (Fig. \ref{2} (b)). In Fig. \ref{2} (c) we assumed unit values for $H_{2}$ and $H_{1}$, and vary $H_{0}$.
\begin{figure}[h!]
\begin{center}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig4.eps}
\label{2a}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig5.eps}
\label{2b}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig6.eps}
\label{2c}}
\caption{\small{Behavior of the swampland dS conjecture ($c$) in term of $n_{s}$ in units of constant parameters, consistent to the Eq. (\ref{24}).}}
\label{2}
\end{center}
\end{figure}
As shown in the figures, we plotted the values associated with the swampland dS conjecture and the different values obtained for each of the cosmological parameters, such as the scalar spectral index $n_{s}$ in Fig. \ref{1} and Fig. \ref{2} from equations (\ref{23}) and (\ref{24}) as well as the tensor-to-scalar ratio ($r$) in Fig. \ref{3} and Fig. \ref{4} consistent with the change to any of the Hubble parameters. In each plot, except for each mentioned Hubble parameter's changes, we assume other parameters such as $M_{pl}$ as a unit constant positive value. The range associated with these parameters is well determined. As you can see in Fig. \ref{1} and Fig. \ref{2}, the swampland conjecture is well behaved concerning the scalar-spectrum-index for the different values of the Hubble parameter in Fig. \ref{1}, which is derived from the equation (\ref{23}), and shows acceptable values. Similarly, swampland conjectures regarding the tensor-to-scalar ratio as well as the various values of the Hubble parameter per equation (\ref{26}) in Fig. \ref{4} is well defined and shows more acceptable values than by Fig. \ref{3}.\\
\begin{figure}[h!]
\begin{center}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig7.eps}
\label{3a}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig8.eps}
\label{3b}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig9.eps}
\label{3c}}
\caption{\small{Behavior of the swampland dS conjecture ($c$) in term of tensor-to-scalar ratio $r$ in units of constant parameters, consistent to the Eq. (\ref{25}).}}
\label{3}
\end{center}
\end{figure}
It is clear from Fig. \ref{4} that $c$ parameter is increasing function of $r$ as well as Hubble parameters.\\
\begin{figure}[h!]
\begin{center}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig10.eps}
\label{4a}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig11.eps}
\label{4b}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig12.eps}
\label{4c}}
\caption{\small{Behavior of the swampland dS conjecture ($c$) in term of tensor-to-scalar ratio $r$ in units of constant parameters, consistent to the Eq. (\ref{26}).}}
\label{4}
\end{center}
\end{figure}
Next, by using the equations (\ref{21}) and (\ref{22}), we plot the cosmic parameters $r$ in terms of $n_{s}$ by Fig. \ref{5}. The range of each of these parameters is determined. As you can see in the Fig. \ref{5}, these values are comparable to observable data. In these figures, we examined the change of the two cosmological parameters of the scalar-spectrum-index $(n_{s})$ and the tensor-to-scalar ratio $r$ for differences of the Hubble parameters. We assumed that the other values were positive of a unit order. It is interesting to note that the range and values obtained for single-field inflation models \cite{das,86} are more accurate than the two-field model according to the Swampland conditions and are consistent with observable data.\\
\begin{figure}[h!]
\begin{center}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig16.eps}
\label{6a}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig17.eps}
\label{6b}}
\subfigure[]{
\includegraphics[height=4cm,width=4cm]{fig18.eps}
\label{6c}}
\caption{\small{The $(r-n_{s})$ plan in units of constant parameters. (a) by variation of $H_{2}$; (b) by variation of $H_{1}$; and (c) by variation of $H_{0}$.}}
\label{6}
\end{center}
\end{figure}
The allowable area for the scalar-spectral-index and the tensor-to-scalar ratio determined according to the Hubble parameter. As mentioned above, the obtained range is somewhat comparable to the observable data \cite{77}. Here this model compared to single-field inflation \cite{86} with the slow-roll condition concerning swampland conjecture, it is less accurate. So, the single-field inflation with swampland conjecture can be more critical in cosmological studies. This paper introduces a two-field inflation model, and we obtained each of the cosmological parameters separately with analytical and numerical analysis. According to the swampland dS conjecture, we examined the inflation model. We determined the range associated with each of the cosmological parameters by plotting some figures consistent with the observable data. In the next works, we will evaluate other implications of this cosmological model according to different conditions.
\section{Conclusions}
Different inflation models according to various conditions such as slow-roll, ultra-slow-roll, constant-roll, and other conditions already investigated in literatures, where many cosmological implications evaluated. In this paper, we investigated a new perspective of the two-field inflation model with respect to the new swampland dS conjecture. Therefore, we studied the two-field inflation model and some cosmological parameters. Then, we obtained the mentioned parameter such as the scalar spectral index $n_{s}$ and tensor-to-scalar ratio ($r$). We applied the mentioned swampland dS conjecture in this paper to our model concerning the observational data like Planck 2018 \cite{1807.06209}.\\
Finally, we compared these values with the observable data by plotting some figures. As mentioned above, the obtained range is somewhat comparable to the observable data. Here, this model compared to single-field inflation with the slow-roll condition concerning swampland conjecture, it is less accurate. So, the single-field inflation with swampland conjecture can be more important in cosmological studies. In general, the concepts expressed in this article can be examined for the multi-field inflation model by using other ideas such as slow-roll and constant-roll conditions. Also, it is interesting to apply two-field scenario to the M-theory compactifications \cite{88}. In the future works, we will discuss these concepts.
|
1,116,691,499,388 | arxiv |
\section{Introduction}
\label{sec:intro}
\input{introduction}
\section{Pure exploration, confidence interval, and p-value}
\label{sec:background}
\input{background}
\section{Advances in best-arm identification and hypothesis testing}
\label{sec:advances}
\input{advances}
\section{Challenges of adaptive online testing}
\label{sec:difficulty}
\input{difficulty}
\section{Future research directions}
We briefly discuss several promising directions that may contribute greatly to decision-making with adaptive online testing for IR.
\begin{itemize}[leftmargin=*]
\item Extension to the fixed budget setting. The fixed budget setting may be more practical for certain IT productions such as advertising \cite{sen2017identifying}. While the best-arm identification algorithms can often switch between the fixed confidence and fixed budget setting \cite{gabillon2012best}, the focus of hypothesis testing may vary. Increasing the power of a experiment design, rather than controlling the false discovery rate, might be more demanding for fixed budget problems. More research is needed to incorporate the new objective.
\item \emph{Thompson Sampling} (TS) for exploration. The Bayesian idea behind TS's exploration strategy is as powerful as the CB method \cite{agrawal2012analysis}, and \citet{russo2016simple} recently developed TS-driven best-arm identification algorithm. A distinctive advantage of TS is to include prior knowledge about the experimentation (e.g. offline testing results). Although hypothesis testing is strictly a frequentist's choice, there is possibility to seek collaboration especially when TS involves only in exploration.
\item Involving \emph{contextual information}. Just like how contextual information serves multi-armed bandit, they can also be used to optimize adaptive online testing. For instance, when an online testing is designed to select the best banner size, the unknown rewards for each arm may be size-related. If so, an extra learning step can lead to better exploration. \citet{kato2021role} and \citet{chen2020contextual} has done some preliminary work in this direction, but this field remains largely unstudied.
\item Real-world examination. Most of the methods in this short paper have only been tested via simulation or small-scale studies. Their efficacy in large-scale IT productions remains to be told.
The real-time computation and updates required by adaptive online testing will certainly put a test on the infrastructure at the serving end. Business requirements may also complicate the design and analysis of adaptive online testing. Applied research with deployment solutions will be highly appreciated.
\end{itemize}
\bibliographystyle{ACM-Reference-Format}
|
1,116,691,499,389 | arxiv | \section{\label{sec:introduction}{INTRODUCTION}}
Free energy is a significant physical property for estimating thermodynamic stability.
It is desirable to estimate free energy as accurately as possible.
Such free energy estimation is becoming important in a variety of research fields;
in particular, biological molecules including proteins or interfaces of nano-scale materials
have been raised as a target for free energy calculations\textcolor{black}{\cite{Yamamori2013,Sinko2012,
Konig2014}}.
It is thus desirable to develop methods that improve the accuracy and efficiency
in free energy calculations using molecular simulation.
The free energy in molecular systems has often been evaluated for a given constraint
(reaction pass).\cite{Ciccotti2004}
Such a constraint is usually specified by using a set of reaction
coordinates,\cite{Sprik1998} for example, distances
between molecules, bond angles, and dihedral angles, etc.
In order to get free energy landscapes, various techniques
[thermodynamic integration (TI)\cite{Kirkwood1935},
free energy perturbation,\cite{Zwanzig1954}
umbrella sampling,\cite{Torrie1977} and so on] have been developed so far.
Although TI is derived from the statistical mechanics faithfully,
some difficulties have been pointed out;
poor sampling which could come from a breakdown of the ergodicity and
numerical integration as postprocessing.
To overcome these difficulties, free energy calculation methods
based on mean force dynamics (MFD) have been proposed.\cite{Rosso2002,Laio2002}
In MFD, a set of $L$ reaction coordinates (collective variables),
${\bf X} \equiv \{X_{1}, X_{2}, \cdots, X_{L} \}$, is regarded as a set of fictitious
dynamical variables, and their trajectories are designed to be generated
by hypothetical dynamical equations.
Morishita {\it et al.}\cite{Morishita2012,Morishita2013}
have recently introduced a logarithmic form of the free energy along ${\bf X}$ [$F({\bf X})$]
to enable us to easily sample rare events in MFD calculations.
This method is called logarithmic mean force dynamics (LogMFD),
in which the free energy can be estimated on-the-fly.
The evaluation of mean force (MF), i.e., slope of $F({\bf X})$ with respect to ${\bf X}$,
in such a method based on MFD can be
improved
by incorporating first-principles (FP) molecular dynamics
(MD),\cite{Car1985} replacing the classical MD using empirical force fields.
FPMD allows us to include effects of the electronic state explicitly;
for example, bond-formation or bond-breaking,
which may considerably influence the free energy profiles in molecular systems.
In this paper,
we have developed
first-principles MFD in the framework of LogMFD, namely,
first-principles LogMFD (FP-LogMFD).
We reconstruct the free energy landscape
for a molecular system of glycine dipeptide using FP-LogMFD.
This demonstration indicates that FPMD can be
incorporated into LogMFD of
multi-dimensional ${\bf X}$-systems
and that the scheme developed here is found to be promising
for the free energy reconstruction using {\it ab initio} techniques.
The successful combination of LogMFD and FPMD is indebted to
the efficiency for sampling rare events in LogMFD. The logarithmic form
introduced in LogMFD suppresses the effective energy barriers for the
dynamical variables ${\bf X}$.
This makes it possible to sample configurations with higher energy,
as frequently as those with much lower energy.
This feature also makes it possible
to improve the accuracy of the MF by increasing the number of statistical samples (FPMD steps).
In the next section, we review the LogMFD method briefly and
demonstrate how to incorporate FPMD into LogMFD.
In Sec. III, we present the free energy profile with respect
to the dihedral angles in glycine dipeptide molecule.
In Sec. IV, we will discuss the entropic contribution and numerical accuracy
in the present results
by comparing it
with the \textcolor{black}{classical MD}
result previously obtained using
an empirical force field. Finally, Sec. V summarizes this paper.
\section{\label{sec:theory}Methods}
\subsection{\label{subsec:theory}Equations of mean force dynamics}
We present a brief review for LogMFD.
This review would be a good introduction to our new scheme which employs a non-empirical
approach.
We consider a system of $N$ atoms with a given temperature $T_{\rm ext}$,
and aim to reconstruct the free energy profile $F({\bf X})$ with respect to ${\bf X}$.
Each reaction coordinate $X_{p}(\{ {\bf R}_{I} \})$ is generally a function of
the atomic coordinates $\{ {\bf R}_{I}\}$,
where $p$ and $I$ specify the $p$'th reaction coordinate and
the $I$'th atom, respectively.
In MFD, however, ${\bf X}$ are regarded as dynamical variables, being independent of $\{ {\bf R}_{I}\}$.
We now consider the following postulated Hamiltonian for ${\bf X}$;
\begin{eqnarray}
H_{\rm MFD} & = &
\sum_{p}^{L}\frac{1}{2}M_{p} \dot{X}_{p}^{2} + F({\bf X}),
\label{HMFD}
\end{eqnarray}
where the first and second terms on the right-hand-side are the kinetic
and potential energies, respectively, for $X_{p}$ ($\dot{X}_{p}$ means the
velocity $d X_{p}/dt$) and $M_{p}$ is the fictitious mass for $X_{p}$.
The equation of motion for ${X_{p}}$ is thus obtained as,
\begin{eqnarray}
M_{p} \ddot{X}_{p} &=& -
\frac{\partial F({\bf X})}{\partial X_{p}},
\label{eqmf}
\end{eqnarray}
where $-{\partial F({\bf X})/\partial X_{p}}$ is the MF.
The solution for this equation of motion fulfills the conservation law, i.e.,
$H_{\rm MFD}$ can be seen as a constant of motion, as long as the MF
is accurately evaluated.
Several methods based on MFD have been proposed thus far, which provide
us free energy profiles with respect to reaction coordinates and allow us
to discuss many kinds of physics involving the reaction coordinates.
Metadynamics\cite{Laio2002} has been introduced utilizing the concept of MFD,
and has been applied to a variety of systems including biosystems
to sample rare events and to reconstruct free energy profiles.
Morishita {\it et al.}\cite{Morishita2012,Morishita2013}
have proposed LogMFD in which $F({\bf X})$ in Eq. (\ref{HMFD})
is replaced with a logarithmic form of $F({\bf X})$, and have demonstrated
several improvements in the free energy calculation.
In LogMFD,
the following Hamiltonian is introduced instead of Eq. (\ref{HMFD});
\begin{eqnarray}
H_{\rm LogMFD} & = & \sum_{p}^{L} \frac{1}{2} M_{p}\dot{X}_{p}^{2} +
\gamma{\mathrm {log}} \{ \alpha F({\bf X})+ 1 \},
\label{HMFD1}
\end{eqnarray}
where $\gamma$ and $\alpha$ are positive constant parameters,
which are chosen to effectively reduce the energy barriers experienced
by ${\bf X}$. The resultant equation of motion for $X_{p}$ is given as,
\begin{eqnarray}
M_{p} \ddot{X}_{p} &=& - \left(\frac{\alpha \gamma}{\alpha F({\bf X})+1}\right)
\frac{\partial F({\bf X})}{\partial X_{p}}.
\label{eqmfdeta0}
\end{eqnarray}
In practice, ${\bf X}$ can be thermostatted in LogMFD calculations,
and the equation of motion is slightly modified as follows;
\begin{eqnarray}
M_{p} \ddot{X}_{p} &=& - \left(\frac{\alpha \gamma}{\alpha F({\bf X})+1}\right)
\frac{\partial F({\bf X})}{\partial X_{p}}
- M_{p} \dot{X}_{p} \dot{\eta},
\label{eqmfd} \\
Q_{\eta} \ddot{\eta} &=& \sum_{p} M_{p} \dot{X}_{p}^{2} -
L k_{\rm B} T_{X},
\label{eqmfdeta}
\end{eqnarray}
where $\eta$ is the thermostat variable which controls the temperature of
${\bf X}\ $\textcolor{black}{
($T_{X}$)},
$Q_{\eta}$ is the mass for $\eta$,
and $k_{\rm B}$ is Boltzmann's constant.
With a single Nos\'e-Hoover thermostat \cite{Nose1984,Hoover1985} as in Eqs. (\ref{eqmfd}) and (\ref{eqmfdeta}),
the following pseudo Hamiltonian is a constant of motion instead of $H_{\rm LogMFD}$;\cite{Morishita2012,Morishita2013}
\begin{eqnarray}
\hat{H}_{\rm LogMFD} & = & \sum_{p}^{L} \frac{1}{2} M_{p}\dot{X}_{p}^{2} +
\gamma{\mathrm {log}} \{ \alpha F({\bf X})+ 1 \}
\nonumber \\
& & +\frac{1}{2}Q_{\eta} \dot{\eta}^{2}+ L k_{\rm B}T_{X} \eta.
\label{HMFD2}
\end{eqnarray}
Note that $T_{X}$ is not necessarily the same as
the temperature for atoms, $T_{\rm ext}$.
The heights of the energy barriers on
$\gamma \log \{ \alpha F({\bf X})+1 \}$
are much lower than those on $F({\bf X})$.
This reduction of the barrier height enables the coordinate $X_{p}$
to easily cross the barriers at a moderate temperature of $T_{X}$,
allowing us to evaluate the free energy associated with rare events.
$\partial F({\bf X})/\partial X_p$ is obtained as an ensemble average and,
practically, can be estimated as a time-averaged quantity from a thermostatted MD or
Monte Carlo (MC) simulation at a given temperature $T_{\rm ext}$ with a given potential
$\Phi$ for the $N$-atom system and a set of fixed reaction coordinates ${\bf X}$;
\begin{eqnarray}
\frac{\partial F({\bf X})}{\partial X_{p}}
& = &
\frac{1}{Z}\int d{\bf R}
\left[
\frac{\partial \Phi({\bf R}) }{\partial X_{p}}
\right]_{\bf X}
e^{-\Phi({\bf R})/k_{\rm B}T_{\rm ext}}
\nonumber \\
& \simeq &
\frac{1}{\tau} \int^{\tau}_{0} dt
\left[
\frac{\partial \Phi({\bf R}(t)) }{\partial X_{p}}
\right]_{\bf X},
\label{canonical}
\end{eqnarray}
\noindent
where
\begin{eqnarray}
Z & = & \int d{\bf R} e^{-\Phi({\bf R})/k_{\rm B} T_{\rm ext}}.
\label{pf}
\end{eqnarray}
Here, $\tau$ is the simulation time period, and
the $[\ \ \ ]_{\bf X}$ represents the ensemble average under
the set of constraints.
In the MF estimation, it is expected that the canonical
MD or MC simulation provides
the canonical distribution under the constraint on ${\bf X}$.
The MF is, in our approach, evaluated using thermostatted FPMD.
The potential energy $\Phi({\bf R})$ and
the details of the MF evaluation will be discussed later on.
We need to know $F({\bf X})$ to calculate the force on $X_{p}$
in Eq. (\ref{eqmfd}), however, $F({\bf X})$ itself is the quantity we want
to obtain.
This problem can be solved using the conserved quantity, $H_{\rm LogMFD}$ or
$\hat{H}_{\rm LogMFD}$ [Eq. (\ref{HMFD1}) or (\ref{HMFD2})].
Using this conservation law, $F({\bf X})$ can be directly evaluated
with $\hat{H}_{\rm LogMFD}$ (when we employ a single Nos\'e-Hoover thermostat)
whose value needs to be set at the beginning of the LogMFD run;\cite{Morishita2012,Morishita2013}
\begin{eqnarray}
F({\bf X}) & = & \frac{1}{\alpha}
\left[ {\rm exp}
\left\{ \frac{1}{\gamma}
\left(\hat{H}_{\rm LogMFD} - \sum_{p}^{L}
\frac{1}{2}M_{p} \dot{X}_{p}^{2}
\right.
\right.
\right.
\nonumber \\
& & \left.
\left.
\left.
- \frac{1}{2}
\textcolor{black}{
Q_{\eta}
}
\dot {\eta}^{2} - L k_{\rm B}T_{X} \eta
\right)
\right\}
- 1
\right] .
\label{freeenergy}
\end{eqnarray}
It is required that $F({\bf X}) > 0$ at any ${\bf X}$
to enhance the sampling in the ${\bf X}$ subspace.
This requirement can be actually fulfilled by using appropriate values for $\hat{H}_{\rm LogMFD}$;
$\hat{H}_{\rm LogMFD}$ should be larger than the sum of the initial kinetic energy
for ${\bf X}$ and the initial terms for $\eta$.
See Ref. \onlinecite{Morishita2013} for details.
Equation (\ref{freeenergy}) indicates that the $F({\bf X})$ is successively obtained along the dynamics
of $\{X_{p}\}$, namely, ``on-the-fly".\textcolor{black}{
This means that we need not perform
any postprocessing unlike in TI,
}
which overcomes some drawbacks of the TI
method.
\textcolor{black}{In TI, we need to}
decompose the ${\bf X}$ subspace into many bins with a finite width,
implying a possible missing of remarkable characters
in the free energy due to a discretized mesh.
In contrast, LogMFD provides $F({\bf X})$ with much higher resolution than TI,
since LogMFD generates almost continuous {\bf X}-trajectories and
the $F({\bf X})$ trajectories (this will be illustrated in Fig. \ref{every}).
\textcolor{black}{Summarizing LogMFD, it allows us to sample higher energy states
efficiently and to evaluate the free energy at the local point of reaction coordinates
without any postprocessing.}
The flow chart of the LogMFD method is displayed in Fig.~\ref{chart}.
To update $\{ X_{p} \}$, the most important quantity is the MF,
which is evaluated using thermostatted FPMD in our approach.
Details of our FPMD approach are presented in the next subsection;
Car-Parrinello molecular dynamics (CP-FPMD)\cite{Car1985}
with double Nos\'e-Hoover thermostats.\cite{Blochl1992,Morishita1999}
\begin{figure}
\includegraphics[width=5cm]{fig1.eps}
\caption{\label{chart}
Flow chart of first-principles LogMFD.}
\end{figure}
\subsection{\label{subsec:meanforce}First-principles mean force}
In MFD methods, the MF needs to be estimated as accurately as possible
at the temperature $T_{\rm ext}$. The MF from first-principles
could improve the accuracy of the free energy profiles.
We now address two technical issues associated with the evaluation
of the MF in our approach.
Firstly, the constraint on ${\bf X}$ during the FPMD run is discussed.
In order to impose a constraint on atomic coordinates, one may employ
the SHAKE method in which a holonomic constraint is realized,\cite{SHAKE}
although complex equations should be solved for the Lagrange multiplier.
Alternatively, the harmonic potential method can also be utilized, allowing us to
use Eq. (\ref{canonical}) without any correction terms.
In this work, we chose the latter method.
\textcolor{black}{
Secondly,
}
to keep a given temperature for the system and to generate the canonical
distribution for the atomic trajectories, we employ thermostats.
Newly developed thermostats,\cite{Morishita2010} as well as the original
thermostat,\cite{Nose1984,Hoover1985} can also be used in conjunction with FPMD in which
the Born-Oppenheimer(BO) surface is strictly searched
in the time evolution\cite{Payne1992} or CP-FPMD.\cite{Car1985}
In our FPMD approach, the following energy is considered:
\begin{eqnarray}
E_{\rm tot} & = & E_{\rm BP} + E_{\rm hp},
\label{etot} \\
E_{\rm hp} & = & \sum_{p} \frac{1}{2} k_{p} (\tilde{X}_{p}( \{ {\bf R}_{I} \} ) - X_{p})^{2},
\label{hp} \\
E_{\rm BP} & = & \sum_{i} m_{\varphi}
\langle
\dot{\varphi}_{i} | \dot{\varphi}_{i}
\rangle
+ \frac{1}{2} \sum_{I} M_{I} \dot{{\bf R}}_{I}^{2}
\nonumber \\
& & + \frac{1}{2} Q_{\rm e}\dot{x}_{\rm e}^{2} + 2E_{\rm kin}^{0} x_{\rm e}
\nonumber \\
& & + \frac{1}{2} Q_{R}\dot{x}_{R}^{2} + g k_{\rm B}T x_{R}
\nonumber \\
& & + E_{\rm fp}[\{ \varphi_{i} \},\{ {\bf R}_{I} \}],
\label{bp}
\end{eqnarray}
where $E_{\rm BP}$ and $E_{\rm hp}$ are the energy
in the Bl\"ochl-Parrinello(BP) method\cite{Blochl1992}
and
the harmonic potentials for the constraint, respectively, and $E_{\rm fp}$ represents
the potential energy in the system of electrons and ions (see Eq. (5) in Ref.
\onlinecite{Blochl1992}).
$x_{R}$ ($x_{\rm e}$) is the dynamical variable for the thermostat and
$Q_{R}$ ($Q_{\rm e}$) is the corresponding mass for $x_{R}$ ($x_{\rm e}$).
$g$ is the number of ionic degrees of freedom.
The quantity $\tilde{X}_{p}( \{ {\bf R}_{I} \} )$ in Eq. (\ref{hp}) is
constructed from the current atomic coordinates,
which is tightly constrained to $X_{p}$ according to $E_{\rm hp}$ [Eq. (12)].
To this end, the constant $k_{p}$ is chosen
to be a large value.
The atomic forces come from the contributions of $E_{\rm fp}$, the thermostat, and
the constraint $E_{\rm hp}$. These contributions result in the following equation of motion
for ${\bf R}_{I}$;
\begin{eqnarray}
M_{I}\ddot{\bf R}_{I} &=& {\bf F}_{I}^{\rm fp} - M_{I}\dot{\bf R}_{I}\dot{x}_{R}
\nonumber \\ & & - \sum_{p} k_{p} (\tilde{X}_{p}( \{ {\bf R}_{I} \} )-X_{p})
\frac{\partial \tilde{X}_{p}( \{ {\bf R}_{I} \} )}
{\partial {\bf R}_{I}}.
\label{cp4}
\end{eqnarray}
The equations of motion for the wavefunction ($\varphi_{i}$) and the
heat baths ($x_{\rm e}$ and $x_{R}$) are not changed from the original BP method
by introducing the constraint, implying a less effort
for converting a conventional computational code to the present one.
According to Eq. (\ref{canonical}), the first-principles mean force is obtained
as a time average of
$-\partial E_{\rm tot}/\partial X_{p}$:
\begin{eqnarray}
-\frac{\partial F({\bf X})}{\partial X_{p}} & \sim & k_{p}
\langle
\tilde{X}_{p}( \{ {\bf R}_{I} \} )-X_{p}
\rangle ,
\label{mf}
\end{eqnarray}
where $\langle \ \ \ \rangle$ represents a canonical ensemble or a time average.
This formula is general as far as the atomic configuration samples the canonical distribution
under the required constraint. This implies that
the relation of Eq. (\ref{mf}) is also useful in the TI method, as explained in
Appendix A.
In order to show an achievement of the constraint and the temperature control,
we present typical time evolution of the reaction coordinate
$X_{p}$ in the next section.
In CP-FPMD, the fictitious kinetic energy of the wave function,
namely, the first term in Eq. (\ref{bp}), should follow the dynamics of $\{ {\bf R}_{I} \}$
as quickly as possible.\cite{Blochl1992}
To this end, it is important to find an appropriate $E_{\rm kin}^{0}$.
For a given atomic configuration, (i) we start a CP-FPMD run with the system exactly
on the BO surface. (We converge the electronic state to the BO surface beforehand.)
Then (ii) we perform the CP-FPMD run for a few tens of MD steps without the heat baths
(may be better, without the constraint).
During this period, the system slightly leaves the exact BO surface.
(iii) If the temperature of the system reaches $T_{\rm st}$
within the given period, then we set the value of the kinetic energy of
the wave functions at the moment as $E_{\rm kin}^{0}$. Due to a practical reason for stabilizing
the simulation, $T_{\rm st}$ is taken as $\sim 0.85 T_{\rm ext}$.
(iv) When the system does not reach an appropriate temperature, we introduce a new atomic
configuration by distorting the previous atomic configuration.
We restart the process from (i).
After setting $E_{\rm kin}^{0}$, we switch on the thermostats in
the BP method accompanied with the constraint of Eq. (\ref{hp}).
The rest of the FPMD steps are used for the MF evaluation of Eq. (\ref{mf}).
In the computation mentioned above, the initial atomic configurations for each of the series
of the CP-FPMD runs were taken from
the atomic configuration in the CP-FPMD runs previously done.
We will detail the procedure to perform FP-LogMFD calculations,
including parameter settings, in the next section.
\begin{figure}
\includegraphics[width=6cm]{fig2.eps}
\caption{\label{glycine}
Atomic structure of the glycine dipeptide molecule
with atomic specification.
Two dihedral angles,
$\phi$ and $\psi$, are formed by the atomic series C(2)-N(1)-C(3)-C(4)
and N(1)-C(3)-C(4)-N(2), respectively.
The figure displays the atomic configuration with
\textcolor{black}{
$(\phi, \psi)=(180^\circ , 180^\circ)$}
.}
\end{figure}
\section{\label{sec:demonstration}NUMERICAL DEMONSTRATION}
\subsection{\label{subsec:molecule}Molecular configuration}
To illustrate our {\it ab initio} approach to free energy reconstruction, we consider the free
energy profile of glycine dipeptide molecule
[2-(Acetylamino)-N-methylacetamide] in vacuum,
as shown in Fig.~\ref{glycine}. The atoms are specified by
the symbol with numbering of C(1), C(2), $\cdots$ , O(1), O(2), .. etc..
from the left-hand-side of the figure.
The two dihedral angles are labeled as $\phi$ and $\psi$, which are formed by the atomic
series C(2)-N(1)-C(3)-C(4) and N(1)-C(3)-C(4)-N(2), respectively.
In other words, these angles are formed by the plane of N(1)-C(3)-C(4) and the plane
associated with the peptide bond (-OCNH-). In nature, the latter plane in proteins
has usually observed as the {\it trans}-form rather than
as the {\it cis}-form.\cite{text-book}
Actually, in our calculation, the {\it cis}-form of the right-hand-side of the peptide bonds
in Fig.~\ref{glycine} is higher in energy by 2.1 kcal/mol than the form
presented in Fig. \ref{glycine}.
In this section, we demonstrate the application of FP-LogMFD to the
glycine dipeptide molecule.
We have obtained the free energy landscape $F(\phi, \psi)$ with respect to
the dihedral angles $\phi$ and $\psi$ at room temperature 300 K ($=T_{\rm ext}$).
First, FP-LogMFD with the fixed dihedral angle $\phi$ was
performed to set the parameters required and to obtain the one-dimensional
free energy profile along $\psi$.
Then, the FP-LogMFD runs in the $\phi$-$\psi$ space were performed, revealing
the details of the two-dimensional free energy landscape.
We have also performed TI calculation to reconstruct the one-dimensional
free energy profile, which is compared to the FP-LogMFD result for benchmarking.
\subsection{\label{subsec:parameter}Parameter setting}
For the CP-FPMD runs, we have used the plane wave basis set and
density functional theory with the generalized gradient
approximation(GGA).\cite{KS,Perdew92}
The energy cutoffs of 25 and 250 Ry are taken for electronic wave function and charge density,
respectively.\cite{Pasquarello1992} The ultrasoft pseudopotentials are
used.\cite{Vanderbilt1990} The $\Gamma$-point sampling is adopted for the molecular
system placed in a cubic box with the dimension of 20 a.u.(10.58 \AA).
For the canonical FPMD simulation in the framework of the BP method,
the time step is set to 10 a.u. ($\sim$ 0.24 fs).
This is a typical value for the CP method.\cite{Car1985}
The parameters for $Q_{R}$, $Q_{\rm e}$, and $m_{\varphi}$ are set to
$5 \times 10^{5}$ a.u.,
$5 \times 10^{3}$ a.u.,\textcolor{black}{\cite{Oda2002}} and 200 a.u.,\textcolor{black}{\cite{Oda2004}} respectively.
$E_{\rm kin}^{0}$ in Eq. (\ref{bp}) is automatically determined by the anzatz
described before (see Sec. \ref{subsec:meanforce}).
Figure \ref{kineticenergy} presents the time evolution of the kinetic energy
of the wave functions and the instantaneous temperature
of the molecular system (proportional to the kinetic energy of
atoms).
In the present calculations, $E_{\rm kin}^{0}$ was set to be 0.0049 a.u.
\begin{figure}
\includegraphics[width=7.5cm]{fig3.eps}
\caption{\label{kineticenergy}
Time evolution of the instantaneous temperature (green full curve, left scale) and
the kinetic energy of the wave functions (red dashed curve, right scale) in the FPMD run
at $T_{\rm ext}=300$ K with the constraints
\textcolor{black}{
$\phi=-80.0^\circ$ and $\psi=-76.8^\circ$.
}
The horizontal line indicates the values of
$T_{\rm ext}$ and $E_{\rm kin}^{0}$.
The arrow indicates the time step at which the
constraint on the dihedral angles and the temperature control is turned on.}
\end{figure}
In this demonstration, the following harmonic potentials are employed to constraint $\tilde{\phi}$ and $\tilde{\psi}$;
\begin{eqnarray}
E_{\rm hp} & = & \frac{1}{2} k_{\phi}
(\tilde{\phi}( \{ {\bf R}_{I} \} ) - \phi)^{2}
+\frac{1}{2} k_{\psi}
(\tilde{\psi}( \{ {\bf R}_{I} \} ) - \psi)^{2},
\label{harmonicpotential}
\end{eqnarray}
where $\phi$ and $\psi$ are the target dihedral angles [$X_{p}$ in Eq. (\ref{hp})]
and $\tilde{\phi}$ and $\tilde{\psi}$ are the temporal ones determined
from the instantaneous molecular configuration [$\tilde{X}_{p}$ in Eq. (\ref{hp})].
Both of $k_{\phi}$ and $k_{\psi}$ are taken to be 2.4 a.u./${\rm rad}^{2}$
(\textcolor{black}{0.46 kcal/mol/${\rm deg}^{2}$}).
Figure \ref{dihed} shows typical time evolution of $\tilde{\phi}$
and $\tilde{\psi}$
with \textcolor{black}{$(\phi, \psi)=(-80.0^\circ, -76.8^\circ)$}.
This figure indicates that the temporal $\tilde{\phi}$ and $\tilde{\psi}$
fluctuate around the respective given value, implying the constraint
to be imposed correctly.
\begin{figure}
\includegraphics[width=7.5cm]{fig4-1.eps}
\includegraphics[width=7.5cm]{fig4-2.eps}
\caption{\label{dihed}
Time evolution of
the instantaneous dihedral angles,
(a) $\tilde{\phi}$ and (b) $\tilde{\psi}$,
in the FPMD run
with the constraints,
\textcolor{black}{
$\phi=-80.0^\circ$ and $\psi=-76.8^\circ$}
(horizontal lines).}
\end{figure}
\begin{figure}
\includegraphics[width=7.5cm]{fig5-1.eps}
\includegraphics[width=7.5cm]{fig5-2.eps}
\caption{\label{meanforce}
Time evolution of the instantaneous force,
(a) $k_{\phi}(\tilde{\phi}-\phi)$ and (b) $k_{\psi}(\tilde{\psi}-\psi)$,
accompanied with the horizontal lines which indicates
the mean forces, $-\partial F/\partial \phi$
and $-\partial F/\partial \psi$.}
\end{figure}
The time evolution of the temporal force is presented in Fig.~\ref{meanforce}.
Averaging over 500 steps (from the 21th step to 520th step),
the mean forces acting on $\phi$ and $\psi$ were estimated to be
\textcolor{black}{
$-$0.06863 and 0.02220 (kcal/mol)/deg},
respectively
(the fluctuations are
limited to
\textcolor{black}{
$\pm 0.87$ (kcal/mol)/deg
}
).
The accuracy of
the MF strongly depends on the number of MD steps, defined as $N_{\rm BP}$.
In fact, we found that the decrease of $N_{\rm BP}$ (from 500 steps to 300 steps)
deteriorated the MF, and the resultant free energy profile became much worse,
compared to those by $N_{\rm BP}=500$.
$N_{\rm BP}$ was thus set to 500 steps
in the present study.
Evaluation of the MF is also needed in the TI method.
The MF at each of the grid points in the TI calculation was obtained by
averaging the instantaneous forces from a set of many FPMD runs with
Eqs. (\ref{etot}) and (\ref{hp}).
The successive simulation started with random atomic distortions from the
previous atomic configuration. In the present study, we have performed
120 FPMD runs, each consisting of 600 FPMD steps, i.e., 72,000 FPMD steps in total
for each grid point of the reaction coordinate.
60,000 FPMD steps out of the 72,000 steps were devoted to estimation of the MF
at a single grid point.
For a set of given coordinates $(\phi, \psi)$, as shown in Fig.~\ref{chart},
$-\partial F/\partial \phi$ and $-\partial F/\partial \psi$
were estimated for the hypothetical dynamics given
by Eqs. (\ref{eqmfd}) and (\ref{eqmfdeta}) with $T_{X}=300$ K in FP-LogMFD.
A single Nos\'e-Hoover thermostat\cite{Nose1984,Morishita2010,Hoover1985}
was used.
In the FP-LogMFD runs,
the variables of $\phi$, $\psi$, and $\eta$ were updated
using a time step of 1 $\tau$, with the masses of \textcolor{black}{$M_{\phi(\psi)}=1.7\times 10^{4}$ (kcal/mol)/(deg/$\tau^{2}$)}
and $Q_{\eta}=L k_{\rm B}T_{\rm X} \tau_{\eta}^{2}$ with $\tau_{\eta}=50$ $\tau$,
where $\tau$ represents the time unit.
[the time unit can, in fact, be arbitrarily chosen, e.g., $\tau$=1 fs,
since the dynamics of $\phi$ has nothing to do with the resultant
$F(\phi)$.]
The parameters of $\alpha$ and $\gamma$, which determines the degree
of the effective reduction of the free energy
barriers, were taken as $\alpha=3$ (kcal/mol)$^{-1}$ and $\gamma=1/\alpha$,
with this value of $\gamma$ corresponding to 170 K.
After solving Eqs. (\ref{eqmfd}) and (\ref{eqmfdeta}),
the conversion to $F(\phi, \psi)$ was performed
using Eq. (\ref{freeenergy}) with
$\hat{H}_{\rm LogMFD}=1$ kcal/mol.
$\hat{H}_{\rm LogMFD}$ should be
set to ensure $\alpha F_{\rm min}(\phi, \psi) + 1 > 0$,
where $F_{\rm min}$ is the minimum of the free energy.
Note however that there is, in principle, no upper limit for
the value of $\hat{H}_{\rm LogMFD}$.\cite{Morishita2013}
The validity of the LogMFD results mainly depends
on the accuracy of the MF,
which influences the conservation of $\hat{H}_{\rm LogMFD}$ [Eq. (\ref{HMFD2})].
As was already mentioned, the quality of the MF
can be controlled by $N_{\rm BP}$ and
the mass parameter $M_{\phi(\psi)}$.\cite{Morishita2013}
The increase of
$M_{\phi(\psi)}$, which reduces (suppresses) the velocity of
the dynamical variables,
results in a more accurate profile for the MF, and thus, the free energy profile.
We found, by decreasing the $M_{\phi(\psi)}$ by the factor ten,
that the difference between the LogMFD and TI results becomes
0.22 kcal/mol from 0.18 kcal/mol on average.
In the present system, the periodicity with respect to $\phi$ and $\psi$ can be available
for checking the accuracy of simulations.
\subsection{\label{subsec:1d}One dimensional profile}
\begin{figure}
\includegraphics[width=7.5cm]{fig6-1.eps}
\includegraphics[width=7.5cm]{fig6-2.eps}
\caption{\label{every}
(a) MF
with respect to the dihedral angle $\psi$
in the glycine dipeptide molecule, constraining the other dihedral
angle $\phi$ to
\textcolor{black}{
$-$80$^{\circ}$
}.
The blue curve and red symbols indicate the MF calculated
from the LogMFD and TI calculations, respectively.
(b) The magnified profile of the MF (vibrational curve)
around
\textcolor{black}{
$\psi=-76.8^\circ$},
with a smooth curve showing
the profile obtained
by averaging over ten MFD time steps.
The arrow indicates the width of the mesh used in the
TI calculation,
showing that the MF around this $\psi$ range is approximated
by only a single grid-point result.
}
\end{figure}
\begin{figure}
\includegraphics[width=7.5cm]{fig7.eps}
\caption{\label{logmfd-ti}
Free energy profiles
with respect to the dihedral angle $\psi$ in the glycine
dipeptide molecule, constraining the other dihedral angle
$\phi$ to
\textcolor{black}{
$-$80$^{\circ}$},
obtained from
the LogMFD (blue curve) and TI
(red dots) calculations.
The logarithmic energy ($\gamma \log (\alpha F(\psi)+1$) )
(magenta curve) is also presented for comparison, indicating
a substantial reduction of the free energy barrier.}
\end{figure}
For demonstrating the free energy evaluation using FP-LogMFD,
we performed FP-LogMFD simulations for the dynamical
variable $\psi$ while
keeping $\phi$ to be
\textcolor{black}{$-$80$^{\circ}$}.
In Fig.~\ref{every}(a), the MF profiles from the LogMFD and
TI calculations are presented, showing the LogMFD result
is in good agreement with the TI result.
Figure \ref{every}(a) also shows that there are regions where
the MF drastically varies
in a narrow range, e.g.,
\textcolor{black}{
$\psi = -115^\circ \sim -57.3^\circ$}.
In Fig.~\ref{every}(b), the magnified profile in the range of
\textcolor{black}{
$-78.5^\circ \leq \psi \leq -74.9^\circ$}
indicates
that, although the data by LogMFD shows a
vibrational behavior,
the MF averaged over 10 MFD steps varies smoothly.
This behavior of the MF in LogMFD is remarkable
when the profile
exhibits a rapid variation.
As shown in Fig.~\ref{every}(b),
a set of uniformly sparse grid points is only used
in the TI method due to a limited computational resources.
LogMFD thus can provide missing data in between
each of the grid points
in the TI calculations without much additional
computational cost.
Figure~\ref{logmfd-ti} shows the free energy profiles
obtained by the LogMFD and TI methods.
Each of the free energy profiles is shifted to have the same value
(5 kcal/mol) at
\textcolor{black}{
$\psi=-180^\circ$}
for comparison in Fig. ~\ref{logmfd-ti}.
LogMFD runs were initiated at
\textcolor{black}{
$\psi=57.3^\circ$}
(around the minimum) to either direction (with increasing or
decreasing $\psi$) with $T_{\rm X}=300$ K and
were ended at
\textcolor{black}{
$\psi=92^\circ$
}
after passing through the periodic
boundary at
\textcolor{black}{
$180^\circ$ or $-180^\circ$}.
It should be remarked that the value of
\textcolor{black}{
$F(\psi=92^\circ)$}
estimated when
\textcolor{black}{
$\psi=92^\circ$}
was sampled for the first time is almost the same
as the
\textcolor{black}{
$F(\psi=92^\circ)$
}
estimated when
\textcolor{black}{
$\psi=92^\circ$} was sampled the second time,
indicating the energy dissipation, which degrades the accuracy of $F(\psi)$, is negligible.
There is the maximum at
\textcolor{black}{
$\psi=-77.9^\circ$},
and the minimum at
\textcolor{black}{
$\psi=63.0^\circ$}
in the profile,
as shown in Fig.~\ref{logmfd-ti}.
We stress here that the
dynamics for the reaction coordinate $\psi$ was very smooth,
even the large energy barrier exists.
The difference between the minimum and maximum free
energy approximately amounts to 8 kcal/mol,
corresponding to about 4024 K.
This energy difference was entirely suppressed by
the logarithmic form.
Figure~\ref{logmfd-ti} also shows the effective potential curve
of $\gamma \log ( \alpha F(\psi) + 1 )$,
indicating that the actual energy barrier for $\psi$
became $\sim$ 0.8 kcal/mol, comparable to 402 K.
Such a substantial reduction of the energy barrier can be
controlled by the parameters ($\alpha$ and $\gamma$).
\begin{figure}
\includegraphics[width=7.5cm]{fig8.eps}
\caption{\label{distance}
Atomic distances for O(1)-H(7), O(1)-O(2), and O(1)-N(2)
as a function of $\psi$.}
\end{figure}
Before proceeding to the two dimensional landscape, we discuss
the one-dimensional free energy profile in more detail.
As pointed out, there are the minimum and maximum
in the profile.
The former and latter are related to a hydrogen bond and
a rendezvous of a pair of the oxygen atoms in the peptide bonds, respectively.
Other characteristic properties are found around
\textcolor{black}{
$\psi=0^\circ$ and $140^\circ$},
where the free energy shows
\textcolor{black}{
a profile with zero curvature}.
We consider that this is due to
breaking of the hydrogen bond which is formed
around
\textcolor{black}{
$\psi=57.3^\circ$}.
This consideration is supported by the fact that
the MF in the corresponding part
\textcolor{black}{is}
almost zero
(see Fig.~\ref{every}).
Atomic distances as a function of $\psi$ are shown
in Fig.~\ref{distance}. From this figure,
the free energy minimum in Fig.~\ref{logmfd-ti}
is found to appear around the minimum distance of O(1)-H(7) and
O(1)-N(2), while the energy maximum appears around the minimum
distance of O(1)-O(2). The latter case may correspond to a
large electric dipole state for the molecule.
The distance of 2.5 $\sim$ 3 \AA\ for O(1)-H(7) at
\textcolor{black}{
$\psi=0^\circ$ and $120^\circ$}
is out of the range of the hydrogen bonding,
where the MF is $\sim$ 0.
From this, we consider
that an energy of about 3 kcal/mol is gained by the hydrogen bond
(see Fig.~\ref{logmfd-ti}).
This energy is comparable to a typical bonding energy of the hydrogen
bond (3 $\sim$ 10 kcal/mol) reported
in a literature.\cite{text-book}
\subsection{\label{subsec:2d}Two dimensional profile}
\begin{figure}
\includegraphics[width=8.0cm]{fig9.eps}
\caption{\label{freemap2}
(a) Free energy contour map $F(\phi,\psi)$
and (b)--(e) typical atomic configurations at three stable
states (C5 and C7 atomic configurations) and an unstable state.
The white bullets indicate the positions for the stable states.
A pass way that approximately connects these stable states
with a straight line is displayed in a white dashed line
(see also Fig. \ref{feC5C7}).
}
\end{figure}
\begin{figure}
\includegraphics[width=7.5cm]{fig10.eps}
\caption{\label{feC5C7}
The one dimensional free energy profile along the line
that approximately connects the C5 and C7 states.
The full and dotted curves represents the bare and symmetrized plot of
$F(\phi,\psi)$.}
\end{figure}
For constructing the two dimensional free energy profile, the dynamical
equations for both of
$\phi$ and $\psi$ were used.
Supposing
that the free energy minimum in the $\phi$-$\psi$ space may be lower than
that in the one-dimensional $\psi$ space at
\textcolor{black}{
$\phi=-80^\circ$},
$\hat{H}_{\rm LogMFD}$ was
increased to 1.2 kcal/mol to
shift the baseline of the free energy landscape.
The temperature $T_{X}$ and $\alpha$ were chosen to be
\textcolor{black}{
small values};
$T_{X}=200$ K and $\alpha$ =2 (kcal/mol)$^{-1}$
\textcolor{black}{
to suppress numerical errors in longer simulations},
while
the other parameters took the same values as used
in the one dimensional FP-LogMFD calculations.
The two-dimensional FP-LogMFD runs were started from the minimum of the one dimensional
profile of $F(\psi)$
and were extended to four directions.
The branch off can be performed from one simulation to others.
These simulations can be performed independently, implying
that parallel treatment is highly effective in LogMFD.\cite{Morishita2013}
Figure~\ref{freemap2} shows the free energy contour map
in the $\phi$-$\psi$ plane with typical molecular configurations.
The glycine dipeptide molecule has an intrinsic mirror symmetry in its
atomic geometry. Atomic structures which are related to each other
by the mirror operation with respect to the N(1)-C(3)-C(4) plane
has the same energy in gas phase. This feature should also be seen in
the free energy landscape $F(\phi, \psi)$.
Therefore, the statistical errors can be reduced by symmetrizing
the two-dimensional free energy
with respect to the point of
\textcolor{black}{
$(\phi, \psi)=(180^\circ, 180^\circ)$}
(there is the inversion symmetry in the map).
A non-symmetrized free energy profile
along the white dashed line in Fig. \ref{freemap2}
is presented in the last paragraph in this subsection.
The free energy landscape (Fig. \ref{freemap2}) shows that there are three stable states
(three energy valleys) and a series of unstable states (energy mountains).
The most stable state appears around
\textcolor{black}{
$(\phi, \psi)=(180^\circ, 180^\circ)$} ,
whose atomic configuration is presented in Fig.~\ref{glycine}
(or Fig.~\ref{freemap2}(d)).
This is assigned to the C5 configuration\cite{Cheam1989} and
is stabilized by the hydrogen bond, the five-membered ring,
and the configuration with separated oxygen atoms (almost zero electric dipole).
The other two stable states, found around
\textcolor{black}{
$(\phi, \psi)=(288^\circ, 88^\circ),
(72^\circ, 272^\circ)$},
are assigned to the C7 configuration and are also
stabilized with the hydrogen bond, the seven-membered ring,
and the configuration with moderately separated oxygen atoms
(small electric dipole).
The free energy for the C7 configuration is higher by 0.58 kcal/mol than that
for the C5 configuration.
This energy difference is quite small and comparable to 290 K.
The total (internal) energy computation also indicates that the C5
configuration is either lower in energy than the C7 by 0.38 kcal/mol, while
the work by the quantum chemistry calculation reports that
the C5 is \textcolor{black}{either lower than the C7 by 0.58 kcal/mol,\cite{Quentin1992} or higher by 1
and 0.58 kcal/mol.\cite{Fujitani2009,Klimkowski1985}
}
The atomic configuration of the most unstable state is presented
in Fig.~\ref{freemap2}(b).
This instability comes from an assemble of oxygen atoms in the molecule
(implying a large electric dipole).
The energy barrier measured from the bottom of the free energy landscape
(highest energy mountain)
amounts to 26 kcal/mol, corresponding to
13000 K and to 730 K with
$ \gamma {\rm log} (\alpha F+1)$. Again,
LogMFD enables to sample such higher energy configuration
in the same footing used around the ground state.
It is interesting to see the transition from the most stable state
to another stable state. The one dimensional free energy profile
roughly linking the C7 and C5 configurations
is presented in Fig.~\ref{feC5C7}, as a typical energy profile.
For simplicity,
the pass way of reaction coordinate was assumed to
be along the straight line which connects the two states near
the C5 and C7 states in the $\phi$-$\psi$ plane, as specified
in Fig.~\ref{freemap2}.
From Fig. \ref{feC5C7}, the energy barrier between C5 and C7
configurations is estimated to be about 3 kcal/mol
when measured from the C5 configuration.
The energy differences between the C5 and C7 states shown in Fig.~\ref{feC5C7}
are about 1 kcal/mol. These values are slightly larger
than the value reported above (0.58 kcal/mol)
because of the approximate pass way (this approximation causes the uncertainty
of about 0.4 kcal/mol).
\subsection{\label{subsec:efficiency}Computational efficiency}
In constructing the free energy profile (Fig. \ref{logmfd-ti}),
4 $\times$ 10$^{6}$ FPMD steps were devoted in
the FP-LogMFD calculation,
while 7.2 $\times$ 10$^{6}$ FPMD steps were needed in the TI calculation.
About 45 \% of the computational cost was saved.
This demonstrates a good efficiency of LogMFD
in the computational cost.
In addition, in the course of the construction of the two dimensional
profile (Fig.~\ref{freemap2}), we carried out a set of LogMFD runs which,
in total, sampled
1.2 $\times$ 10$^{8}$ FPMD steps (configurations).
Even though the accuracy in the two dimensional profile may be slightly reduced,
the computational cost is only 30 times larger than
that in the one dimensional calculation.
\section{\label{sec:results}DISCUSSIONS}
\begin{figure}
\includegraphics[width=7.5cm]{fig11.eps}
\caption{\label{dft}
Total (internal) energy
for the glycine dipeptide molecule as a function
of $\psi$ keeping
\textcolor{black}{
$\phi=-80^{\circ}$},
which was
obtained from the density functional theory (DFT)
calculation
with (purple triangle symbols) or without (green circle symbols)
the van der Waals(vdW) correction.
The blue asterisks denote the difference between the free energy
obtained by LogMFD and that by TI, while the red squares
denote the difference between the free energy
by LogMFD and the internal energy without the vdW correction.
}
\end{figure}
As mentioned in Sec. \ref{subsec:molecule}, the peptide bond takes
the {\it trans}- or {\it cis}-form. In our simulations,
the {\it trans}-form has been entirely observed and the statistical sampling
of the {\it cis}-form has been missed. This is because the barrier between
the {\it trans}- and {\it cis}-forms may be extremely high,
and also because the reaction coordinates chosen in the present LogMFD
calculations may not be suitable for sampling the {\it cis}-form.
If one needs to sample the {\it cis}-form, incorporation of additional
reaction coordinates is of use, which is easily realized in LogMFD.
It is interesting to see the contribution of the entropy in the free energy.
We have calculated the total energy (the internal energy)
as a function of $\psi$ keeping
\textcolor{black}{
$\phi=-80^{\circ}$},
as shown in Fig.~\ref{dft}.
The grid points used in the calculation of the internal energy
are the same as used in the TI calculation.
The internal energy profile is very similar to the free energy profile
(Fig.~\ref{logmfd-ti}).
The difference between the free energy and the internal energy
is found to be within 0.5 kcal/mol
(if the energy scale is adjusted to give zero entropy at
\textcolor{black}{
$\psi=-180^\circ$}), implying a small contribution from the entropy.
We roughly estimated the uncertainty of the free energy as $\sim$ 0.4 kcal/mol,
which is comparable to the variation of the entropy with $\psi$.
It is thus considered that the entropic contribution is hardly changed
with $\psi$.
This is not surprising because
the number
of possible conformations in the present system is relatively small, which
does not significantly depend on the dihedral angles.
Also, the glycine dipeptide molecule is in vacuum, not in a solvent.
We however stress that LogMFD is able to unveil the variation of the entropy, if any,
which is, for example, seen in our preliminary calculations for a model system
of protein-G consisting of 56 amino acids.\cite{Isobe2001}
The free energy profile for the glycine dipeptide molecule
was previously obtained using \textcolor{black}{classical LogMFD with} an empirical
force field.\cite{Morishita2012,Morishita2013}
The profile is similar to that obtained using FP-LogMFD in this work,
indicating the validity of the empirical force-field to some extent.
There are, however, some differences in the profile.
As pointed out in Sec. \ref{subsec:1d}, we observe the
\textcolor{black}{
zero curvature}
around
\textcolor{black}{
$\psi=0^\circ$ and $140^\circ$}.
This behavior is also seen in the internal energy profiles
(Fig. \ref{dft}) in the FP-LogMFD approach.
In fact, the explicit inclusion of the van der Waals
interaction\cite{Dion2004,Cooper2010,Obata2013}
into
\textcolor{black}{
the DFT(GGA)}
calculations
does not change the overall profile of the internal energy
\textcolor{black}{
(note that the binding energy is underestimated using GGA)}.
It is thus considered that the
\textcolor{black}{
zero curvature}
is not attributed to an inappropriate DFT description,
while the linear behavior observed around
\textcolor{black}{
$\psi=0^\circ$ and $140^\circ$}
in the previous results
may come from insufficient transferability of the empirical force field.
\section{\label{sec:summary}SUMMARY}
We have demonstrated that the {\it ab initio} based MF
can be incorporated
into the LogMFD method, which improves the reliability
and accuracy in the free energy calculation.
FP-LogMFD has been applied to reconstruction
of the free energy landscapes of the glycine dipeptide
molecule, and the C5 and C7
conformations have been identified as the ground and metastable conformations,
respectively.
It has been confirmed that the substantial reduction of the free energy barriers,
thanks to the logarithmic form,
enables us to efficiently reconstruct the free energy profile,
which was found to agree well with that
obtained by the TI method.
The free energy profile from the first-principles approach
indicates that the empirical force field for the glycine dipeptide molecule
is sufficient to obtain the overall profile of the free energy landscape.
The LogMFD method allows us not only to easily sample rare events,
but also to reconstruct the free energy profile ``{\it on-the-fly}"
without suffering from the problems such as how to arrange the grid points
or how to perform the numerical integration (as postprocessing) in TI.
It has been demonstrated in the present study that
free energy profiles using {\it ab initio} force field can be
reconstructed with less computational cost than is needed in
the TI method.
The FP-LogMFD method developed here is thus a promising tool for
reconstructing free energy profiles, especially those
in which accurate descriptions for interatomic interactions are required.
\begin{acknowledgments}
The computation in this work was done using the facilities of
the Supercomputer Center, Institute for Solid State Physics,
University of Tokyo and the facilities of the Research Center for
Computational Science, National Institutes of Natural Sciences, Okazaki, Japan.
This work was partly supported by Grant-in-Aid for Scientific
Research from JSPS/MEXT (Grant Nos. 22104012, 22340106, 23510120 and 24740297)
and the Computational Materials Science Initiative (CMSI), Japan.
\end{acknowledgments}
|
1,116,691,499,390 | arxiv | \subsubsection{The Second-Level Disengagement}
\begin{figure}[t]
\centering
\subfigure[Clusters $Y_{(a,a')}$ and its child-cluster $Y_{(b,b')}$ (clusters $C_a$ and $C_{a'+1}$ do not belong to $Y_{(a,a')}$).]{\scalebox{0.15}{\includegraphics{figs/instance_for_level_2_1.jpg}}\label{fig: path_cluster_H}
}
\subfigure[The layout of graph $G_{Y_{(a,a')}}$.]{\scalebox{0.15}{\includegraphics{figs/instance_for_level_2_2.jpg}}\label{fig: path_cluster_G}}
\caption{An illustration of an instance $G_{Y_{(a,a')}}$ after the first-level disengagement.
\end{figure}
\subsubsection*{Step 1. Constructing Inner and Outer Paths}
\subsubsection*{Step 2. Constructing Sub-Instances}
\newpage
\section{Proofs Omitted from \Cref{sec: high level}}
\subsection{Proof of \Cref{claim: bound by level}}
\label{Appx: inductive bound proof}
The proof is by induction on $h(I)$. The base case is when $h(I)=0$, so $v(I)$ is a leaf vertex of $T^*$, and hence of $T$. Denote $I=(G,\Sigma)$. From \Cref{obs: leaf}, either $|E(G)|\leq \mu^{c''}$; or $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$; or $\mathsf{OPT}_{\mathsf{cnwrs}}(I)> |E(G)|^2/\mu^{c''}$.
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$, then the algorithm returns a solution of cost $0$. Otherwise, $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq 1$, and, if $|E(G)|\leq \mu^{c''}$, then the algorithm returns the trivial solution, of cost at most $|E(G)|^2\leq |E(G)|\cdot \mu^{c''}$. Lastly, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)>|E(G)|^2/\mu^{c''}$, then, since the trivial solution $\phi'$ is considered by the algorithm, it returns a solution of cost at most $|E(G)|^2\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \mu^{c''}$.
Assume now that the claim holds for all vertices $v(I)$ of $T^*$ with $h(I)<q$, for some $0<q\leq \mathsf{dep}(T)$. Consider any vertex $v(I)$ of the tree $T^*$ with $h(I)=q$. Let $v(I_1),\ldots,v(I_k)$ be the child vertices of $v(I)$ in the tree $T^*$. Denote $I=(G,\Sigma)$ and $|E(G)|=m$. Additionally, for all $1\leq r\leq k$, denote $I_r=(G_r,\Sigma_r)$ and $m_r=|E(G_r)|$.
Since instance $I$ is not a leaf instance of $T$, $|E(G)|\geq \mu^{c''}$ must hold.
Since we have assumed that event ${\cal{E}}$ does not happen, either $\mathsf{OPT}_{\mathsf{cnwrs}}(I)> |E(G)|^2/\mu^{c''}$, or $\sum_{r=1}^k\mathsf{OPT}_{\mathsf{cnwrs}}(I_r)\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot 2^{c_g(\log m)^{3/4}\log\log m}$ must hold.
In the former case, the algorithm is guaranteed to return a solution to $I$ whose cost is at most $|E(G)|^2\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \mu^{c''}$, since the trivial solution $\phi'$ is considered as one of the possible solutions. From now on we focus on latter case. For all $1\leq r\leq k$, let $\phi_r$ be the solution to instance $I_r$ that the algorithm computes recursively. From the induction hypothesis, for all $1\leq r\leq k$:
\[\mathsf{cr}(\phi_r)\leq 2^{\tilde c\cdot h(I_r)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c''\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I_r)+(\log m^*)^{4c_g h(I_r)}\mu^{2c''\cdot \tilde c}\cdot m_r.\]
Notice that, for all $1\leq r\leq k$, $h(I_r)\leq q-1$. Moreover, from \Cref{thm: main}, $\sum_{r=1}^km_r\leq m\cdot(\log m)^{c_g}\leq m\cdot(\log m^*)^{c_g}$. Lastly, as noted already, $\sum_{r=1}^k\mathsf{OPT}_{\mathsf{cnwrs}}(I_r)\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot 2^{c_g(\log m)^{3/4}\log\log m}$ must hold.
Altogether, we get that:
\[
\begin{split}
\sum_{r=1}^k\mathsf{cr}(\phi_r)&\leq 2^{\tilde c\cdot (q-1)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c''\cdot c_g}\cdot \sum_{r=1}^k\mathsf{OPT}_{\mathsf{cnwrs}}(I_r)+(\log m^*)^{4c_g (q-1)}\mu^{2c''\cdot \tilde c}\cdot \sum_{r=1}^km_r
\\
&\leq 2^{\tilde c\cdot (q-1)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c''\cdot c_g}\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot 2^{c_g(\log m)^{3/4}\log\log m} \\
&\hspace{6cm}+(\log m^*)^{4c_g (q-1)}\mu^{2c''\cdot \tilde c}\cdot m\cdot(\log m^*)^{c_g}\\
&\leq 2^{\tilde c\cdot (q-0.5)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c''\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ (\log m^*)^{4c_gq-3c_g}\mu^{2c''\cdot \tilde c}\cdot m\\
&\hspace{6cm}+2^{\tilde c\cdot (q-0.5)\cdot (\log m^*)^{3/4}\log\log m^*} \cdot\mu^{c''\cdot c_g}\cdot m.
\end{split}\]
Since $q\leq \mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$ from \Cref{obs: few recursive levels},
the last term is bounded by:
\[ 2^{\tilde c\cdot (\log m^*)^{7/8}/c^*} \cdot\mu^{c''\cdot c_g}\cdot m \leq \mu^{2c''\cdot c_g}\cdot m\]
(since $\mu =2^{c^*(\log m^*)^{7/8}\log\log m^*}$). Therefore, we get that:
\[\sum_{r=1}^k\mathsf{cr}(\phi_r)\leq 2^{\tilde c\cdot (q-0.5)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c''\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ (\log m^*)^{4c_gq-2c_g}\mu^{2c''\cdot \tilde c}\cdot m. \]
The solution that our algorithm returns for instance $I$ is obtained by applying Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace from \Cref{thm: main} to solutions $\phi_1,\ldots,\phi_k$ to instances $I_1,\ldots,I_k$ (or some other solution the algorithm considers, if its cost is smaller). Since event ${\cal{E}}$ does not happen, the cost of the resulting solution is bounded by:
\[
\begin{split}
&c_g\cdot\bigg(\sum_{r=1}^k\mathsf{cr}(\phi_r)\bigg) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{c_g}\\
&\hspace{1cm}\leq c_g\cdot 2^{\tilde c\cdot (q-0.5)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c''\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+\mu^{c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)\\
&\hspace{3cm}+ c_g\cdot(\log m^*)^{4c_gq-2c_g}\mu^{2c''\cdot \tilde c}\cdot m+\mu^{c_g}\cdot m\\
&\hspace{1cm}\leq 2^{\tilde c\cdot q\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c''\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+(\log m^*)^{4c_gq}\mu^{2c''\cdot \tilde c}\cdot m,
\end{split}
\]
as required.
\subsection{Proof of \Cref{claim: combine drawings}}
\label{Appx: Proof of combine drawings}
We construct the solution $\phi(I)$ to instance $I$ in three steps.
In the first step, we compute a solution $\phi(I')$ to every instance $I'\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$, as follows. Consider any instance $I'=(G',\Sigma')\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$. Recall that we are given a solution $\phi(I'')$ to every instance $I''\in \overline{\mathcal{I}}(I')$. Recall also that $\overline{\mathcal{I}}(I')$ is a $\nu$-decomposition of instance $I'$. We apply the efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace(\overline{\mathcal{I}}(I'))$ from the definition of $\nu$-decomposition to the drawings in set $\set{\phi(I'')}_{I''\in \overline{\mathcal{I}}(I')}$, to compute a feasible solution $\phi(I')$ to instance $I'$, of cost $\mathsf{cr}(\phi(I'))\leq O\textsf{left} (\sum_{I''\in \overline{\mathcal{I}}(I')}\mathsf{cr}(\phi(I''))\textsf{right} )$.
Overall, we get that:
\begin{equation}\label{eq: solutoins to narrow instances}
\sum_{I'\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}}\mathsf{cr}(\phi(I'))\leq \sum_{I''\in {\mathcal{I}}^*}O(\mathsf{cr}(\phi(I''))).
\end{equation}
We have now obtained a solution $\phi(I')$ to every instance $I'\in \big(\hat{{\mathcal{I}}}_{\textsf {small}}\cup \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}_{\textsf {small}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}\big)$.
In the second step, we compute a solution $\phi(I')$ to every instance $I'\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}$.
Consider any such instance $I'\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}$.
Recall that we applied the algorithm from \Cref{lem: many paths} to instance $I'$, to obtain a collection $\tilde {\mathcal{I}}(I')$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace. Every instance in the resulting collection belongs to $\tilde {\mathcal{I}}_{\textsf{small}}$ or to $ \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$.
We use Algorithm $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace'$, that is guaranteed from \Cref{lem: many paths}, to compute a solution $\phi(I')$ to instance $I'$.
Since we have assumed that event ${\cal{E}}_2$ did not happen, the cost of the solution is bounded by:
$\mathsf{cr}(\phi(I')) \leq \sum_{\tilde I=(\tilde G, \tilde \Sigma)\in \tilde {\mathcal{I}}(I')}\mathsf{cr}(\phi(\tilde I)) + \mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot\mu^{c'_g}$.
Overall, we get that:
\begin{equation}\label{eq: solutions to wide not connected}
\sum_{I'\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}}\mathsf{cr}(\phi(I'))\leq \sum_{I''\in {\mathcal{I}}^*}O(\mathsf{cr}(\phi(I''))+\sum_{I'\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot\mu^{c'_g}.
\end{equation}
We now describe the third step. We have so far obtained a solution $\phi(I')$ to every instance $I'\in \big(\hat{{\mathcal{I}}}_{\textsf {small}}\cup \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup
\hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}\big)$, that is, a solution to every instance in $\hat {\mathcal{I}}$.
Recall that $\hat {\mathcal{I}}$ is a $\nu_1$-decomposition of the input instance $I$. Since we have assumed that Event ${\cal{E}}_1$ did not happen, $\sum_{I'\in \hat{\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 100\cdot\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \nu_1$. By combining Inequalities \ref{eq: solutoins to narrow instances} and \ref{eq: solutions to wide not connected}, we get that:
\begin{equation}\label{eq: solution for step 1}
\begin{split}
\sum_{I'\in \hat {\mathcal{I}}}\mathsf{cr}(\phi(I'))&\leq \sum_{I''\in {\mathcal{I}}^*}O(\mathsf{cr}(\phi(I''))+\sum_{I'\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot\mu^{c'_g}\\
&\leq \sum_{I''\in {\mathcal{I}}^*}O(\mathsf{cr}(\phi(I''))+100\cdot\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \nu_1\cdot \mu^{c'_g}\\
&\leq \sum_{I''\in {\mathcal{I}}^*}O(\mathsf{cr}(\phi(I''))+\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \mu^{O(1)},
\end{split}
\end{equation}
since $\nu_1= 2^{O((\log m)^{3/4}\log\log m)}$ and $\mu\gg \nu_1$.
Lastly, we apply the efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace(\hat {\mathcal{I}})$ that is guaranteed by the definition of $\nu_1$-decomposition to the solutions $\set{\phi(I')}_{I'\in \hat {\mathcal{I}}}$, to obtain a feasible solution $\phi(I)$ to instance $I$.
The cost of the solution is bounded by $\sum_{I'\in \hat {\mathcal{I}}}O(\mathsf{cr}(\phi(I')))\leq
\sum_{I''\in {\mathcal{I}}^*}O(\mathsf{cr}(\phi(I''))+\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \mu^{O(1)}$, as required.
\iffalse
If events ${\cal{E}}_1,{\cal{E}}_2,{\cal{E}}_3$ did not happen, then the cost of the solution is bounded by
\[\begin{split}
&\mathsf{cr}(\phi(I)) \leq O\bigg(\sum_{I'\in \hat {\mathcal{I}}}\mathsf{cr}(\phi(I'))\bigg) \\
&\leq O\bigg(\sum_{I'\in \hat {\mathcal{I}}_{\textsf {small}}}\mathsf{cr}(\phi(I'))+\sum_{I'\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}}\mathsf{cr}(\phi(I'))+\sum_{I'\in \hat {\mathcal{I}}^{(w)}_{\textsf {large}}}\mathsf{cr}(\phi(I'))\bigg)\\
&\leq O\bigg(\sum_{I'\in \hat {\mathcal{I}}_{\textsf {small}}}\mathsf{cr}(\phi(I'))+\sum_{I'\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}}\mathsf{cr}(\phi(I'))+\sum_{I'\in \hat {\mathcal{I}}^{(w)}_{\textsf {large}}}\bigg(\sum_{\tilde I\in \tilde
{\mathcal{I}}(I')}\mathsf{cr}(\phi(\tilde I))+\mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot\mu^{c'_g}\bigg)\bigg)\\
&\leq O\bigg(\sum_{I'\in \hat {\mathcal{I}}_{\textsf {small}}}\mathsf{cr}(\phi(I'))+\sum_{I'\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}}\mathsf{cr}(\phi(I'))+\sum_{I'\in \hat {\mathcal{I}}^{(w)}_{\textsf {large}}}\sum_{\tilde I\in \tilde
{\mathcal{I}}(I')}\mathsf{cr}(\phi(\tilde I))\bigg)+100(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{c'_g}\cdot\nu_1\\
&\leq O\bigg(\sum_{I'\in \hat {\mathcal{I}}_{\textsf {small}}}\mathsf{cr}(\phi(I'))+\sum_{\tilde I\in \tilde
{\mathcal{I}}_{\textsf {small}}}\mathsf{cr}(\phi(\tilde I))+\sum_{I'\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}\cup \tilde {\mathcal{I}}^{(n)}_{\textsf {large}}}\mathsf{cr}(\phi(I'))\bigg)+100(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{c'_g}\cdot\nu_1\\
&\leq O\bigg(\sum_{I'\in \hat {\mathcal{I}}_{\textsf {small}}}\mathsf{cr}(\phi(I'))+\sum_{\tilde I\in \tilde
{\mathcal{I}}_{\textsf {small}}}\mathsf{cr}(\phi(\tilde I))+\sum_{I'\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}\cup \tilde {\mathcal{I}}^{(n)}_{\textsf {large}}}O\bigg(\sum_{I''\in \overline{\mathcal{I}}(I')}\mathsf{cr}(\phi(I''))\bigg)\bigg)+100(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{c'_g}\cdot\nu_1\\
&\leq O\textsf{left} (\sum_{I''\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I'))\textsf{right} ) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{O(1)}.
\end{split}
\]
\fi
\subsection{Proof of \Cref{obs: bound sum of opts}}
\label{Appx: Proof of bound sum of opts}
Throughout the proof, we assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$ and bad event ${\cal{E}}$ did not happen.
Since event ${\cal{E}}_1$ does not happen:
\begin{equation}\label{eq: bound on opts step 1}
\sum_{I'\in \hat{\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 100\nu_1\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m).
\end{equation}
Recall that $\hat {\mathcal{I}}=\hat{{\mathcal{I}}}_{\textsf {small}}\cup \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup
\hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}$. In Step 2 of the algorithm, applied the algorithm from \Cref{lem: many paths} to every instance $I'=(G',\Sigma')\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}$, to compute a collection $\tilde {\mathcal{I}}(I')$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace. Consider now any such instance $I'\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}$. Since we have assumed that Event ${\cal{E}}_2$ did not happen:
\[
\sum_{\tilde I\in \tilde {\mathcal{I}}(I')}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)\le \mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot (\log |E(G')|)^{c'_g}\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot (\log m)^{c'_g}.
\]
Recall that we have defined $\tilde {\mathcal{I}}=\bigcup_{I'\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}}\tilde {\mathcal{I}}(I')$. Combining the above inequality with Equation \ref{eq: bound on opts step 1}, and recalling that $\nu_1=2^{O((\log m)^{3/4}\log\log m)}$, we get that:
\begin{equation}\label{eq: bound opts second}
\sum_{\tilde I\in \tilde {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)\le \sum_{I'\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot (\log m)^{c'_g}\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot 2^{O((\log m)^{3/4}\log\log m)}.
\end{equation}
Consider now an instance $I'=(G',\Sigma')\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$. Since we have assumed that event ${\cal{E}}_3$ does not happen, from the definition of a $\nu$-decomposition:
\[\expect{\sum_{I''\in \overline{\mathcal{I}}(I')}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} )\cdot \nu. \]
Recall that $\overline{\mathcal{I}}_{\textsf{small}}=\bigcup_{I'\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}}\overline{\mathcal{I}}(I')$. Recall also that, from Inequality \ref{eq: num of edges step 1},
$\sum_{I'=(G',\Sigma')\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}}|E(G')|\le m\cdot (\log m)^{c'_g}$,
and from Inequality \ref{ineq: total edges step 2},
$\sum_{I'=(G',\Sigma')\in \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}}|E(G')|\leq 2m\cdot (\log m)^{c'_g}$. Altogether, we get that:
\begin{equation}\label{eq: bound ops last step}
\begin{split}
\expect{\sum_{I''\in \overline{\mathcal{I}}_{\textsf{small}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}&\leq \sum_{I'=(G',\Sigma')\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}\cup \tilde {\mathcal{I}}^{(n)}_{\textsf {large}}} \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} )\cdot \nu\\
&\leq \sum_{I'\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}\cup \tilde {\mathcal{I}}^{(n)}_{\textsf {large}}} \mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot \nu +4m\cdot (\log m)^{c'_g}\cdot \nu\\
&\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot 2^{O((\log m)^{3/4}\log\log m)}
\end{split}
\end{equation}
(we have used Equations \ref{eq: bound on opts step 1} and \ref{eq: bound opts second} in order to bound $\sum_{I'\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')$ and $\sum_{I'\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')$, respectively, and the fact that $\nu_1,\nu\leq 2^{O((\log m)^{3/4}\log\log m)}$).
Finally, by combining Equations \ref{eq: bound on opts step 1}, \ref{eq: bound opts second} and \ref{eq: bound ops last step}, we get that:
\[
\begin{split}
\expect{\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}&\leq \expect{\sum_{I''\in \hat {\mathcal{I}}_{\textsf {small}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')+ \sum_{I''\in \tilde {\mathcal{I}}_{\textsf {small}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')+\sum_{I''\in \overline {\mathcal{I}}_{\textsf {small}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}\\
&\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot 2^{O((\log m)^{3/4}\log\log m)}.
\end{split}
\]
We denote this expectation by $\eta'$.
Let $\hat {\cal{E}}$ be the bad event that
$\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')>100\eta'$. From Markov inequality, $\prob{\hat {\cal{E}}\mid \neg{\cal{E}}}<1/100$.
\section{Proofs Omitted from \Cref{sec: high level}}
\subsection{Proof of \Cref{claim: bound by level}}
\label{Appx: inductive bound proof}
The proof is by induction on $h(I)$. The base case is when $h(I)=0$, so $v(I)$ is a leaf vertex of $T^*$, and hence of $T$. Denote $I=(G,\Sigma)$.
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$, then the algorithm returns a solution of cost $0$. Otherwise, $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq 1$, and, if $|E(G)|\leq \mu^{c'}$, then the algorithm returns the trivial solution, of cost at most $|E(G)|^2\leq |E(G)|\cdot \mu^{c'}\leq |E(G)|\cdot \mu^{\tilde c}$, assuming that $\tilde c\geq c'$. Lastly, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)>|E(G)|^2/\mu^{c'}$, then, since the trivial solution $\phi'$ is considered by the algorithm, it returns a solution of cost at most $|E(G)|^2\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \mu^{c'}$.
We assume that the claim holds for all vertices $v(I)$ of $T^*$ with $h(I)<q$, for some $q\geq 0$, and consider any vertex $v(I)$ of the tree $T^*$ such that $h(I)=q$. Let $v(I_1),\ldots,v(I_k)$ be the child vertices of $v(I)$ in the tree $T^*$. Denote $I=(G,\Sigma)$ and $|E(G)|=m$. Additionally, for all $1\leq r\leq k$, denote $I_r=(G_r,\Sigma_r)$ and $m_r=|E(G_r)|$.
Since $I$ is not a leaf instance of $T$, $|E(G)|\geq \mu^{c'}$ must hold.
Since we have assumed that ${\cal{E}}$ does not happen, from \Cref{thm: main}, either $\mathsf{OPT}_{\mathsf{cnwrs}}(I)> |E(G)|^2/\mu^{c'}$, or $\sum_{r=1}^k\mathsf{OPT}_{\mathsf{cnwrs}}(I_r)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot 2^{c_g(\log m)^{3/4}\log\log m} +m\cdot\mu^{c_g}$, and the algorithm $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace$, when given as input a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}^i_j$, returns a solution $\phi$ to instance $I$ with $\mathsf{cr}(\phi)\le c_g\cdot (\sum_{I'\in {\mathcal{I}}_j}\mathsf{cr}(\phi(I')) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{c_g}$. Notice that, in the former case, the algorithm is guaranteed to return a solution to $I$ whose cost is at most $|E(G)|^2\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \mu^{c'}$, since the trivial solution $\phi'$ is considered as one of the possible solutions. From now on we focus on latter case. For all $1\leq r\leq k$, let $\phi_r$ be the solution to instance $I_r$ that the algorithm computes recursively. From the induction hypothesis, for all $1\leq r\leq k$:
\[\mathsf{cr}(\phi_r)\leq 2^{\tilde c\cdot h(I_r)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c'\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I_r)+(\log m^*)^{4c_g h(I_r)}\mu^{2c'\cdot \tilde c}\cdot m_r.\]
Notice that, for all $1\leq r\leq k$, $h(I_r)\leq q-1$. Moreover, from \Cref{thm: main}, $\sum_{r=1}^km_r\leq m\cdot(\log m)^{c_g}\leq m\cdot(\log m^*)^{c_g}$. Lastly, as noted already, for some constant $\tilde c'$, $\sum_{r=1}^k\mathsf{OPT}_{\mathsf{cnwrs}}(I_r)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot 2^{\tilde c'(\log m)^{3/4}\log\log m} +m\cdot\mu^{c_g}$ must hold.
We assume that $\tilde c$ is a large enough constant, so that $\tilde c\geq 2\tilde c'$.
Altogether, we get that:
\[
\begin{split}
\sum_{r=1}^k\mathsf{cr}(\phi_r)&\leq 2^{\tilde c\cdot (q-1)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c'\cdot c_g}\cdot \sum_{r=1}^k\mathsf{OPT}_{\mathsf{cnwrs}}(I_r)+(\log m^*)^{4c_g (q-1)}\mu^{2c'\cdot \tilde c}\cdot \sum_{r=1}^km_r
\\
&\leq 2^{\tilde c\cdot (q-1)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c'\cdot c_g}\cdot \textsf{left} ( \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot 2^{\tilde c'(\log m)^{3/4}\log\log m} +m\cdot\mu^{c_g}\textsf{right} )\\
&\hspace{6cm}+(\log m^*)^{4c_g (q-1)}\mu^{2c'\cdot \tilde c}\cdot m\cdot(\log m^*)^{c_g}\\
&\leq 2^{\tilde c\cdot (q-0.5)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c'\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ (\log m^*)^{4c_gq-3c_g}\mu^{2c'\cdot \tilde c}\cdot m\\
&\hspace{6cm}+2^{\tilde c\cdot (q-1)\cdot (\log m^*)^{3/4}\log\log m^*} \cdot\mu^{c_g}\cdot m.
\end{split}\]
Since $q\leq \mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$ from \Cref{obs: few recursive levels},
the last term is bounded by:
\[ 2^{\tilde c\cdot (\log m^*)^{7/8}/c^*} \cdot\mu^{c_g}\cdot m \leq \mu^{2c'\tilde c}\]
(since $\mu =2^{c^*(\log m^*)^{7/8}\log\log m^*}$). Therefore, we get that:
\[\sum_{r=1}^k\mathsf{cr}(\phi_r)\leq 2^{\tilde c\cdot (q-0.5)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c'\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ (\log m^*)^{4c_gq-2c_g}\mu^{2c'\cdot \tilde c}\cdot m. \]
The solution that our algorithm returns for instance $I$ is obtained by applying Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace from \Cref{thm: main} to solutions $\phi_1,\ldots,\phi_k$ to instances $I_1,\ldots,I_r$. Since event ${\cal{E}}$ does not happen, the cost of the resulting solution is bounded by:
\[
\begin{split}
&c_g\cdot\bigg(\sum_{r=1}^k\mathsf{cr}(\phi_r)\bigg) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{c_g}\\
&\hspace{1cm}\leq c_g\cdot 2^{\tilde c\cdot (q-0.5)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c'\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+\mu^{c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)\\
&\hspace{3cm}+ c_g\cdot(\log m^*)^{4c_gq-2c_g}\mu^{2c'\cdot \tilde c}\cdot m+\mu^{c_g}\cdot m\\
&\hspace{1cm}\leq 2^{\tilde c\cdot q\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c'\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+(\log m^*)^{4c_gq}\mu^{2c'\cdot \tilde c}\cdot m,
\end{split}
\]
as required.
\subsection{Phase 3: Petal-Based Disengagement and the Final Family of Instances}
\label{subsec: phase 3}
In this subsection we first compute a collection ${\mathcal{I}}_3$ of subinstances of the current instance $I=(G,\Sigma)$. These subinstances will ``almost'' have all properties required in \Cref{lem: decomposing problematic instances by petal}, except that we will not be able to guarantee that for each resulting subinstance $\tilde I=(\tilde G,\tilde E)$, $|E(\tilde G)|\leq m/(2\mu)$. However, we will guarantee that each such resulting graph $\tilde G$ does not have vertices whose degree is at least $m/\mu^4$. This fact will be used in the second part of this subsection in order to further decompose each subinstance of ${\mathcal{I}}_3$ into smaller subinstances. This will be done using an algorithm similar to that from Phase 1, except that now, since the instances we apply the algorithm to do not have high-degree vertices, we will not obtain any flower clusters, and so each subinstance obtained in this final phase will be sufficiently small.
The intuition for the current phase is that we would like to define a set of clusters in the current graph $G$, using the set ${\mathcal{X}}=\set{X_1,\ldots,X_k}$ of petals of the flower cluster, and then perform basic disengagement, described in \Cref{subsec: basic disengagement} on instance $I$ with the set ${\mathcal{X}}$ of clusters. Unfortunately, the set ${\mathcal{X}}$ of clusters is not laminar, as the clusters in ${\mathcal{X}}$ all share vertex $u^*$. In order to overcome this obstacle, the algorithm in this phase consists of three steps. In the first step we ``split'' the vertex $u^*$, by creating new vertices $u_1,\ldots,u_k$, each of which is then added to a distinct cluster $X_i$. We show that the optimal solution value to the resulting split instance that we construct is bounded by $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$, and that any solution to this new split instance can be efficiently transformed into a solution to the original instance, by only slightly increasing the solution cost. In the second step, we perform a basic disengagement of the new split instance using the modified set ${\mathcal{X}}$ of clusters. We show that each of the resulting instances does not contain vertices of degree at least $m/\mu^4$. We also bound the total solution cost and the total number of edges in the new instances. In the third and the final step we further decompose each resulting instance, exploiting the low degrees of its vertices. We now describe each of the steps in turn.
\subsubsection{Step 1: the Split Instance}
Recall that we are given an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, and a (modified) flower cluster $C\subseteq G$ with center $u^*$, and a set ${\mathcal{X}}=\set{X_1,\ldots,X_k}$ of petals, such that each petal is routable in $G$.
For all $1\leq i\leq r$, we let $E_i=E(X_i)\cap \delta_G(u^*)$, and we denote $E_i=(e_{i,1},\ldots,e_{i,q_i})$, where the edges are indexed according to their order in the ordering ${\mathcal{O}}_{u^*}\in \Sigma$; in other words, the ordering of the set $\delta_G(u^*)$ of edges in $\Sigma$ is: ${\mathcal{O}}_{u^*}=(e_{1,1},\ldots,e_{1,q_1},e_{2,1},\ldots,e_{2,q_2},\ldots,e_{k,1},\ldots,e_{k,q_k})$. For all $1\leq i\leq r$, we also let $\hat E_i=\delta_G(X_i)\setminus\delta_G(u^*)$, and we denote $|\hat E_i|=\hat q_i$; see \Cref{fig: original_petal} for an illustration.
Recall that, from Property \ref{prop: flower cluster routing} of a flower cluster, there is a set ${\mathcal{Q}}_i$ of edge-disjoint paths routing the edges of $\hat E_i$ to the edges of $E_i$, such that every inner vertex on every path lies in $X_i$, and, since petal $X_i$ is routable in $G$, there is a set ${\mathcal{Q}}'_i$ of paths in graph $G$, routing the edges of $\hat E_i$ to vertex $u^*$ such that the paths are internally disjoint from $X_i$ and cause congestion at most $3000$.
\begin{figure}[h]
\centering
\subfigure[An original petal $X_i$. Paths of ${\mathcal{Q}}_i$ are shown in pink and paths of ${\mathcal{Q}}'_i$ are shown in orange.]{\scalebox{0.4}{\includegraphics[scale=0.35]{figs/original_petal.jpg}}\label{fig: original_petal}
}
\hspace{0.4cm}
\subfigure[New cluster $X'_i$.
]{
\scalebox{0.34}{\includegraphics[scale=0.5]{figs/new_petal}}\label{fig: new_petal}}
\caption{Construction of a split instance $I'=(G',\Sigma')$.
}
\end{figure}
In order to define the new split instance $I'=(G',\Sigma')$, we start with a graph $G'=G\setminus\delta_G(u^*)$. We then add $k$ new vertices $u_1,\ldots,u_k$ to $G'$. Next, we process each index $1\leq i\leq k$ one by one. When index $i$ is processed,
we add a collection $A'_i=\set{a'_{i,1},\ldots,a'_{i, \hat q_i}}$ of $ \hat q_i$ parallel edges connecting $u^*$ to $u_i$ (recall that $\hat q_i=|\hat E_i|$). Additionally, for every edge $e_{i,j}=(u^*,x_{i,j})\in E_i$, we add a new edge $a_{i,j}=(u_i,x_{i,j})$ to graph $G'$; we view $a_{i,j}$ as a copy of edge $e_{i,j}$, and we will not distinguish between these edges. We denote $A_i=\set{a_{i,j}\mid 1\leq j\leq q_i}$. In order to complete the construction of graph $G'$, for every edge $e=(u,v)\not\in \delta_G(u^*)$ of the graph $G$ whose endpoints lie in different petals, we subdivide the edge $e$ with a new vertex $y_e$.
For all $1\leq i\leq k$, we let $X'_i$ be the subgraph of $G'$ induced by $(V(X_i)\setminus \set{u^*})\cup \set{u_i}$. Notice that graph $X'_i$ is completely identical to graph $X_i$, except that vertex $u^*$ is replaced by vertex $u_i$.
For all $1\leq i\leq k$, we denote by $\hat A_i=\delta_{G'}(X'_i)\setminus A'_i$, where $A'_i$ is the set of parallel edges connecting $u_i$ to $u^*$ (see \Cref{fig: new_petal}). It is easy to see that there is a one-to-one correspondence between edges of $\hat A_i$ in graph $G'$ and edges of $\hat E_i$ in graph $G$.
In order to complete the definition of the split instance $I'$, we need to define its corresponding rotation system $\Sigma'$. It is easy to verify that, every vertex $v\in V(G')\setminus \set{u^*,u_1,\ldots,u_k}$ whose degree in $G'$ is greater than $2$, $\delta_{G'}(v)=\delta_G(v)$ holds (we do not distinguish here between edges whose endpoints lie in different petals of $G$ and their subdivided counterparts). For each such vertex, we set the ordering ${\mathcal{O}}'_v\in \Sigma'$ of the edges of $\delta_{G'}(v)$ to be the same as the ordering ${\mathcal{O}}_v\in \Sigma$ of the edges of $\delta_{G}(v)$. Note that $\delta_{G'}(u^*)=A'_1\cup\cdots\cup A'_k$. We set the ordering ${\mathcal{O}}'_{u^*} \in \Sigma'$ of the edges of $\delta_{G'}(u^*)$ to be $(a'_{1,1},\ldots,a'_{1, \hat q_1},a'_{2,1},\ldots,a'_{2, \hat q_2},\ldots,a'_{k,1},\ldots,a'_{k, \hat q_k} )$. In other words, edges in sets $A'_1,\ldots,A'_k$ appear in this order of their sets, and within each set $A'_i$, the edges of $\set{a'_{i,j}}_{j=1}^{ \hat q_i}$ are ordered in the increasing order of index $j$.
Lastly, for all $1\leq i\leq k$, we define the ordering ${\mathcal{O}}'_{u_i}\in \Sigma'$ of the edges of $\delta_{G'}(u_i)=A'_i\cup A_i$ to be: $(a'_{i,1},a'_{i,2},\ldots,a'_{i, \hat q_i}, a_{i, q_i}, a_{i, q_i-1},\ldots, a_{i,1})$.
\begin{figure}[h]
\centering
\subfigure[Schematic view of graph $G'$ when $C$ is a $4$-petal flower cluster.]{\scalebox{0.37}{\includegraphics[scale=0.4]{figs/split_flower_1.jpg}
}
\hspace{0.1cm}
\subfigure[The ordering ${\mathcal{O}}'_{u^*}\in \Sigma'$ of edges of $\delta_{G'}(u^*)$.]{
\scalebox{0.4}{\includegraphics[scale=0.444]{figs/split_flower_3.jpg}}\label{fig: rotation at u^*}}
\caption{Split instance $I'=(G',\Sigma')$.
}\label{fig: split_flower}
\end{figure}
This completes the definition of the new split instance $I'=(G',\Sigma')$;
see \Cref{fig: split_flower} for an illustration.
We now establish some of its properties. We start with the following easy observation:
\begin{observation}\label{obs: split instance cost}
$|E(G')|\leq 4|E(G)|$, and $\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)$.
\end{observation}
The proof of \Cref{obs: split instance cost} is immediate. The first statement is immediate to see. For the second statement, given any solution $\phi$ to instance $I$, we can obtain a solution $\phi'$ to instance $I'$ by splitting the vertex $u^*$ to obtain images of vertices $u_1,\ldots,u_k$ and images of the edges in sets $A_1',\ldots,A_k'$ in a natural way (see \Cref{fig: split_drawing}), and subdividing images of edges whose endpoints lie in different petals of $G$.
\begin{figure}[h]
\centering
\subfigure[Before: the images of the original vertex $u^*$ and its incident edges in $\phi$
]{\scalebox{0.45}{\includegraphics[scale=0.53]{figs/split_drawing_1.jpg}
}
\hspace{2cm}
\subfigure[After: the images of the new vertices $u, u_1,\ldots,u_k$ and their incident edges in $\phi'$.
]{
\scalebox{0.45}{\includegraphics[scale=0.5]{figs/split_drawing_2.jpg}}}
\caption{Transforming a solution for instance $I$ into a solution for instance $I'$.}\label{fig: split_drawing}
\end{figure}
The next lemma shows that a solution to instance $I'$ can be transformed into a solution to instance $I$ while only slightly increasing the solution cost.
The proof uses arguments similar to those used in basic and advanced disengagement, but is somewhat tedious, and is deferred to \Cref{apd: Proof of solution to split to solution to original}.
\begin{lemma}\label{lem: solution to split to solution to original}
There is an efficient algorithm that, given a solution $\phi'$ to instance $I'$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq O(\mathsf{cr}(\phi'))$.
\end{lemma}
\subsubsection{Step 2: Disengagement of the Petals}
In this step, we consider the split instance $I'=(G',\Sigma')$ that was constructed in Step 1 of the current phase, and we will apply Algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace from \Cref{subsec: basic disengagement} to this instance, together with the family ${\mathcal{L}}=\set{X'_1,\ldots,X'_k}$ of clusters in order to perform a basic disengagement of these clusers, with a parameter $\beta=c(\log m)^{18}$, where $c$ is a large enough constant. Note that the clusters of ${\mathcal{L}}$ are disjoint, so ${\mathcal{L}}$ is a laminar family of clusters. In order to be able to use \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace, we need to define, for each cluster $X'_i$, a distribution ${\mathcal{D}}'(X'_i)$ over the set $\Lambda'(X'_i)$ of external routers for $X'_i$.
Consider some cluster $X'_i\in {\mathcal{L}}$. Recall that petal $X_i$ is routable in $G$, and so there is a set ${\mathcal{Q}}'_i$ of paths in $G$, routing the edges of $\hat E_i=\delta_G(X_i)\setminus\delta_G(u^*)$ to vertex $u^*$, such that the paths in ${\mathcal{Q}}'_i$ cause congestion at most $3000$, and they are internally disjoint from $X_i$. By suitably subdividing the first edge of every path in ${\mathcal{Q}}'_i$, and by replacing the last edge $e_{i',j}$ on each such path by the corresponding edge $a_{i',j}$, we obtain a collection ${\mathcal{Q}}''_i$ of paths in graph $G'$, routing the edges of $\hat A_i$ to vertices of $\set{u_{i'}}_{i'\neq i}$, such that the paths in ${\mathcal{Q}}''_i$ are internally disjoint from $X'_i$, and cause congestion at most $3000$. Note that, for all $1\leq i'\leq k$ with $i'\neq i$, the number of paths terminating at vertex $u_{i'}$ is at most $3000|\hat A'_{i'}|\leq 3000\hat q_i\leq 3000|A'_{i'}|$. Therefore, by appending an edge of $A'_{i'}$ at the end of each such path, for all indices $i'\neq i$, we obtain a set ${\mathcal{P}}'_i=\set{P(\hat a)\mid \hat a\in \hat A_i}$ of paths in graph $G'$, that cause congestion at most $3000$, such that for each edge $\hat a\in \hat A_i$, path $P(\hat a)$ has $\hat a$ as its first edge, terminates at vertex $u^*$, and is internally disjoint from $X'_i$. Lastly, for every edge $a'_{i,j}\in A'_i$, we define a path $P(a'_{i,j})$ consisting of only the edge $a'_{i,j}$ itself, and add that path to set ${\mathcal{P}}'_i$. We have now obtained a set ${\mathcal{P}}'_i$ of paths in graph $G'$, routing the edges of $\delta_{G'}(X'_i)$ to vertex $u^*$, such that the paths are internally disjoint from $X'_i$. Therefore, ${\mathcal{P}}'_i\in \Lambda'(X'_i)$. We then let the distribution ${\mathcal{D}}'(X'_i)$ choose the path set ${\mathcal{P}}'_i$ with probability $1$.
We add each such cluster $X'_i$ to the set ${\mathcal{L}}^{\operatorname{light}}$ of light clusters, and define, for each such cluster $X'_i$, a distribution ${\mathcal{D}}(X'_i)$ over the set $\Lambda(X'_i)$ of internal routers for $X'_i$, such that $X'_i$ is $\beta$-light with respect to ${\mathcal{D}}(X'_i)$.
In fact, the disctibution ${\mathcal{D}}(X'_i)$ will select a single set ${\mathcal{P}}_i\in \Lambda(X'_i)$ of paths with probability $1$. The set ${\mathcal{P}}_i$ of paths is constructed as follows. From the properties of the flower cluster, there is a set ${\mathcal{Q}}_i$ of edge-disjoint paths in graph $G$, routing the edges of $\delta_G(X_i)\setminus \delta_G(u^*)$ to vertex $u^*$, such that every inner vertex on every path lies in $X_i$. Since cluster $X'_i$ can be obtained from $X_i$ by replacing vertex $u_i$ with vertex $u^*$, we obtain a collection ${\mathcal{P}}_i$ of edge-disjoint paths, routing the edges of $\hat A_i$ to vertex $u_i$, such that every inner vertex on every path lies in $X_i'$. For every edge $a'_{i,j}\in A'_i$, we add a path $P(a'_{i,j})$, consisting of the edge $a'_{i,j}$ only, to set ${\mathcal{P}}_i$. We then obtain a set ${\mathcal{P}}_i$ of edge-disjoint paths, routing the edges of $\delta_{G'}(X'_i)$ to vertex $u_i$ inside $X'_i$, that is, ${\mathcal{P}}_i\in \Lambda(X_i)$.
Consider the set ${\mathcal{I}}_3$ of subinstances of $I'$, that is obtained by performing a basic disengagement of instance $I'$ via the tuple $({\mathcal{L}},{\mathcal{L}}^{\operatorname{bad}}, {\mathcal{L}}^{\operatorname{light}}, \set{{\mathcal{D}}'(X'_i)}_{i=1}^k,
\set{{\mathcal{D}}(X'_i)}_{i=1}^k$) (here, we set ${\mathcal{L}}^{\operatorname{bad}}=\emptyset$).
Recall that family ${\mathcal{I}}_3$ of instances contains a single global instance $\hat I=(\hat G,\hat \Sigma)$, where graph $\hat G$ is obtained from graph $G'$ by contracting, for all $1\leq i\leq k$, the vertices of $X'_i$ into a supernode.
Additionally, for every cluster $X'_i\in {\mathcal{L}}$, we obtain an instance $I(X'_i)=(G_i,\Sigma_i)$, where graph $G_i$ is obtained from graph $G_i$, by contracting all vertices of $V(G')\setminus V(X'_i)$ into a supernode.
We summarize the properties of the resulting family ${\mathcal{I}}_3$ of instances in the following claim.
\begin{claim}\label{claim: properties of instances from petal-based disengagement}
\begin{itemize}
\item $\sum_{\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}_3}|E(\tilde G)|\leq O(|E(G)|)$;
\item $\expect{\sum_{\tilde I\in {\mathcal{I}}_3}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)}\leq O
\textsf{left} ((\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot \log^{36}m\textsf{right} )$;
\item There is an efficient algorithm, that, given, for each instance $\tilde I\in {\mathcal{I}}_3$, a solution $\phi(\tilde I)$, computes a solution to instance $I$ of cost at most $O\textsf{left} (\sum_{\tilde I\in {\mathcal{I}}_3}\mathsf{cr}(\phi(\tilde I))\textsf{right} )$.
\end{itemize}\end{claim}
\begin{proof}
For the first assertion, recall that, from \Cref{lem: number of edges in all disengaged instances} $\sum_{\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}_3}|E(\tilde G)|\leq O(|E(G' )|)$. Since, from the construction of the split instance, $|E(G')|\leq O(|E(G)|)$, the assertion follows.
In order to prove the second assertion, recall that, from \Cref{lem: disengagement final cost}, $\expect{\sum_{\tilde I\in {\mathcal{I}}_3}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)}\leq O\textsf{left} (\beta^2\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|)\textsf{right} )$. Since, as discussed above, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)$, $|E(G')|\leq O(|E(G)|)$, and $\beta\leq O(\log^{18}m)$, the assertion follows.
In order to prove the last assertion, we use the algorithm from \Cref{lem: basic disengagement combining solutions}, that, given, for each instance $\tilde I\in {\mathcal{I}}_3$, a solution $\phi(\tilde I)$, computes a solution $\phi'$ for instance $I'$ of cost at most $\sum_{\tilde I\in {\mathcal{I}}_3}\mathsf{cr}(\phi(\tilde I))$. We then use
the algorithm from \Cref{lem: solution to split to solution to original} in order to compute a solution $\phi$ for instance $I$ of cost at most $O(\mathsf{cr}(\phi'))\leq O\textsf{left} (\sum_{\tilde I\in {\mathcal{I}}_3}\mathsf{cr}(\phi(\tilde I))\textsf{right} )$.
\end{proof}
Consider now the global instance $\hat I=(\hat G,\hat \Sigma)$. Since graph $\hat G$ is obtained from $G'$ by contracting every cluster $X'_i$ into a supernode, for every edge $e\in E(\hat G)$, either $e$ is incident to $u^*$, or it corresponds to an edge of $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$, where ${\mathcal{C}}$ is the initial collection of clusters that we computed in Phase 1. Recall that, from \Cref{eq: num of edges between clusters}, $
|E^{\textnormal{\textsf{out}}}({\mathcal{C}})|\leq m/(80\mu)$. Recall that $\deg_{G'}(u^*)=\sum_{i=1}^k\hat q_i=\sum_{i=1}^k|\hat E_i|$. From Modified Property \ref{prop: flower cluster small boundary size} of the flower cluster, $\sum_{i=1}^k|\hat E_i|\leq 200m/\mu^{42}$. Therefore, overall, $|E(\hat G)|\leq |E^{\textnormal{\textsf{out}}}({\mathcal{C}})|+\deg_{G'}(u^*)\leq m/(40\mu)$.
Next, we consider petal-based instances, and we prove that for each such instance, the maximum vertex degree is small.
\begin{claim}\label{claim: degree of petal instance}
For all $1\leq i\leq k$, if $I(X'_i)=(G_i,\Sigma_i)$ is the instance of $\tilde {\mathcal{I}}$ associated with cluster $X'_i$, then every vertex degree in graph $G_i$ is less than $m/\mu^4$.
\end{claim}
\begin{proof}
Recall that graph $G_i$ is obtained from graph $G'$ by contracting all vertices of $V(G')\setminus V(X'_i)$ into a supernode, that we denote by $u'$. Recall that graph $X'_i$ is identical to the petal $X_i$, except that we replace vertex $u^*$ with vertex $u_i$. From the definition of a flower cluster, every vertex of $X_i$, except for vertex $u^*$, has degree less than $m/\mu^4$ in $G$. The degree of vertex $u_i$ in the new graph is bounded by $q_i+\hat q_i$. Here, $q_i=|A_i|=|E_i|\leq m/(2\mu^4)$ from Property \ref{prop: flower cluster edges near center partition} of the flower cluster, and $\hat q_i\leq 200m/\mu^{42}$ from Modified Property \ref{prop: flower cluster small boundary size} of the flower cluster. Therefore, the degree of $u_i$ in graph $G_i$ is less than $m/\mu^4$. It now remains to bound the degree of the supernode $u'$ in graph $G_i$. The edges incident to $u'$ are the edges of $A'_i\cup \hat A_i$, and their number is bounded by $2\hat q_i$, which, from the above discussion, is bounded by $400m/\mu^{42}$.
\end{proof}
\subsubsection{Step 3: Final Decomposition}
In this step, we consider each petal-based instance $I(X'_i)=(G_i,\Sigma_i)$ in which $|E(G_i)|>m/(2\mu)$. We further decompose each such instance into subinstances, by exploiting the fact that graph $G_i$ does not have high-degree vertices, using the following lemma.
\begin{lemma}\label{lem: last decomposition}
There is an efficient randomized algorithm, that, given an instance $\tilde I=(\tilde G,\tilde \Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace and parameters $m,\mu$, such that $m$ is greater than a large enough constant, $m/(2\mu)<|E(\tilde G)|\leq 3m$, $\mu\geq 2^{\Omega(\sqrt{\log m})}$, and maximum vertex degree in $\tilde G$ is less than $m/\mu^4$, either correctly establishes that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \Omega \textsf{left}( \frac{m^2}{ \mu^{5.5}} \textsf{right} )$, or
computes a $\nu_3$-decomposition $\tilde {\mathcal{I}}'$ of $\tilde I$, for $\nu_3=2^{O((\log m)^{3/4}\log\log m)}$, such that, for every instance $\tilde I'=(\tilde G',\tilde \Sigma')\in \tilde {\mathcal{I}}'$, $|E(\tilde G')|\leq m/(2\mu)$.
\end{lemma}
We prove the lemma below, after we complete the proof of \Cref{lem: decomposing problematic instances by petal} using it.
Consider some index $1\leq i\leq k$. If $|E(G_i)|< m/(2\mu)$, then we let the set ${\mathcal{I}}(X'_i)$ of subinstances of $I(X'_i)$ consist of a single instance -- instance $I(X'_i)$. Otherwise,
we apply the algorithm from \Cref{lem: last decomposition} to instance $I(X'_i)=(G_i,\Sigma_i)\in {\mathcal{I}}_3$. If the algorithm from \Cref{lem: last decomposition} computes a $\nu_3$-decomposition $\tilde {\mathcal{I}}$ of $I(X'_i)$, such that, for every instance $\tilde I'=(\tilde G',\tilde \Sigma')\in \tilde {\mathcal{I}}$, $|E(\tilde G')|\leq m/(2\mu)$, then we set ${\mathcal{I}}(X'_i)=\tilde {\mathcal{I}}$. Otherwise, we terminate the algorithm and return FAIL.
If, every time \Cref{lem: last decomposition} is invoked, it returns a $\nu_3$-decomposition of the corresponding instance $I(X'_i)$, then we output a collection $\set{\hat I}\cup \textsf{left} (\bigcup_{i=1}^k{\mathcal{I}}(X'_i)\textsf{right})$ of instances. From \Cref{claim: compose algs}, it is immediate to verify that this algorithm produces a $\nu_2$-decomposition of instance $I$ for $\nu_2=O(\nu_3)$ (since the family ${\mathcal{I}}_3$ of subinstances of $I$ computed in Step 2 of the current phase is an $O(\log^{36}m)$-decomposition of instance $I$, from \Cref{claim: properties of instances from petal-based disengagement}), and the graph associated with each instance has at most $m/(2\mu)$ edges.
Assume now that
$\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)<\frac{ m^2}{c'' \mu^{13}}$ for some large enough constant $c''$.
Recall that $|E(G)|\leq O(m)$, and, from the statement of \Cref{lem: not many paths}, $m\geq \mu^{50}$, so $|E(G)|<\frac{m^2}{\mu^{13}}$. Therefore, $(\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)+|E(G)|) <\frac{2m^2}{c'' \mu^{13}}$.
Recall that, from \Cref{claim: properties of instances from petal-based disengagement}, $\expect{\sum_{\tilde I\in {\mathcal{I}}_3}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)}\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)$.
We say that a bad event ${\cal{E}}''$ happens if $\sum_{i=1}^k\mathsf{OPT}_{\mathsf{cnwrs}}(I(X'_i))>c\mu^5(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)$ for some large enough constant $c$. From Markov's inequality, $\prob{{\cal{E}}''}\leq 1/(8\mu^4)$.
If $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)<\frac{ m^2}{c'' \mu^{13}}$, and the bad event ${\cal{E}}''$ did not appen, then for all $1\leq i\leq k$,
$\mathsf{OPT}_{\mathsf{cnwrs}}(I(X'_i))<c\mu^5(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\leq \frac{2cm^2}{c''\mu^8}$. Note that our algorithm may only return FAIL if there is some index $1\leq i\leq k$, such that $|E(G_i)|\geq m/(2\mu)$, and $\mathsf{OPT}_{\mathsf{cnwrs}}(I(X'_i))\geq \Omega \textsf{left}( \frac{|E(G_i)|^2}{ \mu^{5.5}} \textsf{right} )\geq \Omega \textsf{left}(\frac{ m^2}{ \mu^{7.5}} \textsf{right} )$. From the above discussion, and since we can choose $c''$ to be a large enough constant compared to $c$, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)<\frac{ m^2}{c'' \mu^{13}}$, then the algorithm may only return FAIL if ${\cal{E}}''$ happens, which happens with probability at most $1/(8\mu^4)$.
\iffalse
Therefore, if
If ${\cal{E}}''$ did not happen, then $\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\geq \Omega \textsf{left}(\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(I(X'_i))}{\mu}\textsf{right} )\geq \Omega \textsf{left}(\frac{ m^2}{ \mu^8} \textsf{right} )$. Since we are guaranteed that $|E(G)|\leq O(m)$, and, from the statement of \Cref{lem: not many paths}, $m\geq \mu^{10}$, we conclude that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \Omega \textsf{left}(\frac{ m^2}{ \mu^8} \textsf{right} )$. Therefore, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<\textsf{left}(\frac{c'' m^2}{ \mu^8} \textsf{right} )$ for some large enough constant $c''$, then the only way the algorithm may return FAIL is if event ${\cal{E}}''$ happens, which may only happen with probability at most $1/(2\mu)$.
\fi
In order to complete \Cref{lem: decomposing problematic instances by petal}, and \Cref{lem: not many paths}, it is now enough to prove \Cref{lem: last decomposition}.
\begin{proofof}{\Cref{lem: last decomposition}}
In order to simplify the notation, we denote instance $\tilde I=(\tilde G,\tilde \Sigma)$ by $I=(G,\Sigma)$. We will essentially repeat the algorithm from Phase 1, except that, since there are no high-degree vertices in $G$, we do not need to deal with flower cluster, and all instances that we will obtain in the final decomposition will be small.
We start by applying the algorithm from \Cref{lem: decomposition into small clusters} to graph $H=G$, with terminal set $T=\emptyset$, parameter $\tau=2\mu^{1.1}$, and the parameter $m$ replaced with $3m$. Recall that the maximum vertex degree in $G$ is less than $\frac{m}{\mu^4}< \frac{3m}{\check c\tau^3 \log^5 (3m)}$, as required.
Assume first that the algorithm from \Cref{lem: decomposition into small clusters} establishes that that $\mathsf{OPT}_{\mathsf{cr}}(G)\geq \Omega \textsf{left}( \frac{m^2}{ \tau^4\log^5 m} \textsf{right} )\geq \Omega \textsf{left}( \frac{m^2}{ \mu^{5.5}} \textsf{right} )$. We then terminate the algorithm and report that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \Omega \textsf{left}( \frac{m^2}{ \mu^{5.5}} \textsf{right} )$.
Therefore, we assume from now on that the algorithm from \Cref{lem: decomposition into small clusters} computes a collection ${\mathcal{C}}'$ of disjoint clusters of $G$, such that every cluster $C\in {\mathcal{C}}'$ has the $\alpha'$-bandwidth property, where $\alpha'=\Omega\textsf{left}(\frac{1}{\log^{1.5}m}\textsf{right} )$.
Since $m\geq \mu^4$, we then get that every cluster in ${\mathcal{C}}'$ has the $\alpha_0=1/\log^3m$-bandwidth property.
Additionally, we are guaranteed that, for each such cluster $C\in {\mathcal{C}}'$, $|E(C)|\leq m/\tau\leq m/(4\mu)$,
$\bigcup_{C\in {\mathcal{C}}'}V(C)=V(G)$, and
$|\bigcup_{C\in {\mathcal{C}}'}\delta_G(C)|\leq m/\tau=m/(2\mu^{1.1})$. Notice that in particular, the number of edges of $G$ with endpoints in different clusters is $|E^{\textnormal{\textsf{out}}}({\mathcal{C}}')|\leq m/(2\mu^{1.1})$. Since we have assumed that $|E(G)|\geq m/(2\mu)$, we get that $\sum_{C\in {\mathcal{C}}'}|\delta_G(C)|\leq |E(G)|/\mu^{0.1}$.
We apply the algorithm from \Cref{thm: disengagement - main} to instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, and the set ${\mathcal{C}}'$ of clusters. Let $\tilde {\mathcal{I}}'$ be the resulting collection of subinstances of $I$ that the algorithm computes. Recall that the algorithm guarantees that $\tilde {\mathcal{I}}'$ is a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition of $I$, and moreover, for each instance $\tilde I'=(\tilde G',\tilde \Sigma')\in \tilde {\mathcal{I}}'$, there is at most one cluster $C\in {\mathcal{C}}'$ with $E(C)\subseteq E(\tilde G')$, and all other edges of $\tilde G'$ lie in set $E^{\textnormal{\textsf{out}}}({\mathcal{C}}')$. Since $|E^{\textnormal{\textsf{out}}}({\mathcal{C}}')|\leq m/(2\mu^{1.1})$, and, for every cluster $C\in {\mathcal{C}}'$, $|E(C)|\leq m/(4\mu)$, we are guaranteed that, for every instance $\tilde I'=(\tilde G',\tilde \Sigma')\in \tilde {\mathcal{I}}'$, $|E(\tilde G')|\leq m/(2\mu)$.
\end{proofof}
\subsubsection*{Step 1: Separating high-degree vertices}
\label{sec:size_reduction}
In the first step, we recursively decompose $G'$ into a set ${\mathcal{C}}_1$ of clusters, such that each cluster contains at most one high-degree vertex, as follows.
Throughout, we maintain a set $\hat{\mathcal{C}}$ of clusters of $G$. Initially, $\hat{\mathcal{C}}=\set{G}$.
While there exists some cluster $C\in \hat{\mathcal{C}}$ that contains at least two high-degree vertices, algorithm continues to be executed as follows.
We pick arbitrarily a cluster $C\in \hat{\mathcal{C}}$ with at least two high-degree vertices. Let $u,u'$ be two high-degree vertices of $C$. We compute a $u$-$u'$ min-cut of $C$, and let $C_u, C_{u'}$ be the two clusters obtained from $C$ by deleting the edges in the min-cut. We then replace the cluster $C$ in $\hat{\mathcal{C}}$ with clusters $C_u, C_{u'}$ and continue to the next iteration. This completes the description of the first step. Let ${\mathcal{C}}_1$ be the set of clusters that we obtain ta the end of the first step. Clearly, each cluster of ${\mathcal{C}}_1$ contains at most one high-degree vertex.
We now show that $|E^{\textsf{out}}({\mathcal{C}}_1)|=O(m/\mu^2)$. Since a high-degree vertex has degree at least $m/\mu^8$, initially $G$ contains at most $\mu^8$ high-degree vertices. Therefore, in the first step we performed at most $\mu^8$ min-cut computations. On the other hand, since $G$ is not $\mu$-interesting, all min-cuts that we computed in the first step has size at most $m/\mu^{12}$. It follows that $|E({\mathcal{C}}_1)|\le O(\mu^8)\cdot O(m/\mu^{12})=O(m/\mu^4)$.
\subsubsection*{Step 2: Trimming clusters with a high-degree vertex}
In this step, we further process the clusters of ${\mathcal{C}}_1$ that contains a high-degree vertex one-by-one, as follows.
Let $C$ be a cluster in ${\mathcal{C}}_1$ that contains a unique high-degree vertex $u$.
We compute a min-cut $(A,B)$ of $\tilde C$ separating $u$ from all vertices of $\Gamma(C)$ (recall that $\Gamma(C)$ the set of endpoints of edges in $\delta(C)$ that do not lie in $C$), with $u\in A$. Let $C'$ be the subgraph of $C$ induced by vertices of $A$.
Therefore, $|E(C',C\setminus C')|\le |\delta(C)|$.
We decompose the cluster $C$ into $C'$ and $C\setminus C'$.
Note that $C'$ is connected and still contains a unique high-degree vertex $u$, and there is a set of edge-disjoint paths in $C'$ connecting $\delta(C')$ to $u$.
Also note that and the subgraph $C\setminus C'$ does not contain any high-degree vertex, and it may not be connected.
We replace the cluster $C$ in $\hat{\mathcal{C}}$ by cluster $C'$ and connected components of $C\setminus C'$.
We further process the cluster $C'$ as follows.
Let $d=m/\mu^{10}$.
Denote $\delta(u)=\set{e_1,\ldots,e_{\deg(u)}}$, where the edges are indexed according to the ordering ${\mathcal{O}}_u$ in $\Sigma$.
Denote $r=\lceil\frac{\deg(u)}{d}\rceil$.
For each $1\le i\le r-1$, let $E_i=\set{e_{(i-1)d+1},\ldots,e_{id}}$, and let $E_{r}=\set{e_{(r-1)d+1},\ldots,e_{\deg(u)}}$.
Clearly, sets $E_1,E_2,\ldots,E_r$ are consecutive subsets of edges in $\delta(u)$, each containing at most $d$ edges.
Let the cluster $\tilde C'$ be obtained from $C'$ by replacing the vertex $u$ with $r$ distinct new vertices $u_1,\ldots,u_r$, such that for each $1\le i\le r$, the edges of $E_i$ are incident to $u_i$.
Denote $U=\set{u_1,\ldots,u_r}$.
For each $1\le i\le r$, we compute a min-cut $(A_i,B_i)$ of $\tilde C'$, that separates $u_i$ from all vertices in $U\setminus \set{u_i}$, with $u_i\in A_i$.
From Lemma~\ref{lem:min-cut_structure}, the sets $\set{A_i}_{1\le i\le r}$ are mutually disjoint.
We denote by $X_i$ the subgraph of $C'$ induced by vertices of $A_i$, so the clusters $X_1,\ldots,X_r$ are mutually vertex-disjoint. Note that, since $G$ is not $\mu$-interesting, for each $1\le i\le r$, $|E(X_i,C'\setminus X_i)|\le m/\mu^{12}$, so $\sum_{1\le i\le r} |E(X_i,C'\setminus X_i)|\le O(\deg(u)/d)\cdot O(m/\mu^{12})\le O(\deg(u)/\mu^2)$.
For each $1\le i\le r$, we denote $\tilde E_i=\delta(X_i)$.
We use the following observation.
\begin{observation}
\label{obs: pedal-well-linked}
For each $1\le i\le r$, there is a set ${\mathcal{Q}}_i$ of paths in $X_i$ connecting edges of $\tilde E_i$ to $u_i$, such that $\cong_{X_i}({\mathcal{Q}}_i)\le O(1)$.
\end{observation}
\begin{proof}
The set $\tilde E_i$ can be partitioned into two subsets: set $\tilde E'_i$ of edges connecting a vertex of $X_i$ to some vertex of $\tilde C\setminus X_i$, and set $\tilde E''_i$ of edges connecting a vertex of $X_i$ to some vertex of $G\setminus C'$.
We first show that there is a set ${\mathcal{Q}}'_i$ of edge-disjoint paths in $X_i$ connecting edges of $\tilde E'_i$ to $u_i$.
Assume not, then there exists a cut $(A'_i, B'_i)$ in $X_i$ separating all vertices of $\Gamma(X_i)$ from $u_i$, such that $|E(A'_i,B'_i)|\le |\Gamma(X_i)|$. Clearly, the cut $(A'_i,V(\tilde C')\setminus A'_i)$ also separates $u_i$ from all vertices in $U\setminus\set{u_i}$, causing a contradiction to the minimality of the cut $(A_i,B_i)$.
We then show that there is a set ${\mathcal{Q}}''_i$ of paths in $X_i$ connecting edges of $\tilde E'_i$ to $u_i$, such that $\cong_{X_i}({\mathcal{Q}}'')\le 2$.
Note that $\tilde E''_i\subseteq\delta(C')$, and recall that there is a set ${\mathcal{P}}$ of edge-disjoint paths connecting $\delta(C')$ to $u$.
Let ${\mathcal{P}}_i\subseteq {\mathcal{P}}$ be the subset of paths connecting edges of $\tilde E''_i$ to $u$. Consider a path $P_e\in {\mathcal{P}}_i$ that connects the edge $e\in \tilde E''_i$ to $u$. Either it lies entirely in cluster $X_i$, or it starts at $e$, goes over some edges of $X_i$, and leaves cluster $X_i$ via some edge $e'\in \tilde E'_i$.
In the latter case, let $P'_e$ be path obtained by concatenating (i) the maximal beginning subpath of $P$ (the subpath containing $e$) that lies in $X_i$; and (ii) the path of ${\mathcal{Q}}'_i$ that connects the edge $e'$ to $u$ minus the edge $e'$. Clearly, $P'_e$ connects edge $e$ to $u$ and lies entirely in $X_i$. We let the set ${\mathcal{Q}}''_i$ contains (i) all paths of ${\mathcal{P}}_i$ that lies entirely in $X_i$; and (ii) for each path $P_e\in {\mathcal{P}}_i$ that connects the edge $e$ to $u$, the path $P'_e$. It remains to show that $\cong_{X_i}({\mathcal{Q}}'')\le 2$.
Since the paths in ${\mathcal{P}}_i$ are edge-disjoint, each path in ${\mathcal{P}}_i$ contains a distinct edge of $\tilde E'_i$, if any.
Therefore, for each edge $e\in X_i$, $\cong_{{\mathcal{Q}}''_i}(e)\le \cong_{{\mathcal{P}}_i}(e)+\cong_{{\mathcal{Q}}'_i}(e)\le 2$.
Observation~\ref{obs: pedal-well-linked} now follows by letting ${\mathcal{Q}}_i={\mathcal{Q}}'_i\cup {\mathcal{Q}}''_i$.
\end{proof}
\begin{figure}[h]
\includegraphics[scale=0.3]{figs/flower_edge.jpg}
\caption{An illustration of the edge sets in a flower cluster.\label{fig:flower_edge}}
\end{figure}
Let $C''$ be the cluster obtained by first taking the union of clusters $X_1,\ldots,X_r$, and then unifying the vertices $v_1,\ldots,v_r$ back to the original vertex $v$.
Put in other words, $C''$ is the subgraph of $C'$ induced by vertices of $\bigcup_{1\le i\le r}V(X_i)$.
See Figure~\ref{fig:flower_edge} for an illustration.
From Observation~\ref{obs: pedal-well-linked} and the definitions of good cluster and flower cluster: if $E(C'')|>m/\mu^2$, $C''$ is a flower cluster; otherwise, it is a good cluster.
We decompose the cluster $C'$ into $C''$ and $C'\setminus C''$ by deleting from $C'$ the edges of $E(C'',C'\setminus C'')$.
Note that and the subgraph $C'\setminus C''$ does not contain any high-degree vertex, and it may not be connected. We replace the cluster $C'$ in $\hat{\mathcal{C}}$ by cluster $C''$ and connected components of $C'\setminus C''$.
Let ${\mathcal{C}}_2$ be the set of clusters that we obtain after processing every cluster of ${\mathcal{C}}_1$ that contains a high-degree node in this way. We now show that $|E^{\textsf{out}}({\mathcal{C}}_2)|=O(m/\mu)$.
Let $C$ be any cluster in ${\mathcal{C}}_1$ that contains a high-degree vertex. Recall that $C$ is first decomposed into $C'$ and $C\setminus C'$, and then $C'$ is further decomposed into $C''$ and $C'\setminus C''$. Therefore, from the above discussion, the number of edges between sub-clusters of $C$ in ${\mathcal{C}}_2$ is $|E(C',C\setminus C')|+|E(C'',C'\setminus C'')|\le |\delta(C)|+O(\deg(u)/\mu)$. Summing over all clusters $C\in {\mathcal{C}}_1$ that contains a high-degree vertex, we get that $|E^{\textsf{out}}({\mathcal{C}}_2)|\le O(|E^{\textsf{out}}({\mathcal{C}}_1)|)+O(|E(C)|/\mu^2)\le O(m/\mu^2)$.
\subsubsection*{Step 3: Size reduction via balanced cuts}
The set ${\mathcal{C}}_2$ of clusters we obtain after the second step can be further decomposed into three subsets: set ${\mathcal{C}}^f_2$ of flower clusters, set ${\mathcal{C}}^g_2$ of good clusters, and the set ${\mathcal{C}}'_2$ that contains all other clusters. Note that no clusters in ${\mathcal{C}}'_2$ contain high-degree vertices. In this step, we will further decompose clusters of ${\mathcal{C}}'_2$ into good clusters.
We first iteratively decompose clusters of ${\mathcal{C}}'_2$ that contains more than $m/\mu^2$ edges into smaller clusters, as follows.
Throughout, we maintain a set $\hat{\mathcal{C}}$ of clusters of $G$. Initially, $\hat{\mathcal{C}}={\mathcal{C}}'_2$. The algorithm in this step proceeds in iterations and continues to be executed as long as $\hat{\mathcal{C}}$ contains a cluster with more than $m/\mu^2$ edges.
In each iteration, we pick arbitrarily a cluster $C$ in $\hat {\mathcal{C}}$ with $|E(C)|\ge m/\mu^2$. We apply the algorithm in Theorem~\ref{thm:most_balanced_sparsest-cut} to $C$ and obtain a cut $(A,B)$ of $C$, and check the size of $|E(A,B)|$. If $|E(A,B)|\ge m\cdot \log n/\mu^4$, then we report $\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\ge |E(G')|^2/\mu^{10}$ and the algorithm is aborted. If $|E(A,B)|\le O(m\cdot \log n/\mu^4)$, then replace the cluster $C$ with the subgraphs of $C$ induced by $A$ and $B$, respectively.
This completes the description of the algorithm in this step.
We first show that, whenever we claim that $\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\ge |E(G')|^2/\mu^{10}$, the claim is correct.
\begin{observation}
\label{obs: abort_correctly}
Let $(A,B)$ be the cut of $C$ that we obtained in some iteration of the algorithm. If $|E(A,B)|\ge m\cdot \log n/\mu^4$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\ge |E(G')|^2/\mu^{10}$.
\end{observation}
\begin{proof}
Assume the contrast that $\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\le |E(G')|^2/\mu^{10}$. Note that, if we denote by $\Sigma_C$ the rotation system on $C$ induced by $\Sigma'$, then $\mathsf{OPT}_{\mathsf{cr}}(C)\le \mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\le \mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')$. Recall that $C$ has no high-degree vertices, so $\Delta(C)\le m/\mu^8$.
Therefore, from Lemma~\ref{lem:min_bal_cut}, the size of the minimum balanced cut in $C$ is at most $O\big(\sqrt{\mathsf{OPT}_{\mathsf{cr}}(C)+|E(C)|\cdot\Delta(C)}\big)\le O\big( \sqrt{m^2/\mu^{10}+m\cdot(m/\mu^8)}\big)\leq O(m/\mu^{4})$. Then from Theorem~\ref{thm:most_balanced_sparsest-cut}, the size of the cut $(A,B)$ is at most $O(\sqrt{\log n}\cdot m/\mu^{4})<m\cdot \log n/\mu^4$, a contradiction.
\end{proof}
Let ${\mathcal{C}}'_3$ be the set of clusters that we obtain from decomposing the clusters of ${\mathcal{C}}'_2$ in this step. Denote ${\mathcal{C}}_3={\mathcal{C}}^f_2 \cup{\mathcal{C}}^g_2 \cup {\mathcal{C}}'_3$.
We now show that, if the algorithm is not aborted, then $|E^{\textsf{out}}({\mathcal{C}}_3)|=O(m\log n/\mu^2)$.
Note that $E^{\textsf{out}}({\mathcal{C}}_3)\setminus E^{\textsf{out}}({\mathcal{C}}_2)=E^{\textsf{out}}({\mathcal{C}}'_3)\setminus E^{\textsf{out}}({\mathcal{C}}'_2)$, namely the new edges in $E^{\textsf{out}}({\mathcal{C}}_3)$ come from decomposing the clusters in $E^{\textsf{out}}({\mathcal{C}}'_2)$. From the algorithm in this step, each resulting cluster in $E^{\textsf{out}}({\mathcal{C}}_3)$ has size at least $\Omega(m/\mu^2)$. Therefore, the number of balanced-cut computations that we have performed in this step is at most $\mu^2$. From Observation~\ref{obs: abort_correctly}, if the algorithm is not aborted, then each cut that we computed has size at most $m\cdot \log n/\mu^4$. Therefore, the total number of inter-cluster edges that we have created in this step is $O(m\cdot \log n/\mu^4)\cdot O(\mu^2)=O(m\cdot \log n/\mu^2)$. As a result, $|E^{\textsf{out}}({\mathcal{C}}_3)|\le |E^{\textsf{out}}({\mathcal{C}}_2)|+O(m\cdot \log n/\mu^2)=O(m\cdot \log n/\mu^2)$.
\subsubsection*{Step 4: Well-linked decomposition}
Recall that after the third step, we obtain a set ${\mathcal{C}}_3$ of clusters with $|E^{\textsf{out}}({\mathcal{C}}_3)|\le O(m\cdot \log n/\mu^2)$, that can be further decomposed into
${\mathcal{C}}_3={\mathcal{C}}^f_2 \cup{\mathcal{C}}^g_2 \cup {\mathcal{C}}'_3$, where the set ${\mathcal{C}}^f_2$ contains flower clusters, the set ${\mathcal{C}}^g_2$ contains good clusters, and the set ${\mathcal{C}}'_3$ contains clusters with at most $m/\mu^2$ edges and no high-degree vertices.
In this step, we further decompose clusters in ${\mathcal{C}}'_3$ to get good clusters. Specifically, for each cluster $C$ of ${\mathcal{C}}'_3$, we apply the algorithm of Theorem~\ref{thm:well_linked_decomposition} to it, and obtain a set $\set{C_1,\ldots,C_k}$ of clusters. We then replace the cluster $C$ in ${\mathcal{C}}'_3$ with clusters $C_1,\ldots,C_k$. This completes the description of this step.
Let ${\mathcal{C}}^g_4$ be the set of clusters that we obtain from decomposing the clusters of ${\mathcal{C}}'_2$ in this step. From Theorem~\ref{thm:well_linked_decomposition}, every cluster of ${\mathcal{C}}^g_4$ is $\alpha$-boundary-well-linked. Since every cluster of ${\mathcal{C}}^g_4$ contains at most $m/\mu^2$ edges, by definition, every cluster of ${\mathcal{C}}^g_4$ is a good cluster. Therefore, if we denote by ${\mathcal{C}}^*$ the set of clusters that we obtain after the four steps, then each cluster of ${\mathcal{C}}^*$ is either a good cluster or a flower cluster.
It remains to show that $|E^{\textsf{out}}({\mathcal{C}}^*)|\le O(m/\mu)$. Note that $E^{\textsf{out}}({\mathcal{C}}^*)\setminus E^{\textsf{out}}({\mathcal{C}}_3)=E^{\textsf{out}}({\mathcal{C}}^g_4)\setminus E^{\textsf{out}}({\mathcal{C}}'_3)$.
From Theorem~\ref{thm:well_linked_decomposition}, $|E^{\textsf{out}}({\mathcal{C}}^g_4)\setminus E^{\textsf{out}}({\mathcal{C}}'_3)|\le O(|E^{\textsf{out}}({\mathcal{C}}_3)|)\le O(m\cdot \log n/\mu^2)$. Therefore, $|E^{\textsf{out}}({\mathcal{C}}^*)|\le O(|E^{\textsf{out}}({\mathcal{C}}_3)|)\le O(m\cdot \log n/\mu^2)$.
This completes the proof of Lemma~\ref{lem:main_decomposition}.
\iffalse
\begin{theorem}[Theorem 3 of \cite{chuzhoy2012vertex}]
There is an efficient algorithm, that given any graph $G=(V,E)$ and any cluster $S\subseteq V$ of vertices with $|\delta(S)|=z$, computes a set ${\mathcal{R}}$ of clusters of $S$, such that,
\begin{itemize}
\item the vertex sets $\set{V(R)}_{R\in {\mathcal{R}}}$ partitions $V(S)$;
\item for each cluster $R\subseteq{\mathcal{R}}$, $|\delta(R)|\le |\delta(S)|$;
\item for each cluster $R\subseteq{\mathcal{R}}$, $R$ is $\Omega(\log^{-3/2}z)$-boundary-well-linked;
\item $\sum_{R\in {\mathcal{R}}}|\delta(R)|\le 1.2|\delta(S)|$.
\end{itemize}
\begin{lemma}
\label{lem:min-cut_structure}
Let $G$ be any graph, and let $x,y,s,t$ be vertices of $G$. Let $(X,Y)$ be the minimum $x$-$y$ cut, and let $(S,T)$ be the minimum $s$-$t$ cut. Then one of the following must hold:
\begin{itemize}
\item $S\subseteq X, Y\subseteq T$; or
\item $X\subseteq S,T\subseteq Y$; or
\item $T\subseteq X, Y\subseteq S$; or
\item $X\subseteq T, S\subseteq Y$.
\end{itemize}
In other words, the two cuts are laminar, and one side of the first cut must be contained in one side of the second cut and vice versa.
\end{lemma}
\fi
\subsection{Decomposition into Nice Instances -- Proof of \Cref{thm: advanced disengagement get nice instances}}
\label{sec: advanced disengagement - get nice instances}
This subsection is dedicated to the proof of \Cref{thm: advanced disengagement get nice instances}. The main idea of the proof is to carefully construct a laminar family ${\mathcal{L}}$ of clusters of $G$, whose depth is $2^{O((\log m)^{3/4}\log\log m)}$, and then apply Algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace from \Cref{subsec: basic disengagement}, to compute a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition ${\mathcal{I}}_1$ of instance $I$ via basic disengagement, using the laminar family ${\mathcal{L}}$. The main challenge is to construct the laminar family ${\mathcal{L}}$ in such a way that each resulting instance $I'=(G',\Sigma')\in {\mathcal{I}}_1$ is a nice subinstance of $I$ with respect to ${\mathcal{C}}$, and to compute a nice witness structure for each such graph $G'$.
We will construct the laminar family gradually, in the top-bottom fashion, using the notion of \emph{legal clustering}.
In order to define the legal clustering, we consider a graph $G'$, together with a special vertex $v^*\in V(G)$. Intuitively, graph $G'$ represents some cluster $S\in {\mathcal{L}}$ that we have constructed already, and it is a graph that is obtained from $G$ by contracting all vertices of $G\setminus S$ into the special vertex $v^*$. We will also consider the subset ${\mathcal{C}}'\subseteq{\mathcal{C}}$ of all clusters $C\in {\mathcal{C}}$ with $C\subseteq S$. Intuitively, our goal is to construct a collection ${\mathcal{R}}$ of disjoint clusters of $G'$, each of which must be a subgraph of $S$, that will then be added to ${\mathcal{L}}$. Recall that, if ${\mathcal{L}}$ is a laminar family of clusters of graph $G$, and ${\mathcal{I}}_1$ is a collection of subinstances of $I$ obtained by decomposing $I$ via basic disengagement, then every cluster $S\in {\mathcal{L}}$ has a subinstance $I(S)=(G(S),\Sigma(S))\in {\mathcal{I}}_1$ associated with it. Graph $G(S)$ is obtained from graph $G$ as follows. First, we contract the vertices of $V(G)\setminus V(S)$ into a supernode $v^*$, obtaining graph $G'$. Next, for every child-cluster $R\in {\mathcal{L}}$ of $S$, we contract $R$ into a supernode $v_R$. Therefore, if ${\mathcal{R}}$ is the set of child-clusters of $S$, then $G(S)=G'_{|{\mathcal{R}}}$. Recall that we need to ensure that instance $I(S)$ is a nice instance. Given a graph $G'$, a special vertex $v^*$ in $G'$, and a collection ${\mathcal{C}}'$ of disjoint basic clusters of $G'$, the notion of {legal clustering} of $G'$ with respect to $v^*$ and ${\mathcal{C}}'$ is designed to ensure that every instance in our final decomposition ${\mathcal{I}}_1$ of $I$, created via the process described above, is a nice instance.
Consider a graph $G'$ with a special vertex $v^*\in G'$.
We will consider clusters $R\subseteq G'\setminus \set{u^*}$. Recall we have defined a collection $\Lambda'_{G'}(R)$ of external routers for $R$, where each router ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$ is a collection of paths routing all edges of $\delta_{G'}(R)$ to a single vertex of $G'\setminus R$, such that all paths in ${\mathcal{Q}}'(R)$ are internally disjoint from $R$. We start by defining the notion of \emph{helpful clustering}, which will be used in the definition of legal clustering. We fix two parameters that will be used throughout this section: $\alpha_1=(\alpha_0)^2=1/\log^6m$ and $\beta=\log^{18}m$.
\begin{definition}[Helpful Clustering]
Let $G'$ be a graph with a special vertex $v^*\in V(G')$, and let ${\mathcal{C}}'$ be a collection of disjoint vertex-induced subgraphs of $G'\setminus\set{v^*}$, that we call \emph{basic clusters}. Let ${\mathcal{R}}$ be another collection of disjoint clusters of $G'$, and assume that for every cluster $R\in {\mathcal{R}}$, we are given a distribution ${\mathcal{D}}'(R)$ over the external routers in $\Lambda'_{G'}(R)$. We say that $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ is a \emph{helpful clustering} of $G'$ with respect to $v^*$ and ${\mathcal{C}}'$, iff the following conditions hold:
\begin{itemize}
\item vertex $v^*$ does not belong to any of the clusters in ${\mathcal{R}}$;
\item for every basic cluster $C'\in {\mathcal{C}}$, and for every cluster $R\in {\mathcal{R}}$, either $C'\subseteq R$, or $V(C')\cap V(R)=\emptyset$;
\item every cluster $R\in {\mathcal{R}}$ has the $\alpha_1$-bandwidth property in $G'$; and
\item for every cluster $R\in {\mathcal{R}}$, for every edge $e\in E(G')\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}'(R)]{\cong_{G'}({\mathcal{Q}}'(R),e)}\leq \beta$.
\end{itemize}
\end{definition}
Consider again a graph $G'$ with a special vertex $v^*\in V(G')$, and some cluster $R\subseteq G'$.
We say that an external $R$-router ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$ is \emph{careful} with respect to the special vertex $v^*$, if each edge of $\delta_{G'}(v^*)$ belongs to at most one path in ${\mathcal{Q}}'(R)$ (note that in general paths in ${\mathcal{Q}}'(R)$ may cause an arbitrarily large congestion in $G'$). We denote by $\Lambda''_{G'}(R)\subseteq \Lambda'_{G'}(R)$ the collection of all external $R$-routers ${\mathcal{Q}}'(R)$ that are careful with respect to $v^*$. We say that a distribution ${\mathcal{D}}'(R)$ over the collection $\Lambda'_{G'}(R)$ of external $R$-routers is \emph{careful} with respect to $v^*$, if every router ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$ to which ${\mathcal{D}}'(R)$ assigns a non-zero probability lies in $\Lambda''_{G'}(R)$.
We will consider two different types of legal clustering.
We start by defining the first, and the simpler type of legal clusterings.
\begin{definition}[Type-1 Legal Clustering]
Let $G'$ be a graph with a special vertex $v^*\in V(G')$, and let ${\mathcal{C}}'$ be a collection of disjoint vertex-induced subgraphs of $G'\setminus\set{v^*}$, that we call {basic clusters}. Let ${\mathcal{R}}$ be another collection of disjoint clusters of $G'$, and assume that for every cluster $R\in {\mathcal{R}}$, we are given a distribution ${\mathcal{D}}'(R)$ over the external routers in $\Lambda'_{G'}(R)$. We say that $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ is a \emph{type-1 legal clustering} of $G'$ with respect to $v^*$ and ${\mathcal{C}}'$, if the following conditions hold:
\begin{itemize}
\item $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ is a helpful clustering of $G'$ with respect to $v^*$ and ${\mathcal{C}}'$;
\item there is at most one cluster $C\in {\mathcal{C}}'$, that is contained in $G'\setminus \textsf{left}(\bigcup_{R\in {\mathcal{R}}}R\textsf{right} )$;
and
\item for every cluster $R\in {\mathcal{R}}$, distribution ${\mathcal{D}}'(R)$ over external routers is careful with respect to $v^*$.
\end{itemize}
\end{definition}
While type-1 legal clustering would be ideal in order to construct the laminar family ${\mathcal{L}}$ and to perform a basic disengagement of instance $I$ via ${\mathcal{L}}$, we may not always succeed in computing a type-1 legal clustering of a given graph $G'$, and we may need to employ type-2 legal clustering, that is defined below, instead. Before we define the type-2 legal clustering formally, we provide some intuition. Type-2 legal clustering is defined somewhat similarly to type-1 legal clustering, except that we no longer require that, for every cluster $R\in {\mathcal{R}}$, the distribution ${\mathcal{D}}'(R)$ is careful with respect to $v^*$. We also no longer require that at most one cluster of ${\mathcal{C}}'$ is contained in $G'\setminus \bigcup_{R\in {\mathcal{R}}}R$. However, we require that, additionally, the decomposition provides a nice witness structure for the graph $G'_{|{\mathcal{R}}}$ with respect to the set ${\mathcal{C}}'(G'_{|{\mathcal{R}}})$ of clusters (all clusers of ${\mathcal{C}}'$ that are contained in graph $G'_{|{\mathcal{R}}}$).
Unfortunately, the relaxation of the requirement that the distributions ${\mathcal{D}}'(R)$ for clusters $R\in {\mathcal{R}}$ is careful with respect to $v^*$ creates some major difficulties. For intuition, recall that we will construct the laminar family ${\mathcal{L}}$ of clusters of $G$ gradually, in the top-bottom fashion. Assume that $S$ is some cluster of the current laminar family ${\mathcal{L}}$, such that no cluster of ${\mathcal{L}}$ is strictly contained in $S$. Let $G'$ be the graph obtained from $G$ by contracting all vertices of $V(G)\setminus V(S)$ into the special vertex $v^*$, and let ${\mathcal{C}}'$ be the set of all clusters of ${\mathcal{C}}$ that are contained in $S$. The idea of our algorithm is to compute a type-1 or a type-2 legal clustering ${\mathcal{R}}$ in graph $G'$; assume that we compute a type-2 legal clustering. We then add the clusters of ${\mathcal{R}}$ to the laminar family ${\mathcal{L}}$, and continue to the next iteration. From the discussion so far, for each such cluster $R\in {\mathcal{R}}$, the type-2 legal clustering provides a distribution ${\mathcal{D}}'(R)$ over the collection $\Lambda'_{G'}(R)$ of external routers in graph $G'$. However, in order to execute the basic disengagement via the laminar family ${\mathcal{L}}$ (see \Cref{subsec: basic disengagement}), we need the distribution ${\mathcal{D}}'(R)$ to be supported over the collection $\Lambda'_G(R)$ of external routers in graph $G$. In other words, the problem is that paths in sets ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$ that are assigned non-zero probability by ${\mathcal{D}}'(R)$ may contain the special vertex $v^*$, which is not a vertex of $G$. Recall however that special vertex $v^*$ represents the cluster $G\setminus S$, and so edges incident to $v^*$ in $G'$ are precisely the edges of $\delta_G(S)$. Therefore, we could exploit the distribution ${\mathcal{D}}'(S)$ over the external routers for cluster $S$ in $G$, in order to get rid of the special vertex $v^*$ on the paths of ${\mathcal{Q}}'(R)$, where ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$. In other words, by composing the distributions ${\mathcal{D}}'(R)$ and ${\mathcal{D}}'(S)$, we could obtain the desired distribution over the set $\Lambda'_{G}(R)$ of external routers for cluster $R$ in the original graph $G$. Unfortunately, this kind of recursive composition of distributions may lead to an explosion in the congestion of the resulting sets of paths. Even if the depth of the laminar family ${\mathcal{L}} $ is quite modest (say $O(\log m)$), we may obtain distributions ${\mathcal{D}}''(R)$ over the routers in $\Lambda'_G(R)$, for which the maximum expected congestion on an edge of $G$ may be as large as $|\delta_G(R)|$, which is unacceptable. If we could ensure that the distributions ${\mathcal{D}}'(R)$ obtained in type-2 legal clustering are careful with respect to $v^*$, then this accumulation of congestion could be avoided, but unfortunately we do not know how to ensure that. In order to overcome this difficulty, we will carefully alternate between type-1 and type-2 legal clusterings. Specifically, we will require that a type-2 legal clustering contains a single distinguished cluster $R^*$, whose corresponding distribution ${\mathcal{D}}'(R^*)$ is careful with respect to $v^*$, and that $R^*$ contains a very large fraction of clusters of ${\mathcal{C}}'$. We will also require that a type-1 legal clustering ${\mathcal{R}}'$ of the graph associated with cluster $R^*$ is provided, and that for each cluster $R'\in {\mathcal{R}}'$, the number of clusters of ${\mathcal{C}}'$ contained in $R'$ is relatively small. This carefull alternation between type-1 and type-2 legal clusterings will allow us to compute distributions ${\mathcal{D}}'(S)$ over the routers of $\Lambda'_G(S)$ for each cluster $S\in {\mathcal{L}}$ of the laminar family that we construct, such that the expected congestion on every edge of $G$ due to the router drawn from the distribution is not too large. We now formally define a type-2 legal clustering.
\begin{definition}[Type-2 Legal Clustering]
Let $G'$ be a graph with a special vertex $v^*\in V(G')$, and let ${\mathcal{C}}'$ be a collection of disjoint vertex-induced subgraphs of $G'\setminus\set{v^*}$, that we call {basic clusters}.
A \emph{type-2 legal clustering} of $G'$ with respect to $v^*$ and ${\mathcal{C}}'$ consists of the following four ingredients:
\begin{enumerate}
\item a helpful clustering $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ of $G'$ with respect to $v^*$ and ${\mathcal{C}}'$;
\item a nice witness structure for the graph $G'_{|{\mathcal{R}}}$ with respect to the set ${\mathcal{C}}''$ of clusters, where ${\mathcal{C}}''$ contains every cluster $C\in {\mathcal{C}}'$ with $C\subseteq G'\setminus\textsf{left}(\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right})$;
\item a distinguished cluster $R^*\in {\mathcal{R}}$, that contains at least $\floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )|{\mathcal{C}}'|}$ clusters of ${\mathcal{C}}'$, such that the distribution ${\mathcal{D}}'(R^*)$ is careful with respect to $v^*$; and
\item a type-1 legal clustering $({\mathcal{R}}',\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}'})$ of
graph $G^*$, with respect to special vertex $v^{**}$, and cluster set ${\mathcal{C}}^*$, where $G^*$ is the graph that is obtained from graph $G'$ by contracting all vertices of $G'\setminus R^*$ into the special vertex $v^{**}$, and ${\mathcal{C}}^*$ contains all clusters $C\in {\mathcal{C}}'$ with $C\subseteq R^*$. We also require that every cluster $R'\in {\mathcal{R}}'$ contain at most $\floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )|{\mathcal{C}}'|}$ clusters of ${\mathcal{C}}'$.
\end{enumerate}
\end{definition}
The key ingerdient of the proof of \Cref{thm: advanced disengagement get nice instances} is the following theorem, that will allow us to gradually construct the desired laminar family ${\mathcal{L}}$ of clusters.
\begin{theorem}\label{thm: construct one level of laminar family}
There is an efficient randomized algorithm, whose input consists of:
\begin{itemize}
\item a graph $G'$, and a parameter $m$ that is greater than a sufficiently large constant, such that $|V(G')|,|E(G')|\leq m$;
\item a special vertex $v^*\in V(G')$, such that the cluster $G'\setminus\set{v^*}$ has the $\alpha_1$-bandwidth property in $G'$; and
\item a collection ${\mathcal{C}}'$ of disjoint vertex-induced subgraphs of $G'\setminus \set{v^*}$ called \emph{basic clusters}, such that every cluster $C\in {\mathcal{C}}'$ has the $\alpha_0$-bandwidth property, and $|{\mathcal{C}}'|\geq 2$.
\end{itemize}
The algorithm either returns FAIL, or computes a type-1 or a type-2 legal clustering of $G'$ with respect to $v^*$ and ${\mathcal{C}}'$. The probability that the algorithm returns FAIL is $1/m^8$. Moreover, if the algorithm computes a type-1 legal clustering $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$, then every cluster $R\in {\mathcal{R}}$ contains at most $\floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )|{\mathcal{C}}'|}$ clusters of ${\mathcal{C}}'$.
\end{theorem}
We prove \Cref{thm: construct one level of laminar family} in the remainder of this subsection, after we complete
the proof of \Cref{thm: advanced disengagement get nice instances} using it.
We will construct a laminar family ${\mathcal{L}}$ of clusters of graph $G$, in a top-down manner. For every cluster $R\in {\mathcal{L}}$, we will define a distribution ${\mathcal{D}}''(R)$ over external routers in $\Lambda'_G$, such that, for every edge $e\in E(G)\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}''(R)]{\cong_{G}({\mathcal{Q}}'(R),e)}\leq \beta^{O((\log m)^{3/4})}$. We will also define a partition $({\mathcal{L}}^{\operatorname{light}},{\mathcal{L}}^{\operatorname{light}})$ of the clusters of ${\mathcal{L}}$, and we will define, for each cluster $R\in {\mathcal{L}}^{\operatorname{light}}$ a distribution ${\mathcal{D}}(R)$ over the internal routers in $\Lambda_G(R)$, such that cluster $R$ is $\hat \beta$-light with respect to ${\mathcal{D}}(R)$, for $\hat \beta=2^{O((\log m)^{3/4}\log\log m)}$.
We will ensure that, with high probability, every cluser in ${\mathcal{L}}^{\operatorname{bad}}$ is $\hat \beta$-bad. Once we complete consructing the laminar family ${\mathcal{L}}$, we will apply algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace from \Cref{subsec: basic disengagement} to the resulting tuple $({\mathcal{L}},{\mathcal{L}}^{\operatorname{bad}}, {\mathcal{L}}^{\operatorname{light}}, \set{{\mathcal{D}}''(R)}_{R\in {\mathcal{L}}},\set{{\mathcal{D}}(R)}_{R\in {\mathcal{L}}^{\operatorname{light}}})$ to obtain the final collection ${\mathcal{I}}_1$ of instances. We will also provide a nice witness structure for each such resulting instance. We now proceed to describe tha algorithm for constructing the laminar family ${\mathcal{L}}$ of clusters.
Initially, we start with the laminar family ${\mathcal{L}}$ containing a single cluster -- graph $G$. Since $\Lambda'_G(G)=\emptyset$ (as $\delta_G(G)=\emptyset$), the distribution ${\mathcal{D}}'(G)$ is defined in a trivial way (e.g. it selects $\emptyset$ with probability $1$). We also let ${\mathcal{L}}^{\operatorname{light}}$ contain a single cluster -- the cluster $G$, whose distribution ${\mathcal{D}}(G)$ over internal routers is defined in a similar trivial way. Lastly, we set ${\mathcal{L}}^{\operatorname{bad}}=\emptyset$.
We simultaneously consider the partitioning tree $\tau({\mathcal{L}})$ associated with the laminar family ${\mathcal{L}}$ (see \Cref{subsubsec: laminar} for a definition). Initially, tree $\tau({\mathcal{L}})$ consists of a single vertex $v(G)$, associated with the cluster $G$. The algorithm then performs iterations, as long as there is some cluster $R\in {\mathcal{L}}$, whose corresponding vertex $v(R)$ is a leaf vertex in the tree $\tau({\mathcal{L}})$, and there are at least two clusters of ${\mathcal{C}}$ that are contained in $R$. We will ensure that every cluster $R'\in {\mathcal{L}}$ has the $\alpha_1$-bandwidth property in $G$. Notice that this trivially holds for the initial cluster $G$.
We now describe an interation for processing a cluster $R\in {\mathcal{L}}$. We assume that $v(R)$ is a leaf vertex in the current partitioning tree $\tau({\mathcal{L}})$, and that there are at least two clusters of ${\mathcal{C}}$ that are contained in $R$.
In order to process cluster $R$, we construct a graph $G'$, with a special vertex $v^*$, as follows. If $R=G$, then we let $G'$ be a graph that is obtained from $G$, by adding a new special vertex $v^*$ to it, that connects with an edge to an arbitrary fixed vertex $v_0\in V(G)$. Otherwise, if $R\subsetneq G$, then we let $G'$ be the graph that is obtained from $G$ by contracting all vertices of $V(G)\setminus V(R)$ into the special vertex $v^*$. Note that, since we are guaranteed that cluster $R$ has the $\alpha_1$-bandwidth property in $G$, cluster $G'\setminus\set{v^*}$ of $G'$ must have the $\alpha_1$-bandwidth property in $G'$ (in case where $R=G$ this property holds trivially, as $\delta_G'(G)$ contains a single edge). We let ${\mathcal{C}}'\subseteq {\mathcal{C}}$ be the set of all basic clusters $C\in {\mathcal{C}}$ with $C\subseteq R$. We then apply the algorithm from
\Cref{thm: construct one level of laminar family} to graph $G'$, special vertex $v^*$, and set ${\mathcal{C}}'$ of clusters; parameter $m$ remains the same as in the input to \Cref{thm: advanced disengagement get nice instances}. If the algorithm returns FAIL, then we terminate our algorithm with a FAIL. Otherwise, consider the legal clustering that the algorithm produces (which may be a type-1 or a type-2 legal clustering), and let ${\mathcal{R}}$ be the resulting set of clusters.
We add every cluster $R'\in {\mathcal{R}}$ to the laminar family ${\mathcal{L}}$, where it becomes a child cluster of cluster $R$. Recall that, from the properties of a helpful clustering, vertex $v^*$ may not lie in $R'$, so $R'\subseteq R$ must hold. Moreover, $R'$ must have the $\alpha_1$-bandwidth property in $G'$, and hence in $G$.
Recall that we also obtain a distribution ${\mathcal{D}}'(R')$ over external routers in $\Lambda'_{G'}(R')$, such that, for every edge $e\in E(G')\setminus E(R')$, $\expect[{\mathcal{Q}}'(R')\sim{\mathcal{D}}'(R')]{\cong_{G'}({\mathcal{Q}}'(R'),e)}\leq \beta$. Unfortunately, this distribution is not sufficiently good for us, since we need the distribution ${\mathcal{D}}'(R')$ to be over the collection $\Lambda'_G(R')$ of external routers in graph $G$, and not in graph $G'$. We show how to modify this distribution later.
Next, we process each cluster $R'\in {\mathcal{R}}$ one by one. Consider any such cluster $R'$.
We apply Algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace from \Cref{thm:algclassifycluster} to instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and cluster $R'$, that has the $\alpha_1$-bandwidth property in $G$, together with parameter $p=1/m^{10}$. Recall that the running time of the algorithm is $O(\operatorname{poly}(m\log m))$. If the algorithm returns FAIL, then we add $R'$ to the set ${\mathcal{L}}^{\operatorname{bad}}$ of clusters. Otherwise, the algorithm computes a distribution ${\mathcal{D}}(R')$ over internal routers in $\Lambda_G(R')$, such that cluster $R'$ is $\beta^*$-light with respect to ${\mathcal{D}}(R')$, where $\beta^*=2^{O(\sqrt{\log m}\cdot \log\log m)}$. We then add cluster $R'$ to set ${\mathcal{L}}^{\operatorname{light}}$.
Recall that, if cluster $R'$ is not $\eta^*$-bad, for $\eta^*=2^{O((\log m)^{3/4}\log\log m)}$, then the probability that the algorithm returns FAIL (that is, the algorithm \emph{errs}), is at most $1/m^{10}$. If the clustering ${\mathcal{R}}$ is a type-1 legal clustering, then we also \emph{mark} the vertex $v(R')$ in the decomposition tree $\tau({\mathcal{L}})$, to indicate that the distribution ${\mathcal{D}}'(R')$ is careful with respect to $v^*$. Otherwise, ${\mathcal{R}}'$ is a type-2 legal clustering, and we only mark vertex $v(R')$ if $R'=R^*$, where $R^*$ is the distinguished cluster. Recall that in this case, the distribution ${\mathcal{D}}'(R^*)$ over external routers of $R^*$ is also careful with respect to $v^*$.
If the algorithm from \Cref{thm: construct one level of laminar family} returned a type-2 legal clustering, then we also consider the type-1 legal clustering ${\mathcal{R}}'$ of $R^*$, that is given as part of the type-2 legal clustering of $R$. We process every cluster $R''\in {\mathcal{R}}'$ one by one. When cluster $R''$ is processed, we add it to the laminar family ${\mathcal{L}}$ and we add vertex $v(R'')$ to the partitioning tree $\tau({\mathcal{L}})$ as a child of vertex $v(R^*)$; we also mark vertex $v(R'')$ in the tree, to indicate that the distribution ${\mathcal{D}}'(R'')$ over the external routers of $R''$ is careful with respect to $v^{**}$. As before, we apply the Algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace from \Cref{thm:algclassifycluster} to instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and cluster $R''$, that has the $\alpha_1$-bandwidth property in $G$, together with parameter $p=1/m^{10}$, As before, if the algorithm returns FAIL, then we add $R''$ to ${\mathcal{L}}^{\operatorname{bad}}$, and otherwise we add it to ${\mathcal{L}}^{\operatorname{light}}$, together with the distribution ${\mathcal{D}}(R'')$ over internal routers in $\Lambda_G(R'')$, such that cluster $R''$ is $\beta^*$-light with respect to ${\mathcal{D}}(R'')$. This completes the description of the algorithm for constructing the laminar family ${\mathcal{L}}$ of clusters. We now establish some of its useful properties.
The following claim, whose proof appears in \Cref{subsec: bounding tree height} will be used to bound the height of the tree $\tau$, and the number of marked vertices on any root-to-leaf path.
\begin{claim}\label{claim: path length in decomposition tree}
Consider any root-to-leaf path $P$ in the decomposition tree $\tau({\mathcal{L}})$. Then $P$ contains at most $2^{O((\log m)^{3/4})}$ marked vertices, and at most $O(\log^{3/4} m)$ unmarked vertices. In particular, the depth of the tree $\tau({\mathcal{L}})$ is at most $2^{O((\log m)^{3/4})}$.
\end{claim}
Next, we provide an algorithm for computing, for each cluster $R\in {\mathcal{L}}$, the desired distribution ${\mathcal{D}}''(R)$ over the external routers in $\Lambda'_G(R)$. The proof of the following claim is somewhat technical, and is deferred to \Cref{subsec: external routers}.
\begin{claim}\label{claim: compose distributions}
There is an efficient algorithm that, given a cluster $R\in {\mathcal{L}}$, computes a distribution ${\mathcal{D}}''(R)$ over the external $R$-routers in $\Lambda'_G(R)$, such that, for every edge $e\in E(G)\setminus E(R)$ $$\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}''(R)]{\cong_{G}({\mathcal{Q}}'(R),e)}\leq \beta^{i+1},$$
where $i$ is the total number of unmarked vertices on the unique path in tree $\tau({\mathcal{L}})$, connecting $v(R)$ to the root of the tree.
\end{claim}
To summarize, if our algorithm did not return FAIL, we have now obtained a laminar family ${\mathcal{L}}$ of clusters of graph $G$, with $G\in {\mathcal{L}}$, so that the depth of the family ${\mathcal{L}}$ is at most $2^{O((\log m)^{3/4})}$.
We have also computed a partition $({\mathcal{L}}^{\operatorname{light}},{\mathcal{L}}^{\operatorname{bad}})$ of clusters in ${\mathcal{L}}$, and, for each cluster $R\in {\mathcal{L}}^{\operatorname{light}}$, a distribution ${\mathcal{D}}(R)$ over internal routers in $\Lambda_G(R)$, such that cluster $R$ is $\beta^*$-light with respect to ${\mathcal{D}}(R)$, for $\beta^*=2^{O(\sqrt{\log m}\cdot \log\log m)}$. We are also guaranteed that, with probability at least $1-1/m^9$, every cluster $R\in {\mathcal{L}}^{\operatorname{bad}}$ is $\eta^*$-bad, for $\eta^*=2^{O((\log m)^{3/4}\log\log m)}$. Lastly, we have computed, for every cluster $R\in {\mathcal{L}}$, a distribution ${\mathcal{D}}''(R)$ over external routers in $\Lambda'_G(R)$, such that, for every edge $e\in E(G)\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}''(R)]{\cong_{G}({\mathcal{Q}}'(R),e)}\leq \beta^{i+1}$, where $i$ is the total number of unmarked vertices on the unique path in tree $\tau({\mathcal{L}})$, connecting $v(R)$ to the root of the tree. Since, from \Cref{claim: path length in decomposition tree}, the number of such vertices is bounded by $O(\log^{3/4} m)$, we get that, for every edge $e\in E(G)\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}''(R)]{\cong_{G}({\mathcal{Q}}'(R),e)}\leq \beta^{O((\log m)^{3/4})}\leq 2^{O((\log m)^{3/4}\log\log m)}$, since $\beta=\log^{18}m$.
We apply algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace from \Cref{subsec: basic disengagement} to the resulting tuple $({\mathcal{L}}={\mathcal{L}}^{\operatorname{bad}}\cup {\mathcal{L}}^{\operatorname{light}}, \set{{\mathcal{D}}''(R)}_{R\in {\mathcal{L}}},\set{{\mathcal{D}}(R)}_{R\in {\mathcal{L}}^{\operatorname{light}}})$, to obtain the final collection ${\mathcal{I}}_1$ of subinstances of $I$.
Let $\hat \beta=2^{c(\log m)^{3/4}\log\log m}$ for some large enough constant $c$. We are then guaranteed that every cluster $R\in {\mathcal{L}}^{\operatorname{light}}$ is $\hat\beta$-light with respect to the distribution ${\mathcal{D}}(R)$, and that, for every cluster $R\in {\mathcal{L}}$ and every edge $e\in E(G)\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}''(R)]{\cong_{G}({\mathcal{Q}}'(R),e)}\leq \hat \beta$. We can set $c$ to be large enough so that $\eta^*\leq \hat \beta$ holds. We say that a bad event ${\cal{E}}$ happens if some cluster $R\in {\mathcal{L}}^{\operatorname{bad}}$ is not $\hat \beta$-bad. From the above discussion, the probability of ${\cal{E}}$ happening is at most $1/m^9$. If Event ${\cal{E}}$ does not happen, then, from \Cref{lem: disengagement final cost},
$\expect{\sum_{I'\in {\mathcal{I}}_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq O(\mathsf{dep}({\mathcal{L}})\cdot\hat \beta^2\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|))\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)) $.
If Event ${\cal{E}}$ happens (which happens with probability at most $1/m^9$), then clearly
$\expect{\sum_{I'\in {\mathcal{I}}_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq \sum_{I'=(G'',\Sigma'')\in {\mathcal{I}}_1}|E(G'')|^2\leq m^3$.
Therefore, overall, $\expect{\sum_{I'\in {\mathcal{I}}_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)) $
Additionally, from \Cref{lem: basic disengagement combining solutions},
there is an efficient algorithm, that, given, for each instance $I'\in {\mathcal{I}}$, a solution $\phi(I')$, computes a solution for instance $I$ of value at most $\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))$.
In order to prove that the algorithm computes a valid $2^{O((\log m)^{3/4}\log\log m)}$-decomposition of instance $I$, it is now sufficient to prove that $\sum_{I'=(G'',\Sigma'')\in {\mathcal{I}}_1}|E(G'')|\leq O(|E(G)|)$. We do so in the next claim, whose proof is deferred to \Cref{subsec:appx few edges}.
\begin{claim}\label{claim: few edges}
$\sum_{I'=(G'',\Sigma'')\in {\mathcal{I}}_1}|E(G'')|\leq O(|E(G)|)$.
\end{claim}
From the above discussion, if our algorithm did not return FAIL, it computed a valid $2^{O((\log m)^{3/4}\log\log m)}$-decomposition of instance $I$. Consider now any resulting instance $\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}_1$. From the definition of basic disengagment, this instance must correspond to some cluster $R\in {\mathcal{L}}$. Assume first that $R\neq G$, and let ${\mathcal{R}}$ be the set of all child clusters of $R$ in ${\mathcal{L}}$ (if $R$ corresponds to a leaf vertex of $\tau({\mathcal{L}})$, then ${\mathcal{R}}=\emptyset$). Recall that graph $\tilde G$ is obtained from graph $G$ by first contracting all vertices of $G\setminus R$ into a supernode $v^*$; denote the resulting graph by $G'$. We then contract every cluster $R'\in {\mathcal{R}}$ into a supernode, obtaining graph $\tilde G$. In other words, $\tilde G=G'_{|{\mathcal{R}}}$.
We now consider three cases. The first case is when ${\mathcal{R}}$ is a type-1 legal clustering for cluster $R$. In this case, there is at most one cluster $C\in {\mathcal{C}}$ that is contained in $G'\setminus \bigcup_{R'\in {\mathcal{R}}}R'$, from the definition of type-1 legal clustering. Therefore, at most one cluster $C$ of ${\mathcal{C}}$ is contained in $\tilde G$. We define the nice witness structure for graph $\tilde G$, with respect to the set ${\mathcal{C}}'$ of basic clusters, where ${\mathcal{C}}'=\set{C}$ if there is a cluster $C\in {\mathcal{C}}$ that is contained in $\tilde G$, and ${\mathcal{C}}'=\emptyset$ otherwise. We let $\tilde {\mathcal{S}}=\set{\tilde S_1}$, where $\tilde S_1=\hat G$, and we let $\tilde {\mathcal{S}}'=\set{\tilde S_1'}$, where $\tilde S_1'=C$ if ${\mathcal{C}}'=\set{C}$, and $\tilde S_1'=\set{v}$, where $v$ is an arbitrary vertex of $\tilde G$ otherwise. Note that, under these definitions, $\hat E=\emptyset$, and so we can set $\hat {\mathcal{P}}=\emptyset$. It is easy to verify that $(\tilde {\mathcal{S}},\tilde {\mathcal{S}}',\hat {\mathcal{P}})$ is a nice witness structure for $\tilde G$.
The second case is when cluster $R$ corresponds to a leaf vertex of tree $\tau({\mathcal{L}})$. From our algorithm, this means that there is at most one cluster $C\in {\mathcal{C}}$ with $C\subseteq R$. In this case, we define the nice witness structure for graph $\tilde G$ similarly to the first case.
Lastly, in the third case, when the algorithm from \Cref{thm: construct one level of laminar family} was applied to cluster $R$, it returned a type-2 legal clustering, with the corresponding cluster set ${\mathcal{R}}$. In this case, the algorithm also must produce a nice witness structure for the graph $G'_{|{\mathcal{R}}}=\tilde G$, with respect to the set ${\mathcal{C}}''$ of clusters, that contains every cluster $C\in {\mathcal{C}}$ with $C\subseteq G'\setminus\textsf{left}(\bigcup_{R'\in {\mathcal{R}}}V(R')\textsf{right})$. In other words, ${\mathcal{C}}''={\mathcal{C}}(\tilde G)$.
It remains to consider the case where $R=G$. As before, we let ${\mathcal{R}}$ denote the set of all child-clusters of cluster $R$. Recall that in this case, graph $\tilde G$ is obtained from graph $G$ by contracting every cluster $R'\in {\mathcal{R}}$ into a supernode. Graph $G'$ was obtained from graph $G$ by adding a special vertex $v^*$ that connects to some vertex $v_0\in V(G)$ with an edge. Therefore, graph $G'_{|{\mathcal{R}}}$ is a graph that is obtained from $\tilde G$ by adding a special vertex $v^*$ to it and connecting it to some vertex of $\tilde G$. Recall that vertex $v^*$ may not lie in any cluster of ${\mathcal{R}}$. We start by defining a nice structure for graph $G'_{|{\mathcal{R}}}$, with respect to the collection ${\mathcal{C}}''$ of clusters, that contains every cluster $C\in {\mathcal{C}}$ with $C\subseteq G'_{|{\mathcal{R}}}$ exactly as before (when we assumed that $R\neq G$). Since vertex $v^*$ has degree $1$, we can assume that no path in $\hat {\mathcal{P}}$ contains $v^*$. Moreover, if $v^*\in \tilde S'_i$, for some $\tilde S'_i\in \tilde {\mathcal{S}}'_i$, then cluster $\tilde S'_i\setminus\set{v^*}$ still has the $\alpha^*$-bandwidth property. By deleting vertex $v^*$ from the cluster of $\tilde {\mathcal{S}}$ to which it belongs, and also from a cluster of $\tilde {\mathcal{S}}'$ to which it belongs (if such a cluster exists), we obtain a nice witness structure for graph $\tilde G$, with respect to cluster set ${\mathcal{C}}''$, as required.
The algorithm may return FAIL if any application of the algorithm from \Cref{thm: construct one level of laminar family} returned FAIL. Since $|{\mathcal{L}}|\leq m$, and the probability that a singel application of the algorithm from \Cref{thm: construct one level of laminar family} returns FAIL is at most $1/m^8$, overall, the probability that the algorithm returns FAIL is at most $1/m^6$.
In order to complete the proof of \Cref{thm: advanced disengagement get nice instances} it is now enough to prove \Cref{thm: construct one level of laminar family}, which we do next.
\input{level-1-advanced-disengagement}
\input{level-2-advanced-disengagement}
\input{level-3-advanced-disengagement}
\subsection{Proof of \Cref{claim: compose algs}}
\label{apd: Proof of compose algs}
Denote $I=(G,\Sigma)$ and $m=|E(G)|$. Recall that from Property \ref{prop: few edges}, $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\leq m\cdot (\log m)^{O(1)}$, so in particular, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}'$, $|E(G')|\leq m\cdot (\log m)^{O(1)}$.
From the same property, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}'$, $\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}''(I')}|E(G'')|\leq |E(G')|\cdot (\log(|E(G')|))^{O(1)}\leq |E(G')|\cdot (\log m)^{O(1)}$. Therefore, altogether, we get that:
\[
\begin{split}
\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}}|E(G'')|& = \sum_{I'\in {\mathcal{I}}'}\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}''(I')}|E(G'')|\\
& \leq \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\cdot (\log m)^{O(1)}\\
& \leq m\cdot (\log m)^{O(1)},\\
\end{split}
\]
establishing Property \ref{prop: few edges}.
Next, we establish Property \ref{prop: modified expectation}, using the same property of Algorithms $\ensuremath{\mathsf{Alg}}\xspace_1$ and $\ensuremath{\mathsf{Alg}}\xspace_2$:
\iffalse{split form}
\[\begin{split}
\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}''}|E(G'')|&= \\
&=\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}''(I')}|E(G'')|\\
&\leq \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}O(|E(G')|)\\
&\leq O(|E(G)|).
\end{split}\]
\fi
\[\begin{split}
\expect{\sum_{I''\in {\mathcal{I}}''}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}&=\sum_{I'\in {\mathcal{I}}'}\expect{\sum_{I''\in {\mathcal{I}}''(I')}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}\\
&\leq \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\expect{\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\bigg)\cdot \nu''}\\
&= \nu''\cdot\textsf{left}(\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\textsf{right})+ \nu''\cdot\expect{\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\\
&\leq O(\nu''\cdot m\cdot (\log m)^{O(1)})+\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot (\nu'\nu'')\\
&\leq \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \nu''\cdot\max\set{2\nu',(\log m)^{O(1)}}\\
&\leq \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \nu.
\end{split}\]
Lastly, we establish Property \ref{prop: alg to put together} of Algorithm $\ensuremath{\mathsf{Alg}}\xspace$, using the same property of algorithms $\ensuremath{\mathsf{Alg}}\xspace_1$ and $\ensuremath{\mathsf{Alg}}\xspace_2$.
Assume that we are given, for every instance $I''\in {\mathcal{I}}''$, a feasible solution $\phi(I'')$ to $I''$.
We process instances $I'\in {\mathcal{I}}'$ one by one. For each such instance, we apply Algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}}''(I'))$ that is given by Property \ref{prop: alg to put together} of the decomposition ${\mathcal{I}}''(I')$ to solutions $\phi(I'')$ of instances $I''\in {\mathcal{I}}''(I')$, to obtain a solution $\phi(I')$ to instance $I'$ of cost at most $O\textsf{left} (\sum_{I''\in {\mathcal{I}}''(I')}\mathsf{cr}(\phi(I''))\textsf{right} )$.
We then apply the algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}}')$, given by Property \ref{prop: alg to put together} of the decomposition ${\mathcal{I}}'$ of $I$, to the resulting solutions $\phi(I')$ for instances $I'\in {\mathcal{I}}'$, to obtain a solution $\phi(I)$ to instance $I$, whose cost is at most:
$$O\textsf{left} (\sum_{ I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I'))\textsf{right} )\le O\textsf{left} (\sum_{ I''\in {\mathcal{I}}}\mathsf{cr}(\phi(I''))\textsf{right} ).$$
\section{Proof of Lemma~\ref{lem:local_non_int}}
\label{apd:locally_non_interfering}
In this section we provide the proof of Lemma~\ref{lem:local_non_int}.
We view the paths in $\tilde{\mathcal{P}}$ as originating from $v$ and ending at $v'$.
First, we perform the standard re-routing to the paths of $\tilde{\mathcal{P}}$ and obtain another set $\tilde{\mathcal{P}}'$ of paths, such that $E(\tilde{\mathcal{P}}')\subseteq E(\tilde{\mathcal{P}})$, and the graph consisting of directed paths in $\tilde{\mathcal{P}}'$ is a DAG.
This defines a topological order on vertices of $G$.
We will sequentially perform, for each vertex $u\in V(G)\setminus\set{v,v'}$ according to this topological order, a \emph{route-swap} operation to all the paths that contains $u$, that is defined as follows.
Consider the graph $G[\tilde{\mathcal{P}}',u]$ and think of the edges in $G[\tilde{\mathcal{P}}',u]$ as directed edges, so each edge is an \emph{in-edge} if it is directed towards $u$, and is an \emph{out-edge} otherwise. We first recompute a matching between the in-edges and the out-edges, such that the paths formed by the new pairs of edges are locally non-interfering at $u$. Let $(e_{in}, e_{out})$ be a pair in the matching, where $e_{in}$ originally belongs to the path $P$ of $\tilde{{\mathcal{P}}'}$ and $e_{out}$ originally belongs to the path $P$ of $\tilde{{\mathcal{P}}'}$. We define a new path from this pair $(e_{in}, e_{out})$ as the concatenation of the subpath of $P$ between $v$ and $u$, and the subpath of $P'$ between $u$ and $v'$. We replace the paths in $\tilde{\mathcal{P}}'$ that contains $u$ with all the new paths that we defined according the the matching. This completes the description of a route-swap operation.
See Figure~\ref{fig:route_swap} for an illustration.
\begin{figure}[h]
\label{fig:route_swap}
\centering
\subfigure[Before: the paths are not locally non-interfering at $u$.]
{\scalebox{0.33}{\includegraphics{figs/route_swap_1}}}
\hspace{0cm}
\subfigure[After: the paths are locally non-interfering at $u$.]
{\scalebox{0.33}{\includegraphics{figs/route_swap_2}}}
\caption{An illustration of a route-swap operation.}
\end{figure}
It is not hard to see that, after the route-swap operation at $u$, the paths in $\tilde{\mathcal{P}}'$ are locally non-interfering at $u$. Therefore, if we denote by ${\mathcal{P}}$ the resulting set of paths that we obtain, then the paths in ${\mathcal{P}}$ are locally non-interfering at all vertices of $V(G)\setminus\set{v,v'}$. This completes the proof of Lemma~\ref{lem:local_non_int}.
\section{Proofs Omitted from \Cref{sec: many paths}}
\label{Proofs Omitted from many paths}
\subsection{Proof of \Cref{obs: combine solutions for split}}
\label{subsec: combine solutions for split}
We denote $I_F=(G_F,\Sigma_F)$, $I_{F_1}=(G_{F_1},\Sigma_{F_1}), I_{F_2}=(G_{F_2},\Sigma_{F_2})$. We also denote the skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$, the augmneting structure ${\mathcal{A}}=(\Pi_F=\set{P_1,P_2},\set{b_u}_{u\in V(K\cup P_1\cup P_2)}, \psi')$, and $K'=K\cup P_1\cup P_2$.
We construct the solution $\phi$ to instance $I_F$ gradually. We start by letting
$\phi$ be the drawing $\psi'$ of graph $K'$, and then gradually modify it.
Consider the clean solution $\phi_1$ to the $F_1$-instance $I_{F_1}$. From the definition of augmentation, there must be a single connected component $C\in {\mathcal{C}}(F_1)$, with $P_1\cup P_2\subseteq C$. We let $\sigma$ be a closed curve that follows the boundary of $C$ just inside the face $F_1$, and we let $D$ be the disc whose boundary is $\sigma$, and whose interior contains the images of all vertices of $G_{F_1}$. We can define a similar disc $D'$ in the current drawing $\phi$ of $K'$. We then move the contents of the disc $D$ in drawing $\phi_1$ to disc $D'$ in drawing $\phi$, so that the boundaries of both discs coinside. In this way, we ``plant'' the drawing $\phi_1$ of $G_{F_1}$ inside face $F_1$ in $\phi$ (we need to extend the images of the edges incident to the vertices of $C$ so that they terminate at their endpoints in $C$, similarly to their drawing in $\phi_1$). We use a similar procedure to ``plant'' the drawing $\phi_2$ of $G_{F_2}$ inside face $F_2$ of $\phi$.
So far we have constructed a drawing $\phi$ of graph $G_F\setminus E^{\mathsf{del}}$, in which all vertices and edges are drawn in face $F$, the rotation system induced by $\Sigma_F$ is respected, and the number of crossings is $\mathsf{cr}(\phi_1)+\mathsf{cr}(\phi_2)$. In order to complete the drawing $\phi$ of graph $G_F$, it remains to ``insert'' the images of the edges of $E^{\mathsf{del}}$ into this drawing.
We do so using the following lemma from \cite{chuzhoy2020towards}.
\begin{lemma}[Lemma 9.2 of \cite{chuzhoy2020towards}]
There is an efficient algorithm, that, given an instance $(H,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, a subset $E'\subseteq E(H)$ of edges of $H$, and a drawing $\phi$ of graph $H\setminus E'$ that respects $\Sigma$, compute a drawing $\phi'$ of $H$ that respects $\Sigma$, such that $\mathsf{cr}(\phi')\le \mathsf{cr}(\phi)+|E'|\cdot |E(H)|$.
\end{lemma}
There is one subtlety in using the above lemma directly in order to insert the edges of $E^{\mathsf{del}}$ into the drawing $\phi$: we need to ensure that the drawing remains clean, so the images of the newly inserted edges may not cross the images of the edges of $\partial(F)$. This is easy to achieve, for example, by making a large number copies of every edge in $\partial(F)$, so that the copy of each such edge $e$ that is furthest from the boundary of $F$ will not be crossed by any newly inserted edges (as otherwise we would create a crossing with every copy of $e$, resulting in too many crossings). Another equivalent approach is to first contract every connected component of $\partial(F)$ into a supernode, insert the edges of $E^{\mathsf{del}}$ into the resulting drawing of the contracted instance, and then un-contract the supernodes. In either case, we obtain a clean drawing $\phi$ of instance $I_F$, in which the number of crossings is bounded by $\mathsf{cr}(\phi_1)+\mathsf{cr}(\phi_2)+|E^{\mathsf{del}}|\cdot |E(G_F)|$.
\subsection{Proof of \Cref{claim: find potential augmentors}}
\label{subsec: proof of finding potential augmentors}
Consider the contracted instance $\hat I_F=(\hat G_F,\hat \Sigma_F)$. Since this instance is wide, there is a vertex $v\in V(\hat G_F)$, a partition $(E_1,E_2)$ of the edges of $\delta_{\hat G_F}(v)$, such that the edges of $E_1$ appear consequently in the rotation ${\mathcal{O}}_v\in \hat \Sigma_F$, and so do the edges of $E_2$. Additionally, there is a collection ${\mathcal{P}}$ of at least $\floor{m/\mu^{{50}}}$ simple edge-disjoint cycles in $\hat G_F$, such that every cycle $P\in {\mathcal{P}}$ contains one edge of $E_1$ and one edge of $E_2$.
Note that we can compute the vertex $v$, the partition $(E_1,E_2)$ of the edges of $\delta_{\hat G_F}(v)$, and the set ${\mathcal{P}}$ of cycles with the above properties efficiently. We partition the cycles of ${\mathcal{P}}$ into three subsets: set ${\mathcal{P}}^0$ contains all cycles $P\in {\mathcal{P}}$, such that no vertex of $O$ is a supernode. Set ${\mathcal{P}}^1$ contains all cycles $P\in{\mathcal{P}}$, such that exactly one vertex of $P$ is a supernode. Set ${\mathcal{P}}^2$ contains all remaining cycles. Clearly, at least one of the three cycle sets ${\mathcal{P}}^1,{\mathcal{P}}^2,{\mathcal{P}}^3$ must contain at least $\frac{|E(G_F)|}{6\mu^{50}}$, cycles. We now consider three cases, depending on which of the three sets contains this many cycles.
\paragraph{Case 1: $|{\mathcal{P}}^0|\geq \frac{|E(G_F)|}{6\mu^{50}}$.}
In this case, every path in ${\mathcal{P}}^0$ is also a path in graph $G_F$, and path set ${\mathcal{P}}^0$ is a type-1 potential $F$-augmenting set for ${\mathcal K}$. We return path set ${\mathcal{P}}^0$.
\paragraph{Case 2: $|{\mathcal{P}}^1|\geq \frac{|E(G_F)|}{6\mu^{50}}$.}
Assume first that vertex $v$ is a supernode, that is, for some connected component $C\in {\mathcal{C}}(F)$, $v=v_C$. Note that $\delta_{\hat G_F}(C)\subseteq E^F(C)$. Recall that the ordering ${\mathcal{O}}_{v_C}\in \hat\Sigma_F$ of the edges of $\delta_{\hat G_F}(C)$ is induced by the ordering ${\mathcal{O}}^F(C)$ of the edges in $E^F(C)$. Therefore, there is a partition $(E_1',E_2')$ of the edges of $E^F(C)$ with $E_1\subseteq E_1'$, $E_2\subseteq E_2'$, such that the edges of $E_1'$ appear consecutively in the ordering ${\mathcal{O}}^F(C)$. We fix any such partition $(E_1',E_2')$. Every cycle in ${\mathcal{P}}^1$ then naturally defines a path in graph $G_F$, that has an edge of $E_1'$ as its first edge, and an edge of $E_2'$ as its last edge. Since no other supernodes may lie on the paths of ${\mathcal{P}}^1$, we obtain a valid type-2 potential $F$-augmenting set of cardinality at least
$\frac{|E(G_F)|}{6\mu^{50}}$.
Assume now that $v$ is not a supernode.
For every connected component $C\in {\mathcal{C}}(F)$, let ${\mathcal{P}}^1(C)\subseteq {\mathcal{P}}^1$ be the subset of cycles containing the supernode $v_C$.
Since $|{\mathcal{C}}(F)|=r$, there is some component $C\in {\mathcal{C}}(F)$ with $|{\mathcal{P}}^1(C)|\geq \frac{|E(G_F)|}{6r\mu^{50}}$. We fix this connected component $C$, and we denote ${\mathcal{Q}}={\mathcal{P}}^1(C)$.
Note that every cycle $Q\in {\mathcal{Q}}$ can be thought of as a concatenation of two edge-disjoint paths, both of which connect $v_C$ to $v$. We denote by $Q_1$ the path containing an edge of $E_1$, and by $Q_2$ the path containing an edge of $E_2$. We then denote ${\mathcal{Q}}_1=\set{Q_1\mid Q\in {\mathcal{Q}}}$ and ${\mathcal{Q}}_2=\set{Q_2\mid Q\in {\mathcal{Q}}}$. Let $E'_1$ denote the set of all edges of $\hat G_F$ that are incident to vertex $v_C$, and lie on the paths of ${\mathcal{Q}}_1$. We define edge set $E'_2$ for the paths of ${\mathcal{Q}}_2$ similarly. Note that the edges of $E'_1\cup E'_2$ in graph $\hat G_F$ correspond to a subset of edges of $E^F(C)$ in graph $G_F$. We say that a partition $(E_1'',E_2'')$ of the edges of $E^F(C)$ is a \emph{good partition} iff the following hold:
\begin{itemize}
\item the edges of $E_1''$ appear consecutively in the ordering ${\mathcal{O}}^F(C)$;
\item $|E_1''\cap E_1'|\geq |E_1'|/2$; and
\item $|E_2''\cap E_2'|\geq 3|E_2'|/4$.
\end{itemize}
Note that we can efficiently compute a good partition of the edges in $E^F(C)$ if such a partition exists, and we can also efficiently determine whether a good partition of the edges in $E^F(C)$ exists.
Assume first that a good partition of the edges in $E^F(C)$ exists, and let $(E_1'',E_2'')$ be a good partition that we have computed. Let ${\mathcal{Q}}^*\subseteq {\mathcal{Q}}$ be the set of all cycles $Q\in {\mathcal{Q}}$, such that path $Q_1$ contains an edge of $E_1''$, and path $Q_2$ contains an edge of $E_2''$. From the definition of a good partition, it is easy to verify that $|{\mathcal{Q}}^*|\geq |{\mathcal{Q}}|/4\geq \frac{|E(G_F)|}{24r\mu^{50}}$ must hold. We then return path set ${\mathcal{Q}}^*$, which must be a valid type-2 potential $F$-augmenting set. If we establish that no good partition of the edges in $E^F(C)$ exists, then we report that in every solution $\phi$ to instance $I_F$ with $\mathsf{cr}(\phi)\leq \frac{|E(G_F)|^2}{2^{20}\cdot \mu^{250}\cdot r^2}$, either some pair of edges in $\partial(F)$ cross, or the number of crossings in which edges of $\partial(F)$ participate is at least $\frac{|E(G_F)|}{64\cdot \mu^{50}\cdot r}$. We now prove that in this case, this indeed must indeed be true.
\begin{observation}\label{obs: no augmenting set}
If no good partition of the edges in $E^F(C)$ exists, then in every solution $\phi$ to instance $I_F$ in every solution $\phi$ to instance $I_F$ with $\mathsf{cr}(\phi)\leq \frac{|E(G_F)|^2}{2^{20}\cdot \mu^{250}\cdot r^2}$, either some pair of edges in $\partial(F)$ cross, or the number of crossings in which edges of $\partial(F)$ participate is at least $\frac{|E(G_F)|}{64\cdot \mu^{50}\cdot r}$.
\end{observation}
\begin{proof}
Denote $|{\mathcal{Q}}|=k$, and denote the edges of $E_1$ by $e_1,\ldots,e_k$, and the edges of $E_2$ by $e_{k+1,\ldots,e_{2k}}$, so that the edges $e_1,\ldots,e_{2k}$ appear in the rotation ${\mathcal{O}}_v\in \hat \Sigma_F$ in the order of their indices (recall that the edges in $E_1$ must appear consecutively in ${\mathcal{O}}_v$).
Assume for contradiction that there exists a drawing $\phi$ of $I_F$ with $\mathsf{cr}(\phi)\leq \frac{|E(G_F)|^2}{2^{20}\cdot \mu^{250}\cdot r^2}$, such that no pair of edges of $\partial(F)$ cross, and the edges of $\partial(F)$ participate in fewer than $\frac{|E(G_F)|}{64r\mu^{50}}$ crossings.
We say that a path $Q_1\in {\mathcal{Q}}_1$ is \emph{good} iff the edges of $Q_1$ participate in fewer than $k/32$ crossings in $\phi$. Let ${\mathcal{Q}}'_1\subseteq {\mathcal{Q}}_1$ be the set containing all good paths. Observe that, $|{\mathcal{Q}}'_1|\geq 7k/8$ must hold. Indeed, otherwise, since the paths in ${\mathcal{Q}}_1$ are edge-disjoint we get that:
\[\mathsf{cr}(\phi')\geq |{\mathcal{Q}}_1\setminus{\mathcal{Q}}'_1|\cdot \frac k {32}\geq \frac{k}{8}\cdot \frac{k}{256}>\frac{|E(G_F)|^2}{2^{20}\cdot \mu^{250}\cdot r^2}, \]
since $k\geq \frac{|E(G_F)|}{6r\mu^{50}}$.
We conclude that $|{\mathcal{Q}}'_1|\geq 7k/8$. Let ${\mathcal{Q}}''_1\subseteq {\mathcal{Q}}'_1$ be the set of all good paths $Q_1\in {\mathcal{Q}}'_1$, such that no crossing $(e,e')_p$ in $\phi$ exists with $e\in E(Q_1)$ and $e'\in E(K)$. Since we have assumed that the edges of $\partial(F)$ participate in at most $\frac{|E(G_F)|}{64r\mu^{50}}$ crossings in $\phi$, $|{\mathcal{Q}}''_1|\geq 3k/4$.
Let $1\leq i\leq k$ be the smallest index such that the path $Q_1\in {\mathcal{Q}}''_1$ contains the edge $e_i\in E_1$, and let $1\leq j\leq k$ be the largest index such that the path $Q_1'\in {\mathcal{Q}}''_1$ contains the edge $e_j\in E_1$.
Since $|{\mathcal{Q}}''_1|\geq 3k/4$, $j-i\geq 3k/4-1$ must hold.
Since paths $Q_1,Q_1'$ are good, there are at most $k/16$ paths $Q_1''\in {\mathcal{Q}}''_1$, such that there is a crossing in $\phi$ between an edge of $E(Q_1'')$ and an edge of $E(Q_1)\cup E(Q_1')$. Let $\tilde {\mathcal{Q}}_1\subseteq {\mathcal{Q}}_1''\setminus\set{Q_1,Q_1'}$ be the set of all paths $Q_1''$, such that there is no crossing in $\phi$ between an edge of $E(Q_1'')$ and an edge of $E(Q_1)\cup E(Q_1')$. From the above discussion, $|\tilde {\mathcal{Q}}_1|\geq k/2$.
Let $\tilde {\mathcal{Q}}_2\subseteq {\mathcal{Q}}_2$ be the set of all paths $Q_2\in {\mathcal{Q}}_2$, such that there is no crossing in $\phi$ between an edge of $E(Q_2)$ and an edge of $E(Q_1)\cup E(Q_1')\cup E(C)$. Using the same reasoning as above, $|\tilde {\mathcal{Q}}_2|\geq |{\mathcal{Q}}_2|-k/16-\frac{|E(G_F)|}{64r\mu^{50}}\geq 3k/4$.
Let $\psi$ be a drawing that is obtained from $\phi$ by deleting the images of all vertices and edges from it, except for the vertices and edges of $Q_1\cup Q_1'\cup C$.
Since we have assumed that the edges of $\partial(F)$ cannot cross each other in $\phi$, and since the edges of $Q_1\cup Q_1'$ may not cross the edges of $C$, the only possible crossings in $\psi$ are between edges of $E(Q_1)\cup E(Q_1')$.
Note that vertex $v$ lies on the boundary of exactly two faces of drawing $\psi$, that we denote by $F_1$ and $F_2$, respectively. For every edge $e_{\ell}\in E_1$ with $i<\ell<j$, the segment of $\phi(e)$ that is contained in the tiny $v$-disc $D_{\phi}(v)$ must then lie in $F_1$, and for every edge $e\in E_2$, the segment of $\phi(e)$ that is contained in the tiny $v$-disc $D_{\phi}(v)$ must then lie in $F_2$ (up to switching the names of the two faces). \mynote{would be good to add a figure here.}
Therefore, for every path $\tilde Q_1\in \tilde {\mathcal{Q}}_1$, the image of the first edge of $\tilde Q_1$ in $\phi$ must intersect $F_1$, and for every path $\tilde Q_2\in \tilde {\mathcal{Q}}_2$, the image of the first edge of $\tilde Q_2$ in $\phi$ must intersect $F_2$. Since the edges that belong to the paths in $\tilde {\mathcal{Q}}_1\cup \tilde {\mathcal{Q}}_2$ may not cross edges of $E(Q_1)\cup E(Q'_1)\cup E(C)$ in drawing $\phi$, we get that, for every path $\tilde Q_1\in \tilde {\mathcal{Q}}_1$, the image of $\tilde Q_1$ in $\phi$ must be contained in $F_1$, and for every path $\tilde Q_2\in \tilde {\mathcal{Q}}_2$, the image of $\tilde Q_2$ in $\phi$ must be contained in $F_2$.
Let $\tilde E_1$ be the set of all edges of $E^F(C)$ that lie on the paths in $\tilde {\mathcal{Q}}_1$, and let $\tilde E_2$ be defined similarly for $\tilde {\mathcal{Q}}_2$. Since the drawing of $C$ induced by $\phi$ is planar, there must be a partition $(E_1'',E_2'')$ of the edges of $E^F(C)$, such that the edges of $E''_1$ appear consecutively in the ordering ${\mathcal{O}}^F(C)$; $\tilde E_1\subseteq E''_1$, and $\tilde E_2\subseteq E_2''$. Since $|E''_1|\geq |\tilde E_1|\geq k/2$, while $|E''_2|\geq |\tilde E_2|\geq 3k/4$, we get that $(E_1'',E_2'')$ is a good partition of $E^F(C)$.
\end{proof}
\paragraph{Case 3: $|{\mathcal{P}}^2|\geq \frac{|E(G_F)|}{6\mu^{50}}$.}
Consider some cycle $P\in {\mathcal{P}}^2$, and recall that at least two vertices of $P$ are supernodes. We can then mark all supernode vertices on cycle $P$, and use them to partition $P$ into a collection ${\mathcal{R}}(P)$ of edge-disjoint paths, such that, for every path $R\in {\mathcal{R}}(P)$, the endpoints of $R$ are two distinct supernodes, and none of the inner vertices of $R$ is a supernode. Note that $|{\mathcal{R}}(P)|\geq 2$ must hold. Let ${\mathcal{R}}=\bigcup_{P\in {\mathcal{P}}^2}{\mathcal{R}}(P)$, so $|{\mathcal{R}}|\geq 2|{\mathcal{P}}_2|$. Then there must be two distinct connected component $C,C'\in {\mathcal{C}}(F)$, and a subset ${\mathcal{R}}(C)\subseteq {\mathcal{R}}$ of at least $\frac{|{\mathcal{R}}|}{r^2|}\geq \frac{|E(G_F)|}{6r^2\cdot \mu^{50}}$ paths, such that every path $R\in {\mathcal{R}}(C)$ originates at vertex $v_C$, terminates at vertex $v_{C'}$, and contains no supernodes as inner vertices. Since the paths in ${\mathcal{R}}(C)$ do not contain supernodes as inner vertices, they naturally define a collection ${\mathcal{R}}'$ of edge-disjoint paths in graph $G_F$, where every path $R\in {\mathcal{R}}'$ connects a vertex of $C$ to a vertex of $C'$ and does not contain vertices of $\partial(F)$ as inner vertices. Therefore, ${\mathcal{R}}'$ is a type-3 potential $F$-augmenting path set, whose cardinality is at least $\frac{|E(G_F)|}{6r^2\cdot \mu^{50}}$, as required.
\section{Proofs Omitted from \Cref{sec: many paths}}
\label{Proofs Omitted from many paths}
\subsection{Proof of \Cref{claim: find potential augmentors}}
\label{subsec: proof of finding potential augmentors}
Consider the ${\mathcal{J}}$-contracted instance $\hat I =(\hat G ,\hat \Sigma )$. Since it is a wide instance, there is a high-degree vertex $v^*\in V(\hat G )$, a partition $(E_1,E_2)$ of the edges of $\delta_{\hat G }(v^*)$, such that the edges of $E_1$ appear consequently in the rotation ${\mathcal{O}}_{v^*}\in \hat \Sigma $, and a collection ${\mathcal{R}}$ of at least $\floor{|E(\hat G)|/\mu^{50}}=\floor{\hat m(I)/\mu^{50}}$ simple edge-disjoint cycles in $\hat G$, such that every cycle $P\in {\mathcal{R}}$ contains one edge of $E_1$ and one edge of $E_2$.
Note that we can compute the vertex $v^*$, the partition $(E_1,E_2)$ of the edges of $\delta_{\hat G }(v^*)$, and the set ${\mathcal{R}}$ of cycles with the above properties efficiently.
Assume first that $v^*=v_J$. In this case, $\delta_{\hat G}(v^*)=\delta_G(J)$, so $(E_1,E_2)$ is also a partition of the edges of $\delta_G(J)$. From the definition of a ${\mathcal{J}}$-contracted instance, the rotation ${\mathcal{O}}_{v^*}\in \hat \Sigma$ is identical to the ordering ${\mathcal{O}}(J)$ of the edges of $\delta_{\hat G}(v^*)=\delta_G(J)$. Therefore, edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}(J)$.
The set ${\mathcal{R}}$ of cycles in graph $\hat G$ then naturally defines a set ${\mathcal{P}}$ of simple paths in graph $G$, where every path $P'\in {\mathcal{P}}$ has an edge of $E_1$ as its first edge, an edge of $E_2$ as its last edge, and it is internally disjoint from $J$. We discard arbitrary paths from ${\mathcal{P}}$ until $|{\mathcal{P}}|=\floor{\frac{\hat m(I)}{\mu^{50}}}$ holds, obtaining the desired set of promising paths.
From now on we assume that $v^*\neq v_J$, that is, $v^*$ is a vertex of $G$. From the definition of a high-degree vertex,
$\deg_G(v)\geq \frac{\hat m(I)}{\mu^4}$. Therefore, there is a collection ${\mathcal{Q}}$ of at least $\ceil{\frac{2\hat m(I)}{\mu^{50}}}$ edge-disjoint paths in $G$ connecting $v^*$ to vertices of $J$, and we can compute such a collection of paths efficiently using standard Maximum $s$-$t$ Flow. We can assume w.l.o.g. that every path in ${\mathcal{Q}}$ is internally disjoint from $J$, and we view the paths in ${\mathcal{Q}}$ as being directed away from $v^*$. We can then compute a partition $(E_1',E_2')$ of the edges of $\delta_G(J)$, such that the edges of $E_1'$ appear consecutively in the ordering ${\mathcal{O}}(J)$, and there are two subsets ${\mathcal{Q}}_1,{\mathcal{Q}}_2\subseteq {\mathcal{Q}}$ of paths of cardinality $\floor{\frac{\hat m(I)}{\mu^{50}}}$ each, such that the last edge of every path in ${\mathcal{Q}}_1$ lies in $E_1'$, and the last edge of every path in ${\mathcal{Q}}_2$ lies in $E_2'$. By arbitrarily matching the paths in ${\mathcal{Q}}_1$ to the paths in ${\mathcal{Q}}_2$ and concatenating the pairs of matched paths, we obtain a collection ${\mathcal{P}}$ of edge-disjoint paths, each of which has an edge of $E_1'$ as its first edge, an edge of $E_2'$ as its last edge, and is internally disjoint from $J$. We discard paths from ${\mathcal{Q}}$ until $|{\mathcal{Q}}|=\floor{\frac{\hat m(I)}{\mu^{50}}}$ holds, obtaining the desired promising set of paths.
\subsection{Proof of \Cref{obs: combine solutions for split}}
\label{subsec: combine solutions for split}
We denote the input instances by $I=(G,\Sigma)$, $I_{1}=(G_{1},\Sigma_{1})$, and $I_{2}=(G_{2},\Sigma_{2})$. We denote the core structure by ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$, and the ${\mathcal{J}}$-enhancement structure by ${\mathcal{A}}=\set{P,\set{b_u}_{u\in V(J')},\rho'}$, where $J'=J\cup P$.
We let $({\mathcal{J}}_1,{\mathcal{J}}_2)$ be the split of the core structure ${\mathcal{J}}$ via the enhancement structure ${\mathcal{A}}$, and we denote ${\mathcal{J}}_1=(J_1,\set{b_u}_{u\in V(J_1)},\rho_{J_1}, F_1)$ and ${\mathcal{J}}_2=(J_2,\set{b_u}_{u\in V(J_2)},\rho_{J_2}, F_2)$.
Let $E^{\mathsf{del}}=E(G)\setminus(E(G_1)\cup E(G_2))$, let $G'=G\setminus E^{\mathsf{del}}$, and let $\Sigma'$ be the rotation system for $G'$ that is induced by $\Sigma$. We first compute a ${\mathcal{J}}$-clean solution $\phi'$ to instance $I'=(G',\Sigma')$ of \ensuremath{\mathsf{MCNwRS}}\xspace with $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi_1)+\mathsf{cr}(\phi_2)$, and then insert the edges of $E^{\mathsf{del}}$ into this drawing, to obtain the final solution $\phi$ to instance $I$.
We now describe the construction of the solution $\phi'$ to instance $I'$.
Consider the drawing $\rho'$ of graph $J'$, and the faces $F_1,F_2$ of this drawing that were used to define the split $({\mathcal{J}}_1,{\mathcal{J}}_2)$ of the core structure ${\mathcal{J}}$.
Recall that the drawing of graph $J_1$ induced by $\rho'$ is precisely $\rho_{J_1}$, and the drawing of graph $J_2$ induced by $\rho'$ is precisely $\rho_{J_2}$.
Consider now the ${\mathcal{J}}_1$-clean drawing $\phi_1$ of graph $G_1$ on the sphere. The drawing of $J_1$ induced by $\phi_1$ is identical to $\rho_{J_1}$, and the images of all edges and vertices of $G_1$ are contained in region $F_1$ of this drawing. We plant the drawing $\phi_1$ of $G_1$ into the face $F_1$ of drawing $\rho'$, so that the images of the edges and the vertices of $J_1$ in both drawings coincide, and the images of all vertices and edges of $G_1$ appear in face $F_1$. We similarly plant drawing $\phi_2$ of $G_2$ inside face $F_2$ of $\rho'$, obtaining a solution $\phi'$ to instance $I'$, whose cost is bounded by $\mathsf{cr}(\phi_1)+\mathsf{cr}(\phi_2)$. Since the images of all edges and vertices of $G'$ are contained in region $F^*_{\rho_J}=F_1\cup F_2$, this drawing is ${\mathcal{J}}$-clean.
In order to complete the construction of the solution $\phi$ to instance $I$, it remains to ``insert'' the images of the edges of $E^{\mathsf{del}}$ into $\phi'$.
We do so using \Cref{lem: edge insertion}.
There is, however, one subtlety in using this lemma directly in order to insert the edges of $E^{\mathsf{del}}$ into the drawing $\phi'$: we need to ensure that the drawing remains ${\mathcal{J}}$-clean, so the images of the newly inserted edges may not cross the images of the edges of $J$. This is easy to achieve, for example, by first contracting core $J$ into a supernode and modifying the drawing $\phi$ to obtain a drawing of the resulting graph in a natural way. We then insert the edges of $E^{\mathsf{del}}$ into this drawing of the contracted instance using the algorithm from \Cref{lem: edge insertion}, and then un-contract the supernode $v_J$. We obtain a ${\mathcal{J}}$-clean solution $\phi$ to instance $I$, whose number of crossings is bounded by $\mathsf{cr}(\phi_1)+\mathsf{cr}(\phi_2)+|E^{\mathsf{del}}|\cdot |E(G)|$.
\subsection{Proof of \Cref{claim: curves orderings crossings}}
\label{subsec: curves orderings crossings}
The proof of \Cref{claim: curves orderings crossings} is similar to the proof of Claim 9.9 in \cite{chuzhoy2020towards}.
For all $1\leq i\leq 4k+2$, we denote by $\gamma_i$ the image of path $P_i$ in $\phi$, that is, $\gamma_i$ is the concatenation of the images of the edges of $P_i$. Clearly, all curves in the resulting set $\Gamma=\set{\gamma_1,\ldots,\gamma_{4k+2}}$ connect $\phi(u)$ to $\phi(v)$, and they enter the image of $u$ in $\phi$ in the order of their indices.
Notice that the curves $\gamma_i\in \Gamma$ are not necessarily simple: if a pair of edges lying on path $P_i$ cross at some point $p$, then curve $\gamma_i$ crosses itself at point $p$.
In such a case, point $p$ may not lie on any other curve in $\Gamma$.
Next, we will slightly modify the curves in $\Gamma$ by ``nudging'' them in the vicinity of their common vertices. In order to do so, we consider every vertex $x\in V(G)\setminus\set{u,v}$ that belongs to at least two paths of ${\mathcal{P}}$ one by one.
We now describe an iteration when a vertex $x\in V(G)\setminus\set{u,v}$ is processed.
Let ${\mathcal{P}}^x\subseteq {\mathcal{P}}$ be the set of all paths containing vertex $x$. Note that $x$ must be an inner vertex on each such path. For convenience, we denote ${\mathcal{Q}}^x=\set{P_{i_1},\ldots,P_{i_z}}$. Consider the tiny $x$-disc $D(x)=D_{\phi}(x)$. For all $1\leq j\leq z$, denote by $s_j$ and $t_j$ the two points on the curve $\gamma_{i_j}$ that lie on the boundary of disc $D(x)$. We use the algorithm from \Cref{claim: curves in a disc} to compute a collection $\set{\sigma_1,\ldots,\sigma_z}$ of curves, such that, for all $1\leq j\leq z$, curve $\sigma_j$ connects $s_j$ to $t_j$, and the interior of the curve is contained in the interior of $D(x)$. Recall that every pair of resulting curves crosses at most once, and every point in the interior of $D(x)$ may be contained in at most two curves. Moreover, a pair $\sigma_r,\sigma_q$ of such curves may only cross if the two pairs $(s_r,t_r),(s_q,t_q)$ of points on the boundary of $D(x)$ cross. This, in turn, may only happen if paths $P_{i_r},P_{i_q}$ have a transversal intersection at vertex $x$, which is impossible. Therefore, the curves $\sigma_1,\ldots,\sigma_z$ do not cross each other.
For all $1\leq j\leq z$, we modify the curve $\gamma_{i_j}$, by replacing the segment of the curve that is contained in disc $D(x)$ with $\sigma_j$.
Once every vertex $x\in V(G)\setminus\set{u,v}$ is processed, we obtain the final set $\Gamma'=\set{\gamma'_1,\ldots,\gamma'_{4k+2}}$ of curves.
Notice that for any pair $1\leq j<j'\leq 4k+2$ of indices, curves $\gamma'_j,\gamma'_{j'}$ cross if and only if there is a crossing $(e,e')_{p}$ in $\phi$ with $e\in E(P_j)$ and $e'\in E(P_{j'})$.
It is now enough to prove that curves $\gamma'_1$ and $\gamma'_{2k+1}$ do not cross. Assume for contradiction that the two curves cross. Among all crossing points between these two curves, let $p'$ be the point that is closest to $\phi(u)$ on $\gamma'_1$. Let $\lambda$ be the segment of $\gamma'_1$ from $\phi(u)$ to $p'$, and let $\lambda'$ be defined similarly for $\gamma'_{2k+1}$. We modify curves $\lambda$ and $\lambda'$ to remove all their self-loops, so the curves become simple. Note that curves $\lambda$ and $\lambda'$ both originate at $\phi(u)$ and terminate at $p'$, and they do not cross. Let $\lambda^*$ be the simple closed curve obtained from the union of $\lambda$ and $\lambda'$. Curve $\lambda^*$ partitions the sphere into two internally disjoint discs, that we denote by $D$ and $D'$.
Since the curves $\gamma'_1,\ldots,\gamma'_{4k+2}$ enter the image of $u$ in $\phi$ in the order of their indices, either (i) for all $1< i<2k+1$, the intersection of $\gamma'_i$ with the tiny $u$-disc $D_{\phi}(u)$ lies in $D$, and for all $2k+1< i\leq 4k+2$, the intersection of $\gamma'_i$ with $D_{\phi}(u)$ lies in $D'$, or (ii) the opposite is true. We assumme without loss of generality that it is the former.
Note that point $\phi(v)$ must lie in the interior of one of these discs -- we assume without loss of generality that it is $D'$.
Consider now some index $1<i<2k+1$. Recall that the segment of $\gamma'_i$ that is contained in $D_{\phi}(u)$ is contained in $D$, while point $\phi(v)$, that is an endpoint of $\gamma'_i$, lies in the interior of $D'$. Therefore, there must be some point $r$ that lies on $\gamma_i$ and on $\lambda^*$, such that the two curves have a transversal intersection at $r$.
This point may not be $p'$, since curves $\gamma'_1$ and $\gamma'_{2k+1}$ have a crossing at $p'$, and this crossing corresponds to a crossing between an edge of $P_1$ and an edge of $P_{2k+1}$ in $\phi$. Therefore, $\gamma_i'$ has a transversal crossing with either $\lambda$ or $\lambda'$. In the former case, there is a crossing between an edge of $P_1$ and an edge of $P_i$, while in the latter case there is a crossing between an edge of $P_{2k+1}$ and an edge of $P_i$. We conclude that for all $1<i<2k+1$, some edge of $P_i$ must cross an image of an edge of $P_1$ or of $P_{2k+1}$ in $\phi$. Since the edges of $P_1$ participate in at most $k$ crossings, and so do the edges of $P_{2k+1}$, and since we have assumed that an edge of $P_1$ crosses an edge of $P_{2k+1}$, this is impossible.
\iffalse
Let $\gamma$ be a curve on the sphere. If $p$ is a point at which $\gamma$ crosses itself, then we say that $p$ is a \emph{self-crossing point} of $\gamma$.
Consider now a pair $\gamma,\gamma'$ of curves, such that the number of points that lie on both curves (that is, their crossing points) is finite. Let $p$ be a point lying on both $\gamma$ and $\gamma'$, and assume that $p$ is not a self-crossing point for either of the two curves. Consider the tiny $p$-disc $D$. Let $s,t$ denote the two points of $\gamma$ that lie on the boundary of $D$, and let $s',t'$ denote the two points of $\gamma'$ lying on the boundary of $D$. We say that the crossing of curves $\gamma,\gamma'$ at $p$ is \emph{transversal} iff points $s,s',t',t'$ appear in this circular order on the boundary of $D$. Otherwise, we say that the crossing is \emph{non-transversal}. If a crossing of $\gamma$ and $\gamma'$ at point $p$ is non-transversal, we sometimes say that $\gamma$ and $\gamma'$ \emph{touch} at point $p$. If the crossing is transversal, we may say that $\gamma$ and $\gamma'$ \emph{cross but do not touch} at point $p$.
We will use the following simple auxiliary claim several times. The proof is similar to the proof of Claim 9.9 in \cite{chuzhoy2020towards} and is deferred to Section \ref{subsec: curves orderings crossings} of Appendix.
\begin{claim}\label{claim: curves orderings crossings}
Let $\Gamma=\set{\gamma_1,\ldots,\gamma_{4k+2}}$ be a collection of curves on the sphere for which the following hold:
\begin{itemize}
\item the number of points $r$ that belong to two or more curves of $\Gamma$ is finite, and so is the number of self-crossing points of each curve;
\item all curves in $\Gamma$ originate at point $p$ and terminate at point $q$;
\item points $p,q$ may not be inner points on any curve in $\Gamma$;
\item curves $\gamma_1,\ldots,\gamma_{4k+2}$ enter point $p$ in the order of their indices;
\item if two distinct curves $\gamma_i,\gamma_j\in \Gamma$ have a transversal intersection at point $r$, then $r$ does not belong to any curve in $\Gamma\setminus\set{\gamma_i,\gamma_j}$; and
\item if point $p$ is a self-crossing point of some curve $\gamma\in \Gamma$, then $p$ may not lie on another curve of $\Gamma$.
\end{itemize}
Assume further that curve $\gamma_1$ has at most $k$ transversal crossings with other curves in $\Gamma$, and the same is true for curve $\gamma_{2k+1}$. Then there are no transversal crossings between $\gamma_1$ and $\gamma_{2k+1}$.
\end{claim}
\subsection{Proof of \Cref{claim: curves orderings crossings}}
\label{subsec: curves orderings crossings}
The proof of \Cref{claim: curves orderings crossings} is similar to the proof of Claim 9.9 in \cite{chuzhoy2020towards}.
Assume for contradiction that there is some point $p'$, such that curves $\gamma_1$ and $\gamma_{2k+1}$ have a transversal crossing at $p'$. Among all such points $p'$, we choose the one that is closest to point $p$ on $\gamma_1$. Let $\gamma$ be the segment of $\gamma_1$ from $p$ to $p'$, and let $\gamma'$ be defined similarly for $\gamma_{2k+1}$. We modify curves $\gamma$ and $\gamma'$ to remove all their self-loops, so the curves become simple. Note that curves $\gamma$ and $\gamma'$ both originate at $p$ and terminate at $p'$, and any crossing between these two curves must be non-transversal.
We will further slightly modify the curves $\gamma$ and $\gamma'$, to eliminate all crossing points between them. In order to do so, we iteratively consider every crossing point $r$ between $\gamma$ and $\gamma'$. Let $D(r)$ be a tiny $r$-disc. Denote by $s,t$ the points of $\gamma$ lying on the boundary of $D(r)$, and assume that $s$ appears closer to $p$ than $t$ on $\gamma$. Denote by $s',t'$ the points of $\gamma'$ lying on the boundary of $D(r)$, and assume that $s'$ appears closer to $p$ than $t'$ on $\gamma$. Since the crossing of $\gamma$ and $\gamma'$ at $r$ is non-transversal, we can define two disjoint segments of the boundary of $D(r)$: segemnt $\sigma$ connecting $s$ to $t$, and segment $\sigma'$ connecting $s'$ to $t'$. We modify curve $\gamma$ by replacing the segment of $\gamma$ between $s$ and $t$ with $\sigma$, and we similarly modify curve $\gamma'$ by replacing the segment of $\gamma'$ between $s'$ and $t'$ with $\sigma'$ (see Figure~\ref{fig: curve_separation}).
\begin{figure}[h]
\centering
\subfigure[The curves $\gamma$ and $\gamma'$ touch at the image of $u$ in $\tilde\phi$.]{\scalebox{0.13}{\includegraphics{figs/curve_separation_before.jpg}}}
\hspace{0.1cm}
\subfigure[The curves $\hat \gamma,\hat \gamma'$ share only endpoints $p$ and $\tilde\phi(v')$.]{
\scalebox{0.13}{\includegraphics{figs/curve_separation_after.jpg}}}
\caption{Modifying curves $\gamma,\gamma'$ around their crossing point $r$. \mynote{could you please update this figure so it matches the notation of this claim? and also delete the green boundary around the blue circles?}. \label{fig: curve_separation}}
\end{figure}
Once every crossing point of $\gamma$ and $\gamma'$ is processed, we let $\gamma^*$ be the simple closed curve obtained from the union of $\gamma$ and $\gamma'$. Note that $\gamma^*$ partitions the sphere into two internally disjoint discs, that we denote by $D$ and $D'$.
Since the curves $\gamma_1,\ldots,\gamma_{4k+2}$ enter point $p$ in the order of their indices, either (i) for all $1< i<2k+1$, the intersection of $\gamma_i$ with the tiny $p$-disc $D(p)$ lies in $D$, and for all $2k+1< i\leq 4k+2$, the intersection of $\gamma_i$ with $D(p)$ lies in $D'$, or (ii) the opposite is true. We assumme without loss of generality that it is the former.
Note that point $q$ must lie in the interior of one of these discs -- we assume without loss of generality that it is $D'$.
We now prove that for all $1<i<2k+1$, there must be a transversal intersection between $\gamma_i$ and at least one of the curves $\gamma_1,\gamma_{2k+1}$. This will lead to contradiction, since one of the curves $\gamma_1,\gamma_{2k+1}$ must participate in more than $k$ transversal crossings (including the crossing between these two curves). In order to complete the proof of \Cref{claim: curves orderings crossings}, it is now enough to prove the following observation.
\begin{observation}
For all $1<i<2k+1$, there is a transversal intersection between $\gamma_i$ and at least one of the curves $\gamma_1,\gamma_{2k+1}$.
\end{observation}
\begin{proof}
Fix an index $1<i<2k+1$. Recall that the segment of $\gamma_i$ that is contained in $D(p)$ is contained in $D$, while point $q$, that is an endpoint of $\gamma_i$ lies in the interior of $D'$. Therefore, there must be some point $z$ that lies on $\gamma_i$ and on $\gamma^*$, such that the two curves have a transversal intersection at $z$. We let $z$ be such a point that is closest to $p$ on $\gamma_i$.
Note that $z\neq p$ must hold, since point $p$ may not be an inner point of $\gamma_i$. It is also impossible that $z=p'$, since curves $\gamma_1,\gamma_{2k+1}$ have a transversal intersection at point $p'$, so $p'$ may not lie on $\gamma_i$.
If point $z$ lies on $\gamma_1\cap \gamma$ or $\gamma_{2k+1}\cap \gamma'$, then $\gamma_i$ must have a transversal intersection with $\gamma_1$ or $\gamma_{2k+1}$.
Assume now that this is not the case. Then there must be some point $r$ at which $\gamma_1$ and $\gamma_{2k+1}$ touch, with point $z$ lying on the boundary of disc $D(r)$, either on segment $\sigma$ or on segment $\sigma'$ that we have defined. Assume w.l.o.g. that $z\in \sigma$.
We will show that $\gamma_i$ and $\gamma_1$ must have a tranversal crossing at point $r$.
Assume for contradiction that this is not the case.
As before, let $s,t$ be the points of $\gamma_1$ lying on the boundary of $D(r)$, with $s$ lying closer to $p$ than $t$ on $\gamma_1$. Similarly, let $s',t'$ the points of $\gamma_{2k+1}$ lying on the boundary of $D(r)$, and assume that $s'$ appears closer to $p$ than $t'$ on $\gamma_{2k+1}$. Recall that $\sigma$ and $\sigma'$ are two disjoint segments of the boundary of $D(R)$, with $\sigma$ connecting $s$ to $t$, and $\sigma'$ connecting $s'$ to $t'$. Since $\gamma_i$ intersects $D(r)$, point $r$ must lie on $\gamma_i$. Let $s''$ and $t''$ denote the two points on $\gamma_i$ that lie on the boundary of $D(r)$. Since $\gamma_i$ contains a point of $\sigma$, and since we have assumed that the crossing of $\gamma_1$ and $\gamma_i$ at $r$ is non-transversal, points $s,t,s'',t''$ must appear on the boundary of $D(r)$ in one of the following two orderings: either $(s,s'',t'',t)$, or $(s,t'',s'',t)$. In other words, both $s'',t''\in \sigma$.
Let $\lambda_1$ be the segment of $\gamma$ from $p$ to $s$. Let $\lambda_2$ be the segment of the boundary of $D(r)$ between $s$ and $s'$ that is internally disjoint from $\sigma$. Let $\lambda_3$ be the segment of $\gamma'$ from $s'$ to $p$, and let $\lambda^*$ be a simple closed curve x
\end{proof}
\fi
\subsection{Proof of \Cref{claim: bound unlucky paths}}
\label{sec: bound unlucky paths}
Assume for contradiction that there is a vertex $x\in V(G)\setminus V(J)$, and a set ${\mathcal{Q}}\subseteq {\mathcal{P}}^*$ of $\ceil{\frac{512\mu^{13b}\mathsf{cr}(\phi)}{m}}$ good paths that are unlucky with respect to $x$. We denote ${\mathcal{Q}}=\set{Q_1,\ldots, Q_{\lambda}}$, where ${\lambda}=\ceil{\frac{512\mu^{13b}\mathsf{cr}(\phi)}{m}}$. Let ${\lambda}'=4\cdot \ceil{{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}}$, so ${\lambda}/{\lambda}'\geq 16$. For all $1\leq i\leq {\lambda}$, we denote by $\hat e_i$ the first edge on path $Q_i$ that is incident to vertex $x$, and by $\hat e_i'$ the edge following $\hat e_i$ on path $Q_i$. We assume that the paths in ${\mathcal{Q}}$ are indexed so that the edges $\hat e_1,\ldots,\hat e_\lambda$ appear in the rotation ${\mathcal{O}}_x\in \Sigma$ in the order of their indices.
Recall that we are given a partition $(E_1,E_2)$ of the edges of $\delta_G(J)$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}(J)$, and every path in ${\mathcal{P}}$ has an edge of $E_1$ as its first edge, and an edge of $E_2$ as its last edge. From our construction, every path in ${\mathcal{P}}^*$, and hence in ${\mathcal{Q}}$, has an edge of $E_1$ as its first edge, and an edge of $E_2$ as its last edge. Consider the solution $\phi$ to instance $I$, that is ${\mathcal{J}}$-valid. Let $\phi'$ be the drawing that is obtained from $\phi$ after we delete all edges and vertices from it, except for those lying in $J$, and on the paths of ${\mathcal{Q}}$. Since all paths in ${\mathcal{Q}}$ are good, there are no crossings in $\phi'$ in which the edges of $J$ are involved. Let $D(J)$ be the disc associated with core $J$ in $\phi$. This disc contains the image of $J$ in $\phi'$ in its interior, and its boundary follows closely the drawing of $J$.
Notice that the image of every path $Q\in {\mathcal{Q}}$ in $\phi$ must intersect the interior of the region $F^*$, from the definition of a valid core structure (see \Cref{def: valid core 2}), and since each such path contains edges incident to vertices of $J$. Therefore, the image of every path $Q\in {\mathcal{Q}}$ in $\phi'$ is contained in the region $F^*$. We can then ensure that no crossing points of $\phi'$ are contained in disc $D(J)$; the only vertices and edges whose images in $\phi'$ are contained in $D(J)$ are the vertices and edges of $J$; and the only other edges whose images intersect $D(J)$ in $\phi'$ are the edges of $\delta_G(J)$ that lie on the paths of ${\mathcal{Q}}$. Moreover, for each such edge $e$, the intersection of $\phi'(e)$ and $D(J)$ is a simple curve.
We partition the boundary of disc $D(J)$ into two segment $\sigma$ and $\sigma'$, such that $\sigma$ contains all intersection points of the boundary of $D(J)$ with the images of the edges in $E_1$ in $\phi$, while $\sigma'$ contains all intersection points of the boundary of $D(J)$ with the images of the edges in $E_2$ in $\phi$ (see \Cref{fig: proof520_3}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.13]{figs/proof520_3.jpg}
\caption{Partitioning the boundary of disc $D(J)$ into segments $\sigma$ (black) and $\sigma'$ (red).
Edges of $E_1$ are shown in light blue and edges of $E_2$ are shown in green.}\label{fig: proof520_3}
\end{figure}
Let $\phi''$ be the drawing that is obtained from $\phi'$ by deleting the images of the vertices and the edges of $J$ from it, and deleting the segment of the image of every edge $e\in \delta_G(J)$ that is contained in the interior of $D(J)$. We then contract segment $\sigma$ into a point $p$, and segment $\sigma'$ into a point $q$, so that the image of every path $Q\in {\mathcal{Q}}$ in the resulting drawing connects $p$ to $q$, and for every edge $e\in \bigcup_{Q\in {\mathcal{Q}}}E(Q)$, the number of crossings in which $e$ participates in $\phi''$ is bounded by the number of crossings in which $e$ participates in $\phi'$.
For all $1\leq i\leq {\lambda}$, we let $R_i$ be the subpath of $Q_i$ from its first vertex to $x$, so $R_i$ contains an edge of $E_1$, and let ${\mathcal{R}}=\set{R_i\mid 1\leq i\leq {\lambda}}$.
We also denote ${\mathcal{Q}}'=\set{Q_{i\cdot {\lambda}'}\mid 1\leq i\leq 16}$, and, for all $1\leq i\leq 16$, we let $\tilde R_i=R_{i\cdot {\lambda}'}$ -- the subpath of path $Q_{i\lambda'}$, from its first endpoint to vertex $x$. Let $\tilde {\mathcal{R}}=\set{\tilde R_1,\ldots,\tilde R_{16}}$.
Recall that all paths in ${\mathcal{Q}}$ are good, and so each such path participates in at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}$ crossings. Since for every pair $P,P'\in {\mathcal{P}}^*$ of paths, for every vertex $v\in V(P)\cap V(P')$ with $v\not\in V(J)$, the intersection of $P$ and $P'$ at $v$ is non-transversal with respect to $\Sigma$, the paths in set ${\mathcal{R}}$ are non-transversal with respect to $\Sigma$, so are the paths in $\tilde {\mathcal{R}}$. Therefore, from \Cref{claim: curves orderings crossings}, for any pair $2\cdot \ceil{\frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}}< i<j\leq {\lambda}$ of indices with $j-i> 2\cdot\ceil{ \frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}}$, there is no crossing in $\phi''$ between an edge of $R_i$ and an edge of $R_j$. In particular, since ${\lambda}'=4\cdot \ceil{{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}}$, there are no crossings in $\phi''$ between pairs of edges lying in distinct paths of $\tilde {\mathcal{R}}$.
For $1\leq i\leq 16$, we let $\gamma_i$ be the curve that is obtained from the image of path $\tilde R_i$ in $\phi''$, after removing all self-loops. Consider the resulting collection $\Gamma=\set{\gamma_1,\ldots,\gamma_{16}}$ of curves. All curves in $\Gamma$ originate at point $p$ and terminate at the image of $x$ in $\phi''$. The curves do not cross themselves or each other, and they enter the image of $x$ in the order of their indices. The curves in $\Gamma$ partition the sphere into $16$ regions, that we denote by $\tilde F_1,\ldots,\tilde F_{16}$. Region $\tilde F_1$ has curves $\gamma_1$ and $\gamma_{16}$ as its boundaries, and for $1<i\leq 16$, region $\tilde F_i$ has curves $\gamma_{i-1}$ and $\gamma_i$ as its boundaries. Note that point $q$ must be contained in the interior of one of these regions. We assume without loss of generality that it is $\tilde F_1$ (as otherwise we could re-index the paths of ${\mathcal{Q}}$ accordingly).
Next, we consider the path $Q^*=Q_{8\lambda'+1}$, and we prove that path $Q^*$ is not unlucky for vertex $x$, reaching a contradiction.
\begin{observation}\label{obs: not unlucky}
Path $Q^*$ is not unlucky for vertex $x$.
\end{observation}
\begin{proof}
Let $e^*$, $e^{**}$ be the edges of $Q^*$ that are incident to $x$, with $e^*$ lying before $e^{**}$ on the path.
Let $\hat E_1(x)\subseteq \delta_G(x)$ be the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies between $e^*$ and $e^{**}$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$. Let $\hat E_2(x)\subseteq \delta_G(x)$ be the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies between $e^{**}$ and $e^{*}$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$. It is enough to prove that $|\hat E_1(x)|,|\hat E_2(x)|\geq \frac{\mathsf{cr}(\phi)\mu^{13b}}{m}$.
For all $2\leq i\leq 16$, let ${\mathcal{Q}}_i=\set{Q_j\mid (i-1){\lambda}'<j< i\lambda'}$. Recall that ${\lambda}'=4\cdot \ceil{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}$, and every path $\tilde R_i\in \tilde {\mathcal{R}}$ participates in at most $ \frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}$ crossings in $\phi''$, since the paths in ${\mathcal{Q}}$ are good. Therefore, there must be a subset ${\mathcal{Q}}'_i\subseteq {\mathcal{Q}}_i$ of at least $2\cdot \ceil{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}$ paths, such that for every path $Q_j\in {\mathcal{Q}}'_i$, the image of $Q_j$ in $\phi$ does not cross the curves $\gamma_{i-1}$ and $\gamma_i$. In particular, the image of every path in set $\set{R_j\mid Q_j\in {\mathcal{Q}}'_i}$ is contained in region $\tilde F_i$ in $\phi''$.
Recall that we have defined a set ${\mathcal{Q}}'_4\subseteq {\mathcal{Q}}_4$ of at least $2\cdot \ceil{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}$ paths, where for each path $Q_j\in {\mathcal{Q}}'_4$, the image of the corresponding path $R_j$ is contained in $\tilde F_{4}$. Recall that, for every path $Q_j\in {\mathcal{Q}}$, we denoted by
$\hat e_j$ the first edge on path $Q_j$ that is incident to $x$.
We denote by $E^L=\set{\hat e_j\mid Q_j\in {\mathcal{Q}}'_4}$. Clearly, for every edge $\hat e_j\in E^L$, the image of $\hat e_j$ in $\phi''$ lies in region $\tilde F_4$.
Similarly, we have defined a set
${\mathcal{Q}}'_{14}\subseteq {\mathcal{Q}}_{14}$ of at least $2\cdot \ceil{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}$ paths, where for each path $Q_j\in {\mathcal{Q}}'_{14}$, the image of the corresponding path $R_j$ is contained in $\tilde F_{14}$.
We denote by $E^R=\set{\hat e_j\mid Q_j\in {\mathcal{Q}}'_{14}}$. Clearly, for every edge $\hat e_j\in E^R$, the image of $\hat e_j$ in $\phi''$ lies in region $\tilde F_{14}$.
It is also easy to see that the imgage of edge $e^*$ must be contained in $\tilde F_7\cup \tilde F_8\cup \tilde F_9\cup \tilde F_{10}$ (as otherwise path $Q^*$ would need to cross more than $\frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}$ other paths in ${\mathcal{Q}}$, for example, the paths of ${\mathcal{Q}}'_7$, or the paths of ${\mathcal{Q}}'_{10}$).
Lastly, we show that the intersection of the image of edge $e^{**}$ and the tiny $x$-disc $D_{\phi''}(x)$, that we denote by $\sigma(e^{**})$, must be contained in $\tilde F_2\cup \tilde F_1\cup \tilde F_{16}$. Note that, if this is the case, then either (i) $E^L\subseteq \hat E_1(x)$ and $E^R\subseteq \hat E_2(x)$; or (ii) $E^R\subseteq \hat E_1(x)$ and $E^L\subseteq \hat E_2(x)$ hold. In either case, since $|E^R|,|E^L|\geq
2\cdot \ceil{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}$, path $Q^*$ is not unlucky for $x$.
It now remains to prove that $\sigma(e^{**})$ is contained in $\tilde F_2\cup \tilde F_1\cup \tilde F_{16}$. Assume otherwise. From the definition of tiny $x$-disc, $\sigma(e^{**})$ must be contained in some region $\tilde F_i$, for $3\leq i\leq 15$. Recall that we have defined a subset ${\mathcal{Q}}'_2\subseteq {\mathcal{Q}}_2$ of at least $2\cdot \ceil{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}$ paths, such that, for every path $Q_j\in {\mathcal{Q}}'_2$, the image of the corresponding path $R_j\in {\mathcal{R}}$ is contained in region $\tilde F_2$. We have also defined a collection ${\mathcal{Q}}'_{16}\subseteq {\mathcal{Q}}_{16}$ of at least
$2\cdot \ceil{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}$ paths, such that, for every path $Q_j\in {\mathcal{Q}}'_{16}$, the image of the corresponding path $R_j\in {\mathcal{R}}$ is contained in region $\tilde F_{16}$.
Consider now the segment $\gamma^*$ of the image of path $Q^{*}$ in $\phi''$ from vertex $x$ to point $q$. Since $\sigma(e^{**})$ is contained in $\bigcup_{3\leq i\leq 15}\tilde F_i$, while point $q$ is contained in $\tilde F_1$, curve $\gamma^*$ has to either cross the image of every path in $\set{R_j\mid Q_j\in {\mathcal{Q}}'_2}$, or it has to cross the image of every path in $\set{R_j\mid Q_j\in {\mathcal{Q}}'_{16}}$. In either case, since the paths of ${\mathcal{Q}}$ are non-transversal with respect to $\Sigma$, the edges of path $Q^*$ must participate in at least $2\cdot \ceil{\frac{4\mu^{13b}\mathsf{cr}(\phi)}{m}}$ crossings, contradicting the fact that $Q^*$ is a good path.
\end{proof}
\subsection{Proof of \Cref{claim: new drawing}}
\label{sec:getting new drawing}
We start with $\phi'$ being the drawing of graph $G'$ that is induced by $\phi$.
Since bad event ${\cal{E}}_1$ did not happen, drawing $\phi'$ does not contain crossings between edges of $E(P^*)$ and edges of $E(J)$, but it may contain crossings between pairs of edges in $E(P^*)$. We next show how to modify this drawing in order to eliminate all such crossings.
Let $\gamma$ be the image of the path $P^*$ in $\phi'$. Notice that $ \gamma$ is either a closed or an open curve, that may cross itself in a number of points.
Recall that path $P^*$ is internally disjont from $J$, and there are no crossings in $\phi'$ between edges of $P^*$ and edges of $J$. Moreover, from the definition of valid core structure (see \Cref{def: valid core 2}), and since $\phi$ is a ${\mathcal{J}}$-valid drawing of $G$, $\gamma$ must intersect the interior of the region $F^*$. Therefore, $\gamma\subseteq F^*$ must hold. In order to obtain the desired final drawing of graph $G'$, we will only modify the images of the edges and vertices that lie on $P^*$, with the new images contained in $F^*$, so that the resulting drawing of $G'$ is $\phi$-compatible.
We can partition the curve $\gamma$ into a collection $\Gamma$ of curves, for which the following hold. First, there is a single special curve $\gamma^*\in \Gamma$, which is either a simple closed or a simple open curve. In the former case, $\gamma^*$ contains the image of exactly one vertex of $J$, and in the latter case, the endpoints of $\gamma^*$ are images of two distinct vertices of $J$. All other curves in $\Gamma$ are simple closed curves. For every pair $\gamma_1,\gamma_2\in \Gamma$ of distinct curves, $\gamma_1$ and $\gamma_2$ may share at most one point, and that point must be a crossing point between a pair of edges of $E(P^*)$ in $\phi'$. We ensure that every point of $ \gamma$ lies on at least one curve of $\Gamma$. Since all curves in $\Gamma$ are simple, no curve in $\Gamma$ may cross itself.
We need the following observation.
\begin{observation}\label{obs: no heavy vertices on loops}
If neither of the Events ${\cal{E}}_1,{\cal{E}}_3$ happenned, then for every curve $\gamma'\in \Gamma\setminus\set{\gamma^*}$, every vertex $x$ with $\phi'(x)\in \gamma'$ is a light vertex.
\end{observation}
\begin{proof}
Assume otherwise. Let $\gamma'\in \Gamma\setminus\set{\gamma^*}$ be a curve, and $x$ a vertex with $\phi'(x)\in \gamma'$, such that $x$ is a heavy vertex. Denote $\phi'(x)$ by $p$, and notice that point $p$ may not lie on any other curves in $\Gamma$ (since every point shared by a pair of curves in $\Gamma$ is a crossing point between a pair of edges from $E(P^*)$.)
Let $D=D_{\phi'}(x)$ be a tiny $x$-disc. For every edge $\hat e$ that is incident to $x$, we denote by $\sigma(\hat e)$ the intersection of $\phi'(\hat e)$ with $D$. Let $e,e'$ be the two edges of $P^*$ that are incident to $x$.
Denote by $\hat E_1(x)\subseteq \delta_G(x)$ the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies strictly between $e$ and $e'$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$. Let $\hat E_2(x)\subseteq \delta_G(x)$ be the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies strictly between $e'$ and $e$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$. Since Event ${\cal{E}}_3$ did not happen, path $P$ may not be unlucky with respect to $x$, so $|\hat E_1(x)|,|\hat E_2(x)|\geq \frac{\mathsf{cr}(\phi)\mu^{13b}}{m}$ holds.
Curve $\gamma'$ partitions the sphere into two regions, that we denote by $\tilde F$ and $\tilde F'$. Since the curves $\sigma(e),\sigma(e')$ are contained in $\gamma'$, it must be the case that either (i) for every edge $\hat e\in \hat E_1(x)$, $\sigma(\hat e)\subseteq \tilde F$, and for every edge $\hat e\in \hat E_2(x)$, $\sigma(\hat e)\subseteq \tilde F'$, or (ii) the opposite is true. We assume w.l.o.g. that it is the former. Note that, since $\gamma$ is disjoint from the image of the core $J$ in $\phi'$ (except for its endpoints), the image of $J$ in $\phi'$ must be contained either in the interior of $\tilde F$, or in the interior of $\tilde F'$; we assume w.l.o.g. that it is the former. Consider now some edge $\hat e\in \hat E_2(x)$, and let $\hat P\in {\mathcal{P}}^*$ be a path that contains $\hat e$. Observe that $\sigma(\hat e)\subseteq \tilde F'$, while both endpoints of $\hat P$ belong to $J$, whose image lies in the interior of $\tilde F$. Therefore, the image of path $\hat P$ in $\phi'$ must cross the curve $\gamma'$. Let $q$ be a point of $\gamma'$ that lies on the image of $\hat P$, such that the image of $\hat P$ and $\gamma'$ have a transversal intersection at $q$. Note that $q$ may not be the image of a vertex of $G$, since for every vertex $v\in V(G)\setminus V(J)$, for every pair $P,P'\in {\mathcal{P}}^*$ of paths containing $v$, the intersection of $P$ and $P'$ at $v$ is non-transversal with respect to $\Sigma$. Therefore, $q$ is a crossing point between an edge of $\hat P$ and an edge of $P^*$. We conclude that the edges of $P^*$ participate in at least $|\hat E_2(x)|\geq \frac{\mathsf{cr}(\phi)\mu^{13b}}{m}$ crossings. But since we have assumed that Event ${\cal{E}}_1$ did not happen, path $P^*$ must be good, a contradiction. (Note that it is possible that, for some good path $\hat P\in {\mathcal{P}}^*$, both edges of $\hat P$ that are incident to $x$ lie in $\hat E_2(x)$. But in that case, the image of $\hat P$ must cross $\gamma$ twice, since both endpoints of $\hat P$ lie in $J$).
\end{proof}
Let $p_1,\ldots,p_z$ denote the points on the curve $\gamma^*$ that correspond to crossing points between pairs of edges of $P^*$, and assume that these points appear on $\gamma^*$ in this order. For all $1\leq i\leq z$, let $e_i,e'_i$ be the pair of edges of $P^*$ that cross at point $p_i$, with edge $e_i$ appearing before edge $e'_i$ on path $P^*$.
For all $1\leq i\leq z$, let $Q_i$ be the subpath of path $P^*$ from edge $e_i$ to edge $e'_{i}$, and we denote by $y_i,y'_i$ the second and the penultimate vertices of $Q_i$, respectively. Let $Q'_i$ be the subpath of $Q_i$ connecting $y_i$ to $y'_i$. From \Cref{obs: no heavy vertices on loops}, if Events ${\cal{E}}_1,{\cal{E}}_3$ did not happen, the every vertex of $Q'_i$ is a light vertex. Since we have deleted all edges of $E'$ from $G$ to obtain graph $G'$, every vertex of $Q'_i$ is incident to exactly two edges in $G'$, and these edges lie on path $P^*$.
We let $D_i$ be a tiny $p_i$-disc in the drawing $\phi'$. Let $s_i$ be the point on the image of edge $e_i$ lying on the boundary of $D_i$, and let $t_i$ be the point on the image of edge $e'_i$ lying on the boundary of $D_i$. We modify the drawing $\phi'$ in order to ``straighten'' the loop corresponding to the image of the path $Q'_i$, as follows. First, we truncate the images of the edges $e_i$ and $e'_i$, by deleting the segment of $\phi'(e_i)$ between $s_i$ and $\phi'(y_i)$, and similarly deleting the segment of $\phi'(e'_i)$ between $t_i$ and $\phi'(y'_i)$. We then delete the images of all vertices and edges of $Q'_i$ from $\phi'$. We place the new image of $y_i$ at point $s_i$, and the new image of $y'_i$ at point $t_i$. We then add an image of the path $Q'_i$ as a simple curve with endpoints $s_i$ and $t_i$, that is contained in $D_i$, so that the image of $Q'_i$ is contained in $\gamma^*\cap D_i$ (see \Cref{fig: proof522}).
\begin{figure}[h]
\centering
\subfigure[Before: tiny $p_i$-disc $D_i$ is shown in gray, and the original image of path $Q_i$ is shown in blue.]{
\scalebox{0.42}{\includegraphics[scale=0.25]{figs/proof522_1.jpg}}}
\hspace{0.5cm}
\subfigure[After: the new images of vertices $y_i,y'_i$ are shown in brown, and the new image of path $Q_i$ is shown in blue.]{\scalebox{0.42} {\includegraphics[scale=0.25]{figs/proof522_2.jpg}}
}
\caption{Modifying the image of path $Q_i$.
}\label{fig: proof522}
\end{figure}
Clearly, this modification does not increase the number of crossings, and it is local to region $F^*$. Once every point $p_1,\ldots,p_z$ is processed, we obtain the final solution $\phi'$ to instance $I'$ that is compatible with $\phi$, with $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$. Note that our transformation step does not introduce any new crossings. Therefore, if $(e_1,e_2)_p$ is a crossing in drawing $\phi'$, then there is a crossing between edges $e_1$ and $e_2$ at point $p$ in drawing $\phi$.
\subsection{Proof of \Cref{claim: cut set small case2}}
\label{subsec: small cut set in case 2}
Assume for contradiction that Event ${\cal{E}}$ did not happen, but $|E''|> \frac{2\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$. From the Maximum Flow - Minimum Cut Theorem, there is a collection ${\mathcal{Q}}$ of $\ceil{ \frac{2\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}}+|\chi^{\mathsf{dirty}}(\phi)|$ edge-disjont paths connecting $s$ to $t$ in $H$. Note that every edge in $H$ corresponds to some distinct edge in graph $G$. We do not distinguish between these edges. We show that, for every path $Q\in {\mathcal{Q}}$, there is a crossing between an edge of $Q$ and an edge of $E(P^*)\cup E(J)$ in $\phi'$. From \Cref{claim: new drawing}, and since the number of crossings in which the edges of $J$ may participate is bounded by $|\chi^{\mathsf{dirty}}(\phi)|$, it then follows that the edges of $P^*$ participate in at least $\ceil{ \frac{2\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}}$ crossings in $\phi$. But, since we have assumed that Event ${\cal{E}}_1$ did not happen, path $P^*$ is good, so its edges may particpate in at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}$ crossings, a contradiction. It now remains to prove that, for every path $Q\in {\mathcal{Q}}$, there is a crossing between an edge of $Q$ and an edge of $E(P^*)\cup E(J)$ in $\phi'$.
Consider any path $Q\in {\mathcal{Q}}$. This path naturally defines a path $Q'$ in graph $G$, whose first edge, denoted by $e(Q)$, lies in $\tilde E_1$, and last edge, denoted by $e'(Q)$, lies in $\tilde E_2$. Then the image of edge $e(Q)$ in $\phi'$ must intersect the interior of region $F_1$, while the image of edge $e'(Q)$ in $\phi'$ must intersect the interior of region $F_2$. Therefore, the image of the path $Q'$ in $\phi'$ must cross the boundary of the face $F_1$.
Since path $Q'$ is internally disjoint from $V(J')$, the image of some edge on path $Q'$ must cross the image of some edge of $E(J')=E(P^*)\cup E(J)$ in $\phi'$.
\subsection{Proof of \Cref{obs: few edges in split graphs case2}}
\label{subsec: few edges in split Case 2}
We start by recalling how the enhancement path $P^*$ was selected. Recall that initial set ${\mathcal{P}}^*$ of paths had cardinality
$k\geq \frac{15m}{16\mu^{b}}$.
We denoted by $E^*_1=\set{e_1,\ldots,e_{k}}\subseteq E_1$ the subset of edges that belong to the paths of ${\mathcal{P}}^*$, where the edges are indexed so that $e_1,\ldots,e_{k}$ appear consecutively, in the order of their indices in the ordering ${\mathcal{O}}(J)$. For all $1\leq j\leq k$, we denote by $P_j\in {\mathcal{P}}^*$ the unique path originating at the edge $e_j$. We then selected an index $\floor{k/3}<j^*<\ceil{2k/3}$ uniformly at random, and we let $P^*=P_{j^*}$.
Let $e'\in E_2$ be the edge of $P^*$ lying in $E_2$. Let $\tilde E_1$ be the set of edges lying between $e_{j^*}$ and $e'$ in ${\mathcal{O}}(J)$, and let $\tilde E_2$ be the set of edges lying between $e'$ and $e_{j^*}$ in ${\mathcal{O}}(J)$. Then one of the sets $\tilde E_1, \tilde E_2$ of edges must contain all edges in $\set{e_1,\ldots,e_{\floor{k/3}-1}}$, while the other must contain all edges in $\set{e_{\ceil{2k/3}+1},\ldots,e_{k}}$. Therefore, $|\tilde E_1|, |\tilde E_2|\geq \frac {k} 6\geq \frac{m}{12\mu^{b}}$.
Notice that graph $G_1$ may only contain edges from one of the sets $\tilde E_1, \tilde E_2$, and so $|E(G_1)|\leq m-\frac{m}{32\mu^{b}}$. Using similar reasoning, $|E(G_2)|\leq m-\frac{m}{32\mu^{b}}$.
\input{appx-avoid-guiding-curves}
\input{appx-tunnels}
\section{Proofs Omitted from Section~\ref{sec:long prelim}}
\label{sec: apd_prelim}
\subsection{Proof of \Cref{lem: find reordering}}
\label{subsec: compute reordering}
Suppose we are given a graph $G$, and, for every vertex $v\in V(G)$, an oriented circular ordering $({\mathcal{O}}_v,b_v)$ of edges in $\delta_G(v)$.
We say that a drawing $\phi$ of $G$ \emph{respects the oriented orderings of the vertices of $G$} iff for every vertex $v\in V(G)$, the oriented circular order in which the images of the edges of $\delta_G(v)$ enter $v$ is $({\mathcal{O}}_v,b_v)$.
We use the following theorem from~\cite{pelsmajer2011crossing}.
\begin{theorem}[Corollary 5.6 of~\cite{pelsmajer2011crossing}]
\label{thm: compute_reordering_curves}
There is an efficient algorithm, that, given a two-vertex loopless multigraph $G$ (so $V(G)=\set{v,v'}$ and $E(G)$ only contains parallel edges connecting $v$ to $v'$), and, for each vertex $v\in V(G)$, an oriented ordering $({\mathcal{O}}_v,b_v)$ of its incident edges, computes a drawing $\phi$ of $G$ that respects the given oriented orderings, such that $\mathsf{cr}(\phi)$ is at most twice the minimum number of crossings of any drawing of $G$ that respects the oriented orderings.
\end{theorem}
The proof of Lemma~\ref{lem: find reordering} easily follows \Cref{thm: compute_reordering_curves}. Recall that we are given a pair $({\mathcal{O}},b)$, $({\mathcal{O}}',b')$ of oriented orderings of a collection $U$ of elements. We construct a two-vertex loopless graph $G$ with oriented ordering on its vertices, as follows. Denote $U=\set{u_1,\ldots,u_r}$.
The vertex set of $G$ is $\set{v,v'}$. The edge set of $G$ consists of $r$ parallel edges connecting $v$ to $v'$, that we denote by $e_{u_1}, e_{u_2},\ldots e_{u_r}$, respectively.
The oriented ordering $({\mathcal{O}},b)$ of the elements of $U$ naturally defines an oriented ordering $(\hat {\mathcal{O}},b)$ of the edges of $G$, and similarly, the oriented ordering $({\mathcal{O}},b')$ of the elements of $U$ defiens an oriented ordering $(\hat {\mathcal{O}},b')$ of the edges of $G$.
We define the oriented orderings for the vertices of $G$ as follows: $({\mathcal{O}}_v,b_v)=(\hat {\mathcal{O}},b)$ and $({\mathcal{O}}_{v'},b_{v'})=(\hat {\mathcal{O}}',- b')$.
Consider any drawing $\phi$ of $G$ on the sphere that respects the oriented orderings for $v,v'$ defined above. Let $D'=D_{\phi}(v')$ be a tiny $v'$-disc. For all $1\leq i\leq r$, we denote the unique point on the image of edge $e_{u_i}$ that lies on the boundary of $D'$ by by $p'_i$.
Similarly, we let $\hat D=D_{\phi}(v)$ be a tiny $v$-disc, and, for all $1\leq i\leq r$, we denote the unique point on the image of edge $e_{u_i}$ that lies on the boundary of $\hat D$ by $p_i$. Let $D$ be the disc whose boundary is the same as the boundary of $\hat D$, but whose interior is disjoint from that of $\hat D$. Then $D'\subseteq D$, and the boundaries of the two discs are disjoint. Furthermore, points $p_1,\ldots,p_r$ appear on the boundary of $D$ according to the oriented ordering $({\mathcal{O}},b)$, and points $p'_1,\ldots,p'_r$ appear on the boundary of $D'$ according to the oriented ordering $({\mathcal{O}}',b')$.
For all $1\leq i\leq r$, let $\gamma_i$ be the segment of the image of the edge $e_{u_i}$ between points $p_i$ and $p'_i$. Then $\set{\gamma_i\mid 1\leq i\leq r}$ is a set of reordering curves for the orderings $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$, and moreover, the cost of this curve set is exactly the number of crossings in $\phi$.
Using a similar reasoning, any set $\Gamma$ of reordering curves for the orderings $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$ can be converted into a drawing of graph $G$ that respects the oriented orderings at vertices $v$ and $v'$, in which the number of crossings is exactly the cost of $\Gamma$. Therefore, there is a drawing of $G$ that respects the oriented orderings at $v$ and $v'$, whose number of crossings is bounded by $\mbox{\sf dist}(({\mathcal{O}},b),({\mathcal{O}}',b'))$. We Apply the algorithm from Theorem~\ref{thm: compute_reordering_curves} to graph $G$, and then compute a set of reordering curves from the resulting drawing of $G$ as described above. From the above discussion, the cost of the resulting set of curves is at most $2\cdot \mbox{\sf dist}(({\mathcal{O}},b),({\mathcal{O}}',b'))$.
\subsection{Proof of \Cref{lem: ordering modification}}
\label{apd: Proof of find reordering}
The proof easily follows from \Cref{lem: find reordering}.
We denote $\delta_G(v)=\set{e_1,\ldots,e_r}$, and, for all $1\leq i\leq r$, we let $p_i$ be the unique point of $\phi(e_i)$ lying on the boundary of the disc $D$. Let $\sigma_i$ be the segment of $\phi(e_i)$ that is disjoint from the interior of the disc $D$. In other words, if $e_i=(v,u_i)$, then $\sigma_i$ is a curve connecting $\phi(u_i)$ to $p_i$.
Note that the points $p_1,\ldots,p_r$ appear on the boundary of $D$ according to the circular ordering ${\mathcal{O}}'_v$ of their corresponding edges.
We assume w.l.o.g. that the orientation of the ordering is $b_v'=-1$.
Let $D'$ be another disc, that is contained in $D$, with $\phi(v)$ lying in the interior of $D'$, such that the boundaries of $D$ and $D'$ are disjoint.
We place points $p_1',\ldots,p'_r$ on the boundary of the disc $D'$, so that all resulting points are distinct, and they appear on the boundary of $D'$ in the order ${\mathcal{O}}_v$ of their corresponding edges, using a positive orientation of the ordering. For all $1\leq i\leq r$, we can compute a simple curve $\gamma_i$, connecting $\phi(v)$ to $p'_i$, such that $\gamma_i$ is contained in $D'$ and only intersects the boundary of $D'$ at its endpoint $p'_i$. We also ensure that all resulting curves $\gamma_1,\ldots,\gamma_r$ are mutually internally disjoint. Using the algorithm from \Cref{lem: find reordering}, we compute a collection $\Gamma=\set{\gamma_1',\ldots,\gamma'_r}$ of {reordering curves}, where for $1\leq i\leq r$, curve $\gamma'_i$ connects $p_i$ to $p'_i$, is contained in $D$, and is disjoint from the interior of $D'$. Note that the total number of crossings between the curves in $\Gamma$ is at most $2\mbox{\sf dist}{(({\mathcal{O}}_v,1),({\mathcal{O}}'_v,-1))}$. For all $1\leq i\leq r$, we define a new image of the edge $e_i$ to be the concatenation of the curves $\sigma_i,\gamma'_i$, and $\gamma_i$. The images of all remaining edges and vertices of $G$ remain unchanged. Denote the resulting drawing of the graph $G$ by $\phi'$. It is immediate to verify that the edges of $\delta_G(v)$ enter the image of $v$ in the order ${\mathcal{O}}_v$ in $\phi'$, and that the drawings of $\phi$ and $\phi'$ are identical except for the segments of the images of the edges in $\delta_G(v)$ that lie inside the disc $D$. It is also immediate to verify that $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)+
2\mbox{\sf dist}{(({\mathcal{O}}_v,1),({\mathcal{O}}'_v,-1))}$.
We repeat the same algorithm again, only this time the points $p_1',\ldots,p'_r$ are placed on the boundary of disc $D'$ in the order ${\mathcal{O}}_v$ of their corresponding edges, using a negative orientation of the ordering. The remainder of the algorithm remains unchanged, and produces a drawing $\phi''$ of $G$. As before, the edges of $\delta_G(v)$ enter the image of $v$ in the order ${\mathcal{O}}_v$ in $\phi''$, and that the drawings of $\phi$ and $\phi''$ are identical except for the segments of the images of the edges in $\delta_G(v)$ that lie inside the disc $D$. Moreover, $\mathsf{cr}(\phi'')\leq \mathsf{cr}(\phi)+
2\mbox{\sf dist}{(({\mathcal{O}}_v,-1),({\mathcal{O}}'_v,-1))}$. Let $\phi^*$ be the drawing with smaller number of crossings, among $\phi'$ and $\phi''$. Our algorithm returns the drawing $\phi^*$ as its final outcome. From the above discussion, $\mathsf{cr}(\phi^*)\leq \mathsf{cr}(\phi)+
2\mbox{\sf dist}{({\mathcal{O}}_v,{\mathcal{O}}'_v)}$.
\iffalse
Assume without loss of generality that the curves of $\Gamma$ enter $z$ in the oriented ordering $({\mathcal{O}},0)$. We first apply the algorithm from \Cref{lem: find reordering} to compute (i) a $2$-approximation $a_0$ of $\mbox{\sf dist}(({\mathcal{O}},0),({\mathcal{O}}',0))$, and (ii) a $2$-approximation $a_1$ of $\mbox{\sf dist}(({\mathcal{O}},0),({\mathcal{O}}',1))$.
Assume without loss of generality that $a_0\le a_1$, so $a_0\le 2\cdot \mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
For each $u\in U$, we denote by $p_u$ the intersection between the curve $\gamma_u$ and the boundary of $D$.
We erase the segments of curves of $\Gamma$ inside the disc $D$. We then place another disc $D'$ around $z$ inside the disc $D$, and let $\set{p'_u}_{u\in U}$ be a set of points appearing on the boundary of disc $D'$ in the order $({\mathcal{O}}',0)$. We use \Cref{lem: find reordering} to compute a set $\tilde\Gamma=\set{\tilde\gamma_u\mid u\in U}$ of reordering curves, where for each $u\in U$, the curve $\tilde\gamma_u$ connects $p_u$ to $p'_u$.
We then define, for each $u\in U$, the curve $\gamma'_u$ as the union of (i) the subcurve of $\gamma_u$ outside the disc $D$ (connecting its endpoint outside $D$ to $p_u$); (ii) the curve $\tilde\gamma_u$ (connecting $p_u$ to $p'_u$); and (iii) the straight line-segment connecting $p'_u$ to $z$. It is clear that (i) the curves $\set{\gamma'_u\mid u\in U}$ enter $z$ in the order $({\mathcal{O}}',0)$; (ii) for each $u\in U$, the curve $\gamma_u$ differs from the curve $\gamma'_u$ only within some tiny disc $D$ that contains $z$; and (iii) the number of crossings between curves of $\Gamma'$ within disc $D$ is at most $2\cdot \mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
\fi
\iffalse{some other proof}
\begin{proof}
This is a sketch, needs to be formalized. Let's look at the optimal set $\Gamma^*$ of reordering curves, and let $\gamma'_i$ be the curve participating in fewest crossings. Let's say this number of crossings is $N$. Then the cost of $\Gamma$ is at least $Nr/2$. The main claim is that there exists a collection $\Gamma'$ of reordering curves, of cost at most $2\operatorname{cost}(\Gamma^*)$, in which $\gamma_i$ does not participate in crossings. Assume first that the claim is correct. Finding such a drawing is straightforward. We rotate the discs so that $p'_i,p_i$ appear aligned, and connect them with a straight line, that becomes the drawing of $\Gamma_i$. Cutting $D\setminus D'$ along this line, we obtain a rectangle, whose vertical sides are the cut line, top horizontal line is boundary of $D$, and bottom horizontal line is boundary of $D'$. In this rectangle we just connect $p_j,p_j'$ with a straight line, adjusting the drawings as needed so no three curves meet at the same point. It is easy to see that resulting curves $\gamma_j,\gamma_k$ will only cross if the left-to-right ordering of $p_j,p_k$ is opposite from that of $p'_j,p'_k$. In any drawing where no curve crosses $\gamma_i$ any such pair of curves will cross as well. So we obtain a drawing of cost at most $\operatorname{cost}(\Gamma')\leq 2\operatorname{cost}(\Gamma^*)$.
Next, we show that there exists a collection $\Gamma'$ of curves of cost at most $2\operatorname{cost}(\Gamma^*)$ in which no curve crosses $\gamma_i$. In fact we compute $\Gamma'$ exactly as shown above. We just need to argue that the cost of this set of curves is at most $\Gamma^*$. In order to do so, we partition $\Gamma^*\setminus\set{\gamma_i}$ into two subsets: set $\Gamma_1$ containing all curves that don't cross $\gamma_i$, and set $\Gamma_2$ containing remaining curves. Consider now any pair $\gamma_j,\gamma_k$ of curves in $\Gamma'$, and assume that they cross (the drawing that we compute ensures that every pair of curves crosses at most once). in the new drawing. As observed above, this can only happen if the left-to-right ordering of $p_j,p_k$ on the top edge of the rectangle is opposite from that of $p'_j,p'_k$ on the bottom edge.
We now consider three cases. First, if the original curves routing elements $u_i,u_j$ lied in $\Gamma_1$, then the two corresponding curves had to cross in $\Gamma^*$ as well. Second, if the two original curves lied in $\Gamma_2$, then, as we will show later, they also had to cross before. So the case that remains is when one of the curves is in $\Gamma_1$ and another is in $\Gamma_2$. But because $|\Gamma_2|\leq N$, we will have at most $Nr$ such new crossings. Because $\operatorname{cost}(\Gamma^*)\geq Nr/2$, we get that $\operatorname{cost}(\Gamma')\leq \operatorname{cost}(\Gamma^*)+Nr\leq 5\operatorname{cost}(\Gamma')$. It now remains to prove the following claim.
\begin{claim}
Consider two elements $u_j,u_k$, such that the left-to-right ordering of $p_j,p_k$ on the top edge of the rectangle is opposite from the left-to-right ordering of $p'_j,p'_k$ on the bottom edge. Let $\gamma'_j,\gamma'_k$ be the two curves of $\Gamma^*$ corresponding to these elements, and assume that $\gamma'_i,\gamma'_k\in \Gamma_2$ (so both curves cross $\gamma_i$). Then $\gamma'_j,\gamma'_k$ must cross.
\end{claim}
\begin{proof}
Assume that $p_j$ appears to the left of $p_k$ on the top edge of the rectangle.
Consider the annulus $A=D\setminus D'$, and the drawings of $\gamma_i,\gamma'_j$ in it. We can assume that the two curves cross at most once. Consider a face $F$ of the resulting drawing whose boundary contains $p_k$. The three boundary edges of this face are: the segment of the top boundary of the rectangle from $p_j$ to $p_i$; segment of $\gamma'_i$ from $p_i$ to its meeting point $q$ with $\gamma'_j$; and segment of $\gamma'_j$ from $p_j$ to $q$. In particular, $p'_k$ does not lie on the boundary of $F$.
Assume for contradiction that $\gamma'_j,\gamma'_k$ do not cross. Then curve $\gamma'_k$ has to cross the boundary of $F$, which can only done by crossing the segment $\sigma$ of the boundary that lies on $\gamma'_i$. Denote this crossing point by $q'$. This is the only crossing between $\gamma'_k$ and $\gamma'_i$. If we denote by $F'$ the unique face that shares the boundary segment $\sigma$ with $F$, then that face does not contain $p'_k$ on its boundary (a figure is needed). Therefore, to reach $p'_k$, curve $\gamma'_k$ either needs to cross $\gamma'_j$, or to cross $\gamma'_i$ once again.
\end{proof}
\end{proof}
\fi
\iffalse
\mynote{previous proof, should keep only one}
We now prove \Cref{lem: ordering modification} using \Cref{lem: find reordering}.
Assume without loss of generality that the curves of $\Gamma$ enter $z$ in the oriented ordering $({\mathcal{O}},0)$. We first apply the algorithm from \Cref{lem: find reordering} to compute (i) a $2$-approximation $a_0$ of $\mbox{\sf dist}(({\mathcal{O}},0),({\mathcal{O}}',0))$, and (ii) a $2$-approximation $a_1$ of $\mbox{\sf dist}(({\mathcal{O}},0),({\mathcal{O}}',1))$.
Assume without loss of generality that $a_0\le a_1$, so $a_0\le 2\cdot \mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
For each $u\in U$, we denote by $p_u$ the intersection between the curve $\gamma_u$ and the boundary of $D$.
We erase the segments of curves of $\Gamma$ inside the disc $D$. We then place another disc $D'$ around $z$ inside the disc $D$, and let $\set{p'_u}_{u\in U}$ be a set of points appearing on the boundary of disc $D'$ in the order $({\mathcal{O}}',0)$. We use \Cref{lem: find reordering} to compute a set $\tilde\Gamma=\set{\tilde\gamma_u\mid u\in U}$ of reordering curves, where for each $u\in U$, the curve $\tilde\gamma_u$ connects $p_u$ to $p'_u$.
We then define, for each $u\in U$, the curve $\gamma'_u$ as the union of (i) the subcurve of $\gamma_u$ outside the disc $D$ (connecting its endpoint outside $D$ to $p_u$); (ii) the curve $\tilde\gamma_u$ (connecting $p_u$ to $p'_u$); and (iii) the straight line-segment connecting $p'_u$ to $z$. It is clear that (i) the curves $\set{\gamma'_u\mid u\in U}$ enter $z$ in the order $({\mathcal{O}}',0)$; (ii) for each $u\in U$, the curve $\gamma_u$ differs from the curve $\gamma'_u$ only within some tiny disc $D$ that contains $z$; and (iii) the number of crossings between curves of $\Gamma'$ within disc $D$ is at most $2\cdot \mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
\fi
\subsection{Proof of Theorem~\ref{thm: type-1 uncrossing}}
\label{apd: type-1 uncrossing}
Let $Z$ be the set of crossings points between the curves of $\Gamma$. For each such crossing point $z\in Z$, we consider a tiny disc $D_z$, that contains the point $z$ in its interior. We select the discs $D_z$ to ensure that all such discs are disjoint, and, moreover, if $z$ is a crossing point between curves $\gamma_1,\gamma_2$, then for every curve $\gamma\in \Gamma\setminus\set{\gamma_1,\gamma_2}$, $\gamma\cap D_z=\emptyset$, while for every curve $\gamma\in \set{\gamma_1\cup \gamma_2}$, $\gamma\cap D_z$ is a simple open curve whose endpoints lie on the boundary of $D_z$.
We start with $\Gamma_1'=\Gamma_1$, and then iteratively modify the curves in $\Gamma_1'$, as long as there is a pair of distinct curves $\gamma_1,\gamma_2\in \Gamma$ that cross more than once.
Each iteration is executed as follows. Let $\gamma_1,\gamma_2\in \Gamma_1'$ be a pair of curves that cross more than once, and let $z,z'$ be two crossing points between $\gamma_1,\gamma_2$, that appear consecutively on $\gamma_1$. In other words, no other point that appears between $z$ and $z'$ on $\gamma_1$ may belong to $\gamma_2$.
We denote by $s_1,t_1$ the endpoints of $\gamma_1$, such that $z$ appears closer to $s_1$ than $z'$ on $\gamma_1$.
Similarly, we denote by $s_2,t_2$ the endpoints of $\gamma_2$, such that $z$ appears closer to $s_2$ than $z'$ on $\gamma_2$.
We denote by $x_1,x_2$ the two points of $\gamma_1$ that lie on the boundary of disc $D_z$, and denote by $x_3,x_4$ the two points of $\gamma_1$ lying on the boundary of disc $D_{z'}$,
such that the points $x_1,z,x_2,x_3,z',x_4$ appear on $\gamma_1$ in this order.
We define points $y_1,y_2,y_3,y_4$ on $\gamma_2$ similarly (see \Cref{fig:non_crossing_representation1}).
\begin{figure}[h]
\centering
\subfigure[Before: Curve $\gamma_1$ is shown in blue and curve $\gamma_2$ is shown in red. The disc on the left is $D_z$, and the disc on the right is $D_{z'}$.]{\scalebox{0.45}{\includegraphics{figs/type_1_uncross_proof_1.jpg}} \label{fig:non_crossing_representation1}
}
\hspace{0cm}
\subfigure[After: Curve $\gamma'_1$ is shown in blue and curve $\gamma'_2$ is shown in red.]{
\scalebox{0.45}{\includegraphics{figs/type_1_uncross_proof_2.jpg}}\label{fig:non_crossing_representation2}}
\caption{An illustration of an iteration of type-1 uncrossing.}\label{fig: type_1_uncross_proof}
\end{figure}
In order to execute the iteration, we slightly modify the curves $\gamma_1, \gamma_2$, by ``swapping'' their segments between points $x_2,x_3$ and $y_2,y_3$, respectively, and slightly ``nudging'' them inside the discs $D_z,D_{z'}$, as show in \Cref{fig: type_1_uncross_proof}. We now describe the construction of the new curves $\gamma_1',\gamma_2'$ more formally.
Note that the points $x_1,x_2,y_1,y_2$ appear on the boundary of $D_z$ clockwise in either the order $(x_1,y_1,x_2,y_2)$ or the order $(x_1,y_2,x_2,y_1)$.
Therefore, we can find two disjoint simple curves $\eta_1$ and $\eta_2$ that are contained in disc $D_z$, with $\eta_1$ connecting $x_1$ to $y_2$, and $\eta_2$ connecting $y_1$ to $x_2$.
Similarly, we compute two disjoint simple curves curves $\eta'_1,\eta'_2$, that are contained in disc $D_{z'}$, with $\eta_1'$ connecting $y_3$ to $x_4$, and $\eta_2'$ connecting $x_3$ to $y_4$ (see \Cref{fig:non_crossing_representation2}).
We let $\gamma_1'$ be a curve, that is constructed by concatenating the following five curves: (1) the segment of $\gamma_1$ from $s_1$ to $x_1$; (2) curve $\eta_1$; (3) the segment of $\gamma_2$ from $y_2$ to $y_3$; (4) curve $\eta'_1$; and (5) the segment of $\gamma_1$ from $x_4$ to $t_1$. Similarly, let $\gamma_2'$ be a curve, that is constructed by concatenating the following five curves: (1) the segment of $\gamma_2$ from $s_2$ to $y_1$; (2) curve $\eta_2$; (3) the segment of $\gamma_1$ from $x_2$ to $x_3$; (4) curve $\eta'_2$; and (5) the segment of $\gamma_2$ from $y_4$ to $t_2$. We then remove any self loops from the two curves, to obtain the final curves $\gamma'_1,\gamma_2'$, that replace the curves $\gamma_1,\gamma_2$ in $\Gamma_1'$. Note that $\gamma'_1$ has the same endpoints as $\gamma_1$, and the same is true for $\gamma'_2$ and $\gamma_2$. It is also easy to verify that, at the end of the iteration, the number of crossings between the curves of $\Gamma_1'\cup \Gamma_2$ strictly decreases, and the number of crossings between the curves of $\Gamma_1'$ and the curves of $\Gamma_2$, that we denoted by $\chi(\Gamma_1',\Gamma_2)$, does not grow. Moreover, for every curve $\gamma\in \Gamma_2$, the number of crossings of $\gamma$ with the curves in $\Gamma_1'$ may not grow either. Once the algorithm terminates, we obtain the desired set $\Gamma_1'$ of curves, in which every pair of distinct curves crosses at most once. From the above discussion, it is immediate to verify that the curves in $\Gamma_1$ have all required properties. Since the curves in $\Gamma$ are in general position, the number of iterations is bounded by the number of crossing points between the curves.
\subsection{Proof of \Cref{claim: curves in a disc}}
\label{apd: Proof of curves in a disc}
We start by constructing a collection $\Gamma'=\set{\gamma'_i\mid 1\leq i\leq k}$ of curves that are in general position, such that for all $1\leq i\leq k$, curve $\gamma'_i$ has $s_i$ and $t_i$ as its endpoints, and is contained in disc $D$. In order to construct the set $\Gamma'$ of curves, we let $p$ be any point in the interior of the disc $D$, and $r>0$ be some value, such that a circle of radius $r$ that is centered at point $p$ is contained in the disc $D$. For all $1\leq i\leq k$, let $\ell_i$ be a straight line, connecting point $s_i$ to $p$, and $\ell'_i$ a straight line, connecting point $t_i$ to $p$. We can assume that both lines are contained in the disc $D$, by stretching the disc as needed. For all $1\leq i\leq k$, we choose a radius $0<r_i<r$, so that $0<r_1<\cdots<r_k<r$ holds. For an index $1\leq i\leq k$, we let $C_i$ be the boundary of a circle of radius $r_i$ that is centered at point $p$, and we let $q_i,q'_i$ be the points on lines $\ell_i$ and $\ell'_i$, respectively, that lie on $C_i$. We let curve $\gamma'_i$ be a concatenation of three curves: the segment of $\ell_i$ from $s_i$ to $q_i$; a segment of $C_i$ between $q_i$ and $q'_i$; and the segment of $\ell'_i$ from $q'_i$ to $t_i$. Consider the resulting set
$\Gamma'=\set{\gamma'_i\mid 1\leq i\leq k}$ of curves. Clearly, for all $1\leq i\leq k$, curve $\gamma'_i$ has $s_i$ and $t_i$ as its endpoints, and is contained in disc $D$.
It is also easy to verify that curves of $\Gamma'$ are in general position.
Next, we use the algorithm from \Cref{thm: type-1 uncrossing} to perform a type-1 uncrossing of the curves in $\Gamma'$. Specifically, we set $\Gamma_1=\Gamma'$ and $\Gamma_2=\emptyset$. We denote by $\Gamma=\Gamma_1'=\set{\gamma_i\mid 1\leq i\leq k}$ the set of curves that the algorithm outputs. Recall that, for all $1\leq i\leq k$, curve $\gamma_i$ has $s_i$ and $t_i$ as its endpoints; the curves in $\Gamma$ are in general position; and every pair of curves in $\Gamma$ cross at most once. From the description of the type-1 uncrossing operation it is easy to verify that all curves in $\Gamma$ are contained in the disc $D$.
Consider now two pairs $(s_i,t_i),(s_j,t_j)$ of points, with $i\neq j$. Note that curve $\gamma_i$ partitions the disc $D$ into two regions, that we denote by $F$ and $F'$. If the two pairs $(s_i,t_i),(s_j,t_j)$ cross, then $s_j,t_j$ may not lie on the boundary of the same region, and so curve $\gamma_j$ must cross curve $\gamma_i$ exactly once. If the two pairs do not cross, then $s_j,t_j$ either both lie on the boundary of $F$, or they both lie on the boundary of $F'$. It is then impossible that curves $\gamma_i,\gamma_j$ cross exactly once, and, since every pair of curves crosses at most once, they cannot cross.
\iffalse
We first set, for each $1\le i\le k$, a curve $\gamma'_i$ to be the straight line segment connecting $s_i$ to $t_i$. It is easy to see that every pair of curves in $\Gamma'=\set{\gamma'_i\mid 1\le i\le k}$ cross at most once, and a pair $\gamma'_i,\gamma'_j$ of curves cross iff the two pairs $(s_i,t_i), (s_j,t_j)$ of points cross.
If the curves of $\Gamma'$ are in general position (namely no point belongs to at least three curves of $\Gamma'$), then we set, for each $1\le i \le k$, $\gamma_i=\gamma'_i$, and return the collection $\Gamma=\set{\gamma_i\mid 1\le i\le k}$ of curves. It is easy to verify that $\Gamma$ satisfies the required property.
If the curves $\Gamma'$ are not in general position, then we let, for each $1\le i\le k$, a curve $\gamma_i$ be obtained by slightly perturbing the curve $\gamma'_i$. See \Cref{fig: perturb} for an illustration. We then return the collection $\Gamma=\set{\gamma_i\mid 1\le i\le k}$ of curves. It is easy to verify that $\Gamma$ satisfies the required property.
\begin{figure}[h]
\centering
\subfigure[Before: Curves $\gamma'_1,\gamma'_2,\gamma'_3$ contain the same point.]{\scalebox{1.1}{\includegraphics[scale=0.4]{figs/perturb_before.jpg}}}
\hspace{0.7cm}
\subfigure[After: Curves $\gamma_1,\gamma_2,\gamma_3$ are in general position.]{\scalebox{1.1}{\includegraphics[scale=0.4]{figs/perturb_after.jpg}}}
\caption{An illustration of slight perturbation of curves in the proof of \Cref{claim: curves in a disc}.\label{fig: perturb}}
\end{figure}
\fi
\subsection{Proof of Theorem~\ref{thm: new type 2 uncrossing}}
\label{apd: new type 2 uncrossing}
We start with an initial set $\Gamma'=\set{\gamma'(Q)\mid Q\in {\mathcal{Q}}}$ of curves, where, for each path $Q\in {\mathcal{Q}}$, $\gamma'(Q)$ is the image of the path $Q$ in $\phi$. In other words, $\gamma'(Q)$ is a concatenation of the images of the edges of $Q$ in $\phi$. Note however that the resulting set $\Gamma'$ of curves is not in general position. This is since a vertex $v\in V(G)$ may serve as an inner vertex on more than two paths of ${\mathcal{Q}}$, and in such a case its image $\phi(v)$ serves as an inner point on more than two curves in $\Gamma'$.
Let $V'\subseteq V(G)$ be the set of all vertices $v\in V(G)$, such that more than two paths in ${\mathcal{Q}}$ contain $v$.
In our first step, we transform the set $\Gamma'$ of curves so that the resulting curves are in general position, while ensuring that the endpoints of each curve $\gamma'(Q)$ remain unchanged, and each such curve $\gamma'(Q)$ remains aligned with the graph $\bigcup_{Q'\in {\mathcal{Q}}}Q'$. We do so by performing \emph{nudging operation} around every vertex $v\in V_1$, as follows.
Consider any vertex $v\in V_1$, and let ${\mathcal{Q}}^v\subseteq {\mathcal{Q}}$ be the set of all paths containing vertex $v$. Note that $v$ must be an inner vertex on each such path. For convenience, we denote ${\mathcal{Q}}^v=\set{Q_1,\ldots,Q_z}$. Consider the tiny $v$-disc $D=D_{\phi}(v)$. For all $1\leq i\leq z$, denote by $a_i$ and $b_i$ the two points on curve $\gamma(Q_i)$ that lie on the boundary of disc $D$. Use the algorithm from \Cref{claim: curves in a disc} to compute a collection $\set{\sigma_1,\ldots,\sigma_z}$ of curves, such that, for all $1\leq i\leq z$, curve $\sigma_i$ connects $a_i$ to $b_i$, and the interior of the curve is contained in the interior of $D$. Recall that every pair of resulting curves crosses at most once, and every point in the interior of $D$ may be contained in at most two curves.
For all $1\leq i\leq z$, we modify the curve $\gamma(Q_i)$, by replacing the segment of the curve that is contained in disc $D$ with $\sigma_i$.
Once every vertex $v\in V_1$ is processed, we obtain a collection $\Gamma''=\set{\gamma''(Q)\mid Q\in {\mathcal{Q}}}$ of curves, where for all $Q\in {\mathcal{Q}}$, curve $\gamma''(Q)$ connects $\phi(s(Q))$ to $\phi(t(Q))$. Moreover, it is easy to verify that each resulting curve $\gamma''(Q)\in \Gamma''$ is aligned with the graph $\bigcup_{Q'\in {\mathcal{Q}}}Q'$, and that the curves in $\Gamma''$ are in general position.
o
First, for each path $Q\in {\mathcal{Q}}$, we define curve $\gamma'(Q)$ to be the concatenation of all curves representing images of edges in $E(Q)$ in $\phi$. Clearly, the curve $\gamma'(Q)$ connects $\phi(s_Q)$ to $\phi(t_Q)$. We designate $\phi(s_Q)$ as its first endpoint $s(\gamma'(Q))$ and $\phi(t_Q)$ as its last endpoint $t(\gamma'(Q))$.
We then denote $\Gamma'=\set{\gamma'(Q)}$.
In the second step, we iteratively modify the curves in $\Gamma'$ as follows.
Throughout, we maintain (i) a set $\hat{\Gamma}$ of curves, that is initialized to be $\Gamma'$; and (ii) a set $Z$ of points that lie in at least two curves of $\hat\Gamma$ and is an inner point of at least one of these two curves.
The algorithm continues to be executed as long as set $Z$ is not empty.
We now describe an iteration.
We first arbitrarily pick a point $z$ from $Z$. Clearly, $z$ is either a crossing between two edges of $E({\mathcal{Q}})$ in $\phi$, or is the image of some vertex of $V({\mathcal{Q}})$ in $\phi$.
Let $D_z$ be a tiny disc around the point $z$.
Let $\gamma_{1},\ldots,\gamma_{k}$ be the curves in $\hat \Gamma$ that contains $z$.
For each $1\le i\le k$, if $z$ is an inner point of $\gamma_{i}$, then recall that $s(\gamma_i)$ the first endpoint of $\gamma_i$ and $t(\gamma_i)$ is the last endpoint of $\gamma_i$, and we denote by $s'_i, t'_i$ the intersections between curve $\gamma_i$ and the boundary $D_z$, so the points $s(\gamma_i),s'_i,z,t'_i,t(\gamma_i)$ appear on the directed curve $\gamma_i$ in this order.
If $z$ is the first endpoint of $\gamma_i$, then we denote $s'_i=z$ and denote by $t'_i$ the intersection between $\gamma_i$ and the boundary of $D_z$, so the points $s'_i,z,t'_i,t(\gamma_i)$ appear on the directed curve $\gamma_i$ in this order.
If $z$ is the last endpoint of $\gamma_i$, then we denote $t'_i=z$ and denote by $s'_i$ the intersection between $\gamma_i$ and the boundary of $D_z$, so the points $s(\gamma_i),s'_i,z,t'_i$ appear on the directed curve $\gamma_i$ in this order.
\begin{figure}[h]
\centering
\subfigure[Before: Curves $\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5$ are shown in red, yellow, pink, black and green respectively.]{\scalebox{0.33}{\includegraphics[scale=0.4]{figs/type_2_uncross_proof_1.jpg}
}
\hspace{0.1cm}
\subfigure[Curves of $\set{\eta^i_z\mid 1\le i\le k}$ connecting points of $S'$ to points of $T'$ are shown in dash blue lines.]{
\scalebox{0.33}{\includegraphics[scale=0.4]{figs/type_2_uncross_proof_2.jpg}}}
\hspace{0.1cm}
\subfigure[After: New curves $\hat\gamma_1,\hat\gamma_2,\hat\gamma_3,\hat\gamma_4,\hat\gamma_5$ are shown in red, yellow, pink, black and green respectively.]{
\scalebox{0.33}{\includegraphics[scale=0.4]{figs/type_2_uncross_proof_3.jpg}}}
\caption{An illustration of an iteration of the iterative algorithm for modifying the curves in $\hat \Gamma$.}\label{fig: type_2_uncross_proof}
\end{figure}
We define the multisets $S'=\set{s'_i\mid 1\le i\le k}$ and $T'=\set{t'_i\mid 1\le i\le k}$.
We now modify the curves $\gamma_1,\ldots,\gamma_k$ as follows.
We move the point $z$ (along with all points $s'_i=z$ and $t'_j=z$) to the boundary of $D_z$. Now all points of $S'\cup T'$ lie on the boundary of $D_z$.
We then construct a set of curves within $D_{z}$ in rounds as follows. In each round, pick a pair $s_i,t_j$ of vertices, where $s_i\in S'$ and $t_j\in T'$, that appears consecutively on the boundary of $D_z$. Clearly such a pair of vertices exist. We then draw a curve connecting $s_i$ to $t_j$ without crossing all curves that are constructed before, and then delete $s_i,t_j$ from sets $S',T'$, respectively. It is easy to verify that, at the end of this process, we will obtain a set $\set{\eta^{i}_z\mid 1\le i\le k}$ of curves, such that
\begin{itemize}
\item for each $1\le i\le k$, the curve $\eta_z^i$ connects $s'_i$ to a distinct point $t'_{x_i}$ of $T'$;
\item for each $1\le i\le k$, the curve $\eta_z^i$ lies entirely in the disc $D_z$; and
\item the curves in $\set{\eta^{i}_z\mid 1\le i\le k}$ are mutually disjoint, except that they may share $z$ as an endpoint.
\end{itemize}
See Figure~\ref{fig: type_2_uncross_proof} for an illustration.
For each $1\le i\le k$, we denote by $\gamma^s_i$ the subcurve of $\gamma_i$ between $s_i$ and $s'_i$, and by $\gamma^t_i$ the subcurve of $\gamma_i$ between $t'_i$ and $t_i$.
We now move the point $z$ (along with all points $s'_i=z$ and $t'_j=z$) to back to its original position.
We then define, for each $1\le i\le k$, a new curve $\hat \gamma_i$ to be the sequential concatenation of curves $\gamma^s_i, \eta^i_z, \gamma^t_{x_i}$.
See Figure~\ref{fig: type_2_uncross_proof} for an illustration.
We then replace the curves $\gamma_1,\ldots,\gamma_k$ in $\hat\Gamma$ with the new curves $\hat\gamma_1,\ldots,\hat\gamma_k$. This completes the description of an iteration.
Let $\Gamma''$ be the set of curves that we obtain at the end of the algorithm.
It is easy to verify that, in the iteration described above, for each $1\le i\le k$, the first endpoint of the new curve $\hat\gamma_i$ is identical to that of the original curve $\gamma_i$, and similarly the multiset of the last endpoints of new curves $\hat\gamma_1,\ldots,\hat\gamma_k$ is identical to that of the original curves $\gamma_1,\ldots,\gamma_k$.
It is also easy to verify that the point $z$ is no longer contains in any of the new curves $\hat\gamma_1,\ldots,\hat\gamma_k$ as an inner vertex (and so it does not belong to set $Z$ anymore), while other points in $Z$ remains in $Z$, and there are no new points added to $Z$.
So in each iteration, one point is removed from set $Z$, and therefore the algorithm of processing $\Gamma'$ will terminate in $O(\chi(\Gamma'))$ rounds.
Moreover, from the terminating criterion of the algorithm, the curves in the resulting set $\Gamma''$ no longer cross with each other.
In the last step, we remove self-crossings of all curves of $\Gamma''$. Let $\Gamma$ be the set of curves we obtain.
We designate the curve in $\Gamma$ with first endpoint $\phi(s(Q))$ as $\gamma(Q)$. We now show that the set $\Gamma$ of curves satisfy all properties required in \Cref{thm: new type 2 uncrossing}.
From the above discussion, it is clear that the second, the fourth and the fifth properties are satisfied. We now show that the curves in $\Gamma$ are aligned with graph $\bigcup_{Q'\in {\mathcal{Q}}}Q'$.
Clearly, in each iteration of the above curve modification process, we re-draw the curve only within a tiny disc around a crossing of two edges or around the image of a vertex of $V({\mathcal{Q}})$, and then (possibly) change the course of curves.
In the self-loop removing step, it is clear that the self-crossing has to be the crossing of two edges in $E({\mathcal{Q}})$.
Therefore, those re-drawn segments and the self-crossing points can be viewed as segments $\sigma_1,\ldots,\sigma_{r}$ of the curves, and since we start from the curves in $\Gamma'$ which only contain images of edges of $E(Q)$, the other segments $\sigma'_1,\ldots,\sigma'_{r-1}$ are all contiguous segments of non-zero length of the image of some edge in $\phi$.
We last show that the third property is satisfied. In fact, if $u^*$ is the common first endpoint of all curves in $\Gamma'$, then point $u^*$ never belongs to set $Z$ and is therefore not processed. This implies that the first segments of all curves of $\Gamma'$ has not been changed, and therefore, for each path $Q\in {\mathcal{Q}}$, curve $\gamma(Q)$ contains the segment of the image of $e_1(Q)$ that lies in the disc $D_{\phi}(u^*)$.
This completes the proof of \Cref{thm: new type 2 uncrossing}.
\subsection{Proof of \Cref{cor: new type 2 uncrossing}}
\label{apd: cor new type 2 uncrossing}
For each edge $e\in E({\mathcal{Q}})$, we denote by $\phi(e)$ the curve that represents the image of $e$ in $\phi$, and then we create a set $\Pi_e$ of $\cong_{G}({\mathcal{Q}},e)$ mutually internally disjoint curves connecting the endpoints of $e$ in $\phi$, that lies in an arbitrarily thin strip around $\phi(e)$.
We next assign, for each edge $e\in E({\mathcal{Q}})$ and for each path in ${\mathcal{Q}}$ that contains the edge $e$, a distinct curve in $\Pi_e$ to this path.
Therefore, each curve in $\bigcup_{e\in E({\mathcal{Q}})}\Pi_e$ is assigned to exactly one path of ${\mathcal{Q}}$,
and each path $Q\in{\mathcal{Q}}$ is assigned with, for each edge $e\in E(Q)$, a curve in $\Pi_e$. Let $\gamma'(Q)$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E({\mathcal{Q}})}\Pi_e$ that are assigned to path $Q$, so $\gamma'(Q)$ connects the the image $\phi(s_Q)$ of the first endpoint $s_Q$ of $Q$ to the image $\phi(t_Q)$ of the last endpoint $t_Q$ of $Q$ in $\phi$.
We then perform the iterative curve-modification algorithm presented in \Cref{apd: new type 2 uncrossing} to set $\Gamma'=\set{\gamma'(Q)}$ of curves and obtain a set $\Gamma''$ of curves. We then remove self-crossings in the same way as \Cref{apd: new type 2 uncrossing}. Let $\Gamma$ be the set of curves we obtain.
We designate the curve in $\Gamma$ with first endpoint $\phi(s(Q))$ as $\gamma(Q)$. We now show that the set $\Gamma$ of curves satisfy all properties required in \Cref{cor: new type 2 uncrossing}.
From the analysis in \Cref{apd: new type 2 uncrossing}, it is clear that the first, the second, and the third properties are satisfied. We now show that the fourth property is also satisfied.
Note that, although the curves in $\Gamma$ are not rigorously graph-aligned curves, they can be partitioned into consecutive segments $(\sigma_1,\sigma_1',\sigma_2,\sigma_2',\ldots,\sigma'_{r-1},\sigma_r)$, such that each $\sigma'_1$, although not necessarily a contiguous segment of the image of some edge of $E({\mathcal{Q}})$ in ${\mathcal{Q}}$, but it lies entirely within an arbitrarily thin strip of a contiguous segment of the image of some edge of $E({\mathcal{Q}})$ in ${\mathcal{Q}}$. Moreover, for each edge $e'\in E(G\setminus C)$, the number of segments of curves in $\Gamma$ that lie entirely within the thin strip of $\phi(e)$ is bounded by $\cong_G({\mathcal{Q}},e')$.
From the fact that the curves in $\Gamma$ do not cross each other, for every edge $e\in E(C)$, the number of crossings between the image of edge $e$ in $\phi$ and the curves in $\Gamma$ is bounded by the sum of, over all edges $e'\in E(G\setminus C)$, the number of crossings between $e$ and $e'$ in $\phi$, times $\cong_G({\mathcal{Q}}, e)$, which is in turn bounded by $\chi_{\phi}(e, G\setminus C)\cdot \cong_G({\mathcal{Q}})$.
This completes the proof of \Cref{cor: new type 2 uncrossing}.
\subsection{Proof of \Cref{obs: splicing}}
\label{apd: Proof of splicing}
We first show that $S({\mathcal{P}}')=S({\mathcal{P}})$ and $T({\mathcal{P}}')=T({\mathcal{P}})$.
We denote by $s$ and $t$ the first and the last endpoints of $P$, respectively, and by $s'$ and $t'$ the first and the last endpoints of $P'$, respectively.
From the construction, the first endpoint of $\tilde P$ is $s$ and the last endpoint of $\tilde P$ is $t'$; and the first endpoint of $\tilde P'$ is $s'$ and the last endpoint of $\tilde P'$ is $t$.
Since the first endpoints and the last endpoints of other paths are not changed, $S({\mathcal{P}}')=S({\mathcal{P}})$ and $T({\mathcal{P}}')=T({\mathcal{P}})$.
Assume that both $\tilde P,\tilde P'$ are simple paths. We will show that $|\Pi^T({\mathcal{P}}')|<|\Pi^T({\mathcal{P}})|$.
Specifically, we will construct a mapping $f$ that maps each triple in $\Pi^T({\mathcal{P}}')$ to a triple in $\Pi^T({\mathcal{P}})$, such that $f$ is injective but not surjective.
Note that ${\mathcal{P}}'\setminus\set{\tilde P, \tilde P'}={\mathcal{P}}\setminus\set{P, P'}$. We denote $\hat {\mathcal{P}}={\mathcal{P}}\setminus\set{P, P'}$.
Since the set $\Pi^T({\mathcal{P}})$ contains the triple $(P,P',v)$, $v$ must be an inner vertex of both $P$ and $P'$.
Since we have assumed that paths $\tilde P, \tilde P'$ are simple paths, $v$ is the only vertex shared by paths $\tilde P$ and $\tilde P'$, and $v$ is the only vertex shared by paths $P$ and $P'$. From the construction, the intersection of paths $\tilde P,\tilde P'$ at $v$ is non-transversal with respect to $\Sigma$.
Therefore, each triple in $\Pi^T({\mathcal{P}}')$ is of one of the following types:
\begin{itemize}
\item $(Q,Q',v')$ where $Q,Q'\in \hat{\mathcal{P}}$;
\item $(Q,\tilde P ,v')$ or $(Q,\tilde P' ,v')$ where $Q\in \hat{\mathcal{P}}$ and $v'\ne v$;
\item $(Q,\tilde P ,v)$ or $(Q,\tilde P' ,v)$ where $Q\in \hat{\mathcal{P}}$.
\end{itemize}
First, it is easy to see that, for every triple $(Q,Q',v')\in \Pi^T({\mathcal{P}}')$ such that $Q,Q'\in {\mathcal{P}}'\setminus\set{\tilde P, \tilde P'}$, the triple $(Q,Q',v')$ also belongs to the set $\Pi^T({\mathcal{P}})$.
We then map such a triple to itself.
Second, consider a triple $(Q,\tilde P,v')$ in $\Pi^T({\mathcal{P}}')$ such that $Q\in \hat {\mathcal{P}}$ and $v'\ne v$. If vertex $v'$ belongs to the subpath of $\tilde P$ from $s$ to $v$, then it is easy to verify that the triple $(Q,P,v')$ belongs to $\Pi^T({\mathcal{P}})$, and we map the triple $(Q,\tilde P,v')$ to the triple $(Q,P,v')$.
If $v'$ belongs to the subpath of $P$ from $v$ to $t'$, then it is easy to verify that the triple $(Q,P',v')$ belongs to $\Pi^T({\mathcal{P}})$, and we map the triple $(Q,\tilde P,v')$ to the triple $(Q,P',v')$.
Similarly, for each triple $(Q,\tilde P',v')\in\Pi^T({\mathcal{P}}')$, either the triple $(Q,P,v')$ or the triple $(Q,P',v')$ belongs to $\Pi^T({\mathcal{P}})$, and we map the triple $(Q,\tilde P',v')$ to either $(Q,P,v')$ or $(Q,P',v')$ accordingly.
Last, we consider the triples in $\Pi^T({\mathcal{P}}')$ of the third type.
We denote by $e_a,e_b$ the edges of $P$ incident to $v$, where $e_a$ precedes $e_b$ in $P$ (recall that $v$ is an inner vertex of $P$). We define edges $e'_a,e'_b$ similarly for path $P'$.
Assume that $\delta(v)=\set{e_a,e^1_1,\ldots,e^1_{r_1},e_b,e^2_1,\ldots,e^2_{r_2},e_a',e^3_1,\ldots,e^1_{r_3},e_b',e^4_1,\ldots,e^1_{r_4}}$, where edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
We denote $E_1=\set{e^1_1,\ldots,e^1_{r_1}}$, $E_2=\set{e^2_1,\ldots,e^2_{r_2}}$,
$E_3=\set{e^3_1,\ldots,e^3_{r_3}}$ and
$E_4=\set{e^4_1,\ldots,e^4_{r_4}}$.
See \Cref{fig: splicing_1} for an illustration.
Consider now a path $Q\in \hat {\mathcal{P}}$, and we denote by $e^Q_a,e^Q_b$ the paths of $Q$ incident to $v$, where $e^Q_a$ precedes $e^Q_b$ in $Q$.
We distinguish between the following cases.
\textbf{Case 1. $(Q,P,v), (Q,P',v)\in \Pi^T({\mathcal{P}})$.} In this case, if set $\Pi^T({\mathcal{P}}')$ contains the triple $(Q,\tilde P, v)$, then we map it to the triple $(Q,P,v)$, and if set $\Pi^T({\mathcal{P}}')$ contains the triple $(Q,\tilde P', v)$, then we map it to the triple $(Q,P',v)$.
\textbf{Case 2. $(Q,P,v), (Q,P',v)\notin \Pi^T({\mathcal{P}})$.} In this case, it is easy to verify that both edges $e'_a,e'_b$ belong to the same set of $E_1,E_2,E_3,E_4$. Therefore, the intersection between $Q$ and $\tilde P$ at $v$ is non-transversal, namely the set $\Pi^T({\mathcal{P}}')$ does not contain the triple $(Q,\tilde P,v)$. Similarly, the set $\Pi^T({\mathcal{P}}')$ does not contain the triple $(Q,\tilde P' ,v)$, either.
\textbf{Case 3. $(Q,P,v)\in \Pi^T({\mathcal{P}}), (Q,P',v)\notin \Pi^T({\mathcal{P}})$.} In this case, it is easy to verify that either $e^Q_a\in E_1$ and $e^Q_b\in E_4$ (or symmetrically $e^Q_a\in E_4$ and $e^Q_b\in E_1$), or $e^Q_a\in E_2$ and $e^Q_b\in E_3$ (or symmetrically $e^Q_a\in E_3$ and $e^Q_b\in E_2$). In the former case (see \Cref{fig: splicing_2}), the intersection between paths $Q$ and $\tilde P$ at $v$ is transversal, while the intersection between paths $Q$ and $\tilde P'$ at $v$ is non-transversal. In other words, the set $\Pi^T({\mathcal{P}}')$ contains the triple $(Q,\tilde P,v)$ but not the triple $(Q,\tilde P' ,v)$. We then map $(Q,\tilde P,v)$ to the triple $(Q,P,v)$.
In the latter case (see \Cref{fig: splicing_3}), the intersection between paths $Q$ and $\tilde P'$ at $v$ is transversal, while the intersection between paths $Q$ and $\tilde P$ at $v$ is non-transversal. In other words, the set $\Pi^T({\mathcal{P}}')$ contains the triple $(Q,\tilde P',v)$ but not the triple $(Q,\tilde P ,v)$. We then map triple $(Q,\tilde P',v)$ to the triple $(Q,P,v)$.
\begin{figure}[h]
\centering
\subfigure[A schematic view of edges $e_a$, $e_b$, $e'_a$, $e'_b$ and edge sets $\set{E_i}_{1\le i\le 4}$.]{\scalebox{0.9}{\includegraphics[scale=0.4]{figs/splicing_1.jpg}}\label{fig: splicing_1}
}
\hspace{0.3cm}
\subfigure[An illustration of the former case in Case 3.]{\scalebox{0.9}{\includegraphics[scale=0.4]{figs/splicing_2.jpg}}\label{fig: splicing_2}}
\hspace{0.3cm}
\subfigure[An illustration of the latter case in Case 3.]{
\scalebox{0.9}{\includegraphics[scale=0.4]{figs/splicing_3.jpg}}\label{fig: splicing_3}}
\caption{An illustration of edges and edge sets in the proof of \Cref{obs: splicing}.}
\end{figure}
It is easy to verify that the mapping we construct is injecture. We now show that it is surjective. In fact, since the intersection between paths $P,P'$ at $v$ is transversal, the set $\Pi^T({\mathcal{P}})$ contains the triple $(P,P',v)$. However, it is easy to verify that no triple in $\Pi^T({\mathcal{P}}')$ is mapped to the triple $(P,P',v)$. Therefore, the mapping we construct is surjective. This completes the proof of \Cref{obs: splicing}.
\subsection{Proof of Lemma~\ref{lem: non_interfering_paths}}
\label{apd: Proof of non_interfering_paths}
We note that the proof of a similar version of Lemma~\ref{lem: non_interfering_paths} was included in the proof of Lemma 9.5 in \cite{chuzhoy2020towards}.
For completeness, we provide its proof here.
We first pre-process the set ${\mathcal{R}}$ of paths by removing self-loops of all paths of ${\mathcal{R}}$.
Let ${\mathcal{R}}^*$ be the set of paths we get.
The algorithm is iterative. Throughout, we maintain a set $\hat {\mathcal{R}}$ of paths in $G$, that is initialized to be ${\mathcal{R}}^*$. The algorithm proceeds in iterations and continues to be executed as long as the set $\Pi^T(\hat{\mathcal{R}})$ for the current path set $\hat{\mathcal{R}}$ is non-empty.
We now describe an iteration of the algorithm.
Let $(P,P',v)$ be any triple in $\Pi^T(\hat{\mathcal{R}})$. We perform the path splicing to paths $P,P'$ at vertex $v$. Let $\tilde P,\tilde P'$ be the paths we get.
We let path $\hat P$ be obtained from $\tilde P$ by removing self-loops of $\tilde P$, and we define path $\hat P'$ similarly. So $\hat P,\hat P'$ are simple paths. We then replace the paths $P,P'$ in $\hat{\mathcal{R}}$ by paths $\hat P,\hat P'$ and continue to the next iteration. This completes the description of the algorithm.
Let ${\mathcal{R}}'$ be the set of paths we obtain at the end of the algorithm.
We first show that the algorithm indeed terminates after $|V(G)|\cdot |E(G)|\cdot |{\mathcal{R}}|^3$ rounds. We let $x(\hat {\mathcal{R}})=|\Pi^T(\hat{\mathcal{R}})|$ and $y(\hat{\mathcal{R}})=\sum_{e\in E(G)}\cong_{G}(\hat {\mathcal{R}}, e)$. Clearly, over the course of the algorithm, $x(\hat {\mathcal{R}})\le |V(G)|\cdot |{\mathcal{R}}|^2$, since all paths in $\hat {\mathcal{R}}$ are simple and every pair of simple paths in $G$ share at most $|V(G)|$ vertices, and $y(\hat {\mathcal{R}})\le |E(G)|\cdot |{\mathcal{R}}|$, since $|\hat{\mathcal{R}}|=|{\mathcal{R}}|$ always holds and every edge appears in at most $|{\mathcal{R}}|$ paths in $\hat {\mathcal{R}}$. Note that, from \Cref{lem: non_interfering_paths}, in one iteration of the algorithm, either the value of $y(\hat{\mathcal{R}})$ decreases by at least $1$ (if the pair $\tilde P,\tilde P'$ of paths we obtained in the splicing step are not both simple and we need to remove self-loops of at least one of them afterwards); or the value of $y(\hat{\mathcal{R}})$ stays the same and the value of $x(\hat{\mathcal{R}})$ decreases by at least $1$ (if the pair $\tilde P,\tilde P'$ of paths we obtained in the splicing step are both simple and the size of $|\Pi^T(\hat{{\mathcal{R}}})|$ decreases by at least $1$). Also note that the algorithm will terminate whenever $x(\hat{\mathcal{R}})=0$. Altogether, we get that the algorithm must terminate after at most $|V(G)|\cdot |E(G)|\cdot |{\mathcal{R}}|^3$ rounds.
We now show that the set ${\mathcal{R}}'$ satisfies the desired properties in \Cref{lem: non_interfering_paths}. First, note we have pre-processed the set ${\mathcal{R}}$ of paths into the set ${\mathcal{R}}^*$ of simple paths, and from the description of the algorithm, at any time, the set $\hat {\mathcal{R}}$ will only contain simple paths. We then conclude that the resulting set ${\mathcal{R}}'$ only contains simple paths as well.
Second, note that the pre-processing step of removing self-loops does not change the first or the last endpoint of any path in ${\mathcal{R}}$, so $S({\mathcal{R}}^*)=S({\mathcal{R}})$ and $T({\mathcal{R}}^*)=T({\mathcal{R}})$. Additionally, from \Cref{obs: splicing}, the path-splicing step does not change the set of first endpoints or the set of first endpoints, so $S({\mathcal{R}}')=S({\mathcal{R}}^*)$ and $T({\mathcal{R}}')=T({\mathcal{R}}^*)$. Altogether, we get that $S({\mathcal{R}}')=S({\mathcal{R}})$ and $T({\mathcal{R}}')=T({\mathcal{R}})$. Third, since the algorithm does not terminate as long as the set $\hat{{\mathcal{R}}}$ of paths is not non-transversal with respect to $\Sigma$, the resulting path set ${\mathcal{R}}'$ is non-transversal with respect to $\Sigma$.
Fourth, it is easy to observe that, in either a self-loop-removing step or a path-splicing step, we do not increase the congestion on any edge. Therefore, for the resulting path set ${\mathcal{R}}'$, it must hold that, for each edge $e\in E(G)$, $\cong_G({\mathcal{R}}',e)\le \cong_G({\mathcal{R}},e)$.
\iffalse{for edge-disjoint path}
Let $H=\bigcup_{P\in {\mathcal{R}}}P$ be the subgraph of $\hat G$ consisting of paths of ${\mathcal{R}}$.
Our first step is to compute a set $\hat{\mathcal{R}}$ of $|\hat {\mathcal{R}}|=|{\mathcal{R}}|$ edge-disjoint paths connecting $v$ to $v'$, using standard max-flow algorithms.
We view the paths in $\hat{\mathcal{R}}$ as being directed from $v$ to $v'$. Let $\hat H$ be the directed graph obtained by taking the union of the directed paths in $\hat{\mathcal{R}}$, so $\hat H$ is a subgraph of $H$. From the max-flow algorithm, we can further assume there is an ordering $\Omega$ of its vertices, such that, for every pair $x,y$ of vertices in $\hat H$, if there is a path of $\hat {\mathcal{R}}$ in which $x$ appears before $y$, then $x$ also appears before $y$ in the ordering $\Omega$.
For each vertex $u\in V(\hat H)$, we denote by $\delta^+(u)$ the set of edges leaving $u$ in $\hat H$, and by $\delta^-(u)$ the set of edges entering $u$ in $\hat H$. Clearly, if $u\ne v,v'$, then $|\delta^+(u)|=|\delta^-(u)|$.
We use the following simple observation.
\begin{observation}
\label{obs:rerouting_matching}
We can efficiently compute, for each vertex $u\in V(\hat H)\setminus \set{v,v'}$, a perfect matching $M(u)\subseteq \delta^-(u)\times \delta^+(u)$ between the edges of $\delta^-(u)$ and the edges of $\delta^+(u)$, such that, for each pair of matched pairs $(e^-_1,e^+_1)$ and $(e^-_2,e^+_2)$ in $M(u)$, the intersection of the path that consists of the edges $e^-_1,e^+_1$ and the path that consists of edges $e^-_2,e^+_2$ at vertex $u$ is non-transversal with respect to $\hat\Sigma$.
\end{observation}
\begin{proof}
We start with $M(u)=\emptyset$ and perform $|\delta^-(u)|$ iterations. In each iteration, we select a pair $e^-\in \delta^-(u), e^+\in \delta^+(u)$ of edges that appear consecutively in the rotation ${\mathcal{O}}_v$ of $\tilde\Sigma$. We add $(e^-,e^+)$ to $M(u)$, delete them from $\delta^-(u)$ and $\delta^+(u)$ respectively, and then continue to the next iteration. It is immediate to verify that the resulting matching $M(u)$ satisfies the desired properties.
\end{proof}
We now gradually modify the set $\hat{\mathcal{R}}$ of paths in order to obtain a set ${\mathcal{R}}'$ of edge-disjoint paths connecting $v$ to $v'$, that is non-transversal with respect to $\hat \Sigma$.
We process all vertices of $V(\hat H)\setminus \set{v,v'}$ one-by-one, according to the ordering $\Omega$. We now describe an iteration in which a vertex $u$ is processed. Let $\hat {\mathcal{R}}_u\subseteq \hat {\mathcal{R}}$ be the set of paths containing $u$. For each path $P\in \hat {\mathcal{R}}_u$, we delete the unique edge of $\delta^+(u)$ that lies on this path, thereby decomposing $P$ into two-subpaths: path $P^-$ connecting $v$ to $u$; and path $P^+$ connecting some vertex $u'$ that is the endpoint of an edge of $\delta^+(u)$ to $v'$. Define $\hat {\mathcal{R}}^-_u=\set{P^-\mid P\in \hat {\mathcal{R}}_u}$ and
$\hat {\mathcal{R}}^+_u=\set{P^+\mid P\in \hat {\mathcal{R}}_u}$.
We will glue these paths together using the edges in $\delta^+(u)$ and the matching $M(u)$ produced in Observation \ref{obs:rerouting_matching}. Specifically, we construct a new set ${\mathcal{R}}'$ of paths that contains, for each path $P\in \hat {\mathcal{R}}$, a new path $P'$, as follows. Consider a path $P\in \hat {\mathcal{R}}$. If $P\notin \hat {\mathcal{R}}_u$, then we let $P'=P$. Otherwise, consider the unique path $P^-\in \hat {\mathcal{R}}^-_u$ that is a subpath of $P$, and let $e^-_P$ be the last edge on this path. Let $e^+$ be the edge in $\delta^+(u)$ that is matched with $e^-_P$ in $M(u)$, and let $\hat P^+$ be the unique path in $\hat {\mathcal{R}}^+_u$ that contains $e^+$.
We then define the new path $P'$ to be the concatenation of the path $P^-$, the edge $e^+$, and the path $\hat P^+$. This finishes the description of an iteration.
It is easy to verify that $|{\mathcal{R}}'|=|\hat {\mathcal{R}}|=|{\mathcal{R}}|$, $E({\mathcal{R}}')\subseteq E(\hat{\mathcal{R}})\subseteq E({\mathcal{R}})$, and the set ${\mathcal{R}}'$ of paths is locally non-interfering with respect to $\hat\Sigma$.
\fi
\iffalse
Our first step is to compute another one-to-one routing $\hat{\mathcal{R}}$ of vertice of $S$ to vertices of $T$, via $|\hat {\mathcal{R}}|=|{\mathcal{R}}|$ simple paths, such that for every $e\in E( G)$, $\cong_{ G}(\hat{\mathcal{R}},e)\le \cong_{ G}({\mathcal{R}},e)$, using standard max-flow algorithms.
We first construct a graph $H$ as follows.
We start from the graph $\bigcup_{P\in {\mathcal{R}}}P$.
We then replace each edge $e$ of $E({\mathcal{R}})$ with $\cong_{G}({\mathcal{R}},e)$ copies of it, and let each path of ${\mathcal{R}}$ that contains $e$ takes a distinct copy.
Finally, we add two new vertices $s,t$, and then add, for each vertex $v\in S$, an edge $(s,v)$; and for each vertex $v'\in S$, an edge $(v',t)$.
This finishes the definition of $H$.
Now paths of ${\mathcal{R}}$ are edge-disjoint paths in $H$.
For each vertex $v\in V(H)\setminus \set{s,t}$, we define its rotation ${\mathcal{O}}'_v$ as follows. We start with its oritinal rotation ${\mathcal{O}}_v\in \Sigma$, and then replace, for each edge $e$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $\cong_{G}({\mathcal{R}},e)$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$, and the ordering among the copies is arbitrary.
Now we compute the a set $\hat {\mathcal{R}}$ of $|\hat {\mathcal{R}}|=|{\mathcal{R}}|$ edge-disjoint paths connecting $s$ to $t$ in $H$. It is easy to see that such a set of path exists.
We view these paths as directed from $s$ to $t$.
From the max-flow algorithm, we can further assume there is an ordering $\Omega$ of its vertices, such that, for every pair $x,y$ of vertices in $H$, if there is a path of $\hat{\mathcal{R}}$ in which $x$ appears before $y$, then $x$ also appears before $y$ in the ordering $\Omega$.
For each vertex $u\in V(H)$, we denote by $\delta^+(u)$ the set of edges leaving $u$ in $H$, and by $\delta^-(u)$ the set of edges entering $u$ in $H$. Clearly, if $u\ne s,t$, then $|\delta^+(u)|=|\delta^-(u)|$.
We now gradually modify the set $\hat{\mathcal{R}}$ of paths in order to obtain a set $\hat{\mathcal{R}}'$ of edge-disjoint paths in $H$ connecting $s$ to $t$, that is non-transversal with respect to the new orderings in $\set{{\mathcal{O}}'_v\mid v\in V(H)\setminus\set{s,t}}$.
We process all vertices of $V(H)\setminus \set{s,t}$ one-by-one, according to the ordering $\Omega$. We now describe an iteration in which a vertex $u$ is processed. Let $\hat {\mathcal{R}}_u\subseteq \hat {\mathcal{R}}$ be the set of paths containing $u$. For each path $P\in \hat {\mathcal{R}}_u$, we delete the unique edge of $\delta^+(u)$ that lies on this path, thereby decomposing $P$ into two-subpaths: path $P^-$ connecting $s$ to $u$; and path $P^+$ connecting some vertex $u'$ that is the endpoint of an edge of $\delta^+(u)$ to $t$. Define $\hat {\mathcal{R}}^-_u=\set{P^-\mid P\in \hat {\mathcal{R}}_u}$ and
$\hat {\mathcal{R}}^+_u=\set{P^+\mid P\in \hat {\mathcal{R}}_u}$.
We will glue these paths together using the edges in $\delta^+(u)$ and the matching $M(u)$ produced in \Cref{obs:rerouting_matching_cong}. Specifically, we construct a new set $\hat{\mathcal{R}}'$ of paths that contains, for each path $P\in \hat {\mathcal{R}}$, a new path $P'$, as follows. Consider a path $P\in \hat {\mathcal{R}}$. If $P\notin \hat {\mathcal{R}}_u$, then we let $P'=P$. Otherwise, consider the unique path $P^-\in \hat {\mathcal{R}}^-_u$ that is a subpath of $P$, and let $e^-_P$ be the last edge on this path. Let $e^+$ be the edge in $\delta^+(u)$ that is matched with $e^-_P$ in $M(u)$, and let $\hat P^+$ be the unique path in $\hat {\mathcal{R}}^+_u$ that contains $e^+$.
We then define the new path $P'$ to be the concatenation of the path $P^-$, the edge $e^+$, and the path $\hat P^+$. This finishes the description of an iteration.
It is easy to verify that $|\hat{\mathcal{R}}'|=|\hat {\mathcal{R}}|=|{\mathcal{R}}|$, $E_{H}(\hat {\mathcal{R}}')\subseteq E_{H}(\hat{\mathcal{R}})$, and the set $\hat{\mathcal{R}}'$ of paths is non-transversal with respect to the orderings in $\set{{\mathcal{O}}'_v\mid v\in V(H)\setminus\set{s,t}}$.
Finally, we let the set ${\mathcal{R}}'$ contains, for each path $P'\in \hat{\mathcal{R}}'$, the original path of $P'\setminus\set{s,t}$ in $H$ (namely, if $P'\setminus\set{s,t}$ contains a copy of $e$, then its original path contains the edge $e$).
It is easy to verify that $|{\mathcal{R}}'|=|{\mathcal{R}}|$, for each $e\in G$, $\cong_{G}({\mathcal{R}}',e)\le \cong_{G}({\mathcal{R}},e)$, and the set ${\mathcal{R}}'$ of paths is non-transversal with respect to $\Sigma$.
\fi
\iffalse
\subsection{Proof of \Cref{obs:rerouting_matching_cong}}
\label{apd: Proof of rerouting_matching_cong}
Denote $E=\delta_G(v)$. We first define a multiset $E'$ and an ordering ${\mathcal{O}}'$ on $E'$ as follows. Set $E'$ contains, for each edge $e\in E$, $(n^-_e+n^+_e)$ copies of $e$, in which $n^-_e$ copies are labelled $-$, and other $n^+_e$ copies are labelled $+$. The ordering ${\mathcal{O}}'$ on $E'$ is obtained from the ordering ${\mathcal{O}}_v$ on $E$ by replacing, for each $e\in E'$, the edge $e$ in the ordering ${\mathcal{O}}_v$ with its $(n^-_e+n^+_e)$ copies in $E'$, that appear consecutively at the location of $e$ in ${\mathcal{O}}_v$, and the ordering among these copies is arbitrary.
Since $\sum_{e\in E}n^-_e=\sum_{e\in E}n^+_e$, the number of edges in $E'$ that are labelled $-$ is the same as the number of edges in $E'$ that are labelled $+$.
We now iteratively construct a set $M'$ of ordered pairs as follows.
We start with $M'=\emptyset$ and perform iterations.
In each iteration, we find a pair of edges in $E'$ that appears consecutively in ${\mathcal{O}}'$, such that one of them (denoted by $e'_1$) is labelled $-$ and the other (denoted by $e'_2$) is labelled $+$. It is clear that such a pair always exists. Note that it is possible that $e'_1$ and $e'_2$ are distinct copies of the same edge of $E$.
We then add the ordered pair $(e'_1,e'_2)$ to $M'$, and delete $e'_1,e'_2$ from $E'$ and ${\mathcal{O}}'$.
It is clear that after this iteration, the number of remaining edges in $E'$ (and in ${\mathcal{O}}'$) that are labelled $-$ is still the same as the number of remaining edges in $E'$ (and in ${\mathcal{O}}'$) that are labelled $+$. Therefore, at the end of the process, $E'$ and ${\mathcal{O}}'$ become empty and $M'$ contains $\sum_{e\in E}n^+_e$ ordered pairs of edges in $E'$. Moreover, for each pair of ordered pairs in $M'$, the intersection between the paths consisting of corresponding pairs of edges is non-transversal with respect to ${\mathcal{O}}'$.
We then let set $M$ contains, for each pair $(e'_1,e'_2)\in M$, a pair $(e_1,e_2)$, where $e_1$ is the original edge of $e'_1$ in $E$ and $e_2$ is the original edge of $e'_2$ in $E$. It is clear that for each pair of ordered pairs in $M$, the intersection between the paths consisting of corresponding pairs of edges is non-transversal with respect to ${\mathcal{O}}'$.
\fi
\iffalse
\subsection{Proof of \Cref{lem: non transversal cost of cycles bounded by cr}}
\label{apd: Proof of non transversal cost of cycles bounded by cr}
Consider a pair $R,R'$ of edge-disjoint cycles with $R\in {\mathcal{R}}$ and $R'\in {\mathcal{R}}'$. We denote by $\gamma$ the closed curve obtained by taking the union of the images of all edges of $R$ in $\phi$, and we define the closed curve $\gamma'$ similarly. Since $R$ and $R'$ are edge-disjoint, and since $\phi$ is a feasible solution to instance $I=(G,\Sigma)$, curves $\gamma$ and $\gamma'$ are in general position. Let $X(\gamma,\gamma')$ be the set of crossings between curves $\gamma,\gamma'$ that are not vertex-images in $\phi$, so $|X(\gamma,\gamma')|=\chi(\gamma,\gamma')$. We will show that, if the intersection between $R$ and $R'$ are transversal at exactly one of their shared vertices, then $X(\gamma,\gamma')\ne \emptyset$.
Assume for contradiction that $X(\gamma,\gamma')=\emptyset$. Let $v_1,\ldots,v_k$ be the shared vertices between $R$ and $R'$, and let $x_1,\ldots,x_k$ be the images of $v_1,\ldots,v_k$ in $\phi'$, respectively. Assume without loss of generality that vertices $v_1,\ldots,v_k$ appear sequentially on cycle $R$, and the intersection between $R$ and $R'$ is transversal at $v_1$.
We give a direction for the closed curve $\gamma$ as follows: $x_1\to x_2\to\ldots\to x_k\to x_1$.
Clearly, the curve $\gamma$ partitions the sphere into internally-disjoint discs, and since $R$ is a simple cycle, vertex $x_1$ lies on the boundary of exactly two of these discs, that we denote by $D_{+}$ and $D_{-}$.
Since cycles $R,R'$ are disjoint and the intersection of them is transversal at $v_1$, if we denote by $\gamma'_+$ the tiny segment of curve $\gamma'$ that has $x_1$ as an endpoint and goes a little in one direction of $\gamma'$, and denote by $\gamma'_-$ the tiny segment of curve $\gamma'$ that has $x_1$ as an endpoint and goes a little in the other direction of $\gamma'$, then one of segments $\gamma'_+, \gamma'_+$ belong to $D_{+}$ and the other belongs to $D_{-}$. Therefore, since $\gamma'$ is a closed curve, $\gamma'$ has to intersect the boundary of disc $D_{+}$ at some other point $x'$. However, since $R$ and $R'$ are edge-disjoint and the intersections of $R$ and $R'$ are non-transversal at $v_2,\ldots,v_k$, $x$ cannot be any of $\set{x_2,\ldots,x_k}$. Therefore, $x'$ has to be some non-vertex-image crossings between $\gamma$ and $\gamma'$.
We now complete the proof of \Cref{lem: non transversal cost of cycles bounded by cr}.
For every cycle $R\in {\mathcal{R}}$, we denote by $\gamma_R$ the closed curve obtained by taking the union of the images of all edges of $R$ in $\phi$, and for each cycle $R\in {\mathcal{R}}'$, we define the closed curve $\gamma_R'$ similarly.
Then it is easy to verify that every crossing $(e,e')$ in drawing $\phi$ belongs to at most $\bigg(\cong_G({\mathcal{R}},e)\cdot \cong_G({\mathcal{R}}',e')+\cong_G({\mathcal{R}},e')\cdot \cong_G({\mathcal{R}}',e)\bigg)$ sets among all sets of $\set{X(\gamma_R,\gamma_{R'})\mid R\in {\mathcal{R}}, R'\in {\mathcal{R}}'}$.
Since for every pair $R, R'$ of edge-disjoint cycles with $R\in {\mathcal{R}}, R'\in {\mathcal{R}}'$, the intersection of $R$ and $R'$ is transversal at at most one of their common vertices, we get that
\[
\begin{split}
\mathsf{cost}_{\mathsf {NT}}({\mathcal{R}},{\mathcal{R}}';\Sigma)\le & \sum_{R\in {\mathcal{R}}, R'\in {\mathcal{R}}'} |X(\gamma_{R}, \gamma_{R'})|\\
\le & \sum_{e,e'\in E(G)}\chi_{\phi}(e,e')\cdot \bigg(\cong_G({\mathcal{R}},e)\cdot \cong_G({\mathcal{R}}',e')+\cong_G({\mathcal{R}},e')\cdot \cong_G({\mathcal{R}}',e)\bigg).
\end{split}
\]
\subsection{Proof of \Cref{thm: new nudging}}
\label{apd: Proof of curve_manipulation}
We first prove the following lemma.
\begin{lemma}[Nudging]
\label{obs: curve_manipulation}
There exists an efficient algorithm, that, given a point $p$ in the plane, a disc $D$ containing $p$, and a set $\Gamma$ of curves, such that every curve $\gamma$ of $\Gamma$ contains the point $p$ and intersects the boundary of $D$ at two distinct points $s_{\gamma},t_{\gamma}$, and $p$ is the only point in $D$ that belongs to at least two curves of $\Gamma$, computes, for each curve $\gamma\in \Gamma$, a curve $\gamma'$, such that
\begin{itemize}
\item for each curve $\gamma\in \Gamma$, the curve $\gamma'$ does not contain $p$, and is identical to the curve $\gamma$ outside the disc $D$;
\item the segments of curves of $\set{\gamma'\mid \gamma\in \Gamma}$ inside disc $D$ are in general position;
\item for each pair $\gamma_1,\gamma_2$ of curves in $\Gamma$, the new curves $\gamma_1,\gamma_2$ cross inside $D$ iff the order in which the points $s_{\gamma_1},t_{\gamma_1},s_{\gamma_2},t_{\gamma_2}$ appear on the boundary of $D$ is either $(s_{\gamma_1},s_{\gamma_2},t_{\gamma_1},t_{\gamma_2})$ or $(s_{\gamma_1},t_{\gamma_2},t_{\gamma_1},s_{\gamma_2})$.
\end{itemize}
\end{lemma}
\begin{proof}
We first compute, for each $\gamma\in \Gamma$, a curve $\tilde\gamma$ inside disc $D$ connecting $s_{\gamma}$ to $t_{\gamma}$, such that (i) the curves in $\set{\tilde\gamma\mid \gamma\in \Gamma}$ are in general position; and (ii) for each pair $\gamma_1,\gamma_2$ of curves in $\Gamma$, the curves $\tilde\gamma_1$ and $\tilde\gamma_2$ intersects iff the order in which the points $s_{\gamma_1},t_{\gamma_1},s_{\gamma_2},t_{\gamma_2}$ appear on the boundary of $D$ is either $(s_{\gamma_1},s_{\gamma_2},t_{\gamma_1},t_{\gamma_2})$ or $(s_{\gamma_1},t_{\gamma_2},t_{\gamma_1},s_{\gamma_2})$.
It is clear that this can be achieved by first setting, for each curve $\gamma$, the curve $\tilde \gamma$ to be the line segment connecting $s_{\gamma}$ to $t_{\gamma}$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\tilde \gamma\mid \gamma\in \Gamma}$.
We now define, for each $\gamma\in \Gamma$, the curve $\tilde\gamma$ to be the union of the part of $\gamma$ outside $D$ and the curve $\tilde\gamma$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, all conditions of \Cref{obs: curve_manipulation} are satisfied.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of $\Gamma$ are shown in distinct colors. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of $\set{\tilde\gamma\mid \gamma\in \Gamma}$ are shown in dash lines.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the proof of \Cref{obs: curve_manipulation}.}\label{fig: curve_con}
\end{figure}
We now use \Cref{obs: curve_manipulation} to prove \Cref{thm: new nudging}.
Denote $G'=G\setminus C$ and denote $\hat {\mathcal{Q}}={\mathcal{Q}}\cup {\mathcal{Q}}'$.
For each $e\in E(G')$, we let set $\Pi_{e}$ contain $\cong_{G'}(\hat {\mathcal{Q}},e)$ curves connecting the endpoints of $e$ that are internally disjoint and lying inside an arbitrarily thin strip around the curve $\phi(e)$. We then assign, for each path $Q\in \hat {\mathcal{Q}}$ and each edge $e\in E(Q)$, a distinct curve of $\Pi_{e}$ to $Q$, so each curve in $\bigcup_{e\in E(G')}\Pi_{e}$ is assigned to exactly one path of $Q$.
For each $e\in \delta_G(C)$, let $\zeta_Q$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E(G')}\Pi_{e}$ that are assigned to $Q$, so $\zeta_Q$ connects $\phi(s_Q)$ to $\phi(t^*)$.
We denote $Z=\set{\zeta_Q\mid Q\in \hat {\mathcal{Q}}}$.
In fact, when we assign curves of $\bigcup_{e\in \delta_{G'}(t^*)}\Pi_{e}$ to path of $\hat{\mathcal{Q}}$, we additionally ensure that the order in which the curves of $\set{\zeta_Q\mid Q\in \hat {\mathcal{Q}}}$ enter $\phi(t^*)$ is identical to ${\mathcal{O}}^{\operatorname{guided}}(\hat{\mathcal{Q}},{\mathcal{O}}_{t^*})$, the ordering guided by the set $\hat{\mathcal{Q}}$ of paths and the ordering ${\mathcal{O}}_{t^*}$. Note that this can be easily achieved according the the definition of ${\mathcal{O}}^{\operatorname{guided}}(\hat{\mathcal{Q}},{\mathcal{O}}_{t^*})$.
We then modify the curves of $Z$ as follows.
For each vertex $v\in V(G')$, we denote by $x_v$ the point that represents the image of $v$ in $\phi$, and we let $X$ contain all points of $\set{x_v\mid v\in V(G')}$ that are crossings between curves in $Z$, so $x_{t^*}\notin X$.
We then iteratively process all points of $X$ as follows.
Consider a point $x_v$ of $X$ and let $D_{\phi}(v)$ be a tiny $v$-disc.
Let $Z(v)$ be the subset of all curves in $Z$ that contains $x_v$.
Let $Z'(v)$ be the set of curves we obtain by applying the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $Z(v)$ within disc $D_{\phi}(v)$. We then replace curves of $Z(v)$ in $Z$ by curves of $Z'(v)$, and continue to the next iteration. Let $\Gamma'$ be the set of curves that we obtain after processing all points of $X$ in this way.
Note that the curves of $\Gamma'$ might be non-simple, and a pair of curves in $\Gamma'$ may cross more than once. We then remove self-loops from all curves of $\Gamma'$, and perform type-1 uncrossing (the algorithm from \Cref{thm: type-1 uncrossing}) to curves of $\Gamma'$, and finally denote by $\Gamma=\set{\gamma(Q)\mid Q\in \hat{\mathcal{Q}}}$ the set of curves we obtained. This finishes the construction of the set $\Gamma$ of curves.
From \Cref{obs: curve_manipulation} and the construction of $\Gamma$, it is easy to verify that the first and the second properties are satisfied.
We now show that $\Gamma$ satisfies the fourth property.
Since our operations at vertex-images only modify the image of curves within tiny discs around inner-vertex-images but does not change the course of any curves, and since type-1 uncrossing does not change the last segment of curves in $\Gamma'$ (and therefore does not change the ordering in which curves of $\Gamma'$ enter $\phi(t^*)$), the curves of $\Gamma$ enter $\phi(t^*)$ in the order ${\mathcal{O}}^{\operatorname{guided}}(\hat{\mathcal{Q}},{\mathcal{O}}_{t^*})$.
We next show that $\Gamma$ satisfies the fifth property.
Note that the curves of $\Gamma$ all lie within thin strips of the image of edges of $E(\hat{\mathcal{Q}})$ in $\phi$, and for each edge $e'\in E(G')$, the number of segments in $\Gamma$ that lie entirely within the thin strip of $\phi(e')$ is bounded by $\cong_{G'}(\hat{\mathcal{Q}},e')$. Therefore, the number of crossings between the image of any edge $e\in E(C)$ and the curves of $\Gamma$ is bounded by the sum, over all edges $e'\in E(G')$, the number of crossings between $e$ and $e'$ in $\phi$, times $\cong_G(\hat{\mathcal{Q}}, e')$, which is in turn bounded by $\chi_{\phi}(e, G\setminus C)\cdot \cong_G(\hat{\mathcal{Q}})$.
It remains to show that $\Gamma$ satisfies the third property, and this is the only place where we treat path sets ${\mathcal{Q}}$ and ${\mathcal{Q}}'$ differently. First, since we performed type-1 uncrossing to the set $\Gamma'$ of curves in the last step, the number of crossings between curves of $\Gamma({\mathcal{Q}}')=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}'}$ is at most $|{\mathcal{Q}}'|^2$.
Second, we bound the number of crossings between curves of $\Gamma({\mathcal{Q}})=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$, as follows.
On the one hand, it is easy to verify that, if we denote $Z({\mathcal{Q}})=\set{\zeta_Q\mid Q\in {\mathcal{Q}}}$, then the number of non-vertex-image crossings between curves of $Z({\mathcal{Q}}))$ is at most $\chi^2_{\phi}({\mathcal{Q}})$.
Note that
\[
\begin{split}
\chi^2_{\phi}({\mathcal{Q}})
\le & \sum_{e,e'\in E(G)}\chi_{\phi}(e,e')\cdot \cong_G({\mathcal{Q}},e)\cdot \cong_G({\mathcal{Q}},e')\\
\le & \sum_{e,e'\in E(G)}\chi_{\phi}(e,e')\cdot \frac{(\cong_G({\mathcal{Q}},e))^2+ (\cong_G({\mathcal{Q}},e'))^2}{2}\\
\le & \sum_{e\in E(G)}\chi_{\phi}(e)\cdot(\cong_G({\mathcal{Q}},e))^2.
\end{split}
\]
On the other hand, in the step where we repeatedly apply the algorithm from \Cref{obs: curve_manipulation} to the set $Z$ of curves, it is not hard to verify that, if a crossing between two curves $\gamma(Q)$ and $\gamma(Q')$ (where $Q,Q'\in {\mathcal{Q}}$) is introduced in this step, then either they share an edge, or their intersection is transversal at a common inner vertex. The number of pairs of paths in ${\mathcal{Q}}$ that share an edge is bounded by $\sum_{e\in E(G)}(\cong_G({\mathcal{Q}},e))^2$. The number of pairs of paths in ${\mathcal{Q}}$ whose intersection is transversal at some of their common inner vertex is bounded by $\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}};\Sigma)$.
Third, using similar arguments, we can show that the number of crossings between a curve of $\Gamma({\mathcal{Q}})$ and a curve of $\Gamma({\mathcal{Q}}')$ is at most $$\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}},{\mathcal{Q}}';\Sigma)+\sum_{e\in E(G)} \cong_G({\mathcal{Q}},e)\cdot\cong_G({\mathcal{Q}}',e)+
\sum_{e,e'\in E(G)}
\chi_{\phi}(e,e')\cdot \cong_G({\mathcal{Q}},e)\cdot\cong_G({\mathcal{Q}}',e').$$
Altogether, they imply that $\Gamma$ satisfies the third property.
This completes the proof of \Cref{thm: new nudging}.
\subsection{Proof of \Cref{lem: curves from non-transversal paths}}
\label{apd: Proof of curves from non-transversal paths}
We construct a feasible solution $\phi'$ of $(G',\Sigma')$ as follows.
Recall that we are given a solution $\phi$ of instance $(G,\Sigma)$. We start with drawing $\phi(G\setminus E({\mathcal{P}}))$, the drawing of subgraph $G\setminus E({\mathcal{P}})$ induced by $\phi$. It remains to add the images of $\set{e_P\mid P\in {\mathcal{P}}}$ to $\phi(G\setminus E({\mathcal{P}}))$, which we do as follows.
For each path $P\in {\mathcal{P}}$, we denote $\gamma_P=\phi(P)$ the image of $P$ in $\phi$, so $\gamma_P$ is a curve (possibly non-simple) connecting $\phi(s)$ to $\phi(t)$.
Denote $\Gamma=\set{\gamma_P\mid P\in {\mathcal{P}}}$.
We iteratively modify the curves in $\Gamma$, by processing all vertices of $V({\mathcal{P}})$ one-by-one as follows.
Consider a vertex $v\in V(P)$. Denote $\Gamma_v=\set{\gamma_P\mid v\in V(P)}$. We apply \Cref{obs: curve_manipulation} to curves in $\Gamma_v$ within an arbitrarily small disc $D_v$ around $\phi(v)$, and get a new set $\Gamma'_v=\set{\gamma'_P\mid \gamma_P\in \Gamma_v}$ of curves. We then replace the curves of $\Gamma_v$ in $\Gamma$ with curves of $\Gamma_v$. This completes the iteration of processing vertex $v$.
Note that, since the paths of ${\mathcal{P}}$ are edge-disjoint and non-transversal with respect to $\Sigma$, the curves of $\Gamma_v$ do not cross in $D_v$.
Let $\Gamma^*=\set{\gamma_P\mid P\in {\mathcal{P}}}$ be the set of curves we get after processing all vertices of $V(P)$ in this way.
Note that curves of $\Gamma^*$ may be non-simple. We then delete self-loops of for each curve of $\Gamma^*$. We then add curves of $\Gamma^*$ to the drawing $\phi(G\setminus E({\mathcal{P}}))$, where for each $P\in {\mathcal{P}}$, the curve $\gamma^*_{P}$ is designated as the image of $e_P$.
Let $\phi'$ be the resulting drawing. It is easy to verify that the $\phi'$ is a feasible solution of the instance $(G',E')$.
Since we have not created additional crossings in the process of obtaining the curves of $\Gamma^*$, any crossing in $\phi'$ is also a crossing in $\phi$. We then conclude that $\mathsf{cr}(\phi')\le \mathsf{cr}(\phi)$.
\fi
\subsection{Proof of \Cref{claim: remove congestion}}
\label{apd: Proof of remove congestion}
Denote $k=|{\mathcal{P}}|$.
We define an $s$-$t$ flow network $H$, as follows. We start with the graph $G$, and set the capacity of every edge in $G$ to be $1$. Add a source vertex $s$, and a destination vertex $t$, and, for every vertex $v\in V(G)$, add an edge $(s,v)$ of capacity $n_S(v)$, and an edge $(t,v)$ of capacity $n_T(v)$ to $H$. Notice that the set ${\mathcal{P}}$ of paths naturally defines a valid $s$-$t$ flow of value $k/\rho$ in this network, where we send $1/\rho$ flow units on each path in ${\mathcal{P}}$. From the integrality of flow, since all edge capacities in $H$ are integral, there is an integral $s$-$t$ flow in $H$, of value at least $k/\rho$. This integral flow immediately defines the desired collection ${\mathcal{P}}'$ of edge-disjoint paths in graph $G$.
\iffalse
\subsection{Proof of \Cref{lem: opt of contracted instance}}
\label{apd: Proof of opt of contracted instance}
\znote{To Complete.}
\fi
\iffalse{previous type-2 unucrossing}
\subsection{Proof of Theorem~\ref{thm: type-2 uncrossing}}
\label{apd: type-2 uncrossing}
We use similar arguments as in the proof of Theorem~\ref{thm: type-1 uncrossing}.
Let $Z$ be the set of points that lie on at least two curves of $\Gamma$. For each point $z\in Z$, let $D_z$ be an arbitrarily small disc around point $z$, such that, if $z$ lies on curves $\gamma_{1},\ldots,\gamma_{k}$, then
\begin{itemize}
\item the disc $D_z$ is disjoint from all curves of $\Gamma\setminus\set{\gamma_{1},\ldots,\gamma_{k}}$;
\item for each $1\le i\le k$, the intersection between disc $D_z$ and curve $\gamma_i$ is a simple subcurve of $\gamma_i$; and
\item for each $1\le i\le k$, the curve $\gamma_i$ intersects the boundary of $D_z$ at two distinct points.
\end{itemize}
Moreover, all discs in $\set{D_z}_{z\in Z}$ are mutually disjoint.
We now process the sets $\Gamma_1,\ldots,\Gamma_r$ of curves one-by-one.
We will describe the algorithm for processing the set $\Gamma_1$, and the algorithms for processing sets $\Gamma_2,\ldots,\Gamma_r$ are the same.
The algorithm for processing the set $\Gamma_1$ is iterative.
Throughout, we maintain (i) a set $\hat{\Gamma}_1$ of curves, that is initialized to be $\Gamma_1$; and (ii) a set $\hat Z$ of points that lie in at least two curves of $\hat\Gamma_1\cup (\Gamma\setminus \Gamma_1)$, that is initialized to be $Z$.
The algorithm continues to be executed as long as there is still a point that lies on at least two curves of $\hat \Gamma_1$ and is an inner point of each of these curves.
We now describe an iteration.
We first arbitrarily pick such a point $z$, and let $\gamma_{1},\ldots,\gamma_{k}$ be the curves that contains $z$.
For each $1\le i\le k$, we denote by $s_i$ the first endpoint of $\gamma_i$ and by $t_i$ the last endpoint of $\gamma_i$, and denote by $s'_i, t'_i$ the intersections between curve $\gamma_i$ and the boundary of disc $D_z$, such that the points $s_i,s'_i,z,t'_i,t_i$ appear on the directed curve $\gamma_i$ in this order.
\begin{figure}[h]
\centering
\subfigure[Before: Curves $\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5$ are shown in red, yellow, pink, black and green respectively. The points of $S'\cup T'$ appear on the boundary of disc $D_z$.]{\scalebox{0.33}{\includegraphics[scale=1.0]{figs/type_2_uncross_proof_1.jpg}
}
\hspace{0.1cm}
\subfigure[The set $\set{\eta^i_z\mid 1\le i\le k}$ of curves connecting distinct vertices of $S'$ to distinct vertices of $T'$ are shown in dash purple lines.]{
\scalebox{0.33}{\includegraphics[scale=1.0]{figs/type_2_uncross_proof_2.jpg}}}
\hspace{0.1cm}
\subfigure[After: New curves $\hat\gamma_1,\hat\gamma_2,\hat\gamma_3,\hat\gamma_4,\hat\gamma_5$ are shown in red, yellow, pink, black and green respectively. Point $z$ no longer lies in any of these curves.]{
\scalebox{0.33}{\includegraphics[scale=1.0]{figs/type_2_uncross_proof_3.jpg}}}
\caption{An illustration of an iteration in type-1 uncrossing.}\label{fig: type_2_uncross_proof}
\end{figure}
We now modify the curves $\gamma_1,\ldots,\gamma_k$ as follows. Denote $S'=\set{s'_i\mid 1\le i\le k}$ and $T'=\set{t'_i\mid 1\le i\le k}$. We first use the algorithm in~\Cref{obs:rerouting_matching_cong} to compute a set $\set{\eta^{i}_z\mid 1\le i\le k}$ of curves, such that
\begin{itemize}
\item for each $1\le i\le k$, the curve $\eta_z^i$ connects $s'_i$ to a distinct point $t'_{x_i}$ of $T'$;
\item for each $1\le i\le k$, the curve $\eta_z^i$ lies entirely in the disc $D_z$; and
\item the curves in $\set{\eta^{i}_z\mid 1\le i\le k}$ are mutually disjoint.
\end{itemize}
Note that the algorithm in~\Cref{obs:rerouting_matching_cong} only gives us a matching between points in $S'$ and points $T'$, but it can be easily tranformed into a set of curves satisfying the above properties.
See Figure~\ref{fig: type_2_uncross_proof} for an illustration.
For each $1\le i\le k$, we denote by $\gamma^s_i$ the subcurve of $\gamma_i$ between $s_i$ and $s'_i$, and by $\gamma^t_i$ the subcurve of $\gamma_i$ between $t'_i$ and $t_i$.
We then define, for each $1\le i\le k$, a new curve $\hat \gamma_i$ to be the sequential concatenation of curves $\gamma^s_i, \eta^i_z, \gamma^t_{x_i}$.
See Figure~\ref{fig: type_2_uncross_proof} for an illustration.
We then replace the curves $\gamma_1,\ldots,\gamma_k$ in $\hat\Gamma_1$ with the new curves $\hat\gamma_1,\ldots,\hat\gamma_k$. This completes the description of an iteration.
Let $\Gamma'_1$ be the set of curves that we obtain at the end of the algorithm, and let $Z'$ be set of points that lie on at least two curves of $\Gamma'_1\cup (\Gamma\setminus \Gamma_1)$.
It is easy to verify that, in the iteration described above, the multiset of the first endpoints of new curves $\hat\gamma_1,\ldots,\hat\gamma_k$ is identical to that of the original curves $\gamma_1,\ldots,\gamma_k$ and similarly the multiset of the last endpoints of new curves $\hat\gamma_1,\ldots,\hat\gamma_k$ is identical to that of the original curves $\gamma_1,\ldots,\gamma_k$.
Therefore, $S(\Gamma'_1)=S(\Gamma_1)$ and $T(\Gamma'_1)=T(\Gamma_1)$.
It is also easy to verify that the point $z$ no longer lies in any of the new curves $\hat\gamma_1,\ldots,\hat\gamma_k$, while other points in $\hat Z$ remains in $\hat Z$, and there are no new points added to $\hat Z$.
So in each iteration, one point is removed from $\hat Z$, and therefore the algorithm of processing $\Gamma_1$ will terminate in $O(\chi(\Gamma_1))$ rounds.
Moreover, from the terminating criterion of the algorithm, the curves in the resulting set $\Gamma_1'$ no longer cross with each other.
For each $1\le i\le r$, let $\Gamma'_i$ be the set of curves that we obtain from processing the set $\Gamma_i$ of curves using the same algorithm.
We now show that the sets $\Gamma_1,\ldots,\Gamma_r$ satisfy the conditions in \Cref{thm: type-2 uncrossing}.
First, from the above discussion, for each $1\le i\le r$, $S(\Gamma'_i)=S(\Gamma_i)$ and $T(\Gamma'_i)=T(\Gamma_i)$.
Second, if we denote by $Z'$ the set of points that lie in at least two curves of $\Gamma'=\Gamma_0\cup(\bigcup_{1\le i\le r}\Gamma'_i)$, then clearly $Z'\subseteq Z$, and for each $1\le i\le r$, $Z'$ does not contain a point lying in at least two curves of $\Gamma'_i$.
It follows that for each $1\le i\le r$, curves in $\Gamma_i$ do not cross each other; for every curve $\gamma\in \Gamma_0$, $\chi(\gamma,\Gamma'\setminus \Gamma_0)\le \chi(\gamma,\Gamma\setminus \Gamma_0)$; and
$\chi(\Gamma')\le \chi(\Gamma)$.
Last, it is easy to verify from the algorithm that,
if a point is an endpoint of a curve in $\Gamma'$, then it may not be an inner point of any other curve in $\Gamma'$.
Moreover, if a point of $Z$ lie in more than 2 curves in $\Gamma'$, then it has to be an endpoint of every curve that contains it.
Therefore, curves in $\Gamma'$ are in general position.
This completes the proof of \Cref{thm: type-2 uncrossing}.
{previous type-2 uncrossing}
\fi
\subsection{Proof of Corollary~\ref{cor: approx_balanced_cut}}
\label{apd: Proof of approx_balanced_cut}
Denote $G=(V,E)$.
Let $H$ be the graph obtained from $G$ by subdividing each edge $e$ of $G$ with a new vertex $x_e$.
Note that, even if $G$ is a multigraph, $H$ is a simple graph.
We denote $\Upsilon=\set{x_e\mid e\in E}$, so $V(H)=V\cup \Upsilon$.
Let $(S,T)$ be a cut in $G$. We say that a cut $(S',T')$ in $H$ \emph{extends $(S,T)$}, iff $S\subseteq S'$, $T\subseteq T'$, $x_e\in S'$ for all edges $e\in E_G(S)$, and $x_e\in T'$ for all edges $e\in E_G(T)$.
We use the following observation.
\begin{observation}
\label{obs: balanced_vertex_edge_cut}
Let $(S,T)$ be a cut in $G$ and $(S',T')$ a cut in $H$ that extends $(S,T)$, then
\begin{itemize}
\item $|E_G(S,T)|=|E_H(S',T')|$;
\item if $(S,T)$ is a $\theta$-edge-balanced cut in $G$ for some real number $0<\theta<1/2$, then the cut $(S',T')$ is a $(\theta/3)$-vertex-balanced cut in $H$; and
\item if $(S',T')$ is a $\theta$-vertex-balanced cut in $H$ for some real number $0<\theta<1/2$, then the cut $(S,T)$ is a $(\theta/2)$-edge-balanced cut in $G$.
\end{itemize}
\end{observation}
\begin{proof}
From the construction of $H$, since the cut $(S',T')$ extends $(S,T)$, if an edge $e'$ belong to $E_H(S',T')$, then one endpoint of $e'$ has to be $x_e$ for some $e\in E_G(S,T)$, and the other endpoint is an endpoint of $e$ in either $S$ or $T$, depending on whether the vertex $x_e$ belongs to $S'$ or $T'$.
On the other hand, for each edge $e\in E_G(S,T)$ with $e=(u,u')$ and $u\in S, u'\in T$, consider the edges $(u,x_e)$ and $(u',x_e)$ of $H$. It is easy to see that, if the cut $(S',T')$ extends $(S,T)$, then exactly one of edges $(u,x_e), (u',x_e)$ belongs to $E_H(S',T')$. Altogether, $|E_G(S,T)|=|E_H(S',T')|$.
If $(S,T)$ is a $\theta$-edge-balanced cut in $G$, namely $|E_G(S)|,|E_G(T)|\ge \theta\cdot|E|$, note that $|V(S')|\ge |E_G(S)|+|S|$, and $|V(H)|=|E|+|V|$.
Since $G$ is connected, $|E|\ge |V|-1\ge |V|/2$, so $|E|\ge (|E|+|V|)/3$.
Therefore,
$$|V(S')|\ge |E_G(S)|\ge \theta\cdot |E|\ge \theta\cdot (|E|+|V|)/3\ge
(\theta/3)\cdot|V(H)|.$$
Similarly, $|V(T')|\ge(\theta/3)\cdot|V(H)|$, and therefore $(S',T')$ is a $(\theta/3)$-vertex-balanced cut in $H$.
If $(S',T')$ is a $\theta$-vertex-balanced cut in $G$,
namely $|S'|,|T'|\ge \theta\cdot |V(H)|$,
then
\[
|E_G(S)|\ge |S'|/2 \ge (\theta/2)\cdot|V(H)| \ge (\theta/2)\cdot|E|.
\] Similarly, $|E_G(T)|\ge (\theta/2)\cdot|E|$, and therefore $(S,T)$ is a $(\theta/2)$-edge-balanced cut in $G$.
\end{proof}
We now use Lemma~\ref{obs: balanced_vertex_edge_cut} to prove Corollary~\ref{cor: approx_balanced_cut}. Recall that we are given a constant $\hat c$ with $0<\hat c< 1/2$.
We first apply Theorem~\ref{thm: ARV} with parameter $c=\hat c/3$ and obtain a constant $c'$ with $0<c'<c$, and an algorithm ${\mathcal{A}}^{c,c'}_{\text{vertex}}$ that, given any connected simple graph $\tilde G$, computes a $c'$-vertex-balanced cut of $\tilde G$, whose size is at most $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)$ times the size of a minimum $c$-vertex-balanced cut of $\tilde G$.
We set the constant $\hat c'=c'/4$, and construct an algorithm ${\mathcal{A}}^{\hat c,\hat c'}_{\text{edge}}$ that, given any connected multigraph $G$, computes a $\hat c'$-edge-balanced cut of $G$, whose size is at most $O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n))$ times the size of a minimum $\hat c$-edge-balanced cut of $G$, as follows.
Let $G$ be the input connected multigraph. We first construct the graph $H$ by subdividing each edge of $G$ as described above. Note that $H$ is a simple graph. We then apply the algorithm ${\mathcal{A}}^{c,c'}_{\text{vertex}}$ to graph $H$ and obtain a $c'$-vertex-balanced cut $(A,B)$ of $H$, such that $|E_H(A,B)|$ is at most $O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)\cdot X)$, where $X$ is the size of a minimum $c$-vertex-balanced cut in $H$.
We will process the cut $(A,B)$ of $H$ to obtain a cut $(A^*,B^*)$ of $G$, in several steps.
\paragraph{Step 1.}
We start with some definitions.
Given a cut $(\hat A,\hat B)$ of $H$, we say that a vertex $x_e\in \Upsilon$ is \emph{$\hat A$-isolated} iff $x_e\in \hat A$, and if we denote by $u,v$ the endpoints of $e$ as an edge in $E$, then $u,v$ belong to $\hat B$ as vertices of $V(H)$.
Similarly, we say that a vertex $x_e\in \Upsilon$ is \emph{$\hat B$-isolated} iff $x_e\in \hat B$, and if $e=(u,v)$, then $u,v\in \hat A$.
In the first step, we will process the cut $(A,B)$ given by the algorithm ${\mathcal{A}}^{c,c'}_{\text{vertex}}$ and obtain another cut $(A_1,B_1)$ of $H$, such that there are either only $A_1$-isolated vertices or only $B_1$-isolated vertices, but not both. Specifically, we start with the cut $(A,B)$, and while there is a $A$-isolated vertex $u$ and a $B$-isolated vertex $u'$, we move $u$ from $A$ to $B$ and move $u'$ from $B$ to $A$.
It is clear that, after such a round of ``exchanging isolated vertices'', the number of $A$-isolated vertices and the number of $B$-isolated vertices are both decreased by $1$, the sizes of $A$ and $B$ do not change, and the number of edges in the cut $(A,B)$ is decreased by $4$. Therefore, this step will terminate after $O(n^2)$ rounds.
Let $(A_1, B_1)$ be the cut obtained at the end of this step, then it is clear that there are either only $A_1$-isolated vertices or only $B_1$-isolated vertices.
It is also clear that $|E(A_1,B_1)|\le |E(A,B)|$, and $|A_1|=|A|$ and $|B_1|=|B|$. Therefore, $|A_1|,|B_1|\ge c'\cdot|V(H)|$.
We assume without loss of generality that there are only $B_1$-isolated vertices.
\paragraph{Step 2.}
We start with some definitions. Given a cut $(\hat A,\hat B)$ of $H$ such that there is no $\hat A$-isolated vertices, we say that a vertex $u\in V$ is $\hat A$-dominated, iff $u\in \hat A$, and among the edges $e_1,\ldots,e_d$ of $\delta_G(u)$ (where $d=\deg_G(u)$), at least $(c'/2)$-fraction of the corresponding vertices $x_{e_1},\ldots,x_{e_d}$ are $\hat B$-isolated. We use the following claim.
\begin{claim}
\label{clm: dominated_vertex}
If the number of $B_1$-isolated vertices is at least $(c'/2)\cdot |E|$, then there is at least one $A_1$-dominated vertex.
\end{claim}
\begin{proof}
Assume the constrast that the number of $B_1$-isolated vertices is less than $(c'/2)\cdot V(H)$, while there is no $A_1$-dominated vertex. Then we have
\[
\begin{split}
\big|\set{e\mid x_e \text{ is $B_1$-dominated}}\big|
& = \frac{1}{2}\cdot \sum_{v\in V}\big|\set{e\in \delta_G(v)\mid x_e \text{ is $B_1$-dominated}}\big|\\
& < \frac{1}{2}\cdot \sum_{v\in V}\frac{c'}{2}\cdot |\delta_G(v)| =\frac{c'}{4}\cdot 2\cdot|E|=\frac{c'}{2}\cdot |E|,
\end{split}
\]
a contradiction to the fact that number of $B_1$-isolated vertices is at least $(c'/2)\cdot |E|$.
\end{proof}
We now describe the algorithm in this step. We start with the cut $(A_1,B_1)$. Recall that we have assumed that there are only $B_1$-isolated vertices.
We proceed in iterations, and terminate as long as the number of $B_1$-isolated vertices is at most $|B_1|/2$.
While the number of $B_1$-isolated vertices is at least $|B_1|/2$, since $|B_1|\ge c'\cdot |V(H)|\ge c'\cdot |E|$, the number of $B_1$-isolated vertices is at least $(c'/2)\cdot |E|$. From Claim~\ref{clm: dominated_vertex}, there is at least one $A_1$-dominated vertex. We then find an $A_1$-dominated vertex $u$ (by examining each vertex of $V$), so $u\in A_1$. Denote $d=\deg_G(u)$ and $\delta_G(u)=\set{e_1,\ldots,e_d}$, and assume that the vertex $x_{e_1}$ is a $B_1$-isolated vertex.
We then move $u$ from $A_1$ to $B_1$ and move $x_{e_1}$ from $B_1$ to $A_1$. This completes the description of an iteration. Note that, after each iteration of processing the $A_1$-dominated vertex $u$, the size of $A_1$ and $B_1$ remain the same, the number of $B_1$-isolated vertices is decreased by at least $(c'/2)\cdot d$, and the size of $|E(A_1,B_1)|$ is increased by at most $(1-c'/2)\cdot d$.
Therefore, the algorithm in this step will terminate in $O(n^2)$ rounds.
Let $(A_2,B_2)$ be the cut obtained at the end of this step. It is easy to see that $|A|=|A_1|=|A_2|$, $|B|=|B_1|=|B_2|$, the number of $B_2$-isolated vertices is at least $|B_2|/2$, and
$|E(A_2,B_2)|\le \frac{1-c'/2}{c'/2}\cdot |E(A_1,B_1)|$.
We define the cut $(A_3,B_3)$ from the cut $(A_2,B_2)$ by moving all $B_2$-isolated vertices from $B_2$ to $A_2$. Then $|A_3|\ge |A_2|$, $|B_3|\ge |B_2|/2$ and $|E(A_3,B_3)|\le |E(A_2,B_2)|$. Lastly we define $A^*=A_3\cap V$ and $B^*=B_3\cap V$, so $(A^*, B^*)$ is a cut of $G$ with no $A^*$-isolated vertices or $B^*$-isolated vertices, and therefore the cut $(A_3,B_3)$ in $H$ extends $(A^*,B^*)$.
We then let the cut $(A^*,B^*)$ be the output of our algorithm ${\mathcal{A}}_{\text{edge}}^{\hat c,\hat c'}$. This completes the description of the algorithm.
It remains to prove the correctness of the algorithm. On one hand, since $(A,B)$ is a $c'$-vertex balanced cut of $H$, $|A_3|\ge |A_2|=|A|\ge c'\cdot |V(H)|$ and $|B_3|\ge |B_2|/2=|B|/2\ge (c'/2)\cdot |V(H)|$. Therefore $(A_3,B_3)$ is a $(c'/2)$-vertex balanced cut of $H$. From
Observation~\ref{obs: balanced_vertex_edge_cut}, $(A^*, B^*)$ is a $(c'/4)$-edge-balanced cut in $G$. On the other hand, let $(\tilde A,\tilde B)$ be a minimum $\hat c$-edge-balanced cut in $G$, and denote its size by $Y$. Then from Observation~\ref{obs: balanced_vertex_edge_cut}, any cut in $H$ that extends $(\tilde A,\tilde B)$ is a $(\hat c/3)$-vertex-balanced cut in $H$, so $Y\ge X$ (recall that $\hat c/3=c$). Since $c'$ is a constant,
\[
|E(A^*,B^*)|=|E(A_3,B_3)|\le |E(A_2,B_2)|\le \frac{1-c'/2}{c'/2}\cdot |E(A,B)|= O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n) \cdot X)\le O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n) \cdot Y),
\]
and it follows that our cut $(A^*,B^*)$ has size at most $O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n))$ times the size of a minimum $\hat c$-edge-balanced cut of $G$.
This completes the proof of Corollary~\ref{cor: approx_balanced_cut}.
\iffalse
\begin{observation}
For any constant $c$, there is a minimum $c$-vertex-balanced cut that is a regular cut.
\end{observation}
\begin{proof}
Let $(A,B)$ be a minimum $c$-vertex-balanced cut. We construct another cut $(\hat A, \hat B)$ as follows. Let $V_A=A\cap V, V_B=A\cap V$.
We first add vertices of $V_A$ to $\hat A$ and add vertices of $V_B$ to $\hat B$.
For each edge $e\in E_G(V_A)$, we add all vertices of $\Gamma_e$ to $\hat A$. Similarly, for each edge $e\in E_G(V_B)$, we add all vertices of $\Gamma_e$ to $\hat B$. Next, for each edge $e=(u,u')$ with $u\in A$ and $u'\in B$, denote $\gamma_e=\set{u_1,\ldots,u_{\ell-1}}$, and we add vertices $u_1,\ldots,u_{|\Gamma_e\cap A|}$ to $\hat A$, and the rest of $\Gamma_e$ to $\hat B$.
\end{proof}
We now set $\hat c=$ and $\hat c'=$, and use the algorithm in Theorem~\ref{thm: ARV} to compute a $\hat c$-vertex balanced cut of $H$, that we denote by $(A,B)$. From Theorem~\ref{thm: ARV}, the value of $|E(A,B)|$ is at most $O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}})$ times the size of a minimum $\hat c'$-vertex-balanced of $G$.
\fi
\subsection{Proof of Theorem~\ref{lem:min_bal_cut}}
\label{apd: Proof of min_bal_cut}
We use the following theorem from~\cite{lipton1979separator}.
\begin{theorem}[Theorem 4 from~\cite{lipton1979separator}]
\label{thm: weighted_planar_separator}
Let $G$ be any $n$-vertex \textbf{simple} planar graph having nonnegative vertex costs summing to no more than one. Then the vertices of $G$ can be partitioned into three sets $A,B,C$, such that no edge joins a vertex in $A$ with a vertex in $B$, neither $A$ nor $B$ has total cost exceeding $2/3$, and $C$ contains no more than $\sqrt{8n}$ vertices.
\end{theorem}
Let $G=(V,E)$ be the input multigraph and let $\phi$ be an optimal drawing of $G$ in the plane.
We define a graph $G'$ in three steps as follows.
We start with graph $G$.
In the first step, we subdivide each edge $e$ of $G$ with a new vertex $y_e$. Namely, every edge $e=(u,u')$ of $G$ is replaced by a path $(u,y_e,u')$ of length $2$, and we denote $\Upsilon=\set{y_e\mid e\in E}$.
Let $\hat G_1$ be the resulting graph after this step, so $\hat G_1$ is a simple graph and $V(\hat G_1)=V\cup \Upsilon$ and $|V(\hat G_1)|=n+|E|$. Moreover, the drawing $\phi$ of $G$ naturally induces a drawing $\hat \phi_1$ of $\hat G_1$ with $\mathsf{cr}(\phi)=\mathsf{cr}(\hat{\phi}_1)$.
We say that the edge $e=(u,u')$ in $E$ \emph{gives birth to} the edges $(u,y_e)$ and $(u',y_e)$ of $\hat G_1$.
In the second step, for every crossing in $\hat\phi_1$ caused by a pair $e,e'$ of edges of $\hat G_1$, where $e=(u,v)$ and $e'=(u',v')$, we add a new node $x_{e,e'}$ to $\hat G_1$, and replace the edges $e,e'$ with four new edges, connecting $x_{e,e'}$ to $u,u',v,v'$, respectively.
Let $\hat G_2$ be the resulting graph after this step.
Denote $X=\set{x_{e,e'}\mid \text{there is a crossing in $\hat\phi_1$ caused by }e,e'}$, so $V(\hat G_2)=V\cup\Upsilon\cup X$ and $|V(\hat G_2)|=\mathsf{OPT}_{\mathsf{cr}}(G)+n+|E|$.
Clearly, $\hat G_2$ is a simple planar graph, and the drawing $\hat\phi_1$ of $\hat G_1$ naturally induces a planar drawing $\hat\phi_2$ of $\hat G_2$.
In the third step, we iteratively process every vertex $v$ of $\hat G_2$ with $\deg_{\hat G_2}(v)>4$.
Note that any vertex $v$ with $\deg_{\hat G_2}(v)>4$ has to be some vertex $v\in V$ with $\deg_G(v)=\deg_{\hat G_2}(v)>4$.
We now describe an iteration of processing such a vertex $v$.
Denote $d=\deg_{\hat G_2}(v)$ and denote $\delta_{\hat G_2}(v)=\set{e_1,\ldots,e_d}$.
We replace the vertex $v$ with a $\deg_{\hat G_2}(v)\times \deg_{\hat G_2}(v)$-grid, that we denote by $H_v$, where the vertex appearing at the intersection of the $i$-th row and the $j$-th column is denoted by $v_{i,j}$, for each $1\le i,j\le d$. We then let, for each $1\le t\le d$, the edge $e_t$ be incident to the vertex $v_{1,t}$.
This completes the description of an iteration, and also completes the description of this step.
Let $G'=(V',E')$ be the resulting graph after this step.
Clearly, $G'$ is a simple planar graph, since we can obtain a planar drawing of $G'$ from the planar drawing $\hat \phi_2$ of $\hat G_2$ by replacing the image of each vertex $v$ with $\deg_{\hat G_2}(v)>4$ by the image of the grid $H_v$.
We denote $V_{\le 4}=\set{v\in V\mid \deg_G(v)\le 4}$ and $V_{>4}=\set{v\in V\mid \deg_G(v)> 4}$, then $V'=V_{\le 4}\cup \Upsilon\cup X\cup (\bigcup_{v\in V_{>4}}V(H_v))$.
Therefore, $|V'|\le \mathsf{OPT}_{\mathsf{cr}}(G)+|E|+\sum_{v\in V(G)}(\deg_{G}(v))^2\le \mathsf{OPT}_{\mathsf{cr}}(G)+2\Delta|E|$, where $\Delta$ is the maximum vertex degree of $G$.
Note that the maximum vertex degree of $G'$ is $4$, since every vertex of $\hat G_2$ with degree greater than $4$ is replaced with a grid, whose maximum vertex degree is $4$.
We now assign weight $1/|\Upsilon|$ to all vertices of $\Upsilon$ in $G'$, and weight $0$ to all other vertices of $G'$. Then we apply Theorem~\ref{thm: weighted_planar_separator} to $G'$ with the above vertex weights. Let $(A,B,C)$ be the partition of $V'$ that we obtain, so $|\Upsilon\cap A|\le (2/3)\cdot |\Upsilon|$, $|\Upsilon\cap B|\le (2/3)\cdot |\Upsilon|$, and $|C|\le \sqrt{8|V'|}=\sqrt{8\cdot\mathsf{OPT}_{\mathsf{cr}}(G)+16\Delta\cdot |E|}$.
We now construct a cut of the original graph $G$ based on the partition $(A,B,C)$ of $V'$, as follows.
We construct a set $\tilde E$ of edges of $G$ as follows. Initially we set $\tilde E=\emptyset$. We first add, for each vertex $v\in C\cap V_{\le 4}$, all edges of $\delta_G(v)$ to $\tilde E$. Note that there are at most $4\cdot|C\cap V_{\le 4}|$ such edges.
We then add, for each vertex $y_e\in C\cap \Upsilon$ (where $e$ represents an edge of $E$), the edge $e$ to $\tilde E$. Note that there are at most $|C\cap\Upsilon|$ such edges.
Next, for each vertex $x_{e,e'}\in C\cap X$ where $e,e'$ are edges of $\hat G_1$, let $\hat e$ and $\hat e'$ be the edges of $E$ that give birth to edges $e$ and $e'$ respectively, and we add edges $\hat e,\hat e'$ to $\tilde E$. Note that there are at most $2\cdot|C\cap X|$ such edges.
Lastly, we process the vertices of $V_{> 4}$ one-by-one to complete the construction of $\tilde E$, as follows.
Consider a vertex $v\in V_{>4}$ and the $(d\times d)$-grid $H_v$, where $d=\deg_G(v)$ and we denote $\delta_G(v)=\set{e_1,\ldots,e_d}$. Recall that for each $1\le t\le d$, the edge $e_t$ is incident to the vertex $v_{1,t}$.
Denote by $V_1$ the set of vertices on the first row of $H_v$, namely $V_1=\set{v_{1,1},\ldots,v_{1,d}}$. Denote $A_1=A\cap V_1$, $B_1=B\cap V_1$, and $C_1=C\cap V_1$. If $|A_1|\le |B_1|$, then we add the edges of $\set{e_t\mid v_{1,t}\in A_1\cup C_1}$ to $\tilde E$, otherwise we add the edges of $\set{e_t\mid v_{1,t}\in B_1\cup C_1}$ to $\tilde E$. This completes the description of processing a vertex $v\in V_{>4}$, and also completes the construction of $\tilde E$. We use the following claim.
\begin{claim}
\label{clm: grid well-linked}
If $|A_1|\le |B_1|$, then $\big|\set{e_t\mid v_{1,t}\in A_1\cup C_1}\big|=|A_1|+|C_1|\le |C\cap V(H_v)|$.
\end{claim}
\begin{proof}
Define the graph $H'_v$ to be the graph obtained from $H_v$ by deleting all vertices of $C_1$ and their incident edges.
Clearly, $|C\cap V(H_v)|=|C_1|+|C\cap V(H'_v)|$, and therefore it suffices to show that $|C\cap V(H'_v)|\ge |A_1|$.
From Theorem~\ref{thm: weighted_planar_separator}, no edge of $G'$ joins a vertex in $A$ with a vertex in $B$. Therefore, the vertices of $C\cap V(H'_v)$ forms a vertex cut in $H'_{v}$ separating vertices of $A_1$ from vertices of $B_1$. We now show that there is a set of $|A_1|$ node-disjoint paths in $H'_v$, each connecting a distinct vertex of $A_1$ to a vertex of $B_1$. Note that, from the max-flow mincut-theorem, this implies that $|C_1|\ge |A_1|$, and the claim follows.
To construct a set ${\mathcal{P}}$ of $|A_1|$ node-disjoint paths connecting vertices of $A_1$ to vertices of $B_1$, we consider the index set $I=\set{i\mid v_{1,i}\in A_1\cup B_1}$. While there is a pair $i,i'$ of indices of $I$, such that $i<i'$, $v_{i},v_{i'}$ belongs to different sets of $A_1,B_1$, and there is no other $j\in I$ with $i<j<i'$, we define the path $P_{i,i'}$ to be the sequential concatenation of the path $(v_{1,i},v_{2,i},\ldots,v_{i',i})$, the path $(v_{i',i},v_{i',i+1},\ldots,v_{i',i'})$, and the path $(v_{i',i'},v_{i'-1,i'},\ldots,v_{1,i'})$. We then add the path $P_{i,i'}$ to ${\mathcal{P}}$, delete $i,i'$ from $I$ and continue to the next iteration.
It is clear that, at the end of the process, $I$ does not contain any index $i$ with $v_{1,i}\in A_1$, and for each $v_{1,j}\in A_1$, there is a path $P_{j,j'}$ connecting $v_{1,j}$ to some vertex $v_{1,j'}$ of $B_1$, so $|{\mathcal{P}}|=|A_1|$. It is also easy to verify that the paths of ${\mathcal{P}}$ are node-disjoint.
\end{proof}
Similarly, we can show that if $|A_1|> |B_1|$, then $\big|\set{e_t\mid v_{1,t}\in B_1\cup C_1}\big|=|B_1|+|C_1|\le |C\cap V(H_v)|$. Therefore, the number of edges that we added to $\tilde E$ when processing a vertex $v\in V_{>4}$ is at most $|C\cap V(H_v)|$.
Altogether,
\[
|\tilde E|\le
4\cdot|C\cap V_{\le 4}|+|C\cap\Upsilon|+2\cdot|C\cap X|+\sum_{v\in V_{>4}}|C\cap V(H_v)|\le 4\cdot |C|=\sqrt{2^{7}\cdot\mathsf{OPT}_{\mathsf{cr}}(G)+2^8\Delta\cdot |E|}.
\]
Let $(\tilde{A},\tilde{B})$ be the cut obtained by removing the edges of $\tilde E$ from $G$, such that the corresponding vertex $y_e\in \Upsilon$ for any edge $e\in E_G(\tilde A)$ lies in $A$, and the corresponding vertex $y_e\in \Upsilon$ for any edge $e\in E_G(\tilde B)$ lies in $B$.
It is easy to see that there is no edge in $G\setminus \tilde E$ connecting a vertex of $\tilde A$ with a vertex of $\tilde B$.
It suffices to show that $|E_G(\tilde A)|\ge |E|/4$ and $|E_G(\tilde B)|\ge |E|/4$.
We now show that $|E_G(\tilde A)|\ge |E|/4$, and the proof for $|E_G(\tilde B)|\ge |E|/4$ is symmetric.
Denote $E^A=\set{e \in E\mid y_e\in (\Upsilon\cap A)}$. It is easy to see that $(E^A\setminus \tilde E)\subseteq E_G(\tilde A)$.
Recall that $|\Upsilon\cap B|\le (2/3)\cdot|\Upsilon|$, so $|\Upsilon\cap C|+|\Upsilon\cap A|\ge |\Upsilon|/3=|E|/3$.
Therefore,
$$|E_G(\tilde A)|\ge |E^A\setminus \tilde E|\ge |E^A|-|\tilde E|\ge (|E|/3-|\Upsilon\cap C|)-|\tilde E|\ge |E|/3-2\sqrt{2^{7}\cdot\mathsf{OPT}_{\mathsf{cr}}(G)+2^8\Delta|E|}
\ge |E|/4,
$$
where the last inequality follows from $\Delta\le |E|^2/2^{100}$ and $\mathsf{OPT}_{\mathsf{cr}}(G)\le |E|/2^{100}$. This completes the proof of
Theorem~\ref{lem:min_bal_cut}.
\subsection{Proof of Theorem~\ref{thm: bandwidth_means_boundary_well_linked}}
\label{apd: Proof of bandwidth_means_boundary_well_linked}
We define an instance of the max-flow problem as follows.
We contract all vertices of $T_1$ into a single vertex $u_1$, and contract all vertices of $T_2$ into a single vertex $u_2$.
Let $H$ be the resulting graph.
The capacities on all edges of $E(H)$ are defined to be $\ceil{1/\alpha}$.
Since the capacity for each edge is integral, to complete the proof of Theorem~\ref{thm: bandwidth_means_boundary_well_linked}, it suffices to show that the maximum $u_1$-$u_2$ flow in $H$ has value $|T_1|$.
From the max-flow min-cut theorem, it suffices to show that the minimum $u_1$-$u_2$ cut in $H$ has value at least $|T_1|$.
Let $(A,B)$ be any in $H$ with $u_1\in A$ and $u_2\in B$. It is clear that $((A\setminus\set{u_1})\cup T_1, (B\setminus\set{u_2})\cup T_2)$ is a cut in $G$ separating vertices of $T_1$ from vertices of $T_2$, and all edges in $E((A\setminus\set{u_1})\cup T_1, (B\setminus\set{u_2})\cup T_2)$ also belong to $E_H(A,B)$ as an edge in $H$.
Since vertices of $T$ are $\alpha$-well-linked in $G$,
$|E((A\setminus\set{u_1})\cup T_1, (B\setminus\set{u_2})\cup T_2)|\ge\alpha\cdot|T_1|$, and therefore the capacity of the cut $(A,B)$ in $H$ has capacity at least $\alpha\cdot|T_1|\cdot \ceil{1/\alpha} \ge |T_1|$ (recall that $|T_1|=|T_2|$).
This completes the proof of Theorem~\ref{thm: bandwidth_means_boundary_well_linked}.
\iffalse
We subdivide every edge $e\in \delta(S)$ by a vertex $x_e$, and we denote $\Upsilon=\set{x_e\mid e\in \delta(S)}$.
We then let $H$ be the subgraph of the resulting graph induced by vertices of $V(S)\cup \Upsilon$. Since $S$ has the $\alpha$-bandwidth property, vertices of $\Upsilon$ are $\alpha$-well-linked in $H$.
We contract all vertices of $\set{x_e\mid e\in E_1}$ into a single vertex $u_1$, and contract all vertices of $\set{x_e\mid e\in E_2}$ into a single vertex $u_2$.
Let $H'$ be the resulting graph.
The capacities on all edges of $\delta_{H'}(u_1)$ and all edges of $\delta_{H'}(u_2)$ are defined to be $1$.
The capacities on all edges of $E(S)$ are defined to be $\ceil{1/\alpha}$.
Since the capacity for each edge is integral, to complete the proof of Theorem~\ref{thm: bandwidth_means_boundary_well_linked}, it suffices to show that the maximum $u_1$-$u_2$ flow in $H'$ has value $|E_1|$.
From the max-flow min-cut theorem, it suffices to show that the minimum $u_1$-$u_2$ cut in $H'$ has value at least $|E_1|$.
Consider any cut $(A,B)$ in $H'$, with $u_1\in A$ and $u_2\in B$. Assume that $a$ edges of $\delta(u_1)$ and $b$ edges of $\delta(u_2)$ belong to the cut $E(A,B)$. Since vertices of $\Upsilon$ are $\alpha$-well-linked in $H$, $|E(A,B)|\ge a+b+\alpha\cdot \ceil{1/\alpha}\cdot\min\set{|E_1|-a,|E_2|-b}\ge |E_1|$ (recall that $|E_1|=|E_2|$). This completes the proof of Theorem~\ref{thm: bandwidth_means_boundary_well_linked}.
\fi
\subsection{Proof of Theorem~\ref{thm:well_linked_decomposition}}
\label{apd: Proof of well_linked_decomposition}
We use the following lemma.
\begin{lemma}
\label{lem: decrease in total budget}
For any integers $a,b,c>0$ such that $c=O\big(\frac{\min\set{a,b}}{\log^{1.5}n}\big)$,
$$(a+b)\cdot\log_{3/2} (2(a+b))> (a+c)\cdot\log_{3/2} (2(a+c))+(b+c)\cdot\log_{3/2} (2(b+c)).$$
\end{lemma}
\begin{proof} Assume without loss of generality that $a\ge b$, we have
\[\begin{split}
& \text{ }(a+b)\cdot\log_{3/2} (2(a+b))- (a+c)\cdot\log_{3/2} (2(a+c))-(b+c)\cdot\log_{3/2} (2(b+c))\\
\ge & \text{ } (a+b)\cdot\log_{3/2} (2(a+b))- (a+c)\cdot\log_{3/2} (2(a+b))-(b+c)\cdot\log_{3/2} (2(b+c))\\
= & \text{ } (b-c)\cdot\log_{3/2} (2(a+b))-(b+c)\cdot\log_{3/2} (2(b+c))\\
= & \text{ } (b-c)\cdot\bigg(\log_{3/2} (2(a+b))-(1+\frac{2c}{b-c})\cdot\log_{3/2} (2(b+c))\bigg)\\
= & \text{ } (b-c)\cdot\bigg(\log_{3/2} (2(b+c))+1-(1+\frac{2c}{b-c})\cdot\log_{3/2} (2(b+c))\bigg)\\
= & \text{ } (b-c)\cdot\bigg(\log_{3/2} (2(b+c))+1-(1+\frac{2}{\log^{1.5} n})\cdot\log_{3/2} (2(b+c))\bigg)\\
= & \text{ } (b-c)\cdot\bigg(1-\frac{\log_{3/2} (2(b+c))}{\log^{1.5} n}\bigg)> 0
\end{split}\]
\end{proof}
We decompose the cluster $S$ into small clusters and compute routing paths from the boundary edges of these small clusters to the boundary of $S$ as follows. Throughout, we maintain (i) a set $\hat{\mathcal{R}}$ of sub-clusters of $S$, that is initialized to be $\hat{\mathcal{R}}=\set{S}$; and (ii) for each edge $e\in E(S)\cup \delta_G(S)$, a real number $b(e)$ called the \emph{budget} of edge $e$, that is defined as follows. If $e$ is incident to a vertex $v\in V(S)$ such that $|\delta(v)\cap \delta(S)|\ge 0.9\cdot|\delta(S)|$, then the budget of $e$ is defined to be $1$ and will stay unchanged throughout the algorithm. If $e$ is not incident to such a vertex, then
\begin{itemize}
\item if $e\in E^{\textnormal{\textsf{out}}}(\hat{{\mathcal{R}}})$, then $b(e)=\alpha\cdot\bigg(\log_{3/2}(2|\delta_G(R)|)+\log_{3/2}(2|\delta_G(R')|)\bigg)$, where $R,R'$ are the two clusters of $\hat {\mathcal{R}}$ that contain an endpoint of $e$;
\item if $e\in \delta_G(S)$, then $b(e)=\alpha\cdot\log_{3/2}(2|\delta_G(R)|)$, where $R$ is the unique cluster of $\hat {\mathcal{R}}$ that contains an endpoint of $e$; and
\item if $e\notin (E^{\textnormal{\textsf{out}}}(\hat{{\mathcal{R}}})\cup \delta_G(S))$, then $b(e)=0$.
\end{itemize}
Denote $z=|\delta_G(S)|$, so if there is no vertex $v\in V(S)$ such that $|\delta(v)\cap \delta(S)|\ge 0.9\cdot|\delta(S)|$, then initially the total budget on all edges is $\alpha\cdot z\cdot\log_{3/2}(2z)$, otherwise the total initial budget is $|\delta(v)|+ \alpha(|\delta(S)|-|\delta(v)|)\cdot \log_{3/2}(2(|\delta(S)|-|\delta(v)|))$.
The algorithm proceeds in iterations, and will be terminated if the following two conditions are satisfied: (i) all clusters of $\hat{\mathcal{R}}$ have the $\alpha$-bandwidth property; and (ii) for each cluster $R\in \hat {\mathcal{R}}$, we have computed a set ${\mathcal{P}}(R)$ of paths routing edges of $\delta_G(R)$ to edges of $\delta_G(S)$ as required in \Cref{thm:well_linked_decomposition}.
We now describe the execution of an iteration. Assume first that $\hat {\mathcal{R}}$ contains a cluster that does not have the $\alpha$-bandwidth property. Let $R$ be such a cluster.
Consider the graph $R^+$ (see \Cref{def: Graph C^+}). We apply the algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace to graph $R^+$ and terminal set $T(R)$, to obtain an $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)$-approximate sparsest cut $(X,Y)$ in graph $R^+$ with respect to the set $T(R)$ of terminals.
Since $R$ does not have the $\alpha$-bandwidth property, $|E(X,Y)|\le \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T(R)|,|Y\cap T(R)}$.
Denote $X'=X\setminus T(R)$ and $Y'=Y\setminus T(R)$, so $(X',Y')$ is a partition of $R$. We then replace the cluster $R$ in $\hat {\mathcal{R}}$ with clusters $X',Y'$ and continue to the next iteration.
We now show that in this case, the total budget of all edges decrease after this iteration. Denote $z_{X'}=|\delta_G(X')|$, $z_{Y'}=|\delta_G(Y')|$ and $z'=|E(X,Y)|$. Assume first that no vertex $v$ of $R$ satisfies that $|\delta(v)\cap \delta(S)|\ge 0.9\cdot|\delta(S)|$.
In this case, the decrease in budget of all edges of $\delta_G(R)\cap \delta_G(X')$ is $\alpha z_{X'}\cdot(\log_{3/2}(2(z_{X'}+z_{Y'}))-\log_{3/2}(2(z_{X'}+z')))$.
Similarly, the decrease in budget of all edges of $\delta_G(R)\cap \delta_G(Y')$ is $\alpha z_{Y'}\cdot(\log_{3/2}(2(z_{X'}+z_{Y'}))-\log_{3/2}(2(z_{Y'}+z')))$.
On the other hand, the increase in budget of all edges of $E(X,Y)$ is $\alpha\cdot z'\cdot (\log_{3/2}(2(z_{X'}+z'))+\log_{3/2}(2(z_{Y'}+z')))$.
Therefore, the decrease in budget of all edges is
\[
\alpha\cdot\bigg((z_{X'}+z_{Y'})\cdot\log_{3/2}(2(z_{X'}+z_{Y'}))-(z_{X'}+z')\log_{3/2}(2(z_{X'}+z'))-(z_{Y'}+z')\log_{3/2}(2(z_{Y'}+z'))\bigg).
\]
Since $\alpha=O(\frac{1}{\log^2 m})$, $\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}=O(\frac{1}{\log^{1.5} m})$ and so $z'=O(\frac{\min\set{z_{X'},z_{Y'}}}{\log^{1.5} m})$.
From \Cref{lem: decrease in total budget}, the above value is positive, so the total budget decrease after this iteration.
Assume now that there exists a vertex $v\in V(R)$ satisfies that $|\delta(v)\cap \delta(S)|\ge 0.9\cdot|\delta(S)|$. Clearly, there can be at most one such vertex. Assume without loss of generality that $v\in X'$, then since $z'=O(\frac{\min\set{z_{X'},z_{Y'}}}{\log^{1.5} m})$, the total budget of all new edges of $E(X,Y)$ is at most $2z'\cdot \log (2m)$, which is smaller than the decrease of total budget of all edges in $\delta_G(Y')$. Altogether, total budget decreases after this iteration.
Assume now that $\hat {\mathcal{R}}$ contains a cluster $R$ that does not yet have the set ${\mathcal{P}}(R)$ of paths routing edges of $\delta_G(R)$ to edges of $\delta_G(S)$ as required in \Cref{thm:well_linked_decomposition}.
Let graph $G'$ be obtained from $G$ by first contracting all vertices of $R$ into a single vertex $v_R$ and then contracting vertices of $V\setminus S$ into a single vertex $v^*$. Observe that $\delta_{G'}(v_R)=\delta_G(R)$ and $\delta_{G'}(v^*)=\delta_G(S)$.
We now compute the min-cut in $G'$ separating $v_R$ from $v^*$. If the min-cut has capacity at least $|\delta_{G'}(v_R)|/100$, then from the max-flow min-cut theorem, we can efficiently find a set of $|\delta_{G'}(v_R)|$ paths connecting $v_R$ to $v^*$ in $G'$, causing congestion at most $100$ in $G'$, and this set of paths can be immediately converted to a set ${\mathcal{P}}(R)$ of paths routing edges of $\delta_G(R)$ to edges of $\delta_G(S)$ as required in \Cref{thm:well_linked_decomposition}, causing congestion at most $100$.
Assume now that min-cut has capacity less than $|\delta_{G'}(v_R)|/100$. Let $(X,Y)$ be the min-cut where $v_R\in X$. Let $X'$ be the cluster obtained from $X$ by uncontracting the vertex $v_R$ back to the cluster $R$. We replace the cluster $R$ in $\hat{{\mathcal{R}}}$ by cluster $X'$, and the for every other cluster $R'\in \hat {\mathcal{R}}$ such that $R'\cap X'\ne \emptyset$ but $R'\not\subseteq X'$, we replace $R'$ by cluster $R'\setminus X'$. This completes the description of this iteration.
We now show that in this case, the total budget of all edges decrease after this iteration.
Consider the set $\hat {\mathcal{R}}$ of clusters before this iteration. Let $\hat{\mathcal{R}}_1\subseteq \hat {\mathcal{R}}$ be the set of clusters $R'$ such that $R'\subseteq X'$.
Let $\hat{\mathcal{R}}_2\subseteq \hat {\mathcal{R}}$ be the set of clusters $R'$ such that $R'\cap X'\ne \emptyset$ but $R'\not\subseteq X'$.
First consider the cluster $R$. Denote $z_R=|\delta_G(R)|$, then at least $0.99z_R$ edges of $\delta_G(R)$ loses their budget, so the budget decrease on these edges is at least $\alpha\cdot 0.99z_R\log_{3/2}(2z_R)$.
Second consider the cluster $X'$. Since $|\delta_G(X')|\le |\delta_G(R)|/100$, so the budget increase on edges of $\delta_G(X')$ is at most $\alpha\cdot0.01z_R\log_{3/2}(0.02z_R)$.
Third, it is easy to verify that the budget increase of a cluster $R'\in \hat{\mathcal{R}}_1$ is already counted in the budget increase on edges of $\delta_G(X')$.
Last, consider a cluster $R'\in \hat{\mathcal{R}}_2$. We denote $R'_X=R'\cap X'$ and $R'_Y=R'\setminus R'_X$.
We denote $Z'_1=|E(R'_X,X'\setminus R'_X)|$, $Z'_2=|E(R'_Y,X'\setminus R'_X)|$,
$Z'_3=|E(R'_Y,S\setminus (X'\cup R'_X))|$,
$Z'_4=|E(R'_X,S\setminus (X'\cup R'_X))|$,
and $Z^*=|E(R'_X,R'_Y)|$.
We denote by $z'_1,z'_2,z'_3,z'_4,z^*$ the cardinalities of these sets, and denote $z'=z'_1+z'_2+z'_3+z'_4$.
From the minimality of the cut $(X,Y)$, we get that $z'_1\ge z^*+z_4$ and $z'_3\ge z^*+z_2$. Observe that the budget of edges in $Z'_2\cup Z'_4$ are already counted in the budget increase on edges of $\delta_G(X')$. Also observe that the total budget of edges in $Z'_3$ decreases (since $z'_3+z'_2+z^*\le z'$). Moreover, the increase in budget of edges in $Z^*$ that are not counted in the budget increase on edges of $\delta_G(X')$ is $\alpha\cdot z^*\log (2(z'_3+z'_2+z^*))$, while the the decrease in budget of edges in $Z'_1$ that are not counted in the budget decrease on edges of $\delta_G(R)$ is $\alpha\cdot z'_1\log (2z')$, so the total budget change of all edges incident to $R'$ is negative. Altogether, the total budget decreases.
Therefore, we can now conclude that, when the algorithm terminates, the number of edges connecting different clusters of $\hat{{\mathcal{R}}}$ is at most $\alpha \cdot z\cdot\log_{3/2}(2z)$, and therefore $\sum_{R\in {\mathcal{R}}}|\delta(R)|\le |\delta(S)|\cdot\textsf{left}(1+O(\alpha\cdot \log m)\textsf{right})$.
It is easy to verify that all other properties of \Cref{thm:well_linked_decomposition} are also satisfied.
\iffalse{the proof below does not find the routing paths}
Consider the following algorithm of decomposing the cluster $S$ with the parameter $\alpha$. Throughout, we maintain a set $\hat{\mathcal{R}}$ of sub-clusters of $S$, that is initialized to be $\hat{\mathcal{R}}=\set{S}$. The algorithm proceeds in iterations, and is terminated whenever all clusters of $\hat{\mathcal{R}}$ have the $\hat\alpha$-bandwidth property. In each iteration, we find a cluster $R\in \hat{\mathcal{R}}$ that does not have the $\alpha$-bandwidth property, compute a cut $(A,B)$ of $R$ with $|E(A,B)|\le \alpha\cdot \min\set{|\delta(A)\cap\delta(R)|,|\delta(B)\cap\delta(R)|}$, replace the cluster $R$ in $\hat{\mathcal{R}}$ with the subgraph of $R$ induced by $A$ and the subgraph of $R$ induced by $B$, and then proceed to the next iteration.
We use the following theorem, which is a corollary of Theorem 2.8 in~\cite{chuzhoy2012routing}.
\begin{theorem}[Corollary of Theorem 2.8 in \cite{chuzhoy2012routing}]
Assume $\alpha<1/(48\log n)$ and let $\hat{\mathcal{R}}$ be the partition of $S$ produced by the above algorithm, then $\sum_{R\in {\mathcal{R}}}|\delta(R)|\le |\delta(S)|\cdot(1+32\alpha\log n)$.
\end{theorem}
However, the above algorithm is not efficient, since it is NP-hard to determine if a cluster $R$ has the $\alpha$-bandwidth property.
We modify the above algorithm as follows. When checking if a cluster $R$ has the $\alpha$-bandwidth property, we set up an instance of the sparsest cut problem on a graph $H_R$, that is obtained as follows. We subdivide each edge $e\in \delta(R)$ by a vertex $x_e$, and let $H$ be the subgraph of the resulting graph induced by $V(R)\cup \set{x_e\mid e\in \delta(R)}$. We then compute the sparsest cut in $H_R$, with the set $\set{x_e\mid e\in \delta(R)}$ of terminals, using the algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace.
If the sparsity of the cut that we obtain is greater than $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)\cdot\alpha$, then from the algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace, the cluster $R$ has the bandwidth property. Otherwise, we proceed as if we obtain an $\alpha$-sparse cut. Namely, if $(A,B)$ is the sparsest cut we get from \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace, then we replace the cluster $R$ in $\hat{\mathcal{R}}$ with the subgraph of $R$ induced by $A$ and the subgraph of $R$ induced by $B$, and then proceed to the next iteration.
Since the algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace is efficient, our modified algorithm is efficient.
Let ${\mathcal{R}}$ be the resulting set of clusters of $S$.
We now show that the set ${\mathcal{R}}$ satisfies the properties of Theorem~\ref{thm:well_linked_decomposition}.
It is immediate from the algorithm that the vertex sets $\set{V(R)}_{R\in {\mathcal{R}}}$ partitions $V(S)$, and each cluster $R\in {\mathcal{R}}$ has the $\alpha$-bandwidth property. Since $\alpha<1/(48\log^2n)$, $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)\cdot\alpha<1$, so every time in the algorithm when a cluster $R$ is decomposed into two smaller clusters $R'$ and $R''$, $|\delta(R)|>|\delta(R')|$ and $|\delta(R)|>|\delta(R'')|$. Therefore, for each $R\in {\mathcal{R}}$, $|\delta(R)|\le |\delta(S)|$. The last property follows from the proof of Theorem 2.8 in \cite{chuzhoy2012routing}, and the fact that we only decompose a cluster $R\in \hat {\mathcal{R}}$ when the sparsity of the cut that we obtained is below $\alpha\cdot\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)$.
\fi
\subsection{Proof of \Cref{lem: routing path extension}}
\label{apd: Proof of routing path extension}
Denote $z=\ceil{\frac{|T|}{|T'|}}$. We arbitrarily partition the vertices of $T\setminus T'$ into $(z-1)$ subsets $T_1,\ldots,T_{z-1}$ of cardinality at most $|T'|$ each.
Consider some index $1\leq i\leq z-1$. Since vertices of $T$ are $\alpha$-well-linked in $G$, using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{J}}'_i$ of paths in graph $G$, routing vertices of $T_i$ to vertices of $T'$, such that $\cong_G({\mathcal{J}}'_i)\le \ceil{1/\alpha}$, and each vertex of $T'\cup T_i$ is the endpoint of at most one path in ${\mathcal{J}}'_i$.
By concatenating the paths in ${\mathcal{J}}'_i$ with paths in ${\mathcal{P}}'$, we obtain a collection ${\mathcal{J}}_i$ of paths in graph $G$ routing vertices of $T_i$ to $x$, that cause edge-congestion $\ceil{1/\alpha}$.
Define ${\mathcal{P}}={\mathcal{P}}'\cup(\bigcup_{i=1}^{z-1}{\mathcal{J}}_i)$.
It is clear that the paths in ${\mathcal{P}}$ route the vertices of $T$ to $x$.
Moreover, for every edge $e\in E(G)$,
\[
\begin{split}
\cong_G({\mathcal{P}},e) & \le
\cong_G({\mathcal{P}}',e)+\sum_{1\le i\le z-1}\cong_G({\mathcal{J}}_i,e)\\
& = z\cdot \cong_G({\mathcal{P}}',e)+\sum_{1\le i\le z-1}\cong_G({\mathcal{J}}'_i,e)\\
& \le \ceil{\frac{|T|}{|T'|}}\bigg(\cong_G({\mathcal{P}}',e)+\ceil{1/\alpha}\bigg).
\end{split}
\]
\input{appx-layered-wld.tex}
\subsection{Proof of Claim~\ref{claim: embed expander}}
\label{apd: Proof of embed expander}
We use the cut-matching game of Khandekar, Rao and Vazirani~\cite{khandekar2009graph}, defined as follows. The game is played between two players, called the cut player and the matching player. The input to the game is an even integer $N$.
The game is played in iterations. We start with a graph $W$ with vertex set $V$ of cardinality $N$ and an empty edge set. In every iteration, some edges are added to $W$. The game ends when $W$ becomes a $\ensuremath{\frac{1}{2}}$-expander.
The goal of the cut player is to construct a $\ensuremath{\frac{1}{2}}$-expander in as few iterations as
possible, whereas the goal of the matching player is to prevent the construction of the expander for as long as possible. The iterations proceed as follows. In every iteration $j$, the cut player chooses a partition $(Z_j, Z'_j)$ of $V$ with $|Z_j| = |Z'_j|$, and the
matching player chooses a perfect matching $M_j$ that matches the nodes of $Z_j$ to the nodes of $Z'_j$. The edges of $M_j$ are then
added to $W$. Khandekar, Rao, and Vazirani~\cite{khandekar2009graph} showed that there is a randomized efficient algorithm for the cut player (that is, an algorithm that, in every iteration $j$, given the current graph $W$, computes a partition $(Z_j,Z'_j)$ of $V$ with $|Z_j| = |Z'_j|$), that guarantees that after
$O(\log^2{N})$ iterations, with high probability, the graph $W$ is a $(1/2)$-expander, regardless of the specific matchings chosen by the matching player.
We use the above cut-matching game in order to compute an expander $W$ with vertex set $T$, and to embed it into $G$, using standard techniques.
If $|T|$ is an even integer, then we start with the graph $W$ containing the vertices of $T$; otherwise, we let $t\in T$ be an arbitrary vertex, and we start with $V(W)=T\setminus\set{t}$. Initially, $E(W)=\emptyset$. We then perform iterations. In the $i$th iteration, we apply the algorithm of the cut player to the current graph $W$, and obtain a partition $(Z_i,Z'_i)$ of its vertices with $|Z_j| = |Z'_j|$. Using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we compute a collection ${\mathcal{P}}_i$ of paths in graph $G$, routing vertices of $Z_i$ to vertices of $Z'_i$, so that every vertex of $Z_i\cup Z'_i$ is an endpoint of exactly one path in ${\mathcal{P}}_i$, and the paths in ${\mathcal{P}}_i$ cause congestion at most $\ceil{1/\alpha}\leq 2/\alpha$ in $G$. Let $M_i$ be a perfect matching between vertices of $Z_i$ and vertices of $Z'_i$ defined by the set ${\mathcal{P}}_i$ of paths: that is, we add to $M_i$ a pair $(t,t')$ of vertices iff some path in ${\mathcal{P}}_i$ has endpoints $t$ and $t'$. We then treat $M_i$ as the response of the matching player, and add the edges of $M_i$ to $W$, completing the current iteration of the game.
Let $W$ be the graph obtained after $i^*=O(\log^2k)$ iterations, that is guaranteed to be a $1/2$-expander with high probability. We then set ${\mathcal{P}}=\bigcup_{i=1}^{i^*}{\mathcal{P}}_i$. It is immediate to verify that ${\mathcal{P}}$ is an embedding of $W$ into $G$. Since each path set ${\mathcal{P}}_i$ causes congestion $O(1/\alpha)$, and $i^*\leq O(\log^2k)$, the paths in ${\mathcal{P}}$ cause congestion $O((\log^2 k)/\alpha)$. If $|T|$ is even, then we have constructed the desired expander and its embedding into $G$ as required. If $|T|$ is odd, then we add the terminal $t$ to the graph $W$. Let $P$ be any path in graph $G$, connecting $t$ to any terminal $t'\in T\setminus\set{t}$; such a path must exist since the set of terminals is $\alpha$-well-linked in $G$. We then add edge $(t,t')$ to graph $W$, and we let its embedding be $P(e)=P$; we add path $P$ to ${\mathcal{P}}$. It is easy to verify that this final graph $W$ is $1/4$ expander, provided that the original graph $W$ obtained at the end of the cut-matching game was a $1/2$-expander. We have also obtained an embedding of $W$ into $G$ with congestion $O((\log^2k)/\alpha)$.
\subsection{Proof of Observation~\ref{obs: cr of exp}}
\label{apd: Proof of cr of exp}
Let $W'$ be the graph obtained from $W$ by discarding parallel edges. It is immediate to verify that graph $W'$ is a $\phi$-expander, for $\phi=\Omega(1/\log^2 k)$.
From \Cref{lem:min_bal_cut}, the minimum $1/4$-edge-balanced cut in graph $W'$ has size at most:
\[O\textsf{left}(\sqrt{\mathsf{OPT}_{\mathsf{cr}}(W')+|E(W')|\cdot \log^2k} \textsf{right})\leq O\textsf{left}(\sqrt{\mathsf{OPT}_{\mathsf{cr}}(W)+k\cdot \log^4k} \textsf{right}).\]
On the other hand, since $W'$ is a $\phi$-expander, and its vertices have degrees $O(\log^2k)$, the size of the minimum $1/4$-edge-balanced cut is $\Omega(k/\log^4k)$. We conclude that $\mathsf{OPT}_{\mathsf{cr}}(W)\geq \Omega(k^2/\log^8k)$.
\subsection{Proof of Corollary~\ref{cor: routing well linked vertex set}}
\label{apd: Proof of routing well linked vertex set}
The proof relies on known results for routing on expanders, that are summarized in the next claim, that is well-known, and follows immediately from the results of \cite{leighton1999multicommodity}. A proof can be found, e.g. in \cite{chuzhoy2012routing}.
\begin{claim}[Corollary C.2 in \cite{chuzhoy2012routing}]\label{claim: routing on expander}
There is an efficient randomized algorithm that, given as input an $n$-vertex $\phi$-expander $H$, and any partial matching $M$ over the vertices of $H$, computes, for every pair $(u,v)\in M$, a path $P(u,v)$ connecting $u$ to $v$ in $H$, such that with high probability, the set $\set{P(u,v)\mid (u,v)\in M}$ of paths causes congestion $O(\log^2 n/\phi)$ in $H$.
\end{claim}
We start by computing a graph $W$ with $V(W)=T$, and its embedding ${\mathcal{P}}=\set{P(e)\mid e\in E(W)}$ into $G$ with congestion $O((\log^2k)/\alpha)$ using the algorithm from \Cref{claim: embed expander} (recall that, with high probability, $W$ is an $(1/4)$-expander). Next,
we use the algorithm from \Cref{claim: routing on expander} to compute a collection ${\mathcal{R}}'=\set{P(u,v)\mid (u,v)\in M}$ of paths in graph $W$, where for all $(u,v)\in M$, a path $P(u,v)$ connects $u$ to $v$ in $W$, such that the congestion of the set ${\mathcal{R}}'$ of paths in $W$ is $O(\log^2k)$ with high probability.
Lastly, we consider the paths $P(u,v)\in {\mathcal{R}}'$ one-by-one. We transform each such path $P(u,v)$ into a path $R(u,v)$ connecting $u$ to $v$ in graph $G$ by replacing, for every edge $e\in P(u,v)$, the edge $e$ with the path $P(e)\in {\mathcal{P}}$ embedding the edge $e$ into $G$. Once every edge of $P(u,v)$ is processed, let $R(u,v)$ denote the final path. Since the paths in ${\mathcal{R}}'$ cause congestion at most $O(\log^2k)$ in $W$ (with high probability), while the paths in ${\mathcal{P}}$ cause congestion $O(\log^2k/\alpha)$ in $G$, we get that with high probability, the paths in the resulting set ${\mathcal{R}}=\set{R(u,v)\mid (u,v)\in M}$ cause congestion $O(\log^4k/\alpha)$ in $G$.
\subsection{Proof of Corollary~\ref{cor: embed complete graph}}
\label{apd: Proof of embed complete graph}
We partition the set $E(K)$ of edges into $3z$ matchings $M_1,\ldots,M_{3z}$, and then use \Cref{cor: routing well linked vertex set} to compute, for each $1\leq i\leq 3z$, a set $\tilde {\mathcal{R}}_i=\set{\tilde P(e)\mid e\in M_i}$ of paths in graph $G$, where for all $e=(t,t')\in M_i$, path $\tilde P(e)$ connects $t$ to $t'$, and with high probability, the paths in $\tilde {\mathcal{P}}_i$ cause edge-congestion $O(\log^4z/\alpha)$ in graph $G$. Let $\tilde {\mathcal{P}}=\bigcup_{i=1}^{3z}\tilde {\mathcal{P}}_i$. Then $\tilde {\mathcal{P}}$ is an embedding of $K_z$ into $G$, and with high probability, the congestion of this embedding is $O(z\log^4z)/\alpha$.
\subsection{Proof of Claim~\ref{clm: contracted_graph_well_linkedness}}
\label{apd: Proof of contracted_graph_well_linkedness}
Let $(A,B)$ be an arbitrary cut in $G$, and we denote $T_A=T\cap A$ and $T_B=T\cap B$. Assume without loss of generality that $|T_A|\le |T_B|$. It suffices to show that $|E_G(A,B)|\ge (\alpha_1\alpha_2)\cdot |T_A|$.
Denote $H=G_{|{\mathcal{C}}}$. From the definition of $H$, its vertices can be partitioned into two sets: set $V'$ contains all vertices that does not belong to any cluster of ${\mathcal{C}}$, and set $V''$ that contains all super-nodes that correspond to clusters in ${\mathcal{C}}$. So $V'=V(G)\setminus (\bigcup_{C\in {\mathcal{C}}}V(C))$ and $V''=\set{v_C\mid C\in {\mathcal{C}}}$.
Moreover, since no vertices of $T$ lies in any cluster of ${\mathcal{C}}$, $T\subseteq V'$.
We construct a cut $(A',B')$ in $H$ based on the cut $(A,B)$ in $G$ as follows.
For each vertex $v\in V'$, we add $v$ to $A'$ iff $v$ belongs to $A$ as a vertex in $G$, and we add $v$ to $B'$ iff $v$ belongs to $B$ as a vertex in $G$.
It remains to specify whether the super-node $v_C$ belongs to $A'$ or $B'$ for each cluster $C\in {\mathcal{C}}$.
Consider a cluster $C\in {\mathcal{C}}$, the set $\delta_G(C)$ of edges can be partitioned into four subsets: $\hat E_A, \hat E_B, \hat E_{AB}, \hat E_{BA}$ as follows. Let $e=(u,u')$ be an edge of $\delta_G(C)$, where $u\in C$ and $u'\notin C$.
If $u,u'\in A$, then $e\in \hat E_A$.
If $u,u'\in B$, then $e\in \hat E_B$.
If $u\in A,u'\in B$, then $e\in \hat E_{AB}$.
If $u\in B,u'\in A$, then $e\in \hat E_{BA}$.
If $|\hat E_A|+|\hat E_{AB}|\le |\hat E_B|+|\hat E_{BA}|$, then the supernode $v_C$ is added to $B'$; otherwise the supernode $v_C$ is added to $A'$. This completes the definition of the cut $(A',B')$. Clearly, $T\cap A'=T_A$ and $T\cap B'=T_B$. Since $T$ is $\alpha_2$-well-linked in $H$, $|E_{H}(A',B')|\ge \alpha_2 \cdot|T_A|$.
We claim that $|E_G(A,B)|\ge \alpha_1\cdot |E_{H}(A',B')|$. Note that, if this is true, then it follows that $|E_G(A,B)|\ge (\alpha_1\alpha_2)\cdot |T_A|$ and the proof of Claim~\ref{clm: contracted_graph_well_linkedness} is completed.
It remains to show that $|E_G(A,B)|\ge \alpha_1\cdot |E_{H}(A',B')|$. On one hand, from the definition of $H$, $V'\cap A=V'\cap A'$ and $V'\cap B=V'\cap B'$, so an edge in $E_{H}(V'\cap A', V'\cap B')$ is also an edge in $E_{G}(A, B)$. On the other hand, let $C$ be any cluster of ${\mathcal{C}}$. Assume first that $v_C\in B'$, then $|\hat E_A|+|\hat E_{AB}|\le |\hat E_B|+|\hat E_{BA}|$. Since cluster $C$ has $\alpha_1$-bandwidth property, $|E_G(A\cap C,B\cap C)|\ge \alpha_1\cdot (|\hat E_A|+|\hat E_{AB}|)=\alpha_1\cdot |\delta_{H}(v_C)\cap E_H(A',B')|$. Similarly, if $v_C\in A'$, then
$|E_G(A\cap C,B\cap C)|\ge \alpha_1\cdot |\delta_{H}(v_C)\cap E_H(A',B')|$.
Altogether,
\[
\begin{split}
|E_{G}(A,B)| &\ge |E_{G}(A\cap V',B\cap V')|+\sum_{C\in {\mathcal{C}}}|E_{G}(A\cap C,B\cap C)|\\
&\ge |E_{H}(A'\cap V',B'\cap V')|+\sum_{C\in {\mathcal{C}}}\alpha_1\cdot |\delta_{H}(v_C)\cap E_H(A',B')|\\
&\ge \alpha_1\cdot|E_{H}(A',B')|.
\end{split}
\]
\subsection{Proof of \Cref{cor: contracted_graph_well_linkedness}}
\label{apd: Proof of cor contracted_graph_well_linkedness}
Let $G^+$ be the graph obtained from $G$ by subividing each edge $e\in \delta_G(R)$ with a new vertex $t_e$.
Denote $T=\set{t_e\mid e\in \delta_G(R)}$.
Recall that the augmentation $R^+$ of cluster $R$ in $G$ is defined to be the subgraph of $G^+$ induced by all vertices of $V(R)\cup T$.
Denote $\hat G^+ = G^+_{\mid {\mathcal{C}}}$ and $\hat R^+ = R^+_{\mid {\mathcal{C}}}$. Recall that $\hat R=R_{\mid {\mathcal{C}}}$. It is easy to verify that $\delta_{\hat G}(\hat R)=T$, and $\hat R^+$ is the augmentation of cluster $\hat R$ in graph $\hat G$. Therefore, since $\hat R$ has the $\alpha_2$-bandwidth property in $\hat G$, $T$ is $\alpha_2$-well-linked in $\hat G^+$.
Note that ${\mathcal{C}}$ is a collection of disjoint clusters of $G$, and $T\cap (\bigcup_{C\in {\mathcal{C}}}V(C))=\emptyset$.
Moreover, note that each cluster $C\in {\mathcal{C}}$ has the $\alpha_1$-bandwidth property, and the set $T$ of vertices is $\alpha_2$-well-linked in $\hat G^+$.
From \Cref{clm: contracted_graph_well_linkedness}, we get that $T$ is $(\alpha_1\alpha_2)$-well-linked in graph $G^+$. Equivalently, the cluster $R$ has $(\alpha_1\alpha_2)$-bandwidth property in $G$.
\subsection{Proof of \Cref{claim: routing in contracted graph}}
\label{apx: contracted graph routing}
For convenience, we denote $G_{|{\mathcal{C}}}$ by $\hat G$, and we denote $|T|=k$.
We assume w.l.o.g. that the paths in ${\mathcal{P}}$ are simple, and we direct each such path towards $x$. We then graduately modify the paths in ${\mathcal{P}}$, by processing the clusters of ${\mathcal{C}}$ one-by-one.
Consider now any such cluster $C\in {\mathcal{C}}$, and let ${\mathcal{P}}(C)\subseteq {\mathcal{P}}$ be the subset of paths that contain the vertex $v(C)$. For each such path $P\in {\mathcal{P}}(C)$, let $e_P(C)$ and $e_P'(C)$ denote the edges appearing immediately before and immediately after $v(C)$ on $P$. We then denote $E_1(C)=\set{e_P(C)\mid P\in {\mathcal{P}}(C)}$ and $E_2(C)=\set{e'_P(C)\mid P\in {\mathcal{P}}(C)}$. Since cluster $C$ has the $\alpha$-bandwidth property, from \Cref{cor: bandwidth_means_boundary_well_linked}, there is a collection ${\mathcal{R}}(C)$ of paths in graph $C\cup\delta(C)$ of cardinality $|{\mathcal{P}}(C)|$, where each path connects a distinct edge of $E_1(C)$ to a distinct edge of $E_2(C)$, and every edge in $C$ participates in at most $\ceil{1/\alpha}$ such paths. We modify the paths in set ${\mathcal{P}}(C)$ as follows. First, for each path $P\in {\mathcal{P}}(C)$, we delete the vertex $v(C)$ from $P$, together with its two incident edges. Let $P_1,P_2$ be the two resulting sub-paths of $P$. Next, we let ${\mathcal{P}}_1(C)=\set{P_1\mid P\in {\mathcal{P}}(C)}$, and ${\mathcal{P}}_2(C)=\set{P_2\mid P\in {\mathcal{P}}(C)}$. Lastly, let ${\mathcal{P}}'(C)$ be the set of paths obtained by concatenating the paths in ${\mathcal{P}}_1(C),{\mathcal{R}}(C)$ and ${\mathcal{P}}_2(C)$. We then delete from ${\mathcal{P}}$ the paths that belong to ${\mathcal{P}}(C)$, and add the paths of ${\mathcal{P}}'(C)$ instead. It is easy to verify that the resulting set ${\mathcal{P}}$ of paths still routes the vertices of $T$ to $x$.
Once we complete processing every cluster $C\in {\mathcal{C}}$, we obtain a collection ${\mathcal{P}}'$ of $k$ paths, routing the vertices of $T$ to the vertex $x$. Since the paths of ${\mathcal{P}}$ at the beginning of the algorithms are edge-disjoint paths in $G_{|{\mathcal{C}}}$, it is easy to see that, for each edge $e\in \bigcup_{C\in {\mathcal{C}}}E(C)$, $\cong_G({\mathcal{P}}',e)\le 1$, and for each edge $e\notin \bigcup_{C\in {\mathcal{C}}}E(C)$, $\cong_{G}({\mathcal{P}}',e)\leq \ceil{1/\alpha}\leq 2/\alpha$. From \Cref{claim: remove congestion}, we conclude that there is a collection ${\mathcal{P}}''$ of at least $\alpha k/2$ edge-disjoint paths in graph $G$, where each path in ${\mathcal{P}}''$ connects a distinct vertex of $T$ to $x$. We can compute such a collection ${\mathcal{P}}''$ of paths via standard maximum $s$-$t$ flow computation.
\subsection{Proof of \Cref{lem: crossings in contr graph}}
\label{apd: Proof of crossings in contr graph}
Let $\phi^*$ be an optimal solution to instance $I$. Let $G'$ be the graph that is obtained from $G$ by subdividing every edge $e\in \bigcup_{C\in {\mathcal{C}}}\delta_G(C)$ with a vertex $t_e$, and let $T=\set{t_e\mid e\in \bigcup_{C\in {\mathcal{C}}}\delta_G(C)}$ be the resulting set of new vertices. For every cluster $C\in {\mathcal{C}}$, we denote by $T_C=\set{t_e\mid e\in\delta_G(C)}$, and we let $C'$ be the sub-graph of $G'$ induced by $C\cup T_C$. Clearly, the set $T_C$ of vertices is $\alpha$-well-linked in $C'$. Observe that drawing $\phi^*$ of $G$ naturally defines a drawing $\phi'$ of graph $G'$, with $\mathsf{cr}(\phi^*)=\mathsf{cr}(\phi')$.
Consider now some cluster $C\in {\mathcal{C}}$.
Recall that \Cref{lem: simple guiding paths} provides a randomized algorithm that computes a distribution ${\mathcal{D}}$ over the set of all internal $C$-routers. We then let ${\mathcal{Q}}=\set{Q(e)\mid e\in \delta_G(C)}$ be a (random) set of paths sampled from distribution ${\mathcal{D}}$. From \Cref{lem: simple guiding paths}, for each edge $e\in E(C)$, $\expect{\cong({\mathcal{Q}},e)}\leq O(\log^4m/\alpha)$.
Let $u$ be the common endpoints of paths in ${\mathcal{Q}}$.
We slightly modify $u$ and ${\mathcal{Q}}$, as follows: if $u\not\in T_C$, then we set $u_C=u$ and ${\mathcal{Q}}(C)={\mathcal{Q}}$. Assume now that $u\in T_C$. Recall that, from the definition of graph $C'$, every vertex in $T_C$ has degree exactly $1$ in $C'$. We then let $u_C$ be the unique neighbor of $u$ in $C'$. We modify each path $Q_t\in {\mathcal{Q}}$ as follows: if $t\neq u$, then we truncate the path at vertex $u_C$, so it connects $t$ to $u_C$; otherwise, we let $Q_t$ be the path consisting of a single edge $(u,u_C)$. We denote the final set of paths by ${\mathcal{Q}}(C)$; each path in ${\mathcal{Q}}(C)$ connects a distinct vertex of $T_C$ to $u_C$, and it is easy to verify that
for every edge $e\in E(C')$, $\expect{\cong({\mathcal{Q}}(C),e)}\leq O(\log^4m/\alpha)$.
We denote ${\mathcal{Q}}=\bigcup_{C\in {\mathcal{C}}}{\mathcal{Q}}(C)$, so for every edge $e\in E(G')$, $\expect{\cong({\mathcal{Q}}(C),e)}\leq O(\log^4m/\alpha)$.
For each $C\in {\mathcal{C}}$, we then apply the algorithm from \Cref{cor: new type 2 uncrossing} to graph $G'$, drawing $\phi'$, cluster $C'$ and the set ${\mathcal{Q}}(C)$ of paths, where $u_C$ is viewed as the last endpoint of every path in ${\mathcal{Q}}(C)$. Let $\Gamma_C=\set{\gamma_C(e)\mid e\in \delta_{G'}(C)}$ be the set of curves we get, where the curve $\gamma_C(e)$ corresponds to the path in ${\mathcal{Q}}(C)$ that has $t_e$ as its first endpoint. We now view, for each cluster $C\in{\mathcal{C}}$ and each edge $e\in \delta_{G'}(C)$, the curve $\gamma_C(e)$ as the image of $e$. Then we obtain a drawing $\phi''$ of $G'$, and moreover, from
\Cref{cor: new type 2 uncrossing},
\[
\begin{split}
\expect[]{\mathsf{cr}(\phi'')}\le
&\text{ }\mathsf{cr}(\phi')+\sum_{e,e' \text{ from different clusters of }{\mathcal{C}}}\chi_{\phi'}(e,e')\cdot\expect[]{\cong({\mathcal{Q}},e)}\cdot\expect[]{\cong({\mathcal{Q}},e')}\\
\le &\text{ } O\textsf{left}( \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \frac{\log^8 n}{\alpha^2}\textsf{right}).
\end{split}
\]
Clearly, drawing $\phi''$ naturally defines a drawing $\phi'''$ of $G$, with $\mathsf{cr}(\phi''')=\mathsf{cr}(\phi'')$, such that
for every verteex $x\in V(G_{\mid {\mathcal{C}}}\cap V(G))$, the ordering of the edges of $\delta_G(x)$ as they enter $x$ in $\phi'''$ is consistent with the the ordering of the edges of $\delta_G(x)$ as they enter $x$ in $\phi^*$, which is ${\mathcal{O}}_x\in \Sigma$. This completes the proof of \Cref{lem: crossings in contr graph}.
\iffalse
Let ${\mathcal{C}}=\set{C_1,\ldots,C_r}$.
We partition the set $E(G')$ of edges into subsets $E_0,E_1,\ldots,E_r$ as follows: for all $1\leq i\leq r$, $E_i=E(C'_i)$, and $E_0$ contains all edges that do not lie in any cluster $C'_1,\ldots,C'_r$. We let $\chi_1$ be the set of all crossings $(e,e')$ in $\phi'$, with $e,e'$ lying in the same set $E_i$, for $0\leq i\leq r$, and we let $\chi_2$ be the set of all crossings $(e,e')$ in $\phi'$ where $e\in E_i$, $e'\in E_j$, and $i\neq j$.
Consider now some index $1\leq i\leq r$, and some edge $e\in E(C'_i)$. If $e$ does not participate in any path in ${\mathcal{Q}}(C_i)$, then we set $n_e=1$; otherwise, we let $n_e=\cong({\mathcal{Q}}(C_i),e)$ be the number of paths in ${\mathcal{Q}}(C_i)$, in which the edge $e$ participates. For each edge $e\in E_0$, we set $n_e=1$.
Let $G''$ be the graph obtained from $G'$, where we replace every edge $e\in E(G')$ with $n_e$ parallel copies. For all $0\leq i\leq r$, we denote by $E'_i$ the set of edges of $G''$ corresponding to the edges of $E_i$. Drawing $\phi'$ of $G'$ naturally defines a drawing $\phi''$ of $G''$: for every edge $e\in E(G')$, we draw the $n_e$ parellel copies of the edge $e$ next to each other, in parallel to the original drawing of the edge $e$ and very close to it.
We denote by $\chi_1'$ the set of all crossings $(e,e')$ in the resulting drawing $\phi''$, where both $e$ and $e'$ lie in the same set $E'_i$ of edges, for any $0\leq i\leq r$, and we let $\chi_2'$ be the set of all crossings $(e,e')$ in $\phi''$, where $e\in E_i'$, $e'\in E_j'$, and $0\leq i<j\leq r$. We use the following simple observation to bound the number of crossigns in $\chi_2'$.
\begin{observation}\label{obs: number of type 2 crossings}
\[\expect{|\chi_2'|}\leq |\chi_2|\cdot O(\log^8n/\alpha^2).\]
\end{observation}
\begin{proof}
Consider some crossing $(e_1,e_2)\in \chi_2'$. Then there must be a crossing $(e_1',e_2')\in \chi_2$, where $e_1$ is a copy of $e_1'$ and $e_2$ is a copy of $e_2'$. We say that crossing $(e_1',e_2')$ is responsible for the crossing $(e_1,e_2)$. We have now assigned, to each crossing in $\chi_2'$, a crossing in $\chi_2$ that is responsible for it.
Consider now some crossing $(e,e')\in \chi_2$. The number of crossings in $\chi_2'$ that it is responsible for is precisely $n_e\cdot n_{e'}$. Assume that $e\in E_i,e'\in E_j$, for some $0\leq i,j\leq r$, with $i\neq j$. Since $i\neq j$, $n_e$ and $n_{e'}$ are independent random variables. Therefore, $\expect{n_e\cdot n_{e'}}=\expect{n_e}\cdot \expect{n_{e'}}\leq O(\log^8n/\alpha^2)$.
Altogether, we get that:
\[\expect{|\chi_2'|}\leq \expect{\sum_{(e,e')\in \chi_2}n_e\cdot n_{e'}}\leq \sum_{(e,e')\in \chi_2}\expect{n_e\cdot n_{e'}}\leq |\chi_2|\cdot O(\log^8n/\alpha^2).\]
\end{proof}
Let $\chi_0$ be the set of all crossings $(e,e')$ in the drawing $\phi'$ in which $e,e'\in E_0$.
For all $1\leq i\leq r$, let $C''_i$ be the subgraph of $G''$ induced by the vertices of $C'_i$. We can now define a set ${\mathcal{Q}}'(C_i)$ of paths in graph $C''_i$, routing the vertices of $T_{C_i}$ to $u_C$, such that the paths in ${\mathcal{Q}}'(C_i)$ are edge-disjoint. In order to do so, we simply assign, for each edge $e\in E_i$, a distinct copy of $e$ to every path $Q_t\in {\mathcal{Q}}(C_i)$ that contains $e$.
Next, we define a collection $\Gamma_0,\Gamma_1,\ldots,\Gamma_r$ of curves, as follows. For each edge $e\in E_0$, we add a curve $\gamma(e)$ to $\Gamma_0$, where $\gamma(e)$ is the image of the edge $e$ in the drawing $\phi''$.
Consider now some index $1\leq i\leq r$. We start by letting $\Gamma_i$ contain the set of images of all paths in ${\mathcal{Q}}'(C_i)$ in the drwaing $\phi'$; in other words, we can denote $\Gamma_i=\set{\gamma_i(t)\mid t\in T_C}$, where $\gamma_i(t)$ is the image of the unique path $Q'_t\in {\mathcal{Q}}'(C_i)$, connecting $t$ to $u(C_i)$. We direct all curves in $\Gamma_i$ towards the image of $u(C_i)$, and we delete loops from each resulting curve, so that all curves in set $\Gamma_i$ become simple.
Consider the resulting collection $\Gamma=\Gamma_0\cup\Gamma_1\cup\cdots \Gamma_r$ of curves. Let $p$ be any point that lies on more than two such curves.
Assume first that $p$ is an image of some vertex $v\in V(G'')$. If there is some index $1\leq i\leq r$, such that $v\in V(C_i)$, then every curve $\gamma\in \Gamma$ that contains $v$ must belong to $\Gamma_i$. Otherwise, for every curve $\gamma\in \Gamma$ with $p\in \gamma$, point $p$ is an endpoint of $\gamma$.
Assume now that $p$ is not an image of a vertex in $V(G'')$, so it is an inner point on the image of some edge of $G''$. I this case, from the definition of a graph drawing, $p$ may lie on images of at most two edges of $G''$, and so it may not belong to more than three curves of $\Gamma$.
To conclude, if a point $p$ lies on more than $2$ curves of $\Gamma$, then either it is an endpoint of every curve $\gamma$ containing $p$, or there is some index $1\leq i\leq r$, such that every curve $\gamma$ containing $p$ belongs to $\Gamma_i$.
Therefore,
we then use the algorithm from \Cref{thm: type-2 uncrossing} in order to uncross each set $\Gamma_i$ of curves, for $1\leq i\leq r$. Recall that we obtain new sets $\Gamma'_1,\ldots,\Gamma'_r$ of curves. For each $1\leq i\leq r$ and $t\in T_C$, we denote by $\gamma'_i(t)$ the unique curve in $\Gamma'_i$ that originates at the image of $t$ and terminates at the image of $u(C_i)$.
\mynote{from here on needs to be fixed once the statement of \Cref{thm: type-2 uncrossing} is fixed}
Recall that \Cref{thm: type-2 uncrossing}, we are guaranteed that, if $p$ is a point lying on more than two curves in $\Gamma_0\cup \Gamma'_1\cup\cdots\cup\Gamma'_r$, then $p$ is an endpoint of all these curves, and that the total number of crossings between all curves in $\Gamma_0\cup \Gamma'_1\cup\cdots\cup\Gamma'_r$ is bounded by $|\chi_2'|+|\chi_0|$.
We are now ready to define the drawing $\phi$ of graph $G_{|{\mathcal{C}}}$. We start from the drawing $\phi'$ of $G'$.
For every cluster $C_i\in {\mathcal{C}}$, we delete the images of all vertices of $C$, except for $u_C$, and of all edges of $E(C')$. The image of $u_C$ becomes the image of the vertex $v(C)$ representing the cluster $C$ in the contracted graph. We then add the curves in $\Gamma'_i$ to this drawing; recall that each such curve connects $u(C)$ to a distinct vertex $t\in T_C$. Once all clusters $C_i\in {\mathcal{C}}$ are processed, we obtain a drawing of a graph $\hat G$, that is obtained from $G_{|{\mathcal{C}}}$ by subdividing every edge $e\in \bigcup_{C\in {\mathcal{C}}}\delta(v(C))$ with a vertex $t_e$. The set of images of the edges of $\hat G$ in this drawing is precisely $\Gamma_0\cup \Gamma'_1\cup\cdots\cup\Gamma'_r$, and so the number of crossings in this drawing is bounded by $|\chi_2'|+|\chi_0|$. This drawing of $\hat G$ immediately defines a drawing of $G_{|{\mathcal{C}}}$, by surpressing the images of all vertices $\set{t_e\mid e\in \bigcup_{C\in {\mathcal{C}}}\delta(v(C))}$. The number of crossings in this final drawing is bounded by $|\chi_2'|+|\chi_0|$. Clearly, $|\chi_0|\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)$, since we did not modify the drawings of the edges in $E_0$, and each edge $e\in E_0$ has $n_e=1$.
From
\Cref{obs: number of type 2 crossings},
$\expect{|\chi_2'|}\leq |\chi_2|\cdot O(\log^8n/\alpha^2)$. Moreover, since we did not modify the drawings of the edges in $E_0$, it is easy to verify that for every vertex $x\in V(G_{|{\mathcal{C}}})\cap V(G)$, the ordering of the edges of $\delta_G(x)$ as they enter $x$ in the final drawing $\phi$ is consistent with the ordering ${\mathcal{O}}_x\in \Sigma$.
\znote{to modify using new type-2 uncrossing}
\fi
\iffalse
\subsection{Proof of \Cref{lem: convert path}}
\label{apd: Proof of convert path}
\znote{To add}.
\fi
\subsection{Proof of \Cref{lem: GH tree path vs contraction}}
\label{apd: Proof of GH tree path vs contraction}
For each $1\le i\le n-1$, we denote $S_{i}=\set{v_1,\ldots,v_i}$ and $\overline{S}_{i}=\set{v_{i+1},\ldots,v_n}$.
Let $v_C,v_{C'}$ be two vertices that appear consecutively on the contracted path $\tau_{{\mathcal{C}}}$, where $v_C$ corresponds to cluster $C$ and $v_{C'}$ corresponds to cluster $C'$ ($C,C'\in {\mathcal{C}}$). We denote $C=\set{v_p,\ldots,v_r}$ and $C'=\set{v_{r+1},\ldots,v_q}$.
Since path $\tau$ is a Gomory-Hu tree of $G$, then the cut $(S_r,\overline{S}_r)$ is a min-cut in $G$ separating $v_r$ from $v_{r+1}$.
On the other hand, note that any cut separating $v_C$ from $v_{C'}$ in the contracted graph $G_{{\mathcal{C}}}$ corresponds to a cut in the original graph $G$ separating $v_r$ from $v_{r+1}$, with the same set of edges. Moreover, if we denote by $S,\overline{S}$ the components of $\tau$ obtained by removing the edge $(v_C,v_{C'})$ from $\tau$, then the cut $(S,\overline{S})$ in $G_{{\mathcal{C}}}$ corresponds to the cut $S_r,\overline{S}_r$ in $G$. Therefore, the cut $(S,\overline{S})$ is a min-cut in $G_{{\mathcal{C}}}$ separating $v_C$ from $v_{C'}$. It follows that $\tau_{|{\mathcal{C}}}$ is a Gomory-Hu tree of the contracted graph $G_{|{\mathcal{C}}}$.
\subsection{Proof of Lemma~\ref{lem: multiway cut with paths sets}}
\label{apd: Proof of multiway cut with paths sets}
We use the following lemma.
\begin{lemma}
\label{lem: cut_uncrossing}
Let $S=\set{s_1,\ldots,s_k}$ be a set of vertices of $G$. For any pair $1\le i,j\le k$, if $(U_i,\overline{U_i})$ is a minimum cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$ with $s_i\in U_i$, and $(U_j,\overline{U_j})$ is a minimum cut in $G$ separating $\set{s_j}$ from $S\setminus \set{s_j}$ with $s_j\in U_j$, then $(U_i\setminus U_j, \overline{U_i\setminus U_j})$ is a minimum cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$, and $(U_j\setminus U_i, \overline{U_j\setminus U_i})$ is a minimum cut in $G$ separating $\set{s_j}$ from $S\setminus \set{s_j}$.
\end{lemma}
\begin{proof}
If $U_i\cap U_j=\emptyset$ then there is nothing to prove. We now assume that $U_i\cap U_j\ne\emptyset$.
We denote $A=U_i\setminus U_j$, $B=U_j\setminus U_i$, $C=U_i\cap U_j$, and $D=\overline{U_i\cup U_j}$.
Note that $s_i\in U_i$ and $s_i\notin U_j$, so $A\ne \emptyset$, and similarly $B\ne \emptyset$.
Note that the cut $(A,B\cup C\cup D)$ is also a cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$.
Since $(U_i,\overline{U_i})$ is a minimum cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$,
\[
|E(A,B)|+|E(A,D)|+|E(C,B)|+|E(C,D)|\le |E(A,B)|+|E(A,C)|+|E(A,D)|.
\]
We get that $|E(C,B)|+|E(C,D)|\le |E(A,C)|$.
Similarly, note that the cut $(B,A\cup C\cup D)$ is also a cut in $G$ separating $\set{s_j}$ from $S\setminus \set{s_j}$, while $(U_j,\overline{U_j})$ is a minimum cut in $G$ separating $\set{s_j}$ from $S\setminus \set{s_j}$. Therefore,
\[
|E(A,B)|+|E(D,B)|+|E(A,C)|+|E(D,C)|\le |E(B,A)|+|E(B,C)|+|E(B,D)|.
\]
We get that $|E(A,C)|+|E(C,D)|\le |E(B,C)|$.
Altogether, $|E(C,D)|=0$ and $|E(A,C)|=|E(B,C)|$. Therefore,
$|E(A,B\cup C\cup D)|=|E(U_i,\overline{U_i})|$ and
$|E(B,A\cup C\cup D)|=|E(U_j,\overline{U_j})|$, which means that
$(U_i\setminus U_j, \overline{U_i\setminus U_j})$ is a minimum cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$, and $(U_j\setminus U_i, \overline{U_j\setminus U_i})$ is a minimum cut in $G$ separating $\set{s_j}$ from $S\setminus \set{s_j}$.
\end{proof}
We now complete the proof of Lemma~\ref{lem: multiway cut with paths sets} using Lemma~\ref{lem: cut_uncrossing}.
Recall that we are given a set $S=\set{s_1,\ldots,s_k}$ of vertices of $G$. We first compute, for each $1\le i\le k$, a minimum cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$, that we denote by $(U_i,\overline{U_i})$, with $s_i\in U_i$. We then let $A_i=U_i\setminus (\bigcup_{1\le j\le k, j\ne i}U_j)$, for each $1\le i\le k$.
It is clear that, for all $1\le i\le k$, $S\cap A_i=\set{s_i}$, and for all $1\le i<j\le k$, $A_i\cap A_j=\emptyset$.
We claim that the cut $(A_i,\overline{A_i})$ is a min-cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$.
To see that, we start with $A^*_i=U_i$, and then sequentially for $j=1,\ldots, i-1,i+1,\ldots,k$, we delete from $A^*_i$ the elements that belong to $U_j$. From Lemma~\ref{lem: cut_uncrossing}, after each round, the cut $(A^*_i,\overline{A^*_i})$ remains a min-cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$, and at the end, $A^*_i=U_i\setminus (\bigcup_{1\le j\le k, j\ne i}U_j)$.
It remains to compute, for each $1\le i\le k$, a set ${\mathcal{Q}}_i$ of paths connecting edges of $\delta(A_i)$ to $s_i$. We set up an instancee of max-flow problem as follows.
We define the graph $H_i$ to be the graph obtained from $G[A_i]\cup \delta_G(A_i)$, (where $G[A_i]$ is the subgraph of $G$ induced by vertices of $A_i$) by contracting all vertices that do not belong to $A_i$ but are connected by an edge to some vertex of $A_i$ into a single vertex, that we denote by $t_i$. We then compute in graph $H_i$ the max-flow from $s_i$ to $t_i$, with each edge of $H_i$ assigned with unit capacity. It is easy to see that we can obtain a set of $|\delta_G(A_i)|$ paths in $H_i$ connecting edges of $\delta_G(A_i)$ to $u$, and we denote this set by ${\mathcal{Q}}_i$. Since otherwise, from the max-flow min-cut theorem, we will obtain a cut (other than the cut $(A_i,\set{t_i})$) in $H_i$ separating $s_i$ from $t_i$ that contains strictly less edges than $(A_i,\set{t_i})$, causing a contradiction to the fact that the cut $(A_i,\overline{A_i})$ is a min-cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$.
\subsection{Proof of \Cref{obs: grid 1st row well-linked}}
\label{apd: Proof grid 1st row well-linked}
We denote by $T$ the first row of the $(r\times r)$-grid.
Consider any partition $(A,B)$ of the vertex set of the $(r\times r)$-grid.
Denote $A_1=T\cap A$ and $B_1=T\cap B$, and assume without loss of generality that $|A_1|\le |B_1|$.
We will show that there exists a set of $|A_1|$ node-disjoint paths connecting vertices of $A_1$ to vertices of $B_1$, which implies that $|E(A,B)|\ge \min\set{|A\cap T|,|B\cap T|}$.
The node-disjoint paths are constructed similarly as in
the proof of \Cref{clm: grid well-linked}.
We maintain an index set $I$, initializing $I=\set{1,\ldots,r}$.
While there is a pair $i,i'$ of indices of $I$, such that $i<i'$, $v_{i},v_{i'}$ belongs to different sets of $A_1,B_1$, and there is no other $j\in I$ with $i<j<i'$, we define the path $P_{i,i'}$ to be the sequential concatenation of the path $(v_{1,i},v_{2,i},\ldots,v_{i',i})$, the path $(v_{i',i},v_{i',i+1},\ldots,v_{i',i'})$, and the path $(v_{i',i'},v_{i'-1,i'},\ldots,v_{1,i'})$. We then add the path $P_{i,i'}$ to ${\mathcal{P}}$, delete $i,i'$ from $I$ and continue to the next iteration.
It is clear that, at the end of the process, $I$ does not contain any index $i$ with $v_{1,i}\in A_1$, and for each $v_{1,j}\in A_1$, there is a path $P_{j,j'}$ connecting $v_{1,j}$ to some vertex $v_{1,j'}$ of $B_1$, so $|{\mathcal{P}}|=|A_1|$. It is also easy to verify that the paths of ${\mathcal{P}}$ are node-disjoint.
\section{Proofs Omitted from Section~\ref{sec:long prelim}}
\label{sec: apd_prelim}
\input{apx-prelims-paths}
\input{apx-prelims-cuts}
\input{appx-layered-wld.tex}
\input{appx-prelims-expanders}
\input{apx-prelims-curves}
\input{appx-prelims-contracted}
\iffalse
\subsection{Proof of \Cref{obs:rerouting_matching_cong}}
\label{apd: Proof of rerouting_matching_cong}
Denote $E=\delta_G(v)$. We first define a multiset $E'$ and an ordering ${\mathcal{O}}'$ on $E'$ as follows. Set $E'$ contains, for each edge $e\in E$, $(n^-_e+n^+_e)$ copies of $e$, in which $n^-_e$ copies are labelled $-$, and other $n^+_e$ copies are labelled $+$. The ordering ${\mathcal{O}}'$ on $E'$ is obtained from the ordering ${\mathcal{O}}_v$ on $E$ by replacing, for each $e\in E'$, the edge $e$ in the ordering ${\mathcal{O}}_v$ with its $(n^-_e+n^+_e)$ copies in $E'$, that appear consecutively at the location of $e$ in ${\mathcal{O}}_v$, and the ordering among these copies is arbitrary.
Since $\sum_{e\in E}n^-_e=\sum_{e\in E}n^+_e$, the number of edges in $E'$ that are labelled $-$ is the same as the number of edges in $E'$ that are labelled $+$.
We now iteratively construct a set $M'$ of ordered pairs as follows.
We start with $M'=\emptyset$ and perform iterations.
In each iteration, we find a pair of edges in $E'$ that appears consecutively in ${\mathcal{O}}'$, such that one of them (denoted by $e'_1$) is labelled $-$ and the other (denoted by $e'_2$) is labelled $+$. It is clear that such a pair always exists. Note that it is possible that $e'_1$ and $e'_2$ are distinct copies of the same edge of $E$.
We then add the ordered pair $(e'_1,e'_2)$ to $M'$, and delete $e'_1,e'_2$ from $E'$ and ${\mathcal{O}}'$.
It is clear that after this iteration, the number of remaining edges in $E'$ (and in ${\mathcal{O}}'$) that are labelled $-$ is still the same as the number of remaining edges in $E'$ (and in ${\mathcal{O}}'$) that are labelled $+$. Therefore, at the end of the process, $E'$ and ${\mathcal{O}}'$ become empty and $M'$ contains $\sum_{e\in E}n^+_e$ ordered pairs of edges in $E'$. Moreover, for each pair of ordered pairs in $M'$, the intersection between the paths consisting of corresponding pairs of edges is non-transversal with respect to ${\mathcal{O}}'$.
We then let set $M$ contains, for each pair $(e'_1,e'_2)\in M$, a pair $(e_1,e_2)$, where $e_1$ is the original edge of $e'_1$ in $E$ and $e_2$ is the original edge of $e'_2$ in $E$. It is clear that for each pair of ordered pairs in $M$, the intersection between the paths consisting of corresponding pairs of edges is non-transversal with respect to ${\mathcal{O}}'$.
\fi
\iffalse
\subsection{Proof of \Cref{lem: non transversal cost of cycles bounded by cr}}
\label{apd: Proof of non transversal cost of cycles bounded by cr}
Consider a pair $R,R'$ of edge-disjoint cycles with $R\in {\mathcal{R}}$ and $R'\in {\mathcal{R}}'$. We denote by $\gamma$ the closed curve obtained by taking the union of the images of all edges of $R$ in $\phi$, and we define the closed curve $\gamma'$ similarly. Since $R$ and $R'$ are edge-disjoint, and since $\phi$ is a feasible solution to instance $I=(G,\Sigma)$, curves $\gamma$ and $\gamma'$ are in general position. Let $X(\gamma,\gamma')$ be the set of crossings between curves $\gamma,\gamma'$ that are not vertex-images in $\phi$, so $|X(\gamma,\gamma')|=\chi(\gamma,\gamma')$. We will show that, if the intersection between $R$ and $R'$ are transversal at exactly one of their shared vertices, then $X(\gamma,\gamma')\ne \emptyset$.
Assume for contradiction that $X(\gamma,\gamma')=\emptyset$. Let $v_1,\ldots,v_k$ be the shared vertices between $R$ and $R'$, and let $x_1,\ldots,x_k$ be the images of $v_1,\ldots,v_k$ in $\phi'$, respectively. Assume without loss of generality that vertices $v_1,\ldots,v_k$ appear sequentially on cycle $R$, and the intersection between $R$ and $R'$ is transversal at $v_1$.
We give a direction for the closed curve $\gamma$ as follows: $x_1\to x_2\to\ldots\to x_k\to x_1$.
Clearly, the curve $\gamma$ partitions the sphere into internally-disjoint discs, and since $R$ is a simple cycle, vertex $x_1$ lies on the boundary of exactly two of these discs, that we denote by $D_{+}$ and $D_{-}$.
Since cycles $R,R'$ are disjoint and the intersection of them is transversal at $v_1$, if we denote by $\gamma'_+$ the tiny segment of curve $\gamma'$ that has $x_1$ as an endpoint and goes a little in one direction of $\gamma'$, and denote by $\gamma'_-$ the tiny segment of curve $\gamma'$ that has $x_1$ as an endpoint and goes a little in the other direction of $\gamma'$, then one of segments $\gamma'_+, \gamma'_+$ belong to $D_{+}$ and the other belongs to $D_{-}$. Therefore, since $\gamma'$ is a closed curve, $\gamma'$ has to intersect the boundary of disc $D_{+}$ at some other point $x'$. However, since $R$ and $R'$ are edge-disjoint and the intersections of $R$ and $R'$ are non-transversal at $v_2,\ldots,v_k$, $x$ cannot be any of $\set{x_2,\ldots,x_k}$. Therefore, $x'$ has to be some non-vertex-image crossings between $\gamma$ and $\gamma'$.
We now complete the proof of \Cref{lem: non transversal cost of cycles bounded by cr}.
For every cycle $R\in {\mathcal{R}}$, we denote by $\gamma_R$ the closed curve obtained by taking the union of the images of all edges of $R$ in $\phi$, and for each cycle $R\in {\mathcal{R}}'$, we define the closed curve $\gamma_R'$ similarly.
Then it is easy to verify that every crossing $(e,e')$ in drawing $\phi$ belongs to at most $\bigg(\cong_G({\mathcal{R}},e)\cdot \cong_G({\mathcal{R}}',e')+\cong_G({\mathcal{R}},e')\cdot \cong_G({\mathcal{R}}',e)\bigg)$ sets among all sets of $\set{X(\gamma_R,\gamma_{R'})\mid R\in {\mathcal{R}}, R'\in {\mathcal{R}}'}$.
Since for every pair $R, R'$ of edge-disjoint cycles with $R\in {\mathcal{R}}, R'\in {\mathcal{R}}'$, the intersection of $R$ and $R'$ is transversal at at most one of their common vertices, we get that
\[
\begin{split}
\mathsf{cost}_{\mathsf {NT}}({\mathcal{R}},{\mathcal{R}}';\Sigma)\le & \sum_{R\in {\mathcal{R}}, R'\in {\mathcal{R}}'} |X(\gamma_{R}, \gamma_{R'})|\\
\le & \sum_{e,e'\in E(G)}\chi_{\phi}(e,e')\cdot \bigg(\cong_G({\mathcal{R}},e)\cdot \cong_G({\mathcal{R}}',e')+\cong_G({\mathcal{R}},e')\cdot \cong_G({\mathcal{R}}',e)\bigg).
\end{split}
\]
\subsection{Proof of \Cref{thm: new nudging}}
\label{apd: Proof of curve_manipulation}
We first prove the following lemma.
\begin{lemma}[Nudging]
\label{obs: curve_manipulation}
There exists an efficient algorithm, that, given a point $p$ in the plane, a disc $D$ containing $p$, and a set $\Gamma$ of curves, such that every curve $\gamma$ of $\Gamma$ contains the point $p$ and intersects the boundary of $D$ at two distinct points $s_{\gamma},t_{\gamma}$, and $p$ is the only point in $D$ that belongs to at least two curves of $\Gamma$, computes, for each curve $\gamma\in \Gamma$, a curve $\gamma'$, such that
\begin{itemize}
\item for each curve $\gamma\in \Gamma$, the curve $\gamma'$ does not contain $p$, and is identical to the curve $\gamma$ outside the disc $D$;
\item the segments of curves of $\set{\gamma'\mid \gamma\in \Gamma}$ inside disc $D$ are in general position;
\item for each pair $\gamma_1,\gamma_2$ of curves in $\Gamma$, the new curves $\gamma_1,\gamma_2$ cross inside $D$ iff the order in which the points $s_{\gamma_1},t_{\gamma_1},s_{\gamma_2},t_{\gamma_2}$ appear on the boundary of $D$ is either $(s_{\gamma_1},s_{\gamma_2},t_{\gamma_1},t_{\gamma_2})$ or $(s_{\gamma_1},t_{\gamma_2},t_{\gamma_1},s_{\gamma_2})$.
\end{itemize}
\end{lemma}
\begin{proof}
We first compute, for each $\gamma\in \Gamma$, a curve $\tilde\gamma$ inside disc $D$ connecting $s_{\gamma}$ to $t_{\gamma}$, such that (i) the curves in $\set{\tilde\gamma\mid \gamma\in \Gamma}$ are in general position; and (ii) for each pair $\gamma_1,\gamma_2$ of curves in $\Gamma$, the curves $\tilde\gamma_1$ and $\tilde\gamma_2$ intersects iff the order in which the points $s_{\gamma_1},t_{\gamma_1},s_{\gamma_2},t_{\gamma_2}$ appear on the boundary of $D$ is either $(s_{\gamma_1},s_{\gamma_2},t_{\gamma_1},t_{\gamma_2})$ or $(s_{\gamma_1},t_{\gamma_2},t_{\gamma_1},s_{\gamma_2})$.
It is clear that this can be achieved by first setting, for each curve $\gamma$, the curve $\tilde \gamma$ to be the line segment connecting $s_{\gamma}$ to $t_{\gamma}$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\tilde \gamma\mid \gamma\in \Gamma}$.
We now define, for each $\gamma\in \Gamma$, the curve $\tilde\gamma$ to be the union of the part of $\gamma$ outside $D$ and the curve $\tilde\gamma$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, all conditions of \Cref{obs: curve_manipulation} are satisfied.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of $\Gamma$ are shown in distinct colors. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of $\set{\tilde\gamma\mid \gamma\in \Gamma}$ are shown in dash lines.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the proof of \Cref{obs: curve_manipulation}.}\label{fig: curve_con}
\end{figure}
We now use \Cref{obs: curve_manipulation} to prove \Cref{thm: new nudging}.
Denote $G'=G\setminus C$ and denote $\hat {\mathcal{Q}}={\mathcal{Q}}\cup {\mathcal{Q}}'$.
For each $e\in E(G')$, we let set $\Pi_{e}$ contain $\cong_{G'}(\hat {\mathcal{Q}},e)$ curves connecting the endpoints of $e$ that are internally disjoint and lying inside an arbitrarily thin strip around the curve $\phi(e)$. We then assign, for each path $Q\in \hat {\mathcal{Q}}$ and each edge $e\in E(Q)$, a distinct curve of $\Pi_{e}$ to $Q$, so each curve in $\bigcup_{e\in E(G')}\Pi_{e}$ is assigned to exactly one path of $Q$.
For each $e\in \delta_G(C)$, let $\zeta_Q$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E(G')}\Pi_{e}$ that are assigned to $Q$, so $\zeta_Q$ connects $\phi(s_Q)$ to $\phi(t^*)$.
We denote $Z=\set{\zeta_Q\mid Q\in \hat {\mathcal{Q}}}$.
In fact, when we assign curves of $\bigcup_{e\in \delta_{G'}(t^*)}\Pi_{e}$ to path of $\hat{\mathcal{Q}}$, we additionally ensure that the order in which the curves of $\set{\zeta_Q\mid Q\in \hat {\mathcal{Q}}}$ enter $\phi(t^*)$ is identical to ${\mathcal{O}}^{\operatorname{guided}}(\hat{\mathcal{Q}},{\mathcal{O}}_{t^*})$, the ordering guided by the set $\hat{\mathcal{Q}}$ of paths and the ordering ${\mathcal{O}}_{t^*}$. Note that this can be easily achieved according the the definition of ${\mathcal{O}}^{\operatorname{guided}}(\hat{\mathcal{Q}},{\mathcal{O}}_{t^*})$.
We then modify the curves of $Z$ as follows.
For each vertex $v\in V(G')$, we denote by $x_v$ the point that represents the image of $v$ in $\phi$, and we let $X$ contain all points of $\set{x_v\mid v\in V(G')}$ that are crossings between curves in $Z$, so $x_{t^*}\notin X$.
We then iteratively process all points of $X$ as follows.
Consider a point $x_v$ of $X$ and let $D_{\phi}(v)$ be a tiny $v$-disc.
Let $Z(v)$ be the subset of all curves in $Z$ that contains $x_v$.
Let $Z'(v)$ be the set of curves we obtain by applying the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $Z(v)$ within disc $D_{\phi}(v)$. We then replace curves of $Z(v)$ in $Z$ by curves of $Z'(v)$, and continue to the next iteration. Let $\Gamma'$ be the set of curves that we obtain after processing all points of $X$ in this way.
Note that the curves of $\Gamma'$ might be non-simple, and a pair of curves in $\Gamma'$ may cross more than once. We then remove self-loops from all curves of $\Gamma'$, and perform type-1 uncrossing (the algorithm from \Cref{thm: type-1 uncrossing}) to curves of $\Gamma'$, and finally denote by $\Gamma=\set{\gamma(Q)\mid Q\in \hat{\mathcal{Q}}}$ the set of curves we obtained. This finishes the construction of the set $\Gamma$ of curves.
From \Cref{obs: curve_manipulation} and the construction of $\Gamma$, it is easy to verify that the first and the second properties are satisfied.
We now show that $\Gamma$ satisfies the fourth property.
Since our operations at vertex-images only modify the image of curves within tiny discs around inner-vertex-images but does not change the course of any curves, and since type-1 uncrossing does not change the last segment of curves in $\Gamma'$ (and therefore does not change the ordering in which curves of $\Gamma'$ enter $\phi(t^*)$), the curves of $\Gamma$ enter $\phi(t^*)$ in the order ${\mathcal{O}}^{\operatorname{guided}}(\hat{\mathcal{Q}},{\mathcal{O}}_{t^*})$.
We next show that $\Gamma$ satisfies the fifth property.
Note that the curves of $\Gamma$ all lie within thin strips of the image of edges of $E(\hat{\mathcal{Q}})$ in $\phi$, and for each edge $e'\in E(G')$, the number of segments in $\Gamma$ that lie entirely within the thin strip of $\phi(e')$ is bounded by $\cong_{G'}(\hat{\mathcal{Q}},e')$. Therefore, the number of crossings between the image of any edge $e\in E(C)$ and the curves of $\Gamma$ is bounded by the sum, over all edges $e'\in E(G')$, the number of crossings between $e$ and $e'$ in $\phi$, times $\cong_G(\hat{\mathcal{Q}}, e')$, which is in turn bounded by $\chi_{\phi}(e, G\setminus C)\cdot \cong_G(\hat{\mathcal{Q}})$.
It remains to show that $\Gamma$ satisfies the third property, and this is the only place where we treat path sets ${\mathcal{Q}}$ and ${\mathcal{Q}}'$ differently. First, since we performed type-1 uncrossing to the set $\Gamma'$ of curves in the last step, the number of crossings between curves of $\Gamma({\mathcal{Q}}')=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}'}$ is at most $|{\mathcal{Q}}'|^2$.
Second, we bound the number of crossings between curves of $\Gamma({\mathcal{Q}})=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$, as follows.
On the one hand, it is easy to verify that, if we denote $Z({\mathcal{Q}})=\set{\zeta_Q\mid Q\in {\mathcal{Q}}}$, then the number of non-vertex-image crossings between curves of $Z({\mathcal{Q}}))$ is at most $\chi^2_{\phi}({\mathcal{Q}})$.
Note that
\[
\begin{split}
\chi^2_{\phi}({\mathcal{Q}})
\le & \sum_{e,e'\in E(G)}\chi_{\phi}(e,e')\cdot \cong_G({\mathcal{Q}},e)\cdot \cong_G({\mathcal{Q}},e')\\
\le & \sum_{e,e'\in E(G)}\chi_{\phi}(e,e')\cdot \frac{(\cong_G({\mathcal{Q}},e))^2+ (\cong_G({\mathcal{Q}},e'))^2}{2}\\
\le & \sum_{e\in E(G)}\chi_{\phi}(e)\cdot(\cong_G({\mathcal{Q}},e))^2.
\end{split}
\]
On the other hand, in the step where we repeatedly apply the algorithm from \Cref{obs: curve_manipulation} to the set $Z$ of curves, it is not hard to verify that, if a crossing between two curves $\gamma(Q)$ and $\gamma(Q')$ (where $Q,Q'\in {\mathcal{Q}}$) is introduced in this step, then either they share an edge, or their intersection is transversal at a common inner vertex. The number of pairs of paths in ${\mathcal{Q}}$ that share an edge is bounded by $\sum_{e\in E(G)}(\cong_G({\mathcal{Q}},e))^2$. The number of pairs of paths in ${\mathcal{Q}}$ whose intersection is transversal at some of their common inner vertex is bounded by $\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}};\Sigma)$.
Third, using similar arguments, we can show that the number of crossings between a curve of $\Gamma({\mathcal{Q}})$ and a curve of $\Gamma({\mathcal{Q}}')$ is at most $$\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}},{\mathcal{Q}}';\Sigma)+\sum_{e\in E(G)} \cong_G({\mathcal{Q}},e)\cdot\cong_G({\mathcal{Q}}',e)+
\sum_{e,e'\in E(G)}
\chi_{\phi}(e,e')\cdot \cong_G({\mathcal{Q}},e)\cdot\cong_G({\mathcal{Q}}',e').$$
Altogether, they imply that $\Gamma$ satisfies the third property.
This completes the proof of \Cref{thm: new nudging}.
\subsection{Proof of \Cref{lem: curves from non-transversal paths}}
\label{apd: Proof of curves from non-transversal paths}
We construct a feasible solution $\phi'$ of $(G',\Sigma')$ as follows.
Recall that we are given a solution $\phi$ of instance $(G,\Sigma)$. We start with drawing $\phi(G\setminus E({\mathcal{P}}))$, the drawing of subgraph $G\setminus E({\mathcal{P}})$ induced by $\phi$. It remains to add the images of $\set{e_P\mid P\in {\mathcal{P}}}$ to $\phi(G\setminus E({\mathcal{P}}))$, which we do as follows.
For each path $P\in {\mathcal{P}}$, we denote $\gamma_P=\phi(P)$ the image of $P$ in $\phi$, so $\gamma_P$ is a curve (possibly non-simple) connecting $\phi(s)$ to $\phi(t)$.
Denote $\Gamma=\set{\gamma_P\mid P\in {\mathcal{P}}}$.
We iteratively modify the curves in $\Gamma$, by processing all vertices of $V({\mathcal{P}})$ one-by-one as follows.
Consider a vertex $v\in V(P)$. Denote $\Gamma_v=\set{\gamma_P\mid v\in V(P)}$. We apply \Cref{obs: curve_manipulation} to curves in $\Gamma_v$ within an arbitrarily small disc $D_v$ around $\phi(v)$, and get a new set $\Gamma'_v=\set{\gamma'_P\mid \gamma_P\in \Gamma_v}$ of curves. We then replace the curves of $\Gamma_v$ in $\Gamma$ with curves of $\Gamma_v$. This completes the iteration of processing vertex $v$.
Note that, since the paths of ${\mathcal{P}}$ are edge-disjoint and non-transversal with respect to $\Sigma$, the curves of $\Gamma_v$ do not cross in $D_v$.
Let $\Gamma^*=\set{\gamma_P\mid P\in {\mathcal{P}}}$ be the set of curves we get after processing all vertices of $V(P)$ in this way.
Note that curves of $\Gamma^*$ may be non-simple. We then delete self-loops of for each curve of $\Gamma^*$. We then add curves of $\Gamma^*$ to the drawing $\phi(G\setminus E({\mathcal{P}}))$, where for each $P\in {\mathcal{P}}$, the curve $\gamma^*_{P}$ is designated as the image of $e_P$.
Let $\phi'$ be the resulting drawing. It is easy to verify that the $\phi'$ is a feasible solution of the instance $(G',E')$.
Since we have not created additional crossings in the process of obtaining the curves of $\Gamma^*$, any crossing in $\phi'$ is also a crossing in $\phi$. We then conclude that $\mathsf{cr}(\phi')\le \mathsf{cr}(\phi)$.
\fi
\iffalse
\subsection{Proof of \Cref{lem: opt of contracted instance}}
\label{apd: Proof of opt of contracted instance}
\znote{To Complete.}
\fi
\iffalse{previous type-2 unucrossing}
\subsection{Proof of Theorem~\ref{thm: type-2 uncrossing}}
\label{apd: type-2 uncrossing}
We use similar arguments as in the proof of Theorem~\ref{thm: type-1 uncrossing}.
Let $Z$ be the set of points that lie on at least two curves of $\Gamma$. For each point $z\in Z$, let $D_z$ be an arbitrarily small disc around point $z$, such that, if $z$ lies on curves $\gamma_{1},\ldots,\gamma_{k}$, then
\begin{itemize}
\item the disc $D_z$ is disjoint from all curves of $\Gamma\setminus\set{\gamma_{1},\ldots,\gamma_{k}}$;
\item for each $1\le i\le k$, the intersection between disc $D_z$ and curve $\gamma_i$ is a simple subcurve of $\gamma_i$; and
\item for each $1\le i\le k$, the curve $\gamma_i$ intersects the boundary of $D_z$ at two distinct points.
\end{itemize}
Moreover, all discs in $\set{D_z}_{z\in Z}$ are mutually disjoint.
We now process the sets $\Gamma_1,\ldots,\Gamma_r$ of curves one-by-one.
We will describe the algorithm for processing the set $\Gamma_1$, and the algorithms for processing sets $\Gamma_2,\ldots,\Gamma_r$ are the same.
The algorithm for processing the set $\Gamma_1$ is iterative.
Throughout, we maintain (i) a set $\hat{\Gamma}_1$ of curves, that is initialized to be $\Gamma_1$; and (ii) a set $\hat Z$ of points that lie in at least two curves of $\hat\Gamma_1\cup (\Gamma\setminus \Gamma_1)$, that is initialized to be $Z$.
The algorithm continues to be executed as long as there is still a point that lies on at least two curves of $\hat \Gamma_1$ and is an inner point of each of these curves.
We now describe an iteration.
We first arbitrarily pick such a point $z$, and let $\gamma_{1},\ldots,\gamma_{k}$ be the curves that contains $z$.
For each $1\le i\le k$, we denote by $s_i$ the first endpoint of $\gamma_i$ and by $t_i$ the last endpoint of $\gamma_i$, and denote by $s'_i, t'_i$ the intersections between curve $\gamma_i$ and the boundary of disc $D_z$, such that the points $s_i,s'_i,z,t'_i,t_i$ appear on the directed curve $\gamma_i$ in this order.
\begin{figure}[h]
\centering
\subfigure[Before: Curves $\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5$ are shown in red, yellow, pink, black and green respectively. The points of $S'\cup T'$ appear on the boundary of disc $D_z$.]{\scalebox{0.33}{\includegraphics[scale=1.0]{figs/type_2_uncross_proof_1.jpg}
}
\hspace{0.1cm}
\subfigure[The set $\set{\eta^i_z\mid 1\le i\le k}$ of curves connecting distinct vertices of $S'$ to distinct vertices of $T'$ are shown in dash purple lines.]{
\scalebox{0.33}{\includegraphics[scale=1.0]{figs/type_2_uncross_proof_2.jpg}}}
\hspace{0.1cm}
\subfigure[After: New curves $\hat\gamma_1,\hat\gamma_2,\hat\gamma_3,\hat\gamma_4,\hat\gamma_5$ are shown in red, yellow, pink, black and green respectively. Point $z$ no longer lies in any of these curves.]{
\scalebox{0.33}{\includegraphics[scale=1.0]{figs/type_2_uncross_proof_3.jpg}}}
\caption{An illustration of an iteration in type-1 uncrossing.}\label{fig: type_2_uncross_proof}
\end{figure}
We now modify the curves $\gamma_1,\ldots,\gamma_k$ as follows. Denote $S'=\set{s'_i\mid 1\le i\le k}$ and $T'=\set{t'_i\mid 1\le i\le k}$. We first use the algorithm in~\Cref{obs:rerouting_matching_cong} to compute a set $\set{\eta^{i}_z\mid 1\le i\le k}$ of curves, such that
\begin{itemize}
\item for each $1\le i\le k$, the curve $\eta_z^i$ connects $s'_i$ to a distinct point $t'_{x_i}$ of $T'$;
\item for each $1\le i\le k$, the curve $\eta_z^i$ lies entirely in the disc $D_z$; and
\item the curves in $\set{\eta^{i}_z\mid 1\le i\le k}$ are mutually disjoint.
\end{itemize}
Note that the algorithm in~\Cref{obs:rerouting_matching_cong} only gives us a matching between points in $S'$ and points $T'$, but it can be easily tranformed into a set of curves satisfying the above properties.
See Figure~\ref{fig: type_2_uncross_proof} for an illustration.
For each $1\le i\le k$, we denote by $\gamma^s_i$ the subcurve of $\gamma_i$ between $s_i$ and $s'_i$, and by $\gamma^t_i$ the subcurve of $\gamma_i$ between $t'_i$ and $t_i$.
We then define, for each $1\le i\le k$, a new curve $\hat \gamma_i$ to be the sequential concatenation of curves $\gamma^s_i, \eta^i_z, \gamma^t_{x_i}$.
See Figure~\ref{fig: type_2_uncross_proof} for an illustration.
We then replace the curves $\gamma_1,\ldots,\gamma_k$ in $\hat\Gamma_1$ with the new curves $\hat\gamma_1,\ldots,\hat\gamma_k$. This completes the description of an iteration.
Let $\Gamma'_1$ be the set of curves that we obtain at the end of the algorithm, and let $Z'$ be set of points that lie on at least two curves of $\Gamma'_1\cup (\Gamma\setminus \Gamma_1)$.
It is easy to verify that, in the iteration described above, the multiset of the first endpoints of new curves $\hat\gamma_1,\ldots,\hat\gamma_k$ is identical to that of the original curves $\gamma_1,\ldots,\gamma_k$ and similarly the multiset of the last endpoints of new curves $\hat\gamma_1,\ldots,\hat\gamma_k$ is identical to that of the original curves $\gamma_1,\ldots,\gamma_k$.
Therefore, $S(\Gamma'_1)=S(\Gamma_1)$ and $T(\Gamma'_1)=T(\Gamma_1)$.
It is also easy to verify that the point $z$ no longer lies in any of the new curves $\hat\gamma_1,\ldots,\hat\gamma_k$, while other points in $\hat Z$ remains in $\hat Z$, and there are no new points added to $\hat Z$.
So in each iteration, one point is removed from $\hat Z$, and therefore the algorithm of processing $\Gamma_1$ will terminate in $O(\chi(\Gamma_1))$ rounds.
Moreover, from the terminating criterion of the algorithm, the curves in the resulting set $\Gamma_1'$ no longer cross with each other.
For each $1\le i\le r$, let $\Gamma'_i$ be the set of curves that we obtain from processing the set $\Gamma_i$ of curves using the same algorithm.
We now show that the sets $\Gamma_1,\ldots,\Gamma_r$ satisfy the conditions in \Cref{thm: type-2 uncrossing}.
First, from the above discussion, for each $1\le i\le r$, $S(\Gamma'_i)=S(\Gamma_i)$ and $T(\Gamma'_i)=T(\Gamma_i)$.
Second, if we denote by $Z'$ the set of points that lie in at least two curves of $\Gamma'=\Gamma_0\cup(\bigcup_{1\le i\le r}\Gamma'_i)$, then clearly $Z'\subseteq Z$, and for each $1\le i\le r$, $Z'$ does not contain a point lying in at least two curves of $\Gamma'_i$.
It follows that for each $1\le i\le r$, curves in $\Gamma_i$ do not cross each other; for every curve $\gamma\in \Gamma_0$, $\chi(\gamma,\Gamma'\setminus \Gamma_0)\le \chi(\gamma,\Gamma\setminus \Gamma_0)$; and
$\chi(\Gamma')\le \chi(\Gamma)$.
Last, it is easy to verify from the algorithm that,
if a point is an endpoint of a curve in $\Gamma'$, then it may not be an inner point of any other curve in $\Gamma'$.
Moreover, if a point of $Z$ lie in more than 2 curves in $\Gamma'$, then it has to be an endpoint of every curve that contains it.
Therefore, curves in $\Gamma'$ are in general position.
This completes the proof of \Cref{thm: type-2 uncrossing}.
{previous type-2 uncrossing}
\fi
\iffalse
Let ${\mathcal{C}}=\set{C_1,\ldots,C_r}$.
We partition the set $E(G')$ of edges into subsets $E_0,E_1,\ldots,E_r$ as follows: for all $1\leq i\leq r$, $E_i=E(C'_i)$, and $E_0$ contains all edges that do not lie in any cluster $C'_1,\ldots,C'_r$. We let $\chi_1$ be the set of all crossings $(e,e')$ in $\phi'$, with $e,e'$ lying in the same set $E_i$, for $0\leq i\leq r$, and we let $\chi_2$ be the set of all crossings $(e,e')$ in $\phi'$ where $e\in E_i$, $e'\in E_j$, and $i\neq j$.
Consider now some index $1\leq i\leq r$, and some edge $e\in E(C'_i)$. If $e$ does not participate in any path in ${\mathcal{Q}}(C_i)$, then we set $n_e=1$; otherwise, we let $n_e=\cong({\mathcal{Q}}(C_i),e)$ be the number of paths in ${\mathcal{Q}}(C_i)$, in which the edge $e$ participates. For each edge $e\in E_0$, we set $n_e=1$.
Let $G''$ be the graph obtained from $G'$, where we replace every edge $e\in E(G')$ with $n_e$ parallel copies. For all $0\leq i\leq r$, we denote by $E'_i$ the set of edges of $G''$ corresponding to the edges of $E_i$. Drawing $\phi'$ of $G'$ naturally defines a drawing $\phi''$ of $G''$: for every edge $e\in E(G')$, we draw the $n_e$ parellel copies of the edge $e$ next to each other, in parallel to the original drawing of the edge $e$ and very close to it.
We denote by $\chi_1'$ the set of all crossings $(e,e')$ in the resulting drawing $\phi''$, where both $e$ and $e'$ lie in the same set $E'_i$ of edges, for any $0\leq i\leq r$, and we let $\chi_2'$ be the set of all crossings $(e,e')$ in $\phi''$, where $e\in E_i'$, $e'\in E_j'$, and $0\leq i<j\leq r$. We use the following simple observation to bound the number of crossigns in $\chi_2'$.
\begin{observation}\label{obs: number of type 2 crossings}
\[\expect{|\chi_2'|}\leq |\chi_2|\cdot O(\log^8n/\alpha^2).\]
\end{observation}
\begin{proof}
Consider some crossing $(e_1,e_2)\in \chi_2'$. Then there must be a crossing $(e_1',e_2')\in \chi_2$, where $e_1$ is a copy of $e_1'$ and $e_2$ is a copy of $e_2'$. We say that crossing $(e_1',e_2')$ is responsible for the crossing $(e_1,e_2)$. We have now assigned, to each crossing in $\chi_2'$, a crossing in $\chi_2$ that is responsible for it.
Consider now some crossing $(e,e')\in \chi_2$. The number of crossings in $\chi_2'$ that it is responsible for is precisely $n_e\cdot n_{e'}$. Assume that $e\in E_i,e'\in E_j$, for some $0\leq i,j\leq r$, with $i\neq j$. Since $i\neq j$, $n_e$ and $n_{e'}$ are independent random variables. Therefore, $\expect{n_e\cdot n_{e'}}=\expect{n_e}\cdot \expect{n_{e'}}\leq O(\log^8n/\alpha^2)$.
Altogether, we get that:
\[\expect{|\chi_2'|}\leq \expect{\sum_{(e,e')\in \chi_2}n_e\cdot n_{e'}}\leq \sum_{(e,e')\in \chi_2}\expect{n_e\cdot n_{e'}}\leq |\chi_2|\cdot O(\log^8n/\alpha^2).\]
\end{proof}
Let $\chi_0$ be the set of all crossings $(e,e')$ in the drawing $\phi'$ in which $e,e'\in E_0$.
For all $1\leq i\leq r$, let $C''_i$ be the subgraph of $G''$ induced by the vertices of $C'_i$. We can now define a set ${\mathcal{Q}}'(C_i)$ of paths in graph $C''_i$, routing the vertices of $T_{C_i}$ to $u_C$, such that the paths in ${\mathcal{Q}}'(C_i)$ are edge-disjoint. In order to do so, we simply assign, for each edge $e\in E_i$, a distinct copy of $e$ to every path $Q_t\in {\mathcal{Q}}(C_i)$ that contains $e$.
Next, we define a collection $\Gamma_0,\Gamma_1,\ldots,\Gamma_r$ of curves, as follows. For each edge $e\in E_0$, we add a curve $\gamma(e)$ to $\Gamma_0$, where $\gamma(e)$ is the image of the edge $e$ in the drawing $\phi''$.
Consider now some index $1\leq i\leq r$. We start by letting $\Gamma_i$ contain the set of images of all paths in ${\mathcal{Q}}'(C_i)$ in the drwaing $\phi'$; in other words, we can denote $\Gamma_i=\set{\gamma_i(t)\mid t\in T_C}$, where $\gamma_i(t)$ is the image of the unique path $Q'_t\in {\mathcal{Q}}'(C_i)$, connecting $t$ to $u(C_i)$. We direct all curves in $\Gamma_i$ towards the image of $u(C_i)$, and we delete loops from each resulting curve, so that all curves in set $\Gamma_i$ become simple.
Consider the resulting collection $\Gamma=\Gamma_0\cup\Gamma_1\cup\cdots \Gamma_r$ of curves. Let $p$ be any point that lies on more than two such curves.
Assume first that $p$ is an image of some vertex $v\in V(G'')$. If there is some index $1\leq i\leq r$, such that $v\in V(C_i)$, then every curve $\gamma\in \Gamma$ that contains $v$ must belong to $\Gamma_i$. Otherwise, for every curve $\gamma\in \Gamma$ with $p\in \gamma$, point $p$ is an endpoint of $\gamma$.
Assume now that $p$ is not an image of a vertex in $V(G'')$, so it is an inner point on the image of some edge of $G''$. I this case, from the definition of a graph drawing, $p$ may lie on images of at most two edges of $G''$, and so it may not belong to more than three curves of $\Gamma$.
To conclude, if a point $p$ lies on more than $2$ curves of $\Gamma$, then either it is an endpoint of every curve $\gamma$ containing $p$, or there is some index $1\leq i\leq r$, such that every curve $\gamma$ containing $p$ belongs to $\Gamma_i$.
Therefore,
we then use the algorithm from \Cref{thm: type-2 uncrossing} in order to uncross each set $\Gamma_i$ of curves, for $1\leq i\leq r$. Recall that we obtain new sets $\Gamma'_1,\ldots,\Gamma'_r$ of curves. For each $1\leq i\leq r$ and $t\in T_C$, we denote by $\gamma'_i(t)$ the unique curve in $\Gamma'_i$ that originates at the image of $t$ and terminates at the image of $u(C_i)$.
\mynote{from here on needs to be fixed once the statement of \Cref{thm: type-2 uncrossing} is fixed}
Recall that \Cref{thm: type-2 uncrossing}, we are guaranteed that, if $p$ is a point lying on more than two curves in $\Gamma_0\cup \Gamma'_1\cup\cdots\cup\Gamma'_r$, then $p$ is an endpoint of all these curves, and that the total number of crossings between all curves in $\Gamma_0\cup \Gamma'_1\cup\cdots\cup\Gamma'_r$ is bounded by $|\chi_2'|+|\chi_0|$.
We are now ready to define the drawing $\phi$ of graph $G_{|{\mathcal{C}}}$. We start from the drawing $\phi'$ of $G'$.
For every cluster $C_i\in {\mathcal{C}}$, we delete the images of all vertices of $C$, except for $u_C$, and of all edges of $E(C')$. The image of $u_C$ becomes the image of the vertex $v(C)$ representing the cluster $C$ in the contracted graph. We then add the curves in $\Gamma'_i$ to this drawing; recall that each such curve connects $u(C)$ to a distinct vertex $t\in T_C$. Once all clusters $C_i\in {\mathcal{C}}$ are processed, we obtain a drawing of a graph $\hat G$, that is obtained from $G_{|{\mathcal{C}}}$ by subdividing every edge $e\in \bigcup_{C\in {\mathcal{C}}}\delta(v(C))$ with a vertex $t_e$. The set of images of the edges of $\hat G$ in this drawing is precisely $\Gamma_0\cup \Gamma'_1\cup\cdots\cup\Gamma'_r$, and so the number of crossings in this drawing is bounded by $|\chi_2'|+|\chi_0|$. This drawing of $\hat G$ immediately defines a drawing of $G_{|{\mathcal{C}}}$, by surpressing the images of all vertices $\set{t_e\mid e\in \bigcup_{C\in {\mathcal{C}}}\delta(v(C))}$. The number of crossings in this final drawing is bounded by $|\chi_2'|+|\chi_0|$. Clearly, $|\chi_0|\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)$, since we did not modify the drawings of the edges in $E_0$, and each edge $e\in E_0$ has $n_e=1$.
From
\Cref{obs: number of type 2 crossings},
$\expect{|\chi_2'|}\leq |\chi_2|\cdot O(\log^8n/\alpha^2)$. Moreover, since we did not modify the drawings of the edges in $E_0$, it is easy to verify that for every vertex $x\in V(G_{|{\mathcal{C}}})\cap V(G)$, the ordering of the edges of $\delta_G(x)$ as they enter $x$ in the final drawing $\phi$ is consistent with the ordering ${\mathcal{O}}_x\in \Sigma$.
\znote{to modify using new type-2 uncrossing}
\fi
\iffalse
\subsection{Proof of \Cref{lem: convert path}}
\label{apd: Proof of convert path}
\znote{To add}.
\fi
\iffalse
\subsection{Proof of \Cref{lem: GH tree path vs contraction}}
\label{apd: Proof of GH tree path vs contraction}
For each $1\le i\le n-1$, we denote $S_{i}=\set{v_1,\ldots,v_i}$ and $\overline{S}_{i}=\set{v_{i+1},\ldots,v_n}$.
Let $v_C,v_{C'}$ be two vertices that appear consecutively on the contracted path $\tau_{{\mathcal{C}}}$, where $v_C$ corresponds to cluster $C$ and $v_{C'}$ corresponds to cluster $C'$ ($C,C'\in {\mathcal{C}}$). We denote $C=\set{v_p,\ldots,v_r}$ and $C'=\set{v_{r+1},\ldots,v_q}$.
Since path $\tau$ is a Gomory-Hu tree of $G$, then the cut $(S_r,\overline{S}_r)$ is a min-cut in $G$ separating $v_r$ from $v_{r+1}$.
On the other hand, note that any cut separating $v_C$ from $v_{C'}$ in the contracted graph $G_{{\mathcal{C}}}$ corresponds to a cut in the original graph $G$ separating $v_r$ from $v_{r+1}$, with the same set of edges. Moreover, if we denote by $S,\overline{S}$ the components of $\tau$ obtained by removing the edge $(v_C,v_{C'})$ from $\tau$, then the cut $(S,\overline{S})$ in $G_{{\mathcal{C}}}$ corresponds to the cut $S_r,\overline{S}_r$ in $G$. Therefore, the cut $(S,\overline{S})$ is a min-cut in $G_{{\mathcal{C}}}$ separating $v_C$ from $v_{C'}$. It follows that $\tau_{|{\mathcal{C}}}$ is a Gomory-Hu tree of the contracted graph $G_{|{\mathcal{C}}}$.
\fi
\section{Proofs Omitted from \Cref{sec: short_prelim}}
\label{sec: apd_short_prelim}
\subsection{Proof of Theorem~\ref{thm: crwrs_planar}}
\label{apd: Proof of crwrs_planar}
For every vertex $u\in V(G)$, we denote $d_u=\deg_G(v)$, and we denote $\delta_G(u)=\set{e_1(u),\ldots,e_{d_u}(u)}$, where the edges are indexed according to their order in the rotation ${\mathcal{O}}_u\in \Sigma$.
In order to prove the theorem, we construct a new graph $G'$, that is obtained from $G$ by replacing every vertex $u\in V(G)$ with the $(d_u\times d_u)$-grid. We show that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$, then graph $G'$ is planar. We then provide an algorithm, that, given a planar drawing of $G'$, computes a solution to instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace whose cost is $0$.
We start by defining the graph $G'$. For every vertex $u\in V(G)$, we let $H_u$ be the $(d_u\times d_u)$ grid. We denote the vertices that appear on the first row of grid $H_u$ by $x_1(u),\ldots,x_{d_u}(u)$, in the natural order of their appearance. In order to construct graph $G'$, we start with the disjoint union of the graphs in $\set{H_u}_{u\in V(G)}$. We then consider every edge $e\in E(G)$ one by one. Let $e=(u,u')$ be any such edge, and assume that $e=e_i(u)=e_j(u')$ (that is, $e$ is the $i$th edge incident to $u$, and the $j$th edge incident to $u'$). We then add edge $e'=(x_i(u),x_j(u'))$ to graph $G'$, and we view edge $e'$ as \emph{representing} the edge $e\in E(G)$. This completes the construction of the graph $G'$. We call the edges of $G'$ that lie in set $\set{e'\mid e\in E(G)}$ \emph{primary edges}, and the remaining edges of $G'$ \emph{secondary edges}. Notice that, from our construction, a vertex of $G'$ may be incident to at most one primary edge.
We use the following two observations.
\begin{observation}\label{obs: G' planar}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$, then graph $G'$ is planar.
\end{observation}
\begin{proof}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$, and let $\phi$ be a solution to instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with $\mathsf{cr}(\phi)=0$. We transform drawing $\phi$ to obtain a planar drawing $\psi$ of the graph $G'$.
In order to do so, for every vertex $u\in V(G)$, we consider the tiny $u$-disc $D(u)=D_{\phi}(u)$. For every edge $e_i(u)\in \delta_G(u)$, we denote by $p_i(u)$ the unique point of the image of $e_i(u)$ in $\phi$ that lies on the boundary of the disc $D(u)$. Note that points $p_1(u),\ldots,p_{d_u}(u)$ must appear on the boundary of disc $D(u)$ in this circular order. If they are encountered in this order as we traverse the boundary of $D(u)$ in counter-clock-wise direction, then we say that vertex $u$ is positive; otherwise we say that it is negative. We let $D'(u)$ be a disc that is contained in $D(u)$, such that the boundaries of $D(u)$ and $D'(u)$ are disjoint.
Consider now a vertex $u\in V(G)$, and let $\psi_u$ be the standard drawing of the grid $H_u$. We let $\tilde D(u)$ be a disc in the drawing $\psi_u$, such that the image of the grid $H_u$ is contained in $\tilde D(u)$, and the images of vertices $x_1(u),\ldots,x_{d_u}(u)$, that we denote by $p'_1(u),\ldots,p'_{d_u}(u)$ lie on the boundary of $\tilde D(u)$, and are encountered in this order as we traverse the boundary of $\tilde D(u)$ in the counter-clock-wise direction.
We also ensure that the only points of $\psi_u(H_u)$ that lie on the boundary of $\tilde D(u)$ are $p'_1(u),\ldots,p'_{d_u}(u)$.
In order to define a planar drawing $\psi$ of graph $G'$, we process every vertex $u\in V(G)$ one by one. Consider any such vertex $u$. If $u$ is a positive vertex, then we plant the drawing $\psi_u$ of $H_u$ inside the disc $D'(u)$ that we defined before, so that the discs $\tilde D(u)$ and $D'(u)$ coincide. Observe that, in this case, points $p_1(u),\ldots,p_{d_u}(u)$ are encountered in this order on the boundary of $D(u)$ as we traverse it in counter-clock-wise direction; and similarly, points $p'_1(u),\ldots,p'_{d_u}(u)$ are encountered in this order on the boundary of $D'(u)$ as we traverse it in counter-clock-wise direction. For all $1\leq i\leq d_u$, we can then define a curve $\gamma_i(u)$ connecting points $p_i(u)$ and $p'_i(u)$, that is contained in $D(u)$, and is internally disjoint from $D'(u)$. Moreover, we can ensure that all curves in $\set{\gamma_i(u)\mid 1\leq i\leq d_u}$ are disjoint from each other and are internally disjoint from the boundary of $D_u$.
If $u$ is a negative vertex, then we repeat the same process, except that we plant a mirror image of the drawing $\psi_u$ of $H_u$ inside the disc $D'(u)$. This allows us to define the set $\set{\gamma_i(u)\mid 1\leq i\leq d_u}$ of disjoint curves as before, where for $1\leq i\leq d_u$, curve $\gamma_i(u)$ connects points $p_i(u)$ and $p'_i(u)$, is contained in $D(u)$, and is internally disjoint from $D'(u)$.
So far, for every vertex $u\in V(G)$, we have defined the images of the vertices and the edges of the grid $H_u$ in $\psi$. In order to complete the drawing $\psi$ of graph $G'$, we process the edges $e\in E(G)$ one by one. Consider any such edge $e=(u,u')$, and assume that $e=e_i(u)=e_{j}(u')$. Note that the image $\phi(e)$ of edge $e$ contains points $p_i(u)$ and $p_j(u')$. Let $\sigma(e)$ be the segment of $\phi(e)$ between these two points. Notice that, by construction, $\sigma(e)$ is internally disjoint from all discs in $\set{D(u'')}_{u''\in V(G)}$. Recall that graph $G'$ contains an edge $e'=(x_i(u),x_j(u'))$ representing edge $e$. We let the image of edge $e'$ in $\psi$ be the concatenation of curves $\gamma_i(u)$ (that connects the image of $x_i(u)$ to point $p_i(u)$; $\sigma(e)$ (connecting $p_i(u)$ to $p_j(u)$); and $\gamma_j(u')$ (connecting $p_j(u')$ to the image of $x_j(u')$).
This completes the definition of the drawing $\psi$ of $G'$. It is immediate to verify that it is a planar drawing.
\end{proof}
\begin{observation}
\label{obs:grid_expansion_2}
There is an efficient algorithm, that, given a planar drawing $\phi'$ of graph $G'$, computes a feasible solution $\phi$ to instance $I$ of \textnormal{\textsf{MCNwRS}}\xspace with no crossings.
\end{observation}
\begin{proof}
Consider the planar drawing $\phi'$ of graph $G'$ on the sphere.
Recall that for all $r\geq 1$, the $(r\times r)$-grid graph has a unique planar drawing. Therefore, for every vertex $u\in V(G)$, the drawing of grid $H_u$ that is induced by $\phi'$ is the standard drawing $\psi_u$ of the grid. Recall that the boundary of the grid $H_u$ is a simple cycle. Let $\gamma_u$ be the closed curve, that is obtained by taking the union of the images of all edges of the boundary of $H_u$. Notice that $\gamma_u$ must be a simple curve, and, moreover, for every pair $u',u''$ of distinct vertices of $G'$, $\gamma_{u'}\cap \gamma_{u''}=\emptyset$. For a vertex $u\in V(G)$, let $D'(u)$ be the disc whose boundary is $\gamma_u$, such that the drawing of $H_u$ in $\phi'$ is contained in $D'(u)$. We denote by $p^*(u)$ the image of vertex $v_{d_u,d_u}$ of the grid $H_u$, and, for $1\le i\leq r$, by $p_i(u)$ the image of vertex $x_i(u)$. Note that points $p_1(u),\ldots,p_{d_u}(u),p^*(u)$ appear in this circular order on the boundary of $D'(u)$. Notice also that it is possible that, for a pair $u'\neq u''$ of vertices of $G$, $D'(u')\subseteq D'(u'')$.
Let $\Gamma$ denote the set of curves that contains, for every primary edge $e'$ of $G'$, its image $\phi'(e')$.
We use the following claim.
\begin{claim}\label{claim: construct curves along paths}
There is an efficient algorithm that constructs, for each vertex $u\in V(G)$ and index $1\leq i\leq d_u$, a curve $\gamma_i(u)$ that is contained in $D'(u)$ and connects $p_i(u)$ to $p^*(u)$. Moreover, for every pair $\gamma,\gamma'$ of distinct curves in set $\Gamma\cup \set{\gamma_i(u)\mid u\in V(G); 1\leq i\leq d_u}$, every point $p\in \gamma\cap \gamma'$ must be an endpoint of both curves.
\end{claim}
We prove the claim below, after we complete the proof of \Cref{obs:grid_expansion_2} using it. We define a drawing $\phi$ of graph $G$ as follows. For every vertex $u\in V(G)$, the image $\phi(u)$ is defined to be $p^*(u)$. Consider now some edge $e=(u,u')\in E(G)$, and assume that $e=e_i(u)=e_j(u')$. We then let the image of $e$ in $\phi$ be the concatenation of three curves: (i) curve $\gamma_i(u)$, connecting $p^*(u)$ to $p_i(u)$; (ii) the image of edge $e'=(v_i(u),v_j(u'))\in E(G')$ in drawing $\phi'$, that connects $p_i(u)$ to $p_j(u')$; and (iii) curve $\gamma_j(u')$, connecting $p_j(u')$ to $p^*(u')$. Notice that the resulting curve connects $\phi(u)$ to $\phi(u')$, as required. This completes the definition of the drawing $\phi$ of $G$.
We now show that this is a legal drawing, and that the number of crossings in this drawing is $0$. Indeed, assume for contradition that there are two edges $e_1,e_2\in E(G)$, and that some point $p$ lies in $\phi(e_1)\cap \phi(e_2)$. Note that the endpoints of $\phi(e_2)$ may not be inner points of $\phi(e_1)$ and vice versa. Therefore, $p$ is an inner point on both $\phi(e_1)$ and $\phi(e_2)$.
From our construction, there must be two curves $\gamma,\gamma'\in \Gamma\cup \set{\gamma_i(u)\mid u\in V(G); 1\leq i\leq d_u}$, with $\gamma\subseteq \phi(e_1)$ and $\gamma'\subseteq \phi(e_2)$ that contain $p$.
From \Cref{claim: construct curves along paths}, point $p$ must be an endpoint of both curves. Assume that $e_1=(u_1,u'_1)$, and that $e_1=e_i(u_1)=e_j(u'_1)$. Then, from our construction, $p=p_i(u_1)$ or $p=p_j(u_1')$ must hold. Similarly, assuming that $e_2=(u_2,u'_2)$, and that $e_2=e_{i'}(u_2)=e_{j'}(u'_2)$, we get that $p=p_{i'}(u_2)$ or $p=p_{j'}(u_2')$ must hold. This may only happen if two distinct primary edges of $G'$ are incident to the same vertex of $G'$, which is impossible from our construction.
We conclude that $\phi$ is a valid drawing of $G$ with $0$ crossings.
Next, we show that $\phi$ obeys the rotation system $\Sigma$. Consider some vertex $u\in V(G)$, and a tiny $u$-disc $D_{\phi}(u)$. For $1\leq i\leq d_u$, let $\tilde p_i(u)$ be the point on the boundary of $D_{\phi}(u)$ that lies on the image of edge $e_i(u)$ in $\phi$. In particular, point $\tilde p_i(u)$ belongs to the curve $\gamma_i(u)$, whose endpoints are $p_i(u),p^*(u)$. Since points $p_1(u),\ldots,p_{d_u}(u)$ appear on the boundary of disc $D'(u)$ in this circular order, and the curves $\gamma_1(u),\ldots,\gamma_{d_u}(u)$ are internally disjoint, points
$\tilde p_1(u),\ldots,\tilde p_{d_u}(u)$ must appear on the boundary of disc $ D_{\phi}(u)$ in this circular order. We conclude that drawing $\phi$ of $G$ obeys the rotation system $\Sigma$.
In order to complete the proof of \Cref{obs:grid_expansion_2}, it is now enough to prove \Cref{claim: construct curves along paths}, which we do next.
\begin{proofof}{\Cref{claim: construct curves along paths}}
Consider a vertex $u\in V(G)$.
For convenience, for $1\leq i,j\leq d_u$, we denote by $v_{i,j}(u)$ the unique vertex of the grid $H_u$ lying in the intersection of its $i$th row and $j$th column.
Let $A(u)=\set{a_1(u),\ldots,a_{d_u-1}(u)}$ be the sequence of edges on the last row of the grid $H_u$. Recall that, for $1\leq i<d_u$, curve $\phi'(a_i(u))$ is contained in the boundary of the disc $D'(u)$. We
denote $\sigma_i(u)=\phi'(a_i(u))$, and
draw another curve $\sigma'_i(u)$, whose endpoints are the same as those of $\sigma_i(u)$, such that $\sigma'_i(u)$ is contained in the interior of $D'(u)$; is internally disjoint from $\sigma_i(u)$ and the images of all edges of $G'$ in $\phi'$; and it is drawn in parallel to $\sigma_i(u)$ right next to it. Next, we let $\hat D_i(u)$ be the disc, whose boundary is the union of the curves $\sigma_i(u)$ and $\sigma_i'(u)$
(see \Cref{fig:extra_1}).
Lastly, we let $\hat D(u)\subseteq D'(u)$ be smallest disc, whose interior contains, for all $1\leq i\leq d_u-1$, the disc $\hat D_i(u)$, and, for all $1\leq i\leq d_u$, the intersection of the tiny $v_{d_u,i}(u)$-disc $D_{\phi'}(v_{d_u,i}(u))$ and the disc $D'(u)$ (see \Cref{fig:extra_2}).
\begin{figure}[h]
\centering
\subfigure[Curves in $\set{\sigma_i(u),\sigma'_i(u)}_i$, and the corresponding discs $\set{\hat D_i(u)}_i$. The boundary of disc $D'(u)$ is shown in black.]{\scalebox{0.1}{\includegraphics{figs/extra_1.jpg}}\label{fig:extra_1}
}
\hspace{0.5cm}
\subfigure[Disc $\hat D(u)$ is shown in red.]{\scalebox{0.1}{\includegraphics{figs/extra_2.jpg}}\label{fig:extra_2}}
\caption{Illustration of curves in $\set{\sigma_i(u),\sigma'_i(u)}_i$ and the corresponding discs.}
\end{figure}
From our construction, the only vertices of $G'$ whose images are contained in disc $\hat D(u)$ are the vertices lying in the last row of the grid $H_u$. The only edges of $G'$ that may have a non-empty intersection with disc $\hat D(u)$ are the edges of $H_u$ that are incident to the vertices of $H_u$ lying in the last row of the grid.
Consider now some index $1\leq i\leq d_u$. Let $\gamma'_i(u)$ be the curve obtained by concatenating the images of all edges that lie on the $i$th column of grid $H_u$. We truncate the curve $\gamma_i'(u)$, so that it terminates at a point on the boundary of disc $\hat D(u)$, and is internally disjoint from disc $\hat D(u)$. We denote by $p'_i(u)$ the point on the boundary of $\hat D(u)$ that lies on $\gamma_i'(u)$. Note that curve $\gamma_i'(u)$ connects points $p_i(u)$ and $p'_i(u)$; it is contained in disc $D'(u)$, and it is internally disjoint from disc $\hat D(u)$. It is easy to verify that, since drawing $\phi'$ of $G'$ is planar, curves $\gamma'_i(u),\ldots,\gamma'_{d_u}(u)$ are disjoint from each other. For all $1\leq i\leq d_u$, we then let $\gamma''_i(u)$ be any simple curve connecting $p_i'(u)$ to $p^*(u)$, that is contained in disc $\hat D(u)$; we construct the curves $\gamma''_1(u),\ldots,\gamma''_{d_u}(u)$ so that they are internally disjoint from each other. For all $1\leq i\leq d_u$, we then let $\gamma_i(u)$ be the curve obtained by concatenating $\gamma'_i(u)$ and $\gamma''_i(u)$. It is immediate to verify that curve $\gamma_i(u)$ is contained in $D'(u)$, and it connects $p_i(u)$ to $p^*(u)$. From our construction, it is easy to verify that, for any pair $\gamma,\gamma'$ of distinct curves in set $\Gamma\cup \set{\gamma_i(u)\mid u\in V(G); 1\leq i\leq d_u}$, every point $p\in \gamma\cap \gamma'$ must be an endpoint of both curves.
\end{proofof}
\end{proof}
We are now ready to complete the proof of Theorem~\ref{thm: crwrs_planar}. We construct the graph $G'$ as described above, and use the algorithm from \Cref{thm: testing planarity} to test whether graph $G'$ is planar, and if so, to compute a planar drawing $\phi'$ of $G'$.
If $G'$ is not planar, then we correctly establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\ne 0$, from \Cref{obs: G' planar}.
If $G'$ is planar, then we apply the algorithm from \Cref{obs:grid_expansion_2} to graph $G'$ and its planar drawing $\phi'$, to compute a valid solution $\phi$ to instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace with $\mathsf{cr}(\phi)=0$.
\iffalse
First, we process all vertices of $G$ with degree at least $4$, as follows.
Consider a vertex $v\in V(G)$ with $d=\deg_G(v)\ge 4$, and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
Since $d\ge 4$, $H_v$ is $3$-vertex-connected. Therefore, from~\cite{whitney1992congruent}, the the image of the grid $H_v$ in $\phi'$ is identical to the unique planar drawing of $H_v$ shown in~\ref{fig: grid_flip_after}. We denote by $F^{out}_v$ the face of this drawing whose boundary contains vertices $v_{1,1},\ldots,v_{1,d}$, and denote by $F^{i}_v$ the face of whose boundary contains vertices $v_{1,i},v_{1,i+1},v_{2,i},v_{2,i+1}$.
Since $\phi'$ is planar, the image of $e_i$ either lies entirely in $F^{out}$, or lies entirely in $F^{i}_v$, or lies entirely in $F^{i-1}_v$.
Now consider the drawing of $\phi'$ inside the face $F^i_v$. If the images of $e_i,e_{i+1}$ do not lie in $F^i_v$, then since $\phi'$ is planar, the interior of $F^i_v$ does not contain any image (or segments of image) of vertices and edges of $G'$.
If the image of $e_i$ lie in $F^i_v$, but the image of $e_{i+1}$ does not lie in $F^i_v$, then $G\setminus e_i$ is not connected, and the interior of $F^i_v$ contains the image of the connected component of $G\setminus e_i$ that does not contain $v$. We ``flip'' the image inside the face $F^i_v$ into the face $F^{out}_v$ as in Figure~\ref{fig: grid_flip}, such that the new image lies in an arbitrarily small disc $\eta(v_{1,i})$ of the image of $v_{1,i}$.
\begin{figure}[h]
\centering
\subfigure[Before flipping.]{\scalebox{0.36}{\includegraphics{figs/grid_flip_1.jpg}
}
\hspace{0.76cm}
\subfigure[After flipping.]{
\scalebox{0.36}{\includegraphics{figs/grid_flip_2.jpg}}
}
\caption{An illustration of flipping the image inside the face $F^i_v$ into the disc $\eta(v_{1,i})$.
}\label{fig: grid_flip}
\end{figure}
The case where the image of $e_{i+1}$ lie in $F^i_v$ but the image of $e_{i}$ does not lie in $F^i_v$ is treated similarly.
Assume now that the images of both $e_i$ and $e_{i+1}$ lie in $F^i_v$, then $G\setminus v$ is not connected, and the interior of $F^i_v$ contains the image of the connected component of $G\setminus v$ that contains the edges $e_i,e_{i+1}$. We ``flip'' the image inside the face $F^i$ into the face $F^{out}_v$ as in Figure~\ref{fig: another_grid_flip}, such that the new image lies in an arbitrarily thin strip around the image of the edge $(v_{1,i},v_{1,i+1})$.
\begin{figure}[h]
\centering
\subfigure[Before flipping.]{\scalebox{0.37}{\includegraphics{figs/grid_flip_3.jpg}
}
\hspace{0.76cm}
\subfigure[After flipping.]{
\scalebox{0.37}{\includegraphics{figs/grid_flip_4.jpg}}}
\caption{An illustration of flipping the image inside $F^i$ into the thin strip around edge $(v_{1,i},v_{1,i+1})$.
}\label{fig: another_grid_flip}
\end{figure}
After processing all vertices in this way, we obtain a planar drawing $\phi''$ of $G'$, such that, for each vertex $v$ with $\deg_G(v)\ge 4$, in the image of the grid $H_v$ in $\phi''$, only the interior of the face $F^{out} _v$ contains image of vertices and edges of $G'$. Therefore, we can contract, for each such vertex, the image of $H_v$ into a single point $y_v$. Let $\phi$ be the resulting drawing. Note that, if we view the point $y_v$ as the image of $v$, then $\phi$ is a planar drawing of $G$ and moreover, from our way of contracting the image of $H_v$, $\phi$ is a feasible solution to the instance $(G,\Sigma)$.
\end{proof}
\fi
\iffalse
Assume without loss of generality that $G$ is connected; otherwise, we process each connected component of $G$ separately.
We start by constructing a graph $G'$, that is obtained from $G$, as follows.
Initially, we set $G'=G$, and then process every vertex $v$ of $G'$ whose current degree is greater is at least $4$ one by one.
We now describe an iteration for processing a vertex $v$ of $G'$.
Denote $d=\deg_G(v)$, and let $e_1,\ldots,e_d$ be the edges in $\delta_G(v)$, indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$. For each $1\le t\le d$, denote by $u_t$ the endpoint of edge $e_t$ that is different from $v$.
We delete vertex $v$ from $G'$, and instead add a $d\times d$-grid $H_v$ graph, denoting, for all $1\leq i,j\leq d$, the vertex at the intersection of row $i$ and column $j$ by $v_{i,j}$. For all $1\leq t\leq d$, we add a new edge $(u_t,v_{1,t})$ to graph $G'$, that represents the original edge $e_t$.
Note that this transformation does not increase the degree of any vertex that lied in $G'$ before the current iteration.
See Figure~\ref{fig: grid_expansion} for an illustration of an iteration execution.
Once every vertex of $G'$ has degree at most $4$, we obtain the final graph $G'$.
\begin{figure}[h]
\centering
\subfigure[Vertex $v$ and its incident edges $e_1,\ldots, e_d$ ordered according to the ordering ${\mathcal{O}}_v$.]{\scalebox{0.4}{\includegraphics{figs/grid_expansion_1.jpg}
}
\hspace{0.1cm}
\subfigure[The grid $H_v$ and the reorganization of edges $e_1,\ldots, e_d$.]{
\scalebox{0.4}{\includegraphics{figs/grid_expansion_2.jpg}}\label{fig: grid_flip_after}}
\caption{An illustration of an iteration of processing vertex $v$ in constructing the graph $G'$. \mynote{is it possible to modify the two figures so that vertices that are different from $v$ are called $u_1,\ldots,u_d$, and the circles representing the vertices have the same outline color as the inner color? Thanks!}
}\label{fig: grid_expansion}
\end{figure}
We use the following observations.
\begin{observation}
\label{obs:grid_expansion_1}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$, then graph $G'$ is planar.
\end{observation}
\begin{proof}
Let $\phi$ be a feasible solution to instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with $\mathsf{cr}(\phi)=0$. We modify $\phi$ to construct a drawing of graph $G'$ with no crossings. The modification follows the process that we used in order to modify graph $G$ to obtain graph $G'$.
Specifically, we let $H=G$, and then process every vertex of $H$ whose degree is greater than $4$ one by one. Consider an iteration where some vertex $v$ is processed, and denote by $e_1,\ldots,e_d$ the set of edges that are incident to vertex $v$ in graph $G$,
Consider a vertex $v\in V(G)$ with $d=\deg_G(v)\ge 4$, and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
Since $\deg_G(v)\ge 4$, $H_v$ is $3$-vertex-connected. Since the grid $H_v$ is a planar graph, and it was proved by Whitney in~\cite{whitney1992congruent} that every $3$-vertex-connected graph has a unique planar drawing, we know that $H_v$ has a unique planar drawing, which is shown in Figure~\ref{fig: grid_flip_after}.
Let $\eta(v)$ be an arbitrarily small disc around the image $\phi(v)$ of $v$ in $\phi$, such that the boundary of $\eta(v)$ only intersects with the image of edges $e_1,e_2\ldots,e_d$, and the intersections are denoted by $x_1,\ldots,x_d$, respectively.
Note that, since $\phi$ is a feasible solution to the instance $(G,\Sigma)$, the intersections $x_1,\ldots,x_d$ appear on the boundary of $\eta(v)$ in this order.
We erase the drawing of $\phi$ inside the disc $\eta(v)$, and put the unique planar drawing of the grid $H_v$ inside $\eta(v)$, with vertices $v_{1,1},\ldots,v_{1,d}$ appearing on the outer face.
We then connect, for each $1\le i\le d$, the image of vertex $v_{1,i}$ to the intersection $x_i$ via a simple curve in $\eta(v)$, such that all these curves are mutually disjoint and internally disjoint from the image of $H_v$. It is clear that the drawing obtained this way is a planar drawing of $G'$.
\end{proof}
\fi
\subsection{Proof of Theorem~\ref{thm: crwrs_uncrossing}}
\label{apd: Proof of crwrs_uncrossing}
We start with any feasible solution $\phi$ to instance $I$, and then gradually modify it to ensure that every pair of edges in $G$ cross at most once.
As long as there is a pair $e,e'\in E(G)$ of distinct edges, whose images cross at least twice in ${\phi}$, we perform the following modification step. Let $p,q$ be two crossing points between curves $\phi(e),\phi(e')$, that appear consecutively on $\phi(e)$; in other words, the segment of $\phi(e)$ between $a$ and $b$ contains no other point that lies on $\phi(e')$. We ``uncross'' the images of edges $e$ and $e'$, as shown in Figure~\ref{fig:uncross-curves}.
(In Sections \ref{subsec: uncrossing type 1} and \ref{apd: type-1 uncrossing} we provide a more formal description of this uncrossing process, that we refer to as \emph{type-1 uncrossing}).
It is easy to see that, after this uncrossing step, the new drawing remains a feasible solution to instance $I$, and the number of crossings in the drawing decreases by at least $2$.
We continue this process until every pair of edges of $G$ cross at most once in $\phi$. It is clear that the resulting drawing contains at most $|E(G)|^2$ crossings.
\begin{figure}[h]
\centering
\subfigure[Before: Curves $\phi(e)$ (red) and $\phi(e')$ (blue) cross at $p$ and $q$.]{\scalebox{0.12}{\includegraphics{figs/uncross-1.jpg}
}
\hspace{1.5cm}
\subfigure[After: The modified curves no longer cross at $p$ or at $q$.]{
\scalebox{0.12}{\includegraphics{figs/uncross-2.jpg}}}
\caption{Uncrossing two curves.
}\label{fig:uncross-curves}
\end{figure}
\iffalse
We start with any feasible solution $\phi$ of the instance $(G,\Sigma)$. Let $\Gamma_1$ be the set of curves in the drawing $\hat\phi$. We apply the algorithm of Theorem~\ref{thm: type-1 uncrossing} to the set $\Gamma_1$ of curves, with $\Gamma_2=\emptyset$. Let $\Gamma'_1$ be the set of curves that we obtain. From Theorem~\ref{thm: type-1 uncrossing}, it is clear that the curves in $\Gamma'_1$ form a drawing $\phi'$ of $G$ that respects the rotation system $\Sigma$. Moreover, in the drawing $\phi'$, any pair of curves cross at most once. Therefore, $\mathsf{cr}(\phi')\le |E(G)|^2$.
\fi
\section{Proofs Omitted from \Cref{sec: main disengagement}}
\subsection{Proof of \Cref{claim: path length in decomposition tree}}
\label{subsec: bounding tree height}
Consider any cluster $R\in {\mathcal{L}}$. Let $N(R)$ denote the number of clusters $C\in {\mathcal{C}}$ with $C\subseteq R$. Clearly, $N(R)\leq |{\mathcal{C}}|\leq m$ must hold.
Assume first that $R\neq G$, and vertex $v(R)$ is unmarked in the tree $\tau({\mathcal{L}})$. Let $R'$ be the parent-cluster of $R$, that is $v(R')$ is the parent vertex of $v(R)$ in $\tau({\mathcal{L}})$. In this case, the algorithm from \Cref{thm: construct one level of laminar family}, when applied to the graph $G'$ corresponding to cluster $R'$, returned a type-2 legal clustering ${\mathcal{R}}$ of $G'$, with $R\in {\mathcal{R}}$, and moreover, $R$ is not the distinguished cluster $R^*$. Let ${\mathcal{C}}'$ denote the set of all clusters $C\in {\mathcal{C}}$ with $C\subseteq R'$, so that $N(R')=|{\mathcal{C}}'|$. Recall that, from the definition of type-2 legal clustering, the distinguished cluster $R^*$ must contain at least $\floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )|{\mathcal{C}}'|}$ clusters of ${\mathcal{C}}'$. Therefore, if $N(R')\geq 2^{(\log m)^{3/4}}$, then $R$ may contain at most $2N(R')/2^{(\log m)^{3/4}}$ clusters of ${\mathcal{C}}'$, that is, $N(R)\leq 2N(R')/2^{(\log m)^{3/4}}$. Otherwise, $N(R)\leq 1$ must hold. Consider now any root-to-leaf path $P$ in three $\tau({\mathcal{L}})$, and assume that $R_1,R_2,\ldots,R_z$ is the sequence of unmarked clusters whose corresponding vertices appear on the path in this order. Then, for all $1\leq i<z$, $N(R_{i+1})\leq \ceil{2N(R_i)/2^{(\log m)^{3/4}}}$, and so $z\leq O\textsf{left} (\log^{3/4}m\textsf{right} )$ must hold.
Next, we consider a cluster $R\in {\mathcal{L}}\setminus\set{G}$, whose corresponding vertex $v(R)$ in tree $\tau({\mathcal{L}})$ is marked. Let $R'$ be the parent-cluster of $R$. Note that two cases are possible. The first case is that the algorithm from \Cref{thm: construct one level of laminar family} was applied to the graph corresponding to $R'$, and it returned a type-1 legal clustering ${\mathcal{R}}$ with $R\in {\mathcal{R}}$. In this case, the theorem guarantees that $N(R)\leq $\floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )N(R')}$ $. In the second case, there is a parent-cluster $R''$ of cluster $R'$, to which the algorithm from \Cref{thm: construct one level of laminar family} was applied, and it returned a type-2 legal clustering ${\mathcal{R}}$, with $R'\in {\mathcal{R}}$, such that $R'=R^*$ is the distinguished cluster of the decomposition, and cluster $R$ lies in the type-1 legal clustering ${\mathcal{R}}'$ of the graph corresponding to cluster $R'$. In this latter case, from the definition of type-2 legal clustering, we are guaranteed that $N(R)\leq \floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )N(R'')}$. Therefore, if we consider any root-to-leaf path $P$ in the tree $\tau({\mathcal{L}})$, and we let $R_1',R_2',\ldots, R_{y}'$ be the sequence of marked clusters whose corresponding vertices lie on $P$ in this order, then for all $1< i<\floor{y/2}$, $N(R'_{2i})\leq \textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )N(R'_{2i-2})$. Since $N(G)\leq m$, we get that $y\leq O\textsf{left}(2^{(\log m)^{3/4}}\cdot \log m\textsf{right} )\leq 2^{O((\log m)^{3/4})}$.
\subsection{Proof of \Cref{claim: compose distributions}}
\label{subsec: external routers}
The proof is by induction on the distance from $v(R)$ to the root of the tree $\tau({\mathcal{L}})$. Recall that, for $R=G$, we set ${\mathcal{D}}''(R)$ to assign probability $1$ to an empty set of paths.
If $v(R)$ is the child vertex of $v(G)$, then consider the graph $G'$, that is obtained from $G$ by adding a new special vertex $v^*$ to it, that connects with an edge to some arbitrary vertex $v_0$. Recall that, when we applied the algorithm from \Cref{thm: construct one level of laminar family} to this graph $G'$, it computed a legal clustering ${\mathcal{R}}$ of $G'$, with $R\in {\mathcal{R}}$, together with a distribution ${\mathcal{D}}'(R)$ over the sets of paths in $\Lambda'_{G'}(R)$, such that, for every edge $e\in E(G')\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}'(R)]{\cong_{G'}({\mathcal{Q}}'(R),e)}\leq \beta$.
Consider any external router ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$ that is assigned a non-zero probabilty. Let $u$ be the vertex at which every path of ${\mathcal{Q}}'(R)$ terminates. If $u\neq v^*$, then, since vertex $v^*$ has degree $1$ in $G'$, no path in ${\mathcal{Q}}'(R)$ contains the vertex $v^*$, and so the paths of ${\mathcal{Q}}'(R)$ lie in $G$. Otherwise, by removing the last vertex from each path in ${\mathcal{Q}}'(R)$, we obtain a new set ${\mathcal{Q}}''(R)$ of paths in $\Lambda'_G(R)$, such that, for every edge $e\in E(G)$, $\cong_G({\mathcal{Q}}''(R),e)\leq \cong_G({\mathcal{Q}}'(R),e)$. Therefore, we can transform ${\mathcal{D}}'(R)$ into a distribution ${\mathcal{D}}''(R)$ over the family $\Lambda'_{G}(R)$ of external $R$-routers, such that, for every edge $e\in E(G)\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}''(R)]{\cong_{G}({\mathcal{Q}}'(R),e)}\leq \beta$.
Next, we consider any cluster $R\in {\mathcal{L}}$, such that the distance from $v(R)$ to $v(G)$ in tree $\tau({\mathcal{L}})$ is greater than $1$. Let $R'$ be the parent-cluster of $R$, and let $G'$ be the graph obtained from $G$ by contracting all vertices of $V(G)\setminus V(R')$ into a special vertex $v^*$.
Recall that, from \Cref{thm: construct one level of laminar family}, we have obtained a distribution ${\mathcal{D}}'(R)$ over external routers in $\Lambda'_{G'}(R)$, such that, for every edge $e\in E(G')\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}'(R)]{\cong_{G'}({\mathcal{Q}}'(R),e)}\leq \beta$. Recall that, if $v(R)$ is a marked vertex, every router ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$, to which ${\mathcal{D}}'(R)$ assigns a non-zero probability, is careful with respect to $v^*$, that is, the paths in ${\mathcal{Q}}'(R)$ cause congestion at most $1$ on every edge $e\in \delta_{G'}(v^*)$.
Let $i$ denote the number of unmarked vertices on the path connecting $v(R')$ to the root of $\tau({\mathcal{L}})$. From the induction hypothesis, we have computed a distribution ${\mathcal{D}}''(R')$ over the external routers in $\Lambda'_G(R')$, such that, for every edge $e\in E(G)\setminus E(R')$, $\expect[{\mathcal{Q}}'(R')\sim{\mathcal{D}}''(R')]{\cong_{G}({\mathcal{Q}}'(R'),e)}\leq \beta^{i+1}$.
We now compute the desired distribution a distribution ${\mathcal{D}}''(R)$ over the external routers in $\Lambda'_G(R)$, such that, for every edge $e\in E(G)\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}''(R)]{\cong_{G}({\mathcal{Q}}'(R),e)}\leq \beta^{j+1}$, where $j=i$ if vertex $v(R)$ is marked, and $j=i+1$ otherwise. We provide the distribution implicitly, by providing an efficient algorithm for drawing a set $\tilde {\mathcal{Q}}'(R)$ of paths from the distribution. The algorithm for drawing an external router from the distribution ${\mathcal{D}}''(R)$ proceeds as follows.
First, the algorithm draws an external router ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$ for cluster $R$ in graph $G'$. If the paths in ${\mathcal{Q}}'(R)$ do not contain the vertex $v^*$, then this is the set of paths that we return. We now assume that at least one path in set ${\mathcal{Q}}'(R)$ contains vertex $v^*$. We denote by $u'$ the vertex of $G'$ that serves as the last vertex on every path in ${\mathcal{Q}}'(R)$.
Next, the algorithm draws a router ${\mathcal{Q}}'(R')\in \Lambda'_{G}(R')$ from the distribution ${\mathcal{D}}''(R')$ that we have constructed by the induction hypothesis. We denote by $u''$ the vertex of $G$ that serves as the last vertex on every path in ${\mathcal{Q}}'(R')$. The final set $\tilde {\mathcal{Q}}'(R)$ of paths that the algorithm returns is constructed by combining the sets ${\mathcal{Q}}'(R)$ and ${\mathcal{Q}}'(R')$ of paths, as follows.
We consider two cases. The first case is when $u'=v^*$. In this case, for every edge $e\in \delta_{G'}(R)=\delta_G(R)$, the unique path $Q(e)\in {\mathcal{Q}}'(R)$ that as $e$ as its first edge terminates at vertex $v^*$. We denote by $e'$ the last edge on path $Q(e)$. Note that edge $e'$, that is incident to vertex $v^*$ in graph $G'$, corresponds to an edge of $\delta_G(R')$ in graph $G$; we do not distinguish between the two edges. Therefore, there is some path $Q'(e)\in {\mathcal{Q}}'(R')$, whose first edge is $e'$, and last vertex is $u''$. By concatenating the paths $Q(e)$ and $Q'(e)$, we obtain path $Q^*(e)$ in graph $G$, connecting edge $e$ to vertex $u''$. We then let $\tilde {\mathcal{Q}}'(R)=\set{Q^*(e)\mid e\in \delta_G(R)}$ be the final set of paths that the algorithm outputs.
The second case is when $u'\neq v^*$. In this case, some paths in ${\mathcal{Q}}'(R)$ may contain vertex $v^*$ as an inner vertex. Consider any path $Q\in {\mathcal{Q}}'(R)$ that contains the vertex $v^*$, and let $e\in \delta_{G'}(v^*)$ be any edge that is incident to $v^*$ that the path contains. Recall that $e\in \delta_G(R)$, and so there is some path $Q'(e)\in {\mathcal{Q}}'(R')$, whose first edge is $e$, and last vertex is $u''$. We replace edge $e$ with path $Q'(e)$ on path $Q$. Note that originally path $Q$ must have contained two edges that are incident to $v^*$; denote them by $e$ and $e'$. We have replaced edge $e$ with a path connecting $e$ to vertex $u''$ in graph $G$, and we replace edge $e'$ with a path connecting $e'$ to $u''$ in graph $G$, but we reverse the direction of the path. In this way, we can glue the two paths to each other via the vertex $u''$. Once every path of ${\mathcal{Q}}'(R)$ containing $v^*$ is processed in this manner, we obtain the final set $\tilde {\mathcal{Q}}'(R)$ of paths in graph $G$.
This completes the definition of the distribution ${\mathcal{D}}''(R)$ over the set $\Lambda'_G(R)$ of external routers for $R$ in $G$. It now remains to analyze the expected congestion on each edge of $E(G)\setminus E(R)$. Fix any edge $e\in E(G)\setminus E(R)$. First, if edge $e$ lies in graph $R'\cup \delta_G(R')$, then $\cong_{G}(\tilde {\mathcal{Q}}'(R),e)=\cong_{G'}({\mathcal{Q}}'(R),e)$, and so, from the definition of helpful clustering, $\expect[\tilde {\mathcal{Q}}'(R)\sim{\mathcal{D}}''(R)]{\cong_{G}(\tilde {\mathcal{Q}}'(R),e)}\leq \expect[ {\mathcal{Q}}'(R)\sim{\mathcal{D}}'(R)]{\cong_{G'}( {\mathcal{Q}}'(R),e)} \leq \beta$.
Assume now that $e\in E(G)\setminus (E(R')\cup \delta_G(R'))$. Note that, if the set ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$ that was drawn from distribution ${\mathcal{D}}'(R)$ is careful with respect to vertex $v^*$, then every edge of $\delta_{G'}(v^*)$ may lie on at most one path of ${\mathcal{Q}}'(R)\in \Lambda'_{G'}(R)$, and so every path in the set ${\mathcal{Q}}'(R')$ that was drawn from distribution ${\mathcal{D}}''(R')\in \Lambda'_G(R')$ is used by at most one path in $\tilde {\mathcal{Q}}'(R)$. Therefore:
$\expect[\tilde {\mathcal{Q}}'(R)\sim{\mathcal{D}}''(R)]{\cong_{G}(\tilde {\mathcal{Q}}'(R),e)}\leq \expect[ {\mathcal{Q}}'(R')\sim{\mathcal{D}}''(R')]{\cong_{G}( {\mathcal{Q}}'(R'),e)} \leq \beta^{i+1}$ in this case.
Recall that, if vertex $v(R)$ is marked, then every router ${\mathcal{Q}}'(R)$ that has non-zero probability to be drawn from distribution ${\mathcal{D}}'(R)$ is careful with respect to $v^*$, while the number of unmarked vertices on the unique path connecting $v(R)$ to $v(G)$ in tree $\tau({\mathcal{L}})$ is $i$.
It remains to consider the case where vertex $v(R)$ is unmarked.
Consider the following two-step process for drawing a router $\tilde {\mathcal{Q}}'(R)\in \Lambda_G(R)$ from the distribution ${\mathcal{D}}''(R)$, that is equivalent to the one described above. In the first step, we select a router ${\mathcal{Q}}'(R')\in \Lambda_G(R')$ from the distribution ${\mathcal{D}}''(R')$. Then, in the second step, we select a router ${\mathcal{Q}}'(R)\in \Lambda_{G'}(R)$ from distribution ${\mathcal{D}}'(R)$. Lastly, composing the two sets of paths as described above, we obain the final router $\tilde {\mathcal{Q}}'(R)$.
Fix an edge $e\in E(G)\setminus (E(R')\cup \delta_G(R'))$, and assume that the set of paths ${\mathcal{Q}}'(R')\in \Lambda_G(R')$ that was chosen from the distribution ${\mathcal{D}}''(R')$ causes congestion $z$ on edge $e$. We denote ${\mathcal{Q}}'(R')=\set{Q(e')\mid e'\in \delta_G(R')}$, where path $Q(e')$ originates at edge $e'$ and terminates at vertex $u''$. Let $E'\subseteq \delta_G(R')$ be the set of all edges $e'$, whose corresponding path $Q(e')$ contains the edge $e$, so $|E'|=z$. Denoting $E'=\set{e_1,\ldots,e_z}$, and assuming that the path set ${\mathcal{Q}}'(R')$ is fixed, we can now write:
\[
\expect[{\mathcal{Q}}'(R)\sim {\mathcal{D}}'(R)]{\cong_{G}(\tilde {\mathcal{Q}}'(R),e)}=\expect[{\mathcal{Q}}'(R)\sim {\mathcal{D}}'(R)]{\sum_{i=1}^z\cong_{G'}({\mathcal{Q}}'(R),e_i)} \leq \beta z.
\]
Recall that $z$ is the congestion caused by the set ${\mathcal{Q}}'(R')$ of paths on edge $e$. Therefore, overall:
\[\begin{split}
\expect[\tilde {\mathcal{Q}}'(R) \sim {\mathcal{D}}''(R)]{\cong_G(\tilde {\mathcal{Q}}'(R),e)}
&\leq \beta\cdot \expect[{\mathcal{Q}}'(R')\sim {\mathcal{D}}''(R')]{\cong_G({\mathcal{Q}}'(R'),e)}\leq \beta^{i+2},
\end{split}
\]
from the induction hypothesis. Since, in the case that $v(R)$ is unmarked, the number of unmarked vertices on the path connecting $v(R)$ to $v(G)$ in tree $\tau({\mathcal{L}})$ is $i+1$, this completes the proof of the claim.
\subsection{Proof of \Cref{claim: few edges}}
\label{subsec:appx few edges}
Consider some cluster $R\in {\mathcal{L}}$. Let ${\mathcal{R}}$ be the set of child-clusters of $R$. Consider the graph $\tilde R=R\setminus\textsf{left}(\bigcup_{R'\in {\mathcal{R}}}R'\textsf{right} )$. Note that, from the definition of the laminar family, every edge of $E(G)$ may lie in at most one graph in the collection $\set{\tilde R\mid R\in {\mathcal{L}}}$.
Observe that collection ${\mathcal{I}}_1$ of instances can be defined as ${\mathcal{I}}_1=\set{I(R)\mid R\in {\mathcal{L}}}$, where $I(R)=(G(R),\Sigma(R))$ is the instance associated with cluster $R$. Graph $G(R)$ is obtained from graph $G$, by first contracting the vertices of $V(G)\setminus V(R)$ into a supernode $v^*$, and then contracting each child cluster $R'$ of $R$ into a supernode $v(R')$. We partition the set of edges of $G(R)$ into two subsets: the first subset, that we call \emph{internal edges}, and denote by $E_1(R)$, is the edge set $ E(\tilde R)$. The second subset, that we call \emph{external edges}, and denote by $E_2(R)$, is the set of all edges that are incident to the supernodes of $G(R)$. From the above discussion, $\sum_{R\in {\mathcal{L}}}|E_1(R)|\leq |E(G)|$, as every edge may serve as an internal edge for at most one graph $G(R)$. It now remains to bound the total number of external edges in all graphs in $\set{G(R)\mid R\in {\mathcal{L}}}$.
For all $1\leq i\leq \mathsf{dep}({\mathcal{L}})$, we denote by ${\mathcal{L}}_i\subseteq {\mathcal{L}}$ the set of all clusters $R\in {\mathcal{L}}$, such that vertex $v(R)$ lies at distance exactly $i$ from the root of the tree $\tau({\mathcal{L}})$. Note that for each cluster $R\in {\mathcal{L}}_i$, every external edge $e\in E_2(R)$ corresponds to some edge of the original graph $G$ that has at least one endpoint in cluster $R$. Moreover, since every basic cluster $C\in {\mathcal{C}}$ is either contained in $R$ or is disjoint from $R$, each such edge must lie in $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$. Since the clusters in set ${\mathcal{L}}_i$ are disjoint from each other, we get that $\sum_{R\in {\mathcal{L}}_i}|E_2(R)|\leq \sum_{C\in {\mathcal{C}}}|\delta_G(C)|\leq |E(G)|/\mu^{0.1}$, from the statement of \Cref{thm: advanced disengagement get nice instances}.
Since, from \Cref{claim: path length in decomposition tree}, $\mathsf{dep}({\mathcal{L}})\leq 2^{O((\log m)^{3/4})}$, while
$\mu\geq 2^{c^*(\log m )^{7/8}\log\log m}$ for a large enough constant $c^*$, we get that, overall:
\[\sum_{R\in {\mathcal{L}}}|E_2(R)|\leq \mathsf{dep}({\mathcal{L}})\cdot \frac{|E(G)|}{\mu^{0.1}}\leq \frac{2^{O((\log m)^{3/4})}\cdot |E(G)|}{2^{0.1\cdot c^*(\log m)^{7/8}\log\log m}}\leq |E(G)|.
\]
\subsection{Proof of \Cref{obs: subtree to cluster}}
\label{appx: subtree to cluster}
Let $u'$ be the parent-vertex of $u$ in $\tau$. Denote $U=V(\tau_u)$, and $U'=V(\hat H)\setminus U$. Recall that we have denoted by $S$ the cluster of $H$ that is defined by vertex set $U\subseteq V(\hat H)$.
Let ${\mathcal{R}}'\subseteq {\mathcal{R}}$ contain all clusters $R$ with $v_R\in U$. Notice that, from \Cref{thm: GH tree properties}, $(U,U')$ is the minimum cut in graph $\hat H$ separating $u$ from $u'$. Let $E'=E_{\hat H}(U,U')$. Observe that, equivalently, $E'=\delta_H(S)$. From the properties of minimum cut, there is a set ${\mathcal{P}}$ of edge-disjoint paths in graph $\hat H$, routing the edges of $E'=\delta_{\hat H}(U)$ to vertex $u$, such that all internal vertices on every path of ${\mathcal{P}}$ lie in $U$. Similarly, there is a set ${\mathcal{P}}'$ of edge-disjoint paths in graph $\hat H$, routing the edges of $E'=\delta_{\hat H}(U')$ to vertex $u'$, such that all internal vertices on every path of ${\mathcal{P}}'$ lie in $U'$.
The existence of the set ${\mathcal{P}}$ of paths in $\hat H[U]$ immediately implies that cluster $\hat H[U]$ has the $1$-bandwidth property in graph $\hat H$. Since graph $\hat H[U]$ is precisely the contracted graph of $S$ with respect to cluster set ${\mathcal{R}}'$, that is, $\hat H[U]=S_{|{\mathcal{R}}'}$, and since every cluster in ${\mathcal{R}}'$ has the $\alpha$-bandwidth property, from \Cref{cor: contracted_graph_well_linkedness}, cluster $S$ has the $\alpha$-bandwidth property in graph $H$.
Next, we show an algorithm to construct the desired distribution ${\mathcal{D}}'(S)$ over the external routers in $\Lambda'_{H}(S)$. We start with the set ${\mathcal{P}}'$ of paths routing the edges of $E'$ to vertex $u'$ in graph $\hat H[U']$.
Assume first that $u'$ is not a supernode.
Let $\hat H'$ be the graph obtained as follows: we first subdivide every edge $e\in E'$ with a terminal vertex $t_e$, and we let $T=\set{t_e\mid e\in E'}$ be the resulting set of terminals. We then let $\hat H'$ be the subgraph of the resulting graph induced by vertex set $U'\cup T$. Let ${\mathcal{R}}''\subseteq {\mathcal{R}}$ be the set of all clusters $R$ with $v_R\in U'$.
We apply the algorithm from \Cref{claim: routing in contracted graph} to graph $\hat H'$, cluster set ${\mathcal{R}}''$, and set ${\mathcal{P}}'$ of paths, to obtain a set ${\mathcal{P}}''$ of paths in graph $H$, such that, for every edge $e\in \bigcup_{R\in {\mathcal{R}}''}E(R)$, the paths of ${\mathcal{P}}''$ cause congestion at most $ \ceil{1/\alpha}$, and for every edge $e\in E(H\setminus U)\setminus \textsf{left}(\bigcup_{R\in {\mathcal{R}}''}E(R)\textsf{right} )$, the paths of ${\mathcal{P}}''$ cause congestion at most $1$.
We are also guaranteed that the paths in ${\mathcal{P}}''$ route the edges of $\delta_H(S)$ to vertex $u'$, and all internal vertices on every path in ${\mathcal{P}}''$ are disjoint from $U$. Lastly, since the edges incident to the special vertex $v^*$ do not lie in the clusters of ${\mathcal{R}}''$, the set ${\mathcal{P}}''$ of paths is careful with respect to $v^*$. From the above discussion, ${\mathcal{P}}''\in \Lambda'_H(S)$. We then let distribution ${\mathcal{D}}'(S)$ assign probability $1$ to the set ${\mathcal{P}}''$ of paths.
Assume now that vertex $u'$ is a supernode, and $u'=v_R$ for some cluster $R\in {\mathcal{R}}$. We repeat the algorithm from above, except that, when we apply the algorithm from \Cref{claim: routing in contracted graph} to graph $\hat H'$, we use cluster set ${\mathcal{R}}''\setminus \set{R}$ instead of ${\mathcal{R}}''$. The resulting set of paths ${\mathcal{P}}''$ then routes the edges of $E'$ to the edges of $\delta_G(R)$, and they remain internally disjoint from cluster $S$. As before, the set ${\mathcal{P}}''$ of edges is careful with respect to $v^*$, and it causes edge-congestion at most $\ceil{1/\alpha}$. Moreover, every edge of $\delta_G(R)$ participates in at most one path in ${\mathcal{P}}''$. We then use the algorithm from \Cref{lem: simple guiding paths} in order to compute a distribution ${\mathcal{D}}(R)$ over the internal $R$-routers in $\Lambda_H(R)$, such that, for every edge $e\in E(R)$, $\expect[{\mathcal{Q}}\in {\mathcal{D}}(R)]{\cong({\mathcal{Q}},e)}\leq O(\log^4m/\alpha)$ (in order to use the lemma we subdivide every edge in $\delta_G(R)$ with a terminal, and apply the lemma to the augmented cluster $R^+$ together with the resulting set of terminals). In order to define the distribution ${\mathcal{D}}'(S)$, we first select a set ${\mathcal{Q}}\in \Lambda_G(R)$ of paths from the distribution ${\mathcal{D}}(R)$, and then concatenate the paths in ${\mathcal{P}}''$ with the paths in ${\mathcal{Q}}$. It is immediate to verify that the resulting distribution ${\mathcal{D}}'(S)$ is supported over the external $S$-routers in $\Lambda'_H(S)$, which are careful with respect to $v^*$, and that $\expect[{\mathcal{Q}}'(S)\sim{\mathcal{D}}'(S)]{\cong_{H}({\mathcal{Q}}'(S),e)}\leq O(\log^4m/\alpha)$.
\subsection{Proof of \Cref{obs:J wl}}
\label{subsec:J-clusters well-linked}
Consider some cluster $J\in {\mathcal{J}}$, and let $u_0$ be the center node of cluster $J$. It is enough to show that there is a set ${\mathcal{P}}$ of paths (internal $J$-router) in graph $\hat H'$, routing all edges of $\delta_{\hat H'}(J)$ to vertex $u_0$, so that every inner vertex on every path in ${\mathcal{P}}$ lies in $J$, and the congestion of ${\mathcal{P}}$ is at most $O(\log m)$.
Let ${\mathcal{P}}^*$ be the set of all paths $P$ in graph $\hat H'$, such that the first edge on $P$ lies in $\delta_{\hat H'}(J)$, the last vertex of $P$ is $u_0$, and all inner vertices of $P$ lie in $J$.
Recall that a flow $f$ defined over a set ${\mathcal{P}}'$ of (directed) paths is an assigment of a flow value $f(P)$ to every path $P\in {\mathcal{P}}'$. Given such a flow $f$, we say that an edge $e$ sends one flow unit iff the total amount of flow $f(P)$, for all paths $P\in {\mathcal{P}}'$ that originate at edge $e$, is $1$. The congestion caused by flow $f$ is the maximum, over all edges $e'$, of $\sum_{\stackrel{P\in {\mathcal{P}}':}{e\in P}}f(P)$.
We show below that there is a flow $f$, defined over the set ${\mathcal{P}}^*$ of paths, in which every edge $e\in \delta_{H'}(J)$ sends one flow unit, and the flow causes congestion at most $O(\log m)$. From the integrality of flow, it then follows that that there is a set ${\mathcal{P}}$ of paths routing the edges of $\delta_{\hat H'}(J)$ to vertex $u_0$, so that for every path of ${\mathcal{P}}$, every inner vertex on the path lies in $J$, and the congestion of ${\mathcal{P}}$ is at most $O(\log m)$, and so cluster $J$ has $\Omega(1/\log m)$-bandwidth property. From now on we focus on defining the flow $f$.
For $1\leq i\leq h$, let $L'_i=L_i\cap V(J)$, and let $J_i$ be the subgraph of $J$ induced by vertex set $\set{u_0}\cup L'_1\cup\cdots\cup L'_{i}$. We also let $J_0$ be the graph that consists of a single vertex -- vertex $u_0$. For all $0\leq i\leq h$, we denote $E_i=\delta_{\hat H'}(J_i)$, and we denote by ${\mathcal{P}}^*_i$ the set of all paths $P$ in graph $\hat H'$, such that the first edge of $P$ lies in $E_i$, the last vertex of $P$ is $u_0$, and all inner vertices of $P$ are contained in $J_i$.
Additionally, we let $\tilde E_i\subseteq E_i$ be the set of all the edges $e\in E_i$, such that, for some vertex $v\in V(J_i)\setminus\set{u_0}$, $e\in \delta^{\operatorname{up}}(v)$. We let $\tilde {\mathcal{P}}^*_i\subseteq {\mathcal{P}}^*_i$ be the set of all paths whose first edge lies in $\tilde E_i$. Note that any flow $f_i$ defined over the set ${\mathcal{P}}^*_i$ of paths immediately defines a flow $f'_i$ over the set $\tilde {\mathcal{P}}^*_i$ of paths, by setting, for every path $P\in \tilde {\mathcal{P}}^*_i$, $f'(P)=f(P)$, and setting the flow on all other paths to $0$. We call $f'_i$ the \emph{restriction of flow $f_i$ to the set $\tilde {\mathcal{P}}^*_i$ of paths}.
We prove the following claim.
\begin{claim}\label{claim: route level by level}
For all $1\leq i\leq h$, there is a flow $f_i$, defined over the set ${\mathcal{P}}^*_i$ of paths, in which every edge of $E_i$ sends one flow unit, and the total congestion is bounded by $2^{512}\cdot i$. Moreover, if we let $f'_i$ be the restriction of $f_i$ to the set $\tilde {\mathcal{P}}^*_i$ of paths, then the congestion caused by $f'_i$ is at most $\textsf{left} (1+\frac{256}{\log m}\textsf{right})^i$.
\end{claim}
Notice that the proof of \Cref{obs:J wl} immediately follows from \Cref{claim: route level by level}, since $J_h=J$, and flow $f_h$ defines the desired flow in graph $J$, that causes congestion at most $O(h)\leq O(\log m)$. It now remains to prove \Cref{claim: route level by level}.
\begin{proofof}{\Cref{claim: route level by level}}
The proof is by induction on $i$. The base is when $i=0$. In this case, $\delta_{\hat H'}(J_0)=\delta_{\hat H'}(u_0)$. The set ${\mathcal{P}}^*_0$ of paths contains, for every edge $e\in \delta_{\hat H'}(u_0)$, a path $P(e)$ that only consists of the edge $e$ itself. We obtain flow $f_0$ by sending one flow unit on each such path $P(e)$. Note that $\tilde E_0=\emptyset$ in this case, and the resulting flow has congestion $1$.
Assume now that the claim holds for some $0\leq i< h$. We now prove it for $i+1$.
We partition the set $E_{i+1}$ of edges into two subsets. The first subset, $E'_{i+1}$ is $E_{i+1}\cap E_i$. The second subset, $E''_{i+1}$ contains all remaining edges of $E_{i+1}$. It is easy to verify that, for every edge $e\in E''_{i+1}$, there is some vertex $v\in L'_{i+1}$, with $e\in \delta_{\hat H'}(v)\setminus E_i$.
For every edge $e\in E'_{i+1}$, the flow on the paths that originate from $e$ remains unchanged from $f_i$. In other words, for every path $P\in {\mathcal{P}}^*_i$ that starts with edge $e\in E'_{i+1}$ (and hence $P\in {\mathcal{P}}^*_{i+1}$), we set $f_{i+1}(P)=f_i(P)$.
Consider now some vertex $v\in L'_i$. Note that, when vertex $v$ was added to cluster $J$, the number of edges connecting $v$ to vertices that belonged to $J$ at that time was at least $|\delta_{\hat H'}(v)|/128$. From the definition of layered well-linked decomposition, $|\delta^{\operatorname{up}}(v)|<|\delta^{\operatorname{down}}(v)|/\log m\leq |\delta_{\hat H'}(v)|/\log m$. Therefore, at the time when $v$ was added to $J$, there were at least $|\delta_{\hat H'}(v)|/256$ edges in $\delta^{\operatorname{down}}(v)$, that connected $v$ to the vertices of $J$. It is immediate to verify that each such edge must lie in $E_i$, and moreover, it must lie in $\tilde E_i$. To conclude, there is a set $E'(v)\subseteq \delta^{\operatorname{down}}(v)$ of at least $|\delta_{\hat H'}(v)|/256$ edges, that lie in $\tilde E_i$. Note that none of these edges may lie in $E_{i+1}$. We now define the flow $f_{i+1}$ that originates at the edges of $\delta_{\hat H'}(v)\cap E''_{i+1}$.
Every edge $e\in \delta_{\hat H'}(v)\setminus E'(v)$ that lies in $E''_{i+1}$, spreads one unit of flow evenly among the edges of $E'(v)$, and then uses the flow that each of these edges sends in $f_i$, in order to reach $u_0$. In other words, for every edge $e\in \delta_{\hat H'}(v)\cap E_{i+1}''$, for every edge $e'\in E'(v)$, and for every path $P\in {\mathcal{P}}^*_i$ whose first edge is $e'$, we consider the path $P'\in {\mathcal{P}}^*_{i+1}$, that is obtained by appending the edge $e$ at the beginning of path $P$, and we set $f_{i+1}(P')=f_i(P)/|E'(v)|$. Since each edge $e\in E'(v)$ sends one flow unit in $f_i$, each edge $e\in \delta_{\hat H'}(v)\cap E_{i+1}''$ now sends one flow unit in $f_{i+1}$.
This completes the definition of the flow $f_{i+1}$. We now analyze the congestion caused by this flow.
First, the flow on the paths originating at the edges of $E'_{i+1}$ remains unchanged, and causes congestion at most $2^{512}i$. Next, we consider on flow on paths originating at edges of $E''_{i+1}$. Consider again some vertex $v\in L_{i+1}'$, and recall that $|E'(v)|\geq |\delta_{\hat H'}(v)|/256$. Observe that edges of $E'(v)$ must lie in edge set $\tilde E_i$. Since every edge $e\in \delta_{\hat H'}(v)\cap E''_{i+1}$ spreads one unit of flow evenly among the edges of $E'(v)$, each edge of $E'(v)$ is responsible for sending at most $\frac{|\delta_{\hat H'}(v)|}{|E'(v)|}\leq 256$ flow units. In other words, for each edge $e\in E'(v)$, the flow originating at $e$ in $f_i$ is scaled by at most factor $256$ in order to obtain flow $f_{i+1}$. Therefore, the flow $f_{i+1}$ originating at edges of $E''_{i+1}$ causes congestion at most $256\cdot \textsf{left} (1+\frac{256}{\log m}\textsf{right})^i$. Overall, flow $f_i$ causes congestion at most $2^{512}i+256\cdot \textsf{left} (1+\frac{256}{\log m}\textsf{right})^i\leq 2^{512}\cdot (i+1)$, since $i+1\leq h\leq \log m$.
Lastly, we bound the congestion of the flow $f'_{i+1}$, which is the restriction of the flow $f_{i+1}$ to the set $\tilde {\mathcal{P}}_i^*$ of paths. We partition the edges of $\tilde E_{i+1}$ into two subsets: $\tilde E'_{i+1}=\tilde E_i\cap \tilde E_{i+1}$, and $\tilde E_{i+1}''$ containing all remaining edges. Observe that for each edge $e\in \tilde E_{i+1}''$, there is some vertex $v\in L'_{i+1}$ with $e\in \delta^{\operatorname{up}}(v)$.
The flow $f'_{i+1}$ that originates at the edges of $\tilde E'_{i+1}$ remains unchanged from $f'_i$, and causes congestion at most $\textsf{left} (1+\frac{256}{\log m}\textsf{right})^i$. In order to bound the congestion caused by the flow $f'_{i+1}$ originating from edges of $\tilde E'_{i+1}$, consider some vertex $v\in L'_{i+1}$. Recall that $|E'(v)| \geq |\delta_{\hat H'}(v)|/256$, while
$|\delta^{\operatorname{up}}(v)|<|\delta_{\hat H'}(v)|/\log m\leq 256|E'(v)| /\log m$. Since every edge $e\in \delta^{\operatorname{up}}(v)\setminus E'(v)$ spreads one unit of flow evenly among the edges of $E'(v)$, each edge of $E'(v)$ is responsible for sending at most $\frac{|\delta^{\operatorname{up}}(v)|}{|E'(v)|}\leq \frac{256}{\log m}$ flow units. In other words, for each edge $e\in E'(v)$, the flow originating at $e$ in $f'_i$ is scaled by at most factor $256/\log m$ in order to obtain flow $f'_{i+1}$. Therefore, the flow $f'_{i+1}$ originating at edges of $\tilde E''_{i+1}$ causes congestion at most $\frac{256}{\log m}\cdot \textsf{left} (1+\frac{256}{\log m}\textsf{right})^i$. Overall, flow $f'_{i+1}$ causes congestion at most $\textsf{left} (1+\frac{256}{\log m}\textsf{right})^i+ \frac{256}{\log m}\cdot \textsf{left} (1+\frac{256}{\log m}\textsf{right})^i\leq \textsf{left} (1+\frac{256}{\log m}\textsf{right} )^{i+1}$.
\end{proofof}
\subsection{Proof of \Cref{claim: simplifying cluster is enough}}
\label{subsec: simplifying cluster is enough}
We denote by $E'=\delta_{\check H}(S)$. We define a cluster $S'$ in graph $G$, corresponding to cluster $S$, as usual: First, we define vertex set $V(S')$, and then we let $S'$ be the subgraph of $G$ induced by $V(S')$. Vertex set $V(S')$ contains every regular vertex of $S$. Additionally, for every $R$-node $v_{R}\in S$, it contains all vertices of $R$, and for every $J$-node $v_{J'}\in S$, it contains all vertices of the cluster $J'\in {\mathcal{J}}'$.
We start with the following simple observation.
\begin{observation}\label{obs: compute external paths for S'}
There is an efficient algorithm to compute a distribution ${\mathcal{D}}'(S')$ over the set $\Lambda'_G(S')$ of external $S'$-routers in $G$, such that, for every edge $e\in E(G)\setminus E(S')$,
$\expect[{\mathcal{Q}}'(S')\sim {\mathcal{D}}'(S')]{\cong({\mathcal{Q}}'(S'),e)}\leq O(\log^{14.5}m)$.
\end{observation}
\begin{proof}
Recall that every cluster
$R\in {\mathcal{R}}$ has the $\alpha_1=1/\log^6m$-bandwidth property in graph $G$, and every cluster $J'\in {\mathcal{J}}'$ has the $\Omega(1/\log^{9.5} m)$-bandwidth property in $G$. Let $x$ be the vertex of graph $\check H$ that serves as the last vertex on every path of ${\mathcal{P}}(S)$. Assume first that $x$ is a regular vertex, that is, $x\in V(G)$. Then we can use the algorithm from \Cref{claim: routing in contracted graph} (by first subdividing every edge of $E'$ with a terminal) to obtain a collection ${\mathcal{P}}(S')$ of paths in graph $G$, that is an external $S'$-router, such that the paths in ${\mathcal{P}}(S')$ cause congestion $O\textsf{left} (\log^{10.5}m\textsf{right} )$. We define a distribution ${\mathcal{D}}'(S')$ over the set $\Lambda'_G(S')$ of external $S'$-routers, that assigns probability $1$ to the router ${\mathcal{P}}(S')$.
Assume now that $x$ is a supernode, that corresponds to some cluster $A\in {\mathcal{R}}''\cup{\mathcal{J}}'$. Again, applying the algorithm from \Cref{claim: routing in contracted graph} (but this time replacing the graph $G$ with the graph that is obtained from $G$ by contracting cluster $A$ into a supernode $x$), we obtain a collection ${\mathcal{P}}(S')$ of paths in graph $G$, routing the edges of $E'=\delta_G(S')$ to the edges of $\delta_G(A)$, with congestion $O( \log^{10.5}m )$, such that the paths in ${\mathcal{P}}(S')$ are internally disjoint from both $S'$ and $A$. Moreover, from \Cref{claim: routing in contracted graph} the paths in ${\mathcal{P}}(S')$ cause congestion at most $\beta'=O(\log m)$ on the edges of $\delta_G(A)$.
Recall that cluster $A$ must have the $\Omega(1/\log^{9.5} m)$-bandwidth property in $G$. By applying the algorithm from \Cref{lem: simple guiding paths} to cluster $A$, we obtain a distribution ${\mathcal{D}}(A)$ over the set $\Lambda_G(A)$ of internal $A$-routers, such that, for every edge $e\in E(A)$, $\expect[{\mathcal{Q}}(A)\sim {\mathcal{D}}(A)]{\cong({\mathcal{Q}}(A),e)}\leq O((\log^4m)\cdot (\log^{9.5}m))=O(\log^{13.5}m)$. We now define the distribution ${\mathcal{D}}'(S')$ over the set $\Lambda'_G(S')$ of external $S'$-routers, by providing an algorithm to draw a set of paths from the distribution. In order to do so, we first choose a set ${\mathcal{Q}}(A)\in \Lambda_G(A)$ of paths from the distribution ${\mathcal{D}}(A)$.
Denote ${\mathcal{Q}}(A)=\set{Q(e)\mid e\in \delta_G(A)}$, where path $Q(e)$ has $e$ as its first edge. Let ${\mathcal{Q}}'$ be a multi-set of paths, in which, for every edge $e\in \Lambda_G(A)$, the path $Q(e)$ is included $\cong_G({\mathcal{P}}(S'),e)\leq O(\log m)$ times.
We then concatenate the paths in ${\mathcal{P}}(S')$ with the paths in ${\mathcal{Q}}'$, obtaining an external $S'$-router ${\mathcal{Q}}'(S')\in \Lambda_{G}(S')$. From the above discussion, it is immediate to verify that, for every edge $e\in E(G)\setminus E(S')$,
$\expect[{\mathcal{Q}}'(S')\sim {\mathcal{D}}'(S')]{\cong({\mathcal{Q}}'(S'),e)}\leq O(\log^{14.5}m)$.
\end{proof}
We now consider three cases, depending on whether cluster $S$ contains any $J$-node, and whether the cluster $J'\in {\mathcal{J}}$ corresponding to the $J$-node has a cluster of ${\mathcal{C}}$ or of ${\mathcal{W}}'$ as its center cluster.
\paragraph{Case 1.} The first case happens if there is at least one $J$-node $v_{J'}\in S$, such that the center-cluster of the cluster $J'\in {\mathcal{J}}$ is a cluster of ${\mathcal{C}}$, that we denote by $C^*$ (see \Cref{fig: NF6}).
Let ${\mathcal{C}}^*\subseteq {\mathcal{C}}$ be the set of all clusters $C\in {\mathcal{C}}$ with $C\subseteq S'$, and let ${\mathcal{R}}^*\subseteq {\mathcal{R}}$ be the set of all clusters $R\in {\mathcal{R}}$ with $R\subseteq S'$. Observe that cluster $C^*$ may not be contained in any cluster of ${\mathcal{R}}$, from the definition of cluster set ${\mathcal{J}}$. We will modify the set ${\mathcal{R}}$ of clusters, by deleting the clusters of ${\mathcal{R}}^*$ from it, and adding a new set ${\mathcal{R}}^{**}$ of clusters instead, so that the resulting cluster set $\tilde {\mathcal{R}}=({\mathcal{R}}\setminus{\mathcal{R}}^*)\cup {\mathcal{R}}^{**}$ is a helpful clustering that is better than ${\mathcal{R}}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF6.jpg}
\caption{Case $1$: $S'$ contains at least one $J'$-cluster (shown in brown) whose center cluster $C^*\in {\mathcal{C}}$ is marked with $*$. Clusters of ${\mathcal{R}}^*$ are shown in red.}\label{fig: NF6}
\end{figure}
In order to define the new set ${\mathcal{R}}^{**}$ of clusters, we start with the augmented cluster $X=(S')^+$; that is, we subdivide every edge $e\in E'$ in graph $G$ with a terminal $t_e$, and we let $T=\set{t_e\mid e\in E'}$ be the resulting set of terminals. We then let $X$ be the subgraph of the resulting graph induced by $T\cup V(S')$. Next, we obtain a graph $X'$ from $X$, by contracting every basic cluster $C\in {\mathcal{C}}^*$ into a supernode $v_C$, so $X'=X_{|{\mathcal{C}}^*}$. We apply the algorithm from \Cref{thm:well_linked_decomposition} to graph $X'$, its cluster $X'\setminus T$, and parameter $\alpha_0=1/\log^3m$ (so the requirement that $\alpha_0< \min\set{\frac 1 {64\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot\log m},\frac 1 {48\log^2 m}}$ is satisfied). The algorithm computes a collection ${\mathcal{Y}}$ of clusters of $X'\setminus T$ (the well-linked decomosition), such that the vertex sets $\set{V(Y)}_{Y\in {\mathcal{Y}}}$ partition $V(X')\setminus T$, and every cluster $Y\in {\mathcal T}$ has the $\alpha_0$-bandwidth property, with $|\delta_{X'}(Y)|\leq |T|=|E'|$. We are also guaranteed that:
\begin{equation}\label{eq: few edges in WLD}
\sum_{Y\in {\mathcal{Y}}}|\delta_{X'}(Y)|\le |T|\cdot\textsf{left}(1+O(\alpha_0\cdot \log^{1.5} m)\textsf{right})= |E'|\cdot \textsf{left}(1+O(\alpha_0\cdot \log^{1.5} m)\textsf{right}).
\end{equation}
Recall that, additionally, the algorithm computes,
for every cluster $Y\in {\mathcal{Y}}$, a set ${\mathcal{P}}(Y)=\set{P(e)\mid e\in \delta_{X'}(Y)}$ of paths, such that, for every edge $e\in \delta_{X'}(Y)$, path $P(e)$ has $e$ as its first edge, and some terminal of $T$ as its last vertex, with all inner vertices of $P(e)$ lying in $V(X')\setminus V(Y)$. We are also guaranteed that, for every cluster $Y\in {\mathcal{Y}}$, the set ${\mathcal{P}}(Y)$ of paths causes edge-congestion at most $100$ in $X'$.
We now define the set ${\mathcal{R}}^{**}$ of clusters, that will be added to ${\mathcal{R}}$ instead of the clusters of ${\mathcal{R}}^*$. For every cluster $Y\in {\mathcal{Y}}$, we let $R_Y$ be the cluster of graph $G$ corresponding to cluster $Y$. Intuitively, $R_Y$ is obtained from $Y$ by uncontracting all basic clusters whose corresponding supernode lies in $Y$. Formally, we denote by ${\mathcal{C}}(Y)\subseteq {\mathcal{C}}$ the set of all basic clusters $C$ whose corresponding supernode $v_C$ lies in $Y$. We define vertex set $V(R_Y)$ to contain all regular vertices lying in $Y$ (that is, vertices of $V(Y)\cap V(G)$), and all vertices of $\bigcup_{C\in {\mathcal{C}}(Y)}V(C)$. We then let $R_Y$ be the subgraph of $G$ induced by $V(R_Y)$. Note that $Y=(R_Y)_{|{\mathcal{C}}(Y)}$. Since every cluster in ${\mathcal{C}}$ has the $\alpha_0$-bandwidth property, and every cluster $Y\in {\mathcal{Y}}$ has the $\alpha_0$-bandwidth property, from \Cref{cor: contracted_graph_well_linkedness}, cluster $R_Y$ has the $\alpha_0^2=\alpha_1$-bandwidth property. Consider now the set ${\mathcal{P}}(Y)$ of paths in graph $X'$, routing the edges of $\delta_{X'}(Y)$ to the vertices of $T$, with congestion at most $100$, and recall that the paths of ${\mathcal{P}}(Y)$ are internally disjoint from $Y$. Since $Y=(R_Y)_{|{\mathcal{C}}(Y)}$, and every cluster in ${\mathcal{C}}$ has the $\alpha_0$-bandwidth property, we can use the algorithm from \Cref{claim: routing in contracted graph}, to obtain a collection ${\mathcal{P}}(R_Y)$ of paths in graph $G$, routing the edges of $\delta_{G}(R_Y)$ to the edges of $E'$, with congestion at most $200/\alpha_0$, such that the paths in ${\mathcal{P}}(R_Y)$ are internally disjoint from $R_Y$, and they cause congestion at most $100$ on edges of $E'$.
We are now ready to define a distribution ${\mathcal{D}}'(R_Y)$ over the set $\Lambda'_G(R_Y)$ of external $R_Y$-routers in $G$, by providing an algorithm to draw an external router ${\mathcal{Q}}'(R_Y)$ from the distribution. In order to draw a set ${\mathcal{Q}}'(R_Y)$ of paths from distribution ${\mathcal{D}}'(R_Y)$, we start by drawing an external $S'$-router ${\mathcal{Q}}'(S')\in \Lambda_G(S')$ from the distribution ${\mathcal{D}}'(S')$ given by \Cref{obs: compute external paths for S'}. Recall that paths in ${\mathcal{Q}}'(S')$ route the edges of $E'$ to some vertex $u\not\in S'$, they are internally disjoint from $S'$, and $\expect[{\mathcal{Q}}'(S')\sim {\mathcal{D}}'(S')]{\cong({\mathcal{Q}}'(S'),e)}\leq O(\log^{14.5}m)$.
We denote ${\mathcal{Q}}'(S)=\set{Q(e)\mid e\in \delta_G(S')}$, where for all $e\in \delta_G(S')$, path $Q(e)$ has $e$ as its first edge.
We let ${\mathcal{Q}}''$ be a multi-set of paths obtained by including, for each edge $e\in \delta_G(S')$, exactly $\cong_G({\mathcal{P}}(R_Y),e)\leq 100$ copies of the path $Q(e)$.
By concatenating the paths in ${\mathcal{P}}(R_Y)$ with the paths in ${\mathcal{Q}}''$, we obtain a set ${\mathcal{Q}}'(R_Y)$ of paths (an external $R_Y$-router), routing the edges of $\delta_G(R_Y)$ to vertex $u$ in $G$, such that the paths in the set are inernally disjoint from $R_Y$. For every edge in $E(S')$, the congestion caused by the paths in ${\mathcal{Q}}'(R_Y)$ is at most $200/\alpha_0\leq O(\log^3m)$, while for every edge $e\in E(G\setminus S')$, $\cong_G({\mathcal{Q}}'(R_Y),e)\leq 100\cong_G({\mathcal{Q}}'(S'),e)$. This completes the definition of the distribution ${\mathcal{D}}'(R_Y)$ over the set $\Lambda'_G(R_Y)$ of external $R_Y$-routers. From the above discussion, for every edge $e\in E(G)\setminus E(R_Y)$, $\expect[{\mathcal{Q}}'(R_Y)\sim {\mathcal{D}}'(R_Y)]{\cong({\mathcal{Q}}'(R_Y),e)}\leq O(\log^{15.5}m)\leq \beta$, since $\beta=\log^{18}m$.
We set ${\mathcal{R}}^{**}=\set{R_Y\mid Y\in {\mathcal{Y}}}$, and we define a new clustering $\tilde {\mathcal{R}}=({\mathcal{R}}\setminus {\mathcal{R}}^*)\cup {\mathcal{R}}^{**}$.
It is immediate to verify that all clusters in $\tilde {\mathcal{R}}$ are disjoint from each other and every cluster $R\in \tilde {\mathcal{R}}$ has $\alpha_1$-bandwidth property. From the construction of the graph $X'$, we are guaranteed that, for every basic cluster $C\in {\mathcal{C}}$, and for every cluster $R\in {\mathcal{R}}$, either $C\subseteq R$, or $C\cap R=\emptyset$ must hold. We have also defined, for every cluster $R\in {\mathcal{R}}$, a distribution
${\mathcal{D}}'(R)$ over external $R$-routers in $\Lambda'_{G}(R)$, such that, for every edge $e\in E(G)\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}'(R)]{\cong_{G}({\mathcal{Q}}'(R),e)}\leq \beta$. Since we have ensured that vertices $v^*,u^*$ do not lie in set $S$, we are guaranteed that $R^*\in {\mathcal{R}}\setminus {\mathcal{R}}^*$, and so $R^*\in \tilde {\mathcal{R}}$. Similarly, the special vertex $v^*$ does not lie in any cluster of $\tilde {\mathcal{R}}$. Therefore, $(\tilde {\mathcal{R}},\set{{\mathcal{D}}'(R)_{R\in \tilde {\mathcal{R}}}})$ is a {helpful clustering} of $G$ with respect to $v^*$ and ${\mathcal{C}}$, with $R^*\in \tilde {\mathcal{R}}$. It now only remains to show that it is better than the original clustering ${\mathcal{R}}$.
Observe that the only basic clusters that may be contained in the clusters of ${\mathcal{R}}^*$ are the clusters of ${\mathcal{C}}^*$. From the definition of the set ${\mathcal{J}}$ of clusters, since cluster $C^*\in {\mathcal{C}}^*$ is a center-cluster of some cluster $J'\in {\mathcal{J}}'$, we are guaraneed that cluster $C^*$ is not contained in any cluster of ${\mathcal{R}}$. But, since the vertices of $\set{V(Y)}_{Y\in {\mathcal{Y}}}$ partition vertex set $V(X')\setminus T$, we are guaranteed that every cluster of ${\mathcal{C}}^*$ is contained in some cluster of ${\mathcal{R}}^{**}$. Therefore, the number of basic clusters of ${\mathcal{C}}$ contained in $G\setminus\textsf{left}(\bigcup_{R\in {\mathcal{R}}}R\textsf{right} )$ is strictly greater than the number of
basic clusters of ${\mathcal{C}}$ contained in $G\setminus\textsf{left}(\bigcup_{R\in \tilde{\mathcal{R}}}R\textsf{right} )$. We conclude that clustering $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$.
\paragraph{Case 2.}
We now consider the second case, where cluster $S$ does not contain any $J$-node, so every vertex of $S$ is either an $R$-node or a regular vertex. In this case, we proceed exactly as before: we define the augmented cluster $X=(S')^+$ and its contracted version $X'=X_{|{\mathcal{C}}^*}$. We then compute a well-linked decomposition ${\mathcal{Y}}$ of $X'\setminus T$, and the corresponding set ${\mathcal{R}}^{**}=\set{R_Y\mid Y\in {\mathcal{Y}}}$ of clusters of $G$ exactly as before. We also define a new clustering $\tilde {\mathcal{R}}$ and the distributions ${\mathcal{D}}'(R_Y)$ over sets of $R_Y$-routers for clusters $R_Y\in {\mathcal{R}}^{**}$ exactly as before. Using the same reasoning as in Case 1, the final clustering $(\tilde {\mathcal{R}},\set{{\mathcal{D}}'(R)_{R\in \tilde {\mathcal{R}}}})$ is a {helpful clustering} of $G$ with respect to $v^*$ and ${\mathcal{C}}$, with $R^*\in \tilde {\mathcal{R}}$. However, since we are no longer guaranteed that $S$ contains a $J$-node (which in turn contains a cluster of ${\mathcal{C}}$ as a center cluster), we need to employ a different argument in order to prove that $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$. Since $S$ does not contain any $J$-node, from the definition of a simplifying cluster, $|E_{\check H}(S)|\geq |\delta_{\check H}(S)|/\log m=|E'|/\log m$ must hold. Notice that every edge of $E_{\check H}(S)$ corresponds to an edge of $S'_{|{\mathcal{R}}^*}$, and so $|E(S'_{|{\mathcal{R}}^*})|\geq |E'|/\log m$. On the other hand, from \Cref{eq: few edges in WLD}, $|E(S'_{|{\mathcal{R}}^{**}})|\leq \sum_{Y\in {\mathcal{Y}}}|\delta_{X'}(Y)|-|E'|\le O(|E'|\alpha_0\cdot \log^{1.5} m)< |E'|/\log m$, since $\alpha_0= 1/\log^3m$.
From our definition of the set $\tilde {\mathcal{R}}$ of clusters, $|E(G_{|\tilde {\mathcal{R}}})|=|E(G_{|{\mathcal{R}}})|-|E(S'_{|{\mathcal{R}}^*})|+|E(S'_{|{\mathcal{R}}^{**}})|<|E(G_{|{\mathcal{R}}})|$.
We conclude that the new helpful clustering $\tilde {\mathcal{R}}$ is better than ${\mathcal{R}}$.
\paragraph{Case 3.}
It remains to consider the third case, where $S$ contains at least one $J$-node, and for each such $J$-node $v_{J'}$, the center-cluster of the cluster $J'\in {\mathcal{J}}'$ lies in ${\mathcal{W}}'$. We fix any $J$-node $v_{J'}\in V(S)$, and we denote by $W'\in {\mathcal{W}}'$ the center-cluster of $J'$.
There is a difficulty with following the approach used in Cases 1 and 2 in this case: it is possible that every cluster of ${\mathcal{C}}^*$ is contained in some cluster of ${\mathcal{R}}^*$, and, additionally, it is possible that the total number of edges in $S'_{|{\mathcal{R}}^*}$ is quite small compared to $|E'|$. While we could still obtain a new helpful clustering $\tilde {\mathcal{R}}$ in the same way as in Cases 1 and 2, it may no longer be the case that $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$. In order to overcome this difficulty, we will replace cluster $S'$ of $G$ with a different cluster $\tilde S'\subseteq S'$ that has similar properties to cluster $S'$, except that, if we denote by $\tilde {\mathcal{R}}^*$ the set of all clusters $R\in {\mathcal{R}}^*$ with $R\subseteq \tilde S'$, then $|E(S'_{|\tilde {\mathcal{R}}^*})|$ is sufficiently large compared to $|E_G(\tilde S')|$. We construct the cluster $\tilde S'$ using the following claim.
\begin{claim} \label{claim: case 3}
There is an efficient algorithm, that, if Case 3 happenned, computes a cluster $\tilde S'\subseteq S'$ in graph $G$, such that for every cluster $R\in {\mathcal{R}}^*$, either $R\subseteq \tilde S'$ or $R\cap \tilde S'=\emptyset$ holds, and similarly, for every cluster $C\in {\mathcal{C}}^*$, either $C\subseteq \tilde S'$ or $C\cap \tilde S'=\emptyset$ holds. Moreover, if we denote by $\tilde {\mathcal{R}}^*$ the set of all clusters $R\in {\mathcal{R}}^*$ with $R\subseteq \tilde S'$, then $|E(\tilde S'_{|\tilde {\mathcal{R}}^*})|\geq |\delta_G(\tilde S')|/(64\log m)$. Additionally, the algorithm computes a distribution ${\mathcal{D}}'(\tilde S')$ over the set $\Lambda'_G(\tilde S')$ of external $\tilde S'$-routers in graph $G$, such that, for every edge $e\in E(G\setminus \tilde S')$,
$\expect[{\mathcal{Q}}'(\tilde S')\sim {\mathcal{D}}'(\tilde S')]{\cong({\mathcal{Q}}'(\tilde S'),e)}\leq O(\log^{14.5}m)$.
\end{claim}
We provide the proof of \Cref{claim: case 3} below, after completing the proof of \Cref{claim: simplifying cluster is enough} using it. We employ the algorithm from Cases 1 and 2, except that we apply it to cluster $\tilde S'$ of $G$ instead of $S'$, and we replace the set ${\mathcal{R}}^*$ of clusters with $\tilde {\mathcal{R}}^{*}$. Let $\tilde {\mathcal{C}}^*\subseteq {\mathcal{C}}^*$ be the set of all basic clusters $C\in {\mathcal{C}}^*$ with $C\subseteq \tilde S'$. Let $\tilde {\mathcal{R}}^{**}$ be the set of clusters that the algorithm from Cases 1 and 2 computes (that were denoted by ${\mathcal{R}}^{**}$ before), when applied to cluster $\tilde S'$. For every cluster $R\in \tilde {\mathcal{R}}^{**}$, the algorithm obtains a distribution ${\mathcal{D}}'(R)$ over the set $\Lambda'_G(R)$ of external $R$-routers.
Let $\tilde {\mathcal{R}}=({\mathcal{R}}\setminus \tilde{\mathcal{R}}^*)\cup \tilde {\mathcal{R}}^{**}$. Using the same arguments as in Cases 1 and 2, $(\tilde {\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in \tilde {\mathcal{R}}})$ is a helpful clustering in $G$ with respect to $v^*$ and ${\mathcal{C}}$, with $R^*\in \tilde {\mathcal{R}}$. It now only remains to show that $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$.
As in Case 2, the only basic clusters of ${\mathcal{C}}$ that the clusters of $\tilde {\mathcal{R}}^*$ may contain are the clusters of $\tilde {\mathcal{C}}^*$. As in Case 2, each such cluster is guaranteed to be contained in some cluster of $\tilde {\mathcal{R}}^{**}$. Therefore, the number of clusters of ${\mathcal{C}}$ that are contained in $G\setminus(\bigcup_{R\in{\mathcal{R}}}R)$ is greater than or equal to the number of clusters of ${\mathcal{C}}$ that are contained in $G\setminus (\bigcup_{R\in \tilde {\mathcal{R}}}R)$. In order to prove that $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$, it is now enough to prove that $|E(G_{|\tilde {\mathcal{R}}})|<|E(G_{|{\mathcal{R}}})|$.
We denote $\delta_G(\tilde S')$ by $E''$. On the one hand,
from \Cref{claim: case 3}, $|E(\tilde S'_{|\tilde {\mathcal{R}}^*})|\geq |E''|/(64\log m)$. On the other hand, from \Cref{eq: few edges in WLD}, $|E(\tilde S'_{|\tilde {\mathcal{R}}^{**}})|\leq
\sum_{Y\in {\mathcal{Y}}}|\delta_{X'}(Y)|-|E''| \le |E''|\cdot O(\alpha_0\cdot \log^{1.5} m))<|E''|/(64\log m)$, since $\alpha_0=1/\log^3m$.
As in Case 2, $|E(G_{|\tilde {\mathcal{R}}})|=|E(G_{|{\mathcal{R}}})|-|E(S'_{|\tilde {\mathcal{R}}^*})|+|E(S'_{|\tilde {\mathcal{R}}^{**}})|<|E(G_{|{\mathcal{R}}})|$.
Therefore, $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$. It now remains to complete the proof of \Cref{claim: case 3}, which we do next.
\begin{proofof}{\Cref{claim: case 3}}
Recall that we have fixed a $J$-node $v_{J'}\in V(S)$, and a center-cluster $W'\in {\mathcal{W}}'$ of the cluster $J'\in {\mathcal{J}}'$.
We let $\tilde S'$ be a vertex-induced subgraph of $S'$ with the following properties:
\begin{itemize}
\item $W'\subseteq \tilde S'$;
\item for every cluster $R\in {\mathcal{R}}^*$, either $R\subseteq \tilde S'$, or $R\cap \tilde S'=\emptyset$;
\item for every cluster $C\in {\mathcal{C}}^*$, either $C\subseteq \tilde S'$ or $C\cap \tilde S'=\emptyset$; and
\item $|E(\tilde S')|$ is minimized among all graphs $\tilde S'$ for which the above conditions hold.
\end{itemize}
Such a graph $\tilde S'$ can be computed via standard minimum cut computation: we start with graph $G$, and we let $G_1$ be the graph obtained from $G$ by contracting the cluster $W'$ into a source $s$, and contracting $G\setminus S'$ into a destination $t$. Next, we obtain a graph $G_2$ from $G$, by contracting every cluster $R\in {\mathcal{R}}$ with $R\subseteq S'\setminus W'$ into a supernode $v_R$, and similarly contracting every basic cluster $C\in {\mathcal{C}}'$ with $C\subseteq S'\setminus W'$ into a supernode $v_C$. Let $(Z,Z')$ be the minimum $s$-$t$ cut in $G_2$, and denote $E''=E_{G_2}(Z,Z')$. Observe that, from the max-flow / min-cut theorem, there is a collection ${\mathcal{Q}}$ of edge-disjoint path in graph $G_2$, routing the edges of $E''$ to $t$, such that all paths in ${\mathcal{Q}}$ are internally disjoint from $Z$. We let $\tilde S'\subseteq S'$ be the subgraph of $\tilde S'$ that is defined by $Z$ (that is, we un-contract every cluster of ${\mathcal{R}}\cup{\mathcal{C}}$ whose corresponding supernode lies in $Z$). Note that $\delta_G(\tilde S')=E''$, and $W'\subseteq \tilde S'$.
Let $W\in {\mathcal{W}}$ be the $W$-cluster in graph $\hat H=G_{|{\mathcal{C}}'\cup {\mathcal{R}}}$ that corresponds to $W'$; in other words, cluster $W'$ was obtained from $W$ by un-contracting its supernodes.
Let $\tilde {\mathcal{R}}^*$ the set of all clusters $R\in {\mathcal{R}}^*$ with $R\subseteq \tilde S'$.
Clearly, every edge of $W$ is an edge of $S'_{|\tilde {\mathcal{R}}^*}$. From the definition of a valid set of $W$-clusters,
$|E(S'_{|\tilde {\mathcal{R}}^*})|\geq |E_{\hat H}(W)|\geq |\delta_{\hat H}(W)|/(64\log m)\geq |E''|/(64\log m)$ (we have used the fact that $|E(Z,Z')|\leq |\delta_G(W')|=|\delta_{\hat H}(W)|$ must hold, from the definition of minimum cut).
It now remains to define the distribution ${\mathcal{D}}'(\tilde S')$ over the set $\Lambda'_G(\tilde S')$ of external $\tilde S'$-routers in graph $G$. As before, we will provide an efficient algorithm to draw a set ${\mathcal{Q}}'(S')$ of paths from the distribution. As observed already, there is a set ${\mathcal{Q}}$ of edge-disjoint paths in graph $G_2$ routing the edges of $\delta_{G_2}(Z)$ to vertex $t$, so that the paths are internally disjoint from cluster $Z$. Note that every edge $e\in E''$ has exactly one path $Q(e)\in {\mathcal{Q}}$ whose first edge is $e$, and the last edge of $Q(e)$ lies in $E'$. Since every cluster of ${\mathcal{R}}$ has the $\alpha_1=1/\log^6m$-bandwidth property, and every cluster $C\in {\mathcal{C}}$ has the $\alpha_0=1/\log^3m$-bandwidth property, we can use the algorithm from \Cref{claim: routing in contracted graph} to compute a collection ${\mathcal{P}}=\set{P(e) \mid e\in E''}$ of paths in graph $G_1$, where for every edge $e\in E''$, path $P(e)$ has $e$ as its first edge and some edge of $E'$ as its last edge. Moreover, the paths in $ {\mathcal{P}}$ cause edge-congestion at most $2/\alpha_1\leq 2/\log^6m$ in graph $G_1$, and congestion at most $1$ on edges of $E'$, and they are internally disjoint from $\tilde S'$. Note that the paths of $ {\mathcal{P}}$ are also contained in the original graph $G$. We are now ready to define the distribution ${\mathcal{D}}'(\tilde S')$. In order to draw an external $\tilde S'$-router ${\mathcal{Q}}'(\tilde S')\in \Lambda'_G(\tilde S')$ from the distribution, we first
draw a set ${\mathcal{Q}}'(S')\in \Lambda_G(S')$ of paths from the distribution ${\mathcal{D}}'(S')$, given by \Cref{obs: compute external paths for S'}. Recall that paths in ${\mathcal{Q}}'(S')$ route the edges of $E'$ to some vertex $u\not\in S'$, they are internally disjoint from $S'$, and $\expect[{\mathcal{Q}}'(S')\sim {\mathcal{D}}'(S')]{\cong({\mathcal{Q}}'(S'),e)}\leq O(\log^{14.5}m)$. By concatenating the paths in ${\mathcal{P}}$ with the paths in ${\mathcal{Q}}'(S')$, we obtain a set ${\mathcal{Q}}'(\tilde S')$ of paths, routing the edges of $\delta_G(\tilde S')$ to vertex $u$ in $G$, such that the paths in the set are inernally disjoint from $\tilde S'$. Moreover, for every edge of $ S'\cup E'$, the congestion caused by the paths in ${\mathcal{Q}}'(\tilde S')$ is at most $O(\log^6m)$, while for every edge of $G\setminus S'$, $\cong_G({\mathcal{Q}}'(\tilde S'),e)\leq \cong_G({\mathcal{Q}}'(S'),e)$. This completes the definition of the distribution ${\mathcal{D}}'(\tilde S')$ over the set $\Lambda'_G(\tilde S')$ of external $\tilde S'$-routers. Clearly, for every edge $e\in E(G\setminus \tilde S')$, $\expect[{\mathcal{Q}}'(\tilde S')\sim {\mathcal{D}}'(\tilde S')]{\cong({\mathcal{Q}}'(\tilde S'),e)}\leq O(\log^{14.5}m)$.
\end{proofof}
\subsection{Proof of \Cref{obs: left and right down-edges}}
\label{subsec: left and right down-edges}
Consider any vertex $v\in V(\check H)\setminus\textsf{left}(\bigcup_{i=1}^rS'_i\textsf{right} )$, and assume that it lies in $S_i\setminus S'_i$, for some $1\leq i\leq r$.
Recall that $|\delta^{\operatorname{up}}(v)|\leq |\delta_{\check H}(v)|/\log m$ must hold, and, since $v$ was not added to $S'_i$, $|\delta^{\operatorname{down},\operatorname{straight}''}(v)|\leq |\delta_{\check H}(v)|/128$.
Therefore, $|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{left}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq 63|\delta_{\check H}(v)|/64$ holds.
Assume now that $i>1$, and consider the Gomory-Hu tree $\tau$ of the graph $\check H$, and the two connected components of the graph obtained from $\tau$ after the edge $(u_{i-1},u_i)$ is removed from $\tau$. Denote by $A$ the set of all vertices lying in the connected component containing $u_{i-1}$, and by $B$ the set of all vertices lying in the other connected component. From the definition of cluster $S_i$, $v\in B$, and moreover, $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut in graph $\check H$. Therefore, if we let $A'=A\cup \set{v}$ and $B'=B\setminus\set{v}$, then $|E_{\check H}(A',B')|\geq |E_{\check H}(A,B)|$ must hold. Observe that the only difference between the edge sets $E_{\check H}(A',B')$ and $E_{\check H}(A,B)$ is that the edges of $\delta^{\operatorname{down},\operatorname{left}}(v)$ contribute to $E_{\check H}(A,B)$ but not to $E_{\check H}(A',B')$; the edges of $\delta^{\operatorname{down},\operatorname{right}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)\cup \delta^{\operatorname{down},\operatorname{straight}''}(v)$ contribute to $E_{\check H}(A',B')$ but not to $E_{\check H}(A,B)$; and the edges of $\delta^{\operatorname{up}}(v)$ may contribute to either cut (see \Cref{fig: NF7}).
Therefore, it must be the case that:
$$|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq |\delta^{\operatorname{down},\operatorname{right}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|+|\delta^{\operatorname{down},\operatorname{straight}''}(v)|+|\delta^{\operatorname{up}}(v)|.$$
\begin{figure}[h]
\centering
\includegraphics[scale=0.15]{figs/NF7.jpg}
\caption{Illustration for the proof of \Cref{obs: left and right down-edges}. The edges of $\delta^{\operatorname{down},\operatorname{left}}(v)$ are shown in blue, the edges of $\delta^{\operatorname{down},\operatorname{straight}''}(v)$ are shown in red, the edges of $\delta^{\operatorname{down},\operatorname{straight}'}(v)$ are shown in green, and the edges of $\delta^{\operatorname{down},\operatorname{right}}(v)$ are shown in brown. The blue edges lie in $E(A,B)\setminus E(A',B')$. The edges crossing the pink dashed line lie in $E(A',B')\setminus E(A,B)$. Additionally, edges of $\delta^{\operatorname{up}}(v)$ may lie in set $E(A',B')\setminus E(A,B)$.}\label{fig: NF7}
\end{figure}
From the definition of the layers $L_1,\ldots,L_h$, $|\delta^{\operatorname{up}}(v)|\leq |\delta(v)|/\log m$, and, since $v$ was not added to $S'_i$, $|\delta^{\operatorname{down},\operatorname{straight}''}(v)|\leq |\delta(v)|/128$. Therefore:
$$|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq |\delta^{\operatorname{down},\operatorname{right}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|+|\delta(v)|/64.$$
If we assume, for contradiction, that $|\delta^{\operatorname{down},\operatorname{left}}(v)|> 2(|\delta^{\operatorname{down},\operatorname{right}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|)$, then, combining this with the above inequality, we will get that $|\delta^{\operatorname{down},\operatorname{right}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|<|\delta(v)|/64$, and therefore, $|\delta^{\operatorname{down},\operatorname{left}}(v)|<|\delta(v)|/32$, contradicting the fact that
$|\delta^{\operatorname{down},\operatorname{left}}(v)|+|\delta^{\operatorname{down},\operatorname{right}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq 63|\delta(v)|/64$.
We conclude that $|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{right}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|)$ must hold.
We now show that $S_r=S'_r$. Assume for contradiction that this is not the case, and let $v\in S_r\setminus S'_r$ be a vertex that minimizes the index $j$ for which $v\in L'_j$. Notice that, from the definition, $\delta^{\operatorname{down},\operatorname{right}}(v)$, $\delta^{\operatorname{down},\operatorname{straight}'}(v)=\emptyset$ must hold.
Since we have established that $|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{right}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|)$, we get that $\delta^{\operatorname{down},\operatorname{left}}=\emptyset$ as well. Altogether, we get that $\delta^{\operatorname{down}}(v)=\delta^{\operatorname{down},\operatorname{straight}''}(v)$. Since $|\delta^{\operatorname{up}}(v)|\leq |\delta_{\check H}(v)|/\log m$, we get that the number of edges connecting $v$ to vertices of $S'_r$, $|\delta^{\operatorname{down},\operatorname{straight}''}(v)|>|\delta_{\check H}(v)|/2$, and so $v$ should have been added to set $S'_r$. Therefore, $S_r=S'_r$ must hold.
The proof that $|\delta^{\operatorname{down},\operatorname{right}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{left}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|)$ is similar, except that now we need to consider the cut obtained by deleting the edge $(u_i,u_{i+1})$ from the tree $\tau$. The proof that $S_1=S'_1$ is symmetric to the proof that $S_r=S'_r$ (in fact, if we reverse the order of the vertices $u_1,\ldots,u_r$ on path $P^*$, by rooting the tree $\tau$ at vertex $u_r=u^*$, then the definitions of sets $S_i,S'_i$ will remain unchanged, and we can simply repeat the proof from above).
\subsection{Proof of \Cref{obs: left and right mappings}}
\label{subsec: left and right mappings}
Consider any vertex $v\in V(\check H)\setminus\textsf{left}(\bigcup_{i=1}^rS'_i\textsf{right} )$. From \Cref{obs: left and right down-edges}, $|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|)$. Therefore:
$$|\delta^{\operatorname{down},\operatorname{left}}(v)|+|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\leq 3(|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|).$$
On the other hand, since $|\delta^{\operatorname{down},\operatorname{straight}''}(v)|\leq |\delta_{\check H}(v)|/128$, and $|\delta^{\operatorname{up}}(v)|\leq |\delta_{\check H}(v)|/\log m$, we get that:
$$|\delta^{\operatorname{down},\operatorname{left}}(v)|+|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq \textsf{left} (\frac {127}{128} -\frac 1 {\log m}\textsf{right} )\cdot |\delta_{\check H}(v)|.$$
By combining the two inequalities, we get that:
\[ |\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq \textsf{left} (\frac {127} {384} -\frac 1 {3\log m}\textsf{right} )\cdot |\delta_{\check H}(v)|>|\delta^{\operatorname{down},\operatorname{straight}''}(v)|+|\delta^{\operatorname{up}}(v)|. \]
Therefore, we can define a mapping $f^{\operatorname{right}}(v)$, that maps every edge of $\delta^{\operatorname{down},\operatorname{straight}''}(v)\cup\delta^{\operatorname{up}}(v)$ to a distinct edge of $\delta^{\operatorname{down},\operatorname{right}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$.
Using exactly the same reasoning, we get that:
\[ |\delta^{\operatorname{down},\operatorname{left}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|>|\delta^{\operatorname{down},\operatorname{straight}''}(v)|+|\delta^{\operatorname{up}}(v)|. \]
Therefore, we can define a mapping $f^{\operatorname{left}}(v)$, that maps every edge of $\delta^{\operatorname{down},\operatorname{straight}''}(v)\cup\delta^{\operatorname{up}}(v)$ to a distinct edge of $\delta^{\operatorname{down},\operatorname{left}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$.
\input{level-3-monotone-paths}
\renewcommand{\textbf{E}'}{\tilde E}
\subsection{Proof of \Cref{obs: bad inded structure}}
\label{subsec: edges between Sis}
Fix an index $1<i<r$. Consider the two connected components of the Gomory-Hu tree $\tau$ that are obtained after the edge $(u_{i-1},u_i)$ is deleted from it. Denote the sets of vertices of the two components by $A$ and $B$, where $u_{i-1}\in A$. Recall that $(A,B)$ is the minimum cut separating $u_{i-1}$ from $u_{i}$ in $\check H$. Observe that $A=V(S_1)\cup \cdots\cup V(S_{i-1})$, while $B=V(S_i)\cup \cdots\cup V(S_r)$. Note also that:
\[|E(A,B)|\geq |E_{i-1}'|+|\textbf{E}'_i^{\operatorname{left}}|+|\textbf{E}'_{i+1}^{\operatorname{left}}|+|\textbf{E}'_i^{\operatorname{over}}|\]
(see \Cref{fig: NF10a}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF10a.jpg}
\caption{Illustration for the Proof of \Cref{obs: bad inded structure}. Cut $(A,B)$ is shown in a pink dashed line.}\label{fig: NF10a}
\end{figure}
Next, we consider another $u_{i-1}$--$u_i$ cut $(X,Y)$ in $\check H$, where $Y=V(S_i)$, and $Xß=V(\check H)\setminus Y$. Observe that:
\[|E(X,Y)|= |E_{i-1}'|+|\textbf{E}'_i^{\operatorname{left}}|+ |\textbf{E}'_i^{\operatorname{right}}|+|E_i'|\]
(see \Cref{fig: NF10b}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF10b.jpg}
\caption{Illustration for the Proof of \Cref{obs: bad inded structure}. Cut $(X,Y)$ is shown in a pink dashed line.}\label{fig: NF10b}
\end{figure}
Since $|E(A,B)|\leq |E(X,Y)|$ must hold, we conclude that:
\begin{equation}\label{eq: bound from left}
|\textbf{E}'_{i+1}^{\operatorname{left}}|+|\textbf{E}'_i^{\operatorname{over}}|\leq |\textbf{E}'_i^{\operatorname{right}}|+|E_i'|.
\end{equation}
We now repeat the same reasoning with cuts separating $u_{i+1}$ from $u_{i+2}$.
Consider the two connected components of the Gomory-Hu tree $\tau$ that are obtained after the edge $(u_{i+1},u_{i+2})$ is deleted from it. Denote the sets of vertices of the two components by $A'$ and, $B'$, where $u_{i+1}\in A'$. Recall that $(A',B')$ is the minimum cut separating $u_{i+1}$ from $u_{i+2}$ in $\check H$. Note that $A'=V(S_1)\cup \cdots\cup V(S_{i+1})$, and:
\[|E(A',B')|\geq |E_{i+1}'|+|\textbf{E}'_{i+1}^{\operatorname{right}}|+|\textbf{E}'_{i}^{\operatorname{right}}|+|\textbf{E}'_i^{\operatorname{over}}|\]
(see \Cref{fig: NF11a}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF11a.jpg}
\caption{Illustration for the proof of \Cref{obs: bad inded structure}, with cut $(A,B)$ shown in a pink dashed line
}\label{fig: NF11a}
\end{figure}
We now consider another $u_{i+1}$--$u_{i+2}$ cut $(X',Y')$ in $\check H$, where $X'=V(S_{i+1})$, and $Y'=V(\check H)\setminus X'$. Observe that:
\[|E(X',Y')= |E_{i+1}'|+|\textbf{E}'_{i+1}^{\operatorname{right}}|+|E_{i}'|+|\textbf{E}'_{i+1}^{\operatorname{left}}|\]
(see \Cref{fig: NF11b}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF11b.jpg}
\caption{Illustration for the proof of \Cref{obs: bad inded structure}, with cut $(X,Y)$ shown in a pink dashed line.}\label{fig: NF11b}
\end{figure}
Since $|E(A',B')|\leq |E(X',Y')|$ must hold, we conclude that:
\begin{equation}\label{eq: bound from right}
|\textbf{E}'_{i}^{\operatorname{right}}|+|\textbf{E}'_i^{\operatorname{over}}|\leq
|E_{i}'|+|\textbf{E}'_{i+1}^{\operatorname{left}}|.
\end{equation}
By adding \Cref{eq: bound from left} and \Cref{eq: bound from right}, we conclude that $|\textbf{E}'_i^{\operatorname{over}}|\leq
|E_{i}'|$. We also immediately get that $|\textbf{E}'_{i+1}^{\operatorname{left}}|\leq
|E_{i}'|+|\textbf{E}'_{i}^{\operatorname{right}}|$ from \Cref{eq: bound from left} and $|\textbf{E}'_{i}^{\operatorname{right}}|\leq
|E_{i}'|+|\textbf{E}'_{i+1}^{\operatorname{left}}|$ from \Cref{eq: bound from right}.
\subsection{Proof of \Cref{claim: bound S' to S'' edges}}
\label{subsec: bound S' to S'' edges}
Fix an index $1<i<r$. We prove that $|\downedges{i}|\leq 0.1|\rightedges{i}|$, and show the existence of the set ${\mathcal{P}}^{\operatorname{right}}$ of paths. The proof that $|\downedges{i}|\leq 0.1|\leftedges{i}|$ and the proof of the existence of the set ${\mathcal{P}}^{\operatorname{left}}$ of paths is symmetric.
Let ${\mathcal{P}}$ be the set of all paths $P$ in graph $\check H$, such that the first edge of $P$ lies in $\downedges{i}$, the last edge lies in $\rightedges{i}$, and all inner vertices of $P$ lies in $S''_i$. We show a flow $f$, defined over the set ${\mathcal{P}}$ of paths, in which every edge in $\downedges{i}$ sends one flow unit, every edge in $\rightedges{i}$ receives at most $1/10$ flow unit, and each edge of $E(S''_i)$ carries at most one flow unit. The existence of such a flow will then prove that $|\downedges{i}|\leq 0.1|\rightedges{i}|$, and will imply the existence of the path set ${\mathcal{P}}^{\operatorname{right}}$ with the required properties from the integrality of flow.
In order to define the flow $f$, we consider the vertices of $S''_i$ in the decreasing order of their levels $L'_h,\ldots,L'_1$. When we consider level $L'_j$, for $1\leq j\leq h$, we assume that, for every vertex $v\in L'_j\cap S''_i$, for every edge $e\in \delta^{\operatorname{up}}(v)$, connecting $v$ to another vertex of $S''_i$, the flow on edge $e$ is fixed already, and that the only edges that are incident to $v$ that may carry non-zero flow are edges of $\delta^{\operatorname{up}}(v)\cup\delta^{\operatorname{down},\operatorname{straight}''}(v)$. Initially, for every edge $e\in \downedges{i}$, we set $f(e)=1$. Note that for every edge $e\in \downedges{i}$, if $v$ is the endpoint of $e$ lying in $S''_i$, then $e\in \delta^{\operatorname{down},\operatorname{straight}''}(v)$.
We start with the level $L'_h$. Consider any vertex $v\in L'_h$. From the definition, $\delta^{\operatorname{up}}(v)=\emptyset$. Recall that, from \Cref{obs: left and right down-edges}, $|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{left}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq 63|\delta(v)|/64$, and additionally, $|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|)$.
Therefore, $|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq 21|\delta(v)|/64$ must hold. At the same time, $|\delta^{\operatorname{down},\operatorname{straight}''}(v)|\leq |\delta(v)|/128$. We now consider two cases. The first case is when $|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq |\delta(v)|/64$. In this case, $|\delta^{\operatorname{down},\operatorname{straight}''}(v)|\leq |\delta^{\operatorname{down},\operatorname{straight}'}(v)|$ holds. We assign, to each edge $e\in \delta^{\operatorname{down},\operatorname{straight}''}(v)$, a distinct edge $e'\in \delta^{\operatorname{down},\operatorname{straight}'}(v)$, and we set $f(e')=f(e)$. Consider now the second case, where $|\delta^{\operatorname{down},\operatorname{right}}(v)|\geq |\delta(v)|/8$ must hold. In this case, $|\delta^{\operatorname{down}, \operatorname{straight}''}(v)|\leq |\delta^{\operatorname{down},\operatorname{straight}'}(v)|/10$. We can now assign, to each edge $e\in \delta^{\operatorname{down},\operatorname{straight}''}(v)$ ten distinct edges $e_1,\ldots,e_{10}\in \delta^{\operatorname{down},\operatorname{right}}(v)$, such that each edge of $\delta^{\operatorname{down},\operatorname{right}}(v)$ is assigned to at most one edge of $\delta^{\operatorname{down},\operatorname{straight}''}(v)$. If edge $e'\in \delta^{\operatorname{down},\operatorname{right}}(v)$ is assigned to edge $e\in \delta^{\operatorname{down},\operatorname{straight}''}(v)$, then we set $f(e')=f(e)/10$.
Assume now that we have processed levels $L'_h,\ldots,L'_{j+1}$, and consider some level $L'_j$. Let $v\in L'_j\cap S''_i$ be any vertex. The processing of vertex $v$ is similar to the one above, except that now some edges of $\delta^{\operatorname{up}}(v)$ may carry flow, and we need to forward this flow to edges in $\delta^{\operatorname{down}}(v)$. As before, from \Cref{obs: left and right down-edges}, $|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{left}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq 63|\delta(v)|/64$, and additionally, $|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|)$.
Therefore, $|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq 21|\delta(v)|/64$ must hold. At the same time, $|\delta^{\operatorname{down},\operatorname{straight}''}(v)|\leq |\delta(v)|/128$, and $\delta^{\operatorname{up}}(v)\leq \delta^{\operatorname{down}}(v)/\log m$. We again consider two cases. The first case is when $|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq |\delta(v)|/64$. In this case, $|\delta^{\operatorname{down}''}(v)|+|\delta^{\operatorname{up}}(v)|\leq |\delta^{\operatorname{down},\operatorname{straight}'}(v)|$ holds. We assign, to each edge $e\in \delta^{\operatorname{down},\operatorname{straight}''}(v)\cup \delta^{\operatorname{up}}(v)$, a distinct edge $e'\in \delta^{\operatorname{down},\operatorname{straight}'}(v)$, and we set $f(e')=f(e)$. Consider now the second case, where $|\delta^{\operatorname{down},\operatorname{right}}(v)|\geq |\delta(v)|/8$ must hold. In this case, $|\delta^{\operatorname{down}, \operatorname{straight}''}(v)|+|\delta^{\operatorname{up}}(v)|\leq |\delta^{\operatorname{down},\operatorname{straight}'}(v)|/10$. As before, we can assign, to each edge $e\in \delta^{\operatorname{down},\operatorname{straight}''}(v)\cup \delta^{\operatorname{up}}(v)$ ten distinct edges $e_1,\ldots,e_{10}\in \delta^{\operatorname{down},\operatorname{right}}(v)$, such that each edge of $\delta^{\operatorname{down},\operatorname{right}}(v)$ is assigned to at most one edge of $\delta^{\operatorname{down},\operatorname{straight}''}(v)\cup \delta^{\operatorname{up}}(v)$. If edge $e'\in \delta^{\operatorname{down},\operatorname{right}}(v)$ is assigned to edge $e\in \delta^{\operatorname{down},\operatorname{straight}''}(v)\cup \delta^{\operatorname{up}}(v)$, then we set $f(e')=f(e)/10$.
Notice that, when vertices $v\in L'_1$ are processed, we are guaranteed that $\delta^{\operatorname{down}'}(v)=\emptyset$, and so eventually all flow originating at the edges of $\downedges{i}$ reaches the edges of $\rightedges{i}$. This completes the definition of the flow $f$. It is immediate to verify that $f$ is defined over the set ${\mathcal{P}}$ of paths; every edge in $\downedges{i}$ sends one flow unit; every edge in $\rightedges{i}$ receives at most $1/10$ flow units; and each edge of $E(S''_i)$ carries at most one flow unit.
\subsection{Proof of \Cref{claim: bound left and right for S''}}
\label{subsec: bound left and right for S''}
We prove that $|\leftedges{i}|\leq 1.1|\rightedges{i}|$; the proof that $|\rightedges{i}|\leq 1.1|\leftedges{i}|$ is symmetric.
Consider the cut $(A,B)$ in graph $\check H$, where $A=V(S_1)\cup\cdots\cup V(S_{i-1})$, and $B=V(S_i)\cup\cdots\cup V(S_r)$ (see \Cref{fig: NF14a}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF14a.jpg}
\caption{Illustration for the proof of \Cref{claim: bound left and right for S''}, with cut $(A,B)$ shown in a pink dashed line.}\label{fig: NF14a}
\end{figure}
Note that $A$ and $B$ are precisely the sets of vertices of the two connected components of the graph $\tau\setminus\set{(u_{i-1},u_i)}$, where $\tau$ is the Gomory-Hu tree of graph $\check H$. Therefore, $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut in $\check H$. We now consider another $u_{i-1}$-$u_i$ cut $(A',B')$ in $\check H$, where $A'=A\cup V(S''_i)$, and $B'=B\setminus V(S''_i)$ (see \Cref{fig: NF14b}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF14b.jpg}
\caption{Illustration for the proof of \Cref{claim: bound left and right for S''}, with cut $(A',B')$ shown in a pink dashed line. Edges of $\delta^{\operatorname{down}}(S''_{i})$ are shown in black.}\label{fig: NF14b}
\end{figure}
Observe that edges of $\leftedges{i}$ contribute to $E(A,B)$ but not to $E(A',B')$, while edges of $\rightedges{i}\cup \downedges{i}$ contribute to $E(A',B')$ but not to $E(A,B)$, and this is the only difference between the two edge sets. Since $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut, we get that $|\leftedges{i}|\leq |\rightedges{i}|+|\downedges{i}|$ must hold. Since, from \Cref{claim: bound S' to S'' edges}, $|\downedges{i}|\leq 0.1|\rightedges{i}|$, we conclude that $|\leftedges{i}|\leq 1.1|\rightedges{i}|$.
\subsection{Proof of \Cref{claim left edges for S' and S''}}
\label{subsec: left edges for S' and S''}
Fix an index $1<i<r$. We start by proving that $|\rightedges{i}|\leq 1.3|E_{i}|+1.3|\rightCedges{i}|$.
As before, we consider the cut $(A,B)$ in graph $\check H$, where $A=V(S_1)\cup\cdots\cup V(S_{i-1})$, and $B=V(S_i)\cup\cdots\cup V(S_r)$ (see \Cref{fig: NF16a}). As before, $A$ and $B$ are precisely the sets of vertices of the two connected components of the graph $\tau\setminus\set{(u_{i-1},u_i)}$, where $\tau$ is the Gomory-Hu tree of graph $\check H$, and so $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut in $\check H$. We now consider another $u_{i-1}$-$u_i$ cut $(A',B')$ in $\check H$, where $B'=V(S'_i)$, and $A'=V(\check H)\setminus V(S'_i)$ (see \Cref{fig: NF16b}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF16a.jpg}
\caption{Illustration for the proof of \Cref{claim left edges for S' and S''}, with cut $(A,B)$ shown in a pink dashed line. Edges of $\delta^{\operatorname{left}}(S'_{i})$ are shown in light green, and edges of $\delta^{\operatorname{left}}(S''_{i})$ are shown in blue.
}\label{fig: NF16a}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF16b.jpg}
\caption{Illustration for the proof of \Cref{claim left edges for S' and S''}, with cut $(A',B')$ shown in a pink dashed line. Edges of $\delta^{\operatorname{right}}(S'_{i})$ are shown in brown, and edges of $\delta^{\operatorname{left}}(S'_{i})$ are shown in green.}\label{fig: NF16b}
\end{figure}
Note that $|E(A,B)|\geq |E_{i-1}|+|\leftCedges{i}|+|\leftedges{i}|$, while $|E(A',B')|=|E_{i-1}|+|\leftCedges{i}|+|\downedges{i}|+|E_{i}|+|\rightCedges{i}|$. From the fact that $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut, it must be the case that:
\[
|\leftedges{i}|\leq |\downedges{i}|+|E_{i}|+|\rightCedges{i}|
\]
Since, from \Cref{claim: bound left and right for S''}, $|\rightedges{i}|\leq 1.1|\leftedges{i}|$, and, from \Cref{claim: bound S' to S'' edges}, $|\downedges{i}|\leq 0.1|\rightedges{i}|$, we get that:
\[
\begin{split}
|\rightedges{i}|&\leq 1.1|\leftedges{i}|\\ &\leq 1.1|\downedges{i}|+1.1|E_{i}|+1.1|\rightCedges{i}|\\ &\leq 0.11|\rightedges{i}|+1.1|E_{i}|+1.1|\rightCedges{i}|,
\end{split}
\]
and so $|\rightedges{i}|\leq 1.3|E_{i}|+1.3|\rightCedges{i}|$.
The proof that $|\leftedges{i+1}|\leq 1.3|E_{i}|+1.3|\leftCedges{i+1}|$ is symmetric.
We consider the cut $(X,Y)$ in graph $\check H$, where $X=V(S_1)\cup\cdots\cup V(S_{i+1})$, and $Y=V(S_{i+2})\cup\cdots\cup V(S_r)$ (see \Cref{fig: NF22a}). Note that $X$ and $Y$ are precisely the sets of vertices of the two connected components of the graph $\tau\setminus\set{(u_{i+1},u_{i+2})}$, where $\tau$ is the Gomory-Hu tree of graph $\check H$, and so $(X,Y)$ is the minimum $u_{i+1}$-$u_{i+2}$ cut in $\check H$. We now consider another $u_{i+1}$-$u_{i+2}$ cut $(X',Y')$ in $\check H$, where $X'=V(S'_{i+1})$, and $Y'=V(\check H)\setminus V(S'_{i+1})$ (see \Cref{fig: NF22b}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF22a.jpg}
\caption{Illustration for the proof of \Cref{claim left edges for S' and S''}, with cut $(X,Y)$ shown in a pink dashed line. Edges of $\delta^{\operatorname{right}}(S'_{i+1})$ are shown in light green, and edges of $\delta^{\operatorname{right}}(S''_{i+1})$ are shown in dark green.}\label{fig: NF22a}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF22b.jpg}
\caption{Illustration for the proof of \Cref{claim left edges for S' and S''}, with cut $(X',Y')$ shown in a pink dashed line. Edges of $\delta^{\operatorname{right}}(S'_{i+1})$ are shown in brown, and edges of $\delta^{\operatorname{left}}(S'_{i+1})$ are shown in green.}\label{fig: NF22b}
\end{figure}
Note that $|E(X,Y)|\geq |E_{i+1}|+|\rightCedges{i+1}|+|\rightedges{i+1}|$, while $|E(X',Y')|=|E_{i+1}|+|\rightCedges{i+1}|+|\leftCedges{i+1}|+|\downedges{i+1}|+|E_{i}|$. From the fact that $(X,Y)$ is the minimum $u_{i}$-$u_{i+1}$ cut, it must be the case that:
\[
|\rightedges{i+1}|\leq |\leftCedges{i+1}|+|\downedges{i+1}|+|E_{i}|
\]
Since, from \Cref{claim: bound left and right for S''}, $|\leftedges{i+1}|\leq 1.1|\rightedges{i+1}|$, and, from \Cref{claim: bound S' to S'' edges}, $|\downedges{i+1}|\leq 0.1|\leftedges{i+1}|$, we get that:
\[
\begin{split}
|\leftedges{i+1}|&\leq 1.1|\rightedges{i+1}|\\ &\leq 1.1|\downedges{i+1}|+1.1|E_{i}|+1.1|\leftCedges{i+1}|\\ &\leq 0.11|\leftedges{i+1}|+1.1|E_{i}|+1.1|\leftCedges{i}|,
\end{split}
\]
and so $|\leftedges{i+1}|\leq 1.3|E_{i}|+1.3|\leftCedges{i+1}|$.
\subsection{Proof of \Cref{claim left edges for S' only}}
\label{subsec: left edges for S' only}
Fix an index $1\leq i<r$.
As before, we consider the cut $(A,B)$ in graph $\check H$, where $A=V(S_1)\cup\cdots\cup V(S_{i-1})$, and $B=V(S_i)\cup\cdots\cup V(S_r)$. As before, $A$ and $B$ are precisely the sets of vertices of the two connected components of the graph $\tau\setminus\set{(u_{i-1},u_i)}$, where $\tau$ is the Gomory-Hu tree of graph $\check H$, and so $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut in $\check H$ (see \Cref{fig: NF17a}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF17a.jpg}
\caption{Illustration for the proof of \Cref{claim left edges for S' only}, with cut $(A,B)$ shown in pink dashed line. Edges of $\delta^{\operatorname{left}}(S'_{i})$ are shown in green.}\label{fig: NF17a}
\end{figure}
We now consider another $u_{i-1}$-$u_i$ cut $(A',B')$ in $\check H$, where $B'=V(S'_i)$, and $A'=V(\check H)\setminus V(S'_i)$ (see \Cref{fig: NF17b}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF17b.jpg}
\caption{Illustration for the proof of \Cref{claim left edges for S' only}, with cut $(A',B')$ shown in pink dashed line. Edges of $\delta^{\operatorname{left}}(S'_{i})$ are shown in green, and edges of $\delta^{\operatorname{right}}(S'_{i})$ are shown in brown.}\label{fig: NF17b}
\end{figure}
Note that $|E(A,B)|\geq |E_{i-1}|+|\leftCedges{i}|+|\tilde E_{i+1}^{\operatorname{left}}|$, while $|E(A',B')|=|E_{i-1}|+|\leftCedges{i}|+|\downedges{i}|+|E_{i}|+|\rightCedges{i}|$. From the fact that $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut, it must be the case that:
\[
|\tilde E_{i+1}^{\operatorname{left}}|\leq |\downedges{i}|+|E_{i}|+|\rightCedges{i}|
\]
Recall that, from \Cref{claim: bound S' to S'' edges}, $|\downedges{i}|\leq 0.1|\rightedges{i}|$, and, from \Cref{claim left edges for S' and S''}, $|\rightedges{i}|\leq 1.3|E_{i}|+1.3|\rightCedges{i}|$.
Therefore, $\downedges{i}\leq 0.13|E_i|+0.13|\rightCedges{i}|$, and:
\begin{equation}\label{eq: left bound }
|\tilde E_{i+1}^{\operatorname{left}}|\leq 1.13|E_{i}|+1.13|\rightCedges{i}|.
\end{equation}
Consider now the set $\leftCedges{i+1}$ of edges (see \Cref{fig: NF18}).
Recall that this set contains every edge $e=(u,v)$ with $u\in V(S'_{i+1})$, and $v$ either lying in $V(S_1)\cup\cdots\cup V(S_{i-1})$, or in $V(S''_i)$. In the former case, $e\in \tilde E_{i+1}^{\operatorname{left}}$, while in the latter case, $e\in \rightedges{i}$. Therefore, $|\leftCedges{i+1}|\leq |\tilde E_{i+1}^{\operatorname{left}}|+|\rightedges{i}|$. From \Cref{claim left edges for S' and S''}: $|\rightedges{i}|\leq 1.3|E_{i}|+1.3|\rightCedges{i}|$.
Combining this with \Cref{eq: left bound }, we get that $|\leftCedges{i+1}|\leq |\tilde E_{i+1}^{\operatorname{left}}|+|\rightedges{i}|\leq 2.5|E_{i}|+2.5|\rightCedges{i}|$, as required.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF18.jpg}
\caption{Set $\delta^{\operatorname{left}}(S'_{i+1})$ of edges (shown in green).}\label{fig: NF18}
\end{figure}
We now employ a symmetric argument in order to bound $|\rightCedges{i}|$: consider the cut $(X,Y)$ in graph $\check H$, where $X=V(S_1)\cup\cdots\cup V(S_{i+1})$, and $Y=V(S_{i+2})\cup\cdots\cup V(S_r)$ (see \Cref{fig: NF19a}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF19a.jpg}
\caption{Illustration for the proof of \Cref{claim left edges for S' only}, with cut $(X,Y)$ shown in pink dashed line. Edges of $\delta^{\operatorname{right}}(S'_{i+1})$ are shown in brown.}\label{fig: NF19a}
\end{figure}
As before, $X$ and $Y$ are precisely the sets of vertices of the two connected components of the graph $\tau\setminus\set{(u_{i+1},u_{i+2})}$, where $\tau$ is the Gomory-Hu tree of graph $\check H$, and so $(X,Y)$ is the minimum $u_{i+1}$-$u_{i+2}$ cut in $\check H$. We now consider another $u_{i+1}$-$u_{i+2}$ cut $(X',Y')$ in $\check H$, where $X'=V(S'_{i+1})$, and $Y'=V(\check H)\setminus V(S'_{i+1})$
(see \Cref{fig: NF19b}).
Note that $|E(X,Y)|\geq |E_{i+1}|+|\rightCedges{i+1}|+|\tilde E_i^{\operatorname{right}}|$, while $|E(X',Y')|=|E_{i+1}|+|\rightCedges{i+1}|+|E_{i}|+|\leftCedges{i+1}|+|\downedges{i+1}|$. From the fact that $(X,Y)$ is the minimum $u_{i+1}$-$u_{i+2}$ cut, it must be the case that:
\[
|\tilde E_i^{\operatorname{right}}|\leq |E_{i}|+|\leftCedges{i+1}|+|\downedges{i+1}|.
\]
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF19b.jpg}
\caption{Illustration for the proof of \Cref{claim left edges for S' only}, with cut $(X',Y')$ shown in pink dashed line. Edges of $\delta^{\operatorname{left}}(S'_{i+1})$ are shown in green, and edges of $\delta^{\operatorname{right}}(S'_{i+1})$ are shown in brown.}\label{fig: NF19b}
\end{figure}
As before, from \Cref{claim: bound S' to S'' edges}, $|\downedges{i+1}|\leq 0.1\cdot |\leftedges{i+1}|$, and, from \Cref{claim left edges for S' and S''}, $|\leftedges{i+1}|\leq 1.3|E_{i}|+1.3|\leftCedges{i+1}|$.
Therefore, $|\downedges{i}|\leq 0.13 |E_{i}|+0.13|\leftCedges{i+1}|$, and:
\begin{equation}\label{eq: right bound}
|\tilde E_i^{\operatorname{right}}|\leq 1.13|E_{i}|+1.13|\leftCedges{i+1}|.
\end{equation}
Consider now edge set $\rightCedges{i}$. Recall that it contains every edge $e=(u,v)$ with $u\in V(S_i)$, such that either $v\in V(S_{i+2})\cup\cdots \cup V(S_r)$, or $v\in V(S''_{i+1})$. In the former case, $e\in \tilde E_i^{\operatorname{right}}$, and in the latter case, $v\in \leftedges{i+1}$ holds. Therefore, $|\rightCedges{i}|\leq |\tilde E_i^{\operatorname{right}}|+|\leftedges{i+1}|$. From \Cref{claim left edges for S' and S''}: $|\leftedges{i+1}|\leq 1.3|E_{i}|+1.3|\leftCedges{i+1}|$. Therefore, altogether, $|\rightCedges{i}|\leq |\tilde E_i^{\operatorname{right}}|+|\leftedges{i+1}|\leq 2.5|E_{i}|+2.5|\leftCedges{i+1}|$.
\subsection{Proof of \Cref{claim: non-J-node-boundary size}}
\label{subsec: non-J-node boundary size}
Fix some index $1<i<r$, and recall how graph $S'_i\subseteq S_i$ was constructed. Initially, we set $V(S'_i)=\set{u_i}$. As long as there was any vertex $x\in S_i\setminus S'_i$, such that the number of edges connecting $x$ to vertices of $V(S'_i)$ was at least $|\delta(x)|/128$, we added $x$ to $V(S'_i)$. Recall that $|\delta^{\operatorname{up}}(x)|\leq |\delta(x)|/\log m$. Therefore, at least $|\delta(x)|\textsf{left}(\frac 1 {128}-\frac{1}{\log m}\textsf{right} )$ edges that connect $x$ to vertices of $V(S'_i)$ lie in $\delta^{\operatorname{down}}(x)$. We then let $S'_i$ be the subgraph of $\check H$ induced by $V(S'_i)$.
Denote $V(S'_i)=\set{u_i=x_0,x_1,\ldots,x_z}$, where the vertices $x_1,\ldots,x_z$ were added to $S'_i$ in the order of their indices.
Notice that for all $1\leq a\leq z$, if $x_a\in L'_{j_a}$, then, from the above discussion, at least one vertex in $\set{x_0,\ldots,x_{a-1}}$ must lie in $L'_1\cup L'_2\cup\cdots\cup L'_{j_a-1}$. It is then easy to see (by induction on $a$), that, if $u_i\in L'_j$, for some $1\leq j\leq h$, then every vertex of $S'_i\setminus\set{u_i}$ lies in $L'_{j+1}\cup\cdots\cup L'_h$.
For all $0\leq a\leq z$, we consider the vertex set $X_a=\set{x_0,\ldots,x_a}$, and we define a weight $w_a(e)$ of every edge $e\in E(\check H)$ with respect to $X_a$ as follows. First, we let $\delta^{\operatorname{up}}(X_a)$ be the set of all edges $e=(x,y)$ with $x\in X_a$, $y\not\in X_a$, such that $e\in \delta^{\operatorname{up}}(x)$. Every edge $e\in \delta^{\operatorname{up}}(X_a)$ is assigned weight $w_a(e)=130$. Every edge $e\in \textsf{left} ( \bigcup_{v\in X_a}\delta(v)\textsf{right} )\setminus \delta^{\operatorname{up}}(X_a)$ is assigned weight $w_a(e)=1$. All other edges $e\in E(\check H)$ are assigned weight $w_a(e)=0$. We denote $W_a=\sum_{e\in E(\check H)}w_a(e)$. Observe that $W_0=|\delta^{\operatorname{down}}(u_i)|+130|\delta^{\operatorname{up}}(u_i)|\leq |\delta(u_i)|\cdot \textsf{left}(1+\frac{130}{\log m}\textsf{right} )$. Additionally, $W_z\geq |\bigcup_{v\in X_z}\delta(v)|=|\bigcup_{v\in S'_i}\delta(v)|$. We now show that for all $1\leq a\leq z$, $W_a\leq W_{a-1}$. Notice that, if this is the case, then $|\delta(S'_i)|\leq W_z\leq W_0\leq |\delta(u_i)|\cdot \textsf{left}(1+\frac{130}{\log m}\textsf{right} )$. Therefore, in order to complete the proof of \Cref{claim: non-J-node-boundary size}, it is enough to prove the following observation.
\begin{observation}
For all $1\leq a\leq z$, $W_a\leq W_{a-1}$.
\end{observation}
\begin{proof}
Consider some index $1\leq a\leq z$. Recall that $X_a=X_{a-1}\cup \set{x_a}$. Recall that, as observed above, at least $|\delta(x_a)|\textsf{left}(\frac 1 {128}-\frac{1}{\log m}\textsf{right} )$ edges that connect $x_a$ to vertices of $X_{a-1}$ lie in $\delta^{\operatorname{down}}(x_a)$. Denote this edge set by $E^*\subseteq \delta^{\operatorname{up}}(X_a)$. Each edge $e\in E^*$ has $w_{a-1}(e)=130$, and $w_a(e)=1$. Additionally, every edge $e\in \delta^{\operatorname{down}}(x_a)\setminus E^*$ has $w_{a-1}(e)=0$ and $w_a(e)=1$. Finally, each edge $e\in \delta^{\operatorname{up}}(x_a)$ has $w_{a-1}(e)=0$ and $w_a(e)=130$. For all other edges $e'\in E(\check H)$, $w_{a-1}(e')=w_a(e')$.
Therefore, altogether, we get that:
\[ \begin{split}
W_{a}&=W_{a-1}-129|E^*|+|\delta^{\operatorname{down}}(x_a)\setminus E^*| +130|\delta^{\operatorname{up}}(x_a)|\\
&=W_{a-1}-130|E^*|+|\delta^{\operatorname{down}}(x_a)|+130|\delta^{\operatorname{up}}(x_a)|\\
&\leq W_{a-1}-130\cdot |\delta(x_a)|\textsf{left}(\frac 1 {128}-\frac{1}{\log m}\textsf{right} )+|\delta(x_a)|\cdot \textsf{left}(1 +\frac{130}{\log m}\textsf{right} )\\
&\leq W_{a-1}-\frac{|\delta(x_a)|}{64}+\frac{|\delta(x_a)|\cdot 260}{\log m}\\
&\leq W_{a-1}.
\end{split} \]
(we have used the fact that $|E^*|\geq |\delta(x_a)|\textsf{left}(\frac 1 {128}-\frac{1}{\log m}\textsf{right} )$ and that $m$ is sufficiently large).
\end{proof}
\subsection{Proof of \Cref{claim: many edges left right large}}
\label{subsec:many edges left right large}
Assume that there is an index $1\leq a\leq r$, such that at least $|\delta(u_{i^*})|/16$ edges connect $u_{i^*}$ to vertices of $S''_a$. We start by proving that $|\rightedges{a}|\geq |\delta(u_{i^*})|\cdot\frac {\log m}{256}$.
Denote by $E'$ the set of all edges connecting $u_{i^*}$ to vertices of $S''_a$, so $|E'|\geq |\delta(u_{i^*})|/16$. We denote by $E''=E'\cap \delta^{\operatorname{down}}(u_{i^*})$.
Since $|\delta^{\operatorname{up}}(u_{i^*})|\leq |\delta(u_{i^*})|/\log m$, $|E''|\geq |\delta(u_{i^*})|/32$.
Consider now a set ${\mathcal{P}}$ of paths, defined as follows: ${\mathcal{P}}$ contains every path $P$ of $\check H$, whose first edge lies in $E''$, last edge lies in $\rightedges{a}$, and all inner vertices lie in $S''_a$. We will show a flow $f$ defined over the paths in ${\mathcal{P}}$, in which every edge in $E''$ sends $\frac{\log m}{8}$ flow units, every edge in $\rightedges{a}$ receives at most one flow unit, and every edge $e\in E(S''_a)$ carries at most one flow unit. Clearly, this will prove that $|\rightedges{a}|\geq |E''|\cdot \frac{\log m}{8}\geq |\delta(u_{i^*})|\cdot\frac {\log m}{256}$.
We now focus on defining the flow $f$. Initially, we set the flow on every edge $e\in E''$ to $\frac{\log m}{8}$, and for every edge of $e'\in E(\check H)\setminus E''$, we set $f(e')=0$. Note that, if $e=(u_{i^*},v)\in E''$, then $e\in \delta^{\operatorname{up}}(v)$ must hold, since $e\in \delta^{\operatorname{down}}(u_{i^*})$ from the definition of $E''$. Next, we consider indices $j=h,h-1,\ldots,1$ one by one. We assume that, when index $j$ is considered, for every vertex $v\in L'_j\cap S''_a$, for every edge $e\in \delta^{\operatorname{up}}(v)$, the flow $f(e)$ is fixed, and $f(e)\leq \frac{\log m}{8}$. During the iteration when index $j$ is processed we will finalize the flow values $f(e')$ for every edge $e'\in \set{\delta^{\operatorname{down}}(v)\mid v\in L'_j\cap S''_a}$.
We now describe the iteration when index $j$ is processed. Consider any vertex $v\in L'_j\cap S''_a$. Recall that $|\delta^{\operatorname{up}}(v)|\leq |\delta(v)|/\log m$, and so the total flow that the edges of $\delta^{\operatorname{up}}(v)$ carry is bounded by $\frac{|\delta(v)|}{\log m}\cdot \frac{\log m}{8}\leq \frac{|\delta(v)|}{8}$.
Recall that, from \Cref{obs: left and right down-edges},
$|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{left}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq 63|\delta(v)|/64$, and
$|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|)$.
By combining the two inequalities, we get that $|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq 21|\delta(v)|/64$.
We now define the flow on every edge $e'\in \delta^{\operatorname{down}}(v)$, as follows. If $e'\in \delta^{\operatorname{down},\operatorname{left}}(v)\cup\delta^{\operatorname{down},\operatorname{straight}''}(v)$, then we set $f(e')=0$. Otherwise, $e'\in \delta^{\operatorname{down},\operatorname{right}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$. We then set $f(e')=\frac{\sum_{e\in \delta^{\operatorname{up}}(v)}f(e)}{|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|}$. From the above discussion, we are guaranteed that for every edge $e\in \delta^{\operatorname{down}}(v)$, $f(e)\leq 1$. This completes the description of the interation where index $j$ is processed. Once al indices $j=r,r-1,\ldots,1$ are processed, we obtain a final flow $f$.
From the construction of the flow $f$, flow conservation constraints hold for every vertex $v\in S''_a$. The only edges that carry non-zero flow are edges of $E''\cup E(S''_a)\cup \rightedges{a}$. Moreover, each edge in $E''$ carries $\frac{\log m}{8}$ flow units, and every edge in $\rightedges{a}$ carries at most one flow unit. We can then apply standard flow-paths decomposition of the flow $f$, to obtain a flow that is defined over the set ${\mathcal{P}}$ of paths, where every edge of $E''$ sends $\frac{\log m}{8}$ flow units, and every edge in $\rightedges{a}$ receives at most one flow unit. We conclude that $|\rightedges{a}|\geq |E''|\cdot \frac{\log m}{8}\geq |\delta(u_{i^*})|\cdot\frac {\log m}{256}$.
The proof that $|\leftedges{a}|\geq |\delta(u_{i^*})|\cdot\frac {\log m}{256}$ is symmetric.
\iffalse
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF20.jpg}
\caption{An illustration of edges set $\delta^{\operatorname{right}}(S_i')$ (shown in brown).\znote{this is Figure 20, pointer needed}}\label{fig: NF20}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF21a.jpg}
\caption{An illustration of cut $(A,B)$ (shown in pink dash line). Edges of $\delta^{\operatorname{right}}(S'_{i})$ are shown in brown, and edges of $\delta^{\operatorname{right}}(S''_{i})$ are shown in dark green.\znote{this is Figure 21a, pointer needed}}\label{fig: NF21a}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF21b.jpg}
\caption{An illustration of cut $(A',B')$ (shown in pink dash line). Edges of $\delta^{\operatorname{left}}(S'_{i})$ are shown in green, and edges of $\delta^{\operatorname{right}}(S'_{i})$ are shown in brown. \znote{this is Figure 21b, pointer needed}}\label{fig: NF21b}
\end{figure}
\fi
Lastly, we prove that $u_a$ is a $J$-node. Observe first that, since $S''_a\neq\emptyset$, from \Cref{obs: left and right down-edges}, $a\not\in\set{1,r}$. Conisder a cut $(A,B)$ in graph $\check H$, where $A=S_1\cup\cdots\cup S_{a-1}$, and $B=S_a\cup S_{a+1}\cup\cdots\cup S_r$ (see \Cref{fig: NF24a}). From the construction of the Gomory-Hu tree $\tau$, $(A,B)$ is a minimum $u_{a-1}$-$u_a$ cut in graph $\check H$. Since $\leftedges{a}\subseteq E(A,B)$, we get that $|E(A,B)|\geq |\leftedges{a}|\geq |\delta(u_{i^*})|\cdot\frac {\log m}{256}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF24a.jpg}
\caption{Illustration for the Proof of \Cref{claim: many edges left right large}, with cut $(A,B)$ shown in pink dashed line. Edges of $\delta^{\operatorname{left}}(S''_{a})$ are shown in green.}\label{fig: NF24a}
\end{figure}
Consider another $u_{a-1}$-$u_a$ cut $(A',B')$ in graph $\check H$, where $B'=S'_a$ and $A'=V(\check H)\setminus S'_a$ (see \Cref{fig: NF24b}). Then $|E(A',B')|\geq |E(A,B)|\geq |\delta(u_{i^*})|\cdot\frac {\log m}{256}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF24b.jpg}
\caption{Illustration for the Proof of \Cref{claim: many edges left right large}, with cut $(A',B')$ shown in pink dashed line.}\label{fig: NF24b}
\end{figure}
Assume now for contradiction that $u_a$ is not a $J$-node. Then from \Cref{claim: non-J-node-boundary size}, $|E(A',B')|=|\delta(S'_a)|\leq \textsf{left}(1+\frac{130}{\log m}\textsf{right} )|\delta(u_a)|\leq 2|\delta(u_a)|$. Therefore, we conclude that $|\delta(u_a)|\geq \frac{|E(A',B')|}{2}\geq
|\delta(u_{i^*})|\cdot\frac {\log m}{512}>|\delta(u_{i^*})|$, contradicting the choice of the index $i^*$ (we have used the fact that $m$ is sufficiently large).
\subsection{Proof of \Cref{claim: simplifying cluster case 1}}
\label{subsec: simplifying cluster Case 1}
We consider two cuts in graph $\check H$.
The first cut, $(A_1,B_1)$ is defined as follows:
$A_1=V(S_1)\cup\cdots\cup V(S_{i})$, $B_1=V(S_{i+1})\cup \cdots\cup V(S_r)$.
From the defintion of the Gomory-Hu tree $\tau$, and from \Cref{cor: G-H tree_edge_cut}, $(A_1,B_1)$ is a minimum $u_i$--$u_{i+1}$ cut in graph $\check H$, and moreover, there is a set ${\mathcal{P}}_1=\set{P(e)\mid e\in E(A_1,B_1)}$ of edge-disjoint paths in graph $\check H$, where, for each edge $e\in E(A_1,B_1)$, path $P(e)$ has $e$ as its first edge, vertex $u_{i}$ as its last vertex, and is internally disjoint from $B_1$, and hence from $S^*$.
We can define another $u_i$--$u_{i+1}$ cut $(A'_1,B'_1)$ in graph $\check H$, where $B'_1=V(S'_{i+1})$, and $A'_1=V(\check H)\setminus B'_1$. Since $(A_1,B_1)$ is a minimum $u_i$--$u_{i+1}$ cut, we get that:
\[|E(A_1,B_1)|\leq |E(A'_1,B_1')|=|\delta(S'_{i+1})|\leq 2|\delta(u_{i+1})|\leq 2|\delta(u_i)|\]
(we have used \Cref{claim: non-J-node-boundary size} for the penultimate inequality, and the definition of the index $i=i^*$ for the last inequality).
Similarly, we consider a second cut $(A_2,B_2)$, that is defined as follows: $A_2=V(S_1)\cup\cdots\cup V(S_{i+2})$, $B_2=V(S_{i+3})\cup \cdots\cup V(S_r)$. As before, from the defintion of the Gomory-Hu tree $\tau$, and from \Cref{cor: G-H tree_edge_cut}, $(A_2,B_2)$ is a minimum $u_{i+2}$--$u_{i+3}$ cut in graph $\check H$, and moreover, there is a set ${\mathcal{P}}_2=\set{P'(e)\mid e\in E(A_2,B_2)}$ of edge-disjoint paths in graph $\check H$, where, for each edge $e\in E(A_2,B_2)$, path $P'(e)$ has $e$ as its first edge, vertex $u_{i+3}$ as its last vertex, and is internally disjoint from $A_2$, and hence from $S^*$.
As before, we can define another $u_{i+2}$--$u_{i+3}$ cut $(A'_2,B'_2)$ in graph $\check H$, setting $A'_2=V(S'_{i+2})$, and $B'_2=V(\check H)\setminus A'_2$. Since $(A_2,B_2)$ is a minimum $u_{i+2}$--$u_{i+3}$ cut, we get that:
\[|E(A_2,B_2)|\leq |E(A'_2,B_2')|=|\delta(S'_{i+2})|\leq 2|\delta(u_{i+2})|\leq 2|\delta(u_i)|.\]
Observe that $\delta(S^*)=E(A_1,B_1)\cup E(A_2,B_2)$ (see \Cref{fig: NF25}). Therefore, from the above discussion, $|\delta(S^*)|\leq 4|\delta(u_i)|$. On the other hand, all
edges connecting $u_{i}$ to vertices of $\bigcup_{a>i+2}S_a$ lie in set
$\tilde E^{\operatorname{over}}_{i+1}$ (see \Cref{fig: NF25}), and so, from \Cref{obs: bad inded structure}, $|E_{i+1}'|\geq |\textbf{E}'_{i+1}^{\operatorname{over}}|\geq |\delta(u_{i})|/16$. Recall that
$E'_{i+1}=E(S_{i+1},S_{i+2})$, and so in particular, $E'_{i+1}\subseteq E(S^*)$. We conclude that $|E(S^*)|\geq |E'_{i+1}|\geq |\delta(u_{i})|/16\geq |\delta(S^*)|/64$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.15]{figs/NF25.jpg}
\caption{Illustration for the proof of \Cref{claim: simplifying cluster case 1}. The edges of $E^*$ are shown in red and brown. At least $|\delta(u_i)|/16$ edges of $E^*$ (shown in brown) have an endpoint in $S_{i+1}\cup\ldots,\cup S_r$, and these edges belong to set $E(A_2,B_2)$.}\label{fig: NF25}
\end{figure}
In order to prove that $S^*$ is a simplifying cluster, it is now enough to show a collection ${\mathcal{P}}^*=\set{P^*(e)\mid e\in \delta(S^*)}$ of paths in graph $\check H$, that cause congestion at most $\beta'=O(\log m)$, such that, for every edge $e\in \delta(S^*)$, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+3}$ as its last vertex, and it is internally disjoint from $S^*$.
Recall that we have defined a set
${\mathcal{P}}_2=\set{P'(e)\mid e\in E(A_2,B_2)}$ of edge-disjoint paths in graph $\check H$, where, for each edge $e\in E(A_2,B_2)$, path $P'(e)$ has $e$ as its first edge, vertex $u_{i+3}$ as its last vertex, and is internally $S^*$. For each edge $e\in E(A_2,B_2)$, we set $P^*(e)=P(e)$. Since $\delta(S^*)=E(A_1,B_1)\cup E(A_2,B_2)$, it remains to define the paths $P^*(e)$ for edges $e\in E(A_1,B_1)$.
As observed above, $|E(A_1,B_1)|\leq 2|\delta(u_i)|$.
From \Cref{obs: many through edges},
at least $ |\delta(u_{i})|/16$ edges connect $u_{i}$ to vertices of $\bigcup_{a>i+2}S_a$. Denote this set of edges by $\tilde E^*$. Clearly, $\tilde E^*\subseteq E(A_2,B_2)$ (see \Cref{fig: NF25}).
Since $|E(A_1,B_1)|\leq 2|\delta(u_i)| \leq 32|\tilde E^*|$, we can define a mapping $M$ from the edges of
$E(A_1,B_1)$ to edges of $\tilde E^*$, such that, for every edge $e'\in \tilde E^*$, at most $32$ edges of $E(A_1,B_1)$ are mapped to it. Consider now some edge $e\in E(A_1,B_1)$. The final path $P^*(e)$ is a concatenation of two paths. The first path is $P(e)\in {\mathcal{P}}_1$, that originates at $e$, terminates at $u_i$, and it is internally disjoint from $S^*$. Let $e'=M(e)$ be the edge of $\tilde E^*$ to which edge $e$ is mapped (recall that $e'$ must be incident to $u_i$). The second path is $P'(e')\in {\mathcal{P}}_2$, that starts at edge $e'$ and terminates at vertex $u_{i+3}$. This completes the definition of the set ${\mathcal{P}}^*=\set{P^*(e)\mid e\in \delta(S^*)}$ of paths. From the construction, for every edge $e\in \delta(S^*)$, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+3}$ as its last vertex, and it is internally disjoint from $S^*$. It now remains to bound the congestion of the path set ${\mathcal{P}}^*$. Recall that each of the path sets ${\mathcal{P}}_1,{\mathcal{P}}_2$ causes congestion $1$. Each path in ${\mathcal{P}}_1$ is used once, and each path in ${\mathcal{P}}_2$ may be used by up to $33$ paths. Therefore, the total congestion caused by paths in ${\mathcal{P}}^*$ is at most $34$. We conclude that $S^*$ is a simplfying cluster.
\subsection{Proof of \Cref{claim: simplifying cluster case 2}}
\label{subsec: simplifying cluster Case 2}
Since $u_{i+1}$ is a $J$-node, in order to prove that $S^*$ is a simplifying cluster, it is enough to show a collection ${\mathcal{P}}^*=\set{P^*(e)\mid e\in \delta(u_{i+1})}$
of paths in graph $\check H$, causing congestion at most $\beta'=O(\log m)$, where for each edge $e\in \delta(u_{i+1})$, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+2}$ as its last vertex, and is internally disjoint from $S^*$.
As before, we define two cuts in graph $\check H$: cut $(A_1,B_1)$, with $A_1=V(S_1)\cup \cdots\cup V(S_i)$ and $B_1=V(\check H)\setminus A_1$, and cut $(A_2,B_2)$, with $A_2=V(S_1)\cup\cdots\cup V(S_{i+1})$ and $B_2=V(\check H)\setminus A_2$. As before, from our construction, $(A_1,B_1)$ is a minimum $u_i$--$u_{i+1}$ cut in $\check H$, and so there is a set ${\mathcal{P}}_1=\set{P(e)\mid e\in E(A_1,B_1)}$ of edge-disjoint paths, that are internally disjoint from $B_1$, where for each edge $e\in E(A_1,B_1)$, path $P(e)$ has $e$ as its first edge and vertex $u_i$ as its last vertex. Similarly, $(A_2,B_2)$
is a minimum $u_{i+1}$--$u_{i+2}$ cut in $\check H$, and so there is a set ${\mathcal{P}}_2=\set{P'(e)\mid e\in E(A_2,B_2)}$ of edge-disjoint paths, that are internally disjoint from $A_2$, where for each edge $e\in E(A_2,B_2)$, path $P'(e)$ has $e$ as its first edge and vertex $u_{i+2}$ as its last vertex.
We partition the edge set $\delta(S^*)$ into three subsets (see \Cref{fig: NF26}).
The first subset, that we denote by $\delta_1$, contains all edges of $\delta(S^*)$ that lie in the set $E(A_2,B_2)$. The second set, that we denote by $\delta_2$, contains all edges of $\delta(S^*)$ that lie in $\delta^{\operatorname{down}}(S_{i+1}'')$ -- that is, they connect $u_{i+1}$ to vertices of $S''_{i+1}$. The third set $\delta_3$ contains all remaining edges. Note that every edge of $\delta_3$ must lie in $E_i\cup \leftCedges{i+1}\subseteq E(A_1,B_1)$.
We now consider each of the three sets of edges in turn.
\begin{figure}[h]
\centering
\includegraphics[scale=0.15]{figs/NF26.jpg}
\caption{Partition of the set $\delta(S^*)$ of edges into three subsets: set $\delta_1$ (shown in green), set $\delta_2$ (shown in black), and set $\delta_3$ (shown in red). The pink dashed line on the left shows cut $(A_1,B_1)$, and the pink dashed line on the right shows cut $(A_2,B_2)$.
}\label{fig: NF26}
\end{figure}
First, for every edge $e\in \delta_1$, we let $P^*(e)=P'(e)$, where $P'(e)$ is the path of ${\mathcal{P}}_2$, that has $e$ as its first edge, vertex $u_{i+2}$ as its last vertex, and is internally disjoint from $S^*$. We set ${\mathcal{P}}^*_1=\set{P^*(e)\mid e\in \delta_1}$. Clearly, ${\mathcal{P}}^*_1\subseteq {\mathcal{P}}_2$, and the paths in ${\mathcal{P}}^*_1$ are edge-disjoint.
Next, we consider the edges of $\delta_2=\delta^{\operatorname{down}}(S_{i+1}'')$. Recall that, from \Cref{claim: bound S' to S'' edges}, there is a set ${\mathcal{P}}^{\operatorname{right}}=\set{P^{\operatorname{right}}(e)\mid e\in \delta_2}$ of edge-disjoint paths in $\check H$, where, for each edge $ e\in \delta_2$, path $P^{\operatorname{right}}(e)$ has $e$ as its first edge, some edge of $\rightedges{i+1}$ as its last edge, and all inner vertices of $P^{\operatorname{right}}(e)$ are contained in $S''_{i+1}$, so that the paths of ${\mathcal{P}}^{\operatorname{right}}$ are internally disjoint from $S^*$. Consider an edge $e\in \delta_2$, and let $e'\in \rightedges{i+1}$ be the last edge on the path $P^{\operatorname{right}}(e)$. Then $e'\in E(A_2,B_2)$. We let $P^*(e)$ be the path obtained by concatenating the path $P^{\operatorname{right}}(e)$ with the path $P'(e')\in {\mathcal{P}}_2$. Clearly, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+2}$ as its last vertex, and it is internally disjoint from $S^*$. We set ${\mathcal{P}}^*_2=\set{P^*(e)\mid e\in \delta_2}$. It is easy to verify that the paths of ${\mathcal{P}}^*_2$ are edge-disjoint.
Lastly, we consider the edges of $\delta_3\subseteq E(A_1,B_1)$. Observe that a cut $(A_1',B_1')$, where $A_1'=\set{u_i}$, and $B_1'=V(\check H)\setminus A_1'$ is a $u_i$--$u_{i+1}$-cut in graph $\check H$, and so $|\delta_3|\leq |E(A_1,B_1)|\leq |E(A_1',B_1')|\leq |\delta(u_i)|$. We denote by $\tilde E'$ the set of all edges connecting $u_i$ to vertices of $S''_{i+1}$, and we denote by $\tilde E''$ the set of all edges connecitng $u_i$ to vertices of $S_{i+2}\cup\cdots\cup S_r$.
Recall that set $E^*$ of edges contains all edges connecting $u_i$ to vertices of $\bigcup_{a>i}S_a$.
Let $\check E_i\subseteq E_i$ be the set of edges connecting $u_i$ to $u_{i+1}$. Since $S'_{i+1}=\set{u_{i+1}}$, $E^*=\tilde E'\cup \tilde E''\cup \check E_i$. From our assumtion, $|E^*|\geq |\delta(u_i)|/4$.
Furthermore, since vertex $u_i$ was not added to the $J$-cluster corresponding to vertex $u_{i+1}$, $|\check E_i|\leq |\delta(u_i)|/128$.
Therefore, either $|\tilde E'|\geq |\delta(u_i)|/16$, or $|\tilde E''|\geq |\delta(u_i)|/16$ must hold.
We assume first that it is the latter. Since $|\delta_3|\leq |\delta(u_i)|$, we can define a mapping $M$ from the edges of $\delta_3$ to edges of $\tilde E''$, where, for each edge $e'\in \tilde E''$, at most $16$ edges of $\delta_3$ are mapped to $e'$. Observe that $\tilde E''\subseteq E(A_2,B_2)$. Consider now some edge $e\in \delta_3$. We obtain the path $P^*(e)$ by concatenating two paths: path $P(e)\in {\mathcal{P}}_1$, connecting $e$ to vertex $u_i$, and the path $P'(e')\in {\mathcal{P}}_2$, where $e'$ is the edge of $\tilde E''$ to which $e$ is mapped. Recall that $P'(e')$ has $e'$ as its first edge and $u_{i+2}$ as its last vertex; edge $e'$ is incident to $u_i$. Therefore, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+2}$ as its last vertex, and it is internally disjoint from $S^*$. We then set ${\mathcal{P}}^*_3=\set{P^*(e)\mid e\in \delta_3}$. It is easy to verify that the paths of ${\mathcal{P}}^*_3$ cause edge-congestion at most $17$.
Lastly, we assume that $|\tilde E'|\geq |\delta(u_i)|/16$. As before, we define a mapping $M$ from edges of $\delta_3$ to edges of $\tilde E'$, where, for each edge $e'\in \tilde E'$, at most $16$ edges of $\delta_3$ are mapped to $e'$.
Consider now some edge $e'=(u_i,v)\in \tilde E'$, and recall that $v\in S''_{i+1}$. Recall that the algorithm from \Cref{lem: prefix and suffix path} provided a construction of a right-monotone path $P(e',v)$. This path starts with edge $e'$, and it must terminate at some vertex of $V(S'_{i+2})\cup V(S'_{i+3})\cup\cdots\cup V(S'_r)$. All inner vertices on path $P(e',v)$ must lie in $V(S''_{i+1})\cup V(S''_{i+2})\cup\cdots\cup V(S''_r)$. Therefore, if $e''$ is the first edge of $P(e',v)$ that is not contained in $S''_{i+1}$, then $e''\in E(A_2,B_2)$. We denote by $\tilde P(e')$ the subpath of $P(e',v)$ that starts with edge $e'$ and ends with edge $e''$. From \Cref{lem: prefix and suffix path}, we are guaranteed that all paths in set $\set{\tilde P(e')\mid e'\in \tilde E'}$ cause congestion $O(\log m)$. Consider now some edge $e\in \delta_3$. We let $P^*(e)$ be a path that is a concatenation of three paths. The first path is the path $P(e)\in {\mathcal{P}}_1$, that connects $e$ to $u_i$. Let $e'\in \tilde E'$ be the edge to which $e$ is mapped by $M$. The second path is $\tilde P(e')$, connecting $e'$ to some edge $e''\in E(A_2,B_2)$. The third path is the path $P'(e'')\in{\mathcal{P}}_2$, connecting $e''$ to vertex $u_{i+2}$. It is immediate to verify that the resulting path $P^*(e)$ has $e$ as its first edge, $u_{i+2}$ as its last vertex, and it is internally disjoint from $S^*$.
We then set ${\mathcal{P}}^*_3=\set{P^*(e)\mid e\in \delta_3}$. It is easy to verify that the paths of ${\mathcal{P}}^*_3$ cause congestion at most $O(\log m)$.
Finally, we set ${\mathcal{P}}^*={\mathcal{P}}^*_1\cup {\mathcal{P}}^*_2\cup {\mathcal{P}}^*_3$. From our construction, the set ${\mathcal{P}}^*$ of paths routes the edges of $\delta(S^*)$ to vertex $u_{i+2}$, the paths of ${\mathcal{P}}^*$ are internally disjoint from $S^*$, and they cause edge-congestion $O(\log m)$ as required. We conclude that $S^*$ is a simplifying cluster.
\subsection{Proof of \Cref{claim: simplifying cluster case 3}}
\label{subsec: simplifying cluster Case 3}
Since $u_{i+2}$ is a $J$-node, in order to prove that $S^*$ is a simplifying cluster, it is enough to show a collection ${\mathcal{P}}^*=\set{P^*(e)\mid e\in \delta(u_{i+2})}$
of paths in graph $\check H$, where for each edge $e\in \delta(u_{i+2})$, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+3}$ as its last vertex, and is internally disjoint from $S^*$.
The construction of the set ${\mathcal{P}}^*$ of paths is almost identical to that from Case 2.
As before, we define two cuts in graph $\check H$:
cut $(A_1,B_1)$, with $A_1=V(S_1)\cup \cdots\cup V(S_{i+1})$ and $B_1=V(\check H)\setminus A_1$, and cut $(A_2,B_2)$, with $A_2=V(S_1)\cup\cdots\cup V(S_{i+2})$ and $B_2=V(\check H)\setminus A_2$
(see \Cref{fig: NF27}).
As before, from our construction, $(A_1,B_1)$ is a minimum $u_{i+1}$--$u_{i+2}$ cut in $\check H$, and so there is a set ${\mathcal{P}}_1=\set{P(e)\mid e\in E(A_1,B_1)}$ of edge-disjoint paths, that are internally disjoint from $B_1$, where for each edge $e\in E(A_1,B_1)$, path $P(e)$ has $e$ as its first edge and vertex $u_{i+1}$ as its last vertex. Similarly, $(A_2,B_2)$
is a minimum $u_{i+2}$--$u_{i+3}$ cut in $\check H$, and so there is a set ${\mathcal{P}}_2=\set{P'(e)\mid e\in E(A_2,B_2)}$ of edge-disjoint paths, that are internally disjoint from $A_2$, where for each edge $e\in E(A_2,B_2)$, path $P'(e)$ has $e$ as its first edge and vertex $u_{i+3}$ as its last vertex.
\begin{figure}[h]
\centering
\includegraphics[scale=0.15]{figs/NF27.jpg}
\caption{Illustration for the proof of \Cref{claim: simplifying cluster case 3}. Edges of $\delta_1$ are shown in green, edges of $\delta_2$ are shown in black, and edges of $\delta_3$ are shown in red. The pink dashed line on the left shows cut $(A_1,B_1)$ and the pink dashed line on the right shows cut $(A_2,B_2)$.
}\label{fig: NF27}
\end{figure}
As before, we partition the edge set $\delta(S^*)$ into three subsets. The first subset, that we denote by $\delta_1$, contains all edges of $\delta(S^*)$ that lie in the set $E(A_2,B_2)$. The second set, that we denote by $\delta_2$, contains all edges of $\delta(S^*)$ that lie in $\delta^{\operatorname{down}}(S_{i+2}'')$ -- that is, they connect $u_{i+2}$ to vertices of $S''_{i+2}$. The third set $\delta_3$ contains all remaining edges. As before, $\delta_3$ must lie in $E_{i+1}\cup \leftCedges{i+2}\subseteq E(A_1,B_1)$.
We consider each of the three sets of edges in turn.
The constructions of the path sets ${\mathcal{P}}^*_1=\set{P^*(e)\mid e\in \delta_1}$ and ${\mathcal{P}}^*_2=\set{P^*(e)\mid e\in \delta_2}$ remain exactly the same as in Case 2. We now focus on constructing the set ${\mathcal{P}}^*_3=\set{P^*(e)\mid e\in \delta_3}$ of paths.
Consider the edges of $\delta_3\subseteq E(A_1,B_1)$. Recall that we have assumed that $u_{i+1}$ is not a $J$-node. From the choice of the index $i^*=i$, $|\delta(u_{i+1})|\leq |\delta(u_i)|$. As before, we consider another $u_{i+1}$--$u_{i+2}$ cut $(A_1',B_1')$, where $A_1'=\set{u_{i+1}}$, and $B_1'=V(\check H)\setminus A_1'$. As before, $|\delta_3|\leq |E(A_1,B_1)|\leq |E(A_1',B_1')|\leq |\delta(u_{i+1})|\leq |\delta(u_i)|$.
We denote by $\tilde E'$ the set of all edges connecting $u_i$ to vertices of $S''_{i+2}$, and by $\tilde E''$ the set of all edges connecitng $u_i$ to vertices of $S_{i+3}\cup\cdots\cup S_r$.
Since $u_{i+2}$ is a $J$-node,
$S'_{i+2}=\set{u_{i+2}}$. As before, since vertex $u_i$ was not added to the $J$-cluster corresponding to vertex $u_{i+2}$, the number of edges connecting $u_i$ to $u_{i+2}$ is bounded by $|\delta(u_{i})|/128$.
Recall that we have established that at least $|\delta(u_i)|/8$ edges connect $u_i$ to vertices of $\bigcup_{a>i+1}V(S_a)$. Each such edge either lies in $\tilde E'\cup \tilde E''$, or it connects $u_i$ to $u_{i+2}$. Therefore, either $|\tilde E'|\geq |\delta(u_i)|/32$, or $|\tilde E''|\geq |\delta(u_i)|/32$ must hold.
The remainder of the construction of the paths in ${\mathcal{P}}^*_3$ is very similar to that for Case 2.
We assume first that $|\tilde E''|\geq |\delta(u_i)|/32$. Since $|\delta_3|\leq |\delta(u_i)|$, we can define a mapping $M$ from the edges of $\delta_3$ to edges of $\tilde E''$, where, for each edge $e'\in \tilde E''$, at most $32$ edges of $\delta_3$ are mapped to $e'$. Observe that $\tilde E''\subseteq E(A_1,B_1)$ and $\tilde E''\subseteq E(A_2,B_2)$. Consider now some edge $e\in \delta_3$. We obtain the path $P^*(e)$ by concatenating three paths. The first path is path $P(e)\in {\mathcal{P}}_1$, connecting $e$ to vertex $u_{i+1}$.
Denote by $e'$ the edge of $\tilde E''$ to which edge $e$ is mapped. The second path is path $P(e')\in {\mathcal{P}}_1$ (which we reverse), connecting vertex $u_{i+1}$ to edge $e'$. The third path is path $P'(e')\in {\mathcal{P}}_2$, connecting edge $e'$ to vertex $u_{i+3}$. Clearly, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+3}$ as its last vertex, and it is internally disjoint from $S^*$. We then set ${\mathcal{P}}^*_3=\set{P^*(e)\mid e\in \delta_3}$. It is easy to verify that the paths of ${\mathcal{P}}^*_3$ cause edge-congestion at most $O(1)$.
Finally, we assume that $|\tilde E'|\geq |\delta(u_i)|/32$. Recall that the edges of $\tilde E'$ connect vertex $u_i$ to vertices of $S''_{i+2}$, and in particular $\tilde E'\subseteq E(A_1,B_1)$.
As before, we define a mapping $M$ from edges of $\delta_3$ to edges of $\tilde E'$, where, for each edge $e'\in \tilde E'$, at most $32$ edges of $\delta_3$ are mapped to $e'$.
Consider now some edge $e'=(u_i,v)\in \tilde E'$, and recall that $v\in S''_{i+1}$. Recall that the algorithm from \Cref{lem: prefix and suffix path} provides a construction of a right-monotone path $P(e',v)$. This path starts with edge $e'$, and it must terminate at some vertex of $V(S'_{i+3})\cup V(S'_{i+4})\cup\cdots\cup V(S'_r)$. All inner vertices on path $P(e',v)$ must lie in $V(S''_{i+2})\cup V(S''_{i+3})\cup\cdots\cup V(S''_r)$. Therefore, if $e''$ is the first edge of $P(e',v)$ that is not contained in $S''_{i+2}$, then $e''\in E(A_2,B_2)$. We denote by $\tilde P(e')$ the subpath of $P(e',v)$ that starts with edge $e'$ and ends with edge $e''$. From \Cref{lem: prefix and suffix path}, we are guaranteed that all paths in set $\set{\tilde P(e')\mid e'\in \tilde E'}$ cause congestion $O(\log m)$. Consider now some edge $e\in \delta_3$. We let $P^*(e)$ be a path that is a concatenation of four paths. The first path is the path $P(e)\in {\mathcal{P}}_1$, that connects $e$ to $u_{i+1}$. Let $e'\in \tilde E'$ be the edge to which $e$ is mapped by $M$, and recall that $e'\in E(A_1,B_1)$. The second path is $P(e')\in {\mathcal{P}}_1$, which we reverse, so the path now connects vertex $u_{i+1}$ to edge $e'$.
The third path is
$\tilde P(e')$, connecting $e'$ to some edge $e''\in E(A_2,B_2)$. The fourth and the last path is the path $P'(e'')\in{\mathcal{P}}_2$, connecting $e''$ to vertex $u_{i+3}$. It is immediate to verify that the resulting path $P^*(e)$ has $e$ as its first edge, $u_{i+3}$ as its last vertex, and it is internally disjoint from $S^*$.
We then set ${\mathcal{P}}^*_3=\set{P^*(e)\mid e\in \delta_3}$. From the above discussion, the paths of ${\mathcal{P}}^*_3$ cause congestion at most $O(\log m)$.
Lastly, we set ${\mathcal{P}}^*={\mathcal{P}}^*_1\cup {\mathcal{P}}^*_2\cup {\mathcal{P}}^*_3$. It is easy to verify that the set ${\mathcal{P}}^*$ of paths routes the edges of $\delta(S^*)$ to vertex $u_{i+3}$, the paths are internally disjoint from $S^*$, and they cause congestion $O(\log m)$ as required. We conclude that $S^*$ is a simplifying cluster.
\subsection{Proof of \Cref{claim: simplifying cluster last case}}
\label{subsec: simplifying cluster last}
Since $u_{i+1}$ is a $J$-node, it is enough to show a collection ${\mathcal{P}}^*=\set{P^*(e)\mid e\in \delta(u_{i+1})}$
of paths in graph $\check H$, where for each edge $e\in \delta(u_{i+1})$, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+2}$ as its last vertex, and is internally disjoint from $S^*$.
As before, we define two cuts in graph $\check H$: cut $(A_1,B_1)$, with $A_1=V(S_1)\cup \cdots\cup V(S_i)$ and $B_1=V(\check H)\setminus A_1$, and cut $(A_2,B_2)$, with $A_2=V(S_1)\cup\cdots\cup V(S_{i+1})$ and $B_2=V(\check H)\setminus A_2$. As before, from our construction, $(A_1,B_1)$ is a minimum $u_i$--$u_{i+1}$ cut in $\check H$, and so there is a set ${\mathcal{P}}_1=\set{P(e)\mid e\in E(A_1,B_1)}$ of edge-disjoint paths, that are internally disjoint from $B_1$, where for each edge $e\in E(A_1,B_1)$, path $P(e)$ has $e$ as its first edge and vertex $u_i$ as its last vertex. Similarly, $(A_2,B_2)$
is a minimum $u_{i+1}$--$u_{i+2}$ cut in $\check H$, and so there is a set ${\mathcal{P}}_2=\set{P'(e)\mid e\in E(A_2,B_2)}$ of edge-disjoint paths, that are internally disjoint from $A_2$, where for each edge $e\in E(A_2,B_2)$, path $P'(e)$ has $e$ as its first edge and vertex $u_{i+2}$ as its last vertex.
As before, we partition the set $\delta(S^*)$ of edges into three subsets (see \Cref{fig: NF28}).
The first subset, that we denote by $\delta_1$, contains all edges of $\delta(S^*)$ that lie in the set $E(A_2,B_2)$. The second set, that we denote by $\delta_2$, contains all edges of $\delta(S^*)$ that lie in $\delta^{\operatorname{down}}(S_{i+1}'')$ -- that is, they connect $u_{i+1}$ to vertices of $S''_{i+1}$. The third set, $\delta_3$, contains all remaining edges. As before, very edge of $\delta_3$ must lie in $E_i\cup \leftCedges{i+1}\subseteq E(A_1,B_1)$.
We now consider each of the three sets of edges in turn.
\begin{figure}[h]
\centering
\includegraphics[scale=0.15]{figs/NF28.jpg}
\caption{Illustration for the proof of \Cref{claim: simplifying cluster last case}. Edges of $\delta_1$ are shown in green, edges of $\delta_2$ are shown in black, and edges of $\delta_3$ are shown in red. Additionally, edges of $\tilde E'$ are shown in purple and edges of $\tilde E''$ are shown in brown. The pink dashed line on the left shows cut $(A_1,B_1)$ and the pink dashed line on the right shows cut $(A_2,B_2)$.
}\label{fig: NF28}
\end{figure}
First, for every edge $e\in \delta_1$, we let $P^*(e)=P'(e)$, where $P'(e)$ is the path of ${\mathcal{P}}_2$, that has $e$ as its first edge, vertex $u_{i+2}$ as its last vertex, and is internally disjoint from $S^*$. We set ${\mathcal{P}}^*_1=\set{P^*(e)\mid e\in \delta_1}$. Clearly, ${\mathcal{P}}^*_1\subseteq {\mathcal{P}}_2$, and the paths in ${\mathcal{P}}^*_1$ are edge-disjoint.
Next, we consider the edges of $\delta_2=\delta^{\operatorname{down}}(S_{i+1}'')$. Recall that, from \Cref{claim: bound S' to S'' edges}, there is a set ${\mathcal{P}}^{\operatorname{right}}=\set{P^{\operatorname{right}}(e)\mid e\in \delta_2}$ of edge-disjoint paths in $\check H$, where, for each edge $ e\in \delta_2$, path $P^{\operatorname{right}}(e)$ has $e$ as its first edge, some edge of $\rightedges{i+1}$ as its last edge, and all inner vertices of $P^{\operatorname{right}}(e)$ are contained in $S''_{i+1}$, so that the paths of ${\mathcal{P}}^{\operatorname{right}}$ are internally disjoint from $S^*$. Consider an edge $e\in \delta_2$, and let $e'\in \rightedges{i+1}$ be the last edge on the path $P^{\operatorname{right}}(e)$. Then $e'\in E(A_2,B_2)$. We let $P^*(e)$ be the path obtained by concatenating the path $P^{\operatorname{right}}(e)$ with the path $P'(e')\in {\mathcal{P}}_2$. Clearly, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+2}$ as its last vertex, and it is internally disjoint from $S^*$. We set ${\mathcal{P}}^*_2=\set{P^*(e)\mid e\in \delta_2}$. It is easy to verify that the paths of ${\mathcal{P}}^*_2$ are edge-disjoint.
Lastly, we consider the edges of $\delta_3\subseteq E(A_1,B_1)$.
Clearly, $|\delta_3|\leq |E_i|+|\hat E_i|\leq 8|E_i|+7|\rightCedges{i}|+7|\leftCedges{i+1}|$, from \Cref{eq: bound on hat E}.
Note that, if $i=1$, then, from \Cref{obs: left and right down-edges},
$S'_1=S_1$, and so $\leftCedges{2}=\emptyset$. Otherwise, from
\Cref{claim left edges for S' only}, $|\leftCedges{i+1}|\leq 2.5|E_{i}|+2.5|\rightCedges{i}|$.
In either case, we get that:
$$|\delta_3|\leq 30|E_i|+29|\rightCedges{i}|\leq 30|\rightCedges{i}|,$$
since have assumed that
$|\rightCedges{i}|>64|E_i|$.
We partition the
set $\rightCedges{i}$ into two subsets: set $\tilde E'$, containing all edges $(u_i,v)$, with $v\in S_{i+1}''$, and set $\tilde E''$, containing all remaining edges. Note that, for each edge $(u_i,v)\in \tilde E''$, $v\in S_{i+2}\cup\cdots\cup S_{r}$ must hold.
The remainder of the construction of the set ${\mathcal{P}}^*_3=\set{P^*(e)\mid e\in \delta_3}$ of paths is very similar to the analysis of Case 2 in the proof of \Cref{lem: each ui is a J-node}. Since $|\delta_3|\leq 30|\rightCedges{i}|$, either $|\tilde E'|\geq |\delta_3|/60$ or $|\tilde E''|\geq |\delta_3|/60$ must hold. Assume first that it is the latter. Then we can define a map $M$ from the edges of $\delta_3$ to edges of $\tilde E''$, where, for each edge $e'\in \tilde E''$, at most $60$ edges of $\delta_3$ are mapped to $e'$. Observe that $\tilde E''\subseteq E(A_2,B_2)$. Consider now some edge $e\in \delta_3$. We obtain the path $P^*(e)$ by concatenating two paths: path $P(e)\in {\mathcal{P}}_1$, connecting $e$ to vertex $u_i$, and the path $P'(e')\in {\mathcal{P}}_2$, where $e'$ is the edge of $\tilde E''$ to which $e$ is mapped. Recall that path $P'(e')$ has $e'$ as its first edge and $u_{i+2}$ as its last vertex, and that edge $e'$ is incident to $u_i$. Therefore, path $P^*(e)$ has $e$ as its first edge, vertex $u_{i+2}$ as its last vertex, and it is internally disjoint from $S^*$. We then set ${\mathcal{P}}^*_3=\set{P^*(e)\mid e\in \delta_3}$. It is easy to verify that the paths of ${\mathcal{P}}^*_3$ cause edge-congestion at most $60$.
Lastly, we assume that $|\tilde E'|\geq |\delta_3|/60$. As before, we define a mapping $M$ from edges of $\delta_3$ to edges of $\tilde E'$, where, for each edge $e'\in \tilde E'$, at most $60$ edges of $\delta_3$ are mapped to $e'$.
Consider now some edge $e'=(u_i,v)\in \tilde E'$, and recall that $v\in S''_{i+1}$. Recall that the algorithm from \Cref{lem: prefix and suffix path} provided a construction of a right-monotone path $P(e',v)$. This path starts with edge $e'$, and it must terminate in some vertex of $V(S'_{i+2})\cup V(S'_{i+3})\cup\cdots\cup V(S'_r)$. All inner vertices on path $P(e',v)$ must lie in $V(S''_{i+1})\cup V(S''_{i+2})\cup\cdots\cup V(S''_r)$. Therefore, if $e''$ is the first edge of $P(e',v)$ that is not contained in $S''_{i+1}$, then $e''\in E(A_2,V_2)$. We denote by $\tilde P(e')$ the subpath of $P(e',v)$ that starts with edge $e'$ and ends with edge $e''$. From \Cref{lem: prefix and suffix path}, we are guaranteed that all paths in set $\set{\tilde P(e')\mid e'\in \tilde E'}$ cause congestion $O(\log m)$. Consider now some edge $e\in \delta_3$. We let $P^*(e)$ be a path that is a concatenation of three paths. The first path is the path $P(e)\in {\mathcal{P}}_1$, that connects $e$ to $u_i$. Let $e'\in \tilde E'$ be the edge to which $e$ is mapped by $M$. The second path is $\tilde P(e')$, connecting $e'$ to some edge $e''\in E(A_2,B_2)$. The third path is the path $P'(e'')\in{\mathcal{P}}_2$, connecting $e''$ to vertex $u_{i+2}$. It is immediate to verify that the resulting path $P^*(e)$ has $e$ as its first edge, $u_{i+2}$ as its last vertex, and it is internally disjoint from $S^*$.
We then set ${\mathcal{P}}^*_3=\set{P^*(e)\mid e\in \delta_3}$. From the above discussion, the paths of ${\mathcal{P}}^*_3$ cause congestion at most $O(\log m)$.
Finally, we set ${\mathcal{P}}^*={\mathcal{P}}^*_1\cup {\mathcal{P}}^*_2\cup {\mathcal{P}}^*_3$. It is easy to verify that the set ${\mathcal{P}}^*$ of paths routes the edges of $\delta(S^*)$ to vertex $u_{i+2}$ with congestion $O(\log m)$, and all paths of ${\mathcal{P}}^*$ are internally disjoint from $S^*$. We conclude that $S^*$ is a simplifying cluster, a contradiction.
\renewcommand{\textbf{E}'}{\textbf{E}'}
\subsection{Proof of \Cref{claim: computing out-paths}}
\label{subsec: computing out-paths}
We provide the construction of the set ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}$ of paths; the construction of the set ${\mathcal{P}}^{\mathsf{out},\operatorname{right}}$ of paths is symmetric.
We maintain a set ${\mathcal{R}}=\set{R(e)\mid e\in \hat E}$ of paths, that we gradually modify over the course of the algorithm. We will ensure that, throughout the algorithm, ${\mathcal{R}}$ is a collection of simple paths that contains, for each edge $e\in \hat E$, a path $R(e)$ that originates at $e$ and is a left-monotone path. Additionally, for every vertex $v\in V'$, the number of paths of ${\mathcal{R}}$ terminating at $v$ is $n_1(v)$ throughout the algorithm.
Initially, for each edge $e\in \hat E$, we let $R(e)$ be the path obtained by appending edge $e$ at the beginning of the path $P^1(e)\in \hat{\mathcal{P}}(e)$, and we set ${\mathcal{R}}=\set{R(e)\mid e\in \hat E}$.
Clearly, all invariants hold for the initial set ${\mathcal{R}}$ of paths.
We perform the algorithm as long as there are two paths $R(e),R(e')$ in ${\mathcal{R}}$, and some vertex $v$ that lies on both paths, such that the intersection of $R(e)$ and $R(e')$ at $v$ is transversal. Note that $v$ must be an inner vertex on both $R(e)$ and $R(e')$. Assume that path $R(e)$ terminates at some vertex $u\in V'$, while path $R(e')$ terminates at vertex $u'\in V'$. We perform splicing of paths $R(e),R(e')$ at vertex $v$ (see \Cref{subsec: non-transversal paths and splicing}), obtaining two new paths: path $\tilde R(e)$, whose first edge is $e$ and last vertex is $u'$; and path $\tilde R(e')$, whose first edge is $e'$ and last vertex is $u$. If any of the resulting paths $\tilde R(e),\tilde R(e')$ is non-simple, we remove cycles from it, until it becomes a simple path. We then update the set ${\mathcal{R}}$ of paths by replacing $R(e)$ and $R(e')$ with paths $\tilde R(e)$ and $\tilde R(e')$, respectively. It is easy to verify that, if $R(e),R(e')$ were both left-monotone paths, then so are paths $\tilde R(e)$ and $\tilde R(e')$. It is also immediate to verify that all other invariants hold. Clearly, when the algorithm terminates, the final set ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}={\mathcal{R}}$ of paths has all required properties.
It now remains to show that the algorithm is efficient. From \Cref{obs: splicing}, after every iteration, either (i) $\sum_{R\in {\mathcal{R}}}|E(R)|$ decreases, or (ii) $|\Pi^T({\mathcal{R}})|$ decreases, and $\sum_{R\in {\mathcal{R}}}|E(R)|$ remains fixed. It is then immediate to verify that the algorithm terminates after $\operatorname{poly}(|E(G)|)$ iterations.
\iffalse
We start by constructing a directed flow network $H$.
Initially, we let $V(H)=V(G)$, and then we process every edge $e=(u,v)$ of graph $G$ that lies in edge set $\tilde E''$.
We consider two cases. If there is some index $1\leq i\leq r$, such that $u,v\in \tilde S_i$, then we add two anti-parallel edges $(u,v)$ and $(v,u)$ to graph $H$ (that we view as copies of the edge $e$), and we set the capacity of each such edge to be $\cong_G(\hat {\mathcal{P}}^1,e)$ (that is, the number of paths in $\hat {\mathcal{P}}^1$ that contain $e$).
Otherwise, $e\in \hat E$ must hold. Assume that $u\in S_i,v\in S_j$, and $i<j$. In this case, we add a new vertex $x_e$ to graph $H$, and two directed edges $(v,x_e),(x_e,u)$. The capacity of each of these edges is $\cong_G(\hat {\mathcal{P}}^1,e)$ as before. Denote $X=\set{x_e\mid e\in \hat E}$. We add a source $s$ that connects to every vertex in $X$ with a directed edge of capacity $1$. We also add a destination $t$, and we connect every vertex $v\in V'$ to $t$ with an edge of capacity $n_1(v)$. Observe that the set $\hat {\mathcal{P}}^1$ of paths defines an $s$-$t$ flow in the resulting flow network $H$, whose value is $|X|$. Using standard algorithms for maximum flow, we can compute an $s$-$t$ flow $f$ in graph $H$ of value $|X|$, such that $f$ is integral and acyclic. In other words, if $W$ is any directed cycle in $H$, then there must be some edge $e\in W$ with $f(e)=0$. Let $H'$ be the graph obtained from $H$ by deleting every edge $e$ with $f(e)=0$ from it. Since flow $f$ is acyclic, the resulting graph $H'$ is an acyclic graph. Therefore, we can define an ordering ${\mathcal{O}}^*=(v_1,v_2,\ldots,v_N)$ of its vertices, such that, for every edge $(v_i,v_j)\in E(H')$, $j< i$ holds. From our construction of graph $H$, we can ensure that, for all $1\leq i< r$, all vertices of $S_i$ appear after vertices of $S_{i+1}$ in the ordering ${\mathcal{O}}^*$. We can now compute a standard flow-path decomposition of the flow $f$, to obtain a collection ${\mathcal{R}}=\set{R(e)\mid e\in \hat E}$ of paths in graph $H'$, where for each edge $e\in \hat E$, path $R(e)$ originates at vertex $x_e$, terminates at a vertex of $V'$, and is internally disjoint from $V'$. Moreover, from our construction, every vertex $v\in V'$ is an endpoint of exactly $n_1(v)$ paths in ${\mathcal{R}}$. Since the paths of $\hat {\mathcal{P}}$ cause edge-congestion at most $\eta$ in graph $G$ (from the definition of nice guiding paths), the paths of ${\mathcal{R}}$ cause edge-congestion at most $O(\log^{18}m)$. By suppressing the vertices of $X$ on the paths in ${\mathcal{R}}$, we obtain a collection ${\mathcal{R}}'=\set{R'(e)\mid e\in \hat E}$ of paths in graph $G$, that causes edge-congestion at most $O(\log^{18}m)$, such that, for each edge $e\in \hat E$, path $R'(e)$ originates at edge $e$, and is left-monotone. As before, every vertex $v\in V'$ is the last endpoint of exactly $n_1(v)$ such paths. However, the paths in ${\mathcal{R}}'$ may not be non-transversal with respect to $\Sigma$. We now gradually modify them to make them non-transversal with respect to $\Sigma$.
Let $V''=V(G)\setminus V'$, and let ${\mathcal{O}}''$ be the ordering of the vertices of $V''$ induced by the ordering ${\mathcal{O}}^*$. From our construction, for every path $R'(e)\in {\mathcal{R}}'(e)$, all vertices of $R'(e)$ (except for the last one) lie in $V''$, and the order of their appearance on path $R'(e)$ is consistent with the ordering ${\mathcal{O}}''$.
We consider the vertices of $V''$ one by one, in the order defined by ${\mathcal{O}}''$. We now describe an iteration when vertex $v\in V''$ is processed.
Let ${\mathcal{R}}(v)\subseteq {\mathcal{R}}'$ be the set of all paths containing the vertex $v$. For each path $R'(e)\in {\mathcal{R}}(v)$, we let $R_1(e)$ be the subpath of $R'(e)$ from the edge $e$ up to vertex $v$ (including it), and we let $R_2(e)$ be the subpath of $R'(e)$ from vertex $v$ to the last vertex of $R'(e)$, that must lie in $V'$. Let ${\mathcal{R}}_1(v)=\set{R_1(e)\mid R'(e)\in {\mathcal{R}}(v)}$, and ${\mathcal{R}}_2(v)=\set{R_2(e)\mid R'(e)\in {\mathcal{R}}(v)}$.
For every edge $e'\in \delta_G(v)$, we let $n^-(e')$ be the number of paths in set ${\mathcal{R}}_1(v)$, whose last edge is $e'$, and we let $n^+(e')$ be the number of paths in set ${\mathcal{R}}_2(v)$, whose first edge is $e'$. It is easy to verify that either $n^-(e')=0$ or $n^+(e')=0$ must hold for each such edge $e'$, and, moreover, $\sum_{e\in \delta_G(v)}n^-(e)=\sum_{e\in \delta_G(v)}n^+(e)$.
We use the algorithm from \Cref{obs:rerouting_matching_cong} with the rotation ${\mathcal{O}}_v\in \Sigma$, in order to compute a multiset $M\subseteq \delta_G(v)\times \delta_G(v)$ of $|M|=\sum_{e\in \delta_G(v)}n^+(e)$ ordered pairs of the edges of $\delta_G(v)$, such that, for every edge $e\in \delta_G(v)$, $M$ contains $n^-_e$ pairs $(e^-,e^+)$ of edges with $e^-=e$, and $n^+_e$ pairs $(e^-,e^+)$ of edges with $e^+=e$. Recall that we are also guaranteed that, for every pair $(e^-_1,e^+_1),(e^-_2,e^+_2)\in M$, the intersection between path $P_1=(e^-_1,e^+_1)$ and path $P_2=(e^-_2,e^+_2)$ at vertex $v$ is non-transversal with respect to the rotation ${\mathcal{O}}_v\in \Sigma$.
We associate every pair $(e',e'')\in M$ of edges with a distinct path $R_1(e)\in {\mathcal{R}}_1(v)$ that contains $e'$ as is last edge, and with a distinct path $R_2(\tilde e)\in {\mathcal{R}}_2(v)$ that contains $e''$ as its first edge. Therefore, we can view the set $M$ of edge pairs as defining a matching $M'$ between the paths of ${\mathcal{R}}_1(v)$ and the paths of ${\mathcal{R}}_2(v)$. For every edge $e\in \hat E$ with $R'(e)\in {\mathcal{R}}(v)$, we define a new path $R''(e)$, that is obtained by concatenating the path $R_1(e)$ with the path $R_2(\tilde e)$, where $R_2(\tilde e)$ is the path of ${\mathcal{R}}_2$ that is matched to $R_1(e)$ by $M'$. Let ${\mathcal{R}}'(v)=\set{R''(e)\mid R'(e)\in {\mathcal{R}}(v)}$ be this new set of paths. We then update the path set ${\mathcal{R}}'$ by deleting all paths of ${\mathcal{R}}(v)$ from it, and adding the paths of ${\mathcal{R}}'(v)$ to it instead. Note that the resulting set of path is still guaranteed to only contain left-monotone paths in graph $G$, that route the edges of $\hat E$ to vertices of $V'$, with edge-congestion at most $O(\log^{18}m)$. Note that, for every pair of paths in ${\mathcal{R}}'$ that contain the vertex $v$, their intersection at vertex $v$ is now non-transversal.
Once every vertex of $V''$ is processed, we obtain the final set of paths ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}={\mathcal{R}}'$. It is immediate to verify that each resulting path must be simple (from the fact that $H'$ is acyclic). From the above discussion, the paths in ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}$ cause edge-congestion at most $O(\log^{18}m)$, and, for each edge $e\in \hat E$, there is a unique path $P^{\mathsf{out},\operatorname{left}}(e)\in {\mathcal{P}}^{\mathsf{out},\operatorname{left}}$, whose first edge is $e$, and the path is left-monotone. Moreover, our algorithm ensures that the paths in ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}$ are non-transversal with respect to $\Sigma$, and that, for every vertex $v\in V'$, exactly $n_1(v)$ paths in ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}$ terminate at $v$.
\fi
\subsection{Proof of \Cref{claim: enough segments}}
\label{subsec: enough segments}
In order to prove the claim, we use the following observation.
\begin{observation}\label{obs: intervals}
Let ${\mathcal{I}}$ be a collection of $k$ intervals of non-zero length, where for each interval $I\in {\mathcal{I}}$, $I\subseteq [0,r]$, interval $I$ is closed on the left and open on the right, and the endpoints of $I$ are integers in $\set{0,\ldots,r}$.
Let ${\mathcal{I}}'$ be another collection of $k$ intervals of non-zero length, where for each interval $I'\in {\mathcal{I}}'$, $I'\subseteq [0,r]$, interval $I'$ is closed on the left and open on the right, and the endpoints of $I'$ are integers in $\set{0,\ldots,r}$. Assume further that for every integer $0\leq i\leq r$, the number of intervals of ${\mathcal{I}}$ for which $i$ serves as the left endpoint is equal to the number of intervals of ${\mathcal{I}}'$ for which $i$ serves as the left endpoint, and similarly, the number of intervals of ${\mathcal{I}}$ for which $i$ serves as the right endpoint is equal to the number of intervals of ${\mathcal{I}}'$ for which $i$ serves as the right endpoint. Then for every integer $p\in [0,r)$, the total number of intervals in ${\mathcal{I}}$ containing $p$ is equal to the total number of intervals in ${\mathcal{I}}'$ containing $p$.
\end{observation}
\begin{proof}
Let $p$ be any integer in $[0,r)$.
Consider any interval $I=(a,b)\in {\mathcal{I}}\cup {\mathcal{I}}'$. Clearly, $p\in I$ iff $a\leq p<b$.
Let ${\mathcal{I}}_1\subseteq {\mathcal{I}}$ be the set of all intervals $I=[a,b)\in {\mathcal{I}}$ with $a\leq p$, and let ${\mathcal{I}}_2\subseteq {\mathcal{I}}$ be the set of all intervals $I=[a,b)\in {\mathcal{I}}$ with $b\leq p$. Clearly, ${\mathcal{I}}_2\subseteq {\mathcal{I}}_1$, and an interval $I\in {\mathcal{I}}$ contains the point $p$ iff $I\in {\mathcal{I}}_1\setminus {\mathcal{I}}_2$. We define subsets ${\mathcal{I}}_1',{\mathcal{I}}_2'$ of intervals of ${\mathcal{I}}'$ similarly. As before, an interval $I'\in {\mathcal{I}}'$ contains the point $p$ iff $I'\in {\mathcal{I}}'_1\setminus {\mathcal{I}}'_2$. Since, for every integer $0\leq i\leq p$, the number of intervals in ${\mathcal{I}}$ whose left endpoint is $i$ is equal to the number of intervals in ${\mathcal{I}}'$ whose left endpoint is $i$, we get that $|{\mathcal{I}}_1|=|{\mathcal{I}}_1'|$. Similarly, since, for every integer $0\leq i\leq p$, the number of intervals in ${\mathcal{I}}$ whose right endpoint is $i$ is equal to the number of intervals in ${\mathcal{I}}'$ whose right endpoint is $i$, we get that $|{\mathcal{I}}_2|=|{\mathcal{I}}'_2|$.
Therefore, $|{\mathcal{I}}_1\setminus {\mathcal{I}}_2|=|{\mathcal{I}}_2'\setminus {\mathcal{I}}_2'|$. Since ${\mathcal{I}}_1\setminus {\mathcal{I}}_2$ is precisely the set of all intervals $I\in {\mathcal{I}}$ that contain $p$, and ${\mathcal{I}}_1'\setminus {\mathcal{I}}_2'$ is precisely the set of all intervals $I'\in {\mathcal{I}}'$ that contiain $p$, the observation follows.
\end{proof}
We construct two collections of intervals, ${\mathcal{I}}=\set{I(e)\mid e\in \hat E}$, and ${\mathcal{I}}'=\set{I'(e)\mid e\in \hat E}$, as follows. Consider an edge $e\in \hat E$. Assume that $\operatorname{span}'(e)=\set{i',i'+1,\ldots,j'-1}$, and that $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$. We then let $I(e)=[i',j')$ and $I'(e)=[i'',j'')$.
Let $L$ be the multiset of integers, that serve as the left endpoint of every interval in ${\mathcal{I}}$. Consider an integer $1\leq i\leq r$. The number of times that $i$ appears in set $L$ is equal to the number of edges $e\in \hat E$, such that $i$ is the first element of $\operatorname{span}'(e)$; equivalently, the prefix path $P^1(e)$ must terminate at a vertex of $S_i$. Therefore, the number of times that integer $i$ appears in $L$ is $\sum_{v\in V(S_i)}n_1(v)$. Note that, if $i$ is the left endpoint of some interval $I'(e)\in {\mathcal{I}}'$, then path $P^{\mathsf{out}}(e)$ must originate at a vertex of $S_i$. Therefore, path $P^{\mathsf{out},\operatorname{left}}(e)$ that was constructed by the algorithm from \Cref{claim: computing out-paths} must terminate at a vertex of $S_i$. From \Cref{claim: computing out-paths}, for every vertex $v\in V'$, the number of paths of ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}$ terminating at $v$ is exactly $n_1(v)$. Therefore, the number of intervals of ${\mathcal{I}}'$ for which $i$ serves as the left endpoint is equal to $\sum_{v\in V(S_i)}n_1(v)$, which is exactly the number of intervals of ${\mathcal{I}}$, for which $i$ serves as the left endpoint.
Using similar reasoning, for every integer $1\leq i\leq r$,
the number of intervals of ${\mathcal{I}}'$ for which $i$ serves as the right endpoint is equal to the number of intervals of ${\mathcal{I}}$, for which $i$ serves as the right endpoint.
Note that for each integer $1\leq t\leq r$, the number of intervals of ${\mathcal{I}}$ containing $t$ is precisely $N_t$, while the number of intervals of ${\mathcal{I}}'$ containing $t$ is precisely $N'_t$. We conclude that $N_t=N'_t$.
In order to prove the second assertion, observe that, for every index $1\leq t<r$, for every edge $e\in \hat E $ with $t\in \operatorname{span}'(e)$, the mid-segment $P^2(e)$ of the nice guiding path $P(e)\in \hat {\mathcal{P}}$ must contain some edge of $E_t$. Since the paths in $\hat {\mathcal{P}}$ cause congestion at most $O(\log^{18}m)$, we get that $N_t\leq O(\log^{18}m)\cdot |E_t|$.
\subsection{Proof of \Cref{obs: bound N' values}}
\label{subsec:bound N' values}
Consider an edge $e\in E(G)$. Recall that, if $e$ is a primary edge for an index $1\leq z\leq r$ (that is, $e\in E(\tilde S_z)\cup \delta_G(\tilde S_z)$), then $N'_z(e)=1$. Otherwise, $N'_z(e)$ is the number of paths in set: $$\set{Q(e')\mid e'\in E_{z-1}\cup E_z^{\operatorname{left}}}\cup \set{Q'(e')\mid e'\in E_z\cup E_z^{\operatorname{right}}}$$ that contain $e$. Here, for an edge $e'\in E_{z-1}\cup E_z^{\operatorname{left}}$, $Q(e')$ is the unique path of the internal router ${\mathcal{Q}}(U_{z-1})$ that originates at $e'$, and for an edge $e'\in E_z\cup E_z^{\operatorname{right}}$, $Q'(e')$ is the unique path of the external router ${\mathcal{Q}}'(U_z)$ that originates at $e'$.
If a secondary edge $e\in E(S_{z-1})\cup E(S_{z+1})$, then, from \Cref{obs: bound on num of copies}, $\expect{N'_z(e)}\leq \expect{N_z(e)}\leq \hat \eta$.
Otherwise, $N'_z(e)$ is the total number of auxiliary cycles in set $\set{W(e')\mid e'\in E_z^{\operatorname{left}}\cup E_z^{\operatorname{right}}}$
that contain the edge $e$. Consider now some edge $e'\in \hat E$, and assume that $\operatorname{span}(e')=\set{i,\ldots,j-1}$. Then $e'$ may lie in set $E_z^{\operatorname{left}}\cup E_z^{\operatorname{right}}$ only for $z\in \set{i,j}$. It may also be a primary edge only for indices $z\in \set{i,j}$.
Since, from \Cref{obs: bound congestion of cycles}, edge $e$ may appear on at most $O(\log^{34}m)$ cycles in ${\mathcal{W}}$, and since there are at most $O(1)$ indices $z$, for which $e\in E_{z-1}\cup E_z\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{right}}$, or $e$ is a primary edge, we get that, overall,
\[\expect{\sum_{z=1}^rN'_z(e)}\leq O(\hat \eta)+O(\log^{34}m)\leq O(\hat \eta).\]
\subsection{Proof of \Cref{obs: bound Pi triples}}
\label{subsec: proof of obs bound Pi triples}
Recall that, for $1\leq z\leq r$, $\Pi^T_z$ is the set of all triples $(e,e',v)$, where $e\in E_z^{\operatorname{right}}$, $e'\in \hat E_z$, and $v$ is a vertex that lies on both $W(e)$ and $W(e')$, such that cycles $W(e)$ and $W(e')$ have a transversal intersection at $v$. Note that $E_z^{\operatorname{right}}\subseteq \hat E_z$.
Recall that, from \Cref{obs: auxiliary cycles non-transversal at at most one}, for every pair $e,e'\in \hat E_z$ of edges, there is at most
one vertex $v$, such that $W(e)$ and $W(e')$ have a transversal intersection at vertex $v$. We say that a triple $(e,e',v)\in \Pi^T_z$ is a \emph{type-1 triple} if the cycles $W(e), W(e')$ share an edge. Let $(e,e',v)$ be a type-1 triple of $\Pi^T_z$, and let $e^*$ be an arbitrary edge shared by $W(e)$ and $W(e')$. We say that edge $e^*$ is \emph{responsible} for the triple $(e,e',v)$. If the cycles $W(e),W(e')$ do not share edges, then we say that triple $(e,e',v)$ is a type-2 triple.
We now bound the total number of type-1 triples in $\bigcup_{z=1}^r\Pi^T_z$. Consider an edge $e^*\in E(G)$. From \Cref{obs: bound congestion of cycles}, edge $e^*$ may appear on at most $O(\log^{34}m)$ cycles in ${\mathcal{W}}$. Consider now any such pair $W(e),W(e')$ of cycles. Assume that $\operatorname{span}(e)=\set{i,\ldots,j-1}$, and $\operatorname{span}(e')=\set{i',\ldots,j'-1}$. Recall that a triple $(e,e',v)$ may only lie in a set $\Pi^T_z$ if $e\in E_z^{\operatorname{right}}$, so $z=i$ must hold. Therefore, every pair $e,e'\in \hat E$ of edges, for which $e^*\in E(W(e))\cap E(W(e'))$, contributes at most $O(1)$ triples to set $\bigcup_{z=1}^r\Pi^T_z$. Overall, edge $e^*$ may be responsible for at most $O(\log^{68}m)$ triples in $\bigcup_{z=1}^r\Pi^T_z$, and the total number of type-1 triples in $\bigcup_{z=1}^r\Pi^T_z$ is at most $|E(G)|\cdot O(\log^{68}m)$.
Next, we consider a type-2 triple $(e,e',v)\in \Pi^T_z$. Observe that $v$ is the only vertex at which $W(e)$ and $W(e')$ have a transversal intersection, and cycles $W(e),W(e')$ do not share any edges. It is then easy to see that, in the drawing $\phi^*$ of graph $G$, there must be a crossing between an edge of $W(e)$ and an edge of $W(e')$. We say that this crossing is responsible for the triple $(e,e',v)$.
Consider now some crossing $(e_1,e_2)\in \chi^*$. As before, from \Cref{obs: bound congestion of cycles} edge $e_1$ lies on at most $O(\log^{34}m)$ cycles of ${\mathcal{W}}$, and the same bound holds for edge $e_2$. Therefore, there are at most $O(\log^{68}m)$ pairs $(e,e')\in \hat E$ of edges, with $e_1\in W(e)$ and $e_2\in W(e')$. As before, there is at most one index $z$ for which $e_1\in E_z^{\operatorname{right}}$, and at most one index $z$, for which $e_2\in E_z^{\operatorname{right}}$. Therefore, crossing $(e_1,e_2)$ may be responsible for at most $O(\log^{68}m)$ triples in $\bigcup_{z=1}^r\Pi^T_z$, and the total number of type-2 triples in $\bigcup_{z=1}^r\Pi^T_z$ is at most $|\chi^*|\cdot O(\log^{68}m)$.
\subsection{Proof of \Cref{obs: bound transversal pairs}}
\label{subsec: proof of bound transversal pairs}
From now on, we assume for contradiction that
we assume that $E(H(e_1))\cap E(H(e_2))=\emptyset$, and, additionally, there is no pair of edges $\tilde e_1\in H(e_1)$, $\tilde e_2\in H(e_2)\cup W''(e_2)$, whose images cross in $\phi^*$, and similarly, there is no pair of edges $\tilde e_1'\in H(e_1)\cup W''(e_1)$, $\tilde e_2'\in H(e_2)$, whose images cross in $\phi^*$.
From \Cref{obs: transversal pairs property}, edges $\hat a_{e_1}',\hat a_{e_2}',a'_{e_1},a'_{e_2}$ appear in this order in the rotation ${\mathcal{O}}_{u_{z-1}}\in \Sigma$.
We now define two points in the drawing $\phi^*$ of $G$: point $p$, that is an internal point on the image of edge $\hat a_{e_2}$, that is very close to the image of the vertex $\hat x_{e_2}$, such that the segment of the image of edge $a_{e_2}$ between $p$ and the image of $\hat x_{e_2}$ does not participate in any crossings. The second point, $p'$, is defined similarly on the image of edge $a_{e_2}$, very close to the image of vertex $x_{e_2}$
(see \Cref{fig: NN8}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN8.jpg}
\caption{Illustration for the proof of \Cref{obs: bound transversal pairs}. Edges $\hat a_{e_1}$ and $a_{e_2}$ are shown in red, with points $p$ and $p'$ marked. Path $W'(e_1)$ is the concatenation of the brown path and edges $a_{e_1}, \hat a_{e_1}$. Path $W''(e_1)$ is shown in pink.
}\label{fig: NN8}
\end{figure}
Next, we consider three curves in the drawing $\phi^*$ of $G$. The first curve, $\gamma_1$, is the union of the images of paths $\hat P(e_1)$ and $ P(e_1)$ in $\phi^*$. The second curve, $\gamma_2$, is the image of the path $W'(e_1)$, and the third curve, $\gamma_3$, is the image of the path $W''(e_1)$ in $\phi^*$
(see \Cref{fig: NN9}).
Observe that the endpoints of each of the three curves are the images of vertices $y_{e_1}$ and $\hat y_{e_1}$, and that points $p$ and $p'$ may not lie on any of these curves, as we have assumed that $E(H(e_1))\cap E(H(e_2))=\emptyset$. We will next show that the closed curve obtained by the union of curves $\gamma_1$ and $\gamma_2$ may not separate points $p$ and $p'$; the closed curve obtained by the union of the curves $\gamma_1$ and $\gamma_3$ must separate points $p$ and $p'$; and the closed curve obtained by the union of the curves $\gamma_2$ and $\gamma_3$ may not separate points $p$ and $p'$. We will then show that this is impossible, reaching a contradiction.
\begin{figure}[h]
\centering
\includegraphics[scale=0.1]{figs/NN9.jpg}
\caption{Curves $\gamma_1,\gamma_2$, and $\gamma_3$.}\label{fig: NN9}
\end{figure}
\begin{observation}\label{obs: first two curves}
Let $\tilde \gamma_1$ be the closed curve obtained by the union of the curves $\gamma_2$ and $\gamma_3$. The points $p$ and $p'$ are not separated by $\tilde \gamma_1$. In other words, if we consider the open regions into which curve $\tilde \gamma_1$ partitions the sphere, then points $p,p'$ lie in the same region.
\end{observation}
\begin{proof}
Assume otherwise. Consider another curve $\gamma^*$, which is obtained from the image of the path $W'(e_2)$, by truncating it so it connects point $p$ to point $p'$.
Notice that path $W'(e_2)$ may not share any vertices with $W''(e_1)$ (except possibly for the endpoints of path $W'(e_2)$, whose images do not appear on curve $\gamma^*$).
Moreover, cycles $W(e_1)$ and $W(e_2)$ may not have transversal intersections at a vertex of $V(S_{z-1})$ (from \Cref{obs: auxiliary cycles non-transversal at at most one}).
Since we have assumed that no edge of $W'(e_2)\subseteq H(e_2)$ may cross an edge of $H(e_1)\cup W''(e_1)$, we get that curve $\gamma^*$ may not cross curve $\tilde \gamma_1$, and so points $p$ and $p'$ may not be separated by curve $\tilde \gamma_1$.
\end{proof}
From \Cref{obs: first two curves}, we can define a simple curve $\zeta$, whose endpoints are $p$ and $p'$, such that neither of the curves $\gamma_2,\gamma_3$ crosses $\zeta$.
\begin{observation}\label{obs: second two curves}
Let $\tilde \gamma_2$ be the closed curve obtained by the union of the curves $\gamma_1$ and $\gamma_2$. Then points $p$ and $p'$ are not separated by $\tilde \gamma_2$. In other words, if we consider the open regions into which curve $\tilde \gamma_2$ partitions the sphere, then points $p,p'$ lie in the same region.
\end{observation}
\begin{proof}
Assume otherwise. Consider another curve $\gamma^*$, which is obtained from the image of the path $W''(e_2)$, by appending to it the image of edge $\hat a_{e_2}$ between the image of vertex $\hat y_{e_2}$ and point $p$, and the image of edge $a_{e_2}$ between the image of vertex $y_{e_2}$ and point $p'$.
Notice that path $W''(e_2)$ may not share any vertices with paths $W'(e_1)$, $P(e_1)$, and $P(e_2)$ (except for possibly the endpoints of $W''(e_2)$). Observe also that paths $W'(e_1),P(e_1)$ both contain the edge $a_{e_1}=(x_{e_1},y_{e_1})$, so, even if vertex $y_{e_1}\in W''(e_2)$, curve $\gamma^*$ may not cross curve $\tilde \gamma_2$ at the image of vertex $y_{e_1}$. Similarly, paths $W'(e_1),\hat P(e_1)$ both contain the edge $\hat a_{e_1}=(\hat x_{e_1},\hat y_{e_1})$. Therefore, even if vertex $\hat y_{e_1}\in W''(e_2)$, curve $\gamma^*$ may not cross curve $\tilde \gamma_2$ at the image of vertex $\hat y_{e_1}$. Since we have assumed that no edge of $W''(e_2)$ may cross an edge of $H(e_1)$, we conclude that curve $\gamma^*$ may not cross curve $\gamma_2$. Therefore, points $p$ and $p'$ must lie in the same region defined by $\tilde \gamma_2$.
\end{proof}
From \Cref{obs: second two curves}, we can now define a simple curve $\zeta'$, whose endpoints are $p$ and $p'$, such that neither of the curves $\gamma_1,\gamma_2$ crosses $\zeta'$.
Let $\zeta^*$ be the curve obtained from the union of the curves $\zeta,\zeta'$. Recall that curve $\gamma_2$, that connects the images of the vertices $y_{e_1}$ and $\hat y_{e_1}$, may not cross the curve $\zeta^*$. Therefore, if we denote by $q$ and $q'$ the images of the vertices $y_{e_1}$ and $\hat y_{e_1}$, respectively, then points $q$ and $q'$ do not lie on curve $\zeta^*$, and they are not separated by curve $\zeta^*$. Lastly, we need the following observation.
\begin{observation}\label{obs: third two curves}
Let $\tilde \gamma_3$ be the closed curve obtained by the union of the curves $\gamma_1$ and $\gamma_3$. Then points $p$ and $p'$ are separated by $\tilde \gamma_3$. In other words, if we consider the set ${\mathcal{F}}$ open regions into which curve $\tilde \gamma_3$ partitions the sphere, then points $p,p'$ lie in different regions.
\end{observation}
\begin{proof}
Recall that we have assumed that $E(H(e_1))\cap E(H(e_2))=\emptyset$, and that no edge of $H(e_1)\cup W''(e_1)$ may cross an edge of $P(e_2)\cup \hat P(e_2)\subseteq H(e_2)$. Additionally, path $W''(e_1)$ is internally disjoint from paths $P(e_2)$ and $\hat P(e_2)$, and,
since paths in ${\mathcal{Q}}(S_{z-1})$ are non-transversal with respect to $\Sigma$, paths $P(e_1),\hat P(e_1),P(e_2),\hat P(e_2)$ do not have transversal intersections. Therefore, the image of path $P(e_2)$ beween point $p'$ and the image of vertex $u_{z-1}$ may not cross the curve $\tilde \gamma_3$, and it must be contained in a single region of ${\mathcal{F}}$, that we denote by $F$. Similarly, the image of path $\hat P(e_2)$ beween point $p$ and the image of vertex $u_{z-1}$ may not cross the curve $\tilde \gamma_3$, and it must be contained in a single region of ${\mathcal{F}}$, that we denote by $F'$. Lastly, recall that we have established that the images of edges $\hat a_{e_1}',\hat a_{e_2}',a'_{e_1},a'_{e_2}$ enter the image of edge $u_{z-1}$ in this circular order. Recall that edges $\hat a_{e_1}',a'_{e_1}$ lie on paths $\hat P(e_1),P(e_1)$, respectively, while edges $\hat a_{e_2}',a'_{e_2}$ lie on paths $\hat P(e_2),P(e_2)$, respectively. Since curve $\tilde \gamma_3$ is a closed curve, it must be the case that $F\neq F'$.
\end{proof}
To summarize, so far we have defined two points $p,p'$, and two curves $\zeta,\zeta'$ connecting them. We have also denoted by $\zeta^*$ the closed curve obtained by taking the union of these two curves. We also defined two points $q$ and $q'$, and two curves $\gamma_1,\gamma_3$ connecting them, and we denoted by $\tilde \gamma_3$ the closed curve obtained by taking the union of these two curves. We have established that curve $\zeta^*$ may not separate $q$ and $q'$, while curve $\tilde \gamma_3$ must separate $p$ and $p'$. Lastly, from the definition, curve $\gamma_1$ may not cross $\zeta'$, while curve $\gamma_3$ may not cross $\zeta$. We now show that this is impossible to achieve. We denote by $D$ a disc in the plane, whose boundary lies on curve $\zeta^*$, and whose interior contains the points $q$ and $q'$, and does not contain any point of $\zeta^*$. Such a disc must exist, since points $q$ and $q'$ are not separated by $\zeta^*$. Note that points $p$ and $p'$ do not lie in the interior of the disc $D$, and yet they are separated by curve $\tilde \gamma_3$. this may only happen if curve $\tilde \gamma_3$ intersects both $\zeta$ and $\zeta'$.
Consider curve $\tilde \gamma$ that is obtained from $\tilde \gamma_3$ by deleting the point $q$ from it. On this curve, we can mark two points $a$ and $b$, such that $a$ lies on $\zeta$, $b$ lies on $\zeta'$, and no point of $\tilde \gamma$ that lies between $a$ and $b$ belongs to $\zeta^*$. Denote by $\tilde \gamma'$ the segment of $\tilde \gamma$ between $a$ and $b$. Note that this segment is disjoint from the interior of $D$, so it may not contain the point $q'$. Therefore, either $\tilde \gamma\subseteq \gamma_1$, or $\tilde \gamma\subseteq \gamma_3$. In the former case, we get that $\gamma_1$ crosses $\zeta'$, while in the latter case, we get that $\gamma_3$ crosses $\zeta$, a contradiction.
\iffalse
Consider now the drawing $\phi^*$ of graph $G$. We let $p_1$ be an inner point on the image of edge $\hat a_{e_1}$, that is very close to the image of vertex $\hat x_{e_1}$, and we let $p_1'$ be an inner point on the image of edge $a_{e_1}$, that is very close to the image of vertex $x_{e_1}$. We delete from drawing $\phi^*$ the images of all edges and vertices except for those that belong to graph $H(e_1)$. The resulting drawing partitions the sphere into open regions, and we denote this set of open regions by ${\mathcal{F}}_1$. We denote by $F_1,F_1'\in {\mathcal{F}}_1$ the regions in whose interiors points $p_1$ and $p_1'$ lie, respectively.
Similarly, we let $p_2$ be an inner point on the image of edge $\hat a_{e_2}$, that is very close to the image of vertex $\hat x_{e_2}$, and we let $p_2'$ be an inner point on the image of edge $a_{e_2}$, that is very close to the image of vertex $x_{e_2}$. We delete from drawing $\phi^*$ the images of all edges and vertices except for those that belong to graph $H(e_1)$. The resulting drawing partitions the sphere into open regions, and we denote this set of open regions by ${\mathcal{F}}_2$. We denote by $F_2,F_2'\in {\mathcal{F}}_2$ the regions in whose interiors points $p_1$ and $p_1'$ lie, respectively. We need the following claim:
\begin{claim}\label{claim: points separated} Assume that for every pair of edges $\tilde e_1'\in E(H(e_1)), \tilde e_2'\in E(H(e_2))$, the images of $\tilde e_1'$ and $\tilde e_2'$ do not cross in the drawing $\phi^*$ of $G$. Then
either $F_1\neq F_1'$, or $F_2\neq F_2'$ must hold.
\end{claim}
We prove the claim below, after we complete the proof of \Cref{obs: bound transversal pairs} using it.
It now remains to prove \Cref{obs: bound transversal pairs}.
\begin{proofof}{\Cref{claim: points separated}}
Assume for contradiction that $F_1=F_1'$ and that $F_2=F_2'$. Consider the drawing $\phi^*$ of graph $G$. Then we can draw a curve $\gamma_1$, connecting points $p_1$ and $p_1'$, so that $\gamma_1$ does not cross the image of any vertex or any edge in $H(e_2)$. Let $\gamma^*_1$ be the curve obtained by the union of curve $\gamma_1$, the images of the paths $P(e_1),\hat P(e_1)$, the image of edge $\hat a_{e_1}$ between the image of $\hat x_{e_1}$ and point $p_1$, and the image of edge $a_{e_1}$, bewteen the image of vertex $x_{e_1}$ and point $p_1'$. Since we have assumed that no edge lies in both $H(e_1)$ and $H(e_2)$, and since the paths in ${\mathcal{Q}}(S_z)$ are non-transversal with respect to $\Sigma$, the images of the paths $P(e_2),\hat P(e_2)$ may not cross the curve $\gamma^*_1$, except at the image of vertex $u_{z-1}$.
Similarly, we can draw a curve $\gamma_2$, connecting points $p_2$ and $p_2'$, so that $\gamma_2$ does not cross the image of any vertex or any edge in $H(e_1)$. Let $\gamma^*_2$ be the curve obtained by the union of curve $\gamma_2$, the images of the paths $P(e_2),\hat P(e_2)$, the image of edge $\hat a_{e_2}$ between the image of vertex $\hat x_{e_2}$ and point $p_2$, and the image of edge $a_{e_2}$, between the image of vertex $x_{e_2}$ and point $p_2'$. Since we have assumed that no edge lies in both $H(e_1)$ and $H(e_2)$, and since the paths in ${\mathcal{Q}}(S_z)$ are non-transversal with respect to $\Sigma$, the images of the paths $P(e_1),\hat P(e_1)$ may not cross the curve $\gamma^*_2$, except at the image of vertex $u_{z-1}$.
We now obtained two closed curves, $\gamma^*_1$ and $\gamma^*_2$.
\end{proofof}
\fi
\iffalse
only old stuff below
\subsection{Proof of \Cref{obs: accounting}}
\label{apd: Proof of accounting}
We first prove property \ref{prop_1}.
First, from the construction of ${\mathcal{W}}_i$ and \Cref{obs: edge_occupation in outer and inner paths}, every edge in $E(G)\setminus \big(\bigcup_{1\le i\le r}E(S_i)\big)$ participates in at most $O(\cong({\mathcal{P}}))\le O(\log^{18} m)\le \beta^*$ paths in ${\mathcal{W}}$.
Consider now an index $1\le i\le r$ and consider an edge $e\in E(S_i)$.
Assume first that $i\in Z_{\text{fail}}$. From the construction of ${\mathcal{W}}$, \Cref{obs: edge_occupation in outer and inner paths} and the analysis in Step 1, $$\expect[]{\cong_G({\mathcal{W}},e)}\le O(\log^{18} m)\cdot \expect[]{\cong_G({\mathcal{Q}}_i,e)}\le O(\log^{18} m)\cdot O(\log^4m/\alpha^*)\le \beta^*.$$
Assume now that $i\in Z_{\text{succ}}$. From \Cref{obs: edge_occupation in outer and inner paths},
$$\expect[]{\cong_G({\mathcal{W}},e)}\le O(\log^{18} m)\cdot \expect[]{\cong_G({\mathcal{Q}}_i,e)}\le O(\log^{18} m)\cdot \expect[]{(\cong_G({\mathcal{Q}}_i,e))^2} \le O(\log^{18} m)\cdot\beta^*.$$
We then prove property \ref{prop_2}. Let $1\le i_1<\cdots<i_k\le r$ be all indices in $Z_{\text{fail}}$. We will show that, for each $1\le t\le k-1$, $\sum_{i_t\le i <i_{t+1}} |{\mathcal{W}}^{\operatorname{bad}}_i|\le |\delta(S_{i_t})|\cdot\cong_G({\mathcal{P}})$. In fact, since a bad path in ${\mathcal{W}}_i$ must contain an edge from a bad cluster, from the construction of set ${\mathcal{W}}_i$, for an edge $e\in E(U_i,\overline{U}_i)$, if the path $W_i(e)$ in ${\mathcal{W}}_i$ is bad, then it has to visit the cluster $S_{i_t}$, and therefore contain an edge of $|\delta(S_{i_t})|$. From \Cref{obs: edge_occupation in outer and inner paths}, each edge of $\delta(S_{i_t})$ participates in at most $\cong_G({\mathcal{P}})$ paths in ${\mathcal{W}}$, and it follows that $\sum_{i_t\le i <i_{t+1}} |{\mathcal{W}}^{\operatorname{bad}}_i|\le |\delta(S_{i_t})|\cdot\cong_G({\mathcal{P}})$.
Therefore,
\begin{equation*}
\begin{split}
\sum_{1\le i\le r-1}|{\mathcal{W}}^{\operatorname{bad}}_i|^2 & \text{ }\le
\sum_{1\le t\le k}\bigg(\sum_{i_t\le i<i_{t+1}}|{\mathcal{W}}^{\operatorname{bad}}_i|\bigg)^2\le
\sum_{1\le t\le k}|\delta(S_{i_t})|^2\cdot(\cong_G({\mathcal{P}}))^2\\
&\text{ }\le \sum_{1\le t\le k}O\big(\eta^*\cdot(\cong_G({\mathcal{P}}))^2\cdot (\chi(S_{i_t})+|E(S_{i_t})|)\big)
\le O\big(\eta^*\cdot \log^{36}m\cdot(\mathsf{cr}(\phi^*)+|E(G)|)\big).\\
\end{split}
\end{equation*}
We now prove \ref{prop_3}. Note that the paths in ${\mathcal{W}}^{\operatorname{light}}$ does not intersect any cluster $S_i$ with $i\in Z_{\text{fail}}$. Therefore,
from \Cref{obs: edge_occupation in outer and inner paths}, for each edge $e\notin\bigcup_{S\in {\mathcal{S}}}E(S)$, $\expect[]{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2}\le (\cong_G({\mathcal{P}}))^2\le O(\log^{36}m)$; for each edge $e\in E(S_i)$ where $i\in Z_{\text{fail}}$, $\expect[]{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2}=0$; and for each edge $e\in E(S_i)$ where $i\in Z_{\text{succ}}$, $\expect[]{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2}\le (\cong_G({\mathcal{P}}))^2\cdot\beta^*\le O(\log^{36}m\cdot \beta^*)$.
We then prove property \ref{prop_5}. From \Cref{obs: edge_occupation in outer and inner paths},
\[
\expect[]{\sum_{e\in E(S_i)}(\cong_G({\mathcal{W}},e))^2}\le
O(\log^{36}m)\cdot\expect[]{(\cong_G({\mathcal{Q}}_i,e))^2}\le O(\log^{36}m)\cdot \beta^*.
\]
We then prove property \ref{prop_4}.
In fact,
\[
\begin{split}
& \expect{\sum_{e,e'\in E(G)}\chi(e,e')\cdot \cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{light}},e')}\\
\le & \sum_{e,e'\in E(G)}\chi_{\phi}(e,e')\cdot \expect{\frac{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2+ (\cong_G({\mathcal{W}}^{\operatorname{light}},e'))^2}{2}}
\\
\le & \sum_{e\in E(G)}\bigg(\sum_{e'\in E(G)}\chi_{\phi}(e,e')\bigg)\cdot \expect{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2}
\\
\le & \sum_{e\in E(G)}\chi_{\phi}(e)\cdot\expect{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2}\le O(\beta^*\cdot\log^{36}m\cdot \mathsf{cr}(\phi^*)),
\end{split}
\]
where the last inequality uses property \ref{prop_3}.
\subsection{Proof of \Cref{obs: accounting_2}}
\label{apd: Proof of accounting_2}
We will prove that:
\begin{enumerate}
\item
\label{propnew_1}
if $e\in E_{\text{succ}}$, then $\expect[]{\cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{bad}},e')}\le O\big(\beta^*\cdot \log^{36}m\big)$;
\item
\label{propnew_2}
if $e\in E_{\text{fail}}$, then $\expect[]{\cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{bad}},e')}=0$; and
\item
\label{propnew_3}
if $e\notin E_{\text{succ}}\cup E_{\text{fail}}$, then $\expect[]{\cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{bad}},e')}\le O(\beta^*\cdot \log^{18}m)$.
\end{enumerate}
We first show \ref{propnew_1}. Assume $e\in E(S_i)$ and $e'\in E(S_j)$, so index $i\in Z_{\text{succ}}$. If $i=j$, then $e'\in E_{\text{succ}}$, so
\[
\begin{split}
\expect{ \cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{bad}},e')} &\le \expect{\frac{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2+ (\cong_G({\mathcal{W}}^{\operatorname{bad}},e'))^2}{2}}\\
&\le \expect{\frac{(\cong_G({\mathcal{W}},e))^2}{2}}+\expect{\frac{(\cong_G({\mathcal{W}},e'))^2}{2}}
\le O(\log^{36}m)\cdot \beta^*.
\end{split}
\]
We then show \ref{propnew_2}.
Note that, by definition, a path in ${\mathcal{W}}^{\operatorname{light}}$ may not contain an edge in $E_{\text{fail}}$, so $\cong_G({\mathcal{W}}^{\operatorname{light}},e)=0$, and it follows that $\expect[]{\cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{bad}},e')}=0$.
We next show \ref{propnew_3}. Since $e\notin E_{\text{succ}}\cup E_{\text{fail}}$, from \Cref{obs: edge_occupation in outer and inner paths}, $\cong_G({\mathcal{W}}^{\operatorname{light}},e)\le O(\log^{18}m)$. Therefore, from property \ref{prop_1} in \Cref{obs: accounting},
\[\expect[]{\cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{bad}},e')}\le O(\log^{18}m) \cdot \expect[]{\cong_G({\mathcal{W}}^{\operatorname{bad}},e')}\le O(\beta^*\cdot \log^{36}m).\]
\fi
\subsection{Proof of \Cref{thm: wld all paths congestion}}
\label{subsec: proof of wld cor}
Our algorithm consists of a number of phases. For all $j\geq 1$, the input to phase $j$ consists of a collection ${\mathcal{R}}_j$ of disjoint clusters of $S$, and, for every cluster $R\in {\mathcal{R}}_j$, two sets ${\mathcal{P}}_1(R),{\mathcal{P}}_2(R)$ of paths in graph $G$. We require that ${\mathcal{P}}_1(R)=\set{P_1(e)\mid e\in \delta_G(R)}$, where for every edge $e\in \delta_G(R)$, path $P(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, and all inner vertices of $P(e)$ lie in $V(S)\setminus V(R)$. Additionally, $\cong_G({\mathcal{P}}_1(R))\leq 400/\alpha$. We also require that there is a subset $\hat E_R\subseteq \delta_G(R)$ of at least $\floor{|\delta_G(R)|/64}$ edges, such that ${\mathcal{P}}_2(R)=\set{P_2(e)\mid e\in \hat E_R}$, where for every edge $e\in \hat E_R$, path $P(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, and all inner vertices of $P(e)$ lie in $V(S)\setminus V(R)$.
We denote by $S_j$ the subgraph of $G$ induced by the set $V(S)\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}_j}V(R)\textsf{right} )$ of vertices.
We will ensure that the following ivariants hold:
\begin{properties}{P}
\item for every cluster $R\in{\mathcal{R}}_j$, $|\delta_G(R)|\le |\delta_G(S)|$; \label{prop: small boundary each cluster}
\item every cluster $R\in{\mathcal{R}}_j$ has the $\alpha$-bandwidth property in graph $G$; \label{prop: bw prop each cluster}
\item $\sum_{R\in {\mathcal{R}}_j}|\delta_G(R)|\le 2|\delta_G(S)|\cdot \sum_{j'=0}^{j-1}\frac{1}{2^{j'}}$; \label{prop: total boundary sum}
\item the congestion caused by the set $\bigcup_{R\in {\mathcal{R}}_j}{\mathcal{P}}_2(R)$ of paths is at most $400j/\alpha$; \label{prop: bound on congestion}
\item $|\delta_G(S_j)|\leq |\delta_G(S)|/16^{j-1}$, and there is a set ${\mathcal{Q}}_j$ of paths in graph $G$, routing the edges of $\delta_G(S_j)$ to edges of $\delta_G(S)$, such that for every path in ${\mathcal{Q}}_j$, all inner vertices on the path lie in $V(S)\setminus V(S_j)$, and the paths in ${\mathcal{Q}}_j$ cause congestion at most $2/\alpha$. \label{prop: boundary goes down}
\end{properties}
The algorithm terminates once $\bigcup_{R\in {\mathcal{R}}_j}V(R)=V(S)$.
Notice that, if we ensure that the above properties hold after each phase, the number of phases of the algorithm is $z\leq \ceil{\log m}$ (since $|\delta_G(S_z)|\geq 1$ must hold). Once the algorithm terminates, we return the final set ${\mathcal{R}}_z$ of clusters. It is then
easy to verify that this set of clusters has all required properties.
The input to the first phase is ${\mathcal{R}}_1=\emptyset$, so $S_1=S$. The set ${\mathcal{Q}}_1$ of paths contains, for every edge $e\in \delta_G(S)$, a path $Q(e)$ consisting of the edge $e$ only.
It is easy to verify that all invariants hold for this input.
We now assume that we are given an input ${\mathcal{R}}_j$ to phase $j$, for which Properties \ref{prop: small boundary each cluster} -- \ref{prop: boundary goes down} hold. We now describe the algorithm for executing the $j$th phase.
The algorithm consists of two steps. In the first step, we apply the algorithm from \Cref{thm:well_linked_decomposition} to graphs $G$ and $S_j$ (if $S_j$ is not connected, then we apply the algorithm to every connected component of $S_j$). We let ${\mathcal{R}}'$ be the set of clusters that the algorithm returns. We start by setting ${\mathcal{R}}_{j+1}={\mathcal{R}}_j\cup {\mathcal{R}}'$ (but eventually we will discard some clusters from ${\mathcal{R}}_{j+1}$ in the second step). Before we continue to the second step, we verify that Invariants \ref{prop: small boundary each cluster} -- \ref{prop: total boundary sum} hold for the current set ${\mathcal{R}}_{j+1}$ of clusters.
Recall that, from Invariant \ref{prop: boundary goes down}, $|\delta_G(S_j)|\leq \delta_G(S)/16^{j-1}$. The algorithm from \Cref{thm:well_linked_decomposition} ensures that, for every cluster $R\in {\mathcal{R}}'$, $|\delta_G(R)|\leq |\delta_G(S_j)|\leq |\delta_G(S)|$. It also ensures that every cluster $R\in {\mathcal{R}}'$ has the $\alpha$-bandwidth property in $G$, and that:
\[\sum_{R\in {\mathcal{R}}'}|\delta_G(R)|\leq
|\delta_G(S_{j})|\cdot\textsf{left}(1+O(\alpha\cdot \log^{1.5} m)\textsf{right})\leq 2|\delta_G(S_{j})|\leq 2|\delta_G(S)|/16^{j-1}.
\]
(we have used the fact that $\alpha< \frac 1 {c\log^2 m}$ for a large enough constant $c$). Therefore, Invariants \ref{prop: small boundary each cluster} -- \ref{prop: total boundary sum} hold for the current set ${\mathcal{R}}_{j+1}$ of clusters. Notice that currently $V(S)=\bigcup_{R\in {\mathcal{R}}_{j+1}}V(R)$ holds.
We now describe the second step of the algorithm. Our goal is to discard some clusters from ${\mathcal{R}}_{j+1}$, and to define the sets ${\mathcal{P}}_1(R)$, ${\mathcal{P}}_2(R)$ of paths for each cluster that remains in ${\mathcal{R}}_{j+1}$, so that Invariants \ref{prop: bound on congestion} and \ref{prop: boundary goes down} hold.
In order to do so, we construct a flow network $H$, as follows. We start from graph $G$, and contract all vertices of $V(G)\setminus V(S)$ into a destination vertex $t$. We also contract every cluster $R\in {\mathcal{R}}_{j+1}$ into a vertex $u(R)$. Additionally, we add a source vertex $s$. For every cluster $R\in {\mathcal{R}}'$, we connect $s$ to vertex $u(R)$ with an edge of capacity $|\delta_G(R)|$. All remaining edges of $H$ have capacity $64$. This completes the definition of the flow network $H$.
Next, we compute a minimum $s$-$t$ cut $(X,Y)$ in $H$. We partition the edges of $E_H(X,Y)$ into two subsets: set $E'$ containing all edges incident to $s$, and set $E''$ containing all remaining edges. Recall that the capacity of every edge in $E''$ is $64$. Clearly, the value of the minimum $s$-$t$ cut in $H$ is at most:
$$\sum_{R\in {\mathcal{R}}'}|\delta_G(R)|\leq 2|\delta_G(S)|/16^{j-1},$$
as we could set $X=\set{s}$ and $Y=V(G)\setminus X$. Therefore, the total capacity of all edges in $E''$ is at most $2|\delta_G(S)|/16^{j-1}$, and $|E''|\leq \frac{2|\delta_G(S)|}{16^{j-1}\cdot 64}\leq \frac{|\delta_G(S)|}{16^j}$.
We discard from set ${\mathcal{R}}_{j+1}$ all clusters $R$ with $u(R)\in X$, obtaining the final set ${\mathcal{R}}_{j+1}$ of clusters. Clearly, Invariants
\ref{prop: small boundary each cluster} -- \ref{prop: total boundary sum} continue to hold for this final set of clusters.
Let $S_{j+1}$ the subgraph of $S$ induced by $V(S)\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}_{j+1}}V(R)\textsf{right} )$. Then $\delta_G(S_{j+1})= E''$, and so $|\delta_G(S_{j+1})|\leq |\delta_G(S)|/16^{j}$.
We now show that there exists a set ${\mathcal{Q}}_{j+1}$ of paths in graph $G$, routing the edges of $\delta_G(S_{j+1})$ to edges of $\delta_G(S)$, such that all inner vertices on every path lie in $V(S)\setminus V(S_{j+1})$, and the paths of ${\mathcal{Q}}_{j+1}$ cause congestion at most $2/\alpha$. In order to do so, we first consider the flow network $H$. Let ${\mathcal{P}}^*$ be the set of all paths $P$ in $H$, such that the first edge on $P$ lies in $E''$, the last vertex of $P$ is $t$, and all inner vertices of $P$ lie in $Y$.
From the maximum flow / minimum cut theorem, there is a flow $f$ in $H$, defined over the set ${\mathcal{P}}^*$ of paths, where every edge of $E''$ sends $64$ flow units. Note that the edges of $E'$ (and in particular, all edges incident to $s$) do not carry any flow in $f$. Let $H'$ be the graph obtained from $H$ after we contract all vertices of $X$ into a supernode $s^*$, and delete all edges incident to the orginal source $s$ from this graph. From the above discussion, there is an $s^*$-$t$ flow in the resulting graph, in which every edge of $E''$ carries one flow unit, and all other edges of $H'$ carry at most one flow unit each (the flow is obtained by scaling flow $f$ by factor $64$). Therefore, there is a collection ${\mathcal{Q}}$ of $|E''|$ edge-disjoint paths in graph $H'$, routing the edges of $E''$ to vertex $t$. Let ${\mathcal{R}}^*$ be the set of all clusters $R$ whose corresponding supernode $u(R)$ lies in $Y$. Since every cluster in ${\mathcal{R}}^*$ has $\alpha$-bandwidth property, from \Cref{claim: routing in contracted graph}, there is a collection ${\mathcal{Q}}_{j+1}$ of paths in graph $G$, routing the edges of $E''=\delta_G(S_{j+1})$ to edges of $\delta_G(S)$, with congestion at most $2/\alpha$, such that all inner vertices on every path in ${\mathcal{Q}}_{j+1}$ lie in $V(S)\setminus V(S_{j+1})$.
This establishes Property \ref{prop: boundary goes down}.
For every cluster $R\in {\mathcal{R}}_{j+1}\cap {\mathcal{R}}_j$, we leave the sets ${\mathcal{P}}_1(R)$ and ${\mathcal{P}}_2(R)$ of paths unchanged. This ensures that the congestion caused by the set ${\mathcal{P}}_1(R)$ of paths is at most $400/\alpha$, and that the total congestion caused by the set $\bigcup_{R\in {\mathcal{R}}_{j+1}\cap {\mathcal{R}}_j}{\mathcal{P}}_2(R)$ of paths is at most $400j/\alpha$. Next, we define the sets ${\mathcal{P}}_1(R)$ and ${\mathcal{P}}_2(R)$ of paths for clusters $R\in {\mathcal{R}}_{j+1}\setminus {\mathcal{R}}_j$.
Consider some cluster $R\in {\mathcal{R}}_{j+1}\setminus {\mathcal{R}}_j$, and recal that ${\mathcal{R}}_{j+1}\setminus {\mathcal{R}}_j\subseteq {\mathcal{R}}'$.
Recall that the algorithm from \Cref{thm:well_linked_decomposition} returned a set ${\mathcal{P}}'(R)=\set{P'(e)\mid e\in \delta_G(R)}$ of paths in graph $G$ with $\cong_G({\mathcal{P}}'(R))\leq 100$, such that, for every edge $e\in \delta_G(R)$, path $P'(e)$ has $e$ as its first edge and some edge of $\delta_G(S_j)$ as its last edge, and all inner vertices of $P'(e)$ lie in $V(S_j)\setminus V(R)$. We combine these paths with the set ${\mathcal{Q}}_j$ of paths given by Invariant \ref{prop: boundary goes down} to obtain the desired set ${\mathcal{P}}_1(R)=\set{P_1(e)\mid e\in \delta_G(R)}$ of paths, where for every edge $e\in \delta_G(R)$, path $P(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, and all inner vertices of $P(e)$ lie in $V(S)\setminus V(R)$. Since $\cong_G({\mathcal{P}}'(R))\leq 100$, while $\cong_G({\mathcal{Q}}_j)\leq 2/\alpha$, it is easy to verify that $\cong_G({\mathcal{P}}_1(R))\leq 400/\alpha$.
It now remains to define the sets ${\mathcal{P}}_2(R)$ of paths for all clusters $R\in {\mathcal{R}}_{j+1}\setminus {\mathcal{R}}_j$. In order to do so, we
consider again the flow network $H$. Recall that for every cluster $R\in {\mathcal{R}}_{j+1}\setminus {\mathcal{R}}_j$, $u(R)\in Y$ holds, and moreover, there is an edge $(s,u(R))$ of capacity $|\delta_G(R)|$, that belongs to $E'\subseteq E_H(X,Y)$.
Let ${\mathcal{P}}^{**}$ be the set of all paths $P$ that connect $s$ to $t$ in $H$, such that the first edge on $P$ lies in $E'$.
From the maximum flow/minimum cut theorem, there is a flow $f$ in $H$ over the set ${\mathcal{P}}^{**}$ of paths, in which every edge $e=(s,u(R))\in E'$ sends $|\delta_G(R)|$ flow units (the capacity of the edge $e$). Scaling this flow down by factor $64$, using the integrality of flow, and deleting the first edge from every flow-path, we obtain a collection ${\mathcal{Q}}'$ of edge-disjoint paths in graph $H$, such that, for every cluster $R\in {\mathcal{R}}_{j+1}\setminus {\mathcal{R}}_j$, at least $\floor{|\delta_G(R)|/64}$
paths in ${\mathcal{Q}}'$ originate at edges of $\delta_H(u(R))$, and all paths in ${\mathcal{Q}}'$ terminate at vertex $t$. As before, we use the algorithm from \Cref{claim: routing in contracted graph} in order to obtain a collection ${\mathcal{Q}}''$ of paths in graph $G$, such that, for every cluster $R\in {\mathcal{R}}_{j+1}\setminus{\mathcal{R}}_j$, there is a subset ${\mathcal{Q}}''(R)\subseteq {\mathcal{Q}}''$ of at least $\floor{|\delta_G(R)|/64}$ paths that originate at edges of $\delta_G(R)$, and all paths in ${\mathcal{Q}}''(R)$ terminate at edges of $\delta_G(S)$.
The algorithm ensures that, for every edge $e\in \bigcup_{R\in {\mathcal{R}}_{j+1}\setminus{\mathcal{R}}_j}\delta_G(R)$, at most one path of ${\mathcal{Q}}''$ uses $e$, and the total congestion caused by the paths of ${\mathcal{Q}}''$ is at most $2/\alpha$. Consider now a cluster $R\in {\mathcal{R}}_{j+1}\setminus{\mathcal{R}}_j$, and the corresponding set ${\mathcal{Q}}''(R)$ of paths. Let $Q\in {\mathcal{Q}}''(R)$ be any such path. Observe that path $Q$ may not be internally disjoint from $R$. We let $e$ be the last edge on $Q$ that belongs to $\delta_G(R)$, and we truncate path $Q$, so that it now originates at edge $e$ and terminates at some edge of $\delta_G(S)$. This ensures that path $Q$ is internally disjoint from $R$. We let ${\mathcal{P}}_2(R)$ be the resulting set of paths, obtained after every path of ${\mathcal{Q}}''(R)$ was processed. From the above discussion, the set ${\mathcal{P}}_2(R)$ routes a subset $\hat E_R\subseteq \delta_G(R)$ of at least $\floor{|\delta_G(R)|/64}$ edges to edges of $\delta_G(S)$; all paths in ${\mathcal{P}}_2(R)$ are internally disjoint from $R$; and the total congestion caused by the paths in $\bigcup_{R\in {\mathcal{R}}_{j+1}\setminus {\mathcal{R}}_j}{\mathcal{P}}_2(R)$ is at most $2/\alpha$. Altogether, the paths in $\bigcup_{R\in {\mathcal{R}}_{j+1}}{\mathcal{P}}_2(R)$ cause congestion at most $400(j+1)/\alpha$, establising Property \ref{prop: bound on congestion}.
\subsection{Proof of \Cref{claim: avoid guiding curves}}
\label{subsec: proof of claim avoind guiding curves}
Consider a pair of indices $0\leq j\leq r$ and $0\leq a<2^{r-j}$, and recall that there is a level-$j$ curve $\lambda_{j,a}\in \Lambda_j$ connecting point $p_{a\cdot 2^j}$ to point $p_{(a+1)\cdot 2^j}$. Recall that we have defined a segment $\sigma_{j,a}$ of the boundary of disc $D$, whose endpoints are $p_{a\cdot 2^j}$ and $p_{(a+1)\cdot 2^j}$, where $\sigma_{j,a}$ does not contain the point $p_{(a+1)\cdot 2^j+1}$. It will be convenient for us to view the segment $\sigma_{j,a}$ as closed on one side and open on another side, that is, $p_{a\cdot 2^j}\in \sigma_{j,a}$, and $p_{(a+1)\cdot 2^j}\not\in \sigma_{j,a}$. We let $T_a^j$ be the set of all anchor vertices whose images lie on the curve $\sigma_{j,a}$.
Notice that, for each level $j$, the collections $\set{T_a^j\mid 0\leq a<2^{r-j}}$ of vertices are disjoint from each other. For every pair $0\leq j<j'\leq r$ of levels and indices $0\leq a< 2^{r-j}$ and $0\leq a'< 2^{r-j'}$, either $\sigma_{j,a}\cap \sigma_{j',a'}=\emptyset$, or $\sigma_{j,a}\subseteq \sigma_{j',a'}$. In the former case, $T_a^j\cap T_{a'}^{j'}=\emptyset$, while in the latter case, $T_a^j\subseteq T_{a'}^{j'}$.
For all $0\leq j\leq r$ and $0\leq a<2^{r-j}$, we let $X^j_a$ be the subset of vertices of $G'$ with the following properties:
\begin{itemize}
\item $X^j_a\cap A=T^j_a$;
\item $|\delta_{G'}(X^j_a)|$ is minimized among all sets $X^j_a$ with the above property; and
\item $|X^j_a|$ is minimized among all sets $X^j_a$ with the above two properties.
\end{itemize}
In other words, we let $(X^j_a,V(G')\setminus X^j_a)$ be a minimum cut separating vertices of $T^j_a$ from the remaining vertices of $A$, that minimizes the number of vertices in $X^j_a$. Note that, from \Cref{obs: small boundary cuts}, $|\delta_{G'}(X^j_a)|\leq 4\check m'/\mu^{2b}$.
The following simple observation follows immediately from submodularity of cuts.
\begin{observation}\label{obs: laminar}
For every pair $0\leq j\leq j'\leq r$ of levels and indices $0\leq a< 2^{r-j}$ and $0\leq a'< 2^{r-j'}$, if $T_a^j\cap T_{a'}^{j'}=\emptyset$ then $X_a^j\cap X_{a'}^{j'}=\emptyset$, and if $T_a^j\subseteq T_{a'}^{j'}$, then $X_a^j\subseteq X_{a'}^{j'}$.
\end{observation}
\begin{proof}
Consider a pair $0\leq j\leq j'\leq r$ of levels, and indices $0\leq a< 2^{r-j}$ and $0\leq a'< 2^{r-j'}$. Assume first that $T_a^j\cap T_{a'}^{j'}=\emptyset$, but $X_a^j\cap X_{a'}^{j'}\neq \emptyset$. Let $Y=X_a^j\setminus X_{a'}^{j'}$ and $Y'=X_{a'}^{j'}\setminus X_{a}^{j}$. Since $X_a^j\cap A=T_a^j$ and $X_{a'}^{j'}\cap A=T_{a'}^{j'}$, we get that $Y\cap A=T_a^j$ and $Y'\cap A=T_{a'}^{j'}$. Since $X_a^j\cap X_{a'}^{j'}\neq \emptyset$, $|Y|< |X_{a}^j|$ and $|Y'|<|X_{a'}^{j'}|$ holds. Lastly, from submodularity of cuts:
%
\[|\delta_{G'}(Y)|+|\delta_{G'}(Y')|\leq |\delta_{G'}(X_{a}^j)|+ |\delta_{G'}(X_{a'}^{j'})|. \]
%
Since $|\delta_{G'}(X_{a}^j)|$ minimizes the number of edges in a cut separating the vertices of $T_a^j$ from the remaining vertices of $A$, and similarly $|\delta_{G'}(X_{a'}^{j'})|$ minimizes the number of edges in a cut separating the vertices of $T_{a'}^{j'}$ from the remaining vertices of $A$, $|\delta_{G'}(Y)|=|\delta_{G'}(X_{a}^j)|$ and $|\delta_{G'}(Y')|\leq |\delta_{G'}(X_{a'}^{j'})|$ must hold, a contradiction.
Assume now that $T_a^j\subseteq T_{a'}^{j'}$, but $X_a^j\not\subseteq X_{a'}^{j'}$.
Let $Y=X_a^j\cap X_{a'}^{j'}$, and let $Y'=X_{a'}^{j'}\cup X_{a}^{j}$. It is immediate to verify that $Y\cap A=T_a^j$, $Y'\cap A=T_{a'}^{j'}$, and $|Y|<|X_{a}^{j}|$. From submodularity of cuts:
%
\[|\delta_{G'}(Y)|+|\delta_{G'}(Y')|\leq |\delta_{G'}(X_{a}^j)|+ |\delta_{G'}(X_{a'}^{j'})|. \]
%
Using the same argument as before, $|\delta_{G'}(Y)|=|\delta_{G'}(X_{a}^j)|$ and $|\delta_{G'}(Y')|= |\delta_{G'}(X_{a'}^{j'})|$ must hold. This contradicts the minimality of the cut $X_{a}^{j}$, as $|Y|< |X_{a}^{j}|$.
\end{proof}
We denote, for all $0\leq j\leq r$, ${\mathcal{X}}^j=\set{X^j_a\mid 0\leq a<2^{r-j}}$, and ${\mathcal{X}}=\bigcup_{j=0}^r{\mathcal{X}}^j$. For simplicity, we will refer to the sets of vertices in ${\mathcal{X}}$ as \emph{clusters} (each such vertex set $X^j_a$ indeed naturally defines a cluster $G'[X^j_a]$ of graph $G'$). Note that the set ${\mathcal{X}}$ of clusters is laminar.
We can naturally associate a partitioning tree $\tau$ with the set ${\mathcal{X}}$ of clusters. The set of vertices of the tree $\tau$ is $\set{u(X)\mid X\in {\mathcal{X}}\cup \set{V(G')}}$. The root of the tree is vertex $u(X)$ where $X=V(G')$. This vertex has one child vertex -- $u(X^r_0)$, corresponding to the unique cluster in ${\mathcal{X}}^r$. For every non-root vertex $u(X^j_a)$, there are exactly two level-$(j-1)$ clusters that are contained in $X^j_a$: clusters $X^{j-1}_{a'}$ and $X^{j-1}_{a''}$, where $a'=2a$ and $a''=2a+1$. Vertices $u(X^{j-1}_{a'})$ and $u(X^{j-1}_{a''})$ become child-vertices of $u(X^j_a)$ in the tree; we refer to clusters $X^{j-1}_{a'}$ and $X^{j-1}_{a''}$ as \emph{child-clusters} of $X^j_a$, where $X^{j-1}_{a'}$ is the left child and $X^{j-1}_{a''}$ is the right child. We also say that $X^j_a$ is a \emph{parent-cluster} of $X^{j-1}_{a'}$ and $X^{j-1}_{a''}$. The leaves of the tree $\tau$ are vertices in set $\set{u(X)\mid X\in {\mathcal{X}}^0}$.
It will be convenient for us to subdivide some of the edges of $G'$, in order to ensure the following two properties:
\begin{properties}{P}
\item for every cluster $X\in {\mathcal{X}}$, if $e=(x,y)\in \delta_{G'}(X)$, with $x\in X$, then vertex $y$ lies in the parent-cluster of $X$, and neither $x$ nor $y$ are anchor vertices; \label{prop: subdivision1}
\item for every cluster $X\in {\mathcal{X}}$, if $X'$ and $X''$ are the two child-clusters of $X$, and we denote $Y=X\setminus (X'\cup X'')$, then for every pair $e,e'\in \delta_{G'}(Y)$ of edges, the two edges $e,e'$ do not share endpoints. \label{prop: subdivision2}
\iffalse
\item for every cluster $X\in {\mathcal{X}}$, if $X'$ and $X''$ are the two child-clusters of $X$, then for every pair $e\in \delta_{G'}(X'), e'\in \delta_{G'}(X'')$ of edges, the two edges $e,e'$ do not share endpoints; \label{prop: subdivision1a}
\item for every cluster $X\in {\mathcal{X}}$, for every pair $e,e'\in \delta_{G'}(X)$ of edges, $e$ and $e'$ do not share endpoints;
\item for every pair of clusters $X,X'\in {\mathcal{X}}$, where $X'$ is a child-cluster of $X$, for every pair $e\in \delta_{G'}(X), e'\in \delta_{G'}(X')$ of edges, the two edges $e,e'$ do not share endpoints. \label{prop: subdivision2}
\fi
\end{properties}
In order to achieve the above properties, we will subdivide some edges of $G'$, and we will update the clusters in ${\mathcal{X}}$ accordingly. We will ensure that the clusters remain laminar, and that, for all $0\leq j\leq r$ and $0\leq a<2^{r-j}$, $X^j_a$ remains the smallest cut separating the vertices of $T^j_a$ from the remaining vertices of $A$, with $|\delta_{G'}(X^j_a)|$ remaining unchanged.
In order to perform this transformation, we process the clusters of ${\mathcal{X}}$ in sets ${\mathcal{X}}^0,{\mathcal{X}}^1,\ldots,{\mathcal{X}}^r$ in this order of the sets.
Consider an iteration when some cluster $X^j_a$ is processed, for $0\leq j\leq r$ and $0\leq a<2^{r-j}$. Consider any edge $e=(x,y)\in \delta_{G'}(X^j_a)$, and assume that $x\in X^j_a$. We subdivide edge $e$ with two new vertices, replacing it with a path $(x,t_e,t'_e,y)$. We add vertex $t_e$ to both $X^j_a$ and all its ancestor clusters, and we add vertex $t'_e$ to all ancestor clusters of $X^j_a$ (but not to $X^j_a$). Notice that, after this subdivision step, the updated set ${\mathcal{X}}$ of clusters remains laminar, and for every cluster $X\in {\mathcal{X}}$, $|\delta_{G'}(X)|$ does not grow. Once we process every edge $e\in \delta_{G'}(X^j_a)$, we complete the processing of cluster $X^j_a$. Once every cluster in ${\mathcal{X}}$ is processed, we obtain an updated graph $G'$, with the updated family ${\mathcal{X}}$ of clusters, for which properties \ref{prop: subdivision1} and \ref{prop: subdivision2} hold.
We update the input drawing $\phi$ of graph $G'$ by subdiving the images of its edges appropriately, to obtain a drawing of the current graph $G'$, that we also denote by $\phi$. Note that for all $0\leq j\leq r$ and $0\leq a<2^{r-j}$, every edge $e$ in the current set $\delta_{G'}(X^j_a)$ is obtained by subdiving some edge in the original graph $G'$, and both endpoints of $e$ are new vertices that were used for the subdivision. We can then place the images of the endpoints of $e$ close enough to each other, so that the image of the edge $e$ does not participate in any crossings in the new drawing $\phi$. We will assume from now on that for all $0\leq j\leq r$ and $0\leq a<2^{r-j}$, the edges of $\delta_{G'}(X^j_a)$ do not participate in crossings in $\psi$.
For all $0\leq j\leq r$ and $0\leq a<2^{r-j}$, we define a graph $H_{j,a}$ associated with cluster $X^j_a$, as follows. If $j=0$, then $H_{j,a}=G'[X^j_a]$. Otherwise, let $X^{j-1}_{a'},X^{j-1}_{a''}$ be the two child clusters of cluster $X^j_a$, where $X^{j-1}_{a'}$ is the left child cluster. We let $H_{j,a}$ be the subgraph of $G'$ induced by vertex set $X^j_a\setminus \textsf{left} (X^{j-1}_{a'}\cup X^{j-1}_{a''} \textsf{right} )$. We also define three subsets of vertices of $H_{j,a}$: set $T^{\mathsf{parent}}_{j,a}$ contains all vertices of $H_{j,a}$ that serve as endpoints of the edges of $\delta_{G'}(X^j_a)$; set $T^{\mathsf{lchild}}_{j,a}$ contains all vertices of $H_{j,a}$ that serve as endpoints of the edges of $\delta_{G'}(X^{j-1}_{a'})$; and set $T^{\mathsf{rchild}}_{j,a}$ contains all vertices of $H_{j,a}$ that serve as endpoints of the edges of $\delta_{G'}(X^{j-1}_{a''})$. From Property
\ref{prop: subdivision2}, these three sets of vertices are mutually disjoint.
We denote by $T(H_{j,a})=T^{\mathsf{parent}}_{j,a}\cup T^{\mathsf{lchild}}_{j,a}\cup T^{\mathsf{rchild}}_{j,a}$.
For $j=0$, for all $0\leq a< 2^r$, we define the set $T^{\mathsf{parent}}_{j,a}$ of vertices of graph $H_{j,a}$ similarly. We do not define the sets $T^{\mathsf{lchild}}_{j,a},T^{\mathsf{rchild}}_{j,a}$ of vertices, but instead we use the set $T^j_a$ of anchor vertices that we defined already. From Property \ref{prop: subdivision1}, vertex sets $T^{\mathsf{parent}}_{j,a}$ and $T^j_a$ are disjoint.
We denote $T(H_{j,a})=T^{\mathsf{parent}}_{j,a}\cup T^j_a$.
Lastly, we define a graph $H^*$ -- a subgraph of $G'$ induced by vertex set $V(G')\setminus X^r_0$. We let $T^*$ be the set of all anchor vertices in $A\setminus T^r_0$, and we let $T^{**}$ be the set of all vertices of $H^*$ that serve as endpoints of the edges of $\delta_{G'}(X^r_0)$. We denote $T(H^*)=T^*\cup T^{**}$.
Let ${\mathcal{H}}=\set{H^*}\cup \set{H_{j,a}\mid 0\leq j\leq r, 0\leq a<2^{r-j}}$ be the resulting collection of subgraphs of $G'$. Note that the graphs in ${\mathcal{H}}$ are mutually disjoint from each other and $\bigcup_{H\in {\mathcal{H}}}V(H)=V(G')$.
Next, for every graph $H\in {\mathcal{H}}$, we will define a disc $D(H)$, and we will also define an ordering of the vertices in $T(H)$. We will then modify the current drawing $\phi$ of graph $G'$, so that, for every graph $H\in {\mathcal{H}}$, the image of $H$ lies in disc $D(H)$, with the vertices of $T(H)$ lying on the disc boundary, in the pre-specified order. We will ensure that, for all $0\leq j\leq r$ and $0\leq a< 2^{r-j}$, the only edges whose images cross the curve $\lambda^j_a$ are the edges of $\delta_{G'}(X^j_a)$. Since, as observed above, $|\delta_{G'}(X^j_a)|\leq 4\check m'/\mu^{2b}$, this will ensure that at most $4\check m'/\mu^{2b}$ edges cross each curve $\lambda\in \Lambda$ in the final drawing.
We now consider every graph $H\in {\mathcal{H}}$ in turn, define the corresponding disc $D(H)$, and the ordering of the vertices of $T(H)$.
Consider some pair of indices $0\leq j< r$ and $0\leq a< 2^{r-j}$. Recall that $(X^j_a,V(G')\setminus X^j_a)$ is a minimum cut in the current graph $G'$ separating vertices of $T^j_a$ from the remaining vertices of $A$. Therefore, there is a set ${\mathcal{Q}}_{j,a}=\set{Q_{j,a}(e)\mid e\in \delta_{G'}(X^j_a)}$ of edge-disjoint paths, that are internally disjoint from $X^j_a$, such that, for every edge $e\in \delta_{G'}(X^j_a)$, path $Q_{j,a}(e)$ originates at edge $e$, and terminates at some vertex of $A\setminus T^j_a$. Similarly, there is a set ${\mathcal{Q}}'_{j,a}=\set{Q'_{j,a}(e)\mid e\in \delta_{G'}(X^j_a)}$ of edge-disjoint paths, whose inner vertices are contained in $X^j_a$, such that, for every edge $e\in \delta_{G'}(X^j_a)$, path $Q'_{j,a}(e)$ originates at edge $e$, and terminates at some vertex of $T^j_a$. From \Cref{lem: non_interfering_paths}, we can assume w.l.o.g. that the paths in set ${\mathcal{Q}}_{j,a}$ are non-transversal with respect to the rotation system $\Sigma'$, and the same is true regarding the paths in ${\mathcal{Q}}'_{j,a}$. We define an oriented ordering ${\mathcal{O}}_{j,a}$ of the edges in set $\delta_{G'}(X^j_a)$, as follows. For every edge $e\in \delta_{G'}(X^j_a)$, let $v_e$ be the last vertex on path $Q'_{j,a}(e)$, that must lie in $T^j_a$. From our construction, it is easy to verify that every vertex of $A$ has degree $1$ in $G'$, so all vertices in set $\set{v_e\mid e\in \delta_{G'}(X^j_a)}$ are distinct. We define the oriented ordering ${\mathcal{O}}_{j,a}$ of the edges of $\delta_{G'}(X^j_a)$ to be the order in which their corresponding vertices $v_e$ are encountered along the boundary of the disc $D$, as we traverse it in counter-clock-wise direction. We use this ordering in order to define an oriented ordering ${\mathcal{O}}(T^{\mathsf{parent}}_{j,a})$ of the set $T^{\mathsf{parent}}_{j,a}$ of vertices of graph $H^j_a$: recall that the vertices of $T^{\mathsf{parent}}_{j,a}$ are the endpoints of the edges of $\delta_{G'}(X^j_a)$ that lie in $X^j_a$, and every edge in $\delta_{G'}(X^j_a)$ is incident to a distinct vertex of $T^{\mathsf{parent}}_{j,a}$. We let the oriented ordering ${\mathcal{O}}(T^{\mathsf{parent}}_{j,a})$ of the vertices of $T^{\mathsf{parent}}_{j,a}$ be identical to the oriented ordering ${\mathcal{O}}_{j,a}$ of the edges of $\delta_{G'}(X^j_a)$, except that we reverse the orientation. In other words, in order to obtain the ordering ${\mathcal{O}}(T^{\mathsf{parent}}_{j,a})$, we replace, in ordering ${\mathcal{O}}_{j,a}$ every edge $e\in \delta_{G'}(X^j_a)$ with its endpoint that lies in $T^{\mathsf{parent}}_{j,a}$, and then reverse the orientation of the resulting ordering.
Assume now that cluster $X^j_a$ is the child cluster of some other cluster $X^{j'}_{a'}$. We assume w.l.o.g. that it is the left child cluster; the other case is dealt with similarly. Recall that the set $T^{\mathsf{lchild}}_{j',a'}$ of vertices contains all endpoints of the edges of $\delta_{G'}(X^j_a)$ that lie in $H_{j',a'}$. We define an ordering ${\mathcal{O}}(T^{\mathsf{lchild}}_{j',a'})$ of the vertices of $T^{\mathsf{lchild}}_{j',a'}$ to be identical to the oriented ordering ${\mathcal{O}}_{j,a}$ of the edges of $\delta_{G'}(X^j_a)$. In other words, in order to obtain the ordering ${\mathcal{O}}(T^{\mathsf{lchild}}_{j',a'})$, we replace, in ordering ${\mathcal{O}}_{j,a}$ every edge $e\in \delta_{G'}(X^j_a)$ with its endpoint that lies in $T^{\mathsf{lchild}}_{j',a'}$. If $j=r$ and $a=0$, then cluster $X^r_0$ is the child cluster of $V(G')$. The latter cluster, in turn, corresponds to graph $H^*$. In this case, the set $T^{**}$ of vertices of $H^*$ contains all endpoints of the edges of $\delta_{G'}(H_{r,0})$ that lie in $V(H^*)$. We define an ordering ${\mathcal{O}}(T^{**})$ of the vertices of $T^{**}$ similarly: it is identical to the ordering ${\mathcal{O}}_{r,0}$ of the edges of $\delta_{G'}(X^r_0)$.
Next, we define, for every graph $H\in {\mathcal{H}}$, a corresponding disc $D(H)$. Consider first the graph $H^*$. Let $\lambda'_{r,0}$ be a curve that has the same endpoints as $\lambda_{r,0}$, is internally disjoint from $\lambda_{r,0}$, and follows the curve $\lambda_{r,0}$ closely outside the disc $D^r_0$. Let $\sigma'$ be the segment of the boundary of disc $D$ that connects the two endpoints of $\lambda_{r,0}$, and is internally disjoint from the boundary of disc $D^r_0$. We let $D(H^*)$ be the disc that is contained in $D$, whose boundary is the concatenation of the curves $\lambda'_{r,0}$ and $\sigma'$ (see \Cref{fig: discD_0}).
Consider now indices $0<j\leq r$ and $0\leq a<2^{r-j}$, and the graph $H_{j,a}$. Let $X_{j-1,a'}$ and $X_{j-1,a''}$ be the left and the right child clusters of $X_{j,a}$, respectively. We let $\lambda''_{j,a}$ be a curve whose endpoints are the same as those of $\lambda_{j,a}$, so that $\lambda''_{j,a}$ follows the curve $\lambda_{j,a}$ closely inside disc $D^j_a$. We let $\lambda'_{j-1,a'}$ be a curve whose endpoints are the same as those of $\lambda_{j-1,a'}$, so that $\lambda'_{j-1,a'}$ follows the curve $\lambda_{j-1,a'}$ closely, and is internally disjoint from disc $D^{j-1}_{a'}$. Similarly, we let $\lambda'_{j-1,a''}$ be a curve whose endpoints are the same as those of $\lambda_{j-1,a''}$, so that $\lambda'_{j-1,a''}$ follows the curve $\lambda_{j-1,a''}$ closely, and is internally disjoint from disc $D^{j-1}_{a''}$.
The concatenation of the curves $\lambda''_{j,a}, \lambda'_{j-1,a'}$ and $\lambda'_{j-1,a''}$ is a simple closed curve that is contained in disc $D^j_a$. We let $D(H^j_a)$ be the disc that is contained in $D$, whose boundary is the concatenation of $\lambda''_{j,a}, \lambda'_{j-1,a'}$ and $\lambda'_{j-1,a''}$.
(see \Cref{fig: discD_1}).
\begin{figure}[h]
\centering
\subfigure[Disc $D(H*)$.]{
\scalebox{0.32}{\includegraphics[scale=0.25]{figs/discD_0.jpg}}\label{fig: discD_0}}
\hspace{0.2cm}
\subfigure[Disc $D(H_{j,a})$ for $0<j\le r$.]{
\scalebox{0.32}{\includegraphics[scale=0.25]{figs/discD_1.jpg}}\label{fig: discD_1}}
\hspace{0.2cm}
\subfigure[Disc $D(H_{0,a})$.]{\scalebox{0.32} {\includegraphics[scale=0.25]{figs/discD_2.jpg}}\label{fig: discD_2}
}
\caption{Discs $D(H)$ for graphs $H\in {\mathcal{H}}$}\label{fig: discs}
\end{figure}
Lastly, we consider the index $j=0$, and any index $0\leq a<2^r$.
We let $\lambda''_{0,a}$ be a curve whose endpoints are the same as those of $\lambda_{0,a}$, so that $\lambda''_{0,a}$ follows the curve $\lambda_{0,a}$ closely inside disc $D^0_a$. Recall that $\sigma^0_a$ is a segment of the boundary of disc $D$, whose endpoints are the same as those of $\lambda_{0,a}$, with point $p_{a+1}$ not lying on $\sigma^0_a$. We let $D(H_{0,a})$ be the disc that is contained in $D$, whose boundary is the concatenation of $\lambda''_{0,a}$ and $\sigma^0_a$
(see \Cref{fig: discD_2}).
We note that every anchor vertex in $A$ must belong to one of the graphs in $\set{H^*}\cup\set{H_{0,a}\mid 0\leq a< 2^r}$.
For every graph $H\in {\mathcal{H}}$, we define a set $\chi(H)$ of crossings as follows. For $H=H^*$, $\chi(H)$ contains all crossings in the drawing $\phi$. For a pair of indices $0\leq j\leq r$ and $0\leq a< 2^{r-j}$, $\chi(H_{j,a})$ is the set of all crossings in $\phi$ in which the edges of $G'[X^j_a]$ participate.
The following claim is central to the proof of \Cref{claim: avoid guiding curves}.
\begin{claim}\label{claim: move to discs2}
Consider any graph $H\in {\mathcal{H}}$, let $\Sigma_H$ be the rotation system for $H$ induced by $\Sigma'$, and let $I_H=(H,\Sigma_H)$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. There is a solution $\psi(H)$ to instance $I_H$ with $\mathsf{cr}(\psi(H))\leq O(|\chi(H)|)$, where the image of the graph $H$ is contained in disc $D(H)$. Moreover, the following hold:
\begin{itemize}
\item If $H=H^*$, then the images of the vertices of $T^*=V(H^*)\cap A$ appear on segment $\sigma'$ of the boundary of $D(H^*)$, in the same locations as in $\phi$, and the images of the vertices of $T^{**}$ appear on segment $\lambda'_{r,0}$ of the boundary of $D(H^*)$, in the same order as in ${\mathcal{O}}(T^{**})$, incuding the orientation (that is defined with respect to disc $D(H^*)$).
\item If $H=H_{j,a}$ for $j=0$, then the images of the vertices of $T^j_a=V(H_{j,a})\cap A$ appear on the segment $\sigma^j_a$ of the boundary of disc $D(H_{j,a})$, in the same locations as in $\phi$, and the images of the vertices of $T^{\mathsf{parent}}_{j,a}$ appear on the segment $\lambda''_{j,a}$ of the boundary of disc $D(H_{j,a})$, in the same order as in ${\mathcal{O}}(T^{\mathsf{parent}}_{j,a})$, including the orientation (that is defined with respect disc $D(H_{j,a})$).
\item If $H=H_{j,a}$ for $j>0$, then the images of the vertices of $T^{\mathsf{parent}}_{j,a}$ appear on the segment $\lambda''_{j,a}$ of the boundary of disc $D(H_{j,a})$, in the same order as in ${\mathcal{O}}(T^{\mathsf{parent}}_{j,a})$, including the orientation, and similarly, the images of the vertices in sets $T^{\mathsf{lchild}}_{j,a}$ and $T^{\mathsf{rchild}}_{j,a}$ appear on the segments $\lambda'_{j-1,a'}$ and $\lambda'_{j-1,a''}$ of the boundary of the disc $D(H_{j,a})$, respectively, where $X^{j-1}_{a'}$ is the left child of $X^{j}_{a}$ and $X^{j-1}_{a''}$ is its right child. The ordering of the images of the vertices of $T^{\mathsf{lchild}}_{j,a}$ on $\lambda'_{j-1,a'}$ is identical to ${\mathcal{O}}(T^{\mathsf{lchild}}_{j,a})$, including orientation, and the ordering of the images of the vertices of $T^{\mathsf{rchild}}_{j,a}$ on $\lambda'_{j-1,a''}$ is identical to ${\mathcal{O}}(T^{\mathsf{rchild}}_{j,a})$, including orientation. The orientations of all orderings are with respect to disc $D(H_{j,a})$.
\end{itemize}
\end{claim}
We provide the proof of \Cref{claim: move to discs2} in the following subsection, after we complete the proof of \Cref{claim: avoid guiding curves} using it.
In order to construct the solution $\psi'$ to instance $I'$ we start by planting, for every graph $H\in {\mathcal{H}}$, the image $\psi(H)$ of $H$ into the disc $D(H)$. From \Cref{claim: move to discs2}, and since every vertex of $A$ lies in either $T^*$ or in $\bigcup_{a=0}^{2^r-1}T^0_a$, the images of the vertices of $A$ remain the same as in $\phi$. In order to complete the drawing of graph $G'$, we need to insert the edges of $\delta_{G'}(X^j_a)$ for all $0\leq j\leq r$ and $0\leq a<2^{r-j}$ into the current drawing.
Observe that the endpoints of all such edges have degree $2$ in $G'$ from our construction of graph $G'$.
We now fix an index $0\leq j< r$ and $0\leq a<2^{r-j}$. Assume that cluster $X^j_a$ is a child cluster of some cluster $X^{j+1}_{a'}$, and assume w.l.o.g. that it is a left child cluster (the other case is dealt with similarly). Consider the set $E'=\delta_{G'}(X^j_a)$ of edges. Recall that these edges define a perfect matching between the sets $T^{\mathsf{lchild}}_{j+1,a'}$ and $T^{\mathsf{parent}}_{j,a}$ of vertices. The images of the vertices of $T^{\mathsf{lchild}}_{j+1,a'}$ appear on curve $\lambda'_{j,a}$, while the images of the vertices of $T^{\mathsf{parent}}_{j,a}$ appear on curve $\lambda''_{j,a}$. Let $D^*_{j,a}$ be the disc that is contained in $D$, whose boundary is the concatenation of the curves $\lambda'_{j,a}$ and $\lambda''_{j,a}$. Denote $E'=\set{e_1,e_2,\ldots,e_q}$, where the edges are indexed according to the ordering ${\mathcal{O}}_{j,a}$. For all $1\leq i\leq q$, let $e_i=(x_i,y_i)$, where $x_i\in T^{\mathsf{lchild}}_{j+1,a'}$ and $y_i\in T^{\mathsf{parent}}_{j,a}$. Then the images of vertices $x_1,\ldots,x_q$ appear on curve $\lambda'_{j,a}$ in the order of their indices, and the images of vertices $y_1,\ldots,y_q$ appear on curve $\lambda''_{j,a}$ in the order of their indices, but the orientations of the two orderings are different (the orientation of the first ordering is with respect to $D(H_{j+1,a'})$, while the orientation of the second ordering is with respect to $D(H_{j,a})$). Therefore, the images of vertices $x_1,x_2,\ldots,x_q,y_q,\ldots,y_1$ appear on the boundary of disc $D^*_{j,a}$ in this circular order. We can then define, for all $1\leq i\leq q$, a curve $\gamma_i$ that is contained in disc $D^*_{j,a}$ connecting the image of $x_i$ to the image of $y_i$. We can ensure that no two such curves cross each other, and each curve crosses $\lambda_{j,a}$ exactly once. We then let, for all $1\leq i\leq q$, curve $\gamma_i$ be the image of edge $e_i$. Since, as observed above, $|E'|=|\delta_{G'}(X^j_a)|\leq 4\check m'/\mu^{2b}$, we introduce at most $4\check m'/\mu^{2b}$ crossings between images of edges of $G'$ and curve $\lambda_{j,a}$.
It now only remains to take care of edge set $\delta_{G'}(X^r_0)$. We insert these edges exactly as before. The only difference is that vertex set $T^{\mathsf{lchild}}_{j+1,a}$ is replaced with $T^{**}$.
We have now obtained a solution $\psi'$ to instance $I'$, in which the images of all vertices and edges of $G'$ lie in disc $D$, and the images of all anchor vertices remain the same as in $\phi$. For every curve $\lambda\in \Lambda$, for every vertex $v\in V(G')$, the image of $v$ in $\psi'$ does not lie on an inner point of $\lambda$, and for every edge $e\in E(G')$, the image of $e$ in $\psi'$ may intersect $\lambda$ in at most one point. For every curve $\lambda\in \Lambda$, the total number of edges in $E(G')$ whose images intersect $\lambda$ is at most $4\check m'/\mu^{2b}$.
It now remains to bound the number of crossings in $\psi'$. Since the insertion of the edges of $\delta_{G'}(X^j_a)$ for all $0\leq j\leq r$ and $0\leq a<2^{r-j}$ did not introduce any crossings, from \Cref{claim: move to discs2}, $\mathsf{cr}(\psi')\leq \sum_{H\in {\mathcal{H}}}O(|\chi(H)|)$.
Recall $|\chi(H^*)|=\mathsf{cr}(\phi)$, and, for all $0\leq j\leq r$ and $0\leq a< 2^{r-j}$, $|\chi(H^j_a)|$ is number of crossings in $\phi$ in which the edges of $G'[X^j_a]$ participate.
Observe that vertex sets in $\set{X^j_a \mid 0\leq j\leq r, 0\leq a< 2^{r-j}}$ define a laminar family of depth $O(\log \check m')$. Therefore, every edge $e$ may belong to at most $O(\log \check m')$ graphs in set $\set{G'[X^j_a] \mid 0\leq j\leq r, 0\leq a< 2^{r-j}}$. We conclude that every crossing $(e,e')_p$ of $\phi$ may belong to at most $O(\log \check m')$ sets $\set{\chi(H)\mid H\in {\mathcal{H}}}$. Therefore, overall, $\mathsf{cr}(\psi')\leq \sum_{H\in {\mathcal{H}}}O(|\chi(H)|)\leq \mathsf{cr}(\phi)\cdot O (\log \check m')$.
In order to complete the proof of \Cref{claim: avoid guiding curves}, it remains to prove \Cref{claim: move to discs2}, which we do next.
\subsection{Proof of \Cref{claim: move to discs2}}
Fix a pair of indices $0\leq j\leq r$ and $0\leq a< 2^{r-j}$.
Recall that we have defined two sets of paths associated with the edges of $\delta_{G'}(X^j_a)$. The first set of paths is a set ${\mathcal{Q}}_{j,a}=\set{Q_{j,a}(e)\mid e\in \delta_{G'}(X^j_a)}$ of edge-disjoint paths, that are internally disjoint from $X^j_a$, such that, for every edge $e\in \delta_{G'}(X^j_a)$, path $Q_{j,a}(e)$ originates at edge $e$, and terminates at some vertex of $A\setminus T^j_a$. The second set of paths is a set ${\mathcal{Q}}'_{j,a}=\set{Q'_{j,a}(e)\mid e\in \delta_{G'}(X^j_a)}$ of edge-disjoint paths, that are contained in $G'[X^j_a]$, such that, for every edge $e\in \delta_{G'}(X^j_a)$, path $Q'_{j,a}(e)$ originates at edge $e$, and terminates at some vertex of $T^j_a$. Both sets of paths are non-transversal with respect to $\Sigma'$. We now define two sets of curves: $\Gamma_{j,a}$ (corresponding to the paths in ${\mathcal{Q}}_{j,a}$), and $\Gamma'_{j,a}$ (corresponding to the paths in ${\mathcal{Q}}'_{j,a}$) that will be used in constructing the new drawings for graphs in ${\mathcal{H}}$. We denote by $\sigma'_{j,a}$ the segment of the boundary of disc $D$ that is the complement of $\sigma_{j,a}$. In other words, if the boundary of $D$ is denoted by $\beta$, then $\sigma'_{j,a}=\beta\setminus\sigma_{j,a}$. For every edge $e\in \delta_{G'}(X^j_a)$, we denote $e=(x_e,y_e)$, where $x_e\in X^{j}_a$.
We start with the set $\Gamma'_{j,a}$ of curves. Initially, for every edge $e\in \delta_{G'}(X^j_a)$, we let $\gamma'_{j,a}(e)$ be the image of the path $Q'_{j,a}(e)$.
Note that curve $\gamma'_{j,a}(e)$ connects the image of $y_e$ to some point on $\sigma_{j,a}$, and it contains $\phi(e)$. Let $\Gamma'_{j,a}=\set{\gamma'_{j,a}(e)\mid e\in \delta_{G'}(X^j_a)}$.
From the definition of the ordering ${\mathcal{O}}_{j,a}$, the oriented ordering ${\mathcal{O}}_{j,a}$ of the edges in $\delta_{G'}(X^j_a)$ is identical to the order of the endpoints of their corresponding curves $\gamma'_{j,a}(e)$ along the boundary of the disc $D$. We assume w.l.o.g. that the orientation of the ordering ${\mathcal{O}}_{j,a}$ is counter-clock-wise.
The curves of $\Gamma'_{j,a}$ may not be in general position: if a vertex of $V(G')\setminus X^j_a$ lies on more than two paths in ${\mathcal{Q}}'_{j,a}$, then its image belongs to more than two curves of $\Gamma'_{j,a}$. We perform a nudging procedure by modifying the curves in $\Gamma'_{j,a}$ locally within tiny discs $D_{\phi}(v)$ of vertices $v\in X^j_a$ that belong to at least two paths of ${\mathcal{Q}}'_{j,a}$, using the algorithm from \Cref{claim: curves in a disc} (see also \Cref{sec: curves in a disc} for the definition of a nuding procedure). Since that paths in ${\mathcal{Q}}'_{j,a}$ are non-transversal with respect to $\Sigma'$, this nudging procedure does not introduce any new crossings between the curves in $\Gamma'_{j,a}$. We summarize the properties of the resulting set $\Gamma'_{j,a}$ of curves, that are immediate form our definitions and construction, and the fact that the vertices of $A$ have degree $1$ in $G'$, in the following observation.
\begin{observation}\label{obs: inner curves}
Consider the final set $\Gamma'_{j,a}=\set{\gamma'_{j,a}(e)\mid e\in \delta_{G'}(X^j_a)}$ of curves. For every edge $e\in \delta_{G'}(X^j_a)$, curve $\gamma'_{j,a}(e)$ connects the image of vertex $y_e$ in $\phi$ to some point on $\sigma_{j,a}$, and it contains $\phi(e)$.
The number of crossings between the curves in $\Gamma'_{j,a}$ is at most $|\chi(H_{j,a})|$, and the number of crossings between the curves in $\Gamma'_{j,a}$ and the edges of $G'\setminus X^j_a$ is at most $|\chi(H_{j,a})|$. The oriented ordering ${\mathcal{O}}_{j,a}$ of the edges of $\delta_{G'}(X^j_a)$ is identical to the oriented ordering of the endpoints of the corresponding curves $\gamma'_{j,a}(e)$ on the boundary of disc $D$; we assume that the orientation of the ordering is counter-clock-wise.
\end{observation}
The construction of the set $\Gamma_{j,a}$ of curves is very similar, except that we also perform a type-2 uncrossing for them. In order to obtain the set $\Gamma_{j,a}$ of curves, we simply
apply the algorithm from \Cref{thm: new type 2 uncrossing} to perform a type-2 uncrossing of the set ${\mathcal{Q}}_{j,a}$ of paths. We denote the resulting set of curves by $\Gamma_{j,a}=\set{\gamma_{j,a}(e)\mid e\in \delta_{G'}(X^j_a)}$. We summarize the properties of the resulting set of curves in the following observation, that follows immediately from the discussion so far and \Cref{thm: new type 2 uncrossing}.
\begin{observation}\label{obs: outer curves}
Consider the set $\Gamma_{j,a}=\set{\gamma_{j,a}(e)\mid e\in \delta_{G'}(X^j_a)}$ of curves. For every edge $e\in \delta_{G'}(X^j_a)$, curve $\gamma_{j,a}(e)$ connects the image of vertex $y_e$ in $\phi$ to some point on $\sigma'_{j,a}$.
There are no crossings between the curves in $\Gamma'_{j,a}$, and the number of crossings between the curves in $\Gamma_{j,a}$ and the edges of $G'[X^j_a]$ is at most $|\chi(H_{j,a})|$. The number of crossings between the curves in $\Gamma_{j,a}$ and the curves in $\Gamma'_{j,a}$ is bounded by $|\chi(H_{j,a})|$.
\end{observation}
(The last assertion follows from \Cref{obs: inner curves} and the fact that the curves in $\Gamma_{j,a}$ are aligned with the graph $\bigcup_{Q\in {\mathcal{Q}}_{j,a}}Q$.)
We let ${\mathcal{O}}'_{j,a}$ be the oriented ordering of the edges in $\delta_{G'}(X^j_a)$ defined by the oriented ordering of the endpoints of the corresponding curves in $\set{\gamma_{j,a}(e)\mid e\in \delta_{G'}(X^j_a)}$ on the boundary of disc $D$. We assume w.l.o.g. that the orientation of the ordering is counter-clock-wise.
We need the following obervation:
\begin{observation}\label{obs: bound distance between orderings}
$\mbox{\sf dist}({\mathcal{O}}_{j,a},{\mathcal{O}}'_{j,a})\leq O(|\chi(H_{j,a})|)$.
\end{observation}
Note that the above observation bounds the distance between two {\bf oriented} orderings.
\begin{proof}
For every edge $e\in \delta_{G'}(X^j_a)$, let $\gamma^*(e)$ be the curve obtained by concatenating curves $\gamma_{j,a}(e)$ and $\gamma'_{j,a}(e)$. Let $\Gamma^*=\set{\gamma^*(e)\mid e\in \delta_{G'}(X^j_a)}$ be the resulting set of curves. It is immediate to verify that $\Gamma^*$ is a valid reordering set of curves for the oriented orderings ${\mathcal{O}}_{j,a},{\mathcal{O}}'_{j,a}$. From Observations \ref{obs: inner curves} and \ref{obs: outer curves}, the number of crossings between the curves of $\Gamma$ is at most $O(|\chi(H_{j,a})|)$.
\end{proof}
We are now ready to define a solution $\psi(H)$ for every instance $I_H$ with $H\in {\mathcal{H}}$. We start with a graph $H=H_{j,a}$, where $0<j\leq r$ and $0\leq a< 2^{r-j}$.
Assume that $X^{j-1}_{a'}$ and $X^{j-1}_{a''}$ are the left and the right child clusters of $X^j_a$ respectively.
Recall that the boundary of the disc $D(H_{j,a})$ is the concatenation of the curves $\lambda''_{j,a},\lambda'_{j-1,a'}$ and $\lambda'_{j-1,a''}$. We let $\lambda^*_{j,a}$ be a curve with the same endpoints as $\lambda''_{j,a}$, that is internally disjoint from $\lambda''_{j,a}$, and is contained in disc $D(H_{j,a})$. We let $D'(H_{j,a})$ be the disc that is contained in $D(H_{j,a})$, whose boundary is the concatenation of the curves $\lambda^*_{j,a},\lambda'_{j-1,a'}$ and $\lambda'_{j-1,a''}$.
We obtain an initial drawing $\psi(H_{j,a})$ of graph $H_{j,a}$ as follows. We start with the drawing of the graph $H_{j,a}$ that is induced by $\phi$. Conside now some vertex $t\in T^{\mathsf{parent}}_{j,a}$, and let $e_t$ be the unique edge of $H_{j,a}$ incident to $t$. We replace the current image of $e_t$ with the concatenation of $\phi(e_t)$ and the curve $\gamma_{j,a}(e_t)\in \Gamma_{j,a}$, and we move the image of $t$ to the endpoint of this curve that lies on segment $\sigma'_{j,a}$ of the boundary of disc $D$. Consider now some vertex $t\in T^{\mathsf{lchild}}_{j,a}$, and let $e_t$ be the unique edge of $H_{j,a}$ incident to $t$. We replace the current image of $e_t$ with the curve $\gamma'_{j-1,a'}(e_t)\in \Gamma'_{j-1,a'}$, and we place the image of $t$ on the endpoint of the curve that lies in the segment $\sigma'_{j-1,a'}$ of the boundary of $D$. Lastly, we consider vertices $t\in T^{\mathsf{rchild}}_{j,a}$, and for each such vertex, we let $e_t$ be the unique edge of $H_{j,a}$ incident to $t$. We replace the current image of $e_t$ with the curve $\gamma'_{j-1,a''}(e_t)\in \Gamma'_{j-1,a''}$, and we place the image of $t$ on the endpoint of the curve that lies in the segment $\sigma'_{j-1,a'}$ of the boundary of $D$. We plant the resulting drawing of graph $H_{j,a}$ into the disc $D'(H_{j,a})$, so that the segments $\sigma'_{j,a},\sigma_{j-1,a'}$ and $\sigma_{j-1,a''}$ of the boundary of $D$ coincide with segments $\lambda^*_{j,a},\lambda'_{j-1,a'}$ and $\lambda'_{j-1,a''}$ of the boundary of $D'(H_{j,a})$, respectively. We denote the resulting drawing of $H_{j,a}$ by $\psi'_{j,a}$. It is easy to verify that it is a valid solution to instance $I_{H_{j,a}}$. Observe that, from our construction, the images of the vertices of $T^{\mathsf{parent}}_{j,a}$ appear on curve $\lambda^*_{j,a}$ in drawing $\psi'_{j,a}$, and their (oriented) ordering on this curve (with respect to disc $D'(H_{j,a})$) is identical to the ordering ${\mathcal{O}}'_{j,a}$ of their corresponding edges in $\delta_{G'}(X_{j,a})$ that we have defined above. The vertices
of $T^{\mathsf{lchild}}(j,a)$ appear on curve $\lambda'_{j-1,a'}$, and their (oriented) ordering on this curve (with respect to disc $D'(H_{j,a})$) is identical to ${\mathcal{O}}(T^{\mathsf{lchild}}_{j,a})$. Similarly, the vertices
of $T^{\mathsf{rchild}}(j,a)$ appear on curve $\lambda'_{j-1,a''}$, and their (oriented) ordering on this curve (with respect to disc $D'(H_{j,a})$) is identical to ${\mathcal{O}}(T^{\mathsf{rchild}}_{j,a})$.
We now bound the number of crossings in drawing $\psi'_{j,i}$. For convenience, denote $H'= H_{j,a}\setminus \textsf{left} (T^{\mathsf{parent}}_{j,a}\cup T^{\mathsf{lchild}}_{j,a}\cup T^{\mathsf{rchild}}_{j,a}\textsf{right} )$.
Recall that $\chi(H_{j,a})$ is the set of all crossings in drawing $\psi'$ of $G'$ in which the edges of $G'[X^j_a]$ participate.
First, the total number of crossings between the edges of $E(H')$ is clearly bounded by $|\chi(H_{j,a})|$. There are no crossings between the curves in $\Gamma_{j,a}$. The number of crossings between the curves in $\Gamma_{j,a}$ and the edges of $E(H')$ is bounded by $|\chi(H_{j,a})|$ from \Cref{obs: outer curves}. The number of crossings between the curves in $\Gamma'_{j-1,a'}$ and the edges of $E(H')$ is bounded by $|\chi(H_{j,a})|$, and the number of crossings between the curves in $\Gamma'_{j-1,a'}$ is also bounded by $|\chi(H_{j,a})|$ from \Cref{obs: inner curves}. Similarly, the number of crossings between the curves in $\Gamma'_{j-1,a''}$, and the number of crossings between the curves in $\Gamma'_{j-1,a''}$ and the edges of $E(H')$ is bounded by $|\chi(H_{j,a})|$. It now remains to bound the number of crossings between the curves of $\Gamma'_{j-1,a'}$ and the curves of $\Gamma'_{j-1,a''}$. Notice that each such crossing corresponds to a unique crossing between an edge of $G'[X^{j-1}_{a'}]$ and an edge of $G'[X^{j-1}_{a''}]$. Since $G'[X^{j-1}_{a'}], G'[X^{j-1}_{a''}]\subseteq G'[X^j_a]$, the number of crossings between the curves of $\Gamma'_{j-1,a'}$ and the curves of $\Gamma'_{j-1,a''}$ is also bounded by $|\chi(H_{j,a})|$. Overall, the total number of crossings in $\phi_{j,a}$ is bounded by $O(|\chi(H_{j,a})|)$.
Finally, we need to ``fix'' the current drawing of graph $H_{j,a}$ by reordering the images of the vertices of $T^{\mathsf{parent}}_{j,a}$. Recall that every vertex $t\in T^{\mathsf{parent}}_{j,a}$ is an endpoint of a distinct edge in $\delta_{G'}(X^j_a)$. We have defined two oriented orderings of the edges of $\delta_{G'}(X^j_a)$: ordering ${\mathcal{O}}_{j,a}$ and ordering ${\mathcal{O}}'_{j,a}$. Each of these orderings naturally defines an oriented ordering of the vertices of $T^{\mathsf{parent}}_{j,a}$: we have denoted by ${\mathcal{O}}(T^{\mathsf{parent}}_{j,a})$ the ordering of the vertices of $T^{\mathsf{parent}}_{j,a}$ defined by ${\mathcal{O}}_{j,a}$, after we reverse the ordering. We denote the oriented ordering of the vertices of $T^{\mathsf{parent}}_{j,a}$ corresponding to ${\mathcal{O}}'_{j,a}$ by ${\mathcal{O}}'(T^{\mathsf{parent}}_{j,a})$. Note that the images of the vertices of $T^{\mathsf{parent}}_{j,a}$ appear on the curve $\lambda^*_{j,a}$ in the ordering ${\mathcal{O}}'(T^{\mathsf{parent}}_{j,a})$, as we traverse the boundary of the disc $D'(H_{j,a})$ in the clockwise direction. But we are required to ensure that the imges of the vertices of $T^{\mathsf{parent}}_{j,a}$ appear on the curve $\lambda''_{j,a}$ in the ordering ${\mathcal{O}}(T^{\mathsf{parent}}_{j,a})$, as we traverse the boundary of the disc $D'(H_{j,a})$ in the clockwise direction. Let $D^*$ be the disc that is contained in $D(H_{j,a})$, whose boundary is the concatenation of the curves $\lambda''_{j,a}$ and $\lambda^*_{j,a}$.
We place the images of the vertices of $T^{\mathsf{parent}}_{j,a}$ on curve $\lambda''_{j,a}$, so that they are encountered in the order ${\mathcal{O}}(T^{\mathsf{parent}}_{j,a})$, as we traverse the boundary of $D^*$ in the clockwise direction. Notice that the previous images of the vertices of $T^{\mathsf{parent}}_{j,a}$ appeared on curve $\lambda^*_{j,a}$, and they are encountered on that curve in the order ${\mathcal{O}}'(T^{\mathsf{parent}}_{j,a})$, as we traverse the boundary of $D^*$ in the counter-clockwise direction. Recall that, from \Cref{obs: bound distance between orderings}, the distance between the oriented orderings ${\mathcal{O}}(T^{\mathsf{parent}}_{j,a})$ and ${\mathcal{O}}'(T^{\mathsf{parent}}_{j,a})$ is
$\mbox{\sf dist}({\mathcal{O}}_{j,a},{\mathcal{O}}'_{j,a})\leq O(|\chi(H_{j,a})|)$. Therefore, we can define, for every vertex $t\in T^{\mathsf{parent}}_{j,a}$ a curve $\gamma^*_t$, that is contained in disc $D^*$, and connects the original image of $t$ to the new image of $t$. The total number of crossings between the curves in $\set{\gamma^*_t\mid t\in T^{\mathsf{parent}}_{j,a}}$ is bounded by $O(|\chi(H_{j,a})|)$. In order to obtain the final solution $\psi(H_{j,a})$ to instance $I_{H_{j,a}}$, we start with solution $\psi'_{j,a}$ to the same instance. For every vertex $t\in T^{\mathsf{parent}}_{j,a}$, we consider the unique edge $e_t$ of $H_{j,a}$ that is incident to $t$. We extend the image of $e_t$ by appending the curve $\gamma^*_t$ to it, and we move the image of $t$ to its new location on curve $\lambda''_{j,a}$. This completes the construction of solution $\psi(H_{j,a})$ to instance $I_{H_{j,a}}$, for $j>0$.
Next, we consider a graph $H_{j,a}\in {\mathcal{H}}$ with $j=0$. We first define a curve $\lambda^*_{j,a}$ exactly as before. We then let $D'(H_{j,a})$ be the disc that is contained in $D(H_{j,a})$, whose boundary is the concatenation of the curves $\lambda^*_{j,a}$ and $\sigma_{j,a}$. In order to construct the initial solution $\psi'_{j,a}$ to instance $H_{j,a}$, we start with the drawing of graph $H_{j,a}$ that is induced by $\phi$. We then process every vertex $t\in T^{\mathsf{parent}}_{j,a}$ exactly as before, replacing the image of the unique edge $e_t$ incident to $t$ with the curve $\gamma_{j,a}(e_t)$, and moving the image of $t$ to segment $\sigma'_{j,a}$ of the boundary of $D$. We plant the resulting drawing of graph $H_{j,a}$ inside disc $D'(H_{j,a})$, so that the segments $\sigma_{j,a}$ in disc $D$ and $D'(H_{j,a})$ coincide, and the images of the anchor vertices in set $T_{j,a}$ remain unchanged. We also ensure that segment $\sigma'_{j,a}$ on the bondary of disc $D$ coincides with segment $\lambda^*_{j,a}$ on the boundary of $D'(H_{j,a})$. We then modify the images of the edges incident to the vertices $ T^{\mathsf{parent}}_{j,a}$, and update the images of the vertices of $ T^{\mathsf{parent}}_{j,a}$ exactly as before.
The solution $\psi(H^*)$ to the instance $I_{H^*}$ associated with graph $H^*$ is computed very similarly. The main difference is that this time we do not need to define the curve $\lambda^*_{j,a}$, and instead we can plant the initial image of $H^*$ directly into the disc $D(H^*)$, making sure that the images of the anchor vertices that belong to $H^*$ (the vertices of $T^*$) remain unchanged. We no longer need to take care of the set $T^{\mathsf{parent}}_{j,a}$ of vertices, and the vertices of $T^{**}$ are treated like the vertices of $T^{\mathsf{lchild}}_{j,a}$ or $T^{\mathsf{rchild}}_{j,a}$ in the case where $j>0$.
\subsection{Proof of \Cref{obs: bounds on opt}}
\label{subsec: proof of obs on bounds on opt}
Assume that
$\mathsf{OPT}_{\mathsf{cnwrs}}(I)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}m}$. Then, from \Cref{lem: find ordering of terminals}:
\begin{equation}\label{eq: bound on cr}
\begin{split}
\mathsf{cr}(\phi)\leq &O\textsf{left}(\frac{(\tilde k\tilde \alpha\alpha')^2\eta^2\log^{54}m}{c_1\eta'\alpha^{12}(\alpha')^4}\textsf{right} ) + O\textsf{left} (
\frac{\tilde k \eta\log^{37}m}{\alpha^6(\alpha')^2}\textsf{right} ) \\
&\leq O\textsf{left}(\frac{\tilde k^2\log^{46}m}{c_1\eta^7\alpha^{10}(\alpha')^2}\textsf{right} ) + O\textsf{left} (
\frac{\tilde k \eta\log^{37}m}{\alpha^6(\alpha')^2}\textsf{right} ),
\end{split}
\end{equation}
since $\tilde \alpha=\Theta(\alpha/\log^4m)$ and $\eta'\geq \eta^{13}$ (from the statement of \Cref{thm: find guiding paths}).
Recall that, since we have assumed that Special Case 4 did not happen, $k\geq \eta^6$, and from Property \ref{prop after step 1: number of pseudoterminals},
$\tilde k \geq \Omega(\alpha^3k/\log^8m)$. Therefore, $\tilde k\geq \Omega(\eta^6\alpha^3/\log^8m)$, and $\eta\leq O\textsf{left} (\frac{\tilde k\log^8m}{\eta^5\alpha^3}\textsf{right} )$. We can now bound the second term in Equation \ref{eq: bound on cr} as follows:
\[ O\textsf{left} (
\frac{\tilde k \eta\log^{37}m}{\alpha^6(\alpha')^2}\textsf{right} )
\leq O\textsf{left} (
\frac{\tilde k^2\log^{45}m}{\eta^5\alpha^9(\alpha')^2}\textsf{right} )
\]
Recall that we have assumed that $\log m>c_0'$, for some large enough constant $c_0'$, whose value we can set to be greater than $c_1$.
Therefore, $\mathsf{cr}(\phi)\leq
O\textsf{left}(\frac{\tilde k^2\log^{46}m}{c_1\eta^6\alpha^{10}(\alpha')^2}\textsf{right} )$ must hold.
Lastly, from the conditions of \Cref{thm: find guiding paths}, $\eta\geq c^*\log^{46}m/(\alpha^{10}(\alpha')^2)$. Since we can assume that $c_1$ is a sufficiently large constant, we conclude that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}m}$, then $\mathsf{cr}(\phi)\leq \frac{\tilde k^2}{c_2\eta^5}$ holds, where $c_2$ is an arbitrarily large constant whose value we can set later.
\subsection{Proof of \Cref{obs: transform paths 2}}
\label{subsec: transform paths 2}
In order to obtain the distribution ${\mathcal{D}}'$, for every router ${\mathcal{Q}}\in \Lambda'$, whose probability value in ${\mathcal{D}}$ is $p({\mathcal{Q}})>0$, we construct a router ${\mathcal{Q}}'\in \Lambda(H,\tilde T)$, as follows. Let $y'$ be the center vertex of ${\mathcal{Q}}$, and assume that $y'\in \Pi(y)$ for some vertex $y\in V(H)$. For every terminal $t\in \tilde T$, let $Q_t\in {\mathcal{Q}}$ be the unique path connecting $t$ to $y'$ in $H''$. By suppressing all inner edges on path $Q_t$, we obtain a path $Q'_t$ in graph $H$, connecting $t$ to $y$. We then set ${\mathcal{Q}}'=\set{Q'_t\mid t\in \tilde T}$. It is easy to verify that paths in ${\mathcal{Q}}'$ route $\tilde T$ to $y$ in graph $H$, so ${\mathcal{Q}}'\in \Lambda(H,\tilde T)$. Moreover, for every edge $e\in E(H)$, $\cong_H({\mathcal{Q}}',e)\leq \cong_{H''}({\mathcal{Q}},e)$. We assign to the router ${\mathcal{Q}}'\in \Lambda(H,\tilde T)$ the same probability value $p({\mathcal{Q}})$. Let ${\mathcal{D}}'$ be the resulting distribution over the routers of $\Lambda(H,\tilde T)$. Since every edge $e\in E(H)$ is an outer edge of $H''$, it is immediate to verify that $\expect[{\mathcal{Q}}'\sim {\mathcal{D}}']{(\cong_{H}({\mathcal{Q}}',e))^2}\leq \beta$.
\section{Proofs Omitted from \Cref{sec: routing within a cluster}}
\subsection{Proof of \Cref{thm: basic decomposition of a graph}}
\label{sec: appx-decomposition-good-bad-other}
Note that, since graph $G$ is connected, $|V(G)|\leq m+1\leq 2m$ must hold.
Throughout, we use a parameter $\eta'=c\eta\log_{3/2}m\log_2m$, were $c$ is a large enough constant, whose value we set later.
The algorithm maintains a collection ${\mathcal{C}}$ of disjoint clusters of $G\setminus T$, such that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)\setminus T$. Set ${\mathcal{C}}$ of clusters is partitioned into two subsets: set ${\mathcal{C}}^A$ of active clusters and set ${\mathcal{C}}^I$ of inactive clusters. We will ensure that every cluster $C\in {\mathcal{C}}^I$ has the $\alpha'$-bandwidth property. Set ${\mathcal{C}}^I$ of inactive clusters is, in turn, partitioned into three subsets, ${\mathcal{C}}_1^I,{\mathcal{C}}_2^I$, and ${\mathcal{C}}_3^I$. For every cluster $C\in {\mathcal{C}}_3^I$, we will define a vertex $u(C)\in V(C)$, and an internal $C$-router ${\mathcal{Q}}(C)$, whose center vertex is $u(C)$, such that the paths in ${\mathcal{Q}}(C)$ are edge-disjoint. For every cluster
$C\in {\mathcal{C}}^I_1$, we will ensure that $|E(C)|\leq O(\eta^4\log^8m)\cdot |\delta_G(C)|$ holds.
Lastly, for every cluster $C\in {\mathcal{C}}^I_2$, we will ensure that
$\mathsf{OPT}_{\mathsf{cr}}(C)\geq \Omega(|E(C)|^2/(\eta^2\operatorname{poly}\log m))$, and $|E(C)|> \Omega(\eta^4 |\delta_G(C)|\log^8m)$. We start with ${\mathcal{C}}^I=\emptyset$, and ${\mathcal{C}}^A$ containing a single cluster $G\setminus T$ (note that graph $G\setminus T$ is connected since $G$ is connected and every terminal has degree $1$). The algorithm terminates once ${\mathcal{C}}^A=\emptyset$, and once this happens, we return ${\mathcal{C}}^I$ as the algorithm's outcome.
In order to bound the number of edges in the contracted graph $E(G_{|{\mathcal{C}}})$, we will use edge budgets and vertex budgets, that are defined as follows.
\paragraph{Edge Budgets.}
If an edge $e$ belongs to the boundary $\delta_G(C)$ of a cluster $C\in {\mathcal{C}}$, then, if $C\in {\mathcal{C}}^I$, we set the budget $B_C(e)=1$, and otherwise we set it to be $B_C(e)=\log_{3/2}(|\delta_G(C)|)$. If cluster $C$ is the unique cluster with $e\in \delta_G(C)$, then we set $B(e)=B_C(e)$. If there are two clusters $C\neq C'\in {\mathcal{C}}$ with $e\in \delta_G(C)$ and $e\in \delta_G(C')$, then we set $B(e)=B_C(e)+B_{C'}(e)$. Lastly, if no cluster $C\in {\mathcal{C}}$ with $e\in \delta_G(C)$ exists, then we set $B(e)=0$.
\paragraph{Vertex Budgets.}
Vertex budgets are defined as follows. For every cluster $C\in {\mathcal{C}}^A$, for every vertex $v\in V(C)$, we set the budget $B(v)=\frac{c\deg_C(v)\log_{3/2}m\cdot \log_{2}(|E(C)|)}{8\eta'}$, where $c$ is the constant used in the definition of $\eta'$. The budgets of all other vertices are set to $0$.
\paragraph{Cluster Budgets and Total Budget.}
For a cluster $C\in{\mathcal{C}}$, we define its edge-budget $B^E(C)=\sum_{e\in \delta_G(C)}B_C(e)$, and its vertex-budget $B^V(C)=\sum_{v\in V(C)}B(v)$. The total budget of a cluster $C\in {\mathcal{C}}$ is $B(C)=B^E(C)+B^V(C)$, and the total budget in the system is $B^*=\sum_{C\in {\mathcal{C}}}B(C)=\sum_{e\in E(G)}B(e)+\sum_{v\in V(G\setminus T)}B(v)$.
Notice that at the beginning of the algorithm, the budget of every vertex $v\in V(G)\setminus T$ is bounded by:
$$\frac{c\cdot \deg_{G\setminus T}(v)\cdot \log_{3/2}m\cdot \log_2|E(G)|}{8\eta'}\leq \frac{\deg_G(v)}{8\eta},$$
the budget of every edge incident to a vertex in $T$ is at most $\log_{3/2}(|T|)\leq 16\log m$, while the budget of every other edge is $0$. Therefore, the total budget $B^*$ in the system at the beginning of the algorithm is:
\[ \frac{m}{4\eta}+ 16k\log m\leq \frac{m}{\eta},\]
since $k\leq \frac{m}{16\eta\log m}$ from the statement of \Cref{thm: basic decomposition of a graph}.
We will ensure that, throughout the algorithm, the total budget $B^*$ never increases. Since, from the definition, $B^*\geq \sum_{C\in {\mathcal{C}}}|\delta_G(C)|$, this ensures that, when the algorithm terminates, $|E(G_{|{\mathcal{C}}})|\leq m/\eta$, so the set ${\mathcal{C}}^I$ of clusters and its partition $({\mathcal{C}}_1^I,{\mathcal{C}}_2^I,{\mathcal{C}}_3^I)$ is a valid output of the algorithm.
As mentioned above, the algorithm starts with ${\mathcal{C}}^I=\emptyset$, and ${\mathcal{C}}^A$ contains a single cluster -- cluster $G\setminus T$. As long as ${\mathcal{C}}^A\neq \emptyset$, we perform iterations, where in each iteration we select an arbitrary cluster $C\in {\mathcal{C}}^A$ to process.
We now describe the execution of an iteration in which cluster $C\in {\mathcal{C}}^A$ is processed.
The algorithm for processing cluster $C$ consists of three steps, that we describe next.
\paragraph{Step 1: Bandwidth Property.}
In this step we will either establish that $C$ has the $\alpha'$-bandwidth property, or we will partition it into smaller clusters that will replace $C$ in set ${\mathcal{C}}^A$.
Let $C^+$ be the augmentation of cluster $C$. Recall that $C^+$ is a graph that is obtained as follows. We start with the graph $G$, and we subdivide every edge $e\in \delta_G(C)$ with a vertex $t_e$, letting $T(C)=\set{t_e\mid e\in \delta_G(C)}$ be this new set of vertices. We then let $C^+$ be the subgraph of the resulting graph induced by $V(C)\cup T(C)$. From \Cref{obs: wl-bw}, cluster $C$ has the $\alpha'$-bandwidth property iff the set $T(C)$ of vertices is $\alpha'$-well-linked in $C^+$. We apply the algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace to graph $C^+$ and terminal set $T(C)$, to obtain an $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)=O(\sqrt{\log m})$-approximate sparsest cut $(X,Y)$ in graph $C^+$ with respect to the set $T(C)$ of terminals.
We can assume without loss of generality that, for every vertex $t_e\in T(C)$, if $t_e\in X$, and $e'=(t_e,v)$ is the unique edge that is incident to $t_e$ in $C^+$, then $v\in X$ as well (as otherwise we can move $t_e$ to $Y$, making the cut only sparser). Similarly, if $t_e\in Y$, then $v\in Y$ as well.
We assume w.l.o.g. that $|X\cap T(C)|\leq |Y\cap T(C)|$.
We then consider two cases. First, if $|E(X,Y)|\geq \alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |X\cap T(C)|$, then we are guaranteed that vertex set $T(C)$ is $\alpha'$-well-linked in $C^+$, and therefore cluster $C$ has the $\alpha'$-bandwidth property. Assume now that $|E(X,Y)|< \alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |X\cap T(C)|$.
Let $X'=X\setminus T(C)$ and $Y'=Y\setminus T(C)$, so $(X',Y')$ is a partition of $C$. Note that $|T(C)\cap X|=|\delta_G(C)\cap \delta_G(X')|$ and similarly $|T(C)\cap Y|=|\delta_G(C)\cap \delta_G(Y')|$.
We remove cluster $C$ from ${\mathcal{C}}^A$, and we add all connected components of $C[X']$ and $C[Y']$ to ${\mathcal{C}}^A$ instead. Observe that we are still guaranteed that $\bigcup_{C'\in {\mathcal{C}}}V(C')=V(G)\setminus T$. We now show that the total budget in the system does not increase as the result of this step.
Since $|X'|,|Y'|<|V(C)|$, it is immediate to verify that, for every vertex $v$ of $C$, its budget may only decrease. The only edges whose budget may increase are the edges of $E_C(X',Y')$. The number of such edges is bounded by $\alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |X\cap T(C)|=\alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(C)\cap \delta_G(X')|$, and the budget of each of them increases by at most $2\log_{3/2}m\leq 8\log m$, so the total increase in the budget of all edges due to this step is bounded by:
\[8\alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(C)\cap \delta_G(X')|\cdot \log m\leq |\delta_G(C)\cap \delta_G(X')|,\]
since $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}$.
Consider now some edge $e\in \delta_G(C)\cap \delta_G(X')$. Since we have assumed that $|X\cap T(C)|\leq |Y\cap T(C)|$, it is easy to verify that $|\delta_G(X')|\leq 2|\delta_G(C)|/3$. Therefore, if an endpoint of an edge $e\in \delta_G(C)$ belongs to a new cluster $C'\subseteq G[X']$, then the new budget $B_{C'}(e)$ becomes at most:
\[\log_{3/2}(|\delta_G(X')|)\leq \log_{3/2}(|\delta_G(C)|)-1. \]
The total decrease in the global budget due to the edges of
$\delta_G(X')\cap \delta_G(C)$ is then at least $|\delta_G(C)\cap \delta_G(X')|$. We conclude that overall the global budget does not increase.
We assume from now on that algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace returned a cut $(X,Y)$ of $C^+$ with $|E(X,Y)|\geq \alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |X\cap T(C)|$, and so cluster $C$ has the $\alpha'$-bandwidth property.
Assume now that $|E(C)|\leq (\eta')^4|\delta_G(C)|$. From the definition of $\eta'$, we are then guaranteed that $|E(C)|\leq O(\eta^4\log^8m)\cdot |\delta_G(C)|$.
We then remove cluster $C$ from ${\mathcal{C}}^A$ and add it to the set ${\mathcal{C}}^I$ of inactive clusters, and to the set ${\mathcal{C}}^I_1$ of clusters. Therefore, we assume from now on that
$|E(C)|> (\eta')^4|\delta_G(C)|$.
\iffalse
As a first step, we apply the algorithm from \Cref{thm:well_linked_decomposition} to compute a well-linked decomposition ${\mathcal{W}}$ of $C$, with parameter $\alpha'$. Recall that we are guaranteed that vertex sets
$\set{V(W)}_{W\in {\mathcal{W}}}$ partition $V(C)$;
for each cluster $W\subseteq{\mathcal{W}}$, $|\delta_G(W)|\le |\delta_G(C)|$, and each cluster $W\subseteq{\mathcal{W}}$ has the $\alpha'$-bandwidth property.
We now consider two cases. If ${\mathcal{W}}$ contains more than a single cluster, then we say that the well-linked decomposition step was successful; otherwise we say that it was unsuccessful. We assume first that the well-linked decomposition step was successful. We then remove $C$ from ${\mathcal{C}}^A$, and add all clusters of ${\mathcal{W}}$ to ${\mathcal{C}}^A$ instead. We now show that the tota budget $B^*$ in the system did not increase. Let $B^*$ denote
; and
\item $\sum_{R\in {\mathcal{R}}}|\delta(R)|\le |\delta(S)|\cdot\textsf{left}(1+O(\alpha\cdot \log^{3/2} n)\textsf{right})$
\begin{theorem}
\label{thm:well_linked_decomposition}
There is an efficient algorithm, that, given any graph $G=(V,E)$, any cluster $S$ of $G$ and any parameter $\alpha\le 1/(48\log^2 n)$, computes a set ${\mathcal{R}}$ of clusters of $S$, such that,
\begin{itemize}
\item the vertex sets $\set{V(R)}_{R\in {\mathcal{R}}}$ partition $V(S)$;
\item for each cluster $R\subseteq{\mathcal{R}}$, $|\delta(R)|\le |\delta(S)|$;
\item each cluster $R\subseteq{\mathcal{R}}$ has the $\alpha$-bandwidth property; and
\item $\sum_{R\in {\mathcal{R}}}|\delta(R)|\le |\delta(S)|\cdot\textsf{left}(1+O(\alpha\cdot \log^{3/2} n)\textsf{right})$.
\end{itemize}
\end{theorem}
\fi
\paragraph{Step 2: Sparse Balanced Cut.}
In this step, we apply the algorithm from \Cref{cor: approx_balanced_cut} to graph $C$ with parameter $\hat c=3/4$, to obtain a $\hat c'$-edge-balanced cut $(Z,Z')$ of $C$ (where $1/2<\hat c'<1$), whose value is at most
$O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))$ times the value of a minimum $3/4$-edge-balanced cut of $C$. We say that this step is \emph{successful} if $|E_G(Z,Z')|<|E(C)|/\eta'$. Assume first that the step was successful. Then we remove cluster $C$ from set ${\mathcal{C}}^A$, and add all connected components of graphs $C[Z],C[Z']$ to set ${\mathcal{C}}^A$ insead. We now show that the total budget in the system does not increase as the result of this step. Observe that the budget of every vertex may only decrease, and the same is true for the budget of every edge, except for the edges in set $\delta_G(C)\cup E_G(Z,Z')$. The budget of each such edge may increase by at most $2\log_{3/2}m$, so the total increase in the budgets of all edges is bounded by $(|\delta_G(C)|+|E_G(Z,Z')|)\cdot 2\log_{3/2}m \leq \frac{4|E(C)|\cdot \log_{3/2}m}{\eta'}$ (we have used the fact that $|E(C)|> (\eta')^4|\delta_G(C)|$). We now show that this increase in total budget is compensated by the decrease in the budgets of the vertices of $Z$.
Assume without loss of generality that $|E(Z)|\leq |E(Z')|$.
Recall that for every vertex $v\in Z$, its original budget is:
$B(v)=\frac{c\deg_C(v)\log_{3/2}m\cdot \log_{2}(|E(C)|)}{8\eta'}$.
From our assumption that $|E(Z)|\leq |E(Z')|$, $\log_{2}(|E(Z)|)\leq \log_{2}(|E(C)|)-1$.
The new budget of vertex $v$ is:
$$B'(v)=\frac{c\deg_Z(v)\log_{3/2}m\cdot \log_{2}(|E(Z)|)}{8\eta'}\leq \frac{c\deg_Z(v)\log_{3/2}m\cdot (\log_{2}(|E(C)|)-1)}{8\eta'}.$$
Therefore, for every vertex $v\in Z$, its budget decreases by at least
$\frac{c\deg_Z(v)\log_{3/2}m}{8\eta'}$.
From the definition of a $(3/4)$-edge-balanced cut, $|E(Z')|\leq \hat c'|E(C)|$, for some universal constant $\hat c'$. In particular:
\[
\sum_{v\in Z}\deg_Z(v)\geq |E(C)|-|E(Z')|\geq (1-\hat c')|E(C)|. \label{eq: many edges in Z}
\]
Overall, the budget of the vertices in $Z$ decreases by at least:
\[\frac{c\log_{3/2}m}{8\eta'}\cdot \sum_{v\in Z}\deg_Z(v)\geq \frac{c\log_{3/2}m}{8\eta'}\cdot (1-\hat c')\cdot |E(C)|.\]
Since $\hat c'<1$, and we can set $c$ to be a large enough constant, we can ensure that this is at least $\frac{4|E(C)|\cdot \log_{3/2}m}{\eta'}$, so the overall budget in the system does not increase.
If the current step is successful, then we replace cluster $C$ with the connected components of $C[Z]$ and $C[Z']$ in set ${\mathcal{C}}^A$ and continue to the next iteration. Therefore, we assume from now on that the current step was not successful. In other words, the algorithm from \Cref{cor: approx_balanced_cut} returned a cut$(Z,Z')$ with $|E_G(Z,Z')|\geq |E(C)|/\eta'$. Since this cut is within factor $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)=O(\sqrt{\log m})$ from the minimum $3/4$-edge-balanced cut, we conclude that the value of the minimum $3/4$-edge-balanced cut in $C$ is at least $\rho=\frac{|E(C)|}{\eta'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)}$.
From \Cref{lem:min_bal_cut}, if the maximum vertex degree $\Delta$ in graph $C$ is at most $|E(C)|/2^{40}$ and $\mathsf{OPT}_{\mathsf{cr}}(C)\le |E(C)|^2/2^{40}$,
then graph $C$ has a $(3/4)$-edge-balanced cut of value at most $\tilde c\cdot\sqrt{\mathsf{OPT}_{\mathsf{cr}}(C)+\Delta\cdot|E(C)|}$ where $\tilde c>2^{40}$ is some universal constant.
As the size of the minimum $3/4$-balanced cut in $C$ is at least $\rho$, we conclude that either $\Delta\geq |E(C)|/2^{40}$, or $\mathsf{OPT}_{\mathsf{cr}}(C)> |E(C)|^2/2^{40}$, or
$\sqrt{\mathsf{OPT}_{\mathsf{cr}}(C)+\Delta\cdot|E(C)|}\geq \rho/\tilde c$. The latter can only happen if either $\mathsf{OPT}_{\mathsf{cr}}(C)\geq \rho^2/\tilde c^2$, or $\Delta\geq \rho^2/(\tilde c^2\cdot |E(C)|)$. Substituting the value of $\rho=\frac{|E(C)|}{\eta'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)}$, we conclude that either $\mathsf{OPT}_{\mathsf{cr}}(C)\geq \frac{|E(C)|^2}{(\tilde c \eta'\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))^2} $, or $\Delta\geq \frac{|E(C)|}{(\tilde c\eta'\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))^2}$, or $\Delta\geq \frac{|E(C)|}{2^{40}}$. Note that we can check whether the last two conditions hold efficiently. If they do not hold, then we are guaranteed that $\mathsf{OPT}_{\mathsf{cr}}(C)\geq \frac{|E(C)|^2}{(\tilde c\eta'\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))^2 }\geq \frac{|E(C)|^2}{\eta^2\operatorname{poly}\log m }$. Recall that we are also guaranteed that $|E(C)|> (\eta')^4\cdot |\delta_G(C)|=\Omega(\eta^4|\delta^G(C)|\log^8m)$. We then remove cluster $C$ from the set ${\mathcal{C}}^A$ of active clusters and add it to set ${\mathcal{C}}^I$ of inactive clusters, where it joins the set ${\mathcal{C}}^I_2$ of clusters.
From now on we can assume that
$|E(C)|>\Omega(\eta^4|\delta^G(C)|\log^8m)$, and that either $\Delta\geq \frac{|E(C)|}{(\tilde c \eta'\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))^2}$, or $\Delta\geq \frac{|E(C)|}{2^{40}}$ hold. Note that, since $\eta'=c\eta\log_{3/2}m\log_2 m$, and since we can set $c$ to be a large enough constant, we can ensure that $\frac{|E(C)|}{2^{40}}\geq \frac{|E(C)|}{\tilde c^2(\eta')^2\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)}$. Therefore, from now on we assume that there is some vertex $v^*$ in graph $C$, whose degree in $C$ is at least $\frac{|E(C)|}{(\tilde c \eta'\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))^2}$.
Since we have assumed that $|E(C)|\geq (\eta')^4|\delta_G(C)|$, we get that
$\deg_G(v^*)\geq 8|\delta_G(C)|\eta'$.
\paragraph{Step 3: routing to a vertex}
We consider again the graph $C^+$ that we have defined in Step 1, with the corresponding set $T(C)=\set{t_e\mid e\in \delta_G(C)}$ of terminal vertices.
Using standard Maximum Flow computation, we compute a maximum-cardinality set ${\mathcal{P}}$ of edge-disjoint paths, where each path connects a distinct vertex of $T(C)$ to to vertex $v^*$. We now consider two cases. In the first case, $|{\mathcal{P}}|=|\delta_G(C)|$. In this case, there is a set ${\mathcal{Q}}(C)$ of edge-disjoint paths in cluster $C$, routing edges of $\delta_e(C)$ to vertex $v^*$, which can be easily obtained from ${\mathcal{P}}$. We then remove cluster $C$ from the set ${\mathcal{C}}^A$ of active clusters and add it to set ${\mathcal{C}}^I$ of inactive clusters, where it joins cluster set ${\mathcal{C}}^I_3$.
Assume now that $|{\mathcal{P}}|<|\delta_G(C)|$. From the maximum flow / minimum cut theorem, there is a collection $E'$ of at most $|\delta_G(C)|$ edges in graph $C^+$, such that, in graph $C^+\setminus E'$, there is no path connecting an edge of $\delta_G(C)$ to vertex $u^*$. Let $C'$ be the connected component of $C^+\setminus E'$ containing $v^*$, so $C'\subseteq C$. Note that $\delta_G(C')\subseteq E'$, so there is a set ${\mathcal{Q}}(C')$ of edge-disjoint paths routing the edges of $\delta_G(C')$ to vertex $v^*$, with all inner vertices of the paths in ${\mathcal{Q}}(C')$ lying in $C'$; in other words, ${\mathcal{Q}}(C')$ is an internal $C'$-router. We add cluster $C'$ to set ${\mathcal{C}}^I$ and ${\mathcal{C}}^I_3$. Next, we delete cluster $C$ from ${\mathcal{C}}^A$, and we add every connected component of $C\setminus C'$ as a new cluster to set ${\mathcal{C}}^A$.
It now remains to prove that the total budget in the system does not increase as the result of these changes. Note that the budget of every vertex may only decrease, and the budget of every edge, except for the edges of $\delta_G(C)\cup E'$, may also only decrease. The increase in the budget of every edge in $\delta_G(C)\cup E'$ is bounded by $2\log_{3/2}m$. Therefore, the total increase in the budget is bounded by $(|\delta_G(C)|+|E'|)\cdot 2\log_{3/2}m\leq 4|\delta_G(C)|\cdot \log_{3/2}m$. Note that the budget of vertex $v^*$ was initially at least $\frac{c\deg_C(v)\log_{3/2}m\cdot \log_{2}(|E(C)|)}{8\eta'}$, and it becomes $0$ at the end of this step. Therefore, the decrease in the budget is at least:
\[\frac{c\deg_C(v^*)\log_{3/2}m\cdot \log_{2}(|E(C)|)}{8\eta'}\geq 4|\delta_G(C)|\cdot \log_{3/2}m,\]
since $\deg_C(v^*)\geq 8|\delta_G(C)|\eta'$. Overall, the budget does not grow.
The algorithm terminates once ${\mathcal{C}}^A$ becomes empty, and it returns the set ${\mathcal{C}}^I$ of clusters. From our invariants, it is immediate to verify that this set of clusters has all required properties. It remains to establish that the algorithm is efficient. In every iteration of the algorithm, we either add a cluster to set ${\mathcal{C}}^I$, or we split a single cluster of ${\mathcal{C}}^A$ into at least two clusters. Once a cluster is added to ${\mathcal{C}}^I$, it remains there until the end of the algorithm. It is then easy to verify that the number of iterations is bounded by $O(|V(G)|)\le O(m)$, and every iteration can be executed efficiently.
\subsection{Proof of \Cref{obs: opt is small}}
\label{subsec: proof of obs opt is small}
Let $\phi^*$ be the optimal solution to instance $(\tilde C,\Sigma_{\tilde C})$. We can assume w.l.o.g. that every pair of edges cross at most once in $\phi^*$. Denote by $\chi$ the collection of all pairs $e,e'\in E(\tilde C)$ of edges, such that the images of $e$ and $e'$ cross in $\phi^*$.
Assume for now that, for every cluster $W\in {\mathcal{W}}^{\operatorname{light}}$, the routers ${\mathcal{Q}}(W)$ and $\hat {\mathcal{Q}}(W)$ are fixed, and so the rotation system $\hat \Sigma$ for graph $H$ is fixed as well. For every edge $e\in E(\tilde C)$, we define an integer $N(e)$ as follows. If there is a cluster $W\in {\mathcal{W}}^{\operatorname{light}}$ with $e\in E(W)$, then we let $N(e)$ be the number of paths in $\hat {\mathcal{Q}}(W)$ containing $e$. Therefore, $N(e)=\cong_G(\hat {\mathcal{Q}}(W),e)\leq \cong_G({\mathcal{Q}}(W),e)$. Otherwise, we set $N(e)=1$.
We will prove the following observation:
\begin{observation}\label{obs: opt is small fixed choice}
Suppose the routers $\hat {\mathcal{Q}}(W)$ for all clusters $W\in {\mathcal{W}}^{\operatorname{light}}$ are fixed, and let $\hat \Sigma$ be the corresponding rotation system for $H$. Then:
$$\mathsf{OPT}_{\mathsf{cnwrs}}(H, \hat \Sigma)\leq O\textsf{left}(\sum_{(e,e')\in \chi}N(e)\cdot N(e')\textsf{right} )+O\textsf{left}(\sum_{e\in E(\tilde C)}(N(e))^2\textsf{right} )$$.
\end{observation}
The proof of \Cref{obs: opt is small} immediately follows from \Cref{obs: opt is small fixed choice}. Indeed:
\[
\begin{split}
\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(H, \hat \Sigma)}&\leq \expect{\sum_{(e,e')\in \chi}O(N(e)\cdot N(e'))+\sum_{e\in E(\tilde C)}O((N(e))^2)}\\
&\leq O\textsf{left} (\sum_{(e,e')\in \chi}\expect{N(e)\cdot N(e')}\textsf{right} )+O\textsf{left}( \sum_{e\in E(\tilde C)}\expect{(N(e))^2} \textsf{right} )\\
&\leq O\textsf{left} (\sum_{(e,e')\in \chi}\textsf{left}(\expect{(N(e))^2}+\expect{ (N(e'))^2}\textsf{right} )\textsf{right} )+O\textsf{left}( \sum_{e\in E(\tilde C)}\expect{N(e))^2} \textsf{right} ).
\end{split} \]
Consider now some edge $e\in E(\tilde C)$. Assume first that there is some cluster $W\in {\mathcal{W}}^{\operatorname{light}}$ with $e\in E(W)$. Then, as observed above, $N(e)\leq \cong_G({\mathcal{Q}}(W),e)$. Since ${\mathcal{Q}}(W)$ is a router of $\Lambda_G(W)$ that is drawn from the distribution ${\mathcal{D}}(W)$, and since cluster $W$ is $\beta_i$-light with respect to ${\mathcal{D}}(W)$, we get that:
$$\expect[{\mathcal{Q}}(W)\sim {\mathcal{D}}(W)]{(N(e))^2}\leq \expect[{\mathcal{Q}}(W)\sim{\mathcal{D}}(W)]{(\cong_{G}({\mathcal{Q}}(W),e))^2}\leq \beta_{i}.$$
Otherwise, $e\in E(\tilde C)\setminus\textsf{left}(\bigcup_{W\in {\mathcal{W}}^{\operatorname{light}}}E(W)\textsf{right})$, and so $N(e)=1$ holds. Overall, for every edge $e\in E(\tilde C)$, $\expect{(N(e))^2}\leq \beta_i$. Therefore, we get that:
\[ \expect{\mathsf{OPT}_{\mathsf{cnwrs}}(H, \hat \Sigma)}\leq|\chi|\cdot O(\beta_i)+|E(\tilde C)|\cdot O(\beta_i)= O\textsf{left}(\beta_i\cdot\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde C,\Sigma_{\tilde C})+|E(\tilde C)|\textsf{right} )\textsf{right} ).\]
In order to complete the proof of \Cref{obs: opt is small}, it is now enough to prove \Cref{obs: opt is small fixed choice}.
\newpage
\begin{proofof}{\Cref{obs: opt is small fixed choice}}
The proof uses arguments that are very similar to those used in the proof of \Cref{lem: disengagement final cost}, and more specifically in the proof of \Cref{claim: cost of cluster instance}. Similar argument are used in several places throughout this paper, so we only provide a proof sketch here. We start with the optimal solution $\phi^*$ to instance $(\tilde C,\Sigma_{\tilde C})$, and then gradually transform it to obtain a solution $\hat\psi$ to instance $(H,\hat \Sigma)$.
Let $C^*$ be the graph that is obtained from $\tilde C$ as follows. We let $V(C^*)=V(\tilde C)$. For every edge $e=(u,v)\in E(\tilde C)$ with $N(e)>0$, we add a set $J(e)$ of $N(e)$ parallel edges $(u,v)$ to graph $C^*$. We call the edges of $J(e)$ \emph{copies of edge $e$}. We can modify the solution $\phi^*$ to instance $\tilde C$ to obtain a drawing $\psi$ of graph $C^*$ in a natural way: the images of all vertices of $\tilde C$ remain unchanged. For every edge $e\in E(\tilde C)$, we draw the images of the edges of $J(e)$ in $\psi$ in parallel very close to the original image of edge $e$ in $\phi^*$. It is immediate to verify that the number of crossings in the resulting drawing $\psi$ of graph $C^*$ is bounded by $\sum_{(e,e')\in \chi}N(e)\cdot N(e')$.
Consider now some cluster $W\in {\mathcal{W}}^{\operatorname{light}}$. We will now define, for every edge $e\in \delta_{\tilde C}(W)$, a curve $\gamma^*(e)$, that will serve as the image of $e$ in the solution $\hat \psi$ to instance $(H,\hat \Sigma)$ that we construct. Note that for each edge $e\in \delta_{\tilde C}(W)$, $N(e)=1$, and so set $J(e)$ of edges contains exactly one copy of edge $e$. We do not distinguis between edge $e$ and its unique copy in $J(e)$.
We use the internal $W$-router $\hat {\mathcal{Q}}(W)$ in graph $G$, in order to define a collection $\hat {\mathcal{Q}}'(W)=\set{\hat Q'(e)\mid e\in \delta_{\tilde C}(W)}$ of edge-disjoint paths in graph $C^*\cup \delta_{\tilde C}(W)$, such that, for every edge $e\in \delta_{\tilde C}(W)$, path $\hat Q'(e)$ originates at edge $e$, terminates at vertex $u(W)$ (the center vertex of the router $\hat {\mathcal{Q}}(W)$), and all internal vertices of $\hat Q'(e)$ lie in $V(W)$. In order to obtain the collection $\hat {\mathcal{Q}}'(W)$ of paths from the router $\hat {\mathcal{Q}}(W)$, for every edge $e\in E(W)$ with $N(e)>0$, we assign every copy of $e$ in $J(e)$ to a distinct path of $\hat {\mathcal{Q}}(W)$ that contains the edge $e$.
Consider any edge $e\in \delta_{\tilde C}(W)$, and denote $e=(x_e,y_e)$, where $x_e\in V(W)$. Initially, we let $\gamma(e)$ be the image of the path $\hat Q'(e)$ in the drawing $\psi$ of $C^*$. Let $\Gamma(W)=\set{\gamma(e)\mid e\in \delta_{\tilde C}(W)}$ be the resulting collection of curves. Note that, for every edge $e\in \delta_{\tilde C}(W)$, curve $\gamma(e)$ connects the image of vertex $y_e$ in drawing $\phi^*$ to the image of vertex $u(W)$ in the same drawing.
We are now ready to construct an initial drawing $\psi'$ of the graph $H$. For regular every vertex $v\in V(H)\cap V(\tilde C)$, the image of $v$ in $\psi'$ remains the same as in $\phi^*$ (and in $\psi$). For every edge $e$ of $H$ whose both endpoints are regular vertices, the image of $e$ remains the same as in $\phi^*$ (and the same as in $\psi$). Consider now some cluster $W\in {\mathcal{W}}^{\operatorname{light}}$. The image of the supernode $v_W$ in drawing $\psi'$ is the image of vertex $u(W)$ in $\phi^*$ (and in $\psi$). For every edge $e\in \delta_{\tilde C}(W)$, the initial image of edge $e$ is the curve $\gamma(e)$. Lastly, since the degree of every terminal $t\in T$ is $1$ in $H$, we can add these terminals and their adjacent edges to the current drawing without increasing the number of crossings. We note that the resulting drawing $\psi'$ of graph $H$ may not be a valid drawing. This is since, whenever there is a cluster $W\in {\mathcal{W}}^{\operatorname{light}}$ and a vertex $x\in V(W)$ that lies on more than two paths of $\hat {\mathcal{Q}}(W)$, then point $p=\phi^*(x)$ belongs to more than two curves of $\Gamma(W)$, and hence more than two edges cross at point $p$. In order to overcome this difficulty, for every cluster $W\in {\mathcal{W}}^{\operatorname{light}}$, for every vertex $x\in V(W)$ that lies on at least two paths of $\hat {\mathcal{Q}}(W)$, we perform a nudging operatio of the curves in $\Gamma(W)$ in the vicinity of vertex $x$ (see \Cref{sec: curves in a disc}). The curves are modified locally inside the tiny $x$-disc $D_{\phi^*}(x)$ to ensure that every point of $D_{\phi^*}(x)$ lies on at most two curves. Consider now a pair $e_1,e_2\in \delta_{\tilde C}(W)$ of edges. Let $\hat Q(e_1),\hat Q(e_2)$ be the paths of $\hat {\mathcal{Q}}(W)$ that originate at $e_1$ and $e_2$, respectively, and assume that both paths contain vertex $x$. From the definition of the nudging procedure (see also \Cref{claim: curves in a disc}), and since the paths in $\hat {\mathcal{Q}}(W)$ are non-transversal with respect to $\Sigma_{\tilde C}$, the curves $\gamma(e_1),\gamma(e_2)$ that are obtained at the end of the nudging operation may only cross inside disc $D_{\phi^*}(x)$ if some edge $e^*\in\delta_{\tilde C}(x)$ lies on both $\hat Q(e_1)$ and $\hat Q(e_2)$. We say that edge $e^*$ is \emph{responsible} for this crossing of $\gamma(e_1)$ and $\gamma(e_2)$.
For every edge $e\in \delta_{\tilde C}(W)$, let $\gamma'(e)$ be the curve that is obtained after the nudging operation is performed for every vertex $x\in V(W)$ that belongs to at least two paths of $\hat {\mathcal{Q}}(W)$, and denote $\Gamma'(W)=\set{\gamma'(e)\mid e\in \delta_{\tilde C}(W)}$. For every edge $e\in \delta_{\tilde C}(W)$, we replace the image of edge $e$ in the current drawing with the curve $\gamma'(e)$.
Consider the drawing $\psi'$ of graph $H$ that is obtained after all clusters $W\in {\mathcal{W}}^{\operatorname{light}}$ are processed. It is now easy to verify that $\psi'$ is a valid drawing of graph $H$. We partition the crossigns of drawing $\psi'$ into two types: a crossing is of type 1 if it is present in drawing $\psi$, and it is of type 2 otherwise. Equivalently, a crossing $(e,e')_{p}$ of $\psi'$ is of type 2 if there is a cluster $W\in {\mathcal{W}}^{\operatorname{light}}$, and a vertex $x\in V(W)$, such that the crossing point $p$ lies in $D_{\phi^*}(x)$.
The number of type-1 crossings in $\psi'$ remains bounded by $\mathsf{cr}(\psi)\leq \sum_{(e,e')\in \chi}N(e)\cdot N(e')$. In order to bound the number of type-2 crossings, observe that for every cluster $W\in {\mathcal{W}}^{\operatorname{light}}$, vertex $x\in V(W)$ and edge $e\in \delta_{\tilde C}(x)$, the number of type-2 crossings for which edge $e$ may be responsible is at most $(\cong_{\tilde C}(\hat {\mathcal{Q}}(W),e))^2=(N(e))^2$. Overall, we get that $\mathsf{cr}(\psi')\leq \sum_{(e,e')\in \chi}N(e)\cdot N(e')+\sum_{e\in E(\tilde C)}(N(e))^2$.
Observe that for every regular vertex $x\in V(H)\cap V(\tilde C)$, drawing $\psi'$ of $H$ obeys the rotation ${\mathcal{O}}_x\in \hat \Sigma$ (which is identical to the rotation of $x$ in $\Sigma_{\tilde C}$). However, it is possible that for some supernodes $v_W$, the rotation ${\mathcal{O}}_{v_W}\in \hat \Sigma$ is not obeyed by $\psi'$. We fix this issue by introducing at most $O\textsf{left} (\sum_{e\in E(\tilde C)}(N(e))^2\textsf{right} )$ additional crossings, as follows.
Consider a cluster $W\in {\mathcal{W}}^{\operatorname{light}}$, and denote $\delta_{\tilde C}(u(W))=\set{a_1,a_2,\ldots,a_r}$, where the edges are indexed according to their order in the rotation ${\mathcal{O}}_{u(W)}\in \Sigma_{\tilde C}$. For all $1\leq i\leq r$, let ${\mathcal{Q}}_i\subseteq \hat {\mathcal{Q}}(W)$ be the set of paths whose last edge is $a_i$. Denote by $A_i\subseteq \delta_{\tilde C}(W)$ the set of edges $e$, for which the unique path $\hat Q(e)\in \hat {\mathcal{Q}}(W)$ that originates at $e$ terminates at edge $a_i$; in other words, $\hat Q(e)\in {\mathcal{Q}}_i$. Denote by $\Gamma'_i\subseteq \Gamma'(W)$ the set of the images of the edges of $A_i$ in the current drawing $\psi'$. Let ${\mathcal{O}}'$ be that the circular order of the edges of $\delta_{\tilde C}(W)=A_1\cup A_2\cup\cdots\cup A_r$, in which their images in $\psi'$ enter the image of $v_W$.
Then for all $1\leq i\leq r$, the edges of $A_i$ appear consecutively in ${\mathcal{O}}'$ in some arbitrary order, while the edges lying in different groups of $A_1,A_2,\ldots,A_r$ appear in ${\mathcal{O}}'$ in the natural order of the indices of these groups.
Recall that the rotation ${\mathcal{O}}_{v_W}\in \hat \Sigma$ is a circular ordering of the edges of $\delta_{\tilde C}(W)=A_1\cup A_2\cup\cdots\cup A_r$ that is guided by the paths of $\hat {\mathcal{Q}}(W)$. In this ordering,
for all $1\leq i\leq r$, the edges of $A_i$ appear consecutively in some arbitrary order, while the edges lying in different groups of $A_1,A_2,\ldots,A_r$ appear in ${\mathcal{O}}_{v_W}$ in the natural order of the indices of these groups. In other words, the only difference between the orderings ${\mathcal{O}}'$ and ${\mathcal{O}}_{v_W}$ is that, for all $1\leq i\leq r$, the edges of $A_i$ may appear in different order in the two orderings. Therefore, $\mbox{\sf dist}({\mathcal{O}}',{\mathcal{O}}_{v_W})\leq \sum_{i=1}^r|A_i|^2=\sum_{i=1}^r(\cong_{\tilde C}(\hat {\mathcal{Q}}(W),e_i))^2\leq \sum_{i=1}^r(N(e_i))^2$. For all $1\leq i\leq r$, we slightly modify the curves of $\Gamma'_i$ inside the tiny $v_W$-disc $D_{\psi'}(v_W)$ to ensure that the images of the edges of $\delta_{\tilde C}(W)$ enter the image of vertex $v_W$ in the circular order ${\mathcal{O}}_{v_W}$. From the above discussion, this can be done by introducing at most $\sum_{i=1}^r(N(e_i))^2$ new crossings. Once every cluster $W\in {\mathcal{W}}^{\operatorname{light}}$ is processed, we obtain a valid solution $\hat \psi$ to instance $(H,\hat \Sigma)$, with $\mathsf{cr}(\hat \psi)\leq O\textsf{left}(\sum_{(e,e')\in \chi}N(e)\cdot N(e')+\sum_{e\in E(\tilde C)}(N(e))^2\textsf{right} )$.
\end{proofof}
\section{Proofs Omitted from \Cref{sec: computing the decomposition}}
\subsection{Proof of \Cref{lem: decomposition into small clusters}}
\label{sec: appx-decomposition-small-clusters}
We use a parameter $\tau'=c\tau\log_{3/2}m\log_2m$, were $c$ is a large enough constant, whose value we set later.
The algorithm maintains a collection ${\mathcal{C}}$ of disjoint clusters of $H\setminus T$, with $\bigcup_{C\in {\mathcal{C}}}V(C)=V(H)\setminus T$.
Set ${\mathcal{C}}$ of clusters is partitioned into two subsets: set ${\mathcal{C}}^A$ of active clusters and set ${\mathcal{C}}^I$ of inactive clusters. We will ensure that every cluster $C\in {\mathcal{C}}^I$ has the $\alpha'$-bandwidth property, and $|E(C)|\leq m/\tau$. We start with ${\mathcal{C}}^I=\emptyset$, and ${\mathcal{C}}^A$ containing all connected components of $H\setminus T$. The algorithm terminates once ${\mathcal{C}}^A=\emptyset$, and, when this happens, we return ${\mathcal{C}}^I$ as the algorithm's outcome.
In order to bound the number of edges in $|\bigcup_{C\in {\mathcal{C}}}\delta_H(C)|$, we use edge budgets and vertex budgets, that we define next.
\paragraph{Edge budgets.}
Consider a cluster $C\in {\mathcal{C}}$ and an edge $e\in \delta_H(C)$. If $C\in {\mathcal{C}}^I$, we set the budget $B_C(e)=1$, and otherwise we set it to be $B_C(e)=\log_{3/2}(2|\delta_H(C)|)$. If cluster $C$ is the unique cluster with $e\in \delta_H(C)$, then we set the budget of the edge $e$ to be $B(e)=B_C(e)$. If there are two clusters $C\neq C'\in {\mathcal{C}}$ with $e\in \delta_H(C)$ and $e\in \delta_H(C')$, then we set the budget of the edge $e$ to be $B(e)=B_C(e)+B_{C'}(e)$. Lastly, if no cluster $C\in {\mathcal{C}}$ with $e\in \delta_H(C)$ exists, then we set $B(e)=0$.
\paragraph{Vertex budgets.}
Vertex budgets are defined as follows. For every cluster $C\in {\mathcal{C}}^A$, for every vertex $v\in V(C)$, we set the budget $B(v)=\frac{c\deg_C(v)\log_{3/2}m\cdot \log_{2}(|E(C)|)}{8\tau'}$, where $c$ is the constant used in the definition of $\tau'$. The budgets of all other vertices are set to $0$.
\paragraph{Cluster budgets and total budget.}
For a cluster $C\in{\mathcal{C}}$, we define its edge-budget $B^E(C)=\sum_{e\in \delta_H(C)}B_C(e)$, and its vertex-budget $B^V(C)=\sum_{v\in V(C)}B(v)$. The total budget of a cluster $C\in {\mathcal{C}}$ is $B(C)=B^E(C)+B^V(C)$, and the total budget in the system is $B^*=\sum_{C\in {\mathcal{C}}}B(C)= 2\cdot\sum_{e\in E(G)}B(e)+\sum_{v\in V(G)}B(v)$.
\paragraph{Initial budget.}
At the beginning of the algorithm, the budget of every vertex $v\in V(H)\setminus T$ is at most:
$$\frac{c\deg_{H\setminus T}(v) \log_{3/2}m\log_2|E(H)|}{8\tau'}\leq \frac{\deg_H(v)}{8\tau},$$
the budget of every edge incident to a vertex in $T$ is at most $\log_{3/2}(2|T|)\leq 16\log m$, while the budget of every other edge is $0$. Therefore, the total budget $B^*$ in the system at the beginning of the algorithm is at most:
\[ \frac{m}{4\tau}+ 2\cdot 16k\log m\leq \frac{m}{\tau},\]
since $k\leq m/(64\tau\log m)$.
We will ensure that, throughout the algorithm, the total budget $B^*$ never increases. Since, from the definition, $B^*\geq \sum_{C\in {\mathcal{C}}}|\delta_H(C)|$, this ensures that, when the algorithm terminates, $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|\leq m/\tau$, so the set ${\mathcal{C}}^I$ of clusters is a valid output of the algorithm.
\paragraph{Algorithm execution.}
As mentioned above, the algorithm starts with ${\mathcal{C}}^I=\emptyset$, and ${\mathcal{C}}^A$ containing all connected components of $H\setminus T$, with ${\mathcal{C}}={\mathcal{C}}^A\cup {\mathcal{C}}^I$. As long as ${\mathcal{C}}^A\neq \emptyset$, we perform iterations, where in each iteration we select an arbitrary cluster $C\in {\mathcal{C}}^A$ to process.
We now describe the execution of an iteration in which a cluster $C\in {\mathcal{C}}^A$ is processed.
The algorithm for processing cluster $C$ consists of two parts, that we describe next.
\paragraph{Part 1: bandwidth property.}
In this step we either establish that $C$ has the $\alpha'$-bandwidth property, or we partition it into smaller clusters that will replace $C$ in set ${\mathcal{C}}^A$.
Recall that an augmentation $C^+$ of cluster $C$ is a graph that is obtained from graph $H$, by subdividing every edge $e\in \delta_H(C)$ with a vertex $t_e$, letting $T(C)=\set{t_e\mid e\in \delta_H(C)}$ be this new set of vertices, and then letting $C^+$ be the subgraph of the resulting graph induced by $V(C)\cup T(C)$. Recall that cluster $C$ has the $\alpha'$-bandwidth property iff the set $T(C)$ of vertices is $\alpha'$-well-linked in $C^+$. We apply the algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace to graph $C^+$ and terminal set $T(C)$, to obtain an $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)$-approximate sparsest cut $(X,Y)$ in graph $C^+$ with respect to the set $T(C)$ of terminals. We assume w.l.o.g. that $|X\cap T(C)|\leq |Y\cap T(C)|$.
We consider two cases. First, if $|E(X,Y)|\geq \alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |X\cap T(C)|$, then we are guaranteed that the set $T(C)$ of vertices is $\alpha'$-well-linked in $C^+$, and therefore cluster $C$ has the $\alpha'$-bandwidth property. In this case, we continue to Part 2 of the algorithm. Assume now that $|E(X,Y)|< \alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |X\cap T(C)|$.
We can assume without loss of generality that, for every vertex $t_e\in T(C)$, if $t_e\in X$, and $e'=(t_e,v)$ is the unique edge that is incident to $t_e$ in $C^+$, then $v\in X$ as well (since otherwise we can move vertex $t_e$ to $Y$, only making the cut sparser). Similarly, if $t_e\in Y$, then $v\in Y$ as well.
Let $X'=X\setminus T(C)$ and $Y'=Y\setminus T(C)$, so $(X',Y')$ is a partition of $C$. Note that $|T(C)\cap X|=|\delta_H(C)\cap \delta_H(X')|$ and similarly $|T(C)\cap Y|=|\delta_H(C)\cap \delta_H(Y')|$.
We remove cluster $C$ from ${\mathcal{C}}^A$ and from ${\mathcal{C}}$, and we all connected components of $C[X']$ and $C[Y']$ to ${\mathcal{C}}^A$ and to ${\mathcal{C}}$ instead. Observe that we are still guaranteed that $\bigcup_{C'\in {\mathcal{C}}}V(C')=V(H)\setminus T$. We now show that the total budget in the system does not increase as the result of this step.
Since every cluster that was newly added to ${\mathcal{C}}^A$ is contained in $C$, it is immediate to verify that, for every vertex $v$ of $C$, its budget may only decrease. The only edges whose budget may increase are the edges of $E_C(X',Y')$. The number of such edges is bounded by $\alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |X\cap T(C)|=\alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_H(C)\cap \delta_H(X')|$, and the budget of each such edge increases by at most $\log_{3/2}(2m)\leq 4\log m$, so the total increase in the budget of all edges due to this step is bounded by:
\[4\alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_H(C)\cap \delta_H(X')|\cdot \log m\leq |\delta_H(C)\cap \delta_H(X')|,\]
since $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}$.
Consider now some edge $e\in \delta_H(C)\cap \delta_H(X')$. Since we have assumed that $|X\cap T(C)|\leq |Y\cap T(C)|$, it is easy to verify that $|\delta_H(X')|\leq 2|\delta_H(C)|/3$. Therefore, for every edge $e\in \delta_H(X')\cap \delta_H(C)$, if $C'\subseteq H[X']$ is the new cluster with $e\in \delta(C')$, then:
\[B_{C'}(e)=\log_{3/2}(2|\delta_H(C')|)\leq \log_{3/2}(2|\delta_H(X')|)\leq \log_{3/2}(2|\delta_H(C)|)-1. \]
Therefore, the total decrease in the global budget due to the edges of
$\delta_H(X')\cap \delta_H(C)$ is at least $|\delta_H(C)\cap \delta_H(X')|$. We conclude that overall the budget $B^*$ does not increase.
We assume from now on that algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace returned a cut $(X,Y)$ of $C^+$ with $|E(X,Y)|\geq \alpha'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |X\cap T(C)|$, and so cluster $C$ has the $\alpha'$-bandwidth property.
If $|E(C)|\leq m/\tau$, then we remove cluster $C$ from ${\mathcal{C}}^A$, and add it to the set ${\mathcal{C}}^I$ of inactive clusters. It is easy to verify that total budget $B^*$ may not increase as the result of this step. Therefore, we assume from now on that
$|E(C)|> m/\tau$.
\paragraph{Part 2: sparse balanced cut.}
In this step, we apply the algorithm from \Cref{cor: approx_balanced_cut} to graph $C$ with parameter $\hat c=3/4$, to obtain a $\hat c'$-edge-balanced cut $(Z,Z')$ of $C$ (where $1/2<\hat c'<1$), whose size is at most
$O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))$ times the size of a minimum $3/4$-edge-balanced cut of $C$. We say that this step is \emph{successful} if $|E_H(Z,Z')|<|E(C)|/\tau'$. Assume first that the step was successful. Then we remove cluster $C$ from ${\mathcal{C}}^A$ and from ${\mathcal{C}}$, and add all connected components of $C[Z],C[Z']$ to ${\mathcal{C}}^A$ and to ${\mathcal{C}}$ instead. We now show that the total budget in the system does not increase as the result of this step. Observe that the budget of every vertex may only decrease, and the same is true for the budget of every edge, except for the edges in set $\delta_H(Z)\cup \delta_H(Z')$.
We first bound the increase in the budgets of the edges of $\delta_H(Z)\cup \delta_H(Z')$. We consider two cases. The first case happens if $|\delta_H(C)|\leq \frac{8 |E(C)|}{\tau'}$. Since the budget of every edge is always bounded by $\log_{3/2}(2m)$,
and since $|E_H(Z,Z')|\leq \frac{|E(C)|}{\tau'}$,
so the total increase in the budgets of all edges is bounded by $\frac{10|E(C)|\cdot \log_{3/2}(2m)}{\tau'}$.
Consider now the second case, where $|\delta_H(C)|> \frac{8 |E(C)|}{\tau'}$, and assume without loss of generality that $|\delta_H(Z)|\leq |\delta_H(Z')|$. In this case, since $|E_H(Z,Z')|\leq \frac{|E(C)|}{\tau'}$, $|\delta_H(Z)|\leq |\delta_H(C)|$, so the budgets of the edges in $\delta_H(Z)\cap \delta_H(C)$ may not grow. As before, the budgets of the edges of $E_H(Z,Z')$ may grow by at most $|E_H(Z,Z')|\cdot\log_{3/2}(2m)\leq \frac{|E(C)|\cdot \log_{3/2}(2m)}{\tau'}$. Lastly, for every edge $e\in \delta_H(Z')\cap \delta_H(C)$, the original budget $B_C(e)$ is $\log_{3/2}(2|\delta_H(C)|)$, and, if $C'\subseteq H[Z']$ is the new cluster with $e\in \delta_H(C')$, then the new budget $B_{C'}(e)=
\log_{3/2}(2|\delta_H(C')|)$. Since $|\delta_H(C')|\leq |\delta_H(C)|+|E_H(Z,Z')|$, we get that the increase in the budget of $e$ is bounded by $\log_{3/2}\textsf{left} (\frac{|\delta_H(C)|+|E_H(Z,Z')|}{|\delta_H(C)|}\textsf{right} )=\log_{3/2}\textsf{left} (1+\frac{|E_H(Z,Z')|}{|\delta_H(C)|}\textsf{right} )$.
Since we have assumed that $|\delta_H(C)|> \frac{8 |E(C)|}{\tau'}$, while $|E_H(Z,Z')|<\frac{|E(C)|}{\tau'}$, we get that $\frac{|E_H(Z,Z')|}{|\delta_H(C)|}<1/2$. Since for all $\epsilon\in (0,1/2)$, $\ln(1+\epsilon)\leq \epsilon$, we get that the increase in the budget of $e$ is bounded by $\frac{|E_H(Z,Z')|}{|\delta_H(C)|\cdot \ln(3/2)}\leq \frac{4|E_H(Z,Z')|}{|\delta_H(C)|}$. Overall, we get that the budget of the edges of $\delta_H(Z')\cap \delta_H(C)$ increases by at most:
\[|\delta_H(Z')\cap \delta_H(C)|\cdot \frac{4|E_H(Z,Z')|}{|\delta_H(C)|}\leq 4|E_H(Z,Z')|\leq \frac{4|E(C)|}{\tau'}. \]
To summarize, regardless of which of the above two cases happened, the total increase in the budgets of all edges is bounded by $\frac{10|E(C)|\cdot \log_{3/2}(2m)}{\tau'}$. Next, we show that the total decrease in the budgets of the vertices is high enough to compensate for this increase.
Assume without loss of generality that $|E(Z)|\leq |E(Z')|$. From the definition of edge-balanced cut, $|E(Z')|\leq \hat c'|E(C)|$, for some universal constant $\hat c'$. In particular:
\begin{equation}
\sum_{v\in Z}\deg_Z(v)\geq 2(|E(C)|-|E(Z')|-|E(Z,Z')|)\geq 2(1-\hat c'-1/\tau')\cdot |E(C)|. \label{eq: many edges in Z 2}
\end{equation}
On the other hand, from our assumption that $|E(Z)|\leq |E(Z')|$, $\log_{2}(|E(Z)|)\leq \log_{2}(|E(C)|)-1$.
Recall that for every vertex $v\in Z$, its original vertex budget is:
$B(v)=\frac{c\deg_C(v)\log_{3/2}m\cdot \log_{2}(|E(C)|)}{8\tau'}$, and its new budget is:
$$B'(v)=\frac{c\deg_Z(v)\log_{3/2}m\cdot \log_{2}(|E(Z)|)}{8\tau'}\leq \frac{c\deg_Z(v)\log_{3/2}m\cdot (\log_{2}(|E(C)|)-1)}{8\tau'}.$$
Therefore, for every vertex $v\in Z$, its budget decreases by at least
$\frac{c\deg_Z(v)\log_{3/2}m}{8\tau'}$. Overall, the budget of the vertices in $Z$ decreases by at least:
\[\frac{c\log_{3/2}m}{8\tau'}\cdot \sum_{v\in Z}\deg_Z(v)\geq \frac{c\log_{3/2}m}{4\tau'}\cdot (1-\hat c'-1/\tau')\cdot |E(C)|\]
(from Equation \ref{eq: many edges in Z 2}.) Since $\tau'=c\tau\log_{3/2}m\log_2m$, and since we can set $c$ to be a large enough constant, we can ensure that this is at least $\frac{16|E(C)|\cdot \log_{3/2}(2m)}{\tau'}$, so the overall budget in the system does not increase.
We assume from now on that the current step was not successful. In other words, the algorithm from \Cref{cor: approx_balanced_cut} returned a cut $(Z,Z')$ with $|E_H(Z,Z')|\geq |E(C)|/\tau'$. Since the size of this cut is within factor $O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))$ from the minimum $3/4$-edge-balanced cut, we conclude that the value of the minimum $3/4$-edge-balanced cut in $C$ is at least $\rho=\Omega\textsf{left}(\frac{|E(C)|}{\tau'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)}\textsf{right} )$.
Recall that, from \Cref{lem:min_bal_cut}, if the maximum vertex degree $\Delta$ in graph $C$ is at most $|E(C)|/2^{40}$, and $\mathsf{OPT}_{\mathsf{cr}}(C)\le |E(C)|^2/2^{40}$, then graph $C$ must contain a $(3/4)$-edge-balanced cut of value at most $\tilde c\cdot\sqrt{\mathsf{OPT}_{\mathsf{cr}}(C)+\Delta\cdot|E(C)|}$ where $\tilde c$ is some universal constant.
As the size of the minimum $3/4$-balanced cut in $C$ is at least $\rho$, we conclude that either $\Delta\geq |E(C)|/2^{40}$, or $\mathsf{OPT}_{\mathsf{cr}}(C)> |E(C)|^2/2^{40}$, or
$\sqrt{\mathsf{OPT}_{\mathsf{cr}}(C)+\Delta\cdot|E(C)|}\geq \rho/\tilde c$ must hold.
The latter can only happen if either $\mathsf{OPT}_{\mathsf{cr}}(C)\geq \frac{\rho^2}{2\tilde c^2}$, or $\Delta\geq \frac{\rho^2}{2\tilde c^2\cdot |E(C)|}$. Substituting the value of $\rho=\Omega\textsf{left}(\frac{|E(C)|}{\tau'\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)}\textsf{right} )$,
and recalling that $|E(C)|>m/\tau$, while $\tau'=c\tau\log_{3/2}m\log_2m$,
we conclude that either (i) $\mathsf{OPT}_{\mathsf{cr}}(C)\geq \Omega\textsf{left}(\frac{|E(C)|^2}{2(\tilde c \tau'\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))^2}\textsf{right} ) \geq \Omega \textsf{left}( \frac{|E(C)|^2}{ \tau^2\log^5 m} \textsf{right} ) \geq \Omega \textsf{left}( \frac{m^2}{ \tau^4\log^5 m} \textsf{right} )$; or (ii) $\Delta\geq \Omega\textsf{left}(\frac{|E(C)|}{2(\tilde c\tau'\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))^2}\textsf{right} )\geq \Omega\textsf{left}(\frac{m}{\tau^3\log^5m}\textsf{right} ) $; or (iii) $\Delta\geq \frac{|E(C)|}{2^{40}}\geq \frac{m}{2^{40}\tau} $.
However, since we are guaranteed that $\Delta\leq \frac{m}{c^*\tau^3\log^5m}$ for a large enough constant $c^*$, we can rule out the latter two options, and conclude that $\mathsf{OPT}_{\mathsf{cr}}(H)\geq \mathsf{OPT}_{\mathsf{cr}}(C)\geq \Omega \textsf{left}( \frac{m^2}{ \tau^4\log^5 m} \textsf{right} )$.
If Phase 2 is unsuccessful, then we terminate the algorithm and declare that $\mathsf{OPT}_{\mathsf{cr}}(H)\geq \Omega \textsf{left}( \frac{|E(H)|^2}{ \tau^4\log^5 m} \textsf{right} )$.
\subsection{Proof of \Cref{lem: solution to split to solution to original}}
\label{apd: Proof of solution to split to solution to original}
Let $\phi'$ be a solution to instance $I'$.
Throughout the proof, we denote by $D=D_{\phi'}(u^*)$ the tiny $u^*$-disc in drawing $\phi'$.
Recall that
$\delta_{G'}(u^*)=\set{a'_{1,1},\ldots,a'_{1,\hat q_1},a'_{2,1},\ldots,a'_{2,\hat q_2},\ldots,a'_{k,1},\ldots,a'_{k,\hat q_k}}$. Moreover, the edges of $\delta_{G'}(u^*)$ appear in this circular order in the rotation ${\mathcal{O}}_{u^*}'\in \Sigma'$.
We assume w.l.o.g. that the orientation of this ordering in $\phi'$ is positive. In other words, the edges are encountered in this order as we traverse the boundary of $D$ so that the interior of $D$ always lies to our left (see, e.g. \Cref{fig: positive}, in which the orientation of the ordering ${\mathcal{O}}_{u^*}'$ is positive).
Consider now vertex $u_i$, for some $1\leq i\leq k$. Recall that
the set $\delta_{G'}(u_i)$ of edges is the union of two subsets: set
$A'_i=\set{a'_{i,1},\ldots,a'_{i,\hat q_i}}$ of parallel edges connecting $u_i$ to $u^*$, and set $A_i=\set{a_{i,1},\ldots,a_{i, q_i}}$ of edges corresponding to the edge set $E_i\subseteq E(G)$. Recall that the ordering of the edges in $\delta_{G'}(u_i)$ in the rotation system $\Sigma'$ is: $(a'_{i,1},a'_{i,2},\ldots,a'_{i, \hat q_i}, a_{i, q_i}, a_{i, q_i-1},\ldots, a_{i,1})$. We say that vertex $u_i$ is \emph{synchronized} with $u^*$, if the orientation of the above ordering in $\phi'$ is negative (see \Cref{fig: syn_example}). In other words, if we traverse the boundary of the tiny $u_i$-disc $D_{\phi'}(u_i)$ so that its interior always lies to our right, then we will encounter the edges of $\delta_{G'}(u_i)$ in the order $(a'_{i,1},a'_{i,2},\ldots,a'_{i, \hat q_i}, a_{i, q_i}, a_{i, q_i-1},\ldots, a_{i,1})$.
We need the following simple observation.
\begin{observation}\label{obs: not synchronized}
If, for some index $1\leq i\leq k$, vertex $u_i$ is not synchronized with $u^*$, then there are at least $\hat q_i^2/8$ crossings $(e,e')$ in $\phi'$ with $e,e'\in A_i'$.
\end{observation}
\begin{proof}
We delete from drawing $\phi'$ all vertices and edges except for vertices $u^*,u_i$ and edges of $A'_i$. For all $1\leq j\leq \hat q_i$, let $s_j$ be the point on the boundary of $D$ that lies on the image of edge $e_{i,j}$, let $t_j$ be the point on the boundary of $D_{\phi'}(u_i)$ that lies on the image of $e_{i,j}$, and let $\gamma_{j}$ be the segment of the image of edge $e_{i,j}$ between $s_j$ and $t_j$.
We assume without loss of generality that $\gamma_j$ does not cross itself; if it does, then we remove self loops until $\gamma_j$ does not cross itself. Denote $\Gamma=\set{\gamma_1,\ldots,\gamma_{\hat q_j}}$ the resulting set of curves.
\begin{figure}[h]
\centering
\subfigure[The ordering ${\mathcal{O}}'_{u^*}$ with positive orientation.
]{ \scalebox{0.45}{\includegraphics[scale=0.42]{figs/split_flower_4.jpg}}\label{fig: positive
}
\hspace{0.4cm}
\subfigure[Vertex $u_i$ is synchronized with $u^*$. Note that edges of $A'_i$ may cross each other but they intersect the boundaries of the discs $D_{\phi'}(u^*)$ and $D_{\phi'}(u_i)$ in the order indicated above.]{
\scalebox{0.45}{\includegraphics[scale=0.25]{figs/synchronize_example.jpg}}\label{fig: syn_example}}
\caption{An illustration of the positive orientation of ${\mathcal{O}}'_{u^*}$ and the oriented rotation of the synchronized vertex $u_i$.
}
\end{figure}
From our assumptions, points $s_1,\ldots,s_{\hat q_i}$ appear in this order on the boundary of $D$, when we traverse it so that the interior of $D$ lies to our left, while points $t_1,\ldots,t_{\hat q_i}$ appear in this order on the boundary of $\eta_i$, when we traverse it so that the interior of the disc is to our left (see \Cref{fig: synchronize_curve}).
\begin{figure}[h]
\centering
\subfigure[Points $\set{s_k,t_k}_{1\le k\le \hat q_i}$ and curves $\gamma_1, \gamma',\gamma''$.
]{\scalebox{0.42} {\includegraphics[scale=0.35]{figs/synchronize_curve.jpg}\label{fig: synchronize_curve}}
}
\hspace{0.2cm}
\subfigure[Curves $\sigma,\gamma'',\sigma', \gamma'$ and disc $\tilde D$.
]{
\scalebox{0.35}{\includegraphics[scale=0.3]{figs/closed_curve.jpg}}\label{fig: closed_curve}}
\caption{Illustrations for the proof of \Cref{obs: not synchronized}.}
\end{figure}
Assume for contradiction that there are fewer than $\hat q_i^2/8$ crossings $(e,e')$ in $\phi'$ with $e,e'\in A_i'$. Then there is some curve $\gamma_j\in \Gamma$, whose image crosses fewer than $\hat q_i/8$ curves in $\Gamma$. Assume w.l.o.g. that this curve is $\gamma_1$.
Let $\Gamma'\subseteq \Gamma\setminus\set{\gamma_1}$ be the set of curves that do not cross $\gamma_1$, so $|\Gamma'|\geq |\Gamma|/2$. We next show that every pair distinct of curves in $\Gamma'$ must cross, leading to a conradiction. Indeed, consider any pair $\gamma_x,\gamma_y\in \Gamma'$ of distinct curves in $\Gamma'$, and assume without loss of generality that $x<y$.
Let $\gamma'$ and $\gamma''$ be two curves that follow curve $\gamma_1$ immediately to the left and immediately to the right, respectively. Let $s',s''$ be the endpoints of curves $\gamma'$ and $\gamma''$ lying on the boundary of $D$, respectively, and let $t',t''$ be be the endpoints of curves $\gamma'$ and $\gamma''$ lying on the boundary of $D_{\phi'}(u_i)$, respectively. Notice that points $s',s''$ partition the boundary of $D$ into two segments, whose endpoints are $s'$ and $s''$; we let $\sigma$ be the segment that does not contain $s_1$. Similarly, points $t',t''$ partition the boundary of $D_{\phi'}(u_i)$ into two segments, whose endpoints are $t'$ and $t''$; we let $\sigma'$ be the segment that does not contain $t_1$. Let $\lambda$ be the closed curve obtained by concatenating curves $\sigma,\gamma'',\sigma'$, and $\gamma'$ (see \Cref{fig: closed_curve}), and let $\tilde D$ be the disc whose boundary is $\lambda$, that contains the images of the curves $\gamma_x$ and $\gamma_y$. Then points $s_x,s_y,t_x,t_y$ appear on the boundary of $\eta^*$ in this order. Therefore, curves $\gamma_x,\gamma_y$ must cross. We conclude that every pair of curves in $\Gamma'$ must cross, a contradiction.
\end{proof}
In order to transform the drawing $\phi'$ of $G'$ into a drawing $\phi$ of $G$, we start by considering the tiny $u^*$-disc $D$ in the drawing $\phi'$. For each edge $a'_{i,j}\in \delta_{G'}(u^*)$, the image of the edge in $\phi'$ intersects the boundary of $D$ at exactly one point, that we denote by $p_{i,j}$.
Recall that, from our assumptions, points $ p_{1,1},\ldots, p_{1, \hat q_1}, p_{2,1},\ldots, p_{2, \hat q_2},\ldots, p_{k,1},\ldots, p_{k, \hat q_k}$ appear in this order on the boundary of $D$, if we traverse it so that the interior of $D$ lies to our left.
For all $1\leq i\leq k$, we define a segment $\sigma_i$ of the boundary of $D$, that contains all points $p_{i,1},\ldots, p_{i,\hat q_i}$. Observe that these segments can be defined so that they are mutually disjoint, and they appear on the boundary of $D$ in their natural order $ \sigma_1,\ldots, \sigma_k$, as we traverse the boundary of $D$ so that its interior lies to our left. Next, for each $1\leq i\leq k$, we define a disc $D_i$, that is contained in $D$, such that the intersection of the boundary of $D$ and the boundary of $D_i$ is precisely $\sigma_i$, the image of $u^*$ lies outside $D_i$, and all discs $D_1,\ldots,D_k$ are mutually disjoint.
From the above discussion, for all $1\leq i\leq k$, the points $ p_{i,1},\ldots, p_{i,\hat q_i}$ appear in this order on segment $ \sigma_i$ of the boundary of $D_i$, as we traverse this boundary so that the interior of the disc $D_i$ lies to our left (see \Cref{fig: segments}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.25]{figs/segments.jpg}
\caption{The interior of the disc $D$. Segments $\sigma'_1,\ldots,\sigma'_k$ are shown in purple.
}\label{fig: segments}
\end{figure}
Consider now some index $1\leq i\leq k$.
Let $\sigma'_i$ be any segment of non-zero length on the boundary of disc $D_i$, that is disjoint from segment $\sigma_i$.
Let $p'_{i,1},\ldots,p'_{i,q_i}$ be an arbitrary collection of distinct points on $\sigma'_i$, that appear on $\sigma'_i$ in this order, as we traverse the boundary of $D_i$ so that its interior lies to our right (see \Cref{fig: segments}). We can then define, for each $1\leq i\leq k$ and $1\leq j\leq q_i$, a curve $\zeta_{i,j}$, that originates at the image of $u^*$ and terminates at point $p'_{i,j}$, such that all curves in set $\set{\zeta_{i,j}\mid 1\leq i\leq k, 1\leq j\leq q_i}$ are mutually internally disjoint. From our definitions so far, the circular order in which these curves enter the image of $u^*$ is:
$(\zeta_{1,1},\ldots,\zeta_{1,q_1},\zeta_{2,1},\ldots,\zeta_{2,q_2},\ldots,\zeta_{k,1},\ldots,\zeta_{k,q_k})$ (see \Cref{fig: segments}). For all $1\leq i\leq k$ and $1\leq j\leq q_i$, we will use the curve $\zeta_{i,j}$ in order to draw the edge $e_{i,j}\in E_i$; in fact we will refer to $\zeta_{i,j}$ as the \emph{first segment of the drawing of edge $e_{i,j}$}. We will later define a second segment of the drawing of this edge, and then eventually stitch the two segments together to complete the drawing of the edge.
Notice that so far, for all $1\leq i\leq k$, we have defined a collection $\set{p'_{i,1}, p'_{i,2},\ldots,p'_{i,q_i},p_{i,\hat q_i},p_{i,\hat q_i-1},\ldots, p_{i,1}}$ of points on the boundary of $D_i$, that appear on it in this order, as we traverse the boundary so that the interior of $D_i$ lies to our right (see \Cref{fig: two discs}).
Lastly, for all $1\leq i\leq k$, we define another disc $D'_i\subseteq D_i$, whose boundary is disjoint from the boundary of $D_i$ (see \Cref{fig: two discs}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.17]{figs/two_disc.jpg}
\caption{Discs $D_i$ and $D'_i$. Segments $\sigma_i$ and $\sigma'_i$ on the boundary of $D_i$ are shown in red and purple, respectively.}\label{fig: two discs}
\end{figure}
\iffalse
\begin{figure}[h]
\centering
\subfigure[Discs $D_i$ and $D'_i$. Segments $\sigma_i$ and $\sigma'_i$ on the boundary of $D_i$ are shown in red and purle, respectively. \mynote{There is again a mistake in indexing the $p'_{i,j}$ points. Last index should be $p_{i,q_i}$, not $p_{i,\hat q_i}$. Also, if it's not too much trouble, the $\ldots$ signs should appear outside of the disc $D_i$ in both cases, otherwise it's not clear what it is that they refer to}]{\scalebox{0.2}{\includegraphics[scale=0.40]{figs/two_disc.jpg}\label{fig: two discs}}
}
\hspace{0.2cm}
\subfigure[Schematic view of images of edges of $A'_i$ and $A_i$ and relevant points on the boundary of $D$ and $\tilde D_i$. Images of edges of $A_i$ are shown in red. Image of each edge $a_{i,j}\in A_i$ is shown in purple, with segments $\zeta_{i,j}$ dotted. \mynote{can you please rename $z_{i,j}$ points as $z'_j$ (note: no $i$-index anymore), add the $z_j$ points to the boundary of $D$? (see Figure 10 in the updated figures file). Also, the points $p'_{i,j}$ should become $p''_{i,j}$, and the index of the last point should be $p''_{i,q_i}$, not $\hat q_i$. Can you also move the placement of $\ldots$ signs to correct locations? The pink one should be outside the disc and before the last $z$-vertex, and for the green ones the sign is missing. If you could also move $p_{i,2}$ outside of the disc and move blue $\ldots$ outside of the disc it would be great. }]{
\scalebox{0.52}{\includegraphics[scale=0.2]{figs/image_construction.jpg}}\label{fig: image construction} }
\caption{An illustration of discs and curves used in \Cref{obs: not synchronized}. \mynote{can you please split the figure below into two separate figures. I think this will improve readability, things are confusing enough without putting these two together. Also the caption of the joint figure does not make sense.}}
\end{figure}
\fi
In the remainder of the algorithm, we process each index $1\leq i\leq k$ one by one. We let $G_0=G'$, and for all $1\leq i\leq k$, we let $G_i$ be the graph obtained from $G'$ by contracting vertices $u_1,\ldots,u_i$ into the vertex $u^*$; we delete self-loops but keep parallel edges. Note that graph $G_k$ is identical to graph $G$, except that some of the edges of $G$ are subdivided in $G_k$. Therefore, a drawing of graph $G_k$ immediately gives a drawing of graph $G$. For each $1\leq i\leq k$, the input to the $i$th iteration is a drawing $\phi_{i-1}$ of graph $G_{i-1}$, in which, for all $1\leq i'\leq i-1$, all vertices and edges of $X_{i'}$ are drawn inside the disc $D'_{i'}$. The goal of the $i$th iteration is to produce a drawing $\phi_i$ of graph $G_i$, in which, for all $1\leq i'\leq i$, all vertices and edges of $X_{i'}$ are drawn inside the disc $D'_i$. The final drawing $\phi_k$ of graph $G_k$, obtained at the end of the last iteration immediately provides a drawing of graph $G$. Let $\phi_0=\phi'$ be the given drawing of graph $G_0$, that is a solution to instance $I'$ of \ensuremath{\mathsf{MCNwRS}}\xspace. For all $1\leq i\leq k$, we denote by $\mathsf{cr}_i$ the total number of crossings in $\phi_0$, in which edges of $E(X'_i)\cup A'_i\cup \hat A_i$ participate. Clearly, $\sum_{i=1}^k\mathsf{cr}_i\leq 2\mathsf{cr}(\phi)$.
We will ensure that the following invariants hold, for all $1\leq i\leq k$:
\begin{properties}{Inv}
\item over the course of iteration $i$, we may only change the images of the vertices and edges of $X'_i$, and the images of the edges of $\delta_G(X_i)\cup \delta_{G'}(X'_i)$; the images of the remaining edges and vertices of the graph remain unchanged;
\label{inv: only drawing for Xi changes}
\item for every edge $e\in E(G_{i-1})\setminus (E(X'_i)\cup A'_i\cup\hat A_i)$, the number of crossings in which edge $e$ participates in $\phi_i$ is bounded by the number of crossings in which edge $e$ participated in $\phi_{i-1}$; and \label{inv: num of crossings does not increase}
\label{inv: number of crossings does not increase}
\item $\mathsf{cr}(\phi_i)\leq \mathsf{cr}(\phi)+O(\mathsf{cr}_i)$. \label{inv: small increase in crossings}
\end{properties}
From the above invariants, it is immediate to see that the final drawing $\phi_k$ of graph $G_k$ has at most $O(\mathsf{cr}(\phi))$ crossings.
In order to execute the $i$th iterations, we use the two sets ${\mathcal{Q}}_i,{\mathcal{Q}}'_i$ of paths that we have defined, in order to define two sets $\Gamma_i,\Gamma'_i$ of curves, that will serve as ``guiding curves'' for the transformation of the drawing $\phi_{i-1}$. We also use the current drawing $\phi_{i-1}$ in order to compute a ``nice'' drawing $\psi_i$ of graph $X_i$, together with a partial drawing of edges incident to vertices of $X_i$ in $G$. We then ``plant'' this drawing inside the disc $D'_i$, and then complete the drawings of the edges of $\delta_G(X_i)$.
The input to the first iteration is the initial drawing $\phi_0=\phi'$ of graph $G_0=G'$ on the sphere.
We now fix a single index $1\leq i\leq k$, and describe the iteration in which index $i$ is processed. Our starting point is a drawing $\phi_{i-1}$ of graph $G_{i-1}$.
Note that, from Invariant \ref{inv: number of crossings does not increase}, the total number of crossings in which the edges of $E(X'_i)\cup A'_i\cup \hat A_i$ participate in drawing $\phi_{i-1}$ is at most $\mathsf{cr}_i$. We will use this fact later.
The algorithm for processing index $i$ consists of three stages. In the first stage, we use the set ${\mathcal{Q}}_i$ of paths in order to define the first set $\Gamma_i$, of ``guiding'' curves. In the second stage, we use the set ${\mathcal{Q}}'_i$ of paths in order to define the second set $\Gamma'_i$ of ``guiding'' curves, and we compute a drawing $\psi_i$ of $X_i$, together with a partial drawing of edges of $\delta_{G}(X_i)$. In the third and the final stage stage, we ``plant'' this drawing inside the disc $D'_i$, and complete the drawing $\phi_i$. We now describe each of the three stages in turn.
\subsubsection{Stage 1: First Set of Guiding Curves, and Partial Drawing of Edges of $\hat E_i$}
Recall that we have defined a collection ${\mathcal{Q}}_i=\set{Q(e)\mid e\in \hat E_i}$ of edge-disjoint paths in graph $G$, where for each edge $e\in \hat E_i$, path $Q(e)$ has $e$ as its first edge and $u^*$ as its last vertex, and all its inner vertices are contained in $X_i$. From the definition of graph $X'_i$, and from the fact that there are $|\hat E_i|=\hat q_i$ edges connecting $u_i$ to $u^*$ in $G'$ (the edges of $A'_i$), it is immediate to see that there must be a set $\hat {\mathcal{Q}}_i=\set{\hat Q(\hat a)\mid \hat a\in \hat A_i}$ of edge-disjoint paths in graph $G'_{i-1}$, where for each edge $\hat a\in \hat A_i$, path $\hat Q(\hat a)$ contains $\hat a$ as its first edge and terminates at vertex $u^*$, such that all inner vertices of $\hat Q(\hat a)$ are contained in $X'_i$.
We apply the algorithm from \Cref{thm: new type 2 uncrossing} to perform a type-2 uncrossing of the paths in $\hat {\mathcal{Q}}_i$. The input to this algorithm is graph $G_{i-1}$ and its drawing $\phi_{i-1}$ on the sphere, and the set $\hat {\mathcal{Q}}_i$ of paths, which we view as being directed away from $u^*$. Let $\Gamma_i=\set{\gamma(\hat a)\mid \hat a\in \hat A_i}$ denote the set of curves that the algorithm outputs, that are aligned with the graph $\bigcup_{Q\in \hat Q_i}Q$. For each edge $\hat a\in \hat A_i$, if $y(\hat a)$ is the endpoint of $\hat a$ that does not lie in $X'_i$, then curve $\gamma(\hat a)$ originates at the image of $u^*$ and terminates at the image of $y(\hat a)$. Moreover, the curves in set $\Gamma_i$ do not cross each other.
Recall that, for every path $\hat Q\in \hat {\mathcal{Q}}_i$, the first edge of $\hat Q$ (the edge incident to $u^*$) must be an edge of $A'_i$. From the definition of aligned curves, the theorem guarantees that, for every edge $a'_i\in A'_i$, there is a unique curve $\gamma(\hat a)\in \Gamma_i$ that contains the segment of the image of $a'_i$ that lies inside disc $D$. In particular, for all $1\leq j\leq \hat q_i$, there is a unique curve $\gamma(\hat a)\in \Gamma_i$ containing the point $p_{i,j}$ on the boundary of $D_i$. We denote $\hat A_i=\set{\hat a_{i,1},\ldots,\hat a_{i,\hat q_i}}$, where for all $1\leq j\leq \hat q_i$, edge $\hat a_{i,j}$ is the unique edge whose corresponding curve $\gamma(\hat a_{i,j})$ contains the point $p_{i,j}$. From the definition of aligned curves, each such curve $\gamma(\hat e_{i,j})$ intersects the boundary of $D$ at a unique point - point $p_{i,j}$. For all $1\leq j\leq \hat q_i$, we denote $\hat a_{i,j}=(\hat x_{i,j},\hat y_{i,j})$, where $x_{i,j}\in X'_i$. For all $1\leq j\leq \hat q_j$, we let $\hat \zeta_{i,j}$ be the segment of curve $\gamma(\hat a_{i,j})$ from the image of $y_{i,j}$ to point $p_{i,j}$ on the boundary of disc $D$. We denote the resulting set of curves by $\hat Z_{i}=\set{\hat \zeta_{i,j}\mid 1\leq j\leq \hat q_i}$.
Consider now the drawing $\phi_{i-1}$ of graph $G_{i-1}$. We slightly modify this drawing, as follows. First, we delete from $\phi_{i-1}$ the images of all vertices of $X'_i$ and all edges of $E(X'_i)\cup \hat A_i\cup A'_i$. Next, we add to this drawing the set $Z_i=\set{\zeta_{i,j}\mid 1\leq j\leq q}$ of curves; recall that for all $1\leq j\leq q$, curve $\zeta_{i,j}$ is contained in disc $D$, is internally disjoint from disc $D_i$, and connects the image of $u^*$ to point $p'_{i,j}$ on the boundary of disc $D_i$ (see \Cref{fig: segments}). Recall that we called curve $\zeta_{i,j}$ the first segment in the drawing of edge $e_{i,j}$. Additionally, we add to the current drawing the set $\hat Z_i=\set{\hat \zeta_{i,j}\mid 1\leq j\leq \hat q_i}$ of curves. Recall that, for all $1\leq j\leq \hat q_i$, curve $\hat \zeta_{i,j}$ is internally disjoint from disc $D$, and it connects the image of vertex $y_{i,j}$ (the endpoint of edge $\hat a_{i,j}\in \hat A_i$ lying outside $X'_i$) to point $p_{i,j}$ on the boundary of disc $D_i$ (see also \Cref{fig: two discs}). We refer to curve $\hat \zeta_{i,j}$ as \emph{the first segment in the drawing of edge $\hat a_{i,j}$}.
We denote the resulting drawing by $\phi'_{i-1}$. Note that, since the curves in $\Gamma_i$ are aligned with the graph $\bigcup_{Q\in \hat {\mathcal{Q}}_i}Q$, and, since the paths in set $\hat {\mathcal{Q}}_i$ are edge-disjoint, the total number of crossings in drawing $\phi'_{i-1}$ (including crossings between curves representing edges that were not erased and curves in sets $Z_{i}$ and $\hat Z_i$) is bounded by $\mathsf{cr}(\phi_{i-1})$. Moreover, for every edge $e\in E(G_{i-1})\setminus (E(X'_i)\cup A_i'\cup \hat A_i$), the number of crossings in which $e$ participates in $\phi'_{i-1}$ is bounded by the number of crossings in which $e$ participates in $\phi_{i-1}$, and the image of $e$ is disjoint from disc $D_i$.
This completes the first stage of the algorithm.
\subsubsection{Stage 2: Second Set of Guiding Curves and Drawing of $X_i$}
In this stage we consider again drawing $\phi_{i-1}$ of graph $G_{i-1}$. We will start by defining another set $\Gamma'_i$ of guiding curves in this graph (which we will eventually use in order to draw a second segment of each edge in $\hat A_i$). We then exploit the drawing $\phi_{i-1}$ and the curves in $\Gamma'_i$ in order to compute a drawing $\psi_i$ of graph $X_i$, and, for each edge $e\in \delta_{G}(X_i)$, a drawing of a segment of $e$ that is incident to its endpoint that lies in $X_i$. We will also define a new collection $\Gamma^*_i$ of curves, that will be useful for us in Stage 3.
\paragraph{Set $\Gamma'_i$ of guiding curves.}
We start by defining a set $\Gamma'_i$ of guiding curves. Consider the drawing $\phi_{i-1}$ of $G_{i-1}$.
Recall that petal $X_i$ is routable in graph $G$. Therefore, there is a set ${\mathcal{Q}}'_i$ of paths routing the edges of $\hat E_i$ to vertex $u^*$ in graph $G$, such that the paths in ${\mathcal{Q}}'_i$ are internally disjoint from $X_i$, and cause congestion at most $3000$.
Consider any path $Q=Q(\hat e)\in {\mathcal{Q}}'_i$, whose first edge is $\hat e \in \hat E_i$. Recall that we have subdivided each such edge $\hat e\in \hat E_i$ with a vertex in graph $G'$. Let $e',e''$ denote the two edges that we obtained from subdividing edge $\hat e$, and assume that $e'\in \hat A_i$. We then replace edge $\hat e$ with $\hat e''$ on path $Q$. Assume now that the last edge of $Q$ is $e_{i',j}\in E_{i'}$. Since path $Q$ is internally disjoint from $X_i$, $i'\neq i$ must hold. If $i'>i$, then, by replacing edge $e_{i',j}$ with the corresponding edge $a_{i',j}$, we obtain a new path $Q'$ in graph $G_{i-1}$, whose first vertex is an endpoint of an edge of $\hat A_i$, and last vertex is $u_{i'}$. If $i'<i$, then we set $Q'=Q$. Let ${\mathcal{Q}}''_i=\set{Q'\mid Q\in {\mathcal{Q}}'_i}$ be the resulting set of paths in graph $G_{i-1}$.
Consider now any vertex $u_{i'}$, for $i< i'\leq k$. Every path $Q'\in {\mathcal{Q}}''_i$ that terminates at $u_{i'}$ must contain an edge of $\hat A_{i'}$. Therefore, the number of paths in ${\mathcal{Q}}''_i$ terminating at $u_{i'}$ is bounded by $3000\hat q_{i'}$. Since, for all $1\leq i''\leq k$, $|A'_{i''}|=\hat q_{i''}$, we can then extend all such paths, using the edges of $A'_{i'}$, to ensure that they terminate at vertex $u^*$, such that all such paths cause congestion at most $3000$ in $G_{i-1}$. Therefore, we have established that there is a set ${\mathcal{Q}}'''_i=\set{Q'''_{i,j} \mid 1\leq j\leq \hat q_i}$ of paths in graph $G_{i-1}$ that cause congestion at most $3000$, such that, for all $1\leq j\leq \hat q_i$, path $Q'''_{i,j}$ originates at vertex $\hat y_{i,j}$ (the endpoint of the edge $\hat a_{i,j}\in \hat A_i$ that does not lie in $X'_i$), terminates at vertex $u^*$, and is internally disjoint from $X'_i$.
In order to construct the set $\Gamma'_i$ of curves, we consider a graph $H$, that is obtained as follows. We start with $H=G_{i-1}$. We delete from this graph all edges $e\in E(G_{i-1})\setminus (E(X'_i)\cup \hat A_i\cup A'_i)$ that do not participate in the paths of ${\mathcal{Q}}'''_i$. For all remaining edges $e\in E(G_{i-1})\setminus (E(X'_i)\cup \hat A_i\cup A'_i)$, we replace $e$ with $\cong_{G_{i-1}}({\mathcal{Q}}'''_i,e)$ parallel copies. Lastly, we delete all isolated vertices from the resulting graph. Note that drawing $\phi_{i-1}$ of graph $G_{i-1}$ naturally defines a drawing $\phi'$ of graph $H$: after deleting all edges of $E(G_{i-1})\setminus E(H)$ and all vertices of $V(G_{i-1})\setminus V(H)$ from the drawing, for every remaining edge $e\in E(H)\setminus (E(X'_i)\cup \hat A_i\cup A'_i)$, we draw the parallel copies of $e$ along the original image of edge $e$ in $\phi_{i-1}$.
We can now use the set ${\mathcal{Q}}'''_i$ paths in graph $G_{i-1}$ in order to define a set $\hat {\mathcal{Q}}'_i=\set{\hat Q'_{i,j}\mid 1\leq j\leq \hat q_i}$ of edge-disjoint paths in graph $H$, where for all $1\leq j\leq \hat q_i$, path $\hat Q_{i,j}$ originates at vertex $\hat y_{i,j}$, terminates at vertex $u^*$, and is disjoint from $X_i'$.
We use the algorithm from \Cref{thm: new type 2 uncrossing} in order to compute a type-2 uncrossing of the paths in $\hat {\mathcal{Q}}'_i$. Specifically, the algorithm is applied to graph $H$, its drawing $\phi'$, and the set $\hat {\mathcal{Q}}'_i$ of edge-disjoint paths.
The algorithm returns a set $\hat \Gamma=\set{\hat \gamma_{i,j}\mid 1\leq j\leq \hat q_i}$
of internally disjoint curves, where, for all $1\leq j\leq \hat q_i$, curve $\hat \gamma_{i,j}$ connects the image of $\hat y_j$ to the image of vertex $u^*$ in drawing $\phi'$, and all curves in $\hat \Gamma$ are aligned with the graph $\bigcup_{\hat Q\in \hat {\mathcal{Q}}'_i}Q$. We will also consider the curves in set $\hat \Gamma$ in the drawing $\phi_{i-1}$ of $G_{i-1}$. As before, each curve $\hat \gamma_{i,j}$ connects the image of $\hat y_{i,j}$ to the image of vertex $u^*$ in drawing $\phi_{i-1}$.
Since the paths in the original set ${\mathcal{Q}}'$ caused congestion at most $3000$, it is immediate to verify that the number of crossings between the images of the edges of $E(X'_i)\cup A'_i\cup \hat A_i$ in $\phi_{i-1}$ and the curves of $\hat \Gamma$ is at most $3000\cdot\mathsf{cr}_i$.
Since the curves in $\hat \Gamma$ are aligned with the graph $\bigcup_{\hat Q\in \hat {\mathcal{Q}}'_i}Q$, each curve $\hat \gamma_{i,j}\in \hat \Gamma$ intersects the boundary of the tiny $u^*$-disc $D$ in a single point, that we denote by $z_j$. Sine the paths in $\hat {\mathcal{Q}}'$ may not use the edges of $A'_i$, we are guaranteed that each such point $z_j$ may not lie on the segment $\sigma_i$ (the segment containing the points $p_{i,1},\ldots,p_{i,\hat q_i}$; point $p_{i,j'}$ is the intersection point of the image of edge $a'_{i,j'}$ and the boundary of $D$, see \Cref{fig: segments}).
For convenience, we re-index the points in set $\set{z_j}_{1\leq j\leq \hat q_i}$, so that points $z_1,z_2,\ldots,z_{\hat q_i},p_{i,\hat q_i},\ldots,p_{i,1}$ appear on the boundary of $D$ in this order. For each $1\leq j\leq \hat q_i$, we denote by $\ell(j)$ the unique index such that curve $\hat \gamma_{i,j}$ contains the point $z_{\ell(j)}$. For all $1\leq j\leq \hat q_i$, we let $\gamma'_{i,j}$ be the segment of curve $\hat \gamma_{i,j}$ from the image of vertex $\hat y_{i,j}$ to point $z_{\ell(j)}$. We then set $\Gamma'_i=\set{\gamma'_{i,j}\mid 1\leq j\leq \hat q_i}$. From the above discussion, the total number of crossings
between the images of the edges of $E(X'_i)\cup A'_i\cup \hat A_i$ in $\phi_{i-1}$ and the curves of $\Gamma'_i$ is at most $3000\cdot\mathsf{cr}_i$.
\iffalse
the set of curves that the algorithm outuputs, where $\gamma'_{i,j}$ is the curve that originates at vertex $\hat v_{i,j}$. Note that each curve in $\Gamma'_i$ terminates at the image of vertex $u_i$, and the curves in $\Gamma'_i$ are disjoint from each other.
Since the curves in $\Gamma'_i$ are aligned with graph $\bigcup_{Q\in {\mathcal{Q}}''_i}Q$, the total number of crossings between the curves in $\Gamma'_i$ and the edges of $X'_i$ is bounded by $100\chi_i$. Since the curves in $\Gamma_i$
that are aligned with the graph $\bigcup_{Q\in \hat Q_i}Q$, the total number of crossings between the curves of $\Gamma_i$ and the curves of $\Gamma'_i$ is bounded by $\chi_i+\mathsf{cr}(\hat E_i)$, where $\chi'_i$ is the number of crossings between edges of $\hat E_i$ in drawing $\phi_{i-1}$.
Since the curves in $\Gamma'_i$ are aligned with graph $\bigcup_{Q\in {\mathcal{Q}}''_i}Q$, for every edge $a'_{i,j}\in A'_i$, there is a unique curve $\gamma'_{i,j'}\in \Gamma'_i$ that contains the segment of the image of $a'_{i,j}$ that lies in the disc $\tilde D_i$. In particular, it contains point $z_{i,j}$ on the boundary of disc $\tilde D_i$. Note however that it is possible that $j\neq j'$, that is, the curve connects the image of vertex $v_{i,j'}$ to point $z_{i,j}$, where $j\neq j'$. For convenience, we will denote point $z_{i,j}$ by $s_{j'}$. For all $1\leq j\leq \hat q_i$, we then let $\hat \zeta'_{i,j}$ be obtained by concatenating the image of the edge $(\hat x_{i,j},\hat v_{i,j})$ with the segment of the curve $\gamma'_{i,j}$ connecting the image of $\hat v_{i,j}$ to point $s_j$. We then set $\hat Z'_i=\set{\hat \zeta'_{i,j}\mid 1\leq j\leq \hat q_i}$.
\fi
\paragraph{Set $\Gamma^*_i$ of Auxiliary curves.}
We need to define another set of curves, that we will use in Stage 3. Recall that we have defined a set of points $z_1,z_2,\ldots,z_{\hat q_i},p_{i,\hat q_i},\ldots,p_{i,1}$ that appear on the boundary of disc $D$ in this order. We can consider two orderings of elements of $\set{1,\ldots,\hat q_i}$: the first ordering is their natural ordering, while the second ordering is $\ell(1),\ell(2),\ldots,\ell(\hat q_i)$ -- ordering that is defined by the curves in $\Gamma'_i$. In Stage 3 of our algorithm, we will need to show that the distance between these two orderings is small, in order to combine different segments of the drawings of the edges of $A_i$ to complete their drawing. The set $\Gamma^*_i$ of curves, that we define in the next observation, will be used in order to do so.
\begin{observation}\label{obs: curves of gamma star}
There is an efficient algorithm to construct a collection $\Gamma^*_i=\set{\gamma^*_{i,j}\mid 1\leq j\leq \hat q_i}$ of curves, such that, for all $1\leq j\leq \hat q_i$, curve $\gamma^*_{i,j}$ connects point $p_{i,j}$ on the boundary of disc $D$ to point $z_{\ell(j)}$, and it is internally disjoint from disc $D$. Moreover, the total number of crossings between the curves of $\Gamma^*_i$ is $O(\mathsf{cr}_i)$.
\end{observation}
\begin{proof}
Consider an index $1\leq j\leq \hat q_i$. Recall that we have defined a curve $\hat \zeta_{i,j}$, which is a sub-curve of some curve of $\Gamma_i$, connecting point $p_{i,j}$ to the image of vertex $\hat y_{i,j}$ in $\phi_{i-1}$. We concatenate this curve with curve $\gamma'_{i,j}\in \Gamma'_i$, connecting the image of vertex $\hat y_{i,j}$ to point $z_{\ell(j)}$, obtaining the curve $\gamma^*_{i,j}$, that connects $p_{i,j}$ to $z_{\ell(j)}$. From the construction of curves in $\Gamma_i$ and $\Gamma'_i$, and the alignemnt properties of each such curve, we are guaranteed that each resulting curve is internally disjoint from disc $D$.
In order to bound the number of crossings between the curves of $\Gamma^*$, recall that the total number of crossings
between the images of the edges of $E(X'_i)\cup A'_i\cup \hat A_i$ in $\phi_{i-1}$ and the curves of $\Gamma'_i$ is at most $3000\mathsf{cr}_i$. Since the curves of $\Gamma_i$ are aligned with graph $\bigcup_{Q\in \hat {\mathcal{Q}}_i}Q$, and the paths of ${\mathcal{Q}}_i$ are edge-disjoint and contained in $X'_i\cup A'_i\cup \hat A_i$, we get that the total number of crossings between the curves of $\Gamma^*_i$ is $O(\mathsf{cr}_i)$.
\end{proof}
\paragraph{Partial drawing of edges of $\hat A_i$.}
Next, we define a collection $\hat Z_i'=\set{\hat \zeta'_{i,j}\mid 1\leq j\leq \hat q_i}$, of curves, that we will use in order to obtain partial drawing of the edges of $\hat A_i$. For all $1\leq j\leq \hat q_i$, curve $\hat \zeta'_{i,j}$ will connect the image of vertex $\hat x_{i,j}$ (the endpoint of edge $\hat a_{i,j}$ lying in $X'_i$) to a point on the boundary of the tiny $u_i$-disc in $\phi_{i-1}$. We will then view curve $\hat \zeta'_{i,j}$ as the \emph{second segment in the drawing of edge $\hat a_{i,j}$}, and we will add all such curves to the drawing $\psi_i$ that we compute in this stage.
In order to compute the set $\hat Z_i'$ of curves, we will utilize the cuves of $\Gamma'_i$, the images of the edges of $\hat A_i\cup A'_i$ in drawing $\phi_{i-1}$, and another set of curves that we define next.
Recall that we have defined a set $z_1,z_2,\ldots,z_{\hat q_i},p_{i,\hat q_i},\ldots,p_{i,1}$ of points on the boundary of disc $D$, that appear on the boundary of $D$ in this order. We define, for all $1\leq j\leq \hat q_i$ a curve $\rho_j$, connecting $z_j$ to $p_{i,j}$, such that all curves in $\set{\rho_1,\ldots,\rho_{\hat q_i}}$ are contained in disc $D$ and are disjoint from each other (see \Cref{fig: rho curves}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.15]{figs/rho_curves.jpg}
\caption{An illustration of curves $\rho_1,\ldots,\rho_{\hat q_i}$ in disc $D$.}\label{fig: rho curves}
\end{figure}
We denote by $\tilde D_i=D_{\phi_{i-1}}(u_i)$ the tiny $u_i$-disc in $\phi_{i-1}$. Consider the images of the edges in set $A'_i=\set{a'_{i,j}\mid 1\leq j\leq \hat q_i}$ in drawing $\phi_{i-1}$ (these are the parallel edges connecting $u^*$ to $u_i$). From our definition, for all $1\leq j\leq \hat q_i$, point $p_{i,j}$ is the unique point on the boundary of disc $D$ that lies on the image of edge $a'_{i,j}$. We denote the unique point of the image of $a'_{i,j}$ lying on the boundary of disc $\tilde D_i$ by $z'_{j}$. Recall that, from our assumptions, points $p_{i,1},\ldots,p_{i,\hat q_i},z_{\hat q_i},\ldots,z_1$ appear in this order on the boundary of $D$, as we traverse it so that the interior of the disc lies to our left (see \Cref{fig: image construction}). If $u_i$ is synchronized with $u^*$, then, from the definition of the rotation ${\mathcal{O}}'_{u_i}\in \Sigma'$, points $z'_{1},\ldots,z'_{\hat q_i}$ appear in this order on the boundary of disc $\tilde D_i$, as we traverse it so that the interior of the disc lies to our right; if $u_i$ is not synchronized with $u^*$, then this order is reversed (see \Cref{fig: image construction}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.2]{figs/image_construction.jpg}
\caption{Schematic view of images of edges of $A'_i$ and $A_i$ and relevant points on the boundary of $D$ and $\tilde D_i$. Images of edges of $A_i$ are shown in red. Image of each edge $a_{i,j}\in A_i$ is shown in purple, with segments $\zeta_{i,j}$ dotted.
}\label{fig: image construction}
\end{figure}
We are now ready to define the curves of $\hat Z_i'$. Consider some index $1\leq j\leq \hat q_i$. Curve $\hat \zeta'_{i,j}$ is the concatenation of four curves: (i) the image of the edge $\hat a_i=(\hat x_{i,j},\hat y_{i,j})$ in $\phi_{i-1}$; (ii) curve $\gamma'_{i,j}\in \Gamma'_i$ (that connects the image of $\hat y_{i,j}$ to point $z_\ell(j)$ on the boundary of $D$); (iii) curve $\rho_j$ (that connects $z_\ell(j)$ to $p_{i,j}$ and is contained in $D$); and (iv) a segment of the image of the edge $a'_{i,j}$ in $\phi_{i-1}$, from point $p_{i,j}$ on the boundary of disc $D$, to point $z'_{\ell'(j)}$ on the boundary of disc $\tilde D_i$, where $1\leq \ell'(j)\leq \hat q_i$. Note that, if $u_i$ is synchronized with $u^*$, then $\ell'(j)=\ell(j)$ must hold, while otherwise $\ell'(j)=\hat q_i-\ell(j) +1$. We refer to the resulting curve $\hat \zeta'_{i,j}$ as the \emph{second segment in the drawing of edge $\hat a_{i,j}$}, and we denote $\hat Z_i'=\set{\hat \zeta'_{i,j}\mid 1\leq j\leq \hat q_i}$.
\iffalse
We need the following simple observation.
\begin{observation}\label{obs: outside routing paths}
There exists a set ${\mathcal{Q}}''_i$ of paths in graph $G_{i-1}$, routing the vertices of $\set{\hat v_{i,j}\mid 1\leq j\leq \hat q_i}$ to vertex $u_i$, so that the paths are internally disjoint from $X_i'$, and cause congestion at most $100$. Moreover, every edge of $A'_i$ appears as the last edge on exactly one path in ${\mathcal{Q}}''_i$.
\end{observation}
\begin{proof}
\mynote{this proof is not perfect, should be improved for clarity}
Since petal $X_i$ is routable in graph $G$, there is a set ${\mathcal{Q}}'_i$ of paths routing the edges of $\hat E_i$ to vertex $u^*$ in graph $G$, such that the paths in ${\mathcal{Q}}'_i$ are internally disjoint from $X_i$, and cause congestion at most $100$.
Consider any path $Q=Q(\hat e_{i,j})\in {\mathcal{Q}}'_i$, whose first edge is $\hat e_{i,j}$. We replace the first edge on this path with the edge $(\hat v_{i,j},\hat y_{i,j})$. Assume now that the last edge on $Q$ is $e_{i',j}\in E_{i'}$, for some $i'\neq i$. If $i'>i$, then, by replacing edge $e_{i',j}$ with the corresponding edge $a_{i',j}$, we obtain a new path $Q'$ in graph $G_{i-1}$, whose first endpoint is $\hat v_{i,j}$, and last vertex is $u_{i'}$. If $i'<i$, then we set $Q'=Q$. Let ${\mathcal{Q}}''_i=\set{Q'\mid Q\in {\mathcal{Q}}'_i}$ be the resulting set of paths in graph $G_{i-1}$. Consider now any vertex $u_{i'}$, for $i< i'\leq k$. Every path $Q'\in {\mathcal{Q}}''_i$ that terminates at $u_{i'}$ must contain an edge of $A'_{i'}$. Therefore, the number of paths in ${\mathcal{Q}}''_i$ terminating at $u_{i'}$ is bounded by $100\hat q_{i'}$. Since, for all $1\leq i''\leq k$, $|\hat E_{i''}|=\hat q_{i''}$, we can then extend all such paths, using the edges of $A'_{i'}\cup \hat A_{i}$, to ensure that they terminate at vertex $u_i$. Moreover, since $|{\mathcal{Q}}''_i|=|\hat q_i|=|A_i'|$, we can extend all paths in ${\mathcal{Q}}''_i$ as described above so that each of them terminates at $u_i$, and all such paths cause congestion at most $100$ in $G_{i-1}$, and every edge of $A'_i$ appears as the last edge on exactly one path in ${\mathcal{Q}}''_i$.
To conclude, we have established that there exists a collection ${\mathcal{Q}}''_i$ of paths in graph $G'$, routing the vertices of $\set{\hat v_{i,j}\mid 1\leq j\leq \hat q_i}$ to vertex $u_i$, so that the paths are internally disjoint from $X_i'$, and cause congestion at most $100$.
\end{proof}
Consider a graph $G'$, that is obtained from graph $G_{i-1}$ as follows: for every edge $e\in E(G_{i-1})\setminus E(X'_i)$, we include $\cong_{G_{i-1}}({\mathcal{Q}}''_i,e)$ copies of the edge $e$. If an edge
$e\in E(G_{i-1})\setminus E(X'_i)$ does not participate in any paths in ${\mathcal{Q}}''_i$, then it is deleted from $G'$. We also delete all isolated vertices from $G'$. Note that drawing $\phi_{i-1}$ of $G_{i-1}$ naturally defines a drawing $\phi'$ of graph $G'$, where for each edge $e\in E(G_{i-1})\setminus E(X'_i)$, we draw the copies of $e$ along the original image of edge $e$ in $\phi_{i-1}$.
We can now modify the set ${\mathcal{Q}}''_i$ of paths so that every edge of $G'$ i used by a distinct path in ${\mathcal{Q}}''_i$, that is, the paths become edge-disjoint.
We use the algorithm from \Cref{thm: new type 2 uncrossing} in order to compute a type-2 uncrossing of the paths in ${\mathcal{P}}''_i$. Specifically, the algorithm is applied to graph $G'$, its drawing $\phi'$, with $C=X'_i$, and the set ${\mathcal{Q}}''_i$, which is now a set of edge-disjoint paths in graph $G'$. We denote by $\Gamma'_i=\set{\gamma'_{i,j}\mid 1\leq j\leq \hat q_i}$ the set of curves that the algorithm outuputs, where $\gamma'_{i,j}$ is the curve that originates at vertex $\hat v_{i,j}$. Note that each curve in $\Gamma'_i$ terminates at the image of vertex $u_i$, and the curves in $\Gamma'_i$ are disjoint from each other.
Since the curves in $\Gamma'_i$ are aligned with graph $\bigcup_{Q\in {\mathcal{Q}}''_i}Q$, the total number of crossings between the curves in $\Gamma'_i$ and the edges of $X'_i$ is bounded by $100\chi_i$. Since the curves in $\Gamma_i$
that are aligned with the graph $\bigcup_{Q\in \hat Q_i}Q$, the total number of crossings between the curves of $\Gamma_i$ and the curves of $\Gamma'_i$ is bounded by $\chi_i+\mathsf{cr}(\hat E_i)$, where $\chi'_i$ is the number of crossings between edges of $\hat E_i$ in drawing $\phi_{i-1}$.
Since the curves in $\Gamma'_i$ are aligned with graph $\bigcup_{Q\in {\mathcal{Q}}''_i}Q$, for every edge $a'_{i,j}\in A'_i$, there is a unique curve $\gamma'_{i,j'}\in \Gamma'_i$ that contains the segment of the image of $a'_{i,j}$ that lies in the disc $\tilde D_i$. In particular, it contains point $z_{i,j}$ on the boundary of disc $\tilde D_i$. Note however that it is possible that $j\neq j'$, that is, the curve connects the image of vertex $v_{i,j'}$ to point $z_{i,j}$, where $j\neq j'$. For convenience, we will denote point $z_{i,j}$ by $s_{j'}$. For all $1\leq j\leq \hat q_i$, we then let $\hat \zeta'_{i,j}$ be obtained by concatenating the image of the edge $(\hat x_{i,j},\hat v_{i,j})$ with the segment of the curve $\gamma'_{i,j}$ connecting the image of $\hat v_{i,j}$ to point $s_j$. We then set $\hat Z'_i=\set{\hat \zeta'_{i,j}\mid 1\leq j\leq \hat q_i}$.
\fi
\paragraph{Computing the drawing $\psi_i$.}
Consider the current drawing $\phi_{i-1}$ of graph $G_{i-1}$. We slightly modify this drawing in order to obtain a drawing $\psi_i$ of graph $X_i$, and, for each edge $e\in \delta_{G_i}(X_i)\setminus \delta_{G_i}(u^*)$, a drawing of a segment of $e$ that is incident to its endpoint that lies in $X_i$.
In order to do so, we start with the drawing $\phi_{i-1}$ of graph $G_{i-1}$, and we delete from it the images of all edges and vertices, except for those lying in $X'_i$. We will also make use of the disc $\tilde D_{i-1}$ that we have defined. Next, for all $1\leq j\leq q_i$, we consider the image of edge $a_{i,j}$ in the current drawing. Recall that this image intersects the boundary of $\tilde D_i$ at a single point, that we denote by $p''_{i,j}$. From the definition of the rotation ${\mathcal{O}}'_{u_i}\in \Sigma'$, points $z'_{1},\ldots,z'_{\hat q_i},p''_{i,q_i},\ldots,p''_{i,2},p''_{i,1}$ appear on the boundary of disc $\tilde D_i$ in this order, and, if $u_i$ is synchronized with $u^*$, then they are encountered in this order as we traverse the boundary of $\tilde D_i$ so its interior lies on our right; see \Cref{fig: image construction}.
Consider now some edge $a_{i,j}\in A_i$, for $1\leq j\leq q_i$, and assume that $a_{i,j}=(u_i,x_{i,j})$, where $x_{i,j}\in V(X_i)$. We denote by $\zeta'_{i,j}$ the segment of the image of edge $a_{i,j}$ from the image of $x_{i,j}$ to point $p''_{i,j}$ (see \Cref{fig: image construction}), and we refer to $\zeta'_{i,j}$ \emph{the second segment in the drawing of edge $e_{i,j}$}. We delete, from the current drawing, the portion of the image of $a_{i,j}$ lying in the interior of the disc $\tilde D_i$; in other words, we replace the image of $a_{i,j}$ with the curve $\zeta'_{i,j}$.
Lastly, we add the curves in set $\hat Z'_i=\set{\hat \zeta'_{i,j}\mid 1\leq j\leq \hat q_i}$ to the current drawing. Recall that, for each
$1\leq j\leq \hat q$, curve $\hat \zeta'_{i,j}$ connects the image of vertex $\hat x_{i,j}$ (the endpoint of edge $\hat a_{i,j}\in \hat E_i$ that lies in $X_i$) to point $z'_{\ell'(j)}$ on the boundary of $\tilde D_i$, where $1\leq \ell'(j)\leq \hat q_i$.
This completes the drawing $\psi_i$.
We now bound the number of crossings in this drawing.
In order to bound the number of crossings, recall that every curve $\zeta'_{i,j}\in Z'_{i,j}$ is a segment of the image of edge $a_{i,j}\in A_{i,j}$, and every curve $\hat \zeta'_{i,j}\in \hat Z'_i$ is a concatenation of four curves: the image of the edge $\hat a_{i,j}\in \hat A_i$; the curve $\gamma'_{i,j}\in \Gamma'_i$; the curve $\rho_j$, and the image of the edge $a'_{i,\ell(j)}\in A'_i$. Since the curves in $\Gamma'_i$ are disjoint from each other, and the curves $\rho_1,\ldots,\rho_{\hat q_i}$ are disjoint from each other and are contained in disc $D$, the total number of crossings in $\psi_i$ is bounded by (i) the number of crossings between pairs of edges in $E(X'_i)\cup \hat A_i\cup A'_i$ (which is bounded by $\mathsf{cr}_i$ by definition); and (ii) the number of crossings between edges of $E(X'_i)\cup \hat A_i\cup A'_i$ and curves of $\Gamma'_i$ (which is bounded by $O(\mathsf{cr}_i)$ from the discussion above. We conclude that drawing $\psi_i$ has $O(\mathsf{cr}_i)$ crossings.
We will view the interior of the disc $\tilde D_i$ as the ``outer face'' of the drawing $\psi_i$. If we denote by $\tilde D'_i$ the disc in the sphere whose boundary is the same as the boundary of $\tilde D_i$, but its interior is disjoint from the interior of $\tilde D_i$, then the current drawing $\psi_i$ is contained in $\tilde D'_i$. To summarize, drawing $\psi_i$ consists of: (i) the drawing of all edges and vertices of $X_i\setminus\set{u^*}$; (ii) for every edge $a_{i,j}=(x_{i,j},u^*)\in E_i$, a curve $\zeta'_{i,j}$, connecting the image of $x_{i,j}$ to point $p''_{i,j}$ on the boundary of $\tilde D'_i$; and (iii) for every edge $\hat a_{i,j}=(\hat x_{i,j},\hat y_{i,j})\in \hat A_i$ with $\hat x_{i,j}\in X_i$, a curve $\hat \zeta'_{i,j}$, connecting the image of $\hat x_{i,j}$ to point $z'_{\ell'(j)}$ on the boundary of $\tilde D_i'$, where $1\leq \ell'(j)\leq \hat q_i$. Note that the points $p''_{i,1},\ldots,p''_{i,q_i},z'_{\hat q_i},\ldots,z'_{2},z'_{1}$ appear on the bounary of disc $\tilde D_i'$ in this order, and, if $u_i$ is synchronized with $u^*$, then they are encountered in this order as we traverse the boundary of $\tilde D'_i$ so its interior lies to our right.
\subsubsection{Stage 3: Computing the Drawing $\phi_i$ of $G_i$}
We start with the drawing $\phi_{i-1}'$ that we computed in Stage 1.
We consider two cases. The first case is when vertex $u_i$ is synchronized with vertex $u^*$. In this case, we place the drawing $\psi_i$ that we computed in Stage 2 of the algorithm inside the disc $D'_i$, so that the discs $D'_i$ and $\tilde D'_i$ coincide. In the second case, vertex $u_i$ is not synchronized with vertex $u^*$. In this case, we place a mirror image of the drawing $\phi_i$ inside the disc $D'_i$, so that the boundaries of the discs $D'_i$ and (the mirror image of) disc $\tilde D'_i$ coincide (see \Cref{fig: too_many_points_1}). In either case, we obtain two disjoint segments on the boundary of $D'_i$: segment $\tilde \sigma_i$, containing the points $p''_{i,1},\ldots,p''_{i,q_i}$, and segment $\tilde \sigma'_i$, containing the points $z'_1,\ldots,z'_{\hat q_i}$. Moreover, we are now guaranteed that points $p''_{i,1},\ldots,p''_{i,q_i},z'_{\hat q_i},\ldots,z'_{2},z'_{1}$ are encountered in this order as we traverse the boundary of $D_i'$, so that the interior of $D_i'$ lies to our
right (see \Cref{fig: too_many_points_1}).
Recall that we have defined a collection of points $\set{p'_{i,1}, p'_{i,2},\ldots,p'_{i, q_i},p_{i,\hat q_i},p_{i,\hat q_i-1},\ldots, p_{i,1}}$ on the boundary of $D_i$, that appear in this order on the boundary of $D_i$, as we traverse it so that the interior of $D_i$ lies to our right (see Figures \ref{fig: segments} and \ref{fig: image construction}). Consider now some edge $e_{i,j}=(u^*,x_{i,j})\in E_i$, for $1\leq i\leq q_i$. Recall that we have already defined a curve $\zeta_{i,j}$, that serves as the first segment of the drawing of $e_{i,j}$, and connects the image of $u^*$ to point $p'_{i,j}$ on the boundary of $D_i$, such that curve $\zeta_{i,j}$ is internally disjoint from $D_i$. We have also defined a curve $\zeta'_{i,j}$ inside the disc $D'_i$, that connects the image of vertex $x_{i,j}$ to point $p''_{i,j}$ on the boundary of $D'_{i,j}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.105]{figs/too_many_points_1.jpg}
\caption{Planting disc $\tilde D'_i$ inside $D'_i$
}\label{fig: too_many_points_1}
\end{figure}
Recall that we have also defined another segment $\sigma'_i$ on the boundary of $D_i$, that contains points $p'_{i,1},\ldots, p'_{i, q_i}$, and a segment $\hat \sigma'_{i}$ on the boundary of $D'_i$, containing the points in $z'_1,\ldots,z'_{\hat q_i}$. We can then define two disjoint discs that are both contained in $D_i\setminus D'_i$: one disc, $D^1_i$, with segments $\sigma_i$ and $\tilde \sigma_i$ on its boundary, and another disc, $D^2_i$, with segments $\tilde \sigma'_i,\sigma_i'$ on its boundary (see \Cref{fig: too_many_points_2}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.105]{figs/too_many_points_2.jpg}
\caption{An illustration of discs $D^1_i$ and $D^2_i$.}\label{fig: too_many_points_2}
\end{figure}
Observe that points $\set{p'_{i,1}, p'_{i,2},\ldots,p'_{i,q_i},p''_{i,q_i},\ldots,p''_{i,1}}$ appear in this circular order on the boundary of disc $D^1_i$. Therefore, we can define, for all $1\leq j\leq q_i$, a curve $\zeta''_{i,j}$, connecting point $p'_{i,j}$ to point $p''_{i,j}$, such that the interior of the curve is contained in disc $D^1_i$, and all curves in $\set{\zeta''_{i,j}\mid 1\leq j\leq q_i}$ are mutually disjoint (see \Cref{fig: pp_and_ppp_curves}).
For all $1\leq j\leq q_i$, we then let the image of the edge $e_{i,j}$ be the concatenation of the curves $\zeta_{i,j}$, $\zeta''_{i,j}$, and $\zeta'_{i,j}$.
Lastly, it remains to complete the drawing of the edges of $\hat A_i$.
Recall that for every edge $\hat a_{i,j}=(\hat x_{i,j},\hat y_{i,j})$ (where $\hat x_{i,j}\in X_i$), we have already defined two segments of the drawing of $\hat a_{i,j}$. The first segment, $\hat \zeta_{i,j}$, is internally disjoint from disc $D$, and connects the image of $\hat y_{i,j}$ to point $p_{i,j}$ on the boundary of $D_i$. The second segment, $\hat \zeta'_{i,j}$, is contained in disc $D'_i$, and connect the image of $\hat x_{i,j}$ to point $z'_{\ell'(j)}$ on the boundary of disc $D'_{i}$. In order to complete the drawing of edge $\hat a_{i,j}$, we will define a third curve, $\hat \zeta''_{i,j}$, that is contained in disc $D_i^2$, and connects points $p_{i,j}$ and $z'_{\ell'(j)}$ to each other. See \Cref{fig: p_and_zp_curves} for an illustration.
\begin{figure}[h]
\centering
\subfigure[An illustration of curves $\zeta_{i,1}.\ldots,\zeta_{i,q_i}$.]{
\scalebox{0.42}{\includegraphics[scale=0.37]{figs/pp_and_ppp_curves.jpg}}\label{fig: pp_and_ppp_curves}}
\hspace{0.5cm}
\subfigure[An illustration of disc $D^2_i$ and points on its boundary.]{\scalebox{0.42} {\includegraphics[scale=0.37]{figs/p_and_zp_curves.jpg}\label{fig: p_and_zp_curves}}
}
\caption{Stitching the images of the edgs of $E_i$ and $\hat E_i$
}
\end{figure}
We will use the following observation in order to complete the drawing.
\begin{observation}\label{obs: completing the drawing}
There is an efficient algorithm to compute a collection $\set{\hat \zeta''_{i,j}\mid 1\leq j\leq \hat q_i}$ of curves that are contained in disc $D_i^2$, such that, for all $1\leq j\leq \hat q_i$, curve $\zeta''_{i,j}$ connects points $p_{i,j}$ and $z'_j$, and the number of crossings between the curves in $\set{\hat \zeta''_{i,j}\mid 1\leq j\leq \hat q_i}$ is at most $O(\mathsf{cr}_i)$.
\end{observation}
\begin{proof}
We consider two cases. The first case is when $u^*$ and $u_i$ are not synchonized. In this case, we let $\set{\hat \zeta''_{i,j}\mid 1\leq j\leq \hat q_i}$ of curves that are contained in disc $D_i^2$, such that, for all $1\leq j\leq \hat q_i$, curve $\zeta''_{i,j}$ connects points $p_{i,j}$ and $z'_j$, and every pair of curves cross at most once. In this case, the number of crossings between the curves in $\set{\hat \zeta''_{i,j}\mid 1\leq j\leq \hat q_i}$ is at most $\hat q_i^2$. Since, from \Cref{obs: not synchronized}, there were at least $\hat q_i^2/8$ crossings $(e,e')$ in $\phi'$ with $e,e'\in A_i'$, we get that $\mathsf{cr}_i\geq \Omega(\hat q_i^2)$, and so the the number of crossings between the curves in $\set{\hat \zeta''_{i,j}\mid 1\leq j\leq \hat q_i}$ is at most $O(\mathsf{cr}_i)$ as required.
In the second case, $u^*$ and $u_i$ are synchronized.
Recall that in Stage 2 of the algorithm, in \Cref{obs: curves of gamma star}, we have constructed a collection $\Gamma^*_i=\set{\gamma^*_{i,j}\mid 1\leq j\leq \hat q_i}$ of curves, such that, for all $1\leq j\leq \hat q_i$, curve $\gamma^*_{i,j}$ connects point $p_{i,j}$ on the boundary of disc $D$ to point $z_{\ell(j)}$, and it is internally disjoint from disc $D$; the total number of crossings between the curves of $\Gamma^*_i$ is $O(\mathsf{cr}_i)$. Recall that points $z_1,z_2,\ldots,z_{\hat q_i},p_{i,\hat q_i},\ldots,p_{i,1}$ that appear on the boundary of disc $D$ in this order, while points $z'_1,z'_2,\ldots,z'_{\hat q_i},p_{i,\hat q_i},\ldots,p_{i,1}$ appear on the boundary of disc $D^2_i$ in this order. The key point is that, as observed in Stage 2 of the algorithm, if vertex $u_i$ is synchronized with vertex $u^*$, then for all $1\leq j\leq \hat q_i$, $\ell(j)=\ell(j')$. Therefore, we can copy the collection $\Gamma^*_i$ of curves to the interior of the disc $D^2_i$, such that, for all $1\leq j\leq \hat q_i$, one of the resulting curves, that we denote by $\hat \zeta''_{i,j}$ connects $p_{i,j}$ to $z'_{\ell'(j)}=z'_{\ell(j)}$.
\end{proof}
\iffalse
\subsubsection{Stage 4: Completing the Drawing of the Edges of $\hat E_i$.}
In order to complete the new drawing $\phi_i$, it is now enough to compute a drawing of each edge $e\in \hat E_i$. Consider any such edge $\hat e_{i,j}=(x_j,y_j)$, with $x_j\in X_i$. Recall that we have already defined a curve $\hat \zeta'_{i,j}$, that is contained in disc $D'_i$, and connects the image of $x_j$ to a point $t(\hat e_{i,j})$ on the segment $\pi_i'$ of the boundary of $D'_i$. We have also defined another curve $\hat \zeta_{i,j}$, that is disjoint from the interior of disc $D$, and connects the image of $y_j$ to point $p_{i,j}$ on the boundary of $D_i$. We would now like to ``stitch'' these two segments together, by computing a curve $\hat \zeta''_{i,j}$, that is contained in $D^2_i$, and connects $t(\hat e_{i,j})$ to $p_{i,j}$.
As before, we denote $\hat E_i=\set{\hat e_{i,1},\ldots,\hat e_{i,\hat q_i}}$, where for all $1\leq j\leq \hat q_i$, curve $\hat \zeta_{i,j}$ terminates at point $\hat p_{i,j}$. Let $t_{i,j}=t(\hat e_{i,j})$ be the endpoint of curve $\hat \zeta'_{i,j}$ that lies on the boundary of $D'_i$. The main difficulty is that the ordering of the points $t_{i,1},t_{i,2},\ldots,t_{i,\hat q_i}$ on segment $\hat \sigma'_i$ does not necessarily align with the ordering of the points $p_{i,1},\ldots,p_{i,\hat q_i}$ on the segment $\hat \sigma_i$ of $D_i^2$.
We now consider two cases. The first case happens when $u_i$ is not syncronized with $u^*$.
In this case, from \Cref{obs: not synchronized}, there are at least $\hat q_i^2/8$ crossings $(e,e')$ in $\phi'$ with $e,e'\in \hat E_i'$. We then define, for all $1\leq j\leq \hat q_i$, a curve $\zeta^3_{i,j}$ that is contained in $D^2_i$ and connects point $x_j$ to point $\hat p_{i,j}$, such that every pair of curves in set $\set{\zeta^3_{i,j}\mid 1\leq j\leq \hat q_i}$ cross at most once. For every edge $\hat e_{i,j}\in \hat E_j$, by combining the curves $\hat \zeta'(e_{i,j})$, $\hat \zeta^3_{i,j}$ and $\hat \zeta_{i,j}$, we obtain a single curve connecting the images of the endpoints of $\hat e_{i,j}$, that we view as the image of edge $\hat e_{i,j}$ in the drawing $\phi_i$.
It now remains to consider the case where $u^*$ and $u_i$ are synchronized.
Recall that we have defined disc $\eta^*_i$ and disc $\eta'_i$ in the drawing $\psi_i$, that are completely disjoint except for sharing the segment $\pi_i$ on both their boundaries. Recall that we have planted disc $\eta^*_i$ so that it coinsides with disc $D_i$. We can then consider planting this disc as before, but together with the disc $\eta'_i$, which is now drawn entirly in $D_i\setminus D'_i$, except that it now shares the segment $\pi$ on its boundary with disc $D'_i$. Consider the disc $D^*_i$ that is obtained from the union of discs $D'_i$ and $\eta^*_i$, and consider the set $Z^{**}_i$ of curves that we have defined in drawing $\psi_i$. The curves in $Z^{**}_i$ are contained in disc $D^*_i$, and, for all $1\leq j\leq \hat q_{i}$, curve $\zeta^{**}_{i,j}\in Z^{**}_i$ connects point $t_{i,j}$ to point $s_{i,j}$ on the boundary of $D^*_i$. Moreover, points $s_{1,i},s_{i,2},\ldots,s_{i,\hat q_i}$ appear on the boundary of $D^*_i$ consecutively in this order, as we traverse the boundary of $D^*_i$ so that its interior lies to our right. Note that, if we take a mirror image of the curves in $Z^{**}_i$ and place them inside the disc $D^2_i$, then we can obtain the desired collection of curves $Z_i^3=\set{\zeta''_{i,j}\mid 1\leq j\leq \hat q_j}$, where curve $\zeta''_{i,j}$ connects point $t_{i,j}$ to point $p_{i,j}$ on the boundary of $D^2_i$. The number of crossings between the curves in $Z_i^3$ is bounded by the number of crossings between the curves of $Z^{**}_i$, which is in turn bounded by $\chi_i$. This completes the construction of the drawing $\phi_i$ of graph $G_i$.
\fi
\subsubsection{Analysis}
We now show that, assuming that Invariants \ref{inv: only drawing for Xi changes}--\ref{inv: small increase in crossings} hold at the beginning of the $i$th iteration, they continue to hold at the end of the iteration. Indeed, it is immediate to see that we only change the images of vertices and edges of $X'_i$, and $A'_i\cup \hat A_i$, which establishes Invariant \ref{inv: only drawing for Xi changes}. Consider now some edge
$e\in E(G_{i-1})\setminus (E(X'_i)\cup A'_i\cup\hat A_i)$, and some other edge $e'$ that crosses $e$ in $\phi_i$. If $e'$ is not an edge of $E(X_i)\cup\hat A_i$, then the image of $e'$ was not changed in the current iteration, and the crossing lies in $\phi_{i-1}$ as well. Notice that $e$ may not cross edges of $E(X_i)$, as for each such edge $e'$, either $e'$ is drawn inside disc $D'_i$, or $e'\in E_i$, so the first segemnt of $e'$ is some curve $\zeta_{i,j}$ (that is contained in $D$), and the remainder of the image of $e'$ is contained in $D'_i$. Assume now that $e'\in \hat A_i$. In this case, only the first segment of the drawing $e'$, which is a segment of some curve in $\Gamma_i$, may cross edge $e$, as the remainder of the image of $e'$ lies in disc $D$. Overall, the number of crossings in which edge $e$ participates in drawing $\phi_i$ is bounded by the number of crossings in which edge $e$ participates in drawing $\phi'_{i-1}$ (that was defined in Stage 1), which is in turn bounded by the number of crossings in which $e$ participates in $\phi_{i-1}$. We conclude that Invariant \ref{inv: num of crossings does not increase} continues to hold. Lastly, it remains to establish Invariant \ref{inv: small increase in crossings}. From the above discussion, for each edge $e\in E(G_{i-1})\setminus (E(X'_i)\cup A'_i\cup\hat A_i)$, the number of crossings in which $e$ participates in drawing $\phi_i$ is bounded by the number of crossings in which $e$ participates in drawing $\phi_{i-1}$. From our analysis of Stage 2, the number of crossings in $\psi_i$ is bounded by $O(\mathsf{cr}_i)$. Recall that the curves of $\Gamma_i$ cannot cross each other, and they are internally disjoint from disc $D$. The only additional crossings that we introduced are the crossings between the curves of
$\set{\hat \zeta''_{i,j}\mid 1\leq j\leq \hat q_i}$ that were computed in \Cref{obs: completing the drawing}; from the observation, the number of such crossings is at most $O(\mathsf{cr}_i)$.
This establishes Invariant \ref{inv: small increase in crossings}.
Overall, after $k$ iterations, we obtain a drawing $\phi_k$ of graph $G_k$ with $O(\mathsf{cr}(\phi')+\sum_{i=1}^k\mathsf{cr}_i)\leq O(\mathsf{cr}(\phi'))$ crossings. Since graph $G_k$ can be obtained from graph $G$ by subdividing some of its edges, this immediately provides a drawing of graph $G$ with $O(\mathsf{cr}(\phi'))$ crossings. From our construction it is immediate to verify that the resulting drawing obeys the rotation system $\Sigma$, so we obtain a feasible solution to instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace.
\subsection{Proof of \Cref{lem: high opt or lots of paths}}
\label{sec: few paths high opt}
We assume for contradiction that for every regular vertex $x\in V(\hat H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, and yet $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<\frac{(\tilde k\tilde \alpha \alpha')^2}{c'\eta'\log^{20}m}$, where $c'$ is the constant from the definition of $\tilde k'$.
The high-level idea of the proof is the following.
We use the algorithm from \Cref{claim: embed expander} in order to embed an expander $W$ over the set $\tilde T$ of terminals into $\hat H_1$, which, from \Cref{obs: cr of exp} has a high crossing number. On the other hand, from \Cref{lem: crossings in contr graph}, there is a drawing of the contracted graph $\hat H_1$ with relatively few crossings. We exploit this latter drawing of $\hat H_1$, the embedding of the expander $W$ into $\hat H_1$, and the fact that any set ${\mathcal{J}}'(x)$ of edge-disjoint paths routing a subset of terminals to a vertex $x$ of $\hat H_1\cap V(H_1)$ must have a small cardinality, in order to obtain a drawing of the expander $W$ with relatively few crosings, reaching a contradiction. We now proceed with a formal proof.
\iffalse
\begin{observation}\label{obs: few paths}
Let $x$ be a vertex of $V(\hat H_1)\cap V(H_1)$, and let ${\mathcal{J}}'$ be a collection of edge-disjoint paths in graph $\hat H_1$, routing a subset of terminals to the vertex $x$. Then $|{\mathcal{J}}'|\leq 2\tilde k'/\alpha'$.
\end{observation}
\begin{proof}
Assume otherwise.
Then, from \Cref{claim: routing in contracted graph}, there is a collection of at least $\alpha'|{\mathcal{J}}'|/2\geq \tilde k'$ edge-disjoint paths in graph $H_1$, routing a subset of terminals to $x$, a contradiction.
\end{proof}
\fi
By applying \Cref{claim: embed expander} to graph $\hat H_1$ and the set $\tilde T$ of terminals (that is $\tilde \alpha$-well-linked in $\hat H_1$ from Property \ref{prop after step 1: terminals in H1}), we conclude that there exist a graph $W$ with $V(W)=\tilde T$ and maximum vertex degree at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2\tilde k$, and an embedding ${\hat{\mathcal{P}}}$ of $W$ into $\hat H_1$ with congestion at most $\frac{\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2\tilde k}{\tilde \alpha}$, such that $W$ is a $1/4$-expander.
Moreover, from \Cref{obs: cr of exp}, the crossing number of $W$ is at least $\tilde k^2/(\tilde c\log^8\tilde k)\geq \tilde k^2/(\tilde c\log^8m)$, for some constant $\tilde c$.
Recall that we have assumed for contradiction that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)< \textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{c'\eta'\log^{20}m}\textsf{right} )$ for some large enough constant $c'$. We will exploit this fact in order to show that there exists a drawing of $W$ with fewer than $\tilde k^2/(\tilde c\log^8m)$
crossings, reaching a contradiction.
Recall that, from \Cref{lem: crossings in contr graph}, there is a drawing $\phi$ of the contracted graph $\hat H_1$, whose number of crossings is bounded by:
\[O\textsf{left}(\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8m}{(\alpha')^2}\textsf{right})\leq
O\textsf{left}(\frac{\tilde k^2\tilde \alpha^2}{c'\eta'\log^{12}m}\textsf{right}),\]
from our assumption that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)< \textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{c'\eta'\log^{20}m}\textsf{right} )$.
In the remainder of the proof, we gradually modify the drawing $\phi$ of $\hat H_1$, to transform it into a drawing of the expander $W$ with fewer than $\tilde k^2/(\tilde c\log^8m)$ crossings, leading to a contradiction.
The drawing of $W$ is obtained from the drawing $\phi$ of $\hat H_1$ by exploiting the embedding ${\hat{\mathcal{P}}}$ of $W$ into $\hat H_1$. Intuitively, we would like to use the images of the paths in ${\hat{\mathcal{P}}}$ in $\phi$ in order to draw the edges of the graph $W$. Unfortunately, the curves representing the images of these paths are not in general position. This is since that paths in ${\hat{\mathcal{P}}}$ may share edges, and they may also share vertices other than their endpoints. We modify the drawing $\phi$ in three stages. In the first stage, we create a number of copies of every edge in $\hat H_1$, so that the paths in ${\hat{\mathcal{P}}}$ no longer share edges, and in the second stage, we modify the drawing of the curves corresponding to the images of the paths in ${\hat{\mathcal{P}}}$ in the vicinity of the vertices they share, using a nudging operation. Then in the third and the last stage we define the final drawing of the expander $W$.
\paragraph{Stage 1: shared edges.}
For every edge $e\in E(\hat H_1)$, let $N_e$ be the total number of paths in ${\hat{\mathcal{P}}}$ containing $e$; recall that $N_e\leq O((\log^2\tilde k)/\tilde \alpha)\leq O((\log^2m)/\tilde \alpha)$ must hold. Let $\hat H'$ be the graph obtained from graph $\hat H_1$ by deleting from it all edges that do not participate in paths in ${\hat{\mathcal{P}}}$, and, for each remaining edge $e$, creating $N_e$ parallel copies of edge $e$. Drawing $\phi$ of graph $\hat H_1$ can be easily extended to a drawing $\phi'$ of graph $\hat H'$, by deleting the images of all edges $e$ with $N_e=0$, and, for every edge $e$ with $N_e>1$, drawing the parallel copies of $e$ in parallel to the original image of $e$, at a very small distance from it, so that the images of the copies of $e$ do not cross. Since, for every edge $e$, $N_e\leq O((\log^2m)/\tilde \alpha)$, every crossing in the drawing $\phi$ may give rise to at most $O((\log^4m)/\tilde \alpha^2)$ crossings in the drawing $\phi'$, and so the number of crossings in $\phi'$ is bounded by:
\[\mathsf{cr}(\phi)\cdot O\textsf{left}(\frac{\log^4m}{\tilde \alpha^2}\textsf{right} )\leq O\textsf{left}(\frac{\tilde k^2\tilde \alpha^2}{c'\eta'\log^{12}m}\textsf{right} )\cdot O\textsf{left}(\frac{\log^4m}{\tilde \alpha^2}\textsf{right} )\leq O\textsf{left}(\frac{\tilde k^2}{c'\eta'\log^8m}\textsf{right} ).\]
Notice that the set ${\hat{\mathcal{P}}}$ of paths in graph $\hat H_1$ naturally defines a set ${\hat{\mathcal{P}}}'$ of edge-disjoint paths in graph $\hat H'$, embedding the expander $W$ into $\hat H'$ (where for every edge $e\in E(\hat H)$, every path in ${\hat{\mathcal{P}}}$ containing $e$ now uses a different copy of $e$).
\paragraph{Stage 2: shared vertices.}
We process every vertex $x\in V(\hat H')\setminus \tilde T$ one by one.
Let ${\hat{\mathcal{P}}}'(x)=\set{P_1,\ldots,P_r}$ be the set of all paths in ${\hat{\mathcal{P}}}'$ containing $x$. For each such path $P_i(x)$, let $e(P_i,x)$ and $e'(P_i,x)$ be the edges immediately preceding and immediately following $x$ on path $P_i$. Let $D(x)$ be a very small disc containing the image $x$ in the drawing $\phi'$ in its interior. Consider now some path $P_i\in {\hat{\mathcal{P}}}'(x)$, and let $q_i,q'_i$ be the points where the images of $e(P_i,x)$ and $e'(P_i,x)$ intersect the boundary of $D(x)$, respectively. We define a curve $\gamma(P_i,x)$ inside disc $D(x)$, connecting $q_i$ and $q'_i$, such that, for every pair $P_i,P_j\in {\hat{\mathcal{P}}}'(x)$ of distinct paths, the two corresponding curves $\gamma(P_i,x)$, $\gamma(P_j,x)$ cross at most once, and every point of $D(X)$ lies on at most two such curves.
\paragraph{Stage 3: final drawing of $W$.}
We are now ready to define the final drawing $\phi''$ of the expander $W$. Recall that $V(W)=\tilde T$. For every terminal $t\in \tilde T$, the image of $t$ in $\phi''$ remains the same as the image of $t$ in $\phi'$.
Consider now some edge $\hat e=(t,t')\in E(W)$, and let $P(\hat e)\in {\hat{\mathcal{P}}}'$ be its embedding path. Denote $P=(e_1,e_2,\ldots,e_{\ell})$, and denote the vertices of $P$ by $t=x_0,x_1,x_2,\ldots,x_{\ell-1},x_{\ell}=t'$ in the order of their appearance on $P$. For each edge $e_i$, let $\gamma(e_i)$ be its image in the drawing $\phi'$. If $i>1$, then we delete the portion of $\gamma(e_{i})$ that lies inside the disc $D(x_{i-1})$, and similarly, if $i<\ell$, then we delete the portion of $\gamma(e_i)$ that lies inside the disc $D(x_{i})$. The image of the edge $\hat e$ is obtained by concatenating the curves $\gamma(e_1),\gamma(P,x_1),\gamma(e_2),\ldots,\gamma(P,x_{\ell-1}),\gamma(e_{\ell})$.
This completes the definition of the drawing $\phi''$ of the expander $W$. Our last step is to show that this drawing contains few crossings, leading to a contradiction. The following claim will then complete the proof of \Cref{lem: high opt or lots of paths}.
\begin{claim}
$\mathsf{cr}(\phi'')<\tilde k^2/(\tilde c\log^8m)$.
\end{claim}
\begin{proof}
Recall that drawing $\phi'$ of $\hat H'$ contained at most $O\textsf{left}(\frac{\tilde k^2}{c'\eta'\log^8m}\textsf{right} )$ crossings.
For every vertex $u\in V(\hat H_1)$, let $N_u$ de note the number of paths in $\hat {\mathcal{P}}$ containing $u$.
It is then easy to verify that the total number of crossings in $\phi''$ is at most:
\[\mathsf{cr}(\phi')+ \sum_{u\in V(\hat H_1)}N_u^2\leq O\textsf{left}(\frac{\tilde k^2}{c'\eta'\log^8m}\textsf{right} )+ \sum_{u\in V(\hat H_1)}N_u^2. \]
Assuming that $c'$ is a large enough constant, the above expression is bounded by $
\tilde k^2/(2\tilde c\log^8m)+ \sum_{u\in V(\hat H_1)}N_u^2$. Therefore, it is now enough to prove that $\sum_{u\in V(\hat H_1)}N_u^2\leq \tilde k^2/(2\tilde c\log^8m)$.
We partition the vertices of $\hat H_1$ into two subsets: the set $U=\set{v_C\mid C\in {\mathcal{C}}_X}$ of supernodes, and the set $U'=V(\hat H_1)\setminus U$ of regular vertices, that lie in $H_1$.
Consider first a supernode $v=v_C$. Since the paths in ${\hat{\mathcal{P}}}$ cause edge-congestion at most $O(\log^2m/\tilde \alpha)$ in graph $\hat H_1$, $N_v\leq O(\deg_{\hat H_1}(v)\log^2m/\tilde \alpha)$, and so $N_v^2\leq O(|\delta_{H_1}(C)|^2\log^4m/\tilde \alpha^2)$.
Recall that from Property \ref{prop after step 1: small squares of boundaries}:
\[\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2\leq \frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}m}.\]
Therefore, we conclude that:
\[
\begin{split}
\sum_{v\in U}N_v^2&\leq \sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2 \cdot O\textsf{left}(\frac{\log^4m}{\tilde \alpha^2}\textsf{right})\\
& \leq \frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}m}
\cdot O\textsf{left}(\frac{\log^{4}m}{\tilde \alpha^2}\textsf{right})\\
&\leq \frac{\tilde k^2}{4\tilde c\log^8m},
\end{split}\]
since we have assumed that $c_1$ is a large enough constant.
Lastly, it remains to show that $\sum_{v\in U'}N_v^2\leq \frac{\tilde k^2}{4\tilde c\log^8m}$. We do so using the following claim.
\begin{claim}\label{claim: small degree for non super nodes}
For every vertex $v\in U'$, $N_v< \frac{8\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\tilde k'\log^4m}{\tilde \alpha}$.
\end{claim}
\begin{proof}
Assume for contradiction that there is some vertex $v\in U'$, with $N_v\geq \frac{8\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\tilde k'\log^4m}{\tilde \alpha}$. Then there is a set ${\mathcal{P}}\subseteq {\hat{\mathcal{P}}}$ of at least $\frac{8\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\tilde k'\log^4m}{\tilde \alpha}$ paths in the embedding ${\hat{\mathcal{P}}}$ of $W$ into $\hat H_1$, containing $v$. Recall that for every path $P\in{\hat{\mathcal{P}}}$, both endpoints of $P$ are terminals in $\tilde T$, and that, since the maximum vertex degree in $W$ is
at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2 \tilde k\leq \ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2 m$, every terminal may serve as an endpoint of at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2m$ paths in ${\mathcal{P}}$.
Consider any path $P\in {\mathcal{P}}$, and let $t,t'$ be its two endpoints. Let $P'$ be the subpath of $P$ between $t$ and $v$. We then set ${\mathcal{P}}_1=\set{P'\mid P\in {\mathcal{P}}}$. Furthermore, since every terminal may serve as an endpoint of at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2m$ paths in ${\mathcal{P}}_1$, there is a subset ${\mathcal{P}}_2\subseteq {\mathcal{P}}_1$ of at least $\frac{|{\mathcal{P}}|}{\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2m}$ paths, each of which originates at a distinct terminal of $\tilde T$, and terminates at $v$. Recall that the paths in ${\hat{\mathcal{P}}}$ cause edge-congestion at most $\frac{\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2m}{\tilde \alpha}$ in $\hat H_1$.
Lastly, from \Cref{claim: remove congestion}, there is a collection ${\mathcal{P}}_3$ of edge-disjoint paths in $\hat H_1$, routing a subset of terminals to $v$, such that:
\[|{\mathcal{P}}_3|\geq \frac{\tilde \alpha\cdot |{\mathcal{P}}_2|}{2\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2m}\geq \frac{\tilde \alpha |{\mathcal{P}}|}{2\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\log^4m}>\tilde k',\]
contradicting the fact that the largest number of edge-disjoint paths routing terminals of $\tilde T$ to $v$ is bounded by $\tilde k'$.
\end{proof}
We group the vertices of $U'$ into groups $S_1,\ldots,S_q$, where $q=\ceil{\log \textsf{left} (\frac{8\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\tilde k'\log^4m}{\tilde \alpha}\textsf{right} )}+1$. Set $S_i$ contains all vertices $v\in U'$, with $2^{i-1}\leq N_v<2^i$.
Since paths in ${\hat{\mathcal{P}}}$ cause edge-congestion $O((\log^2m)/\tilde \alpha)$, for all $1\leq i\leq q$, for every vertex $v\in S_i$, $\deg_{\hat H}(v)\geq \Omega(\tilde \alpha N_v/\log^2m)\geq \Omega(\tilde \alpha \cdot 2^i/\log^2m)$. Since the total number of edges in graph $\hat H_1$ is
$|E(\hat H_1)|\leq O(\tilde k\cdot \eta \log^8m/\alpha^3)$ from Property \ref{prop after step 1: few edges},
while $\tilde \alpha=\Omega(\alpha/\log^4m)$ from Property \ref{prop after step 1: terminals in H1},
we get that, for all $1\leq i\leq q$,
\[\sum_{v\in S_i}N_v\leq \sum_{v\in S_i}O\textsf{left} (\frac{\deg_{\hat H}(v)\log^2m}{\tilde \alpha}\textsf{right} )
\leq O\textsf{left}(\frac{|E(\hat H_1)| \log^6m}{\alpha}\textsf{right} )\leq O\textsf{left}(\frac{\tilde k\eta\log^{14}m}{\alpha^4}\textsf{right}). \]
Therefore:
\[\sum_{v\in S_i}N_v^2\leq 2^{i+1}\cdot \sum_{v\in S_i}N_v\leq O\textsf{left}(\frac{2^i\cdot \tilde k\eta\log^{14}m}{\alpha^4}\textsf{right}).\]
Summing up over all sets $S_1,\ldots,S_q$, we get that:
\[
\begin{split}
\sum_{v\in U'}N_v^2& \leq
O\textsf{left}(\frac{2^q\tilde k\eta\log^{14}m}{\alpha^4}\textsf{right}) \\
&\leq O\textsf{left}(\frac{\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\tilde k'\tilde k\eta\log^{18}m}{\alpha^4\tilde \alpha}\textsf{right})\\
&\leq O\textsf{left}(\frac{\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\tilde k^2}{c'\log^8m}\textsf{right}).
\end{split}\]
since
$q=\ceil{\log \textsf{left} (\frac{8\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\tilde k'\log^4m}{\tilde \alpha}\textsf{right} )}+1$,
$\tilde \alpha=\Omega(\alpha/\log^4m)$, and
$\tilde k'=\tilde k\alpha^5/(c'\eta\log^{36}m)$.
Lastly, since we can fix $c'$ to be a large enough constant, we get that $\sum_{v\in U'}N_v^2< \frac{k^2}{4\tilde c\log^8m}$.
To conclude, we have shown that $\mathsf{cr}(\phi'')\leq \frac{\tilde k^2}{\tilde c\log^8m}$, a contradiction.
\end{proof}
\section{Proofs Omitted from \Cref{sec: guiding paths}}
\label{sec: appx guiding paths}
\input{appx-splitting}
\input{appx-few-paths-large-opt}
\input{ordering-of-terminals}
\input{appx-bound-on-opt}
\subsection{Proof of \Cref{thm: layered well linked decomposition}}
\label{sec: layered well linked}
Note that by letting $c$ be a large enough constant, we can ensure that $\alpha < \min\set{\frac 1 {64\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m},\frac 1 {48\log^2 m}}$ holds.
The algorithm starts with layer ${\mathcal{L}}_0$ containing a single subgraph of $H$ -- the subgraph $C$, and then performs iterations. The input to iteration $i$ is layers ${\mathcal{L}}_0,{\mathcal{L}}_1,\ldots,{\mathcal{L}}_{i-1}$, each of which is a collection of disjoint clusters of $H$. We ensure that all clusters in $\bigcup_{i'=0}^{i-1}{\mathcal{L}}_{i'}$ are mutually disjoint, and each cluster $W\in \bigcup_{i'=1}^{i-1}{\mathcal{L}}_{i'}$ has $\alpha$-bandwidth property. We let $S_i$ be the subgraph of $H$ induced by vertex set $\bigcup_{i'=0}^{i-1}\bigcup_{W\in {\mathcal{L}}_{i'}}V(W)$, and we let $E_i=\delta_H(S_i)$. In subsequent iterations, we will create layers ${\mathcal{L}}_i,{\mathcal{L}}_{i+1},\ldots$, each of which will contain clusters that are disjoint from $S_i$. Notice that, for all $1\leq i'<i$, for every cluster $W\in {\mathcal{L}}_{i'}$, the partition of the edges of $\delta_H(W)$ into $\delta^{\operatorname{down}}(W)$ and $\delta^{\operatorname{up}}(W)$ is now settled, since the layers ${\mathcal{L}}_0,\ldots,{\mathcal{L}}_{i'-1}$ will not undergo any changes in subsequent iterations, and the edges of $E_i\cap \delta_H(W)$ are guaranteed to lie in $\delta^{\operatorname{up}}(W)$. We will ensure that, for all $1\leq i'<i$, every cluster in ${\mathcal{L}}_{i'}$ has properties \ref{condition: layered well linked}, \ref{condition: layered decomp each cluster prop} and \ref{condition: layered decomp edge ratio}. We will also ensure that $|E_i|\leq |\delta_H(C)|/2^{i-1}$. The algorithm terminates once $S_i=H$. Since we ensure that for all $i$, $|E_i|\leq |\delta_H(C)|/2^{i-1}$, the number of iterations is bounded by $\log m$.
We now describe the execution of the $i$th iteration. We start by considering the subgraph $S'_i=H\setminus V(S_i)$ of $H$. We apply the algorithm from \Cref{thm:well_linked_decomposition} to graph $H$, its subgraph $S=S'_i$, and the parameter $\alpha=\frac{1}{c\log^{2.5}m}$. As observed already, $\alpha < \min\set{\frac 1 {64\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot\log m},\frac 1 {48\log^2 m}}$ holds. (If graph $S'_i$ is not connected, then we apply the algorithm to every connected component of $S'_i$ separately). Let ${\mathcal{W}}_i$ be the collection of clusters that the algorithm returns.
Recall that we are guaranteed that the sets $\set{V(W)}_{W\in {\mathcal{W}}_i}$ of vertices partition $V(S'_i)$, and for each such cluster $W\in {\mathcal{W}}_i$, $|\delta_H(W)|\le |\delta_H(S'_i)|=|\delta_H(S_i)|\leq |\delta_H(C)|$. We are also guaranteed that each cluster $W\in {\mathcal{W}}_i$ has the $\alpha$-bandwidth property, and that $\sum_{W\in {\mathcal{W}}_i}|\delta_H(W)|\le |\delta_H(S'_i)|\cdot\textsf{left}(1+O(\alpha\cdot \log^{3/2} m)\textsf{right})=|E_i|\cdot \textsf{left}(1+O(\alpha\cdot \log^{3/2} m)\textsf{right})$.
Since $\alpha=\frac{1}{c\log^{2.5}m}$ for a large enough constant $c$, we can ensure that $\sum_{W\in {\mathcal{W}}_i}|\delta_H(W)|\le |E_i|\cdot\textsf{left} (1+\frac{1}{1000\log m}\textsf{right} )$.
If there is a cluster $W\in {\mathcal{W}}_i$ with $|E_H(W)|<|\delta_H(W)|/(64\log m)$, then remove $W$ from ${\mathcal{W}}_i$ and add each of its vertices as a separate cluster to ${\mathcal{W}}_i$. Clearly, we have increased the sum $\sum_{W\in {\mathcal{W}}_i}|\delta_H(W)|$ by a factor of at most $\textsf{left} (1+1/(32\log m)\textsf{right} )$ (since each edge appears in at most two sets of $\set{\delta_H(W)}_{W\in {\mathcal{W}}_i}$). Therefore, for the resulting set ${\mathcal{W}}_i$, we get that:
\[\sum_{W\in {\mathcal{W}}_i}|\delta_H(W)|\le |E_i|\cdot\textsf{left} (1+\frac{1}{1000\log m}\textsf{right} )\cdot\textsf{left} (1+\frac{1}{32\log m}\textsf{right} )\le |E_i|\cdot\textsf{left} (1+\frac{1}{16\log m}\textsf{right}).\]
Recall that the algorithm from \Cref{thm:well_linked_decomposition} also computes, for each cluster $W\in {\mathcal{W}}$, a set ${\mathcal{P}}'(W)$ of paths in graph $H$ routing the edges of $\delta_H(W)$ to edges of $\delta_H(S'_i)=E_i$, such that the paths of
${\mathcal{P}}'(W)$ avoid $W$ and cause congestion at most $100$ in $H$.
We partition the set ${\mathcal{W}}_i$ of clusters into two subsets: set ${\mathcal{W}}_i'$ contains all clusters $W\in {\mathcal{W}}_i$, such that $|\delta_H(W)\setminus E_i|< |\delta_H(W)\cap E_i|/\log m$, and set ${\mathcal{W}}_i''$ contains all remaining clusters. We then set ${\mathcal{L}}_i={\mathcal{W}}_i'$. This finishes the description of the iteration. Recall that we define $S_{i+1}$ to be the subgraph of $H$ induced by the set $V(S_i)\cup \textsf{left} (\bigcup_{W\in {\mathcal{L}}_i}V(W)\textsf{right} )$ of vertices, and $E_{i+1}=\delta_H(S_{i+1})$. We now analyze the iteration.
First, from the algorithm, every cluster $W\in {\mathcal{L}}_i$ satisfies Property \ref{condition: layered decomp each cluster prop}, since the first inequality is guaranteed by \Cref{thm:well_linked_decomposition}, and we have replaced every cluster that does not satisfies the second inequality of Property \ref{condition: layered decomp each cluster prop} by single-vertex clusters.
We now prove the following claim.
\begin{claim} $|E_{i+1}|\leq |E_i|/2$.
\end{claim}
\begin{proof}
We partition the set $E_{i+1}$ of edges into two subsets. The first set, $E'_{i+1}$, contains all edges of $E_{i+1}$ that lie in the sets $\set{\delta_H(W)\setminus E_i}_{W\in {\mathcal{W}}'_i}$. Since, for each cluster $W\in {\mathcal{W}}'_i$, $|\delta_H(W)\setminus E_i|\leq |\delta_H(W)\cap E_i|/\log m$, we get that $|E_{i+1}'|\leq |E_i|/\log m$. The second set, $E''_{i+1}$, contains all remaining edges of $E_{i+1}$. It is easy to verify that every edge of $E''_{i+1}$ belongs to set $\delta_H(W)\cap E_i$ of edges for some cluster $W\in {\mathcal{W}}_i''$.
Recall that $\sum_{W\in {\mathcal{W}}_i}|\delta_H(W)|\le |E_i|\cdot\textsf{left} (1+\frac{1}{16\log m}\textsf{right} )$. Therefore, since $E_i\subseteq \bigcup_{W\in {\mathcal{W}}_i}\delta_H(W)$, we get that $\sum_{W\in {\mathcal{W}}''_i}|\delta_H(W)\setminus E_i|\leq \frac{|E_i|}{16\log m}$. For every cluster $W\in {\mathcal{W}}''_i$, $|\delta_H(W)\setminus E_i|\geq \frac{|\delta_H(W)\cap E_i|}{\log m}$ from the definition of set ${\mathcal{W}}''_i$. Therefore:
\[|E_{i+1}''|=\sum_{W\in {\mathcal{W}}''_i}|\delta_H(W)\cap E_i|\leq (\log m)\cdot \sum_{W\in {\mathcal{W}}''_i}|\delta_H(W)\setminus E_i|\leq \frac{|E_i|}{16}.\]
Altogether, $|E_{i+1}|=|E'_{i+1}|+|E''_{i+1}|\leq |E_{i}|/2$.
\end{proof}
Recall that we have already established that, for every cluster $W\in {\mathcal{L}}_{i}$, $|\delta_H(W)|\leq |\delta_H(C)|$. Consider any such cluster $W\in {\mathcal{L}}_i$. Since layers ${\mathcal{L}}_0,\ldots,{\mathcal{L}}_i$ will remain unchanged in the remainder of the algorithm, the partition of the edge set $\delta_H(W)$ into $\delta^{\operatorname{up}}(W)$ and $\delta^{\operatorname{down}}(W)$ is now settled, and moreover $\delta^{\operatorname{down}}(W)=\delta_H(W)\cap E_i$, while $\delta^{\operatorname{up}}(W)=\delta_H(W)\setminus E_i$. From the definition of cluster set ${\mathcal{W}}'_i={\mathcal{L}}_i$, we get that, for every cluster $W\in {\mathcal{L}}_i$, $|\delta^{\operatorname{up}}(W)|\leq |\delta^{\operatorname{down}}(W)|/\log m$ holds. Therefore, property \ref{condition: layered decomp edge ratio} holds for every cluster $W\in {\mathcal{L}}_i$. Recall that we have already established property \ref{condition: layered well linked} for each such cluster as well.
The algorithm terminates once $S_i=H$. Let $r$ denote the index of the last iteration. Since, for all $i$, $|E_{i+1}|\leq |E_i|/2$, $r\leq \log m$ holds. We let ${\mathcal{W}}=\bigcup_{i=1}^r{\mathcal{L}}_i$ be the final collection of clusters. We now claim that $({\mathcal{W}},({\mathcal{L}}_1,\ldots,{\mathcal{L}}_r))$ is a valid layered $\alpha$-well-linked decomposition of $H$ with respect to $C$. Note that our algorithm immediately guarantees Property \ref{condition: layered decomposition is partition}, and we have already established Properties \ref{condition: layered well linked} -- \ref{condition: layered decomp edge ratio} of the decomposition. In order to establish property \ref{condition: layered decomposition few edges}, observe that for all $1\leq i\leq r$:
\[\sum_{W\in {\mathcal{L}}_i}|\delta_H(W)|=\sum_{W\in {\mathcal{L}}_i}(|\delta^{\operatorname{up}}(W)|+|\delta^{\operatorname{down}}(W)|)\leq \sum_{W\in {\mathcal{L}}_i}2|\delta^{\operatorname{down}}(W)|= \sum_{W\in {\mathcal{L}}_i}2|\delta_H(W)\cap E_i|=2|E_i|.\]
Therefore,
\[\sum_{W\in {\mathcal{W}}}|\delta_H(W)|\leq 2 \sum_{1\le i\le r} |E_i|\leq 4|E_1|=4|\delta_H(C)|.\]
We conclude that property \ref{condition: layered decomposition few edges} holds for the decomposition. Lastly, it remains to establish property \ref{condition: layered decomposition routing}. We do so using the following claim.
\begin{claim}\label{claim: routing the frontier}
For all $0\leq i< r$, there is a collection ${\mathcal{R}}_{i+1}=\set{R(e)\mid e\in E_{i+1}}$ of paths such that, for every edge $e\in E_{i+1}$, path $R(e)$ has $e$ as its first edge, and some edge $e'\in E_i$ as its last edge. Moreover, every edge $e'\in E_i$ participates in at most one path of ${\mathcal{R}}_{i+1}$, the paths in ${\mathcal{R}}_{i+1}$ cause congestion at most $\ceil{1/\alpha}$, and for each path $R(e)\in {\mathcal{R}}_{i+1}$, all inner vertices on $R(e)$ lie in $S_{i+1}\setminus S_i$.
\end{claim}
\begin{proof}
We partition the set $E_{i+1}$ of edges into two subsets: $E'_{i+1}=E_i\cap E_{i+1}$, and $E''_{i+1}=E_{i+1}\setminus E_i$. For each edge $e\in E'_{i+1}$, the path $R(e)$ consists of a single edge -- the edge $e$.
Observe that $E''_{i+1}\subseteq \textsf{left} (\bigcup_{W\in {\mathcal{L}}_{i}}\delta_H(W)\textsf{right} )\setminus E_i$. For every cluster $W\in {\mathcal{L}}_{i}$, we let $\hat E(W)=\delta_H(W)\cap E_{i+1}''$. Clearly, $\hat E(W)\subseteq \delta^{\operatorname{up}}(W)$. Recall that $\delta^{\operatorname{down}}(W)=\delta_H(W)\cap E_i$, and $|\delta^{\operatorname{down}}(W)|>|\delta^{\operatorname{up}}(W)|$. From \Cref{cor: bandwidth_means_boundary_well_linked}, there is a set ${\mathcal{R}}(W)$ of paths, that is a one-to-one routing of edges in $\delta^{\operatorname{up}}(W)$ to a subset of edges in $\delta^{\operatorname{down}}(W)$, such that, for each path $Q\in{\mathcal{R}}(W)$, all its edges, except for the first and the last, belong to $E(W)$, and the paths in ${\mathcal{R}}(W)$ cause congestion at most $\ceil{1/\alpha}$. For each edge $e\in \hat E(W)\subseteq \delta^{\operatorname{up}}(W)$, we let $R(e)\in {\mathcal{R}}(W)$ be the unique path whose first edge is $e$. We have now defined, for each edge $e\in E_{i+1}$, a path $R(e)$, whose first edge is $e$, last edge lies in $E_{i}$, and all inner vertices lie in $S_{i+1}\setminus S_i=\bigcup_{W\in {\mathcal{L}}_i}V(W)$. It is immediate to verify that the resulting set ${\mathcal{R}}_{i+1}$ of paths causes congestion at most $\ceil{1/\alpha}$, and that each edge of $E_i$ participates in at most one such path.
\end{proof}
We obtain the following immediate corollary of \Cref{claim: routing the frontier}.
\begin{corollary}\label{cor: routing the frontier}
For all $0\leq i< r$, there is a collection ${\mathcal{R}}'_{i+1}=\set{R'(e)\mid e\in E_{i+1}}$ of paths such that, for every edge $e\in E_{i+1}$, path $R'(e)$ has $e$ as its first edge, some edge of $\delta_H(C)$ as its last edge, and all its inner vertices are contained in $S_{i+1}\setminus C$. Moreover, every edge in $\delta_H(C)$ may participate in at most one path in ${\mathcal{R}}'_{i+1}$, and the paths in ${\mathcal{R}}'_{i+1}$ cause congestion at most $\ceil{1/\alpha}$.
\end{corollary}
\begin{proof}
The proof is by induction on $i$. For $i=0$, we let the set ${\mathcal{R}}'_1$ of paths contain, for each edge $e\in \delta_H(C)=\delta_H(S_1)$, a path $R'(e)$ that consists of a single edge - the edge $e$.
Assume now that we have defined the sets ${\mathcal{R}}'_1,\ldots,{\mathcal{R}}'_{i}$ of paths. In order to define the set ${\mathcal{R}}'_{i+1}=\set{R'(e)\mid e\in E_{i+1}}$ of paths, consider any edge $e\in E_{i+1}$, and let $R(e)\in {\mathcal{R}}_{i+1}$ be the unique path that has $e$ as its first edge. Denote by $e'\in E_i$ the last edge on path $R(e)$, and consider the path $R'(e')\in {\mathcal{R}}_i$, connecting $e'$ to an edge of $\delta_H(C)$. We obtain the path $R'(e)$ by concatenating $R(e)$ and $R'(e')$. It is easy to verify that the resulting set ${\mathcal{R}}'_{i+1}=\set{R'(e)\mid e\in E_{i+1}}$ of paths has all required properties.
\end{proof}
We are now ready to establish Property \ref{condition: layered decomposition routing} of the decomposition. Consider some layer ${\mathcal{L}}_i$, for $1\leq i\leq r$, and some cluster $W\in {\mathcal{L}}_i$. Recall that, when the algorithm from \Cref{thm:well_linked_decomposition} was applied to cluster $S'_{i}$ in iteration $i$, it returned a set ${\mathcal{P}}'(W)$ of paths in graph $H$, routing the edges of $\delta_H(W)$ to edges of $\delta_H(S'_i)=E_i$, such that the paths of
${\mathcal{P}}'(W)$ avoid $W$ and cause congestion at most $100$ in $H$. We can assume without loss of generality that, if an edge of $E_i$ lies on a path of ${\mathcal{P}}'(W)$, then it is the last edge on that path. Equivalently, no path of ${\mathcal{P}}'(W)$ may contain a vertex of $S_i$ as its inner vertex. Consider now some edge $e\in \delta_H(W)$, and let $P'(e)\in {\mathcal{P}}'(W)$ be the path whose first edge is $e$. Let $e'\in E_i$ be the last edge on path $P'(e)$, and let $R'(e')$ be the unique path in ${\mathcal{R}}'_i$ that has $e'$ as its first edge. Recall that the last edge of ${\mathcal{R}}'_i$ lies in $\delta_H(C)$. Moreover, since all inner vertices on path $R'(e')$ lie in $S_i\setminus C$, no inner vertex of path $R'(e')$ may lie in $W$. By concatenating the paths $P'(e)$ and $R'(e')$, we obtain a path $P(e)$, whose first edge is $e$, and last edge lies in $\delta_H(C)$. From the above discussion, no inner vertex of $P(e)$ lies in $W$. We then let ${\mathcal{P}}(W)=\set{P(e)\mid e\in \delta_H(W)}$. We have now obtained a set of paths routing the edges of $\delta_H(W)$ to edges of $\delta_H(C)$, such that the paths in ${\mathcal{P}}(W)$ avoid $W$. It now remains to analyze the congestion that this set of paths causes in graph $H$. As the set ${\mathcal{P}}'(W)$ of paths causes congestion at most $100$, every edge $e'\in E_i$ may participate in at most $100$ such paths. Since the congestion caused by the set ${\mathcal{R}}'_i$ of paths is at most $\ceil{1/\alpha}$, and since no vertex of $S_i$ may serve as an inner vertex on a path of ${\mathcal{P}}'(W)$, the total congestion caused by paths in ${\mathcal{P}}(W)$ is at most $100\cdot \ceil{\frac 1 {\alpha}}\leq \frac{200}{\alpha}$.
\section{Proof of \Cref{thm: main_result}}
\label{sec: proof of main theorem}
In this section, we provide the proof of \Cref{thm: main_result} from \Cref{thm: main_rotation_system} and \Cref{thm: MCN_to_rotation_system}.
Suppose we are given a simple $n$-vertex graph $G$ with maximum vertex degree $\Delta$. We use the algorithm from \Cref{thm: MCN_to_rotation_system} in order to compute an instance $I=(G',\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, with $m=|E(G')|\leq O\textsf{left}(\mathsf{OPT}_{\mathsf{cr}}(G)\cdot \operatorname{poly}(\Delta\cdot\log n)\textsf{right})$, and $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq O\textsf{left}(\mathsf{OPT}_{\mathsf{cr}}(G)\cdot \operatorname{poly}(\Delta\cdot\log n)\textsf{right} )$.
Notice that, since $G$ is a simple graph, $\mathsf{OPT}_{\mathsf{cr}}(G)\leq |E(G)|^2\leq n^4$, and $\Delta\leq n$. Therefore, $m=|E(G')|\leq \operatorname{poly}(n)$.
We use the algorithm from
\Cref{thm: main_rotation_system} to compute a solution to instance $I$ of \textnormal{\textsf{MCNwRS}}\xspace, such that, w.h.p., the number of crossings in the solution is bounded by $2^{O((\log m)^{7/8}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right})$. Lastly, using the algorithm from \Cref{thm: MCN_to_rotation_system}, we efficiently compute a drawing of graph $G$, with the number of crossings bounded by:
\[\begin{split}
&\textsf{left} (2^{O((\log m)^{7/8}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right}) +\mathsf{OPT}_{\mathsf{cr}}(G)\textsf{right} )\cdot \operatorname{poly}(\Delta\log n) \\
&\quad\quad\quad\quad\leq O\textsf{left} (2^{O((\log n)^{7/8}\log\log n)}\cdot \operatorname{poly}(\Delta)\textsf{right} )\cdot \mathsf{OPT}_{\mathsf{cr}}(G).
\end{split}\]
\subsection{Proof of Claim~\ref{clm: contracted_graph_well_linkedness}}
\label{apd: Proof of contracted_graph_well_linkedness}
Consider any $T$-cut $(A,B)$ in graph $G$, and denote $T_A=T\cap A$ and $T_B=T\cap B$. Assume without loss of generality that $|T_A|\le |T_B|$. It is enough to show that $|E_G(A,B)|\ge (\alpha_1\alpha_2)\cdot |T_A|$. Assume for contradiction that this is not the case.
Denote $H=G_{|{\mathcal{C}}}$. We partition the set $V(H)$ of vertices into two subsets: set $V'=V(H)\cap V(G)$ of regular vertices, and set $V''=\set{v_C\mid C\in {\mathcal{C}}}$ of supernodes.
Note that $T\subseteq V'$ must hold. We use the cut $(A,B)$ in $G$, in order to construct a cut $(A',B')$ in graph $H$, with $A'\cap T=T_A$ and $B'\cap T=T_B$, such that $|E_H(A',B')|<\alpha_2\cdot |T_A|$, contradicting the fact that vertex set $T$ is $\alpha_2$-well-linked in graph $G_C$.
In order to construct the cut $(A',B')$ in $H$, we first process every vertex $v\in V'$ one by one. For each such vertex $v$, if $v\in A$, then we add $v$ to $A'$, and otherwise we add it to $B'$. Notice that this process guarantees that $T_A\subseteq A'$ and $T_B\subseteq B'$.
Next, we process every cluster $C\in {\mathcal{C}}$ one by one. Notice that partition $(A,B)$ of $V(G)$ naturally defines a partition $(A_C,B_C)$ of $V(C)$, where $A_C=A\cap V(C)$ and $B_C=B\cap V(C)$. We denote $E'_C=E_G(A_C,B_C)$, $E_1(C)=\delta_G(A_C)\setminus E'_C$, and $E_2(C)=\delta_G(B_C)\setminus E'_C$. If $|E_1(C)|\leq |E_2(C)|$, then we add supernode $v_C$ to $B'$, and otherwise we add it to $A'$. Assume w.l.o.g. that $v_C$ was added to $B'$, so $|E_1(C)|\leq |E_2(C)|$ holds. From the $\alpha_1$-bandwidth property of $C$, we get that $|E'_C|\geq \alpha_1\cdot |E_1(C)|$. Notice that the edges of $E'_C$ lie in the cut $(A,B)$ in graph $G$, but they do not contribute to the cut $(A',B')$ in graph $H$. On the other hand, edges of $E_1(C)$ may lie in $E_H(A',B')\setminus E_G(A,B)$. We \emph{charge} the edges of $E_1(C)$ to the edges of $E'$. Since $|E'|\geq \alpha_1\cdot |E_1(C)|$, every edge of $E'$ pays at most $1/\alpha_1$ units for the edges of $E_1(C)$, so the total charge to the edges of $E'$ is $|E_1(C)|$.
Once every cluster $C\in {\mathcal{C}}$ is processed, we obtain the final cut $(A',B')$ in graph $H$. For every edge $e\in E_H(A',B')$, either $e\in E_G(A,B)$, or $e$ is charged to some edges of $E_G(A,B)\setminus E_H(A',B')$. Since the charge to every edge of $E_G(A,B)\setminus E_H(A',B')$ is at most $1/\alpha_1$, we get that $|E_H(A',B')|\leq |E_G(A,B)|/\alpha_1$. Since we have assumed that $|E_G(A,B)|< (\alpha_1\alpha_2)\cdot |T_A|$, we get that $|E_H(A',B')|<\alpha_2\cdot |T_A|=\alpha_2\cdot |T\cap A'|$, contradicting the fact that vertex set $T$ is $\alpha_2$-well-linked in $H$.
\subsection{Proof of \Cref{cor: contracted_graph_well_linkedness}}
\label{apd: Proof of cor contracted_graph_well_linkedness}
Let $G^+$ be the graph obtained from $G$ by subividing each edge $e\in \delta_G(R)$ with a new vertex $t_e$.
Denote $T=\set{t_e\mid e\in \delta_G(R)}$.
Recall that the augmentation $R^+$ of cluster $R$ in $G$ is defined to be the subgraph of $G^+$ induced by vertex set $V(R)\cup T$.
It is immediate to verify that every cluster $C\in {\mathcal{C}}$ has the $\alpha_1$-bandwidth property in graph $R^+$. Furthermore, from \Cref{obs: wl-bw}, the set $T$ of vertices is $\alpha_2$-well-linked in graph $R^+_{|{\mathcal{C}}}$. By applying \Cref{clm: contracted_graph_well_linkedness} to graph $R^+$, vertex set $T$ and collection ${\mathcal{C}}$ of clusters, we get that $T$ is $(\alpha_1\cdot\alpha_2)$-well-linked in graph $R^+$. From \Cref{obs: wl-bw}, cluster $R$ has the $(\alpha_1\cdot\alpha_2)$-bandwidth property in $G$.
\subsection{Proof of \Cref{claim: routing in contracted graph}}
\label{apx: contracted graph routing}
For convenience, we denote $|T|=k$.
We assume w.l.o.g. that the paths in ${\mathcal{P}}$ are simple, and we direct each such path towards $x$. We then graduately modify the paths in ${\mathcal{P}}$, by processing the clusters of ${\mathcal{C}}$ one by one.
Consider any cluster $C\in {\mathcal{C}}$, and let ${\mathcal{P}}(C)\subseteq {\mathcal{P}}$ be the subset of paths that contain the supernode $v_C$. For each path $P\in {\mathcal{P}}(C)$, let $e_P(C)$ and $e_P'(C)$ denote the edges appearing immediately before and immediately after $v_C$ on $P$. We denote $E_1(C)=\set{e_P(C)\mid P\in {\mathcal{P}}(C)}$ and $E_2(C)=\set{e'_P(C)\mid P\in {\mathcal{P}}(C)}$. We use the algorithm from \Cref{cor: bandwidth_means_boundary_well_linked}, to compute a collection ${\mathcal{R}}(C)$ of paths that is a one-to-one routing of the edges of $E_1(C)$ to the edges of $E_2(C)$, such that all inner vertices on the paths of ${\mathcal{R}}(C)$ lie in $C$, and every edge in $E(C)$ participates in at most $\ceil{1/\alpha}$ such paths. We modify the paths in set ${\mathcal{P}}(C)$ as follows. First, for each path $P\in {\mathcal{P}}(C)$, we delete the vertex $v_C$ from $P$, together with its two incident edges. Let $P_1,P_2$ be the two resulting subpaths of $P$. We then let ${\mathcal{P}}_1(C)=\set{P_1\mid P\in {\mathcal{P}}(C)}$, and ${\mathcal{P}}_2(C)=\set{P_2\mid P\in {\mathcal{P}}(C)}$. Lastly, let ${\mathcal{P}}'(C)$ be the set of paths obtained by concatenating the paths in ${\mathcal{P}}_1(C),{\mathcal{R}}(C)$ and ${\mathcal{P}}_2(C)$. We delete from ${\mathcal{P}}$ the paths that belong to ${\mathcal{P}}(C)$, and add the paths of ${\mathcal{P}}'(C)$ instead. It is easy to verify that the resulting set ${\mathcal{P}}$ of paths still routes the vertices of $T$ to $x$.
Once we process every cluster $C\in {\mathcal{C}}$, we obtain a collection ${\mathcal{P}}'$ of $k$ paths, routing the vertices of $T$ to vertex $x$ in graph $G$. Since the paths of ${\mathcal{P}}$ at the beginning of the algorithms are edge-disjoint, for each edge $e\in E(G)\setminus\textsf{left}( \bigcup_{C\in {\mathcal{C}}}E(C)\textsf{right} )$, $\cong_G({\mathcal{P}}',e)\le 1$. From our construction, for each edge $e\in \bigcup_{C\in {\mathcal{C}}}E(C)$, $\cong_{G}({\mathcal{P}}',e)\leq \ceil{1/\alpha}\leq 2/\alpha$. Lastly, we apply the algorithm from \Cref{claim: remove congestion}, to graph $G$ and the set ${\mathcal{P}}'$ of paths, to obtain a collection ${\mathcal{P}}''$ of at least $\alpha k/2$ edge-disjoint paths in graph $G$, where each path in ${\mathcal{P}}''$ connects a distinct vertex of $T$ to $x$.
\subsection{Proof of \Cref{lem: crossings in contr graph}}
\label{apd: Proof of crossings in contr graph}
Let $\phi^*$ be an optimal solution to instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace. Let $G'$ be the graph that is obtained from $G$ by subdividing every edge $e\in \bigcup_{C\in {\mathcal{C}}}\delta_G(C)$ with a vertex $t_e$, and let $T=\set{t_e\mid e\in \bigcup_{C\in {\mathcal{C}}}\delta_G(C)}$ be the resulting set of new vertices. For every cluster $C\in {\mathcal{C}}$, we denote by $T_C=\set{t_e\mid e\in\delta_G(C)}$, and we let $C^+$ be the subgraph of $G'$ induced by vertex set $V(C)\cup T_C$. From \Cref{obs: wl-bw}, vertex set $T_C$ is $\alpha$-well-linked in $C^+$. Observe that drawing $\phi^*$ of $G$ naturally defines a drawing $\phi'$ of graph $G'$, with $\mathsf{cr}(\phi')=\mathsf{cr}(\phi^*)$.
We denote ${\mathcal{C}}=\set{C_1,\ldots,C_r}$, where the clusters are indexed arbitrarily. For $1\leq i\leq r$, we let ${\mathcal{C}}_i=\set{C_1,\ldots,C_i}$, and we let $G'_i=G'_{|{\mathcal{C}}_i}$. We also denote $G'_0=G'$. We perform $r$ iterations. The input to the $i$th iteration is a drawing $\phi'_{i-1}$ of the graph $G'_{i-1}$, and the output is a drawing $\phi'_i$ of the graph $G'_i$. We set $\phi'_0=\phi'$.
We now describe the $i$th iteration, for $1\leq i\leq r$. For convenience, we denote $C_i=C$.
\Cref{cor: simple guiding paths} guarantees that there is a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers, such that, for every edge $e\in E(C)$, $\expect[{\mathcal{Q}}(C)\sim {\mathcal{D}}(C)]{\cong({\mathcal{Q}}(C),e)}\leq O((\log |\delta_G(C)|)^4/\alpha)\leq O((\log^4m)/\alpha)$. We let ${\mathcal{Q}}(C)=\set{Q(e)\mid e\in \delta_G(C)}$ be an internal $C$-router sampled from the distribution ${\mathcal{D}}(C)$, and we denote by $u(C)$ the center of the router ${\mathcal{Q}}(C)$. Note that the set ${\mathcal{Q}}(C)$ of paths in graph $G$ naturally defines a set of paths in graph $C^+$, routing the vertices of $T_C$ to vertex $u(C)$. Abusing the notation, we denote this set of paths by ${\mathcal{Q}}(C)$ as well.
Applying the algorithm from \Cref{cor: new type 2 uncrossing} to graph $G'_{i-1}$, its drawing $\phi'_{i-1}$, subgraph $G'_{i-1}\setminus C^+$, and the set ${\mathcal{Q}}(C)$ of paths, we obtain a collection $\Gamma(C)=\set{\gamma(e)\mid e\in \delta_{G'_{i-1}}(C)}$ of curves, such that, for every edge $e\in \delta_{G'_{i-1}}(C)$, curve $\gamma(e)$ originates at the image of the endpoint of $e$ that lies in $T_C$, and terminates at the image of $u(C)$. Furthermore, the curves in $\Gamma$ do not cross each other, and, for every edge $e\in E(G'_{i-1})\setminus E(C^+)$, the number of crossings between $\phi'_{i-1}(e)$ and the curves in $\Gamma(C)$ is bounded by
$\sum_{e'\in E(C^+)}\chi(e,e')\cdot \cong_{G'}({\mathcal{Q}}(C),e')$, where $\chi(e,e')$ is the number of crossings between $\phi'_{i-1}(e)$ and $\phi'_{i-1}(e')$. For every edge $e'\in E(C^+)$, and every crossing $(e,e')_p$ between $e$ and $e'$ in $\phi'_{i-1}$, we \emph{charge} this crossing $\cong_G({\mathcal{Q}}(C),e')$ units, and we say that crossing $(e,e')_p$ is \emph{responsible} for $\cong_{G'}({\mathcal{Q}}(C),e')$ new crossings between the edge $e$ and the curves in $\Gamma(C)$. Therefore, the total charge to all crossings between $e$ and the edges of $E(C^+)$ is at least the total number of crossings between $\phi'_{i-1}(e)$ and the curves in $\Gamma(C)$.
We obtain a drawing $\phi'_i$ of the graph $G_i$ as follows. We start from the drawing $\phi'_{i-1}$ of graph $G'_{i-1}$, and delete all edges and vertices of $C_i^+\setminus T_C$ from it. We place the image of the supernode $v_{C_i}$ at the image of the vertex $u(C_i)$ in $\phi'_{i-1}$.
For every edge $e\in \delta_{G'_i}(v_{C_i})$, we let $\gamma(e)\in \Gamma$ be the new image of the edge $e$. This concludes the definition of the drawing $\phi'_i$ of graph $G'_i$. Note that:
\[ \mathsf{cr}(\phi'_i)-\mathsf{cr}(\phi'_{i-1})\leq \sum_{e\in E(G'_{i-1})\setminus E(C_i^+)}\sum_{e'\in E(C_i^+)} \chi(e,e')\cdot \cong_{G'}({\mathcal{Q}}(C),e'). \]
Once every cluster $C\in {\mathcal{C}}$ is processed in this manner, we obtain the final drawing $\phi$ of $G_{|{\mathcal{C}}}$, by suppressing the images of the vertices of $T$ in the drawing $\phi'_r$ of the graph $G'_r$. Note that for every vertex $x\in V(G_{|{\mathcal{C}}})\cap V(G)$, we did not modify the images of the edges of $\delta_G(x)$ inside the tiny $\phi^*$-disc $D_{\phi^*}(x)$, so the order in which these edges enter the image of $x$ continues to be ${\mathcal{O}}_x$. It now remains to bound the number of crossings in the drawing $\phi$.
We only bound the number of new crossings that were added due to the transformations that we perform.
Consider some crossing $(e,e')_p$ in the drawing $\phi'$ of graph $G'$. If neither of the edges $e,e'$ lie in $\bigcup_{C\in {\mathcal{C}}}E(C^+)$, then no new crossings between these edges were introduced, and this crossing was not charged for any new crossings. Assume next that $e\in E(C_i^+)$, for some cluster $C_i\in {\mathcal{C}}$, and $e'\not\in \bigcup_{C\in {\mathcal{C}}}E(C^+)$. Then crossing $(e,e')_p$ may be responsible for up to $\cong_{G'}({\mathcal{Q}}(C_i),e)$ new crossings. Each of these new crossings is between the image of $e'$ and the images of the edges of $\delta_{G'_i}(v_{C_i})$, and so they cannot be responsible for any additional new crossings. Since $\expect{\cong_{G'}({\mathcal{Q}}(C_i),e)}\leq O((\log^4m)/\alpha)$, the total expected number of crossings for which crossing $(e,e')_p$ is responsible is at most $O((\log^4m)/\alpha)$.
Lastly, assume that $e\in E(C_i^+)$ and $e'\in E(C_j^+)$, for $C_i,C_j\in {\mathcal{C}}$. If $i=j$, then crossing $(e,e')_p$ is not responsible for any new crossings. Assume now without loss of generality that $i<j$. After cluster $C_i$ is processed, crossing $(e,e')_p$ may be responsible for at most
$\cong_{G'}({\mathcal{Q}}(C_i),e)$ new crossings.
All these new crossings are between the images of the edges of $\delta_{G'_i}(v_{C_i})$ and the image of edge $e'$. Once cluster $C_j$ is processed, each of the resulting crossings may in turn be responsible for at most $\cong_{G'}({\mathcal{Q}}(C_j),e')$ new crossings.
Each of these new crossings is between images of edges in $\delta_{G'_j}(v_{C_i})$ and images of edges in $\delta_{G'_j}(v_{C_j})$, so they in turn will not be responsible for any new crossing.
Therefore, overall, crossing
$(e,e')_p$ may be responsible for up to $\cong_{G'}({\mathcal{Q}}(C_i),e)\cdot \cong_{G'}({\mathcal{Q}}(C_j),e')$ new crossings. Since $\cong_{G'}({\mathcal{Q}}(C_i),e)$ and $ \cong_{G'}({\mathcal{Q}}(C_j),e')$ are independent random variables, and the expected value of each of these variables is at most $O((\log^4m)/\alpha)$, the expected number of crossings for which crossing $(e,e')_p$ is responsible is at most $O((\log^8m)/\alpha^2)$. We conclude that $\expect{\mathsf{cr}(\phi)}\leq O((\log^8m)/\alpha^2)\cdot \mathsf{cr}(\phi')\leq O((\log^8m)/\alpha^2)\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)$. Therefore, there exists a drawing $\phi$ of the contracted graph $G_{|{\mathcal{C}}}$, with $\mathsf{cr}(\phi)\leq O((\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8m)/\alpha^2)$, in which, for every regular vertex $x\in V(G_{|{\mathcal{C}}})\cap V(G)$, the ordering of the edges of $\delta_G(x)$ as they enter $x$ in $\phi$ is consistent with the rotation ${\mathcal{O}}_x\in \Sigma$.
\subsection{Proof of Claim~\ref{claim: embed expander}}
\label{apd: Proof of embed expander}
For convenience, we sometimes refer to vertices of $T$ as \emph{terminals}.
We use the cut-matching game of Khandekar, Rao and Vazirani~\cite{khandekar2009graph}, defined as follows. The game is played between two players, called the cut player and the matching player. The input to the game is an even integer $N$.
The game is played in iterations. We start with a graph $W$, whose vertex set $V$ has cardinality $N$, and the edge set is empty. In every iteration, some edges are added to $W$. The game ends when $W$ becomes a $\ensuremath{\frac{1}{2}}$-expander.
The goal of the cut player is to construct a $\ensuremath{\frac{1}{2}}$-expander in as few iterations as
possible, whereas the goal of the matching player is to prevent the construction of the expander for as long as possible. The iterations proceed as follows. In every iteration $j$, the cut player chooses a partition $(Z_j, Z'_j)$ of $V$ with $|Z_j| = |Z'_j|$, and the
matching player chooses a perfect matching $M_j$ that matches the nodes of $Z_j$ to the nodes of $Z'_j$. The edges of $M_j$ are then
added to $W$. Khandekar, Rao, and Vazirani~\cite{khandekar2009graph} showed that there is an efficient randomized algorithm for the cut player (that is, an algorithm that, in every iteration $j$, given the current graph $W$, computes a partition $(Z_j,Z'_j)$ of $V$ with $|Z_j| = |Z'_j|$), that guarantees that after
$O(\log^2{N})$ iterations, with high probability, graph $W$ is a $(1/2)$-expander, regardless of the specific matchings chosen by the matching player.
We use the above cut-matching game in order to compute an expander $W$ with vertex set $T$, and to embed it into $G$, using standard techniques.
If $|T|$ is an even integer, then we start with the graph $W$ containing the vertices of $T$; otherwise, we let $t\in T$ be an arbitrary vertex, and we start with $V(W)=T\setminus\set{t}$. Initially, $E(W)=\emptyset$. We then perform iterations. In the $i$th iteration, we apply the algorithm of the cut player to the current graph $W$, and obtain a partition $(Z_i,Z'_i)$ of its vertices with $|Z_j| = |Z'_j|$. Using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we compute a collection ${\mathcal{P}}_i$ of paths in graph $G$, routing vertices of $Z_i$ to vertices of $Z'_i$, so that every vertex of $Z_i\cup Z'_i$ is an endpoint of exactly one path in ${\mathcal{P}}_i$, and the paths in ${\mathcal{P}}_i$ cause congestion at most $\ceil{1/\alpha}\leq 2/\alpha$ in $G$. Let $M_i$ be the perfect matching between vertices of $Z_i$ and vertices of $Z'_i$ defined by the set ${\mathcal{P}}_i$ of paths: that is, we add to $M_i$ a pair $(t,t')$ of vertices iff some path in ${\mathcal{P}}_i$ has endpoints $t$ and $t'$. We then treat $M_i$ as the response of the matching player, and add the edges of $M_i$ to $W$, completing the current iteration of the game.
Let $W$ be the graph obtained after $i^*=O(\log^2k)$ iterations, that is guaranteed to be a $1/2$-expander with high probability. We then set ${\mathcal{P}}=\bigcup_{i=1}^{i^*}{\mathcal{P}}_i$. It is immediate to verify that ${\mathcal{P}}$ is an embedding of $W$ into $G$. Since each set ${\mathcal{P}}_i$ of paths causes congestion $O(1/\alpha)$, and $i^*\leq O(\log^2k)$, the paths in ${\mathcal{P}}$ cause congestion $O((\log^2 k)/\alpha)$. If $|T|$ is even, then we have constructed the desired expander and its embedding into $G$ as required. If $|T|$ is odd, then we add the terminal $t$ to the graph $W$. Let $P$ be any path in graph $G$, connecting $t$ to any terminal $t'\in T\setminus\set{t}$; such a path must exist since the set of terminals is $\alpha$-well-linked in $G$. We then add edge $(t,t')$ to graph $W$, and we let its embedding be $P(e)=P$; we add path $P$ to ${\mathcal{P}}$. It is easy to verify that this final graph $W$ is $1/4$ expander, provided that the original graph $W$ obtained at the end of the cut-matching game was a $1/2$-expander. We have also obtained an embedding of $W$ into $G$ with congestion $O((\log^2k)/\alpha)$.
Lastly, since the number of iterations in the cut-matching game is $O(\log^2k)$, and the set of edges that is added to $W$ in every iteration is a maching, we get that the maximum vertex degree in $W$ is $O(\log^2k)$.
\subsection{Proof of Observation~\ref{obs: cr of exp}}
\label{apd: Proof of cr of exp}
We assume that $c>2^{120}$ is a large enough constant. Let $\Delta$ denote the maximum vertex degree in $W$. Since $c$ is a large enough constant, we can assume that $\Delta\leq c^{1/8}\log^2k<k/2^{40}$. Assume for contradiction that $\mathsf{OPT}_{\mathsf{cr}}(W)<k^2/(c\log^8k)$. From \Cref{lem:min_bal_cut}, there is a $(3/4)$-edge-balanced cut $(A,B)$ in $W$, with:
\[|E_W(A,B)|\leq O(\sqrt{\mathsf{OPT}_{\mathsf{cr}}(W)+\Delta\cdot |E(W)|})\leq O(\sqrt{\mathsf{OPT}_{\mathsf{cr}}(W)+c^{1/4}\cdot k\cdot \log^4k})\leq O\textsf{left} (\frac{k}{ c^{1/4}\log^4k}\textsf{right} ).\]
(We have used the fact that, since $k>c$, and $c$ is a large enough constant, $\log^4k<\frac{k}{c^{3/4}\log^8k}$.)
Since cut $(A,B)$ is a $(3/4)$-edge-balanced cut, $|E_W(A)|\leq 3|E(W)|/4$. Therefore, $|E_W(B)|\geq |E(W)|/4-|E_W(A,B)|\geq |E(W)|/8$. Since the degree of every vertex in $W$ is at most $\Delta\leq c^{1/8}\log^2k$, we get that $|B|\geq \frac{|E(W)|}{8\Delta}\geq \frac{|E(W)|}{8c^{1/8}\log^2k}$. Using the same reasoning, $|A|\geq \frac{|E(W)|}{8c^{1/8}\log^2k}$. Since graph $W$ is a $\frac 1 4$-expander, $|E_W(A,B)|\geq \frac 1 4\cdot \min\set{|A|,|B|}\geq \frac 1 4 \cdot \frac{|E(W)|}{8c^{1/8}\log^2k}>\frac{k}{32c^{1/8}\log^2k}$ must hold, a contradiction.
\subsection{Proof of Corollary~\ref{cor: routing well linked vertex set}}
\label{apd: Proof of routing well linked vertex set}
The proof relies on known results for routing on expanders, that are summarized in the next claim, that is well-known, and follows immediately from the results of \cite{leighton1999multicommodity}. A proof can be found, e.g. in \cite{chuzhoy2012routing}.
\begin{claim}[Corollary C.2 in \cite{chuzhoy2012routing}]\label{claim: routing on expander}
There is an efficient randomized algorithm that, given as input an $n$-vertex $\alpha$-expander $H$, and any partial matching $M$ over the vertices of $H$, computes, for every pair $(u,v)\in M$, a path $P(u,v)$ connecting $u$ to $v$ in $H$, such that with high probability, the set $\set{P(u,v)\mid (u,v)\in M}$ of paths causes congestion $O(\log^2 n/\alpha)$ in $H$.
\end{claim}
We start by computing a graph $W$ with $V(W)=T$, and its embedding ${\mathcal{P}}=\set{P(e)\mid e\in E(W)}$ into $G$ with congestion $O((\log^2k)/\alpha)$ using the algorithm from \Cref{claim: embed expander} (recall that, with high probability, $W$ is an $(1/4)$-expander). Next,
we use the algorithm from \Cref{claim: routing on expander} to compute a collection ${\mathcal{R}}'=\set{R'(u,v)\mid (u,v)\in M}$ of paths in graph $W$, where for all $(u,v)\in M$, path $R(u,v)$ connects $u$ to $v$ in $W$, such that the congestion of the set ${\mathcal{R}}'$ of paths in $W$ is $O(\log^2k)$ with high probability.
Lastly, we consider the paths $R'(u,v)\in {\mathcal{R}}'$ one by one. We transform each such path $R'(u,v)$ into a path $R(u,v)$ connecting $u$ to $v$ in graph $G$ by replacing, for every edge $e\in R'(u,v)$, the edge $e$ with the path $P(e)\in {\mathcal{P}}$ embedding the edge $e$ into $G$. Since the paths in ${\mathcal{R}}'$ with high probability cause congestion at most $O(\log^2k)$ in $W$, while the paths in ${\mathcal{P}}$ cause congestion $O(\log^2k/\alpha)$ in $G$, we get that with high probability, the paths in the resulting set ${\mathcal{R}}=\set{R(u,v)\mid (u,v)\in M}$ cause congestion $O(\log^4k/\alpha)$ in $G$.
\subsection{Proof of Corollary~\ref{cor: embed complete graph}}
\label{apd: Proof of embed complete graph}
We partition the set $E(K)$ of edges into $3z$ matchings $M_1,\ldots,M_{3z}$, and then use \Cref{cor: routing well linked vertex set} to compute, for each $1\leq i\leq 3z$, a set $\tilde {\mathcal{R}}_i=\set{\tilde P(e)\mid e\in M_i}$ of paths in graph $G$, where for all $e=(t,t')\in M_i$, path $\tilde P(e)$ connects $t$ to $t'$, and with high probability, the paths in $\tilde {\mathcal{P}}_i$ cause edge-congestion $O((\log^4z)/\alpha)$ in graph $G$. Let $\tilde {\mathcal{P}}=\bigcup_{i=1}^{3z}\tilde {\mathcal{P}}_i$. Then $\tilde {\mathcal{P}}$ is an embedding of $K_z$ into $G$, and with high probability, the congestion of this embedding is $O((z\log^4z)/\alpha)$.
\subsection{Proof of \Cref{lem: alg procsplit'}}
\label{subsec: proof of procsplit'}
In order to simplify the notation, we denote instance $I_F=(G_F,\Sigma_F)$ by $I=(G,\Sigma)$, and we denote $|E(G)|$ by $m$. Since instance $I$ is not acceptable, face $F$ may not be a forbidden face of $\tilde {\mathcal{F}}(W)$. Recall that we have computed a set ${\mathcal{S}}(F)\subseteq{\mathcal{C}}(W)$ of connected components, and, for each such component $C\in {\mathcal{S}}(F)$, a subgraph $J_{C,F}$ that is a core. Intuitively, vertices and edges of $J_{C,F}$ must serve as the boundary of face $F$ in any ${\mathcal{W}}$-compatible drawing of $W$, and in particular their images lie on the boundary of the region $F$ in the drawing $\phi$ of graph $\check G'$, given by Property \ref{prop: drawing} of valid input. We have also defined, for each component $C\in {\mathcal{S}}(F)$, a core structure ${\mathcal{J}}_{C,F}$, whose corresponding core is $J_{C,F}$, and we have defined a collection ${\mathcal K}_{ F}=\set{{\mathcal{J}}_{C, F}\mid C\in {\mathcal{S}}( F)}$ of core structures associated with $F$. From the definition of a ${\mathcal{W}}$-decomposition, for all $C\in {\mathcal{S}}(F)$, ${\mathcal{J}}_{C,F}$ is a valid core structure for $I$, and so ${\mathcal K}_F$ is a valid skeleton structure for instance $I$. In order to simplify the notation, we denote ${\mathcal K}_F=\set{{\mathcal{J}}_{C,F}\mid C\in {\mathcal{S}}(F)}$ by ${\mathcal K}=\set{{\mathcal{J}}_1,\ldots,{\mathcal{J}}_z}$, and for all $1\leq i\leq z$, we denote the core associated with the cores structure ${\mathcal{J}}_i$ by $J_i$. Since, from \Cref{lem: compute phase 2 decomposition}, $|\check{\mathcal{K}}|\leq \mu^{100}$, from the definition of a skeleton augmentation, $|{\mathcal{C}}(W)|\leq \mu^{100}$, and so $z\leq \mu^{100}$ holds. We also denote by $K=\bigcup_{i=1}^zJ_i$ the skeleton associated with the skeleton structure ${\mathcal K}$.
As before, we denote $E^{\mathsf{del}}=E(\check G')\setminus\textsf{left} (\bigcup_{G_{F'}\in {\mathcal{G}}}E(G_{F'})\textsf{right})$, $\tilde G=\check G'\setminus E^{\mathsf{del}}$, and let $\tilde \Sigma$ be the rotation system for graph $\tilde G$ induced by $\check \Sigma'$. Let $\tilde I=(\tilde G,\tilde \Sigma)$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Recall that Property \ref{prop: drawing} of valid input to Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$ ensures that there is a solution $\phi$ to instance $\tilde I$, that is ${\mathcal{W}}$-compatible. Additionally, $\mathsf{cr}(\phi)\leq (\check m')^2/\mu^{60b}$, $|\chi^{\mathsf{dirty}}(\phi)|\leq \check m'/\mu^{60b}$, the total number of crossings of $\phi$ in which the edges of $E(W)\setminus E(\check K)$ participate is at most $\mathsf{cr}(\phi)\mu^{26b}/\check m'$.
As in the original procedure \ensuremath{\mathsf{ProcSplit}}\xspace, the algorithm consists of two steps. In the first step, we compute a promising path set and an enhancement of the skeleton ${\mathcal K}$. In the second step we complete the construction of the new skeleton augmenting structure ${\mathcal{W}}'$ and the new ${\mathcal{W}}'$-decompositon ${\mathcal{G}}'$ of graph $\check G'$.
\subsubsection{Step 1: Computing Skeleton Enhancement}
As before, we start by computing a promising path set, that will then be used in order to compute an enhancement of skeleton ${\mathcal K}$. In the algorithm for $\ensuremath{\mathsf{ProcSplit}}\xspace$, the construction of the promising path set exploited the fact that the input instance $I$ was wide. Here, instead, we exploit the fact that instaince $I$ is not acceptable. The resulting promising path set will be somewhat smaller, and it will only be of type 2 or type 3.
From the definition of aceptable subinstances, there must be an index $1\leq i^*\leq z$, and a partition $(E_1,E_2)$ of the edges of $\delta_G(J_{i^*})$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}(J_{i^*})$, and the following holds. Let $(\tilde E',\tilde E'')$ be the partitition of the edges in set $\tilde E=\bigcup_{i=1}^z\delta_G(J_i)$, where $\tilde E'=E_1$, and $\tilde E''=\tilde E\setminus \tilde E'$. As before, let
$\tilde H$ be the graph that is obtained from graph $G$ by, first subdividing every edge $e\in \tilde E$ with a vertex $t_e$, and then deleting all vertices and edges of $K$ from it. Then the minimum cut in graph $\tilde H$ separating the vertices of $\set{t_e\mid e\in \tilde E'}$ from the vertices of $\set{t_e\mid e\in \tilde E''}$ must contain at least $\check m'/\mu^{2b}$ edges. Equivalently, there is a set ${\mathcal{P}}$ of $\ceil{\check m'/\mu^{2b}}$ edge-disjoint paths in graph $\tilde H$, connecting vertices of
$\set{t_e\mid e\in \tilde E'}$ to vertices of $\set{t_e\mid e\in \tilde E''}$. We can assume w.l.o.g. that the paths of ${\mathcal{P}}$ do not contain the vertices of $\set{t_e\mid e\in \tilde E}$ as inner vertices.
Set ${\mathcal{P}}$ of paths in graph $\tilde H$ naturally defines a collection of at least $\ceil{\check m'/\mu^{2b}}$ edge-disjoint paths in graph $G$, where every path has an edge of $\tilde E'=E_1$ as its first edge, and an edge of $\tilde E''$ as its last edge, and it does not contain any vertices of $K$ as inner vertices. Abusing the notation, we denote this path set by ${\mathcal{P}}$ as well. We partition path set ${\mathcal{P}}$ into two $z$ subsets ${\mathcal{P}}^1,\ldots,{\mathcal{P}}^z$, as follows: consider any path $P\in {\mathcal{P}}$. Recall that the last edge of $P$ lies in $\tilde E=\bigcup_{i=1}^z\delta_G(J_i)$. We add path $P$ to set ${\mathcal{P}}^i$ if the last edge of $P$ lies in $\delta_G(Z_i)$. Clearly, there must be an index $1\leq i\leq z$, such that $|{\mathcal{P}}^i|\geq \frac{|{\mathcal{P}}|}z \geq \frac{\check m'}{\mu^{2b}\cdot z}\geq \frac{\check m'}{\mu^{2b+100}}\geq \frac{m}{\mu^{2b+100}}$. We then let ${\mathcal{P}}'$ be that path set ${\mathcal{P}}_i$. Observe that, if $i=i^*$, then ${\mathcal{P}}'$ is a promising path set of type $2$, and every path in ${\mathcal{P}}'$ has an edge of $E_1$ as its first edge, and an edge of $E_2$ as its last edge. Otherwise, ${\mathcal{P}}'$ is a promising path set of type $3$. We let $i^{**}$ be the index for which the last edge of every path in ${\mathcal{P}}'$ lies in $\delta_G(J_{i^{**}})$ (so it is possible that $i^{**}=i^*$).
Note that the existence of the path set ${\mathcal{P}}'$ implies that $m\geq \check m'/\mu^{2b}$ holds. Let ${\mathcal{P}}''\subseteq {\mathcal{P}}'$ be an arbitrary subset of $\ceil{\frac{4m}{\mu^{3b}}}$ paths. We construct an enhancement $\Pi=(P_1,P_2)$ from the path set ${\mathcal{P}}''$ exactly like in the algorithm for \ensuremath{\mathsf{ProcSplit}}\xspace. Recall that the algorithm first constructs a collection ${\mathcal{P}}^*$ of edge-disjoint non-transversal paths using the paths in ${\mathcal{P}}''$, and then selects one or two such paths into the enhancement. The only difference is that the parameter $k'=|{\mathcal{P}}^*|$ is now somewhat smaller, $k'=\ceil{\frac{4m}{\mu^{3b}}}$, and so the probability that a path $P\in {\mathcal{P}}^*$ is selected into the enhancement is now bounded by $2\mu^{3b}/m\leq 2\mu^{5b}/\check m'$ (since, as observed above, $m\geq \check m'/\mu^{2b}$). Another difference is that, since the promising path set ${\mathcal{P}}''$ may only be of type 2 or 3, the resulting enhancement is also guaranteed to be of type 2 or 3. We denote by $P^*_1$ and $P^*_2$ the paths of ${\mathcal{P}}^*$ that were chosen into the enhancement (where it is possible that $P^*_2=P^*_1$), and we denote the enhancement itself by $\Pi=\set{P_1,P_2}$.
Next, we define the bad events and bound their probabilities. This part is almost identical to that in Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. The main differences are that (i) the parameter $k'$ is now smaller by factor $\mu^{\Theta(1)}$, so various parameters need to be scaled by this factor; (ii) the required failure probability is smaller than that from Procedure \ensuremath{\mathsf{ProcSplit}}\xspace by factor $\mu^{\Theta(1)}$; and (iii) the drawing of $I$ induced by the drawing $\phi$ of $\tilde I$ is no longer guaranteed to be semi-clean with respect to $K$ (instead we will exploit the fact that the edges of $W$ participate in relatively few crossings in $\phi$).
\paragraph{Good Paths and Bad Event ${\cal{E}}_1'$.}
We use the following modified definition of good paths.
\begin{definition}[Good path]
We say that a path $P\in {\mathcal{P}}^*$ is \emph{good} if the following happen:
\begin{itemize}
\item the number of crossings in which the edges of $P$ participate in $\phi$ is at most $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$; and
\item there is no crossing in $\phi$ between an edge of $P$ and an edge of $W$.
\end{itemize} A path that is not good is called a \emph{bad path}.
\end{definition}
We now bound the number of bad paths in ${\mathcal{P}}^*$.
\begin{observation}\label{obs: number of bad paths2}
The number of bad paths in ${\mathcal{P}}^*$ is at most $\frac{10m}{\mu^{15b}}$.
\end{observation}
\begin{proof}
Since the paths in ${\mathcal{P}}^*$ are edge-disjoint, and every crossing involves two edges, the number of paths $P\in {\mathcal{P}}^*$ such that there are more than $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ crossings in $\phi$ in which the edges of $P$ participate, is at most $\frac{8m}{\mu^{15b}}$. Additionally, we are guaranteed that $|\chi^{\mathsf{dirty}}(\phi)|\le \frac{\check m'}{\mu^{60b}}\leq \frac m{\mu^{58b}}$ (since $m\geq \frac{\check m'}{\mu^{2b}}$). Therefore, the number of paths $P\in {\mathcal{P}}^*$, for which there is a crossing between an edge of $P$ and an edge of $\check K$ is bounded by $\frac m {\mu^{58b}}$.
Lastly, we are guaranteed that the total number of crossings in which the edges of $E(W)\setminus E(\check K)$ participate is at most $\frac{\mathsf{cr}(\phi)\mu^{26b}}{\check m'}\leq \frac{\check m'}{\mu^{17b}}\leq \frac{m}{\mu^{15b}}$, since $\mathsf{cr}(\phi)\leq \frac{(\check m')^2}{\mu^{60b}}$.
Overall, the number of bad paths in ${\mathcal{P}}^*$ is bounded by $\frac{10m}{\mu^{15b}}$.
\end{proof}
We say that bad event ${\cal{E}}_1$ happens if at least one of the paths $P_1^*,P_2^*$ that was chosen from ${\mathcal{P}}^*$ in order to construct the enhancement is bad. Since the number of bad paths is bounded by $\frac{4m}{\mu^{15b}}$, and a path of ${\mathcal{P}}^*$ is chosen to the enhancement with probability at most $\frac{2\mu^{3b}}{m}$, we immediately get the following observation.
\begin{claim}\label{claim: event 1 prob2}
$\prob{{\cal{E}}_1'}\leq 8/\mu^{12b}$.
\end{claim}
\paragraph{Heavy and Light Vertices, and Bad Event ${\cal{E}}_2'$.}
We use a parameter $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$. We say that a vertex $x\in V(G)$ is \emph{heavy} if at least $h$ paths of ${\mathcal{P}}^*$ contain $x$; otherwise, we say that $x$ is \emph{light}.
As before, we denote by $E'$ the set of all edges $e\in E(G)$, such that $e$ is incident to some light vertex $x$ that lies in $V(P^*_1)\cup V(P^*_2)$, and $e\not\in E(P^*_1)\cup E(P^*_2)$. We say that bad event ${\cal{E}}_2'$ happens if $|E'|>\frac{\mathsf{cr}(\phi)\mu^{48b}}{m}$.
We now bound the probability of bad event ${\cal{E}}_2'$.
\begin{claim}\label{claim: second bad event bound2}
$\prob{{\cal{E}}_2'}\leq 1/\mu^{12}$.
\end{claim}
\begin{proof}
Consider some light vertex $x\in V(G)$. Since $x$ lies on fewer than $h$ paths of ${\mathcal{P}}^*$, and each such path is chosen to the enhancement with probability at most $2\mu^{3b}/m$, the probability that $x$ lies in $V(P_1^*)\cup V(P^*_2)$ is bounded by $\frac{2h\mu^{3b}}{m}$.
Consider now some edge $e=(x,y)\in E(G)$. Edge $e$ may lie in $E'$ only if $x$ is a light vertex lying in $V(P_1^*)\cup V(P^*_2)$, or the same is true for $y$. Therefore, the probability that $e\in E'$ is at most $ \frac{4h\mu^{3b}}{m} $, and $\expect{|E'|}\leq 4h\mu^{3b}\leq \frac{4\mathsf{cr}(\phi)\mu^{35b}}{m}$, since $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$. From Markov's inequality, $\prob{|E'|>\frac{\mathsf{cr}(\phi)\mu^{48b}}{m}}\leq \frac{1}{\mu^{12b}}$.
\end{proof}
\paragraph{Unlucky Paths and Bad Event ${\cal{E}}_3'$.}
The definition of unlucky paths is identical to that in the proof of \Cref{thm: procsplit}, except that we use a slightly different parameter.
\begin{definition}[Unlucky Paths]
Let $x\in V(G)\setminus V(K)$ be a vertex, and let $P\in {\mathcal{P}}^*$ be a good path that contains $x$. Let $e,e'$ be the two edges of $P$ that are incident to $x$. Let $\hat E_1(x)\subseteq \delta_G(x)$ be the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies between $e$ and $e'$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$. Let $\hat E_2(x)\subseteq \delta_G(x)$ be the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies between $e'$ and $e$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$.
We say that path $P$ is \emph{unlucky with respect to vertex $x$} if either $|\hat E_1(x)|<\frac{\mathsf{cr}(\phi)\mu^{15b}}{m}$ or $|\hat E_2(x)|< \frac{\mathsf{cr}(\phi)\mu^{15b}}{m}$ holds.
We say that a path $P\in {\mathcal{P}}^*$ is an \emph{unlucky path} if there is at least one heavy vertex $x\in V(G)\setminus V(K)$, such that $P$ is unlucky with respect to $x$.
\end{definition}
We use the following claim in order to bound the number of unlucky paths. The claim
is an analogue of \Cref{claim: bound unlucky paths}, and its proof is almost identical, with minor changes due to difference in parameters. For completeness, we include a proof in Section \ref{sec: bound unlucky paths2} of Appendix.
\begin{claim}\label{claim: bound unlucky paths2}
Assume that Event ${\cal{E}}_1'$ did not happen. Then for every vertex $x\in V(G)\setminus V(K)$, the total number of good paths in ${\mathcal{P}}^*$ that are unlucky with respect to $x$ is at most $\frac{2^{14}\mathsf{cr}(\phi)\mu^{15b}}{m}$.
\end{claim}
We say that bad event ${\cal{E}}'_3$ happens if at least one of $P_1^*$, $P_2^*$ is an unlucky path.
\begin{claim}\label{claim: third event bound2}
$\prob{{\cal{E}}'_3}\leq 9/\mu^{12b}$.
\end{claim}
\begin{proof}
Clearly, $\prob{{\cal{E}}'_3}\leq \prob{{\cal{E}}'_1}+\prob{{\cal{E}}'_3\mid \neg{\cal{E}}'_1}$. From \Cref{claim: event 1 prob2},
$\prob{{\cal{E}}'_1}\leq 8/\mu^{12b}$. We now bound $\prob{{\cal{E}}_3'\mid \neg{\cal{E}}_1'}$.
Recall that a heavy vertex must have degree at least $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$ in $G$. Therefore, the total number of heavy vertices in $G$ is at most $\frac{2m}{h}\leq \frac{2m^2}{ \mathsf{cr}(\phi)\mu^{31b}}$. From \Cref{claim: bound unlucky paths2}, if Event ${\cal{E}}'_1$ did not happen, then for every heavy vertex $x\in V(G)\setminus V(K)$, there are at most $\frac{2^{14}\mu^{15b}\mathsf{cr}(\phi)}{m}$ paths in ${\mathcal{P}}^*$ that are good and unlucky for $x$. Therefore, the total number of good paths in ${\mathcal{P}}^*$ that are unlucky with respect to some heavy vertex is at most:
%
\[\frac{2m^2}{ \mathsf{cr}(\phi)\mu^{31b}}\cdot \frac{2^{14}\mu^{15b}\mathsf{cr}(\phi)}{m}
\leq \frac{2^{15}\cdot m}{\mu^{16b}}.
\]
%
Since $|{\mathcal{P}}^*|\geq \frac{4m}{\mu^{3b}}$, from \Cref{obs: number of bad paths2}, at least $ \frac{2m}{\mu^{3b}}$ paths in ${\mathcal{P}}^*$ are good.
We select two paths into the enhancement $\Pi$, and, if Event ${\cal{E}}_1'$ did not happen, every good path is equally likely to be selected. Therefore, the probability that a good path that is unlucky is selected into the enhancement, conditioned on the event ${\cal{E}}_1$ not happening is bounded by:
%
\[\frac{2^{15}\cdot m}{\mu^{16b}}\cdot \frac{\mu^{3b}}{m}\leq \frac{2^{15}}{\mu^{13b}}\leq \frac{1}{\mu^{12b}}, \]
%
since $\mu$ is large enough. We conclude that $\prob{{\cal{E}}'_3\mid \neg{\cal{E}}'_1}\leq 1/\mu^{12b}$, and $\prob{{\cal{E}}'_3}\leq 9/\mu^{12b}$.
\end{proof}
Recall that we have denoted by $E'$ the set of all edges that are incident to the light vertices of $P^*_1\cup P^*_2$, excluding the edges of $E(P^*_1)\cup E(P^*_2)$.
Denote $\tilde G'=\tilde G\setminus E'$, and let $\tilde \Sigma'$ be the rotation system for graph $G'$ induced by $\tilde \Sigma$. For convenience, we also denote $G'=G\setminus E'$ and we let $\Sigma'$ be the rotation system for graph $G'$ induced by $\tilde \Sigma$. Denote $\tilde I'=(\tilde G',\tilde \Sigma')$, and $I'-(G',\Sigma')$. Let $\Pi=\set{P_1,P_2}$ be the enhancement of skeleton $K$ that we have constructed. We denote $K'=K\cup P_1\cup P_2$, and we let $W'=W\cup P_1\cup P_2$ be the new augmentation of the original skeleton $\check K$ for graph $\check G'$. Drawing $\phi$ of graph $\tilde G=\check G'\setminus E^{\mathsf{del}}$ naturally defines a drawing of graph $\tilde G'$. Moreover, if Event ${\cal{E}}'_1$ did not happen, then there are no crossings in this drawing between the edges of $E(P_1)\cup E(P_2)$ and the edges of $E(W)$. However, it is possible that this drawing contains crossings between pairs of edges in $E(P_1)\cup E(P_2)$. In the next claim we show that, if events ${\cal{E}}'_1$ and ${\cal{E}}'_3$ did not happen, then drawing $\phi$ can be modified to obtain a drawing $\phi'$ of $\tilde G'$, in which the edges of $W'$ do not cross each other, and the induced drawing of $W$ remains identical to that induced by $\phi$.
We do so in the following claim, that is an analogue of \Cref{claim: new drawing}.
\begin{claim}\label{claim: new drawing2}
Assume that neither of the events ${\cal{E}}'_1$ and ${\cal{E}}'_3$ happened. Then there is a solution $\phi'$ to instance $\tilde I'=(\tilde G',\tilde \Sigma')$, in which there are no crossings between pairs of edges in $E(W')$. Moreover, the drawing of $W$ induced by $\phi'$ is identical to the drawing of $W$ induced by $\phi$, drawing $\phi'$ is ${\mathcal{W}}$-compatible, and, for every edge $e\in E(\tilde G')$, the number of crossings in which $e$ participates in $\phi'$ is bounded by the number of crossings in which $e$ participates in $\phi$. In particular,
$\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$ and $\chi^{\mathsf{dirty}}(\phi')\subseteq \chi^{\mathsf{dirty}}(\phi)$ hold.
\end{claim}
The proof of \Cref{claim: new drawing2} is essentially identical to the proof of \Cref{claim: new drawing}. Recall that the main idea in the proof is to remove, one by one, the loops in the images of the paths $P_1,P_2$ in the drawing $\phi$. The reason these loops can be removed without increasing the number of crossings is that every vertex whose image lies on such a loop must be a light vertex (see, e.g., \Cref{obs: no heavy vertices on loops}). This, in turn, relies on the fact that the parameter used in the definition of unlucky paths is greater than the parameter used in the definition of bad paths. While both parameters are now scaled by factor $\mu^{\Theta(b)}$, this relationship between them continues to hold, and so the analogue of \Cref{obs: no heavy vertices on loops} continues to hold as well. We also need to show that the analogue of \Cref{obs: case 3 not crossing} continues to hold, that is, there are no crossings $(e,e')_p$ in $\phi$ where $e\in E(P_1)$ and $e'\in E(P_2)$. The proof of the observation relies on the fact that that the parameter bounding the number of crossings in which the edges of a good path may participate (denoted by $N$), is less than $k'/100$, and that the number of bad paths in ${\mathcal{P}}^*$ is bounded by $k'/16$. While both parameters $N$ and $k'$ are now different, namely $N=\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ and $k'=4m/\mu^{3b}$, it is still the case that $N<k'/100$ (since, from Condition \ref{prop: drawing}, $\mathsf{cr}(\phi)\leq (\check m')^2/\mu^{60b}$, and $m\geq \check m'/\mu^{2b}$). The number of bad paths is bounded by $\frac{10m}{\mu^{15b}}<k'/16$ from \Cref{obs: number of bad paths2}. The remainder of the proof of \Cref{claim: new drawing2} is identical to the proof of \Cref{claim: new drawing} and is omitted here.
As before, drawing $\phi'$ of $\tilde G'$ is derived from drawing $\phi$ of $\tilde G$, and neither are known to our algorithm. Note that, since ${\mathcal{J}}_{i^*}$ is a valid core structure for instance $I$, the image of the paths $P_1,P_2$ of the enhancement in $\phi'$ must intersect the region $F$. If Event ${\cal{E}}_1'$ does not happen, then the images of paths $P_1,P_2$ do not cross the images of the edges of $W$ in $\phi$, and so the images of both paths are contained in region $F$. This remains true for drawing $\phi'$ of graph $\tilde G'$. Therefore, in the drawing of the new augmented skeleton $W'$ induced by $\phi'$, the images of the edges of $W'$ do not cross each other, and the images of paths $P_1,P_2$ are contained in region $F$. It is easy to verify that the images of the two paths partition face $F$ into two new regions, that we denote by $F_1$ and $F_2$.
\paragraph{Terrible Vertices and Bad Event ${\cal{E}}'_4$.}
As before, for a vertex $x\in V(G)$, we denote by $N(x)$ the number of paths in ${\mathcal{P}}^*$ containing $x$, by $N^{\operatorname{bad}}(x)$ the number of bad paths in ${\mathcal{P}}^*$ containing $x$, and by $N^{\operatorname{good}}(x)$ the number of good paths in ${\mathcal{P}}^*$ containing $x$. The definition of the notion of a terrible vertex remains the same as before:
\begin{definition}[Terrible Vertex]
A vertex $x\in V(G)$ is \emph{terrible} if it is a heavy vertex, and $N^{\operatorname{bad}}(x)\geq N^{\operatorname{good}}(x)/64$.
\end{definition}
As before, we say that a bad event ${\cal{E}}'_4$ happens if any vertex of $P^*_1\cup P^*_2$ is a terrible vertex. We bound the probability of Event ${\cal{E}}'_4$ in the following claim, whose proof is essentially identical to the proof of \Cref{claim: no terrible vertices}. Since we use slightly different parameters, we provide the proof here for completeness.
\begin{claim}\label{claim: no terrible vertices2}
$\prob{{\cal{E}}_4'}\leq \frac{2^{14}\log m}{\mu^{12b}}$.
\end{claim}
\begin{proof}
Let $U$ be the set of all heavy vertices of $G$. We group the vertices $x\in U$ geometrically into classes, using the parameter $N(x)$. Recall that $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$, and, for every heavy vertex $x\in U$, $N(x)\geq h$. Let $q=2\ceil{\log m}$. For $1\leq i\leq q$, we let the class $S_i$ contain all vertices $x\in U$ with $2^{i-1}\cdot h\leq N(x)<2^i\cdot h$.
Let $S'_i\subseteq S_i$ be the set containing all terrible vertices of $S_i$. For every vertex $x\in S'_i$, $N^{\operatorname{bad}}(x)\geq N^{\operatorname{good}}(x)/64$. Since $N(x)=N^{\operatorname{bad}}(x)+N^{\operatorname{good}}(x)$, we get that $N^{\operatorname{bad}}(x)\geq N(x)/65\geq 2^{i-1}\cdot h/65$. From \Cref{obs: number of bad paths2}, the total number of bad paths is at most $\frac{10m}{\mu^{15b}}$, so
%
\[|S'_i|\leq \frac{10m/\mu^{15b}}{2^{i-1}\cdot h/65}\leq \frac{650m}{ 2^{i-1}\cdot h\cdot \mu^{15b}}.\]
Recall that the probability that a path $P\in {\mathcal{P}}^*$ is selected into the enhancement is at most $\frac{2\mu^{3b}}{m}$. A vertex $x\in S'_i$ may lie in $V(P^*_1)\cup V(P^*_2)$ only if at least one of the $N(x)$ paths of ${\mathcal{P}}^*$ containing $x$ is selected into the enhancement. Therefore, $\prob{x\in (V(P^*_1)\cup V(P^*_2))}\leq \frac{2\mu^{3b}\cdot N(x)}{m}\leq \frac{2^{i+1}\cdot h\cdot \mu^{3b}}{m}$.
Overall, the probability that some vertex of $S'_i$ belongs to $V(P^*_1)\cup V(P^*_2)$ is at most
$$|S'_i|\cdot \frac{2^{i+1}\cdot h\cdot \mu^{3b}}{m}\leq \frac{650m}{ 2^{i-1}\cdot h\cdot \mu^{15b}}\cdot \frac{2^{i+1}\cdot h\cdot \mu^{3b}}{m}\leq \frac{2^{12}}{\mu^{12b}}.$$
Using the union bound over all $q=2\ceil{\log m}$ classes, we get that the probability of Event ${\cal{E}}'_4$ is bounded by $\frac{2^{14}\log m}{\mu^{12b}}$.
\end{proof}
\paragraph{Bad Event ${\cal{E}}'$.}
Let ${\cal{E}}'$ be the bad event that at least one of the events ${\cal{E}}'_1,{\cal{E}}'_2,{\cal{E}}'_3,{\cal{E}}'_4$ happen. From the Union Bound and Claims \ref{claim: event 1 prob2}, \ref{claim: second bad event bound2}, \ref{claim: third event bound2} and \ref{claim: no terrible vertices2}, $\prob{{\cal{E}}'}\leq O\textsf{left}( \frac{\log m}{\mu^{12b}}\textsf{right} ) \leq \frac{1}{\mu^{11b}}$, by definition of $\mu$.
\subsubsection{Step 2: Completing the Construction of ${\mathcal{W}}'$ and ${\mathcal{G}}'$}
In this step we complete the construction of the skeleton augmenting structure ${\mathcal{W}}'$ and of the ${\mathcal{W}}'$-decomposition of $\check G'$ ${\mathcal{G}}'$. This step is very similar to the second step of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace, with several key differences. The first difference is that we no longer need to take care of the case where $\Pi$ is a type-1 enhancement. The second difference is that the parameters are now somewhat different. The third difference, which is more major than the first two, is that we will not be able to guarantee that the final drawing of the graph $\tilde G\setminus E^{\mathsf{del}}(F)$ that we obtain, induces drawings of graphs $G_{F_1},G_{F_2}$ that are semi-clean with respect to their skeletons. We will still, however, need to ensure that the drawing is ${\mathcal{W}}'$-compatible, so every core of $\set{J_1,\ldots,J_z}$ is drawn in the ``correct'' face (for example, if $J_i\subseteq G_{F_1}$, then core $J_i$ should be drawn inside). This will require an additional step, in which we disconnect some of the cores from the corresponding graph $G_1$ or $G_2$.
We consider the instance $I'=(G',\Sigma')$, where $G'=\tilde G\setminus E'$, and $\Sigma'$ is the rotation system for $G'$ induced by $\check \Sigma'$. We will also consider the solution $\phi'$ to instance $I'$ given by \Cref{claim: new drawing} (assuming that bad event ${\cal{E}}'$ did not happen).
We start by computing an orientation $b'_u$ for every vertex $u\in V(W')$, and a drawing $\rho_{C'}$ of every connected component $C'\in {\mathcal{C}}(W')$. We then split graph $G$ into two subgraphs $G_1$ and $G_2$, and complete the construction of the skeleton augmenting structure. Lastly, we will remove some additional edges from the resulting graph, to ensure that there is a good drawing of that graph that is ${\mathcal{W}}'$-compatible.
We consider two cases, depending on whether the original promising path ${\mathcal{P}}$ was of type $2$ or $3$.
\subsubsection*{Case 1: ${\mathcal{P}}$ was a type-2 promising path set}
The first case happens when the promising path set ${\mathcal{P}}$ was of type-2. Recall that the endpoints of all paths in ${\mathcal{P}}$ are contained in $V(J)$, and we have computed a partition $(E_1,E_2)$ of the edges of $\delta_G(J)$, so the edges of $E_1$ are consecutive in the ordering ${\mathcal{O}}(J)$. Recall that, in this case, $\Pi=\set{P_1,P_2}$, where $P_1$ is either a path or a cycle that contains an edge of $E_1$ as its first edge, and an edge of $E_2$ as its second edge, and $P_2=\emptyset$. For convenience of notation, we denote $P_1$ by $P$.
Recall that, from \Cref{claim: new drawing2}, if Event ${\cal{E}}'$ does not happen, then drawing $\phi'$ of $G'$ contains no crossings between the edges of $E(W')$.
Throughout, we denote by $\gamma$ the image of the path $P$ in $\phi'$.
In order to construct the skeleton augmenting structure ${\mathcal{W}}'$, we first let $W'=W\cup P$ be the skeleton augmentation associated with ${\mathcal{W}}'$. Next, we define a drawing $\rho_C$ for every component $C\in {\mathcal{C}}(W)$.
Let $C^*\in {\mathcal{C}}(W)$ be the connected component with $J\subseteq C^*$, and denote by $\tilde C=C^*\cup P$. Since path $P$ must be internally disjoint from all vertices of $W$, it is immediate to verify that ${\mathcal{C}}(W')=({\mathcal{C}}(W)\setminus C^*)\cup \tilde C)$. For every component $C\in {\mathcal{C}}(W')\setminus \set{\tilde C}$, its drawing $\rho_C$ in ${\mathcal{W}}'$ remains the same as in ${\mathcal{W}}$. Consider now the component $\tilde C$. Recall that we are given, as part of skeleton augmenting structure ${\mathcal{W}}$ a drawing $\rho_{C^*}$ of graph $C^*$ and an orientation $b'_u$ of every vertex $u\in V(C^*)$. We are also given a face $F'\in {\mathcal{F}}(\rho_{C^*})$, such that the face $F$ in drawing $\phi$ is contained in the region $F'$. Since drawing $\phi'$ is ${\mathcal{W}}$-compatible, and it does not contain any crossings between the edges of $W'$, we are guaranteed that path $P$ is drawn inside the region $F'$ of $\phi'$, in a natural way. Since the orientations of all vertices of $C^*$ in $\phi'$ are consistent with those given by ${\mathcal{W}}$, and since the drawing of $C^*$ induced by $\phi'$ is $\rho_{C^*}$, we can efficiently compute a drawing $\rho_{\tilde C}$ of graph $\tilde C$, that is identical to the drawing of $\tilde C$ induced by $\phi'$. This drawing of $\tilde C$ is a unique planar drawing of the graph that has the following properties:
\begin{itemize}
\item the drawing of $C^*$ induced by $\rho_{\tilde C}$ is $\rho_{C^*}$;
\item the image of path $P$ is contained in region $F'$;
\item the drawing is consistent with the rotation system $\check \Sigma'$; and
\item the orientation of every vertex $u\in V(C^*)$ in the drawing is the orientation $b'_u$ given by ${\mathcal{W}}'$.
\end{itemize}
If $F=F^*(\rho_{C^*})$, then we will designate one of the face $F_1$ or $F_2$ as the outer face of the drawing $\rho_{\tilde C}$ later.
So far we have defined the new skeleton augmentation $W'$ and a drawing $\rho_C$ of every connected component $C\in {\mathcal{C}}(W)$. Next, we define the orientation $b'_u$ of every vertex $u\in V(W')$. For vertices $u\in V(W')\cap V(W)$, the orientation $b'_u$ in ${\mathcal{W}}'$ remains the same as the orientation $b'_u$ in ${\mathcal{W}}$. It now remains to define the orientation $b'_u$ of vertices $u\in V(P)\setminus V(C^*)$. The procedure for computing these orienttions is very similar to the one from Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace$.
We first recall the definition of the circular ordering ${\mathcal{O}}(J)$ of the edges of $\delta_G(J)$. In order to define the ordering, we considered the disc $D(J)$ in the drawing $\rho_{J}$ of $J$ defined by the core structure ${\mathcal{J}}_{i^*}$. Recall that the drawing $\rho_{J}$ is the drawing of $J$ induced by the drawing $\rho_{C^*}$ of $C^*$. In this drawing, the orientation of every vertex $u\in V(J)$ is $b'_u$. We have defined, for every edge $e\in \delta_G(J)$, a point $p(e)$ on the boundary of the disc $D(J)$, and we let ${\mathcal{O}}(J)$ be the circular ordering of the edges of $\delta_G(J)$, in which the points $p(e)$ corresponding to these edges appear on the boundary of the disc $D(J)$, as we traverse it in counter-clock-wise direction. If the drawing of $J$ in $\phi'$ is identical to $\rho_{J}$ (including the orientation), then we say that the orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ is $1$. In this case, we are guaranteed that for every vertex $u\in V(J)$, its orientation in $\phi'$ is $b'_u$. Otherwise, the drawing of $J$ in $\phi'$ is the mirror image of $\rho_{J}$. We then say that the orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ is $-1$. In this case, we are guaranteed that for every vertex $u\in V(J)$, its orientation in $\phi'$ is $-b'_u$.
For every vertex $u\in V(P)\cup V(J)$, we consider the tiny $u$-disc $D_{\phi'}(u)$ in the drawing $\phi'$. For every edge $e\in \delta_{G}(u)\setminus E'$, we denote by $\sigma(e)$ the segment of $\phi'(e)$ that is drawn inside the disc $D_{\phi'}(u)$. Since edge $e$ belongs to graph $G$, $\sigma(e)$ must be contained in the region $F\in {\mathcal{F}}(W)$ of the drawing $\phi'$. Let $\tilde E=\textsf{left} (\bigcup_{u\in V(P)\cup V(J)}\delta_{G'}(u)\textsf{right} )\setminus (E(P)\cup E(J))$.
The image of path $P$ in $\phi'$ splits region $F$ into two regions, that we denote by $F_1$ and $F_2$. We let $e_1\in E_1$ be any edge that does not lie on path $P$, and we assume without loss of generality that $\sigma(e_1)\subseteq F_1$.
We partition edge set $\tilde E$ into a set $\tilde E^{\mathsf{in}}$ of \emph{inner edges} and the set $\tilde E^{\mathsf{out}}$ of outer edges, as before: Edge set $\tilde E^{\mathsf{in}}$ contains all edges $e\in \tilde E$ with $\sigma(e)$ contained in the face $F_1$ of $\tilde \rho$, and $\tilde E^{\mathsf{out}}$ contains all remaining edges (so for every edge $e\in \tilde E^{\mathsf{out}}$, $\sigma(e)$ is contained in $F_2$). As before, we show an algorithm that correctly computes the orientation of every vertex $u\in V(P)$ in the drawing $\phi'$, and the partition $(\tilde E^{\mathsf{in}},\tilde E^{\mathsf{out}})$ of the edges of $\tilde E$.
\paragraph{Computing Vertex Orientations on Path $P$ and the Partition $(\tilde E^{\mathsf{in}},\tilde E^{\mathsf{out}})$.}
The algorithm for computing vertex orientations for vertices of $P$, and the partition $(\tilde E^{\mathsf{in}},\tilde E^{\mathsf{out}})$ of the edges of $\tilde E$ is identical to that from Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. The main difference is the change in the parameters. We provide the algorithm and its analysis for completeness.
Consider any vertex $u\in V(P)$. Let $\hat e(u)$, $\hat e'(u)$ be the two edges of $P$ that are incident to $u$, where we assume that $\hat e(u)$ appears earlier on $P$ (we assume that $P$ is directed from an edge of $E_1$ to an edge of $E_2$). Edges $\hat e(u),\hat e'(u)$ partition the edge set $\delta_{G'}(u)\setminus\set{\hat e(u),\hat e'(u)}$ into two subsets, that we denote by $\hat E_1(u)$ and $\hat E_2(u)$, each of which appears consecutively in the rotation ${\mathcal{O}}_u\in \Sigma'$.
Note that either (i) $\hat E_1(u)\subseteq \tilde E^{\mathsf{in}}$ and $\hat E_2(u)\subseteq \tilde E^{\mathsf{out}}$ holds, or (ii) $\hat E_2(u)\subseteq \tilde E^{\mathsf{in}}$ and $\hat E_1(u)\subseteq \tilde E^{\mathsf{out}}$ holds.
We now construct edge sets $\tilde E_1',\tilde E_2'$, and fix an orientation $b'_u$ for every vertex $u\in V(P)\setminus V(J)$. We then show that $\tilde E_1'=\tilde E^{\mathsf{in}}$, $\tilde E_2=\tilde E^{\mathsf{out}}$, and that the orientations of all vertices of $P$ that we compute are consistent with the drawing $\phi'$.
Recall that we have computed a drawing $\rho_{\tilde C}$ of $\tilde C=C^*\cup P$, that is identical to the drawing of $\tilde C$ induced by $\phi'$. We let $\tilde \rho$ be the drawing of $J\cup P$ that is induced by $\rho_{\tilde C}$.
Since we have assumed that $\sigma(e_1)$ is contained in the face $F_1$, for every edge $e\in \delta_G(J)$, we can correctly determine whether $e\in \tilde E^{\mathsf{in}}$ or $e\in \tilde E^{\mathsf{out}}$. In the former case, we add $e$ to $\tilde E_1'$, and in the latter case we add $e$ to $\tilde E_2'$.
Therefore, for every path $P'\in {\mathcal{P}}^*$, the first and the last edges of $P$ are already added to either $\tilde E_1'$ or $\tilde E_2'$.
Next, we process every inner vertex $u$ on path $P$.
Consider any such vertex $u$. As before, if $u$ is a light vertex, then its orientation $b'_u$ can be chosen arbitrarily, as $\deg_{\tilde G'}(u)=2$.
Assume now that $u$ is a heavy vertex.
In order to decide on the orientation of $u$, we let ${\mathcal{P}}(u)$ contain all paths $P'\in {\mathcal{P}}^*\setminus\set{P}$ with $u\in P'$. We partition the set ${\mathcal{P}}(u)$ of paths into four subsets: set ${\mathcal{P}}_1(u)$ contains all paths $P'$ whose first edge lies in $\tilde E_1'$, and the first edge that is incident to $u$ lies in $\hat E_1(u)$. Set ${\mathcal{P}}_2(u)$ contains all paths $P'$ whose first edge lies in $\tilde E_2'$, and the first edge that is incident to $u$ lies in $\hat E_2(u)$. Similarly, set ${\mathcal{P}}'_1(u)$ contains all paths $P'\in {\mathcal{P}}(u)$ whose first edge lies in $\tilde E_1'$ and the first edge that is incident to $u$ lies in $\hat E_2(u)$, while set ${\mathcal{P}}'_2(u)$ contains all paths $P'\in {\mathcal{P}}'(u)$, whose first edge lies in $\tilde E_2'$ and the first edge that is incident to $u$ lies in $\hat E_1(u)$. We let $w(u)=|{\mathcal{P}}_1|+|{\mathcal{P}}_2|$, and $w'(u)=|{\mathcal{P}}'_1|+|{\mathcal{P}}'_2|$.
If $w(u)\leq w'(u)$, then we set $b'_u=-1$, add the edges of $\hat E_1(u)$ to $\tilde E'_1$, and add the edges of $\hat E_2(u)$ to $\tilde E'_2$. Otherwise we set $b'_u=1$, add the edges of $\hat E_1(u)$ to $\tilde E'_2$, and add the edges of $\hat E_2(u)$ to $\tilde E'_1$.
This completes the algorithm for computing the orientations of the inner vertices of $P$, and of the partition $(\tilde E_1',\tilde E_2')$ of the edge set $\tilde E$.
We use the following claim to show that both are computed correctly.
\begin{claim}\label{claim: orientations and edge split is computed correctly Case2.2}
Assume that Event ${\cal{E}}'$ did not happen. If the orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ is $1$, then for every vertex $u\in V(P)\cup V(C^*)$, the orientation of $u$ in $\phi'$ is $b'_u$.
Otherwise, for every vertex $u\in V(P)\cup V(C^*)$, the orientation of $u$ in $\phi'$ is $-b_u$. In either case,
$\tilde E_1'=\tilde E^{\mathsf{in}}$ and $\tilde E_2'=\tilde E^{\mathsf{out}}$.
\end{claim}
\begin{proof}
We assume without loss of generality that the orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ is $1$. The argument for the other case is symmetric. From the above discussion, for every vertex $u\in V(J)$, the orientation of $u$ in $\phi'$ is $b'_u$. It is now enough to show that, if $u\in P$ is a heavy vertex, then the orientation of $u$ in $\phi'$ is $b_u$.
Recall that we denoted $N(u)=|{\mathcal{P}}(u)|$, and we have denoted by $N^{\operatorname{bad}}(u)$ and $N^{\operatorname{good}}(u)$ the total number of bad and good paths in ${\mathcal{P}}(u)$, respectively. Since we have assumed that bad event ${\cal{E}}_4'$ did not happen, vertex $u$ is not a terrible vertex, that is, $N^{\operatorname{bad}}(u)\leq N^{\operatorname{good}}(u)/64$. Since $N(u)=N^{\operatorname{bad}}(u)+N^{\operatorname{good}}(u)$, we get that $N^{\operatorname{bad}}(u)<N(u)/65$.
Assume first that the orientation of vertex $u$ in $\phi'$ is $-1$. In this case, for every edge $e\in \hat E_1(u)\cup \tilde E_1'$, the segment $\sigma(e)$ lies in face $F_1$ of $\tilde \rho$, while for every edge $e'\in \hat E_2(u)\cup \tilde E_2'$, the segment $\sigma(e')$ lies in face $F_2$.
We claim that in this case $w'(u)>w(u)$ must hold, and so our algorithm sets $b_u=-1$ correctly. Indeed, assume otherwise.
Then $w'(u)\geq N(u)/2$. Let ${\mathcal{Q}}$ denote the set of all good paths in ${\mathcal{P}}'_1(u)\cup {\mathcal{P}}'_2(u)$. Then $|{\mathcal{Q}}|\geq w'(u)-N^{\operatorname{bad}}(u)\geq N(u)/2-N(u)/65\geq N(u)/4\geq h/4$, since $u$ is a heavy vertex.
We now show that, for every path $Q\in {\mathcal{Q}}$, there must be a crossing between an edge of $Q$ and an edge of $P$ in $\phi'$.
Indeed, consider any path $Q\in {\mathcal{Q}}$. Since $Q\in {\mathcal{P}}'_1(u)\cup {\mathcal{P}}'_2(u)$, the image of the path $Q$ must cross the boundary of the face $F_1$ in $\rho$. Since path $Q$ is a good path, and it does not contain vertices of $J$ as inner vertices, no inner point of the image of $Q$ in $\phi'$ may belong to the image of $J$ in $\phi'$. Since the paths in ${\mathcal{P}}$ are transversal, it then follows that there must be a crossing between an edge of $Q$ and an edge of $P$ in $\phi'$. But then the edges of $P$ participate in at least $\frac h 2\geq \frac{\mathsf{cr}(\phi)\mu^{31b}}{2m}$ crossings in $\phi'$, and hence in $\phi$.
However, since we have assumed that bad event ${\cal{E}}_1$ did not happen, $P$ is a good path, and so its edges may participate in at most $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ crossings in $\phi$, a contradiction. Therefore, when the orientation of $u$ in $\phi'$ is $-1$, our algorithm correctly sets $b_u=-1$.
In the case where the orientation of $u$ in $\phi'$ is $1$, the analysis is symmetric.
\end{proof}
In order to complete the construction of the skeleton augmenting structure ${\mathcal{W}}'$, it remains to define, for every pair $C,C'\in {\mathcal{C}}(W')$ of distinct connected components, a face $R_C(C')$ of ${\mathcal{F}}(\rho_C)$. Additionally, if face $F$ is the outer face in the drawing $\rho_{C^*}$ of component $C^*$, then we need to designate one of the faces $F_1,F_2$ as the outer face of the drawing $\rho_{\tilde C}$ of component $\tilde C$. We do so in the next step, once we split the graph $G'$ into two subgraphs, $G_1$ and $G_2$.
\paragraph{Computing the Split.}
We construct a flow network $\tilde H'$ as follows. We start with $\tilde H'=G'$, and then, for all $1\leq i\leq z$ with $i\neq i^*$, we contract all vertices of the core $J_i$ into a supernode $v_{J_i}$. Next, we subdivide every edge $e\in \tilde E$ with a vertex $t_e$, and denote $T_1=\set{t_e\mid e\in \tilde E_1'}$, $T_2=\set{t_e\mid e\in \tilde E_2'}$. We delete all vertices of $P\cup J$ and their adjacent edges from the resulting graph, contract all vertices of $T_1$ into a source vertex $s$, and contract all vertices of $T_2$ into a destination vertex $t$. We then compute a minimum $s$-$t$ cut $(A,B)$ in the resulting flow network $\tilde H'$, and we denote by $E''=E_H(A,B)$. The following claim, that is an analogue of \Cref{claim: cut set small case2}, bounds the cardinality of $E''$.
\begin{claim}\label{claim: cut set small case2.2}
If bad event ${\cal{E}}'$ did not happen, then $|E''|\leq \frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
\end{claim}
The proof of the claim is essentially identical to the proof of \Cref{claim: cut set small case2.2}. We assume for contradiction that Event ${\cal{E}}'$ did not happen, but $|E''|\leq \frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
We can then use Max Flow -- Min Cut theorem to construct a collection ${\mathcal{Q}}$ of edge-disjoint paths in graph $G'$, such that, for every path $Q\in {\mathcal{Q}}$, drawing $\phi'$ must contain at least one crossing between an edge of $Q$ and an edge of $W\cup P$. Since Event ${\cal{E}}'$ did not happen, path $P$ is good, so its edges participate in at most $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ crossings in $\phi$ and hence in $\phi'$. Additionally, edges of $\check K$ participate in at most $|\chi^{\mathsf{dirty}}(\phi)|$ crossings in $\phi$ (and hence in $\phi'$), and edges of $W\setminus K$ participate in at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{26b}}{\check m'}\leq \frac{\mathsf{cr}(\phi)\cdot \mu^{26b}} m$ crossings from Property \ref{prop: drawing}. Overall, there are fewer than $\frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$ crossings in $\phi'$ in which edges of $P\cup W$ participate, a contradiction.
We now construct the initial graphs $G_1,G_2$ associated with the faces $F_1$ and $F_2$, respectively; the final graphs $G_1',G_2'$ will be obtained by deleting some edges from $G_1$ and $G_2$.
In order to do so, we define two vertex sets $A',B'$ in graph $G'$, as before. We start with $A'=A\setminus\set{s}$ and $B'=B\setminus\set{t}$. For every index $1\leq i\leq r$ with $i\neq i^*$, if $v_{J_i}\in A$, then we replace $v_{J_i}$ with vertex set $V(J_i)$ in $A'$, and otherwise we replace $v_{J_i}$ with vertex set $V(J_i)$ in $B'$.
Additionally, we let $J\subseteq J\cup P$ be the graph containing all vertices and edges of $J\cup P$, whose images in $\tilde \rho$ lie on the boundary of $F$. Similarly, we let $J'\subseteq J\cup P$ be the graph containing all vertices and edges of $J\cup P$, whose images in $\tilde \rho$ lie on the boundary of $F_2$.
We then let $G_1$ be the subgraph of $G'$, whose vertex set is $V(A')\cup V(J)$, and edge set contains all edges of $E_{G'}(A')$, $E_{G'}(A',V(J)$, and all edges of $J$. Similarly, we let $G_2$ be the subgraph of $G'$, whose vertex set is $V(B')\cup V(J')$, and edge set contains all edges of $E_{G'}(B')$, $E_{G'}(B',V(J'))$, and all edges of $J'$.
The rotation system $\Sigma_1$ for graph $G_1$ and the rotation system $\Sigma_2$ for graph $G_2$ are induced by $\Sigma'$.
Let $I_1=(G_1,\Sigma_1)$ and $I_2=(G_2,\Sigma_2)$ be the resulting two instances of \ensuremath{\mathsf{MCNwRS}}\xspace.
We need the following observation that, intuitively, shows that graphs $G_1$ and $G_2$ are significantly smaller than the original graph $G$.
\begin{observation}\label{obs: few edges in split graphs case2.2}
If Event ${\cal{E}}$ did not happen, then $|E(G_1)\setminus E(W')|,|E(G_2)\setminus E(W')|\leq |E(G)\setminus E(W)|-\frac{\check m'}{\mu^{5b}}$.
\end{observation}
The proof of the observation is identical to the proof of \Cref{obs: few edges in split graphs case2}, and follows from the fact that, from the definition of the process for computing an enhancement, at least $k'/4$ edges of $E_1$ must lie in $\tilde E_1'$, and at least $k'/4$ edges of $E_1'$ must lie in $\tilde E_2$. Since $k'=\ceil{\frac{4m}{\mu^{3b}}}$ and $m\geq \check m'/\mu^{2b}$, the claim follows.
Let $\tilde G''=\tilde G'\setminus E''=\tilde G\setminus (E'\cup E'')$, and let $\tilde \Sigma''$ be the rotation system for graph $\tilde G''$ induced by $\tilde \Sigma$. We denote $\tilde I''=(\tilde G'',\tilde \Sigma'')$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace.
Let $\phi''$ be the drawing of $\tilde G''$ that is induced by the drawing $\phi'$ of $\tilde G'$. We also denote $G''=G'\setminus E''=G\setminus (E'\cup E'')$. Notice that $E(G_1)\cup E(G_2)=E(G'')$.
We now complete the construction of the skeleton augmenting structure ${\mathcal{W}}'$. For all $1\leq i\leq z$, let $C_i\in {\mathcal{C}}(W)$ be the connected component that contains the core $J_i$, and let $F^i\in {\mathcal{F}}(\rho_{C_i})$ be the face that corresponds to face $F$ of $\tilde {\mathcal{F}}({\mathcal{W}})$ (in other words, face $F$ in the drawing of $W$ induced by $\phi$ is contained in region $F^i$). From the construction of the set $\tilde {\mathcal{F}}({\mathcal{W}})$ of faces, in the decomposition forest $H({\mathcal{W}})$, either vertices $x_{C_1},\ldots,x_{C_z}$ are the roots of the trees in the forest; or one of these vertices is the parent-vertex of the remaining vertices. We assume w.l.o.g. that in the latter case, vertex $x_{C_1}$ is the parent of vertices $x_{C_2},\ldots,x_{C_z}$ in the forest $H({\mathcal{W}})$.
Consider any pair $C,C'\in {\mathcal{C}}(W')$ of distinct connected components. Assume first that $C,C'\neq \tilde C$. In this case, face $R_{C}(C')$ remains the same in ${\mathcal{W}}'$ in the augmenting structure ${\mathcal{W}}$. Since drawing $\phi$ if $\tilde G$ was ${\mathcal{W}}$-compatible, and drawing $\phi''$ is obtained from $\phi$ by deleting some edges that do not belong to $W'$, and possibly making some additional local changes within the region $F$, the image of component $C'$ in $\phi''$ remains contained in face $R_C(C')$.
Assume now that $C'=\tilde C$. In this case, we let the face $R_{C'}(\tilde C)$ in ${\mathcal{W}}'$ be the face $R_{C'}(C_{i^*})$ of ${\mathcal{W}}$. It is easy to verify that the drawing of component $\tilde C$ in $\phi''$ remains in the region $R_{C'}(C_{i^*})$.
Next, we assume that $C=\tilde C$. Recall that component $C_{i^*}$ of $W$ is replaced by component $\tilde C=C_{i^*}\cup P$ in $W'$. Assume first that $C'\not\in\set{C_1,\ldots,C_z}$. Then face $R=R_{C_{i^*}}(C)$ in ${\mathcal{W}}$ is differen from the face $F^{i^*}\in {\mathcal{F}}(\rho_{C_{i^*}})$ (which in turn corresponds to face $F$). Then $R$ remains a face of ${\mathcal{F}}(\rho_{\tilde C})$, and we set $R_{\tilde C}(C)=R$. Using the same reasoning as above, the image of component $C'$ in $\phi''$ is indeed contained in region $R$.
Lastly, assume that $C=\tilde C$ and $C'=C_i$ for some $1\leq i\leq z$ with $i\neq i'$. Assume first that $i^*=1$, and vertex $x_{C_1}$ is the parent vertex of vertices $x_{C_2},\ldots,x_{C_z}$. In this case, face $F^1\in {\mathcal{F}}(\rho_{C_1})$ is not the infinite face $F^*(\rho_{C_1})$, and $R_{C_i}(C_1)=F^*(\rho_{C_i})$ in ${\mathcal{W}}$. Face $F^1$ of $\rho_{C_1}$ is partitioned into two faces in $\rho_{\tilde C}$, that we denote by $F^1_1$ and $F^1_2$, respectively. If $J_i\subseteq G_1$, then we set $R_{C_1}(C_i)=F^1_1$, and otherwise we set $R_{C_1}(C_i)=F^1_2$. (We note that in this case we are not guaranteed that the image of component $C_i$ in $\phi''$ is contained in the ``correct'' region; we fix this later).
If vertices $x_{C_1},\ldots,x_{C_z}$ are the roots of the trees of $H({\mathcal{W}})$, then we define the regions $R_{\tilde C}(C_i)$ for $i\neq i^*$ similarly. In this case, face $F^{i^*}$ is the infinite face of drawing $\rho_{C_{i^*}}$, and we designate one of the two new faces $F^{i^*}_1,F^{i^*}_2$ of drawing $\rho_{\tilde C}$ (that are both contained in $F^{i^*}$) as the infinite face arbitrarily.
Lastly, we assume that $i^*\neq 1$, and vertex $x_{C_1}$ is the parent vertex of vertices $x_{C_2},\ldots,x_{C_z}$. In this case, face $F^{i^*}$ is the infinite face of drawing $\rho_{C_{i^*}}$. As before, face $F^{i^*}$ of $\rho_{C_{i^*}}$ is partitioned into two new faces, $F^{i^*}_1$ and $F^{i^*}_2$ in drawing $\rho_{\tilde C}$. If $J_1\subseteq G_1$, then we let $F^{i^*}_1$ be the infinite face of the drawing $\rho_{\tilde C}$, and otherwise we let $F^{i^*}_2$ be the infinite face of $\rho_{\tilde C}$. For all $1\leq i\leq z$ with $i\neq i^*$, if $J_i\subseteq G_1$ then we let $R_{\tilde C}(C_i)=F^{i^*}_1$, and otherwise we let $R_{\tilde C}(C_i)=F^{i^*}_2$.
This completes the definition of the new skeleton augmneting structure ${\mathcal{W}}'$.
At this point, we could set $E^{\mathsf{del}}=E'\cup E''$, and define ${\mathcal{G}}'=({\mathcal{G}}\setminus G)\cup (G'_1\cup G'_2)$, letting $G'_1$ be associated with face $F_1\in \tilde {\mathcal{F}}({\mathcal{W}}')$, $G'_2$ with face $F_2$, and each remaining graph of ${\mathcal{G}}'$ with the same face as before. The resulting collection ${\mathcal{G}}'$ of graphs would indeed be a valid ${\mathcal{W}}'$-decomposition of graph $\tilde G''$. But unfortunately the drawing $\phi''$ of $\tilde G''$ is not necessarily ${\mathcal{W}}'$-compatible. This is because it is possible that some connected component $C_i$, whose corresponding core $J_i$ is contained in graph $G_1'$ is not drawn in face $F_1$, and similarly, for some connected component $C_{i'}$ with $J_{i'}\subseteq G_2'$, it is possible that $C_{i'}$ is not drawin in $F_2$. In the next step, we ``fix'' this issue, by possibly removing some additional edges from graph $\tilde G'$. We start with the following observation.
\begin{observation}\label{obs: disconnect cores} Assume that Event ${\cal{E}}'$ did not happen.
Consider any index $1\leq i\leq z$ with $i\neq i^*$, such that $J_i\subseteq G_1$ but the image of $C_i$ in $\phi''$ is not contained in $F_1$. Then there is a collection $\hat E_i\subseteq E(G_1)\setminus E(W')$ of at most
$\frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$ edges, such that in graph $G_1\setminus \hat E_i$ there is no path connecting a vertex of $J_i$ to a vertex of $J$. Similarly, if $J_i\subseteq G_2$ but the image of $C_i$ in $\phi''$ is not contained in $F_2$, then there is a collection $\hat E_i\subseteq E(G_2)\setminus E(W')$ of at most
$\frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$ edges, such that in graph $G_2\setminus \hat E_i$ there is no path connecting a vertex of $J_i$ to a vertex of $J'$.
\end{observation}
\begin{proof}
Consider any index $1\leq i\leq z$ with $i\neq i^*$, such that $J_i\subseteq G_1$ but the image of $C_i$ in $\phi''$ is not contained in $F_1$. Assume for contradiction that a minimum-cardinality set $\hat E_i\subseteq E(G_1)\setminus E(W')$ of edges for which graph $G_1\setminus \hat E_i$ has no path connecting a vertex of $J_i$ to a vertex of $J$ has cardinality greater than
$\frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
Let $\tilde H$ be the graph that is obtained from $G_1$ by contracting every core $J_{\ell}\subseteq G_1$ with $\ell\not\in\set{i,i^*}$ into a vertex $v_{\ell}$. From the maximum flow / minimum cut theorem, there is a collection ${\mathcal{Q}}'$ of of at least $\frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$ edge-disjoint paths in graph $\tilde H$, each of which connects a vertex of $J_i$ to a vertex of $J\cup P$. Consider any such path $Q'\in {\mathcal{Q}}'$. Note that every edge of $Q'$ also belongs to graph $G_1$. We claim that there must be a crossing in $\phi''$ between an edge of $Q'$ and an edge of $J\cup P$.
Indeed, path $Q'$ naturally defines a path $Q$ in graph $G_1$, connecting a vertex of $J_i$ to a vertex of $J\cup P$. Consider the last edge $e$ on path $Q$. Then $e\in \tilde E'_1$ must hold, and so the image of $e$ in $\phi''$ must intersect the face $F_1$. Therefore, if we denote by $\gamma$ the image of $J\cup P$ in $\phi''$, then there is some edge $e'\in E(Q)$, whose image in $\phi''$ crosses $\gamma$. Edge $e'$ may not belong to any core $J_{\ell}$ or to path $P$, since the edges of $W\cup P$ do not cross each other in $\phi''$. Therefore, $e'\in E(Q')$ must hold.
We conclude that there are at least $|{\mathcal{Q}}'|\geq \frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$ crossings in $\phi''$ in which edges of $E(P)\cup E(J)$ participate. Since we have assumed that event ${\cal{E}}'$ did not happen, path $P$ is good, and so its edges participate in at most
$\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ crossings in $\phi$ and hence in $\phi''$. Additionally, edges of $\check K$ participate in at most $|\chi^{\mathsf{dirty}}(\phi)|$ crossings in $\phi$ (and hence in $\phi''$), and edges of $W\setminus K$ participate in at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{26b}}{\check m'}\leq \frac{\mathsf{cr}(\phi)\cdot \mu^{26b}} m$ crossings from Property \ref{prop: drawing}. Since $J\subseteq W$, there fewer than $\frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$ crossings in $\phi''$ in which edges of $E(J)\cup E(P)$ participate, a contradiction.
The proof for cores $J_i\subseteq G_2$ with the image of $C_i$ in $\phi''$ drawn outside of $F_2$ is symmetric.
\end{proof}
We now define another collection $E'''\subseteq E(G'')\setminus E(W')$ of edges to be deleted from graph $\tilde G''$, and we define a new graph $\tilde G'''=\tilde G''\setminus E'''$. Initially, we set $\tilde G'''=\tilde G''$ and $E'''=\emptyset$. We then process every core $J_i$, for $1\leq i\leq z$ and $i\neq i^*$ one by one. Consider an interation when a core $J_i$ is processed. Assume w.l.o.g. that $J_i\subseteq G_1$; the case where $J_i\subseteq G_2$ is symmetric. Let $E^i\subseteq E(G_1)\setminus (E'''\cup E(W'))$ be a minimum-cardinality subset of edges whoe removal in $G_1$ disconnects vertices of $J_i$ from vertices of $J\cup P$. such an edge set can be found efficiently by computing a minimum $s$-$t$ cut in a graph obtained from $G_1\setminus E'''$ by contracting each connected component of $W'\cap G_1$ into a supernode, and
letting $s$ and $t$ be the supernodes representing $J_i$ and $J\cup P$ respectively. We now consider two cases. If $|E^i|>\frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$, then, from \Cref{obs: disconnect cores}, the image of $J_i$ in $\phi''$ must appear inside the region $F_1$. Since the drawing of $W'$ induced by $\phi''$ is planar, the image of the connected component $C_i$ of ${\mathcal{C}}(W')$ that contains $J_i$ also appears inside the region $F_1$ in $\phi''$. The second case happens if $|E^i|\leq \frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$. In this case, we add the edges of $E^i$ to set $E'''$, and we delete them from graph $\tilde G''$, and from the drawing $\phi''$ of $\tilde G''$. @@@@
@@@
\subsubsection*{Case 2: ${\mathcal{P}}$ was a type-3 promising path set}
In the second case, ${\mathcal{P}}$ was a type-3 promising path set. We let $1\leq i^*<i^{**}\leq r$ be the indices for which all paths in ${\mathcal{P}}$ originate at vertices of $J$ and terminate at vertices of $J_{i^{**}}$. We denote $E_1=\delta_G(J)$ and $E_2=\delta_G(J)$.
If the two paths chosen from ${\mathcal{P}}^*$ share at least one vertex of $V(G)\setminus V(K)$, then we have constructed a type-2 enhancement whose both endpoints lie in $J$. In this case, the algorithm and its analysis are practically identical to those from Case 2 and are omitted here. We now assume that the two chosen paths are disjoint from each other. For convenience of notation, we denote $P=P^*_1$ and $P'=P^*_2$. Throughout, we denote by $\tilde J$ the graph obtained from the union of $J,J_{i^{**}},P$ and $P'$.
Recall that, from \Cref{claim: new drawing}, if neither of the events ${\cal{E}}_1,{\cal{E}}_3$ happened, drawing $\phi'$ contains no crossings between the edges of $E(K')$.
Throughout, we denote by $\gamma$ the image of path $P$ and by $\gamma'$ the image of path $P'$ in $\phi'$.
Consider now a drawing $\tilde \rho$ of graph $\tilde J$ induced by $\phi'$. This drawing has no crossings, and it has two faces, that we denote by $F_1$ and $F_2$, respectively, whose boundaries contain both curves $\gamma$ and $\gamma'$. Since the drawing of $J$ induced by $\tilde \rho$ is $\rho_{J}$ and the same is true for $J_{i^{**}}$, and since the orientations of all vertices in $V(J)$ are fixed with respect to each other, the orientations of all vertices in $V(J_{i^{**}})$ are fixed with respect to each other, and the paths $P$ and $P'$ do not share inner vertices, we can compute the drawing $\tilde \rho$ efficiently, though we do not know its orientation in $\phi'$.
As in previous two cases, for every vertex $u\in V(\tilde J)$, we consider the tiny $u$-disc $D_{\phi'}(u)$. For every edge $e\in \delta_{G'}(u)$, we denote by $\sigma(e)$ the segment of $\phi'(e)$ that is drawn inside the disc $D_{\phi'}(u)$. Let $\tilde E=\textsf{left} (\bigcup_{u\in V(\tilde J)}\delta_{G'}(u)\textsf{right} )\setminus E(\tilde J)$. We partition edge set $\tilde E$ into a set $\tilde E^{\mathsf{in}}$ of \emph{inner edges} and the set $\tilde E^{\mathsf{out}}$ of outer edges exactly as before: Edge set $\tilde E^{\mathsf{in}}$ contains all edges $e\in \tilde E$ with $\sigma(e)\subseteq F_1$, and $\tilde E^{\mathsf{out}}$ contains all remaining edges. Let $e_1$ be an arbitrary fixed edge of $E_1$ that does not lie on $P$ or $P'$. As before, we assume without loss of generality that $e_1\in \tilde E^{\mathsf{in}}$.
The algorithm for computing the orientation of the inner vertices of $P$ and $P'$ is practically identical to the one used in Case 2, with a minor difference in the analysis that we highlight below.
We define an orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ exactly as in Case 2. As in Case 1, we can now efficiently determine, for every edge $e\in E_1\cup E_2$, whether it belongs to $E^{\mathsf{in}}$ or $E^{\mathsf{out}}$. In the former case, we add the edge to set $\tilde E_1'$, and in the latter case, we add it to $\tilde E_2'$.
The algorithm for computing the orientation of every inner vertex $u$ of $P$ and $P'$ is identical to that used in Case 2.
The proof that the orientations of the vertices are computed correctly with respect to $\phi'$ provided that Event ${\cal{E}}$ does not happen (that is, an analogue of \Cref{claim: orientations and edge split is computed correctly Case2}) is also almost identical. The only difference is that now the images of the paths in sets ${\mathcal{P}}_1(u),{\mathcal{P}}_2(u),{\mathcal{P}}'_1(u),{\mathcal{P}}'_2(u)$ may cross the images of the edges of $J_{i^{**}}$. However, the total number of such paths, whose images cross the edges of $E(J)\cup E(J_{i^{**}})$, remains bounded by $N^{\operatorname{bad}}(u)$, so the same calculations work in Case 3 as well.
\paragraph{Computing the Split.}
The algorithm for computing the drawing $\tilde \rho$ of the graph $K'=K\cup P$ and of the split $(I_1,I_2)$ of the instance $I$ is almost identical to that in Cases 1 and 2. We provide a sketch below, highlighting the differences.
We construct a flow network $H$ as follows. We start with $H=G'$, and then, for all $1\leq i\leq r$ with $i\not\in\set{i^*,i^{**}}$, we contract all vertices of the core $J_i$ into a supernode $v_{J_i}$. Next, we subdivide every edge $e\in \tilde E$ with a vertex $t_e$, and denote $T_1=\set{t_e\mid e\in \tilde E_1'}$, $T_2=\set{t_e\mid e\in \tilde E_2'}$. We delete all vertices of $\tilde J$ and their adjacent edges from the resulting graph, contract all vertices of $T_1$ into a source vertex $s$, and contract all vertices of $T_2$ into a destination vertex $t$. We then compute a minimum $s$-$t$ cut $(A,B)$ in the resulting flow network $H$, and we denote by $E''=E_H(A,B)$. The following claim, that is an analogue of \Cref{claim: cut set small case2} for Case 2. The proof is virtually identical and is omitted here.
\begin{claim}\label{claim: cut set small case3}
If Events ${\cal{E}}$ did not happen, then $|E''|\leq \frac{64\mu^{700}\mathsf{cr}(\phi)}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
\end{claim}
The drawing $\rho'$ of the graph $K'$ is constructed exactly as before: we start with the drawing $\tilde \rho$ of $\tilde J$, and then plant the discs $D_i$ corresponding to the cores $J_i\in {\mathcal K}\setminus\set{J,J_{i^{**}}}$ inside face $F$ or inside face $F'$, depending on whether the corresponding supernode $v_{J_i}$ lies in $A$ or $B$.
Consider now the split $({\mathcal K}_1,{\mathcal K}_2)$ of ${\mathcal K}$ along the resulting enhancement structure ${\mathcal{A}}$.
Then ${\mathcal K}_1$ contains all core structures ${\mathcal{J}}_i$ with $v_{J_i}\in A$, while ${\mathcal K}_2$ contains all core structures ${\mathcal{J}}_i$ with $v_{J_i}\in B$. Additionally, ${\mathcal K}_1$ contains a core structure ${\mathcal{J}}$, whose corresponding core $J$ contains all vertices and edges of $\tilde J$ that lie on the boundary of face $F$ in drawing $\tilde \rho$. The drawing $\rho_J$ associated with $J$ is its induced drawing in $\tilde \rho$, with face $F$ serving as the infinite face. Similarly, ${\mathcal K}_2$ contains a core structure ${\mathcal{J}}'$, whose corresponding core $J'$ contains all vertices and edges of $\tilde J$ that lie on the boundary of face $F'$ in drawing $\tilde \rho$. The drawing $\rho_{J'}$ associated with $J'$ is its induced drawing in $\tilde \rho$, with face $F'$ serving as the infinite face. Notice that $V(J)\cup V(J')=V(\tilde J)$, and $E(J)\cup E(J')=E(\tilde J)$.
The split $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ of instance $I$ along the enhacement structure ${\mathcal{A}}$ is defined exactly as before. We start with $A'=A\setminus\set{s}$ and $B'=B\setminus\set{t}$. For every core structure ${\mathcal{J}}_i\in{\mathcal K}\setminus\set{J,J_{i^{**}}}$, if $v_{J_i}\in A$, then we replace $v_{J_i}$ with vertex set $V(J_i)$ in $A'$, and otherwise we replace $v_{J_i}$ with vertex set $V(J_i)$ in $B'$.
We then let $G_1$ be the subgraph of $G'$, whose vertex set is $V(A')\cup V(J)$, and edge set contains all edges of $E_{G'}(A')$, $E_{G'}(A',V(J))$, and all edges of $J$. Similarly, we let $G_2$ be the subgraph of $G'$, whose vertex set is $V(B')\cup V(J')$, and edge set contains all edges of $E_{G'}(B')$, $E_{G'}(B',V(J'))$, and all edges of $J'$.
The rotation system $\Sigma_1$ for graph $G_1$ and the rotation system $\Sigma_2$ for graph $G_2$ are induced by $\Sigma$.
Let $I_1=(G_1,\Sigma_1)$ and $I_2=(G_2,\Sigma_2)$ be the resulting two instances of \ensuremath{\mathsf{MCNwRS}}\xspace. The proof that $(I_1,I_2)$ is a split of $I$ along ${\mathcal{A}}$ is the same as in Case 2, and is omitted here.
We denote $E^{\mathsf{del}}=E(G)\setminus (E(G_1)\cup E(G_2))$. As before, $E^{\mathsf{del}}=E'\cup E''$, and, if Event ${\cal{E}}$ did not happen, then $|E^{\mathsf{del}}|\leq \frac{2\mathsf{cr}(\phi)\cdot \mu^{2150}}m +|\chi^{\mathsf{dirty}}(\phi)| < \frac{\mathsf{cr}(\phi)\cdot \mu^{2200}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
As in Case 2, it is easy to verify that $|{\mathcal K}_1|,|{\mathcal K}_2|\leq r$ must hold.
The following observation is an analogue of \Cref{obs: few edges in split graphs case2} for Case 3. The proof is virtually identical (with a very slight change in parameters) and is omitted here.
\begin{observation}\label{obs: few edges in split graphs case3}
If Event ${\cal{E}}$ did not happen, then $|E(G_1)|,|E(G_2)|\leq m-\frac{m}{32\mu^{300}}$.
\end{observation}
The next observation is an analogue of \Cref{obs: semi-clean case 2} for Case 3. Its proof is almost identical to that of \Cref{obs: semi-clean case 2}. We provide a proof sketch in Section \ref{subsec: getting semi-clean solution for case 3} of Appendix.
\begin{observation}\label{obs: semi-clean case 3}
If bad event ${\cal{E}}$ does not happen, then
there is a semi-clean solution $\phi_1$ for instance $I_1$ with respect to ${\mathcal K}_1$, and a semi-clean solution $\phi_2$ for instance $I_2$ with respect to ${\mathcal K}_2$, such that $|\chi^*(\phi_1)|+|\chi^*(\phi_2)|\leq |\chi^*(\phi)|$, and $|\chi^{\mathsf{dirty}}(\phi_1)|+ |\chi^{\mathsf{dirty}}(\phi_2)|\leq |\chi^{\mathsf{dirty}}(\phi)|+\frac{64\mu^{700}\mathsf{cr}(\phi)}{m}$.
\end{observation}
Altogether, if Event ${\cal{E}}$ did not happen, then the algorithm produces a valid output for \ensuremath{\mathsf{ProcSplit}}\xspace. Since the probability of Event ${\cal{E}}$ is at most $1/\mu^{399}$, this completes the proof of \Cref{thm: procsplit}.
@@@================
\paragraph{The Math}
\begin{itemize}
\item $\check m'/\mu^{2b}\leq m\leq \check m'$.
\item Goal fail probability for each bad event is $1/\mu^{12b}$
\item A path is selected into enhancement with probability $\frac{2\mu^{3b}}{m}$
\item A bad path participates in $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ crossings, at most $\frac{10m}{\mu^{15b}}$ bad paths.
\item number of edges we can delete: at most $ \frac{\mathsf{cr}(\phi)\cdot \mu^{120b}}{m}$, increase in $N(\phi)$ at most $\mathsf{cr}(\phi)\mu^{17b}/\check m'$
\item unlucky path: at most $\frac{\mathsf{cr}(\phi)\mu^{15b}}{m}$ paths on each side.
\item $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$, bad event ${\cal{E}}_2'$ happens if $|E'|>\frac{\mathsf{cr}(\phi)\mu^{48b}}{m}$.
\item selecting unlucky paths: there are at most $\frac{2^{14}\mathsf{cr}(\phi)\mu^{15b}}{m}$ unucky paths for each heavy vertex. Number of heavy vertices: $m/h$. So total number of unlucky paths: $\frac{2^{14}\mathsf{cr}(\phi)\mu^{15b}}{h}$. A path is selected with probability $\frac{2\mu^{3b}}{m}$. So probability that any unlucky path is selected: $\frac{2^{15}\mathsf{cr}(\phi)\mu^{18b}}{hm}$. This must be at most $1/\mu^{12b}$. So $h=\frac{2^{15}\mathsf{cr}(\phi)\mu^{30b}}{m}$. Let's make it $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$
\end{itemize}
\paragraph{Valid Input for $\ensuremath{\mathsf{ProcSplit}}\xspace'$.}
A valid input to Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$ consists of a skeleton augmenting structure ${\mathcal{W}}$ for $\check{\mathcal{K}}$ (whose corresponding skeleton augmentation we denote by $W$), parameters $g_1,g_2$, a ${\mathcal{W}}$-decomposition ${\mathcal{G}}$ of $\check G'$, and a face $F\in \tilde {\mathcal{F}}({\mathcal{W}})$, for which the following hold. Denote $E^{\mathsf{del}}=E(\check G')\setminus\textsf{left} (\bigcup_{G_{F'}\in {\mathcal{G}}}E(G_{F'})\textsf{right})$, $\tilde G=\check G'\setminus E^{\mathsf{del}}$, and let $\tilde \Sigma$ be the rotation system for graph $\tilde G$ induced by $\check \Sigma'$. Let $\tilde I=(\tilde G,\tilde \Sigma)$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Then:
\mynote{note that I replaced $30$ with $60$ in property C2 twice. This needs to be propagated in this section, and probably also back. I also added $g_1,g_2$ and the conditions on them}
\begin{properties}{C}
\item Instance $I_F=(G_F,\Sigma_F)$ associated with the graph $G_F\in {\mathcal{G}}$ is not acceptable; and \label{prop: instance unacceptable}
\item there exists a solution $\phi$ to instance $\tilde I$, that is semi-clean with respect to $\check{\mathcal{K}}$, and has the following additional properties: \label{prop: drawing}
\begin{itemize}
\item drawing $\phi$ is ${\mathcal{W}}$-compatible;
\item $\mathsf{cr}(\phi)\leq \min\set{g_1,(\check m')^2/\mu^{60b}}$;
\item $|\chi^{\mathsf{dirty}}(\phi)|\leq \set{g_2,\check m'/\mu^{60b}}$ (as before, a crossing of $\phi$ is dirty if it envolves an edge of the skeleton $\check K$); and
\item the total number of crossings of $\phi$ in which the edges of $E(W)\setminus E(\check K)$ participate is at most $\mathsf{cr}(\phi)\mu^{26b}/\check m'$.
\end{itemize}
\end{properties}
As before, the semi-clean solution $\phi$ is not known to the algorithm.
\mynote{this was updated to a new bound on $E^{\mathsf{del}}(F)$. The remainder of this section needs to be updated to reflect all this.}
\paragraph{Valid Output for $\ensuremath{\mathsf{ProcSplit}}\xspace'$.}
A valid output for Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$ consists of
a skeleton augmenting structure ${\mathcal{W}}'$ for $\check{\mathcal{K}}$ (whose corresponding skeleton augmentation we denote by $W'$), an edge set $E^{\mathsf{del}}(F)\subseteq E(G_F)\setminus E(W')$ of cardinality at most $\textsf{left}(\frac{g_1}{|E(G_F)|}+g_2\textsf{right} )\cdot \mu^{120b}$, and a ${\mathcal{W}}'$-decomposition ${\mathcal{G}}'$ of grpah $\check G'$, for which the following hold.
Denote $E^{\mathsf{del}}=E(\check G')\setminus\textsf{left} (\bigcup_{G_{F'}\in {\mathcal{G}}}E(G_{F'})\textsf{right})$, $\tilde G'=\check G'\setminus (E^{\mathsf{del}}\cup E^{\mathsf{del}}(F))$, and let $\tilde \Sigma'$ be the rotation system for graph $\tilde G'$ induced by $\check \Sigma'$. Let $\tilde I'=(\tilde G',\tilde \Sigma')$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Then:
\begin{properties}{P}
\item $W\subseteq W'$, and $\tilde {\mathcal{F}}({\mathcal{W}}')=\textsf{left}(\tilde{\mathcal{F}}({\mathcal{W}})\setminus\set{F}\textsf{right})\bigcup\set{F_1,F_2}$;\label{prop: same faces}
\item for every face $F'\in \tilde {\mathcal{F}}({\mathcal{W}})\setminus\set{F}$, the graph $G_{F'}$ associated with face $F'$ in the decomposition ${\mathcal{G}}$ is identical to the graph $G'_{F'}$ associated with face $F'$ in the decomposition ${\mathcal{G}}'$; \label{prop: same graphs}
\item if $G_F$ is the graph associated with face $F$ in ${\mathcal{G}}$, and $G_{F_1},G_{F_2}$ are the graphs associated with faces $F_1$ and $F_2$, respectively, in ${\mathcal{G}}'$, then $E(G_F)\setminus (E(G_{F_1})\cup E(G_{F_2}))= E^{\mathsf{del}}(F)$. Moreover, $|E(G_{F_1})\setminus E(W')|, |E(G_{F_2})\setminus E(W')|\leq |E(G_{F})\setminus E(W)|-\check m'/\mu^{5b}$; \label{prop: small instances} and
\item there exists a solution $\phi'$ to instance $\tilde I'$, that is semi-clean with respect to $\check{\mathcal{K}}$, and has the following additional properties: \label{prop: drawing after}
\begin{itemize}
\item drawing $\phi'$ is ${\mathcal{W}}'$-compatible;
\item $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$;
\item $|\chi^{\mathsf{dirty}}(\phi)|\leq |\chi^{\mathsf{dirty}}(\phi')|$; and
\item if we denote by $N(\phi)$ the total number of crossings of $\phi$ in which the edges of $E(W)\setminus E(\check K)$ participate, and define $N(\phi')$ similarly for $\phi'$, then $N(\phi')\leq N(\phi)+\mathsf{cr}(\phi)\mu^{17b}/\check m'$
\end{itemize}
\end{properties}
We summarize our algorithm for Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$ in the following lemma.
\begin{lemma}\label{lem: alg procsplit'}
There is an efficient randomized algorithm, that, given a valid input to Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$, with probability at least $1-1/\mu^{10b}$ produces a valid output for Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$.
\end{lemma}
============
\subsection*{The Math}
\begin{itemize}
\item Acceptable instance definition will give a path set containing $\check m'/\mu^{2b}$ paths. In particular, $m\geq \check m'/\mu^{2b}$ and $m\leq \check m'$. So number of paths is at least $m/\mu^{2b}$. This will give promising set containing at least $m/\mu^{2b+100}$ paths. Let's take a subset of $4m/\mu^{3b}$ such paths. So $k'=4m/\mu^{3b}$.
\item A path is chosen into enhancement with probability at most $2\mu^{3b}/m\leq 2\mu^{5b}/\check m'$. Expected number of crossings involving it is $\mathsf{cr}(\psi)\mu^{3b}/m$. So to get failure probability $1/\mu^{12b}$, we'll say that the path is bad if it participates in more than $\mathsf{cr}(\psi)\mu^{15b}/m$ crossings. This is at most $\mathsf{cr}(\psi)\mu^{17b}/\check m'$. This will be times 2 for the 2 paths.
\item So: total number of crossings in $\psi$ involving edges of the skeleton will be bounded by $\mathsf{cr}(\psi)\mu^{21b}/\check m'$. We should make it $\mathsf{cr}(\psi)\mu^{25b}/\check m'$ in step claim statement.
\item what is the probability that a chosen path crosses some edge of the skeleton? At most $\frac{\mathsf{cr}(\psi)\mu^{22b}}{\check m'}\cdot\frac{\mu^{5b}}{\check m'}\leq \frac{\mathsf{cr}(\psi)\mu^{27b}}{(\check m')^2}$. This needs to be less than $1/\mu^{12b}$. OK if $\mathsf{cr}(\psi)\leq (\check m')^2\cdot \mu^{60b}$.
\item in acceptable instance, the cut size is at most $\check m'/\mu^{2b}$. Number of crossings with edges of $W$ is $\mathsf{cr}(\psi)\mu^{25b}/\check m'$. We need $\mathsf{cr}(\psi)\mu^{25b}/\check m'\ll \check m'/\mu^{2b}$, which is OK because $\mathsf{cr}(\psi)\ll (\check m')^2/\mu^{60}$.
\end{itemize}
\subsection*{The definitions (copied from before)}
\paragraph{Skeleton Augmentation.}
Consider the instance $\check I'=(\check G',\check \Sigma')$ and the corresponding skeleton structure $\check {\mathcal K}=({\mathcal{J}}_1,\ldots,{\mathcal{J}}_r)$ that serve as input to \Cref{thm: phase 2}, and the skeleton $\check K$ corresponding to skeleton structure $\check{\mathcal{K}}$.
An augmentation of the skeleton $\check K$ is a subgraph $W\subseteq \check G'$ that has the following properties.
\begin{itemize}
\item $\check K\subseteq W$, and every connected component $C$ of $W$ contains at least one core $J_i$, whose corresponding core structure ${\mathcal{J}}_i$ lies in $\check{\mathcal{K}}$ (we note that $C$ may contain more than one such core); and
\item for every connected component $C$ of $W$, for every edge $e\in E(C)$, graph $C\setminus\set{e}$ is connected.
\end{itemize}
\paragraph{Skeleton Augmenting Structure}
\begin{enumerate}
\item The first ingredient is an augmentation $W$ of the skeleton $\check K$.
\item The second ingredient is an orientaton $b'_u\in \set{-1,1}$ for every vertex $u\in V(W)$, such that the following holds. For all $1\leq i\leq r$, either (i) for every vertex $u\in V(J_i)$, the orientation $b'_u$ is identical to the orientation $b_u$ that is given as part of the core structure ${\mathcal{J}}_i$; or (ii) for every vertex $u\in V(J_i)$, $b'_u=b_u$ holds.
\item The third ingredient is a drawing $\rho_C$ of each connected component $C\in {\mathcal{C}}(W)$ in the plane, that contains no crossings, and is consistent with the rotation system $\Sigma$ and the orientations $\set{b'_u}_{u\in V(C)}$. We additionally require that, for all $1\leq i\leq r$ with $J_i\subseteq C$, there is a disc $D_i$ containing the images of all vertices and edges of $J_i$. The drawing of $J_i$ inside $D_i$ must be identical to the drawing $\rho_{J_i}$ that is given as part of the core structure ${\mathcal{J}}_i$ (up to orientation that may be arbitrary), with disc $D_i$ in $\rho_C$ coinciding with disc $D(J_i)$ in $\rho_{J_i}$. The only vertices of $C$ whose image in $\rho_C$ appears in disc $D_i$ are the vertices of $J_i$. The only edges of $C$ whose image in $\rho_C$ is contained in $D_i$ are the edges of $J_i$. The only edges whose image in $\rho_C$ intersects $D_i$ are edges of $\delta_C(J_i)$, and, for each such edge $e$, the intesection of the image of $e$ with disc $D_i$ is a contiguous curve that may not intersect the interior of any forbidden face $F\in {\mathcal{F}}(\rho_{J_i})$.
As before, we denote by ${\mathcal{F}}(\rho_C)$ the collection of all faces of $\rho_C$, and by $F^*(\rho_C)$ the infinite face of the drawing. Notice that, for all $1\leq i\leq r$ with $J_i\subseteq C$, every forbidden face $F\in {\mathcal{F}}^{\operatorname{X}}(\rho_{J_i})$ is also a face of ${\mathcal{F}}(\rho_C)$. We view $F$ also as a forbidden face of the drawing $\rho_C$, and we denote by ${\mathcal{F}}^{\operatorname{X}}(\rho_C)\subseteq {\mathcal{F}}(\rho_C)$ the set of all forbidden faces for $\rho_C$. In other words, ${\mathcal{F}}^{\operatorname{X}}(\rho_C)$ is the union, over all indices $1\leq i\leq r$ with $J_i\subseteq C$, of ${\mathcal{F}}^{\operatorname{X}}(\rho_{J_i})$.
\item The fourth and the last ingredient in a skeleton augmenting structure is, for every unordered pair $C,C'\in {\mathcal{C}}(W)$ of distinct connected components of $W$, a face $R_C(C')\in {\mathcal{F}}(\rho_C)\setminus{\mathcal{F}}^{\operatorname{X}}(\rho_C)$, and a face $R_{C'}(C)\in {\mathcal{F}}(\rho_{C'})\setminus {\mathcal{F}}^{\operatorname{X}}(\rho_{C'})$, so that either $R_C(C')=F^*(\rho_C)$, or $R_{C'}(C)=F^*(\rho_{C'})$ holds.
We additionally require that, for every triple $C,C',C''\in {\mathcal{C}}(W)$ of distinct components, if $R_C(C')=F^*(\rho_{C})$ and $R_{C'}(C'')=F^*(\rho_{C'})$, then $R_C(C'')=F^*(\rho_{C})$ holds.
\end{enumerate}
\begin{definition}[${\mathcal{W}}$-compatible drawings]
Consider a skeleton augmenting structure \newline
${\mathcal{W}}=(W,\set{b'_u}_{u\in V(W)},\set{\rho_C}_{C\in {\mathcal{C}}(W)}, \set{R_C(C')}_{C,C'\in {\mathcal{C}}(W)})$, and let $\phi$ be a drawing of the augmented skeleton $W$ in the plane. We say that $\phi$ is a ${\mathcal{W}}$-compatible drawing of $W$, if the following hold:
\begin{itemize}
\item drawing $\phi$ contains no crossings;
\item for every connected component $C\in {\mathcal{C}}(W)$, the drawing of $C$ induced by $\phi$ is identical to the drawing $\rho_C$ (but the orientation may be arbitrary); and
\item for every pair $C,C'\in {\mathcal{C}}(W)$ of distinct connected components, the image of $C'$ is contained in the region $R_{C}(C')\in {\mathcal{F}}(\rho_C)$ of the drawing $\phi$, and the image of $C$ is contained in the region $R_{C'}(C)\in {\mathcal{F}}(\rho_{C'})$ of the drawing $\phi$.
\end{itemize}
As before, we denote by ${\mathcal{F}}(\phi)$ the set of all faces of the drawing $\phi$, and by $F^*(\phi)$ the infinite face of the drawing. Note that for every component $C\in {\mathcal{C}}(W)$, for every forbidden face $F\in {\mathcal{F}}^X(\rho_C)$, $F\in {\mathcal{F}}(\phi)$ must hold. Moreover, if $J=\partial_{\rho_C}(F)$, then $J=\partial_{\phi}(F)$. We denote by ${\mathcal{F}}^X(\phi)=\bigcup_{C\in {\mathcal{C}}(W)}{\mathcal{F}}^X(\rho_C)$.
Given any subgraph $G'$ of $\check G'$ and a drawing $\phi'$ of $G'$, we say that $\phi$ is ${\mathcal{W}}$-compatible if the drawing $\phi$ of $W$ induced by $\phi'$ is ${\mathcal{W}}$-compatible, and, moreover, for every forbidden face $F\in {\mathcal{F}}^{\operatorname{X}}(\phi)$, no vertex of $G'$ has its image in the interior of $F$.
\end{definition}
\paragraph{${\mathcal{W}}$-Decomposition of $\check G'$ and Face-Based Instances}
Given a skeleton augmeting structure ${\mathcal{W}}$, a ${\mathcal{W}}$-decomposition of the input graph $\check G'$ is a collection ${\mathcal{G}}=\set{G_{\tilde F}\mid \tilde F\in \tilde {\mathcal{F}}({\mathcal{W}})}$ of subgraphs of $\check G'$, for which the following hold. For every face $\tilde F\in \tilde {\mathcal{F}}({\mathcal{W}})$, let $\Sigma_{\tilde F}$ be the rotation system for the graph $G_{\tilde F}\in {\mathcal{G}}$ induced by $\check \Sigma'$, and let $I_{\tilde F}=(G_{\tilde F},\Sigma_{\tilde F})$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Then:
\begin{itemize}
\item for every face $\tilde F\in \tilde {\mathcal{F}}({\mathcal{W}})$, for every component $C\in {\mathcal{S}}(\tilde F)$, $J_{C,\tilde F}\subseteq G_{\tilde F({\mathcal{W}})}$, and ${\mathcal{J}}_{C,\tilde F}$ is a valid core structure for $I_{\tilde F}$ (see Definitions \ref{def: valid core 1} and \ref{def: valid core 2}). Moreover, if $\tilde F\in \tilde {\mathcal{F}}^{\operatorname{X}}({\mathcal{W}})$, and $C$ is the unique component of ${\mathcal{S}}(\tilde F)$, then $G_{\tilde F}=J_{C,\tilde F}$;
\item for every face $\tilde F\in \tilde {\mathcal{F}}({\mathcal{W}})$, for every vertex $u\in V(G_F)\cap W$, there must be a component $C\in {\mathcal{S}}(\tilde F)$ with $u\in V(J_{C,\tilde F})$; and
\item for every pair $\tilde F,\tilde F'\in \tilde{\mathcal{F}}({\mathcal{W}})$ of distinct faces, $V(G_{\tilde F}\cap V(G_{\tilde F'}\subseteq V(W)$, and $E(G_{\tilde F}\cap E(G_{\tilde F'}\subseteq E()W)$.
\end{itemize}
\paragraph{Acceptable Subinstances.}
Consider a skeleton augmentating structure ${\mathcal{W}}$, a ${\mathcal{W}}$-decomposition ${\mathcal{G}}$ of graph $\check G'$, and a face $\tilde F\in \tilde {\mathcal{F}}({\mathcal{W}})\setminus \tilde {\mathcal{F}}^{\operatorname{X}}({\mathcal{W}})$. Recall that we have defined a collection ${\mathcal K}_{\tilde F}$ of core structures associated with face $\tilde F$, that we now denote, for convenience, by ${\mathcal K}_{\tilde F}=\set{{\mathcal{J}}_1^{\tilde F},\ldots,{\mathcal{J}}^{\tilde F}_z}$. For all $1\leq i\leq z$, let $J^{\tilde F}_i$ be the core associated with the core structure ${\mathcal{J}}_{\tilde F}$.
Recall that ${\mathcal K}_{\tilde F}$ is a valid skeleton structure for instance $I_{\tilde F}=(G_{\tilde F},\Sigma_{\tilde F})$ of \ensuremath{\mathsf{MCNwRS}}\xspace. We denote by $K_{\tilde F}=\bigcup_{i=1}^zJ_i^{\tilde F}$ the skeleton graph associated with ${\mathcal K}_{\tilde F}$.
For all $1\leq i\leq z$,
we denote by $\tilde E_i=\delta_{G_{\tilde F}}(J^{\tilde F}_i)$ the set of all edges of $G_{\tilde F}$ that are incident to the vertices of the core $J_i^{\tilde F}$. Since ${\mathcal{J}}^{\tilde F}_i$ is a valid core structure for instance $I_{\tilde F}$, we can define an ordering ${\mathcal{O}}(J^{\tilde F}_i)$ of the edges of $\tilde E_i$ exactly as before. We then define $\tilde E^{\tilde F}=\bigcup_{i=1}^z\tilde E_i$.
Next, we define interesting partitions of the edges of $\tilde E^{\tilde F}$.
\begin{definition}[Interesting Partitions of Edges.]
Consider some partition $(\tilde E',\tilde E'')$ of the edges of $\tilde E^{\tilde F}$. We say that this partition is \emph{interesting} if there is some index $1\leq i\leq z$, and a partition $(A,A')$ of the edges of $\tilde E_i$ into two subsets, where the edges of $A$ appear consecutively in the ordering ${\mathcal{O}}(J_i^{\tilde F})$ and so do the edges of $A'$, such that $\tilde E'=A$, and $\tilde E''=\tilde E\setminus \tilde E'$ hold.
\end{definition}
We are now ready to define acceptable instances.
\begin{definition}[Acceptable Instance]
Let ${\mathcal{W}}$ be a skeleton augmentating structure, let ${\mathcal{G}}$ be a ${\mathcal{W}}$-decomposition graph $\check G'$, and let $\tilde F\in \tilde {\mathcal{F}}({\mathcal{W}})\setminus \tilde {\mathcal{F}}^{\operatorname{X}}({\mathcal{W}})$ be a face. Consider a graph $H_{\tilde F}$ that is obtained from graph $G_{\tilde F}$ by first subdividing every edge $e\in \tilde E^{\tilde F}$ with a vertex $t_e$, and then deleting all vertices and edges of $K_{\tilde F}$ from it. We say that instance $G_{\tilde F}$ is \emph{acceptable} iff for every interesting partition $(\tilde E',\tilde E'')$ of the edges of $\tilde E^{\tilde F}$, there is a cut $(X,Y)$ in graph $H_{\tilde F}$ with vertex set $\set{t_e\mid e\in \tilde E'}$ contained in $X$, vertex set $\set{t_e\mid e\in \tilde E'}$ contained in $Y$, and $|E_{H_{\tilde F}}(X,Y)|\leq \check m'/\mu^{2b}$.
\end{definition}
\subsection{Proof of \Cref{lem: alg procsplit'}}
\label{subsec: proof of procsplit'}
In order to simplify the notation, we denote instance $I_F=(G_F,\Sigma_F)$ by $I=(G,\Sigma)$, and we denote $|E(G)|$ by $m$. Since instance $I$ is not acceptable, face $F$ may not be a forbidden face of $\tilde {\mathcal{F}}(W)$. Recall that we have computed a set ${\mathcal{S}}(F)\subseteq{\mathcal{C}}(W)$ of connected components, and, for each such component $C\in {\mathcal{S}}(F)$, a subgraph $J_{C,F}$ that is a core. Intuitively, vertices and edges of $J_{C,F}$ must serve as the boundary of face $F$ in any ${\mathcal{W}}$-compatible drawing of $W$, and in particular their images lie on the boundary of the region $F$ in the drawing $\phi$ of graph $\check G'$, given by Property \ref{prop: drawing} of valid input. We have also defined, for each component $C\in {\mathcal{S}}(F)$, a core structure ${\mathcal{J}}_{C,F}$, whose corresponding core is $J_{C,F}$, and we have defined a collection ${\mathcal K}_{ F}=\set{{\mathcal{J}}_{C, F}\mid C\in {\mathcal{S}}( F)}$ of core structures associated with $F$. From the definition of a ${\mathcal{W}}$-decomposition, for all $C\in {\mathcal{S}}(F)$, ${\mathcal{J}}_{C,F}$ is a valid core structure for $I$, and so ${\mathcal K}_F$ is a valid skeleton structure for instance $I$. In order to simplify the notation, we denote ${\mathcal K}_F=\set{{\mathcal{J}}_{C,F}\mid C\in {\mathcal{S}}(F)}$ by ${\mathcal K}=\set{{\mathcal{J}}_1,\ldots,{\mathcal{J}}_z}$, and for all $1\leq i\leq z$, we denote the core associated with the cores structure ${\mathcal{J}}_i$ by $J_i$. Since, from \Cref{lem: compute phase 2 decomposition}, $|\check{\mathcal{K}}|\leq \mu^{100}$, from the definition of a skeleton augmentation, $|{\mathcal{C}}(W)|\leq \mu^{100}$, and so $z\leq \mu^{100}$ holds. We also denote by $K=\bigcup_{i=1}^zJ_i$ the skeleton associated with the skeleton structure ${\mathcal K}$.
As before, we denote $E^{\mathsf{del}}=E(\check G')\setminus\textsf{left} (\bigcup_{G_{F'}\in {\mathcal{G}}}E(G_{F'})\textsf{right})$, $\tilde G=\check G'\setminus E^{\mathsf{del}}$, and let $\tilde \Sigma$ be the rotation system for graph $\tilde G$ induced by $\check \Sigma'$. Let $\tilde I=(\tilde G,\tilde \Sigma)$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Recall that Property \ref{prop: drawing} of valid input to Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$ ensures that there is a solution $\phi$ to instance $\tilde I$, that is ${\mathcal{W}}$-compatible. Additionally, $\mathsf{cr}(\phi)\leq (\check m')^2/\mu^{60b}$, $|\chi^{\mathsf{dirty}}(\phi)|\leq \check m'/\mu^{60b}$, the total number of crossings of $\phi$ in which the edges of $E(W)\setminus E(\check K)$ participate is at most $\mathsf{cr}(\phi)\mu^{26b}/\check m'$.
As in the original procedure \ensuremath{\mathsf{ProcSplit}}\xspace, the algorithm consists of two steps. In the first step, we compute a promising path set and an enhancement of the skeleton ${\mathcal K}$. In the second step we complete the construction of the new skeleton augmenting structure ${\mathcal{W}}'$ and the new ${\mathcal{W}}'$-decompositon ${\mathcal{G}}'$ of graph $\check G'$.
\subsubsection{Step 1: Computing Skeleton Enhancement}
As before, we start by computing a promising path set, that will then be used in order to compute an enhancement of skeleton ${\mathcal K}$. In the algorithm for $\ensuremath{\mathsf{ProcSplit}}\xspace$, the construction of the promising path set exploited the fact that the input instance $I$ was wide. Here, instead, we exploit the fact that instaince $I$ is not acceptable. The resulting promising path set will be somewhat smaller, and it will only be of type 2 or type 3.
From the definition of aceptable subinstances, there must be an index $1\leq i^*\leq z$, and a partition $(E_1,E_2)$ of the edges of $\delta_G(J_{i^*})$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}(J_{i^*})$, and the following holds. Let $(\tilde E',\tilde E'')$ be the partitition of the edges in set $\tilde E=\bigcup_{i=1}^z\delta_G(J_i)$, where $\tilde E'=E_1$, and $\tilde E''=\tilde E\setminus \tilde E'$. As before, let
$\tilde H$ be the graph that is obtained from graph $G$ by, first subdividing every edge $e\in \tilde E$ with a vertex $t_e$, and then deleting all vertices and edges of $K$ from it. Then the minimum cut in graph $\tilde H$ separating the vertices of $\set{t_e\mid e\in \tilde E'}$ from the vertices of $\set{t_e\mid e\in \tilde E''}$ must contain at least $\check m'/\mu^{2b}$ edges. Equivalently, there is a set ${\mathcal{P}}$ of $\ceil{\check m'/\mu^{2b}}$ edge-disjoint paths in graph $\tilde H$, connecting vertices of
$\set{t_e\mid e\in \tilde E'}$ to vertices of $\set{t_e\mid e\in \tilde E''}$. We can assume w.l.o.g. that the paths of ${\mathcal{P}}$ do not contain the vertices of $\set{t_e\mid e\in \tilde E}$ as inner vertices.
Set ${\mathcal{P}}$ of paths in graph $\tilde H$ naturally defines a collection of at least $\ceil{\check m'/\mu^{2b}}$ edge-disjoint paths in graph $G$, where every path has an edge of $\tilde E'=E_1$ as its first edge, and an edge of $\tilde E''$ as its last edge, and it does not contain any vertices of $K$ as inner vertices. Abusing the notation, we denote this path set by ${\mathcal{P}}$ as well. We partition path set ${\mathcal{P}}$ into two $z$ subsets ${\mathcal{P}}^1,\ldots,{\mathcal{P}}^z$, as follows: consider any path $P\in {\mathcal{P}}$. Recall that the last edge of $P$ lies in $\tilde E=\bigcup_{i=1}^z\delta_G(J_i)$. We add path $P$ to set ${\mathcal{P}}^i$ if the last edge of $P$ lies in $\delta_G(Z_i)$. Clearly, there must be an index $1\leq i\leq z$, such that $|{\mathcal{P}}^i|\geq \frac{|{\mathcal{P}}|}z \geq \frac{\check m'}{\mu^{2b}\cdot z}\geq \frac{\check m'}{\mu^{2b+100}}\geq \frac{m}{\mu^{2b+100}}$. We then let ${\mathcal{P}}'$ be that path set ${\mathcal{P}}_i$. Observe that, if $i=i^*$, then ${\mathcal{P}}'$ is a promising path set of type $2$, and every path in ${\mathcal{P}}'$ has an edge of $E_1$ as its first edge, and an edge of $E_2$ as its last edge. Otherwise, ${\mathcal{P}}'$ is a promising path set of type $3$. We let $i^{**}$ be the index for which the last edge of every path in ${\mathcal{P}}'$ lies in $\delta_G(J_{i^{**}})$ (so it is possible that $i^{**}=i^*$).
Note that the existence of the path set ${\mathcal{P}}'$ implies that $m\geq \check m'/\mu^{2b}$ holds. Let ${\mathcal{P}}''\subseteq {\mathcal{P}}'$ be an arbitrary subset of $\ceil{\frac{4m}{\mu^{3b}}}$ paths. We construct an enhancement $\Pi=(P_1,P_2)$ from the path set ${\mathcal{P}}''$ exactly like in the algorithm for \ensuremath{\mathsf{ProcSplit}}\xspace. Recall that the algorithm first constructs a collection ${\mathcal{P}}^*$ of edge-disjoint non-transversal paths using the paths in ${\mathcal{P}}''$, and then selects one or two such paths into the enhancement. The only difference is that the parameter $k'=|{\mathcal{P}}^*|$ is now somewhat smaller, $k'=\ceil{\frac{4m}{\mu^{3b}}}$, and so the probability that a path $P\in {\mathcal{P}}^*$ is selected into the enhancement is now bounded by $2\mu^{3b}/m\leq 2\mu^{5b}/\check m'$ (since, as observed above, $m\geq \check m'/\mu^{2b}$). Another difference is that, since the promising path set ${\mathcal{P}}''$ may only be of type 2 or 3, the resulting enhancement is also guaranteed to be of type 2 or 3. We denote by $P^*_1$ and $P^*_2$ the paths of ${\mathcal{P}}^*$ that were chosen into the enhancement (where it is possible that $P^*_2=P^*_1$), and we denote the enhancement itself by $\Pi=\set{P_1,P_2}$.
Next, we define the bad events and bound their probabilities. This part is almost identical to that in Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. The main differences are that (i) the parameter $k'$ is now smaller by factor $\mu^{\Theta(1)}$, so various parameters need to be scaled by this factor; (ii) the required failure probability is smaller than that from Procedure \ensuremath{\mathsf{ProcSplit}}\xspace by factor $\mu^{\Theta(1)}$; and (iii) the drawing of $I$ induced by the drawing $\phi$ of $\tilde I$ is no longer guaranteed to be semi-clean with respect to $K$ (instead we will exploit the fact that the edges of $W$ participate in relatively few crossings in $\phi$).
\paragraph{Good Paths and Bad Event ${\cal{E}}_1'$.}
We use the following modified definition of good paths.
\begin{definition}[Good path]
We say that a path $P\in {\mathcal{P}}^*$ is \emph{good} if the following happen:
\begin{itemize}
\item the number of crossings in which the edges of $P$ participate in $\phi$ is at most $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$; and
\item there is no crossing in $\phi$ between an edge of $P$ and an edge of $W$.
\end{itemize} A path that is not good is called a \emph{bad path}.
\end{definition}
We now bound the number of bad paths in ${\mathcal{P}}^*$.
\begin{observation}\label{obs: number of bad paths2}
The number of bad paths in ${\mathcal{P}}^*$ is at most $\frac{10m}{\mu^{15b}}$.
\end{observation}
\begin{proof}
Since the paths in ${\mathcal{P}}^*$ are edge-disjoint, and every crossing involves two edges, the number of paths $P\in {\mathcal{P}}^*$ such that there are more than $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ crossings in $\phi$ in which the edges of $P$ participate, is at most $\frac{8m}{\mu^{15b}}$. Additionally, we are guaranteed that $|\chi^{\mathsf{dirty}}(\phi)|\le \frac{\check m'}{\mu^{60b}}\leq \frac m{\mu^{58b}}$ (since $m\geq \frac{\check m'}{\mu^{2b}}$). Therefore, the number of paths $P\in {\mathcal{P}}^*$, for which there is a crossing between an edge of $P$ and an edge of $\check K$ is bounded by $\frac m {\mu^{58b}}$.
Lastly, we are guaranteed that the total number of crossings in which the edges of $E(W)\setminus E(\check K)$ participate is at most $\frac{\mathsf{cr}(\phi)\mu^{26b}}{\check m'}\leq \frac{\check m'}{\mu^{17b}}\leq \frac{m}{\mu^{15b}}$, since $\mathsf{cr}(\phi)\leq \frac{(\check m')^2}{\mu^{60b}}$.
Overall, the number of bad paths in ${\mathcal{P}}^*$ is bounded by $\frac{10m}{\mu^{15b}}$.
\end{proof}
We say that bad event ${\cal{E}}_1$ happens if at least one of the paths $P_1^*,P_2^*$ that was chosen from ${\mathcal{P}}^*$ in order to construct the enhancement is bad. Since the number of bad paths is bounded by $\frac{4m}{\mu^{15b}}$, and a path of ${\mathcal{P}}^*$ is chosen to the enhancement with probability at most $\frac{2\mu^{3b}}{m}$, we immediately get the following observation.
\begin{claim}\label{claim: event 1 prob2}
$\prob{{\cal{E}}_1'}\leq 8/\mu^{12b}$.
\end{claim}
\paragraph{Heavy and Light Vertices, and Bad Event ${\cal{E}}_2'$.}
We use a parameter $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$. We say that a vertex $x\in V(G)$ is \emph{heavy} if at least $h$ paths of ${\mathcal{P}}^*$ contain $x$; otherwise, we say that $x$ is \emph{light}.
As before, we denote by $E'$ the set of all edges $e\in E(G)$, such that $e$ is incident to some light vertex $x$ that lies in $V(P^*_1)\cup V(P^*_2)$, and $e\not\in E(P^*_1)\cup E(P^*_2)$. We say that bad event ${\cal{E}}_2'$ happens if $|E'|>\frac{\mathsf{cr}(\phi)\mu^{48b}}{m}$.
We now bound the probability of bad event ${\cal{E}}_2'$.
\begin{claim}\label{claim: second bad event bound2}
$\prob{{\cal{E}}_2'}\leq 1/\mu^{12}$.
\end{claim}
\begin{proof}
Consider some light vertex $x\in V(G)$. Since $x$ lies on fewer than $h$ paths of ${\mathcal{P}}^*$, and each such path is chosen to the enhancement with probability at most $2\mu^{3b}/m$, the probability that $x$ lies in $V(P_1^*)\cup V(P^*_2)$ is bounded by $\frac{2h\mu^{3b}}{m}$.
Consider now some edge $e=(x,y)\in E(G)$. Edge $e$ may lie in $E'$ only if $x$ is a light vertex lying in $V(P_1^*)\cup V(P^*_2)$, or the same is true for $y$. Therefore, the probability that $e\in E'$ is at most $ \frac{4h\mu^{3b}}{m} $, and $\expect{|E'|}\leq 4h\mu^{3b}\leq \frac{4\mathsf{cr}(\phi)\mu^{35b}}{m}$, since $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$. From Markov's inequality, $\prob{|E'|>\frac{\mathsf{cr}(\phi)\mu^{48b}}{m}}\leq \frac{1}{\mu^{12b}}$.
\end{proof}
\paragraph{Unlucky Paths and Bad Event ${\cal{E}}_3'$.}
The definition of unlucky paths is identical to that in the proof of \Cref{thm: procsplit}, except that we use a slightly different parameter.
\begin{definition}[Unlucky Paths]
Let $x\in V(G)\setminus V(K)$ be a vertex, and let $P\in {\mathcal{P}}^*$ be a good path that contains $x$. Let $e,e'$ be the two edges of $P$ that are incident to $x$. Let $\hat E_1(x)\subseteq \delta_G(x)$ be the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies between $e$ and $e'$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$. Let $\hat E_2(x)\subseteq \delta_G(x)$ be the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies between $e'$ and $e$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$.
We say that path $P$ is \emph{unlucky with respect to vertex $x$} if either $|\hat E_1(x)|<\frac{\mathsf{cr}(\phi)\mu^{15b}}{m}$ or $|\hat E_2(x)|< \frac{\mathsf{cr}(\phi)\mu^{15b}}{m}$ holds.
We say that a path $P\in {\mathcal{P}}^*$ is an \emph{unlucky path} if there is at least one heavy vertex $x\in V(G)\setminus V(K)$, such that $P$ is unlucky with respect to $x$.
\end{definition}
We use the following claim in order to bound the number of unlucky paths. The claim
is an analogue of \Cref{claim: bound unlucky paths}, and its proof is almost identical, with minor changes due to difference in parameters. For completeness, we include a proof in Section \ref{sec: bound unlucky paths2} of Appendix.
\begin{claim}\label{claim: bound unlucky paths2}
Assume that Event ${\cal{E}}_1'$ did not happen. Then for every vertex $x\in V(G)\setminus V(K)$, the total number of good paths in ${\mathcal{P}}^*$ that are unlucky with respect to $x$ is at most $\frac{2^{14}\mathsf{cr}(\phi)\mu^{15b}}{m}$.
\end{claim}
We say that bad event ${\cal{E}}'_3$ happens if at least one of $P_1^*$, $P_2^*$ is an unlucky path.
\begin{claim}\label{claim: third event bound2}
$\prob{{\cal{E}}'_3}\leq 9/\mu^{12b}$.
\end{claim}
\begin{proof}
Clearly, $\prob{{\cal{E}}'_3}\leq \prob{{\cal{E}}'_1}+\prob{{\cal{E}}'_3\mid \neg{\cal{E}}'_1}$. From \Cref{claim: event 1 prob2},
$\prob{{\cal{E}}'_1}\leq 8/\mu^{12b}$. We now bound $\prob{{\cal{E}}_3'\mid \neg{\cal{E}}_1'}$.
Recall that a heavy vertex must have degree at least $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$ in $G$. Therefore, the total number of heavy vertices in $G$ is at most $\frac{2m}{h}\leq \frac{2m^2}{ \mathsf{cr}(\phi)\mu^{31b}}$. From \Cref{claim: bound unlucky paths2}, if Event ${\cal{E}}'_1$ did not happen, then for every heavy vertex $x\in V(G)\setminus V(K)$, there are at most $\frac{2^{14}\mu^{15b}\mathsf{cr}(\phi)}{m}$ paths in ${\mathcal{P}}^*$ that are good and unlucky for $x$. Therefore, the total number of good paths in ${\mathcal{P}}^*$ that are unlucky with respect to some heavy vertex is at most:
%
\[\frac{2m^2}{ \mathsf{cr}(\phi)\mu^{31b}}\cdot \frac{2^{14}\mu^{15b}\mathsf{cr}(\phi)}{m}
\leq \frac{2^{15}\cdot m}{\mu^{16b}}.
\]
%
Since $|{\mathcal{P}}^*|\geq \frac{4m}{\mu^{3b}}$, from \Cref{obs: number of bad paths2}, at least $ \frac{2m}{\mu^{3b}}$ paths in ${\mathcal{P}}^*$ are good.
We select two paths into the enhancement $\Pi$, and, if Event ${\cal{E}}_1'$ did not happen, every good path is equally likely to be selected. Therefore, the probability that a good path that is unlucky is selected into the enhancement, conditioned on the event ${\cal{E}}_1$ not happening is bounded by:
%
\[\frac{2^{15}\cdot m}{\mu^{16b}}\cdot \frac{\mu^{3b}}{m}\leq \frac{2^{15}}{\mu^{13b}}\leq \frac{1}{\mu^{12b}}, \]
%
since $\mu$ is large enough. We conclude that $\prob{{\cal{E}}'_3\mid \neg{\cal{E}}'_1}\leq 1/\mu^{12b}$, and $\prob{{\cal{E}}'_3}\leq 9/\mu^{12b}$.
\end{proof}
Recall that we have denoted by $E'$ the set of all edges that are incident to the light vertices of $P^*_1\cup P^*_2$, excluding the edges of $E(P^*_1)\cup E(P^*_2)$.
Denote $\tilde G'=\tilde G\setminus E'$, and let $\tilde \Sigma'$ be the rotation system for graph $G'$ induced by $\tilde \Sigma$. For convenience, we also denote $G'=G\setminus E'$ and we let $\Sigma'$ be the rotation system for graph $G'$ induced by $\tilde \Sigma$. Denote $\tilde I'=(\tilde G',\tilde \Sigma')$, and $I'-(G',\Sigma')$. Let $\Pi=\set{P_1,P_2}$ be the enhancement of skeleton $K$ that we have constructed. We denote $K'=K\cup P_1\cup P_2$, and we let $W'=W\cup P_1\cup P_2$ be the new augmentation of the original skeleton $\check K$ for graph $\check G'$. Drawing $\phi$ of graph $\tilde G=\check G'\setminus E^{\mathsf{del}}$ naturally defines a drawing of graph $\tilde G'$. Moreover, if Event ${\cal{E}}'_1$ did not happen, then there are no crossings in this drawing between the edges of $E(P_1)\cup E(P_2)$ and the edges of $E(W)$. However, it is possible that this drawing contains crossings between pairs of edges in $E(P_1)\cup E(P_2)$. In the next claim we show that, if events ${\cal{E}}'_1$ and ${\cal{E}}'_3$ did not happen, then drawing $\phi$ can be modified to obtain a drawing $\phi'$ of $\tilde G'$, in which the edges of $W'$ do not cross each other, and the induced drawing of $W$ remains identical to that induced by $\phi$.
We do so in the following claim, that is an analogue of \Cref{claim: new drawing}.
\begin{claim}\label{claim: new drawing2}
Assume that neither of the events ${\cal{E}}'_1$ and ${\cal{E}}'_3$ happened. Then there is a solution $\phi'$ to instance $\tilde I'=(\tilde G',\tilde \Sigma')$, in which there are no crossings between pairs of edges in $E(W')$. Moreover, the drawing of $W$ induced by $\phi'$ is identical to the drawing of $W$ induced by $\phi$, drawing $\phi'$ is ${\mathcal{W}}$-compatible, and, for every edge $e\in E(\tilde G')$, the number of crossings in which $e$ participates in $\phi'$ is bounded by the number of crossings in which $e$ participates in $\phi$. In particular,
$\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$ and $\chi^{\mathsf{dirty}}(\phi')\subseteq \chi^{\mathsf{dirty}}(\phi)$ hold.
\end{claim}
The proof of \Cref{claim: new drawing2} is essentially identical to the proof of \Cref{claim: new drawing}. Recall that the main idea in the proof is to remove, one by one, the loops in the images of the paths $P_1,P_2$ in the drawing $\phi$. The reason these loops can be removed without increasing the number of crossings is that every vertex whose image lies on such a loop must be a light vertex (see, e.g., \Cref{obs: no heavy vertices on loops}). This, in turn, relies on the fact that the parameter used in the definition of unlucky paths is greater than the parameter used in the definition of bad paths. While both parameters are now scaled by factor $\mu^{\Theta(b)}$, this relationship between them continues to hold, and so the analogue of \Cref{obs: no heavy vertices on loops} continues to hold as well. We also need to show that the analogue of \Cref{obs: case 3 not crossing} continues to hold, that is, there are no crossings $(e,e')_p$ in $\phi$ where $e\in E(P_1)$ and $e'\in E(P_2)$. The proof of the observation relies on the fact that that the parameter bounding the number of crossings in which the edges of a good path may participate (denoted by $N$), is less than $k'/100$, and that the number of bad paths in ${\mathcal{P}}^*$ is bounded by $k'/16$. While both parameters $N$ and $k'$ are now different, namely $N=\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ and $k'=4m/\mu^{3b}$, it is still the case that $N<k'/100$ (since, from Condition \ref{prop: drawing}, $\mathsf{cr}(\phi)\leq (\check m')^2/\mu^{60b}$, and $m\geq \check m'/\mu^{2b}$). The number of bad paths is bounded by $\frac{10m}{\mu^{15b}}<k'/16$ from \Cref{obs: number of bad paths2}. The remainder of the proof of \Cref{claim: new drawing2} is identical to the proof of \Cref{claim: new drawing} and is omitted here.
As before, drawing $\phi'$ of $\tilde G'$ is derived from drawing $\phi$ of $\tilde G$, and neither are known to our algorithm. Note that, since ${\mathcal{J}}_{i^*}$ is a valid core structure for instance $I$, the image of the paths $P_1,P_2$ of the enhancement in $\phi'$ must intersect the region $F$. If Event ${\cal{E}}_1'$ does not happen, then the images of paths $P_1,P_2$ do not cross the images of the edges of $W$ in $\phi$, and so the images of both paths are contained in region $F$. This remains true for drawing $\phi'$ of graph $\tilde G'$. Therefore, in the drawing of the new augmented skeleton $W'$ induced by $\phi'$, the images of the edges of $W'$ do not cross each other, and the images of paths $P_1,P_2$ are contained in region $F$. It is easy to verify that the images of the two paths partition face $F$ into two new regions, that we denote by $F_1$ and $F_2$.
\paragraph{Terrible Vertices and Bad Event ${\cal{E}}'_4$.}
As before, for a vertex $x\in V(G)$, we denote by $N(x)$ the number of paths in ${\mathcal{P}}^*$ containing $x$, by $N^{\operatorname{bad}}(x)$ the number of bad paths in ${\mathcal{P}}^*$ containing $x$, and by $N^{\operatorname{good}}(x)$ the number of good paths in ${\mathcal{P}}^*$ containing $x$. The definition of the notion of a terrible vertex remains the same as before:
\begin{definition}[Terrible Vertex]
A vertex $x\in V(G)$ is \emph{terrible} if it is a heavy vertex, and $N^{\operatorname{bad}}(x)\geq N^{\operatorname{good}}(x)/64$.
\end{definition}
As before, we say that a bad event ${\cal{E}}'_4$ happens if any vertex of $P^*_1\cup P^*_2$ is a terrible vertex. We bound the probability of Event ${\cal{E}}'_4$ in the following claim, whose proof is essentially identical to the proof of \Cref{claim: no terrible vertices}. Since we use slightly different parameters, we provide the proof here for completeness.
\begin{claim}\label{claim: no terrible vertices2}
$\prob{{\cal{E}}_4'}\leq \frac{2^{14}\log m}{\mu^{12b}}$.
\end{claim}
\begin{proof}
Let $U$ be the set of all heavy vertices of $G$. We group the vertices $x\in U$ geometrically into classes, using the parameter $N(x)$. Recall that $h=\frac{\mathsf{cr}(\phi)\mu^{31b}}{m}$, and, for every heavy vertex $x\in U$, $N(x)\geq h$. Let $q=2\ceil{\log m}$. For $1\leq i\leq q$, we let the class $S_i$ contain all vertices $x\in U$ with $2^{i-1}\cdot h\leq N(x)<2^i\cdot h$.
Let $S'_i\subseteq S_i$ be the set containing all terrible vertices of $S_i$. For every vertex $x\in S'_i$, $N^{\operatorname{bad}}(x)\geq N^{\operatorname{good}}(x)/64$. Since $N(x)=N^{\operatorname{bad}}(x)+N^{\operatorname{good}}(x)$, we get that $N^{\operatorname{bad}}(x)\geq N(x)/65\geq 2^{i-1}\cdot h/65$. From \Cref{obs: number of bad paths2}, the total number of bad paths is at most $\frac{10m}{\mu^{15b}}$, so
%
\[|S'_i|\leq \frac{10m/\mu^{15b}}{2^{i-1}\cdot h/65}\leq \frac{650m}{ 2^{i-1}\cdot h\cdot \mu^{15b}}.\]
Recall that the probability that a path $P\in {\mathcal{P}}^*$ is selected into the enhancement is at most $\frac{2\mu^{3b}}{m}$. A vertex $x\in S'_i$ may lie in $V(P^*_1)\cup V(P^*_2)$ only if at least one of the $N(x)$ paths of ${\mathcal{P}}^*$ containing $x$ is selected into the enhancement. Therefore, $\prob{x\in (V(P^*_1)\cup V(P^*_2))}\leq \frac{2\mu^{3b}\cdot N(x)}{m}\leq \frac{2^{i+1}\cdot h\cdot \mu^{3b}}{m}$.
Overall, the probability that some vertex of $S'_i$ belongs to $V(P^*_1)\cup V(P^*_2)$ is at most
$$|S'_i|\cdot \frac{2^{i+1}\cdot h\cdot \mu^{3b}}{m}\leq \frac{650m}{ 2^{i-1}\cdot h\cdot \mu^{15b}}\cdot \frac{2^{i+1}\cdot h\cdot \mu^{3b}}{m}\leq \frac{2^{12}}{\mu^{12b}}.$$
Using the union bound over all $q=2\ceil{\log m}$ classes, we get that the probability of Event ${\cal{E}}'_4$ is bounded by $\frac{2^{14}\log m}{\mu^{12b}}$.
\end{proof}
\paragraph{Bad Event ${\cal{E}}'$.}
Let ${\cal{E}}'$ be the bad event that at least one of the events ${\cal{E}}'_1,{\cal{E}}'_2,{\cal{E}}'_3,{\cal{E}}'_4$ happen. From the Union Bound and Claims \ref{claim: event 1 prob2}, \ref{claim: second bad event bound2}, \ref{claim: third event bound2} and \ref{claim: no terrible vertices2}, $\prob{{\cal{E}}'}\leq O\textsf{left}( \frac{\log m}{\mu^{12b}}\textsf{right} ) \leq \frac{1}{\mu^{11b}}$, by definition of $\mu$.
\subsubsection{Step 2: Completing the Construction of ${\mathcal{W}}'$ and ${\mathcal{G}}'$}
In this step we complete the construction of the skeleton augmenting structure ${\mathcal{W}}'$ and of the ${\mathcal{W}}'$-decomposition of $\check G'$ ${\mathcal{G}}'$. This step is very similar to the second step of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace, with several key differences. The first difference is that we no longer need to take care of the case where $\Pi$ is a type-1 enhancement. The second difference is that the parameters are now somewhat different. The third difference, which is more major than the first two, is that we will not be able to guarantee that the final drawing of the graph $\tilde G\setminus E^{\mathsf{del}}(F)$ that we obtain, induces drawings of graphs $G_{F_1},G_{F_2}$ that are semi-clean with respect to their skeletons. We will still, however, need to ensure that the drawing is ${\mathcal{W}}'$-compatible, so every core of $\set{J_1,\ldots,J_z}$ is drawn in the ``correct'' face (for example, if $J_i\subseteq G_{F_1}$, then core $J_i$ should be drawn inside). This will require an additional step, in which we disconnect some of the cores from the corresponding graph $G_1$ or $G_2$.
We consider the instance $I'=(G',\Sigma')$, where $G'=\tilde G\setminus E'$, and $\Sigma'$ is the rotation system for $G'$ induced by $\check \Sigma'$. We will also consider the solution $\phi'$ to instance $I'$ given by \Cref{claim: new drawing} (assuming that bad event ${\cal{E}}'$ did not happen).
We start by computing an orientation $b'_u$ for every vertex $u\in V(W')$, and a drawing $\rho_{C'}$ of every connected component $C'\in {\mathcal{C}}(W')$. We then split graph $G$ into two subgraphs $G_1$ and $G_2$, and complete the construction of the skeleton augmenting structure. Lastly, we will remove some additional edges from the resulting graph, to ensure that there is a good drawing of that graph that is ${\mathcal{W}}'$-compatible.
We consider two cases, depending on whether the original promising path ${\mathcal{P}}$ was of type $2$ or $3$.
\subsubsection*{Case 1: ${\mathcal{P}}$ was a Type-2 Promising Path Set}
The first case happens when the promising path set ${\mathcal{P}}$ is of type 2. Recall that the endpoints of all paths in ${\mathcal{P}}$ are contained in $V(J)$, and that we have computed a partition $(E_1,E_2)$ of the edges of $\delta_G(J)$, so the edges of $E_1$ are consecutive in the ordering ${\mathcal{O}}(J)$, and every path of ${\mathcal{P}}^*$ has an edge of $E_1$ as its first edge and an edge of $E_2$ as its last edge. Recall that, in this case, $\Pi=\set{P_1,P_2}$, where $P_1\in {\mathcal{P}}^*$ is either a path or a cycle, and $P_2=\emptyset$. For convenience of notation, we denote $P_1$ by $P$.
Recall that, from \Cref{claim: new drawing2}, if Event ${\cal{E}}'$ does not happen, then drawing $\phi'$ of $G'=G\setminus E'$ contains no crossings between the edges of $E(W')$.
Throughout, we denote by $\gamma$ the image of the path $P$ in $\phi'$.
In order to construct the skeleton augmenting structure ${\mathcal{W}}'$, we first let $W'=W\cup P$ be the skeleton augmentation associated with ${\mathcal{W}}'$. Next, we define a drawing $\rho_C$ for every component $C\in {\mathcal{C}}(W)$.
Let $C_{i^*}\in {\mathcal{C}}(W)$ be the connected component with $J\subseteq C_{i^*}$, and denote by $\tilde C=C_{i^*}\cup P$. Since path $P$ is internally disjoint from all vertices of $W$, it is immediate to verify that ${\mathcal{C}}(W')=({\mathcal{C}}(W)\setminus C_{i^*})\cup \tilde C)$. For every component $C\in {\mathcal{C}}(W')\setminus \set{\tilde C}$, its drawing $\rho_C$ in ${\mathcal{W}}'$ remains the same as in ${\mathcal{W}}$.
Consider now the component $\tilde C$. Recall that we are given, as part of skeleton augmenting structure ${\mathcal{W}}$ a drawing $\rho_{C_{i^*}}$ of graph $C_{i^*}$ and an orientation $b'_u$ of every vertex $u\in V(C_{i^*})$. We are also given a face $F^{i^*}\in {\mathcal{F}}(\rho_{C_{i^*}})$, such that the face $F$ in drawing $\phi$ is contained in the region $F^{i^*}$. Since drawing $\phi$ is ${\mathcal{W}}$-compatible, and it does not contain any crossings between the edges of $W'$, we are guaranteed that path $P$ is drawn inside the region $F^{i^*}$ of $\phi'$, in a natural way. Since the orientations of all vertices of $C_{i^*}$ in $\phi'$ are consistent with those given by ${\mathcal{W}}$, and since the drawing of $C_{i^*}$ induced by $\phi'$ is $\rho_{C_{i^*}}$, we can efficiently compute a drawing $\rho_{\tilde C}$ of graph $\tilde C$, that is identical to the drawing of $\tilde C$ induced by $\phi'$. This drawing is the unique planar drawing of of $\tilde C$ that has the following properties:
\begin{itemize}
\item the drawing of $C_{i^*}$ induced by $\rho_{\tilde C}$ is $\rho_{C_{i^*}}$;
\item the image of path $P$ is contained in region $F^{i^*}$;
\item the drawing is consistent with the rotation system $\check \Sigma'$; and
\item the orientation of every vertex $u\in V(C_{i^*})$ in the drawing is the orientation $b'_u$ given by ${\mathcal{W}}$.
\end{itemize}
Notice that the set ${\mathcal{F}}(\rho_{\tilde C})$ of faces of this new drawing of graph $\tilde C$ is precisely $({\mathcal{F}}(\rho_{C_{i^*}})\setminus F^{i^*})\cup (F^{i^*}_1,F^{i^*}_2)$, where $F^{i^*}_1$ and $F^{i^*}_2$ are the two new faces that are obtained by partitioning the face $F^{i^*}$ with the image of path $P$. We assume w.l.o.g. that region $F^{i^*}_1$ in the drawing of $W'$ induced by $\phi'$ contains the face $F_1$ of this drawing, and region $F^{i^*}_2$ contains the face $F_2$.
If $F^{i^*}=F^*(\rho_{C_{i^*}})$ is the infinite face of the drawing $\rho_{C_{i^*}}$, then we will designate one of the faces $F^{i^*}_1$ or $F^{i^*}_2$ as the infinite face of the drawing $\rho_{\tilde C}$ later.
So far we have defined the new skeleton augmentation $W'$ and a drawing $\rho_C$ of every connected component $C\in {\mathcal{C}}(W)$. Next, we define the orientation $b'_u$ of every vertex $u\in V(W')$. For vertices $u\in V(W')\cap V(W)$, the orientation $b'_u$ in ${\mathcal{W}}'$ remains the same as the orientation $b'_u$ in ${\mathcal{W}}$. It now remains to define the orientation $b'_u$ of vertices $u\in V(P)\setminus V(C_{i^*})$. The procedure for computing these orienttions is very similar to the one from Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace$.
We first recall the definition of the circular ordering ${\mathcal{O}}(J)$ of the edges of $\delta_G(J)$. In order to define the ordering, we considered the disc $D(J)$ in the drawing $\rho_{J}$ of $J$ defined by the core structure ${\mathcal{J}}_{i^*}$. Recall that the drawing $\rho_{J}$ is the drawing of $J$ induced by the drawing $\rho_{C_{i^*}}$ of $C_{i^*}$. In this drawing, the orientation of every vertex $u\in V(J)$ is $b'_u$. We have defined, for every edge $e\in \delta_G(J)$, a point $p(e)$ on the boundary of the disc $D(J)$, and we let ${\mathcal{O}}(J)$ be the circular ordering of the edges of $\delta_G(J)$, in which the points $p(e)$ corresponding to these edges appear on the boundary of the disc $D(J)$, as we traverse it in counter-clock-wise direction. If the drawing of $J$ in $\phi'$ is identical to $\rho_{J}$ (including the orientation), then we say that the orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ is $1$. In this case, we are guaranteed that for every vertex $u\in V(J)$, its orientation in $\phi'$ is $b'_u$. Otherwise, the drawing of $J$ in $\phi'$ is the mirror image of $\rho_{J}$. We then say that the orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ is $-1$. In this case, we are guaranteed that for every vertex $u\in V(J)$, its orientation in $\phi'$ is $-b'_u$.
For every vertex $u\in V(P)\cup V(J)$, we consider the tiny $u$-disc $D_{\phi'}(u)$ in the drawing $\phi'$. For every edge $e\in \delta_{G}(u)\setminus E'$, we denote by $\sigma(e)$ the segment of $\phi'(e)$ that is drawn inside the disc $D_{\phi'}(u)$. Since edge $e$ belongs to graph $G$, $\sigma(e)$ must be contained in the region $F\in {\mathcal{F}}(W)$ of the drawing $\phi'$. Let $\tilde E=\textsf{left} (\bigcup_{u\in V(P)\cup V(J)}\delta_{G'}(u)\textsf{right} )\setminus (E(P)\cup E(J))$.
The image of path $P$ in $\phi'$ splits region $F$ into two regions, that we denote by $F_1$ and $F_2$. We let $e_1\in E_1$ be any edge that does not lie on path $P$, and we assume without loss of generality that $\sigma(e_1)\subseteq F_1$.
We partition edge set $\tilde E$ into a set $\tilde E^{\mathsf{in}}$ of \emph{inner edges} and the set $\tilde E^{\mathsf{out}}$ of outer edges, as before: Edge set $\tilde E^{\mathsf{in}}$ contains all edges $e\in \tilde E$ with $\sigma(e)$ contained in the region $F_1$ of $\phi'$, and $\tilde E^{\mathsf{out}}$ contains all remaining edges (so for every edge $e\in \tilde E^{\mathsf{out}}$, $\sigma(e)$ is contained in $F_2$). As before, we show an algorithm that correctly computes the orientation of every vertex $u\in V(P)$ in the drawing $\phi'$, and the partition $(\tilde E^{\mathsf{in}},\tilde E^{\mathsf{out}})$ of the edges of $\tilde E$.
\paragraph{Computing Vertex Orientations on Path $P$ and the Partition $(\tilde E^{\mathsf{in}},\tilde E^{\mathsf{out}})$.}
The algorithm for computing vertex orientations for vertices of $P$, and the partition $(\tilde E^{\mathsf{in}},\tilde E^{\mathsf{out}})$ of the edges of $\tilde E$ is identical to that from Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. The main difference is the change in the parameters. We provide the algorithm and its analysis for completeness.
Consider any vertex $u\in V(P)$. Let $\hat e(u)$, $\hat e'(u)$ be the two edges of $P$ that are incident to $u$, where we assume that $\hat e(u)$ appears earlier on $P$ (we assume that $P$ is directed from an edge of $E_1$ to an edge of $E_2$). Edges $\hat e(u),\hat e'(u)$ partition the edge set $\delta_{G'}(u)\setminus\set{\hat e(u),\hat e'(u)}$ into two subsets, that we denote by $\hat E_1(u)$ and $\hat E_2(u)$, each of which appears consecutively in the rotation ${\mathcal{O}}_u\in \Sigma'$.
Note that either (i) $\hat E_1(u)\subseteq \tilde E^{\mathsf{in}}$ and $\hat E_2(u)\subseteq \tilde E^{\mathsf{out}}$ holds, or (ii) $\hat E_2(u)\subseteq \tilde E^{\mathsf{in}}$ and $\hat E_1(u)\subseteq \tilde E^{\mathsf{out}}$ holds.
We now construct edge sets $\tilde E_1',\tilde E_2'$, and fix an orientation $b'_u$ for every vertex $u\in V(P)\setminus V(J)$. We then show that $\tilde E_1'=\tilde E^{\mathsf{in}}$, $\tilde E_2=\tilde E^{\mathsf{out}}$, and that the orientations of all vertices of $P$ that we compute are consistent with the drawing $\phi'$.
Recall that we have computed a drawing $\rho_{\tilde C}$ of $\tilde C=C_{i^*}\cup P$, that is identical to the drawing of $\tilde C$ induced by $\phi'$. We let $\tilde \rho$ be the drawing of $J\cup P$ that is induced by $\rho_{\tilde C}$.
Since we have assumed that $\sigma(e_1)$ is contained in the face $F_1$, for every edge $e\in \delta_G(J)$, we can correctly determine whether $e\in \tilde E^{\mathsf{in}}$ or $e\in \tilde E^{\mathsf{out}}$. In the former case, we add $e$ to $\tilde E_1'$, and in the latter case we add $e$ to $\tilde E_2'$.
Therefore, for every path $P'\in {\mathcal{P}}^*$, the first and the last edges of $P$ are already added to either $\tilde E_1'$ or $\tilde E_2'$.
Next, we process every inner vertex $u$ on path $P$.
Consider any such vertex $u$. As before, if $u$ is a light vertex, then its orientation $b'_u$ can be chosen arbitrarily, as $\deg_{\tilde G'}(u)=2$.
Assume now that $u$ is a heavy vertex.
In order to decide on the orientation of $u$, we let ${\mathcal{P}}(u)$ contain all paths $P'\in {\mathcal{P}}^*\setminus\set{P}$ with $u\in P'$. We partition the set ${\mathcal{P}}(u)$ of paths into four subsets: set ${\mathcal{P}}_1(u)$ contains all paths $P'$ whose first edge lies in $\tilde E_1'$, and the first edge that is incident to $u$ lies in $\hat E_1(u)$. Set ${\mathcal{P}}_2(u)$ contains all paths $P'$ whose first edge lies in $\tilde E_2'$, and the first edge that is incident to $u$ lies in $\hat E_2(u)$. Similarly, set ${\mathcal{P}}'_1(u)$ contains all paths $P'\in {\mathcal{P}}(u)$ whose first edge lies in $\tilde E_1'$ and the first edge that is incident to $u$ lies in $\hat E_2(u)$, while set ${\mathcal{P}}'_2(u)$ contains all paths $P'\in {\mathcal{P}}'(u)$, whose first edge lies in $\tilde E_2'$ and the first edge that is incident to $u$ lies in $\hat E_1(u)$. We let $w(u)=|{\mathcal{P}}_1|+|{\mathcal{P}}_2|$, and $w'(u)=|{\mathcal{P}}'_1|+|{\mathcal{P}}'_2|$.
If $w(u)\leq w'(u)$, then we set $b'_u=-1$, add the edges of $\hat E_1(u)$ to $\tilde E'_1$, and add the edges of $\hat E_2(u)$ to $\tilde E'_2$. Otherwise we set $b'_u=1$, add the edges of $\hat E_1(u)$ to $\tilde E'_2$, and add the edges of $\hat E_2(u)$ to $\tilde E'_1$.
This completes the algorithm for computing the orientations of the inner vertices of $P$, and of the partition $(\tilde E_1',\tilde E_2')$ of the edge set $\tilde E$.
We use the following claim to show that both are computed correctly.
\begin{claim}\label{claim: orientations and edge split is computed correctly Case2.2}
Assume that Event ${\cal{E}}'$ did not happen. If the orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ is $1$, then for every vertex $u\in V(P)\cup V(C_{i^*})$, the orientation of $u$ in $\phi'$ is $b'_u$.
Otherwise, for every vertex $u\in V(P)\cup V(C_{i^*})$, the orientation of $u$ in $\phi'$ is $-b_u$. In either case,
$\tilde E_1'=\tilde E^{\mathsf{in}}$ and $\tilde E_2'=\tilde E^{\mathsf{out}}$.
\end{claim}
\begin{proof}
We assume without loss of generality that the orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ is $1$. The argument for the other case is symmetric. From the above discussion, for every vertex $u\in V(J)$, the orientation of $u$ in $\phi'$ is $b'_u$. It is now enough to show that, if $u\in P$ is a heavy vertex, then the orientation of $u$ in $\phi'$ is $b_u$.
Recall that we denoted $N(u)=|{\mathcal{P}}(u)|$, and we have denoted by $N^{\operatorname{bad}}(u)$ and $N^{\operatorname{good}}(u)$ the total number of bad and good paths in ${\mathcal{P}}(u)$, respectively. Since we have assumed that bad event ${\cal{E}}_4'$ did not happen, vertex $u$ is not a terrible vertex, that is, $N^{\operatorname{bad}}(u)\leq N^{\operatorname{good}}(u)/64$. Since $N(u)=N^{\operatorname{bad}}(u)+N^{\operatorname{good}}(u)$, we get that $N^{\operatorname{bad}}(u)<N(u)/65$.
Assume first that the orientation of vertex $u$ in $\phi'$ is $-1$. In this case, for every edge $e\in \hat E_1(u)\cup \tilde E_1'$, the segment $\sigma(e)$ lies in face $F_1$ of $\tilde \rho$, while for every edge $e'\in \hat E_2(u)\cup \tilde E_2'$, the segment $\sigma(e')$ lies in face $F_2$.
We claim that in this case $w'(u)>w(u)$ must hold, and so our algorithm sets $b_u=-1$ correctly. Indeed, assume otherwise.
Then $w'(u)\geq N(u)/2$. Let ${\mathcal{Q}}$ denote the set of all good paths in ${\mathcal{P}}'_1(u)\cup {\mathcal{P}}'_2(u)$. Then $|{\mathcal{Q}}|\geq w'(u)-N^{\operatorname{bad}}(u)\geq N(u)/2-N(u)/65\geq N(u)/4\geq h/4$, since $u$ is a heavy vertex.
We now show that, for every path $Q\in {\mathcal{Q}}$, there must be a crossing between an edge of $Q$ and an edge of $P$ in $\phi'$.
Indeed, consider any path $Q\in {\mathcal{Q}}$. Since $Q\in {\mathcal{P}}'_1(u)\cup {\mathcal{P}}'_2(u)$, the image of the path $Q$ must cross the boundary of the face $F_1$ in $\rho$. Since path $Q$ is a good path, and it does not contain vertices of $J$ as inner vertices, no inner point of the image of $Q$ in $\phi'$ may belong to the image of $J$ in $\phi'$. Since the paths in ${\mathcal{P}}$ are transversal, it then follows that there must be a crossing between an edge of $Q$ and an edge of $P$ in $\phi'$. But then the edges of $P$ participate in at least $\frac h 2\geq \frac{\mathsf{cr}(\phi)\mu^{31b}}{2m}$ crossings in $\phi'$, and hence in $\phi$.
However, since we have assumed that bad event ${\cal{E}}_1$ did not happen, $P$ is a good path, and so its edges may participate in at most $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ crossings in $\phi$, a contradiction. Therefore, when the orientation of $u$ in $\phi'$ is $-1$, our algorithm correctly sets $b_u=-1$.
In the case where the orientation of $u$ in $\phi'$ is $1$, the analysis is symmetric.
\end{proof}
In order to complete the construction of the skeleton augmenting structure ${\mathcal{W}}'$, it remains to define, for every pair $C,C'\in {\mathcal{C}}(W')$ of distinct connected components, a face $R_C(C')$ of ${\mathcal{F}}(\rho_C)$. Additionally, if face $F^{i^*}\in {\mathcal{F}}(\rho_{C_{i^*}})$ is the outer face of the drawing $\rho_{C_{i^*}}$ of component $C_{i^*}$, then we need to designate one of the faces $F^{i^*}_1$ or $F^{i^*}_2$ as the outer face of the drawing $\rho_{\tilde C}$ of component $\tilde C$. We do so in the next step, once we split the graph $G'$ into two subgraphs, $G_1$ and $G_2$.
\paragraph{Computing the Split.}
We construct a flow network $\tilde H'$ as follows. We start with $\tilde H'=G'$, and then, for all $1\leq i\leq z$ with $i\neq i^*$, we contract all vertices of the core $J_i$ into a supernode $v_{J_i}$. Next, we subdivide every edge $e\in \tilde E$ with a vertex $t_e$, and denote $T_1=\set{t_e\mid e\in \tilde E_1'}$, $T_2=\set{t_e\mid e\in \tilde E_2'}$. We delete all vertices of $P\cup J$ and their adjacent edges from the resulting graph, contract all vertices of $T_1$ into a source vertex $s$, and contract all vertices of $T_2$ into a destination vertex $t$. We then compute a minimum $s$-$t$ cut $(A,B)$ in the resulting flow network $\tilde H'$, and we denote by $E''=E_{\tilde H'}(A,B)$. The following claim, that is an analogue of \Cref{claim: cut set small case2}, bounds the cardinality of $E''$.
\begin{claim}\label{claim: cut set small case2.2}
If bad event ${\cal{E}}'$ did not happen, then $|E''|\leq \frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
\end{claim}
The proof of the claim is essentially identical to the proof of \Cref{claim: cut set small case2.2}. We assume for contradiction that Event ${\cal{E}}'$ did not happen, but $|E''|> \frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
We can then use Max Flow -- Min Cut theorem to construct a collection ${\mathcal{Q}}$ of edge-disjoint paths in graph $G'$, such that, for every path $Q\in {\mathcal{Q}}$, drawing $\phi'$ must contain at least one crossing between an edge of $Q$ and an edge of $W\cup P$. Since Event ${\cal{E}}'$ did not happen, path $P$ is good, so its edges participate in at most $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}$ crossings in $\phi$ and hence in $\phi'$. Additionally, edges of $\check K$ participate in at most $|\chi^{\mathsf{dirty}}(\phi)|$ crossings in $\phi$ (and hence in $\phi'$), and edges of $W\setminus K$ participate in at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{26b}}{\check m'}\leq \frac{\mathsf{cr}(\phi)\cdot \mu^{26b}} m$ crossings from Property \ref{prop: drawing2}. Overall, there are fewer than $\frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$ crossings in $\phi'$ in which edges of $P\cup W$ participate, a contradiction.
We now construct the graphs $G_1,G_2$ associated with the faces $F_1$ and $F_2$, respectively.
In order to do so, we define two vertex sets $A',B'$ of graph $G'$, as before. We start with $A'=A\setminus\set{s}$ and $B'=B\setminus\set{t}$. For every index $1\leq i\leq r$ with $i\neq i^*$, if $v_{J_i}\in A$, then we replace vertex $v_{J_i}$ with vertex set $V(J_i)$ in $A'$, and otherwise we replace $v_{J_i}$ with vertex set $V(J_i)$ in $B'$.
Additionally, we let $J\subseteq J\cup P$ be the graph containing all vertices and edges of $J\cup P$, whose images in $\rho_{\tilde C}$ lie on the boundary of face $F^{i^*}_1$. Similarly, we let $J'\subseteq J\cup P$ be the graph containing all vertices and edges of $J\cup P$, whose images in $\rho_{\tilde C}$ lie on the boundary of face $F^{i^*}_2$.
We then let $G_1$ be the subgraph of $G'$, whose vertex set is $V(A')\cup V(J)$, and edge set contains all edges of $E_{G'}(A')$, $E_{G'}(A',V(J))$, and all edges of $J$. Similarly, we let $G_2$ be the subgraph of $G'$, whose vertex set is $V(B')\cup V(J')$, and edge set contains all edges of $E_{G'}(B')$, $E_{G'}(B',V(J'))$, and all edges of $J'$.
The rotation system $\Sigma_1$ for graph $G_1$ and the rotation system $\Sigma_2$ for graph $G_2$ are induced by $\Sigma'$.
Let $I_1=(G_1,\Sigma_1)$ and $I_2=(G_2,\Sigma_2)$ be the resulting two instances of \ensuremath{\mathsf{MCNwRS}}\xspace.
We need the following observation that, intuitively, shows that graphs $G_1$ and $G_2$ are significantly smaller than the original graph $G$.
\begin{observation}\label{obs: few edges in split graphs case2.2}
If Event ${\cal{E}}'$ did not happen, then $|E(G_1)\setminus E(W')|,|E(G_2)\setminus E(W')|\leq |E(G)\setminus E(W)|-\frac{\check m'}{\mu^{5b}}$.
\end{observation}
The proof of the observation is identical to the proof of \Cref{obs: few edges in split graphs case2}, and follows from the fact that, from the definition of the process for computing an enhancement, at least $k'/4$ edges of $E_1$ must lie in $\tilde E_1'$, and at least $k'/4$ edges of $E_1'$ must lie in $\tilde E_2$. Since $k'=\ceil{\frac{4m}{\mu^{3b}}}$ and $m\geq \check m'/\mu^{2b}$, the claim follows.
Let $E^{\mathsf{del}}(F)=E'\cup E''$. If bad event ${\cal{E}}'$ did not happen, then, from the definition of bad event ${\cal{E}}_2'$, $|E'|\leq \frac{\mathsf{cr}(\phi)\mu^{48b}}{m}$ and, from \Cref{claim: cut set small case2.2}, $|E''|\leq \frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
Since $\mathsf{cr}(\phi)\leq g_1$ and $|\chi^{\mathsf{dirty}}(\phi)|\leq g_2$, we get that, if bad event ${\cal{E}}'$ did not happen, then $|E^{\mathsf{del}}(F)|\leq \frac{g_1\cdot \mu^{48}}{m}+g_2$, as required.
Let $\tilde G''=\tilde G'\setminus E''=\tilde G\setminus E^{\mathsf{del}}$, and let $\tilde \Sigma''$ be the rotation system for graph $\tilde G''$ induced by $\tilde \Sigma$. We denote $\tilde I''=(\tilde G'',\tilde \Sigma'')$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace.
Let $\phi''$ be the drawing of $\tilde G''$ that is induced by the drawing $\phi'$ of $\tilde G'$. We also denote $G''=G'\setminus E''=G\setminus E^{\mathsf{del}}$. Notice that $E(G_1)\cup E(G_2)=E(G'')$.
We now complete the construction of the skeleton augmenting structure ${\mathcal{W}}'$. For all $1\leq i\leq z$, let $C_i\in {\mathcal{C}}(W)$ be the connected component that contains the core $J_i$, and let $F^i\in {\mathcal{F}}(\rho_{C_i})$ be the face that corresponds to face $F$ of $\tilde {\mathcal{F}}({\mathcal{W}})$ (in other words, face $F$ in the drawing of $W$ induced by $\phi$ is contained in region $F^i$). From the construction of the set $\tilde {\mathcal{F}}({\mathcal{W}})$ of faces, in the decomposition forest $H({\mathcal{W}})$, either vertices $x_{C_1},\ldots,x_{C_z}$ are the roots of the trees in the forest; or one of these vertices is the parent-vertex of the remaining vertices. We assume w.l.o.g. that in the latter case, vertex $x_{C_1}$ is the parent of vertices $x_{C_2},\ldots,x_{C_z}$ in the forest $H({\mathcal{W}})$.
Consider any pair $C,C'\in {\mathcal{C}}(W')$ of distinct connected components. Assume first that $C,C'\neq \tilde C$. In this case, face $R_{C}(C')$ remains the same in ${\mathcal{W}}'$ in the augmenting structure ${\mathcal{W}}$. Since drawing $\phi$ of $\tilde G$ was ${\mathcal{W}}$-compatible, and drawing $\phi''$ is obtained from $\phi$ by deleting some edges that do not belong to $W'$, and possibly making some additional local changes within the region $F$, the image of component $C'$ in $\phi''$ remains contained in region $R_C(C')$.
Assume now that $C'=\tilde C$. In this case, we let the face $R_{C'}(\tilde C)$ in ${\mathcal{W}}'$ be the face $R_{C'}(C_{i^*})$ of ${\mathcal{W}}$. It is easy to verify that the drawing of component $\tilde C$ in $\phi''$ remains in the region $R_{C'}(C_{i^*})$.
Next, we assume that $C=\tilde C$. Recall that component $C_{i^*}$ of $W$ is replaced by component $\tilde C=C_{i^*}\cup P$ in $W'$.
Assume first that in ${\mathcal{W}}$, $R_{C_{i^*}}(C')\neq F^{i^*}$. In this case, face $R_{C_{i^*}}(C')$ remains a face of $\rho_{\tilde C}$, and the image of component $C'$ appears in region $R_{C_{i^*}}(C')$ in $\phi'$. We then set $R_{\tilde C}(C')=R_{C_{i^*}}(C')$ in ${\mathcal{W}}'$.
It now remains to consider pairs $C,C'$ of components of $W'$ with $C=\tilde C$, such that $R_{C_{i^*}}(C')=F^{i^*}$ in ${\mathcal{W}}$. We start with the following observation.
\begin{observation}\label{obs: cores drawn correctly} Assume that Event ${\cal{E}}'$ did not happen.
Consider any index $1\leq i\leq z$ with $i\neq i^*$. If $J_i\subseteq G_1$, then the image of $C_i$ appears in region $F_1^{i^*}$ in $\phi''$, and if $J_i\subseteq G_2$, then the image of $C_i$ appears in region $F_2^{i^*}$ in $\phi''$.
\end{observation}
\begin{proof}
Assume for contradiction that Event ${\cal{E}}'$ did not happen, and there is an index $1\leq i\leq z$ with $J_i\subseteq G_1$, such that the image of $C_i$ is not contained in region $F_1^{i^*}$ of $\phi''$.
Let $E^*$ be the set of edges $e\in E(\tilde G'')$, such that the image of $e$ in $\phi''$ crosses the image of some edge of $W'$. Recall that $|\chi^{\mathsf{dirty}}(\phi)|\leq g_2$ and the total number of crossings of $\phi$ in which the edges of $E(W)\setminus E(\check K)$ participate is at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{26b}}{\check m'}$, from Property \ref{prop: drawing}. If Event ${\cal{E}}'$ did not happen, then the number of crossings in which the edges of $P$ participate is at most $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}\leq \frac{\mathsf{cr}(\phi)\mu^{17b}}{\check m}$. Therefore, overall, $|E^*|\leq g_2+\frac{2\mathsf{cr}(\phi)\cdot \mu^{26b}}{\check m'}\leq \textsf{left} (\frac{g_1}{\check m'}+g_2\textsf{right} )\cdot \mu^{100b}$.
From Property \ref{prop: new}, there must be a path $P'$ in graph $\tilde G'\setminus E^*$ connecting a vertex of $C_i$ to a vertex of $C_{i^*}\cup P$. We assume w.l.o.g. that $P'$ does not contain any vertex of $V(C_i)\cup V(C_{i^*})\cup V(P)$ as an inner vertex. We will reach a contradiction by showing that there is some edge $e\in E(P')$, whose image crosses the image of an edge of $J_{i^*}\cup P$ in $\phi''$. This is impossible because then $e$ should have been added to $E^*$.
Let $\psi$ be a drawing obtained from drawing $\phi''$ by deleting the images of all edges of $E^*$ from the drawing. We view $\psi$ as a drawing on the sphere, and we define, for all $1\leq \ell\leq z$, a disc $D(J_{\ell})$, whose boundary follows the image of the core $J_{\ell}$ inside the face $F$, as before. Note that the image of the connected component $C_{\ell}$ of $W$ must be contained in the disc $D(J_{\ell})$.
Since path $P'$ originates at a vertex of $C_i$ and terminates at a vertex of $C_{i^*}\cup P$, its image must intersect region $F$. We denote by $Q_1,Q_2,\ldots,Q_h$ maximal subpaths of the path $P'$, such that for all $1\leq h'\leq h$, the image of the path $Q_{h'}$ in $\psi$ is contained in the interior of region $F$, except for its endpoints that lie on the boundary of $F$. We assume that paths $Q_1,\ldots,Q_h$ appear on path $P'$ in this order. For all $1\leq h'\leq h$, we denote by $1\leq a_{h'}\leq z$ the index for which the first vertex of $Q_{h'}$ belongs to core $J_{a_{h'}}$, so $a_1=i$. From the definition of the paths $Q_1,\ldots,Q_h$, for all $1\leq h'\leq h$, the last vertex of $Q_{h'-1}$ must also belong to core $J_{a_{h'}}$, and the last vertex of $Q_h$ must belong to and the last vertex of $Q_h$ must belong to $J_{i^*}\cup J_{i^{**}}\cup P\cup P'$.
Consider now some index $1\leq h'\leq h$. Let $e_{h'}$ be the first edge on path $Q_{h'}$, and let $e'_{h'}$ be the last edge on path $Q_{h'}$. Since the images of the edges of path $P'$ in $\psi$ may not cross the images of any edges of $W$, and the image of $P'$ intersects the interior of $F$, we get that $\psi(e_{h'})\subseteq F$. Moreover, since an endpoint of $e_{h'}$ belongs to $J_{a_{h'}}$, from the definition of valid core structure (see \Cref{def: valid core 2}), $e_{h'}\in E(G)$ must hold. Using a similar reasoning, $e'_{h'}\in E(G)$. Since no inner vertex of $Q_{h'}$ belongs to $J_{i^*}\cup P$, and since $V(G_1)\cap V(G_2)\subseteq V(J_{i^*})\cup V(P)$, it must be the case that either $Q_{h'}\subseteq G_1$, or $Q_{h'}\subseteq G_2$. Since $J_i=J_{a_1}\subseteq G_1$, and since, for all $1\leq h'< h$ the last vertex of $Q_{h'}$ and the first vertex of $Q_{h'+1}$ belong to the same core $J_{a_{h'}}$, it must be the case that for all $1\leq h'\leq h$, $Q_{h'}\subseteq G_1$.
Recall that the image $\rho_{\tilde C}$ of graph $\tilde C=C_{i^*}\cup P$ has two faces, that we denoted by $ F^{i^*}_1$ and $ F^{i^*}_2$, that correspond to regions $F_1$ and $F_2$ of the drawing of $W'$ induced by $\phi'$, respectively. Clearly, for all $1\leq h'\leq h$, the image of $J_{h'}$ is either contained in $F^{i^*}_1$, or it is contained in $ F^{i^*}_2$. The image of $J_{i}$ is contained by $ F^{i^*}_2$ from our assumption that it is not contained in $F^{i^*}_1$. However, if we denote by $e^*$ the last edge on path $Q_h$, then the image of $e^*$ must be contained in $ F^{i^*}_1$, since $e^*\in E(G_1)$, and so $e^*\in \tilde E_1$ must hold.
We conclude that the image of some path $Q_{h'}$ must intersect both regions $F^{i^*}_1$ and $F^{i^*}_2$, which may only happen if the image of some edge $e\in E(P')$ crosses the image of some edge $e\in E(P\cup C^*)$ in $\psi$, which is impossible.
The proof for the case where $J_i\subseteq G_2$ is symmetric.
\end{proof}
We now consider pairs $C,C'$ of connected components of $W'$, where $C=\tilde C$, and $R_{C_{i^*}}(C')=F^{i^*}$.
We consider three cases. The first case is when vertices $x_{C_1},\ldots,x_{C_z}$ are the roots of the trees of the forest $H({\mathcal{W}})$. In this case, there must be an index $1\leq i\leq z$ with $i\neq i^*$, such that vertex $x_{C'}$ is a descendant of vertex $x_{C_i}$, or $x_{C'}=x_{C_i}$. If $J_i\subseteq G_1$, then we set $R_{\tilde C}(C')=F^{i^*}_1$, and otherwise, we set $R_{\tilde C}(C')=F^{i^*}_2$. From \Cref{obs: cores drawn correctly}, it is easy to verify that the image of component $C'$ in the drawing $\phi''$ is indeed contained in region $R_{\tilde C}(C')$.
The second case is when vertex $x_{C_1}$ is a parent-vertex of vertices $x_{C_2},\ldots,x_{C_z}$ in forest $H({\mathcal{W}})$, and $i^*=1$. In this case, there must be an index $2\leq i\leq z$, such that vertex $x_{C'}$ is a descendant of vertex $x_{C_i}$, or $x_{C'}=x_{C_i}$. We set $R_{\tilde C}(C')$ exactly as in Case 1.
The third and the last case is when vertex $x_{C_1}$ is a parent-vertex of vertices $x_{C_2},\ldots,x_{C_z}$ in forest $H({\mathcal{W}})$, and $i^*=1$.
If there is an index $1\leq i\leq z$ with $i\neq i^*$, such that vertex $x_{C'}$ is a descendant of vertex $x_{C_i}$, or $x_{C'}=x_{C_i}$, then we set the region $R_{\tilde C}(C')$ exactly like in Cases 1 and 2. Otherwise, either $C'=C_1$, or vertex $x_{C'}$ is an ancestor of vertex $x_{C_1}$. In this case, if $C_1\subseteq G_1$, then we set $R_{\tilde C}(C')=F^{i^*}_1$, and otherwise we set $R_{\tilde C}(C')=F^{i^*}_2$. As before, from \Cref{obs: cores drawn correctly}, it is easy to verify that the image of component $C'$ in the drawing $\phi''$ is indeed contained in region $R_{\tilde C}(C')$.
This completes the definition of the regions $R_{C}(C')$ of pairs $C,C'\in {\mathcal{C}}(W')$ of distinct connected components. From the above discussion, for each such pair $C,C'$ of regions, the image of component $C'$ in $\phi''$ is contained in region $R_{C}(C')$.
Finally, we need to consider the case where face $F^{i^*}$ is the infinite face of the drawing $\rho_{C_{i^*}}$. In this case, we need to designate one of the two new faces $F^{i^*}_1$ or $F^{i^*}_2$ as the infinite face of the new drawing $\rho_{\tilde C}$ of graph $\tilde C$, in a way that is consistent with the drawing $\phi''$. If vertices $x_{C_1},\ldots,x_{C_z}$ are the roots of the trees of the forest $H({\mathcal{W}})$, then we designate one of the two faces $F^{i^*}_1$, $F^{i^*}_2$ as the infinite face of $\rho_{\tilde C}$ arbitrarily. Otherwise, vertex $x_{C_1}$ is a parent-vertex of vertices $x_{C_2},\ldots,x_{C_z}$ in forest $H({\mathcal{W}})$, and, since $F^{i^*}=F^*(\rho_{C_{i^*}})$, $i^*\neq 1$ must hold. If $C_1\subseteq G_1$, then we let $F^{i^*}_1$ be the infinite face of $\rho_{\tilde C}$, and otherwise we let $F^{i^*}_2$ be the infinite face of $\rho_{\tilde C}$. It is easy to verify that this choice is consistent with the drawing $\phi''$.
To summarize, we have constructed a skeleton augmenting structure ${\mathcal{W}}'$, whose corresponding skeleton augmentation is denoted by $W'$, and a set $E^{\mathsf{del}}(F)\subseteq E(G)\setminus E(W')$ of edges. If bad event ${\cal{E}}'$ did not happen, then, as we have shown already, $|E^{\mathsf{del}}(F)|\leq \frac{g_1\cdot \mu^{48}}{m}+g_2$ holds.
Our construction guarantees that $W\subseteq W'$ and $\tilde {\mathcal{F}}({\mathcal{W}}')=\textsf{left}(\tilde{\mathcal{F}}({\mathcal{W}})\setminus\set{F}\textsf{right})\bigcup\set{F_1,F_2}$.
Let ${\mathcal{G}}'$ be a collection of graphs constructed as follows. For every face $F'\in \tilde {\mathcal{F}}({\mathcal{W}})\setminus\set{F}$, we add the graph $G_{F'}\in {\mathcal{G}}$ to ${\mathcal{G}}'$. Additionally, we add graph $G_{F_1}=G_1$ that is associated with the new face $F_1\in \tilde {\mathcal{F}}({\mathcal{W}}')$, and graph $G_{F_2}=G_2$, that is associated with face $F_2\in \tilde {\mathcal{F}}({\mathcal{W}}')$.
From our construction, $E(G_F)\setminus (E(G_{F_1})\cup E(G_{F_2}))= E^{\mathsf{del}}(F)$. Moreover, from \Cref{obs: few edges in split graphs case2.2}
If Event ${\cal{E}}'$ did not happen, then $|E(G_{F_1})\setminus E(W')|,|E(G_{F_2})\setminus E(W')|\leq |E(G_F)\setminus E(W)|-\frac{\check m'}{\mu^{5b}}$.
Recall that we denoted $\tilde G''=\tilde G\setminus E^{\mathsf{del}}(F)=\check G'\setminus(E^{\mathsf{del}}\cup E^{\mathsf{del}}(F))$, and we let $\tilde \Sigma''$ be the rotation system for graph $\tilde G''$ induced by $\check \Sigma'$. We have shown that, if bad event ${\cal{E}}'$ does not happen, then there is a solution $\phi''$ to the resulitng instance $\tilde I''=(\tilde G'',\tilde \Sigma'')$ that is ${\mathcal{W}}'$-compatible. Since drawing $\phi$ ws semi-clean with respect to $\check{\mathcal{K}}$, it is easy to verify that drawing $\phi''$ has this property as well. Additionally, from our construction, $\mathsf{cr}(\phi'')\leq \mathsf{cr}(\phi)$ and $|\chi^{\mathsf{dirty}}(\phi)|\leq |\chi^{\mathsf{dirty}}(\phi')|$. Lastly, let $N(\phi)$ be the total number of crossings of $\phi$ in which the edges of $E(W)\setminus E(\check K)$ participate, and define $N(\phi'')$ similarly for $\phi''$, then $N(\phi'')-N(\phi)$ is bounded by the number of crossings in $\phi$, in which the edges of the path $P$ participate. If bad event ${\cal{E}}'$ does not happen, then $N(\phi'')-N(\phi)\leq \frac{\mathsf{cr}(\phi)\cdot \mu^{15b}}{4m}\leq \frac{\mathsf{cr}(\phi)\cdot \mu^{17b}}{\check m'}$, since $m\geq \check m'/\mu^{2b}$ must hold. We conclude that, if bad event ${\cal{E}}'$ does not happen, and path set ${\mathcal{P}}$ was a promising path set of type $2$, then we obtain a valid output for Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$. Since $\prob{{\cal{E}}'}\leq \frac{1}{\mu^{11b}}$, this concludes the analysis of the algorithm for Case 1.
\subsubsection*{Case 2: ${\mathcal{P}}$ was a type-3 promising path set}
In the second case, the promising set ${\mathcal{P}}$ of paths was of type 3. We let $1\leq i^{**}\leq z$ be the index for which all paths in ${\mathcal{P}}$ terminate at vertices of $J_{i^{**}}$.
If the two paths chosen from ${\mathcal{P}}^*$ share at least one vertex of $V(G)\setminus V(K)$, then we have constructed a type-2 enhancement whose both endpoints lie in $J$. In this case, the algorithm and its analysis are practically identical to those from Case 1 and are omitted here. We now assume that the two chosen paths are disjoint from each other. For convenience of notation, we denote $P=P^*_1$ and $P'=P^*_2$. Throughout, we denote by $\tilde J$ the graph obtained from the union of $J,J_{i^{**}},P$ and $P'$. We also let $\tilde C=C_{i^*}\cup C_{i^{**}}\cup P\cup P'$.
Recall that we set $W'=W\cup P\cup P'$.
Recall also that, from \Cref{claim: new drawing2}, if Event ${\cal{E}}'$ does not happen, then drawing $\phi'$ of $G'=G\setminus E'$ contains no crossings between the edges of $E(W')$.
Throughout, we denote by $\gamma$ the image of path $P$ and by $\gamma'$ the image of path $P'$ in $\phi'$.
In order to construct the skeleton augmenting structure ${\mathcal{W}}'$, we first let $W'=W\cup P\cup P'$ be the skeleton augmentation associated with ${\mathcal{W}}'$. Next, we define a drawing $\rho_C$ for every component $C\in {\mathcal{C}}(W)$.
For every component $C\in {\mathcal{C}}(W')\setminus \set{\tilde C}$, its drawing $\rho_C$ in ${\mathcal{W}}'$ remains the same as in ${\mathcal{W}}$.
Consider now the component $\tilde C$. We would like to ensure that the drawing $\rho_{\tilde C}$ of $\tilde C$ that we define is identical to the drawing of $\tilde C$ induced by $\phi'$. Recall that we are given, as part of skeleton augmenting structure ${\mathcal{W}}$, a drawing $\rho_{C_{i^*}}$ of graph $C_{i^*}$, a drawing $\rho_{C_{i^{**}}}$ of graph $C_{i^{**}}$, and an orientation $b'_u$ of every vertex $u\in V(C_{i^*})\cup V(C_{i^*})$. We are also given a face $F^{i^*}\in {\mathcal{F}}(\rho_{C_{i^*}})$, such that the face $F$ in drawing $\phi$ is contained in the region $F^{i^*}$, and a face $F^{i^{**}}\in {\mathcal{F}}(\rho_{C_{i^{**}}})$, such that the face $F$ in drawing $\phi$ is contained in the region $F^{i^{**}}$. Since drawing $\phi$ is ${\mathcal{W}}$-compatible, and it does not contain any crossings between the edges of $W'$, we are guaranteed that paths $P$ and $P'$ are drawn inside the region $F^{i^*}\cap F^{i^{**}} $ of $\phi'$, in a natural way.
However, we do not know the orientations of the drawings $\rho_{C_{i^*}}$ of $C_{i^*}$ and $\rho_{C_{i^{**}}}$ of $C_{i^{**}}$ with respect to each other in $\phi'$. We overcome this difficulty in the same way as in Procedure \ensuremath{\mathsf{ProcSplit}}\xspace.
We recall how the paths $P$ and $P'$ were constructed.
We started with a set ${\mathcal{P}}^*$ of $k'=\ceil{\frac{4m}{\mu^{3b}}}$ simple edge-disjoint paths in graph $G$, with every path originating at a vertex of $J_{i^*}$ and terminating at a vertex of $J_{i^{**}}$. Additionally, the paths in ${\mathcal{P}}^*$ are internally disjoint from the skeleton $K$, and they are non-transversal with respect to $\Sigma$. We denoted by $E^*_1\subseteq \delta(J_{i^{*}})$ the subset of edges that belong to the paths of ${\mathcal{P}}^*$, letting $E^*_1=\set{e_1,\ldots,e_{k'}}$, where the edges are indexed in the order of their appearence in the ordering ${\mathcal{O}}(J_{i^{*}})$. For all $1\leq j\leq k'$, we denoted by $P_j\in {\mathcal{P}}^*$ the unique path originating at the edge $e_j\in E^*_1$. We then selected an index $\floor{k'/8}<j^*<\ceil{k'/4}$ uniformly at random, that determined the chosen paths $P_{j^*}$ and $P_{j^*+\floor{k'/2}}$. Since we assumed that the chosen paths do not share any vertices of $V(G)\setminus V(K)$, we eventually set $P=P_{j^*}$ and $P'=P_{j^*+\floor{k'/2}}$.
We choose another index $j^*< j^{**}<j^*+\floor{k'/2}$ uniformly at random, and we let $P''=P_{j^{**}}$. We say that a bad event ${\cal{E}}'_5$ happens if $P''$ is a bad path, or some edge $e\in E(P'')$ crosses the image of an edge of $P\cup P'$ in the drawing $\phi$. From \Cref{obs: number of bad paths2}, the number of bad paths in ${\mathcal{P}}^*$ is at most $\frac{10m}{\mu^{15b}}$. Moreover, if bad event ${\cal{E}}'$ did not happen, then both paths $P$ and $P'$ are good, and so the number of crossings in which the edges of $E(P)\cup E(P')$ participate in $\phi$ is at most $\frac{\mathsf{cr}(\phi)\mu^{15b}}{4m}\leq \frac{m}{4\mu^{15b}}$ (from Property \ref{prop valid input drawing2} of valid input to Procedure \ensuremath{\mathsf{ProcSplit}}\xspace, and since $m\leq \check m'$). Therefore, if Event ${\cal{E}}'$ did not happen, there are at most $\frac{12m}{\mu^{15b}}$ indices $1\leq i\leq k$, such that path $P_i$ is bad, or some edge of $P_i$ crosses an edge of $E(P)\cup E(P')$ in $\phi$.
Since $k'=\ceil{\frac{4m}{\mu^{3b}}}$, $\prob{{\cal{E}}_5'\mid \neg{\cal{E}}'}\leq \frac{3}{\mu^{12b}}$.
Consider the graph $\tilde C'=C_{i^*}\cup C_{i^{**}}\cup P\cup P'\cup P''$. If bad events ${\cal{E}}',{\cal{E}}_5$ did not happen, then there is a unique planar drawing $\tilde \rho'$ of graph $\tilde C'$ that has the following properties:
\begin{itemize}
\item the drawing of $C_{i^*}$ induced by $\tilde \rho'$ is $\rho_{C_{i^*}}$;
\item the drawing of $C_{i^{**}}$ induced by $\tilde \rho'$ is $\rho_{C_{i^{**}}}$;
\item the images of path $P$, $P'$ and $P''$ are contained in region $F^{i^*}\cap F^{i^{**}}$;
\item the drawing is consistent with the rotation system $\check \Sigma'$;
\item the orientation of every vertex $u\in V(C_{i^*})$ in the drawing is the orientation $b'_u$ given by ${\mathcal{W}}$; and
\item either the orientation of every vertex $u\in V(C_{i^{**}})$ in the drawing is the orientation $b'_u$ given by ${\mathcal{W}}$ (in which case we say that the orientations of $C_{i^*}$ and $C_{i^{**}}$ agree), or the orientation of every vertex $u\in V(C_{i^{**}})$ in the drawing is $-b'_u$ (in which case we say that the orientations of $C_{i^*}$ and $C_{i^{**}}$ disagree).
\end{itemize}
Drawing $\tilde \rho'$ with all above properties can be computed efficiently. We then let $\rho_{\tilde C}$ be the drawing of $\tilde C$ induced by $\tilde \rho'$. From the above discussion, if neither of the events ${\cal{E}}',{\cal{E}}'_5$ happened, then $\rho_{\tilde C}$ is identical to the drawing of $\tilde C$ induced by $\phi'$.
Notice that the set ${\mathcal{F}}(\rho_{\tilde C})$ of faces of this new drawing of graph $\tilde C$ is precisely $\textsf{left} ({\mathcal{F}}(\rho_{C_{i^*}})\cup {\mathcal{F}}(\rho_{C_{i^{**}}}\textsf{right} )\setminus \set{F^{i^*},F^{i^{**}}})\cup (F^{i^*}_1,F^{i^*}_2)$, where $F^{i^*}_1$ and $F^{i^*}_2$ are the two new faces that are obtained by partitioning the region $F^{i^*}\cap F^{i^{**}}$ with the images of paths $P$ and $P'$. We assume w.l.o.g. that region $F^{i^*}_1$ in the drawing of $W'$ induced by $\phi'$ contains the face $F_1$ of this drawing, and region $F^{i^*}_2$ contains the face $F_2$.
Notice that, in the skeleton structure ${\mathcal{W}}$, $R_{C_{i^*}}(C_{i^{**}})=F^{i^*}$ and $R_{C_{i^{**}}}(C_{i^{*}})=F^{i^{**}}$ must hold. Moreover, from the definition of skeleton augmenting structure, either $F^{i^*}=F^*(\rho_{C_{i^*}})$, or $F^{i^{**}}=F^*(\rho_{C_{i^{**}}})$ must hold.
If $F^{i^*}=F^*(\rho_{C_{i^*}})$ but $F^{i^{**}}\neq F^*(\rho_{C_{i^{**}}})$, then face $F^*(\rho_{C_{i^{**}}})$ is also a face of $\rho_{\tilde C}$, and we set this face to be the infinite face $F^*(\rho_{\tilde C})$. Similarly, if $F^{i^{**}}= F^*(\rho_{C_{i^{**}}})$ but $F^{i^*}\neq F^*(\rho_{C_{i^*}})$, then face $F^*(\rho_{C_{i^{*}}})$ is also a face of $\rho_{\tilde C}$, and we set this face to be the infinite face $F^*(\rho_{\tilde C})$. If $F^{i^*}=F^*(\rho_{C_{i^*}})$ and $F^{i^{**}}\neq F^*(\rho_{C_{i^{**}}})$, then we will designate one of the faces $F^{i^*}_1$ or $F^{i^*}_2$ as the infinite face of the drawing $\rho_{\tilde C}$ later.
So far we have defined the new skeleton augmentation $W'$ and a drawing $\rho_C$ of every connected component $C\in {\mathcal{C}}(W)$. Next, we define the orientation $b'_u$ of every vertex $u\in V(W')$. For vertices $u\in (V(W')\cap V(W))\setminus V(C_{i^{**}})$, the orientation $b'_u$ in ${\mathcal{W}}'$ remains the same as the orientation $b'_u$ in ${\mathcal{W}}$.
If the orientations of $C_{i^*}$ and $C_{i^{**}}$ agree in drawing $\tilde \rho'$, then for every vertex $u\in V(C_{i^{**}})$, the orientation $b'_u$ in ${\mathcal{W}}'$ remains the same as the orientation $b'_u$ in ${\mathcal{W}}$; otherwise, for every vertex $u\in V(C_{i^{**}})$, the orientation $b'_u$ is reversed in ${\mathcal{W}}'$.
It now remains to define the orientation $b'_u$ of vertices $u\in (V(P)\cup V(P'))\setminus (V(C_{i^*})\cup V(C_{i^{**}})$. The procedure for computing these orientations is very similar to the one from Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace$.
As in Case 1, for every vertex $u\in V(\tilde C)$, we consider the tiny $u$-disc $D_{\phi'}(u)$. For every edge $e\in \delta_{G}(u)\setminus E'$, we denote by $\sigma(e)$ the segment of $\phi'(e)$ that is drawn inside the disc $D_{\phi'}(u)$. Since edge $e$ belongs to graph $G$, $\sigma(e)$ must be contained in the region $F\in {\mathcal{F}}(W)$ of the drawing $\phi'$. Let $\tilde E=\textsf{left} (\bigcup_{u\in V(P)\cup V(P')\cup V(J)\cup V(J_{i^{**}})}\delta_{G'}(u)\textsf{right} )\setminus (E(P)\cup E(P')\cup E(J)\cup E(J_{i^{**}}))$.
The images of paths $P$ and $P'$ in $\phi'$ split region $F$ into two regions, that we denote by $F_1$ and $F_2$. We let $e_1\in E_1$ be any edge that does not lie in $E(P)\cup E(P')$, and we assume without loss of generality that $\sigma(e_1)\subseteq F_1$.
We partition edge set $\tilde E$ into a set $\tilde E^{\mathsf{in}}$ of \emph{inner edges} and the set $\tilde E^{\mathsf{out}}$ of outer edges, as before: Edge set $\tilde E^{\mathsf{in}}$ contains all edges $e\in \tilde E$ with $\sigma(e)$ contained in the face $F_1$ of $\rho_{\tilde C}$, and $\tilde E^{\mathsf{out}}$ contains all remaining edges (so for every edge $e\in \tilde E^{\mathsf{out}}$, $\sigma(e)$ is contained in $F_2$). As before, we show an algorithm that correctly computes the orientation of every vertex $u\in (V(P)\cup V(P'))\setminus (V(J)\cup V(J_{i^{**}}))$ in the drawing $\phi'$, and the partition $(\tilde E^{\mathsf{in}},\tilde E^{\mathsf{out}})$ of the edges of $\tilde E$.
The algorithm for computing the orientation of the inner vertices of $P$ and $P'$ is practically identical to the one used in Case 1, and modifies it in the same way as the algorithm for Case 3 in \ensuremath{\mathsf{ProcSplit}}\xspace modified the algorithm for Case 2. We omit it here.
In order to complete the construction of the skeleton augmenting structure ${\mathcal{W}}'$, it remains to define, for every pair $C,C'\in {\mathcal{C}}(W')$ of distinct connected components, a face $R_C(C')$ of ${\mathcal{F}}(\rho_C)$. Additionally, if $F^{i^*}=F^*(\rho_{C_{i^*}})$ and $F^{i^{**}}\neq F^*(\rho_{C_{i^{**}}})$, then we need to designate one of the faces $F^{i^*}_1$ or $F^{i^*}_2$ as the outer face of the drawing $\rho_{\tilde C}$ of component $\tilde C$. We do so in the next step, once we split the graph $G'$ into two subgraphs, $G_1$ and $G_2$.
\paragraph{Computing the Split.}
The construction of the flow network $\tilde H'$ an $s$-$t$-cut $(A,B)$ in $\tilde H'$, and edge set $E''=E_{\tilde H'}(A,B)$ are identical to those in Case 3 of Step 2 of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. Using the same arguments as in the proof of \Cref{claim: cut set small case2.2}, if neither of the bad events ${\cal{E}}',{\cal{E}}'_5$ happened, then $|E''|\leq \frac{\mathsf{cr}(\phi)\mu^{30b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
We now construct the graphs $G_1,G_2$ associated with the faces $F_1$ and $F_2$, respectively.
In order to do so, we define two vertex sets $A',B'$ of graph $G'$, as before. We start with $A'=A\setminus\set{s}$ and $B'=B\setminus\set{t}$. For every index $1\leq i\leq r$ with $i\not\in\set{ i^*,i^{**}}$, if $v_{J_i}\in A$, then we replace vertex $v_{J_i}$ with vertex set $V(J_i)$ in $A'$, and otherwise we replace $v_{J_i}$ with vertex set $V(J_i)$ in $B'$.
Additionally, we let $J\subseteq J\cupJ_{i^{**}}\cup P\cup P'$ be the graph containing all vertices and edges of $J\cupJ_{i^{**}}\cup P\cup P'$, whose images in $\rho_{\tilde C}$ lie on the boundary of face $F^{i^*}_1$. Similarly, we let $J'\subseteq J\cup J_{i^{**}}\cup P\cup P'$ be the graph containing all vertices and edges of $J\cupJ_{i^{**}}\cup P\cup P'$, whose images in $\rho_{\tilde C}$ lie on the boundary of face $F^{i^*}_2$.
We then let $G_1$ be the subgraph of $G'$, whose vertex set is $V(A')\cup V(J)$, and edge set contains all edges of $E_{G'}(A')$, $E_{G'}(A',V(J))$, and all edges of $J$. Similarly, we let $G_2$ be the subgraph of $G'$, whose vertex set is $V(B')\cup V(J')$, and edge set contains all edges of $E_{G'}(B')$, $E_{G'}(B',V(J'))$, and all edges of $J'$.
The rotation system $\Sigma_1$ for graph $G_1$ and the rotation system $\Sigma_2$ for graph $G_2$ are induced by $\Sigma'$.
Let $I_1=(G_1,\Sigma_1)$ and $I_2=(G_2,\Sigma_2)$ be the resulting two instances of \ensuremath{\mathsf{MCNwRS}}\xspace.
The following observation is an analogue of \Cref{obs: few edges in split graphs case2.2} and its proof follows from the same arguments.
\begin{observation}\label{obs: few edges in split graphs case2.3}
If Event ${\cal{E}}'$ did not happen, then $|E(G_1)\setminus E(W')|,|E(G_2)\setminus E(W')|\leq |E(G)\setminus E(W)|-\frac{\check m'}{\mu^{5b}}$.
\end{observation}
Let $E^{\mathsf{del}}(F)=E'\cup E''$. As in Case 1, if bad event ${\cal{E}}'$ did not happen, then $|E^{\mathsf{del}}(F)|\leq \frac{g_1\cdot \mu^{48}}{m}+g_2$, as required.
As in Case 1, we let $\tilde G''=\tilde G'\setminus E''=\tilde G\setminus E^{\mathsf{del}}$, and let $\tilde \Sigma''$ be the rotation system for graph $\tilde G''$ induced by $\tilde \Sigma$. We denote $\tilde I''=(\tilde G'',\tilde \Sigma'')$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace.
Let $\phi''$ be the drawing of $\tilde G''$ that is induced by the drawing $\phi'$ of $\tilde G'$. We also denote $G''=G'\setminus E''=G\setminus E^{\mathsf{del}}$. As before, $E(G_1)\cup E(G_2)=E(G'')$.
We now complete the construction of the skeleton augmenting structure ${\mathcal{W}}'$. For all $1\leq i\leq z$, let $C_i\in {\mathcal{C}}(W)$ be the connected component that contains the core $J_i$, and let $F^i\in {\mathcal{F}}(\rho_{C_i})$ be the face that corresponds to face $F$ of $\tilde {\mathcal{F}}({\mathcal{W}})$ (in other words, face $F$ in the drawing of $W$ induced by $\phi$ is contained in region $F^i$). From the construction of the set $\tilde {\mathcal{F}}({\mathcal{W}})$ of faces, in the decomposition forest $H({\mathcal{W}})$, either vertices $x_{C_1},\ldots,x_{C_z}$ are the roots of the trees in the forest; or one of these vertices is the parent-vertex of the remaining vertices. We assume w.l.o.g. that in the latter case, vertex $x_{C_1}$ is the parent of vertices $x_{C_2},\ldots,x_{C_z}$ in the forest $H({\mathcal{W}})$.
Consider any pair $C,C'\in {\mathcal{C}}(W')$ of distinct connected components. Assume first that $C,C'\neq \tilde C$. In this case, face $R_{C}(C')$ remains the same in ${\mathcal{W}}'$ in the augmenting structure ${\mathcal{W}}$. Since drawing $\phi$ of $\tilde G$ was ${\mathcal{W}}$-compatible, and drawing $\phi''$ is obtained from $\phi$ by deleting some edges that do not belong to $W'$, and possibly making some additional local changes within the region $F$, the image of component $C'$ in $\phi''$ remains contained in region $R_C(C')$ in $\phi''$.
Assume now that $C'=\tilde C$. It is easy to verify that, in this case, in ${\mathcal{W}}$, $R_{C'}(C_{i^*})=R_{C'}(C_{i^{**}})$ must hold.
Indeed, if $R_{C'}(C_{i^*})\neq F^*(\rho_{C'})$, then vertex $x_{C'}$ must be an ancestor of vertex $x_{C_{i^*}}$ in forest $H({\mathcal{W}})$. Similarly, if $R_{C'}(C_{i^{**}})\neq F^*(\rho_{C'})$, then vertex $x_{C'}$ must be an ancestor of vertex $x_{C_{i^{**}}}$ in forest $H({\mathcal{W}})$. Since vertices $x_{C_{i^*}}$ and $x_{C_{i^{**}}}$ are either sibling vertices in $H({\mathcal{W}})$, or one of them is a parent vertex of another, if vertex $x_{C'}$ is an ancestor of one of these two vertices, then it must be an ancestor of another. Moreover, since both components $C_{i^*},C_{i^{**}}$ belong to the set ${\mathcal{S}}(F)$ of components corresponding to region $F$, $R_{C'}(C_{i^*})=R_{C'}(C_{i^{**}})$ must hold. We then set face $R_{C'}(\tilde C)$ in ${\mathcal{W}}'$ to be the face $R_{C'}(C_{i^{*}})$ in ${\mathcal{W}}$.
It is easy to verify that the drawing of component $\tilde C$ in $\phi''$ remains in the region $R_{C'}(C_{i^*})$.
Next, we assume that $C=\tilde C$. Recall that components $C_{i^*}, C_{i^{**}}$ of $W$ are replaced by component $\tilde C=C_{i^*}\cup C_{i^{**}}\cup P\cup P'$ in $W'$. We use the following observation, that is an analogue of \Cref{obs: cores drawn correctly} for Case 1. The proof is almost identical and is provided here for completeness.
\begin{observation}\label{obs: cores drawn correctly2} Assume that neither of the Events ${\cal{E}}',{\cal{E}}'_5$ happenned.
Consider any index $1\leq i\leq z$ with $i\not\in\set{ i^*,i^{**}}$. If $J_i\subseteq G_1$, then the image of $C_i$ appears in region $F_1^{i^*}$ in $\phi''$, and if $J_i\subseteq G_2$, then the image of $C_i$ appears in region $F_2^{i^*}$ in $\phi''$.
\end{observation}
\begin{proof}
Assume for contradiction that neither of the Events ${\cal{E}}',{\cal{E}}'_5$ happenned, and there is an index $1\leq i\leq z$ with $J_i\subseteq G_1$, such that the image of $C_i$ is not contained in region $F_1^{i^*}$ of $\phi''$.
Let $E^*$ be the set of edges $e\in E(\tilde G'')$, such that the image of $e$ in $\phi''$ crosses the image of some edge of $W'$. Using the same accounting as in the proof of \Cref{obs: cores drawn correctly},
$|E^*|\leq \textsf{left} (\frac{g_1}{\check m'}+g_2\textsf{right} )\cdot \mu^{100b}$ must hold.
From Property \ref{prop: new}, there must be a path $\tilde P$ in graph $\tilde G'\setminus E^*$, connecting a vertex of $C_i$ to a vertex of $C_{i^*}\cup C_{i^{**}}\cup P\cup P'$. We assume w.l.o.g. that $\tilde P$ does not contain any vertex of $V(C_i)\cup V(C_{i^*})\cup V(C_{i^{**}}) V(P)\cup V(P')$ as an inner vertex. We will reach a contradiction by showing that there is some edge $e\in E(\tilde P)$, whose image crosses the image of an edge of $J_{i^*}\cup J_{i^{**}}\cup P \cup P'$ in $\phi''$. This is impossible because then $e$ should have been added to $E^*$.
Let $\psi$ be a drawing obtained from drawing $\phi''$ by deleting the images of all edges of $E^*$ from the drawing. We view $\psi$ as a drawing on the sphere, and we define, for all $1\leq \ell\leq z$, a disc $D(J_{\ell})$, whose boundary follows the image of the core $J_{\ell}$ inside the face $F$, as before. As before, the image of the connected component $C_{\ell}$ of $W$ must be contained in the disc $D(J_{\ell})$.
Since path $\tilde P$ originates at a vertex of $C_i$ and terminates at a vertex of $C_{i^*}\cup C_{i^{**}}\cup P\cup P'$, its image must intersect the interior of region $F$. We denote by $Q_1,Q_2,\ldots,Q_h$ maximal subpaths of the path $\tilde P$, such that for all $1\leq h'\leq h$, the image of the path $Q_{h'}$ in $\psi$ is contained in the interior of region $F$, except for its endpoints, that lie on the boundary of $F$. We assume that paths $Q_1,\ldots,Q_h$ appear on path $\tilde P$ in this order. For all $1\leq h'\leq h$, we denote by $1\leq a_{h'}\leq z$ the index for which the first vertex of $Q_{h'}$ belongs to core $J_{a_{h'}}$, so $a_1=i$. From the definition of the paths $Q_1,\ldots,Q_h$, for all $1\leq h'\leq h$, the last vertex of $Q_{h'-1}$ must also belong to core $J_{a_{h'}}$, and the last vertex of $Q_h$ must belong to $J_{i^*}\cup J_{i^{**}}\cup P\cup P'$.
Consider now some index $1\leq h'\leq h$. Let $e_{h'}$ be the first edge on path $Q_{h'}$, and let $e'_{h'}$ be the last edge on path $Q_{h'}$. Since the images of the edges of path $\tilde P$ in $\psi$ may not cross the images of any edges of $W$, and the image of $\tilde P$ intersects the interior of $F$, we get that $\psi(e_{h'})\subseteq F$. Moreover, since an endpoint of $e_{h'}$ belongs to $J_{a_{h'}}$, from the definition of valid core structure (see \Cref{def: valid core 2}), $e_{h'}\in E(G)$ must hold. Using a similar reasoning, $e'_{h'}\in E(G)$. Since no inner vertex of $Q_{h'}$ belongs to $J_{i^*}\cup J_{i^{**}}\cup P\cup P'$, and since $V(G_1)\cap V(G_2)\subseteq V(J_{i^*})\cup V(J_{i^{**}})\cup V(J_{i^*})\cup V(P)\cup V(P')$, it must be the case that either $Q_{h'}\subseteq G_1$, or $Q_{h'}\subseteq G_2$. Since $J_i=J_{a_1}\subseteq G_1$, and since, for all $1\leq h'< h$ the last vertex of $Q_{h'}$ and the first vertex of $Q_{h'+1}$ belong to the same core $J_{a_{h'}}$, it must be the case that for all $1\leq h'\leq h$, $Q_{h'}\subseteq G_1$.
Recall that the image $\rho_{\tilde C}$ of graph $\tilde C=C_{i^*}\cup C_{i^{**}}\cup P\cup P'$ has two special faces, that we denoted by $ F^{i^*}_1$ and $ F^{i^*}_2$, which correspond to regions $F_1$ and $F_2$ of the drawing of $W'$ induced by $\phi'$, respectively. Clearly, for all $1\leq h'\leq h$, the image of $J_{h'}$ is either contained in $F^{i^*}_1$, or it is contained in $ F^{i^*}_2$. The image of $J_{i}$ is contained by $ F^{i^*}_2$ from our assumption that it is not contained in $F^{i^*}_1$. However, if we denote by $e^*$ the last edge on path $Q_h$, then the image of $e^*$ must be contained in $ F^{i^*}_1$, since $e^*\in E(G_1)$, and so $e^*\in \tilde E_1$ must hold.
We conclude that the image of some path $Q_{h'}$ must intersect both regions $F^{i^*}_1$ and $F^{i^*}_2$, which may only happen if the image of some edge $e\in E(\tilde P)$ crosses the image of some edge $e\in E(P\cup P'\cup C_{i^*}\cup C_{i^{**}})$ in $\psi$, which is impossible.
The proof for the case where $J_i\subseteq G_2$ is symmetric.
\end{proof}
We now consider some component $C'\in {\mathcal{C}}(W')\setminus \set{\tilde C}$ and compute the region $R=R_{\tilde C}(C')$, such that the image of $C'$ in $\phi''$ is contained in $R$.
We need to consider several cases. Assume first that vertices $x_{C_{i^*}}$ and $x_{C_{i^{**}}}$ are siblings in the forest $H({\mathcal{W}})$.
If vertex $x_{C'}$ is a descenadnt of some vertex $x_{C_i}$, where $1\leq i\leq z$, $i\not\in \set{i^*,i^{**}}$, and $x_i$ is not a parent-vertex of $x_{C_{i^*}}$, then region $R=R_{\tilde C}(C')$ is determined as follows. If $J_i\subseteq G_1$, then we set $R_{\tilde C}(C')=F^{i^*}_1$, and otherwise we set $R_{\tilde C}(C')=F^{i^*}_2$. If $C'=C_i$ for some index $1\leq i\leq z$ with $i\not\in \set{i^*,i^{**}}$, such that and $x_i$ is not a parent-vertex of $x_{C_{i^*}}$, then we let $R_{\tilde C}(C')=F^{i^*}_1$ if $J_i\subseteq G_1$, and we let $R_{\tilde C}(C')=F^{i^*}_2$ otherwise. If vertex $x_{C'}$ is a descendant of $C_{i^*}$, then face $R_{C_{i^*}}(C')$ of ${\mathcal{F}}(\rho_{C_{i^*}})$ is also a face of ${\mathcal{F}}(\rho_{\tilde C})$, and we set $R_{\tilde C}(C')=R_{C_{i^*}}(C')$. If $x_{C'}$ is a descendant of $x_{C_{i^{**}}}$, then we set $R_{\tilde C}(C')$ similarly.
If vertices $x_{C_{i^*}}, x_{C_{i^{**}}}$ are both roots of trees in the forest $H({\mathcal{W}})$, then we designate face $F^{i^*}_1$ of ${\mathcal{F}}(\rho_{\tilde C})$ as the infinite face of the drawing. In this case, for every component $C'\in {\mathcal{C}}(W')\setminus \set{\tilde C}$, vertex $x_{C'}$ is either a descendant of a vertex $x_{C_i}$ for $1\leq i\leq z$, or $x_{C'}\not\in \set{x_{C_1},\ldots,x_{C_z}}$, so region $R_{\tilde C}(C')$ is defined already. Otherwise, vertex $x_{C_1}$ is the parent-vertex of both $x_{C_{i^*}}$ and $ x_{C_{i^{**}}}$. If $J_1\subseteq G_1$, then we designate face $F^{i^*}_1$ of ${\mathcal{F}}(\rho_{\tilde C})$ as the infinite face of the drawing, and, for every component $C'\in {\mathcal{C}}(W')$ such that $x_{C'}$ is not a descendant of $x_{C_1}$, we set $R_{\tilde C}(C')=F^{i^*}_1$. Otherwise, we designate face $F^{i^*}_2$ of ${\mathcal{F}}(\rho_{\tilde C})$ as the infinite face of the drawing, and, for every component $C'\in {\mathcal{C}}(W')$ such that $x_{C'}$ is not a descendant of $x_{C_1}$, we set $R_{\tilde C}(C')=F^{i^*}_2$.
It remains to consider the case where one of the vertices $x_{C_{i^*}}$, $x_{C_{i^{**}}}$ is the parent-vertex of the other in forest $H({\mathcal{W}})$. We assume w.l.o.g. that $x_{C_{i^*}}$ is the parent of $x_{C_{i^{**}}}$. In this case, the infinite face $F^*(\rho_{C_{i^*}})$ is also a face of drawing $\rho_{\tilde C}$ of $\tilde C$, and it is designated as the infinite face of $\rho_{\tilde C}$ of drawing Consider now some component $C'\in {\mathcal{C}}(W')\setminus \tilde{C}$. If vertex $x_{C'}$ is a descendant of any vertex of $\set{x_{C_i}\mid 1\leq i\leq z; i\neq i^*}$, or $x_{C'}\in \set{x_{C_i}\mid 1\leq i\leq z; i\not\in \set{i^*,i^{**}}}$, then we set $R_{\tilde C}(C')$ exactly as before. Otherwise, we set $R_{\tilde C}(C')=F^*(\rho_{\tilde C})=F^*(\rho_{C_{i^*}})$.
This completes the definition of the regions $R_{C}(C')$ of pairs $C,C'\in {\mathcal{C}}(W')$ of distinct connected components. It is easy to verify that, for each such pair $C,C'$ of regions, the image of component $C'$ in $\phi''$ is contained in region $R_{C}(C')$.
We have also defined the infinite face of the drawing $\rho_{\tilde C}$.
To summarize, we have constructed a skeleton augmenting structure ${\mathcal{W}}'$, whose corresponding skeleton augmentation is denoted by $W'$, and a set $E^{\mathsf{del}}(F)\subseteq E(G)\setminus E(W')$ of edges. If neither of the bad events ${\cal{E}}',{\cal{E}}'_5$ happenned, then, as we have shown already, $|E^{\mathsf{del}}(F)|\leq \frac{g_1\cdot \mu^{48}}{m}+g_2$ holds.
Our construction guarantees that $W\subseteq W'$ and $\tilde {\mathcal{F}}({\mathcal{W}}')=\textsf{left}(\tilde{\mathcal{F}}({\mathcal{W}})\setminus\set{F}\textsf{right})\bigcup\set{F_1,F_2}$.
The construction of the ${\mathcal{W}}'$-decomposition ${\mathcal{G}}'$ remains the same as in Case 1, and the remainder of the analysis remains the same as before. We conclude that, if neither of the bad events ${\cal{E}}'$, ${\cal{E}}'_5$ happenned, and path set ${\mathcal{P}}$ was a promising path set of type $3$, then we obtain a valid output for Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$. Since
$\prob{{\cal{E}}'}\leq \frac{1}{\mu^{11b}}$ and $\prob{{\cal{E}}_5'\mid \neg{\cal{E}}'}\leq \frac{3}{\mu^{12b}}$, the probability that at least one of these two events happens is at most $\frac{1}{\mu^{11b}}+\frac{3}{\mu^{12b}}\leq \frac{1}{\mu^{10b}}$. This concludes the analysis of the algorithm for Case 2.
\section{Proofs Omitted from \Cref{sec: guiding paths orderings basic disengagement}}
\label{sec: appendix: proofs from basic disengagement}
\subsection{Proof of \Cref{lem: basic disengagement combining solutions}}
\label{appx: proof of basic disengagement combining solutions}
In this proof, we assume that all drawings are on the sphere.
For every cluster $C\in {\mathcal{L}}$, denote by ${\mathcal{W}}(C)\subseteq {\mathcal{L}}$ the set of all child clusters of $C$, and by ${\mathcal{W}}^*(C)\subseteq {\mathcal{L}}$ the set of all descendant clusters of $C$. We define a new instance $I'_C=(G'_C,\Sigma'_C)$ of \ensuremath{\mathsf{MCNwRS}}\xspace associated with cluster $C$, as follows. If $C=G$, then $G'_C=G$ and $\Sigma'_C=\Sigma$. Otherwise, graph $G'_C$ is obtained from graph $G$, by contracting all vertices of $V(G)\setminus V(C)$ into a supernode $v^*$. Rotation system $\Sigma'_C$ is defined as follows. Note that $\delta_{G'_C}(v^*)=\delta_G(C)$. We define the rotation ${\mathcal{O}}_{v^*}\in \Sigma'_C$ to be ${\mathcal{O}}(C)$. For every other vertex $v\in V(G'_C)$, $\delta_{G'_C}(v)=\delta_G(v)$ holds, and its rotation ${\mathcal{O}}_v\in \Sigma'_C$ remains the same as in $\Sigma$. This completes the definition of instance $I'_C$.
Notice that instance $I_C$ can be obtained from instance $I'_C$ by contracting, for each cluster $C'\in {\mathcal{W}}(C)$, the vertices of $C'$ into a supernode $v_{C'}$, and then setting the rotation of the edges incident to this supernode to ${\mathcal{O}}(C')$.
We prove by induction that there is an efficient algorithm, that, given a cluster $C\in {\mathcal{L}}$, and solutions $\set{\phi(I_{C'})}_{C'\in {\mathcal{W}}^*(C)}$ to instances associated with the descendant clusters of $C$, computes a solution $\phi'(I'_C)$ to instance $I'_C$, of cost at most $\sum_{C'\in {\mathcal{W}}^*(C)}\phi(I_{C'})$. Since $I'_G=I$, this will complete the proof of the lemma.
The proof is by induction of the length of the longest path in the partitioning tree $\tau({\mathcal{L}})$ between $v(C)$ and its descendant. The base of the induction is when cluster $C$ is the leaf of the tree $\tau({\mathcal{L}})$. In this case, $I'_C=I_C$ holds, and we let $\phi'(I'_C)=\phi(I_C)$.
For the induction step, we consider some cluster $C\in {\mathcal{L}}$, whose corresponding vertex $v(C)$ is not a leaf vertex of the tree $\tau({\mathcal{L}})$.
Assume that ${\mathcal{W}}(C)=\set{C_1,\ldots,C_r}$. For convenience, for each $1\leq i\leq r$, we denote the supernode $v_{C_i}$ representing cluster $C_i$ in graph $G_C$ by $v_i$. By applying the induction hypothesis to every cluster $C_i\in {\mathcal{W}}(C)$, we obtain a solution $\phi'_i=\phi'(I'_{C_i})$ to instance $I'_{C_i}$ of \ensuremath{\mathsf{MCNwRS}}\xspace, whose cost is $\mathsf{cr}(\phi'_i)\leq \sum_{C'\in {\mathcal{W}}^*(C_i)}\mathsf{cr}(\phi(I_{C'}))$. It is now enough to show an efficient algorithm that constructs a solution $\phi'(I'_C)$ to instance $I'_C$, whose cost is at most $\mathsf{cr}(\phi(I_C))+\sum_{i=1}^r\mathsf{cr}(\phi'_i)\leq \sum_{C'\in {\mathcal{W}}^*(C)}\mathsf{cr}(\phi(I_{C'}))$.
We start with the solution $\tilde \phi=\phi(I_C)$ to instance $I_C$, and we process the clusters $C_1,\ldots,C_r$ one by one, gradually modifying the drawing $\tilde \phi$. We now describe the iteration when cluster $C_i$ is processed. We denote $\delta_{G'_C}(v_i)=\delta_G(C_i)=\set{e^i_1,\ldots,e^i_{q_i}}$, where the edges are indexed according to their ordering in ${\mathcal{O}}(C_i)$.
For all $1\leq j\leq q_i$, we denote $e^i_j=(x_j,y_j)$, where $x_j\in C_i$.
Let $D_i=D_{\tilde \phi}(v_i)$ be a tiny $v_i$-disc in the drawing $\tilde \phi$. For all $1\leq j\leq q_i$, we denote by $p^i_j$ the unique point on the image of edge $e^i_j$ that lies on the boundary of the disc $D_i$, and we let $\gamma(e^i_j)$ denote the segment of the image of $e^i_j$ that is disjoint from the interior of $D_i$. Therefore, $\gamma(e^i_j)$ connects the image of vertex $y_j$ to point $p^i_j$. Notice that points $p^i_1,\ldots,p^i_{q_i}$ appear on the boundary of $D_i$ in this circular order. If the orientation of this ordering is positive, then we say that vertex $v_i$ is positive, and otherwise we say that it is negative. We erase the parts of the images of all edges in the interior of disc $D_i$, and we erase the image of the vertex $v_i$ from the current drawing. We place another disc $D'_i$ inside $D_i$, so that $D'_i\subseteq D_i$, and the boundaries of both discs are disjoint.
Next, we consider the drawing $\phi'_i$ of the graph $G'_{C_i}$. We let $\hat D_i=D_{\phi'_i}(v^*)$ be a tiny $v^*$-disc in this drawing. Recall that $\delta_{G'_{C_i}}(v^*)=\delta_G(C_i)$. For all $1\leq j\leq q_i$, we denote by $\hat p^i_j$ the unique point on the image of the edge $e^i_j$ in $\phi'_i$ that lies on the boundary of the disc $\hat D_i$. Note that points $\hat p^i_1,\ldots,\hat p^i_{q_i}$ must appear on the boundary of the disc $\hat D_i$ in this circular order, from the definition of the rotation ${\mathcal{O}}_{v^*}\in \Sigma'_{C_i}$. We assume w.l.o.g. that, if vertex $v_i$ is positive, then the orientation of this ordering is negative, and otherwise it is positive (if this is not the case then we simply flip the drawing $\phi'_i$). Let $\hat D'_i$ be the disc that has the same boundary as $\hat D_i$ but whose interior is disjoint from that of $\hat D_i$ (so $\hat D'_i$ is the complement of disc $\hat D_i$; recall that the drawing $\phi'_i$ is on the sphere). For all $1\leq j\leq q_i$, we denote by $\gamma'(e^i_j)$ the segment of the image of edge $e^i_j$ that lies inside $\hat D'_i$. Therefore, $\gamma'(e^i_j)$ connects the image of vertex $x_j$ to point $\hat p^i_j$.
We copy the disc $\hat D'_i$, together with its contents (in $\phi'_i$), to the current drawing $\tilde \phi$, so that the boundaries and the interiors of the discs $\hat D'_i$ and $D'_i$ coincide.
Assume w.l.o.g. that vertex $v_i$ is positive. Then points $p^i_1,\ldots,p^i_{q_i}$ appear on the boundary of disc $D'_i$ in this counter-clock-wise order, while points $\hat p^i_1,\ldots,\hat p^i_{q_i}$ appear on the boundary of disc $\hat D'_i$ in this counter-clock-wise order. Therefore, we can compute a collection $\set{\sigma_1,\ldots,\sigma_{q_i}}$ of mutually disjoint curves, where for all $1\leq j\leq q_i$, curve $\sigma_j$ has endpoints $p_j^i$ and $\hat p_j^i$, and all inner points of $\sigma_j$ lie in $D_i\setminus D'_i$, and are disjoint from the boundary of $D_i$. For all $1\leq j\leq q_i$, we now define the image of the edge $e_j=(x_j,y_j)$ to be the concatenation of the curves $\gamma^i_j,\sigma_j$, and $\hat \gamma^i_j$.
Once every cluster $C_i\in {\mathcal{W}}(C)$ is processed in this manner, we obtain a solution $\phi'(I'_C)$ to instance $I'_C$ of \ensuremath{\mathsf{MCNwRS}}\xspace. It is immediate to verify that the total number of crossings in this solution is at most $\mathsf{cr}(\phi(I_C))+\sum_{i=1}^r\mathsf{cr}(\phi'_i)\leq \sum_{C'\in {\mathcal{W}}^*(C)}\mathsf{cr}(\phi(I_{C'}))$. The lemma follows by letting $\phi$ be the solution $\phi'(I_{C'})$ that we construct for instance $I_{C'}$, where $C'=G$.
\subsection{Proof of \Cref{lem: disengagement final cost}}
\label{subsec: appx basic diseng opt bounds}
Throughout the proof, we denote $|E(G)|=m$.
Consider first a cluster $C\in {\mathcal{L}}^{\operatorname{light}}$. Recall that, in order to define instance $I_C$, we used the distribution ${\mathcal{D}}(C)$ over the internal $C$-routers, where $C$ is $\beta$-light with respect to ${\mathcal{D}}(C)$.
We selected a router ${\mathcal{Q}}(C)$ from the distribution ${\mathcal{D}}(C)$ at random, whose center vertex is denoted by $u(C)$. We then used the algorithm from \Cref{lem: non_interfering_paths} to compute a non-transversal set $\tilde {\mathcal{Q}}(C)$ of paths, routing all edges of $\delta_G(C)$ to vertex $u(C)$, so $\tilde {\mathcal{Q}}(C)$ is also an internal $C$-router. The set $\tilde {\mathcal{Q}}(C)$ of paths was used in order to define the ordering ${\mathcal{O}}(C)$ of the edges of $\delta_G(C)$, which was in turned used in order to define instance $I_C$.
Consider now a cluster $C\in {\mathcal{L}}^{\operatorname{bad}}$. We apply the algorithm from \Cref{cor: simple guiding paths} to $C$, obtaining a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers, such that, for every edge $e\in E(C)$, $\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{\cong({\mathcal{Q}}(C),e)}\leq O(\log^4m/\alpha_0)\leq O(\log^{16}m)$. We then select a router ${\mathcal{Q}}(C)$ from the distribution ${\mathcal{D}}(C)$ at random, and denote by $u(C)$ its center vertex. We view the paths of ${\mathcal{Q}}(C)$ as being directed towards $u(C)$.
Next, we use the algoritm from \Cref{lem: non_interfering_paths} to compute a non-transversal set $\tilde {\mathcal{Q}}(C)$ of paths, routing all edges of $\delta_G(C)$ to vertex $u(C)$, so $\tilde {\mathcal{Q}}(C)$ is also an internal $C$-router. The algorithm ensures that, for every edge $e\in E(G)$, $\cong_G(\tilde {\mathcal{Q}}(C),e)\leq \cong_G({\mathcal{Q}}(C),e)$.
Consider an optimal solution $\phi^*$ to instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace. For every cluster $C\in {\mathcal{L}}$, denote by $\chi(C)$ the set of all crossings $(e,e')_p$ in the drawing $\phi^*$, where at least one of the edges $e,e'$ lies in $E(C)\cup \delta_G(C)$.
Recall that ${\mathcal{I}}=\set{I_C\mid C\in {\mathcal{L}}}$. The proof of the lemma follows from the following claim.
\begin{claim}\label{claim: cost of cluster instance}
For every cluster $C\in {\mathcal{L}}$, $\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I_C)}\leq O(\beta^2\cdot(|\chi(C)|+|E(C)|))$.
\end{claim}
Indeed, for all $1\leq i\leq \mathsf{dep}({\mathcal{L}})$, let ${\mathcal{L}}_i\subseteq {\mathcal{L}}$ be the set of all clusters that lie at level $i$ of the laminar family. Note that all clusters in ${\mathcal{L}}_i$ are mutually disjoint. Therefore, every crossing $(e,e')_p$ of the drawing $\phi^*$ may contribute to the sets $\chi(C)$ of at most four clusters of ${\mathcal{L}}_i$: at most two clusters $C$ with $e\in E(C)\cup \delta_G(C)$, and at most two clusters $C'$ with $e\in E(C')\cup \delta_G(C')$. Therefore:
\[
\sum_{C\in {\mathcal{L}}_i}\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I_C)}\leq \sum_{C\in {\mathcal{L}}_i} O(\beta^2\cdot(|\chi(C)|+|E(C)|)) \leq O(\beta^2\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)).
\]
Summing this up over all $1\leq i\leq \mathsf{dep}({\mathcal{L}})$, we get that $\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq O(\mathsf{dep}({\mathcal{L}})\cdot\beta^2\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|))$. In order to complete the proof of \Cref{lem: disengagement final cost}, it is now enough to prove \Cref{claim: cost of cluster instance}, which we do next.
In the remainder of this proof, we fix a cluster $C\in {\mathcal{L}}$. Recall that there is a distribution ${\mathcal{D}}'(C)$ over the set $\Lambda'(C)$ of external $C$-routers, such that for every edge $e$ of $E(G\setminus C)$, $$\expect[{\mathcal{Q}}'(C)\sim{\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta.$$
We sample an external $C$-router ${\mathcal{Q}}'(C)$ from the distribution ${\mathcal{D}}'(C)$. We view the paths of ${\mathcal{Q}}'(C)$ as being directed towards vertex $u'(C)$, that is the center of the router. We apply the algorithm from \Cref{lem: non_interfering_paths} to obtain a collection $\tilde {\mathcal{Q}}'(C)$ of non-transversal paths, routing the edges of $\delta_G(C)$ to $u'(C)$, such that, for every edge $e\in E(G)$, $\cong_G(\tilde {\mathcal{Q}}'(C),e)\leq \cong_G({\mathcal{Q}}'(C),e)$. In particular, $\tilde {\mathcal{Q}}'(C)$ is also an external $C$-router with center vertex $u'(C)$.
In order to simplify the notation, we denote $\tilde {\mathcal{Q}}'(C)$ by ${\mathcal{Q}}'$, and $\tilde {\mathcal{Q}}(C)$ by ${\mathcal{Q}}$. To summarize, ${\mathcal{Q}}'$ is an external $C$-router, and we are guaranteed that, for every edge $e\in E(G\setminus C)$, $\expect{\cong_G({\mathcal{Q}}',e)}\leq \beta$. Set ${\mathcal{Q}}$ of paths is an internal $C$-router. If $C\in {\mathcal{L}}^{\operatorname{bad}}$, then for every edge $e\in E(C)$,
$\expect{\cong({\mathcal{Q}},e)}\leq \beta$, while, if $C\in {\mathcal{L}}^{\operatorname{light}}$ then, for every edge $e\in E(C)$, $\expect{(\cong_G({\mathcal{Q}},e))^2}\leq \beta$.
We denote the center vertex of router ${\mathcal{Q}}$ by $u$, and the center vertex of router ${\mathcal{Q}}'$ by $u'$.
We denote by ${\mathcal{W}}=\set{C_1,\ldots,C_r}$ the set of child-clusters of $C$. We partition ${\mathcal{W}}$ into two subsets: ${\mathcal{W}}^{\operatorname{bad}}={\mathcal{W}}\cap {\mathcal{L}}^{\operatorname{bad}}$, and ${\mathcal{W}}^{\operatorname{light}}={\mathcal{W}}\cap {\mathcal{L}}^{\operatorname{light}}$. For convenience, for all $1\leq i\leq r$, we denote the internal $C_i$-router $\tilde {\mathcal{Q}}(C_i)$ that we have constructed by ${\mathcal{Q}}_i$, and its center vertex by $u_i$. Recall that, if $C_i\in {\mathcal{L}}^{\operatorname{bad}}$, then for every edge $e\in E(C_i)$,
$\expect{\cong({\mathcal{Q}}_i,e)}\leq \beta$, while, if $C_i\in {\mathcal{L}}^{\operatorname{light}}$ then, for every edge $e\in E(C_i)$, $\expect{(\cong_G({\mathcal{Q}}_i,e))^2}\leq \beta$.
It will be convenient for us to also define another cluster $C_0$ to be the connected component of $G\setminus C$ containing the vertex $u'$ -- the center vertex of the external $C$-router ${\mathcal{Q}}'$. It is immediate to verify that the set ${\mathcal{Q}}'$ of paths is an internal $C_0$-router, and for consistency we denote it by ${\mathcal{Q}}_0$. We also denote the center vertex of this router by $u_0=u'$. Recall that each one of the sets ${\mathcal{Q}}_0,{\mathcal{Q}}_1,\ldots,{\mathcal{Q}}_r$ of paths is non-transversal with respect to $\Sigma$.
For convenience, we denote $R=C\setminus\textsf{left}(\bigcup_{i=1}^rC_i\textsf{right} )$, and ${\mathcal{Q}}^*=\bigcup_{i=0}^r{\mathcal{Q}}_i$.
Lastly, it will be convenient for us to assume that no edge of $G$ has its endpoints in two distinct clusters of $C_0,C_1,\ldots,C_r$. For each such edge $e$, we subdivide the edge with a new vertex $v_e$, that is added to graph $R$ as an isolated vertex.
Note that, for all $0\leq i<j\leq r$, the only vertices that may be shared by paths in ${\mathcal{Q}}_i$ and paths in ${\mathcal{Q}}_j$ are vertices of $V(R)$, which must serve as endpoints of those paths.
The remainder of the proof of \Cref{claim: cost of cluster instance} consists of three steps. In the first step, we will define a new graph $H$ by slightly modifying graph $G$, and compute its drawing $\psi$. In the second step, we use graph $H$ and its drawing $\psi$ in order to construct an initial drawing $\phi'$ of graph $G_C$ (associated with instance $I_C=(G_C,\Sigma_C)\in {\mathcal{I}}$ of \ensuremath{\mathsf{MCNwRS}}\xspace). This drawing, however, may not obey all rotations in $\Sigma_C$. In the third and the last step, we modify drawing $\phi'$ to obtain the final drawing $\phi''$ of $G_C$, that is a valid solution to instance $I_C$ of \ensuremath{\mathsf{MCNwRS}}\xspace. We now describe each of the three steps in turn.
\subsubsection{Step 1: Graph $H$}
We assign, to every edge $e\in E(G)$, an integer $n_e\geq 0$, as follows. For every edge $e\in E(R)\cup \delta_G(R)$, we let $n_e=1$. For every other edge $e\in E(G)\setminus E(R)$, we let $n_e=\cong_G({\mathcal{Q}}^*,e)$.
In order to construct graph $H$, we start with $V(H)=V(G)$. For every edge $e=(u,v)\in E(G)$ with $n_e>0$, we add a set $J(e)$ of $n_e$ parallel edges $(u,v)$ to graph $H$, and we call the edges of $J(e)$ \emph{copies of edge $e$}. This completes the definition of the graph $H$. Note that graph $H$ is a random graph, as values $\set{n_e}_{e\in E(G)}$ are random variables.
Consider the optimal solution $\phi^*$ to instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace. We use $\phi^*$ in order to define a drawing $\psi$ of the graph $H$, in a natural way: for every vertex $v\in V(H)$, its image in $\psi$ remains the same as in $\phi^*$. For every edge $e\in E(G)$ with $n(e)\geq 1$, we draw the edges of $J(e)$ in parallel to the image of $e$ in $\phi^*$, immediately next to it, so that their images do not cross.
Consider the resulting drawing $\psi$ of graph $H$, and let $(e'_1,e'_2)_p$ be a crossing of $\psi$. Assume that $e'_1\in J(e_1)$ and $e'_2\in J(e_2)$. Then the images of the edges $e_1,e_2$ cross in $\phi^*$, at some point $p'$ that is very close to point $p$. We say that crossing $(e_1,e_2)_{p'}$ of $\phi^*$ is \emph{responsible} for the crossing $(e_1',e_2')_p$ of $\psi$.
Next, we classify the crossings of the drawing $\psi$ into three types, and we bound the expected number of crossings of some of the types. Consider some crossing $(e'_1,e'_2)_p$ of $\psi$, and let $(e_1,e_2)_{p'}$ be the crossing of $\phi^*$ that is responsible for $(e_1',e_2')_p$. We say that crossing $(e'_1,e'_2)_p$ is a \emph{type-1 crossing} if there is some cluster $C_i\in {\mathcal{W}}^{\operatorname{light}}$, with $e_1,e_2\in E(C_i)$. We say that it is a \emph{type-2 crossing} if there is an index $0\leq i\leq r$ with $e_1,e_2\in E(C_i)$, and either $i=0$ or $C_i\in{\mathcal{W}}^{\operatorname{bad}}$ holds. We say that $(e_1',e_2')_p$ is a \emph{type-3 crossing} otherwise.
We now bound the expected number of type-1 and type-3 crossings. We do not bound the number of type-2 crossings, as all such crossings will eventually be eliminated.
\paragraph{Type-1 crossings.} Consider some cluster $C_i\in {\mathcal{W}}^{\operatorname{light}}$, and some crossing $(e_1,e_2)_{p'}$ of $\phi^*$, such that $e_1,e_2\in E(C_i)$. Notice that crossing $(e_1,e_2)_{p'}$ lies in $\chi(C_i)$. The number of type-1 crossings in $\phi$ that this crossing is responsible for is $n_{e_1}\cdot n_{e_2}\leq n_{e_1}^2+n_{e_2}^2$. Observe that, for an edge $e\in E(C_i)$, $n_e=\cong_G({\mathcal{Q}}_i,e)$, and so $\expect{n_e^2}=\expect{(\cong_G({\mathcal{Q}}_i,e))^2}\leq \beta$. We conclude that the total expected number of type-1 crossings in $\psi$ is bounded by:
\[ \sum_{C_i\in {\mathcal{W}}^{\operatorname{light}}} \sum_{(e_1,e_2)_{p'}\in \chi(C_i)}\expect{n_{e_1}^2+n_{e_2}^2 }\leq \sum_{C_i\in {\mathcal{W}}^{\operatorname{light}}} O(\beta \cdot |\chi(C_i)|)\leq O(\beta\cdot |\chi(C)|).
\]
\paragraph{Type-3 crossings.} Consider some crossing $(e_1,e_2)_{p'}$ of $\phi^*$. If edges $e_1,e_2$ lie in the same cluster $C_i$, for $0\leq i\leq r$, then this crossing may not be responsible for any type-3 crossings in $\psi$. Assume now that this is not the case. Then the total number of type-3 crossings that $(e_1,e_2)_{p'}$ is responsible for is at most $n_{e_1}\cdot n_{e_2}$. Furthermore $n_{e_1},n_{e_2}$ are independent random variables, each of which has expectation at most $\beta$. Therefore, the expected number of type-3 crossings for which crossing $(e_1,e_2)_{p'}$ is responsible is at most $\beta^2$. Note that, at least one of the edges $e_1,e_2$ must lie in $E(C)\cup \delta_G(C)$, so crossing $(e_1,e_2)_{p'}$ must lie in $\chi(C)$. We conclude that the total expected number of type-3 crossings in $\phi$ is at most $|\chi(C)|\cdot \beta^2$.
Consider now an index $0\leq i\leq r$, and let $E_i=\delta_G(C_i)$. From our definition, for every edge $e\in E_i$, $n_e=1$. Recall that we have defined a set ${\mathcal{Q}}_i=\set{Q(e)\mid e\in E_i}$ of paths in graph $G$, routing all edges of $E_i$ to the vertex $u_i\in V(C_i)$. The paths in ${\mathcal{Q}}_i$ are non-transversal with respect to $\Sigma$, and all their inner vertices are contained in $C_i$. We will now define a corresponding set $\hat {\mathcal{Q}}_i=\set{\hat Q(e)\mid e\in E_i}$ of paths in graph $H$, routing the edges of $E_i$ to the same vertex $u_i$, such that the paths in $\hat {\mathcal{Q}}_i$ are edge-disjoint. In order to do so, we assign, to every path $Q(e)\in {\mathcal{Q}}_i$, for every edge $e'\in E(Q(e))\setminus\set{e}$, a copy of the edge $e'$ from $J(e')$, such that every copy of edge $e'$ is assigned to a distinct path. We will then obtain path $\hat Q(e)$ from path $Q(e)$ by replacing every edge $e'\in E(Q(e))\setminus\set{e}$ with its copy that was assigned to $Q(e)$.
Consider any edge $e'\in E(C_i)$. If $e'$ is not incident to the vertex $u_i$, then we assign each copy of $e'$ to a distinct path in ${\mathcal{Q}}_i$ that contains $e'$ arbitrarily. Assume now that edge $e'$ is incident to vertex $u_i$, and that $C_i\not\in {\mathcal{W}}^{\operatorname{light}}$. In this case, as before, we assign each copy of $e'$ to a distinct path in ${\mathcal{Q}}_i$ that contains $e'$ arbitrarily. It now remains to consider the case where $C_i\in {\mathcal{W}}^{\operatorname{light}}$, and edges that are incident to vertex $u_i$. We need to assign copies of such edges to paths in ${\mathcal{Q}}_i$ more carefully. The goal of this more careful assignment is to achieve the following property: if we denote $E_i=\set{e_1,\ldots,e_{h_i}}$, where the edges are indexed according to the ordering ${\mathcal{O}}(C_i)$, and, for each such edge $e_j$, we denote by $e'_j$ be the last edge on path $\hat Q(e_j)$ (that we are trying to construct), then the images of the edges $\set{e'_1,\ldots,e'_{h_i}}$ enter the image of $u_i$ in the drawing $\psi$ of graph $H$ in this circular order. We now describe the procedure for assigning copies of edges of $\delta_G(u_i)$ to paths in ${\mathcal{Q}}_i$.
We start by revisiting the definition of the ordering ${\mathcal{O}}(C_i)$ of the edges of $E_i=\delta_G(C_i)$, which is an ordering that is guided by the set ${\mathcal{Q}}_i$ of paths and the rotation system $\Sigma$.
Denote $\delta_G(u_i)=\set{a^i_1,\ldots,a^i_{z_i}}$, where the edges are indexed according to their circular ordering ${\mathcal{O}}_{u_i}\in \Sigma$. We assume w.l.o.g. that the orientation of this ordering in the drawing $\phi^*$ of $G$ is negative (or clock-wise). For all $1\leq j\leq z_i$, let ${\mathcal{Q}}^j_i\subseteq {\mathcal{Q}}_i$ the set of paths in ${\mathcal{Q}}_i$ whose last edge is $a^i_j$. We defined an ordering $\hat {\mathcal{O}}_i$ of the paths in ${\mathcal{Q}}_i$, where the paths in sets ${\mathcal{Q}}_i^1,\ldots,{\mathcal{Q}}_i^{z_i}$ appear in the natural order of their indices, and for all $1\leq j\leq z_i$, the ordering of the paths in set ${\mathcal{Q}}_i^j$ is arbitrary. We denote ${\mathcal{Q}}_i^j=\set{Q(e_1^{i,j}),Q(e_2^{i,j}),\ldots,Q(e_{m_{i,j}}^{i,j})}$, and assume that these paths are indexed according to the ordering that we have chosen when defining $\hat {\mathcal{O}}_i$.
Ordering $\hat{\mathcal{O}}_i$ of the paths in ${\mathcal{Q}}_i$ was then used to define the ordering ${\mathcal{O}}(C_i)$ of the edges in $E_i$: we obtain the ordering ${\mathcal{O}}(C_i)$ from $\hat {\mathcal{O}}_i$ by replacing, for every path $Q(e_{\ell}^{i,j})\in {\mathcal{Q}}_i$, the path $Q(e_{\ell}^{i,j})$ in $\hat {\mathcal{O}}$ with the edge $e_{\ell}^{i,j}$ (the first edge of $Q(e_{\ell}^{i,j})$).
Consider now some edge $a^i_j\in \delta_G(u_i)$. Recall that we have defined a set $J(a^i_j)$ of $n_{a^i_j}= m_{i,j}$ copies of the edge $a^i_j$. We denote these copies by $\hat a^{i,j}_1,\ldots,\hat a^{i,j}_{m_{i,j}}$, where the copies are indexed according to the order in which their images enter the image of vertex $u_i$ in the drawing $\psi$ of $H$, in the clock-wise direction. For all $1\leq \ell\leq m_{i,j}$, we assign the copy $\hat a^{i,j}_{\ell}$ of edge $a^i_j$ to the path $Q(e_{\ell}^{i,j})$.
This completes the assignment of the copies of the edges incident to vertex $u_i$ to the paths of ${\mathcal{Q}}_i$.
We now define a set $\hat {\mathcal{Q}}_i=\set{\hat Q(e)\mid e\in E_i}$ of paths in graph $H$, routing the edges of $E_i$ to vertex $u_i$, as follows. For every edge $e\in E_i$, path $\hat Q(e)$ is obtained from the path $Q(e)\in {\mathcal{Q}}_i$ by replacing every edge $e'\in E(Q(e))\setminus \set{e}$ with the copy of $e'$ that was assigned to path $Q(e)$. The following observation summarizes the properties of the path set $\hat {\mathcal{Q}}_i$, that follow immediately from our construction.
\begin{observation}\label{obs: properties of new path set}
Paths in set $\hat {\mathcal{Q}}_i=\set{\hat Q(e)\mid e\in E_i}$ route the edges of $E_i$ to vertex $u_i$ in graph $H$, and all inner vertices on all paths in $\hat {\mathcal{Q}}_i$ lie in $V(C_i)$. Moreover, the paths of $\hat {\mathcal{Q}}_i$ are edge-disjoint. Additionally, if $C_i\in {\mathcal{W}}^{\operatorname{light}}$, then the following holds. Denote $E_i=\set{e_1,\ldots,e_{h_i}}$, where the edges are indexed according to the ordering ${\mathcal{O}}(C_i)$. For each such edge $e_j$, let $e'_j$ be the last edge on path $\hat Q(e_j)$. Then the images of the edges $e'_1,\ldots,e'_{h_i}$ enter the image of $u_i$ in the drawing $\psi$ of graph $H$ in the circular order of their indices.
\end{observation}
\subsubsection{Step 2: Initial Drawing of Graph $G_C$}
In this step we exploit the drawing $\psi$ of graph $H$ that we have constructed in the first step, in order to construct an initial drawing $\phi'$ of graph $G_C$.
In order to construct the drawing $\phi'$ of graph $G_C$, we start with the drawing $\psi$ of graph $H$, and then gradually modify it. We place the image of the vertex $v^*$ in $\phi'$ at point $\psi(u_0)$, and, for all $1\leq i\leq r$, we place the image of the vertex $v_{C_i}$ at point $\psi(u_i)$. Intuitively, the images of the vertices and the edges of $R$ will remain unchanged. For all $0\leq i\leq r$, we will utilize the images of the paths of $\hat{\mathcal{Q}}_i$ in $\psi$ in order to draw the edges of $E_i$. There are two issues with this approach. First, we did not bound the expected number of type-2 crossings in $\psi$, so there may be many crossings between pairs of edges lying on paths of ${\mathcal{Q}}_i$, where $i=0$, or $C_i\in {\mathcal{W}}^{\operatorname{bad}}$. We take care of this issue by performing a type-2 uncrossing for each such path set $\hat {\mathcal{Q}}_i$, to obtain the drawings of the edges in $E_i$. The second problem then remains for indices $i$ with $C_i\in {\mathcal{W}}^{\operatorname{light}}$. Since several paths from $\hat {\mathcal{Q}}_i$ may share the same vertex, there could be points that lie on images of multiple paths of $\hat {\mathcal{Q}}_i$. We take care of this latter issue by employing a nudging procedure. We now describe each of these two operations in turn.
\paragraph{Uncrossing.}
We consider indices $i$ for which either $i=0$ or $C_i\in {\mathcal{W}}^{\operatorname{bad}}$ holds one by one. Consider any such index $i$. We view every path of $\hat {\mathcal{Q}}_i$ as being directed towards the vertex $u_i$. We use the algorithm from \Cref{thm: new type 2 uncrossing} in order to compute a type-2 uncrossing, that produces, for every edge $e\in E_i$, a directed curve $\gamma(e)$, that connects the image of the endpoint of $e$ that lies in $R$ to the image of $u_i$ in $\psi$. Recall that we are guaranteed that the curves in the resulting set $\Gamma_i=\set{\gamma(e)\mid e\in E_i}$ do not cross each other, and each such curve is aligned with the drawing of graph $\bigcup_{\hat Q(e)\in \hat {\mathcal{Q}}_i}\hat Q(e)$ induced by $\psi$.
Let $\psi'$ be a drawing obtained from $\psi$ as follows. For every index $i$ with $i=0$ or $C_i\in {\mathcal{W}}^{\operatorname{bad}}$, we delete the images of all vertices of $V(C_i)$ and all edges with at least one endpoint in $V(C_i)$ from the drawing. If $i=0$, then we place the image of vertex $v^*$ at point $\psi(u_0)$, and otherwise we place the image of vertex $v_{C_i}$ at point $\psi(u_i)$. For every edge $e\in E_i$, we then let $\gamma(e)\in \Gamma_i$ be the image of the edge $e$.
Note that this uncrossing step has eliminated all type-2 crossings, and every crossing in the resulting drawing $\psi'$ corresponds to a distinct type-1 or type-3 crossing of $\psi$. Therefore, the expected number of crossings of $\psi'$ is bounded by $O(\beta^2\cdot |\chi(C)|)$.
We call all crossings that are currently present in drawing $\psi'$ \emph{primary crossings}.
\paragraph{Nudging.}
We now consider the indices $i$ with $C_i\in {\mathcal{W}}^{\operatorname{light}}$ one by one. When such an index $i$ is considered, we delete the images of all vertices of $V(C_i)$ and all edges with at least one endpoint in $V(C_i)$ from the current drawing $\psi'$. We then place the image of vertex $v_{C_i}$ at point $\psi(u_i)$. For every edge $e\in E_i$, we initially let $\gamma(e)$ be the image of the path $\hat Q(e)\in \hat {\mathcal{Q}}_i$ in $\psi$, and we add $\gamma(e)$ to the current drawing as the image of the edge $e$. Note that the curves in $\set{\gamma(e)\mid e\in E_i}$ enter the image of $v_{C_i}$ in the order ${\mathcal{O}}(C_i)$ of their corresponding edges in $E_i$, from \Cref{obs: properties of new path set}. However, it is possible that, for some vertex $x\in V(C_i)$, point $\psi(x)$ lies on more than two curves from
$\set{\gamma(e)\mid e\in E_i}$.
We process each vertex $x\in V(C_i)\setminus \set{u_i}$ one by one. Consider any such vertex $x$, and let ${\mathcal{Q}}^x\subseteq \hat{\mathcal{Q}}_i$ be the set of all paths containing vertex $x$. Note that $x$ must be an inner vertex on each such path. For convenience, we denote ${\mathcal{Q}}^x=\set{Q(e_1),\ldots,Q(e_z)}$. Consider the tiny $x$-disc $D=D_{\psi}(x)$. For all $1\leq j\leq z$, denote by $s_j$ and $t_j$ the two points on the curve $\gamma(e_j)$ that lie on the boundary of disc $D$. We use the algorithm from \Cref{claim: curves in a disc} to compute a collection $\set{\sigma_1,\ldots,\sigma_z}$ of curves, such that, for all $1\leq j\leq z$, curve $\sigma_j$ connects $s_j$ to $t_j$, and the interior of the curve is contained in the interior of $D$. Recall that every pair of resulting curves crosses at most once, and every point in the interior of $D$ may be contained in at most two curves.
Consider now a pair $\sigma_{\ell},\sigma_{\ell'}$ of curves, and assume that these two curves cross. Recall that, from \Cref{claim: curves in a disc}, this may only happen if the two pairs $(s_{\ell},t_{\ell})$, $(s_{\ell'},t_{\ell'})$ of points cross. Denote by $e_1,e_2$ the two edges that lie on path $\hat Q(e_{\ell})$ immediately before and immediately after vertex $x$, and denote by $e_1',e_2'$ the two edges that lie on path $\hat Q(e_{\ell'})$ immediately before and immediately after vertex $x$. We assume that edges $e_1,e_2$ are copies of edges $\hat e_1,\hat e_2$ of $G$, and similarly, $e_1',e_2'$ are copies of edges $\hat e_1',\hat e_2'$ of $G$, respectively. Assume first that there are four distinct edges in set $\set{\hat e_1,\hat e_1',\hat e_2,\hat e_2'}$. From the fact that the two pairs $(s_{\ell},t_{\ell})$, $(s_{\ell'},t_{\ell'})$ of points cross, we get that these four edges must appear in the rotation ${\mathcal{O}}_x\in \Sigma$ in the order $(\hat e_1,\hat e_1',\hat e_2,\hat e_2')$. Since the paths of ${\mathcal{Q}}_i$ are non-transversal with respect to $\Sigma$, this is impossible. Therefore, we conclude that paths $Q(e_{\ell}),Q(e_{\ell'})$ must share an edge that is incident to $x$. If $e^*$ is an edge incident to $x$ that the two paths share, then we say that $e^*$ is \emph{responsible for the crossing between $\sigma_{\ell}$ and $\sigma_{\ell'}$}.
For all $1\leq j\leq z$, we modify the curve $\gamma(e_j)$, by replacing the segment of the curve that is contained in disc $D$ with $\sigma_j$.
Once every vertex $x\in V(C_i)\setminus \set{u_i}$ is processed, we obtain the final set $\Gamma'_i=\set{\gamma'(e)\mid e\in E_i}$ of curves, which are now guaranteed to be in general position. For every edge $e\in E_i$, we modify the image of edge $e$ in the current drawing, by replacing it with the new curve $\gamma'(e)$. As before, curves of $\Gamma'_i$ enter the image of $v_{C_i}$ according to the ordering ${\mathcal{O}}(C_i)$.
Once every index $i$ with $C_i\in {\mathcal{W}}^{\operatorname{light}}$ is processed, we obtain a valid drawing $\phi'$ of the graph $G_C$. In this drawing, for every index $i$ with $C_i\in {\mathcal{W}}^{\operatorname{light}}$, the images of the edges in set $E_i=\delta_{G_C}(v_{C_i})$ enter the image of vertex $v_{C_i}$ according to the ordering ${\mathcal{O}}_{v_{C_i}}\in \Sigma_C$, which is precisely ${\mathcal{O}}(C_i)$. However, for indices $i$ with $C_i\in {\mathcal{W}}^{\operatorname{bad}}$, this property may not hold, and the edges incident to $v^*$ may enter the image of $v^*$ in an arbitrary order. For every other vertex $v$ of $G_C$, the rotation ${\mathcal{O}}_v\in \Sigma_{C}$ is identical to the rotation ${\mathcal{O}}_v\in \Sigma$, and is obeyed by the current drawing $\phi'$. We modify the drawing $\phi'$ to obtain a drawing that is consistent with the rotation system $\Sigma_{C}$ in the third step. Notice however that the nudging operation may have introduced some new crossings. Each such new crossing must be contained in a disc $D_{\psi}(x)$, for some vertex $x$ that must lie in some cluster $C_i\in {\mathcal{W}}^{\operatorname{light}}$. We call all such new crossings \emph{secondary crossings}. We now bound the total number of secondary crossings.
Fix an index $i$ with $C_i\in {\mathcal{W}}^{\operatorname{light}}$, and consider some vertex $x\in V(C_i)$. Every secondary crossing that is contained in ${\mathcal{D}}_{\psi}(x)$ is a crossing between a pair $\sigma_{\ell},\sigma_{\ell'}$ of curves that we have defined when processing vertex $x$, and each such crossing was charged to an edge of $G$ that is incident to $x$, whose copies lie on the corresponding two paths $\hat Q(e_{\ell}),\hat Q(e_{\ell'})\in \hat {\mathcal{Q}}_i$. If $e$ is an edge that is incident to $x$ in $G$, then there are at most $(\cong_G({\mathcal{Q}}_i,e))^2$ pairs of paths in ${\mathcal{Q}}_i$ that contain $e$, and each such pair of paths may give rise to a single secondary crossing in $D_{\psi}(x)$ that is charged to edge $e$. Therefore, the total expected number of secondary crossings that are contained in discs $D_{\psi}(x)$ for vertices $x\in V(C_i)$ is bounded by:
\[ \sum_{e\in E(C_i)}O(\expect{(\cong_G({\mathcal{Q}}_i,e))^2})\leq O(\beta \cdot |E(C_i)|), \]
since $C_i\in {\mathcal{W}}^{\operatorname{light}}$.
We conclude that the total expected number of secondary crossings in $\phi'$ is at most $\sum_{C_i\in{\mathcal{W}}^{\operatorname{light}}}O(\beta \cdot |E(C_i)|)\leq O(\beta\cdot |E(C)|)$, and the total number of all crossings in $\phi'$ is at most $O(\beta^2\cdot (|\chi(C)|+|E(C)|))$.
\subsubsection{Step 3: the Final Drawing}
So far we have obtained a drawing $\phi'$ of graph $G_C$, that obeys the rotations ${\mathcal{O}}_v\in \Sigma_C$ for all vertices $v\in V(G_C)$, except possibly for vertex $v^*$, and vertices $v_{C_i}$, for $C_i\in {\mathcal{W}}^{\operatorname{bad}}$. We now fix this drawing to obtain a final drawing $\phi''$ of $G_C$ that obeys the rotation system $\Sigma_C$.
Let $U=\set{v^*}\cup \set{v_{C_i}\mid C_i\in {\mathcal{W}}^{\operatorname{bad}}}$. For each vertex $x\in U$, we denote by $\hat {\mathcal{O}}(x)={\mathcal{O}}_x\in \Sigma_C$ the rotation associated with vertex $x$ in the rotation system $\Sigma_C$, and by $\hat {\mathcal{O}}'(x)$ the circular order in which the edges of $\delta_{G_C}(x)$ enter the image of $x$ in the current drawing $\phi'$. Note that, for a vertex $x=v_{C_i}$, where $C_i\in {\mathcal{W}}^{\operatorname{bad}}$, if we denote by $\Sigma(C_i)$ the rotation system induced by $\Sigma$ for cluster $C$, then the following must hold:
\[\mbox{\sf dist}(\hat {\mathcal{O}}(x),\hat {\mathcal{O}}'(x))\leq |\delta_{G_C}(x)|^2=|\delta_G(C_i)|^2\leq \beta\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(C_i,\Sigma(C_i))+|E(C_i)|)\leq \beta(|\chi(C_i)|+|E(C_i)|).\]
(we have used the fact that cluster $C_i$ is a $\beta$-bad cluster).
We use the following claim, whose proof appears in Section \ref{subsec: appx bounding distance between rotations}, in order to bound $\mbox{\sf dist}(\hat {\mathcal{O}}(v^*),\hat {\mathcal{O}}'(v^*))$.
\begin{claim}\label{claim: bounding distance between rotations}
$\expect{\mbox{\sf dist}(\hat {\mathcal{O}}(v^*),\hat {\mathcal{O}}'(v^*))}\leq \beta^2\cdot (|\chi(C)|+|E(C)|)$.
\end{claim}
In order to compute the final drawing $\phi''$ of graph $G_C$, we process the vertices $x\in U$ one by one. When vertex $x$ is processed, we apply the algorithm from \Cref{lem: ordering modification} to it. The algorithm modifies the current drawing of graph $G_C$ within the tiny $x$-disc $D(x)$ to ensure that the images of the edges of $\delta_{G_C}(x)$ enter the image of $x$ in the circular order $\hat {\mathcal{O}}(x)$. This modification increases the number of crossings in the current drawing by at most $2\cdot\mbox{\sf dist}(\hat {\mathcal{O}}(x),\hat {\mathcal{O}}'(x))$. Once every vertex of $U$ is processed, we obtain the final drawing $\phi''$ of graph $G_C$, which obeys the rotation system $\Sigma_C$. Moreover, $\mathsf{cr}(\phi'')\leq \mathsf{cr}(\phi')+\sum_{x\in U}2\cdot\mbox{\sf dist}(\hat {\mathcal{O}}(x),\hat {\mathcal{O}}'(x))$. Recall that $\expect{\mathsf{cr}(\phi')}\leq O(\beta^2)\cdot (|\chi(C)|+|E(C)|)$, and that, for every vertex $x=v_{C_i}$ with $C_i\in {\mathcal{W}}^{\operatorname{bad}}$, $\mbox{\sf dist}(\hat {\mathcal{O}}(x),\hat {\mathcal{O}}'(x))\leq \beta(|\chi(C_i)|+|E(C_i)|)$. Combining this with \Cref{claim: bounding distance between rotations}, we get that:
\[\expect{\mathsf{cr}(\phi'')}\leq O(\beta^2)\cdot (|\chi(C)|+|E(C)|)+\sum_{C_i\in {\mathcal{W}}^{\operatorname{bad}}} O(\beta)\cdot (|\chi(C_i)|+|E(C_i)|)\leq O(\beta^2\cdot (|\chi(C)|+|E(C)|)). \]
This completes the proof of \Cref{claim: cost of cluster instance}.
\subsection{Proof of \Cref{claim: bounding distance between rotations}}
\label{subsec: appx bounding distance between rotations}
Clearly, $\mbox{\sf dist}(\hat {\mathcal{O}}(v^*),\hat {\mathcal{O}}'(v^*))\leq |\delta_{G_C}(v^*)|^2=|\delta_G(C)|^2$. If $C\in {\mathcal{L}}^{\operatorname{bad}}$, then cluster $C$ is $\beta$-bad, and so $\mbox{\sf dist}(\hat {\mathcal{O}}(v^*),\hat {\mathcal{O}}'(v^*))\leq |\delta_G(C)|^2\leq \beta\cdot (|\chi(C)|+|E(C)|)$ from the definition of $\beta$-bad clusters. Therefore, we assume from now on that $C\in {\mathcal{L}}^{\operatorname{light}}$.
For convenience of notation, we denote $v^*$ by $u'$, and we denote $\hat {\mathcal{O}}(v^*)$ and $\hat {\mathcal{O}}'(v^*)$ by ${\mathcal{O}}$ and ${\mathcal{O}}'$, respectively. We also denote $E'=\delta_G(C)=\delta_{G_C}(u')$.
In order to prove the claim, we will construct a collection $\Gamma=\set{\gamma(e)\mid e\in E'}$ of curves in the plane, all of which connect two points $p$ and $q$. We will ensure that the order in which the curves enter the point $p$ is precisely ${\mathcal{O}}'$, and the order in which they enter the point $q$ is ${\mathcal{O}}$. We will also ensure that the curves of $\Gamma$ are in general position. By showing that the expected number of crossings between the curves in $\Gamma$ is relatively small, we will obtain the desired bound on $\expect{\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')}$.
In order to construct the curves in $\Gamma$, we consider again the intance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace and its optimal solution $\phi^*$. Recall that we have computed an internal $C$-router ${\mathcal{Q}}=\set{Q(e)\mid e\in E'}$, where for each edge $e\in E'$, path $Q(e)$ originates with edge $e$, terminates at vertex $u$, and all its inner vertices lie in $C$. The paths in ${\mathcal{Q}}$ are non-transversal with respect to $\Sigma$, and, for every edge $e'\in E(C)$, $\expect{(\cong_G({\mathcal{Q}},e'))^2}\leq \beta$. We have also constructed an external $C$-router ${\mathcal{Q}}'=\set{Q'(e)\mid e\in E'}$, where for each edge $e\in E'$, path $Q(e)$ originates with edge $e$, terminates at vertex $u'$, and all its inner vertices are disjoint from $C$. The paths in ${\mathcal{Q}}'$ are non-transversal with respect to $\Sigma$, and, for every edge $e'\in E(G\setminus C)$, $\expect{\cong_G({\mathcal{Q}}',e')}\leq \beta$. We note that path set ${\mathcal{Q}}'$ is exactly the same as path set ${\mathcal{Q}}_0$ -- the internal router for $C_0$, that we used in the first step of the algorithm. Intuitively, we would like to let $p$ be the image of vertex $u'$ and $q$ the image of vertex $u$ in $\phi^*$. For every edge $e\in E'$, we would like to use the concatenation of the images of paths $Q(e)$ and $Q'(e)$ in $\phi^*$ in order to construct the curve $\gamma(e)$. This approach has several problems. First, the paths in sets ${\mathcal{Q}}$ and ${\mathcal{Q}}'$ may share edges and vertices, and so the resulting curves may not be in a general position. Second, there could be many crossings between edges lying on the paths of ${\mathcal{Q}}'$, which may lead to many crossings between curves of $\Gamma$. We take care of all these issues in the following three steps. In the first step, we take care of the congestion issue by constructing a graph $H'$ and its drawing $\psi'$. The construction is somewhat similar to the construction of graph $H$, in that we make several copies of some of the edges of $G$, in a way that allows us to define edge-disjoint paths in graph $H'$ replacing the path sets ${\mathcal{Q}}$ and ${\mathcal{Q}}'$. In the second step, we perform uncrossing of curves corresponding to the paths in ${\mathcal{Q}}'$, in order to eliminate some of the crossings. In the third step we perform nudging of curves corresponding to the paths in ${\mathcal{Q}}$. We now describe each of these steps in turn.
\paragraph{Graph $H'$.}
For each edge $e\in E'$, we set $n'_e=1$. For an edge $e\in E(C)$, we set $n'_e=\cong_G({\mathcal{Q}},e)$, and for an edge $e\in E(G\setminus C)$, we set $n'_e=\cong_G({\mathcal{Q}}',e)$. For every other edge $e$, we set $n'_e=0$. Note that for each edge $e\in E(G\setminus C)$, $n'_e=n_e$ holds, where $n_e$ is the parameter that we have used in the construction of graph $H$ in Step 1 of the algorithm.
In order to construct graph $H'$, we start with $V(H')=V(G)$. For every edge $e=(x,y)\in E(G)$ with $n'_e\neq 0$, we add a new set $J'(e)$ of $n'_e$ parallel edges connecting $x$ to $y$ to graph $H'$, that we view as \emph{copies of edge $e$}. As before, we use the drawing $\phi^*$ of $G$ in order to compute a drawing $\psi'$ of graph $H'$. For every vertex $x\in V(H')$, we let its image in $\psi'$ be $\phi^*(x)$. For every edge $e\in E(G)$ with $n'_e\neq 0$, we draw the edges of $J'(e)$ in parallel to the image of $e$ in $\phi^*$. Recall that for each edge $e\in E(G\setminus C)$, $n'_e=n_e$ holds, and so set $J'(e)$ of copies of $e$ can be thought of as being identical to the set $J(e)$ of copies of $e$ that we have constructed for graph $H$. We ensure that the specific drawing of the edges of $J'(e)$ in $\psi'$ is identical to the drawing of these edges in $\psi$.
We now bound the expected number of crossings in the resulting drawing $\psi'$ of graph $H'$. Consider any such crossing $(e_1',e_2')_{p'}$, and assume that $e_1'\in J'(e_1)$, $e_2'\in J'(e_2)$ holds for some edges $e_1,e_2\in E(G)$. Then there must be some crossing $(e_1,e_2)_p$ in the drawing $\phi^*$ of $G$, with point $p$ lying very close to point $p'$. We say that crossing $(e_1,e_2)_p$ of $\phi^*$ is \emph{responsible} for the crossing $(e_1',e_2')_{p'}$ of $\psi'$. It is immediate to verify that every crossing $(e_1,e_2)_p$ of $\phi^*$ may be responsible for at most $n'_{e_1}\cdot n'_{e_2}$ crossings of $\psi'$.
We classify the crossings of $\psi'$ into three types. Let $(e_1',e_2')_{p'}$ be a crossing of $\psi'$, and let $(e_1,e_2)_p$ be the crossing of $\phi^*$ responsible for it. We say that $(e_1',e_2')_{p'}$ is a \emph{type-1 crossing} if $e_1,e_2\in E(C)$. We say that it is a \emph{type-2 crossing} if $e_1,e_2\in E(G)\setminus(E(C)\cup \delta_G(C))$. Otherwise, we say that it is a type-3 crossing. We now bound the expected number of type-1 and type-3 crossings; type-2 crossings will eventually be eliminated.
In order to bound the expected number of type-1 crossings, consider any crossing $(e_1,e_2)_p$ of $\phi^*$ with $e_1,e_2\in E(C)$. Recall that this crossing may be responsible for at most $n'_{e_1}\cdot n'_{e_2}\leq (n'_{e_1})^2+(n'_{e_2})^2$ type-1 crossings of $\psi'$, and moreover, $(e_1,e_2)_p\in \chi(C)$ must hold. Since we have assumed that $C\in {\mathcal{L}}^{\operatorname{light}}$, for every edge $e\in E(C)$, $\expect{(n'_e)^2}=\expect{(\cong_G({\mathcal{Q}},e))^2} \leq \beta$. Therefore, the expected number of type-1 crossings of $\psi'$ for which $(e_1,e_2)_p$ is responsible for is at most $\expect{(n'_{e_1})^2+(n'_{e_1})^2}\leq 2\beta$.
Overall, the total expected number of type-1 crossings in $\psi'$ is then bounded by $O(\beta\cdot |\chi(C)|)$.
In order to bound the expected number of type-3 crossings, consider any crossing $(e_1,e_2)_p$ of $\phi^*$, and assume that neither $e_1,e_2\in E(C)$ nor $e_1,e_2\in E(G)\setminus (E(C)\cup \delta_G(C))$ holds. Recall that this crossing may be repsonsible for at most $n'_{e_1}\cdot n'_{e_2}$ type-3 crossings of $\psi'$. Moreover, $n'_{e_1},n'_{e_2}$ are independent random variables, and the expected value of each such variable is at most $\beta$. Therefore, the expected number of type-3 crossings of $\psi'$ for which crossing $(e_1,e_2)_p$ of $\phi^*$ is responsible for is at most $\beta^2$.
Notice that one of the edges $e_1,e_2$ lies in $E(C)\cup \delta_G(C)$, and so
$(e_1,e_2)_p\in \chi(C)$ must hold. Overall, the total expected number of type-3 crossings in $\psi'$ is then bounded by $O(\beta^2\cdot |\chi(C)|)$.
We conclude that the total expected number of type-1 and type-3 crossings in $\psi'$ is at most $O(\beta^2\cdot |\chi(C)|)$.
Next, we construct a set $\hat {\mathcal{Q}}'=\set{\hat Q'(e)\mid e\in E'}$ of edge-disjoint paths in graph $H'$, routing the edges of $E'$ to vertex $u'$. In order to do so, we assign, for every path $Q'(e)\in {\mathcal{Q}}'$, for every edge $e'\in E(Q'(e))\setminus\set{e}$, a copy of edge $e'$ to path $Q'(e)$. Observe that path set ${\mathcal{Q}}'$ in graph $G$ was denoted by ${\mathcal{Q}}_0$ in Step 1 of the algorithm, and, as observed before, for every edge $e'\in E(G)\setminus (E(C)\cup \delta_G(C))$, $n_{e'}=n'_{e'}$ and so $J(e')=J'(e')$. For each such edge $e'\in E(G)\setminus (E(C)\cup \delta_G(C))$, we assign a distinct copy of $e'\in J'(e')$ to every path in ${\mathcal{Q}}'$ that contains $e'$. We ensure that this assignment is exactly the same as the assignment done in Step 1 of the algorithm. For each edge $e\in E'$, we then obtain a path $\hat Q'(e)$ in graph $H'$ from path $Q'(e)$ by replacing each edge $e'\in E(Q'(e))\setminus\set{e}$ with the copy of $e'$ that was assigned to $e'$. We then denote $\hat {\mathcal{Q}}'=\set{\hat Q'(e)\mid e\in E'}$. From our construction, $\hat {\mathcal{Q}}'$ is a set of edge-disjoint paths in graph $H'$, routing the edges of $E'$ to vertex $u'$, and all inner vertices on the paths in $\hat {\mathcal{Q}}'$ are disjoint from $C$. Moreover, from our construction, $\hat {\mathcal{Q}}'=\hat {\mathcal{Q}}_0$ -- the set of edge-disjoint paths in $H$ that we have constructed in Step 1 of the algorithm.
We also construct a set $\hat {\mathcal{Q}}=\set{\hat Q(e)\mid e\in E'}$ of edge-disjoint paths in graph $H'$, routing the edges of $E'$ to vertex $u$. In order to do so, we assign, for every path $Q(e)\in {\mathcal{Q}}$, for every edge $e'\in E(Q(e))\setminus\set{e}$, a copy of edge $e'$ to path $Q(e)$.
Consider now any edge $e'\in E(C)$. If edge $e'$ is not incident to vertex $u$, then we assign every copy of $e$ in $J'(e')$ to a distinct path of ${\mathcal{Q}}$ containing $e'$ arbitrarily. If edge $e'$ is incident to vertex $u$, then we perform the assignment more carefully, using the same procedure that we used for every cluster $C_i\in {\mathcal{W}}^{\operatorname{light}}$ in order to assign, for each edge $e'$ incident to $u_i$, copies of $e'$ to paths in ${\mathcal{Q}}_i$; we do not repeat the description of the procedure there. For each edge $e\in E'$, we obtain a path $\hat Q(e)$ in graph $H'$ from path $Q(e)$ by replacing each edge $e'\in E(Q(e))\setminus\set{e}$ with the copy of $e'$ that was assigned to $e'$. We then denote $\hat {\mathcal{Q}}=\set{\hat Q(e)\mid e\in E'}$.
Recall that ${\mathcal{O}}={\mathcal{O}}(C)$ is a circular ordering of the edges of $E'=\delta_G(C)$ that is guided by the set ${\mathcal{Q}}$ of paths and the rotation system $\Sigma$. As in Step 1 of the algorithm, our assignment of edges incident to vertex $u$ ensures the following crucial property. Denote $E'=\set{e_1,\ldots,e_k}$, where the edges are indexed according to the ordering ${\mathcal{O}}$. For each such edge $e_j$, let $e'_j$ be the last edge on path $\hat Q(e_j)$. Then the images of edges $e'_1,\ldots,e'_k$ enter the image of the vertex $u$ in the drawing $\psi'$ of $H'$ in the order of their indices.
\paragraph{Uncrossing.}
We view every path of $\hat {\mathcal{Q}}'$ as being directed towards the vertex $u'$. We use the algorithm from \Cref{thm: new type 2 uncrossing} in order to compute a type-2 uncrossing, that produces, for every edge $e_j\in E'$, a directed curve $\hat \gamma'(e_j)$, that connects the image of the endpoint of $e_j$ lying in $C$ to $u'$. Recall that we are guaranteed that the curves in the resulting set $\hat \Gamma'=\set{\hat \gamma'(e_j)\mid e_j\in E'}$ do not cross each other, and each such curve is aligned with the drawing of graph $\bigcup_{\hat Q'(e_{\ell})\in \hat {\mathcal{Q}}'}\hat Q'(e_{\ell})$ induced by $\psi'$ (which is identical to the drawing induced by $\psi$).
Notice that the steps that we have followed in constructing the set $\hat \Gamma'$ of curves are identical to those we followed in order to construct the set $\Gamma_0$ of curves, and so the resulting two sets of curves are identical. In particular, the order in which the curves of $\hat \Gamma$ enter the image of vertex $u'$ in $\psi'$ is exactly ${\mathcal{O}}'$.
We now construct another set of curves, $\hat \Gamma=\set{\hat \gamma(e_j)\mid e_j\in E'}$, by letting, for each edge $e_j\in E'$, $\hat \gamma(e_j)$ be the image of the path $\hat Q(e_j)\in \hat {\mathcal{Q}}$ in the drawing $\psi'$ of $H'$. From our construction of the set $\hat {\mathcal{Q}}$ of paths, the order in which the curves of $\hat \Gamma$ enter the image of vertex $u$ in $\psi$ is exactly ${\mathcal{O}}$. For every edge $e_j\in E'$, we then let $\gamma(e_j)$ be a curve, connecting the images of $u$ and $u'$ in $\psi'$, obtained by combining the curves $\hat \gamma(e_j)$ and $\hat \gamma'(e_j)$. In order to combine the two curves, let $y_j$ be the endpoint of $e_j$ lying in $C$, and let $p_j$ be the unique point on $\psi'(e_j)$ that lies on the boundary of the tiny $y_j$-disc $D_{\psi'}(y_j)$. We truncate curve $\hat\gamma'(e_j)$ so it connects point $p_j$ to the image of vertex $u'$, and we truncate the curve $\hat \gamma(e_j)$, so it connects point $p_j$ to the image of vertex $u$. We then concatenate the resulting two curves to obtain the curve $\gamma(e_j)$.
Consider the resulting set $\Gamma=\set{\gamma(e_j)\mid e_j\in E'}$ of curves. From the above discussion, the curves enter the image of $u'$ in $\psi'$ according to the ordering ${\mathcal{O}}'$, and they enter the image of $u$ according to the ordering ${\mathcal{O}}$. The total number of crossings between the curves in $\Gamma$ is bounded by $\mathsf{cr}(\psi')$. We call all such crossings \emph{primary crossings}. Recall that the expected number of primary crossings is at most $O(\beta^2\cdot |\chi(C)|)$. However, the curves in $\Gamma$ may not be in general position. This is since some vertices $x\in V(C)$ may lie on a number of paths in $\hat {\mathcal{Q}}$. In the next step we perform ``nudging'' around such vertices, to ensure that the resulting curves are in general position.
\paragraph{Nudging.}
The nudging procedure and its analysis are identical to those from Step 2 of the algorithm. We only need to perform nudging of the curves in $\Gamma$ around vertices $x\in V(C)\setminus\set{u}$.
We process each vertex $x\in V(C)\setminus \set{u}$ one by one. Consider any such vertex $x$, and let ${\mathcal{Q}}^x\subseteq \hat{\mathcal{Q}}$ be the set of all paths containing vertex $x$. Note that $x$ must be an inner vertex on each such path. For convenience, we denote ${\mathcal{Q}}^x=\set{\hat Q(e_1),\ldots,\hat Q(e_z)}$. Consider the tiny $x$-disc $D=D_{\psi'}(x)$. For all $1\leq j\leq z$, denote by $s_j$ and $t_j$ the two points on the curve $\gamma(e_j)$ that lie on the boundary of disc $D$. We use the algorithm from \Cref{claim: curves in a disc} to compute a collection $\set{\sigma_1,\ldots,\sigma_z}$ of curves, such that, for all $1\leq j\leq z$, curve $\sigma_j$ connects $s_j$ to $t_j$, and the interior of the curve is contained in the interior of $D$. Recall that every pair of resulting curves crosses at most once, and every point in the interior of $D$ may be contained in at most two curves.
Consider now a pair $\sigma_{\ell},\sigma_{\ell'}$ of curves, and assume that these two curves cross. Recall that, from \Cref{claim: curves in a disc}, this may only happen if the two pairs $(s_{\ell},t_{\ell})$, $(s_{\ell'},t_{\ell'})$ of points cross. Denote by $e_1,e_2$ the two edges that lie on path $\hat Q(e_{\ell})$ immediately before and immediately after vertex $x$, and denote by $e_1',e_2'$ the two edges that lie on path $\hat Q(e_{\ell'})$ immediately before and immediately after vertex $x$. We assume that edges $e_1,e_2$ are copies of edges $\hat e_1,\hat e_2$ of $G$, and similarly, $e_1',e_2'$ are copies of edges $\hat e_1',\hat e_2'$ of $G$, respectively. Assume first that there are four distinct edges in set $\set{\hat e_1,\hat e_1',\hat e_2,\hat e_2'}$. From the fact that the two pairs $(s_{\ell},t_{\ell})$, $(s_{\ell'},t_{\ell'})$ of points cross, we get that these four edges must appear in ${\mathcal{O}}_x\in \Sigma$ in the order $(\hat e_1,\hat e_1',\hat e_2,\hat e_2')$. Since the paths of ${\mathcal{Q}}$ are non-transversal with respect to $\Sigma$, this is impossible. Therefore, we conclude that paths $Q(e_{\ell}),Q(e_{\ell'})$ must share an edge that is incident to $x$. If $e^*$ is an edge incident to $x$ that the two paths share, then we say that $e^*$ is \emph{responsible for the crossing between $\sigma_{\ell}$ and $\sigma_{\ell'}$}.
For all $1\leq j\leq z$, we modify the curve $\gamma(e_j)$, by replacing the segment of the curve that is contained in disc $D$ with $\sigma_j$.
Once every vertex $x\in V(C)\setminus \set{u}$ is processed, we obtain the final set $\Gamma^*=\set{\gamma^*(e)\mid e\in E'}$ of curves, which are now guaranteed to be in general position. Notice that as before, the curves in $\Gamma^*$ enter the image of $u'$ according to the ordering ${\mathcal{O}}'$, and they enter the image of $u$ according to the ordering ${\mathcal{O}}$. It now only remains to bound the expected number of crossings between the curves of $\Gamma^*$.
Notice that the nudging operation may have introduced some new crossings. Each such new crossing must be contained in a disc $D_{\psi'}(x)$, for some vertex $x\in V(C)\setminus\set{u}$. We call all such new crossings \emph{secondary crossings}. We now bound the expected number of secondary crossings.
Consider some vertex $x\in V(C)\setminus\set{u}$. Every secondary crossing that is contained in ${\mathcal{D}}_{\psi'}(x)$ is a crossing between a pair $\sigma_{\ell},\sigma_{\ell'}$ of curves that we have defined when processing vertex $x$, and each such crossing was charged to an edge of $G$ that is incident to $x$. If $e$ is an edge of $G$ that is incident to $x$, then there are at most $(\cong_G({\mathcal{Q}},e))^2$ pairs of paths in ${\mathcal{Q}}^x$ that contain copies of $e$, and each such pair of paths may give rise to a single secondary crossing in $D_{\psi'}(x)$ that is charged to edge $e$. Therefore, the total expected number of secondary crossings is bounded by:
\[ \sum_{e\in E(C)}O(\expect{(\cong_G({\mathcal{Q}},e))^2})\leq O(\beta \cdot|E(C)|),\]
since we have assumed that $C\in {\mathcal{W}}^{\operatorname{light}}$.
Overall, the expected number of crossings between the curves in $\Gamma^*$ is at most $O(\beta^2\cdot (|\chi(C)|+|E(C)|))$, proving that $\expect{\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')}\leq O(\beta^2\cdot (|\chi(C)|+|E(C)|))$.
\subsection{Proof of \Cref{lem: routing path extension}}
\label{apd: Proof of routing path extension}
Denote $z=\ceil{\frac{|T|}{|T'|}}$. We arbitrarily partition the vertices of $T\setminus T'$ into $(z-1)$ subsets $T_1,\ldots,T_{z-1}$ of cardinality at most $|T'|$ each.
Consider some index $1\leq i\leq z-1$. Since vertices of $T$ are $\alpha$-well-linked in $G$, using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{Q}}'_i$ of paths in graph $G$, routing vertices of $T_i$ to vertices of $T'$, such that $\cong_G({\mathcal{Q}}'_i)\le \ceil{1/\alpha}$, and each vertex of $T'\cup T_i$ is the endpoint of at most one path in ${\mathcal{Q}}'_i$.
By concatenating the paths in ${\mathcal{Q}}'_i$ with paths in ${\mathcal{P}}$, we obtain a collection ${\mathcal{Q}}_i$ of paths in graph $G$ routing vertices of $T_i$ to vertex $x$. For every edge $e\in E(G)$, $\cong_G({\mathcal{Q}}_i,e)\leq \ceil{1/\alpha}+\cong_G({\mathcal{P}},e)$.
Lastly, we set ${\mathcal{P}}'=\bigcup_{i=1}^{z-1}{\mathcal{Q}}_i$.
It is clear that the paths in ${\mathcal{P}}$ route the vertices of $T$ to $x$.
Moreover, for every edge $e\in E(G)$,
\[
\cong_G({\mathcal{P}}',e) \le \sum_{i=1}^z\cong_G({\mathcal{Q}}_i,e)\\
\le \ceil{\frac{|T|}{|T'|}}\bigg(\cong_G({\mathcal{P}},e)+\ceil{1/\alpha}\bigg).
\]
\subsection{Proof of \Cref{lem: splitting}}
\label{sec: splitting}
We denote $\tilde \alpha'=\frac{\tilde \alpha}{c\log^2m}$, where $c$ is a large enough constant whose value we set later.
Throughout the algorithm, we will maintain a collection ${\mathcal{W}}$ of disjoint clusters of $G\setminus T$. We will ensure that this collection ${\mathcal{W}}$ of clusters has some useful properties, that are summarized in the following definition.
\begin{definition}
A collection ${\mathcal{W}}$ of disjoint clusters of $G\setminus T$ is a \emph{legal clustering of $G$} if the following hold:
\begin{itemize}
\item $\bigcup_{W\in {\mathcal{W}}}V(W)=V(G)\setminus T$;
\item every cluster $W\in {\mathcal{W}}$ has the $\tilde \alpha'$-bandwidth property; and
\item for every cluster $W\in {\mathcal{W}}$, $|\delta_G(W)|\leq \tilde \alpha k/64$.
\end{itemize}
\end{definition}
Given a legal clustering ${\mathcal{W}}$ of $G$, we associate with it a contracted graph $\hat G=G_{|{\mathcal{W}}}$; recall that $\hat G$ is obtained from graph $G$ by contracting every cluster $W\in {\mathcal{W}}$ into a supernode $v_W$; we keep parallel edges but delete self loops. Observe that from the definition of a legal clustering, the only regular (non-supernode) vertices of $\hat G$ are the vertices of $T$.
Given a legal clustering ${\mathcal{W}}$ of $G$, we denote by $\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})$ the set of all edges $(u,v)$ of $G$, where $u$ and $v$ belong to different clusters of ${\mathcal{W}}$; equivalently, $\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})=E(\hat G\setminus T)$. We will not distinguish between the edges of $\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})$ and the edges of $\hat G\setminus T$.
We need the following simple claim, whose analogues were proved in
\cite{chuzhoy2012routing,chuzhoy2012polylogarithmic,chekuri2016polynomial,chuzhoy2016improved}.
\begin{claim}\label{claim: contracted graph lots of edges}
Let ${\mathcal{W}}$ be a legal clustering of $G$. If the set $T$ of terminals is $\tilde \alpha$-well-linked in $G$, then
$|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|\geq \tilde \alpha k/4$.
\end{claim}
\begin{proof}
For every cluster $W\in {\mathcal{W}}$, let $T_W\subseteq T$ be the set containing every terminal $t\in T$, such that, if $e=(t,v)$ is the unique edge incident to $t$ in $G$, then $v\in W$. Denote $n_W=|T_W|$. Since we are guaranteed that for every cluster $W\in {\mathcal{W}}$, $|\delta_G(W)|\leq \tilde \alpha k/64$, $n_W\leq \tilde \alpha k/64$ must hold. Note that there is a partition ${\mathcal{W}}_1,{\mathcal{W}}_2$ of ${\mathcal{W}}$, such that $\sum_{W\in {\mathcal{W}}_1}n_W,\sum_{W\in {\mathcal{W}}_2}n_W\geq k/4$. Indeed, we can compute such a partition using a simple greedy algorithm: start with ${\mathcal{W}}_1,{\mathcal{W}}_2=\emptyset$, and process the clusters $W\in {\mathcal{W}}$ one by one. When cluster $W\in {\mathcal{W}}$ is processed, we add it to ${\mathcal{W}}_1$ if $\sum_{W\in {\mathcal{W}}_1}n_W<\sum_{W\in {\mathcal{W}}_2}n_W$, and we add it to ${\mathcal{W}}_2$ otherwise. We are guaranteed that at the end of this procedure, $\textsf{left} |\sum_{W\in {\mathcal{W}}_1}n_W-\sum_{W\in {\mathcal{W}}_2}n_W\textsf{right} |\leq \max_{W\in {\mathcal{W}}}\set{n_W}\leq \tilde \alpha k/64$ holds. It is then immediate to verify that $\sum_{W\in {\mathcal{W}}_1}n_W,\sum_{W\in {\mathcal{W}}_2}n_W\geq k/4$.
We construct a partition $(X,Y)$ of $V(G)$, where $X=\bigcup_{W\in {\mathcal{W}}_1}(V(W)\cup T_W)$, and $Y=\bigcup_{W\in {\mathcal{W}}_2}(V(W)\cup T_W)$. Then $|X\cap T|,|Y\cap T|\geq k/4$, and, since we have assumed that the set $T$ of terminals is $\tilde \alpha$-well-linked in $G$, $|E_G(X,Y)|\geq \tilde \alpha k/4$. Notice that every edge in $E_G(X,Y)$ connects a pair of vertices lying in different clusters of ${\mathcal{W}}$, so $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|\geq |E_G(X,Y)|\geq \tilde \alpha k/4$.
\end{proof}
The following lemma is key to the proof of \Cref{lem: splitting}.
\begin{lemma}\label{lem: better clustering}
There is an efficient algorithm, that, given a legal clustering ${\mathcal{W}}$ of $G$ with $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|\geq \tilde \alpha k/4$, either produces another legal clustering ${\mathcal{W}}'$ of $G$ with $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}')|<|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|$, or it computes two disjoint subgraphs $C_1,C_2$ of $G$, each of which has the $\tilde \alpha'$-bandwidth property, such that, for all $i\in \set{1,2}$, there is a set ${\mathcal{R}}_i$ of at least $\Omega(\tilde \alpha^2k/\log^2m)$ edge-disjoint paths in $G$, routing a subset $T_i\subseteq T$ of terminals to the edges of $\delta_G(C_i)$.
\end{lemma}
We prove \Cref{lem: better clustering} in the following subsection, after we complete the proof of \Cref{lem: splitting} using it.
Throughout the algorithm, we maintain a legal clustering ${\mathcal{W}}$ of $G$. If, at any point in the algorithm's execution, we obtain a legal clustering ${\mathcal{W}}$ with $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|<\tilde \alpha k/4$, then, from \Cref{claim: contracted graph lots of edges}, the set $T$ of terminals is not $\tilde \alpha$-well-linked in $G$. We then terminate the algorithm and return FAIL. Therefore, from now on we assume that every legal clustering ${\mathcal{W}}$ that the algorithm obtains has $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|\geq \tilde \alpha k/4$.
We start with an initial collection ${\mathcal{W}}$ of clusters of $V(G)\setminus T$, where for every vertex $v\in V(G)\setminus T$, we add a cluster $\set{v}$ to ${\mathcal{W}}$. It is easy to verify that ${\mathcal{W}}$ is a legal clustering of $G$, since the degree of every
vertex in $G$ is guaranteed to be at most $\tilde \alpha k/64$. We then perform a number of iterations. In every iteration, we apply the algorithm from \Cref{lem: better clustering} to the current legal clustering ${\mathcal{W}}$. If the outcome of the algorithm is another legal clustering ${\mathcal{W}}'$ of $G$ with $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}')|<|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|$, then we replace ${\mathcal{W}}$ with ${\mathcal{W}}'$ and continue to the next iteration. Assume now that the outcome of the algorithm from \Cref{lem: better clustering} is a pair $C_1,C_2$ of disjoint subgraphs of $G$, each of which has the $\tilde \alpha'$-bandwidth property, such that for all $i\in \set{1,2}$, there is a set ${\mathcal{R}}_i$ of at least $\Omega(\tilde \alpha^2k/\log^2m)$ edge-disjoint paths in $G$, routing a subset $T_i\subseteq T$ of terminals to the edges of $\delta_G(C_i)$.
In this case, we let $(X,Y)$ be a partition of $V(G)$ with $V(C_1)\subseteq X$ and $V(C_2)\subseteq Y$, that minimizes $|E_G(X,Y)|$ among all such partitions. Notice that such a partition $(X,Y)$ can be computed via a standard minimum $s$-$t$ cut computation. Moreover, from the Maximum Flow / Minimum Cut theorem, we are guaranteed that there is a collection ${\mathcal{Q}}$ of $|E_G(X,Y)|$ edge-disjoint paths, each of which connects a vertex of $V(C_1)$ to a vertex of $V(C_2)$; we can assume w.l.o.g. that each path in ${\mathcal{Q}}$ is simple and it does not contain vertices of $V(C_1)\cup V(C_2)$ as inner vertices. Notice that every path in ${\mathcal{Q}}$ must contain exactly one edge of the set $E'=E_G(X,Y)$, and every edge of $E'$ must lie on exactly one such path. For every path $Q\in {\mathcal{Q}}$, we denote by $Q_1$ the subpath of $Q$ from its endpoint that lies in $C_1$ to an edge of $E'$, and we denote by $Q_2$ the subpath of $Q$ from an edge of $E'$ to a vertex of $C_2$. Let ${\mathcal{Q}}_1=\set{Q_1\mid Q\in {\mathcal{Q}}}$ and ${\mathcal{Q}}_2=\set{Q_2\mid Q\in {\mathcal{Q}}}$. Then ${\mathcal{Q}}_1$ is a set of edge-disjoint paths routing the edges of $E'$ to edges of $\delta_G(C_1)$, in graph $G[X]\cup E'$, and similarly, ${\mathcal{Q}}_2$ is a set of edge-disjoint paths routing the edges of $E'$ to edges of $\delta_G(C_2)$ in graph $G[Y]\cup E'$.
We now show that the partition $(X,Y)$ of $V(G)$ has all required properties.
Assume w.l.o.g. that $|X\cap T|\geq |Y\cap T|$.
Recall that there is a set ${\mathcal{R}}_2$ of at least $\Omega(\tilde \alpha^2k/\log^2m)$ edge-disjoint paths in $G$, routing a subset $T_2\subseteq T$ of terminals to the edges of $\delta_G(C_2)$. Assume first that $|T_2\cap X|\geq |T_2|/2$. Let ${\mathcal{R}}_2'\subseteq {\mathcal{R}}_2$ be the set of paths whose endpoint lies in $T_2\cap X$, so $|{\mathcal{R}}_2'|\geq |{\mathcal{R}}_2|/2\geq \Omega(\tilde \alpha^2k/\log^2m)$. Then each path $R\in {\mathcal{R}}_2'$ connects a vertex of $T_2\cap X$ to a vertex of $Y$, so it must contain an edge of $E_G(X,Y)$. By suitably truncating each such path $R\in {\mathcal{R}}_2'$, we obtain a collection ${\mathcal{R}}$ of $\Omega(\tilde \alpha^2k/\log^2m)$ edge-disjoint paths, routing the terminals of $T_2\cap X$ to the edges of $E_{G}(X,Y)$.
Assume now that $|T_2\cap X|<|T_2|/2$. Let $h=\ceil{|T_2|/2}$. Then $|X\cap T|,|Y\cap T|\geq h$ must hold. We apply the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked} to graph $G$ and two arbitrary subsets $T_1'\subseteq X\cap T, T'_2\subseteq Y\cap T$ of terminals, of cardinality $h$ each.
If the set $T$ of terminals is $\tilde \alpha$-well-linked in $G$, the algorithm must return a collection ${\mathcal{R}}'$ of paths in graph $G$, such that ${\mathcal{R}}'$ is an one-to-one routing of vertices of $T_1'$ to vertices of $T_2'$, and $\cong_G({\mathcal{R}}')\leq \ceil{1/\tilde \alpha}$. If the algorithm fails to return such a collection of paths, then we are guaranteed that the set $T$ of terminals is not $\tilde \alpha$-well-linked in $G$. We then terminate the algorithm and return FAIL. Therefore, we assume from now on that the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked} returned a collection ${\mathcal{R}}'$ of paths with $\cong_G({\mathcal{R}}')\leq \ceil{1/\tilde \alpha}$, such that ${\mathcal{R}}'$ is an one-to-one routing of $T_1'$ to $T_2'$. From \Cref{claim: remove congestion}, there is a collection ${\mathcal{R}}''$ of at least $\Omega(h\tilde \alpha)=\Omega(\tilde \alpha^3k/\log^2m)$ edge-disjont paths in graph $G$, routing a subset $T''_1\subseteq T'_1$ of terminals to a subset $T''_2\subseteq T'_2$ of terminals. Each path $R\in {\mathcal{R}}''$ must then contain an edge of $E_G(X,Y)$. By suitably truncating each such path, we obtain a collection ${\mathcal{R}}$ of $\Omega(\tilde \alpha^3k/\log^2m)$ edge-disjoint paths, routing the terminals of $T''_1$ to the edges of $E_G(X,Y)$.
It is now enough to prove that each of the clusters $G[X],G[Y]$ has the $\tilde \alpha'/2$-bandwidth property, which we do in the following claim.
\begin{claim}
Each of the clusters $G[X],G[Y]$ has the $\tilde \alpha'/2$-bandwidth property.
\end{claim}
\begin{proof}
We show this for $G[X]$; the proof for $G[Y]$ is symmetric.
Assume for contradiction that $G[X]$ does not have the $\tilde \alpha'/2$-bandwidth property. Then there must be a partition $(A,B)$ of $X$, such that $|E_G(A,B)|<\tilde \alpha'\cdot \min\set{|\delta_G(X)\cap \delta_G(A)|,|\delta_G(X)\cap \delta_G(B)|}/2$. We assume w.l.o.g. that $|\delta_G(X)\cap \delta_G(A)|\leq |\delta_G(X)\cap \delta_G(B)|$, and we denote $|\delta_G(X)\cap \delta_G(A)|$ by $r$.
Note that partition $(A,B)$ of $X$ naturally defines a partition $(A',B')$ of $V(C_1)$, with $A'=A\cap V(C_1)$ and $B'=B\cap V(C_1)$. Since cluster $C_1$ has the $\tilde \alpha'$-bandwidth property, while $|E_{C_1}(A',B')|\leq |E_G(A,B)|< \tilde \alpha' r/2$,
either $|\delta_G(A')\cap \delta_G(C)|<r/2$, or $|\delta_G(B')\cap \delta_G(C_1)|<r/2$ must hold. Assume w.l.o.g. that it is the former. Recall that $|\delta_G(X)\cap \delta_G(A)|= r$, and there is a set ${\mathcal{Q}}_1$ is a set of edge-disjoint paths routing the edges of $E'=\delta_G(X)$ to edges of $\delta_G(C_1)$. Let ${\mathcal{Q}}'\subseteq {\mathcal{Q}}_1$ be the paths that originate at edges of $|\delta_G(X)\cap \delta_G(A)|$, so $|{\mathcal{Q}}'|\geq r$. Recall that each path in ${\mathcal{Q}}'$ terminates at an edge of $\delta_G(C_1)$. However, since $|\delta_G(A')\cap \delta_G(C_1)|<r/2$, at least $r/2$ of the paths in ${\mathcal{Q}}'$ must contain a vertex of $B$. Therefore, at least $r/2$ of the paths in ${\mathcal{Q}}'$ contain an edge of $E_G(A,B)$, and so $|E_G(A,B)|\geq r/2$, a contradiction.
\end{proof}
\subsection*{Proof of \Cref{lem: better clustering}}
We need the following claim, which is a constructive version of Lemma 5.8 from \cite{chuzhoy2016improved}; the proof is almost identical to that in \cite{chuzhoy2016improved} and is included here for completeness.
\begin{claim}\label{claim: partition into two}
There is an efficient algorithm that, given any graph $G'$ with maximum vertex degree at most $\Delta$, computes a partition $(A,B)$ of $V(G')$, with $|E(A)|,|E(B)|\geq \frac{|E(G')|}{4}-\Delta$.
\end{claim}
\begin{proof}
For every vertex $v\in V(G')$, let $d(v)$ denote its degree in $G'$. For a subset $S\subseteq V(G')$ of vertices, let $\operatorname{vol}(S)=\sum_{v\in S}d(v)$.
We start by computing an initial partition $(A,B)$ of $V(G')$, with $|\operatorname{vol}(A)-\operatorname{vol}(B)|\leq \Delta$, using a simple greedy algorithm: start with $A=B=\emptyset$, and process the vertices of $G'$ one-by-one. When $v$ is processed, add it to $A$ if $\operatorname{vol}(A)<\operatorname{vol}(B)$ currently holds, and add it to $B$ otherwise. It is easy to see that at the end of this procedure, we obtain a partition $(A,B)$ of $V(G')$ with $|\operatorname{vol}(A)-\operatorname{vol}(B)|\leq \Delta$.
We then iterate. The input to iteration $i$ is a partition $(A_i,B_i)$ of $V(G')$ with $|\operatorname{vol}(A_i)-\operatorname{vol}(B_i)|\leq 2\Delta$, where the input to the first iteration is the partition $(A_1,B_1)=(A,B)$ that we have just computed. We assume w.l.o.g. that $\operatorname{vol}(A_i)\geq \operatorname{vol}(B_i)$ holds, so $|E(A_i)|\geq |E(B_i)|$. If $|E(B_i)|< \frac{|E(G')|}{4}-\Delta$, then the outcome of the $i$th iteration is a partition
$(A_{i+1},B_{i+1})$ of $V(G')$ with $|\operatorname{vol}(A_{i+1})-\operatorname{vol}(B_{i+1})|\leq 2\Delta$, and $|E(A_{i+1},B_{i+1})|<|E(A_{i},B_i)|$; otherwise, the algorithm terminates. In the latter case, we get that $|E(A_i)|\geq|E(B_i)|\geq \frac{|E(G')|}{4}-\Delta$, as required.
We now describe the execution of the $i$th iteration, whose input is is a partition $(A_i,B_i)$ of $V(G')$ with $|\operatorname{vol}(A_i)-\operatorname{vol}(B_i)|\leq 2\Delta$, such that $\operatorname{vol}(A_i)\geq \operatorname{vol}(B_i)$ and $|E(B_i)|< \frac{|E(G')|}{4}-\Delta$ hold.
For every vertex $v\in A_i$, let $d_1(v)$ be the number of edges incident to $v$ whose other endpoint belongs to $A_i$, and let $d_2(v)$ be the number of edges incident to $v$ whose other endpoint belongs to $B_i$. As we show later, there must exist a vertex $v\in A_i$ with $d_1(v)< d_2(v)$. Let $v$ be any such vertex. We then define a new partition $(A_{i+1},B_{i+1})$ of $V(G')$ as follows: $A_{i+1}=A_i\setminus\set{v}$ and $B_{i+1}=B_i\cup\set{v}$. It is easy to verify that $|E(A_{i+1},B_{i+1})|<|E(A_{i},B_i)|$, while:
$$|\operatorname{vol}(A_{i+1})-\operatorname{vol}(B_{i+1})|\leq \max\set{|\operatorname{vol}(A_{i})-\operatorname{vol}(B_{i})|,2d(v)}\leq 2\Delta.$$
We then output the partition $(A_{i+1},B_{i+1})$ of $V(G')$ and terminate the iteration.
It now remains to show that, if $|E(B_i)|< \frac{|E(G')|}{4}-\Delta$, there must exist a vertex $v\in A_i$ with $d_1(v)< d_2(v)$.
Indeed, assume for contradiction that for every vertex $v\in A_i$, $d_1(v)\geq d_2(v)$.
Then $|E(A_i)|= \ensuremath{\frac{1}{2}}\sum_{v\in A_i}d_1(v)\geq \ensuremath{\frac{1}{2}}\sum_{v\in A_i}d_2(v)=\ensuremath{\frac{1}{2}} |E(A_i,B_i)|$.
Altogether, $|E(G')|=|E(A_i)|+|E(B_i)|+|E(A_i,B_i)|\leq 4|E(A_i)|$, and so $|E(A_i)|\geq |E(G')|/4$.
On the other hand:
\[|E(B_i)|= \frac {\operatorname{vol}(B_i)-|E(A_i,B_i)|} 2 \geq \frac {\operatorname{vol}(A_i)-2\Delta-|E(A_i,B_i)|} 2 \geq \frac{2|E(A_i)|-2\Delta}2\geq \frac{|E(G')|}{4}-\Delta,\]
a contradiction to our assumption that $|E(B_i)|<\frac{|E(G')|}4-\Delta$.
The algorithm terminates once we obtain a partition $(A_i,B_i)$ of $V(G')$ with $|E(A_i)|,|E(B_i)|\geq \frac{|E(G')|}{4}-\Delta$. From the above discussion, this is guaranteed to happen after at most $|E(G')|$ iterations. Since each iteration can be executed efficiently, the claim follows.
\end{proof}
We apply the algorithm from \Cref{claim: partition into two} to graph $\hat G'=\hat G\setminus T$. Recall that, since for every cluster $W\in {\mathcal{W}}$, $|\delta_G(W)|\leq \tilde \alpha k/64$, every vertex in graph $\hat G'$ has degree at most $\tilde \alpha k/64$. Moreover,
$|E(\hat G')|=|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|\geq \tilde \alpha k/4$. Therefore, we obtain a partition $(A,B)$ of $V(\hat G')$ with $|E_{\hat G'}(A)|,|E_{\hat G'}(B)|\geq |E(\hat G')|/4-\tilde \alpha k/64\geq |E(\hat G')|/8\geq \tilde \alpha k/32$.
Let ${\mathcal{W}}_1$ be the set of all clusters $W\in {\mathcal{W}}$ with $v_W\in A$, and let ${\mathcal{W}}_2$ be the set of all clusters $W\in {\mathcal{W}}$ with $v_W\in B$. Clearly, $({\mathcal{W}}_1,{\mathcal{W}}_2)$ is a partition of ${\mathcal{W}}$. Our next step is summarized in the following claim.
\begin{claim}\label{claim: processing one side}
There is an efficient algorithm, that, given any subset ${\mathcal{C}}\subseteq {\mathcal{W}}$ of clusters, such that the total number of edges $e=(u,v)\in E(G)$ where $u$ and $v$ lie in distinct clusters of ${\mathcal{C}}$ is at least $|E(\hat G')|/8$, outputs one of the following:
\begin{itemize}
\item either a legal clustering ${\mathcal{W}}'$ of $G$ with $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}')|<|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|$; or
\item a cluster $C$ with $V(C)\subseteq \bigcup_{C'\in {\mathcal{C}}}V(C')$, such that $C$ has the $\tilde \alpha'$-bandwidth property, and there exists a collection ${\mathcal{R}}$ of at least $\tilde \alpha \cdot \tilde \alpha' k/256$ edge-disjoint paths in $G$ routing a subset of terminals to the edges of $\delta_G(C)$.
\end{itemize}
\end{claim}
Observe that \Cref{claim: processing one side} finishes the proof of \Cref{lem: better clustering}, as follows.
Let $A'=\bigcup_{W\in {\mathcal{W}}_1}V(W)$, and let $B'=\bigcup_{W\in {\mathcal{W}}_2}V(W)$; clearly, $A'\cap B'=\emptyset$.
We aply the algorithm from \Cref{claim: processing one side} to the set ${\mathcal{W}}_1$ of clusters, and then separately to the set ${\mathcal{W}}_2$ of clusters. If the outcome of any of the two algorithms is a legal clustering ${\mathcal{W}}'$ of $G$ with $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}')|<|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|$, then we return this legal clustering ${\mathcal{W}}'$ and terminate the algorithm. Therefore, we assume from now on that the outcome of the algorithm from \Cref{claim: processing one side} when applied to cluster set ${\mathcal{W}}_1$ is a cluster $C_1$ with $V(C_1)\subseteq A'$, such that $C_1$ has the $\tilde \alpha'$-bandwidth property, and there exists a collection ${\mathcal{R}}_1$ of at least $\tilde \alpha \cdot \tilde \alpha' k/256=\Omega(\tilde \alpha^2k/\log^2m)$ edge-disjoint paths in $G$ routing some subset $T_1\subseteq T$ of terminals to the edges of $\delta_G(C_1)$. Similarly, the outcome of the algorithm from \Cref{claim: processing one side} when applied to cluster set ${\mathcal{W}}_2$ is a cluster $C_2$ with $V(C_2)\subseteq B'$, such that $C_2$ has the $\tilde \alpha'$-bandwidth property, and there exists a collection ${\mathcal{R}}_2$ of at least $\tilde \alpha \cdot \tilde \alpha' k/256=\Omega(\tilde \alpha^2k/\log^2m)$ edge-disjoint paths routing some subset $T_2\subseteq T$ of terminals to the edges of $\delta_G(C_2)$. We then return $C_1$ and $C_2$. From the above discussion, $C_1$ and $C_2$ are both disjoint and have the required properties. In order to complete the proof of
\Cref{lem: better clustering}, it is now enough to prove \Cref{claim: processing one side}, which we do next.
\begin{proofof}{\Cref{claim: processing one side}}
Let $S\subseteq V(G)$ be a set of vertices that contains, for every cluster $W\in {\mathcal{C}}$, the vertices of $W$. Since every edge that is incident to a vertex of $S$ either has a terminal of $T$ as its other endpoint, or belongs to $\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})$, from \Cref{claim: contracted graph lots of edges}, we get that $|\delta_G(S)|\leq k+|E(\hat G')|\leq 8|E(\hat G')|/\tilde \alpha$, since, from \Cref{claim: contracted graph lots of edges}, $|E(\hat G')|=|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|\geq \tilde \alpha k/4$.
We apply the algorithm from \Cref{thm:well_linked_decomposition} to graph $G$ and its cluster $G[S]$, with parameter $\alpha=\tilde \alpha'$ (recall that $\tilde \alpha'=\frac{\tilde{\alpha}}{c\log^2 m}$ for a large enough constant $c$, so the requirement that $\tilde \alpha'< \min\set{\frac 1 {64\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\log m},\frac 1 {48\log^2 m}}$ is satisfied), to obtain a collection ${\mathcal{C}}'$ of disjoint clusters of $G[S]$ (if grah $G[S]$ is not connected, then we apply the algorithm from \Cref{thm:well_linked_decomposition} to each connected component of $G[S]$ separately; this does not affect the remainder of the analysis).
Recall that $\set{V(C')\mid C'\in {\mathcal{C}}'}$ partitions $S$; every cluster $C'\in {\mathcal{C}}'$ has the $\tilde \alpha'$-bandwidth property, and:
\[\begin{split}
\sum_{C'\in {\mathcal{C}}'}|\delta_G(C')| & \le |\delta_G(S)|\cdot\textsf{left}(1+O(\tilde \alpha'\cdot \log^{3/2} m)\textsf{right})\\
&\leq |\delta_G(S)|+O\textsf{left}(\frac{8|E(\hat G')|\tilde \alpha'\log^{3/2}m}{\tilde \alpha}\textsf{right} )\\
&\leq |\delta_G(S)|+O\textsf{left}(\frac{|E(\hat G')|}{c\log^{1/2}m}\textsf{right} ) \\
&\leq |\delta_G(S)|+\frac{|E(\hat G')|}{64},
\end{split} \]
since
$\tilde \alpha'=\tilde \alpha/c\log^2m$, and $c$ is a large enough constant.
We consider now a new clustering ${\mathcal{W}}'$ of $G$, that is obtained as follows: start from ${\mathcal{W}}'={\mathcal{W}}\setminus {\mathcal{C}}$, and then add the clusters of ${\mathcal{C}}'$ to ${\mathcal{W}}'$. It is easy to verify that the clusters in ${\mathcal{W}}'$ are all mutually disjoint, and that $\bigcup_{W'\in {\mathcal{W}}}V(W)=V(G)\setminus T$. Moreover, every clsuter $W\in {\mathcal{W}}$ has the $\tilde \alpha'$-bandwidth property. Next, we show that $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}')|< |\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|$.
Indeed, we can partition the edge set $\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})$ into three subsets: set $E_1$ contains all edges $e=(u,v)$, where $u$ and $v$ lie in different clusters of ${\mathcal{W}}\setminus {\mathcal{C}}$; set $E_2$ contains all edges $e=(u,v)$, where $u$ lies in a cluster of ${\mathcal{W}}\setminus {\mathcal{C}}$, and $v$ lies in a cluster of ${\mathcal{C}}$; lastly, $E_3$ contains all edges $e=(u,v)$, where $u$ and $v$ lie in different clusters of ${\mathcal{C}}$.
We can partition $\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}')$ into three subsets $E_1',E_2',E_3'$ similarly, using cluster set ${\mathcal{C}}'$ instead of ${\mathcal{C}}$. Clearly, $E_1=E_1'$, and $E_2=E_2'=\delta_G(S)$. From the statement of \Cref{claim: processing one side}, $|E_3|\geq |E(\hat G')|/8$. On the other hand, since we have established that $\sum_{C\in {\mathcal{C}}'}|\delta_G(C)|\leq |\delta_G(S)|+\frac{|E(\hat G')|}{64}$, and since $E_2=\delta_G(S)\subseteq \bigcup_{C\in {\mathcal{C}}'}\delta_G(C)$, we get that $|E_3'|\leq \frac{|E(\hat G')|}{64}$. We conclude that $|E_3'|<|E_3|$, and $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}')|<|\hat E^{\textnormal{\textsf{out}}}(W)|$. Note however that ${\mathcal{W}}'$ may not be a legal clustering of $G$, since we do not guarantee that for every cluster $C\in {\mathcal{C}}'$, $\delta_G(C)\leq \tilde \alpha k/64$ (this property is guaranteed to hold for every cluster of ${\mathcal{W}}\setminus {\mathcal{C}}'$ though, since ${\mathcal{W}}$ is a legal clustering). In the remainer of the algorithm, we will attempt to ``fix'' the clustering ${\mathcal{W}}'$ so it becomes a legal clustering of $G$, and, if we fail to do so, we will produce the desired cluster $C$.
In the remainder of the algorithm, we will maintain a set ${\mathcal{W}}^*$ of clusters, starting with ${\mathcal{W}}^*={\mathcal{W}}'$, and we will ensure that the following invariants hold for ${\mathcal{W}}^*$ at all times:
\begin{properties}{I}
\item all clusters in ${\mathcal{W}}^*$ are disjoint from each other, and $\bigcup_{W\in {\mathcal{W}}^*}V(W)=V(G)\setminus T$; \label{inv: partition}
\item every cluster $W\in {\mathcal{W}}^*$ has the $\tilde \alpha'$-bandwidth property; \label{inv: bandwidth prop}
\item $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^*)|< |\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|$; \label{inv: boundaries dont grow} and
\item if $|\delta_G(W)|>\tilde \alpha k/64$ for some cluster $W\in {\mathcal{W}}^*$, then $V(W)\subseteq S$. \label{inv: large cluster in S}
\end{properties}
Note that all these invariants hold for the initial setting ${\mathcal{W}}^*={\mathcal{W}}'$. The algorithm performs a number of iterations, as long as there is some cluster $W\in {\mathcal{W}}^*$ with $|\delta_G(W)|>\tilde \alpha k/64$. We now describe the execution of a single iteration.
Let $W\in {\mathcal{W}}^*$ be any cluster with $|\delta_G(W)|>\tilde \alpha k/64$. Using the standard max-flow computation, we compute a maximum-cardinality set ${\mathcal{R}}$ of edge-disjoint paths in graph $G$, where each path in ${\mathcal{R}}$ connects a distinct terminal of $T$ to a distinct edge of $\delta_G(W)$.
We now consider two cases. The first case happens if $|{\mathcal{R}}|\geq \tilde \alpha \tilde \alpha' k/256$. In this case, we terminate the algorithm and return the cluster $W$. Notice that, from Invariant \ref{inv: large cluster in S}, we are guaranteed that $V(W)\subseteq S=\bigcup_{C'\in {\mathcal{C}}}V(C)$, and from Invariant \ref{inv: bandwidth prop}, $W$ has the $\tilde \alpha'$-bandwidth property.
Assume now that the second case happens, that is, $|{\mathcal{R}}|<\tilde \alpha \tilde \alpha' k/256$. From the maximum flow / minimum cut theorem, there is a partition $(A,B)$ of $V(G)$, with $V(W)\subseteq A$, $T\subseteq B$, and $|E_G(A,B)|\leq |{\mathcal{R}}|<\tilde \alpha \tilde \alpha' k/256$.
We slightly modify the cut $(A,B)$ in graph $G$, to compute a new cut $(A',B')$ with $W\subseteq A'$, $T\subseteq V(B')$, such that for every cluster $C\in {\mathcal{W}}^*$, either $V(C)\subseteq A'$, or $V(C)\subseteq B'$ holds. In order to do so, we process every cluster $C\in {\mathcal{W}}^*\setminus\set{W}$ one by one. Consider an iteration when cluster $C$ is processed. If $V(C)\subseteq A$, or $V(C)\subseteq B$, then no further updates for cluster $C$ are necessary. Otherwise, we denote by $E'(C)=E_G(A,B)\cap E(C)$ -- the set of edges that cluster $C$ contributes to $E_G(A,B)$.
We partition the set $\delta_G(C)$ of edges into two subsets: set $\delta^A(C)$, $\delta^B(C)$, as follows. Let $e=(u,v)\in \delta_G(C)$ be any such edge, and assume that $v\in V(C)$. If $v\in A$, then we add $e$ to $\delta^A(C)$, and otherwise we add $e$ to $\delta^B(C)$. If $|\delta^A(C)|<|\delta^B(C)|$, then we move all vertices of $V(C)\cap A$ to $B$. Notice that in this case, since $C$ has the $\tilde \alpha'$-bandwidth property, $|\delta^A(C)|\leq |E'(C)|/\tilde \alpha'$. Once the vertices of $V(C)\cap A$ are moved to $B$, the edges of $E'(C)$ no longer lie in the cut $E_G(A,B)$, and the only new edges that may have been added to the cut $E_G(A,B)$ are the edges of $\delta^A(C)$. Since $|\delta^A(C)|\leq |E'(C)|/\tilde \alpha'$, we charge every edge of $E'(C)$ $1/\tilde \alpha'$ units for the edges of $\delta^A(C)$. Note that the edges of $E'(C)$ will never be charged again by the algorithm. Otherwise, $|\delta^B(C)|\leq |\delta^A(C)|$ holds, and we move the vertices of $B\cap V(C)$ to $A$. As before, $|\delta^B(C)|\leq |E'(C)|/\tilde \alpha'$ must hold, and the edges of $E'(C)$ no longer belong to the cut $E_G(A,B)$. The only new edges that may have been added to the cut are the edges of $\delta^B(C)$. As before, we charge every edge of $E'(C)$ $1/\tilde \alpha'$ units for the edges of $\delta^B(C)$.
Once every cluster in ${\mathcal{W}}^*\setminus \set{W}$ is processed, we obtain a new partition $(A',B')$ of $V(G)$, with $W\subseteq A'$, and $T\subseteq V(B')$, such that for every cluster $C\in {\mathcal{W}}^*$, either $V(C)\subseteq A'$, or $V(C)\subseteq B'$ holds. Moreover, since every edge in $E_G(A',B')\setminus E_G(A,B)$ is charged to some edge of $E_G(A,B)\setminus E_G(A',B')$, and the charge to each edge of $E_G(A,B)$ is at most $1/\tilde \alpha'$, we get that $|E_G(A',B')|\leq |E_G(A,B)|/\tilde \alpha'\leq \tilde \alpha k/256$.
We modify the clustering ${\mathcal{W}}^*$ in two steps. In the first step, we remove from ${\mathcal{W}}^*$ all clusters $W'$ with $V(W')\subseteq A'$, and we add to it cluster $G[A']$ (notice that we can assume w.l.o.g. that graph $G[A']$ is connected, since otherwise, we can move the sets of vertices corresponding to all connected components of $G[A']$ that are disjoint from $W$ to $B'$). Let ${\mathcal{W}}^{**}$ denote this new clustering. It is immediate to verify that Invariants \ref{inv: partition} and \ref{inv: large cluster in S} continue to hold in ${\mathcal{W}}^{**}$, since $|\delta_G(A')|<\tilde \alpha k/64$.
Note also that the edges of $\delta_G(W)\setminus \delta_G(A')$ no longer belong to $\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^{**})$, while no new edges were added to
$\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^{**})$.
Since $|\delta_G(W)|>\tilde \alpha k/64$, while $|\delta_G(A')|=|E_G(A',B')|\leq \tilde \alpha k/256$, we get that $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^{**})|\leq |\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^{*})|-\tilde \alpha k/64$.
Note that Invariant \ref{inv: bandwidth prop} is guaranteed to hold for every cluster of ${\mathcal{W}}^{**}$ except for possibly $G[A']$. In our last step,
we apply the algorithm from \Cref{thm:well_linked_decomposition} to compute a well-linked decomposition of the cluster $G[A']$ with parameter $\tilde \alpha'$. Recall that the algorithm computes a collection ${\mathcal{C}}^*$ of clusters, such that vertex sets $\set{V(C')\mid C'\in {\mathcal{C}}^*}$ partition $A'$, each cluster $C'\in {\mathcal{C}}^*$ has the $\tilde \alpha'$-bandwidth property, and $\delta_G(C')\leq \delta_G(A')<\tilde \alpha k/64$ for all $C'\in {\mathcal{C}}^*$. Moreover, we are guaranteed that:
\[\sum_{C'\in {\mathcal{C}}^*}|\delta_G(C')|\le |\delta_G(A')|\cdot\textsf{left}(1+O(\tilde \alpha'\cdot \log^{3/2} m)\textsf{right})\leq 2|\delta_G(A')|\leq \tilde \alpha k/128.
\]
We let ${\mathcal{W}}^{***}$ be obtained from ${\mathcal{W}}^{**}$ by replacing $G[A']$ with the collection ${\mathcal{C}}^*$ of clusters. From the above discussion, it is immediate to verify that Invariants \ref{inv: partition}, \ref{inv: bandwidth prop} and \ref{inv: large cluster in S} hold for ${\mathcal{W}}^{***}$. Moreover, $\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^{***})\subseteq \hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^{**})\cup \textsf{left}(\bigcup_{C'\in {\mathcal{C}}^*}\delta_G(C')\textsf{right} )$. Therefore, $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^{***})|\leq |\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^{**})|+\sum_{C'\in {\mathcal{C}}^*}|\delta_G(C')|\leq |\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^*)|$. This establishes Invariant \ref{inv: boundaries dont grow} for ${\mathcal{W}}^{***}$. We then set ${\mathcal{W}}^*={\mathcal{W}}^{***}$, and continue to the next iteration.
If the algorithm never terminates with a cluster $W$ and a collection ${\mathcal{R}}$ of at least $\tilde \alpha\talpha'k/256$ edge-disjoint paths routing a subset of terminals to the edges of $\delta_G(W)$, then the algorithm terminates once every cluster $W'\in {\mathcal{W}}^*$ has $|\delta_G(W')|\leq \tilde \alpha k/64$. We are then guaranteed that ${\mathcal{W}}^*$ is a legal clustering of $G$, and moreover, $|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}}^*)|<|\hat E^{\textnormal{\textsf{out}}}({\mathcal{W}})|$. We return the clustering ${\mathcal{W}}^*$ as the output of the algorithm.
\end{proofof}
\subsection{Proof of \Cref{obs: tunnels}}
\label{subsec: appx tunnels}
We start with the following simple observation.
\begin{observation}\label{obs: nice tunnel}
Let $p_i,p_{i'}$ be a pair of distinct points of $\Pi$, and assume that, for some integers $0\leq j\leq r$ and $0\leq a\leq 2^{r-j}$, $i'=a\cdot 2^j$. Assume further that $|i'-i|\leq 2^j$. Then there is a tunnel of length at most $j+1$ connecting $p_i$ to $p_{i'}$.
\end{observation}
\begin{proof}
We assume w.l.o.g. that $i'<i$; the other case is symmetric. The proof is by induction on $j$. If $j=0$, then $i=i'+1$, and we can let tunnel $L$ consist of a single level-$0$ curve $\lambda_{0,i'}$, that connects $p_{i'}$ to $p_i$.
Consider now some integer $j>0$, and assume that the claim holds for all integers $0\leq \tilde j<j$. We prove that the claim holds for $j$. If $i'-i=2^j$, then there is a single level-$j$ curve $\lambda_{j,a}$ that connects $p_{i'}$ to $p_i$, and we let the tunnel $L$ consist of this one curve. We assume from now on that $i'-i<2^j$. Let $j'$ be the largest integer so that $2^{j'}\leq i'-i$, so $0\leq j'<j$ holds. Then there must be a level-$j'$ curve $\lambda_{j',a'}$, whose endpoints are $p_{i'}$ and $p_{i''}$, where $i''=i'+2^{j'}$. Notice that $i'<i''\leq i$ must hold, and moreover, $|i-i''|\leq 2^{j'}<2^j$ must hold. If $p_{i''}=p_i$, then we let the tunnel $L$ consist of the curve $\lambda_{j',a'}$. Otherwise, from the induction hypothesis, there is a tunnel $L'$, of length at most $j'<j$, that connects $p_{i''}$ to $p_i$. We let $L$ be a tunnel that is obtained by appending curve $\lambda_{j',a'}$ at the beginning of tunnel $L'$.
\end{proof}
We are now ready to complete the proof of \Cref{obs: tunnels}. Consider a pair $p_i,p_{i'}$ of distinct points of $\Pi$, and assume w.l.o.g. that $i'<i$. Let $j$ be the largest integer, so that at least two points lie in $\set{p_{a\cdot 2^j}\mid 0\leq a<2^{r-j}}\cap \set{p_{i''}\mid i'\leq i''\leq i}$. Then there must be a pair of points $p_x,p_y$, with $i'\leq x<y\leq i$, such that $x=2^j\cdot a$ for some integer $a$, and $y=2^j\cdot (a+1)$. Note that there is a level-$j$ curve $\lambda_{j,a}$ in $\Lambda$ connecting $p_x$ and $p_y$. Moreover, $i-y\leq 2^j$ and $x-i'\leq 2^j$. From \Cref{obs: nice tunnel}, there is a tunnel $L_1$ of length at most $j$ connecting $p_{i'}$ to $p_x$, and a tunnel $L_2$ of length at most $j$ connecting $p_y$ to $p_{i}$. We then let $L$ be a tunnel obtained by concatenating tunnel $L_1$, curve $\lambda_{j,a}$, and tunnel $L_2$. Note that tunnel $L$ connects $p_{i'}$ to $p_{i}$, and its length is at most $2j+3\leq 2r+3\leq O(\log \check m')$.
\subsection{Proof of Theorem~\ref{thm:well_linked_decomposition}}
\label{apd: Proof of well_linked_decomposition}
\mynote{I am rewriting this proof right now. Please don't change this file.}
Our algorithm maintains a collection ${\mathcal{R}}$ of clusters of $S$, that is initialized to ${\mathcal{R}}=\set{S}$. Throughout the algorithm, we ensure that the clusters in ${\mathcal{R}}$ are disjoint, and that $\bigcup_{R\in {\mathcal{R}}}V(R)=V(S)$. For a given collection ${\mathcal{R}}$ of clusters with the above properties, we define a \emph{budget} $b(e)$ for every edge $e\in E(G)$, as follows. If $e\in \delta_G(S)$, and its endpoint $v$ that lies in $S$ belongs to a cluster $R\in{\mathcal{R}}$, then we set the budget $b(e)=1+8\alpha \cdot \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log_{3/2}(|\delta_G(R)|)$. If edge $e$ has its endpoints in two distinct clusters $R,R'\in {\mathcal{R}}$, then we set $b(e)=2+8\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log_{3/2}(|\delta_G(R)|)+8\alpha \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log_{3/2}(|\delta_G(R')|)$. Otherwise, we set $b(e)=0$.
Notice that, for every edge $e\in E(G)$, $b(e)\leq 3$ always holds. Additionally, for every edge $e\in \bigcup_{R\in {\mathcal{R}}}\delta_G(R)$, $b(e)\geq 2$ if the endpoints of $e$ lie in two different clusters of ${\mathcal{R}}$, and $b(e)\geq 1$ if $e\in \delta_G(S)$. Therefore, if we denote by $B=\sum_{e\in E(G)}b(e)$ the total budget in the system, then, throughout the algorithm, $B\geq \sum_{R\in {\mathcal{R}}}|\delta_G(R)|$ holds. Lastly, observe that, at the beginning of the algorithm, $B\leq |\delta_G(S)|\cdot (1+O(\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m))$. Throughout the algorithm, we will modify the clusters in set ${\mathcal{R}}$, which will lead to changes in budgets of the edges of $G$. We will ensure however that the total budget $B$ never increases, and so, if ${\mathcal{R}}$ is the final set of clusters that we obtain, then $\sum_{R\in {\mathcal{R}}}|\delta_G(R)|\leq B\leq (1+O(\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m))=(1+O(\alpha\cdot \log^{1.5} m))$ holds.
Throughout the algorithm, we maintain a partition of the set ${\mathcal{R}}$ of clusters into two subsets: set ${\mathcal{R}}^A$ of \emph{active} clusters, and set ${\mathcal{R}}^I$ of \emph{inactive} clusters. We will ensure that every cluster $R\in {\mathcal{R}}^I$ has the $\alpha$-bandwidth property. Additionally, we will store, with every inactive cluster $R\in {\mathcal{R}}^I$, a set ${\mathcal{P}}(R)=\set{P(e)\mid e\in \delta_G(R)}$ of paths in graph $G$, (that we refer to as \emph{witness path set for $R$}), that causes edge-congestion at most $100$, and, for every edge $e\in \delta_G(R)$, path $P(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, and all inner vertices of $P(e)$ lie in $V(S)\setminus V(R)$. At the beginning of the algorithm, we set ${\mathcal{R}}^A={\mathcal{R}}=\set{S}$ and ${\mathcal{R}}^I=\emptyset$. The algorithm proceeds in iterations, as long as ${\mathcal{R}}^A\neq \emptyset$.
We now describe a single iteration. The iteration selects an arbitrary cluster $R\in {\mathcal{R}}^A$ to process. It then either establishes that $R$ has the $\alpha$-bandwidth property in graph $G$, and computes a witness path set for $R$ (in which case $R$ is moved from ${\mathcal{R}}^A$ to ${\mathcal{R}}^I$); or it modifies the set ${\mathcal{R}}$ of clusters in a way that ensures that the total budget of all edges decreases by at least $1/m$. An iteration that processes a cluster $R\in {\mathcal{R}}^A$ consists of two steps. The purpose of the first step is to either establish the $\alpha$-bandwidth property of cluster $R$, or to replace it with a collection of smaller clusters in ${\mathcal{R}}$. The purpose of the second step is to either compute the witness set ${\mathcal{P}}(R)$ of paths for cluster $R$, or to modify the set ${\mathcal{R}}$ of clusters in a way that decreases the total budget of all edges. We now describe each of the two steps in turn.
\paragraph{Step 1: Bandwidth Property.}
Let $R^+$ be the augmentation of the cluster $R$ in graph $G$. Recall that $R^+$ is a graph that is obtained from $G$ through the following process. First, we subdivide every edge $e\in \delta_G(R)$ with a vertex $t_e$, and we let $T=\set{t_e\mid e\in \delta_G(R)}$ be the resulting set of vertices. We then let $R^+$ be the subgraph of the resulting graph induced by vertex set $V(R)\cup T$. We apply the algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace for computing approximate sparsest cut to graph $R^+$, with the set $T$ of vertices, to obtain a $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)$-approximate sparsest cut $(X,Y)$ in graph $R^+$ with respect to vertex set $T$.
We now consider two cases. The first case happens if $|E(X,Y)|> \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T}$. In this case, we are guaranteed that the smallest sparsity of any $T$-cut in graph $R^+$ is at least $\alpha$, or equivalently, set $T$ of vertices is $\alpha$-well-linked in $R^+$. From \Cref{obs: wl-bw}, cluster $R$ has the $\alpha$-bandwidth property in graph $G$. In this case, we proceed to the second step of the algorithm.
Assume now that $|E(X,Y)|\leq \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T}$.
Since $\alpha< \min\set{1/\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m),1/(48\log^2 m)}$, we get that the sparsity of the cut $(X,Y)$ is less than $1$. Consider now any vertex $t\in T$, and let $v$ be the unique neighbor of $t$ in $R^+$. We can assume w.l.o.g. that either $t,v$ both lie in $X$, or they both lie in $Y$. Indeed, if $t\in X$ and $v\in Y$, then moving vertex $t$ from $X$ to $Y$ does not increase the sparsity of the cut $(X,Y)$. This is because, for any two reals $1\leq a<b$, $\frac{a-1}{b-1}\leq \frac a b$. Similarly, if $t\in Y$ and $v\in X$, then moving $t$ from $Y$ to $X$ does not increase the sparsity of the cut $(X,Y)$. Therefore, we assume from now on, that for every vertex $t\in T$, if $v$ is the unique neighbor of $t$ in $R^+$, then either $v,t\in X$, or $v,t\in Y$ holds.
Consider now the partition $(X',Y')$ of $V(R)$, where $X'=X\setminus T$ and $Y'=Y\setminus T$. It is easy to verify that $|\delta_G(R)\cap \delta_G(X')|=|X\cap T|$, and $|\delta_G(R)\cap \delta_G(Y')|=|Y\cap T|$. Let $E'=E_G(X',Y')$, and assume w.l.o.g. that $|\delta_G(R)\cap \delta_G(X')|\leq |\delta_G(R)\cap \delta_G(Y')|$. Then $|E'|\leq \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|$ must hold. We remove cluster $R$ from sets ${\mathcal{R}}$ and ${\mathcal{R}}^A$, and we add instead every connected component of graphs $G[X']$ and $G[Y']$ to both sets. It is immediate to verify that ${\mathcal{R}}$ remains a collection of disjoint clusters of $G$, and that $\bigcup_{R'\in {\mathcal{R}}}V(R')=V(G)$. We now show that the total budget $B$ decreases by at least $1/m$ as the result of this operation.
Note that the only edges whose budgets may change as the result of this operation are edges of $\delta_G(R)\cup E'$. Observe that, for each edge $e\in \delta_G(R)\cap \delta_G(Y')$, its budget $b(e)$ may not increase. Since we have assumed that $|\delta_G(R)\cap \delta_G(X')|\leq |\delta_G(R)\cap \delta_G(Y')|$, and since $|E'|<|\delta_G(R)|/8$, we get that $|\delta_G(X')|\leq 2|\delta_G(R)|/3$. Therefore, for every edge $e\in \delta_G(X')\cap \delta_G(R)$, its budget $b(e)$ decreases by at least $8\alpha \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot\log_{3/2}(|\delta_G(R)|)-8\alpha \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot\log_{3/2}(|\delta_G(X')|)$. Since $|\delta_G(X')|\leq 2|\delta_G(R)|/3$, we get that $ \log_{3/2}(|\delta_G(R)|)\leq \log_{3/2}(3|\delta_G(X')|/2)\leq 1+\log_{3/2}(|\delta_G(X')|$. We conclude that the budget $b(e)$ of each edge $e\in \delta_G(X')\cap \delta_G(R)$ decreases by at least $8\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)$.
On the other hand, the budget of every edge $e\in E'$ increases by at most $3$. Since $|E'|\leq \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|$, we get that the decrease in the budget $B$ is at least:
\[
\begin{split}
8\alpha\cdot |\delta_G(X')\cap \delta_G(R)|-3|E'|&\geq 8\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(X')\cap \delta_G(R)|- \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|
\\&\geq 7 \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|\\&>1/m,\end{split}\]
since $\alpha\geq 1/m$.
To conclude, if $|E(X,Y)|\leq \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T}$, then we have modified the set ${\mathcal{R}}$ of clusters, so that the total budget $B$ decreases by at least $1/m$, and all invatiants hold. In this case, we terminate the current iteration.
From now on we assume that $|E(X,Y)|> \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T}$, which, as observed already, implies that cluster $R$ has the $\alpha$-bandwidth property. We now proceed to describe the second step of the algorithm.
\paragraph{Step 2: Witness Path Set.}
In the second step, we attempt to compute a witness path set ${\mathcal{P}}(R)$ for cluster $R$. If we succeed in doing so, we will move cluster $R$ from ${\mathcal{R}}^A$ to ${\mathcal{R}}^I$. Otherwise, we will further modify cluster set ${\mathcal{R}}$ so that all invariants will continue to hold and the total budget will decrease by at least $1/m$.
We construct the following flow network. Starting from graph $G$, we contract all vertices of $R$ into a source vertex $s$, and we contract all vertices of $V(G)\setminus V(S)$ into a destination vertex $t$. Denote the resulting graph by $H$, and observe that $\delta_H(s)=\delta_G(R)$, and $\delta_H(t)=\delta_G(S)$. We set the capacity $c(e)$ of every edge incident to $s$ to $1$, and the capacity of every other edge in graph $H$ to $100$. We then compute the maximum $s$-$t$ flow $f$ in the resulting flow network.
We consider two cases. The first case is when the value of the flow $f$ is $|\delta_H(s)|$. Since all edge capacities are inegral, we can assume that flow $f$ is integral as well. Note that in this case, for every path $P$ connecting $s$ to $t$, either $f(P)=0$ or $f(P)=1$ must hold, as the capacities of all edges incident to $s$ are $1$. Therefore, flow $f$ naturally defines a collection ${\mathcal{P}}'(R)$ of $s$-$t$ paths, that cause congestion at most $100$ in $H$, and each edge $e\in \delta_G(s)$ serves as the first edge of exactly one such path. Path set ${\mathcal{P}}'(R)$ then naturally defines a witness path set ${\mathcal{P}}(R)=\set{P(e)\mid e\in \delta_G(R)}$ for cluster $R$ in graph $G$, that cause edge-congestion at most $100$, and, for every edge $e\in \delta_G(R)$, path $P(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, with all inner vertices of $P(e)$ lying in $V(S)\setminus V(R)$. We then move cluster $R$ from ${\mathcal{R}}^A$ to ${\mathcal{R}}^I$ and terminate the current iteration. It is easy to verify that all invariants continue to hold, and the total budget $B$ does not change.
It remains to consider the second case, where the value of the flow $f$ is less than $|\delta_H(s)|$. We compute a minimum $s$-$t$ cut $(A',B')$ in graph $H$, whose value is less than $|\delta_H(s)|$. We partition the edge set $E(A',B')$ into two subsets: set $E'=E(A',B')\cap \delta_H(s)$, and set $E''=E(A',B')\setminus E'$. Recall that the capacity of every edge in $E'$ is $1$, while the capacity of every edge in $E''$ is $100$. Therefore, $|E'|+100|E''|<|\delta_G(R)|$.
Observe that cut $(A',B')$ in $H$ naturally defines cut $(A,B)$ of graph $S$: we let $A=(A'\setminus \set{s})\cup V(R)$, and $B=V(S)\setminus A)$. Notice that $E_G(A,B)\subseteq E_H(A',B')$ (as some edges in $\delta_G(S)$ may have contributed to cut $(A',B')$ but no longer contribute to cut $(A,B)$. We perform the following modifications to cluster sets ${\mathcal{R}},{\mathcal{R}}^I$, and ${\mathcal{R}}^A$. First, we remove cluster $R$ from ${\mathcal{R}}$ and from ${\mathcal{R}}^A$, and we add every connected component of $S[A]$ to both sets instead. Next, we consider every cluster $R'\in {\mathcal{R}}$ with $R'\cap A\neq \emptyset$. We remove cluster $R'$ from ${\mathcal{R}}$, and we add instead every connected component of $S[V(R')\setminus A]$. We also remove cluster $R'$ from the cluster set in $\set{{\mathcal{R}}^I,{\mathcal{R}}^A}$ to which it belongs, and we add every connected component of $S[V(R')\setminus A]$ to set ${\mathcal{R}}^I$. This completes the description of the modification of cluster sets ${\mathcal{R}},{\mathcal{R}}^I,{\mathcal{R}}^A$. Note that all clusters in ${\mathcal{R}}$ remain disjoint, and $\bigcup_{R''\in {\mathcal{R}}'}V(R'')=V(S)$ continues to hold. It remains to show that the total budget $B$ decreases by at least $1/m$.
Observe first that every edge in $\delta_G(A)$ belongs to the edge set $E_{H}(A',B')$. Since $|E_{H}(A',B')|<|\delta_H(s)|=|\delta_G(R)|$, we get that $|\delta_G(A)|<|\delta_G(R)|$. Therefore, the budgets of the edges in $\delta_G(A)\cap \delta_G(R)$ did not increase. All remaining edges of $\delta_G(A)$ lie in edge set $E''$. Therefore,
the only edges whose budget may have increased are the edges of $E_G(A,B)\cap E''$.
For each edge $e\in E_{G}(A,B)\cap E''$, its budget may have grown from $0$ to at most $3$, while the number of all such edges is $|E''|<(|\delta_G(R)|-|E'|)/100$. Therefore, the total increase in the budget $B$ due to the edges of $E_{G}(A,B)\cap E''$ is at most $(|\delta_G(R)|-|E'|)/30$. We show that this increase is compensated by the decrease in the budgets of the edges of $\delta_G(R)\setminus E'$. Consider any edge $e\in \delta_G(R)\setminus E'$. Edge $e$ had budget at least $1$ originally, but after the current iteration, since the endpoints of $e$ both lie in $A$, its budget becomes $0$. Therefore, the decease in the budget $B$ due to the edges of $\delta_G(R)\setminus E'$ is at least $|\delta_G(R)\setminus E'|$. Overall, we get that the decrease in the budget $B$ is at least:
\[ |\delta_G(R)\setminus E'|- (|\delta_G(R)|-|E'|)/100\geq 1/m.\]
This concludes the description of the iteration. The algorithm terminates when $|{\mathcal{R}}^I|=\emptyset$ holds, at which point from the invariants we obtain the final cluster set ${\mathcal{R}}$, together with witness path sets $\set{{\mathcal{P}}(R)}_{R\in {\mathcal{R}}}$ with all required properties.
In particular, as observed above, since the budget of every edge in $\bigcup_{R\in {\mathcal{R}}}\delta_G(R)$ is at least $1$, and the total budget $B$ never increases, $\sum_{R\in {\mathcal{R}}}|\delta_G(R)|\leq B\leq (1+O(\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m))=(1+O(\alpha\cdot \log^{1.5} m))$ holds.
It remains to prove that the algorithm is efficient. Clearly, the algorithm for executing each iteration is efficient. We now show that the number of iterations is bounded by $O(m^2)$. Consider any iteration $i$ of the algorithm. Recall that, as the result of iteration $i$, either the budget $B$ decreased by at least $1/m$ (in which case we say that $i$ is a type-$1$ iteration); or budget $B$ did not change, but the number of clusters in set ${\mathcal{R}}^I$ decreased by $1$ (in which case we say that $i$ is a type-$2$ iteration). It is then immediate to see that the number of type-$1$ iterations, over the course of the algorithm, is bounded by $O(m)$. Since every cluster of ${\mathcal{R}}$ must contain at least one vertex, and $|V(S)|\leq m$ (because $S$ is a connected graph), the number of type-$2$ iterations between every consecutive pair of type-$1$ iterations is bounded by $O(m)$. Therefore, the total number of iterations of the algorithm is $O(m^2)$, and so the algorithm is efficient.
\subsection{Proof of \Cref{lem: find reordering}}
\label{subsec: compute reordering}
Suppose we are given a graph $G$, and, for every vertex $v\in V(G)$, an oriented circular ordering $({\mathcal{O}}_v,b_v)$ of edges in $\delta_G(v)$.
We say that a drawing $\phi$ of $G$ \emph{obeys the oriented orderings $\set{({\mathcal{O}}_v,b_v)}_{v\in V}$ at the vertices of $G$} if, for every vertex $v\in V(G)$, the oriented circular order in which the images of the edges of $\delta_G(v)$ enter $v$ in $\phi$ is $({\mathcal{O}}_v,b_v)$.
We use the following theorem from~\cite{pelsmajer2011crossing}.
\begin{theorem}[Corollary 5.6 of~\cite{pelsmajer2011crossing}]
\label{thm: compute_reordering_curves}
There is an efficient algorithm, that, given a two-vertex loopless multigraph $G$ (so $V(G)=\set{v,v'}$ and $E(G)$ only contains parallel edges connecting $v$ to $v'$), and, for each vertex $v\in V(G)$, an oriented ordering $({\mathcal{O}}_v,b_v)$ of its incident edges, computes a drawing $\phi$ of $G$ that obeys the given oriented orderings, such that $\mathsf{cr}(\phi)$ is at most twice the minimum number of crossings of any drawing of $G$ that obeys the given oriented orderings.
\end{theorem}
The proof of Lemma~\ref{lem: find reordering} easily follows \Cref{thm: compute_reordering_curves}. Recall that we are given a pair $({\mathcal{O}},b)$, $({\mathcal{O}}',b')$ of oriented orderings of a collection $U$ of elements. We construct a two-vertex loopless graph $G$ with oriented ordering on its vertices, as follows. Denote $U=\set{u_1,\ldots,u_r}$.
The vertex set of $G$ is $\set{v,v'}$. The edge set of $G$ consists of $r$ parallel edges connecting $v$ to $v'$, that we denote by $e_{u_1}, e_{u_2},\ldots e_{u_r}$, respectively.
The oriented ordering $({\mathcal{O}},b)$ of the elements of $U$ naturally defines an oriented ordering $(\hat {\mathcal{O}},b)$ of the edges of $G$, and similarly, the oriented ordering $({\mathcal{O}},b')$ of the elements of $U$ defines an oriented ordering $(\hat {\mathcal{O}},b')$ of the edges of $G$.
We define the oriented orderings for the vertices of $G$ as follows: $({\mathcal{O}}_v,b_v)=(\hat {\mathcal{O}},-b)$ and $({\mathcal{O}}_{v'},b_{v'})=(\hat {\mathcal{O}}',b')$.
Consider any drawing $\phi$ of $G$ on the sphere that obeys the oriented orderings for $v,v'$ defined above. Let $D'=D_{\phi}(v')$ be a tiny $v'$-disc. For all $1\leq i\leq r$, we denote the unique point on the image of edge $e_{u_i}$ that lies on the boundary of $D'$ by by $p'_i$.
Similarly, we let $\hat D=D_{\phi}(v)$ be a tiny $v$-disc, and, for all $1\leq i\leq r$, we denote the unique point on the image of edge $e_{u_i}$ that lies on the boundary of $\hat D$ by $p_i$. Let $D$ be the disc whose boundary is the same as the boundary of $\hat D$, but whose interior is disjoint from that of $\hat D$. Then $D'\subseteq D$, and the boundaries of the two discs are disjoint. Furthermore, points $p_1,\ldots,p_r$ appear on the boundary of $D$ according to the oriented ordering $({\mathcal{O}},b)$, and points $p'_1,\ldots,p'_r$ appear on the boundary of $D'$ according to the oriented ordering $({\mathcal{O}}',b')$.
For all $1\leq i\leq r$, let $\gamma_i$ be the segment of the image of the edge $e_{u_i}$ between points $p_i$ and $p'_i$. Then $\set{\gamma_i\mid 1\leq i\leq r}$ is a set of reordering curves for the orderings $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$, and moreover, the cost of this curve set is exactly the number of crossings in $\phi$.
Using a similar reasoning, any set $\Gamma$ of reordering curves for the orderings $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$ can be converted into a drawing of graph $G$ that obeys the oriented orderings at vertices $v$ and $v'$, in which the number of crossings is exactly the cost of $\Gamma$. Therefore, there is a drawing of $G$ that obeys the oriented orderings at $v$ and $v'$, whose number of crossings is bounded by $\mbox{\sf dist}(({\mathcal{O}},b),({\mathcal{O}}',b'))$. We apply the algorithm from Theorem~\ref{thm: compute_reordering_curves} to graph $G$, and then compute a set of reordering curves from the resulting drawing of $G$ as described above. From the above discussion, the cost of the resulting set of curves is at most $2\cdot \mbox{\sf dist}(({\mathcal{O}},b),({\mathcal{O}}',b'))$.
\subsection{Proof of \Cref{lem: ordering modification}}
\label{apd: Proof of find reordering}
The proof easily follows from \Cref{lem: find reordering}.
We denote $\delta_G(v)=\set{e_1,\ldots,e_r}$, and, for all $1\leq i\leq r$, we let $p_i$ be the unique point of $\phi(e_i)$ lying on the boundary of the disc $D$. Let $\sigma_i$ be the segment of $\phi(e_i)$ that is disjoint from the interior of the disc $D$. In other words, if $e_i=(v,u_i)$, then $\sigma_i$ is a curve connecting $\phi(u_i)$ to $p_i$.
Note that the points $p_1,\ldots,p_r$ appear on the boundary of $D$ according to the circular ordering ${\mathcal{O}}'_v$ of their corresponding edges.
We assume w.l.o.g. that the orientation of the ordering is $b_v'=-1$.
Let $D'$ be another disc, that is contained in $D$, with $\phi(v)$ lying in the interior of $D'$, such that the boundaries of $D$ and $D'$ are disjoint.
We place points $p_1',\ldots,p'_r$ on the boundary of the disc $D'$, so that all resulting points are distinct, and they appear on the boundary of $D'$ in the order ${\mathcal{O}}_v$ of their corresponding edges, using a positive orientation of the ordering. For all $1\leq i\leq r$, we can compute a simple curve $\gamma_i$, connecting $\phi(v)$ to $p'_i$, such that $\gamma_i$ is contained in $D'$ and only intersects the boundary of $D'$ at its endpoint $p'_i$. We also ensure that all resulting curves $\gamma_1,\ldots,\gamma_r$ are mutually internally disjoint. Using the algorithm from \Cref{lem: find reordering}, we compute a collection $\Gamma=\set{\gamma_1',\ldots,\gamma'_r}$ of {reordering curves}, where for $1\leq i\leq r$, curve $\gamma'_i$ connects $p_i$ to $p'_i$, is contained in $D$, and is disjoint from the interior of $D'$. Note that the total number of crossings between the curves in $\Gamma$ is at most $2\cdot\mbox{\sf dist}{(({\mathcal{O}}_v,1),({\mathcal{O}}'_v,-1))}$. For all $1\leq i\leq r$, we define a new image of the edge $e_i$ to be the concatenation of the curves $\sigma_i,\gamma'_i$, and $\gamma_i$. The images of all remaining edges and vertices of $G$ remain unchanged. Denote the resulting drawing of the graph $G$ by $\phi'$. It is immediate to verify that the edges of $\delta_G(v)$ enter the image of $v$ in the order ${\mathcal{O}}_v$ in $\phi'$, and that the drawings of $\phi$ and $\phi'$ are identical except for the segments of the images of the edges in $\delta_G(v)$ that lie inside the disc $D$. It is also immediate to verify that $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)+
2\cdot\mbox{\sf dist}{(({\mathcal{O}}_v,1),({\mathcal{O}}'_v,-1))}$.
We repeat the same algorithm again, only this time the points $p_1',\ldots,p'_r$ are placed on the boundary of disc $D'$ in the order ${\mathcal{O}}_v$ of their corresponding edges, using a negative orientation of the ordering. The remainder of the algorithm remains unchanged, and produces a drawing $\phi''$ of $G$. As before, the edges of $\delta_G(v)$ enter the image of $v$ in the order ${\mathcal{O}}_v$ in $\phi''$, and the drawings of $\phi$ and $\phi''$ are identical except for the segments of the images of the edges in $\delta_G(v)$ that lie inside the disc $D$. Moreover, $\mathsf{cr}(\phi'')\leq \mathsf{cr}(\phi)+
2\cdot\mbox{\sf dist}{(({\mathcal{O}}_v,-1),({\mathcal{O}}'_v,-1))}$. Let $\phi^*$ be the drawing with smaller number of crossings, among $\phi'$ and $\phi''$. Our algorithm returns the drawing $\phi^*$ as its final outcome. From the above discussion, $\mathsf{cr}(\phi^*)\leq \mathsf{cr}(\phi)+
2\cdot\mbox{\sf dist}{({\mathcal{O}}_v,{\mathcal{O}}'_v)}$.
\iffalse
Assume without loss of generality that the curves of $\Gamma$ enter $z$ in the oriented ordering $({\mathcal{O}},0)$. We first apply the algorithm from \Cref{lem: find reordering} to compute (i) a $2$-approximation $a_0$ of $\mbox{\sf dist}(({\mathcal{O}},0),({\mathcal{O}}',0))$, and (ii) a $2$-approximation $a_1$ of $\mbox{\sf dist}(({\mathcal{O}},0),({\mathcal{O}}',1))$.
Assume without loss of generality that $a_0\le a_1$, so $a_0\le 2\cdot \mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
For each $u\in U$, we denote by $p_u$ the intersection between the curve $\gamma_u$ and the boundary of $D$.
We erase the segments of curves of $\Gamma$ inside the disc $D$. We then place another disc $D'$ around $z$ inside the disc $D$, and let $\set{p'_u}_{u\in U}$ be a set of points appearing on the boundary of disc $D'$ in the order $({\mathcal{O}}',0)$. We use \Cref{lem: find reordering} to compute a set $\tilde\Gamma=\set{\tilde\gamma_u\mid u\in U}$ of reordering curves, where for each $u\in U$, the curve $\tilde\gamma_u$ connects $p_u$ to $p'_u$.
We then define, for each $u\in U$, the curve $\gamma'_u$ as the union of (i) the subcurve of $\gamma_u$ outside the disc $D$ (connecting its endpoint outside $D$ to $p_u$); (ii) the curve $\tilde\gamma_u$ (connecting $p_u$ to $p'_u$); and (iii) the straight line-segment connecting $p'_u$ to $z$. It is clear that (i) the curves $\set{\gamma'_u\mid u\in U}$ enter $z$ in the order $({\mathcal{O}}',0)$; (ii) for each $u\in U$, the curve $\gamma_u$ differs from the curve $\gamma'_u$ only within some tiny disc $D$ that contains $z$; and (iii) the number of crossings between curves of $\Gamma'$ within disc $D$ is at most $2\cdot \mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
\fi
\iffalse{some other proof}
\begin{proof}
This is a sketch, needs to be formalized. Let's look at the optimal set $\Gamma^*$ of reordering curves, and let $\gamma'_i$ be the curve participating in fewest crossings. Let's say this number of crossings is $N$. Then the cost of $\Gamma$ is at least $Nr/2$. The main claim is that there exists a collection $\Gamma'$ of reordering curves, of cost at most $2\operatorname{cost}(\Gamma^*)$, in which $\gamma_i$ does not participate in crossings. Assume first that the claim is correct. Finding such a drawing is straightforward. We rotate the discs so that $p'_i,p_i$ appear aligned, and connect them with a straight line, that becomes the drawing of $\Gamma_i$. Cutting $D\setminus D'$ along this line, we obtain a rectangle, whose vertical sides are the cut line, top horizontal line is boundary of $D$, and bottom horizontal line is boundary of $D'$. In this rectangle we just connect $p_j,p_j'$ with a straight line, adjusting the drawings as needed so no three curves meet at the same point. It is easy to see that resulting curves $\gamma_j,\gamma_k$ will only cross if the left-to-right ordering of $p_j,p_k$ is opposite from that of $p'_j,p'_k$. In any drawing where no curve crosses $\gamma_i$ any such pair of curves will cross as well. So we obtain a drawing of cost at most $\operatorname{cost}(\Gamma')\leq 2\operatorname{cost}(\Gamma^*)$.
Next, we show that there exists a collection $\Gamma'$ of curves of cost at most $2\operatorname{cost}(\Gamma^*)$ in which no curve crosses $\gamma_i$. In fact we compute $\Gamma'$ exactly as shown above. We just need to argue that the cost of this set of curves is at most $\Gamma^*$. In order to do so, we partition $\Gamma^*\setminus\set{\gamma_i}$ into two subsets: set $\Gamma_1$ containing all curves that don't cross $\gamma_i$, and set $\Gamma_2$ containing remaining curves. Consider now any pair $\gamma_j,\gamma_k$ of curves in $\Gamma'$, and assume that they cross (the drawing that we compute ensures that every pair of curves crosses at most once). in the new drawing. As observed above, this can only happen if the left-to-right ordering of $p_j,p_k$ on the top edge of the rectangle is opposite from that of $p'_j,p'_k$ on the bottom edge.
We now consider three cases. First, if the original curves routing elements $u_i,u_j$ lied in $\Gamma_1$, then the two corresponding curves had to cross in $\Gamma^*$ as well. Second, if the two original curves lied in $\Gamma_2$, then, as we will show later, they also had to cross before. So the case that remains is when one of the curves is in $\Gamma_1$ and another is in $\Gamma_2$. But because $|\Gamma_2|\leq N$, we will have at most $Nr$ such new crossings. Because $\operatorname{cost}(\Gamma^*)\geq Nr/2$, we get that $\operatorname{cost}(\Gamma')\leq \operatorname{cost}(\Gamma^*)+Nr\leq 5\operatorname{cost}(\Gamma')$. It now remains to prove the following claim.
\begin{claim}
Consider two elements $u_j,u_k$, such that the left-to-right ordering of $p_j,p_k$ on the top edge of the rectangle is opposite from the left-to-right ordering of $p'_j,p'_k$ on the bottom edge. Let $\gamma'_j,\gamma'_k$ be the two curves of $\Gamma^*$ corresponding to these elements, and assume that $\gamma'_i,\gamma'_k\in \Gamma_2$ (so both curves cross $\gamma_i$). Then $\gamma'_j,\gamma'_k$ must cross.
\end{claim}
\begin{proof}
Assume that $p_j$ appears to the left of $p_k$ on the top edge of the rectangle.
Consider the annulus $A=D\setminus D'$, and the drawings of $\gamma_i,\gamma'_j$ in it. We can assume that the two curves cross at most once. Consider a face $F$ of the resulting drawing whose boundary contains $p_k$. The three boundary edges of this face are: the segment of the top boundary of the rectangle from $p_j$ to $p_i$; segment of $\gamma'_i$ from $p_i$ to its meeting point $q$ with $\gamma'_j$; and segment of $\gamma'_j$ from $p_j$ to $q$. In particular, $p'_k$ does not lie on the boundary of $F$.
Assume for contradiction that $\gamma'_j,\gamma'_k$ do not cross. Then curve $\gamma'_k$ has to cross the boundary of $F$, which can only done by crossing the segment $\sigma$ of the boundary that lies on $\gamma'_i$. Denote this crossing point by $q'$. This is the only crossing between $\gamma'_k$ and $\gamma'_i$. If we denote by $F'$ the unique face that shares the boundary segment $\sigma$ with $F$, then that face does not contain $p'_k$ on its boundary (a figure is needed). Therefore, to reach $p'_k$, curve $\gamma'_k$ either needs to cross $\gamma'_j$, or to cross $\gamma'_i$ once again.
\end{proof}
\end{proof}
\fi
\iffalse
\mynote{previous proof, should keep only one}
We now prove \Cref{lem: ordering modification} using \Cref{lem: find reordering}.
Assume without loss of generality that the curves of $\Gamma$ enter $z$ in the oriented ordering $({\mathcal{O}},0)$. We first apply the algorithm from \Cref{lem: find reordering} to compute (i) a $2$-approximation $a_0$ of $\mbox{\sf dist}(({\mathcal{O}},0),({\mathcal{O}}',0))$, and (ii) a $2$-approximation $a_1$ of $\mbox{\sf dist}(({\mathcal{O}},0),({\mathcal{O}}',1))$.
Assume without loss of generality that $a_0\le a_1$, so $a_0\le 2\cdot \mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
For each $u\in U$, we denote by $p_u$ the intersection between the curve $\gamma_u$ and the boundary of $D$.
We erase the segments of curves of $\Gamma$ inside the disc $D$. We then place another disc $D'$ around $z$ inside the disc $D$, and let $\set{p'_u}_{u\in U}$ be a set of points appearing on the boundary of disc $D'$ in the order $({\mathcal{O}}',0)$. We use \Cref{lem: find reordering} to compute a set $\tilde\Gamma=\set{\tilde\gamma_u\mid u\in U}$ of reordering curves, where for each $u\in U$, the curve $\tilde\gamma_u$ connects $p_u$ to $p'_u$.
We then define, for each $u\in U$, the curve $\gamma'_u$ as the union of (i) the subcurve of $\gamma_u$ outside the disc $D$ (connecting its endpoint outside $D$ to $p_u$); (ii) the curve $\tilde\gamma_u$ (connecting $p_u$ to $p'_u$); and (iii) the straight line-segment connecting $p'_u$ to $z$. It is clear that (i) the curves $\set{\gamma'_u\mid u\in U}$ enter $z$ in the order $({\mathcal{O}}',0)$; (ii) for each $u\in U$, the curve $\gamma_u$ differs from the curve $\gamma'_u$ only within some tiny disc $D$ that contains $z$; and (iii) the number of crossings between curves of $\Gamma'$ within disc $D$ is at most $2\cdot \mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
\fi
\subsection{Proof of Theorem~\ref{thm: type-1 uncrossing}}
\label{apd: type-1 uncrossing}
Let $Z$ be the set of crossings points between the curves of $\Gamma$. For each such crossing point $z\in Z$, we consider a tiny disc $D_z$, that contains the point $z$ in its interior. We select the discs $D_z$ to ensure that all such discs are disjoint, and, moreover, if $z$ is a crossing point between curves $\gamma_1,\gamma_2$, then for every curve $\gamma\in \Gamma\setminus\set{\gamma_1,\gamma_2}$, $\gamma\cap D_z=\emptyset$, while for every curve $\gamma\in \set{\gamma_1\cup \gamma_2}$, $\gamma\cap D_z$ is a simple open curve whose endpoints lie on the boundary of $D_z$.
We start with $\Gamma_1'=\Gamma_1$, and then iteratively modify the curves in $\Gamma_1'$, as long as there is a pair of distinct curves $\gamma_1,\gamma_2\in \Gamma_1'$ that cross more than once.
Each iteration is executed as follows. Let $\gamma_1,\gamma_2\in \Gamma_1'$ be a pair of curves that cross more than once, and let $z,z'$ be two crossing points between $\gamma_1,\gamma_2$, that appear consecutively on $\gamma_1$. In other words, no other point that appears between $z$ and $z'$ on $\gamma_1$ may belong to $\gamma_2$.
We denote by $s_1,t_1$ the endpoints of $\gamma_1$, such that $z$ appears closer to $s_1$ than $z'$ on $\gamma_1$.
Similarly, we denote by $s_2,t_2$ the endpoints of $\gamma_2$, such that $z$ appears closer to $s_2$ than $z'$ on $\gamma_2$.
We denote by $x_1,x_2$ the two points of $\gamma_1$ that lie on the boundary of disc $D_z$, and denote by $x_3,x_4$ the two points of $\gamma_1$ lying on the boundary of disc $D_{z'}$,
such that the points $x_1,z,x_2,x_3,z',x_4$ appear on $\gamma_1$ in this order.
We define points $y_1,y_2,y_3,y_4$ on $\gamma_2$ similarly (see \Cref{fig:non_crossing_representation1}).
\begin{figure}[h]
\centering
\subfigure[Before: Curve $\gamma_1$ is shown in blue and curve $\gamma_2$ is shown in red. The disc on the left is $D_z$, and the disc on the right is $D_{z'}$.]{\scalebox{0.1}{\includegraphics{figs/type_1_uncross_proof_1.jpg}} \label{fig:non_crossing_representation1}
}
\hspace{0.5cm}
\subfigure[After: Curve $\gamma'_1$ is shown in blue and curve $\gamma'_2$ is shown in red.]{
\scalebox{0.1}{\includegraphics{figs/type_1_uncross_proof_2.jpg}}\label{fig:non_crossing_representation2}}
\caption{An iteration of the algorithm for performing a type-1 uncrossing.}\label{fig: type_1_uncross_proof}
\end{figure}
In order to execute the iteration, we slightly modify the curves $\gamma_1, \gamma_2$, by ``swapping'' their segments between points $x_2,x_3$ and $y_2,y_3$, respectively, and slightly nudging them inside the discs $D_z,D_{z'}$, as show in \Cref{fig: type_1_uncross_proof}. We now describe the construction of the new curves $\gamma_1',\gamma_2'$ more formally.
Note that the points $x_1,x_2,y_1,y_2$ appear on the boundary of $D_z$ clockwise in either the order $(x_1,y_1,x_2,y_2)$ or the order $(x_1,y_2,x_2,y_1)$.
Therefore, we can find two disjoint simple curves $\eta_1$ and $\eta_2$ that are contained in disc $D_z$, with $\eta_1$ connecting $x_1$ to $y_2$, and $\eta_2$ connecting $y_1$ to $x_2$.
Similarly, we compute two disjoint simple curves curves $\eta'_1,\eta'_2$, that are contained in disc $D_{z'}$, with $\eta_1'$ connecting $y_3$ to $x_4$, and $\eta_2'$ connecting $x_3$ to $y_4$ (see \Cref{fig:non_crossing_representation2}).
We let $\gamma_1'$ be a curve, that is constructed by concatenating the following five curves: (1) the segment of $\gamma_1$ from $s_1$ to $x_1$; (2) curve $\eta_1$; (3) the segment of $\gamma_2$ from $y_2$ to $y_3$; (4) curve $\eta'_1$; and (5) the segment of $\gamma_1$ from $x_4$ to $t_1$. Similarly, let $\gamma_2'$ be a curve, that is constructed by concatenating the following five curves: (1) the segment of $\gamma_2$ from $s_2$ to $y_1$; (2) curve $\eta_2$; (3) the segment of $\gamma_1$ from $x_2$ to $x_3$; (4) curve $\eta'_2$; and (5) the segment of $\gamma_2$ from $y_4$ to $t_2$. We then remove any self loops from the two curves, to obtain the final curves $\gamma'_1,\gamma_2'$, that replace the curves $\gamma_1,\gamma_2$ in $\Gamma_1'$. Note that $\gamma'_1$ has the same endpoints as $\gamma_1$, and the same is true for $\gamma'_2$ and $\gamma_2$. It is also easy to verify that, at the end of the iteration, the number of crossings between the curves of $\Gamma_1'\cup \Gamma_2$ strictly decreases, and the number of crossings between the curves of $\Gamma_1'$ and the curves of $\Gamma_2$, that we denoted by $\chi(\Gamma_1',\Gamma_2)$, does not grow. Moreover, for every curve $\gamma\in \Gamma_2$, the number of crossings of $\gamma$ with the curves in $\Gamma_1'$ may not grow either. Once the algorithm terminates, we obtain the desired set $\Gamma_1'$ of curves, in which every pair of distinct curves crosses at most once. From the above discussion, it is immediate to verify that the curves in $\Gamma_1'$ have all required properties. Since the curves in $\Gamma$ are in general position, the number of iterations is bounded by the number of crossing points between the curves.
\subsection{Proof of \Cref{claim: curves in a disc}}
\label{apd: Proof of curves in a disc}
We start by constructing a collection $\Gamma'=\set{\gamma'_i\mid 1\leq i\leq k}$ of curves that are in general position, such that for all $1\leq i\leq k$, curve $\gamma'_i$ has $s_i$ and $t_i$ as its endpoints, and is contained in disc $D$. In order to construct the set $\Gamma'$ of curves, we let $p$ be any point in the interior of the disc $D$, and $r>0$ be some real number, such that a radius-$r$ circle centered at point $p$ is contained in the disc $D$. For all $1\leq i\leq k$, let $\ell_i$ be a straight line, connecting point $s_i$ to $p$, and $\ell'_i$ a straight line, connecting point $t_i$ to $p$. We can assume that both lines are contained in the disc $D$, by stretching the disc as needed. For all $1\leq i\leq k$, we choose a radius $0<r_i<r$, so that $0<r_1<\cdots<r_k<r$ holds. For an index $1\leq i\leq k$, we let $C_i$ be the boundary of a radius-$r_i$ circle centered at point $p$, and we let $q_i,q'_i$ be the points on lines $\ell_i$ and $\ell'_i$, respectively, that lie on $C_i$. We let curve $\gamma'_i$ be a concatenation of three curves: the segment of $\ell_i$ from $s_i$ to $q_i$; a segment of $C_i$ between $q_i$ and $q'_i$; and the segment of $\ell'_i$ from $q'_i$ to $t_i$. Consider the resulting set
$\Gamma'=\set{\gamma'_i\mid 1\leq i\leq k}$ of curves. Clearly, for all $1\leq i\leq k$, curve $\gamma'_i$ has $s_i$ and $t_i$ as its endpoints, and is contained in disc $D$.
It is also easy to verify that curves of $\Gamma'$ are in general position.
Next, we use the algorithm from \Cref{thm: type-1 uncrossing} to perform a type-1 uncrossing of the curves in $\Gamma'$. Specifically, we set $\Gamma_1=\Gamma'$ and $\Gamma_2=\emptyset$. We denote by $\Gamma=\Gamma_1'=\set{\gamma_i\mid 1\leq i\leq k}$ the set of curves that the algorithm outputs. Recall that, for all $1\leq i\leq k$, curve $\gamma_i$ has $s_i$ and $t_i$ as its endpoints; the curves in $\Gamma$ are in general position; and every pair of curves in $\Gamma$ cross at most once. From the description of the type-1 uncrossing operation, it is easy to verify that all curves in $\Gamma$ are contained in the disc $D$.
Consider now two pairs $(s_i,t_i),(s_j,t_j)$ of points, with $i\neq j$. Note that curve $\gamma_i$ partitions the disc $D$ into two regions, that we denote by $F$ and $F'$. If the two pairs $(s_i,t_i),(s_j,t_j)$ cross, then $s_j,t_j$ may not lie on the boundary of the same region, and so curve $\gamma_j$ must cross curve $\gamma_i$ exactly once. If the two pairs do not cross, then $s_j,t_j$ either both lie on the boundary of $F$, or they both lie on the boundary of $F'$. It is then impossible that curves $\gamma_i,\gamma_j$ cross exactly once, and, since every pair of curves cross at most once, they cannot cross.
\iffalse
We first set, for each $1\le i\le k$, a curve $\gamma'_i$ to be the straight line segment connecting $s_i$ to $t_i$. It is easy to see that every pair of curves in $\Gamma'=\set{\gamma'_i\mid 1\le i\le k}$ cross at most once, and a pair $\gamma'_i,\gamma'_j$ of curves cross iff the two pairs $(s_i,t_i), (s_j,t_j)$ of points cross.
If the curves of $\Gamma'$ are in general position (namely no point belongs to at least three curves of $\Gamma'$), then we set, for each $1\le i \le k$, $\gamma_i=\gamma'_i$, and return the collection $\Gamma=\set{\gamma_i\mid 1\le i\le k}$ of curves. It is easy to verify that $\Gamma$ satisfies the required property.
If the curves $\Gamma'$ are not in general position, then we let, for each $1\le i\le k$, a curve $\gamma_i$ be obtained by slightly perturbing the curve $\gamma'_i$. See \Cref{fig: perturb} for an illustration. We then return the collection $\Gamma=\set{\gamma_i\mid 1\le i\le k}$ of curves. It is easy to verify that $\Gamma$ satisfies the required property.
\begin{figure}[h]
\centering
\subfigure[Before: Curves $\gamma'_1,\gamma'_2,\gamma'_3$ contain the same point.]{\scalebox{1.1}{\includegraphics[scale=0.4]{figs/perturb_before.jpg}}}
\hspace{0.7cm}
\subfigure[After: Curves $\gamma_1,\gamma_2,\gamma_3$ are in general position.]{\scalebox{1.1}{\includegraphics[scale=0.4]{figs/perturb_after.jpg}}}
\caption{An illustration of slight perturbation of curves in the proof of \Cref{claim: curves in a disc}.\label{fig: perturb}}
\end{figure}
\fi
\subsection{Proof of Theorem~\ref{thm: new type 2 uncrossing}}
\label{apd: new type 2 uncrossing}
We start with an initial set $\Gamma'=\set{\gamma'(Q)\mid Q\in {\mathcal{Q}}}$ of curves, where, for each path $Q\in {\mathcal{Q}}$, $\gamma'(Q)$ is the image of the path $Q$ in $\phi$. In other words, $\gamma'(Q)$ is the concatenation of the images of the edges of $Q$ in $\phi$. Note however that the resulting set $\Gamma'$ of curves may not be in general position. This is since a vertex $v\in V(G)$ may serve as an inner vertex on more than two paths of ${\mathcal{Q}}$, and in such a case its image $\phi(v)$ serves as an inner point of more than two curves in $\Gamma'$.
Let $V'\subseteq V(G)$ be the set of all vertices $v\in V(G)$, such that more than two paths in ${\mathcal{Q}}$ contain $v$.
In our first step, we transform the set $\Gamma'$ of curves so that the resulting curves are in general position, while ensuring that the endpoints of each curve $\gamma'(Q)$ remain unchanged, and each such curve $\gamma'(Q)$ remains aligned with the graph $\bigcup_{Q'\in {\mathcal{Q}}}Q'$. We do so by performing a \emph{nudging operation} around every vertex $v\in V'$, as follows.
Consider any vertex $v\in V'$, and let ${\mathcal{Q}}(v)\subseteq {\mathcal{Q}}$ be the set of all paths containing vertex $v$. Note that $v$ must be an inner vertex on each such path. For convenience, we denote ${\mathcal{Q}}(v)=\set{Q_1,\ldots,Q_z}$. Consider the tiny $v$-disc $D=D_{\phi}(v)$. For all $1\leq i\leq z$, denote by $a_i$ and $b_i$ the two points on curve $\gamma(Q_i)$ that lie on the boundary of disc $D$. We use the algorithm from \Cref{claim: curves in a disc} to compute a collection $\set{\sigma_1,\ldots,\sigma_z}$ of curves, such that, for all $1\leq i\leq z$, curve $\sigma_i$ connects $a_i$ to $b_i$, and the interior of the curve is contained in the interior of $D$. Recall that every pair of resulting curves crosses at most once, and every point in the interior of $D$ may be contained in at most two curves.
For all $1\leq i\leq z$, we modify the curve $\gamma(Q_i)$, by replacing the segment of the curve that is contained in disc $D$ with $\sigma_i$.
Once every vertex $v\in V'$ is processed, we obtain a collection $\Gamma''=\set{\gamma''(Q)\mid Q\in {\mathcal{Q}}}$ of curves, where for every path $Q\in {\mathcal{Q}}$, curve $\gamma''(Q)$ connects $\phi(s(Q))$ to $\phi(t(Q))$. Moreover, it is easy to verify that each resulting curve $\gamma''(Q)\in \Gamma''$ is aligned with the drawing of the graph $\bigcup_{Q'\in {\mathcal{Q}}}Q'$ induced by $\phi$, and that the curves in $\Gamma''$ are in general position.
We let $S$ be the multiset of points that contains, for every curve $\gamma''(Q)\in \Gamma''$ its first endpoint $\phi(s(Q))$, and we let $T$ be the multiset of points contianing the last endpoint of each such curve.
We initially let, for each path $Q\in {\mathcal{Q}}$, $\gamma(Q)$ be the curve obtained by deleting all loops from $\gamma''(Q)$, and we denote by $\Gamma=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$ the resulting set of curves. We gradually modify the curves in $\Gamma$ in order to eliminate all crossings between them. Throughout the algorithm, we ensure that for each path $Q\in {\mathcal{Q}}$, curve $\gamma(Q)$ originates at point $\phi(s(Q))$, and moreover, if $e_1(Q)$ is the first edge on path $Q$, then segment $\phi(e_1(Q))\cap D_{\phi}(s(Q))$ is contained in $\gamma(Q)$. We also ensure that the multiset containing the last point on every curve of $\Gamma$ remains unchanged throughout the algorithm. Let $P$ be the collection of all points $p$, such that at least two curves of $\Gamma$ contain $p$ as an inner point.
We perform iterations, as long as $P\neq\emptyset$. Each iteration is executed as follows. Let $p\in P$ be any point, and let $Q,Q'\in {\mathcal{Q}}$ be two paths whose corresponding curves $\gamma(Q),\gamma(Q')$ contain the point $p$. Let $x,x'$ be the two points of $\gamma(Q)$ that lie on the boundary of the tiny $p$-disc $D(p)$, with $x$ appearing before $x'$ on $\gamma(Q)$. Let $y,y'$ be the two points of $\gamma(Q')$, that lie on the boundary of $D(p)$, with $y$ appearing before $y'$ on $\gamma(Q')$. We now consider two cases. In the first case, the circular clock-wise ordering of points $x,x',y,y'$ on the boundary of $D(p)$ is either $(x,y,x',y')$, or $(x,y',x',y)$. In this case, the two pairs $(x,y')$ and $(y,x')$ of points on the boundary of $D(p)$ do not cross. Therefore, from \Cref{claim: curves in a disc}, we can construct two disjoint curves $\sigma,\sigma'$ that are contained in $D(p)$, with $\sigma$ connecting $x$ to $y'$ and $\sigma'$ connecting $y$ to $x'$. We let $\gamma'(Q)$ be a curve that is obtained by concatenating the segment of $\gamma(Q)$ from its first endpoint to $x$; the curve $\sigma$; and the segment of $\gamma(Q')$ from $y'$ to its last endpoint. Similarly, we let $\gamma'(Q')$ be a curve that is obtained by concatenating the segment of $\gamma(Q')$ from its first endpoint to $y$; the curve $\sigma'$; and the segment of $\gamma(Q)$ from $x'$ to its last endpoint. We then replace $\gamma(Q)$ with $\gamma'(Q)$ and $\gamma(Q')$ with $\gamma'(Q')$ in $\Gamma$. In the second case, the circular clock-wise ordering of points $x,x',y,y'$ on the boundary of $D(p)$ must be either $(x,x',y,y')$, or $(x',x,y,y')$, or $(x,x',y',y)$, or $(x',x,y',y)$. In either case, the two pairs $(x,x')$ and $(y,y')$ of points on the boundary of $D(p)$ do not cross. Therefore, from \Cref{claim: curves in a disc}, we can construct two disjoint curves $\sigma,\sigma'$ that are contained in $D(p)$, with $\sigma$ connecting $x$ to $x'$ and $\sigma'$ connecting $y$ to $y'$. We modify curve $\gamma(Q)$ by replacing its segment that is contained in $D(p)$ with $\sigma$, and we similarly replace the segment of $\gamma(Q')$ that is contained in $D(p)$ with $\sigma'$. If either of the new curves $\gamma(Q),\gamma(Q')$ has loops, we turn the corresponding curve into a simple one by removing all loops from it. We also update the set $P$ of points, by removing from it points that no longer belong to two curves in $\Gamma$. This finishes the description of an iteration. It is easy to verify that no new crossing points between curves in $\Gamma$ are created, the curves in $\Gamma$ remain in general position, and each such curve is aligned with the drawing of the graph $\bigcup_{Q''\in {\mathcal{Q}}}Q''$ induced by $\phi$. It is also easy to verify that the invariants continue to hold.
Consider the set $\Gamma$ of curves that we obtain at the end of the algorithm. Clearly, the curves in $\Gamma$ do not cross with each other, and they are aligned with the drawing of the graph $\bigcup_{Q\in {\mathcal{Q}}}Q$ induced by $\phi$. It is also immediate to verify that they have all remaining required properties.
Since $|P|$ strictly decreases from iteration to iteration, the number of iterations is bounded by the number of crossings of the drawing $\phi$ of $G$, which is in turn bounded by the input size. Each iteration can be executed in time polynomial in the input size, so the algorithm is efficient.
\subsection{Proof of \Cref{cor: new type 2 uncrossing}}
\label{apd: cor new type 2 uncrossing}
For each edge $e\in E(G)\setminus E(C)$, we let $n_e=\cong_G({\mathcal{Q}},e)$. Let $H$ be a new graph, with $V(H)=V(G)$, whose edge set consists of the set $E(C)$ of edges, and, for each edge $e\in E(G)\setminus E(C)$, a set $J(e)$ of $n_e$ parallel copies of the edge $e$. Note that the drawing $\phi$ of graph $G$ naturally defines a drawing $\phi'$ of graph $H$. In order to obtain the drawing $\phi'$ of $H$, we start with the drawing $\phi$ of $G$, and then, for every edge $e\in E(G)\setminus E(C)$ with $n_e>0$, we draw the edges of $J(e)$ in parallel to $\phi(e)$, very close to it. We also delete the images of all edges $e\in E(G)\setminus E(C)$ with $n_e=0$. Note that, for every edge $e\in E(C)$, the number of crossings between $\phi'(e)$ and the images of the edges of $E(H)\setminus E(C)$ in the drawing $\phi'$ is at most $\sum_{e'\in E(G)\setminus E(C)}\chi(e,e')\cdot \cong_G({\mathcal{Q}},e')$, where $\chi(e,e')$ is the number of crossings between $\phi(e)$ and $\phi(e')$.
The set ${\mathcal{Q}}$ of paths in graph $G$ naturally defines a set ${\mathcal{Q}}' $ of edge-disjoint paths in graph $H$, where, for each edge $e\in E(G)\setminus E(C)$, for every path $Q\in {\mathcal{Q}}$ containing the edge $e$, we replace $e$ with a distinct edge of $J(e)$ on path $Q$. In particular, the multisets $S({\mathcal{Q}})$, $S({\mathcal{Q}}')$ of vertices containing the first vertex of every path in set ${\mathcal{Q}}$ and ${\mathcal{Q}}'$, respectively, remain unchanged, and the same is true regarding the multisets $T({\mathcal{Q}}),T({\mathcal{Q}}')$ of paths, containing the last vertex of every path in set ${\mathcal{Q}}$ and ${\mathcal{Q}}'$, respectively.
We apply the algorithm from \Cref{thm: new type 2 uncrossing} to graph $H$, the drawing $\phi'$ of $H$, and the set ${\mathcal{Q}}'$ of edge-disjoint paths in $H$. Let $\Gamma'=\set{\gamma'(Q')\mid Q'\in {\mathcal{Q}}'}$ be the resulting set of curves. Recall that, for every path $Q\in {\mathcal{Q}}$, there is a distinct path $Q'\in {\mathcal{Q}}'$, that is obtained from $Q$ by replacing each edge $e\in E(Q)$ with one of its copies. For each path $Q\in {\mathcal{Q}}$, we then let $\gamma(Q)=\gamma'(Q')$, and we consider the resulting set $\Gamma=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$ of curves. The algorithm from \Cref{thm: new type 2 uncrossing} ensures that the curves in $\Gamma$ do not cross each other, and that, for every path $Q\in {\mathcal{Q}}$, $s(\gamma(Q))=\phi(s(Q))$. It also guarantees that the multiset $T(\Gamma)$ is precisely the multiset $\set{\phi(t(Q))\mid Q\in {\mathcal{Q}}}$.
Lastly, consider any edge $e\in E(C)$. Since the curves in set $\Gamma'$ are aligned with the drawing of the graph $\bigcup_{Q'\in {\mathcal{Q}}'}Q'$ induced by $\phi'$, the number of crossings between $\phi'(e)=\phi(e)$ and the curves in set $\Gamma'=\Gamma$ is bounded by the number of crossings between $\phi'(e)$ and the images of the edges of $E(H)\setminus E(C)$ in drawing $\phi'$ of $H$, which is, in turn, bounded by
$\sum_{e'\in E(G)\setminus E(C)}\chi(e,e')\cdot \cong_G({\mathcal{Q}},e')$.
\subsection{Proof of Lemma~\ref{lem: multiway cut with paths sets}}
\label{apd: Proof of multiway cut with paths sets}
We use the following claim.
\begin{claim}
\label{lem: cut_uncrossing}
Let $G$ be a graph, $S$ a subset of vertices in $G$, and $x,y\in S$ two distinct vertices. Assume that $(A,B)$ is a minimum cut separating $x$ from $S\setminus\set{x}$ with $x\in A$, and $(A',B')$ is a minimum cut separating $y$ from $S\setminus \set{y}$ with $y\in A'$. Consider another cut $(\hat A,\hat B)$, where $\hat A=A\setminus A'$, and $\hat B=V(G)\setminus \hat A$. Then $(\hat A,\hat B)$ is a minimum cut separating $x$ from $S\setminus \set{x}$ in $G$.
\end{claim}
\begin{proof}
Since $(A',B')$ is a cut separating $y$ from $S\setminus\set{y}$ with $y\in A'$, we get that $A'\cap S=\set{y}$. Similarly, $A\cap S=\set{x}$. Therefore, $\hat A\cap S=\set{x}$, and so $(\hat A,\hat B)$ is indeed a cut separating $x$ from $S\setminus\set{x}$. It now remains to show that $|E(\hat A,\hat B)|\leq |E(A,B)|$.
Denote $\hat A'=A'\setminus A$, and $\hat B'=V(G)\setminus \hat A'$. Using the same argument as above, $(\hat A',\hat B')$ is a cut separating $y$ from $S\setminus \set{y}$.
From submodularity of cuts, for any pair $X,Y$ of vertex subsets in a graph $G$, $|\delta_G(X)|+|\delta_G(Y)|\geq |\delta_G(X\setminus Y)|+|\delta_G(Y\setminus X)|$.
Therefore:
$$|\delta_G(A)|+|\delta_G(A')|\geq |\delta_G(A\setminus A')|+ |\delta_G(A'\setminus A)|=|\delta_G(\hat A)|+|\delta_G(\hat A')|.$$
Notice however that $(A,B)$ is a minimum cut separating $x$ from $S\setminus \set{x}$, so $|\delta_G(A)|=|E(A,B)|\leq |E(\hat A,\hat B)|=|\delta_G(\hat A)|$. Similarly, since $(A',B')$ is a minimum cut separating $y$ from $S\setminus \set{y}$, we get that $|\delta_G(A')|=|E(A',B')|\leq |E(\hat A',\hat B')|=|\delta_G(\hat A')|$. We conclude that $|\delta_G(A)|+|\delta_G(A')|=|\delta_G(\hat A)|+|\delta_G(\hat A')|$ must hold. If we assume for contradiction that $(\hat A,\hat B)$ is not a minimum cut separating $x$ from $S\setminus\set{x}$, then $|\delta_G(A)|<|\delta_G(\hat A)|$ must hold, and so $|\delta_G(A')|>|\delta_G(\hat A')|$, a contradiction to the minimality of the cut $(A',B')$. We conclude that $(\hat A,\hat B)$ is a minimum cut separating $x$ from $S\setminus\set{x}$.
\end{proof}
\iffalse
If $U_i\cap U_j=\emptyset$ then there is nothing to prove. We now assume that $U_i\cap U_j\ne\emptyset$.
We denote $A=U_i\setminus U_j$, $B=U_j\setminus U_i$, $C=U_i\cap U_j$, and $D=\overline{U_i\cup U_j}$.
Note that $s_i\in U_i$ and $s_i\notin U_j$, so $A\ne \emptyset$, and similarly $B\ne \emptyset$.
Note that the cut $(A,B\cup C\cup D)$ is also a cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$.
Since $(U_i,\overline{U_i})$ is a minimum cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$,
\[
|E(A,B)|+|E(A,D)|+|E(C,B)|+|E(C,D)|\le |E(A,B)|+|E(A,C)|+|E(A,D)|.
\]
We get that $|E(C,B)|+|E(C,D)|\le |E(A,C)|$.
Similarly, note that the cut $(B,A\cup C\cup D)$ is also a cut in $G$ separating $\set{s_j}$ from $S\setminus \set{s_j}$, while $(U_j,\overline{U_j})$ is a minimum cut in $G$ separating $\set{s_j}$ from $S\setminus \set{s_j}$. Therefore,
\[
|E(A,B)|+|E(D,B)|+|E(A,C)|+|E(D,C)|\le |E(B,A)|+|E(B,C)|+|E(B,D)|.
\]
We get that $|E(A,C)|+|E(C,D)|\le |E(B,C)|$.
Altogether, $|E(C,D)|=0$ and $|E(A,C)|=|E(B,C)|$. Therefore,
$|E(A,B\cup C\cup D)|=|E(U_i,\overline{U_i})|$ and
$|E(B,A\cup C\cup D)|=|E(U_j,\overline{U_j})|$, which means that
$(U_i\setminus U_j, \overline{U_i\setminus U_j})$ is a minimum cut in $G$ separating $\set{s_i}$ from $S\setminus \set{s_i}$, and $(U_j\setminus U_i, \overline{U_j\setminus U_i})$ is a minimum cut in $G$ separating $\set{s_j}$ from $S\setminus \set{s_j}$.
\end{proof}
\fi
We now complete the proof of Lemma~\ref{lem: multiway cut with paths sets} using Claim~\ref{lem: cut_uncrossing}.
Recall that we are given a set $S=\set{s_1,\ldots,s_k}$ of vertices of graph $G$. We first compute, for all $1\le i\le k$, a minimum cut separating $\set{s_i}$ from $S\setminus \set{s_i}$ in $G$, that we denote by $(U_i,\overline{U_i})$, with $s_i\in U_i$. For each $1\leq i\leq k$, we then let $A_i=U_i\setminus (\bigcup_{1\le j\le k, j\ne i}U_j)$. Clearly, for all $1\leq i<j\leq k$, $A_i\cap A_j=\emptyset$.
Consider now some index $1\leq i\leq k$. We claim that $(A_i,V(G)\setminus A_i)$ is a minimum cut separating $s_i$ from $S\setminus \set{s_i}$ in graph $G$. For convenience, assume that $i=k$ (the other cases are symmetric). For all $1\leq j<k$, let $Z_j=U_k\setminus (U_1\cup U_2\cup\cdots\cup U_j)$, so that $Z_{k-1}=A_k$. Set $Z_0=U_k$. By applying
\Cref{lem: cut_uncrossing} to each of the sets $Z_0,\ldots,Z_{k-1}$ in turn, and using the fact that, for all $1\leq j\leq k-1$, $Z_j=Z_{j-1}\setminus U_j$,
we get that, for all $0\leq j\leq k-1$, $(Z_j,V(G)\setminus Z_j)$ is a minimum cut separating $s_k$ from $S\setminus\set{s_k}$ in $G$.
It remains to compute, for each $1\le i\le k$, a set ${\mathcal{Q}}_i$ of paths routing the edges of $\delta(A_i)$ to $s_i$. Fix an index $1\leq i\leq k$. We construct a flow network as follows. Let $H_i$ be the graph obtained from $G[A_i]\cup \delta_G(A_i)$, by contracting all vertices that do not belong to $A_i$ into a single vertex, that we denote by $t_i$. We set the capacity of every edge in $H_i$ to be $1$, and compute a maximum $s_i$-$t_i$ flow in the resulting network. From the max-flow / min-cut theorem, the value of the resulting flow must be $|\delta_G(A_j)|$, and from the integrality of flow we can ensure that the resulting flow is integral. We can then use this flow to obtain a set ${\mathcal{Q}}_i=\set{Q_i(e)\mid e\in \delta_G(e)}$ of edge-disjoint paths, where, for all $e\in \delta_G(e)$, path $Q_i(e)$ has $e$ as its first edge, $s_i$ as its last vertex, and all inner vertices of $Q_i(e)$ are contained in $A_i$.
\subsection{Proof of \Cref{cor: approx_balanced_cut}}
\label{apd: Proof of approx_balanced_cut}
Let $0< \eta <1$ be some parameter.
In order to avoid confusion, throughout this proof, we will refer to $\eta$-balanced cuts as $\eta$-edge-balanced cuts. We now define the notion of $\eta$-vertex-balanced cuts, that will be used in this proof.
We say that a cut $(A,B)$ in a graph $G$ is \emph{$\eta$-vertex-balanced} if $|A|,|B|\leq \eta\cdot |V(G)|$. We say that a cut $(A,B)$ is a \emph{minimum $\eta$-vertex-balanced cut in $G$} if $(A,B)$ is an $\eta$-vertex-balanced cut of minimum value $|E(A,B)|$. We need the following theorem.
\begin{theorem}[Corollary 2 in~\cite{ARV}]
\label{thm: ARV}
For every constant $1/2<\eta<1$, there is another constant $\eta<\eta'<1$, and an efficient algorithm, that, given any \textbf{simple} connected graph $G$ with $n$ vertices, computes an $\eta'$-vertex-balanced cut $(A,B)$ in $G$, whose value $|E(A,B)|$ is at most $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)$ times the value of a minimum $\eta$-vertex-balanced cut of $G$.
\end{theorem}
We now turn to prove \Cref{cor: approx_balanced_cut}.
Let $G$ be the input graph, with $|E(G)|=m$.
For every vertex $v\in V(G)$, we denote by $d_v=\deg_G(v)$ the degree of $v$ in $G$. For each such vertex $v\in V(G)$, we denote $\delta_G(v)=\set{e_1(v),\ldots,e_{d_v}(v)}$, where the edges are indexed arbitrarily, and we let $K_v$ be a complete graph on $d_v$ vertices. We denote $V(K_v)=\set{x_1(v),\ldots,x_{d_v}(v)}$.
We construct a new graph $H$ as follows. First, we let $H$ be a disjoint union of graphs $K_v$, for all $v\in V(G)$. We call all edges in $\bigcup_{v\in V(G)}E(K_v)$ \emph{internal edges}. Next, we consider the edges of the graph $G$ one by one. Consider any such edge $e=(v,v')$, and assume that $e=e_i(v)=e_j(v')$. In other words, $e$ is the $i$th edge incident to $v$ and the $j$th edge incident to $v'$. We add the edge $e'=(x_i(v),x_j(v'))$ to graph $H$, and we view this edge as the \emph{copy of the edge $e$}. We call the resulting set $\set{e'\mid e\in E(H)}$ of edges \emph{external edges of $H$}. This completes the definition of the graph $H$. Note that $|V(H)|=\sum_{v\in V(G)}d_v=2m$, and every vertex of $H$ is incident to exactly one external edge.
Consider now any cut $(A',B')$ in graph $H$. We say that cut $(A',B')$ is \emph{canonical} if, for every vertex $v\in V(G)$, either $V(K_v)\subseteq A'$, or $V(K_v)\subseteq B'$.
Let $(X,Y)$ be a minimum $\hat \eta$-edge-balanced cut in graph $G$, and let $\rho=|E_G(X,Y)|$ denote its value. We start with the following observation.
\begin{observation}\label{small balanced cut in G}
There is an $\eta_1$-vertex-balanced cut in graph $H$ of value at most $\rho$, for $\eta_1=\frac{1+\hat \eta}{2}$.
\end{observation}
\begin{proof}
We construct a cut $(X',Y')$ in graph $H$ using the cut $(X,Y)$ in $G$, as follows. We start with $X',Y'=\emptyset$. For every vertex $v\in V(H)$, if $v\in X$, then we add all vertices of $K_v$ to $X'$, and otherwise we add them to $Y'$. It is immediate to verify that the value of the resulting cut $(X',Y')$ is $|E_H(X',Y')|=|E_G(X,Y)|=\rho$.
We now show that cut $(X',Y')$ is $\eta_1$-vertex-balanced. In order to do so, it is enough to show that $|X'|,|Y'|\leq \eta_1\cdot |V(H)|$. We show that $|X'|\leq \eta_1\cdot |V(H)|$. The proof for $Y'$ is symmetric.
Indeed:
%
\[|X'|=\sum_{v\in X}|V(K_v)|=\sum_{v\in X}d_v=2|E_G(X)|+|E_G(X,Y)|\leq |E_G(X)|+m\leq \hat \eta m+m\leq \frac{1+\hat \eta}{2}\cdot |V(H)|,
\]
%
which is bounded by $\eta_1|V(H)|$ (we have used the fact that $|V(H)|=2m$).
\end{proof}
We can now use the algorithm from \Cref{thm: ARV} to compute an $\eta_2$-vertex-balanced cut $(X',Y')$ in graph $H$, whose value is at most $\rho'=\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(2m)\cdot \rho$. Here, $\eta_1<\eta_2<1$ is some constant. Note that, if cut $(X',Y')$ were canonical, we could immediately obtain a corresponding cut $(A,B)$ in graph $G$, whose value is at most $\rho'$, with the guarantee that $(A,B)$ is a $\hat \eta'$-edge-balanced cut, for some constant $\hat \eta'$. We use the following observation in order to convert the cut into a canonical one.
\begin{observation}\label{obs: make canonical}
There is an efficient algorithm, that, given an $\eta'$-vertex-balanced cut $(X',Y')$ in graph $H$ of value $\rho'$, for some $0<\eta'<1$, computes a canonical $\eta^*$-vertex-balanced cut $(X^*,Y^*)$ in graph $H$ of value $\rho^*\leq O(\rho')$, for $\eta^*=\max\set{\frac{1+\eta'}2,0.95}$.
\end{observation}
\begin{proof}
For every vertex $v\in V(G)$, we denote $X_v=X'\cap V(K_v)$ and $Y_v=Y'\cap V(K_v)$. Notice that graph $K_v$ contributes $|X_v|\cdot |Y_v|$ edges to the cut $(X',Y')$. We say that vertex $v$ is \emph{indecisive} if $|X_v|,|Y_v|\geq \frac{1-\eta'}{2}\cdot d_v$, and we say that it is \emph{decisive} otherwise.
We modify the cut $(X',Y')$ in two steps. In the first step, we construct a new cut $(X'',Y'')$ in graph $H$ as follows. We start from $(X'',Y'')=(X',Y')$. We then consider every decisive vertex $v\in V(G)$ one by one. Consider any such vertex $v$, and recall that either $|X_v|<\frac{1-\eta'}{2}\cdot d_v$ holds, or $|Y_v|<\frac{1-\eta'}{2}\cdot d_v$. In the former case, we move the vertices of $X_v$ to $Y''$, while in the latter case we move the vertices of $Y_v$ to $X''$. Notice that $|X_v|\cdot |Y_v|$ edges of $K_v$ lie in the cut $(X',Y')$. At the end of the current iteration, no internal edges of $K_v$ contribute to the cut $(X'',Y'')$, but we may have added new external edges to the cut: if vertices of $X_v$ were moved to $Y''$, then we may have added up to $|X_v|$ such new edges (edges incident to vertices of $X_v$), and otherwise we may have added up to $|Y_v|$ such new edges. In either case, it is easy to see that $|E(X'',Y'')|$ may not grow as the result of the current iteration.
The first step terminates once every decisive vertex of $G$ is processed. Notice that the total number of new vertices that we may have added to set $X''$ over the course of this step is at most:
%
\[ \frac{1-\eta'}{2}\cdot\sum_{v\in V(G)}d_v\leq (1-\eta') m.\]
%
Since we are guaranteed that $|X'|\leq \eta'\cdot (2m)$, we get that, at the end of the current step, $|X''|\leq \eta'\cdot (2m)+ (1-\eta') m\leq \frac{1+\eta'}{2}\cdot (2m)$ holds. Similarly, $|Y''|\leq \frac{1+\eta'}{2}\cdot (2m)$. We conclude that $(X'',Y'')$ is an $\eta''$-vertex-balanced cut in $H$, of value at most $\rho'$, where $\eta''=\frac{1+\eta'}{2}$.
In the second step, we construct the final cut $(X^*,Y^*)$ in $H$ by taking care of indecisive vertices. Assume first that there is some indecisive vertex $v\in V(H)$ with $d_v\geq m/10$. Notice that, in this case, the number of edges that graph $K_v$ contributes to cut $(X',Y')$ is at least $|X_v|\cdot |Y_v|\geq \frac{1-\eta'}{4}\cdot (d_v)^2\geq \frac{1-\eta'}{400}\cdot m^2$. Therefore, $\rho'>\frac{1-\eta'}{400}\cdot m^2$ must hold. Consider now a new cut $(X^*,Y^*)$ in graph $H$, where $X^*=V(K_v)$ and $Y^*=V(H)\setminus X^*$. Notice that $|X^*|=d_v\leq m$ and $|Y^*|\leq 2m-|X^*|\leq 2m\cdot 0.95$ holds. Therefore, cut $(X^*,Y^*)$ is $0.95$-vertex-balanced. Additionally, $|E_H(X^*,Y^*)|\leq d_v\leq m\leq O(\rho')$. We then return the cut $(X^*,Y^*)$ as the outcome of the algorithm. We assume from now on that for every indecisive vertex $v\in V(H)$, $d_v< m/10$.
We start with $(X^*,Y^*)=(X'',Y'')$, and then process every indecisive vertex $v\in V(G)$ one by one. Consider an iteration when vertex $v$ is processed. Recall that graph $K_v$ contributes at least $|X_v|\cdot |Y_v|\geq \max\set{|X_v|,|Y_v|}$ edges to the cut $(X^*,Y^*)$. If $|X^*|<|Y^*|$, then we move the vertices of $Y_v$ from $Y^*$ to $X^*$. Notice that, after this transformation, the inner edges of $K_v$ no longer contribute to the cut, and at most $|Y_v|$ new outer edges are added to the cut. Therefore, the value of the cut does not increase. Otherwise, $|X^*|\geq |Y^*|$, and we move the vertices of $X_v$ from $X^*$ to $Y^*$. Using the same argument as before, the value of the cut does not increase. This completes the description of an iteration. Consider the cut $(X^*,Y^*)$ that is obtained at the end of the algorithm, after all indecisive vertices are processed.
We now show that $|X^*|,|Y^*|\leq \eta^*\cdot (2m)$. We prove this for $X^*$, and the proof for $Y^*$ is symmetric. We consider two cases. First, if no new vertices were added to $X^*$ over the course of the second step, then $|X^*|\leq |X''|\leq \frac{1+\eta'}{2}\cdot (2m)$. Assume now that some vertices were added to $X^*$, and let $v$ be the last indecisive vertex of $G$, for which the vertices of $K_v$ were added to $X^*$. Then before vertex $v$ was processed, $|X^*|\leq |Y^*|$ held. Since $d_v\leq m/10$ from our assumption, at the end of the iteration when $v$ was processed, $|X^*|\leq 1.1m$ held. Since no new vertices were added to $X^*$ in subsequent iterations, we get that $|X^*|\leq 1.1m\leq \eta^*\cdot (2m)$ holds at the end of the algorithm. We conclude that $(X^*,Y^*)$ is an $\eta^*$-balanced cut, of value at most $O(\rho')$.
\end{proof}
By applying the algorithm from \Cref{obs: make canonical} to the $\eta'$-vertex-balanced cut $(X',Y')$ in graph $H$, we obtain a canonical $\eta^*$-vertex-balanced cut $(X^*,Y^*)$ in graph $H$, with $\eta^*=\max\set{\frac{1+\eta'}2,0.95}$, whose value is $\rho^*\leq O(\rho')=O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))\cdot \rho$. We use this cut in order to construct a cut $(A,B)$ in $G$ as follows: for every vertex $v\in V(G)$, if $V(K_v)\subseteq X^*$, then vertex $v$ is added to $A$, and otherwise it is added to $B$. Notice that $|E_G(A)|\leq \sum_{v\in A}d_v/2\leq |X^*|/2\leq \eta^*\cdot m$. Similarly, $|E_G(B)|\leq \eta^*\cdot m$. Therefore, cut $(A,B)$ is $\eta^*$-edge-balanced. Additionally, $|E_G(A,B)|\leq |E_H(A^*,B^*)|\leq O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))\cdot \rho$.
\subsection{Proof of Theorem~\ref{lem:min_bal_cut}}
\label{apd: Proof of min_bal_cut}
We use the following theorem from~\cite{lipton1979separator}.
\begin{theorem}[Theorem 4 from~\cite{lipton1979separator}]
\label{thm: weighted_planar_separator}
Let $G$ be any \textbf{simple} $n$-vertex planar graph with weights $w_v\geq 0$ on its vertices $v\in V(G)$, such that $\sum_{v\in V(G)}w_v\leq 1$. Then there is a partition $(A,B,C)$ of $V(G)$, such that no edge connects a vertex of $A$ to a vertex in $B$; $\sum_{v\in A}w_v,\sum_{v\in B}w_v\leq 2/3$; and $|C|\leq \sqrt{8n}$.
\end{theorem}
In order to prove \Cref{lem:min_bal_cut}, we define a new simple planar graph $G'$ that is obtained by modifying graph $G$, using its optimal drawing. We then apply \Cref{thm: weighted_planar_separator} to graph $G'$, and transform the resulting partition $(A,B,C)$ of $V(G')$ into a $(3/4)$-edge-balanced cut of graph $G$, whose value is at most $O(\sqrt{\mathsf{OPT}_{\mathsf{cr}}(G)+\Delta\cdot m})$.
In order to define graph $G'$, we first define an intermediate graph $G_1$. Consider the input graph $G$ and its optimal drawing $\phi$ in the plane. For every vertex $v\in V(G)$, we denote $d_v=\deg_G(v)$, and we denote $\delta_G(v)=\set{e_1(v),\ldots,e_{d_v}(v)}$, where the edges are indexed according to the order in which their images enter the image of $v$ in $\phi$, in the counter-clock-wise direction. We let $H_v$ be the $(d_v\times d_v)$-grid, and we denote the set of the vertices on the first row of the grid by $X(v)=\set{x_1(v),\ldots,x_{d_v}(v)}$, where the vertices are indexed in their natural order. In order to define graph $G_1$, we start with the disjoint union of all grids in $\set{H_v}_{v\in V(G)}$. We refer to the edges that lie in these grids as \emph{internal edges}. Next, we process every edge $e\in E(G)$ one by one. Consider any such edge $e=(v,v')$, and assume that $e=e_i(v)=e_j(v')$, that is, $e$ is the $i$th edge incident to $v$ and the $j$th edge incident to $v'$. We then add an edge $e'=(x_i(v),x_j(v'))$ to graph $G_1$. We think of edge $e'$ as the \emph{copy} of the edge $e$ in $G_1$. The edges in set $\set{e'\mid e\in E(G)}$ are called \emph{external edges} of graph $G_1$. We say that a cut $(A,B)$ in graph $G_1$ is \emph{canonical} if, for every vertex $v\in V(G)$, either $V(H_v)\subseteq A$, or $V(H_v)\subseteq B$ holds. Note that a canonical cut $(A,B)$ in graph $G_1$ naturally defines a cut $(A',B')$ of the same value on graph $G$, where a vertex $v\in V(G)$ is added to $A'$ if $V(H_v)\subseteq A$, and it is added to $B'$ otherwise. Lastly, note that the optimal drawing $\phi$ of $G$ defines a drawing $\phi_1$ of $G_1$ with the same number of crossings. In order to obtain drawing $\phi_1$ of $G_1$, we start with the drawing $\phi$ of $G$, and then inflate the image of every vertex $v\in V(G)$, so that it becomes a disc $D(v)$. We place another smaller disc $D'(v)$ inside $D(v)$, so that the boundaries of both discs are disjoint. We then place the standard drawing of the grid $H_v$ inside disc $D'(v)$, so that vertices $x_1(v),\ldots,x_{d_v}(v)$ appear on the boundary of the disc $D'(v)$ in this counter-clock-wise order. By slightly extending the images of the edges $e_1(v),\ldots,e_{d_v}(v)$ inside $D(v)\setminus D'(v)$, we can ensure that the image of each such edge $e_i(v)$ terminates at the image of the vertex $x_i(v)$. Once all vertices of $V(G)$ are processed in this manner, we obtain a drawing $\phi_1$ of graph $G_1$, in which the number of crossings is bounded by $\mathsf{cr}(\phi)=\mathsf{OPT}_{\mathsf{cr}}(G)$.
In order to obtain the final graph $G'$, we start with $G'=G_1$, and we denote $V(G_1)=X$. Next, for every {\bf external} edge $e'\in E(G_1)$, we subdivide the edge with a new vertex $u_{e'}$. In other words, if $e'=(x_i(v),x_j(v'))$, then we replace the edge with a path consisting of two edges: $(x_i(v),u_{e'})$, and $(u_{e'},x_j(v'))$. We denote this new set of vertices representing the external edges of $G_1$ by $U=\set{u_{e'}\mid e\in E(G)}$. Note that drawing $\phi_1$ of graph $G_1$ can be easily transformed into a drawing of this new graph, without increasing the number of crossings. Denote the resulting drawing by $\phi_2$. In our last step, for every crossing point $p$ between a pair $a,a'$ of edges in drawing $\phi_2$, we replace point $p$ with a new vertex $y_p$. In other words, if $a=(s,t)$ and $a'=(s',t')$, then we add a new vertex $y_p$ to the graph. We then replace edge $a=(s,t)$ with two new edges, $(s,y_p)$ and $(y_p,t)$, and we similarly replace edge $a'$ with two new edges, $(s',y_p)$ and $(y_p,t')$. We continue processing every crossing point in drawing $\phi_2$ one by one in this manner, until no more crossings remain. We denote this new set of vertices, that represent all crossing points in the original drawing $\phi_2$, by $Y$. Note that $|Y|=\mathsf{OPT}_{\mathsf{cr}}(G)$. This completes the definition of the graph $G'$. Observe that $V(G')=X\cup Y\cup U$, and so:
\[|V(G')|=\sum_{v\in V(G)}(d_v)^2+m+\mathsf{OPT}_{\mathsf{cr}}(G)\leq \Delta\cdot \sum_{v\in V(G)}d_v+m+\mathsf{OPT}_{\mathsf{cr}}(G)\leq 3\Delta m+\mathsf{OPT}_{\mathsf{cr}}(G). \]
It is also immediate to verify that $G'$ is a simple planar graph, and that the maximum vertex degree in $G'$ is at most $4$. We now assign weights $w_v$ to vertices $v\in V(G')$, as follows: every vertex $u_{e'}\in U$ is assigned weight $1/m$, and all other vertices are assigned weight $0$. It is immediate to verify that $\sum_{v\in V(G')}w_v=1$.
From \Cref{thm: weighted_planar_separator}, there is a partition $(A,B,C)$ of $V(G')$, such that no edge connects a vertex of $A$ to a vertex in $B$; $\sum_{v\in A}w_v,\sum_{v\in B}w_v\leq 2/3$; and $|C|\leq \sqrt{8|V(G')|}\leq \sqrt{24\Delta m+8\mathsf{OPT}_{\mathsf{cr}}(G)}$.
We convert this partition of vertices of $G'$ into a $(3/4)$-balanced cut in graph $G$ in three steps. In the first step, we use the partition $(A,B,C)$ of $V(G')$ in order to construct a cut $(A_1,B_1)$ in graph $G_1$. In the second step, we transform this cut into a canonical cut $(A_2,B_2)$ in graph $G_1$. Lastly, in the third step, we use this canonical cut in order to define the final cut $(A^*,B^*)$ in graph $G$.
We now describe each of the steps in turn.
\paragraph{Step 1: Cut in Graph $G_1$.}
We define a cut $(A_1,B_1)$ in graph $G_1$ as follows.
We start with $A_1=B_1=\emptyset$, and then process every vertex $v\in V(G)$ one by one. When vertex $v\in V(G)$ is processed, we consider every vertex $x\in V(H_v)$. If $x\in A\cup C$, then we add $x$ to $A_1$, and otherwise we add $x$ to $B_1$.
Consider now the resulting cut $(A_1,B_1)$ in graph $G_1$. We first claim that $|E_{G_1}(A_1,B_1)|\leq 4|C|$.
In order to prove this, we assign, to every edge $e\in E_{G_1}(A_1,B_1)$, some vertex $x\in C$ that is \emph{responsible} for $e$, and we will ensure that every vertex of $C$ is responsible for at most $4$ edges of $ E_{G_1}(A_1,B_1)$.
Consider some edge $e\in E_{G_1}(A_1,B_1)$.
If either of the endpoints of $e$ lies in set $C$, then we assign $e$ to that endpoint. Otherwise, there must be some vertex $x$ of graph $G'$ that subdivided the edge $e$ (so either $x\in U$ or $x\in Y$ holds), and $x\in C$. In this case, we assign $e$ to this vertex $x$. Since the degree of every vertex in $G'$ is at most $4$, every vertex of $C$ may be assigned to at most $4$ edges of $E_{G_1}(A_1,B_1)$, and so we conclude that $|E_{G_1}(A_1,B_1)|\leq 4|C|\leq 4\cdot \sqrt{24\Delta m+8\mathsf{OPT}_{\mathsf{cr}}(G)}$.
Next, we bound the total number of external edges in $E_{G_1}(A_1)$ and in $E_{G_1}(B_1)$. For convenience, denote by $E'$ the set of all external edges in graph $G_1$. Consider some external edge $e'=(x_i(v ),x_j(v'))\in E'$. Let $P(e')$ be the path that replaced the edge $e'$ in graph $G'$. Recall that path $P(e')$ is a path connecting $x_i(v)$ to $x_j(v')$, it contains the vertex $u_{e'}$ representing the edge $e'$, and possibly additional vertices representing the crossing points of edge $e'$ with other edges.
We claim that either vertex $y_{e'}$ lies in $A$, or some vertex of $P(e')$ (including possibly $x_i(v)$ or $x_j(v')$) must lie in $C$. Indeed, assume that $y_{e'}\not \in A$. If none of the vertices of $P(e')$ lie in $C$, then $y_{e'}\in B$, while $x_i(v),x_j(v')\in A$ must hold. This is impossible since there are no edges connecting vertices of $A$ to vertices of $B$. Therefore, either $y_{e'}\in A$, or at least one vertex on $P(e')$ lies in $C$. Since every vertex of $G'$ has degree at most $4$, every vertex of $C$ may lie on at most $4$ paths in $\set{P(e')\mid e\in E(G)}$. Since the weight of every vertex in $U$ is $1/m$, we get that:
\[|E'\cap E_{G_1}(A_1)|\leq |U\cap A|+4|C|\leq m\cdot \sum_{v\in A}w_v+4|C|\leq 2m/3+ 4\cdot \sqrt{24\Delta m+8\mathsf{OPT}_{\mathsf{cr}}(G)}.\]
Using similar reasoning, $|E'\cap E_{G_1}(B_1)|\leq 2m/3+ 4\cdot \sqrt{24\Delta m+8\mathsf{OPT}_{\mathsf{cr}}(G)}$.
\paragraph{Step 2: Canonical Cut in Graph $G_1$.}
In this step we construct a cut $(A_2,B_2)$ in graph $G_1$ that is canonical, by gradually modifying the cut $(A_1,B_1)$. For every vertex $v\in V(G)$, we denote $n_A(v)=|X(v)\cap A_1|$, and $n_B(v)=|X(v)\cap B_1|$. We use the following simple observation.
\begin{observation}\label{obs: cut in grid}
For every vertex $v\in V(G)$, $|E(H_v)\cap E_{G_1}(A_1,B_1)|\geq\min\set{n_A(v),n_B(v)}$.
\end{observation}
\begin{proof}
We partition the columns of the grid $H_v$ into two subsets, ${\mathcal{W}}',{\mathcal{W}}''$, as follows. For $1\leq i\leq d_v$, the $i$th column of the grid is added to set ${\mathcal{W}}'$ if vertex $x_i(v)$ (the vertex of the $i$th column that lies on the first row of the grid) lies in $A_1$. Otherwise, the $i$th column is added to ${\mathcal{W}}''$.
We now consider three cases. The first case happens if, for every row $R$ of the grid $H_v$, at least one edge of $R$ lies in $E_{G_1}(A_1,B_1)$. Clearly, in this case, $|E(H_v)\cap E_{G_1}(A_1,B_1)|\geq d_v\geq \min\set{n_A(v),n_B(v)}$. The second case happens if, for every column $W\in {\mathcal{W}}'$, at least one edge of $W$ lies in $E_{G_1}(A_1,B_1)$. In this case, $|E(H_v)\cap E_{G_1}(A_1,B_1)|\geq n_A(v)\geq \min\set{n_A(v),n_B(v)}$. Lastly, the third case happens if, for every column $W\in {\mathcal{W}}''$, at least one edge of $W$ lies in $E_{G_1}(A_1,B_1)$. In this case, $|E(H_v)\cap E_{G_1}(A_1,B_1)|\geq n_B(v)\geq \min\set{n_A(v),n_B(v)}$.
We now claim that at least one of the above three cases has to happen. Indeed, assume otherwise. Then there is some row $R$ of the grid, and two columns $W\in {\mathcal{W}}'$, $W'\in {\mathcal{W}}''$, such that no edge of $E(R)\cup E(W')\cup E(W'')$ lies in $E_{G_1}(A_1,B_1)$. Assume that $W'$ is the $i$th column and $W''$ is the $j$th column of the grid $H_v$. Since $x_i(v)\in A_1$, $x_j(v)\in B_1$, and $R\cup W'\cup W''$ is a connected graph, this is impossible.
\end{proof}
We say that a vertex $v\in V(G)$ is \emph{indecisive} iff $n_A,n_B\geq d_v/32$; otherwise we say that vertex $v$ is \emph{decisive}. We start with $(A_2,B_2)=(A_1,B_1)$, and then gradually modify this cut, by processing the decisive vertices one by one. When such a vertex $v$ is processed, if $n_A<d_v/32$, then we move the vertices of $V(H_v)\cap A_2$ from $A_2$ to $B_2$. Notice that this transformation adds up to $n_A$ new external edges to cut $E_{G_1}(A_2,B_2)$ -- the edges incident to the vertices of $X(v)\cap A_2$. However, from \Cref{obs: cut in grid}, at least $n_A=|X(v)\cap A_2|$ edges of $H_v$ contributed to the cut $E_{G_1}(A_2,B_2)$ before this transformation, and they no longer contribute to the cut after the transformation. Therefore, $|E_{G_1}(A_2,B_2)|$ does not increase as the result of this transformation. If $n_A\geq d_v/32$, then $n_B<d_v/32$ must hold, and we move the vertices of $V(H_v)\cap B_2$ from $B_2$ to $A_2$. Using the same arguments as before, $|E_{G_1}(A_2,B_2)|$ does not increase.
Consider the cut $(A_2,B_2)$ of $G_1$ that is obtained after all decisive vertices are processed. From the above discussion, $|E_{G_1}(A_2,B_2)|\leq |E_{G_1}(A_1,B_1)|\leq 4\cdot \sqrt{24\Delta m+8\mathsf{OPT}_{\mathsf{cr}}(G)}$. Moreover, the total number of
new edges of $E'$ that were added to $E_{G_1}(A_2)$ is bounded by $\sum_{v\in V(G)}d_v/32\leq m/16$.
Since $|E'\cap E_{G_1}(A_1)|\leq 2m/3+ 4\cdot \sqrt{24\Delta m+8\mathsf{OPT}_{\mathsf{cr}}(G)}$, if $\mathsf{OPT}_{\mathsf{cr}}(G)\le m^2/2^{40}$ and $\Delta\leq m/2^{40}$, then
\[|E'\cap E_{G_1}(A_2)|\leq m/16+2m/3+ 4\cdot \sqrt{24\Delta m+8\mathsf{OPT}_{\mathsf{cr}}(G)}\leq 3m/4.\]
Using the same reasoning, if $\mathsf{OPT}_{\mathsf{cr}}(G)\le m^2/2^{40}$, then $|E'\cap E_{G_1}(B_2)|\leq 3m/4$.
Next, we construct a canonical cut $(A_3,B_3)$ in graph $G_1$, by starting with $(A_3,B_3)=(A_2,B_2)$, and processing every indecisive vertex $v\in V(G)$ one by one. When vertex $v$ is processed, if $|E'\cap E_{G_1}(A_3)|\leq |E'\cap E_{G_1}(B_3)|$, then we move all vertices of $H_v\cap B_3$ from $B_3$ to $A_3$, and otherwise we move all vertices of $H_v\cap A_3$ from $A_3$ to $B_3$. Note that in either case, the total number of external edges that are added to cut $(A_3,B_3)$ is bounded by $d_v$. From \Cref{obs: cut in grid}, the number of edges of $H_v$ that contributed to the cut $(A_3,B_3)$ before this transformation is at least $\min\set{n_A,n_B}$. Since vertex $v$ is indecisive, $n_A,n_B\geq d_v/32$. Once every indecisive vertex of $G$ is processed, we obtain the final cut $(A_3,B_3)$ in graph $G_1$, that must be canonical. From the above discussion:
\[|E_{G_1}(A_3,B_3)|\leq 32\cdot |E_{G_1}(A_2,B_2)|\leq 32|E_{G_1}(A_1,B_1)|
\leq O\textsf{left}(\sqrt{\Delta m+\mathsf{OPT}_{\mathsf{cr}}(G)}\textsf{right} ).\]
We claim that $|E'\cap E_{G_1}(A_3)|,|E'\cap E_{G_1}(B_3)|\leq 3m/4$. We prove this for $A_3$, as the proof for $B_3$ is symmetric. If no new vertices were added to set $A_3$, then $|E'\cap E_{G_1}(A_3)|\leq |E'\cap E_{G_1}(A_2)|\leq 3m/4$ holds. Assume now that new vertices were added to $A_3$, and let $v$ be the last indecisive vertex that was processed by the algorithm, for which vertices of $H_v$ were added to $A_3$. Then, before vertex $v$ was processed, $|E'\cap E_{G_1}(A_3)|\leq |E'\cap E_{G_1}(B_3)|$ held; therefore, $|E'\cap E_{G_1}(A_3)|\leq m/2$. Note that moving the vertices of $V(H_v)\cap B_3$ from $B_3$ to $A_3$ could have added at most $d_v\leq \Delta\leq m/2^{40}$ new edges to $E'\cap E_{G_1}(A_3)$, and so $|E'\cap E_{G_1}(A_3)|\leq 3m/4$ must hold at the end of this iteration. Since the iteration when $v$ was processed is the last iteration when vertices were added to $A_3$, we conclude that $|E'\cap E_{G_1}(A_3)|\leq 3m/4$ holds at the end of the algorithm. Using the same reasoning, $|E'\cap E_{G_1}(B_3)|\leq 3m/4$ holds as well.
\paragraph{Step 3: Balanced Cut in $G$.}
We are now ready to define the final cut $(A^*,B^*)$ in graph $G$. We add to $A^*$ every vertex $v\in V(G)$ with $V(H_v)\subseteq A_3$, and we add all remaining vertices of $V(G)$ to $B^*$. It is easy to verify that $|E_G(A^*,B^*)|=|E_{G_1}(A_3,B_3)|\leq O\textsf{left}(\sqrt{\Delta m+\mathsf{OPT}_{\mathsf{cr}}(G)}\textsf{right} )$. Additionally, $|E_G(A^*)|\leq |E'\cap E_{G_1}(A_3)|\leq 3m/4$, and similarly $|E_G(B^*)|\leq |E'\cap E_{G_1}(B_3)|\leq 3m/4$. We conclude that $(A^*,B^*)$ is a $(3/4)$-edge-balanced cut in graph $G$, whose value is at most $O(\sqrt{\mathsf{OPT}_{\mathsf{cr}}(G)+\Delta\cdot m})$.
\subsection{Proof of \Cref{obs: grid 1st row well-linked}}
\label{apd: Proof grid 1st row well-linked}
The proof is practically identical to the proof of \Cref{obs: cut in grid}. Consider a cut $(A,B)$ in graph $H$, and denote $n_A=|S\cap A|$ and $n_B=|S\cap B|$. It is enough to show that $|E(A,B)|\geq \min\set{n_A,n_B}$.
We partition the columns of the grid graph $H$ into two subsets, ${\mathcal{W}}',{\mathcal{W}}''$, as follows. For $1\leq i\leq r$, the $i$th column of the grid is added to set ${\mathcal{W}}'$ if the unique vertex of $S$ lying in the $i$th column belongs to $A$. Otherwise, the $i$th column is added to ${\mathcal{W}}''$.
We now consider three cases. The first case happens if, for every row $R$ of the grid $H$, at least one edge of $R$ lies in $E(A,B)$. Clearly, in this case, $|E(A,B)|\geq r\geq \min\set{n_A,n_B}$. The second case happens if, for every column $W\in {\mathcal{W}}'$, at least one edge of $W$ lies in $E(A,B)$. In this case, $|E(A,B)|\geq n_A\geq \min\set{n_A,n_B}$. Lastly, the third case happens if, for every column $W\in {\mathcal{W}}''$, at least one edge of $W$ lies in $E(A,B)$. In this case, $| E(A,B)|\geq n_B\geq \min\set{n_A,n_B}$.
We now claim that at least one of the above three cases has to happen. Indeed, assume otherwise. Then there is some row $R$ of the grid, and two columns $W\in {\mathcal{W}}'$, $W'\in {\mathcal{W}}''$, such that no edge of $E(R)\cup E(W')\cup E(W'')$ lies in $E(A,B)$. But the unique vertex of $S\cap V(W)$ lies in $A$, the unique vertex of $S\cap V(W')$ lies in $B$, and $R\cup W'\cup W''$ is a connected graph, a contradiction.
\subsection{Proof of Theorem~\ref{thm: bandwidth_means_boundary_well_linked}}
\label{apd: Proof of bandwidth_means_boundary_well_linked}
We construct an $s$-$t$ flow network, as follows. We start with the graph $G$, and then add a new source vertex $s$, that connects to every vertex in $T_1$ with an edge. We also add a new destination vertex $t$, and connect every vertex of $T_2$ to $t$ with an edge.
Denote the resulting graph by $H$. For every edge $e\in E(H)$, if $e$ is incident to $s$ or to $t$, then we set its capacity $c(e)=1$, and otherwise we set $c(e)=\ceil{1/\alpha}$.
Note that the capacity of every edge in the resulting flow network is integral.
We show below that the value of the maximum $s$-$t$ flow in the resulting flow network is $k=|T_1|$. From the integrality of flow, we can then compute an integral $s$-$t$ flow $f$ of value $k$ in $H$. Let ${\mathcal{P}}$ be the set of all $s$-$t$ paths in graph $H$. Since flow $f$ is integral, and since the capacity of every edge incident to $s$ and to $t$ is $1$, for every path $P\in {\mathcal{P}}$, $f(P)=0$ or $f(P)=1$ holds. Moreover, if ${\mathcal{P}}'\subseteq {\mathcal{P}}$ is the set of all paths $P$ with $f(P)=1$, then $|{\mathcal{P}}'|=k$.
Since the capacity of every edge in $\set{(s,x)\mid x\in T_1}$, and the capacity of every edge in $\set{(y,x)\mid y\in T_1}$ is $1$, each such edge belongs to exactly one path in ${\mathcal{P}}'$. Therefore,
set ${\mathcal{P}}'$ of paths naturally defines a one-to-one routing ${\mathcal{Q}}$ of vertices of $T_1$ to vertices of $T_2$ in graph $G$, with $\cong_G({\mathcal{Q}})\leq \ceil{1/\alpha}$. In order to complete the proof of the theorem, it is now enough to show that the value of the maximum $s$-$t$ flow in graph $H$ is at least $k$.
Assume for contradiction that this is not the case. Consider a minimum $s$-$t$ cut $(A,B)$ in graph $H$. From our assumption, the value of the cut is less than $k$. We partition the set $T_1$ of vertices into two subsets: set $T_1'=T_1\cap A$ and set $T_1''=T_1\cap B$. Note that, for every vertex $x\in T_1''$, its corresponding edge $(s,x)$ belongs to the cut $E(A,B)$. We denote by $E_1=\set{(s,x)\mid x\in T_1''}$ the corresponding set of edges. We also partition the set $T_2$ of vertices into two subsets: set $T_2'=T_2\cap B$ and set $T_2''=T_2\cap A$. Note that, for every vertex $y\in T_2''$, its corresponding edge $(y,t)$ belongs to the cut $E(A,B)$. We denote by $E_2=\set{(y,t)\mid y\in T_2''}$ the corresponding set of edges.
Lastly, we denote by $E'=E(A,B)\setminus (E_1\cup E_2)$ the set of the remaining edges in the cut $(A,B)$. Note that each edge in $E'$ has capacity $\ceil{1/\alpha}$, while each edge in $E_1\cup E_2$ has capacity $1$. Since we have assumed that the value of the cut $(A,B)$ is less than $k$, we get that:
\begin{equation}\label{eq: bound the cut}
|T_1''|+|T_2''|+|E'|\cdot \ceil{1/\alpha}= |E_1|+|E_2|+|E'|\cdot \ceil{1/\alpha}=\sum_{e\in E_H(A,B)}c(e)<k
\end{equation}
We define a cut $(A',B')$ in graph $G$ using cut $(A,B)$ as follows: $A'=A\setminus \set{s}$ and $B'=B\setminus \set{t}$. Notice that $|E_G(A',B')|=|E'|$. From \Cref{eq: bound the cut}, we then get that:
\[|E_G(A',B')|=|E'| < \alpha\cdot (k-|T_1''|-|T_2''|) \]
Since $|T_1'|=k-|T_1''|$ and $|T_2'|=k-|T_2''|$, we get that:
\[|E_G(A',B')| < \alpha\cdot \min\set{|T_1'|,|T_2'|}.\]
Lastly, since $T_1'\subseteq A'$ and $T_2'\subseteq B'$, we get that $|E_G(A',B')| < \alpha\cdot \min\set{|T\cap A'|,|T\cap B'|}$, contradicting the fact that the set $T$ of vertices is $\alpha$-well-linked in $G$.
\subsection{Proof of Theorem~\ref{thm:well_linked_decomposition}}
\label{apd: Proof of well_linked_decomposition}
Assume first that $0<\alpha\le 1/m$. Then we simply let ${\mathcal{R}}=\set{S}$. Since $\alpha\le 1/m$, and $S$ is a connected graph, it is easy to verify that it has the $\alpha$-bandwidth property in graph $G$. For each edge $e\in \delta_G(S)$, we simply let $P(e)$ be the path that contains the single edge $e$. It is easy to verify that cluster set ${\mathcal{R}}=\set{S}$, and the set ${\mathcal{P}}(S)=\set{P(e)\mid e\in \delta_G(S)}$ of paths have all required properties.
We assume from now on that $\frac 1 m <\alpha< \min\set{\frac 1 {64\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m},\frac 1 {48\log^2 m}}$ holds.
Our algorithm maintains a collection ${\mathcal{R}}$ of clusters of $S$, that is initialized to ${\mathcal{R}}=\set{S}$. Throughout the algorithm, we ensure that the following invariants hold:
\begin{properties}{I}
\item all clusters in ${\mathcal{R}}$ are mutually disjoint; \label{inv1: disjointness}
\item $\bigcup_{R\in {\mathcal{R}}}V(R)=V(S)$; and \label{inv2: partition}
\item for every cluster $R\in {\mathcal{R}}$, $|\delta_G(R)|\leq |\delta_G(S)|$. \label{inv3: small boundary}
\end{properties}
For a given collection ${\mathcal{R}}$ of clusters with the above properties, we define a \emph{budget} $b(e)$ for every edge $e\in E(G)$, as follows. If $e\in \delta_G(S)$, and the endpoint of $e$ that lies in $S$ belongs to a cluster $R\in{\mathcal{R}}$, then we set the budget $b(e)=1+8\alpha \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log_{3/2}(|\delta_G(R)|)$. If edge $e$ has its endpoints in two distinct clusters $R,R'\in {\mathcal{R}}$, then we set $b(e)=2+8\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log_{3/2}(|\delta_G(R)|)+8\alpha \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log_{3/2}(|\delta_G(R')|)$. Otherwise, we set $b(e)=0$.
Notice that, for every edge $e\in E(G)$, $b(e)\leq 3$ always holds. Additionally, for every edge $e\in \bigcup_{R\in {\mathcal{R}}}\delta_G(R)$, $b(e)\geq 2$ if the endpoints of $e$ lie in two different clusters of ${\mathcal{R}}$, and $b(e)\geq 1$ if $e\in \delta_G(S)$. Therefore, if we denote by $B=\sum_{e\in E(G)}b(e)$ the total budget in the system, then, throughout the algorithm, $B\geq \sum_{R\in {\mathcal{R}}}|\delta_G(R)|$ holds. Lastly, observe that, at the beginning of the algorithm, $B\leq |\delta_G(S)|\cdot (1+O(\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m))$. Throughout the algorithm, we will modify the clusters in set ${\mathcal{R}}$, leading to changes in the budgets of the edges of $G$. We will ensure however that the total budget $B$ never increases, and so, if ${\mathcal{R}}$ is the final set of clusters that we obtain, then $\sum_{R\in {\mathcal{R}}}|\delta_G(R)|\leq B\leq (1+O(\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m))=(1+O(\alpha\cdot \log^{1.5} m))$ holds.
Throughout the algorithm, we maintain a partition of the set ${\mathcal{R}}$ of clusters into two subsets: set ${\mathcal{R}}^A$ of \emph{active} clusters, and set ${\mathcal{R}}^I$ of \emph{inactive} clusters. We will ensure that the following additional invariant holds:
\begin{properties}[3]{I}
\item every cluster $R\in {\mathcal{R}}^I$ has the $\alpha$-bandwidth property.\label{inv: last - bw}
\end{properties}
Additionally, we will store, with every inactive cluster $R\in {\mathcal{R}}^I$, a set ${\mathcal{P}}(R)=\set{P(e)\mid e\in \delta_G(R)}$ of paths in graph $G$, (that we refer to as \emph{witness set of paths for $R$}), such that $\cong_G({\mathcal{P}}(R))\leq 100$, and, for every edge $e\in \delta_G(R)$, path $P(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, and all inner vertices of $P(e)$ lie in $V(S)\setminus V(R)$. At the beginning of the algorithm, we set ${\mathcal{R}}^A={\mathcal{R}}=\set{S}$ and ${\mathcal{R}}^I=\emptyset$. Clearly, all invariants hold at the beginning of the algorithm. We then proceed in iterations, as long as ${\mathcal{R}}^A\neq \emptyset$.
In order to execute an iteration, we select an arbitrary cluster $R\in {\mathcal{R}}^A$ to process. We will either establish that $R$ has the $\alpha$-bandwidth property in graph $G$ and compute a witness set ${\mathcal{P}}(R)$ of paths for $R$ (in which case $R$ is moved from ${\mathcal{R}}^A$ to ${\mathcal{R}}^I$); or we will modify the set ${\mathcal{R}}$ of clusters so that the total budget of all edges decreases by at least $1/m$. An iteration that processes a cluster $R\in {\mathcal{R}}^A$ consists of two steps. The purpose of the first step is to either establish the $\alpha$-bandwidth property of cluster $R$, or to replace it with a collection of smaller clusters in ${\mathcal{R}}$. The purpose of the second step is to either compute the witness set ${\mathcal{P}}(R)$ of paths for cluster $R$, or to modify the set ${\mathcal{R}}$ of clusters so that the total budget of all edges decreases. We now describe each of the two steps in turn.
\paragraph{Step 1: Bandwidth Property.}
Let $R\in {\mathcal{R}}^A$ be any active cluster, and
let $R^+$ be the augmentation of $R$ in graph $G$. Recall that $R^+$ is a graph that is obtained from $G$ through the following process. First, we subdivide every edge $e\in \delta_G(R)$ with a vertex $t_e$, and we let $T=\set{t_e\mid e\in \delta_G(R)}$ be the resulting set of vertices. We then let $R^+$ be the subgraph of the resulting graph induced by vertex set $V(R)\cup T$. We apply Algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace for computing approximate sparsest cut to graph $R^+$, with the set $T$ of vertices, to obtain a $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)$-approximate sparsest cut $(X,Y)$ in graph $R^+$ with respect to vertex set $T$.
We now consider two cases. The first case happens if $|E(X,Y)|\geq \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T|}$. In this case, we are guaranteed that the minimum sparsity of any $T$-cut in graph $R^+$ is at least $\alpha$, or equivalently, set $T$ of vertices is $\alpha$-well-linked in $R^+$. From \Cref{obs: wl-bw}, cluster $R$ has the $\alpha$-bandwidth property in graph $G$. In this case, we proceed to the second step of the algorithm.
Assume now that $|E(X,Y)|< \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T|}$.
Since $\alpha\leq \min\set{\frac 1 {64\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m},\frac 1 {48\log^2 m}}$, we get that the sparsity of the cut $(X,Y)$ is less than $1$. Consider now any vertex $t\in T$, and let $v$ be the unique neighbor of $t$ in $R^+$. We can assume w.l.o.g. that either $t,v$ both lie in $X$, or they both lie in $Y$. Indeed, if $t\in X$ and $v\in Y$, then moving vertex $t$ from $X$ to $Y$ does not increase the sparsity of the cut $(X,Y)$. This is because, for any two real numbers $1\leq a<b$, $\frac{a-1}{b-1}\leq \frac a b$. Similarly, if $t\in Y$ and $v\in X$, then moving $t$ from $Y$ to $X$ does not increase the sparsity of the cut $(X,Y)$. Therefore, we assume from now on, that for every vertex $t\in T$, if $v$ is the unique neighbor of $t$ in $R^+$, then either both $v,t\in X$, or both $v,t\in Y$.
Consider now the partition $(X',Y')$ of $V(R)$, where $X'=X\setminus T$ and $Y'=Y\setminus T$. It is easy to verify that $|\delta_G(R)\cap \delta_G(X')|=|X\cap T|$, and $|\delta_G(R)\cap \delta_G(Y')|=|Y\cap T|$. Let $E'=E_G(X',Y')$, and assume w.l.o.g. that $|\delta_G(R)\cap \delta_G(X')|\leq |\delta_G(R)\cap \delta_G(Y')|$. Then $|E'|< \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|$ must hold. We remove cluster $R$ from sets ${\mathcal{R}}$ and ${\mathcal{R}}^A$, and we add instead every connected component of graphs $G[X']$ and $G[Y']$ to both sets. It is immediate to verify that ${\mathcal{R}}$ remains a collection of disjoint clusters of $G$, and that $\bigcup_{R'\in {\mathcal{R}}}V(R')=V(G)$. Since $|E'|<\min\set{|\delta_G(R)\cap \delta_G(X')|,|\delta_G(R)\cap \delta_G(Y')|}$, we get that for every cluster $C$ that we just added to ${\mathcal{R}}$, $|\delta_G(C)|\leq |\delta_G(R)|\leq |\delta_G(S)|$ (from Invariant \ref{inv3: small boundary}). Therefore, all invariants continue to hold. We now show that the total budget $B$ decreases by at least $1/m$ as the result of this operation.
Note that the only edges whose budgets may change as the result of this operation are edges of $\delta_G(R)\cup E'$. Observe that, for each edge $e\in \delta_G(R)\cap \delta_G(Y')$, its budget $b(e)$ may not increase. Since we have assumed that $|\delta_G(R)\cap \delta_G(X')|\leq |\delta_G(R)\cap \delta_G(Y')|$, and since $|E'|<|\delta_G(R)|/8$, we get that $|\delta_G(X')|\leq 2|\delta_G(R)|/3$. Therefore, for every edge $e\in \delta_G(X')\cap \delta_G(R)$, its budget $b(e)$ decreases by at least $8\alpha \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot\log_{3/2}(|\delta_G(R)|)-8\alpha \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot\log_{3/2}(|\delta_G(X')|)$. Since $|\delta_G(X')|\leq 2|\delta_G(R)|/3$, we get that $ \log_{3/2}(|\delta_G(R)|)\leq \log_{3/2}(3|\delta_G(X')|/2)\leq 1+\log_{3/2}(|\delta_G(X')|$. We conclude that the budget $b(e)$ of each edge $e\in \delta_G(X')\cap \delta_G(R)$ decreases by at least $8\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)$.
On the other hand, the budget of every edge $e\in E'$ increases by at most $3$. Since $|E'|\leq \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|$, we get that the decrease in the budget $B$ is at least:
\[
\begin{split}
&8\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(X')\cap \delta_G(R)|-3|E'|\\&\hspace{3cm}\geq 8\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(X')\cap \delta_G(R)|- 3\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|
\\&\hspace{3cm}\geq 5 \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|\\&\hspace{3cm}>1/m,\end{split}\]
since $\alpha\geq 1/m$.
To conclude, if $|E(X,Y)|< \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T|}$, then we have modified the set ${\mathcal{R}}$ of clusters, so that all invariants continue to hold, and the total budget $B$ decreases by at least $1/m$. In this case, we terminate the current iteration.
From now on we assume that $|E(X,Y)|> \alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T|}$, which, as observed already, implies that cluster $R$ has the $\alpha$-bandwidth property. We now proceed to describe the second step of the algorithm.
\paragraph{Step 2: Witness Set of Paths.}
In the second step, we attempt to compute a witness set ${\mathcal{P}}(R)$ of paths for cluster $R$. If we succeed in doing so, we will move cluster $R$ from ${\mathcal{R}}^A$ to ${\mathcal{R}}^I$. Otherwise, we will further modify the set ${\mathcal{R}}$ of clusters, so that all invariants continue to hold, and the total budget decreases by at least $1/m$.
We construct the following flow network. Starting from graph $G$, we contract all vertices of $R$ into a source vertex $s$, and we contract all vertices of $V(G)\setminus V(S)$ into a destination vertex $t$. Denote the resulting graph by $H$, and observe that $\delta_H(s)=\delta_G(R)$, and $\delta_H(t)=\delta_G(S)$. We set the capacity $c(e)$ of every edge incident to $s$ to $1$, and the capacity of every other edge in graph $H$ to $100$. We then compute the maximum $s$-$t$ flow $f$ in the resulting flow network.
We consider two cases. The first case is when the value of the flow $f$ is $|\delta_H(s)|$. Since all edge capacities are integral, we can assume that flow $f$ is integral as well. Note that in this case, for every path $P$ connecting $s$ to $t$, either $f(P)=0$ or $f(P)=1$ must hold, as the capacities of all edges incident to $s$ are $1$. Therefore, flow $f$ naturally defines a collection ${\mathcal{P}}'(R)$ of $s$-$t$ paths, with $\cong_H({\mathcal{P}}'(R))\leq 100$, where each edge $e\in \delta_G(s)$ serves as the first edge of exactly one such path. Set ${\mathcal{P}}'(R)$ of paths then naturally defines a witness set ${\mathcal{P}}(R)=\set{P(e)\mid e\in \delta_G(R)}$ of paths for cluster $R$ in graph $G$, with $\cong_G({\mathcal{P}}(R))\leq 100$, where, for every edge $e\in \delta_G(R)$, path $P(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, with all inner vertices of $P(e)$ lying in $V(S)\setminus V(R)$. We then move cluster $R$ from ${\mathcal{R}}^A$ to ${\mathcal{R}}^I$ and terminate the current iteration. It is easy to verify that all invariants continue to hold, and the total budget $B$ does not change.
It remains to consider the second case, where the value of the flow $f$ in $H$ is less than $|\delta_H(s)|$. We compute a minimum $s$-$t$ cut $(A',B')$ in graph $H$, whose value is less than $|\delta_H(s)|$. We partition the set $E(A',B')$ of edges into two subsets: set $E'=E(A',B')\cap \delta_H(s)$, and set $E''=E(A',B')\setminus E'$. Recall that the capacity of every edge in $E'$ is $1$, while the capacity of every edge in $E''$ is $100$. Therefore, $|E'|+100|E''|<|\delta_G(R)|$.
Observe that cut $(A',B')$ in $H$ naturally defines cut $(A,B)$ of graph $S$: we let $A=(A'\setminus \set{s})\cup V(R)$, and $B=V(S)\setminus A$. Notice that $\delta_G(A)= E_H(A',B')$.
Let ${\mathcal{A}}$ denote the set of all connected components of graph $S[A]=G[A]$.
Let ${\mathcal{X}}$ denote the set of all clusers $R'\in {\mathcal{R}}$ with $R'\cap A\neq \emptyset$. For each such cluster $R'\in {\mathcal{X}}$, let ${\mathcal{Y}}(R')$ be the set of all connected components of $R'\setminus A$. We need the following observation.
\begin{observation}\label{obs: small boundary for new clusters}
For every cluster $C\in {\mathcal{A}}$, $|\delta_G(C)|\leq |\delta_G(R)|$. Additionally,
for every cluster $R'\in {\mathcal{X}}$, and every cluster $R''\in {\mathcal{Y}}(R')$, $|\delta_G(R'')|\leq |\delta_G(R')|$.
\end{observation}
\begin{proof}
Consider first some cluster $C\in {\mathcal{A}}$. Clearly, $\delta_G(C)\subseteq \delta_G(A)\subseteq E_H(A',B')$. Since $|E_{H}(A',B')|<|\delta_H(s)|=|\delta_G(R)|$, we get that $|\delta_G(C)|\leq |\delta_G(R)|$.
Consider now some cluster $R'\in {\mathcal{X}}$. Denote by $R'_A$ the subgraph of $R'$ induced by $V(R')\cap A$, and denote by $R'_B=R'\setminus A$. Let $E_1=\delta_G(R')\cap \delta_G(R'_A)$, $E_2=\delta_G(R')\cap \delta_G(R'_B)$, and $\hat E=E_G(R'_A,R'_B)$. We show below that $|\hat E|\leq |E_1|$ must hold. Assume for now that this is true, and consider any cluster $R''\in {\mathcal{Y}}(R')$. Since $R''$ is a connected component of $R'_B$, we get that $\delta_G(R'')\subseteq E_2\cup \hat E$. Therefore, if $|\hat E|\leq |E_1|$, then $|\delta_G(R'')|\leq |E_2|+|\hat E|\leq |E_1|+|E_2|=|\delta_G(R')|$ holds.
It now remains to prove that $|\hat E| \leq |E_1|$. Consider the cut $(A',B')$ in graph $H$, and recall that it is a minimum $s$-$t$ cut. From the definition of the cut $(A,B)$, the edges of $\hat E$ belong to the edge set $E_H(A',B')$. Since none of these edges is incident to $s$, the capacity of every edge in $\hat E$ is $100$. Consider now a new $s$-$t$ cut $(A'',B'')$ in graph $H$, where $A''=A'\setminus V(R'_A)$ and $B''=B'\cup V(R'_B)$. Note that the edges of $\hat E$ no longer contribute to this cut, and the only new edges that were added to this cut are the edges of $E_1$, each of which has a capacity that is either $1$ or $100$. Therefore, $\sum_{e\in E_H(A'',B'')}c(e)\leq \sum_{e\in E_H(A',B')}c(e)-100|\hat E|+100|E_1|$. Since $(A',B')$ is a minimum $s$-$t$ cut in graph $H$, $|\hat E|\leq |E_1|$ must hold.
\end{proof}
We perform the following modifications to the sets ${\mathcal{R}},{\mathcal{R}}^I$ and ${\mathcal{R}}^A$ of clusters. First, we remove cluster $R$ from ${\mathcal{R}}$ and from ${\mathcal{R}}^A$, and we add every cluster of ${\mathcal{A}}$ to both sets instead. Next, we consider every cluster $R'\in {\mathcal{X}}$ one by one. We remove each such cluster $R'$ from ${\mathcal{R}}$, and we add instead every cluster in ${\mathcal{Y}}(R')$ to ${\mathcal{R}}$. We also remove cluster $R'$ from the cluster set in $\set{{\mathcal{R}}^I,{\mathcal{R}}^A}$ to which it belongs, and we add every cluster of ${\mathcal{Y}}(R')$ to set ${\mathcal{R}}^I$. This completes the description of the modification of the sets ${\mathcal{R}},{\mathcal{R}}^I,{\mathcal{R}}^A$ of clusters. Note that all clusters in ${\mathcal{R}}$ remain disjoint, and $\bigcup_{R''\in {\mathcal{R}}'}V(R'')=V(S)$ continues to hold.
Moreover, from \Cref{obs: small boundary for new clusters}, combined with Invariant \ref{inv3: small boundary}, for every cluster $R'\in {\mathcal{R}}$, $|\delta_G(R')|\leq |\delta_G(S)|$ continues to hold.
It remains to show that the total budget $B$ decreases by at least $1/m$.
Consider any edge $e\in E(G)\setminus E''$, whose budget, at the end of the current step, is non-zero. Assume first that the budget of $e$ was non-zero at the beginning of the current step. Then, from \Cref{obs: small boundary for new clusters}, the budget of $e$ could not have increased as the result of the current step. If the budget of an edge $e$ was $0$ at the beginning of the current step and is non-zero at the end of the current step, then $e\in E''$ must hold.
Therefore, the only edges whose budget may have increased as the result of the current step are edges of $E''$.
For each edge $e\in E''$, its budget may have grown from $0$ to at most $3$, while the number of all such edges is $|E''|<(|\delta_G(R)|-|E'|)/100$. Therefore, the total increase in the budget $B$ due to the edges of $E''$ is at most $(|\delta_G(R)|-|E'|)/30$. We show that this increase is compensated by the decrease in the budgets of the edges of $\delta_G(R)\setminus E'$. Consider any edge $e\in \delta_G(R)\setminus E'$. Edge $e$ had budget at least $1$ originally, but after the current iteration, since the endpoints of $e$ both lie in $A$, its budget becomes $0$. Therefore, the decrease in the budget $B$ due to the edges of $\delta_G(R)\setminus E'$ is at least $|\delta_G(R)\setminus E'|$. Overall, we get that the decrease in the budget $B$ is at least:
\[ |\delta_G(R)\setminus E'|- (|\delta_G(R)|-|E'|)/30\geq 1/2.\]
This concludes the description of an iteration. The algorithm terminates when ${\mathcal{R}}^I=\emptyset$ holds, at which point we obtain the final set ${\mathcal{R}}$ of clusters, together with the witness sets $\set{{\mathcal{P}}(R)}_{R\in {\mathcal{R}}}$ of paths, that, from the invariants, have all required properties.
In particular, as observed above, since $B\geq\sum_{R\in {\mathcal{R}}}|\delta_G(R)|$, and $B$ never increases over the course of the algorithm, $\sum_{R\in {\mathcal{R}}}|\delta_G(R)|\leq B\leq (1+O(\alpha\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m))=(1+O(\alpha\cdot \log^{1.5} m))$ holds.
It remains to prove that the algorithm is efficient. Clearly, the algorithm for executing every iteration is efficient. We now show that the number of iterations is bounded by $O(m^3)$. Consider any iteration $i$ of the algorithm. Recall that, as the result of iteration $i$, either the budget $B$ decreased by at least $1/m$ (in which case we say that $i$ is a type-$1$ iteration); or budget $B$ did not change, but the number of clusters in set ${\mathcal{R}}^I$ decreases by $1$ (in which case we say that $i$ is a type-$2$ iteration). It is then immediate to see that the number of type-$1$ iterations, over the course of the algorithm, is bounded by $O(m^2)$. Since every cluster of ${\mathcal{R}}$ must contain at least one vertex, and $|V(S)|\leq m$ (because $S$ is a connected graph), the number of type-$2$ iterations executed between every consecutive pair of type-$1$ iterations is bounded by $O(m)$. Therefore, the total number of iterations of the algorithm is $O(m^3)$, and so the algorithm is efficient.
\subsection{Proof of \Cref{claim: remove congestion}}
\label{apd: Proof of remove congestion}
Denote $k=|{\mathcal{P}}|$ and $\rho =\cong_G({\mathcal{P}})$.
We define an undirected $s$-$t$ flow network $H$, as follows. We start with the graph $G$, and set the capacity of every edge in $G$ to be $1$. We then add a source vertex $s$, that connects to every vertex $v\in V(G)$ with an edge of capacity $n_S(v)$, and a destination vertex $t$, that connects to every vertex $v\in V(G)$ with an edge of capacity $n_T(v)$. Notice that, by sending $1/\rho$ flow units on every path $P\in {\mathcal{P}}$, we obtain an $s$-$t$ flow of value $k/\rho$ in this network. From the integrality of flow, since all edge capacities in $H$ are integral, there is an integral $s$-$t$ flow in $H$, of value at least $k/\rho$. This integral flow defines the desired collection ${\mathcal{P}}'$ of at least $k/\rho$ edge-disjoint paths in graph $G$. We can use standard algorithms for computing maximum $s$-$t$ flow in order to obtain the set ${\mathcal{P}}'$ of paths with these properties.
\subsection{Proof of \Cref{obs: splicing}}
\label{apd: Proof of splicing}
We first show that $S({\mathcal{P}}')=S({\mathcal{P}})$ and $T({\mathcal{P}}')=T({\mathcal{P}})$.
We denote by $s$ and $t$ the first and the last endpoints of $P$, respectively, and by $s'$ and $t'$ the first and the last endpoints of $P'$, respectively.
From the construction, the first endpoint of $\tilde P$ is $s$, the last endpoint of $\tilde P$ is $t'$, the first endpoint of $\tilde P'$ is $s'$, and the last endpoint of $\tilde P'$ is $t$. It is then immediate to verify that $S({\mathcal{P}}')=S({\mathcal{P}})$ and $T({\mathcal{P}}')=T({\mathcal{P}})$.
We now prove the second assertion. In order to do so, we assume that both $\tilde P,\tilde P'$ are simple paths, and we will show that $|\Pi^T({\mathcal{P}}')|<|\Pi^T({\mathcal{P}})|$.
For every vertex $u\in V(G)$, let $N_1(u)$ be the number of triples of $\Pi^T({\mathcal{P}})$ in which $u$ participates, and let $N_2(u)$ be the number of triples of $\Pi^T({\mathcal{P}}')$ in which $u$ participates. It is enough to show that, for every vertex $u\in V(G)\setminus\set{v}$, $N_2(u)\leq N_1(u)$, and that $N_2(v)<N_1(v)$.
Consider some vertex $u\in V(G)\setminus\set{v}$. We will assign, to every triple $(Q,Q',u)\in \Pi^T({\mathcal{P}}')$, a unique triple in $\Pi^T({\mathcal{P}})$ that is responsible to it, and we will ensure that every triple in $\Pi^T({\mathcal{P}})$ is responsible for at most one such triple.
Consider some triple $(Q,Q',u)\in \Pi^T({\mathcal{P}}')$. If neither of the two paths $Q,Q'$ lies in $\set{\tilde P,\tilde P'}$, then triple $(Q,Q',u)$ lies in $\Pi^T({\mathcal{P}})$ as well, and we make $(Q,Q',u)$ responsible for itself. If $Q=\tilde P$ and $Q'=\tilde P'$ (or the other way around), then either $Q$ is a subpath of $P$ and $Q'$ is a subpath of $P'$, or the other way around (we use the fact that paths $P,P'$ are simple, so $Q,Q'$ may not be subpaths of the same path). In either case, it is easy to see that triple $(P,P',u)$ lied in $\Pi^T({\mathcal{P}})$. We make the triple $(P,P',u)$ responsible for triple $(Q,Q,u)$. The last case is when exactly one of the paths $Q,Q'$ is in $\set{\tilde P,\tilde P'}$. We assume w.l.o.g. that $Q=\tilde P$, and $Q'\not\in\set{\tilde P, \tilde P'}$. If $u$ lies on path $P$ between its first vertex and $v$, then triple $(P,Q',u)$ lies in $\Pi^T({\mathcal{P}})$, and we make it responsible for $(Q,Q',u)$. Otherwise, triple $(P',Q',u)$ lies in $\Pi^T({\mathcal{P}})$, and we make it responsible for $(Q,Q',u)$.
It is easy to see that every triple $(\hat Q,\hat Q',u)\in \Pi^T({\mathcal{P}})$ is responsible for at most one triple in $\Pi^T({\mathcal{P}}')$. Indeed, if neither of $\hat Q,\hat Q'$ lies in $\set{P,P'}$, then triple $(\hat Q,\hat Q',u)$ may only be responsible for itself. If both $\hat Q,\hat Q'\in \set{P,P'}$, then triple $(P,P',u)$ may only be responsible for triple $(\tilde P,\tilde P',u)$. If exactly one of $\hat Q,\hat Q'$ lies in $\set{P,P'}$, for example, $\hat Q=P$, then two cases are possible: if vertex $u$ lies between the first endpoint of $P$ and $v$, then triple $(P,Q',u)$ may only be responsible for triple $(\tilde P,Q',u)$, and otherwise it may only be responsible for triple $(\tilde P',Q',u)$. We conclude that $N_2(u)\leq N_1(u)$.
Consider now the case where $u=v$, and consider some triple $(Q,Q',v)\in \Pi^T({\mathcal{P}}')$. If neither of the two paths $Q,Q'$ lies in $\set{\tilde P,\tilde P'}$, then triple $(Q,Q',v)$ lies in $\Pi^T({\mathcal{P}})$, and we make $(Q,Q',v)$ responsible for itself. Note that, in case where $u=v$, it is impossible that the triple $(\tilde P,\tilde P',v)$ lies in $\Pi^T({\mathcal{P}}')$. Therefore, it remains to consider the triples $(Q,Q',v)$, where exactly one of the paths $Q,Q'$ lies in $\set{\tilde P,\tilde P'}$.
We call such triples \emph{problematic triples}, and we assume w.l.o.g. that in each such triple, $Q\not \in \set{\tilde P,\tilde P'}$. If path $Q$ participates in a problematic triple, then we say that path $Q$ is a \emph{problematic path}.
We denote by $e_a,e'_{a}$ the two edges on path $P$ that are incident to vertex $v$, and we assume that $e_a$ appears before $e'_{a}$ on $P$. We denote by $e_b,e'_{b}$ the two edges on path $P'$ that are incident to $v$, and we assume that $e_b$ appears before $e'_{b}$ on $P'$. Recall that path $\tilde P$ contains edges $e_a$ and $e'_{b}$, while path $\tilde P'$ contains edges $e_{b}$ and $e'_{a}$. Recall that edges $e_a,e_b,e'_{a},e'_{b}$ must appear in this circular order in ${\mathcal{O}}_v\in \Sigma$, since paths $P$ and $P'$ are transversal (recall that the ordering is unoriented). We use the edges of $\set{e_a,e_b,e'_{a},e'_{b}}$ to partition the edge set $\delta_G(v)\setminus \set{e_a,e_b,e'_{a},e'_{b}}$ into four subsets: set $E_1$ of edges appearing between $e_a$ and $e_b$ in ${\mathcal{O}}_v$; set $E_2$ of edges appearing between $e_b$ and $e'_a$; set $E_3$ of edges appearing between $e'_a$ and $e'_b$, and set $E_4$ of all remaining edges, that must appear between $e'_b$ and $e_a$ (see \Cref{fig: splicing_1}).
\begin{figure}[h]
\centering
\scalebox{0.8}{\includegraphics[scale=0.1]{figs/splicing_1.jpg}}
\caption{A schematic view of edges $e_a$, $e_b$, $e'_a$, $e'_b$ and edge sets $\set{E_i}_{1\le i\le 4}$.}\label{fig: splicing_1}
\end{figure}
Consider now some problematic path $Q$, and denote by $e(Q),e'(Q)$ the two edges that lie on $Q$ and are incident to $v$. Note that $e(Q),e'(Q)$ must lie in different sets of $\set{E_1,\ldots,E_4}$. Since path $\tilde P$ contains edges $e_a,e'_b$, while path $\tilde P'$ contains edges $e'_a,e_b$, in order for path $Q$ to be problematic, at least one of the two edges $e(Q),e'(Q)$ must lie in one of the sets $E_2,E_4$. We assume w.l.o.g. that $e(Q)\in E_2$. If $e'(Q)\in E_4$, then both $(Q,\tilde P,v)$ and $(Q,\tilde P',v)$ are problematic pairs. But in this case, both $(Q,P,v)$ and $(Q,P',v)$ lied in $\Pi^T({\mathcal{P}})$. We make triple $(Q,P,v)$ responsible for $(Q,\tilde P,v)$, and we make triple $(Q,P',v)$ responsible for $(Q,\tilde P',v)$. Otherwise, $e'_Q\in E_1$ or $e'_Q\in E_3$ must hold. In the either case, the only problematic triple involving path $Q$ is $(Q,\tilde P',v)$. In the former case, $(Q,P',v)\in \Pi^T({\mathcal{P}})$, and we make this triple responsible for $(Q,\tilde P',v)$, while in the latter case, $(Q,P,v)\in \Pi^T({\mathcal{P}})$, and we make this triple responsible for $(Q,\tilde P',v)$.
So far, we have assigned, to every triple $(Q,Q',v)\in \Pi^T({\mathcal{P}}')$, a distinct triple $(\hat Q,\hat Q',v)\in \Pi^T({\mathcal{P}})$ that is responsible for it. Note that triple $(P,P',v)$ is not responsible for any triple $(Q,Q',v)\in \Pi^T({\mathcal{P}}')$, so $N_2(v)<N_1(v)$. We conclude that $|\Pi^T({\mathcal{P}}')|<|\Pi^T({\mathcal{P}})|$.
\iffalse
We assume w.l.o.g. that $Q=\tilde P$ and $Q'\not\in \set{\tilde P,\tilde P'}$.
If $u=v$, then triple $(P,P',v)\in \Pi^T({\mathcal{P}})$ is not responsible for any triples in $\Pi^T({\mathcal{P}}')$, so $N_2(v)<N_1(v)$.
Specifically, we will construct a mapping $h$, that maps each triple in $\Pi^T({\mathcal{P}}')$ to a triple in $\Pi^T({\mathcal{P}})$, such that at most one triple is mapped to every triple in $\Pi^T({\mathcal{P}})$, and there is some triple in $f$ is injective but not surjective.
Note that ${\mathcal{P}}'\setminus\set{\tilde P, \tilde P'}={\mathcal{P}}\setminus\set{P, P'}$. We denote $\hat {\mathcal{P}}={\mathcal{P}}\setminus\set{P, P'}$.
Since the set $\Pi^T({\mathcal{P}})$ contains the triple $(P,P',v)$, $v$ must be an inner vertex of both $P$ and $P'$.
Since we have assumed that paths $\tilde P, \tilde P'$ are simple paths, $v$ is the only vertex shared by paths $\tilde P$ and $\tilde P'$, and $v$ is the only vertex shared by paths $P$ and $P'$. From the construction, the intersection of paths $\tilde P,\tilde P'$ at $v$ is non-transversal with respect to $\Sigma$.
Therefore, each triple in $\Pi^T({\mathcal{P}}')$ is of one of the following types:
\begin{itemize}
\item $(Q,Q',v')$ where $Q,Q'\in \hat{\mathcal{P}}$;
\item $(Q,\tilde P ,v')$ or $(Q,\tilde P' ,v')$ where $Q\in \hat{\mathcal{P}}$ and $v'\ne v$;
\item $(Q,\tilde P ,v)$ or $(Q,\tilde P' ,v)$ where $Q\in \hat{\mathcal{P}}$.
\end{itemize}
First, it is easy to see that, for every triple $(Q,Q',v')\in \Pi^T({\mathcal{P}}')$ such that $Q,Q'\in {\mathcal{P}}'\setminus\set{\tilde P, \tilde P'}$, the triple $(Q,Q',v')$ also belongs to the set $\Pi^T({\mathcal{P}})$.
We then map such a triple to itself.
Second, consider a triple $(Q,\tilde P,v')$ in $\Pi^T({\mathcal{P}}')$ such that $Q\in \hat {\mathcal{P}}$ and $v'\ne v$. If vertex $v'$ belongs to the subpath of $\tilde P$ from $s$ to $v$, then it is easy to verify that the triple $(Q,P,v')$ belongs to $\Pi^T({\mathcal{P}})$, and we map the triple $(Q,\tilde P,v')$ to the triple $(Q,P,v')$.
If $v'$ belongs to the subpath of $P$ from $v$ to $t'$, then it is easy to verify that the triple $(Q,P',v')$ belongs to $\Pi^T({\mathcal{P}})$, and we map the triple $(Q,\tilde P,v')$ to the triple $(Q,P',v')$.
Similarly, for each triple $(Q,\tilde P',v')\in\Pi^T({\mathcal{P}}')$, either the triple $(Q,P,v')$ or the triple $(Q,P',v')$ belongs to $\Pi^T({\mathcal{P}})$, and we map the triple $(Q,\tilde P',v')$ to either $(Q,P,v')$ or $(Q,P',v')$ accordingly.
Last, we consider the triples in $\Pi^T({\mathcal{P}}')$ of the third type.
We denote by $e_a,e_b$ the edges of $P$ incident to $v$, where $e_a$ precedes $e_b$ in $P$ (recall that $v$ is an inner vertex of $P$). We define edges $e'_a,e'_b$ similarly for path $P'$.
Assume that $\delta(v)=\set{e_a,e^1_1,\ldots,e^1_{r_1},e_b,e^2_1,\ldots,e^2_{r_2},e_a',e^3_1,\ldots,e^1_{r_3},e_b',e^4_1,\ldots,e^1_{r_4}}$, where edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
We denote $E_1=\set{e^1_1,\ldots,e^1_{r_1}}$, $E_2=\set{e^2_1,\ldots,e^2_{r_2}}$,
$E_3=\set{e^3_1,\ldots,e^3_{r_3}}$ and
$E_4=\set{e^4_1,\ldots,e^4_{r_4}}$.
See \Cref{fig: splicing_1} for an illustration.
Consider now a path $Q\in \hat {\mathcal{P}}$, and we denote by $e^Q_a,e^Q_b$ the paths of $Q$ incident to $v$, where $e^Q_a$ precedes $e^Q_b$ in $Q$.
We distinguish between the following cases.
\textbf{Case 1. $(Q,P,v), (Q,P',v)\in \Pi^T({\mathcal{P}})$.} In this case, if set $\Pi^T({\mathcal{P}}')$ contains the triple $(Q,\tilde P, v)$, then we map it to the triple $(Q,P,v)$, and if set $\Pi^T({\mathcal{P}}')$ contains the triple $(Q,\tilde P', v)$, then we map it to the triple $(Q,P',v)$.
\textbf{Case 2. $(Q,P,v), (Q,P',v)\notin \Pi^T({\mathcal{P}})$.} In this case, it is easy to verify that both edges $e'_a,e'_b$ belong to the same set of $E_1,E_2,E_3,E_4$. Therefore, the intersection between $Q$ and $\tilde P$ at $v$ is non-transversal, namely the set $\Pi^T({\mathcal{P}}')$ does not contain the triple $(Q,\tilde P,v)$. Similarly, the set $\Pi^T({\mathcal{P}}')$ does not contain the triple $(Q,\tilde P' ,v)$, either.
\textbf{Case 3. $(Q,P,v)\in \Pi^T({\mathcal{P}}), (Q,P',v)\notin \Pi^T({\mathcal{P}})$.} In this case, it is easy to verify that either $e^Q_a\in E_1$ and $e^Q_b\in E_4$ (or symmetrically $e^Q_a\in E_4$ and $e^Q_b\in E_1$), or $e^Q_a\in E_2$ and $e^Q_b\in E_3$ (or symmetrically $e^Q_a\in E_3$ and $e^Q_b\in E_2$). In the former case (see \Cref{fig: splicing_2}), the intersection between paths $Q$ and $\tilde P$ at $v$ is transversal, while the intersection between paths $Q$ and $\tilde P'$ at $v$ is non-transversal. In other words, the set $\Pi^T({\mathcal{P}}')$ contains the triple $(Q,\tilde P,v)$ but not the triple $(Q,\tilde P' ,v)$. We then map $(Q,\tilde P,v)$ to the triple $(Q,P,v)$.
In the latter case (see \Cref{fig: splicing_3}), the intersection between paths $Q$ and $\tilde P'$ at $v$ is transversal, while the intersection between paths $Q$ and $\tilde P$ at $v$ is non-transversal. In other words, the set $\Pi^T({\mathcal{P}}')$ contains the triple $(Q,\tilde P',v)$ but not the triple $(Q,\tilde P ,v)$. We then map triple $(Q,\tilde P',v)$ to the triple $(Q,P,v)$.
\begin{figure}[h]
\centering
\subfigure[A schematic view of edges $e_a$, $e_b$, $e'_a$, $e'_b$ and edge sets $\set{E_i}_{1\le i\le 4}$.]{\scalebox{0.9}{\includegraphics[scale=0.4]{figs/splicing_1.jpg}}\label{fig: splicing_1}
}
\hspace{0.3cm}
\subfigure[An illustration of the former case in Case 3.]{\scalebox{0.9}{\includegraphics[scale=0.4]{figs/splicing_2.jpg}}\label{fig: splicing_2}}
\hspace{0.3cm}
\subfigure[An illustration of the latter case in Case 3.]{
\scalebox{0.9}{\includegraphics[scale=0.4]{figs/splicing_3.jpg}}\label{fig: splicing_3}}
\caption{An illustration of edges and edge sets in the proof of \Cref{obs: splicing}.}
\end{figure}
It is easy to verify that the mapping we construct is injecture. We now show that it is surjective. In fact, since the intersection between paths $P,P'$ at $v$ is transversal, the set $\Pi^T({\mathcal{P}})$ contains the triple $(P,P',v)$. However, it is easy to verify that no triple in $\Pi^T({\mathcal{P}}')$ is mapped to the triple $(P,P',v)$. Therefore, the mapping we construct is surjective. This completes the proof of \Cref{obs: splicing}.
\fi
\subsection{Proof of Lemma~\ref{lem: non_interfering_paths}}
\label{apd: Proof of non_interfering_paths}
We first preprocess the set ${\mathcal{R}}$ of paths by removing cycles from the paths, to obtain a collection ${\mathcal{R}}$ of simple paths.
The algorithm is iterative. Throughout the algorithm, we maintain a set $\hat {\mathcal{R}}$ of paths in $G$, that is initialized to be ${\mathcal{R}}$. The algorithm proceeds in iterations, as long as $\Pi^T(\hat{\mathcal{R}})\neq \emptyset$. An iteration is executed as follows.
Let $(P,P',v)$ be any triple in $\Pi^T(\hat{\mathcal{R}})$. We perform path splicing of $P$ and $P'$ at vertex $v$, obtaining two new paths $\tilde P$ and $\tilde P'$. We then remove cycles from $\tilde P$ and $\tilde P'$, to obtain two simple paths, which are then added to $\hat {\mathcal{R}}$, replacing the paths $P$ and $P'$. Note that, from \Cref{obs: splicing}, multisets $S(\hat {\mathcal{R}}),T(\hat {\mathcal{R}})$ remain unchanged after the execution of the iteration. It is also easy to verify, from the definition of the splicing procedure, that, for every edge $e\in E(G)$, $\cong_G(\hat {\mathcal{R}},e)$ may not increase after the iteration execution. Moreover, if the paths $\tilde P,\tilde P'$ obtained after the splicing procedure are simple, then $|\Pi^T({\mathcal{R}})|$ is guaranteed to decrease after the current iteration, while otherwise, $\sum_{R\in \hat{\mathcal{R}}}|E(R)|$ must decrease. We conclude that, after every iteration of the algorithm, either $\sum_{R\in \hat{\mathcal{R}}}|E(R)|$ decreases, or $\sum_{R\in \hat{\mathcal{R}}}|E(R)|$ remains unchanged and $|\Pi^T(\hat {\mathcal{R}})|$ decreases. Since $|\Pi^T(\hat {\mathcal{R}})|\leq |\hat {\mathcal{R}}|^2\cdot |V(G)|$, the number of iterations in the algorithm is bounded by $|\hat {\mathcal{R}}|^2\cdot |V(G)|\cdot |E(G)|$, and so the algorithm is efficient. The output ${\mathcal{R}}'$ of the algorithm is the set $\hat {\mathcal{R}}$ of paths that is obtained when the algorithm terminates. From the above discussion, we get that $S({\mathcal{R}}')=S({\mathcal{R}})$ and $T( {\mathcal{R}}')=T({\mathcal{R}})$. Once the algorithm terminates, the paths in set ${\mathcal{R}}'=\hat {\mathcal{R}}$ are non-transversal with respect to $\Sigma$. Lastly, from the above discussion, for every edge $e\in E(G)$, $\cong_G(\hat {\mathcal{R}},e)$ may not increase over the course of the algorithm, and so $\cong_G({\mathcal{R}}',e)\le \cong_G({\mathcal{R}},e)$ must hold.
\iffalse{for edge-disjoint path}
Let $H=\bigcup_{P\in {\mathcal{R}}}P$ be the subgraph of $\hat G$ consisting of paths of ${\mathcal{R}}$.
Our first step is to compute a set $\hat{\mathcal{R}}$ of $|\hat {\mathcal{R}}|=|{\mathcal{R}}|$ edge-disjoint paths connecting $v$ to $v'$, using standard max-flow algorithms.
We view the paths in $\hat{\mathcal{R}}$ as being directed from $v$ to $v'$. Let $\hat H$ be the directed graph obtained by taking the union of the directed paths in $\hat{\mathcal{R}}$, so $\hat H$ is a subgraph of $H$. From the max-flow algorithm, we can further assume there is an ordering $\Omega$ of its vertices, such that, for every pair $x,y$ of vertices in $\hat H$, if there is a path of $\hat {\mathcal{R}}$ in which $x$ appears before $y$, then $x$ also appears before $y$ in the ordering $\Omega$.
For each vertex $u\in V(\hat H)$, we denote by $\delta^+(u)$ the set of edges leaving $u$ in $\hat H$, and by $\delta^-(u)$ the set of edges entering $u$ in $\hat H$. Clearly, if $u\ne v,v'$, then $|\delta^+(u)|=|\delta^-(u)|$.
We use the following simple observation.
\begin{observation}
\label{obs:rerouting_matching}
We can efficiently compute, for each vertex $u\in V(\hat H)\setminus \set{v,v'}$, a perfect matching $M(u)\subseteq \delta^-(u)\times \delta^+(u)$ between the edges of $\delta^-(u)$ and the edges of $\delta^+(u)$, such that, for each pair of matched pairs $(e^-_1,e^+_1)$ and $(e^-_2,e^+_2)$ in $M(u)$, the intersection of the path that consists of the edges $e^-_1,e^+_1$ and the path that consists of edges $e^-_2,e^+_2$ at vertex $u$ is non-transversal with respect to $\hat\Sigma$.
\end{observation}
\begin{proof}
We start with $M(u)=\emptyset$ and perform $|\delta^-(u)|$ iterations. In each iteration, we select a pair $e^-\in \delta^-(u), e^+\in \delta^+(u)$ of edges that appear consecutively in the rotation ${\mathcal{O}}_v$ of $\tilde\Sigma$. We add $(e^-,e^+)$ to $M(u)$, delete them from $\delta^-(u)$ and $\delta^+(u)$ respectively, and then continue to the next iteration. It is immediate to verify that the resulting matching $M(u)$ satisfies the desired properties.
\end{proof}
We now gradually modify the set $\hat{\mathcal{R}}$ of paths in order to obtain a set ${\mathcal{R}}'$ of edge-disjoint paths connecting $v$ to $v'$, that is non-transversal with respect to $\hat \Sigma$.
We process all vertices of $V(\hat H)\setminus \set{v,v'}$ one-by-one, according to the ordering $\Omega$. We now describe an iteration in which a vertex $u$ is processed. Let $\hat {\mathcal{R}}_u\subseteq \hat {\mathcal{R}}$ be the set of paths containing $u$. For each path $P\in \hat {\mathcal{R}}_u$, we delete the unique edge of $\delta^+(u)$ that lies on this path, thereby decomposing $P$ into two-subpaths: path $P^-$ connecting $v$ to $u$; and path $P^+$ connecting some vertex $u'$ that is the endpoint of an edge of $\delta^+(u)$ to $v'$. Define $\hat {\mathcal{R}}^-_u=\set{P^-\mid P\in \hat {\mathcal{R}}_u}$ and
$\hat {\mathcal{R}}^+_u=\set{P^+\mid P\in \hat {\mathcal{R}}_u}$.
We will glue these paths together using the edges in $\delta^+(u)$ and the matching $M(u)$ produced in Observation \ref{obs:rerouting_matching}. Specifically, we construct a new set ${\mathcal{R}}'$ of paths that contains, for each path $P\in \hat {\mathcal{R}}$, a new path $P'$, as follows. Consider a path $P\in \hat {\mathcal{R}}$. If $P\notin \hat {\mathcal{R}}_u$, then we let $P'=P$. Otherwise, consider the unique path $P^-\in \hat {\mathcal{R}}^-_u$ that is a subpath of $P$, and let $e^-_P$ be the last edge on this path. Let $e^+$ be the edge in $\delta^+(u)$ that is matched with $e^-_P$ in $M(u)$, and let $\hat P^+$ be the unique path in $\hat {\mathcal{R}}^+_u$ that contains $e^+$.
We then define the new path $P'$ to be the concatenation of the path $P^-$, the edge $e^+$, and the path $\hat P^+$. This finishes the description of an iteration.
It is easy to verify that $|{\mathcal{R}}'|=|\hat {\mathcal{R}}|=|{\mathcal{R}}|$, $E({\mathcal{R}}')\subseteq E(\hat{\mathcal{R}})\subseteq E({\mathcal{R}})$, and the set ${\mathcal{R}}'$ of paths is locally non-interfering with respect to $\hat\Sigma$.
\fi
\iffalse
Our first step is to compute another one-to-one routing $\hat{\mathcal{R}}$ of vertice of $S$ to vertices of $T$, via $|\hat {\mathcal{R}}|=|{\mathcal{R}}|$ simple paths, such that for every $e\in E( G)$, $\cong_{ G}(\hat{\mathcal{R}},e)\le \cong_{ G}({\mathcal{R}},e)$, using standard max-flow algorithms.
We first construct a graph $H$ as follows.
We start from the graph $\bigcup_{P\in {\mathcal{R}}}P$.
We then replace each edge $e$ of $E({\mathcal{R}})$ with $\cong_{G}({\mathcal{R}},e)$ copies of it, and let each path of ${\mathcal{R}}$ that contains $e$ takes a distinct copy.
Finally, we add two new vertices $s,t$, and then add, for each vertex $v\in S$, an edge $(s,v)$; and for each vertex $v'\in S$, an edge $(v',t)$.
This finishes the definition of $H$.
Now paths of ${\mathcal{R}}$ are edge-disjoint paths in $H$.
For each vertex $v\in V(H)\setminus \set{s,t}$, we define its rotation ${\mathcal{O}}'_v$ as follows. We start with its oritinal rotation ${\mathcal{O}}_v\in \Sigma$, and then replace, for each edge $e$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $\cong_{G}({\mathcal{R}},e)$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$, and the ordering among the copies is arbitrary.
Now we compute the a set $\hat {\mathcal{R}}$ of $|\hat {\mathcal{R}}|=|{\mathcal{R}}|$ edge-disjoint paths connecting $s$ to $t$ in $H$. It is easy to see that such a set of path exists.
We view these paths as directed from $s$ to $t$.
From the max-flow algorithm, we can further assume there is an ordering $\Omega$ of its vertices, such that, for every pair $x,y$ of vertices in $H$, if there is a path of $\hat{\mathcal{R}}$ in which $x$ appears before $y$, then $x$ also appears before $y$ in the ordering $\Omega$.
For each vertex $u\in V(H)$, we denote by $\delta^+(u)$ the set of edges leaving $u$ in $H$, and by $\delta^-(u)$ the set of edges entering $u$ in $H$. Clearly, if $u\ne s,t$, then $|\delta^+(u)|=|\delta^-(u)|$.
We now gradually modify the set $\hat{\mathcal{R}}$ of paths in order to obtain a set $\hat{\mathcal{R}}'$ of edge-disjoint paths in $H$ connecting $s$ to $t$, that is non-transversal with respect to the new orderings in $\set{{\mathcal{O}}'_v\mid v\in V(H)\setminus\set{s,t}}$.
We process all vertices of $V(H)\setminus \set{s,t}$ one-by-one, according to the ordering $\Omega$. We now describe an iteration in which a vertex $u$ is processed. Let $\hat {\mathcal{R}}_u\subseteq \hat {\mathcal{R}}$ be the set of paths containing $u$. For each path $P\in \hat {\mathcal{R}}_u$, we delete the unique edge of $\delta^+(u)$ that lies on this path, thereby decomposing $P$ into two-subpaths: path $P^-$ connecting $s$ to $u$; and path $P^+$ connecting some vertex $u'$ that is the endpoint of an edge of $\delta^+(u)$ to $t$. Define $\hat {\mathcal{R}}^-_u=\set{P^-\mid P\in \hat {\mathcal{R}}_u}$ and
$\hat {\mathcal{R}}^+_u=\set{P^+\mid P\in \hat {\mathcal{R}}_u}$.
We will glue these paths together using the edges in $\delta^+(u)$ and the matching $M(u)$ produced in \Cref{obs:rerouting_matching_cong}. Specifically, we construct a new set $\hat{\mathcal{R}}'$ of paths that contains, for each path $P\in \hat {\mathcal{R}}$, a new path $P'$, as follows. Consider a path $P\in \hat {\mathcal{R}}$. If $P\notin \hat {\mathcal{R}}_u$, then we let $P'=P$. Otherwise, consider the unique path $P^-\in \hat {\mathcal{R}}^-_u$ that is a subpath of $P$, and let $e^-_P$ be the last edge on this path. Let $e^+$ be the edge in $\delta^+(u)$ that is matched with $e^-_P$ in $M(u)$, and let $\hat P^+$ be the unique path in $\hat {\mathcal{R}}^+_u$ that contains $e^+$.
We then define the new path $P'$ to be the concatenation of the path $P^-$, the edge $e^+$, and the path $\hat P^+$. This finishes the description of an iteration.
It is easy to verify that $|\hat{\mathcal{R}}'|=|\hat {\mathcal{R}}|=|{\mathcal{R}}|$, $E_{H}(\hat {\mathcal{R}}')\subseteq E_{H}(\hat{\mathcal{R}})$, and the set $\hat{\mathcal{R}}'$ of paths is non-transversal with respect to the orderings in $\set{{\mathcal{O}}'_v\mid v\in V(H)\setminus\set{s,t}}$.
Finally, we let the set ${\mathcal{R}}'$ contains, for each path $P'\in \hat{\mathcal{R}}'$, the original path of $P'\setminus\set{s,t}$ in $H$ (namely, if $P'\setminus\set{s,t}$ contains a copy of $e$, then its original path contains the edge $e$).
It is easy to verify that $|{\mathcal{R}}'|=|{\mathcal{R}}|$, for each $e\in G$, $\cong_{G}({\mathcal{R}}',e)\le \cong_{G}({\mathcal{R}},e)$, and the set ${\mathcal{R}}'$ of paths is non-transversal with respect to $\Sigma$.
\fi
\subsubsection{Bad Indices and their Properties}
We denote the clusters in ${\mathcal{C}}$ by $C_1,C_2,\ldots,C_r$. For convenience, for each $1\leq i\leq r$, we denote the vertex $v(C_i)$ of the graph $H=G_{|{\mathcal{C}}}$ by $x_i$. Recall that we have assumed that the Gomory-Hu tree $\tau$ of $H$ is a path. We assume without loss of generality that the clusters are indexed according to their appearance on the path $\tau$. For $1\leq i\leq r$, we let $S_i=\set{x_1,\ldots,x_i}$ and let $\overline{S}_i=\set{x_{i+1},\ldots,x_r}$. Recall that, by the definition of the Gomory-Hu tree, $(S_i, \overline{S}_i)$ is a minimum cut in graph $H$ separating vertex $x_i$ from vertex $x_{i+1}$. From the properties of minimum cut, there is a set of paths in graph $H$, routing the edges of $\delta_H(S_i)$ to vertex $x_{i}$ inside $S_i$, and there is a set of paths in graph $H$, routing the edges of $\delta_H(\overline{S}_i)$ to vertex $x_{i+1}$ inside $\overline{S}_i$.
Recall that every edge in graph $H$ corresponds to some edge in $E^{\textsf{out}}({\mathcal{C}})$, and we do not distinguish between these edges. In particular, for all $1\le i,j\le r$, we will also view edges of $E(C_i,C_j)$ as parallel edges connecting vertex $x_i$ to vertex $x_j$ in graph $H$.
For each $1\le i\le k$, we denote
$\hat E_i=E(C_i,C_{i+1})$,
$E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$, and
$E_i^{\operatorname{over}}=\bigcup_{i'<i,j'>i+1}E(C_{i'},C_{j'})$.
We need the following observation.
\begin{observation}\label{obs: bad inded structure}
For all $1\leq i<r$, the following hold:
\begin{itemize}
\item $|\hat E_i|\geq |E_i^{\operatorname{over}}|$;
\item $|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_{i+1}^{\operatorname{left}}|-|E_i^{\operatorname{right}}|$; and
\item $|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$.
\end{itemize}
\end{observation}
\begin{proof}
Consider the cut $(\set{x_{i}},V(H)\setminus \set{x_i})$ in $H$. Its size is $|\hat E_i|+|\hat E_{i-1}|+ |E_i^{\operatorname{right}}|+|E_{i}^{\operatorname{left}}|$. Note that this cut separates $x_i$ from $x_{i-1}$. Since the minimum cut separating $x_i$ from $x_{i-1}$ in $H$ is $(S_{i-1},\overline{S}_{i-1})$, and $|E(S_{i-1},\overline{S}_{i-1})|=|\hat E_{i-1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|$, we get that $|\hat E_{i-1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|\hat E_{i-1}|+ |E_i^{\operatorname{right}}|+|E_{i}^{\operatorname{left}}|$,
and so
\begin{equation}
\label{eqn1}
|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+ |E_i^{\operatorname{right}}|.
\end{equation}
Consider the cut $(\set{x_{i+1}},V(H)\setminus \set{x_{i+1}})$ in $H$. Its size is $|\hat E_i|+|\hat E_{i+1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i+1}^{\operatorname{right}}|$. Note that this cut separates $x_{i+1}$ from $x_{i+2}$. Since the minimum cut separating $x_{i+1}$ from $x_{i+2}$ in $H$ is $(S_{i+1},\overline{S}_{i+1})$, and $|E(S_{i+1},\overline{S}_{i+1})|=|\hat E_{i+1}|+ |E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{right}}|+|E_i^{\operatorname{over}}|$, we get that $|\hat E_{i+1}|+ |E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{right}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|\hat E_{i+1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i+1}^{\operatorname{right}}|$,
and so
\begin{equation}
\label{eqn2}
|E_i^{\operatorname{right}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|E_{i+1}^{\operatorname{left}}|.
\end{equation}
Combining \Cref{eqn1,eqn2}, we get that $|\hat E_i|\geq |E_i^{\operatorname{over}}|$. By rearranging the sides of the two inequalities, we get that
$|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_{i+1}^{\operatorname{left}}|-|E_i^{\operatorname{right}}|$, and
$|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$.
\end{proof}
\begin{definition}
We say that index $i$ is \emph{bad} iff $|E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|>10000|\hat E_i|$.
\end{definition}
We prove the following observation.
\begin{observation}\label{obs: bad instance}
If index $i$ is bad, then $|E_{i+1}^{\operatorname{left}}|-|\hat E_i| \leq |E_i^{\operatorname{right}}|\leq |E_{i+1}^{\operatorname{left}}|+|\hat E_i|$, and moreover $4998|E_{i+1}^{\operatorname{left}}|/4999 \leq |E_i^{\operatorname{right}}|\leq 4999|E_{i+1}^{\operatorname{left}}|/4998$.
If index $i$ is not bad, then $|\hat E_i|\ge |E_{i+1}^{\operatorname{left}}|/5001$ and $|\hat E_i|\ge |E_{i}^{\operatorname{right}}|/5001$.
\end{observation}
\begin{proof}
Assume first that $i$ is bad. Since $|E_i^{\operatorname{over}}|\leq |\hat E_i|$, we get that either $|E_i^{\operatorname{right}}|\geq 4999|\hat E_i|$, or $|E_{i+1}^{\operatorname{left}}|\geq 4999|\hat E_i|$. Assume without loss of generality that $|E_i^{\operatorname{right}}|\geq |E_{i+1}^{\operatorname{left}}|$, so $|E_i^{\operatorname{right}}|\geq 4999 |\hat E_i|$. Since, from \Cref{obs: bad inded structure},
$|\hat E_i| \geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$, we get that $|E_{i+1}^{\operatorname{left}}|\geq |E_i^{\operatorname{right}}|-|\hat E_i|\geq 4998|E_i^{\operatorname{right}}|/4999$.
Assume now that $i$ is not bad. By definition, $10000|\hat E_i|\ge |E^\operatorname{right}_i|+|E^\operatorname{left}_{i+1}|+|E^{\operatorname{over}}_i|$. Combined with \Cref{obs: bad inded structure}, we get that $|\hat E_i|\ge |E^\operatorname{left}_{i+1}|/5001$ and $|\hat E_i|\ge |E^\operatorname{right}_{i}|/5001$.
\end{proof}
Let $1\le i\le r$ be some index.
We define indices $\mathsf{LM}(i)$ and $\mathsf{RM}(i)$ as follows.
Recall that $E^{\operatorname{right}}_{i}$ is the set of all edges $(x_{i},x_z)$ with $z>i+1$. We let $\mathsf{RM}(i)$ be the smallest integer $z>i+1$, such that at least half the edges in $E^{\operatorname{right}}_{i}$ have their endpoints in $\set{x_{i+2},\ldots,x_z}$. It is easy to see that at least half the edges in $E^{\operatorname{right}}_{i}$ must have their endpoints in $\set{x_{z},\ldots,x_r}$. Similarly, we let $\mathsf{LM}(i)$ be the largest integer $z<i$, such that at least half the edges in $E^{\operatorname{left}}_{i+1}$ have their endpoints in $\set{x_{z},\ldots,x_{i-1}}$. It is easy to see that at least half the edges in $E^{\operatorname{left}}_{i+1}$ must have their endpoints in $\set{x_{1},\ldots,x_z}$. We need the following crucial observation and claim.
\begin{observation}\label{obs: large capacity up to midpoint}
Let $1\leq i\leq r$ be an index. Then for every index $i<z<\mathsf{RM}(i)-1$, $|\hat E_z|\geq |E^{\operatorname{right}}_{i}|/2$; similarly, for every index $\mathsf{LM}(i)<z'< i$, $|\hat E_{z'}|\geq |E^{\operatorname{left}}_{i+1}|/2$.
\end{observation}
\begin{proof}
Fix an index $1\leq i\leq r$ and another index $i<z<\mathsf{RM}(i)-1$. Recall that there is a subset $E'\subseteq E^{\operatorname{right}}_{i}$ of at least $|E^{\operatorname{right}}_{i}|/2$ edges that have an endpoint in $\set{x_{\mathsf{RM}(i)},\ldots,x_r}$. It is easy to verify that $E'\subseteq E_{z}^{\operatorname{over}}$. From \Cref{obs: bad inded structure}, $|\hat E_z|\geq |E_{z}^{\operatorname{over}}|\geq |E^{\operatorname{right}}_{i}|/2$. The proof for index $\mathsf{LM}(i)<z'< i$ is symmetric.
\end{proof}
\begin{claim}\label{claim: inner and outer paths}
Let $a$ be some bad index and let $z\in \set{1,\ldots,r}$ be another index such that $a<z<\mathsf{RM}(a)$.
If we denote by $X$ the cluster of $H$ induced by vertices $x_{a+1},\ldots,x_z$, then (i) there is a set $\hat{\mathcal{Q}}(X)$ of paths routing the edges of $\delta_H(X)$ to vertex $x_z$ in $X$ with congestion $O(1)$; and (ii) there is a set $\hat{\mathcal{P}}(X)$ of paths routing the edges of $\delta_H(X)$ to vertex $x_{z+1}$ outside $X$ with congestion $O(1)$.
Similarly,
let $a'$ be some bad index and let $z'\in \set{1,\ldots,r}$ be another index such that $\mathsf{LM}(a')<z'<a$.
If we denote by $X'$ the cluster of $H$ induced by vertices $x_{z'},\ldots,x_{a'-1}$, then (i) there is a set $\hat{\mathcal{Q}}(X')$ of paths routing the edges of $\delta_H(X')$ to vertex $x_{z'}$ in $X'$ with congestion $O(1)$; and (ii) there is a set $\hat{\mathcal{P}}(X')$ of paths routing the edges of $\delta_H(X')$ to vertex $x_{z'-1}$ outside $X$ with congestion $O(1)$.
\end{claim}
\begin{proof}
We only prove the claim for cluster $X$, and the proof for cluster $X'$ is symmetric.
We first prove (i). We partition the set $\delta_H(X)$ into two subsets: set $E_1$ contains all edges of $\delta_H(X)$ with an endpoint in $S_a$ (recall that $S_a=\set{x_1,\ldots,x_a}$); and set $E_2$ contains all edges of $\delta_H(X)$ with an endpoint in $\overline{S}_z$ (recall that $\overline{S}_z=\set{x_{z+1},\ldots,x_q}$).
Recall that, from the properties of minimum cut, there is a set ${\mathcal{R}}$ of edge-disjoint paths routing the edges of $E_2$ to vertex $x_z$ inside $S_z$.
Let ${\mathcal{R}}^{in}\subseteq {\mathcal{R}}$ be the subset of all paths of ${\mathcal{R}}$ that are entirely contained in $X$, and we denote ${\mathcal{R}}^{out}={\mathcal{R}}\setminus {\mathcal{R}}^{in}$.
For each path $R\in {\mathcal{R}}^{out}$, we view $x_z$ as its last vertex and the edge of $E_2$ as its first edge, and we denote by $e_R$ the first edge of $E_1$ that appears on $R$. It is clear that edges in $\set{e_{R}\mid R\in {\mathcal{R}}^{out}}$ are mutually distinct.
On the other hand, since $E_1\subseteq \hat E_a\cup E_{a+1}^{\operatorname{left}}\cup E_a^{\operatorname{over}}$ and $a$ is a bad index, combined with \Cref{obs: bad inded structure}, we get that $|E_1|\leq |\hat E_a|+| E_{a+1}^{\operatorname{left}}|+|E_a^{\operatorname{over}}|\leq 2|E_a^{\operatorname{right}}|$. Moreover, from \Cref{obs: large capacity up to midpoint}, for all $a<z'\leq z$, $|\hat E_{z'}|\geq |E_a^{\operatorname{right}}|/2\ge |E_1|/4$.
Therefore,
there is a set $\hat{\mathcal{Q}}_1=\set{\hat Q_e\mid e\in E_1}$ of paths, routing edges of $E_1$ to $x_z$ in $X$, using only edges of $\bigcup_{a< z'< z}\hat E_{z'}$, such that $\cong_{X}(\hat{\mathcal{Q}}_1)\le 4$; and similarly,
there is a set $\hat{{\mathcal{R}}}^{out}=\set{\hat R\mid R\in {\mathcal{R}}^{out}}$ of paths, routing edges of $\set{e_R\mid R\in {\mathcal{R}}^{out}}$ to $x_z$ in $X$, using only edges of $\bigcup_{a< z'< z}\hat E_{z'}$, such that $\cong_{X}(\hat{{\mathcal{R}}}^{out})\le 4$.
We now define $\hat{\mathcal{Q}}^{out}$ to be the set that contains, for each path $R\in {\mathcal{R}}^{out}$, the concatenation of (i) the subpath of $R$ between its first edge (in $E_2$) and edge $e_R$ (not included); and (ii) the path $\hat R\setminus \set{e_R}$.
We now let $\hat{\mathcal{Q}}_2={\mathcal{R}}^{in}\cup\hat{\mathcal{Q}}^{out}$ and $\hat{\mathcal{Q}}(X)=\hat{\mathcal{Q}}_1\cup \hat{\mathcal{Q}}_2$.
It is clear that the set $\hat{\mathcal{Q}}(X)$ contains, for each edge of $\delta_H(X)$, a path routing the edge to $x_z$ in $X$, and
$$\cong_{X}(\hat{\mathcal{Q}}(X))\le \cong_{X}(\hat{\mathcal{Q}}_1)+\cong_{X}(\hat{\mathcal{Q}}_2)\le 4+\cong_{X}({\mathcal{R}}^{in})+\cong_{X}(\hat{\mathcal{Q}}^{out})=4+1+(1+4)=10.$$
\begin{figure}[h]
\centering
\subfigure[Cluster $X$ and its relevant vertices and edge sets.]{\scalebox{0.3}{\includegraphics{figs/prob_index_1.jpg}}\label{fig: path_cluster_H}
}
\hspace{4pt}
\subfigure[Cluster $X'$ and its relevant vertices and edge sets.]{\scalebox{0.3}{\includegraphics{figs/prob_index_2.jpg}}\label{fig: path_cluster_G}}
\caption{An illustration of clusters $X,X'$ in $H$.
\end{figure}
We now prove (ii).
From the properties of minimum cuts, there is a set ${\mathcal{R}}_1$ of edge-disjoint paths routing edges of $E_1$ to $x_{a}$ in $S_{a}$ (and therefore outside $X$).
Similarly, there is a set ${\mathcal{R}}_2$ of edge-disjoint paths routing the edges of $E_2$ to $x_{z+1}$ inside $\overline{S}_{z}$ (and therefore outside $X$). As observed before, $|E_1|\leq 2|E_a^{\operatorname{right}}|$. Moreover, there is a set $E'\subseteq E_a^{\operatorname{right}}$ of at least $|E_a^{\operatorname{right}}|/2$ edges, each of which connects $x_a$ to a vertex in $\set{x_{z+1},\ldots,x_r}$.
For each edge $e\in E_1$, we assign to it an edge $e'$ of $E'$, such that each edge of $E'$ is assigned to at most $4$ edges of $E'_1$.
Observe that $E'\subseteq \delta_H(\overline{S}_{z})$, so there is a set ${\mathcal{R}}_3$ of $|E'|\geq |E_a^{\operatorname{right}}|/2$ edge-disjoint paths routing the edges of $E'$ to $x_{z+1}$, that are disjoint from $X$.
We now define $\hat{{\mathcal{P}}}_1$ to be the set that contains, for each edge $e\in E_1$, the union of (i) the path in ${\mathcal{R}}_1$ routing $e$ to $x_a$ in $S_a$; and (ii) the path in ${\mathcal{R}}_3$ that routing $e'$ to $x_{z+1}$ in $\overline{S}_z$.
We let $\hat{{\mathcal{P}}}(X)={\mathcal{R}}_2\cup \hat{{\mathcal{P}}}_1$.
It is clear that the set $\hat{\mathcal{P}}(X)$ contains, for each edge of $\delta_H(X)$, a path routing the edge to vertex $x_{z+1}$ outside $X$, and
$$\cong_{H}(\hat{\mathcal{Q}}(X))\le
\cong_{H}({\mathcal{R}}_2)+\cong_{H}(\hat{\mathcal{P}}_1)\le 1+\cong_{H}({\mathcal{R}}_1)+4\cdot\cong_{H}({\mathcal{R}}_3)=1+(1+4)=6.$$
\end{proof}
\subsubsection{Problematic Indices and Inner Paths}
\iffalse
We say that an index $1\le i\le r$ is \emph{left-problematic}, iff $i$ is bad and
$|E(x_{\mathsf{LM}(i)},x_{i+1})|\ge 0.99\cdot|E^{\operatorname{left}}_{i+1}|$;
and similarly, we say that $i$ is \emph{right-problematic}, iff $i$ is bad and
$|E(x_{i},x_{\mathsf{RM}(i)})|\ge 0.99\cdot|E^{\operatorname{right}}_{i}|$.
\fi
\iffalse
\begin{definition}
We say that an index $i$ is \emph{left-problematic} iff (i) index $i$ is bad; (ii) index $\mathsf{LM}(i)$ is bad; and (iii) $|E(x_{i},x_{\mathsf{LM}(i)})|\ge 0.99\cdot|E^{\operatorname{left}}_{i+1}|$.
Similarly, we say that an index $i$ is \emph{right-problematic} iff (i) index $i$ is bad; (ii) index $\mathsf{RM}(i)$ is bad; and (iii) $|E(x_{i},x_{\mathsf{LM}(i)})|\ge 0.99\cdot|E^{\operatorname{right}}_{i}|$.
\end{definition}
\fi
\begin{definition}
We say that an ordered pair $(i,i')$ (with $i<i'$) of indices is \emph{problematic}, iff
\begin{itemize}
\item both indices $i,i'$ are bad;
\item $i=\mathsf{LM}(i')$ and $i'+1=\mathsf{RM}(i)$; and
\item $|E(x_{i},x_{i'+1})|\ge 0.99\cdot\max\set{|E^{\operatorname{left}}_{i'+1}|,|E^{\operatorname{right}}_{i}|}$.
\end{itemize}
\end{definition}
We say that an index $i$ is \emph{left-problematic}, iff there exists an index $i'<i$, such that the pair $(i',i)$ is problematic. Similarly, we say that an index $i$ is \emph{right-problematic}, iff there exists an index $i'>i$, such that the pair $(i,i')$ is problematic.
Define $E'=E(H)\setminus (\bigcup_{1\le i\le r-1}\hat E_i)$.
In other words, $E'$ is the subset of edges in $E^{\mathsf{out}}({\mathcal{C}})$ connecting a pair $C_i,C_j$ of clusters of ${\mathcal{C}}$, with $j\ge i+2$, and if we view $E'$ as a set of edges in $H$, then $E'$ contains all edges connecting a pair $x_i,x_j$ of vertices in $H$ with $j\ge i+2$.
For an edge $e\in E'$ connecting $x_i$ to $x_j$, we say that a path $P$ in $H$ is a \emph{left inner path} of $e$, iff the endpoints of $P$ are $x_i$ and $x_{i+1}$, and $P$ only contains vertices of $S_{i+1}$; we say that a path $P$ in $H$ is a \emph{right inner path} of $e$, iff the endpoints of $P$ are $x_{j-1}$ and $x_{j}$, and $P$ only contains vertices of $\overline{S}_{j-2}$; and we say that a path $P$ in $H$ is a \emph{middle inner path} of $e$, iff path $P$ sequentially visits vertices $x_{i+1}, x_{i+2},\dots,x_{j-1}$.
See \Cref{fig: LMR_inner} for an illustration.
\begin{figure}[h]
\centering
\includegraphics[scale=0.20]{figs/LMR_inner.jpg}
\caption{An illustration of left inner path (orange), middle inner path (purple) and right inner path (green) of edge $(x_i,x_j)\in E'$.}\label{fig: LMR_inner}
\end{figure}
We use the following observations and claims.
\begin{observation}
\label{obs: middle_inner_paths}
We can efficiently construct, for each edge $e\in E'$, a middle inner path $P^{\textsf{Mid}}_e$ of $e$, such that the paths in
${\mathcal{P}}^{\textsf{Mid}}=\set{P^{\textsf{Mid}}_e\mid e\in E'}$ are edge-disjoint.
\end{observation}
\begin{proof}
From the definition of middle inner paths, for each index $1\le i\le r$, the only edges in $E'$ whose middle inner paths contain an edge of $\hat E_i$ are edges of $E^{\operatorname{over}}_i$. From \Cref{obs: bad inded structure}, $|E^{\operatorname{over}}_i|\le \hat E_i$. Therefore, we can assign a distinct edge of $\hat E_i$ to each edge of $E^{\operatorname{over}}_i$. It is clear that, for each edge $e\in E'$, the union of all edges that are assigned to it forms a middle inner path of $e$. \Cref{obs: middle_inner_paths} then follows.
\end{proof}
\begin{claim}
If index $i$ is not left-problematic, then we can efficiently construct a set ${\mathcal{P}}^{\operatorname{left}}_i$ that contains, for each edge $e\in E^{\operatorname{right}}_i$, a left inner path $P^{\operatorname{left}}_e$ of $e$, such that $\cong_H({\mathcal{P}}^{\operatorname{left}}_i)\le O(1)$. Similarly, if index $i$ is not right-problematic, then we can efficiently construct a set ${\mathcal{P}}^{\operatorname{right}}_i$ that contains, for each edge $e\in E^{\operatorname{left}}_{i+1}$, a right inner path $P^{\operatorname{right}}_e$ of $e$, such that $\cong_H({\mathcal{P}}^{\operatorname{right}}_i)=O(1)$.
\end{claim}
\begin{proof}
We only consider the case where index $i$ is not left-problematic. The proof of the case where index $i$ is not right-problematic is symmetric. From the definition of a left-problematic index, either index $i$ is not bad, or $|E(x_{i+1},x_{\mathsf{LM}(i)})|\le 0.99\cdot |E^{\operatorname{left}}_{i+1}|$, or $|E(x_{i+1},x_{\mathsf{LM}(i)})|\le 0.99\cdot |E^{\operatorname{right}}_{\mathsf{LM}(i)}|$, or index $i$ is bad, $|E(x_{i+1},x_{\mathsf{LM}(i)})|> 0.99\cdot |E^{\operatorname{left}}_{i+1}|$, but index $\mathsf{LM}(i)$ is not bad. We consider each of these cases in turn.
\paragraph{Case 1. Index $i$ is not bad.}
From \Cref{obs: problematic instance}, $|\hat E_i|\ge |E^\operatorname{right}_{i}|/5001$. Therefore, we simply let set ${\mathcal{P}}^{\operatorname{left}}_i$ contain, for each edge $e\in \hat E_{i}$, at most $5001$ paths that contain the single edge $e$, such that $|{\mathcal{P}}^{\operatorname{left}}_i|=|E^\operatorname{right}_i|$. It is clear that these paths have endpoints $x_i$ and $x_{i+1}$, and $\cong_H({\mathcal{P}}^{\operatorname{left}}_i)\le O(1)$.
\paragraph{Case 2. $|E(x_{i+1},x_{\mathsf{LM}(i)})|\le 0.99\cdot |E^{\operatorname{left}}_{i+1}|$.}
Since set $E^{\operatorname{left}}_{i+1}$ is the union of sets $E(x_{i+1},\set{x_{\mathsf{LM}(i)+1},\ldots,x_{i-1}})$, $E(x_{i+1},x_{\mathsf{LM}(i)})$ and $E(x_{i+1},S_{\mathsf{LM}(i)-1})$, either $|E(x_{i+1},\set{x_{\mathsf{LM}(i)+1},\ldots,x_{i-1}})|\ge 0.005 |E^{\operatorname{left}}_{i+1}|$ holds, or $|E(x_i,S_{\mathsf{LM}(i)-1})|\ge 0.005 |E^{\operatorname{left}}_{i+1}|$ holds. We consider these cases in turn.
\paragraph{Case 2.1. $|E(x_{i+1},\set{x_{\mathsf{LM}(i)+1},\ldots,x_{i-1}})|\ge 0.005 \cdot|E^{\operatorname{left}}_{i+1}|$.}
Denote $\tilde E_i=E(x_{i+1},\set{x_{\mathsf{LM}(i)+1},\ldots,x_{i-1}})$.
Recal that from \Cref{obs: large capacity up to midpoint}, for each $\mathsf{LM}(i)< z < i$, $|\hat E_{z}|\ge |E^{\operatorname{left}}_{i+1}|/2$.
We first process all edges of $\tilde E_i$ iteratively as follows. Throughout, for each edge $\hat e\in \bigcup_{\mathsf{LM}(i)+1\le t\le i-1}\hat E_t$, we maintain an integer $z_{\hat e}$ indicating how many times the edge $\hat e$ has been used, that is initialized to be $0$.
Consider an iteration of processing an edge $e\in \tilde E_i$. Assume $e$ connects $x_{i+1}$ to $x_j$, for some $\mathsf{LM}(i)\le t\le i-1$, we then pick, for each $j\le t\le i-1$, an edge $\hat e$ of $\hat E_t$ with minimum $z_{\hat e}$ over all edges of $\hat E_t$, and let $P^{\operatorname{left}}_e$ be the path obtained by taking the union of $e$ and all picked edges.
It is easy to see that the path $P^{\operatorname{left}}_e$ sequentially visits nodes $x_{i+1}, x_j, x_{j+1},\ldots,x_i$ (and so have endpoints $x_{i+1},x_i$).
We then increase the value of $z_{\hat e}$ by $1$ for all picked edges $\hat e$, and proceed to the next iteration. Let $\tilde{\mathcal{P}}^{\operatorname{left}}_i$ be the set of left inner paths after we processed all edges of $E'$ in this way, so $|\tilde{\mathcal{P}}^{\operatorname{left}}_i|=|\tilde E_i|\ge 0.005|E^{\operatorname{left}}_i|$,
and it is clear that $\cong_H(\tilde{\mathcal{P}}^{\operatorname{left}}_i)\le 2$.
We then let set ${\mathcal{P}}^{\operatorname{left}}_i$ contain, at most $200$ copies of each path in $\tilde{\mathcal{P}}^{\operatorname{left}}_i$, so left-inner paths in set ${\mathcal{P}}^{\operatorname{left}}_i$ connect $x_{i+1}$ to $x_i$, and $\cong_H({\mathcal{P}}^{\operatorname{left}}_i)=O(1)$.
\paragraph{Case 2.2. $|E(x_{i+1},S_{\mathsf{LM}(i)-1})|\ge 0.005 |E^{\operatorname{left}}_{i+1}|$.}
Note that $E(x_{i+1},S_{\mathsf{LM}(i)-1})\subseteq E^{\operatorname{over}}_{\mathsf{LM}(i)}$, so $|\hat E_{\mathsf{LM}(i)}|\ge |E(x_{i+1},S_{\mathsf{LM}(i)-1})|\ge 0.005 |E^{\operatorname{left}}_{i+1}|$, and we can construct the set ${\mathcal{P}}^{\operatorname{left}}_i$ of left-inner paths in the same way as Case 2.1, by processing edges of $\tilde E_i\cup E(x_{i+1},x_{\mathsf{LM}(i)})$, with $\cong_H({\mathcal{P}}^{\operatorname{left}}_i)=O(1)$.
\paragraph{Case 3. $|E(x_{i+1},x_{\mathsf{LM}(i)})|\le 0.99\cdot |E^{\operatorname{right}}_{\mathsf{LM}(i)}|$.}
Since set $E^{\operatorname{right}}_{\mathsf{LM}(i)}$ is the union of three disjoint edge sets $E(x_{\mathsf{LM}(i)},\set{x_{\mathsf{LM}(i)+2},\ldots,x_{i}})$, $E(x_{i+1},x_{\mathsf{LM}(i)})$ and $E(x_{\mathsf{LM}(i)},\overline{S}_{i+1})$, either $|E(x_{\mathsf{LM}(i)},\set{x_{\mathsf{LM}(i)+2},\ldots,x_{i}})|\ge 0.005 |E^{\operatorname{left}}_{i+1}|$ holds, or $|E(x_{\mathsf{LM}(i)},\overline{S}_{i+1})|\ge 0.005 |E^{\operatorname{left}}_{i+1}|$ holds.
It is easy to see that, in the former case, we can construct the set ${\mathcal{P}}^{\operatorname{left}}_i$ of left-inner paths in the same way as Case 2.1, with $\cong_H({\mathcal{P}}^{\operatorname{left}}_i)=O(1)$.
Assume now that the latter case happens
Note that $E(x_{\mathsf{LM}(i)},\overline{S}_{i+1})\subseteq E^{\operatorname{over}}_i$, so $|\hat E_i|\ge |E^{\operatorname{over}}_i|\ge |E(x_{\mathsf{LM}(i)},\overline{S}_{i+1})|\ge 0.005 |E^{\operatorname{left}}_{i+1}|$, and therefore index $i$ is not bad. Therefore we can can construct the set ${\mathcal{P}}^{\operatorname{left}}_i$ of left-inner paths in the same way as Case 1.
\paragraph{Case 4. Index $i$ is bad, $|E(x_{i},x_{\mathsf{LM}(i)})| > 0.99\cdot |E^{\operatorname{left}}_{i+1}|$, but index $\mathsf{LM}(i)$ is not bad.}
We first show that, for each index $\mathsf{LM}(i)\le z < i$, $|\hat E_{z}|\ge |E^{\operatorname{left}}_{i+1}|/6000$.
First, from \Cref{obs: large capacity up to midpoint}, for each $\mathsf{LM}(i)< z < i$, $|\hat E_{z}|\ge |E^{\operatorname{left}}_{i+1}|/2$.
Since $\mathsf{LM}(i)$ is not bad, from \Cref{obs: problematic instance}, $|\hat E_{\mathsf{LM}(i)}|\ge |E^\operatorname{right}_{\mathsf{LM}(i)}|/5001$.
Note that $E(x_{i},x_{\mathsf{LM}(i)})\subseteq E^{\operatorname{right}}_{\mathsf{LM}(i)}$,
$|\hat E_{\mathsf{LM}(i)}|\ge |E(x_{i},x_{\mathsf{LM}(i)})|/5001\ge |E^{\operatorname{left}}_{i+1}|/6000$.
Altogether, we get that for each $\mathsf{LM}(i)\le z < i$, $|\hat E_{z}|\ge |E^{\operatorname{left}}_{i+1}|/6000$.
It is easy to see that we can construct the set ${\mathcal{P}}^{\operatorname{left}}_i$ of left-inner paths in the same way as in Case 2.1, with $\cong_H({\mathcal{P}}^{\operatorname{left}}_i)=O(1)$.
\end{proof}
\begin{observation}
\label{obs: problematic pair non-crossing}
Problematic pairs do not interleave.
In other words, if $(i_1,i'_1)$, $(i_2,i'_2)$ are problematic pairs, with $i_1<i_2$, then either $i_1<i_2<i'_2<i'_1$, or $i_1<i'_1<i_2<i'_2$.
\end{observation}
\begin{proof}
Assume for the sake of contradiction that $i_1<i_2<i'_1<i'_2$.
Note that $E(x_{i_1},x_{i'_1+1})\subseteq E^{\operatorname{over}}_{i_2}$ and $E(x_{i_2},x_{i'_2+1})\subseteq E^{\operatorname{over}}_{i'_1}$.
Therefore,
\[|E(x_{i_1},x_{i'_1+1})| \le |E^{\operatorname{over}}_{i_2}| \le |\hat E_{i_2}|
\le \frac{|E(x_{i_2},x_{i'_2+1})|}{0.99\times 4998}
\le \frac{|E^{\operatorname{over}}_{i'_1}|}{0.99\times 4998}
\le \frac{|\hat E_{i'_1}|}{0.99\times 4998}
\le \frac{|E(x_{i_1},x_{i'_1+1})|}{(0.99\times 4998)^2},\]
a contradiction.
\end{proof}
\begin{observation}
\label{obs: nested problematic pair}
If index pairs $(i_1,i'_1), (i_2,i'_2)$ are problematic and satisfy that $i_1<i_2<i'_2<i'_1$, then $|E(x_{i_1},x_{i'_1+1})|\le |E(x_{i_2},x_{i'_2+1})|/4000$.
\end{observation}
\begin{proof}
Note that $E(x_{i_1},x_{i'_1+1})\subseteq E^{\operatorname{over}}_{i_2}$, so
$|E(x_{i_1},x_{i'_1+1})| \le |E^{\operatorname{over}}_{i_2}| \le |\hat E_{i_2}|
\le |E(x_{i_2},x_{i'_2+1})|/(0.99\cdot 4998)
\le |E(x_{i_2},x_{i'_2+1})|/4000$.
\end{proof}
\begin{claim}
\label{claim: alpha cut property}
Let $(a,z)$ be some problematic pair of indices.
Define $H'$ to be the graph obtained from $H$ by contracting vertices $x_1,\ldots,x_{a},x_{z+1},\ldots,x_r$ into a single vertex $v^*$.
Then for each $a< z'< z$,
$$|E_{H'}(x_{z'},x_{z'+1})|\ge \frac{|E_{H'}(\set{x_{a+1},\ldots,x_{z'}},\set{v^*})|}{6}.$$
\end{claim}
\begin{figure}[h]
\centering
\subfigure[Edge sets $E'_1,E'_2$ in graph $H$.]{\scalebox{0.12}{\includegraphics{figs/cut_property_1.jpg}}}
\hspace{4pt}
\subfigure[Edge sets $E'_1,E'_2$ in graph $H'$.]{\scalebox{0.12}{\includegraphics{figs/cut_property_2.jpg}}}
\caption{An illustration of edge sets in $H$ and $H'$.}\label{fig: cut_property}
\end{figure}
\begin{proof}
Fix some $z'$ with $a<z'< z$. We denote $S'_{z'}=\set{x_{a+1},\ldots,x_{z'}}$ and $\overline{S}_{z'}'=\set{x_{z'+1},\ldots,x_z}$.
Note that vertices of $S'_{z'},\overline{S}'_{z'}$ are also vertices of $V(H)$.
We denote $E'_1=E_H(S'_{z'},\overline{S}_z)$ and $E'_2=E_H(S'_{z'},S_a)$. Note that edges of $E'_1,E'_2$ are also edges of $H'$, and $E_{H'}(\set{x_{a+1},\ldots,x_{z'}},\set{v^*})=E'_1\cup E'_2$.
See \Cref{fig: cut_property} for an illustration.
On the one hand, $E'_1\subseteq E^{\operatorname{left}}_{z+1}\cup E^{\operatorname{over}}_z$.
Since $(a,z)$ is a problematic pair, $|E(x_a,x_{z+1})|\ge 0.99\cdot |E^{\operatorname{left}}_{z+1}|$.
Therefore,
$$|E'_1|\le |E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_z|\le 1.01\cdot|E^{\operatorname{left}}_{z+1}|\le 1.03\cdot|E(x_a,x_{z+1})|\le 1.03\cdot|\hat E_{z'}|= 1.03\cdot|E_{H'}(x_{z'},x_{z'+1})|.$$
On the other hand, from \Cref{obs: bad inded structure}, $|E'_2|\le |E^{\operatorname{right}}_{a}|+|E^{\operatorname{left}}_{a+1}|+|E^{\operatorname{over}}_{a}|\le 2\cdot|E^{\operatorname{right}}_{a}|$, and from \Cref{obs: large capacity up to midpoint}, $|E_{H'}(x_{z'},x_{z'+1})|=|\hat E_{z'}|\ge |E^{\operatorname{right}}_{a}|/2$.
Therefore, $|E'_2|\le 4\cdot |E_{H'}(x_{z'},x_{z'+1})|$. Altogether,
$|E_1'|+|E_2'|\le 6\cdot |E_{H'}(x_{z'},x_{z'+1})|$.
\iffalse
Consider now the cut $(S_z,\set{v^*})$ in $H$.
We define sets $E',E'_1,E'_2,E'_3$ similarly. Clearly, $E'_1=\emptyset$. Observe that $E_H(x_z,x_{z+1})\subseteq E(S_z,\set{v^*})$, so it suffices to show that $|E'_2|+|E'_3|\le O(|E(x_z,x_{z+1})|)$.
Since $z$ is not a bad index, $|E^{\operatorname{right}}_{z}|+|E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_{z}|\le 100\cdot|E(x_{z},x_{z+1})|$.
Note that
$$|E'_2|\le 2\cdot|E^{\operatorname{right}}_{a}|\le 2(|E(x_a,x_{z+1})|+|E^{\operatorname{over}}_{z}|)\le 2(|E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_{z}|),$$
and
$$|E'_3|\le |E(S'_z,x_{z+1})|+|E^{\operatorname{over}}_{z}|\le |E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_{z}|.$$
So in this case $|E'|\le 300\cdot |E(x_{z'},x_{z'+1})|\le 300\cdot|E(S_z,\set{v^*})|$.
This completes the proof that $H'$ has the $\alpha$-cut property with respect to path $\tau'$, for $\alpha=300$.
\fi
\end{proof}
\subsubsection{Bad Indices and their Basic Properties}
In this subsection we define bad indices and analyze its properties.
For each $1\leq i\leq r$, we define $S_i=\set{x_1,\ldots,x_i}$ and $\overline{S}_i=\set{x_{i+1},\ldots,x_r}$. Recall that, by the definition of the Gomory-Hu tree, $(S_i, \overline{S}_i)$ is a minimum cut in graph $H$ separating vertex $x_i$ from vertex $x_{i+1}$. From the properties of minimum cut, there is a set of edge-disjoint paths in graph $H$, routing the edges of $\delta_H(S_i)$ to vertex $x_{i}$ inside $S_i$, and there is a set of edge-disjoint paths in graph $H$, routing the edges of $\delta_H(\overline{S}_i)$ to vertex $x_{i+1}$ inside $\overline{S}_i$.
\begin{definition}
We say that index $i$ is \emph{bad} iff $|E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|>10000|\hat E_i|$.
\end{definition}
\begin{observation}\label{obs: bad inded structure}
For all $1\leq i<r$, the following hold:
\begin{itemize}
\item $|\hat E_i|\geq |E_i^{\operatorname{over}}|$;
\item $|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_{i+1}^{\operatorname{left}}|-|E_i^{\operatorname{right}}|$; and
\item $|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$.
\end{itemize}
\end{observation}
\begin{proof}
Consider the cut $(\set{x_{i}},V(H)\setminus \set{x_i})$ in $H$. Its size is $|\hat E_i|+|\hat E_{i-1}|+ |E_i^{\operatorname{right}}|+|E_{i}^{\operatorname{left}}|$. Note that this cut separates $x_i$ from $x_{i-1}$. Since the minimum cut separating $x_i$ from $x_{i-1}$ in $H$ is $(S_{i-1},\overline{S}_{i-1})$, and $|E(S_{i-1},\overline{S}_{i-1})|=|\hat E_{i-1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|$, we get that $|\hat E_{i-1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|\hat E_{i-1}|+ |E_i^{\operatorname{right}}|+|E_{i}^{\operatorname{left}}|$,
and so
\begin{equation}
\label{eqn1}
|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+ |E_i^{\operatorname{right}}|.
\end{equation}
Consider the cut $(\set{x_{i+1}},V(H)\setminus \set{x_{i+1}})$ in $H$. Its size is $|\hat E_i|+|\hat E_{i+1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i+1}^{\operatorname{right}}|$. Note that this cut separates $x_{i+1}$ from $x_{i+2}$. Since the minimum cut separating $x_{i+1}$ from $x_{i+2}$ in $H$ is $(S_{i+1},\overline{S}_{i+1})$, and $|E(S_{i+1},\overline{S}_{i+1})|=|\hat E_{i+1}|+ |E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{right}}|+|E_i^{\operatorname{over}}|$, we get that $|\hat E_{i+1}|+ |E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{right}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|\hat E_{i+1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i+1}^{\operatorname{right}}|$,
and so
\begin{equation}
\label{eqn2}
|E_i^{\operatorname{right}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|E_{i+1}^{\operatorname{left}}|.
\end{equation}
Combining \Cref{eqn1,eqn2}, we get that $|\hat E_i|\geq |E_i^{\operatorname{over}}|$. By rearranging the sides of the two inequalities, we get that
$|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_{i+1}^{\operatorname{left}}|-|E_i^{\operatorname{right}}|$, and
$|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$.
\end{proof}
We prove the following observation.
\begin{observation}\label{obs: bad instance}
If $i$ is a bad index, then $|E_{i+1}^{\operatorname{left}}|-|\hat E_i| \leq |E_i^{\operatorname{right}}|\leq |E_{i+1}^{\operatorname{left}}|+|\hat E_i|$, and moreover $4998|E_{i+1}^{\operatorname{left}}|/4999 \leq |E_i^{\operatorname{right}}|\leq 4999|E_{i+1}^{\operatorname{left}}|/4998$.
If index $i$ is not bad, then $|\hat E_i|\ge |E_{i+1}^{\operatorname{left}}|/5001$ and $|\hat E_i|\ge |E_{i}^{\operatorname{right}}|/5001$.
\end{observation}
\begin{proof}
Assume first that index $i$ is bad. Since $|E_i^{\operatorname{over}}|\leq |\hat E_i|$, we get that either $|E_i^{\operatorname{right}}|\geq 4999|\hat E_i|$, or $|E_{i+1}^{\operatorname{left}}|\geq 4999|\hat E_i|$. Assume without loss of generality that $|E_i^{\operatorname{right}}|\geq |E_{i+1}^{\operatorname{left}}|$, so $|E_i^{\operatorname{right}}|\geq 4999 |\hat E_i|$. Since, from \Cref{obs: bad inded structure},
$|\hat E_i| \geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$, we get that $|E_{i+1}^{\operatorname{left}}|\geq |E_i^{\operatorname{right}}|-|\hat E_i|\geq 4998|E_i^{\operatorname{right}}|/4999$.
Assume now that $i$ is not bad. By definition, $10000|\hat E_i|\ge |E^\operatorname{right}_i|+|E^\operatorname{left}_{i+1}|+|E^{\operatorname{over}}_i|$. Combined with \Cref{obs: bad inded structure}, we get that $|\hat E_i|\ge |E^\operatorname{left}_{i+1}|/5001$ and $|\hat E_i|\ge |E^\operatorname{right}_{i}|/5001$.
\end{proof}
\paragraph{Nice pair of clusters.} Let $a<b<c$ be indices. We say that the pair $C_{(a,b]}, C_{(b,c]}$ of clusters are \emph{nice}, iff $|E(C_{(a,b]},C_{(b,c]})|\ge |E(C_{(a,b]},C_{(c,r]})|/1000$
and $|E(C_{(a,b]},C_{(b,c]})|\ge |E(C_{(b,c]},C_{(0,a]})|/10000$.
In other words, if we let ${\mathcal{C}}'$ be the set of clusters obtained from ${\mathcal{C}}$ by replacing clusters $C_{a+1},\ldots,C_c$ with clusters $C_{(a,b]}, C_{(b,c]}$ then the Gomory-Hu tree $\tau'$ of the contracted subugraph $G_{\mid {\mathcal{C}}'}$ is a path where the index corresponding to cluster $C_{(a,b]}$ is not a bad index.
We use the following claim and its immediate corollary.
\begin{claim}
\label{clm: contraction does not create new bad indices}
Let $a<b<c$ be indices where $b$ is not bad. Then the pair $C_{(a,b]}, C_{(b,c]}$ of clusters is nice.
\end{claim}
\begin{proof}
Note that $E(C_{(a,b]},C_{(c,r]})\subseteq \big(E(C_{(a,b-1]},C_{(c,r]})\cup E(C_{b},C_{(c,r]})\big)\subseteq \big(E^{\operatorname{over}}_b\cup E^{\operatorname{right}}_b\big)$, and note that $\hat E_b\subseteq E(C_{(a,b]},C_{(b,c]})$. Since index $b$ is not bad, from \Cref{obs: bad instance} and \Cref{obs: bad inded structure}, we get that $|\hat E_b|\ge |E^{\operatorname{over}}_b|$ and $|\hat E_b|\ge |E^{\operatorname{right}}_b|/5001$, and therefore $|\hat E_b|\ge |E^{\operatorname{over}}_b\cup E^{\operatorname{right}}_b|/10000$. It follows that $|E(C_{(a,b]},C_{(b,c]})|\ge |E(C_{(a,b]},C_{(c,r]})|/1000$. Similarly, we can show that $|E(C_{(a,b]},C_{(b,c]})|\ge |E(C_{(b,c]},C_{(0,a]})|/10000$.
\end{proof}
\begin{corollary}
\label{cor: nice pairs imply no bad index}
Let ${\mathcal{Y}}$ be a ${\mathcal{C}}$-respecting partition of $G$ into a sequence ${\mathcal{Y}}=\set{Y_1,\ldots,Y_k}$ of clusters, such that every pair $Y_s,Y_{s+1}$ of clusters is nice. Then Gomory-Hu tree of the contracted graph $H'=G_{\mid {\mathcal{Y}}}$ is the path $(y_1,\ldots,y_k)$ (where $y_s$ is the vertex of $H'$ that corresponds to $Y_s$), and it has no bad indices.
\end{corollary}
\paragraph{Left and right midpoints.}
Let $1\le i\le r$ be some index.
We define indices $\mathsf{LM}(i)$ and $\mathsf{RM}(i)$ as follows.
Recall that $E^{\operatorname{right}}_{i}$ is the set of all edges $(x_{i},x_z)$ with $z>i+1$. We let $\mathsf{RM}(i)$ be the smallest integer $z>i+1$, such that at least half the edges in $E^{\operatorname{right}}_{i}$ have their endpoints in $\set{x_{i+2},\ldots,x_z}$. It is easy to see that at least half the edges in $E^{\operatorname{right}}_{i}$ must have their endpoints in $\set{x_{z},\ldots,x_r}$. Similarly, we let $\mathsf{LM}(i)$ be the largest integer $z<i$, such that at least half the edges in $E^{\operatorname{left}}_{i+1}$ have their endpoints in $\set{x_{z},\ldots,x_{i-1}}$. It is easy to see that at least half the edges in $E^{\operatorname{left}}_{i+1}$ must have their endpoints in $\set{x_{1},\ldots,x_z}$. We need the following crucial observation and claim.
\begin{observation}\label{obs: large capacity up to midpoint}
Let $1\leq i\leq r$ be an index. Then for every index $i<z<\mathsf{RM}(i)-1$, $|\hat E_z|\geq |E^{\operatorname{right}}_{i}|/2$; similarly, for every index $\mathsf{LM}(i)<z'< i$, $|\hat E_{z'}|\geq |E^{\operatorname{left}}_{i+1}|/2$.
\end{observation}
\begin{proof}
Fix an index $1\leq i\leq r$ and another index $i<z<\mathsf{RM}(i)-1$. Recall that there is a subset $E'\subseteq E^{\operatorname{right}}_{i}$ of at least $|E^{\operatorname{right}}_{i}|/2$ edges that have an endpoint in $\set{x_{\mathsf{RM}(i)},\ldots,x_r}$. It is easy to verify that $E'\subseteq E_{z}^{\operatorname{over}}$. From \Cref{obs: bad inded structure}, $|\hat E_z|\geq |E_{z}^{\operatorname{over}}|\geq |E^{\operatorname{right}}_{i}|/2$. The proof for index $\mathsf{LM}(i)<z'< i$ is symmetric.
\end{proof}
For each pair $1\le j< j'\le r$, we define $X_{[j,j']}$ as the subgraph of $H$ induced by all vertices $x_t$ with index $t$ belonging to the interval $[j,j']$. We define clusters
$X_{(j,j']}$,
$X_{[j,j')}$, and
$X_{(j,j')}$ similarly. We use the following claim.
\begin{claim}\label{claim: inner and outer paths}
Let $a,z$ be indices such that $a$ is bad and $a<z<\mathsf{RM}(a)$.
Then (i) there is a set $\hat{\mathcal{Q}}(X_{(a,z]})$ of paths routing the edges of $\delta_H(X_{(a,z]})$ to vertex $x_z$ in $X_{(a,z]}$ with congestion $O(1)$; and (ii) there is a set $\hat{\mathcal{P}}(X)$ of paths routing the edges of $\delta_H(X_{(a,z]})$ to vertex $x_{z+1}$ outside $X_{(a,z]}$ with congestion $O(1)$.
Similarly, let $a',z'$ be indices such that $a'$ is bad and $\mathsf{LM}(a')<z'<a'$.
Then there exists (i) a set $\hat{\mathcal{Q}}(X_{[z',a')})$ of paths routing edges of $\delta_H(X_{[z',a')})$ to $x_{z'}$ in $X_{[z',a')}$ with congestion $O(1)$; and (ii) a set $\hat{\mathcal{P}}(X_{[z',a')})$ of paths routing edges of $\delta_H(X_{[z',a')})$ to $x_{z'-1}$ outside $X_{[z',a')}$ with congestion $O(1)$.
\end{claim}
\begin{proof}
We only prove the claim for cluster $X_{(a,z]}$, and the proof for cluster $X_{[z',a')}$ is symmetric.
Denote $X=X_{(a,z]}$. We first prove (i). We partition the set $\delta_H(X)$ into two subsets: set $E_1$ contains all edges of $\delta_H(X)$ with an endpoint in $S_a$ (recall that $S_a=\set{x_1,\ldots,x_a}$); and set $E_2$ contains all edges of $\delta_H(X)$ with an endpoint in $\overline{S}_z$ (recall that $\overline{S}_z=\set{x_{z+1},\ldots,x_q}$).
Recall that, from the properties of minimum cut, there is a set ${\mathcal{R}}$ of edge-disjoint paths routing the edges of $E_2$ to vertex $x_z$ inside $S_z$.
Let ${\mathcal{R}}^{in}\subseteq {\mathcal{R}}$ be the subset of all paths of ${\mathcal{R}}$ that are entirely contained in $X$, and we denote ${\mathcal{R}}^{out}={\mathcal{R}}\setminus {\mathcal{R}}^{in}$.
For each path $R\in {\mathcal{R}}^{out}$, we view $x_z$ as its last vertex and the edge of $E_2$ as its first edge, and we denote by $e_R$ the first edge of $E_1$ that appears on $R$. It is clear that edges in $\set{e_{R}\mid R\in {\mathcal{R}}^{out}}$ are mutually distinct.
On the other hand, since $E_1\subseteq \hat E_a\cup E_{a+1}^{\operatorname{left}}\cup E_a^{\operatorname{over}}$ and $a$ is a bad index, combined with \Cref{obs: bad inded structure}, we get that $|E_1|\leq |\hat E_a|+| E_{a+1}^{\operatorname{left}}|+|E_a^{\operatorname{over}}|\leq 2|E_a^{\operatorname{right}}|$. Moreover, from \Cref{obs: large capacity up to midpoint}, for all $a<z'\leq z$, $|\hat E_{z'}|\geq |E_a^{\operatorname{right}}|/2\ge |E_1|/4$.
Therefore,
there is a set $\hat{\mathcal{Q}}_1=\set{\hat Q_e\mid e\in E_1}$ of paths, routing edges of $E_1$ to $x_z$ in $X$, using only edges of $\bigcup_{a< z'< z}\hat E_{z'}$, such that $\cong_{X}(\hat{\mathcal{Q}}_1)\le 4$; and similarly,
there is a set $\hat{{\mathcal{R}}}^{out}=\set{\hat R\mid R\in {\mathcal{R}}^{out}}$ of paths, routing edges of $\set{e_R\mid R\in {\mathcal{R}}^{out}}$ to $x_z$ in $X$, using only edges of $\bigcup_{a< z'< z}\hat E_{z'}$, such that $\cong_{X}(\hat{{\mathcal{R}}}^{out})\le 4$.
We now define $\hat{\mathcal{Q}}^{out}$ to be the set that contains, for each path $R\in {\mathcal{R}}^{out}$, the concatenation of (i) the subpath of $R$ between its first edge (in $E_2$) and edge $e_R$ (not included); and (ii) the path $\hat R\setminus \set{e_R}$.
We now let $\hat{\mathcal{Q}}_2={\mathcal{R}}^{in}\cup\hat{\mathcal{Q}}^{out}$ and $\hat{\mathcal{Q}}(X)=\hat{\mathcal{Q}}_1\cup \hat{\mathcal{Q}}_2$.
It is clear that the set $\hat{\mathcal{Q}}(X)$ contains, for each edge of $\delta_H(X)$, a path routing the edge to $x_z$ in $X$, and
$$\cong_{X}(\hat{\mathcal{Q}}(X))\le \cong_{X}(\hat{\mathcal{Q}}_1)+\cong_{X}(\hat{\mathcal{Q}}_2)\le 4+\cong_{X}({\mathcal{R}}^{in})+\cong_{X}(\hat{\mathcal{Q}}^{out})=4+1+(1+4)=10.$$
\begin{figure}[h]
\centering
\subfigure[Cluster $X$ and its relevant vertices and edge sets.]{\scalebox{0.3}{\includegraphics{figs/prob_index_1.jpg}}\label{fig: path_cluster_H}
}
\hspace{4pt}
\subfigure[Cluster $X'$ and its relevant vertices and edge sets.]{\scalebox{0.3}{\includegraphics{figs/prob_index_2.jpg}}\label{fig: path_cluster_G}}
\caption{An illustration of clusters $X,X'$ in $H$.
\end{figure}
We now prove (ii).
From the properties of minimum cuts, there is a set ${\mathcal{R}}_1$ of edge-disjoint paths routing edges of $E_1$ to $x_{a}$ in $S_{a}$ (and therefore outside $X$).
Similarly, there is a set ${\mathcal{R}}_2$ of edge-disjoint paths routing the edges of $E_2$ to $x_{z+1}$ inside $\overline{S}_{z}$ (and therefore outside $X$). As observed before, $|E_1|\leq 2|E_a^{\operatorname{right}}|$. Moreover, there is a set $E'\subseteq E_a^{\operatorname{right}}$ of at least $|E_a^{\operatorname{right}}|/2$ edges, each of which connects $x_a$ to a vertex in $\set{x_{z+1},\ldots,x_r}$.
For each edge $e\in E_1$, we assign to it an edge $e'$ of $E'$, such that each edge of $E'$ is assigned to at most $4$ edges of $E'_1$.
Observe that $E'\subseteq \delta_H(\overline{S}_{z})$, so there is a set ${\mathcal{R}}_3$ of $|E'|\geq |E_a^{\operatorname{right}}|/2$ edge-disjoint paths routing the edges of $E'$ to $x_{z+1}$, that are disjoint from $X$.
We now define $\hat{{\mathcal{P}}}_1$ to be the set that contains, for each edge $e\in E_1$, the union of (i) the path in ${\mathcal{R}}_1$ routing $e$ to $x_a$ in $S_a$; and (ii) the path in ${\mathcal{R}}_3$ that routing $e'$ to $x_{z+1}$ in $\overline{S}_z$.
We let $\hat{{\mathcal{P}}}(X)={\mathcal{R}}_2\cup \hat{{\mathcal{P}}}_1$.
It is clear that the set $\hat{\mathcal{P}}(X)$ contains, for each edge of $\delta_H(X)$, a path routing the edge to vertex $x_{z+1}$ outside $X$, and
$$\cong_{H}(\hat{\mathcal{Q}}(X))\le
\cong_{H}({\mathcal{R}}_2)+\cong_{H}(\hat{\mathcal{P}}_1)\le 1+\cong_{H}({\mathcal{R}}_1)+4\cdot\cong_{H}({\mathcal{R}}_3)=1+(1+4)=6.$$
\end{proof}
\iffalse
\begin{claim}
\label{claim: alpha cut property}
Let $a$ be a bad index and let $z$ be an index such that $a<z<\mathsf{RM}(a)$.
Define $H'$ to be the graph obtained from $H$ by contracting vertices $x_1,\ldots,x_{a},x_{z+1},\ldots,x_r$ into a single vertex $v^*$.
Then for each $a< z'< z$,
$$|E_{H'}(x_{z'},x_{z'+1})|\ge \frac{|E_{H'}(\set{x_{a+1},\ldots,x_{z'}},\set{v^*})|}{6}.$$
\end{claim}
\begin{figure}[h]
\centering
\subfigure[Edge sets $E'_1,E'_2$ in graph $H$.]{\scalebox{0.12}{\includegraphics{figs/cut_property_1.jpg}}}
\hspace{4pt}
\subfigure[Edge sets $E'_1,E'_2$ in graph $H'$.]{\scalebox{0.12}{\includegraphics{figs/cut_property_2.jpg}}}
\caption{An illustration of edge sets in $H$ and $H'$.}\label{fig: cut_property}
\end{figure}
\begin{proof}
Fix some $z'$ with $a<z'< z$. We denote $S'_{z'}=\set{x_{a+1},\ldots,x_{z'}}$ and $\overline{S}_{z'}'=\set{x_{z'+1},\ldots,x_z}$.
Note that vertices of $S'_{z'},\overline{S}'_{z'}$ are also vertices of $V(H)$.
We denote $E'_1=E_H(S'_{z'},\overline{S}_z)$ and $E'_2=E_H(S'_{z'},S_a)$. Note that edges of $E'_1,E'_2$ are also edges of $H'$, and $E_{H'}(\set{x_{a+1},\ldots,x_{z'}},\set{v^*})=E'_1\cup E'_2$.
See \Cref{fig: cut_property} for an illustration.
On the one hand, $E'_1\subseteq E^{\operatorname{left}}_{z+1}\cup E^{\operatorname{over}}_z$.
Since $(a,z)$ is a problematic pair, $|E(x_a,x_{z+1})|\ge 0.99\cdot |E^{\operatorname{left}}_{z+1}|$.
Therefore,
$$|E'_1|\le |E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_z|\le 1.01\cdot|E^{\operatorname{left}}_{z+1}|\le 1.03\cdot|E(x_a,x_{z+1})|\le 1.03\cdot|\hat E_{z'}|= 1.03\cdot|E_{H'}(x_{z'},x_{z'+1})|.$$
On the other hand, from \Cref{obs: bad inded structure}, $|E'_2|\le |E^{\operatorname{right}}_{a}|+|E^{\operatorname{left}}_{a+1}|+|E^{\operatorname{over}}_{a}|\le 2\cdot|E^{\operatorname{right}}_{a}|$, and from \Cref{obs: large capacity up to midpoint}, $|E_{H'}(x_{z'},x_{z'+1})|=|\hat E_{z'}|\ge |E^{\operatorname{right}}_{a}|/2$.
Therefore, $|E'_2|\le 4\cdot |E_{H'}(x_{z'},x_{z'+1})|$. Altogether,
$|E_1'|+|E_2'|\le 6\cdot |E_{H'}(x_{z'},x_{z'+1})|$.
\iffalse
Consider now the cut $(S_z,\set{v^*})$ in $H$.
We define sets $E',E'_1,E'_2,E'_3$ similarly. Clearly, $E'_1=\emptyset$. Observe that $E_H(x_z,x_{z+1})\subseteq E(S_z,\set{v^*})$, so it suffices to show that $|E'_2|+|E'_3|\le O(|E(x_z,x_{z+1})|)$.
Since $z$ is not a bad index, $|E^{\operatorname{right}}_{z}|+|E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_{z}|\le 100\cdot|E(x_{z},x_{z+1})|$.
Note that
$$|E'_2|\le 2\cdot|E^{\operatorname{right}}_{a}|\le 2(|E(x_a,x_{z+1})|+|E^{\operatorname{over}}_{z}|)\le 2(|E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_{z}|),$$
and
$$|E'_3|\le |E(S'_z,x_{z+1})|+|E^{\operatorname{over}}_{z}|\le |E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_{z}|.$$
So in this case $|E'|\le 300\cdot |E(x_{z'},x_{z'+1})|\le 300\cdot|E(S_z,\set{v^*})|$.
This completes the proof that $H'$ has the $\alpha$-cut property with respect to path $\tau'$, for $\alpha=300$.
\fi
\end{proof}
\fi
\begin{observation}
\label{obs: bad index non-interleaving}
Let $i < i'$ be two bad indices, then either $\mathsf{RM}(i)\le i'+1$, or $\mathsf{LM}(i')\ge i$.
\end{observation}
\begin{proof}
Assume for contradiction that $\mathsf{LM}(i')<i<i'<\mathsf{RM}(i)-1$.
Note that $E(x_{i},\bar S_{\mathsf{RM}(i)-1})\subseteq E^{\operatorname{over}}_{i'}$ and $E(x_{i'},S_{\mathsf{LM}(i')})\subseteq E^{\operatorname{over}}_{i}$.
Therefore,
\[|E(x_{i},\bar S_{\mathsf{RM}(i)-1})| \le |E^{\operatorname{over}}_{i'}| \le |\hat E_{i'}|
\le \frac{|E(x_{i'},S_{\mathsf{LM}(i')})|}{0.5\times 4998}
\le \frac{|E^{\operatorname{over}}_{i}|}{0.5\times 4998}
\le \frac{|\hat E_{i}|}{0.5\times 4998}
\le \frac{|E(x_{i_1},\bar S_{\mathsf{RM}(i)-1})|}{(0.5\times 4998)^2},\]
a contradiction.
\end{proof}
\subsection{Basic Cluster Disengagement}
\label{subsec: basic disengagement}
The input to the \textsf{Basic Cluster Disengagement}\xspace procedure consists of an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace and a laminar family ${\mathcal{L}}$ of clusters of $G$. Recall that, by the definition, $G\in {\mathcal{L}}$ must hold. We further assume that we are given a partition $({\mathcal{L}}^{\operatorname{light}},{\mathcal{L}}^{\operatorname{bad}})$ of the clusters of ${\mathcal{L}}\setminus \set{G}$, and, for every cluster $C\in {\mathcal{L}}^{\operatorname{light}}$, a distribution ${\mathcal{D}}(C)$ over its internal $C$-routers (that may be given implicitly). The output of the procedure is a collection ${\mathcal{I}}=\set{I_C\mid C\in {\mathcal{L}}}$ of subinstances of $I$. In order to define the instances of ${\mathcal{I}}$, we will define, for every cluster $C\in {\mathcal{L}}$, an ordering ${\mathcal{O}}(C)$ of the edges of $\delta_G(C)$. Family ${\mathcal{I}}$ of subinstances of $I$ is then constructed by disengaging instance $I$ via the laminar family ${\mathcal{L}}$ and the collection $\set{{\mathcal{O}}(C)}_{C\in {\mathcal{L}}}$ of orderings.
In order to describe the algorithm for computing the collection ${\mathcal{I}}$ of subinstances of $I$, it is now enough to describe an algorithm that computes, for every cluster $C\in {\mathcal{L}}$, an ordering ${\mathcal{O}}(C)$ of the edges of $\delta_G(C)$.
Consider any such cluster $C\in {\mathcal{L}}$. If $C=G$, then $\delta_G(C)=\emptyset$, and the ordering ${\mathcal{O}}(C)$ is trivial. If $C\in {\mathcal{L}}^{\operatorname{bad}}$, then we let ${\mathcal{O}}(C)$ be an arbitrary ordering of the edges of $\delta_G(C)$.
Lastly, consider a cluster $C\in {\mathcal{L}}^{\operatorname{light}}$. We select an internal router ${\mathcal{Q}}(C)\in \Lambda_G(C)$ from the given distribution ${\mathcal{D}}(C)$ at random.
We view the paths in ${\mathcal{Q}}(C)$ as being directed towards the center vertex $u(C)$ of the router. We use the algorithm from \Cref{lem: non_interfering_paths} to compute a non-transversal path set $\tilde {\mathcal{Q}}(C)$, routing all edges of $\delta_G(C)$ to vertex $u(C)$, so $\tilde {\mathcal{Q}}(C)$ is also an internal $C$-router. The algorithm ensures that, for every edge $e\in E(G)$, $\cong_G(\tilde {\mathcal{Q}}(C),e)\leq \cong_G({\mathcal{Q}}(C),e)$.
We then let the ordering ${\mathcal{O}}(C)$ of the edges of $\delta_G(C)$ be an ordering guided by the set $\tilde {\mathcal{Q}}(C)$ of paths in graph $G$, and the rotation system $\Sigma$, so ${\mathcal{O}}(C)={\mathcal{O}}^{\operatorname{guided}}(\tilde {\mathcal{Q}}(C),\Sigma)$.
This completes the description of the algorithm for selecting an ordering ${\mathcal{O}}(C)$ of the edges in $\delta_G(C)$ for each cluster $C\in {\mathcal{L}}$; note that this algorithm is randomized. This also completes the description of the algorithm for performing \textsf{Basic Cluster Disengagement}\xspace, that we refer to as \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace in the remainder of the paper. Since this algorithm essentially performs disengagement via a laminar family of clusters of $G$, \Cref{lem: number of edges in all disengaged instances} and \Cref{lem: basic disengagement combining solutions} continue to hold for the resulting collection ${\mathcal{I}}$ of instances. But we can now show that, under some conditions, we can bound the expected value of $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')$.
When using the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace for performing \textsf{Basic Cluster Disengagement}\xspace of an instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace via a laminar family ${\mathcal{L}}$, we will typically require that the following properties hold, for some parameter $\beta$:
\begin{properties}{P}
\item every cluster $C\in {\mathcal{L}}^{\operatorname{bad}}$ is $\beta$-bad, and has the $\alpha_0$-bandwidth property in $G$, for some $\alpha_0\geq \Omega(1/\log^{12}m)$; \label{prop: for bad clusters}
\item every cluster $C\in {\mathcal{L}}^{\operatorname{light}}$ is $\beta$-light with respect to the given distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of its internal routers; and \label{prop: for light clusters}
\item for every cluster $C\in {\mathcal{L}}$, there is a distribution ${\mathcal{D}}'(C)$ over the set $\Lambda'(C)$ of external $C$-routers, such that for every edge $e\in E(G\setminus C)$, $\expect[{\mathcal{Q}}'(C)\sim{\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta$.\label{prop: external routers}
\end{properties}
Observe that the algorithm for computing the family ${\mathcal{I}}$ of clusters is randomized.
We show in the following lemma that, if all the above conditions hold, then the expected value of $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')$ is suitably bounded. The proof is somewhat technical, and is deferred to Section \ref{subsec: appx basic diseng opt bounds} of Appendix.
\begin{lemma}\label{lem: disengagement final cost}
Let $I=(G,\Sigma)$ be an instance of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, ${\mathcal{L}}$ a laminar family of clusters of $G$, $({\mathcal{L}}^{\operatorname{light}},{\mathcal{L}}^{\operatorname{bad}})$ a partition of cluster set ${\mathcal{L}}\setminus \set{G}$, and, for every cluster $C\in {\mathcal{L}}^{\operatorname{light}}$, ${\mathcal{D}}(C)$ a distribution over internal $C$-routers. Let ${\mathcal{I}}$ be the collection of subinstances of $I$ obtained by applying Algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace to instance $I$, with laminar family ${\mathcal{L}}$, cluster sets ${\mathcal{L}}^{\operatorname{light}},{\mathcal{L}}^{\operatorname{bad}}$, and distributions $\set{{\mathcal{D}}(C)}_{C\in {\mathcal{L}}^{\operatorname{light}}}$. Assume further that Properties \ref{prop: for bad clusters} -- \ref{prop: external routers} hold for some parameter $\beta\geq c(\log |E(G)|)^{18}$, where $c$ is a large enough constant. Then:
%
\[\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq O(\mathsf{dep}({\mathcal{L}})\cdot\beta^2\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)).\]
\end{lemma}
We note that the distributions $\set{{\mathcal{D}}'(C)}_{C\in {\mathcal{L}}}$ over external routers of the clusters play no role in constructing the collection ${\mathcal{I}}$ of subinstances of $I$, but they are essential in order to ensure that the expectation of $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}(I')$ is suitably bounded. Since this property is essential to us, we will only use \textsf{Basic Cluster Disengagement}\xspace when such distributions are given. Therefore, abusing the notation, we refer to the family ${\mathcal{I}}$ of subinstances of $I$ computed via \textsf{Basic Cluster Disengagement}\xspace of instance $I$ as described above, as a \textsf{Basic Cluster Disengagement}\xspace of $I$ via the tuple $({\mathcal{L}},{\mathcal{L}}^{\operatorname{bad}},{\mathcal{L}}^{\operatorname{light}}, \set{{\mathcal{D}}'(C)}_{C\in {\mathcal{L}}},\set{{\mathcal{D}}(C)}_{C\in {\mathcal{L}}^{\operatorname{light}}})$.
\subsection{First Key Tool: Basic Disengagement of Clusters}
\label{subsec: basic disengagement}
Throughout this subsection, we assume that we are given an instance $(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, and we fix its optimal solution $\phi^*$.
For brevity of notations, we will omit subscript $\phi^*$ when we use the notations $\chi_{\phi^*}(\cdot),\chi^1_{\phi^*}(\cdot),\chi^2_{\phi^*}(\cdot),\chi_{\phi^*}(\cdot,\cdot)$.
Let $C$ be a cluster of $G$, and let ${\mathcal{Q}}$ be a set of paths in $C$.
Recall that for a pair $Q,Q'\in {\mathcal{Q}}$ of paths, we have denoted by $\chi(Q,Q')$ be the total number of crossings $(e,e')_p$ in the drawing $\phi^*$, where $(e,e')$ is an ordered pair of edges, with $e\in E(Q)$ and $e'\in E(Q')$.
Note that the paths $Q,Q'$ are allowed to share edges, so a crossing $(e,e')_p$ with $e,e'\in E(Q)\cap E(Q')$ is counted twice in $\chi(Q,Q')$.
For a subgraph $H$ of $G$ that is not necessarily disjoint with $C$, we denote $\chi({\mathcal{Q}},H)=\sum_{Q\in {\mathcal{Q}}}\chi(Q,H)$.
\iffalse
\mynote{last sentence is not ideal. It sounds like you need to define this notion for *every* subgraph $H$, when in fact what you are saying is "for a subgraph $H$ of $G$, and a set ${\mathcal{Q}}$ of paths...."}
Define $C^+=C\cup\delta(C)$. \mynote{this is not consistent with how we define $C^+$ in other places (subdivide edges in $\delta_G(C)$ by terminals, let $C^+$ be the subgraph induced by $V(C)$ plus terminals. Better to use the same definition or name it something else.}
The cost of the set ${\mathcal{Q}}$ of paths with respect to cluster $C$
is defined as: $\operatorname{cost}_C({\mathcal{Q}})=\chi({\mathcal{Q}},G\!\setminus\! C^+)+\min\set{|{\mathcal{Q}}|^2,\sum_{\stackrel{Q,Q'\in {\mathcal{Q}}:}{Q\neq Q'}}\chi(Q,Q')}$.
\fi
\iffalse{backup definitions}
\paragraph{Routing Paths and Orderings.}
Given a cluster $C$ of $G$, let $u(C)$ be an arbitrary vertex of $C$. We say that a path set ${\mathcal{Q}}(C)$ is a \emph{set of paths routing the edges of $\delta_G(C)$ to $u(C)$ inside $C$} iff ${\mathcal{Q}}(C)=\set{Q_C(e)\mid e\in \delta_G(C)}$, and for all $e\in \delta(C)$, path $Q_C(e)$ has $e$ as its first edge and $u(C)$ as its last vertex, and moreover $Q(e)\setminus\set{e}$ is contained in $C$. Similarly, given a vertex $u'(C)\not\in C$, we say that path set ${\mathcal{P}}(C)$ is a \emph{set of paths routing the edges of $\delta(C)$ to $u'(C)$ outside of $C$} iff ${\mathcal{P}}(C)=\set{P_C(e)\mid e\in \delta_G(C)}$, and for all $e\in \delta(C)$, path $P_C(e)$ has $e$ as its first edge, $u'(C)$ as its last vertex, and, except for its first vertex, is completely disjoint from $C$.
Assume now that we are given a cluster $C\subseteq G$, a vertex $u(C)\in V(C)$, and a set ${\mathcal{Q}}(C)$ of paths routing the edges of $\delta_G(C)$ to $u(C)$ inside $C$, such that the paths in ${\mathcal{Q}}(C)$ are non-transversal with respect to $\Sigma$.
Recall that we are given an ordering ${\mathcal{O}}_{u(C)}\in \Sigma$ of the edges in $\delta_G(u(C))$. We use this ordering and the set ${\mathcal{Q}}(C)$ of paths in order to define a circular ordering of the edges of $\delta_G(C)$, that we denote by $\hat {\mathcal{O}}(C,{\mathcal{Q}})$. Intuitively, this is the ordering in which the paths in ${\mathcal{Q}}(C)$ enter $u(C)$.
Formally, for every path $Q\in {\mathcal{Q}}(C)$, let $e^*(Q)$ be the last edge lying on path $Q$, that must belong to $\delta_G(C)$. We first define a circular ordering of the paths in ${\mathcal{Q}}(C)$, as follows: the paths are ordered according to the circular ordering of their last edges $e^*(Q)$ in ${\mathcal{O}}_{u(C)}\in \Sigma$, breaking ties arbitrarily. Since every path $Q\in {\mathcal{Q}}(C)$ is associated with a unique edge in $\delta_G(C)$, that serves as the first edge on path $Q$, this ordering of the paths in ${\mathcal{Q}}(C)$ immediately defines a circular ordering $\hat {\mathcal{O}}(C,{\mathcal{Q}}(C))$ of the edges of $\delta_G(C)$.
\fi
\paragraph{Basic Disengagement.}
Assume now that we are given a collection ${\mathcal{C}}$ of disjoint sub-graphs of $G$ that we call \emph{clusters}, and that, for each such cluster $C\in {\mathcal{C}}$, we are given (i) a vertex $u(C)\in V(C)$, and a set ${\mathcal{Q}}(C)=\set{Q_C(e)\mid e\in \delta_G(C)}$ of paths that are non-transversal with respect to $\Sigma$, routing the edges of $\delta_G(C)$ to $u(C)$ inside $C$; and
(ii) a vertex $u'(C)\notin V(C)$, and a set ${\mathcal{Q}}'(C)=\set{Q'_C(e)\mid e\in \delta_G(C)}$ of paths that are non-transversal with respect to $\Sigma$, routing the edges of $\delta_G(C)$ to $u'(C)$ outside of $C$.
Denote ${\mathcal{Q}}=\bigcup_{C\in {\mathcal{C}}}{\mathcal{Q}}(C)$, and ${\mathcal{Q}}'=\bigcup_{C\in {\mathcal{C}}}{\mathcal{Q}}'(C)$.
We construct a collection ${\mathcal{I}}({\mathcal{C}},{\mathcal{Q}},{\mathcal{Q}}')$ of sub-instances of instance $I$, that we refer to as the \emph{decomposition of instance $I$ via $({\mathcal{C}},{\mathcal{Q}},{\mathcal{Q}}')$}.
First, for every cluster $C\in {\mathcal{C}}$, we construct an instance $I_C=(G_C,\Sigma_C)$. Graph $G_C$ is obtained from graph $C^+$ by unifying all endpoints of edges in $\delta_G(C)$ that do not lie in $C$ into a single vertex $v^*_C$. Note that for every vertex $x\in V(C)\setminus \set{v^*(C)}$, $\delta_{G_C}(x)=\delta_G(x)$. The ordering of the edges in $\delta_{G_C}(x)$ in $\Sigma_C$ remains the same as in $\Sigma$. For vertex $v^*(C)$, the set of its incident edges in $G_C$ is precisely $\delta_G(C)$. We define the ordering ${\mathcal{O}}_{v^*(C)}$ of these edges in $\Sigma_C$ to be $\hat{\mathcal{O}}(C,{\mathcal{Q}}(C))$.
In addition to the instances $\set{I_C}_{C\in {\mathcal{C}}}$, we define one more instance $\hat I({\mathcal{C}},{\mathcal{Q}})=(\hat G,\hat \Sigma)$. We define the graph $\hat G=G_{|{\mathcal{C}}}$, and we denote the vertex in $\hat G$ representing the cluster $C$ by $u^*_C$.
Note that for every vertex $x\in V(\hat G)\setminus \set{u^*_C}_{C\in {\mathcal{C}}}$, $\delta_{\hat G}(x)=\delta_G(x)$, and the ordering ${\mathcal{O}}_x$ of the edges incident to $x$ remains the same as in $\Sigma$. For a vertex $u^*_C$, where $C\in {\mathcal{C}}$, the set of its incident edges is $\delta_G(C)$. We define the ordering ${\mathcal{O}}_{u^*_C}$ in $\hat \Sigma$ to be the canonical ordering $\hat{\mathcal{O}}(C,{\mathcal{Q}}(C),\Sigma)$ (see definition in \Cref{sec: canonical ordering}).
We let ${\mathcal{I}}({\mathcal{C}},{\mathcal{Q}},{\mathcal{Q}}')=\set{\hat I({\mathcal{C}},{\mathcal{Q}})}\cup \set{I_C}_{C\in {\mathcal{C}}}$ be the resulting collection of instances.
We start by proving the following lemma.
\begin{lemma}\label{lem: low solution costs}
$\sum_{I'\in {\mathcal{I}}({\mathcal{C}},{\mathcal{Q}},{\mathcal{Q}}')}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)+2\cdot\chi^2({\mathcal{Q}})+2\cdot\sum_{e'\in E(G)}(\cong({\mathcal{Q}},e'))^2+\\
\chi({\mathcal{Q}},G\setminus \bigcup_{C\in {\mathcal{C}}}C)+\sum_{C\in {\mathcal{C}}}\bigg(\chi({\mathcal{Q}}'(C),C)+\chi({\mathcal{Q}}'(C),{\mathcal{Q}}(C))\bigg)$.
\end{lemma}
\begin{proof}
Recall that $\phi^*$ is an optimal solution of instance $I$.
We show that we can construct
\begin{itemize}
\item for each cluster $C\in {\mathcal{C}}$, a solution $\phi_{C}$ of instance $I_C$ from $\phi^*$, such that \\
$\mathsf{cr}(\phi_C)\le \chi^2(C)+ \chi({\mathcal{Q}}'(C),C)+\chi({\mathcal{Q}}'(C),{\mathcal{Q}}(C))+\sum_{e'\in E(C)}\bigg(\cong({\mathcal{Q}}(C),e')\bigg)^2+\chi^2({\mathcal{Q}}(C))$;
\item a solution $\hat \phi$ of instance $\hat I({\mathcal{C}},{\mathcal{Q}})$ from $\phi^*$, such that\\
$\mathsf{cr}(\hat \phi)\le \chi^2(G\setminus \bigcup_{C\in{\mathcal{C}}}C)+\chi^2({\mathcal{Q}})+\chi({\mathcal{Q}},G\setminus \bigcup_{C\in {\mathcal{C}}}C)+\sum_{e'\in E(G)}\bigg(\cong({\mathcal{Q}},e')\bigg)^2$.
\end{itemize}
Since $\chi^2(G\setminus \bigcup_{C\in{\mathcal{C}}}C)+\sum_{C\in {\mathcal{C}}}\chi^2(C)\le \mathsf{cr}(\phi^*)=\mathsf{OPT}_{\mathsf{cnwrs}}(I)$, it is not hard to see that the above bounds implies \Cref{lem: low solution costs}.
\iffalse{with temp}
Note that this implies \Cref{lem: low solution costs}, since
\[
\begin{split}
\sum_{I'\in {\mathcal{I}}({\mathcal{C}},{\mathcal{Q}},{\mathcal{Q}}')}\mathsf{OPT}_{\mathsf{cnwrs}}(I') = & \text{ }
\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I({\mathcal{C}},{\mathcal{Q}}))+\sum_{C\in {\mathcal{C}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I_C)\\
\leq & \text{ } \mathsf{cr}(\hat \phi)+\sum_{C\in {\mathcal{C}}}\mathsf{cr}(\phi_C)\\
\leq & \text{ }O(\mathsf{cr}(\phi^*)\cdot\beta)+\sum_{C\in {\mathcal{C}}}\bigg(\chi({\mathcal{Q}}'(C),C)+O(\chi^2(C)\cdot\beta)\bigg)\\
\leq & \text{ } O\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\beta+\sum_{C\in {\mathcal{C}}}\chi({\mathcal{Q}}'(C),C)\bigg),
\end{split}
\]
where the last inequality is due to $\sum_{C\in {\mathcal{C}}}\chi^2(C)\le \mathsf{cr}(\phi^*)=\mathsf{OPT}_{\mathsf{cnwrs}}(I)$.
\fi
We first construct the solution $\hat \phi$ for the instance $\hat I({\mathcal{C}},{\mathcal{Q}})$, as follows.
We start with drawing $\phi^*$, and process the clusters of ${\mathcal{C}}$ one-by-one.
We now describe the iteration of processing a cluster $C$.
Recall that we are given, for each cluster $C\in{\mathcal{C}}$, a vertex $u(C)\in V(C)$ and a set ${\mathcal{Q}}(C)$ of paths routing the edges of $\delta_G(C)$ inside $C$ that are non-transversal with respect to $\Sigma$.
For each $e'\in E(C)$, we denote by $\pi_{e'}$ the curve that represents the image of $e'$ in $\phi^*$, and we let set $\Pi_{e'}$ contain $\cong_C({\mathcal{Q}}(C),e')$ curves connecting the endpoints of $e'$ that are internally mutually non-crossing and lying inside an arbitrarily thin strip around $\pi_{e'}$. We then assign, for each edge $e\in \delta_G(C)$ and each edge $e'\in E(Q_C(e))$, a distinct curve of $\Pi_{e'}$ to $e$. So each curve in $\bigcup_{e'\in E(C)}\Pi_{e'}$ is assigned to exactly one edge of $\delta_G(C)$.
For each $e\in \delta_G(C)$, let $\gamma_e$ be the curve obtained by concatenating all curves in $\bigcup_{e'\in E(C)}\Pi_{e'}$ that are assigned to $e$, so $\gamma_e$ connects the endpoint of $e$ outside $C$ to $z_C$, where $z_C$ is the image of $u(C)$ in $\phi^*$.
We denote $\Gamma_C=\set{\gamma_e\mid e\in \delta_G(C)}$.
In fact, when we assign curves of $\bigcup_{e'\in \delta_{G_C}(u(C))}\Pi_{e'}$ to edges of $\delta_G(C)$, we additionally ensure that, if we view $\gamma_e$ as the image of $e$ for each $e\in \delta_G(C)$, then the order in which the curves of $\set{\gamma_e\mid e\in \delta_G(C)}$ enter $z_C$ is identical to $\hat {\mathcal{O}}(C,{\mathcal{Q}}(C))$. Note that this can be easily achieved according the the definition of $\hat {\mathcal{O}}(C,{\mathcal{Q}}(C))$.
For each vertex $v\in V(C)$, we denote by $x_v$ the point that represents the image of $v$ in $\phi^*$ (namely $x_v=\phi^*(v)$), and we let $X$ contains all points of $\set{x_v\mid v\in V(C)}$ that are crossings between curves in $\Gamma_C$.
We then process all points of $X$ as follows.
Consider a point $x_v$ of $X$ and let $D_v$ be an arbitrarily small disc around $x_v$.
Let $\Gamma_C(v)$ be the subset of all curves in $\Gamma_C$ that contains $x_v$.
Let $\tilde\Gamma_C(v)$ be the set of curves we obtain by applying the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $\Gamma_C(v)$ within an arbitrarily small disc around $x_v$. We then replace curves of $\Gamma_C(v)$ in $\Gamma_C$ by curves of $\tilde\Gamma_C(v)$, and continue to the next iteration. Let $\tilde\Gamma_C=\set{\tilde\gamma_e\mid e\in \delta_G(C)}$ be the set of curves that we obtain after processing all points in $X$ in this way.
Note that the curves of $\tilde\Gamma_C$ might be non-simple. Let $\tilde\Gamma'_C=\set{\tilde\gamma'_e\mid e\in \delta_G(C)}$ be the set of curves that we obtain after removing self-loops from all curves of $\tilde\Gamma_C$.
From \Cref{obs: curve_manipulation}, curves of $\tilde\Gamma'_C$ are in general position. We now view $z_C$ as the image of $u^*_C$, and for each edge $e\in \delta_{G}(C)$, the curve $\tilde\gamma'_e$ as the image of $e$.
This completes the iteration of processing cluster $C$.
Define $G'=G\setminus \bigcup_{C\in {\mathcal{C}}}C$.
It is easy to see that
\iffalse{analysis with temp}
\[
\begin{split}
\chi(\tilde\Gamma'_C)
= & \sum_{(e,e'):\text{ } e,e'\text{ cross in }\phi^*}\cong_{G}({\mathcal{Q}}(C),e)\cdot \cong_{G}({\mathcal{Q}}(C),e')\\
\le &
\sum_{(e,e'): \text{ } e,e'\text{ cross in }\phi^*}\frac{(\cong_{G}({\mathcal{Q}}(C),e))^2+(\cong_{G}({\mathcal{Q}}(C),e'))^2}{2}\\
= & \text{ } \frac{1}{2}\cdot \sum_{e\in E({\mathcal{Q}}(C))}\chi(e,C)\cdot(\cong_{G}({\mathcal{Q}}(C),e))^2
\le O(\chi^2(C)\cdot\beta).
\end{split}
\]
and
\[
\begin{split}
\chi(\tilde\Gamma'_C,\phi^*(G\setminus \bigcup_{C\in {\mathcal{C}}}C))
= & \sum_{e\in E(C^+)}\sum_{e'\in E(G\setminus C^+)}
\mathbf{1}[e,e'\text{ cross in }\phi^*]\cdot \cong_{G}({\mathcal{Q}}(C),e)\\
= & \sum_{e\in E(C^+)}\chi(e,G\setminus \bigcup_{C\in {\mathcal{C}}}C)\cdot \cong_{G}({\mathcal{Q}}(C),e).
\end{split}
\]
\fi
\[
\chi(\tilde\Gamma'_C)
\le \sum_{e'\in E(C)}\bigg(\cong({\mathcal{Q}}(C),e')\bigg)^2+\sum_{Q,Q'\in {\mathcal{Q}}(C)}\chi(Q,Q').
\]
and
\iffalse{analysis with temp}
\[
\begin{split}
\chi(\tilde\Gamma'_C,\phi^*(G\setminus \bigcup_{C\in {\mathcal{C}}}C))
= & \sum_{e\in E(C)}\sum_{e'\in E(G\setminus C)}
\mathbf{1}[e,e'\text{ cross in }\phi^*]\cdot \cong_{G}({\mathcal{Q}}(C),e)\\
= & \sum_{e\in E(C)}\chi(e,G\setminus \bigcup_{C\in {\mathcal{C}}}C)\cdot \cong_{G}({\mathcal{Q}}(C),e)\\
= & \text{ }\chi^1(C)\cdot M_1.
\end{split}
\]
\fi
\[
\chi(\tilde\Gamma'_C,\phi^*(G'))\le \sum_{Q\in {\mathcal{Q}}(C)}\chi(Q, G').
\]
Let $\set{\tilde\Gamma'_C\mid C\in {\mathcal{C}}}$ be the sets of curves that we obtain after processing all clusters of ${\mathcal{C}}$ in the same way. It is clear that, if we view (i) for each $C\in {\mathcal{C}}$, the point $\phi^*(u(C))$ as the image of $u^*_C$; and (ii) for each $C\in {\mathcal{C}}$ and for each $e\in \delta_G(C)$, the curve $\tilde\gamma'_e\in \tilde\Gamma'_C$ as the image of $e$, then we obtain a drawing of $\hat G$ that respects the rotation system $\Sigma$, that we denote by $\hat\phi$. Moreover,
\iffalse{analysis with temp}
\[
\begin{split}
\mathsf{cr}(\hat \phi)
= & \text{ }\chi^2(G\setminus \bigcup_{C\in {\mathcal{C}}}C)+\sum_{C\in {\mathcal{C}}}(\chi(\tilde\Gamma'_C)+\chi(\tilde\Gamma'_C, \phi^*(G\setminus \bigcup_{C\in {\mathcal{C}}}C))) + \sum_{C,C'\in {\mathcal{C}}, C\ne C'}\chi(\tilde\Gamma'_C, \tilde\Gamma'_{C'})\\
\le &\text{ }\sum_{(e,e'): \text{ } e,e'\text{ cross in }\phi^*}\max\set{\frac{\cong_{G}({\mathcal{Q}},e)^2}{2},1}+\max\set{\frac{\cong_{G}({\mathcal{Q}},e')^2}{2},1}\\
= & \text{ } \sum_{e\in E(G)}\chi(e)\cdot\max\set{\frac{\cong_{G}({\mathcal{Q}},e)^2}{2},1}\\
\le & \sum_{e\in E(G)}\chi(e)\cdot O(\beta)= O(\mathsf{cr}(\phi^*)\cdot\beta).
\end{split}
\]
\fi
\[
\begin{split}
\mathsf{cr}(\hat \phi)
= & \text{ }\chi^2(G')+\sum_{C\in {\mathcal{C}}}(\chi(\tilde\Gamma'_C)+\chi(\tilde\Gamma'_C, \phi^*(G'))) + \sum_{C,C'\in {\mathcal{C}}, C\ne C'}\chi(\tilde\Gamma'_C, \tilde\Gamma'_{C'})\\
\le &\text{ }\chi^2(G')+\sum_{C\in {\mathcal{C}}}\bigg(\chi^2({\mathcal{Q}}(C))+\chi({\mathcal{Q}}(C),G')+\sum_{e'\in E(C)}\cong({\mathcal{Q}}(C),e')^2\bigg) +\sum_{C,C'\in {\mathcal{C}}, C\ne C'}\chi({\mathcal{Q}}(C), {\mathcal{Q}}(C'))\\
= & \text{ }\chi^2(G')+\chi^2({\mathcal{Q}})+\chi({\mathcal{Q}},G')+\sum_{e'\in E(G)}\cong({\mathcal{Q}},e')^2.
\end{split}
\]
We now fix a cluster $C\in {\mathcal{C}}$ and construct the solution $\phi_C$ as follows.
Recall that $V(G_C)=V(C)\cup\set{v^*_C}$ and $E(G_C)=E_G(C)\cup \delta_{G_C}(v^*_C)$.
Also recall that we are given a vertex $u'(C)\notin C$ and a set ${\mathcal{Q}}'(C)$ of paths routing edges of $\delta_G(C)$ to $u'(C)$ outside of $C$. We start with the drawing of $C\cup E({\mathcal{Q}}'(C))$ induced by $\phi^*$. We do not modify the drawing of $C$, and we construct the drawings of edges of $\delta_{G_C}(v^*_C)$ as follows.
For each $e'\in E({\mathcal{Q}}'(C))$, we denote by $\pi_{e'}$ the curve that represents the image of $e'$ in $\phi^*$, and we let set $\Pi_{e'}$ contain $\cong_G({\mathcal{Q}}'(C),e')$ curves connecting the endpoints of $e'$ lying inside an arbitrarily thin strip around $\pi_{e'}$. We then assign, for each edge $e\in \delta_G(C)$ and for each edge $e'\in E(Q'_C(e))$, a distinct curve in $\Pi_{e'}$ to $e$. So each curve in $\bigcup_{e'\in E({\mathcal{Q}}'(C))}\Pi_{e'}$ is assigned to exactly one edge of $\delta_G(C)$. For each edge $e\in \delta_G(C)$, let $\gamma_e$ be the curve obtained by concatenating all curves in $\bigcup_{e'\in E({\mathcal{Q}}'(C))}\Pi_{e'}$ that are assigned to $e$, so $\gamma_e$ connects the endpoint of $e$ in $C$ to $z'_C$, where $z'_C$ is the image of $u'(C)$ in $\phi^*$. We denote $\Gamma'_C=\set{\gamma'_e\mid e\in \delta_G(C)}$, and we view $z'_C$ as the last endpoint of curves in $\Gamma'_C$. We let $\Gamma_0$ contains all curves representing the image of some edge in $E(C)$, and then apply \Cref{thm: type-2 uncrossing} to sets $\Gamma_0,\Gamma'_C$ of curves. Let $\Gamma''_C=\set{\gamma''_e\mid e\in \delta_G(C)}$ be the set of curves that we obtain. Now if we view the curve $\gamma''_e$ as the image of $e$ for each edge $e\in \delta_G(C)$, then combined with the drawing of $C$ induced by the drawing $\phi^*$, we obtain a drawing of $G_C$, that we denote by $\phi'_C$. Clearly, the drawing $\phi'_C$ respects the rotation defined in $\Sigma_C$ at all vertices of $C$. However, it may not respect the rotation ${\mathcal{O}}_{v^*_C}$, namely the image of edges of $\delta_{G_C}(v^*_C)$ may not enter $z'_C$ in the order ${\mathcal{O}}_{v^*_C}$. To obtain a feasible solution to the instance $I_C$, we further modify the drawing $\phi'_C$ as follows.
Let $D_{z'_C}$ be an arbitrarily small disc around point $z'_C$. For each $e\in \delta_{G_C}(v^*_C)$, we denote by $p_e$ the intersection between the curve $\gamma''_e$ and the boundary of $D_{z'_C}$.
We erase the drawing of $\phi'_C$ inside the disc $D_{z'_C}$. We then place another disc $D'_{z'_C}$ inside the disc $D_{z'_C}$, and let $\set{p'_e}_{e\in \delta_{G_C}(v^*_C)}$ be a set of points appearing on the boundary of disc $D'_{z'_C}$ in the order ${\mathcal{O}}_{v^*_C}$. We use \Cref{lem: find reordering} to compute a set $\set{\zeta_e}_{e\in \delta_{G_C}(v^*_C)}$ of reordering curves, where for each $e\in \delta_{G_C}(v^*_C)$, the curve $\zeta_e$ connects $p_e$ to $p'_e$.
We then define, for each $e\in \delta_{G_C}(v^*_C)$, the curve $\eta_e$ as the union of (i) the subcurve of $\gamma''_e$ outside the disc $D_{z'_C}$ (connecting its endpoint in $C$ to $p_e$); (ii) the curve $\zeta_e$ (connecting $p_e$ to $p'_e$); and (iii) the straight line-segment connecting $p'_e$ to $z'_C$. We now view the curve $\eta_e$ as the image of $e$ for all edges $e\in \delta_{G_C}(v^*_C)$. We denote by $\phi_C$ the drawing obtained by the union of all curves in $\set{\eta_e}_{e\in \delta_{G_C}(v^*_C)}$ and the image of $C$ in $\phi'_C$. Clearly, the drawing $\phi_C$ is a feasible solution to the instance $I_C$.
It remains to show that
$$\mathsf{cr}(\phi_C)\le \chi^2(C)+ \chi({\mathcal{Q}}'(C),C)+\chi({\mathcal{Q}}'(C),{\mathcal{Q}}(C))+\sum_{e'\in E(C)}\bigg(\cong({\mathcal{Q}}(C),e')\bigg)^2+\chi^2({\mathcal{Q}}(C)).$$
First, since we have not modified the drawing of $C$, the number of crossings between edges of $E_G(C)$ are bounded by $\chi^2(C)$.
Second, from the definition of curves of $\Gamma'_C$, the number of crossings between image of $C$ and the curves of $\Gamma'_C$ is at most $\chi({\mathcal{Q}}'(C),C)$. From~\Cref{thm: type-2 uncrossing} and the construction of curves in $\set{\eta_e}_{e\in \delta_G(C)}$, the number of crossings between image of $C$ and the curves of $\set{\eta_e}_{e\in \delta_G(C)}$ is also at most $\chi({\mathcal{Q}}'(C),C)$.
Third, from~\Cref{thm: type-2 uncrossing},$\chi(\Gamma''_C)=0$. Therefore, all the crossings between curves of $\set{\eta_e}_{e\in \delta_G(C)}$ lie inside the disc $D_{z'}$.
We denote by $({\mathcal{O}},b'_C)$ the oriented ordering in which the curves of $\Gamma''_C$ enter $z'$. From \Cref{lem: find reordering}, the number of crossings inside $D_{z'}$ is at most $2\cdot\mbox{\sf dist}(({\mathcal{O}},b'_C),({\mathcal{O}}_{v^*_C},b_C))$, where $b_C$ is the orientation for the unoriented circular ordering at vertex $u(C)$.
Recall that ${\mathcal{O}}_{v^*_C}=\hat{\mathcal{O}}({\mathcal{Q}}(C),C)$.
We now show that $\mbox{\sf dist}(({\mathcal{O}},b'_C),(\hat{\mathcal{O}}(C,{\mathcal{Q}}(C)),b_C))=O(\chi({\mathcal{Q}}(C))\cdot\beta+\chi({\mathcal{Q}}'(C),C))$.
In fact, we will construct a set of reordering curves for $({\mathcal{O}},b'_C)$ and $({\mathcal{O}}_{v^*_C},b_C)$.
Recall that we have constructed, for each $e\in \delta_G(C)$, a curve $\tilde\gamma'_e$ connecting $e$ to $z$. We define, for each $e\in \delta_G(C)$, the curve $\xi_e$ to be the union of $\eta_e$ and $\tilde \gamma'_e$, so all curves of $\set{\xi_e}_{e\in \delta_G(C)}$ connects $z$ to $z'$. Note that the curves of $\Xi_C=\set{\xi_e}_{e\in \delta_G(C)}$ enter $z$ in the order $(\hat{\mathcal{O}}(C,{\mathcal{Q}}(C)),b_C)$ and enter $z'$ in the order $({\mathcal{O}},b'_C)$, so $\mbox{\sf dist}(({\mathcal{O}},b'_C),(\hat{\mathcal{O}}(C,{\mathcal{Q}}(C)),b_C))\le \chi(\Xi_C)$.
Note that
$$\chi(\Xi_C)
\le \chi({\mathcal{Q}}'(C),{\mathcal{Q}}(C))+ \chi(\tilde\Gamma'_C)
\le \chi({\mathcal{Q}}'(C),{\mathcal{Q}}(C))+ \sum_{e'\in E(C)}\bigg(\cong({\mathcal{Q}}(C),e')\bigg)^2+\sum_{Q,Q'\in {\mathcal{Q}}(C)}\chi(Q,Q'),$$
so $\mathsf{cr}(\phi_C)\le \chi^2(C)+ \chi({\mathcal{Q}}'(C),C)+\chi({\mathcal{Q}}'(C),{\mathcal{Q}}(C))+\sum_{e'\in E(C)}\bigg(\cong({\mathcal{Q}}(C),e')\bigg)^2+\chi^2({\mathcal{Q}}(C))$.
\end{proof}
$\ $
Lastly, we show that we can combine solutions to instances in ${\mathcal{I}}({\mathcal{C}},{\mathcal{Q}},{\mathcal{Q}}')$, to obtain a solution to instance $I$ without increasing the total number of crossings.
\begin{lemma}\label{lem: combine solutions}
There is an efficient algorithm, that, given, for each instance $I'\in {\mathcal{I}}({\mathcal{C}},{\mathcal{Q}},{\mathcal{Q}}')$, a solution $\phi(I')$, computes a solution for instance $I$ of value at most $\sum_{I'\in {\mathcal{I}}({\mathcal{C}},{\mathcal{Q}},{\mathcal{Q}}')}\mathsf{cr}(\phi(I'))$.
\end{lemma}
\begin{proof}
For each $C\in {\mathcal{C}}$, we rename the solution $\phi(I_C)$ to the instance $I_C$ by $\phi_C$, and we rename the solution $\phi(\hat I({\mathcal{C}},{\mathcal{Q}}))$ to the instance $\hat I({\mathcal{C}},{\mathcal{Q}})$ by $\hat \phi$. We will construct a solution $\phi$ for instance $I$, from the solutions $\hat{\phi}$ and $\set{\phi_C}_{C\in {\mathcal{C}}}$, as follows.
We start with the drawing $\hat \phi$, and process clusters of ${\mathcal{C}}$ one-by-one. Recall that the instance $I({\mathcal{C}},{\mathcal{Q}})=(\hat G,\hat \Sigma)$, where graph $\hat G$ contains a vertex $u^*_C$ for every cluster $C\in {\mathcal{C}}$. Moreover, the incident edges of $u^*_C$ in $G$ is $\delta_G(C)$, and ${\mathcal{O}}_{u^*_C}=\hat {\mathcal{O}}({\mathcal{Q}}(C),C)$.
Also recall that, for each cluster $C\in {\mathcal{C}}$, the graph $G_C$ contains a vertex $v^*_C$ with $\delta_{G_C}(v^*_C)=\delta_G(C)$, and ${\mathcal{O}}_{v^*_C}=\hat {\mathcal{O}}({\mathcal{Q}}(C),C)$.
We now describe an iteration of processing cluster $C$.
Let $D_C$ be an arbitrarily small disc around the image of $u^*_C$ in $\hat\phi$. For each edge $e\in \delta_G(C)$, we denote by $p_e$ the intersection of the image of $e$ in $\hat \phi$ with the boundary of $D_C$, so the points $\set{p_e}_{e\in \delta_G(C)}$ appear on the boundary of $D_C$ in the order $\hat{\mathcal{O}}(C,{\mathcal{Q}}(C))$.
Similarly, let $D'_C$ be an arbitrarily small disc around the image of $v^*_C$ in $\phi_C$, and we define points $\set{p'_e}_{e\in \delta_G(C)}$ similarly, so they appear on the boundary of $D'_C$ in the order $\hat{\mathcal{O}}(C,{\mathcal{Q}}(C))$.
We now erase the part of the drawing $\hat\phi$ inside the disc $D_C$, and place the drawing of $\phi_C$ outside the disc $D'_C$ inside $D_C$, such that the boundary of $D_C$ coincides with the boundary of $D'_C$, while the interior of $D_C$ coincides with the exterior of $D'_C$, and for each edge $e\in \delta_G(C)$, the points $p_e$ and $p'_e$ coincide (note that this can be achieved since points in $\set{p_e}_{e\in \delta_G(C)}$ appear on the boundary of $D_C$ in the same order as points in $\set{p_e}_{e\in \delta_G(C)}$ appear on the boundary of $D_C$). This completes the iteration of processing $C$.
For each $e\in \delta_G(C)$, we view the union of (i) the subcurve of the image of $e$ in $\hat \phi$ between the image of its non-$u^*_C$ endpoint and the point $p_e$; and (ii) the subcurve of the image of $e$ in $\phi_C$ between the image of its non-$v^*_C$ endpoint and the point $p'_e$, as the image of $e$.
Clearly, after the iteration of processing $C$, we have not created any additional crossings.
Let $\phi$ be the drawing that we obtain after processing all clusters of ${\mathcal{C}}$ in this way. It is easy to verify that $\phi$ is a feasible solution to the instance $I$. Since we have not created additional crossings in each iteration, $\mathsf{cr}(\phi)\le \mathsf{cr}(\hat \phi)+\sum_{C\in {\mathcal{C}}}\mathsf{cr}(\phi_C)$.
This completes the proof of \Cref{lem: combine solutions}.
\end{proof}
\iffalse
Recall that given an instance $(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, we say that another instance $(G',\Sigma')$ is its \emph{sub-instance} iff graph $G'$ is obtained from a sub-graph of $G$ by contracting some of its vertex sets into super-nodes. The ordering ${\mathcal{O}}(u)$ for the edges incident to super-nodes may be arbitrary, but the ordering of edges incident to other vertices is determined by the ordering in $\Sigma$.
Suppose we are given any collection ${\mathcal{C}}$ of disjoint sub-graphs of $G$, such that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$. We denote by $E^{\textsf{out}}_{{\mathcal{C}}}$ the set of all edges of $G$ whose endpoints lie in different clusters, and by $E^{\textsf{in}}_{{\mathcal{C}}}$ the set of all edges of $G$ whose endpoints lie in the same cluster, so $E^{\textsf{out}}_{{\mathcal{C}}}\cup E^{\textsf{in}}_{{\mathcal{C}}}=E(G)$.
The main theorem of this subsection is the following.
\mynote{maybe define the sub-instances before the theorem statement to make it more concrete?}
\znote{Yes. I am about to suggest this as well.}
\begin{theorem}[Basic Disengagement of Clusters]\label{thm: disengagement}
There is an efficient algorithm, that, given an instance $(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, and a collection ${\mathcal{C}}$ of disjoint sub-graphs of $G$, such that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$, and, for every cluster $C\in {\mathcal{C}}$, a vertex $u(C)\in C$, and a vertex $u'(C)\not \in C$, together with a collection ${\mathcal{Q}}(C)$ of paths routing the edges of $\delta(C)$ to $u(C)$ inside $C$ with congestion $\eta$, and a collection ${\mathcal{P}}(C)$ of edge-disjoint paths routing the edges of $\delta(C)$ to $u'(C)$ outside of $C$ with congestion $\eta$, computes a collection ${\mathcal{I}}$ of sub-instances of $(G,\Sigma)$, with the following properties:
\begin{itemize}
\item For every cluster $C\in {\mathcal{C}}$, there is a unique instance $(H_C,\Sigma_C)$, with $C\subseteq H_C$. Moreover, $|E(H_C)|\leq |E(C)|+|E^{\textsf{out}}_{{\mathcal{C}}}|$;
\item $\sum_{(H,\Sigma')\in {\mathcal{I}}}|E(H)|\leq \sum_{C\in {\mathcal{C}}}|E(C)|+2|E^{\textsf{out}}_{{\mathcal{C}}}|$; and
\item $\sum_{(H,\Sigma')\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma')\leq \eta^2\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\operatorname{poly}\log n$.
\end{itemize}
Moreover, there is an efficient algorithm, that, given, for each instance $I'\in {\mathcal{I}}$, a feasible solution $\phi_I$, computes a feasible solution $\phi$ to instance $I$ with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\phi_I)\textsf{right} ) \cdot \operatorname{poly}\log n$.
\end{theorem}
The proof of the theorem is somewhat involved and technical, and is deferred to Section \ref{sec:disengagement}.
\mynote{this theorem stands on its own and the proof can be done without taking care of the path case. We use it later.}
\fi
\subsection{Proof of \Cref{claim: bound distance between rotations}}
\label{subsec: bound distance between rotations}
Assume first that $S_{z+1}\in {\mathcal{S}}^{\operatorname{bad}}$. Clearly, $\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')\leq |\delta_{G_z}(v^{**})|^2\leq |\delta_G(U_z)|^2$.
From
\Cref{cor: few edges crossing cuts},
$|\delta_G(U_{z})|\leq |\delta_G(S_{z+1})|\cdot O(\log^{34}m)$. If the bad event ${\cal{E}}$ does not happen, then, from \Cref{obs: congestion square of internal routers},
$\mathsf{OPT}_{\mathsf{cnwrs}}(S_{z+1},\Sigma(S_{z+1}))+|E(S_{z+1})|\geq \frac{|\delta_G(S_{z+1})|^2}{\hat \eta}$,
where $\Sigma(S_{z+1})$ is the rotation system for graph $S_{z-1}$ induced by $\Sigma$. Therefore:
\[
\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')\leq |\delta_G(U_z)|^2\leq O(\log^{68}m)\cdot |\delta_G(S_{z+1})|^2\leq \hat \eta^2\cdot \textsf{left} (|\chi^*_{z+1}|+|E(S_{z+1})|\textsf{right} ).
\]
Assume now that $S_{z}\in {\mathcal{S}}^{\operatorname{bad}}$. As before, $\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')\leq |\delta_G(U_z)|^2$.
From
\Cref{cor: few edges crossing cuts}, $|\delta_G(U_{z})|\leq |\delta_G(S_{z})|\cdot O(\log^{34}m)$. If the bad event ${\cal{E}}$ does not happen, then, from \Cref{obs: congestion square of internal routers},
$\mathsf{OPT}_{\mathsf{cnwrs}}(S_{z},\Sigma(S_{z}))+|E(S_{z})|\geq \frac{|\delta_G(S_{z})|^2}{\hat \eta}$,
where $\Sigma(S_{z})$ is the rotation system for graph $S_{z}$ induced by $\Sigma$. Therefore:
\[
\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}') \leq |\delta_G(U_z)|^2\leq O(\log^{68}m)\cdot |\delta_G(S_{z})|^2\leq \eta^2\cdot \textsf{left} (|\chi^*_{z}|+|E(S_{z})|\textsf{right} ).
\]
We assume from now on that $S_z,S_{z+1}\in {\mathcal{S}}^{\operatorname{light}}$.
In order to complete the proof of \Cref{claim: bound distance between rotations},
we will define, for every edge $e\in \delta_G(U_z)$, a curve $\gamma(e)$, such that all curves in the resulting set $\Gamma^*=\set{\gamma(e)\mid e\in \delta_G(U_z)}$ are in general position; each one of the curves originates at the image of vertex $v^{**}$ in the drawing $\phi''_z$ of $G_z$; and each one of the curves terminates at the image of vertex $u_z$ in the drawing $\phi''_z$. We will ensure that the order in which the curves in set $\Gamma^*$ enter the image of $v^{**}$ is precisely the ordering ${\mathcal{O}}$ of their corresponding edges, while the order in which they enter the image of $u_z$ is precisely the ordering ${\mathcal{O}}'$ of their corresponding edges. We will then bound the number of crossings between the curves in $\Gamma^*$, thereby bounding $\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
In order to define the set $\Gamma^*$ of curves, we define, for every edge $e\in \delta_G(U_z)$, a path $\tilde R(e)$ in graph $G_z$, that connects vertex $v^{**}$ to vertex $u_{z}$, and originates at edge $e$.
For each edge $e\in \delta_G(U_z)$, the curve $\gamma(e)$ is then obtained by slightly altering the image of the path $\tilde R(e)$ in the drawing $\phi''_z$, in order to ensure that all resulting curves in $\Gamma^*$ are in general position.
We start by defining the set $\tilde {\mathcal{R}}=\set{\tilde R(e)\mid e\in \delta_G(U_z)}$ of paths.
\paragraph{Set $\tilde {\mathcal{R}}=\set{\tilde R(e)\mid e\in \delta_G(U_z)}$ of paths.}
Consider an edge $e\in \delta_G(U_z)$. Assume first that $e\in E_z$. Denote $e=(u,v)$, where $u\in S_z$ and $v\in S_{z+1}$ (see \Cref{fig: NN4}).
Note that edge $e$ belongs to graph $G_z$, where it connects vertex $u$ to vertex $v^{**}$. Let $\tilde Q(e)$ be the unique path of the internal $U_z$-router ${\mathcal{Q}}(U_z)$ that originates at edge $e$; recall that the path terminates at vertex $u_z$, and, from the construction of the path set ${\mathcal{Q}}(U_z)$, path $\tilde Q(e)$ is also the unique path of the internal $S_z$-router ${\mathcal{Q}}(S_z)$ that originates at edge $e$. Therefore, all internal vertices of path $\tilde Q(e)$ lie in $S_z$, and path $\tilde Q(e)$ is contained in graph $G_z$. We then let $\tilde R(e)$ be the path $\tilde Q(e)$ in graph $G_z$ (that now connects vertex $u_z$ to vertex $v^{**}$.)
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN4.jpg}
\caption{Definition of path $\tilde R(e)$ when $e\in E_z$.}\label{fig: NN4}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN5.jpg}
\caption{Construction of path $\tilde R(e)$ when $e\in E_z^{\operatorname{over}}$. Path $\tilde Q(e)$ is shown in green.}\label{fig: NN5}
\end{figure}
Next, we consider an edge $e\in E^{\operatorname{over}}_z$. Assume that $e=(u,v)$, where $u\in \bigcup_{i<z}V(\tilde S_i)$ and $v\in \bigcup_{i>z}V(\tilde S_i)$ (see \Cref{fig: NN5}).
Consider the unique path $\tilde Q(e)\in {\mathcal{Q}}(U_z)$ that originates at edge $e$. Recall that the path terminates at vertex $u_z$, and it must contain some edge $a_e\in \delta_G(S_z)$ (in fact it may only contain one such edge). Note that, in graph $G_z$, all vertices of $U_{z-1}$ were contracted into vertex $v^*$, and so both edges $e$ and $a_e$ are incident to vertex $v^*$ in $G_z$. The subpath of the path $\tilde Q(e)$ from edge $a_e$ to vertex $u_z$ is precisely the unique path of the internal router ${\mathcal{Q}}(S_z)$ that originates at edge $a_e$, which we denote by $\tilde Q'(e)$. We then let $\tilde R(e)$ be the path obtained by appending the edge $e$ at the beginning of the path $\tilde Q'(e)$. Note that path $\tilde R(e)$ is contained in graph $G_z$, and it connects vertex $v^{**}$ to vertex $u_z$.
In fact, path $\tilde R(e)$ is a concatenation of edge $(v^*,v^{**})$ and path $\tilde Q'(e)$.
Lastly, we consider an edge $e\in E^{\operatorname{right}}_z$. Assume that $e=(u,v)$, where $u\in V(\tilde S_z)$, and $v\in \bigcup_{i>z}V(\tilde S_i)$. Note that edge $e$ is also present in graph $G_z$, where it now connects vertex $u$ to vertex $v^{**}$.
Consider the unique path $\tilde Q(e)\in {\mathcal{Q}}(U_z)$ that originates at edge $e$, and recall that this path terminates at vertex $u_z$.
We now consider two cases. First, if path $\tilde Q(e)$ is contained in cluster $\tilde S_z$, then it is contained in the current graph $G_z$, except that now it connects vertex $v^{**}$ to vertex $u_z$. We then set $\tilde R(e)=\tilde Q(e)$ (see \Cref{fig: NN6a}). We denote by $a_e$ the unique edge of $\tilde R(e)$ that lies in $\delta_G(S_z)$.
Otherwise, let $u''$ be the first vertex on path $\tilde Q(e)$ that does not belong to $\tilde S_z$, and let $u'$ be the vertex preceding $u''$ on the path (see \Cref{fig: NN6b}).
Denote $a^*_e=(u',u'')$. Note that edge $a^*_e$ also lies in graph $G_z$, where it connects vertex $u'$ to vertex $v^*$.
Moreover, path $\tilde Q(e)$ must now contain some edge $a_e\in \delta_G(S_z)$. Since, in graph $G_z$, all vertices of $U_{z-1}$ were contracted into the vertex $v^*$, edge $a_e$ is now incident to vertex $v^*$. The subpath of the path $\tilde Q(e)$ from edge $a_e$ to vertex $u_z$, that we denote by $\tilde Q'(e)$, is precisely the unique path of the internal $S_z$-router ${\mathcal{Q}}(S_z)$ that originates at edge $a_e$. We then let $\tilde R(e)$ be the path obtained by concatenating the subpath of $\tilde Q(e)$ from edge $e$ to edge $a^*_e$ (that, in graph $G_z$, connects $v^{**}$ to $v^*$), and the path $\tilde Q'(e)$ (that originates at $v^*$ in $G_z$). Note that path $\tilde R(e)$ is contained in graph $G_z$, and it connects vertex $v^{**}$ to vertex $u_z$.
Since $\delta_G(U_z)=E_z\cup E_z^{\operatorname{over}}\cup E_z^{\operatorname{right}}$, we have now constructed a path $\tilde R(e)$ for each edge $e\in \delta_G(U_z)$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN6a.jpg}
\caption{Construction of path $\tilde R(e)$ when $e\in E_z^{\operatorname{right}}$ and $\tilde Q(e)\subseteq \tilde S_z$. Path $\tilde Q(e)$ is shown in green.}\label{fig: NN6a}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN6b.jpg}
\caption{Construction of path $\tilde R(e)$ when $e\in E_z^{\operatorname{right}}$ and $\tilde Q(e)$ is not contained in $\tilde S_z$. Path $\tilde Q(e)$ is shown in green.}\label{fig: NN6b}
\end{figure}
Consider the final set $\tilde {\mathcal{R}}=\set{\tilde R(e)\mid e\in \delta_G(U_z)}$ of paths that we have defined in graph $G_z$. Notice that, for each edge $e\in \delta_G(U_z)$, there is some edge $a_e\in \delta_G(S_z)$ that lies both on the path $\tilde {\mathcal{Q}}(e)\in {\mathcal{Q}}(U_z)$, and on path $\tilde R(e)$ (in the case where $e\in E_z$, we set $a_e=e$; for the other two cases, we have defined the edge $a_e$ explicitly). Moreover, the unique path of the internal $S_z$-router ${\mathcal{Q}}(S_z)$ that originates at edge $a_e$ is a subpath of $\tilde R(e)$. Therefore, the
last edge on path $\tilde R(e)$ is identical to the last edge on the unique path $\tilde Q(e)\in {\mathcal{Q}}(U_z)$ that originates at edge $e$. Recall that we have defined the ordering ${\mathcal{O}}=\tilde {\mathcal{O}}_z$ of the edges of $\delta_G(U_z)=\delta_{G_z}(v^{**})$ to be
${\mathcal{O}}^{\operatorname{guided}}({\mathcal{Q}}(U_z),\Sigma)$ -- the ordering that is guided by the internal $U_z$-router ${\mathcal{Q}}(U_z)$ (see definition in \Cref{subsec: guiding paths rotations}). Since the rotation ${\mathcal{O}}_{u_z}$ in $\Sigma$ and $\Sigma_z$ is identical, equivalently, ${\mathcal{O}}'={\mathcal{O}}^{\operatorname{guided}}(\tilde{\mathcal{R}},\Sigma_z)$, that is, ordering ${\mathcal{O}}'$ can be defined as an ordering that is guided by the set $\tilde {\mathcal{R}}$ of paths in graph $G_z$, with respect to the rotation system $\Sigma_z$.
Notice that for every edge $e\in \delta_{G}(U_z)$, an edge $e'$ may lie on path $\tilde R(e)$ only if $e'$ lies on path $\tilde Q(e)\in {\mathcal{Q}}(U_z)$.
From our construction, if $e'\in \delta_{G_z}(U_z)$, then $e'$ may lie on at most one path of $\tilde {\mathcal{R}}$ -- the path $\tilde R(e')$. From \Cref{obs: bound congestion of routers}, an edge $e'\in E(G_z)\setminus E(S_z)$ may participate in at most $O(\log^{34}m)$ paths of $\tilde {\mathcal{R}}$, and an edge $e'\in E(S_z)$ may participate in at most $O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_z),e')$ paths of $\tilde {\mathcal{R}}$.
Next, we construct an auxiliary graph $H'_z$, by replicating some edges of $G_z$ and deleting some other edges, similarly to our construction of graph $H_z$. We will use the paths of $\tilde {\mathcal{R}}$ in order to define a collection of edge-disjoint paths $\tilde {\mathcal{R}}'$ in the resulting graph $H'_z$, which will in turn be used in order to construct the collection $\Gamma^*$ of curves.
\paragraph{Graph $H'_z$ and its drawing $\psi'_z$}
For every edge $e\in E(G_z)$, we let $\tilde N(e)$ be the number of paths in $\tilde {\mathcal{R}}$ that contain the edge $e$.
From the discussion so far, we obtain the following immediate observation.
\begin{observation}\label{obs: bound tilde N}
For each edge $e\in E(S_z)$, $\tilde N(e)\leq O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_z),e')$, and for each edge $e\in E(G_z)\setminus E(S_z)$, $\tilde N(e)\leq O(\log^{34}m)$.
\end{observation}
In order to construct the graph $H'_z$, we start with the set $V(H'_z)=V(G_z)$ of vertices. For every edge $e\in E(G_z)$ with $\tilde N(e)>0$, we add a collection $J'(e)$ of $\tilde N(e)$ parallel copies of the edge $e$ to graph $H'_z$. We also assign each copy of edge $e$ in set $J'(e)$ to a distinct path of $\tilde {\mathcal{R}}$ that contains the edge $e$, arbitrarily. As in Step 1 of the algorithm for computing a drawing of graph $G_z$, we can now define a collection $\tilde {\mathcal{R}}'=\set{\tilde R'(e)\mid e\in \delta_G(U_z)}$ of edge-disjoint paths in graph $H'_z$, as follows: for each edge $e\in \delta_G(U_z)$, path $\tilde R'(e)$ is obtained from path $\tilde R(e)$ by replacing each edge $e'\in \tilde R(e)$ with the copy of edge $e'$ that is assigned to path $\tilde R(e)$.
Drawing $\phi''_z$ of graph $G_z$ naturally defines drawing $\psi'_z$ of graph $H'_z$: for each edge $e\in E(G_z)$ with $\tilde N(e)>0$, we draw all copies of $e$ to appear in parallel to the image of $e$, without crossing each other.
As in Step 1 of the algorithm for constructing a drawing for graph $G_z$, we can assign the copies of the edges incident to vertex $u_z$ more carefully, to ensure that the images of the paths in $\tilde {\mathcal{R}}'$ enter the image of $u_z$ according to the ordering $\tilde {\mathcal{O}}_z={\mathcal{O}}'$. In other words, if we denote $\delta_G(U_z)=\set{\tilde a'_1,\tilde a'_2,\ldots\tilde a'_h}$, and the edges are indexed in the order of their appearance in the ordering $\tilde {\mathcal{O}}_z={\mathcal{O}}'$, and if, for all $1\leq i\leq h$, we denote by $\tilde a''_i$ the last edge on the path $\tilde R'(\tilde a'_i)$, then the images of the edges $\tilde a''_1,\ldots,\tilde a''_h$ enter the image of vertex $u_z$ in the natural order of their indices. Note that the images of edges $\tilde a_1',\ldots,\tilde a_h'$ enter the image of $v^{**}$ in the ordering ${\mathcal{O}}$.
We now bound the number of crossings in the drawing $\psi'_z$ of graph $H'_z$. Consider any crossing in $\psi'_z$ between a pair of edges $e_1',e_2'$. Assume that $e_1'$ is a copy of edge $e_1\in E(G_z)$, and $e_2'$ is a copy of edge $e_2\in E(G_z)$. Clearly, the images of the edges $e_1$ and $e_2$ must cross in drawing $\phi_z''$, and we say that this crossing is responsible for the crossing $(e_1',e_2')$. It is easy to see that a crossing $(e_1,e_2)$ in drawing $\phi_z$ may be responsible for at most $\tilde N(e_1)\cdot \tilde N(e_2)$ crossings in $\psi'_z$. If neither of the edges $e_1,e_2$ lie in $E(S_z)$, then, from \Cref{obs: bound tilde N}, $\tilde N(e_1),\tilde N(e_2)\leq O(\log^{34}m)$. Therefore, the total number of crossings of $\psi_{z}'$, for which crossings $(e_1,e_2)$ of $\phi''_z$ with $e_1,e_2\not\in E(S_z)$ are responsible is at most:
$O(\log^{68}m)\cdot \mathsf{cr}(\phi''_z)$.
If exactly one of the two edges (say $e_1$) lies in $E(S_z)$, then, from \Cref{obs: bound tilde N}, $\tilde N(e_1)\leq O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_z),e_1)$, while $\tilde N(e_2)\leq O(\log^{34}m)$. Moreover, crossing $(e_1,e_2)$ must be a type-1 primary crossing of $\phi''_z$. Recall that the expected number of type-1 crossings in $\phi''_z$ is bounded by $|\chi^*_z|\cdot \hat\eta$.
Recall also that the random variable corresponding to the total number of type-1 primary crossings only depends on the random choices of the internal routers ${\mathcal{Q}}(S_{z-1})$ and ${\mathcal{Q}}(S_{z+1})$, and it is independent of the random choice of the internal router ${\mathcal{Q}}(S_{z})$. For each edge $e\in E(S_z)$, $\expect{\cong_G({\mathcal{Q}}(S_z),e)}\leq \hat \eta$ (from \Cref{obs: congestion square of internal routers} and our assumption that $S_z\in {\mathcal{S}}^{\operatorname{light}}$), and random variable $\cong_G({\mathcal{Q}}(S_z),e)$ only depends on the selection of the internal $S_z$-router ${\mathcal{Q}}(S_z)$. Since the random variable representing the number of type-1 primary crossings of $\phi''_z$ is independent from the random variables $\set{\cong_G({\mathcal{Q}}(S_z),e)}_{e\in E(S_z)}$, we get that the total expected number of crossings of $\psi_z''$, for which crossings $(e_1,e_2)$ of $\phi''_z$, with exactly one of $e_1,e_2$ lying in $E(S_z)$ are responsible, is at most: $|\chi^*_z|\cdot \hat \eta^{O(1)}$.
Lastly, assume that both edges $e_1,e_2\in E(S_z)$. Then, from \Cref{obs: bound tilde N}, $\tilde N(e_1)\leq O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_z),e_1)$ and $\tilde N(e_2)\leq O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_z),e_2)$. The number of crossigns in drawing $\psi_z'$ for which crossing $(e_1,e_2)$ is responsible is then bounded by:
\[\begin{split}& O(\log^{68}m)\cdot \cong_G({\mathcal{Q}}(S_z),e_1)\cdot \cong_G({\mathcal{Q}}(S_z),e_2)\\&\quad\quad\quad\quad\leq O(\log^{68}m)\textsf{left}( (\cong_G({\mathcal{Q}}(S_z),e_1))^2+ (\cong_G({\mathcal{Q}}(S_z),e_2))^2 \textsf{right} ) .
\end{split} \]
From \Cref{obs: congestion square of internal routers}, and since we have assumed that $S_z\in {\mathcal{S}}^{\operatorname{light}}$, for every edge $e\in E(S_z)$, $\expect{\textsf{left} (\cong_{G}({\mathcal{Q}}(S_z),e)\textsf{right} )^2}\le \hat \eta$. As before, random variable $\textsf{left} (\cong_{G}({\mathcal{Q}}(S_z),e)\textsf{right} )^2$ only depends on the random selection of the internal $S_z$-router ${\mathcal{Q}}(S_z)$.
Clearly, the expected number of crossings of $\psi'_z$ for which crossing $(e_1,e_2)$ is responsible is at most $O(\hat \eta^2)$.
Note also that crossing $(e_1,e_2)$ must a type-1 primary crossing of $\phi''_z$, and the expected number of such crossings is bounded by $|\chi^*_z|\cdot \hat \eta$. As before, the random variable corresponding to the number of type-1 primary crossings of $\phi''_z$ does not depend on the selection of the internal $S_z$-router ${\mathcal{Q}}(S_z)$. Therefore, the total expected number of crossings of $\psi_z''$, for which crossings $(e_1,e_2)$ of $\phi''_z$, with $e_1,e_2\in E(S_z)$ are responsible is at most: $|\chi^*_z|\cdot \hat \eta^{O(1)}$.
Overall, the total expected number of crossings in drawing $\psi'_z$ of graph $H'_z$ is bounded by:
\[ \mathsf{cr}(\phi''_z)\cdot O(\log^{68}m)+|\chi^*_z|\cdot \hat \eta^{O(1)}.\]
\paragraph{Constructing the set $\Gamma^*$ of curves.}
For every edge $e\in \delta_G(U_z)$, we initially let $\gamma(e)$ be the image of the path $\tilde R'(e)$ in the drawing $\psi'_z$ of graph $H'_z$. From our construction, the curves in set $\Gamma^*=\set{\gamma(e)\mid e\in \delta_G(U_z)}$ all originate at the image of vertex $v^{**}$, and terminate at the image of vertex $u_z$ in $\psi'_z$. Moreover, from our construction, the order in which the curves of $\Gamma^*$ enter the image of $v^{**}$ is according to the ordering ${\mathcal{O}}$ of the edges of $\delta_G(U_z)$, while the order in which the curves of $\Gamma^*$ enter the image of $u_z$ is according to the ordering ${\mathcal{O}}'$ of the edges of $\delta_G(U_z)$. However, the curves of $\Gamma^*$ are not in general position, as a point $p$ may serve as an inner point on more than $2$ such curves; this, however, may only happen if $p$ is an image of some vertex $v\in V(G_z)\setminus\set{u_z,v^{**}}$.
We will now ``nudge'' the curves in the vicinity of each such vertex to ensure that the resulting set of curves is in general position. The nudging procedure is identical to that we have employed in Step 2 (see \Cref{subsubsec: step 2}).
We process every vertex $v\in V(G_z)\setminus\set{u_z,v^{**}}$ one by one. Consider an iteration when any such vertex $v$ is processed. Let $A(v)\subseteq \delta_G(U_{z})$ be the set of all edges $e\in \delta_G(U_{z})$, such that curve $\gamma(e)$ contains the image of vertex $v$ (in $\psi'_z$). We denote $A(v)=\set{a_1,\ldots,a_k}$. Consider the tiny $v$-disc $D(v)=D_{\psi'_z}(v)$ in the drawing $\psi'_z$ of graph $H'_z$. For all $1\leq i\le k$, we let $s_i,t_i$ be the two points at which curve $\gamma(a_i)$ intersects the boundary of the disc $D(v)$. Note that all points $s_1,t_1,\ldots,s_k,t_k$ must be distinct. We use the algorithm from \Cref{claim: curves in a disc} in order to construct a collection $\set{\gamma'_1,\ldots,\gamma'_k}$ of curves, such that, for all $1\leq i\leq k$, curve $\gamma'_i$ has $s_i$ and $t_i$ as its endpoints, and is completely contained in $D(v)$. Recall that the claim ensures that, for every pair $1\leq i<j\leq k$ of indices, if the two pairs $(s_i,t_i),(s_j,t_j)$ of points cross, then curves $\gamma_i,\gamma_j$ intersect at exactly one point; otherwise, curves $\gamma_i,\gamma_j$ do not intersect.
For all $1\leq i\leq k$, we modify the curve $\Gamma(a_i)$ as follows: we replace the segment of the curve between points $s_i,t_i$ with the curve $\gamma_i$.
Once every vertex $v\in V(G_z)\setminus\set{u_z,v^{**}}$ is processed, we obtain the final collection $\Gamma^*=\set{\gamma(e)\mid e\in \delta_G(U_z)}$ of curves, which are now in general position. The order in which these curves enter the images of vertices $u_{z}$ and $v^{**}$ did not change, but we may have added some new crossings over the course of this modification of the curves in $\Gamma^*$. For convenience, we say that a crossing between a pair of curves in $\Gamma^*$ is \emph{primary} if this crossing existed before this last modification, and otherwise it is called \emph{secondary}. For each point $p$ corresponding to a secondary crossing, there must be a vertex $v$, with $p\in D(v)$.
The expected number of all primary crossings remains unchanged, and is bounded by $\mathsf{cr}(\phi''_z)\cdot O(\log^{68}m)+|\chi^*_z|\cdot \hat \eta^{O(1)}$. We now bound the expected number of all secondary crossings.
Consider a pair $e_1,e_2\in \delta_G(U_z)$ of distinct edges, and assume that there is some vertex $v\in V(G_z)\setminus\set{u_z,v^{**}}$, such that the curves $\gamma(e_1),\gamma(e_2)$ cross at some point $p\in D(v)$. Using the same arguments as before, this may only happen in one of two cases: either (i) some edge $e\in \delta_{G_z}(v)$ lies on both $\tilde R(e_1)$ and $\tilde R(e_2)$; or (ii) paths $\tilde R(e_1),\tilde R(e_2)$ have a transversal intersection at vertex $v$. In the former case, we say that the crossing is a type-1 secondary crossing, and that edge $e$ is responsible for it, while in the second case we say that the crossing is a type-2 secondary crossing, and that the transversal intersection of paths $\tilde R(e_1),\tilde R(e_2)$ at vertex $v$ is responsible for it.
Clearly, for every edge $e\in E(G_z)$, the total number of type-1 secondary intersections for which $e$ may be responsible is at most $(\tilde N(e))^2$. If $e\in \delta_{G_z}(v^{**})$, then $\tilde N(e)=1$, and $e$ may not be responsible for any type-1 secondary crossings. If $e\in E(G_z)\setminus (E(S_z)\cup \delta_G(v^{**}))$, then, from \Cref{obs: bound tilde N}, $\tilde N(e)\leq O(\log^{34}m)$. Otherwise, if $e\in E(S_z)$, then, from \Cref{obs: bound tilde N}, $\tilde N(e)\leq O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_{z}),e)$.
Moreover, since we have assumed that $S_z\in {\mathcal{S}}^{\operatorname{light}}$, from \Cref{obs: congestion square of internal routers}, $\expect{\textsf{left} (\cong_{G}({\mathcal{Q}}(S_z),e)\textsf{right} )^2}\le \hat \eta$.
Overall, the total expected number of type-1 secondary crossings is bounded by: $\hat\eta\cdot |E(G_z)\setminus \delta_{G_z}(v^{**})|\leq \hat \eta \cdot (|E(\tilde S_z)|+|\delta_G(U_{z-1}|)$. Since, from \Cref{cor: few edges crossing cuts},
$|\delta_G(U_{z-1})|\leq |\delta_G(S_{z})|\cdot O(\log^{34}m)$, and $\delta_G(S_z)\subseteq E(\tilde S_z)\cup \delta_G(\tilde S_z)$, we get that the total expected number of type-1 secondary crossings is at most:
$\hat \eta \cdot \textsf{left} (|E(\tilde S_z)|+| \delta_G(\tilde S_z)|\textsf{right} )$.
We now turn to bound the number of type-2 secondary crossings.
Consider any such crossing between a pair of curves $\gamma(e_1),\gamma(e_2)$, and assume that this crossing is charged to transversal intersection of the paths $\tilde R(e_1),\tilde R(e_2)$ at vertex $v$ with respect to $\Sigma_z$. We claim that in this case, $v=v^{*}$ must hold. Indeed, assume otherwise. As vertex $v^{**}$ may not serve as an inner vertex on any path of $\tilde {\mathcal{R}}'$, it must be the case that $v\in V(\tilde S_z)$. If $v\in V(S_z)$, then there must be two paths $Q,Q'\in {\mathcal{Q}}(S_z)$, such that $Q\subseteq \tilde R(e_1)$ and $Q'\subseteq \tilde R(e_2)$. From the construction of the internal router ${\mathcal{Q}}(S_z)$, all paths in ${\mathcal{Q}}(S_z)$ are non-transversal with respect to $\Sigma$, so it is impossible that $Q$ and $Q'$ have a transversal intersection at $v$, and the same is true for paths $\tilde R(e_1)$ and $\tilde R(e_2)$. Otherwise, $v\in V(\tilde S_z)\setminus V(S_z)$. In this case, $v$ must be a vertex that lies on each of the two paths $\tilde Q(e_1),\tilde Q(e_2)$ of the internal router ${\mathcal{Q}}(U_z)$, and moreover, the two paths must have a transversal intersection at $v$. But that is impossible from \Cref{obs: inner non-transversal}. Therefore, $v=v^{*}$ must hold.
We denote by $\Pi$ the set of all pairs $(e_1,e_2)\in \delta_G(U_z)$ of edges, such that paths $\tilde R(e_1),\tilde R(e_2)$ have a transversal intersection at vertex $v^*$, with respect to $\Sigma_Z$. Clearly, the number of type-2 secondary crossings between the curves of $\Gamma^*$ is bounded by $|\Pi|$.
We use the following claim, whose proof appears in \Cref{subsec: bound the Pi}, in order to bound $\expect{|\Pi|}$.
\begin{claim}\label{claim: bound on Pi}
\[\begin{split}
\expect{|\Pi|}&\leq \hat \eta^2\cdot \textsf{left}(\sum_{e\in E(G)}\expect{N'_z(e)}+\sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N'_z(e)}+\expect{N'_z(e')} \textsf{right})\textsf{right} )\\
&+\hat \eta^2\cdot \textsf{left} (|E(S_{z-1})|+|E(\tilde S_z)|+|\delta_G(S_{z-1})|+|\chi^*_{z-1}|+|\chi^*_z|\textsf{right})+|\Pi_z^T|.
\end{split}\]
\end{claim}
In order to complete the proof \Cref{claim: bound distance between rotations}, it now remains to bound the expected number of crossings between the curves of $\Gamma^*$.
Recall that the expected number of all primary crossings between the curves of $\Gamma^*$ is bounded by $\mathsf{cr}(\phi''_z)\cdot O(\log^{68}m)+|\chi^*_z|\cdot \hat \eta^{O(1)}$, while the expected number of type-1 secondary crossings is at most $\hat \eta \cdot \textsf{left} (|E(\tilde S_z)|+| \delta_G(\tilde S_z)|\textsf{right} )$. The expected number of type-2 secondary crossings is bounded by $\expect{|\Pi|}$. We conclude that, if $S_z,S_{z+1}\in {\mathcal{S}}^{\operatorname{light}}$, then the expected number of all crossings between the curves in $\Gamma^*$ is bounded by:
\[\begin{split} &\hat \eta^{O(1)}\cdot \textsf{left} (\sum_{e\in E(G)}\expect{N'_z(e)}+\sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N'_z(e)}+\expect{N'_z(e')} \textsf{right})\textsf{right} )\\
&\quad\quad\quad\quad+\eta^{O(1)}\cdot \textsf{left} (|\chi^*_{z-1}|+|\chi^*_z|+|E(S_{z-1})|+|E(\tilde S_z)|+| \delta_G(\tilde S_z)|+|\delta_G(S_{z-1})|\textsf{right} )\\
&\quad\quad\quad\quad+\hat \eta \cdot \mathsf{cr}(\phi''_z)+|\Pi_z^T|.
\end{split}
\]
In order to complete the proof of \Cref{claim: bound distance between rotations}, it now remains to prove \Cref{claim: bound on Pi}, which we do next.
\subsection{Proof of \Cref{claim: bound on Pi}}
\label{subsec: bound the Pi}
Assume first that $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$. Since we have assumed that Event ${\cal{E}}$ did not happen, from \Cref{obs: congestion square of internal routers}, $\mathsf{OPT}_{\mathsf{cnwrs}}(S_{z-1},\Sigma(S_{z-1}))+|E(S_{z-1})|\geq \frac{|\delta_G(S_{z-1})|^2}{\hat \eta}$, where $\Sigma(S_{z-1})$ is the rotation system for graph $S_{z-1}$ induced by $\Sigma$. We then get that $|\delta_G(S_{z-1})|^2\leq \hat \eta\cdot (|\chi^*_{z-1}|+|E(S_{z-1})|)$.
On the other hand, if $(e_1,e_2)\in \Pi$, then paths $\tilde Q(e_1),\tilde Q(e_2)\in {\mathcal{Q}}(U_z)$ must each contain an edge of $\delta_G(S_{z-1})$. Since, from \Cref{obs: bound congestion of routers}, each edge of $\delta_G(S_{z-1})$ may appear on at most $O(\log^{34}m)$ paths of ${\mathcal{Q}}(U_{z-1})$, we get that $|\Pi|\leq O(\log^{68}m)\cdot |\delta_G(S_{z-1})|^2\leq \hat \eta^2\cdot (|\chi^*_{z-1}|+|E(S_{z-1})|)$.
From now on we assume that $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$.
Consider a pair of edges $(e_1,e_2)\in \Pi$. Note that both $\tilde R(e_1)$ and $\tilde R(e_2)$ must contain the vertex $v^*$, and $e_1,e_2\in \delta_G(U_z)$ must hold. Recall that $\delta_G(U_z)=E_z\cup E_z^{\operatorname{right}}\cup E_z^{\operatorname{over}}$, and that, for each edge $e\in E_z$, path $\tilde R(e)$ may not contain vertex $v^*$. Therefore, $e_1,e_2\in E_z^{\operatorname{right}}\cup E_z^{\operatorname{over}}$ must hold.
Note that, for an edge $e\in E_z^{\operatorname{right}}$, it is possible that path $\tilde R(e)$ does not contain the vertex $v^*$. For convenience, we denote by $E_z^{\operatorname{right}'}$ the set of all edges $e\in E_z^{\operatorname{right}}$ for which $v^*\in \tilde R(e)$.
We denote by $\Pi^1\subseteq \Pi$ the set of all pairs $(e_1,e_2)$ where at least one of the two edges $e_1,e_2$ lies in $E_z^{\operatorname{right}'}$, and we denote by $\Pi^2=\Pi\setminus \Pi^1$. Clearly, for every pair $(e_1,e_2)\in \Pi^2$, $e_1,e_2\in E_z^{\operatorname{over}}$ must hold.
We will now define, for each edge $e\in E_z^{\operatorname{right}'}\cup E^{\operatorname{over}}_z$ three special edges $a^*_e,a_e$, and $\hat a_e$ associated with $e$, a new cycle $\hat W(e)$ in graph $G$, and some additional structures.
Consider first an edge $e\in E_z^{\operatorname{over}}$. Denote $e=(u,v)$, and assume that $u$ is the left endpoint of the edge. Then, from the definition of edge set $E_z^{\operatorname{over}}$, $u\in \bigcup_{i<z}V(\tilde S_i)$ and $v\in \bigcup_{i>z}V(\tilde S_i)$ must hold (see \Cref{fig: NN7a}).
In particular, $e\in \delta_G(U_{z-1})\cap \delta_G(U_z)$. We denote $a^*_e=e$. Clearly, $v^*\in \tilde R(e)$ must hold. We let $a_e$ be the edge immediately following vertex $v^*$ on path $\tilde R(e)$. Observe that $a_e$ is also an edge of graph $G$, where it must belong to edge set $E_{z-1}$ (see \Cref{fig: NN7a} and \Cref{fig: NN5}).
We denote the endpoints of edge $a_e$ graph $G$ by $a_e=(x_e,y_e)$, with $x_e\in V(S_{z-1})$ and $y_e\in V(S_z)$.
Since edge $a_e$ lies on path $\tilde Q(e)$, that path visits the cluster $S_{z-1}$. We denote by $\hat x_e$ the first vertex on path $Q(e)$ that belongs to cluster $S_{z-1}$, by $\hat a_e$ the edge preceding vertex $\hat x_e$ on the path, and by $\hat y_e$ the other endpoint of edge $\hat a_e$ (see \Cref{fig: NN7a}).
From the construction of the set ${\mathcal{Q}}(U_{z-1})$ of paths, and
the auxiliary cycles in ${\mathcal{W}}$, edges $\hat a_e$ and $a_e$ must lie on the cycle $W(e)$. We denote by $W'(e)$ the subpath of the auxiliary cycle $W(e)$ that connects vertex $\hat y_e$ to vertex $y_e$, such that all inner vertices of $W'(e)$ lie in $S_{z-1}$.
We denote by $\hat W^{\operatorname{left}}(e)$ the subpath of $W(e)$ from $\hat y_e$ to $v$, that is internally disjoint from $W'(e)$, and by $\hat W^{\operatorname{right}}(e)$ the subpath of $W(e)$ from $u$ to $y_e$ that is internally disjoint from $W'(e)$. Observe that edge $e$ lies on both $\hat W^{\operatorname{right}}(e)$ and $\hat W^{\operatorname{left}}(e)$.
Let $P(e)$ be the path of the internal $S_{z-1}$-router ${\mathcal{Q}}(S_{z-1})$ that originates at edge $a_e$,
and let $\hat P(e)$ be the path of ${\mathcal{Q}}(S_{z-1})$ that originates at edge $\hat a_e$.
We now define a new cycle $\hat W(e)$ associated with the edge $e$ to be the concatenation of the paths $\hat W^{\operatorname{left}},\hat P(e),P(e)$, and $\hat W^{\operatorname{right}}$ (after deleting the extra copy of the edge $e$, so that we obtain a cycle).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN7a.jpg}
\caption{Definition of edges $a_e, \hat a_e$ and $a^*_e$ when edge $e\in E_z^{\operatorname{over}}$. Path $W'_e$ is the concatenation of the brown path and edges $a_e, \hat a_e$. Paths $\hat W^{\operatorname{left}}(e)$ (connecting $e$ to $\hat y_e$) and $\hat W^{\operatorname{right}}(e)$ (connecting $e$ to $y_e$) are shown in pink; both these paths also contain edge $e=a^*_e$.}\label{fig: NN7a}
\end{figure}
Next, we consider an edge $e\in E_z^{\operatorname{right}'}$. Denote $e=(u,v)$, and assume that $u$ is the left endpoint of the edge. Then, from the definition of edge set $E_z^{\operatorname{right}}$, $u\in V(\tilde S_z)$ and $v\in \bigcup_{i>z}V(\tilde S_i)$ must hold (see \Cref{fig: NN7b}).
We let $a^*_e$ be the first edge on path $\tilde R(e)$ that is not contained in $E(\tilde S_z)$. Note that $a^*_e$ is also an edge of graph $G$, where it connects a vertex of $V(\tilde S_z)$, that we denote by $y^*_e$, to a vertex of $\bigcup_{i<z}V(\tilde S_i)$, that we denote by $x^*_e$. It is easy to see that edge $a^*_e$ must lie on the auxiliary cycle $W(e)$, and that it belongs to $\delta_G(U_{z-1})$, and more specifically to $E_z^{\operatorname{left}}$. We will say that edge $e$ \emph{owns} edge $a^*_e$, and that edge $a^*_e$ \emph{belongs} to edge $e$. Note that an edge of $E_z^{\operatorname{left}}$ may belong to a number of edges of $E_z^{\operatorname{right}'}$.
Since edge $a^*_e$ lies on the auxiliary cycle $W(e)$, it must be the case that $z-1\in \operatorname{span}''(e)$, and so cycle $W(e)$ must contain an edge of $E_{z-1}$, that we denote by $a_e$ (see \Cref{fig: NN7b}).
As before, we denote the endpoints of edge $a_e$ in graph $G$ by $a_e=(x_e,y_e)$, with $x_e\in V(S_{z-1})$ and $y_e\in V(S_z)$.
We also denote by $P(e)$ the unique path of the internal $S_{z-1}$-router ${\mathcal{Q}}(S_{z-1})$ that originates at edge $a_e$. We define the path $\hat W^{\operatorname{right}}(e)$ to be the subpath of the auxiliary cycle $W(e)$, between vertices $x^*_e$ and $y_e$, that is disjoint from cluster $S_{z-1}$. Notice that this path contains both edges $a^*_e$ and $e$. Path $\hat P(e)$ and edge $\hat e_a$ are defined slightly differently. Recall again that $a^*_e\in \delta_G(U_{z-1})$. We consider the unique path $Q(a^*_e)$ of the internal $U_{z-1}$-router ${\mathcal{Q}}(U_{z-1})$ that originates at edge $a^*_e$. Recall that path $Q(a^*_e)$ terminates at vertex $u_{z-1}$, so it must contain some edge of $\delta_G(S_{z-1})$, and, from the definition of the internal router ${\mathcal{Q}}(U_{z-1})$, exactly one edge of $\delta_G(S_{z-1})$ lies on path $Q(a^*_e)$. We denote that edge by $\hat a_e=(\hat x_e,\hat y_e)$, where $\hat x_e$ is the endpoint of the edge that lies in $S_{z-1}$. We let $\hat W^{\operatorname{left}}(e)$ be the subpath of $Q(a^*(e))$ from vertex $y^*_e$ to vertex $\hat y_e$. Observe that $a^*_e\in W^{\operatorname{left}}(e)$; and that path $W^{\operatorname{left}}(e)$ is a subpath of both the auxiliary cycle $W(a^*(e))$, and of path $W^{\mathsf{out},\operatorname{left}}(a^*(e))$. We denote by $\hat P(e)$ the unique path of the internal $S_{z-1}$-router ${\mathcal{Q}}(S_{z-1})$ that originates at edge $\hat a_e$. Lastly, we define a new cycle $\hat W(e)$ associated with edge $e$, to be the union of paths $\hat W^{\operatorname{left}}(e),\hat P(e),P(e)$, and $\hat W^{\operatorname{right}}(e)$, after we delete the extra copy of edge $a^*_e$ (see \Cref{fig: NN7b}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN7b.jpg}
\caption{Definitions of edges $a_e,\hat a_e$ and $a^*_e$ when edge $e\in E_z^{\operatorname{right}'}$. Path $\hat W^{\operatorname{left}}(e)$ is the concatenation of the purple path and edge $a^*_e$. Path $\hat W^{\operatorname{right}}(e)$ is the concatenation of edges $a^*_e, e, a_e$ and the two brown paths.}\label{fig: NN7b}
\end{figure}
For consistency of notation, for an edge $e=(u,v)\in E_z^{\operatorname{over}}$, where $u$ is the left endpoint of $e$, we will also say that $e$ owns the edge $a^*_e=e$, and that edge $a^*_e$ belongs to $e$. We will also denote $x^*_e=u$ and $y^*_e=v$.
For every edge $e\in E^{\operatorname{over}}_z\cup E_z^{\operatorname{right}'}$, we have now defined two paths $P(e),\hat P(e)\in {\mathcal{Q}}(S_{z-1})$. We denote by $a_e'$ the last edge on path $P(e)$, and by $\hat a_e'$ the last edge on path $\hat P(e)$; both these edges are incident to $u_{z-1}$ (see \Cref{fig: NN7a} and \Cref{fig: NN7b}).
We now provide several observations that will be useful for us later.
\begin{observation}\label{obs: transversal pairs property}
For every pair $(e_1,e_2)\in \Pi$, either edge set $\set{a'_{e_1},\hat a'_{e_1},a'_{e_2},\hat a'_{e_2}}$ contains fewer than four distinct edges, or edges $\hat a'_{e_1},\hat a'_{e_2},a'_{e_1},a'_{e_2}$ appear in this order in the rotation ${\mathcal{O}}_{u_{z-1}}\in \Sigma$.
\end{observation}
\begin{proof}
Since $(e_1,e_2)\in \Pi$, the paths $\tilde R(e_1),\tilde R(e_2)$ (that lie in graph $G_z$) have a transversal intersection at vertex $v^*$. Since vertex $v^*$ was obtained by contracting all vertices of $U_{z-1}$ in graph $G$, it is easy to verify that the edges of path $\tilde R(e_1)$ that immediately precede and follow vertex $v^*$ on the path are $a^*_{e_1}$ and $a_{e_1}$, respectively (see \Cref{fig: NN5} and \Cref{fig: NN6b}).
Similarly,
the edges of path $\tilde R(e_2)$ that immediately precede and follow vertex $v^*$ on the path are $a^*_{e_2}$ and $a_{e_2}$, respectively.
Since the paths $\tilde R(e_1),\tilde R(e_2)$ have a transversal intersection at vertex $v^*$, edges $a^*_{e_1},a^*_{e_2},a_{e_1},a_{e_2}$ appear in this order in the circular ordering ${\mathcal{O}}_{v^*}\in \Sigma_z$ (up to reversing the ordering). We now recall how the ordering ${\mathcal{O}}_{v^*}\in \Sigma_z$ was constructed.
Recall that $\delta_{G_z}(v^*)=\delta_G(U_{z-1})$, and in particular, $a^*_{e_1},a_{e_1},a^*_{e_2},a_{e_2}\in \delta_G(U_{z-1})$. The ordering
${\mathcal{O}}_{v^*}\in \Sigma_z$ was defined to be identical to the ordering $\tilde {\mathcal{O}}_{z-1}$ of the edges of $\delta_G(U_{z-1})$, which, in turn, is the ordering guided by the set ${\mathcal{Q}}(U_{z-1})$ of paths.
From our construction, it is immediate to verify that the last edge of the unique path in ${\mathcal{Q}}(U_{z-1})$ that originates at edge $a^*_{e_1}$ is $\hat a_{e_1}'$, and the last edge of the unique path in ${\mathcal{Q}}(U_{z-1})$ that originates at edge $a_{e_1}$ is $a_{e_1}'$ (see \Cref{fig: NN7a} and \Cref{fig: NN7b}).
Similarly, the last edge of the unique path in ${\mathcal{Q}}(U_{z-1})$ that originates at edge $a^*_{e_2}$ is $\hat a_{e_2}'$, and the last edge of the unique path in ${\mathcal{Q}}(U_{z-1})$ that originates at edge $a_{e_2}$ is $a_{e_2}'$. From the definition of the ordering $\tilde {\mathcal{O}}_{z-1}$, it must be the case that either set $\set{a_{e_1},\hat a_{e_1},a_{e_2},\hat a_{e_2}}$ contains fewer than four distinct edges, or edges $\hat a'_{e_1},\hat a'_{e_2},a'_{e_1},a'_{e_2}$ appear in this order in the rotation ${\mathcal{O}}_{u_{z-1}}\in \Sigma$.
\end{proof}
We will use the following simple observation in order to bound the congestion that is caused by the set $\hat {\mathcal{W}}=\set{\hat W(e)\mid e\in E_z^{\operatorname{over}}\cup E_z^{\operatorname{right}'}}$ of cycles.
\begin{observation}\label{obs: bound cong due to new cycles}
Each edge $e\in E_z^{\operatorname{left}}$ may belong to at most $O(\log^{34}m)$ edges of $E_z^{\operatorname{right}'}$. Additionally, an edge $e\in E(G)\setminus E(S_{z-1})$ may lie on at most $O(\log^{68}m)$ cycles of $\hat {\mathcal{W}}$, while an edge $e\in E(S_{z-1})$ may lie on at most $O(\log^{68}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$ cycles of $\hat {\mathcal{W}}$. Lastly, an edge $e\in E(G)\setminus \textsf{left} ( E(\tilde S_z)\cup E(S_{z-1})\textsf{right} )$ may lie on at most $O(\log^{34}m)\cdot N'_z(e)$ cycles of $\set{\hat W(e')\mid e'\in E_z^{\operatorname{right}'}}$.
\end{observation}
\begin{proof}
In order to prove the first assertion, consider any edge $e\in E_z^{\operatorname{left}}$. From our construction, $e$ may belong to an edge $e'\in E_z^{\operatorname{right}'}$ only if $e\in W(e')$. From \Cref{obs: bound congestion of cycles}, edge $e$ may lie on at most $O(\log^{34}m)$ cycles of ${\mathcal{W}}$, and so $e$ may belong to at most $O(\log^{34}m)$ edges of $E_z^{\operatorname{right}'}$.
Consider now some edge $e\in E(G)\setminus E(S_{z-1})$. Notice that, if $e$ lies on a cycle $\hat W(e')$ for some edge $e'\in E_z^{\operatorname{right}'}\cup E_z^{\operatorname{over}}$, then either $e\in W(e')$, or $e\in W(a^*_{e'})$ must hold. Since, from \Cref{obs: bound congestion of cycles}, edge $e$ may lie on at most $O(\log^{34}m)$ cycles of ${\mathcal{W}}$, and, as we have shown, every edge $e''\in E_z^{\operatorname{left}}$ may belong to at most $O(\log^{34}m)$ edges of $E_z^{\operatorname{right}}$, we get that $e$ may lie on at most $O(\log^{68}m)$ cycles of $\hat {\mathcal{W}}$.
Consider now an edge $e\in E(S_{z-1})$. Notice that, if $e$ lies on a cycle $\hat W(e')$ for some edge $e'\in E_z^{\operatorname{right}'}\cup E_z^{\operatorname{over}}$, then either $e\in P(e')$, or $e\in \hat P(e')$ must hold. Consider some path $P\in {\mathcal{Q}}(S_{z-1})$ that contains the edge $e$, and let $a$ be the first edge on path $P$. Consider any edge $e'\in E_z^{\operatorname{right}'}\cup E_z^{\operatorname{over}}$, for which $P=P(e')$ or $P=\hat P(e')$ holds. Then $a=a_{e'}$ or $a=\hat a_{e'}$ must hold, and in particular, edge $a$ must lie on $\hat W(e')$. As we have shown, every edge $e\in \delta_G(S_{z-1})$ may lie on at most $O(\log^{68}m)$ cycles of $\hat {\mathcal{W}}$. Therefore, there are at most $O(\log^{68}m)$ edges $e'\in E_z^{\operatorname{right}'}\cup E_z^{\operatorname{over}}$, for which $P=P(e')$ or $P=\hat P(e')$ holds. We conclude that $e$
may lie on at most $O(\log^{68}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$ cycles of $\hat W$.
It now remains to prove the last assertion. Consider an edge $e\in E(G)\setminus \textsf{left} ( E(\tilde S_z)\cup E(S_{z-1})\textsf{right} )$. Assume that $e\in \hat W(e')$ for some edge $e'\in E_z^{\operatorname{right}'}$. From our construction, this may only happen if either $e$ lies on the unique path of ${\mathcal{Q}}(U_z)$ that originates at $e'$; or $e$ lies on the unique path of ${\mathcal{Q}}(U_{z-1})$ that originates at $a^*_{e'}$.
Recall that, in the latter case, $a^*_{e'}\in E_z^{\operatorname{left}}$ must hold. Recall that $N'_z(e)$ is the total number of paths in ${\mathcal{Q}}(U_{z-1})\cup {\mathcal{Q}}'(U_z)$ that originate at edges of $E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{right}}\cup E_z$ and contain $e$.
Since each edge $a^*\in E_z^{\operatorname{left}}$ may belong to at most $O(\log^{34}m)$ edges of $E_z^{\operatorname{right}'}$, we get that, overall, edge $e$ may lie on at most $O(\log^{34}m)\cdot N'_z(e)$ cycles of $\set{\hat W(e')\mid e'\in E_z^{\operatorname{right}'}}$.
\end{proof}
Recall that we have denoted by $\Pi^1\subseteq \Pi$ the set of all edge pairs $(e_1,e_2)\in \Pi$, where at least one of the two edges lies in $E_z^{\operatorname{right}'}$. We will always assume w.l.o.g. that $e_1\in E_z^{\operatorname{right}'}$ for each such pair. We bound the expected cardinalities of sets $\Pi^1$ and $\Pi^2$ separately in the following two claims.
\begin{claim}\label{claim: bound Pi1}
The expected cardinality of set $\Pi^1$ is at most:
\[ \hat \eta^2\cdot \textsf{left}(\sum_{e\in E(G)}\expect{N'_z(e)}+\sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N'_z(e)}+\expect{N'_z(e')}\textsf{right} )+|E(S_{z-1}|+|E(\tilde S_z)|+|\chi^*_{z-1}|+|\chi^*_z|\textsf{right} )+ |\Pi^T_z|.\]
\end{claim}
\begin{proof}
We denote by $\Pi^1_1\subseteq \Pi^1$ the set of all pairs $(e_1,e_2)\in \Pi^1$, for which cycles $\hat W(e_1),\hat W(e_2)$ share at least one edge. We let $e$ be any edge in $E(\hat W(e_1))\cap E(\hat W(e_2))$, and we say that $e$ is responsible for the pair $(e_1,e_2)$.
Consider now any pair of edges $(e_1,e_2)\in \Pi^1\setminus \Pi^1_1$, and their two corresponding cycles $\hat W(e_1),\hat W(e_2)$. Note that the two cycles do not share edges, and, from \Cref{obs: transversal pairs property}, they have a transversal intersection at vertex $u_{z-1}$. Therefore, either there is a pair of edges $e_1'\in E(\hat W(e_1))$ and $e_2'\in E(\hat W(e_2))$ that cross in the drawing $\phi^*$ of $G$; or there is a vertex $v\neq u_{z-1}$, such that $\hat W(e_1),\hat W(e_2)$ have a transversal intersection at $v$. In the former case, we say that crossing $(e_1',e_2')$ is responsible for the edge pair $(e_1,e_2)$. In the latter case, we say that the transversal intersection of $\hat W(e_1),\hat W(e_2)$ at $v$ is responsible for the pair $(e_1,e_2)$. We denote by $\Pi^1_2\subseteq \Pi^1\setminus \Pi^1_1$ the set of all pairs $(e_1,e_2)$, such that some crossing $(e_1',e_2')$ is responsible for $(e_1,e_2)$, and we denote by $\Pi^1_3=\Pi^1\setminus (\Pi^1_1\cup \Pi^1_2)$. We now bound the number of pairs in each one of the three sets one by one in the following three observations.
\begin{observation}\label{obs: bound first set of pairs}
$\expect{|\Pi_1^1|}\leq \hat \eta^2\cdot \textsf{left} (\sum_{e\in E(G)}\expect{N'_z(e)}+|E(S_{z-1}|+|E(\tilde S_z)|\textsf{right} )$.
\end{observation}
\begin{proof}
Consider an edge $e\in E(G)$. We will bound the expected number of pairs $(e_1,e_2)\in \Pi^1_1$ of edges, for which edge $e$ is responsible. For each such pair, we assume w.l.o.g. that $e_1\in E_z^{\operatorname{right}'}$. We distinguish between three cases.
The first case is when $e\in E(G)\setminus \textsf{left} ( E(\tilde S_z)\cup E(S_{z-1})\textsf{right} )$. From \Cref{obs: bound cong due to new cycles}, $e$ may lie on at most $O(\log^{34}m)\cdot N'_z(e)$ cycles of $\set{\hat W(e')\mid e'\in E_z^{\operatorname{right}'}}$, and on at most $O(\log^{68}m)$ cycles of $\hat {\mathcal{W}}$. Therefore, such an edge may be responsible for at most $O(\log^{102}m)\cdot N'_z(e)\leq \hat \eta \cdot N'_z(e)$ edge pairs in $\Pi_1^1$.
The second case is when $e\in E(\tilde S_z)$. In this case, from \Cref{obs: bound cong due to new cycles}, edge $e$ lies on at most $O(\log^{68}m)$ cycles of $\hat {\mathcal{W}}$. Therefore, such an edge may be responsible for at most $O(\log^{136}m)\leq \hat \eta$ edge pairs in $\Pi_1^1$, and overall, the edges of $\tilde S_z$ may be responsible for at most $\hat \eta\cdot |E_G(\tilde S_z)|$ edge pairs in $\Pi^1_1$.
The third and the last case is when $e\in E(S_{z-1})$. From \Cref{obs: bound cong due to new cycles}, such an edge may lie on at most $O(\log^{68}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$ cycles of $\hat W$, and so it may be responsible for at most $O(\log^{138}m)\cdot \textsf{left}(\cong_G({\mathcal{Q}}(S_{z-1}),e)\textsf{right})^2\leq \hat \eta \cdot \textsf{left}(\cong_G({\mathcal{Q}}(S_{z-1}),e)\textsf{right})^2$ edge pairs in $\Pi_1^1$. Since we have assumed that $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, from \Cref{obs: congestion square of internal routers}, $\expect{\textsf{left} (\cong_{G}({\mathcal{Q}}(S_{z-1}),e)\textsf{right} )^2}\le \hat \eta$. Therefore, the expected number of edge pairs in $\Pi_1^1$ for which edge $e$ is responsible is at most $\hat \eta^2$, and the total expected number of edge pairs in $\Pi_1^1$ for which edges of $E(S_{z-1})$ are responsible is at most $\hat \eta^2\cdot |E(S_{z-1})|$. The bound now follows.
\end{proof}
\begin{observation}\label{obs: bound second set of pairs}
$\expect{|\Pi_2^1|}\leq \hat \eta^2\cdot \textsf{left}(\sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N'_z(e)}+\expect{N'_z(e')}\textsf{right} )+|\chi^*_{z-1}|+|\chi^*_z|\textsf{right} )$.
\end{observation}
\begin{proof}
Consider a crossing $(e,e')\in \phi^*$. We bound the number of pairs $(e_1,e_2)\in \Pi_2^1$ with $e_1\in E_z^{\operatorname{right}'}$, for which the crossing $(e,e')$ is responsible.
Recall that, if crossing $(e,e')$ is responsible for a pair $(e_1,e_2)\in \Pi_2^1$, then $e\in \hat W(e_1)$ and $e'\in \hat W(e_2)$ must hold.
We first consider the case where neither of the edges $e,e'$ lie in $E(\tilde S_z)\cup E(S_{z-1})$. In this case, from \Cref{obs: bound cong due to new cycles}, edge $e'$ may lie on at most $O(\log^{68}m)$ cycles of $\hat {\mathcal{W}}$, while edge $e$ may lie on at most $O(\log^{34}m)\cdot N'_z(e)$ cycles of $\set{\hat W(e')\mid e'\in E_z^{\operatorname{right}'}}$. Therefore, crossing $(e,e')$ may be responsible for at most $O(\log^{102}m)\cdot N'_z(e)\leq \hat \eta \cdot N'_z(e)$ edge pairs in $\Pi_2^1$. Note that, if either of the edges $e,e'$ lies in $E(\tilde S_z)\cup E(S_{z-1})$, then crossing $(e,e')$ belongs to $\chi^*_{z-1}\cup \chi^*_{z}$. We conclude overall, all crossings $(e,e')\in \chi^*\setminus (\chi^*_{z-1}\cup \chi^*_{z})$ may be responsible for at most $\sum_{(e,e')\in \chi^*}\hat \eta \cdot \textsf{left} (N'_z(e)+N'_z(e')\textsf{right} )$ pairs in $\Pi_2^1$.
Next, we consider the case where at least one of the edges $e,e'$ lies in $E(\tilde S_z)\cup E(S_{z-1})$, so crossing $(e,e')$ belongs to $\chi^*_{z-1}\cup \chi^*_{z}$. We let $\hat N(e)$ be the random variable indicating the number of cycles in $\hat {\mathcal{W}}$ containing edge $e$, and we define random variable $\hat N(e')$ for edge $e'$ similarly. Notice that random variables $\hat N(e),\hat N(e')$ may not be independent, if $e,e'\in E(S_{z-1})$. The total number of edge pairs in $\Pi^1_2$ for which crossing $(e,e')$ is responsible is bounded by $\hat N(e)\cdot \hat N(e')\leq (\hat N(e))^2+(\hat N(e'))^2$. From \Cref{obs: bound cong due to new cycles}, combined with \Cref{obs: congestion square of internal routers} and our assumption that $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, we get that $\expect{(\hat N(e))^2}, \expect{(\hat N(e'))^2)}\leq \hat \eta^2$. We conclude that the expected number of pairs in $\Pi^1_2$ for which all crossings $(e,e')\in \chi^*_{z-1}\cup \chi^*_{z}$ are responsible is at most $\hat \eta^2\cdot (|\chi^*_{z-1}|+|\chi^*_z|)$.
\end{proof}
\begin{observation}\label{obs: bound third set of pairs}
$|\Pi_3^1|\leq |\Pi^T_z|$.
\end{observation}
\begin{proof}
Consider an edge pair $(e,e')\in \Pi^1_3$. Recall that there must be a vertex $v\neq u_{z-1}$, such that cycles $\hat W(e),\hat W(e')$ have a transversal intersection at $v$. Recall also that $e\in E_z^{\operatorname{right}'}\subseteq E_z^{\operatorname{right}}$. We claim that $v\in V(\overline{U}_z)$ must hold, and moreover, auxiliary cycles $W(e),W(e')$ must have a transversal intersection at vertex $v$.
Indeed, assume first that $v\in V(S_{z-1})\setminus\set {u_{z-1}}$. In this case, vertex $v$ must lie on $P(e)\cup \hat P(e)$, and on $P(e')\cup \hat P(e')$. Since paths $P(e),\hat P(e),P(e'),\hat P(e')$ all belong to the internal $S_{z-1}$-router ${\mathcal{Q}}(S_{z-1})$, they cannot have a transversal intersection at any vertex.
Assume now that $v\in V(U_{z-1})\setminus V(S_{z-1})$. In this case, by our construction, $v$ lies on the auxiliary cycle $W(a^*_e)$ of the edge $a^*_e\in E_z^{\operatorname{left}}$ that belongs to $e$, and similarly, $v$ lies on the auxiliary cycle $W(a^*_{e'})$.
Moreover, cycles $W(a^*_e),W(a^*_{e'})$ must have a transversal intersection at vertex $v$.
From \Cref{obs: auxiliary cycles non-transversal at at most one}, this is only possible if $v\in S_j$ for some index $1< j<r$, and either $j-1$ is the last index in both $\operatorname{span}''(a^*_e),\operatorname{span}''(a^*_{e'})$, or $j-1$ is the last index in one of these sets, while $j$ belongs to another. This is impossible, since $v\in V(U_{z-1})$, and $a^*_e,a^*_{e'}\in \delta_G(U_{z-1})$.
We conclude that vertex $v$ may not lie in $U_{z-1}$. But then, from the construction of the cycles $\hat W(e),\hat W(e')$, $v$ must lie on both the auxiliary cycles $W(e),W(e')$, and the two cycles must have a transversal intersection at $v$. From
\Cref{obs: auxiliary cycles non-transversal at at most one}, this is only possible if $v\in S_j$ for some index $1< j<r$, and either $j-1$ is the last index in both $\operatorname{span}''(e),\operatorname{span}''(e')$, or $j-1$ is the last index in one of these sets, while $j$ belongs to another. Since $e,e'\in \delta_G(U_z)$, we conclude that $v\in V(\overline{U}_z)$ must hold, and, from the above discusison, $W(e),W(e')$ must have a transversal intersection at vertex $v$. Recall that $\Pi^T_z$ is the set of triples $(\tilde e,\tilde e',\tilde v)$, where $\tilde e\in E_z^{\operatorname{right}}$, $\tilde e'\in \hat E_z$, and cycles $W(\tilde e)$ and $W(\tilde e')$ have a transversal intersection at $\tilde v$.
We conclude that, if the transversal crossing of cycles $\hat W(e),\hat W(e')$ at vertex $v$ is responsible for edge pair $(e,e')$, then triple $(e,e',v)$ must lie in $\Pi_z^T$, and so $|\Pi^1_3|\leq |\Pi^T_z|$.
\end{proof}
Combining the bounds from Observations \ref{obs: bound first set of pairs}--\ref{obs: bound third set of pairs}, we get that the expected cardinality of set $\Pi^1$ is at most:
\[ \hat \eta^2\cdot \textsf{left}(\sum_{e\in E(G)}\expect{N'_z(e)}+\sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N'_z(e)}+\expect{N'_z(e')}\textsf{right} )+|E(S_{z-1}|+|E(\tilde S_z)|+|\chi^*_{z-1}|+|\chi^*_z|\textsf{right} )+ |\Pi^T_z|.\]
\end{proof}
We use the following claim to bound the expected cardinality of $\Pi^2$.
\begin{claim}\label{claim: bound Pi2}
$$\expect{|\Pi^2|}\leq \hat \eta^2\cdot \textsf{left}(|E(S_{z-1})|+|\delta_G(S_{z-1})|+ |\chi^*_{z-1}|\textsf{right}).$$
\end{claim}
\begin{proof}
Recall that set $\Pi^2$ contains all edge pairs $(e,e')\in \Pi$, with $e,e'\in E_z^{\operatorname{over}}$. Consider any edge $e\in E_z^{\operatorname{over}}$. Recall that we denoted by $W'(e)$ the subpath of the auxiliary cycle $W(e)$ between vertices $y_e$ and $\hat y_e$ that intersects cluster $S_{z-1}$. We denote by $W''(e)$ the subpath of $W(e)$ between vertices $y_e$ and $\hat y_e$ that does not share edges with $W'(e)$. Equivalently, $W''(e)$ is the concatenation of the paths $\hat W^{\operatorname{left}}(e)$ and $\hat W^{\operatorname{right}}(e)$ (after the extra copy of edge $e$ is deleted). We denote by $H(e)$ the
graph obtained from the union of the paths $W'(e),P(e)$ and $\hat P(e)$.
The following observation, whose proof is deferred to \Cref{subsec: proof of bound transversal pairs} is central to the proof of \Cref{claim: bound Pi2}.
\begin{observation}\label{obs: bound transversal pairs}
Let $(e_1,e_2)$ be a pair of edges in $\Pi^2$. Then one of the following must happen: either (i) some edge lies in both $H(e_1)$ and $H(e_2)$; or (ii) there is a pair of edges $\tilde e_1\in H(e_1)$, $\tilde e_2\in H(e_2)\cup W''(e_2)$, whose images cross in the drawing $\phi^*$ of graph $G$; or (iii)
there is a pair of edges $\tilde e_1'\in H(e_1)\cup W''(e_1)$, $\tilde e_2'\in H(e_2)$, whose images cross in the drawing $\phi^*$ of graph $G$.
\end{observation}
As before, we partition the set $\Pi^2$ of edge pairs into two subsets. The first set, $\Pi^2_1$, containing all pairs $(e_1,e_2)\in \Pi^2$, such that there is an edge $e\in E(H_1)\cap E(H_2)$. In this case, we say that edge $e$ is responsible for the pair $(e_1,e_2)$. Set $\Pi^2_2$ contains all remaining edge pairs $(e_1,e_2)\in \Pi^2$. From \Cref{obs: bound cong due to new cycles}, for each such pair $(e_1,e_2)\in \Pi^2_2$, there must be a crossing in $\chi^*$ between a pair of edges
$\tilde e_1$ and $\tilde e_2$, such that either (i) $\tilde e_1\in H(e_1)$ and $\tilde e_2\in H(e_2)\cup W''(e_2)$; or (ii) $\tilde e_1'\in H(e_1)\cup W''(e_1)$ and $\tilde e_2'\in H(e_2)$. For convenience, we will always assume that it is the former. Since $H(e_1)\subseteq S_{z-1}\cup \delta_G(S_{z-1})$, crossing $(e_1,e_2)$ must lie in $\chi^*_{z-1}$, and we say that this crossing is responsible for the pair $(e_1,e_2)\in \Pi$.
We bound the expected cardinalities of the sets $\Pi^2_1,\Pi^2_2$ separately, as before.
In order to bound $\expect{|\Pi^2_1|}$, consider some edge $e\in E(S_{z-1})\cup \delta_G(S_{z-1})$. Note that edge $e$ may only lie in graph $H(e')$, for an edge $e'\in E_z^{\operatorname{over}}$, if $e\in W(e')$, or $e\in \hat W(e')$.
From \Cref{obs: bound congestion of cycles},
edge $e$ may appear on at most $O(\log^{34}m)$ auxiliary cycles of ${\mathcal{W}}$, and, from \Cref{obs: bound cong due to new cycles}, $e$ may lie on at most $O(\log^{68}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$ cycles of $\hat W$. From \Cref{obs: congestion square of internal routers}, since we have assumed that $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, we get that $\expect{\textsf{left} (\cong_{G}({\mathcal{Q}}(S_{z-1}),e)\textsf{right} )^2}\le \hat \eta$. Therefore, the expected number of pairs in $\Pi^2_1$, for which edge $e$ is responsible is at most:
\[O(\log^{136}m)\cdot \expect{(\cong_G({\mathcal{Q}}(S_{z-1},e)))^2}\leq \hat \eta^2. \]
We conclude that $\expect{|\Pi^2_1|}\leq \hat \eta^2\cdot \textsf{left}(|E(S_{z-1})|+|\delta_G(S_{z-1})|\textsf{right})$.
\iffalse
Note that, if $e\in \hat P(e')$, then $e$ lies on the unique path of the internal router ${\mathcal{Q}}(S_{z-1})$ that originates at the edge $\hat a_{e'}$, which, in turn, lies on the cycle $W(e')$. Since edge $e$ lies on at most $\cong_G({\mathcal{Q}}(S_{z-1}),e)$ paths of ${\mathcal{Q}}(S_{z-1})$, and each edge in $\delta_G(S_{z-1})$ belongs to at most $O(\log^{34}m)$ cycles of ${\mathcal{W}}$, there can be at most $O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$
edges $e'\in \delta_G(U_{z-1})$ with $e\in \hat P(e')$. Similarly, if $e\in P(e')$, then $e$ lies on the unique path of the internal router router ${\mathcal{Q}}(S_{z-1})$ that originates at the edge $ a_{e'}\in W(e')$. Using similar arguments, there can be at most $O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$
edges $e'\in \delta_G(U_{z-1})$ with $e\in P(e')$. Overall, we get that there are at most $O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$
edges $e'\in \delta_G(U_{z-1})$ with $e\in H(e')$. The number of pairs $(e_1,e_2)\in \Pi$ for which edge $e$ may be responsible is then bounded by
$O(\log^{68}m)\cdot \textsf{left} (\cong_G({\mathcal{Q}}(S_{z-1}),e)\textsf{right} )^2$. Since we have assumed that $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, from \Cref{obs: congestion square of internal routers}, $\expect{\textsf{left} (\cong_{G}({\mathcal{Q}}(S_i),e)\textsf{right} )^2}\le 2^{O(\sqrt{\log m}\cdot \log\log m)}$. Therefore, the expected number of pairs of $\Pi$ for which edge $e$ is responsible is bounded by $2^{O(\sqrt{\log m}\cdot \log\log m)}$. Overall, the total expected number of type-1 pairs in $\Pi$ is at most:
$$2^{O(\sqrt{\log m}\cdot \log\log m)}\cdot \textsf{left}(|E(S_{z-1})|+|\delta_G(S_{z-1})|\textsf{right}).$$
\fi
In order to bound the expected cardinality of the set $\Pi_2^2$, consider some edge pair $(e_1,e_2)\in \Pi_2^2$, and the crossing $(\tilde e_1,\tilde e_2)$ that is responsible for it, where $\tilde e_1\in H(e_1)$ and $\tilde e_2\in H(e_2)\cup W''(e_2)$. Recall that $H(e_1)\subseteq E(S_{z-1})\cup \delta_G(S_{z-1})$, and that crossing $(\tilde e_1,\tilde e_2)$ must lie in $\chi^*_{z-1}$. Consider now any crossing $(e,e')\in \chi^*_{z-1}$, and assume w.l.o.g. that $e\in E(S_{z-1})\cup \delta_G(S_{z-1})$.
If $e'\in E(S_{z-1})\cup \delta_G(S_{z-1})$ as well, then for every pair $(e_1,e_2)\in \Pi_2^2$ for which crossing $(e,e')$ is responsible, $e\in H(e_1)$ and $e'\in H(e_2)$ must hold. As observed above, the total number of edges $e_1\in \delta_G(U_{z-1})$ with $e\in H(e_1)$ is $O(\log^{68}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$, and similarly, the total number of edges $e_2\in \delta_G(U_{z-1})$ with $e'\in H(e_2)$ is at most $O(\log^{68}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e')$. Therefore, the total number of edge pairs $(e_1,e_2)\in \Pi_2^2$ for which crossing $(e,e')$ is responsible is bounded by:
\[
\begin{split}
O(\log^{136}m)&\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e')\\ &\leq O(\log^{136}m)\cdot \textsf{left} ( (\cong_G({\mathcal{Q}}(S_{z-1}),e))^2+(\cong_G({\mathcal{Q}}(S_{z-1}),e'))^2\textsf{right} ).
\end{split}
\]
As before, from \Cref{obs: congestion square of internal routers} and the assumption that $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$: $$\expect{\textsf{left} (\cong_{G}({\mathcal{Q}}(S_{z-1}),e)\textsf{right} )^2}, \expect{\textsf{left} (\cong_{G}({\mathcal{Q}}(S_{z-1}),e')\textsf{right} )^2}\le \hat \eta.$$
Therefore, the expected number of edge pairs $(e_1,e_2)\in \Pi_2^2$ for which crossing $(e,e')$ is responsible is at most $\hat \eta^2$.
Lastly, we consider a crossing $(e,e')\in \chi^*_{z-1}$ with $e\in E(S_{z-1})\cup \delta_G(S_{z-1})$ and $e'\not\in E(S_{z-1})\cup \delta_G(S_{z-1})$. In this case, for every pair $(e_1,e_2)\in \Pi_2^2$ for which crossing $(e,e')$ is responsible, $e\in H(e_1)$, and $e'\in W''(e_2)\subseteq W(e_2)$ must hold. As before, the total number of edges $e_1\in \delta_G(U_{z-1})$ with $e\in H(e_1)$ is at most $O(\log^{68}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$, and, from \Cref{obs: bound congestion of cycles}, edge $e'$ appears on at most $O(\log^{34}m)$ cycles of ${\mathcal{W}}$.
Therefore, the total expected number of edge pairs $(e_1,e_2)\in \Pi_2^2$ for which crossing $(e,e')$ is responsible is bounded by:
\[ O(\log^{102}m)\cdot \expect{\cong_G({\mathcal{Q}}(S_{z-1}),e)}\leq \hat \eta^2, \]
from \Cref{obs: congestion square of internal routers}. Overall, we get that $\expect{|\Pi_2^2|}\leq \eta^2\cdot |\chi^*_{z-1}|$, and:
$$\expect{|\Pi^2|}\leq \hat \eta^2\cdot \textsf{left}(|E(S_{z-1})|+|\delta_G(S_{z-1})|+ |\chi^*_{z-1}|\textsf{right}).$$
\end{proof}
Combining the bounds from Claims \ref{claim: bound Pi1} and \ref{claim: bound Pi2}, we get that:
\[\begin{split}
\expect{|\Pi|}&\leq \hat \eta^2\cdot \textsf{left}(\sum_{e\in E(G)}\expect{N'_z(e)}+\sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N'_z(e)}+\expect{N'_z(e')} \textsf{right})\textsf{right} )\\
&+\hat \eta^2\cdot \textsf{left} (|E(S_{z-1})|+|E(\tilde S_z)|+|\delta_G(S_{z-1})|+|\chi^*_{z-1}|+|\chi^*_z|\textsf{right})+|\Pi_z^T|,
\end{split}\]
\iffalse
\[\expect{|\Pi|}\leq \hat \eta^2\cdot \textsf{left}(\sum_{e\in E(G)}N'_z(e)+\sum_{(e,e')\in \chi^*} N'_z(e)+|E(S_{z-1})|+|E(\tilde S_z)|+|\delta_G(S_{z-1})|+|\chi^*_{z-1}|+|\chi^*_z|\textsf{right})+|\Pi_z^T|,\]
\fi
completing the proof of \Cref{claim: bound on Pi}.
\subsubsection{Step 1: Computing Auxiliary Graph $H_z$ and Its Drawing}
Recall that graph $G_z$ is obtained from graph $G$ by contracting all vertices of $\bigcup_{1\leq i<z}V(\tilde S_i)$ into the special vertex $v^*_z$, and then contracting all vertices of $\bigcup_{z<i\leq r}$ into the special vertex $v^{**}_z$. Clearly, every edge of $G_z$ corresponds to some edge of $G$, and we do not distinguish between these edges. In order to simplify the notation, when the index $z$ is fixed, we denote vertices $v^*_z$ and $v^{**}_z$ by $v^*$ and $v^{**}$, respectively. Notice that $\delta_{G_z}(v^*)=\delta_G(U_{z-1})=E_{z-1}\cup \hat E_{z-1}$, while $\delta_{G_z}(v^{**})=\delta_G(U_z)=E_{z}\cup \hat E_z$. In order to obtain the drawing $\phi_z$ of graph $G_z$, we will exploit the internal $U_{z-1}$-router ${\mathcal{Q}}(U_{z-1})$, that routes the edges of $\delta_G(U_{z-1})$ to vertex $u_{z-1}$, and the external $U_z$-router ${\mathcal{Q}}'(U_z)$, that routes the edges of $\delta_G(U_z)$ to vertex $u_{z+1}$. For each edge $e\in \delta_G(U_{z-1})$, we denote by $Q(e)$ the unique path of ${\mathcal{Q}}(U_{z-1})$ whose first edge is $e$, and for each edge $e\in \delta_G(U_z)$, we denote by $Q'(e)$ the unique path of ${\mathcal{Q}}'(U_z)$ whose first edge is $e$.
Denote ${\mathcal{Q}}^*_z={\mathcal{Q}}(U_{z-1})\cup {\mathcal{Q}}(U_z)$. For every edge $e\in E(G)$, we define a value $N_z(e)$, as follows. If $e\in E(G) \setminus E(G_z)$, then $N_z(e)=\cong_G({\mathcal{Q}}^*_z,e)$ -- the number of paths in ${\mathcal{Q}}^*_z$ that contain the edge $e$. For each edge $e\in \delta_G(U_{z-1})\cup \delta_G(U_z)\cup E(\tilde S_z)$, we set $N_z(e)=1$.
The will use the following observation.
\begin{observation}\label{obs: bound on num of copies}
Let $e$ be an edge of $ E(G) \setminus E(G_z)$. If $e\not\in E(S_{z-1})\cup E(S_{z+1})$, then $N_z(e)\leq O(\log^{34}m)$; otherwise, $\expect{N_z(e)}\leq \hat \eta$. Moreover, if $e\in E(S_{z-1})$ and $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, then $\expect{(N_z(e))^2}\leq \hat \eta$. Similarly, if $e\in E(S_{z+1})$ and $S_{z+1}\in {\mathcal{S}}^{\operatorname{light}}$, then $\expect{(N_z(e))^2}\leq \hat \eta^2$. (All expectations here are over the selections of the internal routers ${\mathcal{Q}}(S_{z-1})$ and ${\mathcal{Q}}(S_{z+1})$).
\end{observation}
\begin{proof}
Consider an edge $e\in E(G)\setminus E(G_z)$. Notice that either $e\in E(U_{z-1})$, or $e\in E(\overline{U}_z)$ must hold. We assume that it is the former; the other case is symmetric. In this case, $N_z(e)=\cong_G({\mathcal{Q}}(U_{z-1}),e)$, and, from \Cref{obs: bound congestion of routers}, $N_z(e)\leq O(\log^{34}m)$.
Assume now that $e\in E(U_{z-1})$.
From \Cref{obs: bound congestion of routers}, edge $e$ may appear on at most
$\cong_G({\mathcal{Q}}(S_{z-1}),e)\cdot O(\log^{34}m)$ paths of ${\mathcal{Q}}(U_{z-1})$, that is, $N_z(e)\leq \cong_G({\mathcal{Q}}(S_{z-1}),e)\cdot O(\log^{34}m)$.
From \Cref{obs: congestion square of internal routers}, for $1\leq i\leq r$, if $S_i\in {\mathcal{S}}^{\operatorname{light}}$, then
$\expect{\textsf{left} (\cong_{G}({\mathcal{Q}}(S_i),e)\textsf{right} )^2}\le \hat \eta$, while, if $S_i\in {\mathcal{S}}^{\operatorname{bad}}$, then
$\expect{\cong({\mathcal{Q}}(S_i),e)}\leq O(\log^{16}m)\leq \hat \eta$. The observation now follows immediately.
\end{proof}
We now construct an auxiliary graph $H_z$, and its drawing $\psi_z$. In order to do so, we start with $H_z=G$, and $\psi_z=\phi^*$. We call the edges of $E(\tilde S_z)\cup \delta_G(\tilde S_z)$ \emph{primary edges}, and the remaining edges of $G$ \emph{secondary edges}. We now process every secondary edge $e$ one by one. If edge $e$ does not participate in any path of ${\mathcal{Q}}^*_z$ (that is, $N_z(e)=0$), then we delete $e$ from $H_z$ and we delete its image from $\psi_z$. Otherwise, we replace $e$ with a set $J(e)$ of $N_z(e)$ parallel copies of $e$ in graph $H_z$, and we replace the image of $e$ in $\psi_z$ with images of these copies, that follow the original image of $e$ in parallel to it, without crossing each other. For convenience, for each edge $e\in E(\tilde S_z)\cup \delta_G(U_{z-1})\cup \delta_G(U_z)$, we define $J(e)=\set{e}$, and we think of the graph $H_z$ as having a single copy of the edge $e$ (the edge $e$ itself).
This completes the definition of the graph $H_z$ and its drawing $\psi_z$.
For every edge $e\in E(G)\setminus E(G_z)$, we can now assign, to every path of ${\mathcal{Q}}^*_z$ containing $e$, a distinct copy of this edge from $J(e)$. If edge $e\not\in \delta_G(u_{z-1})$, we assign each copy of $e$ in $J(e)$ to a distinct path of ${\mathcal{Q}}^*_z$ containing $e$ arbitrarily. If edge $e\in \delta_G(u_{z-1})$, then we perform the assignment more carefully. Intuitively, this assignment is performed in a way that is consistent with the ordering $\tilde {\mathcal{O}}_{z-1}$ of the edges of $\delta_G(U_{z-1})$ that we have defined, and the ordering of the paths of set ${\mathcal{Q}}(U_{z-1})=\set{Q(e)\mid e\in \delta_G(U_{z-1})}$ that it induces.
\paragraph{Assigning the copies of edges of $\delta_G(u_{z-1})$ to paths.}
Consider the set $\delta_G(U_{z-1})$ of edges. Recall that we have defined an ordering $\tilde {\mathcal{O}}_{z-1}$ of the edges of $\delta_G(U_{z-1})$, which is precisely the ordering ${\mathcal{O}}^{\operatorname{guided}}({\mathcal{Q}}(U_{z-1}),\Sigma)$, that is guided by the internal $U_{z-1}$-router ${\mathcal{Q}}(U_{z-1})$. Denote $\delta_G(U_{z-1})=\set{\hat a_1,\hat a_2,\ldots,\hat a_q}$, where the edges are indexed according to the ordering $\tilde {\mathcal{O}}_{z-1}$.
Recall the procedure that we used in order to define the ordering $\tilde {\mathcal{O}}_{z-1}$ of the edges of $\delta_G(U_{z-1})$ (for convenience we omit the superscript $z-1$): we have denoted $\delta_G(u_{z-1})=\set{e_1,\ldots,e_{|\delta(u_{z-1})|}}$, where the edges are indexed according to their order in the rotation ${\mathcal{O}}_{u_{z-1}}\in \Sigma$. For all $1\leq j\leq |\delta(u_{z-1})|$, we denoted by $A_j\subseteq \delta_G(U_{z-1})$ the set of all edges $e'\in \delta_G(U_{z-1})$, such that the unique path $Q(e')\in {\mathcal{Q}}(U_{z-1})$ originating at edge $e'$ terminates at edge $e_j$.
We have defined the ordering $\tilde {\mathcal{O}}_{z-1}=(\hat a_1,\hat a_2,\ldots,\hat a_q) $ of the edges of $\delta_G(U_{z-1})$ as follows: the edges that lie in sets $A_1,A_2,\ldots,A_{|\delta(u_{z-1})|}$ appear in the order of the indices of their sets, and, for each $1\leq j\leq |\delta(u_{z-1})|$, the ordering of the edges within each set $A_j$ is arbitrary;
denote this latter ordering by $\hat {\mathcal{O}}_j=\set{a^{j}_1,a^{j}_2,\ldots,a^{j}_{q_j}}$.
The current drawing $\psi_z$ of graph $H_z$ naturally defines a circular ordering of the edges of $\delta_H(u_{z-1})$, which is precisely the order in which the images these edges enter the image of $u_{z-1}$. In this circular ordering, the edges of each set $J(e_1),J(e_2),\ldots,J(e_{|\delta_G(u_{z-1})|})$ appear consecutively, in the order of the indices of their sets. For each index $1\leq j\leq |\delta_G(u_{z-1})|$, the above circular ordering defines an ordering $\hat {\mathcal{O}}'_j$ of the edges of $J(e_j)$.
Consider now some edge $e_j\in \delta_G(u_{z-1})$, and assume that $|J(e_j)|=q_j$. On the one hand, we have defined the ordering $\hat {\mathcal{O}}'_j$ of the edges of $J(e_j)$ -- the order in which the images of these edges in $\psi_z$ enter the image of $u_{z-1}$. On the other hand, we have defined an ordering $\hat {\mathcal{O}}_j=\set{a^{j}_1,a^{j}_2,\ldots,a^{j}_{q_j}}$ of the edges of $A_j$ -- that is, the edges $e'\in \delta_G(u_{z-1})$, whose corresponding path $Q(e')$ contains edge $e_j$. For all $1\leq h\leq q_j$, we then assign the $h$th edge of $J(e_j)$ in the ordering $\hat {\mathcal{O}}'_j$ to path $Q(a^{j}_{h})$. This completes the assignment of edges of $H_z$ that are incident to vertex $u_{z-1}$ to the paths of ${\mathcal{Q}}(U_{z-1})$.
For every edge $\hat a_i\in \delta_G(U_{z-1})$, we can now obtain a path $\hat Q(\hat a_i)$ in graph $H_z$, that originates at edge $\hat a_i$ and terminates at vertex $u_{z-1}$, with all inner vertices of $\hat Q(a_i)$ lying in $V(U_{z-1})$, by starting from the path $Q(\hat a_i)\in {\mathcal{Q}}(U_{z-1})$, and replacing every edge $e'\in E(G)\setminus E(G_z)$ with the copy of $e'$ that is assigned to path $Q(\hat a_i)$. Denote the resulting set of paths in graph $H_z$ by $\hat {\mathcal{Q}}_z=\set{\hat Q(\hat a_i)\mid \hat a_i\in \delta_G(U_{z-1})}$. For each edge $\hat a_i\in \delta_G(U_{z-1})$, denote by $\hat a'_i$ the last edge on path $\hat Q(\hat a_i)$. Then the paths of $\hat {\mathcal{Q}}_z$ are mutually edge-disjoint, and they route the edges of $\delta_{G}(U_{z-1})$ to vertex $u_{i-1}$ in $H_z$.
All inner vertices on the paths of $\hat {\mathcal{Q}}_z$ lie in $V(U_{z-1})$.
Moreover, the images of edges $\hat a'_1,\ldots,\hat a'_q$ enter the image of $u_{i-1}$ in the drawing $\psi_z$ of $H_z$ in the circular order of their indices (and recall that edges $\hat a_1,\ldots,\hat a_q$ are indexed in the order of their appearance in $\tilde {\mathcal{O}}_{z-1}$).
Similarly, for every edge $a\in \delta_G(U_z)$, we can now obtain a path $\hat Q'(a)$ in graph $H_z$, that originates at edge $a$ and terminates at vertex $u_{z+1}$, with all inner vertices of $\hat Q'(a)$ lying in $V(\overline{U}_{z})$, by starting from the path $Q'( a)\in {\mathcal{Q}}'(U_{z})$, and replacing every edge $e'\in E(G)\setminus E(G_z)$ with the copy of $e'$ that is assigned to path $Q(a)$. Denote the resulting set of paths in graph $H_z$ by $\hat {\mathcal{Q}}'_z=\set{\hat Q'(a)\mid a\in \delta_G(U_{z})}$.
Then the paths of $\hat {\mathcal{Q}}'_z$ are mutually edge-disjoint, and they route the edges of $\delta_{G}(U_{z})$ to vertex $u_{i+1}$. All inner vertices on paths of $\hat {\mathcal{Q}}'_z$ lie in $V(\overline{U}_z)$.
This completes the construction of graph $H_z$ and its drawing $\psi_z$.
We now analyze the number of crossings in this graph.
\paragraph{Bounding the Number of Crossings in $\psi_z$.}
Recall that $\delta_G(U_{z-1})=E_{z-1}\cup \hat E_{z-1}$, while $\delta_G(U_{z})=E_{z}\cup \hat E_{z}$.
We denote $E_z^{\operatorname{over}}=\hat E_{z-1}\cap \hat E_z$; note that every edge $e\in E_z^{\operatorname{over}}$ has one endpoint in $\bigcup_{1\leq i<z}V(\tilde S_i)$, and another endpoint in $\bigcup_{z< i\leq r}V(\tilde S_i)$ (see \Cref{fig: NN1}).
We also denote by $E_z^{\operatorname{left}}=\hat E_{z-1}\setminus E_z^{\operatorname{over}}$, and by $E_z^{\operatorname{right}}=\hat E_z\setminus E_z^{\operatorname{over}}$. Notice that every edge $e\in E_z^{\operatorname{left}}$ has one endpoint in $\bigcup_{1\leq i<z}V(\tilde S_i)$, and another endpoint in $V(\tilde S_z)$, while every edge $e\in E_z^{\operatorname{right}}$ has one endpoint in $V(\tilde S_z)$ and another endpoint in $\bigcup_{z< i\leq r}V(\tilde S_i)$ (see \Cref{fig: NN1}).
From the above definitions, $\delta_{G_z}(v^*)=\delta_G(U_{z-1})=E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{over}}$, and $\delta_{G_z}(v^{**})=\delta_G(U_{z})=E_{z}\cup E_z^{\operatorname{right}}\cup E_z^{\operatorname{over}}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN1.jpg}
\caption{Set $E^{\operatorname{over}}_z$ of edges is shown in red, set $E^{\operatorname{right}}_z$ in blue, and set $E^{\operatorname{left}}_z$ in green. The left pink dashed line shows the cut $(U_{z-1}, V\setminus U_{z-1})$, and the right pink dashed line shows the cut $(U_{z}, V\setminus U_{z})$.}\label{fig: NN1}
\end{figure}
We now reorganize the paths in $\hat {\mathcal{Q}}_z\cup \hat {\mathcal{Q}}'_z$ as follows. We let ${\mathcal{R}}_1'=\set{\hat Q(e)\mid e\in E_{z-1}\cup E_z^{\operatorname{left}}}$, ${\mathcal{R}}_1''=\set{\hat Q'(e)\mid e\in E_{z}\cup E_z^{\operatorname{right}}}$, and ${\mathcal{R}}_1^{(z)}={\mathcal{R}}_1'\cup {\mathcal{R}}_1''$. For each edge $e\in E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{right}}\cup E_z$, we denote by $R(e)\in {\mathcal{R}}^{(z)}_1$ the unique path that has edge $e$ as its first edge. For every edge $e\in E_z^{\operatorname{over}}$, we let $R(e)$ be the concatenation of the paths $\hat Q(e)\in \hat {\mathcal{Q}}_z$ and $\hat Q'(e)\in \hat {\mathcal{Q}}'_z$, so that path $R(e)$ is a simple path connecting vertices $u_{z-1}$ and $u_{z+1}$, and it contains the edge $e$. We then set ${\mathcal{R}}^{(z)}_2=\set{R(e)\mid e\in E_z^{\operatorname{over}}}$.
For every secondary edge $e'$ in graph $G$, we denote by $N'_z(e')$ the number of paths in set ${\mathcal{R}}_1^{(z)}$ that contain a copy of $e'$, and we denote by $N''_z(e')$ the number of paths in set ${\mathcal{R}}_2^{(z)}$ that contain a copy of $e'$. Note that, equivalently, $N'_z(e')$ is the total number of paths in ${\mathcal{Q}}(U_{z-1})\cup {\mathcal{Q}}'(U_z)$ that originate at edges of $E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{right}}\cup E_z$ and contain $e'$, while $N''_z(e')$ is the total number of paths in ${\mathcal{Q}}(U_{z-1})\cup {\mathcal{Q}}'(U_z)$ that originate at edges of $E_{z}^{\operatorname{over}}$ and contain $e'$. For a primary edge $e'$, we set $N'_z(e')=1$ and $N''_z(e')=0$. Clearly, $N_z(e')=N'_z(e')+N''_z(e')$ holds for every edge $e'$. Intuitively, for each edge $e'$, the value $\sum_{z=1}^rN'_z(e')$ is relatively small, while the value $\sum_{z=1}^rN''_z(e')$ may be quite large. Indeed,
recall that the paths in set ${\mathcal{Q}}(U_{z-1})\cup {\mathcal{Q}}'(U_z)$ can be thought of as constructed by composing subpaths of cycles of $\set{W(e')\mid e'\in \hat E_{z-1}\cup \hat E_z}$ with the internal routers ${\mathcal{Q}}(S_{z-1})$ and ${\mathcal{Q}}(S_{z+1})$.
Consider an edge $e\in \hat E$, and the corresponding cycle $W(e)$. Assume that $\operatorname{span}(e)=\set{i,\ldots,j-1}$. Then there is only one index $z$ for which $e\in E^{\operatorname{left}}_z$ -- index $z=j$. Similarly, there is only one index $z$ for which $e\in E^{\operatorname{right}}(z)$ -- index $z=i$. Therefore,
cycle $W(e)$ contributes its subpath to set ${\mathcal{R}}_1^{(z)}$ only for indices $z=i$ and $z=j$. On the other hand, cycle $W(e)$ may contribute a subpath to set ${\mathcal{R}}_2^{(z)}$ for every index $i<z<j$.
Because of this, we will try to bound the number of crossings in the final drawing $\phi_z$ that we construct for instance $I_z$ in terms of the values $\set{N'_z(e')}_{e'\in E(G)}$. For convenience, when the index $z$ is fixed, we omit the superscript $(z)$ in the notation ${\mathcal{R}}_1^{(z)}$ and ${\mathcal{R}}_2^{(z)}$.
Notice that, from our assumption about drawing $\phi^*$, no pair of edges in drawing $\psi_z$ of $H_z$ may cross more than once, and no edge has its image cross itself. Consider any crossing $(e_1,e_2)$ in drawing $\psi_z$. Assume that $e_1$ is a copy of edge $e_1'\in E(G)$, and that $e_2$ is a copy of edge $e_2'\in E(G)$. Then the images of edges $e_1',e_2'$ must cross in $\phi^*$, and we say that
crossing $(e'_1,e_2')$ in $\phi^*$ is \emph{responsible} for crossing $(e_1,e_2)$ in $\psi_z$.
We classify the crossings in drawing $\psi_z$ into several types, and we bound the number of crossings of each of these types separately.
Consider now a crossing $(e_1,e_2)$ in drawing $\psi_z$ of graph $H_z$. Let $e_1',e_2'$ be the edges of $G$, such that $e_1\in J(e_1')$ and $e_2\in J(e_2')$, so crossing $(e_1',e_2')$ of $\phi^*$ is responsible for crossing $(e_1,e_2)$.
\paragraph{Type-1 Crossings.}
We say that crossing $(e_1,e_2)$ in $\psi_z$ is a \emph{type-1 crossing} if at least one of the two edges $e_1',e_2'$ lies in $E(\tilde S_z)\cup \delta_G(\tilde S_z)$. We assume w.l.o.g. that $e_1'\in E(\tilde S_z)\cup \delta_G(\tilde S_z)$. Notice that crossing $(e_1',e_2')$ of $\phi^*$ may be responsible for at most $N_z(e_2')$ type-1 crossings in $\psi_z$. From \Cref{obs: bound on num of copies}, $\expect{N_z(e_2')}\leq \hat \eta$. We note that random variable $N_z(e_2')$ may only depend on the random selections of the internal routers ${\mathcal{Q}}(S_{z-1})$ and ${\mathcal{Q}}(S_{z+1})$, and in particular it is independent of the random selection of the internal router ${\mathcal{Q}}(S_{z})$.
The expected number of crossings for which crossing $(e_1',e_2')$ is responsible is then bounded by $\hat \eta$. From our definition, crossing $(e_1',e_2')$ must lie in $\chi^*_z$. Therefore, the total expected number of type-1 crossings is bounded by $|\chi^*_z|\cdot \hat\eta$.
We note that the random variable corresponding to the total number of type-1 crossings only depends on the random selections of the internal routers ${\mathcal{Q}}(S_{z-1})$ and ${\mathcal{Q}}(S_{z+1})$, and it is independent of the random selection of the internal router ${\mathcal{Q}}(S_{z})$. We will use this fact later.
\paragraph{Type-2 Crossings.}
We say that a crossing $(e_1,e_2)$ in $\psi_z$ is a \emph{type-2 crossing} if it is not a type-1 crossing, and, additionally, one of the two edges (say $e_1$) lies on a path of ${\mathcal{R}}'_1$, while the other edge (edge $e_2$) lies on a path of ${\mathcal{R}}''_1$. Notice that, in this case, $e_1\in E(U_{z-1})$ and $e_2\in E(\overline{U}_z)$ must hold. A crossing $(e_1',e_2')$ of $\psi^*$ may be responsible for at most $N_z'(e_1')\cdot N_z'(e_2')$ type-2 crossings of $\psi_z$. Moreover, the random variables $N_z'(e_1'), N_z'(e_2')$ are independent from each other. From \Cref{obs: bound on num of copies}, we can bound $\expect{N_z'(e_1')\cdot N_z'(e_2')}\leq \expect{N_z'(e_1')}\cdot \hat \eta$.
Therefore, we get that the total expected number of type-2 crossings is bounded by:
\[\sum_{(e_1',e_2')\in \chi^*} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\cdot \hat \eta.\]
\paragraph{Type-3 Crossings.}
We say that a crossing $(e_1,e_2)$ in $\psi_z$ is a \emph{type-3 crossing} if it is not a type-1 or a type-2 crossing, and, additionally, one of the two edges (say $e_1$) lies on a path of ${\mathcal{R}}'_1$, while the other edge (edge $e_2$) lies on a path of ${\mathcal{R}}'_1\cup {\mathcal{R}}_2$.
We denote by $\tilde \chi_{z-1}$ the set of all crossings $(e_1',e_2')$ of $\phi^*$, where $e_1',e_2'\in E(S_{z-1})$.
Consider any crossing $(e_1',e_2')$ of $\phi^*$ that does not lie in $\tilde \chi_{z-1}$.
This crossing may be responsible for at most $N_z'(e_1')\cdot N_z(e_2')+N_z(e_1')\cdot N'_z(e_2')$ type-3 crossings in $\psi_z$. Notice that random variables $N_z'(e_1')$, $N_z(e_2')$ are independent from each other, as are random variables $N_z(e_1')$, $N_z'(e_2')$. From \Cref{obs: bound on num of copies}, we can bound the expected number of type-3 crossings for which crossing
$(e_1',e_2')$ of $\phi^*$ is responsible by $\expect{N_z'(e_1')}\cdot \hat \eta+\expect{N_z'(e_2')}\cdot \hat \eta$.
Therefore, the total number of type-3 crossings, for which crossings of $\chi^*\setminus \tilde \chi_{z-1}$ are responsible is bounded by:
\[\sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z-1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\cdot \hat \eta.\]
A crossing $(e_1',e_2')\in \tilde \chi_{z-1}$ may be responsible for up to $N_z'(e_1')\cdot N_z(e_2')+N_z(e_1')\cdot N'_z(e_2')\leq 2N_z(e_1')\cdot N_z(e_2')$ type-3 crossings of $\psi_z$. But now the random variables
$N_z(e_1'), N_z(e_2')$ are no longer indepdendent. We can, however, bound $N_z(e_1')\cdot N_z(e_2')\leq (N_z(e_1'))^2+(N_z(e_2'))^2$.
From \Cref{obs: bound congestion of routers}, for an edge $e\in E(S_{z-1})$, $N_z(e)\leq O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$.
Therefore, the total expected number of type-3 crossings is at most:
\[
\begin{split}
&\hat \eta\cdot \sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z-1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\\
&\quad\quad\quad\quad+ O(\log^{68}m)\cdot\sum_{(e_1',e_2')\in \tilde \chi_{z-1}}\textsf{left}( \expect{ (\cong_G({\mathcal{Q}}(S_{z-1}),e_1')^2}+\expect{(\cong_G({\mathcal{Q}}(S_{z-1}),e_2')^2}\textsf{right} ).
\end{split}\]
\iffalse
From \Cref{obs: bound on num of copies}, if $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, $\expect{(N_z(e_1'))^2}, \expect{(N_z(e_2'))^2}\leq
2^{O(\sqrt{\log m}\cdot \log\log m)}$. Therefore, if $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, then
the total number of type-3 crossings, for which crossings of $\tilde \chi_{z-1}$ are responsible is bounded by:
\[
\sum_{(e_1',e_2')\in \chi_{z-1}} \textsf{left} (\expect{(N_z(e_1'))^2}+\expect{(N_z(e_2'))^2}\leq
|\chi_{z-1}|\textsf{right} )\cdot2^{O(\sqrt{\log m}\cdot \log\log m)}\leq |\chi^*_{z-1}| \cdot2^{O(\sqrt{\log m}\cdot \log\log m)}.\]
\fi
\paragraph{Type-4 Crossings.}
We say that a crossing $(e_1,e_2)$ in $\psi_z$ is a \emph{type-4 crossing} if it is not a crossing of one of the first three types, and, additionally, one of the two edges (say, edge $e_1$) lies on a path of ${\mathcal{R}}''_1$, while the other edge (edge $e_2$) lies on a path of ${\mathcal{R}}''_1\cup {\mathcal{R}}_2$.
We denote by $\tilde \chi_{z+1}$ the set of all crossings $(e_1',e_2')$ of $\phi^*$, where $e_1',e_2'\in E(S_{z+1})$.
Consider any crossing $(e_1',e_2')$ of $\phi^*$ that does not lie in $\tilde \chi_{z+1}$.
As before, this crossing may be responsible for at most $N_z'(e_1')\cdot N_z(e_2')+N_z(e_1')\cdot N'_z(e_2')$ type-4 crossings in $\psi_z$.
Using the same analysis as for type-3 crossings, we can bound the expected number of type-4 crossings for which crossing
$(e_1',e_2')$ of $\phi^*\setminus \tilde\chi_{z+1}$ is responsible by $\expect{N_z'(e_1')}\cdot \hat \eta+\expect{N_z'(e_2')}\cdot \hat \eta$.
The total number of type-4 crossings, for which crossings of $\chi^*\setminus \tilde \chi_{z+1}$ are responsible is bounded by:
\[\sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z+1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\cdot \hat \eta.\]
As before, a crossing $(e_1',e_2')\in \tilde \chi_{z+1}$ may be responsible for up to $N_z'(e_1')\cdot N_z(e_2')+N_z(e_1')\cdot N'_z(e_2')\leq 2N_z(e_1')\cdot N_z(e_2')\leq 2(N_z(e_1'))^2+2( N_z(e_2'))^2$ type-4 crossings of $\psi_z$. Using the same reasoning as in type-3 crossings, the total expected number of type-4 crossings is bounded by:
\[
\begin{split}
&\hat \eta\cdot
\sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z+1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\\
&\quad\quad+ O(\log^{68}m)\cdot\sum_{(e_1',e_2')\in \tilde \chi_{z+1}}\textsf{left}( \expect{ (\cong_G({\mathcal{Q}}(S_{z-1}),e_1')^2}+\expect{(\cong_G({\mathcal{Q}}(S_{z+1}),e_2')^2}\textsf{right} ).
\end{split}\]
\paragraph{Type-5 Crossings}
All remaining crossings of $\psi_z$ are type-5 crossings. For each such crossing $(e_1,e_2)$, it must be the case that each of the edges $e_1,e_2$ belongs to a path of ${\mathcal{R}}_2$. We do not bound the number of type-5 crossings, as we will eventually eliminate all such crossings.
\iffalse
For convenience, we denote $\eta''=2^{O(\sqrt{\log m}\cdot \log\log m)}$. We also define $$\Upsilon_{z-1}=
\sum_{(e_1',e_2')\in \tilde \chi_{z-1}} \textsf{left} (\expect{(N_z(e_1'))^2}+\expect{(N_z(e_2'))^2}\textsf{right} ),$$
and we define $\Upsilon_{z+1}$ similarly for the set $\tilde \chi_{z+1}$ of crossings. As noted already, if $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, then $\Upsilon_{z-1}\leq
|\chi_{z-1}| \cdot2^{O(\sqrt{\log m}\cdot \log\log m)}$, and similarly, if $S_{z+1}\in {\mathcal{S}}^{\operatorname{light}}$, then $\Upsilon_{z+1}\leq
|\tilde \chi_{z+1}|\cdot2^{O(\sqrt{\log m}\cdot \log\log m)}$.
We can now bound the expected number of crossings of types $1$--$4$ in drawing $\psi_z$ by the following expression:
\begin{equation}\label{eq: bound primary crossings}
|\psi^*_z|\cdot \eta'' +\sum_{(e_1',e_2')\in \chi^*} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\cdot \eta''+\Upsilon_{z-1}+\Upsilon_{z+1},
\end{equation}
\fi
\iffalse
Clearly, a crossing $(e_1',e_2')$ of $\psi^*$ may be responsible for at most $N_z'(e_1')\cdot N_z(e_2')$ type-2 crossings of $\psi_z$, and in all such crossing the primary edge is a copy of edge $e_1'$. Moroever, the random variables $N_z'(e_1'), N_z'(e_2')$ are independent from each other. From \Cref{obs: bound on num of copies}, we can bound $\expect{N_z'(e_1')\cdot N_z'(e_2')}\leq \expect{N_z'(e_1')}\cdot 2^{O(\sqrt{\log m}\cdot \log\log m)}$.
Therefore, we get that the total expected number of type-2 crossings is bounded by:
\[\sum_{(e_1',e_2')\in \chi^*} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\cdot 2^{O(\sqrt{\log m}\cdot \log\log m)}.\]
\fi
\iffalse
We call the crossings in the drawing $\psi_z$ \emph{type-1 crossings}, to distinguish them between crossings that will be introduced in subsequent steps.
Consider any such crossing $(e_1,e_2)$. Let $e_1'$ be an edge of $G$, such that $e_1$ is a copy of $e_1'$, and define $e_2'$ similarly for $e_2$. Then edges $e_1',e_2'$ must cross in the optimal solution $\phi^*$ to intstance $G$. We say that crossing $(e_1',e_2')$ of $\phi^*$ is \emph{responsible} for the crossing $(e_1,e_2)$.
We now partition all type-1 crossings into three subsets.
The first subset, that we call \emph{type 1a crossings} contains all crossings $(e_1,e_2)$ in $\psi_z$, where at least one of the edges $e_1,e_2$ lies in $E(\tilde S_i)$. In this case, the crossing $(e_1',e_2')$ that is responsible for crossing $(e_1,e_2)$ lies in $\chi^*_z$.
Consider now any crossing $(e_1',e_2')\in \chi^*_z$.
If both $e_1'$ and $e_2'$ lie in $E(\tilde S_i)$, then only one copy of each of the two edges $e_1',e_2'$ belong to graph $H_z$, and so crossing $(e_1',e_2')$ may only be responsible for at most one type-1a crossing in $\psi_z$. Assume now that exactly one of the two edges (say $e_1'$) lies in $E(\tilde S_i)$. Then crossing $(e_1',e_2')$ is responsible for $N_z(e_2')+N'_z(e_2')$ crossings in $\psi_z$.
The second set of crossings of $\psi_z$, that we call \emph{type 1b crossings} contains all crossings between pairs $e_1,e_2$ of edges, such that $(e_1,e_2)$ is not a crossing of type $1a$, and at least one of the two edges lies on a path of $\set{\hat Q(e)\mid e\in E_{z-1}\cup E_z^{\operatorname{left}}}\cup \set{\hat Q'(e)\mid e\in E_{z}\cup E_z^{\operatorname{right}}}$. As before,
\fi
\subsubsection{Step 2: Initial Drawing of $G_z$}
\label{subsubsec: step 2}
For every edge $e\in E_{z-1}\cup E_z^{\operatorname{left}}$, we denote by $\Gamma(e)$ the curve corresponding to the image of path $R(e)\in {\mathcal{R}}'_1$ in the drawing $\psi_z$ of $H_z$. Note that, if $v$ is an endpoint of the path $R(e)$ that lies in $V(\tilde S_z)$, then curve $\Gamma(e)$ connects the image of $v$ to the image of vertex $u_{z-1}$ in $\psi_z$. For every edge $e\in E_{z}\cup E_z^{\operatorname{right}}$, we denote by $\Gamma(e)$ the curve corresponding to the image of the path $R(e)\in {\mathcal{R}}''_1$ in the drawing $\psi_z$ of $H_z$. Note that, if $v$ is an endpoint of $e$ that lies in $V(\tilde S_z)$, then curve $\Gamma(e)$ connects the image of $v$ to the image of vertex $u_{z+1}$ in $\psi_z$.
Lastly, for every edge $e\in E_z^{\operatorname{over}}$, we let $\Gamma(e)$ be the image of the path $R(e)\in {\mathcal{R}}_2$ in $\psi_z$. Notice that curve $\Gamma(e)$ connects the image of $u_{z-1}$ to the image of $u_{z+1}$ in $\psi_z$. From the constructions of the paths in ${\mathcal{R}}'_1\cup {\mathcal{R}}_2$, if we denote $\delta_G(U_{z-1})=\set{\hat a_1,\ldots,\hat a_q}$, where the edges are indexed in the order of their apperance in the ordering $\tilde {\mathcal{O}}_{z-1}$, the curves $\Gamma(\hat a_1),\ldots,\Gamma(\hat a_q)$ enter the image of vertex $u_{z-1}$ in this circular order.
In order to obtain the initial drawing $\phi'_z$ of $G_z$, we start with the drawing $\psi_z$ of graph $H_z$, and we delete from it the images of all vertices except those lying in $V(\tilde S_z)\cup \set{u_{z-1},u_{z+1}}$, and the images of all edges except those lying in $E(\tilde S_z)$; we view the image of vertex $u_{z-1}$ as the image of the special vertex $v^*$, and the image of vertex $u_{z+1}$ as the image of the secial vertex $v^{**}$. We then add to this drawing the curves in $\set{\Gamma(e)\mid e\in \delta_G(U_{z-1})\cup \delta_G(U_z)}$. Each such curve $\Gamma(e)$ becomes an image of the corresponding edge $e$. Notice that the edges of $\delta_G(U_{z-1})$ become incident to $v^*$ in graph $G_z$; the edges of $\delta_G(U_z)$ become incident to $v^{**}$, and the edges of $E_z^{\operatorname{over}}$ connect $v^*$ to $v^{**}$. From the above discussion, the circular order in which the images of the edges in $\delta_{G_z}(v^*)$ enter the image of $v^*$ in the current drawing is exactly the ordering $\tilde {\mathcal{O}}_{z-1}$, which is precisely the ordering ${\mathcal{O}}_{v^*}\in \Sigma_z$. However, the images of the edges of $\delta_{G_{z}}(v^{**})$ may not enter the image of vertex $v^{**}$ in the correct order. We will fix this in subsequent steps. There is one major problem with the current drawing of the graph $G_z$: it is possible that some point $p$ lies on a large number of curves in set $\set{\Gamma(e)\mid e\in \delta_G(U_{z-1})\cup \delta_G(U_z)}$, and it is an inner point on each such curve. This may only happen if $p$ corresponds to an image of some vertex $v$, where $v\in V(U_{z-1})\setminus\set{u_{z-1}}$, or $v\in V(\overline U_{z})\setminus\set{u_{z+1}}$. We will now ``fix'' the images of the edges of $\delta_{G_z}(v^*)\cup \delta_{G_z}(v^{**})$ by slightly ``nudging'' them in the vicinity of each such vertex, to ensure that all resulting curves are in general position. We do so by performing a nudging operation (see \Cref{sec: curves in a disc}).
We process every vertex $v\in \textsf{left} (V(U_{z-1})\cup V(\overline{U}_z)\textsf{right})\setminus\set{u_{z-1},u_{z+1}}$ one by one. Consider an iteration when any such vertex $v$ is processed. Let $A(v)$ be the set of all edges $e\in \delta_G(U_{z-1})\cup \delta_G(U_z)$, such that curve $\Gamma(e)$ contains the image of vertex $v$ (in $\psi_z$). We denote $A(v)=\set{a_1,\ldots,a_k}$. Consider the tiny $v$-disc $D(v)=D_{\psi_z}(v)$ in the drawing $\psi_z$ of graph $H_z$. For all $1\leq i\le k$, we let $s_i,t_i$ be the two points at which curve $\Gamma(a_i)$ intersects the boundary of the disc $D(v)$. Note that all points $s_1,t_1,\ldots,s_k,t_k$ must be distinct. We use the algorithm from \Cref{claim: curves in a disc} in order to construct a collection $\set{\gamma_1,\ldots,\gamma_k}$ of curves, such that, for all $1\leq i\leq k$, curve $\gamma_i$ has $s_i$ and $t_i$ as its endpoints, and is completely contained in $D(v)$. Recall that the claim ensures that, for every pair $1\leq i<j\leq k$ of indices, if the two pairs $(s_i,t_i),(s_j,t_j)$ of points cross, then curves $\gamma_i,\gamma_j$ intersect at exactly one point; otherwise, curves $\gamma_i,\gamma_j$ do not intersect.
For all $1\leq i\leq k$, we modify the curve $\Gamma(a_i)$ as follows: we replace the segment of the curve between points $s_i,t_i$ with the curve $\gamma_i$.
Once every vertex $v\in \textsf{left} (V(U_{z-1})\cup V(\overline{U}_z)\textsf{right})\setminus\set{u_{z-1},u_{z+1}}$ is processed in this way, the curves in set $\set{\Gamma(e)\mid e\in \delta_G(U_{z-1})\cup \delta_G(U_z)}$ are in general position, and we obtain a valid drawing of the graph $G_z$, that we denote by $\phi'_z$. The modification of the curves in $\set{\Gamma(e)\mid e\in \delta_G(U_{z-1})\cup \delta_G(U_z)}$ do not affect the endpoints of the curves, and so, for every vertex $x\in V(G_z)\setminus \set{v^{**}}$, the images of the edges of $\delta_{G_z}(x)$ enter the image of $x$ in the order consistent with the rotation ${\mathcal{O}}_x\in \Sigma_z$. We now bound the number of crossings in drawing $\phi'_z$.
Consider some pair of edges $e,e'\in E(G_z)$ that cross at some point $p$ in the drawing $\phi'_z$. We say that this crossing is \emph{primary} iff point $p$ does not belong to any of the discs in the set $$\set{D(v)\mid v\in \textsf{left} (V(U_{z-1})\cup V(\overline{U}_z)\textsf{right})\setminus\set{u_{z-1},u_{z+1}}};$$ otherwise we say that the crossing is \emph{secondary}.
Notice that every primary crossing in $\phi'_z$ corresponds to a unique crossing in the drawing $\psi_z$ of the graph $H_z$. Recall that we have partitioned all such crossings into five types, and we have bounded the number of crossings of each of the first four types. This partition naturally defines a partition of all primary crossings in $\phi'_z$ into five types. Specifically, primary crossings of type 1 are all crossings $(e,e')$ where at least one of the edges $e,e'$ lies in $E(\tilde S_z)\cup \delta_G(\tilde S_z)$. Primary crossings of type 2 are primary crossings between curves $\Gamma(e),\Gamma(e')$, where $e\in E_{z-1}\cup E_z^{\operatorname{left}}$, while $e'\in E_{z}\cup E_z^{\operatorname{right}}$. Primary crossings of type $3$ are primary crossings between curves
$\Gamma(e),\Gamma(e')$, where $e\in E_{z-1}\cup E_z^{\operatorname{left}}$ and $e'\in E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{over}}$, while primary crossings of type 4 are primary crossings between curves $\Gamma(e),\Gamma(e')$, where $e\in E_{z}\cup E_z^{\operatorname{right}}$, while $e'\in E_{z}\cup E_z^{\operatorname{right}}\cup E_z^{\operatorname{over}}$. Lastly, primary crossings of type 5 are primary crossings between curves $\Gamma(e),\Gamma(e')$, where $e,e'\in E_z^{\operatorname{over}}$.
The number of primary crossings of the first four types is bounded as before.
We now consider secondary crossings of $\phi_z$. Notice that each such crossing must be between a pair of curves $\Gamma(e),\Gamma(e')$, where $e,e'\in \delta_G(U_{z-1})\cup \delta_G(U_z)$.
Consider a pair of curves $\Gamma(e),\Gamma(e')$, where $e,e'\in \delta_G(U_{z-1})\cup \delta_G(U_z)$, and some point $p$ at which the curves cross, such that the crossing is secondary. Let $v\in \textsf{left} (V(U_{z-1})\cup V(\overline{U}_z)\textsf{right})\setminus\set{u_{z-1},u_{z+1}}$ be the vertex such that $p$ lies in the interior of disc $D(v)$. Denote by $s,t$ the points on the boundary of $D$ that lie on $\Gamma(e)$, and define $s',t'$ similary for $\Gamma(e')$. From \Cref{claim: curves in a disc}, curves $\Gamma(e),\Gamma(e')$ may only cross inside the disc $D(v)$ if the pairs $(s,t),(s',t')$ of points on the boundary of $D$ cross. Consider now the paths $R(e)\in {\mathcal{R}}_1\cup {\mathcal{R}}_2$ and $R(e')\in {\mathcal{R}}_1\cup {\mathcal{R}}_2$. Denote by $e_1,e_2$ the two edges on path $R(e)$ that immediately precede and immediately follow vertex $v$, and define edges $e_1',e_2'$ similarly for path $R(e')$. We now consider three cases.
First, if $v\in V(S_{z-1})$, then there must be two paths $Q,Q'\in {\mathcal{Q}}(S_{z-1})$, such that $Q\subseteq R(e)$ and $Q'\subseteq R(e')$, where $Q,Q'$ both contain the vertex $v$. Since the paths of ${\mathcal{Q}}(S_{z-1})$ are non-transversal with respect to $\Sigma$, the only way for the two pairs $(s,t),(s',t')$ of points to cross is if the set $\set{e_1,e_2,e_1',e_2'}$ contains copies of fewer than four distinct edges of $\delta_G(v)$. In other words, for some edge $e^*\in \delta_G(v)$, both $R(e)$ and $R(e')$ contain a copy of $e^*$. In this case, we say that edge $e^*$ is \emph{responsible} for this secondary crossing between $\Gamma(e)$ and $\Gamma(e')$.
The second case is when $v\in V(S_{z+1})$. The analysis of this case is similar to the previous case: there must be an edge $e^*\in \delta_G(v)$ such that both $R(e)$ and $R(e')$ contain a copy of $e^*$. We say that $e^*$ is responsible for this crosisng.
We now consider the third case, when $v\not\in V(S_{z-1})\cup V(S_{z+1})$. We consider the paths $R(e),R(e')\in {\mathcal{R}}_1\cup {\mathcal{R}}_2$, and we define the edges $e_1,e_2,e_1',e_2'$ as before. Since $v\not\in V(S_{z-1})\cup V(S_{z+1})$,
it must be the case that $v\in W(e)\cap W(e')$. If there is an edge $e^*\in \delta_G(v)$, such that both $R(e)$ and $R(e')$ contain a copy of $e^*$, then we designate $e^*$ to be responsible for this crossing as before. Otherwise, the edges in set $\set{e_1,e_2,e_1',e_2'}$ are copies of four distinct edges of $\delta_G(V)$. In this case, the cycles $W(e)$ and $W(e')$ must have a transversal intersection at vertex $v$, from \Cref{obs: inner non-transversal}, and $e,e'\in \hat E_z$ must hold.
In this case, we say that the transversal intersection of $W(e)$ and $W(e')$ at $v$ is responsible for the crossing.
We now classify the secondary crossings into three types and bound the number of crossings of the first two types. We will eventually eliminate all crossings of the third type.
\paragraph{Type-1 secondary crossing.}
Consider a secondary crossing between a pair $\Gamma(e),\Gamma(e')$ of curves, for $e,e'\in \delta_G(U_{z-1})\cup \delta_G(U_z)$, and a secondary crossing of the two curves at some point $p$. We say that the crossing is of type 1 if
$e\in E_{z-1}\cup E_z^{\operatorname{left}}$ and $e'\in E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{over}}$.
\paragraph{Type-2 secondary crossing.}
We say that a secondary crossing between a pair $\Gamma(e),\Gamma(e')$ of curves, for $e,e'\in \delta_G(U_{z-1})\cup \delta_G(U_z)$, is of type 2 if
$e\in E_{z}\cup E_z^{\operatorname{right}}$ and $e'\in E_z\cup E_z^{\operatorname{right}}\cup E_z^{\operatorname{over}}$.
\paragraph{Type-3 secondary crossing.} All remaining secondary crossings are of tye 3. Consider any such crossing between a pair $\Gamma(e),\Gamma(e')$ of curves, for $e,e'\in \delta_G(U_{z-1})\cup \delta_G(U_z)$. Notice that it is impossible that one of the two edges $e,e'$ lies in $E_{z-1}\cup E_z^{\operatorname{left}}$, while the other lies in $E_{z}\cup E_z^{\operatorname{right}}$, since, in such a case, paths $R(e),R(e')$ cannot share any edges. Therefore, $e,e'\in E_z^{\operatorname{over}}$ must hold.
We now bound the expected number of type-1 secondary crossing. Consider any such crossing between a pair $\Gamma(e),\Gamma(e')$ of curves, and assume that the crossing point $p$ lies in disc $D(v)$, for some vertex $v$. From the definition of a type-1 crossing, $v\in V(U_{z-1})\setminus \set{u_{z-1}}$ must hold.
In this case, it is impossible that a pair of auxiliary cycles $W(e),W(e')$ with $e,e'\in \hat E_z$ have a transversal intersection at vertex $v$, from \Cref{obs: auxiliary cycles non-transversal at at most one}. Therefore, some edge of $E_G(U_{z-1})$ must be responsible for this crossing. It is immediate to verify that every edge $e\in E_G(U_{z-1})$ may be responsible for at most $N'_z(e)\cdot N_z(e)$ type-1 secondary crossings. If $e\not\in E(S_{z-1})$, then, from \Cref{obs: bound on num of copies}, $N_z(e)\leq O(\log^{34}m)$. If $e\in E(S_{z-1})$, then, from \Cref{obs: bound congestion of routers}, $N_z(e)\leq O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_{z-1}),e)$. Therefore, the total expected number of type-1 secondary crossings is bounded by:
\[O(\log^{34}m)\cdot \sum_{e\in E(U_{z-1})\setminus E(S_{z-1})}N'_z(e) +O(\log^{68}m)\cdot \sum_{e\in E(S_{z-1})}\expect{\textsf{left} (\cong _G({\mathcal{Q}}(S_{z-1}),e)\textsf{right} )^2}.\]
Next, we bound the expected number of type-2 secondary crossings. This time some of the crossigns are charged to individual edges (that is, some edge of $E(\overline{U}_z)$ is responsible for the crossing), and some crossings are charged to transversal intersections of pairs of cycles $W(e'),W(e'')$, where $e',e''\in \hat E_z$. The expected number of edges of the former type is bounded using the same reasoning as for type-1 crossings, and their expected number is at most:
\[O(\log^{34}m)\cdot \sum_{e\in E(\overline{U}_{z})\setminus E(S_{z+1})}N'_z(e) +O(\log^{68}m)\cdot \sum_{e\in E(S_{z+1})}\expect{\textsf{left} (\cong _G({\mathcal{Q}}(S_z),e)\textsf{right} )^2}.\]
Let $\Pi^T_z$ denote the set of triples $(e,e',v)$, where $e\in E_z^{\operatorname{right}}$, $e'\in \hat E_z$, and $v$ is a vertex that lies on both $W(e)$ and $W(e')$, such that cycles $W(e)$ and $W(e')$ have a transversal intersection at $v$. Clearly, the number of type-2 secondary crossings that are charged to transversal intersections of pairs of cycles is bounded by $|\Pi^T_z|$. Overall, we get that the total expected number of type-2 secondary crossings is bounded by:
\[O(\log^{34}m)\cdot \sum_{e\in E(\overline{U}_{z})\setminus E(S_{z+1})}N'_z(e) +O(\log^{68}m)\cdot \sum_{e\in E(S_{z+1})}\expect{\textsf{left} (\cong _G({\mathcal{Q}}(S_z),e)\textsf{right} )^2}+|\Pi^T_z|.\]
This completes the analysis of the initial drawing $\phi'_z$ of graph $G_z$. Notice that we did not analyze the number of type-5 primary crossings and the number of type-3 secondary crossings. These are all crossings between the images of the edges of $E_z^{\operatorname{over}}$. Unfortunately, a crossing of the original drawing $\phi^*$ of $G$ may give rise to many crossings between edges of $E_z^{\operatorname{over}}$ in drawings $\phi'_z$ of graphs $G_z$, for $1\leq z\leq r$. In the next step, we will slightly modify the drawing $\phi'_z$ in order to eliminate all such crossings. Notice that our current bounds on the expected number of crossings in $\phi'_z$ contain terms like $\sum_{e\in E(S_{z-1})}\expect{\textsf{left} (\cong_G({\mathcal{Q}}(S_{z-1}),e)\textsf{right} )^2}$. If cluster $S_{z-1}$ lies in set ${\mathcal{S}}^{\operatorname{light}}$, then this expression can be bounded by $|E(S_{z-1})|\cdot \hat \eta$. However, if $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$ then this bound may no longer be valid. In such a case we will perform an additional uncrossing operation of the images of edges of $\delta_G(U_{z-1})$ in order to decrease this number of crossings. We also perform such an operation on the images of the edges of $\delta_G(U_z)$ if $S_{z+1}\in {\mathcal{S}}^{\operatorname{bad}}$.
\subsubsection{Step 3: Modified Drawing of $G_z$}
In this step we modify the drawing $\phi'_z$ of $G_z$ to obtain a new modified drawing $\phi''_z$, by performing one or more uncrossing operations.
We first consider the cases where $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$ or $S_{z+1}\in {\mathcal{S}}^{\operatorname{bad}}$ hold, and perform some initial uncrossings to decrease the number of type-3 primary and type-1 secondary crossings (in case where $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$), and the number type-4 primary and type-2 secondary crossings (in case where $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$). After that we perform one more uncrossing operation that will eliminate all type-5 primary and type-3 secondary crossings.
Recall that the expected number of type-3 primary crossings and type-1 secondary crossings (that is, all crossings between images of edge pairs $e,e'$ where $e\in E_{z-1}\cup E_z^{\operatorname{left}}$ and $e'\in E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{over}}$) is at most:
\[
\begin{split}&\hat \eta \cdot \sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z-1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\\
&\quad +O(\log^{34}m)\cdot \sum_{e\in E(U_{z-1})\setminus E(S_{z-1})}N'_z(e) \\
&\quad\quad+ O(\log^{68}m)\cdot\sum_{(e_1',e_2')\in \tilde \chi_{z-1}}\textsf{left}( \expect{ (\cong_G({\mathcal{Q}}(S_{z-1}),e_1')^2}+\expect{(\cong_G({\mathcal{Q}}(S_{z-1}),e_2')^2}\textsf{right} )\\
&\quad +O(\log^{68}m)\cdot \sum_{e\in E(S_{z-1})}\expect{\textsf{left} (\cong _G({\mathcal{Q}}(S_{z-1}),e)\textsf{right} )^2}.
\end{split}
\]
From \Cref{obs: congestion square of internal routers}, if $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, then for every edge $e\in E(S_{z-1})$, $\expect{(\cong_G({\mathcal{Q}}(S_{z-1}),e)^2}\leq
\hat \eta$. Recalling that $\tilde \chi_{z-1}\subseteq \chi^*_{z-1}$, we get that, if $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$, then the expected number of type-3 primary and type-1 secondary crossings is bounded by:
\begin{equation}\label{eq: type 3 bound for light}
\begin{split}&\hat \eta\cdot \sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z-1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\\
&\quad\quad\quad\quad +O(\log^{34}m)\cdot \sum_{e\in E(U_{z-1})\setminus E(S_{z-1})}N'_z(e) \\
&\quad\quad \quad\quad+O(\hat \eta^2)\cdot (|\chi^*_{z-1}|+|E(S_{z-1}|).
\end{split}
\end{equation}
\iffalse
\begin{equation}\label{eq: type 3 bound for light}
\hat \eta^2\cdot \sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z-1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} ) +O(\log^{34}m)\cdot \sum_{e\in E(U_{z-1})\setminus E(S_{z-1})}N'_z(e)+ \hat \eta\cdot (|\chi^*_{z-1}|+|E(S_{z-1}|).
\end{equation}
\fi
Recall that the expected number of type-4 primary crossings and type-2 secondary crossings (that is, all crossings between images of edge pairs $e,e'$ where $e\in E_{z}\cup E_z^{\operatorname{right}}$ and $e'\in E_z\cup E_z^{\operatorname{right}}\cup E_z^{\operatorname{over}}$) is at most:
\[
\begin{split}
&\hat\eta \cdot
\sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z+1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\\
&\quad\quad\quad\quad
+O(\log^{34}m)\cdot \sum_{e\in E(\overline{U}_{z})\setminus E(S_{z+1})}N'_z(e) \\
&\quad\quad\quad\quad+ O(\log^{68}m)\cdot\sum_{(e_1',e_2')\in \tilde \chi_{z+1}}\textsf{left}( \expect{ (\cong_G({\mathcal{Q}}(S_{z+1}),e_1')^2}+\expect{(\cong_G({\mathcal{Q}}(S_{z+1}),e_2')^2}\textsf{right} )\\
&\quad\quad\quad\quad +O(\log^{68}m)\cdot \sum_{e\in E(S_{z+1})}\expect{\textsf{left} (\cong _G({\mathcal{Q}}(S_z),e)\textsf{right} )^2}+|\Pi^T_z|
\end{split}\]
Here, $\Pi^T_z$ is the set of triples $(e,e',v)$, where $e\in E_z^{\operatorname{right}}$, $e'\in \hat E_z$, and cycles $W(e)$ and $W(e')$ have a transversal intersection at vertex $v$.
From \Cref{obs: congestion square of internal routers}, if $S_{z+1}\in {\mathcal{S}}^{\operatorname{light}}$, then for every edge $e\in E(S_{z+1})$, $\expect{(\cong_G({\mathcal{Q}}(S_{z+1}),e)^2}\leq
\hat \eta$. Recalling that $\tilde \chi_{z+1}\subseteq \chi^*_{z+1}$, we get that, if $S_{z+1}\in {\mathcal{S}}^{\operatorname{light}}$, then the expected number of type-4 primary and type-2 secondary crossings is bounded by:
\begin{equation}\label{eq: type 4 bound for light}
\begin{split}&\hat \eta\cdot \sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z+1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\\
&\quad\quad\quad\quad +O(\log^{34}m)\cdot \sum_{e\in E(\overline{U}_z)\setminus E(S_{z+1})}N'_z(e) \\
&\quad\quad\quad\quad +O(\hat \eta^2)\cdot (|\chi^*_{z+1}|+|E(S_{z+1})|)+|\Pi^T_z|.
\end{split}
\end{equation}
We now consider four cases, depending on whether the clusters $S_{z-1},S_{z+1}$ lie in ${\mathcal{S}}^{\operatorname{light}}$ or ${\mathcal{S}}^{\operatorname{bad}}$.
\paragraph{Case 1: $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$ and $S_{z+1}\in {\mathcal{S}}^{\operatorname{light}}$.}
When $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$, we no longer have a bound on $\expect{(\cong_G({\mathcal{Q}}(S_{z-1}),e)^2}$ for edges $e\in E(S_{z-1})$, and so the bound from \Cref{eq: type 3 bound for light} on the number of type-3 primary and type-1 secondary crossings is no longer valid. Instead, we will perform a type-1 uncrossing of the images of the edges of $E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{over}}$. Let $\Gamma_1$ denote the set of curves reprsenting the images of these edges in $\phi_z'$. Let $\Gamma_2$ denote the set of curves reprsenting the images of all remaining edges of $G_z$ in $\phi_z'$. We apply the algorithm from \Cref{thm: type-1 uncrossing} to compute a new collection $\Gamma_1'$ of curves, where, for each edge $e\in E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{over}}$, there is a curve $\gamma(e)\in \Gamma_1'$ connecting the images of the endpoints of $e$. Intuitively, the algortihm for type-1 uncrossing proceeds in iterations, as long as there is a pair $\gamma_1,\gamma_2\in \Gamma_1$ of curves that cross at least twice. Assume that $p$ and $q$ are two points lying on both $\gamma_1$ and $\gamma_2$. The algorithm then uncrosses the two curves, by ``swapping'' the segments of these curves that connect $p$ and $q$ (see Figure~\ref{fig: type_1_uncross_proof} in \Cref{apd: type-1 uncrossing} for an illustration.)
At the end of this procedure, every pair of curves in $\Gamma_1'$ may cross each other at most once. For each curve $\gamma\in \Gamma_2$, the number of crossings between $\gamma$ and the curves in $\Gamma_1'$ is no higher than the number of crossings between $\gamma$ and the curves in $\Gamma_1$. We modify the images of the edges in $E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{over}}$, so that for each such edge $e$, its new image is the curve $\gamma(e)\in \Gamma_1'$. Observe that the total number of primary crossings of types 1,2, and 4 does not increase, and neither does the number of secondary crossings of type 2. We note however that a primary crossing of type 2 (a crossing between images of edges $e,e'$ where $e\in E_{z-1}\cup E_z^{\operatorname{left}}$ and $e'\in E_z\cup E_z^{\operatorname{right}}$) may become a primary crossing of type 4 (a crossing between images of edges $e,e'$ where $e'\in E_z\cup E_z^{\operatorname{right}}$ and $e\in E_z^{\operatorname{over}}$), and vice versa.
The total number of type-3 primary crossings and of type-1 secondary crossings is now bounded by:
$\textsf{left} (|E_{z-1}|+ |E_z^{\operatorname{left}}|+|E_z^{\operatorname{over}}|\textsf{right} )^2\leq |\delta_G(U_{z-1})|^2$.
From \Cref{cor: few edges crossing cuts},
$|\delta_G(U_{z-1})|\leq |\delta_G(S_{z-1})|\cdot O(\log^{34}m)$.
Recall that, if $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$, and the bad event ${\cal{E}}$ does not happen, then, from \Cref{obs: congestion square of internal routers},
$\mathsf{OPT}_{\mathsf{cnwrs}}(S_{z-1},\Sigma(S_{z-1}))+|E(S_{z-1})|\geq \frac{|\delta_G(S_{z-1})|^2}{\hat \eta}$,
where $\Sigma(S_{z-1})$ is the rotation system for graph $S_{z-1}$ induced by $\Sigma$. Therefore,
$|\delta_G(S_{z-1})|^2\leq\hat \eta \cdot \textsf{left} (|\chi^*_{z-1}|+|E(S_{z-1})|\textsf{right} )$ must hold.
Overall, if $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$, and event ${\cal{E}}$ did not happen, then the total number of type-3 primary crossings and of type-1 secondary crossings is now bounded by:
\[|\delta_G(U_{z-1})|^2\leq \hat \eta^2\cdot |\delta_G(S_{z-1})|^2\leq \hat \eta^2\cdot \textsf{left} (|\chi^*_{z-1}|+|E(S_{z-1})|\textsf{right} ).\]
Combining this with the bound from Equation \ref{eq: type 3 bound for light}, we get that, regardless of whether Case 1 happened or not, if event ${\cal{E}}$ did not happen, then, after the current modification, the total expected number of type-3 primary crossings and of type-1 secondary crossings is bounded by:
\begin{equation}\label{eq: type 3 final}
\begin{split}&\hat \eta\cdot \sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z-1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\\
&\quad\quad \quad\quad+O(\log^{34}m)\cdot \sum_{e\in E(U_{z-1})\setminus E(S_{z-1})}N'_z(e) \\
&\quad\quad\quad\quad+ \hat \eta^2 \cdot (|\chi^*_{z-1}|+|E(S_{z-1}|).
\end{split}
\end{equation}
\paragraph{Case 2: $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$ and $S_{z+1}\in {\mathcal{S}}^{\operatorname{bad}}$.}
We now consider the case where $S_{z-1}\in {\mathcal{S}}^{\operatorname{light}}$ and $S_{z+1}\in {\mathcal{S}}^{\operatorname{bad}}$. The modification that we perform is almost identical to that performed in the case where $S_{z-1}\in {\mathcal{S}}^{\operatorname{bad}}$, except that now we uncross the images of the edges in $\delta_G(U_z)$.
As before, when $S_{z+1}\in {\mathcal{S}}^{\operatorname{bad}}$, we no longer have a bound on $\expect{(\cong_G({\mathcal{Q}}(S_{z+1}),e)^2}$ for edges $e\in E(S_{z+1})$, and so the bound from \Cref{eq: type 4 bound for light} on the number of type-4 primary and type-2 secondary crossings is no longer valid. Instead, we will perform a type-1 uncrossing of the images of the edges of $E_{z}\cup E_z^{\operatorname{right}}\cup E_z^{\operatorname{over}}$, similarly to the first case. Let $\Gamma_1$ denote the set of curves reprsenting the images of these edges in the current drawing $\phi_z'$. Let $\Gamma_2$ denote the set of curves reprsenting the images of all remaining edges of $G_z$ in $\phi_z'$. We apply the algorithm from \Cref{thm: type-1 uncrossing} to compute a new collection $\Gamma_1'$ of curves, where, for each edge $e\in E_{z}\cup E_z^{\operatorname{right}}\cup E_z^{\operatorname{over}}$, there is a curve $\gamma(e)\in \Gamma_1'$ connecting the images of the endpoints of $e$.
Recall that every pair of curves in $\Gamma_1'$ may cross at most once, and, for each curve $\gamma\in \Gamma_2$, the number of crossings between $\gamma$ and the curves in $\Gamma_1'$ is no higher than the number of crossings between $\gamma$ and the curves in $\Gamma_1$. We modify the images of the edges in $E_{z}\cup E_z^{\operatorname{right}}\cup E_z^{\operatorname{over}}$, so that for each such edge $e$, its new image is the curve $\gamma(e)\in \Gamma_1'$. As before, the total number of primary crossings of types 1,2, and 3 does not increase, and neither does the number of secondary crossings of type 1. As before, a primary crossing of type 2 (a crossing between images of edges $e,e'$ where $e\in E_{z-1}\cup E_z^{\operatorname{left}}$ and $e'\in E_{z}\cup E_z^{\operatorname{right}}$) may become a primary crossing of type 3 (a crossing between images of edges $e,e'$ where $e'\in E_{z-1}\cup E_z^{\operatorname{left}}$ and $e\in E_z^{\operatorname{over}}$), and vice versa.
The total number of type-4 primary crossings and of type-2 secondary crossings is now bounded by:
$\textsf{left} (|E_{z}|+ |E_z^{\operatorname{right}}|+|E_z^{\operatorname{over}}|\textsf{right} )^2\leq |\delta_G(U_{z})|^2$.
Using the same arguments as in the first case, and the second statement from
\Cref{cor: few edges crossing cuts}, we conclude that $|\delta_G(U_{z})|\leq |\delta_G(S_{z+1})|\cdot O(\log^{34}m)$. As before, if $S_{z+1}\in {\mathcal{S}}^{\operatorname{bad}}$, and the bad event ${\cal{E}}$ does not happen, then, from \Cref{obs: congestion square of internal routers},
$\mathsf{OPT}_{\mathsf{cnwrs}}(S_{z+1},\Sigma(S_{z+1}))+|E(S_{z+1})|\geq \frac{|\delta_G(S_{z-1})|^2}{\hat\eta}$,
where $\Sigma(S_{z+1})$ is the rotation system for graph $S_{z+1}$ induced by $\Sigma$. As before, we get that
$|\delta_G(S_{z+1})|^2\leq \hat \eta\cdot \textsf{left} (|\chi^*_{z+1}|+|E(S_{z+1})|\textsf{right} )$.
Overall, if $S_{z+1}\in {\mathcal{S}}^{\operatorname{bad}}$, and event ${\cal{E}}$ did not happen, then the total number of type-4 primary type-2 secondary crossings is now bounded by:
\[|\delta_G(U_{z+1})|^2\leq O(\log^{68}m)\cdot |\delta_G(S_{z+1})|^2\leq \hat\eta^2\cdot \textsf{left} (|\chi^*_{z+1}|+|E(S_{z+1})|\textsf{right} ).\]
Combining this with the bound from Equation \ref{eq: type 4 bound for light}, we get that, regardless of whether Case 2 happened or not, if event ${\cal{E}}$ did not happen, then, after the current modification, the total expected number of type-4 primary crossings and of type-2 secondary crossings is bounded by:
\begin{equation}\label{eq: type 4 final}
\begin{split}&\hat \eta \cdot \sum_{(e_1',e_2')\in \chi^*\setminus \tilde \chi_{z+1}} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} )\\
&\quad\quad\quad\quad +O(\log^{34}m)\cdot \sum_{e\in E(\overline{U}_z)\setminus E(S_{z+1})}N'_z(e) \\
&\quad\quad\quad\quad +\hat \eta^2\cdot (|\chi^*_{z+1}|+|E(S_{z+1})|)+|\Pi^T_z|.
\end{split}
\end{equation}
\paragraph{Case 3: $S_{z-1},S_{z+1}\in {\mathcal{S}}^{\operatorname{light}}$, and accounting so far.}
If $S_{z-1},S_{z+1}\in {\mathcal{S}}^{\operatorname{light}}$, then we do not perform any modifications for now. We now bound the total number of crossings in the current drawing $\phi'_z$ of graph $G_z$ for cases 1--3, excluding the crossings between pairs of edges in $E_z^{\operatorname{over}}$. If Case 3 happened, then the number of crossings did not increase in this step. If Case 1 happened, then the total number of primary crossings of types 1,2 and 4, and secondary crossings of type 2 did not change, and the number of primary crossings of type 3 and secondary crossings of type 1 is bounded by \Cref{eq: type 3 final}. Similarly, If Case 2 happened, then the total number of primary crossings of types 1,2 and 3, and secondary crossings of type 1 did not change, and the number of primary crossings of type 4 and secondary crossings of type 2 is bounded by \Cref{eq: type 4 final}.
Therefore, if any of the cases 1--3 happened, and event ${\cal{E}}$ did not happen, then the total expected number of crossings in the current drawing $\phi'_z$ of graph $G_z$, excluding the crossings between pairs of edges in $E_z^{\operatorname{over}}$, is at most:
\begin{equation}\label{eq: bound all crossings}
\begin{split}
&\hat \eta^2\textsf{left} ( |\chi^*_{z-1}|+|\chi^*_{z}| +|\chi^*_{z+1}|+|E(S_{z-1})|+|E(S_{z+1})|\textsf{right} )\\
&\quad\quad\quad\quad+ \hat \eta \sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} ) \\
&\quad\quad\quad\quad+ \hat \eta\cdot \sum_{e\in E(U_{z-1})\setminus E(S_{z-1})}N'_z(e)+ \hat \eta\cdot \sum_{e\in E(\overline{U}_{z})\setminus E(S_{z+1})}N'_z(e)+|\Pi^T_z|.
\end{split}
\end{equation}
\paragraph{Case 4: $S_{z-1},S_{z+1}\in {\mathcal{S}}^{\operatorname{bad}}$.}
In this case, we will perform a type-1 uncrossing of the images of the edges of $E_{z-1}\cup E_z^{\operatorname{left}}\cup E_z^{\operatorname{over}}\cup E_z^{\operatorname{right}}\cup E_z=\delta_G(U_{z-1})\cup \delta_G(U_z)$. Let $\Gamma_1$ denote the set curves reprsenting the images of these edges in $\phi_z'$. Let $\Gamma_2$ denote the set of curves reprsenting the images of all remaining edges of $G_z$ in $\phi_z'$. We apply the algorithm from \Cref{thm: type-1 uncrossing} to compute a new collection $\Gamma_1'$ of curves, where, for each edge $e\in \delta_G(U_{z-1})\cup \delta_G(U_z)$, there is a curve $\gamma(e)\in \Gamma_1'$ connecting the images of the endpoints of $e$.
We are guaranteed that every pair of curves in $\Gamma_1'$ may cross each other at most once, and, for each curve $\gamma\in \Gamma_2$, the number of crossings between $\gamma$ and the curves in $\Gamma_1'$ is no greater than the number of crossings between $\gamma$ and the curves in $\Gamma_1$. We modify the images of the edges in $\delta_G(U_{z-1})\cup \delta_G(U_z)$, so that for each such edge $e$, its new image is the curve $\gamma(e)\in \Gamma_1'$.
Note that the total number of type-1 primary crossings does not change. The total number of all other crossings is bounded by $(|\delta_G(U_{z-1})|+| \delta_G(U_z)|)^2\leq O(|\delta_G(U_{z-1})|^2)+O(|\delta_G(U_{z})|^2)$. Using the same reasoning as in Cases 1 and 2, if event ${\cal{E}}$ did not happen,
then:
$$|\delta_G(U_{z-1})|^2\leq \hat \eta^2\cdot \textsf{left} (|\chi^*_{z-1}|+|E(S_{z-1})|\textsf{right} ),$$
and
$$|\delta_G(U_{z})|^2\leq \hat \eta^2\cdot \textsf{left} (|\chi^*_{z+1}|+|E(S_{z+1})|\textsf{right} ),$$
Therefore, if event ${\cal{E}}$ did not happen, the total expected number of crossings in the current drawing is bounded by:
\begin{equation}\label{eq: case 4 final}
\hat \eta^2\cdot \textsf{left}(|\chi^*_{z+1}|+|\chi^*_z|+ |\chi^*_{z+1}|+|E(S_{z-1})|+|E(S_{z+1})|\textsf{right}).
\end{equation}
\paragraph{Uncrossing the Edges of $E_z^{\operatorname{over}}$.}
So far we have constructed a drawing $\phi'_z$ of graph $G_z$ and bounded the expected number of crossings in $\phi'_z$, excluding the crossings between the images of the edges in $E_z^{\operatorname{over}}$. In this step, we eliminate all crossings of the latter type, by performing a type-2 uncrossing of the images of the edges in $E_z^{\operatorname{over}}$.
Specifically, we let $\tilde {\mathcal{Q}}$ be the set of paths in graph $G_z$ that contains, for each edge $e\in E_z^{\operatorname{over}}$, a path $\tilde Q(e)$, that consists of the edge $e$ only. Recall that each edge $e\in E_z^{\operatorname{over}}$ connects the special vertices $v^*$, $v^{**}$ to each other. We view each such path $\tilde Q(e)$ as being directed from $v^*$ to $v^{**}$. We then apply the algorithm from \Cref{thm: new type 2 uncrossing} that performs a type-2 uncrossing on the images of the paths in $\tilde {\mathcal{Q}}$. Let $\Gamma$ be the resulting set of curves that it produces. Recall that, for every edge $e\in E_z^{\operatorname{over}}$, there must be a curve $\gamma(e)\in \Gamma$, that contains the segment of the image of edge $e$ that lies in the disc $D(v^*)$. We replace the current image of the edge $e$ with the curve $\gamma(e)$. Once the images of all edges $e\in E_z^{\operatorname{over}}$ are modified, we obtain the final modified drawing $\phi''_z$ of graph $G_z$. The algorithm from \Cref{thm: new type 2 uncrossing} ensures that the images of the edges in $E_z^{\operatorname{over}}$ do not cross each other. Since the curves in $\Gamma$ are aligned with the graph that consists of the edges of $E_z^{\operatorname{over}}$, we are guaranteed that, for each edge $e\in E(G_z)\setminus E^{\operatorname{over}}_z$, the number of crossings in which edge $e$ participates does not increase.
The algorithm from \Cref{thm: new type 2 uncrossing}, and the type-1 uncrossings that we performed in Cases 1 -- 3 ensure that the order in which the images of the edges of $\delta_{G_z}(v^*)$ enter the image of $v^*$ does not change, and remain consistent with the rotation ${\mathcal{O}}_{v^*}\in \Sigma_z$. To summarize, we have obtained a drawing $\phi''_z$ of graph $G_z$, such that, for every vertex $v\in V(G_z)\setminus\set{v^{**}}$, the images of the edges of $\delta_{G_z}(v)$ enter the image of $v$ in the order consistent with the rotation ${\mathcal{O}}_v\in\Sigma_z$, and the total expected number of crossings in $\phi''_z$ is bounded by:
\iffalse
\begin{equation}\label{eq: bound all crossings 2}
\begin{split}
\hat \eta\textsf{left} ( |\chi^*_{z-1}|+|\chi^*_{z}| +|\chi^*_{z+1}|+|E(S_{z-1})|+|E(S_{z+1})|+ \sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} ) \textsf{right} ) +|\Pi^T_z|.
\end{split}
\end{equation}
where $\hat \eta=2^{O((\log m)^{3/4}\log\log m)}$.
\fi
\begin{equation}\label{eq: bound all crossings final}
\begin{split}
&\hat \eta^2\textsf{left} ( |\chi^*_{z-1}|+|\chi^*_{z}| +|\chi^*_{z+1}|+|E(S_{z-1})|+|E(S_{z+1})|\textsf{right} )\\
&\quad\quad\quad\quad+ \hat \eta\cdot \sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} ) \\
&\quad\quad\quad\quad+ \hat \eta\cdot \sum_{e\in E(U_{z-1})\setminus E(S_{z-1})}N'_z(e)+ \hat \eta\cdot \sum_{e\in E(\overline{U}_{z})\setminus E(S_{z+1})}N'_z(e)+|\Pi^T_z|.
\end{split}
\end{equation}
In the next and the final step, we obtain the final solution $\phi_z$ to instance $I_z=(G_z,\Sigma_z)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, by modifying the current drawing $\phi''_z$ of graph $G_z$ inside the tiny $v^{**}$-disc $D(v^{**})$.
\subsubsection{Step 4: the Final Drawing of Graph $G_z$}
In this step we slightly modify the current drawing $\phi''_z$ of graph $G_z$ in order to obtain the final drawing $\phi_z$ of $G_z$, which is a valid solution to instance $I_z=(G_z,\Sigma_Z)$ of \ensuremath{\mathsf{MCNwRS}}\xspace.
Consider the tiny $v^{**}$-disc $D=D_{\phi''_z}(v^{**})$. Denote $\delta_{G_z}(v^{**})=\set{e_1,\ldots,e_h}$, and, for all $1\leq i\leq h$, let $p_i$ be the point on the image of the edge $e_i$ in $\phi''_z$ that lies on the boundary of the disc $D$. We assume that the edges are indexed so that the points $p_1,\ldots,p_h$ are encountered in this order when traversing the boundary of $D$ in the clock-wise direction. We denote by ${\mathcal{O}}$ this ordering of the edges $e_1,\ldots,e_h$. Let ${\mathcal{O}}'$ be the ordering ${\mathcal{O}}_{v^{**}}\in \Sigma_z$ of the edges of $\delta_{G_z}(v^{**})$.
We use the algorithm from \Cref{lem: ordering modification} to compute a collection $\Gamma=\set{\gamma(e_i)\mid 1\leq i\leq h}$ of curves, such that, for each edge $e_i$, curve $\gamma(e_i)$ only differs from the image of the edge $e_i$ in the current drawing $\phi''_z$ of $G_z$ inside the disc $D$, and the curves of $\Gamma$ enter the image of $v^{**}$ in the order ${\mathcal{O}}'$. We then replace, for each edge $e_i\in \delta_{G_z}(v^{**})$, the current image of the edge $e_i$ with the curve $\gamma(e_i)$. As the result, we obtain a valid solution $\phi_z$ to instance $I_z=(G_z,\Sigma_z)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, as the images of the edges in $\delta_{G_z}(v^{**})$ now enter the image of $v^{**}$ in the correct order. \Cref{lem: ordering modification} guarantees that the number of crossings between the curves in $\Gamma$ within the disc $D$ is bounded by $O(\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}'))$, and these are the only new crossings. Therefore, the number of crossings grows by at most $O(\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}'))$. In the next claim we bound $\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$.
\begin{claim}\label{claim: bound distance between rotations}
If event ${\cal{E}}$ did not happen, then the expectation of $\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')$ is bounded by:
\[\begin{split} &\hat \eta^{O(1)}\cdot \textsf{left} (\sum_{e\in E(G)}\expect{N'_z(e)}+\sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N'_z(e)}+\expect{N'_z(e')}\textsf{right} )\textsf{right} )\\
&\quad\quad\quad\quad+\hat \eta^{O(1)}\cdot \textsf{left} (|\chi^*_{z-1}|+|\chi^*_z|+|\chi^*_{z+1}|+|E(S_{z-1})|+|E(\tilde S_z)|+|E(S_{z+1})|+| \delta_G(\tilde S_z)|+|\delta_G(S_{z-1})|\textsf{right} )\\
&\quad\quad\quad\quad+\hat \eta \cdot \mathsf{cr}(\phi''_z)+|\Pi_z^T|.
\end{split}
\]
\end{claim}
We prove \Cref{claim: bound distance between rotations} below, after we complete the proof of \Cref{claim: existence of good solutions special} using it.
For convenience, we denote by $E^*_z= E(S_{z+1})\cup E(S_{z-1})\cup E(\tilde S_z)\cup \delta_G(\tilde S_z)\cup \delta_G(S_{z-1})$.
Combining the bound from \Cref{eq: bound all crossings final} with the bound from \Cref{claim: bound distance between rotations}, we get that, if Event ${\cal{E}}$ did not happen, then $\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I_z)}$ is bounded by:
\iffalse
\begin{equation}\label{eq: bound all crossings final}
\begin{split}
&\hat \eta^{O(1)}\textsf{left} ( |\chi^*_{z-1}|+|\chi^*_{z}| +|\chi^*_{z+1}|+|E^*_z|\textsf{right} )\\
&\quad\quad\quad\quad+ \hat \eta^{O(1)}\sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} ) \\
&\quad\quad\quad\quad+ \hat \eta^{O(1)}\cdot \textsf{left} ( \sum_{e\in E(U_{z-1})\setminus E(S_{z-1})}N'_z(e)+ \sum_{e\in E(\overline{U}_{z})\setminus E(S_{z+1})}N'_z(e)+|\Pi^T_z|\textsf{right} ).
\end{split}
\end{equation}
\fi
\iffalse
\begin{equation}\label{eq: bound all crossings final}
\begin{split}
\hat \eta^{O(1)}\textsf{left} ( |\chi^*_{z-1}|+|\chi^*_{z}| +|\chi^*_{z+1}|+|E^*_z|+\sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} ) + \sum_{e\in E(U_{z-1})\setminus E(S_{z-1})}N'_z(e)+ \sum_{e\in E(\overline{U}_{z})\setminus E(S_{z+1})}N'_z(e)+|\Pi^T_z|\textsf{right} ).
\end{split}
\end{equation}
\fi
\[
\begin{split}
\hat \eta^{O(1)}\textsf{left} ( |\chi^*_{z-1}|+|\chi^*_{z}| +|\chi^*_{z+1}|+|E^*_z|+ \sum_{(e,e')\in \chi^*} \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} ) +\sum_{e\in E(G)}\expect{N'_z(e)}+|\Pi^T_z|\textsf{right} ).
\end{split}
\]
Note that an edge $e\in E(G)$ may belong to at most $O(1)$ sets in $\set{E^*_z}_{z=1}^r$. Also, a crossing $(e,e')\in \chi^*$ may belong to at most two sets in $\set{\chi^*_z}_{z=1}^r$ Therefore, we get that:
\begin{equation}\label{eq: final bound}
\begin{split}
\expect{\sum_{z=1}^r\mathsf{OPT}_{\mathsf{cnwrs}}(I_z)}&\leq \hat \eta^{O(1)}(|E(G)|+|\chi^*|)\\
&+\hat \eta^{O(1)}\cdot \sum_{(e,e')\in \chi^*}\sum_{z=1}^r \textsf{left} (\expect{N_z'(e_1')}+\expect{N_z'(e_2')}\textsf{right} ) \\
&+ \hat \eta^{O(1)}\cdot\sum_{e\in E(G)}\sum_{z=1}^r\expect{N'_z(e)} \\
&+ \hat \eta^{O(1)}\cdot\sum_{z=1}^r|\Pi^T_z|
\end{split}
\end{equation}
We use the following two observations, whose proofs appear in \Cref{subsec:bound N' values} and \Cref{subsec: proof of obs bound Pi triples}, respectively, in order to complete the proof of \Cref{claim: existence of good solutions special}.
\begin{observation}\label{obs: bound N' values}
For every edge $e\in E(G)$, $\sum_{z=1}^r\expect{N'_z(e)}\leq O(\hat \eta)$.
\end{observation}
\begin{observation}\label{obs: bound Pi triples}
$\sum_{z=1}^r|\Pi^T_z|\leq \textsf{left}(|E(G)|+|\chi^*|\textsf{right} )\cdot O(\log^{68}m)$.
\end{observation}
Combining \Cref{eq: final bound} with Observations \ref{obs: bound N' values} and \ref{obs: bound Pi triples}, and recalling that $\hat \eta=2^{O((\log m)^{3/4}\log\log m)}$, we get that, if event ${\cal{E}}$ did not happen:
\[
\expect{\sum_{z=1}^r\mathsf{OPT}_{\mathsf{cnwrs}}(I_z)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot (|E(G)|+|\chi^*|).
\]
Recall that $\prob{{\cal{E}}}\leq 1/m^{99}$, and, if ${\cal{E}}$ happens, $\sum_{z=1}^r\mathsf{OPT}_{\mathsf{cnwrs}}(I_z)\leq m^3$ must hold. Therefore, overall, $\expect{\sum_{z=1}^r\mathsf{OPT}_{\mathsf{cnwrs}}(I_z)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot (|E(G)|+|\chi^*|)$.
In order to complete the proof of \Cref{claim: existence of good solutions special}, it is now enough to prove \Cref{claim: bound distance between rotations}, which we do next.
\subsection{Proof of \Cref{claim: existence of good solutions special}}
\label{subsec: bound opt costs}
Notice that graphs $G_1,G_r$ have a somewhat different structure than graphs of $\set{G_z}_{1<z<r}$: specifically, graph $G_1$ does not contain vertex $v^*_1$, and graph $G_r$ does not contain vertex $v^{**}_r$. It would be convenient for us to modify these two graphs so that we can treat all resulting graphs uniformly. In order to do so, we add a new dummy vertex $v^*_1$ to graph $G_1$, and connect it with an edge to an arbitrary vertex $v_1\in S_1$. We modify the rotation ${\mathcal{O}}_{v_1}\in \Sigma_1$ to include the new edge $(v_1,v^*_1)$ at an arbitrary position in the rotation. Notice that any solution to the resulting new instance $I_1=(G_1,\Sigma_1)$ of \ensuremath{\mathsf{MCNwRS}}\xspace immediately provides a solution to the original instance $I_1=(G_1,\Sigma_1)$, of the same cost. We similarly modify graph $G_r$, by adding a new dummy vertex $v^{**}_r$, which is connected with an edge to an arbitrary vertex $v_r\in S_r$. We modify the rotation ${\mathcal{O}}_{v_r}\in \Sigma_r$ as before. In order to be consistent, we also add the vertices $v^*_1,v^{**}_r$, and edges $(v^*_1,v_1)$ and $(v^{**}_r,v_r)$ to the original graph $G$, and modify the rotations ${\mathcal{O}}_{v_1},{\mathcal{O}}_{v_r}\in {\mathcal{O}}$, so that they remain consistent with the rotations ${\mathcal{O}}_{v_1}\in \Sigma_1$ and ${\mathcal{O}}_{v_r}\in \Sigma_r$, respectively. Notice that this modification does not increase $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$.
We also modify the nice witness structure, by adding two new clusters $\tilde S_0=\set{v^*_1}$ and $\tilde S_{r+1}=\set{v^{**}_r}$ to $\tilde {\mathcal{S}}$, and their subclusters $S_0=\set{v^*_1}$ and $ S_{r+1}=\set{v^{**}_{r}}$. We note that all the above modifications are only performed for ease of exposition and are not strictly necessary.
Let $\phi^*$ be an optimal solution to instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace. We can assume that no pair of edges cross twice, and that the image of each edge does not cross itself in $\phi^*$.
We denote by $\chi^*$ the set of all unordered pairs $(e,e')$ of edges of $G$, such that the images of $e$ and $e'$ cross in $\phi^*$.
For all $1\leq z\leq r$, we denote by $\chi^*_z\subseteq \chi^*$ the set of all unordered pairs $(e,e')$ of edges of $G$ whose images cross, such that either $e\in E(\tilde S_z)\cup \delta_G(\tilde S_z)$, or $e'\in E(\tilde S_z)\cup \delta_G(\tilde S_z)$, or both. We will use the drawing $\phi^*$ in order to construct, for each $1\leq z\leq r$, a solution $\phi_z$ to instance $I_z=(G_z,\Sigma_z)$
For each $1\leq z\leq r$, we construct a solution $\phi_z$ to instance $I_z$, and then argue that the total expected costs of all these solutions is relatively small. We now fix an index $1\leq z\leq r$, and focus on constructing solution $\phi_z$ to instance $I_z$.
The construction of the solution consists of four steps. In the first step, we construct an auxiliary graph $H_z$ and its drawing $\psi_z$. This graph and its drawing are used in the second step, in order to construct an initial drawing $\phi'_z$ of graph $G_z$. The number of crossings in drawing $\phi'_z$ may be quite large, and we modify the drawing in order to lower the number of crossings in the third step. The resulting drawing, $\phi''_z$, will have a sufficiently low expected number of crossings, but unfortunately it may not obey the rotation ${\mathcal{O}}_{v^{**}_z}\in \Sigma_z$. In the fourth and the last step, we modify this drawing in order to obtain a feasible solution $\phi_z$ to instance $I_z$ of \ensuremath{\mathsf{MCNwRS}}\xspace, while only slightly increasing the number of crossings.
We now fix an index $1\leq z\leq r$, and describe a construction of a solution $\phi_z$ for instance $I_z$ of \ensuremath{\mathsf{MCNwRS}}\xspace step by step.
\input{bound-opts-step1}
\input{bound-opts-step2}
\input{bound-opts-step3}
\input{bound-opts-step4}
\input{bound-opts-reordering}
\iffalse @@@@@@@@
Old stuff below
The remainder of this subsection is dedicated to the proof of Claim~\ref{claim: existence of good solutions special}.
Let $\phi^*$ be an optimal drawing of the instance $(G,\Sigma)$. We can assume that no pair of edges cross twice in $\phi^*$, and that the image of each edge does not cross itself in $\phi^*$. We denote by $\chi^*$ the set of all pairs $(e,e')$ of edges of $G$, such that the images of $e$ and $e'$ cross in $\phi^*$.
For all $1\leq z\leq r$, we denote by $\chi^*_z\subseteq \chi^*$ the set of all pairs $(e,e')$ of edges of $G$ whose images cross, such that either $e\in E(\tilde S_z)$, or $e'\in E(\tilde S_z)$, or both.
We will construct, for each $1\le i\le r$, a solution $\phi_i$ to the instance $(G_i,\Sigma_i)$ using $\phi^*$, such that $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)\le O\textsf{left}(\bigg(\eta^*+(\beta^*)^2\bigg)\cdot \log^{36}m\cdot (\mathsf{cr}(\phi^*)+|E(G)|)\textsf{right})$, and \Cref{claim: existence of good solutions special} then follows.
\paragraph{Constructing the external routers.} Before constructing the drawings for subinstances $I_1,\ldots,I_r$, we first construct, for each $1\le i\le r-1$, an external router ${\mathcal{W}}'_i$ for cluster $U_i$, as follows.
For each edge $e\in E(U_i, \overline{U}_i)$, we define path $W'_i(e)$ as the subpath of cycle $W(e)$ between (including) the edge $e$ and vertex $u_{i+1}\in S_{i+1}$, that lies entirely within $\overline{U}_i$ (see \Cref{obs: outer_path_monotone}).
We denote ${\mathcal{W}}'_i=\set{W'_i(e)\mid e\in E(U_i, \overline{U}_i)}$.
\paragraph{Drawings $\phi_2,\ldots,\phi_{r-1}$.}
First we fix some index $2\le i\le r-1$, and describe the construction of the drawing $\phi_i$.
Recall that $E(G_i)=E_G(\tilde S_i)\cup (\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$.
Also recall that we have defined a set ${\mathcal{W}}_{i-1}$ of paths that contains, for each edge $e\in E(U_{i-1},\overline{U}_{i-1})=\hat E_{i-1}\cup E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}$, a path $W_{i-1}(e)$ routing $e$ to $u_{i-1}$ in $U_{i-1}$, and a set ${\mathcal{W}}'_i$ of paths that contains, for each edge $e\in E(U_i,\overline{U}_i)=\hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$, a path $W'_i(e)$ routing $e$ to $u_{i+1}$ in $\overline{U}_i$.
Clearly, all paths in ${\mathcal{W}}_{i-1}$ and ${\mathcal{W}}'_{i}$ are internally disjoint from $\tilde S_i$. We will view $u_{i-1}$ as the last endpoint of all paths in ${\mathcal{W}}_{i-1}$ and view $u_{i+1}$ as the last endpoint of all paths in ${\mathcal{W}}'_{i}$.
We start with the image of cluster $\tilde S_i$ in $\phi^*$. We then add to it the image of vertices $v'_i, v^*_i$ and edges of $(\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$ as follows.
First, we apply the algorithm from \Cref{cor: new type 2 uncrossing} to graph $G$, drawing $\phi^*$ and the set ${\mathcal{W}}'_{i}$ of paths, and obtain a set $\Gamma'_i=\set{\gamma'_e\text{ }\big|\text{ } e\in \big(\hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}\big)}$ of curves.
Since all paths of ${\mathcal{W}}'_{i}$ share the last endpoint $u_{i+1}$, all curves in $\Gamma'_i$ also share the last endpoint, that we denote by $z^*_i$.
We denote by ${\mathcal{O}}^*_i$ the ordering in which curves of $\Gamma'_i$ enter $z^*_i$.
Similarly, we apply the algorithm from \Cref{thm: new nudging} to graph $G$, drawing $\phi^*$ and the set ${\mathcal{W}}_{i-1}$ of paths, and obtain a set $\Gamma_{i-1}=\set{\gamma_e\text{ }\big|\text{ } e\in \big(\hat E_{i-1}\cup E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\big)}$ of curves. Since all paths of ${\mathcal{W}}_{i-1}$ share the last endpoint $u_{i-1}$, all curves in $\Gamma_{i-1}$ also share the last endpoint, that we denote by $z'_i$.
Lastly, we apply the algorithm from \Cref{lem: ordering modification} to the set $\hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$ of edges (as elements), set $\Gamma'_i=\set{\gamma'_e\mid e\in \big(\hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}\big)}$ of curves, and the ordering $\tilde {\mathcal{O}}_{i}$. We rename the set of curves we obtain by $\Gamma'_i$.
We now view $z^*_i$ as the image of $v^*_i$, view $z'_i$ as the image of $v'_i$, and view (i) for each $e\in \hat E_{i}\cup E_i^{\operatorname{right}}$, the curve $\gamma'_e$ as the image of $e$; (ii) for each $e\in \hat E_{i-1}\cup E_i^{\operatorname{left}}$, the curve $\gamma_e$ as the image of $e$; and (iii) for each $e\in E_i^{\textsf{thr}}$, the concatenation of curves $\gamma_e$ and $\gamma'_e$ as the image of $e$. In this way, we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is easy to verify that $\phi_i$ respects the rotation system $\Sigma_i$.
\iffalse{previous rotation modification}
Let ${\mathcal{D}}$ be a tiny disc around $z^*_{i}$, and let ${\mathcal{D}}'$ be another small disc around $z^*_{i}$ that is strictly contained in ${\mathcal{D}}$.
We first erase the image of all edges of $\delta(v^*_i)$ inside the disc ${\mathcal{D}}$, and for each edge $e\in \delta(v^{*}_i)$, we denote by $p_{e}$ the intersection between the curve representing the current image of $e$ and the boundary of ${\mathcal{D}}$.
We then place, for each edge $e\in \delta(v^{*}_i)$, a point $p'_e$ on the boundary of ${\mathcal{D}}'$, such that the order in which the points in $\set{p'_e\mid e\in \delta(v^{*}_i)}$ appearing on the boundary of ${\mathcal{D}}'$ is precisely ${\mathcal{O}}^{*}_{i}$.
We then apply \Cref{lem: find reordering} to compute a set of reordering curves, connecting points of $\set{p_e\mid e\in \delta(v^{*}_i)}$ to points $\set{p'_e\mid e\in \delta(v^{*}_i)}$.
Finally, for each edge $e\in \delta(v^{*}_i)$, let $\zeta_e$ be the concatenation of (i) the current image of $e$ outside the disc ${\mathcal{D}}$; (ii) the reordering curve connecting $p_e$ to $p'_e$; and (iii) the straight line segment connecting $p'_e$ to $z^{*}_i$ in ${\mathcal{D}}'$.
We view $\zeta_e$ as the image of edge $e$, for each $e\in \delta(v^{*}_i)$.
We denote the resulting drawing of $G_i$ by $\phi_i$. It is clear that $\phi_i$ respects the rotation ${\mathcal{O}}^{*}_i$ at $v^{*}_i$, and therefore it respects the rotation system $\Sigma_i$.
\fi
\paragraph{Final Accounting.} We now upper bound the number of crossings in the drawing $\phi_i$ that we have constructed.
In the remainder of this subsection, we will omit the subscript $\phi^*$ in the notations $\chi_{\phi^*}(\cdot),\chi^1_{\phi^*}(\cdot),\chi^2_{\phi^*}(\cdot), \chi_{\phi^*}(\cdot,\cdot)$ (see \Cref{sec: prelim_crossing_notations} for definitions).
Recall that index set $Z_{\text{fail}}$ contains all indices $i$ of $\set{1,\ldots,r}$ such that the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace returns fail on $S_{i}$, and $Z_{\text{succ}}$ contains all other indices.
Assume events ${\cal{E}}_1,\ldots,{\cal{E}}_r$ do not happen, then for each $i\in Z_{\text{succ}}$, $S_i$ is a $\beta^*$-light cluster, and for each $i\in Z_{\text{fail}}$, $S_i$ is a $\eta^*$-bad cluster.
We denote $E_{\text{fail}}=\bigcup_{i\in Z_{\text{fail}}}E(S_i)$ and $E_{\text{succ}}=\bigcup_{i\in Z_{\text{succ}}}E(S_i)$.
We define ${\mathcal{W}}=\bigcup_{1\le i\le r-1}({\mathcal{W}}_i\cup{\mathcal{W}}'_i)$. Then for each path $W\in {\mathcal{W}}$, we say that it is \emph{bad} iff $W$ contains an edge of $E_{\text{fail}}$, otherwise we say $W$ is \emph{light}. We denote by ${\mathcal{W}}^{\operatorname{light}}$ the set of all good paths in ${\mathcal{W}}$ and denote by ${\mathcal{W}}^{\operatorname{bad}}$ the set of all bad paths in ${\mathcal{W}}$, so sets ${\mathcal{W}}^{\operatorname{light}},{\mathcal{W}}^{\operatorname{bad}}$ partition ${\mathcal{W}}$.
Similarly, for each $1\le i\le r-1$, we define sets ${\mathcal{W}}^{\operatorname{light}}_i={\mathcal{W}}^{\operatorname{light}}\cap {\mathcal{W}}_i$ and ${\mathcal{W}}^{\operatorname{bad}}_i={\mathcal{W}}^{\operatorname{bad}}\cap {\mathcal{W}}_i$.
We first prove the following observations, whose proofs are deferred to \Cref{apd: Proof of accounting} and \Cref{apd: Proof of accounting_2}, respectively.
\begin{observation}
\label{obs: accounting}
The following hold.
\begin{enumerate}
\item \label{prop_1}
For each $e\in E(G)$, $\expect[]{\cong_G({\mathcal{W}},e)}\le O(\log^{18}m)\cdot \beta^*$;
\item \label{prop_2}
$\sum_{1\le i\le r-1}|{\mathcal{W}}^{\operatorname{bad}}_i|^2\le O\big(\eta^*\cdot \log^{36}m\cdot (\mathsf{cr}(\phi^*)+|E(G)|)\big)$;
\item \label{prop_3}
For each $e\in E(G)$, $\expect[]{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2}\le O(\log^{36}m)\cdot \beta^*$;
\item \label{prop_5}
For each $i\in Z_{\text{succ}}$, $\expect[]{\sum_{e\in E(S_i)}(\cong_G({\mathcal{W}},e))^2}\le O(\log^{36}m)\cdot \beta^*$;
\item \label{prop_4}
$\expect[]{\sum_{e,e'\in E(G)}\chi(e,e')\cdot \bigg(\cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{light}},e')\bigg)}\le O\big(\beta^*\cdot \log^{36}m\cdot \mathsf{cr}(\phi^*)\big)$.
\end{enumerate}
\end{observation}
\begin{observation}
\label{obs: accounting_2}
For every pair $e,e'$ of edges in $E(G)$, $$\expect[]{\cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{bad}},e')}\le O\big(\beta^*\cdot \log^{36}m\big).$$
\end{observation}
Combine property \ref{prop_4} in \Cref{obs: accounting} and \Cref{obs: accounting_2} with \Cref{lem: non transversal cost of cycles bounded by cr}, we immediately obtain the following two corollaries.
\begin{corollary}
\label{cor: NTcost_1}
$\expect[]{\sum_{1\le i\le r}\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_i;\Sigma)}\le O\big(\beta^*\cdot \log^{36}m\cdot \mathsf{cr}(\phi^*)\big)$.
\end{corollary}
\begin{corollary}
\label{cor: NTcost_2}
$\expect[]{\sum_{1\le i\le r}\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_i,{\mathcal{W}}^{\operatorname{bad}}_{i};\Sigma)}\le O\big(\beta^*\cdot \log^{36}m\cdot \mathsf{cr}(\phi^*)\big)$.
\end{corollary}
Fix now an index $2\le i\le r-1$.
First, since we have not modified the drawing of $\tilde S_i$, the number of crossings between edges of $E_G(\tilde S_i)$ are bounded by $\chi^2(\tilde S_i)$.
Second, from the definition of curves of $\Gamma'_i$, the number of crossings between image of $\tilde S_i$ and the curves of $\Gamma'_i$ is at most $\chi({\mathcal{W}}'_i,\tilde S_i)$, and similarly the number of crossings between image of $\tilde S_i$ and the curves of $\Gamma_{i-1}$ is at most $\chi({\mathcal{W}}_{i-1},\tilde S_i)$.
Third, from \Cref{thm: new nudging}, the number of crossings between curves of $\Gamma_{i-1}$ is bounded by
\[
\begin{split}
\chi(\Gamma_{i-1})\le & \text{ } |{\mathcal{W}}^{\operatorname{bad}}_{i-1}|^2+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_{i-1};\Sigma)+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_{i-1},{\mathcal{W}}^{\operatorname{bad}}_{i-1};\Sigma)+\sum_{e\in E(G)}(\chi_{\phi}(e)+1)\cdot (\cong_G({\mathcal{W}}^{\operatorname{light}}_{i-1},e))^2\\
& +\sum_{e\in E(G)} \cong_G({\mathcal{W}}^{\operatorname{light}}_{i-1},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{i-1},e)+\sum_{e,e'\in E(G)}
\chi_{\phi}(e,e')\cdot \cong_G({\mathcal{W}}^{\operatorname{light}}_{i-1},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{i-1},e).
\end{split}
\]
Fourth, the number of crossings between curves of $\Gamma_{i-1}$ and curves of $\Gamma'_{i}$ is at most $\chi({\mathcal{W}}'_i,{\mathcal{W}}_{i-1})$.
Lastly, from \Cref{lem: ordering modification}, the number of crossings in drawing $\phi_i$ inside the tiny $z^*_i$-disc is $O(\mbox{\sf dist}({\mathcal{O}}^*_i,\tilde{\mathcal{O}}_i))$. We now give an upper bound of $\mbox{\sf dist}({\mathcal{O}}^*_i,\tilde{\mathcal{O}}_i)$ as follows.
Recall that for each $e\in \hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$, $W_i(e)$ is the subpath of cycle $W(e)$ between (including) the edge $e$ and vertex $u_i$, that lies entirely within $U_i$.
We apply the algorithm from \Cref{thm: new nudging} to graph $G$, drawing $\phi^*$, cluster $\tilde S_{i+1}$ and the set ${\mathcal{W}}_i$ of paths. Let $\Xi_i=\set{\xi_e\mid e\in E(U_i,\overline{U}_i)}$ be the set of curves we get.
From \Cref{thm: new nudging}, the curves of $\Xi_i$ enter $y_i=\phi^*(u_i)$ in the order $\tilde{\mathcal{O}}_i$, and we have
\[\begin{split}
\chi(\Xi_i)\le & \text{ } |{\mathcal{W}}^{\operatorname{bad}}_{i}|^2+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_{i};\Sigma)+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_{i},{\mathcal{W}}^{\operatorname{bad}}_{i};\Sigma)+\sum_{e\in E(G)}(\chi_{\phi}(e)+1)\cdot (\cong_G({\mathcal{W}}^{\operatorname{light}}_{i},e))^2\\
& +\sum_{e\in E(G)} \cong_G({\mathcal{W}}^{\operatorname{light}}_{i},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{i},e)+\sum_{e,e'\in E(G)}
\chi_{\phi}(e,e')\cdot \cong_G({\mathcal{W}}^{\operatorname{light}}_{i},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{i},e).
\end{split}\]
We define, for each edge $e\in E(U_i,\overline{U}_i)$, the curve $\zeta_e$ to be the union of $\xi_e$ and $\gamma'_e$. It is easy to verify that $\set{\zeta_e\mid e\in E(U_i,\overline{U}_i)}$ is a set of reordering curves for ${\mathcal{O}}^*_i$ and $\tilde{\mathcal{O}}_i$, and so
\[
\mbox{\sf dist}({\mathcal{O}}^*_i,\tilde{\mathcal{O}}_i)\le \chi\big(\set{\zeta_e\mid e\in E(U_i,\overline{U}_i)}\big)\le \chi(\Xi_i)+\chi({\mathcal{W}}_i,{\mathcal{W}}'_{i-1}).
\]
\iffalse{previous analysis of crossings}
\begin{claim}
\label{clm: rerouting_crossings}
The number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\end{claim}
\begin{proof}
Denote by ${\mathcal{O}}^*$ the ordering in which the curves $\set{\gamma'_{W_e}\mid e\in \delta_{G_i}(v_i^{\operatorname{right}})}$ enter $z_{\operatorname{right}}$, the image of $u_{i+1}$ in $\phi'_i$.
From~\Cref{lem: find reordering} and the algorithm in Step 4 of modifying the drawing within the disc ${\mathcal{D}}$, the number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is at most $O(\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}}))$. Therefore, it suffices to show that $\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}})=O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
In fact, we will compute a set of curves connecting the image of $u_i$ and the image of $u_{i+1}$ in $\phi^*_i$, such that each curve is indexed by some edge $e\in\delta_{G_i}(v_i^{\operatorname{right}})$ these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$ and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$, and the number of crossings between curves of $Z$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
For each $e\in E^{\textsf{thr}}_i$, we denote by $\eta_e$ the curve obtained by taking the union of (i) the curve $\gamma'_{W_e}$ (that connects $u_{i+1}$ to $u_{i-1}$); and (ii) the curve representing the image of the subpath of $P_e$ in $\phi^*_i$ between $u_i$ and $u_{i-1}$.
Therefore, the curve $\eta_e$ connects $u_i$ to $u_{i+1}$.
We then modify the curves of $\set{\eta_e\mid e\in E^{\textsf{thr}}_i}$, by iteratively applying the algorithm from \Cref{obs: curve_manipulation} to these curves at the image of each vertex of $S_{i-1}\cup S_{i+1}$. Let $\set{\zeta_e\mid e\in E^{\textsf{thr}}_i}$ be the set of curves that we obtain. We call the obtained curves \emph{red curves}. From~\Cref{obs: curve_manipulation}, the red curves are in general position. Moreover, it is easy to verify that the number of intersections between the red curves is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1}))$.
We call the curves in $\set{\gamma'_{W_e}\mid e\in \hat E_i}$ \emph{yellow curves}, call the curves in $\set{\gamma'_{W_e}\mid e\in E^{\operatorname{right}}_i}$ \emph{green curves}. See \Cref{fig: uncrossing_to_bound_crossings} for an illustration.
From the construction of red, yellow and green curves, we know that these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$, and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$.
Moreover, we are guaranteed that the number of intersections between red, yellow and green curves is at most $\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/uncross_to_bound_crossings.jpg}
\caption{An illustration of red, yellow and green curves.}\label{fig: uncrossing_to_bound_crossings}
\end{figure}
\end{proof}
\fi
From the above discussion, for each $2\le i\le r-1$,
\begin{equation*}
\mathsf{cr}(\phi_i)\le \text{ }\chi^2(\tilde S_i)+\chi({\mathcal{W}}'_i\cup {\mathcal{W}}_{i-1},\tilde S_i)+\chi({\mathcal{W}}_i,{\mathcal{W}}'_i)+\chi({\mathcal{W}}_{i-1},{\mathcal{W}}'_{i})+ \chi(\Gamma_{i-1}) +\chi(\Xi_i).
\end{equation*}
\paragraph{Drawings $\phi_1$ and $\phi_{r}$.}
The drawings $\phi_1$ and $\phi_{r}$ are constructed similarly. We only describe the construction of $\phi_1$, and the construction of $\phi_r$ is symmetric.
Recall that graph $G_1$ contains only one super-node $v_1^{*}$, that is obtained by contracting $\overline{U}_1$, and $\delta_{G_1}(v_1^{*})=\hat E_1\cup E^{\operatorname{right}}_1$.
We construct the external router ${\mathcal{W}}'_1$ in the same way as sets ${\mathcal{W}}'_2,\ldots,{\mathcal{W}}'_{r-1}$. Then we construct the drawing $\phi_1$ in a similar way as constructing drawings $\phi_2,\ldots,\phi_{r-1}$.
From similar analysis,
\begin{equation*}
\begin{split}
\mathsf{cr}(\phi_1)\le\text{ } & \chi^2(\tilde S_1)+ \chi({\mathcal{W}}'_1,\tilde S_1)+\chi({\mathcal{W}}'_1,{\mathcal{Q}}_1)+|{\mathcal{W}}^{\operatorname{bad}}_1|^2\\
& +\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_{1};\Sigma)+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_{1},{\mathcal{W}}^{\operatorname{bad}}_{1};\Sigma)+\sum_{e\in E(G)}(\chi_{\phi}(e)+1)\cdot (\cong_G({\mathcal{W}}^{\operatorname{light}}_{1},e))^2\\
& +\sum_{e\in E(G)} \cong_G({\mathcal{W}}^{\operatorname{light}}_{1},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{1},e)+\sum_{e,e'\in E(G)}
\chi_{\phi}(e,e')\cdot \cong_G({\mathcal{W}}^{\operatorname{light}}_{1},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{1},e).
\end{split}
\end{equation*}
Similarly, the drawing $\phi_r$ that we obtained in the same way satisfies that
\begin{equation*}
\begin{split}
\mathsf{cr}(\phi_r)\le \text{ } & \chi^2(\tilde S_r)+ \chi({\mathcal{W}}_{r-1},\tilde S_r)+\chi({\mathcal{W}}_{r-1},{\mathcal{Q}}_r)+|{\mathcal{W}}^{\operatorname{bad}}_{r-1}|^2\\
& +\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_{r-1};\Sigma)+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_{r-1},{\mathcal{W}}^{\operatorname{bad}}_{r-1};\Sigma)+\sum_{e\in E(G)}(\chi_{\phi}(e)+1)\cdot (\cong_G({\mathcal{W}}^{\operatorname{light}}_{r-1},e))^2\\
& +\sum_{e\in E(G)} \cong_G({\mathcal{W}}^{\operatorname{light}}_{r-1},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{r-1},e)+\sum_{e,e'\in E(G)}
\chi_{\phi}(e,e')\cdot \cong_G({\mathcal{W}}^{\operatorname{light}}_{r-1},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{r-1},e).
\end{split}
\end{equation*}
\iffalse{previous construction of phi_1}
We start with the drawing of $C_1\cup E({\mathcal{W}}_1)$ induced by $\phi^*$, that we denote by $\phi^*_1$. We will not modify the image of $S_i$ in $\phi^*_i$ and will construct the image of edges in $\delta(v_1^{\operatorname{right}})$.
We perform similar steps as in the construction of drawings $\phi_2,\ldots,\phi_{r-1}$.
We first construct, for each path $W\in {\mathcal{W}}_1$, a curve $\gamma_W$ connecting its endpoint in $C_1$ to the image of $u_2$ in $\phi^*$, as in Step 1.
Let $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
We then process all intersections between curves of $\Gamma_1$ as in Step 2. Let $\Gamma'_1=\set{\gamma'_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
Since $\Gamma^{\textsf{thr}}_1=\emptyset$, we do not need to perform Steps 3 and 4. If we view the image of $u_2$ in $\phi^*_1$ as the image of $v^{\operatorname{right}}_1$, and for each edge $e\in \delta(v^{\operatorname{right}}_1)$, we view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is clear that this drawing respects the rotation system $\Sigma_1$. Moreover,
\[\mathsf{cr}(\phi_1)=\chi^2(C_1)+O\textsf{left}(\hat\chi_1({\mathcal{Q}}_2)+\sum_{W\in {\mathcal{W}}_1}\hat\chi_1(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_1,e)^2\textsf{right}).\]
Similarly, the drawing $\phi_k$ that we obtained in the similar way satisfies that
\[\mathsf{cr}(\phi_k)=\chi^2(C_k)+O\textsf{left}(\hat\chi_k({\mathcal{Q}}_{r-1})+\sum_{W\in {\mathcal{W}}_k}\hat\chi_k(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_k,e)^2\textsf{right}).\]
\fi
We now complete the proof of \Cref{claim: existence of good solutions special}. It remains to estimate $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
We will upper-bound every term in the above formulas.
First note that $\sum_{1\le i\le r}\chi^2(\tilde S_i)\le O(\mathsf{cr}(\phi^*))$.
Then from \ref{prop_2} in \Cref{obs: accounting},
$$\sum_{1\le i\le r-1}|{\mathcal{W}}^{\operatorname{bad}}_i|^2\le O\big(\eta^*\cdot \log^{36}m\cdot (\mathsf{cr}(\phi^*)+|E(G)|)\big).$$
Next, from \ref{prop_1} in \Cref{obs: accounting}, we get that
$$\sum_{1\le i\le r}\expect{\chi({\mathcal{W}}'_i,S_i)+\chi({\mathcal{W}}_{i-1},S_i)} \le \sum_{1\le i\le r}\expect{\chi({\mathcal{W}},S_i)} \le \sum_{1\le i\le r}\beta^*\cdot \chi(S_i)\le O(\beta^*\cdot\mathsf{cr}(\phi^*));$$
and notice that paths in ${\mathcal{W}}_i$ are edge-disjoint and constructed independently from paths in ${\mathcal{W}}'_i$,
\begin{equation*}
\begin{split}
\sum_{1\le i\le r}\expect{\chi({\mathcal{W}}_i,{\mathcal{W}}'_i)} &\text{ }\le
\sum_{1\le i\le r}\sum_{e\in E(U_i)}\sum_{e'\in E(\overline{U}_i)}\chi(e,e')\cdot\expect[]{\cong_G({\mathcal{W}}_i,e)\cdot \cong_G({\mathcal{W}}'_i,e')}\\
&\text{ }\le
\sum_{1\le i\le r}\sum_{e\in E(U_i)}\sum_{e'\in E(\overline{U}_i)}\chi(e,e')\cdot\expect[]{\cong_G({\mathcal{W}}_i,e)}\cdot \expect[]{\cong_G({\mathcal{W}}'_i,e')}\\
&\text{ }\le \sum_{e,e'\in E(G)}\chi(e,e')\cdot\expect[]{\cong_G({\mathcal{W}},e)}\cdot \expect[]{\cong_G({\mathcal{W}},e')}\\
&\text{ } \le O((\beta^*)^2\cdot\mathsf{cr}(\phi^*)),
\end{split}
\end{equation*}
and similarly $\sum_{1\le i\le r}\expect{\chi({\mathcal{W}}_{i-1},{\mathcal{W}}'_i)}\le O((\beta^*)^2\cdot\mathsf{cr}(\phi^*))$.
Recall that, from \Cref{cor: NTcost_1} and \Cref{cor: NTcost_2}, we have
$$\expect[]{\sum_{1\le i\le r}\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_i;\Sigma)}\le O\big(\beta^*\cdot \log^{36}m\cdot \mathsf{cr}(\phi^*)\big).$$
and
$$\expect[]{\sum_{1\le i\le r}\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}^{\operatorname{light}}_i,{\mathcal{W}}^{\operatorname{bad}}_{i};\Sigma)}\le O\big(\beta^*\cdot \log^{36}m\cdot \mathsf{cr}(\phi^*)\big).$$
Next, from \ref{prop_3} in \Cref{obs: accounting}, we get that
\begin{equation*}
\begin{split}
&\expect[]{\sum_{\le i\le r}\sum_{e\in E(G)}(\chi(e)+1)\text{ }\cdot(\cong_G({\mathcal{W}}^{\operatorname{light}}_{i},e))^2}
\le \sum_{e\in E(G)}(\chi(e)+1)\cdot\expect[]{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2}\\
&\le \sum_{e\in E(G)}\chi(e)\cdot\expect[]{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2}
+ \sum_{e\in E(G)}\expect[]{(\cong_G({\mathcal{W}}^{\operatorname{light}},e))^2}\\
&\text{ }\le O(\log^{36}m\cdot\beta^*\cdot (\mathsf{cr}(\phi^*)+|E(G)|)).
\end{split}
\end{equation*}
Next, from \Cref{obs: accounting_2}, we get that
\begin{equation*}
\begin{split}
& \expect[]{\sum_{1\le i\le r}\sum_{e\in E(G)}\cong_G({\mathcal{W}}^{\operatorname{light}}_{i},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{bad}}_{i},e)}= \sum_{e\in E(G)}\expect[]{\sum_{1\le i\le r}\cong_G({\mathcal{W}}^{\operatorname{light}}_{i},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{i},e')}\\
& \text{ }
\le \sum_{e\in E(G)}\expect[]{\cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}},e')}\le O\bigg(\beta^*\cdot \log^{36}m\cdot |E(G)|\bigg).
\end{split}
\end{equation*}
Lastly, from \Cref{obs: accounting_2}, we get that
\begin{equation*}
\begin{split}
& \expect[]{\sum_{1\le i\le r}\sum_{e\in E(G)}\sum_{e'\in E(G)}\chi(e,e') \cdot\cong_G({\mathcal{W}}^{\operatorname{light}}_{i},e)\cdot \cong_G({\mathcal{W}}^{\operatorname{bad}}_{i},e')}\\
& \text{ }= \sum_{e\in E(G)}\sum_{e'\in E(G)}\chi(e,e')\cdot\expect[]{\sum_{1\le i\le r}\cong_G({\mathcal{W}}^{\operatorname{light}}_{i},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}}_{i},e')}\\
& \text{ }
\le \sum_{e\in E(G)}\sum_{e'\in E(G)}\chi(e,e')\cdot\expect[]{\cong_G({\mathcal{W}}^{\operatorname{light}},e)\cdot\cong_G({\mathcal{W}}^{\operatorname{bad}},e')}\le O\bigg(\beta^*\cdot \log^{36}m\cdot \mathsf{cr}(\phi^*)\bigg).
\end{split}
\end{equation*}
Altogether, they imply that
$$\sum_{1\le i\le r}\mathsf{cr}(\phi_i) \le O\textsf{left}(\bigg(\mathsf{cr}(\phi^*)+|E(G)|\bigg)\cdot \bigg(\eta^*+(\beta^*)^2\bigg)\cdot \log^{36}m\textsf{right}).$$
This completes the proof of \Cref{claim: existence of good solutions special}.
\iffalse
{backup from here}
Next, we if
if none of the bad events ${\cal{E}}_1,\ldots,{\cal{E}}_r$ happens, then for each $1\le i\le r$, $e\in E(G)$, $\expect[]{(\cong_G(\tilde{\mathcal{W}},e))^2}\le O(\beta)$.
for each edge $e\in E(G)$, $\expect[]{(\cong_G(\tilde{\mathcal{W}},e))^2}\le O(\beta)$. Therefore, on the one hand,
\[
\sum_{1\le i\le r}\sum_{e\in E(G)}\expect[]{\cong_G({\mathcal{W}}_i,e)^2+\cong_G({\mathcal{W}}'_i,e)^2}
\le \sum_{e\in E(G)} \expect[]{\cong_G(\tilde{\mathcal{W}},e)^2}\le O\big(|E(G)|\cdot\beta\big),
\]
and on the other hand,
\[
\begin{split}
\sum_{1\le i\le r}\expect{\chi^2({\mathcal{W}}_i)+\chi({\mathcal{W}}_i,{\mathcal{W}}'_i)} & = \text{ } \expect{\sum_{e,e'\in E(G)}\chi(e,e')\cdot\cong_{G}(\tilde{\mathcal{W}},e)\cdot\cong_{G}(\tilde{\mathcal{W}},e')}\\
& \text{ } \le \expect{\sum_{e,e'\in E(G)}\chi(e,e')\cdot\frac{\cong_{G}(\tilde{\mathcal{W}},e)^2+\cong_{G}(\tilde{\mathcal{W}},e')^2}{2}}\\
& \text{ } \le \expect{\sum_{e\in E(G)}\chi(e,G)\cdot\frac{\cong_{G}(\tilde{\mathcal{W}},e)^2}{2}}\le O(\beta\cdot \mathsf{cr}(\phi^*)).
\end{split}\]
From similar arguments, we can show that
$\sum_{1\le i\le r}\sum_{e\in E(S_i)}\expect{\cong_G({\mathcal{Q}}_i,e)^2} \le O(\beta\cdot |E(G)|)$ and
$\sum_{1\le i\le r}\expect{\chi^2({\mathcal{Q}}_i)} \le O(\beta\cdot\mathsf{cr}(\phi^*))$.
Lastly, we show that $\sum_{1\le i\le r-1}\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}_i)\le O(\beta\cdot \mathsf{cr}(\phi^*))$.
Denote by ${\mathcal{W}}=\set{W(e)\mid e\in \hat E}$ the set of all auxiliary cycles constructed in \Cref{sec: guiding and auxiliary paths}. Note that each path in $\bigcup_{1\le i\le r-1}{\mathcal{W}}_i$ is a subpath of a distinct cycle in ${\mathcal{W}}$. Therefore, $\sum_{1\le i\le r-1}\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}_i,\Sigma)\le\mathsf{cost}_{\mathsf {NT}}(\bigcup_{1\le i\le r-1}{\mathcal{W}}_i,\Sigma) \le \mathsf{cost}_{\mathsf {NT}}({\mathcal{W}},\Sigma)$.
Then from \Cref{obs: edge_occupation in outer and inner paths}, for every edge $e\in E(G')$, $\expect[]{(\cong_{G'}({\mathcal{W}},e))^2}\le O(\beta\cdot (\cong_{G'}({\mathcal{P}}))^2)$. Therefore, from \Cref{lem: non transversal cost of cycles bounded by cr}, we get that
\[\expect[]{\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}},\Sigma)}\le \mathsf{cr}(\phi^*)\cdot O(\beta \cdot (\cong_{G'}({\mathcal{P}}))^2)=\mathsf{cr}(\phi^*)\cdot 2^{O((\log m)^{3/4}\log\log m)}.\]
Altogether, they imply that $\sum_{1\le i\le r}\mathsf{cr}(\phi_i) \le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot 2^{O((\log m)^{3/4}\log\log m)})$.
This completes the proof of \Cref{claim: existence of good solutions special}.
\fi
\iffalse{previous bad-chain centered setting assumptions}
\paragraph{Assumption 1: The Gomory-Hu tree $\tau$ of graph $H$ is a path.}
Let $H=G_{\mid{\mathcal{S}}}$ be the contracted graph.
We denote the clusters in ${\mathcal{S}}$ by $C_1,C_2,\ldots,C_r$. For convenience, for each $1\leq i\leq r$, we denote by $x_i$ the vertex of graph $H$ that represents the cluster $S'_i$. Throughout this subsection, we assume that the Gomory-Hu tree $\tau$ of graph $H$ is a path, and we assume without loss of generality that the clusters are indexed according to their appearance on the path $\tau$. Note that each edge in $E^{\textnormal{\textsf{out}}}({\mathcal{S}})$ corresponds to an edge in $H$, and we do not distinguish between them.
\paragraph{Assumption 2: Bad Chains of Single-Vertex Clusters.}
We allow some clusters of ${\mathcal{S}}\setminus\set{C_1,C_r}$ to contain only a single vertex. Let indices $1< i\le j< n$ be such that clusters $S'_{i-1},S'_{j+1}$ are not single-vertex clusters, while all clusters $S'_{i},\ldots,S'_{j}$ are single-vertex clusters.
We call the union of clusters $S'_{i},\ldots,S'_{j}$ a \emph{bad chain}, and we denote $\mathsf{BC}[i,j]=\bigcup_{i\le t\le j}S'_t$. We call index $i$ the \emph{left endpoint} and index $j$ the \emph{right endpoint} of the bad chain.
Consider the graph $H=G_{\mid{\mathcal{S}}}$. Note that each bad chain $\mathsf{BC}[i,j]$ belongs to graph $H$. Consider a bad chain $\mathsf{BC}[i,j]$, we say that a path $P$ is contained in the bad chain, iff $V(P)\subseteq \mathsf{BC}[i,j]$.
Let $x_s, x_t$ be two vertices (with $s<t$) that do not belong to any bad chains, and let $\mathsf{BC}[i_1,j_1],\ldots,\mathsf{BC}[i_k,j_k]$ be all bad chains whose endpoints lie in $\set{s+1,\ldots,t-1}$. Assume without loss of generality that $i_1\le j_1< i_2\le j_2 <\ldots <i_k\le j_k$.
We say that a path connecting $x_{s}$ to $x_{t}$ is \emph{nice}, iff it is the sequential concatenation of: a path that sequentially visits vertices $x_s,x_{s+1}\ldots,x_{i_1}$, a path connecting $x_{i_1}$ to $x_{j_1}$ that is contained in the bad chain $\mathsf{BC}[i_1,j_1]$, a path that sequentially visits vertices $x_{j_1},x_{j_1+1}\ldots,x_{i_2}$, a path connecting $x_{i_2}$ to $x_{j_2}$ that is contained in the bad chain $\mathsf{BC}[i_2,j_2]$, \ldots , and a path that sequentially visits vertices $x_{j_k},x_{j_k+1}\ldots,x_{t}$.
See \Cref{fig: nice path} for an illustration.
If vertex $x_s$ belongs to some bad chain $\mathsf{BC}[i_0,j_0]$, then for an $x_s$-$x_t$ path to be nice, the path must start with a path connecting $x_s$ to $x_{j_0}$ that is contained in the bad chain $\mathsf{BC}[i_0,j_0]$.
Similarly, if vertex $x_t$ belongs to some bad chain $\mathsf{BC}[i_{k+1},j_{k+1}]$, then a nice $x_s$-$x_t$ path must end with a path connecting $x_{i_{k+1}}$ to $x_{t}$ that is contained in the bad chain $\mathsf{BC}[i_{k+1},j_{k+1}]$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.23]{figs/innerpath.jpg}
\caption{An illustration of a nice path connecting $x_s$ to $x_t$. Bad chains are shown in pink.}\label{fig: nice path}
\end{figure}
\paragraph{Assumption 3: Auxiliary Paths.}
Let $E'$ be the subset of edges in $E^{\mathsf{out}}({\mathcal{S}})$ that contains all edges connecting a pair $S'_i,S'_j$ of clusters in ${\mathcal{S}}$, with $j\ge i+2$, and when $E'$ is viewed as a set of edges in $H$, then $E'$ contains all edges connecting a pair $x_i,x_j$ of vertices in $H$ with $j\ge i+2$.
We additionally assume that, for each edge $e\in E'$ that, when viewed as an edge in $H$, connects $x_i$ to $x_j$, we are given a nice path $P_e$ in $H$ connecting $x_i$ to $x_j$, that we call the \emph{auxiliary path} of $e$.
Additionally, all paths in $\set{P_e}_{e\in E'}$ cause congestion $O(1)$ in graph $H$.
We now start to describe the algorithm in the following subsections.
\subsubsection{Computing Internal Routers and Auxiliary Cycles}
\label{sec: guiding and auxiliary paths}
We first apply the algorithm from \Cref{thm:algclassifycluster} to each non-single-vertex cluster of ${\mathcal{S}}$.
In particular, for each non-single-vertex cluster $S'_i$, we apply the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace to the instance $(G,\Sigma)$ and the cluster $S'_i$ (recall that cluster $S'_i$ has the $\alpha_0$-bandwidth property, where $\alpha_0=1/\log^3m$) for $100\log n$ times. If any application of \ensuremath{\mathsf{AlgClassifyCluster}}\xspace returns a distribution ${\mathcal{D}}(S'_i)$ over the set of internal $S'_i$-routers, such that $S'_i$ is $\beta^*$-light with respect to ${\mathcal{D}}(S'_i)$, then we continue to sample a set of guiding paths from ${\mathcal{D}}(S'_i)$. Let ${\mathcal{Q}}_i$ be the set we obtain, and let $u_i$ be the vertex of $S'_i$ that serves as the common endpoint of all paths in ${\mathcal{Q}}_i$. If all applications of \ensuremath{\mathsf{AlgClassifyCluster}}\xspace return FAIL, then consider the graph $S'_i^+$ (see \Cref{def: Graph C^+}). We apply the algorithm from \Cref{lem: simple guiding paths} to cluster $S^+_i$ and and set $T(S'_i)$, and let $u_i$ be the vertex and ${\mathcal{Q}}_i$ be the set of paths that we obtain. Clearly, ${\mathcal{Q}}_i$ can be also viewed as a set of paths routing edges of $\delta_G(S'_i)$ to $u_i$. From \Cref{lem: simple guiding paths}, for each edge $e\in E(S'_i)$, $\expect{\cong({\mathcal{Q}}_i,e)}\leq O(\log^4m/\alpha_0)$. Finally, for each $1\le i\le r$ and for each $e\in \delta(S'_i)$, we denote by $Q_i(e)$ the path of ${\mathcal{Q}}_i$ that routes $e$ to $u_i$.
We then apply the algorithm from \Cref{lem: non_interfering_paths} to each set ${\mathcal{Q}}_i$ of paths and the rotation system $\Sigma$, and rename the sets of paths we obtain as ${\mathcal{Q}}_1,\ldots,{\mathcal{Q}}_r$. Now we are guaranteed that, for each $1\le i\le r$, the set ${\mathcal{Q}}_i$ of paths is non-transversal with respect to $\Sigma$.
\textbf{Bad Event $\xi_i$.} For each index $1\le i\le r$, we say that the event $\xi_i$ happens, iff $S'_i$ is a non-single-vertex cluster, $S'_i$ is not an $\eta^*$-bad cluster, and all the $100\log n$ applications of \ensuremath{\mathsf{AlgClassifyCluster}}\xspace return FAIL.
From \Cref{thm:algclassifycluster}, $\Pr[\xi_i]\le (1/2)^{100\log n}=O(n^{-50})$. Then from the union bound over all $1\le i\le r$, $\Pr[\bigcup_{1\le i\le r}\xi_i]\le O(n^{-49})$.
We then compute a set ${\mathcal{R}}^*$ of cycles in $G$, using the set $\set{P_e}_{e\in E'}$ of inner paths and the path sets $\set{{\mathcal{Q}}_i}_{1\le i\le r}$, as follows.
Consider an edge $e\in E'$ and its inner path $P_e$ in $H$.
We denote $P_e=(e_1,\ldots,e_k)$, and for each $1\le j\le k$, we denote $e_j=(x_{t_{j-1}}, x_{t_{j}})$ (and so $e=(x_{t_0},x_{t_k})$ as an edge in $H$).
Recall that the set ${\mathcal{Q}}_i$ contains, for each edge $e\in\delta(S'_i)$, a path $Q_i(e)$ routing edge $e$ to $u_i$.
We then define path $P^*_e$ as the sequential concatenation of paths $Q_{t_{0}}(e_1),Q_{t_{1}}(e_1), Q_{t_{1}}(e_2),Q_{t_{2}}(e_2),\ldots, Q_{t_{r-1}}(e_k),Q_{t_{r}}(e_k)$.
It is clear that path $P^*_e$ sequentially visits the vertices $u_{t_0},u_{t_1},\ldots,u_{t_k}$ in $G$.
Since edge $e$, as an edge of $G$, connects a vertex of $S'_{t_0}$ to a vertex of $S'_{t_k}$. We then define path $P^{**}_e$ to be the union of paths $Q_{t_{k}}(e)$ and $Q_{t_{0}}(e)$, so path $P^{**}_e$ connects $u_{t_k}$ to $u_{t_0}$.
Finally, we define cycle $R^*_e$ to be the union of paths $P^*_e$ and paths $P^{**}_e$, and we denote ${\mathcal{R}}^*=\set{R^*_e\mid e\in E'}$.
\iffalse
\begin{figure}[h]
\centering
\includegraphics[scale=0.24]{figs/auxiliary_path.jpg}
\caption{An illustration of the auxiliary cycle $R^*_e$ of an edge $e$ (solid red): left auxiliary path shown in orange, middle auxiliary path shown in purple and right auxiliary path shown in green.}\label{fig: LMR_auxi}
\end{figure}
\fi
Recall that the set $\set{P_e}_{e\in E'}$ of inner paths is guaranteed to cause congestion at most $O(1)$ in graph $H$, then from the construction of cycles in ${\mathcal{R}}^*$, it is easy to see that, for each edge $e\in E'$, $\cong_{G}({\mathcal{R}}^*,e)\le O(1)$, and
for each edge $e\in \bigcup_{1\le i\le r}E(S'_i)$, $\expect[]{\cong_{G}({\mathcal{R}}^*,e)^2}\le O(\beta^*)$.
\iffalse{should be moved to the next subsection}
We further modify the cycles in ${\mathcal{R}}^*$ to obtain a new set ${\mathcal{R}}$ of cycles, such that the intersection of every pair of cycles in ${\mathcal{R}}$ is non-transversal with respect to $\Sigma$ at no more than one of their shared vertices.
We will use the following lemma.
\znote{To complete the following lemma}
\begin{lemma}[De-interfering]
There is an efficient algorithm, that given a graph $G$, a rotation system $\Sigma$, and a set ${\mathcal{P}}$ of paths that share an endpoint $u$, computes a set ${\mathcal{P}}'=\set{P'\mid P\in {\mathcal{P}}}$ of paths, such that
\end{lemma}
\begin{proof}
Let $R,R'$ be a pair of cycles in ${\mathcal{R}}^*$ and let $v$ be a shared vertex of $R$ and $R'$. We say that the tuple $(R,R',v)$ is \emph{bad}, iff the intersection of cycles $R,R'$ at $v$ is transversal with respect to $\Sigma$. We iteratively process the set ${\mathcal{R}}^*$ of cycles as follows. While there exist a pair $R,R'$ of cycles in ${\mathcal{R}}^*$ and two vertices $v_1,v_2\in V(R)\cap V(R')$, such that the tuples $(R,R',v)$ and $(R,R',v)$ are bad, we process cycles $R,R'$ as follows.
Eventually, for each $e\in E'$, we obtain a cycle $R_e$ in $G$ that is called the \emph{auxiliary cycle} of edge $e$.
\end{proof}
\fi
\znote{Todo: de-interfere auxiliary cycles to get cycles $\set{R_e}_{e\in E'}$.}
\subsubsection{Constructing Sub-Instances}
\label{sec: compute advanced disengagement}
In this step we will construct, for each non-single vertex cluster $S'_i\in {\mathcal{S}}$, a sub-instance $I_i=(G_i,\Sigma_i)$ of $(G,\Sigma)$, and for each bad chain $\mathsf{BC}[i,j]$, a sub-instance $I_{\mathsf{BC}[i,j]}$ of $(G,\Sigma)$, such that the set of all created sub-instances satisfy the properties in \Cref{thm: disengagement - main}.
For each index $1\le i\le r$, we define edge sets
$\hat E_i=E(S'_i,S'_{i+1})$, $E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(S'_i,S'_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(S'_i,S'_{j'})$, and $E^{\textsf{thr}}_i=\bigcup_{i'\le i-1,j'\ge i+1}E(S'_{i'},S'_{j'})$.
Let $\mathsf{BC}[i,j]$ be a bad chain. Similarly we define edge sets
$E_{[i,j]}^{\operatorname{right}}=\bigcup_{k>j+1}E(\mathsf{BC}[i,j],C_k)$, $E_{[i,j]}^{\operatorname{left}}=\bigcup_{k<i-1}E(\mathsf{BC}[i,j],C_k)$,
$\hat E_{[i,j]}^{\operatorname{right}}=E(\mathsf{BC}[i,j],S'_{j+1})$, $\hat E_{[i,j]}^{\operatorname{left}}=E(\mathsf{BC}[i,j],S'_{i-1})$, and $E^{\textsf{thr}}_{[i,j]}=\bigcup_{i'\le i-1,j'\ge j+1}E(S'_{i'},S'_{j'})$.
\paragraph{Cluster-based Instance $I_i=(G_i,\Sigma_i)$ ($2\le i\le r$).}
Let $S'_i$ be some non-single-vertex cluster with $2\le i\le r$.
We define the instance $I_i=(G_i,\Sigma_i)$ as follows.
The graph $G_i$ is obtained from $G$ by first contracting clusters $C_1,\ldots,S'_{i-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_i$, and then contracting clusters $S'_{i+1},\ldots,C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_i$, and finally deleting self-loops on the super-nodes $v^{\operatorname{left}}_i$ and $v^{\operatorname{right}}_i$. So $V(G_i)=V(S'_i)\cup \set{ v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$. See \Cref{fig: disengaged instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edge sets in $G$. Edges of $E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}$ are shown in green, and edges of $E^{\textsf{thr}}_{i}$ are shown in red. ]{\scalebox{0.35}{\includegraphics{figs/disengaged_instance_1.jpg}
}
\hspace{5pt}
\subfigure[Graph $G_i$. $\delta_{G_i}(v_i^{\operatorname{right}})=\hat E_i \cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, and $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1} \cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_{i}$.]{
\scalebox{0.28}{\includegraphics{figs/disengaged_instance_2.jpg}}}
\caption{An illustration of the construction of sub-instance $(G_i,\Sigma_i)$.}\label{fig: disengaged instance}
\end{figure}
We now define the orderings in $\Sigma_i$. First, for each vertex $v\in V(S'_i)$, the ordering on its incident edges is defined to be ${\mathcal{O}}_v$, the rotation on vertex $v$ in the given rotation system $\Sigma$.
It remains to define the rotations of super-nodes $v^{\operatorname{left}}_i,v^{\operatorname{right}}_i$.
We first consider $v^{\operatorname{left}}_i$.
Note that $\delta_{G_i}(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$.
For each edge $\hat e\in \hat E_{i-1}$, recall that $Q_{i-1}(\hat e)$ is the path in $S'_{i-1}$ routing edge $\hat e$ to $u_{i-1}$.
For each edge $e\in E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i$, we denote by $W^{\operatorname{left}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i-1}$ and does not contain $u_i$.
We then denote
$${\mathcal{W}}^{\operatorname{left}}_i=\set{W^{\operatorname{left}}_i(e)\mid e\in E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i}\cup \set{Q_{i-1}(\hat e)\mid \hat e\in \hat E_{i-1}}.$$
Intuitively, the rotation on vertex $v^{\operatorname{left}}_i$ is defined to be the ordering in which the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ enter $u_{i-1}$.
Formally, for every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$, let $e^*_W$ be the unique edge of path $W$ that is incident to $u_{i-1}$. We first define a circular ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$, as follows: the paths are ordered according to the circular ordering of their corresponding edges $e^*_W$ in ${\mathcal{O}}_{u_{i-1}}\in \Sigma$, breaking ties arbitrarily. Since every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$ is associated with a unique edge in $\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$, this ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ immediately defines a circular ordering of the edges of $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$, that we denote by ${\mathcal{O}}^{\operatorname{left}}_i$. See Figure~\ref{fig: v_left rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i-1}=\set{\hat e_1,\ldots,\hat e_4}$, $E^{\operatorname{left}}_{i}=\set{e^g_1,e^g_2}$ and $E^{\textsf{thr}}_{i}=\set{e^r_1,e^r_2}$.
Paths of ${\mathcal{W}}^{\operatorname{left}}_i$ excluding their first edges are shown in dash lines.]{\scalebox{0.13}{\includegraphics{figs/rotation_left_1.jpg}}}
\hspace{3pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{left}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. Set $\delta_{G_i}(v^{\operatorname{left}}_i)=\set{\hat e_1,\hat e_2,\hat e_3,\hat e_4,e^g_1, e^g_2, e^r_1,e^r_2}$. The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on them around $v^{\operatorname{left}}_i$ is shown above.]{
\scalebox{0.16}{\includegraphics{figs/rotation_left_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_left rotation}
\end{figure}
The rotation ${\mathcal{O}}^{\operatorname{right}}_{i}$ on vertex $v^{\operatorname{right}}_i$ is defined similarly.
Note that $\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$.
For each edge $\hat e'\in \hat E_{i}\cup E_i^{\operatorname{right}}$, recall that $Q_{i}(\hat e')$ is the path in $S'_{i}$ routing edge $\hat e'$ to vertex $u_{i}$.
For each edge $e\in E^{\textsf{thr}}_i$, we denote by $W^{\operatorname{right}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i}$ and contains $u_{i-1}$.
We then denote
$${\mathcal{W}}^{\operatorname{right}}_i=\set{W^{\operatorname{right}}_i(e)\text{ }\big|\text{ } e\in E^{\textsf{thr}}_i}\cup \set{Q_{i}(\hat e')\text{ }\big|\text{ } \hat e'\in \hat E_{i}\cup E^{\operatorname{right}}_i}.$$
The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ is then defined in a similar way as the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$, according to the paths of ${\mathcal{W}}^{\operatorname{right}}_i$ and the rotation ${\mathcal{O}}_{u_{i}}\in \Sigma$.
See Figure~\ref{fig: v_right rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i}=\set{\hat e_1',\ldots,\hat e_4'}$, $E^{\operatorname{right}}_{i}=\set{\tilde e^g_1,\tilde e^g_2}$ and $E^{\textsf{thr}}_{i}=\set{\tilde e^r_1,\tilde e^r_2}$.
Paths of ${\mathcal{W}}^{\operatorname{right}}_i$ excluding their first edges are shown in dash lines. ]{\scalebox{0.13}{\includegraphics{figs/rotation_right_1.jpg}
}
\hspace{3pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. Set $\delta_{G_i}(v^{\operatorname{right}}_i)=\set{\hat e_1',\hat e_2',\hat e_3',\hat e_4',\tilde e^g_1,\tilde e^g_2,\tilde e^r_1,\tilde e^r_2}$. The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on them around $v^{\operatorname{right}}_i$ is shown above.]{
\scalebox{0.17}{\includegraphics{figs/rotation_right_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_right rotation}
\end{figure}
\paragraph{Instances $I_1=(G_1,\Sigma_1)$ and $I_r=(G_r,\Sigma_r)$.}
The instances $(G_1,\Sigma_1)$ and $(G_r,\Sigma_r)$ are defined similarly to a cluster-centered instance, but instead of two super-nodes, the graphs $G_1$ and $G_r$ contain one super-node each.
In particular, graph $G_1$ is obtained from $G$ by contracting clusters $C_2,\ldots, C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_1$, and then deleting self-loops on it.
So $V(G_1)=V(C_1)\cup \set{v^{\operatorname{right}}_{1}}$ and $\delta_{G_1}(v^{\operatorname{right}}_{1})=\hat E_1\cup E^{\operatorname{right}}_1$.
The rotation of a vertex $v\in V(C_1)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{right}}_1$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{right}}_i$ for any index $2\le i\le r-1$.
Graph $G_r$ is obtained from $G$ by contracting clusters $C_1,\ldots, S'_{r-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_r$, and then deleting self-loops on it.
So $V(G_r)=V(C_r)\cup \set{v^{\operatorname{left}}_{r}}$ and $\delta_{G_r}(v^{\operatorname{left}}_{r})=\hat E_{r-1}\cup E^{\operatorname{left}}_r$.
The rotation of a vertex $v\in V(C_r)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{left}}_r$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{left}}_i$ for any index $2\le i\le r-1$.
\paragraph{Bad Chain-based Instance $I_{\mathsf{BC}[i,j]}$.} Let $\mathsf{BC}[i,j]$ be a bad chain. We define the instance $I_{\mathsf{BC}[i,j]}=(G_{[i,j]},\Sigma_{[i,j]})$ as follows.
The graph $G_{[i,j]}$ is obtained from $G$ by first contracting clusters $C_1,\ldots,S'_{i-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_{[i,j]}$, and then contracting clusters $S'_{j+1},\ldots,C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_{[i,j]}$, and finally deleting self-loops on the super-nodes $v^{\operatorname{left}}_{[i,j]}$ and $v^{\operatorname{right}}_{[i,j]}$. So $V(G_i)=\mathsf{BC}[i,j]\cup \set{v^{\operatorname{left}}_{[i,j]},v^{\operatorname{right}}_{[i,j]}}$. See \Cref{fig: bad instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Edges of $\hat E^{\operatorname{left}}_{[i,j]}\cup \hat E^{\operatorname{right}}_{[i,j]}$ are in orange, edges of $E^{\operatorname{left}}_{[i,j]}$ and $E^{\operatorname{right}}_{[i,j]}$ are in green and blue respectively and edges of $E^{\textsf{thr}}_{[i,j]}$ are in red. ]{\scalebox{0.12}{\includegraphics{figs/badchain_instance_1.jpg}
}
\hspace{0pt}
\subfigure[Graph $G_{[i,j]}$. $\delta(v_{[i,j]}^{\operatorname{right}})=\hat E_{[i,j]}^{\operatorname{right}} \cup E_{[i,j]}^{\operatorname{right}} \cup E^{\textsf{thr}}_{[i,j]}$, and $\delta(v_{[i,j]}^{\operatorname{left}})=\hat E_{[i,j]}^{\operatorname{left}} \cup E_{[i,j]}^{\operatorname{left}} \cup E^{\textsf{thr}}_{[i,j]}$.]{
\scalebox{0.14}{\includegraphics{figs/badchain_instance_2.jpg}}}
\caption{An illustration of the construction of sub-instance $(G_{[i,j]},\Sigma_{[i,j]})$.}\label{fig: bad instance}
\end{figure}
We now define the orderings in $\Sigma_i$. First, for each vertex $v\in \mathsf{BC}[i,j]$, the ordering on its incident edges is defined to be ${\mathcal{O}}_v$, the rotation on vertex $v$ in the given rotation system $\Sigma$.
It remains to define the rotations of super-nodes $v^{\operatorname{left}}_{[i,j]},v^{\operatorname{right}}_{[i,j]}$.
We first consider $v^{\operatorname{left}}_{[i,j]}$.
Note that $\delta_{G_i}(v^{\operatorname{left}}_{[i,j]})=\hat E^{\operatorname{left}}_{[i,j]}\cup E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$.
For each edge $\hat e\in \hat E^{\operatorname{left}}_{[i,j]}$, recall that $Q_{i-1}(\hat e)$ is the path in $S'_{i-1}$ routing edge $\hat e$ to $u_{i-1}$.
For each edge $e\in E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$, we denote by $W^{\operatorname{left}}_{[i,j]}(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i-1}$ and does not contain $u_i$.
We then denote
$${\mathcal{W}}^{\operatorname{left}}_i=\set{W^{\operatorname{left}}_i(e)\mid e\in E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}}\cup \set{Q_{i-1}(\hat e)\mid \hat e\in \hat E^{\operatorname{left}}_{[i,j]}}.$$
Intuitively, the rotation on vertex $v^{\operatorname{left}}_{[i,j]}$ is defined to be the ordering in which the paths in ${\mathcal{W}}^{\operatorname{left}}_{[i,j]}$ enter $u_{i-1}$.
Formally, for every path $W\in {\mathcal{W}}^{\operatorname{left}}_{[i,j]}$, let $e^*_W$ be the unique edge of path $W$ that is incident to $u_{i-1}$. We first define a circular ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_{[i,j]}$, as follows: the paths are ordered according to the circular ordering of their corresponding edges $e^*_W$ in ${\mathcal{O}}_{u_{i-1}}\in \Sigma$, breaking ties arbitrarily. Since every path $W\in {\mathcal{W}}^{\operatorname{left}}_{[i,j]}$ is associated with a unique edge in $\hat E^{\operatorname{left}}_{[i,j]}\cup E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$, this ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_{[i,j]}$ immediately defines a circular ordering of the edges of $\delta_{G_{[i,j]}}(v_{[i,j]}^{\operatorname{left}})=\hat E^{\operatorname{left}}_{[i,j]}\cup E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$, that we denote by ${\mathcal{O}}^{\operatorname{left}}_{[i,j]}$. See Figure~\ref{fig: bad_v_left rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[$\hat E^{\operatorname{left}}_{[i,j]}=\set{\hat e_1,\hat e_2}$, $E^{\operatorname{left}}_{[i,j]}=\set{e^g_1}$ and $E^{\textsf{thr}}_{[i,j]}=\set{e^r_1}$.
Paths of ${\mathcal{W}}^{\operatorname{left}}_{[i,j]}$ excluding their first edges are shown in dash lines.]{\scalebox{0.14}{\includegraphics{figs/badinstance_rotation_left_1.jpg}}}
\hspace{3pt}
\subfigure[$\delta(v^{\operatorname{left}}_{[i,j]})=\set{\hat e_1,\hat e_2,e^g_1,e^r_1}$. The rotation ${\mathcal{O}}^{\operatorname{left}}_{[i,j]}$ on them around $v^{\operatorname{left}}_{[i,j]}$ is shown above.]{
\scalebox{0.18}{\includegraphics{figs/badinstance_rotation_left_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{left}}_{[i,j]}$ on vertex $v^{\operatorname{left}}_{[i,j]}$ in the instance $(G_{[i,j]}, \Sigma_{[i,j]})$.}\label{fig: bad_v_left rotation}
\end{figure}
The rotation ${\mathcal{O}}^{\operatorname{right}}_{[i,j]}$ on vertex $v^{\operatorname{right}}_{[i,j]}$ is defined similarly.
Note that $\delta_{G_i}(v^{\operatorname{right}}_{[i,j]})=\hat E^{\operatorname{right}}_{[i,j]}\cup E^{\operatorname{right}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$.
For each edge $\hat e'\in \delta_{G_i}(v^{\operatorname{right}}_{[i,j]})$, we denote by $W^{\operatorname{right}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{j}$ and does not contains $u_{j+1}$.
We then denote
$${\mathcal{W}}^{\operatorname{right}}_i=\set{W^{\operatorname{right}}_i(e)\text{ }\big|\text{ } e\in \hat E^{\operatorname{right}}_{[i,j]}\cup E^{\operatorname{right}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}}.$$
The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ is then defined in a similar way as the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$, according to the paths of ${\mathcal{W}}^{\operatorname{right}}_i$ and the rotation ${\mathcal{O}}_{u_{i}}\in \Sigma$.
See Figure~\ref{fig: bad v_right rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[$\hat E^{\operatorname{right}}_{[i,j]}=\set{\hat e'_1,\hat e'_2}$, $E^{\operatorname{right}}_{[i,j]}=\set{e^b_1}$ and $E^{\textsf{thr}}_{[i,j]}=\set{e^r_1}$.
Paths of ${\mathcal{W}}^{\operatorname{right}}_{[i,j]}$ excluding their first edges are shown in dash lines. ]{\scalebox{0.14}{\includegraphics{figs/badinstance_rotation_right_1.jpg}
}
\hspace{3pt}
\subfigure[$\delta(v^{\operatorname{right}}_i)=\set{\hat e'_1,\hat e'_2,e^b_1,e^r_1}$. The rotation ${\mathcal{O}}^{\operatorname{right}}_{[i,j]}$ on them around $v^{\operatorname{right}}_{[i,j]}$ is shown above.]{
\scalebox{0.18}{\includegraphics{figs/badinstance_rotation_right_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{right}}_{[i,j]}$ on vertex $v^{\operatorname{right}}_{[i,j]}$ in the instance $(G_{[i,j]}, \Sigma_{[i,j]})$.}\label{fig: bad v_right rotation}
\end{figure}
We will use the following claims later for completing the proof of \Cref{thm: disengagement - main} in the special case.
\znote{the following two observations to modify}
\begin{observation}
\label{obs: disengaged instance size}
The total number of edges in all sub-instances we have defined is $O(|E(G)|)$.
\end{observation}
\begin{proof}
\iffalse
Note that, in the sub-instances $\set{(G_i,\Sigma_i)}_{1\le i\le r}$, each graph of $\set{G_i}_{1\le i\le r}$ is obtained from $G$ by contracting some sets of clusters of ${\mathcal{S}}$ into a single super-node, so each edge of $G_i$ corresponds to an edge in $E(G)$.
Therefore, for each $1\le i\le r$,
$$|E(G_i)|=|E_G(S'_i)|+|\delta_G(S'_i)|\le |E_G(S'_i)|+|E^{\textsf{out}}({\mathcal{S}})|\le m/(100\mu)+m/(100\mu)\le m/\mu.$$
\fi
%
Note that $E(G_i)=E(S'_i)\cup \hat E_i\cup \hat E_{i-1}\cup E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$.
First, each edge of $\bigcup_{1\le i\le r}E(S'_i)$ appears in exactly one graphs of $\set{G_i}_{1\le i\le r}$.
Second, each edge of $\bigcup_{1\le i\le r}\hat E_i$ appears in exactly two graphs of $\set{G_i}_{1\le i\le r}$.
Consider now an edge $e\in E'$. If $e$ connects a vertex of $S'_i$ to a vertex of $S'_j$ for some $j\ge i+2$, then $e$ will appear as an edge in $E_i^{\operatorname{right}}\subseteq E(G_i)$ and an edge in $E_j^{\operatorname{left}}\subseteq E(G_j)$, and it will appear as an edge in $E_k^{\textsf{thr}}\subseteq E(G_k)$ for all $i<k<j$.
On one hand, we have $\sum_{1\le i\le r}|E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}|\le 2|E(G)|$.
On the other hand, note that $E_k^{\textsf{thr}}=E(S'_{k-1},S'_{k+1})\cup E^{\operatorname{over}}_{k-1}\cup E^{\operatorname{over}}_{k}$, and each edge of $e$ appears in at most two graphs of $\set{G_i}_{1\le i\le r}$ as an edge of $E(S'_{k-1},S'_{k+1})$.
Moreover, from \Cref{obs: bad inded structure}, $|E^{\operatorname{over}}_{k-1}|\le |\hat E_{k-1}|$ and $|E^{\operatorname{over}}_{k}|\le |\hat E_{k}|$.
Altogether, we have
\begin{equation}
\begin{split}
\sum_{1\le i\le r}|E(G_i)| & =
\sum_{1\le i\le r}\textsf{left}(
|E(S'_i)|+ |\hat E_i|+ |\hat E_{i-1}|+ |E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|+|E_i^{\textsf{thr}}|
\textsf{right})\\
& = \sum_{1\le i\le r}
|E(S'_i)|+ \sum_{1\le i\le r} \textsf{left}(|E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|\textsf{right})+ \sum_{1\le i\le r}\textsf{left}(|E_i^{\textsf{thr}}|+ |\hat E_i|+ |\hat E_{i-1}|\textsf{right})\\
& \le |E(G)|+ 2\cdot |E(G)| + \sum_{1\le i\le r}\textsf{left}(|E(S'_{i-1},S'_{i+1})|+ 2|\hat E_i|+ 2|\hat E_{i-1}|\textsf{right})\\
& \le 8\cdot |E(G)|.
\end{split}
\end{equation}
This completes the proof of \Cref{obs: disengaged instance size}.
\end{proof}
\begin{observation}
\label{obs: rotation for stitching}
For each $1\le i\le r-1$, if we view the edge in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$ as edges of $E(G)$, then $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$, and moreover, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{observation}
\begin{proof}
Recall that for each $1\le i\le r-1$,
$\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})=\hat E_{i}\cup E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}$. From the definition of sets $E_i^{\textsf{thr}},E_{i+1}^{\textsf{thr}}, E^{\operatorname{right}}_i, E^{\operatorname{left}}_{i+1}$,
\[
\begin{split}
E_i^{\textsf{thr}}\cup E^{\operatorname{right}}_i
= & \set{e\in E(S'_{i'},S'_{j'})\mid i'<i<j'\text{ or }i'=i<j'}\\
= & \set{e\in E(S'_{i'},S'_{j'})\mid i'\le i<j'}\\
= & \set{e\in E(S'_{i'},S'_{j'})\mid i'< i+1\le j'}\\
= & \set{e\in E(S'_{i'},S'_{j'})\mid i'<i+1<j'\text{ or }i'<i+1=j'}=E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}.
\end{split}
\]
Therefore, $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Moreover, from the definition of path sets ${\mathcal{W}}^{\operatorname{right}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_{i+1}$, it is not hard to verify that, for every edge $e\in \delta_{G_i}(v^{\operatorname{right}}_i)$, the path in ${\mathcal{W}}^{\operatorname{right}}_i$ that contains $e$ as its first edge is identical to the path in ${\mathcal{W}}^{\operatorname{left}}_{i+1}$ that contains $e$ as its first edge. According to the way that rotations ${\mathcal{O}}^{\operatorname{right}}_i,{\mathcal{O}}^{\operatorname{left}}_{i+1}$ are defined, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{proof}
\fi
\iffalse{backup: original analysis of total cost of subinstances}
Specifically, we use the following two claims, whose proofs will be provided later.
\begin{claim}
\label{claim: existence of good solutions special}
$\expect{\sum_{1\le i\le r}\mathsf{OPT}_{\mathsf{cnwrs}}(G_i,\Sigma_i)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$.
\end{claim}
\begin{claim}
\label{claim: stitching the drawings together}
There is an efficient algorithm, that given, for each $1\le i\le r$, a feasible solution $\phi_i$ to the instance $(G_i,\Sigma_i)$, computes a solution to the instance $(G,\Sigma)$, such that $\mathsf{cr}(\phi)\le \sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
\end{claim}
We use the algorithm described in~\Cref{sec: guiding and auxiliary paths} and~\Cref{sec: compute advanced disengagement}, and return the disengaged instances $(G_1,\Sigma_1),\ldots,(G_r,\Sigma_r)$ as the collection of sub-instances of $(G,\Sigma)$.
From the previous subsections, the algorithm for producing the sub-instances is efficient.
On the other hand, it follows immediately from \Cref{obs: disengaged instance size}, \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together} that the sub-instances $(G_1,\Sigma_1),\ldots,(G_r,\Sigma_r)$ satisfy the properties in \Cref{thm: disengagement - main}.
This completes the proof of \Cref{thm: disengagement - main}.
We now provide the proofs of \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together}.
$\ $
\begin{proofof}{Claim~\ref{claim: existence of good solutions special}}
Let $\phi^*$ be an optimal drawing of the instance $(G,\Sigma)$.
We will construct, for each $1\le i\le r$, a drawing $\phi_i$ of $G_i$ that respects the rotation system $\Sigma_i$, based on the drawing $\phi^*$, such that $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)\le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot 2^{O((\log m)^{3/4}\log\log m)})$, and \Cref{claim: existence of good solutions special} then follows.
\paragraph{Drawings $\phi_2,\ldots,\phi_{r-1}$.}
First we fix some index $2\le i\le r-1$, and describe the construction of the drawing $\phi_i$. We start with some definitions.
Recall that $E(G_i)=E_G(S'_i)\cup (\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$.
We define the auxiliary path set ${\mathcal{W}}_i={\mathcal{W}}^{\operatorname{left}}_i\cup {\mathcal{W}}^{\operatorname{right}}_i$, so
$${\mathcal{W}}_i=\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}\cup \set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}},$$
where for each $e\in E^{\operatorname{left}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$; for each $e\in E^{\operatorname{right}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between $u_{i+1}$ and its last endpoint; and for each $e\in E^{\textsf{thr}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$, the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$, and the subpath of $P_e$ between $u_{i+1}$ and its last endpoint.
\iffalse
We use the following observation.
\begin{observation}
\label{obs: wset_i_non_interfering}
The set ${\mathcal{W}}_i$ of paths are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
Recall that the paths in ${\mathcal{Q}}_{i-1}$ only uses edges of $E(S'_{i-1})\cup \delta(S'_{i-1})$, and they are non-transversal.
And similarly, the paths in ${\mathcal{Q}}_{i+1}$ only uses edges of $E(S'_{i+1})\cup \delta(S'_{i+1})$, and they are non-transversal.
Therefore, the paths in $\set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}}$ are non-transversal.
From \Cref{obs: non_transversal_1} and \Cref{obs: non_transversal_2}, the paths in $\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}$ are non-transversal. Therefore, it suffices to show that, the set ${\mathcal{W}}_i$ of paths are non-transversal at all vertices of $S'_{i-1}$ and all vertices of $S'_{i+1}$. Note that, for each edge $e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i})$, $W_e\cap (S'_{i-1}\cup \delta(S'_{i-1}))$ is indeed a path of ${\mathcal{Q}}_{i-1}$. Therefore, the paths in ${\mathcal{W}}_i$ are non-transversal at all vertices of $S'_{i-1}$. Similarly, they are also non-transversal at all vertices of $S'_{i+1}$. Altogether, the paths of ${\mathcal{W}}_i$ are non-transversal with respect to $\Sigma$.
\end{proof}
\fi
For uniformity of notations, for each edge $\hat e\in \hat E_i$, we rename the path $Q_{i+1}(\hat e)$ by $W_{\hat e}$, and similarly for each edge $\hat e\in \hat E_{i-1}$, we rename the path $Q_{i-1}(\hat e)$ by $W_{\hat e}$. Therefore, ${\mathcal{W}}_i=\set{W_e\mid e\in E(G_i)\setminus E(S'_i)}$.
Put in other words, the set ${\mathcal{W}}_i$ contains, for each edge $e$ in $G_i$ that is incident to $v^{\operatorname{left}}_i$ or $v^{\operatorname{right}}_i$, a path named $W_e$.
It is easy to see that all paths in ${\mathcal{W}}_i$ are internally disjoint from $S'_i$.
We further partition the set ${\mathcal{W}}_i$ into two sets: ${\mathcal{W}}_i^{\textsf{thr}}=\set{W_e\mid e\in E^{\textsf{thr}}_i}$ and $\tilde {\mathcal{W}}_i={\mathcal{W}}_i\setminus {\mathcal{W}}_i^{\textsf{thr}}$.
We are now ready to construct the drawing $\phi_i$ for the instance $(G_i,\Sigma_i)$.
Recall that $\phi^*$ is an optimal drawing of the input instance $(G,\Sigma)$.
We start with the drawing of $S'_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$, that we denote by $\phi^*_i$. We will not modify the image of $S'_i$ in $\phi^*_i$, but will focus on constructing the image of edges in $E(G_i)\setminus E(S'_i)$, based on the image of edges in $E({\mathcal{W}}_i)$ in $\phi^*_i$.
Specifically, we proceed in the following four steps.
\paragraph{Step 1.}
For each edge $e\in E({\mathcal{W}}_i)$, we denote by $\pi_e$ the curve that represents the image of $e$ in $\phi^*_i$. We create a set of $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting the endpoints of $e$ in $\phi^*_i$, that lies in an arbitrarily thin strip around $\pi_e$.
We denote by $\Pi_e$ the set of these curves.
We then assign, for each edge $e\in E({\mathcal{W}}_i)$ and for each path in ${\mathcal{W}}_i$ that contains the edge $e$, a distinct curve in $\Pi_e$ to this path.
Therefore, each curve in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ is assigned to exactly one path of ${\mathcal{W}}_i$,
and each path $W\in {\mathcal{W}}_i$ is assigned with, for each edge $e\in E(W)$, a curve in $\Pi_e$. Let $\gamma_W$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ that are assigned to path $W$, so $\gamma_W$ connects the endpoints of path $W$ in $\phi^*_i$.
In fact, when we assign curves in $\bigcup_{e\in \delta(u_{i-1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{left}}_i$ (recall that $\delta(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\operatorname{left}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{left}}_i)}$), we additionally ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{left}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{left}}_i)$ enter $u_{i-1}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
And similarly, when we assign curves in $\bigcup_{e\in \delta(u_{i+1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{right}}_i$ (recall that $\delta(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\operatorname{right}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{right}}_i)}$), we ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{right}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{right}}_i)$ enter $u_{i+1}$ in the same order as ${\mathcal{O}}^{\operatorname{right}}_i$.
Note that this can be easily achieved according to the definition of ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$.
We denote $\Gamma_i=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$, and we further partition the set $\Gamma_i$ into two sets: $\Gamma_i^{\textsf{thr}}=\set{\gamma_{W}\mid W\in {\mathcal{W}}^{\textsf{thr}}_i}$ and $\tilde \Gamma_i=\Gamma_i\setminus \Gamma_i^{\textsf{thr}}$.
We denote by $\hat \phi_i$ the drawing obtained by taking the union of the image of $S'_i$ in $\phi^*_i$ and all curves in $\Gamma_i$.
For every path $P$ in $G_i$, we denote by $\hat{\chi}_i(P)$ the number of crossings that involves the ``image of $P$'' in $\hat \phi_i$, which is defined as the union of, for each edge $e\in E(\tilde{\mathcal{W}}_i)$, an arbitrary curve in $\Pi_e$.
Clearly, for each edge $e\in E({\mathcal{W}}_i)$, all curves in $\Pi_e$ are crossed by other curves of $(\Gamma_i\setminus \Pi_e)\cup \phi^*_i(S'_i)$ same number of times.
Therefore, $\hat{\chi}_i(P)$ is well-defined.
For a set ${\mathcal{P}}$ of paths in $G_i$, we define $\hat{\chi}_i({\mathcal{P}})=\sum_{P\in {\mathcal{P}}}\hat{\chi}_i(P)$.
\iffalse
We use the following observation.
\znote{maybe remove this observation?}
\begin{observation}
\label{obs: curves_crossings}
The number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$,
and the number of intersections between a curve in $\tilde\Gamma_i$ and the image of edges of $S'_i$ in $\phi^*_i$, are both $O(\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W))$.
\end{observation}
\begin{proof}
We first show that the number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Note that, from the construction of curves in $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i}$, every crossing between a pair $W,W'$ of curves in $\tilde\Gamma_i$ must be the intersection between a curve in $\Pi_e$ for some $e\in E(W)$ and a curve in $\Pi_{e'}$ for some $e'\in E(W')$, such that the image $\pi_e$ for $e$ and the image $\pi_{e'}$ for $e'$ intersect in $\phi^*$.
Therefore, for each pair $W,W'$ of paths in $\tilde{\mathcal{W}}_i$, the number of points that belong to only curves $\gamma_W$ and $\gamma_{W'}$ is at most the number of crossings between the image of $W$ and the image of $W'$ in $\phi^*$.
It follows that the number of points that belong to exactly two curves of $\tilde\Gamma_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Altogether, the number of intersections between curves in $\tilde\Gamma_i$ is at most $|V(\tilde {\mathcal{W}}_i)|+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
We now show that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $S'_i$ in $\phi^*_i$ that are not vertex-image is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Let $W$ be a path of $\tilde {\mathcal{W}}_i$ and consider the curve $\gamma_W$. Note that $\gamma_W$ is the union of, for each edge $e\in E(W)$, a curve that lies in an arbitrarily thin strip around $\pi_e$.
Therefore, the number of crossings between $\gamma_W$ and the image of $S'_i$ in $\phi^*_i$ is identical to the number of crossings the image of path $W$ and the image of $S'_i$ in $\phi^*_i$, which is at most $\hat\mathsf{cr}(W)$.
It follows that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $S'_i$ in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
\end{proof}
\fi
\paragraph{Step 2.}
For each vertex $v\in V({\mathcal{W}}_i)$, we denote by $x_v$ the point that represents the image of $v$ in $\phi^*_i$, and we let $X$ contains all points of $\set{x_v\mid v\in V({\mathcal{W}}_i)}$ that are intersections between curves in $\Gamma_i$.
We now manipulate the curves in $\set{\gamma_W\mid W\in {\mathcal{W}}_i}$ at points of $X$, by processing points of $X$ one-by-one, as follows.
Consider a point $x_v$ that is an intersection between curves in $\Gamma_i$, where $v\in V({\mathcal{W}}_i)$, and let $D_v$ be an arbitrarily small disc around $x_v$.
We denote by ${\mathcal{W}}_i(v)$ the set of paths in ${\mathcal{W}}_i$ that contains $v$, and further partition it into two sets: ${\mathcal{W}}^{\textsf{thr}}_i(v)={\mathcal{W}}_i(v)\cap {\mathcal{W}}^{\textsf{thr}}_i$ and $\tilde{\mathcal{W}}_i(v)={\mathcal{W}}_i(v)\cap \tilde{\mathcal{W}}_i$. We apply the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $\set{\gamma_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ within disc $D_v$. Let $\set{\gamma'_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ be the set of new curves that we obtain. From \Cref{obs: curve_manipulation},
(i) for each path $W\in \tilde{\mathcal{W}}_i(v)$, the curve $\gamma'_W$ does not contain $x_v$, and is identical to the curve $\gamma_W$ outside the disc $D_v$;
(ii) the segments of curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc $D_v$ are in general position; and
(iii) the number of icrossings between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside $D_v$ is bounded by $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\iffalse{just for backup}
\begin{proof}
Denote $d=\deg_G(v)$ and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
For each path $W\in \tilde{\mathcal{W}}_i(v)$, we denote by $p^{-}_W$ and $p^{+}_W$ the intersections between the curve $\gamma_W$ and the boundary of ${\mathcal{D}}_v$.
We now compute, for each $W\in W\in \tilde{\mathcal{W}}_i(v)$, a curve $\zeta_W$ in ${\mathcal{D}}_v$ connecting $p^{-}_W$ to $p^{+}_W$, such that (i) the curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ are in general position; and (ii) for each pair $W,W'$ of paths, the curves $\zeta_W$ and $\zeta_{W'}$ intersects iff the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{-}_{W'},p^{+}_{W},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_{W'},p^{-}_{W},p^{+}_{W'})$.
It is clear that this can be achieved by first setting, for each $W$, the curve $\zeta_W$ to be the line segment connecting $p^{-}_W$ to $p^{+}_W$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
We now define, for each $W$, the curve $\gamma'_W$ to be the union of the part of $\gamma_W$ outside ${\mathcal{D}}_v$ and the curve $\zeta_W$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, the first and the second condition of \Cref{obs: curve_manipulation} are satisfied.
It remains to estimate the number of intersections between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$, which equals the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$.
Since the paths in $\tilde{\mathcal{W}}_i(v)$ are non-transversal with respect to $\Sigma$ (from \Cref{obs: wset_i_non_interfering}), from the construction of curves $\set{\gamma_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$, if a pair $W,W'$ of paths in $\tilde {\mathcal{W}}_i(v)$ do not share edges of $\delta(v)$, then the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_W,p^{-}_{W'},p^{+}_{W'})$, and therefore the curves $\zeta_{W}$ and $\zeta_{W'}$ will not intersect in ${\mathcal{D}}_v$.
Therefore, only the curves $\zeta_W$ and $\zeta_{W'}$ intersect iff $W$ and $W'$ share an edge of $\delta(v)$.
Since every such pair of curves intersects at most once, the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$ is at most $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are shown in black, and curves of $\tilde{\mathcal{W}}_i(v)$ are shown in blue, red, orange and green. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are not modified, while curves of $\tilde{\mathcal{W}}_i(v)$ are re-routed via dash lines within disc ${\mathcal{D}}_v$.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the step of processing $x_v$.}\label{fig: curve_con}
\end{figure}
\fi
We then replace the curves of $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ in $\Gamma_i$ by the curves of $\set{\gamma'_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
This completes the description of the iteration of processing point the point $x_v\in X$. Let $\Gamma'_i$ be the set of curves that we obtain after processing all points in $X$ in this way. Note that we have never modified the curves of $\Gamma^{\textsf{thr}}_i$, so $\Gamma^{\textsf{thr}}_i\subseteq\Gamma'_i$, and we denote $\tilde\Gamma'_i=\Gamma'_i\setminus \Gamma^{\textsf{thr}}_i$.
We use the following observation.
\begin{observation}
\label{obs: general_position}
Curves in $\tilde\Gamma'_i$ are in general position, and if a point $p$ lies on more than two curves of $\Gamma'_i$, then either $p$ is an endpoint of all curves containing it, or all curves containing $p$ belong to $\Gamma^{\textsf{thr}}_i$.
\end{observation}
\begin{proof}
From the construction of curves in $\Gamma_i$, any point that belong to at least three curves of $\Gamma_i$ must be the image of some vertex in $\phi^*$. From~\Cref{obs: curve_manipulation}, curves in $\tilde\Gamma'_i$ are in general position; curves in $\tilde\Gamma'_i$ do not contain any vertex-image in $\phi^*$ except for their endpoints; and they do not contain any intersection of a pair of paths in $\Gamma_i^{\textsf{thr}}$. \Cref{obs: general_position} now follows.
\end{proof}
\paragraph{Step 3.}
So far we have obtained a set $\Gamma'_i$ of curves that are further partitioned into two sets $\Gamma'_i=\Gamma^{\textsf{thr}}_i\cup \tilde\Gamma'_i$,
where set $\tilde\Gamma'_i$ contains, for each path $W\in \tilde {\mathcal{W}}_i$, a curve $\gamma'_W$ connecting the endpoints of $W$, and the curves in $\tilde\Gamma'_i$ are in general position;
and set $\Gamma^{\textsf{thr}}_i$ contains, for each path $W\in {\mathcal{W}}^{\textsf{thr}}_i$, a curve $\gamma_W$ connecting the endpoints of $W$.
Recall that all paths in ${\mathcal{W}}^{\textsf{thr}}_i$ connects $u_{i-1}$ to $u_{i+1}$.
Let $z_{\operatorname{left}}$ be the point that represents the image of $u_{i-1}$ in $\phi_i^*$ and let $z_{\operatorname{right}}$ be the point that represents the image of $u_{i+1}$ in $\phi_i^*$.
Then, all curves in $\Gamma^{\textsf{thr}}_i$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
We view $z_{\operatorname{left}}$ as the first endpoint of curves in $\Gamma^{\textsf{thr}}_i$ and view $z_{\operatorname{right}}$ as their last endpoint.
We then apply the algorithm in \Cref{thm: type-2 uncrossing}, where we let $\Gamma=\Gamma^{\textsf{thr}}_i$ and let $\Gamma_0$ be the set of all other curves in the drawing $\phi^*_i$. Let $\Gamma^{\textsf{thr}'}_i$ be the set of curves we obtain. We then designate, for each edge $e\in E^{\textsf{thr}}_i$, a curve in $\Gamma^{\textsf{thr}'}_i$ as $\gamma'_{W_e}$, such that the curves of $\set{\gamma'_{W_e}\mid e\in \hat E_{i-1}\cup E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i}$ enters $z_{\operatorname{left}}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
Recall that ${\mathcal{W}}_i=\set{W_e\mid e\in (E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\cup E_i^{\operatorname{right}}\cup \hat E_{i-1}\cup \hat E_i)}$, and, for each edge $e\in E_i^{\operatorname{left}}\cup \hat E_{i-1}$, the curve $\gamma'_{W_e}$ connects its endpoint in $S'_i$ to $z_{\operatorname{left}}$; for each edge $e\in E_i^{\operatorname{right}}\cup \hat E_{i}$, the curve $\gamma'_{W_e}$ connects the endpoint of $e$ to $z_{\operatorname{right}}$; and for each edge $e\in E_i^{\textsf{thr}}$, the curve $\gamma'_{W_e}$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
Therefore, if we view $z_{\operatorname{left}}$ as the image of $v^{\operatorname{left}}_i$, view $z_{\operatorname{right}}$ as the image of $v^{\operatorname{right}}_i$, and for each edge $e\in E(G_i)\setminus E(S'_i)$, view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi'_i$.
It is clear from the construction of curves in $\set{\gamma'_{W_e}\mid e\in E(G_i)\setminus E(S'_i)}$ that this drawing respects all rotations in $\Sigma_i$ on vertices of $V(S'_i)$ and vertex $v^{\operatorname{left}}_i$.
However, the drawing $\phi'_i$ may not respect the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$. We further modify the drawing $\phi'_i$ at $z_{\operatorname{right}}$ in the last step.
\paragraph{Step 4.}
Let ${\mathcal{D}}$ be an arbitrarily small disc around $z_{\operatorname{right}}$ in the drawing $\phi'_i$, and let ${\mathcal{D}}'$ be another small disc around $z_{\operatorname{right}}$ that is strictly contained in ${\mathcal{D}}$.
We first erase the drawing of $\phi'_i$ inside the disc ${\mathcal{D}}$, and for each edge $e\in \delta(v^{\operatorname{right}}_i)$, we denote by $p_{e}$ the intersection between the curve representing the image of $e$ in $\phi'_i$ and the boundary of ${\mathcal{D}}$.
We then place, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, a point $p'_e$ on the boundary of ${\mathcal{D}}'$, such that the order in which the points in $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ appearing on the boundary of ${\mathcal{D}}'$ is precisely ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We then apply \Cref{lem: find reordering} to compute a set of reordering curves, connecting points of $\set{p_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ to points $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$.
Finally, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, let $\gamma_e$ be the concatenation of (i) the image of $e$ in $\phi'_i$ outside the disc ${\mathcal{D}}$; (ii) the reordering curve connecting $p_e$ to $p'_e$; and (iii) the straight line segment connecting $p'_e$ to $z_{\operatorname{right}}$ in ${\mathcal{D}}'$.
We view $\gamma_e$ as the image of edge $e$, for each $e\in \delta(v^{\operatorname{right}}_i)$.
We denote the resulting drawing of $G_i$ by $\phi_i$. It is clear that $\phi_i$ respects the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$, and therefore it respects the rotation system $\Sigma_i$.
We use the following claim.
\begin{claim}
\label{clm: rerouting_crossings}
The number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\end{claim}
\begin{proof}
Denote by ${\mathcal{O}}^*$ the ordering in which the curves $\set{\gamma'_{W_e}\mid e\in \delta_{G_i}(v_i^{\operatorname{right}})}$ enter $z_{\operatorname{right}}$, the image of $u_{i+1}$ in $\phi'_i$.
From~\Cref{lem: find reordering} and the algorithm in Step 4 of modifying the drawing within the disc ${\mathcal{D}}$, the number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is at most $O(\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}}))$. Therefore, it suffices to show that $\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}})=O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
In fact, we will compute a set of curves connecting the image of $u_i$ and the image of $u_{i+1}$ in $\phi^*_i$, such that each curve is indexed by some edge $e\in\delta_{G_i}(v_i^{\operatorname{right}})$ these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$ and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$, and the number of crossings between curves of $Z$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
For each $e\in E^{\textsf{thr}}_i$, we denote by $\eta_e$ the curve obtained by taking the union of (i) the curve $\gamma'_{W_e}$ (that connects $u_{i+1}$ to $u_{i-1}$); and (ii) the curve representing the image of the subpath of $P_e$ in $\phi^*_i$ between $u_i$ and $u_{i-1}$.
Therefore, the curve $\eta_e$ connects $u_i$ to $u_{i+1}$.
We then modify the curves of $\set{\eta_e\mid e\in E^{\textsf{thr}}_i}$, by iteratively applying the algorithm from \Cref{obs: curve_manipulation} to these curves at the image of each vertex of $S'_{i-1}\cup S'_{i+1}$. Let $\set{\zeta_e\mid e\in E^{\textsf{thr}}_i}$ be the set of curves that we obtain. We call the obtained curves \emph{red curves}. From~\Cref{obs: curve_manipulation}, the red curves are in general position. Moreover, it is easy to verify that the number of intersections between the red curves is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1}))$.
We call the curves in $\set{\gamma'_{W_e}\mid e\in \hat E_i}$ \emph{yellow curves}, call the curves in $\set{\gamma'_{W_e}\mid e\in E^{\operatorname{right}}_i}$ \emph{green curves}. See \Cref{fig: uncrossing_to_bound_crossings} for an illustration.
From the construction of red, yellow and green curves, we know that these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$, and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$.
Moreover, we are guaranteed that the number of intersections between red, yellow and green curves is at most $\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/uncross_to_bound_crossings.jpg}
\caption{An illustration of red, yellow and green curves.}\label{fig: uncrossing_to_bound_crossings}
\end{figure}
\end{proof}
From the above discussion and Claim~\ref{clm: rerouting_crossings}, for each $2\le i\le r-1$,
\[
\mathsf{cr}(\phi_i)=\chi^2(S'_i)+O\textsf{left}(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right}).
\]
\iffalse
We now estimate the number of crossings in $\phi_i$ in the next claim.
\begin{claim}
\label{clm: number of crossings in good solutions}
$\mathsf{cr}(\phi_i)=\chi^2(S'_i)+O\textsf{left}(\sum_{W\in \tilde{\mathcal{W}}_i}\mathsf{cr}(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})$.
\end{claim}
\begin{proof}
$2\cdot \sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2=\sum_{v\in V(G)}\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2.$
\znote{need to redefine the orderings ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$ to get rid of $\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$ here, which we may not be able to afford.}
\end{proof}
\fi
\paragraph{Drawings $\phi_1$ and $\phi_{r}$.}
The drawings $\phi_1$ and $\phi_{r}$ are constructed similarly. We describe the construction of $\phi_1$, and the construction of $\phi_1$ is symmetric.
Recall that the graph $G_1$ contains only one super-node $v_1^{\operatorname{right}}$, and $\delta_{G_1}(v_1^{\operatorname{right}})=\hat E_1\cup E^{\operatorname{right}}_1$. We define ${\mathcal{W}}_1=\set{W_e\mid e\in E^{\operatorname{right}}_1}\cup \set{Q_2(\hat e)\mid \hat e\in \hat E_1}$. For each $\hat e\in \hat E_1$, we rename the path $Q_2(\hat e)$ by $W_e$, so ${\mathcal{W}}_1$ contains, for each edge $e\in \delta_{G_1}(v^1_{\operatorname{right}})$, a path named $W_e$ connecting its endpoints to $u_2$. Via similar analysis in \Cref{obs: wset_i_non_interfering}, it is easy to show that the paths in ${\mathcal{W}}_1$ are non-transversal with respect to $\Sigma$.
We start with the drawing of $C_1\cup E({\mathcal{W}}_1)$ induced by $\phi^*$, that we denote by $\phi^*_1$. We will not modify the image of $S'_i$ in $\phi^*_i$ and will construct the image of edges in $\delta(v_1^{\operatorname{right}})$.
We perform similar steps as in the construction of drawings $\phi_2,\ldots,\phi_{r-1}$.
We first construct, for each path $W\in {\mathcal{W}}_1$, a curve $\gamma_W$ connecting its endpoint in $C_1$ to the image of $u_2$ in $\phi^*$, as in Step 1.
Let $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
We then process all intersections between curves of $\Gamma_1$ as in Step 2. Let $\Gamma'_1=\set{\gamma'_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
Since $\Gamma^{\textsf{thr}}_1=\emptyset$, we do not need to perform Steps 3 and 4. If we view the image of $u_2$ in $\phi^*_1$ as the image of $v^{\operatorname{right}}_1$, and for each edge $e\in \delta(v^{\operatorname{right}}_1)$, we view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is clear that this drawing respects the rotation system $\Sigma_1$. Moreover,
\[\mathsf{cr}(\phi_1)=\chi^2(C_1)+O\textsf{left}(\hat\chi_1({\mathcal{Q}}_2)+\sum_{W\in {\mathcal{W}}_1}\hat\chi_1(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_1,e)^2\textsf{right}).\]
Similarly, the drawing $\phi_k$ that we obtained in the similar way satisfies that
\[\mathsf{cr}(\phi_k)=\chi^2(C_k)+O\textsf{left}(\hat\chi_k({\mathcal{Q}}_{r-1})+\sum_{W\in {\mathcal{W}}_k}\hat\chi_k(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_k,e)^2\textsf{right}).\]
We now complete the proof of \Cref{claim: existence of good solutions special}, for which it suffices to estimate $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
Recall that, for each $1\le i\le r$, $\tilde {\mathcal{W}}_i=\set{W_e\mid e\in \hat E_{i-1}\cup \hat E_{i-1}\cup E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}}$, where $E^{\operatorname{left}}_{i}=E(S'_i,\bigcup_{1\le j\le i-2}S'_j)$, and
$E^{\operatorname{right}}_{i}=E(S'_i,\bigcup_{i+2\le j\le r}S'_j)$. Therefore, for each edge $e\in E'\cup (\bigcup_{1\le i\le r-1}\hat E_i)$, the path $W_e$ belongs to exactly $2$ sets of $\set{\tilde{\mathcal{W}}_i}_{1\le i\le r}$.
Recall that the path $W_e$ only uses edges of the inner path $P_e$ and the outer path $P^{\mathsf{out}}_e$.
Let $\tilde{\mathcal{W}}=\bigcup_{1\le i\le r}\tilde{\mathcal{W}}_i$, from \Cref{obs: edge_occupation in outer and inner paths},
for each edge $e\in E'\cup (\bigcup_{1\le i\le r-1}\hat E_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(1)$, and
for $1\le i\le r$ and for each edge $e\in E(S'_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(\cong_G({\mathcal{Q}}_i,e))$.
Therefore, on one hand,
\[
\begin{split}
\sum_{1\le i\le r}\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} 2\cdot \cong_G(\tilde {\mathcal{W}},e)\cdot\cong_G(\tilde {\mathcal{W}},e')\\
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} \textsf{left}(\cong_G(\tilde {\mathcal{W}},e)^2+\cong_G(\tilde {\mathcal{W}},e')^2\textsf{right})\\
& \le
\sum_{e\in E(G)} \chi(e)\cdot \cong_G(\tilde {\mathcal{W}},e)^2
= O(\mathsf{cr}(\phi^*)\cdot\beta),
\end{split}
\]
and on the other hand,
\[
\begin{split}
\sum_{1\le i\le r}\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2
& \le \sum_{e\in E(G)} \textsf{left}(\sum_{1\le i\le r} \cong_G(\tilde {\mathcal{W}}_i,e)\textsf{right})^2\\
& \le O\textsf{left}(\sum_{e\in E(G)} \cong_G(\tilde {\mathcal{W}},e)^2 \textsf{right})\\
& \le O\textsf{left}(|E(G)|+\sum_{1\le i\le r}\sum_{e\in E(S'_i)} \cong_G({\mathcal{Q}}_i,e)^2\textsf{right})=O(|E(G)|\cdot\beta).
\end{split}
\]
Moreover, $\sum_{1\le i\le r}\chi^2(S'_i)\le O(\mathsf{cr}(\phi^*))$, and $\sum_{1\le i\le r}\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})\le O(\mathsf{cr}(\phi^*)\cdot\beta)$.
Altogether,
\[
\begin{split}
\sum_{1\le i\le r}\mathsf{cr}(\phi_i)
& \le
O\textsf{left}(\sum_{1\le i\le r}
\textsf{left}(
\chi^2(S'_i)+ \hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})
\textsf{right})\\
& \le O(\mathsf{cr}(\phi^*))+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(|E(G)|\cdot\beta)\\
& \le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot\beta).
\end{split}
\]
This completes the proof of \Cref{claim: existence of good solutions special}.
\end{proofof}
\fi
\iffalse{incorrect analysis of non-interfering}
We use the following observation.
\begin{observation}
\label{obs: wset_i_non_interfering}
The set ${\mathcal{W}}_i$ of paths are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
Recall that the paths in ${\mathcal{Q}}_{i-1}$ only uses edges of $E(S'_{i-1})\cup \delta(S'_{i-1})$, and they are non-transversal.
And similarly, the paths in ${\mathcal{Q}}_{i+1}$ only uses edges of $E(S'_{i+1})\cup \delta(S'_{i+1})$, and they are non-transversal.
Therefore, the paths in $\set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}}$ are non-transversal.
From \Cref{obs: non_transversal_1} and \Cref{obs: non_transversal_2}, the paths in $\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}$ are non-transversal. Therefore, it suffices to show that, the set ${\mathcal{W}}_i$ of paths are non-transversal at all vertices of $S'_{i-1}$ and all vertices of $S'_{i+1}$. Note that, for each edge $e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i})$, $W_e\cap (S'_{i-1}\cup \delta(S'_{i-1}))$ is indeed a path of ${\mathcal{Q}}_{i-1}$. Therefore, the paths in ${\mathcal{W}}_i$ are non-transversal at all vertices of $S'_{i-1}$. Similarly, they are also non-transversal at all vertices of $S'_{i+1}$. Altogether, the paths of ${\mathcal{W}}_i$ are non-transversal with respect to $\Sigma$.
\end{proof}
\fi
\iffalse {previous nudging and uncrossing steps}
We further partition ${\mathcal{W}}_i$ into two sets: ${\mathcal{W}}_i^{\textsf{thr}}=\set{W_i(e)\mid e\in E^{\textsf{thr}}_i}$ and $\tilde {\mathcal{W}}_i={\mathcal{W}}_i\setminus {\mathcal{W}}_i^{\textsf{thr}}$.
We are now ready to construct the drawing $\phi_i$ for the instance $(G_i,\Sigma_i)$.
Recall that $\phi^*$ is an optimal drawing of the input instance $(G,\Sigma)$.
We start with the drawing of $S'_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$, that we denote by $\phi^*_i$. We will not modify the image of $S'_i$ in $\phi^*_i$, but will focus on constructing the image of edges in $E(G_i)\setminus E(S'_i)$, based on the image of edges in $E({\mathcal{W}}_i)$ in $\phi^*_i$.
Specifically, we proceed in the following four steps.
\paragraph{Step 1.}
For each edge $e\in E({\mathcal{W}}_i)$, we denote by $\pi_e$ the curve that represents the image of $e$ in $\phi^*_i$. We create a set of $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting the endpoints of $e$ in $\phi^*_i$, that lies in an arbitrarily thin strip around $\pi_e$.
We denote by $\Pi_e$ the set of these curves.
We then assign, for each edge $e\in E({\mathcal{W}}_i)$ and for each path in ${\mathcal{W}}_i$ that contains the edge $e$, a distinct curve in $\Pi_e$ to this path.
Therefore, each curve in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ is assigned to exactly one path of ${\mathcal{W}}_i$,
and each path $W\in {\mathcal{W}}_i$ is assigned with, for each edge $e\in E(W)$, a curve in $\Pi_e$. Let $\gamma_W$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ that are assigned to path $W$, so $\gamma_W$ connects the endpoints of path $W$ in $\phi^*_i$.
In fact, when we assign curves in $\bigcup_{e\in \delta(u_{i-1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{left}}_i$ (recall that $\delta(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\operatorname{left}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{left}}_i)}$), we additionally ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{left}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{left}}_i)$ enter $u_{i-1}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
And similarly, when we assign curves in $\bigcup_{e\in \delta(u_{i+1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{right}}_i$ (recall that $\delta(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\operatorname{right}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{right}}_i)}$), we ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{right}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{right}}_i)$ enter $u_{i+1}$ in the same order as ${\mathcal{O}}^{\operatorname{right}}_i$.
Note that this can be easily achieved according to the definition of ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$.
We denote $\Gamma_i=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$, and we further partition the set $\Gamma_i$ into two sets: $\Gamma_i^{\textsf{thr}}=\set{\gamma_{W}\mid W\in {\mathcal{W}}^{\textsf{thr}}_i}$ and $\tilde \Gamma_i=\Gamma_i\setminus \Gamma_i^{\textsf{thr}}$.
We denote by $\hat \phi_i$ the drawing obtained by taking the union of the image of $S'_i$ in $\phi^*_i$ and all curves in $\Gamma_i$.
For every path $P$ in $G_i$, we denote by $\hat{\chi}_i(P)$ the number of crossings that involves the ``image of $P$'' in $\hat \phi_i$, which is defined as the union of, for each edge $e\in E(\tilde{\mathcal{W}}_i)$, an arbitrary curve in $\Pi_e$.
Clearly, for each edge $e\in E({\mathcal{W}}_i)$, all curves in $\Pi_e$ are crossed by other curves of $(\Gamma_i\setminus \Pi_e)\cup \phi^*_i(S'_i)$ same number of times.
Therefore, $\hat{\chi}_i(P)$ is well-defined.
For a set ${\mathcal{P}}$ of paths in $G_i$, we define $\hat{\chi}_i({\mathcal{P}})=\sum_{P\in {\mathcal{P}}}\hat{\chi}_i(P)$.
\iffalse
We use the following observation.
\znote{maybe remove this observation?}
\begin{observation}
\label{obs: curves_crossings}
The number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$,
and the number of intersections between a curve in $\tilde\Gamma_i$ and the image of edges of $S'_i$ in $\phi^*_i$, are both $O(\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W))$.
\end{observation}
\begin{proof}
We first show that the number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Note that, from the construction of curves in $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i}$, every crossing between a pair $W,W'$ of curves in $\tilde\Gamma_i$ must be the intersection between a curve in $\Pi_e$ for some $e\in E(W)$ and a curve in $\Pi_{e'}$ for some $e'\in E(W')$, such that the image $\pi_e$ for $e$ and the image $\pi_{e'}$ for $e'$ intersect in $\phi^*$.
Therefore, for each pair $W,W'$ of paths in $\tilde{\mathcal{W}}_i$, the number of points that belong to only curves $\gamma_W$ and $\gamma_{W'}$ is at most the number of crossings between the image of $W$ and the image of $W'$ in $\phi^*$.
It follows that the number of points that belong to exactly two curves of $\tilde\Gamma_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Altogether, the number of intersections between curves in $\tilde\Gamma_i$ is at most $|V(\tilde {\mathcal{W}}_i)|+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
We now show that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $S'_i$ in $\phi^*_i$ that are not vertex-image is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Let $W$ be a path of $\tilde {\mathcal{W}}_i$ and consider the curve $\gamma_W$. Note that $\gamma_W$ is the union of, for each edge $e\in E(W)$, a curve that lies in an arbitrarily thin strip around $\pi_e$.
Therefore, the number of crossings between $\gamma_W$ and the image of $S'_i$ in $\phi^*_i$ is identical to the number of crossings the image of path $W$ and the image of $S'_i$ in $\phi^*_i$, which is at most $\hat\mathsf{cr}(W)$.
It follows that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $S'_i$ in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
\end{proof}
\fi
\paragraph{Step 2.}
For each vertex $v\in V({\mathcal{W}}_i)$, we denote by $x_v$ the point that represents the image of $v$ in $\phi^*_i$, and we let $X$ contains all points of $\set{x_v\mid v\in V({\mathcal{W}}_i)}$ that are intersections between curves in $\Gamma_i$.
We now manipulate the curves in $\set{\gamma_W\mid W\in {\mathcal{W}}_i}$ at points of $X$, by processing points of $X$ one-by-one, as follows.
Consider a point $x_v$ that is an intersection between curves in $\Gamma_i$, where $v\in V({\mathcal{W}}_i)$, and let $D_v$ be an arbitrarily small disc around $x_v$.
We denote by ${\mathcal{W}}_i(v)$ the set of paths in ${\mathcal{W}}_i$ that contains $v$, and further partition it into two sets: ${\mathcal{W}}^{\textsf{thr}}_i(v)={\mathcal{W}}_i(v)\cap {\mathcal{W}}^{\textsf{thr}}_i$ and $\tilde{\mathcal{W}}_i(v)={\mathcal{W}}_i(v)\cap \tilde{\mathcal{W}}_i$. We apply the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $\set{\gamma_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ within disc $D_v$. Let $\set{\gamma'_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ be the set of new curves that we obtain. From \Cref{obs: curve_manipulation},
(i) for each path $W\in \tilde{\mathcal{W}}_i(v)$, the curve $\gamma'_W$ does not contain $x_v$, and is identical to the curve $\gamma_W$ outside the disc $D_v$;
(ii) the segments of curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc $D_v$ are in general position; and
(iii) the number of icrossings between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside $D_v$ is bounded by $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\iffalse{just for backup}
\begin{proof}
Denote $d=\deg_G(v)$ and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
For each path $W\in \tilde{\mathcal{W}}_i(v)$, we denote by $p^{-}_W$ and $p^{+}_W$ the intersections between the curve $\gamma_W$ and the boundary of ${\mathcal{D}}_v$.
We now compute, for each $W\in W\in \tilde{\mathcal{W}}_i(v)$, a curve $\zeta_W$ in ${\mathcal{D}}_v$ connecting $p^{-}_W$ to $p^{+}_W$, such that (i) the curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ are in general position; and (ii) for each pair $W,W'$ of paths, the curves $\zeta_W$ and $\zeta_{W'}$ intersects iff the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{-}_{W'},p^{+}_{W},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_{W'},p^{-}_{W},p^{+}_{W'})$.
It is clear that this can be achieved by first setting, for each $W$, the curve $\zeta_W$ to be the line segment connecting $p^{-}_W$ to $p^{+}_W$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
We now define, for each $W$, the curve $\gamma'_W$ to be the union of the part of $\gamma_W$ outside ${\mathcal{D}}_v$ and the curve $\zeta_W$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, the first and the second condition of \Cref{obs: curve_manipulation} are satisfied.
It remains to estimate the number of intersections between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$, which equals the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$.
Since the paths in $\tilde{\mathcal{W}}_i(v)$ are non-transversal with respect to $\Sigma$ (from \Cref{obs: wset_i_non_interfering}), from the construction of curves $\set{\gamma_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$, if a pair $W,W'$ of paths in $\tilde {\mathcal{W}}_i(v)$ do not share edges of $\delta(v)$, then the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_W,p^{-}_{W'},p^{+}_{W'})$, and therefore the curves $\zeta_{W}$ and $\zeta_{W'}$ will not intersect in ${\mathcal{D}}_v$.
Therefore, only the curves $\zeta_W$ and $\zeta_{W'}$ intersect iff $W$ and $W'$ share an edge of $\delta(v)$.
Since every such pair of curves intersects at most once, the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$ is at most $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are shown in black, and curves of $\tilde{\mathcal{W}}_i(v)$ are shown in blue, red, orange and green. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are not modified, while curves of $\tilde{\mathcal{W}}_i(v)$ are re-routed via dash lines within disc ${\mathcal{D}}_v$.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the step of processing $x_v$.}\label{fig: curve_con}
\end{figure}
\fi
We then replace the curves of $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ in $\Gamma_i$ by the curves of $\set{\gamma'_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
This completes the description of the iteration of processing point the point $x_v\in X$. Let $\Gamma'_i$ be the set of curves that we obtain after processing all points in $X$ in this way. Note that we have never modified the curves of $\Gamma^{\textsf{thr}}_i$, so $\Gamma^{\textsf{thr}}_i\subseteq\Gamma'_i$, and we denote $\tilde\Gamma'_i=\Gamma'_i\setminus \Gamma^{\textsf{thr}}_i$.
We use the following observation.
\begin{observation}
\label{obs: general_position}
Curves in $\tilde\Gamma'_i$ are in general position, and if a point $p$ lies on more than two curves of $\Gamma'_i$, then either $p$ is an endpoint of all curves containing it, or all curves containing $p$ belong to $\Gamma^{\textsf{thr}}_i$.
\end{observation}
\begin{proof}
From the construction of curves in $\Gamma_i$, any point that belong to at least three curves of $\Gamma_i$ must be the image of some vertex in $\phi^*$. From~\Cref{obs: curve_manipulation}, curves in $\tilde\Gamma'_i$ are in general position; curves in $\tilde\Gamma'_i$ do not contain any vertex-image in $\phi^*$ except for their endpoints; and they do not contain any intersection of a pair of paths in $\Gamma_i^{\textsf{thr}}$. \Cref{obs: general_position} now follows.
\end{proof}
\paragraph{Step 3.}
So far we have obtained a set $\Gamma'_i$ of curves that are further partitioned into two sets $\Gamma'_i=\Gamma^{\textsf{thr}}_i\cup \tilde\Gamma'_i$,
where set $\tilde\Gamma'_i$ contains, for each path $W\in \tilde {\mathcal{W}}_i$, a curve $\gamma'_W$ connecting the endpoints of $W$, and the curves in $\tilde\Gamma'_i$ are in general position;
and set $\Gamma^{\textsf{thr}}_i$ contains, for each path $W\in {\mathcal{W}}^{\textsf{thr}}_i$, a curve $\gamma_W$ connecting the endpoints of $W$.
Recall that all paths in ${\mathcal{W}}^{\textsf{thr}}_i$ connects $u_{i-1}$ to $u_{i+1}$.
Let $z_{\operatorname{left}}$ be the point that represents the image of $u_{i-1}$ in $\phi_i^*$ and let $z_{\operatorname{right}}$ be the point that represents the image of $u_{i+1}$ in $\phi_i^*$.
Then, all curves in $\Gamma^{\textsf{thr}}_i$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
We view $z_{\operatorname{left}}$ as the first endpoint of curves in $\Gamma^{\textsf{thr}}_i$ and view $z_{\operatorname{right}}$ as their last endpoint.
We then apply the algorithm in \Cref{thm: type-2 uncrossing}, where we let $\Gamma=\Gamma^{\textsf{thr}}_i$ and let $\Gamma_0$ be the set of all other curves in the drawing $\phi^*_i$. Let $\Gamma^{\textsf{thr}'}_i$ be the set of curves we obtain. We then designate, for each edge $e\in E^{\textsf{thr}}_i$, a curve in $\Gamma^{\textsf{thr}'}_i$ as $\gamma'_{W_e}$, such that the curves of $\set{\gamma'_{W_e}\mid e\in \hat E_{i-1}\cup E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i}$ enters $z_{\operatorname{left}}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
Recall that ${\mathcal{W}}_i=\set{W_e\mid e\in (E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\cup E_i^{\operatorname{right}}\cup \hat E_{i-1}\cup \hat E_i)}$, and, for each edge $e\in E_i^{\operatorname{left}}\cup \hat E_{i-1}$, the curve $\gamma'_{W_e}$ connects its endpoint in $S'_i$ to $z_{\operatorname{left}}$; for each edge $e\in E_i^{\operatorname{right}}\cup \hat E_{i}$, the curve $\gamma'_{W_e}$ connects the endpoint of $e$ to $z_{\operatorname{right}}$; and for each edge $e\in E_i^{\textsf{thr}}$, the curve $\gamma'_{W_e}$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
Therefore, if we view $z_{\operatorname{left}}$ as the image of $v^{\operatorname{left}}_i$, view $z_{\operatorname{right}}$ as the image of $v^{\operatorname{right}}_i$, and for each edge $e\in E(G_i)\setminus E(S'_i)$, view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi'_i$.
It is clear from the construction of curves in $\set{\gamma'_{W_e}\mid e\in E(G_i)\setminus E(S'_i)}$ that this drawing respects all rotations in $\Sigma_i$ on vertices of $V(S'_i)$ and vertex $v^{\operatorname{left}}_i$.
However, the drawing $\phi'_i$ may not respect the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$. We further modify the drawing $\phi'_i$ at $z_{\operatorname{right}}$ in the last step.
\fi
\fi
\subsection{The Cut Property}
We first introduce a useful definition that will be repeatedly used in this section.
Let $G$ be a graph and let $P$ be a path with $V(P)=V(G)$. Denote $P=(v_1,\ldots,v_n)$, where vertices appear on $P$ in this order. For each $1\le i\le n-1$, we denote $S_i=\set{v_1,\ldots,v_i}$ and $\overline{S}_i=\set{v_{i+1},\ldots,v_n}$.
We say that graph $G$ has the \emph{$\alpha$-cut property with respect to $P$}, iff for each $1\le i\le n-1$, we have $|E_G(v_i,v_{i+1})|\ge \alpha\cdot |E_G(S_i,\overline{S}_i)|$.
We use the following simple observations.
\begin{observation}
If graph $G$ has the $\alpha$-cut property with respect to $P$, then there exist, for each $1\le i\le n-1$, a set ${\mathcal{R}}_i^{\operatorname{left}}$ of paths routing edges of $E_G(S_i,\overline{S}_i)$ to $v_i$ using only edges of $\bigcup_{1\le t\le i-1}E(v_t,v_{t+1})$, and a set ${\mathcal{R}}_i^{\operatorname{right}}$ of paths routing edges of $E_G(S_i,\overline{S}_i)$ to $v_{i+1}$ using only edges of $\bigcup_{i\le t\le n-1}E(v_t,v_{t+1})$, such that the congestion caused by paths in $\bigcup_i({\mathcal{R}}_i^{\operatorname{left}}\cup {\mathcal{R}}_i^{\operatorname{right}})$ is at most $\ceil{1/\alpha}$.
\end{observation}
\section{An Algorithm for Narrow Instances -- Proof of \Cref{lem: not many paths}}
\label{sec: computing the decomposition}
We assume that we are given a narrow instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem. Throughout this section, we denote $|E(G)|=m$. We fix some optimal solution $\phi^*$ to instance $I$.
We will gradually construct the desired family ${\mathcal{I}}$ of instances, over the course of three phases. We will employ partitions of graphs into clusters, that are defined as follows.
\begin{definition}[Partition into Clusters]
Let $H$ be a graph, and let ${\mathcal{C}}$ be a collection of subgraphs of $H$. We say that ${\mathcal{C}}$ is a \emph{partition of $H$ into clusters}, if each subgraph $C\in {\mathcal{C}}$ is a connected vertex-induced subgraph (cluster) of $H$, $\bigcup_{C\in {\mathcal{C}}}V(C)=V(H)$, and for every pair $C,C'\in {\mathcal{C}}$ of distinct subgraphs, $V(C)\cap V(C')=\emptyset$.
\end{definition}
Recall that a vertex $v\in V(G)$ is a high-degree vertex, if $\deg_G(v)\geq m/\mu^4$.
It will be convenient for us to assume that, if $u$ is a neighbor vertex of a high-degree vertex, then the degree of $u$ is $2$, and that no vertex is a neighbor of two high-degree vertices. In order to achieve this, we simply subdivide every edge that is incident to a high-degree vertex with a single vertex; if, for an edge $e=(u,u')$, both its endpoints are high-degree vertices, then we subdivide this edge with two new vertices.
Let $G'$ denote the resulting graph, and let $\Sigma'$ be the rotation system associated with $G'$, that is naturally defined from rotation system $\Sigma$ for $G$: for every vertex $v\in V(G)\cap V(G')$, the circular ordering of the edges of $\delta_G(v)=\delta_{G'}(v)$ remains the same, and for every vertex $v\in V(G')\setminus V(G)$, $|\delta_{G'}(v)|=2$, so its rotation is trivial. We denote by $I'=(G',\Sigma')$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Assume that we compute an $\eta$-decomposition ${\mathcal{I}}'$ of instance $I'$.
It is easy to verify that ${\mathcal{I}}'$ is also an $O(\eta)$-decomposition of instance $I$. This is since $|E(G')|\leq O(|E(G)|)$, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')=\mathsf{OPT}_{\mathsf{cnwrs}}(I)$, and,
if we are given, for every instance $\tilde I\in {\mathcal{I}}$, a solution $\phi(\tilde I)$, then we can efficiently compute a solution $\phi(\tilde I')$ of the same cost to the corresponding instance $\tilde I'\in {\mathcal{I}}'$. Lastly, a solution to instance $I'$ can be efficiently transformed into a solution to instance $I$ of the same cost. Therefore, from now on we will focus on decomposing instance $I'$ into a collection ${\mathcal{I}}'$ of subinstances with required properties. To simplify the notation, we denote $G'$ by $G$, $\Sigma'$ by $\Sigma$, and $I'$ by $I$. Note that $|E(G)|\leq 3m$ now holds; every neighbor of a high-degree vertex in $G$ has degree at most $2$; and no vertex is a neighbor of two high-degree vertices.
Intuitively, in order to prove \Cref{lem: not many paths}, it is enough to compute a partition ${\mathcal{C}}$ of the input graph $G$ into clusters, such that, for every cluster $C\in {\mathcal{C}}$, $|E(C)|\leq m/\mu^2$ holds; we refer to such clusters as \emph{small} clusters. Additionally, we require that the total number of edges with endpoints in different clusters is small, namely, $|E^{\textsf{out}}({\mathcal{C}})|\leq m/\mu^2$.
Once such a collection ${\mathcal{C}}$ of clusters is computed, we can use the algorithm from \Cref{thm: disengagement - main} in order to compute the desired decomposition ${\mathcal{I}}$ of instance $I$. Unfortunately, we are unable to compute such a decomposition ${\mathcal{C}}$ of graph $G$ into small clusters directly. The main obstacle to computing such a decomposition via standard techniques is that graph $G$ may contain high-degree vertices. In order to overcome this difficulty, we define a new type of clusters, called \emph{flower clusters}. Flower clusters will be used in order to isolate high-degree vertices in graph $G$. Eventually, we will compute a decomposition ${\mathcal{C}}$ of $G$ into clusters, where each cluster in ${\mathcal{C}}$ is either a small cluster or a flower cluster, and the number of edges whose endpoints lie in different clusters of ${\mathcal{C}}$ is sufficiently small. We now formally define flower clusters (see \Cref{fig: flower_cluster} for an illustration).
\begin{definition}[Flower Cluster]
We say that a subgraph $C\subseteq G$ is a \emph{flower cluster} with a center vertex $u(C)\in V(C)$ and a set ${\mathcal{X}}(C)=\set{X_1,\ldots X_k}$ of petals, if the following hold:
\begin{properties}{F}
\item $C$ is a connected vertex-induced subgraph of $G$, and for all $1\leq i\leq k$, $X_i$ is a vertex-induced subgraph of $C$; \label{prop: flower cluster vertex induced petals too}
\item $\bigcup_{i=1}^kV(X_i)=V(C)$, and for every pair $X_i,X_j\in {\mathcal{X}}$ of clusters with $i\neq j$, $X_i\cap X_j=\set{u(C)}$; \label{prop: flower cluster petal intersection}
\item the degree of $u(C)$ in $G$ is at least $m/\mu^4$, and all other vertices of $C$ have degrees below $m/\mu^4$ in $G$; \label{prop: flower center has large degree, everyone else no}
\item $|\delta_G(C)|\leq 96m/\mu^{42}$, and the total number of edges $e=(u,v)$ with $u\in V(X_i)\setminus \set{u(C)}$ and $v\in V(X_j)\setminus \set{u(C)}$ for all $1\leq i< j\leq k$ is at most $96m/\mu^{42}$; \label{prop: flower cluster small boundary size}
\item there is a partition $E_1,\ldots,E_k$ of all edges of $\delta_C(u(C))$ into subsets, such that, for all $1\leq i\leq k$, edges in $E_i$ are consecutive in the ordering ${\mathcal{O}}_{u(C)}\in \Sigma$, $|E_i|\leq m/(2\mu^4)$, and $E_i\subseteq E(X_i)$ (so in particular $\delta_C(u(C))=\delta_G(u(C))$; and \label{prop: flower cluster edges near center partition}
\item for every cluster $X_i\in {\mathcal{X}}$, there is a set ${\mathcal{Q}}_i$ of edge-disjoint paths, routing all edges of $\delta_G(X_i)\setminus \delta_G(u(C))$ to $u(C)$, such that all inner vertices on all paths of ${\mathcal{Q}}_i$ are contained in $X_i$. \label{prop: flower cluster routing}
\end{properties}
\end{definition}
\begin{figure}[h]
\centering
\subfigure[A schematic view of the petals $X_1,X_2,X_3,X_4$.]{\scalebox{0.17}{\includegraphics{figs/flower_1.jpg}}}
\hspace{30pt}
\subfigure[Paths in set ${\mathcal{Q}}_2$ for petal $X_2$ are shown in pink.]{\scalebox{0.15}{\includegraphics{figs/flower_2.jpg}}}
\caption{An illustration of a $4$-petal flower cluster.}\label{fig: flower_cluster}
\end{figure}
Notice that ${\mathcal{Q}}=\bigcup_{i=1}^k{\mathcal{Q}}_i$ is a set of edge-disjoint paths, routing all edges of $\delta_G(C)$ to vertex $u(C)$ inside $C$, and its existence certifies that a flower cluster must have $1$-bandwidth property.
The remainder of the proof of \Cref{lem: not many paths} consists of three phases. In the first phase, we compute a decomposition ${\mathcal{C}}$ of the graph $G$ into clusters, such that the total number of edges with endpoints lying in different clusters is small, and every cluster in ${\mathcal{C}}$ is either a small cluster or a flower cluster. We then use the algorithm from \Cref{thm: disengagement - main} in order to compute an initial collection ${\mathcal{I}}_1$ of subinstances of instance $I$. This collection will have all required properties, except that for some instances $\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}_1$, $|E(\tilde G)|\leq m/(2\mu)$ may not hold. We call such instances \emph{problematic}. Each such problematic instance consists of a single flower cluster $C\in {\mathcal{C}}$, and possibly some additional edges that lie in $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ (recall that $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ is the set of all edges whose endpoints belong to different clusters of ${\mathcal{C}}$). In the subsequent two phases, we consider each of the problematic instances in ${\mathcal{I}}_1$ separately, and further decompose it into smaller subinstances. Specifically, suppose we are given some problematic instance $\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}_1$ with corresponding flower cluster $C\subseteq \tilde G$, whose center is vertex $u$ and the set of petals is ${\mathcal{X}}$. Consider any petal $X\in {\mathcal{X}}$. We say that petal $X$ is \emph{routable} in graph $\tilde G$ if there is a collection ${\mathcal{P}}(X)=\set{P(e)\mid e\in \delta_{\tilde G}(X)\setminus \delta_{\tilde G}(u)}$ of paths in $\tilde G$ that cause congestion at most $4000$, such that for each edge $e\in \delta_{\tilde G}(X)\setminus \delta_{\tilde G}(u)$, path $P(e)$ has $e$ as its first edge and $u$ as its last vertex, and all inner vertices of $P(e)$ are disjoint from $X$. We show that, if every petal of a flower cluster $C$ is routable in $\tilde G$, then we can further decompose instance $\tilde I$ into a collection ${\mathcal{I}}'(\tilde I)$ smaller subinstances, using reasoning similar to that in the basic disengagement procedure (see \Cref{subsec: basic disengagement}). It is still however possible that for some resulting instance $\tilde I'=(\tilde G',\tilde \Sigma')\in {\mathcal{I}}'(\tilde I)$, $|E(\tilde G')|>m/(2\mu)$ holds. However, the disengagement procedure ensures that such an instance may not contain high-degree vertices. Therefore, applying the algorithm from Phase 1 to each such instance $\tilde I'$ will yield a decomposition of the initial instance $I$ into subinstances with all required properties. One difficulty with the approach we have just outlined is that,
if $\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}_1$ is a problematic instance with corresponding flower cluster $C\subseteq \tilde G$, then we are not guaranteed that the petals of $C$ are routable in $\tilde G$. In order to overcome this difficulty, in Phase 2, we consider each such problematic instance $\tilde I\in {\mathcal{I}}$ separately. By performing a layered well-linked decomposition and an additional disengagement step, we decompose each such instance into a collection of subinstances, such that at most one resulting subinstance contains the flower cluster, and we further modify this flower cluster to ensure that each of its petals is routable in the resulting graph. In the third phase, we perform further disengagement on instances containing flower clusters, that exploits the fact that now every petal of the flower cluster is routable. Some of the resulting instances may still contain too many edges, but, as we show, they may not contain high-degree vertices. We then perform one final disengagement on such instances in order to obtain the final decomposition. We now describe each of the three phases in turn.
\input{decomposition-phase1.tex}
\input{phase2-decomposition}
\input{Phase3-decomposition}
\subsection{Phase 1: Flower Clusters, Small Clusters, and Initial Disengagement}
\label{subsec: phase 2 flower vs small cluster partition}
The first phase consists of three steps. In the first step, we carve flower clusters out of the graph $G$. In the second step, we decompose the remainder of the graph $G$ into small clusters. Lastly, we perform a disengagement step for all resulting clusters in the third step.
\subsubsection{Step 1: Carving out Flower Clusters}
Let $S=\set{s_1,\ldots,s_k}$ be the set of all vertices of $G$ that have degree at least $m/\mu^4$. Notice that, since we have assumed that no edge connects a pair of high-degree vertices, and $|E(G)|\leq 3m$, $k\leq 3\mu^4$ must hold.
In this step, we use with the following lemma, in order to carve flower clusters out of graph $G$.
\begin{lemma}\label{lemma: carve out flower clusters}
There is an efficient algorithm that computes a collection ${\mathcal{C}}^f=\set{C^f_1,\ldots,C^f_k}$ of disjoint clusters of $G$, such that for all $1\leq i\leq k$, $C^f_i$ is a flower cluster with center $s_i$. The algorithm also computes, for each resulting flower cluster $C^f_i$, the corresponding set ${\mathcal{X}}(C^f_i)$ of petals.
\end{lemma}
\begin{proof}
We start with the following simple claim that allows us to compute an initial collection $\set{C_1,\ldots,C_k}$ of clusters with some useful properties. Eventually, for all $1\leq i\leq k$, we will define a flower cluster $C_i^f\subseteq C_i$.
\begin{claim}\label{claim: flowers initial}
There is an efficient algorithm to compute $k$ disjoint vertex-induced subgraphs $C_1,\ldots, C_k$, such that for all $1\leq i\leq k$, $s_i\in V(C_i)$, $\delta_G(s_i)\subseteq E(C_i)$, and $|\delta_G(C_i)|\leq 3m/\mu^{46}$. Additionally, the algorithm computes, for all $1\leq i\leq k$, a set ${\mathcal{Q}}_i$ of edge-disjoint paths routing the edges of $\delta_G(C_i)$ to $s_i$ inside $C_i$.
\end{claim}
\begin{proof}
We use \Cref{lem: multiway cut with paths sets} to compute, for all $1\leq i\leq k$, a set $A_i$ of vertices of $G$, such that $S\cap A_i=\set{s_i}$, and $(A_i,V(G)\setminus A_i)$ is a minimum cut separating $s_i$ from the vertices of $S\setminus\set{s_i}$ in $G$, and the vertex sets $A_1,\ldots,A_k$ are all mutually disjoint from each other. Recall that the algorithm also computes,
for all $1\leq i\leq k$, a set ${\mathcal{Q}}_i$ of edge-disjoint paths, routing the edges of $\delta_G(A_i)$ to vertex $s_i$ inside $G[A_i]$.
Recall that we have assumed that, for all $1\leq i\leq r$, for every edge $e=(s_i,v_e)\in \delta_G(s_i)$, the degree of vertex $v_e$ in $G$ is at most $2$, and $v_e$ is the neighbor of at most one vertex in $S$ -- vertex $s_i$. If such a vertex $v_e$ does not lie in $A_i$, then we move it to $A_i$ (if $v_e$ lies in some other vertex set $A_{i'}$, we remove it from that vertex set). Since the degree of $v_e$ in $G$ is $2$, this does not increase the cardinalities of edge sets $\delta_G(A_{i'})$ for any $1\leq i'\leq k$. Moreover, we can adjust the sets of paths $\set{{\mathcal{Q}}_i}_{i=1}^k$, such that, for all $1\leq i\leq k$, set ${\mathcal{Q}}_i$ remains a set of edge-disjoint paths routing the edges of $\delta_G(A_i)$ to $s_i$ inside $G[A_i]$.
For all $1\leq i\leq k$, we denote $C_i=G[A_i]$. Note that, from the maximum flow / minimum cut theorem, for each $i$, there is a collection ${\mathcal{P}}_i$ of at least $|\delta_G(C_i)|$ edge-disjoint paths connecting $s_i$ to vertices of $S\setminus \set{s_i}$. Since $|S|=k\leq 3\mu^4$, there is some vertex $s_j\in S\setminus\set{s_i}$, such that at least $|\delta_G(C_i)|/k\geq |\delta_G(C_i)|/(3\mu^4)$ of the paths in ${\mathcal{P}}_i$ connect $s_i$ to $s_j$. Since instance $I$ is non-interesting, the number of such paths must be bounded by $m/\mu^{50}$, and so for all $1\leq i\leq k$, $|\delta_G(C_i)|\leq 3m/\mu^{46}$.
\end{proof}
Next, for all $1\leq i\leq k$, we will compute a flower cluster $C^f_i\subseteq C_i$, with center $s_i$, together with a set ${\mathcal{X}}_i$ of its petals, such that $|\delta_G(C_i^f)|\leq 32m/\mu^{42}$. The following claim will complete the proof of \Cref{lemma: carve out flower clusters}.
\begin{claim}\label{claim: compute flowers}
There is an efficient algorithm, that, for all $1\leq i\leq k$, computes a flower cluster $C^f_i\subseteq C_i$ with center $s_i$, together with a set ${\mathcal{X}}_i$ of petals.
\end{claim}
\begin{proof}
Fix an index $1\leq i\leq k$. For simplicity of notation, in the remainder of the proof, we denote $C_i$ by $C$, $s_i$ by $s$, and the set ${\mathcal{Q}}_i$ of paths by ${\mathcal{Q}}$. Recall that $|\delta_G(C)|\leq 3m/\mu^{46}$.
Since $\deg_G(s)\geq m/\mu^4$, and $\delta_G(s)=\delta_C(s)$, we get that $\deg_C(s)\geq m/\mu^4$. Let $r=\floor{d_C(s)\cdot \frac{8\mu^4}{m}}$. Since $m/\mu^4 \leq \deg_C(s)\leq m$, we get that $8\leq r\leq 8\mu^4$. We compute a partition $E_1,\ldots,E_r$ of the edges of $\delta_C(s)$ into disjoint subsets, each of which contains at most $\floor{m/(2\mu^4)}$ edges that appear in the ordering ${\mathcal{O}}_{s}\in \Sigma$ consecutively. From our choice of $r$, we get that $r\cdot\floor{m/(2\mu^4)}\geq d_G(s)$, so such a partition must exist.
Consider now a graph $H$ that is defined as follows. We start with the graph $C\cup \delta_G(C)$. Let $R$ be the set of all endpoints of the edges of $\delta_G(C)$ that do not lie in $C$. We unify all such vertices into a vertex $a_0$. Next, we subdivide every edge $e\in \delta_C(s)$ with a new vertex $v_e$, and delete vertex $s$ from the resulting graph. Finally, for all $1\leq j\leq r$, we unify the vertices in set $V_j=\set{v_e\mid e\in E_j}$ into a single vertex $a_j$, obtaining the graph $H$.
Denote $Z=\set{a_0,a_1,\ldots,a_r}$. Note that $d_H(a_0)=|\delta_G(C)|\leq 3m/\mu^{46}$, so for all $1\leq j\leq r$, there are at most $3m/\mu^{46}$ edge-disjoint paths connecting $a_0$ to $a_j$.
From the definition of interesting instances, for all $1\leq j<j'\leq r$, the maximum number of edge-disjoint paths connecting $a_j$ to $a_{j'}$ is at most $m/\mu^{50}$. Therefore, for all $1\leq j\leq r$, there is a cut separating $a_{j}$ from all vertices of $Z\setminus a_j$, containing at most $3m/\mu^{46}+rm/\mu^{50}\leq 12m/\mu^{46}$ edges.
We apply the algorithm from \Cref{lem: multiway cut with paths sets} to graph $H$ and vertex set $Z$. For all $1\leq j\leq r$, we denote by $A'_j$ the set of vertices that the algorithm returns for vertex $a_j$. Recall that, if $e=(s,v)$ is an edge of $E_j$, then vertex $v$ must have degree at most $2$ in graph $G$. We can then assume without loss of generality that $v$ lies in $A_j'$; if it lies in another set $A_{j'}'$, or it does not lie in any such set, we can simply move it to $A_j'$; this will not increase the values of the cuts $|E(A'_{j'},V(C)\setminus A'_{j'})|$ for any $j'$. We can also adjust the sets of paths ${\mathcal{Q}}'_j$ that the algorithm from \Cref{lem: multiway cut with paths sets} returns for each $1\leq j\leq r$, so that paths in ${\mathcal{Q}}'_j$ are edge-disjoint and route edges of $\delta_H(A'_j)$ to vertex $a_j$ inside the cluster $H[A'_j]$.
For all $1\leq j\leq k$, let $X_j$ be the subgraph of $C$ induced by the vertex set $(A'_j\setminus\set{a_j})\cup \set{s}]$. We then let $C^f$ be the subgraph of $G$ induced by vertex set $\bigcup_{j=1}^kV(X_j)$. We claim that $C^f$ is a flower cluster with center $s$ and the set ${\mathcal{X}}=\set{X_1,\ldots,X_r}$ of petals.
We now verify that $C^f$ and ${\mathcal{X}}$ have all required properties.
First, it is immediate to verify that $C^f$ is a vertex-induced subgraph of $G$, for all $1\leq j\leq r$, $X_j$ is a vertex-induced subgraph of $C^f$, $\bigcup_{j=1}^rV(X_j)=V(C^f)$, and for all $1\leq j<j'\leq r$, $X_j\cap X_{j'}=\set{u}$. This establishes Properties \ref{prop: flower cluster vertex induced petals too} and \ref{prop: flower cluster petal intersection}.
Consider now some index $1\leq j\leq r$. Since we have assumed that, for every edge $e=(u,v)\in E_j$, $v\in A_j'$ holds, we get that $E_j\subseteq X_j$. Since $\delta_G(s)\subseteq \delta_C(s)$, and edge sets $E_1,\ldots,E_r$ partition $\delta_C(s)$, this establishes Property \ref{prop: flower cluster edges near center partition}. $\delta_G(s)\subseteq C^f$.
Recall that for all $1\leq j\leq r$, $|\delta_C(X_j)\setminus \delta_C(s)|\le 12m/\mu^{46}$ (since $(A'_j,V(H)\setminus A'_j)$ is a minimum cut in $H$ separating $a_{j}$ from all vertices of $Z\setminus a_j$, whose value we have bounded above.) Therefore, $|\delta_G(C^f)|\leq r\cdot 12m/\mu^{46}\leq (8\mu^4)\cdot (12m/\mu^{46})\leq 96m/\mu^{42}$, and moreover the total number of edges $e=(u,v)$ with $u,v$ lying in different sets $X_1,\ldots,X_r$ is also bounded by $96m/\mu^{42}$. This establishes Property \ref{prop: flower cluster small boundary}. Since no vertex of $S$ may lie in $C^f$ (as $C^f\subseteq C$), all vertices of $V(C)\setminus \set{s}$ have degrees at most $m/\mu^4$ in $G$. This establishes Property \ref{prop: flower center has large degree, everyone else no}.
Lastly, we need to establish Property \ref{prop: flower cluster routing}. Consider any petal $X_j\in {\mathcal{X}}$.
From the definition of graph $H$, and since vertex $a_0$ may not belong to set $A'_j$, we get that $\delta_H(A'_j)=\delta_G(X_j)\setminus \delta_G(s)$. Recall that the algorithm from \Cref{lem: multiway cut with paths sets} provided a set ${\mathcal{Q}}'_j$ of edge-disjoint paths that route the edges of $\delta_H(A'_j)$ to vertex $a_j$ inside the cluster $H[A'_j]$. By replacing vertex $a_j$ with vertex $s$ on each such path, we obtain a collection ${\mathcal{Q}}''_j$ of edge-disjoint paths in graph $C^f$, that route the edges of $\delta_G(X_j)$ to $s$ inside $X_j$.
We conclude that $C^f$ is a valid flower cluster with center $s$ and set ${\mathcal{X}}$ of petals.
\end{proof}
\end{proof}
\subsubsection{Step 2: Small Clusters}
Let $C_0$ be the cluster that is obtained from graph $G$ after we delete all vertices lying in $\bigcup_{C\in {\mathcal{C}}^f}V(C)$ from it, that is, $C_0=G\setminus(\bigcup_{C\in {\mathcal{C}}^f}V(C))$.
Note that, since $|{\mathcal{C}}^f|\leq 3\mu^4$, and, for all $C\in {\mathcal{C}}^f$, $|\delta_G(C)|\leq 96m/\mu^{42}$, we get that $|\bigcup_{C\in {\mathcal{C}}^f}|\delta_G(C)|\leq (3\mu^4)\cdot (96m/\mu^{42})\leq 288m/\mu^{38}$, and so $|\delta_G(C_0)|\leq 288m/\mu^{38}$.
In this step, our goal is to further decompose cluster $C_0$ into a collection ${\mathcal{C}}^s$ of clusters, that we refer to as \emph{small clusters}. We will require that each cluster $C\in {\mathcal{C}}^s$ has the $\alpha_0$-bandwidth property, for an appropriately chosen parameter $\alpha_0$, and that $|E(C)|\leq |E(G)|/\mu^2$. Moreover, we will require that the total number of edges whose endpoints lie in different clusters of ${\mathcal{C}}^f\cup {\mathcal{C}}^s$ is relatively small. We show an algorithm that either computes such a decomposition of $C_0$, or establishes that $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)$ is sufficiently large, by utilizing the following lemma; the lemma will also be used later in this section in a slightly different setting, so it is stated for a more general setting than what is needed here.
\begin{lemma}\label{lem: decomposition into small clusters}
There is an efficient algorithm, whose input consists of a graph $H$, a set $T\subseteq V(H)$ of $k$ vertices called terminals (where possibly $T=\emptyset$), and parameters $m,\tau\geq 0$, such that $ |E(H)|\leq m$, $k\leq m/(32\tau\log m)$; every vertex in $T$ has degree $1$ in $H$; and maximum vertex degree in $H$ is at most $\frac{m}{c^*\tau^3 \log^5 m}$, for a large enough constant $c^*$.
The algorithm
either correctly certifies that $\mathsf{OPT}_{\mathsf{cr}}(H)\geq \Omega \textsf{left}( \frac{m^2}{ \tau^4\log^5 m} \textsf{right} )$, or computes a collection ${\mathcal{C}}$ of disjoint vertex-induced subgraphs of $H\setminus T$ called clusters, with the following properties:
\begin{itemize}
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property, where $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}=\Omega\textsf{left}(\frac{1}{\log^{1.5}m}\textsf{right} )$;
\item $\bigcup_{C\in {\mathcal{C}}}V(C)=V(H)\setminus T$;
\item for every cluster $C\in {\mathcal{C}}$, $|E(C)|\leq m/\tau$; and
\item $|\bigcup_{C\in {\mathcal{C}}}\delta_H(C)|\leq m/\tau$.
\end{itemize}
\end{lemma}
The proof of the lemma is very similar to the proof of \Cref{thm: basic decomposition of a graph} and is delayed to Section \ref{sec: appx-decomposition-small-clusters} of Appendix.
We consider the graph $C_0^+$, that is an augmentation of cluster $C_0$. Recall that $C_0^+$ is obtained from graph $G$, by first subdividing every edge $e\in \delta_G(C_0)$ with a vertex $t_e$, setting $T=\set{t_e\mid e\in \delta_G(C_0)}$, and letting $C_0^+$ be the subgraph of the resulting graph induced by $T\cup V(C_0)$.
We apply the algorithm from \Cref{lem: decomposition into small clusters} to graph $H=C_0^+$, the set $T$ of terminals, and parameter $\tau=160\mu$.
Recall that, since $C_0$ contains no high-degree vertices, maximum vertex degree in $C_0$ is bounded by $\frac{m}{\mu^4}\leq \frac{m}{c^*\tau^3 \log^5 m}$. Recall also that $|T|=|\delta_G(C_0)|\leq 288m/\mu^{38}\leq m/(32\tau\log m)$.
If the algorithm from \Cref{lem: decomposition into small clusters} certifies that $\mathsf{OPT}_{\mathsf{cr}}(C_0^+)\geq \Omega \textsf{left}( \frac{m^2}{ \tau^4\log^5 m} \textsf{right} )$, then we terminate the algorithm and return FAIL. Notice that we are guaranteed that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \Omega \textsf{left}( \frac{m^2}{ \mu^5} \textsf{right} )$.
Therefore, we assume from now on that the algorithm from \Cref{lem: decomposition into small clusters} computed a collection ${\mathcal{C}}^s$ of disjoint clusters, such that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(C_0)$, every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property in $G$, where $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}=\Omega\textsf{left}(\frac{1}{\log^{1.5}m}\textsf{right} )$, for every cluster $C\in {\mathcal{C}}^s$, $|E(C)|\leq m/(160\mu)$; and
$|\bigcup_{C\in {\mathcal{C}}^s}\delta_G(C)|\leq m/(160\mu)$.
We refer to clusters in set ${\mathcal{C}}^s$ as \emph{small clusters}.
Let ${\mathcal{C}}={\mathcal{C}}^s\cup {\mathcal{C}}^f$. Recall that $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ is the set of all edges whose endpoints lie in different clusters of ${\mathcal{C}}$.
Since $|\bigcup_{C\in {\mathcal{C}}^s}\delta_G(C)|\leq m/(160\mu)$,
$|{\mathcal{C}}^f|\leq 3\mu^4$, and, for all $C\in {\mathcal{C}}^f$, $|\delta_G(C)|\leq 96m/\mu^{42}$, we get that:
\begin{equation}\label{eq: num of edges between clusters}
E^{\textnormal{\textsf{out}}}({\mathcal{C}})|\leq \frac m {160\mu}+(3\mu^4)\cdot \frac{96m}{\mu^{42}}\leq \frac m{80\mu}.
\end{equation}
\subsubsection{Step 3: Initial Disengagement}
In this step, we consider the set ${\mathcal{C}}={\mathcal{C}}^s\cup {\mathcal{C}}^f$ of clusters. Recall that all clusters in ${\mathcal{C}}$ are disjoint and $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$. Moreover, every cluster in ${\mathcal{C}}^s$ has the $\alpha'$-bandwidth property, for $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}$, while every cluster in ${\mathcal{C}}^f$ has $1$-bandwidth property. Since $m\geq \mu^4$, we then get that every cluster in ${\mathcal{C}}$ has the $\alpha_0=1/\log^3m$-bandwidth property.
\mynote{needs fixing -- need to restate in terms of $\nu$-decomposition}
We apply the algorithm from \Cref{thm: disengagement - main} to instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, and the set ${\mathcal{C}}$ of clusters. Let ${\mathcal{I}}_1$ be the resulting collection of subinstances of $I$ that the algorithm computes. Recall that we are guaranteed that, for each subinstance $I'=(G',\Sigma')\in {\mathcal{I}}_1$, there is at most one cluster $C\in {\mathcal{C}}$ with $E(C)\subseteq E(G')$, and all other edges of $G'$ lie in set $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$. We are also guaranteed that $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_1}|E(G')|\leq O(|E(G)|)$, and $\expect{\sum_{I'\in {\mathcal{I}}_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$.
Let ${\mathcal{A}}_1$ denote the efficient algorithm, that, given a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}_1$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{I'\in {\mathcal{I}}_1}\mathsf{cr}(\phi(I'))\textsf{right} )$.
We have therefore shown an efficient randomized algorithm that computes a $\nu_0$-decomposition ${\mathcal{I}}_1$ of instance $I$, for $\nu_0= 2^{O((\log m)^{3/4}\log\log m)}$.
We partition instances in ${\mathcal{I}}_1$ into two subsets, set ${\mathcal{I}}'_1$ containing all instances $I'=(G',\Sigma')$ with $|E(G')|\leq m/\mu$, and set ${\mathcal{I}}''_1$ containing all remaining instances. We refer to instances in ${\mathcal{I}}''_1$ as \emph{problematic instances}. Consider now any problematic instance $I'=(G',\Sigma')\in {\mathcal{I}}''_1$.
Recall that, from the guarantees of \Cref{thm: disengagement - main}, there is at most one cluster $C\in {\mathcal{C}}$ with $C\subseteq G'$, and all edges of $G'$ lie in $E(C')\cup E^{\textnormal{\textsf{out}}}({\mathcal{C}})$.
Since we are guaranteed that $|\bigcup_{C\in {\mathcal{C}}}\delta(C)|\leq |\bigcup_{C\in {\mathcal{C}}^s}\delta(C)|+|\bigcup_{C\in {\mathcal{C}}^f}\delta(C)|\leq m/(160\mu)+288m/\mu^{38}\leq m/(80\mu)$, we get that $|E(G')\setminus E(C')|\leq m/(80\mu)$, and $|E(C')|\geq m/\mu-m/(80\mu)$. In particular, cluster $C'$ may not be a small cluster, so it must be a flower cluster. We say that $C'$ is a flower cluster associated with the problematic instance $I'\in {\mathcal{I}}''_1$. In the remaining phases, we will further decompose each problematic instance into subinstances, proving the following theorem.
\newpage
\begin{theorem}\label{thm: decomposing problematic instances}
There is an efficient randomized algorithm, that, given a problematic instance $I'=(G',\Sigma')\in {\mathcal{I}}''_1$, either returns FAIL, or computes a $\nu_1$-decomposition $\tilde {\mathcal{I}}(I')$ of $I'$, where $\nu_1=2^{O((\log m)^{3/4}\log\log m)}$, such that, for each instance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}(I')$, $|E(\tilde G)|\leq m/\mu$. Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I')< m^2/\textsf{left} (\mu^{13}\cdot 2^{c'(\log m)^{3/4}\log\log m}\textsf{right} )$ for some large enough constant $c'$, then the probability that the algorithm returns FAIL is at most $1/\mu^3$.
\end{theorem}
Before providing the proof of \Cref{thm: decomposing problematic instances}, we show that the proof of \Cref{lem: not many paths} follows from it.
We apply the algorithm from \Cref{thm: decomposing problematic instances} to every problematic instance $I'\in {\mathcal{I}}''_1$. Assume first that, for each such instance $I'$, the algortihm returns a $\nu_1$-decomposition $\tilde {\mathcal{I}}(I')$ of $I'$, such that, for each instance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}(I')$, $|E(\tilde G)|\leq m/\mu$. In this case, we return the collection
${\mathcal{I}}={\mathcal{I}}'_1\cup \textsf{left}(\bigcup_{I'\in {\mathcal{I}}''_1}\tilde {\mathcal{I}}(I')\textsf{right} )$ of instances. From
from \Cref{claim: compose algs}, since $\nu_1\cdot\nu_2= 2^{O((\log m)^{3/4}\log\log m)}$, ${\mathcal{I}}$ is indeed a $\nu$-decomposition of $I$, and, from the above discussion, we are guaranteed that, for every instance $\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}$, $|E(\tilde G)|\leq m/\mu$.
If, for any problematic instance $I'\in {\mathcal{I}}''_1$, the algorithm from \Cref{thm: decomposing problematic instances} returned FAIL, then our algorithm returns FAIL as well.
Assume now that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\textsf{left} (\mu^{14}\cdot 2^{c^*(\log m)^{3/4}\log\log m}\textsf{right} )$ for some large enough constant $c^*$. We will show that in this case, the probability that our algorithm returns FAIL is at most $1/\mu$. Since we have assumed that $m>\mu^{30}$ (from the statement of \Cref{lem: not many paths}), and $m^*$ is large enough, we can assume that $\sqrt{m}>2^{c^*(\log m)^{3/4}\log\log m}$, and so $|E(G)|<m^2/\textsf{left} (\mu^{14}\cdot 2^{c^*(\log m)^{3/4}\log\log m}\textsf{right} )$.
Recall that ${\mathcal{I}}_1$ is a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition of instance $I$, and so:
$$\expect{\sum_{I'\in {\mathcal{I}}_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} ).$$
In particular, there is some constant $c$, such that $\expect{\sum_{I'\in {\mathcal{I}}_1''}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq 2^{c(\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$. Let ${\cal{E}}$ be the bad event that $\sum_{I'\in {\mathcal{I}}_1''}\mathsf{OPT}_{\mathsf{cnwrs}}(I')> 8\mu\cdot 2^{c(\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$. From Markov's inequality, the probability that ${\cal{E}}$ happens is at most $1/(8\mu)$.
Assume now that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\textsf{left} (\mu^{14}\cdot 2^{c^*(\log m)^{3/4}\log\log m}\textsf{right} )$, and that the bad event ${\cal{E}}$ does not happen. Then for every problematic instance $I'\in {\mathcal{I}}''_1$:
\[
\begin{split}
\mathsf{OPT}_{\mathsf{cnwrs}}(I')&\leq 8\mu\cdot 2^{c(\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\\
&\leq 8\mu\cdot 2^{c(\log m)^{3/4}\log\log m)}\cdot \frac{2m^2}{\mu^{14}\cdot 2^{c^*(\log m)^{3/4}\log\log m}}\\
&\leq \frac{m^2}{\mu^{13}\cdot 2^{c'(\log m)^{3/4}\log\log m}},
\end{split}
\]
where $c'$ is the constant from \Cref{thm: decomposing problematic instances}, assuming that $c^*$ is a large enough constant. Therefore, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\textsf{left} (\mu^{14}\cdot 2^{c^*(\log m)^{3/4}\log\log m}\textsf{right} )$ and Event ${\cal{E}}$ does not happen, then, for every problematic instance $I'\in {\mathcal{I}}''_1$, the probability that the algorithm from
\Cref{thm: decomposing problematic instances} returns FAIL is at most $1/(\mu^3)$. Since $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_1}|E(G')|\leq O(|E(G)|)$, and, for every problematic instance $I''=(G',\Sigma')\in {\mathcal{I}}''_1$, $|E(G')|\geq m/\mu$, we get that $|{\mathcal{I}}''_1|\leq O(\mu)$. Therefore, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\textsf{left} (\mu^{14}\cdot 2^{c^*(\log m)^{3/4}\log\log m}\textsf{right} )$ and ${\cal{E}}$ does not happen, then the probability that the algorithm from
\Cref{thm: decomposing problematic instances} returns FAIL for any problematic instance $I'\in {\mathcal{I}}''_2$ is at most $1/(2\mu)$. Since the probability that event ${\cal{E}}$ happens is at most $1/(8\mu)$, we get that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\textsf{left} (\mu^{14}\cdot 2^{c^*(\log m)^{3/4}\log\log m}\textsf{right} )$, then the probability that our algorithm returns FAIL is at most $1/\mu$.
\iffalse
\mynote{ please restate the thm in terms of computing a $\nu'$-decomposition}
\begin{theorem}\label{thm: decomposing problematic instances}
There is an efficient randomized algorithm, that, given a problematic instance $I'=(G',\Sigma')\in {\mathcal{I}}''_1$, computes a collection $\tilde {\mathcal{I}}(I')$ of subinstances of $I'$, with the following properties:
\begin{itemize}
\item $\sum_{\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}(I')}|E(\tilde G)|\leq O(|E(G')|)$;
\item for each subinstance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}$, $|E(\tilde G)|\leq m/\mu$;
and
\item $\expect{\sum_{\tilde I\in \tilde {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} )$.
\end{itemize}
Additionally, there is an efficient algorithm ${\mathcal{A}}(I')$, that, given a solution $\phi(\tilde I)$ to each instance $\tilde I\in \tilde {\mathcal{I}}$, computes a solution $\phi$ to instance $I'$, with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{\tilde I\in \tilde {\mathcal{I}}(I')}\mathsf{cr}(\phi(\tilde I))\textsf{right} )$.
\end{theorem}
\fi
\iffalse
Indeed, we let ${\mathcal{I}}={\mathcal{I}}'_1\cup \textsf{left} (\bigcup_{I'\in {\mathcal{I}}''_1}\tilde {\mathcal{I}}(I')\textsf{right} )$. It is immediate to verify that for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\leq m/\mu$. Additionally:
\[\begin{split}
\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|&=\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'_1}|E(G')|+\sum_{I'\in {\mathcal{I}}''_1}\sum_{\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}(I')}|E(\tilde G)|\\
&\leq \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'_1}|E(G')|+\sum_{I'=(G',\Sigma')\in {\mathcal{I}}''_1}O(|E(G')|)\\
&\leq O(|E(G)|).
\end{split}\]
Similarly:
\[\begin{split}
\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}&=\sum_{I'\in {\mathcal{I}}'_1}\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I')}+\sum_{I'\in {\mathcal{I}}''_1}\sum_{\tilde I\in \tilde {\mathcal{I}}(I')}\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)}\\
&\leq \sum_{I'\in {\mathcal{I}}'_1}\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I')} +\sum_{I'=(G',\Sigma')\in {\mathcal{I}}''_1}\textsf{left} (\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I')}+|E(G')|\textsf{right} )\cdot 2^{O((\log m)^{3/4}\log\log m)}\\
&\leq \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot 2^{O((\log m)^{3/4}\log\log m)}.
\end{split}\]
Lastly, assume that we are given, for each instance $I'\in {\mathcal{I}}_1$, a feasible solution $\phi(I')$ to $I'$. For every instance $I''\in {\mathcal{I}}''_1$, we use Algorithm ${\mathcal{A}}(I'')$, that computes a solution $\phi(I'')$ to instance $I''$ of cost at most $O\textsf{left} (\sum_{\tilde I\in \tilde {\mathcal{I}}(I'')}\mathsf{cr}(\phi(\tilde I))\textsf{right} )$. We then apply algorithm ${\mathcal{A}}_1$ to the resulting solutions to all instances in ${\mathcal{I}}_1$, to obtain a solution to instance $I$ of cost at most $O\textsf{left} (\sum_{ I'\in {\mathcal{I}}_1}\mathsf{cr}(\phi(I))\textsf{right} )$.
\fi
In order to complete the proof of \Cref{lem: not many paths}, it is now enough to prove \Cref{thm: decomposing problematic instances}. From now on, we fix a single a problematic instance $I'=(G',\Sigma')\in {\mathcal{I}}''_1$, and its corresponding flower cluster $C'\in{\mathcal{C}}^f$, and provide an algorithm to compute a decomposition of $I'$ into subclusters with required properties.
\subsection{Phase 1: Flower Clusters, Small Clusters, and Initial Disengagement}
\label{subsec: phase 2 flower vs small cluster partition}
The first phase consists of three steps. In the first step, we carve flower clusters out of the graph $G$. In the second step, we decompose the remainder of the graph $G$ into small clusters. Lastly, we perform a disengagement step for all resulting clusters in the third step.
\subsubsection{Step 1: Carving out Flower Clusters}
Let $S=\set{s_1,\ldots,s_k}$ be the set of all vertices of $G$ that have degree at least $m/\mu^4$. Notice that, since we have assumed that no edge connects a pair of high-degree vertices, and $|E(G)|\leq 3m$, $k\leq 3\mu^4$ must hold.
In this step, we use with the following lemma, in order to carve flower clusters out of graph $G$.
\begin{lemma}\label{lemma: carve out flower clusters}
There is an efficient algorithm that computes a collection ${\mathcal{C}}^f=\set{C^f_1,\ldots,C^f_k}$ of disjoint clusters of $G$, such that for all $1\leq i\leq k$, $C^f_i$ is a flower cluster with center $s_i$. The algorithm also computes, for each resulting flower cluster $C^f_i$, the corresponding set ${\mathcal{X}}(C^f_i)$ of petals.
\end{lemma}
\begin{proof}
We start with the following simple claim that allows us to compute an initial collection $\set{C_1,\ldots,C_k}$ of clusters with some useful properties. Eventually, for all $1\leq i\leq k$, we will define a flower cluster $C_i^f\subseteq C_i$.
\begin{claim}\label{claim: flowers initial}
There is an efficient algorithm to compute $k$ disjoint clusters $C_1,\ldots, C_k$ of $G$, such that for all $1\leq i\leq k$, $s_i\in V(C_i)$, $\delta_G(s_i)\subseteq E(C_i)$, and $|\delta_G(C_i)|\leq 9m/\mu^{46}$. Additionally, the algorithm computes, for all $1\leq i\leq k$, a set ${\mathcal{Q}}_i$ of edge-disjoint paths routing the edges of $\delta_G(C_i)$ to $s_i$, such that all inner vertices of every path lie in $C_i$.
\end{claim}
\begin{proof}
We use the algorithm from \Cref{lem: multiway cut with paths sets} to compute, for all $1\leq i\leq k$, a set $A_i$ of vertices of $G$, such that $S\cap A_i=\set{s_i}$, and $(A_i,V(G)\setminus A_i)$ is a minimum cut separating $s_i$ from the vertices of $S\setminus\set{s_i}$ in $G$, and the vertex sets $A_1,\ldots,A_k$ are mutually disjoint. Recall that the algorithm also computes,
for all $1\leq i\leq k$, a set ${\mathcal{Q}}_i$ of edge-disjoint paths, routing the edges of $\delta_G(A_i)$ to vertex $s_i$, with all inner vertices on every path of ${\mathcal{Q}}_i$ lying in $A_i$.
Recall that we have assumed that, for all $1\leq i\leq r$, for every edge $e=(s_i,v_e)\in \delta_G(s_i)$, the degree of vertex $v_e$ in $G$ is at most $2$, and $v_e$ is the neighbor of at most one vertex in $S$ -- vertex $s_i$. If such a vertex $v_e$ does not lie in $A_i$, then we move it to $A_i$ (if $v_e$ lies in some other vertex set $A_{i'}$, we remove it from that vertex set). Since the degree of $v_e$ in $G$ is $2$, this does not increase the cardinalities of edge sets $\delta_G(A_{i'})$ for any $1\leq i'\leq k$. Moreover, we can adjust the sets of paths $\set{{\mathcal{Q}}_i}_{i=1}^k$, such that, for all $1\leq i\leq k$, set ${\mathcal{Q}}_i$ remains a set of edge-disjoint paths routing the edges of $\delta_G(A_i)$ to $s_i$, with all inner vertices on every path lying in $A_i$.
For all $1\leq i\leq k$, we let $C_i=G[A_i]$. Note that, from the max-flow / min-cut theorem, for each $i$, there is a collection ${\mathcal{P}}_i$ of at least $|\delta_G(C_i)|$ edge-disjoint paths connecting $s_i$ to vertices of $S\setminus \set{s_i}$. Since $|S|=k\leq 3\mu^4$, there is some vertex $s_j\in S\setminus\set{s_i}$, such that at least $|\delta_G(C_i)|/k\geq |\delta_G(C_i)|/(3\mu^4)$ of the paths in ${\mathcal{P}}_i$ connect $s_i$ to $s_j$. Since the original instance $I$ that served as input to \Cref{lem: not many paths} is narrow, from \Cref{obs: narrow prop 2}, the number of such paths must be bounded by $2\ceil{m/\mu^{50}}\le 3m/\mu^{50}$, and so for all $1\leq i\leq k$, $|\delta_G(C_i)|\leq 9m/\mu^{46}$.
\end{proof}
The following claim will complete the proof of \Cref{lemma: carve out flower clusters}.
\begin{claim}\label{claim: compute flowers}
There is an efficient algorithm, that, for all $1\leq i\leq k$, computes a flower cluster $C^f_i\subseteq C_i$ with center $s_i$, together with a set ${\mathcal{X}}_i$ of petals.
\end{claim}
\begin{proof}
Fix an index $1\leq i\leq k$. For simplicity of notation, in the remainder of the proof, we denote $C_i$ by $C$, $s_i$ by $s$, and the set ${\mathcal{Q}}_i$ of paths by ${\mathcal{Q}}$. Recall that $|\delta_G(C)|\leq 9m/\mu^{46}$.
Since $\deg_G(s)\geq m/\mu^4$, and $\delta_G(s)=\delta_C(s)$, we get that $\deg_C(s)\geq m/\mu^4$. Let $r=\floor{\deg_C(s)\cdot \frac{6\mu^4}{m}}$. Since $m/\mu^4 \leq \deg_C(s)\leq m$, we get that $6\leq r\leq 6\mu^4$. We compute a partition $E_1,\ldots,E_r$ of the edges of $\delta_C(s)$ into disjoint subsets, each of which contains at most $\floor{m/(2\mu^4)}$ edges that appear in the ordering ${\mathcal{O}}_{s}\in \Sigma$ consecutively. From our choice of $r$, we get that $r\cdot\floor{m/(2\mu^4)}\geq \deg_G(s)$, so such a partition must exist.
Consider now a graph $H$ that is defined as follows. We start with the graph $C\cup \delta_G(C)$. Let $R$ be the set of all endpoints of the edges of $\delta_G(C)$ that do not lie in $C$. We unify all vertices of $R$ into a vertex $a_0$. Next, we subdivide every edge $e\in \delta_C(s)$ with a new vertex $v_e$, and delete vertex $s$ from the resulting graph. Finally, for all $1\leq j\leq r$, we unify the vertices in set $V_j=\set{v_e\mid e\in E_j}$ into a single vertex $a_j$, obtaining the graph $H$.
Denote $Z=\set{a_0,a_1,\ldots,a_r}$. Note that $\deg_H(a_0)=|\delta_G(C)|\leq 9m/\mu^{46}$, so for all $1\leq j\leq r$, there are at most $9m/\mu^{46}$ edge-disjoint paths connecting $a_0$ to $a_j$.
Since the original instance $I$ that served as input to \Cref{lem: not many paths} is a narrow instance, from \Cref{obs: narrow prop 2}, for all $1\leq j<j'\leq r$, the maximum number of edge-disjoint paths connecting $a_j$ to $a_{j'}$ is at most $2m/\mu^{50}$. Therefore, for all $1\leq j\leq r$, there is a cut separating $a_{j}$ from all vertices of $Z\setminus a_j$, containing at most $9m/\mu^{46}+2rm/\mu^{50}\leq 16m/\mu^{46}$ edges.
We apply the algorithm from \Cref{lem: multiway cut with paths sets} to graph $H$ and vertex set $Z$. For all $1\leq j\leq r$, we denote by $A'_j$ the set of vertices that the algorithm returns for vertex $a_j$. Recall that, if $e=(s,x)$ is an edge of $E_j$, then vertex $x$ must have degree at most $2$ in graph $G$. We can then assume without loss of generality that $x$ lies in $A_j'$; if it lies in another set $A_{j'}'$, or it does not lie in any such set, we can simply move it to $A_j'$; this will not increase the values of the cuts $|E(A'_{j'},V(C)\setminus A'_{j'})|$ for any $j'$. We can also adjust the sets $\set{{\mathcal{Q}}'_j}$ of paths that the algorithm from \Cref{lem: multiway cut with paths sets} returns for each $1\leq j\leq r$, so that the paths in ${\mathcal{Q}}'_j$ are edge-disjoint and route edges of $\delta_H(A'_j)$ to vertex $a_j$, with all inner vertices on every path lying in $A'_j$.
For all $1\leq j\leq k$, let $X_j$ be the subgraph of $C$ induced by the vertex set $(A'_j\setminus\set{a_j})\cup \set{s}$. We then let $C^f$ be the subgraph of $G$ induced by vertex set $\bigcup_{j=1}^kV(X_j)$. We claim that $C^f$ is a flower cluster with center $s$ and set ${\mathcal{X}}=\set{X_1,\ldots,X_r}$ of petals.
We now verify that $C^f$ and ${\mathcal{X}}$ have all required properties.
First, it is immediate to verify that $C^f$ is a vertex-induced subgraph of $G$, and that for all $1\leq j\leq r$, $X_j$ is a vertex-induced subgraph of $C^f$ with $\bigcup_{j=1}^rV(X_j)=V(C^f)$, and, for all $1\leq j<j'\leq r$, $X_j\cap X_{j'}=\set{u}$. Additionally, we can assume that $C$ is connected: if this is not the case, then connected components of $C$ that are disjoint from $u$ can be discarded. This establishes Properties \ref{prop: flower cluster vertex induced petals too} and \ref{prop: flower cluster petal intersection}.
Consider now some index $1\leq j\leq r$. Since we have assumed that, for every edge $e=(u,v)\in E_j$, $v\in A_j'$ holds, we get that $E_j\subseteq X_j$. Since $\delta_G(s)\subseteq \delta_C(s)$, and edge sets $E_1,\ldots,E_r$ partition $\delta_C(s)$, this establishes Property \ref{prop: flower cluster edges near center partition}. $\delta_G(s)\subseteq E(C^f)$.
Recall that for all $1\leq j\leq r$, $|\delta_C(X_j)\setminus \delta_C(s)|\le 16m/\mu^{46}$ (since $(A'_j,V(H)\setminus A'_j)$ is a minimum cut in $H$ separating $a_{j}$ from all vertices of $Z\setminus a_j$, whose value we have bounded above). Therefore, $|\delta_G(C^f)|\leq r\cdot 16m/\mu^{46}\leq (6\mu^4)\cdot (16m/\mu^{46})\leq 96m/\mu^{42}$, and moreover the total number of edges $e=(u,v)$ with $u,v$ lying in different sets $X_1,\ldots,X_r$ is also bounded by $96m/\mu^{42}$. This establishes Property \ref{prop: flower cluster small boundary size}. Since no vertex of $S$ may lie in $C^f$ (as $C^f\subseteq C$), all vertices of $V(C)\setminus \set{s}$ have degrees at most $m/\mu^4$ in $G$. This establishes Property \ref{prop: flower center has large degree, everyone else no}.
Lastly, we need to establish Property \ref{prop: flower cluster routing}. Consider any petal $X_j\in {\mathcal{X}}$.
From the definition of graph $H$, and since vertex $a_0$ may not belong to set $A'_j$, we get that $\delta_H(A'_j)=\delta_G(X_j)\setminus \delta_G(s)$. Recall that the algorithm from \Cref{lem: multiway cut with paths sets} provided a set ${\mathcal{Q}}'_j$ of edge-disjoint paths that route the edges of $\delta_H(A'_j)$ to vertex $a_j$, with all inner vertices on the paths lying in $A'_j$. By replacing vertex $a_j$ with vertex $s$ on each such path, we obtain a collection ${\mathcal{Q}}''_j$ of edge-disjoint paths in graph $C^f$, that route the edges of $\delta_G(X_j)$ to $s$, with all inner vertices on all paths lying in $X_j$.
We conclude that $C^f$ is a valid flower cluster with center $s$ and set ${\mathcal{X}}$ of petals.
\end{proof}
\end{proof}
\subsubsection{Step 2: Small Clusters}
Let $C_0$ be the cluster that is obtained from graph $G$ after we delete all vertices lying in $\bigcup_{C\in {\mathcal{C}}^f}V(C)$ from it, that is, $C_0=G\setminus(\bigcup_{C\in {\mathcal{C}}^f}V(C))$.
Note that, since $|{\mathcal{C}}^f|\leq 3\mu^4$, and, for all $C\in {\mathcal{C}}^f$, $|\delta_G(C)|\leq 96m/\mu^{42}$, we get that $|\bigcup_{C\in {\mathcal{C}}^f}\delta_G(C)|\leq (3\mu^4)\cdot (96m/\mu^{42})\leq 288m/\mu^{38}$, and so $|\delta_G(C_0)|\leq 288m/\mu^{38}$.
In this step, our goal is to further decompose cluster $C_0$ into a collection ${\mathcal{C}}^s$ of clusters, that we refer to as \emph{small clusters}. We will require that each cluster $C\in {\mathcal{C}}^s$ has the $\alpha_0$-bandwidth property, for an appropriately chosen parameter $\alpha_0$, and that $|E(C)|\leq |E(G)|/\mu^2$. Moreover, we will require that the total number of edges whose endpoints lie in different clusters of ${\mathcal{C}}^f\cup {\mathcal{C}}^s$ is relatively small. We show an algorithm that either computes such a decomposition of $C_0$, or establishes that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is sufficiently large, by utilizing the following lemma; the lemma will also be used later in this section in a slightly different setting, so it is stated for a more general setting than what is needed here.
\begin{lemma}\label{lem: decomposition into small clusters}
There is an efficient algorithm, whose input consists of a graph $H$, a set $T\subseteq V(H)$ of $k$ vertices called terminals (where possibly $T=\emptyset$), and parameters $m,\tau\geq 0$, such that $ |E(H)|\leq m$,
$k\leq m/(64\tau\log m)$; every vertex in $T$ has degree $1$ in $H$; and maximum vertex degree in $H$ is at most $\frac{m}{\check c\tau^3 \log^5 m}$, for a large enough constant $\check c$.
The algorithm
either correctly certifies that $\mathsf{OPT}_{\mathsf{cr}}(H)\geq \Omega \textsf{left}( \frac{m^2}{ \tau^4\log^5 m} \textsf{right} )$, or computes a collection ${\mathcal{C}}$ of disjoint clusters of $H\setminus T$ with the following properties:
\begin{itemize}
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property, where $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}=\Omega\textsf{left}(\frac{1}{\log^{1.5}m}\textsf{right} )$;
\item $\bigcup_{C\in {\mathcal{C}}}V(C)=V(H)\setminus T$;
\item for every cluster $C\in {\mathcal{C}}$, $|E(C)|\leq m/\tau$; and
\item $|\bigcup_{C\in {\mathcal{C}}}\delta_H(C)|\leq m/\tau$.
\end{itemize}
\end{lemma}
The proof of the lemma is very similar to the proof of \Cref{thm: basic decomposition of a graph} and is deferred to Section \ref{sec: appx-decomposition-small-clusters} of Appendix.
We consider the graph $C_0^+$, that is an augmentation of cluster $C_0$. Recall that $C_0^+$ is obtained from graph $G$, by first subdividing every edge $e\in \delta_G(C_0)$ with a vertex $t_e$, setting $T=\set{t_e\mid e\in \delta_G(C_0)}$, and letting $C_0^+$ be the subgraph of the resulting graph induced by $T\cup V(C_0)$.
We apply the algorithm from \Cref{lem: decomposition into small clusters} to graph $H=C_0^+$, the set $T$ of terminals, and parameter $\tau=160\mu^{1.1}$.
Recall that, since $C_0$ contains no high-degree vertices, maximum vertex degree in $C_0$ is bounded by $\frac{m}{\mu^4}\leq \frac{m}{\check c\tau^3 \log^5 m}$. Recall also that $|T|=|\delta_G(C_0)|\leq 288m/\mu^{38}\leq m/(64\tau\log m)$.
If the algorithm from \Cref{lem: decomposition into small clusters} certifies that $\mathsf{OPT}_{\mathsf{cr}}(C_0^+)\geq \Omega \textsf{left}( \frac{m^2}{ \tau^4\log^5 m} \textsf{right} )$, then we terminate the algorithm and return FAIL. Notice that we are guaranteed that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \Omega \textsf{left}( \frac{m^2}{ \mu^5} \textsf{right} )$.
Therefore, we assume from now on that the algorithm from \Cref{lem: decomposition into small clusters} computed a collection ${\mathcal{C}}^s$ of disjoint clusters, such that $\bigcup_{C\in {\mathcal{C}}^s}V(C)=V(C_0)$, every cluster $C\in {\mathcal{C}}^s$ has the $\alpha'$-bandwidth property in $G$, where $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}=\Omega\textsf{left}(\frac{1}{\log^{1.5}m}\textsf{right} )$, for every cluster $C\in {\mathcal{C}}^s$, $|E(C)|\leq m/(160\mu^{1.1})$; and
$|\bigcup_{C\in {\mathcal{C}}^s}\delta_G(C)|\leq m/(160\mu^{1.1})$.
We refer to clusters in set ${\mathcal{C}}^s$ as \emph{small clusters}.
Let ${\mathcal{C}}={\mathcal{C}}^s\cup {\mathcal{C}}^f$. Recall that $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ is the set of all edges whose endpoints lie in different clusters of ${\mathcal{C}}$.
Since $|\bigcup_{C\in {\mathcal{C}}^s}\delta_G(C)|\leq m/(160\mu^{1.1})$,
$|{\mathcal{C}}^f|\leq 3\mu^4$, and, for all $C\in {\mathcal{C}}^f$, $|\delta_G(C)|\leq 96m/\mu^{42}$, we get that:
\begin{equation}\label{eq: num of edges between clusters}
|E^{\textnormal{\textsf{out}}}({\mathcal{C}})|\leq \frac m {160\mu^{1.1}}+(3\mu^4)\cdot \frac{96m}{\mu^{42}}\leq \frac m{80\mu^{1.1}}.
\end{equation}
\subsubsection{Step 3: Initial Disengagement}
In this step, we consider the set ${\mathcal{C}}={\mathcal{C}}^s\cup {\mathcal{C}}^f$ of clusters. Recall that all clusters in ${\mathcal{C}}$ are disjoint and $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$. Moreover, every cluster in ${\mathcal{C}}^s$ has the $\alpha'$-bandwidth property, for $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}$, while every cluster in ${\mathcal{C}}^f$ has $1$-bandwidth property (which follows from the definition of flower clusters). Since $m\geq \mu^4$, we then get that every cluster in ${\mathcal{C}}$ has the $\alpha_0$-bandwidth property, for $\alpha_0=1/\log^3m'$, where $m'=|E(G)|\leq 3m$.
We apply the algorithm from \Cref{thm: disengagement - main} to instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, with parameter $m'$ replacing $m$, and the set ${\mathcal{C}}$ of clusters; parameter $\mu$ remains unchanged. Let ${\mathcal{I}}_1$ be the resulting collection of instances that the algorithm computes, that is a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition of instance $I$.
Recall that we are guaranteed that, for each instance $I'=(G',\Sigma')\in {\mathcal{I}}_1$, $I'$ is a subinstance of $I$, and there is at most one cluster $C\in {\mathcal{C}}$ with $E(C)\subseteq E(G')$, and all other edges of $G'$ lie in set $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$.
We have therefore shown an efficient randomized algorithm that computes a $\nu_0$-decomposition ${\mathcal{I}}_1$ of instance $I$, for $\nu_0= 2^{O((\log m)^{3/4}\log\log m)}$.
We partition instances in ${\mathcal{I}}_1$ into two subsets, set ${\mathcal{I}}'_1$ containing all instances $I'=(G',\Sigma')$ with $|E(G')|\leq m/(2\mu)$, and set ${\mathcal{I}}''_1$ containing all remaining instances. We refer to instances in ${\mathcal{I}}''_1$ as \emph{problematic instances}. Consider now any problematic instance $I'=(G',\Sigma')\in {\mathcal{I}}''_1$.
Recall that, from the guarantees of \Cref{thm: disengagement - main}, there is at most one cluster $C'\in {\mathcal{C}}$ with $C'\subseteq G'$, and all edges of $G'$ lie in $E(C')\cup E^{\textnormal{\textsf{out}}}({\mathcal{C}})$.
Since we are guaranteed that
$$\bigg|\bigcup_{C\in {\mathcal{C}}}\delta(C)\bigg|\leq \bigg|\bigcup_{C\in {\mathcal{C}}^s}\delta(C)\bigg|+\bigg|\bigcup_{C\in {\mathcal{C}}^f}\delta(C)\bigg|\leq m/(160\mu)+288m/\mu^{38}\leq m/(80\mu),$$ we get that $|E(G')\setminus E(C')|\leq m/(80\mu)$, and $|E(C')|\geq m/(2\mu)-m/(80\mu)$. In particular, cluster $C'$ may not be a small cluster, so it must be a flower cluster. We say that $C'$ is the flower cluster associated with the problematic instance $I'\in {\mathcal{I}}''_1$. In the remaining phases, we will further decompose each problematic instance into subinstances, proving the following theorem.
\begin{theorem}\label{thm: decomposing problematic instances}
There is an efficient randomized algorithm, that, given a problematic instance $I'=(G',\Sigma')\in {\mathcal{I}}''_1$, either returns FAIL, or computes a $\nu_1$-decomposition $\tilde {\mathcal{I}}(I')$ of $I'$, where $\nu_1=2^{O((\log m)^{3/4}\log\log m)}$, such that, for each instance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}(I')$, $|E(\tilde G)|\leq m/(2\mu)$. Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I')< m^2/\textsf{left} (\mu^{18}\cdot 2^{c'(\log m)^{3/4}\log\log m}\textsf{right} )$ for some large enough constant $c'$, then the probability that the algorithm returns FAIL is at most $1/\mu^4$. (Here, $m=|E(G)|$, where $G$ is associated with the original instance $I$, that serves as input to \Cref{lem: not many paths}).
\end{theorem}
Before providing the proof of \Cref{thm: decomposing problematic instances}, we show that the proof of \Cref{lem: not many paths} follows from it.
We apply the algorithm from \Cref{thm: decomposing problematic instances} to every problematic instance $I'\in {\mathcal{I}}''_1$. Assume first that, for each such instance $I'$, the algortihm returns a $\nu_1$-decomposition $\tilde {\mathcal{I}}(I')$ of $I'$, such that, for each instance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}(I')$, $|E(\tilde G)|\leq m/(2\mu)$. In this case, we return the collection
${\mathcal{I}}={\mathcal{I}}'_1\cup \textsf{left}(\bigcup_{I'\in {\mathcal{I}}''_1}\tilde {\mathcal{I}}(I')\textsf{right} )$ of instances. From
\Cref{claim: compose algs}, since $\nu_1\cdot\nu_2= 2^{O((\log m)^{3/4}\log\log m)}$, ${\mathcal{I}}$ is indeed a $\nu$-decomposition of $I$, and, from the above discussion, we are guaranteed that, for every instance $\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}$, $|E(\tilde G)|\leq m/(2\mu)$.
If, for any problematic instance $I'\in {\mathcal{I}}''_1$, the algorithm from \Cref{thm: decomposing problematic instances} returned FAIL, then our algorithm returns FAIL as well.
Recall that $\mu=2^{\check c(\log m^*)^{7/8}\log\log m^*}$, where $m^*\ge m$.
Assume now that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{21}$.
We will show that in this case, the probability that our algorithm returns FAIL is at most $O(1/\mu^2)$. Since we have assumed that $m>\mu^{50}$ (from the statement of \Cref{lem: not many paths}),
$|E(G)|<m^2/\mu^{20}$.
Recall that ${\mathcal{I}}_1$ is a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition of instance $I$, and so:
$$\expect{\sum_{I'\in {\mathcal{I}}_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} ).$$
In particular, there is some constant $c$, such that $\expect{\sum_{I'\in {\mathcal{I}}_1''}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq 2^{c(\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$. Let ${\cal{E}}$ be the bad event that $\sum_{I'\in {\mathcal{I}}_1''}\mathsf{OPT}_{\mathsf{cnwrs}}(I')> 8\mu^2\cdot 2^{c(\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$. From Markov's inequality, the probability that ${\cal{E}}$ happens is at most $1/(8\mu^2)$.
Assume now that
$\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{21}$, and that the bad event ${\cal{E}}$ does not happen. Then for every problematic instance $I'\in {\mathcal{I}}''_1$:
%
\[
\begin{split}
\mathsf{OPT}_{\mathsf{cnwrs}}(I')&\leq 8\mu^2\cdot 2^{c(\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\\
&\leq 8\mu^2\cdot 2^{c(\log m)^{3/4}\log\log m)}\cdot \frac{2m^2}{\mu^{21}}\\
&\leq \frac{m^2}{\mu^{18}\cdot 2^{c'(\log m)^{3/4}\log\log m}},
\end{split}
\]
where $c'$ is the constant from \Cref{thm: decomposing problematic instances}. Therefore, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{21}$ and Event ${\cal{E}}$ does not happen, then, for every problematic instance $I'\in {\mathcal{I}}''_1$, the probability that the algorithm from
\Cref{thm: decomposing problematic instances} returns FAIL is at most $1/\mu^4$. Since $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_1}|E(G')|\leq |E(G)|\cdot (\log m)^{O(1)}$, and, for every problematic instance $I''=(G',\Sigma')\in {\mathcal{I}}''_1$, $|E(G')|\geq m/(2\mu)$, we get that $|{\mathcal{I}}''_1|\leq\mu\cdot (\log m)^{O(1)}$. Therefore, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{21}$ and ${\cal{E}}$ does not happen, then the probability that the algorithm from
\Cref{thm: decomposing problematic instances} returns FAIL for any problematic instance $I'\in {\mathcal{I}}''_2$ is at most $O(1/\mu^2)$. Since the probability that event ${\cal{E}}$ happens is at most $1/(8\mu^2)$, we get that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{21}$, then the probability that our algorithm returns FAIL is at most $O(1/\mu^2)$.
\iffalse
\mynote{ please restate the thm in terms of computing a $\nu'$-decomposition}
\begin{theorem}\label{thm: decomposing problematic instances}
There is an efficient randomized algorithm, that, given a problematic instance $I'=(G',\Sigma')\in {\mathcal{I}}''_1$, computes a collection $\tilde {\mathcal{I}}(I')$ of subinstances of $I'$, with the following properties:
\begin{itemize}
\item $\sum_{\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}(I')}|E(\tilde G)|\leq O(|E(G')|)$;
\item for each subinstance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}$, $|E(\tilde G)|\leq m/\mu$;
and
\item $\expect{\sum_{\tilde I\in \tilde {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} )$.
\end{itemize}
Additionally, there is an efficient algorithm ${\mathcal{A}}(I')$, that, given a solution $\phi(\tilde I)$ to each instance $\tilde I\in \tilde {\mathcal{I}}$, computes a solution $\phi$ to instance $I'$, with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{\tilde I\in \tilde {\mathcal{I}}(I')}\mathsf{cr}(\phi(\tilde I))\textsf{right} )$.
\end{theorem}
\fi
\iffalse
Indeed, we let ${\mathcal{I}}={\mathcal{I}}'_1\cup \textsf{left} (\bigcup_{I'\in {\mathcal{I}}''_1}\tilde {\mathcal{I}}(I')\textsf{right} )$. It is immediate to verify that for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\leq m/\mu$. Additionally:
\[\begin{split}
\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|&=\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'_1}|E(G')|+\sum_{I'\in {\mathcal{I}}''_1}\sum_{\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}(I')}|E(\tilde G)|\\
&\leq \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'_1}|E(G')|+\sum_{I'=(G',\Sigma')\in {\mathcal{I}}''_1}O(|E(G')|)\\
&\leq O(|E(G)|).
\end{split}\]
Similarly:
\[\begin{split}
\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}&=\sum_{I'\in {\mathcal{I}}'_1}\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I')}+\sum_{I'\in {\mathcal{I}}''_1}\sum_{\tilde I\in \tilde {\mathcal{I}}(I')}\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)}\\
&\leq \sum_{I'\in {\mathcal{I}}'_1}\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I')} +\sum_{I'=(G',\Sigma')\in {\mathcal{I}}''_1}\textsf{left} (\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I')}+|E(G')|\textsf{right} )\cdot 2^{O((\log m)^{3/4}\log\log m)}\\
&\leq \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot 2^{O((\log m)^{3/4}\log\log m)}.
\end{split}\]
Lastly, assume that we are given, for each instance $I'\in {\mathcal{I}}_1$, a feasible solution $\phi(I')$ to $I'$. For every instance $I''\in {\mathcal{I}}''_1$, we use Algorithm ${\mathcal{A}}(I'')$, that computes a solution $\phi(I'')$ to instance $I''$ of cost at most $O\textsf{left} (\sum_{\tilde I\in \tilde {\mathcal{I}}(I'')}\mathsf{cr}(\phi(\tilde I))\textsf{right} )$. We then apply algorithm ${\mathcal{A}}_1$ to the resulting solutions to all instances in ${\mathcal{I}}_1$, to obtain a solution to instance $I$ of cost at most $O\textsf{left} (\sum_{ I'\in {\mathcal{I}}_1}\mathsf{cr}(\phi(I))\textsf{right} )$.
\fi
In order to complete the proof of \Cref{lem: not many paths}, it is now enough to prove \Cref{thm: decomposing problematic instances}. From now on, we fix a single a problematic instance $I'=(G',\Sigma')\in {\mathcal{I}}''_1$, and its corresponding flower cluster $C'\in{\mathcal{C}}^f$, and provide an algorithm to compute a decomposition of $I'$ into subclusters with required properties.
\subsection{Splitting an $F$-Subinstance}
\label{subsec: decomposition step}
In this subsection we describe our main subroutine, called \ensuremath{\mathsf{ProcSplit}}\xspace. The goal of the subroutine is to split a given $F$-instance into two. In order to simplify the statement of the main result of this subsection, we start by defining a valid input and a valid output of the subroutine. Recall that $\check I=(\check G,\check \Sigma)$ is the input instance of $\ensuremath{\mathsf{MCNwRS}}\xspace$ for \Cref{lem: many paths}, and $|E(\check G)|=\check m$.
\paragraph{Valid Input for \ensuremath{\mathsf{ProcSplit}}\xspace.}
A valid input for \ensuremath{\mathsf{ProcSplit}}\xspace consists of an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for $I$, and face $F\in {\mathcal{F}}({\mathcal K})$, such that $I=(G,\Sigma)$ is a valid $F$-subinstance of $I$. In other words, every vertex of ${\mathcal K}$ must lie in $\partial(F)$. Additionally, we require that:
\begin{itemize}
\item Graph $G$ is a subgraph of $\check G$, with $|E(G)|>\check m/\mu$, and $\Sigma$ is the rotation system for $G$ induced by $\check \Sigma$; and
\item The contracted instance $\hat I$ corresponding to instance $I$ is narrow.
\end{itemize}
We denote $r=|{\mathcal{C}}(F)|$.
\paragraph{Valid Output for \ensuremath{\mathsf{ProcSplit}}\xspace.}
A valid output for \ensuremath{\mathsf{ProcSplit}}\xspace consists of an $F$-augmenting structure ${\mathcal{A}}=(\Pi_F=\set{P_1,P_2},\set{b_u}_{u\in V(K\cup P_1\cup P_2)}, \psi')$ and a split $(I_{F_1}=(G_{F_1},\Sigma_{F_1}), I_{F_2}=(G_{F_2},\Sigma_{F_2}))$ of instance $I$ along ${\mathcal{A}}$.
The main theorem of this subsection summarizes the properties of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace.
\begin{theorem}\label{thm: procsplit}
There is an efficient randomized algorithm, that, given a valid input $I,{\mathcal K},F$ to procedure \ensuremath{\mathsf{ProcSplit}}\xspace, either returns FAIL, or produces a valid output ${\mathcal{A}}, I_{F_1}=G_{F_1},\Sigma_{F_1}), I_{F_2}=(G_{F_2},\Sigma_{F_2})$ to the procedure, such that the following hold. If there is a semi-clean solution $\phi$ to the $F$-instance $I$ with $\mathsf{cr}(\phi)\leq \frac{|E(G)|^2}{r^2\cdot \mu^{c''}}$ and $|\chi^{\mathsf{dirty}}(\phi)|\leq \frac{|E(G)|}{ \mu^{c''}\cdot r}$, where $c''$ is a large enough constant, then:
\begin{itemize}
\item $|E(G)\setminus (E(G_1)\cup E(G_2))|\leq \frac{\mathsf{cr}(\phi)\cdot \mu^{O(1)}}{|E(G)|}$; and
\item there are semi-clean solutions $\phi_1$ and $\phi_2$ to instances $I_{F_1}$ and $I_{F_2}$ respectively, with $|\chi^{\mathsf{dirty}}(\phi_1)|\leq X$, $|\chi^{\mathsf{dirty}}(\phi_2)|\leq x$, and $\expect{\mathsf{cr}^*(\phi_1)+\mathsf{cr}^*(\phi_2)}\leq \mathsf{cr}^*(\phi)$.
\end{itemize}
Lastly, if there is a semi-clean solution $\phi$ to the $F$-instance $I$ with $\mathsf{cr}(\phi)\leq \frac{|E(G)|^2}{r^2\cdot \mu^{c''}}$ and $|\chi^{\mathsf{dirty}}(\phi)|\leq \frac{|E(G)|}{ \mu^{c''}\cdot r}$, then the probability that the algorithm returns FAIL is at most $1/\mu^{32}$.
\end{theorem}
We will refer to the algorithm from \Cref{thm: procsplit} as \ensuremath{\mathsf{ProcSplit}}\xspace. The remainder of this subsection is dedicated to proving \Cref{thm: procsplit}. The algorithm consists of three steps. In the first step, we compute an augmentation $\Pi_F=\set{P_1,P_2}$, and analyze its properties. In the second step, we fix the orientations $b_u$ for vertices $u\in (V(P_1)\cup V(P_2))\setminus V(K)$. We will also show that the semi-clean solution $\phi$ to instance $I$ can be modified to obey these orientations. Then in the third step we compute both the split $(I_{F_1},I_{F_2})$, and the drawing $\psi'$ of $K\cup P_1\cup P_2$. We now describe each of the steps in turn.
\subsubsection{Step 1: Computing an $F$-Augmentation}
In order to compute an $F$-augmentation, we will first compute a large \emph{potential $F$-augmenting set}. Intuitively, a potential $F$-augmenting set is a large collection of edge-disjoint cycles or paths, each of which is a valid $F$-augmentation. As before, we partition the potential $F$-augmenting sets into three types, that will give rise to the three different types of $F$-augmnetations. The reason we do it here is that different types of $F$-augmentations will be used in Phase 1 and Phase 2 of the algorithm, and we would like our partitioning subroutine to be general enough so it can be used for both.
\begin{definition}[Potential $F$-Augmenting Path Set]
Given an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for $I$, a face $F\in {\mathcal{F}}({\mathcal K})$, a valid $F$-subinstance $I_F=(G_F,\Sigma_F)$ of $I$,
and a set ${\mathcal{P}}$ of simple edge-disjoint paths in graph $G_F$, that are internally disjoint from $\partial(F)$, we say that ${\mathcal{P}}$ is
a \emph{potential $F$-augmenting set} for the skeleton structure ${\mathcal K}$, if one of the following holds:
\begin{itemize}
\item (type-1 potential $F$-augmenting set): there is a vertex $v\in V(G_F)$ and a partition $(E_1,E_2)$ of the edges of $\delta_{G_F}(v)$ into two subsets, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma_F$, and every path in ${\mathcal{P}}$ is a cycle that contains exactly one edge of $E_1$ and one edge of $E_2$ (we will view vertex $v$ as the endpoint of the cycle); or
\item (type-2 potential $F$-augmenting set): there is a connected component $C\in \partial(F)$ and a partition $(E_1,E_2)$ of the edges of $E^F(C)$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}^F(C)$, and every path in ${\mathcal{P}}$ has an edge of $E_1$ as its first edge, and an edge of $E_2$ as its last edge; or
\item (type-3 potential $F$-augmenting path set): there are two distinct connected components $C,C'\in {\mathcal{C}}(F)$, such that each path in ${\mathcal{P}}$ originates at a vertex of $C$ and terminates at a vertex of $C'$.
\end{itemize}
\end{definition}
We note that, as with type-2 $F$-augmentations, some paths in type-2 potential $F$-augmenting set may be cycles.
In our partitioning procedure, we will start with a given potential $F$-augmenting set ${\mathcal{P}}$, and we will construct an $F$-augmentation of ${\mathcal K}$ by simply selecting a path of ${\mathcal{P}}$ uniformly at random.
The following observation is immediate from the above definitions.
\begin{observation}\label{obs: potential to actual augmentors}
For all $1\leq i\leq 3$, if ${\mathcal{P}}$ is a type-$i$ potential $F$-augmenting set, then every path $P\in {\mathcal{P}}$ is a type-$i$ $F$-augmentation of ${\mathcal K}$.
\end{observation}
Next, we show that if a contracted instance associated with a valid $F$-subinstance $I_F$ is wide, then we can efficiently compute a wide enough potential $F$-augmenting path set. The proof is deferred to Section \ref{subsec: proof of finding potential augmentors} of Appendix.
\begin{claim}\label{claim: find potential augmentors}
There is an efficient algorithm, whose input is
an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for $I$, a face $F\in {\mathcal{F}}({\mathcal K})$ with $|{\mathcal{C}}(F)|=r$, and a valid $F$-subinstance $I_F=(G_F,\Sigma_F)$ of $I$, such that the corresponding contracted instance $\hat I_F=(\hat G_F,\hat \Sigma_F)$ is wide. The algorithm either computes a potential $F$-augmenting path set ${\mathcal{P}}$ of cardinality $\ceil{\frac{|E(G_F)|}{24\mu^{50}\cdot r^2}}$, or
correctly establishes that
in every solution $\phi$ to instance $I_F$ with $\mathsf{cr}(\phi)\leq \frac{|E(G_F)|^2}{2^{20}\cdot \mu^{250}\cdot r^2}$, either some pair of edges in $\partial(F)$ cross, or the number of crossings in which edges of $\partial(F)$ participate is at least $\frac{|E(G_F)|}{64\cdot \mu^{50}\cdot r}$.
\end{claim}
==========================================
\begin{theorem}\label{lem: many paths}
There is an efficient randomized algorithm, that, given an interesting subinstance $I=(G,\Sigma)$ of $I^*$ with $m=|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$, computes a non-empty collection ${\mathcal{I}}$ of subinstances of $I$, such that:
\begin{itemize}
\item for each instance $I'=(G',\Sigma')\in {\mathcal{I}}$, if $I'$ is an interesting instance, then $|E(G')|\le m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\le |E(G)|$; and
\item
there is an efficient algorithm, that, given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq \sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) + (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}$. \znote{constants subject to change}
\end{itemize}
Additionally, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, then with probability at least $31/32$:
$$\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{O(1)}.$$
\end{theorem}
\subsection{Splitting a Subinstance: Procedure \ensuremath{\mathsf{ProcSplit}}\xspace}
\label{subsec: decomposition step}
In this subsection we describe the main subroutine that we use in our algorithm, called \ensuremath{\mathsf{ProcSplit}}\xspace. The goal of the subroutine is to split a given subinstance of the input instance $\check I$ into two. In order to simplify the statement of the main result of this subsection, we start by defining a valid input and a valid output of the subroutine. Recall that $\check I=(\check G,\check \Sigma)$ is the instance of $\ensuremath{\mathsf{MCNwRS}}\xspace$ that was obtained by subdividing the instance that serves as input to \Cref{lem: many paths}.
\paragraph{Valid Input for \ensuremath{\mathsf{ProcSplit}}\xspace.}
A valid input to Procedure \ensuremath{\mathsf{ProcSplit}}\xspace consists of a subgraph $G$ of $\check G$, a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for the subinstance $I=(G,\Sigma)$ of $\check I$ defined by $G$, and a promising set ${\mathcal{P}}$ of paths for $I$ and ${\mathcal{J}}$ of cardinality $\floor{\frac{|E(G)|}{\mu^b}}$ for some constant $b$, such that
there exists a solution $\phi$ to instance $I$ that is ${\mathcal{J}}$-valid, with $\mathsf{cr}(\phi)\leq \frac{|E(G)|^2}{\mu^{60b}}$ and $|\chi^{\mathsf{dirty}}(\phi)|\leq \frac{|E(G)|}{\mu^{60b}}$.
\iffalse
for which the following hold:
\begin{properties}{C}
\item graph $G$ is a subgraph of $\check G$, and $\Sigma$ is the rotation system for $G$ induced by $\check \Sigma$; and \label{prop valid input size}
\item there exists a solution $\phi$ to instance $I$ that is ${\mathcal{J}}$-valid, with $\mathsf{cr}(\phi)\leq \frac{|E(G)|^2}{\mu^{60b}}$ and $|\chi^{\mathsf{dirty}}(\phi)|\leq \frac{|E(G)|}{\mu^{60b}}$. \label{prop valid input drawing}
\end{properties} \fi
We emphasize that the solution $\phi$ to instance $I$ is not given as part of input and it is not known to the algorithm.
\paragraph{Valid Output for \ensuremath{\mathsf{ProcSplit}}\xspace.}
A valid output for \ensuremath{\mathsf{ProcSplit}}\xspace consists of a
${\mathcal{J}}$-enhancement structure ${\mathcal{A}}$, and
a split $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ of $I$ along ${\mathcal{A}}$. Let $P^*$ be the enhancement path of ${\mathcal{A}}$, and let $({\mathcal{J}}_1,{\mathcal{J}}_2)$ be the split of the core structure ${\mathcal{J}}$ along ${\mathcal{A}}$.
Denote $E^{\mathsf{del}}(I)=E(G)\setminus (E(G_1)\cup E(G_2))$. Let $G'=G\setminus E^{\mathsf{del}}(I)$, and let $I'=(G',\Sigma')$ be the subinstance of $\check I$ defined by graph $G'$. We require that the following properties hold:
\begin{properties}{P}
\item $|E^{\mathsf{del}}(I)|\leq \frac{2\mathsf{cr}(\phi)\cdot \mu^{38b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$, where $m=|E(G)|$;\label{prop output deleted edges}
\item $|E(G_1)|,|E(G_2)|\leq |E(G)|-\frac{|E(G)|}{32\mu^{b}}$; and \label{prop: smaller graphs}
\item there is a ${\mathcal{J}}$-valid solution $\phi'$ for instance $I'$ that has the following properties:
\begin{itemize}
\item drawing $\phi'$ is compatible with $\phi$;
\item the images of the edges of $E(J)\cup E(P^*)$ do not cross each other in $\phi'$;
\item $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$;
\item the number of crossings in which the edges of $P^*$ participate in $\phi'$ is at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}$;
\item if we let $\phi_1$ be the solution to instance $I_1$ induced by $\phi'$, then drawing $\phi_1$ is ${\mathcal{J}}_1$-valid, and similarly, if we let $\phi_2$ be the solution to instance $I_2$ induced by $\phi'$, then drawing $\phi_2$ is ${\mathcal{J}}_2$-valid.
\end{itemize} \label{prop output drawing}
\end{properties}
The main theorem of this subsection summarizes the properties of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace.
\begin{theorem}\label{thm: procsplit}
There is an efficient randomized algorithm, that, given a valid input $(G,{\mathcal{J}},{\mathcal{P}})$ to procedure \ensuremath{\mathsf{ProcSplit}}\xspace, with probability at least $1-\frac{2^{20}}{\mu^{10b}}$ computes a valid output for the procedure.
\end{theorem}
We will refer to the algorithm from \Cref{thm: procsplit} as \ensuremath{\mathsf{ProcSplit}}\xspace. The remainder of this subsection is dedicated to the proof of \Cref{thm: procsplit}. Throughout the proof, we denote by $I=(G,\Sigma)$ the subinstance of $\check I$ defined by graph $G$, by ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ the given core structure for $I$, and by $m=|E(G)|$. The algorithm consists of two steps. In the first step, we compute an enhancement $P^*$ of the core structure ${\mathcal{J}}$ and analyze its properties. In the second step, we complete the construction of the enhancement structure ${\mathcal{A}}$ and of the split $(I_1,I_2)$ of instance $I$ along ${\mathcal{A}}$. We now describe each of the steps in turn. In order to simplify the exposition, throughout this subsection, we use ``enhancement'' and ``enhancement structure'' when we refer to an enhancement of ${\mathcal{J}}$ and enhancement structure for ${\mathcal{J}}$. For simplicity of notation, we also denote the drawing $\rho_J$ of the core $J$ by $\rho$, and the face $F^*(\rho_J)$ of this drawing by $F^*$.
\input{interesting-phase1-step1}
\input{interesting-phase1-step2}
\subsection{Splitting a Subinstance: Procedure \ensuremath{\mathsf{ProcSplit}}\xspace}
\label{subsec: decomposition step}
In this subsection we provide our main procedure, called $\ensuremath{\mathsf{ProcSplit}}\xspace$, whose goal is to split a given subinstance of the input instance $\check I=(\check G,\check \Sigma)$ into two via a core enhancement structure.
\paragraph{Valid Input for \ensuremath{\mathsf{ProcSplit}}\xspace.}
A valid input to Procedure \ensuremath{\mathsf{ProcSplit}}\xspace consists of an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for $I$, and a promising set ${\mathcal{P}}$ of paths for $I$ and ${\mathcal{J}}$ of cardinality $\floor{\frac{|E(G)|}{\mu^b}}$ for some constant $b$, for which the following hold:
\begin{properties}{C}
\item graph $G$ is a subgraph of $\check G$, and $\Sigma$ is the rotation system for $G$ induced by $\check \Sigma$; and \label{prop valid input size}
\item there exists a solution $\phi$ to instance $I$ that is ${\mathcal{J}}$-valid, such that $\mathsf{cr}(\phi)\leq \frac{|E(G)|^2}{\mu^{60b}}$ and $|\chi^{\mathsf{dirty}}(\phi)|\leq \frac{|E(G)|}{ \mu^{60b}}$. \label{prop valid input drawing}
\end{properties}
We emphasize that the solution $\phi$ is not given as part of input and it is not known to the algorithm.
\paragraph{Valid Output for \ensuremath{\mathsf{ProcSplit}}\xspace.}
A valid output for \ensuremath{\mathsf{ProcSplit}}\xspace consists of a
${\mathcal{J}}$-enhancement structure ${\mathcal{A}}$, and
a split $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ of $I$ along ${\mathcal{A}}$. Let $P^*$ be the enhancement path of ${\mathcal{A}}$, and let $({\mathcal{J}}_1,{\mathcal{J}}_2)$ be the split of the core structure ${\mathcal{J}}$ along ${\mathcal{A}}$.
Denote $E^{\mathsf{del}}(I)=E(G)\setminus (E(G_1)\cup E(G_2))$. Let $G'=G\setminus E^{\mathsf{del}}(I)$, let $\Sigma'$ be the rotation system for $G'$ induced by $\Sigma$, and let $I'=(G',\Sigma')$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. We require that the following properties hold:
\begin{properties}{P}
\item $|E^{\mathsf{del}}(I)|\leq \frac{\mathsf{cr}(\phi)\cdot \mu^{2200}}{|E(G)|}+|\chi^{\mathsf{dirty}}(\phi)|$ \mynote{??};\label{prop output deleted edges}
\item $|E(G_1)|,|E(G_2)|\leq |E(G)|-\frac{|E(G)|}{32\mu^{b}}$; and \label{prop: smaller graphs}
\item there is a ${\mathcal{J}}$-valid solution $\phi'$ for instance $I'$ that is compatible with $\phi$, in which the images of the edges of $E(J)\cup E(P^*)$ do not cross each other, $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$, and the number of crosings in which the edges of $P^*$ participate is at most \mynote{?? $\mathsf{cr}(\phi)\mu^{26b}/\check m'$}.\label{prop output drawing}
\end{properties}
The main theorem of this subsection summarizes the properties of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace.
\begin{theorem}\label{thm: procsplit}
There is an efficient randomized algorithm, that, given a valid input $I=(G,\Sigma)$, ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$, ${\mathcal{P}}$ to procedure \ensuremath{\mathsf{ProcSplit}}\xspace, with probability at least $1-1/\mu^{10b}$ computes a valid output for the procedure.
\end{theorem}
We will refer to the algorithm from \Cref{thm: procsplit} as \ensuremath{\mathsf{ProcSplit}}\xspace. The remainder of this subsection is dedicated to the proof of \Cref{thm: procsplit}. Throughout the proof, we denote $m=|E(G)|$. The algorithm consists of two steps. In the first step, we compute an enhancement $P$ of the core structure ${\mathcal{J}}$ and analyze its properties. In the second step, we comlete the construction of the enhancement structure ${\mathcal{A}}$ and of the split $(I_1,I_2)$ of instance $I$ along ${\mathcal{A}}$. We now describe each of the steps in turn. In order to simplify the exposition, throughout this subsection, we use ``enhancement'' and ``enhancement structure'' when we refer to an enhancement of ${\mathcal{J}}$ and enhancement structure for ${\mathcal{J}}$. For simplicity of notation, we also denote the drawing $\rho_J$ of the core $J$ by $\rho$, and the face $F^*_{\rho_J}$ of this drawing by $F^*$.
\input{interesting-phase1-step1}
\input{interesting-phase1-step2}
\section{Proof of Lemma \ref{lem: not many paths} -- Computing the Decomposition}
\input{decomposition-new}
\iffalse
Assume we are given a valid sub-instance $(G',\Sigma')$, where $G'$ is not $\mu$-interesting. We denote $m=|E(G')|$.
We say that a vertex $v$ is a \emph{high-degree vertex}, if its degree is at least $m/\mu^8$.
We say that a cluster $C$ of $G$ is a \emph{good cluster}, if $|E(C)|\le m/\mu^2$, and $C$ is $\alpha$-boundary-well-linked.
We say that a cluster $C$ of $G$ is a \emph{flower cluster}, iff $|E(C)|> m/\mu^2$, $C$ contains a unique high-degree vertex $u$, and there is a set $\set{C_1,\ldots,C_k}$ of clusters of $C$, such that
\begin{itemize}
\item for each $1\le i\le k$, $u\in V(C_i)$;
\item the vertex sets $\set{V(C_i)\setminus \set{u}}_{1\le i\le k}$ forms a partition of $V(C)\setminus \set{u}$;
\item for each $1\le i\le k$, there is a set ${\mathcal{Q}}_i$ of paths in $C_i$ connecting edges of $\delta(C_i)$ to $u$, such that the paths of ${\mathcal{Q}}_i$ cause congestion $O(1)$.
\end{itemize}
\fi
\newpage
The main result of this section is the following lemma.
\begin{lemma}
\label{lem:main_decomposition}
There is an efficient algorithm, that, given any sub-instance $(G',\Sigma')$ (with $|E(G')|=m$),
either correctly claims that $\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\ge |E(G')|^2/\mu^{10}$,
or computes a set ${\mathcal{C}}$ of vertex-disjoint clusters of $G'$, such that each cluster of ${\mathcal{C}}$ is either a good cluster or a flower cluster, the vertex sets $\set{V(C)}_{C\in {\mathcal{C}}}$ partitions $V(G')$, and $|E^{\textsf{out}}({\mathcal{C}})|= O(m/\mu)$.
\end{lemma}
In the remainder of this section we provide the proof of Lemma~\ref{lem:main_decomposition}.
\subsection{Special Case: No Problematic Indices}
\label{subsec: no prob edges}
\mynote{A quick note that may be useful later. Suppose we have clusters $C_1,\ldots,C_r$ organized on a path, and there is no bad index. Suppose now we apply *basic disengagement* procedure (exactly as described in Sec 6.2) iteratively, first disengaging $C_1$, and getting graph $G_1$, where $C_1$ becomes a single vertex, then disengaging $C_2$ from $G_1$, and so on. As the result, instances that we obtain are *exactly* the instances created in this section, including the specific rotation around the two contracted vertices. This means that we don't need to worry about combining the solutions of the resulting instances together - the algorithm for basic disengagement does it (applying Lemma 6.3 as is). But the algorithm for basic disengagement won't give us the bound that we need on the total solution costs of the new instances (if we use Lemma 6.2 as is, the cost may be too high, because we'll be charging to same crossings again and again). This bound has to rely on the fact that we can route every edge connecting two non-consecutive clusters via a central path.
One question that I'm wondering about is whether it is critical that, for an edge $e$ connecting $C_i$ to $C_j$, is routing path $P(e)$ has to go through the central path. I believe that there must be a portion of $P(e)$ between $C_i$ and $C_j$ that goes between the two clusters on a central path. But if before that it jumps to some clusters that lie before $C_i$ or after $C_j$ I think it does not matter to us. This may be good to understand because this may allow us to weaken the condition of "no bad index". If we decide to extend this algorithm to the case of a path-of-stars, this understanding may be helpful.}
In this subsection we show the proof of Lemma~\ref{lem: path case}, with the additional assumption that no index $i$ is problematic. That is, for each $1\le i\le k-1$, $|E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\le 100|\hat E_i|$.
Recall that we are given, for each $1\le i\le k$, a vertex $u_i$ of $C_i$, and a set ${\mathcal{Q}}_i$ of paths in $C_i$ connecting the edges of $\delta(C_i)$ to $u_i$, such that for every edge $e\in E(C_i)$, $\mathbb{E}[\cong_{C_i}({\mathcal{Q}}_i,e)^2]\le \beta$.
Recall that $E^{\textsf{out}}({\mathcal{C}})$ is the set of edges connecting distinct clusters of ${\mathcal{C}}$.
We denote by $E'$ the subset of edges in $E^{\textsf{out}}({\mathcal{C}})$ connecting a pair $C_i,C_j$ of clusters of ${\mathcal{C}}$, with $j\ge i+2$.
Therefore, $E(G)=E'\cup\textsf{left}(\bigcup_{1\le i\le k-1}\hat E_i\textsf{right})\cup \textsf{left}(\bigcup_{1\le i\le k}E(C_i)\textsf{right})$.
Recall that $H=G_{|{\mathcal{C}}}$, and the node in $H$ that represents the cluster $C_i$ in $G$ is denoted by $x_i$.
Therefore, each edge in $E'\cup \textsf{left}(\bigcup_{1\le i\le k-1}\hat E_i\textsf{right})$ corresponds to an edge in $H$, and we do not distinguish between them.
We now start to describe the algorithm that construct the sub-instances $(G_1,\Sigma_1),\ldots,(G_k,\Sigma_k)$.
The algorithm proceeds in two stages. In the first stage, we compute, for each edge of $E'$, two auxiliary paths in $G$, that we call its \emph{inner path} and \emph{outer path} respectively. In the second stage, we use these auxiliary paths to define the sub-instances and show that they satisfy the properties of \Cref{lem: path case}.
In the remainder of this section, we describe the first stage in \Cref{sec: inner and outer paths} and the second stage in \Cref{sec: disengaged instances}, and then complete the proof of \Cref{lem: path case} in \Cref{sec: path case with no problematic index}.
\subsubsection{Compute Inner and Outer Paths}
\label{sec: inner and outer paths}
In the first stage, we will compute, for each pair $i,j$ of indices (with $i\le j-2$) and for each edge $e\in E'$ that connects a vertex of $C_i$ to a vertex of $C_j$, a path $P_e$ connecting $u_i$ to $u_j$ that only uses edges of $\bigcup_{i\le t\le j-1}\hat E_t$ and edges of $\bigcup_{i\le t\le j}E(C_t)$, and a path $P^{\mathsf{out}}_e$ connecting $u_i$ to $u_j$ that only uses edge $e$ and edges of $E(C_i)\cup E(C_j)$. We call path $P_e$ the \emph{inner path} of $e$ and path $P^{\mathsf{out}}_e$ the \emph{outer path} of $e$.
We first compute the outer paths. Let $e$ be an edge of $E'$ connecting a vertex of $C_i$ to a vertex of $C_j$, so $e\in \delta(C_i)$ and $e\in \delta(C_j)$. Recall that the set ${\mathcal{Q}}_i$ contains a path $Q_i(e)$, connecting $e$ to $u_i$, and the set ${\mathcal{Q}}_j$ contains a path $Q_j(e)$, connecting $e$ to $u_j$.
We simply define its outer path $P^{\mathsf{out}}_e$ to be the union of path $Q_i(e)$ and path $Q_j(e)$.
It is clear that path $P^{\mathsf{out}}_e$ connects $u_i$ to $u_j$, and only uses edge $e$ and edges of $E(C_i)\cup E(C_j)$.
Throughout the section, we view $u_j$ as the first endpoint of $P^{\mathsf{out}}_e$ and $u_i$ as its last endpoint. We denote ${\mathcal{P}}^{\mathsf{out}}=\set{P_e^{\mathsf{out}}\mid e\in E'}$, and for each $1\le i\le k$, we denote ${\mathcal{P}}^{\mathsf{out}}_{*,i}=\set{P^{\mathsf{out}}_e \mid e\in E^{\operatorname{left}}_{i}}$ and ${\mathcal{P}}^{\mathsf{out}}_{i,*}=\set{P^{\mathsf{out}}_e \mid e\in E^{\operatorname{right}}_{i}}$.
We now compute the set ${\mathcal{P}}=\set{P_e}_{e\in E'}$ of inner paths in three steps. In the first step, we compute a set $\tilde {\mathcal{P}}$ of paths in $H$. In the second step, we transform the set $\tilde {\mathcal{P}}$ of paths in $H$ into a set ${\mathcal{P}}^*$ of paths in $G$.
In the third step, we further process paths in ${\mathcal{P}}^*$ to obtain the set ${\mathcal{P}}$ of inner paths for edges of $E'$.
We now describe each step in more details.
\paragraph{Step 1.}
In this step we process edges in $E'$ one-by-one. Throughout, for each edge $\hat e\in \bigcup_{1\le i\le k-1}\hat E_i$, we maintain an integer $z_{\hat e}$ indicating how many times the edge $\hat e$ has been used, that is initialized to be $0$.
Consider an iteration of processing an edge $e\in E'$. Assume $e$ connects $C_i$ to $C_j$ in $G$, we then pick, for each $i\le t\le j-1$, an edge $\hat e$ of $\hat E_t$ with minimum $z_{\hat e}$ over all edges of $\hat E_t$, and let $\tilde P_e$ be the path obtained by taking the union of all picked edges.
Note that, in $H$, each edge of $\hat E_t$ connects $x_t$ to $x_{t+1}$, so the path $\tilde P_e$ sequentially visits nodes $x_i, x_{i+1},\dots,x_j$.
We then increase the value of $z_{\hat e}$ by $1$ for all picked edges $\hat e$, and proceed to the next iteration. After processing all edges of $E'$, we obtain a set $\tilde{\mathcal{P}}=\set{\tilde P_e}_{e\in E'}$ of paths in $H$.
We use the following observation.
\begin{observation}
\label{obs:central_congestion}
For each edge $\hat e\in \bigcup_{1\le i\le k-1}\hat E_i$, $\cong_{H}(\tilde{\mathcal{P}},\hat e)\le 100$.
\end{observation}
\begin{proof}
From the algorithm, for each $1\le i\le k-1$, the paths of $\tilde{\mathcal{P}}$ that contains an edge of $\hat E_i$ are $\set{\tilde P_e\mid e\in \big(E_i^{\operatorname{right}} \cup E_{i+1}^{\operatorname{left}} \cup E_i^{\operatorname{over}}\big)}$.
Since $|E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\le 100|\hat E_i|$, each edge of $\hat E_i$ is used is used at most $100$ times. Observation~\ref{obs:central_congestion} then follows.
\end{proof}
\iffalse
We now further process the set $\tilde{\mathcal{P}}$ of paths to obtain another set $\tilde{\mathcal{P}}'$ of paths with some additional properties, as follows.
\paragraph{Step 2.1.} Let $\tilde G'$ be the graph obtained from $\tilde G$ by replacing each edge $e$ with $\beta$ parallel copies $e_1,\ldots,e_\beta$ connecting its endpoints. Now for each path $\tilde P\in \tilde{{\mathcal{P}}}$, we define a path $\tilde P^*$ as the union of, for each edge $e\in \tilde P$, a copy $e_t$ of $e$, such that, for all paths of $\tilde{\mathcal{P}}$ that contains the edge $e$, their corresponding paths in $\set{\tilde P^*\mid e\in \tilde P}$ contain distinct copies of $e$. We denote $\tilde{\mathcal{P}}^*=\set{\tilde P^*\mid \tilde P\in \tilde {\mathcal{P}}}$. From the above discussion, the paths of $\tilde{\mathcal{P}}^*$ are edge-disjoint.
\fi
\paragraph{Step 2.}
In this step we compute a set ${\mathcal{P}}^*$ of paths in $G$, using the paths in $\tilde{\mathcal{P}}$ and path sets $\set{{\mathcal{Q}}_i}_{1\le i\le k}$, as follows.
Recall that the set ${\mathcal{Q}}_i$ contains, for each edge $e\in\delta(C_i)$, a path $Q_i(e)$ connecting $e$ to $u_i$.
Also recall that, in graph $G$, an edge $\hat e\in \hat E_i$ belongs to both $\delta(C_i)$ and $\delta(C_{i+1})$.
Therefore, for an edge $\hat e\in \hat E_i$, the union of path $Q_i(\hat e)$ and path $Q_{i-1}(\hat e)$ is a path connecting $u_i$ to $u_{i+1}$.
Now consider an edge $e\in E'$ connecting a vertex of $C_i$ to a vertex of $C_{j}$ (with $i\le j-2$), and the corresponding path $\tilde P_e=(\hat e_i,\hat e_{i+1},\ldots,\hat e_{j-1})$ in $\tilde{\mathcal{P}}$, where $\hat e_t\in \hat E_t$ for each $i\le t\le j-1$. We define the path $P^*_e$ as the sequential concatenation of paths $Q_{i}(\hat e_i)\cup Q_{i+1}(\hat e_i), Q_{i+1}(\hat e_{i+1})\cup Q_{i+2}(\hat e_{i+1}),\ldots,Q_{j-1}(\hat e_{j-1})\cup Q_{j}(\hat e_{j-1})$.
It is clear that path $P^*_e$ only contains edges of $\bigcup_{i\le t\le j-1}\hat E_t$ and edges of $\bigcup_{i\le t\le j}E(C_t)$, and sequentially visits vertices $u_i,u_{i+1},\ldots,u_{j}$.
Denote ${\mathcal{P}}^*=\set{P^*_e\mid e\in E'}$.
From Observation~\ref{obs:central_congestion}, it is easy to see that for each edge $e\in \bigcup_{1\le i\le k}E(C_i)$, $\cong_{G}({\mathcal{P}}^*,e)\le 100\cdot\cong_{G}({\mathcal{Q}}_i,e)$.
\iffalse
It would be convenient to state our algorithm if the core paths of ${\mathcal{P}}$ are edge-disjoint.
For which, we construct another instance $(G',\Sigma')$ of \textnormal{\textsf{MCNwRS}}\xspace, as follows.
Denote ${\mathcal{Q}}=\bigcup_{1\le i\le k}{\mathcal{Q}}_i$.
We start with the instance $(G,\Sigma)$.
For each edge $e\in E({\mathcal{Q}})$, we replace $e$ with $2\cdot\cong_{{\mathcal{Q}}}(e)$ parallel copies.
For each edge $e\in E({\mathcal{P}})\setminus E({\mathcal{Q}})$, we replace $e$ with $2$ parallel copies.
For each vertex $v$ such that $\delta(v)\cap E({\mathcal{P}}\cup{\mathcal{Q}})= \emptyset$, we define its rotation ${\mathcal{O}}'_v$ to be identical to the its rotation ${\mathcal{O}}_v$ in $\Sigma$.
For each vertex $v$ such that $\delta(v)\cap E({\mathcal{P}}\cup{\mathcal{Q}})\ne \emptyset$, we define its rotation ${\mathcal{O}}'_v$ as follows. We start with the rotation ${\mathcal{O}}_v$, and then (i) replace, for each edge $e\in \delta(v)\cap E({\mathcal{Q}})$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $2\cdot\cong_{{\mathcal{Q}}}(e)$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$,
and the ordering among the copies is arbitrary; and
(ii) replace, for each edge $e\in \delta(v)\cap E({\mathcal{P}})$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $2$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$,
and the ordering among the copies is arbitrary.
We use the following observation.
\begin{observation}
$\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\le \mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\le O(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot\operatorname{poly}\log n)$.
Moreover, given a drawing $\phi\in \Phi(G,\Sigma)$, we can efficiently compute a drawing $\phi'\in \Phi(G',\Sigma')$, with $\mathsf{cr}(\phi')\le O(\mathsf{cr}(\phi)\cdot\operatorname{poly}\log n)$.
\end{observation}
\begin{proof}
\end{proof}
Now for each path $P\in {\mathcal{P}}$, we define a path $P'$ in $G'$ as the union of, for each edge $e\in P$, a copy $e_t$ of $e$, such that for all paths of ${\mathcal{P}}$ that contains the edge $e$, their corresponding paths in $\set{P'\mid P\in {\mathcal{P}}, e\in P}$ contain distinct copies of $e$. We denote ${\mathcal{P}}'=\set{P'\mid P\in {\mathcal{P}}}$.
From the above discussion, the paths of ${\mathcal{P}}'$ are edge-disjoint.
Moreover, note that $V(G)=V(G')$, and if a path $P\in {\mathcal{P}}$ connects $u_i$ to $u_j$ in $G$, then the corresponding path $P'$ connects $u_i$ to $u_j$ in $G'$.
\fi
\paragraph{Step 3.}
In this step we further process paths in ${\mathcal{P}}^*=\set{P^*_e\mid e\in E'}$ to obtain the set ${\mathcal{P}}=\set{P_e\mid e\in E'}$ of inner paths for edges of $E'$.
\iffalse
\begin{itemize}
\item $|{\mathcal{P}}|=|\tilde{\mathcal{P}}'|$;
\item for each pair $1\le i,j\le k$, the number of paths in ${\mathcal{P}}'$ connecting $u_i$ to $u_j$ is the same as that of ${\mathcal{P}}^*$; and
\item the paths in ${\mathcal{P}}^*$ are non-transversal with respect to $\Sigma'$.
\end{itemize}
\fi
For each path $P^*\in {\mathcal{P}}^*$ connecting $u_i$ to $u_{j}$ (recall that such a path visits vertices $u_i,u_{i+1},\ldots,u_j$ sequentially), we define, for each $i\le t\le j-1$, the path $P^*_{(i)}$ to be the subpath of $P^*$ between $u_{i}$ and $u_{i+1}$. For each $1\le i\le k-1$, we denote ${\mathcal{P}}^*_{(i)}=\set{P^*_{(i)}\text{ } \bigg|\text{ } u_i,u_{i+1}\in V(P^*),P^*\in {\mathcal{P}}^*}$, so all paths in ${\mathcal{P}}^*_{(i)}$ connects $u_i$ to $u_{i+1}$.
We then apply the algorithm in Lemma~\ref{lem: non_interfering_paths} to the set ${\mathcal{P}}^*_{(i)}$ of paths, and obtain a set ${\mathcal{R}}^*_{(i)}$ of $|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|$ paths connecting $u_i$ to $u_{i+1}$, such that for every $e\in E(G)$, $\cong_{G}({\mathcal{R}}^*_{(i)},e)\le \cong_{G}({\mathcal{P}}^*_{(i)},e)$, and the set ${\mathcal{R}}^*_{(i)}$ of paths are non-transversal with respect to $\Sigma$.
We now re-organize the paths in sets ${\mathcal{R}}^*_{(1)},\ldots, {\mathcal{R}}^*_{(k-1)}$ to obtain the set ${\mathcal{P}}$ of inner paths.
Note that, if we select, for each $i\le t\le j-1$, a path of ${\mathcal{R}}^*_{(i)}$, then, by taking the union of them, we can obtain a path $\hat P$ connecting $u_i$ to $u_j$.
We call the selected path of ${\mathcal{R}}^*_{(t)}$ the \emph{$t$-th segment} of $\hat P$. Therefore, a path connecting $u_i$ to $u_j$ obtained in this way is the union of its $i$-th segment, its $(i+1)$-th segment, $\ldots$ , and its $(j-1)$-th segment.
We now incrementally construct the set ${\mathcal{P}}$ of paths.
Intuitively, we will silmutaneously construct all paths of ${\mathcal{P}}$ in $k-1$ iterations, where in the $i$-th iteration, we will determine the $i$-th segment of all paths in ${\mathcal{P}}$.
Throughout, we will maintain a set $\hat{{\mathcal{P}}}=\set{\hat P_e\mid e\in E'}$ of paths, where each path $\hat P_e$ is indexed by an edge $e$ of $E'$.
Assume that the edge $e$ connects a vertex of $C_i$ to a vertex of $C_j$ (with $j\ge i+2$), then the path $\hat P_e$ is supposed to originate at $u_i$ and terminate at $u_j$.
We call $u_i$ the \emph{destined origin} of $\hat P_e$ and call $u_i$ the \emph{destined terminal} of $\hat P_e$.
Initially, $\hat{{\mathcal{P}}}$ contains $|E'|$ paths, and all paths in $\hat{{\mathcal{P}}}$ contain no edge.
We will sequentially process vertices $u_1,\ldots,u_{k-1}$, and, for each $1\le i\le k-1$, upon processing vertex $u_i$, determine which path of ${\mathcal{R}}^*_{(i)}$ serves as the $i$-th segment of which path of $\hat{\mathcal{P}}$.
We now fix an index $1\le i\le k-1$ and describe the iteration of processing vertex $u_i$. The current set $\hat{\mathcal{P}}$ of paths can be partitioned into four sets: set $\hat{\mathcal{P}}^{o}_i$ contains all paths of $\hat{\mathcal{P}}$ whose destined origin is $u_i$; set $\hat{\mathcal{P}}^{t}_i$ contains all paths of $\hat{\mathcal{P}}$ whose destined terminal is $u_i$; set $\hat{\mathcal{P}}^{\textsf{thr}}_i$ contains all paths whose destined origin is $u_{i'}$ for some index $i'<i$ and whose destined terminal is $u_{j'}$ for some index $j'>i$; and set $\hat{\mathcal{P}}\setminus (\hat{\mathcal{P}}^{o}_i\cup \hat{\mathcal{P}}^{t}_i\cup \hat{\mathcal{P}}^{\textsf{thr}}_i)$ contains all other paths.
Note that the paths in set $\hat{\mathcal{P}}^{t}_i$ and set $\hat{\mathcal{P}}\setminus (\hat{\mathcal{P}}^{o}_i\cup \hat{\mathcal{P}}^{t}_i\cup \hat{\mathcal{P}}^{\textsf{thr}}_i)$ do not contain an $i$-th segment, so in this iteration we will determine the $i$-th segment of paths in the sets $\hat{\mathcal{P}}^{o}_i$ and $\hat{\mathcal{P}}^{\textsf{thr}}_i$.
Note that the paths in $\hat{\mathcal{P}}^{o}_i$ currently contain no edge,
and the paths in $\hat{\mathcal{P}}^{\textsf{thr}}_i$ currently contain up to its $(i-1)$-th segment.
We then denote
\begin{itemize}
\item by $L^-_i$ the multi-set of the current last edges of paths in $\hat{\mathcal{P}}^{\textsf{thr}}_i$;
\item by $L^+_i$ the multi-set of the first edges of paths in ${\mathcal{R}}^*_{(i)}$ (note that these paths are currently not designated as the $i$-th segment of any path in $\hat{{\mathcal{P}}}$); and
\item by $L^{\mathsf{out}}_i$ the multi-set of last edges of paths in ${\mathcal{P}}^{\mathsf{out}}_{i,*}=\set{P^{\mathsf{out}}_e \mid e\in E^{\operatorname{right}}_i}$.
\end{itemize}
Clearly, elements in sets $L^-_i, L^+_i,L^{\mathsf{out}}_i$ are edges of $\delta(u_i)$.
We then define, for each $e\in \delta(u_i)$, $n^-_e=n_{L^-_i}(e)+n_{L^{\mathsf{out}}_i}(e)$ and
$n^+_e=n_{L^+_i}(e)$.
We use the following simple observation.
\begin{observation}
For each edge $e\in \delta(u_i)$, $\sum_{e\in \delta(u_i)}n^-_e=\sum_{e\in \delta(u_i)}n^+_e$.
\end{observation}
\begin{proof}
On one hand, $\sum_{e\in \delta(u_i)}n_{L^{\mathsf{out}}_i}(e)=|\bigcup_{j>i+1}E(C_i,C_j)|=|\hat{\mathcal{P}}^{o}_i|$, and $\sum_{e\in \delta(u_i)}n_{L^{-}_i}(e)=|L^-_i|=|\hat{\mathcal{P}}^{\textsf{thr}}_i|$.
On the other hand, recall that $|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|$, ${\mathcal{P}}^*_{(i)}=\set{P^*_{(i)}\text{ } \bigg|\text{ } u_i,u_{i+1}\in V(P^*),P^*\in {\mathcal{P}}^*}$ and the only edges in $E'$ such that path $P^*_e$ contains vertices $u_i,u_{i+1}$ are edges of $E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$. Therefore,
$\sum_{e\in \delta(u_i)}n_{L^{+}_i}(e)=|L^+_i|=|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|=|E^{\textsf{thr}}_i|+|E^{\operatorname{right}}_i|=|\hat{\mathcal{P}}^{\textsf{thr}}_i|+|\hat{\mathcal{P}}^{o}_i|$.
\end{proof}
We apply the algorithm in \Cref{obs:rerouting_matching_cong} to graph $G$, vertex $u_i$, rotation ${\mathcal{O}}_{u_i}$ and integers $\set{n^-_e,n^+_{e}}_{e\in \delta(u_i)}$.
Let $M$ be the multi-set of ordered pairs of the edges of $\delta(u_i)$ that we obtain.
We then designate:
\begin{itemize}
\item for each path $\hat P_e\in \hat{\mathcal{P}}^{\textsf{thr}}_i$ with $e^-$ as its current last edge, a path of ${\mathcal{R}}^*_{(i)}$ that contains the edge $e^+$ as its first edge with $(e^-,e^+)\in M$, as the $i$-th segment of $\hat P_e$; and
\item for each path $P^{\mathsf{out}}_e\in{\mathcal{P}}^{\mathsf{out}}_{i,*}$ with $e^-$ as its last edge (recall that we view $u_i$ as the last endpoint of such a path), a path of ${\mathcal{R}}^*_{(i)}$ that contains the edge $e^+$ as its first edge with $(e^-,e^+)\in M$, as the $i$-th segment of $\hat P_e$;
\end{itemize}
such that each path of ${\mathcal{R}}^*_{(i)}$ is assigned to exactly one path of $\hat {\mathcal{P}}^{\textsf{thr}}_i\cup {\mathcal{P}}^{\mathsf{out}}_{i,*}$.
This completes the description of the $i$-th iteration.
See Figure~\ref{fig: inner_path} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges in $\delta(u_i)$: only the last/first edges of paths are shown. Here ${\mathcal{P}}^{\mathsf{out}}_{*,i}=\set{P^{\mathsf{out}}_e \mid e\in E^{\operatorname{left}}_{i}}$.]{\scalebox{0.32}{\includegraphics{figs/inner_path_1.jpg}}}
\hspace{1pt}
\subfigure[Sets $L^-_i, L^+_i,L^{\mathsf{out}}_i$ and the pairing (shown in dash pink lines) given by the algorithm in \Cref{obs:rerouting_matching_cong}.]{
\scalebox{0.32}{\includegraphics{figs/inner_path_2.jpg}}}
\caption{An illustration of an iteration in Step 3 of computing inner paths.}\label{fig: inner_path}
\end{figure}
Let ${\mathcal{P}}$ be the set of paths that we obtain after $k-1$ iterations of processings vertices $u_1,\ldots,u_{k-1}$, and we rename the path $\hat P_e$ at the end of the algorithm by $P_e$, and say that $P_e$ is the inner path of $e$, for every edge $e\in E'$.
It is clear that $|{\mathcal{P}}|=|E'|$. It is also easy to see from the algorithm that, for each edge $e\in E'$ connecting $C_i$ to $C_j$ (with $j\ge i+2$), the inner path $P_e$ starts at $u_i$ and ends at $u_j$ (as it is supposed to), and it contains, for each $i\le t\le j-1$, a path of ${\mathcal{R}}^*_{(i)}$ as its $t$-th segment.
Therefore, path $P_e$ visits vertices $u_i,u_{i+1},\ldots,u_j$ sequentially.
The following observations are immediate from our algorithm of constructing the inner paths.
\begin{observation}
\label{obs: non_transversal_1}
For each $1\le i\le k-1$, if we denote, for each $e\in E_i^{\operatorname{right}}$, by $R_e$ the path consisting of the last edge of path $P^{\mathsf{out}}_e$ and the first edge of path $P_e$, then the paths in $\set{P_e\mid e\in \bigcup_{ i'<i<j'}E(C_{i'},C_{j'})}$ and $\set{R_e\mid e\in E_i^{\operatorname{right}}}$ are non-transversal at $u_i$.
\end{observation}
\begin{observation}
\label{obs: non_transversal_2}
The inner paths in ${\mathcal{P}}$ are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
On one hand, note that for each $1\le i\le k-1$, the set ${\mathcal{R}}^*_{(i)}$ of paths are non-transversal with respect to $\Sigma$.
Since the $i$-th segments of paths in ${\mathcal{P}}$ are paths in ${\mathcal{R}}^*_{(i)}$, it follows that the paths of ${\mathcal{P}}$ are non-transversal at all vertices of $V(G)\setminus\set{u_1,\ldots,u_k}$.
On the other hand, from the algorithm and \Cref{obs:rerouting_matching_cong}, it is easy to verify that the paths of ${\mathcal{P}}$ are also non-transversal at $u_1,\ldots,u_k$.
\end{proof}
\begin{observation}
\label{obs: edge_occupation in outer and inner paths}
Each edge $e\in E'$ belongs to exactly one outer path and no inner paths, each edge of $\bigcup_{1\le i\le k-1}\hat E_i$ belongs to no outer paths and $O(1)$ inner paths; and each edge of $\bigcup_{1\le i\le k}E(C_i)$ belongs to no outer paths and $O(\cong_G({\mathcal{Q}}_i,e))$ inner paths.
\end{observation}
\subsubsection{Construct Sub-Instances}
\label{sec: disengaged instances}
Recall that ${\mathcal{C}}=\set{C_1,\ldots,C_k}$.
In this subsection we will construct, for each cluster $C_i\in {\mathcal{C}}$, an sub-instance $I_i=(G_i,\Sigma_i)$ of $(G,\Sigma)$, such that the instances ${\mathcal{I}}=\set{I_1,\ldots,I_k}$ satisfy the properties in \Cref{lem: path case}.
For each $1\le i\le k$, recall that $E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$. We denote $E^{\textsf{thr}}_i=\bigcup_{i'\le i-1,j'\ge i+1}E(C_{i'},C_{j'})$.
\iffalse
\begin{claim}
\label{clm: rotation_distance}
There exist integers $b_1,\ldots,b_k\in \set{0,1}$, such that $\sum_{1\le i\le k-1}\mbox{\sf dist}(({\mathcal{O}}^{\operatorname{right}}_i,b_i), ({\mathcal{O}}^{\operatorname{left}}_{i+1},b_{i+1}))\le O(\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\cdot\operatorname{poly}\log n)$.
\end{claim}
\begin{proof}
\end{proof}
\fi
\iffalse
We first construct instance $(G_k,\Sigma_k)$ as follows.
The graph $G_k$ is obtained from $G$ by contracting all vertices in $G\setminus C_k$ into a single vertex, that we denote by $v^{\operatorname{left}}_k$.
Notice that the edges incident to $v^{\operatorname{left}}_k$ correspond to the edges of $\delta(C_k)$.
The ordering of a vertex of $v\ne v^{\operatorname{left}}_k$ in $\Sigma_k$ is identical to the ordering ${\mathcal{O}}_v$ of $v$ in the given rotation system $\Sigma$.
Note that $\delta_{G_k}(v^{\operatorname{left}}_k)=\delta_G(C_k)=\delta_H(u_k)$,
and there is an one-to-one correspondence between edges in $\delta_H(u_k)$ and paths in ${\mathcal{P}}^*_{(i-1)}$.
The ordering of the vertex $v^{\operatorname{left}}_k$ in $\Sigma_k$ is then defined to be ${\mathcal{O}}^{\operatorname{right}}_{k-1}$ (which is an ordering on the set ${\mathcal{P}}^*_{(i-1)}$, and is therefore also an ordering on $\delta_{G_k}(v^{\operatorname{left}}_k)$).
The instance $(G_1,\Sigma_1)$ is defined similarly.
The graph $G_1$ is obtained from $G$ by contracting all vertices in $G\setminus C_1$ into a single vertex, that we denote by $v^{\operatorname{right}}_1$.
We will not distinguish between edges incident to $v^{\operatorname{left}}_1$ and edges of $\delta(C_k)$.
The ordering of a vertex $v\ne v^{\operatorname{right}}_1$ in $\Sigma_1$ is identical to the ordering ${\mathcal{O}}_v$ in $\Sigma$.
Note that $\delta_{G_1}(v^{\operatorname{right}}_1)=\delta_G(C_1)=\delta_H(u_1)$,
and there is an one-to-one correspondence between edges in $\delta_H(u_1)$ and paths in ${\mathcal{P}}^*_{(1)}$.
The ordering of the vertex $v^{\operatorname{right}}_1$ in $\Sigma_1$ is then defined to be ${\mathcal{O}}^{\operatorname{left}}_2$ (which is an ordering on the set ${\mathcal{P}}^*_{(1)}$, and is therefore also an ordering on $\delta_{G_1}(v^{\operatorname{right}}_1)$).
\fi
\paragraph{Instances $(G_2,\Sigma_2),\ldots,(G_{k-1},\Sigma_{k-1})$.}
We first fix some index $2\le i\le k-1$ and define the instance $(G_i,\Sigma_i)$ as follows.
The graph $G_i$ is obtained from $G$ by first contracting clusters $C_1,\ldots,C_{i-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_i$, and then contracting clusters $C_{i+1},\ldots,C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_i$, and finally deleting self-loops on super-nodes $v^{\operatorname{left}}_i$ and $v^{\operatorname{right}}_i$. So $V(G_i)=V(C_i)\cup \set{ v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$. See \Cref{fig: disengaged instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edge sets in $G$. Edges of $E^{\operatorname{left}}_{i}$ and $E^{\operatorname{right}}_{i}$ are shown in green. Edges of $E^{\textsf{thr}}_{i}$ are shown in red. ]{\scalebox{0.32}{\includegraphics{figs/disengaged_instance_1.jpg}
}
\hspace{1pt}
\subfigure[Graph $G_i$. The edges incident to $v_i^{\operatorname{right}}$ are $\hat E_i \cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, and the edges incident to $v_i^{\operatorname{left}}$ are $\hat E_{i-1} \cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_{i}$.]{
\scalebox{0.32}{\includegraphics{figs/disengaged_instance_2.jpg}}}
\caption{An illustration of the disengaged instance.}\label{fig: disengaged instance}
\end{figure}
The ordering on the incident edges of a vertex $v\in V(C_i)$ in $\Sigma_i$ is defined to be ${\mathcal{O}}_v$, the rotation on vertex $v$ in the given rotation system $\Sigma$.
It remains to define the orderings on the incident edges of super-nodes $v^{\operatorname{left}}_i,v^{\operatorname{right}}_i$.
We first consider the super-node $v^{\operatorname{left}}_i$.
Note that $\delta_{G_i}(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$.
For each edge $\hat e\in \hat E_{i-1}$, recall that $Q_{i-1}(\hat e)$ is the path in $C_{i-1}$ connecting $\hat e$ to $u_{i-1}$.
For each edge $e\in E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$ that connects a vertex of $C_{i'}$ to a vertex of $C_{j'}$ with $i'<i<j'$, we denote by $W_e$ the path obtained by concatenating
(i) the subpath of $P^{\mathsf{out}}_e$ between (including) edge $e$ and its last endpoint $u_{i'}$; and
(ii) the subpath of $P_e$ between its first endpoint $u_{i'}$ and the vertex $u_{i-1}$.
Clearly, the path $W_e$ defined above connects $e$ to $u_{i-1}$.
We denote ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_e\mid e\in E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i}\cup \set{Q_{i-1}(\hat e)\mid \hat e\in \hat E_{i-1}}$, and we now define the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$ according to the paths of ${\mathcal{W}}^{\operatorname{left}}_i$ and the rotation ${\mathcal{O}}_{u_{i-1}}$, in a similar way that we define the circular ordering on $\delta_G(C)$ according to the paths in ${\mathcal{Q}}(C)$ and the ordering ${\mathcal{O}}_{u(C)}$ in Section~\ref{subsec: basic disengagement}.
Intuitively, the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ is the ordering in which the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ enter $u_{i-1}$.
Formally, for every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$, let $e^*_W$ be the last edge lying on path $W$, that must belong to $\delta_{G}(u_{i-1})$. We first define a circular ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$, as follows: the paths are ordered according to the circular ordering of their last edges $e^*_W$ in ${\mathcal{O}}_{u_{i-1}}\in \Sigma$, breaking ties arbitrarily. Since every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$ is associated with a unique edge in $\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, that serves as the first edge on path $W$, this ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ immediately defines a circular ordering of the edges of $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, that we denote by ${\mathcal{O}}^{\operatorname{left}}_i$. See Figure~\ref{fig: v_left rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i-1}=\set{\hat e_1,\ldots,\hat e_4}$, $E^{\operatorname{left}}_{i}=\set{e^g_1,e^g_2}$ and $E^{\textsf{thr}}_{i}=\set{e^r_1,e^r_2}$.
Paths of ${\mathcal{W}}^{\operatorname{left}}_i$ excluding their first edges are shown in dash lines.]{\scalebox{0.36}{\includegraphics{figs/rotation_left_1.jpg}}}
\hspace{1pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{left}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. $\delta_{G_i}(v^{\operatorname{left}}_i)=\set{\hat e_1,\hat e_2,\hat e_3,\hat e_4,e^g_1,e^g_2, e^r_1,e^r_2}$, and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on them around $v^{\operatorname{left}}_i$ is shown above.]{
\scalebox{0.28}{\includegraphics{figs/rotation_left_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_left rotation}
\end{figure}
The rotation ${\mathcal{O}}^{\operatorname{right}}_{i}$ on vertex $v^{\operatorname{right}}_i$ is defined similarly.
Note that $\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$.
For each edge $\hat e'\in \hat E_{i}$, recall that $Q_{i}(\hat e')$ is the path in $C_{i}$ connecting $\hat e'$ to $u_{i}$.
For each edge $e\in E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$ that connects a vertex of $C_{i'}$ to a vertex of $C_{j'}$ with $i'\le i<j'$, we denote by $W_e$ the path obtained by concatenating
(i) the subpath of $P^{\mathsf{out}}_e$ between (including) edge $e$ and its last endpoint $u_{i'}$; and
(ii) the subpath of $P_e$ between its first endpoint $u_{i'}$ and vertex $u_{i}$ (note that for an edge $e\in E^{\operatorname{right}}_{i}$, such a subpath only contains a single node $u_i$).
Clearly, the path $W_e$ defined above connects $e$ to $u_{i}$.
We denote ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_e\text{ }\big|\text{ } e\in E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i}\cup \set{Q_{i}(\hat e')\text{ }\big|\text{ } \hat e'\in \hat E_{i}}$.
The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ is defined in a similar way as the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$, according to the paths of ${\mathcal{W}}^{\operatorname{right}}_i$ and the rotation ${\mathcal{O}}_{u_{i}}$.
See Figure~\ref{fig: v_right rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i}=\set{\hat e_1',\ldots,\hat e_4'}$, $E^{\operatorname{left}}_{i}=\set{\tilde e^g_1,\tilde e^g_2}$ and $E^{\textsf{thr}}_{i}=\set{e^r_1,e^r_2}$.
Paths of ${\mathcal{W}}^{\operatorname{right}}_i$ excluding their first edges are shown in dash lines. ]{\scalebox{0.36}{\includegraphics{figs/rotation_right_1.jpg}
}
\hspace{1pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. $\delta_{G_i}(v^{\operatorname{right}}_i)=\set{\hat e_1',\hat e_2',\hat e_3',\hat e_4',\tilde e^g_1,\tilde e^g_2, e^r_1,e^r_2}$, and the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on them around $v^{\operatorname{right}}_i$ is shown above.]{
\scalebox{0.28}{\includegraphics{figs/rotation_right_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_right rotation}
\end{figure}
\paragraph{Instances $(G_1,\Sigma_1)$ and $(G_k,\Sigma_k)$.}
The instances $(G_1,\Sigma_1)$ and $(G_k,\Sigma_k)$ are defined similarly, but instead of two super-nodes, the graphs $G_1$ and $G_k$ contain one super-node each.
In particular, graph $G_1$ is obtained from $G$ by contracting clusters $C_2,\ldots, C_k$ into a super-node, that we denote by $v^{\operatorname{right}}_1$, and then deleting self-loops on it.
So $V(G_1)=V(C_1)\cup \set{v^{\operatorname{right}}_{1}}$ and $\delta_{G_1}(v^{\operatorname{right}}_{1})=\hat E_1\cup E^{\operatorname{right}}_1$.
The rotation of a vertex $v\in V(C_1)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{right}}_1$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{right}}_i$ for any index $2\le i\le k-1$.
Graph $G_k$ is obtained from $G$ by contracting clusters $C_1,\ldots, C_{k-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_k$, and then deleting self-loops on it.
So $V(G_k)=V(C_k)\cup \set{v^{\operatorname{left}}_{k}}$ and $\delta_{G_k}(v^{\operatorname{left}}_{k})=\hat E_{k-1}\cup E^{\operatorname{left}}_k$.
The rotation of a vertex $v\in V(C_k)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{left}}_k$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{left}}_i$ for any index $2\le i\le k-1$.
We will use the following claims later for completing the proof of \Cref{lem: path case} in the special case where there is no problematic index $i$.
\begin{observation}
\label{obs: disengaged instance size}
$\sum_{1\le i\le k}|E(G_i)|= 2m$, and for each $1\le i\le k$, $|E(G_i)|\le m/\mu$.
\end{observation}
\begin{proof}
Note that, in the sub-instances $\set{(G_i,\Sigma_i)}_{1\le i\le k}$, each graph of $\set{G_i}_{1\le i\le k}$ is obtained from $G$ by contracting some sets of clusters of ${\mathcal{C}}$ into a single super-node. Therefore, each edge of $G_i$ corresponds to an edge in $E(G)$ (and we do not distinguish between them).
So we can write $E(G_i)=E_G(C_i)\cup E_G(C_i,\overline{C_i})$.
Therefore, for each $1\le i\le k$, $|E(G_i)|=|E_G(C_i)|+|E_G(C_i,\overline{C_i})|\le |E_G(C_i)|+|E^{\textsf{out}}({\mathcal{C}})|\le m/(100\mu)+m/(100\mu)\le m/\mu$.
On the other hand, since clusters $C_1,\ldots,C_k$ are vertex-disjoint, every edge of $E(G)$ appears twice in the graphs $\set{G_i}_{1\le i\le k}$. It follows that $\sum_{1\le i\le k}|E(G_i)|= 2m$.
\end{proof}
\begin{observation}
\label{obs: rotation for stitching}
For each $1\le i\le k-1$, if we view edge sets $\delta_{G_i}(v^{\operatorname{right}}_i),\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$ as subsets of $E(G)$, then $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$, and
${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{observation}
\begin{proof}
Recall that for each $1\le i\le k-1$,
$\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})=\hat E_{i}\cup E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}$. From the definition of sets $E_i^{\textsf{thr}},E_{i+1}^{\textsf{thr}}, E^{\operatorname{right}}_i, E^{\operatorname{left}}_{i+1}$,
\[
\begin{split}
E_i^{\textsf{thr}}\cup E^{\operatorname{right}}_i
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i<j'\text{ or }i'=i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'\le i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'< i+1\le j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i+1<j'\text{ or }i'<i+1=j'}=E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}.
\end{split}
\]
Therefore, $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Moreover, from the definition of path sets ${\mathcal{W}}^{\operatorname{right}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_{i+1}$, it is not hard to verify that, for every edge $e\in \delta_{G_i}(v^{\operatorname{right}}_i)$, the path in ${\mathcal{W}}^{\operatorname{right}}_i$ that contains $e$ as its first edge is identical to the path in ${\mathcal{W}}^{\operatorname{left}}_{i+1}$ that contains $e$ as its first edge. According to the way that rotations ${\mathcal{O}}^{\operatorname{right}}_i,{\mathcal{O}}^{\operatorname{left}}_{i+1}$ are defined, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{proof}
\subsubsection{Completing the Proof of \Cref{lem: path case} in the Special Case}
\label{sec: path case with no problematic index}
In this section we complete the proof of \Cref{lem: path case} in the special case where there is no problematic index. Specifically, we use the following two claims, whose proofs will be provided later.
\begin{claim}
\label{claim: existence of good solutions special}
$\sum_{1\le i\le k}\mathsf{OPT}_{\mathsf{cnwrs}}(G_i,\Sigma_i)\le O((\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+|E(G)|)\cdot \beta)$.
\end{claim}
\begin{claim}
\label{claim: stitching the drawings together}
There is an efficient algorithm, that given, for each $1\le i\le k$, a feasible solution $\phi_i$ to the instance $(G_i,\Sigma_i)$, computes a solution to the instance $(G,\Sigma)$, such that $\mathsf{cr}(\phi)\le \sum_{1\le i\le k}\mathsf{cr}(\phi_i)$.
\end{claim}
We use the algorithm described in~\Cref{sec: disengaged instances} and~\Cref{sec: inner and outer paths}, and return the disengaged instances $(G_1,\Sigma_1),\ldots,(G_k,\Sigma_k)$ as the collection of sub-instances of $(G,\Sigma)$.
From the previous subsections, the algorithm for producing the sub-instances is efficient.
On the other hand, it follows immediately from \Cref{obs: disengaged instance size}, \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together} that the sub-instances $(G_1,\Sigma_1),\ldots,(G_k,\Sigma_k)$ satisfy the properties in \Cref{lem: path case}.
This completes the proof of \Cref{lem: path case}.
We now provide the proofs of \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together}.
$\ $
\begin{proofof}{Claim~\ref{claim: existence of good solutions special}}
We will construct, for each $1\le i\le k$, a drawing $\phi_i$ of $G_i$ that respects the rotation system $\Sigma_i$, based on the optimal drawing $\phi^*$ of the instance $(G,\Sigma)$, such that $\sum_{1\le i\le k}\mathsf{cr}(\phi_i)\le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot \beta)$. The \Cref{claim: existence of good solutions special} then follows.
\paragraph{Drawings $\phi_2,\ldots,\phi_{k-1}$.}
First we fix some index $2\le i\le k-1$, and describe the construction of the drawing $\phi_i$. We start with some definitions.
Recall that $E(G_i)=E_G(C_i)\cup (\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$.
We define the auxiliary path set ${\mathcal{W}}_i={\mathcal{W}}^{\operatorname{left}}_i\cup {\mathcal{W}}^{\operatorname{right}}_i$, so
$${\mathcal{W}}_i=\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}\cup \set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}},$$
where for each $e\in E^{\operatorname{left}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$; for each $e\in E^{\operatorname{right}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between $u_{i+1}$ and its last endpoint; and for each $e\in E^{\textsf{thr}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$, the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$, and the subpath of $P_e$ between $u_{i+1}$ and its last endpoint.
We use the following observation.
\begin{observation}
\label{obs: wset_i_non_interfering}
The set ${\mathcal{W}}_i$ of paths are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
Recall that the paths in ${\mathcal{Q}}_{i-1}$ only uses edges of $E(C_{i-1})\cup \delta(C_{i-1})$, and they are non-transversal.
And similarly, the paths in ${\mathcal{Q}}_{i+1}$ only uses edges of $E(C_{i+1})\cup \delta(C_{i+1})$, and they are non-transversal.
Therefore, the paths in $\set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}}$ are non-transversal.
From \Cref{obs: non_transversal_1} and \Cref{obs: non_transversal_2}, the paths in $\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}$ are non-transversal. Therefore, it suffices to show that, the set ${\mathcal{W}}_i$ of paths are non-transversal at all vertices of $C_{i-1}$ and all vertices of $C_{i+1}$. Note that, for each edge $e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i})$, $W_e\cap (C_{i-1}\cup \delta(C_{i-1}))$ is indeed a path of ${\mathcal{Q}}_{i-1}$. Therefore, the paths in ${\mathcal{W}}_i$ are non-transversal at all vertices of $C_{i-1}$. Similarly, they are also non-transversal at all vertices of $C_{i+1}$. Altogether, the paths of ${\mathcal{W}}_i$ are non-transversal with respect to $\Sigma$.
\end{proof}
For uniformity of notations, for each edge $\hat e\in \hat E_i$, we rename the path $Q_{i+1}(\hat e)$ by $W_{\hat e}$, and similarly for each edge $\hat e\in \hat E_{i-1}$, we rename the path $Q_{i-1}(\hat e)$ by $W_{\hat e}$. Therefore, ${\mathcal{W}}_i=\set{W_e\mid e\in E(G_i)\setminus E(C_i)}$.
Put in other words, the set ${\mathcal{W}}_i$ contains, for each edge $e$ in $G_i$ that is incident to $v^{\operatorname{left}}_i$ or $v^{\operatorname{right}}_i$, a path named $W_e$.
It is easy to see that all paths in ${\mathcal{W}}_i$ are internally disjoint from $C_i$.
We further partition the set ${\mathcal{W}}_i$ into two sets: ${\mathcal{W}}_i^{\textsf{thr}}=\set{W_e\mid e\in E^{\textsf{thr}}_i}$ and $\tilde {\mathcal{W}}_i={\mathcal{W}}_i\setminus {\mathcal{W}}_i^{\textsf{thr}}$.
We are now ready to construct the drawing $\phi_i$ for the instance $(G_i,\Sigma_i)$.
Recall that $\phi^*$ is an optimal drawing of the input instance $(G,\Sigma)$.
We start with the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$, that we denote by $\phi^*_i$. We will not modify the image of $C_i$ in $\phi^*_i$, but will focus on constructing the image of edges in $E(G_i)\setminus E(C_i)$, based on the image of edges in $E({\mathcal{W}}_i)$ in $\phi^*_i$.
Specifically, we proceed in the following four steps.
\paragraph{Step 1.}
For each edge $e\in E({\mathcal{W}}_i)$, we denote by $\pi_e$ the curve that represents the image of $e$ in $\phi^*_i$. We create a set of $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting the endpoints of $e$ in $\phi^*_i$, that lies in an arbitrarily thin strip around $\pi_e$.
We denote by $\Pi_e$ the set of these curves.
We then assign, for each edge $e\in E({\mathcal{W}}_i)$ and for each path in ${\mathcal{W}}_i$ that contains the edge $e$, a distinct curve in $\Pi_e$ to this path.
Therefore, each curve in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ is assigned to exactly one path of ${\mathcal{W}}_i$,
and each path $W\in {\mathcal{W}}_i$ is assigned with, for each edge $e\in E(W)$, a curve in $\Pi_e$. Let $\gamma_W$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ that are assigned to path $W$, so $\gamma_W$ connects the endpoints of path $W$ in $\phi^*_i$.
In fact, when we assign curves in $\bigcup_{e\in \delta(u_{i-1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{left}}_i$ (recall that $\delta(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\operatorname{left}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{left}}_i)}$), we additionally ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{left}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{left}}_i)$ enter $u_{i-1}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
And similarly, when we assign curves in $\bigcup_{e\in \delta(u_{i+1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{right}}_i$ (recall that $\delta(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\operatorname{right}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{right}}_i)}$), we ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{right}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{right}}_i)$ enter $u_{i+1}$ in the same order as ${\mathcal{O}}^{\operatorname{right}}_i$.
Note that this can be easily achieved according to the definition of ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$.
We denote $\Gamma_i=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$, and we further partition the set $\Gamma_i$ into two sets: $\Gamma_i^{\textsf{thr}}=\set{\gamma_{W}\mid W\in {\mathcal{W}}^{\textsf{thr}}_i}$ and $\tilde \Gamma_i=\Gamma_i\setminus \Gamma_i^{\textsf{thr}}$.
We denote by $\hat \phi_i$ the drawing obtained by taking the union of the image of $C_i$ in $\phi^*_i$ and all curves in $\Gamma_i$.
For every path $P$ in $G_i$, we denote by $\hat{\chi}_i(P)$ the number of crossings that involves the ``image of $P$'' in $\hat \phi_i$, which is defined as the union of, for each edge $e\in E(\tilde{\mathcal{W}}_i)$, an arbitrary curve in $\Pi_e$.
Clearly, for each edge $e\in E({\mathcal{W}}_i)$, all curves in $\Pi_e$ are crossed by other curves of $(\Gamma_i\setminus \Pi_e)\cup \phi^*_i(C_i)$ same number of times.
Therefore, $\hat{\chi}_i(P)$ is well-defined.
For a set ${\mathcal{P}}$ of paths in $G_i$, we define $\hat{\chi}_i({\mathcal{P}})=\sum_{P\in {\mathcal{P}}}\hat{\chi}_i(P)$.
\iffalse
We use the following observation.
\znote{maybe remove this observation?}
\begin{observation}
\label{obs: curves_crossings}
The number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$,
and the number of intersections between a curve in $\tilde\Gamma_i$ and the image of edges of $C_i$ in $\phi^*_i$, are both $O(\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W))$.
\end{observation}
\begin{proof}
We first show that the number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Note that, from the construction of curves in $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i}$, every crossing between a pair $W,W'$ of curves in $\tilde\Gamma_i$ must be the intersection between a curve in $\Pi_e$ for some $e\in E(W)$ and a curve in $\Pi_{e'}$ for some $e'\in E(W')$, such that the image $\pi_e$ for $e$ and the image $\pi_{e'}$ for $e'$ intersect in $\phi^*$.
Therefore, for each pair $W,W'$ of paths in $\tilde{\mathcal{W}}_i$, the number of points that belong to only curves $\gamma_W$ and $\gamma_{W'}$ is at most the number of crossings between the image of $W$ and the image of $W'$ in $\phi^*$.
It follows that the number of points that belong to exactly two curves of $\tilde\Gamma_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Altogether, the number of intersections between curves in $\tilde\Gamma_i$ is at most $|V(\tilde {\mathcal{W}}_i)|+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
We now show that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ that are not vertex-image is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Let $W$ be a path of $\tilde {\mathcal{W}}_i$ and consider the curve $\gamma_W$. Note that $\gamma_W$ is the union of, for each edge $e\in E(W)$, a curve that lies in an arbitrarily thin strip around $\pi_e$.
Therefore, the number of crossings between $\gamma_W$ and the image of $C_i$ in $\phi^*_i$ is identical to the number of crossings the image of path $W$ and the image of $C_i$ in $\phi^*_i$, which is at most $\hat\mathsf{cr}(W)$.
It follows that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
\end{proof}
\fi
\paragraph{Step 2.}
For each vertex $v\in V({\mathcal{W}}_i)$, we denote by $x_v$ the point that represents the image of $v$ in $\phi^*_i$, and we let $X$ contains all points of $\set{x_v\mid v\in V({\mathcal{W}}_i)}$ that are intersections between curves in $\Gamma_i$.
We now manipulate the curves in $\set{\gamma_W\mid W\in {\mathcal{W}}_i}$ at points of $X$, by processing points of $X$ one-by-one, as follows.
Consider a point $x_v$ that is an intersection between curves in $\Gamma_i$, where $v\in V({\mathcal{W}}_i)$, and let $D_v$ be an arbitrarily small disc around $x_v$.
We denote by ${\mathcal{W}}_i(v)$ the set of paths in ${\mathcal{W}}_i$ that contains $v$, and further partition it into two sets: ${\mathcal{W}}^{\textsf{thr}}_i(v)={\mathcal{W}}_i(v)\cap {\mathcal{W}}^{\textsf{thr}}_i$ and $\tilde{\mathcal{W}}_i(v)={\mathcal{W}}_i(v)\cap \tilde{\mathcal{W}}_i$. We apply the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $\set{\gamma_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ within disc $D_v$. Let $\set{\gamma'_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ be the set of new curves that we obtain. From \Cref{obs: curve_manipulation},
(i) for each path $W\in \tilde{\mathcal{W}}_i(v)$, the curve $\gamma'_W$ does not contain $x_v$, and is identical to the curve $\gamma_W$ outside the disc $D_v$;
(ii) the segments of curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc $D_v$ are in general position; and
(iii) the number of icrossings between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside $D_v$ is bounded by $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\iffalse{just for backup}
\begin{proof}
Denote $d=\deg_G(v)$ and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
For each path $W\in \tilde{\mathcal{W}}_i(v)$, we denote by $p^{-}_W$ and $p^{+}_W$ the intersections between the curve $\gamma_W$ and the boundary of ${\mathcal{D}}_v$.
We now compute, for each $W\in W\in \tilde{\mathcal{W}}_i(v)$, a curve $\zeta_W$ in ${\mathcal{D}}_v$ connecting $p^{-}_W$ to $p^{+}_W$, such that (i) the curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ are in general position; and (ii) for each pair $W,W'$ of paths, the curves $\zeta_W$ and $\zeta_{W'}$ intersects iff the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{-}_{W'},p^{+}_{W},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_{W'},p^{-}_{W},p^{+}_{W'})$.
It is clear that this can be achieved by first setting, for each $W$, the curve $\zeta_W$ to be the line segment connecting $p^{-}_W$ to $p^{+}_W$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
We now define, for each $W$, the curve $\gamma'_W$ to be the union of the part of $\gamma_W$ outside ${\mathcal{D}}_v$ and the curve $\zeta_W$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, the first and the second condition of \Cref{obs: curve_manipulation} are satisfied.
It remains to estimate the number of intersections between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$, which equals the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$.
Since the paths in $\tilde{\mathcal{W}}_i(v)$ are non-transversal with respect to $\Sigma$ (from \Cref{obs: wset_i_non_interfering}), from the construction of curves $\set{\gamma_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$, if a pair $W,W'$ of paths in $\tilde {\mathcal{W}}_i(v)$ do not share edges of $\delta(v)$, then the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_W,p^{-}_{W'},p^{+}_{W'})$, and therefore the curves $\zeta_{W}$ and $\zeta_{W'}$ will not intersect in ${\mathcal{D}}_v$.
Therefore, only the curves $\zeta_W$ and $\zeta_{W'}$ intersect iff $W$ and $W'$ share an edge of $\delta(v)$.
Since every such pair of curves intersects at most once, the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$ is at most $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are shown in black, and curves of $\tilde{\mathcal{W}}_i(v)$ are shown in blue, red, orange and green. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are not modified, while curves of $\tilde{\mathcal{W}}_i(v)$ are re-routed via dash lines within disc ${\mathcal{D}}_v$.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the step of processing $x_v$.}\label{fig: curve_con}
\end{figure}
\fi
We then replace the curves of $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ in $\Gamma_i$ by the curves of $\set{\gamma'_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
This completes the description of the iteration of processing point the point $x_v\in X$. Let $\Gamma'_i$ be the set of curves that we obtain after processing all points in $X$ in this way. Note that we have never modified the curves of $\Gamma^{\textsf{thr}}_i$, so $\Gamma^{\textsf{thr}}_i\subseteq\Gamma'_i$, and we denote $\tilde\Gamma'_i=\Gamma'_i\setminus \Gamma^{\textsf{thr}}_i$.
We use the following observation.
\begin{observation}
\label{obs: general_position}
Curves in $\tilde\Gamma'_i$ are in general position, and if a point $p$ lies on more than two curves of $\Gamma'_i$, then either $p$ is an endpoint of all curves containing it, or all curves containing $p$ belong to $\Gamma^{\textsf{thr}}_i$.
\end{observation}
\begin{proof}
From the construction of curves in $\Gamma_i$, any point that belong to at least three curves of $\Gamma_i$ must be the image of some vertex in $\phi^*$. From~\Cref{obs: curve_manipulation}, curves in $\tilde\Gamma'_i$ are in general position; curves in $\tilde\Gamma'_i$ do not contain any vertex-image in $\phi^*$ except for their endpoints; and they do not contain any intersection of a pair of paths in $\Gamma_i^{\textsf{thr}}$. \Cref{obs: general_position} now follows.
\end{proof}
\paragraph{Step 3.}
So far we have obtained a set $\Gamma'_i$ of curves that are further partitioned into two sets $\Gamma'_i=\Gamma^{\textsf{thr}}_i\cup \tilde\Gamma'_i$,
where set $\tilde\Gamma'_i$ contains, for each path $W\in \tilde {\mathcal{W}}_i$, a curve $\gamma'_W$ connecting the endpoints of $W$, and the curves in $\tilde\Gamma'_i$ are in general position;
and set $\Gamma^{\textsf{thr}}_i$ contains, for each path $W\in {\mathcal{W}}^{\textsf{thr}}_i$, a curve $\gamma_W$ connecting the endpoints of $W$.
Recall that all paths in ${\mathcal{W}}^{\textsf{thr}}_i$ connects $u_{i-1}$ to $u_{i+1}$.
Let $z_{\operatorname{left}}$ be the point that represents the image of $u_{i-1}$ in $\phi_i^*$ and let $z_{\operatorname{right}}$ be the point that represents the image of $u_{i+1}$ in $\phi_i^*$.
Then, all curves in $\Gamma^{\textsf{thr}}_i$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
We view $z_{\operatorname{left}}$ as the first endpoint of curves in $\Gamma^{\textsf{thr}}_i$ and view $z_{\operatorname{right}}$ as their last endpoint.
We then apply the algorithm in \Cref{thm: type-2 uncrossing}, where we let $\Gamma=\Gamma^{\textsf{thr}}_i$ and let $\Gamma_0$ be the set of all other curves in the drawing $\phi^*_i$. Let $\Gamma^{\textsf{thr}'}_i$ be the set of curves we obtain. We then designate, for each edge $e\in E^{\textsf{thr}}_i$, a curve in $\Gamma^{\textsf{thr}'}_i$ as $\gamma'_{W_e}$, such that the curves of $\set{\gamma'_{W_e}\mid e\in \hat E_{i-1}\cup E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i}$ enters $z_{\operatorname{left}}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
Recall that ${\mathcal{W}}_i=\set{W_e\mid e\in (E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\cup E_i^{\operatorname{right}}\cup \hat E_{i-1}\cup \hat E_i)}$, and, for each edge $e\in E_i^{\operatorname{left}}\cup \hat E_{i-1}$, the curve $\gamma'_{W_e}$ connects its endpoint in $C_i$ to $z_{\operatorname{left}}$; for each edge $e\in E_i^{\operatorname{right}}\cup \hat E_{i}$, the curve $\gamma'_{W_e}$ connects the endpoint of $e$ to $z_{\operatorname{right}}$; and for each edge $e\in E_i^{\textsf{thr}}$, the curve $\gamma'_{W_e}$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
Therefore, if we view $z_{\operatorname{left}}$ as the image of $v^{\operatorname{left}}_i$, view $z_{\operatorname{right}}$ as the image of $v^{\operatorname{right}}_i$, and for each edge $e\in E(G_i)\setminus E(C_i)$, view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi'_i$.
It is clear from the construction of curves in $\set{\gamma'_{W_e}\mid e\in E(G_i)\setminus E(C_i)}$ that this drawing respects all rotations in $\Sigma_i$ on vertices of $V(C_i)$ and vertex $v^{\operatorname{left}}_i$.
However, the drawing $\phi'_i$ may not respect the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$. We further modify the drawing $\phi'_i$ at $z_{\operatorname{right}}$ in the last step.
\paragraph{Step 4.}
Let ${\mathcal{D}}$ be an arbitrarily small disc around $z_{\operatorname{right}}$ in the drawing $\phi'_i$, and let ${\mathcal{D}}'$ be another small disc around $z_{\operatorname{right}}$ that is strictly contained in ${\mathcal{D}}$.
We first erase the drawing of $\phi'_i$ inside the disc ${\mathcal{D}}$, and for each edge $e\in \delta(v^{\operatorname{right}}_i)$, we denote by $p_{e}$ the intersection between the curve representing the image of $e$ in $\phi'_i$ and the boundary of ${\mathcal{D}}$.
We then place, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, a point $p'_e$ on the boundary of ${\mathcal{D}}'$, such that the order in which the points in $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ appearing on the boundary of ${\mathcal{D}}'$ is precisely ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We then apply \Cref{lem: find reordering} to compute a set of reordering curves, connecting points of $\set{p_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ to points $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$.
Finally, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, let $\gamma_e$ be the concatenation of (i) the image of $e$ in $\phi'_i$ outside the disc ${\mathcal{D}}$; (ii) the reordering curve connecting $p_e$ to $p'_e$; and (iii) the straight line segment connecting $p'_e$ to $z_{\operatorname{right}}$ in ${\mathcal{D}}'$.
We view $\gamma_e$ as the image of edge $e$, for each $e\in \delta(v^{\operatorname{right}}_i)$.
We denote the resulting drawing of $G_i$ by $\phi_i$. It is clear that $\phi_i$ respects the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$, and therefore it respects the rotation system $\Sigma_i$.
We use the following claim.
\begin{claim}
\label{clm: rerouting_crossings}
The number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\end{claim}
\begin{proof}
Denote by ${\mathcal{O}}^*$ the ordering in which the curves $\set{\gamma'_{W_e}\mid e\in \delta_{G_i}(v_i^{\operatorname{right}})}$ enter $z_{\operatorname{right}}$, the image of $u_{i+1}$ in $\phi'_i$.
From~\Cref{lem: find reordering} and the algorithm in Step 4 of modifying the drawing within the disc ${\mathcal{D}}$, the number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is at most $O(\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}}))$. Therefore, it suffices to show that $\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}})=O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
In fact, we will compute a set of curves connecting the image of $u_i$ and the image of $u_{i+1}$ in $\phi^*_i$, such that each curve is indexed by some edge $e\in\delta_{G_i}(v_i^{\operatorname{right}})$ these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$ and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$, and the number of crossings between curves of $Z$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
For each $e\in E^{\textsf{thr}}_i$, we denote by $\eta_e$ the curve obtained by taking the union of (i) the curve $\gamma'_{W_e}$ (that connects $u_{i+1}$ to $u_{i-1}$); and (ii) the curve representing the image of the subpath of $P_e$ in $\phi^*_i$ between $u_i$ and $u_{i-1}$.
Therefore, the curve $\eta_e$ connects $u_i$ to $u_{i+1}$.
We then modify the curves of $\set{\eta_e\mid e\in E^{\textsf{thr}}_i}$, by iteratively applying the algorithm from \Cref{obs: curve_manipulation} to these curves at the image of each vertex of $C_{i-1}\cup C_{i+1}$. Let $\set{\zeta_e\mid e\in E^{\textsf{thr}}_i}$ be the set of curves that we obtain. We call the obtained curves \emph{red curves}. From~\Cref{obs: curve_manipulation}, the red curves are in general position. Moreover, it is easy to verify that the number of intersections between the red curves is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1}))$.
We call the curves in $\set{\gamma'_{W_e}\mid e\in \hat E_i}$ \emph{yellow curves}, call the curves in $\set{\gamma'_{W_e}\mid e\in E^{\operatorname{right}}_i}$ \emph{green curves}. See \Cref{fig: uncrossing_to_bound_crossings} for an illustration.
From the construction of red, yellow and green curves, we know that these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$, and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$.
Moreover, we are guaranteed that the number of intersections between red, yellow and green curves is at most $\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/uncross_to_bound_crossings.jpg}
\caption{An illustration of red, yellow and green curves.}\label{fig: uncrossing_to_bound_crossings}
\end{figure}
\end{proof}
From the above discussion and Claim~\ref{clm: rerouting_crossings}, for each $2\le i\le k-1$,
\[
\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right}).
\]
\iffalse
We now estimate the number of crossings in $\phi_i$ in the next claim.
\begin{claim}
\label{clm: number of crossings in good solutions}
$\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\sum_{W\in \tilde{\mathcal{W}}_i}\mathsf{cr}(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})$.
\end{claim}
\begin{proof}
$2\cdot \sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2=\sum_{v\in V(G)}\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2.$
\znote{need to redefine the orderings ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$ to get rid of $\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$ here, which we may not be able to afford.}
\end{proof}
\fi
\paragraph{Drawings $\phi_1$ and $\phi_{k}$.}
The drawings $\phi_1$ and $\phi_{k}$ are constructed similarly. We describe the construction of $\phi_1$, and the construction of $\phi_1$ is symmetric.
Recall that the graph $G_1$ contains only one super-node $v_1^{\operatorname{right}}$, and $\delta_{G_1}(v_1^{\operatorname{right}})=\hat E_1\cup E^{\operatorname{right}}_1$. We define ${\mathcal{W}}_1=\set{W_e\mid e\in E^{\operatorname{right}}_1}\cup \set{Q_2(\hat e)\mid \hat e\in \hat E_1}$. For each $\hat e\in \hat E_1$, we rename the path $Q_2(\hat e)$ by $W_e$, so ${\mathcal{W}}_1$ contains, for each edge $e\in \delta_{G_1}(v^1_{\operatorname{right}})$, a path named $W_e$ connecting its endpoints to $u_2$. Via similar analysis in \Cref{obs: wset_i_non_interfering}, it is easy to show that the paths in ${\mathcal{W}}_1$ are non-transversal with respect to $\Sigma$.
We start with the drawing of $C_1\cup E({\mathcal{W}}_1)$ induced by $\phi^*$, that we denote by $\phi^*_1$. We will not modify the image of $C_i$ in $\phi^*_i$ and will construct the image of edges in $\delta(v_1^{\operatorname{right}})$.
We perform similar steps as in the construction of drawings $\phi_2,\ldots,\phi_{k-1}$.
We first construct, for each path $W\in {\mathcal{W}}_1$, a curve $\gamma_W$ connecting its endpoint in $C_1$ to the image of $u_2$ in $\phi^*$, as in Step 1.
Let $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
We then process all intersections between curves of $\Gamma_1$ as in Step 2. Let $\Gamma'_1=\set{\gamma'_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
Since $\Gamma^{\textsf{thr}}_1=\emptyset$, we do not need to perform Steps 3 and 4. If we view the image of $u_2$ in $\phi^*_1$ as the image of $v^{\operatorname{right}}_1$, and for each edge $e\in \delta(v^{\operatorname{right}}_1)$, we view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is clear that this drawing respects the rotation system $\Sigma_1$. Moreover,
\[\mathsf{cr}(\phi_1)=\chi^2(C_1)+O\textsf{left}(\hat\chi_1({\mathcal{Q}}_2)+\sum_{W\in {\mathcal{W}}_1}\hat\chi_1(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_1,e)^2\textsf{right}).\]
Similarly, the drawing $\phi_k$ that we obtained in the similar way satisfies that
\[\mathsf{cr}(\phi_k)=\chi^2(C_k)+O\textsf{left}(\hat\chi_k({\mathcal{Q}}_{k-1})+\sum_{W\in {\mathcal{W}}_k}\hat\chi_k(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_k,e)^2\textsf{right}).\]
We now complete the proof of \Cref{claim: existence of good solutions special}, for which it suffices to estimate $\sum_{1\le i\le k}\mathsf{cr}(\phi_i)$.
Recall that, for each $1\le i\le k$, $\tilde {\mathcal{W}}_i=\set{W_e\mid e\in \hat E_{i-1}\cup \hat E_{i-1}\cup E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}}$, where $E^{\operatorname{left}}_{i}=E(C_i,\bigcup_{1\le j\le i-2}C_j)$, and
$E^{\operatorname{right}}_{i}=E(C_i,\bigcup_{i+2\le j\le k}C_j)$. Therefore, for each edge $e\in E'\cup (\bigcup_{1\le i\le k-1}\hat E_i)$, the path $W_e$ belongs to exactly $2$ sets of $\set{\tilde{\mathcal{W}}_i}_{1\le i\le k}$.
Recall that the path $W_e$ only uses edges of the inner path $P_e$ and the outer path $P^{\mathsf{out}}_e$.
Let $\tilde{\mathcal{W}}=\bigcup_{1\le i\le k}\tilde{\mathcal{W}}_i$, from \Cref{obs: edge_occupation in outer and inner paths},
for each edge $e\in E'\cup (\bigcup_{1\le i\le k-1}\hat E_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(1)$, and
for $1\le i\le k$ and for each edge $e\in E(C_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(\cong_G({\mathcal{Q}}_i,e))$.
Therefore, on one hand,
\[
\begin{split}
\sum_{1\le i\le k}\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} 2\cdot \cong_G(\tilde {\mathcal{W}},e)\cdot\cong_G(\tilde {\mathcal{W}},e')\\
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} \textsf{left}(\cong_G(\tilde {\mathcal{W}},e)^2+\cong_G(\tilde {\mathcal{W}},e')^2\textsf{right})\\
& \le
\sum_{e\in E(G)} \chi(e)\cdot \cong_G(\tilde {\mathcal{W}},e)^2
= O(\mathsf{cr}(\phi^*)\cdot\beta),
\end{split}
\]
and on the other hand,
\[
\begin{split}
\sum_{1\le i\le k}\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2
& \le \sum_{e\in E(G)} \textsf{left}(\sum_{1\le i\le k} \cong_G(\tilde {\mathcal{W}}_i,e)\textsf{right})^2\\
& \le O\textsf{left}(\sum_{e\in E(G)} \cong_G(\tilde {\mathcal{W}},e)^2 \textsf{right})\\
& \le O\textsf{left}(|E(G)|+\sum_{1\le i\le k}\sum_{e\in E(C_i)} \cong_G({\mathcal{Q}}_i,e)^2\textsf{right})=O(|E(G)|\cdot\beta).
\end{split}
\]
Moreover, $\sum_{1\le i\le k}\chi^2(C_i)\le O(\mathsf{cr}(\phi^*))$, and $\sum_{1\le i\le k}\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})\le O(\mathsf{cr}(\phi^*)\cdot\beta)$.
Altogether,
\[
\begin{split}
\sum_{1\le i\le k}\mathsf{cr}(\phi_i)
& \le
O\textsf{left}(\sum_{1\le i\le k}
\textsf{left}(
\chi^2(C_i)+ \hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})
\textsf{right})\\
& \le O(\mathsf{cr}(\phi^*))+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(|E(G)|\cdot\beta)\\
& \le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot\beta).
\end{split}
\]
This completes the proof of \Cref{claim: existence of good solutions special}.
\end{proofof}
$\ $
\iffalse
We denote by $\hat{\phi_i}$ the resulting drawing, and it is clear that $\hat{\phi_i}$ is a drawing of the instance $(\hat G_i,\hat\Sigma_i)$, such that $\mathsf{cr}(\hat{\phi_i})\le \mathsf{cr}(C_i)+O(\operatorname{cost}({\mathcal{W}}_i)\cdot\operatorname{poly}\log n)$.
We first define an instance $(\hat G_i,\hat\Sigma_i)$ as follows. We start with the subgraph of $G$ induced by all edges of $E_G(C_i)$ and $E({\mathcal{W}}_i)$, and then create, for each edge $e$ in the subgraph, $\cong_{{\mathcal{W}}_i}(e)$ parallel copies of it. This finishes the description of graph $\hat G_i$.
For each vertex $v$ in $\hat G_i$, we define its rotation $\hat{\mathcal{O}}_v$ as follows. We start with the rotation ${\mathcal{O}}_v\in \Sigma$, and then replace, for each edge $e\in E({\mathcal{W}}_i)$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $\cong_{{\mathcal{W}}_i}(e)$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$, where the ordering among the copies is arbitrary.
On the one hand, in graph $\hat G_i$, we can make paths of ${\mathcal{W}}_i$ edge-disjoint by letting, for each edge $e\in E({\mathcal{W}}_i)$, each path of ${\mathcal{W}}_i$ that contains $e$ now take a distinct copy of $e$ in $\hat G_i$.
On the other hand, a drawing $\hat \phi_i$ of the instance $(\hat G_i,\hat\Sigma_i)$ can be easily computed from $\phi^*$, as follows.
We start with $\hat\phi'_i$, the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$. Then for each edge of ${\mathcal{W}}_i$, let $\gamma_e$ be the curve that represents the image of $e$, and we create $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting endpoints of $e$, that lies in an arbitrarily thin strip around $\gamma_e$. We denote by $\hat{\phi_i}$ the resulting drawing, and it is clear that $\hat{\phi_i}$ is a drawing of the instance $(\hat G_i,\hat\Sigma_i)$, such that $\mathsf{cr}(\hat{\phi_i})\le \mathsf{cr}(C_i)+O(\operatorname{cost}({\mathcal{W}}_i)\cdot\operatorname{poly}\log n)$.
For each edge $e\in E_G(C_i)$, we denote by $\gamma_e$ the curve in $\hat \phi_i$ that represents the image of $e$; and for each path $W\in {\mathcal{W}}_i$, we denote by $\gamma_W$ the curve in $\hat \phi_i$ that represents the image of $W$.
We define $\Gamma_0=\set{\gamma_e\mid e\in E_G(C_i)}$ and $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$. It is immediate to verify that the sets $\Gamma_0,\Gamma_1$ of curves satisfy the condition of \Cref{thm: type-2 uncrossing}. We then apply the algorithm in \Cref{thm: type-2 uncrossing} to $\Gamma_0,\Gamma_1$, and let $\Gamma_1'$ be the set of curves that we obtain.
From \Cref{thm: type-2 uncrossing}, the curves in $\Gamma_1'$ do not intersect internally between each other, and have the same sets of first endpoints and last endpoints.
We now show that we can obtain a drawing $\phi_i$ of the instance $(G_i,\Sigma_i)$ using the curves $\Gamma_0,\Gamma_1'$.
For each edge $e\in E_G(C_i)$, we still let the curve $\gamma_e\in \Gamma_0$ be the image of $e$.
For each edge path $W\in {\mathcal{W}}$, from \Cref{thm: type-2 uncrossing}, there is a curve $\gamma'\in \Gamma_1'$ connecting the endpoint of $W$ to we still let the curve $\gamma_e\in \Gamma_0$ be the image of $e$.
\znote{maybe we need another type of uncrossing here.}
\fi
\begin{proofof}{Claim~\ref{claim: stitching the drawings together}}
We first define a set $\set{(U_i,\Sigma'_i)\mid 1\le i\le k}$ of instances of \textnormal{\textsf{MCNwRS}}\xspace as follows.
We define $U_1=G$, and for each $2\le i\le k$, we define $U_i$ to be the graph obtained from $G$ by contracting the clusters $C_1,\ldots, C_{i-1}$ into a vertex $v^{\operatorname{left}}_i$.
Note that each edge in $U_i$ is also an edge in $G$, and we do not distinguish between them.
Note that $U_1=G$, we define the rotation system $\Sigma'_1$ on $U_1$ to be $\Sigma$.
For each $2\le i\le k$, we define the rotation system $\Sigma'_i$ on $U_i$ as follows.
For each vertex $v\in \bigcup_{i\le t\le k}V(C_t)$, note that its incident edges in $U_i$ are the edges of $\delta_G(v)$, and its rotation in $\Sigma'_i$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the input rotation system $\Sigma$.
For vertex $v^{\operatorname{left}}_i$, note that its incident edges are the edges of $\delta_{G_i}(v^{\operatorname{left}}_i)$, and its rotation in $\Sigma'_i$ is defined to be ${\mathcal{O}}^{\operatorname{left}}_i$, the rotation on $v^{\operatorname{left}}_i$ of instance $(G_i,\Sigma_i)$.
Note that $U_k=G_k$ and $\Sigma'_k=\Sigma_k$, so the drawing $\phi_k$ of the instance $(G_k,\Sigma_k)$ is also a drawing of the instance $(U_k,\Sigma'_k)$. For clarity, when we view this drawing as a solution to the instance $(U_k,\Sigma'_k)$, we rename it by $\psi_k$.
We will sequentially, for $i=k-1,\ldots,1$, compute a drawing of the instance $(U_i,\Sigma'_i)$ using the drawing $\psi_{i+1}$ of $(U_{i+1},\Sigma'_{i+1})$ and the drawing $\phi_i$ of $(G_{i},\Sigma_{i})$, and eventually, we return the drawing $\psi_1$ of $(U_{1},\Sigma'_{1})$ as the solution to the instance $(G,\Sigma)$.
We now fix an index $1\le i< k-1$ and construct the drawing $\psi_i$ to the instance $(U_{i},\Sigma'_{i})$, assuming that we have computed a drawing $\psi_{i+1}$ to the instance $(U_{i+1},\Sigma'_{i+1})$, as follows.
Recall that $V(G_i)=V(C_i)\cup\set{v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$ if $i\ge 2$ and $V(G_i)=V(C_i)\cup\set{v^{\operatorname{right}}_i}$ if $i=1$, and $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{U_{i+1}}(v^{\operatorname{left}}_{i+1})$. Moreover, from \Cref{obs: rotation for stitching} and the definition of instance $(U_{i+1},\Sigma'_{i+1})$, the rotation on $v^{\operatorname{right}}_i$ in $\Sigma_{i}$ is identical to the rotation on $v^{\operatorname{left}}_{i+1}$ in $\Sigma'_{i+1}$.
Denote $F=\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{U_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Let ${\mathcal{D}}$ be an arbitrarily small disc around the image of $v_{i+1}^{\operatorname{left}}$ in $\psi_{i+1}$. For each edge $e\in F$, we denote by $p_e$ the intersection between the image of $e$ with the boundary of ${\mathcal{D}}$.
Therefore, the order in which the points $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}$ is ${\mathcal{O}}^{\operatorname{left}}_{i+1}$.
We erase the drawing of $\psi_{i+1}$ inside the disc ${\mathcal{D}}$, and view the area inside the disc ${\mathcal{D}}$ as the outer face of the drawing.
Similarly, let ${\mathcal{D}}'$ be an arbitrarily small disc around the image of $v_{i}^{\operatorname{right}}$ in $\phi_{i}$. For each edge $e\in F$, we denote by $p'_e$ the intersection between the image of $e$ with the boundary of ${\mathcal{D}}'$.
Therefore, the order in which the points $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}'$ is ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We erase the drawing of $\phi_{i}$ inside the disc ${\mathcal{D}}'$, and let ${\mathcal{D}}''$ be another disc that is strictly contained in ${\mathcal{D}}'$. We now place the drawing of $\psi_{i+1}$ inside ${\mathcal{D}}'$ (after we erase part of it inside ${\mathcal{D}}$), so that the boundary of ${\mathcal{D}}$ in $\psi_{i+1}$ coincide with the boundary of ${\mathcal{D}}''$ in $\phi_{i}$, while the interior of ${\mathcal{D}}$ coincides with the exterior of ${\mathcal{D}}''$. We then compute a set $\set{\zeta_e\mid e\in F}$ of curves lying in ${\mathcal{D}}'\setminus {\mathcal{D}}''$, where for each $e\in F$, the curve $\zeta_e$ connects $p'_e$ to $p_e$, such that all curves of $\set{\zeta_e\mid e\in F}$ are mutually disjoint. Note that this can be done since the order in which $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}'$ is identical to $\set{p'_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}$.
We denote by $\psi_i$ the resulting drawing we obtained.
See Figure~\ref{fig: stitching} for an illustration.
Clearly, $\psi_i$ is a drawing of $U_i$ that respects $\Sigma'_i$ if we view, for each edge $e\in F$, the union of (i) the image of $e$ in $\phi_i$ outside the disc ${\mathcal{D}}'$; (ii) the curve $\zeta_e$; and (iii) the image of $e$ in $\psi_{i+1}$ inside the disc ${\mathcal{D}}''$, as the image of $e$.
\begin{figure}[h]
\centering
\subfigure[The drawing $\psi_{i+1}$, where the boundary of ${\mathcal{D}}$ is shown in dash black.]{\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_1.jpg}
}
\hspace{0.45cm}
\subfigure[The drawing $\psi_{i+1}$ after we erase its part inside ${\mathcal{D}}$ and view the interior of ${\mathcal{D}}$ as the outer face.]{
\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_2.jpg}}}
\hspace{0.45cm}
\subfigure[The drawing $\psi_i$, where the curves of $\set{\zeta_e\mid e\in F}$ are shown in dash line segments.]{
\scalebox{0.36}{\includegraphics[scale=1.0]{figs/stitching_3.jpg}}}
\caption{An illustration of an iteration in type-1 uncrossing.}\label{fig: stitching}
\end{figure}
Clearly, any crossing in the drawing $\psi_i$ is either a crossing of $\phi_i$ or a crossing of $\psi_{i+1}$, so $\mathsf{cr}(\psi_i)\le \mathsf{cr}(\psi_{i+1})+\mathsf{cr}(\phi_{i})$.
Therefore, if we rename the drawing $\psi_1$ of the instance $(U_1,\Sigma'_1)$ by $\phi$, then $\phi$ is a drawing of $G$ that respects $\Sigma$, and $\mathsf{cr}(\phi)\le \sum_{1\le i\le k}\mathsf{cr}(\phi_i)$.
\end{proofof}
\subsection{Special Case: No Problematic Indices}
\label{subsec: no prob edges}
\mynote{A quick note that may be useful later. Suppose we have clusters $C_1,\ldots,C_r$ organized on a path, and there is no bad index. Suppose now we apply *basic disengagement* procedure (exactly as described in Sec 6.2) iteratively, first disengaging $C_1$, and getting graph $G_1$, where $C_1$ becomes a single vertex, then disengaging $C_2$ from $G_1$, and so on. As the result, instances that we obtain are *exactly* the instances created in this section, including the specific rotation around the two contracted vertices. This means that we don't need to worry about combining the solutions of the resulting instances together - the algorithm for basic disengagement does it (applying Lemma 6.3 as is). But the algorithm for basic disengagement won't give us the bound that we need on the total solution costs of the new instances (if we use Lemma 6.2 as is, the cost may be too high, because we'll be charging to same crossings again and again). This bound has to rely on the fact that we can route every edge connecting two non-consecutive clusters via a central path.
One question that I'm wondering about is whether it is critical that, for an edge $e$ connecting $C_i$ to $C_j$, is routing path $P(e)$ has to go through the central path. I believe that there must be a portion of $P(e)$ between $C_i$ and $C_j$ that goes between the two clusters on a central path. But if before that it jumps to some clusters that lie before $C_i$ or after $C_j$ I think it does not matter to us. This may be good to understand because this may allow us to weaken the condition of "no bad index". If we decide to extend this algorithm to the case of a path-of-stars, this understanding may be helpful.}
In this subsection we show the proof of Lemma~\ref{lem: path case}, with the additional assumption that no index $i$ is problematic. That is, for each $1\le i\le k-1$, $|E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\le 100|\hat E_i|$.
Recall that we are given, for each $1\le i\le k$, a vertex $u_i$ of $C_i$, and a set ${\mathcal{Q}}_i$ of paths in $C_i$ connecting the edges of $\delta(C_i)$ to $u_i$, such that for every edge $e\in E(C_i)$, $\mathbb{E}[\cong_{C_i}({\mathcal{Q}}_i,e)^2]\le \beta$.
Recall that $E^{\textsf{out}}({\mathcal{C}})$ is the set of edges connecting distinct clusters of ${\mathcal{C}}$.
We denote by $E'$ the subset of edges in $E^{\textsf{out}}({\mathcal{C}})$ connecting a pair $C_i,C_j$ of clusters of ${\mathcal{C}}$, with $j\ge i+2$.
Therefore, $E(G)=E'\cup\textsf{left}(\bigcup_{1\le i\le k-1}\hat E_i\textsf{right})\cup \textsf{left}(\bigcup_{1\le i\le k}E(C_i)\textsf{right})$.
Recall that $H=G_{|{\mathcal{C}}}$, and the node in $H$ that represents the cluster $C_i$ in $G$ is denoted by $x_i$.
Therefore, each edge in $E'\cup \textsf{left}(\bigcup_{1\le i\le k-1}\hat E_i\textsf{right})$ corresponds to an edge in $H$, and we do not distinguish between them.
We now start to describe the algorithm that construct the sub-instances $(G_1,\Sigma_1),\ldots,(G_k,\Sigma_k)$.
The algorithm proceeds in two stages. In the first stage, we compute, for each edge of $E'$, two auxiliary paths in $G$, that we call its \emph{inner path} and \emph{outer path} respectively. In the second stage, we use these auxiliary paths to define the sub-instances and show that they satisfy the properties of \Cref{lem: path case}.
In the remainder of this section, we describe the first stage in \Cref{sec: inner and outer paths} and the second stage in \Cref{sec: disengaged instances}, and then complete the proof of \Cref{lem: path case} in \Cref{sec: path case with no problematic index}.
\subsubsection{Compute Inner and Outer Paths}
\label{sec: inner and outer paths}
In the first stage, we will compute, for each pair $i,j$ of indices (with $i\le j-2$) and for each edge $e\in E'$ that connects a vertex of $C_i$ to a vertex of $C_j$, a path $P_e$ connecting $u_i$ to $u_j$ that only uses edges of $\bigcup_{i\le t\le j-1}\hat E_t$ and edges of $\bigcup_{i\le t\le j}E(C_t)$, and a path $P^{\mathsf{out}}_e$ connecting $u_i$ to $u_j$ that only uses edge $e$ and edges of $E(C_i)\cup E(C_j)$. We call path $P_e$ the \emph{inner path} of $e$ and path $P^{\mathsf{out}}_e$ the \emph{outer path} of $e$.
We first compute the outer paths. Let $e$ be an edge of $E'$ connecting a vertex of $C_i$ to a vertex of $C_j$, so $e\in \delta(C_i)$ and $e\in \delta(C_j)$. Recall that the set ${\mathcal{Q}}_i$ contains a path $Q_i(e)$, connecting $e$ to $u_i$, and the set ${\mathcal{Q}}_j$ contains a path $Q_j(e)$, connecting $e$ to $u_j$.
We simply define its outer path $P^{\mathsf{out}}_e$ to be the union of path $Q_i(e)$ and path $Q_j(e)$.
It is clear that path $P^{\mathsf{out}}_e$ connects $u_i$ to $u_j$, and only uses edge $e$ and edges of $E(C_i)\cup E(C_j)$.
Throughout the section, we view $u_j$ as the first endpoint of $P^{\mathsf{out}}_e$ and $u_i$ as its last endpoint. We denote ${\mathcal{P}}^{\mathsf{out}}=\set{P_e^{\mathsf{out}}\mid e\in E'}$, and for each $1\le i\le k$, we denote ${\mathcal{P}}^{\mathsf{out}}_{*,i}=\set{P^{\mathsf{out}}_e \mid e\in E^{\operatorname{left}}_{i}}$ and ${\mathcal{P}}^{\mathsf{out}}_{i,*}=\set{P^{\mathsf{out}}_e \mid e\in E^{\operatorname{right}}_{i}}$.
We now compute the set ${\mathcal{P}}=\set{P_e}_{e\in E'}$ of inner paths in three steps. In the first step, we compute a set $\tilde {\mathcal{P}}$ of paths in $H$. In the second step, we transform the set $\tilde {\mathcal{P}}$ of paths in $H$ into a set ${\mathcal{P}}^*$ of paths in $G$.
In the third step, we further process paths in ${\mathcal{P}}^*$ to obtain the set ${\mathcal{P}}$ of inner paths for edges of $E'$.
We now describe each step in more details.
\paragraph{Step 1.}
In this step we process edges in $E'$ one-by-one. Throughout, for each edge $\hat e\in \bigcup_{1\le i\le k-1}\hat E_i$, we maintain an integer $z_{\hat e}$ indicating how many times the edge $\hat e$ has been used, that is initialized to be $0$.
Consider an iteration of processing an edge $e\in E'$. Assume $e$ connects $C_i$ to $C_j$ in $G$, we then pick, for each $i\le t\le j-1$, an edge $\hat e$ of $\hat E_t$ with minimum $z_{\hat e}$ over all edges of $\hat E_t$, and let $\tilde P_e$ be the path obtained by taking the union of all picked edges.
Note that, in $H$, each edge of $\hat E_t$ connects $x_t$ to $x_{t+1}$, so the path $\tilde P_e$ sequentially visits nodes $x_i, x_{i+1},\dots,x_j$.
We then increase the value of $z_{\hat e}$ by $1$ for all picked edges $\hat e$, and proceed to the next iteration. After processing all edges of $E'$, we obtain a set $\tilde{\mathcal{P}}=\set{\tilde P_e}_{e\in E'}$ of paths in $H$.
We use the following observation.
\begin{observation}
\label{obs:central_congestion}
For each edge $\hat e\in \bigcup_{1\le i\le k-1}\hat E_i$, $\cong_{H}(\tilde{\mathcal{P}},\hat e)\le 100$.
\end{observation}
\begin{proof}
From the algorithm, for each $1\le i\le k-1$, the paths of $\tilde{\mathcal{P}}$ that contains an edge of $\hat E_i$ are $\set{\tilde P_e\mid e\in \big(E_i^{\operatorname{right}} \cup E_{i+1}^{\operatorname{left}} \cup E_i^{\operatorname{over}}\big)}$.
Since $|E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\le 100|\hat E_i|$, each edge of $\hat E_i$ is used is used at most $100$ times. Observation~\ref{obs:central_congestion} then follows.
\end{proof}
\iffalse
We now further process the set $\tilde{\mathcal{P}}$ of paths to obtain another set $\tilde{\mathcal{P}}'$ of paths with some additional properties, as follows.
\paragraph{Step 2.1.} Let $\tilde G'$ be the graph obtained from $\tilde G$ by replacing each edge $e$ with $\beta$ parallel copies $e_1,\ldots,e_\beta$ connecting its endpoints. Now for each path $\tilde P\in \tilde{{\mathcal{P}}}$, we define a path $\tilde P^*$ as the union of, for each edge $e\in \tilde P$, a copy $e_t$ of $e$, such that, for all paths of $\tilde{\mathcal{P}}$ that contains the edge $e$, their corresponding paths in $\set{\tilde P^*\mid e\in \tilde P}$ contain distinct copies of $e$. We denote $\tilde{\mathcal{P}}^*=\set{\tilde P^*\mid \tilde P\in \tilde {\mathcal{P}}}$. From the above discussion, the paths of $\tilde{\mathcal{P}}^*$ are edge-disjoint.
\fi
\paragraph{Step 2.}
In this step we compute a set ${\mathcal{P}}^*$ of paths in $G$, using the paths in $\tilde{\mathcal{P}}$ and path sets $\set{{\mathcal{Q}}_i}_{1\le i\le k}$, as follows.
Recall that the set ${\mathcal{Q}}_i$ contains, for each edge $e\in\delta(C_i)$, a path $Q_i(e)$ connecting $e$ to $u_i$.
Also recall that, in graph $G$, an edge $\hat e\in \hat E_i$ belongs to both $\delta(C_i)$ and $\delta(C_{i+1})$.
Therefore, for an edge $\hat e\in \hat E_i$, the union of path $Q_i(\hat e)$ and path $Q_{i-1}(\hat e)$ is a path connecting $u_i$ to $u_{i+1}$.
Now consider an edge $e\in E'$ connecting a vertex of $C_i$ to a vertex of $C_{j}$ (with $i\le j-2$), and the corresponding path $\tilde P_e=(\hat e_i,\hat e_{i+1},\ldots,\hat e_{j-1})$ in $\tilde{\mathcal{P}}$, where $\hat e_t\in \hat E_t$ for each $i\le t\le j-1$. We define the path $P^*_e$ as the sequential concatenation of paths $Q_{i}(\hat e_i)\cup Q_{i+1}(\hat e_i), Q_{i+1}(\hat e_{i+1})\cup Q_{i+2}(\hat e_{i+1}),\ldots,Q_{j-1}(\hat e_{j-1})\cup Q_{j}(\hat e_{j-1})$.
It is clear that path $P^*_e$ only contains edges of $\bigcup_{i\le t\le j-1}\hat E_t$ and edges of $\bigcup_{i\le t\le j}E(C_t)$, and sequentially visits vertices $u_i,u_{i+1},\ldots,u_{j}$.
Denote ${\mathcal{P}}^*=\set{P^*_e\mid e\in E'}$.
From Observation~\ref{obs:central_congestion}, it is easy to see that for each edge $e\in \bigcup_{1\le i\le k}E(C_i)$, $\cong_{G}({\mathcal{P}}^*,e)\le 100\cdot\cong_{G}({\mathcal{Q}}_i,e)$.
\iffalse
It would be convenient to state our algorithm if the core paths of ${\mathcal{P}}$ are edge-disjoint.
For which, we construct another instance $(G',\Sigma')$ of \textnormal{\textsf{MCNwRS}}\xspace, as follows.
Denote ${\mathcal{Q}}=\bigcup_{1\le i\le k}{\mathcal{Q}}_i$.
We start with the instance $(G,\Sigma)$.
For each edge $e\in E({\mathcal{Q}})$, we replace $e$ with $2\cdot\cong_{{\mathcal{Q}}}(e)$ parallel copies.
For each edge $e\in E({\mathcal{P}})\setminus E({\mathcal{Q}})$, we replace $e$ with $2$ parallel copies.
For each vertex $v$ such that $\delta(v)\cap E({\mathcal{P}}\cup{\mathcal{Q}})= \emptyset$, we define its rotation ${\mathcal{O}}'_v$ to be identical to the its rotation ${\mathcal{O}}_v$ in $\Sigma$.
For each vertex $v$ such that $\delta(v)\cap E({\mathcal{P}}\cup{\mathcal{Q}})\ne \emptyset$, we define its rotation ${\mathcal{O}}'_v$ as follows. We start with the rotation ${\mathcal{O}}_v$, and then (i) replace, for each edge $e\in \delta(v)\cap E({\mathcal{Q}})$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $2\cdot\cong_{{\mathcal{Q}}}(e)$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$,
and the ordering among the copies is arbitrary; and
(ii) replace, for each edge $e\in \delta(v)\cap E({\mathcal{P}})$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $2$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$,
and the ordering among the copies is arbitrary.
We use the following observation.
\begin{observation}
$\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\le \mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\le O(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot\operatorname{poly}\log n)$.
Moreover, given a drawing $\phi\in \Phi(G,\Sigma)$, we can efficiently compute a drawing $\phi'\in \Phi(G',\Sigma')$, with $\mathsf{cr}(\phi')\le O(\mathsf{cr}(\phi)\cdot\operatorname{poly}\log n)$.
\end{observation}
\begin{proof}
\end{proof}
Now for each path $P\in {\mathcal{P}}$, we define a path $P'$ in $G'$ as the union of, for each edge $e\in P$, a copy $e_t$ of $e$, such that for all paths of ${\mathcal{P}}$ that contains the edge $e$, their corresponding paths in $\set{P'\mid P\in {\mathcal{P}}, e\in P}$ contain distinct copies of $e$. We denote ${\mathcal{P}}'=\set{P'\mid P\in {\mathcal{P}}}$.
From the above discussion, the paths of ${\mathcal{P}}'$ are edge-disjoint.
Moreover, note that $V(G)=V(G')$, and if a path $P\in {\mathcal{P}}$ connects $u_i$ to $u_j$ in $G$, then the corresponding path $P'$ connects $u_i$ to $u_j$ in $G'$.
\fi
\paragraph{Step 3.}
In this step we further process paths in ${\mathcal{P}}^*=\set{P^*_e\mid e\in E'}$ to obtain the set ${\mathcal{P}}=\set{P_e\mid e\in E'}$ of inner paths for edges of $E'$.
\iffalse
\begin{itemize}
\item $|{\mathcal{P}}|=|\tilde{\mathcal{P}}'|$;
\item for each pair $1\le i,j\le k$, the number of paths in ${\mathcal{P}}'$ connecting $u_i$ to $u_j$ is the same as that of ${\mathcal{P}}^*$; and
\item the paths in ${\mathcal{P}}^*$ are non-transversal with respect to $\Sigma'$.
\end{itemize}
\fi
For each path $P^*\in {\mathcal{P}}^*$ connecting $u_i$ to $u_{j}$ (recall that such a path visits vertices $u_i,u_{i+1},\ldots,u_j$ sequentially), we define, for each $i\le t\le j-1$, the path $P^*_{(i)}$ to be the subpath of $P^*$ between $u_{i}$ and $u_{i+1}$. For each $1\le i\le k-1$, we denote ${\mathcal{P}}^*_{(i)}=\set{P^*_{(i)}\text{ } \bigg|\text{ } u_i,u_{i+1}\in V(P^*),P^*\in {\mathcal{P}}^*}$, so all paths in ${\mathcal{P}}^*_{(i)}$ connects $u_i$ to $u_{i+1}$.
We then apply the algorithm in Lemma~\ref{lem: non_interfering_paths} to the set ${\mathcal{P}}^*_{(i)}$ of paths, and obtain a set ${\mathcal{R}}^*_{(i)}$ of $|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|$ paths connecting $u_i$ to $u_{i+1}$, such that for every $e\in E(G)$, $\cong_{G}({\mathcal{R}}^*_{(i)},e)\le \cong_{G}({\mathcal{P}}^*_{(i)},e)$, and the set ${\mathcal{R}}^*_{(i)}$ of paths are non-transversal with respect to $\Sigma$.
We now re-organize the paths in sets ${\mathcal{R}}^*_{(1)},\ldots, {\mathcal{R}}^*_{(k-1)}$ to obtain the set ${\mathcal{P}}$ of inner paths.
Note that, if we select, for each $i\le t\le j-1$, a path of ${\mathcal{R}}^*_{(i)}$, then, by taking the union of them, we can obtain a path $\hat P$ connecting $u_i$ to $u_j$.
We call the selected path of ${\mathcal{R}}^*_{(t)}$ the \emph{$t$-th segment} of $\hat P$. Therefore, a path connecting $u_i$ to $u_j$ obtained in this way is the union of its $i$-th segment, its $(i+1)$-th segment, $\ldots$ , and its $(j-1)$-th segment.
We now incrementally construct the set ${\mathcal{P}}$ of paths.
Intuitively, we will silmutaneously construct all paths of ${\mathcal{P}}$ in $k-1$ iterations, where in the $i$-th iteration, we will determine the $i$-th segment of all paths in ${\mathcal{P}}$.
Throughout, we will maintain a set $\hat{{\mathcal{P}}}=\set{\hat P_e\mid e\in E'}$ of paths, where each path $\hat P_e$ is indexed by an edge $e$ of $E'$.
Assume that the edge $e$ connects a vertex of $C_i$ to a vertex of $C_j$ (with $j\ge i+2$), then the path $\hat P_e$ is supposed to originate at $u_i$ and terminate at $u_j$.
We call $u_i$ the \emph{destined origin} of $\hat P_e$ and call $u_i$ the \emph{destined terminal} of $\hat P_e$.
Initially, $\hat{{\mathcal{P}}}$ contains $|E'|$ paths, and all paths in $\hat{{\mathcal{P}}}$ contain no edge.
We will sequentially process vertices $u_1,\ldots,u_{k-1}$, and, for each $1\le i\le k-1$, upon processing vertex $u_i$, determine which path of ${\mathcal{R}}^*_{(i)}$ serves as the $i$-th segment of which path of $\hat{\mathcal{P}}$.
We now fix an index $1\le i\le k-1$ and describe the iteration of processing vertex $u_i$. The current set $\hat{\mathcal{P}}$ of paths can be partitioned into four sets: set $\hat{\mathcal{P}}^{o}_i$ contains all paths of $\hat{\mathcal{P}}$ whose destined origin is $u_i$; set $\hat{\mathcal{P}}^{t}_i$ contains all paths of $\hat{\mathcal{P}}$ whose destined terminal is $u_i$; set $\hat{\mathcal{P}}^{\textsf{thr}}_i$ contains all paths whose destined origin is $u_{i'}$ for some index $i'<i$ and whose destined terminal is $u_{j'}$ for some index $j'>i$; and set $\hat{\mathcal{P}}\setminus (\hat{\mathcal{P}}^{o}_i\cup \hat{\mathcal{P}}^{t}_i\cup \hat{\mathcal{P}}^{\textsf{thr}}_i)$ contains all other paths.
Note that the paths in set $\hat{\mathcal{P}}^{t}_i$ and set $\hat{\mathcal{P}}\setminus (\hat{\mathcal{P}}^{o}_i\cup \hat{\mathcal{P}}^{t}_i\cup \hat{\mathcal{P}}^{\textsf{thr}}_i)$ do not contain an $i$-th segment, so in this iteration we will determine the $i$-th segment of paths in the sets $\hat{\mathcal{P}}^{o}_i$ and $\hat{\mathcal{P}}^{\textsf{thr}}_i$.
Note that the paths in $\hat{\mathcal{P}}^{o}_i$ currently contain no edge,
and the paths in $\hat{\mathcal{P}}^{\textsf{thr}}_i$ currently contain up to its $(i-1)$-th segment.
We then denote
\begin{itemize}
\item by $L^-_i$ the multi-set of the current last edges of paths in $\hat{\mathcal{P}}^{\textsf{thr}}_i$;
\item by $L^+_i$ the multi-set of the first edges of paths in ${\mathcal{R}}^*_{(i)}$ (note that these paths are currently not designated as the $i$-th segment of any path in $\hat{{\mathcal{P}}}$); and
\item by $L^{\mathsf{out}}_i$ the multi-set of last edges of paths in ${\mathcal{P}}^{\mathsf{out}}_{i,*}=\set{P^{\mathsf{out}}_e \mid e\in E^{\operatorname{right}}_i}$.
\end{itemize}
Clearly, elements in sets $L^-_i, L^+_i,L^{\mathsf{out}}_i$ are edges of $\delta(u_i)$.
We then define, for each $e\in \delta(u_i)$, $n^-_e=n_{L^-_i}(e)+n_{L^{\mathsf{out}}_i}(e)$ and
$n^+_e=n_{L^+_i}(e)$.
We use the following simple observation.
\begin{observation}
For each edge $e\in \delta(u_i)$, $\sum_{e\in \delta(u_i)}n^-_e=\sum_{e\in \delta(u_i)}n^+_e$.
\end{observation}
\begin{proof}
On one hand, $\sum_{e\in \delta(u_i)}n_{L^{\mathsf{out}}_i}(e)=|\bigcup_{j>i+1}E(C_i,C_j)|=|\hat{\mathcal{P}}^{o}_i|$, and $\sum_{e\in \delta(u_i)}n_{L^{-}_i}(e)=|L^-_i|=|\hat{\mathcal{P}}^{\textsf{thr}}_i|$.
On the other hand, recall that $|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|$, ${\mathcal{P}}^*_{(i)}=\set{P^*_{(i)}\text{ } \bigg|\text{ } u_i,u_{i+1}\in V(P^*),P^*\in {\mathcal{P}}^*}$ and the only edges in $E'$ such that path $P^*_e$ contains vertices $u_i,u_{i+1}$ are edges of $E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$. Therefore,
$\sum_{e\in \delta(u_i)}n_{L^{+}_i}(e)=|L^+_i|=|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|=|E^{\textsf{thr}}_i|+|E^{\operatorname{right}}_i|=|\hat{\mathcal{P}}^{\textsf{thr}}_i|+|\hat{\mathcal{P}}^{o}_i|$.
\end{proof}
We apply the algorithm in \Cref{obs:rerouting_matching_cong} to graph $G$, vertex $u_i$, rotation ${\mathcal{O}}_{u_i}$ and integers $\set{n^-_e,n^+_{e}}_{e\in \delta(u_i)}$.
Let $M$ be the multi-set of ordered pairs of the edges of $\delta(u_i)$ that we obtain.
We then designate:
\begin{itemize}
\item for each path $\hat P_e\in \hat{\mathcal{P}}^{\textsf{thr}}_i$ with $e^-$ as its current last edge, a path of ${\mathcal{R}}^*_{(i)}$ that contains the edge $e^+$ as its first edge with $(e^-,e^+)\in M$, as the $i$-th segment of $\hat P_e$; and
\item for each path $P^{\mathsf{out}}_e\in{\mathcal{P}}^{\mathsf{out}}_{i,*}$ with $e^-$ as its last edge (recall that we view $u_i$ as the last endpoint of such a path), a path of ${\mathcal{R}}^*_{(i)}$ that contains the edge $e^+$ as its first edge with $(e^-,e^+)\in M$, as the $i$-th segment of $\hat P_e$;
\end{itemize}
such that each path of ${\mathcal{R}}^*_{(i)}$ is assigned to exactly one path of $\hat {\mathcal{P}}^{\textsf{thr}}_i\cup {\mathcal{P}}^{\mathsf{out}}_{i,*}$.
This completes the description of the $i$-th iteration.
See Figure~\ref{fig: inner_path} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges in $\delta(u_i)$: only the last/first edges of paths are shown. Here ${\mathcal{P}}^{\mathsf{out}}_{*,i}=\set{P^{\mathsf{out}}_e \mid e\in E^{\operatorname{left}}_{i}}$.]{\scalebox{0.32}{\includegraphics{figs/inner_path_1.jpg}}}
\hspace{1pt}
\subfigure[Sets $L^-_i, L^+_i,L^{\mathsf{out}}_i$ and the pairing (shown in dash pink lines) given by the algorithm in \Cref{obs:rerouting_matching_cong}.]{
\scalebox{0.32}{\includegraphics{figs/inner_path_2.jpg}}}
\caption{An illustration of an iteration in Step 3 of computing inner paths.}\label{fig: inner_path}
\end{figure}
Let ${\mathcal{P}}$ be the set of paths that we obtain after $k-1$ iterations of processings vertices $u_1,\ldots,u_{k-1}$, and we rename the path $\hat P_e$ at the end of the algorithm by $P_e$, and say that $P_e$ is the inner path of $e$, for every edge $e\in E'$.
It is clear that $|{\mathcal{P}}|=|E'|$. It is also easy to see from the algorithm that, for each edge $e\in E'$ connecting $C_i$ to $C_j$ (with $j\ge i+2$), the inner path $P_e$ starts at $u_i$ and ends at $u_j$ (as it is supposed to), and it contains, for each $i\le t\le j-1$, a path of ${\mathcal{R}}^*_{(i)}$ as its $t$-th segment.
Therefore, path $P_e$ visits vertices $u_i,u_{i+1},\ldots,u_j$ sequentially.
The following observations are immediate from our algorithm of constructing the inner paths.
\begin{observation}
\label{obs: non_transversal_1}
For each $1\le i\le k-1$, if we denote, for each $e\in E_i^{\operatorname{right}}$, by $R_e$ the path consisting of the last edge of path $P^{\mathsf{out}}_e$ and the first edge of path $P_e$, then the paths in $\set{P_e\mid e\in \bigcup_{ i'<i<j'}E(C_{i'},C_{j'})}$ and $\set{R_e\mid e\in E_i^{\operatorname{right}}}$ are non-transversal at $u_i$.
\end{observation}
\begin{observation}
\label{obs: non_transversal_2}
The inner paths in ${\mathcal{P}}$ are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
On one hand, note that for each $1\le i\le k-1$, the set ${\mathcal{R}}^*_{(i)}$ of paths are non-transversal with respect to $\Sigma$.
Since the $i$-th segments of paths in ${\mathcal{P}}$ are paths in ${\mathcal{R}}^*_{(i)}$, it follows that the paths of ${\mathcal{P}}$ are non-transversal at all vertices of $V(G)\setminus\set{u_1,\ldots,u_k}$.
On the other hand, from the algorithm and \Cref{obs:rerouting_matching_cong}, it is easy to verify that the paths of ${\mathcal{P}}$ are also non-transversal at $u_1,\ldots,u_k$.
\end{proof}
\begin{observation}
\label{obs: edge_occupation in outer and inner paths}
Each edge $e\in E'$ belongs to exactly one outer path and no inner paths, each edge of $\bigcup_{1\le i\le k-1}\hat E_i$ belongs to no outer paths and $O(1)$ inner paths; and each edge of $\bigcup_{1\le i\le k}E(C_i)$ belongs to no outer paths and $O(\cong_G({\mathcal{Q}}_i,e))$ inner paths.
\end{observation}
\subsubsection{Construct Sub-Instances}
\label{sec: disengaged instances}
Recall that ${\mathcal{C}}=\set{C_1,\ldots,C_k}$.
In this subsection we will construct, for each cluster $C_i\in {\mathcal{C}}$, an sub-instance $I_i=(G_i,\Sigma_i)$ of $(G,\Sigma)$, such that the instances ${\mathcal{I}}=\set{I_1,\ldots,I_k}$ satisfy the properties in \Cref{lem: path case}.
For each $1\le i\le k$, recall that $E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$. We denote $E^{\textsf{thr}}_i=\bigcup_{i'\le i-1,j'\ge i+1}E(C_{i'},C_{j'})$.
\iffalse
\begin{claim}
\label{clm: rotation_distance}
There exist integers $b_1,\ldots,b_k\in \set{0,1}$, such that $\sum_{1\le i\le k-1}\mbox{\sf dist}(({\mathcal{O}}^{\operatorname{right}}_i,b_i), ({\mathcal{O}}^{\operatorname{left}}_{i+1},b_{i+1}))\le O(\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\cdot\operatorname{poly}\log n)$.
\end{claim}
\begin{proof}
\end{proof}
\fi
\iffalse
We first construct instance $(G_k,\Sigma_k)$ as follows.
The graph $G_k$ is obtained from $G$ by contracting all vertices in $G\setminus C_k$ into a single vertex, that we denote by $v^{\operatorname{left}}_k$.
Notice that the edges incident to $v^{\operatorname{left}}_k$ correspond to the edges of $\delta(C_k)$.
The ordering of a vertex of $v\ne v^{\operatorname{left}}_k$ in $\Sigma_k$ is identical to the ordering ${\mathcal{O}}_v$ of $v$ in the given rotation system $\Sigma$.
Note that $\delta_{G_k}(v^{\operatorname{left}}_k)=\delta_G(C_k)=\delta_H(u_k)$,
and there is an one-to-one correspondence between edges in $\delta_H(u_k)$ and paths in ${\mathcal{P}}^*_{(i-1)}$.
The ordering of the vertex $v^{\operatorname{left}}_k$ in $\Sigma_k$ is then defined to be ${\mathcal{O}}^{\operatorname{right}}_{k-1}$ (which is an ordering on the set ${\mathcal{P}}^*_{(i-1)}$, and is therefore also an ordering on $\delta_{G_k}(v^{\operatorname{left}}_k)$).
The instance $(G_1,\Sigma_1)$ is defined similarly.
The graph $G_1$ is obtained from $G$ by contracting all vertices in $G\setminus C_1$ into a single vertex, that we denote by $v^{\operatorname{right}}_1$.
We will not distinguish between edges incident to $v^{\operatorname{left}}_1$ and edges of $\delta(C_k)$.
The ordering of a vertex $v\ne v^{\operatorname{right}}_1$ in $\Sigma_1$ is identical to the ordering ${\mathcal{O}}_v$ in $\Sigma$.
Note that $\delta_{G_1}(v^{\operatorname{right}}_1)=\delta_G(C_1)=\delta_H(u_1)$,
and there is an one-to-one correspondence between edges in $\delta_H(u_1)$ and paths in ${\mathcal{P}}^*_{(1)}$.
The ordering of the vertex $v^{\operatorname{right}}_1$ in $\Sigma_1$ is then defined to be ${\mathcal{O}}^{\operatorname{left}}_2$ (which is an ordering on the set ${\mathcal{P}}^*_{(1)}$, and is therefore also an ordering on $\delta_{G_1}(v^{\operatorname{right}}_1)$).
\fi
\paragraph{Instances $(G_2,\Sigma_2),\ldots,(G_{k-1},\Sigma_{k-1})$.}
We first fix some index $2\le i\le k-1$ and define the instance $(G_i,\Sigma_i)$ as follows.
The graph $G_i$ is obtained from $G$ by first contracting clusters $C_1,\ldots,C_{i-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_i$, and then contracting clusters $C_{i+1},\ldots,C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_i$, and finally deleting self-loops on super-nodes $v^{\operatorname{left}}_i$ and $v^{\operatorname{right}}_i$. So $V(G_i)=V(C_i)\cup \set{ v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$. See \Cref{fig: disengaged instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edge sets in $G$. Edges of $E^{\operatorname{left}}_{i}$ and $E^{\operatorname{right}}_{i}$ are shown in green. Edges of $E^{\textsf{thr}}_{i}$ are shown in red. ]{\scalebox{0.32}{\includegraphics{figs/disengaged_instance_1.jpg}
}
\hspace{1pt}
\subfigure[Graph $G_i$. The edges incident to $v_i^{\operatorname{right}}$ are $\hat E_i \cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, and the edges incident to $v_i^{\operatorname{left}}$ are $\hat E_{i-1} \cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_{i}$.]{
\scalebox{0.32}{\includegraphics{figs/disengaged_instance_2.jpg}}}
\caption{An illustration of the disengaged instance.}\label{fig: disengaged instance}
\end{figure}
The ordering on the incident edges of a vertex $v\in V(C_i)$ in $\Sigma_i$ is defined to be ${\mathcal{O}}_v$, the rotation on vertex $v$ in the given rotation system $\Sigma$.
It remains to define the orderings on the incident edges of super-nodes $v^{\operatorname{left}}_i,v^{\operatorname{right}}_i$.
We first consider the super-node $v^{\operatorname{left}}_i$.
Note that $\delta_{G_i}(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$.
For each edge $\hat e\in \hat E_{i-1}$, recall that $Q_{i-1}(\hat e)$ is the path in $C_{i-1}$ connecting $\hat e$ to $u_{i-1}$.
For each edge $e\in E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$ that connects a vertex of $C_{i'}$ to a vertex of $C_{j'}$ with $i'<i<j'$, we denote by $W_e$ the path obtained by concatenating
(i) the subpath of $P^{\mathsf{out}}_e$ between (including) edge $e$ and its last endpoint $u_{i'}$; and
(ii) the subpath of $P_e$ between its first endpoint $u_{i'}$ and the vertex $u_{i-1}$.
Clearly, the path $W_e$ defined above connects $e$ to $u_{i-1}$.
We denote ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_e\mid e\in E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i}\cup \set{Q_{i-1}(\hat e)\mid \hat e\in \hat E_{i-1}}$, and we now define the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$ according to the paths of ${\mathcal{W}}^{\operatorname{left}}_i$ and the rotation ${\mathcal{O}}_{u_{i-1}}$, in a similar way that we define the circular ordering on $\delta_G(C)$ according to the paths in ${\mathcal{Q}}(C)$ and the ordering ${\mathcal{O}}_{u(C)}$ in Section~\ref{subsec: basic disengagement}.
Intuitively, the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ is the ordering in which the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ enter $u_{i-1}$.
Formally, for every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$, let $e^*_W$ be the last edge lying on path $W$, that must belong to $\delta_{G}(u_{i-1})$. We first define a circular ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$, as follows: the paths are ordered according to the circular ordering of their last edges $e^*_W$ in ${\mathcal{O}}_{u_{i-1}}\in \Sigma$, breaking ties arbitrarily. Since every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$ is associated with a unique edge in $\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, that serves as the first edge on path $W$, this ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ immediately defines a circular ordering of the edges of $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, that we denote by ${\mathcal{O}}^{\operatorname{left}}_i$. See Figure~\ref{fig: v_left rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i-1}=\set{\hat e_1,\ldots,\hat e_4}$, $E^{\operatorname{left}}_{i}=\set{e^g_1,e^g_2}$ and $E^{\textsf{thr}}_{i}=\set{e^r_1,e^r_2}$.
Paths of ${\mathcal{W}}^{\operatorname{left}}_i$ excluding their first edges are shown in dash lines.]{\scalebox{0.36}{\includegraphics{figs/rotation_left_1.jpg}}}
\hspace{1pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{left}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. $\delta_{G_i}(v^{\operatorname{left}}_i)=\set{\hat e_1,\hat e_2,\hat e_3,\hat e_4,e^g_1,e^g_2, e^r_1,e^r_2}$, and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on them around $v^{\operatorname{left}}_i$ is shown above.]{
\scalebox{0.28}{\includegraphics{figs/rotation_left_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_left rotation}
\end{figure}
The rotation ${\mathcal{O}}^{\operatorname{right}}_{i}$ on vertex $v^{\operatorname{right}}_i$ is defined similarly.
Note that $\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$.
For each edge $\hat e'\in \hat E_{i}$, recall that $Q_{i}(\hat e')$ is the path in $C_{i}$ connecting $\hat e'$ to $u_{i}$.
For each edge $e\in E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$ that connects a vertex of $C_{i'}$ to a vertex of $C_{j'}$ with $i'\le i<j'$, we denote by $W_e$ the path obtained by concatenating
(i) the subpath of $P^{\mathsf{out}}_e$ between (including) edge $e$ and its last endpoint $u_{i'}$; and
(ii) the subpath of $P_e$ between its first endpoint $u_{i'}$ and vertex $u_{i}$ (note that for an edge $e\in E^{\operatorname{right}}_{i}$, such a subpath only contains a single node $u_i$).
Clearly, the path $W_e$ defined above connects $e$ to $u_{i}$.
We denote ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_e\text{ }\big|\text{ } e\in E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i}\cup \set{Q_{i}(\hat e')\text{ }\big|\text{ } \hat e'\in \hat E_{i}}$.
The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ is defined in a similar way as the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$, according to the paths of ${\mathcal{W}}^{\operatorname{right}}_i$ and the rotation ${\mathcal{O}}_{u_{i}}$.
See Figure~\ref{fig: v_right rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i}=\set{\hat e_1',\ldots,\hat e_4'}$, $E^{\operatorname{left}}_{i}=\set{\tilde e^g_1,\tilde e^g_2}$ and $E^{\textsf{thr}}_{i}=\set{e^r_1,e^r_2}$.
Paths of ${\mathcal{W}}^{\operatorname{right}}_i$ excluding their first edges are shown in dash lines. ]{\scalebox{0.36}{\includegraphics{figs/rotation_right_1.jpg}
}
\hspace{1pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. $\delta_{G_i}(v^{\operatorname{right}}_i)=\set{\hat e_1',\hat e_2',\hat e_3',\hat e_4',\tilde e^g_1,\tilde e^g_2, e^r_1,e^r_2}$, and the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on them around $v^{\operatorname{right}}_i$ is shown above.]{
\scalebox{0.28}{\includegraphics{figs/rotation_right_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_right rotation}
\end{figure}
\paragraph{Instances $(G_1,\Sigma_1)$ and $(G_k,\Sigma_k)$.}
The instances $(G_1,\Sigma_1)$ and $(G_k,\Sigma_k)$ are defined similarly, but instead of two super-nodes, the graphs $G_1$ and $G_k$ contain one super-node each.
In particular, graph $G_1$ is obtained from $G$ by contracting clusters $C_2,\ldots, C_k$ into a super-node, that we denote by $v^{\operatorname{right}}_1$, and then deleting self-loops on it.
So $V(G_1)=V(C_1)\cup \set{v^{\operatorname{right}}_{1}}$ and $\delta_{G_1}(v^{\operatorname{right}}_{1})=\hat E_1\cup E^{\operatorname{right}}_1$.
The rotation of a vertex $v\in V(C_1)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{right}}_1$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{right}}_i$ for any index $2\le i\le k-1$.
Graph $G_k$ is obtained from $G$ by contracting clusters $C_1,\ldots, C_{k-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_k$, and then deleting self-loops on it.
So $V(G_k)=V(C_k)\cup \set{v^{\operatorname{left}}_{k}}$ and $\delta_{G_k}(v^{\operatorname{left}}_{k})=\hat E_{k-1}\cup E^{\operatorname{left}}_k$.
The rotation of a vertex $v\in V(C_k)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{left}}_k$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{left}}_i$ for any index $2\le i\le k-1$.
We will use the following claims later for completing the proof of \Cref{lem: path case} in the special case where there is no problematic index $i$.
\begin{observation}
\label{obs: disengaged instance size}
$\sum_{1\le i\le k}|E(G_i)|= 2m$, and for each $1\le i\le k$, $|E(G_i)|\le m/\mu$.
\end{observation}
\begin{proof}
Note that, in the sub-instances $\set{(G_i,\Sigma_i)}_{1\le i\le k}$, each graph of $\set{G_i}_{1\le i\le k}$ is obtained from $G$ by contracting some sets of clusters of ${\mathcal{C}}$ into a single super-node. Therefore, each edge of $G_i$ corresponds to an edge in $E(G)$ (and we do not distinguish between them).
So we can write $E(G_i)=E_G(C_i)\cup E_G(C_i,\overline{C_i})$.
Therefore, for each $1\le i\le k$, $|E(G_i)|=|E_G(C_i)|+|E_G(C_i,\overline{C_i})|\le |E_G(C_i)|+|E^{\textsf{out}}({\mathcal{C}})|\le m/(100\mu)+m/(100\mu)\le m/\mu$.
On the other hand, since clusters $C_1,\ldots,C_k$ are vertex-disjoint, every edge of $E(G)$ appears twice in the graphs $\set{G_i}_{1\le i\le k}$. It follows that $\sum_{1\le i\le k}|E(G_i)|= 2m$.
\end{proof}
\begin{observation}
\label{obs: rotation for stitching}
For each $1\le i\le k-1$, if we view edge sets $\delta_{G_i}(v^{\operatorname{right}}_i),\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$ as subsets of $E(G)$, then $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$, and
${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{observation}
\begin{proof}
Recall that for each $1\le i\le k-1$,
$\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})=\hat E_{i}\cup E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}$. From the definition of sets $E_i^{\textsf{thr}},E_{i+1}^{\textsf{thr}}, E^{\operatorname{right}}_i, E^{\operatorname{left}}_{i+1}$,
\[
\begin{split}
E_i^{\textsf{thr}}\cup E^{\operatorname{right}}_i
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i<j'\text{ or }i'=i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'\le i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'< i+1\le j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i+1<j'\text{ or }i'<i+1=j'}=E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}.
\end{split}
\]
Therefore, $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Moreover, from the definition of path sets ${\mathcal{W}}^{\operatorname{right}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_{i+1}$, it is not hard to verify that, for every edge $e\in \delta_{G_i}(v^{\operatorname{right}}_i)$, the path in ${\mathcal{W}}^{\operatorname{right}}_i$ that contains $e$ as its first edge is identical to the path in ${\mathcal{W}}^{\operatorname{left}}_{i+1}$ that contains $e$ as its first edge. According to the way that rotations ${\mathcal{O}}^{\operatorname{right}}_i,{\mathcal{O}}^{\operatorname{left}}_{i+1}$ are defined, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{proof}
\subsubsection{Completing the Proof of \Cref{lem: path case} in the Special Case}
\label{sec: path case with no problematic index}
In this section we complete the proof of \Cref{lem: path case} in the special case where there is no problematic index. Specifically, we use the following two claims, whose proofs will be provided later.
\begin{claim}
\label{claim: existence of good solutions special}
$\sum_{1\le i\le k}\mathsf{OPT}_{\mathsf{cnwrs}}(G_i,\Sigma_i)\le O((\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+|E(G)|)\cdot \beta)$.
\end{claim}
\begin{claim}
\label{claim: stitching the drawings together}
There is an efficient algorithm, that given, for each $1\le i\le k$, a feasible solution $\phi_i$ to the instance $(G_i,\Sigma_i)$, computes a solution to the instance $(G,\Sigma)$, such that $\mathsf{cr}(\phi)\le \sum_{1\le i\le k}\mathsf{cr}(\phi_i)$.
\end{claim}
We use the algorithm described in~\Cref{sec: disengaged instances} and~\Cref{sec: inner and outer paths}, and return the disengaged instances $(G_1,\Sigma_1),\ldots,(G_k,\Sigma_k)$ as the collection of sub-instances of $(G,\Sigma)$.
From the previous subsections, the algorithm for producing the sub-instances is efficient.
On the other hand, it follows immediately from \Cref{obs: disengaged instance size}, \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together} that the sub-instances $(G_1,\Sigma_1),\ldots,(G_k,\Sigma_k)$ satisfy the properties in \Cref{lem: path case}.
This completes the proof of \Cref{lem: path case}.
We now provide the proofs of \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together}.
$\ $
\begin{proofof}{Claim~\ref{claim: existence of good solutions special}}
We will construct, for each $1\le i\le k$, a drawing $\phi_i$ of $G_i$ that respects the rotation system $\Sigma_i$, based on the optimal drawing $\phi^*$ of the instance $(G,\Sigma)$, such that $\sum_{1\le i\le k}\mathsf{cr}(\phi_i)\le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot \beta)$. The \Cref{claim: existence of good solutions special} then follows.
\paragraph{Drawings $\phi_2,\ldots,\phi_{k-1}$.}
First we fix some index $2\le i\le k-1$, and describe the construction of the drawing $\phi_i$. We start with some definitions.
Recall that $E(G_i)=E_G(C_i)\cup (\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$.
We define the auxiliary path set ${\mathcal{W}}_i={\mathcal{W}}^{\operatorname{left}}_i\cup {\mathcal{W}}^{\operatorname{right}}_i$, so
$${\mathcal{W}}_i=\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}\cup \set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}},$$
where for each $e\in E^{\operatorname{left}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$; for each $e\in E^{\operatorname{right}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between $u_{i+1}$ and its last endpoint; and for each $e\in E^{\textsf{thr}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$, the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$, and the subpath of $P_e$ between $u_{i+1}$ and its last endpoint.
We use the following observation.
\begin{observation}
\label{obs: wset_i_non_interfering}
The set ${\mathcal{W}}_i$ of paths are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
Recall that the paths in ${\mathcal{Q}}_{i-1}$ only uses edges of $E(C_{i-1})\cup \delta(C_{i-1})$, and they are non-transversal.
And similarly, the paths in ${\mathcal{Q}}_{i+1}$ only uses edges of $E(C_{i+1})\cup \delta(C_{i+1})$, and they are non-transversal.
Therefore, the paths in $\set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}}$ are non-transversal.
From \Cref{obs: non_transversal_1} and \Cref{obs: non_transversal_2}, the paths in $\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}$ are non-transversal. Therefore, it suffices to show that, the set ${\mathcal{W}}_i$ of paths are non-transversal at all vertices of $C_{i-1}$ and all vertices of $C_{i+1}$. Note that, for each edge $e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i})$, $W_e\cap (C_{i-1}\cup \delta(C_{i-1}))$ is indeed a path of ${\mathcal{Q}}_{i-1}$. Therefore, the paths in ${\mathcal{W}}_i$ are non-transversal at all vertices of $C_{i-1}$. Similarly, they are also non-transversal at all vertices of $C_{i+1}$. Altogether, the paths of ${\mathcal{W}}_i$ are non-transversal with respect to $\Sigma$.
\end{proof}
For uniformity of notations, for each edge $\hat e\in \hat E_i$, we rename the path $Q_{i+1}(\hat e)$ by $W_{\hat e}$, and similarly for each edge $\hat e\in \hat E_{i-1}$, we rename the path $Q_{i-1}(\hat e)$ by $W_{\hat e}$. Therefore, ${\mathcal{W}}_i=\set{W_e\mid e\in E(G_i)\setminus E(C_i)}$.
Put in other words, the set ${\mathcal{W}}_i$ contains, for each edge $e$ in $G_i$ that is incident to $v^{\operatorname{left}}_i$ or $v^{\operatorname{right}}_i$, a path named $W_e$.
It is easy to see that all paths in ${\mathcal{W}}_i$ are internally disjoint from $C_i$.
We further partition the set ${\mathcal{W}}_i$ into two sets: ${\mathcal{W}}_i^{\textsf{thr}}=\set{W_e\mid e\in E^{\textsf{thr}}_i}$ and $\tilde {\mathcal{W}}_i={\mathcal{W}}_i\setminus {\mathcal{W}}_i^{\textsf{thr}}$.
We are now ready to construct the drawing $\phi_i$ for the instance $(G_i,\Sigma_i)$.
Recall that $\phi^*$ is an optimal drawing of the input instance $(G,\Sigma)$.
We start with the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$, that we denote by $\phi^*_i$. We will not modify the image of $C_i$ in $\phi^*_i$, but will focus on constructing the image of edges in $E(G_i)\setminus E(C_i)$, based on the image of edges in $E({\mathcal{W}}_i)$ in $\phi^*_i$.
Specifically, we proceed in the following four steps.
\paragraph{Step 1.}
For each edge $e\in E({\mathcal{W}}_i)$, we denote by $\pi_e$ the curve that represents the image of $e$ in $\phi^*_i$. We create a set of $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting the endpoints of $e$ in $\phi^*_i$, that lies in an arbitrarily thin strip around $\pi_e$.
We denote by $\Pi_e$ the set of these curves.
We then assign, for each edge $e\in E({\mathcal{W}}_i)$ and for each path in ${\mathcal{W}}_i$ that contains the edge $e$, a distinct curve in $\Pi_e$ to this path.
Therefore, each curve in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ is assigned to exactly one path of ${\mathcal{W}}_i$,
and each path $W\in {\mathcal{W}}_i$ is assigned with, for each edge $e\in E(W)$, a curve in $\Pi_e$. Let $\gamma_W$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ that are assigned to path $W$, so $\gamma_W$ connects the endpoints of path $W$ in $\phi^*_i$.
In fact, when we assign curves in $\bigcup_{e\in \delta(u_{i-1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{left}}_i$ (recall that $\delta(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\operatorname{left}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{left}}_i)}$), we additionally ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{left}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{left}}_i)$ enter $u_{i-1}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
And similarly, when we assign curves in $\bigcup_{e\in \delta(u_{i+1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{right}}_i$ (recall that $\delta(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\operatorname{right}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{right}}_i)}$), we ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{right}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{right}}_i)$ enter $u_{i+1}$ in the same order as ${\mathcal{O}}^{\operatorname{right}}_i$.
Note that this can be easily achieved according to the definition of ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$.
We denote $\Gamma_i=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$, and we further partition the set $\Gamma_i$ into two sets: $\Gamma_i^{\textsf{thr}}=\set{\gamma_{W}\mid W\in {\mathcal{W}}^{\textsf{thr}}_i}$ and $\tilde \Gamma_i=\Gamma_i\setminus \Gamma_i^{\textsf{thr}}$.
We denote by $\hat \phi_i$ the drawing obtained by taking the union of the image of $C_i$ in $\phi^*_i$ and all curves in $\Gamma_i$.
For every path $P$ in $G_i$, we denote by $\hat{\chi}_i(P)$ the number of crossings that involves the ``image of $P$'' in $\hat \phi_i$, which is defined as the union of, for each edge $e\in E(\tilde{\mathcal{W}}_i)$, an arbitrary curve in $\Pi_e$.
Clearly, for each edge $e\in E({\mathcal{W}}_i)$, all curves in $\Pi_e$ are crossed by other curves of $(\Gamma_i\setminus \Pi_e)\cup \phi^*_i(C_i)$ same number of times.
Therefore, $\hat{\chi}_i(P)$ is well-defined.
For a set ${\mathcal{P}}$ of paths in $G_i$, we define $\hat{\chi}_i({\mathcal{P}})=\sum_{P\in {\mathcal{P}}}\hat{\chi}_i(P)$.
\iffalse
We use the following observation.
\znote{maybe remove this observation?}
\begin{observation}
\label{obs: curves_crossings}
The number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$,
and the number of intersections between a curve in $\tilde\Gamma_i$ and the image of edges of $C_i$ in $\phi^*_i$, are both $O(\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W))$.
\end{observation}
\begin{proof}
We first show that the number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Note that, from the construction of curves in $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i}$, every crossing between a pair $W,W'$ of curves in $\tilde\Gamma_i$ must be the intersection between a curve in $\Pi_e$ for some $e\in E(W)$ and a curve in $\Pi_{e'}$ for some $e'\in E(W')$, such that the image $\pi_e$ for $e$ and the image $\pi_{e'}$ for $e'$ intersect in $\phi^*$.
Therefore, for each pair $W,W'$ of paths in $\tilde{\mathcal{W}}_i$, the number of points that belong to only curves $\gamma_W$ and $\gamma_{W'}$ is at most the number of crossings between the image of $W$ and the image of $W'$ in $\phi^*$.
It follows that the number of points that belong to exactly two curves of $\tilde\Gamma_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Altogether, the number of intersections between curves in $\tilde\Gamma_i$ is at most $|V(\tilde {\mathcal{W}}_i)|+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
We now show that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ that are not vertex-image is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Let $W$ be a path of $\tilde {\mathcal{W}}_i$ and consider the curve $\gamma_W$. Note that $\gamma_W$ is the union of, for each edge $e\in E(W)$, a curve that lies in an arbitrarily thin strip around $\pi_e$.
Therefore, the number of crossings between $\gamma_W$ and the image of $C_i$ in $\phi^*_i$ is identical to the number of crossings the image of path $W$ and the image of $C_i$ in $\phi^*_i$, which is at most $\hat\mathsf{cr}(W)$.
It follows that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
\end{proof}
\fi
\paragraph{Step 2.}
For each vertex $v\in V({\mathcal{W}}_i)$, we denote by $x_v$ the point that represents the image of $v$ in $\phi^*_i$, and we let $X$ contains all points of $\set{x_v\mid v\in V({\mathcal{W}}_i)}$ that are intersections between curves in $\Gamma_i$.
We now manipulate the curves in $\set{\gamma_W\mid W\in {\mathcal{W}}_i}$ at points of $X$, by processing points of $X$ one-by-one, as follows.
Consider a point $x_v$ that is an intersection between curves in $\Gamma_i$, where $v\in V({\mathcal{W}}_i)$, and let $D_v$ be an arbitrarily small disc around $x_v$.
We denote by ${\mathcal{W}}_i(v)$ the set of paths in ${\mathcal{W}}_i$ that contains $v$, and further partition it into two sets: ${\mathcal{W}}^{\textsf{thr}}_i(v)={\mathcal{W}}_i(v)\cap {\mathcal{W}}^{\textsf{thr}}_i$ and $\tilde{\mathcal{W}}_i(v)={\mathcal{W}}_i(v)\cap \tilde{\mathcal{W}}_i$. We apply the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $\set{\gamma_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ within disc $D_v$. Let $\set{\gamma'_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ be the set of new curves that we obtain. From \Cref{obs: curve_manipulation},
(i) for each path $W\in \tilde{\mathcal{W}}_i(v)$, the curve $\gamma'_W$ does not contain $x_v$, and is identical to the curve $\gamma_W$ outside the disc $D_v$;
(ii) the segments of curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc $D_v$ are in general position; and
(iii) the number of icrossings between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside $D_v$ is bounded by $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\iffalse{just for backup}
\begin{proof}
Denote $d=\deg_G(v)$ and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
For each path $W\in \tilde{\mathcal{W}}_i(v)$, we denote by $p^{-}_W$ and $p^{+}_W$ the intersections between the curve $\gamma_W$ and the boundary of ${\mathcal{D}}_v$.
We now compute, for each $W\in W\in \tilde{\mathcal{W}}_i(v)$, a curve $\zeta_W$ in ${\mathcal{D}}_v$ connecting $p^{-}_W$ to $p^{+}_W$, such that (i) the curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ are in general position; and (ii) for each pair $W,W'$ of paths, the curves $\zeta_W$ and $\zeta_{W'}$ intersects iff the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{-}_{W'},p^{+}_{W},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_{W'},p^{-}_{W},p^{+}_{W'})$.
It is clear that this can be achieved by first setting, for each $W$, the curve $\zeta_W$ to be the line segment connecting $p^{-}_W$ to $p^{+}_W$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
We now define, for each $W$, the curve $\gamma'_W$ to be the union of the part of $\gamma_W$ outside ${\mathcal{D}}_v$ and the curve $\zeta_W$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, the first and the second condition of \Cref{obs: curve_manipulation} are satisfied.
It remains to estimate the number of intersections between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$, which equals the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$.
Since the paths in $\tilde{\mathcal{W}}_i(v)$ are non-transversal with respect to $\Sigma$ (from \Cref{obs: wset_i_non_interfering}), from the construction of curves $\set{\gamma_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$, if a pair $W,W'$ of paths in $\tilde {\mathcal{W}}_i(v)$ do not share edges of $\delta(v)$, then the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_W,p^{-}_{W'},p^{+}_{W'})$, and therefore the curves $\zeta_{W}$ and $\zeta_{W'}$ will not intersect in ${\mathcal{D}}_v$.
Therefore, only the curves $\zeta_W$ and $\zeta_{W'}$ intersect iff $W$ and $W'$ share an edge of $\delta(v)$.
Since every such pair of curves intersects at most once, the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$ is at most $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are shown in black, and curves of $\tilde{\mathcal{W}}_i(v)$ are shown in blue, red, orange and green. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are not modified, while curves of $\tilde{\mathcal{W}}_i(v)$ are re-routed via dash lines within disc ${\mathcal{D}}_v$.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the step of processing $x_v$.}\label{fig: curve_con}
\end{figure}
\fi
We then replace the curves of $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ in $\Gamma_i$ by the curves of $\set{\gamma'_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
This completes the description of the iteration of processing point the point $x_v\in X$. Let $\Gamma'_i$ be the set of curves that we obtain after processing all points in $X$ in this way. Note that we have never modified the curves of $\Gamma^{\textsf{thr}}_i$, so $\Gamma^{\textsf{thr}}_i\subseteq\Gamma'_i$, and we denote $\tilde\Gamma'_i=\Gamma'_i\setminus \Gamma^{\textsf{thr}}_i$.
We use the following observation.
\begin{observation}
\label{obs: general_position}
Curves in $\tilde\Gamma'_i$ are in general position, and if a point $p$ lies on more than two curves of $\Gamma'_i$, then either $p$ is an endpoint of all curves containing it, or all curves containing $p$ belong to $\Gamma^{\textsf{thr}}_i$.
\end{observation}
\begin{proof}
From the construction of curves in $\Gamma_i$, any point that belong to at least three curves of $\Gamma_i$ must be the image of some vertex in $\phi^*$. From~\Cref{obs: curve_manipulation}, curves in $\tilde\Gamma'_i$ are in general position; curves in $\tilde\Gamma'_i$ do not contain any vertex-image in $\phi^*$ except for their endpoints; and they do not contain any intersection of a pair of paths in $\Gamma_i^{\textsf{thr}}$. \Cref{obs: general_position} now follows.
\end{proof}
\paragraph{Step 3.}
So far we have obtained a set $\Gamma'_i$ of curves that are further partitioned into two sets $\Gamma'_i=\Gamma^{\textsf{thr}}_i\cup \tilde\Gamma'_i$,
where set $\tilde\Gamma'_i$ contains, for each path $W\in \tilde {\mathcal{W}}_i$, a curve $\gamma'_W$ connecting the endpoints of $W$, and the curves in $\tilde\Gamma'_i$ are in general position;
and set $\Gamma^{\textsf{thr}}_i$ contains, for each path $W\in {\mathcal{W}}^{\textsf{thr}}_i$, a curve $\gamma_W$ connecting the endpoints of $W$.
Recall that all paths in ${\mathcal{W}}^{\textsf{thr}}_i$ connects $u_{i-1}$ to $u_{i+1}$.
Let $z_{\operatorname{left}}$ be the point that represents the image of $u_{i-1}$ in $\phi_i^*$ and let $z_{\operatorname{right}}$ be the point that represents the image of $u_{i+1}$ in $\phi_i^*$.
Then, all curves in $\Gamma^{\textsf{thr}}_i$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
We view $z_{\operatorname{left}}$ as the first endpoint of curves in $\Gamma^{\textsf{thr}}_i$ and view $z_{\operatorname{right}}$ as their last endpoint.
We then apply the algorithm in \Cref{thm: type-2 uncrossing}, where we let $\Gamma=\Gamma^{\textsf{thr}}_i$ and let $\Gamma_0$ be the set of all other curves in the drawing $\phi^*_i$. Let $\Gamma^{\textsf{thr}'}_i$ be the set of curves we obtain. We then designate, for each edge $e\in E^{\textsf{thr}}_i$, a curve in $\Gamma^{\textsf{thr}'}_i$ as $\gamma'_{W_e}$, such that the curves of $\set{\gamma'_{W_e}\mid e\in \hat E_{i-1}\cup E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i}$ enters $z_{\operatorname{left}}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
Recall that ${\mathcal{W}}_i=\set{W_e\mid e\in (E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\cup E_i^{\operatorname{right}}\cup \hat E_{i-1}\cup \hat E_i)}$, and, for each edge $e\in E_i^{\operatorname{left}}\cup \hat E_{i-1}$, the curve $\gamma'_{W_e}$ connects its endpoint in $C_i$ to $z_{\operatorname{left}}$; for each edge $e\in E_i^{\operatorname{right}}\cup \hat E_{i}$, the curve $\gamma'_{W_e}$ connects the endpoint of $e$ to $z_{\operatorname{right}}$; and for each edge $e\in E_i^{\textsf{thr}}$, the curve $\gamma'_{W_e}$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
Therefore, if we view $z_{\operatorname{left}}$ as the image of $v^{\operatorname{left}}_i$, view $z_{\operatorname{right}}$ as the image of $v^{\operatorname{right}}_i$, and for each edge $e\in E(G_i)\setminus E(C_i)$, view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi'_i$.
It is clear from the construction of curves in $\set{\gamma'_{W_e}\mid e\in E(G_i)\setminus E(C_i)}$ that this drawing respects all rotations in $\Sigma_i$ on vertices of $V(C_i)$ and vertex $v^{\operatorname{left}}_i$.
However, the drawing $\phi'_i$ may not respect the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$. We further modify the drawing $\phi'_i$ at $z_{\operatorname{right}}$ in the last step.
\paragraph{Step 4.}
Let ${\mathcal{D}}$ be an arbitrarily small disc around $z_{\operatorname{right}}$ in the drawing $\phi'_i$, and let ${\mathcal{D}}'$ be another small disc around $z_{\operatorname{right}}$ that is strictly contained in ${\mathcal{D}}$.
We first erase the drawing of $\phi'_i$ inside the disc ${\mathcal{D}}$, and for each edge $e\in \delta(v^{\operatorname{right}}_i)$, we denote by $p_{e}$ the intersection between the curve representing the image of $e$ in $\phi'_i$ and the boundary of ${\mathcal{D}}$.
We then place, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, a point $p'_e$ on the boundary of ${\mathcal{D}}'$, such that the order in which the points in $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ appearing on the boundary of ${\mathcal{D}}'$ is precisely ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We then apply \Cref{lem: find reordering} to compute a set of reordering curves, connecting points of $\set{p_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ to points $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$.
Finally, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, let $\gamma_e$ be the concatenation of (i) the image of $e$ in $\phi'_i$ outside the disc ${\mathcal{D}}$; (ii) the reordering curve connecting $p_e$ to $p'_e$; and (iii) the straight line segment connecting $p'_e$ to $z_{\operatorname{right}}$ in ${\mathcal{D}}'$.
We view $\gamma_e$ as the image of edge $e$, for each $e\in \delta(v^{\operatorname{right}}_i)$.
We denote the resulting drawing of $G_i$ by $\phi_i$. It is clear that $\phi_i$ respects the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$, and therefore it respects the rotation system $\Sigma_i$.
We use the following claim.
\begin{claim}
\label{clm: rerouting_crossings}
The number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\end{claim}
\begin{proof}
Denote by ${\mathcal{O}}^*$ the ordering in which the curves $\set{\gamma'_{W_e}\mid e\in \delta_{G_i}(v_i^{\operatorname{right}})}$ enter $z_{\operatorname{right}}$, the image of $u_{i+1}$ in $\phi'_i$.
From~\Cref{lem: find reordering} and the algorithm in Step 4 of modifying the drawing within the disc ${\mathcal{D}}$, the number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is at most $O(\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}}))$. Therefore, it suffices to show that $\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}})=O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
In fact, we will compute a set of curves connecting the image of $u_i$ and the image of $u_{i+1}$ in $\phi^*_i$, such that each curve is indexed by some edge $e\in\delta_{G_i}(v_i^{\operatorname{right}})$ these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$ and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$, and the number of crossings between curves of $Z$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
For each $e\in E^{\textsf{thr}}_i$, we denote by $\eta_e$ the curve obtained by taking the union of (i) the curve $\gamma'_{W_e}$ (that connects $u_{i+1}$ to $u_{i-1}$); and (ii) the curve representing the image of the subpath of $P_e$ in $\phi^*_i$ between $u_i$ and $u_{i-1}$.
Therefore, the curve $\eta_e$ connects $u_i$ to $u_{i+1}$.
We then modify the curves of $\set{\eta_e\mid e\in E^{\textsf{thr}}_i}$, by iteratively applying the algorithm from \Cref{obs: curve_manipulation} to these curves at the image of each vertex of $C_{i-1}\cup C_{i+1}$. Let $\set{\zeta_e\mid e\in E^{\textsf{thr}}_i}$ be the set of curves that we obtain. We call the obtained curves \emph{red curves}. From~\Cref{obs: curve_manipulation}, the red curves are in general position. Moreover, it is easy to verify that the number of intersections between the red curves is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1}))$.
We call the curves in $\set{\gamma'_{W_e}\mid e\in \hat E_i}$ \emph{yellow curves}, call the curves in $\set{\gamma'_{W_e}\mid e\in E^{\operatorname{right}}_i}$ \emph{green curves}. See \Cref{fig: uncrossing_to_bound_crossings} for an illustration.
From the construction of red, yellow and green curves, we know that these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$, and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$.
Moreover, we are guaranteed that the number of intersections between red, yellow and green curves is at most $\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/uncross_to_bound_crossings.jpg}
\caption{An illustration of red, yellow and green curves.}\label{fig: uncrossing_to_bound_crossings}
\end{figure}
\end{proof}
From the above discussion and Claim~\ref{clm: rerouting_crossings}, for each $2\le i\le k-1$,
\[
\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right}).
\]
\iffalse
We now estimate the number of crossings in $\phi_i$ in the next claim.
\begin{claim}
\label{clm: number of crossings in good solutions}
$\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\sum_{W\in \tilde{\mathcal{W}}_i}\mathsf{cr}(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})$.
\end{claim}
\begin{proof}
$2\cdot \sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2=\sum_{v\in V(G)}\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2.$
\znote{need to redefine the orderings ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$ to get rid of $\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$ here, which we may not be able to afford.}
\end{proof}
\fi
\paragraph{Drawings $\phi_1$ and $\phi_{k}$.}
The drawings $\phi_1$ and $\phi_{k}$ are constructed similarly. We describe the construction of $\phi_1$, and the construction of $\phi_1$ is symmetric.
Recall that the graph $G_1$ contains only one super-node $v_1^{\operatorname{right}}$, and $\delta_{G_1}(v_1^{\operatorname{right}})=\hat E_1\cup E^{\operatorname{right}}_1$. We define ${\mathcal{W}}_1=\set{W_e\mid e\in E^{\operatorname{right}}_1}\cup \set{Q_2(\hat e)\mid \hat e\in \hat E_1}$. For each $\hat e\in \hat E_1$, we rename the path $Q_2(\hat e)$ by $W_e$, so ${\mathcal{W}}_1$ contains, for each edge $e\in \delta_{G_1}(v^1_{\operatorname{right}})$, a path named $W_e$ connecting its endpoints to $u_2$. Via similar analysis in \Cref{obs: wset_i_non_interfering}, it is easy to show that the paths in ${\mathcal{W}}_1$ are non-transversal with respect to $\Sigma$.
We start with the drawing of $C_1\cup E({\mathcal{W}}_1)$ induced by $\phi^*$, that we denote by $\phi^*_1$. We will not modify the image of $C_i$ in $\phi^*_i$ and will construct the image of edges in $\delta(v_1^{\operatorname{right}})$.
We perform similar steps as in the construction of drawings $\phi_2,\ldots,\phi_{k-1}$.
We first construct, for each path $W\in {\mathcal{W}}_1$, a curve $\gamma_W$ connecting its endpoint in $C_1$ to the image of $u_2$ in $\phi^*$, as in Step 1.
Let $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
We then process all intersections between curves of $\Gamma_1$ as in Step 2. Let $\Gamma'_1=\set{\gamma'_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
Since $\Gamma^{\textsf{thr}}_1=\emptyset$, we do not need to perform Steps 3 and 4. If we view the image of $u_2$ in $\phi^*_1$ as the image of $v^{\operatorname{right}}_1$, and for each edge $e\in \delta(v^{\operatorname{right}}_1)$, we view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is clear that this drawing respects the rotation system $\Sigma_1$. Moreover,
\[\mathsf{cr}(\phi_1)=\chi^2(C_1)+O\textsf{left}(\hat\chi_1({\mathcal{Q}}_2)+\sum_{W\in {\mathcal{W}}_1}\hat\chi_1(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_1,e)^2\textsf{right}).\]
Similarly, the drawing $\phi_k$ that we obtained in the similar way satisfies that
\[\mathsf{cr}(\phi_k)=\chi^2(C_k)+O\textsf{left}(\hat\chi_k({\mathcal{Q}}_{k-1})+\sum_{W\in {\mathcal{W}}_k}\hat\chi_k(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_k,e)^2\textsf{right}).\]
We now complete the proof of \Cref{claim: existence of good solutions special}, for which it suffices to estimate $\sum_{1\le i\le k}\mathsf{cr}(\phi_i)$.
Recall that, for each $1\le i\le k$, $\tilde {\mathcal{W}}_i=\set{W_e\mid e\in \hat E_{i-1}\cup \hat E_{i-1}\cup E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}}$, where $E^{\operatorname{left}}_{i}=E(C_i,\bigcup_{1\le j\le i-2}C_j)$, and
$E^{\operatorname{right}}_{i}=E(C_i,\bigcup_{i+2\le j\le k}C_j)$. Therefore, for each edge $e\in E'\cup (\bigcup_{1\le i\le k-1}\hat E_i)$, the path $W_e$ belongs to exactly $2$ sets of $\set{\tilde{\mathcal{W}}_i}_{1\le i\le k}$.
Recall that the path $W_e$ only uses edges of the inner path $P_e$ and the outer path $P^{\mathsf{out}}_e$.
Let $\tilde{\mathcal{W}}=\bigcup_{1\le i\le k}\tilde{\mathcal{W}}_i$, from \Cref{obs: edge_occupation in outer and inner paths},
for each edge $e\in E'\cup (\bigcup_{1\le i\le k-1}\hat E_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(1)$, and
for $1\le i\le k$ and for each edge $e\in E(C_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(\cong_G({\mathcal{Q}}_i,e))$.
Therefore, on one hand,
\[
\begin{split}
\sum_{1\le i\le k}\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} 2\cdot \cong_G(\tilde {\mathcal{W}},e)\cdot\cong_G(\tilde {\mathcal{W}},e')\\
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} \textsf{left}(\cong_G(\tilde {\mathcal{W}},e)^2+\cong_G(\tilde {\mathcal{W}},e')^2\textsf{right})\\
& \le
\sum_{e\in E(G)} \chi(e)\cdot \cong_G(\tilde {\mathcal{W}},e)^2
= O(\mathsf{cr}(\phi^*)\cdot\beta),
\end{split}
\]
and on the other hand,
\[
\begin{split}
\sum_{1\le i\le k}\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2
& \le \sum_{e\in E(G)} \textsf{left}(\sum_{1\le i\le k} \cong_G(\tilde {\mathcal{W}}_i,e)\textsf{right})^2\\
& \le O\textsf{left}(\sum_{e\in E(G)} \cong_G(\tilde {\mathcal{W}},e)^2 \textsf{right})\\
& \le O\textsf{left}(|E(G)|+\sum_{1\le i\le k}\sum_{e\in E(C_i)} \cong_G({\mathcal{Q}}_i,e)^2\textsf{right})=O(|E(G)|\cdot\beta).
\end{split}
\]
Moreover, $\sum_{1\le i\le k}\chi^2(C_i)\le O(\mathsf{cr}(\phi^*))$, and $\sum_{1\le i\le k}\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})\le O(\mathsf{cr}(\phi^*)\cdot\beta)$.
Altogether,
\[
\begin{split}
\sum_{1\le i\le k}\mathsf{cr}(\phi_i)
& \le
O\textsf{left}(\sum_{1\le i\le k}
\textsf{left}(
\chi^2(C_i)+ \hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})
\textsf{right})\\
& \le O(\mathsf{cr}(\phi^*))+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(|E(G)|\cdot\beta)\\
& \le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot\beta).
\end{split}
\]
This completes the proof of \Cref{claim: existence of good solutions special}.
\end{proofof}
$\ $
\iffalse
We denote by $\hat{\phi_i}$ the resulting drawing, and it is clear that $\hat{\phi_i}$ is a drawing of the instance $(\hat G_i,\hat\Sigma_i)$, such that $\mathsf{cr}(\hat{\phi_i})\le \mathsf{cr}(C_i)+O(\operatorname{cost}({\mathcal{W}}_i)\cdot\operatorname{poly}\log n)$.
We first define an instance $(\hat G_i,\hat\Sigma_i)$ as follows. We start with the subgraph of $G$ induced by all edges of $E_G(C_i)$ and $E({\mathcal{W}}_i)$, and then create, for each edge $e$ in the subgraph, $\cong_{{\mathcal{W}}_i}(e)$ parallel copies of it. This finishes the description of graph $\hat G_i$.
For each vertex $v$ in $\hat G_i$, we define its rotation $\hat{\mathcal{O}}_v$ as follows. We start with the rotation ${\mathcal{O}}_v\in \Sigma$, and then replace, for each edge $e\in E({\mathcal{W}}_i)$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $\cong_{{\mathcal{W}}_i}(e)$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$, where the ordering among the copies is arbitrary.
On the one hand, in graph $\hat G_i$, we can make paths of ${\mathcal{W}}_i$ edge-disjoint by letting, for each edge $e\in E({\mathcal{W}}_i)$, each path of ${\mathcal{W}}_i$ that contains $e$ now take a distinct copy of $e$ in $\hat G_i$.
On the other hand, a drawing $\hat \phi_i$ of the instance $(\hat G_i,\hat\Sigma_i)$ can be easily computed from $\phi^*$, as follows.
We start with $\hat\phi'_i$, the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$. Then for each edge of ${\mathcal{W}}_i$, let $\gamma_e$ be the curve that represents the image of $e$, and we create $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting endpoints of $e$, that lies in an arbitrarily thin strip around $\gamma_e$. We denote by $\hat{\phi_i}$ the resulting drawing, and it is clear that $\hat{\phi_i}$ is a drawing of the instance $(\hat G_i,\hat\Sigma_i)$, such that $\mathsf{cr}(\hat{\phi_i})\le \mathsf{cr}(C_i)+O(\operatorname{cost}({\mathcal{W}}_i)\cdot\operatorname{poly}\log n)$.
For each edge $e\in E_G(C_i)$, we denote by $\gamma_e$ the curve in $\hat \phi_i$ that represents the image of $e$; and for each path $W\in {\mathcal{W}}_i$, we denote by $\gamma_W$ the curve in $\hat \phi_i$ that represents the image of $W$.
We define $\Gamma_0=\set{\gamma_e\mid e\in E_G(C_i)}$ and $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$. It is immediate to verify that the sets $\Gamma_0,\Gamma_1$ of curves satisfy the condition of \Cref{thm: type-2 uncrossing}. We then apply the algorithm in \Cref{thm: type-2 uncrossing} to $\Gamma_0,\Gamma_1$, and let $\Gamma_1'$ be the set of curves that we obtain.
From \Cref{thm: type-2 uncrossing}, the curves in $\Gamma_1'$ do not intersect internally between each other, and have the same sets of first endpoints and last endpoints.
We now show that we can obtain a drawing $\phi_i$ of the instance $(G_i,\Sigma_i)$ using the curves $\Gamma_0,\Gamma_1'$.
For each edge $e\in E_G(C_i)$, we still let the curve $\gamma_e\in \Gamma_0$ be the image of $e$.
For each edge path $W\in {\mathcal{W}}$, from \Cref{thm: type-2 uncrossing}, there is a curve $\gamma'\in \Gamma_1'$ connecting the endpoint of $W$ to we still let the curve $\gamma_e\in \Gamma_0$ be the image of $e$.
\znote{maybe we need another type of uncrossing here.}
\fi
\begin{proofof}{Claim~\ref{claim: stitching the drawings together}}
We first define a set $\set{(U_i,\Sigma'_i)\mid 1\le i\le k}$ of instances of \textnormal{\textsf{MCNwRS}}\xspace as follows.
We define $U_1=G$, and for each $2\le i\le k$, we define $U_i$ to be the graph obtained from $G$ by contracting the clusters $C_1,\ldots, C_{i-1}$ into a vertex $v^{\operatorname{left}}_i$.
Note that each edge in $U_i$ is also an edge in $G$, and we do not distinguish between them.
Note that $U_1=G$, we define the rotation system $\Sigma'_1$ on $U_1$ to be $\Sigma$.
For each $2\le i\le k$, we define the rotation system $\Sigma'_i$ on $U_i$ as follows.
For each vertex $v\in \bigcup_{i\le t\le k}V(C_t)$, note that its incident edges in $U_i$ are the edges of $\delta_G(v)$, and its rotation in $\Sigma'_i$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the input rotation system $\Sigma$.
For vertex $v^{\operatorname{left}}_i$, note that its incident edges are the edges of $\delta_{G_i}(v^{\operatorname{left}}_i)$, and its rotation in $\Sigma'_i$ is defined to be ${\mathcal{O}}^{\operatorname{left}}_i$, the rotation on $v^{\operatorname{left}}_i$ of instance $(G_i,\Sigma_i)$.
Note that $U_k=G_k$ and $\Sigma'_k=\Sigma_k$, so the drawing $\phi_k$ of the instance $(G_k,\Sigma_k)$ is also a drawing of the instance $(U_k,\Sigma'_k)$. For clarity, when we view this drawing as a solution to the instance $(U_k,\Sigma'_k)$, we rename it by $\psi_k$.
We will sequentially, for $i=k-1,\ldots,1$, compute a drawing of the instance $(U_i,\Sigma'_i)$ using the drawing $\psi_{i+1}$ of $(U_{i+1},\Sigma'_{i+1})$ and the drawing $\phi_i$ of $(G_{i},\Sigma_{i})$, and eventually, we return the drawing $\psi_1$ of $(U_{1},\Sigma'_{1})$ as the solution to the instance $(G,\Sigma)$.
We now fix an index $1\le i< k-1$ and construct the drawing $\psi_i$ to the instance $(U_{i},\Sigma'_{i})$, assuming that we have computed a drawing $\psi_{i+1}$ to the instance $(U_{i+1},\Sigma'_{i+1})$, as follows.
Recall that $V(G_i)=V(C_i)\cup\set{v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$ if $i\ge 2$ and $V(G_i)=V(C_i)\cup\set{v^{\operatorname{right}}_i}$ if $i=1$, and $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{U_{i+1}}(v^{\operatorname{left}}_{i+1})$. Moreover, from \Cref{obs: rotation for stitching} and the definition of instance $(U_{i+1},\Sigma'_{i+1})$, the rotation on $v^{\operatorname{right}}_i$ in $\Sigma_{i}$ is identical to the rotation on $v^{\operatorname{left}}_{i+1}$ in $\Sigma'_{i+1}$.
Denote $F=\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{U_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Let ${\mathcal{D}}$ be an arbitrarily small disc around the image of $v_{i+1}^{\operatorname{left}}$ in $\psi_{i+1}$. For each edge $e\in F$, we denote by $p_e$ the intersection between the image of $e$ with the boundary of ${\mathcal{D}}$.
Therefore, the order in which the points $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}$ is ${\mathcal{O}}^{\operatorname{left}}_{i+1}$.
We erase the drawing of $\psi_{i+1}$ inside the disc ${\mathcal{D}}$, and view the area inside the disc ${\mathcal{D}}$ as the outer face of the drawing.
Similarly, let ${\mathcal{D}}'$ be an arbitrarily small disc around the image of $v_{i}^{\operatorname{right}}$ in $\phi_{i}$. For each edge $e\in F$, we denote by $p'_e$ the intersection between the image of $e$ with the boundary of ${\mathcal{D}}'$.
Therefore, the order in which the points $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}'$ is ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We erase the drawing of $\phi_{i}$ inside the disc ${\mathcal{D}}'$, and let ${\mathcal{D}}''$ be another disc that is strictly contained in ${\mathcal{D}}'$. We now place the drawing of $\psi_{i+1}$ inside ${\mathcal{D}}'$ (after we erase part of it inside ${\mathcal{D}}$), so that the boundary of ${\mathcal{D}}$ in $\psi_{i+1}$ coincide with the boundary of ${\mathcal{D}}''$ in $\phi_{i}$, while the interior of ${\mathcal{D}}$ coincides with the exterior of ${\mathcal{D}}''$. We then compute a set $\set{\zeta_e\mid e\in F}$ of curves lying in ${\mathcal{D}}'\setminus {\mathcal{D}}''$, where for each $e\in F$, the curve $\zeta_e$ connects $p'_e$ to $p_e$, such that all curves of $\set{\zeta_e\mid e\in F}$ are mutually disjoint. Note that this can be done since the order in which $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}'$ is identical to $\set{p'_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}$.
We denote by $\psi_i$ the resulting drawing we obtained.
See Figure~\ref{fig: stitching} for an illustration.
Clearly, $\psi_i$ is a drawing of $U_i$ that respects $\Sigma'_i$ if we view, for each edge $e\in F$, the union of (i) the image of $e$ in $\phi_i$ outside the disc ${\mathcal{D}}'$; (ii) the curve $\zeta_e$; and (iii) the image of $e$ in $\psi_{i+1}$ inside the disc ${\mathcal{D}}''$, as the image of $e$.
\begin{figure}[h]
\centering
\subfigure[The drawing $\psi_{i+1}$, where the boundary of ${\mathcal{D}}$ is shown in dash black.]{\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_1.jpg}
}
\hspace{0.45cm}
\subfigure[The drawing $\psi_{i+1}$ after we erase its part inside ${\mathcal{D}}$ and view the interior of ${\mathcal{D}}$ as the outer face.]{
\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_2.jpg}}}
\hspace{0.45cm}
\subfigure[The drawing $\psi_i$, where the curves of $\set{\zeta_e\mid e\in F}$ are shown in dash line segments.]{
\scalebox{0.36}{\includegraphics[scale=1.0]{figs/stitching_3.jpg}}}
\caption{An illustration of an iteration in type-1 uncrossing.}\label{fig: stitching}
\end{figure}
Clearly, any crossing in the drawing $\psi_i$ is either a crossing of $\phi_i$ or a crossing of $\psi_{i+1}$, so $\mathsf{cr}(\psi_i)\le \mathsf{cr}(\psi_{i+1})+\mathsf{cr}(\phi_{i})$.
Therefore, if we rename the drawing $\psi_1$ of the instance $(U_1,\Sigma'_1)$ by $\phi$, then $\phi$ is a drawing of $G$ that respects $\Sigma$, and $\mathsf{cr}(\phi)\le \sum_{1\le i\le k}\mathsf{cr}(\phi_i)$.
\end{proofof}
\subsection{A Special Case}
In this section we provide the proof of \Cref{thm: disengagement - main} in a special case. Specifically, we will assume that the Gomory-Hu tree of the contracted graph $G_{\mid{\mathcal{C}}}$ is a path, and we are additionally given a set of paths called the \emph{inner paths} that will be defined later.
Let $H=G_{\mid{\mathcal{C}}}$ be the contracted graph.
We denote the clusters in ${\mathcal{C}}$ by $C_1,C_2,\ldots,C_r$. For convenience, for each $1\leq i\leq r$, we denote by $x_i$ the vertex of graph $H$ that represents the cluster $C_i$. Throughout this subsection, we assume that the Gomory-Hu tree $\tau$ of graph $H$ is a path, and we assume without loss of generality that the clusters are indexed according to their appearance on the path $\tau$. Note that each edge in $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ corresponds to an edge in $H$, and we do not distinguish between them.
For each $1\le i\le r$, we define edge sets
$\hat E_i=E(C_i,C_{i+1})$,
$E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$, $E_i^{\textsf{thr}}=\bigcup_{i'<i<j'}E(C_{i'},C_{j'})$, $E_i^{\operatorname{over}}=\bigcup_{i'<i,j'>i+1}E(C_{i'},C_{j'})$ and we define vertex sets $S_i=\set{x_1,\ldots,x_i}$ and $\overline{S}_i=\set{x_{i+1},\ldots,x_r}$.
We need the following observation.
\begin{observation}\label{obs: bad inded structure}
For all $1\leq i<r$, the following hold:
\begin{itemize}
\item $|\hat E_i|\geq |E_i^{\operatorname{over}}|$;
\item $|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_{i+1}^{\operatorname{left}}|-|E_i^{\operatorname{right}}|$; and
\item $|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$.
\end{itemize}
\end{observation}
\begin{proof}
Consider the cut $(\set{x_{i}},V(H)\setminus \set{x_i})$ in $H$. Its size is $|\hat E_i|+|\hat E_{i-1}|+ |E_i^{\operatorname{right}}|+|E_{i}^{\operatorname{left}}|$. Note that this cut separates $x_i$ from $x_{i-1}$. Since the minimum cut separating $x_i$ from $x_{i-1}$ in $H$ is $(S_{i-1},\overline{S}_{i-1})$, and $|E(S_{i-1},\overline{S}_{i-1})|=|\hat E_{i-1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|$, we get that $|\hat E_{i-1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|\hat E_{i-1}|+ |E_i^{\operatorname{right}}|+|E_{i}^{\operatorname{left}}|$,
and so
\begin{equation}
\label{eqn1}
|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+ |E_i^{\operatorname{right}}|.
\end{equation}
Consider the cut $(\set{x_{i+1}},V(H)\setminus \set{x_{i+1}})$ in $H$. Its size is $|\hat E_i|+|\hat E_{i+1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i+1}^{\operatorname{right}}|$. Note that this cut separates $x_{i+1}$ from $x_{i+2}$. Since the minimum cut separating $x_{i+1}$ from $x_{i+2}$ in $H$ is $(S_{i+1},\overline{S}_{i+1})$, and $|E(S_{i+1},\overline{S}_{i+1})|=|\hat E_{i+1}|+ |E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{right}}|+|E_i^{\operatorname{over}}|$, we get that $|\hat E_{i+1}|+ |E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{right}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|\hat E_{i+1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i+1}^{\operatorname{right}}|$,
and so
\begin{equation}
\label{eqn2}
|E_i^{\operatorname{right}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|E_{i+1}^{\operatorname{left}}|.
\end{equation}
Combining \Cref{eqn1,eqn2}, we get that $|\hat E_i|\geq |E_i^{\operatorname{over}}|$. By rearranging the sides of the two inequalities, we get that
$|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_{i+1}^{\operatorname{left}}|-|E_i^{\operatorname{right}}|$, and
$|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$.
\end{proof}
Define $E'=E(H)\setminus (\bigcup_{1\le i\le r-1}\hat E_i)$.
Equivalently, $E'$ is the subset of $E^{\mathsf{out}}({\mathcal{C}})$ that contains all edges connecting a pair $C_i,C_j$ of clusters in ${\mathcal{C}}$, with $j\ge i+2$, and when $E'$ is viewed as a set of edges in $H$, then $E'$ contains all edges connecting a pair $x_i,x_j$ of vertices in $H$ with $j\ge i+2$.
\paragraph{Inner Paths.} For an edge $e\in E'$ connecting $x_i$ to $x_j$ in $H$, we say that a path $P$ in $H$ is a \emph{left inner path} of $e$, iff the endpoints of $P$ are $x_i$ and $x_{i+1}$, and $P$ only contains vertices of $S_{i+1}$; we say that a path $P$ in $H$ is a \emph{right inner path} of $e$, iff the endpoints of $P$ are $x_{j-1}$ and $x_{j}$, and $P$ only contains vertices of $\overline{S}_{j-2}$; and we say that a path $P$ in $H$ is a \emph{middle inner path} of $e$, iff path $P$ sequentially visits vertices $x_{i+1}, x_{i+2},\dots,x_{j-1}$.
We say that a path $P$ is an \emph{inner path} of $e$, iff $P$ is the union of (i) a left inner path of $e$; (ii) a middle inner path of $e$; and (iii) a right inner path of $e$.
See \Cref{fig: LMR_inner} for an illustration.
\begin{figure}[h]
\centering
\includegraphics[scale=0.20]{figs/LMR_inner.jpg}
\caption{An illustration of left inner path (orange), middle inner path (purple) and right inner path (green) of edge $(x_i,x_j)\in E'$.}\label{fig: LMR_inner}
\end{figure}
In the remainder of this subsection, we will additionally assume that we are given, for each edge $e\in E'$, an inner path $P_e$ of $e$ in graph $H$, such that the set $\set{P_e}_{e\in E'}$ of paths causes congestion at most $\eta$, for some $\eta=2^{O((\log m)^{3/4}\log\log m)}$.
We now provide the proof of \Cref{thm: disengagement - main} with these additional assumptions. We will use the parameters $\eta=2^{O((\log m)^{3/4}\log\log m)}$, $\beta^*=2^{O(\sqrt{\log m}\cdot \log\log m)}$, and $\eta^*=2^{O((\log m)^{3/4}\log\log m)}$.
\subsubsection{Computing Guiding Paths and Auxiliary Cycles}
\label{sec: guiding and auxiliary paths}
We first apply the algorithm from \Cref{thm:algclassifycluster} to each cluster of ${\mathcal{C}}$.
In particular, for each $1\le i\le r$, we apply the \ensuremath{\mathsf{AlgClassifyCluster}}\xspace to the instance $(G,\Sigma)$ and the cluster $C_i$ (recall that cluster $C_i$ has the $\alpha_0$-bandwidth property, where $\alpha_0=1/\log^3m$) for $100\log n$ times. If any application of \ensuremath{\mathsf{AlgClassifyCluster}}\xspace returns a distribution ${\mathcal{D}}(C_i)$ over sets of guiding paths in $\Lambda(C_i)$, such that $C_i$ is $\beta^*$-light with respect to ${\mathcal{D}}(C_i)$, then we continue to sample a set of guiding paths from ${\mathcal{D}}(C_i)$. Let ${\mathcal{Q}}_i$ be the set we obtain, and let $u_i$ be the vertex of $C_i$ that serves as the common endpoint of all paths in ${\mathcal{Q}}_i$. If all applications of \ensuremath{\mathsf{AlgClassifyCluster}}\xspace return FAIL, then consider the graph $C_i^+$ (see \Cref{def: Graph C^+}). We apply the algorithm from \Cref{lem: simple guiding paths} to cluster $C^+_i$ and and set $T(C_i)$, and let $u_i$ be the vertex and ${\mathcal{Q}}_i$ be the set of paths that we obtain. Clearly, ${\mathcal{Q}}_i$ can be also viewed as a set of paths routing edges of $\delta_G(C_i)$ to $u_i$. From \Cref{lem: simple guiding paths}, for each edge $e\in E(C_i)$, $\expect{\cong({\mathcal{Q}}_i,e)}\leq O(\log^4m/\alpha_0)$. Finally, for each $1\le i\le r$ and for each $e\in \delta(C_i)$, we denote by $Q_i(e)$ the path of ${\mathcal{Q}}_i$ that routes $e$ to $u_i$.
We then apply the algorithm from \Cref{lem: non_interfering_paths} to each set ${\mathcal{Q}}_i$ of paths and the rotation system $\Sigma$, and rename the sets of paths we obtain as ${\mathcal{Q}}_1,\ldots,{\mathcal{Q}}_r$. Now we are guaranteed that, for each $1\le i\le r$, the set ${\mathcal{Q}}_i$ of paths is non-transversal with respect to $\Sigma$.
\paragraph{Bad Event $\xi_i$.} For each index $1\le i\le r$, we say that the event $\xi_i$ happens, iff $C_i$ is not an $\eta^*$-bad cluster, but all the $100\log n$ applications of \ensuremath{\mathsf{AlgClassifyCluster}}\xspace return FAIL.
From \Cref{thm:algclassifycluster}, $\Pr[\xi_i]\le (1/2)^{100\log n}=O(n^{-50})$. Then from the union bound over all $1\le i\le r$, $\Pr[\bigcup_{1\le i\le r}\xi_i]\le O(n^{-49})$.
We then compute a set ${\mathcal{R}}^*$ of cycles in $G$, using the set $\set{P_e}_{e\in E'}$ of inner paths and the path sets $\set{{\mathcal{Q}}_i}_{1\le i\le r}$, as follows.
Consider an edge $e\in E'$ and its inner path $P_e$ in $H$.
We denote $P_e=(e_1,\ldots,e_k)$, and for each $1\le j\le k$, we denote $e_j=(x_{t_{j-1}}, x_{t_{j}})$ (and so $e=(x_{t_0},x_{t_k})$ as an edge in $H$).
Recall that the set ${\mathcal{Q}}_i$ contains, for each edge $e\in\delta(C_i)$, a path $Q_i(e)$ routing edge $e$ to $u_i$.
We then define path $P^*_e$ as the sequential concatenation of paths $Q_{t_{0}}(e_1),Q_{t_{1}}(e_1), Q_{t_{1}}(e_2),Q_{t_{2}}(e_2),\ldots, Q_{t_{r-1}}(e_k),Q_{t_{r}}(e_k)$.
It is clear that path $P^*_e$ sequentially visits the vertices $u_{t_0},u_{t_1},\ldots,u_{t_k}$ in $G$.
Since edge $e$, as an edge of $G$, connects a vertex of $C_{t_0}$ to a vertex of $C_{t_k}$. We then define path $P^{**}_e$ to be the union of paths $Q_{t_{k}}(e)$ and $Q_{t_{0}}(e)$, so path $P^{**}_e$ connects $u_{t_k}$ to $u_{t_0}$.
Finally, we define cycle $R^*_e$ to be the union of paths $P^*_e$ and paths $P^{**}_e$, and we denote ${\mathcal{R}}^*=\set{R^*_e\mid e\in E'}$.
See \Cref{fig: LMR_auxi} for an illustration.
\begin{figure}[h]
\centering
\includegraphics[scale=0.24]{figs/auxiliary_path.jpg}
\caption{An illustration of the auxiliary cycle $R^*_e$ of an edge $e$ (solid red): left auxiliary path shown in orange, middle auxiliary path shown in purple and right auxiliary path shown in green.}\label{fig: LMR_auxi}
\end{figure}
Recall that the set $\set{P_e}_{e\in E'}$ of inner paths is guaranteed to cause congestion at most $\beta$ in graph $H$, then from the construction of cycles in ${\mathcal{R}}^*$, it is easy to see that, for each edge $e\in E'$, $\cong_{G}({\mathcal{R}}^*,e)\le O(\beta)$, and
for each edge $e\in \bigcup_{1\le i\le r}E(C_i)$, $\expect[]{\cong_{G}({\mathcal{R}}^*,e)^2}\le O(\beta^2\cdot\beta^*)$.
\znote{to modify from here, define left/mid/right auxiliary path}
We further modify the cycles in ${\mathcal{R}}^*$ to obtain a new set ${\mathcal{R}}$ of cycles, such that the intersection of every pair of cycles in ${\mathcal{R}}$ is non-transversal with respect to $\Sigma$ at at most one of their shared vertices.
Let $R,R'$ be a pair of cycles in ${\mathcal{R}}^*$ and let $v$ be a shared vertex of $R$ and $R'$. We say that the tuple $(R,R',v)$ is \emph{bad}, iff the intersection of cycles $R,R'$ at $v$ is transversal with respect to $\Sigma$. We iteratively process the set ${\mathcal{R}}^*$ of cycles as follows. While there exist a pair $R,R'$ of cycles in ${\mathcal{R}}^*$ and two vertices $v_1,v_2\in V(R)\cap V(R')$, such that the tuples $(R,R',v)$ and $(R,R',v)$ are bad, we process cycles $R,R'$ as follows.
Eventually, for each $e\in E'$, we obtain a cycle $R_e$ in $G$ that is called the \emph{auxiliary cycle} of edge $e$.
\subsubsection{Constructing Sub-Instances}
\label{sec: compute advanced disengagement}
In this step we will construct, for each cluster $C_i\in {\mathcal{C}}$, an sub-instance $I_i=(G_i,\Sigma_i)$ of $(G,\Sigma)$, such that the instances ${\mathcal{I}}=\set{I_1,\ldots,I_r}$ satisfy the properties in \Cref{thm: disengagement - main}.
Recall that, for each $1\le i\le r$, $E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$, and $E^{\textsf{thr}}_i=\bigcup_{i'\le i-1,j'\ge i+1}E(C_{i'},C_{j'})$.
\paragraph{Instances $(G_2,\Sigma_2),\ldots,(G_{r-1},\Sigma_{r-1})$.}
We first fix some index $2\le i\le r-1$ and define the instance $(G_i,\Sigma_i)$ as follows.
The graph $G_i$ is obtained from $G$ by first contracting clusters $C_1,\ldots,C_{i-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_i$, and then contracting clusters $C_{i+1},\ldots,C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_i$, and finally deleting self-loops on the super-nodes $v^{\operatorname{left}}_i$ and $v^{\operatorname{right}}_i$. So $V(G_i)=V(C_i)\cup \set{ v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$. See \Cref{fig: disengaged instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edge sets in $G$. Edges of $E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}$ are shown in green, and edges of $E^{\textsf{thr}}_{i}$ are shown in red. ]{\scalebox{0.32}{\includegraphics{figs/disengaged_instance_1.jpg}
}
\hspace{5pt}
\subfigure[Graph $G_i$. Note that $\delta_{G_i}(v_i^{\operatorname{right}})=\hat E_i \cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, and $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1} \cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_{i}$.]{
\scalebox{0.32}{\includegraphics{figs/disengaged_instance_2.jpg}}}
\caption{An illustration of the construction of sub-instance $(G_i,\Sigma_i)$.}\label{fig: disengaged instance}
\end{figure}
We now define the orderings in $\Sigma_i$. First, for each vertex $v\in V(C_i)$, the ordering on its incident edges is defined to be ${\mathcal{O}}_v$, the rotation on vertex $v$ in the given rotation system $\Sigma$.
It remains to define the rotations of super-nodes $v^{\operatorname{left}}_i,v^{\operatorname{right}}_i$.
We first consider $v^{\operatorname{left}}_i$.
Note that $\delta_{G_i}(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$.
For each edge $\hat e\in \hat E_{i-1}$, recall that $Q_{i-1}(\hat e)$ is the path in $C_{i-1}$ routing edge $\hat e$ to $u_{i-1}$.
For each edge $e\in E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i$, we denote by $W^{\operatorname{left}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i-1}$ and contains its entire left auxiliary path.
We then denote
$${\mathcal{W}}^{\operatorname{left}}_i=\set{W^{\operatorname{left}}_i(e)\mid e\in E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i}\cup \set{Q_{i-1}(\hat e)\mid \hat e\in \hat E_{i-1}}.$$
\znote{need to treat the red edges that enter $C_{i-1}$ specially}
Intuitively, the rotation on vertex $v^{\operatorname{left}}_i$ is defined to be the ordering in which the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ enter $u_{i-1}$.
Formally, for every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$, let $e^*_W$ be the unique edge of path $W$ that is incident to $u_{i-1}$. We first define a circular ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$, as follows: the paths are ordered according to the circular ordering of their corresponding edges $e^*_W$ in ${\mathcal{O}}_{u_{i-1}}\in \Sigma$, breaking ties arbitrarily. Since every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$ is associated with a unique edge in $\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$, this ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ immediately defines a circular ordering of the edges of $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$, that we denote by ${\mathcal{O}}^{\operatorname{left}}_i$. See Figure~\ref{fig: v_left rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i-1}=\set{\hat e_1,\ldots,\hat e_4}$, $E^{\operatorname{left}}_{i}=\set{e^g_1}$ and $E^{\textsf{thr}}_{i}=\set{e^r_1}$.
Paths of ${\mathcal{W}}^{\operatorname{left}}_i$ excluding their first edges are shown in dash lines.]{\scalebox{0.13}{\includegraphics{figs/rotation_left_1.jpg}}}
\hspace{3pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{left}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. Set $\delta_{G_i}(v^{\operatorname{left}}_i)=\set{\hat e_1,\hat e_2,\hat e_3,\hat e_4,e^g_1, e^r_1}$. The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on them around $v^{\operatorname{left}}_i$ is shown above.]{
\scalebox{0.16}{\includegraphics{figs/rotation_left_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_left rotation}
\end{figure}
The rotation ${\mathcal{O}}^{\operatorname{right}}_{i}$ on vertex $v^{\operatorname{right}}_i$ is defined similarly.
Note that $\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$.
For each edge $\hat e'\in \hat E_{i}\cup E_i^{\operatorname{right}}$, recall that $Q_{i}(\hat e')$ is the path in $C_{i}$ routing edge $\hat e'$ to vertex $u_{i}$.
For each edge $e\in E^{\textsf{thr}}_i$, we denote by $W^{\operatorname{right}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i}$ and contains its entire left auxiliary path.
We then denote
$${\mathcal{W}}^{\operatorname{right}}_i=\set{W^{\operatorname{right}}_i(e)\text{ }\big|\text{ } e\in E^{\textsf{thr}}_i}\cup \set{Q_{i}(\hat e')\text{ }\big|\text{ } \hat e'\in \hat E_{i}\cup E^{\operatorname{right}}_i}.$$
The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ is then defined in a similar way as the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$, according to the paths of ${\mathcal{W}}^{\operatorname{right}}_i$ and the rotation ${\mathcal{O}}_{u_{i}}\in \Sigma$.
See Figure~\ref{fig: v_right rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i}=\set{\hat e_1',\ldots,\hat e_4'}$, $E^{\operatorname{right}}_{i}=\set{\tilde e^g_1}$ and $E^{\textsf{thr}}_{i}=\set{\tilde e^r_1}$.
Paths of ${\mathcal{W}}^{\operatorname{right}}_i$ excluding their first edges are shown in dash lines. ]{\scalebox{0.13}{\includegraphics{figs/rotation_right_1.jpg}
}
\hspace{3pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. Set $\delta_{G_i}(v^{\operatorname{right}}_i)=\set{\hat e_1',\hat e_2',\hat e_3',\hat e_4',\tilde e^g_1,\tilde e^r_1}$. The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on them around $v^{\operatorname{right}}_i$ is shown above.]{
\scalebox{0.16}{\includegraphics{figs/rotation_right_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_right rotation}
\end{figure}
\paragraph{Instances $(G_1,\Sigma_1)$ and $(G_r,\Sigma_r)$.}
The instances $(G_1,\Sigma_1)$ and $(G_r,\Sigma_r)$ are defined similarly, but instead of two super-nodes, the graphs $G_1$ and $G_r$ contain one super-node each.
In particular, graph $G_1$ is obtained from $G$ by contracting clusters $C_2,\ldots, C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_1$, and then deleting self-loops on it.
So $V(G_1)=V(C_1)\cup \set{v^{\operatorname{right}}_{1}}$ and $\delta_{G_1}(v^{\operatorname{right}}_{1})=\hat E_1\cup E^{\operatorname{right}}_1$.
The rotation of a vertex $v\in V(C_1)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{right}}_1$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{right}}_i$ for any index $2\le i\le r-1$.
Graph $G_r$ is obtained from $G$ by contracting clusters $C_1,\ldots, C_{r-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_r$, and then deleting self-loops on it.
So $V(G_r)=V(C_r)\cup \set{v^{\operatorname{left}}_{r}}$ and $\delta_{G_r}(v^{\operatorname{left}}_{r})=\hat E_{r-1}\cup E^{\operatorname{left}}_r$.
The rotation of a vertex $v\in V(C_r)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{left}}_r$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{left}}_i$ for any index $2\le i\le r-1$.
We will use the following claims later for completing the proof of \Cref{thm: disengagement - main} in the special case.
\begin{observation}
\label{obs: disengaged instance size}
$\sum_{1\le i\le r}|E(G_i)|\le O(|E(G)|)$, and for each $1\le i\le r$, $|E(G_i)|\le m/\mu$. \znote{parameter?}
\end{observation}
\begin{proof}
Note that, in the sub-instances $\set{(G_i,\Sigma_i)}_{1\le i\le r}$, each graph of $\set{G_i}_{1\le i\le r}$ is obtained from $G$ by contracting some sets of clusters of ${\mathcal{C}}$ into a single super-node, so each edge of $G_i$ corresponds to an edge in $E(G)$.
Therefore, for each $1\le i\le r$,
$$|E(G_i)|=|E_G(C_i)|+|\delta_G(C_i)|\le |E_G(C_i)|+|E^{\textsf{out}}({\mathcal{C}})|\le m/(100\mu)+m/(100\mu)\le m/\mu.$$
Note that $E(G_i)=E(C_i)\cup \hat E_i\cup \hat E_{i-1}\cup E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$.
First, each edge of $\bigcup_{1\le i\le r}E(C_i)$ appears in exactly one graphs of $\set{G_i}_{1\le i\le r}$.
Second, each edge of $\bigcup_{1\le i\le r}\hat E_i$ appears in exactly two graphs of $\set{G_i}_{1\le i\le r}$.
Consider now an edge $e\in E'$. If $e$ connects a vertex of $C_i$ to a vertex of $C_j$ for some $j\ge i+2$, then $e$ will appear as an edge in $E_i^{\operatorname{right}}\subseteq E(G_i)$ and an edge in $E_j^{\operatorname{left}}\subseteq E(G_j)$, and it will appear as an edge in $E_k^{\textsf{thr}}\subseteq E(G_k)$ for all $i<k<j$.
On one hand, we have $\sum_{1\le i\le r}|E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}|\le 2|E(G)|$.
On the other hand, note that $E_k^{\textsf{thr}}=E(C_{k-1},C_{k+1})\cup E^{\operatorname{over}}_{k-1}\cup E^{\operatorname{over}}_{k}$, and each edge of $e$ appears in at most two graphs of $\set{G_i}_{1\le i\le r}$ as an edge of $E(C_{k-1},C_{k+1})$.
Moreover, from \Cref{obs: bad inded structure}, $|E^{\operatorname{over}}_{k-1}|\le |\hat E_{k-1}|$ and $|E^{\operatorname{over}}_{k}|\le |\hat E_{k}|$.
Altogether, we have
\begin{equation}
\begin{split}
\sum_{1\le i\le r}|E(G_i)| & =
\sum_{1\le i\le r}\textsf{left}(
|E(C_i)|+ |\hat E_i|+ |\hat E_{i-1}|+ |E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|+|E_i^{\textsf{thr}}|
\textsf{right})\\
& = \sum_{1\le i\le r}
|E(C_i)|+ \sum_{1\le i\le r} \textsf{left}(|E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|\textsf{right})+ \sum_{1\le i\le r}\textsf{left}(|E_i^{\textsf{thr}}|+ |\hat E_i|+ |\hat E_{i-1}|\textsf{right})\\
& \le |E(G)|+ 2\cdot |E(G)| + \sum_{1\le i\le r}\textsf{left}(|E(C_{i-1},C_{i+1})|+ 2|\hat E_i|+ 2|\hat E_{i-1}|\textsf{right})\\
& \le 8\cdot |E(G)|.
\end{split}
\end{equation}
This completes the proof of \Cref{obs: disengaged instance size}.
\end{proof}
\begin{observation}
\label{obs: rotation for stitching}
For each $1\le i\le r-1$, if we view the edge in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$ as edges of $E(G)$, then $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$, and moreover, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{observation}
\begin{proof}
Recall that for each $1\le i\le r-1$,
$\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})=\hat E_{i}\cup E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}$. From the definition of sets $E_i^{\textsf{thr}},E_{i+1}^{\textsf{thr}}, E^{\operatorname{right}}_i, E^{\operatorname{left}}_{i+1}$,
\[
\begin{split}
E_i^{\textsf{thr}}\cup E^{\operatorname{right}}_i
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i<j'\text{ or }i'=i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'\le i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'< i+1\le j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i+1<j'\text{ or }i'<i+1=j'}=E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}.
\end{split}
\]
Therefore, $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Moreover, from the definition of path sets ${\mathcal{W}}^{\operatorname{right}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_{i+1}$, it is not hard to verify that, for every edge $e\in \delta_{G_i}(v^{\operatorname{right}}_i)$, the path in ${\mathcal{W}}^{\operatorname{right}}_i$ that contains $e$ as its first edge is identical to the path in ${\mathcal{W}}^{\operatorname{left}}_{i+1}$ that contains $e$ as its first edge. According to the way that rotations ${\mathcal{O}}^{\operatorname{right}}_i,{\mathcal{O}}^{\operatorname{left}}_{i+1}$ are defined, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{proof}
\subsubsection{Completing the Proof of \Cref{thm: disengagement - main} in the Special Case}
\label{sec: path case with no problematic index}
In this section we complete the proof of \Cref{thm: disengagement - main} in the special case where the Gomory-Hu tree of $H=G_{\mid {\mathcal{C}}}$ is a path and we are additionally given, for each edge $e\in E'$, an inner path $P_e$ in graph $H$, such that set $\set{P_e}_{e\in E'}$ of paths causes congestion at most $\eta$ in $H$, for some $\eta=2^{O((\log m)^{3/4}\log\log m)}$.
Specifically, we use the following two claims, whose proofs will be provided later.
\begin{claim}
\label{claim: existence of good solutions special}
$\expect{\sum_{1\le i\le r}\mathsf{OPT}_{\mathsf{cnwrs}}(G_i,\Sigma_i)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$.
\end{claim}
\begin{claim}
\label{claim: stitching the drawings together}
There is an efficient algorithm, that given, for each $1\le i\le r$, a feasible solution $\phi_i$ to the instance $(G_i,\Sigma_i)$, computes a solution to the instance $(G,\Sigma)$, such that $\mathsf{cr}(\phi)\le \sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
\end{claim}
We use the algorithm described in~\Cref{sec: guiding and auxiliary paths} and~\Cref{sec: compute advanced disengagement}, and return the disengaged instances $(G_1,\Sigma_1),\ldots,(G_r,\Sigma_r)$ as the collection of sub-instances of $(G,\Sigma)$.
From the previous subsections, the algorithm for producing the sub-instances is efficient.
On the other hand, it follows immediately from \Cref{obs: disengaged instance size}, \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together} that the sub-instances $(G_1,\Sigma_1),\ldots,(G_r,\Sigma_r)$ satisfy the properties in \Cref{thm: disengagement - main}.
This completes the proof of \Cref{thm: disengagement - main}.
We now provide the proofs of \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together}.
$\ $
\begin{proofof}{Claim~\ref{claim: existence of good solutions special}}
Let $\phi^*$ be an optimal drawing of the instance $(G,\Sigma)$.
We will construct, for each $1\le i\le r$, a drawing $\phi_i$ of $G_i$ that respects the rotation system $\Sigma_i$, based on the drawing $\phi^*$, such that $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)\le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot 2^{O((\log m)^{3/4}\log\log m)})$, and \Cref{claim: existence of good solutions special} then follows.
\paragraph{Drawings $\phi_2,\ldots,\phi_{r-1}$.}
First we fix some index $2\le i\le r-1$, and describe the construction of the drawing $\phi_i$. We start with some definitions.
Recall that $E(G_i)=E_G(C_i)\cup (\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$.
We define the auxiliary path set ${\mathcal{W}}_i={\mathcal{W}}^{\operatorname{left}}_i\cup {\mathcal{W}}^{\operatorname{right}}_i$, so
$${\mathcal{W}}_i=\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}\cup \set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}},$$
where for each $e\in E^{\operatorname{left}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$; for each $e\in E^{\operatorname{right}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between $u_{i+1}$ and its last endpoint; and for each $e\in E^{\textsf{thr}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$, the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$, and the subpath of $P_e$ between $u_{i+1}$ and its last endpoint.
\iffalse
We use the following observation.
\begin{observation}
\label{obs: wset_i_non_interfering}
The set ${\mathcal{W}}_i$ of paths are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
Recall that the paths in ${\mathcal{Q}}_{i-1}$ only uses edges of $E(C_{i-1})\cup \delta(C_{i-1})$, and they are non-transversal.
And similarly, the paths in ${\mathcal{Q}}_{i+1}$ only uses edges of $E(C_{i+1})\cup \delta(C_{i+1})$, and they are non-transversal.
Therefore, the paths in $\set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}}$ are non-transversal.
From \Cref{obs: non_transversal_1} and \Cref{obs: non_transversal_2}, the paths in $\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}$ are non-transversal. Therefore, it suffices to show that, the set ${\mathcal{W}}_i$ of paths are non-transversal at all vertices of $C_{i-1}$ and all vertices of $C_{i+1}$. Note that, for each edge $e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i})$, $W_e\cap (C_{i-1}\cup \delta(C_{i-1}))$ is indeed a path of ${\mathcal{Q}}_{i-1}$. Therefore, the paths in ${\mathcal{W}}_i$ are non-transversal at all vertices of $C_{i-1}$. Similarly, they are also non-transversal at all vertices of $C_{i+1}$. Altogether, the paths of ${\mathcal{W}}_i$ are non-transversal with respect to $\Sigma$.
\end{proof}
\fi
For uniformity of notations, for each edge $\hat e\in \hat E_i$, we rename the path $Q_{i+1}(\hat e)$ by $W_{\hat e}$, and similarly for each edge $\hat e\in \hat E_{i-1}$, we rename the path $Q_{i-1}(\hat e)$ by $W_{\hat e}$. Therefore, ${\mathcal{W}}_i=\set{W_e\mid e\in E(G_i)\setminus E(C_i)}$.
Put in other words, the set ${\mathcal{W}}_i$ contains, for each edge $e$ in $G_i$ that is incident to $v^{\operatorname{left}}_i$ or $v^{\operatorname{right}}_i$, a path named $W_e$.
It is easy to see that all paths in ${\mathcal{W}}_i$ are internally disjoint from $C_i$.
We further partition the set ${\mathcal{W}}_i$ into two sets: ${\mathcal{W}}_i^{\textsf{thr}}=\set{W_e\mid e\in E^{\textsf{thr}}_i}$ and $\tilde {\mathcal{W}}_i={\mathcal{W}}_i\setminus {\mathcal{W}}_i^{\textsf{thr}}$.
We are now ready to construct the drawing $\phi_i$ for the instance $(G_i,\Sigma_i)$.
Recall that $\phi^*$ is an optimal drawing of the input instance $(G,\Sigma)$.
We start with the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$, that we denote by $\phi^*_i$. We will not modify the image of $C_i$ in $\phi^*_i$, but will focus on constructing the image of edges in $E(G_i)\setminus E(C_i)$, based on the image of edges in $E({\mathcal{W}}_i)$ in $\phi^*_i$.
Specifically, we proceed in the following four steps.
\paragraph{Step 1.}
For each edge $e\in E({\mathcal{W}}_i)$, we denote by $\pi_e$ the curve that represents the image of $e$ in $\phi^*_i$. We create a set of $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting the endpoints of $e$ in $\phi^*_i$, that lies in an arbitrarily thin strip around $\pi_e$.
We denote by $\Pi_e$ the set of these curves.
We then assign, for each edge $e\in E({\mathcal{W}}_i)$ and for each path in ${\mathcal{W}}_i$ that contains the edge $e$, a distinct curve in $\Pi_e$ to this path.
Therefore, each curve in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ is assigned to exactly one path of ${\mathcal{W}}_i$,
and each path $W\in {\mathcal{W}}_i$ is assigned with, for each edge $e\in E(W)$, a curve in $\Pi_e$. Let $\gamma_W$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ that are assigned to path $W$, so $\gamma_W$ connects the endpoints of path $W$ in $\phi^*_i$.
In fact, when we assign curves in $\bigcup_{e\in \delta(u_{i-1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{left}}_i$ (recall that $\delta(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\operatorname{left}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{left}}_i)}$), we additionally ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{left}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{left}}_i)$ enter $u_{i-1}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
And similarly, when we assign curves in $\bigcup_{e\in \delta(u_{i+1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{right}}_i$ (recall that $\delta(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\operatorname{right}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{right}}_i)}$), we ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{right}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{right}}_i)$ enter $u_{i+1}$ in the same order as ${\mathcal{O}}^{\operatorname{right}}_i$.
Note that this can be easily achieved according to the definition of ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$.
We denote $\Gamma_i=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$, and we further partition the set $\Gamma_i$ into two sets: $\Gamma_i^{\textsf{thr}}=\set{\gamma_{W}\mid W\in {\mathcal{W}}^{\textsf{thr}}_i}$ and $\tilde \Gamma_i=\Gamma_i\setminus \Gamma_i^{\textsf{thr}}$.
We denote by $\hat \phi_i$ the drawing obtained by taking the union of the image of $C_i$ in $\phi^*_i$ and all curves in $\Gamma_i$.
For every path $P$ in $G_i$, we denote by $\hat{\chi}_i(P)$ the number of crossings that involves the ``image of $P$'' in $\hat \phi_i$, which is defined as the union of, for each edge $e\in E(\tilde{\mathcal{W}}_i)$, an arbitrary curve in $\Pi_e$.
Clearly, for each edge $e\in E({\mathcal{W}}_i)$, all curves in $\Pi_e$ are crossed by other curves of $(\Gamma_i\setminus \Pi_e)\cup \phi^*_i(C_i)$ same number of times.
Therefore, $\hat{\chi}_i(P)$ is well-defined.
For a set ${\mathcal{P}}$ of paths in $G_i$, we define $\hat{\chi}_i({\mathcal{P}})=\sum_{P\in {\mathcal{P}}}\hat{\chi}_i(P)$.
\iffalse
We use the following observation.
\znote{maybe remove this observation?}
\begin{observation}
\label{obs: curves_crossings}
The number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$,
and the number of intersections between a curve in $\tilde\Gamma_i$ and the image of edges of $C_i$ in $\phi^*_i$, are both $O(\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W))$.
\end{observation}
\begin{proof}
We first show that the number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Note that, from the construction of curves in $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i}$, every crossing between a pair $W,W'$ of curves in $\tilde\Gamma_i$ must be the intersection between a curve in $\Pi_e$ for some $e\in E(W)$ and a curve in $\Pi_{e'}$ for some $e'\in E(W')$, such that the image $\pi_e$ for $e$ and the image $\pi_{e'}$ for $e'$ intersect in $\phi^*$.
Therefore, for each pair $W,W'$ of paths in $\tilde{\mathcal{W}}_i$, the number of points that belong to only curves $\gamma_W$ and $\gamma_{W'}$ is at most the number of crossings between the image of $W$ and the image of $W'$ in $\phi^*$.
It follows that the number of points that belong to exactly two curves of $\tilde\Gamma_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Altogether, the number of intersections between curves in $\tilde\Gamma_i$ is at most $|V(\tilde {\mathcal{W}}_i)|+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
We now show that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ that are not vertex-image is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Let $W$ be a path of $\tilde {\mathcal{W}}_i$ and consider the curve $\gamma_W$. Note that $\gamma_W$ is the union of, for each edge $e\in E(W)$, a curve that lies in an arbitrarily thin strip around $\pi_e$.
Therefore, the number of crossings between $\gamma_W$ and the image of $C_i$ in $\phi^*_i$ is identical to the number of crossings the image of path $W$ and the image of $C_i$ in $\phi^*_i$, which is at most $\hat\mathsf{cr}(W)$.
It follows that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
\end{proof}
\fi
\paragraph{Step 2.}
For each vertex $v\in V({\mathcal{W}}_i)$, we denote by $x_v$ the point that represents the image of $v$ in $\phi^*_i$, and we let $X$ contains all points of $\set{x_v\mid v\in V({\mathcal{W}}_i)}$ that are intersections between curves in $\Gamma_i$.
We now manipulate the curves in $\set{\gamma_W\mid W\in {\mathcal{W}}_i}$ at points of $X$, by processing points of $X$ one-by-one, as follows.
Consider a point $x_v$ that is an intersection between curves in $\Gamma_i$, where $v\in V({\mathcal{W}}_i)$, and let $D_v$ be an arbitrarily small disc around $x_v$.
We denote by ${\mathcal{W}}_i(v)$ the set of paths in ${\mathcal{W}}_i$ that contains $v$, and further partition it into two sets: ${\mathcal{W}}^{\textsf{thr}}_i(v)={\mathcal{W}}_i(v)\cap {\mathcal{W}}^{\textsf{thr}}_i$ and $\tilde{\mathcal{W}}_i(v)={\mathcal{W}}_i(v)\cap \tilde{\mathcal{W}}_i$. We apply the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $\set{\gamma_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ within disc $D_v$. Let $\set{\gamma'_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ be the set of new curves that we obtain. From \Cref{obs: curve_manipulation},
(i) for each path $W\in \tilde{\mathcal{W}}_i(v)$, the curve $\gamma'_W$ does not contain $x_v$, and is identical to the curve $\gamma_W$ outside the disc $D_v$;
(ii) the segments of curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc $D_v$ are in general position; and
(iii) the number of icrossings between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside $D_v$ is bounded by $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\iffalse{just for backup}
\begin{proof}
Denote $d=\deg_G(v)$ and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
For each path $W\in \tilde{\mathcal{W}}_i(v)$, we denote by $p^{-}_W$ and $p^{+}_W$ the intersections between the curve $\gamma_W$ and the boundary of ${\mathcal{D}}_v$.
We now compute, for each $W\in W\in \tilde{\mathcal{W}}_i(v)$, a curve $\zeta_W$ in ${\mathcal{D}}_v$ connecting $p^{-}_W$ to $p^{+}_W$, such that (i) the curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ are in general position; and (ii) for each pair $W,W'$ of paths, the curves $\zeta_W$ and $\zeta_{W'}$ intersects iff the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{-}_{W'},p^{+}_{W},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_{W'},p^{-}_{W},p^{+}_{W'})$.
It is clear that this can be achieved by first setting, for each $W$, the curve $\zeta_W$ to be the line segment connecting $p^{-}_W$ to $p^{+}_W$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
We now define, for each $W$, the curve $\gamma'_W$ to be the union of the part of $\gamma_W$ outside ${\mathcal{D}}_v$ and the curve $\zeta_W$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, the first and the second condition of \Cref{obs: curve_manipulation} are satisfied.
It remains to estimate the number of intersections between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$, which equals the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$.
Since the paths in $\tilde{\mathcal{W}}_i(v)$ are non-transversal with respect to $\Sigma$ (from \Cref{obs: wset_i_non_interfering}), from the construction of curves $\set{\gamma_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$, if a pair $W,W'$ of paths in $\tilde {\mathcal{W}}_i(v)$ do not share edges of $\delta(v)$, then the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_W,p^{-}_{W'},p^{+}_{W'})$, and therefore the curves $\zeta_{W}$ and $\zeta_{W'}$ will not intersect in ${\mathcal{D}}_v$.
Therefore, only the curves $\zeta_W$ and $\zeta_{W'}$ intersect iff $W$ and $W'$ share an edge of $\delta(v)$.
Since every such pair of curves intersects at most once, the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$ is at most $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are shown in black, and curves of $\tilde{\mathcal{W}}_i(v)$ are shown in blue, red, orange and green. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are not modified, while curves of $\tilde{\mathcal{W}}_i(v)$ are re-routed via dash lines within disc ${\mathcal{D}}_v$.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the step of processing $x_v$.}\label{fig: curve_con}
\end{figure}
\fi
We then replace the curves of $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ in $\Gamma_i$ by the curves of $\set{\gamma'_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
This completes the description of the iteration of processing point the point $x_v\in X$. Let $\Gamma'_i$ be the set of curves that we obtain after processing all points in $X$ in this way. Note that we have never modified the curves of $\Gamma^{\textsf{thr}}_i$, so $\Gamma^{\textsf{thr}}_i\subseteq\Gamma'_i$, and we denote $\tilde\Gamma'_i=\Gamma'_i\setminus \Gamma^{\textsf{thr}}_i$.
We use the following observation.
\begin{observation}
\label{obs: general_position}
Curves in $\tilde\Gamma'_i$ are in general position, and if a point $p$ lies on more than two curves of $\Gamma'_i$, then either $p$ is an endpoint of all curves containing it, or all curves containing $p$ belong to $\Gamma^{\textsf{thr}}_i$.
\end{observation}
\begin{proof}
From the construction of curves in $\Gamma_i$, any point that belong to at least three curves of $\Gamma_i$ must be the image of some vertex in $\phi^*$. From~\Cref{obs: curve_manipulation}, curves in $\tilde\Gamma'_i$ are in general position; curves in $\tilde\Gamma'_i$ do not contain any vertex-image in $\phi^*$ except for their endpoints; and they do not contain any intersection of a pair of paths in $\Gamma_i^{\textsf{thr}}$. \Cref{obs: general_position} now follows.
\end{proof}
\paragraph{Step 3.}
So far we have obtained a set $\Gamma'_i$ of curves that are further partitioned into two sets $\Gamma'_i=\Gamma^{\textsf{thr}}_i\cup \tilde\Gamma'_i$,
where set $\tilde\Gamma'_i$ contains, for each path $W\in \tilde {\mathcal{W}}_i$, a curve $\gamma'_W$ connecting the endpoints of $W$, and the curves in $\tilde\Gamma'_i$ are in general position;
and set $\Gamma^{\textsf{thr}}_i$ contains, for each path $W\in {\mathcal{W}}^{\textsf{thr}}_i$, a curve $\gamma_W$ connecting the endpoints of $W$.
Recall that all paths in ${\mathcal{W}}^{\textsf{thr}}_i$ connects $u_{i-1}$ to $u_{i+1}$.
Let $z_{\operatorname{left}}$ be the point that represents the image of $u_{i-1}$ in $\phi_i^*$ and let $z_{\operatorname{right}}$ be the point that represents the image of $u_{i+1}$ in $\phi_i^*$.
Then, all curves in $\Gamma^{\textsf{thr}}_i$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
We view $z_{\operatorname{left}}$ as the first endpoint of curves in $\Gamma^{\textsf{thr}}_i$ and view $z_{\operatorname{right}}$ as their last endpoint.
We then apply the algorithm in \Cref{thm: type-2 uncrossing}, where we let $\Gamma=\Gamma^{\textsf{thr}}_i$ and let $\Gamma_0$ be the set of all other curves in the drawing $\phi^*_i$. Let $\Gamma^{\textsf{thr}'}_i$ be the set of curves we obtain. We then designate, for each edge $e\in E^{\textsf{thr}}_i$, a curve in $\Gamma^{\textsf{thr}'}_i$ as $\gamma'_{W_e}$, such that the curves of $\set{\gamma'_{W_e}\mid e\in \hat E_{i-1}\cup E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i}$ enters $z_{\operatorname{left}}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
Recall that ${\mathcal{W}}_i=\set{W_e\mid e\in (E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\cup E_i^{\operatorname{right}}\cup \hat E_{i-1}\cup \hat E_i)}$, and, for each edge $e\in E_i^{\operatorname{left}}\cup \hat E_{i-1}$, the curve $\gamma'_{W_e}$ connects its endpoint in $C_i$ to $z_{\operatorname{left}}$; for each edge $e\in E_i^{\operatorname{right}}\cup \hat E_{i}$, the curve $\gamma'_{W_e}$ connects the endpoint of $e$ to $z_{\operatorname{right}}$; and for each edge $e\in E_i^{\textsf{thr}}$, the curve $\gamma'_{W_e}$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
Therefore, if we view $z_{\operatorname{left}}$ as the image of $v^{\operatorname{left}}_i$, view $z_{\operatorname{right}}$ as the image of $v^{\operatorname{right}}_i$, and for each edge $e\in E(G_i)\setminus E(C_i)$, view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi'_i$.
It is clear from the construction of curves in $\set{\gamma'_{W_e}\mid e\in E(G_i)\setminus E(C_i)}$ that this drawing respects all rotations in $\Sigma_i$ on vertices of $V(C_i)$ and vertex $v^{\operatorname{left}}_i$.
However, the drawing $\phi'_i$ may not respect the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$. We further modify the drawing $\phi'_i$ at $z_{\operatorname{right}}$ in the last step.
\paragraph{Step 4.}
Let ${\mathcal{D}}$ be an arbitrarily small disc around $z_{\operatorname{right}}$ in the drawing $\phi'_i$, and let ${\mathcal{D}}'$ be another small disc around $z_{\operatorname{right}}$ that is strictly contained in ${\mathcal{D}}$.
We first erase the drawing of $\phi'_i$ inside the disc ${\mathcal{D}}$, and for each edge $e\in \delta(v^{\operatorname{right}}_i)$, we denote by $p_{e}$ the intersection between the curve representing the image of $e$ in $\phi'_i$ and the boundary of ${\mathcal{D}}$.
We then place, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, a point $p'_e$ on the boundary of ${\mathcal{D}}'$, such that the order in which the points in $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ appearing on the boundary of ${\mathcal{D}}'$ is precisely ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We then apply \Cref{lem: find reordering} to compute a set of reordering curves, connecting points of $\set{p_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ to points $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$.
Finally, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, let $\gamma_e$ be the concatenation of (i) the image of $e$ in $\phi'_i$ outside the disc ${\mathcal{D}}$; (ii) the reordering curve connecting $p_e$ to $p'_e$; and (iii) the straight line segment connecting $p'_e$ to $z_{\operatorname{right}}$ in ${\mathcal{D}}'$.
We view $\gamma_e$ as the image of edge $e$, for each $e\in \delta(v^{\operatorname{right}}_i)$.
We denote the resulting drawing of $G_i$ by $\phi_i$. It is clear that $\phi_i$ respects the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$, and therefore it respects the rotation system $\Sigma_i$.
We use the following claim.
\begin{claim}
\label{clm: rerouting_crossings}
The number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\end{claim}
\begin{proof}
Denote by ${\mathcal{O}}^*$ the ordering in which the curves $\set{\gamma'_{W_e}\mid e\in \delta_{G_i}(v_i^{\operatorname{right}})}$ enter $z_{\operatorname{right}}$, the image of $u_{i+1}$ in $\phi'_i$.
From~\Cref{lem: find reordering} and the algorithm in Step 4 of modifying the drawing within the disc ${\mathcal{D}}$, the number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is at most $O(\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}}))$. Therefore, it suffices to show that $\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}})=O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
In fact, we will compute a set of curves connecting the image of $u_i$ and the image of $u_{i+1}$ in $\phi^*_i$, such that each curve is indexed by some edge $e\in\delta_{G_i}(v_i^{\operatorname{right}})$ these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$ and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$, and the number of crossings between curves of $Z$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
For each $e\in E^{\textsf{thr}}_i$, we denote by $\eta_e$ the curve obtained by taking the union of (i) the curve $\gamma'_{W_e}$ (that connects $u_{i+1}$ to $u_{i-1}$); and (ii) the curve representing the image of the subpath of $P_e$ in $\phi^*_i$ between $u_i$ and $u_{i-1}$.
Therefore, the curve $\eta_e$ connects $u_i$ to $u_{i+1}$.
We then modify the curves of $\set{\eta_e\mid e\in E^{\textsf{thr}}_i}$, by iteratively applying the algorithm from \Cref{obs: curve_manipulation} to these curves at the image of each vertex of $C_{i-1}\cup C_{i+1}$. Let $\set{\zeta_e\mid e\in E^{\textsf{thr}}_i}$ be the set of curves that we obtain. We call the obtained curves \emph{red curves}. From~\Cref{obs: curve_manipulation}, the red curves are in general position. Moreover, it is easy to verify that the number of intersections between the red curves is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1}))$.
We call the curves in $\set{\gamma'_{W_e}\mid e\in \hat E_i}$ \emph{yellow curves}, call the curves in $\set{\gamma'_{W_e}\mid e\in E^{\operatorname{right}}_i}$ \emph{green curves}. See \Cref{fig: uncrossing_to_bound_crossings} for an illustration.
From the construction of red, yellow and green curves, we know that these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$, and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$.
Moreover, we are guaranteed that the number of intersections between red, yellow and green curves is at most $\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/uncross_to_bound_crossings.jpg}
\caption{An illustration of red, yellow and green curves.}\label{fig: uncrossing_to_bound_crossings}
\end{figure}
\end{proof}
From the above discussion and Claim~\ref{clm: rerouting_crossings}, for each $2\le i\le r-1$,
\[
\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right}).
\]
\iffalse
We now estimate the number of crossings in $\phi_i$ in the next claim.
\begin{claim}
\label{clm: number of crossings in good solutions}
$\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\sum_{W\in \tilde{\mathcal{W}}_i}\mathsf{cr}(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})$.
\end{claim}
\begin{proof}
$2\cdot \sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2=\sum_{v\in V(G)}\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2.$
\znote{need to redefine the orderings ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$ to get rid of $\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$ here, which we may not be able to afford.}
\end{proof}
\fi
\paragraph{Drawings $\phi_1$ and $\phi_{r}$.}
The drawings $\phi_1$ and $\phi_{r}$ are constructed similarly. We describe the construction of $\phi_1$, and the construction of $\phi_1$ is symmetric.
Recall that the graph $G_1$ contains only one super-node $v_1^{\operatorname{right}}$, and $\delta_{G_1}(v_1^{\operatorname{right}})=\hat E_1\cup E^{\operatorname{right}}_1$. We define ${\mathcal{W}}_1=\set{W_e\mid e\in E^{\operatorname{right}}_1}\cup \set{Q_2(\hat e)\mid \hat e\in \hat E_1}$. For each $\hat e\in \hat E_1$, we rename the path $Q_2(\hat e)$ by $W_e$, so ${\mathcal{W}}_1$ contains, for each edge $e\in \delta_{G_1}(v^1_{\operatorname{right}})$, a path named $W_e$ connecting its endpoints to $u_2$. Via similar analysis in \Cref{obs: wset_i_non_interfering}, it is easy to show that the paths in ${\mathcal{W}}_1$ are non-transversal with respect to $\Sigma$.
We start with the drawing of $C_1\cup E({\mathcal{W}}_1)$ induced by $\phi^*$, that we denote by $\phi^*_1$. We will not modify the image of $C_i$ in $\phi^*_i$ and will construct the image of edges in $\delta(v_1^{\operatorname{right}})$.
We perform similar steps as in the construction of drawings $\phi_2,\ldots,\phi_{r-1}$.
We first construct, for each path $W\in {\mathcal{W}}_1$, a curve $\gamma_W$ connecting its endpoint in $C_1$ to the image of $u_2$ in $\phi^*$, as in Step 1.
Let $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
We then process all intersections between curves of $\Gamma_1$ as in Step 2. Let $\Gamma'_1=\set{\gamma'_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
Since $\Gamma^{\textsf{thr}}_1=\emptyset$, we do not need to perform Steps 3 and 4. If we view the image of $u_2$ in $\phi^*_1$ as the image of $v^{\operatorname{right}}_1$, and for each edge $e\in \delta(v^{\operatorname{right}}_1)$, we view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is clear that this drawing respects the rotation system $\Sigma_1$. Moreover,
\[\mathsf{cr}(\phi_1)=\chi^2(C_1)+O\textsf{left}(\hat\chi_1({\mathcal{Q}}_2)+\sum_{W\in {\mathcal{W}}_1}\hat\chi_1(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_1,e)^2\textsf{right}).\]
Similarly, the drawing $\phi_k$ that we obtained in the similar way satisfies that
\[\mathsf{cr}(\phi_k)=\chi^2(C_k)+O\textsf{left}(\hat\chi_k({\mathcal{Q}}_{r-1})+\sum_{W\in {\mathcal{W}}_k}\hat\chi_k(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_k,e)^2\textsf{right}).\]
We now complete the proof of \Cref{claim: existence of good solutions special}, for which it suffices to estimate $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
Recall that, for each $1\le i\le r$, $\tilde {\mathcal{W}}_i=\set{W_e\mid e\in \hat E_{i-1}\cup \hat E_{i-1}\cup E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}}$, where $E^{\operatorname{left}}_{i}=E(C_i,\bigcup_{1\le j\le i-2}C_j)$, and
$E^{\operatorname{right}}_{i}=E(C_i,\bigcup_{i+2\le j\le r}C_j)$. Therefore, for each edge $e\in E'\cup (\bigcup_{1\le i\le r-1}\hat E_i)$, the path $W_e$ belongs to exactly $2$ sets of $\set{\tilde{\mathcal{W}}_i}_{1\le i\le r}$.
Recall that the path $W_e$ only uses edges of the inner path $P_e$ and the outer path $P^{\mathsf{out}}_e$.
Let $\tilde{\mathcal{W}}=\bigcup_{1\le i\le r}\tilde{\mathcal{W}}_i$, from \Cref{obs: edge_occupation in outer and inner paths},
for each edge $e\in E'\cup (\bigcup_{1\le i\le r-1}\hat E_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(1)$, and
for $1\le i\le r$ and for each edge $e\in E(C_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(\cong_G({\mathcal{Q}}_i,e))$.
Therefore, on one hand,
\[
\begin{split}
\sum_{1\le i\le r}\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} 2\cdot \cong_G(\tilde {\mathcal{W}},e)\cdot\cong_G(\tilde {\mathcal{W}},e')\\
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} \textsf{left}(\cong_G(\tilde {\mathcal{W}},e)^2+\cong_G(\tilde {\mathcal{W}},e')^2\textsf{right})\\
& \le
\sum_{e\in E(G)} \chi(e)\cdot \cong_G(\tilde {\mathcal{W}},e)^2
= O(\mathsf{cr}(\phi^*)\cdot\beta),
\end{split}
\]
and on the other hand,
\[
\begin{split}
\sum_{1\le i\le r}\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2
& \le \sum_{e\in E(G)} \textsf{left}(\sum_{1\le i\le r} \cong_G(\tilde {\mathcal{W}}_i,e)\textsf{right})^2\\
& \le O\textsf{left}(\sum_{e\in E(G)} \cong_G(\tilde {\mathcal{W}},e)^2 \textsf{right})\\
& \le O\textsf{left}(|E(G)|+\sum_{1\le i\le r}\sum_{e\in E(C_i)} \cong_G({\mathcal{Q}}_i,e)^2\textsf{right})=O(|E(G)|\cdot\beta).
\end{split}
\]
Moreover, $\sum_{1\le i\le r}\chi^2(C_i)\le O(\mathsf{cr}(\phi^*))$, and $\sum_{1\le i\le r}\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})\le O(\mathsf{cr}(\phi^*)\cdot\beta)$.
Altogether,
\[
\begin{split}
\sum_{1\le i\le r}\mathsf{cr}(\phi_i)
& \le
O\textsf{left}(\sum_{1\le i\le r}
\textsf{left}(
\chi^2(C_i)+ \hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})
\textsf{right})\\
& \le O(\mathsf{cr}(\phi^*))+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(|E(G)|\cdot\beta)\\
& \le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot\beta).
\end{split}
\]
This completes the proof of \Cref{claim: existence of good solutions special}.
\end{proofof}
$\ $
\iffalse
We denote by $\hat{\phi_i}$ the resulting drawing, and it is clear that $\hat{\phi_i}$ is a drawing of the instance $(\hat G_i,\hat\Sigma_i)$, such that $\mathsf{cr}(\hat{\phi_i})\le \mathsf{cr}(C_i)+O(\operatorname{cost}({\mathcal{W}}_i)\cdot\operatorname{poly}\log n)$.
We first define an instance $(\hat G_i,\hat\Sigma_i)$ as follows. We start with the subgraph of $G$ induced by all edges of $E_G(C_i)$ and $E({\mathcal{W}}_i)$, and then create, for each edge $e$ in the subgraph, $\cong_{{\mathcal{W}}_i}(e)$ parallel copies of it. This finishes the description of graph $\hat G_i$.
For each vertex $v$ in $\hat G_i$, we define its rotation $\hat{\mathcal{O}}_v$ as follows. We start with the rotation ${\mathcal{O}}_v\in \Sigma$, and then replace, for each edge $e\in E({\mathcal{W}}_i)$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $\cong_{{\mathcal{W}}_i}(e)$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$, where the ordering among the copies is arbitrary.
On the one hand, in graph $\hat G_i$, we can make paths of ${\mathcal{W}}_i$ edge-disjoint by letting, for each edge $e\in E({\mathcal{W}}_i)$, each path of ${\mathcal{W}}_i$ that contains $e$ now take a distinct copy of $e$ in $\hat G_i$.
On the other hand, a drawing $\hat \phi_i$ of the instance $(\hat G_i,\hat\Sigma_i)$ can be easily computed from $\phi^*$, as follows.
We start with $\hat\phi'_i$, the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$. Then for each edge of ${\mathcal{W}}_i$, let $\gamma_e$ be the curve that represents the image of $e$, and we create $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting endpoints of $e$, that lies in an arbitrarily thin strip around $\gamma_e$. We denote by $\hat{\phi_i}$ the resulting drawing, and it is clear that $\hat{\phi_i}$ is a drawing of the instance $(\hat G_i,\hat\Sigma_i)$, such that $\mathsf{cr}(\hat{\phi_i})\le \mathsf{cr}(C_i)+O(\operatorname{cost}({\mathcal{W}}_i)\cdot\operatorname{poly}\log n)$.
For each edge $e\in E_G(C_i)$, we denote by $\gamma_e$ the curve in $\hat \phi_i$ that represents the image of $e$; and for each path $W\in {\mathcal{W}}_i$, we denote by $\gamma_W$ the curve in $\hat \phi_i$ that represents the image of $W$.
We define $\Gamma_0=\set{\gamma_e\mid e\in E_G(C_i)}$ and $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$. It is immediate to verify that the sets $\Gamma_0,\Gamma_1$ of curves satisfy the condition of \Cref{thm: type-2 uncrossing}. We then apply the algorithm in \Cref{thm: type-2 uncrossing} to $\Gamma_0,\Gamma_1$, and let $\Gamma_1'$ be the set of curves that we obtain.
From \Cref{thm: type-2 uncrossing}, the curves in $\Gamma_1'$ do not intersect internally between each other, and have the same sets of first endpoints and last endpoints.
We now show that we can obtain a drawing $\phi_i$ of the instance $(G_i,\Sigma_i)$ using the curves $\Gamma_0,\Gamma_1'$.
For each edge $e\in E_G(C_i)$, we still let the curve $\gamma_e\in \Gamma_0$ be the image of $e$.
For each edge path $W\in {\mathcal{W}}$, from \Cref{thm: type-2 uncrossing}, there is a curve $\gamma'\in \Gamma_1'$ connecting the endpoint of $W$ to we still let the curve $\gamma_e\in \Gamma_0$ be the image of $e$.
\znote{maybe we need another type of uncrossing here.}
\fi
\begin{proofof}{Claim~\ref{claim: stitching the drawings together}}
We first define a set $\set{(U_i,\Sigma'_i)\mid 1\le i\le r}$ of instances of \textnormal{\textsf{MCNwRS}}\xspace as follows.
We define $U_1=G$, and for each $2\le i\le r$, we define $U_i$ to be the graph obtained from $G$ by contracting the clusters $C_1,\ldots, C_{i-1}$ into a vertex $v^{\operatorname{left}}_i$.
Note that each edge in $U_i$ is also an edge in $G$, and we do not distinguish between them.
Note that $U_1=G$, we define the rotation system $\Sigma'_1$ on $U_1$ to be $\Sigma$.
For each $2\le i\le r$, we define the rotation system $\Sigma'_i$ on $U_i$ as follows.
For each vertex $v\in \bigcup_{i\le t\le r}V(C_t)$, note that its incident edges in $U_i$ are the edges of $\delta_G(v)$, and its rotation in $\Sigma'_i$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the input rotation system $\Sigma$.
For vertex $v^{\operatorname{left}}_i$, note that its incident edges are the edges of $\delta_{G_i}(v^{\operatorname{left}}_i)$, and its rotation in $\Sigma'_i$ is defined to be ${\mathcal{O}}^{\operatorname{left}}_i$, the rotation on $v^{\operatorname{left}}_i$ of instance $(G_i,\Sigma_i)$.
Note that $U_k=G_k$ and $\Sigma'_k=\Sigma_k$, so the drawing $\phi_k$ of the instance $(G_k,\Sigma_k)$ is also a drawing of the instance $(U_k,\Sigma'_k)$. For clarity, when we view this drawing as a solution to the instance $(U_k,\Sigma'_k)$, we rename it by $\psi_k$.
We will sequentially, for $i=r-1,\ldots,1$, compute a drawing of the instance $(U_i,\Sigma'_i)$ using the drawing $\psi_{i+1}$ of $(U_{i+1},\Sigma'_{i+1})$ and the drawing $\phi_i$ of $(G_{i},\Sigma_{i})$, and eventually, we return the drawing $\psi_1$ of $(U_{1},\Sigma'_{1})$ as the solution to the instance $(G,\Sigma)$.
We now fix an index $1\le i< r-1$ and construct the drawing $\psi_i$ to the instance $(U_{i},\Sigma'_{i})$, assuming that we have computed a drawing $\psi_{i+1}$ to the instance $(U_{i+1},\Sigma'_{i+1})$, as follows.
Recall that $V(G_i)=V(C_i)\cup\set{v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$ if $i\ge 2$ and $V(G_i)=V(C_i)\cup\set{v^{\operatorname{right}}_i}$ if $i=1$, and $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{U_{i+1}}(v^{\operatorname{left}}_{i+1})$. Moreover, from \Cref{obs: rotation for stitching} and the definition of instance $(U_{i+1},\Sigma'_{i+1})$, the rotation on $v^{\operatorname{right}}_i$ in $\Sigma_{i}$ is identical to the rotation on $v^{\operatorname{left}}_{i+1}$ in $\Sigma'_{i+1}$.
Denote $F=\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{U_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Let ${\mathcal{D}}$ be an arbitrarily small disc around the image of $v_{i+1}^{\operatorname{left}}$ in $\psi_{i+1}$. For each edge $e\in F$, we denote by $p_e$ the intersection between the image of $e$ with the boundary of ${\mathcal{D}}$.
Therefore, the order in which the points $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}$ is ${\mathcal{O}}^{\operatorname{left}}_{i+1}$.
We erase the drawing of $\psi_{i+1}$ inside the disc ${\mathcal{D}}$, and view the area inside the disc ${\mathcal{D}}$ as the outer face of the drawing.
Similarly, let ${\mathcal{D}}'$ be an arbitrarily small disc around the image of $v_{i}^{\operatorname{right}}$ in $\phi_{i}$. For each edge $e\in F$, we denote by $p'_e$ the intersection between the image of $e$ with the boundary of ${\mathcal{D}}'$.
Therefore, the order in which the points $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}'$ is ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We erase the drawing of $\phi_{i}$ inside the disc ${\mathcal{D}}'$, and let ${\mathcal{D}}''$ be another disc that is strictly contained in ${\mathcal{D}}'$. We now place the drawing of $\psi_{i+1}$ inside ${\mathcal{D}}'$ (after we erase part of it inside ${\mathcal{D}}$), so that the boundary of ${\mathcal{D}}$ in $\psi_{i+1}$ coincide with the boundary of ${\mathcal{D}}''$ in $\phi_{i}$, while the interior of ${\mathcal{D}}$ coincides with the exterior of ${\mathcal{D}}''$. We then compute a set $\set{\zeta_e\mid e\in F}$ of curves lying in ${\mathcal{D}}'\setminus {\mathcal{D}}''$, where for each $e\in F$, the curve $\zeta_e$ connects $p'_e$ to $p_e$, such that all curves of $\set{\zeta_e\mid e\in F}$ are mutually disjoint. Note that this can be done since the order in which $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}'$ is identical to $\set{p'_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}$.
We denote by $\psi_i$ the resulting drawing we obtained.
See Figure~\ref{fig: stitching} for an illustration.
Clearly, $\psi_i$ is a drawing of $U_i$ that respects $\Sigma'_i$ if we view, for each edge $e\in F$, the union of (i) the image of $e$ in $\phi_i$ outside the disc ${\mathcal{D}}'$; (ii) the curve $\zeta_e$; and (iii) the image of $e$ in $\psi_{i+1}$ inside the disc ${\mathcal{D}}''$, as the image of $e$.
\begin{figure}[h]
\centering
\subfigure[The drawing $\psi_{i+1}$, where the boundary of ${\mathcal{D}}$ is shown in dash black.]{\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_1.jpg}
}
\hspace{0.45cm}
\subfigure[The drawing $\psi_{i+1}$ after we erase its part inside ${\mathcal{D}}$ and view the interior of ${\mathcal{D}}$ as the outer face.]{
\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_2.jpg}}}
\hspace{0.45cm}
\subfigure[The drawing $\psi_i$, where the curves of $\set{\zeta_e\mid e\in F}$ are shown in dash line segments.]{
\scalebox{0.36}{\includegraphics[scale=1.0]{figs/stitching_3.jpg}}}
\caption{An illustration of constructing the drawing $\psi_i$ from $\psi_{i+1}$.}\label{fig: stitching}
\end{figure}
Clearly, any crossing in the drawing $\psi_i$ is either a crossing of $\phi_i$ or a crossing of $\psi_{i+1}$, so $\mathsf{cr}(\psi_i)\le \mathsf{cr}(\psi_{i+1})+\mathsf{cr}(\phi_{i})$.
Therefore, if we rename the drawing $\psi_1$ of the instance $(U_1,\Sigma'_1)$ by $\phi$, then $\phi$ is a drawing of $G$ that respects $\Sigma$, and $\mathsf{cr}(\phi)\le \sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
\end{proofof}
\subsection{Disengagement of Nice Instances -- Proof of \Cref{thm: advanced disengagement - disengage nice instances}}
\label{subsec: disengagement with bad chain}
\newcommand{\operatorname{mid}}{\operatorname{mid}}
In this section we provide the proof of \Cref{thm: advanced disengagement - disengage nice instances}.
Recall that we are given as input an instance $I'=(G',\Sigma')$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, that we will denote by $I=(G,\Sigma)$, in order to simplify the notation. Additionally, we are given a set ${\mathcal{C}}'$ of disjoint clusters of $G'$; in order to simplify the notation, we will denote ${\mathcal{C}}'$ by ${\mathcal{C}}$. Lastly, we are given a nice witness structure $(\tilde{\mathcal{S}},\tilde{\mathcal{S}}',\hat{\mathcal{P}})$ for graph $G$ with respect to the set ${\mathcal{C}}$ of clusters, where $\tilde {\mathcal{S}}=\set{\tilde S_1,\ldots, \tilde S_r}$ is the backbone of the witness structure, with vertex sets in $\set{V(\tilde S_i)}_{1\le i\le r}$ partitioning $V(G)$. For convenience, we will denote the set
$\tilde {\mathcal{S}}'=\set{\tilde S'_1,\ldots, \tilde S'_r}$ of the vertebrae of the nice witness structure by ${\mathcal{S}}=\set{S_1,\ldots,S_r}$. Recall that each cluster $S_i\in {\mathcal{S}}$ has the $\alpha^*$-bandwidth property, for $\alpha^*=\Omega(1/\log^{12}m)$.
Recall that we are given a partition of the edges of $G$ into two subsets: set $\tilde E'$, containing all edges of $\bigcup_{1\leq i\leq r}E(S_i)$, and all edges of $\bigcup_{1\leq i<r}E(S_i,S_{i+1})$; and set $\tilde E''=E(G)\setminus \tilde E'$.
Recall that set $\hat E\subseteq \tilde E''$ contains all edges $(v,u)\in \tilde E''$, where $v$ and $u$ lie in different clusters of $\tilde{\mathcal{S}}$, and the set $\hat{\mathcal{P}}$ of paths contains, for each edge $e\in \hat E$, a path $ P(e)$ that consists of three subpaths: $P^1(e), P^2(e)$, and $P^3(e)$, that are called the prefix, the mid-part and the suffix of $P(e)$, respectively.
We denote by $\hat {\mathcal{P}}^1=\set{P^1(e)\mid e\in \hat E}$, $\hat{\mathcal{P}}^2=\set{P^2(e)\mid e\in \hat E}$, and $\hat {\mathcal{P}}^3=\set{P^3(e)\mid e\in \hat E}$, the sets of paths containing all prefixes, all mid-parts, and all suffixes of the paths in $\hat {\mathcal{P}}$, respectively.
Throughout, we will use a parameter $\hat \eta=2^{O((\log m)^{3/4}\log\log m)}$.
In order to compute a decomposition of instance $I$ into subinstances, we need to define, for every edge $e\in \hat E$, a cycle $W(e)$, called an \emph{auxiliary cycle} that has some useful properties. As an intuition, we could obtain a cycle $W(e)$ by taking the union of the nice guiding path $P(e)\in \hat {\mathcal{P}}$ with the edge $e$. The structure of the nice guiding paths ensures that the cycle $W(e)$ has a single contiguous segment $P^2(e)$ that visits a contiguous subset of the vertebrae in the order of their indices. The resulting set $\set{W(e)\mid e\in \hat E}$ of cycles is close to having the properties that we need, except that we would like to ensure that these cycles are non-transeversal (or close to being non-transversal) with respect to $\Sigma$. We discuss the construction of the family $\set{W(e)\mid e\in \hat E}$ of cycles with these properties below.
Next, we define a laminar family ${\mathcal{L}}=\set{U_1,\ldots,U_r}$ of clusters of graph $G$, where for all $1\leq i\leq r$, $U_i$ is the subgraph of $G$ induced by vertex set $V(S_1)\cup\cdots\cup V(S_i)$. We define, for every vertebra $S_i\in {\mathcal{S}}$, an internal $S_i$-router ${\mathcal{Q}}(S_i)$, and use these routers, together with the auxiliary cycles in $\set{W(e)\mid e\in \hat E}$ in order to define an internal $U_i$-router and an external $U_i$-router for every cluster $U_i\in {\mathcal{L}}$. The final decomposition ${\mathcal{I}}_2$ of instance $I$ into subinstances is simply a decomposition via the laminar family ${\mathcal{L}}$ defined in \Cref{subsec: laminar-based decomposition}. Recall that for each cluster $U_z\in {\mathcal{I}}_2$, there is a unique instance $I_z=(G_z,\Sigma_z)\in {\mathcal{I}}_2$, where graph $G_z$ is obtained from $G$ by contracting all vertices of $S_1\cup\cdots \cup S_{z-1}$ into a special vertex $v_z^*$, and all vertices of $S_{z+1}\cup\cdots\cup S_r$ into a special vertex $v_z^{**}$ (for $z=1$, $G_1$ is obtained from $G$ by contracting all vertices of $S_2\cup\cdots \cup S_r$ into a special vertex $v_1^{**}$, and for $z=r$, graph $G_z$ is obtained from $G$ by contracting all vertices of $S_1\cup\cdots\cup S_{r-1}$ into a special vertex $v_r^*$). For each $1\leq z<r$, we use the internal $U_z$-router that we computed, in order to define a circular ordering of the edges of $\delta_G(U_z)$, that will in turn be used in order to define the rotation systems $\set{\Sigma_z}_{z=1}^r$ associated with each subinstance. The techniques developed in \Cref{subsec: laminar-based decomposition} prove that there is an efficient algorithm that combines solutions to the resulting subinstances into a solution to instance $I$ that has a relatively low cost. However, since the depth of the laminar family ${\mathcal{L}}$ may be quite high, we cannot use the tools from \Cref{sec: guiding paths orderings basic disengagement} in order to bound $\sum_{z=1}^r|E(G_z)|$ and $\sum_{z=1}^r\mathsf{OPT}_{\mathsf{cnwrs}}(I_z)$. Instead, we show a simple direct bound on $\sum_{z=1}^r|E(G_z)|$, and a more involved proof for bounding $\sum_{z=1}^r\mathsf{OPT}_{\mathsf{cnwrs}}(I_z)$. The latter proof exploits the internal and external $U_i$-routers that we construct, for $1\leq i\leq r$, which in turn are based on the auxiliary cycles $\set{W(e)\mid e\in \hat E}$, in order to show the existence of a low-cost solution to each instance $I_z\in {\mathcal{I}}_2$. In order to ensure that the costs of these solutions is sufficiently low, it is crucial that the cycles in $\set{W(e)\mid e\in \hat E}$ are \emph{almost} non-transversal with respect to $\Sigma$: that is, for every pair $W(e),W(e')$ of cycles, there is at most one vertex $v$, such that $W(e)$ and $W(e')$ intersect transversally at $v$. In order to define the set ${\mathcal{W}}=\set{W(e)\mid e\in \hat E}$ of cycles, we define two collections of paths: path set ${\mathcal{P}}^{\mathsf{out}}=\set{P^{\mathsf{out}}(e)\mid e\in \hat E}$, which is obtained by modifying the paths of $\hat {\mathcal{P}}^1\cup \hat {\mathcal{P}}^3$, and path set ${\mathcal{P}}^{\mathsf{in}}=\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$. For every edge $e\in \hat E$, the first and the last edges on paths $P^{\mathsf{out}}(e)$ and $P^{\mathsf{in}}(e)$ are identical. Path $P^{\mathsf{out}}(e)$ contains the edge $e$, and all its edges lie in $\tilde E''$. All inner edges of path $P^{\mathsf{in}}(e)$ lie in $\tilde E'$, and the path visits a consequtive subset of clusters of ${\mathcal{S}}$ in their natural order.
The auxiliary cycle $W(e)$ is obtained by taking the union of the paths $P^{\mathsf{out}}(e)$ and $P^{\mathsf{in}}(e)$.
The remainder of the proof of \Cref{thm: advanced disengagement - disengage nice instances} consists of four steps. In the first step, we construct the set ${\mathcal{P}}^{\mathsf{out}}=\set{P^{\mathsf{out}}(e)\mid e\in \hat E}$ of paths. In the second step, we construct the set ${\mathcal{P}}^{\mathsf{in}}=\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$ of paths and the collection ${\mathcal{W}}=\set{W(e)\mid e\in \hat E}$ of auxiliary cycles. In the third step, we construct the laminar family ${\mathcal{L}}=\set{U_1,\ldots,U_r}$ of clusters, and, for all $1\leq z\leq r$, an internal $U_z$-router ${\mathcal{Q}}(U_z)$ and an external $U_z$-router ${\mathcal{Q}}'(U_z)$. In the fourth and the final step, we compute the collection ${\mathcal{I}}_2$ of subinstances of $I$ and analyze its properties. We now describe each of the steps in turn.
\input{path-decomposition-outer-paths}
\input{path-decomposition-inner-paths}
\input{path-decomposition-internal-routers}
\input{path-decomposition-subinstances}
\input{bound-opts}
\iffalse
old stuff below
Our algorithm consists of two stages. In the first stage, we compute the desired set of cycles $\set{W(e)\mid e\in \hat E}$, that we call \emph{auxiliary cycles}, by modifying the set $\hat {\mathcal{P}}$ of nice guiding paths. In the second stage, we construct the subinstances of the $2^{O((\log m)^{3/4}\log\log m)}$-decomposition, using the auxiliary cycles constructed in the first stage.
The algorithms for the first and the second stages appear in Sections \ref{sec: guiding and auxiliary paths} and \ref{subsec: instances}, respectively. We then show that the resulting instances admit solutions of low cost, and that their solutions can be efficiently combined together to produce a low-cost solution of the original instance $I$,
in \Cref{subsec: proof of disengagement with routing}.
\subsubsection{Stage 1: Computing the Auxiliary Cycles}
\label{sec: guiding and auxiliary paths}
\mynote{This needs rewriting}
The goal of the first stage is to compute, for each edge $e\in \hat E$, a cycle $W(e)$ in graph $G$.
We will first compute, for every vertebra $S_i\in {\mathcal{S}}$ of the nice witness structure, a center vertex $u_i\in S_i$.
Consider now some edge $e=(v,u)\in \hat E$, with $v\in \tilde S_i$ and $u\in \tilde S_j$, and assume that $i<j$. From the definition of the nice witness structure, prefix $P^1(e)$ of the nice guiding path $P(e)\in \hat {\mathcal{P}}$ connects vertex $v$ to some vertex $v'$, where $v'\in S_{i'}$ for some index $i'\le i$, and the suffix $P^3(e)$ connects vertex $u$ to some vertex $u'$, where $u'\in S_{j'}$ for some index $j'\ge j$.
The cycle $W(e)$ that we construct for edge $e$ is a union of two paths, that we denote by $P^{\mathsf{in}}(e)$ and $P^{\mathsf{out}}(e)$.
Path $P^{\mathsf{in}}(e)$ will connect the center vertex $u_{i'}$ of the cluster $S_{i'}$ to the center vertex $u_{j'}$ of $S_{j'}$. Additionally, we will ensure that there is a partition of the path $P^{\mathsf{in}}(e)$ into edge-disjoint subpaths $P^{\mathsf{in}}_{i'}(e)$, $P^{\mathsf{in}}_{i'+1}(e),\ldots,P^{\mathsf{in}}_{j'-1}(e)$, such that, for each $i'\le t\le j'-1$, subpath $P^{\mathsf{in}}_{t}(e)$ connects $u_{t}$ to $u_{t+1}$, and it contains an edge $e_t\in E(S_t,S_{t+1})$. Moreover, if we delete edge $e_t$ from path $P^{\mathsf{in}}_{t}(e)$, then one of the two resulting paths is contained in $S_t$, while the other path is contained in $ S_{t+1}$.
We will also ensure that the path $P^{\mathsf{out}}(e)$ has endpoints $u_{i'}$ and $u_{j'}$, and it can be partitioned into three edge-disjoint subpaths $P^{\mathsf{out}}_{\operatorname{left}}(e)$, $P^{\mathsf{out}}_{\operatorname{mid}}(e)$ and $P^{\mathsf{out}}_{\operatorname{right}}(e)$, such that $P^{\mathsf{out}}_{\operatorname{left}}(e)\subseteq S_{i'}$, $P^{\mathsf{out}}_{\operatorname{right}}(e)\subseteq S_{j'}$, and path $P^{\mathsf{out}}_{\operatorname{mid}}(e)$ contains the edge $e$ and is internally disjoint from all clusters of ${\mathcal{S}}$.
Moreover, if we delete edge $e$ from the path $P^{\mathsf{out}}_{\operatorname{mid}}(e)$, then one of the two resulting paths must have all its vertices in $\bigcup_{1\le t\le i}\tilde V(S_t)$, while the other path must have all its vertices in $\bigcup_{j\le t\le r}\tilde V(S_t)$.
The first stage consists of four steps. In the first step, we utilize the bandwidth property of clusters in ${\mathcal{S}}$ to compute, for each cluster $S_i\in {\mathcal{S}}$, an internal router ${\mathcal{Q}}_i\in \Lambda_G(S_i)$ for $S_i$.
In the second step, we construct the path set ${\mathcal{P}}^{\mathsf{out}}=\set{P^{\mathsf{out}}(e)\mid e\in \hat E}$.
In the third step, we construct collections of paths that connect center vertices of consecutive vertebrae clusters, by using the paths in set ${\mathcal{P}}^2$. In the last step, we rcomplete the construction of the path set ${\mathcal{P}}^{\mathsf{in}}=\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$, and of the auxiliary cycles.
We now describe each of the four steps in turn.
We will use the parameters
$\beta^*=2^{O(\sqrt{\log m}\cdot \log\log m)}$ and $\eta^*=2^{O((\log m)^{3/4}\log\log m)}$ from \Cref{thm:algclassifycluster}.
\iffalse
\subsubsection*{Step 2. Segmenting the paths of ${\mathcal{P}}^2$}
Consider an edge $e=(v,u)\in \hat E$, and assume that $e$ connects $v\in S_i$ to $u\in S_j$, where $j\ge i+2$.
We start with the path $P^2(e)$, the mid-part of path $P(e)$. From the definition, the path $P^2(e)$ have endpoints $v',u'$, where $v'\in S_{i'}$ for some $i'\le i$ and $u'\in S_{j'}$ for some $j'\ge j$. Moreover, for each $i'\le t\le j'$, the intersection between $P^2(e)$ and $S'_t$ is a subpath $P^2_t(e)$ of $P^2(e)$, where the first subpath $P^2_{i'}(e)$ contains the endpoint $v'$ and the last subpath $P^2_{i'}(e)$ contains the endpoint $u'$.
We view the path $P^2(e)$ as being directed from $v'$ to $u'$. For each $i'\le t\le j'$, we denote by $e_t$ the edge of $P(e)$ right before the subpath $P^2_t(e)$ and by $e'_t$ the edge of $P(e)$ right after the subpath $P^2_t(e)$, so the edges $e_t,e'_t$ belong to $\delta_G(S'_t)$. We define $\tilde P^2(e)$ as the path obtained from $P^2(e)$ by replacing each subpath $P^2_t(e)$ by the path $Q_t(e_t)\cup Q_t(e'_t)$, where the path $Q_t(e_t)$ is the guiding path of ${\mathcal{Q}}_t$ that routes $e_t$ to $u_t$ in $S'_t$, and path $Q_t(e'_t)$ is the guiding path of ${\mathcal{Q}}_t$ that routes $e'_t$ to $u_t$ in $S'_t$. Clearly, the path $\tilde P^2(e)$ obtained in this way is still a path connecting $v'$ to $u'$, and the intersection between $\tilde P^2(e)$ and every cluster $S'_t$ is a subpath.
\fi
\fi
\section{Disengagement of Clusters -- Proof of Theorem \ref{thm: disengagement}}
\label{sec:disengagement}
This section is dedicated to the proof of Theorem \ref{thm: disengagement}. We start by defining a laminar family of cuts, and stating two main theorem for disengaging clusters in such a family.
\subsection{Laminar Family of Cuts}
\mynote{A useful trick: For each cluster $C\in {\mathcal{C}}$, make $n_e$ extra copies of each edge $e$, where $n_e$ is the number of times $e$ participates in paths in ${\mathcal{Q}}(C)$. Because the cost of ${\mathcal{Q}}(C)$ is small, the number of new crossings introduced will be small. This is done before the disengagement. But from now on we can assume that all paths sets ${\mathcal{Q}}(C)$ (and also ${\mathcal{P}}(C)$ because of how they are constructed) are edge-disjoint. So we don't need to deal with $\operatorname{cost}({\mathcal{P}},{\mathcal{Q}})$ which is tricky to figure out, and instead can just do $\mathsf{cr}({\mathcal{P}},{\mathcal{Q}})$ which is easier. In the following subsections we are still using costs and we are not assuming that paths sets ${\mathcal{P}}(S),{\mathcal{Q}}(S)$ are edge-disjoint, but it will be better to switch to this to make things cleaner and simpler.}
Suppose we are given a cluster $S$, a vertex $u\in S$, and a set ${\mathcal{Q}}$ of paths connecting $\delta(S)$ to $u$ inside $S$, such that the paths in ${\mathcal{Q}}$ are locally non-interfering. We define a circular ordering ${\mathcal{O}}_{{\mathcal{Q}}}(\delta(S))$ of the edges in $\delta(S)$, as follows. Recall that ${\mathcal{Q}}=\set{Q(e)\mid e\in \delta(S)}$, and path $Q(e)$ connects the endpoint of edge $e$ lying in $S$ to the vertex $u$. Denote by $\hat e$ the last edge on path $Q(e)$. Consider the multi-set $\hat E=\set{\hat e\mid e\in \delta(S)}$ of edges (this set is a multi-set, since it is possible that, for a pair $e,e'\in \delta(S)$ of edges, $e\neq e'$ but $\hat e=\hat e'$). We define a circular ordering $\hat {\mathcal{O}}$ of the edges of $\hat E$ as follows: the ordering is consistent with the ordering ${\mathcal{O}}_u\in \Sigma$ of the edges of $\delta(u)$, where ties (between copies of the same edge) are broken arbitrarily \znote{maybe not arbitrarily? if $e\ne e'$ and $\hat e=\hat e'$, then probably we need to go to the other endpoint of $\hat e=\hat e'$ (let's call it $u'$) and look at what edges of $\delta(u')$ the paths $Q(e)$ and $Q(e')$ uses and then determine how to break this tie according to the ordering ${\mathcal{O}}_{u'}$} \mynote{It is OK to make the ordering arbitrary because the costs of paths in ${\mathcal{Q}}$ will be low, so reordering paths terminating at the same edge is fine. In the laminar family = path case we won't have a single $u'$ vertex so we may be forced to use this sort of ordering. Also it's not immediately clear that we can order according to $u'$ because the orientations are not fixed. HOwever: if we do the fix with introducing many copies of each edge, making the paths in ${\mathcal{Q}}$ edge-disjoint and fixing some ordering of resulting edges around each vertex, this issue won't arise}. The ordering $\hat {\mathcal{O}}$ of the edges in $\hat E$ immediately defines the circular ordering ${\mathcal{O}}_{{\mathcal{Q}}}(\delta(S))$ of the edges in $\delta(S)$: if circular ordering of the edges in $\hat E$ is $\hat e_1,\hat e_2,\ldots,\hat e_r$, then the corresponding ordering of the edges in $\delta(S)$ is $e_1,e_2,\ldots,e_r$. We say that ${\mathcal{O}}_{{\mathcal{Q}}}(\delta(S))$ is the \emph{ordering of the edges of $\delta(S)$ determined by ${\mathcal{Q}}$}. Intuitively, the ordering ${\mathcal{O}}_{{\mathcal{Q}}}(\delta(S))$ is determined by the order in which the images of the paths in ${\mathcal{Q}}$ enter the vertex $u$ in any feasible solution to instance $(G,\Sigma)$ of the problem.
\subsection{Disengagement Algorithm}
Suppose we are given an instance $(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, and a laminar family ${\mathcal{L}}$ of sub-graphs of $G$, with $G\in {\mathcal{L}}$. Assume further that for every cluster $S\in {\mathcal{L}}$, graph $S\setminus \bigcup_{S'\in {\mathcal{W}}}S'$ is non-empty, and that we are given a vertex $u(S)\in S\setminus\bigcup_{S'\in {\mathcal{W}}}S'$, together with a set ${\mathcal{Q}}(S)$ of paths, routing the edges of $\delta(S)$ to $u(S)$ inside $S$, so that the paths in ${\mathcal{Q}}(S)$ are locally non-interfering. Assume further that for every cluster $S\in {\mathcal{L}}\setminus\set{G}$, we are given a set ${\mathcal{P}}(S)$ of paths, routing the edges of $\delta(S)$ to $u'(S)$ outside $S$, where $S'$ is the parent-cluster of $S$, so that the paths in ${\mathcal{P}}(S)$ are locally non-interfering.
We construct a new family ${\mathcal{F}}=\set{(G_S,\Sigma_S)\mid S\in {\mathcal{L}}}$ of instances of \textnormal{\textsf{MCNwRS}}\xspace, that are sub-instances of $(G,\Sigma)$, as follows.
Consider the cluster $S\in {\mathcal{L}}$.
In order to construct the instance $(G_S,\Sigma_S)$, we start with the graph $S'$, where $S'$ is the parent-cluster of $S$, if $S\neq G$, and $S'=G$ otherwise. If $S\neq G$, then we create a super-node $v^*$, by contracting all vertices in $S'\setminus S$ into a single vertex. Notice that the edges incident to $v^*$ correspond to the edges of $\delta(S)$ (for simplicity, we will not distinguish between them). The circular ordering of the edges incident to $v^*$ is ${\mathcal{O}}_{{\mathcal{Q}}(S)}(\delta(S))$ (the order in which the corresponding paths in ${\mathcal{Q}}(S)$ enter $u(S)$).
Next, we denote ${\mathcal{W}}(S)=\set{S_1,\ldots,S_q}$. For all $1\leq i\leq q$, we contract all vertices of $S_i$ into a super-node $v_i$. Notice that the edges incident to $v_i$ correspond to the edges of $\delta(S_i)$; we do not distinguish between them. The ordering of the edges incident to $v_i$ in the new instance is determined by ${\mathcal{O}}_{{\mathcal{Q}}(S_i)}(\delta(S_i))$ -- the order in which the corresponding paths in ${\mathcal{Q}}(S_i)$ enter $u(S_i)$.
This completes the definition of the family ${\mathcal{F}}$ of instances. We need the following observation.
\begin{observation}\label{obs: new graph}
For all $S\in {\mathcal{L}}$, graph $G_S$ contains all edges of $S\setminus\bigcup_{i=1}^qS_i$, and the total number of all additional edges in $G_S$ is bounded by $|\delta(S')|+\sum_{i=1}^q|\delta(S_i)|$.
Moreover, $\sum_{S\in {\mathcal{L}}}|E(S)|\leq |E(G)|+\sum_{S\in {\mathcal{L}}}|\delta(S)|$.
\end{observation}
Next, we show that the sum of optimal solution costs of the resulting instances is bounded.
\begin{lemma}\label{lem: disengagement - type 1}
$\sum_{S\in {\mathcal{L}}}\mathsf{OPT}_{\mathsf{cnwrs}}(G_S,\Sigma_S)\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma))+O\textsf{left} (\sum_{S\in {\mathcal{L}}}\operatorname{cost}({\mathcal{Q}}(S))+\operatorname{cost}({\mathcal{Q}}(S),{\mathcal{P}}(S))\textsf{right} )$.
\end{lemma}
The proof of the lemma is somewhat technical and deferred to Section \ref{subsec: type 1 disengagement}.
Lastly, we show that the solutions to instances in ${\mathcal{F}}$ can be combined together to obtain a solution to instance $(G,\Sigma)$.
\begin{lemma}\label{lem: disengagement putting together}
There is an efficient algorithm, that, given, for every instance $(G_S,\Sigma_S)\in {\mathcal{F}}$, a solution $\phi_S$ of cost $\chi_S$, constructs a solution to instance $(G,\Sigma)$ of cost at most $\sum_{(S,\Sigma)\in {\mathcal{F}}}\chi_S$.
\end{lemma}
The proof of the lemma appears in Section \ref{sec: disengagement putting together}.
\subsection{Cost of Disengagement: Proof of Lemma \ref{lem: disengagement - type 1}}
\label{subsec: type 1 disengagement}
\mynote{this is copied from the log}
We start with the graph $G$, and we consider the child clusters of $G$ one-by-one. Starting with the optimal drawing of $G$, we then ``pull'' the drawing of each child $S$ of $G$ out of $G$ (just like we did with the clusters in ${\mathcal{C}}$). Then we repeat the procedure with each child cluster of $G$, pulling out the grandchildren.
In order to analyze this, we need to look closer at the procedure of pulling the drawing of a single cluster. So let's consider the child cluster $S$ of $G$. We assume that we are given some current drawing $\phi$ of $G$ (where possibly we have already pulled the drawings of some clusters out). We call this procedure the \emph{disengagement of $S$ from $G$}.
Let us look at this disengagement procedure closer, and let us analyze the increase in the number of crossings due to it. The procedure consists of these steps. For brevity, we denote the edge set $\delta(S)$ by $E'$, its endpoints lying in $S$ by $X$, and its endpoints lying in $G\setminus S$ by $Y$.
\begin{itemize}
\item Take the current drawing $\phi$ of $G$. Make a disc $D$ around $u_S$. Delete from $\phi$ the drawing of $S$ and the drawing of $E'$. Make a small disc $D'$ containing $D$.
\item Add back the paths in ${\mathcal{Q}}\cup E'$, so that they terminate at the boundary of $D'$. For every edge $e\in E'$, let $\gamma(e)$ be the drawing of the path in ${\mathcal{Q}}$ that used to terminate at an endpoint of $e$ that lies in $X$. We \emph{uncross} the curves $\gamma(e)$ so they don't cross each other. Therefore, the number of crossings that curves in $\Gamma=\set{\gamma(e)\mid e\in E'}$ cause can be bounded by the following:
\begin{itemize}
\item $\mathsf{cr}({\mathcal{Q}},G\setminus (S\cup E'\cup {\mathcal{P}})$;
\item $\mathsf{cr}({\mathcal{Q}},{\mathcal{P}})$;
\item $\mathsf{cr}(E',G\setminus (S\cup E'\cup {\mathcal{P}}))$;
\item $\mathsf{cr}(E',{\mathcal{P}})$.
\end{itemize}
This is because we do not allow the drawings of the curves in $\Gamma$ to cross themselves.
\item Consider the original optimal drawing $\phi^*$ of $G$. Delete from this drawing everything, except for $S,E',{\mathcal{P}}$. For every edge $e\in E'$, the curve obtained by concatenating the drawing of the edge $e$ of $E'$ incident to $x$, and the corresponding path of ${\mathcal{P}}$ is denote by $\gamma'(e)$. Let $\Gamma'=\set{\gamma'(e)\mid e\in E'}$. We uncross the curves $\Gamma'$ as before. Then we place the resulting drawing in disc $D$, so that the curves of $\Gamma'$ terminate on its boundary. The total number of crossings caused by the curves in $\Gamma'$ is bounded by the following;
\begin{itemize}
\item $\mathsf{cr}({\mathcal{P}}, S\setminus {\mathcal{Q}})$;
\item $\mathsf{cr}({\mathcal{P}},{\mathcal{Q}})$;
\item $\mathsf{cr}(E',S\setminus {\mathcal{Q}})$;
\item $\mathsf{cr}(E',{\mathcal{Q}})$.
\end{itemize}
So far increase in the total number of crossings that we have obtained
can be bounded by $\mathsf{cr}({\mathcal{P}},{\mathcal{Q}})$. Now it remains to reorder the curves in $\Gamma$ and in $\Gamma'$ so the order in which they hit the boundary of $D$ or $D'$ is the same as the order of the paths in ${\mathcal{Q}}$ around $u_S$.
\item First reordering: Consider the part of the drawing inside the disc $D$. By concatenating the paths in ${\mathcal{Q}}$ with the curves in $\Gamma'$, we get a collection $\Gamma_1$ of curves connecting $u$ to the boundary of $D$. The order in which these paths leave $u$ and the order in which they arrive at the boundary of $D$ may be different, because these curves may cross. The number of such crossings though is at most: $\mathsf{cr}({\mathcal{Q}},{\mathcal{Q}})+\mathsf{cr}({\mathcal{Q}},E')+\mathsf{cr}({\mathcal{Q}},{\mathcal{P}})$, because the curves in $\Gamma'$ are not allowed to cross each other. By doing the same crossings in the opposite order, we can ensure that the ordering of the curves in $\Gamma_1$ as they hit the boundary of $D$ is the same as their ordering in which they leave $u$. This increases the number of crossings by an additional $\mathsf{cr}({\mathcal{Q}},{\mathcal{Q}})+\mathsf{cr}({\mathcal{Q}},E')+\mathsf{cr}({\mathcal{Q}},{\mathcal{P}})$.
\item Second reordering: now we consider the part of the drawing outside the disc $D$. By concatenating the images of the paths in ${\mathcal{P}}$ and the curves in $\Gamma$, we obtain the set $\Gamma_2$ of curves. Recall that we have uncrossed the curves in $\Gamma$. If we did not uncross them, then the order in which the curves in $\Gamma_2$ hit the boundary of $D'$ would be exactly the right order -- the order in which the paths in ${\mathcal{Q}}$ hit $u$. So we could redo all these crossings to achieve the right order. The total number of such crossings is bounded by $\mathsf{cr}({\mathcal{Q}},E')+\mathsf{cr}({\mathcal{Q}},{\mathcal{Q}})+\mathsf{cr}(E',E')$. (note that the term $\mathsf{cr}(E',E')$ is not an actual increase in the number of crossings, because these crossings are present in the drawing of $G$ and we haven't used them yet).
\end{itemize}
Summing everything up, the total increase in the number of crossings due to the disengagement of $S$ is bounded by:
\[ 2\mathsf{cr}({\mathcal{P}},{\mathcal{Q}})+2\mathsf{cr}({\mathcal{Q}},{\mathcal{Q}})+\mathsf{cr}({\mathcal{Q}},E') \]
This is bounded by a constant times the number of crossings in which the edges of ${\mathcal{Q}}$ participate. So if every edge of $G$ participates in at most $\operatorname{poly}\log n$ sets ${\mathcal{Q}}_S$, the total increase due to disengagement of all clusters is at most $\mathsf{cr}\operatorname{poly}\log n$.
\iffalse
\subsection{Step 1: Computing a Laminar Family of Clusters}
Let ${\mathcal{L}}$ be a laminar family of clusters of $G$, with $G\in {\mathcal{L}}$. We say that ${\mathcal{L}}$ is a \emph{nice} family, iff
\mynote{I think the right thing to do it is not to make any requirements from ${\mathcal{Q}}_S$ and ${\mathcal{P}}_S$, just say that we assume that we are given two such sets of paths. Then say what is the increase in the cost of disengagement as a function of the costs of these sets. By the way, the definition of $\mathsf{cr}({\mathcal{P}},{\mathcal{Q}})$ ignores the fact that the paths in ${\mathcal{Q}}$ may not be edge-disjoint. It should be defined differently (every crossing between the edge of ${\mathcal{Q}}$ and the edge of ${\mathcal{P}}$ must contribute the product of its congestion in ${\mathcal{P}}$ and in ${\mathcal{Q}}$.)}
\begin{itemize}
\item for every cluster $S\in {\mathcal{L}}$, we are given a vertex $u_S\in S$, and a set ${\mathcal{Q}}_S$ of paths connecting the edges of $\delta(S)$ to $u_S$, such that the cost of the paths in ${\mathcal{Q}}_S$ is at most $\mathsf{cr}(S,S) \operatorname{poly}\log n$. Important: the paths in ${\mathcal{Q}}_S$ do not contain the edges of $\delta(S)$.
\item for every cluster $S\in {\mathcal{L}}$, we are given a vertex $u'_S\not\in S$, and a set ${\mathcal{P}}_S$ of paths connecting the edges of $\delta(S)$ to $u'_S$, such that the total number of crossings between the paths in ${\mathcal{P}}_S$ and the edges of $S$ is at most $\mathsf{cr}(S,G)\operatorname{poly}\log n$. Again, the paths in ${\mathcal{P}}_S$ do not contain the edges of $\delta(S)$.
\item the total cost of all edges in $\bigcup_S{\mathcal{Q}}_S$
is bounded by $\mathsf{cr}\operatorname{poly}\log n$.
\end{itemize}
For every cluster $S\in {\mathcal{L}}$, we define a \textnormal{\textsf{MCNwRS}}\xspace instance $(G_S,\Sigma_S)$, as follows. We start with $S$, and, if $S\neq G$, we add another vertex $v^*$ that represents the parent cluster $S^*$ of $S$. We add an edge $(v^*,u)$ for every edge $(v,u)\in E(G)$ with $v\in S^*$, $u\in S$. Then, for every child cluster $S_i$ of $S$, we contract $S_i$ into a single vertex $v_i$. The ordering of edges around $v^*$ is defined according to the ordering in which the paths in ${\mathcal{Q}}_S$ hit $u_S$. For each $i$, the ordering of edges around $v_i$ is defined according to the ordering in which the paths in ${\mathcal{Q}}_{S_i}$ hit $u_{S_i}$.
The main result of this section is the following lemma.
\begin{lemma}
\label{lem:laminar}
Let ${\mathcal{L}}$ be a laminar family of clusters of $G$, then
\begin{enumerate}
\item $\sum_{S\in {\mathcal{L}}}\mathsf{OPT}_{\mathsf{cnwrs}}(G_S,\Sigma_S)\le (\log n)^{O(1)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)$;
\item if we are given, for each $1\le i\le k$, a drawing $\phi_i\in \Phi(G_i,\Sigma_i)$, then we can efficiently compute a drawing $\phi\in \Phi(G,\Sigma)$, such that $\mathsf{cr}(\phi)\le\sum_{1\le i\le k}\mathsf{cr}(\phi_i)$.
\end{enumerate}
\end{lemma}
Note that, if we can further guarantee that $|E(G_S)|\le |E(G)|/\mu$ for each $S\in {\mathcal{L}}$, we will be done.
The remainder of this section is devoted to prove Lemma~\ref{lem:laminar}.
\fi
\section{Third Main Tool - Advanced Disengagement}
\label{sec: main disengagement}
The goal of this section is to prove the following theorem that allows us to perform disengagement in a more general setting than that from basic disengagement.
\begin{theorem}\label{thm: disengagement - main}
There is an efficient randomized algorithm, called \ensuremath{\mathsf{AlgAdvancedDisengagement}}\xspace, whose input consists of an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, parameters $m$ and $\mu\geq 2^{c^*(\log m)^{7/8}\log\log m}$ for some large enough constant $c^*$, and a collection ${\mathcal{C}}$ of disjoint clusters of $G$, for which the following hold:
\begin{itemize}
\item $|V(G)|,|E(G)|\leq m$, and $m$ is greater than a sufficiently large constant;
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha_0$-bandwidth property, for $\alpha_0=1/\log^3m$;
\item $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$; and
\item $\sum_{C\in {\mathcal{C}}}|\delta_G(C)|\leq |E(G)|/\mu^{0.1}$.
\end{itemize}
The algorithm computes a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition ${\mathcal{I}}$ of instance $I$, such that every instance $I'\in {\mathcal{I}}$ is a subinstance of $I$. Moreover, for each resulting instance $I'=(G',\Sigma')\in {\mathcal{I}}$, there is at most one cluster $C\in {\mathcal{C}}$ with $E(C)\subseteq E(G')$. If such a cluster exists, then $E(G')\subseteq E(C)\cup E^{\textnormal{\textsf{out}}}({\mathcal{C}})$, and otherwise $E(G')\subseteq E^{\textnormal{\textsf{out}}}({\mathcal{C}})$.
\end{theorem}
\iffalse{previous statement}
a collection ${\mathcal{I}}$ of subinstances of $I$ with the following properties:
\begin{itemize}
\item for each subinstance $I'=(G',\Sigma')\in {\mathcal{I}}$, there is at most one cluster $C\in {\mathcal{C}}$ with $E(C)\subseteq E(G')$; all other edges of $G'$ lie in set $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq O(|E(G)|)$; and
\item $\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$.
\end{itemize}
\fi
The remainder of this section is dedicated to the proof of \Cref{thm: disengagement - main}.
\iffalse
\begin{definition}[Derivative Graph and Canonical Derivative Graph]
We say that graph $G'$ is a \emph{derivative} of graph $G$ \mynote{other term?} iff there is a subgraph $\tilde G\subseteq G$, and a collection ${\mathcal{R}}$ of mutually disjoint subsets of vertices of $\tilde G$, such that graph $G'$ can be obtained from $\tilde G$ by contracting, for all $R\in {\mathcal{R}}$, vertex set $R$ into a supernode $v_R$; we keep parallel edges but remove self-loops. We do not distiguish between the edges of $G'$ incident to supernodes and their corresponding edges in the original graph $G$. We call the non-supernode vertices of $G'$ \emph{regular vertices}.
We say that graph $G'$ is a \emph{canonical derivative} of graph $G$, if, additionally, for every cluster $C\in {\mathcal{C}}$, either $C\subseteq G'$, or $V(C')\cap V(G')=\emptyset$. Equivalently, for every cluster $C$, either all vertices of $C$ are contracted in $G'$, or none of them are.
\end{definition}
\fi
Over the course of the proof, we will consider subinstances of the input instance $I$.
Recall that an instance $I'=(G',\Sigma')$ of \ensuremath{\mathsf{MCNwRS}}\xspace is a \emph{subinstance} of instance $I=(G,\Sigma)$ (see Definition \ref{def: subinstance}), if there is a subgraph $\tilde G\subseteq G$, and a collection ${\mathcal{R}}$ of mutually disjoint subsets of vertices of $\tilde G$, such that graph $G'$ can be obtained from $\tilde G$ by contracting, for all $R\in {\mathcal{R}}$, vertex set $R$ into a supernode $v_R$; we keep parallel edges but remove self-loops. We do not distiguish between edges of $G'$ incident to supernodes and their corresponding edges in the original graph $G$. We call the non-supernode vertices of $G'$ \emph{regular vertices}.
We also require that, for every regular vertex $v\in V(G')\cap V(G)$, its rotation ${\mathcal{O}}'_v$ in $\Sigma'$ is the same as the rotation ${\mathcal{O}}_v\in \Sigma$. For each supernode $v_R$, its rotation ${\mathcal{O}}'_{v_R}$ can be defined arbitrarily.
We will consider special types of subinstances of a given instance, that we call \emph{canonical} subinstances.
\begin{definition}[Canonical Subinstances]
Let $I'=(G',\Sigma')$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace, and let ${\mathcal{C}}'$ be a collection of disjoint clusters $G'$. We say that instance $I''=(G'',\Sigma'')$ is a \emph{canonical} subinstance of $I'$ with respect to ${\mathcal{C}}'$ if $I''$ is a subinstance of $I'$, and moreover, if $\tilde G\subseteq G'$, and ${\mathcal{R}}$ is a collection of disjoint subsets of vertices of $\tilde G$, such that $G''$ is obtained from $\tilde G$ by contracting every vertex set $R\in {\mathcal{R}}$ into a supernode $v_R$, then the following holds:
For every cluster $C\in {\mathcal{C}}'$, either (i) there is some vertex set $R\in {\mathcal{R}}$ with $V(C)\subseteq R$ (in which case we say that $C$ is \emph{contracted} in graph $G''$); or (ii) $C\subseteq \tilde G$, and for every vertex set $R\in {\mathcal{R}}$, $R\cap V(C)=\emptyset$ (in which case we say that $C$ is \emph{not contracted} in $G''$); or (iii) $V(C)\cap V(\tilde G)=\emptyset$.
\end{definition}
\iffalse
We say that graph $G'$ is a \emph{canonical derivative} of graph $G$, if, additionally, for every cluster $C\in {\mathcal{C}}$, either $C\subseteq G'$, or $V(C')\cap V(G')=\emptyset$. Equivalently, for every cluster $C$, either all vertices of $C$ are contracted in $G'$, or none of them are..
Equivalently, they can be defined as follows. Instance $I'=(G',\Sigma')$ is a subinstance of instance $I=(G,\Sigma)$ iff graph $G'$ is a derivative of graph $G$, and, for every vertex $v\in V(G')\cap V(G)$, its rotation ${\mathcal{O}}'_v$ in $\Sigma'$ is the same as the rotation ${\mathcal{O}}_v\in \Sigma$. For each supernode $u(R)$, its rotation ${\mathcal{O}}'_{u(R)}$ can be defined arbitrarily. We say that a subinstance $I'=(G',\Sigma')$ is a canonical subinstance of instance $I=(G,\Sigma)$ iff $G'$ is a canonical derivative of graph $G$.
\fi
We will consider canonical subinstances of instance $I$ with additional useful properties. We call such subinstances \emph{nice subinstances}. For each such subinstance, we will use a specific \emph{witness structure} to certify that it is indeed a nice subinstance. We will provide two theorems: the first theorem will be used to decompose the input instance $I$ into a collection of nice subinstances, and the second theorem will further decompose each resulting nice subinstance into a collection of subinstances that have the properties required by \Cref{thm: disengagement - main}. In the next subsection, we define the witness structure and the nice subinstances, and then provide the statements of the two theorems that will allow us to complete the proof of \Cref{thm: disengagement - main}.
\subsection{Nice Witness Structure, Nice Subinstances, and Statements of Main Theorems}
Let $G'$ be a graph, let $m>0$ be an integer with $|E(G')|\leq m$, and let ${\mathcal{C}}'$ be a collection of disjoint clusters of $G'$. A \emph{nice witness structure} for $G'$ with respect to ${\mathcal{C}}'$ consists of the following three main ingredients (see \Cref{fig: witness_structure}):
\begin{enumerate}
\item The first ingredient is a sequence $\tilde {\mathcal{S}}=\set{\tilde S_1,\ldots,\tilde S_r}$ of disjoint vertex-induced subgraphs of $G'$, such that $\bigcup_{i=1}^rV(\tilde S_i)=V(G')$, and, for all $1\leq i\le r$, a cluster $\tilde S'_i\subseteq \tilde S_i$ that has the $\alpha^*=\Omega(1/\log^{12}m)$-bandwidth property in $G'$. We require that, for all $1\leq i\leq r$, there is at most one cluster $C \in{\mathcal{C}}'$ with $C\subseteq \tilde S'_i$. Moreover, if such cluster $C$ exists then $E(\tilde S_i)\subseteq E(C)\cup E(G'_{|{\mathcal{C}}'})$ must hold, and otherwise $E(\tilde S_i)\subseteq E(G'_{|{\mathcal{C}}'})$. We also require that for each cluster $C\in {\mathcal{C}}'$, there is an index $1\leq i\leq r$, such that $C\subseteq \tilde S_i'$. We refer to the sequence $\tilde {\mathcal{S}}=\set{\tilde S_1,\ldots,\tilde S_r}$ as \emph{the backbone} of the nice witness structure, and to the clusters in $\tilde {\mathcal{S}}'= \set{\tilde S'_1,\ldots,\tilde S'_r}$ as its \emph{verterbrae}.
\item The second ingredient is a partition of the edges of $E(G')$ into two disjoint subsets, $\tilde E'$ and $\tilde E''$. Set $\tilde E'$ contains all edges of $\bigcup_{i=1}^rE(\tilde S_i')$, and, additionally, for all $1\leq i<r$, it contains every edge $e=(u,v)$ with $u\in \tilde S_i'$, $v\in \tilde S_{i+1}'$.
Set $\tilde E''$ contains all remaining edges of $E(G')$.
Additionally, we let $\hat E\subseteq \tilde E''$ be the set of all edges $(u,v)\in \tilde E''$, where $u$ and $v$ lie in different clusters of $\set{\tilde S_1,\ldots,\tilde S_r}$.
\item The third ingerdient is a set $\hat {\mathcal{P}}=\set{P(e)\mid e\in \hat E}$ of paths, that cause congestion at most $O(\log^{18} m)$ in $G'$ that we call \emph{nice guiding paths}. For each edge $e=(v,u)\in \hat E$, if we assume that $v\in \tilde S_i$, $u\in \tilde S_j$, and $i<j$, then path $P(e)$ connects vertex $v$ to vertex $u$, does not contain the edge $e$, and consists of three subpaths $P^1(e)$, $P^2(e)$ and $P^3(e)$, that have the following properties:
\begin{itemize}
\item There is an index $i'\leq i$, such that path $P^1(e)$ originates at vertex $v$ and terminates at some vertex $v'\in \tilde S'_{i'}$. Path $P^1(e)$ must be simple, and no vertex of $P^1(e)\setminus\set{v'}$ may lie in $\bigcup_{z=1}^rV(\tilde S_z')$. Moreover, if we denote the sequence of vertices on path $P^1(e)$ by $v=v_0,v_1,\ldots,v_q=v'$, and, for all $0\leq z\leq q$, we assume that $v_z\in \tilde S_{i_z}$, then $i=i_0\geq i_1\geq\cdots\geq i_q=i'$. In other words, the path visits the sets $\tilde S_{a}$ in the non-increasing order of index $a$, possibly skipping over some of the indices.
\item Similarly, there is an index $j'\geq j$, such that path $P^3(e)$ originates at vertex $u$ and terminates at some vertex $u'\in \tilde S'_{j'}$. Path $P^3(e)$ must be simple, and no vertex of $P^3(e)\setminus\set{u'}$ may lie in $\bigcup_{z=1}^rV(\tilde S_z')$. Moreover, if we denote the sequence of vertices on path $P^3(e)$ by $u=u_0,u_1,\ldots,u_{q'}=u'$, and, for all $0\leq z'\leq q'$, we assume that $u_{z'}\in \tilde S_{j_{z'}}$, then $j=j_0\leq j_1\leq\cdots\leq j_{q'}=j'$. In other words, the path visits the sets $\tilde S_{a}$ in the non-decreasing order of index $a$, possibly skipping over some of the indices.
\item Lastly, path $P^2(e)$ connects $v'$ to $u'$. It may only use edges of $\tilde E'$, and it can be partitioned into disjoint subpaths $Q_{i'}(e),Q_{i'+1}(e),\ldots,Q_{j'}(e)$, where for all $i'\leq x\leq j'$, $Q_x(e)\subseteq \tilde S'_x$, and $\bigcup_{i'\leq x\leq j'}V(Q_x(e))=V(P^2(e))$.
\end{itemize}
\end{enumerate}
Note that, by definition, for every edge $e\in \hat E$, the paths $P^1(e)$ and $P^3(e)$ only use edges of $\tilde E''$, while path $P^2(e)$ only uses edges of $\tilde E'$. If $e=(v,u)\in \hat E$ is an edge for which $v\in \tilde S'_i$ holds, then $i'=i$ and $P^1(e)=\set{v}$ must hold. Similarly, if $u\in \tilde S'_j$, then $j'=j$ and $P^2(e)=\set{u}$.
\iffalse
Let $G'$ be a any graph, and let ${\mathcal{C}}'$ be a collection of disjoint vertex-induced subgraphs of $G'$ called clusters. A \emph{nice witness structure} for $G'$ with respect to ${\mathcal{C}}'$ consists of the following three main ingredients:
\begin{enumerate}
\item The first ingredient is a collection ${\mathcal{S}}=\set{S_1,\ldots,S_r}$ of disjoint vertex-induced subgraphs of $G'$, such that $\bigcup_{i=1}^rV(S_i)=V(G')$. We require that, for all $1\leq i\leq r$, there is at most one cluster $C_i\in{\mathcal{C}}'$ with $C_i\subseteq S_i$. Moreover, for all $1\leq i\leq r$, if such cluster $C_i$ exists then $E(S_i)\subseteq E(C_i)\cup E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ must hold, and otherwise $E(S_i)\subseteq E^{\textnormal{\textsf{out}}}({\mathcal{C}})$.
\item The second ingredient is a partition of the edges of $E(G')$ into two disjoint subsets, $E'$ and $E''$, such that for all $C\in{\mathcal{C}}'$, $E(C)\subseteq E'$.
In order to define the third ingerdient, let $\hat E$ denote the set of all edges of $E^{\textnormal{\textsf{out}}}({\mathcal{S}})$, that is, edges whose endpoints lie in different clusters of ${\mathcal{S}}$.
\item The third ingerdient is a set ${\mathcal{P}}=\set{P(e)\mid e\in \hat E}$ of paths, that cause edge-congestion at most $O(\operatorname{poly}\log n)$ \mynote{to be set precisely} that we call \emph{nice guiding paths}. For each edge $e=(u,v)\in \hat E$, if we assume that $u\in S_i$, $v\in S_j$, and $i<j$, then path $P(e)$ connects vertex $u$ to vertex $v$, and consists of three subpaths $P^1(e)$, $P^2(e)$ and $P^3(e)$, that have the following properties:
\begin{itemize}
\item There is an index $i'\leq i$, such that path $P^1(e)$ originates at vertex $v$ and terminates at some vertex $v'\in C_{i'}$. All edges of path $P^1(e)$ must belong to set $E''$. Moreover, if we denote the sequence of vertices on path $P^1(e)$ by $v=v_0,v_1,\ldots,v_q=v'$, and, for all $1\leq z\leq q$, we assume that $v_z\in S_{i_z}$, then $i=i_0\geq i_1\geq\cdots\geq i_q=i'$. In other words, the path visits the sets $S_{a}$ in the decreasing order of index $a$, possibly skipping over some of the indices.
\item Similarly, there is an index $j'\geq j$, such that path $P^3(e)$ originats at vertex $u$ and terminates at some vertex $u'\in C_{j'}$. All edges of path $P^3(e)$ must belong to set $E''$. Moreover, if we denote the sequence of vertices on path $P^3(e)$ by $u=u_0,u_1,\ldots,u_{q'}=u'$, and, for all $1\leq z'\leq q'$, we assume that $v_{z'}\in S_{j_{z'}}$, then $j=j_0\geq j_1\geq\cdots\geq j_{q'}=j'$. In other words, the path visits the sets $S_{a}$ in the increasing order of index $a$, possibly skipping over some of the indices.
\item Lastly, path $P^2(e)$ may only use edges of $E'$, and it can be partitioned into disjoint subpaths $Q_{i'}(e),Q_{i'+1}(e),\ldots,Q_{j'}(e)$, where for all $i'\leq x\leq j'$, all vertices of $Q_x(e)$ lie in $S_x$. Moreover, for all $i'\leq x\leq j'$, there must be a non-empty subpath $Q'_x(e)\subseteq Q_x(e)$ with $Q'_x(e)\subseteq C_x$, and all vertices of $Q_x(e) \setminus Q'_x(e)$ must lie in $V(S_x)\setminus V(C_x)$.
\end{itemize}
\end{enumerate}
\fi
\begin{figure}[h]
\centering
\includegraphics[scale=0.23]{figs/witness_structure.jpg}
\caption{An illustration of a nice witness structure and a nice guiding path. An edge $e\in \hat E$ is shown in red. The prefix $P^1(e)$ and the suffix $P^3(e)$ of the nice guiding path $P(e)$ are shown in green, and the mid-part $P^2(e)$ is shown in blue.}\label{fig: witness_structure}
\end{figure}
Clearly, the edge sets $\tilde E',\tilde E'',\hat E$ in the nice witness structure are completely determined by the sequences $\tilde {\mathcal{S}}=\set{\tilde S_1,\ldots,\tilde S_r}$ and $\tilde {\mathcal{S}}'=\set{\tilde S'_1,\ldots,\tilde S'_r}$ of clusters. Therefore, the nice witness structure is completely determined by $\tilde {\mathcal{S}},\tilde{\mathcal{S}}'$, and the set $\hat {\mathcal{P}}=\set{P(e)\mid e\in \hat E}$ of nice guiding paths. We will use the shorthand $(\tilde {\mathcal{S}},\tilde {\mathcal{S}}',\hat {\mathcal{P}})$ for a nice witness structure. For a path $P(e)\in \hat {\mathcal{P}}$, we sometimes refer to $P^1(e),P^3(e)$ and $P^2(e)$ as the \emph{prefix}, the \emph{suffix}, and the \emph{mid-part} of path $P(e)$, respectively.
This completes the definition of a nice witness structure. Next, we define nice subinstances of instance $I$.
Consider a subinsance $I'=(G',\Sigma')$ of the input instance $I$, and assume that $I'$ is a canonical subinstance of $I$ with respect to the set ${\mathcal{C}}$ of clusters. Recall that, from the definition of canonical subinstances, we are guaranteed that for every cluster $C\in {\mathcal{C}}$, either $C\subseteq G'$, or $V(C)\cap V(G')=\emptyset$. We denote by ${\mathcal{C}}(G')$ the set of all clusters $C\in {\mathcal{C}}$ with $C\subseteq G'$.
Lastly, we say that a subinstance $I'=(G',\Sigma')$ of $I$ is a \emph{nice subinstance} of $I$ with respect to ${\mathcal{C}}$, if it is a canonical subinstance with respect to ${\mathcal{C}}$, and there is a nice witness structure for graph $G'$ with respect to the set ${\mathcal{C}}'={\mathcal{C}}(G')$ of its clusters.
The remainder of the proof of \Cref{thm: disengagement - main} uses the following two theorems. The first theorem allows us to decompose a given instance $I$ into a collection of nice subinstances.
\begin{theorem}\label{thm: advanced disengagement get nice instances}
There is an efficient randomized algorithm, whose input consists of an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, parameters $m$ and $\mu\geq 2^{c^*(\log m)^{7/8}\log\log m}$ for some large enough constant $c^*$, and a collection ${\mathcal{C}}$ of disjoint clusters of $G$, for which the following hold:
\begin{itemize}
\item $|V(G)|,|E(G)|\leq m$, and $m$ is greater than a sufficiently large constant;
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha_0$-bandwidth property, for $\alpha_0=1/\log^3m$;
\item $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$; and
\item $\sum_{C\in {\mathcal{C}}}|\delta_G(C)|\leq |E(G)|/\mu^{0.1}$.
\end{itemize}
The algorithm either returns FAIL, or it computes a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition ${\mathcal{I}}_1$ of instance $I$, such that each resulting instance $I'=(G',\Sigma')\in {\mathcal{I}}_1$ is a nice subinstance of $I$ with respect to ${\mathcal{C}}$. In the latter case, the algorithm also computes, for each instance $(G',\Sigma')\in {\mathcal{I}}_1$, a nice witness structure for graph $G'$ with respect to the set ${\mathcal{C}}(G')$ of clusters. The probability that the algorithm returns FAIL is at most $1/m^6$.
\end{theorem}
The second theorem allows us to further decompose nice subinstances of instance $I$ into subinstances that have the desired properties.
\begin{theorem}\label{thm: advanced disengagement - disengage nice instances}
There is an efficient randomized algorithm, whose input consists of:
\begin{itemize}
\item an instance $I'=(G',\Sigma')$ of \ensuremath{\mathsf{MCNwRS}}\xspace;
\item a parameter $m$, such that $|E(G')|\leq m$;
\item a collection ${\mathcal{C}}'$ of disjont clusters of $G'$; and
\item a nice witness structure $(\tilde {\mathcal{S}},\tilde {\mathcal{S}}',\hat{\mathcal{P}})$ for graph $G'$ with respect to the set ${\mathcal{C}}'$ of clusters.
\end{itemize}
The algorithm either returns FAIL, or computes a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition ${\mathcal{I}}_2(I')$ of instance $I'$, such that each resulting instance $I''=(G'',\Sigma'')\in {\mathcal{I}}_2(I')$ is a subinstance of $I'$, and moreover, there is at most one cluster $C\in {\mathcal{C}}'$ with $E(C)\subseteq E(G'')$; if such a cluster exists then $E(G'')\subseteq E(C)\cup E(G'_{|{\mathcal{C}}'})$ holds, and otherwise $E(G'')\subseteq E(G'_{|{\mathcal{C}}'})$. The probability that the algorithm returns FAIL is $1/m^{6}$.
\end{theorem}
Note that \Cref{thm: disengagement - main} immediately follows from \Cref{thm: advanced disengagement get nice instances} and \Cref{thm: advanced disengagement - disengage nice instances}. Indeed, we start by applying the algorithm from \Cref{thm: advanced disengagement get nice instances} to the input instance $I$ and the collection ${\mathcal{C}}$ of its clusters. Assume for now that the algorithm did not return FAIL. Then we obtain a collection ${\mathcal{I}}_1$ of nice subinstances of $I$, and, for each instance $I'=(G',\Sigma')\in {\mathcal{I}}_1$, a nice witness structure for $G'$ with respect to cluster set ${\mathcal{C}}(G')$.
From the definition of a nice subinstance, for every cluster $C\in {\mathcal{C}}(G')$, $C\subseteq G'$, and for every cluster $C\in {\mathcal{C}}\setminus{\mathcal{C}}(G')$, $V(C)\cap V(G')=\emptyset$, so $E(G')\subseteq \textsf{left} (\bigcup_{C\in {\mathcal{C}}(G')}E(C)\textsf{right} )\cup E^{\textnormal{\textsf{out}}}({\mathcal{C}})$.
We then apply
the algorithm from \Cref{thm: advanced disengagement - disengage nice instances} to each such instance $I'=(G',\Sigma')\in {\mathcal{I}}_1$ and the corresponding nice witness structure. Assume for now that this algorithm did not return FAIL. Then we obtain a collection ${\mathcal{I}}_2(I')$ of subinstances of $I'$. We are guaranteed that, for each resulting instance $I''=(G'',\Sigma'')\in {\mathcal{I}}_2(I')$, there is at most one cluster $C\in {\mathcal{C}}(G')$ with $E(C)\subseteq E(G'')$. If such a cluster exists, then $E(G'')\subseteq E(C)\cup E(G'_{|{\mathcal{C}}(G')})\subseteq E(C)\cup E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ holds, and otherwise $E(G'')\subseteq E(G'_{|{\mathcal{C}}(G')})\subseteq E^{\textnormal{\textsf{out}}}({\mathcal{C}}')$, since $E(G')\subseteq \textsf{left} (\bigcup_{C'\in {\mathcal{C}}(G')}E(C')\textsf{right} )\cup E^{\textnormal{\textsf{out}}}({\mathcal{C}})$.
If the algorithm from \Cref{thm: advanced disengagement get nice instances} did not return FAIL, and neither application of the algorithm from \Cref{thm: advanced disengagement - disengage nice instances} returned FAIL, then we return the collection of instances ${\mathcal{I}}=\bigcup_{I'\in {\mathcal{I}}_1}{\mathcal{I}}_2(I')$. From \Cref{claim: compose algs}, we obtain a randomized algorithm that computes a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition ${\mathcal{I}}$ of the input instance $I$.
It now remains to consider a case where the algorithm from \Cref{thm: advanced disengagement get nice instances} or any of the applications of the algorithm from \Cref{thm: advanced disengagement - disengage nice instances} returned FAIL (which may only happen with probability at most $1/m^4$). In this case, we construct the collection ${\mathcal{I}}$ of subinstances of $I$ directly, as follows. For every cluster $C\in {\mathcal{C}}$, we let ${\mathcal{O}}(C)$ be an arbitrary circular ordering of the edges of $\delta_G(C)$. Set ${\mathcal{I}}$ will contain one global instance $\hat I=(\hat G,\hat \Sigma)$, and, for each cluster $C\in {\mathcal{C}}$, a cluster-based instance $I_C=(G_C,\Sigma_C)$. Consider first a cluster $C\in {\mathcal{C}}$. We let $G_C$ be the graph obtained from $G$ by contracting all vertices of $V(G)\setminus V(C)$ into a supernode $u_C$. We define the rotation system $\Sigma_C$ for graph $G_C$ as follows: for every vertex $v\in V(C)$, its rotation ${\mathcal{O}}_v$ in $\Sigma_C$ remains the same as that in $\Sigma$. Observe that $\delta_{G_C}(u_C)=\delta_G(C)$. The rotation ${\mathcal{O}}_{u_C}$ of vertex $u_C$ in $\Sigma_C$ is defined to be ${\mathcal{O}}(C)$. This completes the definition of the cluster-based instance $I_C=(G_C,\Sigma_C)$. We now define the global instance $\hat I=(\hat G,\hat \Sigma)$. Graph $\hat G$ is obtained from graph $G$ by contracting, for every cluster $C\in {\mathcal{C}}$, the set $V(C)$ of vertices into a supernode $u'_C$. Notice that the set of edges incident to $u'_C$ in $\hat G$ is precisely $\delta_G(C)$. We then define a rotation of $u'_C$ in $\hat \Sigma$ to be ${\mathcal{O}}(C)$. This completes the definition of the global instance $\hat I$. Consider now the resulting collection ${\mathcal{I}}$ of subinstances of $I$. It is immediate to verify that $\sum_{(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq O(|E(G)|)$.
Assume now that we are given, for each instance $I'\in {\mathcal{I}}$, a feasible solution $\phi(I')$. We can combine these solutions together to obtain a solution $\phi$ to instance $I$, of cost at most
$O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$, by employing an algorithm similar to that from \Cref{lem: basic disengagement combining solutions} (the algorithm that was used for basic disengagement). Lastly, from \Cref{thm: crwrs_uncrossing}, it is easy to verify that
$\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\le O(m^2)$. Since the probability that the algorithm from \Cref{thm: advanced disengagement get nice instances}, or any of the applications of the algorithm from \Cref{thm: advanced disengagement - disengage nice instances} return FAIL is at most $1/m^4$, overall we have obtained a randomized algorithm that computes a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition ${\mathcal{I}}$ of the input instance $I$ with required properties.
In order to complete the proof of \Cref{thm: disengagement - main}, it is now enough to prove \Cref{thm: advanced disengagement get nice instances} and \Cref{thm: advanced disengagement - disengage nice instances}, which we do in Sections \ref{sec: advanced disengagement - get nice instances} and \ref{subsec: disengagement with bad chain}, respectively.
\input{advanced-disengagement-get-nice}
\input{disengage_with_routing_new}
\subsubsection{Proof of \Cref{claim: rearrange drawing in disc}}
We fix a face $F\in \tilde{\mathcal{F}}'$. For convenience, we denote graph $\textbf{G}_F$ by $G$, and the corresponding instance $\tilde I_F=(\textbf{G}_F,\tilde \Sigma_F)$ of \ensuremath{\mathsf{MCNwRS}}\xspace by $I=(G,\Sigma)$. We also denote the solution to instance $I$ induced by drawing $\tilde \phi$ of $\tilde G$ by $\phi$, so $|\chi(F)|\geq \mathsf{cr}(\phi)$. Recall that we are given a disc $D(F)$, that we denote by $D$, and a collection $A_F$ of anchor vertices (that we denote by $A$). The images of all vertices of $A$ in $\phi$ lie on the boundary of the disc $D$, and the images of all other vertices of $G$ lie in the interior of the disc. The set of edges of $G$ is partitioned into two subsets: set $E^{\operatorname{bad}}$ of bad edges, and set $E^{\operatorname{good}}$ of all remaining edges, that we refer to as good edges. The images of all good edges in $\phi$ are contained in disc $D$. For every bad edge $e\in E^{\operatorname{bad}}$, the endpoints of $e$ (which must be anchor vertices) lie on the boundary of disc $D$.
Our goal is to show that there exists another solution $\psi$ to instance $I$, in which the images of the anchor vertices remain unchanged from $\phi$, the images of all vertices and edges of $G$ are contained in disc $D$, and $\mathsf{cr}(\psi)\leq \textsf{left} (\mathsf{cr}(\phi)+|E^{\operatorname{bad}}|^2+|E^{\operatorname{bad}}|\cdot \frac{\check m'}{\mu^{2b}}\textsf{right} )\cdot (\log \check m')^{O(1)}$. Recall that, from \Cref{obs: small boundary cuts}, for any partition $(S,T)$ of the vertices of $A$, so that the vertices of $S$ appear consecutively on the boundary of $D$ in $\phi$, there is a set $E'$ of at most $4\check m'/\mu^{2b}$ edges in $G$, so that there is no path connecting a vertex of $S$ to a vertex of $T$ in $G\setminus E'$.
We let $A'\subseteq A$ be the set of vertices that serve as endpoints of the bad edges. Note that the edges of $E^{\operatorname{bad}}$ define a perfect matching between the vertices of $A'$. We then let $\Pi=\set{\phi(v)\mid v\in A'}$ be the set of points that serve as images of the vertices of $A'$. Let $r$ be the smallest integer, so that $|\Pi|\leq 2^r$. Clearly, $2^r\leq 4|E^{\operatorname{bad}}|\leq 4\check m'$, and $r\leq \log(4\check m')$. We add additional arbitrary points on the boundary of the disc $D$ to set $\Pi$ until $\Pi$ contains $2^r+1$ distinct points. We denote $\Pi=\set{p_0,p_1,\ldots,p_{2^r}}$, and we assume that the points appear on the boundary of disc $D$ in this order, as we traverse the boundary in counter-clock-wise direction.
Next, we define a number of guiding curves, that we call \emph{corridors}. We will ensure that all these curves are disjoint, except for possibly sharing their endpoints. The curves are partitioned into $r$ levels.
The set $\Lambda_0$ of level-$0$ curves contains, for all $0\leq i<2^r$, a curve $\lambda_{0,i}$, that connects point $p_i$ to point $p_{i+1}$, and is contained in the interior of disc $D$ (except for its two endpoints that lie on the disc boundary). We ensure that all curves in set $\Lambda_0$ are disjoint from each other. Note that for all $0\leq i<2^r$, points $p_i$ and $p_{i+1}$ partition the boundary of the disc $D$ into two segments. Let $\sigma_{0,i}$ be the segment that is disjoint from point $p_{i+2}$. Then we can define a disc $D_{0,i}\subseteq D$ that corresponds to curve $\lambda_{0,i}$, whose boundary is the concatentation of curves $\sigma_{0,i}$ and $\lambda_{0,i}$.
For a level $0<j< r$, we consider the points in $\set{p_{i\cdot 2^j}\mid 0\leq i\leq 2^{r-j}}$, and we connect every consecutive pair of such points with a curve. Specifically, the set $\Lambda_j$ of level-$j$ curves contains, for all $0\leq i<2^{r-j}$, a curve $\lambda_{j,i}$, that connects point $p_{i\cdot 2^j}$ to point $p_{(i+1)\cdot 2^j}$. We draw these curves so that they are internally disjoint from each other and from the curves in $\Lambda_0\cup\cdots\cup\Lambda_{j-1}$, and every curve is contained in the interior of the disc $D$ (except for its endpoints that lie on the disc's boundary).
As before, for every index
$0\leq i<2^{r-j}$, points $p_{i\cdot 2^j}$, $p_{(i+1)\cdot 2^j}$ partition
the boundary of the disc $D$ into two segments. We denote by $\sigma_{j,i}$ the segment that does not contain the point $p_{(i+1)\cdot 2^j+1}$. We let $D_{j,i}$ be the disc that is contained in $D$, whose boundary is the concatenation of curves $\sigma_{j,i}$ and $\lambda_{j,i}$ (see \Cref{fig: smalldiscs}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.13]{figs/discsD.jpg}
\caption{Level-$j$ curves $\lambda_{j,i}$ and $\lambda_{j,i+1}$, with disc $D_{j,i+1}$ shown in red, and a level-$(j-1)$ curve $\lambda_{j-1,2i}$, with disc $D_{j-1,2_i}$ shown in green
}\label{fig: smalldiscs}
\end{figure}
Lastly, the set $\Lambda_r$ of level-$r$ curves contains a single curve $\lambda_{r,0}$ that connects points $p_0$ and $p_{2^r}$. We ensure that this curve is internally disjoint from all curves in $\Lambda_0\cup\cdots\cup\Lambda_{r-1}$, and is contained in the interior of $D$, except for its endpoints that lie on the boundary of the disc.
We define a disc $D_{r,0}$ associated with this curve exactly like before, so points $p_0,\ldots,p_r$ lie on the boundary of $D_{r,0}$.
We denote by $\Lambda=\bigcup_{j=0}^r\Lambda_j$. Let $G'=G\setminus E^{\operatorname{bad}}$, and let $I'=(G',\Sigma')$ be the subinstance of $I$ that is defined by graph $G'$.
The crux of the proof of \Cref{claim: rearrange drawing in disc} is the following claim, that allows us to ``rearrange'' the image of graph $G'$, so that each guiding curve only crosses a small number of edges.
\begin{claim}\label{claim: avoid guiding curves}
There is a solution $\psi'$ to instance $I'$, with $\mathsf{cr}(\psi')\leq \mathsf{cr}(\phi)\cdot (\log \check m')^{O(1)}$, so that the images of all vertices and edges of $G'$ lie in disc $D$, and the images of the anchor vertices in set $A$ remain the same as in $\phi$. Moreover, for every curve $\lambda\in \Lambda$, for every vertex $v\in V(G')$, the image of $v$ in $\psi'$ does not lie on an inner point of $\lambda$; for every edge $e\in E(G')$, the image of $e$ in $\psi'$ may intersect $\lambda$ in at most one point; and the total number of edges in $E(G')$ whose images intersect $\lambda$ is at most $4\check m'/\mu^{2b}$.
\end{claim}
We provide the proof of \Cref{claim: avoid guiding curves} in Section \ref{subsec: proof of claim avoind guiding curves} of Appendix. Given two points $p_i,p_{i'}\in \Pi$, a \emph{tunnel} connecting $p_i$ to $p_{i'}$ is a sequence $L=(\lambda^1,\ldots,\lambda^z)$ of curves of $\Lambda$, such that the concatenation of the curves in $L$ is a simple curve connecting points $p_i$ and $p_{i'}$. The \emph{length} of the tunnel is $z$ -- the number of curves in the sequence. We also need the following simple observation, whose proof appears in Section \ref{subsec: appx tunnels} of Appendix.
\begin{observation}\label{obs: tunnels}
For every pair $p_i,p_{i'}$ of distinct points of $\Pi$, there is a tunnel of length $O(\log \check m')$ connecting $p_i$ to $p_{i'}$.
\end{observation}
The proof of \Cref{claim: rearrange drawing in disc} easily follows from \Cref{claim: avoid guiding curves} and \Cref{obs: tunnels}. We start with the solution $\psi'$ to instance $I'$, that is given by \Cref{claim: avoid guiding curves}. Recall that $\mathsf{cr}(\psi')\leq \mathsf{cr}(\phi)\cdot (\log \check m')^{O(1)}$, the image of $G'$ is contained in disc $D$, and the images of the anchor vertices in set $A$ in $\psi'$ are identical to those in $\phi$. Next, we consider the bad edges one by one, and insert them into the drawing $\psi'$. Consider any such bad edge $e=(x,y)\in E^{\operatorname{bad}}$. Recall that there are two points $p_i,p_{i'}\in \Pi$, such that $\psi'(x)=\phi(x)=p_{i}$, and $\psi'(y)=\phi(x)=p_{i'}$. From our construction of graph $\textbf{G}$, vertices $x$ and $y$ each have degree $2$ in $G$. Let $L=(\lambda^1,\ldots,\lambda^z)$ be a tunnel connecting $p_i$ to $p_{i'}$, with $z\leq O(\log \check m')$, that is given by \Cref{obs: tunnels}. We let the image of the edge $e$ to be a simple curve that connect $p_i$ to $p_{i'}$, and closely follows the image of the curve $\gamma(L)$, obtained by concatenating all curves in $L$, next to this curve. From \Cref{claim: avoid guiding curves}, this new image of edge $e$ crosses the images of at most $O\textsf{left}(\frac{\check m'\cdot \log \check m'}{\mu^{2b}}\textsf{right} )$ edges of $G'$. We allow the images of the edges of $E^{\operatorname{bad}}$ to cross arbitrarily.
Once all bad edges are processed, we obtain a solution $\psi$ to instance $G$, where the images of all vertices and edges are contained in $D$, and the images of the anchor vertices in $A$ are identical to those in $\phi$. The number of crossings between pairs of edges in $E(G')$ is at most $ \mathsf{cr}(\phi)\cdot (\log \check m')^{O(1)}$; the number of crossings between edges of $E^{\operatorname{bad}}$ and edges of $E(G')$ is at most
$O\textsf{left}(\frac{|E^{\operatorname{bad}}|\cdot \check m'\cdot \log \check m'}{\mu^{2b}}\textsf{right} )$; and the number of crossings between the edges of $E^{\operatorname{bad}}$ may be arbitrary.
As our last step, we perform a type-1 uncrossing of the images of the bad edges (see \Cref{thm: type-1 uncrossing}). This procedure locally modifies the images of the bad edges by swapping segments between pairs of images of these edges (see \Cref{fig:type_1_uncrossing}). At the end of this procedure, we are guaranteed that every pair of edges in $E^{\operatorname{bad}}$ cross at most once, and the number of crossings between the new images of the edges in $E^{\operatorname{bad}}$ and the images of the edges in $E(G')$ does not grow. Therefore, the number of crossings in this final solution to instance $I$ is bounded by $\mathsf{cr}(\phi)\cdot (\log \check m')^{O(1)}+O\textsf{left}(\frac{|E^{\operatorname{bad}}|\cdot \check m'\cdot \log \check m'}{\mu^{2b}}\textsf{right} )+|E^{\operatorname{bad}}|^2$, as required.
\subsubsection{Completing the Proof of \Cref{lem: many paths}}
\label{subsubsec: finish the proof}
Given an input instance $\check I=(\check G,\check \Sigma)$, we first apply the algorithm from \Cref{thm: phase 1} to this input. If the algorithm fails, then we terminate the algorithm and return FAIL as well. Recall that, from \Cref{thm: phase 1}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/\mu^3$. Assume now that the algorithm from \Cref{thm: phase 1} did not fail. In this case, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then with probability at least $(1-1/\mu^3)$, the algorithm is successful. We denote by $\tilde {\cal{E}}_1'$ the bad event that the application of this algorithm is unsuccessful, so $\prob{\tilde {\cal{E}}_1'}\leq 1/\mu^3$. Let ${\mathcal{I}}$ be the collection of subinstances of $\check I$ computed by the algorithm from \Cref{thm: phase 1}.
Throughout this subsection, we use the universal constant $b^*=2400$.
Recall that, if Event $\tilde {\cal{E}}'_1$ did not happen, then for every instance $I\in {\mathcal{I}}$, there is a solution $\psi(I)$ to $I$, that is semi-clean with respect to ${\mathcal K}(I)$, such that
$\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. We set the parameter $b$ to be a large enough constant, so that $b\geq 4b^*$ holds.
Recall that, for every instance $I=(G,\Sigma)\in {\mathcal{I}}$, we have denoted by $\hat I=(\hat G,\hat \Sigma)$ the corresponding ${\mathcal K}(I)$-contracted instance, and by $\hat m(I)=|E(\hat G)|$. We say that instance $I$ is \emph{small} if $\hat m(I)\leq \frac{\check m}{\mu^{b^*}}$, and otherwise it is \emph{large}.
We now proceed in $z= 2\log m$ phases, where in the $j$th phase, we compute the $j$th collection ${\mathcal{I}}^{(j)}$ of subinstances of $I$, such that the properties required in \Cref{lem: many paths} are satisfied.
Consider now an index $1\le j\le 2\log m$ and we describe the $j$th phase. Throughout the phase, we use the parameters $g^j_1=2^j$ and $g^j_2=g^j_1\cdot \mu^{b^*}/\check m$.
We intend to apply the algorithm from \Cref{thm: phase 2} to large instances in ${\mathcal{I}}$ with parameters $g^j_1,g^j_2$. But note that the algorithm requires that the input instance $I=(G,\Sigma)$ has a skeleton $K(I)$ that is $g_3(I)$-connected in the input graph $G$ (where $g_3(I)=\big(\frac{g_1^j}{|E(G)|}+g_2^j\big)\cdot\mu^{300b}$), which is not necessarily true for the large instances in ${\mathcal{I}}$. Therefore, we first pre-process instances in ${\mathcal{I}}$, such that the resulting instances are either small or satisfying this connectedness property, as follows.
\newcommand{{\mathcal{I}}_{\sf temp}}{{\mathcal{I}}_{\sf temp}}
\newcommand{\hat{{\mathcal{I}}}_{\sf temp}}{\hat{{\mathcal{I}}}_{\sf temp}}
\subsubsection*{Pre-processing instances in ${\mathcal{I}}$}
The pre-processing procedure is iterative. Throughout, we use a parameter $\tilde g=\big(\frac{g^j_1\mu^{b^*}}{\check m}+g^j_2\big)\cdot\mu^{300b}$.
The procedure maintains a collection ${\mathcal{I}}_{\sf temp}$ of subinstances of $I$, that is initialized to be ${\mathcal{I}}_{\sf temp}={\mathcal{I}}$. We denote $\hat{{\mathcal{I}}}_{\sf temp}=\set{\hat I\mid I\in {\mathcal{I}}_{\sf temp}}$.
Each instance in ${\mathcal{I}}_{\sf temp}$ is either marked \emph{active} or \emph{inactive}. Initially, all large instances are marked active and all small instances are marked inactive.
We will maintain the following invariant throughout the algorithm.
\begin{properties}{IP}
\item for each inactive instance $I=(G,\Sigma)\in {\mathcal{I}}_{\sf temp}$, either $I$ is small, or $I$ satisfies that, if we denote by $K(I)$ the skeleton of $I$, then $K(I)$ is $\tilde g$-connected in $G$;
\label{prop: inactive}
\item for each instance $\hat I=(\hat G,\hat \Sigma)\in \hat{{\mathcal{I}}}_{\sf temp}$, either $|E(\hat G)|\le \check m/\mu$, or $\hat I$ is not wide;
\label{prop: wide or not}
\item $\sum_{I\in {\mathcal{I}}_{\sf temp}}\hat m(I)\le \check m$; and
\label{prop: number of edges}
\item there exists, for every instance $I\in {\mathcal{I}}_{\sf temp}$, a solution $\psi(I)$ to $I$ that is semi-clean with respect to ${\mathcal K}(I)$, such that
$\sum_{I\in {\mathcal{I}}_{\sf temp}}\mathsf{cr}(\psi(I))\le 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}_{\sf temp}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$.
\label{prop: good drawings}
\end{properties}
From \Cref{thm: phase 1} and our initial setting, these invariant holds at the beginning of the pre-processing procedure.
The procedure continues to be executed as long as the collection ${\mathcal{I}}_{\sf temp}$ contains an active instance.
We now describe an iteration of the procedure.
Let $I=(G,\Sigma)$ be an active instance in ${\mathcal{I}}_{\sf temp}$. If instance $I$ is small, then we mark it inactive and terminate the iteration. Clearly all invariants continue to hold after this iteration in this case.
Assume now that instance $I$ is large.
Let ${\mathcal K}(I)=\set{{\mathcal{J}}_1,\ldots,{\mathcal{J}}_{r}}$ be the skeleton structure for $I$, and for each $1\le i\le r$, let $J_i$ be the core associated with ${\mathcal{J}}_i$.
Consider its ${\mathcal K}(I)$-contracted instance $\hat I=(\hat G, \hat \Sigma)$. Recall that graph $\hat G$ is obtained from graph $G$ by contracting, for each $1\le i\le r$, all vertices of the core $J_i$ into a supernode $v_{J_i}$. We now compute, for every pair $1\le i<i'\le r$, a minimum $v_{J_i}$-$v_{J_{i'}}$ cut in $\hat G$.
If for all pairs $1\le i<i'\le r$, the $v_{J_i}$-$v_{J_{i'}}$ minimum-cut in $\hat G$ has value at least $\tilde g$, then we simply mark the instance $I$ in ${\mathcal{I}}$ inactive, and terminate this iteration.
Clearly all invariants continue to hold after this iteration in this case.
Assume now that there exists a pair $1\le i<i'\le r$, such that the $v_{J_i}$-$v_{J_{i'}}$ minimum-cut in $\hat G$ has value less than $\tilde g$.
Let $\hat G_1,\ldots,\hat G_s$ be the graphs obtained from $\hat G$ by deleting the edge of the minimum-cut. Consider now an $1\le i\le s$. Assume that supernodes corresponding to cores $J_{p_1},\ldots, J_{p_t}$ belong to graph $\hat G_i$.
We define instance $I_i=(G_i,\Sigma_i)$ as follows. Graph $G_i$ is the graph obtained from $\hat G_i$ by un-contracting every supernode $v_{J_{p_q}}$ back to its corresponding core $J_{p_q}$, so $G_i$ is a subgraph of $G$, and $\Sigma_i$ is the rotation system on $G_i$ that is induced by $\Sigma$. We then define its core structure to be ${\mathcal K}(I_i)=\set{{\mathcal{J}}_{p_1},\ldots,{\mathcal{J}}_{p_q}}$.
We then replace the instance $I$ in ${\mathcal{I}}_{\sf temp}$ with new instances $I_1,\ldots,I_s$, all marked active, and terminate this iteration.
We now verify that all invariants continue to hold after this iteration in this case.
First, since we marked all new instance that are added to collection ${\mathcal{I}}_{\sf temp}$ active, Invariant \ref{prop: inactive} holds.
Second, since for every new instance $I_i=(G_1,\Sigma_i)$, graph $G_i$ is a subgraph of $G$, and $\hat G_i$ is a subgraph of $\hat G$. If $|E(\hat G)|\le \check m/\mu$, then $|E(\hat G_i)|\le \check m/\mu$; if $\hat I$ is node wide, then $\hat I_i$ is also not wide. Therefore,
\ref{prop: wide or not} holds.
Third, since $\hat G_1\cup\ldots\cup\hat G_s\subseteq \hat G$, $\sum_{1\le i\le s}\hat m(I_i)\le \hat m(I)$ and so
\ref{prop: number of edges} holds.
Last, note that the Invariant \ref{prop: good drawings} holds before this iteration. Let $\psi(I)$ be the semi-clean solution associated with instance $I$. We simply define, for each $1\le i\le s$, $\psi(I_i)$ to be the drawing of $G_i$ induced by drawing $\psi(I)$. It is easy to verify that $\psi(I_i)$ is a semi-clean solution to $I_i$ with respect to its skeleton structure. Moreover, since $G_1,\ldots,G_s$ are disjoint subgraphs of $G$,
$\sum_{1\le i\le s}\mathsf{cr}(\psi(I_i))\le \mathsf{cr}(\psi(I))$, and $\sum_{1\le i\le s}|\chi^{\mathsf{dirty}}(\psi(I_i))|\leq |\chi^{\mathsf{dirty}}(\psi(I))|$.
Therefore, \ref{prop: good drawings} also holds.
Note that the collection of all cores in all skeleton structures is never changed, and initially, the number of cores in all active instances in ${\mathcal{I}}$ is at most $\mu^{b^*}\cdot \mu^{100}$ (since the number of active instances is at most $\mu^{b^*}$ and each such instances has at most $\mu^{100}$ cores). When the pre-processing process terminates, every resulting instance contains at least one core.
Therefore, the number of iterations is at most $\mu^{b^*}\cdot \mu^{100}$. Note that in each iteration we have deleted at most $\tilde g$ edges. Therefore, if we denote by $\hat E_{\sf temp}$ the set of all deleted edges throughout this procedure, then
$|\hat E_{\sf temp}|\le \mu^{b^*}\cdot \mu^{100}\cdot \tilde g$.
Let ${\mathcal{I}}^*$ be the resulting collection we obtained at the end of the pre-processing procedure. Since all instances in ${\mathcal{I}}^*$ are inactive, from Invariant \ref{prop: inactive}, for each instance $I=(G,\Sigma)\in {\mathcal{I}}^*$, either $I$ is small, or its skeleton $K(I)$ is $\tilde g$-connected in $G$.
Moreover, it is easy to verify that, for each instance $I\in {\mathcal{I}}^*$, its skeleton structure has size at most $\mu^{100}$.
Denote $\hat {\mathcal{I}}^*=\set{\hat I\mid I\in {\mathcal{I}}^*}$. We prove the following observation.
\begin{observation}
\label{obs: algo combine drawings}
There is an efficient algorithm, that, given a solution $\phi(\hat I)$ to each instance $\hat I\in \hat {\mathcal{I}}^*$, computes a solution $\check \phi$ to instance $\check I$, with $\mathsf{cr}(\check \phi)\le \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{5000}+g_1^j\cdot\mu^{3b^*+300b}$.
\end{observation}
\begin{proof}
It would be convenient for us to associate a (rooted) partitioning tree $\tau$ for the pre-processing procedure, whose node correspond to all instance that every appeared in the collection ${\mathcal{I}}_{\sf temp}$, as follows.
Initially, $\tau$ contains a root denoted by $v^*$, and, for each instance $I\in {\mathcal{I}}$, a node $v(I)$ representing instance $I$.
Consider now an iteration that some instance $I$ is processed and eventually replaced by new instances $I_1,\ldots,I_s$. We add to $\tau$ nodes $v(I_1),\ldots,v(I_s)$ to them, that become the child nodes of $v(I)$ in $\tau$. We continue until the end of the algorithm. This completes the construction of tree $\tau$. It is easy to see that the instances in the resulting collection ${\mathcal{I}}^*$ correspond to leaf nodes in $\tau$.
We now describe the algorithm for computing a solution $\check \phi$ to instance $\check I$. Assume now we are given, for each instance $I\in {\mathcal{I}}^*$ that correspond to some leaf node in $\tau$, a solution $\phi(\hat I)$ to instance $\hat I$.
We iteratively process non-root nodes of $\tau$ in a bottom-up fashion, as follows.
In each iteration, we take a node $v(I)$ corresponding to instance $I$, such that all its child nodes $v(I_1),\ldots,v(I_s)$ are either leaf nodes or have been processed.
Consider the iteration of the pre-processing procedure in which some instance $I$ is processed and eventually replaced by new instances $I_1,\ldots,I_s$. We denote by $\hat I=(\hat G,\hat \Sigma), \hat I_1=(\hat G_1,\hat \Sigma_1),\ldots,\hat I_s=(\hat G_s,\hat \Sigma_s)$ their contracted instances, respectively.
From the algorithm, graph $\hat G_1\cup \ldots\cup \hat G_s$ are the connected components obtained from $\hat G$ by deleting a set $\hat E$ of edges. We first obtain a drawing of $\hat G\setminus \hat E$ by placing drawings $\phi(\hat I_1),\ldots,\phi(\hat I_s)$ in mutually disjoint discs in the plane.
We then use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $\hat E$ into the drawing of $\hat G\setminus \hat E$, obtaining a solution $\phi(\hat I)$ to instance $\hat I$, such that $\mathsf{cr}(\phi(\hat I))\le \sum_{1\le i\le s}\mathsf{cr}(\phi(\hat I_s))+|E(\hat G)|\cdot |\hat E|$.
In this way we process all non-root nodes, such that at the end, we obtained, for each instance $I\in {\mathcal{I}}$, a solution $\phi(\hat I)$ to its contracted instance $\hat I$, with $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\phi(\hat I))\le \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I))+\check m \cdot \hat E_{\sf temp}$.
Finally, we apply the algorithm from \Cref{thm: phase 1} to the solutions $\phi(\hat I)$ for instances $I\in{\mathcal{I}}$ and obtain a solution $\check\phi$ for $\check I$ with $\mathsf{cr}(\check \phi)\le \sum_{\hat I\in \hat {\mathcal{I}}}\mathsf{cr}(\phi(\hat I))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{5000}$.
Since $|\hat E_{\sf temp}|\le \mu^{b^*}\cdot \mu^{100}\cdot \tilde g$, altogether we get that
$$\mathsf{cr}(\check \phi)
\le \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{5000}+\check m\cdot\mu^{b^*}\cdot \mu^{100}\cdot \tilde g
\le \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{5000}+g_1^j\cdot\mu^{3b^*+300b},$$
as $b^*>100$.
\end{proof}
\subsubsection*{The algorithm for comuting the collection ${\mathcal{I}}^{(j)}$}
We now partition the resulting collection ${\mathcal{I}}^*$ into two subsets: set ${\mathcal{I}}^*_1$ contains all small instances in ${\mathcal{I}}^*$; and set ${\mathcal{I}}^*_2$ contains all remaining instances.
We denote $\hat {\mathcal{I}}_1=\set{\hat I\mid I\in
{\mathcal{I}}^*_1}$.
Since every instance in ${\mathcal{I}}^*_2$ is large (that is, $\hat m(I)\ge \check m/\mu^{b^*}$), from Invariant \ref{prop: number of edges}, we get that $|{\mathcal{I}}^*_2|\le \mu^{b^*}$.
From the pre-processing step, for each instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$, its skeleton $K(I)$ is $\tilde g$-connected in $G$. Since $|E(G)|\ge \check m/\mu^{b^*}$, $\tilde g\ge \textsf{left}(
\frac{g_1^j}{|E(G)|}+g_2^j \textsf{right})$, so its skeleton $K(I)$ is $g_3(I)$-connected in $G$, for $g_3(I)=\textsf{left}(
\frac{g_1^j}{|E(G)|}+g_2^j \textsf{right})$.
For each instance $I\in {\mathcal{I}}^*$, we denote by $\psi(I)$ the solution to $I$ guaranteed by Invariant \ref{prop: good drawings}.
We use the following observation.
\begin{observation}
\label{obs: not too large}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\le \check m^2/\mu^{61b-2b^*}$, then for each instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$, the drawing $\psi(I)$ satisfies that $\mathsf{cr}(\psi(I))\le |E(G)|^2/\mu^{60b}$ and $|\chi^{\mathsf{dirty}}(\psi(I))|\le |E(G)|/\mu^{60b}$.
\end{observation}
\begin{proof}
Let $I=(G,\Sigma)$ be an instance in ${\mathcal{I}}^*_2$, so $|E(G)|\ge \check m/\mu^{b^*}$.
Recall that, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\le 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, so $\mathsf{cr}(\psi(I))\le 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$. Therefore, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\le \check m^2/\mu^{61b-2b^*}$, then
\[
\mathsf{cr}(\psi(I))\le 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I) \le \frac{2\cdot \check m^2}{\mu^{61b-2b^*}}\le
\frac{ \check m^2}{\mu^{60b-2b^*}}
= \frac{ (\check m/\mu^{b^*})^2}{\mu^{60b}}
\le \frac{|E(G)|^2}{\mu^{60b}}.
\]
Also recall that, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\le \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, so $|\chi^{\mathsf{dirty}}(\psi(I))| < \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. Therefore,
\[
|\chi^{\mathsf{dirty}}(\psi(I))|\le \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}
\le \frac{\check m\cdot \mu^{b^*}}{\mu^{61b-2b^*}}
\le \frac{\check m}{\mu^{60b+b^*}}
= \frac{(\check m/\mu^{b^*})}{\mu^{60b}}
\le \frac{|E(G)|}{\mu^{60b}},
\]
since $b\ge 4b^*$.
\end{proof}
Finally, we consider every instance $I\in {\mathcal{I}}^*_2$ one by one.
For each such instance $I$, we apply the algorithm from \Cref{thm: phase 2} to instance $I$, skeleton structure ${\mathcal K}(I)$, the parameters $b,g^j_1,g^j_2$ defined above.
We denote by $I'=(G',\Sigma')$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace, and by $\hat I'$ the ${\mathcal K}(I)$-contraction of this instance. We then denote $\hat {\mathcal{I}}'_2=\set{\hat I'\mid I\in {\mathcal{I}}^*_2}$. The final output of our algorithm is the collection ${\mathcal{I}}^{(j)}=\hat{\mathcal{I}}_1\cup\hat {\mathcal{I}}'_2$ of subinstances of $\check I$.
\subsubsection*{The analysis of the algorithm}
We now verify that the output collections ${\mathcal{I}}^{(1)},\ldots,{\mathcal{I}}^{(z)}$ of subinstances of $\check I$ have all required properties.
Consider an index $1\le j\le z$ and consider the algorithm for constructing collection ${\mathcal{I}}^{(j)}$.
First, from Invariant \ref{prop: wide or not}, for every instance $I\in {\mathcal{I}}^*$, if the corresponding contracted instance $\hat I=(\hat G,\hat \Sigma)$ is a wide instance, then $|E(\hat G)|\le \check m/\mu$.
If instance $I$ lies in ${\mathcal{I}}^*_2$, and $\hat I'=(\hat G',\hat\Sigma')$ is the ${\mathcal K}(I)$-contraction of instance $I'$, then $|E(\hat G')|\leq |E(\hat G)|$, and, if $\hat I$ is not a wide instance, then neither is $\hat I'$. This is since graph $\hat G'$ can be obtained from graph $\hat G$ by deleting the edges of $E^{\mathsf{del}}(I)$ from it. Therefore, we are guaranteed that, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}^{(j)}$, if $I'$ is a wide instance, then $|E(G')|\le \check m/\mu$.
Second, from Invariant \ref{prop: number of edges},
$\sum_{I\in {\mathcal{I}}^*}\hm(I)\le \check m$.
Since, for every instance $I\in {\mathcal{I}}^*_2$, the corresponding resulting instance $\hat I'=(\hat G',\Sigma')\in \hat {\mathcal{I}}_2'$ has $|E(\hat G')|\leq \hat m(I)$, we get that $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}^{(j)}}|E(G')|\le \check m$.
Third, we construct an algorithm that, given a solution to each instance in ${\mathcal{I}}^{(j)}$, computes a solution $\phi$ to instance $\check I$.
For each instance $I\in {\mathcal{I}}^*_2$, we let $\hat I=(\hat G,\hat \Sigma)$ be the corresponding ${\mathcal K}(I)$-contracted instance, and let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal K}(I)$-contracted instance corresponding to the instance $I'$. Note that $V(\hat G)=V(\hat G')$ and $E(\hat G')=E(\hat G)\setminus E^{\mathsf{del}}(I)$. We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{del}}(I)$ into the solution $\phi(\hat I')$ to instance $\hat I'$, obtaining a solution $\phi(\hat I)$ to instance $\hat I$.
Then we use the algorithm from \Cref{obs: algo combine drawings} to compute a drawing of $\check I$. It is clear that the algorithm is efficient.
From now on, we let $j^*$ be the unique integer that $2^{j^*-1}< 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\le 2^{j^*}$. Since $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\le \check m^2$, $j^*\le 2\log m$, so $1\le j^*\le z$. We will show that the collection ${\mathcal{I}}^{(j^*)}$ satisfies the property required in the last bullet of \Cref{lem: many paths}. Consider now the algorithm for constructing the collection ${\mathcal{I}}^{(j^*)}$.
Recall that we have set $g_1^{j^*}=2^{j^*}$ and $g_2^{j^*}=g_1^{j^*}\cdot \mu^{b^*}/\check m$.
Therefore, from Invariant \ref{prop: good drawings}, the semi-clean solutions $\psi(I)$ to $I$ for instances $I\in {\mathcal{I}}^*$ satisfy that
$\sum_{I\in {\mathcal{I}}_{\sf temp}}\mathsf{cr}(\psi(I))\le g_1^{j^*}$ and $\sum_{I\in {\mathcal{I}}_{\sf temp}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq g_2^{j^*}$.
So for each instance $I\in {\mathcal{I}}^*$, $\mathsf{cr}(\psi(I))\le g_1^{j^*}$ and $|\chi^{\mathsf{dirty}}(\psi(I))| \leq g_2^{j^*}$.
Combined with \Cref{obs: not too large}, we get that, for each instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$, both $\mathsf{cr}(\psi(I))\le \min\set{g_1^*,|E(G)|^2/\mu^{60b}}$ and $|\chi^{\mathsf{dirty}}(\psi(I))|\le \min\set{g_2^*, |E(G)|/\mu^{60b}}$ hold.
(We have used the fact that \fbox{$c'>61b-2b^*$}.)
For each instance $I\in {\mathcal{I}}^*_2$, we denote by $\tilde {\cal{E}}_2(I)$ the bad event that the application of the algorithm from \Cref{thm: phase 2} was unsuccessful, so $\prob{\tilde {\cal{E}}_2(I)}\leq 1/\mu^{2b}$.
Let $\tilde {\cal{E}}_2'$ be the bad event that any of the events in $\set{\tilde {\cal{E}}_2(I)\mid I\in {\mathcal{I}}^*_2}$ happened. Since, from the above discussion, $|{\mathcal{I}}^*_2| \leq \mu^{b^*}$, and, for every instance $I\in {\mathcal{I}}^*_2$, $\prob{\tilde {\cal{E}}_2(I)}\leq 1/\mu^{2b}$, while $b>2b^*$, from the Union Bound, we get that $\prob{\tilde {\cal{E}}_2'}<1/\mu^4$. Lastly, we let $\tilde {\cal{E}}'$ be the bad event that either $\tilde {\cal{E}}_1'$ or $\tilde {\cal{E}}_2'$ happened. Clearly, $\prob{\tilde {\cal{E}}'}\leq \prob{\tilde {\cal{E}}_1'}+\prob{\tilde {\cal{E}}_2'}\leq 1/\mu^2$.
We use the following two observations in order to complete the proof of \Cref{lem: many paths}.
\begin{observation}\label{obs: composing contracted solutions}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and that event $\tilde {\cal{E}}'$ did not happen. Then there is an efficient algorithm, that, given a solution $\phi(I')$ for every instance $I'\in {\mathcal{I}}^{(j^*)}$, computes a solution $\check \phi$ to instance $\check I$, with $\mathsf{cr}(\check\phi)\leq \sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{cr}(\phi(I')) + \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}$.
\end{observation}
\begin{proof}
We assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, Event $\tilde {\cal{E}}'$ did not happen, and that we are given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}^{(j^*)}$. We now show an efficient algorithm to compute a solution $\check \phi$ to instance $\check I$. In order to do so, we first consider every instance $I\in {\mathcal{I}}^*_2$ one by one, and compute a solution $\phi(\hat I)$ to instance $\hat I$, from the solution $\phi(\hat I')$ to instance $\hat I'$.
Consider now some instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$. Let $\hat I=(\hat G,\hat \Sigma)$ be the corresponding ${\mathcal K}(I)$-contracted instance, and let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal K}(I)$-contracted instance corresponding to the instance $I'$. Note that $V(\hat G)=V(\hat G')$ and $E(\hat G')=E(\hat G)\setminus E^{\mathsf{del}}(I)$. We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{del}}(I)$ into the solution $\phi(\hat I')$ to instance $\hat I'$, obtaining a solution $\phi(\hat I)$ to instance $\hat I$, whose cost is at most $\mathsf{cr}(\phi(\hat I'))+|E^{\mathsf{del}}(I)|\cdot |E(G)|$. If Event $\tilde {\cal{E}}'$ does not happen, then $|E^{\mathsf{del}}(I)|\leq \big(\frac{g_1^{j^*}}{|E(G)|}+g_2^{j^*}\big)\cdot\mu^{O(1)}=\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}}{|E(G)|}$, and so $\mathsf{cr}(\phi(\hat I))\leq \mathsf{cr}(\phi(\hat I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{O(1)}$.
Now using the algorithm from \Cref{obs: algo combine drawings},
we obtain a solution
$\check\phi$ to instance $\check I$, whose cost is bounded by:
\[
\begin{split}
\mathsf{cr}(\check \phi)&\leq \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I)) + \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}\\
&\leq \sum_{I\in {\mathcal{I}}^*_1}\mathsf{cr}(\phi(\hat I))+\sum_{I\in {\mathcal{I}}^*_2}\textsf{left} (\mathsf{cr}(\phi(\hat I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{O(1)}\textsf{right} )+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}\\
&\le \sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{cr}(\phi(I'))+|{\mathcal{I}}^*_2|\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}\\
&\le \sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{cr}(\phi(I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)},
\end{split}
\]
as $|{\mathcal{I}}^*_2|=\mu^{O(1)}$.
\end{proof}
\begin{observation}\label{obs: cheap solutions to final instances}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and that event $\tilde {\cal{E}}'$ did not happen. Then $\sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log \check m)^{O(1)}$.
\end{observation}
\begin{proof}
We bound $\sum_{I\in {\mathcal{I}}^*_1}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)$ and $\sum_{I\in {\mathcal{I}}^*_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')$ separately.
From \Cref{thm: phase 2}, if Event $\tilde {\cal{E}}'$ did not happen, then, for every instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$,
there is a solution $\psi(I')$ to the corresponding instance $I'$, that is clean with respect to ${\mathcal K}(I)$, with $\mathsf{cr}(\psi(I'))\leq \mathsf{cr}(\psi(I))\cdot (\log \check m)^{O(1)}+|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(G)|}{\mu^b}$.
From \Cref{obs: clean solution to contracted}, there is a solution to the corresponding contracted instance $\hat I'$, of cost at most $\mathsf{cr}(\psi(I'))$. Altogether, we get that:
%
\[\begin{split}
\sum_{I\in {\mathcal{I}}^*_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')&\leq \sum_{I\in {\mathcal{I}}^*_2}\textsf{left}(\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}^*_2}|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(G)|}{\mu^b}\textsf{right} )\cdot (\log \check m)^{O(1)}.\\
&\leq \sum_{I\in {\mathcal{I}}^*_2}\mathsf{cr}(\psi(I))\cdot (\log \check m)^{O(1)}+\textsf{left} (\sum_{I\in {\mathcal{I}}^*_2}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} )^2\cdot (\log \check m)^{O(1)}\\
&\hspace{4cm}+ \frac{\check m\cdot (\log \check m)^{O(1)}}{\mu^b}\cdot\textsf{left}(\sum_{I\in {\mathcal{I}}^*_2}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} ).
\end{split}
\]
%
From Invariant \ref{prop: good drawings}, $\sum_{I\in {\mathcal{I}}^*}\mathsf{cr}(\psi(I))\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$ and $\sum_{I\in {\mathcal{I}}^*}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. Additionally, since we have assumed that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$ for a large enough constant $c'>2b^*$,
$$\textsf{left} (\sum_{I\in {\mathcal{I}}^*_2}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} )^2\leq \frac{(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))^2 \mu^{2b^*}}{\check m^2}\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I).$$ Altogether, we get that:
%
\[
\begin{split}
\sum_{I\in {\mathcal{I}}^*_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot (\log \check m)^{O(1)}+\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}\cdot (\log \check m)^{O(1)}}{\mu^b}\\
&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot (\log \check m)^{O(1)}, \end{split}\]
since $b>4b^*$.
Next, we bound $\sum_{I\in {\mathcal{I}}^*_1}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)$. Consider some instance $I=(G,\Sigma)\in {\mathcal{I}}^*_1$ and the solution $\psi(I)$ that is semi-clean with respect to ${\mathcal K}(I)$. Let $K(I)$ be the skeleton associated with the skeleton structure ${\mathcal K}(I)$. Let $E^{\mathsf{dirty}}(I)\subseteq E(G)\setminus E(K(I))$ be the set of all edges $e$, such that the image of $e$ in $\psi(I)$ crosses the image of some edge of $K(I)$. Let $\hat I=(\hat G,\hat \Sigma)$ be the ${\mathcal K}(I)$-contracted instance corresponding to instance $I$.
Denote $G'=G\setminus E^{\mathsf{dirty}}(I)$, and let $\Sigma'$ be the rotation system for graph $G'$ that is induced by $\Sigma$. Observe that ${\mathcal K}(I)$ is a valid skeleton structure for the resulting instance $I'=(G',\Sigma')$. Let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal K}(I)$-contracted instance associated with $I'$. Observe that the drawing $\psi(I)$ of instance $I$ induces a drawing $\psi(I')$ of instance $I'$ that is clean with respect to ${\mathcal K}(I)$. From \Cref{obs: clean solution to contracted}, there is a solution $\psi(\hat I')$ to the contracted instance $\hat I'$ with $\mathsf{cr}(\psi(\hat I'))\leq \mathsf{cr}(\psi(I'))\leq \mathsf{cr}(\psi(I))$.
We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{dirty}}(I)$ into the drawing $\psi(\hat I')$ to obtain a drawing $\psi(\hat I)$ of instance $\hat I$, with the number of crossings bounded by $\mathsf{cr}(\psi(\hat I'))+|E^{\mathsf{dirty}}(I)|\cdot |E(\hat G)|\leq \mathsf{cr}(\psi(I))+|E^{\mathsf{dirty}}(I)|\cdot |E(\hat G)|$.
Recall that $|E(\hat G)|\leq \frac{\check m}{\mu^{b^*}}$ (since instance $I$ is small), so $\mathsf{cr}(\psi(\hat I))
\leq \mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(I)|\cdot \frac{\check m}{\mu^{b^*}}$.
We then get that:
\[\sum_{I\in {\mathcal{I}}^*_1} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq
\sum_{I\in {\mathcal{I}}^*_1}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}^*_1}|\chi^{\mathsf{dirty}}(I)|\cdot \frac{\check m}{\mu^{b^*}}.
\]
From Invariant \ref{prop: good drawings}, $\sum_{I\in {\mathcal{I}}^*}\mathsf{cr}(\psi(I))\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$ and $\sum_{I\in {\mathcal{I}}^*}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$.
Therefore,
$$\sum_{I\in {\mathcal{I}}^*_1} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\frac{\check m}{\mu^{b^*}} \cdot \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)).$$
\iffalse{analysis for previous iset_2, which should be an empty set}
Assume now that $I\in {\mathcal{I}}_2$. We partition the set ${\mathcal{I}}_2$ of instances into two subsets: set ${\mathcal{I}}_2'$ containing all instances $I=(G,\Sigma)$ with $\mathsf{cr}(\psi(I))> |E(G)|^2/\mu^{60b}$, and set ${\mathcal{I}}_2''$ containing all remaining instances.
As before, for every instance $I=(G,\Sigma)\in {\mathcal{I}}_2$, we denote by $\hat m(I)$ the number of edges in the contracted graph $\hat I$.
Next, we bound $\sum_{I\in {\mathcal{I}}_2'}\mathsf{cr}(\psi(\hat I'))$. Recall that for every instace $I=(G,\Sigma)\in {\mathcal{I}}_2'$:
\[\mathsf{cr}(\psi(I))> \frac{|E(G)|^2}{\mu^{60b}}\geq \frac{(\hat m(I))^2}{\mu^{60b}}\geq \frac{\hat m(I)\cdot \check m}{\mu^{60b+b^*}},\]
since instance $I$ must be large. Since,
from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq 2\check m^2/\mu^{c'}$, we get that $\sum_{I\in {\mathcal{I}}_2'}\hat m(I)\leq \frac{2\check m}{\mu^{c'-60b-b^*}}$. Lastly, since, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, we get that:
\[\begin{split}
\sum_{I\in {\mathcal{I}}_2'} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)&\leq
\sum_{I\in {\mathcal{I}}_2'}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}_2'}|\chi^{\mathsf{dirty}}(I)|\cdot \hm(I)\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\textsf{left} ( \sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))| \textsf{right} )\cdot \textsf{left} ( \sum_{I\in {\mathcal{I}}_2'}\hat m(I) \textsf{right} )\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\frac{2\check m}{\mu^{c'-60b-b^*}}\cdot \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\\
&\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)).\end{split}
\]
It now remains to consider instances $I\in {\mathcal{I}}_2''$. Recall that, for each such instance $I=(G,\Sigma)$, $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{2b}>\hat m(I)/\mu^{60b}$.
Since, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, we get that $\sum_{I\in {\mathcal{I}}_2''}\hat m(I)<\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*+60b}}{\check m}$.
Therefore:
\[\begin{split}
\sum_{I\in {\mathcal{I}}_2''} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)&\leq
\sum_{I\in {\mathcal{I}}_2''}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}_2''}|\chi^{\mathsf{dirty}}(I)|\cdot \hm(I)\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\textsf{left} ( \sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))| \textsf{right} )\cdot \textsf{left} ( \sum_{I\in {\mathcal{I}}_2''}\hat m(I) \textsf{right} )\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I) + \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*+60b}}{\check m}\cdot \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\\
&\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))
,\end{split}
\]
%
since $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, and $c'$ is a large enough constant.
\fi
Altogether, we get that
$$\sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')=\sum_{I\in {\mathcal{I}}^*_1}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)+\sum_{I\in {\mathcal{I}}^*_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log \check m)^{O(1)}.$$
\end{proof}
\iffalse
The calculations
---------------------------------------------------
\begin{itemize}
\item $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$
\item $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$
\item in ${\mathcal{I}}_2$: either $\mathsf{cr}(\psi(I))> |E(G)|/\mu^{2b}$, or $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{2b}$.
\item when pulling edges back, we will pay $\sum_{I\in {\mathcal{I}}_2}\hm(I)\cdot |\chi^{\mathsf{dirty}}(\psi(I))|$.
\item let's say that ${\mathcal{I}}_2'$ contains all instances with $\mathsf{cr}(\psi(I))> |E(G)|/\mu^{2b}>\hat m(I)/\mu^{2b}$. But $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)<\check m/\mu^{c'}$. So $\sum_{I\in {\mathcal{I}}_2'}\hat m(I)\leq \mu^{2b}\cdot \check m/\mu^{c'}$. The increase is this multiplied by $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, so total increase in number of crossings is less than $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$.
\item Now ${\mathcal{I}}_2''$ contains all instances with $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{2b}$. But $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. So $\sum_{I\in {\mathcal{I}}_2}\hat m(I)<\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*+b}}{\check m}$. The increase is this multiplied by $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, so the total increase in number of crossings is at most $\frac{(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))^2\cdot \mu^{2b^*+b}}{(\check m)^2}$. Assuming that $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)<\check m^2/\mu^{c'}$ it's at most $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$.
\end{itemize}
----------------------------------------------
\fi
$\ $
In the remainder of this section we focus on the proof of \Cref{thm: phase 2}. Throughout the proof, we denote the instance $I=(G,\Sigma)$ that serves as the input to the algorithm by $\check I'=(\check G',\check \Sigma')$, with $|E(\check G')|$ denoted by $\check m'$. We denote the skeleton structure ${\mathcal K}(I)$ by $\check {\mathcal K}=({\mathcal{J}}_1,\ldots,{\mathcal{J}}_r)$, where $r\leq \mu^{100}$. For all $1\leq i\leq r$, we denote ${\mathcal{J}}_i=(J_i,\set{b_u}_{u\in V(J_i)},\rho_{J_i})$.
We start with the intuition for the proof. We denote by $\check K=\bigcup_{i=1}^rJ_i$ the skeleton of $\check G'$ associated with the skeleton structure $\check {\mathcal K}$. We can assume that there exists a solution $\check \psi$ to instance $\check I$ that is semi-clean with respect to $\check{\mathcal K}$.
\subsubsection{Completing the Proof of \Cref{lem: many paths}}
\label{subsubsec: finish the proof}
Given an input instance $\check I=(\check G,\check \Sigma)$, we first apply the algorithm from \Cref{thm: phase 1} to this input. If the algorithm fails, then we terminate the algorithm and return FAIL as well. Recall that, from \Cref{thm: phase 1}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/\mu^3$. Assume now that the algorithm from \Cref{thm: phase 1} did not fail. In this case, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then with probability at least $(1-1/\mu^3)$, the algorithm is successful. We denote by $\tilde {\cal{E}}_1'$ the bad event that the application of this algorithm is unsuccessful, so $\prob{\tilde {\cal{E}}_1'}\leq 1/\mu^3$. Let ${\mathcal{I}}$ be the collection of subinstances of $\check I$ computed by the algorithm from \Cref{thm: phase 1}.
Throughout this subsection, we use the universal constant $b^*=2400$.
Recall that, if Event $\tilde {\cal{E}}'_1$ did not happen, then for every instance $I\in {\mathcal{I}}$, there is a solution $\psi(I)$ to $I$, that is semi-clean with respect to ${\mathcal K}(I)$, such that
$\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. We set the parameter $b$ to be a large enough constant, so that $b\geq 4b^*$ holds.
Recall that, for every instance $I=(G,\Sigma)\in {\mathcal{I}}$, we have denoted by $\hat I=(\hat G,\hat \Sigma)$ the corresponding ${\mathcal K}(I)$-contracted instance, and by $\hat m(I)=|E(\hat G)|$. We say that instance $I$ is \emph{small} if $\hat m(I)\leq \frac{\check m}{\mu^{b^*}}$, and otherwise it is \emph{large}.
We now proceed in $z= 2\log m$ phases, where in the $j$th phase, we compute the $j$th collection ${\mathcal{I}}^{(j)}$ of subinstances of $I$, such that the properties required in \Cref{lem: many paths} are satisfied.
Consider now an index $1\le j\le 2\log m$ and we describe the $j$th phase. Throughout the phase, we use the parameters $g^j_1=2^j$ and $g^j_2=g^j_1\cdot \mu^{b^*}/\check m$.
We intend to apply the algorithm from \Cref{thm: phase 2} to large instances in ${\mathcal{I}}$ with parameters $g^j_1,g^j_2$. But note that the algorithm requires that the input instance $I=(G,\Sigma)$ has a skeleton $K(I)$ that is $g_3(I)$-connected in the input graph $G$ (where $g_3(I)=\big(\frac{g_1^j}{|E(G)|}+g_2^j\big)\cdot\mu^{300b}$), which is not necessarily true for the large instances in ${\mathcal{I}}$. Therefore, we first pre-process instances in ${\mathcal{I}}$, such that the resulting instances are either small or satisfying this connectedness property, as follows.
\newcommand{{\mathcal{I}}_{\sf temp}}{{\mathcal{I}}_{\sf temp}}
\newcommand{\hat{{\mathcal{I}}}_{\sf temp}}{\hat{{\mathcal{I}}}_{\sf temp}}
\subsubsection*{Pre-processing instances in ${\mathcal{I}}$}
The pre-processing procedure is iterative. Throughout, we use a parameter $\tilde g=\big(\frac{g^j_1\mu^{b^*}}{\check m}+g^j_2\big)\cdot\mu^{300b}$.
The procedure maintains a collection ${\mathcal{I}}_{\sf temp}$ of subinstances of $I$, that is initialized to be ${\mathcal{I}}_{\sf temp}={\mathcal{I}}$. We denote $\hat{{\mathcal{I}}}_{\sf temp}=\set{\hat I\mid I\in {\mathcal{I}}_{\sf temp}}$.
Each instance in ${\mathcal{I}}_{\sf temp}$ is either marked \emph{active} or \emph{inactive}. Initially, all large instances are marked active and all small instances are marked inactive.
We will maintain the following invariant throughout the algorithm.
\begin{properties}{IP}
\item for each inactive instance $I=(G,\Sigma)\in {\mathcal{I}}_{\sf temp}$, either $I$ is small, or $I$ satisfies that, if we denote by $K(I)$ the skeleton of $I$, then $K(I)$ is $\tilde g$-connected in $G$;
\label{prop: inactive}
\item for each instance $\hat I=(\hat G,\hat \Sigma)\in \hat{{\mathcal{I}}}_{\sf temp}$, either $|E(\hat G)|\le \check m/\mu$, or $\hat I$ is not wide;
\label{prop: wide or not}
\item $\sum_{I\in {\mathcal{I}}_{\sf temp}}\hat m(I)\le \check m$; and
\label{prop: number of edges}
\item there exists, for every instance $I\in {\mathcal{I}}_{\sf temp}$, a solution $\psi(I)$ to $I$ that is semi-clean with respect to ${\mathcal K}(I)$, such that
$\sum_{I\in {\mathcal{I}}_{\sf temp}}\mathsf{cr}(\psi(I))\le 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}_{\sf temp}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$.
\label{prop: good drawings}
\end{properties}
From \Cref{thm: phase 1} and our initial setting, these invariant holds at the beginning of the pre-processing procedure.
The procedure continues to be executed as long as the collection ${\mathcal{I}}_{\sf temp}$ contains an active instance.
We now describe an iteration of the procedure.
Let $I=(G,\Sigma)$ be an active instance in ${\mathcal{I}}_{\sf temp}$. If instance $I$ is small, then we mark it inactive and terminate the iteration. Clearly all invariants continue to hold after this iteration in this case.
Assume now that instance $I$ is large.
Let ${\mathcal K}(I)=\set{{\mathcal{J}}_1,\ldots,{\mathcal{J}}_{r}}$ be the skeleton structure for $I$, and for each $1\le i\le r$, let $J_i$ be the core associated with ${\mathcal{J}}_i$.
Consider its ${\mathcal K}(I)$-contracted instance $\hat I=(\hat G, \hat \Sigma)$. Recall that graph $\hat G$ is obtained from graph $G$ by contracting, for each $1\le i\le r$, all vertices of the core $J_i$ into a supernode $v_{J_i}$. We now compute, for every pair $1\le i<i'\le r$, a minimum $v_{J_i}$-$v_{J_{i'}}$ cut in $\hat G$.
If for all pairs $1\le i<i'\le r$, the $v_{J_i}$-$v_{J_{i'}}$ minimum-cut in $\hat G$ has value at least $\tilde g$, then we simply mark the instance $I$ in ${\mathcal{I}}$ inactive, and terminate this iteration.
Clearly all invariants continue to hold after this iteration in this case.
Assume now that there exists a pair $1\le i<i'\le r$, such that the $v_{J_i}$-$v_{J_{i'}}$ minimum-cut in $\hat G$ has value less than $\tilde g$.
Let $\hat G_1,\ldots,\hat G_s$ be the graphs obtained from $\hat G$ by deleting the edge of the minimum-cut. Consider now an $1\le i\le s$. Assume that supernodes corresponding to cores $J_{p_1},\ldots, J_{p_t}$ belong to graph $\hat G_i$.
We define instance $I_i=(G_i,\Sigma_i)$ as follows. Graph $G_i$ is the graph obtained from $\hat G_i$ by un-contracting every supernode $v_{J_{p_q}}$ back to its corresponding core $J_{p_q}$, so $G_i$ is a subgraph of $G$, and $\Sigma_i$ is the rotation system on $G_i$ that is induced by $\Sigma$. We then define its core structure to be ${\mathcal K}(I_i)=\set{{\mathcal{J}}_{p_1},\ldots,{\mathcal{J}}_{p_q}}$.
We then replace the instance $I$ in ${\mathcal{I}}_{\sf temp}$ with new instances $I_1,\ldots,I_s$, all marked active, and terminate this iteration.
We now verify that all invariants continue to hold after this iteration in this case.
First, since we marked all new instance that are added to collection ${\mathcal{I}}_{\sf temp}$ active, Invariant \ref{prop: inactive} holds.
Second, since for every new instance $I_i=(G_1,\Sigma_i)$, graph $G_i$ is a subgraph of $G$, and $\hat G_i$ is a subgraph of $\hat G$. If $|E(\hat G)|\le \check m/\mu$, then $|E(\hat G_i)|\le \check m/\mu$; if $\hat I$ is node wide, then $\hat I_i$ is also not wide. Therefore,
\ref{prop: wide or not} holds.
Third, since $\hat G_1\cup\ldots\cup\hat G_s\subseteq \hat G$, $\sum_{1\le i\le s}\hat m(I_i)\le \hat m(I)$ and so
\ref{prop: number of edges} holds.
Last, note that the Invariant \ref{prop: good drawings} holds before this iteration. Let $\psi(I)$ be the semi-clean solution associated with instance $I$. We simply define, for each $1\le i\le s$, $\psi(I_i)$ to be the drawing of $G_i$ induced by drawing $\psi(I)$. It is easy to verify that $\psi(I_i)$ is a semi-clean solution to $I_i$ with respect to its skeleton structure. Moreover, since $G_1,\ldots,G_s$ are disjoint subgraphs of $G$,
$\sum_{1\le i\le s}\mathsf{cr}(\psi(I_i))\le \mathsf{cr}(\psi(I))$, and $\sum_{1\le i\le s}|\chi^{\mathsf{dirty}}(\psi(I_i))|\leq |\chi^{\mathsf{dirty}}(\psi(I))|$.
Therefore, \ref{prop: good drawings} also holds.
Note that the collection of all cores in all skeleton structures is never changed, and initially, the number of cores in all active instances in ${\mathcal{I}}$ is at most $\mu^{b^*}\cdot \mu^{100}$ (since the number of active instances is at most $\mu^{b^*}$ and each such instances has at most $\mu^{100}$ cores). When the pre-processing process terminates, every resulting instance contains at least one core.
Therefore, the number of iterations is at most $\mu^{b^*}\cdot \mu^{100}$. Note that in each iteration we have deleted at most $\tilde g$ edges. Therefore, if we denote by $\hat E_{\sf temp}$ the set of all deleted edges throughout this procedure, then
$|\hat E_{\sf temp}|\le \mu^{b^*}\cdot \mu^{100}\cdot \tilde g$.
Let ${\mathcal{I}}^*$ be the resulting collection we obtained at the end of the pre-processing procedure. Since all instances in ${\mathcal{I}}^*$ are inactive, from Invariant \ref{prop: inactive}, for each instance $I=(G,\Sigma)\in {\mathcal{I}}^*$, either $I$ is small, or its skeleton $K(I)$ is $\tilde g$-connected in $G$.
Moreover, it is easy to verify that, for each instance $I\in {\mathcal{I}}^*$, its skeleton structure has size at most $\mu^{100}$.
Denote $\hat {\mathcal{I}}^*=\set{\hat I\mid I\in {\mathcal{I}}^*}$. We prove the following observation.
\begin{observation}
\label{obs: algo combine drawings}
There is an efficient algorithm, that, given a solution $\phi(\hat I)$ to each instance $\hat I\in \hat {\mathcal{I}}^*$, computes a solution $\check \phi$ to instance $\check I$, with $\mathsf{cr}(\check \phi)\le \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{5000}+g_1^j\cdot\mu^{3b^*+300b}$.
\end{observation}
\begin{proof}
It would be convenient for us to associate a (rooted) partitioning tree $\tau$ for the pre-processing procedure, whose node correspond to all instance that every appeared in the collection ${\mathcal{I}}_{\sf temp}$, as follows.
Initially, $\tau$ contains a root denoted by $v^*$, and, for each instance $I\in {\mathcal{I}}$, a node $v(I)$ representing instance $I$.
Consider now an iteration that some instance $I$ is processed and eventually replaced by new instances $I_1,\ldots,I_s$. We add to $\tau$ nodes $v(I_1),\ldots,v(I_s)$ to them, that become the child nodes of $v(I)$ in $\tau$. We continue until the end of the algorithm. This completes the construction of tree $\tau$. It is easy to see that the instances in the resulting collection ${\mathcal{I}}^*$ correspond to leaf nodes in $\tau$.
We now describe the algorithm for computing a solution $\check \phi$ to instance $\check I$. Assume now we are given, for each instance $I\in {\mathcal{I}}^*$ that correspond to some leaf node in $\tau$, a solution $\phi(\hat I)$ to instance $\hat I$.
We iteratively process non-root nodes of $\tau$ in a bottom-up fashion, as follows.
In each iteration, we take a node $v(I)$ corresponding to instance $I$, such that all its child nodes $v(I_1),\ldots,v(I_s)$ are either leaf nodes or have been processed.
Consider the iteration of the pre-processing procedure in which some instance $I$ is processed and eventually replaced by new instances $I_1,\ldots,I_s$. We denote by $\hat I=(\hat G,\hat \Sigma), \hat I_1=(\hat G_1,\hat \Sigma_1),\ldots,\hat I_s=(\hat G_s,\hat \Sigma_s)$ their contracted instances, respectively.
From the algorithm, graph $\hat G_1\cup \ldots\cup \hat G_s$ are the connected components obtained from $\hat G$ by deleting a set $\hat E$ of edges. We first obtain a drawing of $\hat G\setminus \hat E$ by placing drawings $\phi(\hat I_1),\ldots,\phi(\hat I_s)$ in mutually disjoint discs in the plane.
We then use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $\hat E$ into the drawing of $\hat G\setminus \hat E$, obtaining a solution $\phi(\hat I)$ to instance $\hat I$, such that $\mathsf{cr}(\phi(\hat I))\le \sum_{1\le i\le s}\mathsf{cr}(\phi(\hat I_s))+|E(\hat G)|\cdot |\hat E|$.
In this way we process all non-root nodes, such that at the end, we obtained, for each instance $I\in {\mathcal{I}}$, a solution $\phi(\hat I)$ to its contracted instance $\hat I$, with $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\phi(\hat I))\le \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I))+\check m \cdot \hat E_{\sf temp}$.
Finally, we apply the algorithm from \Cref{thm: phase 1} to the solutions $\phi(\hat I)$ for instances $I\in{\mathcal{I}}$ and obtain a solution $\check\phi$ for $\check I$ with $\mathsf{cr}(\check \phi)\le \sum_{\hat I\in \hat {\mathcal{I}}}\mathsf{cr}(\phi(\hat I))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{5000}$.
Since $|\hat E_{\sf temp}|\le \mu^{b^*}\cdot \mu^{100}\cdot \tilde g$, altogether we get that
$$\mathsf{cr}(\check \phi)
\le \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{5000}+\check m\cdot\mu^{b^*}\cdot \mu^{100}\cdot \tilde g
\le \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{5000}+g_1^j\cdot\mu^{3b^*+300b},$$
as $b^*>100$.
\end{proof}
\subsubsection*{The algorithm for comuting the collection ${\mathcal{I}}^{(j)}$}
We now partition the resulting collection ${\mathcal{I}}^*$ into two subsets: set ${\mathcal{I}}^*_1$ contains all small instances in ${\mathcal{I}}^*$; and set ${\mathcal{I}}^*_2$ contains all remaining instances.
We denote $\hat {\mathcal{I}}_1=\set{\hat I\mid I\in
{\mathcal{I}}^*_1}$.
Since every instance in ${\mathcal{I}}^*_2$ is large (that is, $\hat m(I)\ge \check m/\mu^{b^*}$), from Invariant \ref{prop: number of edges}, we get that $|{\mathcal{I}}^*_2|\le \mu^{b^*}$.
From the pre-processing step, for each instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$, its skeleton $K(I)$ is $\tilde g$-connected in $G$. Since $|E(G)|\ge \check m/\mu^{b^*}$, $\tilde g\ge \textsf{left}(
\frac{g_1^j}{|E(G)|}+g_2^j \textsf{right})$, so its skeleton $K(I)$ is $g_3(I)$-connected in $G$, for $g_3(I)=\textsf{left}(
\frac{g_1^j}{|E(G)|}+g_2^j \textsf{right})$.
For each instance $I\in {\mathcal{I}}^*$, we denote by $\psi(I)$ the solution to $I$ guaranteed by Invariant \ref{prop: good drawings}.
We use the following observation.
\begin{observation}
\label{obs: not too large}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\le \check m^2/\mu^{61b-2b^*}$, then for each instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$, the drawing $\psi(I)$ satisfies that $\mathsf{cr}(\psi(I))\le |E(G)|^2/\mu^{60b}$ and $|\chi^{\mathsf{dirty}}(\psi(I))|\le |E(G)|/\mu^{60b}$.
\end{observation}
\begin{proof}
Let $I=(G,\Sigma)$ be an instance in ${\mathcal{I}}^*_2$, so $|E(G)|\ge \check m/\mu^{b^*}$.
Recall that, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\le 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, so $\mathsf{cr}(\psi(I))\le 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$. Therefore, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\le \check m^2/\mu^{61b-2b^*}$, then
\[
\mathsf{cr}(\psi(I))\le 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I) \le \frac{2\cdot \check m^2}{\mu^{61b-2b^*}}\le
\frac{ \check m^2}{\mu^{60b-2b^*}}
= \frac{ (\check m/\mu^{b^*})^2}{\mu^{60b}}
\le \frac{|E(G)|^2}{\mu^{60b}}.
\]
Also recall that, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\le \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, so $|\chi^{\mathsf{dirty}}(\psi(I))| < \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. Therefore,
\[
|\chi^{\mathsf{dirty}}(\psi(I))|\le \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}
\le \frac{\check m\cdot \mu^{b^*}}{\mu^{61b-2b^*}}
\le \frac{\check m}{\mu^{60b+b^*}}
= \frac{(\check m/\mu^{b^*})}{\mu^{60b}}
\le \frac{|E(G)|}{\mu^{60b}},
\]
since $b\ge 4b^*$.
\end{proof}
Finally, we consider every instance $I\in {\mathcal{I}}^*_2$ one by one.
For each such instance $I$, we apply the algorithm from \Cref{thm: phase 2} to instance $I$, skeleton structure ${\mathcal K}(I)$, the parameters $b,g^j_1,g^j_2$ defined above.
We denote by $I'=(G',\Sigma')$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace, and by $\hat I'$ the ${\mathcal K}(I)$-contraction of this instance. We then denote $\hat {\mathcal{I}}'_2=\set{\hat I'\mid I\in {\mathcal{I}}^*_2}$. The final output of our algorithm is the collection ${\mathcal{I}}^{(j)}=\hat{\mathcal{I}}_1\cup\hat {\mathcal{I}}'_2$ of subinstances of $\check I$.
\subsubsection*{The analysis of the algorithm}
We now verify that the output collections ${\mathcal{I}}^{(1)},\ldots,{\mathcal{I}}^{(z)}$ of subinstances of $\check I$ have all required properties.
Consider an index $1\le j\le z$ and consider the algorithm for constructing collection ${\mathcal{I}}^{(j)}$.
First, from Invariant \ref{prop: wide or not}, for every instance $I\in {\mathcal{I}}^*$, if the corresponding contracted instance $\hat I=(\hat G,\hat \Sigma)$ is a wide instance, then $|E(\hat G)|\le \check m/\mu$.
If instance $I$ lies in ${\mathcal{I}}^*_2$, and $\hat I'=(\hat G',\hat\Sigma')$ is the ${\mathcal K}(I)$-contraction of instance $I'$, then $|E(\hat G')|\leq |E(\hat G)|$, and, if $\hat I$ is not a wide instance, then neither is $\hat I'$. This is since graph $\hat G'$ can be obtained from graph $\hat G$ by deleting the edges of $E^{\mathsf{del}}(I)$ from it. Therefore, we are guaranteed that, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}^{(j)}$, if $I'$ is a wide instance, then $|E(G')|\le \check m/\mu$.
Second, from Invariant \ref{prop: number of edges},
$\sum_{I\in {\mathcal{I}}^*}\hm(I)\le \check m$.
Since, for every instance $I\in {\mathcal{I}}^*_2$, the corresponding resulting instance $\hat I'=(\hat G',\Sigma')\in \hat {\mathcal{I}}_2'$ has $|E(\hat G')|\leq \hat m(I)$, we get that $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}^{(j)}}|E(G')|\le \check m$.
Third, we construct an algorithm that, given a solution to each instance in ${\mathcal{I}}^{(j)}$, computes a solution $\phi$ to instance $\check I$.
For each instance $I\in {\mathcal{I}}^*_2$, we let $\hat I=(\hat G,\hat \Sigma)$ be the corresponding ${\mathcal K}(I)$-contracted instance, and let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal K}(I)$-contracted instance corresponding to the instance $I'$. Note that $V(\hat G)=V(\hat G')$ and $E(\hat G')=E(\hat G)\setminus E^{\mathsf{del}}(I)$. We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{del}}(I)$ into the solution $\phi(\hat I')$ to instance $\hat I'$, obtaining a solution $\phi(\hat I)$ to instance $\hat I$.
Then we use the algorithm from \Cref{obs: algo combine drawings} to compute a drawing of $\check I$. It is clear that the algorithm is efficient.
From now on, we let $j^*$ be the unique integer that $2^{j^*-1}< 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\le 2^{j^*}$. Since $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\le \check m^2$, $j^*\le 2\log m$, so $1\le j^*\le z$. We will show that the collection ${\mathcal{I}}^{(j^*)}$ satisfies the property required in the last bullet of \Cref{lem: many paths}. Consider now the algorithm for constructing the collection ${\mathcal{I}}^{(j^*)}$.
Recall that we have set $g_1^{j^*}=2^{j^*}$ and $g_2^{j^*}=g_1^{j^*}\cdot \mu^{b^*}/\check m$.
Therefore, from Invariant \ref{prop: good drawings}, the semi-clean solutions $\psi(I)$ to $I$ for instances $I\in {\mathcal{I}}^*$ satisfy that
$\sum_{I\in {\mathcal{I}}_{\sf temp}}\mathsf{cr}(\psi(I))\le g_1^{j^*}$ and $\sum_{I\in {\mathcal{I}}_{\sf temp}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq g_2^{j^*}$.
So for each instance $I\in {\mathcal{I}}^*$, $\mathsf{cr}(\psi(I))\le g_1^{j^*}$ and $|\chi^{\mathsf{dirty}}(\psi(I))| \leq g_2^{j^*}$.
Combined with \Cref{obs: not too large}, we get that, for each instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$, both $\mathsf{cr}(\psi(I))\le \min\set{g_1^*,|E(G)|^2/\mu^{60b}}$ and $|\chi^{\mathsf{dirty}}(\psi(I))|\le \min\set{g_2^*, |E(G)|/\mu^{60b}}$ hold.
(We have used the fact that \fbox{$c'>61b-2b^*$}.)
For each instance $I\in {\mathcal{I}}^*_2$, we denote by $\tilde {\cal{E}}_2(I)$ the bad event that the application of the algorithm from \Cref{thm: phase 2} was unsuccessful, so $\prob{\tilde {\cal{E}}_2(I)}\leq 1/\mu^{2b}$.
Let $\tilde {\cal{E}}_2'$ be the bad event that any of the events in $\set{\tilde {\cal{E}}_2(I)\mid I\in {\mathcal{I}}^*_2}$ happened. Since, from the above discussion, $|{\mathcal{I}}^*_2| \leq \mu^{b^*}$, and, for every instance $I\in {\mathcal{I}}^*_2$, $\prob{\tilde {\cal{E}}_2(I)}\leq 1/\mu^{2b}$, while $b>2b^*$, from the Union Bound, we get that $\prob{\tilde {\cal{E}}_2'}<1/\mu^4$. Lastly, we let $\tilde {\cal{E}}'$ be the bad event that either $\tilde {\cal{E}}_1'$ or $\tilde {\cal{E}}_2'$ happened. Clearly, $\prob{\tilde {\cal{E}}'}\leq \prob{\tilde {\cal{E}}_1'}+\prob{\tilde {\cal{E}}_2'}\leq 1/\mu^2$.
We use the following two observations in order to complete the proof of \Cref{lem: many paths}.
\begin{observation}\label{obs: composing contracted solutions}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and that event $\tilde {\cal{E}}'$ did not happen. Then there is an efficient algorithm, that, given a solution $\phi(I')$ for every instance $I'\in {\mathcal{I}}^{(j^*)}$, computes a solution $\check \phi$ to instance $\check I$, with $\mathsf{cr}(\check\phi)\leq \sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{cr}(\phi(I')) + \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}$.
\end{observation}
\begin{proof}
We assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, Event $\tilde {\cal{E}}'$ did not happen, and that we are given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}^{(j^*)}$. We now show an efficient algorithm to compute a solution $\check \phi$ to instance $\check I$. In order to do so, we first consider every instance $I\in {\mathcal{I}}^*_2$ one by one, and compute a solution $\phi(\hat I)$ to instance $\hat I$, from the solution $\phi(\hat I')$ to instance $\hat I'$.
Consider now some instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$. Let $\hat I=(\hat G,\hat \Sigma)$ be the corresponding ${\mathcal K}(I)$-contracted instance, and let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal K}(I)$-contracted instance corresponding to the instance $I'$. Note that $V(\hat G)=V(\hat G')$ and $E(\hat G')=E(\hat G)\setminus E^{\mathsf{del}}(I)$. We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{del}}(I)$ into the solution $\phi(\hat I')$ to instance $\hat I'$, obtaining a solution $\phi(\hat I)$ to instance $\hat I$, whose cost is at most $\mathsf{cr}(\phi(\hat I'))+|E^{\mathsf{del}}(I)|\cdot |E(G)|$. If Event $\tilde {\cal{E}}'$ does not happen, then $|E^{\mathsf{del}}(I)|\leq \big(\frac{g_1^{j^*}}{|E(G)|}+g_2^{j^*}\big)\cdot\mu^{O(1)}=\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}}{|E(G)|}$, and so $\mathsf{cr}(\phi(\hat I))\leq \mathsf{cr}(\phi(\hat I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{O(1)}$.
Now using the algorithm from \Cref{obs: algo combine drawings},
we obtain a solution
$\check\phi$ to instance $\check I$, whose cost is bounded by:
\[
\begin{split}
\mathsf{cr}(\check \phi)&\leq \sum_{\hat I\in \hat {\mathcal{I}}^*}\mathsf{cr}(\phi(\hat I)) + \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}\\
&\leq \sum_{I\in {\mathcal{I}}^*_1}\mathsf{cr}(\phi(\hat I))+\sum_{I\in {\mathcal{I}}^*_2}\textsf{left} (\mathsf{cr}(\phi(\hat I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{O(1)}\textsf{right} )+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}\\
&\le \sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{cr}(\phi(I'))+|{\mathcal{I}}^*_2|\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}\\
&\le \sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{cr}(\phi(I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)},
\end{split}
\]
as $|{\mathcal{I}}^*_2|=\mu^{O(1)}$.
\end{proof}
\begin{observation}\label{obs: cheap solutions to final instances}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and that event $\tilde {\cal{E}}'$ did not happen. Then $\sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log \check m)^{O(1)}$.
\end{observation}
\begin{proof}
We bound $\sum_{I\in {\mathcal{I}}^*_1}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)$ and $\sum_{I\in {\mathcal{I}}^*_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')$ separately.
From \Cref{thm: phase 2}, if Event $\tilde {\cal{E}}'$ did not happen, then, for every instance $I=(G,\Sigma)\in {\mathcal{I}}^*_2$,
there is a solution $\psi(I')$ to the corresponding instance $I'$, that is clean with respect to ${\mathcal K}(I)$, with $\mathsf{cr}(\psi(I'))\leq \mathsf{cr}(\psi(I))\cdot (\log \check m)^{O(1)}+|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(G)|}{\mu^b}$.
From \Cref{obs: clean solution to contracted}, there is a solution to the corresponding contracted instance $\hat I'$, of cost at most $\mathsf{cr}(\psi(I'))$. Altogether, we get that:
%
\[\begin{split}
\sum_{I\in {\mathcal{I}}^*_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')&\leq \sum_{I\in {\mathcal{I}}^*_2}\textsf{left}(\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}^*_2}|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(G)|}{\mu^b}\textsf{right} )\cdot (\log \check m)^{O(1)}.\\
&\leq \sum_{I\in {\mathcal{I}}^*_2}\mathsf{cr}(\psi(I))\cdot (\log \check m)^{O(1)}+\textsf{left} (\sum_{I\in {\mathcal{I}}^*_2}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} )^2\cdot (\log \check m)^{O(1)}\\
&\hspace{4cm}+ \frac{\check m\cdot (\log \check m)^{O(1)}}{\mu^b}\cdot\textsf{left}(\sum_{I\in {\mathcal{I}}^*_2}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} ).
\end{split}
\]
%
From Invariant \ref{prop: good drawings}, $\sum_{I\in {\mathcal{I}}^*}\mathsf{cr}(\psi(I))\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$ and $\sum_{I\in {\mathcal{I}}^*}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. Additionally, since we have assumed that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$ for a large enough constant $c'>2b^*$,
$$\textsf{left} (\sum_{I\in {\mathcal{I}}^*_2}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} )^2\leq \frac{(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))^2 \mu^{2b^*}}{\check m^2}\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I).$$ Altogether, we get that:
%
\[
\begin{split}
\sum_{I\in {\mathcal{I}}^*_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot (\log \check m)^{O(1)}+\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}\cdot (\log \check m)^{O(1)}}{\mu^b}\\
&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot (\log \check m)^{O(1)}, \end{split}\]
since $b>4b^*$.
Next, we bound $\sum_{I\in {\mathcal{I}}^*_1}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)$. Consider some instance $I=(G,\Sigma)\in {\mathcal{I}}^*_1$ and the solution $\psi(I)$ that is semi-clean with respect to ${\mathcal K}(I)$. Let $K(I)$ be the skeleton associated with the skeleton structure ${\mathcal K}(I)$. Let $E^{\mathsf{dirty}}(I)\subseteq E(G)\setminus E(K(I))$ be the set of all edges $e$, such that the image of $e$ in $\psi(I)$ crosses the image of some edge of $K(I)$. Let $\hat I=(\hat G,\hat \Sigma)$ be the ${\mathcal K}(I)$-contracted instance corresponding to instance $I$.
Denote $G'=G\setminus E^{\mathsf{dirty}}(I)$, and let $\Sigma'$ be the rotation system for graph $G'$ that is induced by $\Sigma$. Observe that ${\mathcal K}(I)$ is a valid skeleton structure for the resulting instance $I'=(G',\Sigma')$. Let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal K}(I)$-contracted instance associated with $I'$. Observe that the drawing $\psi(I)$ of instance $I$ induces a drawing $\psi(I')$ of instance $I'$ that is clean with respect to ${\mathcal K}(I)$. From \Cref{obs: clean solution to contracted}, there is a solution $\psi(\hat I')$ to the contracted instance $\hat I'$ with $\mathsf{cr}(\psi(\hat I'))\leq \mathsf{cr}(\psi(I'))\leq \mathsf{cr}(\psi(I))$.
We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{dirty}}(I)$ into the drawing $\psi(\hat I')$ to obtain a drawing $\psi(\hat I)$ of instance $\hat I$, with the number of crossings bounded by $\mathsf{cr}(\psi(\hat I'))+|E^{\mathsf{dirty}}(I)|\cdot |E(\hat G)|\leq \mathsf{cr}(\psi(I))+|E^{\mathsf{dirty}}(I)|\cdot |E(\hat G)|$.
Recall that $|E(\hat G)|\leq \frac{\check m}{\mu^{b^*}}$ (since instance $I$ is small), so $\mathsf{cr}(\psi(\hat I))
\leq \mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(I)|\cdot \frac{\check m}{\mu^{b^*}}$.
We then get that:
\[\sum_{I\in {\mathcal{I}}^*_1} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq
\sum_{I\in {\mathcal{I}}^*_1}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}^*_1}|\chi^{\mathsf{dirty}}(I)|\cdot \frac{\check m}{\mu^{b^*}}.
\]
From Invariant \ref{prop: good drawings}, $\sum_{I\in {\mathcal{I}}^*}\mathsf{cr}(\psi(I))\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$ and $\sum_{I\in {\mathcal{I}}^*}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$.
Therefore,
$$\sum_{I\in {\mathcal{I}}^*_1} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\frac{\check m}{\mu^{b^*}} \cdot \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)).$$
\iffalse{analysis for previous iset_2, which should be an empty set}
Assume now that $I\in {\mathcal{I}}_2$. We partition the set ${\mathcal{I}}_2$ of instances into two subsets: set ${\mathcal{I}}_2'$ containing all instances $I=(G,\Sigma)$ with $\mathsf{cr}(\psi(I))> |E(G)|^2/\mu^{60b}$, and set ${\mathcal{I}}_2''$ containing all remaining instances.
As before, for every instance $I=(G,\Sigma)\in {\mathcal{I}}_2$, we denote by $\hat m(I)$ the number of edges in the contracted graph $\hat I$.
Next, we bound $\sum_{I\in {\mathcal{I}}_2'}\mathsf{cr}(\psi(\hat I'))$. Recall that for every instace $I=(G,\Sigma)\in {\mathcal{I}}_2'$:
\[\mathsf{cr}(\psi(I))> \frac{|E(G)|^2}{\mu^{60b}}\geq \frac{(\hat m(I))^2}{\mu^{60b}}\geq \frac{\hat m(I)\cdot \check m}{\mu^{60b+b^*}},\]
since instance $I$ must be large. Since,
from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq 2\check m^2/\mu^{c'}$, we get that $\sum_{I\in {\mathcal{I}}_2'}\hat m(I)\leq \frac{2\check m}{\mu^{c'-60b-b^*}}$. Lastly, since, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, we get that:
\[\begin{split}
\sum_{I\in {\mathcal{I}}_2'} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)&\leq
\sum_{I\in {\mathcal{I}}_2'}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}_2'}|\chi^{\mathsf{dirty}}(I)|\cdot \hm(I)\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\textsf{left} ( \sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))| \textsf{right} )\cdot \textsf{left} ( \sum_{I\in {\mathcal{I}}_2'}\hat m(I) \textsf{right} )\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\frac{2\check m}{\mu^{c'-60b-b^*}}\cdot \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\\
&\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)).\end{split}
\]
It now remains to consider instances $I\in {\mathcal{I}}_2''$. Recall that, for each such instance $I=(G,\Sigma)$, $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{2b}>\hat m(I)/\mu^{60b}$.
Since, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, we get that $\sum_{I\in {\mathcal{I}}_2''}\hat m(I)<\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*+60b}}{\check m}$.
Therefore:
\[\begin{split}
\sum_{I\in {\mathcal{I}}_2''} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)&\leq
\sum_{I\in {\mathcal{I}}_2''}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}_2''}|\chi^{\mathsf{dirty}}(I)|\cdot \hm(I)\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\textsf{left} ( \sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))| \textsf{right} )\cdot \textsf{left} ( \sum_{I\in {\mathcal{I}}_2''}\hat m(I) \textsf{right} )\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I) + \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*+60b}}{\check m}\cdot \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\\
&\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))
,\end{split}
\]
%
since $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, and $c'$ is a large enough constant.
\fi
Altogether, we get that
$$\sum_{I'\in {\mathcal{I}}^{(j^*)}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')=\sum_{I\in {\mathcal{I}}^*_1}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)+\sum_{I\in {\mathcal{I}}^*_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log \check m)^{O(1)}.$$
\end{proof}
\iffalse
The calculations
---------------------------------------------------
\begin{itemize}
\item $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$
\item $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$
\item in ${\mathcal{I}}_2$: either $\mathsf{cr}(\psi(I))> |E(G)|/\mu^{2b}$, or $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{2b}$.
\item when pulling edges back, we will pay $\sum_{I\in {\mathcal{I}}_2}\hm(I)\cdot |\chi^{\mathsf{dirty}}(\psi(I))|$.
\item let's say that ${\mathcal{I}}_2'$ contains all instances with $\mathsf{cr}(\psi(I))> |E(G)|/\mu^{2b}>\hat m(I)/\mu^{2b}$. But $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)<\check m/\mu^{c'}$. So $\sum_{I\in {\mathcal{I}}_2'}\hat m(I)\leq \mu^{2b}\cdot \check m/\mu^{c'}$. The increase is this multiplied by $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, so total increase in number of crossings is less than $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$.
\item Now ${\mathcal{I}}_2''$ contains all instances with $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{2b}$. But $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. So $\sum_{I\in {\mathcal{I}}_2}\hat m(I)<\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*+b}}{\check m}$. The increase is this multiplied by $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, so the total increase in number of crossings is at most $\frac{(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))^2\cdot \mu^{2b^*+b}}{(\check m)^2}$. Assuming that $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)<\check m^2/\mu^{c'}$ it's at most $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$.
\end{itemize}
----------------------------------------------
\fi
$\ $
In the remainder of this section we focus on the proof of \Cref{thm: phase 2}. Throughout the proof, we denote the instance $I=(G,\Sigma)$ that serves as the input to the algorithm by $\check I'=(\check G',\check \Sigma')$, with $|E(\check G')|$ denoted by $\check m'$. We denote the skeleton structure ${\mathcal K}(I)$ by $\check {\mathcal K}=({\mathcal{J}}_1,\ldots,{\mathcal{J}}_r)$, where $r\leq \mu^{100}$. For all $1\leq i\leq r$, we denote ${\mathcal{J}}_i=(J_i,\set{b_u}_{u\in V(J_i)},\rho_{J_i})$.
We start with the intuition for the proof. We denote by $\check K=\bigcup_{i=1}^rJ_i$ the skeleton of $\check G'$ associated with the skeleton structure $\check {\mathcal K}$. We can assume that there exists a solution $\check \psi$ to instance $\check I$ that is semi-clean with respect to $\check{\mathcal K}$.
\subsubsection{Completing the Proof of \Cref{lem: many paths}}
\label{subsubsec: finish the proof}
Given an input instance $\check I^*=(\check G^*,\check \Sigma^*)$, we first apply the algorithm from \Cref{thm: phase 1} to this input. If the algorithm from \Cref{thm: phase 1} fails, then we terminate the algorithm and return FAIL as well.
We denote by $\tilde {\cal{E}}_1'$ the bad event that the application of this algorithm is unsuccessful. Recall that, from \Cref{thm: phase 1}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\leq \check m^2/\mu^{c'}$, then $\prob{\tilde {\cal{E}}_1'}\leq 1/\mu^{200}$, and, if bad event ${\cal{E}}_1'$ does not happen, then the algorithm does not fail.
We assume from now on that the algorithm from \Cref{thm: phase 1} did not fail. Let ${\mathcal{I}}$ be the collection of subinstances of $\check I$ computed by the algorithm from \Cref{thm: phase 1}. Recall that, if Event $\tilde {\cal{E}}'_1$ did not happen, then for every instance $I\in {\mathcal{I}}$, there is a solution $\psi(I)$ to $I$, that is ${\mathcal{J}}(I)$-valid, such that:
$\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}}{\check m}$. We let $b$ be a large enough constant, so that $b\geq 4000$. We assume that the parameter $c'$ from the statement of \Cref{lem: many paths} is sufficiently large compared to $b$, for example, $c'\geq 400b$.
Recall that, for every instance $I=(G,\Sigma)\in {\mathcal{I}}$, we have denoted by $\hat I=(\hat G,\hat \Sigma)$ the corresponding ${\mathcal{J}}(I)$-contracted instance, and by $\hat m(I)=E(\hat G)$. We say that instance $I$ is \emph{small} if $\hat m(I)\leq \frac{\check m}{\mu^{1000}}$, and otherwise it is \emph{large}. We partition the set ${\mathcal{I}}$ of instances into two subsets: set ${\mathcal{I}}_1$ containing all small instances, and set ${\mathcal{I}}_2$ containing all large instances. We let $\hat {\mathcal{I}}_1=\set{\hat I\mid I\in {\mathcal{I}}_1}$ contain the set of all contracted instances corresponding to the instances of ${\mathcal{I}}_1$, and we define set $\hat {\mathcal{I}}_2$ of instances corresponding to the instances of ${\mathcal{I}}_2$ similarly.
We need the following obsevation.
\begin{observation}\label{obs: have invariants}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\leq \check m^2/\mu^{c'}$, and that bad event $\tilde{\cal{E}}'$ did not happen. Then for every instance $I=(G,\Sigma)\in {\mathcal{I}}_2$, there is a solution $\psi(I)$ to instance $I$ that is ${\mathcal{J}}(I)$-valid, with $\mathsf{cr}(\psi(I))\leq m^2/\mu^{240b}$, and $|\chi^{\mathsf{dirty}}(\psi(I))|\leq m/\mu^{240b}$, where $m=|E(G)|$.
\end{observation}
\begin{proof}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)=\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\leq \check m^2/\mu^{c'}$, and that bad event $\tilde {\cal{E}}'$ did not happen. Recall that the algorithm from \Cref{thm: phase 1} ensures that, for every instance $I\in {\mathcal{I}}$,
there is a solution $\psi(I)$ to $I$, that is ${\mathcal{J}}(I)$-valid, with
$\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}}{\check m}$.
Consider now some instance $I=(G,\Sigma)\in {\mathcal{I}}_2$, and denote $|E(G)|=m$. From the above discussion, $\mathsf{cr}(\psi(I))\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \frac{\check m^2}{\mu^{c'}}$ must hold. Since, from definition of set ${\mathcal{I}}_2$ of instances, $m\geq \hat m(I)>\frac{\check m}{\mu^{1000}}$ holds, we get that $\mathsf{cr}(\psi(I))\leq \frac{m^2}{\mu^{c'-2000}}\leq \frac{m^2}{\mu^{240b}}$, since we can set $c'$ to be a large enough constant.
Similarly, $|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}}{\check m}\leq \frac{\check m}{\mu^{c'-900}}$, since $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$. Using the fact that $m\geq \frac{\check m}{\mu^{1000}}$, we get that:
\[ |\chi^{\mathsf{dirty}}(\psi(I))|\leq\frac{m}{\mu^{c'-1900}}\leq \frac{m}{\mu^{240b}}. \]
\end{proof}
We process every instance $I\in {\mathcal{I}}_2$ one by one.
For each such instance $I$, we apply the algorithm from \Cref{thm: phase 2} to instance $I$, core structure ${\mathcal{J}}(I)$, and the constant parameter $b$ defined above. Let $\tilde {\cal{E}}'_2(I)$ be the bad event that this application of the algorithm was unsuccessful.
From \Cref{obs: have invariants} and \Cref{thm: phase 2}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\leq \check m^2/\mu^{c'}$, then $\prob{\tilde {\cal{E}}'_2(I)\mid \neg\tilde {\cal{E}}'_1}\leq 1/\mu^{2b}$. We denote by $I'=(G',\Sigma')$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace, and by $\hat I'$ the corresponding ${\mathcal{J}}(I)$-contracted instance. We then denote $\hat {\mathcal{I}}_2'=\set{\hat I'\mid I\in {\mathcal{I}}_2}$. The final output of our algorithm is the collection ${\mathcal{I}}'=\hat{\mathcal{I}}_1\cup\hat {\mathcal{I}}_2'$ of subinstances of $\check I$.
We now verify that the collection ${\mathcal{I}}'$ of instances has all required properties. First, the algorithm from \Cref{thm: phase 1} ensures that, for every instance $I\in {\mathcal{I}}$, if the corresponding contracted instance $\hat I=(\hat G,\hat \Sigma)$ is a wide instance, then $|E(\hat G)|\le \check m/\mu$. If instance $I$ lies in ${\mathcal{I}}_2$, and $\hat I'=(\hat G',\hat\Sigma')$ is the ${\mathcal{J}}(I)$-contracted instance corresponding to $I'$, then $|E(\hat G')|\leq |E(\hat G)|$, and, if $\hat I$ is not a wide instance, then neither is $\hat I'$. This is since graph $\hat G'$ can be obtained from graph $\hat G$ by deleting the edges of $E^{\mathsf{del}}(I)$ from it\footnote{This is slightly imprecise, since it is possible that $|E(\hat G')|<\hat m(I)$. Therefore, a vertex $v$ may be a high-degree vertex for $\hat G'$ but not for graph $\hat G$. It is therefore possible that $\hat I$ is narrow but $\hat I'$ is not, due to difference in the parameters $|E(\hat G')|$ and $|E(\hat G)|$. However, we can easily fix this issue by adding $\hat m(I)-|E(\hat G')|$ new vertices to graph $\hat G'$, and connecting each of these vertices to a vertex whose degree in $\hat G'$ is smaller than in $\hat G$, so that for every vertex $v\in V(\hat G)$, $\deg_{\hat G'}(v)\leq \deg_{\hat G}(v)$ and $|E(\hat G')|=|E(\hat G)|$ holds. This ensures that, if $\hat I$ is a narrow instance, then so is $\hat I'$. Adding degree-1 vertices to an instance of \ensuremath{\mathsf{MCNwRS}}\xspace does not increase its optimal solution value.}. Therefore, we are guaranteed that, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}'$, if $I'$ is a wide instance, then $|E(G')|\le \check m/\mu$.
The algorithm from \Cref{thm: phase 1} also guarantees that $\sum_{I\in {\mathcal{I}}}\hm(I)\le 2\check m$. Since, for every instance $I\in {\mathcal{I}}_2$, the corresponding instance $\hat I'=(\hat G',\Sigma')\in \hat {\mathcal{I}}_2'$ has $|E(\hat G')|\leq \hat m(I)$, we get that $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\le 2\check m$.
Let $\tilde {\cal{E}}_2'$ be the bad event that any of the events in $\set{\tilde {\cal{E}}'_2(I)\mid I\in {\mathcal{I}}_2}$ happened.
From the definition of the set ${\mathcal{I}}_2$ of instances, for all $I\in {\mathcal{I}}_2$, $\hat m(I)\geq \frac{\check m}{\mu^{1000}}$. Since, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\hat m(I)\le 2\check m$, we get that $|{\mathcal{I}}_2|\leq 2\mu^{1000}$.
Therefore, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\leq \check m^2/\mu^{c'}$, then:
$$\prob{\tilde {\cal{E}}'_2\mid \neg\tilde {\cal{E}}'_1}\leq \sum_{I\in {\mathcal{I}}_2} \prob{\tilde {\cal{E}}'_2(I)\mid \neg\tilde {\cal{E}}'_1}\leq \frac{2\mu^{1000}}{\mu^{2b}}\leq \frac{1}{\mu^4},$$
since $b\geq 4000$.
Lastly, we let $\tilde {\cal{E}}'$ be the bad event that either of the events $\tilde {\cal{E}}'_1$ or $\tilde {\cal{E}}'_2$ happened. Then $\prob{\tilde {\cal{E}}'}\leq \prob{\tilde {\cal{E}}'_1}+\prob{\tilde {\cal{E}}'_2\mid \neg\tilde {\cal{E}}'_1}$. From the above discussion, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\leq \check m^2/\mu^{c'}$, then $\prob{\tilde {\cal{E}}'}\leq \frac{1}{\mu^{200}}+\frac{1}{\mu^4}\leq \frac{1}{\mu^3}$.
We use the following two observations in order to complete the proof of \Cref{lem: many paths}.
\begin{observation}\label{obs: composing contracted solutions}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\leq \check m^2/\mu^{c'}$, and that event $\tilde {\cal{E}}'$ did not happen. Then there is an efficient algorithm, that, given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}'$, computes a solution $\check \phi$ to instance $\check I^*$, with $\mathsf{cr}(\check\phi)\leq \sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I')) + \mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\cdot\mu^{O(1)}$.
\end{observation}
\begin{proof}
We assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\leq \check m^2/\mu^{c'}$, that Event $\tilde {\cal{E}}'$ did not happen, and that we are given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}'$. We show an efficient algorithm to compute a solution $\check \phi$ to instance $\check I$. In order to do so, we consider every instance $I\in {\mathcal{I}}_2$ one by one, and compute a solution $\phi(\hat I)$ to instance $\hat I$, from the solution $\phi(\hat I')$ to instance $\hat I'$.
Consider now some instance $I=(G,\Sigma)\in {\mathcal{I}}_2$. Let $\hat I=(\hat G,\hat \Sigma)$ be the corresponding ${\mathcal{J}}(I)$-contracted instance, and let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal{J}}(I)$-contracted instance corresponding to the instance $I'$. Note that $V(\hat G)=V(\hat G')$ and $E(\hat G')=E(\hat G)\setminus E^{\mathsf{del}}(I)$. We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{del}}(I)$ into the solution $\phi(\hat I')$ to instance $\hat I'$, obtaining a solution $\phi(\hat I)$ to instance $\hat I$, whose cost is at most $\mathsf{cr}(\phi(\hat I'))+|E^{\mathsf{del}}(I)|\cdot |E(\hat G)|$.
Recall that, from \Cref{thm: phase 2}:
$$|E^{\mathsf{del}}(I)|\leq \textsf{left} (\frac{\mathsf{cr}(\psi(I))}{|E(G)|}+|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} ) \cdot \mu^{O(b)}.$$
%
Therefore:
\[
\begin{split}
\mathsf{cr}(\phi(\hat I))&\leq \mathsf{cr}(\phi(\hat I'))+\textsf{left} (\mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(\hat G)|\textsf{right} ) \cdot \mu^{O(b)} \\ &\leq \mathsf{cr}(\phi(\hat I'))+\textsf{left} (\mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(\psi(I))|\cdot \check m\textsf{right} )\cdot \mu^{O(b)}.
\end{split}
\]
Lastly, using the algorithm from \Cref{thm: phase 1}, we obtain a solution
$\check\phi$ to instance $\check I$, whose cost is bounded by:
\[
\begin{split}
\mathsf{cr}(\check \phi)&\leq \sum_{\hat I\in \hat {\mathcal{I}}}\mathsf{cr}(\phi(\hat I)) + \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{8000}\\
&\leq \sum_{I\in {\mathcal{I}}_1}\mathsf{cr}(\phi(\hat I))+\sum_{I\in {\mathcal{I}}_2}\textsf{left} (\mathsf{cr}(\phi(\hat I'))+\mathsf{cr}(\psi(I))\cdot \mu^{O(b)}+|\chi^{\mathsf{dirty}}(\psi(I))|\cdot \check m\cdot \mu^{O(b)}\textsf{right} )+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}\\
&=\sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}+\sum_{I\in {\mathcal{I}}_2}\mathsf{cr}(\psi(I))\cdot \mu^{O(1)}+\sum_{I\in {\mathcal{I}}_2}|\chi^{\mathsf{dirty}}(\psi(I))|\cdot \check m\cdot \mu^{O(1)}.
\end{split}
\]
From \Cref{thm: phase 1}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\leq \check m^2/\mu^{c'}$, and Event $\tilde {\cal{E}}'$ did not happen, then
$\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}}{\check m}$.
Therefore, we get that $\mathsf{cr}(\check \phi)\leq \sum_{I'\in \hat {\mathcal{I}}'}\mathsf{cr}(\phi(I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}$. By suppressing the vertices that were used to subdivide the edges of graph $\check G^*$ to obtain graph $\check G$, we obtain a solution to the original instanc $\check I^*$ of the same cost.
\end{proof}
Lastly, the following observation will complete the proof of \Cref{lem: many paths}.
\begin{observation}\label{obs: cheap solutions to final instances}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)\leq \check m^2/\mu^{c'}$, and that event $\tilde {\cal{E}}'$ did not happen. Then $\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\cdot (\log \check m)^{O(1)}$.
\end{observation}
\begin{proof}
We bound $\sum_{I\in {\mathcal{I}}_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)$ and $\sum_{I\in {\mathcal{I}}_1}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')$ separately.
From \Cref{thm: phase 2}, if Event $\tilde {\cal{E}}'$ did not happen, then, for every instance $I=(G,\Sigma)\in {\mathcal{I}}_2$,
there is a solution $\psi(I')$ to the corresponding instance $I'$, that is clean with respect to ${\mathcal{J}}(I)$, with $\mathsf{cr}(\psi(I'))\leq \textsf{left} (\mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(G)|}{\mu^b}\textsf{right} )\cdot (\log \check m)^{O(1)}$.
From \Cref{obs: clean solution to contracted}, there is a solution to the corresponding contracted instance $\hat I'$, of cost at most $\mathsf{cr}(\psi(I'))$. Altogether, we get that:
%
\[\begin{split}
&\sum_{I\in {\mathcal{I}}_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')\leq \sum_{I\in {\mathcal{I}}_2}\textsf{left}(\mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot \check m}{\mu^b}\textsf{right} )\cdot (\log \check m)^{O(1)}.\\
&\leq \sum_{I\in {\mathcal{I}}_2}\mathsf{cr}(\psi(I))\cdot (\log \check m)^{O(1)}+\textsf{left} (\sum_{I\in {\mathcal{I}}_2}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} )^2\cdot (\log \check m)^{O(1)}+ \frac{\check m\cdot (\log \check m)^{O(1)}}{\mu^b}\cdot\textsf{left}(\sum_{I\in {\mathcal{I}}_2}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} ).
\end{split}
\]
%
From \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$ and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}}{\check m}$. Additionally, since we have assumed that $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)=\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$ for a large enough constant $c'$, $\textsf{left} (\sum_{I\in {\mathcal{I}}_2}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} )^2\leq \frac{(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))^2\cdot \mu^{1800}}{\check m^2}\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$. Altogether, we get that:
%
\[
\begin{split}
\sum_{I\in {\mathcal{I}}_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot (\log \check m)^{O(1)}+\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}\cdot (\log \check m)^{O(1)}}{\mu^b}\\
&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot (\log \check m)^{O(1)}, \end{split}\]
since $b\geq 4000$.
Next, we bound $\sum_{I\in {\mathcal{I}}_1}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)$. Consider some instance $I=(G,\Sigma)\in {\mathcal{I}}_1$ and the solution $\psi(I)$ that is ${\mathcal{J}}(I)$-valid. Let $J(I)$ be the core associated with the core structure ${\mathcal{J}}(I)$. Let $E^{\mathsf{dirty}}(I)\subseteq E(G)\setminus E(J(I))$ be the set of all edges $e$, such that the image of $e$ in $\psi(I)$ crosses the image of some edge of $J(I)$. Let $\hat I=(\hat G,\hat \Sigma)$ be the ${\mathcal{J}}(I)$-contracted instance corresponding to instance $I$.
Denote $G'=G\setminus E^{\mathsf{dirty}}(I)$, and let $\Sigma'$ be the rotation system for graph $G'$ that is induced by $\Sigma$. Observe that ${\mathcal{J}}(I)$ is a valid core structure for the resulting instance $I'=(G',\Sigma')$. Let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal{J}}(I)$-contracted instance associated with $I'$.
Observe that we can easily modify the solution $\psi(I)$ to instance $I$ to obtain a solution $\psi(I')$ to instance $I'$ that is clean with respect to ${\mathcal{J}}(I)$, with $\mathsf{cr}(\psi(I'))\leq \mathsf{cr}(\psi(I))$. Indeed, denote ${\mathcal{J}}(I)=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*)$.
Let $\psi'(I')$ be the solution to instance $I'$ induced by $\psi(I)$.
Since $G'=G\setminus E^{\mathsf{dirty}}(I)$, for every connected component $C$ of $G'$, either the images of all edges and vertices of $C$ in $\psi'(I')$ are contained in the region $F^*$ of the drawing, or the images of all edges and vertices of $C$ in $\psi'(I')$ are disjoint from $F^*$ (note that, if $E(C)\cap \delta_G(J)\neq \emptyset$, then the image of $C$ must be contained in $F^*$, since the image of $C$ must intersect the interior of $F^*$, from the definition of a valid core structure (see \Cref{def: valid core 2})). If the images of all edges and vertices of $C$ in $\psi'(I')$ are disjoint from $F^*$, then $C\cap J=\emptyset$ must hold, and so we can simply move the image of $C$ to lie in the interior of the region $F^*$ without changing the drawing of $C$ itself, and without introducing any new crossings. Once we move the image of each such connected component to lie inside region $F^*$, we obtain a solution $\psi(I')$ to instance $I'$ that is clean with respect to ${\mathcal{J}}(I)$, and $\mathsf{cr}(\psi(I'))\leq \mathsf{cr}(\psi(I))$.
From \Cref{obs: clean solution to contracted}, there is a solution $\psi(\hat I')$ to the contracted instance $\hat I'$ with $\mathsf{cr}(\psi(\hat I'))\leq \mathsf{cr}(\psi(I'))\leq \mathsf{cr}(\psi(I))$.
We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{dirty}}(I)$ into the drawing $\psi(\hat I')$ to obtain a solution $\psi(\hat I)$ of instance $\hat I$, with the number of crossings bounded by $\mathsf{cr}(\psi(\hat I'))+|E^{\mathsf{dirty}}(I)|\cdot |E(\hat G)|\leq \mathsf{cr}(\psi(I))+|E^{\mathsf{dirty}}(I)|\cdot |E(\hat G)|$.
Since $I\in {\mathcal{I}}_1$, $|E(\hat G)|\leq \frac{\check m}{\mu^{1000}}$, so $\mathsf{cr}(\psi(\hat I))
\leq \mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(I)|\cdot \frac{\check m}{\mu^{1000}}$. We then get that:
\[\sum_{I\in {\mathcal{I}}_1} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq
\sum_{I\in {\mathcal{I}}_1}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}_1}|\chi^{\mathsf{dirty}}(I)|\cdot \frac{\check m}{\mu^{1000}}.
\]
From \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$ and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}}{\check m}$.
Therefore:
\[ \sum_{I\in {\mathcal{I}}_1} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\frac{\check m}{\mu^{1000}} \cdot \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}}{\check m} \leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)).\]
Overall, we get that $\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')=\sum_{I\in {\mathcal{I}}_1} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)+\sum_{I\in {\mathcal{I}}_2} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot (\log \check m)^{O(1)}=\mathsf{OPT}_{\mathsf{cnwrs}}(\check I^*)\cdot (\log \check m)^{O(1)}$.
\end{proof}
In the remainder of this section we focus on the proof of \Cref{thm: phase 2}. Throughout the proof, we denote the instance $I=(G,\Sigma)$ that serves as the input to the algorithm by $\check I'=(\check G',\check \Sigma')$, with $|E(\check G')|$ denoted by $\check m'$. We denote the core structure ${\mathcal{J}}(I)$ by $\check{\mathcal{J}}=(\check J,\set{b_u}_{u\in V(\check J)},\rho_{\check J}, F^*)$. We can assume that there is a $\check{\mathcal{J}}$-valid solution $\check \psi$ to instance $\check I'$ with $\mathsf{cr}(\check \psi)\leq (\check m')^2/\mu^{240b}$, and $|\chi^{\mathsf{dirty}}(\check \psi)|\leq \check m'/\mu^{240b}$, since otherwise we can set $E^{\mathsf{del}}(\check I')=E(\check G')\setminus E(\check J)$, which trivially satisfies the requirements of the theorem. From now on we fix a $\check{\mathcal{J}}$-valid solution $\check \psi$ to instance $\check I'$, with $\mathsf{cr}(\check \psi)\leq (\check m')^2/\mu^{240b}$ and $|\chi^{\mathsf{dirty}}(\check \psi)|\leq \check m'/\mu^{240b}$. We emphasize that solution $\check \psi$ is not known to the algorithm.
\subsubsection{Completing the Proof of \Cref{lem: many paths}}
\label{subsubsec: finish the proof}
Given an input instance $\check I=(\check G,\check \Sigma)$, we first apply the algorithm from \Cref{thm: phase 1} to this input. If the algorithm fails, then we terminate the algorithm and return FAIL as well. Recall that, from \Cref{thm: phase 1}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/\mu^3$. Assume now that the algorithm from \Cref{thm: phase 1} did not fail. In this case, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then with probability at least $(1-1/\mu^3)$, the algorithm is successful. We denote by $\tilde {\cal{E}}_1'$ the bad event that the application of this algorithm is unsuccessful, so $\prob{\tilde {\cal{E}}_1'}\leq 1/\mu^3$. Let ${\mathcal{I}}$ be the collection of subinstances of $\check I$ computed by the algorithm from \Cref{thm: phase 1}. Recall that, if Event $\tilde {\cal{E}}'_1$ did not happen, then there is some constant $b^*$, and, for every instance $I\in {\mathcal{I}}$, there is a solution $\psi(I)$ to $I$, that is semi-clean with respect to ${\mathcal K}(I)$, such that
$\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. We set the parameter $b$ to be a large enough constant, so that $b\geq 2b^*$ holds.
Recall that, for every instance $I=(G,\Sigma)\in {\mathcal{I}}$, we have denoted by $\hat I=(\hat G,\hat \Sigma)$ the corresponding ${\mathcal K}(I)$-contracted instance, and by $\hat m(I)=E(\hat G)$. We say that instance $I$ is \emph{small} if $\hat m(I)\leq \frac{\check m}{\mu^{b^*}}$, and otherwise it is \emph{large}. We partition the set ${\mathcal{I}}$ of instances into three subsets: set ${\mathcal{I}}_1$ containing all small instances; set ${\mathcal{I}}_2$ containing all large instances $I=(G,\Sigma)\in {\mathcal{I}}$ for which either $\mathsf{cr}(\psi(I))> |E(G)|^2/\mu^{60b}$, or $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{60b}$ holds, and set ${\mathcal{I}}_3$ containing all remaining instances. Since the algorithm from \ref{thm: phase 1} guarantees that $\sum_{I\in {\mathcal{I}}}\hat m(I)\le \check m$, we get that $|{\mathcal{I}}_2\cup {\mathcal{I}}_3|\leq \mu^{b^*}$. We let $\hat {\mathcal{I}}_1=\set{\hat I\mid I\in {\mathcal{I}}_1}$ contain the set of all contracted instances corresponding to the instances of ${\mathcal{I}}_1$, and we define set $\hat {\mathcal{I}}_2$ of instances corresponding to the instances of ${\mathcal{I}}_2$ similarly.
\znote{We do not know $\mathsf{cr}(\psi(I))$ and $\chi^{\mathsf{dirty}}(\psi(I))$ so we cannot distinguish ${\mathcal{I}}_2$ from ${\mathcal{I}}_3$?}
Next, we consider every instance $I\in {\mathcal{I}}_3$ one by one. For each such instance $I$, we apply the algorithm from \Cref{thm: phase 2} to instance $I$, skeleton structure ${\mathcal K}(I)$, and the parameter $b$ defined above. Let $\tilde {\cal{E}}_2(I)$ be the bad event that this application of the algorithm was unsuccessful, so $\prob{\tilde {\cal{E}}_2(I)}\leq 1/\mu^{2b}$. We denote by $I'=(G',\Sigma')$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace, and by $\hat I'$ the ${\mathcal K}(I)$-contraction of this instance. We then denote $\hat {\mathcal{I}}_3'=\set{\hat I'\mid I\in {\mathcal{I}}_3}$. The final output of our algorithm is the collection ${\mathcal{I}}'=\hat{\mathcal{I}}_1\cup\hat {\mathcal{I}}_2\cup \hat {\mathcal{I}}_3'$ of subinstances of $\check I$.
We now verify that ${\mathcal{I}}'$ has all required properties. First, the algorithm from \Cref{thm: phase 1} ensures that, for every instance $I\in {\mathcal{I}}$, if the corresponding contracted instance $\hat I=(\hat G,\hat \Sigma)$ is a wide instance, then $|E(\hat G)|\le \check m/\mu$. If instance $I$ lies in ${\mathcal{I}}_3$, and $\hat I'=(\hat G',\hat\Sigma')$ is the ${\mathcal K}(I)$-contraction of instance $I'$, then $|E(\hat G')|\leq |E(\hat G)|$, and, if $\hat I$ is not a wide instance, then neither is $\hat I'$. This is since graph $\hat G'$ can be obtained from graph $\hat G$ by deleting the edges of $E^{\mathsf{del}}(I)$ from it. Therefore, we are guaranteed that, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}'$, if $I'$ is a wide instance, then $|E(G')|\le \check m/\mu$.
The algorithm from \Cref{thm: phase 1} also guarantees that $\sum_{I\in {\mathcal{I}}}\hm(I)\le \check m$. Since, for every instance $I\in {\mathcal{I}}_3$, the corresponding instance $\hat I'=(\hat G',\Sigma')\in \hat {\mathcal{I}}_3'$ has $|E(\hat G')|\leq \hat m(I)$, we get that $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\le \check m$.
Let $\tilde {\cal{E}}_2'$ be the bad event that any of the events in $\set{\tilde {\cal{E}}_2(I)\mid I\in {\mathcal{I}}_3}$ happened. Since, from the above discussion, $|{\mathcal{I}}_3| \leq \mu^{b^*}$, and, for every instance $I\in {\mathcal{I}}_3$, $\prob{\tilde {\cal{E}}_2(I)}\leq 1/\mu^{2b}$, while $b>2b^*$, from the Union Bound, we get that $\prob{\tilde {\cal{E}}_2'}<1/\mu^4$. Lastly, we let $\tilde {\cal{E}}'$ be the bad event that either $\tilde {\cal{E}}_1'$ or $\tilde {\cal{E}}_2'$ happened. Clearly, $\prob{\tilde {\cal{E}}'}\leq \prob{\tilde {\cal{E}}_1'}+\prob{\tilde {\cal{E}}_2'}\leq 1/\mu^2$.
We use the following two observations in order to complete the proof of \Cref{lem: many paths}.
\begin{observation}\label{obs: composing contracted solutions}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and that event $\tilde {\cal{E}}'$ did not happen. Then there is an efficient algorithm, that, given a solution $\phi(I')$ for every instance $I'\in {\mathcal{I}}'$, computes a solution $\check \phi$ to instance $\check I$, with $\mathsf{cr}(\check\phi)\leq \sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I')) + \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}$.
\end{observation}
\begin{proof}
We assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, Event $\tilde {\cal{E}}'$ did not happen, and that we are given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}'$. We now show an efficient algorithm to compute a solution $\check \phi$ to instance $\check I$. In order to do so, we consider every instance $I\in {\mathcal{I}}_3$ one by one, and compute a solution $\phi(\hat I)$ to instance $\hat I$, from the solution $\phi(\hat I')$ to instance $\hat I'$.
Consider now some instance $I=(G,\Sigma)\in {\mathcal{I}}_3$. Let $\hat I=(\hat G,\hat \Sigma)$ be the corresponding ${\mathcal K}(I)$-contracted instance, and let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal K}(I)$-contracted instance corresponding to the instance $I'$. Note that $V(\hat G)=V(\hat G')$ and $E(\hat G')=E(\hat G)\setminus E^{\mathsf{del}}(I)$. We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{del}}(I)$ into the solution $\phi(\hat I')$ to instance $\hat I'$, obtaining a solution $\phi(\hat I)$ to instance $\hat I$, whose cost is at most $\mathsf{cr}(\phi(\hat I'))+|E^{\mathsf{del}}(I)|\cdot |E(G)|$. If Event $\tilde {\cal{E}}'$ does not happen, then $|E^{\mathsf{del}}(I)|\leq \frac{\mathsf{cr}(\psi(I))\cdot \mu^{O(1)}}{|E(G)|}$, and so $\mathsf{cr}(\phi(\hat I))\leq \mathsf{cr}(\phi(\hat I'))+\mathsf{cr}(\psi(I))\cdot \mu^{O(1)}$.
Lastly, using the algorithm from \Cref{thm: phase 1}, we obtain a solution
$\check\phi$ to instance $\check I$, whose cost is bounded by:
\[
\begin{split}
\mathsf{cr}(\check \phi)&\leq \sum_{\hat I\in \hat {\mathcal{I}}}\mathsf{cr}(\phi(\hat I)) + \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}\\
&\leq \sum_{I\in {\mathcal{I}}_1\cup {\mathcal{I}}_2}\mathsf{cr}(\phi(\hat I))+\sum_{I\in {\mathcal{I}}_3}\textsf{left} (\mathsf{cr}(\phi(\hat I'))+\mathsf{cr}(\psi(I))\cdot \mu^{O(1)}\textsf{right} )+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}\\
&=\sum_{I'\in \hat {\mathcal{I}}'}\mathsf{cr}(\phi(I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}+\sum_{I\in {\mathcal{I}}_3}\mathsf{cr}(\psi(I))\cdot \mu^{O(1)}.
\end{split}
\]
From \Cref{thm: phase 1}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, and Event $\tilde {\cal{E}}'$ did not happen, then
$\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$.
Therefore, we get that $\mathsf{cr}(\check \phi)\leq \sum_{I'\in \hat {\mathcal{I}}'}\mathsf{cr}(\phi(I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{O(1)}$.
\end{proof}
Lastly, the following observation will complete the proof of \Cref{lem: many paths}.
\begin{observation}\label{obs: cheap solutions to final instances}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and that event $\tilde {\cal{E}}'$ did not happen. Then $\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log \check m)^{O(1)}$.
\end{observation}
\begin{proof}
We bound $\sum_{I\in {\mathcal{I}}_1\cup {\mathcal{I}}_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)$ and $\sum_{I\in {\mathcal{I}}_3}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')$ separately.
From \Cref{thm: phase 2}, if Event $\tilde {\cal{E}}'$ did not happen, then, for every instance $I=(G,\Sigma)\in {\mathcal{I}}_3$,
there is a solution $\psi(I')$ to the corresponding instance $I'$, that is clean with respect to ${\mathcal K}(I)$, with $\mathsf{cr}(\psi(I'))\leq \mathsf{cr}(\psi(I))\cdot (\log \check m)^{O(1)}+|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(G)|}{\mu^b}$.
From \Cref{obs: clean solution to contracted}, there is a solution to the corresponding contracted instance $\hat I'$, of cost at most $\mathsf{cr}(\psi(I'))$. Altogether, we get that:
%
\[\begin{split}
\sum_{I\in {\mathcal{I}}_3}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')&\leq \sum_{I\in {\mathcal{I}}_3}\textsf{left}(\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}_3}|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(G)|}{\mu^b}\textsf{right} )\cdot (\log \check m)^{O(1)}.\\
&\leq \sum_{I\in {\mathcal{I}}_3}\mathsf{cr}(\psi(I))\cdot (\log \check m)^{O(1)}+\textsf{left} (\sum_{I\in {\mathcal{I}}_3}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} )^2\cdot (\log \check m)^{O(1)}\\
&\hspace{4cm}+ \frac{\check m\cdot (\log \check m)^{O(1)}}{\mu^b}\cdot\textsf{left}(\sum_{I\in {\mathcal{I}}_3}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} ).
\end{split}
\]
%
From \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$ and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq O\textsf{left}(\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\textsf{right} )$. Additionally, since we have assumed that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$ for a large enough constant $c'$, $\textsf{left} (\sum_{I\in {\mathcal{I}}_3}|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} )^2\leq \frac{(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))^\cdot \mu^{2b^*}}{\check m^2}\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$. Altogether, we get that:
%
\[
\begin{split}
\sum_{I\in {\mathcal{I}}_3}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot (\log \check m)^{O(1)}+\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}\cdot (\log \check m)^{O(1)}}{\mu^b}\\
&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot (\log \check m)^{O(1)}, \end{split}\]
since $b>b^*$.
Next, we bound $\sum_{I\in {\mathcal{I}}_1\cup {\mathcal{I}}_2}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)$. Consider some instance $I=(G,\Sigma)\in {\mathcal{I}}_1$ and the solution $\psi(I)$ that is semi-clean with respect to ${\mathcal K}(I)$. Let $K(I)$ be the skeleton associated with the skeleton structure ${\mathcal K}(I)$. Let $E^{\mathsf{dirty}}(I)\subseteq E(G)\setminus E(K(I))$ be the set of all edges $e$, such that the image of $e$ in $\psi(I)$ crosses the image of some edge of $K(I)$. Let $\hat I=(\hat G,\hat \Sigma)$ be the ${\mathcal K}(I)$-contracted instance corresponding to instance $I$.
Denote $G'=G\setminus E^{\mathsf{dirty}}(I)$, and let $\Sigma'$ be the rotation system for graph $G'$ that is induced by $\Sigma$. Observe that ${\mathcal K}(I)$ is a valid skeleton structure for the resulting instance $I'=(G',\Sigma')$. Let $\hat I'=(\hat G',\hat \Sigma')$ be the ${\mathcal K}(I)$-contracted instance associated with $I'$. Observe that the drawing $\psi(I)$ of instance $I$ induces a drawing $\psi(I')$ of instance $I'$ that is clean with respect to ${\mathcal K}$. From \Cref{obs: clean solution to contracted}, there is a solution $\psi(\hat I')$ to the contracted instance $\hat I'$ with $\mathsf{cr}(\psi(\hat I'))\leq \mathsf{cr}(\psi(I'))\leq \mathsf{cr}(\psi(I))$.
We use the algorithm from \Cref{lem: edge insertion} in order to insert the edges of $E^{\mathsf{dirty}}(I)$ into the drawing $\psi(\hat I')$ to obtain a drawing $\psi(\hat I)$ of instance $\hat I$, with the number of crossings bounded by $\mathsf{cr}(\psi(\hat I'))+|E^{\mathsf{dirty}}(I)|\cdot |E(\hat G)|\leq \mathsf{cr}(\psi(I))+|E^{\mathsf{dirty}}(I)|\cdot |E(\hat G)|$.
We now consider three cases. First, if $I\in {\mathcal{I}}_1$, then $|E(\hat G)|\leq \frac{\check m}{\mu^{b^*}}$, so $\mathsf{cr}(\psi(\hat I))
\leq \mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(I)|\cdot \frac{\check m}{\mu^{b^*}}$ (since $|E(\hat G)|\leq \frac{\check m}{\mu^{b^*}}$). We then get that:
\[\sum_{I\in {\mathcal{I}}_1} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq
\sum_{I\in {\mathcal{I}}_1}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}_1}|\chi^{\mathsf{dirty}}(I)|\cdot \frac{\check m}{\mu^{b^*}}.
\]
From \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$ and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq O\textsf{left}(\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\textsf{right} )$.
Therefore, $\sum_{I\in {\mathcal{I}}_1} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\frac{\check m}{\mu^{b^*}} \cdot O\textsf{left}(\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\textsf{right} )\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))$.
Assume now that $I\in {\mathcal{I}}_2$. We partition the set ${\mathcal{I}}_2$ of instances into two subsets: set ${\mathcal{I}}_2'$ containing all instances $I=(G,\Sigma)$ with $\mathsf{cr}(\psi(I))> |E(G)|^2/\mu^{60b}$, and set ${\mathcal{I}}_2''$ containing all remaining instances.
As before, for every instance $I=(G,\Sigma)\in {\mathcal{I}}_2$, we denote by $\hat m(I)$ the number of edges in the contracted graph $\hat I$.
Next, we bound $\sum_{I\in {\mathcal{I}}_2'}\mathsf{cr}(\psi(\hat I'))$. Recall that for every instace $I=(G,\Sigma)\in {\mathcal{I}}_2'$:
\[\mathsf{cr}(\psi(I))> \frac{|E(G)|^2}{\mu^{60b}}\geq \frac{(\hat m(I))^2}{\mu^{60b}}\geq \frac{\hat m(I)\cdot \check m}{\mu^{60b+b^*}},\]
since instance $I$ must be large. Since,
from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq 2\check m^2/\mu^{c'}$, we get that $\sum_{I\in {\mathcal{I}}_2'}\hat m(I)\leq \frac{2\check m}{\mu^{c'-60b-b^*}}$. Lastly, since, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, we get that:
\[\begin{split}
\sum_{I\in {\mathcal{I}}_2'} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)&\leq
\sum_{I\in {\mathcal{I}}_2'}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}_2'}|\chi^{\mathsf{dirty}}(I)|\cdot \hm(I)\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\textsf{left} ( \sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))| \textsf{right} )\cdot \textsf{left} ( \sum_{I\in {\mathcal{I}}_2'}\hat m(I) \textsf{right} )\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\frac{2\check m}{\mu^{c'-60b-b^*}}\cdot \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\\
&\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)).\end{split}
\]
It now remains to consider instances $I\in {\mathcal{I}}_2''$. Recall that, for each such instance $I=(G,\Sigma)$, $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{2b}>\hat m(I)/\mu^{60b}$.
Since, from \Cref{thm: phase 1}, $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, we get that $\sum_{I\in {\mathcal{I}}_2''}\hat m(I)<\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*+60b}}{\check m}$.
Therefore:
\[\begin{split}
\sum_{I\in {\mathcal{I}}_2''} \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)&\leq
\sum_{I\in {\mathcal{I}}_2''}\mathsf{cr}(\psi(I))+\sum_{I\in {\mathcal{I}}_2''}|\chi^{\mathsf{dirty}}(I)|\cdot \hm(I)\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)+\textsf{left} ( \sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))| \textsf{right} )\cdot \textsf{left} ( \sum_{I\in {\mathcal{I}}_2''}\hat m(I) \textsf{right} )\\
&\leq 2\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(\check I) + \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*+60b}}{\check m}\cdot \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}\\
&\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))
,\end{split}
\]
%
since $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, and $c'$ is a large enough constant.
Overall, we get that $\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')=\sum_{I\in {\mathcal{I}}_1}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)+\sum_{I\in {\mathcal{I}}_2'}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)+\sum_{I\in {\mathcal{I}}_2''}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)+\sum_{I\in {\mathcal{I}}_3}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log \check m)^{O(1)}$.
\end{proof}
\iffalse
The calculations
---------------------------------------------------
\begin{itemize}
\item $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$
\item $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$
\item in ${\mathcal{I}}_2$: either $\mathsf{cr}(\psi(I))> |E(G)|/\mu^{2b}$, or $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{2b}$.
\item when pulling edges back, we will pay $\sum_{I\in {\mathcal{I}}_2}\hm(I)\cdot |\chi^{\mathsf{dirty}}(\psi(I))|$.
\item let's say that ${\mathcal{I}}_2'$ contains all instances with $\mathsf{cr}(\psi(I))> |E(G)|/\mu^{2b}>\hat m(I)/\mu^{2b}$. But $\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq 2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)<\check m/\mu^{c'}$. So $\sum_{I\in {\mathcal{I}}_2'}\hat m(I)\leq \mu^{2b}\cdot \check m/\mu^{c'}$. The increase is this multiplied by $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, so total increase in number of crossings is less than $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$.
\item Now ${\mathcal{I}}_2''$ contains all instances with $|\chi^{\mathsf{dirty}}(\psi(I))|> |E(G)|/\mu^{2b}$. But $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$. So $\sum_{I\in {\mathcal{I}}_2}\hat m(I)<\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*+b}}{\check m}$. The increase is this multiplied by $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{b^*}}{\check m}$, so the total increase in number of crossings is at most $\frac{(\mathsf{OPT}_{\mathsf{cnwrs}}(\check I))^2\cdot \mu^{2b^*+b}}{(\check m)^2}$. Assuming that $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)<\check m^2/\mu^{c'}$ it's at most $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$.
\end{itemize}
----------------------------------------------
\fi
In the remainder of this section we focus on the proof of \Cref{thm: phase 2}. Throughout the proof, we denote the instance $I=(G,\Sigma)$ that serves as the input to the algorithm by $\check I'=(\check G',\check \Sigma')$, with $|E(\check G')|$ denoted by $\check m'$. We denote the skeleton structure ${\mathcal K}(I)$ by $\check {\mathcal K}=({\mathcal{J}}_1,\ldots,{\mathcal{J}}_r)$, where $r\leq \mu^{100}$. For all $1\leq i\leq r$, we denote ${\mathcal{J}}_i=(J_i,\set{b_u}_{u\in V(J_i)},\rho_{J_i})$.
We start with the intuition for the proof. We denote by $\check K=\bigcup_{i=1}^rJ_i$ the skeleton of $\check G'$ associated with the skeleton structure $\check {\mathcal K}$. We can assume that there exists a solution $\check \psi$ to instance $\check I$ that is semi-clean with respect to $\check{\mathcal K}$ (as otherwise we can let $E^{\mathsf{del}}(\check I')=E(\check G)\setminus E(\check K)$ \znote{what does this mean?} ).
\subsection{Phase 3: Handling Augmented Flower Clusters}
\subsubsection*{Step 1: Moving the Vertices}
The goal of the first step is to prove the following claim.
\begin{claim}\label{claim: semi-clean drawing}
There is a solution $\psi_1$ to instance $I$ that is ${\mathcal K}$-valid,
such that, for every face $F\in \tilde {\mathcal{F}}'$, the images of all vertices of $G_F$ lie in region $F$ of the drawing. Additionally, $\mathsf{cr}(\psi_1)\leq (\mathsf{cr}(\phi)+N^2)\cdot (\log \check m')^{O(1)}$, $|\chi^{*}(\psi_1)|\leq |\chi^{*}(\phi)| \cdot (\log \check m')^{O(1)}$, and the total number of crossings in which the edges of $K$ participate in $\psi_1$ is at most $N\cdot (\log \check m')^{O(1)}$.
\end{claim}
\begin{proof}
Consider some face $F\in \tilde {\mathcal{F}}'$.
Recall that we have defined a core structure ${\mathcal{J}}_F$ associated with face $F$; we denote by $J_F$ the corresponding core graph, so $J_F\subseteq K$. We also denote by $I_F=(G_F,\Sigma_F)$ the instance associated with face $F$, where $G_F\in {\mathcal{G}}$.
Let $E'_F\subseteq E(G_F)$ be the set of edges $e=(x,y)\in E(G_F)$, such that either (i) the image of one of the vertices $x,y$ lies in region $F$ in $\phi$, and the image of the other vertex lies outside of $F$; or (ii) the images of both vertices lie outside region $F$ in $\phi$, and the image of $e$ crosses the boundary of $F$. Since ${\mathcal{J}}_F$ is a valid core structure for instance $I_F$ (see \Cref{def: valid core 2}), for every edge $e\in E(G_F)$ that is incident to a vertex $x\in V(J_F)$, a segment of $\phi(e)$ that contains $\phi(x)$ must be contained in region $F$ in $\phi$. Therefore, for every edge $e\in E'_F$, its image $\phi(e)$ intersects the interior of $F$, and it must cross the image of some edge of $J_F$.
Since the total number of crossings in which the edges of $K$ participate in $\phi$ is bounded by $N$, we get the following immediate observation:
\begin{observation}\label{obs: few boundary edges}
$\sum_{F\in \tilde {\mathcal{F}}'}|E'_F|\leq N$.
\end{observation}
Consider again some face $F\in \tilde {\mathcal{F}}'$.
It will be convenient for us to subdivide every edge of $E'_F$ with one or two vertices, and to adjust the drawing $\phi$ of $G$ to include these new vertices, as follows. Consider an edge $e=(x,y)\in E'_F$. Assume first that for both $x$ and $y$, their images in $\phi$ lie outside the region $F$. In this case, we replace edge $e$ in both $G$ and $G_F$ with a path $(x,t_e,t'_e,y)$. Consider now the image $\phi(e)$ of edge $e$ in drawing $\phi$, and direct it from $\phi(x)$ to $\phi(y)$. Let $p$ be the first point on $\phi(e)$ that belongs to the boundary of face $F$, and let $p'$ be the last point on $\phi(e)$ lying on the boundary of $F$. We place the image of the new vertex $t_e$ on curve $\phi(e)$ immediately next to point $p$, in the interior of face $F$. Similarly, we place the image of the new vertex $t'_e$ on curve $\phi(e)$ immediately next to point $p'$, in the interior of face $F$. Assume now that the image of one of the vertices $x,y$ lies in $F$ (for example, vertex $y$), and the image of the other vertex (vertex $x$) lies outside of $F$. We direct $\phi(e)$ from $\phi(x)$ to $\phi(y)$, and we let $p$ be the first point on $\phi(e)$ that lies on the boundary of $F$. We replace edge $e$ with a path $(x,t_e,y)$ in graph $G$ and in graph $G_F$, and we place the image of vertex $t_e$ on $\phi(e)$, next to point $p$, in the interior of face $F$. For convenience, the graph that is obtained from $G$ after these modifications is still denoted by $G$, and for every face
$F\in \tilde {\mathcal{F}}'$, the resulting graph associated with the face is still denoted by $G_F$. Since the newly added vertices all have degree $2$ in $G$, it is easy to extend the rotation system $\Sigma$ for graph $G$ to include these vertices, and we can similarly extend the rotation system $\Sigma_F$ for each graph $G_F\in {\mathcal{G}}$.
For a face $F\in \tilde {\mathcal{F}}'$, let $H_F\subseteq G_F$ be the graph whose vertex set contains every vertex $x\in V(G_F)$ with $\phi(x)\not \in F$, and every edge $e\in E(G_F)$ whose image in $\phi$ is disjoint from $F$. Let $E''_F\subseteq E(G_F)$ be the set of all edges $e\in E(G_F)$ with one endpoint in $H_F$ and another in $G_F\setminus H_F$. Observe that each such edge $e\in E''_F$ was obtained by subdividing some edge of $E'_F$, so $|E''_F|\leq 2|E'_F|$. For every edge $e=(x,y)\in E''_F$ with $x\in V(H_F)$, the intersection of $\phi(e)$ with the region $F$ is a very short segment of $\phi(e)$ that is incident to $\phi(x)$ (recall that $\phi(x)$ lies inside $F$ very close to its boundary). We denote by $Z_F$ the set of all endpoints of edges $e\in E''_F$ whose image lies in region $F$. We also note that graph $H_F$ is a subgraph of $G_F$ induced by $V(H_F)$. Indeed, if $e=(x,y)$ is an edge of $G_F$ with $x,y\in V(H_F)$, then the image of edge $e$ must be entirely disjoint from region $F$ (otherwise it would have been subdivided).
We need a slight modification of the algorithm for computing a well-linked decomposition from \Cref{thm:well_linked_decomposition}, that is summarized in the following theorem.
The proof is deferred to Section \ref{subsec: proof of wld cor} of Appendix.
\begin{theorem}\label{thm: wld all paths congestion}
There is an efficient algorithm, whose input is a graph $G$, a vertex-induced subgraph $S$ of $G$, and parameters $m$ and $\alpha$, for which $|E(G)|\leq m$ and $0<\alpha< \frac 1 {c\log^2 m}$ hold, for a large enough constant $c$.
The algorithm computes a collection ${\mathcal{R}}$ of vertex-disjoint clusters of $S$, and, for every cluster $R\in {\mathcal{R}}$, two sets ${\mathcal{P}}_1(R)$, ${\mathcal{P}}_2(R)$ of paths, such that the following hold:
\begin{itemize}
\item $\bigcup_{R\in {\mathcal{R}}}V(R)=V(S)$;
\item for every cluster $R\in{\mathcal{R}}$, $|\delta_G(R)|\le |\delta_G(S)|$;
\item every cluster $R\in{\mathcal{R}}$ has the $\alpha$-bandwidth property in graph $G$;
\item $\sum_{R\in {\mathcal{R}}}|\delta_G(R)|\le 4|\delta_G(S)|$;
\item for every cluster $R\in {\mathcal{R}}$, ${\mathcal{P}}_1(R)=\set{P_1(e)\mid e\in \delta_G(R)}$, where for every edge $e\in \delta_G(R)$, path $P_1(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, and all inner vertices of $P_1(e)$ lie in $V(S)\setminus V(R)$. Additionally, $\cong_G({\mathcal{P}}_1(R))\leq 400/\alpha$; and
\item for every cluster $R\in {\mathcal{R}}$, there is a subset $\hat E_R\subseteq \delta_G(R)$ of at least $\floor{|\delta_G(R)|/64}$ edges, such that ${\mathcal{P}}_2(R)=\set{P_2(e)\mid e\in \hat E_R}$, where for every edge $e\in \hat E_R$, path $P_2(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, and all inner vertices of $P_2(e)$ lie in $V(S)\setminus V(R)$. Moreover, $\cong_G\textsf{left}(\bigcup_{R\in {\mathcal{R}}}{\mathcal{P}}_2(R)\textsf{right} )\leq O\textsf{left} (\frac{\log m}{\alpha}\textsf{right} )$.
\end{itemize}
\end{theorem}
We apply the algorithm from \Cref{thm: wld all paths congestion} to graph $G_F$, subgraph $S=H_F$ of $G_F$, parameter $\check m'$ and $\alpha=\frac{1}{\log^4\check m'}$, to compute a collection ${\mathcal{R}}_F$ of vertex-disjoint clusters of $H_F$, such that $\bigcup_{R\in {\mathcal{R}}_F}V(R)=V(H_F)$, $\sum_{R\in {\mathcal{R}}_F}|\delta_{G_F}(R)|\le 4|E''_F|\leq 8|E'_F|$, and every cluster $R\in{\mathcal{R}}_F$ has the $\alpha$-bandwidth property in graph $G_F$.
Additionally, the algorithm computes,
for every cluster $R\in {\mathcal{R}}_F$, a set ${\mathcal{P}}_1(R)=\set{P_1(e)\mid e\in \delta_{G_F}(R)}$ of paths in graph $G_F$, with $\cong_{G_F}({\mathcal{P}}_1(R))\leq 400/\alpha\leq O(\log^4\check m')$, such that, for every edge $e\in \delta_{G_F}(R)$, path $P_1(e)$ has $e$ as its first edge and some edge of $E''_F$ as its last edge, and all inner vertices of $P_1(e)$ lie in $V(H_F)\setminus V(R)$.
It also computes, for every cluster $R\in {\mathcal{R}}_F$,
a subset $\hat E_R\subseteq \delta_G(R)$ of at least $\floor{|\delta_G(R)|/64}$ edges, and another set ${\mathcal{P}}_2(R)=\set{P_2(e)\mid e\in \hat E_R}$ of paths, where for every edge $e\in \hat E_R$, path $P_2(e)$ has $e$ as its first edge and some edge of $E''_F$ as its last edge, such that all inner vertices of $P_2(e)$ lie in $V(H_F)\setminus V(R)$, and the total congestion caused by the paths in $\bigcup_{R\in {\mathcal{R}}}{\mathcal{P}}_2(R)$ is at most $ O\textsf{left} (\frac{\log \check m'}{\alpha}\textsf{right} )\leq O(\log^5\check m')$.
We let $E'''_F=\bigcup_{R\in {\mathcal{R}}_F}\delta_{G_F}(R)$, so $E''_F\subseteq E'''_F$.
It will be convenient for us to further slightly modify graph $G$, by subdividing some of its edges, as follows. Consider a face $F\in \tilde {\mathcal{F}}'$ and an edge $e=(x,y)\in E'''_F$. If there are two distinct clusters $R,R'\in {\mathcal{R}}_F$ with $x\in R$ and $y\in R'$, then we replace edge $e$ with a path $(x,t_{e}^R,t_{e}^{R'},y)$ in $G$, and we denote edge $\tilde e=(t_e^R,t_{e}^{R'})$. We also modify the current drawing $\phi$ by placing the images of the newly added vertices $t_e^R,t_e^{R'}$ on $\phi(e)$. Otherwise, there must be a cluster $R\in {\mathcal{R}}_F$, such that one endpoint of $e$ (say $x$) lies in $R$, and the other endpoint (vertex $y$) lies in $Z_F$. In this case, we replace edge $e$ with a path $(x,t_e^R,y)$ in graph $G$, and we denote edge $\tilde e=(t_e^R,y)$. We place the image of vertex $t_e^R$ on $\phi(e)$, outside the region $F$.
Let $G'$ denote the final graph that is obtained from $G$ once every face $F\in \tilde {\mathcal{F}}'$ and every edge $e\in E'''_F$ is processed. For each face $F\in \tilde {\mathcal{F}}'$ we define a subgraph $G'_F\subseteq G'$ similarly, by subdividing the edges of $E'''_F$ as before, and we denote $\tilde E_F=\set{\tilde e\mid e\in E_F'''}$. We similarly update graph $H_F$, to obtain a new graph $H'_F\subseteq G'_F$, as follows. First, for every edge $e\in E(H_F)$, whose endpoints lie in different clusters of ${\mathcal{R}}_F$, we subdivide edge $e$ with two vertices as before. Additionally, for every edge $e=(x,y)\in E(G_F)$ with $x\in V(H_F)$ and $y\in Z_F$, we add the new edge $(x,t_e^R)$ that was obtained by subdividing $e$ to graph $H'_F$ (here, $R\in {\mathcal{R}}_F$ is the cluster containing $x$).
For every cluster $R\in {\mathcal{R}}_F$, the set ${\mathcal{P}}_1(R)$ of paths naturally defines a set ${\mathcal{P}}'_1(R)$ of paths in graph $G'_F$, where ${\mathcal{P}}'_1(R)=\set{P'_1(e)\mid e\in \delta_{G'_F}(R)}$, with $\cong_{G'_F}({\mathcal{P}}_1'(R))\leq O(\log^4\check m')$, such that, for every edge $e\in \delta_{G'_F}(R)$, path $P_1'(e)$ has $e$ as its first edge, and it terminates at some vertex of $Z_F$. Furthermore, all inner vertices of $P_1'(e)$ lie in $V(H'_F)\setminus V(R)$.
Similarly, set ${\mathcal{P}}_2(R)$ of paths naturally defines a set ${\mathcal{P}}'_2(R)$ of paths in graph $G'_F$, where each path in ${\mathcal{P}}'_2(R)$ starts at a distinct edge of $\delta_{G'_F}(R)$, terminates at some vertex of $Z_F$, and has all its inner vertices contained in $V(H'_F)\setminus V(R)$. As before, $|{\mathcal{P}}'_2(R)|\geq \floor{ |\delta_{G'_F}(R)|/64}$, and all paths in $\bigcup_{R\in {\mathcal{R}}_F}{\mathcal{P}}'_2(R)$ cause congestion at most $O(\log^5\check m')$.
Notice that drawing $\phi$ of $G$ naturally defines a drawing $\phi'$ of graph $G'$. We denote $\tilde E=\bigcup_{F\in \tilde {\mathcal{F}}'}\tilde E_F$. Observe that we have never subdivided the edges of the skeleton $K$, so $K\subseteq G'$ still holds. We can naturally extend the rotation system $\Sigma$ to graph $G'$, to obtain a rotation system $\Sigma'$, and we denote by $I'=(G',\Sigma')$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace.
Let $\phi''$ be any drawing of graph $G'$, and let $(e,e')_p$ be a crossing of $\phi''$. We say that crossing $(e,e')_p$ is \emph{uninteresting} if both $e,e'\in \tilde E$, and we say that this crossing is \emph{interesting} otherwise. We prove the following weaker analogue of \Cref{claim: semi-clean drawing}.
\begin{claim}\label{claim: semi-clean drawing2}
There is a solution $\psi_2$ to instance $I'=(G',\Sigma')$ that is ${\mathcal K}$-valid,
such that, for every face $F\in \tilde {\mathcal{F}}'$, the images of all vertices of $G'_F$ lie in region $F$ of the drawing. Additionally, the number of interesting crossings in $\psi_2$ is bounded by $\mathsf{cr}(\phi)\cdot (\log \check m')^{O(1)}$; $|\chi^{*}(\psi_2))|\leq |\chi^{*}(\phi)| \cdot (\log \check m')^{O(1)}$; and the total number of crossings in which the edges of $K$ participate in $\psi_2$ is at most $N\cdot (\log \check m')^{O(1)}$.
\end{claim}
The proof of \Cref{claim: semi-clean drawing} easily follows from \Cref{claim: semi-clean drawing2}. Indeed, consider the solution $\psi_2$ to instance $I'$. We apply type-1 uncrossing operation to the images of the edges in $\tilde E$ (see \Cref{subsec: uncrossing type 1} and \Cref{thm: type-1 uncrossing} for a formal description). The operation repeatedly selects pairs $e,e'\in \tilde E$ of edges that cross more than once, and then eliminates at least one of the crossings between these edges by a local uncrossing operation that ``swaps'' segments of images of these two edges without affecting the rest of the drawing. Therefore, if $\psi_3$ is the drawing of graph $G'$ obtained at the end of this procedure, then $\psi_3$ is a valid solution to instance $I'$ that remains ${\mathcal K}$-valid. Since the edges of the core $\check J$ may not belong to $\tilde E$, $|\chi^{*}(\psi_3))|\leq |\chi^{*}(\psi_2))|\leq |\chi^{*}(\phi))| \cdot (\log \check m')^{O(1)}$. The number of interesting crossings in $\psi_3$ is bounded by the number of interesting crossings in $\psi_2$, which, in turn, is bounded by $\mathsf{cr}(\phi)\cdot (\log \check m')^{O(1)}$. Since the edges of the skeleton $K$ may not lie in $\tilde E$, the total number of crossings in which the edges of $K$ participate in $\psi_3$ remains at most $N\cdot (\log \check m')^{O(1)}$. As before, for every face $F\in \tilde {\mathcal{F}}'$, for every vertex $x\in V(G'_F)$, $\psi_3(x)\in F$. The number of uninteresting crossings in $\psi_3$ is now bounded by $|\tilde E|^2$, as every pair of edges in $\tilde E$ may now cross at most once.
For every face $F\in \tilde F'$, $|\tilde E_F|=|E'''_F|=\sum_{R\in {\mathcal{R}}_F}|\delta_{G_F}(R)|\leq 8|E'_F|$. From \Cref{obs: few boundary edges},
$\sum_{F\in \tilde {\mathcal{F}}'}|E'_F|\leq N$. Therefore, $|\tilde E|\leq \sum_{F\in \tilde {\mathcal{F}}'}|\tilde E_F|\leq 8N$. The number of uninteresting crossings in $\psi_3$ is then bounded by $|\tilde E|^2\leq 64N^2$. We conclude that the total number of crossings in $\psi_3$ is bounded by
$(\mathsf{cr}(\phi)+N^2)\cdot (\log \check m')^{O(1)}$. Lastly, we can modify solution $\psi_3$ to instance $I'$ to obtain a solution $\psi_1$ to instance $I$ by suppressing the degree-$2$ vertices that we used to subdivide some of the edges of graph $G$. It is immediate to verify that this drawing has all required properties. In order to complete the proof of \Cref{claim: semi-clean drawing}, it is now enough to prove \Cref{claim: semi-clean drawing2}, which we do next.
Consider a face $F\in \tilde {\mathcal{F}}'$. We partition the edges of $\tilde E_F$ into two subsets: set $\tilde E'_F$ containing all edges $(x,y)$ with $x\in V(H'_F)$ and $y\in Z_F$, and set $\tilde E''_F$ containing all remaining edges.
For a cluster $R\in {\mathcal{R}}_F$, we denote $R^+=R\cup \delta_{G'_F}(R)$ the augmentation of cluster $R$ with respect to graph $G'_F$. We also denote by $T_R=\set{t_e^R\mid e\in \delta_{G_F}(R)}$ the set of vertices that serve as endpoints of the edges of $\tilde E_F$ and lie in $R^+$.
Note that every edge $e\in \tilde E''_F$ connects a vertex of $R^+_1$ to a vertex of $R^+_2$ for some pair $R_1,R_2\in {\mathcal{R}}_F$ of distinct clusters.
For each face $F\in \tilde {\mathcal{F}}'$ and cluster $R\in {\mathcal{R}}_F$, we denote by $\chi(R)$ the set of all crossings in the drawing $\phi'$ of $G'$ in which the edges of $R^+$ participate.
We first use the drawing $\phi'$ of $G'$ to compute, for each cluster $R\in {\mathcal{R}}_F$, a drawing $\psi_{R^+}$ of graph $R^+$ inside a disc $D(R)$, with the images of the vertices of $T_R$ lying on the boundary of the disc. We then select a location inside the region $F$, next to its boundary, into which we plant the disc $D(R)$ together with the drawing $\psi_{R^+}$ that is contained in it. Lastly, we modify the images of the edges of $\tilde E$ so that they connect the new images of their endpoints. All these modifications exploit the sets ${\mathcal{P}}_1'(R)$ and ${\mathcal{P}}_2'(R)$ of paths that we have defined for every cluster $R\in {\mathcal{R}}_F$ and face $F\in \tilde {\mathcal{F}}'$.
For a face $F\in \tilde {\mathcal{F}}'$ and a cluster $R\in {\mathcal{R}}_F$, we can now think of the paths in set ${\mathcal{P}}_1'(R)$ as routing the vertices of $T_R$ to vertices of $Z_F$ in graph $G'_F$ (after we discard the first edge from each such path), and similarly we can think of paths in ${\mathcal{P}}_2'(R)$ as routing a subset of at least $\floor{|T_R|/64}$ vertices of $T_R$ to vertices of $Z_F$ in graph $G'_F$. Recall that the paths in ${\mathcal{P}}'_1(R)\cup {\mathcal{P}}'_2(R)$ are internally disjoint from $V(R)$, and we can ensure that they are internally disjoint from $Z_F$. The paths in each set ${\mathcal{P}}_1'(R)$ cause congestion at most $O(\log^4\check m')$, and the paths in $\bigcup_{R\in {\mathcal{R}}(F)}{\mathcal{P}}_2'(R)$ cause congestion at most $O(\log^5\check m')$ in graph $G'_F$. Recall also that our transformation of the graph $G$ and the initial drawing $\phi$ ensures that the image of every vertex $t\in Z_F$ in $\phi'$ appears in the interior of the region $F$, very close to its boundary. If $e$ is the unique edge of $\tilde E'$ incident to $t$, then only a small segment of $\phi'(e)$ that is incident to $\phi'(t)$ is contained in $F$, and that segment does not participate in any crossings.
\paragraph{Computing the Drawings $\psi_{R^+}$.}
Consider a face $F\in \tilde {\mathcal{F}}'$ and a cluster $R\in {\mathcal{R}}_F$. We view the drawing $\phi'$ of $G'$ as a drawing on the sphere. Recall that, from the definition of graph $H_F$, for every edge $e\in E(H_F)$, the image of $e$ in $\phi'$ is disjoint from $F$.
Consider the disc $D(J_F)$, that is associated with the core $J_F$. Recall that disc $D(J_F)$ is a disc that contains the image of the core $J_F$ in its interior, and the boundary of $D(J_F)$ closely follows the boundary of the region $F$ inside the region (see \Cref{fig: core_disc_2}). We can assume w.l.o.g. that the images of all vertices of $Z_F$ lie on the boundary of the disc $D(J_F)$. We let $D(R)$ be the disc that is the complement of disc $D(J_F)$, that is, the boundaries of both discs are identical, but their interiors are disjoint. Note that, from the definition of the graph $H_F$, the images of all vertices and edges of graph $R^+$ lie in disc $D(R)$ in drawing $\phi'$.
Consider a graph $\tilde H_R$, containing all edges and vertices of $R^+$, and all edges and vertices that lie on the paths of ${\mathcal{P}}_1'(R)$. Let $\phi'(R)$ be the drawing of $\tilde H_R$ that is induced by the drawing $\phi'$ of $G'$.
We apply the algorithm from \Cref{cor: new type 2 uncrossing} to perform a type-2 uncrossing of the images of the paths in ${\mathcal{P}}_1'(R)$. The input to the algorithm is graph $\tilde H_R$, its subgraph $C=R^+$, and a set ${\mathcal{Q}}={\mathcal{P}}_1'(R)$ of paths, together with the drawing $\phi'(R)$ of $\tilde H_R$. We direct all paths in ${\mathcal{P}}'_1(R)$ away from the vertices of $T_R$. The algorithm computes a collection $\Gamma=\set{\gamma(t)\mid t\in T_R}$ of curves, where, for every vertex $t\in T_R$, curve $\gamma(t)$ originates at the image of $t$ in $\phi'(R)$, and terminates at the image of some vertex of $Z_F$, which lies on the boundary of disc $D(R)$.
For every pair $e,e'\in E(\tilde H_R)$ of distinct edges, let $N(e,e')$ denote the number of crossings in the drawing $\phi'(R)$ between edges $e$ and $e'$.
The algorithm from \Cref{cor: new type 2 uncrossing} also ensures that the curves in $\Gamma$ do not cross each other, and, for every edge $e\in E(R^+)$, the number of crossings between the image of $e$ in $\phi'(R)$ and the curves in $\Gamma$ is bounded by $\sum_{e'\in E(G_R)\setminus E(R^+)}N(e,e')\cdot \cong_{G_F}({\mathcal{P}}_1'(R),e')\leq (\log \check m')^{O(1)}\cdot \sum_{e'\in E(G_R)\setminus E(R^+)} N(e,e')$.
We are now ready to define the drawing $\psi_{R^+}$ of graph $R^+$. Recall that for every vertex $t\in T_R$, there is exactly one edge in $R^+$ that is incident to $t$, and we denote this edge by $e_t=(x_t,t)$, where $x_t\in V(R)$. The images of all vertices and edges of $R$ in $\psi_{R^+}$ remain the same as in $\phi'(R)$ (which are in turn identical to those in $\phi'$). For every vertex $t\in T_R$, the image of edge $e_t$ is obtained by concatenating the image of edge $e_t$ in $\phi'(R)$ and the curve $\gamma_t\in \Gamma$. The resulting curve connects the image of vertex $x_t$ in the current drawing, to some point $p$ on the boundary of disc $D(R)$. The image of vertex $t$ becomes that point $p$. We note that the image of $e_t$ is contained in disc $D(R)$. This completes the definition of the drawing $\psi_{R^+}$ of graph $R^+$. This drawing obeys the rotation system $\Sigma'$, is contained in disc $D(R)$, with the vertices of $T_R$ lying on the boundary of the disc. From the above discussion, the total number of crossings in this drawing is bounded by $(\log \check m')^{O(1)}\cdot |\chi(R)|$. For convenience, we define another disc $D'(R)$, that contains $D(R)$, so that the boundaries of both discs are disjoint.
\paragraph{Modifying the Drawing $\phi'$ of $G'$}
In this step we select, for every face $F\in \tilde {\mathcal{F}}'$ and every cluster $R\in {\mathcal{R}}_F$, a small disc $D''(R)$ in the interior of the region $F$ in $\phi'$. We will then copy the contents of disc $D'(R)$ (including the drawing $\psi_{R^+}$) into the disc $D''(R)$, and extend the images of all edges of $\tilde E_F$ that are incident to the vertices of $T_R$, so that they terminate at the boundary of the disc $D''(R)$. We will then ``stitch'' the images of these edges inside the region $D''(R)\setminus D(R)$, so that the image of each edge terminates at the image of its endpoint.
Consider a face $F\in \tilde {\mathcal{F}}'$, and a cluster $R\in {\mathcal{R}}_F$. Since cluster $R$ has
the $\alpha$-bandwidth property, for $\alpha=\frac{1}{\log^4\check m'}$, from \Cref{obs: wl-bw}, the set $T_R$ of vertices is $\alpha$-well-linked in $R^+$. We apply the algorithm from \Cref{lem: simple guiding paths} to compute, for every vertex $t\in T$, a set ${\mathcal{Q}}_t=\set{Q_t(t')\mid t'\in T_R\setminus\set{t}}$ of paths, where, for all $t'\in T_R\setminus\set{t}$, path $Q_t(t')$ connects $t'$ to $t$. Let $\hat T_R\subseteq T_R$ be the set containing all vertices $t\in T_R$, such that some path of ${\mathcal{P}}'_2(R)$ originates from $t$. We then select a vertex $t_R\in \hat T_R$ uniformly at random, and we let ${\mathcal{Q}}_R=\set{Q_t(t')\mid t'\in T_R\setminus\set{t_R}}$ be a collection of path connecting every vertex in $T_R\setminus\set{t_R}$ to $t_R$.
We need the following observation.
\begin{observation}\label{obs: low congestion outer routing paths}
For every edge $e\in E(R^+)$, $\expect{\cong({\mathcal{Q}}_R,e)}\leq O(\log^8\check m')$.
\end{observation}
\begin{proof}
Fix an edge $e\in E(R^+)$.
From \Cref{lem: simple guiding paths}, if we were to select a vertex $t\in T_R$ uniformly at random, then $\expect{\cong({\mathcal{Q}}_t,e)}\leq O\textsf{left} (\frac{\log^4\check m'}{\alpha}\textsf{right} )\leq O(\log^8\check m')$. Clearly, in the above process, a vertex $t\in T_R$ is selected with probability $1/|T_R|$. Our algorithm instead selects a vertex $t_R\in \hat T_R$ uniformly at random, so a vertex $t\in \hat T_R$ is selected with probability $\frac{1}{|\hat T_R|}\leq \frac{128}{|T_R|}$, since $|\hat T_R|\geq \floor{|T_R|}{64}$. Therefore, $\expect{\cong({\mathcal{Q}}_R,e)}\leq 128\expect[t\sim T_R]{\cong({\mathcal{Q}}_t,e)}\leq O(\log^8\check m')$.
\end{proof}
We construct another set ${\mathcal{Q}}'_R=\set{Q'(t')\mid t'\in T_R}$ of paths in graph $G'_F$, as follows. Consider the unique path $P_2'(t_R)\in {\mathcal{P}}_2'(R)$ that originates at vertex $t_R$. We denote by $z_R\in Z_F$ the other endpoint of path $P_2'(t_R)$; recall that the image of $z_R$ lies in region $F$, very close to its boundary. For every vertex $t'\in T_R\setminus \set{t_R}$, we let $Q'(t')$ be the
path obtained by concatenating path $Q(t')\in {\mathcal{Q}}_R$ with path $P'_2(t_R)$. For vertex $t_R$, we simply set $Q'(t_R)=P'_2(t_R)$. The resulting set ${\mathcal{Q}}'_R=\set{Q'(t')\mid t'\in T_R}$ of paths is contained in graph $G'_F$, and connects every vertex of $T_R$ to vertex $z_R\in Z_F$. Moreover, the only vertex of $G'_F\setminus H'_F$ that lies on the paths of ${\mathcal{Q}}'_R$ is vertex $z_R$. We assume w.l.o.g. that the paths in ${\mathcal{Q}}'_R$ are simple.
For every face $F\in \tilde {\mathcal{F}}'$, we denote ${\mathcal{Q}}(F)=\bigcup_{R\in {\mathcal{R}}_F}{\mathcal{Q}}'_R$.
We need the following observation.
\begin{observation}\label{obs: bound expected congestion}
For every face $F\in \tilde {\mathcal{F}}'$, for every edge $e\in E(H'_F)\cup \tilde E'_F$, $\expect{\cong_{G'_F}({\mathcal{Q}}(F),e)}\leq O(\log^8\check m')$.
\end{observation}
\begin{proof}
Consider some edge $e\in E(H'_F)\cup \tilde E'_F$.
If there is some cluster $R\in {\mathcal{R}}_F$ with $e\in E(R)$, then denote $R_e=R$; otherwise we let $R_e$ be undefined.
Recall that there are at most $O(\log^5\check m')$ paths in $\bigcup_{R\in {\mathcal{R}}_F}{\mathcal{P}}_2'(R)$ that contain the edge $e$. Let $S(e)$ be the collection of pairs $(R,t)$, where $R\in {\mathcal{R}}_F\setminus\set{R_e}$, $t\in \hat T_R$, and the uniue path of ${\mathcal{P}}_2'(R)$ that originates at $t$ contains the edge $e$. From the above discussion, $|S(e)|\leq O(\log^5\check m')$. Consider now a pair $(R,t)\in S(e)$. The probability that vertex $t$ is selected as vertex $t_R$ is bounded by $\frac{1}{|
\hat T_R|}\leq \frac{128}{|T_R|}$, since $|\hat T_R|\geq \floor{\frac{|T_R|}{64}}$. If vertex $t$ is selected as $t_R$, then every path in set ${\mathcal{Q}}'_R$ may contain the edge $e$, and the number of such paths is $|T_R|$. Therefore, the expected number of paths in $\bigcup_{R\in {\mathcal{R}}_F\setminus\set{R_e}}{\mathcal{Q}}'_R$ that contain edge $e$ is at most $O(\log^5\check m')$. If cluster $R_e$ is defined, then $\expect{\cong({\mathcal{Q}}'_{R_e},e)}\leq \expect{\cong({\mathcal{Q}}_{R_e},e)}\leq O(\log^8\check m')$. Therefore, overall, $\expect{\cong_{G'_F}({\mathcal{Q}}(F),e)}\leq O(\log^8\check m')$.
\end{proof}
For every face $F\in \tilde {\mathcal{F}}'$ and cluster $R\in {\mathcal{R}}_F$, we let $D''(R)$ be a very small disc lying in the interior of region $F$ of $\phi'$, right next to the image of vertex $z_R$. Notice that it is possible that for several distinct clusters $R,R'\in {\mathcal{R}}_F$, $z_R=z_{R'}$ holds. We ensure that all such discs in $\set{D''(R)}_{R\in {\mathcal{R}}_F}$ are mutually disjoint. Eventually, for every component $R\in {\mathcal{R}}_F$, we will plant the drawing $\psi_{R^+}$ inside disc $D''(R)$, so that the discs $D'(R)$ and $D''(R)$ will coincide. In order to modify the images of the edges of $\tilde E_F$, we define, for every cluster $R\in {\mathcal{R}}_F$, for every vertex $t\in T_R$, a curve $\gamma(t)$ connecting the image of $t$ in drawing $\phi'$ to the boundary of disc $D''(R)$. Consider now some edge $e\in \tilde E_F$. Assume first that $e\in \tilde E''_F$, and that $e=(t,t')$, with $t\in T_R$ and $t'\in T_{R'}$, for some clusters $R,R'\in {\mathcal{R}}_F$. In order to define a new image of edge $e$, we start by concatenating the curve $\gamma(t)$ with the image of edge $e$ in $\phi'$, and curve $\gamma(t')$, thereby obtaining a curve connecting a point on the boundary of disc $D''(R)$ to a point on the boundary of disc $D''(R')$. We then extend the curve within $D''(R)\setminus D(R)$ so that it originates at the new image of vertex $t$, and we similarly extend the curve within $D''(R')\setminus D(R')$ so that it terminates at the new image of vertex $t'$. Notice that all crossings between the images of the edges of $\tilde E$ are uninteresting crossings, and \Cref{claim: semi-clean drawing2} allows us to introduce arbitrary number of such crossings. However, we need to ensure that the number of new crossings between the new images of the edges of $\tilde E$ and the remaining edges of $G'$ is small. In order to do so, we need to ensure that the curves in sets $\set{\gamma(t)\mid t\in T_R}$, for all $F\in \tilde {\mathcal{F}}'$ and $R\in {\mathcal{R}}_F$ have few crossings with the images of edges of $G'$.
We now proceed to define the curves $\gamma(t)$, wich is the main component in the remainder of the proof. Intuitively, these curves will follow the images of the paths in ${\mathcal{Q}}(F)$, for all $F\in \tilde {\mathcal{F}}'$.
In order to do so, for every face $F\in \tilde {\mathcal{F}}'$, for every edge $e\in E(H'_F)\cup \tilde E'_F$, let $N_e$ denote the number of paths in set ${\mathcal{Q}}(F)$ containing the edge $e$.
Let $G''$ be a new graph that is obtained from $G'$ as follows. For every face $F\in \tilde {\mathcal{F}}'$, for every edge $e\in E(H'_F)\cup \tilde E'_F$, we add a collection $J(e)$ of $N_e+1$ parallel copies of edge $e$ to the graph. We then let $\phi''$ be a drawing of graph $G''$ that is obtained from the drawing $\phi'$ of graph $G'$ in a natural way: for every face $F\in \tilde {\mathcal{F}}'$ and edge $e\in E(H'_F)\cup \tilde E'_F$, we add images of $N_e+1$ copies of edge $e$ in parallel to the original drawing of edge $e$, very close to it.
Consider a crossing $(e,e')_p$ in this new drawing $\phi''$. We say that the crossing is of \emph{type 1}, if there is some face $F\in \tilde {\mathcal{F}}'$, such that both $e$ and $e'$ are copies of edges that lie in $E(H'_F)\cup \tilde E'_F$. Otherwise, we say that the crossing is of \emph{type 2}.
If a crossing $(e_1,e_2)_p$ in $\phi''$ is of type 2, then there is a crossing $(e_1',e_2')_{p'}$ in $\phi'$ in the vicinity of point $p$, such that either $e_1'=e_1$, or $e_1$ is a copy of edge $e_1'$, and similarly, either $e_2'=e_2$, or $e_2$ is a copy of edge $e_2'$. We say that crossing $(e_1',e_2')_{p'}$ in $\phi'$ is \emph{responsible} for crossing $(e_1,e_2)_p$ in $\phi''$.
Since,
from \Cref{obs: bound expected congestion}, for every face $F\in \tilde {\mathcal{F}}'$ and edge $e\in E(H'_F)\cup \tilde E'_F$, $\expect{\cong_{G'_F}({\mathcal{Q}}(F),e)}\leq O(\log^8\check m')$, and since the random choices made when computing sets ${\mathcal{Q}}(F)$ of paths for different faces $F\in \tilde {\mathcal{F}}'$ are independent, the expected number of type-2 crossings in $\phi''$ for which a single crossing in $\phi'$ is responsible is bounded by $O(\log^{16}\check m')$. Therefore, the expected number of type-2 crossings in $\phi''$ is at most $\mathsf{cr}(\phi')\cdot O(\log^{16}\check m')$.
Consider a face $F\in \tilde {\mathcal{F}}'$. We can use the set ${\mathcal{Q}}(F)$ of paths in graph $G'$ in a natural way, in order to define a set $\tilde {\mathcal{Q}}(F)=\set{\tilde Q(t)\mid t\in \bigcup_{R\in {\mathcal{R}}_F}T_R}$ of edge-disjoint paths in $G''$, where for every cluster $R\in {\mathcal{R}}_F$, for every vertex $t\in T_R$, path $\tilde Q(t)$ connects the image of $t$ in $\phi'$ to vertex $z_R$. Since, for every edge $e\in E(H'_F)\cup \tilde E'_F$, we have added $N_e+1$ copies of edge $e$ to $G''$, for every edge $e\in \tilde E_F$, there is a copy $e^*\in J(e)$ of edge $e$ (that we call \emph{distinguished copy}), that does not belong to any path in $\tilde {\mathcal{Q}}(F)$.
For every component $R\in {\mathcal{R}}_F$, for every vertex $t\in T_R$, we let $\gamma(t)$ be the image of path $\tilde Q(t)$ in $\phi''$. Notice that curve $\gamma(t)$ connects the image of $t$ in $\phi'$ to the image of vertex $z_R$. We slightly modify the final segment of $\gamma(t)$ so that it terminates at the boundary of disc $D''(R)$ (while ensuring that each such curve terminates at a different point on the boundary of the disc). We note that we are allowed to introduce arbitrary number of crossings between the curves in set $\set{\gamma(t)\mid t\in \bigcup_{R\in {\mathcal{R}}_F}T_R}$, as all such crossings will become unimportant crossings in the final drawing of graph $G'$ that we construct.
Consider now some edge $e\in \tilde E''_F$. Assume that $e=(t,t')$, with $t\in T_R$, $t'\in T_{R'}$, where $R,R'\in {\mathcal{R}}_F$ are two distinct clusters. We define a curve $\gamma'(e)$ representing the edge $e$ by concatenating the curve $\gamma(t)$, the image of the distinguished copy $e^*$ of $e$ in $\phi''$; and curve $\gamma(t')$. Notice that the resulting curve $\gamma'(e)$ connects a point on the boundary of disc $D''(t)$ to a point on the boundary of disc $D''(t')$.
Next, we consider some edge $e\in \tilde E'_F$. Assume that $e=(t,z)$, with $z\in Z_F$, and $t\in T_R$, for some cluster $R\in {\mathcal{R}}_F$. We let $\gamma'(e)$ be the curve obtained by concatenating the image of the distinguished copy $e^*$ of $e$ in $\phi''$ with the curve $\gamma(t)$. Therefore, curve $\gamma'(e)$ connects the image of vertex $z$ to a point on the boundary of disc $D''(t)$.
Consider the resulting set $\Gamma_F=\set{\gamma'(e)\mid e\in \tilde E_F}$ of curves. Notice that these curves may not be in general position, since it is possible that for some vertex $v\in V(H'_F)$, the point $\phi''(v)$ lies on more than two such curves (for example, this can happen when $v$ lies on several paths in ${\mathcal{Q}}(F)$). In order to overcome this difficulty, for every vertex $v\in V(H'_F)$ with point $\phi''(v)$ lying on more than two curves of $\Gamma_F$, we modify the curves of $\Gamma_F$ containing point $\phi'(v)$ within the tiny $v$-disc $D_{\phi'}(v)$, for example, as described in the nudging procedure (see \Cref{sec: curves in a disc}). This may introduce new crossings between the curves in $\Gamma_F$, but since these crossings will eventually become unimportant crossings of drawing $\psi_2$, we can afford to introduce an arbitrary number of such crossings.
We are now ready to define the solution $\psi_2$ to instance $I'$. Let $\tilde G=G'\setminus \bigcup_{F\in \tilde{\mathcal{F}}'}(H'_F\cup \tilde E'_F)$. The drawing of graph $\tilde G$ in $\psi_2$ remains the same as in $\phi'$. Consider now some face $F\in \tilde{\mathcal{F}}'$ and cluster $R\in {\mathcal{R}}_F$. We plant drawing $\psi_{R^+}$ inside disc $D''(R)$, so that the boundaries of discs $D'(R)$ and $D''(R)$ coincide. Recall that, in drawing $\psi_{R^+}$, the image of every vertex $t\in T_R$ appears on the boundary of the disc $D(R)$, and the images of all other vertices, and of all edges, are disjoint from $D'(R)\setminus D(R)$. It now remains to add the images of the edges in $\tilde E$ to this drawing. Consider again a face $F\in \tilde{\mathcal{F}}'$, and an edge $e\in \tilde E$. Initially, we let the image of $e$ be the curve $\gamma'(e)\in \Gamma_F$. We now need to modify this curve slightly so it connects the images of the endpoints of edge $e$. Assume first that $e\in \tilde E'_F$, and denote $e=(t,z)$, with $z\in Z_F$, and $t\in T_R$, for some cluster $R\in {\mathcal{R}}_F$. Then curve $\gamma'(e)$ connects the image of $z$ to a point on the boundary of disc $D''(R)$. Recall that the image of $t$ appears on the boundary of disc $D(R)$. We extend curve $\gamma'(e)$ within the region $D''(R)\setminus D(R)$, so that it terminates at the image of vertex $t$. Assume now that $e\in \tilde E''_F$, and denote $e=(t,t')$, where $t\in T_R$, $t'\in T_{R'}$, for some distinct clusters $R,R'\in {\mathcal{R}}_F$. Initially, we let the image of edge $e$ be the curve $\gamma'(e)$ that connects a point on the boundary of $D''(R)$ to a point on the boundary of $D''(R')$. We extend the curve inside the regions $D''(R)\setminus D(R)$ and $D''(R')\setminus D(R')$, so that it connects the image of $t$ to the image of $t'$. The extensions to the curves in $\Gamma_F$ can be performed so that they remain in general position; we may introduce an arbitrary number of new crossings between these curves, but we will not introduce any crossings between these curves and the images of the edges of $\tilde G$. This completes the definition of the solution $\psi_2$ to instance $I'$. We now ensure that this drawing has the required properties. Observe first that every interesting crossing in $\psi_2$ is either between a pair of edges in $\tilde G$ (and so it must be a type-$2$ crossing in drawing $\phi''$ of $G''$); or it is a crossing between a pair of edges of graph $R^+$ for some cluster $R\in \bigcup_{F\in \tilde{\mathcal{F}}'}{\mathcal{R}}_F$ (in which case it exists in drawing $\psi_{R^+}$); or it is a crossing between a curve in $\bigcup_{F\in \tilde{\mathcal{F}}'}\Gamma_F$ and an image of an edge of $\tilde G$ (in which case it corresponds to some type-2 crossing in drawing $\phi''$ of $G''$). Therefore, if we denote by $\chi_2(\phi'')$ the set of all type-2 crossings in drawing $\phi''$, then we get that the number of interesting crossings in $\psi_2$ is bounded by:
\[|\chi_2(\phi'')|+\sum_{F\in \tilde{\mathcal{F}}'}\sum_{R\in {\mathcal{R}}_F}\mathsf{cr}(\psi_{R^+}). \]
Recall that for every cluster $R\in \bigcup_{F\in \tilde{\mathcal{F}}'}{\mathcal{R}}_F$, $\mathsf{cr}(\psi_{R^+})\leq |\chi(R)|$, where $\chi(R)$ the set of all crossings in the drawing $\phi'$ of $G'$ in which the edges of $R^+$ participate. Therefore,
$\sum_{F\in \tilde{\mathcal{F}}'}\sum_{R\in {\mathcal{R}}_F}\mathsf{cr}(\psi_{R^+})\leq 2\mathsf{cr}(\phi')\leq 2\mathsf{cr}(\phi)$.
Since, from the above discussion, the expected number of type-2 crossings in $\phi''$ is bounded by $c\cdot \mathsf{cr}(\phi')\cdot\log^{16}\check m'\leq c\cdot\mathsf{cr}(\phi)\cdot\log^{16}\check m'$ for some large enough constant $c$, we get that the expected number of type-2 crossings in $\phi''$ is at most $ 4c\cdot\mathsf{cr}(\phi)\cdot\log^{16}\check m'$. We say that a bad event ${\cal{E}}_1$ happens if the number of type-2 crossings in $\phi''$ is greater than $ 16c\cdot\mathsf{cr}(\phi)\cdot\log^{16}\check m'$. From Markov's inequality, $\prob{{\cal{E}}_1}\leq 1/4$.
Next, we bound $|\chi^*(\psi_2)|$ -- the number of crossings in which the edges of the core $\check J$ participate. Notice that the edges of the core $\check J$ may not lie in $\bigcup_{F\in \tilde{\mathcal{F}}'}(H'_F\cup \tilde E'_F)$. Consider any crossing $(e,e')_p\in \chi^*(\psi_2)$, and assume that $e\in E(\check J)$. Then either $e'\in E(\tilde G)$ must hold (in which case crossing $(e,e')_p$ is a type-2 crossing in drawing $\phi''$ of $G''$), or $e'\in \tilde E$ (in which case curve $\gamma'(e')$ crosses the image of $e$). Since curve $\gamma'(e')$ was constructed by following the images of one or two paths in $\bigcup_{F\in \tilde{\mathcal{F}}'}\tilde Q(F)$ in $\phi''$, and using the image of edge $e'$ in $\phi''$, there is a crossing $(e,e'')_p$ in drawing $\phi''$, such that the image of edge $e''$ has a non-zero length intersection with curve $\gamma'(e')$.
Therefore, $|\chi^*(\psi_2)|\leq |\chi^*(\phi'')|$. It now remains to bound the number of crossings in which the edges of $\check J$ participate in $\phi''$. Consider any such crossing $(e,e'')_p$, with $e\in E(\check J)$. Then either $e''\in E(\tilde G)$, and crossing $(e,e'')_p$ also exists in drawing $\phi'$ of $G'$; or $e''$ is a copy of some edge $e'\in \bigcup_{F\in \tilde{\mathcal{F}}'}(H_F\cup \tilde E_F)$, and crossing $(e,e')_p$ also exists in drawing $\phi'$ of $G'$. In the former case, we say that crossing $(e,e'')_p$ in $\phi'$ is responsible for crossing $(e,e'')_p$ in $\phi''$, while in the latter case we say that crossing $(e,e')_p$ in $\phi'$ is responsible for crossing $(e,e'')_p$ in $\phi''$. Consider now some crossing $(e,e')_p$ in drawing $\phi'$. If $e'\in E(\tilde G)$, then this crossing may be responsible for at most one crossing in $\phi''$. Otherwise, there must be a face $F\in \tilde{\mathcal{F}}'$, with $e'\in E(H_F)\cup \tilde E_F$. In this case, crossing $(e,e')_p$ may be responsible for at most $N_{e'}+1$ crossings in $\phi''$. Since, from \Cref{obs: bound expected congestion}, for every face $F\in \tilde {\mathcal{F}}'$, for every edge $e'\in E(H'_F)\cup \tilde E'_F$, $\expect{N_{e'}+1}=\expect{\cong_{G'_F}({\mathcal{Q}}(F),e')+1}\leq O(\log^8\check m')$, we get that $\expect{|\chi^*(\psi_2)|}\leq c|\chi^*(\phi')|\log^8\check m'\leq c|\chi^*(\phi)|\log^8\check m'$ for some large enough constant $c$. We say that bad event ${\cal{E}}_2$ happens if $|\chi^*(\psi_2)|>4c|\chi^*(\phi)|\log^8\check m'$. As before, from Markov inequality, $\prob{{\cal{E}}_2}\leq 1/4$.
Lastly, we need to bound the number of crossings in which the edges of $E(K)$ participate in the drawing $\psi_2$. We denote by $N(\phi')$ and $N(\psi_2)$ the number of crossings in which the edges of $E(K)$ participate in drawings $\phi'$ and $\psi_2$, respectively. Recall that $N(\phi')$ is bounded by the number of crossings in which the edges of $K$ participate in $\phi$, which was denoted by $N$. From the definition, the edges of $E(K)$ may not lie in $\bigcup_{F\in \tilde{\mathcal{F}}'}(H'_F\cup \tilde E'_F)$. We can use the same argument that we used in bounding $|\chi^*(\psi_2)|$ to show that every crossing $(e,e'')_p$ in $\phi''$ in which $e\in E(K)$ can be mapped to some crossing $(e,e')_p$ in $\phi'$, such that the total number of crossings mapped to a single crossing $(e,e')_p$ is $1$ if $e'\in E(\tilde G)$ and $N_{e'}+1$ otherwise. Using the same reasoning as above, we get that $\expect{N(\phi'')}\leq c\cdot N(\phi')\cdot \log^8\check m'\leq c\cdot N\cdot \log^8\check m'$, where $c$ is a large enough constant. We say that bad event ${\cal{E}}_3$ happens if $N(\psi_2)>4c\cdot N\cdot \log^8\check m'$. As before, from Markov inequality, $\prob{{\cal{E}}_3}\leq 1/4$.
Using the Union Bound, with probability at least $1/4$ neither of the bad events ${\cal{E}}_1,{\cal{E}}_2,{\cal{E}}_3$ happens. Therefore, there must exist a solution $\psi_2$ to instance $I'=(G',\Sigma')$ that is ${\mathcal K}$-valid,
such that, for every face $F\in \tilde {\mathcal{F}}'$, the images of all vertices of $G'_F$ lie in region $F$ of the drawing, the number of interesting crossings in $\psi_2$ is bounded by $4c\cdot\mathsf{cr}(\phi)\cdot \log^{16} \check m'$, $|\chi^{*}(\psi_2)|\leq 4c\cdot |\chi^{*}(\phi)| \cdot \log^8\check m'$, and $N(\psi_2)\leq 4c\cdot N\cdot \log^8 \check m'$.
This completes the proof of \Cref{claim: semi-clean drawing2} and of \Cref{claim: semi-clean drawing}.
\end{proof}
\subsection*{Step 2: Moving the Bad Edges}
In this step we consider the ${\mathcal K}$-valid solution $\psi_1$ to instance $I$ that is guaranteed to exist from \Cref{claim: semi-clean drawing}. Recall that for every face $F\in \tilde {\mathcal{F}}'$, the images of all vertices of $G_F$ lie in region $F$ of $\psi_1$. Additionally, $\mathsf{cr}(\psi_1)\leq (\mathsf{cr}(\phi)+N^2)\cdot (\log \check m')^{O(1)}$, $|\chi^{*}(\psi_1))|\leq |\chi^{*}(\phi)| \cdot (\log \check m')^{O(1)}$, and the total number of crossings in which the edges of $K$ participate in $\psi_1$ is at most $N\cdot (\log \check m')^{O(1)}$.
For every face $F\in \tilde{\mathcal{F}}'$, we define a disc $D(F)$, that is contained in region $F$ of $\psi_1$, so that the boundary of disc $D(F)$ closely follows the boundary of the region $F$. Equivalently, we can think of disc $D(F)$ as the complement of the disc $D(J_F)$ associated with the core $J_F$ (see e.g. \Cref{def: valid drawing}); the boundaries of both discs coincide, and their interiors are disjoint. Note that, if we traverse the boundary of the disc $D(F)$ in counter-clock direction, we encounter the images of the edges $\delta_{G_F}(J_F)$ in the oriented circular ordering ${\mathcal{O}}(J_F)$. Notice however that images of additional edges of $G$ may cross the boundary of disc $D(F)$ -- edges whose images cross the image of some edge in $E(J_F)$. As before, we say that an edge of $G$ is \emph{bad} if its image in $\psi_1$ crosses the image of some edge of the original core $\check J$. Note that for every bad edge $e=(u,v)$, there must be a face $F\in \tilde{\mathcal{F}}'$ with $e\in E(G_F)$. Intuitively, in this step, for each face $F\in \tilde{\mathcal{F}}'$, we will modify the drawing that is contained in disc $D(F)$ in $\psi_1$, and add the images of the bad edges $e\in E(G_F)$ to this drawing. In order to do so in a modular fashion, so that the resulting drawings for different faces $F$ can be ``glued'' together, we will define a new graph that is associated with the face $F$, that, intuitively, will contain all vertices and (segments of) edges that are drawn inside disc $D(F)$ in $\psi_1$. We will exclude all bad edges that do not lie in $G_F$, and include all bad edges that lie in $G_F$. We will also subdivide the image of every edge that crosses the boundary of disc $D(F)$ with a new vertex, whose image will be placed on the boundary of disc $D(F)$. We view these new vertices as ``anchors''. Intuitively, we will allow the image of the graph associated with face $F$ to be modified arbitrarily, as long as it is contained in disc $D(F)$, and as long as the images of the anchors remain unchanged. This will allow us to replace the part of the drawing of graph $G$ that is contained in $D(F)$ with a new drawing, to which the bad edges that lie in $G_F$ are added.
We start by constructing a new graph $\tilde G$, that is obtained from graph $G$, by subdividing some edges of $G$. Notice that, if graph $\tilde G$ is obtained from graph $G$ in this way, then the rotation system $\Sigma$ for $G$ naturally defines a rotation system $\tilde \Sigma$ for $\textbf{G}$. We then obtain an instance $\tilde I=(\textbf{G},\tilde \Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace problem that we call \emph{instance defined by graph $\textbf{G}$}. We will also define a solution $\tilde \phi$ to instance $\tilde I$, that will be obtained from solution $\psi_1$ to instance $I$ in a natural way. Initially, we start with $\tilde G=G$ and $\tilde \phi=\psi_1$.
For every face $F\in \tilde{\mathcal{F}}'$, we also construct a set $A_F$ of \emph{anchor vertices}, whose image in $\tilde \phi$ appears on the boundary of the disc $D(F)$. Initially, we set $A_F=\emptyset$ for all $F\in \tilde{\mathcal{F}}'$.
Consider some face $F\in \tilde{\mathcal{F}}'$, an an edge $e=(x,y)$ that is incident to some vertex of the core $J_F$. Recall that exactly one endpoint of $e$ (say $x$) belongs to the core $J_F$. We subdivide edge $e$ with a new vertex $t_e$, so that edge $e$ is replaced with a path $(x,t_e,y)$. We also subdivide the image of edge $e$ in $\tilde \phi$ by placing the image of vertex $t_e$ on the first intersection point of $\tilde \phi(e)$ with the boundary of the disc $D(F)$, as we traverse $\tilde \phi(e)$ from $x$ to $y$. We add vertex $t_e$ to the set $A_F$ of anchor vertices. In our final graph $\textbf{G}$, we will delete the edge $(x,t_e)$, and we will view edge $(t_e,y)$ as representing the original edge $e$.
Consider the current graph $\textbf{G}$, and its current drawing $\tilde \phi$. We say that an edge $e\in E(\textbf{G})$ is \emph{good} if its image in $\tilde \phi$ does not cross the image of any edge in $K$. Since we have subdivided the edges incident to the vertices of $K$ that do not lie in $E(K)$, for every face $F\in \tilde{\mathcal{F}}'$, every edge of $\textbf{G}$ that is incident to a vertex of $J_F$ is a good edge. We say that an edge $e\in E(\textbf{G})$ is \emph{bad} if its image in the current drawing $\tilde \phi$ crosses the image of some edge of the original core $\check J$. Notice that the total number of bad edges is bounded by $|\chi^*(\tilde \phi)|\leq |\chi^*(\psi_1)|\leq |\chi^*(\phi)|\cdot (\log \check m')^{O(1)}$ from \Cref{claim: semi-clean drawing}. Notice also that for every bad edge $e\in E(\textbf{G})$, there must be a face $F\in \tilde{\mathcal{F}}'$, such that the images of both endpoints of $e$ lie in the disc $D(F)$. Lastly, we say that an edge $e\in E(\textbf{G})$ is a \emph{migrating edge} if it is neither good nor bad. In this case, the image of $e$ must cross the image of some edge in $E(K)\setminus E(\check J)$. Next, we process all bad edges and all migrating edges.
Consider first a bad edge $e=(x,y)$, and assume that the images of both $x$ and $y$ lie inside disc $D(F)$ for some face $F\in \tilde{\mathcal{F}}'$. Then the image of edge $e$ in $\tilde \phi$ must intersect the boundary of the disc $D(F)$ in at least two points. We direct the image of $e$ in $\tilde \phi$ from $x$ to $y$, denote by $p(e)$ the first point on the boundary of disc $D(F)$ that lies on $\tilde \phi(e)$, and by $p'(e)$ the last point on the boundary of disc $D(F)$ that lies on $\tilde \phi(e)$. If $p(e)$ is the image of the vertex $x$, then we denote $t_e=x$ (in this case, $x$ is already added to the set $A_F$ of anchor vertices for face $F$). Otherwise, we subdivide the edge $e$ with a new vertex $t_e$, whose image is placed at point $p(e)$, and we add $t_e$ to the set $A_F$ of anchor vertices. Similarly, if $p'(e)$ is the image of the vertex $y$ in $\tilde \phi$, then we denote $t'_e=y$. Otherwise, we subdivide the edge $e$ with a new vertex $t'_e$, whose image is placed at point $p'(e)$, and we add $t'_e$ to the set $A_F$ of anchor vertices. If edge $e$ has been subdivided twice, then we have replaced it with path $(x,t_e,t'_e,y)$. If $x\neq t_e$, then edge $(x,t_e)$ now becomes a good edge. This edge does not represent any edge in the original graph $G_F$, so we call it an \emph{extra edge}. Similarly, if $y\neq t'_e$, then edge $(y,t'_e)$ now becomes a good edge, and we also call it an extra edge. The edge $(t_e,t'_e)$ is a bad edge, and we view it as representing the bad edge $e$. Note that the images of both endpoints of this edge now appear on the boundary of disc $D(F)$. We say that face $F$ \emph{owns} this bad edge.
Lastly, we consider a migrating edge $e=(x,y)$. Note that there must be a face $F\in \tilde{\mathcal{F}}'$, such that the images of both $x$ and $y$ lie in disc $D(F)$ in $\tilde \phi$. We direct the image of the edge $e$ in $\tilde \phi$ from $x$ to $y$. Denote by $F_1,F_2,\ldots,F_r$ the sequence of the regions of $\tilde{\mathcal{F}}'$ that the image of the edge $e$ visits, in the order in which it visits them, so $F_1=F$ and $F_r=F$.
For all $1<i<r$, we let $\sigma_i$ be the maximal segment on the image of $e$ that is contained in disc $D(F_i)$, and we denote by $p_i(e)$ and $p'_i(e)$ the first and the last endpoints of $\sigma_i$, respectively, that must lie on the boundary of disc $D(F_i)$. We also let $\sigma_1$ the segment of $\tilde \phi(e)$ from the image of $x$ to the first point that lies on the boundary of disc $D(F)$, denoting by $p'_1(e)$ the endpoint of $\sigma_1$ that is different from the image of $x$. Similarly, we let $\sigma_r$ be the segment of $\tilde \phi(e)$ from the last point that lies on the boundary of disc $D(F)$ to the image of $y$, denoting by $p_r(e)$ the endpoint of $\sigma_r$ that is different from the image of $y$. Note that $\tilde \phi(e)\setminus \textsf{left}(\bigcup_{i=1}^r\sigma_i\textsf{right} )$ is a collection of short segments, each of which lies outside of $\bigcup_{F\in \tilde{\mathcal{F}}'}D(F')$, and crosses some edge of $K$.
We subdivide edge $e$, replacing it with a path $(x,t'_1,t_2,t'_2,\ldots,t_r,y)$. For all $1<i\leq r$, we place the image of the new vertex $t_i$ at point $p_i(e)$, and for all $1\leq i<r$, we place the image of the new vertex $t'_i$ at point $p'_i(e)$. Denote $t_1=x$ and $t'_r=y$. For $1\leq i\leq r$, denote by $e_i$ the new edge $(t_i,t'_i)$, and for $1\leq i<r$, denote by $e'_i$ the new edge $(t'_i,t_{i+1})$. Notice that, for all $1\leq i\leq r$, the image of edge $e_i$ is precisely the segment $\sigma_i$, which is contained in disc $D(F_i)$. Both endpoints of the edge are drawn on the boundary of the disc $D(F_i)$ (except that, for $i=1$, the first endpoint of $\sigma_1$ is the image of $x$ that may lie in the interior of disc $D(F)$, and, for $i=r$, the last endpoint of $\sigma_r$ is the image of vertex $y$ that may lie in the interior of disc $D(F)$). For all $1\leq i\leq r$, edge $e_i$ now becomes a good edge, and we also call it an \emph{extra edge}. As discussed above, the images of edges $e'_1,\ldots,e'_{r-1}$ are short segments that lie outside of $\bigcup_{F'\in \tilde{\mathcal{F}}'}D(F')$ (except for their endpoints that lie on the boundaries of the discs), and each such segment crosses some edge of $K$. For all $1< i<r$, we add the new vertices $t_i$ and $t'_i$ to the set $A_{F_i}$ of anchor vertices for face $F_i$. Additionally, we add $t'_1$ and $t_r$ to $A_F$.
Consider the final graph $\textbf{G}$ obtained after all bad and migrating edges have been procesed, and the resulting solution $\tilde \phi$ to the instance $\tilde I$ defined by $\textbf{G}$. From the above discussion, the total number of extra edges in $\textbf{G}$ is bounded by
$2N(\psi_1)+2|\chi^*(\psi_1)|$, where $N(\psi_1)$ is the number of crossings in which the edges of the skeleton $K$ participate in $\psi_1$. Recall that $N(\psi_1)\leq N\cdot (\log \check m')^{O(1)}\leq \frac{\check m'}{\mu^{3b}}\cdot (\log \check m')^{O(1)}$, and $|\chi^*(\psi_1)|\leq |\chi^{*}(\psi)| \cdot (\log \check m')^{O(1)}\leq \frac{\check m'}{ \mu^{240b}}\cdot (\log \check m')^{O(1)}$. Therefore, the total number of exta edges in
graph $\textbf{G}$ is bounded by $\frac{\check m'}{\mu^{2b}}$.
For every face $F\in \tilde{\mathcal{F}}'$, we now define a subgraph $\textbf{G}_F$ of graph $\textbf{G}$ associated with face $F$. The set of vertices of $\textbf{G}_F$ contains all vertices whose images appear in disc $D(F)$ in the current drawing $\tilde \phi$. Notice that this includes all vertices in the original graph $G_F$ (except for vertices that belong to the core $J_F$), and, additionally, new vertices whose images were added to the boundary of $D(F)$, and were added to the set $A_F$ of anchor vertices. Therefore, $V(\textbf{G}_F)=(V(G_F)\setminus V(J_F))\cup A_F$. The set of edges consists of all edges of $\textbf{G}$ whose image in $\tilde \phi$ is contained in the disc $D(F)$, and all bad edges that belong to face $F$ (that is, all bad edges whose both endpoints lie in $A_F$). Notice that, if $e$ is an edge of $\textbf{G}_F$, then either $e$ is an edge of $G_F$, or it was obtained by subdividing some edge of $G_F$, or $e$ was obtained by subdividing some migrating edge $e'$ that lied in some graph $G_{F'}$ for $F'\neq F$. In the latter case, the endpoints of $e'$ belong to the set $A_F$ of anchors, and edge $e'$ with its endpoints forms a separate connected component in graph $\textbf{G}_F$.
We need the following observation.
\begin{observation}\label{obs: small boundary cuts}
Let $F\in \tilde{\mathcal{F}}'$ be a face, and let $(S,T)$ be a partition of the set $A_F$ of the anchor vertices, so that the images of the vertices of $S$ in $\tilde \phi$ appear consecutively on the boundary of disc $D(F)$. Then there is a collection $E'$ of at most $4\check m'/\mu^{2b}$ edges in graph $\textbf{G}_F$, so that there is no path connecting a vertex of $S$ to a vertex of $T$ in $\textbf{G}_F\setminus E'$.
\end{observation}
\begin{proof}
Assume otherwise. From the max-flow min-cut theorem, there is a collection ${\mathcal{P}}$ of $\ceil{4\check m'/\mu^{2b}}$ edge-disjoint paths in graph $\textbf{G}_F$ connecting vertices of $S$ to vertices of $T$. Since there are at most $\check m'/\mu^{2b}$ extra edges in graph $\textbf{G}$, there is a subset ${\mathcal{P}}'\subseteq {\mathcal{P}}$ of at least $\ceil{2\check m'/\mu^{2b}}$ paths that do not contain extra edges. Therefore, every path in ${\mathcal{P}}'$ only contains edges that were obtained by subdividing the edges of $G_F$. Observe that the anchor vertices representing the edges in $\delta_{G_F}(J_F)$ appear on the boundary of disc $D(F)$ in drawing $\tilde \phi$ according to the ordering ${\mathcal{O}}(J_F)$. Therefore, partition $(S,T)$ of vertices of $A_F$ naturally induces a partition $(E_1,E_2)$ of the set $\delta_{G_F}(J_F)$ of edges, where the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}(J_F)$. But then the existence of the set ${\mathcal{P}}'$ of paths contradicts the fact that $I_F$ is an acceptable instance (see \Cref{def: acceptable instance}).
\end{proof}
For a face $F\in \tilde{\mathcal{F}}'$, we denote by $\chi(F)$ the set of all crossings in the drawing $\tilde \phi$ in which the edges of $\textbf{G}_F$ participate, and we denote by $E^{\operatorname{bad}}(F)$ the set of all bad edges that face $F$ owns.
The proof of \Cref{lem: computed decomposition is good enough} follows from the following claim.
\begin{claim}\label{claim: rearrange drawing in disc}
For every face $F\in \tilde{\mathcal{F}}'$, there is a solution $\psi_F$ to instance $\tilde I_F$, with:
$$\mathsf{cr}(\psi_F)\leq \textsf{left}(|\chi(F)|+|E^{\operatorname{bad}}(F)|^2+|E^{\operatorname{bad}}(F)|\cdot \frac{\check m'}{\mu^{2b}}\textsf{right} )\cdot (\log \check m')^{O(1)},$$ such that the drawing of the graph $\textbf{G}_F$ is contained in disc $D(F)$, and, for every anchor vertex $t\in A_F$, the image of $t$ in $\psi_F$ is identical to its image in $\tilde \phi$.
\end{claim}
We provide the proof of \Cref{claim: rearrange drawing in disc} below, after we complete the proof of \Cref{lem: computed decomposition is good enough} using it. We start from the solution $\tilde \phi$ to instance $\tilde I$. For every face $F\in \tilde{\mathcal{F}}'$, we delete the contents of the disc $D(F)$, and replace them with the contents of disc $D(F)$ in drawing $\psi_F$ of graph $\textbf{G}_F$. Since the images of the anchor vertices of $A_F$ remain unchanged, once every face $F\in \tilde{\mathcal{F}}'$ is processed, we obtain a valid solution to instance $\tilde I$, that we denote by $\tilde \psi'$. Since, for every face $F\in \tilde{\mathcal{F}}'$, for every bad edge $e\in E^{\operatorname{bad}}(F)$, the image of $e$ now lies in disc $D(F)$, for every bad face $F'\in \tilde{\mathcal{F}}^{\operatorname{X}}$, the image of every edge of $\textbf{G}$ in $\tilde \psi'$ is disjoint from the interior of $F'$.
Since graph $\textbf{G}$ was obtained from graph $G$ by subdividing some of its edges, solution $\tilde \psi'$ to instance $\tilde I$ naturally defines a solution $\psi$ to instance $I$, by suppressing the images of vertices that were used to subdivided edges. From the above discussion, the resulting solution $\psi$ to instance $I$ is clean with respect to $\check{\mathcal{J}}$. Moreover:
\[
\begin{split}
\mathsf{cr}(\psi)&\leq \mathsf{cr}(\tilde \phi')\\
&\leq \mathsf{cr}(\tilde \phi)+\sum_{F\in \tilde{\mathcal{F}}'}\mathsf{cr}(\psi_F)\\
&\leq\mathsf{cr}(\tilde \phi)+\sum_{F\in \tilde{\mathcal{F}}'}\textsf{left}(|\chi(F)|+|E^{\operatorname{bad}}(F)|^2+|E^{\operatorname{bad}}(F)|\cdot \frac{\check m'}{\mu^{2b}}\textsf{right} )\cdot (\log \check m')^{O(1)}\\
&\leq \mathsf{cr}(\tilde \phi)\cdot (\log \check m')^{O(1)} +\sum_{F\in \tilde{\mathcal{F}}'}|E^{\operatorname{bad}}(F)|^2\cdot (\log \check m')^{O(1)}+|\chi^*(\psi_1)|\cdot \frac{\check m'\cdot (\log \check m')^{O(1)}}{\mu^{2b}}\\
&\leq (\mathsf{cr}(\phi)+N^2)\cdot (\log \check m')^{O(1)}+|\chi^*(\psi_1)|^2\cdot (\log \check m')^{O(1)}+ |\chi^{*}(\phi)|\cdot \frac{\check m'\cdot (\log \check m')^{O(1)}}{\mu^{2b}}\\
&\leq \textsf{left}(\mathsf{cr}(\phi)+N^2+|\chi^*(\phi)|^2+ \frac{\check m'\cdot|\chi^{*}(\phi)|}{\mu^{2b}}\textsf{right} ) (\log \check m')^{O(1)}.
\end{split}
\]
In order to complete the proof of \Cref{lem: computed decomposition is good enough}, it is now enough to prove \Cref{claim: rearrange drawing in disc}, which we do next.
\section{Third Key Tool: Computing Guiding Paths}
\label{subsec: guiding paths}
We start with intuition. Consider an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, and let $C\subseteq G$ be a subgraph of $G$ that we refer to as a cluster. Fix an optimal solution $\phi^*$ to instance $I$; we define $\mathsf{cr}(C)$ as usual with respect to $\phi^*$ -- the number of crossings in $\phi^*$ in which the edges of $C$ participate. For a set ${\mathcal{Q}}$ of paths in $G$, the cost $\operatorname{cost}({\mathcal{Q}})$ is also defined as before, with respect to $\phi^*$.
Assume that the boundary of $C$ is $\alpha$-well-linked in $C$, and denote $|E(C)|/|\delta(C)|=\eta$. We show below a randomized algorithm that either computes a vertex $u\in V(C)$ and a set ${\mathcal{Q}}$ of paths routing $\delta(C)$ to $u$, whose expected cost is comparable to $\mathsf{cr}(C)$; or with high probability correctly establishes that $\mathsf{cr}(C)>|\delta(C)|^2/\operatorname{poly}(\eta)$.
We say that cluster $C$ is \emph{good} if the former outcome happens, and we say that it is \emph{bad} otherwise.
Intuitively, given our input instance $I=(G,\Sigma)$, we can compute a decomposition ${\mathcal{C}}$ of $G$ into clusters, such that for each cluster $C\in {\mathcal{C}}$, the boundary of $C$ is $1/\operatorname{poly}\log m$-well-linked in $C$, and moreover $|\delta(C)|\geq |E(C)|/\operatorname{poly}(\mu)$. In order to compute the decomposition, we start with ${\mathcal{C}}$ containing a single cluster $G$, and the interate. In every iteration, we select a cluster $C\in {\mathcal{C}}$, and then either compute a sparse balanced cut $(A,B)$ in $C$, or compute a cut $(A,B)$ that is sparse with respect to the edges of $\delta(C)$ (that is, $|E(A,B)|<\min\set{|\delta(C)\cap \delta(A)|,|\delta(C)\cap \delta(B)|}/\operatorname{poly}\log n$; or return FAIL. In the former two cases, we replace $C$ with $C[A]$ and $C[B]$ in ${\mathcal{C}}$, while in the latter case we declare the cluster $C$ \emph{settled}; we will show that in this case the boundary of $C$ is $1/\operatorname{poly}\log n$-well-linked in $C$ and $|\delta(C)|\geq |E(C)|/\operatorname{poly}(\mu)$. The algorithm terminates once every cluster in ${\mathcal{C}}$ is settled. We show that each of the resulting clusters $C\in {\mathcal{C}}$ is sufficiently small (that is, $|E(C)|\leq m/\mu$), and the total number of edges connecting different clusters is also sufficiently small (at most $m/\mu$). If each resulting cluster is a good cluster, then we obtain the desired decomposition ${\mathcal{C}}$, together with the desired collections $\set{{\mathcal{Q}}(C)}_{C\in {\mathcal{C}}}$ of paths. Unfortunately, some of the clusters $C\in {\mathcal{C}}$ may be bad. For each such cluster we are guaranteed with high probability that $\mathsf{cr}(C)>|\delta(C)|^2/\operatorname{poly}(\mu)$, but unfortunately the sum of values of $|\delta(C)|^2$ over all bad clusters $C$ may be quite small, significantly smaller than say $|E(G)|^2$, so we cannot conclude that $\mathsf{OPT}(I)\geq |E(G)|^2/\operatorname{poly}(\mu)$.
In order to overcome this difficulty, we partition the set ${\mathcal{C}}$ of clusters into a set ${\mathcal{C}}'$ of good clustes, and a set ${\mathcal{C}}''$ of bad clusters. Let $G'=G\setminus (\bigcup_{C\in {\mathcal{C}}'}C)$, and let $G''=G'_{|{\mathcal{C}}''}$ be obtained from $G'$ by contracting each bad cluster into a super-node. We then apply the same decomposition algorithm to graph $G''$. We continue this process until no bad clusters remain, so we have obtained a decomposition of $G$ into a collection of good clusters, or we compute a bad cluster $C$ with $|\delta(C)|\geq |E(G)|/\operatorname{poly}(\mu)$, so we can conclude that $\mathsf{OPT}(I)\geq|E(G)|^2/\operatorname{poly}(\mu)$.
In order to be able to carry out this process, we need an algorithm that, given a cluster $C$, either computes the desired set ${\mathcal{Q}}$ of paths routing the edges of $\delta(C)$ to some vertex $u$ of $C$, such that the cost of ${\mathcal{Q}}$ is low; or establishes that $\mathsf{cr}(C)>|\delta(C)|^2/\operatorname{poly}(\mu)$. But because we will need to apply this algorithm to graphs obtained from sub-graphs of $G$ by contracting some bad clusters, we need a more general algorithm, that is summarized in the following theorem.
Suppose we are given a graph $H$ and a set $T$ of its vertices called terminals. Let $\Lambda(H,T)$ denote the set of all pairs $({\mathcal{Q}},x)$, where $x$ is a vertex of $H$, and ${\mathcal{Q}}$ is a collection of paths routing the vertices of $T$ to $x$ in ${\mathcal{Q}}$. A \emph{distribution} ${\mathcal{D}}$ over pairs in $\Lambda(H,T)$ is an assignment, to every pair $({\mathcal{Q}},x)\in \Lambda(H,T)$, of a probability value $p({\mathcal{Q}},x)\geq 0$, such that $\sum_{({\mathcal{Q}},x)\in \Lambda(H,T)}p({\mathcal{Q}},x)=1$. The distribution ${\mathcal{D}}$ is specified by listing all pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$ of pairs with $p({\mathcal{Q}},x)>0$, together with the corresponding probability value $p({\mathcal{Q}},x)$.
\begin{theorem}\label{thm: find guiding paths}
There is a large enough constant $c_0$, and an efficient randomized algrorithm, that receives as input an instance $I=(H,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, where $|V(H)|=n$, a set $T\subseteq V(H)$ of $k$ vertices of $H$ called terminals, and a collection ${\mathcal{C}}$ of disjoint subgraphs of $H$, such that for some parameters $\eta,\eta'\geq 1$ and $0\leq \alpha,\alpha'\leq 1$, the following conditions hold:
\begin{itemize}
\item $|T|\geq |E(H_{|{\mathcal{C}}})|/\eta$;
\item every terminal $t\in T$ has degree $1$ in $H$;
\item for all $C\in {\mathcal{C}}$, $V(C)\cap T=\emptyset$;
\item set $T$ of terminals is $\alpha$-well-linked in the contracted graph $H_{|{\mathcal{C}}}$;
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property; and
\item for every cluster $C\in {\mathcal{C}}$, if $\Sigma_C$ is the rotation system for $C$ that is consistent with $\Sigma$, then $\mathsf{OPT}_{\mathsf{cr}}(C,\Sigma_C)\geq \frac{|\delta(C)|^2}{\eta'}$.
\end{itemize}
The algorithm either returns FAIL, or computes a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$, such that for all $e\in E(C)$, $\expect{(\cong({\mathcal{Q}},e))^2}\leq O(\log^8k/(\alpha\alpha')^2)$ \mynote{OK to make smaller}. Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq
\frac{(k\alpha^4 \alpha')^2}{c_0(\eta^4+\eta')\log^{28}k}$, then the probability that the algorithm returns FAIL is at most $1/2$.
\mynote{need to be careful with this. OK to put $\eta'+\operatorname{poly}\eta$ on the bottom, not OK to put $\eta'\eta$. This is because instance size in the recursion will decrease by factor $\eta$ only.}
\end{theorem}
The remainder of this section is dedicated to the proof of \Cref{thm: find guiding paths}.
For conveninence, we denote the contracted graph $H_{|{\mathcal{C}}}$ by $\hat H$, and we denote $|E(\hat H)|=\hm$. From the statement of \Cref{thm: find guiding paths}, $k\geq \hm/\eta$. Observe that, from \Cref{clm: contracted_graph_well_linkedness}, the set $T$ of terminals is $(\alpha\alpha')$-well-linked in $H$.
We start with some intuition. Assume first that graph $H$ contains a grid (or a grid minor) of size $(\Omega(k\alpha\alpha'/\operatorname{poly}\log n)\times (k\alpha \alpha'/\operatorname{poly}\log n))$, and a collection ${\mathcal{P}}$ of paths connecting every terminal to a distinct vertex on the first row of the grid, such that the paths in ${\mathcal{P}}$ cause a low edge-congestion. For this special case, the algorithm of \cite{Tasos-comm} (see also Lemma D.10 in the full version of \cite{chuzhoy2011algorithm}) provides the distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$ with the required properties. Moreover, if $H$ is a bounded-degree planar graph, with a set $T$ of terminals that is $(\alpha\alpha')$-well-linked,
then there is an efficient algorithm to compute such a grid minor, together with the required collection ${\mathcal{P}}$ of paths. If $H$ is planar but no longer bounded-degree, we can still compute a grid-like structure in it, and apply the same arguments as in \cite{Tasos-comm} in order to compute the desired distribution ${\mathcal{D}}$. The difficulty in our case is that the graph $H$ may be far from being planar, and, even though, from the Excluded Grid theorem of Robertson and Seymour \mynote{add references to the original thm proof and our new proofs}, it must contain a large grid-like structure, without having a drawing of $H$ in the plane with a small number of crossing, we do not know how to compute such a structure.
The proof of the theorem consists of five steps. In the first step, we will either establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is sufficiently large (so the algorithm can return FAIL), or compute a subgraph $\hat H'\subseteq \hat H$, and a partition $(X,Y)$ of $V(\hat H')$, such that each of the clusters $\hat H'[X],\hat H'[Y]$ has the $\hat \alpha$-bandwidth property, for $\hat \alpha=\Omega(\alpha/\log^4n)$, together with a large collection of edge-disjoint paths routing the terminals to the edges of $E_{\hat H'}(X,Y)$ in graph $\hat H'$. Intuitively, we will view from this point onward the edges of $E_{\hat H'}(X,Y)$ as a new set of terminals, that we denote by $\tilde T$ (more precisely, we subdivide each edge of $E_{\hat H'}(X,Y)$ with a new vertex that becomes a new terminal). We show that it is sufficient to prove an analogue of \Cref{thm: find guiding paths} for this new set $\tilde T$ of terminals. The clusters $\hat H'[X],\hat H'[Y]$ of graph $\hat H'$ naturally define a partition $(H_1,H_2)$ of the graph $H$ into two disjoint clusters. In the second step, we either establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is suffciently large
(so the algorithm can return FAIL), or
compute some vertex $x$ of $H_1$, and a collection ${\mathcal{P}}$ of paths in graph $H_1$, routing the terminals of $\tilde T$ to $x$, such that the paths in ${\mathcal{P}}$ cause a relatively low edge-congestion.
We exploit this set ${\mathcal{P}}$ of paths in order to define an ordering of the termianls in $\tilde T$, which is in turn exploited in the third step in order to compute a ``skeleton'' for the grid-like structure. We compute the grid-like structure itself in the fourth step. In the fifth and the final step, we generalize the arguments from \cite{Tasos-comm} and \cite{chuzhoy2011algorithm} in order to obtain the desired distribution ${\mathcal{D}}$.
Before we proceed, we need to consider two simple special cases. In the first case, $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2$ is large. In the second case, we can route a large subset of the terminals to a single vertex of $V(\hat H)\cap V(H)$ in the graph $\hat H$ via edge-disjont paths.
\paragraph{Special Case 1: $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2$ is large.}
We consider the special case where $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2\geq \frac{(k\alpha^4\alpha')^2}{c_0\log^{50}n}$, where $c_0$ is the constant from the statement of \Cref{thm: find guiding paths}.
In this case, since we are guaranteed that, for every cluster $C\in {\mathcal{C}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq |\delta(C)|^2/\eta'$, we get that:
\[\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \sum_{C\in {\mathcal{C}}}\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq \sum_{C\in {\mathcal{C}}}\frac{|\delta(C)|^2}{\eta'}\geq \frac{(k\alpha^4\alpha')^2}{c_0\eta'\log^{50}n}.\]
Therefore, if $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2\geq (k\alpha^4\alpha')^2/(c_0\log^{50}n)$, the algorithm returns FAIL and terminates. We assume from now on that:
\begin{equation}\label{eq: boundaries squared sum bound}
\sum_{C\in {\mathcal{C}}}|\delta(C)|^2< \frac{(k\alpha^4\alpha')^2}{c_0\log^{50}n}
\end{equation}
\paragraph{Special Case 2: Routing of terminals to a single vertex.}
The second special case happens if there exists a collection ${\mathcal{P}}_0$ of at least $\frac{k\alpha\alpha'}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\log^4k}$ edge-disjoint paths in graph $H$ routing some subset $T_0\subseteq T$ of terminals to some vertex $x$ (here, $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}$ is the constant from \Cref{claim: embed expander}).
Note that, if Special Case 1 did not happen, then $x$ may not be a supernode. Indeed, assume that $x=v_C$ for some cluster $C\in {\mathcal{C}}$. Then:
\[(\delta_H(C))^2\geq \Omega\textsf{left}(\frac{(k\alpha \alpha')^2}{\log^8k}\textsf{right} )\geq \frac{(k\alpha^4\alpha')^2}{c_0\log^{28}n},\]
assuming that $c_0$ is a large enough constant, a contradiction. Therefore, we can assume that $x$ is not a supernode.
Denote $z=|{\mathcal{P}}_0|\geq \frac{k\alpha\alpha'}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\log^4k}$.
We compute a single set ${\mathcal{Q}}$ of paths routing the set $T$ of terminals to $x$, with congestion $O(\log^4k/(\alpha \alpha')^2)$, as follows. We partition the set $T\setminus T_0$ of terminals into $q=\ceil{k/z}\leq O(\log^4k/(\alpha\alpha'))$ subsets, each of which contains at most $z$ vertices.
Consider now some index $1\leq i\leq q$. Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in $H$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{P}}_i'$ of paths in graph $H$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in ${\mathcal{P}}'_i$ cause edge-congestion $O(1/(\alpha\alpha'))$, and each vertex of $T_0\cup T_i$ is the endpoint of at most one path in ${\mathcal{P}}_i'$. By concatenating the paths in ${\mathcal{P}}_i'$ with paths in ${\mathcal{P}}_0$, we obtain a collection ${\mathcal{P}}_i$ of paths in graph $H$, connecting every terminal of $T_i$ to $x$, that cause edge-congestion $O(1/(\alpha\alpha'))$. Let ${\mathcal{Q}}=\bigcup_{i=0}^q{\mathcal{P}}_i$ be the resulting set of paths. Observe that set ${\mathcal{Q}}$ contains $k$ paths, routing the terminals in $T$ to the vertex $x$ in graph $H$, with $\cong_H({\mathcal{Q}})\leq O(q/(\alpha\alpha'))\leq O(\log^4k/(\alpha\alpha')^2)$. We return a distribution ${\mathcal{D}}$ consisting of a single pair $({\mathcal{Q}},u)$, and terminate the algorithm.
In the remainder of the algorithm, we assume that neither of the two special cases happened. We now describe each step of the algorithm in turn.
\subsection{Step 1: Splitting the Contracted Graph}
In this step, we split the contracted graph $\hat H$, using the algorithm summarized in the following theorem.
\begin{theorem}\label{thm: splitting}
There is an efficient randomized algorithm that with probability at most $1/\operatorname{poly}(n)$ returns FAIL, and, if it does not return FAIL, then it computes a subgraph $\hat H'\subseteq \hat H$ and a partition $(X,Y)$ of $V(\hat H')$ such that:
\begin{itemize}
\item each of the clusters $\hat H'[X]$ and $\hat H'[Y]$ has the $\hat \alpha'$-bandwidth property, for $\hat \alpha'=\Omega(\alpha/\log^4n)$; and
\item there is a set ${\mathcal{R}}$ of at least $\Omega(\alpha^3k/\log^8n)$ edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to edges of $E_{\hat H'}(X,Y)$.
\end{itemize}
\end{theorem}
\begin{proof}
We start by applying the algorithm from \Cref{claim: embed expander} to graph $\hat H$ and the set $T$ of terminals, to obtain a graph $W$ with $V(W)=T$ and maximum vertex degree $O(\log^2k)$, and an embedding ${\hat{\mathcal{P}}}$ of $W$ into $\hat H$ with congestion $O((\log^2k)/\alpha)$. Let $\hat{\cal{E}}$ be the bad event that $W$ is not a $1/4$-expander. Then $\prob{\hat {\cal{E}}}\leq 1/\operatorname{poly}(k)$.
Let $\hat H'$ be the graph that is obtained from $\hat H$ by deleting from it all edges and vertices except those that participate in the paths in ${\hat{\mathcal{P}}}$. Equivalently, graph $\hat H'$ is obtained from the union of all paths in ${\hat{\mathcal{P}}}$. We need the following observation.
\begin{observation}\label{obs: expansion and degree}
If event $\hat {\cal{E}}$ did not happen, then the set $T$ of vertices is $\hat \alpha$-well-linked in $\hat H'$, for $\hat \alpha=\frac{\alpha}{4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$, and the maximum vertex degree in $\hat H'$ is at most $d=\frac{\alpha k}{512\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$.
\end{observation}
\begin{proof}
Assume that Event $\hat {\cal{E}}$ did not happen. We first prove that the set $T$ of terminals is $\hat \alpha$-well-linked in $\hat H$. Consider any paritition $(A,B)$ of vertices of $\hat H'$, and denote $T_A=T\cap A$, $T_B=T\cap B$. Assume w.l.o.g. that $|T_A|\leq |T_B|$. Then it is sufficient to show that $|E_{\hat H'}(A,B)|\geq \hat \alpha\cdot |T_A|$.
Consider the partition $(T_A,T_B)$ of the vertices of $W$, and denote $E'=E_{W}(T_A,T_B)$. Since $W$ is a $1/4$-expander, $|E'|\geq |T_A|/4$ must hold. Consider now the set $\hat {\mathcal{R}}\subseteq {\hat{\mathcal{P}}}$ of paths containing the embeddings $P(e)$ of every edge $e\in E'$. Each path $R\in \hat {\mathcal{R}}$ connects a vertex of $T_A$ to a vertex of $T_B$, so it must contain an edge of $|E_{\hat H}(A,B)|$. Since $|\hat{\mathcal{R}}|\geq |T_A|/4$, and the paths in ${\hat{\mathcal{P}}}$ cause edge-congestion at most $(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)/\alpha$, we get that $|E_{\hat H}(A,B)|\geq \alpha\cdot |T_A|/(4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)\geq \hat \alpha |T_A|$.
Assume now that maximum vertex degree in $\hat H'$ is greater than $d$, and let $x$ be a vertex whose degree is at least $d$. Let $\hat{\mathcal{Q}}\subseteq \tilde{\mathcal{P}}$ be the set of all paths containing the vertex $x$.
Consider any such path $Q\in \hat{\mathcal{Q}}$. The endpoints of this path are two distinct terminals $t,t'\in T$. We let $Q'\subseteq Q$ be the sub-path of $Q$ between the terminal $t$ and the vertex $x$, and we let ${\mathcal{Q}}'=\set{Q'\mid Q\in \hat{\mathcal{Q}}}$.
Recall that every vertex in $W$ has degree at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$, and so a terminal in $T$ may be an endpoint of at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$ paths in ${\hat{\mathcal{P}}}$. Therefore, there is a subset ${\mathcal{Q}}''\subseteq {\mathcal{Q}}'$ of at least $d/(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)$ paths, each of which originates at a distinct terminal.
But then, from \Cref{claim: routing in contracted graph}, there is a collection ${\mathcal{Q}}'''$ of edge-disjoint paths in graph $H$, routing a subset of terminals to $x$ of cardinality at least:
\[\frac{\alpha'|{\mathcal{Q}}''|} 2\geq \frac{\alpha' d}{2\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\geq \frac{\alpha \alpha' k}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^4k},\]
contradicting the fact that Special Case 2 did not happen.
\end{proof}
Next, we use the following lemma that allows us to compute the required sets $X$, $Y$ of vertices. The proof follows immediately from techniques that were introduced in \cite{chuzhoy2012routing} and then refined in \cite{chuzhoy2012polylogarithmic,chekuri2016polynomial,chuzhoy2016improved}. Unfortunately, all these proofs assumed that the input graph has a bounded vertex degree, and additionally the proofs are somewhat more involved than the proof that we need here (this is because these proofs could only afford a $\operatorname{poly}\log k$ loss in the cardinality of the path set ${\mathcal{R}}$ relatively to $|T|$, while we can afford a $\operatorname{poly}\log n$ loss). Therefore, we provide a complete proof of the lemma in Section \ref{sec: splitting} of the Appendix for completeness.
\begin{lemma}\label{lem: splitting}
There is an efficient algorithm that, given as input an $n$-vertex graph $G$, and a subset $T$ of $k$ vertices of $G$ called terminals, together with a parameter $0<\tilde \alpha<1$, such that the maximum vertex degree in $G$ is at most $\tilde \alpha k/64$, and every vertex of $T$ has degree $1$ in $G$, either returns FAIL, or computes a partition $(X,Y)$ of $V(G)$, such that:
\begin{itemize}
\item each of the clusters $G[X]$, $G[Y]$ has the $\tilde \alpha'$-bandwidth property, for $\tilde \alpha'=\Omega(\tilde \alpha/\log^2n)$; and
\item there is a set ${\mathcal{R}}$ of at least $\Omega(\tilde \alpha^3k/\log^2n)$ edge-disjoint paths in graph $G$, routing a subset of terminals to edges of $E_G(X,Y)$.
\end{itemize}
Moreover, if the vertex set $T$ is $\tilde \alpha$-well-linked in $G$, then the algorithm never returns FAIL.
\end{lemma}
We apply the algorithm from \Cref{lem: splitting} to graph $\hat H'$, the set $T$ of terminals, and parameter $\tilde \alpha=\hat \alpha=\frac{\alpha}{4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$. Recall that we are guaranteed that maximum vertex degree in $\hat H'$ is at most $d=\frac{\alpha k}{512\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\leq \frac{\tilde \alpha k}{64}$.
Note that the algorithm from \Cref{lem: splitting} may only return FAIL if the set $T$ of terminals is not $\hat \alpha$-well-linked in $\hat H'$, which, from \Cref{obs: expansion and degree}, may only happen if Event $\hat {\cal{E}}$ happened, which in turn may only happen with probability $1/\operatorname{poly}(k)$. If the algorithm from \Cref{lem: splitting} returned FAIL, then we restart the algorithm that we have described so far from scratch. We do so at most $1/\operatorname{poly}\log n$ times, and if, in every iteration, the algorithm from \Cref{lem: splitting} returns FAIL, then we return FAIL as the outcome of the algorithm for \Cref{thm: splitting}. Clearly, this may only happen with probability at most $1/\operatorname{poly}(n)$. Therefore, we assume that in one of the iterations, the algorithm from \Cref{lem: splitting} did not return FAIL. From now on we consider the outcome of that iteration.
Let $(X,Y)$ be the partition of $V(\hat H')$ that the algorithm returns. We are then guaranteed that each of the clusters $\hat H'[X],\hat H'[Y]$ has the $\tilde \alpha'$-bandwidth property, where $\tilde \alpha'=\Omega(\hat \alpha/\log^2n)=\Omega(\alpha/\log^4n)$. The algorithm also ensures that there is a collection ${\mathcal{R}}$ of edge-disjoint paths in $\hat H'$, routing a subset of the terminals to edges of $E_{\hat H'}(X,Y)$, with:
\[ |{\mathcal{R}}|\geq \Omega(\tilde \alpha^3k/\log^2n)=\Omega(\alpha^3k/\log^8n). \]
This completes the proof of \Cref{thm: splitting}.
\end{proof}
If the algorithm from \Cref{thm: splitting} returned FAIL (which may only happen with probability at most $1/\operatorname{poly}(n)$), then we terminate the algorithm and return FAIL as well. Therefore, we assume from now on that the algorithm from \Cref{thm: splitting} returned a subgraph $\hat H'\subseteq \hat H$ and a partition $(X,Y)$ of $V(\hat H')$ such that each of the clusters $\hat H'[X]$ and $\hat H'[Y]$ has the $\hat \alpha'$-bandwidth property, for $\hat \alpha'=\Omega(\alpha/\log^4n)$, and
there is a set ${\mathcal{R}}$ of at least $\Omega(\alpha^3k/\log^8n)$ edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to edges of $E_{\hat H'}(X,Y)$.
Notice that we can compute the path set ${\mathcal{R}}$ with the above properties efficiently, using standard Maximum $s$-$t$ flow.
Let $E'\subseteq E_{\hat H'}(X,Y)$ be the subset of edges containing the last edge on every path in ${\mathcal{R}}$, so, by reversing the direction of the paths in ${\mathcal{R}}$, we can view the set ${\mathcal{R}}$ of paths as routing the edges of $E'$ to the terminals.
In the remainder of this step, we will slightly modify the graphs $H$ and $\hat H$, and we will continue working with the modified graphs in the following steps.
Let $\hat H''\subseteq\hat H'$ be the graph obtained from $\hat H'$ by first deleting all edges of $E_{\hat H'}(X,Y)\setminus E'$ from it, and then subdividing every edge $e\in E'$ with a vertex $t_e$. We denote $\tilde T=\set{t_e\mid e\in E'}$, and we refer to vertices of $\tilde T$ as \emph{pseudo-terminals}. Recall that $|\tilde T|=|{\mathcal{R}}|=\Omega(\alpha^3k/\log^8n)$, and there is a set ${\mathcal{R}}'$ of edge-disjoint paths in the resulting graph $\hat H''$, routing the vertices of $\tilde T$ to the vertices of $T$.
We let $\hat H_1$ be the subgraph of $\hat H''$, that is induced by the set $X\cup \tilde T$ of vertices, and we similarly define $\hat H_2=\hat H''[Y\cup \tilde T]$. From the $\hat \alpha'$-bandwidth property of the clusters $\hat H'[X]$ and $\hat H'[Y]$, we are guaranteed that the vertices of $\tilde T$ are $ \hat \alpha'$-well-linked in both $\hat H_1$ and in $\hat H_2$, where $\hat \alpha'= \Omega(\alpha/\log^4n)$. Let ${\mathcal{C}}'\subseteq {\mathcal{C}}$ be the subset of all clusters $C$ whose corresponding supernode $v_C$ lies in grpah $\hat H''$.
For convenience, we also subdivide, in graph $H$, every edge $e\in E'$, with the vertex $t_e$, so graph $\hat H''$ can be now viewed as a subgraph of the contracted graph $H_{|{\mathcal{C}}}$.
Next, we let $H'\subseteq H$ be the subgraph of $H$ that corresponds to the graph $\hat H''$: namely, graph $H'$ is obtained from $\hat H''$ by replacing every supernode $v_C$ with the corresponding cluster $C\in {\mathcal{C}}'$. Equivalently, we can obtain graph $H'$ from $H$, by deleting every edge of $E(\hat H)\setminus E(\hat H'')$ and every regular (non-supernode) vertex of $V(\hat H)\setminus V(\hat H'')$. Additionally, for every cluster $C\in {\mathcal{C}}\setminus {\mathcal{C}}'$, we delete all edges and vertices of $C$ from $H'$. We also define a rotation system $\Sigma'$ for graph $H'$, which is identical to $\Sigma$ (vertices $t_e\in \tilde T$ all have degree $2$, so their corresponding ordering ${\mathcal{O}}_{t_e}$ of incident edges can be set arbitrarily).
We partition the set ${\mathcal{C}}'$ of clusters into two subsets: set ${\mathcal{C}}_X$ contains all clusters $C\in {\mathcal{C}}'$ with $v_C\in X$, and set ${\mathcal{C}}_Y$ contains all clusters $C\in {\mathcal{C}}'$ with $v_C\in Y$. We can similarly define the graphs $H_1,H_2\subseteq H'$, that correspond to the contracted graphs $\hat H_1$ and $\hat H_2$, respectively: let $X'$ contain all vertices $x\in V(H')$, such that either $x\in C$ for some cluster $C\in {\mathcal{C}}_X$, or $x$ is a regular vertex of $\hat H''$ lying in $X$. Similarly, we let $Y'$ contain all vertices $y\in V(H)$, such that either $y\in C$ for some cluster $C\in {\mathcal{C}}_Y$, or $y$ is a regular vertex of $\hat H''$ lying in $Y$. We then let $H_1=H'[X\cup \tilde T]$, and $H_2=H'[Y\cup \tilde T]$.
The following observation, summarizing some properties of graph $H'$, is immediate:
\begin{observation}\label{obs: properties of new graph}
\
\begin{itemize}
\item $\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')\leq \mathsf{OPT}_{\mathsf{cnwrs}} (H,\Sigma)$;
\item $\hat H''=H'_{|{\mathcal{C}}'}$;
\item $|E(\hat H'')|\leq 2|E(\hat H)|\leq 2\eta k\leq O(|\tilde T|\eta \log^8k/\alpha^3)$; and
\item graph $\hat H_1$ is a contracted graph of $H_1$ with respect to ${\mathcal{C}}_X$, and graph $H_2$ is a contracted graph of $H_2$ with respect to ${\mathcal{C}}_Y$. In other words, $\hat H_1=(H_1)_{|{\mathcal{C}}_X}$, and $\hat H_2=(H_2)_{|{\mathcal{C}}_Y}$.
\end{itemize}
\end{observation}
(for third assertion we have used the fact that $k\geq |E(\hat H)|/\eta$ from the statement of \Cref{thm: find guiding paths}, and $|\tilde T|\geq \Omega(\alpha^3k/\log^8n)$.)
Recall that $\Lambda(H',\tilde T)$ denotes the set of all pairs $({\mathcal{Q}},x)$, where $x$ is a vertex of $H'$, and ${\mathcal{Q}}$ is a collection of paths in graph $H'$, routing the vertices of $\tilde T$ to $x$. In the following observation, we show that it is now enough to compute a distribution ${\mathcal{D}}'$ over pairs $({\mathcal{Q}},x)\in \Lambda(H',\tilde T)$, such that for every edge $e\in E(H')$, $\expect{\cong_{H'}({\mathcal{Q}},e)}$ is low, in order to obtain the desired distribution ${\mathcal{D}}$ over pairs in $\Lambda(H,T)$.
\begin{observation}\label{obs: convert distributions}
There is an efficient algorithm, that, given a distribution ${\mathcal{D}}'$ over pairs $({\mathcal{Q}}',x')\in \Lambda(H',\tilde T)$,
such that for every edge $e'\in E(H')$, $\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e'))^2}\leq \beta$ holds,
computes a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$, such that for every edge $e\in E(H)$, $\expect[({\mathcal{Q}},x)\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left}(\frac{\beta \log^{16}n}{\alpha^8(\alpha')^4}\textsf{right} )$.
\end{observation}
\begin{proof}
Recall that there is a set ${\mathcal{R}}'$ of edge-disjoint paths in graph $\hat H''$, routing the vertices of $\tilde T$ to the vertices of $T$, and moreover, such a path set can be found efficiently via a standard maximum $s$-$t$ flow computation.
Since $\hat H''=H'_{|{\mathcal{C}}'}$, from \Cref{claim: routing in contracted graph}, we can efficiently compute a set ${\mathcal{R}}_0$ of edge-disjoint paths in graph $H''$, routing a subset $T_0\subseteq T$ of terminals to $\tilde T$, with $|{\mathcal{R}}_0|\geq \alpha'\cdot |{\mathcal{R}}'|=\alpha'\cdot |\tilde T|=\Omega(\alpha'\alpha^3k/\log^8n)$.
Let $z=\ceil{k/|T_0|}=O(\log^{8}n/(\alpha^3\alpha')$. Next, we partition the terminals of $T\setminus T_0$ into $z$ subsets $T_1,\ldots,T_z$, of cardinality at most $|T_0|$ each. Consider now some index $1\leq i\leq z$. Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in graph $H$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{R}}_i'$ of paths in graph $H$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in ${\mathcal{R}}'_i$ cause edge-congestion $O(1/(\alpha\alpha'))$, and each vertex of $T_0\cup T_i$ is the endpoint of at most one path in ${\mathcal{R}}'_i$. By concatenating the paths in ${\mathcal{R}}'_i$ with paths in ${\mathcal{R}}_0$, we obtain a collection ${\mathcal{R}}_i$ of paths in graph $H$, routing the terminals in $T_i$ to vertices of $\tilde T$, such that every vertex of $\tilde T\cup {\mathcal{R}}_0$ is an endpoint of at most one path in ${\mathcal{R}}_i$. Let ${\mathcal{R}}^*=\bigcup_{i=0}^z{\mathcal{R}}_i$ be the resulting set of paths. Observe that set ${\mathcal{R}}^*$ contains $k$ paths, each of which connects a distinct terminal of $T$ to a vertex of $\tilde T$, and every vertex of $\tilde T$ serves as an endpoint of at most $z=O(\log^{8}n/(\alpha^3\alpha')$ such paths. The paths of ${\mathcal{R}}^*$ cause congestion at most $O(z/\alpha\alpha')\leq O(\log^8k/(\alpha^4(\alpha')^2))$ in graph $H$.
Recall that $H'\subseteq H$. Consider some pair $({\mathcal{Q}}',x')\in \Lambda(H',\tilde T)$ with probability $p({\mathcal{Q}}',x')>0$ in the distribution ${\mathcal{D}}'$. We compute another path set ${\mathcal{Q}}$ in graph $H$, routing all terminals in $T$ to $x'$, and we will assign to the pair $({\mathcal{Q}},x')\in \Lambda(H,T)$ the same probability value $p({\mathcal{Q}}',x')$. For ever pseudoterminal $\tilde t\in \tilde T$, let $Q'_{\tilde t}\in {\mathcal{Q}}'$ be the unique path originating at $\tilde t$. For every terminal $t\in T$, let $R'_t\in {\mathcal{R}}^*$ be the unique path originating at terminal $t$, and let $\tilde t\in \tilde T$ be the other endpoint of path $R'_t$. We let $Q_t$ be the path obtained by concatenating the paths $R'_t$ and $Q'_{\tilde t}$, so path $Q_t$ connects $t$ to vertex $x'$. We then let ${\mathcal{Q}}=\set{Q_t\mid t\in T}$. Note that ${\mathcal{Q}}$ is a set of paths in graph $H$ routing the set $T$ of terminals to vertex $x'$, so $({\mathcal{Q}},x')\in \Lambda(H,T)$. We assign to the pair $({\mathcal{Q}},x')$ probability $p({\mathcal{Q}}',x')$. Note that, since every pseudoterminal $\tilde t\in \tilde T$ may seve as an endpoint of at most $O(\log^{8}n/(\alpha^3\alpha')$ paths in ${\mathcal{R}}^*$, and since the paths in ${\mathcal{R}}^*$ cause edge-congestion at most $O(\log^8k/(\alpha^4(\alpha')^2))$ in graph $H$, for every edge $e\in E(H)$, we get that:
\[\cong_H({\mathcal{Q}},e)\leq O\textsf{left}(\frac{\cong_{H'}({\mathcal{Q}}',e) \cdot \log^8k}{\alpha^3 \alpha'}+\frac{\log^8k}{\alpha^4(\alpha')^2}\textsf{right} ). \]
Therefore, altogether, for every edge $e\in E(H)$:
\[\expect[({\mathcal{Q}},x')\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left}(\frac{\log^{16}k}{\alpha^8(\alpha')^4}\textsf{right} )\cdot \textsf{left} (\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e))^2}+1\textsf{right} ). \]
\end{proof}
From the above discussion, in order to complete the proof of \Cref{thm: find guiding paths}, it is now enough to design a randomized algorithm, that either computes a distribution ${\mathcal{D}}'$ over pairs in $\Lambda(H',\tilde T)$ with $\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e))^2}\leq xxx$, or returns FAIL. It is enough to ensure that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')\geq
\frac{(k\alpha^4 \alpha')^2}{c_0(\eta^4+\eta')\log^{28}k}$, then the probability that the algorithm returns FAIL is at most $1/2$.
In the remainder of the proof we focus on the above goal. It would be convenient for us to simplify the notation, by denoting $H'$ by $H$, $\hat H''$ by $\hat H$, $\tilde T$ by $T$, and $\hat \alpha'$ by $\tilde \alpha$. We also denote ${\mathcal{C}}'$ by ${\mathcal{C}}$. We now summarize all properties of the new graphs $H,\hat H$ that we have established so far, and in the remainder of the proof of \Cref{thm: find guiding paths} we will only work with these new graphs.
\paragraph{Summary of the outcome of Step 1.}
We can assume from now on that we are given a graph $H$, a rotation system $\Sigma$ for $H$, a set $\tilde T$ of terminals in graph $H$, and a collection ${\mathcal{C}}$ of disjoint subgraphs (clusters) of $H\setminus \tilde T$. We denote $|\tilde T|=\tilde k$. The corresponding contracted graph is denoted by $\hat H=H_{|{\mathcal{C}}}$.
We are also given a partition $(X,Y)$ of $V(H)\setminus \tilde T$ (note that for convenience of notation, $X$ and $Y$ are now subsets of vertices of $H$, and not of $\hat H$), and a parition ${\mathcal{C}}_X,{\mathcal{C}}_Y$ of ${\mathcal{C}}$, such that each cluster $C\in {\mathcal{C}}_X$ has $V(C)\subseteq X$, and each cluster $C\in {\mathcal{C}}_Y$ has $V(C)\subseteq Y$. We denote $H_1=H[X\cup \tilde T]$ and $H_2=H[Y\cup \tilde T]$. We also denote by $\hat H_1=(H_1)_{|{\mathcal{C}}_X}$ the contracted graph of $H_1$ with respect to ${\mathcal{C}}_X$, and similarly by $\hat H_2=(H_2)_{|{\mathcal{C}}_Y}$ the contracted graph of $H_2$ with respect to ${\mathcal{C}}_Y$. We have the following additional properties:
\begin{properties}{P}
\item $\tilde k\geq \Omega(\alpha^3k/\log^8n)$;
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property;
\item $|E(\hat H)|\leq O(\tilde k\cdot \eta \log^8k/\alpha^3)$; \label{prop after step 1: few edges}
\item every vertex of $\tilde T$ has degree $1$ in $H_1$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_1$, for $\tilde \alpha=\Omega(\alpha/\log^4n)$;
\item similarly, every vertex of $\tilde T$ has degree $1$ in $H_2$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_2$; and
\item $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2<\frac{(k\alpha^4\alpha')^2}{c_0\log^{50}n}\leq \frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}$, where $c_1$ is some large enough constant, whose value we can set later. \label{prop after step 1: small squares of boundaries}
\end{properties}
Our goal is
to design an efficient randomized algorithm, that either computes a distribution ${\mathcal{D}}$ over pairs in $\Lambda(H,\tilde T)$ with $\expect[({\mathcal{Q}},x)\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq xxx$ for every edge $e\in E(H)$, or returns FAIL.
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1(\eta^4+\eta')\log^{20}n}$ for some large enough constant $c_1$. Then, since we can set $c_0$ to be a sufficiently large constant compared to $c_1$, $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(k\alpha^4 \alpha')^2}{c_0(\eta^4+\eta')\log^{50}n}$ must hold.
Therefore, it is enough to ensure that, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1(\eta^4+\eta')\log^{20}n}$ for some large enough constant $c_1$,
then the probability that the algorithm returns FAIL is at most $1/2$.
\iffalse
\subsection{Step 2: a Modified Graph}
In the remainder of the proof, it would be convenient for us to work with a low-degree equivalent of the graph $\hat H$, that we denote by $H^+$. In order to define the graph $\hat H^+$, we start by defining a rotation system $\hat \Sigma $ for graph $\hat H$, as follows. Let $x\in V(\hat H)$ be a vertex. If $x$ is a supernode, that is, $x=v(C)$ for some cluster $C\in {\mathcal{C}}$, then we define the circular ordering $\hat{\mathcal{O}}_x\in \hat \Sigma$ of the edges of $\delta_{\hat H}(x)$ to be arbitrary.
Otherwise, $x$ is a regular vertex, and it is a vertex of the original graph $H$. We then let $\hat {\mathcal{O}}_x\in \hat \Sigma$ be identical to the ordering ${\mathcal{O}}_x\in \Sigma$, where $\Sigma$ is the original rotation system for graph $H$.
We are now ready to define the modified graph $H^+$. We start with $H^+=\emptyset$, and then process every vertex $u\in V(\hat H)$ one-by-one. We denote by $d(u)$ the degree of the vertex $u$ in graph $\hat H$. We now describe an iteration when a vertex $u\in V(\hat H)$ is processed. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that incident to $u$ in $\hat H$, indexed according to their ordering in $\hat {\mathcal{O}}_u\in \hat \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H^+$. We refer to the edges in the grids $\Pi(u)$ as \emph{inner edges}.
Once every vertex $u\in V(\hat H)$ is processed, we add a collection of \emph{outer edges} to graph $H^+$, as follows. Consider any edge $e=(x,y)\in E(H)$. Assume that $e$ is the $i$th edge of $x$ and the $j$th edge of $y$, that is, $e=e_i(x)=e_j(y)$. Then we add an edge $e'=(s_i(x),s_j(y))$ to graph $H^+$, and we view this edge as the copy of the edge $e\in E(\hat H)$. We will not distinguish between the edge $e$ of $\hat H$ (and the corresponding edge of $H$), and the edge $e'$ of $H^+$.
We note that every terminal $t\in T$ has degree $1$ in $\hat H$, so its corresponding grid $\Pi(t)$ consists of a single vertex, that we also denote by $t$. Therefore, set $T$ of terminals in $H$ naturally corresponds to a set of $k$ terminals in $H^+$, that we denote by $T$ as before.
This completes the definition of the graph $H^+$. Note that the maximum vertex degree in $H^+$ is $4$. We also define a rotation system $\Sigma^+$ for the graph $H^+$ in a natural way: for every vertex $u\in V(H)$,
consider the standard drawing of the grid $\Pi(u)$, to which we add the drawings of the edges that are incident to vertices $s_1(u),\ldots,s_{d(u)}(u)$, so that the edges are drawn on the grid's exterior in a natural way (\mynote{add figure}). This layout defines an ordering ${\mathcal{O}}^+(v)$ of the edges incident to every vertex $v\in \Gamma(u)$.
We start with the following simple claim.
\begin{claim}\label{claim: cheap solution to modified instance} There is a solution to the \textnormal{\textsf{MCNwRS}}\xspace problem instance $(H^+,\Sigma^+)$ of cost at most ..., such that no inner edge of $H^+$ participates in any crossings in the solution.
\end{claim}
\begin{proof}
We start by showing that $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat H,\hat \Sigma)\leq ...$. Recall that, from \Cref{lem: crossings in contr graph}, there is a drawing $\hat \phi$ of graph $\hat H$ with at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2$ crossings, such that for every vertex $x\in V(\hat H)\cap V(H)$, the ordering of the edges of $\delta_{\hat H}(x)$ as they enter $x$ in $\hat \phi$ is consistent with the ordering ${\mathcal{O}}_x\in \Sigma$, and hence with the ordering $\hat {\mathcal{O}}_x\in \hat \Sigma$. Drawing $\hat \phi$ of $\hat H$ may not be a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace since for some supernodes $v(C)$, the ordering of the edges that are incident to $v(C)$ in $\hat H$ as they enter the image of $v(C)$ in $\hat \phi$ may be different from $\hat {\mathcal{O}}_{v(C)}$. For each such vertex $v(C)$, we may need to \emph{reorder} the images of the edges of $\delta_{\hat H}(v(C))=\delta_H(C)$ near the image of $v(C)$, so that they enter the image of $v(C)$ in the correct order. This can be done by introducing at most $|\delta_H(C)|^2$ crossings for each such supernode $v(C)$. The resulting drawing $\hat \phi'$ of $\hat \phi$ is a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, whose cost is bounded by:
\[O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2+\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2+ \]
\end{proof}
\fi
\subsection{Step 2: Routing the Terminals to a Single Vertex, and an Expanded Graph}
In this step we start by considering the graph $H_1$ and the set $\tilde T$ of terminals in it. Our goal is to compute a collection ${\mathcal{J}}$ of paths in graph $H_1$, routing all terminals of $\tilde T$ to a single vertex, such that the paths in ${\mathcal{J}}$ cause a relatively low congestion in graph $H_1$. We show that, if such a collection of paths does not exist, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)$ is high. Intuitively, we will use the set ${\mathcal{J}}$ of paths in order to define an ordering of the terminals in $\tilde T$, which will in turn be used in order to compute a grid-like structure in graph $H_2$. Once we compute the desired set ${\mathcal{J}}$ of paths, we will replace the graph $H$ with its low-degree analogue $H^+$, that we refer to as the \emph{expanded graph}. The remaining steps in the proof of \Cref{thm: find guiding paths} will use this expanded graph only.
We now proceed to describe the algorithm for Step 2. We consider every vertex $x\in V(H_1)$ one-by-one. For each such vertex $x$, we compute a set ${\mathcal{J}}(x)$ of paths in graph $H_1$, with the following properties:
\begin{itemize}
\item every path in ${\mathcal{J}}(x)$ originates at a distinct vertex of $\tilde T$ and terminates at $x$;
\item the paths in ${\mathcal{J}}(x)$ are edge-disjoint; and
\item ${\mathcal{J}}(x)$ is a maximum-cardinality set of paths in $H_1$ with the above two properties.
\end{itemize}
Note that such a set ${\mathcal{J}}(x)$ of paths can be computed via a standard maximum $s$-$t$ flow computation.
Throughout, we use a parameter $\tilde k'=\tilde k\alpha'\alpha^5/(c_2\eta\log^{36}n)$, where $c_2$ is a large enough constant whose value we set later.
If, for every vertex $x\in V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then we reurn FAIL and terminate the algorithm. In the following lemma, whose proof is deferred to Section \ref{sec: few paths high opt} of Appendix we show that, in this case, $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$ must hold.
Note that, since we can set $c_1$ to be a large enough constant, we can ensure that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1(\eta^4+\eta')\log^{20}n}$ holds in this case.
\begin{lemma}\label{lem: high opt or lots of paths}
If, for every vertex $x\in V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$.
\end{lemma}
From now on we assume that there is some vertex $x\in V(H_1)$, for which $|{\mathcal{J}}(x)|\geq \tilde k'$. We denote ${\mathcal{J}}_0={\mathcal{J}}(x)$, and we let $T_0\subseteq \tilde T$ be the set of terminals that serve as endpoints of paths in ${\mathcal{J}}_0$, so $|T_0|=|{\mathcal{J}}_0|=\tilde k'$.
Let $z=\ceil{\tilde k/\tilde k'}=O\textsf{left}(\frac{\eta\log^{36}n}{\alpha^5\alpha'}\textsf{right} )$. Next, we arbitrarily partition the terminals of $\tilde T\setminus T_0$ into $z$ subsets $T_1,\ldots,T_z$, of cardinality at most $\tilde k'$ each. Consider now some index $1\leq i\leq z$. Since the set $\tilde T$ of terminals is $(\tilde \alpha\alpha')$-well-linked in $H_1$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{J}}_i'$ of paths in graph $H_1$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in $T_i$ cause edge-congestion $O(1/(\tilde \alpha\alpha'))$, and each vertex of $T_i$ is the endpoint of at most one path in ${\mathcal{J}}_i'$. By concatenating the paths in ${\mathcal{J}}_i'$ with paths in ${\mathcal{J}}_0$, we obtain a collection ${\mathcal{J}}_i$ of paths in graph $H_1$, connecting every terminal of $T_i$ to $x$, that cause edge-congestion $O(1/(\tilde \alpha\alpha'))$. Let ${\mathcal{J}}=\bigcup_{i=0}^z{\mathcal{J}}_i$ be the resulting set of paths. Observe that set ${\mathcal{J}}$ contains $\tilde k$ paths, routing the terminals in $\tilde T$ to the vertex $x$ in graph $H_1$, with $\cong_H({\mathcal{J}})\leq O\textsf{left}(\frac z{\tilde \alpha\alpha'}\textsf{right} )\leq O\textsf{left}(\frac{\eta\log^{40}n}{\alpha^6(\alpha')^2}\textsf{right} )$. We denote by $\rho=O\textsf{left}(\frac{\eta\log^{40}n}{\alpha^6(\alpha')^2}\textsf{right} )$ this bound on $\cong_{H_1}({\mathcal{P}}')$. We assume w.l.o.g. that the paths in ${\mathcal{J}}$ are simple. Since every terminal in $\tilde T$ has degree $1$ in $H_1$, no path in ${\mathcal{J}}$ may contain a terminal in $\tilde T$ as its inner vertex.
In the remainder of the proof, it would be convenient for us to work with a low-degree equivalent of the graph $H$, that we call \emph{expanded graph}, and denote by $H^+$.
For every edge $e\in E(H)$, let $N_e=\cong_H({\mathcal{J}},e)$. Recall that for every edge $e\in E(H_1\setminus\tilde T)$, $N_e\leq \rho$, and for every other edge $e$ of $H$, $N_e\leq 1$.
Let $H'$ be obtained from graph $H$ by adding, for every edge $e\in E(H)$ with $N_e>1$, $N_e$ parallel copies of the edge $e$. We extend the rotation system $\Sigma$ for graph $H$ to a rotation system $\Sigma'$ for the resulting graph $H'$ in a natural way: consider any vertex $v\in V(H)$, and let $\delta_H(v)=\set{e_1,\ldots,e_r}$, where the edges are indexed in the order consistent with the ordering ${\mathcal{O}}_v\in \Sigma$. For all $1\leq i\leq r$, let $E_i\subseteq E(H')$ be the set of parallel edges corresponding to edge $e_i$. We define an ordering ${\mathcal{O}}'_v$ of edges in $\delta_{H'}(v)$, where for all $1\leq i\leq r$, the edges of $E_i$ appear consecutively in an arbitrary order, and the edges belonging to different sets are ordered according to the natural ordering $E_1,E_2,\ldots,E_r$ of their correspondng subsets. Consider now the resulting instance $(H',\Sigma')$ of \textnormal{\textsf{MCNwRS}}\xspace. We obtain the following immediate observation:
\begin{observation}\label{obs: crossing number of modified graph}
$\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')\leq O(\rho^2)\cdot\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)+|E(H)|\textsf{right} )$.
\end{observation}
\begin{proof}
Let $\phi$ be the optimal solution to instance $(H,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace. We modify $\phi$ to obtain a feasible solution to instance $(H',\Sigma')$, as follows. First, for every edge $e\in E(H)$ with $N_e>1$, we create $N_e$ additional parallel copies of the edge $e$, that are drawn next to the image of edge $e$, in parallel to it. Notice that, if edges $e$ and $e'$ cross in drawing $\phi$, then each such crossing may give rise to at most $\rho^2$ crossings in this new drawing, as $N_e,N_{e'}\leq \rho^2$. The resulting drawing is not necessarily a feasible solution to instance $(H',\Sigma')$, since it may not respect the rotation system $\Sigma'$. But it is easy to transform it into a feasible solution to instance $(H',\Sigma')$ by introducing at most $\sum_{e\in E(H)}2|N_e|^2$ new crossings (this is since for every vertex $v\in V(H)$, if $\delta_H(v)=\set{e_1,\ldots,e_r}$, we may need to reorder the edges in each of the corresponding sets $E_1,\ldots,E_r$ as they enter $e$; this can be done by introducing at most $\sum_{i=1}^r|E_i|^2=\sum_{i=1}^rN_{e_i}^2$ new crossings).
Altogether, we obtain a feasible solution to instance $(H',\Sigma')$ of \textnormal{\textsf{MCNwRS}}\xspace with at most $O(\rho^2)\cdot\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)+|E(H)|\textsf{right} )$ crossings.
\end{proof}
In order to define the graph $H^+$, we start by defining a rotation system $\hat \Sigma $ for graph $\hat H$, as follows. Let $x\in V(\hat H)$ be a vertex. If $x$ is a supernode, that is, $x=v(C)$ for some cluster $C\in {\mathcal{C}}$, then we define the circular ordering $\hat{\mathcal{O}}_x\in \hat \Sigma$ of the edges of $\delta_{\hat H}(x)$ to be arbitrary.
Otherwise, $x$ is a regular vertex, and it is a vertex of the original graph $H$. We then let $\hat {\mathcal{O}}_x\in \hat \Sigma$ be identical to the ordering ${\mathcal{O}}_x\in \Sigma$, where $\Sigma$ is the original rotation system for graph $H$.
Note that, if if
$\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1(\eta^4+\eta')\log^{20}n}$, then:
\[\begin{split}
\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')&<\rho^2\textsf{left}(\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1(\eta^4+\eta')\log^{20}n}+|E(H)|\textsf{right} )\\
&\leq
\end{split}
\]
since
$\rho=O\textsf{left}(\frac{\eta\log^{40}n}{\alpha^6(\alpha')^2}\textsf{right} )$, and
$|E(\hat H)|\leq O(\tilde k\cdot \eta \log^8k/\alpha^3)$ (from Property \ref{prop after step 1: few edges}).
We are now ready to define the modified graph $H^+$. We start with $H^+=\emptyset$, and then process every vertex $u\in V(\hat H)$ one-by-one. We denote by $d(u)$ the degree of the vertex $u$ in graph $\hat H$. We now describe an iteration when a vertex $u\in V(\hat H)$ is processed. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that incident to $u$ in $\hat H$, indexed according to their ordering in $\hat {\mathcal{O}}_u\in \hat \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H^+$. We refer to the edges in the grids $\Pi(u)$ as \emph{inner edges}.
Once every vertex $u\in V(\hat H)$ is processed, we add a collection of \emph{outer edges} to graph $H^+$, as follows. Consider any edge $e=(x,y)\in E(H)$. Assume that $e$ is the $i$th edge of $x$ and the $j$th edge of $y$, that is, $e=e_i(x)=e_j(y)$. Then we add an edge $e'=(s_i(x),s_j(y))$ to graph $H^+$, and we view this edge as the copy of the edge $e\in E(\hat H)$. We will not distinguish between the edge $e$ of $\hat H$ (and the corresponding edge of $H$), and the edge $e'$ of $H^+$.
We note that every terminal $t\in T$ has degree $1$ in $\hat H$, so its corresponding grid $\Pi(t)$ consists of a single vertex, that we also denote by $t$. Therefore, set $T$ of terminals in $H$ naturally corresponds to a set of $k$ terminals in $H^+$, that we denote by $T$ as before.
This completes the definition of the graph $H^+$. Note that the maximum vertex degree in $H^+$ is $4$. We also define a rotation system $\Sigma^+$ for the graph $H^+$ in a natural way: for every vertex $u\in V(H)$,
consider the standard drawing of the grid $\Pi(u)$, to which we add the drawings of the edges that are incident to vertices $s_1(u),\ldots,s_{d(u)}(u)$, so that the edges are drawn on the grid's exterior in a natural way (\mynote{add figure}). This layout defines an ordering ${\mathcal{O}}^+(v)$ of the edges incident to every vertex $v\in \Gamma(u)$.
We start with the following simple claim.
\begin{claim}\label{claim: cheap solution to modified instance} There is a solution to the \textnormal{\textsf{MCNwRS}}\xspace problem instance $(H^+,\Sigma^+)$ of cost at most ..., such that no inner edge of $H^+$ participates in any crossings in the solution.
\end{claim}
\begin{proof}
We start by showing that $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat H,\hat \Sigma)\leq ...$. Recall that, from \Cref{lem: crossings in contr graph}, there is a drawing $\hat \phi$ of graph $\hat H$ with at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2$ crossings, such that for every vertex $x\in V(\hat H)\cap V(H)$, the ordering of the edges of $\delta_{\hat H}(x)$ as they enter $x$ in $\hat \phi$ is consistent with the ordering ${\mathcal{O}}_x\in \Sigma$, and hence with the ordering $\hat {\mathcal{O}}_x\in \hat \Sigma$. Drawing $\hat \phi$ of $\hat H$ may not be a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace since for some supernodes $v(C)$, the ordering of the edges that are incident to $v(C)$ in $\hat H$ as they enter the image of $v(C)$ in $\hat \phi$ may be different from $\hat {\mathcal{O}}_{v(C)}$. For each such vertex $v(C)$, we may need to \emph{reorder} the images of the edges of $\delta_{\hat H}(v(C))=\delta_H(C)$ near the image of $v(C)$, so that they enter the image of $v(C)$ in the correct order. This can be done by introducing at most $|\delta_H(C)|^2$ crossings for each such supernode $v(C)$. The resulting drawing $\hat \phi'$ of $\hat \phi$ is a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, whose cost is bounded by:
\[O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2+\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2+ \]
\end{proof}
\paragraph{Summary of Step 2.}
If Step 2 did not return FAIL, then it has computed a set ${\mathcal{P}}=\set{P_t\mid t\in \tilde T}$ of simple paths in graph $H_1$, routing the vertices of $\tilde T$ to some vertex $x$, with congestion at most $\rho=O\textsf{left}(\frac{\eta\log^{30}n}{\alpha^6(\alpha')^2}\textsf{right} )$, so that the paths in ${\mathcal{P}}$ are locally non-interfering with respect to the rotation system $\Sigma$ for graph $H$.
We used this set ${\mathcal{P}}$ of paths in a natural way in order to define a circular ordering ${\mathcal{O}}^*$ of the terminals in $\tilde T$. The ordering is consistent with the order ${\mathcal{O}}_x\in \Sigma$ of the edges incident to $x$, which can be viewed as defining an ordering of the paths in ${\mathcal{P}}$, as the last edge on each such path is incident to $x$.
\subsection{Step 3: Constructing a Grid Skeleton}
\mynote{do this for $H$, not $\hat H$, but for clusters, external edges are those entering cluster, the rest is internal.}
In the remainder of the proof, it would be convenient for us to work with a low-degree equivalent of the graph $\hat H$, that we denote by $H^+$. In order to define the graph $\hat H^+$, we start by defining a rotation system $\hat \Sigma $ for graph $\hat H$, as follows. Let $x\in V(\hat H)$ be a vertex. If $x$ is a supernode, that is, $x=v(C)$ for some cluster $C\in {\mathcal{C}}$, then we define the circular ordering $\hat{\mathcal{O}}_x\in \hat \Sigma$ of the edges of $\delta_{\hat H}(x)$ to be arbitrary.
Otherwise, $x$ is a regular vertex, and it is a vertex of the original graph $H$. We then let $\hat {\mathcal{O}}_x\in \hat \Sigma$ be identical to the ordering ${\mathcal{O}}_x\in \Sigma$, where $\Sigma$ is the original rotation system for graph $H$.
We are now ready to define the modified graph $H^+$. We start with $H^+=\emptyset$, and then process every vertex $u\in V(\hat H)$ one-by-one. We denote by $d(u)$ the degree of the vertex $u$ in graph $\hat H$. We now describe an iteration when a vertex $u\in V(\hat H)$ is processed. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that incident to $u$ in $\hat H$, indexed according to their ordering in $\hat {\mathcal{O}}_u\in \hat \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H^+$. We refer to the edges in the grids $\Pi(u)$ as \emph{inner edges}.
Once every vertex $u\in V(\hat H)$ is processed, we add a collection of \emph{outer edges} to graph $H^+$, as follows. Consider any edge $e=(x,y)\in E(H)$. Assume that $e$ is the $i$th edge of $x$ and the $j$th edge of $y$, that is, $e=e_i(x)=e_j(y)$. Then we add an edge $e'=(s_i(x),s_j(y))$ to graph $H^+$, and we view this edge as the copy of the edge $e\in E(\hat H)$. We will not distinguish between the edge $e$ of $\hat H$ (and the corresponding edge of $H$), and the edge $e'$ of $H^+$.
We note that every terminal $t\in T$ has degree $1$ in $\hat H$, so its corresponding grid $\Pi(t)$ consists of a single vertex, that we also denote by $t$. Therefore, set $T$ of terminals in $H$ naturally corresponds to a set of $k$ terminals in $H^+$, that we denote by $T$ as before.
This completes the definition of the graph $H^+$. Note that the maximum vertex degree in $H^+$ is $4$. We also define a rotation system $\Sigma^+$ for the graph $H^+$ in a natural way: for every vertex $u\in V(H)$,
consider the standard drawing of the grid $\Pi(u)$, to which we add the drawings of the edges that are incident to vertices $s_1(u),\ldots,s_{d(u)}(u)$, so that the edges are drawn on the grid's exterior in a natural way (\mynote{add figure}). This layout defines an ordering ${\mathcal{O}}^+(v)$ of the edges incident to every vertex $v\in \Gamma(u)$.
We can uset the ${\mathcal{P}}$ of paths in graph $H$ in order to define a set ${\mathcal{P}}'$ of paths in graph $H^+$, routing all terminals to vertices of $\Pi(x)$, with edge-congestion $\rho$, such that the paths in ${\mathcal{P}}'$ are locally non-interfering.
\mynote{explain?}
We now consider two cases. The first case happens, if there is some vertex $x\in V(H)$, with $|{\mathcal{R}}(x)|\geq \tilde k'$, then we fix this vertex $x$, denote ${\mathcal{P}}_0={\mathcal{R}}(x)$, and we denote by $T_0\subseteq \tilde T$ the set of terminals that serve as endpoints of paths in ${\mathcal{P}}_0$.
We assume w.l.o.g. that the paths in ${\mathcal{P}}'$ are simple. Note that, since every terminal in $\tilde T$ has degree $1$ in $H_1$, a vertex of $\tilde T\cup \set{x}$ cannot serve as an inner vertex on any path in ${\mathcal{P}}'$. We then use the algorithm from \Cref{lem: non_interfering_paths} in order to obtain another set ${\mathcal{P}}$ of simple paths routing the vertices of $\tilde T$ to $x$ in graph $H_1$ with congestion at most $\rho$, so that the paths in ${\mathcal{P}}$ are locally non-interfering with respect to the input rotation system $\Sigma$.
We exploit the paths in ${\mathcal{P}}$ in order to define a circular ordering ${\mathcal{O}}^*$ of the terminals in $\tilde T$, as follows.
Recall that we are given, as part of the definition of the instance $I=(H,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, a circular ordering ${\mathcal{O}}_x$ of the edges of $\delta_H(x)$. We use this ordering ${\mathcal{O}}_x$, in order to define an ordering ${\mathcal{O}}^*$ of the terminals in $\tilde T$, in a natural way. Let $\delta_{H_1}(x)=\set{e_1,\ldots,e_{\lambda}}$, where the edges are indexed according to the ordering ${\mathcal{O}}_x$. For all $1\leq j\leq \lambda$, let $\hat T_j\subseteq \tilde T$ be the set of all terminals $t$, such that the last edge on path $P_t\in {\mathcal{P}}$ is $e_j$, where $P_t$ is the unique path in ${\mathcal{P}}$ that originates from $t$. We order the terminals in $\tilde T$ so that terminals belonging to different subsets $\hat T_j$ appear in the natural order $\hat T_1,\hat T_2,\ldots,\hat T_{\lambda}$ of these subsets, while the ordering between the terminals lying in the same set $\hat T_j$ is chosen arbitrarily.
Intuitively, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is significantly smaller than $(\tilde k/\rho)^2$, then there are not that many crossings between images of different paths in ${\mathcal{P}}$. We will exploit this fact in order to construct a grid-like structure in the graph $H_2$, by exploiting the ordering ${\mathcal{O}}^*$ of the terminals that we have just defined in the following steps.
Let ${\mathcal{R}}$ be any collection of paths in this new graph $H^+$. We say that ${\mathcal{R}}$ is a \emph{legal routing} iff $|{\mathcal{R}}|=k$, and there is some vertex $u\in V(H)$, such that the paths in ${\mathcal{R}}$ route the set $T$ of terminals to vertices of $\Gamma(u)$. Note that a legal routing ${\mathcal{R}}$ naturally defines a collection ${\mathcal{R}}'$ of paths in graph $H$ routing the set $T$ of terminals to vertex $u$, such that for every edge $e\in E(H)$, $\cong_{H}({\mathcal{R}},e)\leq \cong_{H^+}({\mathcal{R}},e)$; the set ${\mathcal{R}}'$ of paths is obtained from the paths in ${\mathcal{R}}$ by contracting all outer edges.
Assume now that we compute a distribution ${\mathcal{D}}'$ over legal routings ${\mathcal{R}}$, such that, for every outer edge $e\in E(H^+)$, $\expect{\cong_{H^+}({\mathcal{R}},e)}\leq x$. Then we immediately obtain a distribution ${\mathcal{D}}$ over @@@
\subsection{Plan for the rest of the proof}
\begin{itemize}
\item select 1 path in $z$, where for example $z=\eta$, though can also use $\eta^2$. Will get a grid skeleton of size $z\times z$.
\item before that, throw away all paths that have more than $\eta/\alpha$ outer edges on them. Most paths will stay.
\item now if we look at a vertical path, it has few outer edges, so for most cells it goes through it does not use an outer edge. Which means that all horizontal paths going through that cell meet in a single vertex (or go through a single cluster). Call such a cell good.
\item at least half the cells are good. In a type-1 good cell all horizontal paths go through a single regular vertex. In a type-2 good cell they all go through 1 cluster. The hope is that there are few type-2 good cells.
\item why: we have about $z^2$ good cells. Inside all good cells there are about $k^2/z^2$ inner edges (can make a cell good if both all horizontal paths meet at some vertex, and all vertical paths meet at some vertex, and then it should be the same vertex). The hope is that the clusters can only contribute $\sum_C|\delta(C)|^2$ vertices to the grid. This is because, if we look at a cluster $C$, the total number of vertical/horizontal paths entering $C$ is at most $\delta(C)$. Even if all pairs have their intersections inside this cluster, it can't contribute more than $|\delta(C)|^2$ such intersecting pairs.
\end{itemize}
\newpage
\iffalse
contracted graph $\hat H=H_{|{\mathcal{C}}}$. Let $U=\set{v(C)\mid C\in {\mathcal{C}}}$ be the set of super-node vertices of $\hat H$, and let $U'$ be the set of all remaining vertices. We define a rotation system $\hat \Sigma$ for graph $\hat H$ as follows: for a super-node $v(C)\in U$, we let ${\mathcal{O}}_{v(C)}$ be an arbitrary ordering of the edges incident to $v(C)$, and for a regular vertex $u\in U'$, we let ${\mathcal{O}}_u$ be identical to the ordering in the rotation system $\Sigma$ in instance $I$. Recall that, from \Cref{lem: crossings in contr graph}, there is a drawing $\phi$ of $\hat H$, containing at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)$ crossings, such that for every regular vertex $u\in U'$, the ordering of the edges of $\delta_{\hat H}(u)$ as they enter $u$ in $\phi$ is consistent with ${\mathcal{O}}_u\in \Sigma$. We can use this drawing in order to define a solution $\phi'$ to instance $\hat I=(\hat H,\hat \Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, by reordering the edges entering every super-node $v(C)$ as needed. This can be done so that $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)+\sum_{C\in {\mathcal{C}}'}|\delta_H(C)|^2\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)+k^2/x$. Therefore, $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)+k^2/x$.
We are now ready to define graph $\hat H^+$. In order to do so, we start with graph $\hat H$, and process every vertex $u\in V(\hat H)\setminus T$ (that may be a regular vertex or a super-node one-by-one). Consider any such vertex $u$, and let $e_1,\ldots,e_{d(u)}$ be the edges incident to $u$ in $\hat H$, indexed according to their ordering in ${\mathcal{O}}_u\in \hat \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ vrid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$. We replace vertex $u$ with the grid $\Pi(u)$, and, for all $1\leq i\leq d(u)$, if $e_i=(u',u)$, then we replace $e_i$ with a new edge $e'_i=(u',s_i(u))$. Once every vertex of $V(\hat H)\setminus T$ is processed, we obtain the final graph $\hat H^+$. For every vertex $u$, we clall the edges of the grid $\Pi(u)$ \emph{inner edges}. We call all edges of $E(\hat H^+)$ that are not inner edges \emph{outer edges}. Notice that there is a $1$-to-$1$ correspondence between the outer edges and the edges of graph $\hat H$. The following observation is immediate.
\begin{observation}\label{obs: expanded contracted graph}
The set $T$ of terminals is $(\alpha\alpha')$-well-linked in $\hat H^+$, and there is a drawing $\phi''$ of graph $\hat H^+$ with at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)+k^2/x$ crossings, such that the inner edges do not participate in any crossings in $\phi''$.
\end{observation}
\fi
\section{Third Key Tool: Computing Guiding Paths}
\label{subsec: guiding paths}
We start with intuition. Consider an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, and let $C\subseteq G$ be a subgraph of $G$ that we refer to as a cluster. Fix an optimal solution $\phi^*$ to instance $I$; we define $\mathsf{cr}(C)$ as usual with respect to $\phi^*$ -- the number of crossings in $\phi^*$ in which the edges of $C$ participate. For a set ${\mathcal{Q}}$ of paths in $G$, the cost $\operatorname{cost}({\mathcal{Q}})$ is also defined as before, with respect to $\phi^*$.
Assume that the boundary of $C$ is $\alpha$-well-linked in $C$, and denote $|E(C)|/|\delta(C)|=\eta$. We show below a randomized algorithm that either computes a vertex $u\in V(C)$ and a set ${\mathcal{Q}}$ of paths routing $\delta(C)$ to $u$, whose expected cost is comparable to $\mathsf{cr}(C)$; or with high probability correctly establishes that $\mathsf{cr}(C)>|\delta(C)|^2/\operatorname{poly}(\eta)$.
We say that cluster $C$ is \emph{good} if the former outcome happens, and we say that it is \emph{bad} otherwise.
Intuitively, given our input instance $I=(G,\Sigma)$, we can compute a decomposition ${\mathcal{C}}$ of $G$ into clusters, such that for each cluster $C\in {\mathcal{C}}$, the boundary of $C$ is $1/\operatorname{poly}\log m$-well-linked in $C$, and moreover $|\delta(C)|\geq |E(C)|/\operatorname{poly}(\mu)$. In order to compute the decomposition, we start with ${\mathcal{C}}$ containing a single cluster $G$, and the interate. In every iteration, we select a cluster $C\in {\mathcal{C}}$, and then either compute a sparse balanced cut $(A,B)$ in $C$, or compute a cut $(A,B)$ that is sparse with respect to the edges of $\delta(C)$ (that is, $|E(A,B)|<\min\set{|\delta(C)\cap \delta(A)|,|\delta(C)\cap \delta(B)|}/\operatorname{poly}\log n$; or return FAIL. In the former two cases, we replace $C$ with $C[A]$ and $C[B]$ in ${\mathcal{C}}$, while in the latter case we declare the cluster $C$ \emph{settled}; we will show that in this case the boundary of $C$ is $1/\operatorname{poly}\log n$-well-linked in $C$ and $|\delta(C)|\geq |E(C)|/\operatorname{poly}(\mu)$. The algorithm terminates once every cluster in ${\mathcal{C}}$ is settled. We show that each of the resulting clusters $C\in {\mathcal{C}}$ is sufficiently small (that is, $|E(C)|\leq m/\mu$), and the total number of edges connecting different clusters is also sufficiently small (at most $m/\mu$). If each resulting cluster is a good cluster, then we obtain the desired decomposition ${\mathcal{C}}$, together with the desired collections $\set{{\mathcal{Q}}(C)}_{C\in {\mathcal{C}}}$ of paths. Unfortunately, some of the clusters $C\in {\mathcal{C}}$ may be bad. For each such cluster we are guaranteed with high probability that $\mathsf{cr}(C)>|\delta(C)|^2/\operatorname{poly}(\mu)$, but unfortunately the sum of values of $|\delta(C)|^2$ over all bad clusters $C$ may be quite small, significantly smaller than say $|E(G)|^2$, so we cannot conclude that $\mathsf{OPT}(I)\geq |E(G)|^2/\operatorname{poly}(\mu)$.
In order to overcome this difficulty, we partition the set ${\mathcal{C}}$ of clusters into a set ${\mathcal{C}}'$ of good clustes, and a set ${\mathcal{C}}''$ of bad clusters. Let $G'=G\setminus (\bigcup_{C\in {\mathcal{C}}'}C)$, and let $G''=G'_{|{\mathcal{C}}''}$ be obtained from $G'$ by contracting each bad cluster into a super-node. We then apply the same decomposition algorithm to graph $G''$. We continue this process until no bad clusters remain, so we have obtained a decomposition of $G$ into a collection of good clusters, or we compute a bad cluster $C$ with $|\delta(C)|\geq |E(G)|/\operatorname{poly}(\mu)$, so we can conclude that $\mathsf{OPT}(I)\geq|E(G)|^2/\operatorname{poly}(\mu)$.
In order to be able to carry out this process, we need an algorithm that, given a cluster $C$, either computes the desired set ${\mathcal{Q}}$ of paths routing the edges of $\delta(C)$ to some vertex $u$ of $C$, such that the cost of ${\mathcal{Q}}$ is low; or establishes that $\mathsf{cr}(C)>|\delta(C)|^2/\operatorname{poly}(\mu)$. But because we will need to apply this algorithm to graphs obtained from sub-graphs of $G$ by contracting some bad clusters, we need a more general algorithm, that is summarized in the following theorem.
Suppose we are given a graph $H$ and a set $T$ of its vertices called terminals. Let $\Lambda(H,T)$ denote the set of all pairs $({\mathcal{Q}},x)$, where $x$ is a vertex of $H$, and ${\mathcal{Q}}$ is a collection of paths routing the vertices of $T$ to $x$ in ${\mathcal{Q}}$. A \emph{distribution} ${\mathcal{D}}$ over pairs in $\Lambda(H,T)$ is an assignment, to every pair $({\mathcal{Q}},x)\in \Lambda(H,T)$, of a probability value $p({\mathcal{Q}},x)\geq 0$, such that $\sum_{({\mathcal{Q}},x)\in \Lambda(H,T)}p({\mathcal{Q}},x)=1$. The distribution ${\mathcal{D}}$ is specified by listing all pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$ of pairs with $p({\mathcal{Q}},x)>0$, together with the corresponding probability value $p({\mathcal{Q}},x)$.
\begin{theorem}\label{thm: find guiding paths}
There is a large enough constant $c_0$, and an efficient randomized algrorithm, that receives as input an instance $I=(H,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, where $|V(H)|=n$, a set $T\subseteq V(H)$ of $k$ vertices of $H$ called terminals, and a collection ${\mathcal{C}}$ of disjoint subgraphs of $H$, such that for some parameters $0\leq \alpha,\alpha'\leq 1$ and $\eta,\eta'\geq 1$ with $\eta'\geq \eta^8$, the following conditions hold:
\begin{itemize}
\item $k\geq |E(H_{|{\mathcal{C}}})|/\eta$ and $k\geq \eta^6$;
\item $\eta\geq \frac{c^*\log^{46}n}{\alpha^{10}(\alpha')^2}$ for some large enough constant $c^*$;
\item every terminal $t\in T$ has degree $1$ in $H$;
\item for all $C\in {\mathcal{C}}$, $V(C)\cap T=\emptyset$;
\item set $T$ of terminals is $\alpha$-well-linked in the contracted graph $H_{|{\mathcal{C}}}$;
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property; and
\item for every cluster $C\in {\mathcal{C}}$, if $\Sigma_C$ is the rotation system for $C$ that is consistent with $\Sigma$, then $\mathsf{OPT}_{\mathsf{cr}}(C,\Sigma_C)\geq \frac{|\delta(C)|^2}{\eta'}$.
\end{itemize}
The algorithm either returns FAIL, or computes a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$, such that for all $e\in E(C)$, $\expect{(\cong({\mathcal{Q}},e))^2}\leq O(\log^8k/(\alpha\alpha')^2)$ \mynote{OK to make smaller}. Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq
\frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}n}$, then the probability that the algorithm returns FAIL is at most $1/\operatorname{poly}(n)$.
\mynote{need to be careful with this. OK to put $\eta'+\operatorname{poly}\eta$ on the bottom, not OK to put $\eta'\eta$. This is because instance size in the recursion will decrease by factor $\eta$ only.}
\end{theorem}
The remainder of this section is dedicated to the proof of \Cref{thm: find guiding paths}.
For conveninence, we denote the contracted graph $H_{|{\mathcal{C}}}$ by $\hat H$, and we denote $|E(\hat H)|=\hm$. From the statement of \Cref{thm: find guiding paths}, $k\geq \hm/\eta$. Observe that, from \Cref{clm: contracted_graph_well_linkedness}, the set $T$ of terminals is $(\alpha\alpha')$-well-linked in $H$.
We start with some intuition. Assume first that graph $H$ contains a grid (or a grid minor) of size $(\Omega(k\alpha\alpha'/\operatorname{poly}\log n)\times (k\alpha \alpha'/\operatorname{poly}\log n))$, and a collection ${\mathcal{P}}$ of paths connecting every terminal to a distinct vertex on the first row of the grid, such that the paths in ${\mathcal{P}}$ cause a low edge-congestion. For this special case, the algorithm of \cite{Tasos-comm} (see also Lemma D.10 in the full version of \cite{chuzhoy2011algorithm}) provides the distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$ with the required properties. Moreover, if $H$ is a bounded-degree planar graph, with a set $T$ of terminals that is $(\alpha\alpha')$-well-linked,
then there is an efficient algorithm to compute such a grid minor, together with the required collection ${\mathcal{P}}$ of paths. If $H$ is planar but no longer bounded-degree, we can still compute a grid-like structure in it, and apply the same arguments as in \cite{Tasos-comm} in order to compute the desired distribution ${\mathcal{D}}$. The difficulty in our case is that the graph $H$ may be far from being planar, and, even though, from the Excluded Grid theorem of Robertson and Seymour \mynote{add references to the original thm proof and our new proofs}, it must contain a large grid-like structure, without having a drawing of $H$ in the plane with a small number of crossing, we do not know how to compute such a structure.
The proof of the theorem consists of five steps. In the first step, we will either establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is sufficiently large (so the algorithm can return FAIL), or compute a subgraph $\hat H'\subseteq \hat H$, and a partition $(X,Y)$ of $V(\hat H')$, such that each of the clusters $\hat H'[X],\hat H'[Y]$ has the $\hat \alpha$-bandwidth property, for $\hat \alpha=\Omega(\alpha/\log^4n)$, together with a large collection of edge-disjoint paths routing the terminals to the edges of $E_{\hat H'}(X,Y)$ in graph $\hat H'$. Intuitively, we will view from this point onward the edges of $E_{\hat H'}(X,Y)$ as a new set of terminals, that we denote by $\tilde T$ (more precisely, we subdivide each edge of $E_{\hat H'}(X,Y)$ with a new vertex that becomes a new terminal). We show that it is sufficient to prove an analogue of \Cref{thm: find guiding paths} for this new set $\tilde T$ of terminals. The clusters $\hat H'[X],\hat H'[Y]$ of graph $\hat H'$ naturally define a partition $(H_1,H_2)$ of the graph $H$ into two disjoint clusters. In the second step, we either establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is suffciently large
(so the algorithm can return FAIL), or
compute some vertex $x$ of $H_1$, and a collection ${\mathcal{P}}$ of paths in graph $H_1$, routing the terminals of $\tilde T$ to $x$, such that the paths in ${\mathcal{P}}$ cause a relatively low edge-congestion.
We exploit this set ${\mathcal{P}}$ of paths in order to define an ordering of the termianls in $\tilde T$, which is in turn exploited in the third step in order to compute a ``skeleton'' for the grid-like structure. We compute the grid-like structure itself in the fourth step. In the fifth and the final step, we generalize the arguments from \cite{Tasos-comm} and \cite{chuzhoy2011algorithm} in order to obtain the desired distribution ${\mathcal{D}}$.
Before we proceed, we need to consider two simple special cases. In the first case, $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2$ is large. In the second case, we can route a large subset of the terminals to a single vertex of $V(\hat H)\cap V(H)$ in the graph $\hat H$ via edge-disjont paths.
\paragraph{Special Case 1: $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2$ is large.}
We consider the special case where $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2\geq \frac{(k \alpha^4 \alpha')^2}{c_0\log^{50}n}$, where $c_0$ is the constant from the statement of \Cref{thm: find guiding paths}.
In this case, since we are guaranteed that, for every cluster $C\in {\mathcal{C}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq |\delta(C)|^2/\eta'$, we get that:
\[\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \sum_{C\in {\mathcal{C}}}\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq \sum_{C\in {\mathcal{C}}}\frac{|\delta(C)|^2}{\eta'}\geq \frac{(k \alpha^4 \alpha')^2}{c_0\eta'\log^{50}n}.\]
Therefore, if $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2\geq \frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}n}$, the algorithm returns FAIL and terminates. We assume from now on that:
\begin{equation}\label{eq: boundaries squared sum bound}
\sum_{C\in {\mathcal{C}}}|\delta(C)|^2<\frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}n}
\end{equation}
\paragraph{Special Case 2: Routing of terminals to a single vertex.}
The second special case happens if there exists a collection ${\mathcal{P}}_0$ of at least $\frac{k\alpha\alpha'}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\log^4k}$ edge-disjoint paths in graph $H$ routing some subset $T_0\subseteq T$ of terminals to some vertex $x$ (here, $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}$ is the constant from \Cref{claim: embed expander}).
Note that, if Special Case 1 did not happen, then $x$ may not be a supernode. Indeed, assume that $x=v_C$ for some cluster $C\in {\mathcal{C}}$. Then:
\[(\delta_H(C))^2\geq \Omega\textsf{left}(\frac{(k\alpha \alpha')^2}{\log^8k}\textsf{right} )\geq \frac{(k\eta' \alpha^4 \alpha')^2}{c_0\eta'\log^{50}n},\]
assuming that $c_0$ is a large enough constant, a contradiction (we have used the fact that $\eta'\geq \eta^8$). Therefore, we can assume that $x$ is not a supernode.
Denote $z=|{\mathcal{P}}_0|\geq \frac{k\alpha\alpha'}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\log^4k}$.
We compute a single set ${\mathcal{Q}}$ of paths routing the set $T$ of terminals to $x$, with congestion $O(\log^4k/(\alpha \alpha')^2)$, as follows. We partition the set $T\setminus T_0$ of terminals into $q=\ceil{k/z}\leq O(\log^4k/(\alpha\alpha'))$ subsets, each of which contains at most $z$ vertices.
Consider now some index $1\leq i\leq q$. Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in $H$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{P}}_i'$ of paths in graph $H$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in ${\mathcal{P}}'_i$ cause edge-congestion $O(1/(\alpha\alpha'))$, and each vertex of $T_0\cup T_i$ is the endpoint of at most one path in ${\mathcal{P}}_i'$. By concatenating the paths in ${\mathcal{P}}_i'$ with paths in ${\mathcal{P}}_0$, we obtain a collection ${\mathcal{P}}_i$ of paths in graph $H$, connecting every terminal of $T_i$ to $x$, that cause edge-congestion $O(1/(\alpha\alpha'))$. Let ${\mathcal{Q}}=\bigcup_{i=0}^q{\mathcal{P}}_i$ be the resulting set of paths. Observe that set ${\mathcal{Q}}$ contains $k$ paths, routing the terminals in $T$ to the vertex $x$ in graph $H$, with $\cong_H({\mathcal{Q}})\leq O(q/(\alpha\alpha'))\leq O(\log^4k/(\alpha\alpha')^2)$. We return a distribution ${\mathcal{D}}$ consisting of a single pair $({\mathcal{Q}},u)$, and terminate the algorithm.
In the remainder of the algorithm, we assume that neither of the two special cases happened. We now describe each step of the algorithm in turn.
\subsection{Step 1: Splitting the Contracted Graph}
In this step, we split the contracted graph $\hat H$, using the algorithm summarized in the following theorem.
\begin{theorem}\label{thm: splitting}
There is an efficient randomized algorithm that with probability at most $1/\operatorname{poly}(n)$ returns FAIL, and, if it does not return FAIL, then it computes a subgraph $\hat H'\subseteq \hat H$ and a partition $(X,Y)$ of $V(\hat H')$ such that:
\begin{itemize}
\item each of the clusters $\hat H'[X]$ and $\hat H'[Y]$ has the $\hat \alpha'$-bandwidth property, for $\hat \alpha'=\Omega(\alpha/\log^4n)$; and
\item there is a set ${\mathcal{R}}$ of at least $\Omega(\alpha^3k/\log^8n)$ edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to edges of $E_{\hat H'}(X,Y)$.
\end{itemize}
\end{theorem}
\begin{proof}
We start by applying the algorithm from \Cref{claim: embed expander} to graph $\hat H$ and the set $T$ of terminals, to obtain a graph $W$ with $V(W)=T$ and maximum vertex degree $O(\log^2k)$, and an embedding ${\hat{\mathcal{P}}}$ of $W$ into $\hat H$ with congestion $O((\log^2k)/\alpha)$. Let $\hat{\cal{E}}$ be the bad event that $W$ is not a $1/4$-expander. Then $\prob{\hat {\cal{E}}}\leq 1/\operatorname{poly}(k)$.
Let $\hat H'$ be the graph that is obtained from $\hat H$ by deleting from it all edges and vertices except those that participate in the paths in ${\hat{\mathcal{P}}}$. Equivalently, graph $\hat H'$ is obtained from the union of all paths in ${\hat{\mathcal{P}}}$. We need the following observation.
\begin{observation}\label{obs: expansion and degree}
If event $\hat {\cal{E}}$ did not happen, then the set $T$ of vertices is $\hat \alpha$-well-linked in $\hat H'$, for $\hat \alpha=\frac{\alpha}{4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$, and the maximum vertex degree in $\hat H'$ is at most $d=\frac{\alpha k}{512\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$.
\end{observation}
\begin{proof}
Assume that Event $\hat {\cal{E}}$ did not happen. We first prove that the set $T$ of terminals is $\hat \alpha$-well-linked in $\hat H$. Consider any paritition $(A,B)$ of vertices of $\hat H'$, and denote $T_A=T\cap A$, $T_B=T\cap B$. Assume w.l.o.g. that $|T_A|\leq |T_B|$. Then it is sufficient to show that $|E_{\hat H'}(A,B)|\geq \hat \alpha\cdot |T_A|$.
Consider the partition $(T_A,T_B)$ of the vertices of $W$, and denote $E'=E_{W}(T_A,T_B)$. Since $W$ is a $1/4$-expander, $|E'|\geq |T_A|/4$ must hold. Consider now the set $\hat {\mathcal{R}}\subseteq {\hat{\mathcal{P}}}$ of paths containing the embeddings $P(e)$ of every edge $e\in E'$. Each path $R\in \hat {\mathcal{R}}$ connects a vertex of $T_A$ to a vertex of $T_B$, so it must contain an edge of $|E_{\hat H}(A,B)|$. Since $|\hat{\mathcal{R}}|\geq |T_A|/4$, and the paths in ${\hat{\mathcal{P}}}$ cause edge-congestion at most $(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)/\alpha$, we get that $|E_{\hat H}(A,B)|\geq \alpha\cdot |T_A|/(4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)\geq \hat \alpha |T_A|$.
Assume now that maximum vertex degree in $\hat H'$ is greater than $d$, and let $x$ be a vertex whose degree is at least $d$. Let $\hat{\mathcal{Q}}\subseteq \tilde{\mathcal{P}}$ be the set of all paths containing the vertex $x$.
Consider any such path $Q\in \hat{\mathcal{Q}}$. The endpoints of this path are two distinct terminals $t,t'\in T$. We let $Q'\subseteq Q$ be the sub-path of $Q$ between the terminal $t$ and the vertex $x$, and we let ${\mathcal{Q}}'=\set{Q'\mid Q\in \hat{\mathcal{Q}}}$.
Recall that every vertex in $W$ has degree at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$, and so a terminal in $T$ may be an endpoint of at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$ paths in ${\hat{\mathcal{P}}}$. Therefore, there is a subset ${\mathcal{Q}}''\subseteq {\mathcal{Q}}'$ of at least $d/(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)$ paths, each of which originates at a distinct terminal.
But then, from \Cref{claim: routing in contracted graph}, there is a collection ${\mathcal{Q}}'''$ of edge-disjoint paths in graph $H$, routing a subset of terminals to $x$ of cardinality at least:
\[\frac{\alpha'|{\mathcal{Q}}''|} 2\geq \frac{\alpha' d}{2\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\geq \frac{\alpha \alpha' k}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^4k},\]
contradicting the fact that Special Case 2 did not happen.
\end{proof}
Next, we use the following lemma that allows us to compute the required sets $X$, $Y$ of vertices. The proof follows immediately from techniques that were introduced in \cite{chuzhoy2012routing} and then refined in \cite{chuzhoy2012polylogarithmic,chekuri2016polynomial,chuzhoy2016improved}. Unfortunately, all these proofs assumed that the input graph has a bounded vertex degree, and additionally the proofs are somewhat more involved than the proof that we need here (this is because these proofs could only afford a $\operatorname{poly}\log k$ loss in the cardinality of the path set ${\mathcal{R}}$ relatively to $|T|$, while we can afford a $\operatorname{poly}\log n$ loss). Therefore, we provide a complete proof of the lemma in Section \ref{sec: splitting} of the Appendix for completeness.
\begin{lemma}\label{lem: splitting}
There is an efficient algorithm that, given as input an $n$-vertex graph $G$, and a subset $T$ of $k$ vertices of $G$ called terminals, together with a parameter $0<\tilde \alpha<1$, such that the maximum vertex degree in $G$ is at most $\tilde \alpha k/64$, and every vertex of $T$ has degree $1$ in $G$, either returns FAIL, or computes a partition $(X,Y)$ of $V(G)$, such that:
\begin{itemize}
\item each of the clusters $G[X]$, $G[Y]$ has the $\tilde \alpha'$-bandwidth property, for $\tilde \alpha'=\Omega(\tilde \alpha/\log^2n)$; and
\item there is a set ${\mathcal{R}}$ of at least $\Omega(\tilde \alpha^3k/\log^2n)$ edge-disjoint paths in graph $G$, routing a subset of terminals to edges of $E_G(X,Y)$.
\end{itemize}
Moreover, if the vertex set $T$ is $\tilde \alpha$-well-linked in $G$, then the algorithm never returns FAIL.
\end{lemma}
We apply the algorithm from \Cref{lem: splitting} to graph $\hat H'$, the set $T$ of terminals, and parameter $\tilde \alpha=\hat \alpha=\frac{\alpha}{4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$. Recall that we are guaranteed that maximum vertex degree in $\hat H'$ is at most $d=\frac{\alpha k}{512\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\leq \frac{\tilde \alpha k}{64}$.
Note that the algorithm from \Cref{lem: splitting} may only return FAIL if the set $T$ of terminals is not $\hat \alpha$-well-linked in $\hat H'$, which, from \Cref{obs: expansion and degree}, may only happen if Event $\hat {\cal{E}}$ happened, which in turn may only happen with probability $1/\operatorname{poly}(k)$. If the algorithm from \Cref{lem: splitting} returned FAIL, then we restart the algorithm that we have described so far from scratch. We do so at most $1/\operatorname{poly}\log n$ times, and if, in every iteration, the algorithm from \Cref{lem: splitting} returns FAIL, then we return FAIL as the outcome of the algorithm for \Cref{thm: splitting}. Clearly, this may only happen with probability at most $1/\operatorname{poly}(n)$. Therefore, we assume that in one of the iterations, the algorithm from \Cref{lem: splitting} did not return FAIL. From now on we consider the outcome of that iteration.
Let $(X,Y)$ be the partition of $V(\hat H')$ that the algorithm returns. We are then guaranteed that each of the clusters $\hat H'[X],\hat H'[Y]$ has the $\tilde \alpha'$-bandwidth property, where $\tilde \alpha'=\Theta(\hat \alpha/\log^2n)=\Theta(\alpha/\log^4n)$. The algorithm also ensures that there is a collection ${\mathcal{R}}$ of edge-disjoint paths in $\hat H'$, routing a subset of the terminals to edges of $E_{\hat H'}(X,Y)$, with:
\[ |{\mathcal{R}}|\geq \Omega(\tilde \alpha^3k/\log^2n)=\Omega(\alpha^3k/\log^8n). \]
This completes the proof of \Cref{thm: splitting}.
\end{proof}
If the algorithm from \Cref{thm: splitting} returned FAIL (which may only happen with probability at most $1/\operatorname{poly}(n)$), then we terminate the algorithm and return FAIL as well. Therefore, we assume from now on that the algorithm from \Cref{thm: splitting} returned a subgraph $\hat H'\subseteq \hat H$ and a partition $(X,Y)$ of $V(\hat H')$ such that each of the clusters $\hat H'[X]$ and $\hat H'[Y]$ has the $\hat \alpha'$-bandwidth property, for $\hat \alpha'=\Omega(\alpha/\log^4n)$, and
there is a set ${\mathcal{R}}$ of at least $\Omega(\alpha^3k/\log^8n)$ edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to edges of $E_{\hat H'}(X,Y)$.
Notice that we can compute the path set ${\mathcal{R}}$ with the above properties efficiently, using standard Maximum $s$-$t$ flow.
Let $E'\subseteq E_{\hat H'}(X,Y)$ be the subset of edges containing the last edge on every path in ${\mathcal{R}}$, so, by reversing the direction of the paths in ${\mathcal{R}}$, we can view the set ${\mathcal{R}}$ of paths as routing the edges of $E'$ to the terminals.
In the remainder of this step, we will slightly modify the graphs $H$ and $\hat H$, and we will continue working with the modified graphs in the following steps.
Let $\hat H''\subseteq\hat H'$ be the graph obtained from $\hat H'$ by first deleting all edges of $E_{\hat H'}(X,Y)\setminus E'$ from it, and then subdividing every edge $e\in E'$ with a vertex $t_e$. We denote $\tilde T=\set{t_e\mid e\in E'}$, and we refer to vertices of $\tilde T$ as \emph{pseudo-terminals}. Recall that $|\tilde T|=|{\mathcal{R}}|=\Omega(\alpha^3k/\log^8n)$, and there is a set ${\mathcal{R}}'$ of edge-disjoint paths in the resulting graph $\hat H''$, routing the vertices of $\tilde T$ to the vertices of $T$.
We let $\hat H_1$ be the subgraph of $\hat H''$, that is induced by the set $X\cup \tilde T$ of vertices, and we similarly define $\hat H_2=\hat H''[Y\cup \tilde T]$. From the $\hat \alpha'$-bandwidth property of the clusters $\hat H'[X]$ and $\hat H'[Y]$, we are guaranteed that the vertices of $\tilde T$ are $ \hat \alpha'$-well-linked in both $\hat H_1$ and in $\hat H_2$, where $\hat \alpha'= \Omega(\alpha/\log^4n)$. Let ${\mathcal{C}}'\subseteq {\mathcal{C}}$ be the subset of all clusters $C$ whose corresponding supernode $v_C$ lies in grpah $\hat H''$.
For convenience, we also subdivide, in graph $H$, every edge $e\in E'$, with the vertex $t_e$, so graph $\hat H''$ can be now viewed as a subgraph of the contracted graph $H_{|{\mathcal{C}}}$.
Next, we let $H'\subseteq H$ be the subgraph of $H$ that corresponds to the graph $\hat H''$: namely, graph $H'$ is obtained from $\hat H''$ by replacing every supernode $v_C$ with the corresponding cluster $C\in {\mathcal{C}}'$. Equivalently, we can obtain graph $H'$ from $H$, by deleting every edge of $E(\hat H)\setminus E(\hat H'')$ and every regular (non-supernode) vertex of $V(\hat H)\setminus V(\hat H'')$. Additionally, for every cluster $C\in {\mathcal{C}}\setminus {\mathcal{C}}'$, we delete all edges and vertices of $C$ from $H'$. We also define a rotation system $\Sigma'$ for graph $H'$, which is identical to $\Sigma$ (vertices $t_e\in \tilde T$ all have degree $2$, so their corresponding ordering ${\mathcal{O}}_{t_e}$ of incident edges can be set arbitrarily).
We partition the set ${\mathcal{C}}'$ of clusters into two subsets: set ${\mathcal{C}}_X$ contains all clusters $C\in {\mathcal{C}}'$ with $v_C\in X$, and set ${\mathcal{C}}_Y$ contains all clusters $C\in {\mathcal{C}}'$ with $v_C\in Y$. We can similarly define the graphs $H_1,H_2\subseteq H'$, that correspond to the contracted graphs $\hat H_1$ and $\hat H_2$, respectively: let $X'$ contain all vertices $x\in V(H')$, such that either $x\in C$ for some cluster $C\in {\mathcal{C}}_X$, or $x$ is a regular vertex of $\hat H''$ lying in $X$. Similarly, we let $Y'$ contain all vertices $y\in V(H)$, such that either $y\in C$ for some cluster $C\in {\mathcal{C}}_Y$, or $y$ is a regular vertex of $\hat H''$ lying in $Y$. We then let $H_1=H'[X\cup \tilde T]$, and $H_2=H'[Y\cup \tilde T]$.
The following observation, summarizing some properties of graph $H'$, is immediate:
\begin{observation}\label{obs: properties of new graph}
\
\begin{itemize}
\item $\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')\leq \mathsf{OPT}_{\mathsf{cnwrs}} (H,\Sigma)$;
\item $\hat H''=H'_{|{\mathcal{C}}'}$;
\item $|E(\hat H'')|\leq 2|E(\hat H)|\leq 2\eta k\leq O(|\tilde T|\eta \log^8k/\alpha^3)$; and
\item graph $\hat H_1$ is a contracted graph of $H_1$ with respect to ${\mathcal{C}}_X$, and graph $H_2$ is a contracted graph of $H_2$ with respect to ${\mathcal{C}}_Y$. In other words, $\hat H_1=(H_1)_{|{\mathcal{C}}_X}$, and $\hat H_2=(H_2)_{|{\mathcal{C}}_Y}$.
\end{itemize}
\end{observation}
(for third assertion we have used the fact that $k\geq |E(\hat H)|/\eta$ from the statement of \Cref{thm: find guiding paths}, and $|\tilde T|\geq \Omega(\alpha^3k/\log^8n)$.)
Recall that $\Lambda(H',\tilde T)$ denotes the set of all pairs $({\mathcal{Q}},x)$, where $x$ is a vertex of $H'$, and ${\mathcal{Q}}$ is a collection of paths in graph $H'$, routing the vertices of $\tilde T$ to $x$. In the following observation, we show that it is now enough to compute a distribution ${\mathcal{D}}'$ over pairs $({\mathcal{Q}},x)\in \Lambda(H',\tilde T)$, such that for every edge $e\in E(H')$, $\expect{\cong_{H'}({\mathcal{Q}},e)}$ is low, in order to obtain the desired distribution ${\mathcal{D}}$ over pairs in $\Lambda(H,T)$.
\begin{observation}\label{obs: convert distributions}
There is an efficient algorithm, that, given a distribution ${\mathcal{D}}'$ over pairs $({\mathcal{Q}}',x')\in \Lambda(H',\tilde T)$,
such that for every edge $e'\in E(H')$, $\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e'))^2}\leq \beta$ holds,
computes a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$, such that for every edge $e\in E(H)$, $\expect[({\mathcal{Q}},x)\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left}(\frac{\beta \log^{16}n}{\alpha^8(\alpha')^4}\textsf{right} )$.
\end{observation}
\begin{proof}
Recall that there is a set ${\mathcal{R}}'$ of edge-disjoint paths in graph $\hat H''$, routing the vertices of $\tilde T$ to the vertices of $T$, and moreover, such a path set can be found efficiently via a standard maximum $s$-$t$ flow computation.
Since $\hat H''=H'_{|{\mathcal{C}}'}$, from \Cref{claim: routing in contracted graph}, we can efficiently compute a set ${\mathcal{R}}_0$ of edge-disjoint paths in graph $H''$, routing a subset $T_0\subseteq T$ of terminals to $\tilde T$, with $|{\mathcal{R}}_0|\geq \alpha'\cdot |{\mathcal{R}}'|=\alpha'\cdot |\tilde T|=\Omega(\alpha'\alpha^3k/\log^8n)$.
Let $z=\ceil{k/|T_0|}=O(\log^{8}n/(\alpha^3\alpha')$. Next, we partition the terminals of $T\setminus T_0$ into $z$ subsets $T_1,\ldots,T_z$, of cardinality at most $|T_0|$ each. Consider now some index $1\leq i\leq z$. Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in graph $H$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{R}}_i'$ of paths in graph $H$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in ${\mathcal{R}}'_i$ cause edge-congestion $O(1/(\alpha\alpha'))$, and each vertex of $T_0\cup T_i$ is the endpoint of at most one path in ${\mathcal{R}}'_i$. By concatenating the paths in ${\mathcal{R}}'_i$ with paths in ${\mathcal{R}}_0$, we obtain a collection ${\mathcal{R}}_i$ of paths in graph $H$, routing the terminals in $T_i$ to vertices of $\tilde T$, such that every vertex of $\tilde T\cup {\mathcal{R}}_0$ is an endpoint of at most one path in ${\mathcal{R}}_i$. Let ${\mathcal{R}}^*=\bigcup_{i=0}^z{\mathcal{R}}_i$ be the resulting set of paths. Observe that set ${\mathcal{R}}^*$ contains $k$ paths, each of which connects a distinct terminal of $T$ to a vertex of $\tilde T$, and every vertex of $\tilde T$ serves as an endpoint of at most $z=O(\log^{8}n/(\alpha^3\alpha')$ such paths. The paths of ${\mathcal{R}}^*$ cause congestion at most $O(z/\alpha\alpha')\leq O(\log^8k/(\alpha^4(\alpha')^2))$ in graph $H$.
Recall that $H'\subseteq H$. Consider some pair $({\mathcal{Q}}',x')\in \Lambda(H',\tilde T)$ with probability $p({\mathcal{Q}}',x')>0$ in the distribution ${\mathcal{D}}'$. We compute another path set ${\mathcal{Q}}$ in graph $H$, routing all terminals in $T$ to $x'$, and we will assign to the pair $({\mathcal{Q}},x')\in \Lambda(H,T)$ the same probability value $p({\mathcal{Q}}',x')$. For ever pseudoterminal $\tilde t\in \tilde T$, let $Q'_{\tilde t}\in {\mathcal{Q}}'$ be the unique path originating at $\tilde t$. For every terminal $t\in T$, let $R'_t\in {\mathcal{R}}^*$ be the unique path originating at terminal $t$, and let $\tilde t\in \tilde T$ be the other endpoint of path $R'_t$. We let $Q_t$ be the path obtained by concatenating the paths $R'_t$ and $Q'_{\tilde t}$, so path $Q_t$ connects $t$ to vertex $x'$. We then let ${\mathcal{Q}}=\set{Q_t\mid t\in T}$. Note that ${\mathcal{Q}}$ is a set of paths in graph $H$ routing the set $T$ of terminals to vertex $x'$, so $({\mathcal{Q}},x')\in \Lambda(H,T)$. We assign to the pair $({\mathcal{Q}},x')$ probability $p({\mathcal{Q}}',x')$. Note that, since every pseudoterminal $\tilde t\in \tilde T$ may seve as an endpoint of at most $O(\log^{8}n/(\alpha^3\alpha')$ paths in ${\mathcal{R}}^*$, and since the paths in ${\mathcal{R}}^*$ cause edge-congestion at most $O(\log^8k/(\alpha^4(\alpha')^2))$ in graph $H$, for every edge $e\in E(H)$, we get that:
\[\cong_H({\mathcal{Q}},e)\leq O\textsf{left}(\frac{\cong_{H'}({\mathcal{Q}}',e) \cdot \log^8k}{\alpha^3 \alpha'}+\frac{\log^8k}{\alpha^4(\alpha')^2}\textsf{right} ). \]
Therefore, altogether, for every edge $e\in E(H)$:
\[\expect[({\mathcal{Q}},x')\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left}(\frac{\log^{16}k}{\alpha^8(\alpha')^4}\textsf{right} )\cdot \textsf{left} (\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e))^2}+1\textsf{right} ). \]
\end{proof}
From the above discussion, in order to complete the proof of \Cref{thm: find guiding paths}, it is now enough to design a randomized algorithm, that either computes a distribution ${\mathcal{D}}'$ over pairs in $\Lambda(H',\tilde T)$ with $\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e))^2}\leq xxx$, or returns FAIL. It is enough to ensure that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')\geq
\frac{(k\alpha^4 \alpha')^2}{c_0(\eta^4+\eta')\log^{28}k}$, then the probability that the algorithm returns FAIL is at most $1/\operatorname{poly}(n)$.
In the remainder of the proof we focus on the above goal. It would be convenient for us to simplify the notation, by denoting $H'$ by $H$, $\hat H''$ by $\hat H$, $\tilde T$ by $T$, and $\hat \alpha'$ by $\tilde \alpha$. We also denote ${\mathcal{C}}'$ by ${\mathcal{C}}$. We now summarize all properties of the new graphs $H,\hat H$ that we have established so far, and in the remainder of the proof of \Cref{thm: find guiding paths} we will only work with these new graphs.
\paragraph{Summary of the outcome of Step 1.}
We can assume from now on that we are given a graph $H$, a rotation system $\Sigma$ for $H$, a set $\tilde T$ of terminals in graph $H$, and a collection ${\mathcal{C}}$ of disjoint subgraphs (clusters) of $H\setminus \tilde T$. We denote $|\tilde T|=\tilde k$. The corresponding contracted graph is denoted by $\hat H=H_{|{\mathcal{C}}}$.
We are also given a partition $(X,Y)$ of $V(H)\setminus \tilde T$ (note that for convenience of notation, $X$ and $Y$ are now subsets of vertices of $H$, and not of $\hat H$), and a parition ${\mathcal{C}}_X,{\mathcal{C}}_Y$ of ${\mathcal{C}}$, such that each cluster $C\in {\mathcal{C}}_X$ has $V(C)\subseteq X$, and each cluster $C\in {\mathcal{C}}_Y$ has $V(C)\subseteq Y$. We denote $H_1=H[X\cup \tilde T]$ and $H_2=H[Y\cup \tilde T]$. We also denote by $\hat H_1=(H_1)_{|{\mathcal{C}}_X}$ the contracted graph of $H_1$ with respect to ${\mathcal{C}}_X$, and similarly by $\hat H_2=(H_2)_{|{\mathcal{C}}_Y}$ the contracted graph of $H_2$ with respect to ${\mathcal{C}}_Y$.
Note that $\frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}n}\leq O\textsf{left}( \frac{(\tilde k\tilde \alpha\alpha')^2}{c_0\log^{20}n}\textsf{right} )$. We will use a constant $c_1$, that is a large enough constant, whose value will be set later, and we set $c_0=c_1^2$. We can then assume that $\frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}n}\leq \frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}$. In particular, from Equation \ref{eq: boundaries squared sum bound}, we get that $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2<\frac{(k\alpha^4\alpha')^2}{c_0\log^{50}n}\leq \frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}$. Additionally, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}n}$.
We now summarize the properties of the graphs that we have defined and the relationships between the main parameters.
\begin{properties}{P}
\item $\tilde k\geq \Omega(\alpha^3k/\log^8n)$;\label{prop after step 1: number of pseudoterminals}
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property; \label{prop after step 1: bandwidth property}
\item $|E(\hat H)|\leq O(\tilde k\cdot \eta \log^8k/\alpha^3)$; \label{prop after step 1: few edges}
\item every vertex of $\tilde T$ has degree $1$ in $H_1$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_1$, for $\tilde \alpha=\Theta(\alpha/\log^4n)$; \label{prop after step 1: terminals in H1}
\item similarly, every vertex of $\tilde T$ has degree $1$ in $H_2$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_2$; and \label{prop after step 1: terminals in H1}
\item $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}
$, where $c_1$ is some large enough constant, whose value we can set later. \label{prop after step 1: small squares of boundaries}
\end{properties}
Our goal is
to design an efficient randomized algorithm, that either computes a distribution ${\mathcal{D}}$ over pairs in $\Lambda(H,\tilde T)$ with $\expect[({\mathcal{Q}},x)\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq xxx$ for every edge $e\in E(H)$, or returns FAIL.
It is enough to ensure that, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$,
then the probability that the algorithm returns FAIL is at most $1/\operatorname{poly}(n)$.
\iffalse
\subsection{Step 2: a Modified Graph}
In the remainder of the proof, it would be convenient for us to work with a low-degree equivalent of the graph $\hat H$, that we denote by $H^+$. In order to define the graph $\hat H^+$, we start by defining a rotation system $\hat \Sigma $ for graph $\hat H$, as follows. Let $x\in V(\hat H)$ be a vertex. If $x$ is a supernode, that is, $x=v(C)$ for some cluster $C\in {\mathcal{C}}$, then we define the circular ordering $\hat{\mathcal{O}}_x\in \hat \Sigma$ of the edges of $\delta_{\hat H}(x)$ to be arbitrary.
Otherwise, $x$ is a regular vertex, and it is a vertex of the original graph $H$. We then let $\hat {\mathcal{O}}_x\in \hat \Sigma$ be identical to the ordering ${\mathcal{O}}_x\in \Sigma$, where $\Sigma$ is the original rotation system for graph $H$.
We are now ready to define the modified graph $H^+$. We start with $H^+=\emptyset$, and then process every vertex $u\in V(\hat H)$ one-by-one. We denote by $d(u)$ the degree of the vertex $u$ in graph $\hat H$. We now describe an iteration when a vertex $u\in V(\hat H)$ is processed. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that incident to $u$ in $\hat H$, indexed according to their ordering in $\hat {\mathcal{O}}_u\in \hat \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H^+$. We refer to the edges in the grids $\Pi(u)$ as \emph{inner edges}.
Once every vertex $u\in V(\hat H)$ is processed, we add a collection of \emph{outer edges} to graph $H^+$, as follows. Consider any edge $e=(x,y)\in E(H)$. Assume that $e$ is the $i$th edge of $x$ and the $j$th edge of $y$, that is, $e=e_i(x)=e_j(y)$. Then we add an edge $e'=(s_i(x),s_j(y))$ to graph $H^+$, and we view this edge as the copy of the edge $e\in E(\hat H)$. We will not distinguish between the edge $e$ of $\hat H$ (and the corresponding edge of $H$), and the edge $e'$ of $H^+$.
We note that every terminal $t\in T$ has degree $1$ in $\hat H$, so its corresponding grid $\Pi(t)$ consists of a single vertex, that we also denote by $t$. Therefore, set $T$ of terminals in $H$ naturally corresponds to a set of $k$ terminals in $H^+$, that we denote by $T$ as before.
This completes the definition of the graph $H^+$. Note that the maximum vertex degree in $H^+$ is $4$. We also define a rotation system $\Sigma^+$ for the graph $H^+$ in a natural way: for every vertex $u\in V(H)$,
consider the standard drawing of the grid $\Pi(u)$, to which we add the drawings of the edges that are incident to vertices $s_1(u),\ldots,s_{d(u)}(u)$, so that the edges are drawn on the grid's exterior in a natural way (\mynote{add figure}). This layout defines an ordering ${\mathcal{O}}^+(v)$ of the edges incident to every vertex $v\in \Pi(u)$.
We start with the following simple claim.
\begin{claim}\label{claim: cheap solution to modified instance} There is a solution to the \textnormal{\textsf{MCNwRS}}\xspace problem instance $(H^+,\Sigma^+)$ of cost at most ..., such that no inner edge of $H^+$ participates in any crossings in the solution.
\end{claim}
\begin{proof}
We start by showing that $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat H,\hat \Sigma)\leq ...$. Recall that, from \Cref{lem: crossings in contr graph}, there is a drawing $\hat \phi$ of graph $\hat H$ with at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2$ crossings, such that for every vertex $x\in V(\hat H)\cap V(H)$, the ordering of the edges of $\delta_{\hat H}(x)$ as they enter $x$ in $\hat \phi$ is consistent with the ordering ${\mathcal{O}}_x\in \Sigma$, and hence with the ordering $\hat {\mathcal{O}}_x\in \hat \Sigma$. Drawing $\hat \phi$ of $\hat H$ may not be a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace since for some supernodes $v(C)$, the ordering of the edges that are incident to $v(C)$ in $\hat H$ as they enter the image of $v(C)$ in $\hat \phi$ may be different from $\hat {\mathcal{O}}_{v(C)}$. For each such vertex $v(C)$, we may need to \emph{reorder} the images of the edges of $\delta_{\hat H}(v(C))=\delta_H(C)$ near the image of $v(C)$, so that they enter the image of $v(C)$ in the correct order. This can be done by introducing at most $|\delta_H(C)|^2$ crossings for each such supernode $v(C)$. The resulting drawing $\hat \phi'$ of $\hat \phi$ is a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, whose cost is bounded by:
\[O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2+\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2+ \]
\end{proof}
\fi
\iffalse
\subsection{Step 2: Routing the Terminals to a Single Vertex, and the Epanded Graph}
\mynote{need to redo this: the paths set should be in the contracted graph $\hat H_1$}
In this step we start by considering the graph $H_1$ and the set $\tilde T$ of terminals in it. Our goal is to compute a collection ${\mathcal{J}}$ of paths in graph $H_1$, routing all terminals of $\tilde T$ to a single vertex, such that the paths in ${\mathcal{J}}$ cause a relatively low congestion in graph $H_1$. We show that, if such a collection of paths does not exist, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)$ is high. Intuitively, we will use the set ${\mathcal{J}}$ of paths in order to define an ordering of the terminals in $\tilde T$, which will in turn be used in order to compute a grid-like structure in graph $H_2$. Once we compute the desired set ${\mathcal{J}}$ of paths, we will replace the graph $H$ with its low-degree analogue $H^+$, that we refer to as the \emph{expanded graph}. The remaining steps in the proof of \Cref{thm: find guiding paths} will use this expanded graph only.
We now proceed to describe the algorithm for Step 2. We consider every \emph{regular} vertex (that is, a vertex that is not a supernode) $x\in V(\hat H_1)$ one by one. For each such vertex $x$, we compute a set ${\mathcal{J}}(x)$ of paths in graph $\hat H_1$, with the following properties:
\begin{itemize}
\item every path in ${\mathcal{J}}(x)$ originates at a distinct vertex of $\tilde T$ and terminates at $x$;
\item the paths in ${\mathcal{J}}(x)$ are edge-disjoint; and
\item ${\mathcal{J}}(x)$ is a maximum-cardinality set of paths in $H_1$ with the above two properties.
\end{itemize}
Note that such a set ${\mathcal{J}}(x)$ of paths can be computed via a standard maximum $s$-$t$ flow computation.
Throughout, we use a parameter $\tilde k'=\ceil{\tilde k\alpha^5/(c_2\eta\log^{36}n)}$, where $c_2$ is a large enough constant whose value we set later.
If, for every regular vertex $x\in V(\hat H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then we reurn FAIL and terminate the algorithm. In the following lemma, whose proof is deferred to Section \ref{sec: few paths high opt} of Appendix we show that, in this case, $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$ must hold.
Note that, since we can set $c_1$ to be a large enough constant, we can ensure that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$ holds in this case.
\begin{lemma}\label{lem: high opt or lots of paths}
If, for every regular vertex $x\in V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$.
\end{lemma}
\mynote{need to redo from here}
From now on we assume that there is some vertex $x\in V(H_1)$, for which $|{\mathcal{J}}(x)|\geq \tilde k'$. We denote ${\mathcal{J}}_0={\mathcal{J}}(x)$, and we let $T_0\subseteq \tilde T$ be the set of terminals that serve as endpoints of paths in ${\mathcal{J}}_0$, so $|T_0|=|{\mathcal{J}}_0|=\tilde k'$.
Let $z=\ceil{\tilde k/\tilde k'}=O\textsf{left}(\frac{\eta\log^{36}n}{\alpha^5\alpha'}\textsf{right} )$. Next, we arbitrarily partition the terminals of $\tilde T\setminus T_0$ into $z$ subsets $T_1,\ldots,T_z$, of cardinality at most $\tilde k'$ each. Consider now some index $1\leq i\leq z$. Since the set $\tilde T$ of terminals is $(\tilde \alpha\alpha')$-well-linked in $H_1$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{J}}_i'$ of paths in graph $H_1$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in $T_i$ cause edge-congestion $O(1/(\tilde \alpha\alpha'))$, and each vertex of $T_i$ is the endpoint of at most one path in ${\mathcal{J}}_i'$. By concatenating the paths in ${\mathcal{J}}_i'$ with paths in ${\mathcal{J}}_0$, we obtain a collection ${\mathcal{J}}_i$ of paths in graph $H_1$, connecting every terminal of $T_i$ to $x$, that cause edge-congestion $O(1/(\tilde \alpha\alpha'))$. Let ${\mathcal{J}}=\bigcup_{i=0}^z{\mathcal{J}}_i$ be the resulting set of paths. Observe that set ${\mathcal{J}}$ contains $\tilde k$ paths, routing the terminals in $\tilde T$ to the vertex $x$ in graph $H_1$, with $\cong_H({\mathcal{J}})\leq O\textsf{left}(\frac z{\tilde \alpha\alpha'}\textsf{right} )\leq O\textsf{left}(\frac{\eta\log^{40}n}{\alpha^6(\alpha')^2}\textsf{right} )$. We denote by $\rho=O\textsf{left}(\frac{\eta\log^{40}n}{\alpha^6(\alpha')^2}\textsf{right} )$ this bound on $\cong_{H_1}({\mathcal{P}}')$. We assume w.l.o.g. that the paths in ${\mathcal{J}}$ are simple. Since every terminal in $\tilde T$ has degree $1$ in $H_1$, no path in ${\mathcal{J}}$ may contain a terminal in $\tilde T$ as its inner vertex.
In the remainder of the proof, it would be convenient for us to work with a low-degree equivalent of the graph $H$, that we call \emph{expanded graph}, and denote by $H^+$.
For every edge $e\in E(H)$, let $N_e=\cong_H({\mathcal{J}},e)$. Recall that for every edge $e\in E(H_1\setminus\tilde T)$, $N_e\leq \rho$, and for every other edge $e$ of $H$, $N_e\leq 1$.
\fi
\subsection{Step 2: Routing the Terminals to a Single Vertex, and an Expanded Graph}
In this step we start by considering the graph $H_1$ and the set $\tilde T$ of terminals in it. Our goal is to compute a collection ${\mathcal{J}}$ of paths in graph $H_1$, routing all terminals of $\tilde T$ to a single vertex, such that the paths in ${\mathcal{J}}$ cause a relatively low congestion in graph $H_1$. We show that, if such a collection of paths does not exist, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)$ is high. Intuitively, we will use the set ${\mathcal{J}}$ of paths in order to define an ordering of the terminals in $\tilde T$, which will in turn be used in order to compute a grid-like structure in graph $H_2$. Once we compute the desired set ${\mathcal{J}}$ of paths, we will replace the graph $H$ with its low-degree analogue $H^+$, that we refer to as the \emph{expanded graph}. The remaining steps in the proof of \Cref{thm: find guiding paths} will use this expanded graph only.
\subsubsection{Routing the Terminals to a Single Vertex}
We consider every vertex $x\in V(H_1)$ one-by-one. For each such vertex $x$, we compute a set ${\mathcal{J}}(x)$ of paths in graph $H_1$, with the following properties:
\begin{itemize}
\item every path in ${\mathcal{J}}(x)$ originates at a distinct vertex of $\tilde T$ and terminates at $x$;
\item the paths in ${\mathcal{J}}(x)$ are edge-disjoint; and
\item ${\mathcal{J}}(x)$ is a maximum-cardinality set of paths in $H_1$ with the above two properties.
\end{itemize}
Note that such a set ${\mathcal{J}}(x)$ of paths can be computed via a standard maximum $s$-$t$ flow computation.
Throughout, we use a parameter $\tilde k'=\ceil{\tilde k\alpha'\alpha^5/(c_2\eta\log^{36}n)}$, where $c_2$ is a large enough constant whose value we set later.
If, for every vertex $x\in V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then we reurn FAIL and terminate the algorithm. In the following lemma, whose proof is deferred to Section \ref{sec: few paths high opt} of Appendix we show that, in this case, $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$ must hold.
Note that, since we can set $c_1$ to be a large enough constant, we can ensure that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1(\eta^4+\eta')\log^{20}n}$ holds in this case.
\begin{lemma}\label{lem: high opt or lots of paths}
If, for every vertex $x\in V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$.
\end{lemma}
From now on we assume that there is some vertex $x\in V(H_1)$, for which $|{\mathcal{J}}(x)|\geq \tilde k'$.
\subsubsection{The Expanded Graph}
From now on we fix the vertex $x\in V(H_1)$. Let ${\mathcal{J}}={\mathcal{J}}(x)$ be a set of at least $\tilde k'$ paths in graph $H_1$, routing a subset $\tilde T_0\subseteq \tilde T$ of terminals to vertex $x$.
We are now ready to define the modified graph $H^+$. We start with $H^+=\emptyset$, and then process every vertex $u\in V(H_1)\setminus \tilde T$ one by one. We denote by $d(u)$ the degree of the vertex $u$ in graph $H_1$. We now describe an iteration when a vertex $u\in V(H_1)\setminus \tilde T$ is processed. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that are incident to $u$ in $H_1$, indexed according to their ordering in $ {\mathcal{O}}_u\in \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$ indexed in their natural left-to-right order.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H^+$. We refer to the edges in the resulting grids $\Pi(u)$ as \emph{inner edges}.
Once every vertex $u\in V(H_1)\setminus \tilde T$ is processed, we add
the vertices of $\tilde T$ to the graph $H^+$. Recall that every terminal $t\in \tilde T$ has degree $1$ in $H_1$. We denote the unique edge $e_t$ incident to $t$ by $e_1(t)$, and we denote $s_1(t)=t$.
Next, we add
a collection of \emph{outer edges} to graph $H^+$, as follows. Consider any edge $e=(u,v)\in E(H_1)$. Assume that $e$ is the $i$th edge of $u$ and the $j$th edge of $v$, that is, $e=e_i(u)=e_j(v)$. Then we add an edge $e'=(s_i(u),s_j(v))$ to graph $H^+$, and we view this edge as the \emph{copy of the edge $e\in E(H_1)$}. We will not distinguish between the edge $e$ of $H_1$, and the edge $e'$ of $H^+$.
Our last step is to add vertex $x$ to graph $H^+$, that connects to every terminal $t\in \tilde T$ with an edge $(x,t)$, that is also viewed as an outer edge. The following lemma, whose proof is deferred to Section \ref{sec:ordering of terminals} of Appendix, allows us to compute an ordering $\tilde {\mathcal{O}}$ of the terminals, such that the graph $H^+$ has a drawing $\phi$ with few crossings, in which the inner edges do not participate in any crossings, and the images of edges incident to $x$ enter $x$ in order consistent with $\tilde {\mathcal{O}}$.
\iffalse
Our last step is to add a $(\tilde k\times\tilde k)$-grid $\Pi(x)$ to graph $H^+$ corresponding to the vertex $x$. As before, the edges of the grid are inner edges for graph $H^+$, and we denote the vertices on the first row of the grid by $s_1(x),\ldots,s_{\tilde k}(x)$, indexed in their natural left-to-right order. For all $1\leq i\leq \tilde k$, we add an outer edge $(t_i,s_i(x))$ to graph $H^+$. This completes the definition of the graph $H^+$, provided that we are given an ordering $\tilde {\mathcal{O}}$ of the terminals. The following lemma, whose proof is deferred to Section \ref{sec:ordering of terminals} of the Appendix, allows us to compute an ordering $\tilde {\mathcal{O}}$ of the terminals, such that the resulting graph $H^+$ has a drawing $\phi$ with few crossings, in which the inner edges do not participate in any crossings.
\fi
\begin{lemma}\label{lem: find ordering of terminals}
There is an efficient algorithm to compute an ordering $\tilde {\mathcal{O}}$ of the terminals in $\tilde T$, such that there is a drawing $\phi$ of graph $H^+$ with at most $O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot\frac{\eta^2\log^{74}n}{\alpha^{12}(\alpha')^4}\textsf{right} ) +O \textsf{left} (
\frac{\tilde k \eta\log^{37}n}{\alpha^6(\alpha')^2}\textsf{right} )$ crossings, in which all crossings in $\phi$ are between pairs of outer edges. Moreover, if we denote $\tilde T=\set{t_1,\ldots,t_{\tilde k}}$, where the terminals are indexed according to the ordering $\tilde {\mathcal{O}}$, and, for each $1\leq i\leq t_{\tilde k}$, denote by $e_i=(t_i,x)$ the edge of $H^+$ connecting $t_i$ to $x$, then the images of the edges $e_1,\ldots,e_{\tilde k}$ enter the image of $x$ in this circular order in the drawing $\phi$.
\end{lemma}
From now on we fix the ordering $\tilde {\mathcal{O}}$ of the terminals in $\tilde T$, and the drawing $\phi$ of $H^+$ (which is not known to us).
It will be convenient for us to slightly modify the graph $H^+$ as follows. We denote
the terminals by $\tilde T=\set{t_1,\ldots,t_{\tilde k}}$, where the terminals are indexed according to the circular ordering $\tilde {\mathcal{O}}$. Let $H'$ be a graph obtained from $H^+$, by first deleting the vertex $x$ from it, and then adding, for all $1\leq i<\tilde k$, an edge $e^*_i=(t_i,t_{i+1})$, and another edge $e^*_{\tilde k}=(t_{\tilde k},t_1)$. We denote this set of the newly added edges by $E^*$, and we view them as inner edges. Note that the edges of $E^*$ form a simple cycle, that we denote by $L^*$. We also denote $H''=H'\setminus E^*$.
We note that the drawing $\phi$ can be easily extended to obtain the drawing of $H'$ in the plane, so that the inner edges of $H'$ do not participate in any crossings, and the image of the cycle $L^*$ (which must be a simple closed curve) is the boundary of the outer face.
In order to do so, we start with the drawing $\phi$ of $H^+$ on the sphere, and then draw a small disc $D$ with $x$ lying in its interior, and denote by $\eta$ its boundary.
For every terminal $t_i\in \tilde T$, we denote by $e_i$ the unique edge incident to $t_i$ in $H''$, and by $e'_i=(t_i,x)$. We also denote by $\gamma_i,\gamma'_i$ the images of the edges $e_i,e'_i$ in the current drawing. Let $p_i$ be the unique point on the intersection of $\gamma'_i$ and $\eta$. We move the image of terminal $t_i$ to point $p_i$. We then modify the image of the edge $e_i$, so that it becomes a concatenation of $\gamma_i$, and the portion of $\gamma'_i$ lying outside the interior of $D$. Lastly, we draw the edges of $E^*$ in a natural way, where edge $e^*_i$ is simply a segment of $\eta$ between the images of $t_i$ and $t_{i+1}$, so that all resulting segments are mutually internally disjoint. Once we delete the vertex $x$ from this drawing, no part of the resulting drawing is contaiend in the interior of the disc $D$, and the image of the cycle $L^*$ is precisely $\eta$, so we can view the resuting drawing as a drawing in the plane, with $D$ being the outer face. Note that this transformation does not increase the number of crossings.
Observe that, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$, then:
\begin{equation}\label{eq: bound on cr}
\begin{split}
\mathsf{cr}(\phi)\leq &O\textsf{left}(\frac{(\tilde k\tilde \alpha\alpha')^2\eta^2\log^{54}n}{c_1\eta'\alpha^{12}(\alpha')^4}\textsf{right} ) + O\textsf{left} (
\frac{\tilde k \eta\log^{37}n}{\alpha^6(\alpha')^2}\textsf{right} ) \\
&\leq O\textsf{left}(\frac{\tilde k^2\log^{46}n}{c_1\eta^6\alpha^{10}(\alpha')^2}\textsf{right} ) + O\textsf{left} (
\frac{\tilde k \eta\log^{37}n}{\alpha^6(\alpha')^2}\textsf{right} ),
\end{split}
\end{equation}
since $\tilde \alpha=\Theta(\alpha/\log^4n)$ and $\eta'\geq \eta^8$ (from the statement of \Cref{thm: find guiding paths}).
Recall that, from the statement of
\Cref{thm: find guiding paths}, $k\geq \eta^6$, and from Property \ref{prop after step 1: number of pseudoterminals},
$\tilde k \geq \Omega(\alpha^3k/\log^8n)$. Therefore, $\tilde k\geq \Omega(\eta^6\alpha^3/\log^8n)$, and $\eta\leq O\textsf{left} (\frac{\tilde k\log^8n}{\eta^5\alpha^3}\textsf{right} )$. We can now bound the second term in Equation \ref{eq: bound on cr} as follows:
\[ O\textsf{left} (
\frac{\tilde k \eta\log^{37}n}{\alpha^6(\alpha')^2}\textsf{right} )
\leq O\textsf{left} (
\frac{\tilde k^2\log^{45}n}{\eta^5\alpha^9(\alpha')^2}\textsf{right} )
\]
We can assume that $\log n>c_1$, since otherwise, $n$, and therefore $k$, is bounded by a constant $2^{c_1}$. We can then use an arbitrary spanning tree $\tau$ of the graph $H$,
rooted at an arbitrary vertex $y$, in order to define a collection ${\mathcal{Q}}$ of paths routing all terminals of $\tilde T$ to $y$, where for each terminal $t\in \tilde T$, the corresponding path $Q_t\in {\mathcal{Q}}$ is the unique path connecting $t$ to $y$ in the tree $\tau$. Since $|\tilde T|$ is bounded by a constant, for every edge $e\in E(H)$, $\cong_H({\mathcal{Q}},e)\leq O(1)$. We then return a distribution ${\mathcal{D}}$ consisting of a single pair $(y,{\mathcal{Q}})$, that has probability value $1$. Therefore, we assume from now on that $\log n>c_1$. From the above discussion, we conclude that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$, then $\mathsf{cr}(\phi)\leq
O\textsf{left}(\frac{\tilde k^2\log^{46}n}{c_1\eta^6\alpha^{10}(\alpha')^2}\textsf{right} )$.
\mynote{@@ will need to redo this part to get fewer crossings as needed}
Recall that, from the conditions of \Cref{thm: find guiding paths}, $\eta\geq c^*\log^{46}n/(\alpha^{10}(\alpha')^4)$. Therefore, we can assume that, for some large enough constant $c_2$, if $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$, then $\mathsf{cr}(\phi)\leq \frac{\tilde k^2}{\eta^5}$.
It is therefore enough for us to ensure that, if $\mathsf{cr}(\phi)>
\frac{\tilde k^2}{\eta^5}$, then the probability that the algorithm returns FAIL is at most $1/\operatorname{poly}(n)$.
Let $\Lambda'$ denote all pairs $({\mathcal{Q}},y)$, where $y$ is a vertex in graph $H$, and ${\mathcal{Q}}$ is a set of paths in graph $H^+\setminus\set{x}$, routing the set $\tilde T$ of terminals to vertices of $\Pi(y)$.
Assume now that we compute a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},y)\in \Lambda'$, such that for every {\bf outer} edge $e\in E(H^+\setminus \set{x})$, $\expect[({\mathcal{Q}},y)\in_{{\mathcal{D}}}\Lambda']{(\cong_{H^+\setminus\set{x}}({\mathcal{Q}},e))^2}\leq \beta$. We then immediately obtain a distribution ${\mathcal{D}}'$ over pairs in $\Lambda(H,\tilde T)$, where for every edge $e\in E(H)$, $\expect[({\mathcal{Q}}',y)\in_{{\mathcal{D}}'} \Lambda(H,\tilde T)]{(\cong_{H}({\mathcal{Q}}',e))^2}\leq \beta$.
In order to obtain the distribution ${\mathcal{D}}'$, for every pair $({\mathcal{Q}},y)\in \Lambda(H^+\setminus\set{x},\tilde T)$, whose probability value in ${\mathcal{D}}$ is $p({\mathcal{Q}},y)>0$, we construct a pair $({\mathcal{Q}}',y)\in \Lambda(H,\tilde T)$, as follows. For every terminal $t\in \tilde T$, let $Q_t\in {\mathcal{Q}}$ be the unique path connecting $t$ to a vertex of $\Pi(y)$. By suppressing all inner edges on path $Q_t$, we obtain a path $Q'_t$ in graph $H$, connecting $t$ to $y$. We then set ${\mathcal{Q}}'=\set{Q'_t\mid t\in \tilde T}$. It is easy to verify that paths in ${\mathcal{Q}}'$ route $\tilde T$ to $y$ in graph $H$, and for every edge $e\in E(H)$, $\cong_H({\mathcal{Q}}',e)\leq \cong_{H^+}({\mathcal{Q}},e)$. We assign to the pair $({\mathcal{Q}}',y)$ the same probability value $p({\mathcal{Q}},y)$. Let ${\mathcal{D}}'$ be the resulting distribution over pairs in $\Lambda(H,\tilde T)$. It is immediate to verify that $\expect[({\mathcal{Q}}',y)\in_{{\mathcal{D}}'} \Lambda(H,\tilde T)]{(\cong_{H}({\mathcal{Q}}',e))^2}\leq \beta$.
\subsubsection{Summary of Step 2}
\label{step 2 summary}
In the remainder of the proof of \Cref{thm: find guiding paths} we will work with graph $H'$ only. Recall that graph $H'$ contains a set $E^*=\set{e_1,\ldots,e_{\tilde k}}$ of edges (that are considered to be inner edges), where for all $1\leq i\leq \tilde k$, $e_i=(t_i,t_{i+1})$ (we will use indices modulo $\tilde k$). The set $E^*$ of edges defines a cycle $L^*=(t_1,\ldots,t_{\tilde k})$ in graph $H'$. We also denoted $H''=H'\setminus E^*$. Recall that graph $H''$ is obtained from a subgraph $H_2\subseteq H$, by replacing every vertex $v\in V(H_2)\setminus \tilde T$ with a grid $\Pi(v)$. All edges lying in the resulting rids $\Pi(v)$, and the edges of $E^*$ are inner edges, while all other edges of $H'$ are outer edge. Each outer edge of $H'$ corresponds to some edge of graph $H_2$, and we do not distinguish between these edges. Note that in graph $H'$, all vertices have degrees at most $4$. We will also use the clustering ${\mathcal{C}}_Y$ of graph $H_2$, and the fact that, from Property \ref{prop after step 1: small squares of boundaries}:
\begin{equation}\label{eq: sum of squares}
\sum_{C\in {\mathcal{C}}_Y}|\delta_H(C)|^2<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}. \end{equation}
We further partition the outer edges of graph $H''$ into two subsets: type-1 outer edges and type-2 outer edges. Consider any outer edge $e$ in graph $H''$, and let $e'=(u,v)$ be the corresponding edge in graph $H$. If $u$ and $v$ both lie in the same cluster $C\in {\mathcal{C}}$, then we say that $e$ is a \emph{type-2} outer edge, and otherwise it is a type-1 outer edge.
Intuitively, for each type-1 outer edge, there is a corresponding edge in the contracted graph $\hat H$. From Property \ref{prop after step 1: few edges}, we obtain the following observation.
\begin{observation}\label{obs: few outer edges}
There is a constant $c$, such that the total number of type-1 outer edges in $H''$ is bounded by $\highlightf{c\tilde k\cdot \eta \log^8k/\alpha^3}$.
\end{observation}
======================
\highlightf[purple]{Calculations:}
final number of paths: at least $\alpha^*\tilde k/256$. Supergrid dimensions: $\lambda\times \lambda$. Want: at most $\lambda^2/32$ cells have an outer edge on each path. Number of paths per cell: $\alpha^*\tilde k/(256\lambda)$. So we need that:
\[\frac{\alpha^*\tilde k}{256\lambda}\cdot \frac{\lambda^2}{32}>\frac{c\tilde k\cdot \eta \log^8k}{\alpha^3}. \]
To ensure this, enough to set:
\[\lambda=\frac{2^{16}c\cdot \eta \log^8n}{\alpha^*\alpha^3}. \]
(and this is tight to within constants).
Because $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, we get that $\lambda=O\textsf{left}( \frac{\eta \log^{12}n}{\alpha^4\alpha'} \textsf{right} )$.
If we ensure that $\highlightf{\eta>\frac{c^* \log^{12}n}{\alpha^4\alpha'}}$, then $\lambda<\eta^2$ will hold.
Group size: start with:
\[\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}=\floor{\frac{\alpha^3(\alpha^*)^2\tilde k}{2^{32}c\eta\log^8n}}.\]
This is $\Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^8n} \textsf{right})$. If $\highlightf{\eta>\frac{c^* \log^{8}n}{\alpha^5(\alpha')^2}}$, we get that $\psi>\frac{16\tilde k}{\eta^2}$.
Overall, for this part, we need $\highlightf{\eta>\frac{c^* \log^{12}n}{\alpha^5(\alpha')^2}}$, which so far is ensured.
======================
Recall that from Property \ref{prop after step 1: terminals in H1}, every vertex of $\tilde T$ has degree $1$ in $H_2$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_2$. Combining this with the $\alpha'$-bandwidth property of every cluster $C\in {\mathcal{C}}_Y$ from Property \ref{prop after step 1: bandwidth property}, from \Cref{clm: contracted_graph_well_linkedness}, the set $\tilde T$ of terminals is $\tilde \alpha\cdot \alpha'$-well-linked in $H_2$. Lastly, using the fact that each graph in $\set{\Pi(v)\mid v\in V(H_2)}$ has the $1$-bandwidth property, from \Cref{clm: contracted_graph_well_linkedness}, we get the following observation.
\begin{observation}\label{obs: terminals well linked in H''}
The set $\tilde T$ of terminals is $\alpha^*$-well-linked in $H''$, where $\highlightf{\alpha^*=\tilde \alpha\cdot\alpha'=\Theta(\alpha\alpha'/\log^4n)}$. Moreover, each terminal in $\tilde T$ has degree $1$ in $H''$ and degree $2$ in $H'$.
\end{observation}
(we have used the fact that
$\tilde \alpha=\Theta(\alpha/\log^4n)$ (see Property \ref{prop after step 1: terminals in H1})).
\begin{definition}[Legal drawing of $H'$]
We say that a darwing of graph $H'$ in the plane is \emph{legal} iff it has the following properties:
\begin{itemize}
\item no inner edge of $H'$ participates in any crossing of $\phi^*$, and in particular the image of the cycle $L^*$ is a simple closed curve, denoted by $\eta$; and
\item $\eta$ is the boundary of the outer face in the drawing.
\end{itemize}
\end{definition}
We let $\phi^*$ be a legal drawing of $H'$ with smallest number of crossings, and we denote by $\mathsf{cr}^*$ the number of crossings in $\phi^*$.
Denote $\tilde T=\set{t_1,\ldots,t_{\tilde k}}$, where the terminals arae indexed according their ordering in $\tilde {\mathcal{O}}$. We partition the set $\tilde T$ of terminals into four subsets $T_1,\ldots,T_4$, where $T_1,T_2,T_3$ contain $\floor{\tilde k/4}$ consecutive terminals from $\tilde T$ each, and $T_4$ contains the remaining terminals, in a natural way using the ordering $\tilde {\mathcal{O}}$, that is, $T_1=\set{t_1,\ldots,t_{\floor{\tilde k/4}}}$, $T_2=\set{t_{\floor{\tilde k/4}+1},\ldots,t_{2\floor{\tilde k/4}}}$, $T_3=\set{t_{2\floor{\tilde k/4}+1},\ldots,t_{3\floor{\tilde k/4}}}$, and $T_4=\set{t_{3\floor{\tilde k/4}+1},\ldots,t_{\tilde k}}$. Clearly, each of the four sets contains at least $\floor{\tilde k/4}$ terminals.
For all $1\leq i\leq 4$, we let $\tilde {\mathcal{O}}_i$ be the ordering of the terminals in $T_i$ consistent with their ordering on the boundary of the rectangle (where each ordering ${\mathcal{O}}_i$ is no longer circular), so that the terminals in sets $T_1$ and in $T_3$ are indexed in their bottom-to-top order, and the terminals in $T_2$ and $T_4$ are indexed in their left-to-right order (so $\tilde {\mathcal{O}}$ is obtained by concatenationg ${\mathcal{O}}_1,{\mathcal{O}}_2$, the reversed ordering ${\mathcal{O}}_3$, and the reversed ordering ${\mathcal{O}}_4$).
Recall that $\Lambda'$ is the set of all pairs $({\mathcal{Q}},y)$, where $y\in V(H)$, and ${\mathcal{Q}}$ is a collection of paths in graph $H''$, routing the set $\tilde T$ of terminals to the vertices of $\Pi(y)$.
Our goal is to design a randomized algorithm, that either computes a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},y)\in \Lambda'$, such that for every outer edge $e\in E(H')$, $\expect[({\mathcal{Q}},y)\in_{{\mathcal{D}}} \Lambda']{(\cong_{H'}({\mathcal{Q}},e))^2}\leq xxx$, or returns FAIL.
It is enough to ensure that, if $\mathsf{cr}(\phi)>
\frac{\tilde k^2}{c_2\eta^5}$ for some large enough constant $c_2$, then the probability that the algorithm returns FAIL is at most $1/\operatorname{poly}(n)$.
\subsection{Step 3: Constructing a Grid Skeleton}
In this and the following step we will construct a grid-like structure in graph $H''$. Recall that the set $\tilde T$ of terminals is $\alpha^*$-well-linked in grpah $H''$. From \Cref{thm: bandwidth_means_boundary_well_linked} there is a set ${\mathcal{P}}'$ of paths in $H''$, routing all terminals of $T_1$ to terminals of $T_3$, with edge-congestion at most $\ceil{1/\alpha^*}$, such that the routing is one-to-one. From \Cref{claim: remove congestion}, there is a collection ${\mathcal{P}}''$ of at least $|T_1|/\ceil{1/\alpha^*}=\floor{\tilde k/4}/\ceil{1/\alpha^*}\geq \alpha^*\tilde k/8$ edge-disjoint paths in $H''$, routing some subset of terminals of $T_1$ to a subset of terminal of $T_3$, in graph $H''$. Moreover, since graph $H''$ has maximum vertex degree at most $3$, using arguments similar to those in the proof of \Cref{claim: remove congestion}, there is a collection ${\mathcal{P}}$ of $\floor{\alpha^*\tilde k/32}$ {\bf node-disjoint} paths in graph $H''$, routing some subset $A\subseteq T_1$ of terminals, to some subset $A'\subseteq T_3$ of terminals. We can compute such a set ${\mathcal{P}}$ of paths efficiently using standard maximum $s$-$t$ flow algorithms.
Using similar reasoning, we can compute a collection ${\mathcal{R}}$ of $\floor{\alpha^*\tilde k/32}$ node-disjoint paths in graph $H''$, routing some subset $B\subseteq T_2$ of terminals, to some subset $B'\subseteq T_4$ of terminals.
Intuitively, after we discard a small subset of paths from each of the sets ${\mathcal{P}},{\mathcal{R}}$, the remaining paths will be used in order to construct a grid-like structure, where paths in ${\mathcal{P}}$ will serve as horizontal paths of the grid, and paths in ${\mathcal{R}}$ will serve as vertical paths. If the paths in the resulting sets do not form a grid-like structure, then we will terminate the algorithm with a FAIL. We will prove that, if $\mathsf{cr}^*<...$, then we will construct the grid-like structure successfully with probability at least $1-1/\operatorname{poly}(n)$.
We denote ${\mathcal{P}}_0={\mathcal{P}}$ and ${\mathcal{R}}_0={\mathcal{R}}$. Recall that so far, $\highlightf{|{\mathcal{P}}_0|,|{\mathcal{R}}_0|\geq \floor{\alpha^*\tilde k/32}}$.
Intuitively, if the dimensions of the grid-like structure that we construct are $(h\times h)$, then we need $h$ to be quite close to $\tilde k$, since this grid-like structure will be exploited in order to define the distribution ${\mathcal{D}}$ over pairs in $\Lambda'$.
We will first construct a smaller grid-like structure, that we call a \emph{grid skeleton}. We then extend this grid skeleton to construct a large enough grid-like structure.
We will use two additional parameters. The first parameter is:
\[\lambda=\frac{2^{24}c\cdot \eta \log^8n}{\alpha^*\alpha^3},\]
where $c$ is the constant from \Cref{obs: few outer edges}. Notice that, since $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, we get that $\lambda=O\textsf{left}( \frac{\eta \log^{12}n}{\alpha^4\alpha'} \textsf{right} )$. Moreover, since $\eta>\frac{c^* \log^{12}n}{\alpha^4\alpha'}$ for a large enough constant $c^*$, $\lambda<\eta^2$. The supergrid that we construct will have dimensions $(\Theta(\lambda)\times \Theta(\lambda))$. The second parameter is:
\[\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}=\floor{\frac{\alpha^3(\alpha^*)^2\tilde k}{2^{30}c\eta\log^8n}}.\]
Clearly, $|{\mathcal{R}}_0|,|{\mathcal{P}}_0|\geq \lambda\psi$.
Note that, since $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, $\psi\geq \Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^8n} \textsf{right})$. Since $\highlightf{\eta>\frac{c^* \log^{8}n}{\alpha^5(\alpha')^2}}$, we get that $\psi>\frac{16\tilde k}{\eta^2}$. Every cell of the supergrid will be associated with a collection of $\Theta(\psi)$ horizontal paths and $\Theta(\psi)$ vertical paths, that will help us formt he grid-like structure.
We discard paths from ${\mathcal{P}}_0$ and from ${\mathcal{R}}_0$ arbitrarily, until $|{\mathcal{P}}_0|=|{\mathcal{R}}_0|=\lambda\phi$ holds.
We denote by $A_0\subseteq T_1,A'_0\subseteq T_3$ the endpoints of the paths in ${\mathcal{P}}_0$, and we denote by $B_0\subseteq T_2$, $B'_0\subseteq T_4$ the endpoints of the paths in ${\mathcal{R}}_0$.
\subsubsection*{Grid Skeleton Construction}
\iffalse
Let $c$ be a constant, such that the number of type-1 outer edges is at most $c\tilde k\cdot \eta \log^8k/\alpha^3$.
We say that a path $P\in {\mathcal{P}}_0\cup {\mathcal{R}}_0$ is \emph{long} if it contains at least $128c\eta\log^8k/(\alpha^*\alpha^3)$ type-1 outer edges, and otherwise we call it short.
Since the total number of type-1 outer edges is at most $c\tilde k\cdot \eta \log^8k/\alpha^3$, and the paths in ${\mathcal{P}}_0$ are edge-disjoint, at most $\alpha^*\tilde k/128$ paths in ${\mathcal{P}}_0$ may be long. We let ${\mathcal{P}}_1\subseteq {\mathcal{P}}_0$ be the set of all short paths, so $|{\mathcal{P}}_1|\geq \alpha^*\tilde k/128$. Similarly, we let ${\mathcal{R}}_1\subseteq {\mathcal{R}}_0$ be the set of all short paths, so $|{\mathcal{R}}_1|\geq \alpha^*\tilde k/128$. We denote by $A_1\subseteq A$, $A_1'\subseteq A'$ the sets of endpoints of the paths in ${\mathcal{P}}_1$, and we similarly denote by $B_1\subseteq B$, $B_1'\subseteq B'$ the sets of endpoints o fthe paths in ${\mathcal{R}}_1$.
\fi
We view the paths in ${\mathcal{P}}_0$ as directed from vertices of $A_0$ to vertices of $A'_0$. Recall that $A_0\subseteq T_1$, so the ordering ${\mathcal{O}}_1$ of the terminals in $T_1$ defines an ordering ${\mathcal{O}}_{A_0}=\set{a_1,\ldots,a_{\lambda\psi}}$ of the terminals in $A_0$. This ordering in turn defines an ordering ${\mathcal{O}}_{{\mathcal{P}}_0}$ of the paths in ${\mathcal{P}}_0$, as follows: if, for all $1\leq i\leq \lambda\psi$, $P_i\in {\mathcal{P}}_0$ is the path originating from $a_i$, then ${\mathcal{O}}_{{\mathcal{P}}_0}=\set{P_1,\ldots,P_{\lambda\psi}}$.
Similarly, we view the paths in ${\mathcal{R}}_0$ as directed from vertices of $B_0$ to vertices of $B_0'$. Ordering ${\mathcal{O}}_2$ of terminals in $T_2$ defines an ordering ${\mathcal{O}}_{B_0}=\set{b_1,\ldots,b_{\lambda\psi}}$ of the vertices in $B_0$, which in turn defines an ordering ${\mathcal{O}}_{{\mathcal{R}}_0}=\set{R_1,\ldots,R_{\lambda\psi}}$ of paths in ${\mathcal{R}}_0$, where for all $i$, path $R_i$ originates at vertex $b_i$.
We partition the set ${\mathcal{P}}_0$ of paths into groups ${\mathcal U}_1,\ldots,{\mathcal U}_{\lambda}$ of cardinality $\psi$ each, using the ordering ${\mathcal{O}}_{{\mathcal{P}}_0}$, so for $1\leq i<\lambda$, set ${\mathcal U}_i$ is the $i$th set of $\psi$ consecutive paths of ${\mathcal{P}}_0$.
Let $\lambda'= \floor{(\lambda-1)/2}$.
For all $1\leq i\leq \lambda'$, we let $ P^*_i$ be a path that is chosen uniformly at random from set ${\mathcal U}_{2i}$. Let ${\mathcal{P}}^*=\set{ P^*_1,\ldots, P^*_{\lambda'}}$ be the resulting set of chosen paths. Intuitively, the path in ${\mathcal{P}}^*$ will serve as the horizontal paths in the grid skeleton that we construct.
We then let ${\mathcal{P}}_1\subseteq {\mathcal{P}}_0$ be the set containing all paths in sets $\set{{\mathcal U}_{2i-1}}_{i=1}^{\lambda'}$.
We perform similar computation on the set ${\mathcal{R}}_0$ of paths. First, we partition ${\mathcal{R}}_0$ into groups ${\mathcal U}'_1,\ldots,{\mathcal U}'_{\lambda}$ of cardinality $\psi$ each, using the ordering ${\mathcal{O}}_{{\mathcal{R}}_0}$, so for $1\leq i<\lambda$, set ${\mathcal U}'_i$ is the $i$th set of $\psi$ consecutive paths of ${\mathcal{R}}_0$.
For all $1\leq i\leq \lambda'$, we let $ R^*_i$ be a path that is chosen uniformly at random from set ${\mathcal U}_{2i}$. Let ${\mathcal{R}}^*=\set{ R^*_1,\ldots,\tilde R^*_{\lambda'}}$ be the resulting set of chosen paths. Intuitively, the path in ${\mathcal{R}}^*$ will serve as the vertical paths in the grid skeleton that we construct.
We then let ${\mathcal{R}}_1\subseteq {\mathcal{R}}_0$ be the set containing all paths in sets $\set{{\mathcal U}'_{2i+1}}_{i=1}^{\lambda'-1}$.
We let ${\cal{E}}_1$ be the bad event that there are two distinct paths $Q,Q'\in {\mathcal{R}}^*\cup {\mathcal{P}}^*$, and two distinct edges $e\in E(Q)$, $e'\in E(Q')$, such that the images of $e$ and $e'$ cross.
\begin{observation}\label{obs: first bad event}
If $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then $\prob{{\cal{E}}_1}\leq 1/64$.
\end{observation}
\begin{proof}
Consider any crossing $(e,e')$ in the drawing $\phi^*$. We say that crossing $(e,e')$ is \emph{selected} iff there are two distinct paths $Q,Q'\in {\mathcal{R}}^*\cup {\mathcal{P}}^*$ with $e\in E(Q)$, $e'\in E(Q')$. Notice that $e$ may belong to at most two paths in ${\mathcal{R}}_0\cup {\mathcal{P}}_0$ (one path in each set), and the same is true for $e'$. Each path of ${\mathcal{R}}_0\cup{\mathcal{P}}_0$ is chosen to ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ with probability at most $1/\psi$. Therefore, the probability that a path containing $e$, and a path containing $e'$ are chosen to ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ is at most $4/\psi^2$. Since ${\cal{E}}_1$ can only happen if at least one crossing is chosen, from the union bound,
$\prob{{\cal{E}}_1}\leq 4\mathsf{cr}^*/\psi^2$. Since $\psi>\frac{16\tilde k}{\eta^2}$, if $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then:
\[\prob{{\cal{E}}_1}\leq \frac{4\mathsf{cr}^*}{\psi^2}\leq \frac{1}{64c_2\eta}\leq \frac{1}{64}.\]
\end{proof}
We say that a path $Q\in {\mathcal{R}}_0\cup {\mathcal{P}}_0$ is \emph{heavy} iff there are at least $\frac{\psi}{64\lambda}$ crossings $(e,e')$ in $\phi^*$, such that at least one of the edges $e,e'$ lies on path $Q$. We say that a bad event ${\cal{E}}_2$ happens iff at least one path in ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ is heavy.
\begin{observation}\label{obs: first bad event}
If $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then $\prob{{\cal{E}}_2}\leq 1/64$.
\end{observation}
\begin{proof}
Note that every edge of $H''$ may lie on at most two paths of ${\mathcal{R}}_0\cup {\mathcal{P}}_0$, and every crossing $(e,e')$ involves two edges. Therefore, the total number of heavy paths in ${\mathcal{R}}_0\cup {\mathcal{P}}_0$ is bounded by $\frac{4\mathsf{cr}^*}{\psi/(64\lambda)}=\frac{16\lambda\mathsf{cr}^*}{\psi}$. Assuming that $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, and using the fact that $\psi=\Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^8n} \textsf{right})$ and $\lambda=O\textsf{left}( \frac{\eta \log^{12}n}{\alpha^4\alpha'} \textsf{right} )$, we get that the total number of heavy paths in ${\mathcal{R}}_0\cup {\mathcal{P}}_0$ is bounded by:
\[ \frac{16\lambda\mathsf{cr}^*}{\psi}\leq O\textsf{left}( \frac{\tilde k^2}{c_2\eta^5}\cdot \frac{\eta \log^{12}n}{\alpha^4\alpha'}\cdot\frac{\eta}{\tilde k}\cdot \frac{\log^8n}{\alpha^5(\alpha')^2} \textsf{right} )
\leq O\textsf{left}( \frac{\tilde k\log^{20}n}{c_2\eta^3\alpha^9(\alpha')^3} \textsf{right} ).
\]
Note that each heavy path may be selected to ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ with probability at most $1/\psi$. Therfore, using the union bound and the fact that $\psi=\Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^8n} \textsf{right})$, we get that:
\[\prob{{\cal{E}}_2}\leq
O\textsf{left}( \frac{\tilde k\log^{20}n}{\psi\cdot c_2\eta^3\alpha^9(\alpha')^3}\textsf{right} )\leq
O\textsf{left}( \frac{\log^{20}n}{\cdot c_2\eta^2\alpha^{14}(\alpha')^5}\textsf{right} ).\]
Recall that, from the conditions of \Cref{thm: find guiding paths}, $\eta\geq c^*\log^{46}n/(\alpha^{10}(\alpha')^4)$, where $c^*$ is a sufficiently large constant. Therefore, if
$\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then $\prob{{\cal{E}}_2}\leq 1/64$.
\end{proof}
Let ${\mathcal{R}}'\subseteq {\mathcal{R}}_1$, ${\mathcal{P}}'\subseteq {\mathcal{P}}_1$ be the sets containing all paths $Q$, such that, in drawing $\phi^*$, some edge of $Q$ crosses some edge lying on the paths of ${\mathcal{R}}^*\cup {\mathcal{P}}^*$. We will also use the following observation:
\begin{observation}\label{obs: few bad paths}
If ${\cal{E}}_2$ did not happen, then $|{\mathcal{R}}'|,|{\mathcal{P}}'|\leq \psi/32$.
\end{observation}
\begin{proof}
Recall that $|{\mathcal{R}}^*|+|{\mathcal{P}}^*|\leq \lambda$. If bad event ${\cal{E}}_2$ did not happen, then for each path $Q\in {\mathcal{R}}^*\cup {\mathcal{P}}^*$, there are at most $\frac{\psi}{64\lambda}$ crossings in $\phi^*$, in which edges of $Q$ participate. Therefore, if event ${\cal{E}}_2$ did not happen, there are in total at most $\psi/64$ crossings $(e,e')$ in the drawing $\phi^*$, where at least one of the edges $e,e'$ lies on a path of ${\mathcal{P}}^*\cup {\mathcal{Q}}^*$. Let $E'\subseteq E(H'')$ be the set of all edges $e$, such that there is an edge $e'$ lying on some path of ${\mathcal{P}}^*\cup {\mathcal{Q}}^*$, and crossing $(e,e')$ is present in $\phi^*$. Then $|E'|\leq \psi/32$. Each path in ${\mathcal{R}}'\cup {\mathcal{P}}'$ must contain an edge of $E'$. As the paths in each of the sets ${\mathcal{R}}',{\mathcal{P}}'$ are disjoint, $|{\mathcal{R}}'|,|{\mathcal{P}}'|\leq \psi/32$ must hold.
\end{proof}
\paragraph{Summary of Step 3.}
In this step we have constructed a grid skeleton, that consists of two sets of paths: ${\mathcal{P}}^*=\set{ P^*_1,\ldots, P^*_{\lambda'}}$, and ${\mathcal{R}}^*=\set{ R^*_1,\ldots,\tilde R^*_{\lambda'}}$, where $\lambda'= \floor{(\lambda-1)/2}$. Recall that ${\mathcal{P}}^*\subseteq {\mathcal{P}}_0$, and the paths in ${\mathcal{P}}^*$ are indexed according to their order in ${\mathcal{O}}_{{\mathcal{P}}_0}$. Recall that we have also defined the set ${\mathcal{P}}_1\subseteq {\mathcal{P}}_0$ of paths, containing all paths in sets $\set{{\mathcal U}_{2i-1}}_{i=1}^{\lambda'+1}$. It would be convinient for us to re-index the groups ${\mathcal U}_i$ as follows: for $1\leq i\leq \lambda'$, set ${\mathcal U}_i={\mathcal U}_{2i-1}$. In other words, the paths of ${\mathcal U}_0$ lie before path $P^*_1$ in the ordering ${\mathcal{O}}_{{\mathcal{P}}_0}$, the paths of ${\mathcal U}_{\lambda'+1}$ lie after $P^*_{\lambda'}$ in this ordering, and, for $1\leq i<\lambda'$, the paths of ${\mathcal U}_i$ lie between paths $P^*_i$ and $P^*{i+1}$.
Similarly, ${\mathcal{R}}^*\subseteq {\mathcal{R}}_0$, and the paths in ${\mathcal{R}}^*$ are indexed according to their order in ${\mathcal{O}}_{{\mathcal{R}}_0}$. We have also defined the set ${\mathcal{R}}_1\subseteq {\mathcal{R}}_0$ of paths, containing all paths in sets $\set{{\mathcal U}'_{2i-1}}_{i=1}^{\lambda'+1}$. As before, we re-index them as follows: for $1\leq i\leq\lambda'$, we set ${\mathcal U}'_i={\mathcal U}'_{2i-1}$. Therefore, the paths of ${\mathcal U}'_0$ lie before path $R^*_1$ in the ordering ${\mathcal{O}}_{{\mathcal{R}}_0}$, the paths of ${\mathcal U}'_{\lambda'+1}$ lie after $R^*_{\lambda'}$ in this ordering, and, for $1\leq i<\lambda'$, the paths of ${\mathcal U}'_i$ lie between paths $R^*_i$ and $R^*{i+1}$.
From our definition, if ${\cal{E}}_1$ did not happen, then for every pair $Q,Q'\in {\mathcal{P}}^*\cup {\mathcal{Q}}^*$ of distinct paths, their images in $\phi^*$ do not cross (but note that the image of a single path may cross itself).
We have also defined a set ${\mathcal{P}}'\subseteq {\mathcal{P}}_1$ and a set ${\mathcal{R}}'\subseteq {\mathcal{R}}_1$ of paths, containing all paths $Q$ whose image crosses the image of some path in ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ in drawing $\phi^*$. From \Cref{obs: few bad paths}, if Event ${\cal{E}}_2$ does not happen, then $|{\mathcal{P}}'|,|{\mathcal{R}}'|\leq \psi/32$.
It will be convenient for us to consider the $((\lambda'+1)\times (\lambda'+1))$-grid $\Pi^*$. We view the columns of the grid as corresponding to the left boundary of the bounding box $B^*$, the paths in $\set{R^*_1,\ldots,R^*_{\lambda'}}$, and the right boundary of the bounding box $B^*$.
For convenience, we index the columns of the grid from $0$ to $\lambda+1$, so the left boundary of the bounding box corresponds to column $0$, and, for $1\leq i\leq \lambda_i$, path $P^*_i$ represents the $i$th column of the grid, with the right boundary of $B^*$ repersenting the last column. Similarly, we view the bottom boundary of $B^*$, the paths in $\set{P^*_1,\ldots,P^*_{\lambda'}}$, and the top boundary of $B^*$ as representing the rows of the grid, in the bottom-to-top order. As before, we index the rows of the grid so that the botommost row has index $0$ and the topmost row has index $\lambda'+1$. Notice however that the union of the paths in ${\mathcal{P}}^*\cup {\mathcal{R}}^*$ does not necessarily form a proper grid graph, as it is possible that, for a pair $P\in {\mathcal{P}}^*$, $R\in {\mathcal{R}}^*$ of paths, $P\cap R$ is a collection of several disjoint paths.
We will now consider the drawing $\phi^*$ of $H''$, and we will use it to define vertical and horizontal strips corresponding to paths in ${\mathcal{P}}^*$ and ${\mathcal{R}}^*$, respectively. We will also associate, with each cell of the grid $\Pi^*$, some region of the plane. We assume in the following definitions that Event ${\cal{E}}_1$ did not happen.
Consider first the image $\gamma_i$ of some path $P^*_i\in {\mathcal{P}}^*$ in the drawing $\phi^*$. Note that $\gamma_i$ is not necessarily a simple curve.
We define two simple curves, $\gamma^t_i$ and $\gamma^b_i$, where $\gamma^t_i$ follows the image of $\gamma_i$ from the top, and $\gamma^b_i$ follows it from the bottom. In other words, we let $\gamma^b_i$ be a simple curve, contained in $\gamma_i$, with the same endpoints as $\gamma_i$, such that the following holds: if $K^b_i$ is the area of the bounding box $B^*$ below $\gamma_i$, including $\gamma_i^t$, then $\gamma_i\subseteq K^b_i$. We define the other curve, $\gamma^b_i$ symmetrically, so curve $\gamma_i$ is contained in the disc whose boundary is $\gamma_i^t\cup \gamma_i^b$. For convenience, we let $\gamma_0^t$ be the bottom boundary of the bounding box $B^*$, and $\gamma_{\lambda'+1}^b$ be the top boundary of the bounding box $B^*$.
We now define, for all $0\leq i\leq \lambda'$, a region of the plane that we call the $i$th horizontal strip, and denote by $\mathsf{HStrip}_i$. This strip is simply the closed region of the bounding box between the curves $\gamma_i^t$ and $\gamma_{i-1}^b$.
For every vertical path $R^*_i\in {\mathcal{R}}^*$, we also define two curves, $\gamma^{\ell}_i$ and $\gamma^r_i$, that follow the image $\gamma_i$ of $R^*_i$ in $\phi^*$ on its left and on its right, respectively. We denote by $\gamma_0^r$ the left boundary of the bounding box $B^*$, and by $\gamma_{\lambda'+1}^{\ell}$ its right boundary. For all $0\leq i\leq \lambda'$, we define a vertical strip $\mathsf{VStrip}_i$ to be the closed region of the bounding box $B^*$ betwen $\gamma_i^r$ and $\gamma_i^{\ell}$.
The following observation is immediate.
\begin{observation}\label{obs: must cross chosen paths}
If $R\in {\mathcal{R}}_1$ is a path whose image in $\phi^*$ intersects the interior of more than one vertical strip in $\set{\mathsf{VStrip}_0,\ldots,\mathsf{VStrip}_{\lambda'+1}}$, then $R\in {\mathcal{R}}'$. Similarly, if $P\in {\mathcal{P}}_1$ is a path whose image in $\phi^*$ intersects the interior of more than one horizontal strip in $\set{\mathsf{HStrip}_0,\ldots,\mathsf{HStrip}_{\lambda'+1}}$, then $P\in {\mathcal{P}}'$.
\end{observation}
Lastly, for all $0\leq i,j\leq \lambda'$, we let $\mathsf{CellRegion}_{i,j}=\mathsf{HStrip}_i\cap \mathsf{VStrip}_j$ be a closed region of the plane that we associate with cell $\mathsf{Cell}_{i,j}$ of the grid $\Pi^*$.
====================================
\highlightf{Calculations: heavy paths}
Path is heavy if it participates in $x$ crossings. All paths of skeleton will give $\lambda x$ crossings. Using $\lambda<\eta^2$, this is less than $x\eta^2$ crossings. This should be less than $\psi/16$. Because $\psi>\frac{16\tilde k}{\eta^2}$, enough that $x\eta^2<\tilde k/\eta^2$, or that $x<\tilde k/\eta^4$.
If we fix $x=\tilde k/\eta^4$. Then number of bad paths is at most:
\[\frac{4\mathsf{cr}^*}{x}< \frac{\tilde k^2}{c_2\eta^5}/\frac{\tilde k}{\eta^4}=\frac{\tilde k}{c_2\eta}. \]
A heavy path is selected with probability $1/\psi$. Probability that any heavy path is selected (from union bound) is at most:
\[ \frac{\tilde k}{c_2\eta\psi}. \]
If we use $\psi>\frac{16\tilde k}{\eta^2}$, this is not good enough.
===============================
More detained calculation: all paths of skeleton will give $\lambda x$ crossings, and we want it to be less than $\psi/16$. so $x=\frac{\psi}{16\lambda}$ is fine. Now number of bad paths is at most:
\[\frac{4\mathsf{cr}^*}{x}< \frac{64\mathsf{cr}^*\lambda}{\psi}.\]
A heavy path is selected with probability $1/\psi$. Probability that any heavy path is selected (from union bound) is at most:
\[\frac{64\mathsf{cr}^*\lambda}{\psi^2}\]
So for us it is enough that $\mathsf{cr}^*<\psi^2\cdot 2^{12}/\lambda$.
We have:$ \psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$, so $\frac{\psi}{\lambda}\geq \frac{\alpha^*\tilde k}{128}$. Using $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, we get that $\frac{\psi}{\lambda}\geq \Omega(\frac{\alpha\alpha'\tilde k}{\log^4n})$.
So now $\psi^2/\lambda=\Omega (\frac{\psi\alpha\alpha'\tilde k}{\log^4n} )$.
Can use now $\psi>\frac{16\tilde k}{\eta^2}$, so the expression becomes:
\[\Omega (\frac{\tilde k^2\alpha\alpha'}{\eta^2\log^4n} ) \]
Lastly, because $\eta>\log^4n/(c^*\alpha\alpha')$, we get that $\mathsf{cr}^*<\tilde k^2/(c_2\eta^5)$ is good enough.
=====================================
\subsection{Step 4: Constructing a Grid-Like Structure}
In this step we furthere delete some paths from sets ${\mathcal{R}}_1$ and ${\mathcal{P}}_1$ to ensure that the resulting paths form a grid-like structure. This is done in three stages. In the first stage, we discard some paths to ensure that every remaining path in ${\mathcal{R}}_1$ intersects the paths in ${\mathcal{P}}^*$ ``in order'' (we formally define what it means later). In the second stage, we associate, with every cell of the grid $\Pi^*$ a collection of horizontal and a collection of vertical paths. In the third stage, we ensure that for every cell of the grid $\Pi^*$, there are many inersections between its corresponding horizontal and vertical paths.
Before we continue, we discard some paths of ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ that must lie in ${\mathcal{R}}'\cup {\mathcal{P}}'$. Specifically, consider some path $P\in {\mathcal{P}}^*$, and assume that it lies in group ${\mathcal U}_i$, for $0\leq i\leq \lambda'+1$. Let $(a,a')$ be the endpoints of path $P$, with $a\in T_1$ and $a'\in T_3$. Notice that from the definition, if $i>0$, then $a$ must lie, in the ordering ${\mathcal{O}}_1$ of the terminals of $T_1$, after the endpoint of the path $P^*_i$ that belongs to $T_1$. Similarly, if $i<\lambda'+1$, then $a$ must lie before the endpoin of the path $P^*_{i+1}$ that belongs to $T_1$ in the same ordering. In particular, we are guaranteed that in the drawing $\phi^*$, the image of $P$ must intersect the interior of the horizontal strip $\mathsf{HStrip}_i$. Consider now the endpoint $a'$ of $P$. If $i>0$, let $a'_i$ be the endpoint of path $P_i^*$ that lies in $T_3$, and if $i<\lambda'+1$, let $a'_{i+1}$ be the endpoint of path $P_{i'+1}$ that lies in $T_3$. Note that, if $a'$ lies before $a'_i$ in the ordering ${\mathcal{O}}_3$ of $T_3$, or if $a'$ lies after $a'_{i+1}$ in the ordering ${\mathcal{O}}_3$, then the image of $P$ has to intersect the interior of an additional horizontal strip, and, from \Cref{obs: must cross chosen paths}, path $P$ must lie in ${\mathcal{P}}'$. We discard each such path from set ${\mathcal{P}}_1$ (and from the corresponding set $U_i$). This ensures that, if $P\in U_i$, then its endpoint $a'$ must lie between $a'_i$ and $a'_{i+1}$ in ${\mathcal{O}}_3$, if $1\leq i\leq \lambda'$; it must lie before $a'_{i+1}$ if $i=0$, and it must lie after $a'_i$ if $i=\lambda'+1$.
We process the paths in ${\mathcal{R}}_1$ similarly, discarding paths as needed. Notice that so far all paths that we have discarded from ${\mathcal{P}}_1\cup {\mathcal{R}}_1$ lie in ${\mathcal{P}}'\cup {\mathcal{R}}'$.
\subsubsection{In-Order Intersection}
In this stage we discard some additional paths from ${\mathcal{P}}_1\cup {\mathcal{R}}_1$, to ensure that every remaining path in ${\mathcal{P}}_1$ interesects the paths in ${\mathcal{R}}^*$ in-order (notion that we define below); we do the same for paths in ${\mathcal{R}}_1$. We will ensure that all paths discarded at this stage lie in ${\mathcal{P}}'\cup {\mathcal{R}}'$.
Since the definitions and the algorithms for the paths in ${\mathcal{P}}_1$ and for the paths in ${\mathcal{R}}_1$ are symmetric, we only show how to process the paths in ${\mathcal{P}}_1$ here.
Let $P\in {\mathcal{P}}_1$ be any path, that we view as directed from its endpoint that lies in $T_1$ to its endpoint lying in $T_3$. Let $X(P)=\set{x_1,\ldots,x_r}$ denote all vertices of $P$ lying on paths in $R^*$, that is, $X(P)=V(P)\cap \textsf{left}(\bigcup_{i=1}^{\lambda'}V(R^*_i)\textsf{right} )$. We assume that the vertices of $X(P)$ are indexed in the order of their appearance on $P$. For each such vertex $x_j$, let $i_j$ be the index of the path $R^*_{i_j}\in {\mathcal{R}}^*$ containing $x_j$.
\begin{definition}[In-order intersection]
We say that path $P$ intersects the paths of ${\mathcal{R}}^*$ in-order, iff $r\geq \lambda'$, $i_1=1$, $i_r=\lambda'$, and, for $1\leq j<r$, $|i_j-i_{j+1}|\leq 1$.
\end{definition}
Notice that the definition requires that path $P$ intersects every path of ${\mathcal{R}}^*$ at least once; the first path of ${\mathcal{R}}^*$ that it intersects must be $R_1^*$, and the last path must be $R_{\lambda'}^*$, and for every consecutive pair $x_j,x_{j+1}$ of vertices in $X(P)$, either both vertices lie on the same path of ${\mathcal{R}}^*$, or they lie on consecutive paths of ${\mathcal{R}}^*$. Notice that path $P$ is still allowed to intersect a path of ${\mathcal{R}}^*$ many times, and may go back and forth across all these paths several times.
\begin{observation}\label{obs: not in order intersection}
Assume that Event ${\cal{E}}_1$ did not happen. Let $P\in {\mathcal{P}}_1$ be a path that intersect the paths of ${\mathcal{R}}^*$ not in-order. Then $P\in {\mathcal{P}}'$ must hold.
\end{observation}
\begin{proof}
Assume first that $i_1\neq 1$, that is, vertex $x_1$ lies on some path $R^*_i$ with $i\neq 1$. Let $p$ be a point on the image of path $P$ in $\phi^*$ that is very close to its first endpoint, so $p$ lies in the interior of the horizontal strip $\mathsf{HStrip}_1$, and let $p'$ be the image of the point $x_1$. Clearly, $p'$ does not lie in the interior or on the boundary of $\mathsf{HStrip}_1$, so the image of path $P$ must cross the top boundayr of $\mathsf{HStrip}_1$, which means that the image of some edge $P$ and the image of some edge of $R_1^*$ cross.
The cases where $i_{r}\neq \lambda'$, or there is an index $1\leq j<r$ with $i_j-i_{j+1}> 1$ are treated similarly, as is the case when $r<\lambda'$.
\end{proof}
We discard from ${\mathcal{P}}_1$ all paths $P$ that intersect the paths of ${\mathcal{R}}^*$ not in-order. We denote by ${\mathcal{P}}_2\subseteq {\mathcal{P}}_1$ the set of remaining paths. We also update the groups ${\mathcal U}_0,\ldots,{\mathcal U}_{\lambda'+1}$ accordingly. Observe that so far all paths that we have discarded lie in ${\mathcal{P}}'$.
From \Cref{obs: few bad paths}, assuming that Events ${\cal{E}}_1$ and ${\cal{E}}_2$ did not happen, the number of paths that we have discarded so far from ${\mathcal{P}}_1$ is at most $\psi/32$. In particular, for all $0\leq i\leq \lambda'+1$, $|{\mathcal U}_i|\geq 31\psi/32$ still holds.
We perform the same transformation on set ${\mathcal{R}}_1$ of paths, obtaining a new set ${\mathcal{R}}_2$ of paths, each of which intersects the paths of ${\mathcal{P}}^*$ in-order. We also update the groups ${\mathcal U}'_1,\ldots,{\mathcal U}'_{\lambda'+1}$. As before, for all $0\leq i\leq \lambda'+1$, $|{\mathcal U}'_i|\geq 31\psi/32$ still holds.
\subsubsection{Definining Paths Associated with Grid Cells}
For every path $P\in {\mathcal{P}}_2$, for all $1\leq i\leq \lambda'$, we denote by $v_i(P)$ the last vertex on path $P$ that lies on the vertical path $R^*_i$; note that, from the definition of in-order intersection, such a path must exist. For all $1\leq i< \lambda'$, we define the $i$th segment of $P$, $\sigma_i(P)$, to be the subpath of $P$ between $v_i(P)$ and $v_{i+1}(P)$. We also let $\sigma_0(P)$ be the subpath of $P$ from its first vertex (which must be a terminal of $T_1$) to $v_1(P)$, and by $\sigma_{\lambda'}(P)$ the subpath of $P$ from $v_{\lambda'}(P)$ to the last vertex of $P$ (which must be a terminal of $T_3)$.
Siilarly, for every path $R\in {\mathcal{R}}_2$, for all $1\leq i\leq \lambda'$, we denote by $v_i(R)$ the first vertex on path $R$ that lies on the horizontal path $P^*_i$. For all $1\leq i< \lambda'$, we define the $i$th segment of $P$, $\sigma_i(R)$, to be the subpath of $R$ between $v_i(R)$ and $v_{i+1}(R)$. We also let $\sigma_0(R)$ be the subpath of $R$ from its first vertex (which must be a terminal of $T_2$) to $v_1(R)$, and by $\sigma_{\lambda'}(R)$ the subpath of $R$ from $v_{\lambda'}(R)$ to the last vertex of $R$ (which must be a terminal of $T_3)$.
Consider now some cell $\mathsf{Cell}_{i,j}$ of the grid $\Pi^*$, for some $0\leq i,j\leq \lambda'+1$. We define the set ${\mathcal{P}}^{i,j}$ of horizontal paths, and the set ${\mathcal{R}}^{i,j}$ of vertical paths associated with cell $\mathsf{Cell}_{i,j}$, as follows. In order to define the set ${\mathcal{P}}^{i,j}$ of horizontal paths, we consider the group $U_i$, and, for every path $P\in U_i$, we include its $j$th segment $\sigma_j(P)$ in ${\mathcal{P}}^{i,j}$, so that:
\[{\mathcal{P}}^{i,j}=\set{\sigma_j(P)\mid P\in {\mathcal U}_i}.\]
Similarly, we define:
\[{\mathcal{R}}^{i,j}=\set{\sigma_i(R)\mid R\in {\mathcal U}'_j}.\]
Note that the definition of the sets ${\mathcal{P}}^{i,j},{\mathcal{R}}^{i,j}$ of paths depends on the definition of the direction of the paths in ${\mathcal{P}}_2,{\mathcal{R}}_2$. For example, recall that we think of the vertices of $T_1$ as lying on the left boundary of the bounding box $B^*$, and the vertices of $T_3$ as lying on its right boundary, with the paths in ${\mathcal{P}}_2$ directed from left to right. If we were to flip the bounding box $B^*$, so that the vertices of $T_3$ appear on its left boundary and the vertices of $T_1$ on its right boundary, with the paths in ${\mathcal{P}}$ directed from left to right, then the definition of the sets ${\mathcal{P}}^{i,j}$ of paths may change (as a path in ${\mathcal{P}}_2$ may intersect a path of ${\mathcal{R}}^*$ numerous times).
However, if we flip the bounding box $B^*$, so that the vertices of $T_1$ appear on its bottom boundary, the vertices of $T_3$ on its top boundary, and the vertices of $T_2$ and $T_4$ on its left and right boundaries, respectively, directing the paths in ${\mathcal{P}}$ from bottom to top, and the paths in ${\mathcal{R}}$ from left to right, the definition of path sets $\set{{\mathcal{P}}^{i,j},{\mathcal{R}}^{i,j}}_{0\leq i,j\leq \lambda'+1}$ will now change (but now paths of ${\mathcal{P}}$ become vertical and paths of ${\mathcal{R}}$ become horizontal). This transformation corresponds to flipping the grid $\Pi^*$ along its diagonal (from bottom left to top right), and so the cells that lied in the top right quandrant of the grid remain in the top right quadrant. We may need to use this transformation later, but for now we stay with the original notation.
We also need the following observation.
\begin{observation}\label{obs: paths in cells don't cross}
Let $P\in {\mathcal U}_i,R\in {\mathcal U}'_j$ be a pair of paths, for some $0\leq i,j\leq \lambda'+1$, and assume that their subpaths $\sigma_j(P)\subseteq P,\sigma_i(R)\subseteq R$ do not share any vertices. Then either $P\in {\mathcal{P}}'$, or $R\in {\mathcal{R}}'$, or the images of $\sigma_j(P)$ and $\sigma_i(R)$ cross in the drawing $\phi^*$.
\end{observation}
\begin{proof}
Assume that $P\not\in {\mathcal{P}}'$ and $R\not\in {\mathcal{R}}'$, that is, the images of the paths $P,R$ do not cross the images of the paths in ${\mathcal{P}}^*\cup {\mathcal{R}}^*$ in $\phi^*$. From the definition of set ${\mathcal U}_i$, the image of $P$ intersects the interior of the horizontal strip $\mathsf{HStrip}_i$, and path $P$ does not share any vertices with the paths of ${\mathcal{R}}^*$. Therefore, the image of $P$ must be contained in the strip $\mathsf{HStrip}_i$, and it is disjoint from its top and bottom boundaries $\gamma^b_i,\gamma^t_{i+1}$. Using similar reasoning, the image of $R$ is contained in the strip $\mathsf{VStrip}_j$, and it is disjoint from its left and right boundaries, $\gamma^{\ell}_j,\gamma^r_{j+1}$. Consider now the segment $\sigma_j(P)$ of $P$, whose endpoints lie on $R^*_j$ and $R^*_{j+1}$, respectively. Let $\sigma'_j(P)\subseteq \sigma_j(P)$ be the shortest subpath of $\sigma_j(P)$ whose first endpoint lies on $R^*_j$, and whose last endpoint lies on $R^*_{j+1}$; such a path must exist because we can let $\sigma'_j(P)=\sigma_j(P)$. From the definition of in-order intersection, no inner vertex of $\sigma'_j$ may lie on any path of ${\mathcal{R}}^*$. It is then easy to verify that the image of $\sigma'_j(P)$ in $\phi^*$ must be contained in $\mathsf{CellRegion}_{i,j}$, and must split this region into two subregions: one whose top boundary contains a segment of $\gamma^b_{j+1}$, and one whose bottom boundary contains a segment of $\gamma^t_j$.
Using the same reasoning as above, we can select a segment $\sigma'_i(R)$, whose first endpoint lies on $P^*_i$, last endpoint lies on $P^*_{i+1}$, and all inner vertices are disjoint from the vertices lying on the paths in ${\mathcal{P}}^*$.
As before, the image of $\sigma'(R)$ must be contained in $\mathsf{CellRegion}_{i,j}$, but it connects a point on its top boundary to a point on its bottom boundary. Therefore, the image of $\sigma'_i(R)$ must cross the image of $\sigma'_j(P)$.
\end{proof}
\subsubsection{Completing the Construction of the Grid-Like Structure}
In order to complete the construction of the grid-like structure, we need to ensure that for every pair $0\leq i,j\leq \lambda'+1$ of indices, there are many intersection between the sets ${\mathcal{P}}^{i,j}$ and ${\mathcal{R}}^{i,j}$ of paths. More specifically, we need to ensure that every path $\sigma\in {\mathcal{P}}^{i,j}$ intersects many paths in ${\mathcal{R}}^{i,j}$, and vice versay. This is done to ensure well-linkedness properties: namely, that the collection of vertices containing the fist and the last vertex on every path of ${\mathcal{P}}^{i,j}$ is sufficiently well-linked in the graph obtained from the union of the paths in ${\mathcal{R}}^{i,j}\cup {\mathcal{P}}^{i,j}$. This property, in turn, will be exploit in order to construct the sets ${\mathcal{Q}}$ of paths routing the terminals to sets of vertices $V(\Pi(v)))$ over which the distribution ${\mathcal{D}}$ will be defined. This motivates the following definition.
\begin{definition}
For a pair $0\leq i,j\leq \lambda'+1$ of indices, we say that a path $P\in {\mathcal U}_i$ is \emph{bad for cell $\mathsf{Cell}_{i,j}$} iff there are at least $\psi/16$ paths in ${\mathcal{R}}^{i,j}$ that are disjoint from $\sigma_j(P)$. Similarly, we say that a path $R\in {\mathcal U}'_j$ is bad for cell $\mathsf{Cell}_{i,j}$ iff there are at least $\psi/16$ paths in ${\mathcal{P}}^{i,j}$ that are disjoint from $\sigma_i(R)$.
\end{definition}
Consider now some index $0\leq i\leq \lambda'+1$. We say that a path $P\in {\mathcal U}_i$ is \emph{bad} iff it is bad for at least one cell in $\set{\mathsf{Cell}_{i,j}\mid 0\leq j\leq \lambda'+1}$. Similarly, for an index $0\leq j\leq \lambda'+1$, a path $P\in {\mathcal U}'_j$ is bad iff it is bad for at least one cell in $\set{\mathsf{Cell}_{i,j}\mid 0\leq i\leq \lambda'+1}$. The following observation bounds the number of bad paths in each group ${\mathcal U}_i$ of horizontal paths, and in each group ${\mathcal U}'_j$ of vertical paths.
\begin{observation}\label{obs: few bad paths in each group}
Assume that $\mathsf{cr}^*\leq \frac{\tilde k^2}{c_2\eta^5}$, and that events ${\cal{E}}_1,{\cal{E}}_2$ did not happen. Then for all $0\leq i\leq \lambda'+1$, at most $\psi/16$ paths in ${\mathcal U}_i$ are bad. Similarly, for all $0\leq j\leq \lambda'+1$, at most $\psi/16$ paths in ${\mathcal U}'_j$ are bad.
\end{observation}
\begin{proof}
Fix an index $0\leq i\leq \lambda'+1$, and the corresponding set ${\mathcal U}_i\subseteq {\mathcal{P}}_2$. We partition the set of all bad paths in ${\mathcal U}_i$ into two subsets: set ${\mathcal{B}}_1$ contains all bad paths lying in ${\mathcal{P}}'$, and set ${\mathcal{B}}_2$ contains all remaining bad paths. From \Cref{obs: few bad paths}, $|{\mathcal{B}}_1|\leq \psi/32$.
We further partition the set ${\mathcal{B}}_2$ of bad paths into subsets $\set{{\mathcal{B}}_2^j\mid 0\leq j\leq \lambda'}$, where a path $P$ lies in ${\mathcal{B}}_2^j$ if it is bad for cell $\mathsf{Cell}_{i,j}$ (if path $P$ is bad for several cells, we add it to any of the corresponding sets). Consider now some index $0\leq j\leq \lambda'$, and some path $P\in {\mathcal{B}}_2^j$. From the definition, there is a set $\Sigma'\subseteq {\mathcal{R}}^{i,j}$ of at least $\psi/16$ paths that do not share any vertices with $P$. From \Cref{obs: few bad paths}, at most $\psi/32$ of these paths may lie in ${\mathcal{R}}'$. let $\Sigma''\subseteq \Sigma'$ be the collection of the remaining paths, whose cardinality is at least $\psi/32$. From \Cref{obs: paths in cells don't cross}, for every path $\sigma'\in \Sigma'$, the images of $\sigma_j(P)$, and of $\sigma'$ must cross. We let $\chi_j(P)$ denote the set of all crossings $(e,e')$, where $e\in \sigma_j(P)$, and $e'$ is an edge on a path in $\Sigma'$, so $|\chi_j(P)|\geq \psi/32$. We then let $\chi_{i,j}=\bigcup_{P\in {\mathcal{B}}_2^j}\chi_j(P)$, so $|\chi_{j}|\geq |{\mathcal{B}}_2^j|\cdot \psi/32$. Lastly, we let $\chi=\bigcup_{j=0}^{\lambda'+1}\chi_j$. Notice that set $\chi$ contains at least $|{\mathcal{B}}_2|\cdot \psi/32$ distinct crossings in the drawing $\phi^*$. Assume for contradiction that $|{\mathcal{B}}_2|>\psi/32$. Then:
\[\mathsf{cr}^*>\frac{\psi^2}{2^{10}}>\frac{\tilde k^2}{4\eta^4}>\frac{\tilde k^2}{c_2\eta^5},\]
since $\psi>\frac{16\tilde k}{\eta^2}$, a contradiction.
Therefore, $|{\mathcal{B}}_2|\leq \psi/32$, and overall there are at most $\psi/16$ bad paths in ${\mathcal U}_i$.
The proof for path sets ${\mathcal U}'_j\subseteq {\mathcal{R}}_2$ is identical.
\end{proof}
For all $0\leq i\leq \lambda'+1$, we discard every bad path from ${\mathcal U}_i$. If $|{\mathcal U}_i| <\ceil{7\psi/8}$ for any $i$, then we terminate the algorithm and return FAIL. Notice that in this case, from \Cref{obs: few bad paths in each group}, if $\mathsf{cr}^*\geq \frac{\tilde k^2}{c_2\eta^5}$, then at least one of the events ${\cal{E}}_1,{\cal{E}}_2$ must have happened, and the probability for this is at most $1/2$. Therefore, we assume that for all
$0\leq i\leq \lambda'+1$, $|{\mathcal U}_i|\geq \ceil{7\psi/8}$ holds. We discard additional arbitrary paths from ${\mathcal U}_i$, until $|{\mathcal U}_i|= \ceil{7\psi/8}$.
We then let ${\mathcal{P}}_3=\bigcup_{i=0}^{\lambda'+1}{\mathcal U}_i$ denote the resulting set of paths.
Similarly, for all $0\leq j\leq \lambda'+1$, we discard every bad path from ${\mathcal U}'_j$. If, as the result, $|{\mathcal U}'_j|$ falls below $\ceil{7\psi/8}$, we terminate the algorithm and return FAIL. Otherwise, we discard additional arbitrary paths as needed, so that $|{\mathcal U}'_i|=\ceil{7\psi/8}$ holds. We also let ${\mathcal{R}}_3=\bigcup_{j=0}^{\lambda'+1}{\mathcal U}'_j$
For all $0\leq i,j\leq \lambda'$, we also update the path sets ${\mathcal{P}}^{i,j}$ and ${\mathcal{R}}^{i,j}$ accordingly, discarding the paths that are no longer subpaths of paths in ${\mathcal{P}}_3\cup {\mathcal{R}}_3$. Since we are still guaranteed that $|{\mathcal{P}}^{i,j}|,|{\mathcal{R}}^{i,j}|= \ceil{7\psi/8}$, and since every path that is bad for cell $\mathsf{Cell}_{i,j}$ was discarded, we are guaranteed that every path in ${\mathcal{P}}^{i,j}$ intersects at least $\frac{7\psi}{8}-\frac{\psi}{16}=\frac{13\psi}{16}$ paths of ${\mathcal{R}}^{i,j}$ and vice versa.
Since we use this fact later, we summarize it in the following observation.
\begin{observation}\label{obs: paths for cells}
For all $0\leq i,j\leq \lambda'$, $|{\mathcal{P}}^{i,j}|,|{\mathcal{R}}^{i,j}|= \ceil{7\psi/8}$. Every path in
${\mathcal{P}}^{i,j}$ intersects at least $\frac{13\psi}{16}$ paths of ${\mathcal{R}}^{i,j}$ and vice versa.
\end{observation}
This concludes the construction of the grid-like structure.
\subsection{Step 5: the Routing}
Recall that we have denoted by $\Lambda'$ the set of all pairs $({\mathcal{Q}},y)$, where $y$ is a vertex in graph $H$, and ${\mathcal{Q}}$ is a set of paths in graph $H''$, routing the set $\tilde T$ of terminals to vertices of $\Pi(y)$. In this final step we show an efficient algorithm to compute distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},y)\in \Lambda'$, such that for every outer edge $e\in E(H'')$, $\expect[({\mathcal{Q}},y)\in_{{\mathcal{D}}}\Lambda']{(\cong_{H''}({\mathcal{Q}},e))^2}\leq xxx$.
Our algorithm closely follows the arguments of
\cite{Tasos-comm} (see also Lemma D.10 in the full version of \cite{chuzhoy2011algorithm}), who showed a similar result for a grid graph. In order to provide intuition, we first present their algorithm. Assume that we are given a $(q\times q)$ grid graph $G$ for some integer $q$, and let $T$ be the set of vertices lying on the first row of the grid. For convenience, assume that $q$ is an integral power of $2$. Our goal is to compute a distribution ${\mathcal{D}}'$ over pairs in $\Lambda(G,T)$. In other words, the distribution is over pairs $({\mathcal{Q}},v)$, where $v$ is a vertex of $G$, and ${\mathcal{Q}}$ is a set of paths routing the terminals to $v$. We need to ensure that for every edge $e\in E(G)$, the expectation $\expect[({\mathcal{Q}},v)\in_{{\mathcal{D}}'}\Lambda(G,T)]{(\cong_G({\mathcal{Q}},e))^2}\leq O(\log q)$.
For every vertex $v$ in the top right quadrant of the grid, we will define a set ${\mathcal{Q}}(v)$ of paths in $G$, routing the terminals in $T$ to $v$. Our distribution ${\mathcal{D}}$ then assigns, to every pair $({\mathcal{Q}}(v),v)$, where $v$ is a vertex of $G$ lying in the top right quadrant of the grid, the same probability value $4/q^2$.
We now fix a vertex $v$ in the top right quadrant of the grid, and define the set ${\mathcal{Q}}(v)$ of paths. Let $r=\log(q/4)$. For $1\leq i\leq r$, let $S_i$ be a square subgrid of $G$, of size $(2^i\times 2^i)$, whose upper right $v_i$ has the same column-index as vertex $v$, and the same row-index as the bottom left corner of $S_{i-1}$ (we think of $S_1$ as a $(1\times 1)$-grid consisting only of vertex $v$).
We refer to the subgrids $S_i$ of $G$ as \emph{squares}, and specifically to square $S_i$ as \emph{level-$i$ square}.
For all $1\leq i\leq r$, we denote by $T_i$ the set of vertices lying on the bottom boundary of $S_i$. Using the well-linkedness of the grids, it is easy to show that for all $1\leq i\leq r$, there is a collection ${\mathcal{P}}_i$ of paths in graph $S_i$, routing vertices of $T_i$ to verticse of $T_{i-1}$ with congestion at most $2$, such that every vertex of $T_{i-1}$ serves as endpoint of at most two such paths. For $1\leq i\leq r$, let
Let ${\mathcal{P}}'_i$ be a multipset obtained from set ${\mathcal{P}}_i$ by creating $2^{r-i+1}$ copies of every path in ${\mathcal{P}}_i$.
Let $T_{r+1}\subseteq T$ be a set of $|T_r|$ vertices lying on the bottom boundary of the grid $G$, that contains, for every vertex $t\in T_r$, a vertex $t'$ on the bottom boundary of $G$ with the same column index as $t$. Let $P_t$ be the subpath of the corresponding column of $G$ connecting $t$ to $t'$, and denote ${\mathcal{P}}'_{r+1}=\set{P_t\mid t\in T_r}$.
By concatenating the paths in ${\mathcal{P}}_1',\ldots,{\mathcal{P}}'_{r+1}$, we obtain a collection ${\mathcal{Q}}'(v)$ of paths in grid $G$, routing the terminals in $T_{r+1}$ to vertex $v$.
Notice that for $1\leq i\leq r$, for every edge $e$ lying in $S_i$, the congestion on
edge $e$ due to paths in ${\mathcal{Q}}(v)$ is at most $2^{r-i+2}$.
The key in analyzing the expectation $\expect[({\mathcal{Q}},v)\in_{{\mathcal{D}}'}\Lambda(G,T)]{(\cong_G({\mathcal{Q}},e))^2}$
is to notice that, for all $1\leq i\leq r$, square $S_i$ is a $(2^i\times 2^i)$-subgrid of $G$, whose upper left corner is chosen uniformly at random from a set of $q^2/4$ possible points. The total number of subgrids of $G$ of size $(2^i\times 2^i)$ that contain $e$ is $2^{2i}$, so the probability that any of them is selected is bounded by $2^{2i+2}/q^2$. Therefore, for all $1\leq i\leq r$, with probability at most $2^{2i+2-2r}$, edge $e$ belongs to square $S_i$, and in this case, $\cong_G({\mathcal{Q}},e)\leq 2^{r-i+2}$. Therefore, we get that:
\[\expect[({\mathcal{Q}},v)\in_{{\mathcal{D}}'}\Lambda(G,T)]{(\cong_G({\mathcal{Q}},e))^2}
\leq \sum_{i=1}^{r+1}2^{2i+2-2r}\cdot 2^{2r-2i+4}\leq O(r)=O(\log q).
\]
Using the well-linkedness of the terminals in $T$, it is immediate to extend the set ${\mathcal{Q}}'(v)$ of paths to a set ${\mathcal{Q}}(v)$ routing all terminals in $T$ to $v$, while increasing the congestion on every edge of $G$ by at most an additive constant. This provides the final distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}}(v),v)\in \Lambda(G,T)$.
We will simulate a similar process on the grid $\Pi^*$, and its corresponding grid-like structure that we have constructed. Notice however that $\Pi^*$ is only a $(\lambda'\times\lambda')$-grid (where $\eta\leq \lambda'\leq \eta^2$), while the number of terminals that we need to route is much larger (comparable to $|{\mathcal{R}}_3|$). Therefore, we will attempt to route all terminals to a single cell $\mathsf{Cell}_{i,j}$ in the top right quadrant of the grid (in other words, we will route them to vertices lying on paths in ${\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$). This in itself is not sufficient, since we need to route them to a set $V(\Pi(y))$ of vertices, corresponding to a single vertex of the original graph $y$. This means that we may need to perform some routing within the cell $\mathsf{Cell}_{i,j}$, that is, within the graph obtained from the union of the paths in ${\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$. While generally such a routing may be difficult to compute, we will select a large collection of cells (called good cells) in the top right quadrawnt of the grid, in which such routing is easy to obtain. We will then define, for each good cell, the corresponding set of paths routing the terminals to vertices of $V(\Pi(y))$, for some $y\in V(H)$. We do so by simulating the process described above: we define square subgrids $\set{S_i}$ of the grid $\Pi^*$, and we associate these subgrids with sets of horizontal and vertical paths (subpaths of some paths in ${\mathcal{P}}_3\cup {\mathcal{R}}_3$), so that the desired well-linkedness properties of graphs corresponding to each subgrid $S_i$ are preserved. Eventually, the distribution ${\mathcal{D}}$ chooses one of the good squares uniformly at random, and uses the associated pair $({\mathcal{Q}},y)\in \Lambda'$ in order to route the terminals to vertices of $\Pi(y)$. The analysis of expected congestion squared on every edge is very similar to the one outlined above.
We start by defining the notion of good cells of the grid $\Pi^*$, and showing that a large enough number of such cells exists in the upper right quadrant of $\Pi^*$. We will then define square subgrids of $\Pi^*$ and associate sets of paths with each such subgrid to ensure the required well-linkedness properties. Lastly, we show how to construct the desired routing ${\mathcal{Q}}$ for each good cell.
\subsubsection{Good Cells}
Fix a pair of indices $0\leq i,j\leq \lambda'$, and consider the cell $\mathsf{Cell}_{i,j}$ of the grid $\Pi^*$, and the two corresponding sets ${\mathcal{P}}^{i,j}$, ${\mathcal{R}}^{i,j}$ of paths.
\begin{definition}[Good cells]
A path $\sigma\in {\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$ is \emph{good} for cell $\mathsf{Cell}_{i,j}$ iff $\sigma$ contains no outer edges.
We say that cell $\mathsf{Cell}_{i,j}$ is \emph{horizontally good} iff some path $\sigma \in {\mathcal{P}}^{i,j}$ is good for $\mathsf{Cell}_{i,j}$, and we say that it is \emph{vertically good} iff some path $\sigma\in {\mathcal{R}}^{i,j}$ is good for $\mathsf{Cell}_{i,j}$. If cell $\mathsf{Cell}_{i,j}$ is horizontally or vertically good, then we say that it is a good cell, and otherwise we say that it is a bad cell.
\end{definition}
Assume that cell $\mathsf{Cell}_{i,j}$ is horizontally good, and let $\sigma\in {\mathcal{P}}^{i,j}$ be any horizontal path that is good for this cell. Since $\sigma$ contains no outer edges, there must be a vertex $y\in V(H)$, such that $V(\sigma)\subseteq V(\Pi(y))$. Recall that $|{\mathcal{R}}^{i,j}|=\ceil{7\psi/8}$, and that $\sigma$ intersects at least $13\psi/16$ paths of ${\mathcal{R}}^{i,j}$. Let $\hat {\mathcal{R}}^{i,j}\subseteq {\mathcal{R}}^{i,j}$ be a set of $\ceil{13\psi/16}$ paths, each of which shares at least one vertex with $\sigma$. Note that each such path then must contain a vertex of $\Pi(y)$. We denot by $\mathsf{Portals}_{i,j}$ a set of vertices that contains, for every path $\sigma'\in \hat{\mathcal{R}}^{i,j}$, the first vertex of $\sigma'$ (by definition, each such vertex must lie on path $P^*_i$ if $i>0$, or belong to $T_2$ otherwise). For convenience, we denote vertex $y$ by $y_{i,j}$.
Similarly, if cell $\mathsf{Cell}_{i,j}$ is vertically good (bot not horizontally good), let $\sigma'\in {\mathcal{R}}^{i,j}$ be any vertical path that is good for this cell. As before, $\sigma'$ contains no outer edges, there must be a vertex $y_{i,j}\in V(H)$, such that $V(\sigma')\subseteq V(\Pi(y_{i,j}))$. Recall that $\sigma'$ intersects at least $13\psi/16$ paths of ${\mathcal{P}}^{i,j}$. Let $\hat {\mathcal{P}}^{i,j}\subseteq {\mathcal{P}}^{i,j}$ be a set of $\ceil{13\psi/16}$ paths, each of which shares at least one vertex with $\sigma'$. As before, every path in $\hat {\mathcal{P}}^{i,j}$ must contain a vertex of $\Pi(y_{i,j})$. We denot by $\mathsf{Portals}_{i,j}$ a set of vertices that contains, for every path $\sigma\in \hat{\mathcal{P}}^{i,j}$, the first vertex of $\sigma$.
Let $A$ be the set of all pairs of indices $\floor{\lambda'/2}\leq i,j\leq \lambda'+1$, such that $\mathsf{Cell}_{i,j}$ is good. Next, we show that $|A|$ is sufficiently large. Our routing algorithm will then choose a pair $(i,j)$ of indices from $A$ uniformly at random, and then route the terminals to the vertices in set $\mathsf{Portals}_{i,j}$, from where they will be routed to vertices of $\Pi(y_{i,j})$.
\begin{claim}\label{claim: many good cells}
$|A|\geq (\lambda')^2/16$.
\end{claim}
\begin{proof}
Let ${\mathcal{B}}$ be a collection of all bad cells $\mathsf{Cell}_{i,j}$ lying in the top right quadrant, that is, $\floor{\lambda'/2}\leq i,j\leq \lambda'+1$. It is enough to show that $|{\mathcal{B}}|<(\lambda')^2/16$.
Consider now some bad cell $\mathsf{Cell}_{i,j}\in {\mathcal{B}}$. Consider now any path $Q\in {\mathcal{R}}^{i,j}\cup {\mathcal{P}}^{i,j}$. Since cell $\mathsf{Cell}_{i,j}$ is bad, $Q$ must contain at least one outer edge. We say that $Q$ is a \emph{type-1} bad path for cell $\mathsf{Cell}_{i,j}$ if it contains at least one type-1 outer edge (recall that a type-1 outer edge $e$ corresponds to some edge in graph $H$ that is {\bf not} contained in any cluster in ${\mathcal{C}}$). Otherwise, every outer edge on path $Q$ is a type-2 outer edge, and in this case we say that $Q$ is a type-2 bad cluster for $\mathsf{Cell}_{i,j}$. We say that cell $\mathsf{Cell}_{i,j}$ is \emph{type-1 bad} iff at least $\psi/32$ paths of ${\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$ are type-1 bad for this cell, and otherwise it is type-2 bad. We partition the set ${\mathcal{B}}$ of bad cells into two subsets: set ${\mathcal{B}}_1$ contains all type-1 bad cells, and set ${\mathcal{B}}_2$ contains all type-2 bad cells. It is now enough to prove that $|{\mathcal{B}}_1|,|{\mathcal{B}}_2|<(\lambda')^2/32$, which we do in the following two observations.
\begin{observation}\label{obs: few type-1 bad cells}
$|{\mathcal{B}}_1|< (\lambda')^2/32$.
\end{observation}
\begin{proof}
Consider a type-1 bad cell $\mathsf{Cell}_{i,j}\in {\mathcal{B}}_1$, and let ${\mathcal{Q}}^{i,j}\subseteq {\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$ be a set of $\ceil{\psi/32}$ paths that are type-1 bad paths for cell $\mathsf{Cell}_{i,j}$. Each path in ${\mathcal{Q}}^{i,j}$ must contain at least one type-1 bad edge. Since the paths in ${\mathcal{Q}}^{i,j}$ cause edge-congestion at most $2$, there is a set $E^{i,j}$ of at least $\psi/64$ type-1 outer edges of $H''$, lying on paths of ${\mathcal{Q}}^{i,j}$. Since every edge of $H''$ may lie on at most two paths in ${\mathcal{P}}\cup {\mathcal{R}}$, the total number of outer edges in $H''$ must be at least:
\[\frac{|{\mathcal{B}}_1|\cdot \psi}{64}\geq \frac{(\lambda')^2\cdot \psi}{2^{11}}\geq \frac{\lambda^2\cdot \psi}{2^{15}} ,\]
as $\lambda'=\floor{(\lambda-1)/2}\geq \lambda/4$.
Recall that $\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$ and $\lambda=\frac{2^{24}c\cdot \eta \log^8n}{\alpha^*\alpha^3}$, where $c$ is the constant from \Cref{obs: few outer edges}. Therefore, we get that the total number of outer edges in $H''$ is at least $\frac{2c\tilde k \eta \log^8n}{\alpha^3}$, contradicting \Cref{obs: few outer edges}.
\end{proof}
\begin{observation}\label{obs: few type-1 bad cells}
$|{\mathcal{B}}_2|< (\lambda')^2/32$.
\end{observation}
\begin{proof}
For a cluster $C\in {\mathcal{C}}$, let $X(C)=\bigcup_{y\in V(C)}V(\Pi(y))$. Note that all terminals of $H$ lie outside of the clusters in ${\mathcal{C}}$, and so $X(C)\cap \tilde T=\emptyset$. If a path $Q\in {\mathcal{P}}_3\cup {\mathcal{R}}_3$ contains a vertex of $X(C)$, then it must contain at least one edge of $\delta_H(C)$. As the paths in ${\mathcal{P}}\cup {\mathcal{R}}$ cause edge-congestion at most $2$, the total number of paths $Q\in {\mathcal{P}}\cup {\mathcal{R}}$ with a non-empty intersection with $X(C)$ is at most $2\delta_H(C)$.
Let $\mathsf{IntPairs}\subseteq {\mathcal{P}}_3\times{\mathcal{R}}_3$ be the collection of all pairs of paths $P\in {\mathcal{P}}_3$, $R\in {\mathcal{R}}_3$, such that $P$ and $R$ share at least one vertex.
For a cluster $C\in {\mathcal{C}}$, let $\mathsf{IntPairs}'_C\subseteq \mathsf{IntPairs}$ denote denote the collection of all pairs $(P,R)\in \mathsf{IntPairs}$ of paths, such that some vertex $v\in X(C)$ lies on both $P$ and $R$. Clearly, if $(P,R)\in \mathsf{IntPairs}_C'$, then each of the paths $P$, $R$ must contain at least one edge of $\delta_H(C)$. Therefore, from the above discussion, $|\mathsf{IntPairs}'_C|\leq 4|\delta_H(C)|^2$. Let $\mathsf{IntPairs}'=\bigcup_{C\in {\mathcal{C}}}\mathsf{IntPairs}'_C$. Then:
\[ |\mathsf{IntPairs}'|\leq \sum_{C\in {\mathcal{C}}}|\mathsf{IntPairs}'_C|\leq 4\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2. \]
From Equation \ref{eq: sum of squares} (see \Cref{step 2 summary}), $
\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}$, so we get that:
\begin{equation}\label{eq: bounding num of intersection pairs}
|\mathsf{IntPairs}'|<\frac{4(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n},
\end{equation}
where $c_1$ is an arbitrary large enough constant.
In the remainder of the proof, we assume for contradiction that $|{\mathcal{B}}_2|\geq (\lambda')^2/32$, and we will show that $|\mathsf{IntPairs}'|\geq \frac{4(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}$ must hold, contradicting Equation \ref{eq: bounding num of intersection pairs}.
Consider a type-2 bad cell $\mathsf{Cell}_{i,j}\in {\mathcal{B}}_2$. Recall that every path in ${\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$ contains at least one outer edge, and at most $\psi/32$ such paths contain a type-1 bad edge. Since, from \Cref{obs: paths for cells}, $|{\mathcal{P}}^{i,j}|= \ceil{7\psi/8}$,
there is a collection $\Sigma\subseteq {\mathcal{P}}^{i,j}$ of at least $3\psi/4$ paths $P$, such that all edges on $P$ are either inner edges, or type-2 outer edges. Therefore, if $P\in \Sigma$ is any such path, then there is some cluster $C\in {\mathcal{C}}$ with $V(P)\subseteq X(C)$. Recall that, from \Cref{obs: paths for cells}, each path $P\in \Sigma$ intersects at least $\frac{13\psi}{16}$ paths of ${\mathcal{R}}^{i,j}$. Clearly, if a path $R\in {\mathcal{R}}^{i,j}$ intersects a path $P\in \Sigma$, then $(P,R)\in \mathsf{IntPairs}'$. Therefore, intersections between pairs of paths in ${\mathcal{P}}^{i,j}\times {\mathcal{R}}^{i,j}$ contribute at least $\frac{13\psi}{16}\cdot \frac{3\psi}{4}\geq \frac{\psi^2}{2}$ pairs to set $\mathsf{IntPairs}'$.
Therfore, if we denote by $\mathsf{IntPairs}'_{i,j}$ the collection of all pairs $(P,R)\in \mathsf{IntPairs}'$, where a subpath $\sigma$ of $P$ lies in ${\mathcal{P}}^{i,j}$, and a subpath $\sigma'$ of $R$ lies in ${\mathcal{P}}^{i,j}$, and $\sigma,\sigma'$ contain a vertex $v\in X(C)$, for some cluster $C\in {\mathcal{C}}$, then, from the above discussion, $|\mathsf{IntPairs}'_{i,j}|\geq \frac{\psi^2}{2}$. We claim that for every pair $(P,R)\in \mathsf{IntPairs}'$ of paths, there is at most one pair of indices $0\leq i,j\leq \lambda'+1$, such that $(P,R)\in \mathsf{IntPairs}_{i,j}'$. Indeed, assume that $P\in {\mathcal U}_i$ and $R\in {\mathcal U}'_j$. For a pair $0\leq i',j'\leq \lambda'+1$ of indices, ${\mathcal{P}}^{i',j'}$ contains a subpath of $P$ iff $i'=i$, and ${\mathcal{R}}^{i',j'}$ contains a subpath of $R$ iff $j'=j$. So the only pair $(i',j')$ of indices for which $(P,R) \in \mathsf{IntPairs}'_{i',j'}$ may hold is $(i,j)$. Overall, we get that $|\mathsf{IntPairs}'|\geq |{\mathcal{B}}_2|\cdot \psi^2/2$. Assuming that $|{\mathcal{B}}_2|\geq (\lambda')^2/32$, since $\lambda'=\floor{(\lambda-1)/2}\geq \lambda/4$, we get that $|\mathsf{IntPairs}'|\geq \frac{\lambda^2\psi^2}{32}$.
Recall that $\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$, and $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$.
We conclude that:
\[|\mathsf{IntPairs}'|\geq \frac{(\alpha^*)^2\tilde k^2}{2^{17}}\geq \Omega\textsf{left}( \frac{(\alpha\alpha'\tilde k)^2}{\log^8n} \textsf{right} ). \]
Since we can choose $c_1$ to be a sufficiently large constant, this contradicts Equation \ref{eq: bounding num of intersection pairs}.
\end{proof}
\end{proof}
\mynote{looks like there is no need to rotate. can define type-1 bad cell w.r.t. horizontal paths only. Same for good/bad (horizontally good is good enough). Will need to update.}
\iffalse
Good cell part:
will only look at cells in top left quadrant. May need to swap horizontal and vertical. Not completely symmetric b/c of bottom-up / left-right direction of paths (the segment are defined w.r.t. this direction).
A horizontal path $P$ is good for cell $(i,j)$ iff its segment contains no outer edges, so it is contained in $\Pi(v)$ of some vertex $v$. A cell is good if it has at least one good path. This means that all vertical paths in the cell also visit $\Pi(v)$ in the cell. We want to prove that at least $0.8$-fraction of the cells are good.
In a bad cell, partition paths into two types: type-1 path for cell $(i,j)$ contains an outer edge that is not contained in any cluster (type-1 outer edge), otherwise type-2. A bad cell is type-1 if at least 0.1-frac of the horizontal paths are type-1 for the cell, and type-2 otherwise.
If we have many type-1 cells, then we have about $\psi\lambda^2=\alpha^*\tilde k\lambda$ type-1 outer edges, contradicting Obs. 6.9.
(need to do same analysis with horizontal paths, there is no choice).
So at the end we'll have a constant fraction of cells that are type-2 cells for both vertical and horizontal paths. In each such cell we'll have about $\psi^2$ pairs of paths meet at vertices that are inner to clusters. So overall, we have $(\lambda^2\psi^2)$ pairs of paths meeting at vertices that are inner to clusters. but for a cluster $C$, we have at most $|\delta(C)|$ paths entering it, so at most $|\delta(C)|^2$ pairs of paths can meet in inner vertices of $C$. So total number of such intersections is at most $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2$, which should hopefully be less than $(\lambda\psi)^2=(\alpha^*)^2\tilde k^2$. We currently only have $\sum_C|\delta(C)|^2<(\tilde k\tilde \alpha\alpha')^2/\log^{20}n$. $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, so looks OK.
\subsection{Plan for the rest of the proof}
\begin{itemize}
\item select 1 path in $z$, where for example $z=\eta$, though can also use $\eta^2$. Will get a grid skeleton of size $z\times z$.
\item before that, throw away all paths that have more than $\eta/\alpha$ outer edges on them. Most paths will stay.
\item now if we look at a vertical path, it has few outer edges, so for most cells it goes through it does not use an outer edge. Which means that all horizontal paths going through that cell meet in a single vertex (or go through a single cluster). Call such a cell good.
\item at least half the cells are good. In a type-1 good cell all horizontal paths go through a single regular vertex. In a type-2 good cell they all go through 1 cluster. The hope is that there are few type-2 good cells.
\item why: we have about $z^2$ good cells. Inside all good cells there are about $k^2/z^2$ inner edges (can make a cell good if both all horizontal paths meet at some vertex, and all vertical paths meet at some vertex, and then it should be the same vertex). The hope is that the clusters can only contribute $\sum_C|\delta(C)|^2$ vertices to the grid. This is because, if we look at a cluster $C$, the total number of vertical/horizontal paths entering $C$ is at most $\delta(C)$. Even if all pairs have their intersections inside this cluster, it can't contribute more than $|\delta(C)|^2$ such intersecting pairs.
\end{itemize}
\fi
\newpage
\iffalse
contracted graph $\hat H=H_{|{\mathcal{C}}}$. Let $U=\set{v(C)\mid C\in {\mathcal{C}}}$ be the set of super-node vertices of $\hat H$, and let $U'$ be the set of all remaining vertices. We define a rotation system $\hat \Sigma$ for graph $\hat H$ as follows: for a super-node $v(C)\in U$, we let ${\mathcal{O}}_{v(C)}$ be an arbitrary ordering of the edges incident to $v(C)$, and for a regular vertex $u\in U'$, we let ${\mathcal{O}}_u$ be identical to the ordering in the rotation system $\Sigma$ in instance $I$. Recall that, from \Cref{lem: crossings in contr graph}, there is a drawing $\phi$ of $\hat H$, containing at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)$ crossings, such that for every regular vertex $u\in U'$, the ordering of the edges of $\delta_{\hat H}(u)$ as they enter $u$ in $\phi$ is consistent with ${\mathcal{O}}_u\in \Sigma$. We can use this drawing in order to define a solution $\phi'$ to instance $\hat I=(\hat H,\hat \Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, by reordering the edges entering every super-node $v(C)$ as needed. This can be done so that $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)+\sum_{C\in {\mathcal{C}}'}|\delta_H(C)|^2\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)+k^2/x$. Therefore, $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)+k^2/x$.
We are now ready to define graph $\hat H^+$. In order to do so, we start with graph $\hat H$, and process every vertex $u\in V(\hat H)\setminus T$ (that may be a regular vertex or a super-node one-by-one). Consider any such vertex $u$, and let $e_1,\ldots,e_{d(u)}$ be the edges incident to $u$ in $\hat H$, indexed according to their ordering in ${\mathcal{O}}_u\in \hat \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ vrid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$. We replace vertex $u$ with the grid $\Pi(u)$, and, for all $1\leq i\leq d(u)$, if $e_i=(u',u)$, then we replace $e_i$ with a new edge $e'_i=(u',s_i(u))$. Once every vertex of $V(\hat H)\setminus T$ is processed, we obtain the final graph $\hat H^+$. For every vertex $u$, we clall the edges of the grid $\Pi(u)$ \emph{inner edges}. We call all edges of $E(\hat H^+)$ that are not inner edges \emph{outer edges}. Notice that there is a $1$-to-$1$ correspondence between the outer edges and the edges of graph $\hat H$. The following observation is immediate.
\begin{observation}\label{obs: expanded contracted graph}
The set $T$ of terminals is $(\alpha\alpha')$-well-linked in $\hat H^+$, and there is a drawing $\phi''$ of graph $\hat H^+$ with at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)+k^2/x$ crossings, such that the inner edges do not participate in any crossings in $\phi''$.
\end{observation}
\fi
\section{Constructing Guiding Paths - Proof of \Cref{thm: find guiding paths}}
\label{sec: guiding paths}
For conveninence, we denote the contracted graph $H_{|{\mathcal{C}}}$ by $\hat H$, and we denote $|E(\hat H)|=\hm$. From the statement of \Cref{thm: find guiding paths}, $k\geq \hm/\eta$. Observe that, from \Cref{clm: contracted_graph_well_linkedness}, the set $T$ of terminals is $(\alpha\alpha')$-well-linked in $H$. We will assume in the remainder of the proof that $\log n$ is greater than some large enough constant $c'_0$ (whose value we can set later). If this is not the case, then, $n$, and therefore $k$, is bounded by a constant $2^{c_0'}$. We can then use an arbitrary spanning tree $\tau$ of the graph $H$,
rooted at an arbitrary vertex $y$, in order to define a collection ${\mathcal{Q}}$ of paths routing all terminals of $T$ to $y$, where for each terminal $t\in T$, the corresponding path $Q_t\in {\mathcal{Q}}$ is the unique path connecting $t$ to $y$ in the tree $\tau$. Since $|T|$ is bounded by a constant, for every edge $e\in E(H)$, $\cong_H({\mathcal{Q}},e)\leq O(1)$. We then return a distribution ${\mathcal{D}}$ consisting of a single pair $(y,{\mathcal{Q}})$, that has probability value $1$. Therefore, we assume from now on that $\log n>c_0'$ for some large enough constant $c_0'$ whose value we set later.
We start with some intuition. Assume first that graph $H$ contains a grid (or a grid minor) of size $(\Omega(k\alpha\alpha'/\operatorname{poly}\log n)\times \Omega(k\alpha \alpha'/\operatorname{poly}\log n))$, and a collection ${\mathcal{P}}$ of paths connecting every terminal to a distinct vertex on the first row of the grid, such that the paths in ${\mathcal{P}}$ cause a low edge-congestion. For this special case, the algorithm of \cite{Tasos-comm} (see also Lemma D.10 in the full version of \cite{chuzhoy2011algorithm}) provides the distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$ with the required properties. Moreover, if $H$ is a bounded-degree planar graph, with a set $T$ of terminals that is $(\alpha\alpha')$-well-linked,
then there is an efficient algorithm to compute such a grid minor, together with the required collection ${\mathcal{P}}$ of paths. If $H$ is planar but no longer bounded-degree, we can still compute a grid-like structure in it, and apply the same arguments as in \cite{Tasos-comm} in order to compute the desired distribution ${\mathcal{D}}$. The difficulty in our case is that the graph $H$ may be far from being planar, and, even though, from the Excluded Grid theorem of Robertson and Seymour \cite{robertson1986graph,chuzhoy2019towards},
it must contain a large grid-like structure, without having a drawing of $H$ in the plane with a small number of crossing, we do not know how to compute such a structure.
The proof of \Cref{thm: find guiding paths} consists of five steps. In the first step, we will either establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is sufficiently large (so the algorithm can return FAIL), or compute a subgraph $\hat H'\subseteq \hat H$, and a partition $(X,Y)$ of $V(\hat H')$, such that each of the clusters $\hat H'[X],\hat H'[Y]$ has the $\hat \alpha$-bandwidth property, for $\hat \alpha=\Omega(\alpha/\log^4n)$, together with a large collection of edge-disjoint paths routing the terminals to the edges of $E_{\hat H'}(X,Y)$ in graph $\hat H'$. Intuitively, we will view from this point onward the edges of $E_{\hat H'}(X,Y)$ as a new set of terminals, that we denote by $\tilde T$ (more precisely, we subdivide each edge of $E_{\hat H'}(X,Y)$ with a new vertex that becomes a new terminal). We show that it is sufficient to prove an analogue of \Cref{thm: find guiding paths} for this new set $\tilde T$ of terminals. The clusters $\hat H'[X],\hat H'[Y]$ of graph $\hat H'$ naturally define a partition $(H_1,H_2)$ of the graph $H$ into two disjoint clusters. In the second step, we either establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is suffciently large
(so the algorithm can return FAIL), or
compute some vertex $x$ of $H_1$, and a collection ${\mathcal{P}}$ of paths in graph $H_1$, routing the terminals of $\tilde T$ to $x$, such that the paths in ${\mathcal{P}}$ cause a relatively low edge-congestion.
We exploit this set ${\mathcal{P}}$ of paths in order to define an ordering of the terminals in $\tilde T$, which is in turn exploited in the third step in order to compute a ``skeleton'' for the grid-like structure. We compute the grid-like structure itself in the fourth step. In the fifth and the final step, we generalize the arguments from \cite{Tasos-comm} and \cite{chuzhoy2011algorithm} in order to obtain the desired distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$.
Before we proceed, we need to consider four simple special cases. In the first case, $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2$ is large. In the second case, we can route a large subset of the terminals to a single vertex of $V(\hat H)\cap V(H)$ in the graph $\hat H$ via edge-disjont paths. The third case is when $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)=0$, and the fourth special case is when $k< \eta^6$.
\paragraph{Special Case 1: $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2$ is large.}
We consider the case where $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2\geq \frac{(k \alpha^4 \alpha')^2}{c_0\log^{50}n}$, where $c_0$ is the constant from the statement of \Cref{thm: find guiding paths}.
In this case, since we are guaranteed that, for every cluster $C\in {\mathcal{C}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq |\delta(C)|^2/\eta'$, we get that:
\[\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \sum_{C\in {\mathcal{C}}}\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq \sum_{C\in {\mathcal{C}}}\frac{|\delta(C)|^2}{\eta'}\geq \frac{(k \alpha^4 \alpha')^2}{c_0\eta'\log^{50}n}.\]
Therefore, if $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2\geq \frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}n}$, the algorithm returns FAIL and terminates. We assume from now on that:
\begin{equation}\label{eq: boundaries squared sum bound}
\sum_{C\in {\mathcal{C}}}|\delta(C)|^2<\frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}n}.
\end{equation}
\paragraph{Special Case 2: Routing of terminals to a single vertex.}
The second special case happens if there exists a collection ${\mathcal{P}}_0$ of at least $\frac{k\alpha\alpha'}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^3\log^6k}$ edge-disjoint paths in graph $\hat H$ routing some subset $T_0\subseteq T$ of terminals to some vertex $x$ (here $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}$ is the constant from \Cref{claim: embed expander}).
Note that, if Special Case 1 did not happen, and $c_0$ is a large enough constant, then $x$ may not be a supernode. Indeed, assume that $x=v_C$ for some cluster $C\in {\mathcal{C}}$. Then:
\[|\delta(C)|^2\geq \Omega\textsf{left}(\frac{(k\alpha \alpha')^2}{\log^{12}k}\textsf{right} )\geq \frac{(k \alpha^4 \alpha')^2}{c_0\log^{50}n},\]
which is, assuming that $c_0$ is a large enough constant, a contradiction. Therefore, we can assume that $x$ is not a supernode.
From \Cref{claim: routing in contracted graph}, since the clusters in ${\mathcal{C}}$ have the $\alpha'$-bandwidth property, there is a collection ${\mathcal{P}}'_0$ of paths in graph $H$, routing the vertices of $T_0$ to $x$, with edge-congestion at most $\ceil{1/\alpha'}\leq 2/\alpha'$. Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in graph $H$, from \Cref{lem: routing path extension}, there is a set ${\mathcal{Q}}$ of paths in graph $H$, routing the vertices of $T$ to $x$ with congestion at most:
\[\ceil{\frac{|T|}{|T_0|}}\textsf{left}(\frac 2{\alpha'}+\ceil{\frac 1{\alpha\alpha'}}\textsf{right} )\leq O\textsf{left} ( \frac{\log^6k}{(\alpha\alpha')^2} \textsf{right}).\]
\iffalse
Denote $z=|{\mathcal{P}}_0|\geq \frac{k\alpha\alpha'}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\log^4k}$.
We compute a single set ${\mathcal{Q}}$ of paths routing the set $T$ of terminals to $x$, with congestion $O(\log^4k/(\alpha \alpha')^2)$, as follows. We partition the set $T\setminus T_0$ of terminals into $q=\ceil{k/z}\leq O(\log^4k/(\alpha\alpha'))$ subsets, each of which contains at most $z$ vertices.
Consider now some index $1\leq i\leq q$. Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in $H$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{P}}_i'$ of paths in graph $H$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in ${\mathcal{P}}'_i$ cause edge-congestion $O(1/(\alpha\alpha'))$, and each vertex of $T_0\cup T_i$ is the endpoint of at most one path in ${\mathcal{P}}_i'$. By concatenating the paths in ${\mathcal{P}}_i'$ with paths in ${\mathcal{P}}_0$, we obtain a collection ${\mathcal{P}}_i$ of paths in graph $H$, connecting every terminal of $T_i$ to $x$, that cause edge-congestion $O(1/(\alpha\alpha'))$.
Let ${\mathcal{Q}}=\bigcup_{i=0}^q{\mathcal{P}}_i$ be the resulting set of paths. Observe that set ${\mathcal{Q}}$ contains $k$ paths, routing the terminals in $T$ to the vertex $x$ in graph $H$, with $\cong_H({\mathcal{Q}})\leq O(q/(\alpha\alpha'))\leq O(\log^4k/(\alpha\alpha')^2)$.
\fi
Note that a set ${\mathcal{Q}}$ of paths with the above properties can be computed efficiently via standard maximum flow algorithm.
We return a distribution ${\mathcal{D}}$ consisting of a single pair $({\mathcal{Q}},x)$ with probability value $1$, and terminate the algorithm.
Clearly, for every edge $e\in E(H)$, $\expect{(\cong({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{32}n}{\alpha^{12}(\alpha')^8}\textsf{right} )$.
\paragraph{Special Case 3: $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)=0$.}
Recall that we can efficiently check whether $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)=0$, using the algorithm from \Cref{thm: crwrs_planar}. Assume now that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)=0$. We construct a graph $\tilde H$ from $H$, as follows.
We start with $\tilde H=\emptyset$, and then process every vertex $u\in V(H)\setminus T$ one by one. We now describe an iteration when a vertex $u\in V(H)\setminus T$ is processed. We denote by $d(u)$ the degree of the vertex $u$ in graph $H$. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that are incident to $u$ in $H$, indexed according to their ordering in $ {\mathcal{O}}_u\in \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$ indexed in their natural left-to-right order.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $\tilde H$.
Once every vertex $u\in V(H)\setminus T$ is processed, we add
the vertices of $T$ to the graph $\tilde H$. Recall that every terminal $t\in T$ has degree $1$ in $H$. We denote the unique edge $e_t$ incident to $t$ by $e_1(t)$, and we denote $s_1(t)=t$.
Next, we add another collection of to graph $\tilde H$, as follows. Consider any edge $e=(u,v)\in E(H)$. Assume that $e$ is the $i$th edge of $u$ and the $j$th edge of $v$, that is, $e=e_i(u)=e_j(v)$. Then we add an edge $e'=(s_i(u),s_j(v))$ to graph $\tilde H$, and we view this edge as the \emph{copy of the edge $e\in E(H)$}. From \Cref{clm: contracted_graph_well_linkedness}, the set $T$ of terminals is $(\alpha\cdot \alpha')$-well-linked in $H$. Notice that we can view the graph $H$ as the contracted graph of $\tilde H$ with respect to the set of clusters $\set{\Pi(v)\mid v\in V(H)\setminus T}$. Since each cluster in the set has the $1$-bandwidth property, from \Cref{clm: contracted_graph_well_linkedness}, set $T$ of terminals is $(\alpha\alpha')$-well-linked in $\tilde H$. Consider now the optimal solution $\phi^*$ to instance $(H,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, so drawing $\phi^*$ contains no crossings. It is easy to extend this drawing to a planar drawing of graph $\tilde H$: for every vertex $u\in V(H)\setminus T$, we consider a small disc $\eta(u)$ in $\phi^*$ that contains $u$ in its interior. We then delete the image of $u$ and segments of images of its incident edges lying in $\eta(u)$, and instead place the image of the grid $\Pi(u)$, drawn using its natural layout, so that the images of the vertices $s_1(u),\ldots,s_{d(u)}(u)$ appear on the boundary of the disc. To summarize, graph $\tilde H$ that we have defined is planar, it has maximum vertex degree at most $4$, and the set $T$ of terminals is $(\alpha\alpha')$-well-linked in $\tilde H$. We will use the following theorem from \cite{chuzhoy2019towards}.
\mynote{insert lemma here, finish constructing the distribution}.
\paragraph{Special Case 4: $k< \eta^6$, but $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>0$.}
Assume that $k<\eta^6$. Note that $\frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}n}\leq \frac{\eta^{12}}{\eta'}<1$ in this case (as, from the statemnt of \Cref{thm: find guiding paths}, $\eta'>\eta^{13}$.)
Since we have assumed that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>0$, we get that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq 1>\frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}n}$. We then simply return FAIL and terminate the algorithm.
In the remainder of the algorithm, we assume that neither of the four special cases happened. We now describe each step of the algorithm in detail.
\subsection{Step 1: Splitting the Contracted Graph}
In this step, we split the contracted graph $\hat H$, using the algorithm summarized in the following theorem.
\begin{theorem}\label{thm: splitting}
There is an efficient randomized algorithm that returns FAIL with probability at most $1/\operatorname{poly}(k)$, and, if it does not return FAIL, then it computes a subgraph $\hat H'\subseteq \hat H$ and a partition $(X,Y)$ of $V(\hat H')$ such that:
\begin{itemize}
\item clusters $\hat H'[X]$ and $\hat H'[Y]$ both have the $\hat \alpha'$-bandwidth property in $\hat H'$, for $\hat \alpha'=\Omega(\alpha/\log^4n)$; and
\item there is a set ${\mathcal{R}}$ of $\Omega(\alpha^3k/\log^8n)$ edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to edges of $E_{\hat H'}(X,Y)$.
\end{itemize}
\end{theorem}
\begin{proof}
We start by applying the algorithm from \Cref{claim: embed expander} to graph $\hat H$ and the set $T$ of terminals, to obtain a graph $W$ with $V(W)=T$ and maximum vertex degree at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$, and an embedding ${\hat{\mathcal{P}}}$ of $W$ into $\hat H$ with congestion at most $(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)/\alpha$. Let $\hat{\cal{E}}$ be the bad event that $W$ is not a $1/4$-expander. Then $\prob{\hat {\cal{E}}}\leq 1/\operatorname{poly}(k)$.
Define graph $\hat H'$ as the union of all paths in $\hat{{\mathcal{P}}}$.
We need the following observation.
\begin{observation}\label{obs: expansion and degree}
If event $\hat {\cal{E}}$ did not happen, then the set $T$ of vertices is $\hat \alpha$-well-linked in $\hat H'$, for $\hat \alpha=\frac{\alpha}{4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$, and the maximum vertex degree in $\hat H'$ is at most $d=\frac{\alpha k}{512\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$.
\end{observation}
\begin{proof}
Assume that Event $\hat {\cal{E}}$ did not happen. We first prove that the set $T$ of terminals is $\hat \alpha$-well-linked in $\hat H$. Consider any paritition $(A,B)$ of vertices of $\hat H'$, and denote $T_A=T\cap A$, $T_B=T\cap B$. Assume w.l.o.g. that $|T_A|\leq |T_B|$. Then it is sufficient to show that $|E_{\hat H'}(A,B)|\geq \hat \alpha\cdot |T_A|$.
Consider the partition $(T_A,T_B)$ of the vertices of $W$, and denote $E'=E_{W}(T_A,T_B)$. Since $W$ is a $1/4$-expander, $|E'|\geq |T_A|/4$ must hold. Consider now the set $\hat {\mathcal{R}}\subseteq {\hat{\mathcal{P}}}$ of paths containing the embeddings $P(e)$ of every edge $e\in E'$. Each path $R\in \hat {\mathcal{R}}$ connects a vertex of $T_A$ to a vertex of $T_B$, so it must contain an edge of $|E_{\hat H}(A,B)|$. Since $|\hat{\mathcal{R}}|\geq |T_A|/4$, and the paths in ${\hat{\mathcal{P}}}$ cause edge-congestion at most $(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)/\alpha$, we get that $|E_{\hat H}(A,B)|\geq \alpha\cdot |T_A|/(4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)\geq \hat \alpha |T_A|$.
Assume now that maximum vertex degree in $\hat H'$ is greater than $d$, and let $x$ be a vertex whose degree is at least $d$. Let $\hat{\mathcal{Q}}\subseteq \hat{\mathcal{P}}$ be the set of all paths containing the vertex $x$.
Consider any such path $Q\in \hat{\mathcal{Q}}$. The endpoints of this path are two distinct terminals $t,t'\in T$. We let $Q'\subseteq Q$ be the sub-path of $Q$ between the terminal $t$ and the vertex $x$, and we let ${\mathcal{Q}}'=\set{Q'\mid Q\in \hat{\mathcal{Q}}}$.
Recall that every vertex in $W$ has degree at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$, and so a terminal in $T$ may be an endpoint of at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$ paths in ${\hat{\mathcal{P}}}$. Therefore, there is a subset ${\mathcal{Q}}''\subseteq {\mathcal{Q}}'$ of at least $d/(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)$ paths in $\hat H'$, each of which originates at a distinct terminal.
Since paths in ${\mathcal{Q}}''$ cause congestion at most $(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)/\alpha$, from \Cref{claim: remove congestion}, there is a collection ${\mathcal{Q}}'''$ of edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to $x$ with:
$$|{\mathcal{Q}}'''|\geq |{\mathcal{Q}}''|\cdot \frac{\alpha}{\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\geq \frac{d\alpha}{\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\log^4k}\geq\frac{\alpha^2k}{512\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^3\log^6k},$$
\iffalse
But then, from \Cref{claim: routing in contracted graph}, there is a collection ${\mathcal{Q}}'''$ of edge-disjoint paths in graph $H$, routing a subset of terminals to $x$ of cardinality at least:
\[\frac{\alpha'|{\mathcal{Q}}''|} 2\geq \frac{\alpha' d}{2\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\geq \frac{\alpha \alpha' k}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^4k},\]
\fi
contradicting the fact that Special Case 2 did not happen.
\end{proof}
Next, we use the following lemma to compute the required sets $X$, $Y$ of vertices. The proof follows immediately from techniques that were introduced in \cite{chuzhoy2012routing} and then refined in \cite{chuzhoy2012polylogarithmic,chekuri2016polynomial,chuzhoy2016improved}. Unfortunately, all these proofs assumed that the input graph has a bounded max vertex-degree, and additionally the proofs are somewhat more involved than the proof that we need here (this is because these proofs could only afford a $\operatorname{poly}\log k$ loss in the cardinality of the path set ${\mathcal{R}}$ relatively to $|T|$, while we can afford a $\operatorname{poly}\log n$ loss). Therefore, we provide a complete proof of the lemma in Section \ref{sec: splitting} of the Appendix for completeness.
\begin{lemma}\label{lem: splitting}
There is an efficient algorithm that, given as input an $n$-vertex graph $G$, and a subset $T$ of $k$ vertices of $G$ called terminals, together with a parameter $0<\tilde \alpha<1$, such that the maximum vertex degree in $G$ is at most $\tilde \alpha k/64$, and every vertex of $T$ has degree $1$ in $G$, either returns FAIL, or computes a partition $(X,Y)$ of $V(G)$, such that:
\begin{itemize}
\item each of the clusters $G[X]$, $G[Y]$ has the $\tilde \alpha'$-bandwidth property, for $\tilde \alpha'=\Omega(\tilde \alpha/\log^2n)$; and
\item there is a set ${\mathcal{R}}$ of at least $\Omega(\tilde \alpha^3k/\log^2n)$ edge-disjoint paths in graph $G$, routing a subset of terminals to edges of $E_G(X,Y)$.
\end{itemize}
Moreover, if the vertex set $T$ is $\tilde \alpha$-well-linked in $G$, then the algorithm never returns FAIL.
\end{lemma}
We apply the algorithm from \Cref{lem: splitting} to graph $\hat H'$, the set $T$ of terminals, and parameter $\tilde \alpha=\hat \alpha=\frac{\alpha}{4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$. Recall that we are guaranteed that the maximum vertex-degree in graph $\hat H'$ is at most $d=\frac{\alpha k}{512\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\leq \frac{\tilde \alpha k}{64}$.
Note that the algorithm from \Cref{lem: splitting} may only return FAIL if the set $T$ of terminals is not $\tilde \alpha$-well-linked in $\hat H'$, which, from \Cref{obs: expansion and degree}, may only happen if event $\hat {\cal{E}}$ happened, which in turn may only happen with probability $1/\operatorname{poly}(k)$. If the algorithm from \Cref{lem: splitting} returned FAIL, then we terminate the algorithm and return FAIL as well. Therefore, we assume from now on that the algorithm from \Cref{lem: splitting} did not return FAIL.
\iffalse restart the algorithm that we have described so far from scratch. We do so at most $1/\operatorname{poly}\log n$ times, and if, in every iteration, the algorithm from \Cref{lem: splitting} returns FAIL, then we return FAIL as the outcome of the algorithm for \Cref{thm: splitting}. Clearly, this may only happen with probability at most $1/\operatorname{poly}(n)$. Therefore, we assume that in one of the iterations, the algorithm from \Cref{lem: splitting} did not return FAIL. From now on we consider the outcome of that iteration.\fi
Let $(X,Y)$ be the partition of $V(\hat H')$ that the algorithm returns. We are then guaranteed that each of the clusters $\hat H'[X],\hat H'[Y]$ has the $\tilde \alpha'$-bandwidth property, where $\tilde \alpha'=\Theta(\tilde \alpha/\log^2n)=\Theta(\alpha/\log^4n)$. The algorithm also ensures that there is a collection ${\mathcal{R}}$ of edge-disjoint paths in $\hat H'$, routing a subset of the terminals to edges of $E_{\hat H'}(X,Y)$, with $|{\mathcal{R}}|\geq \Omega(\tilde \alpha^3k/\log^2n)=\Omega(\alpha^3k/\log^8n)$.
This completes the proof of \Cref{thm: splitting}.
\end{proof}
If the algorithm from \Cref{thm: splitting} returned FAIL (which may only happen with probability at most $1/\operatorname{poly}(k)$), then we terminate the algorithm and return FAIL as well. Therefore, we assume from now on that the algorithm from \Cref{thm: splitting} returned a subgraph $\hat H'\subseteq \hat H$ and a partition $(X,Y)$ of $V(\hat H')$ such that each of the clusters $\hat H'[X]$ and $\hat H'[Y]$ has the $\hat \alpha'$-bandwidth property, for $\hat \alpha'=\Theta(\alpha/\log^4n)$, and
there is a set ${\mathcal{R}}$ of $\Omega(\alpha^3k/\log^8n)$ edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to edges of $E_{\hat H'}(X,Y)$.
Notice that we can compute the path set ${\mathcal{R}}$ with the above properties efficiently, using standard maximum flow algorithms. We assume w.l.o.g. that edges of $E_{\hat H}(X,Y)$ do not serve as inner edges on paths in ${\mathcal{R}}$.
Let $E'\subseteq E_{\hat H'}(X,Y)$ be the subset of edges containing the last edge on every path in ${\mathcal{R}}$, so, by reversing the direction of the paths in ${\mathcal{R}}$, we can view the set ${\mathcal{R}}$ of paths as routing the edges of $E'$ to the terminals.
In the remainder of this step, we will slightly modify the graphs $H$ and $\hat H$, and we will continue working with the modified graphs only in the following steps.
Let $\hat H''\subseteq\hat H'$ be the graph obtained from $\hat H'$ by first deleting all edges of $E_{\hat H'}(X,Y)\setminus E'$ from it, and then subdividing every edge $e\in E'$ with a vertex $t_e$. We denote $\tilde T=\set{t_e\mid e\in E'}$, and we refer to vertices of $\tilde T$ as \emph{pseudo-terminals}. Recall that $|\tilde T|=|{\mathcal{R}}|=\Omega(\alpha^3k/\log^8n)$, and there is a set ${\mathcal{R}}'$ of edge-disjoint paths in the resulting graph $\hat H''$, routing the vertices of $\tilde T$ to the vertices of $T$.
We define $\hat H_1=\hat H''[X\cup \tilde T]$, the subgraph of $\hat H''$ induced by the set $X\cup \tilde T$ of vertices, and we define $\hat H_2=\hat H''[Y\cup \tilde T]$ similarly. From the $\hat \alpha'$-bandwidth property of the clusters $\hat H'[X]$ and $\hat H'[Y]$, we are guaranteed that the vertices of $\tilde T$ are $ \hat \alpha'$-well-linked in both $\hat H_1$ and in $\hat H_2$, where $\hat \alpha'= \Theta(\alpha/\log^4n)$. Let ${\mathcal{C}}'\subseteq {\mathcal{C}}$ be the subset of all clusters $C$ whose corresponding supernode $v_C$ lies in graph $\hat H''$.
For convenience, we also subdivide, in graph $H$, every edge $e\in E'$, with the vertex $t_e$, so graph $\hat H''$ can be now viewed as a subgraph of the contracted graph $H_{|{\mathcal{C}}}$.
Next, we let $H'\subseteq H$ be the subgraph of $H$ that corresponds to the graph $\hat H''$. Put in other words, graph $H'$ is obtained from $\hat H''$ by replacing every supernode $v_C$ with the corresponding cluster $C\in {\mathcal{C}}'$. Equivalently, we can obtain graph $H'$ from $H$, by deleting every edge of $E(\hat H)\setminus E(\hat H'')$ and every regular (non-supernode) vertex of $V(\hat H)\setminus V(\hat H'')$. Additionally, for every cluster $C\in {\mathcal{C}}\setminus {\mathcal{C}}'$, we delete all edges and vertices of $C$ from $H'$. We also define a rotation system $\Sigma'$ for graph $H'$, which is identical to $\Sigma$ (vertices $t_e\in \tilde T$ all have degree $2$, so their corresponding ordering ${\mathcal{O}}_{t_e}$ of incident edges can be set arbitrarily).
We partition the set ${\mathcal{C}}'$ of clusters into two subsets: set ${\mathcal{C}}_X$ contains all clusters $C\in {\mathcal{C}}'$ with $v_C\in X$, and set ${\mathcal{C}}_Y$ contains all clusters $C\in {\mathcal{C}}'$ with $v_C\in Y$. We can similarly define the graphs $H_1,H_2\subseteq H'$, that correspond to the contracted graphs $\hat H_1$ and $\hat H_2$, respectively: let $X'$ contain all vertices $x\in V(H')$, such that either $x\in C$ for some cluster $C\in {\mathcal{C}}_X$, or $x$ is a regular vertex of $\hat H''$ lying in $X$. Similarly, we let $Y'$ contain all vertices $y\in V(H)$, such that either $y\in C$ for some cluster $C\in {\mathcal{C}}_Y$, or $y$ is a regular vertex of $\hat H''$ lying in $Y$. We then let $H_1=H'[X\cup \tilde T]$, and $H_2=H'[Y\cup \tilde T]$.
The following observation, summarizing some properties of graph $H'$, is immediate.
\begin{observation}\label{obs: properties of new graph}
Graph $H'$ satisties the following properties:
\begin{itemize}
\item $\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')\leq \mathsf{OPT}_{\mathsf{cnwrs}} (H,\Sigma)$;
\item $\hat H''=H'_{|{\mathcal{C}}'}$;
\item $|E(\hat H'')|\leq 2|E(\hat H)|\leq 2\eta k\leq O(|\tilde T|\eta \log^8n/\alpha^3)$; and
\item graph $\hat H_1$ is a contracted graph of $H_1$ with respect to ${\mathcal{C}}_X$, and graph $H_2$ is a contracted graph of $H_2$ with respect to ${\mathcal{C}}_Y$. In other words, $\hat H_1=(H_1)_{|{\mathcal{C}}_X}$, and $\hat H_2=(H_2)_{|{\mathcal{C}}_Y}$.
\end{itemize}
\end{observation}
For the third assertion we have used the fact that $k\geq |E(\hat H)|/\eta$ from the statement of \Cref{thm: find guiding paths}, and $|\tilde T|\geq \Omega(\alpha^3k/\log^8n)$.
Recall that $\Lambda(H',\tilde T)$ denotes the set of all pairs $({\mathcal{Q}},x)$, where $x$ is a vertex of $H'$, and ${\mathcal{Q}}$ is a collection of paths in graph $H'$, routing the vertices of $\tilde T$ to $x$. In the following observation, we show that, to obtain the desired distribution ${\mathcal{D}}$ over pairs in $\Lambda(H,T)$, it is now sufficient to compute a distribution ${\mathcal{D}}'$ over pairs $({\mathcal{Q}},x)\in \Lambda(H',\tilde T)$, such that for every edge $e\in E(H')$, $\expect{(\cong_{H'}({\mathcal{Q}},e))^2}$ is low.
\begin{observation}\label{obs: convert distributions}
There is an efficient algorithm, that, given a distribution ${\mathcal{D}}'$ over pairs $({\mathcal{Q}}',x')\in \Lambda(H',\tilde T)$,
such that for every edge $e'\in E(H')$, $\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e'))^2}\leq \beta$ holds,
computes a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$, such that for every edge $e\in E(H)$, $\expect[({\mathcal{Q}},x)\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left}(\frac{\beta \log^{16}n}{\alpha^8(\alpha')^4}\textsf{right} )$.
\end{observation}
\begin{proof}
Recall that $H'\subseteq H$. Consider some pair $({\mathcal{Q}}',x')\in \Lambda(H',\tilde T)$, whose probability value in distribution ${\mathcal{D}}'$ is $p({\mathcal{Q}}',x')>0$. We compute another path set ${\mathcal{Q}}$ in graph $H$, routing all terminals in $T$ to $x'$, and we will assign to the pair $({\mathcal{Q}},x')\in \Lambda(H,T)$ the same probability value $p({\mathcal{Q}}',x')$.
Recall that there is a set ${\mathcal{R}}'$ of edge-disjoint paths in graph $\hat H''$, routing the vertices of $\tilde T$ to the vertices of $T$, and moreover, such a path set can be found efficiently via a standard maximum $s$-$t$ flow computation.
Since $\hat H''=H'_{|{\mathcal{C}}'}$, from \Cref{claim: routing in contracted graph}, we can efficiently compute a set ${\mathcal{R}}_0$ of edge-disjoint paths in graph $H'$, routing a subset $T_0\subseteq T$ of terminals to $\tilde T$, with $|{\mathcal{R}}_0|\geq \alpha'\cdot |{\mathcal{R}}'|/2=\alpha'\cdot |\tilde T|/2=\Omega(\alpha'\alpha^3k/\log^8n)$. By concatenating the paths in ${\mathcal{R}}_0$ and the paths in ${\mathcal{Q}}'$, we obtain a collection ${\mathcal{R}}_0'$ of paths in graph $H'$, routing the terminals of $T_0$ to vertex $x'$, such that for every edge $e\in E(H)$, $\cong_H({\mathcal{R}}_0',e)\leq \cong_{H'}({\mathcal{Q}}',e)+1$.
Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in graph $H$ (from \Cref{clm: contracted_graph_well_linkedness}), from \Cref{lem: routing path extension}, there exists a collection ${\mathcal{Q}}$ of paths in graph $H$, routing the terminals in $T$ to vertex $x'$, such that for every edge $e\in E(H)$:
\iffalse
\[
\begin{split}
\cong_{H}({\mathcal{Q}},e)&\leq \ceil{\frac{|T|}{|T_0|}}\textsf{left}(\cong_{H}({\mathcal{R}}'_0,e)+\ceil{\frac{1}{\alpha\alpha'}}\textsf{right} )\\
&\leq O\textsf{left} (\frac{\log^8n}{\alpha'\alpha^3}\textsf{right} )\cdot \textsf{left}(\cong_{H'}({\mathcal{Q}}',e)+\frac{2}{\alpha\alpha'}\textsf{right} )
\end{split}
\]
\fi
\[
\cong_{H}({\mathcal{Q}},e)\leq \ceil{\frac{|T|}{|T_0|}}\textsf{left}(\cong_{H}({\mathcal{R}}'_0,e)+\ceil{\frac{1}{\alpha\alpha'}}\textsf{right} )
\leq O\textsf{left} (\frac{\log^8n}{\alpha'\alpha^3}\textsf{right} )\cdot \textsf{left}(\cong_{H'}({\mathcal{Q}}',e)+\frac{2}{\alpha\alpha'}\textsf{right} ).
\]
\iffalse
Let $z=\ceil{k/|T_0|}=O(\log^{8}n/(\alpha^3\alpha')$. Next, we partition the terminals of $T\setminus T_0$ into $z$ subsets $T_1,\ldots,T_z$, of cardinality at most $|T_0|$ each. Consider now some index $1\leq i\leq z$. Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in graph $H$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{R}}_i'$ of paths in graph $H$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in ${\mathcal{R}}'_i$ cause edge-congestion $O(1/(\alpha\alpha'))$, and each vertex of $T_0\cup T_i$ is the endpoint of at most one path in ${\mathcal{R}}'_i$. By concatenating the paths in ${\mathcal{R}}'_i$ with paths in ${\mathcal{R}}_0$, we obtain a collection ${\mathcal{R}}_i$ of paths in graph $H$, routing the terminals in $T_i$ to vertices of $\tilde T$, such that every vertex of $\tilde T\cup {\mathcal{R}}_0$ is an endpoint of at most one path in ${\mathcal{R}}_i$. Let ${\mathcal{R}}^*=\bigcup_{i=0}^z{\mathcal{R}}_i$ be the resulting set of paths. Observe that set ${\mathcal{R}}^*$ contains $k$ paths, each of which connects a distinct terminal of $T$ to a vertex of $\tilde T$, and every vertex of $\tilde T$ serves as an endpoint of at most $z=O(\log^{8}n/(\alpha^3\alpha')$ such paths. The paths of ${\mathcal{R}}^*$ cause congestion at most $O(z/\alpha\alpha')\leq O(\log^8k/(\alpha^4(\alpha')^2))$ in graph $H$.
For ever pseudoterminal $\tilde t\in \tilde T$, let $Q'_{\tilde t}\in {\mathcal{Q}}'$ be the unique path originating at $\tilde t$. For every terminal $t\in T$, let $R'_t\in {\mathcal{R}}^*$ be the unique path originating at terminal $t$, and let $\tilde t\in \tilde T$ be the other endpoint of path $R'_t$. We let $Q_t$ be the path obtained by concatenating the paths $R'_t$ and $Q'_{\tilde t}$, so path $Q_t$ connects $t$ to vertex $x'$. We then let ${\mathcal{Q}}=\set{Q_t\mid t\in T}$. Note that ${\mathcal{Q}}$ is a set of paths in graph $H$ routing the set $T$ of terminals to vertex $x'$, so $({\mathcal{Q}},x')\in \Lambda(H,T)$. We assign to the pair $({\mathcal{Q}},x')$ probability $p({\mathcal{Q}}',x')$. Note that, since every pseudoterminal $\tilde t\in \tilde T$ may seve as an endpoint of at most $O(\log^{8}n/(\alpha^3\alpha')$ paths in ${\mathcal{R}}^*$, and since the paths in ${\mathcal{R}}^*$ cause edge-congestion at most $O(\log^8k/(\alpha^4(\alpha')^2))$ in graph $H$, for every edge $e\in E(H)$, we get that:
\[\cong_H({\mathcal{Q}},e)\leq O\textsf{left}(\frac{\cong_{H'}({\mathcal{Q}}',e) \cdot \log^8k}{\alpha^3 \alpha'}+\frac{\log^8k}{\alpha^4(\alpha')^2}\textsf{right} ). \]
\fi
Therefore, for every pair $({\mathcal{Q}}',x')\in \Lambda(H',\tilde T)$, whose probability value in distribution ${\mathcal{D}}'$ is $p({\mathcal{Q}}',x')>0$, we have defined a pair $({\mathcal{Q}},x')\in \Lambda(H,T)$, and we have assigned to it the same probability value $p({\mathcal{Q}},x')=p({\mathcal{Q}}',x')$. This completes the definition of the distribution ${\mathcal{D}}$ over pairs in $\Lambda(H,T)$. From the above discussion, for every edge $e\in E(H)$,
\[\expect[({\mathcal{Q}},x')\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left}(\frac{\log^{16}k}{\alpha^8(\alpha')^4}\textsf{right} )\cdot \textsf{left} (\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e))^2}+1\textsf{right}) \leq O\textsf{left}(\frac{\beta \log^{16}n}{\alpha^8(\alpha')^4}\textsf{right} ). \]
\end{proof}
The following immediate corollary is obtained by plugging in the bounds we need into \Cref{obs: convert distributions}.
\begin{corollary}\label{cor: convert distribution fixed values}
There is an efficient algorithm, that, given a distribution ${\mathcal{D}}'$ over pairs $({\mathcal{Q}}',x')\in \Lambda(H',\tilde T)$,
such that for every edge $e'\in E(H')$, $\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e'))^2}\leq O\textsf{left} (\frac{\log^{16}n}{(\alpha\alpha')^4}\textsf{right} )$ holds, produces a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},x)\in \Lambda(H,T)$, such that for every edge $e\in E(H)$: $$\expect[({\mathcal{Q}},x)\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{32}n}{\alpha^{12}(\alpha')^8}\textsf{right} ).$$
\end{corollary}
Recall that $\tilde k\geq \Omega\textsf{left}(\frac{\alpha^3k}{\log^8n}\textsf{right} )$, so $\frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}n}\leq O\textsf{left}( \frac{(\tilde k\hat \alpha'\alpha')^2}{c_0\log^{20}n}\textsf{right} )$. We will use a constant $c_1$, that is a large enough constant, whose value will be set later, and we set $c_0=c_1^2$. We can then assume that $\frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}n}\leq \frac{(\tilde k\hat \alpha'\alpha')^2}{c_1\log^{20}n}$. In particular, from Equation \ref{eq: boundaries squared sum bound}, we get that $\sum_{C\in {\mathcal{C}}}|\delta_{H'}(C)|^2<\frac{(k\alpha^4\alpha')^2}{c_0\log^{50}n}\leq \frac{(\tilde k\hat \alpha'\alpha')^2}{c_1\log^{20}n}$. Additionally, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')>\frac{(\tilde k\hat \alpha'\alpha')^2}{c_1\eta'\log^{20}n}$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}n}$.
In order to complete the proof of \Cref{thm: find guiding paths}, it is now enough to design a randomized algorithm, that either computes a distribution ${\mathcal{D}}'$ over pairs in $\Lambda(H',\tilde T)$ with $\expect[({\mathcal{Q}}',x')\in {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e))^2}\leq O\textsf{left} (\frac{\log^8n}{(\alpha\alpha')^2}\textsf{right} )$, or returns FAIL. It is enough to ensure that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')\leq \frac{(\tilde k\hat \alpha'\alpha')^2}{c_1\eta'\log^{20}n}$, then the probability that the algorithm returns FAIL is at most $1/2$.
In the remainder of the proof we focus on the above goal. It would be convenient for us to simplify the notation, by denoting $H'$ by $H$, $\hat H''$ by $\hat H$, and $\hat \alpha'$ by $\tilde \alpha$. We also denote ${\mathcal{C}}'$ by ${\mathcal{C}}$. We now summarize all properties of the new graphs $H,\hat H$ that we have established so far, and in the remainder of the proof of \Cref{thm: find guiding paths} we will only work with these new graphs.
\paragraph{Summary of the outcome of Step 1.}
We can assume from now on that we are given a graph $H$, a rotation system $\Sigma$ for $H$, a set $\tilde T$ of terminals in graph $H$, and a collection ${\mathcal{C}}$ of disjoint subgraphs (clusters) of $H\setminus \tilde T$. We denote $|\tilde T|=\tilde k$. The corresponding contracted graph is denoted by $\hat H=H_{|{\mathcal{C}}}$.
We are also given a partition $(X,Y)$ of $V(H)\setminus \tilde T$ (note that for convenience of notation, $X$ and $Y$ are now subsets of vertices of $H$, and not of $\hat H$), and a parition ${\mathcal{C}}_X,{\mathcal{C}}_Y$ of ${\mathcal{C}}$, such that each cluster $C\in {\mathcal{C}}_X$ has $V(C)\subseteq X$, and each cluster $C\in {\mathcal{C}}_Y$ has $V(C)\subseteq Y$. We denote $H_1=H[X\cup \tilde T]$ and $H_2=H[Y\cup \tilde T]$. We also denote by $\hat H_1=(H_1)_{|{\mathcal{C}}_X}$ the contracted graph of $H_1$ with respect to ${\mathcal{C}}_X$, and similarly by $\hat H_2=(H_2)_{|{\mathcal{C}}_Y}$ the contracted graph of $H_2$ with respect to ${\mathcal{C}}_Y$.
We now summarize the properties of the graphs that we have defined and the relationships between the main parameters.
\begin{properties}{P}
\item $\tilde k\geq \Omega(\alpha^3k/\log^8n)$;\label{prop after step 1: number of pseudoterminals}
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property; \label{prop after step 1: bandwidth property}
\item $|E(\hat H)|\leq O(\tilde k\cdot \eta \log^8n/\alpha^3)$ (from \Cref{obs: properties of new graph}); \label{prop after step 1: few edges}
\item every vertex of $\tilde T$ has degree $1$ in $H_1$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_1$, for $\tilde \alpha=\Theta(\alpha/\log^4n)$;\label{prop after step 1: terminals in H1}
\item similarly, every vertex of $\tilde T$ has degree $1$ in $H_2$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_2$; and \label{prop after step 1: terminals in H2}
\item $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}
$, where $c_1$ is some large enough constant, whose value we can set later. \label{prop after step 1: small squares of boundaries}
\end{properties}
Our goal is
to design an efficient randomized algorithm, that either computes a distribution ${\mathcal{D}}$ over pairs in $\Lambda(H,\tilde T)$ with $\expect[({\mathcal{Q}},x)\in {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{16}n}{(\alpha\alpha')^4}\textsf{right} )$ for every edge $e\in E(H)$, or returns FAIL.
It is enough to ensure that, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$,
then the probability that the algorithm returns FAIL is at most $1/2$.
\iffalse
\subsection{Step 2: a Modified Graph}
In the remainder of the proof, it would be convenient for us to work with a low-degree equivalent of the graph $\hat H$, that we denote by $H^+$. In order to define the graph $\hat H^+$, we start by defining a rotation system $\hat \Sigma $ for graph $\hat H$, as follows. Let $x\in V(\hat H)$ be a vertex. If $x$ is a supernode, that is, $x=v(C)$ for some cluster $C\in {\mathcal{C}}$, then we define the circular ordering $\hat{\mathcal{O}}_x\in \hat \Sigma$ of the edges of $\delta_{\hat H}(x)$ to be arbitrary.
Otherwise, $x$ is a regular vertex, and it is a vertex of the original graph $H$. We then let $\hat {\mathcal{O}}_x\in \hat \Sigma$ be identical to the ordering ${\mathcal{O}}_x\in \Sigma$, where $\Sigma$ is the original rotation system for graph $H$.
We are now ready to define the modified graph $H^+$. We start with $H^+=\emptyset$, and then process every vertex $u\in V(\hat H)$ one-by-one. We denote by $d(u)$ the degree of the vertex $u$ in graph $\hat H$. We now describe an iteration when a vertex $u\in V(\hat H)$ is processed. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that incident to $u$ in $\hat H$, indexed according to their ordering in $\hat {\mathcal{O}}_u\in \hat \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H^+$. We refer to the edges in the grids $\Pi(u)$ as \emph{inner edges}.
Once every vertex $u\in V(\hat H)$ is processed, we add a collection of \emph{outer edges} to graph $H^+$, as follows. Consider any edge $e=(x,y)\in E(H)$. Assume that $e$ is the $i$th edge of $x$ and the $j$th edge of $y$, that is, $e=e_i(x)=e_j(y)$. Then we add an edge $e'=(s_i(x),s_j(y))$ to graph $H^+$, and we view this edge as the copy of the edge $e\in E(\hat H)$. We will not distinguish between the edge $e$ of $\hat H$ (and the corresponding edge of $H$), and the edge $e'$ of $H^+$.
We note that every terminal $t\in T$ has degree $1$ in $\hat H$, so its corresponding grid $\Pi(t)$ consists of a single vertex, that we also denote by $t$. Therefore, set $T$ of terminals in $H$ naturally corresponds to a set of $k$ terminals in $H^+$, that we denote by $T$ as before.
This completes the definition of the graph $H^+$. Note that the maximum vertex degree in $H^+$ is $4$. We also define a rotation system $\Sigma^+$ for the graph $H^+$ in a natural way: for every vertex $u\in V(H)$,
consider the standard drawing of the grid $\Pi(u)$, to which we add the drawings of the edges that are incident to vertices $s_1(u),\ldots,s_{d(u)}(u)$, so that the edges are drawn on the grid's exterior in a natural way (\mynote{add figure}). This layout defines an ordering ${\mathcal{O}}^+(v)$ of the edges incident to every vertex $v\in \Pi(u)$.
We start with the following simple claim.
\begin{claim}\label{claim: cheap solution to modified instance} There is a solution to the \textnormal{\textsf{MCNwRS}}\xspace problem instance $(H^+,\Sigma^+)$ of cost at most ..., such that no inner edge of $H^+$ participates in any crossings in the solution.
\end{claim}
\begin{proof}
We start by showing that $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat H,\hat \Sigma)\leq ...$. Recall that, from \Cref{lem: crossings in contr graph}, there is a drawing $\hat \phi$ of graph $\hat H$ with at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2$ crossings, such that for every vertex $x\in V(\hat H)\cap V(H)$, the ordering of the edges of $\delta_{\hat H}(x)$ as they enter $x$ in $\hat \phi$ is consistent with the ordering ${\mathcal{O}}_x\in \Sigma$, and hence with the ordering $\hat {\mathcal{O}}_x\in \hat \Sigma$. Drawing $\hat \phi$ of $\hat H$ may not be a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace since for some supernodes $v(C)$, the ordering of the edges that are incident to $v(C)$ in $\hat H$ as they enter the image of $v(C)$ in $\hat \phi$ may be different from $\hat {\mathcal{O}}_{v(C)}$. For each such vertex $v(C)$, we may need to \emph{reorder} the images of the edges of $\delta_{\hat H}(v(C))=\delta_H(C)$ near the image of $v(C)$, so that they enter the image of $v(C)$ in the correct order. This can be done by introducing at most $|\delta_H(C)|^2$ crossings for each such supernode $v(C)$. The resulting drawing $\hat \phi'$ of $\hat \phi$ is a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, whose cost is bounded by:
\[O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2+\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8n/(\alpha')^2+ \]
\end{proof}
\fi
\iffalse
\subsection{Step 2: Routing the Terminals to a Single Vertex, and the Epanded Graph}
\mynote{need to redo this: the paths set should be in the contracted graph $\hat H_1$}
In this step we start by considering the graph $H_1$ and the set $\tilde T$ of terminals in it. Our goal is to compute a collection ${\mathcal{J}}$ of paths in graph $H_1$, routing all terminals of $\tilde T$ to a single vertex, such that the paths in ${\mathcal{J}}$ cause a relatively low congestion in graph $H_1$. We show that, if such a collection of paths does not exist, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)$ is high. Intuitively, we will use the set ${\mathcal{J}}$ of paths in order to define an ordering of the terminals in $\tilde T$, which will in turn be used in order to compute a grid-like structure in graph $H_2$. Once we compute the desired set ${\mathcal{J}}$ of paths, we will replace the graph $H$ with its low-degree analogue $H^+$, that we refer to as the \emph{expanded graph}. The remaining steps in the proof of \Cref{thm: find guiding paths} will use this expanded graph only.
We now proceed to describe the algorithm for Step 2. We consider every \emph{regular} vertex (that is, a vertex that is not a supernode) $x\in V(\hat H_1)$ one by one. For each such vertex $x$, we compute a set ${\mathcal{J}}(x)$ of paths in graph $\hat H_1$, with the following properties:
\begin{itemize}
\item every path in ${\mathcal{J}}(x)$ originates at a distinct vertex of $\tilde T$ and terminates at $x$;
\item the paths in ${\mathcal{J}}(x)$ are edge-disjoint; and
\item ${\mathcal{J}}(x)$ is a maximum-cardinality set of paths in $H_1$ with the above two properties.
\end{itemize}
Note that such a set ${\mathcal{J}}(x)$ of paths can be computed via a standard maximum $s$-$t$ flow computation.
Throughout, we use a parameter $\tilde k'=\ceil{\tilde k\alpha^5/(c'\eta\log^{36}n)}$, where $c'$ is a large enough constant whose value we set later.
If, for every regular vertex $x\in V(\hat H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then we reurn FAIL and terminate the algorithm. In the following lemma, whose proof is deferred to Section \ref{sec: few paths high opt} of Appendix we show that, in this case, $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$ must hold.
Note that, since we can set $c_1$ to be a large enough constant, we can ensure that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$ holds in this case.
The value of the constant $c'$ that is used in the definition of the parameter $\tilde k'$ is set in the proof of the lemma.
\begin{lemma}\label{lem: high opt or lots of paths}
If, for every regular vertex $x\in V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$.
\end{lemma}
\mynote{need to redo from here}
From now on we assume that there is some vertex $x\in V(H_1)$, for which $|{\mathcal{J}}(x)|\geq \tilde k'$. We denote ${\mathcal{J}}_0={\mathcal{J}}(x)$, and we let $T_0\subseteq \tilde T$ be the set of terminals that serve as endpoints of paths in ${\mathcal{J}}_0$, so $|T_0|=|{\mathcal{J}}_0|=\tilde k'$.
Let $z=\ceil{\tilde k/\tilde k'}=O\textsf{left}(\frac{\eta\log^{36}n}{\alpha^5\alpha'}\textsf{right} )$. Next, we arbitrarily partition the terminals of $\tilde T\setminus T_0$ into $z$ subsets $T_1,\ldots,T_z$, of cardinality at most $\tilde k'$ each. Consider now some index $1\leq i\leq z$. Since the set $\tilde T$ of terminals is $(\tilde \alpha\alpha')$-well-linked in $H_1$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{J}}_i'$ of paths in graph $H_1$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in $T_i$ cause edge-congestion $O(1/(\tilde \alpha\alpha'))$, and each vertex of $T_i$ is the endpoint of at most one path in ${\mathcal{J}}_i'$. By concatenating the paths in ${\mathcal{J}}_i'$ with paths in ${\mathcal{J}}_0$, we obtain a collection ${\mathcal{J}}_i$ of paths in graph $H_1$, connecting every terminal of $T_i$ to $x$, that cause edge-congestion $O(1/(\tilde \alpha\alpha'))$. Let ${\mathcal{J}}=\bigcup_{i=0}^z{\mathcal{J}}_i$ be the resulting set of paths. Observe that set ${\mathcal{J}}$ contains $\tilde k$ paths, routing the terminals in $\tilde T$ to the vertex $x$ in graph $H_1$, with $\cong_H({\mathcal{J}})\leq O\textsf{left}(\frac z{\tilde \alpha\alpha'}\textsf{right} )\leq O\textsf{left}(\frac{\eta\log^{40}n}{\alpha^6(\alpha')^2}\textsf{right} )$. We denote by $\rho=O\textsf{left}(\frac{\eta\log^{40}n}{\alpha^6(\alpha')^2}\textsf{right} )$ this bound on $\cong_{H_1}({\mathcal{P}}')$. We assume w.l.o.g. that the paths in ${\mathcal{J}}$ are simple. Since every terminal in $\tilde T$ has degree $1$ in $H_1$, no path in ${\mathcal{J}}$ may contain a terminal in $\tilde T$ as its inner vertex.
In the remainder of the proof, it would be convenient for us to work with a low-degree equivalent of the graph $H$, that we call \emph{expanded graph}, and denote by $H^+$.
For every edge $e\in E(H)$, let $N_e=\cong_H({\mathcal{J}},e)$. Recall that for every edge $e\in E(H_1\setminus\tilde T)$, $N_e\leq \rho$, and for every other edge $e$ of $H$, $N_e\leq 1$.
\fi
\subsection{Step 2: Routing the Terminals to a Single Vertex, and an Expanded Graph}
In this step we start by considering the graph $\hat H_1$ and the set $\tilde T$ of terminals in it. Our goal is to compute a collection ${\mathcal{J}}$ of paths in graph $\hat H_1$, routing all terminals of $\tilde T$ to a single regular vertex, such that the paths in ${\mathcal{J}}$ cause a relatively low congestion in graph $\hat H_1$. We show that, if such a collection of paths does not exist, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)$ is high. Intuitively, we will use the set ${\mathcal{J}}$ of paths in order to define an ordering of the terminals in $\tilde T$, which will in turn be used in order to compute a grid-like structure in graph $H_2$. Once we compute the desired set ${\mathcal{J}}$ of paths, we will replace the graph $H$ with its low-degree analogue $H^+$, that we refer to as the \emph{expanded graph}. The remaining steps in the proof of \Cref{thm: find guiding paths} will use this expanded graph only.
\subsubsection{Routing the Terminals to a Single Vertex}
We process {\bf regular} vertices of $V(\hat H_1)$ (that is, vertices of $V(\hat H_1)\cap V(H_1)$) one-by-one. For each such vertex $x$, we compute a set ${\mathcal{J}}(x)$ of paths in graph $\hat H_1$, with the following properties:
\begin{itemize}
\item every path in ${\mathcal{J}}(x)$ originates at a distinct vertex of $\tilde T$ and terminates at $x$;
\item the paths in ${\mathcal{J}}(x)$ are edge-disjoint; and
\item ${\mathcal{J}}(x)$ is a maximum-cardinality set of paths in $\hat H_1$ with the above two properties.
\end{itemize}
Note that such a set ${\mathcal{J}}(x)$ of paths can be computed via a standard maximum $s$-$t$ flow computation.
Throughout, we use a parameter $\tilde k'=\tilde k\alpha^5/(c'\eta\log^{36}n)$, where $c'$ is a large enough constant whose value we set later.
If, for every vertex $x\in V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then we reurn FAIL and terminate the algorithm. In the following lemma, whose proof is deferred to Section \ref{sec: few paths high opt} of Appendix we show that, in this case, $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$ must hold.
Note that, since we can set $c_1$ to be a large enough constant, we can ensure that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$ holds in this case.
The value of the constant $c'$ that is used in the definition of the parameter $\tilde k'$ is set in the proof of the lemma.
\begin{lemma}\label{lem: high opt or lots of paths}
If, for every vertex $x\in V(\hat H_1)\cap V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}n}\textsf{right} )$.
\end{lemma}
From now on we assume that there is some vertex $x\in V(\hat H_1)\cap V(H_1)$, for which $|{\mathcal{J}}(x)|\geq \tilde k'$.
\subsubsection{The Expanded Graph}
From now on we fix the vertex $x\in V(\hat H_1)\cap V(H_1)$, and we let ${\mathcal{J}}={\mathcal{J}}(x)$ be the set of at least $\tilde k'$ paths in graph $\hat H_1$, routing a subset $\tilde T_0\subseteq \tilde T$ of terminals to vertex $x$.
We are now ready to define the modified graph $H^+$. We start with $H^+=\emptyset$, and then process every vertex $u\in V(H_2)\setminus \tilde T$ one by one. We now describe an iteration when a vertex $u\in V(H_2)\setminus \tilde T$ is processed. We denote by $d(u)$ the degree of the vertex $u$ in graph $H_2$. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that are incident to $u$ in $H_2$, indexed according to their ordering in $ {\mathcal{O}}_u\in \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$ indexed in their natural left-to-right order.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H^+$. We refer to the edges in the resulting grids $\Pi(u)$ as \emph{inner edges}.
Once every vertex $u\in V(H_2)\setminus \tilde T$ is processed, we add
the vertices of $\tilde T$ to the graph $H^+$. Recall that every terminal $t\in \tilde T$ has degree $1$ in $H_2$. We denote the unique edge $e_t$ incident to $t$ by $e_1(t)$, and we denote $s_1(t)=t$.
Next, we add
a collection of \emph{outer edges} to graph $H^+$, as follows. Consider any edge $e=(u,v)\in E(H_2)$. Assume that $e$ is the $i$th edge of $u$ and the $j$th edge of $v$, that is, $e=e_i(u)=e_j(v)$. Then we add an edge $e'=(s_i(u),s_j(v))$ to graph $H^+$, and we view this edge as the \emph{copy of the edge $e\in E(H_2)$}. We will not distinguish between the edge $e$ of $H_2$, and the edge $e'$ of $H^+$.
Our last step is to add vertex $x$ to graph $H^+$, that connects to every terminal $t\in \tilde T$ with an edge $(x,t)$, that is also viewed as an outer edge. The following lemma, whose proof is deferred to Section \ref{sec:ordering of terminals} of Appendix, allows us to compute an ordering $\tilde {\mathcal{O}}$ of the terminals, such that the graph $H^+$ has a drawing $\phi$ with few crossings, in which the inner edges do not participate in any crossings, and the images of edges incident to $x$ enter $x$ in order consistent with $\tilde {\mathcal{O}}$.
\iffalse
Our last step is to add a $(\tilde k\times\tilde k)$-grid $\Pi(x)$ to graph $H^+$ corresponding to the vertex $x$. As before, the edges of the grid are inner edges for graph $H^+$, and we denote the vertices on the first row of the grid by $s_1(x),\ldots,s_{\tilde k}(x)$, indexed in their natural left-to-right order. For all $1\leq i\leq \tilde k$, we add an outer edge $(t_i,s_i(x))$ to graph $H^+$. This completes the definition of the graph $H^+$, provided that we are given an ordering $\tilde {\mathcal{O}}$ of the terminals. The following lemma, whose proof is deferred to Section \ref{sec:ordering of terminals} of the Appendix, allows us to compute an ordering $\tilde {\mathcal{O}}$ of the terminals, such that the resulting graph $H^+$ has a drawing $\phi$ with few crossings, in which the inner edges do not participate in any crossings.
\fi
\begin{lemma}\label{lem: find ordering of terminals}
There is an efficient algorithm to compute an ordering $\tilde {\mathcal{O}}$ of the terminals in $\tilde T$, such that there is a drawing $\phi$ of graph $H^+$ with at most $O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot\frac{\eta^2\log^{74}n}{\alpha^{12}(\alpha')^4}\textsf{right} ) +O \textsf{left} (
\frac{\tilde k \eta\log^{37}n}{\alpha^6(\alpha')^2}\textsf{right} )$ crossings, in which all crossings in $\phi$ are between pairs of outer edges. Moreover, if we denote $\tilde T=\set{t_1,\ldots,t_{\tilde k}}$, where the terminals are indexed according to the ordering $\tilde {\mathcal{O}}$, and, for each $1\leq i\leq t_{\tilde k}$, denote by $e_i=(t_i,x)$ the edge of $H^+$ connecting $t_i$ to $x$, then the images of the edges $e_1,\ldots,e_{\tilde k}$ enter the image of $x$ in this circular order in the drawing $\phi$.
\end{lemma}
From now on we fix the ordering $\tilde {\mathcal{O}}$ of the terminals in $\tilde T$ given by \Cref{lem: find ordering of terminals}, and the drawing $\phi$ of $H^+$ (which is not known to us).
It will be convenient for us to slightly modify the graph $H^+$ as follows. We denote
the terminals by $\tilde T=\set{t_1,\ldots,t_{\tilde k}}$, where the terminals are indexed according to the circular ordering $\tilde {\mathcal{O}}$. Let $H'$ be a graph obtained from $H^+$, by first deleting the vertex $x$ from it, and then adding, for all $1\leq i<\tilde k$, an edge $e^*_i=(t_i,t_{i+1})$, and another edge $e^*_{\tilde k}=(t_{\tilde k},t_1)$. We denote this set of the newly added edges by $E^*$, and we view them as inner edges. Note that the edges of $E^*$ form a simple cycle, that we denote by $L^*$. We also denote $H''=H'\setminus E^*$.
We note that the drawing $\phi$ of $H^+$ can be easily extended to obtain the drawing of $H'$ in the plane, so that the inner edges of $H'$ do not participate in any crossings, and the image of the cycle $L^*$ (which must be a simple closed curve) is the boundary of the outer face.
In order to do so, we start with the drawing $\phi$ of $H^+$ on the sphere, and then draw a small disc $D$ with $x$ lying in its interior, and denote by $\eta$ its boundary.
For every terminal $t_i\in \tilde T$, we denote by $e_i$ the unique edge incident to $t_i$ in $H''$, and by $e'_i=(t_i,x)$. We also denote by $\gamma_i,\gamma'_i$ the images of the edges $e_i,e'_i$ in the current drawing. Let $p_i$ be the unique point on the intersection of $\gamma'_i$ and $\eta$. We move the image of terminal $t_i$ to point $p_i$. We then modify the image of the edge $e_i$, so that it becomes a concatenation of $\gamma_i$, and the portion of $\gamma'_i$ lying outside the interior of $D$. Lastly, we draw the edges of $E^*$ in a natural way, where edge $e^*_i$ is simply a segment of $\eta$ between the images of $t_i$ and $t_{i+1}$, so that all resulting segments are mutually internally disjoint. Once we delete the vertex $x$ from this drawing, no part of the resulting drawing is contained in the interior of the disc $D$, and the image of the cycle $L^*$ is precisely $\eta$, so we can view the resuting drawing as a drawing in the plane, with $D$ being the outer face. Note that this transformation does not increase the number of crossings.
The following observation follows by substituting parameters and bounds that we have already established. The proof is included in Section \ref{subsec: proof of obs on bounds on opt} of Appendix.
\begin{observation}\label{obs: bounds on opt}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}n}$, then $\mathsf{cr}(\phi)\leq
\mathsf{cr}(\phi)\leq \frac{\tilde k^2}{c_2\eta^5}$, for a large enough constant $c_2$.
\end{observation}
It is therefore enough for us to ensure that, if $\mathsf{cr}(\phi)<
\frac{\tilde k^2}{\eta^5}$, then the probability that the algorithm returns FAIL is at most $1/2$.
Let $\Lambda'$ denote all pairs $({\mathcal{Q}},y)$, where $y$ is a vertex in graph $H_2$, and ${\mathcal{Q}}$ is a set of paths in graph $H''$, routing the set $\tilde T$ of terminals to vertices of $\Pi(y)$.
We also need the following simple observation, whose proof is deferred to Section \ref{subsec: transform paths 2} of the Appendix.
\begin{observation}\label{obs: transform paths 2}
There is an efficient algorithm, that, given a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},y)\in \Lambda'$, such that for every {\bf outer} edge $e\in E(H'')$, $\expect[({\mathcal{Q}},y)\in_{{\mathcal{D}}}\Lambda']{(\cong_{H''}({\mathcal{Q}},e))^2}\leq \beta$, computes a distribution ${\mathcal{D}}'$ over pairs in $\Lambda(H,\tilde T)$, where for every edge $e\in E(H)$, $\expect[({\mathcal{Q}}',y)\in_{{\mathcal{D}}'} \Lambda(H,\tilde T)]{(\cong_{H}({\mathcal{Q}}',e))^2}\leq \beta$.
\end{observation}
\subsubsection{Summary of Step 2}
\label{step 2 summary}
In the remainder of the proof of \Cref{thm: find guiding paths} we will work with graph $H'$ only. Recall that graph $H'$ contains a set $E^*=\set{e_1,\ldots,e_{\tilde k}}$ of edges (that are considered to be inner edges), where for all $1\leq i\leq \tilde k$, $e_i=(t_i,t_{i+1})$ (we use indexing modulo $\tilde k$). The set $E^*$ of edges defines a cycle $L^*=(t_1,\ldots,t_{\tilde k})$ in graph $H'$. We also denoted $H''=H'\setminus E^*$. Recall that graph $H''$ is obtained from a subgraph $H_2\subseteq H$, by replacing every vertex $v\in V(H_2)\setminus \tilde T$ with a grid $\Pi(v)$. All edges lying in the resulting rids $\Pi(v)$, and the edges of $E^*$ are inner edges, while all other edges of $H'$ are outer edges. Each outer edge of $H'$ corresponds to some edge of graph $H_2$, and we do not distinguish between these edges. Note that in graph $H'$, all vertices have degrees at most $4$. We will also use the clustering ${\mathcal{C}}_Y$ of graph $H_2$, and the fact that, from Property \ref{prop after step 1: small squares of boundaries}:
\begin{equation}\label{eq: sum of squares}
\sum_{C\in {\mathcal{C}}_Y}|\delta_H(C)|^2<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}. \end{equation}
We further partition the outer edges of graph $H''$ into two subsets: type-1 outer edges and type-2 outer edges. Consider any outer edge $e$ in graph $H''$, and let $e'=(u,v)$ be the corresponding edge in graph $H$. If $u$ and $v$ both lie in the same cluster $C\in {\mathcal{C}}_Y$, then we say that $e$ is a \emph{type-2} outer edge, and otherwise it is a type-1 outer edge.
Intuitively, for each type-1 outer edge, there is a corresponding edge in the contracted graph $\hat H$. From Property \ref{prop after step 1: few edges}, we obtain the following observation.
\begin{observation}\label{obs: few outer edges}
There is a universal constant $c$ (independent of $c_1$ and $c_2$), such that the total number of type-1 outer edges in $H''$ is bounded by ${c\tilde k\cdot \eta \log^8n/\alpha^3}$.
\end{observation}
\iffalse
======================
\highlightf[purple]{Calculations:}
final number of paths: at least $\alpha^*\tilde k/256$. Supergrid dimensions: $\lambda\times \lambda$. Want: at most $\lambda^2/32$ cells have an outer edge on each path. Number of paths per cell: $\alpha^*\tilde k/(256\lambda)$. So we need that:
\[\frac{\alpha^*\tilde k}{256\lambda}\cdot \frac{\lambda^2}{32}>\frac{c\tilde k\cdot \eta \log^8k}{\alpha^3}. \]
To ensure this, enough to set:
\[\lambda=\frac{2^{16}c\cdot \eta \log^8n}{\alpha^*\alpha^3}. \]
(and this is tight to within constants).
Because $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, we get that $\lambda=O\textsf{left}( \frac{\eta \log^{12}n}{\alpha^4\alpha'} \textsf{right} )$.
If we ensure that $\highlightf{\eta>\frac{c^* \log^{12}n}{\alpha^4\alpha'}}$, then $\lambda<\eta^2$ will hold.
Group size: start with:
\[\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}=\floor{\frac{\alpha^3(\alpha^*)^2\tilde k}{2^{32}c\eta\log^8n}}.\]
This is $\Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^8n} \textsf{right})$. If $\highlightf{\eta>\frac{c^* \log^{8}n}{\alpha^5(\alpha')^2}}$, we get that $\psi>\frac{16\tilde k}{\eta^2}$.
Overall, for this part, we need $\highlightf{\eta>\frac{c^* \log^{12}n}{\alpha^5(\alpha')^2}}$, which so far is ensured.
======================
\fi
Recall that from Property \ref{prop after step 1: terminals in H1}, every vertex of $\tilde T$ has $1$ in $H_2$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_2$. Combining this with the $\alpha'$-bandwidth property of every cluster $C\in {\mathcal{C}}_Y$ from Property \ref{prop after step 1: bandwidth property}, from \Cref{clm: contracted_graph_well_linkedness}, the set $\tilde T$ of terminals is $\tilde \alpha\cdot \alpha'$-well-linked in $H_2$. Lastly, using the fact that each graph in $\set{\Pi(v)\mid v\in V(H_2)}$ has the $1$-bandwidth property, from \Cref{clm: contracted_graph_well_linkedness}, we get the following observation.
\begin{observation}\label{obs: terminals well linked in H''}
The set $\tilde T$ of terminals is $\alpha^*$-well-linked in $H''$, where ${\alpha^*=\tilde \alpha\cdot\alpha'=\Theta(\alpha\alpha'/\log^4n)}$. Moreover, each terminal in $\tilde T$ has degree $1$ in $H''$ and degree $3$ in $H'$.
\end{observation}
(we have used the fact that
$\tilde \alpha=\Theta(\alpha/\log^4n)$ (see Property \ref{prop after step 1: terminals in H1})).
We will restrict our attention to special types of drawings of graph $H'$, called \emph{legal drawings}, that we define next.
\begin{definition}[Legal drawing of $H'$]
We say that a drawing of graph $H'$ in the plane is \emph{legal} iff it has the following properties:
\begin{itemize}
\item no inner edge of $H'$ participates in any crossing of $\phi^*$, and in particular the image of the cycle $L^*$ is a simple closed curve, denoted by $\eta$; and
\item $\eta$ is the boundary of the outer face in the drawing.
\end{itemize}
\end{definition}
We let $\phi^*$ be a legal drawing of $H'$ with smallest number of crossings, and we denote by $\mathsf{cr}^*$ the number of crossings in $\phi^*$.
Denote $\tilde T=\set{t_1,\ldots,t_{\tilde k}}$, where the terminals are indexed according their ordering in $\tilde {\mathcal{O}}$. We partition the set $\tilde T$ of terminals into four subsets $T_1,\ldots,T_4$, where $T_1,T_2,T_3$ contain $\floor{\tilde k/4}$ consecutive terminals from $\tilde T$ each, and $T_4$ contains the remaining terminals, in a natural way using the ordering $\tilde {\mathcal{O}}$, that is, $T_1=\set{t_1,\ldots,t_{\floor{\tilde k/4}}}$, $T_2=\set{t_{\floor{\tilde k/4}+1},\ldots,t_{2\floor{\tilde k/4}}}$, $T_3=\set{t_{2\floor{\tilde k/4}+1},\ldots,t_{3\floor{\tilde k/4}}}$, and $T_4=\set{t_{3\floor{\tilde k/4}+1},\ldots,t_{\tilde k}}$. Clearly, each of the four sets contains at least $\floor{\tilde k/4}$ terminals.
Recall that in a legal drawing $\phi$ of $H'$, the image of the cycle $L^*$ is a simple closed curve, that we denoted by $\eta$. It will be convenient for us to view this curve $\eta$ as the boundary of a rectangular area in the plane, that encloses the legal drawing of $H'$. We sometimes refer to this rectangular area as the \emph{bounding box} of the drawing, and denote it by $B^*$. We will think of the terminals in $T_1$ and $T_3$ as appearing on the left and on the right boundaries of $B^*$, respectively, and of the terminals in $T_2$ and $T_4$ as appearing on the top and the bottom boundaries of $B^*$, respectively.
For all $1\leq i\leq 4$, we let $\tilde {\mathcal{O}}_i$ be the ordering of the terminals in $T_i$ consistent with their ordering on the boundary of $B^*$ (where each ordering $\tilde{\mathcal{O}}_i$ is no longer circular), so that the terminals in sets $T_1$ and in $T_3$ are indexed in their bottom-to-top order, and the terminals in $T_2$ and $T_4$ are indexed in their left-to-right order (so $\tilde {\mathcal{O}}$ is obtained by concatenationg $\tilde{\mathcal{O}}_1,\tilde{\mathcal{O}}_2$, the reversed ordering $\tilde{\mathcal{O}}_3$, and the reversed ordering $\tilde{\mathcal{O}}_4$).
Recall that $\Lambda'$ is the set of all pairs $({\mathcal{Q}},y)$, where $y\in V(H_2)$, and ${\mathcal{Q}}$ is a collection of paths in graph $H''$, routing the set $\tilde T$ of terminals to the vertices of $\Pi(y)$.
Our goal is to design a randomized algorithm, that either computes a distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},y)\in \Lambda'$, such that for every outer edge $e\in E(H')$, $\expect[({\mathcal{Q}},y)\in_{{\mathcal{D}}} \Lambda']{(\cong_{H'}({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{16}n}{(\alpha\alpha')^4}\textsf{right} )$, or returns FAIL.
It is enough to ensure that, if $\mathsf{cr}^*\leq
\frac{\tilde k^2}{c_2\eta^5}$ for some large enough constant $c_2$, then the probability that the algorithm returns FAIL is at most $1/4$.
\subsection{Step 3: Constructing a Grid Skeleton}
In this and the following step we will construct a grid-like structure in graph $H''$. Recall that the set $\tilde T$ of terminals is $\alpha^*$-well-linked in grpah $H''$. From \Cref{thm: bandwidth_means_boundary_well_linked} there is a set ${\mathcal{P}}'$ of paths in $H''$, routing all terminals of $T_1$ to terminals of $T_3$, with edge-congestion at most $\ceil{1/\alpha^*}$, such that the routing is one-to-one. From \Cref{claim: remove congestion}, there is a collection ${\mathcal{P}}''$ of at least $|T_1|/\ceil{1/\alpha^*}=\floor{\tilde k/4}/\ceil{1/\alpha^*}\geq \alpha^*\tilde k/8$ edge-disjoint paths in $H''$, routing some subset of terminals of $T_1$ to a subset of terminal of $T_3$, in graph $H''$. Moreover, since graph $H''$ has maximum vertex degree at most $4$, using arguments similar to those in the proof of \Cref{claim: remove congestion}, there is a collection ${\mathcal{P}}$ of $\floor{\alpha^*\tilde k/32}$ {\bf node-disjoint} paths in graph $H''$, routing some subset $A\subseteq T_1$ of terminals, to some subset $A'\subseteq T_3$ of terminals. We can compute such a set ${\mathcal{P}}$ of paths efficiently using standard maximum $s$-$t$ flow algorithms.
Using similar reasoning, we can compute a collection ${\mathcal{R}}$ of $\floor{\alpha^*\tilde k/32}$ node-disjoint paths in graph $H''$, routing some subset $B\subseteq T_2$ of terminals, to some subset $B'\subseteq T_4$ of terminals.
Intuitively, after we discard a small subset of paths from each of the sets ${\mathcal{P}},{\mathcal{R}}$, the remaining paths will be used in order to construct a grid-like structure, where paths in ${\mathcal{P}}$ will serve as horizontal paths of the grid, and paths in ${\mathcal{R}}$ will serve as vertical paths. If the paths in the resulting sets do not form a grid-like structure, then we will terminate the algorithm with a FAIL. We will prove that, if $\mathsf{cr}^*\leq
\frac{\tilde k^2}{c_2\eta^5}$ for a large enough constant $c_2$, then we will construct the grid-like structure successfully with probability at least $3/4$.
We denote ${\mathcal{P}}_0={\mathcal{P}}$ and ${\mathcal{R}}_0={\mathcal{R}}$. Recall that so far, ${|{\mathcal{P}}_0|,|{\mathcal{R}}_0|\geq \floor{\alpha^*\tilde k/32}}$.
Intuitively, if the dimensions of the grid-like structure that we construct are $(h\times h)$, then we need $h$ to be quite close to $\tilde k$, since this grid-like structure will be exploited in order to define the distribution ${\mathcal{D}}$ over pairs in $\Lambda'$.
We will first construct a smaller grid-like structure, that we call a \emph{grid skeleton}. This grid skeleton will be associated with a grid $\Pi^*$ of smaller dimensions, that we sometimes call a \emph{supergrid}. We then extend this grid skeleton to construct a large enough grid-like structure.
We will use two additional parameters. The first parameter is:
\[\lambda=\frac{2^{24}c\cdot \eta \log^8n}{\alpha^*\alpha^3},\]
where $c$ is the constant from \Cref{obs: few outer edges}. Notice that, since $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, we get that $\lambda=O\textsf{left}( \frac{\eta \log^{12}n}{\alpha^4\alpha'} \textsf{right} )$. Moreover, since $\eta>\frac{c^* \log^{12}n}{\alpha^4\alpha'}$ for a large enough constant $c^*$ (from the statement of \Cref{thm: find guiding paths}), $\lambda<\eta^2$. The supergrid that we construct will have dimensions $(\Theta(\lambda)\times \Theta(\lambda))$. The second parameter is:
\[\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}=\floor{\frac{\alpha^3(\alpha^*)^2\tilde k}{2^{30}c\eta\log^8n}}.\]
Clearly, $|{\mathcal{R}}_0|,|{\mathcal{P}}_0|\geq \lambda\psi$.
Note that, since $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, $\psi\geq \Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^{16}n} \textsf{right})$. Since ${\eta>\frac{c^* \log^{16}n}{\alpha^5(\alpha')^2}}$ from the statement of \Cref{thm: find guiding paths}, we get that $\psi>\frac{16\tilde k}{\eta^2}$. Every cell of the supergrid will be associated with a collection of $\Theta(\psi)$ horizontal paths and $\Theta(\psi)$ vertical paths, that will help us form the grid-like structure.
We discard paths from ${\mathcal{P}}_0$ and from ${\mathcal{R}}_0$ arbitrarily, until $|{\mathcal{P}}_0|=|{\mathcal{R}}_0|=\lambda\psi$ holds.
We denote by $A_0\subseteq T_1,A'_0\subseteq T_3$ the endpoints of the paths in ${\mathcal{P}}_0$, and we denote by $B_0\subseteq T_4$, $B'_0\subseteq T_2$ the endpoints of the paths in ${\mathcal{R}}_0$.
\subsubsection*{Grid Skeleton Construction}
\iffalse
Let $c$ be a constant, such that the number of type-1 outer edges is at most $c\tilde k\cdot \eta \log^8k/\alpha^3$.
We say that a path $P\in {\mathcal{P}}_0\cup {\mathcal{R}}_0$ is \emph{long} if it contains at least $128c\eta\log^8k/(\alpha^*\alpha^3)$ type-1 outer edges, and otherwise we call it short.
Since the total number of type-1 outer edges is at most $c\tilde k\cdot \eta \log^8k/\alpha^3$, and the paths in ${\mathcal{P}}_0$ are edge-disjoint, at most $\alpha^*\tilde k/128$ paths in ${\mathcal{P}}_0$ may be long. We let ${\mathcal{P}}_1\subseteq {\mathcal{P}}_0$ be the set of all short paths, so $|{\mathcal{P}}_1|\geq \alpha^*\tilde k/128$. Similarly, we let ${\mathcal{R}}_1\subseteq {\mathcal{R}}_0$ be the set of all short paths, so $|{\mathcal{R}}_1|\geq \alpha^*\tilde k/128$. We denote by $A_1\subseteq A$, $A_1'\subseteq A'$ the sets of endpoints of the paths in ${\mathcal{P}}_1$, and we similarly denote by $B_1\subseteq B$, $B_1'\subseteq B'$ the sets of endpoints o fthe paths in ${\mathcal{R}}_1$.
\fi
We view the paths in ${\mathcal{P}}_0$ as directed from vertices of $A_0$ to vertices of $A'_0$. Recall that $A_0\subseteq T_1$, so the ordering $\tilde{\mathcal{O}}_1$ of the terminals in $T_1$ defines an ordering ${\mathcal{O}}_{A_0}=\set{a_1,\ldots,a_{\lambda\psi}}$ of the terminals in $A_0$. This ordering in turn defines an ordering ${\mathcal{O}}_{{\mathcal{P}}_0}$ of the paths in ${\mathcal{P}}_0$, as follows: if, for all $1\leq i\leq \lambda\psi$, $P_i\in {\mathcal{P}}_0$ is the path originating from $a_i$, then ${\mathcal{O}}_{{\mathcal{P}}_0}=\set{P_1,\ldots,P_{\lambda\psi}}$.
Similarly, we view the paths in ${\mathcal{R}}_0$ as directed from vertices of $B_0$ to vertices of $B_0'$. Ordering $\tilde{\mathcal{O}}_4$ of terminals in $T_4$ defines an ordering ${\mathcal{O}}_{B_0}=\set{b_1,\ldots,b_{\lambda\psi}}$ of the vertices in $B_0$, which in turn defines an ordering ${\mathcal{O}}_{{\mathcal{R}}_0}=\set{R_1,\ldots,R_{\lambda\psi}}$ of paths in ${\mathcal{R}}_0$, where for all $i$, path $R_i$ originates at vertex $b_i$.
We partition the set ${\mathcal{P}}_0$ of paths into groups ${\mathcal U}_1,\ldots,{\mathcal U}_{\lambda}$ of cardinality $\psi$ each, using the ordering ${\mathcal{O}}_{{\mathcal{P}}_0}$, so for $1\leq i<\lambda$, set ${\mathcal U}_i$ is the $i$th set of $\psi$ consecutive paths of ${\mathcal{P}}_0$.
Let $\lambda'= \floor{(\lambda-1)/2}$.
For all $1\leq i\leq \lambda'$, we let $ P^*_i$ be a path that is chosen uniformly at random from set ${\mathcal U}_{2i}$. Let ${\mathcal{P}}^*=\set{ P^*_1,\ldots, P^*_{\lambda'}}$ be the resulting set of chosen paths. Intuitively, the path in ${\mathcal{P}}^*$ will serve as the horizontal paths in the grid skeleton that we construct.
We then let ${\mathcal{P}}_1\subseteq {\mathcal{P}}_0$ be the set containing all paths in sets $\set{{\mathcal U}_{2i-1}}_{i=1}^{\lambda'+1}$.
We perform similar computation on the set ${\mathcal{R}}_0$ of paths. First, we partition ${\mathcal{R}}_0$ into groups ${\mathcal U}'_1,\ldots,{\mathcal U}'_{\lambda}$ of cardinality $\psi$ each, using the ordering ${\mathcal{O}}_{{\mathcal{R}}_0}$, so for $1\leq i<\lambda$, set ${\mathcal U}'_i$ is the $i$th set of $\psi$ consecutive paths of ${\mathcal{R}}_0$.
For all $1\leq i\leq \lambda'$, we let $ R^*_i$ be a path that is chosen uniformly at random from set ${\mathcal U}_{2i}$. Let ${\mathcal{R}}^*=\set{ R^*_1,\ldots, R^*_{\lambda'}}$ be the resulting set of chosen paths. Intuitively, the path in ${\mathcal{R}}^*$ will serve as the vertical paths in the grid skeleton that we construct.
We then let ${\mathcal{R}}_1\subseteq {\mathcal{R}}_0$ be the set containing all paths in sets $\set{{\mathcal U}'_{2i-1}}_{i=1}^{\lambda'+1}$.
We let ${\cal{E}}_1$ be the bad event that there are two distinct paths $Q,Q'\in {\mathcal{R}}^*\cup {\mathcal{P}}^*$, and two distinct edges $e\in E(Q)$, $e'\in E(Q')$, such that the images of $e$ and $e'$ cross in the drawing $\phi^*$ of $H'$.
\begin{observation}\label{obs: first bad event}
If $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then $\prob{{\cal{E}}_1}\leq 1/64$.
\end{observation}
\begin{proof}
Consider any crossing $(e,e')$ in the drawing $\phi^*$. We say that crossing $(e,e')$ is \emph{selected} iff there are two distinct paths $Q,Q'\in {\mathcal{R}}^*\cup {\mathcal{P}}^*$ with $e\in E(Q)$, $e'\in E(Q')$. Notice that $e$ may belong to at most two paths in ${\mathcal{R}}_0\cup {\mathcal{P}}_0$ (one path in each set), and the same is true for $e'$. Each path of ${\mathcal{R}}_0\cup{\mathcal{P}}_0$ is chosen to ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ with probability at most $1/\psi$. Therefore, the probability that a path containing $e$, and a path containing $e'$ are chosen to ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ is at most $4/\psi^2$. Since ${\cal{E}}_1$ can only happen if at least one crossing is chosen, from the union bound,
$\prob{{\cal{E}}_1}\leq 4\mathsf{cr}^*/\psi^2$. Since $\psi>\frac{16\tilde k}{\eta^2}$, if $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then:
\[\prob{{\cal{E}}_1}\leq \frac{4\mathsf{cr}^*}{\psi^2}\leq \frac{1}{64c_2\eta}\leq \frac{1}{64}.\]
\end{proof}
We say that a path $Q\in {\mathcal{R}}_0\cup {\mathcal{P}}_0$ is \emph{heavy} iff there are at least $\frac{\psi}{64\lambda}$ crossings $(e,e')$ in $\phi^*$, such that at least one of the edges $e,e'$ lies on path $Q$. We say that a bad event ${\cal{E}}_2$ happens iff at least one path in ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ is heavy.
\begin{observation}\label{obs: first bad event}
If $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then $\prob{{\cal{E}}_2}\leq 1/64$.
\end{observation}
\begin{proof}
Note that every edge of $H''$ may lie on at most two paths of ${\mathcal{R}}_0\cup {\mathcal{P}}_0$, and every crossing $(e,e')$ involves two edges. Therefore, the total number of heavy paths in ${\mathcal{R}}_0\cup {\mathcal{P}}_0$ is bounded by $\frac{4\mathsf{cr}^*}{\psi/(64\lambda)}=\frac{2^8\lambda\cdot\mathsf{cr}^*}{\psi}$. Assuming that $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, and using the fact that $\psi=\Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^{16}n} \textsf{right})$ and $\lambda=O\textsf{left}( \frac{\eta \log^{12}n}{\alpha^4\alpha'} \textsf{right} )$, we get that the total number of heavy paths in ${\mathcal{R}}_0\cup {\mathcal{P}}_0$ is bounded by:
\[ \frac{2^8\lambda\cdot\mathsf{cr}^*}{\psi}\leq O\textsf{left}( \frac{\tilde k^2}{c_2\eta^5}\cdot \frac{\eta \log^{12}n}{\alpha^4\alpha'}\cdot\frac{\eta}{\tilde k}\cdot \frac{\log^{16}n}{\alpha^5(\alpha')^2} \textsf{right} )
\leq O\textsf{left}( \frac{\tilde k\log^{28}n}{c_2\eta^3\alpha^9(\alpha')^3} \textsf{right} ).
\]
Note that each heavy path may be selected to ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ with probability at most $1/\psi$. Therfore, using the union bound and the fact that $\psi=\Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^{16}n} \textsf{right})$, we get that:
\[\prob{{\cal{E}}_2}\leq
O\textsf{left}( \frac{\tilde k\log^{20}n}{\psi\cdot c_2\eta^3\alpha^9(\alpha')^3}\textsf{right} )\leq
O\textsf{left}( \frac{\log^{44}n}{\cdot c_2\eta^2\alpha^{14}(\alpha')^5}\textsf{right} ).\]
Recall that, from the conditions of \Cref{thm: find guiding paths}, $\eta\geq c^*\log^{46}n/(\alpha^{10}(\alpha')^4)$, where $c^*$ is a sufficiently large constant. Therefore, if
$\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then $\prob{{\cal{E}}_2}\leq 1/64$.
\end{proof}
Let ${\mathcal{R}}'\subseteq {\mathcal{R}}_1$, ${\mathcal{P}}'\subseteq {\mathcal{P}}_1$ be the sets containing all paths $Q$, such that, in drawing $\phi^*$, the image of some edge of $Q$ crosses the image of some edge lying on the paths of ${\mathcal{R}}^*\cup {\mathcal{P}}^*$. Note that the drawing $\phi^*$ is not known to us, and so neither are the path sets ${\mathcal{R}}',{\mathcal{P}}'$. We will also use the following observation:
\begin{observation}\label{obs: few bad paths}
If ${\cal{E}}_2$ did not happen, then $|{\mathcal{R}}'|,|{\mathcal{P}}'|\leq \psi/32$.
\end{observation}
\begin{proof}
Recall that $|{\mathcal{R}}^*|+|{\mathcal{P}}^*|\leq \lambda$. If bad event ${\cal{E}}_2$ did not happen, then for each path $Q\in {\mathcal{R}}^*\cup {\mathcal{P}}^*$, there are at most $\frac{\psi}{64\lambda}$ crossings in $\phi^*$, in which edges of $Q$ participate. Therefore, if event ${\cal{E}}_2$ did not happen, there are in total at most $\psi/64$ crossings $(e,e')$ in the drawing $\phi^*$, where at least one of the edges $e,e'$ lies on a path of ${\mathcal{P}}^*\cup {\mathcal{Q}}^*$. Let $E'\subseteq E(H'')$ be the set of all edges $e$, such that there is an edge $e'$ lying on some path of ${\mathcal{P}}^*\cup {\mathcal{Q}}^*$, and crossing $(e,e')$ is present in $\phi^*$. Then $|E'|\leq \psi/32$. Each path in ${\mathcal{R}}'\cup {\mathcal{P}}'$ must contain an edge of $E'$. As the paths in each of the sets ${\mathcal{R}}',{\mathcal{P}}'$ are disjoint, $|{\mathcal{R}}'|,|{\mathcal{P}}'|\leq \psi/32$ must hold.
\end{proof}
\paragraph{Summary of Step 3.}
In this step we have constructed a grid skeleton, that consists of two sets of paths: ${\mathcal{P}}^*=\set{ P^*_1,\ldots, P^*_{\lambda'}}$, and ${\mathcal{R}}^*=\set{ R^*_1,\ldots,\tilde R^*_{\lambda'}}$, where $\lambda'= \floor{(\lambda-1)/2}$. Recall that ${\mathcal{P}}^*\subseteq {\mathcal{P}}_0$, and the paths in ${\mathcal{P}}^*$ are indexed according to their order in ${\mathcal{O}}_{{\mathcal{P}}_0}$. Recall that we have also defined the set ${\mathcal{P}}_1\subseteq {\mathcal{P}}_0$ of paths, containing all paths in sets $\set{{\mathcal U}_{2i-1}}_{i=1}^{\lambda'+1}$. It would be convinient for us to re-index the groups ${\mathcal U}_i$ as follows: for $1\leq i\leq \lambda'+1$, set ${\mathcal U}_i={\mathcal U}_{2i-1}$. In other words, the paths of ${\mathcal U}_0$ lie before path $P^*_1$ in the ordering ${\mathcal{O}}_{{\mathcal{P}}_0}$, the paths of ${\mathcal U}_{\lambda'+1}$ lie after $P^*_{\lambda'}$ in this ordering, and, for $1\leq i<\lambda'$, the paths of ${\mathcal U}_i$ lie between paths $P^*_i$ and $P^*_{i+1}$.
Similarly, ${\mathcal{R}}^*\subseteq {\mathcal{R}}_0$, and the paths in ${\mathcal{R}}^*$ are indexed according to their order in ${\mathcal{O}}_{{\mathcal{R}}_0}$. We have also defined the set ${\mathcal{R}}_1\subseteq {\mathcal{R}}_0$ of paths, containing all paths in sets $\set{{\mathcal U}'_{2i-1}}_{i=1}^{\lambda'+1}$. As before, we re-index them as follows: for $1\leq i\leq\lambda'$, we set ${\mathcal U}'_i={\mathcal U}'_{2i-1}$. Therefore, the paths of ${\mathcal U}'_0$ lie before path $R^*_1$ in the ordering ${\mathcal{O}}_{{\mathcal{R}}_0}$, the paths of ${\mathcal U}'_{\lambda'+1}$ lie after $R^*_{\lambda'}$ in this ordering, and, for $1\leq i<\lambda'$, the paths of ${\mathcal U}'_i$ lie between paths $R^*_i$ and $R^*_{i+1}$.
From our definition, if ${\cal{E}}_1$ did not happen, then for every pair $Q,Q'\in {\mathcal{P}}^*\cup {\mathcal{Q}}^*$ of distinct paths, their images in $\phi^*$ do not cross (but note that the image of a single path may cross itself).
We have also defined a set ${\mathcal{P}}'\subseteq {\mathcal{P}}_1$ and a set ${\mathcal{R}}'\subseteq {\mathcal{R}}_1$ of paths, containing all paths $Q$ whose image crosses the image of some path in ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ in drawing $\phi^*$. From \Cref{obs: few bad paths}, if Event ${\cal{E}}_2$ does not happen, then $|{\mathcal{P}}'|,|{\mathcal{R}}'|\leq \psi/32$.
Note that the sets ${\mathcal{P}}',{\mathcal{R}}'$ of paths are not known to the algorithm.
It will be convenient for us to consider the $((\lambda'+1)\times (\lambda'+1))$-grid $\Pi^*$. We view the columns of the grid as corresponding to the left boundary of the bounding box $B^*$, the paths in $\set{R^*_1,\ldots,R^*_{\lambda'}}$, and the right boundary of the bounding box $B^*$.
For convenience, we index the columns of the grid from $0$ to $\lambda'+1$, so the left boundary of the bounding box corresponds to column $0$, and, for $1\leq i\leq \lambda'$, path $P^*_i$ represents the $i$th column of the grid, with the right boundary of $B^*$ repersenting the last column. Similarly, we view the bottom boundary of $B^*$, the paths in $\set{P^*_1,\ldots,P^*_{\lambda'}}$, and the top boundary of $B^*$ as representing the rows of the grid, in the bottom-to-top order. As before, we index the rows of the grid so that the botommost row has index $0$ and the topmost row has index $\lambda'+1$. Notice however that the union of the paths in ${\mathcal{P}}^*\cup {\mathcal{R}}^*$ does not necessarily form a proper grid graph, as it is possible that, for a pair $P\in {\mathcal{P}}^*$, $R\in {\mathcal{R}}^*$ of paths, $P\cap R$ is a collection of several disjoint paths.
We will now consider the drawing $\phi^*$ of $H''$, and we will use it to define vertical and horizontal strips corresponding to paths in ${\mathcal{P}}^*$ and ${\mathcal{R}}^*$, respectively. We will also associate, with each cell of the grid $\Pi^*$, some region of the plane. We assume in the following definitions that Event ${\cal{E}}_1$ did not happen.
Consider first the image $\gamma_i$ of some path $P^*_i\in {\mathcal{P}}^*$ in the drawing $\phi^*$. Note that $\gamma_i$ is not necessarily a simple curve.
We define two simple curves, $\gamma^t_i$ and $\gamma^b_i$, where $\gamma^t_i$ follows the image of $\gamma_i$ from the top, and $\gamma^b_i$ follows it from the bottom. In other words, we let $\gamma^b_i$ be a simple curve, whose every point lies on $\gamma_i$, that has the same endpoints as $\gamma_i$, such that the following holds: if $K^b_i$ is the area of the bounding box $B^*$ below $\gamma_i$, including $\gamma_i^t$, then $\gamma_i\subseteq K^b_i$. We define the other curve, $\gamma^b_i$ symmetrically, so curve $\gamma_i$ is contained in the disc whose boundary is $\gamma_i^t\cup \gamma_i^b$ (see \Cref{fig: top_bottom_curves}). For convenience, we let $\gamma_0^t$ be the bottom boundary of the bounding box $B^*$, and $\gamma_{\lambda'+1}^b$ be the top boundary of the bounding box $B^*$.
We now define, for all $0\leq i\leq \lambda'$, a region of the plane that we call the $i$th horizontal strip, and denote by $\mathsf{HStrip}_i$. This strip is simply the closed region of the bounding box between the curves $\gamma_i^t$ and $\gamma_{i-1}^b$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/top_curve.jpg}
\caption{An illustration of curves $\gamma^t_i$ and $\gamma^b_i$. The curve $\gamma_i$ is shown in purple.}\label{fig: top_bottom_curves}
\end{figure}
For every vertical path $R^*_i\in {\mathcal{R}}^*$, we also define two curves, $\gamma^{\ell}_i$ and $\gamma^r_i$, that follow the image $\gamma_i$ of $R^*_i$ in $\phi^*$ on its left and on its right, respectively. We denote by $\gamma_0^r$ the left boundary of the bounding box $B^*$, and by $\gamma_{\lambda'+1}^{\ell}$ its right boundary. For all $0\leq i\leq \lambda'$, we define a vertical strip $\mathsf{VStrip}_i$ to be the closed region of the bounding box $B^*$ betwen $\gamma_i^r$ and $\gamma_i^{\ell}$.
The following observation is immediate from the fact that the paths in ${\mathcal{P}}_0$ are node-disjoint, and so are the paths in ${\mathcal{R}}_0$.
\begin{observation}\label{obs: must cross chosen paths}
If $R\in {\mathcal{R}}_1$ is a path whose image in $\phi^*$ intersects the interior of more than one vertical strip in $\set{\mathsf{VStrip}_0,\ldots,\mathsf{VStrip}_{\lambda'+1}}$, then $R\in {\mathcal{R}}'$. Similarly, if $P\in {\mathcal{P}}_1$ is a path whose image in $\phi^*$ intersects the interior of more than one horizontal strip in $\set{\mathsf{HStrip}_0,\ldots,\mathsf{HStrip}_{\lambda'+1}}$, then $P\in {\mathcal{P}}'$.
\end{observation}
Lastly, for all $0\leq i,j\leq \lambda'$, we let $\mathsf{CellRegion}_{i,j}=\mathsf{HStrip}_i\cap \mathsf{VStrip}_j$ be a closed region of the plane that we associate with cell $\mathsf{Cell}_{i,j}$ of the grid $\Pi^*$.
\iffalse
====================================
\highlightf{Calculations: heavy paths}
Path is heavy if it participates in $x$ crossings. All paths of skeleton will give $\lambda x$ crossings. Using $\lambda<\eta^2$, this is less than $x\eta^2$ crossings. This should be less than $\psi/16$. Because $\psi>\frac{16\tilde k}{\eta^2}$, enough that $x\eta^2<\tilde k/\eta^2$, or that $x<\tilde k/\eta^4$.
If we fix $x=\tilde k/\eta^4$. Then number of bad paths is at most:
\[\frac{4\mathsf{cr}^*}{x}< \frac{\tilde k^2}{c_2\eta^5}/\frac{\tilde k}{\eta^4}=\frac{\tilde k}{c_2\eta}. \]
A heavy path is selected with probability $1/\psi$. Probability that any heavy path is selected (from union bound) is at most:
\[ \frac{\tilde k}{c_2\eta\psi}. \]
If we use $\psi>\frac{16\tilde k}{\eta^2}$, this is not good enough.
===============================
More detained calculation: all paths of skeleton will give $\lambda x$ crossings, and we want it to be less than $\psi/16$. so $x=\frac{\psi}{16\lambda}$ is fine. Now number of bad paths is at most:
\[\frac{4\mathsf{cr}^*}{x}< \frac{64\mathsf{cr}^*\lambda}{\psi}.\]
A heavy path is selected with probability $1/\psi$. Probability that any heavy path is selected (from union bound) is at most:
\[\frac{64\mathsf{cr}^*\lambda}{\psi^2}\]
So for us it is enough that $\mathsf{cr}^*<\psi^2\cdot 2^{12}/\lambda$.
We have:$ \psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$, so $\frac{\psi}{\lambda}\geq \frac{\alpha^*\tilde k}{128}$. Using $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, we get that $\frac{\psi}{\lambda}\geq \Omega(\frac{\alpha\alpha'\tilde k}{\log^4n})$.
So now $\psi^2/\lambda=\Omega (\frac{\psi\alpha\alpha'\tilde k}{\log^4n} )$.
Can use now $\psi>\frac{16\tilde k}{\eta^2}$, so the expression becomes:
\[\Omega (\frac{\tilde k^2\alpha\alpha'}{\eta^2\log^4n} ) \]
Lastly, because $\eta>\log^4n/(c^*\alpha\alpha')$, we get that $\mathsf{cr}^*<\tilde k^2/(c_2\eta^5)$ is good enough.
=====================================
\fi
\subsection{Step 4: Constructing a Grid-Like Structure}
In this step we further delete some paths from sets ${\mathcal{R}}_1$ and ${\mathcal{P}}_1$ to ensure that the resulting paths form a grid-like structure. This is done in three stages. In the first stage, we discard some paths to ensure that every remaining path in ${\mathcal{R}}_1$ intersects the paths in ${\mathcal{P}}^*$ ``in order'' (we formally define this notion later), and process the paths in ${\mathcal{P}}_1$ similarly. In the second stage, we associate, with every cell of the grid $\Pi^*$ a collection of horizontal paths and a collection of vertical paths. In the third stage, we ensure that for every cell of the grid $\Pi^*$, there are many inersections between its corresponding horizontal and vertical paths.
Before we continue, we discard some paths of ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ that must lie in ${\mathcal{R}}'\cup {\mathcal{P}}'$. Specifically, consider some path $P\in {\mathcal{P}}^*$, and assume that it lies in group ${\mathcal U}_i$, for some $0\leq i\leq \lambda'+1$. Let $(a,a')$ be the endpoints of path $P$, with $a\in T_1$ and $a'\in T_3$. Notice that from the definition, if $i>0$, then $a$ must lie, in the ordering $\tilde{\mathcal{O}}_1$ of the terminals of $T_1$, after the endpoint of the path $P^*_i$ that belongs to $T_1$. Similarly, if $i<\lambda'+1$, then $a$ must lie before the endpoin of the path $P^*_{i+1}$ that belongs to $T_1$ in the same ordering. In particular, we are guaranteed that in the drawing $\phi^*$, the image of $P$ must intersect the interior of the horizontal strip $\mathsf{HStrip}_i$. Consider now the endpoint $a'$ of $P$. If $i>0$, let $a'_i$ be the endpoint of path $P_i^*$ that lies in $T_3$, and if $i<\lambda'+1$, let $a'_{i+1}$ be the endpoint of path $P_{i'+1}$ that lies in $T_3$. Note that, if $a'$ lies before $a'_i$ in the ordering $\tilde{\mathcal{O}}_3$ of $T_3$, or if $a'$ lies after $a'_{i+1}$ in the ordering $\tilde{\mathcal{O}}_3$, then the image of $P$ has to intersect the interior of an additional horizontal strip, and, from \Cref{obs: must cross chosen paths}, path $P$ must lie in ${\mathcal{P}}'$. We discard each such path from set ${\mathcal{P}}_1$ (and from the corresponding set $U_i$). This ensures that, if $P\in U_i$, then its endpoint $a'$ must lie between $a'_i$ and $a'_{i+1}$ in $\tilde{\mathcal{O}}_3$, if $1\leq i\leq \lambda'$; it must lie before $a'_{i+1}$ if $i=0$, and it must lie after $a'_i$ if $i=\lambda'+1$.
We process the paths in ${\mathcal{R}}_1$ similarly, discarding paths as needed. Notice that so far all paths that we have discarded from ${\mathcal{P}}_1\cup {\mathcal{R}}_1$ lie in ${\mathcal{P}}'\cup {\mathcal{R}}'$.
\subsubsection{In-Order Intersection}
In this stage we discard some additional paths from ${\mathcal{P}}_1\cup {\mathcal{R}}_1$, to ensure that every remaining path in ${\mathcal{P}}_1$ interesects the paths in ${\mathcal{R}}^*$ in-order (notion that we define below); we do the same for paths in ${\mathcal{R}}_1$. We will ensure that all paths discarded at this stage lie in ${\mathcal{P}}'\cup {\mathcal{R}}'$.
Since the definitions and the algorithms for the paths in ${\mathcal{P}}_1$ and for the paths in ${\mathcal{R}}_1$ are symmetric, we only describe the algorithm to process the paths in ${\mathcal{P}}_1$ here.
Let $P\in {\mathcal{P}}_1$ be any path, that we view as directed from its endpoint that lies in $T_1$ to its endpoint lying in $T_3$. Let $X(P)=\set{x_1,\ldots,x_r}$ denote all vertices of $P$ lying on paths in $R^*$, that is, $X(P)=V(P)\cap \textsf{left}(\bigcup_{i=1}^{\lambda'}V(R^*_i)\textsf{right} )$. We assume that the vertices of $X(P)$ are indexed in the order of their appearance on $P$. For each such vertex $x_j$, let $i_j$ be the index of the path $R^*_{i_j}\in {\mathcal{R}}^*$ containing $x_j$.
\begin{definition}[In-order intersection]
We say that path $P$ intersects the paths of ${\mathcal{R}}^*$ in-order, iff $r\geq \lambda'$, $i_1=1$, $i_r=\lambda'$, and, for $1\leq j<r$, $|i_j-i_{j+1}|\leq 1$.
\end{definition}
Notice that the definition requires that path $P$ intersects every path of ${\mathcal{R}}^*$ at least once; the first path of ${\mathcal{R}}^*$ that it intersects must be $R_1^*$, and the last path must be $R_{\lambda'}^*$, and for every consecutive pair $x_j,x_{j+1}$ of vertices in $X(P)$, either both vertices lie on the same path of ${\mathcal{R}}^*$, or they lie on consecutive paths of ${\mathcal{R}}^*$. Notice that path $P$ is still allowed to intersect a path of ${\mathcal{R}}^*$ many times, and may go back and forth across all these paths several times.
\begin{observation}\label{obs: not in order intersection}
Assume that Event ${\cal{E}}_1$ did not happen. Let $P\in {\mathcal{P}}_1$ be a path that intersect the paths of ${\mathcal{R}}^*$ not in-order. Then $P\in {\mathcal{P}}'$ must hold.
\end{observation}
\begin{proof}
Assume first that $i_1\neq 1$, that is, vertex $x_1$ lies on some path $R^*_i$ with $i\neq 1$. Let $p$ be a point on the image of path $P$ in $\phi^*$ that is very close to its first endpoint, so $p$ lies in the interior of the vertical strip $\mathsf{VStrip}_1$, and let $p'$ be the image of the point $x_1$. Clearly, $p'$ does not lie in the interior or on the boundary of $\mathsf{VStrip}_1$, so the image of path $P$ must cross the right boundary of $\mathsf{VStrip}_1$, which means that the image of some edge of $P$ and the image of some edge of $R_1^*$ cross in $\phi^*$.
The cases where $i_{r}\neq \lambda'$, or there is an index $1\leq j<r$ with $i_j-i_{j+1}> 1$ are treated similarly, as is the case when $r<\lambda'$.
\end{proof}
We discard from ${\mathcal{P}}_1$ all paths $P$ that intersect the paths of ${\mathcal{R}}^*$ not in-order. We denote by ${\mathcal{P}}_2\subseteq {\mathcal{P}}_1$ the set of remaining paths. We also update the groups ${\mathcal U}_0,\ldots,{\mathcal U}_{\lambda'+1}$ accordingly. Observe that so far all paths that we have discarded from ${\mathcal{P}}_1$ lie in ${\mathcal{P}}'$.
From \Cref{obs: few bad paths}, assuming that Events ${\cal{E}}_1$ and ${\cal{E}}_2$ did not happen, the number of paths that we have discarded so far from ${\mathcal{P}}_1$ is at most $\psi/32$. In particular, for all $0\leq i\leq \lambda'+1$, $|{\mathcal U}_i|\geq 31\psi/32$ still holds.
We perform the same transformation on set ${\mathcal{R}}_1$ of paths, obtaining a new set ${\mathcal{R}}_2$ of paths, each of which intersects the paths of ${\mathcal{P}}^*$ in-order. We also update the groups ${\mathcal U}'_0,\ldots,{\mathcal U}'_{\lambda'+1}$. As before, for all $0\leq i\leq \lambda'+1$, $|{\mathcal U}'_i|\geq 31\psi/32$ still holds.
\subsubsection{Definining Paths Associated with Grid Cells}
For every path $P\in {\mathcal{P}}_2$, for all $1\leq i\leq \lambda'$, we denote by $v_i(P)$ the first vertex on path $P$ that belongs to the vertical path $R^*_i$; note that, from the definition of in-order intersection, such a path must exist. For all $1\leq i< \lambda'$, we define the $i$th segment of $P$, $\sigma_i(P)$, to be the subpath of $P$ between $v_i(P)$ and $v_{i+1}(P)$. We also let $\sigma_0(P)$ be the subpath of $P$ from its first vertex (which must be a terminal of $T_1$) to $v_1(P)$, and by $\sigma_{\lambda'}(P)$ the subpath of $P$ from $v_{\lambda'}(P)$ to the last vertex of $P$ (which must be a terminal of $T_3)$. Note that the edges on paths $\sigma_0(P),\ldots,\sigma_{\lambda'}(P)$ partition $E(P)$.
Siilarly, for every path $R\in {\mathcal{R}}_2$, for all $1\leq i\leq \lambda'$, we denote by $v_i(R)$ the first vertex on path $R$ that lies on the horizontal path $P^*_i$. For all $1\leq i< \lambda'$, we define the $i$th segment of $P$, $\sigma_i(R)$, to be the subpath of $R$ between $v_i(R)$ and $v_{i+1}(R)$. We also let $\sigma_0(R)$ be the subpath of $R$ from its first vertex (which must be a terminal of $T_4$) to $v_1(R)$, and by $\sigma_{\lambda'}(R)$ the subpath of $R$ from $v_{\lambda'}(R)$ to the last vertex of $R$ (which must be a terminal of $T_2$).
Consider now some cell $\mathsf{Cell}_{i,j}$ of the grid $\Pi^*$, for some $0\leq i,j\leq \lambda'+1$. We define the set ${\mathcal{P}}^{i,j}$ of horizontal paths, and the set ${\mathcal{R}}^{i,j}$ of vertical paths associated with cell $\mathsf{Cell}_{i,j}$, as follows. In order to define the set ${\mathcal{P}}^{i,j}$ of horizontal paths, we consider the group ${\mathcal U}_i$, and, for every path $P\in {\mathcal U}_i$, we include its $j$th segment $\sigma_j(P)$ in ${\mathcal{P}}^{i,j}$, so that:
\[{\mathcal{P}}^{i,j}=\set{\sigma_j(P)\mid P\in {\mathcal U}_i}.\]
Similarly, we define:
\[{\mathcal{R}}^{i,j}=\set{\sigma_i(R)\mid R\in {\mathcal U}'_j}.\]
\iffalse
Note that the definition of the sets ${\mathcal{P}}^{i,j},{\mathcal{R}}^{i,j}$ of paths depends on the definition of the direction of the paths in ${\mathcal{P}}_2,{\mathcal{R}}_2$. For example, recall that we think of the vertices of $T_1$ as lying on the left boundary of the bounding box $B^*$, and the vertices of $T_3$ as lying on its right boundary, with the paths in ${\mathcal{P}}_2$ directed from left to right. If we were to flip the bounding box $B^*$, so that the vertices of $T_3$ appear on its left boundary and the vertices of $T_1$ on its right boundary, with the paths in ${\mathcal{P}}$ directed from left to right, then the definition of the sets ${\mathcal{P}}^{i,j}$ of paths may change (as a path in ${\mathcal{P}}_2$ may intersect a path of ${\mathcal{R}}^*$ numerous times).
However, if we flip the bounding box $B^*$, so that the vertices of $T_1$ appear on its bottom boundary, the vertices of $T_3$ on its top boundary, and the vertices of $T_2$ and $T_4$ on its left and right boundaries, respectively, directing the paths in ${\mathcal{P}}$ from bottom to top, and the paths in ${\mathcal{R}}$ from left to right, the definition of path sets $\set{{\mathcal{P}}^{i,j},{\mathcal{R}}^{i,j}}_{0\leq i,j\leq \lambda'+1}$ will now change (but now paths of ${\mathcal{P}}$ become vertical and paths of ${\mathcal{R}}$ become horizontal). This transformation corresponds to flipping the grid $\Pi^*$ along its diagonal (from bottom left to top right), and so the cells that lied in the top right quandrant of the grid remain in the top right quadrant. We may need to use this transformation later, but for now we stay with the original notation.
\fi
We need the following observation.
\begin{observation}\label{obs: paths in cells don't cross}
Let $P\in {\mathcal U}_i,R\in {\mathcal U}'_j$ be a pair of paths, for some $0\leq i,j\leq \lambda'+1$, and assume that their subpaths $\sigma_j(P)\subseteq P,\sigma_i(R)\subseteq R$ do not share any vertices. Then either $P\in {\mathcal{P}}'$, or $R\in {\mathcal{R}}'$, or the images of $\sigma_j(P)$ and $\sigma_i(R)$ cross in the drawing $\phi^*$.
\end{observation}
\begin{proof}
Assume that $P\not\in {\mathcal{P}}'$ and $R\not\in {\mathcal{R}}'$, that is, the images of the paths $P,R$ do not cross the images of the paths in ${\mathcal{P}}^*\cup {\mathcal{R}}^*$ in $\phi^*$. From the definition of set ${\mathcal U}_i$, the image of $P$ intersects the interior of the horizontal strip $\mathsf{HStrip}_i$, and path $P$ does not share any vertices with the paths of ${\mathcal{P}}^*$. Therefore, the image of $P$ must be contained in the strip $\mathsf{HStrip}_i$, and it is disjoint from its top and bottom boundaries $\gamma^b_i,\gamma^t_{i+1}$. Using similar reasoning, the image of $R$ is contained in the strip $\mathsf{VStrip}_j$, and it is disjoint from its left and right boundaries, $\gamma^{\ell}_j,\gamma^r_{j+1}$. Consider now the segment $\sigma_j(P)$ of $P$, whose endpoints lie on $R^*_j$ and $R^*_{j+1}$, respectively. Let $\sigma'_j(P)\subseteq \sigma_j(P)$ be the shortest subpath of $\sigma_j(P)$ whose first endpoint lies on $R^*_j$, and whose last endpoint lies on $R^*_{j+1}$; such a path must exist because we can let $\sigma'_j(P)=\sigma_j(P)$. From the definition of in-order intersection, no inner vertex of $\sigma'_j$ may lie on any path of ${\mathcal{R}}^*$. It is then easy to verify that the image of $\sigma'_j(P)$ in $\phi^*$ must be contained in $\mathsf{CellRegion}_{i,j}$, and it must split this region into two subregions: one whose top boundary contains a segment of $\gamma^b_{i+1}$, and one whose bottom boundary contains a segment of $\gamma^t_i$.
Using the same reasoning, we can select a segment $\sigma'_i(R)$, whose first endpoint lies on $P^*_i$, last endpoint lies on $P^*_{i+1}$, and all inner vertices are disjoint from the vertices lying on the paths in ${\mathcal{P}}^*$.
As before, the image of $\sigma'(R)$ must be contained in $\mathsf{CellRegion}_{i,j}$, but it connects a point on its top boundary to a point on its bottom boundary. Therefore, the image of $\sigma'_i(R)$ must cross the image of $\sigma'_j(P)$.
\end{proof}
\subsubsection{Completing the Construction of the Grid-Like Structure}
In order to complete the construction of the grid-like structure, we need to ensure that, for every pair $0\leq i,j\leq \lambda'+1$ of indices, there are many intersection between the sets ${\mathcal{P}}^{i,j}$ and ${\mathcal{R}}^{i,j}$ of paths. More specifically, we need to ensure that every path $\sigma\in {\mathcal{P}}^{i,j}$ intersects many paths in ${\mathcal{R}}^{i,j}$, and vice versa. This is needed in order to ensure well-linkedness properties: namely, that the collection of vertices containing the fist and the last vertex on every path of ${\mathcal{P}}^{i,j}$ is sufficiently well-linked in the graph obtained from the union of the paths in ${\mathcal{R}}^{i,j}\cup {\mathcal{P}}^{i,j}$. This property, in turn, will be exploited in order to construct the sets ${\mathcal{Q}}$ of paths routing the terminals to sets of vertices $V(\Pi(v)))$ over which the distribution ${\mathcal{D}}$ will be defined. This motivates the following definition.
\begin{definition}
For a pair $0\leq i,j\leq \lambda'+1$ of indices, we say that a path $P\in {\mathcal U}_i$ is \emph{bad for cell $\mathsf{Cell}_{i,j}$} iff there are at least $\psi/16$ paths in ${\mathcal{R}}^{i,j}$ that are disjoint from $\sigma_j(P)$. Similarly, we say that a path $R\in {\mathcal U}'_j$ is bad for cell $\mathsf{Cell}_{i,j}$ iff there are at least $\psi/16$ paths in ${\mathcal{P}}^{i,j}$ that are disjoint from $\sigma_i(R)$.
\end{definition}
Consider now some index $0\leq i\leq \lambda'+1$. We say that a path $P\in {\mathcal U}_i$ is \emph{bad} iff it is bad for at least one cell in $\set{\mathsf{Cell}_{i,j}\mid 0\leq j\leq \lambda'+1}$. Similarly, for an index $0\leq j\leq \lambda'+1$, a path $R\in {\mathcal U}'_j$ is bad iff it is bad for at least one cell in $\set{\mathsf{Cell}_{i,j}\mid 0\leq i\leq \lambda'+1}$. The following observation bounds the number of bad paths in each group ${\mathcal U}_i$ of horizontal paths, and in each group ${\mathcal U}'_j$ of vertical paths.
\begin{observation}\label{obs: few bad paths in each group}
Assume that $\mathsf{cr}^*\leq \frac{\tilde k^2}{c_2\eta^5}$, and that events ${\cal{E}}_1,{\cal{E}}_2$ did not happen. Then for all $0\leq i\leq \lambda'+1$, at most $\psi/16$ paths in ${\mathcal U}_i$ are bad. Similarly, for all $0\leq j\leq \lambda'+1$, at most $\psi/16$ paths in ${\mathcal U}'_j$ are bad.
\end{observation}
\begin{proof}
Fix an index $0\leq i\leq \lambda'+1$, and the corresponding set ${\mathcal U}_i\subseteq {\mathcal{P}}_2$. We partition the set of all bad paths in ${\mathcal U}_i$ into two subsets: set ${\mathcal{B}}_1$ contains all bad paths lying in ${\mathcal{P}}'$, and set ${\mathcal{B}}_2$ contains all remaining bad paths. From \Cref{obs: few bad paths}, $|{\mathcal{B}}_1|\leq \psi/32$.
We further partition the set ${\mathcal{B}}_2$ of bad paths into subsets $\set{{\mathcal{B}}_2^j\mid 0\leq j\leq \lambda'}$, where a path $P$ lies in ${\mathcal{B}}_2^j$ if it is bad for cell $\mathsf{Cell}_{i,j}$ (if path $P$ is bad for several cells, we add it to any of the corresponding sets). Consider now some index $0\leq j\leq \lambda'$, and some path $P\in {\mathcal{B}}_2^j$. From the definition, there is a set $\Sigma'\subseteq {\mathcal{R}}^{i,j}$ of at least $\psi/16$ paths that do not share any vertices with $P$. From \Cref{obs: few bad paths}, at most $\psi/32$ of these paths may lie in ${\mathcal{R}}'$. Let $\Sigma''\subseteq \Sigma'$ be the collection of the remaining paths, whose cardinality is at least $\psi/32$. From \Cref{obs: paths in cells don't cross}, for every path $\sigma'\in \Sigma'$, the images of $\sigma_j(P)$, and of $\sigma'$ must cross. We let $\chi_j(P)$ denote the set of all crossings $(e,e')$, where $e\in \sigma_j(P)$, and $e'$ is an edge on a path in $\Sigma'$, so $|\chi_j(P)|\geq \psi/32$. We then let $\chi_{j}=\bigcup_{P\in {\mathcal{B}}_2^j}\chi_j(P)$, so $|\chi_{j}|\geq |{\mathcal{B}}_2^j|\cdot \psi/32$. Lastly, we let $\chi=\bigcup_{j=0}^{\lambda'+1}\chi_j$. Notice that set $\chi$ contains at least $|{\mathcal{B}}_2|\cdot \psi/32$ distinct crossings in the drawing $\phi^*$. Assume for contradiction that $|{\mathcal{B}}_2|>\psi/32$. Then:
\[\mathsf{cr}^*>\frac{\psi^2}{2^{10}}>\frac{\tilde k^2}{4\eta^4}>\frac{\tilde k^2}{c_2\eta^5},\]
since $\psi>\frac{16\tilde k}{\eta^2}$, a contradiction.
Therefore, $|{\mathcal{B}}_2|\leq \psi/32$, and overall there are at most $\psi/16$ bad paths in ${\mathcal U}_i$.
The proof for path sets ${\mathcal U}'_j\subseteq {\mathcal{R}}_2$ is identical.
\end{proof}
For all $0\leq i\leq \lambda'+1$, we discard every bad path from ${\mathcal U}_i$. If $|{\mathcal U}_i| <\ceil{7\psi/8}$ for any $i$, then we terminate the algorithm and return FAIL. Notice that in this case, from \Cref{obs: few bad paths in each group}, if $\mathsf{cr}^*< \frac{\tilde k^2}{c_2\eta^5}$, then at least one of the events ${\cal{E}}_1,{\cal{E}}_2$ must have happened, and the probability for this is at most $1/8$. Therefore, we assume that for all
$0\leq i\leq \lambda'+1$, $|{\mathcal U}_i|\geq \ceil{7\psi/8}$ holds. We discard additional arbitrary paths from ${\mathcal U}_i$, until $|{\mathcal U}_i|= \ceil{7\psi/8}$.
We then let ${\mathcal{P}}_3=\bigcup_{i=0}^{\lambda'+1}{\mathcal U}_i$ denote the resulting set of paths.
Similarly, for all $0\leq j\leq \lambda'+1$, we discard every bad path from ${\mathcal U}'_j$. If, as the result, $|{\mathcal U}'_j|$ falls below $\ceil{7\psi/8}$, we terminate the algorithm and return FAIL. Otherwise, we discard additional arbitrary paths as needed, so that $|{\mathcal U}'_j|=\ceil{7\psi/8}$ holds. We also let ${\mathcal{R}}_3=\bigcup_{j=0}^{\lambda'+1}{\mathcal U}'_j$.
For all $0\leq i,j\leq \lambda'$, we also update the path sets ${\mathcal{P}}^{i,j}$ and ${\mathcal{R}}^{i,j}$ accordingly, discarding the paths that are no longer subpaths of paths in ${\mathcal{P}}_3\cup {\mathcal{R}}_3$. Since we are still guaranteed that $|{\mathcal{P}}^{i,j}|,|{\mathcal{R}}^{i,j}|= \ceil{7\psi/8}$, and since every path that is bad for cell $\mathsf{Cell}_{i,j}$ was discarded, we are guaranteed that every path in ${\mathcal{P}}^{i,j}$ intersects at least $\frac{7\psi}{8}-\frac{\psi}{16}=\frac{13\psi}{16}$ paths of ${\mathcal{R}}^{i,j}$ and vice versa.
Since we use this fact later, we summarize it in the following observation.
\begin{observation}\label{obs: paths for cells}
For all $0\leq i,j\leq \lambda'$, $|{\mathcal{P}}^{i,j}|,|{\mathcal{R}}^{i,j}|= \ceil{7\psi/8}$. Every path in
${\mathcal{P}}^{i,j}$ intersects at least $\frac{13\psi}{16}$ paths of ${\mathcal{R}}^{i,j}$ and vice versa.
\end{observation}
This concludes the construction of the grid-like structure.
\subsection{Step 5: the Routing}
Recall that we have denoted by $\Lambda'$ the set of all pairs $({\mathcal{Q}},y)$, where $y$ is a vertex in graph $H$, and ${\mathcal{Q}}$ is a set of paths in graph $H''$, routing the set $\tilde T$ of terminals to vertices of $\Pi(y)$. In this final step we design an efficient algorithm to compute distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}},y)\in \Lambda'$, such that for every outer edge $e\in E(H'')$, $\expect[({\mathcal{Q}},y)\in_{{\mathcal{D}}}\Lambda']{(\cong_{H''}({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{16}n}{(\alpha\alpha')^4}\textsf{right} )$.
Our algorithm closely follows the arguments of
\cite{Tasos-comm} (see also Lemma D.10 in the full version of \cite{chuzhoy2011algorithm}), who showed a similar result for a grid graph. In order to provide intuition, we first present their algorithm. Assume that we are given a $(q\times q)$ grid graph $G$ for some integer $q$, and let $T$ be the set of vertices lying on the first row of the grid, that we refer to as terminals. For convenience, assume that $q$ is an integral power of $2$. Our goal is to compute a distribution ${\mathcal{D}}'$ over pairs in $\Lambda(G,T)$. In other words, the distribution is over pairs $({\mathcal{Q}},v)$, where $v$ is a vertex of $G$, and ${\mathcal{Q}}$ is a set of paths routing the terminals to $v$. We need to ensure that for every edge $e\in E(G)$, the expectation $\expect[({\mathcal{Q}},v)\in_{{\mathcal{D}}'}\Lambda(G,T)]{(\cong_G({\mathcal{Q}},e))^2}\leq O(\log q)$.
For every vertex $v$ in the top right quadrant of the grid, we will define a set ${\mathcal{Q}}(v)$ of paths in $G$, routing the terminals in $T$ to $v$. Our distribution ${\mathcal{D}}$ then assigns, to every pair $({\mathcal{Q}}(v),v)$, where $v$ is a vertex of $G$ lying in the top right quadrant of the grid, the same probability value $4/q^2$.
We now fix a vertex $v$ in the top right quadrant of the grid, and define the set ${\mathcal{Q}}(v)$ of paths. Let $r=\log(q/4)$. For $1\leq i\leq r$, let $S_i$ be a square subgrid of $G$, of size $(2^i\times 2^i)$, whose upper right corner has the same column-index as vertex $v$, and the same row-index as the bottom left corner of $S_{i-1}$ (we think of $S_1$ as a $(1\times 1)$-grid consisting only of vertex $v$).
We refer to the subgrids $S_i$ of $G$ as \emph{squares}, and specifically to square $S_i$ as \emph{level-$i$ square}.
For all $1\leq i\leq r$, we denote by $T_i$ the set of vertices lying on the bottom boundary of square $S_i$. Using the well-linkedness of the grids, it is easy to show that for all $1\leq i\leq r$, there is a collection ${\mathcal{P}}_i$ of paths in graph $S_i$, routing vertices of $T_i$ to verticse of $T_{i-1}$ with congestion at most $2$, such that every vertex of $T_{i-1}$ serves as endpoint of at most two such paths. For $1\leq i\leq r$, let
Let ${\mathcal{P}}'_i$ be a multipset obtained from set ${\mathcal{P}}_i$ by creating $2^{r-i+1}$ copies of every path in ${\mathcal{P}}_i$.
Let $T_{r+1}\subseteq T$ be a set of $|T_r|$ vertices lying on the bottom boundary of the grid $G$, that contains, for every vertex $t\in T_r$, vertex $t'$ on the bottom boundary of $G$ with the same column index as $t$. Let $P_t$ be the subpath of the corresponding column of $G$ connecting $t$ to $t'$, and denote ${\mathcal{P}}'_{r+1}=\set{P_t\mid t\in T_r}$.
By concatenating the paths in ${\mathcal{P}}_1',\ldots,{\mathcal{P}}'_{r+1}$, we obtain a collection ${\mathcal{Q}}'(v)$ of paths in grid $G$, routing the terminals in $T_{r+1}$ to vertex $v$.
Notice that for $1\leq i\leq r$, for every edge $e$ lying in $S_i$, the congestion on
edge $e$ due to paths in ${\mathcal{Q}}(v)$ is at most $2^{r-i+2}$.
The key in analyzing the expectation $\expect[({\mathcal{Q}},v)\in_{{\mathcal{D}}'}\Lambda(G,T)]{(\cong_G({\mathcal{Q}},e))^2}$
is to notice that, for all $1\leq i\leq r$, square $S_i$ is a $(2^i\times 2^i)$-subgrid of $G$, whose upper right corner is chosen uniformly at random from a set of $q^2/4$ possible points. The total number of subgrids of $G$ of size $(2^i\times 2^i)$ that contain $e$ is $2^{2i}$, so the probability that any of them is selected is bounded by $2^{2i+2}/q^2$. Therefore, for all $1\leq i\leq r$, with probability at most $2^{2i+2-2r}$, edge $e$ belongs to square $S_i$, and in this case, $\cong_G({\mathcal{Q}},e)\leq 2^{r-i+2}$. Therefore, we get that:
\[\expect[({\mathcal{Q}},v)\in_{{\mathcal{D}}'}\Lambda(G,T)]{(\cong_G({\mathcal{Q}},e))^2}
\leq \sum_{i=1}^{r}2^{2i+2-2r}\cdot 2^{2r-2i+4}\leq O(r)=O(\log q).
\]
Using the well-linkedness of the terminals in $T$, it is immediate to extend the set ${\mathcal{Q}}(v)$ of paths to a set ${\mathcal{Q}}'(v)$ routing all terminals in $T$ to $v$, while increasing the congestion on every edge of $G$ by at most an additive constant and a multiplicative constant factor. This provides the final distribution ${\mathcal{D}}$ over pairs $({\mathcal{Q}}(v),v)\in \Lambda(G,T)$.
We will simulate a similar process on the grid $\Pi^*$, and its corresponding grid-like structure that we have constructed. Notice however that $\Pi^*$ is only a $(\lambda'\times\lambda')$-grid (where $\eta\leq \lambda'\leq \eta^2$), while the number of terminals that we need to route is much larger (comparable to $|{\mathcal{R}}_3|$). Therefore, we will attempt to route all terminals to a single cell $\mathsf{Cell}_{i,j}$ in the top right quadrant of the grid (in other words, we will route them to vertices lying on paths in ${\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$). This in itself is not sufficient, since we need to route them to a set $V(\Pi(y))$ of vertices, corresponding to a single vertex $y$ of the original graph $H$. This means that we may need to perform some routing within the cell $\mathsf{Cell}_{i,j}$, that is, within the graph obtained from the union of the paths in ${\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$. While generally such a routing may be difficult to compute, we will select a large collection of cells (called good cells) in the top right quadrant of the grid $\Pi^*$, in which such a routing is easy to obtain. We will then define, for each good cell, the corresponding set of paths routing the terminals to vertices of $V(\Pi(y))$, for some vertex $y\in V(H)$. We do so by simulating the process described above: we define square subgrids $\set{S_i}$ of the grid $\Pi^*$, and we associate these subgrids with sets of horizontal and vertical paths (subpaths of some paths in ${\mathcal{P}}_3\cup {\mathcal{R}}_3$), so that the desired well-linkedness properties of graphs corresponding to each subgrid $S_i$ are achieved. Eventually, the distribution ${\mathcal{D}}$ chooses one of the good squares uniformly at random, and uses the associated pair $({\mathcal{Q}},y)\in \Lambda'$ in order to route the terminals to vertices of $\Pi(y)$. The analysis of expected congestion squared on every edge of $H''$ is very similar to the one outlined above.
We start by defining the notion of good cells of the grid $\Pi^*$, and showing that a large enough number of such cells exist in the upper right quadrant of $\Pi^*$. We will then define square subgrids of $\Pi^*$ and associate sets of paths with each such subgrid to ensure the required well-linkedness properties. Lastly, we show how to construct the desired routing ${\mathcal{Q}}$ for each good cell.
\subsubsection{Good Cells}
Fix a pair of indices $0\leq i,j\leq \lambda'$, and consider the cell $\mathsf{Cell}_{i,j}$ of the grid $\Pi^*$, and the two corresponding sets ${\mathcal{P}}^{i,j}$, ${\mathcal{R}}^{i,j}$ of paths.
\begin{definition}[Good cells]
A path $\sigma\in {\mathcal{P}}^{i,j}$ is \emph{good} for cell $\mathsf{Cell}_{i,j}$ iff $\sigma$ contains no outer edges.
We say that cell $\mathsf{Cell}_{i,j}$ is \emph{good} iff some path $\sigma \in {\mathcal{P}}^{i,j}$ is good for $\mathsf{Cell}_{i,j}$; otherwise we say it is \emph{bad}.
\end{definition}
Assume that cell $\mathsf{Cell}_{i,j}$ is good, and let $\sigma\in {\mathcal{P}}^{i,j}$ be any horizontal path that is good for this cell. Since $\sigma$ contains no outer edges, there must be a vertex $y\in V(H)$, such that $V(\sigma)\subseteq V(\Pi(y))$. Recall that, from \Cref{obs: paths for cells}, $|{\mathcal{R}}^{i,j}|=\ceil{7\psi/8}$, and that $\sigma$ intersects at least $13\psi/16$ paths of ${\mathcal{R}}^{i,j}$. Let $\hat {\mathcal{R}}^{i,j}\subseteq {\mathcal{R}}^{i,j}$ be a set of $\ceil{13\psi/16}$ paths, each of which shares at least one vertex with $\sigma$. Note that each such path then must contain a vertex of $\Pi(y)$. We denote by $\mathsf{Portals}^{i,j}$ the set of vertices that contains, for every path $\sigma'\in \hat{\mathcal{R}}^{i,j}$, the first vertex of $\sigma'$ (by definition, each such vertex must lie on path $P^*_i$ if $i>0$, or belong to $T_4$ otherwise). For convenience, we denote vertex $y$ by $y_{i,j}$.
Let $Z$ be the set of all pairs of indices $\floor{\lambda'/2}\leq i,j\leq \lambda'+1$, such that $\mathsf{Cell}_{i,j}$ is good. Next, we show that $|Z|$ is sufficiently large. Our routing algorithm will then choose a pair $(i,j)$ of indices from $Z$ uniformly at random, and route the terminals to the vertices in set $\mathsf{Portals}^{i,j}$, from where they will be routed to vertices of $\Pi(y_{i,j})$.
\begin{claim}\label{claim: many good cells}
$|Z|\geq (\lambda')^2/16$.
\end{claim}
\begin{proof}
Let ${\mathcal{B}}$ be a collection of all bad cells $\mathsf{Cell}_{i,j}$ lying in the top right quadrant, that is, $\floor{\lambda'/2}\leq i,j\leq \lambda'+1$. It is enough to show that $|{\mathcal{B}}|<(\lambda')^2/16$.
Consider now some bad cell $\mathsf{Cell}_{i,j}\in {\mathcal{B}}$. Consider now any path $Q\in {\mathcal{P}}^{i,j}$. Since cell $\mathsf{Cell}_{i,j}$ is bad, $Q$ must contain at least one outer edge. We say that $Q$ is a \emph{type-1} bad path for cell $\mathsf{Cell}_{i,j}$ if it contains at least one type-1 outer edge (recall that a type-1 outer edge $e$ corresponds to some edge in graph $H$ that is {\bf not} contained in any cluster in ${\mathcal{C}}$). Otherwise, every outer edge on path $Q$ is a type-2 outer edge, and in this case we say that $Q$ is a type-2 bad cluster for $\mathsf{Cell}_{i,j}$. We say that cell $\mathsf{Cell}_{i,j}$ is \emph{type-1 bad} iff at least $\psi/32$ paths of ${\mathcal{P}}^{i,j}$ are type-1 bad for this cell, and otherwise it is type-2 bad. We partition the set ${\mathcal{B}}$ of bad cells into two subsets: set ${\mathcal{B}}_1$ contains all type-1 bad cells, and set ${\mathcal{B}}_2$ contains all type-2 bad cells. It is now enough to prove that $|{\mathcal{B}}_1|,|{\mathcal{B}}_2|<(\lambda')^2/32$, which we do in the following two observations.
\begin{observation}\label{obs: few type-1 bad cells}
$|{\mathcal{B}}_1|< (\lambda')^2/32$.
\end{observation}
\begin{proof}
Assume for contradiction that $|{\mathcal{B}}_1|\geq (\lambda')^2/32$.
Consider a type-1 bad cell $\mathsf{Cell}_{i,j}\in {\mathcal{B}}_1$, and let ${\mathcal{Q}}^{i,j}\subseteq {\mathcal{P}}^{i,j}$ be a set of $\ceil{\psi/32}$ paths that are type-1 bad paths for cell $\mathsf{Cell}_{i,j}$. Each path in ${\mathcal{Q}}^{i,j}$ must contain at least one type-1 bad edge. Since the paths in ${\mathcal{Q}}^{i,j}$ are edge-disjoint, there is a set $E^{i,j}$ of at least $\psi/32$ type-1 outer edges of $H''$, lying on paths of ${\mathcal{Q}}^{i,j}$. Since every edge of $H''$ may lie on at most one path in ${\mathcal{P}}$, the total number of outer edges in $H''$ must be at least:
\[\frac{|{\mathcal{B}}_1|\cdot \psi}{32}\geq \frac{(\lambda')^2\cdot \psi}{2^{10}}\geq \frac{\lambda^2\cdot \psi}{2^{14}} ,\]
as $\lambda'=\floor{(\lambda-1)/2}\geq \lambda/4$.
Recall that $\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$ and $\lambda=\frac{2^{24}c\cdot \eta \log^8n}{\alpha^*\alpha^3}$, where $c$ is the constant from \Cref{obs: few outer edges}. Therefore, we get that the total number of outer edges in $H''$ is at least $\frac{2c\tilde k \eta \log^8n}{\alpha^3}$, contradicting \Cref{obs: few outer edges}.
\end{proof}
\begin{observation}\label{obs: few type-1 bad cells}
$|{\mathcal{B}}_2|< (\lambda')^2/32$.
\end{observation}
\begin{proof}
For a cluster $C\in {\mathcal{C}}$, let $X(C)=\bigcup_{y\in V(C)}V(\Pi(y))$. Note that all terminals of $H$ lie outside of the clusters in ${\mathcal{C}}$, and so $X(C)\cap \tilde T=\emptyset$. If a path $Q\in {\mathcal{P}}_3\cup {\mathcal{R}}_3$ contains a vertex of $X(C)$, then it must contain at least one edge of $\delta_H(C)$. As the paths in ${\mathcal{P}}\cup {\mathcal{R}}$ cause edge-congestion at most $2$, the total number of paths $Q\in {\mathcal{P}}\cup {\mathcal{R}}$ with a non-empty intersection with $X(C)$ is at most $2\delta_H(C)$.
Let $\mathsf{IntPairs}\subseteq {\mathcal{P}}_3\times{\mathcal{R}}_3$ be the collection of all pairs of paths $P\in {\mathcal{P}}_3$, $R\in {\mathcal{R}}_3$, such that $P$ and $R$ share at least one vertex.
For a cluster $C\in {\mathcal{C}}$, let $\mathsf{IntPairs}'_C\subseteq \mathsf{IntPairs}$ denote denote the collection of all pairs $(P,R)\in \mathsf{IntPairs}$ of paths, such that some vertex $v\in X(C)$ lies on both $P$ and $R$. Clearly, if $(P,R)\in \mathsf{IntPairs}_C'$, then each of the paths $P$, $R$ must contain at least one edge of $\delta_H(C)$. Therefore, from the above discussion, $|\mathsf{IntPairs}'_C|\leq 4|\delta_H(C)|^2$. Let $\mathsf{IntPairs}'=\bigcup_{C\in {\mathcal{C}}}\mathsf{IntPairs}'_C$. Then:
%
\[ |\mathsf{IntPairs}'|\leq \sum_{C\in {\mathcal{C}}}|\mathsf{IntPairs}'_C|\leq 4\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2. \]
%
From Equation \ref{eq: sum of squares} (see \Cref{step 2 summary}), $
\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}$, so we get that:
%
\begin{equation}\label{eq: bounding num of intersection pairs}
|\mathsf{IntPairs}'|<\frac{4(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n},
\end{equation}
%
where $c_1$ is an arbitrary large enough constant.
In the remainder of the proof, we assume for contradiction that $|{\mathcal{B}}_2|\geq (\lambda')^2/32$, and we will show that $|\mathsf{IntPairs}'|\geq \frac{4(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}n}$ must hold, contradicting Equation \ref{eq: bounding num of intersection pairs}.
Consider a type-2 bad cell $\mathsf{Cell}_{i,j}\in {\mathcal{B}}_2$. Recall that every path in ${\mathcal{P}}^{i,j}$ contains at least one outer edge, and at most $\psi/32$ such paths contain a type-1 bad edge. Since, from \Cref{obs: paths for cells}, $|{\mathcal{P}}^{i,j}|= \ceil{7\psi/8}$,
there is a collection $\Sigma\subseteq {\mathcal{P}}^{i,j}$ of at least $3\psi/4$ paths $P$, such that all edges on $P$ are either inner edges, or type-2 outer edges. Therefore, if $P\in \Sigma$ is any such path, then there is some cluster $C\in {\mathcal{C}}$ with $V(P)\subseteq X(C)$. Recall that, from \Cref{obs: paths for cells}, each path $P\in \Sigma$ intersects at least $\frac{13\psi}{16}$ paths of ${\mathcal{R}}^{i,j}$. Clearly, if a path $R\in {\mathcal{R}}^{i,j}$ intersects a path $P\in \Sigma$, then $(P,R)\in \mathsf{IntPairs}'$. Therefore, intersections between pairs of paths in ${\mathcal{P}}^{i,j}\times {\mathcal{R}}^{i,j}$ contribute at least $\frac{13\psi}{16}\cdot \frac{3\psi}{4}\geq \frac{\psi^2}{2}$ pairs to set $\mathsf{IntPairs}'$.
Therfore, if we denote by $\mathsf{IntPairs}'_{i,j}$ the collection of all pairs $(P,R)\in \mathsf{IntPairs}'$, where a subpath $\sigma$ of $P$ lies in ${\mathcal{P}}^{i,j}$, and a subpath $\sigma'$ of $R$ lies in ${\mathcal{P}}^{i,j}$, and $\sigma,\sigma'$ contain a vertex $v\in X(C)$, for some cluster $C\in {\mathcal{C}}$, then, from the above discussion, $|\mathsf{IntPairs}'_{i,j}|\geq \frac{\psi^2}{2}$. We claim that for every pair $(P,R)\in \mathsf{IntPairs}'$ of paths, there is at most one pair of indices $0\leq i,j\leq \lambda'+1$, such that $(P,R)\in \mathsf{IntPairs}_{i,j}'$. Indeed, assume that $P\in {\mathcal U}_i$ and $R\in {\mathcal U}'_j$. For a pair $0\leq i',j'\leq \lambda'+1$ of indices, ${\mathcal{P}}^{i',j'}$ contains a subpath of $P$ iff $i'=i$, and ${\mathcal{R}}^{i',j'}$ contains a subpath of $R$ iff $j'=j$. So the only pair $(i',j')$ of indices for which $(P,R) \in \mathsf{IntPairs}'_{i',j'}$ may hold is $(i,j)$. Overall, we get that $|\mathsf{IntPairs}'|\geq |{\mathcal{B}}_2|\cdot \psi^2/2$. Assuming that $|{\mathcal{B}}_2|\geq (\lambda')^2/32$, since $\lambda'=\floor{(\lambda-1)/2}\geq \lambda/4$, we get that $|\mathsf{IntPairs}'|\geq \frac{\lambda^2\psi^2}{1024}$.
Recall that $\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$, and $\alpha^*=\Theta(\tilde \alpha\alpha'/\log^4n)$.
We conclude that:
\[|\mathsf{IntPairs}'|\geq \frac{(\alpha^*)^2\tilde k^2}{2^{22}}\geq \Omega\textsf{left}( \frac{(\tilde \alpha\alpha'\tilde k)^2}{\log^8n} \textsf{right} ). \]
Since we can choose $c_1$ to be a sufficiently large constant, this contradicts Equation \ref{eq: bounding num of intersection pairs}.
\end{proof}
\end{proof}
\subsubsection{Square Subgrids and Corresponding Path Sets}
For integers $1\leq i,j\leq \lambda+1$ and $\ell\leq \min\set{i,j}$, a \emph{square subgrid $S(i,j,\ell)$ of $\Pi^*$} (that we also refer to as a \emph{square}) is defined as the collection of cells $\mathsf{CellSet}(S)=\set{\mathsf{Cell}_{i',j'}\mid i-\ell+1\leq i'\leq i;\quad j-\ell+1\leq j'\leq j}$. Intuitively, $S(i,j,\ell)$ is a subgrid of $\Pi^*$ of size $(\ell\times \ell)$, whose top right corner is the cell $\mathsf{Cell}_{i,j}$.
Given a square $S=S(i,j,\ell)$, we associate with it a collection ${\mathcal{P}}(S)$ of horizontal paths, and ${\mathcal{R}}(S)$ of vertical paths, as follows. Intuitively,
consider the graph obtained by taking the union of all paths ${\mathcal{P}}^{i',j'}$, where $\mathsf{Cell}_{i',j'}\in \mathsf{CellSet}(S)$.
This graph is a collection of disjoint paths, each of which is a subpath of a distinct path in $\bigcup_{i'=i-\ell+1}^i{\mathcal U}_{i'}$; we let ${\mathcal{P}}(S)$ be this set of paths.
Formally, for all $i-\ell+1\leq i'\leq i$, for every path $P\in {\mathcal U}_{i'}$, we include in ${\mathcal{P}}(S)$ the subpath of $P$ from the first vertex of $\sigma_{j-\ell+1}(P)$ to the last vertex of $\sigma_{j}(P)$.
Similarly, set ${\mathcal{R}}(S)$ contains, for all $j-\ell+1\leq j'\leq j$, for every path $R\in {\mathcal U}'_{j'}$, the subpath of $R$ from the first vertex of $\sigma'_{i-\ell+1}(R)$ to the last vertex of $\sigma_{i}(R)$.
Notice that, from \Cref{obs: paths for cells}, $|{\mathcal{P}}(S)|=|{\mathcal{R}}(S)|= \ceil{7\psi/8}\cdot \ell$.
We denote by $\mathsf{EntryPortals}(S)$ the set of all vertices that serve as the first endpoint of the paths in ${\mathcal{P}}(S)$, and by $\mathsf{ExitPortals}(S)$ the set of all vertices that serve as the last endpoint of the paths in ${\mathcal{P}}(S)$. We denote by $G(S)$ the graph obtained by the union of the paths in ${\mathcal{P}}(S)\cup {\mathcal{R}}(S)$.
The following claim will be crucial for our algorithm for computing the routing paths for each good cell.
\begin{claim}\label{claim: routing in square}
Let $S=S(i,j,\ell)$ be a square of $\Pi^*$, for some $1\leq i,j\leq \lambda+1$ and $\ell\leq \min\set{i,j}$, and let $Y\subseteq \mathsf{EntryPortals}(S)$, $Y'\subseteq\mathsf{ExitPortals}(S)$ be two subsets of vertices of cardinality $z$ each, where $z\leq \psi \ell/2$. Then there is a collection ${\mathcal{Q}}$ of edge-disjoint paths in graph $G(S)$, which is a one-to-one routing from $Y$ to $Y'$.
\end{claim}
\begin{proof}
Assume for contradiction that the claim is false. Then, from the maximum flow / minimum cut theorem, there is a collection $E'$ of at most $z-1$ edges in graph $G(S)$, such that $G(S)\setminus E'$ contains no path connecting a vertex of $Y$ to a vertex of $Y'$. Recall that each vertex of $Y$ is an endpoint of a distinct path in ${\mathcal{P}}(S)$, and all paths in ${\mathcal{P}}(S)$ are edge-disjoint. Since $|Y|=z$, while $|E'|\leq z-1$, there is some path $P\in {\mathcal{P}}(S)$, whose endpoint $y$ belongs to $Y$, such that $P$ contains no edge of $E'$. Using the same arguments, there is some path $P'\in {\mathcal{P}}(S)$, whose endpoint $y'$ belongs to $Y'$, that contains no edge of $E'$. Clearly, $P\neq P'$ must hold, as otherwise there is a path in $G(S)\setminus E'$ connecting $y$ to $y'$ -- the path $P$. It is now enough to show that there is some path $R\in {\mathcal{R}}(S)$, that contains no edge of $E'$, but $R\cap P\neq \emptyset$ and $R\cap P'\neq\emptyset$ hold. Indeed, in this case, $P\cup R\cup P'\subseteq G(S)\setminus E'$, and so $y$ remains connected to $y'$ in $G(S)\setminus E'$.
We now show that path $R$ with such properties must exist. Let $\tilde P\in {\mathcal{P}}_0$ be the path with $P\subseteq \tilde P$, and assume that $\tilde P\in {\mathcal U}_{i'}$ Similarly, let $\tilde P'\in {\mathcal{P}}_0$ be the path with $P'\subseteq \tilde P'$, and assume that $\tilde P'\in {\mathcal U}_{i''}$ (where possibly $i'=i''$). Consider some index $j-\ell'+1\leq j'\leq \ell$. Recall
$|{\mathcal U}'_{j'}|=\ceil{7\psi/8}$, and, for every path $R\in {\mathcal U}'_{j'}$, segment $\sigma_{i'}(R)$, lies in ${\mathcal{R}}^{i',j'}$, and segment $\sigma'_{i''}(R)$ lies in ${\mathcal{R}}^{i'',j'}$. Moreover, from \Cref{obs: paths for cells}, path $\sigma_{j'}(\tilde P)$ must intersect at least $\frac{13\psi}{16}$ paths of $\set{\sigma_{i'}(R)\mid {\mathcal U}'_{j'}}$, and
similarly path $\sigma_{j'}(\tilde P')$ must intersect at least $\frac{13\psi}{16}$ paths of $\set{\sigma_{i''}(R)\mid {\mathcal U}'_{j'}}$.
Therefore, there is a subset ${\mathcal U}''_{j'}\subseteq {\mathcal U}'_{j'}$ of at least $\psi/2$ paths $R$, such that both $P$ and $P'$ intersect the subpath of $R$ that belongs to ${\mathcal{R}}(S)$. Overall, there are at least $\ell \psi/2$ paths $R\in {\mathcal{R}}(S)$ that intersect the subpaths of $P$ and of $P'$ that lie in ${\mathcal{P}}(S)$. Since $z\leq \ell\psi/2$, at least one such path is disjoint from $E'$.
\end{proof}
\subsubsection{Routing the Terminals to Good Cells}
We fix some good cell $\mathsf{Cell}_{i,j}$ in the top right quadrant of the grid, that is, $\lambda'/2\leq i,j\leq\lambda'-1$. Recall that we have defined a vertex $y_{i,j}\in V(H)$, and a collection $\hat {\mathcal{R}}^{i,j}\subseteq {\mathcal{R}}^{i,j}$ of $\psi'=\ceil{13\psi/16}$ paths, each of which contains a vertex of $\Pi(y_{i,j})$. We have also defined a set $\mathsf{Portals}^{i,j}$ of vertices that contains, for every path $\sigma'\in \hat {\mathcal{R}}^{i,j}$, the first vertex on $\sigma'$.
We define a set ${\mathcal{Q}}_{i,j}$ of paths in $H''$, routing the terminals of $\tilde T$ to vertices of $\Pi(y_{i,j})$, so $({\mathcal{Q}}_{i,j},y_{i,j})\in \Lambda'$. In order to do so, we first define a path set ${\mathcal{Q}}'_{i,j}$, routing a constant fraction of the terminals of $T_4$ to vertices of $\Pi(y_{i,j})$, and then extend this path set in order to obtain routing of all terminals to vertices of $\Pi(y_{i,j})$.
\paragraph{Routing to $\mathsf{Cell}_{i,j}$.}
In order to define the routing, we let $z=\floor{\log(\lambda'/4)}$, and we define $z+1$ squares $S_0^{i,j},S_1^{i,j},\ldots,S_z^{i,j}$. In order to simplify the notation, we will omit the superscript $i,j$ for now.
Square $S_0$ is $S(i,j,1)$, so it consists of a single cell $\mathsf{Cell}_{i,j}$. We denote by $\mathsf{Portals}_0^{i,j}=\mathsf{Portals}^{i,j}$ the set of $\psi'$ vertices that we have defined. We let ${\mathcal{Q}}_0$ be the set of $\psi'$ paths, containing, for every path $\sigma'\in \hat{\mathcal{R}}^{i,j}$, a subpath of $\sigma'$ between a vertex of $\mathsf{Portals}_0^{i,j}$ and a vertex of $\Pi(y_{i,j})$. Therefore, ${\mathcal{Q}}_0$ is a set of $\psi'$ edge-disjoint paths, routing vertices of $\mathsf{Portals}_0^{i,j}$ to vertices of $\Pi(y_{i,j})$, and all paths of ${\mathcal{Q}}_0$ are contained in ${\mathcal{R}}(S_0)$. We say that cell $\mathsf{Cell}_{i,j}$ is the bottom right corner of square $S_0$.
Fix some index $1\leq r\leq z$, and
assume that we have defined squares $S_0,\ldots,S_{r-1}$. We now define square $S_r$. We let $S_r=(i_r,j,2^r)$, so the length of the side of the square is $2^r$, and the coordinates of the top right corner of $S_r$ are $(i_r,j)$; here, $j$ is the column index of the initial cell $\mathsf{Cell}_{i,j}$, and $i_r$ is the cell immediately under the right bottom corner cell of $S_{r-1}$. In other words, if $S_{r-1}=(i_{r-1},j,2^{r-1})$, then $i_r=i_{r-1}+2^{r-1}$.
We assume that we have also defined a collection $\mathsf{Portals}_{r-1}\subseteq \mathsf{EntryPortals}(S_{r-1})$, containing $2^{r-1}\cdot \psi'$ vertices. Note that the top boundary of square $S_r$ appears immediately under bottom boundary of square $S_{r-1}$, so $\mathsf{EntryPortals}(S_{r-1})\subseteq \mathsf{ExitPortals}(S_r)$, and in particular $\mathsf{Portals}_{r-1}\subseteq \mathsf{ExitPortals}(S_r)$. We select an arbitrary subset $\mathsf{Portals}_r\subseteq \mathsf{EntryPortals}(S_r)$ of $2^r\cdot \psi'$ vertices. By partitioning set $\mathsf{Portals}_r$ into two equal-cardinality subsets $Y_1,Y_2$, and applying \Cref{claim: routing in square} to each of them separately, we obtain two collections ${\mathcal{Q}}_{r}^1,{\mathcal{Q}}_{r}^2$ of edge-disjoint paths in graph $G(S_r)$, routing vertex sets $Y_1$ and $Y_2$, respectively, to vertex set $\mathsf{Portals}_{r-1}$, in a one-to-one routing. Therefore, there is a set ${\mathcal{Q}}_{r}$ of paths in $G(S_r)$, routing vertex set $\mathsf{Portals}_r$ to vertex set $\mathsf{Portals}_{r-1}$ with edge-congestion at most $2$, such that every vertex in $\mathsf{Portals}_{r-1}$ is an endpoint of at most two such paths. Moreover, we can compute such set ${\mathcal{Q}}_r$ of paths efficiently via standard maximum flow.
Lastly, consider the last square $S_z$. We define a subset $T^*\subseteq T_4$ of terminals, as follows. For every vertex $v\in \mathsf{Portals}_z$, let $R_v\in {\mathcal{R}}$ be the vertical path containing $v$, and let $t_v\in T_4$ be the terminal that serves as an endpoint of path $R_v$. We then let $T^*=\set{t_v\mid v\in \mathsf{Portals}_z}$, and we let ${\mathcal{Q}}_{z+1}$ be a set of paths containing, for every vertex $v\in \mathsf{Portals}_z$, the subpath of $R_v$ between $t_v$ and $v$. Therefore, set ${\mathcal{Q}}_{z+1}$ of paths routes terminals of $T^*$ to vertices of $\mathsf{Portals}_z$, and the paths in ${\mathcal{Q}}_{z+1}$ are edge-disjoint. It is also easy to verify that the paths in ${\mathcal{Q}}_{z+1}$ do not contain any edges from graphs $G(S_0)\cup \cdots\cup G(S_z)$.
Note that, since $\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$ and $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$,
\[|T^*|=2^z\cdot \psi'= 2^{\floor{\log(\lambda'/4)}}\cdot \ceil{13\psi/16}=\Omega(\lambda'\cdot \psi)=\Omega(\lambda\psi)=\Omega(\alpha^*\tilde k)=\Omega(\tilde k\alpha\alpha'/\log^4n).
\]
To summarize, we have defined a collection $\set{S_0,\ldots,S_z}$ of squares in the grid $\Pi^*$, where for all $0\leq r\leq z$, square $S_r$ has dimensions $(2^r\times 2^r)$. The squares are aligned on the right, and are stacked on top of each other, with square $S_0$ containing a single cell, $\mathsf{Cell}_{i,j}$. This guarantees that all corresponding graphs $G(S_r)$ are mutually disjoint, except that, for all $0\leq r<z$, $V(S_r)\cap V(S_{r+1})=\mathsf{EntryPortals}(S_r)$. We have defined, for all $0\leq r\leq z$, a set $\mathsf{Portals}_r\subseteq \mathsf{EntryPortals}(S_r)$ of $2^r\cdot \psi'$ vertices, and a set ${\mathcal{Q}}_r$ of paths contained in $G(S_r)$, routing vertices of $\mathsf{Portals}_r$ to vertices of $\mathsf{Portals}_{r-1}$ with edge-congestion at most $2$, so that every vertex of $\mathsf{Portals}_{r-1}$ serves as an endpoint of at most two such paths. Additionally, in graph $G(S_0)$, we have defined a set ${\mathcal{Q}}_0$ of $\psi'$ paths routing vertices of $\mathsf{Portals}_0$ to vertices of $\Pi(y_{i,j})$, and an additional set ${\mathcal{Q}}_{z+1}$ of edge-disjoint paths routing terminals in $T^*$ to vertices of $\mathsf{Portals}_z$ in a one-to-one routing, so that paths in ${\mathcal{Q}}_{z+1}$ do not contain edges of $G(S_0)\cup\cdots\cup G(S_z)$.
We are now ready to define a set ${\mathcal{Q}}'_{i,j}$ of paths, routing terminals of $T^*$ to vertices of $\Pi(y_{i,j})$. In order to do so, for all $0\leq r\leq z$, we let ${\mathcal{Q}}'_r$ be a multi-set of paths, contianing, for every path $\sigma'\in {\mathcal{Q}}_r$, $2^{z-r}$ copies of the path $\sigma'$. Therefore, paths in ${\mathcal{Q}}'_r$ cause edge-congestion $2^{z-r+1}$ in $G(S_r)$. Set ${\mathcal{Q}}'_{i,j}$ of paths is obtained by concatenating paths in sets ${\mathcal{Q}}_{z+1},{\mathcal{Q}}'_z,\ldots,{\mathcal{Q}}'_0$. It is easy to verify that paths in ${\mathcal{Q}}'_{i,j}$ route all terminals in $T^*$ to vertices of $\Pi(y_{i,j})$.
Recall that $|T^*|=\Omega(\tilde k\alpha\alpha'/\log^4n)$, and $|\tilde T|=\tilde k$.
Moreover, from \Cref{obs: terminals well linked in H''},
The set $\tilde T$ of terminals is $\alpha^*$-well-linked in $H''$, where $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$.
From \Cref{lem: routing path extension},
there is a set ${\mathcal{Q}}_{i,j}$ of paths in graph $H''$, routing all vertices of $\tilde T$ to vertices of $\Pi(y_{i,j})$, such that, for every edge $e\in E(H'')$:
\[\cong_{H''}({\mathcal{Q}}_{i,j},e)\le
\ceil{\frac{\tilde k}{|T^*|}}\textsf{left} (\cong_{H''}({\mathcal{Q}}'_{i,j},e)+\ceil{1/\alpha^*}\textsf{right} )\leq
O\textsf{left} (\frac{\log^4n}{\alpha\alpha'}\textsf{right} )\cdot \textsf{left}(\cong_{H''}({\mathcal{Q}}'_{i,j},e)+\frac{\log^4n}{\alpha\alpha'}\textsf{right} ).\]
\paragraph{Distribution ${\mathcal{D}}$ and Analysis.}
The final distribution ${\mathcal{D}}$ over pairs in $\Lambda'$ is defined as follows. For every pair $(i,j)$ of indices in $Z$, pair $({\mathcal{Q}}_{i,j},y_{i,j})$ is assigned the same distribution $1/|Z|$; recall that, from \Cref{claim: many good cells},
$|Z|\geq (\lambda')^2/16$.
We now fix some vertex $e\in E(H'')$, and analyze the expectation $\expect[({\mathcal{Q}}_{i,j},y_{i,j})\in_{\mathcal{D}} \Lambda']{(\cong_{H''}({\mathcal{Q}}_{i,j},e))^2}$.
Recall that there is at most one path $P\in {\mathcal{P}}$ that contains $e$, and at most one path $R\in {\mathcal{R}}$ containing $e$. Moreover, there is at most one pair $(i_1,j_1)$ of indices with $e\in \sigma_{i_1,j_1}(P)$, and at most one pair $(i_2,j_2)$ of indices with $e\in \sigma_{i_2,j_2}(R)$.
We first focus on pair $(i_1,j_1)$ of indices, and the corresponding cell $\mathsf{Cell}_{i_1,j_1}$. Fix some pair $(i,j)\in Z$ of indices, and $0\leq r\leq z$. If $\mathsf{Cell}_{i_1,j_1}\in S_r^{i,j}$, then segment $\sigma_{i_1,j_1}(P)$ of $P$ may lie on at most $2^{z-r+1}$ paths in ${\mathcal{Q}}'_{i,j}$. Notice that there are at most $2^{2r+2}$ square subgrids $S$ of $\Pi^*$ of dimension $(2^r\times 2^r)$, that contain the cell $\mathsf{Cell}_{i_1,j_1}$. For each such square $S$, there is exactly one pair $(i(S),j(S))$ of indices, for which $S_r^{i(S),j(S)}=S$. Since $|Z|\geq (\lambda')^2/16$, the probability that an index $(i,j)\in Z$ is chosen for which $\mathsf{Cell}_{i_1,j_1}$ lies in the square $S_r^{i,j}$ is at most $O(2^{2r+2}/(\lambda')^2)$. Recall that, if $\mathsf{Cell}_{i_1,j_1}\in S_r^{i,j}$, then $\cong_{H''}({\mathcal{Q}}'_{i,j})\leq 2^{z-r+1}\leq O(\lambda'/2^r)$. Moreover, if $\mathsf{Cell}_{i_1,j_1}$ does not lie in any of the squares $S_0^{i,j},\ldots,S_z^{i,j}$, then $\cong_{H''}({\mathcal{Q}}'_{i,j})\leq 1$. The analysis for cell $\mathsf{Cell}_{i_2,j_2}$ is symmetric. Therefore, altogether (now taking into account both the cells $\mathsf{Cell}_{i_1,j_1}$ and $\mathsf{Cell}_{i_2,j_2}$), we get that:
\[\expect[(i,j)\in Z]{(\cong_{H''}({\mathcal{Q}}'_{i,j},e))^2}\leq O(1)+\sum_{r=0}^zO\textsf{left} (\frac{2^{2r+2}}{(\lambda')^2}\cdot\frac{(\lambda')^2}{2^{2r}}\textsf{right} )=O\textsf{left}(
\frac{2^z}{\lambda'}\textsf{right} )=O(z)\leq O(\log n).
\]
Lastly, since $\cong_{H''}({\mathcal{Q}}_{i,j},e)\le
O\textsf{left} (\frac{\log^4n}{\alpha\alpha'}\textsf{right} )\cdot \textsf{left}(\cong_{H''}({\mathcal{Q}}'_{i,j},e)+\frac{\log^4n}{\alpha\alpha'}\textsf{right} )$, we get that:
\[\expect[({\mathcal{Q}}_{i,j},y_{i,j})\in_{\mathcal{D}} \Lambda']{(\cong_{H''}({\mathcal{Q}}_{i,j},e))^2}\leq O\textsf{left} (\frac{\log^{16}n}{(\alpha\alpha')^4}\textsf{right} ).\]
\iffalse
Good cell part:
will only look at cells in top left quadrant. May need to swap horizontal and vertical. Not completely symmetric b/c of bottom-up / left-right direction of paths (the segment are defined w.r.t. this direction).
A horizontal path $P$ is good for cell $(i,j)$ iff its segment contains no outer edges, so it is contained in $\Pi(v)$ of some vertex $v$. A cell is good if it has at least one good path. This means that all vertical paths in the cell also visit $\Pi(v)$ in the cell. We want to prove that at least $0.8$-fraction of the cells are good.
In a bad cell, partition paths into two types: type-1 path for cell $(i,j)$ contains an outer edge that is not contained in any cluster (type-1 outer edge), otherwise type-2. A bad cell is type-1 if at least 0.1-frac of the horizontal paths are type-1 for the cell, and type-2 otherwise.
If we have many type-1 cells, then we have about $\psi\lambda^2=\alpha^*\tilde k\lambda$ type-1 outer edges, contradicting Obs. 6.9.
(need to do same analysis with horizontal paths, there is no choice).
So at the end we'll have a constant fraction of cells that are type-2 cells for both vertical and horizontal paths. In each such cell we'll have about $\psi^2$ pairs of paths meet at vertices that are inner to clusters. So overall, we have $(\lambda^2\psi^2)$ pairs of paths meeting at vertices that are inner to clusters. but for a cluster $C$, we have at most $|\delta(C)|$ paths entering it, so at most $|\delta(C)|^2$ pairs of paths can meet in inner vertices of $C$. So total number of such intersections is at most $\sum_{C\in {\mathcal{C}}}|\delta(C)|^2$, which should hopefully be less than $(\lambda\psi)^2=(\alpha^*)^2\tilde k^2$. We currently only have $\sum_C|\delta(C)|^2<(\tilde k\tilde \alpha\alpha')^2/\log^{20}n$. $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, so looks OK.
\subsection{Plan for the rest of the proof}
\begin{itemize}
\item select 1 path in $z$, where for example $z=\eta$, though can also use $\eta^2$. Will get a grid skeleton of size $z\times z$.
\item before that, throw away all paths that have more than $\eta/\alpha$ outer edges on them. Most paths will stay.
\item now if we look at a vertical path, it has few outer edges, so for most cells it goes through it does not use an outer edge. Which means that all horizontal paths going through that cell meet in a single vertex (or go through a single cluster). Call such a cell good.
\item at least half the cells are good. In a type-1 good cell all horizontal paths go through a single regular vertex. In a type-2 good cell they all go through 1 cluster. The hope is that there are few type-2 good cells.
\item why: we have about $z^2$ good cells. Inside all good cells there are about $k^2/z^2$ inner edges (can make a cell good if both all horizontal paths meet at some vertex, and all vertical paths meet at some vertex, and then it should be the same vertex). The hope is that the clusters can only contribute $\sum_C|\delta(C)|^2$ vertices to the grid. This is because, if we look at a cluster $C$, the total number of vertical/horizontal paths entering $C$ is at most $\delta(C)$. Even if all pairs have their intersections inside this cluster, it can't contribute more than $|\delta(C)|^2$ such intersecting pairs.
\end{itemize}
\fi
\newpage
\iffalse
contracted graph $\hat H=H_{|{\mathcal{C}}}$. Let $U=\set{v(C)\mid C\in {\mathcal{C}}}$ be the set of super-node vertices of $\hat H$, and let $U'$ be the set of all remaining vertices. We define a rotation system $\hat \Sigma$ for graph $\hat H$ as follows: for a super-node $v(C)\in U$, we let ${\mathcal{O}}_{v(C)}$ be an arbitrary ordering of the edges incident to $v(C)$, and for a regular vertex $u\in U'$, we let ${\mathcal{O}}_u$ be identical to the ordering in the rotation system $\Sigma$ in instance $I$. Recall that, from \Cref{lem: crossings in contr graph}, there is a drawing $\phi$ of $\hat H$, containing at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)$ crossings, such that for every regular vertex $u\in U'$, the ordering of the edges of $\delta_{\hat H}(u)$ as they enter $u$ in $\phi$ is consistent with ${\mathcal{O}}_u\in \Sigma$. We can use this drawing in order to define a solution $\phi'$ to instance $\hat I=(\hat H,\hat \Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, by reordering the edges entering every super-node $v(C)$ as needed. This can be done so that $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)+\sum_{C\in {\mathcal{C}}'}|\delta_H(C)|^2\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)+k^2/x$. Therefore, $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)+k^2/x$.
We are now ready to define graph $\hat H^+$. In order to do so, we start with graph $\hat H$, and process every vertex $u\in V(\hat H)\setminus T$ (that may be a regular vertex or a super-node one-by-one). Consider any such vertex $u$, and let $e_1,\ldots,e_{d(u)}$ be the edges incident to $u$ in $\hat H$, indexed according to their ordering in ${\mathcal{O}}_u\in \hat \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ vrid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$. We replace vertex $u$ with the grid $\Pi(u)$, and, for all $1\leq i\leq d(u)$, if $e_i=(u',u)$, then we replace $e_i$ with a new edge $e'_i=(u',s_i(u))$. Once every vertex of $V(\hat H)\setminus T$ is processed, we obtain the final graph $\hat H^+$. For every vertex $u$, we clall the edges of the grid $\Pi(u)$ \emph{inner edges}. We call all edges of $E(\hat H^+)$ that are not inner edges \emph{outer edges}. Notice that there is a $1$-to-$1$ correspondence between the outer edges and the edges of graph $\hat H$. The following observation is immediate.
\begin{observation}\label{obs: expanded contracted graph}
The set $T$ of terminals is $(\alpha\alpha')$-well-linked in $\hat H^+$, and there is a drawing $\phi''$ of graph $\hat H^+$ with at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8n/(\alpha')^2)+k^2/x$ crossings, such that the inner edges do not participate in any crossings in $\phi''$.
\end{observation}
\fi
\section{First Set of Tools: Light Clusters, Bad Clusters, Path-Guided Orderings, and Basic Cluster Disengagement}
\label{sec: guiding paths orderings basic disengagement}
The main goal of this section is to define and analyze the \textsf{Basic Cluster Disengagement}\xspace procedure. Along the way we will define several other central tools that we use throughout the paper, such as light clusters, bad clusters, and path-guided orderings.
We start by defining and analyzing laminar family-based disengagement procedure, which will serve as the basis of the basic disengagement procedure.
\input{laminar-disengagement}
\input{light-bad-guided}
\input{basic-disengagement}
\iffalse
\subsection{Contracted Instance}
\label{subsec: contracted instance}
Assume that we are given an instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, a collection ${\mathcal{C}}$ of disjoint clusters of $G$,
that is partitioned into two subsets, ${\mathcal{C}}^{\operatorname{light}}$ and ${\mathcal{C}}^{\operatorname{bad}}$. Assume further that each cluster $C\in {\mathcal{C}}^{\operatorname{bad}}$ is $\beta$-bad and has the $\alpha_0$-bandwidth property for some $\alpha_0=\Omega(1/\log^{12}m)$, and for each cluster $C\in {\mathcal{C}}^{\operatorname{light}}$ we are given a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers, such that $C$ is $\beta$-light with respect to ${\mathcal{D}}(C)$, for some parameter $\beta$.
We construct a \emph{contracted instance} $\hat I=(\hat G,\hat \Sigma)$ as follow. First, we let $\hat G=G_{|{\mathcal{C}}}$.
Next, we define a rotation system $\hat \Sigma$ for $\hat G$ as follows. For
every regular vertex $v\in V(G)\cap V(\hat G)$, the ordering $\hat {\mathcal{O}}_v\in \hat \Sigma$ of the edges of $\delta_{\hat G}(v)=\delta_G(v)$ is identical to the ordering ${\mathcal{O}}_v\in \Sigma$. For every supernode $v_C$, where $C\in {\mathcal{C}}^{\operatorname{bad}}$, we let the ordering $\hat {\mathcal{O}}_{v_C}\in \hat \Sigma$ of the edges of $\delta_{\hat G}(v_C)$ be arbitrary. Lastly, for each supernode $v_C$ with $C\in {\mathcal{C}}^{\operatorname{light}}$,
we first select an internal router ${\mathcal{Q}}(C)$ for $C$ from the distribution ${\mathcal{D}}(C)$, then apply the algorithm from \Cref{lem: non_interfering_paths} to the set ${\mathcal{Q}}(C)$ of paths and the instance $(G,\Sigma)$ to obtain an new internal router $\hat{\mathcal{Q}}(C)$ for $C$,
and finally we set the ordering $\hat {\mathcal{O}}_{v_C}\in \hat \Sigma$ of the edges of $\delta_{\hat G}(v_C)=\delta_G(C)$ to be ${\mathcal{O}}^{\operatorname{guided}}(\hat{\mathcal{Q}}(C),\Sigma)$, the ordering guided by the set $\hat{\mathcal{Q}}(C)$ of paths and the rotation system $\Sigma$.
Notice that the resulting contracted instance $\hat I=(\hat G,\hat \Sigma)$ is a random instance, in that the construction of the rotation system $\hat \Sigma$ uses random choices. Notice however that, since the algorithm from \Cref{lem: non_interfering_paths} is deterministic, once we fix, for every cluster $C\in {\mathcal{C}}^{\operatorname{light}}$, the choice of the internal $C$-router ${\mathcal{Q}}(C)\in \Lambda(C)$, the rotation system $\hat \Sigma$ is fixed as well.
\begin{theorem}\label{thm: cost of contracted}
Let $I=(G,\Sigma)$ be an instance of \textnormal{\textsf{MCNwRS}}\xspace, let ${\mathcal{C}}={\mathcal{C}}^{\operatorname{bad}}\cup {\mathcal{C}}^{\operatorname{light}}$ be a collection of disjoint clusters of $G$. Assume that each cluster $C\in {\mathcal{C}}^{\operatorname{bad}}$ is $\beta$-bad and has the $\alpha_0$-bandwidth property for some $\alpha_0=\Omega(1/\log^{12}m)$, and for each cluster $C\in {\mathcal{C}}^{\operatorname{light}}$ we are given a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers, such that $C$ is $\beta$-light with respect to ${\mathcal{D}}(C)$, for some parameter $\beta$. Let $\hat I=(\hat G,\hat \Sigma)$ be the corresponding (random) contracted instance. Then:
\[\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)}\leq O(\beta^2\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E({\mathcal{C}})|)).\]
\end{theorem}
\begin{proof}
For each cluster $C\in {\mathcal{C}}^{\operatorname{light}}$, we denote by ${\mathcal{Q}}(C)$ be the internal $C$-router sampled from the distribution ${\mathcal{D}}(C)$, and we denote by $\hat{\mathcal{Q}}(C)$ the internal $C$-router obtained by applying the algorithm from \Cref{lem: non_interfering_paths} to ${\mathcal{Q}}(C)$ and instance $(G,\Sigma)$.
For each cluster $C\in {\mathcal{C}}^{\operatorname{bad}}$, since $C$ has the $\alpha_0$-bandwidth property for some $\alpha_0=\Omega(1/\log^{12}m)$, we apply the algorithm from \Cref{cor: simple guiding paths} to cluster $C$ in graph $G$ and obtain a distribution ${\mathcal{D}}$ over the the set $\Lambda(C)$ of internal $C$-routers. Let ${\mathcal{Q}}(C)$ be an internal $C$-router that is sampled from ${\mathcal{D}}$. We denote by $u(C)$ the target vertex of ${\mathcal{Q}}(C)$. From \Cref{cor: simple guiding paths}, for each edge $e\in E(C)$, $\expect{\cong_G({\mathcal{Q}}(C),e)}\leq O(\log^4 m/\alpha_0)\le \beta$.
Denote $G'=G\setminus (\bigcup_{C\in {\mathcal{C}}}C)$, so $G'$ is a cluster of $G$.
We construct a solution $\hat\phi$ to the instance $\hat I$ as follows.
We start with drawing $\phi^*$, and process the clusters of ${\mathcal{C}}$ one-by-one.
We now describe the iteration of processing a cluster $C$.
Assume first that $C\in {\mathcal{C}}^{\operatorname{light}}$. We apply the algorithm from \Cref{thm: new nudging} to graph $G$, cluster $G'$, set $\hat{\mathcal{Q}}(C)$ of paths and the ordering ${\mathcal{O}}_{u(C)}$, where $u(C)$ is the common last endpoint of all paths in $\hat{\mathcal{Q}}(C)$.
Let $\Gamma_C=\set{\gamma_C(e)\mid e\in \delta_G(C)}$ be the set of curves we get.
Denote $z_C=\phi^*(u(C))$, so all curves of $\Gamma_C$ have the last endpoint $z_C$.
From \Cref{thm: new nudging}, the order in which curves of $\Gamma_C$ enter the point $z_C$ is identical to ${\mathcal{O}}^{\operatorname{guided}}({\mathcal{Q}}(C),\Sigma)$. Moreover, since the set $\hat{\mathcal{Q}}(C)$ is non-transversal with respect to $\Sigma$, from \Cref{lem: non_interfering_paths},
\[\chi(\Gamma_C)\le \chi^2(\hat{\mathcal{Q}}(C))+\sum_{e\in E(C)}(\cong_G(\hat{\mathcal{Q}}(C),e))^2\le \chi^2({\mathcal{Q}}(C))+\sum_{e\in E(C)}(\cong_G({\mathcal{Q}}(C),e))^2.\]
Assume now that $C\in {\mathcal{C}}^{\operatorname{bad}}$.
We apply the algorithm from \Cref{cor: new type 2 uncrossing} to graph $G$, cluster $G'$, and the set ${\mathcal{Q}}(C)$ of paths.
Let $\Gamma'_C=\set{\gamma_C(e)\mid e\in \delta_G(C)}$ be the set of curves we get. We then modify the ordering in which curves of $\Gamma'_C$ enter the point $z_C=\phi^*(u(C))$ by applying the algorithm from \Cref{lem: ordering modification} to $\Gamma'_C$ and the ordering $\hat{\mathcal{O}}_{v_C}$, and denote by $\Gamma_C$ the obtained set of curves.
It is clear that the number of crossings between the set $\Gamma_C$ of curves is bounded by $|\delta_G(C)|^2$.
Let $\set{\Gamma_C\mid C\in {\mathcal{C}}}$ be the sets of curves that we obtain after processing all clusters of ${\mathcal{C}}$ in the same way. It is clear that, if we view, for each $C\in {\mathcal{C}}$, (i) the point $z_C$ as the image of $v_C$; and (ii) for each $e\in \delta_G(C)$, the curve $\gamma_C(e)\in\Gamma_C$ as the image of $e$, then we obtain a drawing of $\hat G$ that respects the rotation system $\hat \Sigma$, that we denote by $\hat\phi$. Moreover,
\iffalse{analysis with temp}
\[
\begin{split}
\mathsf{cr}(\hat \phi)
= & \text{ }\chi^2(G\setminus \bigcup_{C\in {\mathcal{C}}}C)+\sum_{C\in {\mathcal{C}}}(\chi(\tilde\Gamma'_C)+\chi(\tilde\Gamma'_C, \phi^*(G\setminus \bigcup_{C\in {\mathcal{C}}}C))) + \sum_{C,C'\in {\mathcal{C}}, C\ne C'}\chi(\tilde\Gamma'_C, \tilde\Gamma'_{C'})\\
\le &\text{ }\sum_{(e,e'): \text{ } e,e'\text{ cross in }\phi^*}\max\set{\frac{\cong_{G}({\mathcal{Q}},e)^2}{2},1}+\max\set{\frac{\cong_{G}({\mathcal{Q}},e')^2}{2},1}\\
= & \text{ } \sum_{e\in E(G)}\chi(e)\cdot\max\set{\frac{\cong_{G}({\mathcal{Q}},e)^2}{2},1}\\
\le & \sum_{e\in E(G)}\chi(e)\cdot O(\beta)= O(\mathsf{cr}(\phi^*)\cdot\beta).
\end{split}
\]
\fi
\[
\begin{split}
\mathsf{cr}(\hat \phi)
= & \text{ }\chi^2(G')+\sum_{C\in {\mathcal{C}}}(\chi(\Gamma_C)+\chi(\Gamma_C, \phi^*(G'))) + \sum_{C,C'\in {\mathcal{C}}, C\ne C'}\chi(\Gamma_C, \Gamma_{C'})\\
\le &\text{ }\chi^2(G')+\sum_{C\in {\mathcal{C}}^{\operatorname{light}}}\bigg(\chi^2({\mathcal{Q}}(C))+\chi({\mathcal{Q}}(C),G')+\sum_{e\in E(C)}\cong({\mathcal{Q}}(C),e)^2\bigg) \\
& +\sum_{C,C'\in {\mathcal{C}}, C\ne C'}\chi({\mathcal{Q}}(C), {\mathcal{Q}}(C'))
+\sum_{C\in {\mathcal{C}}^{\operatorname{bad}}}\bigg(|\delta_G(C)|^2+\chi({\mathcal{Q}}(C),G')\bigg).
\end{split}
\]
Note that
$\chi^2(G')\le \mathsf{cr}(\phi^* )$,
$$\expect{\sum_{C\in {\mathcal{C}}^{\operatorname{light}}}\sum_{e\in E(C)}\bigg(\cong({\mathcal{Q}}(C),e)\bigg)^2}\le \sum_{C\in {\mathcal{C}}^{\operatorname{light}}}\beta\cdot |E(C)|\le \beta\cdot|E({\mathcal{C}})|,$$
$$\expect{\sum_{C\in {\mathcal{C}}}\chi({\mathcal{Q}}(C),G')}= \expect{\sum_{C\in {\mathcal{C}}}\sum_{e\in E(C)}\chi(e,G')\cdot\cong_{G}({\mathcal{Q}}(C),e)}\le O(\beta\cdot \mathsf{cr}(\phi^*)),$$
\[
\begin{split}\expect{\sum_{C\in {\mathcal{C}}^{\operatorname{light}}}\chi^2({\mathcal{Q}}(C))} & = \expect{\sum_{C\in {\mathcal{C}}^{\operatorname{light}}}\sum_{e,e'\in E(C)}\chi(e,e')\cdot\cong_{G}({\mathcal{Q}}(C),e)\cdot\cong_{G}({\mathcal{Q}}(C),e')}\\
& \le \expect{\sum_{C\in {\mathcal{C}}^{\operatorname{light}}}\sum_{e,e'\in E(C)}\chi(e,e')\cdot\frac{(\cong({\mathcal{Q}}(C),e))^2+(\cong({\mathcal{Q}}(C),e'))^2}{2}}\\
& = \expect{\sum_{C\in {\mathcal{C}}^{\operatorname{light}}}\sum_{e\in E(C)}\chi(e,C\setminus\set{e})\cdot\frac{(\cong({\mathcal{Q}}(C),e))^2}{2}}\\
&\le \beta\cdot \sum_{C\in {\mathcal{C}}^{\operatorname{light}}}\chi^2(C)\le O(\beta\cdot \mathsf{cr}(\phi^*)),
\end{split}\]
\[
\begin{split}
\expect{\sum_{C\ne C'\in {\mathcal{C}}}\chi({\mathcal{Q}}'(C),{\mathcal{Q}}(C))} & = \expect{\sum_{C\ne C'\in {\mathcal{C}}}\sum_{e\in E(C)}\sum_{e'\in E(C')}\chi(e,e')\cdot\cong({\mathcal{Q}}(C),e)\cdot\cong({\mathcal{Q}}(C'),e')}\\
& = \sum_{C\ne C'\in {\mathcal{C}}}\sum_{e\in E(C)}\sum_{e'\in E(C')}\chi(e,e')\cdot\expect{\cong({\mathcal{Q}}(C),e)}\cdot\expect{\cong({\mathcal{Q}}(C'),e')}\\
& = \sum_{C\ne C'\in {\mathcal{C}}}\sum_{e\in E(C)}\sum_{e'\in E(C')}\chi(e,e')\cdot\beta^2\le O(\beta^2\cdot \mathsf{cr}(\phi^*)).
\end{split},\]
and
\[
\sum_{C\in {\mathcal{C}}^{\operatorname{bad}}}|\delta_G(C)|^2 \le \beta\cdot
\sum_{C\in {\mathcal{C}}^{\operatorname{bad}}}(\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)+|E(C)|)\le O(\beta\cdot (\mathsf{cr}(\phi^*)+|E({\mathcal{C}})|)),
\]
Altogether, they imply that $\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)}\leq \expect{\mathsf{cr}(\hat\phi)}\le O(\beta^2\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E({\mathcal{C}})|))$.
\end{proof}
\subsection{Cluster-Based Instance}
\label{subsec: cluster-based instance}
Assume that we are given an instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem,
a cluster $C\subseteq G$, and a distribution ${\mathcal{D}}'(C)$ on the set $\Lambda'_G(C)$ of external $C$-routers.
We now consider two cases. In the first case, we assume that cluster $C$ is a $\beta$-bad cluster, and in the second case we assume that we are given a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers, such that $C$ is $\beta$-light with respect to ${\mathcal{D}}(C)$, for some parameter $\beta$. For each of the two cases, we define a cluster-based instance $I_C=(G_C,\tilde \Sigma_C)$ associated with cluster $C$, as follows.
In each of the two cases, graph $G_C$ is obtained from graph $G$, by contracting all vertices of $V(G)\setminus V(C)$ into a single vertex $v^*$. The rotation system $\tilde \Sigma_C$ is defined as follows. Consider first some vertex $v\in V(G_C)\setminus \set{v^*}$. Note that $\delta_{G_C}(v)=\delta_G(v)$. We set the ordering $\tilde {\mathcal{O}}_v\in \tilde \Sigma_C$ of the edges of $\delta_{G_C}(v)$ to be identical to the ordering ${\mathcal{O}}_v\in \Sigma$. In order to fix the rotation for vertex $v^*$, observe that $\delta_{G_C}(v^*)=\delta_G(C)$. If $C$ is a $\beta$-bad cluster, then we set the ordering $\tilde {\mathcal{O}}_{v^*}\in \tilde \Sigma_C$ to be arbitrary. Otherwise, $C$ is a $\beta$-light cluster, with respect to the distribution ${\mathcal{D}}(C)$. In this case we randomly select an internal router ${\mathcal{Q}}(C)\in \Lambda(C)$ from the distribution ${\mathcal{D}}(C)$, and then apply the algorithm from \Cref{lem: non_interfering_paths} to the set ${\mathcal{Q}}(C)$ of paths and the instance $(G,\Sigma)$ to obtain a new set $\hat{\mathcal{Q}}(C)$ of internal $C$-routers,
and finally we set the ordering $\hat {\mathcal{O}}_{v_C}\in \hat \Sigma$ of the edges of $\delta_{\hat G}(v_C)=\delta_G(C)$ to be ${\mathcal{O}}^{\operatorname{guided}}(\hat{\mathcal{Q}}(C),\Sigma)$, the ordering guided by the set $\hat{\mathcal{Q}}(C)$ of paths and the rotation system $\Sigma$.
\begin{theorem}\label{thm: cost of single cluster}
Let $I=(G,\Sigma)$ be an instance of \textnormal{\textsf{MCNwRS}}\xspace, and let $\phi^*$ be an optimal solution to instance $I$.
Let $C$ be a cluster of $G$, and let ${\mathcal{D}}(C)$ be a distribution over the collection $\Lambda(C)$, such that either $C$ is a $\beta$-bad cluster, or it is a $\beta$-light cluster with respect to ${\mathcal{D}}(C)$.
Assume further that we are given a distribution ${\mathcal{D}}'(C)$ on the set $\Lambda'(C)$ of external $C$-routers, such that for every edge $e\in E(G)\setminus E(C)$, $\expect[{\mathcal{Q}}'(C)\sim{\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta$.
Let $I_C$ be a (random) cluster-based instance for $C$. Then:
\[\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I_C)}\leq O(\beta^2\cdot (\chi(C)+|E(C)|)).\]
\end{theorem}
\begin{proof}
Let ${\mathcal{Q}}'(C)$ be an external $C$-router that is sampled from the distribution ${\mathcal{D}}'(C)$. We denote by $u'(C)$ the vertex of $G\setminus C$ that serves as the common endpoint of paths in ${\mathcal{Q}}'(C)$.
Assume first that $C\in{\mathcal{C}}^{\operatorname{light}}$.
We denote by ${\mathcal{Q}}(C)$ be an internal $C$-router that is sampled from the distribution ${\mathcal{D}}(C)$, and we denote by $\hat{\mathcal{Q}}(C)$ the internal $C$-router obtained by applying the algorithm from \Cref{lem: non_interfering_paths} to ${\mathcal{Q}}(C)$ and instance $(G,\Sigma)$.
We construct a solution $\phi_C$ to the instance $I_C$ as follows.
Recall that $V(G_C)=V(C)\cup\set{v^*}$ and $E(G_C)=E_G(C)\cup \delta_{G_C}(v^*)$.
We start with the drawing of $C\cup E({\mathcal{Q}}'(C))$ induced by $\phi^*$. We do not modify the drawing of $C$, and we construct the images of edges of $\delta_{G_C}(v^*)$ as follows.
We apply the algorithm from \Cref{cor: new type 2 uncrossing} to the graph $G$, drawing $\phi^*$, cluster $C$ and the set ${\mathcal{Q}}'(C)$ of paths, and obtain a set $\Gamma'_C=\set{\gamma'_e\mid e\in \delta_G(C)}$ of curves, so all curves of $\Gamma'_C$ share an endpoint $z'_C=\phi^*(u'(C))$.
Denote by ${\mathcal{O}}'_C$ the ordering in which curves of $\Gamma'_C$ enter $z'_C$.
We then apply the algorithm from \Cref{lem: ordering modification} to the set $\Gamma'_C$ of curves and the ordering $\tilde{\mathcal{O}}_{v^*}$, and rename the obtained set of curves by $\Gamma'_C$. From \Cref{cor: new type 2 uncrossing} and
\Cref{lem: ordering modification},
$\chi(\Gamma'_C)\le \mbox{\sf dist}({\mathcal{O}}'_C,\tilde {\mathcal{O}}_{v^*})$.
\iffalse{previous operations without the new nudging lemma}
For each $e'\in E({\mathcal{Q}}'(C))$, we denote by $\pi_{e'}$ the curve that represents the image of $e'$ in $\phi^*$, and we let set $\Pi_{e'}$ contain $\cong_G({\mathcal{Q}}'(C),e')$ curves connecting the endpoints of $e'$ lying inside an arbitrarily thin strip around $\pi_{e'}$. We then assign, for each edge $e\in \delta_G(C)$ and for each edge $e'\in E(Q'_e)$, a distinct curve in $\Pi_{e'}$ to $e$. So each curve in $\bigcup_{e'\in E({\mathcal{Q}}'(C))}\Pi_{e'}$ is assigned to exactly one edge of $\delta_G(C)$. For each edge $e\in \delta_G(C)$, let $\gamma'_e$ be the curve obtained by concatenating all curves in $\bigcup_{e'\in E({\mathcal{Q}}'(C))}\Pi_{e'}$ that are assigned to $e$, so $\gamma'_e$ connects the endpoint of $e$ in $C$ to $z$, where $z$ is the image of $u'(C)$ in $\phi^*$. We denote $\Gamma'_C=\set{\gamma'_e\mid e\in \delta_G(C)}$, and we view $z$ as the last endpoint of curves in $\Gamma'_C$. We let $\Gamma_0$ contains all curves representing the image of some edge in $E(C)$, and then apply the algorithm from \Cref{thm: type-2 uncrossing} to sets $\Gamma_0,\Gamma'_C$ of curves. Let $\Gamma''_C=\set{\gamma''_e\mid e\in \delta_G(C)}$ be the set of curves that we obtain. Now if we view the curve $\gamma''_e$ as the image of $e$ for each edge $e\in \delta_G(C)$, then combined with the drawing of $C$ induced by the drawing $\phi^*$, we obtain a drawing of $G_C$, that we denote by $\phi'_C$. Clearly, the drawing $\phi'_C$ respects the rotations defined in $\tilde\Sigma_C$ at all vertices of $V(G_C)\setminus \set{v^*}$. However, it may not respect the rotation $\tilde{\mathcal{O}}_{v^*}$, namely the image of edges of $\delta_{G_C}(v^*)$ may not enter $z$ in the order $\tilde{\mathcal{O}}_{v^*}={\mathcal{O}}^{\operatorname{guided}}({\mathcal{Q}}(C),\Sigma)$. To obtain a feasible solution to the instance $I_C$, we further modify the drawing $\phi'_C$ as follows.
Let $D_{z}$ be an arbitrarily small disc around $z$. For each $e\in \delta_{G_C}(v^*)$, we denote by $p_e$ the intersection between the curve $\gamma''_e$ and the boundary of $D_{z}$.
We erase the drawing of $\phi'_C$ inside the disc $D_{z}$. We then place another disc $D'_{z}$ around $z$ inside the disc $D_{z}$, and let $\set{p'_e}_{e\in \delta_{G_C}(v^*)}$ be a set of points appearing on the boundary of disc $D'_{z}$ in the order $\tilde{\mathcal{O}}_{v^*}$. We use \Cref{lem: find reordering} to compute a set $\set{\zeta_e}_{e\in \delta_{G_C}(v^*)}$ of reordering curves, where for each $e\in \delta_{G_C}(v^*)$, the curve $\zeta_e$ connects $p_e$ to $p'_e$.
We then define, for each $e\in \delta_{G_C}(v^*)$, the curve $\eta_e$ as the union of (i) the subcurve of $\gamma''_e$ outside the disc $D_{z}$ (connecting its endpoint in $V(C)$ to $p_e$); (ii) the curve $\zeta_e$ (connecting $p_e$ to $p'_e$); and (iii) the straight line-segment connecting $p'_e$ to $z$. We now view the curve $\eta_e$ as the image of $e$ for all edges $e\in \delta_{G_C}(v^*)$. We denote by $\phi_C$ the drawing obtained by the union of all curves in $\set{\eta_e}_{e\in \delta_{G_C}(v^*)}$ and the image of $C$ in $\phi'_C$. Clearly, the drawing $\phi_C$ is a feasible solution to the instance $I_C$.
\fi
We use the following claim.
\begin{claim}
\label{obs: new crossings}
$\mathsf{cr}(\phi_C)\le \chi^2(C)+ \chi({\mathcal{Q}}'(C),C)+\chi({\mathcal{Q}}(C),{\mathcal{Q}}'(C))+\chi^2({\mathcal{Q}}(C))+\sum_{e\in E(C)}\cong({\mathcal{Q}}(C),e)^2$.
\end{claim}
\begin{proof}
First, since we have not modified the drawing of $C$, the number of crossings between edges of $E_G(C)$ are bounded by $\chi^2(C)$.
Second, from the definition of curves of $\Gamma'_C$, the number of crossings between image of $C$ and the curves of $\Gamma'_C$ is at most $\chi({\mathcal{Q}}'(C),C)$.
It remains to upper bound $\mbox{\sf dist}({\mathcal{O}}'_C,\tilde{\mathcal{O}}_{v^*})$ by constructing a set of reordering curves for ${\mathcal{O}}'_C$ and $\tilde{\mathcal{O}}_{v^*}$ and analyze the number of crossings between them.
Recall that we have constructed, in the proof of \Cref{thm: cost of contracted}, a set $\Gamma_C$ of curves that contains, for each $e\in \delta_G(C)$, a curve $\gamma_e$ connecting $e$ to $z_C=\phi^*(u(C))$, such that the curves of $\Gamma_C$ enter $z_C$ in order ${\mathcal{O}}^{\operatorname{guided}}(\hat{\mathcal{Q}}(C),\Sigma)$ and
$\chi(\Gamma_C)\le \sum_{e'\in E(C)}\big(\cong(\hat{\mathcal{Q}}(C),e')\big)^2+\chi^2(\hat{\mathcal{Q}}(C))$.
We now define, for each $e\in \delta_G(C)$, the curve $\zeta_e$ to be the union of $\gamma_e$ and $\gamma'_e$, so all curves of $\set{\zeta_e}_{e\in \delta_G(C)}$ connects $z_C$ to $z'_C$. Note that the curves of $\Xi_C=\set{\zeta_e}_{e\in \delta_G(C)}$ enter $z_C$ in the order $\tilde{\mathcal{O}}_{v^*}$ and enter $z'_C$ in the order ${\mathcal{O}}'_C$, so $\mbox{\sf dist}({\mathcal{O}}'_C,\tilde{\mathcal{O}}_{v^*})\le \chi(\Xi_C)$.
Note that
$$\chi(\Xi_C)
\le \chi({\mathcal{Q}}'(C),\hat{\mathcal{Q}}(C))+ \sum_{e'\in E(C)}\bigg(\cong(\hat{\mathcal{Q}}(C),e')\bigg)^2+\chi^2(\hat{\mathcal{Q}}(C)),$$
so $\mathsf{cr}(\phi_C)\le \chi^2(C)+ \chi({\mathcal{Q}}'(C),C)+\chi({\mathcal{Q}}'(C),\hat{\mathcal{Q}}(C))+\sum_{e'\in E(C)}\bigg(\cong(\hat{\mathcal{Q}}(C),e')\bigg)^2+\chi^2(\hat{\mathcal{Q}}(C))$.
Lastly, from \Cref{lem: non_interfering_paths}, we get that for each $e\in E(C)$, $\cong_G(\hat{\mathcal{Q}}(C),e)\le \cong_G(\hat{\mathcal{Q}}(C),e)$. \Cref{obs: new crossings} now follows.
\end{proof}
We now use \Cref{obs: new crossings} to show that $\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I_C)}\leq O(\beta^2\cdot (\chi(C)+|E(C)|))$.
Note that $\chi^2(C)\le \chi(C)$,
$$\expect{\sum_{e\in E(C)}\bigg(\cong({\mathcal{Q}}(C),e)\bigg)^2}\le \beta\cdot |E(C)|,$$
$$\expect{\chi({\mathcal{Q}}'(C),C)}= \sum_{e'\in E({\mathcal{Q}}'(C))}\chi(e',C)\cdot \expect[]{\cong_{G}({\mathcal{Q}}'(C),e')}\le \beta\cdot \chi(C),$$
$$\expect{\chi({\mathcal{Q}}'(C),{\mathcal{Q}}(C))} = \expect{\sum_{e\in E({\mathcal{Q}}(C))}\sum_{e'\in E({\mathcal{Q}}'(C))}\chi(e,e')\cdot\cong_{G}({\mathcal{Q}}(C),e)\cdot\cong_{G}({\mathcal{Q}}'(C),e')}\le \beta^2\cdot \chi(C),$$
and
$$\expect{\chi^2({\mathcal{Q}}(C))} = \expect{\sum_{e,e'\in E(C)}\chi(e,e')\cdot\cong_{G}({\mathcal{Q}}(C),e)\cdot\cong_{G}({\mathcal{Q}}(C),e')}\le \beta^2\cdot \chi(C).$$
Altogether, this implies that $\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I_C)}\leq \expect{\mathsf{cr}(\phi_C)}\le O(\beta^2\cdot (\chi(C)+|E(C)|))$.
Assume now that $C\in{\mathcal{C}}^{\operatorname{bad}}$.
We compute the path set $\Gamma''_C$ and then then the drawing $\phi_C$ in the same way as in Case 1.
However, the analysis of the number of crossings in $\phi_C$ is much simpler.
On the one hand, for same reasons as in Case 1, the number of crossings between edges of $E_G(C)$ are bounded by $\chi^2(C)$, and the number of crossings between image of $C$ and the curves of $\Gamma''_C$ is at most $\chi({\mathcal{Q}}'(C),C)$.
On the other hand, if we denote by $({\mathcal{O}}^*,b^*)$ the oriented ordering in which the curves of $\Gamma''_C$ enter $z$, then for each $\tilde b\in \set{0,1}$, $\mbox{\sf dist}(({\mathcal{O}}^*,b^*),(\tilde{\mathcal{O}}_{v^*},\tilde b))\le |\delta_G(C)|^2$. Altogether,
\[\mathsf{cr}(\phi_C)\le \chi^2(C)+ \chi({\mathcal{Q}}'(C),C)+|\delta_G(C)|^2,\]
and it follows that
\[\expect[]{\mathsf{cr}(\phi_C)}\le \chi(C)+\beta\cdot\chi(C)+\beta\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)+|E(C)|).\]
Since $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\le \chi(C)$, we get that
$\expect[]{\mathsf{OPT}_{\mathsf{cnwrs}}(I_C)}\leq O(\beta^2\cdot (\chi(C)+|E(C)|))$.
\end{proof}
\iffalse
Let $G'=G_{|{\mathcal{C}}}$ be the contracted graph of $G$ with respect to $G'$, and let $\Sigma'$ be a rotation system for $G'$ that is defined as follows: for
every regular vertex $v\in V(G)\cap V(G')$, the ordering ${\mathcal{O}}'_v\in \Sigma'$ of the edges of $\delta_{G'}(v)=\delta_G(v)$ is identical to the ordering ${\mathcal{O}}_v\in \Sigma$; for every supernode $v_C$, where $C$ is a cluster in ${\mathcal{C}}$, the ordering ${\mathcal{O}}'_{v(C)}\in \Sigma'$ of the edges of $\delta_{G'}(v(C))$ is defined to be ordering guided by the set ${\mathcal{Q}}(C)$ of paths and the rotation system $\Sigma$. We refer to $\Sigma'$ as the \emph{rotation system guided by $({\mathcal{C}},\set{{\mathcal{Q}}(C)}_{C\in {\mathcal{C}}},\Sigma)$}.
\fi
\iffalse
\mynote{I don't think we need this lemma, but I'm leaving it here in case you need it.}
We use the following lemma, whose proof appears in \Cref{apd: Proof of opt of contracted instance}.
\begin{lemma}
\label{lem: opt of contracted instance}
Let $I=(G,\Sigma)$ be an instance of the \textnormal{\textsf{MCNwRS}}\xspace problem, let $\phi^*$ be an optimal solution for $I$, and let $\chi$ be the set of pairs $(e,e')$ of edges of $G$, whose images cross in $\phi^*$. Assume further that we are given a collection ${\mathcal{C}}$ of disjoint clusters of $G$, and, for every cluster $C\in {\mathcal{C}}$, a guiding pair $(u(C),{\mathcal{Q}}(C))\in \Lambda(C)$. Let $G'=G_{|{\mathcal{C}}}$, and let $\Sigma'$ be the $({\mathcal{C}},\set{{\mathcal{Q}}(C)}_{C\in {\mathcal{C}}},\Sigma)$-guided rotation system for $G'$.
Denote ${\mathcal{Q}}=\bigcup_{C\in {\mathcal{C}}}{\mathcal{Q}}(C)$.
Then:
\[\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\le \mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+\sum_{C\in {\mathcal{C}}}\sum_{e\in E(C)}(\cong_G({\mathcal{Q}},e))^2+\sum_{(e,e')\in \chi}\cong_G({\mathcal{Q}},e)\cdot \cong_G({\mathcal{Q}},e').\]
\end{lemma}
\fi
\iffalse
\mynote{I need the following lemma (this was proved as part of Lemma 7.1, but I need this here as a separate lemma). We are given instance $I=(G,\Sigma)$, a collection ${\mathcal{C}}$ of clusters, and for each cluster $C\in {\mathcal{C}}$, a set ${\mathcal{Q}}(C)$ of paths routing $\delta(C)$ to vertex $u(C)$ inside $C$. We consider $G'=G_{|{\mathcal{C}}}$ the contracted graph, for which we define rotation system $\Sigma'$ that is consistent with $\Sigma$, but for every supernode $v(C)$, we define the rotation to be the order in which the paths in ${\mathcal{Q}}(C)$ enter $u(C)$ (like we do in basic disengagement). The claim is that this instance has a solution whose value is the summation, over all crossings $(e,e')$ in the optimal solution of $\cong(e)\cdot \cong(e')$ plus $\sum_{C\in {\mathcal{C}}}\sum_{e\in E(C)}(\cong(e))^2$, where all congestions are with respect to $\bigcup_{C\in {\mathcal{C}}}{\mathcal{Q}}(C)$.}
\fi
\subsection{Basic Disengagement}
\label{subsec: basic disengagement}
\mynote{this needs to be reorganized. First do the disengagement when the ordering of edges of $\delta_G(C)$ is given. This is enough in order to define the instances. Show that solutions can be combined together, and bound the total number of edges. Only then show how to compute the disengagement when we are given distributions on inner/outer routers, and bound the sum of opts. This is because we fix external routers in disengagement over path, and we need those lemmas there, that don't depend on how we choose the routers.}
In this subsection we design an algorithm, called \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace, that takes as input an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, a laminar ${\mathcal{L}}$ family of clusters of $G$ such that $G\in {\mathcal{L}}$, and some path sets for each cluster in each cluster of ${\mathcal{L}}$, compute a collection ${\mathcal{I}}'$ of subinstances of $I$, that satisfy some good properties.
We now start to describe the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace.
Let $I=(G,\Sigma)$ be an instance of the \textnormal{\textsf{MCNwRS}}\xspace problem, and let ${\mathcal{L}}$ be a laminar family of clusters of $G$. Assume that for every cluster $C\in {\mathcal{L}}$, we are given a distribution ${\mathcal{D}}'(C)$ on the set $\Lambda'(C)$ of external $C$-routers, such that for every edge $e$ of $G\setminus C$, $\expect[{\mathcal{Q}}'(C)\sim{\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta$. Assume further that the family ${\mathcal{L}}$ of clusters is partitioned into two subsets, ${\mathcal{L}}^{\operatorname{light}}$ and ${\mathcal{L}}^{\operatorname{bad}}$, such that each cluster $C\in {\mathcal{L}}^{\operatorname{bad}}$ is $\beta$-bad and has the $\alpha_0$-bandwidth property for some $\alpha_0=\Omega(1/\log^{12}m)$, and for each cluster $C\in {\mathcal{L}}^{\operatorname{light}}$ we are given a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers, such that $C$ is $\beta$-light with respect to ${\mathcal{D}}(C)$.
In particular, we will require that the cluster $G\in {\mathcal{L}}$ belongs to subset ${\mathcal{L}}^{\operatorname{light}}$, and the distributions ${\mathcal{D}}'(G),{\mathcal{D}}(G)$ will give probability $1$ to an empty set of paths.
We denote by $\phi^*$ an optimal solution to instance $I$.
We construct a new family ${\mathcal{I}}$ of instances, that we refer to as the \emph{disengagement of instance $I$} via tuple $({\mathcal{L}}={\mathcal{L}}^{\operatorname{bad}}\cup {\mathcal{L}}^{\operatorname{light}}, \set{{\mathcal{D}}'(C)}_{C\in {\mathcal{L}}},\set{{\mathcal{D}}(C)}_{C\in {\mathcal{L}}^{\operatorname{light}}}$).
In order to construct the family ${\mathcal{I}}$ of instances, we first fix, for every cluster $C\in {\mathcal{L}}^{\operatorname{bad}}$ an arbitrary circular ordering ${\mathcal{O}}^*_C$ on edges of $\delta_G(C)$. Additionally, for every cluster $C\in {\mathcal{L}}^{\operatorname{light}}$, we select an internal $C$-router ${\mathcal{Q}}(C)\in \Lambda(C)$ from the distribution ${\mathcal{D}}(C)$, and we let ${\mathcal{O}}^*_C$ be the ordering of the edges in $\delta_G(C)$ that is guided by the set ${\mathcal{Q}}(C)$ of paths.
The collection ${\mathcal{I}}$ of instances will contain a single global instance $\hat I$, and additionally, for every cluster $C\in {\mathcal{L}}$, an instance $\hat I_C$. Collection ${\mathcal{I}}$ of instances is constructed in two steps.
In the first step, for every cluster $C\in {\mathcal{L}}$, we construct the cluster-based instance $I_C=(G_C,\tilde \Sigma_C)$, described in \Cref{subsec: cluster-based instance}. We set the ordering of the edges incident to vertex $v^*$ in $\tilde \Sigma_C$ to be the fixed ordering ${\mathcal{O}}^*_C$ that we have selected for $C$; recall that, if $C\in {\mathcal{L}}^{\operatorname{light}}$, then this ordering is guided by the the internal $C$-router ${\mathcal{Q}}(C)$, where ${\mathcal{Q}}(C)$ was selected from the distribution ${\mathcal{D}}(C)$, so the construction of the rotation system $\tilde \Sigma_C$ is consistent with the definition of the cluster-based instance. Recall that, from
\Cref{thm: cost of single cluster},
$\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(I_C)}\leq O(\beta^2\cdot (\chi(C)+|E(C)|))$.
In the second step, we consider every resulting cluster instance $I_C=(G_C,\tilde \Sigma_C)$. Fix any such instance $I_C$, and assume that the child-clusters of $C$ in ${\mathcal{L}}$ are $S_1,\ldots,S_r$. We let $\hat I_C=(\hat G_C,\hat\Sigma_C)$ be the contracted instance of $I_C$ with respect to clusters $S_1,\ldots,S_r$. Recall that graph $\hat G_C$ is obtained from graph $G_C$ by contracting, for all $1\leq i\leq r$, cluster $S_i$ into a vertex $v_{S_i}$. We set the ordering of the edges incident to $v_{S_i}$ in $\hat \Sigma_C$ to be precisely the ordering ${\mathcal{O}}^*_{S_i}$ that we have fixed. Recall that, if $S_i\in {\mathcal{L}}^{\operatorname{light}}$, then this ordering is guided by the set ${\mathcal{Q}}(S_i)$ of paths, where the internal $S_i$-router ${\mathcal{Q}}(S_i)\in \Lambda(S_i)$ was sampled from the distribution ${\mathcal{D}}(S_i)$, so the construction of the rotation system $\hat \Sigma_C$ is consistent with the definition of the contracted instance from \Cref{subsec: contracted instance}. This concludes the definition of the instance $\hat I_C$.
From \Cref{thm: cost of contracted}, $\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I_C)}\leq O(\beta^2\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I_C)+|E(C)|))$, and therefore we obtain the following observation.
\begin{figure}[h]
\centering
\subfigure[Layout of graph $G$.]{\scalebox{0.13}{\includegraphics{figs/dis_instance_before}}\label{fig: graph G}}
\hspace{0cm}
\subfigure[The graph $\hat G_C$.]{
\scalebox{0.14}{\includegraphics{figs/dis_instance_after}}\label{fig: C and disc}}
\caption{An illustration of the construction of graph $\hat G_C$.\label{fig: disengaged_instance}}
\end{figure}
\begin{observation}\label{obs: cost of cluster instance}
For every cluster $C\in {\mathcal{L}}$, $\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I_C)}\leq O(\beta^4\cdot(\chi(C)+|E(C)|))$.
\end{observation}
We denote by ${\mathcal{I}}=\set{\hat I_C\mid C\in {\mathcal{L}}}$ the final collection of instances, which is called the \emph{disengagement of instance $I$ via tuple} $({\mathcal{L}}={\mathcal{L}}^{\operatorname{bad}}\cup {\mathcal{L}}^{\operatorname{light}}, \set{{\mathcal{D}}'(C)}_{C\in {\mathcal{L}}},\set{{\mathcal{D}}(C)}_{C\in {\mathcal{L}}^{\operatorname{light}}})$.
We prove the following lemmas.
\begin{lemma}\label{lem: disengagement final cost}
$\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq O(\mathsf{dep}({\mathcal{L}})\cdot\beta^4\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|))$
\end{lemma}
\begin{proof}
From the definition of depth of the laminar family ${\mathcal{L}}$, each edge of $G$ appears in at most $\mathsf{dep}({\mathcal{L}})$ clusters of ${\mathcal{L}}$. Therefore, $\sum_{C\in {\mathcal{L}}}\chi(C)\le O(\mathsf{dep}({\mathcal{L}})\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I))$, and $\sum_{C\in {\mathcal{L}}}|E(C)|\le \mathsf{dep}({\mathcal{L}})\cdot |E(G)|$.
Combined with \Cref{obs: cost of cluster instance}, we get that
$$\expect{\sum_{C\in {\mathcal{L}}}\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I_C)}\le
O\textsf{left}(\beta^4\cdot\bigg(\sum_{C\in {\mathcal{L}}}\chi(C)+|E(C)|\bigg)\textsf{right})\le
O\bigg(\mathsf{dep}({\mathcal{L}})\cdot \beta^4\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\bigg).$$
\end{proof}
\iffalse
\begin{lemma}\label{lem: number of edges in all disengaged instances}
$\sum_{C\in {\mathcal{L}}}|E(G_C)|=O(\mathsf{dep}({\mathcal{L}})\cdot|E(G)|)$. In other words, the total number of edges in all sub-instances in ${\mathcal{I}}$ is $O(\mathsf{dep}({\mathcal{L}})\cdot|E(G)|)$.
\end{lemma}
\begin{proof}
Fix an integer $1\le i\le \mathsf{dep}({\mathcal{L}})$ and denote by $C_1,\ldots,C_k$ all level-$i$ clusters in ${\mathcal{L}}$. We will show that
$\sum_{1\le j\le k}|E(G_{C_j})|\le O(|E(G)|)$.
In fact, for each cluster $C\in {\mathcal{L}}$, from the definition of the cluster-based instance $\hat I_C=(G_C,\Sigma_C)$, $|E(G_C)|\le |E_G(C)|+|\delta_G(C)|$.
Since clusters $C_1,\ldots,C_k$ are mutually vertex-disjoint, each edge of $G$ belongs to at most two sets of $\set{\delta_G(C_j)}_{1\le j\le k}$. Therefore,
$\sum_{1\le j\le k}|E(G_{C_j})|\le \sum_{1\le j\le k}(|E_G(C_j)|+|\delta_G(C_j)|) \le O(|E(G)|)$.
\Cref{lem: number of edges in all disengaged instances} now follows.
\end{proof}
Lastly, we show that solutions to instances in ${\mathcal{I}}$ can be efficiently combined in order to obtain a solution to instance $I$.
\begin{lemma}\label{lem: basic disengagement combining solutions}
There is an efficient algorithm, that, given, for each instance $I'\in {\mathcal{I}}$, a solution $\phi(I')$, computes a solution for instance $I$ of value at most $\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))$.
\end{lemma}
\begin{proof}
Let ${\mathcal{L}}_0\subseteq {\mathcal{L}}$ be the set of child clusters of cluster $G$ in ${\mathcal{L}}$, and we denote by $\hat I=(\hat G,\hat \Sigma)$ the contracted instance of $I=(G,\Sigma)$ with respect to cluster set ${\mathcal{L}}_0$.
We will prove that there is an efficient algorithm, that given a solution $\hat{\phi}$ for the instance $\hat I$, and for each cluster $C\in {\mathcal{L}}_0$, a solution $\phi_C$ for instance $I_C$ (note that $I_C$ is not an instance of ${\mathcal{I}}$), computes a solution for instance $I$ of value at most $\mathsf{cr}(\hat{\phi})+\sum_{C\in {\mathcal{L}}_0}\mathsf{cr}(\phi_C)$.
Note that \Cref{lem: basic disengagement combining solutions} follows easily from applying this algorithm to all clusters in ${\mathcal{L}}$ in a bottom-up fashion.
We start with the drawing $\hat \phi$, and process clusters of ${\mathcal{C}}$ one-by-one. Recall that the instance $\hat I=(\hat G,\hat \Sigma)$, where graph $\hat G$ contains a vertex $v_C$ for every cluster $C\in {\mathcal{C}}$. Moreover, the set of incident edges of $v_C$ in $\hat G$ is $\delta_G(C)$, and $\hat{\mathcal{O}}_{v_C}=\tilde{\mathcal{O}}_{v^*_C}$.
Also recall that, for each cluster $C\in {\mathcal{C}}$, the graph $G_C$ contains a vertex $v^*_C$ with $\delta_{G_C}(v^*_C)=\delta_G(C)$.
We now describe an iteration of processing cluster $C$.
Let $D_C$ be an arbitrarily small disc around the image of $v_C$ in $\hat\phi$. For each edge $e\in \delta_G(C)$, we denote by $p_e$ the intersection of the image of $e$ in $\hat \phi$ with the boundary of $D_C$, so the points $\set{p_e}_{e\in \delta_G(C)}$ appear on the boundary of $D_C$ in the order $\tilde{\mathcal{O}}_{v^*_C}$.
Similarly, let $D'_C$ be an arbitrarily small disc around the image of $v^*_C$ in $\phi_C$, and we define points $\set{p'_e}_{e\in \delta_G(C)}$ similarly, so they appear on the boundary of $D'_C$ in the order $\tilde{\mathcal{O}}_{v^*_C}$.
We now erase the part of the drawing $\hat\phi$ inside the disc $D_C$, and place the drawing of $\phi_C$ outside the disc $D'_C$ inside $D_C$, such that the boundary of $D_C$ coincides with the boundary of $D'_C$, while the interior of $D_C$ coincides with the exterior of $D'_C$, and for each edge $e\in \delta_G(C)$, the points $p_e$ and $p'_e$ coincide (note that this can be achieved since points in $\set{p_e}_{e\in \delta_G(C)}$ appear on the boundary of $D_C$ in the same order as points in $\set{p_e}_{e\in \delta_G(C)}$ appear on the boundary of $D_C$). This completes the iteration of processing $C$.
For each $e\in \delta_G(C)$, we view the union of (i) the subcurve of the image of $e$ in $\hat \phi$ between the image of its non-$v_C$ endpoint and the point $p_e$; and (ii) the subcurve of the image of $e$ in $\phi_C$ between the image of its non-$v^*_C$ endpoint and the point $p'_e$, as the image of $e$.
Clearly, after the iteration of processing $C$, we have not created any additional crossings.
\iffalse
\begin{figure}[h]
\centering
\subfigure[The drawing $\psi_{i+1}$, where the boundary of ${\mathcal{D}}$ is shown in dash black.]{\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_1.jpg}
}
\hspace{0.45cm}
\subfigure[The drawing $\psi_{i+1}$ after we erase its part inside ${\mathcal{D}}$ and view the interior of ${\mathcal{D}}$ as the outer face.]{
\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_2.jpg}}}
\hspace{0.45cm}
\subfigure[The drawing $\psi_i$, where the curves of $\set{\zeta_e\mid e\in F}$ are shown in dash line segments.]{
\scalebox{0.36}{\includegraphics[scale=1.0]{figs/stitching_3.jpg}}}
\caption{An illustration of stitching the drawings $\hat{\phi}$ and $\phi_C$.}\label{fig: stitching}
\end{figure}
\fi
Let $\phi$ be the drawing that we obtain after processing all clusters of ${\mathcal{C}}$ in this way. It is easy to verify that $\phi$ is a feasible solution to the instance $I$. Since we have not created additional crossings in each iteration, $\mathsf{cr}(\phi)\le \mathsf{cr}(\hat \phi)+\sum_{C\in {\mathcal{C}}}\mathsf{cr}(\phi_C)$.
This completes the proof of \Cref{lem: basic disengagement combining solutions}.
\end{proof}
\fi
\iffalse
Recall that given an instance $(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, we say that another instance $(G',\Sigma')$ is its \emph{sub-instance} iff graph $G'$ is obtained from a sub-graph of $G$ by contracting some of its vertex sets into super-nodes. The ordering ${\mathcal{O}}(u)$ for the edges incident to super-nodes may be arbitrary, but the ordering of edges incident to other vertices is determined by the ordering in $\Sigma$.
Suppose we are given any collection ${\mathcal{C}}$ of disjoint sub-graphs of $G$, such that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$. We denote by $E^{\textsf{out}}_{{\mathcal{C}}}$ the set of all edges of $G$ whose endpoints lie in different clusters, and by $E^{\textsf{in}}_{{\mathcal{C}}}$ the set of all edges of $G$ whose endpoints lie in the same cluster, so $E^{\textsf{out}}_{{\mathcal{C}}}\cup E^{\textsf{in}}_{{\mathcal{C}}}=E(G)$.
The main theorem of this subsection is the following.
\mynote{maybe define the sub-instances before the theorem statement to make it more concrete?}
\znote{Yes. I am about to suggest this as well.}
\begin{theorem}[Basic Disengagement of Clusters]\label{thm: disengagement}
There is an efficient algorithm, that, given an instance $(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, and a collection ${\mathcal{C}}$ of disjoint sub-graphs of $G$, such that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$, and, for every cluster $C\in {\mathcal{C}}$, a vertex $u(C)\in C$, and a vertex $u'(C)\not \in C$, together with a collection ${\mathcal{Q}}(C)$ of paths routing the edges of $\delta(C)$ to $u(C)$ inside $C$ with congestion $\eta$, and a collection ${\mathcal{P}}(C)$ of edge-disjoint paths routing the edges of $\delta(C)$ to $u'(C)$ outside of $C$ with congestion $\eta$, computes a collection ${\mathcal{I}}$ of sub-instances of $(G,\Sigma)$, with the following properties:
\begin{itemize}
\item For every cluster $C\in {\mathcal{C}}$, there is a unique instance $(H_C,\Sigma_C)$, with $C\subseteq H_C$. Moreover, $|E(H_C)|\leq |E(C)|+|E^{\textsf{out}}_{{\mathcal{C}}}|$;
\item $\sum_{(H,\Sigma')\in {\mathcal{I}}}|E(H)|\leq \sum_{C\in {\mathcal{C}}}|E(C)|+2|E^{\textsf{out}}_{{\mathcal{C}}}|$; and
\item $\sum_{(H,\Sigma')\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma')\leq \eta^2\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\operatorname{poly}\log n$.
\end{itemize}
Moreover, there is an efficient algorithm, that, given, for each instance $I'\in {\mathcal{I}}$, a feasible solution $\phi_I$, computes a feasible solution $\phi$ to instance $I$ with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\phi_I)\textsf{right} ) \cdot \operatorname{poly}\log n$.
\end{theorem}
The proof of the theorem is somewhat involved and technical, and is deferred to Section \ref{sec:disengagement}.
\mynote{this theorem stands on its own and the proof can be done without taking care of the path case. We use it later.}
\fi
\fi
\subsection{Guiding Paths, Guided Orderings and Guided Rotation System}
\label{subsec: guiding paths and orderings}
\begin{definition}[Routing Paths inside and outside a Cluster]
Let $G$ be a graph and let $C\subseteq G$ be a cluster of $G$. We say that a set ${\mathcal{Q}}(C)=\set{Q_e\mid e\in \delta_G(C)}$ of paths \emph{route the edges of $\delta_G(C)$ inside $C$ to a vertex $v\in V(C)$}, iff for all $e\in \delta_G(C)$, path $Q_e$ has $e$ as its first edge, $v$ as its last vertex, and $Q_e\setminus\set{e}$ is contained in $C$.
We say that a set ${\mathcal{Q}}'(C)=\set{Q_e\mid e\in \delta_G(C)}$ of paths \emph{route the edges of $\delta_G(C)$ outside $C$ to a vertex $v'\in V(G)\setminus V(C)$}, iff for all $e\in \delta_G(C)$, path $Q_e$ has $e$ as its first edge, $v'$ as its last vertex, and $Q_e\setminus\set{e}$ is disjoint from $C$.
\end{definition}
\begin{definition}[Guiding Paths and Guiding Pairs]
Let $(G,\Sigma)$ be an instance of \textnormal{\textsf{MCNwRS}}\xspace, and let $C\subseteq G$ be a cluster of $G$. Let $v\in V(C)$ be any vertex of $C$, and let ${\mathcal{Q}}(C)$ be a set of paths that route the edges of $\delta_G(C)$ to $v$ inside $C$, such that the paths in ${\mathcal{Q}}(C)$ are non-transversal with respect to $\Sigma$. We say that the set ${\mathcal{Q}}(C)$ is a \emph{set of guiding paths}, and $(v,{\mathcal{Q}}(C))$ is a \emph{guiding pair} for cluster $C$. We denote by $\Lambda(C)$ the set of all guiding pairs $(v,{\mathcal{Q}}(C))$.
\end{definition}
\paragraph{Guided Orderings.}
Let $(G,\Sigma)$ be an instance of \textnormal{\textsf{MCNwRS}}\xspace, $C\subseteq G$ a cluster of $G$, and $(v,{\mathcal{Q}})\in \Lambda(C)$ a guiding pair for cluster $C$.
Recall that we are given an ordering ${\mathcal{O}}_{v}\in \Sigma$ of the edges in $\delta_G(v)$. We use this ordering and the set ${\mathcal{Q}}$ of guiding paths in order to define a circular ordering of the edges of $\delta_G(C)$, as follows.
Intuitively, this is the ordering in which the paths in ${\mathcal{Q}}$ enter $v$ under the ordering ${\mathcal{O}}_v$.
Formally, for every path $Q\in {\mathcal{Q}}$, let $e^*(Q)$ be the last edge lying on path $Q$, which by definition belongs to $\delta_G(v)$. We first define a circular ordering of the paths in ${\mathcal{Q}}$, as follows: the paths are ordered according to the circular ordering of their last edges $e^*(Q)$ in ${\mathcal{O}}_{v}\in \Sigma$, breaking ties arbitrarily. Since every path $Q\in {\mathcal{Q}}$ is associated with a unique edge in $\delta_G(C)$, that serves as the first edge on $Q$, this ordering of the paths in ${\mathcal{Q}}$ immediately defines a circular ordering of the edges of $\delta_G(C)$, that we refer to as the ordering guided by ${\mathcal{Q}}$ and $\Sigma$, and denote by ${\mathcal{O}}^{\operatorname{guided}}({\mathcal{Q}},\Sigma)$.
\paragraph{Guided Rotation System in a Contracted Graph.}
Assume that we are given an instance $I=(G,\Sigma)$ be an instance of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, a collection ${\mathcal{C}}$ of disjoint clusters of $G$, and, for every cluster $C\in {\mathcal{C}}$, a vertex $u(C)\in V(C)$, together with a set ${\mathcal{Q}}(C)$ of paths that are non-transversal with respect to $\Sigma$, routing the edges of $\delta_G(C)$ to $u(C)$ inside $C$. Let $G'=G_{|{\mathcal{C}}}$ be the contracted graph of $G$ with respect to $G'$, and let $\Sigma'$ be a rotation system for $G'$ that is defined as follows: for
every regular vertex $v\in V(G)\cap V(G')$, the ordering ${\mathcal{O}}'_v\in \Sigma'$ of the edges of $\delta_{G'}(v)=\delta_G(v)$ is identical to the ordering ${\mathcal{O}}_v\in \Sigma$; for every supernode $v_C$, where $C$ is a cluster in ${\mathcal{C}}$, the ordering ${\mathcal{O}}'_{v(C)}\in \Sigma'$ of the edges of $\delta_{G'}(v(C))$ is defined to be ordering guided by the set ${\mathcal{Q}}(C)$ of paths and the rotation system $\Sigma$. We refer to $\Sigma'$ as the \emph{rotation system guided by $({\mathcal{C}},\set{{\mathcal{Q}}(C)}_{C\in {\mathcal{C}}},\Sigma)$}.
We use the following lemma, whose proof appears in \Cref{apd: Proof of opt of contracted instance}.
\begin{lemma}
\label{lem: opt of contracted instance}
Let $I=(G,\Sigma)$ be an instance of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, let $\phi^*$ be an optimal solution for $I$, and let $\chi$ be the set of pairs $(e,e')$ of edges of $G$, whose images cross in $\phi^*$. Assume further that we are given a collection ${\mathcal{C}}$ of disjoint clusters of $G$, and, for every cluster $C\in {\mathcal{C}}$, a guiding pair $(u(C),{\mathcal{Q}}(C))\in \Lambda(C)$. Let $G'=G_{|{\mathcal{C}}}$, and let $\Sigma'$ be the $({\mathcal{C}},\set{{\mathcal{Q}}(C)}_{C\in {\mathcal{C}}},\Sigma)$-guided rotation system for $G'$.
Denote ${\mathcal{Q}}=\bigcup_{C\in {\mathcal{C}}}{\mathcal{Q}}(C)$.
Then:
\[\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\le \mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+\sum_{C\in {\mathcal{C}}}\sum_{e\in E(C)}(\cong_G({\mathcal{Q}},e))^2+\sum_{(e,e')\in \chi}\cong_G({\mathcal{Q}},e)\cdot \cong_G({\mathcal{Q}},e').\]
\end{lemma}
\mynote{please add proof}
\begin{corollary}\label{cor: bounding crossing number of contracted graph}
Let $I=(G,\Sigma)$ be an instance of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, and let ${\mathcal{C}}$ be a collection of disjoint clusters of $G$. Suppose we are given, for every cluster $C\in {\mathcal{C}}$, a distribution ${\mathcal{D}}(C)$ over guiding pairs $(u(C),{\mathcal{Q}}(C))\in \Lambda(C)$, such that for every edge $e\in E(C)$, $\expect[(u(C),{\mathcal{Q}}(C))\in_{{\mathcal{D}}(C)} \Lambda(C)]{(\cong_G({\mathcal{Q}}(C),e))^2}\leq \beta$, for some $\beta\geq 1$. Let $\Sigma'$ be the the $({\mathcal{C}},\set{{\mathcal{Q}}(C)}_{C\in {\mathcal{C}}},\Sigma)$-guided rotation system for $G'$, where for all $C\in {\mathcal{C}}$, pair $(u(C),{\mathcal{Q}}(C))$ is drawn from the distribution ${\mathcal{D}}(C)$ independently at random. Then:
\[\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')}\le 4\beta \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+|E(G)|\textsf{right} ).\]
\end{corollary}
\iffalse
\mynote{I need the following lemma (this was proved as part of Lemma 7.1, but I need this here as a separate lemma). We are given instance $I=(G,\Sigma)$, a collection ${\mathcal{C}}$ of clusters, and for each cluster $C\in {\mathcal{C}}$, a set ${\mathcal{Q}}(C)$ of paths routing $\delta(C)$ to vertex $u(C)$ inside $C$. We consider $G'=G_{|{\mathcal{C}}}$ the contracted graph, for which we define rotation system $\Sigma'$ that is consistent with $\Sigma$, but for every supernode $v(C)$, we define the rotation to be the order in which the paths in ${\mathcal{Q}}(C)$ enter $u(C)$ (like we do in basic disengagement). The claim is that this instance has a solution whose value is the summation, over all crossings $(e,e')$ in the optimal solution of $\cong(e)\cdot \cong(e')$ plus $\sum_{C\in {\mathcal{C}}}\sum_{e\in E(C)}(\cong(e))^2$, where all congestions are with respect to $\bigcup_{C\in {\mathcal{C}}}{\mathcal{Q}}(C)$.}
\fi
\section{Constructing Internal Routers - Proof of \Cref{thm: find guiding paths}}
\label{sec: guiding paths}
We will repeatedly use the following simple lemma, whose proof is provided in \Cref{apd: Proof of routing path extension}.
\begin{lemma}
\label{lem: routing path extension}
Let $G$ be a graph, let $T$ be a set of vertices that are $\alpha$-well-linked in $G$, for some $0<\alpha<1$, and let $T'$ be a subset of $T$. Suppose we are given a vertex $x\in V(G)$, and a set ${\mathcal{P}}$ of paths in $G$, routing the vertices of $T'$ to $x$. Then there is a set ${\mathcal{P}}'$ of paths routing the vertices of $T$ to $x$, such that, for every edge $e\in E(G)$,
$\cong_G({\mathcal{P}}',e)\le \ceil{\frac{|T|}{|T'|}}(\cong_G({\mathcal{P}},e)+\ceil{1/\alpha})$.
\end{lemma}
For conveninence, we denote the contracted graph $H_{|{\mathcal{C}}}$ by $\hat H$, and we denote $|E(\hat H)|=\hm$. From the statement of \Cref{thm: find guiding paths}, $k\geq \hm/\eta$. Observe that, from \Cref{clm: contracted_graph_well_linkedness}, the set $T$ of terminals is $(\alpha\alpha')$-well-linked in $H$. We will assume in the remainder of the proof that $\log m$ is greater than some large enough constant $c'_0$ (whose value we can set later). If this is not the case, then, $n$, and therefore $k$, is bounded by a constant $2^{c_0'}$. We can then use an arbitrary spanning tree $\tau$ of the graph $H$,
rooted at an arbitrary vertex $y$, in order to define a set ${\mathcal{Q}}$ of paths routing all terminals of $T$ to $y$, where for each terminal $t\in T$, the corresponding path $Q_t\in {\mathcal{Q}}$ is the unique path connecting $t$ to $y$ in the tree $\tau$. Since $|T|$ is bounded by a constant, for every edge $e\in E(H)$, $\cong_H({\mathcal{Q}},e)\leq O(1)$. We then return a distribution ${\mathcal{D}}$ consisting of a single set ${\mathcal{Q}}$ that has probability value $1$. Therefore, we assume from now on that $\log m>c_0'$ for some large enough constant $c_0'$ whose value we set later.
We start with some intuition. Assume first that graph $H$ contains a grid (or a grid minor) of size $(\Omega(k\alpha\alpha'/\operatorname{poly}\log m)\times \Omega(k\alpha \alpha'/\operatorname{poly}\log m))$, and a collection ${\mathcal{P}}$ of paths connecting every terminal to a distinct vertex on the first row of the grid, such that the paths in ${\mathcal{P}}$ cause a low edge-congestion. For this special case, the algorithm of \cite{Tasos-comm} (see also the proof of Lemma D.10 in the full version of \cite{chuzhoy2011algorithm}) provides a distribution ${\mathcal{D}}$ over routers ${\mathcal{Q}}\in \Lambda(H,T)$ with the required properties. Moreover, if $H$ is a bounded-degree planar graph, with a set $T$ of terminals that is $(\alpha\alpha')$-well-linked,
then there is an efficient algorithm to compute such a grid minor, together with the required collection ${\mathcal{P}}$ of paths. If $H$ is planar but no longer bounded-degree, we can still compute a grid-like structure in it, and apply the same arguments as in \cite{Tasos-comm} in order to compute the desired distribution ${\mathcal{D}}$. The difficulty in our case is that the input graph $H$ may be far from being planar, and, even though, from the Excluded Grid theorem of Robertson and Seymour \cite{robertson1986graph},
it must contain a large grid-like structure, without having a drawing of $H$ in the plane with a small number of crossing, we do not know how to compute such a structure\footnote{We note that we need the grid-like structure to have dimensions $(k'\times k')$, where $k'$ is almost linear in $k$. Therefore, we cannot use the known bounds for the Excluded Minor Theorem (e.g. from \cite{chuzhoy2019towards}) for general graphs, and instead we need to use an analogue of the stronger version of the theorem for planar graphs.}.
The proof of \Cref{thm: find guiding paths} consists of five steps. In the first step, we will either establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is sufficiently large (so the algorithm can return FAIL), or compute a subgraph $\hat H'\subseteq \hat H$, and a partition $(X,Y)$ of $V(\hat H')$, such that each of the clusters $\hat H'[X],\hat H'[Y]$ has the $\hat \alpha$-bandwidth property, for $\hat \alpha=\Omega(\alpha/\log^4m)$, together with a large collection of edge-disjoint paths routing the terminals to the edges of $E_{\hat H'}(X,Y)$ in graph $\hat H'$. Intuitively, we will view from this point onward the edges of $E_{\hat H'}(X,Y)$ as a new set of terminals, that we denote by $\tilde T$ (more precisely, we subdivide each edge of $E_{\hat H'}(X,Y)$ with a new vertex that becomes a new terminal). We show that it is sufficient to prove an analogue of \Cref{thm: find guiding paths} for this new set $\tilde T$ of terminals. The clusters $\hat H'[X],\hat H'[Y]$ of graph $\hat H'$ naturally define a partition $(H_1,H_2)$ of the graph $H$ into two disjoint subgraphs. In the second step, we either establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is suffciently large
(so the algorithm can return FAIL), or
compute some vertex $x$ of $H_1$, and a collection ${\mathcal{P}}$ of paths in graph $H_1$, routing the terminals of $\tilde T$ to $x$, such that the paths in ${\mathcal{P}}$ cause a relatively low edge-congestion.
We exploit this set ${\mathcal{P}}$ of paths in order to define an ordering of the terminals in $\tilde T$, which is in turn exploited in the third step in order to compute a ``skeleton'' of the grid-like structure. We compute the grid-like structure itself in the fourth step. In the fifth and the final step, we generalize the arguments from \cite{Tasos-comm} and \cite{chuzhoy2011algorithm} in order to obtain the desired distribution ${\mathcal{D}}$ over routers ${\mathcal{Q}}\in \Lambda(H,T)$, by exploiting this grid-like structure.
Before we proceed, we need to consider four simple special cases. In the first case, $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2$ is large. In the second case, we can route a large subset of the terminals to a single vertex of $V(\hat H)\cap V(H)$ in the graph $\hat H$ via edge-disjont paths. The third case is when $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)=0$, and the fourth special case is when $k< \eta^6$.
\paragraph{Special Case 1: $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2$ is large.}
We consider the case where $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2\geq \frac{(k \alpha^4 \alpha')^2}{c_0\log^{50}m}$, where $c_0$ is the constant from the statement of \Cref{thm: find guiding paths}. For every cluster $C\in {\mathcal{C}}$, let $\Sigma_C$ be the rotation system for $C$ induced by $\Sigma$.
In this case, since we are guaranteed that every cluster $C\in {\mathcal{C}}$ is $\eta'$-bad, that is, $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)+|E(C)|\geq |\delta(C)|^2/\eta'$, we get that:
\[\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(H\setminus T)|\geq \sum_{C\in {\mathcal{C}}}\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)+|E(C)|\textsf{right} )\geq \sum_{C\in {\mathcal{C}}}\frac{|\delta_H(C)|^2}{\eta'}\geq \frac{(k \alpha^4 \alpha')^2}{c_0\eta'\log^{50}m}.\]
Therefore, if $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2\geq \frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}m}$, the algorithm returns FAIL and terminates. We assume from now on that:
\begin{equation}\label{eq: boundaries squared sum bound}
\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2<\frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}m}.
\end{equation}
\paragraph{Special Case 2: Routing of terminals to a single vertex.}
The second special case happens if there exists a collection ${\mathcal{P}}_0$ of at least $\frac{k\alpha^2}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^3\log^6k}$ edge-disjoint paths in graph $\hat H$ routing some subset $T_0\subseteq T$ of terminals to some vertex $x$ (here $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}$ is the constant from \Cref{claim: embed expander}).
Note that, if Special Case 1 did not happen, and $c_0$ is a large enough constant, then $x$ may not be a supernode. Indeed, assume that $x=v_C$ for some cluster $C\in {\mathcal{C}}$. Then:
\[|\delta_H(C)|^2\geq \Omega\textsf{left}(\frac{(k\alpha^2)^2}{\log^{12}k}\textsf{right} )\geq \frac{(k \alpha^4 \alpha')^2}{c_0\log^{50}m},\]
which is, assuming that $c_0$ is a large enough constant, a contradiction. Therefore, we can assume that $x$ is not a supernode.
From \Cref{claim: routing in contracted graph}, since the clusters in ${\mathcal{C}}$ have the $\alpha'$-bandwidth property, there is a collection ${\mathcal{P}}'_0$ of paths in graph $H$, routing the vertices of $T_0$ to $x$, with edge-congestion at most $\ceil{1/\alpha'}\leq 2/\alpha'$. Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in graph $H$, from \Cref{lem: routing path extension}, there is a set ${\mathcal{Q}}$ of paths in graph $H$, routing the vertices of $T$ to $x$ with congestion at most:
\[\ceil{\frac{|T|}{|T_0|}}\textsf{left}(\frac 2{\alpha'}+\ceil{\frac 1{\alpha\alpha'}}\textsf{right} )\leq O\textsf{left} ( \frac{\log^6k}{\alpha^3\alpha'} \textsf{right}).\]
Note that a set ${\mathcal{Q}}$ of paths with the above properties can be computed efficiently via standard maximum flow algorithm.
We return a distribution ${\mathcal{D}}$ consisting of a single router ${\mathcal{Q}}$ with probability value $1$, and terminate the algorithm.
Clearly, for every edge $e\in E(H)$, $\expect{(\cong({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{32}m}{\alpha^{12}(\alpha')^8}\textsf{right} )$.
\paragraph{Special Case 3: $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$.}
Recall that we can efficiently check whether $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$, using the algorithm from \Cref{thm: crwrs_planar}. Assume now that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)=0$.
We use the following theorem from \cite{chuzhoy2020towards}.
\begin{lemma}[Lemma E.2 in \cite{chuzhoy2020towards}]
\label{lem: find_guiding_in_planar}
There is an efficient algorithm, that, given a planar graph $H$ and a subset $T$ of $r$ vertices of $V(H)$ that are $\alpha$-well-linked in $H$ for some $0<\alpha<1$, computes a distribution ${\mathcal{D}}$ over the routers in $\Lambda(H,T)$, such that the distribution has support size $O(r^2)$, and for each edge $e\in E(H)$,
\[\expect[(u^*,{\mathcal{Q}})\sim {\mathcal{D}}]{(\cong_H({\mathcal{Q}},e))^2}=O\bigg(\frac{\log r}{\alpha^4}\bigg).\]
\end{lemma}
Recall that the set $T$ of terminals is $\alpha$-well-linked in the contracted graph $H_{\mid {\mathcal{C}}}$, and every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property.
From \Cref{clm: contracted_graph_well_linkedness}, the set $T$ of terminals is $(\alpha\alpha')$-well-linked in $H$.
We then apply the algorithm from \Cref{lem: find_guiding_in_planar} to graph $H$, terminal set $T$ and parameter $(\alpha\alpha')$. Let ${\mathcal{D}}$ be the distribution over the set $\Lambda(H,T)$ of routers that we obtain. Then:
\[\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_H({\mathcal{Q}},e))^2}=O\bigg(\frac{\log k}{(\alpha\alpha')^4}\bigg) \leq O\textsf{left} (\frac{\log^{32}m}{\alpha^{12}(\alpha')^8}\textsf{right} ).\]
\paragraph{Special Case 4: $k< \eta^6$, but $\mathsf{OPT}_{\mathsf{cnwrs}}(I)>0$.}
Note that $\frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}m}\leq \frac{\eta^{12}}{\eta'}<1$ in this case (as, from the statement of \Cref{thm: find guiding paths}, $\eta'>\eta^{13}$).
Since we have assumed that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)>0$, we get that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq 1>\frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}m}$. We then simply return FAIL and terminate the algorithm.
In the remainder of the proof, we assume that neither of the four special cases happened. We now describe each step of the algorithm in detail.
\subsection{Step 1: Splitting the Contracted Graph}
In this step, we split the contracted graph $\hat H$, using the algorithm summarized in the following theorem.
\begin{theorem}\label{thm: splitting}
There is an efficient randomized algorithm that returns FAIL with probability at most $1/\operatorname{poly}(k)$, and, if it does not return FAIL, then it computes a subgraph $\hat H'\subseteq \hat H$ and a partition $(X,Y)$ of $V(\hat H')$ such that:
\begin{itemize}
\item clusters $\hat H'[X]$ and $\hat H'[Y]$ both have the $\hat \alpha'$-bandwidth property in $\hat H'$, for $\hat \alpha'=\Omega(\alpha/\log^4m)$; and
\item there is a set ${\mathcal{R}}$ of $\Omega(\alpha^3k/\log^8m)$ edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to edges of $E_{\hat H'}(X,Y)$.
\end{itemize}
\end{theorem}
\begin{proof}
We start by applying the algorithm from \Cref{claim: embed expander} to graph $\hat H$ and the set $T$ of terminals, to obtain a graph $W$ with $V(W)=T$ and maximum vertex degree at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$, and an embedding ${\hat{\mathcal{P}}}$ of $W$ into $\hat H$ with congestion at most $(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)/\alpha$. Let $\hat{\cal{E}}$ be the bad event that $W$ is not a $1/4$-expander. Then $\prob{\hat {\cal{E}}}\leq 1/\operatorname{poly}(k)$.
Define graph $\hat H'$ as the union of all paths in $\hat{{\mathcal{P}}}$.
We need the following observation.
\begin{observation}\label{obs: expansion and degree}
If event $\hat {\cal{E}}$ did not happen, then the set $T$ of vertices is $\hat \alpha$-well-linked in $\hat H'$, for $\hat \alpha=\frac{\alpha}{4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$, and the maximum vertex degree in $\hat H'$ is at most $d=\frac{\alpha k}{512\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$.
\end{observation}
\begin{proof}
Assume that Event $\hat {\cal{E}}$ did not happen. We first prove that the set $T$ of terminals is $\hat \alpha$-well-linked in $\hat H$. Consider any paritition $(A,B)$ of vertices of $\hat H'$, and denote $T_A=T\cap A$, $T_B=T\cap B$. Assume w.l.o.g. that $|T_A|\leq |T_B|$. Then it is sufficient to show that $|E_{\hat H'}(A,B)|\geq \hat \alpha\cdot |T_A|$.
Consider the partition $(T_A,T_B)$ of the vertices of $W$, and denote $E'=E_{W}(T_A,T_B)$. Since $W$ is a $1/4$-expander, $|E'|\geq |T_A|/4$ must hold. Consider now the set $\hat {\mathcal{R}}\subseteq {\hat{\mathcal{P}}}$ of paths containing the embeddings $P(e)$ of every edge $e\in E'$. Each path $R\in \hat {\mathcal{R}}$ connects a vertex of $T_A$ to a vertex of $T_B$, so it must contain an edge of $|E_{\hat H}(A,B)|$. Since $|\hat{\mathcal{R}}|\geq |T_A|/4$, and the paths in ${\hat{\mathcal{P}}}$ cause edge-congestion at most $(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)/\alpha$, we get that $|E_{\hat H}(A,B)|\geq \alpha\cdot |T_A|/(4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)\geq \hat \alpha |T_A|$.
Assume now that maximum vertex degree in $\hat H'$ is greater than $d$, and let $x$ be a vertex whose degree is at least $d$. Let $\hat{\mathcal{Q}}\subseteq \hat{\mathcal{P}}$ be the set of all paths containing the vertex $x$.
Consider any such path $Q\in \hat{\mathcal{Q}}$. The endpoints of this path are two distinct terminals $t,t'\in T$. We let $Q'\subseteq Q$ be the subpath of $Q$ between the terminal $t$ and the vertex $x$, and we let ${\mathcal{Q}}'=\set{Q'\mid Q\in \hat{\mathcal{Q}}}$.
Recall that every vertex in $W$ has degree at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$, and so a terminal in $T$ may be an endpoint of at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$ paths in ${\hat{\mathcal{P}}}$. Therefore, there is a subset ${\mathcal{Q}}''\subseteq {\mathcal{Q}}'$ of at least $d/(2\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)$ paths in $\hat H'$, each of which originates at a distinct terminal.
Since paths in ${\mathcal{Q}}''$ cause congestion at most $(\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k)/\alpha$, from \Cref{claim: remove congestion}, there is a collection ${\mathcal{Q}}'''$ of edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to $x$ with:
$$|{\mathcal{Q}}'''|\geq |{\mathcal{Q}}''|\cdot \frac{\alpha}{\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\geq \frac{d\alpha}{2\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^2\log^4k}\geq\frac{\alpha^2k}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}^3\log^6k},$$
\iffalse
But then, from \Cref{claim: routing in contracted graph}, there is a collection ${\mathcal{Q}}'''$ of edge-disjoint paths in graph $H$, routing a subset of terminals to $x$ of cardinality at least:
\[\frac{\alpha'|{\mathcal{Q}}''|} 2\geq \frac{\alpha' d}{2\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\geq \frac{\alpha \alpha' k}{1024\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^4k},\]
\fi
contradicting the fact that Special Case 2 did not happen.
\end{proof}
Next, we use the following lemma to compute the required sets $X$, $Y$ of vertices. The proof follows immediately from techniques that were introduced in \cite{chuzhoy2012routing} and then refined in \cite{chuzhoy2012polylogarithmic,chekuri2016polynomial,chuzhoy2016improved}. Unfortunately, all these proofs assumed that the input graph has a bounded maximum vertex degree, and additionally the proofs are somewhat more involved than the proof that we need here (this is because these proofs could only afford a $\operatorname{poly}\log k$ loss in the cardinality of the set ${\mathcal{R}}$ of paths relatively to $|T|$, while we can afford a $\operatorname{poly}\log m$ loss). Therefore, we provide a proof of the lemma in Section \ref{sec: splitting} of the Appendix for completeness.
\begin{lemma}\label{lem: splitting}
There is an efficient algorithm that, given as input an $m$-edge graph $G$, and a subset $T$ of $k$ vertices of $G$ called terminals, together with a parameter $0<\tilde \alpha<1$, such that the maximum vertex degree in $G$ is at most $\tilde \alpha k/64$, and every vertex of $T$ has degree $1$ in $G$, either returns FAIL, or computes a partition $(X,Y)$ of $V(G)$, such that:
\begin{itemize}
\item each of the clusters $G[X]$, $G[Y]$ has the $\tilde \alpha'$-bandwidth property, for $\tilde \alpha'=\Omega(\tilde \alpha/\log^2m)$; and
\item there is a set ${\mathcal{R}}$ of at least $\Omega(\tilde \alpha^3k/\log^2m)$ edge-disjoint paths in graph $G$, routing a subset of terminals to edges of $E_G(X,Y)$.
\end{itemize}
Moreover, if the set $T$ of vertices is $\tilde \alpha$-well-linked in $G$, then the algorithm never returns FAIL.
\end{lemma}
We apply the algorithm from \Cref{lem: splitting} to graph $\hat H'$, the set $T$ of terminals, and parameter $\tilde \alpha=\hat \alpha=\frac{\alpha}{4\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}$. Recall that we are guaranteed that the maximum vertex degree in graph $\hat H'$ is at most $d=\frac{\alpha k}{512\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k}\leq \frac{\tilde \alpha k}{64}$.
Note that the algorithm from \Cref{lem: splitting} may only return FAIL if the set $T$ of terminals is not $\tilde \alpha$-well-linked in $\hat H'$, which, from \Cref{obs: expansion and degree}, may only happen if event $\hat {\cal{E}}$ happened, which in turn may only happen with probability $1/\operatorname{poly}(k)$. If the algorithm from \Cref{lem: splitting} returned FAIL, then we terminate the algorithm and return FAIL as well. Therefore, we assume from now on that the algorithm from \Cref{lem: splitting} did not return FAIL.
\iffalse restart the algorithm that we have described so far from scratch. We do so at most $1/\operatorname{poly}\log m$ times, and if, in every iteration, the algorithm from \Cref{lem: splitting} returns FAIL, then we return FAIL as the outcome of the algorithm for \Cref{thm: splitting}. Clearly, this may only happen with probability at most $1/\operatorname{poly}(n)$. Therefore, we assume that in one of the iterations, the algorithm from \Cref{lem: splitting} did not return FAIL. From now on we consider the outcome of that iteration.\fi
Let $(X,Y)$ be the partition of $V(\hat H')$ that the algorithm returns. We are then guaranteed that each of the clusters $\hat H'[X],\hat H'[Y]$ has the $\tilde \alpha'$-bandwidth property in $\tilde H$, where $\tilde \alpha'=\Omega(\tilde \alpha/\log^2m)=\Omega(\alpha/\log^4m)$. The algorithm also ensures that there is a collection ${\mathcal{R}}$ of edge-disjoint paths in $\hat H'$, routing a subset of the terminals to edges of $E_{\hat H'}(X,Y)$, with $|{\mathcal{R}}|\geq \Omega(\tilde \alpha^3k/\log^2m)\geq \Omega(\alpha^3k/\log^8m)$.
This completes the proof of \Cref{thm: splitting}.
\end{proof}
If the algorithm from \Cref{thm: splitting} returned FAIL (which may only happen with probability at most $1/\operatorname{poly}(k)$), then we terminate the algorithm and return FAIL as well. Therefore, we assume from now on that the algorithm from \Cref{thm: splitting} returned a subgraph $\hat H'\subseteq \hat H$ and a partition $(X,Y)$ of $V(\hat H')$ such that each of the clusters $\hat H'[X]$ and $\hat H'[Y]$ has the $\hat \alpha'$-bandwidth property, for $\hat \alpha'=\Theta(\alpha/\log^4m)$, and
there is a set ${\mathcal{R}}$ of $\Omega(\alpha^3k/\log^8m)$ edge-disjoint paths in graph $\hat H'$, routing a subset of terminals to edges of $E_{\hat H'}(X,Y)$.
Notice that we can compute the path set ${\mathcal{R}}$ with the above properties efficiently, using standard maximum flow algorithms. We assume w.l.o.g. that edges of $E_{\hat H}(X,Y)$ do not serve as inner edges on paths in ${\mathcal{R}}$.
Let $E'\subseteq E_{\hat H'}(X,Y)$ be the subset of edges containing the last edge on every path in ${\mathcal{R}}$, so, by reversing the direction of the paths in ${\mathcal{R}}$, we can view the set ${\mathcal{R}}$ of paths as routing the edges of $E'$ to the terminals.
In the remainder of this step, we will slightly modify the graphs $H$ and $\hat H$, and we will continue working with the modified graphs only in the following steps.
Let $\hat H''\subseteq\hat H'$ be the graph obtained from $\hat H'$ by first deleting all edges of $E_{\hat H'}(X,Y)\setminus E'$ from it, and then subdividing every edge $e\in E'$ with a vertex $t_e$. We denote $\tilde T=\set{t_e\mid e\in E'}$, and we refer to vertices of $\tilde T$ as \emph{pseudo-terminals}. Recall that $|\tilde T|=|{\mathcal{R}}|=\Omega(\alpha^3k/\log^8m)$, and there is a set ${\mathcal{R}}'$ of edge-disjoint paths in the resulting graph $\hat H''$, routing the vertices of $\tilde T$ to the vertices of $T$.
We define $\hat H_1=\hat H''[X\cup \tilde T]$, the subgraph of $\hat H''$ induced by the set $X\cup \tilde T$ of vertices, and we define $\hat H_2=\hat H''[Y\cup \tilde T]$ similarly. From the $\hat \alpha'$-bandwidth property of the clusters $\hat H'[X]$ and $\hat H'[Y]$ in $\hat H'$, we are guaranteed that the vertices of $\tilde T$ are $ \hat \alpha'$-well-linked in both $\hat H_1$ and in $\hat H_2$, where $\hat \alpha'= \Theta(\alpha/\log^4m)$. Let ${\mathcal{C}}'\subseteq {\mathcal{C}}$ be the subset of all clusters $C$ whose corresponding supernode $v_C$ lies in graph $\hat H''$.
For convenience, we also subdivide, in graph $H$, every edge $e\in E'$, with the vertex $t_e$, so graph $\hat H''$ can be now viewed as a subgraph of the contracted graph $H_{|{\mathcal{C}}}$.
Next, we let $H'\subseteq H$ be the subgraph of $H$ that corresponds to graph $\hat H''$. In other words, graph $H'$ is obtained from $\hat H''$ by replacing every supernode $v_C$ with the corresponding cluster $C\in {\mathcal{C}}'$. Equivalently, we can obtain graph $H'$ from $H$, by deleting every edge of $E(\hat H)\setminus E(\hat H'')$ and every regular (non-supernode) vertex of $V(\hat H)\setminus V(\hat H'')$. Additionally, for every cluster $C\in {\mathcal{C}}\setminus {\mathcal{C}}'$, we delete all edges and vertices of $C$ from $H'$. We also define a rotation system $\Sigma'$ for graph $H'$, which is naturally induced by $\Sigma$ (vertices $t_e\in \tilde T$ all have degree $2$, so their corresponding ordering ${\mathcal{O}}_{t_e}$ of incident edges can be set arbitrarily). Let $I'=(H',\Sigma')$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace.
We partition the set ${\mathcal{C}}'$ of clusters into two subsets: set ${\mathcal{C}}_X$ contains all clusters $C\in {\mathcal{C}}'$ with $v_C\in X$, and set ${\mathcal{C}}_Y$ contains all clusters $C\in {\mathcal{C}}'$ with $v_C\in Y$. We can similarly define the graphs $H_1,H_2\subseteq H'$, that correspond to the contracted graphs $\hat H_1$ and $\hat H_2$, respectively: let $X'$ be the set of vertices of $H'$, containing every vertex $x\in V(H')$, such that either $x\in C$ for some cluster $C\in {\mathcal{C}}_X$, or $x$ is a regular vertex of $\hat H''$ lying in $X$. Similarly, we let $Y'$ contain all vertices $y\in V(H)$, such that either $y\in C$ for some cluster $C\in {\mathcal{C}}_Y$, or $y$ is a regular vertex of $\hat H''$ lying in $Y$. We then let $H_1=H'[X\cup \tilde T]$, and $H_2=H'[Y\cup \tilde T]$.
The following observation, summarizing properties of instance $I'$, is immediate.
\begin{observation}\label{obs: properties of new graph}
Instance $I'=(H',\Sigma')$ of \ensuremath{\mathsf{MCNwRS}}\xspace satisties the following properties:
\begin{itemize}
\item $\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}} (I)$;
\item $\hat H''=H'_{|{\mathcal{C}}'}$;
\item $|E(\hat H'')|\leq 2|E(\hat H)|\leq 2\eta k\leq O(|\tilde T|\eta \log^8m/\alpha^3)$; and
\item graph $\hat H_1$ is a contracted graph of $H_1$ with respect to ${\mathcal{C}}_X$, and graph $\hat H_2$ is a contracted graph of $H_2$ with respect to ${\mathcal{C}}_Y$. In other words, $\hat H_1=(H_1)_{|{\mathcal{C}}_X}$, and $\hat H_2=(H_2)_{|{\mathcal{C}}_Y}$.
\end{itemize}
\end{observation}
For the third assertion we have used the fact that $k\geq |E(\hat H)|/\eta$ from the statement of \Cref{thm: find guiding paths}, and $|\tilde T|\geq \Omega(\alpha^3k/\log^8m)$.
Recall that $\Lambda(H',\tilde T)$ denotes the set of all routers in graph $H'$, with respect to the set $\tilde T$ of terminals. Each such router ${\mathcal{Q}}$ is a set of paths, routing the vertices of $\tilde T$ to some vertex of $H'$.
Intuitively, from now on we would like to work with instance $I'=(H',\Sigma')$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and the new set $\tilde T$ of terminals. To this end, we start by showing that, in order to obtain the desired distribution ${\mathcal{D}}$ over the routers of $\Lambda(H,T)$, it is now sufficient to compute a distribution ${\mathcal{D}}'$ over the routers of $\Lambda(H',\tilde T)$, such that for every edge $e\in E(H')$, $\expect[{\mathcal{Q}}'\sim {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}},e))^2}$ is low.
\begin{observation}\label{obs: convert distributions}
There is an efficient algorithm, that, given an explicit distribution ${\mathcal{D}}'$ over the routers of $\Lambda(H',\tilde T)$,
such that for every edge $e'\in E(H')$, $\expect[{\mathcal{Q}}'\sim{\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e'))^2}\leq \beta$ holds,
computes an explicit distribution ${\mathcal{D}}$ over the routers of $\Lambda(H,T)$, such that for every edge $e\in E(H)$, $\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left}(\frac{\beta \log^{16}m}{\alpha^8(\alpha')^4}\textsf{right} )$.
\end{observation}
\begin{proof}
Recall that $H'\subseteq H$. Consider some router ${\mathcal{Q}}'\in \Lambda(H',\tilde T)$, whose probability value in distribution ${\mathcal{D}}'$ is $p({\mathcal{Q}}')>0$. We compute a router ${\mathcal{Q}}\in \Lambda(H,T)$ corresponding to ${\mathcal{Q}}'$, and we assign to ${\mathcal{Q}}$ the same probability value $p({\mathcal{Q}}')$.
We now show an algorithm for computing a router ${\mathcal{Q}}\in \Lambda(H,T)$ from a router ${\mathcal{Q}}'\in \Lambda(H',\tilde T)$. We denote by $x'$ the vertex that serves as the center of the router ${\mathcal{Q}}'$.
Recall that there is a set ${\mathcal{R}}'$ of edge-disjoint paths in graph $\hat H''$, routing the vertices of $\tilde T$ to the vertices of $T$, and moreover, a set of paths with these properties can be found efficiently via a standard maximum $s$-$t$ flow computation.
Since $\hat H''=H'_{|{\mathcal{C}}'}$, and every cluster in ${\mathcal{C}}'$ has the $\alpha'$-bandwidth property in $H'$, from \Cref{claim: routing in contracted graph}, we can efficiently compute a set ${\mathcal{R}}_0$ of edge-disjoint paths in graph $H'$, routing a subset $T_0\subseteq T$ of terminals to $\tilde T$, with $|{\mathcal{R}}_0|\geq \alpha'\cdot |{\mathcal{R}}'|/2=\alpha'\cdot |\tilde T|/2=\Omega(\alpha'\alpha^3k/\log^8m)$. By concatenating the paths in ${\mathcal{R}}_0$ and the paths in ${\mathcal{Q}}'$, we obtain a collection ${\mathcal{R}}_0'$ of paths in graph $H'$, routing the terminals of $T_0$ to vertex $x'$, such that for every edge $e\in E(H)$, $\cong_{H'}({\mathcal{R}}_0',e)\leq \cong_{H'}({\mathcal{Q}}',e)+1$.
Since the set $T$ of terminals is $(\alpha\alpha')$-well-linked in graph $H$ (from \Cref{clm: contracted_graph_well_linkedness}), from \Cref{lem: routing path extension}, there exists a collection ${\mathcal{Q}}$ of paths in graph $H$, routing the terminals in $T$ to vertex $x'$, such that for every edge $e\in E(H)$:
\[
\cong_{H}({\mathcal{Q}},e)\leq \ceil{\frac{|T|}{|T_0|}}\textsf{left}(\cong_{H}({\mathcal{R}}'_0,e)+\ceil{\frac{1}{\alpha\alpha'}}\textsf{right} )
\leq O\textsf{left} (\frac{\log^8m}{\alpha'\alpha^3}\textsf{right} )\cdot \textsf{left}(\cong_{H'}({\mathcal{Q}}',e)+\frac{2}{\alpha\alpha'}\textsf{right} ).
\]
A set ${\mathcal{Q}}$ of paths with these properties can be computed efficiently via standard maximum $s$-$t$ flow algorithms.
For every router ${\mathcal{Q}}'\in \Lambda(H',\tilde T)$, whose probability value in distribution ${\mathcal{D}}'$ is $p({\mathcal{Q}}')>0$, we have computed a corresponding router ${\mathcal{Q}}\in \Lambda(H,T)$, and we have assigned to it the same probability value $p({\mathcal{Q}})=p({\mathcal{Q}}')$. This completes the definition of the distribution ${\mathcal{D}}$ over the routers in $\Lambda(H,T)$. From the above discussion, for every edge $e\in E(H)$,
\[\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left}(\frac{\log^{16}k}{\alpha^8(\alpha')^4}\textsf{right} )\cdot \textsf{left} (\expect[{\mathcal{Q}}'\sim {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e))^2}+1\textsf{right}) \leq O\textsf{left}(\frac{\beta \log^{16}m}{\alpha^8(\alpha')^4}\textsf{right} ). \]
\end{proof}
The following immediate corollary is obtained by plugging in the bounds required by \Cref{thm: find guiding paths} into \Cref{obs: convert distributions}.
\begin{corollary}\label{cor: convert distribution fixed values}
There is an efficient algorithm, that, given an explicit distribution ${\mathcal{D}}'$ over the routers of $\Lambda(H',\tilde T)$,
such that for every edge $e'\in E(H')$, $\expect[{\mathcal{Q}}'\sim {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e'))^2}\leq O\textsf{left} (\frac{\log^{16}m}{(\alpha\alpha')^4}\textsf{right} )$ holds, produces an explicit distribution ${\mathcal{D}}$ over the routers of $\Lambda(H,T)$, such that, for every edge $e\in E(H)$: $$\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{32}m}{\alpha^{12}(\alpha')^8}\textsf{right} ).$$
\end{corollary}
Denote $\tilde k=|\tilde T|$.
Recall that $\tilde k\geq \Omega\textsf{left}(\frac{\alpha^3k}{\log^8m}\textsf{right} )$, so $\frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}m}\leq O\textsf{left}( \frac{(\tilde k\hat \alpha'\alpha')^2}{c_0\log^{20}m}\textsf{right} )$. We use a large enough constant $c_1$, whose value will be set later, and we set $c_0=c_1^2$. We can then assume that $\frac{(k\alpha^4 \alpha')^2}{c_0\log^{50}m}\leq \frac{(\tilde k\hat \alpha'\alpha')^2}{c_1\log^{20}m}$. In particular, from Equation \ref{eq: boundaries squared sum bound}, we get that $\sum_{C\in {\mathcal{C}}}|\delta_{H'}(C)|^2<\frac{(k\alpha^4\alpha')^2}{c_0\log^{50}m}\leq \frac{(\tilde k\hat \alpha'\alpha')^2}{c_1\log^{20}m}$. Additionally, if
$|E(H'\setminus\tilde T)|+\mathsf{OPT}_{\mathsf{cnwrs}}(I')>\frac{(\tilde k\hat \alpha'\alpha')^2}{c_1\eta'\log^{20}m}$, then $|E(H\setminus T)|+\mathsf{OPT}_{\mathsf{cnwrs}}(I)>\frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}m}$.
In order to complete the proof of \Cref{thm: find guiding paths}, it is now enough to design a randomized algorithm, that either returns FAIL, or computes a distribution ${\mathcal{D}}'$ over the routers in $\Lambda(H',\tilde T)$, such that, for every edge $e\in E(H')$, $\expect[{\mathcal{Q}}'\sim {\mathcal{D}}']{(\cong_{H'}({\mathcal{Q}}',e))^2}\leq O\textsf{left} (\frac{\log^{16}m}{(\alpha\alpha')^4}\textsf{right} )$. It is enough to ensure that, if $|E(H'\setminus \tilde T)|+\mathsf{OPT}_{\mathsf{cnwrs}}(H',\Sigma')\leq \frac{(\tilde k\hat \alpha'\alpha')^2}{c_1\eta'\log^{20}m}$, then the probability that the algorithm returns FAIL is at most $1/2$.
In the remainder of the proof we focus on the above goal. It would be convenient for us to simplify the notation, by denoting $H'$ by $H$, $\Sigma'$ by $\Sigma$, $I'$ by $I$, $\hat H''$ by $\hat H$, and $\hat \alpha'$ by $\tilde \alpha$. We also denote ${\mathcal{C}}'$ by ${\mathcal{C}}$. We now summarize all properties of the new graphs $H,\hat H$ that we have established so far, and in the remainder of the proof of \Cref{thm: find guiding paths} we will only work with these new graphs.
\paragraph{Summary of the Outcome of Step 1.}
We assume from now on that we are given an instance $I=(H,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a set $\tilde T$ of terminals in graph $H$, and a collection ${\mathcal{C}}$ of disjoint subgraphs (clusters) of $H\setminus \tilde T$. We denote $|\tilde T|=\tilde k$. The corresponding contracted graph is denoted by $\hat H=H_{|{\mathcal{C}}}$.
We are also given a partition $(X,Y)$ of $V(H)\setminus \tilde T$ (note that for convenience of notation, $X$ and $Y$ are now subsets of vertices of $H$, and not of $\hat H$), and a parition ${\mathcal{C}}_X,{\mathcal{C}}_Y$ of ${\mathcal{C}}$, such that each cluster $C\in {\mathcal{C}}_X$ has $V(C)\subseteq X$, and each cluster $C\in {\mathcal{C}}_Y$ has $V(C)\subseteq Y$. We denote $H_1=H[X\cup \tilde T]$ and $H_2=H[Y\cup \tilde T]$. We also denote by $\hat H_1=(H_1)_{|{\mathcal{C}}_X}$ the contracted graph of $H_1$ with respect to ${\mathcal{C}}_X$, and similarly by $\hat H_2=(H_2)_{|{\mathcal{C}}_Y}$.
We now summarize the properties of the graphs that we have defined and the relationships between the main parameters.
\begin{properties}{P}
\item $\tilde k\geq \Omega(\alpha^3k/\log^8m)$;\label{prop after step 1: number of pseudoterminals}
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property in $H$; \label{prop after step 1: bandwidth property}
\item $|E(\hat H)|\leq O\textsf{left}(\frac{\tilde k\cdot \eta \log^8m}{\alpha^3}\textsf{right} )$ (from \Cref{obs: properties of new graph}); \label{prop after step 1: few edges}
\item every vertex of $\tilde T$ has degree $1$ in $H_1$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_1$, for $\tilde \alpha=\Theta(\alpha/\log^4m)$;\label{prop after step 1: terminals in H1}
\item similarly, every vertex of $\tilde T$ has degree $1$ in $H_2$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_2$; and \label{prop after step 1: terminals in H2}
\item $\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}m}
$, where $c_1$ is some large enough constant, whose value we can set later. \label{prop after step 1: small squares of boundaries}
\end{properties}
Our goal is
to design an efficient randomized algorithm, that either returns FAIL, or computes a distribution ${\mathcal{D}}$ over the routers in $\Lambda(H,\tilde T)$, such that, for every edge $e\in E(H)$, $\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{16}m}{(\alpha\alpha')^4}\textsf{right} )$.
It is enough to ensure that, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(H\setminus T)|<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}m}$,
then the probability that the algorithm returns FAIL is at most $1/2$.
\iffalse
\subsection{Step 2: a Modified Graph}
In the remainder of the proof, it would be convenient for us to work with a low-degree equivalent of the graph $\hat H$, that we denote by $H^+$. In order to define the graph $\hat H^+$, we start by defining a rotation system $\hat \Sigma $ for graph $\hat H$, as follows. Let $x\in V(\hat H)$ be a vertex. If $x$ is a supernode, that is, $x=v(C)$ for some cluster $C\in {\mathcal{C}}$, then we define the circular ordering $\hat{\mathcal{O}}_x\in \hat \Sigma$ of the edges of $\delta_{\hat H}(x)$ to be arbitrary.
Otherwise, $x$ is a regular vertex, and it is a vertex of the original graph $H$. We then let $\hat {\mathcal{O}}_x\in \hat \Sigma$ be identical to the ordering ${\mathcal{O}}_x\in \Sigma$, where $\Sigma$ is the original rotation system for graph $H$.
We are now ready to define the modified graph $H^+$. We start with $H^+=\emptyset$, and then process every vertex $u\in V(\hat H)$ one-by-one. We denote by $d(u)$ the degree of the vertex $u$ in graph $\hat H$. We now describe an iteration when a vertex $u\in V(\hat H)$ is processed. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that incident to $u$ in $\hat H$, indexed according to their ordering in $\hat {\mathcal{O}}_u\in \hat \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H^+$. We refer to the edges in the grids $\Pi(u)$ as \emph{inner edges}.
Once every vertex $u\in V(\hat H)$ is processed, we add a collection of \emph{outer edges} to graph $H^+$, as follows. Consider any edge $e=(x,y)\in E(H)$. Assume that $e$ is the $i$th edge of $x$ and the $j$th edge of $y$, that is, $e=e_i(x)=e_j(y)$. Then we add an edge $e'=(s_i(x),s_j(y))$ to graph $H^+$, and we view this edge as the copy of the edge $e\in E(\hat H)$. We will not distinguish between the edge $e$ of $\hat H$ (and the corresponding edge of $H$), and the edge $e'$ of $H^+$.
We note that every terminal $t\in T$ has degree $1$ in $\hat H$, so its corresponding grid $\Pi(t)$ consists of a single vertex, that we also denote by $t$. Therefore, set $T$ of terminals in $H$ naturally corresponds to a set of $k$ terminals in $H^+$, that we denote by $T$ as before.
This completes the definition of the graph $H^+$. Note that the maximum vertex degree in $H^+$ is $4$. We also define a rotation system $\Sigma^+$ for the graph $H^+$ in a natural way: for every vertex $u\in V(H)$,
consider the standard drawing of the grid $\Pi(u)$, to which we add the drawings of the edges that are incident to vertices $s_1(u),\ldots,s_{d(u)}(u)$, so that the edges are drawn on the grid's exterior in a natural way (\mynote{add figure}). This layout defines an ordering ${\mathcal{O}}^+(v)$ of the edges incident to every vertex $v\in \Pi(u)$.
We start with the following simple claim.
\begin{claim}\label{claim: cheap solution to modified instance} There is a solution to the \textnormal{\textsf{MCNwRS}}\xspace problem instance $(H^+,\Sigma^+)$ of cost at most ..., such that no inner edge of $H^+$ participates in any crossings in the solution.
\end{claim}
\begin{proof}
We start by showing that $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat H,\hat \Sigma)\leq ...$. Recall that, from \Cref{lem: crossings in contr graph}, there is a drawing $\hat \phi$ of graph $\hat H$ with at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8m/(\alpha')^2$ crossings, such that for every vertex $x\in V(\hat H)\cap V(H)$, the ordering of the edges of $\delta_{\hat H}(x)$ as they enter $x$ in $\hat \phi$ is consistent with the ordering ${\mathcal{O}}_x\in \Sigma$, and hence with the ordering $\hat {\mathcal{O}}_x\in \hat \Sigma$. Drawing $\hat \phi$ of $\hat H$ may not be a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace since for some supernodes $v(C)$, the ordering of the edges that are incident to $v(C)$ in $\hat H$ as they enter the image of $v(C)$ in $\hat \phi$ may be different from $\hat {\mathcal{O}}_{v(C)}$. For each such vertex $v(C)$, we may need to \emph{reorder} the images of the edges of $\delta_{\hat H}(v(C))=\delta_H(C)$ near the image of $v(C)$, so that they enter the image of $v(C)$ in the correct order. This can be done by introducing at most $|\delta_H(C)|^2$ crossings for each such supernode $v(C)$. The resulting drawing $\hat \phi'$ of $\hat \phi$ is a feasible solution to instance $(\hat H,\hat \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, whose cost is bounded by:
\[O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8m/(\alpha')^2+\sum_{C\in {\mathcal{C}}}|\delta_H(C)|^2\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\cdot \log^8m/(\alpha')^2+ \]
\end{proof}
\fi
\iffalse
\subsection{Step 2: Routing the Terminals to a Single Vertex, and the Epanded Graph}
\mynote{need to redo this: the paths set should be in the contracted graph $\hat H_1$}
In this step we start by considering the graph $H_1$ and the set $\tilde T$ of terminals in it. Our goal is to compute a collection ${\mathcal{J}}$ of paths in graph $H_1$, routing all terminals of $\tilde T$ to a single vertex, such that the paths in ${\mathcal{J}}$ cause a relatively low congestion in graph $H_1$. We show that, if such a collection of paths does not exist, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)$ is high. Intuitively, we will use the set ${\mathcal{J}}$ of paths in order to define an ordering of the terminals in $\tilde T$, which will in turn be used in order to compute a grid-like structure in graph $H_2$. Once we compute the desired set ${\mathcal{J}}$ of paths, we will replace the graph $H$ with its low-degree analogue $H^+$, that we refer to as the \emph{expanded graph}. The remaining steps in the proof of \Cref{thm: find guiding paths} will use this expanded graph only.
We now proceed to describe the algorithm for Step 2. We consider every \emph{regular} vertex (that is, a vertex that is not a supernode) $x\in V(\hat H_1)$ one by one. For each such vertex $x$, we compute a set ${\mathcal{J}}(x)$ of paths in graph $\hat H_1$, with the following properties:
\begin{itemize}
\item every path in ${\mathcal{J}}(x)$ originates at a distinct vertex of $\tilde T$ and terminates at $x$;
\item the paths in ${\mathcal{J}}(x)$ are edge-disjoint; and
\item ${\mathcal{J}}(x)$ is a maximum-cardinality set of paths in $H_1$ with the above two properties.
\end{itemize}
Note that such a set ${\mathcal{J}}(x)$ of paths can be computed via a standard maximum $s$-$t$ flow computation.
Throughout, we use a parameter $\tilde k'=\ceil{\tilde k\alpha^5/(c'\eta\log^{36}m)}$, where $c'$ is a large enough constant whose value we set later.
If, for every regular vertex $x\in V(\hat H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then we reurn FAIL and terminate the algorithm. In the following lemma, whose proof is deferred to Section \ref{sec: few paths high opt} of Appendix we show that, in this case, $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}m}\textsf{right} )$ must hold.
Note that, since we can set $c_1$ to be a large enough constant, we can ensure that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)>\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}m}$ holds in this case.
The value of the constant $c'$ that is used in the definition of the parameter $\tilde k'$ is set in the proof of the lemma.
\begin{lemma}\label{lem: high opt or lots of paths}
If, for every regular vertex $x\in V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}m}\textsf{right} )$.
\end{lemma}
\mynote{need to redo from here}
From now on we assume that there is some vertex $x\in V(H_1)$, for which $|{\mathcal{J}}(x)|\geq \tilde k'$. We denote ${\mathcal{J}}_0={\mathcal{J}}(x)$, and we let $T_0\subseteq \tilde T$ be the set of terminals that serve as endpoints of paths in ${\mathcal{J}}_0$, so $|T_0|=|{\mathcal{J}}_0|=\tilde k'$.
Let $z=\ceil{\tilde k/\tilde k'}=O\textsf{left}(\frac{\eta\log^{36}m}{\alpha^5\alpha'}\textsf{right} )$. Next, we arbitrarily partition the terminals of $\tilde T\setminus T_0$ into $z$ subsets $T_1,\ldots,T_z$, of cardinality at most $\tilde k'$ each. Consider now some index $1\leq i\leq z$. Since the set $\tilde T$ of terminals is $(\tilde \alpha\alpha')$-well-linked in $H_1$ (from \Cref{clm: contracted_graph_well_linkedness}), using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{J}}_i'$ of paths in graph $H_1$, routing vertices of $T_i$ to vertices $T_0$, such that the paths in $T_i$ cause edge-congestion $O(1/(\tilde \alpha\alpha'))$, and each vertex of $T_i$ is the endpoint of at most one path in ${\mathcal{J}}_i'$. By concatenating the paths in ${\mathcal{J}}_i'$ with paths in ${\mathcal{J}}_0$, we obtain a collection ${\mathcal{J}}_i$ of paths in graph $H_1$, connecting every terminal of $T_i$ to $x$, that cause edge-congestion $O(1/(\tilde \alpha\alpha'))$. Let ${\mathcal{J}}=\bigcup_{i=0}^z{\mathcal{J}}_i$ be the resulting set of paths. Observe that set ${\mathcal{J}}$ contains $\tilde k$ paths, routing the terminals in $\tilde T$ to the vertex $x$ in graph $H_1$, with $\cong_H({\mathcal{J}})\leq O\textsf{left}(\frac z{\tilde \alpha\alpha'}\textsf{right} )\leq O\textsf{left}(\frac{\eta\log^{40}m}{\alpha^6(\alpha')^2}\textsf{right} )$. We denote by $\rho=O\textsf{left}(\frac{\eta\log^{40}m}{\alpha^6(\alpha')^2}\textsf{right} )$ this bound on $\cong_{H_1}({\mathcal{P}}')$. We assume w.l.o.g. that the paths in ${\mathcal{J}}$ are simple. Since every terminal in $\tilde T$ has degree $1$ in $H_1$, no path in ${\mathcal{J}}$ may contain a terminal in $\tilde T$ as its inner vertex.
In the remainder of the proof, it would be convenient for us to work with a low-degree equivalent of the graph $H$, that we call \emph{expanded graph}, and denote by $H^+$.
For every edge $e\in E(H)$, let $N_e=\cong_H({\mathcal{J}},e)$. Recall that for every edge $e\in E(H_1\setminus\tilde T)$, $N_e\leq \rho$, and for every other edge $e$ of $H$, $N_e\leq 1$.
\fi
\subsection{Step 2: Routing the Terminals to a Single Vertex, and an Expanded Graph}
In this step we start by considering the graph $\hat H_1$ and the set $\tilde T$ of terminals in it. Our goal is to compute a collection ${\mathcal{J}}$ of paths in graph $\hat H_1$, routing all terminals of $\tilde T$ to a single regular vertex, such that the paths in ${\mathcal{J}}$ cause a relatively low congestion in graph $\hat H_1$. We show that, if such a collection of paths does not exist, then $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ is high. Intuitively, we will use the set ${\mathcal{J}}$ of paths in order to define an ordering of the terminals in $\tilde T$, which will in turn be used in order to compute a grid-like structure in graph $H_2$. Once we compute the desired set ${\mathcal{J}}$ of paths, we will replace the graph $H$ with its low-degree analogue $H^*$, that we refer to as the \emph{expanded graph}. The remaining steps in the proof of \Cref{thm: find guiding paths} will use this expanded graph only.
\subsubsection{Routing the Terminals to a Single Vertex in $\hat H_1$}
We process {\bf regular} vertices of $V(\hat H_1)$ (that is, vertices of $V(\hat H_1)\cap V(H_1)$) one by one. For each such vertex $x$, we compute a set ${\mathcal{J}}(x)$ of paths in graph $\hat H_1$, with the following properties:
\begin{itemize}
\item every path in ${\mathcal{J}}(x)$ originates at a distinct vertex of $\tilde T$ and terminates at $x$;
\item the paths in ${\mathcal{J}}(x)$ are edge-disjoint; and
\item ${\mathcal{J}}(x)$ is a maximum-cardinality set of paths in $\hat H_1$ with the above two properties.
\end{itemize}
Note that such a set ${\mathcal{J}}(x)$ of paths can be computed via a standard maximum $s$-$t$ flow computation.
Throughout, we use a parameter $\tilde k'=\tilde k\alpha^5/(c'\eta\log^{36}m)$, where $c'$ is a large enough constant whose value we set later.
If, for every vertex $x\in V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then we reurn FAIL and terminate the algorithm. In the following lemma, whose proof is deferred to Section \ref{sec: few paths high opt} of Appendix we show that, in this case, $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}m}\textsf{right} )$ must hold.
Note that, since we can set $c_1$ to be a large enough constant, we can ensure that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)>\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}m}$ holds in this case.
The value of the constant $c'$ that is used in the definition of the parameter $\tilde k'$ is set in the proof of the lemma.
\begin{lemma}\label{lem: high opt or lots of paths}
If, for every vertex $x\in V(\hat H_1)\cap V(H_1)$, $|{\mathcal{J}}(x)|<\tilde k'$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq
\Omega\textsf{left}(\frac{(\tilde k\tilde \alpha \alpha')^2}{\eta'\log^{20}m}\textsf{right} )$.
\end{lemma}
From now on we assume that there is some vertex $x\in V(\hat H_1)\cap V(H_1)$, for which $|{\mathcal{J}}(x)|\geq \tilde k'$.
\subsubsection{The Expanded Graph}
From now on we fix the vertex $x\in V(\hat H_1)\cap V(H_1)$, and we let ${\mathcal{J}}={\mathcal{J}}(x)$ be a set of at least $\tilde k'$ edge-disjoint paths in graph $\hat H_1$, routing a subset $\tilde T_0\subseteq \tilde T$ of terminals to vertex $x$.
We are now ready to define the expanded graph $H^*$. We start with graph $H^*$ being empty, and then process every vertex $u\in V(H_2)\setminus \tilde T$ one by one. We now describe an iteration when a vertex $u\in V(H_2)\setminus \tilde T$ is processed. We denote by $d(u)$ the degree of the vertex $u$ in graph $H_2$. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that are incident to $u$ in $H_2$, indexed according to their ordering in $ {\mathcal{O}}_u\in \Sigma$. We let $\Pi(u)$ be a $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$ indexed in their natural left-to-right order.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H^*$. We refer to the edges in the resulting grids $\Pi(u)$ as \emph{inner edges}.
Once every vertex $u\in V(H_2)\setminus \tilde T$ is processed, we add
the vertices of $\tilde T$ to the graph $H^*$. Recall that every terminal $t\in \tilde T$ has degree $1$ in $H_2$. We denote the unique edge $e_t$ incident to $t$ by $e_1(t)$, and we denote $s_1(t)=t$.
Next, we add
a collection of \emph{outer edges} to graph $H^*$, as follows. Consider any edge $e=(u,v)\in E(H_2)$. Assume that $e$ is the $i$th edge of $u$ and the $j$th edge of $v$, that is, $e=e_i(u)=e_j(v)$. Then we add an edge $e'=(s_i(u),s_j(v))$ to graph $H^*$, and we view this edge as the \emph{copy of the edge $e\in E(H_2)$}. We will not distinguish between the edge $e$ of $H_2$, and the edge $e'$ of $H^*$.
Our last step is to add vertex $x$ to graph $H^*$, that connects to every terminal $t\in \tilde T$ with an edge $(x,t)$, that is also viewed as an outer edge. The following lemma, whose proof is deferred to Section \ref{sec:ordering of terminals} of Appendix, allows us to compute an ordering $\tilde {\mathcal{O}}$ of the terminals, such that the graph $H^*$ has a drawing $\phi$ with few crossings, in which the inner edges do not participate in any crossings, and the images of the edges incident to $x$ enter $x$ in order consistent with $\tilde {\mathcal{O}}$.
\iffalse
Our last step is to add a $(\tilde k\times\tilde k)$-grid $\Pi(x)$ to graph $H^*$ corresponding to the vertex $x$. As before, the edges of the grid are inner edges for graph $H^*$, and we denote the vertices on the first row of the grid by $s_1(x),\ldots,s_{\tilde k}(x)$, indexed in their natural left-to-right order. For all $1\leq i\leq \tilde k$, we add an outer edge $(t_i,s_i(x))$ to graph $H^*$. This completes the definition of the graph $H^*$, provided that we are given an ordering $\tilde {\mathcal{O}}$ of the terminals. The following lemma, whose proof is deferred to Section \ref{sec:ordering of terminals} of the Appendix, allows us to compute an ordering $\tilde {\mathcal{O}}$ of the terminals, such that the resulting graph $H^*$ has a drawing $\phi$ with few crossings, in which the inner edges do not participate in any crossings.
\fi
\begin{lemma}\label{lem: find ordering of terminals}
There is an efficient algorithm that computes an ordering $\tilde {\mathcal{O}}$ of the terminals in $\tilde T$, such that there is a drawing $\phi$ of graph $H^*$ with at most $O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\frac{\eta^2\log^{74}m}{\alpha^{12}(\alpha')^4}\textsf{right} ) +O \textsf{left} (
\frac{\tilde k \eta\log^{37}m}{\alpha^6(\alpha')^2}\textsf{right} )$ crossings, in which all crossings are between pairs of outer edges. Moreover, if we denote $\tilde T=\set{t_1,\ldots,t_{\tilde k}}$, where the terminals are indexed according to the ordering $\tilde {\mathcal{O}}$, and, for each $1\leq i\leq t_{\tilde k}$, denote by $e_i=(t_i,x)$ the edge of $H^*$ connecting $t_i$ to $x$, then the images of the edges $e_1,\ldots,e_{\tilde k}$ enter the image of $x$ in this circular order in the drawing $\phi$.
\end{lemma}
From now on we fix the ordering $\tilde {\mathcal{O}}$ of the terminals in $\tilde T$ given by \Cref{lem: find ordering of terminals}, and the drawing $\phi$ of $H^*$ (which is not known to the algorithm).
It will be convenient for us to slightly modify the graph $H^*$ as follows. We denote
the terminals by $\tilde T=\set{t_1,\ldots,t_{\tilde k}}$, where the terminals are indexed according to the circular ordering $\tilde {\mathcal{O}}$. Let $H'$ be a graph obtained from $H^*$, by first deleting the vertex $x$ from it, and then adding, for all $1\leq i<\tilde k$, an edge $e^*_i=(t_i,t_{i+1})$, and another edge $e^*_{\tilde k}=(t_{\tilde k},t_1)$. We denote this set of the newly added edges by $E^*$, and we view them as inner edges. Note that the edges of $E^*$ form a simple cycle, that we denote by $L^*$. We also denote $H''=H'\setminus E^*$.
We note that the drawing $\phi$ of $H^*$ can be easily extended to obtain a drawing $\phi'$ of graph $H'$ in the plane, so that the inner edges of $H'$ do not participate in any crossings, and the image of the cycle $L^*$ (which must be a simple closed curve) is the boundary of the outer face.
In order to do so, we start with the drawing $\phi$ of $H^*$ on the sphere, and then consider the tiny $x$-disc $D=D_{\phi}(x)$, denoting its boundary by $\gamma^*$.
For every terminal $t_i\in \tilde T$, we denote by $e_i$ the unique edge incident to $t_i$ in $H''$, and by $e'_i=(t_i,x)$. We also denote by $\gamma_i,\gamma'_i$ the images of the edges $e_i,e'_i$ in drawing $\phi$. Let $p_i$ be the unique point on the intersection of $\gamma'_i$ and $\gamma^*$. We move the image of terminal $t_i$ to point $p_i$. We then modify the image of the edge $e_i$, so that it becomes a concatenation of $\gamma_i$, and the portion of $\gamma'_i$ lying outside the interior of $D$. Lastly, we draw the edges of $E^*$ in a natural way, where edge $e^*_i$ is simply a segment of $\gamma^*$ between the images of $t_i$ and $t_{i+1}$, so that all resulting segments are mutually internally disjoint. Once we delete the vertex $x$ from this drawing, no part of the resulting drawing is contained in the interior of the disc $D$, and the image of the cycle $L^*$ is precisely $\eta$, so we can view the resuting drawing $\phi'$ of $H'$ as a drawing in the plane, with $D$ being its outer face. Note that this transformation does not increase the number of crossings.
The next observation follows by substituting parameters and bounds that we have already established. The proof is included in Section \ref{subsec: proof of obs on bounds on opt} of Appendix.
\begin{observation}\label{obs: bounds on opt}
Let $c_2$ be a large enough constant. We can set the value of constant $c_1$ so that it is large enough, and, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}m}$, then $\mathsf{cr}(\phi')\leq
\mathsf{cr}(\phi)\leq \frac{\tilde k^2}{c_2\eta^5}$.
\end{observation}
We will se the value of constant $c_2$ later, and the value of constant $c_1$ will then be set using \Cref{obs: bounds on opt}.
It is now enough to ensure that, if $\mathsf{cr}(\phi')<
\frac{\tilde k^2}{\eta^5}$, then the probability that the algorithm returns FAIL is at most $1/2$.
Let $\Lambda'=\Lambda(H'',\tilde T)$ be the collection of all routers in graph $H''$ with respect to the set $\tilde T$ of terminals.
We need the following simple observation, whose proof is deferred to Section \ref{subsec: transform paths 2} of the Appendix.
\begin{observation}\label{obs: transform paths 2}
There is an efficient algorithm, that, given an explicit distribution ${\mathcal{D}}$ over the routers of $\Lambda'$, such that for every {\bf outer} edge $e\in E(H'')$, $\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_{H''}({\mathcal{Q}},e))^2}\leq \beta$, computes an explicit distribution ${\mathcal{D}}'$ over the routers in $\Lambda(H,\tilde T)$, where for every edge $e\in E(H)$, $\expect[{\mathcal{Q}}'\sim {\mathcal{D}}']{(\cong_{H}({\mathcal{Q}}',e))^2}\leq \beta$.
\end{observation}
\subsubsection{Summary of Step 2}
\label{step 2 summary}
In the remainder of the proof of \Cref{thm: find guiding paths} we will work with graph $H'$ only. Recall that graph $H'$ contains a set $E^*=\set{e^*_1,\ldots,e^*_{\tilde k}}$ of edges (that are considered to be inner edges), where for all $1\leq i\leq \tilde k$, $e^*_i=(t_i,t_{i+1})$ (we use indexing modulo $\tilde k$). The set $E^*$ of edges defines a cycle $L^*=(t_1,\ldots,t_{\tilde k})$ in graph $H'$. We also denoted $H''=H'\setminus E^*$. Recall that graph $H''$ is obtained from a subgraph $H_2\subseteq H$, by replacing every vertex $v\in V(H_2)\setminus \tilde T$ with a grid $\Pi(v)$. All edges lying in the resulting grids $\Pi(v)$, and the edges of $E^*$ are inner edges, while all other edges of $H'$ are outer edges. Each outer edge of $H'$ corresponds to some edge of graph $H_2$, and we do not distinguish between these edges. Note that in graph $H'$, all vertices have degrees at most $4$. We will also use the clustering ${\mathcal{C}}_Y$ of graph $H_2$, and the fact that, from Property \ref{prop after step 1: small squares of boundaries}:
\begin{equation}\label{eq: sum of squares}
\sum_{C\in {\mathcal{C}}_Y}|\delta_H(C)|^2<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}m}. \end{equation}
We further partition the outer edges of graph $H''$ into two subsets: type-1 outer edges and type-2 outer edges. Consider any outer edge $e$ in graph $H''$, and let $e'=(u,v)$ be the corresponding edge in graph $H$. If $u$ and $v$ both lie in the same cluster $C\in {\mathcal{C}}_Y$, then we say that $e$ is a \emph{type-2} outer edge, and otherwise it is a type-1 outer edge.
Intuitively, for each type-1 outer edge, there is a corresponding edge in the contracted graph $\hat H=H_{|{\mathcal{C}}}$. From Property \ref{prop after step 1: few edges}, we obtain the following observation.
\begin{observation}\label{obs: few outer edges}
There is a universal constant $c$ (independent of $c_1$ and $c_2$), such that the total number of type-1 outer edges in $H''$ is bounded by ${c\tilde k\cdot \eta \log^8m/\alpha^3}$.
\end{observation}
\iffalse
======================
\highlightf[purple]{Calculations:}
final number of paths: at least $\alpha^*\tilde k/256$. Supergrid dimensions: $\lambda\times \lambda$. Want: at most $\lambda^2/32$ cells have an outer edge on each path. Number of paths per cell: $\alpha^*\tilde k/(256\lambda)$. So we need that:
\[\frac{\alpha^*\tilde k}{256\lambda}\cdot \frac{\lambda^2}{32}>\frac{c\tilde k\cdot \eta \log^8k}{\alpha^3}. \]
To ensure this, enough to set:
\[\lambda=\frac{2^{16}c\cdot \eta \log^8n}{\alpha^*\alpha^3}. \]
(and this is tight to within constants).
Because $\alpha^*=\Theta(\alpha\alpha'/\log^4n)$, we get that $\lambda=O\textsf{left}( \frac{\eta \log^{12}n}{\alpha^4\alpha'} \textsf{right} )$.
If we ensure that $\highlightf{\eta>\frac{c^* \log^{12}n}{\alpha^4\alpha'}}$, then $\lambda<\eta^2$ will hold.
Group size: start with:
\[\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}=\floor{\frac{\alpha^3(\alpha^*)^2\tilde k}{2^{32}c\eta\log^8n}}.\]
This is $\Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^8n} \textsf{right})$. If $\highlightf{\eta>\frac{c^* \log^{8}n}{\alpha^5(\alpha')^2}}$, we get that $\psi>\frac{16\tilde k}{\eta^2}$.
Overall, for this part, we need $\highlightf{\eta>\frac{c^* \log^{12}n}{\alpha^5(\alpha')^2}}$, which so far is ensured.
======================
\fi
Recall that from Property \ref{prop after step 1: terminals in H1}, every vertex of $\tilde T$ has $1$ in $H_2$, and vertex set $\tilde T$ is $\tilde \alpha$-well-linked in $\hat H_2$. Combining this with the $\alpha'$-bandwidth property of every cluster $C\in {\mathcal{C}}_Y$ from Property \ref{prop after step 1: bandwidth property}, from \Cref{clm: contracted_graph_well_linkedness}, the set $\tilde T$ of terminals is $\tilde \alpha\cdot \alpha'$-well-linked in $H_2$. Lastly, using the fact that each graph in $\set{\Pi(v)\mid v\in V(H_2)}$ has the $1$-bandwidth property, from \Cref{clm: contracted_graph_well_linkedness}, we get the following observation.
\begin{observation}\label{obs: terminals well linked in H''}
The set $\tilde T$ of terminals is $\alpha^*$-well-linked in $H''$, where ${\alpha^*=\tilde \alpha\cdot\alpha'=\Theta(\alpha\alpha'/\log^4m)}$. Moreover, each terminal in $\tilde T$ has degree $1$ in $H''$ and degree $3$ in $H'$.
\end{observation}
(we have used the fact that
$\tilde \alpha=\Theta(\alpha/\log^4m)$ (see Property \ref{prop after step 1: terminals in H1})).
We will restrict our attention to special types of drawings of graph $H'$, called \emph{legal drawings}, that we define next.
\begin{definition}[Legal drawing of $H'$]
We say that a drawing $\phi^*$ of graph $H'$ in the plane is \emph{legal} if it has the following properties:
\begin{itemize}
\item no inner edge of $H'$ participates in any crossing of $\phi^*$, and in particular the image of the cycle $L^*$ is a simple closed curve, denoted by $\gamma^*$; and
\item $\gamma^*$ is the boundary of the outer face in the drawing.
\end{itemize}
\end{definition}
We let $\phi^*$ be a legal drawing of $H'$ with smallest number of crossings, and we denote by $\mathsf{cr}^*$ the number of crossings in $\phi^*$.
From \Cref{obs: bounds on opt}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\eta'\log^{20}m}$, then $\mathsf{cr}^*\leq
\frac{\tilde k^2}{c_2\eta^5}$.
Denote $\tilde T=\set{t_1,\ldots,t_{\tilde k}}$, where the terminals are indexed according their ordering in $\tilde {\mathcal{O}}$. We partition the set $\tilde T$ of terminals into four subsets $T_1,\ldots,T_4$, where $T_1,T_2,T_3$ contain $\floor{\tilde k/4}$ consecutive terminals from $\tilde T$ each, and $T_4$ contains the remaining terminals, in a natural way using the ordering $\tilde {\mathcal{O}}$, that is, $T_1=\set{t_1,\ldots,t_{\floor{\tilde k/4}}}$, $T_2=\set{t_{\floor{\tilde k/4}+1},\ldots,t_{2\floor{\tilde k/4}}}$, $T_3=\set{t_{2\floor{\tilde k/4}+1},\ldots,t_{3\floor{\tilde k/4}}}$, and $T_4=\set{t_{3\floor{\tilde k/4}+1},\ldots,t_{\tilde k}}$. Clearly, each of the four sets contains at least $\floor{\tilde k/4}$ terminals.
Recall that in a legal drawing $\phi$ of $H'$, the image of the cycle $L^*$ is a simple closed curve, that we denoted by $\gamma^*$. It will be convenient for us to view this curve $\gamma^*$ as the boundary of a rectangular area in the plane, that encloses the legal drawing of $H'$. We sometimes refer to this rectangular area as the \emph{bounding box} of the drawing, and denote it by $B^*$. We will think of the terminals in $T_1$ and $T_3$ as appearing on the left and on the right boundaries of $B^*$, respectively, and of the terminals in $T_2$ and $T_4$ as appearing on the top and the bottom boundaries of $B^*$, respectively.
For all $1\leq i\leq 4$, we let $\tilde {\mathcal{O}}_i$ be the ordering of the terminals in $T_i$ consistent with their ordering on the boundary of $B^*$ (where each ordering $\tilde{\mathcal{O}}_i$ is no longer circular), so that the terminals in sets $T_1$ and in $T_3$ appear in the bottom-to-top order, and the terminals in $T_2$ and $T_4$ appear in their left-to-right order (so $\tilde {\mathcal{O}}$ is obtained by concatenationg $\tilde{\mathcal{O}}_1,\tilde{\mathcal{O}}_2$, the reversed ordering $\tilde{\mathcal{O}}_3$, and the reversed ordering $\tilde{\mathcal{O}}_4$).
Recall that $\Lambda'=\Lambda(H'',\tilde T)$.
Our goal from now on is to design a randomized algorithm, that either computes a distribution ${\mathcal{D}}$ over the routers of $\Lambda'$, such that for every outer edge $e\in E(H'')$, $\expect[{\mathcal{Q}}\sim{\mathcal{D}}]{(\cong_{H'}({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{16}m}{(\alpha\alpha')^4}\textsf{right} )$, or returns FAIL.
It is enough to ensure that, if $\mathsf{cr}^*\leq
\frac{\tilde k^2}{c_2\eta^5}$ for some large enough constant $c_2$, whose value we can set later, then the probability that the algorithm returns FAIL is at most $1/4$.
\subsection{Step 3: Constructing a Grid Skeleton}
In this and the following step we will construct a grid-like structure in graph $H''$. Recall that the set $\tilde T$ of terminals is $\alpha^*$-well-linked in graph $H''$. From \Cref{thm: bandwidth_means_boundary_well_linked} there is a set ${\mathcal{P}}'$ of paths in $H''$, routing all terminals of $T_1$ to terminals of $T_3$, with edge-congestion at most $\ceil{1/\alpha^*}$, such that the routing is one-to-one. From \Cref{claim: remove congestion}, there is a collection ${\mathcal{P}}''$ of at least $|T_1|/\ceil{1/\alpha^*}=\floor{\tilde k/4}/\ceil{1/\alpha^*}\geq \alpha^*\tilde k/8$ edge-disjoint paths in $H''$, routing some subset of terminals of $T_1$ to a subset of terminal of $T_3$, in graph $H''$. Moreover, since graph $H''$ has maximum vertex degree at most $4$, using arguments similar to those in the proof of \Cref{claim: remove congestion}, there is a collection ${\mathcal{P}}$ of $\floor{\alpha^*\tilde k/32}$ {\bf node-disjoint} paths in graph $H''$, routing some subset $A\subseteq T_1$ of terminals, to some subset $A'\subseteq T_3$ of terminals. We can compute such a set ${\mathcal{P}}$ of paths efficiently using standard maximum $s$-$t$ flow algorithms.
Using similar reasoning, we can compute a collection ${\mathcal{R}}$ of $\floor{\alpha^*\tilde k/32}$ node-disjoint paths in graph $H''$, routing some subset $B\subseteq T_2$ of terminals, to some subset $B'\subseteq T_4$ of terminals.
Intuitively, after we discard a small subset of paths from each of the sets ${\mathcal{P}}$ and ${\mathcal{R}}$, the remaining paths will be used in order to construct a grid-like structure, where paths in ${\mathcal{P}}$ will serve as horizontal paths of the grid, and paths in ${\mathcal{R}}$ will serve as vertical paths. If the paths in the resulting sets do not form a grid-like structure, then we will terminate the algorithm with a FAIL. We will prove that, if $\mathsf{cr}^*\leq
\frac{\tilde k^2}{c_2\eta^5}$ for a large enough constant $c_2$, then we will construct the grid-like structure successfully with probability at least $3/4$.
We denote ${\mathcal{P}}_0={\mathcal{P}}$ and ${\mathcal{R}}_0={\mathcal{R}}$. Recall that so far, ${|{\mathcal{P}}_0|,|{\mathcal{R}}_0|\geq \floor{\alpha^*\tilde k/32}}$.
Intuitively, if the dimensions of the grid-like structure that we construct are $(h\times h)$, then we need $h$ to be quite close to $\tilde k$, since this grid-like structure will be exploited in order to define the distribution ${\mathcal{D}}$ over the routers of $\Lambda'$.
We will first construct a smaller grid-like structure, that we call a \emph{grid skeleton}. This grid skeleton will be associated with a grid $\Pi^*$ of smaller dimensions, that we sometimes call a \emph{supergrid}. We then extend this grid skeleton to construct a large enough grid-like structure.
We will use two additional parameters. The first parameter is:
\[\lambda=\frac{2^{24}c\cdot \eta \log^8m}{\alpha^*\alpha^3},\]
where $c$ is the constant from \Cref{obs: few outer edges}. Notice that, since $\alpha^*=\Theta(\alpha\alpha'/\log^4m)$, we get that $\lambda=O\textsf{left}( \frac{\eta \log^{12}m}{\alpha^4\alpha'} \textsf{right} )$. Moreover, since $\eta>\frac{c^* \log^{12}m}{\alpha^4\alpha'}$ for a large enough constant $c^*$ (from the statement of \Cref{thm: find guiding paths}), $\lambda<\eta^2$ holds. The supergrid that we construct will have dimensions $(\Theta(\lambda)\times \Theta(\lambda))$. The second parameter is:
\[\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}=\floor{\frac{\alpha^3(\alpha^*)^2\tilde k}{2^{30}c\eta\log^8m}}.\]
Clearly, $|{\mathcal{R}}_0|,|{\mathcal{P}}_0|\geq \lambda\psi$.
Note that, since $\alpha^*=\Theta(\alpha\alpha'/\log^4m)$, $\psi\geq \Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^{16}m} \textsf{right})$. Since ${\eta>\frac{c^* \log^{16}m}{\alpha^5(\alpha')^2}}$ from the statement of \Cref{thm: find guiding paths}, we get that $\psi>\frac{16\tilde k}{\eta^2}$. Every cell of the supergrid will be associated with a collection of $\Theta(\psi)$ horizontal paths and $\Theta(\psi)$ vertical paths, that will help us form the grid-like structure.
We discard paths from ${\mathcal{P}}_0$ and from ${\mathcal{R}}_0$ arbitrarily, until $|{\mathcal{P}}_0|=|{\mathcal{R}}_0|=\lambda\psi$ holds.
We denote by $A_0\subseteq T_1,A'_0\subseteq T_3$ the endpoints of the paths in ${\mathcal{P}}_0$, and we denote by $B_0\subseteq T_4$, $B'_0\subseteq T_2$ the endpoints of the paths in ${\mathcal{R}}_0$.
\subsubsection*{Grid Skeleton Construction}
We view the paths in ${\mathcal{P}}_0$ as directed from vertices of $A_0$ to vertices of $A'_0$. Recall that $A_0\subseteq T_1$, so the ordering $\tilde{\mathcal{O}}_1$ of the terminals in $T_1$ defines an ordering ${\mathcal{O}}_{A_0}=\set{a_1,\ldots,a_{\lambda\psi}}$ of the terminals in $A_0$. This ordering in turn defines an ordering ${\mathcal{O}}_{{\mathcal{P}}_0}$ of the paths in ${\mathcal{P}}_0$, as follows: if, for all $1\leq i\leq \lambda\psi$, $P_i\in {\mathcal{P}}_0$ is the path originating from $a_i$, then ${\mathcal{O}}_{{\mathcal{P}}_0}=\set{P_1,\ldots,P_{\lambda\psi}}$.
Similarly, we view the paths in ${\mathcal{R}}_0$ as directed from vertices of $B_0$ to vertices of $B_0'$. Ordering $\tilde{\mathcal{O}}_4$ of terminals in $T_4$ defines an ordering ${\mathcal{O}}_{B_0}=\set{b_1,\ldots,b_{\lambda\psi}}$ of the vertices in $B_0$, which in turn defines an ordering ${\mathcal{O}}_{{\mathcal{R}}_0}=\set{R_1,\ldots,R_{\lambda\psi}}$ of paths in ${\mathcal{R}}_0$, where for all $i$, path $R_i$ originates at vertex $b_i$.
We partition the set ${\mathcal{P}}_0$ of paths into groups ${\mathcal U}_1,\ldots,{\mathcal U}_{\lambda}$ of cardinality $\psi$ each, using the ordering ${\mathcal{O}}_{{\mathcal{P}}_0}$, so for $1\leq i<\lambda$, set ${\mathcal U}_i$ is the $i$th set of $\psi$ consecutive paths of ${\mathcal{P}}_0$.
Let $\lambda'= \floor{(\lambda-1)/2}$.
For all $1\leq i\leq \lambda'$, we let $ P^*_i$ be a path that is chosen uniformly at random from set ${\mathcal U}_{2i}$. Let ${\mathcal{P}}^*=\set{ P^*_1,\ldots, P^*_{\lambda'}}$ be the resulting set of chosen paths. Intuitively, the path in ${\mathcal{P}}^*$ will serve as the horizontal paths in the grid skeleton that we construct.
We then let ${\mathcal{P}}_1\subseteq {\mathcal{P}}_0$ be the set containing all paths in sets $\set{{\mathcal U}_{2i-1}}_{i=1}^{\lambda'+1}$.
We perform similar computation on the set ${\mathcal{R}}_0$ of paths. First, we partition ${\mathcal{R}}_0$ into groups ${\mathcal U}'_1,\ldots,{\mathcal U}'_{\lambda}$ of cardinality $\psi$ each, using the ordering ${\mathcal{O}}_{{\mathcal{R}}_0}$, so for $1\leq i<\lambda$, set ${\mathcal U}'_i$ is the $i$th set of $\psi$ consecutive paths of ${\mathcal{R}}_0$.
For all $1\leq i\leq \lambda'$, we let $ R^*_i$ be a path that is chosen uniformly at random from set ${\mathcal U}'_{2i}$. Let ${\mathcal{R}}^*=\set{ R^*_1,\ldots, R^*_{\lambda'}}$ be the resulting set of chosen paths. Intuitively, the path in ${\mathcal{R}}^*$ will serve as the vertical paths in the grid skeleton that we construct.
We then let ${\mathcal{R}}_1\subseteq {\mathcal{R}}_0$ be the set containing all paths in sets $\set{{\mathcal U}'_{2i-1}}_{i=1}^{\lambda'+1}$.
We let ${\cal{E}}_1$ be the bad event that there are two distinct paths $Q,Q'\in {\mathcal{R}}^*\cup {\mathcal{P}}^*$, and two distinct edges $e\in E(Q)$, $e'\in E(Q')$, such that the images of $e$ and $e'$ cross in the drawing $\phi^*$ of $H'$.
\begin{observation}\label{obs: first bad event}
If $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then $\prob{{\cal{E}}_1}\leq 1/64$.
\end{observation}
\begin{proof}
Consider any crossing $(e,e')$ in the drawing $\phi^*$. We say that crossing $(e,e')$ is \emph{selected} if there are two distinct paths $Q,Q'\in {\mathcal{R}}^*\cup {\mathcal{P}}^*$ with $e\in E(Q)$, $e'\in E(Q')$. Notice that $e$ may belong to at most two paths in ${\mathcal{R}}_0\cup {\mathcal{P}}_0$ (one path in each set), and the same is true for $e'$. Each path of ${\mathcal{R}}_0\cup{\mathcal{P}}_0$ is chosen to ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ with probability at most $1/\psi$. Therefore, the probability that a path containing $e$, and a path containing $e'$ are chosen to ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ is at most $4/\psi^2$. Since ${\cal{E}}_1$ can only happen if at least one crossing is chosen, from the union bound,
$\prob{{\cal{E}}_1}\leq 4\mathsf{cr}^*/\psi^2$. Since $\psi>\frac{16\tilde k}{\eta^2}$, if $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then:
\[\prob{{\cal{E}}_1}\leq \frac{4\mathsf{cr}^*}{\psi^2}\leq \frac{1}{64c_2\eta}\leq \frac{1}{64}.\]
\end{proof}
We say that a path $Q\in {\mathcal{R}}_0\cup {\mathcal{P}}_0$ is \emph{heavy} iff there are at least $\frac{\psi}{64\lambda}$ crossings $(e,e')$ in $\phi^*$, such that at least one of the edges $e,e'$ lies on path $Q$. We say that a bad event ${\cal{E}}_2$ happens iff at least one path in ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ is heavy.
\begin{observation}\label{obs: second bad event}
If $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then $\prob{{\cal{E}}_2}\leq 1/64$.
\end{observation}
\begin{proof}
Note that every edge of $H''$ may lie on at most two paths of ${\mathcal{R}}_0\cup {\mathcal{P}}_0$, and every crossing $(e,e')$ involves two edges. Therefore, the total number of heavy paths in ${\mathcal{R}}_0\cup {\mathcal{P}}_0$ is bounded by $\frac{4\mathsf{cr}^*}{\psi/(64\lambda)}=\frac{2^8\lambda\cdot\mathsf{cr}^*}{\psi}$. Assuming that $\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, and using the fact that $\psi=\Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^{16}m} \textsf{right})$ and $\lambda=O\textsf{left}( \frac{\eta \log^{12}m}{\alpha^4\alpha'} \textsf{right} )$, we get that the total number of heavy paths in ${\mathcal{R}}_0\cup {\mathcal{P}}_0$ is bounded by:
\[ \frac{2^8\lambda\cdot\mathsf{cr}^*}{\psi}\leq O\textsf{left}( \frac{\tilde k^2}{c_2\eta^5}\cdot \frac{\eta \log^{12}m}{\alpha^4\alpha'}\cdot\frac{\eta}{\tilde k}\cdot \frac{\log^{16}m}{\alpha^5(\alpha')^2} \textsf{right} )
\leq O\textsf{left}( \frac{\tilde k\log^{28}m}{c_2\eta^3\alpha^9(\alpha')^3} \textsf{right} ).
\]
Note that each heavy path may be selected to ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ with probability at most $1/\psi$. Therfore, using the union bound and the fact that $\psi=\Omega\textsf{left}(\frac{\tilde k}{\eta}\cdot \frac{\alpha^5(\alpha')^2}{\log^{16}m} \textsf{right})$, we get that:
\[\prob{{\cal{E}}_2}\leq
O\textsf{left}( \frac{\tilde k\log^{28}m}{\psi\cdot c_2\eta^3\alpha^9(\alpha')^3}\textsf{right} )\leq
O\textsf{left}( \frac{\log^{44}m}{c_2\eta^2\alpha^{14}(\alpha')^5}\textsf{right} ).\]
Recall that, from the conditions of \Cref{thm: find guiding paths}, $\eta\geq c^*\log^{46}m/(\alpha^{10}(\alpha')^4)$, where $c^*$ is a sufficiently large constant. Therefore, if
$\mathsf{cr}^*<
\frac{\tilde k^2}{c_2\eta^5}$, then $\prob{{\cal{E}}_2}\leq 1/64$.
\end{proof}
Let ${\mathcal{R}}'\subseteq {\mathcal{R}}_1$, ${\mathcal{P}}'\subseteq {\mathcal{P}}_1$ be the sets containing all paths $Q$, such that, in drawing $\phi^*$, the image of some edge of $Q$ crosses the image of some edge lying on the paths of ${\mathcal{R}}^*\cup {\mathcal{P}}^*$. Note that the drawing $\phi^*$ is not known to us, and so neither are the sets ${\mathcal{R}}',{\mathcal{P}}'$ of paths. We will also use the following observation:
\begin{observation}\label{obs: few bad paths}
If ${\cal{E}}_2$ did not happen, then $|{\mathcal{R}}'|,|{\mathcal{P}}'|\leq \psi/32$.
\end{observation}
\begin{proof}
Recall that $|{\mathcal{R}}^*|+|{\mathcal{P}}^*|\leq \lambda$. If bad event ${\cal{E}}_2$ did not happen, then for each path $Q\in {\mathcal{R}}^*\cup {\mathcal{P}}^*$, there are at most $\frac{\psi}{64\lambda}$ crossings in $\phi^*$, in which edges of $Q$ participate. Therefore, if event ${\cal{E}}_2$ did not happen, there are in total at most $\psi/64$ crossings $(e,e')$ in the drawing $\phi^*$, where at least one of the edges $e,e'$ lies on a path of ${\mathcal{P}}^*\cup {\mathcal{Q}}^*$. Let $E'\subseteq E(H'')$ be the set of all edges $e$, such that there is an edge $e'$ lying on some path of ${\mathcal{P}}^*\cup {\mathcal{Q}}^*$, and crossing $(e,e')$ is present in $\phi^*$. Then $|E'|\leq \psi/32$. Each path in ${\mathcal{R}}'\cup {\mathcal{P}}'$ must contain an edge of $E'$. As the paths in each of the sets ${\mathcal{R}}',{\mathcal{P}}'$ are disjoint, $|{\mathcal{R}}'|,|{\mathcal{P}}'|\leq \psi/32$ must hold.
\end{proof}
\paragraph{Summary of Step 3.}
In this step we have constructed a grid skeleton, that consists of two sets of paths: ${\mathcal{P}}^*=\set{ P^*_1,\ldots, P^*_{\lambda'}}$, and ${\mathcal{R}}^*=\set{ R^*_1,\ldots, R^*_{\lambda'}}$, where $\lambda'= \floor{(\lambda-1)/2}$. Recall that ${\mathcal{P}}^*\subseteq {\mathcal{P}}_0$, and the paths in ${\mathcal{P}}^*$ are indexed according to their order in ${\mathcal{O}}_{{\mathcal{P}}_0}$. Recall that we have also defined the set ${\mathcal{P}}_1\subseteq {\mathcal{P}}_0$ of paths, containing all paths in sets $\set{{\mathcal U}_{2i-1}}_{i=1}^{\lambda'+1}$. It would be convinient for us to re-index the groups ${\mathcal U}_i$ as follows: for $0\leq i\leq \lambda'$, set ${\mathcal U}_i={\mathcal U}_{2i+1}$. In other words, the paths of ${\mathcal U}_0$ lie before path $P^*_1$ in the ordering ${\mathcal{O}}_{{\mathcal{P}}_0}$, the paths of ${\mathcal U}_{\lambda'}$ lie after $P^*_{\lambda'}$ in this ordering, and, for $1\leq i<\lambda'$, the paths of ${\mathcal U}_i$ lie between paths $P^*_i$ and $P^*_{i+1}$.
Similarly, ${\mathcal{R}}^*\subseteq {\mathcal{R}}_0$, and the paths in ${\mathcal{R}}^*$ are indexed according to their order in ${\mathcal{O}}_{{\mathcal{R}}_0}$. We have also defined the set ${\mathcal{R}}_1\subseteq {\mathcal{R}}_0$ of paths, containing all paths in sets $\set{{\mathcal U}'_{2i-1}}_{i=1}^{\lambda'+1}$. As before, we re-index them as follows: for $0\leq i\leq\lambda'$, we set ${\mathcal U}'_i={\mathcal U}'_{2i+1}$. Therefore, the paths of ${\mathcal U}'_0$ lie before path $R^*_1$ in the ordering ${\mathcal{O}}_{{\mathcal{R}}_0}$, the paths of ${\mathcal U}'_{\lambda'}$ lie after $R^*_{\lambda'}$ in this ordering, and, for $1\leq i<\lambda'$, the paths of ${\mathcal U}'_i$ lie between paths $R^*_i$ and $R^*_{i+1}$.
From our definition, if ${\cal{E}}_1$ did not happen, then for every pair $Q,Q'\in {\mathcal{P}}^*\cup {\mathcal{Q}}^*$ of distinct paths, their images in $\phi^*$ do not cross (but note that the image of a single path may cross itself).
We have also defined a set ${\mathcal{P}}'\subseteq {\mathcal{P}}_1$ and a set ${\mathcal{R}}'\subseteq {\mathcal{R}}_1$ of paths, containing all paths $Q$ whose image crosses the image of some path in ${\mathcal{R}}^*\cup {\mathcal{P}}^*$ in drawing $\phi^*$. From \Cref{obs: few bad paths}, if Event ${\cal{E}}_2$ does not happen, then $|{\mathcal{P}}'|,|{\mathcal{R}}'|\leq \psi/32$.
Note that the sets ${\mathcal{P}}',{\mathcal{R}}'$ of paths are not known to the algorithm.
It will be convenient for us to consider the $((\lambda'+1)\times (\lambda'+1))$-grid $\Pi^*$. We view the columns of the grid as corresponding to the left boundary of the bounding box $B^*$, the paths in $\set{R^*_1,\ldots,R^*_{\lambda'}}$, and the right boundary of the bounding box $B^*$.
For convenience, we index the columns of the grid from $0$ to $\lambda'+1$, so the left boundary of the bounding box corresponds to column $0$, and, for $1\leq i\leq \lambda'$, path $P^*_i$ represents the $i$th column of the grid, with the right boundary of $B^*$ repersenting the last column. Similarly, we view the bottom boundary of $B^*$, the paths in $\set{P^*_1,\ldots,P^*_{\lambda'}}$, and the top boundary of $B^*$ as representing the rows of the grid, in the bottom-to-top order. As before, we index the rows of the grid so that the botommost row has index $0$ and the topmost row has index $\lambda'+1$. Notice however that the union of the paths in ${\mathcal{P}}^*\cup {\mathcal{R}}^*$ does not necessarily form a proper grid graph, as it is possible that, for a pair $P\in {\mathcal{P}}^*$, $R\in {\mathcal{R}}^*$ of paths, $P\cap R$ is a collection of several disjoint paths.
We will now consider the drawing $\phi^*$ of $H''$, and we will use it to define vertical and horizontal strips corresponding to paths in ${\mathcal{P}}^*$ and ${\mathcal{R}}^*$, respectively. We will also associate, with each cell of the grid $\Pi^*$, some region of the plane. We assume in the following definitions that Event ${\cal{E}}_1$ did not happen.
Consider first the image $\gamma_i$ of some path $P^*_i\in {\mathcal{P}}^*$ in the drawing $\phi^*$. Note that $\gamma_i$ is not necessarily a simple curve.
We define two simple curves, $\gamma^t_i$ and $\gamma^b_i$, where $\gamma^t_i$ follows the image of $\gamma_i$ from the top, and $\gamma^b_i$ follows it from the bottom. In other words, we let $\gamma^b_i$ be a simple curve, whose every point lies on $\gamma_i$, that has the same endpoints as $\gamma_i$, such that the following holds: for every point $p\in \gamma_i$, either $p\in \gamma_i^b$, or $p$ lies above $\gamma_i^b$ in the bounding box $B^*$. We define the other curve, $\gamma^t_i$ symmetrically, so curve $\gamma_i$ is contained in the disc whose boundary is $\gamma_i^t\cup \gamma_i^b$ (see \Cref{fig: top_bottom_curves}). For convenience, we let $\gamma_0^t$ be the bottom boundary of the bounding box $B^*$, and $\gamma_{\lambda'+1}^b$ be the top boundary of the bounding box $B^*$.
We now define, for all $0\leq i\leq \lambda'$, a region of the plane that we call the $i$th horizontal strip, and denote by $\mathsf{HStrip}_i$. This strip is simply the closed region of the bounding box between the curves $\gamma_i^t$ and $\gamma_{i+1}^b$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/top_curve.jpg}
\caption{An illustration of curves $\gamma^t_i$ and $\gamma^b_i$. The curve $\gamma_i$ is shown in purple.}\label{fig: top_bottom_curves}
\end{figure}
For every vertical path $R^*_i\in {\mathcal{R}}^*$, we also define two curves, $\gamma^{\ell}_i$ and $\gamma^r_i$, that follow the image $\gamma_i$ of $R^*_i$ in $\phi^*$ on its left and on its right, respectively. We denote by $\gamma_0^r$ the left boundary of the bounding box $B^*$, and by $\gamma_{\lambda'+1}^{\ell}$ its right boundary. For all $0\leq i\leq \lambda'$, we define a vertical strip $\mathsf{VStrip}_i$ to be the closed region of the bounding box $B^*$ betwen $\gamma_i^r$ and $\gamma_{i+1}^{\ell}$.
The following observation is immediate from the fact that the paths in ${\mathcal{P}}_0$ are node-disjoint, and so are the paths in ${\mathcal{R}}_0$.
\begin{observation}\label{obs: must cross chosen paths}
If $R\in {\mathcal{R}}_1$ is a path whose image in $\phi^*$ intersects the interior of more than one vertical strip in $\set{\mathsf{VStrip}_0,\ldots,\mathsf{VStrip}_{\lambda'+1}}$, then $R\in {\mathcal{R}}'$. Similarly, if $P\in {\mathcal{P}}_1$ is a path whose image in $\phi^*$ intersects the interior of more than one horizontal strip in $\set{\mathsf{HStrip}_0,\ldots,\mathsf{HStrip}_{\lambda'+1}}$, then $P\in {\mathcal{P}}'$.
\end{observation}
Lastly, for all $0\leq i,j\leq \lambda'$, we let $\mathsf{CellRegion}_{i,j}=\mathsf{HStrip}_i\cap \mathsf{VStrip}_j$ be a closed region of the plane that we associate with cell $\mathsf{Cell}_{i,j}$ of the grid $\Pi^*$.
\subsection{Step 4: Constructing a Grid-Like Structure}
In this step we further delete some paths from sets ${\mathcal{R}}_1$ and ${\mathcal{P}}_1$ to ensure that the resulting paths form a grid-like structure. This is done in three stages. In the first stage, we discard some paths to ensure that every remaining path in ${\mathcal{R}}_1$ intersects the paths in ${\mathcal{P}}^*$ ``in order'' (we formally define this notion later), and we process the paths in ${\mathcal{P}}_1$ similarly. In the second stage, we associate, with every cell of the grid $\Pi^*$ a collection of horizontal paths and a collection of vertical paths. In the third stage, we ensure that, for every cell of the grid $\Pi^*$, there are many inersections between its corresponding horizontal and vertical paths.
Before we continue, we discard some paths of ${\mathcal{R}}_1\cup {\mathcal{P}}_1$ that must lie in ${\mathcal{R}}'\cup {\mathcal{P}}'$. Specifically, consider some path $P\in {\mathcal{P}}_1$, and assume that it lies in group ${\mathcal U}_i$, for some $0\leq i\leq \lambda'$. Let $(a,a')$ be the endpoints of path $P$, with $a\in T_1$ and $a'\in T_3$. Notice that from the definition, if $i>0$, then $a$ must lie, in the ordering $\tilde{\mathcal{O}}_1$ of the terminals of $T_1$, after the endpoint of the path $P^*_i$ that belongs to $T_1$. Similarly, if $i<\lambda'$, then $a$ must lie before the endpoint of the path $P^*_{i+1}$ that belongs to $T_1$ in the same ordering. In particular, we are guaranteed that, in the drawing $\phi^*$, the image of $P$ must intersect the interior of the horizontal strip $\mathsf{HStrip}_i$. Consider now the endpoint $a'$ of $P$. If $i>0$, let $a'_i$ be the endpoint of path $P_i^*$ that lies in $T_3$, and if $i<\lambda'$, let $a'_{i+1}$ be the endpoint of path $P_{i'+1}$ that lies in $T_3$. Note that, if $a'$ lies before $a'_i$ in the ordering $\tilde{\mathcal{O}}_3$ of $T_3$, or if $a'$ lies after $a'_{i+1}$ in the ordering $\tilde{\mathcal{O}}_3$, then the image of $P$ has to intersect the interior of an additional horizontal strip, and, from \Cref{obs: must cross chosen paths}, path $P$ must lie in ${\mathcal{P}}'$. We discard each such path from set ${\mathcal{P}}_1$ (and from the corresponding set $U_i$). This ensures that, if $P\in U_i$, then its endpoint $a'$ must lie between $a'_i$ and $a'_{i+1}$ in $\tilde{\mathcal{O}}_3$, if $1\leq i\leq \lambda'$; it must lie before $a'_{i+1}$ if $i=0$, and it must lie after $a'_i$ if $i=\lambda'$.
We process the paths in ${\mathcal{R}}_1$ similarly, discarding paths as needed. Notice that so far all paths that we have discarded from ${\mathcal{P}}_1\cup {\mathcal{R}}_1$ lie in ${\mathcal{P}}'\cup {\mathcal{R}}'$.
\subsubsection{In-Order Intersection}
In this stage we discard some additional paths from ${\mathcal{P}}_1\cup {\mathcal{R}}_1$, to ensure that every remaining path in ${\mathcal{P}}_1$ interesects the paths in ${\mathcal{R}}^*$ in-order (notion that we define below); we do the same for paths in ${\mathcal{R}}_1$. We will ensure that all paths discarded at this stage lie in ${\mathcal{P}}'\cup {\mathcal{R}}'$.
Since the definitions and the algorithms for the paths in ${\mathcal{P}}_1$ and for the paths in ${\mathcal{R}}_1$ are symmetric, we only describe the algorithm to process the paths in ${\mathcal{P}}_1$ here.
Let $P\in {\mathcal{P}}_1$ be any path, that we view as directed from its endpoint that lies in $T_1$ to its endpoint lying in $T_3$. Let $X(P)=\set{x_1,\ldots,x_r}$ denote all vertices of $P$ lying on paths in $R^*$, that is, $X(P)=V(P)\cap \textsf{left}(\bigcup_{i=1}^{\lambda'}V(R^*_i)\textsf{right} )$. We assume that the vertices of $X(P)$ are indexed in the order of their appearance on $P$. For each such vertex $x_j$, let $i_j$ be the index of the path $R^*_{i_j}\in {\mathcal{R}}^*$ containing $x_j$.
\begin{definition}[In-order intersection]
We say that path $P$ intersects the paths of ${\mathcal{R}}^*$ in-order, if $r\geq \lambda'$, $i_1=1$, $i_r=\lambda'$, and, for $1\leq j<r$, $|i_j-i_{j+1}|\leq 1$.
\end{definition}
Notice that the definition requires that path $P$ intersects every path of ${\mathcal{R}}^*$ at least once; the first path of ${\mathcal{R}}^*$ that it intersects must be $R_1^*$, and the last path must be $R_{\lambda'}^*$, and for every consecutive pair $x_j,x_{j+1}$ of vertices in $X(P)$, either both vertices lie on the same path of ${\mathcal{R}}^*$, or they lie on consecutive paths of ${\mathcal{R}}^*$. Notice that path $P$ is still allowed to intersect a path of ${\mathcal{R}}^*$ many times, and may go back and forth across all these paths several times.
\begin{observation}\label{obs: not in order intersection}
Assume that Event ${\cal{E}}_1$ did not happen. Let $P\in {\mathcal{P}}_1$ be a path that intersect the paths of ${\mathcal{R}}^*$ not in-order. Then $P\in {\mathcal{P}}'$ must hold.
\end{observation}
\begin{proof}
Assume first that $i_1\neq 1$, that is, vertex $x_1$ lies on some path $R^*_i$ with $i\neq 1$. Let $p$ be a point on the image of path $P$ in $\phi^*$ that is very close to its first endpoint, so $p$ lies in the interior of the vertical strip $\mathsf{VStrip}_1$, and let $p'$ be the image of the point $x_1$. Clearly, $p'$ does not lie in the interior or on the boundary of $\mathsf{VStrip}_1$, so the image of path $P$ must cross the right boundary of $\mathsf{VStrip}_1$, which means that the image of some edge of $P$ and the image of some edge of $R_1^*$ cross in $\phi^*$.
The cases where $i_{r}\neq \lambda'$, or there is an index $1\leq j<r$ with $i_j-i_{j+1}> 1$ are treated similarly, as is the case when $r<\lambda'$.
\end{proof}
We discard from ${\mathcal{P}}_1$ all paths $P$ that intersect the paths of ${\mathcal{R}}^*$ not in-order. We denote by ${\mathcal{P}}_2\subseteq {\mathcal{P}}_1$ the set of remaining paths. We also update the groups ${\mathcal U}_0,\ldots,{\mathcal U}_{\lambda'}$ accordingly. Observe that so far all paths that we have discarded from ${\mathcal{P}}_1$ lie in ${\mathcal{P}}'$.
From \Cref{obs: few bad paths}, assuming that Events ${\cal{E}}_1$ and ${\cal{E}}_2$ did not happen, the number of paths that we have discarded so far from ${\mathcal{P}}_1$ is at most $\psi/32$. In particular, for all $0\leq i\leq \lambda'$, $|{\mathcal U}_i|\geq 31\psi/32$ still holds.
We perform the same transformation on set ${\mathcal{R}}_1$ of paths, obtaining a new set ${\mathcal{R}}_2$ of paths, each of which intersects the paths of ${\mathcal{P}}^*$ in-order. We also update the groups ${\mathcal U}'_0,\ldots,{\mathcal U}'_{\lambda'}$. As before, for all $0\leq i\leq \lambda'$, $|{\mathcal U}'_i|\geq 31\psi/32$ still holds.
\subsubsection{Definining Paths Associated with Grid Cells}
For every path $P\in {\mathcal{P}}_2$, for all $1\leq i\leq \lambda'$, we denote by $v_i(P)$ the first vertex on path $P$ that belongs to the vertical path $R^*_i$; note that, from the definition of in-order intersection, such a vertex must exist. For all $1\leq i< \lambda'$, we define the $i$th segment of $P$, $\sigma_i(P)$, to be the subpath of $P$ between $v_i(P)$ and $v_{i+1}(P)$. We also let $\sigma_0(P)$ be the subpath of $P$ from its first vertex (which must be a terminal of $T_1$) to $v_1(P)$, and by $\sigma_{\lambda'}(P)$ the subpath of $P$ from $v_{\lambda'}(P)$ to the last vertex of $P$ (which must be a terminal of $T_3)$. Note that the sets of edges that lie on paths $\sigma_0(P),\ldots,\sigma_{\lambda'}(P)$ partition $E(P)$.
Similarly, for every path $R\in {\mathcal{R}}_2$, for all $1\leq i\leq \lambda'$, we denote by $v_i(R)$ the first vertex on path $R$ that lies on the horizontal path $P^*_i$. For all $1\leq i< \lambda'$, we define the $i$th segment of $P$, $\sigma_i(R)$, to be the subpath of $R$ between $v_i(R)$ and $v_{i+1}(R)$. We also let $\sigma_0(R)$ be the subpath of $R$ from its first vertex (which must be a terminal of $T_4$) to $v_1(R)$, and by $\sigma_{\lambda'}(R)$ the subpath of $R$ from $v_{\lambda'}(R)$ to the last vertex of $R$ (which must be a terminal of $T_2$).
Consider now some cell $\mathsf{Cell}_{i,j}$ of the grid $\Pi^*$, for some $0\leq i,j\leq \lambda'$. We define the set ${\mathcal{P}}^{i,j}$ of horizontal paths, and the set ${\mathcal{R}}^{i,j}$ of vertical paths associated with cell $\mathsf{Cell}_{i,j}$, as follows. In order to define the set ${\mathcal{P}}^{i,j}$ of horizontal paths, we consider the group ${\mathcal U}_i\subseteq {\mathcal{P}}_2$, and, for every path $P\in {\mathcal U}_i$, we include its $j$th segment $\sigma_j(P)$ in ${\mathcal{P}}^{i,j}$, so ${\mathcal{P}}^{i,j}=\set{\sigma_j(P)\mid P\in {\mathcal U}_i}$.
Similarly, we define ${\mathcal{R}}^{i,j}=\set{\sigma_i(R)\mid R\in {\mathcal U}'_j}$.
\iffalse
Note that the definition of the sets ${\mathcal{P}}^{i,j},{\mathcal{R}}^{i,j}$ of paths depends on the definition of the direction of the paths in ${\mathcal{P}}_2,{\mathcal{R}}_2$. For example, recall that we think of the vertices of $T_1$ as lying on the left boundary of the bounding box $B^*$, and the vertices of $T_3$ as lying on its right boundary, with the paths in ${\mathcal{P}}_2$ directed from left to right. If we were to flip the bounding box $B^*$, so that the vertices of $T_3$ appear on its left boundary and the vertices of $T_1$ on its right boundary, with the paths in ${\mathcal{P}}$ directed from left to right, then the definition of the sets ${\mathcal{P}}^{i,j}$ of paths may change (as a path in ${\mathcal{P}}_2$ may intersect a path of ${\mathcal{R}}^*$ numerous times).
However, if we flip the bounding box $B^*$, so that the vertices of $T_1$ appear on its bottom boundary, the vertices of $T_3$ on its top boundary, and the vertices of $T_2$ and $T_4$ on its left and right boundaries, respectively, directing the paths in ${\mathcal{P}}$ from bottom to top, and the paths in ${\mathcal{R}}$ from left to right, the definition of path sets $\set{{\mathcal{P}}^{i,j},{\mathcal{R}}^{i,j}}_{0\leq i,j\leq \lambda'+1}$ will now change (but now paths of ${\mathcal{P}}$ become vertical and paths of ${\mathcal{R}}$ become horizontal). This transformation corresponds to flipping the grid $\Pi^*$ along its diagonal (from bottom left to top right), and so the cells that lied in the top right quandrant of the grid remain in the top right quadrant. We may need to use this transformation later, but for now we stay with the original notation.
\fi
We need the following observation.
\begin{observation}\label{obs: paths in cells don't cross}
Let $P\in {\mathcal U}_i,R\in {\mathcal U}'_j$ be a pair of paths, for some $1< i,j< \lambda'$, and assume that their subpaths $\sigma_j(P)\subseteq P,\sigma_i(R)\subseteq R$ do not share any vertices. Then either $P\in {\mathcal{P}}'$, or $R\in {\mathcal{R}}'$, or the images of $\sigma_j(P)$ and $\sigma_i(R)$ cross in the drawing $\phi^*$.
\end{observation}
\begin{proof}
Assume that $P\not\in {\mathcal{P}}'$ and $R\not\in {\mathcal{R}}'$, that is, the images of the paths $P,R$ do not cross the images of the paths in ${\mathcal{P}}^*\cup {\mathcal{R}}^*$ in $\phi^*$. From the definition of set ${\mathcal U}_i$, the image of $P$ intersects the interior of the horizontal strip $\mathsf{HStrip}_i$, and path $P$ does not share any vertices with the paths of ${\mathcal{P}}^*$. Therefore, the image of $P$ must be contained in the strip $\mathsf{HStrip}_i$, and it is disjoint from its top and bottom boundaries $\gamma^t_i,\gamma^b_{i+1}$. Using similar reasoning, the image of $R$ is contained in the strip $\mathsf{VStrip}_j$, and it is disjoint from its left and right boundaries, $\gamma^{r}_j,\gamma^{\ell}_{j+1}$. Consider now the segment $\sigma_j(P)$ of $P$, whose endpoints lie on $R^*_j$ and $R^*_{j+1}$, respectively. Let $\sigma'_j(P)\subseteq \sigma_j(P)$ be the shortest subpath of $\sigma_j(P)$ whose first endpoint lies on $R^*_j$, and whose last endpoint lies on $R^*_{j+1}$; such a path must exist because we can let $\sigma'_j(P)=\sigma_j(P)$. From the definition of in-order intersection, no inner vertex of $\sigma'_j$ may lie on any path of ${\mathcal{R}}^*$. It is then easy to verify that the image of $\sigma'_j(P)$ in $\phi^*$ must be contained in $\mathsf{CellRegion}_{i,j}$, and it must split this region into two subregions: one whose top boundary contains a segment of $\gamma^b_{i+1}$, and one whose bottom boundary contains a segment of $\gamma^t_i$.
Using the same reasoning, we can select a segment $\sigma'_i(R)$, whose first endpoint lies on $P^*_i$, last endpoint lies on $P^*_{i+1}$, and all inner vertices are disjoint from the vertices lying on the paths in ${\mathcal{P}}^*$.
As before, the image of $\sigma'(R)$ must be contained in $\mathsf{CellRegion}_{i,j}$, but it connects a point on its top boundary to a point on its bottom boundary. Therefore, the image of $\sigma'_i(R)$ must cross the image of $\sigma'_j(P)$.
\end{proof}
\subsubsection{Completing the Construction of the Grid-Like Structure}
In order to complete the construction of the grid-like structure, we need to ensure that, for every pair $1< i,j< \lambda'$ of indices, there are many intersection between the sets ${\mathcal{P}}^{i,j}$ and ${\mathcal{R}}^{i,j}$ of paths. More specifically, we need to ensure that every path $\sigma\in {\mathcal{P}}^{i,j}$ intersects many paths in ${\mathcal{R}}^{i,j}$, and vice versa. This is needed in order to ensure well-linkedness properties: namely, that the collection of vertices containing the first and the last vertex on every path of ${\mathcal{P}}^{i,j}$ is sufficiently well-linked in the graph obtained from the union of the paths in ${\mathcal{R}}^{i,j}\cup {\mathcal{P}}^{i,j}$. This property, in turn, will be exploited in order to construct the routers of $\Lambda'$ over which the distribution ${\mathcal{D}}$ will be defined. This motivates the following definition.
\begin{definition}[Bad Paths]
For a pair $0< i,j< \lambda'$ of indices, we say that a path $P\in {\mathcal U}_i$ is \emph{bad for cell $\mathsf{Cell}_{i,j}$} if there are at least $\psi/16$ paths in ${\mathcal{R}}^{i,j}$ that are disjoint from $\sigma_j(P)$. Similarly, we say that a path $R\in {\mathcal U}'_j$ is bad for cell $\mathsf{Cell}_{i,j}$ if there are at least $\psi/16$ paths in ${\mathcal{P}}^{i,j}$ that are disjoint from $\sigma_i(R)$.
Consider now some index $0< i< \lambda'$. We say that a path $P\in {\mathcal U}_i$ is \emph{bad} if it is bad for at least one cell in $\set{\mathsf{Cell}_{i,j}\mid 0< j< \lambda'}$. Similarly, for an index $0< j< \lambda'$, a path $R\in {\mathcal U}'_j$ is bad if it is bad for at least one cell in $\set{\mathsf{Cell}_{i,j}\mid 0< i< \lambda'}$.
\end{definition}
The following observation bounds the number of bad paths in each group ${\mathcal U}_i$ of horizontal paths, and in each group ${\mathcal U}'_j$ of vertical paths.
\begin{observation}\label{obs: few bad paths in each group}
Assume that $\mathsf{cr}^*\leq \frac{\tilde k^2}{c_2\eta^5}$, and that neither of the events ${\cal{E}}_1,{\cal{E}}_2$ happenned. Then for all $0< i< \lambda'$, at most $\psi/16$ paths in ${\mathcal U}_i$ are bad. Similarly, for all $0< j<\lambda'$, at most $\psi/16$ paths in ${\mathcal U}'_j$ are bad.
\end{observation}
\begin{proof}
Fix an index $0< i< \lambda'$, and the corresponding set ${\mathcal U}_i\subseteq {\mathcal{P}}_2$ of paths. We partition the set of all bad paths in ${\mathcal U}_i$ into two subsets: set ${\mathcal{B}}_1$ contains all bad paths lying in ${\mathcal{P}}'$, and set ${\mathcal{B}}_2$ contains all remaining bad paths. From \Cref{obs: few bad paths}, $|{\mathcal{B}}_1|\leq \psi/32$.
We further partition the set ${\mathcal{B}}_2$ of bad paths into subsets $\set{{\mathcal{B}}_2^j\mid 0< j< \lambda'}$, where a path $P$ lies in ${\mathcal{B}}_2^j$ if it is bad for cell $\mathsf{Cell}_{i,j}$ (if path $P$ is bad for several cells, we add it to any of the corresponding sets). Consider now some index $0< j< \lambda'$, and some path $P\in {\mathcal{B}}_2^j$. From the definition, there is a set $\Sigma'\subseteq {\mathcal{R}}^{i,j}$ of at least $\psi/16$ paths that do not share any vertices with $P$. From \Cref{obs: few bad paths}, at most $\psi/32$ of these paths may lie in ${\mathcal{R}}'$. Let $\Sigma''\subseteq \Sigma'$ be the collection of the remaining paths, whose cardinality is at least $\psi/32$. From \Cref{obs: paths in cells don't cross}, for every path $\sigma'\in \Sigma'$, the images of $\sigma_j(P)$, and of $\sigma'$ must cross. We let $\chi_j(P)$ denote the set of all crossings $(e,e')$, where $e\in \sigma_j(P)$, and $e'$ is an edge on a path of $\Sigma''$, so $|\chi_j(P)|\geq \psi/32$. We then let $\chi_{j}=\bigcup_{P\in {\mathcal{B}}_2^j}\chi_j(P)$, so $|\chi_{j}|\geq |{\mathcal{B}}_2^j|\cdot \psi/32$. Lastly, we let $\chi=\bigcup_{j=1}^{\lambda'-1}\chi_j$. Notice that set $\chi$ contains at least $|{\mathcal{B}}_2|\cdot \psi/32$ distinct crossings in the drawing $\phi^*$. Assume for contradiction that $|{\mathcal{B}}_2|>\psi/32$. Then:
\[\mathsf{cr}^*>\frac{\psi^2}{2^{10}}>\frac{\tilde k^2}{4\eta^4}>\frac{\tilde k^2}{c_2\eta^5},\]
since $\psi>\frac{16\tilde k}{\eta^2}$, a contradiction.
Therefore, $|{\mathcal{B}}_2|\leq \psi/32$, and overall there are at most $\psi/16$ bad paths in ${\mathcal U}_i$.
The proof for path sets ${\mathcal U}'_j\subseteq {\mathcal{R}}_2$ is identical.
\end{proof}
For all $0< i< \lambda'$, we discard every bad path from ${\mathcal U}_i$. If $|{\mathcal U}_i| <\ceil{7\psi/8}$ for any $i$, then we terminate the algorithm and return FAIL. Notice that in this case, from \Cref{obs: few bad paths in each group}, if $\mathsf{cr}^*< \frac{\tilde k^2}{c_2\eta^5}$, then at least one of the events ${\cal{E}}_1,{\cal{E}}_2$ must have happened, and the probability for this is at most $1/8$. Therefore, we assume that for all
$0< i< \lambda'$, $|{\mathcal U}_i|\geq \ceil{7\psi/8}$ holds. We discard additional arbitrary paths from ${\mathcal U}_i$, until $|{\mathcal U}_i|= \ceil{7\psi/8}$.
We then let ${\mathcal{P}}_3=\bigcup_{i=1}^{\lambda'}{\mathcal U}_i$ denote the resulting set of paths.
Similarly, for all $0< j< \lambda'$, we discard every bad path from ${\mathcal U}'_j$. If, as the result, $|{\mathcal U}'_j|$ falls below $\ceil{7\psi/8}$, we terminate the algorithm and return FAIL. Otherwise, we discard additional arbitrary paths as needed, so that $|{\mathcal U}'_j|=\ceil{7\psi/8}$ holds. We also let ${\mathcal{R}}_3=\bigcup_{j=1}^{\lambda'}{\mathcal U}'_j$.
For all $0< i,j< \lambda'$, we also update the path sets ${\mathcal{P}}^{i,j}$ and ${\mathcal{R}}^{i,j}$ accordingly, discarding the paths that are no longer subpaths of paths in ${\mathcal{P}}_3\cup {\mathcal{R}}_3$. Since we are still guaranteed that $|{\mathcal{P}}^{i,j}|,|{\mathcal{R}}^{i,j}|= \ceil{7\psi/8}$, and since every path that is bad for cell $\mathsf{Cell}_{i,j}$ was discarded, we are guaranteed that every path in ${\mathcal{P}}^{i,j}$ intersects at least $\frac{7\psi}{8}-\frac{\psi}{16}=\frac{13\psi}{16}$ paths of ${\mathcal{R}}^{i,j}$ and vice versa.
Since we use this fact later, we summarize it in the following observation.
\begin{observation}\label{obs: paths for cells}
For all $0< i,j< \lambda'$, $|{\mathcal{P}}^{i,j}|,|{\mathcal{R}}^{i,j}|= \ceil{7\psi/8}$. Every path in
${\mathcal{P}}^{i,j}$ intersects at least $\frac{13\psi}{16}$ paths of ${\mathcal{R}}^{i,j}$ and vice versa.
\end{observation}
This concludes the construction of the grid-like structure.
\subsection{Step 5: the Routing}
Recall that we have denoted by $\Lambda'=\Lambda(H'',\tilde T)$ the set of all routers in graph $H''$ with respect to the set $\tilde T$ of terminals. In this final step we design an efficient algorithm to compute an explicit distribution ${\mathcal{D}}$ over the routers of $\Lambda'$, such that for every outer edge $e\in E(H'')$, $\expect[{\mathcal{Q}}\sim{\mathcal{D}}]{(\cong_{H''}({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{16}m}{(\alpha\alpha')^4}\textsf{right} )$.
Our algorithm closely follows the arguments of
\cite{Tasos-comm} (see also Lemma D.10 in the full version of \cite{chuzhoy2011algorithm}), who showed a similar result for a grid graph. In order to provide intuition, we first present their algorithm. Assume that we are given a $(q\times q)$ grid graph $G$ for some integer $q$, and let $T$ be the set of vertices lying on the first row of the grid, that we refer to as terminals. For convenience, assume that $q$ is an integral power of $2$. Our goal is to compute a distribution ${\mathcal{D}}'$ over the routers of in $\Lambda(G,T)$. We need to ensure that, for every edge $e\in E(G)$, the expectation $\expect[{\mathcal{Q}}\sim{\mathcal{D}}']{(\cong_G({\mathcal{Q}},e))^2}\leq O(\log q)$.
For every vertex $v$ in the top right quadrant of the grid, we will define a set ${\mathcal{Q}}(v)$ of paths in $G$, routing the terminals in $T$ to $v$. Our distribution ${\mathcal{D}}$ then assigns, to each such router ${\mathcal{Q}}(v)$, the same probability value $4/q^2$.
We now fix a vertex $v$ in the top right quadrant of the grid, and define the router ${\mathcal{Q}}(v)$. Let $r=\log(q/4)$. For $0\leq i\leq r$, let $S_i$ be a square subgrid of $G$, of size $(2^i\times 2^i)$, whose upper right corner has the same column-index as vertex $v$, and the same row-index as the bottom left corner of $S_{i-1}$ (we think of $S_0$ as a $(1\times 1)$-grid consisting only of vertex $v$).
We refer to the subgrids $S_i$ of $G$ as \emph{squares}, and specifically to square $S_i$ as \emph{level-$i$ square}.
For all $0\leq i\leq r$, we denote by $T_i$ the set of vertices lying on the bottom boundary of square $S_i$. Using the well-linkedness of the grids, it is easy to show that for all $1\leq i\leq r$, there is a collection ${\mathcal{P}}_i$ of paths in graph $S_i$, routing vertices of $T_i$ to vertices of $T_{i-1}$ with congestion at most $2$, such that every vertex of $T_{i-1}$ serves as endpoint of at most two such paths. For $1\leq i\leq r$, let
Let ${\mathcal{P}}'_i$ be a multipset obtained from set ${\mathcal{P}}_i$ by creating $2^{r-i+1}$ copies of every path in ${\mathcal{P}}_i$.
Let $T_{r+1}\subseteq T$ be a set of $|T_r|$ vertices lying on the bottom boundary of the grid $G$, that contains, for every vertex $t\in T_r$, vertex $t'$ on the bottom boundary of the grid with the same column index as $t$. Let $P_t$ be the subpath of the corresponding column of $G$ connecting $t$ to $t'$, and denote ${\mathcal{P}}'_{r+1}=\set{P_t\mid t\in T_r}$.
By concatenating the paths in ${\mathcal{P}}_1',\ldots,{\mathcal{P}}'_{r+1}$, we obtain a collection ${\mathcal{Q}}'(v)$ of paths in grid $G$, routing the terminals in $T_{r+1}$ to vertex $v$.
Notice that for all $0\leq i\leq r$, for every edge $e$ lying in $S_i$, the congestion on
edge $e$ due to paths in ${\mathcal{Q}}'(v)$ is at most $2^{r-i+2}$.
The key in analyzing the expectation $\expect[{\mathcal{Q}}\sim{\mathcal{D}}']{(\cong_G({\mathcal{Q}},e))^2}$
is to notice that, for all $1\leq i\leq r$, square $S_i$ is a $(2^i\times 2^i)$-subgrid of $G$, whose upper right corner is chosen uniformly at random from a set of $q^2/4$ possible points. The total number of subgrids of $G$ of size $(2^i\times 2^i)$ that contain $e$ is $2^{2i}$, so the probability that any of them is selected is bounded by $2^{2i+2}/q^2$. Therefore, for all $1\leq i\leq r$, with probability at most $2^{2i+2-2r}$, edge $e$ belongs to square $S_i$, and in this case, $\cong_G({\mathcal{Q}},e)\leq 2^{r-i+2}$. Therefore, we get that:
\[\expect[{\mathcal{Q}}\sim{\mathcal{D}}']{(\cong_G({\mathcal{Q}},e))^2}
\leq \sum_{i=1}^{r}2^{2i+2-2r}\cdot 2^{2r-2i+4}\leq O(r)=O(\log q).
\]
Using the well-linkedness of the terminals in $T$, it is immediate to extend the set ${\mathcal{Q}}'(v)$ of paths to a set ${\mathcal{Q}}^*(v)$ routing all terminals in $T$ to $v$, while increasing the congestion on every edge of $G$ by at most an additive constant and a multiplicative constant factor. This provides the final distribution ${\mathcal{D}}$ over the routers ${\mathcal{Q}}(v)\in \Lambda(G,T)$.
We will simulate a similar process on the grid $\Pi^*$, and its corresponding grid-like structure that we have constructed. Notice however that $\Pi^*$ is only a $(\lambda'\times\lambda')$-grid (where $\eta\leq \lambda'\leq \eta^2$), while the number of terminals that we need to route is much larger (comparable to $|{\mathcal{R}}_3|$). Therefore, we will attempt to route all terminals to a single cell $\mathsf{Cell}_{i,j}$ in the top right quadrant of the grid $\Pi^*$ (in other words, we will route them to vertices lying on paths in ${\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$). This in itself is not sufficient, since we need to route them to a single vertex of $H''$. This means that we may need to perform some routing within the cell $\mathsf{Cell}_{i,j}$, that is, within the graph obtained from the union of the paths in ${\mathcal{P}}^{i,j}\cup {\mathcal{R}}^{i,j}$. While generally such a routing (with low congestion on outer edges) may be difficult to compute, we will select a large collection of cells (called good cells) in the top right quadrant of the grid $\Pi^*$, for which such a routing is easy to obtain. We will then define, for each good cell, the corresponding set of paths routing the terminals to a single vertex $y^*\in V(H'')$. We do so by simulating the process described above: we define square subgrids $\set{S_i}$ of the grid $\Pi^*$, and we associate these subgrids with sets of horizontal and vertical paths (subpaths of some paths in ${\mathcal{P}}_3\cup {\mathcal{R}}_3$), so that the desired well-linkedness properties of graphs corresponding to each subgrid $S_i$ are achieved. Eventually, the distribution ${\mathcal{D}}$ chooses one of the good cells uniformly at random, and uses the associated router ${\mathcal{Q}}\in \Lambda'$ in order to route the terminals to a single vertex of $H''$. The analysis of expected congestion squared on every outer edge of $H''$ is very similar to the one outlined above.
We start by defining the notion of good cells of the grid $\Pi^*$, and showing that a large enough number of such cells exist in the upper right quadrant of $\Pi^*$. We will then define square subgrids of $\Pi^*$ and associate sets of paths with each such subgrid to ensure the required well-linkedness properties. Lastly, we show how to construct the desired routing ${\mathcal{Q}}$ for each good cell.
\subsubsection{Good Cells}
Fix a pair of indices $0< i,j< \lambda'$, and consider the cell $\mathsf{Cell}_{i,j}$ of the grid $\Pi^*$, and the two corresponding sets ${\mathcal{P}}^{i,j}$, ${\mathcal{R}}^{i,j}$ of paths.
\begin{definition}[Good cells]
A path $\sigma\in {\mathcal{P}}^{i,j}$ is \emph{good} for cell $\mathsf{Cell}_{i,j}$ if $\sigma$ contains no outer edges.
We say that cell $\mathsf{Cell}_{i,j}$ is \emph{good} if some path $\sigma \in {\mathcal{P}}^{i,j}$ is good for $\mathsf{Cell}_{i,j}$; otherwise we say it is \emph{bad}.
\end{definition}
Assume that cell $\mathsf{Cell}_{i,j}$ is good, and let $\sigma\in {\mathcal{P}}^{i,j}$ be any horizontal path that is good for this cell. Since $\sigma$ contains no outer edges, there must be a vertex $y\in V(H)$, such that $V(\sigma)\subseteq V(\Pi(y))$. Recall that, from \Cref{obs: paths for cells}, $|{\mathcal{R}}^{i,j}|=\ceil{7\psi/8}$, and that $\sigma$ intersects at least $13\psi/16$ paths of ${\mathcal{R}}^{i,j}$. Let $\hat {\mathcal{R}}^{i,j}\subseteq {\mathcal{R}}^{i,j}$ be a set of $\ceil{13\psi/16}$ paths, each of which shares at least one vertex with $\sigma$. Note that each such path then must contain a vertex of $\Pi(y)$. We denote by $\mathsf{Portals}^{i,j}$ the set of vertices that contains, for every path $\sigma'\in \hat{\mathcal{R}}^{i,j}$, the first vertex of $\sigma'$ (by definition, each such vertex must lie on path $P^*_i$). For convenience, we denote vertex $y$ of $H$ by $y_{i,j}$.
Let $Z$ be the set of all pairs of indices $\floor{\lambda'/2}\leq i,j< \lambda'$, such that $\mathsf{Cell}_{i,j}$ is good. Next, we show that $|Z|$ is sufficiently large. Our routing algorithm will then choose a pair $(i,j)$ of indices from $Z$ uniformly at random, and route the terminals to the vertices in set $\mathsf{Portals}^{i,j}$, from where they will be routed to vertices of $\Pi(y_{i,j})$, and eventually to some specific vertex of $\Pi(y_{i,j})$.
\begin{claim}\label{claim: many good cells}
$|Z|\geq (\lambda')^2/16$.
\end{claim}
\begin{proof}
Let ${\mathcal{B}}$ be a collection of all bad cells $\mathsf{Cell}_{i,j}$ lying in the top right quadrant, that is, $\floor{\lambda'/2}\leq i,j< \lambda'$. It is enough to show that $|{\mathcal{B}}|<(\lambda')^2/16$.
Consider now some bad cell $\mathsf{Cell}_{i,j}\in {\mathcal{B}}$, and any path $Q\in {\mathcal{P}}^{i,j}$. Since cell $\mathsf{Cell}_{i,j}$ is bad, $Q$ must contain at least one outer edge. We say that $Q$ is a \emph{type-1} bad path for cell $\mathsf{Cell}_{i,j}$ if it contains at least one type-1 outer edge (recall that a type-1 outer edge $e$ corresponds to some edge in graph $H$ that is {\bf not} contained in any cluster of ${\mathcal{C}}$). Otherwise, every outer edge on path $Q$ is a type-2 outer edge, and in this case we say that $Q$ is a type-2 bad cluster for $\mathsf{Cell}_{i,j}$. We say that cell $\mathsf{Cell}_{i,j}$ is \emph{type-1 bad} if at least $\psi/32$ paths of ${\mathcal{P}}^{i,j}$ are type-1 bad for this cell, and otherwise it is type-2 bad. We partition the set ${\mathcal{B}}$ of bad cells into two subsets: set ${\mathcal{B}}_1$ contains all type-1 bad cells, and set ${\mathcal{B}}_2$ contains all type-2 bad cells. It is now enough to prove that $|{\mathcal{B}}_1|,|{\mathcal{B}}_2|<(\lambda')^2/32$, which we do in the following two observations.
\begin{observation}\label{obs: few type-1 bad cells}
$|{\mathcal{B}}_1|< (\lambda')^2/32$.
\end{observation}
\begin{proof}
Assume for contradiction that $|{\mathcal{B}}_1|\geq (\lambda')^2/32$.
Consider a type-1 bad cell $\mathsf{Cell}_{i,j}\in {\mathcal{B}}_1$, and let ${\mathcal{Q}}^{i,j}\subseteq {\mathcal{P}}^{i,j}$ be a set of $\ceil{\psi/32}$ paths that are type-1 bad paths for cell $\mathsf{Cell}_{i,j}$. Each path in ${\mathcal{Q}}^{i,j}$ must contain at least one type-1 bad edge. Since the paths in ${\mathcal{Q}}^{i,j}$ are edge-disjoint, there is a set $E^{i,j}$ of at least $\psi/32$ type-1 outer edges of $H''$, lying on paths of ${\mathcal{Q}}^{i,j}$. Since every edge of $H''$ may lie on at most one path in ${\mathcal{P}}$, the total number of outer edges in $H''$ must be at least:
\[\frac{|{\mathcal{B}}_1|\cdot \psi}{32}\geq \frac{(\lambda')^2\cdot \psi}{2^{10}}\geq \frac{\lambda^2\cdot \psi}{2^{14}} ,\]
as $\lambda'=\floor{(\lambda-1)/2}\geq \lambda/4$.
Recall that $\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$ and $\lambda=\frac{2^{24}c\cdot \eta \log^8m}{\alpha^*\alpha^3}$, where $c$ is the constant from \Cref{obs: few outer edges}. Therefore, we get that the total number of outer edges in $H''$ is at least $\frac{2c\tilde k \eta \log^8m}{\alpha^3}$, contradicting \Cref{obs: few outer edges}.
\end{proof}
\begin{observation}\label{obs: few type-2 bad cells}
$|{\mathcal{B}}_2|< (\lambda')^2/32$.
\end{observation}
\begin{proof}
For a cluster $C\in {\mathcal{C}}$, let $X(C)=\bigcup_{y\in V(C)}V(\Pi(y))$. Note that all terminals of $H$ lie outside of the clusters in ${\mathcal{C}}$, and so $X(C)\cap \tilde T=\emptyset$. If a path $Q\in {\mathcal{P}}_3\cup {\mathcal{R}}_3$ contains a vertex of $X(C)$, then it must contain at least one edge of $\delta_H(C)$. As the paths in ${\mathcal{P}}\cup {\mathcal{R}}$ cause edge-congestion at most $2$, the total number of paths $Q\in {\mathcal{P}}\cup {\mathcal{R}}$ with a non-empty intersection with $X(C)$ is at most $2\delta_H(C)$.
Let $\mathsf{IntPairs}\subseteq {\mathcal{P}}_3\times{\mathcal{R}}_3$ be the collection of all pairs of paths $P\in {\mathcal{P}}_3$, $R\in {\mathcal{R}}_3$, such that $P$ and $R$ share at least one vertex.
For a cluster $C\in {\mathcal{C}}$, let $\mathsf{IntPairs}'_C\subseteq \mathsf{IntPairs}$ denote the collection of all pairs $(P,R)\in \mathsf{IntPairs}$ of paths, such that some vertex $v\in X(C)$ lies on both $P$ and $R$. Clearly, if $(P,R)\in \mathsf{IntPairs}_C'$, then each of the paths $P$, $R$ must contain at least one edge of $\delta_H(C)$. Therefore, from the above discussion, $|\mathsf{IntPairs}'_C|\leq 4|\delta_H(C)|^2$. Let $\mathsf{IntPairs}'=\bigcup_{C\in {\mathcal{C}}_Y}\mathsf{IntPairs}'_C$. Then:
%
\[ |\mathsf{IntPairs}'|\leq \sum_{C\in {\mathcal{C}}_Y}|\mathsf{IntPairs}'_C|\leq 4\sum_{C\in {\mathcal{C}}_Y}|\delta_H(C)|^2. \]
%
From Equation \ref{eq: sum of squares} (see \Cref{step 2 summary}), $
\sum_{C\in {\mathcal{C}}_Y}|\delta_H(C)|^2<\frac{(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}m}$, so we get that:
%
\begin{equation}\label{eq: bounding num of intersection pairs}
|\mathsf{IntPairs}'|<\frac{4(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}m},
\end{equation}
%
where $c_1$ is an arbitrarily large constant.
In the remainder of the proof, we assume for contradiction that $|{\mathcal{B}}_2|\geq (\lambda')^2/32$, and we will show that $|\mathsf{IntPairs}'|\geq \frac{4(\tilde k\tilde \alpha\alpha')^2}{c_1\log^{20}m}$ must hold, contradicting Equation \ref{eq: bounding num of intersection pairs}.
Consider a type-2 bad cell $\mathsf{Cell}_{i,j}\in {\mathcal{B}}_2$. Recall that every path in ${\mathcal{P}}^{i,j}$ contains at least one outer edge, and at most $\psi/32$ such paths contain a type-1 bad edge. Since, from \Cref{obs: paths for cells}, $|{\mathcal{P}}^{i,j}|= \ceil{7\psi/8}$,
there is a collection $\Sigma\subseteq {\mathcal{P}}^{i,j}$ of at least $3\psi/4$ paths $P$, such that all edges on $P$ are either inner edges, or type-2 outer edges. Therefore, if $P\in \Sigma$ is any such path, then there is some cluster $C\in {\mathcal{C}}$ with $V(P)\subseteq X(C)$. Recall that, from \Cref{obs: paths for cells}, each path $P\in \Sigma$ intersects at least $\frac{13\psi}{16}$ paths of ${\mathcal{R}}^{i,j}$. Clearly, if a path $R\in {\mathcal{R}}^{i,j}$ intersects a path $P\in \Sigma$, then $(P,R)\in \mathsf{IntPairs}'$. Therefore, intersections between pairs of paths in ${\mathcal{P}}^{i,j}\times {\mathcal{R}}^{i,j}$ contribute at least $\frac{13\psi}{16}\cdot \frac{3\psi}{4}\geq \frac{\psi^2}{2}$ pairs to set $\mathsf{IntPairs}'$.
Therfore, if we denote by $\mathsf{IntPairs}'_{i,j}$ the collection of all pairs $(P,R)\in \mathsf{IntPairs}'$, where a subpath $\sigma$ of $P$ lies in ${\mathcal{P}}^{i,j}$, and a subpath $\sigma'$ of $R$ lies in ${\mathcal{P}}^{i,j}$, and $\sigma,\sigma'$ contain a vertex $v\in X(C)$, for some cluster $C\in {\mathcal{C}}$, then, from the above discussion, $|\mathsf{IntPairs}'_{i,j}|\geq \frac{\psi^2}{2}$. We claim that for every pair $(P,R)\in \mathsf{IntPairs}'$ of paths, there is at most one pair of indices $0< i,j< \lambda'$, such that $(P,R)\in \mathsf{IntPairs}_{i,j}'$. Indeed, assume that $P\in {\mathcal U}_i$ and $R\in {\mathcal U}'_j$. For a pair $0< i',j'< \lambda'$ of indices, ${\mathcal{P}}^{i',j'}$ contains a subpath of $P$ iff $i'=i$, and ${\mathcal{R}}^{i',j'}$ contains a subpath of $R$ iff $j'=j$. So the only pair $(i',j')$ of indices for which $(P,R) \in \mathsf{IntPairs}'_{i',j'}$ may hold is $(i,j)$. Overall, we get that $|\mathsf{IntPairs}'|\geq |{\mathcal{B}}_2|\cdot \psi^2/2$. Assuming that $|{\mathcal{B}}_2|\geq (\lambda')^2/32$, since $\lambda'=\floor{(\lambda-1)/2}\geq \lambda/4$, we get that $|\mathsf{IntPairs}'|\geq \frac{\lambda^2\psi^2}{1024}$.
Recall that $\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$, and, from \Cref{obs: terminals well linked in H''}, $\alpha^*=\Theta(\tilde \alpha\alpha')$.
We conclude that:
\[|\mathsf{IntPairs}'|\geq \frac{(\alpha^*)^2\tilde k^2}{2^{22}}\geq \Omega\textsf{left}( {(\tilde \alpha\alpha'\tilde k)^2} \textsf{right} ). \]
Since we can choose $c_1$ to be a sufficiently large constant, this contradicts Equation \ref{eq: bounding num of intersection pairs}.
\end{proof}
\end{proof}
\subsubsection{Square Subgrids and Corresponding Sets of Paths}
For integers $1\leq i,j< \lambda'$ and $\ell\leq \min\set{i,j}$, a \emph{square subgrid $S=S(i,j,\ell)$ of $\Pi^*$} (that we also refer to as a \emph{square}) is defined as the collection of cells $\mathsf{CellSet}(S)=\set{\mathsf{Cell}_{i',j'}\mid i-\ell+1\leq i'\leq i;\quad j-\ell+1\leq j'\leq j}$. Intuitively, $S(i,j,\ell)$ is a subgrid of $\Pi^*$ of size $(\ell\times \ell)$, whose top right corner is the cell $\mathsf{Cell}_{i,j}$.
Given a square $S=S(i,j,\ell)$, we associate with it a collection ${\mathcal{P}}(S)$ of horizontal paths, and ${\mathcal{R}}(S)$ of vertical paths, as follows. Intuitively,
consider the graph obtained by taking the union of all paths ${\mathcal{P}}^{i',j'}$, where $\mathsf{Cell}_{i',j'}\in \mathsf{CellSet}(S)$.
This graph is a collection of disjoint paths, each of which is a subpath of a distinct path in $\bigcup_{i'=i-\ell+1}^i{\mathcal U}_{i'}$; we let ${\mathcal{P}}(S)$ be this set of paths.
Formally, for all $i-\ell+1\leq i'\leq i$, for every path $P\in {\mathcal U}_{i'}$, we include in ${\mathcal{P}}(S)$ the subpath of $P$ from the first vertex of $\sigma_{j-\ell+1}(P)$ to the last vertex of $\sigma_{j}(P)$.
Similarly, set ${\mathcal{R}}(S)$ contains, for all $j-\ell+1\leq j'\leq j$, for every path $R\in {\mathcal U}'_{j'}$, the subpath of $R$ from the first vertex of $\sigma'_{i-\ell+1}(R)$ to the last vertex of $\sigma'_{i}(R)$.
Notice that, from \Cref{obs: paths for cells}, $|{\mathcal{P}}(S)|=|{\mathcal{R}}(S)|= \ceil{7\psi/8}\cdot \ell$.
We denote by $\mathsf{EntryPortals}(S)$ the set of all vertices that serve as the first endpoint of the paths in ${\mathcal{P}}(S)$, and by $\mathsf{ExitPortals}(S)$ the set of all vertices that serve as the last endpoint of the paths in ${\mathcal{P}}(S)$. We denote by $G(S)$ the graph obtained by the union of the paths in ${\mathcal{P}}(S)\cup {\mathcal{R}}(S)$.
The following claim will be crucial for our algorithm for computing the routing paths for each good cell.
\begin{claim}\label{claim: routing in square}
Let $S=S(i,j,\ell)$ be a square of $\Pi^*$, for some $1\leq i,j< \lambda'$ and $\ell\leq \min\set{i,j}$, and let $Y\subseteq \mathsf{EntryPortals}(S)$, $Y'\subseteq\mathsf{ExitPortals}(S)$ be two subsets of vertices of cardinality $z$ each, where $z\leq \psi \ell/2$. Then there is a collection ${\mathcal{Q}}$ of edge-disjoint paths in graph $G(S)$, which is a one-to-one routing from $Y$ to $Y'$.
\end{claim}
\begin{proof}
Assume for contradiction that the claim is false. Then, from the maximum flow / minimum cut theorem, there is a collection $E'$ of at most $z-1$ edges in graph $G(S)$, such that $G(S)\setminus E'$ contains no path connecting a vertex of $Y$ to a vertex of $Y'$. Recall that each vertex of $Y$ is an endpoint of a distinct path in ${\mathcal{P}}(S)$, and all paths in ${\mathcal{P}}(S)$ are edge-disjoint. Since $|Y|=z$, while $|E'|\leq z-1$, there is some path $P\in {\mathcal{P}}(S)$, whose endpoint $y$ belongs to $Y$, such that $P$ contains no edge of $E'$. Using the same arguments, there is some path $P'\in {\mathcal{P}}(S)$, whose endpoint $y'$ belongs to $Y'$, that contains no edge of $E'$. Clearly, $P\neq P'$ must hold, as otherwise there is a path in $G(S)\setminus E'$ connecting $y$ to $y'$ -- the path $P$. It is now enough to show that there is some path $R\in {\mathcal{R}}(S)$, that contains no edge of $E'$, but $R\cap P\neq \emptyset$ and $R\cap P'\neq\emptyset$ hold. Indeed, in this case, $P\cup R\cup P'\subseteq G(S)\setminus E'$, and so $y$ remains connected to $y'$ in $G(S)\setminus E'$.
We now show that path $R$ with such properties must exist. Let $\tilde P\in {\mathcal{P}}_3$ be the path with $P\subseteq \tilde P$, and assume that $\tilde P\in {\mathcal U}_{i'}$ Similarly, let $\tilde P'\in {\mathcal{P}}_3$ be the path with $P'\subseteq \tilde P'$, and assume that $\tilde P'\in {\mathcal U}_{i''}$ (where possibly $i'=i''$). Consider some index $j-\ell+1\leq j'\leq \ell$. Recall
$|{\mathcal U}'_{j'}|=\ceil{7\psi/8}$, and, for every path $R\in {\mathcal U}'_{j'}$, segment $\sigma'_{i'}(R)$, lies in ${\mathcal{R}}^{i',j'}$, and segment $\sigma'_{i''}(R)$ lies in ${\mathcal{R}}^{i'',j'}$. Moreover, from \Cref{obs: paths for cells}, path $\sigma_{j'}(\tilde P)$ must intersect at least $\frac{13\psi}{16}$ paths of $\set{\sigma'_{i'}(R)\mid {\mathcal U}'_{j'}}$, and
similarly path $\sigma_{j'}(\tilde P')$ must intersect at least $\frac{13\psi}{16}$ paths of $\set{\sigma'_{i''}(R)\mid {\mathcal U}'_{j'}}$.
Therefore, there is a subset ${\mathcal U}''_{j'}\subseteq {\mathcal U}'_{j'}$ of at least $\psi/2$ paths $R$, such that both $P$ and $P'$ intersect the subpath of $R$ that belongs to ${\mathcal{R}}(S)$. Overall, there are at least $\ell \psi/2$ paths $R\in {\mathcal{R}}(S)$ that intersect the subpaths of $P$ and of $P'$ that lie in ${\mathcal{P}}(S)$. Since $z\leq \ell\psi/2$, at least one such path is disjoint from $E'$.
\end{proof}
\subsubsection{Routing the Terminals to Good Cells}
We fix some good cell $\mathsf{Cell}_{i,j}$ in the top right quadrant of the grid, that is, $\floor{\lambda'/2}\leq i,j<\lambda'$. Recall that we have defined a vertex $y_{i,j}\in V(H)$, and a collection $\hat {\mathcal{R}}^{i,j}\subseteq {\mathcal{R}}^{i,j}$ of $\psi'=\ceil{13\psi/16}$ paths, each of which contains a vertex of $\Pi(y_{i,j})$. We have also defined a set $\mathsf{Portals}^{i,j}$ of vertices that contains, for every path $\sigma'\in \hat {\mathcal{R}}^{i,j}$, the first vertex on $\sigma'$. Let $y^*_{i,j}$ be an arbitrary vertex of $\Pi(y_{i,j})$.
We define a set ${\mathcal{Q}}_{i,j}$ of paths in $H''$, routing the terminals of $\tilde T$ to vertex $y^*_{i,j}$, so ${\mathcal{Q}}_{i,j}\in \Lambda'$. In order to do so, we first define a set ${\mathcal{Q}}'_{i,j}$ of paths, routing a constant fraction of the terminals of $T_4$ to vertices of $\Pi(y_{i,j})$, and then extend this path set in order to obtain routing of all terminals to vertex $y^*_{i,j}$.
\paragraph{Routing to $\mathsf{Cell}_{i,j}$.}
In order to define the routing, we let $z=\floor{\log(\lambda'/4)}$, and we define $z+1$ squares $S_0^{i,j},S_1^{i,j},\ldots,S_z^{i,j}$. In order to simplify the notation, we will omit the superscript $i,j$ for now.
Square $S_0$ is $S(i,j,1)$, so it consists of a single cell $\mathsf{Cell}_{i,j}$. We denote by $\mathsf{Portals}_0^{i,j}=\mathsf{Portals}^{i,j}$ the set of $\psi'$ vertices that we have defined. We let ${\mathcal{Q}}_0$ be the set of $\psi'$ paths, containing, for every path $\sigma'\in \hat{\mathcal{R}}^{i,j}$, a subpath of $\sigma'$ between a vertex of $\mathsf{Portals}_0^{i,j}$ and a vertex of $\Pi(y_{i,j})$. Therefore, ${\mathcal{Q}}_0$ is a set of $\psi'$ edge-disjoint paths, routing vertices of $\mathsf{Portals}_0^{i,j}$ to vertices of $\Pi(y_{i,j})$, and all paths of ${\mathcal{Q}}_0$ are contained in ${\mathcal{R}}(S_0)$. We say that cell $\mathsf{Cell}_{i,j}$ is the bottom right corner of square $S_0$.
Fix some index $1\leq r\leq z$, and
assume that we have defined squares $S_0,\ldots,S_{r-1}$. We now define square $S_r$. We let $S_r=(i_r,j,2^r)$, so the length of the side of the square is $2^r$, and the coordinates of the top right corner of $S_r$ are $(i_r,j)$; here, $j$ is the column index of the initial cell $\mathsf{Cell}_{i,j}$, and $i_r$ is the cell immediately under the right bottom corner cell of $S_{r-1}$. In other words, if $S_{r-1}=(i_{r-1},j,2^{r-1})$, then $i_r=i_{r-1}+2^{r-1}$.
We assume that we have also defined a collection $\mathsf{Portals}_{r-1}\subseteq \mathsf{EntryPortals}(S_{r-1})$, containing $2^{r-1}\cdot \psi'$ vertices. Note that the top boundary of square $S_r$ appears immediately under bottom boundary of square $S_{r-1}$, so $\mathsf{EntryPortals}(S_{r-1})\subseteq \mathsf{ExitPortals}(S_r)$, and in particular $\mathsf{Portals}_{r-1}\subseteq \mathsf{ExitPortals}(S_r)$. We select an arbitrary subset $\mathsf{Portals}_r\subseteq \mathsf{EntryPortals}(S_r)$ of $2^r\cdot \psi'$ vertices. By partitioning set $\mathsf{Portals}_r$ into two equal-cardinality subsets $Y_1,Y_2$, and applying \Cref{claim: routing in square} to each of them separately, we obtain two collections ${\mathcal{Q}}_{r}^1,{\mathcal{Q}}_{r}^2$ of edge-disjoint paths in graph $G(S_r)$, routing vertex sets $Y_1$ and $Y_2$, respectively, to vertex set $\mathsf{Portals}_{r-1}$, in a one-to-one routing. Therefore, there is a set ${\mathcal{Q}}_{r}$ of paths in $G(S_r)$, routing vertex set $\mathsf{Portals}_r$ to vertex set $\mathsf{Portals}_{r-1}$ with edge-congestion at most $2$, such that every vertex in $\mathsf{Portals}_{r-1}$ is an endpoint of at most two such paths. Moreover, we can compute such set ${\mathcal{Q}}_r$ of paths efficiently via standard maximum flow.
Lastly, consider the last square $S_z$. We define a subset $T^*\subseteq T_4$ of terminals, as follows. For every vertex $v\in \mathsf{Portals}_z$, let $R_v\in {\mathcal{R}}$ be the vertical path containing $v$, and let $t_v\in T_4$ be the terminal that serves as an endpoint of path $R_v$. We then let $T^*=\set{t_v\mid v\in \mathsf{Portals}_z}$, and we let ${\mathcal{Q}}_{z+1}$ be a set of paths containing, for every vertex $v\in \mathsf{Portals}_z$, the subpath of $R_v$ between $t_v$ and $v$. Therefore, set ${\mathcal{Q}}_{z+1}$ of paths routes terminals of $T^*$ to vertices of $\mathsf{Portals}_z$, and the paths in ${\mathcal{Q}}_{z+1}$ are edge-disjoint. It is also easy to verify that the paths in ${\mathcal{Q}}_{z+1}$ do not contain any edges from graphs $G(S_0)\cup \cdots\cup G(S_z)$.
Note that, since $\psi=\floor{\frac{\alpha^*\tilde k}{64\lambda}}$ and $\alpha^*=\Theta(\alpha\alpha'/\log^4m)$,
\[|T^*|=2^z\cdot \psi'= 2^{\floor{\log(\lambda'/4)}}\cdot \ceil{13\psi/16}=\Omega(\lambda'\cdot \psi)=\Omega(\lambda\psi)=\Omega(\alpha^*\tilde k)=\Omega(\tilde k\alpha\alpha'/\log^4m).
\]
To summarize, we have defined a collection $\set{S_0,\ldots,S_z}$ of squares in the grid $\Pi^*$, where for all $0\leq r\leq z$, square $S_r$ has dimensions $(2^r\times 2^r)$. The squares are aligned on the right, and are stacked on top of each other, with square $S_0$ containing a single cell, $\mathsf{Cell}_{i,j}$. This guarantees that all corresponding graphs $G(S_r)$ are mutually disjoint, except that, for all $0\leq r<z$, $V(S_r)\cap V(S_{r+1})=\mathsf{EntryPortals}(S_r)$. We have defined, for all $0\leq r\leq z$, a set $\mathsf{Portals}_r\subseteq \mathsf{EntryPortals}(S_r)$ of $2^r\cdot \psi'$ vertices, and a set ${\mathcal{Q}}_r$ of paths contained in $G(S_r)$, routing vertices of $\mathsf{Portals}_r$ to vertices of $\mathsf{Portals}_{r-1}$ with edge-congestion at most $2$, so that every vertex of $\mathsf{Portals}_{r-1}$ serves as an endpoint of at most two such paths. Additionally, in graph $G(S_0)$, we have defined a set ${\mathcal{Q}}_0$ of $\psi'$ paths routing vertices of $\mathsf{Portals}_0$ to vertices of $\Pi(y_{i,j})$, and an additional set ${\mathcal{Q}}_{z+1}$ of edge-disjoint paths routing terminals in $T^*$ to vertices of $\mathsf{Portals}_z$ in a one-to-one routing, so that paths in ${\mathcal{Q}}_{z+1}$ do not contain edges of $G(S_0)\cup\cdots\cup G(S_z)$.
We are now ready to define a set ${\mathcal{Q}}'_{i,j}$ of paths, routing terminals of $T^*$ to vertices of $\Pi(y_{i,j})$. In order to do so, for all $0\leq r\leq z$, we let ${\mathcal{Q}}'_r$ be a multi-set of paths, contianing, for every path $\sigma'\in {\mathcal{Q}}_r$, $2^{z-r}$ copies of the path $\sigma'$. Therefore, paths in ${\mathcal{Q}}'_r$ cause edge-congestion $2^{z-r+1}$ in $G(S_r)$. Set ${\mathcal{Q}}'_{i,j}$ of paths is obtained by concatenating paths in sets ${\mathcal{Q}}_{z+1},{\mathcal{Q}}'_z,\ldots,{\mathcal{Q}}'_0$. It is easy to verify that paths in ${\mathcal{Q}}'_{i,j}$ route all terminals in $T^*$ to vertices of $\Pi(y_{i,j})$.
Recall that $|T^*|=\Omega(\tilde k\alpha\alpha'/\log^4m)$, and $|\tilde T|=\tilde k$.
Moreover, from \Cref{obs: terminals well linked in H''},
The set $\tilde T$ of terminals is $\alpha^*$-well-linked in $H''$, where $\alpha^*=\Theta(\alpha\alpha'/\log^4m)$.
From \Cref{lem: routing path extension},
there is a set ${\mathcal{Q}}_{i,j}$ of paths in graph $H''$, routing all vertices of $\tilde T$ to vertices of $\Pi(y_{i,j})$, such that, for every edge $e\in E(H'')$:
\[\cong_{H''}({\mathcal{Q}}_{i,j},e)\le
\ceil{\frac{\tilde k}{|T^*|}}\textsf{left} (\cong_{H''}({\mathcal{Q}}'_{i,j},e)+\ceil{1/\alpha^*}\textsf{right} )\leq
O\textsf{left} (\frac{\log^4m}{\alpha\alpha'}\textsf{right} )\cdot \textsf{left}(\cong_{H''}({\mathcal{Q}}'_{i,j},e)+\frac{\log^4m}{\alpha\alpha'}\textsf{right} ).\]
\paragraph{Distribution ${\mathcal{D}}$ and Analysis.}
The final distribution ${\mathcal{D}}$ over the routers of $\Lambda'$ is defined as follows. For every pair $(i,j)$ of indices in $Z$, we extend the paths in set ${\mathcal{Q}}_{i,j}$ via the inner edges of $\Pi(y_{i,j})$ so that each such path terminates at vertex $y^*_{i,j}$, obtaining a router of $\Lambda'$. Each such resulting router ${\mathcal{Q}}_{i,j}$ is assigned the same distribution $1/|Z|$; recall that, from \Cref{claim: many good cells},
$|Z|\geq (\lambda')^2/16$.
We now fix some outer edge $e\in E(H'')$, and analyze the expectation $\expect[{\mathcal{Q}}_{i,j}\sim {\mathcal{D}}]{(\cong_{H''}({\mathcal{Q}}_{i,j},e))^2}$.
Recall that there is at most one path $P\in {\mathcal{P}}$ that contains $e$, and at most one path $R\in {\mathcal{R}}$ containing $e$. Moreover, there is at most one pair $(i_1,j_1)$ of indices with edge $e$ lying on some path of $ {\mathcal{P}}^{i_1,j_1}$, and at most one pair $(i_2,j_2)$ of indices with edge $e$ lying on a path of ${\mathcal{R}}^{i_2,j_2}$.
We first focus on pair $(i_1,j_1)$ of indices, and the corresponding cell $\mathsf{Cell}_{i_1,j_1}$. Fix some pair $(i,j)\in Z$ of indices, and $0\leq r\leq z$. If $\mathsf{Cell}_{i_1,j_1}\in S_r^{i,j}$, then segment $\sigma_{j_1}(P)$ of $P$ may lie on at most $2^{z-r+1}$ paths in ${\mathcal{Q}}'_{i,j}$. Notice that there are at most $2^{2r+2}$ square subgrids $S$ of $\Pi^*$ of dimension $(2^r\times 2^r)$, that contain the cell $\mathsf{Cell}_{i_1,j_1}$. For each such square $S$, there is exactly one pair $(i(S),j(S))$ of indices, for which $S_r^{i(S),j(S)}=S$. Since $|Z|\geq (\lambda')^2/16$, the probability that an index $(i,j)\in Z$ is chosen for which $\mathsf{Cell}_{i_1,j_1}$ lies in the square $S_r^{i,j}$ is at most $O(2^{2r+2}/(\lambda')^2)$. Recall that, if $\mathsf{Cell}_{i_1,j_1}\in S_r^{i,j}$, then $\cong_{H''}({\mathcal{Q}}'_{i,j})\leq 2^{z-r+1}\leq O(\lambda'/2^r)$. Moreover, if $\mathsf{Cell}_{i_1,j_1}$ does not lie in any of the squares $S_0^{i,j},\ldots,S_z^{i,j}$, then $\cong_{H''}({\mathcal{Q}}'_{i,j})\leq 1$. The analysis for cell $\mathsf{Cell}_{i_2,j_2}$ is symmetric. Therefore, altogether (now taking into account both the cells $\mathsf{Cell}_{i_1,j_1}$ and $\mathsf{Cell}_{i_2,j_2}$), we get that:
\[\expect[(i,j)\in Z]{(\cong_{H''}({\mathcal{Q}}'_{i,j},e))^2}\leq O(1)+\sum_{r=0}^zO\textsf{left} (\frac{2^{2r+2}}{(\lambda')^2}\cdot\frac{(\lambda')^2}{2^{2r}}\textsf{right} )=O(z)\leq O(\log m).
\]
Lastly, since $\cong_{H''}({\mathcal{Q}}_{i,j},e)\le
O\textsf{left} (\frac{\log^4m}{\alpha\alpha'}\textsf{right} )\cdot \textsf{left}(\cong_{H''}({\mathcal{Q}}'_{i,j},e)+\frac{\log^4m}{\alpha\alpha'}\textsf{right} )$, we get that:
\[\expect[{\mathcal{Q}}_{i,j}\sim{\mathcal{D}}]{(\cong_{H''}({\mathcal{Q}}_{i,j},e))^2}\leq O\textsf{left} (\frac{\log^{16}m}{(\alpha\alpha')^4}\textsf{right} ).\]
\subsection{Disengagement of Nice Instances -- Proof of \Cref{thm: advanced disengagement - disengage nice instances}}
\label{subsec: disengagement with bad chain}
In this section we prove \Cref{thm: advanced disengagement - disengage nice instances}.
\znote{later: change the random path sets to distribution}
Assume that we are given a graph $G$ and a set ${\mathcal{C}}=\set{C_1,\ldots,C_r}$ of vertex-disjoint clusters in $G$, such that $\bigcup_{1\le i\le r}V(C_i)=V(G)$.
For each $1\le i\le r$, we define $\hat E_i=E(C_i,C_{i+1})$, $E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$, $E^{\textsf{thr}}_i=\bigcup_{i'\le i-1,j'\ge i+1}E(C_{i'},C_{j'})$, and we denote by $E'$ the subset of $E(G)$ that contains all edges connecting a pair $C_i,C_j$ of distinct clusters with $j\ge i+2$.
For each $1\le i\le r$, we say that cluster $C_i\in {\mathcal{C}}$ is \emph{{helpful}\xspace}, iff its has the $\alpha_0$-bandwidth property for $\alpha_0=1/\log ^3m$.
For each $2\le i\le r-1$, we say that cluster $C_i$ is \emph{{unhelpful}\xspace}, iff there exist two vertices $u^{\operatorname{left}}_i,u^{\operatorname{right}}_i\in V(C_i)$, that we call the \emph{left center} and \emph{right center} of $C_i$, respectively, and three random sets ${\mathcal{Q}}^{\operatorname{left}}_i,{\mathcal{Q}}^{\operatorname{right}}_i,{\mathcal{Q}}^{\textsf{thr}}_i$ of paths, such that
\begin{itemize}
\item the paths of ${\mathcal{Q}}^{\operatorname{left}}_i$ route edges of $\hat E_{i-1}\cup E^{\operatorname{left}}_{i}$ to vertex $u^{\operatorname{left}}_i$ inside $C_{i}\cup C_{i+1}$, and for each edge $e\in E(C_{i}\cup C_{i+1})$, $\expect[]{(\cong_{G}({\mathcal{Q}}_{i}^{\operatorname{left}}, e))^2}\le \beta$;
\item the paths of ${\mathcal{Q}}^{\operatorname{right}}_i$ route edges of $\hat E_{i}\cup E^{\operatorname{right}}_{i}$ to vertex $u^{\operatorname{right}}_i$ inside $C_{i}\cup C_{i-1}$; and for each edge $e\in E(C_{i}\cup C_{i-1})$, $\expect[]{(\cong_{G}({\mathcal{Q}}_{i}^{\operatorname{right}}, e))^2}\le \beta$; and
\item $|{\mathcal{Q}}^{\textsf{thr}}_i|=|E^{\textsf{thr}}_i|$, paths of ${\mathcal{Q}}^{\textsf{thr}}_i$ connect vertex $u^{\operatorname{left}}_i$ to vertex $u^{\operatorname{right}}_i$ in $C_{i}$, and $\cong_{G}({\mathcal{Q}}_{i}^{\textsf{thr}}) \le O(1)$.
\end{itemize}
Consider now the graph $H=G_{\mid {\mathcal{C}}}$,
For each $1\leq i\leq r$, we denote by $x_i$ the vertex of graph $H$ that represents the cluster $C_i$.
Note that each edge in $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ corresponds to an edge in $H$, and we do not distinguish between them.
We say that a path $P$ in $H$ is an \emph{auxiliary path} of an edge $e=(x_i,x_j)$ in $H$ (where $j\ge i+2$), iff $P=(e_i,\ldots,e_{j-1})$, where for each $i\le t\le j-1$, $e_t$ is an edge in $H$ connecting $x_t$ to $x_{t+1}$.
Since such an edge $e=(x_i,x_j)$ is also an edge of $G$, we also say that the path $P$ in $H$ is an auxiliary path for the edge $e$ in $G$.
In this section we prove the following lemma.
\begin{lemma}\label{lem: disengagement with routing}
There is an efficient randomized algorithm, that takes as input an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace with $|E(G)|=m$, a sequence ${\mathcal{C}}=\set{C_1,\ldots,C_r}$ of vertex-disjoint clusters of $G$ such that the vertex sets $\set{V(C)}_{C\in {\mathcal{C}}}$ partition $V(G)$, and a set ${\mathcal{P}}=\set{P_e\mid e\in E'}$ of paths in $H$, that contains, for each edge $e\in E'$, an auxiliary path $P_e$ of $e$, such that
\begin{itemize}
\item $\bigcup_{1\le i\le r}V(C_i)=V$;
\item clusters $C_1,C_r$ are {helpful}\xspace clusters; each of $C_2,\ldots, C_{r-1}$ is either a {helpful}\xspace cluster or an {unhelpful}\xspace cluster; and if $C_i$ is an {unhelpful}\xspace cluster, then both $C_{i-1},C_{i+1}$ are {helpful}\xspace clusters; and
\item $\cong_H({\mathcal{P}})=O(1)$.
\end{itemize}
The algorithm computes an $O(\beta^2)$-decomposition ${\mathcal{I}}$ of instance $I$ with respect to ${\mathcal{C}}$.
Additionally, there is an efficient algorithm ${\mathcal{A}}$, that, given a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$.
\end{lemma}
The algorithm consists of several steps, and will be described in the following subsections.
\znote{to add a high level explanation here and in each subsection}
\subsubsection{Computing Guiding Paths}
\label{sec: guiding and auxiliary paths}
First, for each {helpful}\xspace cluster $C_i\in {\mathcal{C}}$, we apply the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace from \Cref{thm:algclassifycluster} to the instance $(G,\Sigma)$ and cluster $C_i$, with parameter $p=1/n^{100}$.
If it returns a distribution ${\mathcal{D}}(C_i)$ over sets of guiding paths in $\Lambda(C_i)$, such that $C_i$ is $\beta^*$-light with respect to ${\mathcal{D}}(C_i)$ (recall that $\beta^*=2^{O(\sqrt{\log m}\log\log m)}$), then we continue to sample a set of guiding paths from ${\mathcal{D}}(C_i)$. Let ${\mathcal{Q}}_i$ be the set we obtain, and let $u_i$ be the vertex of $C_i$ that serves as the common endpoint of all paths in ${\mathcal{Q}}_i$. If the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace returns FAIL, then consider the augmentation $C_i^+$ of cluster $C_i$ (see \Cref{def: Graph C^+}). We apply the algorithm from \Cref{lem: simple guiding paths} to graph $C^+_i$ and set $T(C_i)$ of its vertices, and let $u_i$ be the vertex and ${\mathcal{Q}}_i$ be the set of paths that we obtain. Clearly, ${\mathcal{Q}}_i$ can be also viewed as a set of paths routing edges of $\delta_G(C_i)$ to $u_i$. From \Cref{lem: simple guiding paths}, for each edge $e\in E(C_i)$, $\expect{\cong({\mathcal{Q}}_i,e)}\leq O(\log^4m/\alpha_0)$. Finally, for each {helpful}\xspace cluster $C_i$, and for each edge $e\in \delta(C_i)$, we denote by $Q_i(e)$ the path of ${\mathcal{Q}}_i$ that routes $e$ to $u_i$.
\textbf{Bad Event $\xi_i$.} For each $1\le i\le r$, we say that the event $\xi_i$ happens, iff $C_i$ is a {helpful}\xspace cluster and not an $\eta^*$-bad cluster, but the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace returns FAIL on it.
From \Cref{thm:algclassifycluster}, $\Pr[\xi_i]\le 1/n^{100}$. Then from the union bound over all $1\le i\le r$, $\Pr[\bigcup_{1\le i\le r}\xi_i]\le O(n^{-99})$.
Then, for each {helpful}\xspace cluster $C_i$, we apply the algorithm from \Cref{lem: non_interfering_paths} to the set ${\mathcal{Q}}_i$ of paths and the rotation system $\Sigma$, and rename the resulting path set by ${\mathcal{Q}}_i$.
Similarly, for each {unhelpful}\xspace cluster $C_i$, we apply the algorithm from \Cref{lem: non_interfering_paths} to the sets ${\mathcal{Q}}_i^{\operatorname{left}}, {\mathcal{Q}}_i^{\operatorname{right}}, {\mathcal{Q}}_i^{\textsf{thr}}$ of paths separately, and rename the resulting path sets by ${\mathcal{Q}}_i^{\operatorname{left}}, {\mathcal{Q}}_i^{\operatorname{right}}, {\mathcal{Q}}_i^{\textsf{thr}}$, respectively.
Now we are guaranteed that, for each {helpful}\xspace cluster $C_i$, the set ${\mathcal{Q}}_i$ of paths is non-transversal with respect to $\Sigma$, and for each {unhelpful}\xspace cluster $C_i$, the sets ${\mathcal{Q}}_i^{\operatorname{left}}, {\mathcal{Q}}_i^{\operatorname{right}}, {\mathcal{Q}}_i^{\textsf{thr}}$ of paths are non-transversal with respect to $\Sigma$.
\subsubsection{Computing Auxiliary Cycles}
\begin{claim}
\label{claim: compute routing cycles}
We can efficiently compute a set ${\mathcal{R}}=\set{R_e\mid e\in E'}$ that contains, for each edge $e\in E'$, a path $R_e$ in $G$, such that
\begin{itemize}
\item for each edge $e$ connects $C_i$ to $C_j$ with $j\ge i+2$, the path $R_e$ has endpoints $u'_i,u'_j$, where:
if $C_i$ is {helpful}\xspace, then $u'_i=u_i$, otherwise $u'_i=u^{\operatorname{right}}_i$; and similarly if $C_i$ is {helpful}\xspace, then $u'_j=u_j$, otherwise $u'_j=u^{\operatorname{left}}_j$;
\iffalse{clearer and longer explanation}
\begin{itemize}
\item if both $C_i, C_j$ are {helpful}\xspace clusters, then $R_e$ has endpoints $u_i,u_j$;
\item if both $C_i, C_j$ are {unhelpful}\xspace clusters, then $R_e$ has endpoints $u^{\operatorname{right}}_i, u^{\operatorname{left}}_j$;
\item if $C_i$ is {helpful}\xspace and $C_j$ is {unhelpful}\xspace, then $R_e$ has endpoints $u_i, u^{\operatorname{left}}_j$;
\item if $C_i$ is {unhelpful}\xspace and $C_j$ is {helpful}\xspace, then $R_e$ has endpoints $u^{\operatorname{right}}_i, u_j$;
\end{itemize}
\fi
\item for each edge $e$ connects $C_i$ to $C_j$ with $j\ge i+2$, the path $R_e$ can be partitioned into subpaths $R_e=(K_i,\hat e_i,K_{i+1},\hat e_{i+1},\ldots,\hat e_{j-1},K_j)$, where for each index $i\le t\le j$, $K_j$ is a subpath that lies entirely in $C_t$, and $\hat e_t$ is an edge of $\hat E_t$;
\item for each $e\in E(G)$, $\expect[]{(\cong_G({\mathcal{R}},e))^2}\le \beta$.
\end{itemize}
\end{claim}
\begin{proof}
For convenience, for an edge $e\in E'$ connecting a vertex of $C_i$ to a vertex of $C_j$ with $j\ge i+2$, we say that $i$ is the \emph{left index} of $e$ (which we sometimes denote by $i_e$) and $j$ is the \emph{right index} of $e$ (which we sometimes denote by $j_e$).
We will gradually construct all paths of $\set{R_e\mid e\in E'}$ simultaneously. We start by setting, for each $e\in E'$, the initial path $R_e$ to be an empty path, and then sequentially process clusters $C_1,\ldots, C_r$. Upon processing the cluster $C_t$, we will construct the subpath $K_t(e)$ and the edge $\hat e_{t+1}$ for every $e\in E'$.
We now describe the iteration of processing cluster $C_t$.
Note that in this iteration, we only need to deal with edges of $E'$ with $i_e\le t\le j_e$.
Assume first that $C_t$ is a {helpful}\xspace cluster.
We denote by $E_t^*$ the set of edges in $E({\mathcal{P}})$ that connects $x_t$ to $x_{t+1}$, namely $E^*_t=E({\mathcal{P}})\cap E_H(x_t,x_{t+1})$. Note that $E^*_t$ is also a set of edges in $G$ connecting $C_t$ to $C_{t+1}$, so $E^*_t\subseteq \hat E_t$.
First, for each edge $e\in E'$ with right index $t$, all its subpaths $K_i(e),\hat e_i,\ldots,\hat e_{t-1}$ have already been constructed. We simply define $K_t(e)$ to be $Q_t(\hat e_{t-1})$, the path in ${\mathcal{Q}}_t$ routing edge $\hat e_{t-1}$ to $u_t$ in $C_t$.
Second, for each edge $e\in E'$ with left index $t$, we simply designate (arbitrarily) a distinct edge of $E^*_i$ as $\hat e_t$, and then define $K_t(e)$ to be $Q_t(\hat e_{t})$, the path in ${\mathcal{Q}}_t$ routing edge $\hat e_{t}$ to $u_t$ in $C_t$.
Finally, consider the set $\set{\hat e_{t-1}\mid i_e < t<j_e}$ and the set of edges in $E^*_t$ that are not yet designated as $\hat e_t$ for any edge of $E'$, that we denote by $E^{**}_t$. Clearly, these two sets are equal-cardinality subsets of $\delta_G(C_t)$.
Since cluster $C_i$ has the $\alpha_0$-bandwidth property, from \Cref{cor: bandwidth_means_boundary_well_linked}, we can efficiently compute an one-to-one routing ${\mathcal K}_i$ of edges of $\set{\hat e_{t-1}\mid i_e < t<j_e}$ to edges of $E^{**}_i$, such that $\cong_G({\mathcal K}_i)=\ceil{1/\alpha_0}$. Now for each edge $e\in E'$ with $i_e < t<j_e$, we designate the path in ${\mathcal K}_i$ that starts at its previously constructed segment $\hat e_{t-1}$ as $K_t(e)$, and designate the last edge of $K_t(e)$ in $E^{**}_i$ as $\hat e_t$.
Assume now that $C_t$ is an {unhelpful}\xspace cluster.
First, for each edge $e\in E'$ with right index $t$, the edge $\hat e_{t-1}$ has been defined in previous iterations, and we simply define $K_t(e)$ to be $Q^{\operatorname{left}}_t(\hat e_{t-1})$, the path in ${\mathcal{Q}}^{\operatorname{left}}_t$ routing edge $\hat e_{t-1}$ to $u_t$ in $C_t$.
Second, for each edge $e\in E'$ with left index $t$, we simply designate (arbitrarily) a distinct edge of $E^*_t$ as $\hat e_t$, and then define $K_t(e)$ to be $Q_t(\hat e_{t})$, the path in ${\mathcal{Q}}_t$ routing edge $\hat e_{t}$ to $u_t$ in $C_t$.
Finally, consider the edges of $E^{\textsf{thr}}_t$ (namely edges $e$ with $i_e < t<j_e$). We denote by $E^{**}_t$ the set of edges in $E^*_t$ that are not designated as $\hat e_t$ for any edge of $E'$. For each edge $e\in E^{\textsf{thr}}_t$, we designate a distinct edge of $E^{**}_t$ as $\hat e_t$, such that every edge of $E^{**}_t$ is assigned to exactly one edge of $ E^{\textsf{thr}}_t$. Recall that we are given three sets of paths ${\mathcal{Q}}^{\operatorname{left}}_i,{\mathcal{Q}}^{\operatorname{right}}_i,{\mathcal{Q}}^{\textsf{thr}}_i$.
For each edge $e\in E^{\textsf{thr}}_t$, we define $K_t(e)$ as the concatenation of (i) $Q^{\operatorname{left}}_t(\hat e_{t-1})$, the path in ${\mathcal{Q}}_{t}^{\operatorname{left}}$ that routes $\hat e$ to $u_t^{\operatorname{left}}$;
(ii) a distinct path in $Q^{\textsf{thr}}_t(\hat e)$ that routes $u_t^{\operatorname{left}}$ to $u_t^{\operatorname{right}}$; and (iii) $Q^{\operatorname{right}}_t(\hat e_t)$, the path in ${\mathcal{Q}}_{t}^{\operatorname{right}}$ that routes $\hat e_t$ to $u_t^{\operatorname{right}}$.
This completes the description of an iteration, and therefore completes the description of the algorithm. Clearly, after processing all clusters $C_1,\ldots, C_r$, we obtain a set ${\mathcal{R}}$ that contains, for each edge $e\in E'$, a path $R_e$. It is not hard to verify that all properties of \Cref{claim: compute routing cycles} are satisfied.
\end{proof}
\iffalse{construct routing path if given routing paths}
\paragraph{Step 1.}
In the first step we will construct a set $\set{R^*_e}_{e\in E'}$ of cycles, using the input auxiliary paths of ${\mathcal{P}}$. Specifically, we will construct, for each edge $e\in E'$ two paths $P^*_e,P^{**}_e$ with the same endpoints, and the cycle $R^*_e$ is then defined to be the union of paths $P^*_e,P^{**}_e$.
Consider now an edge $e\in E'$ and its input auxiliary path $P_e$ in $H$, and assume that $e$ connects $C_i$ to $C_j$ with $j\ge i+2$. Recall that path $P_e$ sequentially visits vertices $x_i,x_{i+1},\ldots, x_j$ in $H$. For convenience, we denote $P=(e_i,e_{i+1},\ldots,e_{j-1})$.
For each {helpful}\xspace cluster $C_t$ with $i<t<j$, we define the path $L_e^t$ as the union of the path $Q_i(e_{t-1})$ in ${\mathcal{Q}}_i$ that routes $e_{t-1}$ to $u_i$, and the path $Q_i(e_{t})$ in ${\mathcal{Q}}_i$ that routes $e_{t-1}$ to $u_i$. So path $L_e^t$ has $e_{t-1}$ as its first edge and $e_{t}$ as its last edge.
For each $i<t<j$ such that $C_t$ is a {unhelpful}\xspace cluster, we define the path $L_e^t$ as follows. Let $P_e^t$ be the subpath of $P_e$ between vertices $z_t$ and $z'_t$. Recall that path $P^t_e$ contains the special vertices $u^{\operatorname{left}}_t,u^{\operatorname{right}}_t$. We define $L_e^t$ as the sequential concatenation of (i) the path in ${\mathcal{Q}}_i^{\operatorname{left}}$ that connects $e_{t-1}$ to $u^{\operatorname{left}}_i$; (ii) the subpath of $P_e^t$ that connects $u^{\operatorname{left}}_t$ to $u^{\operatorname{right}}_t$; and (iii) the path in ${\mathcal{Q}}_i^{\operatorname{right}}$ that connects $e_{t}$ to $u^{\operatorname{right}}_i$. So path $L_e^t$ has $e_{t-1}$ as its first edge and $e_{t}$ as its last edge, and visits $u^{\operatorname{left}}_t$ before $u^{\operatorname{right}}_t$.
Similarly, if $C_i$ is not a {unhelpful}\xspace cluster, then we define $L_e^i$ to be the path $Q_i(e_{i+1})$ that routes $e_{i+1}$ to $u_i$, and if $C_i$ is a {unhelpful}\xspace cluster, then we define $L_e^i$ to be the path in ${\mathcal{Q}}_i^{\operatorname{right}}$ that routes $e_{i+1}$ to $u^{\operatorname{right}}_i$. The path $L_e^j$ is defined similarly.
Finally, the path $P^*_e$ is defined as the sequential concatenation of paths $L_e^i, L_e^{i+1},\ldots, L_e^{j-1},L_e^j$.
\fi
\znote{unhelpful clusters left and right paths currently pass through incorrect clusters}
\znote{to modify until the end of subsection}
We now construct path $P^{**}_e$.
If $C_i$ is not a {unhelpful}\xspace cluster, then we define $L_e^i$ to be the path $Q_i(e)$ that routes $e$ to $u_i$, and if $C_i$ is not a {unhelpful}\xspace cluster, then we define $L_e^i$ to be the path in ${\mathcal{Q}}_i^{\operatorname{right}}$ that routes $e$ to $u^{\operatorname{right}}_i$. The path $L_e^j$ is defined similarly.
The path $P^{**}_e$ is defined as the sequential concatenation of paths $L_e^i, e ,L_e^j$.
See \Cref{fig: auxiliary cycle} for an illustration.
\begin{figure}[h]
\centering
\includegraphics[scale=0.2]{figs/auxiliary_cycle.jpg}
\caption{An illustration of paths $P^*_e$ (shown in red) and $P^{**}_e$ (shown in green) for edge $e=(v,v')$.}\label{fig: auxiliary cycle}
\end{figure}
Note that the given ${\mathcal{P}}$ of paths is random, the sets ${\mathcal{Q}}_i$ for clusters $C_i$ that has the $\alpha_0$-bandwidth property are random, and the sets ${\mathcal{Q}}^{\operatorname{left}}_i,{\mathcal{Q}}^{\operatorname{right}}_i$ for {unhelpful}\xspace clusters $C_i$ are also random, so the set ${\mathcal{R}}^*$ of cycles we constructed is also a random set. We use the following simple observation.
\begin{observation}
For each edge $e\in E'$, $\cong_{G}({\mathcal{R}}^*,e)\le \cong_{G}({\mathcal{P}},e)$, and
for each edge $e\in \bigcup_{1\le i\le r}E(C_i)$, $\expect[]{\cong_{G}({\mathcal{R}}^*,e)^2}\le O\textsf{left}(\beta^2\cdot\expect[]{\cong_{G}({\mathcal{P}},e)^2}\textsf{right})\le O(\beta^4)$.
\end{observation}
\begin{proof}
\znote{To Complete}
\end{proof}
\paragraph{Step 2.} In this step, we further process the set $\set{R^*_e}_{e\in E'}$ of cycles constructed in the first step to obtain the desired set $\set{R_e}_{e\in E'}$ of auxiliary cycles.
\znote{To Complete: de-interfere}
\znote{Need some bound on $\mathsf{cost}_{\mathsf {NT}}({\mathcal{R}},\Sigma)$ here or in prelim?x}
\subsubsection{Constructing the Decomposition into Subinstances}
We use the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace defined in \Cref{subsec: basic disengagement} to construct subinstances of the input instance $(G,\Sigma)$. However, instead of directly applying the analysis in \Cref{subsec: basic disengagement} to bound the total cost of all these subintances, we will use an alternative way of analyzing their total cost, which the required bounds in \Cref{lem: disengagement with routing}.
For each $1\le i\le r$, we define cluster $U_i=\bigcup_{1\le t\le i}C_t$, and $\overline{U}_i=\bigcup_{i+x1\le t\le r}C_t$, so $U_r=G$. We define ${\mathcal{L}}=\set{U_1,\ldots,U_r}$, so ${\mathcal{L}}$ is a laminar family. In order to construct the subinstances using the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace, we need to also specify a distribution on sets of guiding paths for each cluster of $U_1,\ldots,U_{r-1}$, or equivalently construct a random set of guiding paths for each of these clusters, which we do next.
Consider first the cluster $U_1=C_1$. Recall that in \Cref{sec: guiding and auxiliary paths} we have obtained a random set ${\mathcal{Q}}_1$ of guiding paths in $C_1$, that route all edges of $\delta(C_1)$ to a vertex $u_1$ in $C_1$. We simply let the random set ${\mathcal{Q}}_1$ be the random set of guiding paths for $U_1$, that we denote by ${\mathcal{W}}_1$. Consider now an index $i>1$ and the cluster $U_i$.
We construct the random set ${\mathcal{W}}_i$ of guiding paths as follows.
Assume first that $C_i$ is a {helpful}\xspace cluster. For each edge $e\in \hat E_i$, we define $W_i(e)=Q_i(e)$, the path in ${\mathcal{Q}}_i$ that routes $e$ to $u_i$ in $C_i$. For each edge $e\in E(U_i, \overline{U}_i)\setminus \hat E_i$, note that $e\in E'$, and we define path $W_i(e)$ as the subpath of cycle $R_e$ between (including) the edge $e$ and vertex $u_i\in C_i$, that lies entirely in $U_i$. We then let ${\mathcal{W}}_i=\set{W_i(e)\mid e\in E(U_i, \overline{U}_i)}$.
Assume now that $C_i$ is an {unhelpful}\xspace cluster. Similarly, for each edge $e\in \hat E_i$, we define $W_i(e)=Q^{\operatorname{right}}_i(e)$, the path in ${\mathcal{Q}}_i^{\operatorname{right}}$ that routes edge $e$ to $u_i^{\operatorname{right}}$, and for each edge $e\in E(U_i, \overline{U}_i)\setminus \hat E_i$, note that $e\in E'$, and we define path $W_i(e)$ as the subpath of cycle $R_e$ between (including) the edge $e$ and vertex $u^{\operatorname{right}}_i\in C_i$ that lies entirely in cluster $U_i$.
We then let ${\mathcal{W}}_i=\set{W_i(e)\mid e\in E(U_i, \overline{U}_i)}$.
\begin{figure}[h]
\centering
\subfigure[Set $E(U_i,\overline{U}_i)=\hat E_i\cup E^{\operatorname{right}}_{i}\cup E^{\operatorname{left}}_{i+1}\cup E^{\operatorname{over}}_{i}$ where $\hat E_{i}=\set{\hat e_1,\ldots,\hat e_4}$, $E^{\operatorname{right}}_{i}=\set{e^b}$, $E^{\operatorname{left}}_{i+1}=\set{e^g_1,e^g_2}$ and $E^{\operatorname{over}}_{i}=\set{e^r}$.
Paths of ${\mathcal{W}}_i$ excluding their first edges are shown in dash lines.]{\scalebox{0.13}{\includegraphics{figs/in_rotation_1.jpg}}}
\hspace{3pt}
\subfigure[The circular ordering $\tilde{\mathcal{O}}_{i}$ on the edges of $E(U_i,\overline{U}_i)$.]{
\scalebox{0.14}{\includegraphics{figs/in_rotation_2.jpg}}}
\caption{An illustration of path set ${\mathcal{W}}_i$ and the circular ordering $\tilde{\mathcal{O}}_{i}$.}\label{fig: in rotation good}
\end{figure}
We now apply the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace to each cluster $U_i\in {\mathcal{L}}$ and its associated set ${\mathcal{W}}_i$ of guiding paths, and let $I_1,\ldots, I_r$ be the subinstances we obtain. We recall here how the instances are constructed in \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace.
For each $1\le i\le r$, we denote $\tilde {\mathcal{O}}_i={\mathcal{O}}^{\operatorname{guided}}({\mathcal{W}}_i,\Sigma)$, the ordering guided by ${\mathcal{W}}_i$ and $\Sigma$ (see definition in \Cref{subsec: guiding paths rotations}), so $\tilde {\mathcal{O}}_i$ is an ordering on the set $E(U_i, \overline{U}_i)$ of edges. See \Cref{fig: in rotation good} for an illustration.
First, the instance $I_1=(G_1,\Sigma_1)$ is the cluster-based instance associated with cluster $C_1$, namely $(G_1,\Sigma_1)=(G_{C_1},\tilde \Sigma_{C_1})$. Recall that $V(G_{C_1})=V(C_1)\cup \set{v^*_1}$, and the rotation on $v^*_1$, the vertex obtained by contracting $G\setminus C_1$, is $\tilde {\mathcal{O}}_1$ defined above.
Then for each index $2\le i\le r-1$, the instance $I_i=(G_i,\Sigma_i)$ is obtained from the cluster-based instance $(G_{U_i},\tilde \Sigma_{U_i})$, by taking its contracted instance with respect to cluster $U_{i-1}$. See \Cref{fig: subinstance} for an illustration. So $V(G_i)=V(C_i)\cup \set{v^*_i, v'_i}$, where vertex $v^*_i$ is obtained by contracting $G\setminus U_i$ and vertex $v'_i$ is obtained by contracting $U_{i-1}$. The rotation on vertex $v^*_i$ is $\tilde {\mathcal{O}}_i$, and the rotation on vertex $v'_i$ is $\tilde {\mathcal{O}}_{i-1}$.
Finally, the instance $I_r=(G_r,\Sigma_r)$ is the contracted instance of $(G,\Sigma)$ with respect to cluster $U_{r-1}$. Recall that $V(G_r)=V(C_r)\cup \set{v'_r}$, and the rotation on $v'_r$, the vertex obtained from contracting the cluster $U_{r-1}$, is $\tilde {\mathcal{O}}_{r-1}$.
\begin{figure}[h]
\centering
\subfigure[Layout of edge sets in $G$. Edges of $E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}$ are shown in green, and edges of $E^{\textsf{thr}}_{i}$ are shown in red. ]{\scalebox{0.14}{\includegraphics{figs/subinstance_1.jpg}
}
\hspace{5pt}
\subfigure[Graph $G_i$, where $\delta_{G_i}(v_i^{*})=\hat E_i \cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, and $\delta_{G_i}(v_i')=\hat E_{i-1} \cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_{i}$.]{
\scalebox{0.12}{\includegraphics{figs/subinstance_2.jpg}}}
\caption{An illustration of the construction of subinstance $(G_i,\Sigma_i)$.}\label{fig: subinstance}
\end{figure}
\subsubsection{Completing the Proof of \Cref{lem: disengagement with routing}}
\label{subsec: proof of disengagement with routing}
In this section we complete the proof of \Cref{lem: disengagement with routing}.
Since the subinstances are constructed using the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace, \Cref{lem: basic disengagement combining solutions} holds. In particular, if we are given, for each $1\le i\le r$, a solution $\phi_i$ to subinstance $I_i$, then we can efficiently combine them together to obtain a solution $\phi$ for the input instance $I$ of cost at most $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
On the other hand, from the construction of the subinstances $I_1,\ldots,I_r$, it is clear that, for each $1\le i\le r$, $E(C_i)\subseteq E(G_i)$, and all other edges of $G_i$ lie in $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$.
We next show in the following observation that $\sum_{1\le i\le r}|E(G_i)|\le O(|E(G)|)$.
\begin{observation}
\label{obs: disengaged instance size}
$\sum_{1\le i\le r}|E(G_i)|\le O(|E(G)|)$.
\end{observation}
\begin{proof}
\iffalse
Note that, in the sub-instances $\set{(G_i,\Sigma_i)}_{1\le i\le r}$, each graph of $\set{G_i}_{1\le i\le r}$ is obtained from $G$ by contracting some sets of clusters of ${\mathcal{C}}$ into a single super-node, so each edge of $G_i$ corresponds to an edge in $E(G)$.
Therefore, for each $1\le i\le r$,
$$|E(G_i)|=|E_G(C_i)|+|\delta_G(C_i)|\le |E_G(C_i)|+|E^{\textsf{out}}({\mathcal{C}})|\le m/(100\mu)+m/(100\mu)\le m/\mu.$$
\fi
%
Note that $E(G_i)=E(C_i)\cup \hat E_i\cup \hat E_{i-1}\cup E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$.
First, each edge of $\bigcup_{1\le i\le r}E(C_i)$ appears in exactly one graphs of $\set{G_i}_{1\le i\le r}$.
Second, each edge of $\bigcup_{1\le i\le r}\hat E_i$ appears in exactly two graphs of $\set{G_i}_{1\le i\le r}$.
Consider now an edge $e\in E'$. If $e$ connects a vertex of $C_i$ to a vertex of $C_j$ for some $j\ge i+2$, then $e$ will appear as an edge in $E_i^{\operatorname{right}}\subseteq E(G_i)$ and an edge in $E_j^{\operatorname{left}}\subseteq E(G_j)$, and it will appear as an edge in $E_k^{\textsf{thr}}\subseteq E(G_k)$ for all $i<k<j$.
For each $1\le i\le r$, we denote $E^{\operatorname{over}}_{i}=E(U_{i-1},U_{i+1})$.
On one hand, we have $\sum_{1\le i\le r}|E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}|\le 2|E(G)|$.
On the other hand, note that $E_k^{\textsf{thr}}=E(C_{k-1},C_{k+1})\cup E^{\operatorname{over}}_{k-1}\cup E^{\operatorname{over}}_{k}$, and each edge of $e$ appears in at most two graphs of $\set{G_i}_{1\le i\le r}$ as an edge of $E(C_{k-1},C_{k+1})$.
Moreover, from \Cref{obs: bad inded structure}, $|E^{\operatorname{over}}_{k-1}|\le |\hat E_{k-1}|$ and $|E^{\operatorname{over}}_{k}|\le |\hat E_{k}|$.
Altogether, we have
\begin{equation}
\begin{split}
\sum_{1\le i\le r}|E(G_i)| & =
\sum_{1\le i\le r}\textsf{left}(
|E(C_i)|+ |\hat E_i|+ |\hat E_{i-1}|+ |E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|+|E_i^{\textsf{thr}}|
\textsf{right})\\
& = \sum_{1\le i\le r}
|E(C_i)|+ \sum_{1\le i\le r} \textsf{left}(|E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|\textsf{right})+ \sum_{1\le i\le r}\textsf{left}(|E_i^{\textsf{thr}}|+ |\hat E_i|+ |\hat E_{i-1}|\textsf{right})\\
& \le |E(G)|+ 2\cdot |E(G)| + \sum_{1\le i\le r}\textsf{left}(|E(C_{i-1},C_{i+1})|+ 2|\hat E_i|+ 2|\hat E_{i-1}|\textsf{right})\\
& \le 8\cdot |E(G)|.
\end{split}
\end{equation}
This completes the proof of \Cref{obs: disengaged instance size}.
\end{proof}
Therefore, to complete the proof of \Cref{lem: disengagement with routing}, it suffices to prove the following claim.
\begin{claim}
\label{claim: existence of good solutions special}
$\expect{\sum_{1\le i\le r}\mathsf{OPT}_{\mathsf{cnwrs}}(G_i,\Sigma_i)}\leq O(\beta^2\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} ))$.
\end{claim}
The remainder of this subsection is dedicated to the proof of Claim~\ref{claim: existence of good solutions special}.
Let $\phi^*$ be an optimal drawing of the instance $(G,\Sigma)$.
We will construct, for each $1\le i\le r$, a solution $\phi_i$ to the instance $(G_i,\Sigma_i)$ using $\phi^*$, such that $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)\le O(\beta^2\cdot(\mathsf{cr}(\phi^*)+|E(G)|))$, and \Cref{claim: existence of good solutions special} then follows.
\paragraph{Constructing the sets of out-guiding paths.} Before constructing the drawings for subinstances $I_1,\ldots,I_r$, we first construct, for each $1\le i\le r-1$, a random set ${\mathcal{W}}'_i$ of out-guiding paths for cluster $U_i$, as follows.
Consider first the cluster $U_{r-1}=G\setminus C_r$. Recall that in \Cref{sec: guiding and auxiliary paths} we have obtained a random set ${\mathcal{Q}}_r$ of guiding paths in $C_r$, that route all edges of $\delta(C_r)$ to a vertex $u_r$ in $C_r$. We simply let ${\mathcal{W}}'_{r-1}={\mathcal{Q}}_r$. Consider now an index $i<r-1$ and the cluster $U_i$.
We construct the random set ${\mathcal{W}}'_i$ of guiding paths as follows.
Assume first that $C_{i+1}$ is a {helpful}\xspace cluster. For each edge $e\in \hat E_i$, we define $W'_i(e)=Q_{i+1}(e)$, the path in ${\mathcal{Q}}_{i+1}$ that routes $e$ to $u_{i+1}$ within $C_{i+1}$. For each edge $e\in E(U_i, \overline{U}_i)\setminus \hat E_i$, note that $e\in E'$, and we define path $W'_i(e)$ as the subpath of cycle $R_e$ between (including) the edge $e$ and vertex $u_{i+1}\in C_{i+1}$, that lies entirely within $\overline{U}_i$.
Assume now that $C_{i+1}$ is an {unhelpful}\xspace cluster. Similarly, for each edge $e\in \hat E_i$, we define $W'_i(e)=Q^{\operatorname{left}}_{i+1}(e)$, the path in ${\mathcal{Q}}_{i+1}^{\operatorname{left}}$ that routes edge $e$ to $u_{i+1}^{\operatorname{left}}$, and for each edge $e\in E(U_i, \overline{U}_i)\setminus \hat E_i$, note that $e\in E'$, and we define path $W'_i(e)$ as the subpath of cycle $R_e$ between (including) the edge $e$ and vertex $u^{\operatorname{left}}_{i+1}\in C_{i+1}$ that lies entirely within $\overline{U}_i$.
In both cases, wenote ${\mathcal{W}}'_i=\set{W'_i(e)\mid e\in E(U_i, \overline{U}_i)}$.
\paragraph{Drawings $\phi_2,\ldots,\phi_{r-1}$.}
First we fix some index $2\le i\le r-1$, and describe the construction of the drawing $\phi_i$.
Assume first that both $C_{i-1},C_{i+1}$ are {helpful}\xspace clusters.
The cases where $C_{i-1}$ or $C_{i+1}$ is {unhelpful}\xspace can be dealt with similarly.
Recall that $E(G_i)=E_G(C_i)\cup (\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$.
Also recall that we have defined a set ${\mathcal{W}}_{i-1}$ of paths that contains, for each edge $e\in E(U_{i-1},\overline{U}_{i-1})=\hat E_{i-1}\cup E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}$, a path $W_{i-1}(e)$ routing $e$ to $u_{i-1}$ in $U_{i-1}$, and a set ${\mathcal{W}}'_i$ of paths that contains, for each edge $e\in E(U_i,\overline{U}_i)=\hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$, a path $W'_i(e)$ routing $e$ to $u_{i+1}$ in $\overline{U}_i$.
Clearly, all paths in ${\mathcal{W}}_{i-1}$ and ${\mathcal{W}}'_{i}$ are internally disjoint from $C_i$. We will view $u_{i-1}$ as the last endpoint of all paths in ${\mathcal{W}}_{i-1}$ and view $u_{i+1}$ as the last endpoint of all paths in ${\mathcal{W}}'_{i}$.
We start with the image of cluster $C_i$ in $\phi^*$. We then add to it the image of vertices $v'_i, v^*_i$ and edges of $(\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$ as follows.
First, we apply the algorithm from \Cref{cor: new type 2 uncrossing} to graph $G$, drawing $\phi^*$ and the set ${\mathcal{W}}'_{i}$ of paths, and obtain a set $\Gamma'_i=\set{\gamma'_e\text{ }\big|\text{ } e\in \big(\hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}\big)}$ of curves.
Since all paths of ${\mathcal{W}}'_{i}$ share the last endpoint $u_{i+1}$, all curves in $\Gamma'_i$ also share the last endpoint, that we denote by $z^*_i$.
We denote by ${\mathcal{O}}^*_i$ the ordering in which curves of $\Gamma'_i$ enter $z^*_i$.
Similarly, we apply the algorithm from \Cref{thm: new nudging} to graph $G$, drawing $\phi^*$ and the set ${\mathcal{W}}_{i-1}$ of paths, and obtain a set $\Gamma_{i-1}=\set{\gamma_e\text{ }\big|\text{ } e\in \big(\hat E_{i-1}\cup E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\big)}$ of curves. Since all paths of ${\mathcal{W}}_{i-1}$ share the last endpoint $u_{i-1}$, all curves in $\Gamma_{i-1}$ also share the last endpoint, that we denote by $z'_i$.
Lastly, we apply the algorithm from \Cref{lem: ordering modification} to the set $\hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$ of edges (as elements), set $\Gamma'_i=\set{\gamma'_e\mid e\in \big(\hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}\big)}$ of curves, and the ordering $\tilde {\mathcal{O}}_{i}$. We rename the set of curves we obtain by $\Gamma'_i$.
We now view $z^*_i$ as the image of $v^*_i$, view $z'_i$ as the image of $v'_i$, and view (i) for each $e\in \hat E_{i}\cup E_i^{\operatorname{right}}$, the curve $\gamma'_e$ as the image of $e$; (ii) for each $e\in \hat E_{i-1}\cup E_i^{\operatorname{left}}$, the curve $\gamma_e$ as the image of $e$; and (iii) for each $e\in E_i^{\textsf{thr}}$, the concatenation of curves $\gamma_e$ and $\gamma'_e$ as the image of $e$. In this way, we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is easy to verify that $\phi_i$ respects the rotation system $\Sigma_i$.
\iffalse{previous rotation modification}
Let ${\mathcal{D}}$ be a tiny disc around $z^*_{i}$, and let ${\mathcal{D}}'$ be another small disc around $z^*_{i}$ that is strictly contained in ${\mathcal{D}}$.
We first erase the image of all edges of $\delta(v^*_i)$ inside the disc ${\mathcal{D}}$, and for each edge $e\in \delta(v^{*}_i)$, we denote by $p_{e}$ the intersection between the curve representing the current image of $e$ and the boundary of ${\mathcal{D}}$.
We then place, for each edge $e\in \delta(v^{*}_i)$, a point $p'_e$ on the boundary of ${\mathcal{D}}'$, such that the order in which the points in $\set{p'_e\mid e\in \delta(v^{*}_i)}$ appearing on the boundary of ${\mathcal{D}}'$ is precisely ${\mathcal{O}}^{*}_{i}$.
We then apply \Cref{lem: find reordering} to compute a set of reordering curves, connecting points of $\set{p_e\mid e\in \delta(v^{*}_i)}$ to points $\set{p'_e\mid e\in \delta(v^{*}_i)}$.
Finally, for each edge $e\in \delta(v^{*}_i)$, let $\zeta_e$ be the concatenation of (i) the current image of $e$ outside the disc ${\mathcal{D}}$; (ii) the reordering curve connecting $p_e$ to $p'_e$; and (iii) the straight line segment connecting $p'_e$ to $z^{*}_i$ in ${\mathcal{D}}'$.
We view $\zeta_e$ as the image of edge $e$, for each $e\in \delta(v^{*}_i)$.
We denote the resulting drawing of $G_i$ by $\phi_i$. It is clear that $\phi_i$ respects the rotation ${\mathcal{O}}^{*}_i$ at $v^{*}_i$, and therefore it respects the rotation system $\Sigma_i$.
\fi
We now upper bound the number of crossings in the drawing $\phi_i$ that we have constructed.
First, since we have not modified the drawing of $C_i$, the number of crossings between edges of $E_G(C_i)$ are bounded by $\chi^2(C_i)$.
Second, from the definition of curves of $\Gamma'_i$, the number of crossings between image of $C_i$ and the curves of $\Gamma'_i$ is at most $\chi({\mathcal{W}}'_i,C_i)$, and similarly the number of crossings between image of $C_i$ and the curves of $\Gamma_{i-1}$ is at most $\chi({\mathcal{W}}_{i-1},C_i)$.
Third, from \Cref{thm: new nudging}, the number of crossings between curves of $\Gamma_{i-1}$ is bounded by $\chi^2({\mathcal{W}}_{i-1})+\sum_{e\in E(G)}(\cong_G({\mathcal{W}}_{i-1},e))^2+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}_{i-1},\Sigma)$.
Fourth, the number of crossings between curves of $\Gamma_{i-1}$ and curves of $\Gamma'_{i}$ is at most $\chi({\mathcal{W}}'_i,{\mathcal{W}}_{i-1})$.
Lastly, from \Cref{lem: ordering modification}, the number of crossings in drawing $\phi_i$ inside the tiny $z^*_i$-disc is $O(\mbox{\sf dist}({\mathcal{O}}^*_i,\tilde{\mathcal{O}}_i))$. We now give an upper bound of $\mbox{\sf dist}({\mathcal{O}}^*_i,\tilde{\mathcal{O}}_i)$ as follows.
Recall that for each $e\in \hat E_{i}\cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$, $W_i(e)$ is the subpath of cycle $R_e$ between (including) the edge $e$ and vertex $u_i$, that lies entirely within $U_i$.
We apply the algorithm from \Cref{thm: new nudging} to graph $G$, drawing $\phi^*$, cluster $C_{i+1}$ and the set ${\mathcal{W}}_i$ of paths. Let $\Xi=\set{\xi_e\mid e\in E(U_i,\overline{U}_i)}$ be the set of curves we get.
From \Cref{thm: new nudging}, the curves of $\Xi$ enter $y_i=\phi^*(u_i)$ in the order $\tilde{\mathcal{O}}_i$, and we have
$\chi(\Xi)\le \chi^2({\mathcal{W}}_i)+\sum_{e\in E(G)}(\cong_G({\mathcal{W}}_i,e))^2+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}_i,\Sigma)$.
We define, for each edge $e\in E(U_i,\overline{U}_i)$, the curve $\zeta_e$ to be the union of $\xi_e$ and $\gamma'_e$. It is easy to verify that $\set{\zeta_e\mid e\in E(U_i,\overline{U}_i)}$ is a set of reordering curves for ${\mathcal{O}}^*_i$ and $\tilde{\mathcal{O}}_i$, and so
\[
\mbox{\sf dist}({\mathcal{O}}^*_i,\tilde{\mathcal{O}}_i)\le \chi\big(\set{\zeta_e\mid e\in E(U_i,\overline{U}_i)}\big)\le \chi^2({\mathcal{W}}_i)+\sum_{e\in E(G)}(\cong_G({\mathcal{W}}_i,e))^2+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}_i,\Sigma)+\chi({\mathcal{W}}_i,{\mathcal{W}}'_i).
\]
\iffalse{previous analysis of crossings}
\begin{claim}
\label{clm: rerouting_crossings}
The number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\end{claim}
\begin{proof}
Denote by ${\mathcal{O}}^*$ the ordering in which the curves $\set{\gamma'_{W_e}\mid e\in \delta_{G_i}(v_i^{\operatorname{right}})}$ enter $z_{\operatorname{right}}$, the image of $u_{i+1}$ in $\phi'_i$.
From~\Cref{lem: find reordering} and the algorithm in Step 4 of modifying the drawing within the disc ${\mathcal{D}}$, the number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is at most $O(\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}}))$. Therefore, it suffices to show that $\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}})=O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
In fact, we will compute a set of curves connecting the image of $u_i$ and the image of $u_{i+1}$ in $\phi^*_i$, such that each curve is indexed by some edge $e\in\delta_{G_i}(v_i^{\operatorname{right}})$ these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$ and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$, and the number of crossings between curves of $Z$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
For each $e\in E^{\textsf{thr}}_i$, we denote by $\eta_e$ the curve obtained by taking the union of (i) the curve $\gamma'_{W_e}$ (that connects $u_{i+1}$ to $u_{i-1}$); and (ii) the curve representing the image of the subpath of $P_e$ in $\phi^*_i$ between $u_i$ and $u_{i-1}$.
Therefore, the curve $\eta_e$ connects $u_i$ to $u_{i+1}$.
We then modify the curves of $\set{\eta_e\mid e\in E^{\textsf{thr}}_i}$, by iteratively applying the algorithm from \Cref{obs: curve_manipulation} to these curves at the image of each vertex of $C_{i-1}\cup C_{i+1}$. Let $\set{\zeta_e\mid e\in E^{\textsf{thr}}_i}$ be the set of curves that we obtain. We call the obtained curves \emph{red curves}. From~\Cref{obs: curve_manipulation}, the red curves are in general position. Moreover, it is easy to verify that the number of intersections between the red curves is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1}))$.
We call the curves in $\set{\gamma'_{W_e}\mid e\in \hat E_i}$ \emph{yellow curves}, call the curves in $\set{\gamma'_{W_e}\mid e\in E^{\operatorname{right}}_i}$ \emph{green curves}. See \Cref{fig: uncrossing_to_bound_crossings} for an illustration.
From the construction of red, yellow and green curves, we know that these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$, and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$.
Moreover, we are guaranteed that the number of intersections between red, yellow and green curves is at most $\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/uncross_to_bound_crossings.jpg}
\caption{An illustration of red, yellow and green curves.}\label{fig: uncrossing_to_bound_crossings}
\end{figure}
\end{proof}
\fi
From the above discussion, for each $2\le i\le r-1$,
\begin{equation*}
\begin{split}
\mathsf{cr}(\phi_i)= & \text{ }\chi^2(C_i)+\chi({\mathcal{W}}'_i\cup {\mathcal{W}}_{i-1},C_i)+\chi^2({\mathcal{W}}_{i-1})+\sum_{e\in E(G)}(\cong_G({\mathcal{W}}_{i-1},e))^2+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}_{i-1},\Sigma)\\
& + O\bigg(\chi^2({\mathcal{W}}_{i})+\sum_{e\in E(G)}(\cong_G({\mathcal{W}}_{i},e))^2+\mathsf{cost}_{\mathsf {NT}}({\mathcal{W}}_{i},\Sigma)+\chi({\mathcal{W}}_i,{\mathcal{W}}'_i)\bigg).
\end{split}
\end{equation*}
\paragraph{Drawings $\phi_1$ and $\phi_{r}$.}
The drawings $\phi_1$ and $\phi_{r}$ are constructed similarly. We only describe the construction of $\phi_1$, and the construction of $\phi_r$ is symmetric.
Recall that graph $G_1$ contains only one super-node $v_1^{*}$, that is obtained by contracting $\overline{U}_1$, and $\delta_{G_1}(v_1^{*})=\hat E_1\cup E^{\operatorname{right}}_1$.
We construct the set ${\mathcal{W}}'_1$ of out-guiding paths in the same way as sets ${\mathcal{W}}'_2,\ldots,{\mathcal{W}}'_{r-1}$. Then we construct the drawing $\phi_1$ using the algorithm from \Cref{thm: cost of single cluster}.
From \Cref{obs: new crossings},
$$\mathsf{cr}(\phi_1)\le \chi^2(C_1)+ \chi({\mathcal{W}}'_1,C_1)+\chi({\mathcal{W}}'_1,{\mathcal{Q}}_1)+\chi^2({\mathcal{Q}}_1)+\sum_{e\in E(C)}\cong({\mathcal{Q}}_1,e)^2.$$
Similarly, the drawing $\phi_r$ that we obtained in the same way satisfies that
\[\mathsf{cr}(\phi_r)\le \chi^2(C_r)+ \chi({\mathcal{W}}_{r-1},C_r)+\chi({\mathcal{W}}_{r-1},{\mathcal{Q}}_r)+\chi^2({\mathcal{Q}}_r)+\sum_{e\in E(C)}\cong({\mathcal{Q}}_r,e)^2.\]
\iffalse{previous construction of phi_1}
We start with the drawing of $C_1\cup E({\mathcal{W}}_1)$ induced by $\phi^*$, that we denote by $\phi^*_1$. We will not modify the image of $C_i$ in $\phi^*_i$ and will construct the image of edges in $\delta(v_1^{\operatorname{right}})$.
We perform similar steps as in the construction of drawings $\phi_2,\ldots,\phi_{r-1}$.
We first construct, for each path $W\in {\mathcal{W}}_1$, a curve $\gamma_W$ connecting its endpoint in $C_1$ to the image of $u_2$ in $\phi^*$, as in Step 1.
Let $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
We then process all intersections between curves of $\Gamma_1$ as in Step 2. Let $\Gamma'_1=\set{\gamma'_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
Since $\Gamma^{\textsf{thr}}_1=\emptyset$, we do not need to perform Steps 3 and 4. If we view the image of $u_2$ in $\phi^*_1$ as the image of $v^{\operatorname{right}}_1$, and for each edge $e\in \delta(v^{\operatorname{right}}_1)$, we view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is clear that this drawing respects the rotation system $\Sigma_1$. Moreover,
\[\mathsf{cr}(\phi_1)=\chi^2(C_1)+O\textsf{left}(\hat\chi_1({\mathcal{Q}}_2)+\sum_{W\in {\mathcal{W}}_1}\hat\chi_1(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_1,e)^2\textsf{right}).\]
Similarly, the drawing $\phi_k$ that we obtained in the similar way satisfies that
\[\mathsf{cr}(\phi_k)=\chi^2(C_k)+O\textsf{left}(\hat\chi_k({\mathcal{Q}}_{r-1})+\sum_{W\in {\mathcal{W}}_k}\hat\chi_k(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_k,e)^2\textsf{right}).\]
\fi
We now complete the proof of \Cref{claim: existence of good solutions special}, for which it suffices to estimate $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
First note that $\sum_{1\le i\le r}\chi^2(C_i)\le O(\mathsf{cr}(\phi^*))$.
We then define ${\mathcal{W}}=\bigcup_{1\le i\le r}({\mathcal{W}}_i\cup{\mathcal{W}}'_i)$. It is easy to verify that, for each edge $e\in E(G)$, $\expect[]{(\cong_G({\mathcal{W}},e))^2}\le O(\beta)$. Therefore, on the one hand,
\[
\sum_{1\le i\le r}\sum_{e\in E(G)}\expect[]{\cong_G({\mathcal{W}}_i,e)^2+\cong_G({\mathcal{W}}'_i,e)^2}
\le \sum_{e\in E(G)} \expect[]{\cong_G({\mathcal{W}},e)^2}\le O\big(|E(G)|\cdot\beta\big),
\]
and on the other hand,
\[
\begin{split}
\sum_{1\le i\le r}\expect{\chi^2({\mathcal{W}}_i)+\chi({\mathcal{W}}_i,{\mathcal{W}}'_i)} & = \text{ } \expect{\sum_{e,e'\in E(G)}\chi(e,e')\cdot\cong_{G}({\mathcal{W}},e)\cdot\cong_{G}({\mathcal{W}},e')}\\
& \text{ } \le \expect{\sum_{e,e'\in E(G)}\chi(e,e')\cdot\frac{\cong_{G}({\mathcal{W}},e)^2+\cong_{G}({\mathcal{W}},e')^2}{2}}\\
& \text{ } \le \expect{\sum_{e\in E(G)}\chi(e,G)\cdot\frac{\cong_{G}({\mathcal{W}},e)^2}{2}}\le O(\beta\cdot \mathsf{cr}(\phi^*)).
\end{split}\]
Moreover, $\sum_{1\le i\le r}\expect{\chi({\mathcal{W}}'_i,C_i)+\chi({\mathcal{W}}_{i-1},C_i)} \le \sum_{1\le i\le r}\expect{\chi({\mathcal{W}},C_i)} \le O(\beta\cdot\mathsf{cr}(\phi^*))$.
From similar arguments, we can show that
$\sum_{1\le i\le r}\sum_{e\in E(C_i)}\expect{\cong_G({\mathcal{Q}}_i,e)^2} \le O(\beta\cdot |E(G)|)$ and
$\sum_{1\le i\le r}\expect{\chi^2({\mathcal{Q}}_i)} \le O(\beta\cdot\mathsf{cr}(\phi^*))$.
Altogether, they imply that $\sum_{1\le i\le r}\mathsf{cr}(\phi_i) \le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot\beta)$.
This completes the proof of \Cref{claim: existence of good solutions special}.
\iffalse{previous bad-chain centered setting assumptions}
\paragraph{Assumption 1: The Gomory-Hu tree $\tau$ of graph $H$ is a path.}
Let $H=G_{\mid{\mathcal{C}}}$ be the contracted graph.
We denote the clusters in ${\mathcal{C}}$ by $C_1,C_2,\ldots,C_r$. For convenience, for each $1\leq i\leq r$, we denote by $x_i$ the vertex of graph $H$ that represents the cluster $C_i$. Throughout this subsection, we assume that the Gomory-Hu tree $\tau$ of graph $H$ is a path, and we assume without loss of generality that the clusters are indexed according to their appearance on the path $\tau$. Note that each edge in $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ corresponds to an edge in $H$, and we do not distinguish between them.
\paragraph{Assumption 2: Bad Chains of Single-Vertex Clusters.}
We allow some clusters of ${\mathcal{C}}\setminus\set{C_1,C_r}$ to contain only a single vertex. Let indices $1< i\le j< n$ be such that clusters $C_{i-1},C_{j+1}$ are not single-vertex clusters, while all clusters $C_{i},\ldots,C_{j}$ are single-vertex clusters.
We call the union of clusters $C_{i},\ldots,C_{j}$ a \emph{bad chain}, and we denote $\mathsf{BC}[i,j]=\bigcup_{i\le t\le j}C_t$. We call index $i$ the \emph{left endpoint} and index $j$ the \emph{right endpoint} of the bad chain.
Consider the graph $H=G_{\mid{\mathcal{C}}}$. Note that each bad chain $\mathsf{BC}[i,j]$ belongs to graph $H$. Consider a bad chain $\mathsf{BC}[i,j]$, we say that a path $P$ is contained in the bad chain, iff $V(P)\subseteq \mathsf{BC}[i,j]$.
Let $x_s, x_t$ be two vertices (with $s<t$) that do not belong to any bad chains, and let $\mathsf{BC}[i_1,j_1],\ldots,\mathsf{BC}[i_k,j_k]$ be all bad chains whose endpoints lie in $\set{s+1,\ldots,t-1}$. Assume without loss of generality that $i_1\le j_1< i_2\le j_2 <\ldots <i_k\le j_k$.
We say that a path connecting $x_{s}$ to $x_{t}$ is \emph{nice}, iff it is the sequential concatenation of: a path that sequentially visits vertices $x_s,x_{s+1}\ldots,x_{i_1}$, a path connecting $x_{i_1}$ to $x_{j_1}$ that is contained in the bad chain $\mathsf{BC}[i_1,j_1]$, a path that sequentially visits vertices $x_{j_1},x_{j_1+1}\ldots,x_{i_2}$, a path connecting $x_{i_2}$ to $x_{j_2}$ that is contained in the bad chain $\mathsf{BC}[i_2,j_2]$, \ldots , and a path that sequentially visits vertices $x_{j_k},x_{j_k+1}\ldots,x_{t}$.
See \Cref{fig: nice path} for an illustration.
If vertex $x_s$ belongs to some bad chain $\mathsf{BC}[i_0,j_0]$, then for an $x_s$-$x_t$ path to be nice, the path must start with a path connecting $x_s$ to $x_{j_0}$ that is contained in the bad chain $\mathsf{BC}[i_0,j_0]$.
Similarly, if vertex $x_t$ belongs to some bad chain $\mathsf{BC}[i_{k+1},j_{k+1}]$, then a nice $x_s$-$x_t$ path must end with a path connecting $x_{i_{k+1}}$ to $x_{t}$ that is contained in the bad chain $\mathsf{BC}[i_{k+1},j_{k+1}]$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.23]{figs/innerpath.jpg}
\caption{An illustration of a nice path connecting $x_s$ to $x_t$. Bad chains are shown in pink.}\label{fig: nice path}
\end{figure}
\paragraph{Assumption 3: Auxiliary Paths.}
Let $E'$ be the subset of edges in $E^{\mathsf{out}}({\mathcal{C}})$ that contains all edges connecting a pair $C_i,C_j$ of clusters in ${\mathcal{C}}$, with $j\ge i+2$, and when $E'$ is viewed as a set of edges in $H$, then $E'$ contains all edges connecting a pair $x_i,x_j$ of vertices in $H$ with $j\ge i+2$.
We additionally assume that, for each edge $e\in E'$ that, when viewed as an edge in $H$, connects $x_i$ to $x_j$, we are given a nice path $P_e$ in $H$ connecting $x_i$ to $x_j$, that we call the \emph{auxiliary path} of $e$.
Additionally, all paths in $\set{P_e}_{e\in E'}$ cause congestion $O(1)$ in graph $H$.
We now start to describe the algorithm in the following subsections.
\subsubsection{Computing Guiding Paths and Auxiliary Cycles}
\label{sec: guiding and auxiliary paths}
We first apply the algorithm from \Cref{thm:algclassifycluster} to each non-single-vertex cluster of ${\mathcal{C}}$.
In particular, for each non-single-vertex cluster $C_i$, we apply the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace to the instance $(G,\Sigma)$ and the cluster $C_i$ (recall that cluster $C_i$ has the $\alpha_0$-bandwidth property, where $\alpha_0=1/\log^3m$) for $100\log n$ times. If any application of \ensuremath{\mathsf{AlgClassifyCluster}}\xspace returns a distribution ${\mathcal{D}}(C_i)$ over sets of guiding paths in $\Lambda(C_i)$, such that $C_i$ is $\beta^*$-light with respect to ${\mathcal{D}}(C_i)$, then we continue to sample a set of guiding paths from ${\mathcal{D}}(C_i)$. Let ${\mathcal{Q}}_i$ be the set we obtain, and let $u_i$ be the vertex of $C_i$ that serves as the common endpoint of all paths in ${\mathcal{Q}}_i$. If all applications of \ensuremath{\mathsf{AlgClassifyCluster}}\xspace return FAIL, then consider the graph $C_i^+$ (see \Cref{def: Graph C^+}). We apply the algorithm from \Cref{lem: simple guiding paths} to cluster $C^+_i$ and and set $T(C_i)$, and let $u_i$ be the vertex and ${\mathcal{Q}}_i$ be the set of paths that we obtain. Clearly, ${\mathcal{Q}}_i$ can be also viewed as a set of paths routing edges of $\delta_G(C_i)$ to $u_i$. From \Cref{lem: simple guiding paths}, for each edge $e\in E(C_i)$, $\expect{\cong({\mathcal{Q}}_i,e)}\leq O(\log^4m/\alpha_0)$. Finally, for each $1\le i\le r$ and for each $e\in \delta(C_i)$, we denote by $Q_i(e)$ the path of ${\mathcal{Q}}_i$ that routes $e$ to $u_i$.
We then apply the algorithm from \Cref{lem: non_interfering_paths} to each set ${\mathcal{Q}}_i$ of paths and the rotation system $\Sigma$, and rename the sets of paths we obtain as ${\mathcal{Q}}_1,\ldots,{\mathcal{Q}}_r$. Now we are guaranteed that, for each $1\le i\le r$, the set ${\mathcal{Q}}_i$ of paths is non-transversal with respect to $\Sigma$.
\textbf{Bad Event $\xi_i$.} For each index $1\le i\le r$, we say that the event $\xi_i$ happens, iff $C_i$ is a non-single-vertex cluster, $C_i$ is not an $\eta^*$-bad cluster, and all the $100\log n$ applications of \ensuremath{\mathsf{AlgClassifyCluster}}\xspace return FAIL.
From \Cref{thm:algclassifycluster}, $\Pr[\xi_i]\le (1/2)^{100\log n}=O(n^{-50})$. Then from the union bound over all $1\le i\le r$, $\Pr[\bigcup_{1\le i\le r}\xi_i]\le O(n^{-49})$.
We then compute a set ${\mathcal{R}}^*$ of cycles in $G$, using the set $\set{P_e}_{e\in E'}$ of inner paths and the path sets $\set{{\mathcal{Q}}_i}_{1\le i\le r}$, as follows.
Consider an edge $e\in E'$ and its inner path $P_e$ in $H$.
We denote $P_e=(e_1,\ldots,e_k)$, and for each $1\le j\le k$, we denote $e_j=(x_{t_{j-1}}, x_{t_{j}})$ (and so $e=(x_{t_0},x_{t_k})$ as an edge in $H$).
Recall that the set ${\mathcal{Q}}_i$ contains, for each edge $e\in\delta(C_i)$, a path $Q_i(e)$ routing edge $e$ to $u_i$.
We then define path $P^*_e$ as the sequential concatenation of paths $Q_{t_{0}}(e_1),Q_{t_{1}}(e_1), Q_{t_{1}}(e_2),Q_{t_{2}}(e_2),\ldots, Q_{t_{r-1}}(e_k),Q_{t_{r}}(e_k)$.
It is clear that path $P^*_e$ sequentially visits the vertices $u_{t_0},u_{t_1},\ldots,u_{t_k}$ in $G$.
Since edge $e$, as an edge of $G$, connects a vertex of $C_{t_0}$ to a vertex of $C_{t_k}$. We then define path $P^{**}_e$ to be the union of paths $Q_{t_{k}}(e)$ and $Q_{t_{0}}(e)$, so path $P^{**}_e$ connects $u_{t_k}$ to $u_{t_0}$.
Finally, we define cycle $R^*_e$ to be the union of paths $P^*_e$ and paths $P^{**}_e$, and we denote ${\mathcal{R}}^*=\set{R^*_e\mid e\in E'}$.
\iffalse
\begin{figure}[h]
\centering
\includegraphics[scale=0.24]{figs/auxiliary_path.jpg}
\caption{An illustration of the auxiliary cycle $R^*_e$ of an edge $e$ (solid red): left auxiliary path shown in orange, middle auxiliary path shown in purple and right auxiliary path shown in green.}\label{fig: LMR_auxi}
\end{figure}
\fi
Recall that the set $\set{P_e}_{e\in E'}$ of inner paths is guaranteed to cause congestion at most $O(1)$ in graph $H$, then from the construction of cycles in ${\mathcal{R}}^*$, it is easy to see that, for each edge $e\in E'$, $\cong_{G}({\mathcal{R}}^*,e)\le O(1)$, and
for each edge $e\in \bigcup_{1\le i\le r}E(C_i)$, $\expect[]{\cong_{G}({\mathcal{R}}^*,e)^2}\le O(\beta^*)$.
\iffalse{should be moved to the next subsection}
We further modify the cycles in ${\mathcal{R}}^*$ to obtain a new set ${\mathcal{R}}$ of cycles, such that the intersection of every pair of cycles in ${\mathcal{R}}$ is non-transversal with respect to $\Sigma$ at no more than one of their shared vertices.
We will use the following lemma.
\znote{To complete the following lemma}
\begin{lemma}[De-interfering]
There is an efficient algorithm, that given a graph $G$, a rotation system $\Sigma$, and a set ${\mathcal{P}}$ of paths that share an endpoint $u$, computes a set ${\mathcal{P}}'=\set{P'\mid P\in {\mathcal{P}}}$ of paths, such that
\end{lemma}
\begin{proof}
Let $R,R'$ be a pair of cycles in ${\mathcal{R}}^*$ and let $v$ be a shared vertex of $R$ and $R'$. We say that the tuple $(R,R',v)$ is \emph{bad}, iff the intersection of cycles $R,R'$ at $v$ is transversal with respect to $\Sigma$. We iteratively process the set ${\mathcal{R}}^*$ of cycles as follows. While there exist a pair $R,R'$ of cycles in ${\mathcal{R}}^*$ and two vertices $v_1,v_2\in V(R)\cap V(R')$, such that the tuples $(R,R',v)$ and $(R,R',v)$ are bad, we process cycles $R,R'$ as follows.
Eventually, for each $e\in E'$, we obtain a cycle $R_e$ in $G$ that is called the \emph{auxiliary cycle} of edge $e$.
\end{proof}
\fi
\znote{Todo: de-interfere auxiliary cycles to get cycles $\set{R_e}_{e\in E'}$.}
\subsubsection{Constructing Sub-Instances}
\label{sec: compute advanced disengagement}
In this step we will construct, for each non-single vertex cluster $C_i\in {\mathcal{C}}$, a sub-instance $I_i=(G_i,\Sigma_i)$ of $(G,\Sigma)$, and for each bad chain $\mathsf{BC}[i,j]$, a sub-instance $I_{\mathsf{BC}[i,j]}$ of $(G,\Sigma)$, such that the set of all created sub-instances satisfy the properties in \Cref{thm: disengagement - main}.
For each index $1\le i\le r$, we define edge sets
$\hat E_i=E(C_i,C_{i+1})$, $E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$, and $E^{\textsf{thr}}_i=\bigcup_{i'\le i-1,j'\ge i+1}E(C_{i'},C_{j'})$.
Let $\mathsf{BC}[i,j]$ be a bad chain. Similarly we define edge sets
$E_{[i,j]}^{\operatorname{right}}=\bigcup_{k>j+1}E(\mathsf{BC}[i,j],C_k)$, $E_{[i,j]}^{\operatorname{left}}=\bigcup_{k<i-1}E(\mathsf{BC}[i,j],C_k)$,
$\hat E_{[i,j]}^{\operatorname{right}}=E(\mathsf{BC}[i,j],C_{j+1})$, $\hat E_{[i,j]}^{\operatorname{left}}=E(\mathsf{BC}[i,j],C_{i-1})$, and $E^{\textsf{thr}}_{[i,j]}=\bigcup_{i'\le i-1,j'\ge j+1}E(C_{i'},C_{j'})$.
\paragraph{Cluster-based Instance $I_i=(G_i,\Sigma_i)$ ($2\le i\le r$).}
Let $C_i$ be some non-single-vertex cluster with $2\le i\le r$.
We define the instance $I_i=(G_i,\Sigma_i)$ as follows.
The graph $G_i$ is obtained from $G$ by first contracting clusters $C_1,\ldots,C_{i-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_i$, and then contracting clusters $C_{i+1},\ldots,C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_i$, and finally deleting self-loops on the super-nodes $v^{\operatorname{left}}_i$ and $v^{\operatorname{right}}_i$. So $V(G_i)=V(C_i)\cup \set{ v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$. See \Cref{fig: disengaged instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edge sets in $G$. Edges of $E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}$ are shown in green, and edges of $E^{\textsf{thr}}_{i}$ are shown in red. ]{\scalebox{0.35}{\includegraphics{figs/disengaged_instance_1.jpg}
}
\hspace{5pt}
\subfigure[Graph $G_i$. $\delta_{G_i}(v_i^{\operatorname{right}})=\hat E_i \cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, and $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1} \cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_{i}$.]{
\scalebox{0.28}{\includegraphics{figs/disengaged_instance_2.jpg}}}
\caption{An illustration of the construction of sub-instance $(G_i,\Sigma_i)$.}\label{fig: disengaged instance}
\end{figure}
We now define the orderings in $\Sigma_i$. First, for each vertex $v\in V(C_i)$, the ordering on its incident edges is defined to be ${\mathcal{O}}_v$, the rotation on vertex $v$ in the given rotation system $\Sigma$.
It remains to define the rotations of super-nodes $v^{\operatorname{left}}_i,v^{\operatorname{right}}_i$.
We first consider $v^{\operatorname{left}}_i$.
Note that $\delta_{G_i}(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$.
For each edge $\hat e\in \hat E_{i-1}$, recall that $Q_{i-1}(\hat e)$ is the path in $C_{i-1}$ routing edge $\hat e$ to $u_{i-1}$.
For each edge $e\in E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i$, we denote by $W^{\operatorname{left}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i-1}$ and does not contain $u_i$.
We then denote
$${\mathcal{W}}^{\operatorname{left}}_i=\set{W^{\operatorname{left}}_i(e)\mid e\in E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i}\cup \set{Q_{i-1}(\hat e)\mid \hat e\in \hat E_{i-1}}.$$
Intuitively, the rotation on vertex $v^{\operatorname{left}}_i$ is defined to be the ordering in which the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ enter $u_{i-1}$.
Formally, for every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$, let $e^*_W$ be the unique edge of path $W$ that is incident to $u_{i-1}$. We first define a circular ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$, as follows: the paths are ordered according to the circular ordering of their corresponding edges $e^*_W$ in ${\mathcal{O}}_{u_{i-1}}\in \Sigma$, breaking ties arbitrarily. Since every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$ is associated with a unique edge in $\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$, this ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ immediately defines a circular ordering of the edges of $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$, that we denote by ${\mathcal{O}}^{\operatorname{left}}_i$. See Figure~\ref{fig: v_left rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i-1}=\set{\hat e_1,\ldots,\hat e_4}$, $E^{\operatorname{left}}_{i}=\set{e^g_1,e^g_2}$ and $E^{\textsf{thr}}_{i}=\set{e^r_1,e^r_2}$.
Paths of ${\mathcal{W}}^{\operatorname{left}}_i$ excluding their first edges are shown in dash lines.]{\scalebox{0.13}{\includegraphics{figs/rotation_left_1.jpg}}}
\hspace{3pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{left}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. Set $\delta_{G_i}(v^{\operatorname{left}}_i)=\set{\hat e_1,\hat e_2,\hat e_3,\hat e_4,e^g_1, e^g_2, e^r_1,e^r_2}$. The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on them around $v^{\operatorname{left}}_i$ is shown above.]{
\scalebox{0.16}{\includegraphics{figs/rotation_left_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_left rotation}
\end{figure}
The rotation ${\mathcal{O}}^{\operatorname{right}}_{i}$ on vertex $v^{\operatorname{right}}_i$ is defined similarly.
Note that $\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$.
For each edge $\hat e'\in \hat E_{i}\cup E_i^{\operatorname{right}}$, recall that $Q_{i}(\hat e')$ is the path in $C_{i}$ routing edge $\hat e'$ to vertex $u_{i}$.
For each edge $e\in E^{\textsf{thr}}_i$, we denote by $W^{\operatorname{right}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i}$ and contains $u_{i-1}$.
We then denote
$${\mathcal{W}}^{\operatorname{right}}_i=\set{W^{\operatorname{right}}_i(e)\text{ }\big|\text{ } e\in E^{\textsf{thr}}_i}\cup \set{Q_{i}(\hat e')\text{ }\big|\text{ } \hat e'\in \hat E_{i}\cup E^{\operatorname{right}}_i}.$$
The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ is then defined in a similar way as the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$, according to the paths of ${\mathcal{W}}^{\operatorname{right}}_i$ and the rotation ${\mathcal{O}}_{u_{i}}\in \Sigma$.
See Figure~\ref{fig: v_right rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i}=\set{\hat e_1',\ldots,\hat e_4'}$, $E^{\operatorname{right}}_{i}=\set{\tilde e^g_1,\tilde e^g_2}$ and $E^{\textsf{thr}}_{i}=\set{\tilde e^r_1,\tilde e^r_2}$.
Paths of ${\mathcal{W}}^{\operatorname{right}}_i$ excluding their first edges are shown in dash lines. ]{\scalebox{0.13}{\includegraphics{figs/rotation_right_1.jpg}
}
\hspace{3pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. Set $\delta_{G_i}(v^{\operatorname{right}}_i)=\set{\hat e_1',\hat e_2',\hat e_3',\hat e_4',\tilde e^g_1,\tilde e^g_2,\tilde e^r_1,\tilde e^r_2}$. The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on them around $v^{\operatorname{right}}_i$ is shown above.]{
\scalebox{0.17}{\includegraphics{figs/rotation_right_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_right rotation}
\end{figure}
\paragraph{Instances $I_1=(G_1,\Sigma_1)$ and $I_r=(G_r,\Sigma_r)$.}
The instances $(G_1,\Sigma_1)$ and $(G_r,\Sigma_r)$ are defined similarly to a cluster-centered instance, but instead of two super-nodes, the graphs $G_1$ and $G_r$ contain one super-node each.
In particular, graph $G_1$ is obtained from $G$ by contracting clusters $C_2,\ldots, C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_1$, and then deleting self-loops on it.
So $V(G_1)=V(C_1)\cup \set{v^{\operatorname{right}}_{1}}$ and $\delta_{G_1}(v^{\operatorname{right}}_{1})=\hat E_1\cup E^{\operatorname{right}}_1$.
The rotation of a vertex $v\in V(C_1)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{right}}_1$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{right}}_i$ for any index $2\le i\le r-1$.
Graph $G_r$ is obtained from $G$ by contracting clusters $C_1,\ldots, C_{r-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_r$, and then deleting self-loops on it.
So $V(G_r)=V(C_r)\cup \set{v^{\operatorname{left}}_{r}}$ and $\delta_{G_r}(v^{\operatorname{left}}_{r})=\hat E_{r-1}\cup E^{\operatorname{left}}_r$.
The rotation of a vertex $v\in V(C_r)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{left}}_r$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{left}}_i$ for any index $2\le i\le r-1$.
\paragraph{Bad Chain-based Instance $I_{\mathsf{BC}[i,j]}$.} Let $\mathsf{BC}[i,j]$ be a bad chain. We define the instance $I_{\mathsf{BC}[i,j]}=(G_{[i,j]},\Sigma_{[i,j]})$ as follows.
The graph $G_{[i,j]}$ is obtained from $G$ by first contracting clusters $C_1,\ldots,C_{i-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_{[i,j]}$, and then contracting clusters $C_{j+1},\ldots,C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_{[i,j]}$, and finally deleting self-loops on the super-nodes $v^{\operatorname{left}}_{[i,j]}$ and $v^{\operatorname{right}}_{[i,j]}$. So $V(G_i)=\mathsf{BC}[i,j]\cup \set{v^{\operatorname{left}}_{[i,j]},v^{\operatorname{right}}_{[i,j]}}$. See \Cref{fig: bad instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Edges of $\hat E^{\operatorname{left}}_{[i,j]}\cup \hat E^{\operatorname{right}}_{[i,j]}$ are in orange, edges of $E^{\operatorname{left}}_{[i,j]}$ and $E^{\operatorname{right}}_{[i,j]}$ are in green and blue respectively and edges of $E^{\textsf{thr}}_{[i,j]}$ are in red. ]{\scalebox{0.12}{\includegraphics{figs/badchain_instance_1.jpg}
}
\hspace{0pt}
\subfigure[Graph $G_{[i,j]}$. $\delta(v_{[i,j]}^{\operatorname{right}})=\hat E_{[i,j]}^{\operatorname{right}} \cup E_{[i,j]}^{\operatorname{right}} \cup E^{\textsf{thr}}_{[i,j]}$, and $\delta(v_{[i,j]}^{\operatorname{left}})=\hat E_{[i,j]}^{\operatorname{left}} \cup E_{[i,j]}^{\operatorname{left}} \cup E^{\textsf{thr}}_{[i,j]}$.]{
\scalebox{0.14}{\includegraphics{figs/badchain_instance_2.jpg}}}
\caption{An illustration of the construction of sub-instance $(G_{[i,j]},\Sigma_{[i,j]})$.}\label{fig: bad instance}
\end{figure}
We now define the orderings in $\Sigma_i$. First, for each vertex $v\in \mathsf{BC}[i,j]$, the ordering on its incident edges is defined to be ${\mathcal{O}}_v$, the rotation on vertex $v$ in the given rotation system $\Sigma$.
It remains to define the rotations of super-nodes $v^{\operatorname{left}}_{[i,j]},v^{\operatorname{right}}_{[i,j]}$.
We first consider $v^{\operatorname{left}}_{[i,j]}$.
Note that $\delta_{G_i}(v^{\operatorname{left}}_{[i,j]})=\hat E^{\operatorname{left}}_{[i,j]}\cup E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$.
For each edge $\hat e\in \hat E^{\operatorname{left}}_{[i,j]}$, recall that $Q_{i-1}(\hat e)$ is the path in $C_{i-1}$ routing edge $\hat e$ to $u_{i-1}$.
For each edge $e\in E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$, we denote by $W^{\operatorname{left}}_{[i,j]}(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i-1}$ and does not contain $u_i$.
We then denote
$${\mathcal{W}}^{\operatorname{left}}_i=\set{W^{\operatorname{left}}_i(e)\mid e\in E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}}\cup \set{Q_{i-1}(\hat e)\mid \hat e\in \hat E^{\operatorname{left}}_{[i,j]}}.$$
Intuitively, the rotation on vertex $v^{\operatorname{left}}_{[i,j]}$ is defined to be the ordering in which the paths in ${\mathcal{W}}^{\operatorname{left}}_{[i,j]}$ enter $u_{i-1}$.
Formally, for every path $W\in {\mathcal{W}}^{\operatorname{left}}_{[i,j]}$, let $e^*_W$ be the unique edge of path $W$ that is incident to $u_{i-1}$. We first define a circular ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_{[i,j]}$, as follows: the paths are ordered according to the circular ordering of their corresponding edges $e^*_W$ in ${\mathcal{O}}_{u_{i-1}}\in \Sigma$, breaking ties arbitrarily. Since every path $W\in {\mathcal{W}}^{\operatorname{left}}_{[i,j]}$ is associated with a unique edge in $\hat E^{\operatorname{left}}_{[i,j]}\cup E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$, this ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_{[i,j]}$ immediately defines a circular ordering of the edges of $\delta_{G_{[i,j]}}(v_{[i,j]}^{\operatorname{left}})=\hat E^{\operatorname{left}}_{[i,j]}\cup E^{\operatorname{left}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$, that we denote by ${\mathcal{O}}^{\operatorname{left}}_{[i,j]}$. See Figure~\ref{fig: bad_v_left rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[$\hat E^{\operatorname{left}}_{[i,j]}=\set{\hat e_1,\hat e_2}$, $E^{\operatorname{left}}_{[i,j]}=\set{e^g_1}$ and $E^{\textsf{thr}}_{[i,j]}=\set{e^r_1}$.
Paths of ${\mathcal{W}}^{\operatorname{left}}_{[i,j]}$ excluding their first edges are shown in dash lines.]{\scalebox{0.14}{\includegraphics{figs/badinstance_rotation_left_1.jpg}}}
\hspace{3pt}
\subfigure[$\delta(v^{\operatorname{left}}_{[i,j]})=\set{\hat e_1,\hat e_2,e^g_1,e^r_1}$. The rotation ${\mathcal{O}}^{\operatorname{left}}_{[i,j]}$ on them around $v^{\operatorname{left}}_{[i,j]}$ is shown above.]{
\scalebox{0.18}{\includegraphics{figs/badinstance_rotation_left_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{left}}_{[i,j]}$ on vertex $v^{\operatorname{left}}_{[i,j]}$ in the instance $(G_{[i,j]}, \Sigma_{[i,j]})$.}\label{fig: bad_v_left rotation}
\end{figure}
The rotation ${\mathcal{O}}^{\operatorname{right}}_{[i,j]}$ on vertex $v^{\operatorname{right}}_{[i,j]}$ is defined similarly.
Note that $\delta_{G_i}(v^{\operatorname{right}}_{[i,j]})=\hat E^{\operatorname{right}}_{[i,j]}\cup E^{\operatorname{right}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}$.
For each edge $\hat e'\in \delta_{G_i}(v^{\operatorname{right}}_{[i,j]})$, we denote by $W^{\operatorname{right}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{j}$ and does not contains $u_{j+1}$.
We then denote
$${\mathcal{W}}^{\operatorname{right}}_i=\set{W^{\operatorname{right}}_i(e)\text{ }\big|\text{ } e\in \hat E^{\operatorname{right}}_{[i,j]}\cup E^{\operatorname{right}}_{[i,j]}\cup E^{\textsf{thr}}_{[i,j]}}.$$
The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ is then defined in a similar way as the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$, according to the paths of ${\mathcal{W}}^{\operatorname{right}}_i$ and the rotation ${\mathcal{O}}_{u_{i}}\in \Sigma$.
See Figure~\ref{fig: bad v_right rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[$\hat E^{\operatorname{right}}_{[i,j]}=\set{\hat e'_1,\hat e'_2}$, $E^{\operatorname{right}}_{[i,j]}=\set{e^b_1}$ and $E^{\textsf{thr}}_{[i,j]}=\set{e^r_1}$.
Paths of ${\mathcal{W}}^{\operatorname{right}}_{[i,j]}$ excluding their first edges are shown in dash lines. ]{\scalebox{0.14}{\includegraphics{figs/badinstance_rotation_right_1.jpg}
}
\hspace{3pt}
\subfigure[$\delta(v^{\operatorname{right}}_i)=\set{\hat e'_1,\hat e'_2,e^b_1,e^r_1}$. The rotation ${\mathcal{O}}^{\operatorname{right}}_{[i,j]}$ on them around $v^{\operatorname{right}}_{[i,j]}$ is shown above.]{
\scalebox{0.18}{\includegraphics{figs/badinstance_rotation_right_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{right}}_{[i,j]}$ on vertex $v^{\operatorname{right}}_{[i,j]}$ in the instance $(G_{[i,j]}, \Sigma_{[i,j]})$.}\label{fig: bad v_right rotation}
\end{figure}
We will use the following claims later for completing the proof of \Cref{thm: disengagement - main} in the special case.
\znote{the following two observations to modify}
\begin{observation}
\label{obs: disengaged instance size}
The total number of edges in all sub-instances we have defined is $O(|E(G)|)$.
\end{observation}
\begin{proof}
\iffalse
Note that, in the sub-instances $\set{(G_i,\Sigma_i)}_{1\le i\le r}$, each graph of $\set{G_i}_{1\le i\le r}$ is obtained from $G$ by contracting some sets of clusters of ${\mathcal{C}}$ into a single super-node, so each edge of $G_i$ corresponds to an edge in $E(G)$.
Therefore, for each $1\le i\le r$,
$$|E(G_i)|=|E_G(C_i)|+|\delta_G(C_i)|\le |E_G(C_i)|+|E^{\textsf{out}}({\mathcal{C}})|\le m/(100\mu)+m/(100\mu)\le m/\mu.$$
\fi
%
Note that $E(G_i)=E(C_i)\cup \hat E_i\cup \hat E_{i-1}\cup E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$.
First, each edge of $\bigcup_{1\le i\le r}E(C_i)$ appears in exactly one graphs of $\set{G_i}_{1\le i\le r}$.
Second, each edge of $\bigcup_{1\le i\le r}\hat E_i$ appears in exactly two graphs of $\set{G_i}_{1\le i\le r}$.
Consider now an edge $e\in E'$. If $e$ connects a vertex of $C_i$ to a vertex of $C_j$ for some $j\ge i+2$, then $e$ will appear as an edge in $E_i^{\operatorname{right}}\subseteq E(G_i)$ and an edge in $E_j^{\operatorname{left}}\subseteq E(G_j)$, and it will appear as an edge in $E_k^{\textsf{thr}}\subseteq E(G_k)$ for all $i<k<j$.
On one hand, we have $\sum_{1\le i\le r}|E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}|\le 2|E(G)|$.
On the other hand, note that $E_k^{\textsf{thr}}=E(C_{k-1},C_{k+1})\cup E^{\operatorname{over}}_{k-1}\cup E^{\operatorname{over}}_{k}$, and each edge of $e$ appears in at most two graphs of $\set{G_i}_{1\le i\le r}$ as an edge of $E(C_{k-1},C_{k+1})$.
Moreover, from \Cref{obs: bad inded structure}, $|E^{\operatorname{over}}_{k-1}|\le |\hat E_{k-1}|$ and $|E^{\operatorname{over}}_{k}|\le |\hat E_{k}|$.
Altogether, we have
\begin{equation}
\begin{split}
\sum_{1\le i\le r}|E(G_i)| & =
\sum_{1\le i\le r}\textsf{left}(
|E(C_i)|+ |\hat E_i|+ |\hat E_{i-1}|+ |E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|+|E_i^{\textsf{thr}}|
\textsf{right})\\
& = \sum_{1\le i\le r}
|E(C_i)|+ \sum_{1\le i\le r} \textsf{left}(|E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|\textsf{right})+ \sum_{1\le i\le r}\textsf{left}(|E_i^{\textsf{thr}}|+ |\hat E_i|+ |\hat E_{i-1}|\textsf{right})\\
& \le |E(G)|+ 2\cdot |E(G)| + \sum_{1\le i\le r}\textsf{left}(|E(C_{i-1},C_{i+1})|+ 2|\hat E_i|+ 2|\hat E_{i-1}|\textsf{right})\\
& \le 8\cdot |E(G)|.
\end{split}
\end{equation}
This completes the proof of \Cref{obs: disengaged instance size}.
\end{proof}
\begin{observation}
\label{obs: rotation for stitching}
For each $1\le i\le r-1$, if we view the edge in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$ as edges of $E(G)$, then $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$, and moreover, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{observation}
\begin{proof}
Recall that for each $1\le i\le r-1$,
$\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})=\hat E_{i}\cup E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}$. From the definition of sets $E_i^{\textsf{thr}},E_{i+1}^{\textsf{thr}}, E^{\operatorname{right}}_i, E^{\operatorname{left}}_{i+1}$,
\[
\begin{split}
E_i^{\textsf{thr}}\cup E^{\operatorname{right}}_i
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i<j'\text{ or }i'=i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'\le i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'< i+1\le j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i+1<j'\text{ or }i'<i+1=j'}=E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}.
\end{split}
\]
Therefore, $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Moreover, from the definition of path sets ${\mathcal{W}}^{\operatorname{right}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_{i+1}$, it is not hard to verify that, for every edge $e\in \delta_{G_i}(v^{\operatorname{right}}_i)$, the path in ${\mathcal{W}}^{\operatorname{right}}_i$ that contains $e$ as its first edge is identical to the path in ${\mathcal{W}}^{\operatorname{left}}_{i+1}$ that contains $e$ as its first edge. According to the way that rotations ${\mathcal{O}}^{\operatorname{right}}_i,{\mathcal{O}}^{\operatorname{left}}_{i+1}$ are defined, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{proof}
\fi
\iffalse{backup: original analysis of total cost of subinstances}
Specifically, we use the following two claims, whose proofs will be provided later.
\begin{claim}
\label{claim: existence of good solutions special}
$\expect{\sum_{1\le i\le r}\mathsf{OPT}_{\mathsf{cnwrs}}(G_i,\Sigma_i)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$.
\end{claim}
\begin{claim}
\label{claim: stitching the drawings together}
There is an efficient algorithm, that given, for each $1\le i\le r$, a feasible solution $\phi_i$ to the instance $(G_i,\Sigma_i)$, computes a solution to the instance $(G,\Sigma)$, such that $\mathsf{cr}(\phi)\le \sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
\end{claim}
We use the algorithm described in~\Cref{sec: guiding and auxiliary paths} and~\Cref{sec: compute advanced disengagement}, and return the disengaged instances $(G_1,\Sigma_1),\ldots,(G_r,\Sigma_r)$ as the collection of sub-instances of $(G,\Sigma)$.
From the previous subsections, the algorithm for producing the sub-instances is efficient.
On the other hand, it follows immediately from \Cref{obs: disengaged instance size}, \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together} that the sub-instances $(G_1,\Sigma_1),\ldots,(G_r,\Sigma_r)$ satisfy the properties in \Cref{thm: disengagement - main}.
This completes the proof of \Cref{thm: disengagement - main}.
We now provide the proofs of \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together}.
$\ $
\begin{proofof}{Claim~\ref{claim: existence of good solutions special}}
Let $\phi^*$ be an optimal drawing of the instance $(G,\Sigma)$.
We will construct, for each $1\le i\le r$, a drawing $\phi_i$ of $G_i$ that respects the rotation system $\Sigma_i$, based on the drawing $\phi^*$, such that $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)\le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot 2^{O((\log m)^{3/4}\log\log m)})$, and \Cref{claim: existence of good solutions special} then follows.
\paragraph{Drawings $\phi_2,\ldots,\phi_{r-1}$.}
First we fix some index $2\le i\le r-1$, and describe the construction of the drawing $\phi_i$. We start with some definitions.
Recall that $E(G_i)=E_G(C_i)\cup (\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$.
We define the auxiliary path set ${\mathcal{W}}_i={\mathcal{W}}^{\operatorname{left}}_i\cup {\mathcal{W}}^{\operatorname{right}}_i$, so
$${\mathcal{W}}_i=\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}\cup \set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}},$$
where for each $e\in E^{\operatorname{left}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$; for each $e\in E^{\operatorname{right}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between $u_{i+1}$ and its last endpoint; and for each $e\in E^{\textsf{thr}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$, the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$, and the subpath of $P_e$ between $u_{i+1}$ and its last endpoint.
\iffalse
We use the following observation.
\begin{observation}
\label{obs: wset_i_non_interfering}
The set ${\mathcal{W}}_i$ of paths are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
Recall that the paths in ${\mathcal{Q}}_{i-1}$ only uses edges of $E(C_{i-1})\cup \delta(C_{i-1})$, and they are non-transversal.
And similarly, the paths in ${\mathcal{Q}}_{i+1}$ only uses edges of $E(C_{i+1})\cup \delta(C_{i+1})$, and they are non-transversal.
Therefore, the paths in $\set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}}$ are non-transversal.
From \Cref{obs: non_transversal_1} and \Cref{obs: non_transversal_2}, the paths in $\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}$ are non-transversal. Therefore, it suffices to show that, the set ${\mathcal{W}}_i$ of paths are non-transversal at all vertices of $C_{i-1}$ and all vertices of $C_{i+1}$. Note that, for each edge $e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i})$, $W_e\cap (C_{i-1}\cup \delta(C_{i-1}))$ is indeed a path of ${\mathcal{Q}}_{i-1}$. Therefore, the paths in ${\mathcal{W}}_i$ are non-transversal at all vertices of $C_{i-1}$. Similarly, they are also non-transversal at all vertices of $C_{i+1}$. Altogether, the paths of ${\mathcal{W}}_i$ are non-transversal with respect to $\Sigma$.
\end{proof}
\fi
For uniformity of notations, for each edge $\hat e\in \hat E_i$, we rename the path $Q_{i+1}(\hat e)$ by $W_{\hat e}$, and similarly for each edge $\hat e\in \hat E_{i-1}$, we rename the path $Q_{i-1}(\hat e)$ by $W_{\hat e}$. Therefore, ${\mathcal{W}}_i=\set{W_e\mid e\in E(G_i)\setminus E(C_i)}$.
Put in other words, the set ${\mathcal{W}}_i$ contains, for each edge $e$ in $G_i$ that is incident to $v^{\operatorname{left}}_i$ or $v^{\operatorname{right}}_i$, a path named $W_e$.
It is easy to see that all paths in ${\mathcal{W}}_i$ are internally disjoint from $C_i$.
We further partition the set ${\mathcal{W}}_i$ into two sets: ${\mathcal{W}}_i^{\textsf{thr}}=\set{W_e\mid e\in E^{\textsf{thr}}_i}$ and $\tilde {\mathcal{W}}_i={\mathcal{W}}_i\setminus {\mathcal{W}}_i^{\textsf{thr}}$.
We are now ready to construct the drawing $\phi_i$ for the instance $(G_i,\Sigma_i)$.
Recall that $\phi^*$ is an optimal drawing of the input instance $(G,\Sigma)$.
We start with the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$, that we denote by $\phi^*_i$. We will not modify the image of $C_i$ in $\phi^*_i$, but will focus on constructing the image of edges in $E(G_i)\setminus E(C_i)$, based on the image of edges in $E({\mathcal{W}}_i)$ in $\phi^*_i$.
Specifically, we proceed in the following four steps.
\paragraph{Step 1.}
For each edge $e\in E({\mathcal{W}}_i)$, we denote by $\pi_e$ the curve that represents the image of $e$ in $\phi^*_i$. We create a set of $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting the endpoints of $e$ in $\phi^*_i$, that lies in an arbitrarily thin strip around $\pi_e$.
We denote by $\Pi_e$ the set of these curves.
We then assign, for each edge $e\in E({\mathcal{W}}_i)$ and for each path in ${\mathcal{W}}_i$ that contains the edge $e$, a distinct curve in $\Pi_e$ to this path.
Therefore, each curve in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ is assigned to exactly one path of ${\mathcal{W}}_i$,
and each path $W\in {\mathcal{W}}_i$ is assigned with, for each edge $e\in E(W)$, a curve in $\Pi_e$. Let $\gamma_W$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ that are assigned to path $W$, so $\gamma_W$ connects the endpoints of path $W$ in $\phi^*_i$.
In fact, when we assign curves in $\bigcup_{e\in \delta(u_{i-1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{left}}_i$ (recall that $\delta(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\operatorname{left}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{left}}_i)}$), we additionally ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{left}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{left}}_i)$ enter $u_{i-1}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
And similarly, when we assign curves in $\bigcup_{e\in \delta(u_{i+1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{right}}_i$ (recall that $\delta(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\operatorname{right}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{right}}_i)}$), we ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{right}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{right}}_i)$ enter $u_{i+1}$ in the same order as ${\mathcal{O}}^{\operatorname{right}}_i$.
Note that this can be easily achieved according to the definition of ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$.
We denote $\Gamma_i=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$, and we further partition the set $\Gamma_i$ into two sets: $\Gamma_i^{\textsf{thr}}=\set{\gamma_{W}\mid W\in {\mathcal{W}}^{\textsf{thr}}_i}$ and $\tilde \Gamma_i=\Gamma_i\setminus \Gamma_i^{\textsf{thr}}$.
We denote by $\hat \phi_i$ the drawing obtained by taking the union of the image of $C_i$ in $\phi^*_i$ and all curves in $\Gamma_i$.
For every path $P$ in $G_i$, we denote by $\hat{\chi}_i(P)$ the number of crossings that involves the ``image of $P$'' in $\hat \phi_i$, which is defined as the union of, for each edge $e\in E(\tilde{\mathcal{W}}_i)$, an arbitrary curve in $\Pi_e$.
Clearly, for each edge $e\in E({\mathcal{W}}_i)$, all curves in $\Pi_e$ are crossed by other curves of $(\Gamma_i\setminus \Pi_e)\cup \phi^*_i(C_i)$ same number of times.
Therefore, $\hat{\chi}_i(P)$ is well-defined.
For a set ${\mathcal{P}}$ of paths in $G_i$, we define $\hat{\chi}_i({\mathcal{P}})=\sum_{P\in {\mathcal{P}}}\hat{\chi}_i(P)$.
\iffalse
We use the following observation.
\znote{maybe remove this observation?}
\begin{observation}
\label{obs: curves_crossings}
The number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$,
and the number of intersections between a curve in $\tilde\Gamma_i$ and the image of edges of $C_i$ in $\phi^*_i$, are both $O(\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W))$.
\end{observation}
\begin{proof}
We first show that the number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Note that, from the construction of curves in $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i}$, every crossing between a pair $W,W'$ of curves in $\tilde\Gamma_i$ must be the intersection between a curve in $\Pi_e$ for some $e\in E(W)$ and a curve in $\Pi_{e'}$ for some $e'\in E(W')$, such that the image $\pi_e$ for $e$ and the image $\pi_{e'}$ for $e'$ intersect in $\phi^*$.
Therefore, for each pair $W,W'$ of paths in $\tilde{\mathcal{W}}_i$, the number of points that belong to only curves $\gamma_W$ and $\gamma_{W'}$ is at most the number of crossings between the image of $W$ and the image of $W'$ in $\phi^*$.
It follows that the number of points that belong to exactly two curves of $\tilde\Gamma_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Altogether, the number of intersections between curves in $\tilde\Gamma_i$ is at most $|V(\tilde {\mathcal{W}}_i)|+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
We now show that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ that are not vertex-image is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Let $W$ be a path of $\tilde {\mathcal{W}}_i$ and consider the curve $\gamma_W$. Note that $\gamma_W$ is the union of, for each edge $e\in E(W)$, a curve that lies in an arbitrarily thin strip around $\pi_e$.
Therefore, the number of crossings between $\gamma_W$ and the image of $C_i$ in $\phi^*_i$ is identical to the number of crossings the image of path $W$ and the image of $C_i$ in $\phi^*_i$, which is at most $\hat\mathsf{cr}(W)$.
It follows that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
\end{proof}
\fi
\paragraph{Step 2.}
For each vertex $v\in V({\mathcal{W}}_i)$, we denote by $x_v$ the point that represents the image of $v$ in $\phi^*_i$, and we let $X$ contains all points of $\set{x_v\mid v\in V({\mathcal{W}}_i)}$ that are intersections between curves in $\Gamma_i$.
We now manipulate the curves in $\set{\gamma_W\mid W\in {\mathcal{W}}_i}$ at points of $X$, by processing points of $X$ one-by-one, as follows.
Consider a point $x_v$ that is an intersection between curves in $\Gamma_i$, where $v\in V({\mathcal{W}}_i)$, and let $D_v$ be an arbitrarily small disc around $x_v$.
We denote by ${\mathcal{W}}_i(v)$ the set of paths in ${\mathcal{W}}_i$ that contains $v$, and further partition it into two sets: ${\mathcal{W}}^{\textsf{thr}}_i(v)={\mathcal{W}}_i(v)\cap {\mathcal{W}}^{\textsf{thr}}_i$ and $\tilde{\mathcal{W}}_i(v)={\mathcal{W}}_i(v)\cap \tilde{\mathcal{W}}_i$. We apply the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $\set{\gamma_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ within disc $D_v$. Let $\set{\gamma'_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ be the set of new curves that we obtain. From \Cref{obs: curve_manipulation},
(i) for each path $W\in \tilde{\mathcal{W}}_i(v)$, the curve $\gamma'_W$ does not contain $x_v$, and is identical to the curve $\gamma_W$ outside the disc $D_v$;
(ii) the segments of curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc $D_v$ are in general position; and
(iii) the number of icrossings between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside $D_v$ is bounded by $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\iffalse{just for backup}
\begin{proof}
Denote $d=\deg_G(v)$ and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
For each path $W\in \tilde{\mathcal{W}}_i(v)$, we denote by $p^{-}_W$ and $p^{+}_W$ the intersections between the curve $\gamma_W$ and the boundary of ${\mathcal{D}}_v$.
We now compute, for each $W\in W\in \tilde{\mathcal{W}}_i(v)$, a curve $\zeta_W$ in ${\mathcal{D}}_v$ connecting $p^{-}_W$ to $p^{+}_W$, such that (i) the curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ are in general position; and (ii) for each pair $W,W'$ of paths, the curves $\zeta_W$ and $\zeta_{W'}$ intersects iff the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{-}_{W'},p^{+}_{W},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_{W'},p^{-}_{W},p^{+}_{W'})$.
It is clear that this can be achieved by first setting, for each $W$, the curve $\zeta_W$ to be the line segment connecting $p^{-}_W$ to $p^{+}_W$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
We now define, for each $W$, the curve $\gamma'_W$ to be the union of the part of $\gamma_W$ outside ${\mathcal{D}}_v$ and the curve $\zeta_W$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, the first and the second condition of \Cref{obs: curve_manipulation} are satisfied.
It remains to estimate the number of intersections between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$, which equals the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$.
Since the paths in $\tilde{\mathcal{W}}_i(v)$ are non-transversal with respect to $\Sigma$ (from \Cref{obs: wset_i_non_interfering}), from the construction of curves $\set{\gamma_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$, if a pair $W,W'$ of paths in $\tilde {\mathcal{W}}_i(v)$ do not share edges of $\delta(v)$, then the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_W,p^{-}_{W'},p^{+}_{W'})$, and therefore the curves $\zeta_{W}$ and $\zeta_{W'}$ will not intersect in ${\mathcal{D}}_v$.
Therefore, only the curves $\zeta_W$ and $\zeta_{W'}$ intersect iff $W$ and $W'$ share an edge of $\delta(v)$.
Since every such pair of curves intersects at most once, the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$ is at most $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are shown in black, and curves of $\tilde{\mathcal{W}}_i(v)$ are shown in blue, red, orange and green. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are not modified, while curves of $\tilde{\mathcal{W}}_i(v)$ are re-routed via dash lines within disc ${\mathcal{D}}_v$.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the step of processing $x_v$.}\label{fig: curve_con}
\end{figure}
\fi
We then replace the curves of $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ in $\Gamma_i$ by the curves of $\set{\gamma'_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
This completes the description of the iteration of processing point the point $x_v\in X$. Let $\Gamma'_i$ be the set of curves that we obtain after processing all points in $X$ in this way. Note that we have never modified the curves of $\Gamma^{\textsf{thr}}_i$, so $\Gamma^{\textsf{thr}}_i\subseteq\Gamma'_i$, and we denote $\tilde\Gamma'_i=\Gamma'_i\setminus \Gamma^{\textsf{thr}}_i$.
We use the following observation.
\begin{observation}
\label{obs: general_position}
Curves in $\tilde\Gamma'_i$ are in general position, and if a point $p$ lies on more than two curves of $\Gamma'_i$, then either $p$ is an endpoint of all curves containing it, or all curves containing $p$ belong to $\Gamma^{\textsf{thr}}_i$.
\end{observation}
\begin{proof}
From the construction of curves in $\Gamma_i$, any point that belong to at least three curves of $\Gamma_i$ must be the image of some vertex in $\phi^*$. From~\Cref{obs: curve_manipulation}, curves in $\tilde\Gamma'_i$ are in general position; curves in $\tilde\Gamma'_i$ do not contain any vertex-image in $\phi^*$ except for their endpoints; and they do not contain any intersection of a pair of paths in $\Gamma_i^{\textsf{thr}}$. \Cref{obs: general_position} now follows.
\end{proof}
\paragraph{Step 3.}
So far we have obtained a set $\Gamma'_i$ of curves that are further partitioned into two sets $\Gamma'_i=\Gamma^{\textsf{thr}}_i\cup \tilde\Gamma'_i$,
where set $\tilde\Gamma'_i$ contains, for each path $W\in \tilde {\mathcal{W}}_i$, a curve $\gamma'_W$ connecting the endpoints of $W$, and the curves in $\tilde\Gamma'_i$ are in general position;
and set $\Gamma^{\textsf{thr}}_i$ contains, for each path $W\in {\mathcal{W}}^{\textsf{thr}}_i$, a curve $\gamma_W$ connecting the endpoints of $W$.
Recall that all paths in ${\mathcal{W}}^{\textsf{thr}}_i$ connects $u_{i-1}$ to $u_{i+1}$.
Let $z_{\operatorname{left}}$ be the point that represents the image of $u_{i-1}$ in $\phi_i^*$ and let $z_{\operatorname{right}}$ be the point that represents the image of $u_{i+1}$ in $\phi_i^*$.
Then, all curves in $\Gamma^{\textsf{thr}}_i$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
We view $z_{\operatorname{left}}$ as the first endpoint of curves in $\Gamma^{\textsf{thr}}_i$ and view $z_{\operatorname{right}}$ as their last endpoint.
We then apply the algorithm in \Cref{thm: type-2 uncrossing}, where we let $\Gamma=\Gamma^{\textsf{thr}}_i$ and let $\Gamma_0$ be the set of all other curves in the drawing $\phi^*_i$. Let $\Gamma^{\textsf{thr}'}_i$ be the set of curves we obtain. We then designate, for each edge $e\in E^{\textsf{thr}}_i$, a curve in $\Gamma^{\textsf{thr}'}_i$ as $\gamma'_{W_e}$, such that the curves of $\set{\gamma'_{W_e}\mid e\in \hat E_{i-1}\cup E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i}$ enters $z_{\operatorname{left}}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
Recall that ${\mathcal{W}}_i=\set{W_e\mid e\in (E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\cup E_i^{\operatorname{right}}\cup \hat E_{i-1}\cup \hat E_i)}$, and, for each edge $e\in E_i^{\operatorname{left}}\cup \hat E_{i-1}$, the curve $\gamma'_{W_e}$ connects its endpoint in $C_i$ to $z_{\operatorname{left}}$; for each edge $e\in E_i^{\operatorname{right}}\cup \hat E_{i}$, the curve $\gamma'_{W_e}$ connects the endpoint of $e$ to $z_{\operatorname{right}}$; and for each edge $e\in E_i^{\textsf{thr}}$, the curve $\gamma'_{W_e}$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
Therefore, if we view $z_{\operatorname{left}}$ as the image of $v^{\operatorname{left}}_i$, view $z_{\operatorname{right}}$ as the image of $v^{\operatorname{right}}_i$, and for each edge $e\in E(G_i)\setminus E(C_i)$, view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi'_i$.
It is clear from the construction of curves in $\set{\gamma'_{W_e}\mid e\in E(G_i)\setminus E(C_i)}$ that this drawing respects all rotations in $\Sigma_i$ on vertices of $V(C_i)$ and vertex $v^{\operatorname{left}}_i$.
However, the drawing $\phi'_i$ may not respect the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$. We further modify the drawing $\phi'_i$ at $z_{\operatorname{right}}$ in the last step.
\paragraph{Step 4.}
Let ${\mathcal{D}}$ be an arbitrarily small disc around $z_{\operatorname{right}}$ in the drawing $\phi'_i$, and let ${\mathcal{D}}'$ be another small disc around $z_{\operatorname{right}}$ that is strictly contained in ${\mathcal{D}}$.
We first erase the drawing of $\phi'_i$ inside the disc ${\mathcal{D}}$, and for each edge $e\in \delta(v^{\operatorname{right}}_i)$, we denote by $p_{e}$ the intersection between the curve representing the image of $e$ in $\phi'_i$ and the boundary of ${\mathcal{D}}$.
We then place, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, a point $p'_e$ on the boundary of ${\mathcal{D}}'$, such that the order in which the points in $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ appearing on the boundary of ${\mathcal{D}}'$ is precisely ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We then apply \Cref{lem: find reordering} to compute a set of reordering curves, connecting points of $\set{p_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ to points $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$.
Finally, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, let $\gamma_e$ be the concatenation of (i) the image of $e$ in $\phi'_i$ outside the disc ${\mathcal{D}}$; (ii) the reordering curve connecting $p_e$ to $p'_e$; and (iii) the straight line segment connecting $p'_e$ to $z_{\operatorname{right}}$ in ${\mathcal{D}}'$.
We view $\gamma_e$ as the image of edge $e$, for each $e\in \delta(v^{\operatorname{right}}_i)$.
We denote the resulting drawing of $G_i$ by $\phi_i$. It is clear that $\phi_i$ respects the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$, and therefore it respects the rotation system $\Sigma_i$.
We use the following claim.
\begin{claim}
\label{clm: rerouting_crossings}
The number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\end{claim}
\begin{proof}
Denote by ${\mathcal{O}}^*$ the ordering in which the curves $\set{\gamma'_{W_e}\mid e\in \delta_{G_i}(v_i^{\operatorname{right}})}$ enter $z_{\operatorname{right}}$, the image of $u_{i+1}$ in $\phi'_i$.
From~\Cref{lem: find reordering} and the algorithm in Step 4 of modifying the drawing within the disc ${\mathcal{D}}$, the number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is at most $O(\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}}))$. Therefore, it suffices to show that $\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}})=O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
In fact, we will compute a set of curves connecting the image of $u_i$ and the image of $u_{i+1}$ in $\phi^*_i$, such that each curve is indexed by some edge $e\in\delta_{G_i}(v_i^{\operatorname{right}})$ these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$ and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$, and the number of crossings between curves of $Z$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
For each $e\in E^{\textsf{thr}}_i$, we denote by $\eta_e$ the curve obtained by taking the union of (i) the curve $\gamma'_{W_e}$ (that connects $u_{i+1}$ to $u_{i-1}$); and (ii) the curve representing the image of the subpath of $P_e$ in $\phi^*_i$ between $u_i$ and $u_{i-1}$.
Therefore, the curve $\eta_e$ connects $u_i$ to $u_{i+1}$.
We then modify the curves of $\set{\eta_e\mid e\in E^{\textsf{thr}}_i}$, by iteratively applying the algorithm from \Cref{obs: curve_manipulation} to these curves at the image of each vertex of $C_{i-1}\cup C_{i+1}$. Let $\set{\zeta_e\mid e\in E^{\textsf{thr}}_i}$ be the set of curves that we obtain. We call the obtained curves \emph{red curves}. From~\Cref{obs: curve_manipulation}, the red curves are in general position. Moreover, it is easy to verify that the number of intersections between the red curves is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1}))$.
We call the curves in $\set{\gamma'_{W_e}\mid e\in \hat E_i}$ \emph{yellow curves}, call the curves in $\set{\gamma'_{W_e}\mid e\in E^{\operatorname{right}}_i}$ \emph{green curves}. See \Cref{fig: uncrossing_to_bound_crossings} for an illustration.
From the construction of red, yellow and green curves, we know that these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$, and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$.
Moreover, we are guaranteed that the number of intersections between red, yellow and green curves is at most $\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/uncross_to_bound_crossings.jpg}
\caption{An illustration of red, yellow and green curves.}\label{fig: uncrossing_to_bound_crossings}
\end{figure}
\end{proof}
From the above discussion and Claim~\ref{clm: rerouting_crossings}, for each $2\le i\le r-1$,
\[
\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right}).
\]
\iffalse
We now estimate the number of crossings in $\phi_i$ in the next claim.
\begin{claim}
\label{clm: number of crossings in good solutions}
$\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\sum_{W\in \tilde{\mathcal{W}}_i}\mathsf{cr}(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})$.
\end{claim}
\begin{proof}
$2\cdot \sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2=\sum_{v\in V(G)}\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2.$
\znote{need to redefine the orderings ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$ to get rid of $\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$ here, which we may not be able to afford.}
\end{proof}
\fi
\paragraph{Drawings $\phi_1$ and $\phi_{r}$.}
The drawings $\phi_1$ and $\phi_{r}$ are constructed similarly. We describe the construction of $\phi_1$, and the construction of $\phi_1$ is symmetric.
Recall that the graph $G_1$ contains only one super-node $v_1^{\operatorname{right}}$, and $\delta_{G_1}(v_1^{\operatorname{right}})=\hat E_1\cup E^{\operatorname{right}}_1$. We define ${\mathcal{W}}_1=\set{W_e\mid e\in E^{\operatorname{right}}_1}\cup \set{Q_2(\hat e)\mid \hat e\in \hat E_1}$. For each $\hat e\in \hat E_1$, we rename the path $Q_2(\hat e)$ by $W_e$, so ${\mathcal{W}}_1$ contains, for each edge $e\in \delta_{G_1}(v^1_{\operatorname{right}})$, a path named $W_e$ connecting its endpoints to $u_2$. Via similar analysis in \Cref{obs: wset_i_non_interfering}, it is easy to show that the paths in ${\mathcal{W}}_1$ are non-transversal with respect to $\Sigma$.
We start with the drawing of $C_1\cup E({\mathcal{W}}_1)$ induced by $\phi^*$, that we denote by $\phi^*_1$. We will not modify the image of $C_i$ in $\phi^*_i$ and will construct the image of edges in $\delta(v_1^{\operatorname{right}})$.
We perform similar steps as in the construction of drawings $\phi_2,\ldots,\phi_{r-1}$.
We first construct, for each path $W\in {\mathcal{W}}_1$, a curve $\gamma_W$ connecting its endpoint in $C_1$ to the image of $u_2$ in $\phi^*$, as in Step 1.
Let $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
We then process all intersections between curves of $\Gamma_1$ as in Step 2. Let $\Gamma'_1=\set{\gamma'_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
Since $\Gamma^{\textsf{thr}}_1=\emptyset$, we do not need to perform Steps 3 and 4. If we view the image of $u_2$ in $\phi^*_1$ as the image of $v^{\operatorname{right}}_1$, and for each edge $e\in \delta(v^{\operatorname{right}}_1)$, we view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is clear that this drawing respects the rotation system $\Sigma_1$. Moreover,
\[\mathsf{cr}(\phi_1)=\chi^2(C_1)+O\textsf{left}(\hat\chi_1({\mathcal{Q}}_2)+\sum_{W\in {\mathcal{W}}_1}\hat\chi_1(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_1,e)^2\textsf{right}).\]
Similarly, the drawing $\phi_k$ that we obtained in the similar way satisfies that
\[\mathsf{cr}(\phi_k)=\chi^2(C_k)+O\textsf{left}(\hat\chi_k({\mathcal{Q}}_{r-1})+\sum_{W\in {\mathcal{W}}_k}\hat\chi_k(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_k,e)^2\textsf{right}).\]
We now complete the proof of \Cref{claim: existence of good solutions special}, for which it suffices to estimate $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
Recall that, for each $1\le i\le r$, $\tilde {\mathcal{W}}_i=\set{W_e\mid e\in \hat E_{i-1}\cup \hat E_{i-1}\cup E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}}$, where $E^{\operatorname{left}}_{i}=E(C_i,\bigcup_{1\le j\le i-2}C_j)$, and
$E^{\operatorname{right}}_{i}=E(C_i,\bigcup_{i+2\le j\le r}C_j)$. Therefore, for each edge $e\in E'\cup (\bigcup_{1\le i\le r-1}\hat E_i)$, the path $W_e$ belongs to exactly $2$ sets of $\set{\tilde{\mathcal{W}}_i}_{1\le i\le r}$.
Recall that the path $W_e$ only uses edges of the inner path $P_e$ and the outer path $P^{\mathsf{out}}_e$.
Let $\tilde{\mathcal{W}}=\bigcup_{1\le i\le r}\tilde{\mathcal{W}}_i$, from \Cref{obs: edge_occupation in outer and inner paths},
for each edge $e\in E'\cup (\bigcup_{1\le i\le r-1}\hat E_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(1)$, and
for $1\le i\le r$ and for each edge $e\in E(C_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(\cong_G({\mathcal{Q}}_i,e))$.
Therefore, on one hand,
\[
\begin{split}
\sum_{1\le i\le r}\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} 2\cdot \cong_G(\tilde {\mathcal{W}},e)\cdot\cong_G(\tilde {\mathcal{W}},e')\\
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} \textsf{left}(\cong_G(\tilde {\mathcal{W}},e)^2+\cong_G(\tilde {\mathcal{W}},e')^2\textsf{right})\\
& \le
\sum_{e\in E(G)} \chi(e)\cdot \cong_G(\tilde {\mathcal{W}},e)^2
= O(\mathsf{cr}(\phi^*)\cdot\beta),
\end{split}
\]
and on the other hand,
\[
\begin{split}
\sum_{1\le i\le r}\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2
& \le \sum_{e\in E(G)} \textsf{left}(\sum_{1\le i\le r} \cong_G(\tilde {\mathcal{W}}_i,e)\textsf{right})^2\\
& \le O\textsf{left}(\sum_{e\in E(G)} \cong_G(\tilde {\mathcal{W}},e)^2 \textsf{right})\\
& \le O\textsf{left}(|E(G)|+\sum_{1\le i\le r}\sum_{e\in E(C_i)} \cong_G({\mathcal{Q}}_i,e)^2\textsf{right})=O(|E(G)|\cdot\beta).
\end{split}
\]
Moreover, $\sum_{1\le i\le r}\chi^2(C_i)\le O(\mathsf{cr}(\phi^*))$, and $\sum_{1\le i\le r}\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})\le O(\mathsf{cr}(\phi^*)\cdot\beta)$.
Altogether,
\[
\begin{split}
\sum_{1\le i\le r}\mathsf{cr}(\phi_i)
& \le
O\textsf{left}(\sum_{1\le i\le r}
\textsf{left}(
\chi^2(C_i)+ \hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})
\textsf{right})\\
& \le O(\mathsf{cr}(\phi^*))+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(|E(G)|\cdot\beta)\\
& \le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot\beta).
\end{split}
\]
This completes the proof of \Cref{claim: existence of good solutions special}.
\end{proofof}
\fi
\iffalse{incorrect analysis of non-interfering}
We use the following observation.
\begin{observation}
\label{obs: wset_i_non_interfering}
The set ${\mathcal{W}}_i$ of paths are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
Recall that the paths in ${\mathcal{Q}}_{i-1}$ only uses edges of $E(C_{i-1})\cup \delta(C_{i-1})$, and they are non-transversal.
And similarly, the paths in ${\mathcal{Q}}_{i+1}$ only uses edges of $E(C_{i+1})\cup \delta(C_{i+1})$, and they are non-transversal.
Therefore, the paths in $\set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}}$ are non-transversal.
From \Cref{obs: non_transversal_1} and \Cref{obs: non_transversal_2}, the paths in $\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}$ are non-transversal. Therefore, it suffices to show that, the set ${\mathcal{W}}_i$ of paths are non-transversal at all vertices of $C_{i-1}$ and all vertices of $C_{i+1}$. Note that, for each edge $e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i})$, $W_e\cap (C_{i-1}\cup \delta(C_{i-1}))$ is indeed a path of ${\mathcal{Q}}_{i-1}$. Therefore, the paths in ${\mathcal{W}}_i$ are non-transversal at all vertices of $C_{i-1}$. Similarly, they are also non-transversal at all vertices of $C_{i+1}$. Altogether, the paths of ${\mathcal{W}}_i$ are non-transversal with respect to $\Sigma$.
\end{proof}
\fi
\iffalse {previous nudging and uncrossing steps}
We further partition ${\mathcal{W}}_i$ into two sets: ${\mathcal{W}}_i^{\textsf{thr}}=\set{W_i(e)\mid e\in E^{\textsf{thr}}_i}$ and $\tilde {\mathcal{W}}_i={\mathcal{W}}_i\setminus {\mathcal{W}}_i^{\textsf{thr}}$.
We are now ready to construct the drawing $\phi_i$ for the instance $(G_i,\Sigma_i)$.
Recall that $\phi^*$ is an optimal drawing of the input instance $(G,\Sigma)$.
We start with the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$, that we denote by $\phi^*_i$. We will not modify the image of $C_i$ in $\phi^*_i$, but will focus on constructing the image of edges in $E(G_i)\setminus E(C_i)$, based on the image of edges in $E({\mathcal{W}}_i)$ in $\phi^*_i$.
Specifically, we proceed in the following four steps.
\paragraph{Step 1.}
For each edge $e\in E({\mathcal{W}}_i)$, we denote by $\pi_e$ the curve that represents the image of $e$ in $\phi^*_i$. We create a set of $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting the endpoints of $e$ in $\phi^*_i$, that lies in an arbitrarily thin strip around $\pi_e$.
We denote by $\Pi_e$ the set of these curves.
We then assign, for each edge $e\in E({\mathcal{W}}_i)$ and for each path in ${\mathcal{W}}_i$ that contains the edge $e$, a distinct curve in $\Pi_e$ to this path.
Therefore, each curve in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ is assigned to exactly one path of ${\mathcal{W}}_i$,
and each path $W\in {\mathcal{W}}_i$ is assigned with, for each edge $e\in E(W)$, a curve in $\Pi_e$. Let $\gamma_W$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ that are assigned to path $W$, so $\gamma_W$ connects the endpoints of path $W$ in $\phi^*_i$.
In fact, when we assign curves in $\bigcup_{e\in \delta(u_{i-1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{left}}_i$ (recall that $\delta(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\operatorname{left}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{left}}_i)}$), we additionally ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{left}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{left}}_i)$ enter $u_{i-1}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
And similarly, when we assign curves in $\bigcup_{e\in \delta(u_{i+1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{right}}_i$ (recall that $\delta(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\operatorname{right}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{right}}_i)}$), we ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{right}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{right}}_i)$ enter $u_{i+1}$ in the same order as ${\mathcal{O}}^{\operatorname{right}}_i$.
Note that this can be easily achieved according to the definition of ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$.
We denote $\Gamma_i=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$, and we further partition the set $\Gamma_i$ into two sets: $\Gamma_i^{\textsf{thr}}=\set{\gamma_{W}\mid W\in {\mathcal{W}}^{\textsf{thr}}_i}$ and $\tilde \Gamma_i=\Gamma_i\setminus \Gamma_i^{\textsf{thr}}$.
We denote by $\hat \phi_i$ the drawing obtained by taking the union of the image of $C_i$ in $\phi^*_i$ and all curves in $\Gamma_i$.
For every path $P$ in $G_i$, we denote by $\hat{\chi}_i(P)$ the number of crossings that involves the ``image of $P$'' in $\hat \phi_i$, which is defined as the union of, for each edge $e\in E(\tilde{\mathcal{W}}_i)$, an arbitrary curve in $\Pi_e$.
Clearly, for each edge $e\in E({\mathcal{W}}_i)$, all curves in $\Pi_e$ are crossed by other curves of $(\Gamma_i\setminus \Pi_e)\cup \phi^*_i(C_i)$ same number of times.
Therefore, $\hat{\chi}_i(P)$ is well-defined.
For a set ${\mathcal{P}}$ of paths in $G_i$, we define $\hat{\chi}_i({\mathcal{P}})=\sum_{P\in {\mathcal{P}}}\hat{\chi}_i(P)$.
\iffalse
We use the following observation.
\znote{maybe remove this observation?}
\begin{observation}
\label{obs: curves_crossings}
The number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$,
and the number of intersections between a curve in $\tilde\Gamma_i$ and the image of edges of $C_i$ in $\phi^*_i$, are both $O(\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W))$.
\end{observation}
\begin{proof}
We first show that the number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Note that, from the construction of curves in $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i}$, every crossing between a pair $W,W'$ of curves in $\tilde\Gamma_i$ must be the intersection between a curve in $\Pi_e$ for some $e\in E(W)$ and a curve in $\Pi_{e'}$ for some $e'\in E(W')$, such that the image $\pi_e$ for $e$ and the image $\pi_{e'}$ for $e'$ intersect in $\phi^*$.
Therefore, for each pair $W,W'$ of paths in $\tilde{\mathcal{W}}_i$, the number of points that belong to only curves $\gamma_W$ and $\gamma_{W'}$ is at most the number of crossings between the image of $W$ and the image of $W'$ in $\phi^*$.
It follows that the number of points that belong to exactly two curves of $\tilde\Gamma_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Altogether, the number of intersections between curves in $\tilde\Gamma_i$ is at most $|V(\tilde {\mathcal{W}}_i)|+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
We now show that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ that are not vertex-image is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Let $W$ be a path of $\tilde {\mathcal{W}}_i$ and consider the curve $\gamma_W$. Note that $\gamma_W$ is the union of, for each edge $e\in E(W)$, a curve that lies in an arbitrarily thin strip around $\pi_e$.
Therefore, the number of crossings between $\gamma_W$ and the image of $C_i$ in $\phi^*_i$ is identical to the number of crossings the image of path $W$ and the image of $C_i$ in $\phi^*_i$, which is at most $\hat\mathsf{cr}(W)$.
It follows that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
\end{proof}
\fi
\paragraph{Step 2.}
For each vertex $v\in V({\mathcal{W}}_i)$, we denote by $x_v$ the point that represents the image of $v$ in $\phi^*_i$, and we let $X$ contains all points of $\set{x_v\mid v\in V({\mathcal{W}}_i)}$ that are intersections between curves in $\Gamma_i$.
We now manipulate the curves in $\set{\gamma_W\mid W\in {\mathcal{W}}_i}$ at points of $X$, by processing points of $X$ one-by-one, as follows.
Consider a point $x_v$ that is an intersection between curves in $\Gamma_i$, where $v\in V({\mathcal{W}}_i)$, and let $D_v$ be an arbitrarily small disc around $x_v$.
We denote by ${\mathcal{W}}_i(v)$ the set of paths in ${\mathcal{W}}_i$ that contains $v$, and further partition it into two sets: ${\mathcal{W}}^{\textsf{thr}}_i(v)={\mathcal{W}}_i(v)\cap {\mathcal{W}}^{\textsf{thr}}_i$ and $\tilde{\mathcal{W}}_i(v)={\mathcal{W}}_i(v)\cap \tilde{\mathcal{W}}_i$. We apply the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $\set{\gamma_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ within disc $D_v$. Let $\set{\gamma'_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ be the set of new curves that we obtain. From \Cref{obs: curve_manipulation},
(i) for each path $W\in \tilde{\mathcal{W}}_i(v)$, the curve $\gamma'_W$ does not contain $x_v$, and is identical to the curve $\gamma_W$ outside the disc $D_v$;
(ii) the segments of curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc $D_v$ are in general position; and
(iii) the number of icrossings between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside $D_v$ is bounded by $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\iffalse{just for backup}
\begin{proof}
Denote $d=\deg_G(v)$ and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
For each path $W\in \tilde{\mathcal{W}}_i(v)$, we denote by $p^{-}_W$ and $p^{+}_W$ the intersections between the curve $\gamma_W$ and the boundary of ${\mathcal{D}}_v$.
We now compute, for each $W\in W\in \tilde{\mathcal{W}}_i(v)$, a curve $\zeta_W$ in ${\mathcal{D}}_v$ connecting $p^{-}_W$ to $p^{+}_W$, such that (i) the curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ are in general position; and (ii) for each pair $W,W'$ of paths, the curves $\zeta_W$ and $\zeta_{W'}$ intersects iff the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{-}_{W'},p^{+}_{W},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_{W'},p^{-}_{W},p^{+}_{W'})$.
It is clear that this can be achieved by first setting, for each $W$, the curve $\zeta_W$ to be the line segment connecting $p^{-}_W$ to $p^{+}_W$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
We now define, for each $W$, the curve $\gamma'_W$ to be the union of the part of $\gamma_W$ outside ${\mathcal{D}}_v$ and the curve $\zeta_W$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, the first and the second condition of \Cref{obs: curve_manipulation} are satisfied.
It remains to estimate the number of intersections between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$, which equals the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$.
Since the paths in $\tilde{\mathcal{W}}_i(v)$ are non-transversal with respect to $\Sigma$ (from \Cref{obs: wset_i_non_interfering}), from the construction of curves $\set{\gamma_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$, if a pair $W,W'$ of paths in $\tilde {\mathcal{W}}_i(v)$ do not share edges of $\delta(v)$, then the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_W,p^{-}_{W'},p^{+}_{W'})$, and therefore the curves $\zeta_{W}$ and $\zeta_{W'}$ will not intersect in ${\mathcal{D}}_v$.
Therefore, only the curves $\zeta_W$ and $\zeta_{W'}$ intersect iff $W$ and $W'$ share an edge of $\delta(v)$.
Since every such pair of curves intersects at most once, the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$ is at most $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are shown in black, and curves of $\tilde{\mathcal{W}}_i(v)$ are shown in blue, red, orange and green. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are not modified, while curves of $\tilde{\mathcal{W}}_i(v)$ are re-routed via dash lines within disc ${\mathcal{D}}_v$.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the step of processing $x_v$.}\label{fig: curve_con}
\end{figure}
\fi
We then replace the curves of $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ in $\Gamma_i$ by the curves of $\set{\gamma'_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
This completes the description of the iteration of processing point the point $x_v\in X$. Let $\Gamma'_i$ be the set of curves that we obtain after processing all points in $X$ in this way. Note that we have never modified the curves of $\Gamma^{\textsf{thr}}_i$, so $\Gamma^{\textsf{thr}}_i\subseteq\Gamma'_i$, and we denote $\tilde\Gamma'_i=\Gamma'_i\setminus \Gamma^{\textsf{thr}}_i$.
We use the following observation.
\begin{observation}
\label{obs: general_position}
Curves in $\tilde\Gamma'_i$ are in general position, and if a point $p$ lies on more than two curves of $\Gamma'_i$, then either $p$ is an endpoint of all curves containing it, or all curves containing $p$ belong to $\Gamma^{\textsf{thr}}_i$.
\end{observation}
\begin{proof}
From the construction of curves in $\Gamma_i$, any point that belong to at least three curves of $\Gamma_i$ must be the image of some vertex in $\phi^*$. From~\Cref{obs: curve_manipulation}, curves in $\tilde\Gamma'_i$ are in general position; curves in $\tilde\Gamma'_i$ do not contain any vertex-image in $\phi^*$ except for their endpoints; and they do not contain any intersection of a pair of paths in $\Gamma_i^{\textsf{thr}}$. \Cref{obs: general_position} now follows.
\end{proof}
\paragraph{Step 3.}
So far we have obtained a set $\Gamma'_i$ of curves that are further partitioned into two sets $\Gamma'_i=\Gamma^{\textsf{thr}}_i\cup \tilde\Gamma'_i$,
where set $\tilde\Gamma'_i$ contains, for each path $W\in \tilde {\mathcal{W}}_i$, a curve $\gamma'_W$ connecting the endpoints of $W$, and the curves in $\tilde\Gamma'_i$ are in general position;
and set $\Gamma^{\textsf{thr}}_i$ contains, for each path $W\in {\mathcal{W}}^{\textsf{thr}}_i$, a curve $\gamma_W$ connecting the endpoints of $W$.
Recall that all paths in ${\mathcal{W}}^{\textsf{thr}}_i$ connects $u_{i-1}$ to $u_{i+1}$.
Let $z_{\operatorname{left}}$ be the point that represents the image of $u_{i-1}$ in $\phi_i^*$ and let $z_{\operatorname{right}}$ be the point that represents the image of $u_{i+1}$ in $\phi_i^*$.
Then, all curves in $\Gamma^{\textsf{thr}}_i$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
We view $z_{\operatorname{left}}$ as the first endpoint of curves in $\Gamma^{\textsf{thr}}_i$ and view $z_{\operatorname{right}}$ as their last endpoint.
We then apply the algorithm in \Cref{thm: type-2 uncrossing}, where we let $\Gamma=\Gamma^{\textsf{thr}}_i$ and let $\Gamma_0$ be the set of all other curves in the drawing $\phi^*_i$. Let $\Gamma^{\textsf{thr}'}_i$ be the set of curves we obtain. We then designate, for each edge $e\in E^{\textsf{thr}}_i$, a curve in $\Gamma^{\textsf{thr}'}_i$ as $\gamma'_{W_e}$, such that the curves of $\set{\gamma'_{W_e}\mid e\in \hat E_{i-1}\cup E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i}$ enters $z_{\operatorname{left}}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
Recall that ${\mathcal{W}}_i=\set{W_e\mid e\in (E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\cup E_i^{\operatorname{right}}\cup \hat E_{i-1}\cup \hat E_i)}$, and, for each edge $e\in E_i^{\operatorname{left}}\cup \hat E_{i-1}$, the curve $\gamma'_{W_e}$ connects its endpoint in $C_i$ to $z_{\operatorname{left}}$; for each edge $e\in E_i^{\operatorname{right}}\cup \hat E_{i}$, the curve $\gamma'_{W_e}$ connects the endpoint of $e$ to $z_{\operatorname{right}}$; and for each edge $e\in E_i^{\textsf{thr}}$, the curve $\gamma'_{W_e}$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
Therefore, if we view $z_{\operatorname{left}}$ as the image of $v^{\operatorname{left}}_i$, view $z_{\operatorname{right}}$ as the image of $v^{\operatorname{right}}_i$, and for each edge $e\in E(G_i)\setminus E(C_i)$, view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi'_i$.
It is clear from the construction of curves in $\set{\gamma'_{W_e}\mid e\in E(G_i)\setminus E(C_i)}$ that this drawing respects all rotations in $\Sigma_i$ on vertices of $V(C_i)$ and vertex $v^{\operatorname{left}}_i$.
However, the drawing $\phi'_i$ may not respect the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$. We further modify the drawing $\phi'_i$ at $z_{\operatorname{right}}$ in the last step.
\fi
\section{Main Tool: Subinstances and Hierarchical Bad Clusterings}
\label{sec: hierarchnical clustering}
Let $I^*=(G^*,\Sigma^*)$ be the input instance of the \textnormal{\textsf{MCNwRS}}\xspace problem. Throughout this paper, we always denote by $m^*$ the number of edges in $G^*$, and we use a parameter $\mu=2^{(\log m^*)^{3/4}}$.
Our algorithm for solving the \textnormal{\textsf{MCNwRS}}\xspace problem is recursive, and, over the course of the recursion, we will consider some instances $(G,\Sigma)$, that we call \emph{subinstances} of $G^*$.
\subsection{Subinstances of the Input instance}
\label{subsec: subinstances}
A subinstance $I=(G,\Sigma)$ of the input instance $I^*=(G^*,\Sigma^*)$ is determined by a sub-graph $G'\subseteq G^*$, and a collection $\set{S_1,\ldots,S_q}$ of disjoint subsets of vertices of $G'$. Graph $G$ is obtained from $G'$ by unifying, for all $1\leq i\leq q$, all vertices of $S_i$ into a vertex $u_i$. We keep parallel edges but remove self-loops. For every regular vertex $v\in V(G)$ that is also a vertex of $G^*$, its rotation system ${\mathcal{O}}_v$ in $\Sigma$ remains the same as in $\Sigma^*$, while for every vertex $u_i$ corresponding to set $S_i$, we will define its rotation system ${\mathcal{O}}_{u_i}$ explicitly. Notice that the subinstance relation is transitive: if $(G_1,\Sigma_1)$ is a subinstance of $(G_0,\Sigma_0)$, and $(G_2,\Sigma_2)$ is a subinstance of $(G_1,\Sigma_1)$, then $(G_2,\Sigma_2)$ is a subinstance of $(G_0,\Sigma_0)$.
Our algorithm proceeds by first decomposing the input instance $I^*$ of \textnormal{\textsf{MCNwRS}}\xspace into a collection $\set{I_1,\ldots,I_r}$ of subinstances. Each such subinstance is then solved recursively, and can be in turn further decomposed into subinstances.
We will ensure that the total number of recursive levels is bounded by $\ell= \frac{\log(m^*)}{\log \mu}\leq (\log m^*)^{1/4}$.
Each level $i$ of the recursion is then associated with a collection ${\mathcal{J}}_i$ of subinstances of the input instance $I^*$.
\subsection{Main Parameters}
Before we define the hierarchical bad clustering we need to introduce some parameters.
Recall that we have defined the parameter $\mu=2^{(\log m^*)^{3/4}}$, and that the number of the recursive levels is $\ell\leq \frac{\log m^*}{\log \mu}\leq (\log m^*)^{1/4}$.
We will use the following parameters for bandwidth property of clusters: $\alpha^*=1/(\log m^*)^3$, and for $1\leq i\leq \ell$, $\alpha_i=(\alpha^*)^i$. Note that $\alpha_{\ell}=(\alpha^*)^{\ell}\leq 1/(\log m^*)^{3(\log m^*)^{1/4}}\geq 1/ 2^{O((\log m^*)^{1/4}\log\log m^*)}$.
The second central parameter that we use is:
\[\eta^*=\frac{(\log m^*)^{50}}{\alpha_{\ell}^3}\leq 2^{O((\log m^*)^{1/4}\log\log m^*)}.\]
We also define $\eta_1=(\eta^*)^9$, and, for $1\leq i<\ell$, we let $\eta_{i+1}=\frac{\eta_i(\log m^*)^{51}}{\alpha^8\cdot \alpha_{\ell}^2}$.
Note that:
\begin{equation}\label{eq: bound on etaell}
\eta_{\ell}\leq \frac{\eta_1\cdot(\log m^*)^{O(\ell)}}{\alpha_{\ell}^{3\ell}}\leq (\eta^*)^9\cdot 2^{O((\log m^*)^{1/2}\log\log m^*)}\leq 2^{O((\log m^*)^{1/2}\log\log m^*)}.
\end{equation}
\subsection{Hierarchical Bad Clustering}
One of the main tools that our algorithm uses is Hierarchical Bad Clustering. Throughout this subsection, we assume that we are given some subinstance $I=(G,\Sigma)$ of the input instance $I^*$ of \textnormal{\textsf{MCNwRS}}\xspace problem.
Given a connected subgraph $C\subseteq G$ (that we refer to as a \emph{cluster}), we let the $\Sigma_C$ be the rotation system for the cluster $C$ that ins induced by the rotation system $\Sigma$ for $G$. Therefore, we can associate, with a cluster $C\subseteq G$, an instance $(C,\Sigma_C)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem. We denote by $k(C)=|\delta_G(C)|$.
We start by defining a level-1 bad cluster.
\begin{definition}[Level-1 bad cluster]
We say that a connected vertex-induced subgraph $C\subseteq G$ is a \emph{level-1 bad cluster} iff:
\begin{itemize}
\item cluster $C$ has the $\alpha_1$-bandwidth property in $G$;
\item $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq \frac{(k(C))^2}{\eta_1}$; and
\item $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq \frac{(|E(C)|+k(C))^2}{\eta_{\ell}}$.
\end{itemize}
\end{definition}
In order to define a level-$i$ bad cluster, suppose we are given some subgraph $C$ of $G$, and a collection ${\mathcal{B}}$ of disjoint subgraphs of $C$, each of which is a bad cluster from levels $1,\ldots,i-1$. Recall that $C_{|{\mathcal{B}}}$ is a contracted graph, that is obtained from $C$, by contracting every cluster $B\in {\mathcal{B}}$ into a supernode. We denote: $m^{\tiny\textsf{out}}_{{\mathcal{B}}}(C)=|E(C_{|{\mathcal{B}}})|+k(C)$.
Note that $C_{|{\mathcal{B}}}$ is a subgraph of the contracted graph $G_{|{\mathcal{B}}}$.
\begin{definition}[Level-i bad cluster]
Given a connected a connected vertex-induced subgraph $C\subseteq G$, and a collection ${\mathcal{B}}$ of disjoint vertex-induced subgraphs of $C$, such that for all $B\in {\mathcal{B}}$, $B$ is a level-$i'$ bad cluster, for $1\leq i'< i$, we say that $C$ is a \emph{level-i bad cluster} (with respect to ${\mathcal{B}}$) iff:
\begin{itemize}
\item cluster $C_{|{\mathcal{B}}}$ has the $\alpha^*$-bandwidth property in $G_{|{\mathcal{B}}}$;
\item $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq \frac{(k(C))^2}{\eta_i}$; and
\item $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq \frac{(m^{\tiny\textsf{out}}(C))^2}{\eta_{\ell}}$.
\end{itemize}
\end{definition}
We will denote by ${\mathcal{B}}(C)$ the set ${\mathcal{B}}$ of clusters, such that $C$ is a level-$i$ bad cluster with respect to ${\mathcal{B}}$.
Notice that, from the above definition, if cluster $C$ is a level-$i$ bad cluster with respect to cluster set ${\mathcal{B}}(C)$, then it is also a level-$j$ bad cluster, for all $j>i$ (this is true since we allow ${\mathcal{B}}(C')=\emptyset$, in which case $C'_{|{\mathcal{B}}(C')}=C'$. Therefore, we can assume w.l.o.g. that, if $C$ is a level-$i$ bad cluster with respect to a collection ${\mathcal{B}}(C)$ of its subgraphs, then every clusters in ${\mathcal{B}}(C)$ is a level-$(i-1)$ cluster.
We also associate, with each bad cluster $C$, a hierarchical decomposition ${\mathcal{H}}(C)$. If $C$ is a level-$1$ bad cluster, then ${\mathcal{H}}(C)$ contains a single cluster -- the cluster $C$.
If $C$ is a level-$i$ bad cluster with respect to cluster set ${\mathcal{B}}(C)$, then we set ${\mathcal{H}}(C)=\bigcup_{C'\in {\mathcal{B}}(C)}{\mathcal{H}}(C')$.
We say that every cluster in set ${\mathcal{B}}(C)$ is a \emph{child-cluster} of $C$, and all clusters in ${\mathcal{H}}(C)$ are \emph{descendant-clusters} of $C$.
Whenever we say that we are given a level-$i$ bad cluster $C$, we assume that we are given its corresponding hierachical decomposition ${\mathcal{H}}(C)$; note that the corresponding set ${\mathcal{B}}(C)$ of clusters is implicit from ${\mathcal{H}}(C)$.
We will use the following simple observation regarding bad clusters.
\begin{observation}\label{obs: bandwidth property}
Let $I=(G,\Sigma)$ be an instance of \textnormal{\textsf{MCNwRS}}\xspace and let $C\subseteq G$ be a level-$i$ bad cluster in $G$, for some $1\leq i\leq \ell$, with respect to the hierarchical decomposition ${\mathcal{H}}(C)$. Then cluster $C$ has the $(\alpha^*)^i$-bandwidth property in graph $G$.
\end{observation}
\begin{proof}
The proof is by induction on $i$. For $i=1$, if $C$ is a level-$1$ bad cluster, then, from the definition, $C$ has the $\alpha_1$-bandwidth property in $G$, and $\alpha_1=\alpha^*$.
Consider now some cluster $C$, that is a level-$i$ bad cluster with respect to cluster set ${\mathcal{B}}={\mathcal{B}}(C)$. From the induction hypothesis, each cluster $C'\in {\mathcal{B}}$ has the $(\alpha^*)^{i-1}$-bandwidth property in $G$, and from the definition of level-$i$ bad cluster, graph $C_{|{\mathcal{B}}}$ has the $\alpha^*$-bandwidth property in $G_{{\mathcal{B}}}$.
Let $G'$ be the graph obtained from $G$ as follows. First, we subdivide every edge $e\in \delta_G(C)$ with a vertex $t_e$, and set $T=\set{t_e\mid e\in \delta_G(C)}$. We then let $G'$ be the subgraph of the resulting graph induced by $V(C)\cup T$. Consider now the graph $G'_{|{\mathcal{B}}}$.
Alternatively, this graph can be obtained from $G_{|{\mathcal{B}}}$ via a similar procedure: subdivide every edge $e\in \delta_G(C)$ with a vertex $t_e$, and let $G'_{|{\mathcal{B}}}$ be a subgraph of the resulting graph induced by $V(C_{|{\mathcal{B}}})\cup T$. Since cluster $C_{|{\mathcal{B}}}$ has the $\alpha^*$-bandwidth property in $G_{{\mathcal{B}}}$, vertex set $T$ is $\alpha^*$-well-linked in $G'_{{\mathcal{B}}}$. From \Cref{clm: contracted_graph_well_linkedness}, vertex set $T$ is $(\alpha^*)^{i-1}\cdot \alpha^*=(\alpha^*)^{i}$-well-linked in $G'$. It is then easy to verify that cluster $C$ has the $(\alpha^*)^{i}$-bandwidth property in $G$.
\end{proof}
In our recursive algorithm for the \ensuremath{\mathsf{MCNwRS}}\xspace problem, for every instance $I=(G,\Sigma)\in {\mathcal{J}}_i$ at the $i$th recursive level, we will also compute a collection ${\mathcal{B}}$ of bad level-$i$ clusters for $G$.
We denote by $m^{\tiny\textsf{out}}_{{\mathcal{B}}}(G)=|E(G_{|{\mathcal{B}}})|$ the number of edges in the corresponding contracted graph. We will ensure that, for every instance $I=(G,\Sigma)\in {\mathcal{J}}_i$, $m^{\tiny\textsf{out}}_{{\mathcal{B}}}(G)\leq m^*/\mu^{i-1}$. In other words, at every level of the recursion, the number of edges in the contracted graphs that we obtain decreases by factor $\mu$. Since $\mu=2^{(\log m^*)^{3/4}}$, this ensures that the number of the recursive levels is indeed bounded by $\ell=(\log m^*)^{1/4}$.
It will be convenient for us to consider special drawings of the instances $I$, that are \emph{canonical} with respect to their corresponding bad clusterings. In the following subsection we define such drawings and provide some results regarding their existance, and an algorithm for computing such a drawing.
\subsection{Canonical Drawing with Respect to Bad Clusters}
Suppose we are given some subinstance $I=(G,\Sigma)$ of the input instance $I^*$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and a collection ${\mathcal{B}}$ of disjoint level-$i$ bad clusters for $I$. We will consider special types of solutions to instance $I$, that are \emph{canonical} with respect to the clusters in ${\mathcal{B}}$. In this subsection we start by defining such canonical solutions. We then prove that there is an efficient algorithm to compute a canonical solution to instance $I$, whose cost is comparable to $\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m^{\tiny\textsf{out}}_{{\mathcal{B}}}(G)$. Lastly, we show that any solution to instance $I$ can be converted into a canonical solution, with only a slight increase in the solution cost.
We now define canonical drawings.
\begin{definition}[Canonical Drawing with Respect to Level-1 Bad Cluster]
Let $I=(G,\Sigma)$ be an instance of $\ensuremath{\mathsf{MCNwRS}}\xspace$, let $\phi$ be any solution to this instance, and let $C$ be a level-$1$ bad cluster for $I$.
We say that $\phi$ is a \emph{canonical solution with respect to $C$} iff there is a disc $D(C)$ containing the image of all vertices and edges of $C$, such that the images of all vertices and edges of $G\setminus C$ are disjoint from $D(C)$, and for every edge $e\in \delta_G(C)$, the intersection of the image of $e$ with the disc $D(C)$ is a simple curve, so in particular the intersection of the image of $e$ with the boundary of $D$ is a single point.
\end{definition}
\begin{definition}[Canonical Drawing with Respect to Level-i Bad Cluster]
Let $I=(G,\Sigma)$ be an instance of $\ensuremath{\mathsf{MCNwRS}}\xspace$, let $\phi$ be any solution to this instance, and let $C$ be a level-$i$ bad cluster with respect to a collection ${\mathcal{B}}(C)$ of disjoint level-$(i-1)$ bad clusters that are subgraphs of $C$.
We say that $\phi$ is a \emph{canonical solution with respect to $C$} iff:
\begin{itemize}
\item for every cluster $C'\in {\mathcal{B}}(C)$, drawing $\phi$ is canonical with respect to $C'$; and
\item there is a disc $D(C)$ containing the image of all vertices and edges of $C$, such that the images of all vertices and edges of $G\setminus C$ are disjoint from $C$, and for every edge $e\in \delta_G(C)$, the intersection of the image of $e$ with the disc $D(C)$ is a simple curve.
\end{itemize}
\end{definition}
The following simple claim allows us to compute a canonical solution for an instance $(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem with a set ${\mathcal{B}}_i$ of disjoint level-$i$ bad clusters.
\begin{claim}\label{claim: find trivial canonical solution}
There is an efficient algorithm, that, given an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, together with a collection ${\mathcal{B}}_i$ of disjoint level-$i$ bad clusters for $I$, computes a solution $\phi$ to instance $I$, with at most $(m^{\tiny\textsf{out}}_{{\mathcal{B}}_i}(G))^2+O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot \eta_{\ell}\cdot i\textsf{right} )$ crossings.
\end{claim}
\begin{proof}
For all $1\leq i'< i$, we define a collection ${\mathcal{B}}_{i'}$ of level-$i'$ bad clusters, as follows. Note that set ${\mathcal{B}}_i$ of level-$i$ bad clusters is given as part of the input to the algorithm. For $1\leq i'<i$, assume that set ${\mathcal{B}}_{i'+1}$ of level-$(i'+1)$ bad clusters is given. We then set ${\mathcal{B}}_{i'}=\bigcup_{C'\in {\mathcal{B}}_{i'+1}}{\mathcal{B}}(C')$.
For all $1\leq i'\leq i$, we let $G_{i'}=G_{|{\mathcal{B}}_{i'}}$ be the contracted graph constructed with respect to the clusters in ${\mathcal{B}}_{i'}$. We define a rotation system $\Sigma_{i'}$ for graph $G_{i'}$ as follows. For every vertex $v\in V(G_{i'})$ that is also a vertex of $G$, its rotation ${\mathcal{O}}^{i'}_v$ in $\Sigma_{i'}$ remains the same as in $\Sigma$. For a vertex $v\in V(G_{i'})$ that corresponds to some cluster $C'\in {\mathcal{B}}_{i'}$, we let the rotation ${\mathcal{O}}_v^{i'}\in \Sigma^{i'}$ be an arbitrary ordering of its incident edges in $G_{i'}$.
We perform $i+1$ iterations. For $1\leq i'\leq i$, in iteration $i'$, we construct a solution $\phi_{i'}$ to instance $(G_{i'},{\mathcal{O}}_{i'})$ of \ensuremath{\mathsf{MCNwRS}}\xspace.
In the first iteration, we consider instance $(G_i,\Sigma_i)$; recall that $G_i=G_{|{\mathcal{B}}_i}$, and $|E(G_i)|=m^{\tiny\textsf{out}}_{{\mathcal{B}}_i}(G)$. We use the algorithm from \Cref{thm: crwrs_uncrossing} to construct a solution $\phi_i$ to instance $(G_i,\Sigma_i)$, such that the number of crossings in $\phi_i$ is bounded by $|E(G_i)|^2\leq (m^{\tiny\textsf{out}}_{{\mathcal{B}}_i}(G))^2$.
We now fix some index $1\leq i'<i$, and assume that we are given a solution $\phi_{i'+1}$ to instance $(G_{i'+1},\Sigma_{i'+1})$. We show an algorithm to construct a solution $\phi_{i'}$ to instance $(G_{i'},\Sigma_{i'})$.
Consider a cluster $C'\in {\mathcal{B}}_{i'}$, and the corresponding set ${\mathcal{B}}(C')$ of clusters, such that $C'$ is a level-$i'$ bad cluster with respect to ${\mathcal{B}}(C')$. Consider the contracted graph $\tilde C'=C'_{|{\mathcal{B}}(C')}$, and recall that $|E(\tilde C')|+k(C')=m^{\tiny\textsf{out}}_{{\mathcal{B}}(C')}(C)$. Denote $\tilde {\mathcal{B}}_{i'}=\set{\tilde C'\mid C'\in {\mathcal{B}}_{i'}}$. Observe that graph $G_{i'+1}$ is precisely the contracted graph of $G_{i'}$ with respect to ${\mathcal{B}}_{i'}$, that is, $G_{i'+1}=(G_{i'})_{|{\mathcal{B}}_{i'}}$. Intuitively, in order to obtain a solution $\phi_{i'}$ to instance $(G_{i'},\Sigma_{i'})$ from solution $\phi_{i'+1}$ to instance $(G_{i'+1},\Sigma_{i'+1})$, we need to replace, for every cluster $C'\in \beta_{i'}$, the image of the vertex $v(C')$ in $\phi_{i'}$ with a drawing of $\tilde C'$.
We process each cluster $C'\in {\mathcal{B}}_{i'}$ one by one. Consider the interation when cluster $C'$ is processed. We inflate the image of the vertex $v(C')$ in the current drawing, so it becomes a disc, that we denote by $D(C')$. We denote by $\lambda(C')$ the boundary of this disc. We also let $D'(C')$ be a slightly smaller disc lying in the interior of $D(C')$, whose boundary is denoted by $\lambda'(C')$. Using the algorithm from
\Cref{thm: crwrs_uncrossing}, we construct a drawing $\psi(\tilde C')$ of graph $\tilde C'$, with at most $|E(\tilde C')|^2$ crossings. We place this drawing inside the disc $D(C')$. Next, we consider the edges of $\delta_{G_{i'}}(\tilde C')$. Let $e=(u,v)$ be any such edge, with $v\in V(C')$ and $u\not\in V(C')$. Our current drawing contains a curve $\gamma_e$ (the image of the edge $e$ in $\phi_{i'+1}$), connecting the image of $u$ (or a supernode corresponding to a cluster of ${\mathcal{B}}_{i'}$ containing $u$) to some point $p$ on $\lambda$.
We construct a curve $\gamma'_e$ inside the disc $D(C')$, connecting the image of $v$ to point $p$. It is easy to verify that the curves in set $\set{\gamma'_e\mid e\in \delta_{G_{i'}}(\tilde C')}$ can be constructed so that each such curve crosses the image of every edge of $\tilde C'$ at most once, and each pair of such curves crosses at most once. This step then adds at most $|\delta_G(C')|^2+|\delta_G(C')|\cdot |E(\tilde C')|$ new crossings.
\mynote{maybe this should be explained better. Endpoints of edge $e$ may sit deep inside the subclusters.}
Overall, the number of crossings that were added while processing cluster $C'$ is bounded by:
\[ |E(\tilde C')|^2+|\delta_G(C')|^2+|\delta_G(C')|\cdot |E(\tilde C')| \leq (m^{\tiny\textsf{out}}_{{\mathcal{B}}(C')})^2+k(C')\cdot m^{\tiny\textsf{out}}_{{\mathcal{B}}(C')}+(k(C'))^2\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(C',\Sigma_{C'})\cdot \eta_{\ell}) \]
(we have used the definition of type-$i'$ bad clusters for the last inequality.)
Once every cluster $C'\in {\mathcal{B}}_{i'}$ is processed, we obtain a solution $\phi_{i'}$ to instance $(G_{i'},\Sigma_{i'})$. Since $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\geq \sum_{C'\in {\mathcal{B}}_{i'}}\mathsf{OPT}_{\mathsf{cnwrs}}(C',\Sigma_{C'})$, we get that:
\[\mathsf{cr}(\phi_{i'})\leq \mathsf{cr}(\phi_{i'+1})+O\textsf{left}(\eta_{\ell}\cdot \sum_{C'\in {\mathcal{B}}_{i'}}\mathsf{OPT}_{\mathsf{cnwrs}}(C',\Sigma_{C'})\textsf{right} )\leq \mathsf{cr}(\phi_{i'+1})+O\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot \eta_{\ell}\textsf{right} ). \]
In the last iteration, where $i'=0$, for every cluster $C'\in {\mathcal{B}}_1$, we view ${\mathcal{B}}(C')=\emptyset$, and $\tilde C'=C'$. The remainder of the algorithm remains unchanged. At the end of this iteration, we then obtain drawing $\phi_0$ of instance $(G,\Sigma)$. It is immediate to verify that this drawing is canonical with respect to clusters in ${\mathcal{B}}_i$. The number of crossings in drawing $\phi_i$ is bounded by $(m^{\tiny\textsf{out}}_{{\mathcal{B}}_i}(G))^2$, while in each subsequent iteration, the number of crossings grows by at $O\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot \eta_{\ell}\textsf{right} )$. Therefore, the total number of crossings in the final drawing is bounded by $(m^{\tiny\textsf{out}}_{{\mathcal{B}}_i}(G))^2+O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot \eta_{\ell}\cdot i\textsf{right} )$.
\end{proof}
We obtain the following immediate corollary, that follows from the definition of a level-$i$ bad cluster.
Given any cluster $C$ of a graph $G$, we define its corresponding graph $C^+$ as follows. First, we subdivide every edge $e\in \delta_G(C)$ with a vertex $t_e$, and let $T=\set{t_e\mid e\in \delta_G(C)}$ be the set of these new vertices. We then let $C^+$ be the subgraph of the resulting graph induced by $V(C)\cup T$. Rotation system $\Sigma$ for $G$ naturally defines a rotation system $\Sigma_{C^+}$ for $C^+$.
\begin{corollary}\label{claim: find good drawing of type-i cluster}
There is an efficient algorithm, that, given an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, and a level-$i$ bad cluster $C$ for $G$, computes a solution $\phi(C^+)$ to instance $(C^+,\Sigma_{C^+})$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and a disc $D(C)$ containing the drawing $\phi(C)$, such that the vertices of $T$ are drawn on the boundary of $D(C)$, and $\mathsf{cr}(\phi(C^+))\leq O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\cdot \eta_{\ell}\cdot i\textsf{right} )$. Moreover, the drawing $\phi(C^+)$ is canonical with respect to the clusters in ${\mathcal{B}}(C)$.
\end{corollary}
\begin{proof}
The proof easily follows by applying the algorithm from \Cref{claim: find trivial canonical solution} to graph $C^+$.
Let $\phi$ be the resulting drawing of $C^+$, together with the disc $D(C)$ containing the drawing of $C$. For every vertex $t\in T$, let $e_t$ be the unique edge of $C^+$ incident to $t$, and let $p_t$ be the unique point on the intersection of the image of $e_t$ and the boundary of disc $D(C)$. We slighlty modify the drawing $\phi$, by placing the image of every vertex $t\in T$ at point $p_t$, and then erasing the portion of the image of $e_t$ that lies outside the disc $D(C)$. From \Cref{claim: find trivial canonical solution}, the number of crossings in drawing $\phi$ is bounded by:
\[(m^{\tiny\textsf{out}}_{{\mathcal{B}}(C)}(C^+))^2+O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(C^+,\Sigma_{C}^+)\cdot \eta_{\ell}\cdot i\textsf{right} ).\]
Since the vertices of $T$ have degree $1$ in $C^+$, it is easy to verify that $\mathsf{OPT}_{\mathsf{cnwrs}}(C^+,\Sigma_{C}^+)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)$. Moreover, $m^{\tiny\textsf{out}}_{{\mathcal{B}}(C)}(C^+)=m^{\tiny\textsf{out}}_{{\mathcal{B}}(C)}(C)$, and, from the definition of a level-$i$ bad cluster, $(m^{\tiny\textsf{out}}_{{\mathcal{B}}(C)}(C))^2\leq \mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\cdot \eta_{\ell}$. Therefore, altogether,
the number of crossings in the resulting drawing is at most $O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_{C})\cdot \eta_{\ell}\cdot i\textsf{right} )$.
\end{proof}
Lastly, we will use the following corollary to transform an arbitrary drawing of a graph $G$ into a drawing that is canonical with respect to a set ${\mathcal{B}}_i$ of level-$i$ bad clusters of $G$.
\begin{lemma}\label{lemma: transform any drawing into canonical}
There is an efficient algorithm, that, given an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem with $|V(G)|=n$, together with a collection ${\mathcal{B}}_i$ of disjoint level-$i$ bad clusters for $I$, and a solution $\phi$ to instance $I$, computes a solution $\phi'$ to instance $I$ that is canonical with respect to all clusters in ${\mathcal{B}}_i$, such that:
\[\mathsf{cr}(\phi')\leq O\textsf{left} (\frac{\mathsf{cr}(\phi)\cdot \log^8n}{(\alpha^*)^{2i}}\textsf{right} )+O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot \eta_{\ell}\cdot i\textsf{right} ).\]
\end{lemma}
\begin{proof}
For convenience, we denote by $\hat G=G_{|{\mathcal{B}}_i}$ the contracted graph of $G$ with respect to cluster set ${\mathcal{B}}_i$.
From \Cref{obs: bandwidth property}, every cluster in ${\mathcal{B}}_i$ has the $(\alpha^*)^i$-bandwidth property. We can then use the algorithm from \Cref{lem: crossings in contr graph} in order to compute a drawing $\hat \phi$ of $\hat G$, such that, for every vertex $x\in V(\hat G)\cap V(G)$, the ordering of the edges of $\delta_G(x)$ as they enter $x$ in $\hat \phi$ is consistent with the ordering ${\mathcal{O}}_x\in \Sigma$. Moreover, $\mathsf{cr}(\hat \phi)\leq O\textsf{left} (\frac{\mathsf{cr}(\phi)\cdot \log^8n}{(\alpha^*)^{2i}}\textsf{right} )$. Next, we process each cluster $C\in {\mathcal{B}}_i$ one by one. Consider any such cluster $C$, and the image $p_C$ of the vertex $v_C$ representing the cluster $C$ in the contracted graph $\hat G$, in the current drawing.
We inflate point $p_C$ until it becomes a disc, that we denote by $D'(C)$. We denote by $\lambda'(C)$ the boundary of this disc. We also place a smaller disc $D(C)$ in the interior of $D'(C)$, and denote its boundary by $\lambda(C)$. Using the algorithm from
\Cref{claim: find good drawing of type-i cluster}, we compute a solution $\psi(C^+)$ to instance $(C^+,\Sigma_{C^+})$ of \ensuremath{\mathsf{MCNwRS}}\xspace, which is placed inside the disc $D(C)$, such that the images of the vertices in set $\set{t_e\mid e\in \delta_G(C)}$ appear on the boundary of disc $D(C)$. Recall that the drawing $\psi(C^+)$ is canonical with respect to the clusters in ${\mathcal{B}}(C)$, and contains at most $O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\cdot \eta_{\ell}\cdot i\textsf{right} )$ crossings.
Consider now some edge $e\in \delta_G(C)$, and assume that $e=(u,v)$, where $u\in V(C)$ and $v\not\in V(C)$. Let $\gamma_1(e)$ be the original image of the edge $e$; observe that $\gamma_1(e)$ connects the image of the vertex $v$ to some point $p_e$ on the boundary of the disc $D'(C)$. Let $\gamma_2(e)$ be the image of the edge $(u,t_e)$ in drawing $\psi(C^+)$, and the image of $t_e$ lies on the boundary of the disc $D(C)$. We define another curve, $\gamma_3(e)$, connecting the image of $t_e$ to point $p_e$, such that the curve $\gamma_3(e)$ is contained in $D'(C)\setminus D(C)$, except for its one endpoint that lies on the boundary of $D(C)$. We can define the curves in $\set{\gamma_3(e)\mid e\in \delta_G(C)}$ such that every pair of such curves crosses at most once. This incurs at most $(k(C))^2\leq \eta_{\ell}\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)$ additional crossings. Note that the current iteration contributed at most $O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\cdot \eta_{\ell}\cdot i\textsf{right} )$ new crossings to the current drawing.
Once every cluster $C\in {\mathcal{B}}_i$ is processed, we obtain the final drawing $\phi'$, which is a solution to instance $(G,\Sigma)$, that is canonical with respect to all clusters in ${\mathcal{B}}_i$. Since $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\geq \sum_{C\in {\mathcal{B}}_i}\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)$, we get that:
\[
\begin{split}
\mathsf{cr}(\phi')&\leq O\textsf{left} (\frac{\mathsf{cr}(\phi)\cdot \log^8n}{(\alpha^*)^{2i}}\textsf{right} )+O\textsf{left}(\sum_{C\in {\mathcal{B}}_i}\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\cdot \eta_{\ell}\cdot i\textsf{right} )\\
&\leq O\textsf{left} (\frac{\mathsf{cr}(\phi)\cdot \log^8n}{(\alpha^*)^{2i}}\textsf{right} )+O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot \eta_{\ell}\cdot i\textsf{right} ).\end{split}\]
\end{proof}
\subsection{Our Techniques}
\label{subsec: techniques}
In this subsection we provide an overview of the techniques used in the proof of our main technical result, \Cref{thm: main_rotation_system}. For the sake of clarity of exposition, some of the discussion here is somewhat imprecise. Our algorithm relies on the divide-and-conquer technique. Given an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, we compute a collection ${\mathcal{I}}$ of new instances, whose corresponding graphs are significantly smaller than $G$, and then solve each of the resulting new instances separately. Collection ${\mathcal{I}}$ of instances is called a \emph{decomposition of $I$}. We require that the decomposition has several useful properties that will allow us to use it in order to obtain the guarantees from \Cref{thm: main_rotation_system}, by solving the instances in ${\mathcal{I}}$ recursively. Before we define the notion of decomposition of an instance, we need the notion of a \emph{contracted graph}, that we use throughout the paper. Suppose $G$ is a graph, and let ${\mathcal{R}}=\set{R_1,\ldots,R_q}$ be a collection of disjoint subsets of vertices of $G$. The contracted graph of $G$ with respect to ${\mathcal{R}}$, that we denote by $G_{|{\mathcal{R}}}$, is a graph that is obtained from $G$, by contracting, for all $1\leq i\leq q$, the vertices of $R_i$ into a supernode $u_i$. Note that every edge of the resulting graph $G_{|{\mathcal{R}}}$ corresponds to some edge of $G$, and we do not distinguish between them.
The vertices in set $V(G_{|{\mathcal{R}}})\setminus \set{u_1,\ldots,u_q}$ are called \emph{regular vertices}. Each such vertex $v$ also lies in $G$, and moreover, $\delta_{G_{|{\mathcal{R}}}}(v)=\delta_G(v)$. Abusing the notation, given a collection ${\mathcal{C}}=\set{C_1,\ldots,C_r}$ of disjoint subgraphs of $G$, we denote by $G_{|{\mathcal{C}}}$ the contracted graph of $G$ with respect to the collection $\set{V(C_1),\ldots,V(C_r)}$ of subsets of vertices of $G$.
Given a graph $G$ and its drawing $\phi$, we denote by $\mathsf{cr}(\phi)$ the number of crossings in $\phi$.
\noindent{\bf Decomposition of an Instance.}
Given an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, we will informally refer to $|E(G)|$ as the \emph{size of the instance}.
Assume that we are given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$, and some parameter $\eta$ (we will generally use $\eta=2^{O((\log m)^{3/4}\log\log m)}$). Assume further that we are given another collection ${\mathcal{I}}$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace. We say that ${\mathcal{I}}$ is an \emph{$\eta$-decomposition of $I$}, if $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\operatorname{poly}\log m$, and $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot \eta$. Additionally, we require that there is an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace(I)$, that, given a feasible solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}$, computes a feasible solution $\phi$ for instance $I$, with at most $O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$ crossings.
At a high level, our algorithm starts with the input instance $I^*=(G^*,\Sigma^*)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem. Throughout the algorithm, we denote $m^*=|E(G^*)|$, and we use a parameter $\mu=2^{O((\log m^*)^{7/8}\log\log m^*)}$. Over the course of the algorithm, we consider various other instances $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace, but parameters $m^*$ and $\mu$ remain unchanged, and they are defined with respect to the original input instance $I^*$. The main subroutine of the algorithm, that we call \ensuremath{\mathsf{AlgDecompose}}\xspace, receives as input an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and computes an $\eta$-decomposition ${\mathcal{I}}$ of $I$, for $\eta=2^{O((\log m)^{3/4}\log\log m)}$, where $m=|E(G)|$. The subroutine additionally ensures that every instance in the decomposition is sufficiently small compared to $I$, that is, for each instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\leq |E(G)|/\mu$.
We note that this subroutine is in fact randomized, and, instead of ensuring that $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot \eta$, it only ensures this in expectation. We will ignore this minor technicality in this high-level exposition.
It is now easy to complete the proof of \Cref{thm: main_rotation_system} using Algorithm $\ensuremath{\mathsf{AlgDecompose}}\xspace$: we simply apply Algorithm $\ensuremath{\mathsf{AlgDecompose}}\xspace$ to the input instance $I^*$, obtaining a collection ${\mathcal{I}}$ of new instances. We recursively solve each instance in ${\mathcal{I}}$, and then combine the resulting solutions using Algorithm $\ensuremath{\mathsf{Alg}}\xspace(I^*)$, in order to obtain the final solution to instance $I^*$. Since the sizes of the instances decrease by the factor of at least $\mu$ with each application of the algorithm, the depth of the recursion is bounded by $O\textsf{left} ((\log m^*)^{1/8}\textsf{right} )$. At each recursive level, the sum of the optimal solution costs and of the number of edges in all instances at that recursive level increases by at most factor $2^{O((\log m^*)^{3/4}\log\log m^*)}$, leading to the final bound of $2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+m^*\textsf{right})$ on the solution cost.
From now on we focus on the description of Algorithm \ensuremath{\mathsf{AlgDecompose}}\xspace.
We start by describing several technical tools that this algorithm builds on. Throughout, given a graph $G$, we refer to connected vertex-induced subgraphs of $G$ as \emph{clusters}. Given a collection ${\mathcal{C}}$ of disjoint clusters of $G$, we denote by $E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})$ the set of all edges $e\in E(G)$, such that the endpoints of $e$ do not lie in the same cluster of ${\mathcal{C}}$.
We will also use the notion of \emph{subinstances} that we define next.
\paragraph{Subinstances.} Suppose we are given two instances $I=(G,\Sigma)$ and $I'=(G',\Sigma')$ of \ensuremath{\mathsf{MCNwRS}}\xspace. We say that $I'$ is a \emph{subinstance} of instance $I$, if the following hold. First, graph $G'$ must be a graph that is obtained from a subgraph of $G$ by contracting some subsets of its vertices into supernodes. Formally\footnote{We note that this definition closely resembles the notion of graph minors, but, in contrast to the definition of minors, we do not require that the induced subgraphs $\set{G[R_i]}_{1\leq i\leq q}$ are connected.}, there must be a graph $G''\subseteq G$, and a collection ${\mathcal{R}}=\set{R_1,\ldots,R_q}$ of disjoint subsets of vertices of $G''$, such that $G'=G''_{|{\mathcal{R}}}$. For every regular vertex $v$ of $G'$, the rotation ${\mathcal{O}}_v\in \Sigma'$ must be consistent with the rotation ${\mathcal{O}}_v\in \Sigma$ (recall that $\delta_{G'}(v)\subseteq \delta_G(v)$). For every supernode $u_i$ of $G'$, its rotation ${\mathcal{O}}_{u_i}\in \Sigma'$ can be chosen arbitrarily.
Note that the notion of subinstances is transitive: if $I'$ is a subinstance of $I$ and $I''$ is a subinstance of $I'$, then $I''$ is a subinstance of $I$.
The main tool that we use is \emph{disengagement of clusters}.
Intuitively, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and a collection ${\mathcal{C}}$ of disjoint clusters of $G$, the goal is to compute an $\eta$-decomposition ${\mathcal{I}}$ of $I$, such that every instance $I'=(G',\Sigma')\in {\mathcal{I}}$ is a subinstance of $I$, and moreover, there is at most one cluster $C\in {\mathcal{C}}$ that is contained in $G'$, and all edges of $G'$ that do not lie in $C$ must belong to $E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})$. Assume for now that we can design an efficient algorithm for computing such a decomposition. In this case, the high-level plan for implementing Algorithm $\ensuremath{\mathsf{AlgDecompose}}\xspace$ would be as follows. First, we compute a collection ${\mathcal{C}}$ of disjoint clusters of graph $G$, such that, for each cluster $C\in {\mathcal{C}}$, $|E(C)|\leq |E(G)|/(2\mu)$, and $|E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})|\leq |E(G)|/(2\mu)$. Then we perform disengagement of clusters in ${\mathcal{C}}$, obtaining an $\eta$-decomposition of the input instance $I$. We are then guaranteed that each resulting instance in ${\mathcal{I}}$ is sufficiently small. We note that it is not immediately clear how to compute the desired collection ${\mathcal{C}}$ of disjoint clusters of $G$; we discuss this later. For now we focus on algorithms for computing disengagement of clusters. We do not currently have an algorithm to compute the disengagement of clusters in the most general setting described above. In this paper, we design a number of algorithms for computing disengagement of clusters, under some conditions. We start with the simplest algorithm that only works in some restricted settings, and then generalize it to more advanced algorithms that work in more and more general settings. In order to describe the disengagement algorithm for the most basic setting, we need the notion of congestion, and of internal and external routers, that we use throughout the paper, and describe next.
\noindent{\bf Congestion, Internal Routers, and External Routers.}
Given a graph $G$ and a set ${\mathcal{P}}$ of paths in $G$, the \emph{congestion} that the set ${\mathcal{P}}$ of paths causes on an edge $e\in E(G)$, that we denote by $\cong_G({\mathcal{P}},e)$, is the number of paths in ${\mathcal{P}}$ containing $e$. The total congestion caused by the set ${\mathcal{P}}$ of paths in $G$ is $\cong_G({\mathcal{P}})=\max_{e\in E(G)}\set{\cong_G({\mathcal{P}},e)}$.
Consider now a graph $G$ and a cluster $C\subseteq G$. We denote by $\delta_G(C)$ the set of all edges $e\in E(G)$, such that exactly one endpoint of $e$ lies in $C$. An \emph{internal $C$-router} is a collection ${\mathcal{Q}}(C)=\set{Q(e)\mid e\in \delta_G(C)}$ of paths, such that, for each edge $e\in \delta_G(C)$, path $Q(e)$ has $e$ as its first edge, and all its inner vertices lie in $C$. We additionally require that all paths in ${\mathcal{Q}}(C)$ terminate at a single vertex of $C$, that we call the \emph{center vertex of the router}. Similarly, an \emph{external $C$-router}
is a collection ${\mathcal{Q}}'(C)=\set{Q'(e)\mid e\in \delta_G(C)}$ of paths, such that, for each edge $e\in \delta_G(C)$, path $Q'(e)$ has $e$ as its first edge, and all its inner vertices lie in $V(G)\setminus V(C)$. We additionally require that all paths in ${\mathcal{Q}}'(C)$ terminate at a single vertex of $V(G)\setminus V(C)$, that we call the \emph{center vertex of the router}. For a cluster $C\subseteq G$, we denote by $\Lambda_G(C)$ and $\Lambda'_G(C)$ the sets of all internal and all external $C$-routers, respectively.
\noindent{\bf Basic Cluster Disengagement.}
In the most basic setting for cluster disengagement, we are given an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, and a collection ${\mathcal{C}}$ of disjoint clusters of $G$. Additionally, for each cluster $C\in {\mathcal{C}}$, we are given an internal $C$-router $Q(C)$, whose center vertex we denote by $u(C)$, and an external $C$-router $Q'(C)$, whose center vertex we denote by $u'(C)$. The output of the disengagement procedure is a collection ${\mathcal{I}}$ of subinstances of $I$, that consists of a single global instance $\hat I=(\hat G,\hat \Sigma)$, and, for every cluster $C\in {\mathcal{C}}$, an instance $I_C=(G_C,\Sigma_C)$ associated with it. Graph $\hat G$ is the contracted graph of $G$ with respect to ${\mathcal{C}}$; that is, it is obtained from $G$ by contracting every cluster $C\in {\mathcal{C}}$ into a supernode $v_C$. For each cluster $C\in {\mathcal{C}}$, graph $G_C$ is obtained from $G$ by contracting the vertices of $V(G)\setminus V(C)$ into a supernode $v^*_C$. For every cluster $C\in {\mathcal{C}}$, the rotation ${\mathcal{O}}_{v_C}\in \hat \Sigma$ of the supernode $v_C$ in instance $\hat I$ and the rotation ${\mathcal{O}}_{v^*_C}\in \Sigma_C$ of the supernode $v^*_C$ in instance $I_C$ need to be defined carefully, in order to ensure that the sum of the optimal solution costs of all resulting instances is low, and that we can combine the solutions to these instances to obtain a solution to $I$. Observe that the set of edges incident to vertex $v_C$ in $\hat G$ and the set of edges incident to vertex $v^*_C$ in $G_C$ are both equal to $\delta_G(C)$. We define a single ordering ${\mathcal{O}}^C$ of the edge set $\delta_G(C)$, that will serve both as the rotation ${\mathcal{O}}_{v_C}\in \hat \Sigma$, and as the rotation ${\mathcal{O}}_{v^*_C}\in \Sigma_C$. The ordering ${\mathcal{O}}^C$ is defined via the internal $C$-router ${\mathcal{Q}}(C)$, and the order in which the images of the paths of ${\mathcal{Q}}(C)$ enter the image of vertex $u(C)$.
\iffalse
\znote{maybe omit the detailed explanation here?}
Specifically, denote $\delta_G(C)=\set{e_1,\ldots,e_z}$. For all $1\leq i\leq z$, denote by $e'_i$ the last edge on path $Q(e_i)\in {\mathcal{Q}}(C)$. Note that all resulting edges in multi-set $E'=\set{e'_1,\ldots,e'_z}$ are incident to vertex $u(C)\in V(C)$ (set $E'$ is a multi-set because an edge my belong to several paths in ${\mathcal{Q}}(C)$). Note that the rotation ${\mathcal{O}}_{u(C)}\in \Sigma$ naturally defines a circular ordering $\tilde {\mathcal{O}}$ of the edges in multiset $E'$ (where we break ties between copies of the same edge arbitrarily). This ordering $\tilde {\mathcal{O}}$ of the edges of $E'$ then naturally defines an ordering ${\mathcal{O}}^C$ of the edges in $\delta_G(C)$: we obtain the ordering ${\mathcal{O}}^C$ from the ordering $\tilde {\mathcal{O}}$ by replacing, for each $1\leq i\leq z$, edge $e'_i$ with edge $e_i$.
\fi
On the one hand, letting ${\mathcal{O}}_{v_C}={\mathcal{O}}_{v^*_C}$ for every cluster $C\in {\mathcal{C}}$ allows us to easily combine solutions $\phi(I')$ to instances $I'\in {\mathcal{I}}$, in order to obtain a solution to instance $I$, whose cost is at most $O\textsf{left}(\sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I'))\textsf{right} )$. On the other hand, defining ${\mathcal{O}}^C$ via the set ${\mathcal{Q}}(C)$ of paths, for each cluster $C\in {\mathcal{C}}$, allows us to bound $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')$.
We now briefly describe how this latter bound is established, since it will motivate the remainder of the algorithm and clarify the bottlenecks of this approach. We consider an optimal solution $\phi^*$ to instance $I$, and we use it in order to construct, for each instance $I'\in {\mathcal{I}}$, a solution $\psi(I')$, such that $\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\psi(I'))$ is relatively small compared to $\mathsf{cr}(\phi^*)+|E(G)|$.
In order to construct a solution $\psi(\hat I)$ to the global instance $\hat I$, we start with solution $\phi^*$ to instance $I$. We erase from this solution all edges and vertices that lie in the clusters of ${\mathcal{C}}$. For each cluster $C\in {\mathcal{C}}$, we let the image of the supernode $v_C$ coincide with the original image of the vertex $u(C)$ -- the center of the internal $C$-router $Q(C)$. In order to draw the edges that are incident to the supernode $v_C$ in $\hat G$ (that is, the edges of $\delta_G(C)$), we utilize the images of the paths of the internal $C$-router ${\mathcal{Q}}(C)$ in $\phi^*$, that connect, for each edge $e\in \delta_G(C)$, the original image of edge $e$ to the original image of vertex $u(C)$.
Consider now some cluster $C\in {\mathcal{C}}$. In order to construct a solution $\psi(I_C)$ to instance $I_C$, we start again with the solution $\phi^*$ to instance $I$. We erase from it all edges and vertices except for those lying in $C$. We let the image of the supernode $v^*_C$ be the original image of vertex $u'(C)$ -- the center of the external $C$-router ${\mathcal{Q}}'(C)$. In order to draw the edges that are incident to the supernode $v^*_C$ in $G_C$ (that is, the edges of $\delta_G(C)$), we utilize the images of the paths of the external $C$-router ${\mathcal{Q}}'(C)$, that connect, for each edge $e\in \delta_G(C)$, the original image of edge $e$ to the original image of vertex $u'(C)$.
Observe that the only increase in $\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\psi(I'))$, relatively to $\mathsf{cr}(\phi^*)$, is due to the crossings incurred by drawing the edges incident to the supernodes in $\set{v_C}_{C\in {\mathcal{C}}}$ in instance $\hat I$, and for each subinstance $I_C$, drawing the edges incident to supernode $v^*_C$. All such edges are drawn along the images of the paths in $\bigcup_{C\in {\mathcal{C}}}({\mathcal{Q}}(C)\cup {\mathcal{Q}}'(C))$ in $\phi^*$. However, an edge may belong to a number of such paths. With careful accounting we can bound this number of new crossings as follows. Assume that, for every cluster $C\in {\mathcal{C}}$, $\cong_G({\mathcal{Q}}'(C))\leq \beta$. Assume further that, for each cluster $C\in {\mathcal{C}}$, and for each edge $e\in E(C)$, $(\cong_G({\mathcal{Q}}(C),e))^2\leq \beta$. Then $\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\psi(I'))\leq O(\beta^2\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|))$. Therefore, in order to ensure that the collection ${\mathcal{I}}$ of subinstances of $I$ that we have obtained via the cluster disengagement procedure is an $\eta$-decomposition of $I$, we need to ensure that, for every cluster $C\in {\mathcal{C}}$, $\cong_G({\mathcal{Q}}'(C))\leq \beta$, and, for every edge $e\in E(C)$, $(\cong_G({\mathcal{Q}}(C),e))^2\leq \beta$, for $\beta=O(\eta^{1/2})$. This requirement seems impossible to achieve. For example, if maximum vertex degree in graph $G$ is small (say a constant), then some edges incident to the center vertices $\set{u(C),u'(C)}_{C\in {\mathcal{C}}}$ must incur very high congestion. In order to overcome this obstacle, we slightly weaken our requirements. Instead of providing, for every cluster $C\in {\mathcal{C}}$, a single internal $C$-router $Q(C)$, and a single external $C$-router $Q'(C)$, it is sufficient for us to obtain, for each cluster $C\in {\mathcal{C}}$, a \emph{distribution} ${\mathcal{D}}(C)$ over the collection $\Lambda_G(C)$ of internal $C$-routers, such that, for every edge $e\in E(C)$, $\expect[{\mathcal{Q}}(C)\sim {\mathcal{D}}(C)]{(\cong_G({\mathcal{Q}}(C),e))^2}\leq \beta$, and a distribution ${\mathcal{D}}'(C)$ over
the collection $\Lambda'_G(C)$ of external $C$-routers, such that for every edge $e$, $\expect[{\mathcal{Q}}'(C)\sim {\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta$.
To recap, in order to use the \textsf{Basic Cluster Disengagement}\xspace procedure described above to compute an $\eta$-decomposition of the input instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace into sufficiently small instances, it is now enough to design a procedure that, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, computes a collection ${\mathcal{C}}$ of disjoint clusters of $G$, and, for every cluster $C\in {\mathcal{C}}$, a distribution ${\mathcal{D}}(C)$ over the collection $\Lambda_G(C)$ of internal $C$-routers, such that, for every edge $e\in E(C)$, $\expect[{\mathcal{Q}}(C)\sim {\mathcal{D}}(C)]{(\cong_G({\mathcal{Q}}(C),e))^2}\leq \beta$, together with a distribution ${\mathcal{D}}'(C)$ over
the collection $\Lambda'_G(C)$ of external $C$-routers, such that, for every edge $e$, $\expect[{\mathcal{Q}}'(C)\sim {\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta$, for $\beta=O(\sqrt\eta)$.
Additionally, we need to ensure that, for every cluster $C\in {\mathcal{C}}$, $|E(C)|\leq |E(G)|/(2\mu)$, and that $|E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})|\leq |E(G)|/(2\mu)$. While computing a collection ${\mathcal{C}}$ of clusters with the latter two properties seems possible (at least when the maximum vertex degree in $G$ is small), computing the distributions over the internal and the external routers for each cluster $C$ with the required properties seems quite challenging. As a first step towards this goal, we employ the standard notions of well-linkedness and bandwidth property of clusters as a proxy to constructing internal $C$-routers with the required properties. Before we turn to discuss these notions, we note that the \textsf{Basic Cluster Disengagement}\xspace procedure that we have just described can be easily generalized to the more general setting, where the set ${\mathcal{C}}$ of clusters is laminar (instead of only containing disjoint clusters).
This generalization will be useful for us later.
Assume that we are given a laminar family ${\mathcal{C}}$ of clusters (that is, for every pair $C,C'\in {\mathcal{C}}$ of clusters, either $C\subseteq C'$, or $C'\subseteq C$, or $C\cap C'=\emptyset$ holds), with $G\in {\mathcal{C}}$. Assume further that we are given, for each cluster $C\in {\mathcal{C}}$, a distribution ${\mathcal{D}}(C)$ over the collection $\Lambda_G(C)$ of internal $C$-routers, in which, for every edge $e\in E(C)$, $\expect[{\mathcal{Q}}(C)\sim {\mathcal{D}}(C)]{(\cong_G({\mathcal{Q}}(C),e))^2}\leq \beta$, together with a distribution ${\mathcal{D}}'(C)$ over
the collection $\Lambda'_G(C)$ of external $C$-routers, where for every edge $e$, $\expect[{\mathcal{Q}}'(C)\sim {\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta$, for some parameter $\beta$.
The \textsf{Basic Cluster Disengagement}\xspace procedure, when applied to ${\mathcal{C}}$, produces a collection ${\mathcal{I}}=\set{I_C=(G_C,\Sigma_C)\mid C\in {\mathcal{C}}}$ of instances. For every cluster $C\in {\mathcal{C}}$, graph $G_C$ associated with instance $I_C$ is obtained from graph $G$, by first contracting all vertices of $V(G)\setminus V(C)$ into a supernode $v^*_C$, and then contracting, for each child-cluster $C'\in {\mathcal{C}}$ of $C$, the vertices of $V(C')$ into a supernode $v_{C'}$. We define, for every cluster $C$, an ordering of the set $\delta_G(C)$ of edges via an internal $C$-router that is selected from the distribution ${\mathcal{D}}(C)$, and we let the rotation ${\mathcal{O}}_{v^*_C}$ in the rotation system $\Sigma_C$, and the rotation ${\mathcal{O}}_{v_C}$ in the rotation system $\Sigma_{C'}$, where $C'$ is the parent-cluster of $C$, to be identical to this ordering.
Using the same reasoning as in the case where ${\mathcal{C}}$ is a set of disjoint clusters, we show that $\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq O\textsf{left}( \beta^2\cdot \mathsf{dep}({\mathcal{C}})\cdot(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\textsf{right} )$, where $\mathsf{dep}({\mathcal{C}})$ is the depth of the laminar family ${\mathcal{C}}$ of clusters. We then show that ${\mathcal{I}}'$ is an $\eta'$-decomposition of instance $I$, where $\eta'=O(\beta^2\cdot \mathsf{dep}({\mathcal{C}}))$.
As noted already, one of the difficulties in exploiting the \textsf{Basic Cluster Disengagement}\xspace procedure in order to compute an $\eta$-decomposition of the input instance ${\mathcal{I}}$ is the need to compute distributions over the sets of internal and the external $C$-routers for every cluster $C\in{\mathcal{C}}$, with the required properties. We turn instead to the notions of well-linkedness and bandwidth properties of clusters. These notions are extensively studied, and there are many known algorithms for computing a collection ${\mathcal{C}}$ of clusters that have bandwidth property in a graph. We will use this property as a proxy, that will eventually allow us to construct a distribution over the sets of internal $C$-routers for each cluster $C\in {\mathcal{C}}$, with the required properties.
{\bf Well-Linkedness, Bandwidth Property, and Cluster Classification.}
We use the standard graph-theoretic notion of well-linkedness. Let $G$ be a graph, let $T$ be a subset of the vertices of $G$, and let $0<\alpha<1$ be a parameter. We say that the set $T$ of vertices is \emph{$\alpha$-well-linked} in $G$ if for every partition $(A,B)$ of vertices of $G$ into two subsets, $|E_G(A,B)|\geq \alpha\cdot\min\set{|A\cap T|,|B\cap T|}$.
We also use a closely related notion of \emph{bandwidth property} of clusters. Suppose we are given a graph $G$ and a cluster $C\subseteq G$. Intuitively, cluster $C$ has the $\alpha$-bandwidth property (for a parameter $0<\alpha<1$), if the edges of $\delta_G(C)$ are $\alpha$-well-linked in $C$. More formally, we consider the augmentation $C^+$ of cluster $C$, that is defined as follows. We start with the graph $G$, and subdivide every edge $e\in \delta_G(C)$ with a vertex $t_e$, denoting by $T=\set{t_e\mid e\in \delta_G(C)}$ this new set of vertices. The augmentation $C^+$ of $C$ is the subgraph of the resulting graph induced by $V(C)\cup T$. We say that cluster $C$ has the $\alpha$-bandwidth property if set $T$ of vertices is $\alpha$-well-linked in $C^+$.
We note that, if a cluster $C$ has the $\alpha$-bandwidth property, then, using known techniques, we can efficiently construct a distribution ${\mathcal{D}}$ over the set $\Lambda_G(C)$ of internal $C$-routers, such that, for every edge $e\in E(C)$, $\expect[{\mathcal{Q}}(C)\sim {\mathcal{D}}(C)]{\cong({\mathcal{Q}}(C),e)}\leq O(1/\alpha)$.
However, in order to use the \textsf{Basic Cluster Disengagement}\xspace procedure, we need a stronger property: namely, for every edge $e\in E(C)$, we require that $\expect[{\mathcal{Q}}(C)\sim {\mathcal{D}}(C)]{(\cong({\mathcal{Q}}(C),e))^2}\leq \beta$, for some parameter $\beta$. If we are given a distribution ${\mathcal{D}}(C)$ over the set $\Lambda_G(C)$ of internal $C$-routers with this latter property, then we say that cluster $C$ is \emph{$\eta$-light} with respect to ${\mathcal{D}}(C)$. Computing a distribution ${\mathcal{D}}(C)$ for which cluster $C$ is $\eta$-light is a much more challenging task. We come close to achieving it in our Cluster Classification Theorem. Before we describe the theorem, we need one more definition. Let $C$ be a cluster of a graph $G$, and let $\eta'$ be some parameter. Assume that we are given some rotation system $\Sigma$ for graph $G$, and let $\Sigma^C$ be the rotation system for cluster $C$ that is induced by $\Sigma$. Let $I^C=(C,\Sigma^C)$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. We say that cluster $C$ is \emph{$\eta'$-bad} if $\mathsf{OPT}_{\mathsf{cnwrs}}(I^C)\geq |\delta_G(C)|^2/\eta'$.
In the Cluster Classification Theorem, we provide an efficient algorithm, that, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$, and a cluster $C\subseteq G$ that has the $\alpha$-bandwidth property (where $\alpha=\Omega(1/\operatorname{poly}\log m)$), either correctly establishes that cluster $C$ is $\eta'$-bad, for $\eta'=2^{O((\log m)^{3/4}\log\log m)}$, or produces a distribution ${\mathcal{D}}(C)$ over the set $\Lambda_G(C)$ of internal $C$-routers, such that cluster $C$ is $\beta$-light with respect to ${\mathcal{D}}(C)$, for $\beta=2^{O(\sqrt{\log m}\cdot \log \log m)}$.
In fact, the algorithm is randomized, and, with a small probability, it may erroneously classify cluster $C$ as being $\eta'$-bad, when this is not the case. This small technicality is immaterial to this high-level exposition, and we will ignore it here.
The proof of the Cluster Classification Theorem is long and technically involved, and is partially responsible for the high approximation factor that we eventually obtain. It is our hope that a simpler and a cleaner proof of the theorem with better parameters will be discovered in the future. We believe that the theorem is a graph-theoretic result that is interesting in its own right. We now provide a high-level summary of the main challenges in its proof.
At the heart of the proof is an algorithm that we called \ensuremath{\mathsf{AlgFindGuiding}}\xspace. Suppose we are given an instance $I=(H,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and a set $T$ of $k$ vertices of $H$ called terminals, that are $\alpha$-well-linked in $H$, for some parameter $0<\alpha<1$. Denote $C=H\setminus T$ and $|V(H)|=n$. The goal of the algorithm is to either establish that $\mathsf{OPT}_{\mathsf{cnwrs}}(H)+|E(H)|\geq k^2\operatorname{poly}(\alpha/\log n)$; or to compute a distribution ${\mathcal{D}}(C)$ over internal $C$-routers, such that cluster $C$ is $\eta'=\operatorname{poly}(\log n/\alpha)$-light with respect to ${\mathcal{D}}(C)$.
Consider first a much simpler setting, where $H$ is a grid graph, and $T$ is the set of vertices on the first row of the grid. For this special case, the algorithm of \cite{Tasos-comm} (see also Lemma D.10 in the full version of \cite{chuzhoy2011algorithm}) provides the construction of a distribution ${\mathcal{D}}(C)$ over internal $C$-routers with the required properties.
This result can be easily generalized to the case where $H$ is a bounded-degree planar graph, since such a graph must contain a large grid minor. If $H$ is a planar graph, but its maximum vertex degree is no longer bounded, we can still compute a grid-like structure in it, and apply the same arguments as in \cite{Tasos-comm} in order to compute the desired distribution ${\mathcal{D}}(C)$.
The difficulty in our case is that the graph $H$ may be far from being planar, and, even though, from the Excluded Grid theorem of Robertson and Seymour \cite{robertson1986graph,robertson1994quickly},
it must contain a large grid-like structure, without having a drawing of $H$ in the plane with a small number of crossings, we do not know how to compute such a structure\footnote{We note that we need the grid-like structure to have dimensions $(k'\times k')$, where $k'$ is almost linear in $k$. Therefore, we cannot use the known bounds for the Excluded Minor Theorem (e.g. from \cite{chuzhoy2019towards}) for general graphs, and instead we need to use an analogue of the stronger version of the theorem for planar graphs.}. We provide an algorithm that either establishes that $\mathsf{OPT}_{\mathsf{cnwrs}}(H)$ is large compared to $k^2$, or computes a grid-like structure in graph $H$, even if it is not a planar graph. Unfortunately, this algorithm only works in the setting where $|E(H)|$ is not too large comparable to $k$. Specifically, if we ensure that $|E(H)|\leq k\cdot \hat \eta$ for some parameter $\hat \eta$, then the algorithm either computes a distribution ${\mathcal{D}}(C)$ over internal $C$-routers that is $\eta'$-light (with $\eta'=\operatorname{poly}(\log n/\alpha)$ as before), or it establishes that $\mathsf{OPT}_{\mathsf{cnwrs}}(H)+|E(H)|\geq k^2\operatorname{poly}(\alpha/(\hat \eta\log n))$.
Typically, this algorithm would be used in the following setting: we are given a cluster $C$ of a graph $G$, that has the $\alpha$-bandwidth property. We then let $H=C^+$ be the augmentation of $C$, and we let $T$ be the set of vertices of $C^+$ corresponding to the edges of $\delta_H(C)$. In order for this result to be meaningful, we need to ensure that $|E(C)|$ is not too large compared to $|\delta_H(C)|$. Unfortunately, we may need to apply the classification theorem to clusters $C$ for which $|E(C)|\gg |\delta_H(C)|$ holds. In order to overcome this difficulty, given such a cluster $C$, we construct a recursive decomposition of $C$ into smaller and smaller clusters. Let ${\mathcal{L}}$ denote the resulting family of clusters, which is a laminar family of subgraphs of $C$. We ensure that every cluster $C'\in {\mathcal{L}}$ has $\alpha=\Omega(1/\operatorname{poly}\log m)$-bandwidth property, and, additionally, if we let $\hat C'$ be the graph obtained from $C'$ by contracting every child-cluster of $C'$ into a supernode, then the number of edges in $\hat C'$ is comparable to $|\delta_H(C')|$. We consider the clusters of ${\mathcal{L}}$ from smallest to largest. For each such cluster $C'$, we carefully apply Algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace to the corresponding contracted graph $\hat C'$, in order to either classify cluster $\hat C'$ as $\eta(C')$-bad, or to compute a distribution ${\mathcal{D}}(C')$ over internal $C'$-routers, such that $C'$ is $\beta(C')$-light with respect to ${\mathcal{D}}(C')$. Parameters $\eta(C')$ and $\beta(C')$ depend on the height of the cluster $C'$ in the decomposition tree that is associated with the laminar family ${\mathcal{L}}$ of clusters. This recursive algorithm is eventually used to either establish that cluster $C$ is $\eta(C)$-bad, or to compute a distribution ${\mathcal{D}}(C)$ over the set $\Lambda_G(C)$ of internal $C$-routers, such that cluster $C$ is $\beta(C)$-light with respect to ${\mathcal{D}}(C)$. The final parameters $\eta(C)$ and $\beta(C)$ depend exponentially on the height of the decomposition tree associated with the laminar family ${\mathcal{L}}$. This strong dependence on $\mathsf{dep}({\mathcal{L}})$ is one of the reasons for the high approximation factor that our algorithm eventually achieves.
\noindent{\bf Obstacles to Using \textsf{Basic Cluster Disengagement}\xspace.}
Let us now revisit the \textsf{Basic Cluster Disengagement}\xspace routine. We start with an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and denote $|E(G)|=m$. Throughout, we use a parameter $\eta=2^{O((\log m)^{3/4}\log\log m)}$, and $\beta=\eta^{1/8}$.
Recall that the input to the procedure is a collection ${\mathcal{C}}$ of disjoint clusters of $G$. For every cluster $C\in {\mathcal{C}}$, we are also given a distribution ${\mathcal{D}}'(C)$ over the set of external $C$-routers, such that, for every edge $e$, $\expect[{\mathcal{Q}}'(C)\sim {\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta$, and a distribution ${\mathcal{D}}(C)$ over the set of internal $C$-routers, such that cluster $C$ is $\beta$-light with respect to ${\mathcal{D}}(C)$.
We are then guaranteed that the collection ${\mathcal{I}}$ of subinstances of $I$ that is constructed via \textsf{Basic Cluster Disengagement}\xspace is an $\eta$-decomposition of $I$. We can slightly generalize this procedure to handle bad clusters as well.
Specifically, suppose we are given a partition $({\mathcal{C}}^{\operatorname{light}},{\mathcal{C}}^{\operatorname{bad}})$ of the clusters in ${\mathcal{C}}$, and, for each cluster $C\in {\mathcal{C}}^{\operatorname{light}}$, a distribution ${\mathcal{D}}(C)$ over internal $C$-routers, such that cluster $C$ is $\beta$-light with respect to ${\mathcal{D}}(C)$. Assume further that each cluster $C\in {\mathcal{C}}^{\operatorname{bad}}$ is $\beta$-bad. Additionally, assume that we are given, for every cluster $C\in {\mathcal{C}}$, a distribution ${\mathcal{D}}'(C)$ over external $C$-routers, such that, for every edge $e$, $\expect[{\mathcal{Q}}'(C)\sim {\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta$, and that every cluster $C\in {\mathcal{C}}$ has the $\alpha$-bandwidth property, for some $\alpha=\Omega(1/\operatorname{poly}\log m)$. We can then generalize the \textsf{Basic Cluster Disengagement}\xspace procedure to provide the same guarantees as before in this setting, to obtain an $\eta$-decomposition of instance $I$.
Assume now that we are given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with $|E(G)|=m$. For simplicity, assume for now that the maximum vertex degree in $G$ is quite small (it is sufficient, for example, that it is significantly smaller than $m$.) Using known techniques, we can compute a collection ${\mathcal{C}}$ of disjoint clusters of $G$, such that, for every cluster $C\in {\mathcal{C}}$, $|E(C)|\leq m/(2\mu)$; $|E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})|\leq m/(2\mu)$; and every cluster $C\in {\mathcal{C}}$ has $\alpha$-bandwidth property.
If we could, additionally, compute, for each cluster $C\in {\mathcal{C}}$, a distribution ${\mathcal{D}}'(C)$ over external $C$-routers, such that,
for every edge $e$, $\expect[{\mathcal{Q}}'(C)\sim {\mathcal{D}}'(C)]{\cong_G({\mathcal{Q}}'(C),e)}\leq \beta$, then we could use the Cluster Classification Theorem to partition the set ${\mathcal{C}}$ of clusters into subsets ${\mathcal{C}}^{\operatorname{light}}$ and ${\mathcal{C}}^{\operatorname{bad}}$, and to compute, for every cluster $C\in {\mathcal{C}}^{\operatorname{light}}$, a distribution ${\mathcal{D}}(C)$ over the set of its internal routers, such that every cluster in ${\mathcal{C}}^{\operatorname{bad}}$ is $\eta'$-bad, and every cluster $C\in {\mathcal{C}}^{\operatorname{light}}$ is $\eta'$-light with respect to ${\mathcal{D}}(C)$, for some parameter $\eta'$. We could then apply the \textsf{Basic Cluster Disengagement}\xspace procedure in order to compute the desired $\eta$-decomposition of the input instance $I$.
Unfortunately, we currently do not have an algorithm that computes both the collection ${\mathcal{C}}$ of clusters of $G$ with the above properties, and the required distributions over the external $C$-routers for each such cluster $C$. In order to overcome this difficulty, we design \textsf{Advanced Cluster Disengagement}\xspace procedure, that generalizes \textsf{Basic Cluster Disengagement}\xspace, and no longer requires the distribution over external $C$-routers for each cluster $C\in {\mathcal{C}}$.
\noindent{\bf \textsf{Advanced Cluster Disengagement}\xspace.}
The input to the \textsf{Advanced Cluster Disengagement}\xspace procedure is an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and a set ${\mathcal{C}}$ of disjoint clusters of $G$, that we refer to as \emph{basic clusters}. Let $m=|E(G)|$, and $\eta=2^{O((\log m)^{3/4}\log\log m)}$ as before. The output is an $\eta$-decomposition ${\mathcal{I}}$ of $I$, such that every instance $I'=(G',\Sigma')\in {\mathcal{I}}$ is a subinstance of $I$, and moreover, there is at most one basic cluster $C\in {\mathcal{C}}$ with $E(C)\subseteq E(G')$, with all other edges of $G'$ lying in $E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})$. The algorithm for the \textsf{Advanced Cluster Disengagement}\xspace and its analysis are significantly more involved than those of \textsf{Basic Cluster Disengagement}\xspace. We start with some intuition.
Consider the contracted graph $H=G_{|{\mathcal{C}}}$, and its Gomory-Hu tree $T$. This tree naturally defines a laminar family ${\mathcal{L}}$ of clusters of $H$: for every vertex $v\in V(H)$, we add to ${\mathcal{L}}$ the cluster $U_v$, that is the subgraph of $H$ induced by vertex set $V(T_v)$, where $T_v$ is the subtree of $T$ rooted at $v$. From the properties of Gomory-Hu trees, if $v'$ is the parent-vertex of vertex $v$ in $T$, there is an external $U_v$-router ${\mathcal{Q}}'(U_v)$ in graph $H$ with $\cong_H({\mathcal{Q}}'(U_v))=1$. Laminar family ${\mathcal{L}}$ of clusters of $H$ naturally defines a laminar family ${\mathcal{L}}'$ of clusters of the original graph $G$, where for each cluster $U_v\in {\mathcal{L}}$, set ${\mathcal{L}}'$ contains a corresponding cluster $U'_v$, that is obtained from $U_v$, by un-contracting all supernodes that correspond to clusters of ${\mathcal{C}}$. For each such cluster $U'_v\in {\mathcal{L}}'$, we can use the external $U_v$-router ${\mathcal{Q}}'(U_v)$ in graph $H$ in order to construct a distribution ${\mathcal{D}}'(U'_v)$ over external $U'_v$-routers in graph $G$, where for every edge $e$,
$\expect[{\mathcal{Q}}'(U'_v)\sim {\mathcal{D}}'(U'_v)]{\cong_G({\mathcal{Q}}'(U'_v),e)}\leq O(1/\alpha)$. We can then apply the \textsf{Basic Cluster Disengagement}\xspace procedure to the laminar family ${\mathcal{L}}'$ and the distributions $\set{{\mathcal{D}}'(U'_v)}_{U'_v\in {\mathcal{L}}'}$ in order to compute an $\eta^*$-decomposition ${\mathcal{I}}$ of instance $I$, where every instance in ${\mathcal{I}}$ is a subinstance of $I$. Recall that the parameter $\eta^*$ depends on the depth of the laminar family ${\mathcal{L}}'$, which is equal to the depth of the laminar family ${\mathcal{L}}$. Therefore, if $\mathsf{dep}({\mathcal{L}})$ is not too large (for example, it is at most $2^{O((\log m)^{3/4}\log\log m)}$), then we will obtain the desired $\eta$-decomposition of $I$. But unfortunately we have no control over the depth of the laminar family ${\mathcal{L}}$, and in particular the tools described so far do not work when the Gomory-Hu tree $T$ is a path.
Roughly speaking, we would like to design a different disengagement procedure for the case where the tree $T$ is a path, and then reduce the general problem (by exploiting \textsf{Basic Cluster Disengagement}\xspace) to this special case. In fact we follow a similar plan. We define a special type of instances (that we call \emph{nice instances}), that resemble the case where the Gomory-Hu tree of the contracted graph $H=G_{|{\mathcal{C}}}$ is a path. While the motivation behind the definition of nice instances is indeed this special case, the specifics of the definition are somewhat different, in that it is more general in some of its aspects, and more restrictive and well-structured in others. We provide an algorithm for Cluster Disengagment of nice instances, that ensures that, for each resulting subinstance $I'=(G',\Sigma')$, there is at most one cluster $C\in {\mathcal{C}}$ with $C\subseteq G'$, and all other edges of $G'$ lie in $E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})$. We also provide another algorithm, that, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace and a collection ${\mathcal{C}}$ of disjoint basic clusters of graph $G$, computes a decomposition ${\mathcal{I}}'$ of instance $I$, such that each resulting instance $I'=(G',\Sigma')\in {\mathcal{I}}'$ is a subinstance of $I$ and a nice instance, with respect to the subset ${\mathcal{C}}(I')\subseteq {\mathcal{C}}$ of clusters, that contains every cluster $C\in {\mathcal{C}}$ with $C\subseteq G'$.
Combining these two algorithms allows us to compute \textsf{Advanced Cluster Disengagement}\xspace.
\noindent{\bf Algorithm \ensuremath{\mathsf{AlgDecompose}}\xspace.}
Recall that Algorithm
\ensuremath{\mathsf{AlgDecompose}}\xspace, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, needs to compute an $\eta$-decomposition ${\mathcal{I}}$ of $I$, where $\eta=2^{O((\log m)^{3/4}\log\log m)}$ and $m=|E(G)|$, such that, for each instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\leq |E(G)|/\mu$.
We say that a vertex $v$ of $G$ is a \emph{high-degree vertex} if $|\delta_G(v)|\geq m/\operatorname{poly}(\mu)$ (here, $\mu=2^{O((\log m^*)^{7/8}\log\log m^*)}$, and $m^*$ is the number of edges in the original input instance $I^*$ of \ensuremath{\mathsf{MCNwRS}}\xspace).
Consider first the special case where no vertex of $G$ is a high-degree vertex. For this case, it is not hard to generalize known well-linked decomposition techniques to obtain a collection ${\mathcal{C}}$ of disjoint clusters of $G$, such that each cluster $C\in {\mathcal{C}}$ has $\alpha=\Omega(1/\operatorname{poly}\log m)$-bandwidth property, with $|E(C)|<O(m/\mu)$, and, additionally, $|E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})|\leq O(m/\mu)$. We can now apply the \textsf{Advanced Cluster Disengagement}\xspace procedure to the set ${\mathcal{C}}$ of clusters, in order to obtain the desired $\eta$-decomposition of $I$. Recall that we are guaranteed that each resulting instance $I'=(G',\Sigma')\in {\mathcal{I}}$ is a subinstance of $I$, and there is at most one cluster $C\in {\mathcal{C}}$ with $C\subseteq G'$, with all other edges of $G'$ lying in $E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})$. This ensures that $|E(G')|\leq m/\mu$, as required.
In general, however, it is possible that the input instance $I=(G,\Sigma)$ contains high-degree vertices. We then consider two cases. We say that instance $I$ is \emph{wide} if
there is a vertex $v\in V(G)$, a partition $(E_1,E_2)$ of the edges of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and a collection ${\mathcal{P}}$ of at least $m/\operatorname{poly}(\mu)$ simple edge-disjoint cycles in $G$, such that every cycle $P\in {\mathcal{P}}$ contains one edge of $E_1$ and one edge of $E_2$. An instance that is not wide is called \emph{narrow}. We provide separate algorithms for dealing with narrow and wide instances.
\noindent{\bf Narrow Instances.}
The algorithm for decomposing narrow instances relies on and generalizes the algorithm for the special case where $G$ has no high-degree vertices. As a first step, we compute a collection ${\mathcal{C}}$ of disjoint clusters of $G$, such that each cluster $C\in {\mathcal{C}}$ has $\alpha=\Omega(1/\operatorname{poly}\log m)$-bandwidth property, and $|E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})|<O(E(G)/\mu)$. The set ${\mathcal{C}}$ of clusters is partitioned into two subsets: set ${\mathcal{C}}^{s}$ of small clusters, and set ${\mathcal{C}}^{f}$ of \emph{flower clusters}. For each cluster $C\in {\mathcal{C}}^s$, $|E(C)|<O(|E(G)|/\mu)$ holds. If $C$ is a cluster of ${\mathcal{C}}^f$, then we no longer guarantee that $|E(C)|$ is small. Instead, we guarantee that cluster $C$ has a special structure. Specifically, $C$ must contain a single high-degree vertex $u(C)$, that we call the \emph{flower center}, and all other vertices of $C$ must be low-degree vertices. Additionally, there must be a set ${\mathcal{X}}(C)=\set{X_1,\ldots,X_k}$ of subgraphs of $C$, that we call \emph{petals}, such that, for all $1\leq i<j\leq k$, $V(X_i)\cap V(X_j)=\set{u(C)}$. We also require that, for all $1\leq i\leq k$, there is a set $E_i$ of $\Theta(m/\operatorname{poly}(\mu))$ edges of $\delta_G(u(C))$ that are contiguous in the rotation ${\mathcal{O}}_{u(C)}\in \Sigma$, and lie in $X_i$ (see \Cref{fig: flower_cluster intro}). Lastly, we require that, for all $1\leq i\leq k$, there is a set ${\mathcal{Q}}_i$ of edge-disjoint paths, connecting every edge of $\delta_G(X_i)\setminus\delta_G(u(C))$ to vertex $u(C)$, with all inner vertices on the paths lying in $X_i$.
\begin{figure}[h]
\centering
\scalebox{0.13}{\includegraphics{figs/flower_1.jpg}}
\caption{An illustration of a $4$-petal flower cluster.}\label{fig: flower_cluster intro}
\end{figure}
We apply \textsf{Advanced Cluster Disengagement}\xspace to the set ${\mathcal{C}}$ of clusters, in order to compute an initial decomposition ${\mathcal{I}}_1$ of the input instance $I$, such that every instance in ${\mathcal{I}}_1$ is a subinstance of $I$. Unfortunately, it is possible that, for some instances $I'=(G',\Sigma')\in {\mathcal{I}}_1$, $|E(G')|>m/\mu$. For each such instance $I'$, there must be some flower cluster $C\in {\mathcal{C}}^f$ that is contained in $G'$, and all other edges of $G'$ must lie in $E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})$.
We now consider each instance $I'=(G',\Sigma')\in {\mathcal{I}}_1$ with $|E(G')|>m/\mu$ one by one. Assume that $C\in {\mathcal{C}}^f$ is the flower cluster that is contained in $G'$, and ${\mathcal{X}}(C)=\set{X_1,\ldots,X_k}$ is the set of its petals. We further decompose instance $I'$ into a collection ${\mathcal{I}}(C)$ of subinstances, that consists of a single global instance $\hat I(C)$, and $k$ additional instances $I_1(C),\ldots,I_k(C)$. We ensure that the graph $\hat G(C)$ associated with the global instance $\hat I(C)$ only contains edges of $E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})$, so $|E(\hat G(C))|<m/\mu$ holds. For all $1\leq j\leq k$, graph $G_j(C)$ associated with instance $I_j(C)\in {\mathcal{I}}(C)$ contains the petal $X_j$, and all other edges of $G_j(C)$ lie in $E^{\textnormal{\textsf{out}}}_G({\mathcal{C}})$. We note that unfortunately it is still possible that, for some $1\leq j\leq k$, graph $G_j(C)$ contains too many edges (this can only happen if $|E(X_j)|$ is large). However, our construction ensures that, for each such instance $I_j(C)$, no high-degree vertices lie in graph $G_j(C)$. We can then further decompose instance $I_j(C)$ into subinstances using the algorithm that we designed for the case where no vertex of the input graph is a high-degree vertex. After this final decomposition, we are guaranteed that each of the final subinstances of instance $I$ that we obtain contains fewer than $m/\mu$ edges, as required.
\noindent{\bf Wide Instances.}
Suppose we are given a wide instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace. In this case, we compute an $\eta$-decomposition ${\mathcal{I}}$ of instance $I$, such that, for each resulting instance $I'=(G',\Sigma')\in {\mathcal{I}}$, either $|E(G')|<m/\mu$ (in which case we say that $I'$ is a \emph{small instance}), or $I'$ is a narrow instance. We will then further decompose each resulting narrow instance using the algorithm described above.
In order to obtain the decomposition ${\mathcal{I}}$ of $I$, we start with ${\mathcal{I}}=\set{I}$. As long as set ${\mathcal{I}}$ contains at least one wide instance $I'=(G',\Sigma')$ with $|E(G')|\geq m/\mu$, we perform a procedure that ``splits'' instance $I'$ into two smaller subinstances. We now turn to describe this procedure at a high level.
Let $I'=(G',\Sigma')\in {\mathcal{I}}$ be a wide instance with $|E(G')|\geq m/\mu$.
Recall that from the definition of a wide instance, there is a vertex $v\in V(G')$, a partition $(E_1,E_2)$ of the edges of $\delta_{G'}(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma'$, and a collection ${\mathcal{P}}$ of at least $m/\operatorname{poly}(\mu)$ simple edge-disjoint cycles in $G'$, such that every cycle in ${\mathcal{P}}$ contains one edge of $E_1$ and one edge of $E_2$.
Consider the experiment, in which we choose a cycle $W\in {\mathcal{P}}$ uniformly at random. Since $|{\mathcal{P}}|$ is very large, with reasonably high probability, the edges of the cycle $W$ participate in relatively few crossings in the optimal solution to instance $I'$ of \ensuremath{\mathsf{MCNwRS}}\xspace. We show that with high enough probability, there is a near-optimal solution to $I'$, in which cycle $W$ is drawn in the natural way. We use the cycle $W$ in order to partition instance $I'$ into two subinstances $I_1,I_2$ (intuitively, one subinstance corresponds to edges and vertices of $G'$ that are drawn ``inside'' the cycle $W$ in the near-optimal solution to $I'$, and the other subinstance contains all edges and vertices that are drawn ``outside'' $W$). Each of the resulting two instances contains the cycle $W$, and, in order to be able to combine the solutions to the two subinstances to obtain a solution to $I'$, we contract all vertices and edges of $W$, in each of the two instances, into a supernode. Let $I_1',I_2'$ denote these two resulting instances. The main difficulty in the analysis is to show that there is a solution to each resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace, such that the sum of the costs of two solutions is close to $\mathsf{OPT}_{\mathsf{cnwrs}}(I')$. The difficulty arises from the fact that we do not know what the optimal solution to instance $I'$ looks like, and so our partition of $G'$ into two subgraphs that are drawn on different sides of the cycle $W$ in that solution may be imprecise. Instead, we need to ``fix'' the solutions to instances $I_1,I_2$ (that are induced by the optimal solution to $I'$) in order to move all edges and vertices of each subinstance to lie on one side of the cycle $W$. In fact we are unable to do so directly. Instead, we show that we can compute a relatively small collection $E'$ of edges, such that, if we remove the edges of $E'$ from the graphs corresponding to instances $I_1,I_2$, then each of the resulting subinstances has the desired structure: namely, it can be drawn completely inside or completely outside the cycle $W$ with relatively few crossings compared to $\mathsf{OPT}_{\mathsf{cnwrs}}(I')$. After we solve the two resulting subinstances recursively, we combine the resulting solutions, and add the images of the edges of $E'$ back, in order to obtain a solution to instance $I'$.
\subsection{Organization}
We start with preliminaries in \Cref{sec: short_prelim}. We then provide, in \Cref{sec: high level}, the definitions of several main concepts that we use (such as wide and narrow instances), and state three main technical theorems that allow us to decompose wide and narrow instances. We then provide the proof of \Cref{thm: main_rotation_system} using these three theorems. In \Cref{sec:long prelim} we provide additional definitions, notation and summary of known results that we use, together with some easy extensions. This section can be thought of as an expanded version of preliminaries.
We then develop our main technical tools: \textsf{Basic Cluster Disengagement}\xspace in \Cref{sec: guiding paths orderings basic disengagement}, Cluster Classification Theorem in \Cref{sec: routing within a cluster} (with parts of the proof delayed to \Cref{sec: guiding paths}), and \textsf{Advanced Cluster Disengagement}\xspace in \Cref{sec: main disengagement}.
In Sections \ref{sec: not well connected}, \ref{sec: many paths} and \ref{sec: computing the decomposition} we provide the proofs of the three main theorems. Sections \ref{sec: not well connected} and \ref{sec: many paths} deal with decomposing wide instances, and \Cref{sec: many paths} provides an algorithm for decomposing a narrow instance.
\subsubsection{Step 1: Computing an Enhancement}
\label{subsub: step 1 of phase 1 of interesting}
We start by discarding from ${\mathcal{P}}$ all paths that contain more than $32\mu^b$ edges. Since the paths in ${\mathcal{P}}$ are edge-disjoint, the number of the discarded path is bounded by $\frac{m}{32\mu^b}$, and so $|{\mathcal{P}}|\geq \frac{15m}{16\mu^b}$ continues to hold.
In order to compute an enhancement, we slightly modify the promising set ${\mathcal{P}}$ of paths, to ensure that, for every pair $P,P'\in {\mathcal{P}}$ of distinct paths, for every vertex $v\in (V(P)\cap V(P'))\setminus V(J)$, the intersection of $P$ and $P'$ at vertex $v$ is non-transversal.
Throughout, we denote by $(E_1,E_2)$ the partition of the edges of $\delta_G(J)$, with the edges of $E_1$ appearing consecutively in the ordering ${\mathcal{O}}(J)$, suh that every path $P\in {\mathcal{P}}$ has an edge of $E_1$ as its first edge and an edge of $E_2$ as its last edge.
In order to modify the set ${\mathcal{P}}$ of paths, we first subdivide every edge $e\in \delta_G(J)$ with a new vertex $t_e$, denoting $T_1=\set{t_e\mid e\in E_1}$ and $T_2=\set{t_e\mid e\in E_2}$.
Let $H$ be a new graph obtained from $G$ after we delete the vertices and the edges of $J$ from it, contract all vertices of $T_1$ into a new vertex $s$, and contract all vertices of $T_2$ into a new vertex $t$. We define a rotation system $\tilde \Sigma$ for graph $H$ in a natural way: For every vertex $v\in V(H)\setminus\set{s,t}$, its rotation ${\mathcal{O}}_v$ in $\tilde \Sigma$ remains the same as in $\Sigma$. For vertex $s$ its rotation ${\mathcal{O}}_s\in \tilde \Sigma$ is the circular ordering of the edges of $E_1$ induced by the ordering ${\mathcal{O}}(J)$.
Similarly, rotation ${\mathcal{O}}_t\in \tilde \Sigma$ is the circular ordering of the edges of $E_2$ induced by the ordering ${\mathcal{O}}(J)$.
The set ${\mathcal{P}}$ of paths in graph $G$ defines a set ${\mathcal{Q}}$ of $|{\mathcal{P}}|$ edge-disjoint paths in $H$ that connect $s$ to $t$, and are internally disjoint from $s$ and $t$. We apply the algorithm from \Cref{lem: non_interfering_paths} to graph $H$ and the set ${\mathcal{Q}}$ of paths to obtain another set ${\mathcal{Q}}'$ of $|{\mathcal{P}}|$ simple edge-disjoint paths in $H$, where every path connects $s$ to $t$ and is internally disjoint from both $s$ and $t$ as before, but now the paths are non-transversal with respect to $\tilde \Sigma$. Lastly, path set ${\mathcal{Q}}'$ naturally defines a collection ${\mathcal{P}}^*$ of $|{\mathcal{P}}|$ simple edge-disjoint paths in graph $G$, where every path in ${\mathcal{P}}^*$ contains an edge of $E_1$ as its first edge, and an edge of $E_2$ as its last edge, and is internally disjoint from $J$. Moreover, for every vertex $u\in V(G)\setminus V(J)$, for every pair $P,P'\in {\mathcal{P}}^*$ of paths that contain $u$, the intersection of $P$ and $P'$ at $u$ is non-transversal. Notice that ${\mathcal{P}}^*$ remains a promising set of paths of cardinality $|{\mathcal{P}}|$. We view the paths in ${\mathcal{P}}^*$ as being directed from the edges of $E_1$ towards the edges of $E_2$. We denote by $k=|{\mathcal{P}}^*|$, so $k=|{\mathcal{P}}|\geq \frac{15m}{16\mu^b}$.
Let $E^*_1\subseteq E_1$ be the subset of edges that belong to the paths of ${\mathcal{P}}^*$.
We denote $E^*_1=\set{e_1,\ldots,e_{k}}$, where the edges are indexed so that $e_1,\ldots,e_{k}$ appear in the order of their indices in the ordering ${\mathcal{O}}(J)$. For all $1\leq j\leq k$, we denote by $P_j\in {\mathcal{P}}^*$ the unique path originating at the edge $e_j$. We select an index $\floor{k/3}<j^*<\ceil{2k/3}$ uniformly at random, and we let $P^*=P_{j^*}$. Notice that $P^*$ is a valid enhancement for the core structure ${\mathcal{J}}$, and it is either a simple path or a simple cycle. We say that path $P^*$ is \emph{chosen} from set ${\mathcal{P}}^*$.
Notice that the probability that a path of ${\mathcal{P}}^*$ is chosen to be the enhancement path is at most $\frac{4}{k}\leq \frac{4\cdot 16\mu^b}{15m}\leq
\frac{16\mu^{b}}{m}$.
We will now define a number of bad events, and we will show that the probability that either of these events happens is low.
\paragraph{Good Paths and Bad Event ${\cal{E}}_1$.}
We need the following definition.
\begin{definition}[Good path]
We say that a path $P\in {\mathcal{P}}^*$ is \emph{good} if the following hold:
\begin{itemize}
\item the number of crossings in which the edges of $P$ participate in $\phi$ is at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}$; and
\item there are no crossings in $\phi$ between edges of $P$ and edges of $J$.
\end{itemize} A path that is not good is called a \emph{bad path}.
\end{definition}
We now bound the number of bad paths in ${\mathcal{P}}^*$.
\begin{observation}\label{obs: number of bad paths}
The number of bad paths in ${\mathcal{P}}^*$ is at most $\frac{4m}{\mu^{12b}}$.
\end{observation}
\begin{proof}
Since the paths in ${\mathcal{P}}^*$ are edge-disjoint, and every crossing involves two edges, the number of paths $P\in {\mathcal{P}}^*$ such that there are more than $\frac{\mathsf{cr}(\phi)\mu^{12b}}{m}$ crossings in $\phi$ in which the edges of $P$ participate, is at most $\frac{2m}{\mu^{12b}}$. Additionally, we are guaranteed that $|\chi^{\mathsf{dirty}}(\phi)|\le \frac{m}{\mu^{60b}}$. Therefore, the number of paths $P\in {\mathcal{P}}^*$, for which there is a crossing between an edge of $P$ and an edge of $J$, is bounded by $\frac m {\mu^{60b}}$.
Overall, the number of bad paths in ${\mathcal{P}}^*$ is bounded by $\frac{2m}{\mu^{12b}}+\frac m {\mu^{60b}}\leq \frac{4m}{\mu^{12b}}$.
\end{proof}
We say that bad event ${\cal{E}}_1$ happens if path $P^*$ is a bad path. Since the number of bad paths is bounded by $\frac{4m}{\mu^{12b}}$, and a path of ${\mathcal{P}}^*$ is chosen to be the enhancement path with probability at most $\frac{16\mu^{b}}{m}$, we immediately get the following observation.
\begin{claim}\label{claim: event 1 prob2}
$\prob{{\cal{E}}_1}\leq 64/\mu^{11b}$.
\end{claim}
\paragraph{Heavy and Light Vertices, and Bad Event ${\cal{E}}_2$.}
We use a parameter $h=\frac{512\mathsf{cr}(\phi)\cdot \mu^{26b}}{m}$. We say that a vertex $x\in V(G)$ is \emph{heavy} if at least $h$ paths of ${\mathcal{P}}^*$ contain $x$; otherwise, we say that $x$ is \emph{light}. Recall that in order to define an enhancement structure using the enhancement $P^*$, we need to define an orientation for every inner vertex of $P^*$.
Intuitively, we would like this orientation to be consistent with that in the drawing $\phi$ (which is not known to us). If $x$ is a heavy vertex lying on $P^*$, then computing such an orientation is not difficult, as we can exploit the paths of ${\mathcal{P}}^*$ containing $x$ to do so. But if $x$ is a light vertex, we do not have enough information in order to determine its orientation in $\phi$. To get around this problem, we will simply delete all edges incident to the light vertices of $P^*$, except for the edges of $P^*\cup J$, and then let the orientation of each such vertex be arbitrary. We show below that with high probability, the number of edges that we delete is relatively small.
Specifically, we denote by $E'$ the set of all edges $e$, such that $e$ is incident to some light vertex $x\in V(P^*)$, and $e\not\in E(J)\cup E(P^*)$. We say that bad event ${\cal{E}}_2$ happens if $|E'|>\frac{\mathsf{cr}(\phi)\cdot \mu^{38b}}{m}$.
We bound the probability of Event ${\cal{E}}_2$ in the next simple claim.
\begin{claim}\label{claim: third bad event bound}
$\prob{{\cal{E}}_2}\leq 2^{14}/\mu^{11b}$.
\end{claim}
\begin{proof}
Consider some light vertex $x\in V(G)$. Since $x$ lies on fewer than $h= \frac{512\mathsf{cr}(\phi)\cdot \mu^{26b}}{m}$ paths of ${\mathcal{P}}^*$, and each such path is chosen to the enhancement with probability at most $\frac{16\mu^{b}}{m}$, the probability that $x$ lies in $V(P^*)$ is at most $\frac{512\mathsf{cr}(\phi)\mu^{26b}}{m}\cdot \frac{16\mu^{b}}{m}\leq \frac{2^{13}\mathsf{cr}(\phi)\mu^{27b}}{m^2}$.
Consider now some edge $e=(x,y)\in E(G)$. Edge $e$ may lie in $E'$ only if $x$ is a light vertex lying in $V(P^*)$, or the same is true for $y$. Therefore, the probability that $e\in E'$ is at most $\frac{2^{14}\mathsf{cr}(\phi)\mu^{27b}}{m^2}$, and $\expect{|E'|}\leq \frac{2^{14}\mathsf{cr}(\phi)\mu^{27b}}{m}$. From Markov's inequality, $\prob{|E'|>\frac{\mathsf{cr}(\phi)\cdot \mu^{38b}}{m}}\leq \frac{2^{14}}{\mu^{11b}}$.
\end{proof}
\paragraph{Unlucky Paths and Bad Event ${\cal{E}}_3$.}
If Event ${\cal{E}}_1$ does not happen, then we are guaranteed that, in drawing $\phi$, the edges lying on path $P^*$ do not cross the edges of $J$. However, it is still possible that there are crossings between edges that lie on path $P^*$ (that is, the image of path $P^*$ crosses itself). Intuitively, when the image of path $P^*$ crosses itself, then we obtain a loop. If we could show that all vertices lying on this loop are light vertices, then we can ``repair'' the drawing by straightening the loop. This is since we delete all edges incident to the light vertices lying on $P^*$, except for the edges of $E(P^*)\cup E(J)$. Unfortunately, it may happen that some of the vertices lying on these loops are heavy vertices. This may only happen in some limited circumstances, in which case we say that path $P^*$ is \emph{unlucky}. We now define the notion of unlucky paths, and show that the probability that $P^*$ is unlucky is small.
\begin{definition}[Unlucky Paths]
Let $x\in V(G)\setminus V(J)$ be a vertex, and let $P\in {\mathcal{P}}^*$ be a good path that contains $x$. Let $e,e'$ be the two edges of $P$ that are incident to $x$. Let $\hat E_1(x)\subseteq \delta_G(x)$ be the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies between $e$ and $e'$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$. Let $\hat E_2(x)\subseteq \delta_G(x)$ be the set of edges $\hat e\in \delta_G(x)$, such that $\hat e$ lies between $e'$ and $e$ in the rotation ${\mathcal{O}}_x\in \Sigma$ (in clock-wise orientation), and $\hat e$ lies on some good path of ${\mathcal{P}}^*$ (see \Cref{fig: unlucky}).
We say that path $P$ is \emph{unlucky with respect to vertex $x$} if either $|\hat E_1(x)|<\frac{\mathsf{cr}(\phi)\mu^{13b}}{m}$ or $|\hat E_2(x)|< \frac{\mathsf{cr}(\phi)\mu^{13b}}{m}$ holds.
We say that a path $P\in {\mathcal{P}}^*$ is an \emph{unlucky path} if there is at least one heavy vertex $x\in V(G)\setminus V(J)$, such that $P$ is unlucky with respect to $x$.
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[scale=0.13]{figs/unlucky.jpg}
\caption{Definition of the sets $\hat E_1(x)$ and $\hat E_2(x)$ of edges. Path $P$ and its edges $e,e'$ are shown in red.
Edges of $\delta(x)$ are depicted according to the circular order ${\mathcal{O}}_x\in \Sigma$, and the set $\delta(x)\setminus \set{e,e'}$ is split into two subsets (green and blue).
Set $\hat E_1(x)$ contains every green edge that belongs to some good path of ${\mathcal{P}}^*$, and set $\hat E_2(x)$ contains every blue edge that belongs to some good path of ${\mathcal{P}}^*$.
}\label{fig: unlucky}
\end{figure}
We will now show that the total number of good paths in ${\mathcal{P}}^*$ that are unlucky is small, and we will conclude that the probability that an unlucky path was chosen to be the enhancement path $P^*$ is also small. The proof of the following claim is somewhat technical and is defered to Section \ref{sec: bound unlucky paths} of Appendix.
\begin{claim}\label{claim: bound unlucky paths}
For every vertex $x\in V(G)\setminus V(J)$, the total number of good paths in ${\mathcal{P}}^*$ that are unlucky with respect to $x$ is at most $\frac{512\mathsf{cr}(\phi)\cdot \mu^{13b}}{m}$.
\end{claim}
We say that bad event ${\cal{E}}_3$ happens if $P^*$ is an unlucky path.
\begin{claim}\label{claim: third event bound}
$\prob{{\cal{E}}_3}\leq 64/\mu^{10b}$.
\end{claim}
\begin{proof}
Recall that a heavy vertex must have degree at least $h$ in $G$. Therefore, the total number of heavy vertices is at most $\frac{2m}{h}$. From \Cref{claim: bound unlucky paths}, for every heavy vertex $x\in V(G)\setminus V(J)$, there are at most $\frac{512\mathsf{cr}(\phi)\cdot \mu^{13b}}{m}$ paths in ${\mathcal{P}}^*$ that are good and unlucky for $x$. Since $h=\frac{512\mathsf{cr}(\phi)\cdot \mu^{26b}}{m}$, the total number of good paths in ${\mathcal{P}}^*$ that are unlucky with respect to some heavy vertex is at most:
\[ \frac{2m}{h}\cdot \frac{512\mathsf{cr}(\phi)\cdot \mu^{13b}}{m}\leq \frac{2m}{ \mu^{13b}}.\]
Since the probability that a given path $P\in {\mathcal{P}}^*$ is selected is at most $\frac{16\mu^b} m$, the probability that an unlucky path is selected is at most $\frac{32}{\mu^{12b}}$.
\end{proof}
Recall that we have denoted by $E'$ the set of all edges $e$, such that $e$ is incident to some light vertex $x\in V(P^*)$, and $e\not\in E(J)\cup E(P^*)$.
Denote $G'=G\setminus E'$, and let $\Sigma'$ be the rotation system for graph $G'$ induced by $\Sigma$. Denote $I'=(G',\Sigma')$, and denote $J'=J\cup P^*$. Solution $\phi$ to instance $I$ naturally defines a soluton $\phi'$ to instance $I'$, that is compatible with $\phi$, with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\phi')$. Moreover, if Event ${\cal{E}}_1$ did not happen, then there are no crossings in this drawing between the edges of $E(P^*)$ and the edges of $E(J)$. However, it is possible that this drawing contains crossings between pairs of edges in $E(P^*)$. In the next claim we show that, if events ${\cal{E}}_1$ and ${\cal{E}}_3$ did not happen, then drawing $\phi$ can be modified to obtain a solution $\phi'$ to instance $I'$ that is compatible with $\phi$, in which the edges of $J'$ do not cross each other.
The proof of the following claim is deferred to Section \ref{sec:getting new drawing} of Appendix.
\begin{claim}\label{claim: new drawing}
Assume that neither of the events ${\cal{E}}_1$ and ${\cal{E}}_3$ happened. Then there is a solution $\phi'$ to instance $I'=(G',\Sigma')$ that is compatible with $\phi$, with $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$, such that the edges of $E(J')$ do not cross each other in $\phi'$. Moreover, if $(e,e')_p$ is a crossing in drawing $\phi'$, then there is a crossing between edges $e$ and $e'$ at point $p$ in drawing $\phi$.
\end{claim}
We emphasize that drawing $\phi'$ is derived from drawing $\phi$ and neither are known to our algorithm. From now on, we fix the solution $\phi'$ to instance $I'=(G',\Sigma')$ given by \Cref{claim: new drawing}.
\paragraph{Terrible Vertices and Bad Event ${\cal{E}}_4$.}
For a vertex $x\in V(G)$, let $N(x)$ denote the number of paths in ${\mathcal{P}}^*$ containing $x$. We also denote by $N^{\operatorname{bad}}(x)$ the number of bad paths in ${\mathcal{P}}^*$ containing $x$, and by $N^{\operatorname{good}}(x)$ the number of good paths in ${\mathcal{P}}^*$ containing $x$. Next, we define the notion of a terrible vertex.
\begin{definition}[Terrible Vertex]
A vertex $x\in V(G)$ is \emph{terrible} if it is a heavy vertex, and $N^{\operatorname{bad}}(x)\geq N^{\operatorname{good}}(x)/64$.
\end{definition}
We say that a bad event ${\cal{E}}_4$ happens if any vertex of $P^*$ is a terrible vertex. We bound the probability of Event ${\cal{E}}_4$ in the following claim.
\begin{claim}\label{claim: no terrible vertices}
$\prob{{\cal{E}}_4}\leq \frac{2^{18}}{\mu^{10b}}$.
\end{claim}
\begin{proof}
Consider some terrible vertex $x\in V(G)$. Let ${\mathcal{P}}'\subseteq {\mathcal{P}}^*$ be the set of all bad paths in ${\mathcal{P}}^*$ containing $x$, and let ${\mathcal{P}}''\subseteq {\mathcal{P}}^*$ be the set of all good paths in ${\mathcal{P}}^*$ containing $x$. From the definition of a terrible vertex, $|{\mathcal{P}}''|\leq 64|{\mathcal{P}}'|$. Therefore, we can define a mapping $f_x:{\mathcal{P}}''\rightarrow {\mathcal{P}}'$, that maps every path in ${\mathcal{P}}''$ to some path in ${\mathcal{P}}'$, such that, for every path $P\in {\mathcal{P}}'$, at most $64$ paths of ${\mathcal{P}}''$ are mapped to $P$. If, for a pair $P''\in {\mathcal{P}}''$, $P'\in {\mathcal{P}}'$ of paths, $f_x(P'')=P'$, then we say that path $P'$ \emph{tags} path $P''$. Every bad path also tags itself.
Notice that, if path $P\in {\mathcal{P}}^*$ is a bad path, then the total number of paths that it may tag is bounded by $64|V(P)|\leq 2^{12}\mu^b$ (as every path in ${\mathcal{P}}^*$ contains at most $32\mu^b+2$ vertices). Every path of ${\mathcal{P}}^*$ that contains a terrible vertex is now tagged. Since, from \Cref{obs: number of bad paths},
the number of bad paths in ${\mathcal{P}}^*$ is at most $\frac{4m}{\mu^{12b}}$, we conclude that the total number of paths in ${\mathcal{P}}^*$ that contain a terrible vertex is bounded by $\frac{2^{14}m}{\mu^{11b}}$. Lastly, since the probability that a given path $P\in {\mathcal{P}}^*$ is selected is at most $\frac{16\mu^b} m$, the probability that a path containing a terrible vertex is selected is bounded by:
\[ \frac{2^{14}m}{\mu^{11b}}\cdot \frac{16\mu^b} m\leq \frac{2^{18}}{\mu^{10b}}. \]
\end{proof}
\iffalse
\begin{proof}
Let $U$ be the set of all heavy vertices of $G$. We group the vertices $x\in U$ geometrically into classes, using the parameter $N(x)$. Recall that $h=\frac{512\mathsf{cr}(\phi)\cdot \mu^{26b}}{m}$, and, for every heavy vertex $x\in U$, $N(x)\geq h$. Let $q=2\ceil{\log m}$. For $1\leq i\leq q$, we let the class $S_i$ contain all vertices $x\in U$ with $2^{i-1}\cdot h\leq N(x)<2^i\cdot h$.
Let $S'_i\subseteq S_i$ be the set containing all terrible vertices of $S_i$. Recall that, for every vertex $x\in S'_i$, $N^{\operatorname{bad}}(x)\geq N^{\operatorname{good}}(x)/64$. Since $N(x)=N^{\operatorname{bad}}(x)+N^{\operatorname{good}}(x)$, we get that $N^{\operatorname{bad}}(x)\geq N(x)/65\geq 2^{i-1}\cdot h/65$. From \Cref{obs: number of bad paths}, the total number of bad paths is at most $\frac{4m}{\mu^{12b}}$, so:
\[|S'_i|\leq \frac{(4m)/\mu^{12b}}{2^{i-1}\cdot h/65}\leq \frac{2^{10}m}{ 2^{i}\cdot h\cdot \mu^{12b}}.\]
Recall that the probability that a path $P\in {\mathcal{P}}^*$ is selected to be the enhancement path is at most $\frac{16\mu^{b}}{m}$. A vertex $x\in S'_i$ may lie in $V(P^*)$ only if at least one of the $N(x)$ paths of ${\mathcal{P}}^*$ containing $x$ is selected to be the enhancement path. Therefore, $\prob{x\in V(P^*)}\leq \frac{16\mu^{b}\cdot N(x)}{m}\leq \frac{2^{i+4}\cdot h\cdot \mu^{b}}{m}$.
Overall, the probability that some vertex of $S'_i$ belongs to $V(P^*)$ is at most:
$$|S'_i|\cdot \frac{2^{i+4}\cdot h\cdot \mu^{b}}{m}\leq \frac{2^{10}m}{ 2^{i}\cdot h\cdot \mu^{12b}} \cdot \frac{2^{i+4}\cdot h\cdot \mu^{b}}{m}\leq \frac{2^{14}}{\mu^{11b}}.$$
Using the union bound over all $q=2\ceil{\log m}$ classes, we get that the probability of Event ${\cal{E}}_4$ is bounded by $\frac{2^{16}\log m}{\mu^{11b}}$.
\end{proof}
\fi
\paragraph{Bad Event ${\cal{E}}$.}
Let ${\cal{E}}$ be the bad event that either of the events ${\cal{E}}_1,{\cal{E}}_2,{\cal{E}}_3,{\cal{E}}_4$ happens. From the Union Bound and Claims \ref{claim: event 1 prob2}, \ref{claim: third bad event bound}, \ref{claim: third event bound} and \ref{claim: no terrible vertices}, $\prob{{\cal{E}}}\leq \frac{2^{20}}{\mu^{10b}}$.
\subsubsection{Step 2: Computing the Enhancement Structure and the Split}
\label{subsub: step 2 of phase 1 of interesting}
In this step, we compute an orientation $b_u$ for every vertex $u\in V(P^*)\setminus V(J)$, that is identical to the orientation of $u$ in drawing $\phi'$ (though drawing $\phi'$ itself is not known to the algorithm). We will then complete the construction of the enhancement structure ${\mathcal{A}}$, and compute the split of instance $I$ along ${\mathcal{A}}$. Throughout, we denote $J'=J\cup P^*$. Let $\rho'$ be the drawing of graph $J'$ that is induced by drawing $\phi'$ of $G'$. If Event ${\cal{E}}$ did not happen, then drawing $\rho'$ has no crossings, and the image of path $P^*$ is drawn in the region $F^*$. Let $\rho_{J'}$ be the unique drawing of graph $J'$ that has the following properties:
\begin{itemize}
\item drawing $\rho_{J'}$ contains no crossings;
\item drawing $\rho_{J'}$ obeys the rotation system $\Sigma$, and, for every vertex $u\in V(J)$, the orientation of $u$ in $\rho_{J'}$ is the orientation $b_u$ given by ${\mathcal{J}}$;
\item the drawing of graph $J$ induced by $\rho_{J'}$ is precisely $\rho_J$; and
\item the image of path $P^*$ is contained in region $F^*$.
\end{itemize}
Note that there is a unique drawing $\rho_{J'}$ of $J'$ with the above properties, and it can be computed efficiently. Moreover, if event ${\cal{E}}$ did not happen, then $\rho_{J'}=\rho'$ must hold. The image of path $P^*$ partitions the region $F^*$ of $\rho_{J'}$ into two faces, that we denote by $F_1$ and $F_2$. These two faces define regions in drawing $\phi'$ of $G'$, that we denote by $F_1$ and $F_2$, as well.
For every vertex $u\in V(J')$, we consider the tiny $u$-disc $D_{\phi'}(u)$. For every edge $e\in \delta_{G'}(u)$, we denote by $\sigma(e)$ the segment of $\phi'(e)$ that is drawn inside the disc $D_{\phi'}(u)$. Let $\tilde E=\textsf{left} (\bigcup_{u\in V(J')}\delta_{G'}(u)\textsf{right} )\setminus E(J')$. Recall that, from the definiton of a valid core structure, and since the image of path $P^*$ is contained in region $F^*$ of $\phi'$, for every edge $e\in \tilde E$, segment $\sigma(e)$ must be contained in region $F^*$. We partition edge set $\tilde E$ into a set $\tilde E^{\mathsf{in}}$ of \emph{inner edges} and the set $\tilde E^{\mathsf{out}}$ of \emph{outer edges}, as follows. Edge set $\tilde E^{\mathsf{in}}$ contains all edges $e\in \tilde E$ with $\sigma(e)$ contained in region $F_1$ of $\phi'$, and $\tilde E^{\mathsf{out}}$ contains all remaining edges (so for every edge $e\in \tilde E^{\mathsf{out}}$, $\sigma(e)$ is contained in $F_2$). Let $e_1$ be the first edge of $E_1$ in the ordering ${\mathcal{O}}(J)$. We will assume without loss of generality that $e_1\in \tilde E^{\mathsf{in}}$. We now show an algorithm that correctly computes the orientation of every vertex $u\in V(P^*)\setminus V(J)$ in the drawing $\phi'$, and the partition $(\tilde E^{\mathsf{in}},\tilde E^{\mathsf{out}})$ of the edges of $\tilde E$.
Before we describe the algorithm, we recall the definition of the oriented circular ordering ${\mathcal{O}}(J)$ of the edges of $\delta_G(J)$. In order to define the ordering, we considered the disc $D(J)$ in the drawing $\rho_{J}$ of $J$. In this drawing, the orientation of every vertex $u\in V(J)$ is the orientation $b_u$ given by the core structure ${\mathcal{J}}$. We have defined, for every edge $e\in \delta_G(J)$, a point $p(e)$ on the boundary of the disc $D(J)$, and we let ${\mathcal{O}}(J)$ be the circular ordering of the edges of $\delta_G(J)$, in which the points $p(e)$ corresponding to these edges appear on the boundary of the disc $D(J)$, as we traverse the boundary of the disc $D(J)$ in the clock-wise direction. From the definition of a ${\mathcal{J}}$-valid drawing, the drawing of the core $J$ induced by $\phi'$ is identical to $\rho_J$, including its orientation. Additionally, for every vertex $u\in V(J)$, the orientation of $u$ in $\phi'$ is the orientation $b_u$ given by ${\mathcal{J}}$.
\paragraph{Computing Vertex Orientations and the Partition $(\tilde E^{\mathsf{in}},\tilde E^{\mathsf{out}})$.}
Consider any vertex $u\in V(P^*)\setminus V(J)$. Let $\hat e(u)$, $\hat e'(u)$ be the two edges of $P^*$ that are incident to $u$, where we assume that $\hat e(u)$ appears before $\hat e'(u)$ on $P^*$ (we assume that $P^*$ is directed from an edge of $E_1$ to an edge of $E_2$). Edges $\hat e(u),\hat e'(u)$ partition the edge set $\delta_{G'}(u)\setminus\set{\hat e(u),\hat e'(u)}$ into two subsets, that we denote by $\hat E_1(u)$ and $\hat E_2(u)$, each of which appears consecutively in the rotation ${\mathcal{O}}_u\in \Sigma'$.
Note that either (i) $\hat E_1(u)\subseteq \tilde E^{\mathsf{in}}$ and $\hat E_2(u)\subseteq \tilde E^{\mathsf{out}}$ holds, or (ii) $\hat E_2(u)\subseteq \tilde E^{\mathsf{in}}$ and $\hat E_1(u)\subseteq \tilde E^{\mathsf{out}}$ holds.
While we do not know the orientation of vertex $u$ in $\phi'$, once we fix this orientation, we can efficiently determine which of the above two conditions holds. Therefore, we can assume w.l.o.g. that, if the orientation of $u$ in $\phi'$ is $1$ then $\hat E_1(u)\subseteq \tilde E^{\mathsf{in}}$ and $\hat E_2(u)\subseteq \tilde E^{\mathsf{out}}$ hold (as otherwise we can switch the names $\hat E_1(u)$ and $\hat E_2(u)$).
We now construct edge sets $\tilde E_1,\tilde E_2$, and fix an orientation $b_u$ for every vertex $u\in V(P^*)\setminus V(J)$. We then show that $\tilde E_1=\tilde E^{\mathsf{in}}$, $\tilde E_2=\tilde E^{\mathsf{out}}$, and that the orientations of all vertices of $V(P^*)\setminus V(J)$ that we compute are consistent with the drawing $\phi'$.
Consider the drawing $\rho_{J'}$ of graph $J'$ that we have computed. Using this drawing, we can efficiently determine, for every edge $e\in \tilde E$ that is incident to a vertex of $J$, whether $e\in \tilde E^{\mathsf{in}}$ or $e\in \tilde E^{\mathsf{out}}$ holds. In the former case, we add $e$ to $\tilde E_1$, and in the latter case we add it to $\tilde E_2$.
Notice that, for every path $P\in {\mathcal{P}}^*$, the first and the last edges of $P$ are already added to either $\tilde E_1$ or $\tilde E_2$, and so far $\tilde E_1\subseteq \tilde E^{\mathsf{in}}$ and $\tilde E_2\subseteq \tilde E^{\mathsf{out}}$ holds.
Next, we process every inner vertex $u$ on path $P^*$.
Consider any such vertex $u$. If $u$ is a light vertex, then there are exactly two edges that are incident to $u$ in $G'$ -- the edges of the path $P^*$. We can then set the orientation $b_u$ of $u$ to be arbitrary, and we can trivially assume that this orientaiton is identical to the orientation of $u$ in $\phi'$.
Assume now that $u$ is a heavy vertex.
In order to establish the orientation of $u$, we let ${\mathcal{P}}(u)$ contain all paths $P\in {\mathcal{P}}^*\setminus\set{P^*}$ with $u\in P$. We partition the set ${\mathcal{P}}(u)$ of paths into four subsets: set ${\mathcal{P}}_1(u)$ contains all paths $P$ whose first edge lies in $\tilde E_1$, and the first edge of $P$ that is incident to $u$ lies in $\hat E_1(u)$. Set ${\mathcal{P}}_2(u)$ contains all paths $P$ whose first edge lies in $\tilde E_2$, and the first edge of $P$ that is incident to $u$ lies in $\hat E_2(u)$. Similarly, set ${\mathcal{P}}'_1(u)$ contains all paths $P\in {\mathcal{P}}(u)$ whose first edge lies in $\tilde E_1$ and the first edge that is incident to $u$ lies in $\hat E_2(u)$, while set ${\mathcal{P}}'_2(u)$ contains all paths $P\in {\mathcal{P}}'(u)$, whose first edge lies in $\tilde E_2'$ and the first edge that is incident to $u$ lies in $\hat E_1(u)$. We let $w(u)=|{\mathcal{P}}_1|+|{\mathcal{P}}_2|$, and $w'(u)=|{\mathcal{P}}'_1|+|{\mathcal{P}}'_2|$.
If $w(u)\geq w'(u)$, then we set $b_u=1$, add the edges of $\hat E_1(u)$ to $\tilde E_1$, and add the edges of $\hat E_2(u)$ to $\tilde E_2$. Otherwise we set $b_u=-1$, add the edges of $\hat E_1(u)$ to $\tilde E_2$, and add the edges of $\hat E_2(u)$ to $\tilde E_1$.
This completes the algorithm for computing the orientations of the inner vertices of $P^*$, and of the partition $(\tilde E_1,\tilde E_2)$ of the edge set $\tilde E$.
We use the following claim to show that both are computed correctly.
\begin{claim}\label{claim: orientations and edge split is computed correctly Case2}
Assume that Event ${\cal{E}}$ did not happen. Then for every vertex $u\in V(P^*)\setminus V(J)$, the orientation of $u$ in $\phi'$ is $b_u$.
\end{claim}
\begin{proof}
It is now enough to show that, if $u\in V(P^*)\setminus V(J)$ is a heavy vertex, then the orientation of $u$ in $\phi'$ is $b_u$. We now consider any heavy vertex $u\in V(P^*)\setminus V(J)$.
Recall that we denoted $N(u)=|{\mathcal{P}}(u)|$, and we have denoted by $N^{\operatorname{bad}}(u)$ and $N^{\operatorname{good}}(u)$ the total number of bad and good paths in ${\mathcal{P}}(u)$, respectively. Since we have assumed that bad event ${\cal{E}}_4$ did not happen, vertex $u$ is not a terrible vertex, that is, $N^{\operatorname{bad}}(u)< N^{\operatorname{good}}(u)/64$. Since $N(u)=N^{\operatorname{bad}}(u)+N^{\operatorname{good}}(u)$, we get that $N^{\operatorname{bad}}(u)<N(u)/65$.
Assume first that the orientation of vertex $u$ in $\phi'$ is $1$, so $\hat E_1(u)\subseteq \tilde E^{\mathsf{in}}$ and $\hat E_2(u)\subseteq \tilde E^{\mathsf{out}}$.
We claim that in this case $w(u)>w'(u)$ must hold, and so our algorithm sets $b_u=1$ correctly. Indeed, assume otherwise.
Then $w'(u)\geq N(u)/2$. Let ${\mathcal{Q}}$ denote the set of all good paths in ${\mathcal{P}}'_1(u)\cup {\mathcal{P}}'_2(u)$. Then $|{\mathcal{Q}}|\geq w'(u)-N^{\operatorname{bad}}(u)\geq N(u)/2-N(u)/65\geq N(u)/4\geq h/4$, since $u$ is a heavy vertex.
We now show that, for every path $Q\in {\mathcal{Q}}$, there must be a crossing between an edge of $Q$ and an edge of $P^*$ in $\phi'$.
Indeed, consider any path $Q\in {\mathcal{Q}}$. Since $Q\in {\mathcal{P}}'_1(u)\cup {\mathcal{P}}'_2(u)$, either the first edge of $Q$ lies in $\tilde E^{\mathsf{in}}$ and the last edge of $Q$ lies in $\tilde E^{\mathsf{out}}$, or the opposite is true. Therefore, the image of the path $Q$ must cross the boundary of the region $F_1$. Since path $Q$ is a good path, and it does not contain vertices of $J$ as inner vertices, no inner point of the image of $Q$ in $\phi'$ may belong to the image of $J$ in $\phi'$. Since, for every pair $P,P'\in {\mathcal{P}}^*$ of paths, and for every vertex $v\in (V(P)\cap V(P'))\setminus V(J)$, the intersection of $P$ and $P'$ at $v$ is non-transversal, there must be a crossing between an edge of $Q$ and an edge of $P^*$ in $\phi'$. But then the edges of $P^*$ participate in at least $\frac h 4\geq \frac{128\mathsf{cr}(\phi)\cdot \mu^{26b}}{m}$ crossings in $\phi'$, and hence in $\phi$. However, since we have assumed that bad event ${\cal{E}}_1$ did not happen, $P^*$ is a good path, and so its edges may participate in at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}$ crossings in $\phi$, a contradiction. Therefore, when the orientation of $u$ in $\phi'$ is $1$, our algorithm correctly sets $b_u=1$.
In the case where the orientation of $u$ in $\phi'$ is $-1$, the analysis is symmetric. In this case, we consider the set ${\mathcal{Q}}'\subseteq {\mathcal{P}}_1(u)\cup {\mathcal{P}}_2(u)$ containing all good paths. For each such path $P\in {\mathcal{Q}}'$, the image of $P$ in $\phi$ must cross the image of $P^*$. If we assume that $w(u)\geq w'(u)$ in this case, then we reach a contradiction using the same argument as before. Therefore, $w(u)<w'(u)$ must hold, and our algorithm sets $b_u=1$ correctly.
\end{proof}
We have now obtained an enhancement structure
${\mathcal{A}}=(P^*,\set{b_u}_{u\in V(J')},\rho_{J'})$. For every vertex $u\in V(J')$, if $u\in V(J)$, then its orientaiton $b_u$ remains the same as in ${\mathcal{J}}$, and otherwise we let $b_u$ be the orientation that we have computed above. From the above discussion, if Event ${\cal{E}}$ did not happen, then for every vertex $u\in V(J')$ the orientation $b_u$ is identical to its orientation in $\phi'$, and $\rho_{J'}$ is the drawing of $J'$ induced by $\phi'$.
We denote by $({\mathcal{J}}_1,{\mathcal{J}}_2)$ the split of ${\mathcal{J}}$ via the enhancement structure ${\mathcal{A}}$, where ${\mathcal{J}}_1$ is the core structure associated with the face $F_1$. We denote ${\mathcal{J}}_1=(J_1,\set{b_u}_{u\in V(J_1)},\rho_{J_1}, F^*(\rho_{J_1}))$, and ${\mathcal{J}}_2=(J_2,\set{b_u}_{u\in V(J_2)},\rho_{J_2}, F^*(\rho_{J_2}))$, where $F^*(\rho_{J_1})=F_1$ and $F^*(\rho_{J_2})=F_2$.
\paragraph{Computing the Split.}
We now construct a split of instance $I$ along ${\mathcal{A}}$. In order to do so, we construct a flow network $H$ as follows. We start with $H=G'$, and then subdivide every edge $e\in \tilde E$ with a vertex $t_e$, denoting $T_1=\set{t_e\mid e\in \tilde E_1}$ and $T_2=\set{t_e\mid e\in \tilde E_2}$. We delete all vertices of $J'$ and their adjacent edges from the resulting graph, contract all vertices of $T_1$ into a source vertex $s$, and contract all vertices of $T_2$ into a destination vertex $t$. We then compute a minimum $s$-$t$ cut $(A,B)$ in the resulting flow network $H$, and we denote by $E''=E_H(A,B)$. We use the following claim, whose proof is provided in Section \ref{subsec: small cut set in case 2} of Appendix, in order to bound the cardinality of $E''$.
\begin{claim}\label{claim: cut set small case2}
If Event ${\cal{E}}$ did not happen, then $|E''|\leq \frac{2\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
\end{claim}
Let $E^{\mathsf{del}}=E'\cup E''$. If bad event ${\cal{E}}$ did not happen, then $|E'|\leq \frac{\mathsf{cr}(\phi)\cdot \mu^{38b}}{m}$, and, from \Cref{claim: cut set small case2}, $|E''|\leq \frac{2\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
Therefore, overall, if bad event ${\cal{E}}$ did not happen, then $|E^{\mathsf{del}}|\leq \frac{2\mathsf{cr}(\phi)\cdot \mu^{38b}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
Let $G''=G'\setminus E''=G\setminus E^{\mathsf{del}}$, let $\Sigma''$ be the rotation system for graph $G''$ induced by $\Sigma$, and let $I''=(G'',\Sigma'')$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Solution $\phi'$ to instance $I'$ then naturally induces a solution to instance $I''$, that we denote by $\phi''$. From \Cref{claim: new drawing}, this solution is compatible with $\phi$, and $\mathsf{cr}(\phi'')\leq \mathsf{cr}(\phi)$. Moreover, if Event ${\cal{E}}$ did not happen, then the number of crosings in which the edges of $P^*$ participate in $\phi''$ is at most $\frac{\mathsf{cr}(\phi)\cdot \mu^{12b}}{m}$, and the images of edges of $E(J)\cup E(P^*)$ do not cross each other in $\phi'$.
We are now ready to define a split $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ of instance $I$ along the enhacement structure ${\mathcal{A}}$. In order to do so, we define two sets $A',B'$ of vertices in graph $G'$, as follows. We start with $A'=A\setminus\set{s}$ and $B'=B\setminus\set{t}$, where $(A,B)$ is the cut that we have computed in graph $H$. We then add all vertices of the core $J_1$ to $A'$, and all vertices of the core $J_2$ to $B'$.
We let $G_1=G''[A']$ and $G_2=G''[B']$. The rotation system $\Sigma_1$ for graph $G_1$ and the rotation system $\Sigma_2$ for graph $G_2$ are induced by $\Sigma$.
Let $I_1=(G_1,\Sigma_1)$ and $I_2=(G_2,\Sigma_2)$ be the resulting two instances of \ensuremath{\mathsf{MCNwRS}}\xspace. It is immediate to verify that $(I_1,I_2)$ is a valid split of instance $I$ along ${\mathcal{A}}$.
Note that $E(G_1)\cup E(G_2)=E(G'')$.
Let $\phi_1$ be the solution to instance $I_1$ induced by $\phi''$,
and let $\phi_2$ be the solution to instance $I_2$ induced by $\phi'$.
From our construction and \Cref{claim: orientations and edge split is computed correctly Case2}, if bad event ${\cal{E}}$ did not happen,
then drawing $\phi_1$ of $G_1$ is ${\mathcal{J}}_1$-valid, and drawing $\phi_2$ of $G_2$ is ${\mathcal{J}}_2$-valid.
Lastly, we need the following observation, whose proof
appears in Section \ref{subsec: few edges in split Case 2} of Appendix.
\begin{observation}\label{obs: few edges in split graphs case2}
If Event ${\cal{E}}$ did not happen, then $|E(G_1)|,|E(G_2)|\leq m-\frac{m}{32\mu^{b}}$.
\end{observation}
We conclude that, if bad event ${\cal{E}}$ did not happen, then our algorithm computes a valid output for Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. Since $\prob{{\cal{E}}}\leq 2^{20}/\mu^{10b}$, this completes the proof of \Cref{thm: procsplit}.
\section{Introduction}
We study the classical Minimum Crossing Number (\textsf{MCN}\xspace) problem: given an $n$-vertex graph $G$, compute a drawing of $G$ in the plane while minimizing the number of its crossings. Here, a drawing $\phi$ of a graph $G$ is a mapping, that maps every vertex $v\in V(G)$ to some point $\phi(v)$ in the plane, and every edge $e=(u,v)\in E(G)$ to a continuous simple curve $\phi(e)$, whose endpoints are $\phi(u)$ and $\phi(v)$. For a vertex $v\in V(G)$ and an edge $e\in E(G)$, we refer to $\phi(v)$ and to $\phi(e)$ as the \emph{images} of $v$ and of $e$, respectively. We require that, for every vertex $v$ and edge $e$, $\phi(v)\in \phi(e)$ only if $v$ is an endpoint of $e$. We also require that, if some point $p$ belongs to the images of three or more edges, then it must be the image of a shared endpoint of these edges.
A \emph{crossing} in a drawing $\phi$ of $G$ is a point that belongs to the images of two edges of $G$, and is not their common endpoint. The \emph{crossing number} of a graph $G$, denoted by $\mathsf{OPT}_{\mathsf{cr}}(G)$, is the minimum number of crossings in any drawing of $G$ in the plane.
The \textsf{MCN}\xspace problem was initially introduced by Tur\'an \cite{turan_first} in 1944, and has been extensively studied since then (see, e.g., \cite{chuzhoy2011algorithm, chuzhoy2011graph, chimani2011tighter, chekuri2013approximation, KawarabayashiSidi17, kawarabayashi2019polylogarithmic,chuzhoy2020towards}, and also \cite{richter_survey, pach_survey, matousek_book, vrto_biblio, schaefer2012graph} for excellent surveys). The problem is of interest to several communities, including, for example, graph theory and algorithms, and graph drawing.
As such, much effort was invested into studying it from different angles.
But despite all this work, most aspects of the problem are still poorly understood.
In this paper we focus on the algorithmic aspect of \textsf{MCN}\xspace. Since the problem is \mbox{\sf NP}-hard \cite{crossing_np_complete}, and it remains \mbox{\sf NP}-hard even in cubic graphs \cite{Hlineny06a, cabello2013hardness}, it is natural to consider approximation algorithms for it.
Unfortunately, the approximation ratios of all currently known algorithms depend polynomially on $\Delta$, the maximum vertex degree of the input graph. To the best of our knowledge, no non-trivial approximation algorithms are known for the general setting, where $\Delta$ may be arbitrarily large.
One of the most famous results in this area, the Crossing Number Inequality, by Ajtai, Chv\'atal, Newborn and Szemer\'edi \cite{ajtai82} and by Leighton \cite{leighton_book}, shows that, for every graph $G$ with $|E(G)|\geq 4|V(G)|$, the crossing number of $G$ is $\Omega(|E(G)|^3/|V(G)|^2)$.
Since the problem is most interesting when the crossing number of the input graph is low, it is reasonable to focus on low-degree graphs, where the maximum vertex degree $\Delta$ is bounded by either a constant, or a slowly-growing (e.g. subpolynomial) function of $n$. While we do not make such an assumption explicitly, like in all previous work, the approximation factor that we achieve also depends polynomially on $\Delta$.
Even in this setting, there is still a large gap in our understanding of the problem's approximability, and the progress in closing this gap has been slow. On the negative side, only APX-hardness is known \cite{cabello2013hardness,ambuhl2007inapproximability},
that holds even in cubic graphs.
On the positive side, the first non-trivial approximation algorithm for \textsf{MCN}\xspace was obtained by Leighton and Rao in their seminal paper \cite{leighton1999multicommodity}.
Given as input an $n$-vertex graph $G$, the algorithm computes a drawing of $G$ with at most $O((n+\mathsf{OPT}_{\mathsf{cr}}(G)) \cdot \Delta^{O(1)} \log^4n)$ crossings.
This bound was later improved to $O((n+\mathsf{OPT}_{\mathsf{cr}}(G))\cdot \Delta^{O(1)} \log^3n)$ by \cite{even2002improved}, and then to $O((n+\mathsf{OPT}_{\mathsf{cr}}(G)) \cdot \Delta^{O(1)} \log^2n)$ following the improved approximation algorithm of \cite{ARV} for Sparsest Cut. Note that all these algorithms only achieve an $O(n \operatorname{poly}(\Delta\log n)))$-approximation factor. However, their performance improves significantly when the crossing number of the input graph is large.
A sequence of papers~\cite{chuzhoy2011graph,chuzhoy2011algorithm} provided an improved $\tilde O(n^{0.9}\cdot\Delta^{O(1)})$-approximation algorithm for \textsf{MCN}\xspace, followed by a more recent sequence of papers by Kawarabayashi and Sidiropoulos \cite{KawarabayashiSidi17, kawarabayashi2019polylogarithmic}, who obtained an $\tilde O\textsf{left}(\sqrt{n}\cdot \Delta^{O(1)}\textsf{right} )$-approximation algorithm. All of the above results follow the same high-level algorithmic framework, and it was shown by Chuzhoy, Madan and Mahabadi \cite{chuzhoy-lowerbound} (see \cite{chuzhoy2016improved} for an exposition) that this framework is unlikely to yield a better than $O(\sqrt{n})$-approximation.
The most recent result, by
Chuzhoy, Mahabadi and Tan \cite{chuzhoy2020towards}, obtained an $\tilde O(n^{1/2-\epsilon}\cdot \operatorname{poly}(\Delta))$-approximation algorithm for some small fixed constant $\epsilon>0$. This result was achieved by proposing a new algorithmic framework for the problem, that departs from the previous approach. Specifically, \cite{chuzhoy2020towards} reduced the \textsf{MCN}\xspace problem to another problem, called Minimum Crossing Number with Rotation System (\textnormal{\textsf{MCNwRS}}\xspace) that we discuss below, which appears somewhat easier than the \textsf{MCN}\xspace problem, and then provided an algorithm for approximately solving the \ensuremath{\mathsf{MCNwRS}}\xspace problem.
Our main result is a randomized $O\textsf{left}(2^{O((\log n)^{7/8}\log\log n)}\cdot\Delta^{O(1)}\textsf{right} )$-approximation algorithm for \textsf{MCN}\xspace. In order to achieve this result, we design a new algorithm for the \textnormal{\textsf{MCNwRS}}\xspace problem that achieves significantly stronger guarantees than those of \cite{chuzhoy2020towards}. This algorithm, combined with the reduction of \cite{chuzhoy2020towards}, immediately implies the improved approximation for the \textsf{MCN}\xspace problem. We also design several new technical tools that we hope will eventually lead to further improvements. We now turn to discuss the \textnormal{\textsf{MCNwRS}}\xspace problem.
\iffalse
, breaking the barrier via a new framework, which, after Step (i) of the previous framework, adds a step of augmenting the planarizing set $E'$ to a larger set $E''$, such that the graph $G\setminus E''$ has some additional properties, and then implements Steps (ii) and (iii) of the previous framework using the new set $E''$ instead of $E'$.
In particular, using the algorithm in \cite{kawarabayashi2019polylogarithmic} for computing a near-optimal planarizing set, they constructed an efficient algorithm that, giving an planarizing set $E'$ of edges of $G$, computes an augmented planarizing set $E''$ of edges of $G$, such that $|E''|=O\textsf{left} ((|E'|+\mathsf{OPT}_{\mathsf{cr}}(G))\operatorname{poly}(\Delta \log n)\textsf{right} )$, and there exists a drawing of the graph $G$ with at most $O\textsf{left} ((|E'|+\mathsf{OPT}_{\mathsf{cr}}(G))\operatorname{poly}(\Delta \log n)\textsf{right} )$ crossings, where the edges of $G\setminus E''$ do not participate in any crossings. Intuitively, their algorithm showed a way of implementing the first and the added step of the new framework, such that the second and the third steps, if implemented in the best way, will give an $O(\operatorname{poly}(\Delta \log n))$-approximation solution to the input graph $G$.
\fi
In the Minimun Crossing Number with Rotation System (\textnormal{\textsf{MCNwRS}}\xspace) problem, the input consists of a multigraph $G$, and, for every vertex $v\in V(G)$, a circular ordering ${\mathcal{O}}_v$ of edges that are incident to $v$, that we call a \emph{rotation} for vertex $v$.
The set $\Sigma=\set{{\mathcal{O}}_v}_{v\in V(G)}$ of all such rotations is called a \emph{rotation system} for graph $G$. We say that a drawing $\phi$ of $G$ \emph{obeys} the rotation system $\Sigma$, if, for every vertex $v\in V(G)$, the images of the edges in $\delta_G(v)$ enter the image of $v$ in the order ${\mathcal{O}}_v$ (but the \emph{orientation} of the ordering can be either clock-wise or counter-clock-wise). In the \textnormal{\textsf{MCNwRS}}\xspace problem, given a graph $G$ and a rotation system $\Sigma$ for $G$, the goal is to compute a drawing $\phi$ of $G$ that obeys the rotation system $\Sigma$ and minimizes the number of edge crossings. For an instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, we denote by $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ the value of the optimal solution for $I$, that is, the smallest number of crossings in any drawing of $G$ that obeys $\Sigma$. The results of \cite{chuzhoy2020towards} show the following reduction from \textsf{MCN}\xspace to \textnormal{\textsf{MCNwRS}}\xspace: suppose there is an efficient (possibly randomized) algorithm for the \textnormal{\textsf{MCNwRS}}\xspace problem, that, for every instance $I=(G,\Sigma)$, produces a solution whose expected cost is at most $\alpha(m)\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)$, where $m=|E(G)|$. Then there is a randomized $O(\alpha(n)\cdot \operatorname{poly}(\Delta\cdot\log n))$-approximation algorithm for the \textsf{MCN}\xspace problem. Our main technical result is a randomized algorithm, that, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with high probability produces a solution to instance $I$ with at most $2^{O((\log m)^{7/8}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+m\textsf{right})$ crossings, where $m=|E(G)|$. Combining this with the result of \cite{chuzhoy2020towards}, we immediately obtain a randomized $O\textsf{left}(2^{O((\log n)^{7/8}\log\log n)}\cdot\operatorname{poly}(\Delta)\textsf{right} )$-approximation algorithm for the \textsf{MCN}\xspace problem.
The best previous algorithm for the \ensuremath{\mathsf{MCNwRS}}\xspace problem, due to \cite{chuzhoy2020towards}, is a randomized algorithm, that, given an instance $I=(G,\Sigma)$ of the problem, with high probability produces a solution with at most $\tilde O\textsf{left}((\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+m)^{2-\epsilon}\textsf{right})$ crossings, where $\epsilon=1/20$.
A variant of \textnormal{\textsf{MCNwRS}}\xspace was previously studied by Pelsmajer et al.~\cite{pelsmajer2011crossing}, where for each vertex $v$ of the input graph $G$, both the rotation ${\mathcal{O}}_v$ of its incident edges, and the orientation of this rotation (say clock-wise) are fixed.
They showed that this variant of the problem is also \mbox{\sf NP}-hard, and provided an $O(n^4)$-approximation algorithm with running time $O(m^n\log m)$, where $n=|V(G)|$ and $m=|E(G)|$.
They also obtained approximation algorithms with improved guarantees for some special families of graphs.
We introduce a number of new technical tools, that we discuss in more detail in \Cref{subsec: techniques}. Some of these tools require long and technically involved proofs, which resulted in the large length of the paper. We view these tools as laying a pathway towards obtaining better algorithms for the Minimum Crossing Number problem, and it is our hope that these tools will eventually be streamlined and that their proofs will be simplified, leading to a better understanding of the problem and cleaner and simpler algorithms. We believe that some of these tools are interesting in their own right.
\subsection{Our Results}
Throughout this paper, we allow graphs to have parallel edges (but not self-loops); graphs with no parallel edges are explicitly called simple graphs. For convenience, we will assume that the input to the \textsf{MCN}\xspace problem is a simple graph, while graphs serving as inputs to the \ensuremath{\mathsf{MCNwRS}}\xspace problem may have parallel edges. The latter is necessary in order to use the reduction of \cite{chuzhoy2020towards} between the two problems. Note that the number of edges in a graph with parallel edges may be much higher than the number of vertices.
Our main technical contribution is an algorithm for the \ensuremath{\mathsf{MCNwRS}}\xspace problem, that is summarized in the following theorem.
\begin{theorem}
\label{thm: main_rotation_system}
There is an efficient randomized algorithm, that, given an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace with $|E(G)|=m$, computes a drawing of $G$ that obeys the rotation system $\Sigma$. The number of crossings in the drawing is w.h.p. bounded by $2^{O((\log m)^{7/8}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right})$.
\end{theorem}
We rely on the following theorem from \cite{chuzhoy2020towards} in order to obtain an approximation algorithm for the \textsf{MCN}\xspace problem.
\begin{theorem}[Theorem 1.3 in~\cite{chuzhoy2020towards}]
\label{thm: MCN_to_rotation_system}
There is an efficient algorithm, that, given an $n$-vertex graph $G$ with maximum vertex degree $\Delta$, computes an instance $I=(G',\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, with $|E(G')|\leq O\textsf{left}(\mathsf{OPT}_{\mathsf{cr}}(G)\cdot \operatorname{poly}(\Delta\cdot\log n)\textsf{right})$, and $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq O\textsf{left}(\mathsf{OPT}_{\mathsf{cr}}(G)\cdot \operatorname{poly}(\Delta\cdot\log n)\textsf{right} )$. Moreover, there is an efficient algorithm that, given any solution of value $X$ to instance $I$ of \textnormal{\textsf{MCNwRS}}\xspace, computes a drawing of $G$ with the number of crossings bounded by $O\textsf{left} ((X+\mathsf{OPT}_{\mathsf{cr}}(G))\cdot \operatorname{poly}(\Delta\cdot\log n)\textsf{right} )$.
\end{theorem}
Combining \Cref{thm: main_rotation_system} and \Cref{thm: MCN_to_rotation_system}, we immediately obtain the following corollary, whose proof appears in Section \ref{sec: proof of main theorem} of Appendix.
\begin{corollary}
\label{thm: main_result}
There is an efficient randomized algorithm, that, given a simple $n$-vertex graph $G$ with maximum vertex degree $\Delta$, computes a drawing of $G$, such that, w.h.p., the number of crossings in the drawing is at most $O\textsf{left}( 2^{O((\log n)^{7/8}\log\log n)}\cdot\operatorname{poly}(\Delta)\cdot \mathsf{OPT}_{\mathsf{cr}}(G) \textsf{right} )$.
\end{corollary}
\input{high_level_overview}
\subsection{Laminar Family-Based Disengagment}
\label{subsec: laminar-based decomposition}
We start by defining a laminar family of clusters and its associated partitioning tree.
\subsubsection{Laminar Family of Clusters and Partitioning Tree}
\label{subsubsec: laminar}
\begin{definition}[Laminar family of clusters]
Let $G$ be a graph, and let ${\mathcal{L}}$ be a family of clusters of $G$. We say that ${\mathcal{L}}$ is a \emph{laminar family}, if $G\in {\mathcal{L}}$, and additionally, for all $S,S'\in {\mathcal{L}}$, either $S\cap S'=\emptyset$, or $S\subseteq S'$, or $S'\subseteq S$ holds.
\end{definition}
Given a laminar family ${\mathcal{L}}$ of clusters of $G$, we associate a \emph{paritioning tree} $\tau({\mathcal{L}})$ with it, that is defined as follows. The vertex set of the tree is $\set{v(S)\mid S\in {\mathcal{L}}}$; for every cluster $S\in {\mathcal{L}}$, we view vertex $v(S)$ as representing the cluster $S$. The root of the tree is $v(G)$ -- the vertex associated with the graph $G$ itself.
In order to define the edge set, consider a pair $S,S'\in {\mathcal{L}}$ of clusters. If $S\subsetneq S'$, and there is no other cluster $S''\in {\mathcal{L}}$ with $S\subsetneq S''\subsetneq S'$, then we add an edge $(v(S),v(S'))$ to the tree $\tau({\mathcal{L}})$; vertex $v(S)$ becomes a child vertex of $v(S')$ in the tree. We also say that cluster $S$ is a \emph{child cluster} of cluster $S'$, and cluster $S'$ is the \emph{parent cluster} of $S$.
Similarly, we define an ancestor-descendant relation between clusters in a natural way: cluster $S\in {\mathcal{L}}$ is a descendant-cluster of a cluster $S'\in {\mathcal{L}}$ if vertex $v(S')$ lies on the unique path connecting $v(S)$ to $v(G)$ in the tree $\tau({\mathcal{L}})$. If $S$ is a descendant-cluster of $S'$, then $S'$ is an ancestor-cluster of $S$. Notice that every cluster is its own ancestor and its own descendant.
The \emph{depth} of the laminar family ${\mathcal{L}}$ of clusters, denoted by $\mathsf{dep}({\mathcal{L}})$, is the length of the longest root-to-leaf path in tree $\tau({\mathcal{L}})$. We also say that cluster $S$ \emph{lies at level $i$ of the laminar family ${\mathcal{L}}$} iff the distance from $v(S)$ to the root of the tree $\tau({\mathcal{L}})$ is exactly $i$.
\subsubsection{Definition of Laminar Family-Based Disengagement}
\label{subsubsec: laminar based disengagment def}
The input to the Laminar Family-Based Disengegement is an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a laminar family ${\mathcal{L}}$ of clusters of $G$, and, for every cluster $C\in {\mathcal{L}}$, a circular ordering ${\mathcal{O}}(C)$ of the edges of $\delta_G(C)$ (for $C=G$, $\delta_G(C)=\emptyset$ and the ordering ${\mathcal{O}}(C)$ is trivial). The output of the procedure is a collection ${\mathcal{I}}=\set{I_C=(G_C,\Sigma_C)\mid C\in {\mathcal{L}}}$ of subinstances of $I$, that are defined as follows.
Consider a cluster $C\in {\mathcal{L}}$, and denote by ${\mathcal{W}}(C)\subseteq {\mathcal{L}}$ the set of all child-clusters of $C$. In order to construct the graph $G_C$, we start with $G_C=G$. For every cluster $C'\in {\mathcal{W}}(C)$, we contract the vertices of $C'$ into a supernode $v_{C'}$. Additionally, if $C\neq G$, then we contract all vertices of $V(G)\setminus V(C)$ into a supernode $v^*$. This completes the definition of the graph $G_C$ (see \Cref{fig: disengaged_instance}). We now define the rotation system $\Sigma_C$ for $G_C$.
If $C\neq G$, then the set of edges incident to $v^*$ in $G_C$ is exactly $\delta_G(C)$. We set the rotation ${\mathcal{O}}_{v^*}\in \Sigma_C$ to be ${\mathcal{O}}(C)$. For every cluster $C'\in {\mathcal{W}}(C)$, the set of edges incident to $v_{C'}$ in $G_C$ is $\delta_G(C')$. We set the rotation ${\mathcal{O}}_{v_{C'}}\in \Sigma_C$ to be ${\mathcal{O}}(C')$. For every regular vertex $x\in V(G_C)\cap V(G)$, $\delta_{G_C}(v)=\delta_G(v)$ holds, and its rotation ${\mathcal{O}}_v\in \Sigma_C$ remains the same as in $\Sigma$.
\begin{figure}[h]
\centering
\subfigure[Layout of graph $G$ with respect to $C$.]{\scalebox{0.13}{\includegraphics{figs/dis_instance_before}}\label{fig: graph G}}
\hspace{0cm}
\subfigure[Graph $G_C$.]{
\scalebox{0.14}{\includegraphics{figs/dis_instance_after}}\label{fig: C and disc}}
\caption{Construction of graph $G_C$, where $C\in {\mathcal{L}}$ is a cluster with two child-clusters $S_1,S_2$.\label{fig: disengaged_instance}}
\end{figure}
We refer to the resulting collection ${\mathcal{I}}$ of clusters as \emph{disengagement of instance $I$ via the laminar family ${\mathcal{L}}$ and the collection $\set{{\mathcal{O}}(C)}_{C\in {\mathcal{L}}}$ of orderings}.
\subsubsection{Analysis}
We start by showing that the total number of edges in all instances that we obtain via the laminar family-based disengagement procedure is small compared to $|E(G)|$.
\begin{lemma}\label{lem: number of edges in all disengaged instances}
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace, let ${\mathcal{L}}$ be a laminar family of clusters of $G$, and let $\set{{\mathcal{O}}(C)}_{C\in {\mathcal{L}}}$ be a collection of orderings of the edges of $\delta_G(C)$, for every cluster $C\in {\mathcal{L}}$. Consider the collection ${\mathcal{I}}=\set{I_C=(G_C,\Sigma_C)\mid C\in {\mathcal{L}}}$ of subinstances of $I$ obtained by applying the laminar family-based decomposition to instance $I$ via the laminar family ${\mathcal{L}}$ and the orderings in $\set{{\mathcal{O}}(C)}_{C\in {\mathcal{L}}}$. Then
$\sum_{C\in {\mathcal{L}}}|E(G_C)|\leq O(\mathsf{dep}({\mathcal{L}})\cdot|E(G)|)$.
\end{lemma}
\begin{proof}
Fix an integer $1\le i\le \mathsf{dep}({\mathcal{L}})$ and denote by ${\mathcal{L}}_i\subseteq {\mathcal{L}}$ the set of all clusters of ${\mathcal{L}}$ that lie at level $i$ of the partitioning tree.
From the definition of the laminar family ${\mathcal{L}}$ and the partitioning tree, all clusters in set ${\mathcal{L}}_i$ are mutually disjoint.
Consider now some cluster $C\in {\mathcal{L}}_i$, and its corresponding graph $G_C$. Note that every edge of $G_C$ corresponds to some distinct edge of cluster $C$, except for the edges incident to the supernode $v^*$. However, the number of edges incident to $v^*$ is at most $|\delta_G(C)|$. Therefore, overall, $|E(G_C)|\leq |E_G(C)|+|\delta_G(C)|$. Since all clusters in ${\mathcal{L}}_i$ are mutually disjoint, we get that:
\[\sum_{C\in {\mathcal{L}}_i}|E(G_C)|\leq \sum_{C\in {\mathcal{L}}_i}(|E_G(C)|+|\delta_G(C)|)\leq O(|E(G)|).\]
Summing over all indices $1\le i\le \mathsf{dep}({\mathcal{L}})$, we get that $\sum_{C\in {\mathcal{L}}}|E(G_C)|=O(\mathsf{dep}({\mathcal{L}})\cdot|E(G)|)$.
\end{proof}
Next, we show that solutions to the instances in ${\mathcal{I}}$ can be efficiently combined in order to obtain a solution to instance $I$ of relatively low cost. The proof is conceptually simple but somewhat technical, and is deferred to Section \ref{appx: proof of basic disengagement combining solutions} of Appendix.
\begin{lemma}\label{lem: basic disengagement combining solutions}
There is an efficient algorithm, that takes as input an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a laminar family ${\mathcal{L}}$ of clusters of $G$, a collection $\set{{\mathcal{O}}(C)}_{C\in {\mathcal{L}}}$ containing an ordering of the edges of $\delta_G(C)$ for every cluster $C\in {\mathcal{L}}$, and, for every cluster $C\in {\mathcal{L}}$, a solution $\phi(I_C)$ to the instance $I_C\in {\mathcal{I}}$ of \ensuremath{\mathsf{MCNwRS}}\xspace, where ${\mathcal{I}}=\set{I_C\mid C\in {\mathcal{L}}}$ is the collection of subinstances of $I$ obtained via laminar family-based disengagement of $I$ via $({\mathcal{L}},\set{{\mathcal{O}}(C)}_{C\in {\mathcal{L}}})$. The output of the algorithm is a solution to instance $I$ with cost at most $\sum_{I_C\in {\mathcal{I}}}\mathsf{cr}(\phi(I_C))$.
\end{lemma}
So far we have shown that, if ${\mathcal{I}}$ is the collection of instances that is constructed via a laminar family-based disengagement of instance $I$, then the total number of edges in all resulting instances is at most $O(|E(G)|\cdot \mathsf{dep}({\mathcal{L}}))$, and that there is an efficient algorithm for combining the solutions to the resulting instances, in order to obtain a low-cost solution to the original instance $I$. Another highly desirable property of the family ${\mathcal{L}}$ of clusters would be for $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')$ to be small compared to $\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|$. Unfortunately, we cannot show that this property holds, except for some special cases. The \textsf{Basic Cluster Disengagement}\xspace procedure essentially considers one such special case, in which $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')$ can be appropriately bounded. In order to define this procedure formally, we first need to define several central notions that are used throughout our algorithm, namely: light clusters, bad clusters, and path-guided orderings.
\section{Second Main Tool - Disengagement of Clusters}
\label{sec: not many paths}
\iffalse
\subsection{First Tool: Well-Linkedness to Routing}
\begin{theorem}\label{thm: wl to paths}
There is a randomized algorithm, that, given a graph $G$ and a cluster $C\subseteq G$, such that the boundary of $C$ is $\alpha$-well-linked in $C$, computes a vertex $u(C)\in C$, and a collection ${\mathcal{Q}}$ of paths routing $\delta_G(C)$ to $u(C)$ inside $C$, and a tree $\tau$ in $C\cup \delta(C)$ with root $u(C)$, such that the set ${\mathcal{Q}}$ of paths is the collection of all leaf-to-root paths in $\tau$ . Moreover, for every edge $e\in E(C)$ and value $\eta$, the probability that $e$ lies on more than $\eta$ paths in ${\mathcal{Q}}$ is bounded by $O(\frac{\operatorname{poly}\log n}{\alpha \eta})$.
\end{theorem}
\mynote{I have a proof in mind but I'm not 100\% sure. Also need to subdivide the edges of $\delta(C)$ and the resulting vertices should be the leaves of the tree.}
Consider now an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and let ${\mathcal{C}}$ be a collection of disjoint sub-graphs of $G$ called clusters, such that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$. Assume further that the boundary of each cluster $C\in {\mathcal{C}}$ is $\alpha$-well-linked in $C$. For each cluster $C\in {\mathcal{C}}$, let $u(C)$ be a vertex of $C$, and ${\mathcal{Q}}(C)$ a set of paths routing the edges of $\delta(C)$ to $u(C)$. For every edge $e\in E(C)$, let $\eta(e)$ be the total number of paths in ${\mathcal{Q}}(C)$ that contain $e$. If an edge $e$ does not lie in any cluster in ${\mathcal{C}}$, then we set $\eta(e)=0$. Consider now a graph $G'$ that is obtained from $G$ by replacing every edge $e$ with $\eta(e)$ parallel edges if $\eta(e)>0$; otherwise we keep the edge $e$ as is. We refer to the set of parallel edges replacing $e$ as a \emph{bundle} of $e$, and denote it by $\beta(e)$.
If $\eta(e)=0$, then we let $\beta(e)=\set{e}$.
Consider now some vertex $v\in V(G)$, and the set $\set{e_1,\ldots,e_r}$ of its incident edges, indexed in the order consistent with the ordering ${\mathcal{O}}_v\in \Sigma$. Let ${\mathcal{O}}'_v$ be a circular ordering of the edges of $\delta_{G'}(v)$. We say that ${\mathcal{O}}'_v$ is \emph{consistent} with ${\mathcal{O}}(v)$ iff for all $1\leq i\leq r$, all edges of $\beta(e_i)$ appear consecutively in ${\mathcal{O}}'_v$, and the edges belonging to different bundles appear in the ordering $\beta(e_1),\beta(e_2),\ldots,\beta(e_r)$; the ordering of edges within each bundle may be arbitrary. We say that a rotation system $\Sigma'$ for graph $G'$ is \emph{consistent} with rotation system $\Sigma$ for $G$ iff for every vertex $v\in V(G)$, the ordering ${\mathcal{O}}'_v\in \Sigma'$ of the edges of $\delta_{G'}(v)$ is consistent with the ordering ${\mathcal{O}}_v\in \Sigma$ of the edges of $\delta_G(v)$.
\begin{lemma}\label{lem: computing the ordering}
There is an efficient algorithm, that, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a collection ${\mathcal{C}}$ of disjoint sub-graphs of $G$, and, for each cluster $C\in {\mathcal{C}}$, a vertex $u(C)$ and a set ${\mathcal{Q}}(C)$ of paths routing the edges of $\delta_G(C)$ to $u(C)$, constructs a rotation system $\Sigma'$ for the graph $G'$ defined above with respect to $\bigcup_{C\in {\mathcal{C}}}{\mathcal{Q}}(C)$, that is consistent with the rotation system $\Sigma$. Moreover, $\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+\sum_{e\in E(G)}\eta(e)\cdot \mathsf{cr}(e)$
\end{lemma}
\mynote{the above lemma is problematic. will need to think about it some more. We are allowing $\mathsf{cr}({\mathcal{Q}},{\mathcal{Q}})$ crossings. But that means we need to be able to deal with cognestion squared, which we currently can't do.}
\fi
\subsection{Second Key Tool: Advanced Disengagement of Clusters}
\mynote{if possible, please remove $\beta$ from lemma statement and use instead number of crossings between different paths sets, or congestion squared}
\begin{theorem}\label{thm: advanced disengagement of clusters}
There is an efficient algorithm, that, given as input an instance $(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, a collection ${\mathcal{C}}$ of disjoint sub-graphs of $G$, and, for every cluster $C\in {\mathcal{C}}$, a vertex $u(C)$ and a set ${\mathcal{Q}}(C)$ of paths routing the edges of $\delta(C)$ to vertex $u(C)$ inside $C$, such that:
\begin{itemize}
\item $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$;
\item $|E^{\textsf{out}}({\mathcal{C}})|\leq |E(G)|/(100\mu)$;
\item for every cluster $C\in {\mathcal{C}}$, $|E(C)|\leq |E(G)|/(100\mu)$; and
\item for every cluster $C\in {\mathcal{C}}$ and for each edge $e\in C$, $(\cong_C({\mathcal{Q}}(C),e))^2\le \beta$,
\end{itemize}
computes a collection ${\mathcal{I}}=\set{I_1,\ldots,I_k}$ of sub-instances of $(G,\Sigma)$, with the following properties:
\begin{itemize}
\item for all $1\leq i\leq k$, $|E(G_i)|\leq |E(G)|/\mu$;
\item $\sum_{i=1}^k|E(G_i)|\leq O(|E(G)|)$; and
\item $\sum_{1\le i\le k}\mathsf{OPT}_{\mathsf{cnwrs}}(G_i,\Sigma_i)\le O((\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+|E(G)|)\cdot \beta)$.
\end{itemize}
Moreover, there is an efficient algorithm, that, given, for each $1\le i\le k$, a feasible solution $\phi_i$ for instance $(G_i,\Sigma_i)$, computes a feasible solution $\phi$ for instance $(G,\Sigma)$ with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{1\le i\le k}\mathsf{cr}(\phi_i)\textsf{right} )$.
\end{theorem}
The remainder of this subsection is dedicated to the proof of \Cref{thm: advanced disengagement of clusters}. We assume that we are given an instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, and a collection ${\mathcal{C}}$ of disjoint sub-graphs of $G$, like in the statement of the corollary. Let $\phi^*$ be an optimal solution to instance $I$ of \textnormal{\textsf{MCNwRS}}\xspace.
We associate with the graph $G$ and the collection ${\mathcal{C}}$ of clusters a \emph{contracted graph} $H=G_{|{\mathcal{C}}}$, that is defined as follows. Start with $H=G$, and then, for every cluster $C\in {\mathcal{C}}$, contract $C$ into a super-node $u(C)$. We delete all self loops but keep parallel edges.
Next, we compute a Gomory-Hu tree $\tau$ for graph $H$, that we root at an arbitrary vertex. Recall that $V(\tau)=V(H)$, and, for every non-root vertex $v$ in the tree, if $v'$ is the parent of vertex $v$ in the tree, and $\tau_v$ is the sub-tree of $\tau$ rooted at $v$, then, denoting $S=V(\tau_v)$, from Corollary~\ref{cor: G-H tree_edge_cut}, the minimum cut in graph $H$ separating $v$ from $v'$ is precisely $(S,V(H)\setminus S)$.
As it turns out, the most difficult case for the proof of \Cref{thm: advanced disengagement of clusters} is when the graph $\tau$ is a path. In the following subsection, we prove the following lemma for this special case.
The statement of the lemma is almost identical to the statement of \Cref{thm: advanced disengagement of clusters}, except that we now assume that the tree $\tau$ for the graph $H$ is a path.
\begin{lemma}\label{lem: path case}
There is an efficient algorithm, that, given as input an instance $(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, a collection ${\mathcal{C}}$ of disjoint sub-graphs of $G$, and, for every cluster $C\in {\mathcal{C}}$, a vertex $u(C)$ and a set ${\mathcal{Q}}(C)$ of paths routing the edges of $\delta(C)$ to vertex $u(C)$ inside $C$, such that:
\begin{itemize}
\item $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$;
\item $|E^{\textsf{out}}({\mathcal{C}})|\leq |E(G)|/(100\mu)$;
\item for every cluster $C\in {\mathcal{C}}$, $|E(C)|\leq |E(G)|/(100\mu)$;
\item for every cluster $C\in {\mathcal{C}}$ and for each edge $e\in C$, $(\cong_C({\mathcal{Q}}(C),e))^2\le \beta$; and
\item the Gomory-Hu tree $\tau$ for the graph $H=G_{|{\mathcal{C}}}$ is a path,
\end{itemize}
computes a collection ${\mathcal{I}}=\set{I_1,\ldots,I_k}$ of sub-instances of $(G,\Sigma)$, with the following properties:
\begin{itemize}
\item for all $1\leq i\leq k$, $|E(G_i)|\leq |E(G)|/\mu$;
\item $\sum_{i=1}^k|E(G_i)|\leq O(|E(G)|)$; and
\item $\sum_{1\le i\le k}\mathsf{OPT}_{\mathsf{cnwrs}}(G_i,\Sigma_i)\le O((\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+|E(G)|)\cdot \beta)$.
\end{itemize}
Moreover, there is an efficient algorithm, that, given, for each $1\le i\le k$, a feasible solution $\phi_i$ for instance $(G_i,\Sigma_i)$, computes a feasible solution $\phi$ for instance $(G,\Sigma)$ with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{1\le i\le k}\mathsf{cr}(\phi_i)\textsf{right} )$.
\end{lemma}
We defer the proof of \Cref{lem: path case} to the following subsections, and we complete the proof of \Cref{thm: advanced disengagement of clusters} using the lemma here.
\mynote{plan: compute the path decomposition into $\log n$ levels. Then do disengagemnets with the paths as clusters. }
\subsection{Disengagement on a Path -- Proof of \Cref{lem: path case}}
\label{subsec: disengagement on a path}
We denote the clusters in ${\mathcal{C}}$ by $C_1,C_2,\ldots,C_r$. For convenience, for each $1\leq i\leq r$, we denote the vertex $v(C_i)$ of the graph $H=G_{|{\mathcal{C}}}$ by $x_i$. Recall that we have assumed that the tree $\tau$ is a path. We assume without loss of generality that the clusters are indexed according to their appearance on the path $\tau$. For $1\leq i\leq r$, we let $S_i=\set{x_1,\ldots,x_i}$ and let $\overline{S}_i=\set{x_{i+1},\ldots,x_r}$. Recall that, by the definition of the Gomory-Hu tree, $(S_i, \overline{S}_i)$ is a minimum cut in graph $H$ separating vertex $x_i$ from vertex $x_{i+1}$. From the properties of minimum cut, there is a set of paths in graph $H$, routing the edges of $\delta_H(S_i)$ to vertex $x_{i}$ inside $S_i$, and there is a set of paths in graph $H$, routing the edges of $\delta_H(\overline{S}_i)$ to vertex $x_{i+1}$ inside $\overline{S}_i$.
Recall that every edge in graph $H$ corresponds to some edge in $E^{\textsf{out}}({\mathcal{C}})$, and we do not distinguish between these edges. In particular, for all $1\le i,j\le r$, we will also view edges of $E(C_i,C_j)$ as parallel edges connecting vertex $x_i$ to vertex $x_j$ in graph $H$.
For each $1\le i\le k$, we denote
$\hat E_i=E(C_i,C_{i+1})$,
$E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$, and
$E_i^{\operatorname{over}}=\bigcup_{i'<i,j'>i+1}E(C_{i'},C_{j'})$.
We need the following observation.
\begin{observation}\label{obs: bad inded structure}
For all $1\leq i<r$, the following hold:
\begin{itemize}
\item $|\hat E_i|\geq |E_i^{\operatorname{over}}|$;
\item $|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_{i+1}^{\operatorname{left}}|-|E_i^{\operatorname{right}}|$; and
\item $|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$.
\end{itemize}
\end{observation}
\begin{proof}
Consider the cut $(\set{x_{i}},V(H)\setminus \set{x_i})$ in $H$. Its size is $|\hat E_i|+|\hat E_{i-1}|+ |E_i^{\operatorname{right}}|+|E_{i}^{\operatorname{left}}|$. Note that this cut separates $x_i$ from $x_{i-1}$. Since the minimum cut separating $x_i$ from $x_{i-1}$ in $H$ is $(S_{i-1},\overline{S}_{i-1})$, and $|E(S_{i-1},\overline{S}_{i-1})|=|\hat E_{i-1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|$, we get that:
\[ |\hat E_{i-1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|\hat E_{i-1}|+ |E_i^{\operatorname{right}}|+|E_{i}^{\operatorname{left}}|, \]
and so
\[ |E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+ |E_i^{\operatorname{right}}|. \]
Consider the cut $(\set{x_{i+1}},V(H)\setminus \set{x_{i+1}})$ in $H$. Its size is $|\hat E_i|+|\hat E_{i+1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i+1}^{\operatorname{right}}|$. Note that this cut separates $x_{i+1}$ from $x_{i+2}$. Since the minimum cut separating $x_{i+1}$ from $x_{i+2}$ in $H$ is $(S_{i+1},\overline{S}_{i+1})$, and $|E(S_{i+1},\overline{S}_{i+1})|=|\hat E_{i+1}|+ |E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{right}}|+|E_i^{\operatorname{over}}|$, we get that:
\[|\hat E_{i+1}|+ |E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{right}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|\hat E_{i+1}|+ |E_{i+1}^{\operatorname{left}}|+|E_{i+1}^{\operatorname{right}}|, \]
and so
\[ |E_i^{\operatorname{right}}|+|E_i^{\operatorname{over}}|\leq |\hat E_i|+|E_{i+1}^{\operatorname{left}}|. \]
By adding the two inequalities and rearranging the sides, we get that $|\hat E_i|\geq |E_i^{\operatorname{over}}|$. By rearranging the sides of the two inequalities, we get that
$|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_{i+1}^{\operatorname{left}}|-|E_i^{\operatorname{right}}|$, and
$|\hat E_i|-|E_i^{\operatorname{over}}|\geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$.
\end{proof}
\begin{definition}
We say that index $i$ is \emph{problematic} iff $|E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|>100|\hat E_i|$.
\end{definition}
\begin{observation}\label{obs: problematic instance}
If index $i$ is problematic then $|E_{i+1}^{\operatorname{left}}|-|\hat E_i| \leq |E_i^{\operatorname{right}}|\leq |E_{i+1}^{\operatorname{left}}|+|\hat E_i|$, and moreover $48|E_{i+1}^{\operatorname{left}}|/49 \leq |E_i^{\operatorname{right}}|\leq 49|E_{i+1}^{\operatorname{left}}|/48$.
\end{observation}
\begin{proof}
Assume that index $i$ is problematic. Since $|E_i^{\operatorname{over}}|\leq |\hat E_i|$, we get that either $|E_i^{\operatorname{right}}|\geq 49|\hat E_i|$, or $|E_{i+1}^{\operatorname{left}}|\geq 49|\hat E_i|$. Assume without loss of generality that $|E_i^{\operatorname{right}}|\geq |E_{i+1}^{\operatorname{left}}|$, so $|E_i^{\operatorname{right}}|\geq 49 |\hat E_i|$. Since, from \Cref{obs: bad inded structure},
$|\hat E_i| \geq |E_i^{\operatorname{right}}|-|E_{i+1}^{\operatorname{left}}|$, we get that $|E_{i+1}^{\operatorname{left}}|\geq |E_i^{\operatorname{right}}|-|\hat E_i|\geq 48|E_i^{\operatorname{right}}|/49$.
\end{proof}
We start by considering the special case where no index $i$ is problematic, and prove \Cref{lem: path case} for it in the \Cref{subsec: no prob edges}. We then show a reduction from the general case to this special case in \Cref{subsec: path with problematic indices}.
\input{disengage_path.tex}
\subsection{Completing the Proof of \Cref{lem: path case}}
\label{subsec: path with problematic indices}
In this subsection we complete the proof of \Cref{lem: path case}. Recall that we are given a collection ${\mathcal{C}}=\set{C_1,\ldots,C_r}$ of clusters with $\bigcup_{i=1}^rV(C_i)=V(G)$, such that the Gomory-Hu tree $\tau$ for the contracted graph $H=G_{|{\mathcal{C}}}$ is a path, with vertices $x_1,\ldots,x_r$ appearing on the path in this order, where vertex $x_i$ represents cluster $C_i$. We have already show how to complete the proof of \Cref{lem: path case} if no index $1\leq i\leq r$ is problematic, so we assume now that at least one such index is problematic.
Let $i_1<i_2<\cdots<i_q$ be all problematic indices. Consider some problematic index $i_j$.
By definition, $|E_{i_j}^{\operatorname{right}}|+|E_{i_j+1}^{\operatorname{left}}|+|E_{i_j}^{\operatorname{over}}|>100|\hat E_{i_j}|$.
We define indices $\mathsf{LM}(i_j)$ and $\mathsf{RM}(i_j)$ as follows.
Recall that $E^{\operatorname{right}}_{i_j}$ is the set of all edges $(x_{i_j},x_z)$ with $z>i_j+1$. We let $\mathsf{RM}(i_j)$ be the smallest integer $z>i_j+1$, such that at least half the edges in $E^{\operatorname{right}}_{i_j}$ have their endpoints in $\set{x_{i_j+2},\ldots,x_z}$. It is easy to see that at least half the edges in $E^{\operatorname{right}}_{i_j}$ must have their endpoints in $\set{x_{z},\ldots,x_q}$. Similarly, we let $\mathsf{LM}(i_j)$ be the largest integer $z<i_j$, such that at least half the edges in $E^{\operatorname{left}}_{i_j+1}$ have their endpoints in $\set{x_{z},\ldots,x_{i_j-1}}$. It is easy to see that at least half the edges in $E^{\operatorname{left}}_{i_j+1}$ must have their endpoints in $\set{x_{1},\ldots,x_z}$. We need the following crucial observation.
\begin{observation}\label{obs: large capacity up to midpoint}
For all $1\leq j\leq q$, for every index $i_j<z<\mathsf{RM}(i_j)$, $|\hat E_z|\geq |E^{\operatorname{right}}_{i_j}|/2$; and similarly, for every index $\mathsf{LM}(i_j)<z'\leq i_j$, $|\hat E_{z'}|\geq |E^{\operatorname{left}}_{i_j+1}|/2$.
\end{observation}
\begin{proof}
Fix an index $1\leq j\leq q$, and another index index $i_j<z<\mathsf{RM}(i_j)$. Recall that there is a subset $E'\subseteq E^{\operatorname{right}}_{i_j}$ of at least $|E^{\operatorname{right}}_{i_j}|/2$ edges that have an endpoint in $\set{x_{\mathsf{RM}(i_j)},\ldots,x_r}$. It is easy to verify that $E'\subseteq E_{z}^{\operatorname{over}}$. From \Cref{obs: bad inded structure}, $|\hat E_z|\geq |E_{z}^{\operatorname{over}}|\geq |E^{\operatorname{right}}_{i_j}|/2$. The proof for index $\mathsf{LM}(i_j)<z'\leq i_j$ is symmetric.
\end{proof}
\begin{claim}\label{claim: inner and outer paths}
Let $a$ be some problematic index and let $z\in \set{1,\ldots,r}$ be another index such that $a<z<\mathsf{RM}(a)$.
If we denote by $Y$ the cluster of $H$ induced by vertices $x_{a+1},\ldots,x_z$, then (i) there is a set $\hat{\mathcal{Q}}(Y)$ of paths routing the edges of $\delta_H(Y)$ to vertex $x_z$ in $Y$ with congestion $O(1)$; and (ii) there is a set $\hat{\mathcal{P}}(Y)$ of paths routing the edges of $\delta_H(Y)$ to vertex $x_{z+1}$ outside $Y$ with congestion $O(1)$.
Similarly,
let $a'$ be some problematic index and let $z'\in \set{1,\ldots,r}$ be another index such that $\mathsf{LM}(a')<z'<a$.
If we denote by $Y'$ the cluster of $H$ induced by vertices $x_{z'},\ldots,x_{a'-1}$, then (i) there is a set $\hat{\mathcal{Q}}(Y')$ of paths routing the edges of $\delta_H(Y')$ to vertex $x_{z'}$ in $Y'$ with congestion $O(1)$; and (ii) there is a set $\hat{\mathcal{P}}(Y')$ of paths routing the edges of $\delta_H(Y')$ to vertex $x_{z'-1}$ outside $Y$ with congestion $O(1)$.
\end{claim}
\begin{proof}
We only prove the claim for cluster $Y$, and the proof for cluster $Y'$ is symmetric.
We first prove (i). We partition the set $\delta_H(Y)$ into two subsets: set $E_1$ contains all edges of $\delta_H(Y)$ with an endpoint in $S_a$ (recall that $S_a=\set{x_1,\ldots,x_a}$); and set $E_2$ contains all edges of $\delta_H(Y)$ with an endpoint in $\overline{S}_z$ (recall that $\overline{S}_z=\set{x_{z+1},\ldots,x_q}$).
Recall that, from the properties of minimum cut, there is a set ${\mathcal{R}}$ of edge-disjoint paths routing the edges of $E_2$ to vertex $x_z$ inside $S_z$.
Let ${\mathcal{R}}^{in}\subseteq {\mathcal{R}}$ be the subset of all paths of ${\mathcal{R}}$ that are entirely contained in $Y$, and we denote ${\mathcal{R}}^{out}={\mathcal{R}}\setminus {\mathcal{R}}^{in}$.
For each path $R\in {\mathcal{R}}^{out}$, we view $x_z$ as its last vertex and the edge of $E_2$ as its first edge, and we denote by $e_R$ the first edge of $E_1$ that appears on $R$. It is clear that edges in $\set{e_{R}\mid R\in {\mathcal{R}}^{out}}$ are mutually distinct.
On the other hand, since $E_1\subseteq \hat E_a\cup E_{a+1}^{\operatorname{left}}\cup E_a^{\operatorname{over}}$ and $a$ is a problematic index, combined with \Cref{obs: bad inded structure}, we get that $|E_1|\leq |\hat E_a|+| E_{a+1}^{\operatorname{left}}|+|E_a^{\operatorname{over}}|\leq 2|E_a^{\operatorname{right}}|$. Moreover, from \Cref{obs: large capacity up to midpoint}, for all $a<z'\leq z$, $|\hat E_{z'}|\geq |E_a^{\operatorname{right}}|/2\ge |E_1|/4$.
Therefore,
there is a set $\hat{\mathcal{Q}}_1=\set{\hat Q_e\mid e\in E_1}$ of paths, routing edges of $E_1$ to $x_z$ in $Y$, using only edges of $\bigcup_{a< z'< z}\hat E_{z'}$, such that $\cong_{Y}(\hat{\mathcal{Q}}_1)\le 4$; and similarly,
there is a set $\hat{{\mathcal{R}}}^{out}=\set{\hat R\mid R\in {\mathcal{R}}^{out}}$ of paths, routing edges of $\set{e_R\mid R\in {\mathcal{R}}^{out}}$ to $x_z$ in $Y$, using only edges of $\bigcup_{a< z'< z}\hat E_{z'}$, such that $\cong_{Y}(\hat{{\mathcal{R}}}^{out})\le 4$.
We now define $\hat{\mathcal{Q}}^{out}$ to be the set that contains, for each path $R\in {\mathcal{R}}^{out}$, the concatenation of (i) the subpath of $R$ between its first edge (in $E_2$) and edge $e_R$ (not included); and (ii) the path $\hat R\setminus \set{e_R}$.
We now let $\hat{\mathcal{Q}}_2={\mathcal{R}}^{in}\cup\hat{\mathcal{Q}}^{out}$ and $\hat{\mathcal{Q}}(Y)=\hat{\mathcal{Q}}_1\cup \hat{\mathcal{Q}}_2$.
It is clear that the set $\hat{\mathcal{Q}}(Y)$ contains, for each edge of $\delta_H(Y)$, a path routing the edge to $x_z$ in $Y$, and
$$\cong_{Y}(\hat{\mathcal{Q}}(Y))\le \cong_{Y}(\hat{\mathcal{Q}}_1)+\cong_{Y}(\hat{\mathcal{Q}}_2)\le 4+\cong_{Y}({\mathcal{R}}^{in})+\cong_{Y}(\hat{\mathcal{Q}}^{out})=4+1+(1+4)=10.$$
\begin{figure}[h]
\centering
\subfigure[Cluster $Y$ and its relevant vertices and edge sets.]{\scalebox{0.3}{\includegraphics{figs/prob_index_1.jpg}}\label{fig: path_cluster_H}
}
\hspace{4pt}
\subfigure[Cluster $Y'$ and its relevant vertices and edge sets.]{\scalebox{0.3}{\includegraphics{figs/prob_index_2.jpg}}\label{fig: path_cluster_G}}
\caption{An illustration of clusters $Y,Y'$ in $H$.
\end{figure}
We now prove (ii).
From the properties of minimum cuts, there is a set ${\mathcal{R}}_1$ of edge-disjoint paths routing edges of $E_1$ to $x_{a}$ in $S_{a}$ (and therefore outside $Y$).
Similarly, there is a set ${\mathcal{R}}_2$ of edge-disjoint paths routing the edges of $E_2$ to $x_{z+1}$ inside $\overline{S}_{z}$ (and therefore outside $Y$). As observed before, $|E_1|\leq 2|E_a^{\operatorname{right}}|$. Moreover, there is a set $E'\subseteq E_a^{\operatorname{right}}$ of at least $|E_a^{\operatorname{right}}|/2$ edges, each of which connects $x_a$ to a vertex in $\set{x_{z+1},\ldots,x_r}$.
For each edge $e\in E_1$, we assign to it an edge $e'$ of $E'$, such that each edge of $E'$ is assigned to at most $4$ edges of $E'_1$.
Observe that $E'\subseteq \delta_H(\overline{S}_{z})$, so there is a set ${\mathcal{R}}_3$ of $|E'|\geq |E_a^{\operatorname{right}}|/2$ edge-disjoint paths routing the edges of $E'$ to $x_{z+1}$, that are disjoint from $Y$.
We now define $\hat{{\mathcal{P}}}_1$ to be the set that contains, for each edge $e\in E_1$, the union of (i) the path in ${\mathcal{R}}_1$ routing $e$ to $x_a$ in $S_a$; and (ii) the path in ${\mathcal{R}}_3$ that routing $e'$ to $x_{z+1}$ in $\overline{S}_z$.
We let $\hat{{\mathcal{P}}}(Y)={\mathcal{R}}_2\cup \hat{{\mathcal{P}}}_1$.
It is clear that the set $\hat{\mathcal{P}}(Y)$ contains, for each edge of $\delta_H(Y)$, a path routing the edge to vertex $x_{z+1}$ outside $Y$, and
$$\cong_{H}(\hat{\mathcal{Q}}(Y))\le
\cong_{H}({\mathcal{R}}_2)+\cong_{H}(\hat{\mathcal{P}}_1)\le 1+\cong_{H}({\mathcal{R}}_1)+4\cdot\cong_{H}({\mathcal{R}}_3)=1+(1+4)=6.$$
\end{proof}
\znote{to modify}
The plan for the remainder of the proof is the following. First, we will define a family ${\mathcal{X}}$ of clusters of $G$,
and for each cluster $X\in {\mathcal{X}}$, we will define a vertex $u(X)\in X$, with a set ${\mathcal{Q}}(X)$ of paths routing the edges of $\delta_G(X)$ to $u(X)$ inside $X$, and another vertex $u'(X)\not \in X$, with another set ${\mathcal{P}}(X)$ of paths routing the edges of $\delta_G(X)$ to $u'(X)$ outside $X$. Then we will disengage the clusters in ${\mathcal{X}}$ using the basic disengagement (described in \Cref{subsec: basic disengagement}) and obtain, for every cluster $X\in {\mathcal{X}}$, a sub-instance $I_X=(G_X,\Sigma_X)$. We will then show that this sub-instance can be further reduced to an instance $(G'_X,\Sigma'_X)$, such that,
if we denote by ${\mathcal{C}}_X\subseteq {\mathcal{C}}$ the set of clusters in ${\mathcal{C}}$ that are contained in graph $X$, and denote by $H_X=(G'_X)_{|{\mathcal{C}}_X}$ the corresponding contracted graph, and denote by $\tau_X$ the Gomory-Hu tree of $H_X$, then $\tau_X$ is a subpath of $\tau$ that contains no bad indices.
Therefore, we can solve each resulting sub-instance using the algorithm from \Cref{subsec: no prob edges}. This leaves us with one final instance, that is obtained from the graph $G$, by contracting every cluster of $\bigcup_{X\in {\mathcal{X}}}{\mathcal{C}}_X$ into a single vertex. In our last step, we will define the clusters in this final instance in such a way that the Gomory-Hu tree of the corresponding contracted graph is a path with no problematic indices, and we will again use the algorithm from \Cref{subsec: no prob edges} to solve it.
\znote{path, directly with ``problematic indices''?}
\subsubsection{First-Level Disengagement}
\label{subsec: 1st disengagement}
In this step we define a collection ${\mathcal{X}}$ of clusters, and for every cluster $X\in {\mathcal{X}}$, a vertex $u(X)\in X$ with a set ${\mathcal{Q}}(X)$ of paths routing the edges of $\delta_G(X)$ to $u(X)$ inside $X$, and another vertex $u'(X)\not \in X$ with another set ${\mathcal{P}}(X)$ of paths routing the edges of $\delta_G(X)$ to $u'(X)$ outside $X$. Then we perform basic disengagement from \Cref{subsec: basic disengagement} to $G$, using the set ${\mathcal{X}}$ of clusters and their corresponding path sets.
\subsubsection*{Step 1: Defining the Clusters}
We construct the collection ${\mathcal{X}}$ iteratively. We start with ${\mathcal{X}}=\emptyset$, and then process every consecutive pair of problematic indices, as follows.
Consider the index pair $i_j,i_{j+1}$ for some $1\le j\le q-1$.
For brevity, we denote $i_j=a$ and $i_{j+1}=b$.
We now distinguish between the following two cases.
\paragraph{Case 1. $\mathsf{LM}(b)<\mathsf{RM}(a)$.}
By definition, $a<b$, $a<\mathsf{RM}(a)$, and $\mathsf{LM}(b)<b$.
Now since $\mathsf{LM}(b)<\mathsf{RM}(a)$, we get that $\max\set{a,\mathsf{LM}(b)}<\min\set{b,\mathsf{RM}(a)}$.
Let $z$ be any index such that $\max\set{a,\mathsf{LM}(b)}\le z<\min\set{b,\mathsf{RM}(a)}$.
Denote by $Y_j$ the subgraph of $H$ induced by vertices $x_{a+1},\ldots,x_z$, and denote by $Y'_j$ the subgraph of $H$ induced by vertices $x_{z+1},\dots, x_{b}$. See \Cref{fig: path_cluster_H} for an illustration. From \Cref{claim: inner and outer paths}, there is a set $\hat{\mathcal{Q}}(Y_j)$ of paths routing the edges of $\delta_H(Y_j)$ to vertex $x_z$ in $Y_j$ with congestion $O(1)$; and there is a set $\hat{\mathcal{P}}(Y_j)$ of paths routing the edges of $\delta_H(Y_j)$ to vertex $x_{z+1}$ outside $Y_j$ with congestion at most $6$.
Similarly, there is a set $\hat{\mathcal{Q}}(Y'_j)$ of paths routing the edges of $\delta_H(Y'_j)$ to vertex $x_{z+1}$ in $Y'_j$ with congestion $O(1)$; and there is a set $\hat{\mathcal{P}}(Y'_j)$ of paths routing the edges of $\delta_H(Y'_j)$ to vertex $x_{z}$ outside $Y'_j$ with congestion $O(1)$.
\begin{figure}[h]
\centering
\subfigure[Subgraphs $Y_j,Y'_j$ in $H$.]{\scalebox{0.34}{\includegraphics{figs/path_cluster_1.jpg}}\label{fig: path_cluster_H}
}
\hspace{4pt}
\subfigure[Subgraphs $X_j,X'_j$ in $G$.]{\scalebox{0.34}{\includegraphics{figs/path_cluster_2.jpg}}\label{fig: path_cluster_G}}
\caption{An illustration of subgraphs $Y_j,Y'_j$ in $H$ and subgraphs $X_j,X'_j$ in $G$.
\end{figure}
We let $X_j$ be the subgraph of $G$ induced by $\bigcup_{x_k\in Y_j}C_j$, and let $X_j'$ be the subgraph of $G$ induced by $\bigcup_{x_k\in Y'_j}C_k$. We add clusters $X_j,X_j'$ to the set ${\mathcal{X}}$ of subgraphs of $G$. See \Cref{fig: path_cluster_G} for an illustration.
We define vertices $u(X_j),u'(X_j)$ and path sets ${\mathcal{Q}}(X_j),{\mathcal{P}}(X_j)$ as follows.
We set $u(X_j)=u(C_z)$.
For each edge $e\in \delta_G(X_j)$, we define a path $Q_{X_j}(e)$ as follows.
Let $(e_0,x_{\ell_0},e_1,x_{\ell_2},\ldots,x_{\ell_{t-1}},e_t,x_t)$ be the path in $\hat{\mathcal{Q}}(Y_j)$ that routes $e$ to $x_z$, where $e_0=e$ and $x_t=x_z$.
Recall that $e_0,e_1,\ldots,e_t$ are also edges in $G$.
We define path $Q_{X_j}(e)$ to be the sequential concatenation of: the edge $e_0$ in $G$; the path in ${\mathcal{Q}}(C_{\ell_0})$ routing $e_0$ to $u(C_{\ell_0})$; the path in ${\mathcal{Q}}(C_{\ell_0})$ routing $e_1$ to $u(C_{\ell_0})$; the edge $e_1$ in $G$; the path in ${\mathcal{Q}}(C_{\ell_1})$ routing $e_1$ to $u(C_{\ell_1})$, \ldots, the edge $e_t$ in $G$. Clearly, path $Q_{X_j}(e)$ routes $e$ to $u(X_j)$.
We then let set ${\mathcal{Q}}(X_j)$ contains, for each edge $e\in \delta_G(X_j)$, the path $Q_{X_j}(e)$.
We set $u'(X_j)=u(C_{z+1})$, and we define the set ${\mathcal{P}}(X_j)$ of paths using the path set $\hat{\mathcal{P}}(Y_1)$ and the path sets $\set{{\mathcal{Q}}(C_k)\mid k\le a\text{ or }k> z}$ similarly.
We then set $u(X'_j)=u(C_{z+1})$ and $u'(X'_j)=u(C_{z})$, and define path sets ${\mathcal{Q}}(X'_j), {\mathcal{P}}(X'_j)$ similarly.
\iffalse
We set $u(X_j)=u(C_z)$. Using the path sets ${\mathcal{Q}}(C_k)$ for all vertices $x_k\in Y_j$, combined with the set $\hat{\mathcal{Q}}(Y_j)$ of paths that we have constructed, we obtain a set ${\mathcal{Q}}(X_j)$ of paths routing the edges of $\delta_H(X_j)$ to the vertex $u(C_z)$ inside $X_j$. Similarly, we set $u(X_j')=u(C_{z+1})$, and use the path sets ${\mathcal{Q}}(C_k)$ for all vertices $x_k\in Y'_j$, together with the path set $\hat{\mathcal{Q}}(Y_j')$ that we have constructed, to obtain a set ${\mathcal{Q}}(X'_j)$ of paths routing the edges of $\delta_H(X'_j)$ to $u(X'_j)$.
Similarly, we set $u'(X_j)=u(C_{z+1})$. Using the path sets ${\mathcal{Q}}(C_k)$ for all vertices $x_k\in Y_j$, combined with the set $\hat{\mathcal{Q}}(Y_j)$ of paths that we have constructed, we obtain a set ${\mathcal{Q}}(X_j)$ of paths routing the edges of $\delta_H(X_j)$ to the vertex $u(C_z)$ inside $X_j$. Similarly, we set $u(X_j')=u(C_{z+1})$, and use the path sets ${\mathcal{Q}}(C_k)$ for all vertices $x_k\in Y'_j$, together with the path set $\hat{\mathcal{Q}}(Y_j')$ that we have constructed, to obtain a set ${\mathcal{Q}}(X'_j)$ of paths routing the edges of $\delta_H(X'_j)$ to $u(X'_j)$.
We also set $u'(\hat R')=u(C_{z+1})$, and $u'(\hat R'')=u(C_z)$, and define the set ${\mathcal{P}}(\hat R')$ of paths connecting the edges of $\delta_H(\hat R')$ to $u'(\hat R')$ outside $\hat R'$, and the set ${\mathcal{P}}(\hat R'')$ of paths connecting the edges of $\delta_H(\hat R'')$ to $u'(\hat R'')$ outside $\hat R''$, using the paths sets ${\mathcal{P}}'(R')$, ${\mathcal{P}}'(R'')$, $\bigcup_{x_k\in R'}{\mathcal{Q}}(C_k)$ and $\bigcup_{x_k\in R''}{\mathcal{Q}}(C_k)$ similarly.
\fi
\paragraph{Case 2. $\mathsf{RM}(a)\leq \mathsf{LM}(b)$.}
Denote $z=\mathsf{RM}(a)$ and $z'=\mathsf{LM}(b)$.
Note that indices $z,z'$ may not be problematic.
We let $Y_j$ be the subgraph of $H$ induced by vertices $x_{a+1},\ldots,x_z$, and we let $Y_j'$ be the subgraph of $H$ induced by vertices $x_{z'+1},\ldots,x_b$. From \Cref{claim: inner and outer paths}, we can find a set $\hat{\mathcal{Q}}(Y_j)$ of paths routing edges of $\delta_{H}(Y_j)$ to $x_z$ inside $Y_j$, and a set $\hat{\mathcal{P}}(Y_j)$ of paths routing edges of $\delta_{H}(Y_j)$ to $x_{z+1}$ outside $Y_j$. Similarly, we can find a set $\hat{\mathcal{Q}}(Y_j')$ of paths routing edges of $\delta_{H}(Y_j')$ to $x_{z'+1}$ inside $Y_j'$, and a set $\hat{\mathcal{P}}(Y_j')$ of paths routing edges of $\delta_{H'}(Y_j')$ to $x_{z'}$ outside $Y_j'$. We define clusters $X_j,X_j'$ exactly as before, and then add them to set ${\mathcal{X}}$ of clusters. The path sets ${\mathcal{Q}}(X_j)$, ${\mathcal{P}}(X_j)$, ${\mathcal{Q}}(X_j')$, ${\mathcal{P}}(X_j')$ are defined similarly.
Finally, we additionally process the first index and the last index as follows.
For the first index $i_1$, we let $z'=\mathsf{LM}(i_1)$, and define the cluster $Y'_0$ as subgraph of $H$ induced by $x_{z'+1},\ldots,x_{i_1}$. We define the corresponding cluster $X'_0$, vertices $u(X'_0),u'(X'_0)$ and path sets ${\mathcal{Q}}(X'_0),{\mathcal{P}}(X'_0)$ similarly, and add it to the set ${\mathcal{X}}$.
For the last index $i_q$, we let $z'=\mathsf{RM}(i_q)$, and define the cluster $Y_q$ as subgraph of $H$ induced by $x_{i_q+1},\ldots,x_{z'}$. We define the corresponding $X_q$, vertices $u(X_q),u'(X_q)$ and path sets ${\mathcal{Q}}(X_q),{\mathcal{P}}(X_q)$ similarly, and add it to the set ${\mathcal{X}}$.
This completes the construction of the set ${\mathcal{X}}$ of clusters and their affiliated vertices and path sets.
Clearly, the clusters of ${\mathcal{X}}$ are vertex-disjoint subgraphs of $G$. See \Cref{fig: xset_cluster} for an illustration.
\begin{figure}[h]
\centering
\includegraphics[scale=0.55]{figs/xset_clusters.jpg}
\caption{An illustration of all clusters of ${\mathcal{X}}$: clusters $C_i$ with a problematic index $i$ are shown in brown, and clusters $C_i$ that is not contained in any cluster of ${\mathcal{X}}$ are shown in blue. }\label{fig: xset_cluster}
\end{figure}
We use the following observation, which, from the construction of clusters in ${\mathcal{X}}$ and the definition of corresponding path sets, is an easy corollary of \Cref{claim: inner and outer paths}.
\begin{observation}
\label{obs: congestion of basic disengagement paths}
Let $X$ be a cluster of ${\mathcal{X}}$, then for each edge $e$ that belongs to some cluster $C\in {\mathcal{C}}$, $\cong_G({\mathcal{Q}}(X),e)\le O(\cong_G({\mathcal{Q}}(C),e))$; and for each edge $e$ that does not belongs to any cluster of ${\mathcal{C}}$, $\cong_G({\mathcal{Q}}(X),e)\le O(1)$.
\end{observation}
\subsubsection*{Step 2: Computing the Basic Disengagement}
Now for each cluster $X\in {\mathcal{X}}$, we apply the algorithm from \Cref{lem: non_interfering_paths} to the set ${\mathcal{Q}}(X)$ of paths, and obtain a set $\tilde{\mathcal{Q}}(X)$ of paths routing edges of $\delta_G(X)$ to $u(X)$ inside $X$, that are non-transversal with respect to $\Sigma$, such that for each $e\in E(G)$, $\cong_G(\tilde{\mathcal{Q}}(X),e)\le \cong_G({\mathcal{Q}}(X),e)$.
Similarly, let $\tilde{{\mathcal{P}}}(X)$ be the set of paths that we obtained from applying the algorithm from \Cref{lem: non_interfering_paths} to the set ${\mathcal{P}}(X)$ of paths.
Denote $\tilde{{\mathcal{Q}}}=\bigcup_{X\in {\mathcal{X}}}\tilde{\mathcal{Q}}(X)$ and $\tilde{{\mathcal{P}}}=\bigcup_{X\in {\mathcal{X}}}\tilde{\mathcal{P}}(X)$.
We compute the colletion ${\mathcal{I}}({\mathcal{X}},\tilde{{\mathcal{Q}}},\tilde{{\mathcal{P}}})$ of sub-instances of instance $(G,\Sigma)$, using the basic disengagement in \Cref{subsec: basic disengagement}.
From \Cref{lem: low solution costs},
$$\sum_{I'\in {\mathcal{I}}({\mathcal{X}},\tilde{\mathcal{Q}},\tilde{\mathcal{P}})}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq O\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot\beta+\sum_{X\in {\mathcal{X}}}\sum_{\tilde P\in \tilde{\mathcal{P}}(X)}\chi(\tilde P,X)\bigg).$$
From the properties of path set $\tilde{{\mathcal{P}}}$, $\sum_{X\in {\mathcal{X}}}\chi(\tilde {\mathcal{P}},X)\le \sum_{X\in {\mathcal{X}}}\chi({\mathcal{P}},X)$.
Then from \Cref{obs: congestion of basic disengagement paths}, for each cluster $X\in {\mathcal{X}}$,
$\chi({\mathcal{P}},X)\le O(\chi^1(X)\cdot\beta)$. Moreover, since the clusters in ${\mathcal{X}}$ are mutually disjoint, $\sum_{X\in {\mathcal{X}}}O(\chi^1(X)\cdot\beta)\le O(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot\beta)$. Altogether, we get that
$$\sum_{I'\in {\mathcal{I}}({\mathcal{X}},\tilde{\mathcal{Q}},\tilde{\mathcal{P}})}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq O\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot\beta\bigg).$$
\subsubsection*{Step 3: Proving that Instances of $\set{I_X}_{X\in {\mathcal{X}}}$ are Suitable for the Algorithm in \Cref{subsec: no prob edges}}
Recall that ${\mathcal{I}}({\mathcal{X}},\tilde{{\mathcal{Q}}},\tilde{{\mathcal{P}}})=\set{\hat I_{{\mathcal{X}},\tilde{\mathcal{Q}}}}\cup\set{I_X}_{X\in {\mathcal{X}}}$, where for each cluster $X\in {\mathcal{X}}$, instance $I_X=(G_X,\Sigma_X)$ and graph $G_X$ is obtained from $G$ by contracting $G\setminus X$ into a single vertex $v^*$.
We will now show that, for each $X\in {\mathcal{X}}$, we can efficiently construct a set ${\mathcal{C}}_X$ of clusters in $G_X$, such that the contracted graph $H_X=(G_X)_{|{\mathcal{C}}_X}$ has the $\alpha$-cut property with respect to some path $\tau_X$ on vertices of $V(H_X)$.
On the other hand, the instance $\hat I_{{\mathcal{X}},\tilde{\mathcal{Q}}}$ does not necessarily satisfy the above property, so we will need to perform another level of disengagement to it in \Cref{subsec: 2nd disengagement}.
\iffalse
\znote{start here}
First, we consider a cluster $X_j$ that are constructed when processing some consecutive pair $(i_j,i_{j+1})$ of problematic indices in Case 1 and consider the instance $I_X\in {\mathcal{I}}({\mathcal{X}},\tilde{\mathcal{Q}},\tilde{\mathcal{P}})$. The arguments for the cluster $X_j$ is symmetric.
The instance $I_X$
We will construct new graphs $G_{R'}, G_{R''}$ corresponding to the clusters $\hat R',\hat R''$. We will show that after a slight modification, the Gomory-Hu tree of the former graph becomes $\tau_{R'}$, the subpath of $\tau$ consisting of vertices of $R'$, and the Gomory-Hu tree of the latter graph becomes becomes $\tau_{R''}$, the subpath of $\tau$ consisting of vertices of $R''$. As we will show later, paths $\tau_{R'}$ and $\tau_{R''}$ do not contain problematic indices.
\znote{How exactly?}
\fi
\iffalse
We now fix some $1\le j\le q-1$ and consider the clusters $X_j,X'_j$.
Recall that $X_j, X'_j$ are constructed when processing the consecutive pair $(i_j,i_{j+1})$ of problematic indices, based on the clusters $Y_j,Y_j'$ in $H$.
We assume that clusters $X_j, X'_j$ are produced in Case 1 of Step 1.
The arguments for the case where $X_j, X'_j$ are produced in Case 2 of Step 1, and the clusters $X'_0,X_q$ which are obtained by additionally handling the first and the last indices can be handled similarly, are the same.
\fi
We use the following claim.
\begin{claim}
\label{claim: alpha cut property}
Let $a$ be some problematic index and $z$ be another index such that $a<z<\mathsf{RM}(a)$.
Let $X$ be the subgraph of $G$ induced by clusters $C_{a+1},\ldots,C_z$, and define $G'$ as the graph obtained from $G$ by contracting $G\setminus X$ into a single vertex $v^*$.
Define ${\mathcal{C}}_X=\set{C_{a+1},\ldots,C_z,v^*}$ and let $H'=G'_{|{\mathcal{C}}_X}$ be the contracted graph, where $V(H')=\set{x_{a+1},\ldots,x_z,v^*}$.
Define path $\tau'=(x_{a+1},\ldots,x_z,v^*)$. Then the graph $H'$ has the $\alpha$-cut property with respect to path $\tau'$, for some $\alpha=O(1)$.
\end{claim}
\iffalse
We consider the cluster $X_j$, and the arguments for cluster $X'_j$ is symmetric.
For convenience, we rename $X_j$ by $X$, and rename the corresponding cluster $Y_j$ in $H$ by $Y$.
\iffalse
We will construct a new instance $I'_{X}$, such that
\begin{enumerate}
\item\label{pro: opt bound} $\mathsf{OPT}_{\mathsf{cnwrs}}(I'_{X})\le \chi(\tilde{\mathcal{P}}(X),X)+O(\chi^2(X)\cdot\beta)$; and
\item\label{pro: drawing transfer} there is an algorithm, that given a solution $\hat\phi'_X$, efficiently computes a solution $\hat\phi_X$ for instance $I_{X_j}$, such that $\mathsf{cr}(\hat\phi_X)\le 9\cdot \mathsf{cr}(\hat\phi'_X)$.
\end{enumerate}
\fi
Recall that $X=\bigcup_{i_j< i\le z}C_i$, where $z<\mathsf{RM}(i_j)$, and $V(Y)=\set{x_i\mid i_j< i\le z}$.
Also recall that $I_X=(G_X,\Sigma_X)$, where graph $G_X$ is obtained from $G$ by contracting all vertices of $G\setminus X$ into a single vertex, that we denote by $v^*$.
\iffalse
For a vertex $v\in X$, the ordering on $v$ in $\Sigma_X$ is identical to the ordering ${\mathcal{O}}_v\in \Sigma$, and the ordering on vertex $v^*$ is $\hat{{\mathcal{O}}}(\tilde{\mathcal{P}})$, the ordering in which paths of $\tilde{{\mathcal{P}}}$ enters vertex $u(C_{z+1})$.
The new instance $I'_X=(G'_X,\Sigma'_X)$ is defined as follows.
Graph $G'_X$ is obtained from graph $G$ by first contracting all vertices of $\bigcup_{1\le t\le i_j}C_t$ into a single vertex $v^*_{\operatorname{left}}$, and then contracting all vertices of $\bigcup_{z<t\ge r}C_t$ into a single vertex $v^*_{\operatorname{right}}$.
Note that, if we further contract vertices $v^*_{\operatorname{left}}$ and $v^*_{\operatorname{right}}$ into a single vertex, then we obtain the graph $G_X$. The rotation system $\Sigma'_X$ is defined similarly.
For a vertex $v\in X$, the ordering on $v$ in $\Sigma'_X$ is identical to the ordering ${\mathcal{O}}_v\in \Sigma$.
The ordering on vertex $v^*_{\operatorname{left}}$ is $\hat{{\mathcal{O}}}(\tilde{\mathcal{P}})$, the ordering in which paths of $\tilde{{\mathcal{P}}}$ enters vertex $u(C_{a})$, and the ordering on vertex $v^*_{\operatorname{right}}$ is $\hat{{\mathcal{O}}}(\tilde{\mathcal{P}})$, the ordering in which paths of $\tilde{{\mathcal{P}}}$ enters vertex $u(C_{z+1})$. \znote{not rigorous here, to modify the previous definition}
We first prove \ref{pro: opt bound}. Recall in the proof of \Cref{lem: low solution costs}, we obtained from the optimal solution $\phi^*$ of instance $(G,\Sigma)$, a drawing $\phi_X$ of the instance $(G_X,\Sigma_X)$, such that $\mathsf{cr}(\phi_X)\le \chi(\tilde{\mathcal{P}}(X),X)+O(\chi^2(X)\cdot\beta)$.
We now show that this drawing can be easily converted into a feasible solution $\phi'_X$ for instance $I'_X$, with at most same number of crossings. And then Property \ref{pro: opt bound} follows.
Denote $E_{\operatorname{left}}= E_G(X,\bigcup_{1\le t\le i_j}C_t)$ and
$E_{\operatorname{right}}= E_G(X,\bigcup_{z+1\le t\le r}C_t)$.
Recall that drawing $\phi_X$ is obtained from the image $\phi(X\cup E(\tilde {\mathcal{P}}(X)))$, by making copies of the image $\phi(E(\tilde {\mathcal{P}}(X))$ to be concatenated into image $\set{\gamma_P\mid P\in \tilde{\mathcal{P}}(X)}$ of paths in $\tilde{\mathcal{P}}(X)$, and then nudging (applying \Cref{obs: curve_manipulation}) them at all common vertices.
Note that, for each edge $e\in E_G(X,\bigcup_{1\le t\le i_j}C_t)$ the path $\tilde P_e\in \tilde{{\mathcal{P}}}$ contains $u(C_a)$ as its inner vertex.
If we simply ``un-nudge'' the images of paths in ${\mathcal{P}}$ at vertex $u(C_a)$, and then (i) view the image of vertex $u(C_a)$ as the image of $v^*_{\operatorname{left}}$; (ii) view the image of vertex $u(C_{z+1})$ as the image of $v^*_{\operatorname{right}}$ (we denote this point by $z$); (iii) for each edge $e\in E_{\operatorname{left}}$, view the subcurve of $\gamma_{\tilde P_e}$ between its endpoint in $X$ and $z$ as the image of $e$; (iv) for each edge $e\in E_{\operatorname{right}}$, view the curve $\gamma_{\tilde P_e}$ as the image of $e$; and (v) for each edge $e\in E(v^*_{\operatorname{left}},v^*_{\operatorname{right}})$, view the curve $\gamma_{\tilde P_e}$ between $z$ and $u(C_{z+1})$ as the image of $e$.
We now prove \ref{pro: drawing transfer}.
We convert a feasible solution $\hat\phi_{X'}$ of the instance $(G_X,\Sigma_X)$ into a solution $\hat\phi_{X}$ of $(G'_{X},\Sigma'_{X})$ as follows.
We copy the image of each edge of $E(v^*_{\operatorname{left}},v^*_{\operatorname{right}})$ at most $3$ times, concatenate them with the image of $E_{\operatorname{left}}$, nudging them at $v^*_{\operatorname{left}}$ (which creates no additional crossings) and then view the resulting curves as the image of edges of $\delta_{G_X}(v^*)$. It is clear that each crossing in $\hat\phi_{X'}$ gives birth to at most $3^2=9$ crossings in $\hat\phi_{X}$. And \ref{pro: drawing transfer} then follows.
\fi
Let ${\mathcal{C}}_X=\set{C_i\mid i_j<i\le z}\cup \set{v^*}$ be a set of vertex-disjoint clusters of $G_X$ whose vertex sets partition $V(G_X)$.
Let $H_X=(G_X)_{|{\mathcal{C}}_X}$ be the contracted graph, so $V(H_X)=V(Y)\cup \set{v^*}$.
In fact, we can view $H_X$ as obtained from $H$ by contracting $H\setminus Y$ into a single vertex $v^*$.
We define the path $\tau_X=(x_{i_j+1},\ldots,x_z,v^*)$, and prove the following claim.
\begin{claim}
Graph $H_X$ satisfies the $10$-cut property with respect to path $\tau_X$.
\end{claim}
\fi
Recall that ${\mathcal{X}}=\set{X'_0,X_q}\cup\set{X_j,X'_j\mid 1\le j\le q-1}$.
From the construction of clusters of ${\mathcal{X}}$ in the first step,
It is clear that \Cref{claim: alpha cut property} implies that the corresponding instances $\set{I_X}_{X\in {\mathcal{X}}}$ are suitable for the algorithm in \Cref{subsec: no prob edges}. Therefore, it suffices to prove \Cref{claim: alpha cut property}.
\begin{figure}[h]
\centering
\subfigure[Subgraphs $Y_j,Y'_j$ in $H$.]{\scalebox{0.11}{\includegraphics{figs/cut_property_1.jpg}}}
\hspace{4pt}
\subfigure[Subgraphs $X_j,X'_j$ in $G$.]{\scalebox{0.11}{\includegraphics{figs/cut_property_2.jpg}}}
\caption{An illustration of subgraphs $Y_j,Y'_j$ in $H$ and subgraphs $X_j,X'_j$ in $G$. \znote{figure broken}}\label{fig: cut_property}
\end{figure}
\begin{proofof}{\Cref{claim: alpha cut property}}
First fix some $z'$ with $a<z'< z$, so $z'$ is not a problematic index. Denote $S'_{z'}=\set{x_{a+1},\ldots,x_{z'}}$ and $\overline{S}_{z'}'=\set{x_{z'+1},\ldots,x_z}$, and denote by $E'$ the set of edges in the cut $(S'_{z'},\overline{S}_{z'}'\cup\set{v^*})$ of $H$.
Note that graph $H'$ can be viewed as obtained from $H$ (recall that $H=G_{|{\mathcal{C}}}$) by contracting vertices of $\set{x_t\mid 1\le t\le a\text{ or }z< t\le r}$ into a vertex $v^*$, so each edge of $E'$ is also an edge of $H$, and vertices of $S'_{z'},\overline{S}'_{z'}$ are also vertices of $V(H)$.
Clearly, set $E'$, as a set of edges in $H$, can be further partitioned into subsets as follows:
set $E'_1=E_H(S'_{z'},\overline{S}'_{z'})$;
set $E'_2=E_H(S'_{z'},S_a)$; and
set $E'_3=E_H(S'_{z'},\overline{S}_z)$.
See \Cref{fig: cut_property} for an illustration.
Note that $|E'_1|+|E'_3|\le |E^{\operatorname{right}}_{z'}|+|E^{\operatorname{left}}_{z'+1}|+|E^{\operatorname{over}}_{z'}|$. Since $z'$ is not a problematic index, $|E'_1|+|E'_3|\le 100\cdot |E(x_{z'},x_{z'+1})|$.
On the other hand, from \Cref{obs: bad inded structure}, $|E'_2|\le |E^{\operatorname{right}}_{a}|+|E^{\operatorname{left}}_{a+1}|+|E^{\operatorname{over}}_{a}|\le 2\cdot|E^{\operatorname{right}}_{a}|$, and from \Cref{obs: large capacity up to midpoint}, $|E(x_{z'},x_{z'+1})|\ge |E^{\operatorname{right}}_{a}|/2$.
Therefore, $|E'_2|\le 4\cdot |E(x_{z'},x_{z'+1})|$. Altogether,
$|E'|=|E_1'|+|E_2'|+|E_3'|\le 104\cdot |E(x_{z'},x_{z'+1})|$.
Consider now the cut $(S_z,\set{v^*})$ in $H$.
We define sets $E',E'_1,E'_2,E'_3$ similarly. Clearly, $E'_1=\emptyset$. Observe that $E_H(x_z,x_{z+1})\subseteq E(S_z,\set{v^*})$, so it suffices to show that $|E'_2|+|E'_3|\le O(|E(x_z,x_{z+1})|)$.
Since $z$ is not a problematic index, $|E^{\operatorname{right}}_{z}|+|E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_{z}|\le 100\cdot|E(x_{z},x_{z+1})|$.
Note that
$$|E'_2|\le 2\cdot|E^{\operatorname{right}}_{a}|\le 2(|E(x_a,x_{z+1})|+|E^{\operatorname{over}}_{z}|)\le 2(|E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_{z}|),$$
and
$$|E'_3|\le |E(S'_z,x_{z+1})|+|E^{\operatorname{over}}_{z}|\le |E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{over}}_{z}|.$$
So in this case $|E'|\le 300\cdot |E(x_{z'},x_{z'+1})|\le 300\cdot|E(S_z,\set{v^*})|$.
This completes the proof that $H'$ has the $\alpha$-cut property with respect to path $\tau'$, for $\alpha=300$.
\iffalse
It suffices to show that (i) there is a set of paths routing $E'$ to $x_i$ in $S'_i$ with congestion $10$; and (ii) there is a set of paths routing $E'$ to $x_i$ in $\overline{S}'_i\cup\set{v^*}$ with congestion $10$.
We first show (i).
Note that $E'=\delta_{H_X}(S'_i)$.
Similar as the proof of \Cref{claim: inner paths}, from the fact that $a$ is a problematic index (and \Cref{obs: bad inded structure}) and \Cref{obs: large capacity up to midpoint}, we can show that there is a set of paths routing $E'$ to $x_i$ in $S'_i$, with congestion $10$. This is since the proof of \Cref{claim: inner paths} only uses the fact that $z<\mathsf{RM}(a)$, and here we have $i\le z<\mathsf{RM}(a)$.
We now show (ii).
Consider another graph $G'_X$, that is obtained from graph $G$ by contracting clusters $C_1,\ldots,C_{a}$ into a vertex $v^{\operatorname{left}}$ and contracting clusters $C_{z+1},\ldots,C_{r}$ into a vertex $v^{\operatorname{right}}$. Let $H'_X$ be its corresponding contracted graph, so $V(H'_X)=V(Y)\cup \set{v^{\operatorname{left}},v^{\operatorname{right}}}$. Note that graph $G_X$ can be also viewed as obtained from $G'_X$ by contracting $v^{\operatorname{left}},v^{\operatorname{right}}$ into a single vertex $v^*$, and similarly $H_X$ can be viewed as obtained from $H'_X$ by contracting $v^{\operatorname{left}},v^{\operatorname{right}}$ into $v^*$.
We denote $E'_1=E_{H'_X}(v^{\operatorname{left}},S'_i)$ and $E'_2=E_{H'_X}(v^{\operatorname{right}},\overline{S}'_i)\cup E_{H'_X}(S'_i,\overline{S}'_i)$.
From the properties of minimum-cut, there is a set of edge-disjoint paths routing $E'_2$ to $v_{i+1}$ in $\overline{S}'_i\cup v^{\operatorname{right}}$.
Similar as the proof of \Cref{claim: outer paths}, we can show that $|E'_1|\le 2|E(x_a,v^{\operatorname{right}})|$.
Moreover, since $E(x_a,v^{\operatorname{right}})\subseteq \delta_{H'_X}(\overline{S}'_i\cup v^{\operatorname{right}})$, there is a set of edge-disjoint paths routing $E(x_a,v^{\operatorname{right}})$ to $v_{i+1}$, using only edges of $\bigcup_{i<z'<z}\hat{E}_{z'}$.
Therefore, the edges of $E'_1$ can be routed to $v_{i+1}$ in $\overline{S}'_i\cup \set{v^{\operatorname{right}}}$ with congestion $4$.
Altogether, edges of $E'_1\cup E'_2$ can be routed to $v_{i+1}$ in $\overline{S}'_i\cup \set{v^{\operatorname{right}}}$ with congestion $5$ in graph $H'_X$. Therefore, as edges of $H_X$, they can be routed to $v_{i+1}$ in $\overline{S}'_i\cup \set{v^{*}}$ with congestion $5$.
\fi
\end{proofof}
\iffalse
Second, we consider a pair $\hat R',\hat R''$ of clusters that are constructed by performing some consecutive pair $(i_j,i_{j+1})$ of problematic index in Case 2.
As before, we consider the graph $H'$ obtained from $H$ by contracting the vertices of $R'$ into a vertex $v$, and the vertices of $R''$ into a vertex $v'$. The difference this time is that we are guaranteed that at least half the edges in the original set $E_a^{\operatorname{right}}$ have an endpoint in $v$. This ensures that index $a$ stops being problematic. Same happens for index $b$.
Third, we consider the cluster $\hat R''$ obtained from processing the first index $i_1$ and the cluster $\hat R'$ obtained from processing the last index $i_q$.
Once that cluster is contracted, the index will stop being problematic. Last problematic index is treated similarly.
\znote{end here}
\fi
\subsubsection{Second-Level Disengagement}
\label{subsec: 2nd disengagement}
In this step we further process the instance $\hat I({\mathcal{X}},\tilde {\mathcal{Q}})$ of ${\mathcal{I}}({\mathcal{X}},\tilde{\mathcal{Q}},\tilde{\mathcal{P}})$.
Recall that $\hat I({\mathcal{X}},\tilde {\mathcal{Q}})=(\hat G,\hat \Sigma)$, where the graph $\hat G$ is obtained from $G$ by contracting every cluster in ${\mathcal{X}}$ into a single vertex $u^*(X)$.
Denote $J=\bigg[1,\mathsf{LM}(i_1)\bigg]\cup \bigg(\mathsf{RM}(i_q),r\bigg]\cup\text{ }\bigcup_{1\le j\le q-1}\bigg(\mathsf{RM}(i_j), \mathsf{LM}(i_{j+1})\bigg]$,
so
$V(\hat G)=\set{u^*(X)\mid X\in {\mathcal{X}}}\cup (\bigcup_{i\in J}V(C_i)).$
\subsubsection*{Step 1: Defining the Clusters}
We first construct a set $\hat{\mathcal{C}}$ of clusters of $\hat G$.
We start with $\hat{\mathcal{C}}=\emptyset$, and then process every consecutive pair of problematic indices, as follows.
Consider the index pair $i_j,i_{j+1}$ for some $1\le j\le q-1$.
For brevity, we denote $i_j=a$ and $i_{j+1}=b$.
Recall that we have constructed clusters $X_j,X'_j$ in Step 1 of the first-level disengagement in \Cref{subsec: 1st disengagement}, and they correspond to vertices $u^*(X_j),u^*(X_j)$ in $\hat G$.
We now distinguish between the following cases:
\begin{enumerate}
\item \label{blt: cluster construction Case 1} if $\mathsf{LM}(b)<\mathsf{RM}(a)$, then we add the cluster that contain two vertices $u^*(X_j),u^*(X'_j)$ (and edges connecting them) into $\hat{{\mathcal{C}}}$ (see \Cref{fig: new_clusters_1});
\item \label{blt: cluster construction Case 2} if $\mathsf{LM}(b)=\mathsf{RM}(a)=t$, then we add the subgraph of $\hat G$ induced by $C_t\cup \set{u^*(X_j),u^*(X'_j)}$ into $\hat{{\mathcal{C}}}$ (see \Cref{fig: new_clusters_2});
\item \label{blt: cluster construction Case 3} if $\mathsf{LM}(b)>\mathsf{RM}(a)$, then we add (i) the subgraph of $\hat G$ induced by $C_{\mathsf{RM}(a)}\cup \set{u^*(X_j)}$ into $\hat{{\mathcal{C}}}$; (ii) the subgraph of $\hat G$ induced by $C_{\mathsf{LM}(b)}\cup \set{u^*(X'_j)}$ into $\hat{{\mathcal{C}}}$; and (iii) for each $\mathsf{RM}(a)<t<\mathsf{LM}(b)$ (if there is any), the cluster $C_t$ to $\hat{{\mathcal{C}}}$ (see \Cref{fig: new_clusters_3}).
\end{enumerate}
\begin{figure}[h]
\centering
\subfigure[Clusters of $\hat{\mathcal{C}}$ in Case \ref{blt: cluster construction Case 1}.]{\scalebox{0.29}{\includegraphics[scale=1.0]{figs/new_cluster_1.jpg}}\label{fig: new_clusters_1}}
\hspace{0.2cm}
\subfigure[Clusters of $\hat{\mathcal{C}}$ in Case \ref{blt: cluster construction Case 2}.]{
\scalebox{0.29}{\includegraphics[scale=1.0]{figs/new_cluster_2.jpg}}\label{fig: new_clusters_2}}
\hspace{0.2cm}
\subfigure[Clusters of $\hat{\mathcal{C}}$ in Case \ref{blt: cluster construction Case 3}.]{
\scalebox{0.29}{\includegraphics[scale=1.0]{figs/new_cluster_3.jpg}}\label{fig: new_clusters_3}}
\caption{Illustration of constructing clusters in $\hat{{\mathcal{C}}}$.}
\end{figure}
Finally, we additionally process the first index and the last index as follows. For the first index $i_1$, we add the the subgraph of $\hat G$ induced by $C_{\mathsf{LM}(i_1)}\cup \set{u^*(X'_0)}$ into $\hat{{\mathcal{C}}}$; and then for each $1\le t<\mathsf{LM}(b)$ (if there is any), the cluster $C_t$ to $\hat{{\mathcal{C}}}$.
For the last index $i_q$, we add the the subgraph of $\hat G$ induced by $C_{\mathsf{RM}(i_q)}\cup \set{u^*(X_q)}$ into $\hat{{\mathcal{C}}}$; and then for each $\mathsf{RM}(b)<t\le r$ (if there is any), the cluster $C_t$ to $\hat{{\mathcal{C}}}$.
This completes the construction of clusters in ${\mathcal{C}}$.
Recall that $H=G_{|{\mathcal{C}}}$, and the Gomory-Hu tree for $H$ is the path $\tau=(x_1,\ldots,x_r)$.
We define $\hat H$ to be the graph obtained from $H$ by contracting, for each cluster $X\in {\mathcal{X}}$ that we have constructed in the first-level disengagement, the vertices in the corresponding cluster $Y$ in $H$ into a single vertex.
Similarly, we define $\hat\tau$ as the path obtained from $\tau$ by contracting, for each $X\in {\mathcal{X}}$, the vertices in its corresponding cluster $Y$ in $H$ into a single vertex, so $V(\hat\tau)=V(\hat H)$.
Since the vertices in each such cluster $Y$ induce a subpath in $\tau$, from \Cref{lem: GH tree path vs contraction}, the path $\hat\tau$ is a Gomory-Hu tree of $\hat H$.
Note that $V(\hat H)=\set{u^*(X)\mid X\in {\mathcal{X}}}\cup\set{x_t\mid t\in J}$.
Now we define $H'$ as the graph obtained from $\hat H$ by contracting, for each cluster of ${\mathcal{C}}$ that contains a vertex $u^*(X)$ and a cluster $C_t$, the corresponding vertices $u^*(X)$ (or two vertices $u^*(X)$ and $u^*(X')$) and $x_t$ into a single vertex. Clearly, $H'=\hat G_{|\hat{{\mathcal{C}}}}$.
Similarly, we define $\tau'$ as the path obtained from $\hat\tau$ by contracting the same sets of vertices into a single vertex as $H'$, so $V(\hat\tau)=V(\hat H)$. From \Cref{lem: GH tree path vs contraction}, the path $\tau'$ is a Gomory-Hu tree of $H'$.
\iffalse
First, for each $i\in J$, we add cluster $C_i$ to $\hat{{\mathcal{C}}}$.
Second, we add single-vertex clusters $u^*(X'_0), u^*(X_q)$ to $\hat{{\mathcal{C}}}$, and for each $1\le j\le q-1$ such that $\mathsf{RM}(i_j)\leq \mathsf{LM}(i_{j+1})$, we add single-vertex clusters $u^*(X_j),u^*(X'_j)$ to $\hat{\mathcal{C}}$.
Finally, for each $1\le j\le q-1$, we add a cluster that contains only two vertices $u^*(X_j),u^*(X'_j)$ and the edges between them to $\hat{\mathcal{C}}$.
Denote $\hat H=\hat G_{|\hat{\mathcal{C}}}$, so
$V(\hat H)=\set{u^*(X)\mid X\in {\mathcal{X}}}\cup \set{x_i\mid i\in J}$.
It is easy to verify that, the Gomory-Hu tree of the contracted graph $\hat H$, that we denote by $\hat{\tau}$, is a path\znote{add cor in prelim}, where the order in which the vertices appear on $\hat{\tau}$ is consistent with the path $\tau$.
We now show that the path $\hat{\tau}$ does not contain a problematic index.
First, it is easy to verify that every index $i\in J$ is not a problematic index, from \znote{add cor in prelim} and the fact that $x_i$ is not a problematic index in $\tau$.
\fi
\subsubsection*{Step 2: Constructing Paths Sets for Clusters in $\hat{{\mathcal{C}}}$}
In the second step, we compute, for each cluster $C\in \hat{\mathcal{C}}$, a vertex $u(C)$ and a set ${\mathcal{Q}}(C)$ of paths routing edges of $\delta(C)$ to $u(C)$ inside $C$.
We will repeatedly use the following observation.
\begin{observation}
\label{obs: forever not problematic}
Let $a< z$ be indices in $\set{1,\ldots, r}$ such that $a$ is problematic, $z$ is not problematic index and $z<\mathsf{RM}(a)$, then $|\delta_H(\set{x_t\mid a< t\le z})|\le 504 \cdot |E(x_z,x_{z+1})|$.
\end{observation}
\begin{proof}
Since $z$ is not a problematic index, $|E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{right}}_{z}|+|E^{\textsf{thr}}_{z}|\le 100\cdot |E(x_z,x_{z+1})|=100\cdot |\hat E_z|$.
Denote $U=\set{x_t\mid a< t\le z}$.
Note that $U\subseteq S_z$, so
$$|E(U,\overline{S}_z)|\le |E(S_z,\overline{S}_z)|\le |E^{\operatorname{left}}_{z+1}|+|E^{\operatorname{right}}_{z}|+|E^{\textsf{thr}}_{z}|
\le 100\cdot|\hat E_z|.$$
On the other hand, since $z<\mathsf{RM}(a)$, at least half of edges in $E^{\operatorname{right}}_a$ has an endpoint in $\overline{S}_z$. Therefore,
$$|E(U,S_a)|\le |E^{\operatorname{left}}_{a+1}|+|E(x_{a},U)|+|E^{\textsf{thr}}_{a}|\le 2|E^{\operatorname{right}}_a|\le 4|E(x_a,\overline{S}_z)|\le 4(|E^{\operatorname{left}}_{z+1}|+|\hat E_z|)\le 404\cdot |\hat E_z|.$$
Altogether, $|\delta_H(U)|=|E(U,\overline{S}_z)|+|E(U,S_a)|\le 504\cdot |E(x_z,x_{z+1})|$.
\end{proof}
We distinguish between the following cases.
Consider first a cluster $C$ that is constructed in Case \ref{blt: cluster construction Case 1}. From the construction, $C$ contains only two vertices $u^*(X_j),u^*(X'_j)$.
Recall that the corresponding clusters $Y_i,Y'_i$ are obtained by splitting at a non-problematic index $z$ in $H$ (see \Cref{fig: path_cluster_H}).
From \Cref{obs: forever not problematic}, $|\delta_H(Y_j)|\le 504\cdot |E(x_z,x_{z+1})|$.
We set $u(C)=u^*(X'_j)$. Then clearly there is a set ${\mathcal{Q}}(C)$ of paths routing the edges of $\delta(C)$ to $u(C)$ inside $C$, with congestion at most $504$.
Consider now a cluster $C$ that is constructed in Case \ref{blt: cluster construction Case 2}.
From the construction, $C$ contains the cluster $C_t$ and two vertices $u^*(X_j),u^*(X'_j)$.
From \Cref{claim: inner and outer paths}, there is a set $\hat{\mathcal{Q}}(Y_j)$ of paths routing edges of $\delta_H(Y_j)$ to $x_{t-1}$ in $Y_j$, with congestion $O(1)$; and there is a set $\hat{\mathcal{Q}}(Y'_j)$ of paths routing edges of $\delta_H(Y'_j)$ to $x_{t+1}$ in $Y'_j$, with congestion $O(1)$.
Moreover, from \Cref{obs: forever not problematic}, $|\delta_H(Y_j)|\le 504\cdot |E(x_{t-1},x_{t})|$ and
$|\delta_H(Y'_j)|\le 504\cdot |E(x_{t},x_{t+1})|$.
Therefore, if we denote $U=Y_j\cup \set{x_t}\cup Y'_{j}$, then there exists a set $\hat{{\mathcal{Q}}}(U)$ of edges routing edges of $\delta_H(U)$ to $x_t$, with congestion $O(1)$.
We set $u(C)=u(C_t)$, then we can easily construct a set ${\mathcal{Q}}(C)$ of paths routing edges of $\delta(C)$ to $u(C)$, using the paths of $\hat{{\mathcal{Q}}}(U)$ and the paths of ${\mathcal{Q}}(C_t)$ that we are given, such that for each edge $e\in C$, $(\cong_{C}({\mathcal{Q}}(C),e))^2\le O(\beta)$.
Consider now a cluster $C$ that is constructed in Case \ref{blt: cluster construction Case 3}. If $C=C_t$ for some $\mathsf{RM}(i_j)<t<\mathsf{LM}(i_{j+1})$, then we set $u(C)=u(C_t)$ and the path ${\mathcal{Q}}(C_t)$ is the desired path set we want.
If $C$ contains cluster $C_{RM(i_j)}$ and vertex $u^*(X_j)$, then from \Cref{obs: forever not problematic}, $|\delta(u^*(X_j))|\le O(|E(u^*(X_j),C_{RM(i_j)})|)$.
Therefore, if we set $u(C)=u(C_{RM(i_j)})$, then it is easy to construct the desired set ${\mathcal{Q}}(C)$ of paths using the edges of $E(u^*(X_j),C_{RM(i_j)})$ and the path set ${\mathcal{Q}}(C_{RM(i_j)})$.
The way for handling the cluster that contains cluster $C_{LM(i_{j+1})}$ and vertex $u^*(X'_j)$ is symmetric.
The way for handling clusters that are constructed in processing the first and the last indices are the same as that of the clusters constructed in Case \ref{blt: cluster construction Case 3}.
\subsubsection*{Step 3: Proving that graph $H'$ has the $\alpha$-Cut Property for $\alpha=O(1)$}
\iffalse
From the proof of \Cref{claim: outer paths}, for each $1\le j\le q-1$, $|E_H(Y_j,Y'_j)|\ge |\delta_H(Y_j)|/6$. Therefore, there is a set $\hat{{\mathcal{Q}}}_j$ of paths routing the edges of $\delta_{\hat G}(U_j)$ to $u^*(X'_j)$ in $U_j$, that causes congestion at most $6$.
Note that this also immediately implies that the vertices in $\hat{\tau}$ that correspond to clusters $U_1,\ldots,U_{q-1}$ are not problematic indices.
It is then easy to verify that $\hat{\tau}$ does not contain a problematic index.
\fi
We distinguish between the following cases.
Consider first a cluster $C$ that is constructed in Case \ref{blt: cluster construction Case 1}, so $V(C)=\set{u^*(X_j),u^*(X_j)}$.
Denote $a=i_j$ and $b=i_{j+1}$.
Assume first that indices $\mathsf{LM}(b),\mathsf{RM}(a)$ are both greater than $a$ and less than $b+1$ (see \Cref{fig: midpoints inside}). Let $C'$ be the cluster on the right of $C$ in path $\tau'$.
Since $\mathsf{LM}(b)\ge a+1$, at least half the edges of $E^{\operatorname{left}}_{b+1}$ have an endpoint in the cluster $Y_j\cup Y'_j$. Therefore, $|E(Y_j\cup Y'_j,x_{b+1})|\ge |E^{\operatorname{left}}_{b+1}|/2\ge (|E^{\operatorname{left}}_{b+1}|+|E^{\operatorname{right}}_{b}|+|\hat E_{b}|)/6$, and it follows that the vertex in $\tau'$ that corresponds to cluster $C$ is not a problematic index.
Assume now that $\mathsf{LM}(b)<a$ (see \Cref{fig: midpoints outside}).
In this case, we first show that $\mathsf{RM}(a)\le b+1$, since otherwise,
$$|\hat E_a|\le \frac{|E(x_a,\overline{S}_{\mathsf{RM}(a)-1})|}{20}
\le \frac{|E^{\textsf{thr}}_b|}{20}\le \frac{|\hat E_b|}{20}\le \frac{|E(x_{b+1},\overline{S}_{\mathsf{LM}(b)})|}{400}\le \frac{|E^{\textsf{thr}}_a|}{400}\le \frac{|\hat E_a|}{400},$$
a contradiction.
Note that $E(x_{b+1},S_{\mathsf{LM}(B)})\subseteq E^{\textsf{thr}}_a$, so $|E^{\operatorname{left}}_{b+1}|\le 2|\hat E_a|$.
\begin{figure}[h]
\centering
\subfigure[Subgraphs $Y_j,Y'_j$ in $H$.]{\scalebox{0.3}{\includegraphics{figs/cut_prop_1.jpg}}
\label{fig: midpoints inside}}
\hspace{4pt}
\subfigure[Subgraphs $X_j,X'_j$ in $G$.]{\scalebox{0.3}{\includegraphics{figs/cut_prop_2.jpg}}
\label{fig: midpoints outside}}
\caption{An illustration of subgraphs $Y_j,Y'_j$ in $H$ and subgraphs $X_j,X'_j$ in $G$.}\label{fig: cut property case_1}
\end{figure}
Consider now a cluster $C$ that is constructed in Case \ref{blt: cluster construction Case 2}.
Consider now a cluster $C$ that is constructed in Case \ref{blt: cluster construction Case 3}.
We can now apply the result of \Cref{subsec: no prob edges} and obtain a set ${\mathcal{I}}_{\hat G}$ of sub-instances that satisfy the properties of \Cref{lem: path case}, and $\sum_{I'\in {\mathcal{I}}_{\hat G}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\le O((\mathsf{OPT}_{\mathsf{cnwrs}}(\hat G,\hat\Sigma)+|E(\hat G)|)\cdot \beta)$.
\subsubsection*{The General Case.}
We start with some pre-processing steps.
We first apply Lemma~\ref{lem: path set inside a well-linked cluster} to all clusters of ${\mathcal{C}}^*$, and find, for each $C\in {\mathcal{C}}^*$, a vertex $u_C\in C$ and a set ${\mathcal{Q}}^{in}_C$ of paths in $C$, routing edges of $\delta(C)$ to $u_C$, such that $\operatorname{cost}({\mathcal{Q}}_C)\le O(\mathsf{cr}(C)\cdot (\operatorname{poly}\log n/\alpha^2))$.
We then define a circular ordering ${\mathcal{O}}_C$ on the set $\delta(C)$ of edges as the following \znote{?}.
Second, consider a pair $C,C'$ of clusters in ${\mathcal{C}}^*$ such that $S_{C'}$ is the parent cluster of $S_{C}$ in the laminar family ${\mathcal{L}}$.
From the construction of ${\mathcal{L}}$, the cut $(S_C,\overline{S_C})$ is the min-cut in $G'$ separating $C$ from $C'$.
We then apply Lemma~\ref{lem: cut-edge-to-cluster-boundary} to the clusters $C,C'$ and the min-cut $(S_C,\overline{S_C})$ separating them, and obtain a set of edge-disjoint paths in $S_C$ connecting edges of $E(S_C,\overline{S_C})$ to edges of $\delta(C)$, that we denote by ${\mathcal{Q}}^{cut}_C$.
We are now ready to construct the disengaged instance $(H_C,\Sigma_C)$ for clusters in ${\mathcal{C}}^*$.
Consider a cluster $C\in {\mathcal{C}}^*$ and the corresponding cluster $S_C\in {\mathcal{L}}$. The graph $H_C$ is defined as follows.
We start with the graph $G'$. We first contract all vertices in $G\setminus S_{C}$ into a single vertex, that we denote by $v^{out}_C$. Notice that the edges incident to $v^{out}_C$ correspond to the edges of $\delta(S_C)$ (for simplicity, we will not distinguish between them). The circular ordering of the edges incident to $v^{out}_C$ is defined to be ${\mathcal{O}}_C$.
Next, let $S_{C_1},\ldots,S_{C_q}$ be child clusters of $S_C$. For all $1\leq i\leq q$, we contract all vertices of $S_i$ into a super-node $v^{in}_{C,C_i}$. Notice that the edges incident to $v^{in}_{C,C_i}$ correspond to the edges of $\delta(S_{C_i})$; we do not distinguish between them.
This finishes the construction of the graph $H_C$. We now define the rotation system $\Sigma_C$ on it.
From the above discussion, $V(H_C)=V(C)\cup\set{v^{out}_C}\cup\set{v^{in}_{C,C_i}}_{1\le i\le q}$.
The orderings at the vertices of $C$ are identical to their orderings in $\Sigma'$.
The ordering of the edges incident to $v^{out}_C$ is defined to be ${\mathcal{O}}_C$.
The ordering of the edges incident to $v^{in}_{C,C_i}$ is defined to be \znote{?}.
This completes the description of the instance $(H_C,\Sigma_C)$. See Figure~\ref{fig: disengaged_instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of the graph $G$.]{\scalebox{0.3}{\includegraphics{figs/dis_instance_before}}\label{fig: graph G}}
\hspace{1cm}
\subfigure[The graph $H_C$.]{
\scalebox{0.3}{\includegraphics{figs/dis_instance_after}}\label{fig: C and disc}}
\caption{An illustration of the disengagement instance.\label{fig: disengaged_instance}}
\end{figure}
We will now analyze $\sum_{C\in {\mathcal{C}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(H_C,\Sigma_C)$.
Recall that $\tau$ is the structural tree of ${\mathcal{L}}$.
The vertex set $V(\tau)$ can be decomposed into two subsets: set $V_0(\tau)$ contains all nodes with degree $2$ in $\tau$, and set $V_1(\tau)$ contains all other nodes in $\tau$.
We partition ${\mathcal{C}}^*$ into two subsets similarly: set ${\mathcal{C}}^*_0$ contains the clusters in ${\mathcal{C}}^*$ that correspond to nodes in $V_0(\tau)$, and set ${\mathcal{C}}^*_1$ contains the clusters in ${\mathcal{C}}^*$ that correspond to nodes in $V_1(\tau)$.
\begin{claim}
$\sum_{C\in {\mathcal{C}}^*_0}\mathsf{OPT}_{\mathsf{cnwrs}}(H_C,\Sigma_C)\le O(\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\operatorname{poly}\log n)$.
\end{claim}
\subsubsection{Technical Lemmas}
\begin{lemma}
\label{lem: cut-edge-to-cluster-boundary}
There is an efficient algorithm, that, given any pair $C_1,C_2$ vertex-disjoint clusters of $G$, and any min-cut $(A_1,A_2)$ in $G$ separating $C_1$ from $C_2$, with $V(C_1)\subseteq A_1$ and $V(C_2)\subseteq A_2$, computes, for each $i\in \set{1,2}$, there exists a set ${\mathcal{P}}_i$ of edge-disjoint paths in $A_i$ routing the edges of $E(A_1,A_2)$ to edges of $\delta(C_i)$.
\end{lemma}
\begin{lemma}
\label{lem: path set inside a well-linked cluster}
There is an efficient algorithm, that, given any $\alpha$-boundary-linked cluster $C$, computes a vertex $u_C\in C$ and a set ${\mathcal{Q}}_C$ of paths, routing edges of $\delta(C)$ to $u_C$, such that $\operatorname{cost}({\mathcal{Q}}_C)\le O(\mathsf{cr}(C)\cdot (\operatorname{poly}\log n/\alpha^2))$.
\end{lemma}
\subsection{Phase 2: Laminar Family and Disengagement of Clusters}
In this section we assume that the given sub-instance $(G',\Sigma')$ (with $|E(G')|=m$) satisfies that $\mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\ge |E(G')|^2/\mu^{10}$, and in the first phase, we have computed a set ${\mathcal{C}}^*$ of clusters, such that
such that each cluster of ${\mathcal{C}}^*$ is either a good cluster or a flower cluster, the vertex sets $\set{V(C)}_{C\in {\mathcal{C}}^*}$ partitions $V(G')$, and $|E^{\textsf{out}}({\mathcal{C}}^*)|= O(m/\mu)$.
\subsubsection*{Step 1: Compute a laminar family of clusters}
Let $\tilde G$ be the graph obtained from $G'$ by contracting each cluster of ${\mathcal{C}}^*$ into a single vertex, while keeping parallel edges and deleting self-loops. For each cluster $C\in {\mathcal{C}}^*$, let $v_C$ be the node in $\tilde G$ obtained from contracting the cluster $C$.
We compute the Gomory-Hu tree of $\tilde G$, and let $\tilde \tau$ be the tree that we obtain. We root $\tilde \tau$ at an arbitrary node. We now define, based on the structure of the rooted tree $\tilde \tau$, a laminar family $\tilde {\mathcal{L}}$ of clusters in $\tilde G$, as follows.
For each node $v_C$, we define the cluster $\tilde S_C$ to be the subgraph of $\tilde G$ induced by all nodes in the subtree of $\tilde \tau$ rooted at the node $v_C$.
We then let $\tilde {\mathcal{L}}=\set{\tilde S_C\mid C\in {\mathcal{C}}^*}$.
It is easy to verify that the $\tilde {\mathcal{L}}$ is a laminar family of clusters of $\tilde G$.
We define a corresponding laminar family ${\mathcal{L}}$ of clusters in $G$ as follows. For each cluster $\tilde S_C\in \tilde {\mathcal{L}}$, we define the cluster $S_C$ as the subgraph of $\tilde G$ induced by vertices of $\bigcup_{v_{C'}\in \tilde S_C}V(C')$.
It is easy to verify that the ${\mathcal{L}}$ is a laminar family of clusters of $G$.
We denote by $\tau$ the structure tree of ${\mathcal{L}}$.
It is easy to verify that, if we identify the node of $\tau$ that represents the cluster $S_C$ with the node of $\tilde \tau$ that represents the cluster $\tilde S_C$, then $\tau=\tilde \tau$.
\iffalse
We use the following claim.
\begin{claim}
Each cluster $S_C\in {\mathcal{L}}$ is $\alpha$-boundary-well-linked.
\end{claim}
\begin{proof}
Let $S_{C'}$ be the parent cluster of $S_C$.
$(S_C,\overline{S_C})$ is the min-cut in $G'$ separating $C$ from $C'$. Recall that $C$ is $\alpha$-boundary-well-linked. Combined with Lemma~\ref{lem: cut-edge-to-cluster-boundary}, we can derive that $S_C$ is also $\alpha$-boundary-well-linked.
\end{proof}
\fi
\subsubsection*{Step 2: Disengaging the clusters}
In this step we construct, for each cluster $C\in {\mathcal{C}}^*$, an instance $(H_C,\Sigma_C)$ of \textnormal{\textsf{MCNwRS}}\xspace, that we call the \emph{disengaged instance} for cluster $C$.
We first consider the special case where the structure tree $\tau$ of the laminar family ${\mathcal{L}}$ is a path. The algorithm of constructing the disengaged instances in this special case will serve as a building block for the algorithm in the general case.
\subsubsection*{Special Case: $\tau$ is a path.}
\znote{Not ready yet. Please do not read.}
Denote ${\mathcal{C}}^*=\set{C_1,\ldots,C_k}$. Let $S_{C_1},\ldots,S_{C_k}$ be the clusters in ${\mathcal{L}}$, with $S_{C_k}\subseteq S_{C_{k-1}}\subseteq \ldots \subseteq S_{C_1}=G$.
From the construction of ${\mathcal{L}}$, for each $1\le i\le k$, $S_{C_i}=\bigcup_{i\le j\le k}C_j$.
Recall that, for each $1\le i\le k$, the cluster $C_i$ is $\alpha$-boundary-well-linked. We apply Lemma~\ref{lem: path set inside a well-linked cluster} to each cluster $C_i$, and obtain a vertex of $C_i$, which we denote by $u_i$, and a set of paths in $C_i$ connecting the edges of $\delta(C_i)$ to $u_i$, which we denote by ${\mathcal{Q}}_i$.
From Lemma~\ref{lem: path set inside a well-linked cluster}, $\operatorname{cost}({\mathcal{Q}}_i)\le O(\mathsf{cr}(C)\cdot (\operatorname{poly}\log n/\alpha^2))$.
\paragraph{Step 2.1. Compute Central Paths.}
Recall that $E^{\textsf{out}}({\mathcal{C}}^*)$ is the set of edges connecting distinct clusters of ${\mathcal{C}}^*$.
We define $E'$ to be the subset of edges in $E^{\textsf{out}}({\mathcal{C}}^*)$ connecting a pair $C_i,C_j$ of distinct clusters of ${\mathcal{C}}^*$, with $j\ge i+2$.
For each $1\le i\le k$, we denote $E_i=E(C_i,C_{i+1})$.
Therefore, $E(G)=E'\cup\textsf{left}(\bigcup_{1\le i\le k-1}E_i\textsf{right})\cup \textsf{left}(\bigcup_{1\le i\le k}E(C_i)\textsf{right})$.
We will first compute, for each edge $e\in E'$ that connects a vertex of $C_i$ to a vertex of $C_j$, a path $P_e$ connecting $u_i$ to $u_j$, that only uses edges of $\bigcup_{i\le t\le j-1}E_t$ and edges of $\bigcup_{i\le t\le j}E(C_t)$. We call the path $P_e$ the \emph{central path} of $e$.
The central paths $\set{P_e}_{e\in E'}$ are computed as follows. Recall that $\tilde G$ is the graph obtained from $G$ by contracting each cluster of $\set{C_1,\ldots,C_k}$ into a single vertex, while keeping parallel edges and deleting self-loops. Denote by $v_i$ the node in $\tilde G$ that represents the cluster $C_i$ in $G$.
Note that each edge in $E'\cup \textsf{left}(\bigcup_{1\le i\le k-1}E_i\textsf{right})$ corresponds to an edge in $\tilde G$, and we do not distinguish between them.
We now process the edges in $E'$ one-by-one. Throughout, for each edge $\hat e\in \bigcup_{1\le i\le k-1}E_i$, we maintain an integer $x_{\hat e}$ indicating how many times the edge $\hat e$ has been used, that is initialized to be $0$. Consider an iteration of processing an edge $e\in E'$. Assume $e$ connects $v_i$ to $v_j$ in $\tilde G$, we then pick, for each $i\le t\le j-1$, and edge of $E_t$ with minimum $x_{\hat e}$, and let $\tilde P_e$ be the path formed by all picked edges. We then increase the value $x_{\hat e}$ for all picked edges $\hat e$ by $1$, and proceed to the next iteration. After processing all edges of $E'$, we obtain a set $\tilde{\mathcal{P}}=\set{\tilde P_e}_{e\in E'}$ of paths in $\tilde G$.
{\color{red} We further assume that, for each $1\le i\le k$, $|E(S_i,\overline{S_i})|\le \beta\cdot |E_i|$, where we can think of $\beta=\operatorname{poly}\log n$.}
We use the following observation.
\begin{observation}
\label{obs:central_congestion}
For each edge $\hat e\in \bigcup_{1\le i\le k-1}E_i$, $\cong_{\tilde{\mathcal{P}}}(\hat e)\le \beta$.
\end{observation}
\begin{proof}
From the algorithm, for each $1\le i\le k-1$, the paths of $\tilde{\mathcal{P}}$ that contains an edge of $E_i$ are $\set{\tilde P_e\mid e\in E(S_i,\overline{S_i})\setminus E_i}$.
Since we have assumed that $|E(S_i,\overline{S_i})|\le \beta\cdot |E_i|$, each edge of $E_i$ is used at most $\beta$ times by paths of $\tilde{\mathcal{P}}$. The Observation~\ref{obs:central_congestion} follows.
\end{proof}
\iffalse
We now further process the set $\tilde{\mathcal{P}}$ of paths to obtain another set $\tilde{\mathcal{P}}'$ of paths with some additional properties, as follows.
\paragraph{Step 2.1.} Let $\tilde G'$ be the graph obtained from $\tilde G$ by replacing each edge $e$ with $\beta$ parallel copies $e_1,\ldots,e_\beta$ connecting its endpoints. Now for each path $\tilde P\in \tilde{{\mathcal{P}}}$, we define a path $\tilde P^*$ as the union of, for each edge $e\in \tilde P$, a copy $e_t$ of $e$, such that, for all paths of $\tilde{\mathcal{P}}$ that contains the edge $e$, their corresponding paths in $\set{\tilde P^*\mid e\in \tilde P}$ contain distinct copies of $e$. We denote $\tilde{\mathcal{P}}^*=\set{\tilde P^*\mid \tilde P\in \tilde {\mathcal{P}}}$. From the above discussion, the paths of $\tilde{\mathcal{P}}^*$ are edge-disjoint.
\paragraph{Step 2.2.} We uncross
\fi
We now compute the set ${\mathcal{P}}$ of central paths of edges in $E'$, using the paths in $\tilde{\mathcal{P}}$ and sets $\set{{\mathcal{Q}}_i}_{1\le i\le k-1}$.
Recall that in $G$, an edge of $E_i\cup E_{i-1}$ belongs to the set $\delta(C_i)$. Also recall that the set ${\mathcal{Q}}_i$ of paths contains, for each edge $e\in\delta(C_i)$, a path $Q^i_e$ connecting $e$ to $u_i$. Now consider an edge $e\in E'$ connecting a vertex of $C_i$ to a vertex of $C_{j}$, and the corresponding path $\tilde P_e=(e_i,e_{i+1},\ldots,e_{j-1})$ in $\tilde{\mathcal{P}}$, where $e_t\in E_t$ for each $i\le t\le j-1$. We construct the corresponding central path $P_e$ as the sequential concatenation of paths $Q^{i}_{e_i}, e_i, Q^{i+1}_{e_i}, Q^{i+1}_{e_{i+1}},e_{i+1}, Q^{i+2}_{e_{i+1}},\ldots,Q^{j-1}_{e_{j-1}},e_{j-1}, Q^{j}_{e_{j-1}}$.
It is clear that $P_e$ only contains edges of $\bigcup_{i\le t\le j-1}E_t$ and edges of $\bigcup_{i\le t\le j}E(C_t)$.
Denote ${\mathcal{P}}=\set{P_e\mid e\in E'}$.
We use the following observation, that immediately follows from Observation~\ref{obs:central_congestion}.
\begin{observation}
For each edge $\hat e\in \bigcup_{1\le i\le k-1}E_i$, $\cong_{{\mathcal{P}}}(\hat e)\le \beta$. For each edge $\hat e\in \bigcup_{1\le i\le k}E(C_i)$, $\cong_{{\mathcal{P}}}(\hat e)\le \beta\cdot\cong_{{\mathcal{Q}}_i}(\hat e)$.
\end{observation}
We will use the set ${\mathcal{P}}$ of central paths for concatenate drawings of distinct subgraphs of $G$, so it would be convenient to state our algorithm if the paths of ${\mathcal{P}}$ are edge-disjoint.
For which, we construct an instance $(G',\Sigma')$ of \textnormal{\textsf{MCNwRS}}\xspace as follows. We start with the instance $(G,\Sigma)$.
Denote ${\mathcal{Q}}=\bigcup_{1\le i\le k}{\mathcal{Q}}_i$.
For each edge $e\in E({\mathcal{Q}})$, we replace $e$ with $\cong_{{\mathcal{Q}}}(e)\cdot \cong_{{\mathcal{P}}}(e)$ parallel copies.
For each edge $e\in E({\mathcal{P}})\setminus E({\mathcal{Q}})$, we replace $e$ with $\cong_{{\mathcal{P}}}(e)$ parallel copies.
For each vertex $v$ such that $\delta(v)\cap E({\mathcal{P}}\cup{\mathcal{Q}})\ne \emptyset$, its ordering ${\mathcal{O}}'_v$ is defined based on ${\mathcal{O}}_v$, by replacing, for each edge $e\in \delta(v)\cap E({\mathcal{P}}\cup{\mathcal{Q}})$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$, and the ordering among the copies is arbitrary. We use the following observation.
\begin{observation}
$\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\le \mathsf{OPT}_{\mathsf{cnwrs}}(G',\Sigma')\le O(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot\beta^2\cdot\operatorname{poly}\log n)$.
Moreover, given a drawing $\phi\in \Phi(G,\Sigma)$, we can efficiently compute a drawing $\phi'\in \Phi(G',\Sigma')$, with $\mathsf{cr}(\phi')\le O(\mathsf{cr}(\phi)\cdot\beta^2\cdot\operatorname{poly}\log n)$; given a drawing $\phi'\in \Phi(G',\Sigma')$, we can efficiently compute a drawing $\phi\in \Phi(G,\Sigma)$, with $\mathsf{cr}(\phi)\le\mathsf{cr}(\phi')$.
\end{observation}
\begin{proof}
\end{proof}
From now on we will work with the instance $(G',\Sigma')$. Now for each path $P\in {\mathcal{P}}$, we define a path $P'$ in $G'$ as the union of, for each edge $e\in P$, a copy $e_t$ of $e$, such that for all paths of ${\mathcal{P}}$ that contains the edge $e$, their corresponding paths in $\set{P'\mid P\in {\mathcal{P}}, e\in P}$ contain distinct copies of $e$. We denote ${\mathcal{P}}'=\set{P'\mid P\in {\mathcal{P}}}$. From the above discussion, the paths of ${\mathcal{P}}'$ are edge-disjoint.
Moreover, note that $V(G)=V(G')$, and if a path $P\in {\mathcal{P}}$ connects $u_i$ to $u_j$ in $G$, then the corresponding path $P'$ connects $u_i$ to $u_j$ in $G'$.
\paragraph{Step 2.2. De-interfere Central Paths.}
We further process paths in ${\mathcal{P}}'$ to obtain another set ${\mathcal{P}}^*$ of edge-disjoint paths in $G'$, such that
\begin{itemize}
\item $|{\mathcal{P}}'|=|{\mathcal{P}}^*|$;
\item for each pair $1\le i,j\le k$ with $j\ge i+2$, the number of paths in ${\mathcal{P}}'$ connecting $u_i$ to $u_j$ is the same as that of ${\mathcal{P}}^*$; and
\item the paths in ${\mathcal{P}}^*$ are locally non-interfering.
\end{itemize}
For each path $P'\in {\mathcal{P}}'$ originating at $u_i$ and terminating at $u_{j}$ (recall that such a path visits vertices $u_i,u_{i+1},\ldots,u_j$ sequentially), we define, for each $i\le t\le j-1$, the path $P'_{(i)}$ to be the subpath of $P'$ between $u_{i}$ and $u_{i+1}$. For each $1\le i\le k-1$, we denote ${\mathcal{P}}'_{(i)}=\set{P'_{(i)}\text{ } \bigg|\text{ } u_i,u_{i+1}\in V(P')}$.
For each $1\le i\le k-1$, we apply the algorithm in Lemma~\ref{lem: non_interfering_paths} to the set ${\mathcal{P}}'_{(i)}$ of paths (recall that all paths in ${\mathcal{P}}'_{(i)}$ connects $u_i$ to $u_{i+1}$), and obtain a set ${\mathcal{P}}^*_{(i)}$ of locally non-interfering paths connecting $u_i$ to $u_{i+1}$, with $|{\mathcal{P}}^*_{(i)}|=|{\mathcal{P}}'_{(i)}|$ and $E( {\mathcal{P}}^*_{(i)})\subseteq E({\mathcal{P}}'_{(i)})$.
We then concatenate the paths of ${\mathcal{P}}^*_{(1)},{\mathcal{P}}^*_{(2)},\ldots,{\mathcal{P}}^*_{(k-1)}$ to obtain the desired set ${\mathcal{P}}^*$ of paths, using the following observation.
\znote{Need to rethink which definition to include for ``locally non-interfering".}
\begin{observation}[Restatement of Observation F.3 in~\cite{chuzhoy2020towards}]
\label{obs:rerouting_matching}
There is an efficiently algorithm, that, given a graph $\hat G$, a vertex $v$ of $G$, an ordering ${\mathcal{O}}_v$ on the edges of $\delta_G(v)$ and two disjoint subsets $E^-, E^+$ of $\delta_G(v)$ with $|E^-|=|E^+|$, computes a perfect matching $M\subseteq E^-\times E^+$ between the edges of $E^-$ and the edges of $E^+$, such that, for each pair of matched pairs $(e^-_1,e^+_1)$ and $(e^-_2,e^+_2)$ in $M$, the intersection of the path that consists of the edges $e^-_1,e^+_1$ and the path that consists of edges $e^-_2,e^+_2$ at vertex $u$ is non-transversal with respect to ${\mathcal{O}}_v$.
\end{observation}
We now construct the set ${\mathcal{P}}^*$ of paths as follows.
\znote{Concatenation and Designation. Apply Observation~\ref{obs:rerouting_matching} at $u_1,\ldots, u_k$ sequentially.}
Note that we will not compute a drawing of $G'$, we only use the locally non-interfering paths and their relative ordering in $G'$ to help define disengaged instance of $G$.
We then designate, for each edge $e\in E'_{i,j}$, a distinct path $P^*_e$ of ${\mathcal{P}}^*_{i,j}$. Denote by ${\mathcal{O}}^*_{i}$ the ordering of paths in ${\mathcal{P}}^*$ that contains $u_i$, defined by the rotation ${\mathcal{O}}'_{v_i}\in \Sigma'$ at $v$.
\znote{To elaborate.}
Note that
${\mathcal{O}}^*_{i}$ is also an ordering on the edges of $\bigcup_{j_1\le i\le j_2}E'_{j_1,j_2}$.
We use the following observation.
\begin{observation}
\label{lem: rotation_distance}
We can efficiently compute integers $\set{b_i}_{1\le i\le k}$ where $b_i\in \set{0,1}$ for each $1\le i\le k$, such that $\sum_{1\le i\le k-1}\mbox{\sf dist}(({\mathcal{O}}^*_i,b_i), ({\mathcal{O}}^*_{i+1},b_{i+1}))\le O(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot\beta^2\cdot\operatorname{poly}\log n)$.
\end{observation}
\paragraph{Step 2.3. Construct Disengaged Instances.}
We are now ready to construct the disengaged instances.
Recall that ${\mathcal{C}}^*=\set{C_1,\ldots,C_k}$, and the clusters $S_{C_1},\ldots,S_{C_k}$ in the laminar family ${\mathcal{L}}$ satisfies that $S_{C_k}\subseteq S_{C_{k-1}}\subseteq \ldots \subseteq S_{C_1}=G$.
For each $1\le i\le k$, we will construct an instance $(H_i,\Sigma_i)$ of \textnormal{\textsf{MCNwRS}}\xspace.
We first construct instance $(H_k,\Sigma_k)$ as follows.
The graph $H_k$ is obtained from $G$ by contracting all vertices in $G\setminus C_k$ into a single vertex, that we denote by $v^{\operatorname{left}}_k$.
Notice that the edges incident to $v^{\operatorname{left}}_k$ correspond to the edges of $\delta(C_k)$, and we will not distinguish between them.
The ordering of a vertex of $v\ne v^{\operatorname{left}}_k$ in $\Sigma_k$ is identical to the ordering ${\mathcal{O}}_v$ of $v$ in the given rotation system $\Sigma$.
The ordering of the vertex $v^{\operatorname{left}}_k$ in $\Sigma_k$ is defined to be the restricted ordering in ${\mathcal{O}}^*_{k-1}$.
\znote{To elaborate.}
The instance $(H_1,\Sigma_1)$ is defined similarly.
The graph $H_1$ is obtained from $G$ by contracting all vertices in $G\setminus C_1$ into a single vertex, that we denote by $v^{\operatorname{right}}_1$.
We will not distinguish between edges incident to $v^{\operatorname{left}}_1$ and edges of $\delta(C_k)$.
The ordering of a vertex $v\ne v^{\operatorname{right}}_1$ in $\Sigma_1$ is identical to the ordering ${\mathcal{O}}_v$ in $\Sigma$.
The ordering of the vertex $v^{\operatorname{right}}_1$ in $\Sigma_1$ is defined to be the restricted ordering in ${\mathcal{O}}^*_{2}$.
\znote{To elaborate.}
Consider now some index $2\le i\le k-1$. We define the instance $(H_i,\Sigma_i)$ as follows.
The graph $H_i$ is obtained from $G$ by first contracting clusters $C_1,\ldots,C_{i-1}$ into a single vertex, that we denote by $v^{\operatorname{right}}_1$, and then contracting clusters $C_{i+1},\ldots,C_r$ into a single vertex, that we denote by $v^{\operatorname{right}}_i$.
The ordering of a vertex $v\ne v^{\operatorname{left}}_i,v^{\operatorname{right}}_i$ in $\Sigma_1$ is identical to the ordering ${\mathcal{O}}_v$ in $\Sigma$.
The ordering of the vertex $v^{\operatorname{left}}_i$ in $\Sigma_i$ is defined to be ${\mathcal{O}}^*_{i-1}$, and
the ordering of the vertex $v^{\operatorname{right}}_i$ in $\Sigma_i$ is defined to be the ordering of ${\mathcal{O}}^*_{i+1}$.
We prove the following claims.
\begin{claim}
\label{claim: existence of good solutions special}
$\sum_{1\le i\le k}\mathsf{OPT}_{\mathsf{cnwrs}}(H_i,\Sigma_i)\le O(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot\beta^2\cdot\operatorname{poly}\log n)$.
\end{claim}
\begin{claim}
\label{claim: stitching the drawings together}
There is an efficient algorithm, that given, for each $1\le i\le k$, a drawing $\phi_i\in \Phi(H_i,\Sigma_i)$, computes a drawing $\phi\in \Phi(G,\Sigma)$, with
$\mathsf{cr}(\phi)\le \sum_{1\le i\le k}\mathsf{cr}(\phi_i)+O(\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\cdot\beta^2\cdot\operatorname{poly}\log n)$.
\end{claim}
\begin{proofof}{Claim~\ref{claim: existence of good solutions special}}
\end{proofof}
\begin{proofof}{Claim~\ref{claim: stitching the drawings together}}
\end{proofof}
\subsection{Disengagement with respect to a Laminar Family of Clusters}
\label{sec: laminar_disengagement}
Assume we are given an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace problem. Assume we are also given a laminar family ${\mathcal{L}}$ of clusters of $G$, such that $G\in {\mathcal{L}}$, and, for each such cluster $S\in {\mathcal{L}}$, we are additionally given (i) a vertex $u(S)\in V(S)$, and a set ${\mathcal{Q}}(S)=\set{Q_S(e)\mid e\in \delta_G(S)}$ of paths that are non-transversal with respect to $\Sigma$, routing the edges of $\delta_G(S)$ to $u(S)$ inside $S$; and
(ii) a vertex $u'(S)\notin V(S)$, and a set ${\mathcal{P}}(S)=\set{P_S(e)\mid e\in \delta_G(S)}$ of paths that are non-transversal with respect to $\Sigma$, routing the edges of $\delta_G(S)$ to $u'(S)$ outside of $S$.
Denote ${\mathcal{Q}}=\bigcup_{S\in {\mathcal{L}}}{\mathcal{Q}}(S)$, and ${\mathcal{P}}=\bigcup_{S\in {\mathcal{L}}}{\mathcal{P}}(S)$.
We now construct a collection ${\mathcal{J}}({\mathcal{L}},{\mathcal{Q}},{\mathcal{P}})$ of sub-instances of $I$, that we refer to as the \emph{disengagement of instance $I$ with respect to ${\mathcal{L}}$ and path sets ${\mathcal{Q}},{\mathcal{P}}$}.
Consider now a cluster $S\in {\mathcal{L}}$. Let $v(S)$ be the vertex representing $S$ in the partitioning tree $\tau({\mathcal{L}})$ associated with ${\mathcal{L}}$.
Let $v(S_1),\ldots,v(S_q)$ be the children of vertex $v(S)$ in $\tau({\mathcal{L}})$, so clusters $S_1,\ldots,S_q$ are mutually vertex-disjoint subgraphs of $S$.
We construct a sub-instance $J_S=(G_S,\Sigma_S)$ as follows.
The graph $G_S$ is obtained from $G$ by first contracting all vertices and edges of $G\setminus S$ into a super-node, that we denote by $v^{out}_S$, and then contracting, for each $1\leq i\leq q$, all vertices and edges of $S_i$ into a super-node, that we denote by $v^{in}_{S_i}$.
We denote $V'_S=V(S)\setminus \bigcup_{1\le t\le q}V(S_i)$,
so $V(G_S)=V'_S\cup\set{v_S^{out}}\cup\set{v_{S_1}^{in},\ldots,v_{S_q}^{in}}$.
We now define its rotation system $\Sigma_S$ as follows.
Note that for every vertex $v\in V'_S$, $\delta_{G_S}(v)=\delta_{G}(v)$, and the ordering ${\mathcal{O}}_v$ of the edges incident to $v$ remains the same as in $\Sigma$.
For vertex $v^{out}_S$, note that $\delta_{G_S}(v^{out}_S)=\delta_G(S)$, and we define the ordering ${\mathcal{O}}_{v^{out}_S}$ in $\Sigma_S$ to be the canonical ordering $\hat{\mathcal{O}}(S,{\mathcal{Q}}(S))$.
For vertex $v^{in}_{S_i}$, note that $\delta_{G_S}(v^{in}_{S_i})=\delta_G(S_i)$, and we define the ordering ${\mathcal{O}}_{v^{in}_{S_i}}$ in $\Sigma_S$ to be the canonical ordering $\hat{\mathcal{O}}(S_i,{\mathcal{Q}}(S_i))$.
This completes the description of the instance $J_S=(G_S,\Sigma_S)$. See Figure~\ref{fig: disengaged_instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of graph $G$.]{\scalebox{0.13}{\includegraphics{figs/dis_instance_before}}\label{fig: graph G}}
\hspace{1cm}
\subfigure[The graph $G_S$.]{
\scalebox{0.13}{\includegraphics{figs/dis_instance_after}}\label{fig: C and disc}}
\caption{An illustration of the disengaged instance.\label{fig: disengaged_instance}}
\end{figure}
The main result of this subsection is the following lemmas.
\begin{lemma}
$\sum_{S\in {\mathcal{L}},S\ne G}\mathsf{OPT}_{\mathsf{cnwrs}}(G_S,\Sigma_S)\le $\znote{related to the depth of $\tau({\mathcal{L}})$}.
\end{lemma}
\begin{proof}
\znote{to complete}
\end{proof}
\begin{lemma}
There is an efficient algorithm, that, given, for each instance $J\in {\mathcal{J}}({\mathcal{L}},{\mathcal{Q}},{\mathcal{Q}}')$, a solution $\phi(J)$, computes a solution for instance $I$ of value at most $\sum_{J\in {\mathcal{J}}({\mathcal{L}},{\mathcal{Q}},{\mathcal{Q}}')}\mathsf{cr}(\phi(J))$.
\end{lemma}
\begin{proof}
\znote{to complete}
\end{proof}
\section{Second Main Tool - Disengagement of Clusters}
\label{sec: not many paths}
\input{disengagement_path}
\newpage
\subsubsection{Proof of \Cref{thm: construct one level of laminar family}}
\label{subsubsec-construct-one-level-of-laminar}
Throughout the proof, we will consider various graphs, sets of disjoint clusters in these graphs, and the corresponding contracted graphs. Let $H$ be any graph, and let ${\mathcal{R}}$ be any set of disjoint vertex-induced subgraphs (clusters) of graph $H$. Let $\hat H=H_{|{\mathcal{R}}}$ be the contracted graph corresponding to $H$ and ${\mathcal{R}}$, that is obtained from $H$ by contracting every cluster $R\in {\mathcal{R}}$ into a supernode $v_R$. Observe that every subset $\hat U\subseteq V(\hat H)$ of vertices of $\hat H$ naturally defines a vertex-induced subgraph of $H$, which is a subgraph of $H$ induced by vertex set $U=\textsf{left}(\bigcup_{v_R\in \hat U}V(R)\textsf{right} )\cup(V(H)\cap \hat U)$; in other words, $U$ contains all regular vertices of $\hat U$, and the vertices of every cluster $R\in {\mathcal{R}}$ with $v_R\in \hat U$.
We will refer to $H[U]$ as the \emph{subgraph of $H$ (or cluster of $H$) defined by the set $\hat U$ of vertices of $\hat H$}.
Similarly, if $S$ is a cluster of $\hat H$ induced by vertex set $\hat U$, we will refer to $H[U]$ as the cluster of $H$ defined by $S$.
Assume now that we are given any graph $H$, a special vertex $v^*$ in $H$, and a collection ${\mathcal{R}}$ of disjoint clusters of $H$, such that vertex $v^*$ does not lie in any cluster of ${\mathcal{R}}$, and every cluster $R\in {\mathcal{R}}$ has $\alpha$-bandwidth property, for some parameter $0<\alpha<1$. As before, we denote $\hat H=H_{|{\mathcal{R}}}$. Next, we consider a Gomory-Hu tree $\tau$ of the graph $\hat H$
(see \Cref{subsec: GH tree} for a definition). We root the tree $\tau$ at the special vertex $v^*$. For every vertex $u\in V(\tau)$, we let $\tau_u$ be the subtree of $\tau$ rooted at $u$.
We will use the following useful observation multiple times. The proof is deferred to \Cref{appx: subtree to cluster}.
\begin{observation}\label{obs: subtree to cluster}
Let $u\in V(\tau)\setminus \set{v^*}$ be any non-root vertex of the tree $\tau$, and let $S$ be the cluster of $H$ that is defined by the set $V(\tau_u)$ of vertices of $\hat H$. Then cluster $S$ has the $\alpha$-bandwidth property in $H$. Moreover, there is an efficient algorithm to compute a distribution ${\mathcal{D}}'(S)$ over the external routers in $\Lambda'_{H}(S)$, such that distribution ${\mathcal{D}}'(S)$ is careful with respect to $v^*$, and, for every edge $e\in E(H)\setminus E(S)$, $\expect[{\mathcal{Q}}'(S)\sim{\mathcal{D}}'(S)]{\cong_{H}({\mathcal{Q}}'(S),e)}\leq O(\log^4m/\alpha)$.
\end{observation}
For convenience, in the remainder of the proof, we denote graph $G'$ by $G$, and the set ${\mathcal{C}}'$ of clusters by ${\mathcal{C}}$.
We start with the graph $G$ and the set ${\mathcal{C}}$ of basic clusters, and we let $H=G_{|{\mathcal{C}}}$ be the corresponding contracted graph. We consider the Gomory-Hu tree $\tau$ of the graph $H$. We root the tree $\tau$ at the special vertex $v^*$. For every vertex $u\in V(\tau)$, we let $\tau_u$ be the subtree of $\tau$ rooted at $u$, and we let the weight $w(u)$ be the number of supernodes (vertices corresponding to clusters in ${\mathcal{C}}$) in the tree $\tau_u$. Let $u^{*}$ be the vertex of $\tau$ that is furthest from the root $v^*$, such that
$w(u^*)\geq \floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )|{\mathcal{C}}|}$. We now consider two cases.
The first case happens if $u^*=v^*$. In this case, we will compute a type-1 legal clustering of $G$. Let $u_1,\ldots,u_q$ denote all child vertices of $v^*$. For all $1\leq i\leq q$, let $R_i$ be the cluster of the graph $G$ defined by the vertex set $V(\tau_{u_i})$. Denote ${\mathcal{R}}=\set{R_1,\ldots,R_q}$. Since every cluster $C\in {\mathcal{C}}$ has the $\alpha_0$-bandwidth property, and $H=G_{|{\mathcal{C}}}$, from \Cref{obs: subtree to cluster}, each cluster $R_i\in {\mathcal{R}}$ has the $\alpha_0\geq\alpha_1$-bandwidth property. From the construction, vertex $v^*$ may not lie in any of the clusters of ${\mathcal{R}}$, and, for each cluster $R\in {\mathcal{R}}$, and for every basic cluster $C\in {\mathcal{C}}$, either $C\subseteq R$ or $V(C)\cap V(R)=\emptyset$. We use the algorithm from \Cref{obs: subtree to cluster} to construct, for every cluster $R\in {\mathcal{R}}$, a distribution ${\mathcal{D}}'(R)$ over the external $S$-routers in $\Lambda'_{G}(S)$, such that the distribution is careful with respect to $v^*$, and, for every edge $e\in E(G)\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}'(R)]{\cong_{G}({\mathcal{Q}}'(R),e)}\leq O(\log^4m/\alpha_0)\leq \beta$.
Note that $G\setminus\bigcup_{R\in {\mathcal{R}}}R$ consists of only one vertex -- vertex $v^*$. Therefore, ${\mathcal{R}}$ is a legal type-1 clustering of graph $G$. We terminate the algorithm, and return this clustering.
We assume from now on that $u^*\neq v^*$. We will provide an algorithm for computing a type-2 legal clustering of $G$. Let $R^*$ be the subgraph of $G$ defined by vertex set $V(\tau_{u^*})$ of graph $H$.
As before, from
\Cref{obs: subtree to cluster}, cluster $R^*$ has the $\alpha_0\geq\alpha_1$-bandwidth property, it does not contain the verex $v^*$, and, for every basic cluster $C\in {\mathcal{C}}$, either $C\subseteq R^*$ or $V(C)\cap V(R^*)=\emptyset$.
From the definition of vertex $u^*$, the total number of basic clusters of ${\mathcal{C}}$ that are contained in $R^*$ is at least $\floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )|{\mathcal{C}}|}$.
We also use the algorithm from \Cref{obs: subtree to cluster} to construct a distribution ${\mathcal{D}}'(R^*)$ over the external $R^*$-routers in $\Lambda'_{G}(S)$, such that the distribution is careful with respect to $v^*$, and, for every edge $e\in E(G)\setminus E(R^*)$, $\expect[{\mathcal{Q}}'(R^*)\sim{\mathcal{D}}'(R^*)]{\cong_{G}({\mathcal{Q}}'(R^*),e)}\leq O(\log^4m/\alpha_0)\leq \beta$. In the final type-2 legal clustering ${\mathcal{R}}$ for graph $G$ that our algorithm will return, cluster $R^*$ will play the role of the distinguished cluster, and the distribution ${\mathcal{D}}'(R^*)$ over the set of its external routers will remain unchanged.
Let $G^*$ be the graph associated with the cluster $R^*$: that is, graph $G^*$ is obtained from graph $G$ by contracting all vertices of $G\setminus R^*$ into a special vertex, that we denote by $v^{**}$. We also denote by ${\mathcal{C}}^*\subseteq {\mathcal{C}}$ the set of all basic clusters $C\in {\mathcal{C}}$ with $C\subseteq R^*$.
We now construct a type-1 legal clustering ${\mathcal{R}}'$ of $G^*$, which is required as part of definition of type-2 legal clustering of $G$. Denote the child vertices of vertex $u^*$ in the tree $\tau$ by $u_1,\ldots,u_q$. For all $1\leq i\leq q$, let $R_i$ be the cluster of the graph $G$ defined by the vertex set $V(\tau_{u_i})$. Denote ${\mathcal{R}}'=\set{R_1,\ldots,R_q}$. Since every cluster $C\in {\mathcal{C}}$ has the $\alpha_0$-bandwidth property, and $H=G_{|{\mathcal{C}}}$, from \Cref{obs: subtree to cluster}, each cluster $R_i\in {\mathcal{R}}$ has the $\alpha_0\geq\alpha_1$-bandwidth property. From the construction, vertex $v^{**}$ may not lie in any of the clusters of ${\mathcal{R}}'$, and, for each cluster $R\in {\mathcal{R}}'$, and for every basic cluster $C\in {\mathcal{C}}^*$, either $C\subseteq R$ or $V(C)\cap V(R)=\emptyset$ holds. Consider now some cluster $R_i\in {\mathcal{R}}'$, and denote $\delta_G(R_i)=E_i$.
From the properties of the Gomory-Hu tree (see \Cref{thm: GH tree properties}), there is a collection ${\mathcal{Q}}'_i$ of edge-disjoint paths in graph $H$, routing the edges of $E_i$ to vertex $u^*$, that are internally disjoint from $V(\tau_{u_i})$. Let $H^*$ be the graph obtained from $H$, by contracting all vertices of $V(H)\setminus V(\tau_{u^*})$ into a supernode $\hat v^*$. A simple transformation of the paths in ${\mathcal{Q}}'_i$ shows that there is a collection ${\mathcal{Q}}''_i$ of edge-disjoint paths in graph $H^*$, routing the edges of $E_i$ to $u^*$. Observe that graph $H^*$ is precisely the contracted graph of $G^*$ with respect to the set ${\mathcal{C}}^*$ of clusters, that is, $H^*=G^*_{|{\mathcal{C}}^*}$, and recall that each cluster $C\in {\mathcal{C}}^*$ has the $\alpha_0$-bandwidth property.
If vertex $u^*$ is not a supernode, then we apply the algorithm from \Cref{claim: routing in contracted graph} to graph $H^*$, the set ${\mathcal{C}}^*$ of clusters, and the set ${\mathcal{Q}}_i''$ of paths, to obtain a set ${\mathcal{Q}}^*_i$ of paths in graph $G^*$, routing the edges of $E_i$ to vertex $u^*$, such that every path in ${\mathcal{Q}}^*_i$ is internally disjoint from $R_i$. Moreover, for every edge $e\in \bigcup_{C\in {\mathcal{C}}^*}E(C)$, the paths of ${\mathcal{Q}}^*_i$ cause congestion at most $ \ceil{1/\alpha_0}$, while for every edge $e\in E(G^*)\setminus \textsf{left}(\bigcup_{C\in {\mathcal{C}}^*}E(C)\textsf{right} )$, the paths of ${\mathcal{Q}}^*_i$ cause congestion at most $1$. In particular, the set ${\mathcal{Q}}^*_i$ of paths is careful with respect to vertex $v^{**}$. We then define a distribution ${\mathcal{D}}'(R_i)$ over the set $\Lambda'_{G^*}(R_i)$ of external $R_i$-routers to choose the set ${\mathcal{Q}}^*_i$ of paths with probability $1$.
Assume now that vertex $u^*$ is a supernode, and that it represents some cluster $C\in {\mathcal{C}}^*$. We apply the algorithm from \Cref{claim: routing in contracted graph} to graph $H^*$, the set ${\mathcal{C}}^*\setminus\set{C}$ of clusters, and the set ${\mathcal{Q}}_i''$ of paths, to obtain a set ${\mathcal{Q}}^*_i$ of paths in graph $G^*$, routing the edges of $E_i$ to edges of $\delta_{G^*}(C)$, such that every path in ${\mathcal{Q}}^*_i$ is internally disjoint from $R_i$. As before, for every edge $e\in \bigcup_{C'\in {\mathcal{C}}^*}E(C')$, the paths of ${\mathcal{Q}}^*_i$ cause congestion at most $ \ceil{1/\alpha_0}$, while for every edge $e\in E(G^*)\setminus \textsf{left}(\bigcup_{C'\in {\mathcal{C}}^*}E(C')\textsf{right} )$, the paths of ${\mathcal{Q}}^*_i$ cause congestion at most $1$. As before, the set ${\mathcal{Q}}^*_i$ of paths is careful with respect to vertex $v^{**}$.
We use the algorithm from \Cref{lem: simple guiding paths} to compute a distribution ${\mathcal{D}}(C)$ over internal $C$-routers in $\Lambda_{G^*}(C)$, such that, for every edge $e\in E(C)$, $\expect[{\mathcal{Q}}(C)\sim {\mathcal{D}}(C)]{\cong({\mathcal{Q}}(C),e)}\leq \log^4m/\alpha_0$.
We now define a distribution ${\mathcal{D}}'(R_i)$ over the set $\Lambda'_{G^*}(R_i)$ of external $R_i$-routers. In order to draw a router from the distribution, we first choose an internal $C$-router ${\mathcal{Q}}(C)$ from the distribution ${\mathcal{D}}(C)$. Let $x$ be the vertex that serves the center of the router. For every edge $e\in E_i$, we let $\tilde Q(e)$ be the path obtained as follows. First, we let $Q^*(e)$ be the unique path of ${\mathcal{Q}}^*_i$ that originates from edge $e$. We let $e'$ be the last edge on path ${\mathcal{Q}}^*_i$, that must belong to $\delta_{G^*}(C)$. We then let $\tilde Q(e)$ be the path obtained by concatenating path $Q^*(e)$ with the unique path of ${\mathcal{Q}}(C)$ that originates at edge $e$. We let ${\mathcal{Q}}'(R_i)=\set{\tilde Q(e)\mid e\in E_i}$ be the resulting external $R_i$-router, that routes the edges of $E_i$ to $x$. Since every edge of $\delta_{G^*}(C)$ may lie on at most one path of ${\mathcal{Q}}^*_i$, it is immediate to verify that, for every edge $e\in E(C)$, $\cong({\mathcal{Q}}'(R_i),e)\leq \cong({\mathcal{Q}}(C),e)$, and so overall, for every edge $e'$, $\expect[{\mathcal{Q}}'(R_i)\sim {\mathcal{D}}'(R_i)]{\cong_{G^*}({\mathcal{Q}}'(R_i),e')}\leq \frac{\log^4m}{\alpha_0}\leq \beta$,
since $\alpha_0=1/\log^3m$ and $\beta=\log^{18}m$. As before, distribution ${\mathcal{D}}'(R_i)$ is careful with respect to $v^{**}$.
Lastly, observe that at most one cluster $C\in {\mathcal{C}}$ may be contained in graph $R^*\setminus\bigcup_{R\in {\mathcal{R}}'}R$ -- the cluster associated with vertex $u^*$, if $u^*$ is a supernode.
Therefore, $({\mathcal{R}}',\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}'})$ is a type-1 legal clustering of graph $G^*$, with cluster set ${\mathcal{C}}^*$ and special vertex $v^{**}$.
Moreover, from the choice of vertex $u^*$, we are guaranteed that every cluster $R'\in {\mathcal{R}}'$ contain at most $\floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )|{\mathcal{C}}'|}$ clusters of ${\mathcal{C}}$.
The remainder of the algorithm is iterative. We start with a helpful clustering $({\mathcal{R}}=\set{R^*},\set{{\mathcal{D}}'(R^*)})$ of $G$, and we view $R^*$ as the distinguished cluster of ${\mathcal{R}}$. We then iterate. In every iteration, we either establish that the current helpful clustering $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ is a type-2 legal clustering, by computing a nice witness structure for graph $G_{|{\mathcal{R}}}$, with respect to the set ${\mathcal{C}}''$ of clusters, containing every cluster $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$; or we will compute another helpful clustering of $G$ that is ``better'' in some sense, and use it to replace the current helpful clustering $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$. We will ensure that the helpful clustering ${\mathcal{R}}$ that the algorithm maintains always contains the cluster $R^*$ that we defined above, which will always remain the distinguished cluster of ${\mathcal{R}}$. The distribution ${\mathcal{D}}'(R^*)$ over the external $R^*$-routers in $\Lambda'_G(R^*)$, and the type-1 legal clustering $({\mathcal{R}}',\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}'})$ of the graph $G^*$ associated with cluster $R^*$ will remain unchanged throughout the algorithm. We will use the following definition in order to compare different helpful clusterings of $G$.
\begin{definition}[Comparing clusterings]
Let ${\mathcal{R}}_1$, ${\mathcal{R}}_2$ be two helpful clusterings of graph $G$, with respect to special vertex $v^*$ and set ${\mathcal{C}}$ of basic clusters, such that $R^*\in {\mathcal{R}}_1\cap {\mathcal{R}}_2$. Denote by ${\mathcal{C}}_1\subseteq {\mathcal{C}}$ the set of all clusters $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left}(\bigcup_{R\in {\mathcal{R}}_1}R\textsf{right} )$, and define a subset ${\mathcal{C}}_2\subseteq {\mathcal{C}}$ of basic clusters for ${\mathcal{R}}_2$ similarly. We say that clustering ${\mathcal{R}}_2$ is \emph{better} than clustering ${\mathcal{R}}_1$ if one of the following hold:
\begin{itemize}
\item either $|{\mathcal{C}}_2|<|{\mathcal{C}}_1|$; or
\item $|{\mathcal{C}}_1|=|{\mathcal{C}}_2|$, and $|E(G_{|{\mathcal{R}}_2})|<|E(G_{|{\mathcal{R}}_1})|$.
\end{itemize}
\end{definition}
The following lemma is key in the proof of \Cref{thm: construct one level of laminar family}.
\begin{lemma}\label{lemma: better clustering}
There is an efficient randomized algorithm, that, given a helpful clustering $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ of graph $G$ with respect to vertex $v^*$ and set ${\mathcal{C}}$ of basic clusters, such that $R^*\in {\mathcal{R}}$, either (i) establishes that ${\mathcal{R}}$ is a type-2 legal clustering by providing a nice witness structure for graph $G_{|{\mathcal{R}}}$, with respect to the set ${\mathcal{C}}''$ of clusters, containing every cluster $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$, or (ii) computes another helpful clustering $(\tilde{\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in \tilde{\mathcal{R}}})$ of graph $G$ with respect to $v^*$ and ${\mathcal{C}}$ with $R^*\in {\mathcal{R}}$, such that $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$; or (iii) returns FAIL. The latter may only happen with probability at most $1/m^{10}$.
\end{lemma}
It is immediate to complete the proof of \Cref{thm: construct one level of laminar family}.
using \Cref{lemma: better clustering}. Our algorithm starts with the helpful custering $({\mathcal{R}}=\set{R^*},\set{{\mathcal{D}}'(R^*)})$ of $G$, where ${\mathcal{D}}'(R^*)$ is the distribution over external $R^*$-routers that we have computed above, and then iterates. In every iteration, we apply the algorithm from \Cref{lemma: better clustering} to the current helpful clustering $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$. If the algorithm establishes that ${\mathcal{R}}$ is a type-2 legal clustering by providing a nice witness structure for graph for graph $G_{|{\mathcal{R}}}$, with respect to the set ${\mathcal{C}}''$ of clusters, containing every cluster $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$, then we terminate the algorithm with the resulting type-2 legal clustering $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$; we view $R^*$ as the distinguished cluster of ${\mathcal{R}}$, and the type-1 legal clustering $({\mathcal{R}}',\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}'})$ of the graph $G^*$ corresponding to $R^*$ remains unchanged. Otherwise, if the algorithm returns another helpful clustering $(\tilde{\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in \tilde{\mathcal{R}}})$, then we replace $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ with $(\tilde{\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in \tilde{\mathcal{R}}})$ and continue to the next iteration. Lastly, if the algorithm from
\Cref{lemma: better clustering} returns FAIL, then we terminate the algorithm and return FAIL as well.
Let ${\mathcal{C}}'$ be the set of all clusters $C\in{\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left}(\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$, where ${\mathcal{R}}$ is the current helpful clustering.
Since, in every iteration, either $|{\mathcal{C}}'|$ decreases, or $|{\mathcal{C}}'|$ remains the same but the number of edges in graph $G_{|{\mathcal{R}}}$ decreases, the algorithm is guaranteed to terminate after at most $m^2$ iterations.
Since the probability of the algorithm to return FAIL in each iteration is at most $1/m^{10}$, the total probability that the algorithm returns FAIL is at most $1/m^8$. If the algorithm does not return FAIL, then it returns a type-2 clustering of $G$ as required. In order to complete the proof of \Cref{thm: construct one level of laminar family}, it is now enough to prove \Cref{lemma: better clustering}, which we do next.
\subsubsection{Proof of \Cref{lemma: better clustering}}
\label{subsubsec: proof of improving clustering}
Recall that we are given a graph $G$ and a special vertex $v^*$ of $G$. We are also given a collection ${\mathcal{C}}$ of disjoint vertex-induced subgraphs of $G\setminus\set{v^*}$ called basic clusters, such that every basic cluster $C\in {\mathcal{C}}$ has the $\alpha_0$-bandwidth property. Lastly, we are given a helpful clustering $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ of $G$ with respect to $v^*$ and ${\mathcal{C}}$. Recall that vertex $v^*$ may not lie in any cluster of ${\mathcal{R}}$, and every cluster $R\in {\mathcal{R}}$ has the $\alpha_1$-bandwidth property. For every cluster $R\in {\mathcal{R}}$, ${\mathcal{D}}'(R)$ is a distribution over the external $R$-routers in $\Lambda'_G(R)$, and, for every edge $e\in E(G)\setminus E(R)$, $\expect[{\mathcal{Q}}'(R)\sim{\mathcal{D}}'(R)]{\cong_{G'}({\mathcal{Q}}'(R),e)}\leq \beta$.
Additionally, there is a distingiushed cluster $R^*\in {\mathcal{R}}$, whose corresponding distribution ${\mathcal{D}}'(R^*)$ is careful with respect to $v^*$, and $R^*$ contains at least $\floor{\textsf{left}(1-1/2^{(\log m)^{3/4}}\textsf{right} )|{\mathcal{C}}|}$ clusters of ${\mathcal{C}}$.
We denote by ${\mathcal{C}}'$ the set of all clusters $C\in {\mathcal{C}}$, such that $C\subseteq G\setminus\textsf{left}(\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$. Observe that ${\mathcal{R}}\cup {\mathcal{C}}'$ is a set of mutually disjoint clusters of graph $G$ (see \Cref{fig: NF1}).
It will be convenient for us to work with a slightly different contracted graph, that we denote by $\hat H=G_{|({\mathcal{R}}\cup{\mathcal{C}}')}$. Note that every vertex $u\in V(\hat H)$ that is different from a special vertex $v^*$, is either a regular vertex (that is, it is a vertex of $G$), or a supernode corresponding to a cluster of ${\mathcal{C}}'\cup {\mathcal{R}}$. If supernode $u$ represents a cluster of ${\mathcal{C}}'$, then we call it a $C$-node, and otherwise we call it an $R$-node (see \Cref{fig: NF2}).
In order to prove \Cref{lemma: better clustering}, we will mostly work with graph $\hat H$.
Note that $v^*\in V(\hat H)$. We denote by $u^*$ the $R$-node representing the distinguished cluster $R^*\in {\mathcal{R}}$.
We will maintain a collection ${\mathcal{W}}$ of clusters in graph $\hat H$, that we call $W$-clusters, and define next.
\begin{figure}[h]
\centering
\subfigure[A schematic view of graph $G$. Clusters of ${\mathcal{R}}$ are shown in red, clusters of ${\mathcal{C}}$ are shown in blue, clusters of ${\mathcal{C}}'$ are the blue clusters that are disjoint from the red clusters, and vertices of $V(G)\setminus \big( \bigcup_{C\in {\mathcal{C}}}V(C)\big)$ are shown in black.]{\scalebox{0.093}{\includegraphics{figs/NF1.jpg}}\label{fig: NF1}}
\hspace{1.2cm}
\subfigure[A schematic view of graph $\hat H$. Regular vertices are shown in black, $C$-nodes are shown in blue and $R$-nodes are shown in red.]{\scalebox{0.093}{\includegraphics{figs/NF2.jpg}}\label{fig: NF2}}
\caption{Graphs $G$ and $\hat H$.}
\end{figure}
\begin{definition}[Valid set of $W$-clusters]
A set ${\mathcal{W}}$ of disjoint clusters of graph $\hat H$ is a \emph{valid set of $W$-clusters} if:
\begin{itemize}
\item for every cluster $W\in {\mathcal{W}}$, every vertex of $W$ is an $R$-node or a regular vertex, and $W$ does not contain the special vertex $v^*$ or the
$R$-node $u^*$ representing cluster $R^*$;
\item for every cluster $W\in {\mathcal{W}}$, $|E_{\hat H}(W)|\geq |\delta_{\hat H}(W)|/(64\log m)$; and
\item every cluster $W\in {\mathcal{W}}$ has the $\alpha'$-bandwidth property in graph $\hat H$, where $\alpha'=1/(c\log^{2.5}m)$, for some large enough constant $c$.
\end{itemize}
\end{definition}
We will use the following lemma in order to prove \Cref{lemma: better clustering}.
\begin{lemma}\label{lemma: better clustering 2}
There is an efficient randomized algorithm, that, given a valid $W$-clustering ${\mathcal{W}}$ of graph $\hat H$, either (i) establishes that $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ is a type-2 legal clustering of $G$, by providing a nice witness structure for graph $G_{|{\mathcal{R}}}$, with respect to the set ${\mathcal{C}}''$ of clusters, containing every cluster $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$; or (ii) computes another helpful clustering $(\tilde{\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in \tilde {\mathcal{R}}})$ of graph $G$ with respect to vertex $v^*$ and set ${\mathcal{C}}$ of basic clusters, such that $R^*\in {\mathcal{R}}$ and $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$; or (iii) computes a new valid set ${\mathcal{W}}'$ of $W$-clusters in the current graph $\hat H$, such that $|E(\hat H_{|{\mathcal{W}}'})|<|E(\hat H_{|{\mathcal{W}}})|$; or (iv) returns FAIL. The latter may only happen with probability at most $1/m^{11}$.
\end{lemma}
\Cref{lemma: better clustering} easily follows from \Cref{lemma: better clustering 2}. We start with ${\mathcal{W}}=\emptyset$, which is a valid set of $W$-clusters for $\hat H$, and then iterate. In every iteration, we apply the algorithm from \Cref{lemma: better clustering 2} to the current valid set ${\mathcal{W}}$ of $W$-clusters. If the algorithm establishes that ${\mathcal{R}}$ is a type-2 legal clustering of $G$, then we terminate the algorithm and return the correpsonding witness structure for graph $G_{|{\mathcal{R}}}$. If the algorithm computes another helpful clustering $(\tilde{\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in \tilde {\mathcal{R}}})$ of graph $G$ with respect to vertex $v^*$ and set ${\mathcal{C}}$ of basic clusters with $R^*\in {\mathcal{R}}$, such that $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$, then we terminate the algorithm and return the clustering $(\tilde{\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in \tilde {\mathcal{R}}})$. If the algorithm from \Cref{lemma: better clustering 2} returns FAIL, then we terminate and algorithm and return FAIL as well. Otherwise, the algorithm from \Cref{lemma: better clustering 2} computes a valid set ${\mathcal{W}}'$ of $W$-clusers in the current graph $\hat H$, such that $|E(\hat H_{|{\mathcal{W}}'})|<|E(\hat H_{|{\mathcal{W}}})|$. We then replace ${\mathcal{W}}$ with ${\mathcal{W}}'$ and continue to the next iteration. Since the number of edges in graph $\hat H_{|{\mathcal{W}}}$ decreases in every iteration, we are guaranteed that, after at most $m$ iterations the above algorithm terminates.
Since the algorithm from \Cref{lemma: better clustering 2} only returns FAIL with probability at most $1/m^{11}$, the total probability that our algorithm returns FAIL is at most $1/m^{10}$. From now on we focus on the proof of \Cref{lemma: better clustering 2}.
\subsubsection{Proof of \Cref{lemma: better clustering 2}}
\label{subsec: getting nice structure better clustering}
Observe that so far, we have constructed a 3-level hierarchical clustering of the graph $G$. The first level consists of the set ${\mathcal{C}}$ of basic clusters of graph $G$. At the second level, there is a set ${\mathcal{R}}$ of clusters of graph $G$. Recall that, for every basic cluster $C\in {\mathcal{C}}$, either $C\subseteq G\setminus\textsf{left}(\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$, or
there is some cluster $R\in {\mathcal{R}}$, with $C\subseteq R$. As before, we denote by ${\mathcal{C}}'\subseteq {\mathcal{C}}$ the set of all basic clusters $C$ with $C\subseteq G\setminus\textsf{left}(\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$. We can use the valid set ${\mathcal{W}}$ of $W$-clusters in graph $\hat H$, in order to construct another set ${\mathcal{W}}'$ of clusters in the original graph $G$, as follows. Recall that every cluster $W\in {\mathcal{W}}$ may only contain $R$-nodes or regular vertices of $\hat H$. For each such cluster $W$, let ${\mathcal{R}}(W)\subseteq {\mathcal{R}}$ be the set of all clusters $R\in {\mathcal{R}}$ with $v_R\in V(W)$. We then let $W'$ be a subgraph of $G$ induced by vertex set $\textsf{left} (\bigcup_{R\in {\mathcal{R}}(W)}V(R)\textsf{right} )\cup (V(G)\cap V(W))$.
In other words, $V(W')$ contains all regular vertices of $W$, and all vertices lying in clusters of ${\mathcal{R}}(W)$.
We will refer to $W'$ as the \emph{cluster of $G$ defined by $W$}. Finally, let ${\mathcal{W}}'=\set{W'\mid W\in {\mathcal{W}}}$. Note that every basic cluster $C\in {\mathcal{C}}'$ must be disjoint from clusters of ${\mathcal{W}}'$. We denote by ${\mathcal{R}}'={\mathcal{R}}\setminus \textsf{left}(\bigcup_{W\in {\mathcal{W}}}{\mathcal{R}}(W)\textsf{right} )$. Note that each cluster $R\in {\mathcal{R}}'$ is contained in $G\setminus\textsf{left}(\bigcup_{W'\in {\mathcal{W}}'}V(W')\textsf{right} )$, while for each cluster $R\in {\mathcal{R}}\setminus {\mathcal{R}}'$, there is some cluster $W'\in {\mathcal{W}}'$ with $R\subseteq W'$. Therefore, ${\mathcal{C}}'\cup {\mathcal{R}}'\cup {\mathcal{W}}'$ is a collection of disjoint clusters of graph $G$ (see \Cref{fig: NF3}).
Recall that we are guaranteed that every cluster $C\in {\mathcal{C}}'$ has the $\alpha_0$-bandwidth property, where $\alpha_0=1/\log^3m$, and every cluster $R\in {\mathcal{R}}'$ has the $\alpha_1$-bandwidth property, where $\alpha_1=1/\log^6m$. Lastly, every cluster $W\in {\mathcal{W}}$ has the $\alpha'$-bandwidth property (for $\alpha'=1/(c\log^{2.5}m)$, where $c$ is a large enough constant) in graph $\hat H$. From \Cref{cor: contracted_graph_well_linkedness}, every cluster $W'\in {\mathcal{W}}'$ has the $\alpha_1\cdot \alpha'=1/(c\log^{8.5}m)$-bandwidth property in graph $G$.
\begin{figure}[h]
\centering
\subfigure[A schematic view of graph $G$. Clusters of ${\mathcal{W}}'$ are shown in green. Clusters of ${\mathcal{R}}$ are shown in red (with clusters of ${\mathcal{R}}'$ shaded). Clusters of ${\mathcal{C}}$ are shown in blue (with clusters of ${\mathcal{C}}'$ shaded). Regular vertices lying outside of clusters of ${\mathcal{C}}$ are shown in black. Note that, if there exist clusters $C\in {\mathcal{C}}$ and $W'\in {\mathcal{W}}'$ with $C\subseteq W'$, then there exists a cluster $R\in {\mathcal{R}}$ with $C\subseteq R\subseteq W'$. Also, for every cluster $W'\in {\mathcal{W}}'$, $R^*\not\subseteq W'$ and $v^*\notin W'$ hold.]{\scalebox{0.095}{\includegraphics{figs/NF3.jpg}}\label{fig: NF3}}
\hspace{1cm}
\subfigure[A schematic view of graph $\hat H'$. Regular vertices are shown in black, $C$-nodes are shown in blue, $R$-nodes are shown in red, and $W$-nodes are shown in green.]{\scalebox{0.095}{\includegraphics{figs/NF4.jpg}}\label{fig: NF4}}
\caption{Graphs $G$ and $\hat H'$.}
\end{figure}
In order to prove \Cref{lemma: better clustering}, it will be convenient for us to work with graph $\hat H'$, that is a contracted graph of $\hat H$, with respect to set ${\mathcal{W}}$ of clusters, that is, $\hat H'=\hat H_{|{\mathcal{W}}}$. Since graph $\hat H$ is itself a contracted graph of $G$ with respect to ${\mathcal{R}}\cup {\mathcal{C}}'$, it is easy to verify that $\hat H'=G_{|{\mathcal{C}}'\cup{\mathcal{R}}'\cup {\mathcal{W}}'}$ (see \Cref{fig: NF4}).
The vertices of graph $\hat H'$ are partitioned into four types. The first type is regular vertices, which are also the vertices of the original graph $G$; note that the special vertex $v^*$ belongs to graph $\hat H$ as a regular vertex. The second type is supernodes corresponding to clusters of ${\mathcal{C}}'$, that we refer to as $C$-nodes. The third type is supernodes corresponding to clusters of ${\mathcal{R}}'$, that we call $R$-nodes, and it includes the vertex $u^*$, representing the cluster $R^*$. The fourth type is supernodes corresponding to clusters of ${\mathcal{W}}'$, that we call $W$-nodes.
We denote the set of all regular vertices of $\hat H'$, excluding the special vertex $v^*$, by $U^*$. We denote the set of all $R$-nodes, excluding the vertex $u^*$, by $U^{R}$.
The remainder of the proof of \Cref{lemma: better clustering} consists of three steps. In the first step, we perform some manipulations that will allow us to either compute a new valid set $\tilde {\mathcal{W}}$ of $W$-clusters in the current graph $\hat H$, such that $|E(\hat H_{|\tilde {\mathcal{W}}})|<|E(\hat H_{|{\mathcal{W}}})|$, or to organize the vertices of $U^*\cup U^R$ into a nice layered structure. We also define a collection ${\mathcal{J}}$ of clusters of the graph $\hat H'$ in this step. In the second step, we will define another contracted graph $\check H$ with respect to the clustering ${\mathcal{J}}$, and explore some of its properties. In particular, we define the notion of a ``simplifying cluster'' in $\check H$, and show an algorithm that, given a simplifying cluster in $\check H$, produces a helpful clustering $\tilde {\mathcal{R}}$ of $G$ that is better than the current clustering ${\mathcal{R}}$. Lastly, in the third setp, we either compute a nice witness structure in graph $G_{|{\mathcal{R}}}$ as required, or compute a simplifying cluster in graph $\check H$, which in turn allows us to produce a helpful clustering $\tilde {\mathcal{R}}$ in graph $G$ that is better than ${\mathcal{R}}$.
We now describe these steps one by one.
\input{level-3-step-1}
\input{level-3-step-2}
\input{level-3-step-3}
\input{level-3-bad-indices}
\subsubsection{Proof of \Cref{lem: no bad indices}}
\label{subsubsec: no bad indices}
\renewcommand{\textbf{E}'}{\tilde E}
Throughout the proof, we will only consider the graph $\check H$, so we will omit subscript $\check H$ from various notations, such as, for example, $\delta_{\check H}(v)$ for vertices $v\in\check H$.
We start by considering the edges connecting different clusters in $\set{S_1,\ldots,S_r}$, and by establishing some useful relationships between them.
\subsubsection*{Edges Connecting Clusters in $\set{S_1,\ldots,S_r}$}
Fix an index $1\leq i< r$. We denote by $E'_i=E(S_i,S_{i+1})$, and by $\tilde E^{\operatorname{over}}_i$ the set of all edges $e=(u,v)$, such that, if $u\in S_{j},v\in S_{j'}$, then $j<i$ and $j'>i+1$ holds. For all $1<i\leq r$, we denote by $\tilde E_i^{\operatorname{left}}$ the set of all edges $e=(u,v)$ with $u\in S_i$, such that, if $v\in S_j$, then $j<i-1$. Similarly, for all $1\leq i<r$, we denote by $\tilde E_i^{\operatorname{right}}$ the set of all edges $e=(u,v)$ with $u\in S_i$, such that, if $v\in S_j$, then $j>i+1$ holds (see \Cref{fig: NF9}). Notice that $\delta(S_i)=E'_{i-1}\cup E'_i\cup \tilde E_i^{\operatorname{left}}\cup \tilde E_i^{\operatorname{right}}$. Notice also that, by the definition, if $i\in \set{1,2}$, then $E_i^{\operatorname{left}}=\emptyset$; if $i\in \set{1,r-1,r}$, then $\tilde E_i^{\operatorname{over}}=\emptyset$, and, if $i\in \set{r-1,r}$, then $\textbf{E}'_i^{\operatorname{right}}=\emptyset$.
We prove the following observation that helps us relate the sizes of all these edge sets. The proof is deferred to \Cref{subsec: edges between Sis}.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF9.jpg}
\caption{Edge sets $\tilde E^{\operatorname{left}}_i, \tilde E^{\operatorname{left}}_i$ and $\tilde E^{\operatorname{over}}_i$.}\label{fig: NF9}
\end{figure}
\begin{observation}\label{obs: bad inded structure}
For all $1< i<r$, the following hold:
\begin{itemize}
\item $|\textbf{E}'_i^{\operatorname{over}}|\leq |E_{i}'|$.
\item $|\textbf{E}'_{i+1}^{\operatorname{left}}|\leq
|E_{i}'|+|\textbf{E}'_{i}^{\operatorname{right}}|$
\item $|\textbf{E}'_{i}^{\operatorname{right}}|\leq
|E_{i}'|+|\textbf{E}'_{i+1}^{\operatorname{left}}|$.
\end{itemize}
\end{observation}
For all $1\leq i\leq r$, we denote $S''_i=S_i\setminus S'_i$.
\Cref{obs: bad inded structure} allows us to bound the cardinality of the set
$\textbf{E}'_i^{\operatorname{over}}\subseteq \hat E_i$ of edges in terms of the cardinality of the set $E_i'$ of edges. Note that the set $E_i'$ of edges can be thought of as the union of four subsets: set $E_i$, and sets $E(S'_i,S''_{i+1})$, $E(S''_i,S'_{i+1})$, and $E(S''_i,S''_{i+1})$
(see \Cref{fig: NF12}).
The latter three sets are all contained in $\hat E_i$. We will bound the cardinalities of these subsets in terms of $|E_i|$ in turn.
We start by considering edge sets incident to clusters of $\set{S''_i}_{1<i<r}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF12.jpg}
\caption{Edges in set $E'_i$ are shown in black.}\label{fig: NF12}
\end{figure}
\subsubsection*{Edges Incident to Clusters of $\set{S''_i}_{1<i<r}$.}
Consider an index $1< i< r$ (recall that $S''_1=S''_r=\emptyset$ from \Cref{obs: left and right down-edges}). We partition the edges of $\delta(S''_i)$ into three subsets: set $\delta^{\operatorname{down}}(S''_i)=E(S_i',S''_i)$; set $\delta^{\operatorname{left}}(S''_i)$ containing all edges $(u,v)$ with $u\in S''_i$ and $v\in V(S_1)\cup\cdots\cup V(S_{i-1})$; and set $\delta^{\operatorname{right}}(S''_i)$ containing the remaining edges (all edges $(u,v)$ with $u\in S''_i$ and $v\in V(S_{i+1})\cup\cdots\cup V(S_r)$) (see \Cref{fig: NF13}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF13.jpg}
\caption{Edge sets $\delta^{\operatorname{left}}(S''_i),\delta^{\operatorname{right}}(S''_i)$ and $\delta^{\operatorname{down}}(S''_i)$ (shown in black).}\label{fig: NF13}
\end{figure}
\newcommand{\leftedges}[1]{\delta^{\operatorname{left}}(S''_{#1})}
\newcommand{\rightedges}[1]{\delta^{\operatorname{right}}(S''_{#1})}
\newcommand{\downedges}[1]{\delta^{\operatorname{down}}(S''_{#1})}
We next show that for all $1<i<r$, $|\downedges{i}|\leq 0.1\min\set{|\rightedges{i}|,|\leftedges{i}|}$, and that the sizes of the edge sets $\rightedges{i},\leftedges{i}$ are close to each other, in the following two claims, whose proofs are deferred to \Cref{subsec: bound S' to S'' edges} and \Cref{subsec: bound left and right for S''}, respectively.
\begin{claim}\label{claim: bound S' to S'' edges}
For all $1<i<r$, $|\downedges{i}|\leq 0.1\cdot \min\set{|\rightedges{i}|,|\leftedges{i}|}$ holds.
Additionally, there is a set ${\mathcal{P}}^{\operatorname{left}}=\set{P^{\operatorname{left}}(e)\mid e\in \downedges{i}}$ of edge-disjoint paths in $\check H$, where, for each edge $ e\in \downedges{i}$, path $P^{\operatorname{left}}(e)$ has $e$ as its first edge, some edge of $\leftedges{i}$ as its last edge, and all inner vertices of $P^{\operatorname{left}}(e)$ are contained in $S''_i$. Similarly, there is a set ${\mathcal{P}}^{\operatorname{right}}=\set{P^{\operatorname{right}}(e)\mid e\in \downedges{i}}$ of edge-disjoint paths in $\check H$, where, for each edge $ e\in \downedges{i}$, path $P^{\operatorname{right}}(e)$ has $e$ as its first edge, some edge of $\rightedges{i}$ as its last edge, and all inner vertices of $P^{\operatorname{right}}(e)$ are contained in $S''_i$.
\end{claim}
\begin{claim}\label{claim: bound left and right for S''}
For all $1<i<r$, $|\rightedges{i}|\leq 1.1|\leftedges{i}|$, and similarly, $|\leftedges{i}|\leq 1.1|\rightedges{i}|$.
\end{claim}
\newcommand{\leftCedges}[1]{\delta^{\operatorname{left}}(S'_{#1})}
\newcommand{\rightCedges}[1]{\delta^{\operatorname{right}}(S'_{#1})}
Next, we consider edges incident to the clusters of $\set{S'_1,\ldots,S'_r}$.
\subsubsection*{Edges Incident to the Clusters of $\set{S'_1,\ldots,S'_r}$}
Consider some index $1\leq i\leq r$, and consider the edges incident to the cluster $S_i'$ in graph $\check H$ (see \Cref{fig: NF15}).
Recall that we have denoted by $E_{i-1}=E(S_{i-1}',S_i')$, and by $E_i=E(S_i,S_{i+1})$. We have also denoted by $\downedges{i}=E(S_i',S_i'')$. The remaining edges that are incident to $S_i'$ can be partitioned into two subsets: set $\leftCedges{i}$, containing all edges $(u,v)$, with $u\in S'_i$ and $v\in (S_1\cup\cdots\cup S_{i-2})\cup S_{i-1}''$; and set
set $\rightCedges{i}$, containing all edges $(u,v)$, with $u\in S'_i$ and $v\in S_{i+1}''\cup (S_{i+2}\cup\cdots\cup S_{r})$. Next, for all $1<i<r$, we bound the cardinality of edge set $\leftedges{i}$ in terms of the cardinality of $\leftCedges{i}$, and similarly we bound $|\rightedges{i}|$ in terms of $|\rightCedges{i}|$, in the following claim whose proof appears in \Cref{subsec: left edges for S' and S''}.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF15.jpg}
\caption{Edges in set $\delta^{\operatorname{left}}(S'_i)$ are shown in green, edges of $\delta^{\operatorname{right}}(S'_i)$ are shown in brown, and edges of $\delta^{\operatorname{down}}(S''_i)$ in black
}\label{fig: NF15}
\end{figure}
\begin{claim}\label{claim left edges for S' and S''}
For all $1<i<r$:
\begin{itemize}
\item $|\rightedges{i}|\leq 1.3|E_{i}|+1.3|\rightCedges{i}|$; and
\item $|\leftedges{i+1}|\leq 1.3|E_{i}|+1.3|\leftCedges{i+1}|$.
\end{itemize}
\end{claim}
\iffalse
\begin{proof}
Fix an index $1<i<r$.
As before, we consider the cut $(A,B)$ in graph $\check H$, where $A=V(S_1)\cup\cdots\cup V(S_{i-1})$, and $B=V(S_i)\cup\cdots\cup V(S_r)$. As before, $A$ and $B$ are precisely the sets of vertices of the two connected components of the graph $\tau\setminus\set{u_{i-1},u_i}$, where $\tau$ is the Gomory-Hu tree of graph $\check H$, and so $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut in $\check H$. We now consider another $u_{i-1}$-$u_i$ cut $(A',B')$ in $\check H$, where $B'=V(S'_i)$, and $A'=V(\check H)\setminus V(S'_i)$. Note that $|E(A,B)|\geq |\leftCedges{i}|+|\leftCedges{i+1}|+|E_{i-1}|$, while $|E(A',B')|=|E_{i-1}|+|\leftCedges{i}|+|\downedges{i}|+|E_{i}|+|\rightCedges{i}|$. From the fact that $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut, it must be the case that:
\[
|\leftCedges{i+1}|\leq |\downedges{i}|+|E_{i}|+|\rightCedges{i}|
\]
We now employ a symmetric argument: consider the cut $(X,Y)$ in graph $\check H$, where $X=V(S_1)\cup\cdots\cup V(S_{i+1})$, and $Y=V(S_{i+1})\cup\cdots\cup V(S_r)$. As before, $X$ and $Y$ are precisely the sets of vertices of the two connected components of the graph $\tau\setminus\set{u_{i+1},u_{i+2}}$, where $\tau$ is the Gomory-Hu tree of graph $\check H$, and so $(X,Y)$ is the minimum $u_{i+1}$-$u_{i+2}$ cut in $\check H$. We now consider another $u_{i+1}$-$u_{i+2}$ cut $(X',Y')$ in $\check H$, where $X'=V(S'_{i+1})$, and $Y'=V(\check H)\setminus V(S'_{i+1})$. Note that $|E(X,Y)|\geq |\rightCedges{i}|+|\rightCedges{i+1}|+|E_{i+2}|$, while $|E(X',Y')|=|E_{i}|+|\leftCedges{i+1}|+|\downedges{i+1}|+|E_{i+1}|+|\rightCedges{i+1}|$. From the fact that $(X,Y)$ is the minimum $u_{i+1}$-$u_{i+2}$ cut, it must be the case that:
\[
|\rightCedges{i}|+|\rightCedges{i+1}|+|E_{i+2}|\leq |E_{i}|+|\leftCedges{i+1}|+|\downedges{i+1}|+|E_{i+1}|+|\rightCedges{i+1}|.
\]
Since, from \Cref{claim: bound left and right for S''}, $|\rightedges{i}|\leq 1.1|\leftedges{i}|$, and, from \Cref{claim: bound S' to S'' edges}, $|\downedges{i}|\leq 0.1\min\set{|\rightedges{i}|,|\leftedges{i}|}$, we get that:
\[ |\rightedges{i}|\leq 1.1|\downedges{i}|+1.1|E_{i}|+1.1|\rightCedges{i}|\leq 0.11|\rightedges{i}|+1.1|E_{i}|+1.1|\rightCedges{i}|,\]
and so $|\rightedges{i}|\leq 1.3|E_{i}|+1.3|\rightCedges{i}|$.
The proof that $|\leftedges{i}|\leq 1.3|E_{i}|+1.3|\leftCedges{i}|$ is symmetric. The main difference is that we need to consider a $u_i$--$u_{i+1}$ cut $(A,B)$, where $A=V(S_1)\cup\cdots\cup V(S_i)$, and $B=V(S_{i+1})\cup \cdots V(S_r)$, and to compare it with the cut $(A',B')$, where $A'=V(S_i)$ and $B'=V(\check H)\setminus A'$.
\end{proof}
\fi
\subsection*{Accounting So Far}
We now summarize what we have shown so far. Fix some index $1\leq i<r$. Recall that set $\hat E_i$ of edges contains every edge $e=(u,v)\in E(\check H)$, such that, if $u\in S_j$ and $v\in S_{j'}$, then $j\leq i$ and $j'\geq i+1$ holds, but it excludes the edges in the set $E_i=E(S_i',S_{i+1}')$. Therefore, $\hat E_i$ is the union of the following subsets (see \Cref{fig: NF23}):
\begin{itemize}
\item edge set $\textbf{E}'_i^{\operatorname{over}}$, connecting vertices of $V(S_1),\ldots,V(S_{i-1})$ to vertices of $V(S_{i+2}),\ldots,V(S_r)$ (see \Cref{fig: NF9});
\item edges that lie in $\rightedges{i}\cup \leftedges{i+1}$ (see \Cref{fig: NF13});
\item edges that lie in $\rightCedges{i}\cup \leftCedges{i+1}$ (see \Cref{fig: NF15}).
\end{itemize}
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF23.jpg}
\caption{
The set $\hat E_i$ of edges, with the edges of $\tilde E^{\operatorname{over}}_i$ shown in red; the edges of $\delta^{\operatorname{right}}(S_i'')$ and $\delta^{\operatorname{left}}(S_{i+1}'')$ in pink and brown respectively; and the edges of $\delta^{\operatorname{right}}(S_i')$ and $\delta^{\operatorname{left}}(S_{i+1}')$ in green and blue, respectively.
Note that the edges of $E(S''_i, S''_{i+1})$ belong to both $\delta^{\operatorname{right}}(S_i'')$ and $\delta^{\operatorname{left}}(S_i'')$. Also, the edges of $E(S'_i, S''_{i+1})$ belong to both $\delta^{\operatorname{left}}(S_{i+1}'')$ and $\delta^{\operatorname{right}}(S_i')$. Similarly, the edges of $E(S''_i, S'_{i+1})$ belong to both $\delta^{\operatorname{right}}(S_i'')$ and $\delta^{\operatorname{left}}(S_{i+1}')$).
}\label{fig: NF23}
\end{figure}
In \Cref{obs: bad inded structure}, we have established that $|\textbf{E}'_i^{\operatorname{over}}|\leq |E_{i}'|$, where $E_i'=E(S_i,S_{i+1})$. Notice that all edges of $E_i'$ are contained in $E_i\cup \rightedges{i}\cup \leftedges{i+1}$ (see \Cref{fig: NF12}),
so we get that $|\textbf{E}'_i^{\operatorname{over}}|\leq |E_i|+|\rightedges{i}|+|\leftedges{i+1}|$.
From \Cref{claim left edges for S' and S''}, $|\rightedges{i}|\leq 1.3|E_{i}|+1.3|\rightCedges{i}|$, and $|\leftedges{i+1}|\leq 1.3|E_{i}|+1.3|\leftCedges{i+1}|$.
Therefore, altogether, we have shown so far that:
\begin{equation}\label{eq: bound on hat E}
\begin{split}
|\hat E_i|&\leq |\textbf{E}'_i^{\operatorname{over}}|+|\rightedges{i}|+|\leftedges{i+1}| +|\rightCedges{i}|+|\leftCedges{i+1}|\\
&\leq |E_i|+2|\rightedges{i}|+2|\leftedges{i+1}|+|\rightCedges{i}|+|\leftCedges{i+1}|\\
&\leq 7|E_i|+7|\rightCedges{i}|+7|\leftCedges{i+1}|.
\end{split} \end{equation}
Therefore, it now remains to bound $|\rightCedges{i}|$ and $|\leftCedges{i+1}|$ in terms of $|E_i|$. We start with the following claim that allows us to establish some useful connection between the cardinalities of the three edge sets.
The proof appears in \Cref{subsec: left edges for S' only}
\begin{claim}\label{claim left edges for S' only}
For all $1\leq i<r$: $|\rightCedges{i}|\leq 2.5|E_{i}|+2.5|\leftCedges{i+1}|$, and $|\leftCedges{i+1}|\leq 2.5|E_{i}|+2.5|\rightCedges{i}|$.
\end{claim}
\iffalse
\begin{proof}
Fix an index $1\leq i<r$.
As before, we consider the cut $(A,B)$ in graph $\check H$, where $A=V(S_1)\cup\cdots\cup V(S_{i-1})$, and $B=V(S_i)\cup\cdots\cup V(S_r)$. As before, $A$ and $B$ are precisely the sets of vertices of the two connected components of the graph $\tau\setminus\set{u_{i-1},u_i}$, where $\tau$ is the Gomory-Hu tree of graph $\check H$, and so $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut in $\check H$. We now consider another $u_{i-1}$-$u_i$ cut $(A',B')$ in $\check H$, where $B'=V(S'_i)$, and $A'=V(\check H)\setminus V(S'_i)$. Note that, from our definition, all edges of $\leftCedges{i+1}$ contribute to the cut $(A,B)$, in addition to the edges of $\leftCedges{i}$ and $E+{i-1}$. Therefore,
$|E(A,B)|\geq |\leftCedges{i}|+|\leftCedges{i+1}|+|E_{i-1}|$, while $|E(A',B')|=|E_{i-1}|+|\leftCedges{i}|+|\downedges{i}|+|E_{i}|+|\rightCedges{i}|$. From the fact that $(A,B)$ is the minimum $u_{i-1}$-$u_i$ cut, it must be the case that:
\[
\leftCedges{i+1}|\leq |\downedges{i}|+|E_{i}|+|\rightCedges{i}|.
\]
Since, from \Cref{claim: bound left and right for S''}, $|\rightedges{i}|\leq 1.1|\leftedges{i}|$, and, from \Cref{claim: bound S' to S'' edges}, $|\downedges{i}|\leq 0.1\min\set{|\rightedges{i}|,|\leftedges{i}|}$, we get that:
\[ |\rightedges{i}|\leq 1.1|\downedges{i}|+1.1|E_{i}|+1.1|\rightCedges{i}|\leq 0.11|\rightedges{i}|+1.1|E_{i}|+1.1|\rightCedges{i}|,\]
and so $|\rightedges{i}|\leq 1.3|E_{i}|+1.3|\rightCedges{i}|$.
The proof that $|\leftedges{i}|\leq 1.3|E_{i}|+1.3|\leftCedges{i}|$ is symmetric. The main difference is that we need to consider a $u_i$--$u_{i+1}$ cut $(A,B)$, where $A=V(S_1)\cup\cdots\cup V(S_i)$, and $B=V(S_{i+1})\cup \cdots V(S_r)$, and to compare it with the cut $(A',B')$, where $A'=V(S_i)$ and $B'=V(\check H)\setminus A'$.
\end{proof}
\fi
\iffalse
====================
We start by establishing some useful relationships between edges connecting different clusters in $\set{S_1,\ldots,S_r}$. Fix an index $1\leq i\leq r$. We denote by $E'_i=E(S_i,S_{i+1})$, and by $\tilde E^{\operatorname{over}}_i$ the set of all edges $e=(u,v)$, such that, if $u\in S_{j},v\in S_{j'}$, then $j<i$ and $j'>i+1$ holds. We also denote by $\tilde E_i^{\operatorname{left}}$ the set of all edges $e=(u,v)$ with $u\in S_i$, such that, if $v\in S_j$, then $j<i-1$. Similarly, we denote by $\tilde E_i^{\operatorname{right}}$ the set of all edges $e=(u,v)$ with $u\in S_i$, such that, if $v\in S_j$, then $j>i+1$ holds. Notice that $\delta(S_i)=E'_{i-1}\cup E'_i\cup \tilde E_i^{\operatorname{left}}\cup \tilde E_i^{\operatorname{right}}\cup \tilde E_i^{\operatorname{over}}$. Notice that, by the definition, if $i\in \set{1,2}$, then $E_i^{\operatorname{left}}=\emptyset$; if $i\in \set{1,2,r-1,r}$ then $\tilde E_i^{\operatorname{over}}=\emptyset$, and, if $i\in \set{r-1,r}$, then $\textbf{E}'_i^{\operatorname{right}}=\emptyset$.
We prove the following observation that helps us relate the sizes of all these edge sets.
\begin{observation}\label{obs: bad inded structure}
For all $1< i<r$, the following hold:
\begin{itemize}
\item $|\textbf{E}'_i^{\operatorname{over}}|\leq |E_{i}'|$.
\item $|\textbf{E}'_{i}^{\operatorname{left}}|\leq
|E_{i}'|+|\textbf{E}'_{i+1}^{\operatorname{right}}|$
\item $|\textbf{E}'_{i}^{\operatorname{right}}|\leq
|E_{i}'|+|\textbf{E}'_{i+1}^{\operatorname{left}}|$.
\end{itemize}
\end{observation}
\fi
Next, we show that for all $1<i<r$, vertex $u_i$ must be a $J$-node.
\subsubsection*{Proving that $u_2,\ldots,u_{r-1}$ are $J$-nodes.}
We start with the following simple claim, whose proof appears in \Cref{subsec: non-J-node boundary size}.
\begin{claim}\label{claim: non-J-node-boundary size}
Consider an index $1<i<r$, and assume that vertex $u_i$ is not a $J$-node. Then $|\bigcup_{v\in S'_i}\delta(v)|\leq \textsf{left}(1+\frac{130}{\log m}\textsf{right} )|\delta(u_i)|$. Moreover, if $u_i\in L'_j$, for some $1\leq j\leq h$, then every vertex of $S'_i\setminus\set{u_i}$ lies in $L'_{j+1}\cup\cdots\cup L'_h$.
\end{claim}
We are now ready to prove that, for all $1<i<r$, vertex $u_i$ must be a $J$-node.
\begin{lemma}\label{lem: each ui is a J-node}
For all $1<j<r$, vertex $u_i$ is a $J$-node.
\end{lemma}
\begin{proof}
Assume for contradiction that the lemma is false. We fix an index $1<i^*<r$, such that $u_{i*}$ is not a $J$-node, and subject to this, $|\delta(u_{i^*})|$ is maximized, breaking ties arbitrarily.
We first assume that there is some index $a$, such that at least $|\delta(u_{i^*})|/16$ edges connect $u_{i^*}$ to edges of $S''_a$. We show that in this case, $|\leftedges{a}|,|\rightedges{a}|$ are both large, and $u_a$ must be a $J$-node. (Note that it is impossible that $a\in \set{1,r}$, since $S''_1=S''_r=\emptyset$, as we have established in \Cref{obs: left and right down-edges}.)
The proof of the following claim is deferred to \Cref{subsec:many edges left right large}.
\begin{claim}\label{claim: many edges left right large}
Suppose there is an index $1\leq a\leq r$ (where possibly $a=i^*$), such that at least $|\delta(u_{i^*})|/16$ edges connect $u_{i^*}$ to vertices of $S''_a$. Then, $|\leftedges{a}|,|\rightedges{a}|\geq |\delta(u_{i^*})|\cdot\frac {\log m}{256}$, and moreover, $u_a$ is a $J$-node.
\end{claim}
Consider again the vertex $u_{i^*}$, and assume that $u_{i^*}\in L'_j$, for some $1\leq j\leq h$. Recall that, from \Cref{claim: non-J-node-boundary size}, every vertex of $V(S'_{i^*})\setminus\set{u_{i^*}}$ lies in $L'_{j+1}\cup\cdots\cup L'_h$.
Therefore, all edges connecting $u_{i^*}$ to vertices of $V(S'_{i^*})\setminus\set{u_{i^*}}$
lie in $\delta^{\operatorname{up}}(u_{i^*})$, and their number is bounded by $|\delta^{\operatorname{up}}(u_{i^*})|\leq |\delta(u_{i^*})|/\log m$. From \Cref{claim: many edges left right large}, the number of edges connecting $u_{i^*}$ to vertices of $S''_{i^*}$ must be bounded by $|\delta(u_{i^*})|/16$ (as $u_{i^*}$ is not a $J$-node). The remaining edges of $\delta(u_{i^*})$ must connect $u_{i^*}$ to vertices of $\bigcup_{a\neq i^*}V(S_a)$. Denote by $E^*$ the set of all edges connecting $u_{i^*}$ to vertices of $\bigcup_{a>i^*}V(S_a)$, and denote by $E^{**}$ the set of all edges connecting $u_{i^*}$ to vertices of $\bigcup_{a<i^*}V(S_a)$. From the above discusison, $|E^*\cup E^{**}|\geq 7|\delta(u_{i^*})|/8$, and so either $|E^*|\geq |\delta(u_{i^*})|/4$ or $|E^{**}|\geq |\delta(u_{i^*})|/4$ must hold. We assume w.l.o.g. that it is the former. Next we consider three cases. The first case is when neither $u_{i^*+1}$ or $u_{i^*+2}$ are $J$-nodes; the second case is when $u_{i^*+1}$ is a $J$-node; and the third case is when $u_{i^*+2}$ is a $J$-node. We show that neither of these cases is possible, by showing a simplifying cluster that should have been considered by our algorithm. For simplicity of notation, in the remainder of the proof, we denote $i^*$ by $i$.
\paragraph{Case 1: neither of $u_{i+1}$, $u_{i+2}$ is a $J$-node.}
Consider the set $E^*$ of edges; recall that these are all edges connecting $u_{i}$ to vertices of $\bigcup_{a>i}S_a$. We need the following observation:
\begin{observation}\label{obs: many through edges}
At least $ |\delta(u_{i})|/16$ edges connect $u_{i}$ to vertices of $\bigcup_{a>i+2}V(S_a)$.
\end{observation}
(Note that in particular it follows from the observation that $r\geq i+3$ must hold).
\begin{proof}
We partition the edges of $E^*$ into five subsets. The first subset, $E^*_1$, contains all edges of $E^*$ connecting $u_{i}$ to vertices of $S''_{i+1}$, and the second subset, $E^*_2$, contains all edges of $E^*$ connecting $u_{i}$ to vertices of $S''_{i+2}$. Notice that, from \Cref{claim: many edges left right large}, since we have assumed that neither of $u_{i+1}$, $u_{i+2}$ is a $J$-node, $|E^*_1|,|E^*_2|\leq |\delta(u_{i})|/16$. We let $E^*_3$ be the set of all edges of $E^*$ connecting $u_{i}$ to vertices of $\bigcup_{a>i+2}V(S_a)$. Lastly, we let $E^*_4$ and $E^*_5$ be the sets of all edges of $E^*$ connecting $u_{i}$ to vertices of $S'_{i+1}$ and of $S'_{i+2}$, respectively. Assume for contradiction that $|E^*_3|<|\delta(u_{i})|/16$. Then, since $|E^*|\geq |\delta(u_{i})|/4$, either $|E^*_4|\geq |\delta(u_{i})|/32$ or $|E^*_5|\geq |\delta(u_{i})|/32$ must hold. We assume first that $|E^*_4|\geq |\delta(u_{i})|/32$. Since $|\delta^{\operatorname{up}}(u_{i})\leq |\delta(u_{i})|/\log m$, $|E^*_4\cap \delta^{\operatorname{down}}(u_{i})|\geq |\delta(u_{i})|/64$. Notice that, if $e=(u_{i},v)$ is an edge of $E^*_4\cap \delta^{\operatorname{down}}(u_{i})$, then $v\in S'_{i+1}$, and $e\in \delta^{\operatorname{up}}(v)$. Since, for every vertex $v$, $|\delta^{\operatorname{up}}(v)|\leq |\delta(v)|/\log m$, we get that:
\[|\bigcup_{v\in S'_{i+1}}\delta(v)|\geq |E^*_4\cap \delta^{\operatorname{down}}(u_{i})|\cdot \log m\geq \frac{|\delta(u_{i})|\cdot \log m}{32}.\]
On the other hand, from \Cref{claim: non-J-node-boundary size}, $|\bigcup_{v\in S'_{i+1}}\delta(v)|\leq \textsf{left}(1+\frac{130}{\log m}\textsf{right} )|\delta(u_{i+1})|<2|\delta(u_{i+1})|$. Therefore, we get that $|\delta(u_{i+1})|> \frac{|\delta(u_{i})|\cdot \log m}{64}>|\delta(u_{i})|$, contradicting the choice of index $i$.
In the case where $|E^*_5|\geq |\delta(u_{i})|/32$, the analysis is identical.
\end{proof}
Consider a cluster $S^*$, which is a subgraph of $\check H$ induced by $V(S_{i+1})\cup V(S_{i+2})$. In the following claim, whose proof is deferred to \Cref{subsec: simplifying cluster Case 1}, we prove that $S^*$ is a simplifying cluster, reaching a contradiction.
\begin{claim}\label{claim: simplifying cluster case 1}
Cluster $S^*$ is a simplifying cluster.
\end{claim}
\paragraph{Case 2: $u_{i+1}$ is a $J$-node.}
Recall that in this case, from \Cref{obs: central path j-cluster}, $S'_{i+1}=\set{u_{i+1}}$.
We show that cluster $S^*=\set{u_{i+1}}$ is a simplifying cluster in the following claim, whose proof is similar to but slightly more involved than the proof of \Cref{claim: simplifying cluster case 1}, and is deferred to \Cref{subsec: simplifying cluster Case 2}.
\begin{claim}\label{claim: simplifying cluster case 2}
Cluster $S^*$ is a simplifying cluster.
\end{claim}
This is a constradiction, since our algorithm must have identified that $S^*=S'_{i+1}$ is a simplifying cluster.
\paragraph{Case 3: Neither Case 1 nor Case 2 happened.}
Since Cases 1 and 2 did not happen, vertex $u_{i+1}$ is not a $J$-node.
We start with the following simple observation.
\begin{observation}\label{obs: few edges going right}
The number of edges connecting $u_i$ to vertices of $S_{i+1}$ is at most $|\delta(u_i)|/8$.
\end{observation}
\begin{proof}
Assume otherwise.
Since Case 3 happened, $u_{i+1}$ is not a $J$-node, and so, from \Cref{claim: many edges left right large}, at most $|\delta(u_i)|/16$ edges may connect $u_i$ to vertices of $S''_{i+1}$. Therefore, the number of edges connecting $u_i$ to $S'_{i+1}$ must be at least $|\delta(u_i)|/16$.
But then at least $|\delta(u_i)|/32$ edges of $\delta^{\operatorname{down}}(u_i)$ connect $u_i$ to vertices of $S'_{i+1}$. For each such vertex $e'=(u,v)$, $e'\in \delta^{\operatorname{up}}(v)$ must hold. Since, for every vertex $v$, $\delta^{\operatorname{up}}(v)\leq |\delta(v)|/\log m$, we get that $|\bigcup_{v\in S'_{i+1}} \delta(v)|\geq \frac{|\delta(u_i)|\log m}{16}$ must hold. However, from \Cref{claim: non-J-node-boundary size}, $|\delta(u_{i+1})|\geq \frac{|\bigcup_{v\in S'_{i+1}}\delta(v)|}{2}\geq \frac{|\delta(u_i)|\log m}{32}>|\delta(u_i)|$, contradicting the choice of the index $i^*=i$.
\end{proof}
Recall that we have assumed that $|E^*|\geq |\delta(u_i)|/4$, where $E^*$ is the set of all edges connecting $u_{i}$ to vertices of $\bigcup_{a>i}V(S_a)$. Since, from \Cref{obs: few edges going right}, at most $|\delta(u_i)|/8$ edges connect $u_i$ to vertices of $S_{i+1}$, it must be the case that $i+2\leq r$, and at least $|\delta(u_i)|/8$ edges connect $u_i$ to vertices of $\bigcup_{a>i+1}V(S_a)$. Since we have assumed that Case 1 did not happen, vertex $u_{i+2}$ must be a $J$-node, and so, from \Cref{obs: central path j-cluster}, $S'_{i+2}=\set{u_{i+2}}$.
We let $S^*=\set{u_{i+2}}$, and we show, in the next claim, that cluster $S^*$ is a simplifying cluster. The proof of the claim is deferred to
\Cref{subsec: simplifying cluster Case 3}.
\begin{claim}\label{claim: simplifying cluster case 3}
Cluster $S^*$ is a simplifying cluster.
\end{claim}
This is a constradiction, since our algorithm must have identified that $S^*=S'_{i+2}$ is a simplifying cluster.
\end{proof}
In order to complete the proof of \Cref{lem: no bad indices}, it is enough to prove that for all $1\leq i< r$, $|\hat E_i|\leq 1000|E_i|$.
Assume for contradiction that there is some index $1\leq i< r$, for which $|\hat E_i|> 1000|E_i|$ holds.
Recall that we have shown already, in \Cref{eq: bound on hat E}, that:
\[
|\hat E_i|\leq 7|E_i|+7|\rightCedges{i}|+7|\leftCedges{i+1}|.
\]
If $|\hat E_i|> 1000|E_i|$, then either $|\rightCedges{i}|>64|E_i|$, or $|\leftCedges{i+1}|>64|E_i|$. We assume without loss of generality that it is the former; the other case is symmetric. Recall that, since $u_{i}$ is a $J$-node, $S'_{i}=\set{u_{i}}$.
From the definition, set $\rightCedges{i}$ contains all edges $(u_i,v)$ with $v\in S_{i+1}''\cup (S_{i+2}\cup\cdots\cup S_{r})$.
Note that, from \Cref{obs: left and right down-edges},
$S'_r=S_r$, and so $\rightCedges{r-1}=\emptyset$. Therefore, we can assume that $i<r-1$.
We now prove that $S^*=S'_{i+1}=\set{u_{i+1}}$ is a simplifying cluster, in the following claim, whose proof is very similar to the analysis of Case 2 in the proof of \Cref{lem: each ui is a J-node}, and is deferred to \Cref{subsec: simplifying cluster last}.
\begin{claim}\label{claim: simplifying cluster last case}
Cluster $S^*$ is a simplifying cluster.
\end{claim}
We reach a contradiction, since our algorithm must have established that cluster $S^*=S'_{i+1}$ is a simplifying cluster.
\renewcommand{\textbf{E}'}{\textbf{E}'}
\subsection{Constructing the Monotone Paths -- proof of \Cref{lem: prefix and suffix path}}
\label{subsubsec: monotone paths}
For all $1\leq j\leq h$, we define two sets of edges, $E^{\operatorname{left}}_j$ and $E^{\operatorname{right}}_j$, as follows. Start with $E^{\operatorname{left}}_j=E^{\operatorname{right}}_j=\emptyset$. For all $1\leq i\leq r$, for every vertex $v\in U_{i,j}\setminus \set{S'_i}$, we add all edges of $\delta^{\operatorname{down},\operatorname{left}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$ to $E^{\operatorname{left}}_j$, and similarly, we add all edges of $\delta^{\operatorname{down},\operatorname{right}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$ to $E^{\operatorname{right}}_j$. We prove the following claim.
\begin{claim}\label{claim: level by level routing}
There is an efficient algorithm to construct, for all $1\leq j\leq h$, a set ${\mathcal{P}}^{\operatorname{left}}_j=\set{P^{\operatorname{left}}(e)\mid e\in E^{\operatorname{left}}_j}$ of edge-disjoint left-monotone paths, and a set ${\mathcal{P}}^{\operatorname{right}}_j=\set{P^{\operatorname{right}}(e)\mid e\in E^{\operatorname{right}}_j}$ of edge-disjoint right-monotone paths, such that, for every edge $e\in E^{\operatorname{left}}_j$, path $P^{\operatorname{left}}(e)$ starts with edge $e$, and similarly, for every edge $e'\in E^{\operatorname{right}}_j$, path $P^{\operatorname{right}}(e')$ starts with edge $e'$.
\end{claim}
\begin{proof}
The proof is by induction on $j$. The base is when $j=1$. Consider some vertex $v\in U_{i,1}\setminus \set{S_i'}$, for some $1\leq i\leq r$. Note that every edge in $\delta^{\operatorname{down}}(v)$ must connect $v$ to a vertex of $L_0'$, and every vertex of $L_0'$ lies in $\set{u_1,\ldots,u_r}$. Consider now some edge $e=(v,u)$ that is incident to $v$, that lies in $E^{\operatorname{left}}_1$. Note that $e\not \in \delta^{\operatorname{down},\operatorname{straight}'}(v)$, as all edges connecting $u$ to $u_i$ lie in $\delta^{\operatorname{down},\operatorname{straight}''}(v)$, and no edge of $\delta^{\operatorname{down}}(v)$ may connect $v$ to a vertex outside $\set{u_1,\ldots,u_r}$. Therefore, $e\in \delta^{\operatorname{left}}(v)$ must hold. We then let $P^{\operatorname{left}}(e)=(e)$. It is immediate to verify that it is a left-monotone path. We define paths $P^{\operatorname{right}}(e)$ for every edge $e\in \delta_{\check H(v)}\cap E^{\operatorname{right}}_1$ similarly.
We assume now that the claim holds for some index $1\leq j<h$, and we prove it for index $j+1$.
Consider some vertex $v\in U_{i,j+1}\setminus \set{S_i'}$, for some $1\leq i\leq r$, and let $e=(v,u)$ be any edge that is incident to $v$ and lies in $E^{\operatorname{left}}_j$. In this case, $e\in \delta^{\operatorname{down},\operatorname{left}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$ must hold. If we denote by $1\leq i'\leq r$, $1\leq j'\leq h$ the indices for which $u\in U_{i',j'}$, then $i'\leq i$ and $j'\leq j$ must hold.
Assume first that $u\in S'_1\cup\cdots\cup S'_{r}$. In this case, since $e\not\in \delta^{\operatorname{down},\operatorname{straight}''}(v)$, $e\in \delta^{\operatorname{down},\operatorname{left}}(v)$ must hold, and $i'<i$. In this case, we let $P^{\operatorname{left}}(e)=(e)$. It is easy to see that this path is a valid left-monotone path. Otherwise, $u\not \in S'_1\cup\cdots\cup S'_{r}$, and $e\in \delta^{\operatorname{up}}(u)$. We then consider the edge $e'\in \delta^{\operatorname{down},\operatorname{left}}(u)\cup \delta^{\operatorname{down},\operatorname{straight}'}(u)$ to which edge $e$ is mapped by $f^{\operatorname{left}}(u)$. We then let $P^{\operatorname{left}}(e)$ be the path obtained by concatenating the edge $e$ and the path $P^{\operatorname{left}}(e')\in {\mathcal{P}}^{\operatorname{left}}_{j'}$.
Since path $P^{\operatorname{left}}(e')$ is left-monotone, and since $j'\leq j$ and $i\leq i$, path $P^{\operatorname{left}}(e)$ is also left-monotone.
We note that edge $e$, by the definition, may not lie on any path in ${\mathcal{P}}^{\operatorname{left}}_1\cup\cdots\cup {\mathcal{P}}^{\operatorname{left}}_j$. Moreover, edge $e$ is the only edge that is mapped to edge $e'$ by map $f^{\operatorname{left}}(u)$. This ensures that all paths in the resulting set ${\mathcal{P}}^{\operatorname{left}}_j=\set{P^{\operatorname{left}}(e'')\mid e''\in E^{\operatorname{left}}_j}$ are edge-disjoint.
The construction of the set ${\mathcal{P}}^{\operatorname{right}}_{j+1}$ of paths is symmetric.
\end{proof}
We are now ready to complete the proof of \Cref{lem: prefix and suffix path}. Consider an edge $e=(u,v)\in \hat E$, with $u\in S_i$, $v\in S_{i'}$, and $i<i'$.
We describe the construction of path $P(e,u)$; the construction of path $P(e,v)$ is symmetric. If $u\in S'_i$, then path $P(e,u)$ consists only of the vertex $u$. It is a left-monotone path by definition. Therefore, we assume now that $u\not\in S'_i$.
We assume that $u$ lies in layer $L'_j$, for some $1\leq j \leq h$, and $v$ lies in layer $L'_{j'}$, for some $0\leq j'\geq h$ (vertex $u$ may not lie in $L'_0$, since we have assumed that $u\not\in S'_i$).
We now consider two cases. The first case is when $j\leq j'$. In this case, $e\in \delta^{\operatorname{up}}(u)$. We let $e'\in \delta^{\operatorname{down},\operatorname{left}}(u)\cup \delta^{\operatorname{down},\operatorname{straight}'}(u)$ be the edge to which $e$ is mapped by $f^{\operatorname{left}}(u)$. Since $e'\in E^{\operatorname{left}}_j$, there is a left-monotone path $P^{\operatorname{left}}(e')\in {\mathcal{P}}^{\operatorname{left}}_j$. We then let $P(e,u)$ be the path obtained by concatenating edge $e$ with path $P^{\operatorname{left}}(e')$. It is easy to verify that path $P(e,u)$ is left-monotone.
The second case is when $j>j'$. In this case, $e\in \delta^{\operatorname{right}}(u)$. Recall that, from \Cref{obs: left and right down-edges},
$|\delta^{\operatorname{down},\operatorname{right}}(u)|\leq 2(|\delta^{\operatorname{down},\operatorname{left}}(u)|+|\delta^{\operatorname{down},\operatorname{straight}'}(u)|)$. Therefore, we can define another mapping $g^{\operatorname{left}}(u)$, that maps the edges of $\delta^{\operatorname{down},\operatorname{right}}(u)$ to edges of $\delta^{\operatorname{down},\operatorname{left}}(u)\cup\delta^{\operatorname{down},\operatorname{straight}'}(u)$, such that at most two edges are mapped to every edge of $\delta^{\operatorname{down},\operatorname{left}}(u)\cup\delta^{\operatorname{down},\operatorname{straight}'}(u)$. We then let $e'$ be the edge to which $e$ is mapped by $g^{\operatorname{left}}(u)$. As before, $e'\in E^{\operatorname{left}}_j$ must hold, and so there is a left-monotone path $P^{\operatorname{left}}(e')\in {\mathcal{P}}^{\operatorname{left}}_j$. We then let $P(e,u)$ be the path obtained by concatenating edge $e$ with path $P^{\operatorname{left}}(e')$ as before.
This finishes the definition of the path $P(e,u)$. Path $P(e,v)$ is defined symmetrically. Since the paths in $\textsf{left} (\bigcup_{j=1}^h{\mathcal{P}}_j^{\operatorname{right}}\textsf{right} )\cup \textsf{left}(\bigcup_{j=1}^h{\mathcal{P}}_j^{\operatorname{right}}\textsf{right} )$ cause congestion $O(\log m)$, it is easy to verify that the set $\set{P(e,v),P(e,u)\mid e=(u,v)\in \hat E}$ of paths causes congestion $O(\log m)$.
\subsubsection*{Step 1: Layering the Vertices of $U^*\cup U^R$ and Clustering ${\mathcal{J}}$}
\iffalse
Assume first that graph $\hat H'$ contains a pair of vertices $u,u'\in U^*\cup U^R$, such that the number of edges connecting $u$ to $u'$ is at least $\max\set{|\delta_{\hat H'}(u')|,|\delta_{\hat H'}(u')|}/(32\log m)$. Let $S$ be the subgraph of $\hat H'$, induced by the pair $u,u'$ of vertices. It is immediate to verify that $S$ has the $1/(32\log m)\geq \alpha'$-bandwidth property in $\hat H'$, as all edges of $\delta_{\hat H'}(S)$ can be routed to vertex $u$ inside $S$, with edge-congestion at most $32\log m$. Since each of vertices $u$ and $u'$ is either a regular vertex or an $R$-vertex, $S$ is also a cluster of graph $\hat H$, and $S$ has the $\alpha'$-bandwidth property in $\hat H$ as well. Clearly, $|E_{\hat H}(S)|\geq (|\delta_{\hat H'}(u')|+|\delta_{\hat H'}(u')|)/(64\log m)\geq |\delta_{\hat H}(S)|/(64\log m)$. Since $S$ is disjoint from all clusters in ${\mathcal{W}}$, we get that $\tilde {\mathcal{W}}={\mathcal{W}}\cup \set{S}$ is a valid $W$-clustering in graph $\hat H$, and $|E(\hat H_{|\tilde {\mathcal{W}}})|<|E(\hat H_{{\mathcal{W}}})|$. We then return the new $W$-clustering $\tilde {\mathcal{W}}$, and terminate the algorithm. Therefore, we assume from now on that, for every pair of vertices $u,u'\in U^*\cup U^R$, the number of edges connecting $u$ to $u'$ in $\hat H'$ is less than $\max\set{|\delta_{\hat H'}(u')|,|\delta_{\hat H'}(u')|}/(32\log m)$.
\fi
Consider the graph $\hat H'$, and let $C_0$ be the subgraph of $\hat H'$, induced by vertex set $V(\hat H')\setminus (U^*\cup U^R)$. We use the algorithm from \Cref{thm: layered well linked decomposition}, to compute a layered $\alpha'$-well-linked decomposition $({\mathcal{S}},({\mathcal{L}}_1,\ldots,{\mathcal{L}}_h))$ of $\hat H'$ with respect to $C_0$, where $\alpha'=\Theta(1/\log^{2.5}m)$ is the parameter that was used in the definition of a valid set of $W$-clusters, such that $h\leq \log m$. We say that a cluster $S\in {\mathcal{S}}$ is a \emph{singleton cluster}, if it contains a single vertex of $\hat H'$. Assume first that ${\mathcal{S}}$ contains a non-singleton cluster $S$. Recall that $S$ has the $\alpha'$-bandwidth property in $\hat H'$ (from Property \ref{condition: layered well linked} of layered well-linked decomposition; see \Cref{subsec: layered wld}), and it only contains verices of $U^*\cup U^R$. Therefore, $S$ is also a cluster of graph $\hat H$, and it has the $\alpha'$-bandwidth property in $\hat H$.
Moreover, from Property \ref{condition: layered decomp each cluster prop} of the layered well-linked decomposition,
$|E_{\hat H}(S)|=|E_{\hat H'}(S)|\geq |\delta_{\hat H'}(S)|/(64\log m)=|\delta_{\hat H}(S)|/(64\log m)$ holds.
Since $S$ is disjoint from clusters in set ${\mathcal{W}}$, we get that $\tilde {\mathcal{W}}={\mathcal{W}}\cup \set{S}$ is a valid set of $W$-clusters in graph $\hat H$, with $|E(\hat H_{|\tilde{\mathcal{W}}})|<|E(\hat H_{|{\mathcal{W}}})|$. We terminate the algorithm and return the new valid set $\tilde {\mathcal{W}}$ of $W$-clusters.
Therfore, we will assume from now on that every cluster in set ${\mathcal{S}}$ is a singleton cluster. The partition $({\mathcal{L}}_1,\ldots,{\mathcal{L}}_h)$ of the clusters of ${\mathcal{S}}$ into layers then immediately defines a partition $L_1,\ldots,L_h$ of vertices of $U^*\cup U^R$ into layers, where vertex $u$ lies in layer $U_i$ iff cluster $\set{u}\in {\mathcal{S}}$ lies in ${\mathcal{L}}_i$. For convenience, we denote by $L_0=V(\hat H')\setminus (U^*\cup U^R)$. For every vertex $u\in U^*\cup U^R$ that lies in some layer $L_i$, for $1\leq i\leq h$, we partition the edges of $\delta_{\hat H'}(u)$ into two subsets: set $\delta^{\operatorname{down}}(u)$ contains all edges $(u,u')$ with $u'\in L_0\cup L_1\cup \cdots\cup L_{i-1}$, and set $\delta^{\operatorname{up}}(u)$ contains all remaining edges of $\delta_{\hat H'}(u)$. Note that, from Property \ref{condition: layered decomp edge ratio} of the layered well-linked decomposition, for every vertex $u\in U^*\cup U^R$, $|\delta^{\operatorname{up}}(u)|<|\delta^{\operatorname{down}}(u)|/\log m$.
In the remainder of the proof of \Cref{lemma: better clustering}, we will attempt to construct a nice witness structure for graph $G_{|{\mathcal{R}}}$, with respect to the set ${\mathcal{C}}'$ of clusters, containing every cluster $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$. If we fail to do so, then we will compute another helpful clustering $\tilde {\mathcal{R}}$ of graph $G$ with respect to vertex $v^*$ and set ${\mathcal{C}}$ of basic clusters, such that $R^*\in {\mathcal{R}}$ and $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$.
In order to do so, we construct a collection ${\mathcal{J}}$ of clusters in graph $\hat H'$. Every cluster $J\in {\mathcal{J}}$ will contain exactly one vertex that is either a $C$-node or $W$-node, that we refer to as the \emph{center of the cluster}, and possibly a number of additional vertices from $U^*\cup U^R$.
Initially, for every vertex $u$ of $\hat H'$ that is either a $C$-node or a $W$-node, we construct a cluster $J(u)\in {\mathcal{J}}$, that only contains the vertex $u$ as its center node. We then iterate. As long as there exists a vertex $u'\in U^*\cup U^R$, such that at least $|\delta_{\hat H'}(u')|/128$ edges connect $u'$ to the vertices of some cluster $J\in {\mathcal{J}}$, we add vertex $u'$, together with all edges connecting $u'$ to $V(J)$, to cluster $J$. We also delete $u'$ from vertex set $U^*$ or $U^R$ in which it lies. If $u'\in L_i$, for some $1\leq i\leq h$, then we delete $u'$ from $L_i$ and add it to $L_0$.
Consider the set ${\mathcal{J}}$ of clusters in graph $\hat H'$, that is obtained at the end of this procedure. It is immediate to verify that all clusters in ${\mathcal{J}}$ are mutually disjoint; every cluster $J\in {\mathcal{J}}$ contains a single center node that is a $C$-node or a $W$-node, and each remaining vertex of $J$ lies in $U^R\cup U^*$. We need the following observation, whose proof is deferred to Section \ref{subsec:J-clusters well-linked} of Appendix.
\begin{observation}\label{obs:J wl}
Every cluster $J\in {\mathcal{J}}$, has the $\Omega(1/\log m)$-bandwidth property in graph $\hat H'$.
\end{observation}
\subsubsection*{Step 2: New Contracted Graph and Simplifying Clusters}
We start by revisiting the current hierarchical (4-level) clustering of $G$ and defining a new contracted graph. Recall that our starting point is a graph $G$, with a special vertex $v^*$, and a collection ${\mathcal{C}}$ of disjoint basic clusters in $G$, such that $v^*$ does not lie in any cluster of ${\mathcal{C}}$. Recall that every cluster in ${\mathcal{C}}$ has the $\alpha_0$-bandwidth property, where $\alpha_0=1/\log^3m$. This is the first-level clustering.
The second level of clustering is the helpful clustering ${\mathcal{R}}$, which is also a collection of disjoint clusters, each of which has the $\alpha_1$-bandwidth property, where $\alpha_1=1/\log^6m$. Recall that $v^*$ may not lie in any cluster of ${\mathcal{R}}$, and, for every cluster $C\in {\mathcal{C}}$, either $C\subseteq G\setminus\textsf{left}(\bigcup_{R\in {\mathcal{R}}}R\textsf{right} )$; or there is some cluster $R\in {\mathcal{R}}$ with $C\subseteq R$. Recall that we have denoted by ${\mathcal{C}}'\subseteq {\mathcal{C}}$ the set of all clusters $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left}(\bigcup_{R\in {\mathcal{R}}}R\textsf{right} )$. Recall also that we have defined a distinguished cluster $R^*\in {\mathcal{R}}$.
The third level of clustering is a $W$-clustering ${\mathcal{W}}$, that is defined with respect to the contracted graph $\hat H=G_{|{\mathcal{C}}'\cup {\mathcal{R}}}$. Recall that for every cluster $W\in {\mathcal{W}}$, every vertex of $W$ is either a regular vertex or an $R$-node, and $W$ may not contain the special vertex $v^*$ or the $R$-node $u^*$ representing the distinguished cluster $R^*$. We have defined, for every cluster $W\in {\mathcal{W}}$, the corresponding cluster $W'\subseteq G$, that is, intuitively, obtained from $W$ by un-contracting every cluster $R\in {\mathcal{R}}$ with $v_{R}\in V(W)$. We have then set ${\mathcal{W}}'=\set{W'\mid W\in {\mathcal{W}}}$, and we have established that every cluster $W'\in {\mathcal{W}}'$ has the $\Omega(1/\log^{8.5}m)$-bandwidth property in graph $G$. Observe that for every pair $C\in {\mathcal{C}}'$, $W'\in {\mathcal{W}}'$ of clusters, $C\cap W'=\emptyset$ must hold. For every pair $R\in {\mathcal{R}}$, $W'\in {\mathcal{W}}'$ of clusters, either $R\subseteq W'$, or $R\cap W'$ must hold. We denote by ${\mathcal{R}}'$ the set of all custers $R\in {\mathcal{R}}$ with $R\subseteq G\setminus\textsf{left}(\bigcup_{W'\in {\mathcal{W}}'}W'\textsf{right})$. Observe that $R^*\in {\mathcal{R}}'$ must hold, and that ${\mathcal{C}}'\cup {\mathcal{R}}'\cup {\mathcal{W}}'$ is a collection of disjoint clusters in graph $G$. Graph $\hat H'=\hat H_{|{\mathcal{W}}}$ that we used in Step 1 is precisely the graph $G_{|{\mathcal{C}}'\cup {\mathcal{R}}'\cup {\mathcal{W}}'}$.
The fourth and the last level of clustering is defined by the collection ${\mathcal{J}}$ of clusters in graph $\hat H'$ that we have defined in Step 1. Recall that, for every cluster $J\in {\mathcal{J}}$ (that is a subgraph of $\hat H'$), there is a unique center vertex, that is either a $C$-node or a $W$-node, and the remaining vertices of $J$ are regular vertices or $R$-nodes; however, $J$ may not contain the special vertex $v^*$ or the $R$-node $u^*$ representing the distinguished cluster $R^*$. Moreover, every $C$-node and every $W$-node is a center of some cluster in $J$.
As before, we will define, for every cluster $J\in {\mathcal{J}}$, a corresponding cluster $J'$ in graph $G$, in a natural way. We first define the vertex set $V(J')$, and then let $J'$ be the subgraph of $G$ induced by $V(J')$. First, we add to $V(J')$ every regular vertex that lies in $J$ -- each such vertex is a vertex of $G$. Next, for every $R$-node $v_{R}\in J$, we add all vertices of cluster $R$ to $V(J')$; observe that $R\in {\mathcal{R}}'\setminus \set{R^*}$ must hold. Lastly, we consider the unique center vertex of $J$. If that vertex is a $C$-node, corresponding to a cluster $C\in {\mathcal{C}}'$, then we add all vertices of $C$ to $V(J')$. Otherwise, the vertex is a $W$-node, representing some cluster $W'\in {\mathcal{W}}'$. We then add to $V(J')$ all vertices of $V(W')$. Lastly, we set $J'=G[V(J')]$.
We denote by ${\mathcal{J}}'=\set{J'\mid J\in {\mathcal{J}}}$ the set of clusters in graph $G$ corresponding to the cluster set ${\mathcal{J}}$ in $\hat H'$. Observe that for every cluster $C\in {\mathcal{C}}'$, there is a unique cluster $J'(C)\in {\mathcal{J}}'$ containing $C$; we call $C$ the \emph{center-cluster} of $J'(C)$. Similarly, for every cluster $W'\in {\mathcal{W}}'$, there is a unique cluster $J'(W)\in {\mathcal{J}}'$ containing $W'$; we similarly call $W'$ the \emph{center-cluster} of $J'(W)$. Lastly, for every pair of clusters $R\in {\mathcal{R}}'$, $J'\in {\mathcal{J}}'$, either $R\subseteq J'$ or $R\cap J'=\emptyset$ holds. We denote by ${\mathcal{R}}''\subseteq {\mathcal{R}}'$ the set of all custers $R\in {\mathcal{R}}'$, with $R\subseteq G\setminus\textsf{left}(\bigcup_{J'\in {\mathcal{J}}}J'\textsf{right} )$ (see \Cref{fig: NF5}). Note that $R^*\in {\mathcal{R}}''$.
Observe also that ${\mathcal{R}}''\cup {\mathcal{J}}'$ defines a collection of disjoint clusters in graph $G$. Note that the special vertex $v^*$ does not lie in any cluster of ${\mathcal{R}}''\cup {\mathcal{J}}'$, and that every cluster of ${\mathcal{C}}$, ${\mathcal{R}}$, and ${\mathcal{W}}'$ is contained in exactly one cluster of ${\mathcal{R}}''\cup {\mathcal{J}}'$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF5.jpg}
\caption{An illustration of a ${\mathcal{J}}$-clustering. Clusters of ${\mathcal{C}}$ are shown in blue (with clusters of ${\mathcal{C}}'$ shaded).
Clusters of ${\mathcal{R}}$ are shown in red (with clusters of ${\mathcal{R}}'$ shaded).
Clusters of ${\mathcal{W}}'$ are shown in green. Each cluster of ${\mathcal{W}}'$ may contain clusters of ${\mathcal{R}}$, but if a cluster $C\in {\mathcal{C}}$ is contained in $W'$, then there exists $R\in {\mathcal{R}}$ with $C\subseteq R\subseteq W'$.
Vertices of $G$ that do not lie in clusters of ${\mathcal{C}}$ are shown in black.
Clusters of ${\mathcal{J}}'$ are shown in brown. Each cluster of ${\mathcal{J}}'$ contains a cluster of ${\mathcal{W}}'$ or ${\mathcal{C}}'$ as its center cluster (indicated by $*$). Each cluster of ${\mathcal{W}}'\cup {\mathcal{C}}'$ is a center-cluster of some cluster of ${\mathcal{J}}'$. In addition to the center-cluster, a cluster of ${\mathcal{J}}'$ may contain clusters of ${\mathcal{R}}$ and regular vertices. Some clusters of ${\mathcal{R}}'$ and some regular vertices may not lie in any cluster of ${\mathcal{J}}'$.
}\label{fig: NF5}
\end{figure}
Since every cluster in ${\mathcal{C}}$ has the $\alpha_0=1/\log^3m$-bandwidth property; every cluster $W\in {\mathcal{W}}'$ has the $\Omega(1/\log^{8.5}m)$-bandwidth property; and every cluster $R\in {\mathcal{R}}$ has the $1/\log^6m$-bandwidth property in graph $G$, while, from \Cref{obs:J wl},
every cluster $J\in {\mathcal{J}}$, has the $\Omega(1/\log m)$-bandwidth property in graph $\hat H'=G_{|{\mathcal{C}}'\cup {\mathcal{R}}'\cup {\mathcal{W}}'}$, from \Cref{cor: contracted_graph_well_linkedness}, we get that every cluster $J'\in {\mathcal{J}}'$ has the $\Omega(1/\log^{9.5}m)$-bandwidth property. This property will be useful for us later, so we summarize it in the following observation.
\begin{observation}\label{obs: J' clusters wl}
Every cluster $J'\in {\mathcal{J}}'$ has the $\Omega(1/\log^{9.5}m)$-bandwidth property in graph $G$.
\end{observation}
In the remainder of the proof of \Cref{lemma: better clustering 2} we consider the contracted graph $\check H=G_{|{\mathcal{J}}'\cup {\mathcal{R}}''}$, which is exactly the contracted graph of $\hat H'$ with respect to cluster set ${\mathcal{J}}$, that is, $\check H=\hat H'_{|{\mathcal{J}}}$.
The set of vertices of $\check H$ consists of three subsets: the set $V(G)\cap V(\check H)$ of regular vertices; the set $\set{v_{R}\mid R\in {\mathcal{R}}''}$ of supernodes corresponding to clusters of ${\mathcal{R}}''$ that we call $R$-nodes; and the set $\set{v_{J'}\mid J'\in {\mathcal{J}}'}$ of supernodes corresponding to clusters of ${\mathcal{J}}'$ that we call $J$-nodes.
For convenience, we denote by $\hat U^*$ the set of all regular vertices of $\check H$ excluding the special vertex $v^*$, and we denote by $\hat U^R$ the set of all $R$-nodes of $\check H$ excluding the node $u^*$ that represents the distinguished cluster $R^*$.
The algorithm from Step 1 ensures the following property of graph $\check H$:
\begin{properties}{H}
\item for every vertex $u\in \hat U^*\cup \hat U^R$ and $J$-node $v_{J'}$, the number of edges connecting $u$ to $v_{J'}$ in $\check H$ is at most $|\delta_{\check H}(u)|/128$.
\end{properties}
Indeed, if the above property does not hold for a vertex $u\in \hat U^*\cup \hat U^R$ and a $J$-node $v_{J'}$, then vertex $u$ should have been added to the cluster $J\in {\mathcal{J}}$ that corresponds to cluster $J'\in {\mathcal{J}}'$ by the algorithm that constructed the clusters in ${\mathcal{J}}$.
Additionally, the algorithm from Step 1 defines a partition $(L_1,L_2,\ldots,L_h)$ of the set $\hat U^*\cup \hat U^R$ of vertices, with $h\leq \log m$. Let $L_0$ be the set of vertices of $\check H$ containing all $J$-nodes and the special vertices $v^*,u^*$; equivalently, $L_0=V(\check H)\setminus (\hat U^*\cup \hat U^R)$. Recall that for all $1\leq i\leq h$, for every vertex $u\in L_i$, we have partitioned the edge set $\delta_{\check H}(u)$ into two subsets: set $\delta^{\operatorname{down}}(u)$ contains all edges connecting $u$ to vertices of $L_0\cup\cdots\cup L_{i-1}$, while set $\delta^{\operatorname{up}}(u)$ contains all remaining edges of $\delta_{\check H}(u)$. Recall that we have also ensured that the following property holds:
\begin{properties}[1]{H}
\item for every vertex $u\in \hat U^*\cup \hat U^R$, $|\delta^{\operatorname{up}}(u)|<|\delta^{\operatorname{down}}(u)|/\log m$. \label{prop: up and down}
\end{properties}
Next, we define the notion of a simplifying cluster in graph $\check H$. We will then show that, given a simplifying cluster in $\check H$, we can efficiently compute a helpful clustering $\tilde {\mathcal{R}}$ of graph $G$ with respect to vertex $v^*$ and set ${\mathcal{C}}$ of basic clusters, such that $R^*\in \tilde {\mathcal{R}}$ and $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$.
\begin{definition}[Simplifying Cluster]
Let $S$ be a vertex-induced subgraph of $\check H$. We say that $S$ is a \emph{simplifying cluster} if:
\begin{itemize}
\item vertices $v^*,u^*$ do not lie in $S$;
\item there is a set ${\mathcal{P}}(S)$ of paths in graph $\check H$ (external $S$-router), routing the edges of $\delta_{\check H}(S)$ to a single vertex of $\check H\setminus S$, such that all paths in ${\mathcal{P}}(S)$ are internally disjoint from $S$, and they cause congestion at most $\beta'=O(\log m)$; and
\item either $S$ contains at least one $J$-node, or $|E_{\check H}(S)|\geq |\delta_{\check H}(S)|/\log m$.
\end{itemize}
\end{definition}
We will use the following simple observation.
\begin{observation}\label{obs: check simplifying}
There is an efficient algorithm, that, given a cluster $S\subseteq \check H$, establishes whether $S$ is a simplifying cluster.
\end{observation}
\begin{proof}
In order to establish whether $S$ is a simplifying cluster, we need to check whether $S$ contains a $J$-node, or $|E_{\check H}(S)|\geq |\delta_{\check H}(S)|/\log m$ holds, which can be done efficiently. Additionally, we need to check whether there is a set of paths in graph $\check H$, routing the edges of $\delta_{\check H}(S)$ to a single vertex of $\check H\setminus S$, such that the paths are internally disjoint from $S$ and cause congestion at most $\beta'$. The latter can be done efficiently by computing maximum flow between the vertices of $S$ and each vertex of $\check H\setminus S$ in turn.
\end{proof}
In the next claim we show that, if we are given a simplifying cluster $S$ in $\check H$, then
we can efficiently compute a helpful clustering $\tilde {\mathcal{R}}$ of graph $G$ with respect to $v^*$ and ${\mathcal{C}}$, such that $R^*\in \tilde {\mathcal{R}}$ and $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$. The proof of the claim is somewhat technical and is deferred to \Cref{subsec: simplifying cluster is enough}
\begin{claim}\label{claim: simplifying cluster is enough}
There is an efficient algorithm, that, given a simplifying cluster $S$ of $\check H$, computes a helpful clustering $(\tilde {\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in \tilde {\mathcal{R}}})$ of graph $G$ with respect to the special vertex $v^*$ and the set ${\mathcal{C}}$ of basic clusters, such that $R^*\in \tilde{\mathcal{R}}$, and $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$.
\end{claim}
Let $\tau$ be the Gomory-Hu tree of the graph $\check H$. We root the tree at the special vertex $v^*$, and, for every vertex $u\in V(\tau)$, we denote by $\tau_u$ the subtree of $\tau$ rooted at vertex $u$.
Assume first that there is some vertex $u\in V(\tau)$, such that the special $R$-node $u^*$ corresponding to the distinguished cluster $R^*$ does not lie in $\tau_u$, but some $J$-node $v_{J'}$ lies in $\tau_u$.
In this case, we let $S$ be a subgraph of $\check H$ that is induced by $V(\tau_u)$. We claim that $S$ is a simplifying cluster. Indeed, from the construction, neither of $v^*,u^*$ may lie in $S$, and at least one $J$-node lies in $S$. Let $u'$ be the parent-vertex of $u$ in the tree $\tau$. Then from the properties of Gomory-Hu tree (see \Cref{cor: G-H tree_edge_cut}), $(V(S),V(\check H)\setminus V(S)$ is a minimum cut separating $u$ from $u'$ in $\check H$. From the max-flow / min-cut theorem, there is a collection ${\mathcal{P}}$ of $|\delta_{\check H}(S)|$ edge-disjoint paths connecting $u$ to $u'$ in $\check H$. Clearly, each edge $e\in \delta_{\check H}(S)$ is contained in exactly one path of ${\mathcal{P}}$, that we denote by $P(e)$. Let $P'(e)$ be the subpath of $P(e)$ that starts at edge $e$ and terminates at $u'$. Then $P'(e)$ must be internally disjoint from $S$. Therefore, ${\mathcal{P}}(S)=\set{P'(e)\mid e\in \delta_{\check H}(S)}$ is a set of edge-disjoint paths in graph $\check H$, routing the edges of $\delta_{\check H}(S)$ to vertex $u'\in \check H\setminus S$, and the paths in ${\mathcal{P}}(S)$ are internally disjoint from $S$. We conclude that $S$ is a simplifying cluster. We can now use the algorithm from \Cref{claim: simplifying cluster is enough} to compute a helpful clustering $(\tilde {\mathcal{R}},\set{{\mathcal{D}}'_R}_{R\in \tilde {\mathcal{R}}})$ of graph $G$ with respect to the special vertex $v^*$ and the set ${\mathcal{C}}$ of basic clusters, such that $R^*\in \tilde{\mathcal{R}}$, and $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$.
Therefore, we assume from now on that, for every vertex $u\in V(\tau)$, if $u^*$ does not lie in $\tau_u$, then $\tau_u$ does not contain any $J$-node, and so $V(\tau_u)\subseteq \hat U^*\cup \hat U^R$.
Let $P^*$ denote the path connecting $v^*$ to $u^*$ in the tree $\tau$. We denote the sequence of vertices on the path by $v^*=u_1,u_2,\ldots,u_r=u^*$. For all $1\leq i\leq r$, we define a cluster $S_i$ of $\check H$, associated with vertex $u_i$, as follows. We let $S_r$ be the subgraph of $\check H$ induced by the vertices of $\tau_{u_r}$. Consider now some index $1\leq i< r$. Let $\set{x_i^0,x_i^1,x_i^2,\ldots,x^{q_i}_i}$ be the set of all child-vertices of $u_i$ in the tree $\tau$, and assume that $x_i^0=u_{i+1}$. We then let $S_i$ be the subgraph of $\check H$ induced by the set $\set{u_i}\cup V(\tau_{x_i^1})\cup V(\tau_{x_i^2})\cup \cdots\cup V(\tau_{x^{q_i}_i})$ of vertices. In other words, we include in vertex set $V(S_i)$ the vertices lying in all subtrees of the children of $u_i$, except for the vertices lying in the subtree of $u_{i+1}$. From our assumption, for all $1\leq i\leq r$, the only vertex of $S_i$ that may be a $J$-node is the vertex $u_i$; all other vertices of $S_i$ are $R$-nodes or regular vertices (and it is also possible that $u_i$ is an $R$-node or a regular vertex).
For all $1\leq i\leq r$, we also define a subgraph $S'_i\subseteq S_i$, that is constructed as follows. We start by constructing the set $V(S'_i)$ of vertices. Initially, we let $V(S'_i)=\set{u_i}$. While there is any vertex $u\in S_i\setminus S'_i$, such that at the number of edges connecting $u$ to vertices of $V(S'_i)$ is at least $|\delta_{\check H}(u)|/128$, then we add $u$ to $V(S'_i)$. Once this algorithm terminates, we let $S'_i$ be the subgraph of $\check H$ induced by the set $V(S'_i)$ of vertices.
Recall that we have established that, if $v$ is a vertex of $S_i\setminus S_i'$, for some $1\leq i\leq r$, then $v\in \hat U^*\cup \hat U^R$ must hold. The following observation easily follows from the construction of $J$-clusters.
\begin{observation}\label{obs: central path j-cluster}
Consider any index $1<i<r$, for which $u_i$ is a $J$-node. Then $S'_i=\set{u_i}$.
\end{observation}
\begin{proof}
Let $J\in {\mathcal{J}}$ be the cluster of $\hat H'$ that node $u_i$ represents (recall that we can think of graph $\check H$ as a contracted graph of $\hat H'$ with respect to cluster set ${\mathcal{J}}$). Assume for contradiction that $S'_i$ contains at least one vertex in addition to $u_i$, and let $v$ be the first vertex that was added to cluster $S'_i$. Then the number of edges connecting $v$ to $u_i$ is at least $|\delta_{\check H}(v)|/128$. But then $v$ is also a vertex of graph $\hat H'$, in which it serves as either an $R$-node distinct from $u^*$, or a regular vertex distinct from $v^*$. Moreover, the number of edges connecting $v$ to vertices of $J$ is at least $|\delta_{\hat H'}(v)|/128$. Therefore, $v$ should have been added to cluster $J$ when it was constructed, a contradiction.
\end{proof}
Additionally, we get the following observation, whose proof is identical to the proof of \Cref{obs:J wl} and is omitted here.
\begin{observation}\label{obs: S'i wl}
For all $1\leq i\leq r$, cluster $S'_i$ has the $\Omega(1/\log m)$-bandwidth property in graph $\check H$.
\end{observation}
For all $1\leq i\leq r$, we employ the algorithm from \Cref{obs: check simplifying} in order to establish whether $S'_i$ is a simplifying cluster. Additionally, for all $1\leq i<r$, we use the algorithm from \Cref{obs: check simplifying} in order to establish whether the subgraph of $\check H$ induced by vertex set $V(S_i)\cup V(S_{i+1})$ is a simplifying cluster. If the algorithm from \Cref{obs: check simplifying} establishes that any of the above clusters is a simplifying cluster, then we can use the algorithm from \Cref{claim: simplifying cluster is enough} to compute a helpful clustering $(\tilde {\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in \tilde{\mathcal{R}}}$ of graph $G$ with respect to the special vertex $v^*$ and the set ${\mathcal{C}}$ of basic clusters, such that $R^*\in \tilde{\mathcal{R}}$, and $\tilde {\mathcal{R}}$ is a better clustering than ${\mathcal{R}}$.
Therefore, we assume from now on that, for all $1\leq i\leq r$, cluster $S'_i$ is not a simplifying cluster, and for all $1\leq i<r$, the subgraph of $\check H$ induced by $V(S_i)\cup V(S_{i+1})$ is not a simplifying cluster.
We will show, in Step 3, an efficient algorithm that constructs a nice witness structure for graph $G_{|{\mathcal{R}}}$, with respect to the set ${\mathcal{C}}'$ of clusters, that contains every cluster $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$.
\subsubsection*{Step 3: Constructing the Nice Witness Structure}
The goal of this step is to construct a nice witness structure for graph $G_{|{\mathcal{R}}}$, with respect to the set ${\mathcal{C}}''$ of clusters, that contains every cluster $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$.
Intuitively, we will use the clusters $S_1,\ldots,S_r$ that we just defined in order to define the spine $\tilde {\mathcal{S}}=\set{\tilde S_1,\ldots,\tilde S_r}$ of the nice witness structure in the natural way: cluster $\tilde S_i$ will be obtained from $S_i$ by first replacing every $R$-node and every $J$-node of $S_i$ with the corresponding cluster $R\in {\mathcal{R}}''$ or $J'\in {\mathcal{J}}'$, and then contracting the $R$-clusters back. Similarly, we will use the clusters $S'_1,\ldots,S'_r$ in order to define the verterbrae $\tilde S'_1,\ldots,\tilde S'_r$ of the nice witness structure.
We partition the set $E(\check H)$ of edges into two disjoint subsets, $E'$ and $E''$, as follows. Set $E'$ contains all edges of $\bigcup_{i=1}^rE(S_i')$, and, additionally, for all $1\leq i<r$, it contains every edge $e=(u,v)$ with $u\in S_i'$, $v\in S_{i+1}'$. Set $E''$ contains all remaining edges of $E(\check H)$. Additionally, we let $\hat E\subseteq E''$ be the set of all edges $(u,v)\in E''$, where $u$ and $v$ lie in different sets of $\set{ S_1,\ldots, S_r}$.
Next, we develop some tools that will allow us to define the set ${\mathcal{P}}=\set{P(e)\mid e\in \hat E}$ of nice guiding paths for the nice witness structure that we construct.
Recall that in Step 1 of the algorithm, we have partitioned the set $U^*\cup U^R$ of vertices of graph $\hat H'$ into layers $L_1,\ldots,L_h$, where $h\leq \log m$. Recall that $U^*$ is the set of all regular vertices (excluding $v^*$), and $U^R$ is the set of all $R$-nodes (excluding $u^*$) of graph $\hat H'$. Recall that $\check H=\hat H'_{|{\mathcal{J}}}$, that is, graph $\check H$ can be obtained from $\hat H'$ by contracting all clusters of ${\mathcal{J}}$. Therefore, if we denote by $\hat U^*$ the set of all regular vertices of $\check H$ (excluding $v^*$), and by $\hat U^R$ the set of all $R$-nodes of $\check H$ (excluding $u^*$), then $\hat U^*\subseteq U^*$, and $\hat U^R\subseteq U^R$. Therefore, partition $(L_1,\ldots,L_h)$ of $U^*\cup U^R$ naturally defines a partition $(L'_1,\ldots,L_h')$ of $\hat U^R\cup \hat U^*$. Recall that, for all $1\leq i\leq r$, all vertices of $S_i\setminus S'_i$ lie in $\hat U^*\cup \hat U^R$.
We denote by $L'_0=V(\check H)\setminus \textsf{left}(\bigcup_{j=1}^hL'_j\textsf{right} )$.
As before, for all $1\leq j\leq r$, for every vertex $v\in L'_j$, we partition the set $\delta_{\check H}(v)$ of edges into two subsets: set $\delta^{\operatorname{down}}(v)$ containing all edges that connect $v$ to vertices of $L'_0\cup\cdots\cup L'_{j-1}$, and set $\delta^{\operatorname{up}}(v)$ containing all remaining edges incident to $v$. From Property \ref{prop: up and down} of graph $\check H$, for every vertex $v\in \hat U^*\cup \hat U^R$, $|\delta^{\operatorname{up}}(v)|<|\delta^{\operatorname{down}}(v)|/\log m$.
For all $1\leq i\leq r$ and $1\leq j\leq h$, we denote by $U_{i,j}=L'_j\cap V(S_i)$ -- the set of all vertices of $S_i$ that lie in layer $L'_j$.
Consider some pair $1\leq i\leq r$, $1\leq j\leq h$ of indices, and some vertex $v\in U_{i,j}\setminus \set{u_i}$. We partition the edges of $\delta^{\operatorname{down}}(v)$ into four subsets, $\delta^{\operatorname{down},\operatorname{left}}(v),\delta^{\operatorname{down},\operatorname{right}}(v)$, $\delta^{\operatorname{down},\operatorname{straight}'}(v)$, and $\delta^{\operatorname{down},\operatorname{straight}''}(v)$, as follows. Let $e=(u,v)$ be an edge of $\delta^{\operatorname{down}}(v)$, and assume that $u\in U_{i',j'}$. Since $e\in \delta^{\operatorname{down}}(v)$, $j'<j$ must hold. If, additionally, $i'<i$ holds, then we add $e$ to $\delta^{\operatorname{down},\operatorname{left}}(v)$, Similarly, if $i'>i$, then we add $e$ to
$\delta^{\operatorname{down},\operatorname{right}}(v)$. If $i'=i$, and $u\in S'_i$, then $e$ is added to $\delta^{\operatorname{down},\operatorname{straight}''}(v)$, and otherwise it is added to $\delta^{\operatorname{down},\operatorname{straight}'(v)}$. We will use the following simple observation, whose proof appears in \Cref{subsec: left and right down-edges}.
\begin{observation}\label{obs: left and right down-edges}
$S'_1=S_1$, and $S'_r=S_r$. Additionally,
for every vertex $v\in V(\check H)\setminus\textsf{left}(\bigcup_{i=1}^rS'_i\textsf{right} )$:
\begin{itemize}
\item $|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{left}}(v)|+| \delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq 63|\delta(v)|/64$;
\item $|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|)$; and
\item $|\delta^{\operatorname{down},\operatorname{right}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{left}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|)$.
\end{itemize}
\end{observation}
We will also use the following simple observation, whose proof appears in \Cref{subsec: left and right mappings}
\begin{observation}\label{obs: left and right mappings}
There is an efficient algorithm that defines, for every vertex $v\in V(\check H)\setminus\textsf{left}(\bigcup_{i=1}^rS'_i\textsf{right} )$, two mappings: mapping $f^{\operatorname{right}}(v)$, that maps every edge of $\delta^{\operatorname{down},\operatorname{straight}''}(v)\cup\delta^{\operatorname{up}}(v)$ to a distinct edge of $\delta^{\operatorname{down},\operatorname{right}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$, and another mapping $f^{\operatorname{left}}(v)$, that maps every edge of $\delta^{\operatorname{down},\operatorname{straight}''}(v)\cup\delta^{\operatorname{up}}(v)$ to a distinct edge of $\delta^{\operatorname{down},\operatorname{left}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$.
\end{observation}
\iffalse
\subsection{Proof of \Cref{obs: left and right mappings}}
\label{subsec: left and right mappings}
Consider any vertex $v\in V(\check H)\setminus\textsf{left}(\bigcup_{i=1}^rS'_i\textsf{right} )$. From \Cref{obs: left and right down-edges}, $|\delta^{\operatorname{down},\operatorname{left}}(v)|\leq 2(|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|)$. Therefore:
$$|\delta^{\operatorname{down},\operatorname{left}}(v)|+|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\leq 3(|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|).$$
On the other hand, since $|\delta^{\operatorname{down},\operatorname{straight}''}(v)|\leq |\delta_{\check H}(v)|/128$, and $|\delta^{\operatorname{up}}(v)|\leq |\delta_{\check H}(v)|/\log m$, we get that:
$$|\delta^{\operatorname{down},\operatorname{left}}(v)|+|\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq \textsf{left} (\frac {127}{128} -\frac 1 {\log m}\textsf{right} )\cdot |\delta_{\check H}(v)|.$$
By combining the two inequalities, we get that:
\[ |\delta^{\operatorname{down},\operatorname{right}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|\geq \textsf{left} (\frac {127} {384} -\frac 1 {3\log m}\textsf{right} )\cdot |\delta_{\check H}(v)|>|\delta^{\operatorname{down},\operatorname{straight}''}(v)|+|\delta^{\operatorname{up}}(v)|. \]
Therefore, we can define a mapping $f^{\operatorname{right}}(v)$, that maps every edge of $\delta^{\operatorname{down},\operatorname{straight}''}(v)\cup\delta^{\operatorname{up}}(v)$ to a distinct edge of $\delta^{\operatorname{down},\operatorname{right}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$.
Using exactly the same reasoning, we get that:
\[ |\delta^{\operatorname{down},\operatorname{left}}(v)|+|\delta^{\operatorname{down},\operatorname{straight}'}(v)|>|\delta^{\operatorname{down},\operatorname{straight}''}(v)|+|\delta^{\operatorname{up}}(v)|. \]
Therefore, we can define a mapping $f^{\operatorname{left}}(v)$, that maps every edge of $\delta^{\operatorname{down},\operatorname{straight}''}(v)\cup\delta^{\operatorname{up}}(v)$ to a distinct edge of $\delta^{\operatorname{down},\operatorname{left}}(v)\cup \delta^{\operatorname{down},\operatorname{straight}'}(v)$.
\fi
Next, we define the notion of a \emph{left-monotone} and a \emph{right-monotone} path.
\begin{definition}[Left-Monotone and Right-Monotone Paths]
Let $P=(x_1,x_2,\ldots,x_q)$ be a path in graph $\check H$. For all $1\leq a\leq q$, assume that $x_a\in U_{i_a,j_a}$. We say that path $P$ is \emph{left-monotone} if either $q=1$ (that is, $P$ consists of a single vertex), or all of the following conditions holds:
\begin{properties}{M}
\item $j_1>j_2>\cdots>j_q$;\label{prop: monotone levels decrease}
\item for all $1\leq a<q$, vertex $x_a\in S_{i_a}\setminus S'_{i_a}$, and $x_q\in S'_{i_q}$; \label{prop: monotone paths not in kernels} and
\item $i_1\geq i_2\geq\cdots\geq i_q$, and $i_q<i_1$. \label{prop: monotone goes left}
\end{properties}
Similarly, we say $P$ is \emph{right-monotone} if either $q=1$, or properties \ref{prop: monotone levels decrease} and \ref{prop: monotone paths not in kernels} hold for it, together with the following property:
\begin{properties}[2]{M'}
\item $i_1\leq i_2\leq\cdots\leq i_q$, and $i_q>i_1$
\end{properties}
\end{definition}
Observe that the vertices on a left-monotone path must appear in the decreasing order of their layers, and in the non-increasing order of the sets $S_i$ to which they belong.
Similarly, vertices on a right-monotone path appear in the decreasing order of their layers, and in the non-decreasing order of the sets $S_i$ to which they belong.
The following lemma will allow us to construct prefix- and suffix-paths for each edge $e\in \hat E$, by constructing a left-monotone and a right-monotone path for each such edge in graph $\check H$; the proof is deferred to \Cref{subsubsec: monotone paths}.
\begin{lemma}\label{lem: prefix and suffix path}
There is an efficient algorithm that constructs, for every edge $e=(u,v)\in \hat E$ two paths $P(e,u)$ and $P(e,v)$ in graph $\check H$, such that, if $u\in S_i$, $v\in S_{i'}$, and $i<i'$, then path $P(e,u)$ is left-monotone and path $P(e,v)$ is right-monotone. Moreover, the set $\set{P(e,v),P(e,u)\mid e=(u,v)\in \hat E}$ of paths causes congestion $O(\log m)$.
\end{lemma}
Consider now some index $1\leq i< r$. We let $\hat E_i\subseteq \hat E$ contain all edges $e=(u,v)\in \hat E$, such that, if $u\in S_{i'}$, $v\in S_{i''}$, and $i'<i''$, then $i'\leq i$ and $i''\geq i+1$ must hold. We also denote by $E_i\subseteq E'$ the set of all edges $e=(u,v)$ with $u\in S'_i$ and $v\in S'_{i+1}$
(see \Cref{fig: NF8}).
Note that $E_i\cap \hat E_i=\emptyset$.
The next lemma is crucial to the algorithm for constructing a nice witness structure in graph $G_{|{\mathcal{R}}}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NF8.jpg}
\caption{An illustration of edge sets $E_i$ and $\hat E_i$.
}\label{fig: NF8}
\end{figure}
\begin{lemma}\label{lem: no bad indices}
For all $1< i<r$, vertex $u_i$ is a $J$-node, and for all $1\leq i< r$, $|\hat E_i|\leq 1000\cdot |E_i|$.
\end{lemma}
The proof of \Cref{lem: no bad indices} is deferred to \Cref{subsubsec: no bad indices}.
From now on, we denote $H=G_{|{\mathcal{R}}}$, and we denote by ${\mathcal{C}}'\subseteq {\mathcal{C}}$ the set of all basic clusters $C\in {\mathcal{C}}$ with $C\subseteq G\setminus\textsf{left} (\bigcup_{R\in {\mathcal{R}}}V(R)\textsf{right} )$; equivalently, ${\mathcal{C}}'$ contains every cluster $C\in {\mathcal{C}}$ that is contained in $H$. It now remains to construct a nice witness structure in graph $H$ with respect to the set ${\mathcal{C}}'$ of clusters. We start by constructing the backbone and the vertebrae of the witness structure, and by defining the partition $(\tilde E',\tilde E'')$ of the edges of $H$. We then construct the prefix and the suffix of path $P(e)$ for each edge $e\in \hat E$. Lastly, we construct the mid-segment of each such path.
\paragraph{The Backbone and the Vertebrae of the Witness Structure.}
We use the clusters in set ${\mathcal{S}}=\set{S_1,\ldots,S_r}$, and in set ${\mathcal{S}}'=\set{S_1',\ldots,S'_r}$ in order to define the backbone and the vertebrae of the nice witness structure, respectively. Recall that every vertex of graph $\check H$ is either a regular vertex (that is, it lies in both $G$ and $H$); or it is an $R$-node $v_R$ representing some cluster $R\in {\mathcal{R}}$ (in which case it lies in $H=G_{|{\mathcal{R}}}$); or it is a $J$-node $v_{J'}$ for some cluster $J'\in {\mathcal{J}}'$. As we have established, for all $1<i<r$, vertex $u_i$ is a $J$-node, and all other vertices of $\check H$ are either regular vertices of $R$-nodes.
Consider now any such $J$-node $v_{J'}$, and the corresponding cluster $J'\in {\mathcal{J}}'$. Recall that for every cluster $R\in {\mathcal{R}}$, either $R\subseteq J'$, or $R\cap J'=\emptyset$ holds. Denote by ${\mathcal{R}}(J')\subseteq {\mathcal{R}}$ the set of all clusters $R\in {\mathcal{R}}$ with $R\subseteq J'$. Let $J''=J'_{|{\mathcal{R}}(J')}$ be the graph obtained from $J'$ by contracting every cluster $R\in {\mathcal{R}}(J')$ into a supernode. Then $J''\subseteq H$, and we will think of $J''$ as the cluster of $H$ that vertex $v_{J'}\in V(\check H)$ represents. We denote by ${\mathcal{J}}''=\set{J''\mid J'\in {\mathcal{J}}'}$ the resulting set of clusters in graph $H$. Note that equivalently we could define the graph $\check H$ as a graph that is obtained from $H$ by contracting every cluster in ${\mathcal{J}}''$, that is, $\check H=H_{|{\mathcal{J}}''}$. Recall that we have established, in \Cref{obs: J' clusters wl}, that
every cluster $J'\in {\mathcal{J}}'$ has the $\Omega(1/\log^{9.5}m)$-bandwidth property in graph $G$. It then immediately follows that every cluster $J''\in {\mathcal{J}}''$ has the $\Omega(1/\log^{9.5}m)$-bandwidth property in graph $H$.
We start by defining the sequence $\tilde {\mathcal{S}}'=\set{\tilde S'_1,\ldots,\tilde S'_r}$ of the vertebrae of the nice witness structure. Consider an index $1\leq i\leq r$. If $i=1$, then $u_1=v^*$, and so every vertex of set $S'_1$ is a vertex of $H$. We then set $\tilde S'_1=S'_1$. If $i=r$, then $u_r=u^*$. As before, every vertex of $S'_r$ is then a vertex of $H$, and we set $\tilde S_r'=S'_r$. Lastly, assume that $1<i<r$. From \Cref{lem: no bad indices}, $u_i$ is a $J$-node, and, from \Cref{obs: central path j-cluster}, $S'_i=\set{u_i}$. Assume that $u_i=v_{J'}$, where $J'\in {\mathcal{J}}'$. We then let $\tilde S'_i$ be the cluster $J''$ of $H$ that corresponds to vertex $v_{J'}$. This completes the definition of the sequence $\tilde {\mathcal{S}}'=\set{\tilde S'_1,\ldots,\tilde S'_r}$ of the vertebrae of the nice witness structure. Consider
again some index $1\leq i\leq r$. If $i\in \set{1,r}$, then, from the construction, for every cluster $C\in {\mathcal{C}}'$, $C\cap \tilde S'_i=\emptyset$. This is because every cluster of ${\mathcal{C}}'$ must be contained in some cluster of ${\mathcal{J}}'$. Otherwise, assume that $u_i=v_{J'}$, where $J'\in {\mathcal{J}}'$. If the center-cluster of $J'$ is a basic cluster $C\in {\mathcal{C}}'$, then $C\subseteq \tilde S_i'$, and for every other cluster $C'\in {\mathcal{C}}'$, $C\cap \tilde S_i'=\emptyset$. Otherwise, the center-cluster of $J'$ is a cluster $W'\in {\mathcal{W}}'$. In this case, no cluster of ${\mathcal{C}}'$ may be contained in $\tilde S_i'$.
Consider again some index $1\leq i\leq r$.
Recall that we have established, in \Cref{obs: S'i wl}, that cluster $S'_i$ has the $\Omega(1/\log m)$-bandwidth property in graph $\check H$. We have also established above that every cluster in $J''\in {\mathcal{J}}''$ has the $\Omega(1/\log^{9.5}m)$-bandwidth property. From \Cref{cor: contracted_graph_well_linkedness}, cluster $\tilde S'_i$ of $H$ has the $\Omega(1/\log^{10.5}m)\geq \alpha^*$-bandwidth property, since $\alpha^*=\Omega(1/\log^{12}m)$. To conclude, we have shown that, for all $1\leq i\leq r$, cluster $\tilde S'_i$ of $H$ has the $\alpha^*$-bandwidth property. We have also shown that, for all $1\leq i\leq r$, there is at most one cluster $C \in{\mathcal{C}}'$ with $C\subseteq \tilde S'_i$. It is easy to verify that, if such cluster $C$ exists, then $E(\tilde S_i')\subseteq E(C)\cup E(G_{|{\mathcal{C}}'})$, and otherwise $E(\tilde S_i')\subseteq E(G_{|{\mathcal{C}}'})$. This is because for every cluster $C\in {\mathcal{C}}'$, and for all $1\leq i\leq r$, either $C\subseteq \tilde S'_i$, or $C\cap \tilde S'_i=\emptyset$ holds.
\iffalse
We also denote by $U(J')$ the set of all vertices of $J'$ that do not lie in any cluster of ${\mathcal{R}}\cup {\mathcal{C}}\cup{\mathcal{W}}$. If the center-cluster of $J'$ is a basic cluster $C\in {\mathcal{C}}$, then $V(J')=V(C)\cup \textsf{left}(\bigcup_{R\in {\mathcal{R}}(J')}V(R)\textsf{right} )\cup U(J')$. We then denote $C=C(J')$. In this case, we let $J''=J'_{|{\mathcal{R}}(J')}$ be the subgraph of $G_{|{\mathcal{R}}(J')}$ associated with $J'$. Note that there is exactly one cluster $C(J')\in {\mathcal{C}}$ with $C(J')\subseteq J''$, and for every other cluster $C'\in {\mathcal{C}}$, $C'\cap J''=\emptyset$. In the second case, the center-cluster of $J'$ is a cluster $W'\in {\mathcal{W}}'$. Recall that, from the definition of $W$-clusters in graph $\hat H=G_{|{\mathcal{R}}\cup {\mathcal{C}}'}$, every vertex in a $W$-cluster is either a regular vertex or an $R$-node. We denote $W'=W'(J')$, and we denote by $U(W'(J'))$ the set of all vertices of $W'$ that do not lie in clusters of ${\mathcal{R}}$.
Lastly, we denote by ${\mathcal{R}}(W'(J'))$ the set of all clusters $R\in {\mathcal{R}}$ with $R\subseteq W'(J')$. In this case, $V(J')=\bigcup_{R\in {\mathcal{R}}(J')\cup {\mathcal{R}}(W'(J'))}V(R)\cup U(J')\cup U(W'(J'))$. We then let $J''=J'_{|{\mathcal{R}}(J')\cup {\mathcal{R}}(W'(J'))}$ be the subgraph of $G_{|{\mathcal{R}}}$ associated with $J'$. Observe that in this case, for every basic cluster $C\in {\mathcal{C}}$, $C\cap J''=\emptyset$.
As observed at the beginning of Step 2, every cluster $J'\in {\mathcal{J}}'$ has the $\Omega(\alpha_1/\log^{3.5}m)$-bandwidth property. Let ${\mathcal{J}}''=\set{J''\mid J'\in {\mathcal{J}}'}$ be the corresponding set of clusters in graph $G_{|{\mathcal{R}}}$. Clearly, every cluster $J''\in {\mathcal{J}}''$ has the $\Omega(\alpha_1/\log^{3.5}m)$-bandwidth property in $G_{|{\mathcal{R}}}$. Notice also that $\check H$ is the contracted graph of $G_{|{\mathcal{R}}}$ with respect to cluster set ${\mathcal{J}}''$: that is, graph $\check H$ can be obtained from graph $G_{|{\mathcal{R}}}$ by contracting every cluster $J''\in {\mathcal{J}}''$ into a supernode (which becomes a $J$-node).
\fi
We now define the backbone $\tilde {\mathcal{S}}=\set{\tilde S_1,\ldots,\tilde S_r}$ of the nice witness structure. Fix an index $1\leq i\leq r$. If $i\in \set{1,r}$, then, from \Cref{obs: left and right down-edges}, $S'_i=S_i$. We then set $\tilde S_i=\tilde S_i'$. Assume now that $1<i<r$. Recall that in this case, $S'_i=\set{u_i}$ holds, and $u_i$ is a $J$-node, from \Cref{lem: no bad indices}. Note that every vertex of $S_i\setminus\set{u_i}$ is either a regular vertex or an $R$-node, and so it must lie in graph $H$.
We define the set $V(\tilde S_i)$ of vertices to contain all regular vertices and all $R$-nodes that lie in $S_i\setminus\set{u_i}$, and all vertices of $\tilde S_i'$. We then let $\tilde S_i$ be the subgraph of $H$ induced by the set $V(\tilde S_i)$ of vertices. In other words, we can think of cluster $\tilde S_i$ as being obtained from cluster $S_i$ of $\check H$, by un-contracting the $J$-node $u_i$ (into the corresponding cluster of ${\mathcal{J}}''$). This completes the definition of the backbone of the nice witness structure. Since every vertex of $S_i\setminus S'_i$ is either a regular vertex or an $R$-node, either there is a single cluster $C \in{\mathcal{C}}'$ with $C\subseteq \tilde S'_i$, in which case then $E(\tilde S_i)\subseteq E(C)\cup E(G_{|{\mathcal{C}}'})$; or no such cluster exists, in which case $E(\tilde S_i)\subseteq E(G_{|{\mathcal{C}}'})$.
Since vertex sets $V(S_1),\ldots,V(S_r)$ partition $V(\check H)$, it is easy to verify that vertex sets $V(\tilde S_1),\ldots,V(\tilde S_r)$ partition $V(H)$.
\iffalse
------------------------------------
and $\tilde {\mathcal{S}}'=\set{\tilde S_1,\ldots,\tilde S_r}$ of the nice witness structure in graph $G_{{\mathcal{R}}}$. We denote by ${\mathcal{C}}'\subseteq {\mathcal{C}}$ the set of all basic clusters $C$ with $C\subseteq G_{|{\mathcal{R}}}$. Fix an index $1\leq i\leq r$, and consider the cluster $S'_i$. Recall that every vertex of $S'_i$ is either an $R$-node or a regular vertex, except possibly for $u_i$, which may be a $J$-node. If $u_i$ is not a $J$-node, then $S'_i\subseteq G_{|{\mathcal{R}}}$, and we set $\tilde S_i'=S'_i$. Otherwise, if $u_i=v(J')$ for some cluster $J'\in {\mathcal{J}}'$, then we let $\tilde S_i$ be the subgraph of $G_{|{\mathcal{R}}}$ that is induced by vertex set $\textsf{left} (V(S'_i)\setminus \set{u_i}\textsf{right} )\cup V(J'')$, where $J''\in {\mathcal{J}}$ is the cluster corresponding to $J'$. In other words, we obtain $\tilde S'_i$ from $S'_i$ by ``uncontracting'' the cluster $J''$.
Since, for all $1\leq i\leq r$, every vertex of $S_i\setminus S'_i$ is either an $R$-node or a regular vertex, each such vertex lies in $G_{|{\mathcal{R}}}$. We then let $\tilde S_i$ be the subgraph of $G_{|{\mathcal{R}}}$, that is induced by vertex set $\textsf{left} (V(S_i)\setminus V(S'_i)\textsf{right} )\cup V(\tilde S'_i)$. Since vertex sets $V(S_1),V(S_2),\ldots,V(S_r)$ partition $V(\check H)$, we get that $\bigcup_{i=1}^rV(\tilde S_i)=V(G_{|{\mathcal{R}}})$. Note that, from the above discussion, for all $1\leq i\leq r$, if $u_i$ is a $J$-node $u_i=v(J')$, for some cluster $J'\in {\mathcal{J}}'$, whose center-cluster $C(J')\in {\mathcal{C}}$ is a basic cluster, then $C(J')\subseteq \tilde S'_i$, and for any other basic cluster $C'\in {\mathcal{C}}\setminus \set{C(J')}$, $C'\cap \tilde S_i=\emptyset$.
Therefore, $E(\tilde S_i)\subseteq E(C(J'))\cup E^{\textnormal{\textsf{out}}}({\mathcal{C}}')$ in this case. Otherwise, for every basic cluster $C\in {\mathcal{C}}$, $C\cap \tilde S_i=\emptyset$, and $E(\tilde S_i)\subseteq E^{\textnormal{\textsf{out}}}({\mathcal{C}}')$ holds. This completes the definition of the backbone of the nice witness structure.
\fi
Recall that the second ingredient of the nice witness structure is a partition of the edges of $E(H)$ into two disjoint subsets, $\tilde E'$ and $\tilde E''$, that are defined as follows. Set $\tilde E'$ contains all edges of $\bigcup_{i=1}^rE(\tilde S_i')$, and, additionally, for all $1\leq i<r$, it contains every edge $e=(u,v)$ with $u\in \tilde S_i'$, $v\in \tilde S_{i+1}'$.
Since, as observed already, $\check H=H_{|{\mathcal{J}}''}$, it is easy to verify that $E'\subseteq \tilde E'$. Recall that we have denoted, for all $1\leq i<r$, by $E_i\subseteq E'$ the set of all edges $e=(u,v)$ of $\check H$ with $u\in S'_i$ and $v\in S'_{i+1}$. It is easy to verify that $E_i\subseteq E(H)$, and moreover, it is precisely the set of all edges $(u,v)$ in $H$ with $u\in \tilde S'_i$ and $v\in \tilde S'_{i+1}$. In particular, $E_i\subseteq \tilde E'$.
The second edge set in the partition of $E(H)$ contains all remaining edges, $\tilde E''=E(H)\setminus \tilde E'$. From the fact that $\check H=H_{|{\mathcal{J}}''}$, and since, for all $1\leq i\leq r$, $S_i\setminus S'_i$ may only contain regular vertices or $R$-nodes, we get that $E''=\tilde E''$ holds. Lastly, we defined the set $\hat E\subseteq E''$ of all edges $e=(u,v)\in E''$ of graph $\check H$, where $u$ and $v$ lie in different clusters of $\set{ S_1,\ldots, S_r}$. It is immediate to verify that this is exactly the set of edges in graph $\check H$, containing all edges $e=(u,v)\in \tilde E''$ where $u$ and $v$ lie in different clusters of $\set{\tilde S_1,\ldots,\tilde S_r}$.
Recall that we have defined, for all $1\leq i< r$, the set $\hat E_i\subseteq \hat E$ of edges in graph $\check H$, that contains all edges $e=(u,v)\in \hat E$, such that, if $u\in S_{i'}$ and $v\in S_{i''}$ with $i'<i''$, then $i'\leq i$ and $i''\geq i+1$ hold.
It is easy to verify that $\hat E_i$ is also precisely the set of all edges $e=(u,v)\in \hat E$ in graph $H$, such that, if $u\in \tilde S_{i'}$ and $v\in \tilde S_{i''}$ with $i'<i''$, then $i'\leq i$ and $i''\geq i+1$ hold. As before, $E_i\cap \hat E_i=\emptyset$.
In order to complete the construction of the nice witness structure, it now remains to define the paths in set $\hat {\mathcal{P}}=\set{P(e)\mid e\in \hat E}$. Recall that each such path $P(e)$ consists of three subpaths, prefix $P^1(e)$, suffix $P^3(e)$, and mid-segment $P^2(e)$.
We first construct the prefixes and the suffixes of the paths in $\hat {\mathcal{P}}$, and then construct the mid-segment of each such path.
\paragraph{Prefixes and Suffixes of Paths in $\hat {\mathcal{P}}$.}
Consider an edge $e=(u,v)\in \hat E$ in graph $H$. Assume that $u\in \tilde S_i$, $v\in \tilde S_{i'}$, and $i<i'$. We now define vertices $u',v'$ of graph $\check H$ that correspond to $u$ and $v$.
If $u$ is also a vertex of cluster $S_i$ in $\check H$, then we set $u'=u$. Otherwise, $u_i$ must be a $J$-node corresponding to some cluster $J'\in {\mathcal{J}}'$, with vertex $u$ lying in the corresponding cluster $J''\in {\mathcal{J}}''$. In this case, we set $u'=u_i$. We define vertex $v'$ in graph $\check H$, that corresponds to vertex $v$ in graph $H$ similarly, so $v'\in S_{i'}$. Observe that $(u',v')$ is an edge of $\check H$, that lies in the edge set $\hat E$, and it corresponds to edge $e$ in $H$; we do not distinguish between the two edges. Consider now the left-monotone path $P(e,u')$ in graph $\check H$ given by \Cref{lem: prefix and suffix path}, and denote
$P(e,u')=(u'=x_1,x_2,\ldots,x_q)$. For all $1\leq a\leq q$, assume that $x_a\in U_{i_a,j_a}$. Recall that, from the definition of the left-monotone path, $i_1\geq i_2\geq\cdots\geq i_q$, and, if $P(e,u')$ contains more than one vertex, then $i_q<i_1=i$ holds. Additionally, for all $1\leq a<q$, vertex $x_a\not\in S'_{i_a}$, while $x_q\in S'_{i_q}$. In particular, every inner vertex on path $P(e,u')$
is an $R$-node or a regular vertex of $\check H$, and hence it lies in graph $H$. Clearly, every edge of path $P(e,u')$ is an edge of $H$ that lies in edge set $\tilde E''$. Therefore, path $P(e,u')$ is contained in graph $H$. We set the prefix $P^1(e)$ of the path $P(e)$ to be $P(e,u')$. We also denote by $i^{\operatorname{left}}(e)=i_q$, and we denote by $e^{\operatorname{left}}$ the last edge on path $P^1(e)$. Observe that $e^{\operatorname{left}}\in \delta_{H}(\tilde S'_{i^{\operatorname{left}}(e)})$. We define the suffix $P^3(e)$ using the right-monotone path $P(e,v)$ similarly. We denote by $e^{\operatorname{right}}$ the last edge on that path, and by $i^{\operatorname{right}}(e)$ the index $i^*$ such that the last vertex of path $P^3(e)$ belongs to $\tilde S'_{i^*}$. From the definition of monotone paths, if $u\not\in \tilde S_i'$, then $i^{\operatorname{left}}(e)< i$, and, if $v\not\in \tilde S_{i'}'$, then $i^{\operatorname{right}}(e)>i'$.
Lastly, we define the \emph{span} of edge $e$ to be $\operatorname{span}(e)=\set{i^{\operatorname{left}}(e),(i^{\operatorname{left}}(e)+1),\ldots,(i^{\operatorname{right}}(e)-1)}$.
Recall that the congestion caused by the set $\set{P(e,v),P(e,u)\mid e=(u,v)\in \hat E}$ of paths in graph $\check H$ is $O(\log m)$, so the congestion caused by the set $\set{P^1(e),P^3(e)\mid e\in \hat E}$ of paths in graph $H$ is also $O(\log m)$.
\paragraph{Mid-Segments of Paths in $\hat {\mathcal{P}}$.}
We now focus on constructing the mid-segment $P^2(e)$ of the nice guiding path $P(e)$ for every edge $e\in \hat E$.
In order to do so, fix some index $1\leq i<r$, and let $\hat E'_i$ be the set of all edges $e\in \hat E$, such that $i\in \operatorname{span}(e)$. Note that edge $e$ may only lie in $\hat E'_i$ if either (i) $e\in \hat E_i$; or (ii) some edge $e'\in \hat E_i$ belongs to path $P^1(e)$; or (iii) some edge $e''\in \hat E_i$ belongs to path $P^3(e)$. Since the paths in set $\set{P^1(e),P^3(e)\mid e\in \hat E}$ cause congestion at most $O(\log m)$ in graph $H$, from \Cref{lem: no bad indices}, $|\hat E_i'|\leq O(\log m)\cdot |E_i|$. Therefore, we can define an arbitrary mapping $f_i: \hat E_i'\rightarrow E_i$, such that, for every edge $e\in E_i$, at most $O(\log m)$ edges of $\hat E_i'$ are mapped to $e$.
In order to define the mid-segment of every path in $\set{P(e)\mid e\in \hat E}$, we proceed as follows. For all $1\leq i<r$, we will define a collection $M_i$ of pairs of edges in $\delta_{H}(\tilde S'_i)$, so that every edge of $\delta_{H}(\tilde S'_i)$ participates in at most $O(\log m)$ such pairs. We will later exploit the bandwidth property of cluster $\tilde S'_i$ in order to connect every pair of edges in $M_i$ with a path. We start with $M_i=\emptyset$ for all $1\leq i\leq r$, and then gradually add edge pairs to the sets $M_i$.
Consider again some edge $e\in \hat E$, and recall that $\operatorname{span}(e)=\set{i^{\operatorname{left}}(e),(i^{\operatorname{left}}(e)+1),\ldots,(i^{\operatorname{right}}(e)-1)}$.
For convenience, denote $i^{\operatorname{left}}(e)$ by $i'$ and $i^{\operatorname{right}}(e)$ by $i''$.
Recall that the last edge on path $P^1(e)$, that we denoted by $e^{\operatorname{left}}$, is an edge that is incident to cluster $\tilde S_{i'}$ in $H$. Let $e^{i'}$ be the edge of $E_{i'}$ to which edge $e$ is mapped by $f_{i'}$. We then add the pair $(e^{\operatorname{left}},e^{i'})$ to $M_{i'}$.
Consider now any index $i'<i<i''-1$. Let $e^{i-1}\in E_{i-1}$ be the edge to which $e$ is mapped by $f_{i-1}$, and let $e^i\in E_i$ be the edge to which $e$ is mapped by $f_i$. We then add the pair $(e^{i-1},e^i)$ to $M_i$. Lastly, we add the edge pair $(e^{i''-1},e^{\operatorname{right}})$ to $M_{i''}$.
We will define a path $Q^{i'}(e)$ in graph $H$, whose first edge is $e^{\operatorname{left}}$ and last edge is $e^{i'}$, such that all inner vertices of $Q^{i'}(e)$ lie in $\tilde S_{i'}'$. Additionally, for all $i'<i<i''-1$, we will define a path $Q^i(e)$ in graph $H$, whose first edge is $e^{i-1}$ and last edge is $e^i$, such that all inner vertices of $Q^i(e)$ lie in $\tilde S_i'$. Laslty, we will define a path $Q^{i''}(e)$, whose first edge is $e^{i''-1}$, last edge is $e^{\operatorname{right}}$, and all inner vertices lie in $\tilde S_{i''}'$. The final path $P^2(e)$ is then obtained by concatenating the paths $Q^{i'}(e),\ldots,Q^{i''}(e)$, and omitting the first and the last edge from the resulting path.
In order to define the paths of $\set{Q^i(e)\mid e\in \hat E; i^{\operatorname{left}}(e)\leq i<i^{\operatorname{right}}(e)}$, we consider the clusters $\tilde S_i'\in \tilde {\mathcal{S}}'$ one by one. Consider any such cluster $\tilde S_i'$. Recall that we have defined a collection $M_i$ of pairs of edges from $\delta_H(\tilde S_i')$, such that every edge of $\delta_{H}(\tilde S_i')$ appears in at most $O(\log m)$ pairs. Using a standard greedy algorithm, we can compute $z=O(\log m)$ collections $M^1_i,\ldots,M^z_i$ of pairs of edges, such that $\bigcup_{j=1}^zM^j_i=M_i$, and, for all $1\leq j\leq z$, every edge of $\delta_{H}(\tilde S_i')$ participates in at most one pair of $M^j_i$. By applying the algorithm from \Cref{cor: routing well linked vertex set} to the augmented cluster $(\tilde S_i')^+$, we obtain, for each $1\leq j\leq z$, a collection ${\mathcal{Q}}^{j}_i=\set{\hat Q(e,e')\mid (e,e')\in M^j_i}$ of paths, where each path $Q(e,e')$ has $e$ as its first edge, $e'$ as its last edge, and all internal vertices of the path lie in $\tilde S_i'$. Moreover, since cluster $\tilde S_i'$ has $\alpha^*$-bandwidth property,
with high probability, the paths in ${\mathcal{Q}}^j_i$ cause edge-congestion at most $O(\log^4m/\alpha^*)\leq O(\log^{16}m)$, since $\alpha^*=\Omega(1/\log^{12}m)$. By letting ${\mathcal{Q}}_i=\bigcup_{j=1}^z{\mathcal{Q}}^j_i$, we obtain a collection ${\mathcal{Q}}_i=\set{\hat Q(e,e')\mid (e,e')\in M_i}$ of paths, where for every edge pair $(e,e')\in M_i$, path $Q(e,e')$ has $e$ as its first edge, $e'$ as its last edge, and every inner vertex on the path lies in $\tilde S_i'$. The total edge-congestion caused by paths in ${\mathcal{Q}}_i$ is then bounded by $O(\log^{17}m)$ with high probability.
This completes the definition of the nice routing paths $\hat{\mathcal{P}}=\set{P(e) \mid e\in \hat E}$ in graph $H$. From the above discussion, the paths in $\hat {\mathcal{P}}$ cause edge-congestion $O(\log^{18}m)$ with high probability. If the congestion caused by the paths in $\hat {\mathcal{P}}$ is greater than $\Theta(\log^{18}m)$, we return FAIL.
Otherwise, we have established that $({\mathcal{R}},\set{{\mathcal{D}}'(R)}_{R\in {\mathcal{R}}})$ is a type-2 legal clustering in $G$ with respect to $v^*$ and ${\mathcal{C}}'$, by providing a nice witness structure in graph $H=G_{|{\mathcal{R}}}$ with respect to set ${\mathcal{C}}'$ of basic clusters. In order to complete the proof of \Cref{lemma: better clustering 2} and \Cref{thm: advanced disengagement get nice instances}, it now remains to prove \Cref{lem: no bad indices}, which we do next.
\subsection{Light Clusters, Bad Clusters, and Path-Guided Orderings}
\label{subsec: guiding paths rotations}
\begin{definition}[Light Cluster]
\label{def: light cluster}
Let $I=(G,\Sigma)$ be an instance of \emph{\textnormal{\textsf{MCNwRS}}\xspace}, and let $C$ be a cluster of $G$. Assume further that we are given a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers. We say that cluster $C$ is \emph{$\beta$-light with respect to ${\mathcal{D}}(C)$} if, for every edge $e\in E(C)$:
%
$$\expect[{\mathcal{Q}}\sim{\mathcal{D}}(C)]{(\cong_G({\mathcal{Q}},e))^2}\leq \beta.$$
\end{definition}
\begin{definition}[Bad Cluster]\label{def: bad cluster}
Let $I=(G,\Sigma)$ be an instance of \emph{\textnormal{\textsf{MCNwRS}}\xspace}, let $C$ be a cluster of $G$, and let $\Sigma(C)$ be the rotation system for $C$ induced by $\Sigma$. We say that $C$ is a \emph{$\beta$-bad} cluster, if:
\[\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma(C))+|E(C)|\geq \frac{|\delta_G(C)|^2}{\beta}. \]
\end{definition}
\paragraph{Path-Guided Orderings.}
Let $I=(G,\Sigma)$ be an instance of \textnormal{\textsf{MCNwRS}}\xspace, and let $C$ be a cluster of $G$. Consider an internal $C$-router ${\mathcal{Q}}=\set{Q(e)\mid e\in \delta_G(C)}$. Recall that there is some vertex $u\in V(C)$ (the center of the router), such that, for all $e\in \delta_G(C)$, path $Q(e)$ has edge $e$ as its first edge, vertex $u$ as its last vertex, and all inner vertices of $Q(e)$ lie in $C$.
We will use the internal $C$-router ${\mathcal{Q}}$, and the rotation system $\Sigma$ for $G$, in order to define a circular ordering ${\mathcal{O}}$ of the edges of $\delta_G(C)$. We refer to the ordering ${\mathcal{O}}$ as \emph{an ordering guided by ${\mathcal{Q}}$ and $\Sigma$}. Ordering ${\mathcal{O}}$ of the edges of $\delta_G(C)$ is constructed as follows.
Denote $\delta_G(u)=\set{a_1,\ldots,a_r}$, where the edges are indexed according to their circular ordering ${\mathcal{O}}_u\in \Sigma$. For all $1\leq i\leq r$, let ${\mathcal{Q}}_i\subseteq {\mathcal{Q}}$ be the set of paths whose last edge is $a_i$. We first define an ordering $\hat {\mathcal{O}}$ of the paths in ${\mathcal{Q}}$, where the paths in sets ${\mathcal{Q}}_1,\ldots,{\mathcal{Q}}_r$ appear in the natural order of their indices, and for all $1\leq i\leq r$, the ordering of the paths in ${\mathcal{Q}}_i$ is arbitrary. Ordering $\hat{\mathcal{O}}$ of the paths in ${\mathcal{Q}}$ naturally defines the ordering ${\mathcal{O}}$ of the edges of $\delta_G(C)$: we obtain the ordering ${\mathcal{O}}$ from $\hat {\mathcal{O}}$ by replacing, for every path $Q(e)\in {\mathcal{Q}}$, the path $Q(e)$ in $\hat {\mathcal{O}}$ with the edge $e$ (the first edge of $Q(e)$). We refer to ${\mathcal{O}}$ as the ordering of the edges of $\delta_G(C)$ that is guided by ${\mathcal{Q}}$ and $\Sigma$, and we denote it by ${\mathcal{O}}^{\operatorname{guided}}({\mathcal{Q}},\Sigma)$. A convenient way to think of the ordering ${\mathcal{O}}^{\operatorname{guided}}({\mathcal{Q}},\Sigma)$ of the edges of $\delta_G(C)$ is that this order is determined by the order in which the paths of ${\mathcal{Q}}$ enter the vertex $u$, which in turn is determined by the rotation ${\mathcal{O}}_u\in \Sigma$ (as the last edge on each path in ${\mathcal{Q}}$ lies in $\delta_G(u)$).
\subsection{Main Definitions}
\label{subsec: main defs for interesting}
\iffalse
We start by defining $\tau^i$-wide and $\tau^s$-small instances; the former is a generalization of wide instances.
\begin{definition}[$\tau^i$-Wide Instance]
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$, and let $\tau^i$ be a parameter. We say that $I$ is a \emph{$\tau^i$-wide} instance, iff there is a vertex $v\in V(G)$, a partition $(E_1,E_2)$ of the edges of $\delta_G(v)$, such that the edges of $E_1$ appear consequently in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and a collection ${\mathcal{P}}$ of at least $\floor{m/\tau^i}$ simple edge-disjoint cycles in $G$, such that every cycle $P\in {\mathcal{P}}$ contains one edge of $E_1$ and one edge of $E_2$.
If no such cycle set ${\mathcal{P}}$ exists in $G$, then we say that $I$ is a \emph{$\tau^i$-narrow} instance.
\end{definition}
\begin{definition}[$\tau^s$-Small Instances]
We say that a subinstance $I'=(G',\Sigma')$ of an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace is \emph{$\tau^s$-small with respect to $I$}, for a parameter $\tau^s>0$, iff $|E(G')|\le |E(G)/\tau^s$.
\end{definition}
Note that \Cref{lem: many paths} requires that every instance $I'$ in the collection ${\mathcal{I}}$ of subinstances of $\check I$ that we compute is either $\mu^{50}$-narrow, or it is $\mu$-small with respect to $\check I$.
In the remainder of this subsection, we assume that we are given some instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, that we would like to decompose into a collection of instances that are either $\tau^i$-narrow or $\tau^s$-small with respect to $I$. In the first phase of our algorithm, we will let $I$ be the input instance $\check I$, and in the second phase of our algorithm, instance $I$ will be one of the subinstances computed in the first phase. We now define a skeleton graph for $I$, and a skeleton structure.
\fi
\subsubsection{Graph Skeleton and Skeleton Structure}
The first central notion that we use is a skeleton of a graph, and its associated skeleton structure.
\begin{definition}[Skeleton of a Graph]
Let $G$ be a graph, and let $K$ be a subgraph of $G$. We say that $K$ is a \emph{skeleton graph} for $G$ iff for every connected component $C$ of $K$, for every edge $e\in E(C)$, graph $C\setminus\set{e}$ is connected. In particular, $|E(C)|>1$ must hold.
\end{definition}
\begin{definition}[Skeleton Structure]
A \emph{skeleton structure} for an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace consists of:
\begin{itemize}
\item a skeleton graph $K\subseteq G$;
\item an orientation $b_u\in \set{-1,1}$ for every vertex $u\in V(K)$; and
\item a planar drawing $\psi$ of $K$ on the sphere, that is consistent with the rotation system $\Sigma$ and the orientations in $\set{b_u}_{u\in V(K)}$. In other words, for every vertex $u\in V(K)$, the images of the edges of $\delta_K(u)$ enter the image of $u$ in $\psi$ according to their order in the rotation ${\mathcal{O}}_u\in \Sigma$ and orientation $b_u$ (so, e.g. if $b_u=1$ then the orientation is counter-clock-wise).
\end{itemize}
\end{definition}
\subsubsection{Faces of Skeleton Structure and Edge Orderings}
\label{subsubsec: faces and orderings}
Consider now some skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace. Let ${\mathcal{F}}({\mathcal K})$ be the set of all faces in the drawing $\psi$ of $K$ on the sphere. Fix some face $F\in {\mathcal{F}}({\mathcal K})$.
We denote by $\overline F$ the region of the sphere that is disjoint from the face $F$. We view $\overline F$ as an open region, so it does not contain the boundary of $F$.
We also denote by $\partial(F)$ the subgraph of $K$ consisting of all vertices and edges of $K$ that are drawn on the boundary of $F$. Note that graph $\partial(F)$ may not be connected, and we denote by ${\mathcal{C}}(F)$ the set of all connected components of $\partial(F)$.
Consider now any such connected component $C\in {\mathcal{C}}(F)$. From the definition of the skeleton $K$, graph $C$ may not have separator edges. Therefore, we can decompose $C$ into a collection ${\mathcal{J}}(C)$ of simple cycles, each of which is contained in $C$, such that: (i) $\bigcup_{C'\in {\mathcal{J}}(C)}E(C')=E(C)$; (ii) the cycles in ${\mathcal{J}}(C)$ do not share any edges; and (iii) if a vertex $v$ lies on more than one cycle of ${\mathcal{J}}(C)$, then $v$ is a separator vertex for $C$. \mynote{should add figure depicting one such $C$}
\iffalse
Consider now any such connected component $C\in {\mathcal{C}}(F)$. From the definition of the skeleton $K$, graph $C$ may not have separator edges. Let $(e_1,e_2,\ldots,e_r)$ be the sequence of the edges of $C$ in the order in which they are encountered as we traverse the boundary of $F$ along $C$. Note that every edge of $C$ appears exactly once in this sequence. It will be convenient for us to view each such connected component $C$ of $\partial(F)$ as a {\bf non-simple} cycle. From the above discussion, there is a collection ${\mathcal{J}}(C)$ of simple cycles, each of which is contained in $C$, such that: (i) $\bigcup_{C'\in {\mathcal{J}}(C)}E(C')=E(C)$; (ii) the cycles in ${\mathcal{J}}(C)$ do not share any edges; and (iii) if a vertex $v$ lies on more than one cycle of ${\mathcal{J}}(C)$, then $v$ is a separator vertex for $C$. We denote ${\mathcal{J}}_F=\bigcup_{C\in {\mathcal{C}}(F)}{\mathcal{J}}(C)$. For every cycle $C'\in {\mathcal{J}}_F$, we define a disc $D(C')$, whose boundary is the image of $C'$ in $\psi$, and whose interior is disjoint from $F$. We denote by $D^0(C')$ the open disc obtained from $D(C')$ by removing its boundary. Lastly, we let ${\mathcal{D}}(F)=\set{D^0(C')\mid C'\in {\mathcal{J}}_F}$. Note that $\overline F=\bigcup_{C'\in {\mathcal{J}}_F}D^0(C')$. We refer to the discs of ${\mathcal{D}}(F)$ as \emph{forbidden discs for face $F$}. \mynote{to do later: do we need the cycles of ${\mathcal{J}}_F$, and the forbidden discs? if so should we define them later? Also notation ${\mathcal{D}}$ is not good because we used it for distributions.}
\fi
Consider now some vertex $u\in K$, and a tiny $u$-disc $D_{\psi}(u)$ in the drawing $\psi$. Denote $\delta_G(u)=\set{e^u_1,\ldots,e^u_{d_u}}$, where $d_u=\delta_G(u)$, and the edges are indexed according to their ordering in ${\mathcal{O}}_u\in \Sigma$. We can then define a collection $p^u_1,\ldots,p^u_{d_u}$ of distinct points on the boundary of the disc $D_{\psi}(u)$, such that the following conditions hold:
\begin{itemize}
\item when traversing the boundary of $D_{\psi}(u)$ in the direction consistent with the orientation $b_u$, then the points $p^u_1,\ldots,p^u_{d_u}$ are encountered in this order; and
\item for every edge $e_i^u$ that lies in the skeleton $K$, point $p_i^u$ is the unique point on $\psi(e^u_i)$ lying on the boundary of $D_{\psi}(u)$.
\end{itemize}
For all $1\leq i\leq d_u$, we view point $p^u_i$ as representing the edge $e^u_i$. For all $1\leq i\leq d_u$, we draw a simple curve $\gamma^u_i$ connecting the image of $u$ in $\psi$ to point $p^u_i$, that is contained in $D_{\psi}(u)$, such that all resulting curves in $\set{\gamma^u_i\mid 1\leq i\leq d_u}$ are mutually disjoint, and, for every edge $e_i^u$ that lies in the skeleton $K$, the corresponding curve $\gamma^u_i$ is the segment of the image of $e_i^u$ in $\psi$ that is contained in $D_{\psi}(u)$. \mynote{should add a figure showing how these points and curves are computed, and color-code them according to which faces the edges belong to.}
Consider now any face $F\in {\mathcal{F}}({\mathcal K})$ with $u\in \partial(F)$. For every edge $e_i^u$, such that point $p_i^u$ lies in the interior of $F$, we say that edge $e_i^u$ \emph{belongs} to the face $F$. We denote by $E^F$ the set of all edges in $\bigcup_{u\in V(K)}\delta(u)$ that belong to face $F$. Note that the edges of $K$ may not belong to any face, and edge sets $\set{E^F\mid F\in {\mathcal{F}}({\mathcal K})}$ define a partition of the set $\textsf{left}(\bigcup_{u\in V(K)}\delta(u)\textsf{right} )\setminus E(K)$ of edges.
Consider again some face $F\in {\mathcal{F}}({\mathcal K})$, and some connected component $C\in {\mathcal{C}}(F)$ of $\partial(F)$. We denote by $E^F(C)\subseteq E^F$ the subset of edges $e\in E^F$ that are incident to vertices of $C$. Consider now a walk inside the face $F$, along the image of $C$, very close to it. We then let ${\mathcal{O}}^F(C)$ be a circular ordering of the edges of $E^F(C)$, that is defined by the order in which we encounter the curves $\gamma_i^u$ that correspond to these edges on this walk. \mynote{need to add a figure depicting the walk}
\subsubsection{Valid $F$-Subgraphs, Valid $F$-Subinstances, Clean Drawings, and Contracted Instances}
We are now ready to define subinstances of the input instance $I$ that are associated with faces of ${\mathcal{F}}({\mathcal K})$.
\begin{definition}[Valid $F$-Subgraph and Valid $F$-Subinstance]
Let $I=(G,\Sigma)$ be an instance of $\ensuremath{\mathsf{MCNwRS}}\xspace$, and let ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ be a skeleton structure for $I$. Let $F$ be a face of ${\mathcal{F}}({\mathcal K})$, and let $G_F$ be a subgraph of $G$. We say that $G_F$ is a \emph{valid $F$-subgraph} of $G$ the following hold:
\begin{itemize}
\item $\partial(F)\subseteq G_F$; and
\item every edge of $G_F$ that lies in $\textsf{left} (\bigcup_{u\in V(K)}\delta_G(u)\textsf{right} )\setminus E(K)$ belongs to $E^F$.
\end{itemize}
We say that instance $I_F=(G_F,\Sigma_F)$ is a \emph{valid $F$-subinstance} of $I$ iff $G_F$ is a valid $F$-subgraph, and $\Sigma_F$ is the rotation system for $G_F$ that is induced by $\Sigma$.
\end{definition}
Given a valid $F$-subgraph $G_F$ of $G$, we call the vertices and edges of $\partial(F)\cap G_F$ \emph{boundary vertices and edges of $G_F$}, and we call remaining vertices and edges of $G_F$ \emph{inner vertices and edges}. Assume now that we are given a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and, for every face $F\in {\mathcal{F}}({\mathcal K})$, a valid $F$-subgraph $G_F$ of $G$. We say that the graph in $\set{G_F\mid F\in {\mathcal{F}}}$ are \emph{internally disjoint} from each other if no vertex of $G$ serves as an inner vertex of more than one such graph, and no edge of $G$ serves as an inner edge of more than one such graph. In other words, the only vertices and edges that the graphs of $\set{G_F\mid F\in {\mathcal{F}}}$ may share are vertices and edges of $K$.
One of our main subroutines decomposes a single $F$-subinstance of the given instance $I$ into two subinstances. For this subroutine the parts of the sekelton $K$ that do not lie in $\partial(F)$ are irrelevant and can be ignored. Similarly, only the $F$-subinstance of the input graph $G$ is relevant to the decomposition procedure, and the remaining parts of $G$ can be ignored. In other words, we will assume that we are given an instance
$I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for $I$, and a face $F$ of ${\mathcal{F}}({\mathcal K})$, such that $I$ is a valid $F$-subinstance of $I$. This is equivalent to saying that every vertex and edge of $K$ lie in $\partial(F)$.
\paragraph{Clean and Dirty Drawings and Crossings.}
We will consider drawings of $F$-instances that have some special properties, that we now summarize.
\begin{definition}[Clean, Semi-Clean and Dirty Drawings]
Let $I=(G,\Sigma)$ be an instance of $\ensuremath{\mathsf{MCNwRS}}\xspace$, let ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ a skeleton structure for $I$, and $F$ a face of ${\mathcal{F}}({\mathcal K})$. Consider a valid $F$-subinstance $I_F=(G_F,\Sigma_F)$ of $I$ and a solution $\phi$ to instance $I_F$. We say that $\phi$ is a \emph{semi-clean solution} iff the following hold:
\begin{itemize}
\item the drawing of $\partial(F)$ induced by $\phi$ is identical to the drawing $\psi$ given by ${\mathcal K}$; and
\item the image of every vertex of $V(G_F)\setminus V(\partial(F))$ lies in the interior of $F$ in $\phi$ (where $F$ is defined with respect to the drawing $\psi$ of ${\mathcal K}$).
\end{itemize}
If, additionally, the drawing of every edge of $G_F$ is contained in $F$, then we say that $\phi$ is a \emph{clean} solution to $I_F$. A solution that is neither clean nor semi-clean is called \emph{dirty}.
\end{definition}
Consider a semi-clean drawing $\phi$ of an $F$-subinstance $I_F=(G_F,\Sigma_F)$ of $I$, and let $(e,e')_p$ be any crossing in this drawing. Since the drawing is semi-clean, it cannot be the case that both $e,e'\in \partial(F)$. We say that crossing $(e,e')_p$ is \emph{dirty} if exactly one of the two edges $e,e'$ lies in $\partial(F)$, and we say that it is \emph{clean} otherwise. We will denote by $\chi^{\mathsf{dirty}}(\phi)$ and by $\chi^{\mathsf{clean}}(\phi)$ the sets of all dirty and clean crossings in drawing $\phi$, respectively. Generally, we will consider semi-dirty solutions $\phi$ to $F$-instances $F$, in which $|\chi^{\mathsf{dirty}}(\phi)|$ is quite small. This will ensure that the number of crossings $(e,e')_{p}$ in the drawing $\phi$, where point $p$ lies outside of $F$, is also small. Indeed, for each such crossing $(e,e')_{p}$, the images of both edges $e$ and $e'$ must cross the image of $\partial(F)$ (since the endpoints of the edges $e,e'$ are drawn in the interior of $F$). Therefore, in order to bound $\mathsf{cr}(\phi)$, we will focus on crossings $(e,e')_{p}$ with $p\in F$. We denote by $\mathsf{cr}^*(\phi)$ the total number of crossings $(e,e')_{p}$ of $\phi$, where the crossing point $p$ lies in $F$.
\paragraph{Contracted $F$-instance.}
Assume that we are given an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for $I$, a face $F\in{\mathcal K})$, and a valid $F$-subinstance $I_F=(G_F,\Sigma_F)$ of $I$. We now define the \emph{contracted} $F$-instance $\hat I_F=(\hat G_F,\hat \Sigma_F)$ corresponding to instance $I_F$. Graph $\hat G_F$ is obtained from graph $G_F$, by contracting, for every connected component $C\in {\mathcal{C}}(F)$ of $\partial(F)$, all vertices of $C$ into a supernode $v_C$. In order to define the rotation system $\hat \Sigma_F$, for every vertex $u\in V(\hat G_F)\cap V(G_F)$, we let the rotation ${\mathcal{O}}_u$ remain the same as in $\Sigma$, and for every supernode $v_C$ (where $C\in {\mathcal{C}}(F)$), we set the corresponding rotation ${\mathcal{O}}_{v_C}$ to be the ordering of the edges of $\delta_{\hat G_F}(v_C)\subseteq E^F(C)$ induced by ${\mathcal{O}}^F(C)$ -- the ordering of the edges of $E^F(C)$ that we have defined in \Cref{subsubsec: faces and orderings}.
We need the following simple observation.
\begin{observation}\label{obs: clean solution to contracted}
There is an efficient algorithm, whose input is an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for $I$, a face $F$ of ${\mathcal{F}}({\mathcal K})$, a valid $F$-subinstance $I_F=(G_F,\Sigma_F)$ of $I$, and a clean solution $\phi$ to instance $I_F$. The output of the algorithm is a solution $\phi'$ to the corresponding contracted $F$-instance $\hat I_F$, with $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$.
\end{observation}
\begin{proof}
Consider the solution $\phi$ to instance $I_F$.
For every connected component $C\in {\mathcal{C}}(F)$, we draw a disc $D(C)$ that contains the image of $C$ in its interior, such that the only vertices of $G_F$ whose image lies in $D(C)$ are the vertices of $C$, and the only edges whose images intersect $D(C)$ are edges of $E(C)\cup E^F(C)$. \mynote{should add figure depicting the disc} We additionally ensure that no crossing of $\phi$ is contained in $D(C)$, and that, for every edge $e\in E^F(C)$, the intersection of $\phi(e)$ with $D(C)$ is a contiguous segment of $\phi(e)$. For every edge $e\in E^F(C)$, denote by $p_e$ the unique point on the boundary of $D(C)$ that lies on $\phi(e)$. From the definition of the ordering ${\mathcal{O}}^F(C)$ of the edges of $E^F(C)$, it is immediate to verify that the circular ordering of the points of $\set{p_e\mid e\in E^F(C)}$ along the boundary of $D(C)$ is precisely ${\mathcal{O}}^F(C)$. For each edge $e\in E^F(C)$, we erase the segment of $\phi(e)$ that is contained in $D(C)$. We then contract the disc $D(C)$ into a single point, that becomes the image of the supernode $v_C$. Once every connected component $C\in {\mathcal{C}}(F)$ is processed, we obtain a valid solution $\phi'$ to the contracted $F$-instance $\hat I_F$, with $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$.
\end{proof}
We note that the converse of the above observation is also true: given a solution $\hat \phi$ to a contracted $F$-instance $\hat I_F$, we can efficiently construct a clean solution $\phi$ to instance $I_F$, with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\hat \phi)$, by simply un-contracting the supernodes, and replacing each supernode $v_C$ with the corresponding graph $C\in {\mathcal{C}}(F)$.
\subsubsection{Skeleton Augmentation}
\label{subsubsec: sekelton augmentation}
Our main partitioning subroutine starts with a valid $F$-instance $I_F=(G_F,\Sigma_F)$, and splits it into two subinstances. This is done by first augmenting the skeleton structure, that will result in the face $F$ being partitioned into two new faces $F_1$ and $F_2$.
Instance $I_F$ is then split into a valid $F_1$ instance and a valid $F_2$-instance. We now define an augmentation of a skeleton structure. We will use two different types of augmentation. The algorithm for Phase 1 will use all of them, and the algorithm for Phase 2 will only use the last two types.
In order to simplify the notaton and reduce the number of types of augmentations, we will sometimes refer to simple cycles as paths. Given such a simple cycle $W$, we will designate one of the vertices $v\in W$ to be the ``endpoint'' of the path. When referring to two endpoints of the path $W$, we will think of both endpoints being $v$.
\begin{definition}[$F$-Augmentation of a Skeleton Structure]
Given an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for $I$, a face $F\in {\mathcal{F}}({\mathcal K})$, and a valid $F$-subinstance $I_F=(G_F,\Sigma_F)$ of $I$, an \emph{$F$-augmentation} of the skeleton structure ${\mathcal K}$ is one of the following:
\begin{itemize}
\item (type-1 augmentation): a simple cycle $P\subseteq G_F$, that is disjoint from $\partial(F)$; or
\item (type-2 augmentation): a simple path $P\subseteq G_F$, such that the endpoints of $P$ are distinct vertices that belong to the same connected component of $\partial(F)$, while all inner vertices of $P$ are disjoint from $\partial(F)$; or
\item (type-3 augmentation): two internally disjoint paths $P_1$, $P_2$, such that all inner vertices of $P_1$ and of $P_2$ are disjoint from $\partial(F)$, and there are two distinct connected components $C,C'\in \partial(F)$, with $P_1$ and $P_2$ each connecting a vertex of $C$ to a vertex of $C'$.
\end{itemize}
\end{definition}
We note that a type-2 augmnetation may be a cycle, in which case the two endpoints of the path are identical and lie on $\partial(F)$.
\mynote{would be good to add figures showing all four types}
In order to simplify the notation, we will always think of an $F$-augmentation as a pair $\Pi_F=\set{P^*_1,P^*_2}$ of internally disjoint paths, whose inner vertices are disjoint from $\partial(F)$. In case of type-1 augmentation, we can partition the cycle $W$ into two paths; if some vertex $v\in V(W)$ lies in $\partial(F)$, we can define the partition so that $v$ is an endpoint of both paths. In case of a type-2 augmentation, we can either split path $P$ into two internally disjoint subpaths, or, if path $P$ consists of a single edge, let $P_1=P$ and $P_2$ consist of a single vertex that is an endpoint of $P_1$.
We will use the $F$-augmentation $\Pi_F$ in order to augment the skeleton $K$. Additionally, we will also need to augment the corresponding skeleton structure. The next definition allows us to do so.
\begin{definition}[$F$-Augmenting Structure]
Given an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for $I$, a face $F\in {\mathcal{F}}({\mathcal K})$, and a valid $F$-subinstance $I_F=(G_F,\Sigma_F)$ of $I$, an \emph{$F$-augmenting structure} for the skeleton structure ${\mathcal K}$ consists of:
\begin{itemize}
\item an $F$-augmentation $\Pi_F=\set{P^*_1,P^*_2}$;
\item for every vertex $u\in (V(P^*_1)\cup V(P^*_2))\setminus V(K)$, an orientation $b_u\in \set{-1,1}$; and
\item a drawing $\psi'$ of the graph $K\cup P^*_1\cup P^*_2$ on the sphere with no crossings, such that $\psi'$ is consistent with the rotation system $\Sigma$ and the orientations in $\set{b_u}_{u\in V(K\cup P^*_1\cup P^*_2)}$, and the drawing of $K$ induced by $\psi'$ is precisely $\psi$.
\end{itemize}
\end{definition}
We can use an $F$-augmenting structure ${\mathcal{A}}=(\Pi_F=\set{P^*_1,P^*_2},\set{b_u}_{u\in V(K\cup P^*_1\cup P^*_2)}, \psi')$ in order to obtain a new skeleton structure ${\mathcal K}'=(K',\set{b'_u}_{u\in V(K')},\psi')$, where $K'=K\cup P^*_1\cup P^*_2$, for every vertex $u\in V(K)$, its new orientation $b'_u$ remains the same as orientation $b_u$ in ${\mathcal K}$, and for every other vertex $u$ of $K'$, its orientation $b'_u$ is the same as the orientation $b_u$ given by the augmenting structure ${\mathcal{A}}$. Drawing $\psi'$ of $K'$ is the same as the drawing given by the augmenting structure ${\mathcal{A}}$. We denote ${\mathcal K}'={\mathcal K}^{+{\mathcal{A}}}$, and we say that ${\mathcal K}'$ is \emph{obtained by augmenting ${\mathcal K}$ via ${\mathcal{A}}$}.
Note that in drawing $\psi'$ there are now two distinct faces, that we denote by $F_1$ and $F_2$, whose union is $F$.
\subsubsection{A Split of an $F$-Instance}
We now define a \emph{split} of instance $I_F$ along an augmenting structure ${\mathcal{A}}$.
\begin{definition}[A Split of an $F$-Instance]
Suppose we are given an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\psi)$ for $I$, a face $F\in {\mathcal{F}}({\mathcal K})$, and a valid $F$-subinstance $I_F=(G_F,\Sigma_F)$ of $I$. Assume further that we are given an $F$-augmenting tructure ${\mathcal{A}}$, and denote by ${\mathcal K}'={\mathcal K}^{+{\mathcal{A}}}$ the skeleton structure obtained by augmenting ${\mathcal K}$ via ${\mathcal{A}}$. Let $F_1$ and $F_2$ be the two faces of $\psi'$ that are contained in the face $F$ of $\psi$. A \emph{split} of $I_F$ along augmenting structure ${\mathcal{A}}$ is a pair $I_{F_1}=(G_{F_1},\Sigma_{F_1}), I_{F_2}=(G_{F_2},\Sigma_{F_2})$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace, for which the following hold.
\begin{itemize}
\item $I_{F_1}$ is a valid $F_1$-subinstance of $I$, and $I_{F_2}$ is a valid $F_2$-subinstance of $I$, for skeleton structure ${\mathcal K}'$;
\item $E(G_{F_1})\cup E(G_{F_2})\subseteq E(G)$; and
\item graphs $G_{F_1}$ and $G_{F_2}$ are internally disjoint. In other words, they may only share vertices and edges of $\partial(F_1)\cap \partial(F_2)$.
\end{itemize}
\end{definition}
Notice that some edges of graph $G_F$ may not lie in $E(G_{F_1})\cup E(G_{F_2})$. We informally refer to such edges as \emph{deleted edges}, and we will sometimes denote the set of such deleted edges by $E^{\mathsf{del}}$. Typically we will ensure that $|E^{\mathsf{del}}|$ is quite small.
The following observation shows that clean solutions to instances $I_{F_1}$ and $I_{F_2}$ can be combined to obtain a clean solution to instance $I_F$. The proof is deferred to Section \ref{subsec: combine solutions for split} of Appendix.
\begin{observation}\label{obs: combine solutions for split}
There is an efficient algorithm, that, given a split $(I_{F_1},I_{F_2})$ of an $F$-instance $I_F$ along an augmenting structure ${\mathcal{A}}$, a clean solution $\phi_1$ to instance $I_{F_1}$ and a clean solution $\phi_2$ to instance $I_{F_2}$, produces a clean solution $\phi$ to instance $I_F$, with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\phi_1)+\mathsf{cr}(\phi_2)+|E^{\mathsf{del}}|\cdot |E(G_F)|$, where $E^{\mathsf{del}}=E(G_F)\setminus (E(G_{F_1})\cup E(G_{F_2})$.
\end{observation}
\subsubsection{Auxiliary Claim}
We will use the following simple auxiliary claim several times. The proof is similar to the proof of Claim 9.9 in \cite{chuzhoy2020towards} and is deferred to Section \ref{subsec: curves orderings crossings} of Appendix.
\begin{claim}\label{claim: curves orderings crossings}
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace, and let ${\mathcal{P}}=\set{P_1,\ldots,P_{4k+2}}$ be a collection of directed edge-disjoint paths in $G$, that are non-transversal with respect to $\Sigma$. For all $1\leq i\leq 4k+2$, let $e_i$ be the first edge on path $P_i$. Assume that there are two distinct vertices $u,v\in V(G)$, such that all paths in ${\mathcal{P}}$ originate at $u$ and terminate at $v$, and assume further that
edges $e_1,\ldots,e_{4k+2}$ appear in this order in the rotation ${\mathcal{O}}_u\in \Sigma$. Lastly, let $\phi$ be any solution to instance $I$, such that the number of crossings $(e,e')_p$ in $\phi$ with $e$ or $e'$ lying in $E(P_1)$ is at most $k$, and assume that the same is true for $E(P_{2k+1})$. Then $\phi$ does not contain a crossing between an edge of $P_1$ and an edge of $P_{2k+1}$.
\end{claim}
\subsection{Main Definitions}
\label{subsec: main defs for interesting}
Throughout, given a graph $H$ and a drawing $\phi$ of $H$ on the sphere or in the plane, we denote by ${\mathcal{F}}(\phi)$ the set of all faces in drawing $\phi$.
We use the notion of a \emph{subdivided graph}.
\begin{definition}[Subdivided graph]
We say that a graph $G$ is a \emph{subdivided graph}, if $G$ does not contain parallel edges, and additionally, for every edge $e=(u,v)$, either $\deg_G(u)\leq 2$ or $\deg_G(v)\leq 2$ holds.
\end{definition}
Note that, if $G$ is any graph, and $G'$ is a graph obtained by suvdividing every edge of $G$, then $G'$ is a subdivided graph, and so is every subgraph of $G'$. In particular, graph $\check G$ associated with instance $\check I$ of \ensuremath{\mathsf{MCNwRS}}\xspace is subdivided, and so is every subgraph of $\check G$.
\subsubsection{Cores and Core Structures}
The first central notion that we use is that of a core, and its associated core structure.
\begin{definition}[Core and Core Structure]\label{def: valid core 1}
Let $G$ be a subgraph of $\check G$, and let $I=(G,\Sigma)$ be the subinstance of $\check I$ defined by $G$. A \emph{core structure} for instance $I$ consists of the following:
\begin{itemize}
\item a connected subgraph $J$ of $G$ called a \emph{core}, such that, for every edge $e\in E(J)$, graph $J\setminus \set{e}$ is connected (but we also allow $J$ to consist of a single vertex);
\item an orientation $b_u\in \set{-1,1}$ for every vertex $u\in V(J)$;
\item a drawing $\rho_J$ of $J$ on the sphere with no crossings, that is consistent with the rotation system $\Sigma$ and the orientations in $\set{b_u}_{u\in V(J)}$. In other words, for every vertex $u\in V(J)$, the images of the edges in $\delta_J(u)$ enter the image of $u$ in $\rho_J$ according to their order in the rotation ${\mathcal{O}}_u\in \Sigma$ and orientation $b_u$ (so, e.g. if $b_u=1$ then the orientation is counter-clock-wise); and
\item a distinguished face $F^*(\rho_J)\in {\mathcal{F}}(\rho_J)$, such that the image of every vertex $u\in V(J)$, and the image of every edge $e\in E(J)$ is contained in the boundary of face $F^*(\rho_J)$ in drawing $\rho_J$.
\end{itemize}
\end{definition}
We denote a core structure by ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$, and we refer to graph $J$ as the \emph{core associated with ${\mathcal{J}}$}. We denote ${\mathcal{F}}^{\operatorname{X}}(\rho_J)={\mathcal{F}}(\rho_J)\setminus\set{F^*(\rho_J)}$, and we refer to the faces in ${\mathcal{F}}^{\operatorname{X}}(\rho_J)$ as the \emph{forbidden faces} of the drawing $\rho_J$.
The last two requirements in the above definition impose a certain structure on the core graph $J$. Specifically, there must be a collection ${\mathcal{W}}$ of edge-disjoint cycles, with $\bigcup_{W\in {\mathcal{W}}}E(W)=E(J)$, such that every pair $W,W'\in {\mathcal{W}}$ of cycles share at most one vertex, which must be a separator vertex for $J$ (see \Cref{fig: core_disc_1} for an illustration).
Note that, since graph $G$ is subdivided, for every edge $e\in E(G)\setminus E(J)$, at most one endpoint of $e$ may lie in $J$. Indeed, assume that $e=(u,v)$. From the definition of subdivided graphs, either $\deg_G(u)\leq 2$ or $\deg_G(v)\leq 2$ holds. Assume w.l.o.g. that it is the former. If $u\in V(J)$, then graph $J$ contains at most one edge incident to $u$, that we denote by $e'$. But then $J\setminus\set{e'}$ is not a connected graph, contradicting the definition of a core.
\begin{figure}[h]
\centering
\subfigure[A core $J$ and its drawing $\rho_J$, with the separator vertices of $J$ shown in green. The distinguished face $F^*(\rho_J)$ is the infinite face in this drawing.]{\scalebox{0.15}{\includegraphics{figs/core_disc_1.jpg}}\label{fig: core_disc_1}}
\hspace{0.8cm}
\subfigure[Disc $D(J)$ associated with core $J$, and its boundary (shown in pink).
]{\scalebox{0.15}{\includegraphics{figs/core_disc_2.jpg}}\label{fig: core_disc_2}}
\caption{An illustration of a core $J$ and its disc $D(J)$.}\label{fig: core_disc_12}
\end{figure}
Consider now a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J,F^*(\rho_J))$, and specifically the drawing $\rho_J$ of the graph $J$ on the sphere. We define a disc $D(J)$, that contains the drawing of $J$ in its interior, such that the boundary of disc $D(J)$ is contained in face $F^*(\rho_J)$, and it is a simple closed curve that closely follows the boundary of $F^*(\rho_J)$ (see \Cref{fig: core_disc_2}).
Consider some vertex $u\in V(J)$, and the tiny $u$-disc $D(u)=D_{\rho_J}(u)$ in the drawing $\rho_J$. Since vertex $u$ lies on the boundary of face $F^*(\rho_J)$, we can define disc $D(u)$ so that, for every maximal segment $\sigma$ on the boundary of $D(u)$ that is contained in $F^*(\rho_J)$, there is a contiguous curve $\sigma'\subseteq \sigma$ of non-zero length, that is contained in the boundary of the disc $D(J)$ (see \Cref{fig: core_disc_3}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.13]{figs/core_disc_3.jpg}
\caption
Illustration of disc $D(u)$ for a core that contains $u$. Disc $D(u)$ is shown in gray, and disc $D(J)$ is shown in pink. Note that both discs share portions of their boundaries that are contained in face $F^*(\rho_J)$ -- the infinite face in the current drawing.
Edges of $J$ incident to $u$ are shown in blue, and all other edges that are incident to $u$ are shown in brown.
}\label{fig: core_disc_3}
\end{figure}
Denote $\delta_G(u)=\set{e^u_1,\ldots,e^u_{d_u}}$, where $d_u=\deg_G(u)$, so that the edges are indexed according to their ordering in the rotation ${\mathcal{O}}_u\in \Sigma$. We can then define a collection $\set{p^u_1,\ldots,p^u_{d_u}}$ of distinct points on the boundary of the disc $D(u)$, such that the following hold:
\begin{itemize}
\item points $p^u_1,\ldots,p^u_{d_u}$ are encountered in this order when traversing the boundary of $D(u)$ in the direction of the orientation $b_u$;
\item for every edge $e_i^u\in E(J)$, point $p_i^u$ is the unique point on the image of $e_i^u$ in $\rho_J$ lying on the boundary of $D(u)$; and
\item if a point $p^u_i$ lies in the interior of face $F^*(\rho_J)$, then it lies on the boundary of the disc $D(J)$.
\end{itemize}
For all $1\leq i\leq d_u$, we view point $p^u_i$ as representing the edge $e^u_i$.
There is one more property that we require from a core structure.
\begin{definition}[Valid Core Structure]\label{def: valid core 2}
We say that a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J,F^*(\rho_J))$ is \emph{valid} if, for every vertex $u\in V(J)$, for every edge $e_i^u\in \delta_G(u)\setminus E(J)$, the corresponding point $p_i^u$ lies in the interior of face $F^*(\rho_J)$ (and hence on the boundary of the disc $D(J)$).
\end{definition}
In the remainder of this section, whenever we use the term ``core structure", we assume that this core structure is valid.
\paragraph{Ordering ${\mathcal{O}}(J)$ of the Edges of $\delta_G(J)$.}
Consider a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J,F^*(\rho_J))$, the drawing $\rho_J$ of $J$, and its corresponding disc $D(J)$. Recall that, for every vertex $u\in V(J)$ and every edge $e_i^u\in \delta_G(J)$, we have defined a point $p_i^u$ on the boundary of the disc $D(J)$ representing the edge $e_i^u$. Recall that each edge $e\in \delta_G(J)$ has exactly one endpoint in $J$. We define a circular oriented ordering ${\mathcal{O}}(J)$ of the edges of $\delta_G(J)$ to be the circular order in which the points $p_i^u$ corresponding to the edges of $\delta_G(J)$ are encountered, as we traverse the boundary of the disc $D(J)$ in the clock-wise direction.
\subsubsection{Drawings of Graphs}
Next, we define a valid drawing of a graph $G$ with respect to a core structure ${\mathcal{J}}$.
\begin{definition}[A ${\mathcal{J}}$-Valid Solution]\label{def: valid drawing}
Let $G$ be a subgraph of $\check G$, let $I=(G,\Sigma)$ be the subinstance of $\check I$ defined by $G$, and let ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J,F^*(\rho_J))$ be a core structure for $I$. A solution $\phi$ of instance $I$ is \emph{${\mathcal{J}}$-valid} if we can define a disc $D'(J)$ that contains the images of all vertices and edges of the core $J$ in its interior, and the image of the core $J$ in $\phi$ is identical to $\rho_{J}$ (including the orientation), with disc $D'(J)$ in $\phi$ playing the role of the disc $D(J)$ in $\rho_{J}$. We sometimes refer to a ${\mathcal{J}}$-valid solution to instance $I$ as a \emph{${\mathcal{J}}$-valid drawing of graph $G$}.
\end{definition}
Abusing the notation, we will not distinguish between disc $D(J)$ in $\rho_J$ and disc $D'(J)$ in $\phi$, denoting both discs by $D(J)$.
Consider now some solution $\phi$ to instance $I$, that is ${\mathcal{J}}$-valid, with respect to some core structure ${\mathcal{J}}$. The image of graph $J$ in $\phi$ partitions the sphere into regions, each of which corresponds to a unique face of ${\mathcal{F}}(\rho_J)$. We do not distinguish between these regions and faces of ${\mathcal{F}}(\rho_J)$, so we view ${\mathcal{F}}(\rho_J)$ as a collection of regions in the drawing $\phi$ of $G$.
Note that the edges of $J$ may participate in crossings in $\phi$, but no two edges of $J$ may cross each other.
Consider now a crossing $(e,e')_p$ in drawing $\phi$. We say that it is a \emph{dirty} crossing if exactly one of the two edges $e,e'$ lies in $E(J)$. We denote by $\chi^{\mathsf{dirty}}(\phi)$ the set of all dirty crossings of drawing $\phi$. We say that an edge $e\in E(G)\setminus E(J)$ is \emph{dirty} in $\phi$ if it participates in some dirty crossing of $\phi$.
Next, we define special types of ${\mathcal{J}}$-valid drawings, called clean and semi-clean drawings.
\begin{definition}[Clean and Semi-Clean Drawings]\label{def: semiclean drawing}
Let $G$ be a subgraph of $\check G$, let $I=(G,\Sigma)$ be the subinstance of $\check I$ defined by $G$, let ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J,F^*(\rho_J))$ be a core structure for $I$, and let $\phi$ be a solution to instance $I$. We say that $\phi$ is a \emph{semi-clean solution to instance $I$}, or a \emph{semi-clean drawing of $G$}, with respect to ${\mathcal{J}}$, if it is a ${\mathcal{J}}$-valid drawing, and, additionally, the image of every vertex of $V(G)\setminus V(J)$ lies outside of the disc $D(J)$ (so in particular it must lie in the interior of the region $F^*(\rho_J)\in {\mathcal{F}}(\rho_J)$).
If, additionally, the image of every edge of $E(G)\setminus E(J)$
is entirely contained in region $F^*(\rho_J)$, then we say that $\phi$ is a \emph{clean} solution to $I$ with respect to ${\mathcal{J}}$, or that it is a \emph{${\mathcal{J}}$-clean} solution.
\end{definition}
Notice that, from the definition, if $\phi$ is a clean solution to instance $I$ with respect to core structure ${\mathcal{J}}$, then the edges of $J$ may not participate in any crossings in $\phi$.
\paragraph{Drawings of Subgraphs and Compatible Drawings.}
Notice that, if $G'$ is a subgraph of $G$ that contains $J$, then a core structure ${\mathcal{J}}$ for instance $I=(G,\Sigma)$ remains a valid core structure for the subinstance $I'=(G',\Sigma')$ of $\check I$ defined by $G'$. Therefore, ${\mathcal{J}}$-valid drawings are well-defined for every subgraph $G'\subseteq G$.
Assume now that we are given a ${\mathcal{J}}$-valid solution $\phi$ to instance $I$, and a subinstance $I'=(G',\Sigma')$ of $I$ that is defined as above. Intuitively, we will often obtain a ${\mathcal{J}}$-valid solution $\phi'$ to instance $I'$ by slightly modifying the solution $\phi$ to instance $I$. We will, however, restrict the types of modifications that we allow. In particular we do now allow adding any new images of edges (or their segments), or new images of vertices to the forbidden regions in ${\mathcal{F}}^{\operatorname{X}}(\rho_J)$. We now define these restrictions formally.
\begin{definition}[Compatible Drawings.]
\label{def: compatible drawing}
Let $G$ be a subgraph of $\check G$, let $I=(G,\Sigma)$ be the subinstance of $\check I$ defined by $G$, let ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ be a core structure for $I$, and let $\phi$ be a ${\mathcal{J}}$-valid solution to instance $I$. Let $G'$ be a subgraph of $G$ with $J\subseteq G'$, and let $I'=(G',\Sigma')$ be the subinstance of $\check I$ defined by $G'$. Finally, let $\phi'$ be a ${\mathcal{J}}$-valid solution to instance $I'$. We say that drawing $\phi'$ of $G'$ is \emph{compatible} with drawing $\phi$ of $G$ with respect to ${\mathcal{J}}$, if the following hold:
\begin{itemize}
\item the image of the core $J$ and the correpsonding disc $D(J)$ in $\phi'$ are identical to those in $\phi$;
\item if a point $p$ is an inner point of an image of an edge in $\phi'$, then it is an inner point of an image of an edge in $\phi$;
\item if a point $p$ is a crossing point between a pair of edges in $\phi'$, then it is a crossing point between a pair of edges in $\phi$;
\item if a point $p$ is an image of a vertex $v$ in $\phi'$, then either (i) point $p$ is an image of vertex $v$ in $\phi$; or (ii) vertex $v$ has degree $2$ in $G'$, and point $p$ is an inner point on an image of an edge in $\phi$;\;
\item if the image of a vertex $v\in V(G')$ lies outside the region $F^*(\rho_J)$ in $\phi'$, then $\phi'(v)=\phi(v)$; and
\item if $\sigma$ is a maximal segment of an image of an edge $e\in E(G')$ in $\phi'$ that is internally disjoint from region $F^*(\rho_J)$, then $\sigma\subseteq \phi(e)$.
\end{itemize}
\end{definition}
Note that, if we obtain drawing $\phi'$ from drawing $\phi$, then the only changes that are allowed outside of region $F^*(\rho_J)$ is the deletion of images of vertices or (segments of) images of edges. In other words, $\big(\phi'(G')\setminus F^*(\rho_J)\big)\subseteq \big(\phi(G)\setminus F^*(\rho_J)\big)$.
\subsubsection{A ${\mathcal{J}}$-Contracted Instance}
Suppose we are given a subgraph $G$ of $\check G$, together with a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for the subinstance $I$ of $\check I$ defined by $G$.
We now define a \emph{${\mathcal{J}}$-contracted} subinstance $\hat I=(\hat G,\hat \Sigma)$ of instance $I$. Graph $\hat G$ is obtained from graph $G$, by contracting the vertices of the core $J$ into a supernode $v_{J}$. In order to define the rotation system $\hat \Sigma$, for every vertex $u\in V(\hat G)\setminus\set{v_J}$, we let the rotation ${\mathcal{O}}_u$ remain the same as in $\Sigma$, and for the supernode $v_{J}$, we set the corresponding rotation ${\mathcal{O}}_{v_{J}}\in \Sigma'$ to be the ordering ${\mathcal{O}}(J)$ of the edges of $\delta_{\hat G}(v_{J})= \delta_G(J)$ that we have defined above (recall that this is the order in which points $p_i^u$ appear on the boundary of disc $D(J)$, for all $u\in V(J)$ and $1\leq i\leq d_u$).
Throughout our algorithm, we will consider subgraphs $G\subseteq \check G$. Each such subgraph will always be associated with a core structure
${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for the subinstance $I$ of $\check I$ defined by $G$. The ${\mathcal{J}}$-contracted subgraph of $I$ will always be denoted by $\hat I=(\hat G,\hat \Sigma)$. We denote by $\hat m(I)=|E(\hat G)|$ -- the number of edges in the ${\mathcal{J}}$-contracted subinstance of $I$.
We need the following simple observation.
\begin{observation}\label{obs: clean solution to contracted}
There is an efficient algorithm, whose input consists of a subgraph $G$ of $\check G$, a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for the subinstance $I=(G,\Sigma)$ of $\check I$ defined by $G$, and a ${\mathcal{J}}$-clean solution $\phi$ to instance $I$. The output of the algorithm is a solution $\hat \phi$ to the corresponding ${\mathcal{J}}$-contracted instance $\hat I=(\hat G,\hat \Sigma)$, with $\mathsf{cr}(\hat \phi)=\mathsf{cr}(\phi)$.
\end{observation}
\begin{proof}
Consider the solution $\phi$ to instance $I$. From the definition of a clean solution, there is a disc $D(J)$ that contains the drawing of $J$ in $\phi$, which is in turn identical to drawing $\rho_J$. For every vertex $u\in V(G)\setminus V(J)$, its image appears outside the disc $D(J)$ in $\phi$. We are also guaranteed that, for every edge $e\in E(G)\setminus E(J)$, its drawing $\phi(e)$ is contained in region $F^*(\rho_J)$.
By slightly manipulating the boundary of the disc $D(J)$, we can ensure that, for every edge $e\in E(G)\setminus E(J)$, if $e$ is not incident to any vertex of $J$, then $\phi(e)$ does not intersect disc $D(J)$, and otherwise, $\phi(e)\cap D(J)$ is a contiguous curve.
For every edge $e\in \delta_G(J)$, denote by $p_e$ the unique intersection point between the boundary of $D(J)$ and the curve $\phi(e)$.
Since the drawing $\phi$ of $G$ is ${\mathcal{J}}$-valid, the circular ordering of the points of $\set{p_e\mid e\in \delta_G(J)}$ on the boundary of disc $D(J)$ is precisely ${\mathcal{O}}(J)$. For each edge $e\in \delta_G(J)$, we erase the segment of $\phi(e)$ that is contained in $D(J)$. We then contract the disc $D(J)$ into a single point, that becomes the image of the supernode $v_{J}$. We have now obtained a valid solution $\hat \phi$ to the ${\mathcal{J}}$-contracted instance $\hat I$, with $\mathsf{cr}(\hat \phi)= \mathsf{cr}(\phi)$.
\end{proof}
We note that the converse of \Cref{obs: clean solution to contracted} is also true: given a solution $\hat \phi$ to the ${\mathcal{J}}$-contracted instance $\hat I$, we can efficiently construct a clean solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)= \mathsf{cr}(\hat \phi)$, as follows. First, we expand the image of the supernode $v_{J}$ so it becomes a disc, that we denote by $D(J)$. We then plant the drawing $\rho_{J}$ of the core $J$ inside the disc. Note that the circular ordering in which the edges of $\delta_{\hat G}(v_{J})$ enter the boundary of the disc $D(J)$ from the outside is identical to the circular ordering in which the edges of $\delta_G(J)=\delta_{\hat G}(v_{J})$ enter the boundary of the disc $D(J)$ from the inside, and their orientations match. Therefore, we can ``glue'' the corresponding curves to obtain, for each edge $e\in \delta_G(J)$, a valid drawing connecting the images of its endpoints.
\subsubsection{Core Enhancement and Promising Sets of Paths}
\label{subsubsec: sekelton enhancement}
Our main subroutine, called \ensuremath{\mathsf{ProcSplit}}\xspace, starts with a subinstance $I=(G,\Sigma)$ of $\check I$ that is defined by a subgraph $G$ of $\check G$, and a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for it. We ``enhance'' the corresponding core $J$ by adding either one cycle, or one path to it, that we refer to as \emph{core enhancenemt}.
We also decompose instance $I=(G,\Sigma)$ into two subinstances, $I_1=(G_1,\Sigma_1)$ and $I_2=(G_2,\Sigma_2)$, where $G_1,G_2\subseteq G$. Using the enhancement for the core structure ${\mathcal{J}}$, we then construct two new core structures: core structure ${\mathcal{J}}_1$ for instance $I_1$, and core structure ${\mathcal{J}}_2$ for instance $I_2$.
In order to avoid cumbersome notation, we will sometimes refer to simple cycles as paths. Given a simple cycle $W$, we will designate one of the vertices $v\in V(W)$ to be the ``endpoint'' of the cycle. When referring to two endpoints of $W$, we will think of both endpoints as being $v$.
We start by defining the notions of \emph{core enhancement} and \emph{core enhancement structure}.
\begin{definition}[Core Enhancement]
Given a subgraph $G$ of $\check G$, and a core structure \newline ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for the subinstance $I=(G,\Sigma)$ of $\check I$ defined by $G$, an \emph{enhancement} of the core structure ${\mathcal{J}}$ is a simple path $P\subseteq G$ (that may be a simple cycle), whose both endpoints belong to $J$, such that $P$ is internally disjoint from $J$ (see \Cref{fig: enhance}).
\end{definition}
\begin{figure}[h]
\centering
\subfigure[When $P$ is a simple path.]{\scalebox{0.13}{\includegraphics{figs/enhan_1.jpg}}\label{fig: enhance_1}}
\hspace{0.08cm}
\subfigure[When $P$ is a simple cycle.]{\scalebox{0.13}{\includegraphics{figs/enhan_2.jpg}}\label{fig: enhance_2}}
\caption{An illustration of an enhancement of a core structure ${\mathcal{J}}$. The core $J$ is shown in blue, and the enhancement in orange.
}\label{fig: enhance}
\end{figure}
For simplicity of notation, we will sometimes refer to an enhancement of a core structure ${\mathcal{J}}$ as an enhancement of the corresponding core $J$, or as a ${\mathcal{J}}$-enhancement. Next, we define a core enhancement structure.
\begin{definition}[Core Enhancement Structure]
Given a subgraph $G$ of $\check G$, and a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for the subinstance $I=(G,\Sigma)$ of $\check I$ defined by $G$, a \emph{${\mathcal{J}}$-enhancement structure} consists of:
\begin{itemize}
\item a ${\mathcal{J}}$-enhancement $P$;
\item an orientation $b_u\in \set{-1,1}$ for every vertex $u\in V(P)\setminus V(J)$; and
\item a drawing $\rho'$ of the graph $J'=J\cup P$ with no crossings, such that $\rho'$ is consistent with the rotation system $\Sigma$ and the orientations $b_u$ for all vertices $u\in V(J')$ (here, the orientations of vertices of $J$ are determined by ${\mathcal{J}}$), and moreover, $\rho'$ is a clean drawing of $J'$ with respect to ${\mathcal{J}}$.
\end{itemize}
\end{definition}
Intuitively, in the drawing $\rho'$ of graph $J'$, the drawing of graph $J$ should be identical to $\rho_J$, and the path $P$ should be drawn inside the region $F^*(\rho_J)$.
For convenience of notation, we denote a ${\mathcal{J}}$-enhancement by
${\mathcal{A}}=\textsf{left}( P,\set{b_u}_{u\in V(J')},\rho'\textsf{right})$, where $J'=J\cup P$. We will always assume that, for every vertex $u\in V(J)$ its orientation $b_u$ in ${\mathcal{A}}$ is identical to its orientation in ${\mathcal{J}}$.
\paragraph{Promising Set of Paths.}
We now define promising sets of paths, that will be used in order to compute an enhancement of a given core structure.
\begin{definition}[Promising Set of Paths]
Let $G$ be a subgraph of $\check G$, let $I=(G,\Sigma)$ be the subinstance of $\check I$ defined by $G$, let ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ be a core structure for $I$, and let ${\mathcal{P}}$ be a collection of simple edge-disjoint paths in $G$, that are internally disjoint from $J$. We say that ${\mathcal{P}}$ is a \emph{promising set of paths}, if there is a partition $(E_1,E_2)$ of the edges of $\delta_G(J)$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}(J)$, and every path in ${\mathcal{P}}$ has an edge of $E_1$ as its first edge, and an edge of $E_2$ as its last edge. \end{definition}
We note that some paths in a promising path set may be cycles.
We show an efficient algorithm to compute a large set of promising paths for an instance $I$ whose corresponding contracted instance is wide. The proof of the following claim is standard, and it is deferred to Section \ref{subsec: proof of finding potential augmentors} of Appendix.
\begin{claim}\label{claim: find potential augmentors}
There is an efficient algorithm that takes as input a subgraph $G\subseteq \check G$ and a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for the subinstance $I=(G,\Sigma)$ of $\check I$ defined by graph $G$, such that the following properties hold:
\begin{itemize}
\item for every vertex $v\in V(G)$ with $\deg_G(v)\geq\frac{\hat m(I)}{\mu^4}$, there is a collection ${\mathcal{Q}}(v)$ of at least $\frac{2\hat m(I)}{\mu^{50}}$ edge-disjoint paths in $G$ connecting $v$ to the vertices of $J$; and
\item the ${\mathcal{J}}$-contracted subinstance $\hat I$ of $I$ is wide.
\end{itemize}
The algorithm computes a promising set of paths for $I$ and ${\mathcal{J}}$, of cardinality $\floor{\frac{\hat m(I)}{\mu^{50}}}$.
\end{claim}
\subsubsection{Splitting a Core Structure and an Instance via an Enhancement Structure}
\paragraph{Splitting the Core Structure.}
Suppose we are given a subgraph $G$ of $\check G$, a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for the subinstance $I=(G,\Sigma)$ of $\check I$ defined by $G$, and an enhancement structure ${\mathcal{A}}=\textsf{left} (P,\set{b_u}_{u\in V(J')},\rho'\textsf{right} )$ for ${\mathcal{J}}$, where $J'=J\cup P$.
We now show an efficient algorithm that, given ${\mathcal{J}}$ and ${\mathcal{A}}$, splits the core structure ${\mathcal{J}}$ into two new core structures, ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$, using the enhancement structure ${\mathcal{A}}$.
We refer to $({\mathcal{J}}_1,{\mathcal{J}}_2)$ as a \emph{split} of the core structure ${\mathcal{J}}$ via the enhancement structure ${\mathcal{A}}$.
We let $\rho'$ be the drawing of the graph $J'$ on the sphere given by the enhancement structure ${\mathcal{A}}$. Recall that there is a disc $D(J)$ that contains the image of $J$ in $\rho'$, whose drawing in $D(J)$ is identical to $\rho_J$. Additionally, all vertices and edges of $P$ must be drawn in region $F^*(\rho_J)$ of $\rho'$. Therefore, in the drawing $\rho'$ of $J'$, there are two faces incident to the image of the path $P$, that we denote by $F_1$ and $F_2$, respectively, and $F_1\cup F_2=F^*(\rho_J)$ holds.
We let $J_1\subseteq J'$ be the graph containing all vertices and edges, whose images lie on the boundary of face $F_1$ in $\rho'$, and we define graph $J_2\subseteq J'$ similarly for face $F_2$.
We now define the core structure ${\mathcal{J}}_1$, whose corresponding core graph is $J_1$. For every vertex $u\in V(J_1)$, its orientation $b_u$ is the same as in ${\mathcal{A}}$. The drawing $\rho_{J_1}$ of $J_1$ is defined to be the drawing of $J_1$ induced by the drawing $\rho'$ of $J'$. Note that $F_1$ remains a face in the drawing $\rho_{J_1}$. We then let $F^*(\rho_{J_1})=F_1$.
The definition of the core structure ${\mathcal{J}}_2$ is symmetric, except that we use core $J_2$ instead of $J_1$ and face $F_2$ instead of $F_1$ (see \Cref{fig: type_2_enhance} for an illustration). This completes the description of the algorithm for computing a split $({\mathcal{J}}_1,{\mathcal{J}}_2)$ of the core structure ${\mathcal{J}}$ via the enhancement structure ${\mathcal{A}}$.
\begin{figure}[h]
\centering
\subfigure[Before the split. Core $J$ is shown in blue and face $F^*(\rho_J)$ is shown in gray.]{\scalebox{0.09}{\includegraphics{figs/enh_type2_1.jpg}}}
\hspace{0.08cm}
\subfigure[Face $F^*(\rho_J)$ is split into faces $F_1$ and $F_2$ by the image of the enhancement path $P$ (shown in orange).]{\scalebox{0.09}{\includegraphics{figs/enh_type2_2.jpg}}}
\hspace{0.08cm}
\subfigure[New cores $J_1$ (top) and $J_2$ (bottom).]{\scalebox{0.09}{\includegraphics{figs/enh_type2_3.jpg}}}
\caption{Splitting a core structure via an enhancement structure.
}\label{fig: type_2_enhance}
\end{figure}
Next, we define a split of an instance $I$ along a core enhancement structure ${\mathcal{A}}$.
\begin{definition}[Splitting an Instance along an Enhancement Structure]\label{def: split}
Let $G$ be a subgraph of $\check G$, let ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ be a core structure for the subinstance $I=(G,\Sigma)$ of $\check I$ defined by $G$, and let ${\mathcal{A}}=\textsf{left} (P,\set{b_u}_{u\in V(J')},\rho'\textsf{right} )$ be an enhancement structure for ${\mathcal{J}}$, where $J'=J\cup P$. Let $({\mathcal{J}}_1,{\mathcal{J}}_2)$ be the split of ${\mathcal{J}}$ via the enhancement structure ${\mathcal{A}}$, and denote by $J_1,J_2$ the cores of ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$, respectively.
A \emph{split} of instance $I$ along ${\mathcal{A}}$ is a pair $I_1=(G_1,\Sigma_1), I_2=(G_2,\Sigma_2)$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace, for which the following hold.
\begin{itemize}
\item $V(G_1)\cup V(G_2)=V(G)$ and $E(G_1)\cup E(G_2)\subseteq E(G)$;
\item every vertex $v\in V(G_1)\cap V(G_2)$ belongs to $V(J_1)\cap V(J_2)$;
\item instance $I_1$ is the subinstance of $\check I$ defined by $G_1$, and instance $I_2$ is the subinstance of $\check I$ defined by $G_2$; and
\item ${\mathcal{J}}_1$ is a valid core structure for $I_1$, and ${\mathcal{J}}_2$ is a valid core structure for $I_2$.
\end{itemize}
\end{definition}
Notice that some edges of graph $G$ may not lie in $E(G_1)\cup E(G_2)$. We informally refer to such edges as \emph{deleted edges}, and we will sometimes denote the set of such deleted edges by $E^{\mathsf{del}}$. Typically we will ensure that $|E^{\mathsf{del}}|$ is quite small.
The following crucial observation shows that clean solutions to instances $I_1$ and $I_2$ can be combined to obtain a clean solution to instance $I$. The proof is deferred to Section \ref{subsec: combine solutions for split} of Appendix.
\begin{observation}\label{obs: combine solutions for split}
There is an efficient algorithm, whose input consists of a subgraph $G$ of $\check G$, a core structure ${\mathcal{J}}$ for the subinstance $I=(G,\Sigma)$ of $\check I$ defined by $G$, a ${\mathcal{J}}$-enhancement structure ${\mathcal{A}}$, and a split $(I_1,I_2)$ of $I$ along ${\mathcal{A}}$, together with a clean solution $\phi_1$ to instance $I_1$ with respect to ${\mathcal{J}}_1$, and a clean solution $\phi_2$ to instance $I_{2}$ with respect to ${\mathcal{J}}_2$, where $({\mathcal{J}}_1,{\mathcal{J}}_2)$ is the split of ${\mathcal{J}}$ along ${\mathcal{A}}$. The algorithm computes a clean solution $\phi$ to instance $I$ with respect to ${\mathcal{J}}$, with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\phi_1)+\mathsf{cr}(\phi_2)+|E^{\mathsf{del}}|\cdot |E(G)|$, where $E^{\mathsf{del}}=E(G)\setminus (E(G_{1})\cup E(G_{2}))$.
\end{observation}
\subsubsection{Auxiliary Claim}
We will use the following simple auxiliary claim several times. The proof is similar to the proof of Claim 9.9 in \cite{chuzhoy2020towards} and is deferred to Section \ref{subsec: curves orderings crossings} of Appendix.
\begin{claim}\label{claim: curves orderings crossings}
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace, and let ${\mathcal{P}}=\set{P_1,\ldots,P_{4k+2}}$ be a collection of directed simple edge-disjoint paths in $G$, that are non-transversal with respect to $\Sigma$. For all $1\leq i\leq 4k+2$, let $e_i$ be the first edge on path $P_i$. Assume that there are two distinct vertices $u,v\in V(G)$, such that all paths in ${\mathcal{P}}$ originate at $u$ and terminate at $v$, and assume further that
edges $e_1,\ldots,e_{4k+2}$ appear in this order in the rotation ${\mathcal{O}}_u\in \Sigma$. Lastly, let $\phi$ be any solution to instance $I$, such that the number of crossings $(e,e')_p$ in $\phi$ with $e$ or $e'$ lying in $E(P_1)$ is at most $k$, and assume that the same is true for $E(P_{2k+1})$. Then $\phi$ does not contain a crossing between an edge of $P_1$ and an edge of $P_{2k+1}$.
\end{claim}
\section{Proof of \Cref{thm: not well connected}}
\label{sec: not well connected}
We assume that we are given a wide instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with $m=|E(G)|$, such that $\mu^{20}\leq m\leq m^*$.
The high-level idea of the proof is to compute a collection ${\mathcal{C}}$ of disjoint clusters in graph $G$ that have the $\alpha_0$-bandwidth property for $\alpha_0=1/\log^3m$ with $|E(G_{|{\mathcal{C}}})|$ sufficiently small, and then to apply the algorithm for computing advanced disengagement from \Cref{thm: disengagement - main} to ${\mathcal{C}}$, to obtain a $\nu$-decomposition of $I$ into subinstances. In order to ensure that all subinstances have the required properties, we need to ensure that, if $C$ is a cluster of ${\mathcal{C}}$ with $|E(C)|> \frac{m}{\mu}$, then for any pair $u,v$ of disctinct vertices of $C$ whose degree in $C$ is at least $\frac m {\mu^6}$, there are at least $\frac{8m}{\mu^{50}}$ edge-disjoint paths connecting $u$ to $v$ in $C$.
We start with the following lemma that allows us to compute the desired collection ${\mathcal{C}}$ of clusters.
\begin{lemma}\label{lem: initial clusters}
There is an efficient algorithm, that, given a wide instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with $m=|E(G)|$, such that $\mu^{20}\leq m\leq m^*$, computes a collection ${\mathcal{C}}$ of disjoint clusters of $G$, that have the following properties:
\begin{itemize}
\item $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$;
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha_0$-bandwidth property, for $\alpha_0=\frac 1 {\log^3m}$;
\item for every cluster $C\in {\mathcal{C}}$, for every pair $u,v$ of distinct vertices of $C$ with $\deg_G(v),\deg_G(u)\geq \frac{m}{\mu^6}$, there is a collection of at least $\frac{8m}{\mu^{50}}$ edge-disjoint paths in $C$ connecting $u$ to $v$; and
\item $|E^{\textnormal{\textsf{out}}}({\mathcal{C}})|\leq \frac{m}{\mu^{27}}$.
\end{itemize}
\end{lemma}
\begin{proof}
The proof of the lemma uses somewhat standard techniques and is similar, for example, to the proof of \Cref{thm:well_linked_decomposition}. We denote by $U$ the set of all vertices of $G$ whose degree is at least $\frac{m}{\mu^6}$. Clearly, $|U|\leq 2\mu^6$.
Our algorithm maintains a collection ${\mathcal{R}}$ of clusters of $G$. Throughout the algorithm, we ensure that the following invariants hold:
\begin{properties}{I}
\item all clusters in ${\mathcal{R}}$ are mutually disjoint; and \label{inv1: disjointness2}
\item $\bigcup_{R\in {\mathcal{R}}}V(R)=V(G)$. \label{inv2: partition2}
\end{properties}
For a given collection ${\mathcal{R}}$ of clusters with the above properties, we define a \emph{budget} $b(e)$ for every edge $e\in E(G)$, as follows. If both endpoints of $e$ lie in the same cluster of ${\mathcal{C}}$, then we set the budget $b(e)=0$. Assume now that the endpoints of $e$ lie in different clusters $R,R'\in {\mathcal{R}}$.
We define $b_R(e)=1+8\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log_{3/2}(|\delta_G(R)|)$, $b_{R'}(e)=1+8\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log_{3/2}(|\delta_G(R')|)$, and $b(e)=b_R(e)+b_{R'}(e)$. Notice that $b(e)\leq 3$ always holds.
For every cluster $R\in {\mathcal{R}}$, we denote $U_R=U\cap V(R)$. For every vertex $u\in U$, we define a budget $b(u)$ of $u$ as follows. Assume that $u\in V(R)$ for some cluster $R\in {\mathcal{R}}$. Then $b(u)=4|U_R|\cdot \frac{m}{\mu^{40}}$.
We denote by $B=\sum_{e\in E(G)}b(e)+\sum_{u\in U}b(u)$ the \emph{total budget in the system}. Clearly, throughout the algorithm, $B\geq \sum_{R\in {\mathcal{R}}}|\delta_G(R)|$ holds.
At the beginning of the algorithm, we set ${\mathcal{R}}=\set{G}$. Clearly, both invariants hold for ${\mathcal{R}}$. Moreover, the budget of every vertex $u\in U$ is at most $4|U|\cdot \frac{m}{\mu^{40}}$, so the total budget of all vertices in $U$ is at most $4|U|^2\cdot \frac{m}{\mu^{40}}\leq \frac{16m}{\mu^{28}}<\frac{m}{\mu^{27}}$ (since $|U|\leq 2\mu^6$), while the budget of every edge of $G$ is $0$. Therefore, at the beginning of the algorithm, $B\leq \frac{m}{\mu^{27}}$ holds. We will ensure that, throughout the algorithm, the total budget $B$ never increases. Since $B\ge \sum_{R\in {\mathcal{R}}}|\delta_G(R)|$ always holds, this will ensure that, at the end of the algorithm,
$\sum_{R\in {\mathcal{R}}}|\delta_G(R)|\leq \frac{m}{\mu^{27}}$ will hold.
Throughout the algorithm, we maintain a partition of the set ${\mathcal{R}}$ of clusters into two subsets: set ${\mathcal{R}}^A$ of \emph{active} clusters, and set ${\mathcal{R}}^I$ of \emph{inactive} clusters. We will ensure that the following additional invariant holds:
\begin{properties}[2]{I}
\item every cluster $R\in {\mathcal{R}}^I$ has the $\alpha_0$-bandwidth property; and \label{inv: last - bw2}
\item for every cluster $R\in {\mathcal{R}}^I$, for every pair $u,v$ of distinct vertices of $U_R$, there is a collection of at least $\frac{8m}{\mu^{50}}$ edge-disjoint paths in $R$ connecting $u$ to $v$.
\end{properties}
At the beginning of the algorithm, we set ${\mathcal{R}}^A={\mathcal{R}}=\set{G}$ and ${\mathcal{R}}^I=\emptyset$. Clearly, all invariants hold then. We then proceed in iterations, as long as ${\mathcal{R}}^A\neq \emptyset$.
In order to execute an iteration, we select an arbitrary cluster $R\in {\mathcal{R}}^A$ to process. We will either establish that $R$ has the $\alpha_0$-bandwidth property in graph $G$, and that for every pair $u,v\in U_R$ of distinct vertices there is a collection of at least $\frac{8m}{\mu^{50}}$ edge-disjoint paths in $R$ connecting $u$ to $v$ (in which case $R$ is moved from ${\mathcal{R}}^A$ to ${\mathcal{R}}^I$); or we will modify the set ${\mathcal{R}}$ of clusters in a way that ensures that the total budget decreases by at least $1/m$. An iteration that processes a cluster $R\in {\mathcal{R}}^A$ consists of two steps. The purpose of the first step is to either establish the $\alpha_0$-bandwidth property of cluster $R$, or to replace it with a collection of smaller clusters in ${\mathcal{R}}^A$. The purpose of the second step is to either establish that, for every pair $u,v\in U_R$ of distinct vertices there is a collection of at least $\frac{8m}{\mu^{50}}$ edge-disjoint paths in $R$ connecting $u$ to $v$, or to modify the set ${\mathcal{R}}$ of clusters in a way that decreases the total budget by at least $1/m$. We now describe each of the two steps in turn.
\paragraph{Step 1: Ensuring the Bandwidth Property.}
Let $R^+$ be the augmentation of the cluster $R$ in graph $G$. Recall that $R^+$ is a graph that is obtained from $G$ through the following process. First, we subdivide every edge $e\in \delta_G(R)$ with a vertex $t_e$, and we let $T=\set{t_e\mid e\in \delta_G(R)}$ be the resulting set of vertices. We then let $R^+$ be the subgraph of the resulting graph induced by vertex set $V(R)\cup T$. We apply Algorithm \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace for computing approximate sparsest cut to graph $R^+$, with the set $T$ of vertices, to obtain a $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)$-approximate sparsest cut $(X,Y)$ in graph $R^+$ with respect to vertex set $T$.
We now consider two cases. The first case happens if $|E_R(X,Y)|\geq \alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T|}$. In this case, we are guaranteed that the minimum sparsity of any $T$-cut in graph $R^+$ is at least $\alpha_0$, or equivalently, set $T$ of vertices is $\alpha_0$-well-linked in $R^+$. From \Cref{obs: wl-bw}, cluster $R$ has the $\alpha_0$-bandwidth property in graph $G$. In this case, we proceed to the second step of the algorithm.
Assume now that $|E_R(X,Y)|< \alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T|}$.
Since $\alpha_0=1/\log^3m$, and $m$ is larger than a large enough constant (because $m\geq \mu^{20}$), and since $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)=O(\sqrt{\log m})$, we get that the sparsity of the cut $(X,Y)$ is less than $1$. Consider now any vertex $t\in T$, and let $v$ be the unique neighbor of $t$ in $R^+$. We can assume w.l.o.g. that either $t,v$ both lie in $X$, or they both lie in $Y$. Indeed, if $t\in X$ and $v\in Y$, then moving vertex $t$ from $X$ to $Y$ does not increase the sparsity of the cut $(X,Y)$. This is because, for any two real numbers $1\leq a<b$, $\frac{a-1}{b-1}\leq \frac a b$. Similarly, if $t\in Y$ and $v\in X$, then moving $t$ from $Y$ to $X$ does not increase the sparsity of the cut $(X,Y)$. Therefore, we assume from now on, that for every vertex $t\in T$, if $v$ is the unique neighbor of $t$ in $R^+$, then either both $v,t\in X$, or both $v,t\in Y$.
Consider now the partition $(X',Y')$ of $V(R)$, where $X'=X\setminus T$ and $Y'=Y\setminus T$. It is easy to verify that $|\delta_G(R)\cap \delta_G(X')|=|X\cap T|$, and $|\delta_G(R)\cap \delta_G(Y')|=|Y\cap T|$. Let $E'=E_G(X',Y')$, and assume w.l.o.g. that $|\delta_G(R)\cap \delta_G(X')|\leq |\delta_G(R)\cap \delta_G(Y')|$. Then $|E'|< \alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|$ must hold. We remove cluster $R$ from sets ${\mathcal{R}}$ and ${\mathcal{R}}^A$, and we add instead every connected component of graphs $G[X']$ and $G[Y']$ to both sets. It is immediate to verify that ${\mathcal{R}}$ remains a collection of disjoint clusters of $G$, and that $\bigcup_{R'\in {\mathcal{R}}}V(R')=V(G)$. Therefore, all invariants continue to hold. We now show that the total budget $B$ decreases by at least $1/m$ as the result of this operation.
Note that the only edges whose budgets may change as the result of this operation are edges of $\delta_G(R)\cup E'$. Observe that, for each edge $e\in \delta_G(R)\cap \delta_G(Y')$, its budget $b(e)$ may not increase. Since we have assumed that $|\delta_G(R)\cap \delta_G(X')|\leq |\delta_G(R)\cap \delta_G(Y')|$, and since $|E'|<|\delta_G(R)|/8$, we get that $|\delta_G(X')|\leq 2|\delta_G(R)|/3$. Therefore, for every edge $e\in \delta_G(X')\cap \delta_G(R)$, its budget $b(e)$ decreases by at least $8\alpha_0 \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot\log_{3/2}(|\delta_G(R)|)-8\alpha_0 \cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot\log_{3/2}(|\delta_G(X')|)$. Since $|\delta_G(X')|\leq 2|\delta_G(R)|/3$, we get that $ \log_{3/2}(|\delta_G(R)|)\geq \log_{3/2}(3|\delta_G(X')|/2)\geq 1+\log_{3/2}(|\delta_G(X')|$. We conclude that the budget $b(e)$ of each edge $e\in \delta_G(X')\cap \delta_G(R)$ decreases by at least $8\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)$.
On the other hand, the budget of every edge $e\in E'$ increases by at most $3$. Since $|E'|\leq \alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|$, we get that the decrease in the budget $B$ is at least:
%
\[
\begin{split}
&8\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(X')\cap \delta_G(R)|-3|E'|\\&\hspace{3cm}\geq 8\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(X')\cap \delta_G(R)|- 3\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|
\\&\hspace{3cm}\geq 5 \alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot |\delta_G(R)\cap \delta_G(X')|\\&\hspace{3cm}>1/m,\end{split}\]
%
since $\alpha_0\geq 1/m$.
Therefore, the total budget of all edges decreases by at least $1/m$. Since the clusters only become smaller, it is easy to verify that the budgets of the vertices of $U$ do not increase.
To conclude, if $|E_R(X,Y)|< \alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T|}$, then we have modified the set ${\mathcal{R}}$ of clusters, so that all invariants continue to hold, and the total budget $B$ decreases by at least $1/m$. In this case, we terminate the current iteration.
From now on we assume that $|E(X,Y)|> \alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \min\set{|X\cap T|,|Y\cap T|}$, which, as observed already, implies that cluster $R$ has the $\alpha_0$-bandwidth property. We now proceed to describe the second step of the algorithm.
\paragraph{Step 2: Ensuring Connectivity of Vertices of $U$.}
If, for every pair $u,v\in U_R$ of distinct vertices, there is a collection of at least $\frac{8m}{\mu^{50}}$ edge-disjoint paths in $R$ connecting $u$ to $v$, then we move cluster $R$ from ${\mathcal{R}}^A$ to ${\mathcal{R}}^I$ and terminate the current iteration. It is easy to verify that all invariants continue to hold.
Assume now that there is a pair $u,v\in U_R$ of distinct vertices, such that the largest collection of edge-disjoint paths in graph $R$ connecting $u$ to $v$ contains fewer than $\frac{8m}{\mu^{50}}$ paths. From the max-flow / min-cut theorem, there is a cut $(X,Y)$ of $R$, with $u\in X$, $v\in Y$, and $|E_R(X,Y)|<\frac{8m}{\mu^{50}}$. We assume w.l.o.g. that $|X\cap U_R|\leq |Y\cap U_R|$. We delete cluster $R$ from ${\mathcal{R}}$ and from ${\mathcal{R}}^A$, and we add instead every connected component of $G[X]$ and $G[Y]$ to both sets. We now show that the total budget decreases by at least $1/m$ as the result of this procedure.
Notice that for every vertex $x\in U$, the budget of $x$ did not increase. Moreover, if $x$ is a vertex of $X\cap U_R$, then its original budget was $4|U_R|\cdot \frac{m}{\mu^{40}}$, and its new budget is $4|U\cap X|\cdot \frac{m}{\mu^{40}}\leq 2|U_R|\cdot \frac{m}{\mu^{40}}$. Since $U\cap X\neq\emptyset$, we get that $\sum_{x\in U}b(x)$ decreased by at least $\frac{2m}{\mu^{40}}$.
Next, we consider the changes to the budgets of the edges. First, every edge in set $E'=E_R(X,Y)$ had budget $0$ at the beginning of the iteration, and has budget at most $3$ at the end of the iteration. Since $|E_R(X,Y)|\leq \frac{8m}{\mu^{50}}$, the increase in the budget of the edges of $E'$ is bounded by $\frac{24m}{\mu^{50}}$.
We now consider two cases. The first case happens if $|\delta_G(R)|\leq \frac{m}{3\mu^{40}}$. In this case, the increase in the budget of every edge $e\in \delta_G(R)$ is bounded by $3$ (since edge budgets may not exceed $3$), and so the total increase in the budgets of edges $e\in \delta_G(R)$ is bounded by $3|\delta_G(R)|\leq \frac{m}{\mu^{40}}$. The total increase in all edge budgets is then bounded by $\frac{m}{\mu^{40}}+\frac{24m}{\mu^{50}}$, and, since the total budgets of all vertices in $U$ decreases by at least $\frac{2m}{\mu^{40}}$, we get that the total budget $B$ decreases by at least $\frac{m}{4\mu^{40}}\leq \frac{1}{m}$.
Lastly, we assume that $|\delta_G(R)|>\frac{m}{3\mu^{40}}$.
Consider some edge $e\in \delta_G(R)$. Since $|\delta_G(X)|,|\delta_G(Y)|\leq |\delta_G(R)|+|E'|$, the increase in the budget of $e$ is bounded by:
\[8\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot (\log_{3/2}(|\delta_G(R)|+|E'|)-\log_{3/2}(|\delta_G(R)|))\leq 8\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log_{3/2}\textsf{left} (1+\frac{|E'|}{|\delta_G(R)|}\textsf{right} ).\]
Since we have assumed that $|\delta_G(R)|>\frac{m}{3\mu^{40}}$, while $|E'|\leq \frac{3m}{\mu^{50}}$, we get that $\frac{|E'|}{|\delta_G(R)|}<1/2$.
Since for all $\epsilon\in (0,1/2)$, $\ln(1+\epsilon)\leq \epsilon$, we get that the increase in the budget of $e$ is bounded by $8\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \frac{|E'|}{|\delta_G(R)|\cdot \ln(3/2)}\leq 24\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \frac{|E'|}{|\delta_G(R)|}$. The increase in the budget of all edges of $\delta_G(R)$ is then bounded by $24\alpha_0\cdot \ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot|E'|\leq \frac{576m\alpha_0\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)}{\mu^{50}}\leq \frac{m}{\mu^{49}}$. Since the budget of all edges in $E'$ increases by at most $\frac{24m}{\mu^{50}}$, and the budget of the vertices of $U$ decreases by at least $\frac{2m}{\mu^{40}}$, the total budget in the system decreases by at least $\frac{m}{\mu^{40}}\geq \frac 1 m$.
Since the initial budget $B$ is bounded by $\frac{m}{\mu^{27}}$, and in every iteration, either a new cluster is added to set ${\mathcal{R}}^I$, or the budget $B$ decreases by at least $1/m$, the number of iterations is bounded by $\operatorname{poly}(m)$, so the algorithm is efficient. Once the algorithm terminates, ${\mathcal{R}}^I={\mathcal{R}}$ holds. We then return the set ${\mathcal{C}}={\mathcal{R}}$ of clusters as the outcome of the algorithm. From our invarinats, we are guaranteed that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$, every cluster $C\in {\mathcal{C}}$ has the $\alpha_0$-bandwidth property, and for every cluster $C\in {\mathcal{C}}$, for every pair $u,v$ of distinct vertices of $C$ with $\deg_G(v),\deg_G(u)\geq \frac{m}{\mu^6}$, there is a collection of at least $\frac{8m}{\mu^{50}}$ edge-disjoint paths in $C$ connecting $u$ to $v$. Since the total budget $B$ remains bounded by $\frac{m}{\mu^{27}}$, and
$\sum_{C\in {\mathcal{C}}}|\delta_G(C)|\leq B$, we get that $\sum_{C\in {\mathcal{C}}}|\delta_G(C)|\leq \frac{m}{\mu^{27}}$ holds.
\end{proof}
We are now ready to complete the proof of \Cref{thm: not well connected}. We start by applying the algorithm from \Cref{lem: initial clusters} to instance $I$, to obtain a collection ${\mathcal{C}}$ of clusters. We then apply Algorithm \ensuremath{\mathsf{AlgAdvancedDisengagement}}\xspace from \Cref{thm: disengagement - main} to instance $I=(G,\Sigma)$, cluster set ${\mathcal{C}}$, parameter $\mu$ that remains unchanged, and parameter $m=|E(G)|$. Let ${\mathcal{I}}$ be the $2^{O((\log m)^{3/4}\log\log m)}$-decomposition of instance $I$ that the algorithm returns. Recall that every instance $I'\in {\mathcal{I}}$ is a subinstance of $I$.
Consider any instance $I'=(G',\Sigma')\in {\mathcal{I}}$, and assume that $|E(G')|>m/\mu$, and that $I'$ is a wide instance. It is enough to prove that instance $I'$ is well-connected.
Indeed, the algorithm from \Cref{thm: disengagement - main} ensures that there is at most one cluster $C\in {\mathcal{C}}$ with $E(C)\subseteq E(G')$.
If no such cluster exists, then $E(G')\subseteq E^{\textnormal{\textsf{out}}}({\mathcal{C}})$. Since $|E^{\textnormal{\textsf{out}}}({\mathcal{C}})|\leq \frac{m}{\mu^{27}}$, $|E(G')|\leq \frac{m}{\mu^{27}}$ must hold in this case, contradicting our assumption that $|E(G')|>m/\mu$. Therefore, there must be a cluser $C\in {\mathcal{C}}$ with $C\subseteq G'$. The algorithm from \Cref{thm: disengagement - main} then guarantees that $E(G')\subseteq E(C)\cup E^{\textnormal{\textsf{out}}}({\mathcal{C}})$.
Consider some vertex $v\in V(G')$ with $\deg_{G'}(v)\geq \frac{|E(G')|}{\mu^5}$. Since $|E(G')|\geq \frac{m}{\mu}$, $\deg_{G'}(v)\geq\frac{|E(G')|}{\mu^5}\geq \frac{m}{\mu^6}$ must hold. In particular, since $I'$ is a subinstance of $I$, and since $|E^{\textnormal{\textsf{out}}}({\mathcal{C}})|\leq \frac{m}{\mu^{27}}$, vertex $v$ must lie in cluster $C$ (as otherwise all edges incident to $v$ in $G'$ belong to $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$), and so $\deg_{G}(v)\geq \frac{m}{\mu^6}$ must hold as well. The algorithm from \Cref{lem: initial clusters} ensures that, for every pair $u,v$ of vertices of $C$ with $\deg_G(u),\deg_G(v)\geq \frac{m}{\mu^6}$, there is a collection ${\mathcal{P}}$ of at least $\frac{8m}{\mu^{50}}\geq\frac{8|E(G')|}{\mu^{50}}$ edge-disjoint paths in $C$ connecting $u$ to $v$. Since $C\subseteq G'$, every path in ${\mathcal{P}}$ is also contained in $G'$. We conclude that instance $I'$ must be well-connected.
\section{An Algorithm for Interesting Subinstances -- Proof of \Cref{lem: many paths}}
\label{sec: many paths}
\input{interesting-intuition}
\newcommand{\tilde \oset}{\tilde {\mathcal{O}}}
\iffalse
Here is a suggested plan.
\begin{itemize}
\item Given an instance $G'$ on $m'$ edges, and parameters $\tau^s,\tau^i$, define the notion of $\tau^s$-small and $\tau^i$-wide/narrow subinstances of $G$.
\item Define well-structured subinstances and corresponding contracted instances.
\item Define well-structured solutions (maybe call them near-perfect solutions?)
\item Main theorem: efficient randomized algorithm that computes a collection ${\mathcal{I}}$ of subinstances of a given instance $I'=(G',\Sigma')$, such that every instance in ${\mathcal{I}}$ is well-structured, and its corresponding contracted instance is either $\tau^s$-small or $\tau^i$-narrow. For each edge of $G'$, if it is special, it lies in at most two of the instances and if it is non-special it lies in at most one instance. There exists a well-structured solution for each subinstance such that the total cost of these solutions is small. (my guess you can just ensure that all these hold with probability $(1-1/\mu)$ or so?)
\end{itemize}
For now you may want to focus on the above, which is the main tool that we use. We will then use the theorem slightly differently in Phase 1 and Phase 2, but at least we won't need to write the same algorithm twice.
\mynote{to do: introduce parameters $\tau^s,\tau^i$. Define instance $\tau^s$-small and $\tau^i$-wide. Do the decomposition w.r.t. these parameters, so that we could reuse this in second stage. This is from my email from June 1:
-say that an instance is $\tau^i$-wide iff there is a set of $m/\tau^i$ of paths connecting 2 vertices (or one vertex with nested cycles). (I think it’s worth it to give a name to this set of paths, say $\tau^i$-wide set of paths, of type 1 or of type 2).
-say that an instance is $\tau^s$-small iff it contains fewer than $m/\tau^s$ edges.
The claim that you are making: start with a cycle-based instance, and assume that it contains at least x cycles. We can compute another collection of cycle-based subinstances, such that each corresponding contracted instance is either $\tau^i$-wide, or $\tau^s$-small.
(Note: for phase 2 decomposition, the instance that we start decomposing already has some cycles, which is why I want to set it up this way, it shouldn’t make much difference for your part).}
\fi
In this section we provide the proof of \Cref{lem: many paths}.
Recall we are given a wide subinstance $I=(G,\Sigma)$ of $I^*$, the input instance to the \textnormal{\textsf{MCNwRS}}\xspace problem, such that $|E(G)|>\mu^{100{50}}$ and $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<|E(G)|^2/\mu^{100{50}}$.
For convenience, throughout this section, we rename the instance $I$ (the input instance in \Cref{lem: many paths}) by $\check I=(\check G,\check \Sigma)$ and denote $\check m=|E(\check G)|$.
\iffalse{previous wide notion recall}
Recall that, in a wide subinstance $I$, there exists a collection ${\mathcal{P}}$ of at least $m/\mu^{{50}}$ edge-disjoint paths in $G$, that is also called a witness of instance $I$, such that either
\begin{properties}{I}
\item there are two vertices $v',v''\in V(G)$, such that every path in ${\mathcal{P}}$ originates at $v'$ and terminates at $v''$; or
\label{Case_1 many_path}
\item there is a single vertex $v\in V(G)$, and two disjoint subsets $E_1,E_2$ of edges of $\delta_{G}(v)$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and every path in ${\mathcal{P}}$ contains an edge of $E_1$ as its first edge and an edge of $E_2$ as its last edge.
\label{Case_2 many_path}
\end{properties}
\fi
We start with the following definitions.
\begin{definition}[$\tau^i$-wide subinstances]
Let $I=(G,\Sigma)$ be a subinstance of $\check I$ with $|E(G)|=m$. Let $\tau^i>0$ be an integer.
We say that $I$ is a \emph{$\tau^i$-wide} subinstance of $\check I$, iff there exists a collection ${\mathcal{P}}$ of at least $\floor{m/\tau^i}$ edge-disjoint paths in $G$, that is also called a \emph{witness} of instance $I$, such that either
\begin{itemize}
\item there are two vertices $v',v''\in V(G)$, such that every path in ${\mathcal{P}}$ originates at $v'$ and terminates at $v''$; or
\label{Case_1 many_path}
\item there is a single vertex $v\in V(G)$, and two disjoint subsets $E_1,E_2$ of edges of $\delta_{G}(v)$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and every path in ${\mathcal{P}}$ contains an edge of $E_1$ as its first edge and an edge of $E_2$ as its last edge.
\label{Case_2 many_path}
\end{itemize}
We say that the instance $I$ is \emph{$\tau^i$-narrow} iff $I$ is not $\tau^i$-wide.
\end{definition}
\begin{definition}[Small subinstances]
We say that a subinstance $I=(G,\Sigma)$ of $\check I$ is $\tau^s$-small for some integer $\tau^s>0$, iff $|E(G)|\le \tau^s$.
\end{definition}
\subsection{Well-Structured Subinstances, Well-Structured Solutions and a Decomposition Lemma}
\znote{maybe we need the notion "boundary of a face" $\partial F$ back in the prelim}
\paragraph{Well-Structured subinstances and contracted subinstances.}
We say that a subinstance $I=(G,\Sigma)$ of the input instance $\check I$ is \emph{well-structured}, iff $G$ is a subgraph of $\check G$, and we are given
\begin{itemize}
\item a $2$-connected subgraph $K$ of $G$, called the \emph{skeleton} of $I$;
\item a planar drawing $\rho$ of $K$ in the sphere (we denote by ${\mathcal{F}}$ the set of faces in $\rho$).
\item a collection ${\mathcal{G}}=\set{G_F}_{F\in {\mathcal{F}}}$ of subgraphs of $G$, where each subgraph $G_F\in {\mathcal{G}}$ is indexed by a face $F\in{\mathcal{F}}$, such that
\begin{itemize}
\item each vertex $v\notin V(K)$ belongs to exactly one subgraphs in ${\mathcal{G}}$;
\item for each vertex $v\in V(K)$ and each face $F\in {\mathcal{F}}$, subgraph $G_F$ contains vertex $v$ iff $v$ lies on the boundary of face $F$ (namely $v\in \partial F$);
\item for each face $F\in {\mathcal{F}}$ and each vertex $v\in \partial F$, if we denote by $e,e'$ the edges of $\partial F$ incident to $v$, then edges of $\delta_{G_F}(v)$ must be consecutive in the circular ordering ${\mathcal{O}}_v\in \Sigma$ and must lie between $e$ and $e'$. Specifically, if $\delta_G(v)=\set{e,e_1,\ldots,e_s,e',e'_1,\ldots,e'_t}$, where the other edges are indexed according to ${\mathcal{O}}_v$, then either $\delta_{G_F}(v)=\set{e_1,\ldots,e_s}$ or $\delta_{G_F}(v)=\set{e'_1,\ldots,e'_t}$.
\end{itemize}
\end{itemize}
We call the tuple $(K,\rho,{\mathcal{F}},{\mathcal{G}})$ the \emph{structure} of instance $I=(G,\Sigma)$.
Additionally, we call edges of $K$ \emph{special edges} of instance $I$.
Let $F$ be a face in ${\mathcal{F}}$. We define the subinstance $I_F=(G_F,\Sigma_F)$, where the rotation system $\Sigma_F$ is induced by $\Sigma$.
We define the subinstance $\tilde I_F=(\tilde G_F,\tilde \Sigma_F)$ as follows. Since skeleton $K$ is $2$-connected and $F$ is a face in a planar drawing of $K$, graph $\partial F$ is the union of a set of vertex-disjoint cycles.
We denote by ${\mathcal{R}}(F)$ the set of vertex-disjoint cycles in $\partial F$.
Graph $\tilde G_F$ is obtained from $G_F$ by contracting, for each cycle $R$ in ${\mathcal{R}}(F)$, all vertices of $R$ into a supernode, which we denote by $v_R$, so $V(\tilde G)=(V(G)\setminus V(\partial F))\cup \set{v_R\mid R\in {\mathcal{R}}(F)}$.
We now define the rotation system $\tilde \Sigma_F$.
Note that, for every vertex $v\in V(G)\setminus V(\partial F)$, $\delta_{\tilde G_F}(v)=\delta_{\tilde G}(v)$, and we define the ordering $\tilde \oset_v$ in $\tilde \Sigma$ to be ${\mathcal{O}}_v$, the ordering on vertex $v$ in $\Sigma$. It remains to define the orderings $\set{\tilde \oset_{v_{R}}\mid R\in {\mathcal{R}}(F)}$. Consider a cycle $R\in {\mathcal{R}}(F)$ with $R=(v_0,v_1,\ldots,v_{s-1},v_0=v_s)$. For each $0\le i\le s-1$, we denote by $e_i$ the edge in $R$ connecting $v_i$ to $v_{i+1}$, and we denote $\delta_{G_F}(v_i)=\set{e_{i-1}, e^i_1,\ldots,e^i_{d_i},e_i}$, where the edges are indexed according to the ordering ${\mathcal{O}}_{v_i}$ in $\Sigma$. From the definition of graph $\tilde G_F$,
$$\delta_{\tilde G}(v_R)=\bigg(\bigcup_{0\le i\le s-1}\delta_{G_F}(v_i)\bigg)\setminus E(R)=\set{e^0_1,\ldots,e^0_{d_0},e^1_1,\ldots,e^1_{d_1},\ldots,e^{s-1}_1,\ldots,e^{s-1}_{d_{s-1}}}.$$ We then define the ordering $\tilde \oset_{v_{R}}$ to be
$e^0_1,\ldots,e^0_{d_0},e^1_1,\ldots,e^1_{d_1},\ldots,e^{s-1}_1,\ldots,e^{s-1}_{d_{s-1}}$. This completes the definition of instance $\tilde I_F$. We call $\tilde I_F$ the
\emph{contracted subinstance} of $I_F$.
\paragraph{Well-Structured solutions.}
Let $I=(G,\Sigma)$ be a well-structured instance with structure $(K,\rho,{\mathcal{F}},{\mathcal{G}})$. We say that a solution $\phi$ of instance $I$ is a \emph{well-structured solution} (or a \emph{well-structured drawing}) for $I$, iff
\begin{properties}{P}
\item the drawing of $K$ induced by drawing $\phi$ is identical to $\rho$;
\item for each face $F\in {\mathcal{F}}$,
\begin{itemize}
\item all vertices of $G_F\setminus \partial F$ are drawn in the interior of face $F$;
\item for each edge $e\in (G_F\setminus F)$ that is incident to some vertex $v\in F$, the intersection of image of $e$ and the tiny $v$-disc $D_{\phi}(v)$ is entirely contained in face $F$;
and
\end{itemize}
\item the number of edges that crosses a special edge of $I$ in $\phi$ is at most $\mathsf{cr}(\phi)\cdot (\tau^i)^{10}/|E(G)|$.
\label{prop: niubi edge crossing special cycle}
\end{properties}
\newcommand{E^{\sf del}}{E^{\sf del}}
\newcommand{well-structured\xspace}{well-structured\xspace}
The main result in this subsection is the following lemma.
\znote{parameters to be determined.}
\begin{lemma}
\label{lem: many path main}
There is an efficient randomized algorithm, that given a well-structured subinstance $I=(G,\Sigma)$ of $\check I$, with structure $(K,\rho,{\mathcal{F}},{\mathcal{G}})$, for which there exists a well-structured\xspace drawing $\phi$, computes a well-structured subinstance $I'=(G',\Sigma)$ of $\check I$ with structure $(K',\rho',{\mathcal{F}}',{\mathcal{G}}')$, such that
\begin{properties}{C}
\item $G'\subseteq G$, $K\subseteq K'$, and the drawing of $K$ induced by $\rho'$ is identical to $\rho$;
\item for each face $F'\in {\mathcal{F}}'$, the contracted subinstance $\tilde I'_F$ is either $\tau^s$-small or $\tau^i$-narrow;
\label{prop: well-structured and contracted}
\end{properties}
and moreover, with probability $1-O(1/\mu^{a})$,
\begin{properties}[2]{C}
\item $|E(G)\setminus E(G')|\le \mathsf{cr}(\phi)\cdot \operatorname{poly}(\mu)/|E(G)|$;
\label{prop: few deleted edges}
\item for each face $F$ and each edge $e\in G_F$, the segment(s) of curve $\phi(e)$ that lies outside face $F$ is identical to the segment(s) of curve $\phi'(e)$ that lies outside face $F$.
\label{prop: forbidden discs}
\item if $\mathsf{cr}(\phi)\le m^2/\mu^{c}$ for some large enough constant $c>0$, then there exists a well-structured solution $\phi'$ for $I'$, such that $\mathsf{cr}(\phi')\le O(\mathsf{cr}(\phi))$.
\label{prop: number of crossings}
\end{properties}
Moreover, there is an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace(I')$, that given, for each face $F'\in {\mathcal{F}}'$, a feasible solution $\phi(\tilde I'_F)$ to the contracted instance $\tilde I'_F$, computes a well-structured solution $\phi(I')$ for instance $I'$, of cost
$\mathsf{cr}(\phi(I'))\leq O\big(\sum_{F'\in {\mathcal{F}}'}\mathsf{cr}(\phi(\tilde I'_F)\big)$.
\end{lemma}
The remainder of this subsection is dedicated to the proof of \Cref{lem: many path main}.
\iffalse
\begin{properties}{C}
\item each instance $I'\in {\mathcal{I}}$ is a well-structured subinstance;
\label{prop: well-structured instances}
\item for each contracted instance $\tilde I'=(\tilde G',\tilde\Sigma') \in \tilde {\mathcal{I}}$, either $\tilde I'$ is non-wide, or $|E(\tilde G')|\le m/\mu$;\label{prop: no longer wide}
\item the sets $\set{\hat E(I')\mid I'\in {\mathcal{I}}}$ of edges are mutually disjoint;\label{prop: private edges disjoint}
\item $|E^{\sf del}|\le \mathsf{cr}\cdot\mu^{O({50})}/m$, and edges of $E^{\sf del}$ does not belong to any instance of ${\mathcal{I}}$;\label{prop: few deleted edges in phase 1}
\item every edge of $E(G)\setminus \bigg(E^{\sf del}\cup \big(\bigcup_{I'\in {\mathcal{I}}}\hat E(I')\big)\bigg)$ is a special edge of exactly two instances in ${\mathcal{I}}$;\label{prop: few edges}
\item there exists, for every instance $I'\in {\mathcal{I}}$, a well-structured solution $\phi_{I'}$, such that\\
$\expect[]{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot \nu$; and \label{prop: small solution cost}
\item there is an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}})$, that, given, for each contracted instance $\tilde I' \in \tilde{\mathcal{I}}$, a feasible solution $\phi(\tilde I')$, computes a feasible solution $\phi$ for instance $I$, of cost $\mathsf{cr}(\phi^*)\leq O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$. \label{prop: alg to put together}
\end{properties}
\znote{to modify the properties?}
In the second phase, we will compute, for each instance $I'\in {\mathcal{I}}$, a new instance $I''$ obtained from $I'$ by deleting a set of edges from $I'$, such that
\znote{to add more properties}
\fi
\input{many_paths_main_decomposition}
\input{many_paths_main_decomposition_analysis}
\section{An Algorithm for Wide and Well-Connected Instances -- Proof of \Cref{lem: many paths}}
\label{sec: many paths}
\newcommand{\check I}{\check I}
\newcommand{\check G}{\check G}
\newcommand{\check \Sigma}{\check \Sigma}
\newcommand{\check m}{\check m}
\newcommand{m/\mu^{50}}{m/\mu^{50}}
\newcommand{50}{50}
\renewcommand{\bm{\psi}}{\tilde \psi}
\renewcommand{\textbf{G}}{\tilde G}
\newcommand{\tilde I}{\tilde I}
\input{interesting-intuition}
\input{main-defs}
\input{decomposition-step-new}
\input{phase-1-alg}
\input{phase-2-alg}
\section{An Algorithm for Interesting Subinstances -- Proof of \Cref{lem: many paths}}
\label{sec: many paths}
\newcommand{\tilde \oset}{\tilde {\mathcal{O}}}
\iffalse
Here is a suggested plan.
\begin{itemize}
\item Given an instance $G'$ on $m'$ edges, and parameters $\tau^s,\tau^i$, define the notion of $\tau^s$-small and $\tau^i$-wide/narrow subinstances of $G$.
\item Define well-structured subinstances and corresponding contracted instances.
\item Define well-structured solutions (maybe call them near-perfect solutions?)
\item Main theorem: efficient randomized algorithm that computes a collection ${\mathcal{I}}$ of subinstances of a given instance $I'=(G',\Sigma')$, such that every instance in ${\mathcal{I}}$ is well-structured, and its corresponding contracted instance is either $\tau^s$-small or $\tau^i$-narrow. For each edge of $G'$, if it is special, it lies in at most two of the instances and if it is non-special it lies in at most one instance. There exists a well-structured solution for each subinstance such that the total cost of these solutions is small. (my guess you can just ensure that all these hold with probability $(1-1/\mu)$ or so?)
\end{itemize}
For now you may want to focus on the above, which is the main tool that we use. We will then use the theorem slightly differently in Phase 1 and Phase 2, but at least we won't need to write the same algorithm twice.
\mynote{to do: introduce parameters $\tau^s,\tau^i$. Define instance $\tau^s$-small and $\tau^i$-wide. Do the decomposition w.r.t. these parameters, so that we could reuse this in second stage. This is from my email from June 1:
-say that an instance is $\tau^i$-wide iff there is a set of $m/\tau^i$ of paths connecting 2 vertices (or one vertex with nested cycles). (I think it’s worth it to give a name to this set of paths, say $\tau^i$-wide set of paths, of type 1 or of type 2).
-say that an instance is $\tau^s$-small iff it contains fewer than $m/\tau^s$ edges.
The claim that you are making: start with a cycle-based instance, and assume that it contains at least x cycles. We can compute another collection of cycle-based subinstances, such that each corresponding contracted instance is either $\tau^i$-wide, or $\tau^s$-small.
(Note: for phase 2 decomposition, the instance that we start decomposing already has some cycles, which is why I want to set it up this way, it shouldn’t make much difference for your part).}
\fi
In this section we provide the proof of \Cref{lem: many paths}.
Recall we are given a wide subinstance $I=(G,\Sigma)$ of $I^*$, the input instance to the \textnormal{\textsf{MCNwRS}}\xspace problem, such that $|E(G)|>\mu^{100{50}}$ and $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<|E(G)|^2/\mu^{100{50}}$.
For convenience, throughout this section, we rename the instance $I$ (the input instance in \Cref{lem: many paths}) by $\check I=(\check G,\check \Sigma)$ and denote $\check m=|E(\check G)|$.
\iffalse{previous wide notion recall}
Recall that, in a wide subinstance $I$, there exists a collection ${\mathcal{P}}$ of at least $m/\mu^{{50}}$ edge-disjoint paths in $G$, that is also called a witness of instance $I$, such that either
\begin{properties}{I}
\item there are two vertices $v',v''\in V(G)$, such that every path in ${\mathcal{P}}$ originates at $v'$ and terminates at $v''$; or
\label{Case_1 many_path}
\item there is a single vertex $v\in V(G)$, and two disjoint subsets $E_1,E_2$ of edges of $\delta_{G}(v)$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and every path in ${\mathcal{P}}$ contains an edge of $E_1$ as its first edge and an edge of $E_2$ as its last edge.
\label{Case_2 many_path}
\end{properties}
\fi
We start with the following definitions.
\begin{definition}[$\tau^i$-wide subinstances]
Let $I=(G,\Sigma)$ be a subinstance of $\check I$ with $|E(G)|=m$. Let $\tau^i>0$ be an integer.
We say that $I$ is a \emph{$\tau^i$-wide} subinstance of $\check I$, iff there exists a collection ${\mathcal{P}}$ of at least $\floor{m/\tau^i}$ edge-disjoint paths in $G$, that is also called a \emph{witness} of instance $I$, such that either
\begin{itemize}
\item there are two vertices $v',v''\in V(G)$, such that every path in ${\mathcal{P}}$ originates at $v'$ and terminates at $v''$; or
\label{Case_1 many_path}
\item there is a single vertex $v\in V(G)$, and two disjoint subsets $E_1,E_2$ of edges of $\delta_{G}(v)$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and every path in ${\mathcal{P}}$ contains an edge of $E_1$ as its first edge and an edge of $E_2$ as its last edge.
\label{Case_2 many_path}
\end{itemize}
We say that the instance $I$ is \emph{$\tau^i$-narrow} iff $I$ is not $\tau^i$-wide.
\end{definition}
\begin{definition}[Small subinstances]
We say that a subinstance $I=(G,\Sigma)$ of $\check I$ is $\tau^s$-small for some integer $\tau^s>0$, iff $|E(G)|\le \tau^s$.
\end{definition}
\subsection{Well-Structured Subinstances, Well-Structured Solutions and a Decomposition Lemma}
\paragraph{Well-Structured subinstances and their contracted subinstances.}
We say that a subinstance $I=(G,\Sigma)$ of the input instance $\check I$ is \emph{well-structured}, iff $G$ is a subgraph of $\check G$, and there is a set ${\mathcal{R}}$ of edge-disjoint cycles in graph $G$, called the \emph{set of special cycles} of $I$, such that, for each cycle $R\in {\mathcal{R}}$, if we denote $R=(v_0,v_1,v_2,\ldots,v_{s-1},v_{s})$ (where $v_{s}=v_0$) and denote, for each index $0\le i\le s-1$, by $e_i$ the edge of $R$ connecting $v_i$ to $v_{i+1}$, then for each $i$, edges $e_{i-1},e_i$ appear consecutively in the ordering ${\mathcal{O}}_{v_i}\in \Sigma$ (we use the convention that $e_s=e_0$).
We call edges of special cycles in ${\mathcal{R}}$ \emph{special edges} of instance $I$.
Let $I=(G,\Sigma)$ be a well-structured subinstance of $\check I$ with ${\mathcal{R}}$ being its set of special cycles. We define the subinstance $\tilde I=(\tilde G,\tilde \Sigma)$ as follows.
Graph $\tilde G$ is obtained from $G$ by contracting, for each cycle $R\in {\mathcal{R}}$, all vertices of $R$ into a supernode, which we denote by $v_R$, so $V(\tilde G)=(V(G)\setminus V({\mathcal{R}}))\cup \set{v_R\mid R\in {\mathcal{R}}}$.
We now define the rotation system $\tilde \Sigma$.
Note that, for every vertex $v\in V(G)\setminus V({\mathcal{R}})$, $\delta_{G}(v)=\delta_{\tilde G}(v)$, and we define the ordering $\tilde \oset_v$ in $\tilde \Sigma$ to be ${\mathcal{O}}_v$, the ordering on vertex $v$ in $\Sigma$. It remains to define the orderings $\set{\tilde \oset_{v_{R}}\mid R\in {\mathcal{R}}}$. Consider a cycle $R\in {\mathcal{R}}$ with $R=(v_0,v_1,\ldots,v_{s-1},v_0)$, and we define edges $e_0,\ldots,e_{s-1}$ similarly as the last paragraph.
For each index $0\le i\le s-1$, we denote $\delta_{G}(v_i)=\set{e_{i-1}, e^i_1,\ldots,e^i_{d_i},e_i}$, where the edges are indexed according to the ordering ${\mathcal{O}}_{v_i}$ in $\Sigma$. By construction of graph $\tilde G$, $\delta_{\tilde G}(v_R)=\bigg(\bigcup_{0\le i\le s-1}\delta_{G}(v_i)\bigg)\setminus E(R)=\set{e^0_1,\ldots,e^0_{d_0},e^1_1,\ldots,e^1_{d_1},\ldots,e^{s-1}_1,\ldots,e^{s-1}_{d_{s-1}}}$. We then define the ordering $\tilde \oset_{v_{R}}$ to be
$e^0_1,\ldots,e^0_{d_0},e^1_1,\ldots,e^1_{d_1},\ldots,e^{s-1}_1,\ldots,e^{s-1}_{d_{s-1}}$. This completes the definition of instance $\tilde I$. We call $\tilde I$ the
\emph{contracted subinstance} of $I$.
\paragraph{Well-Structured solutions.}
Let $I=(G,\Sigma)$ be a well-structured instance and let ${\mathcal{R}}$ be its set of special cycles. We say that a solution $\phi$ of instance $I$ is a \emph{well-structured solution} (or a \emph{well-structured drawing}) for $I$, iff
\begin{properties}{P}
\item for each cycle $R\in {\mathcal{R}}$, the image of $R$ in $\phi$ is a simple closed curve, that we denote by $\eta_R$;
\label{prop: simple curve}
\item the images of cycles in ${\mathcal{R}}$ do not cross each other in $\phi$, i.e., every intersection between two curves $\eta_R,\eta_{R'}$ (where $R,R'$ are distinct cycles in ${\mathcal{R}}$) is the image of some vertex;
\label{prop: curves not crossing}
\item there exists, for each cycle $R\in {\mathcal{R}}$, a disc $D_R$ whose boundary is the curve $\eta_R$, such that interior of the discs $\set{D_R}_{R\in {\mathcal{R}}}$ are mutually disjoint;
\label{prop: discs interior not crossing}
\item for each edge $e$ that is incident to a vertex $v$ of some special cycle $R\in{\mathcal{R}}$, the intersection of image of $e$ and the tiny $v$-disc $D_{\phi}(v)$ is disjoint from the interior of disc $D_R$;
\label{prop: first segments}
\item for each $R\in {\mathcal{R}}$, the interior of $D_R$ does not contain any vertex image in $\phi$; and
\label{prop: discs interior no vertex}
\item the number of edges that crosses a special edge of $I$ in $\phi$ is at most $\mathsf{cr}(\phi)\cdot (\tau^i)^{10}/|E(G)|$.
\label{prop: niubi edge crossing special cycle}
\end{properties}
We say that a solution $\phi$ for instance $I$ is \emph{almost well-structured} iff it satisfies properties \ref{prop: simple curve}--\ref{prop: first segments}.
\newcommand{E^{\sf del}}{E^{\sf del}}
\newcommand{well-structured\xspace}{well-structured\xspace}
The main result in this subsection is the following lemma.
\znote{parameter $a$ to be determined.}
\begin{lemma}
\label{lem: many path main}
There is an efficient randomized algorithm, that given a well-structured subinstance $I=(G,\Sigma)$ of $\check I$ for which there exists a well-structured\xspace drawing $\phi$, computes (i) a well-structured subinstance $\hat I=(\hat G,\Sigma)$ of $I$ with the same set ${\mathcal{R}}$ of special cycles as $I$; and (ii) a collection ${\mathcal{I}}$ of well-structured subinstances of $\hat I$, such that
\begin{properties}{C}
\item for each instance $I'\in {\mathcal{I}}$, its contracted subinstance $\tilde I'$ is either $\tau^s$-small or $\tau^i$-narrow;
\label{prop: well-structured and contracted}
\item for each edge $e\in E(G)$, either $e\in E(G)\setminus E(\hat G)$, or $e$ is a special edge of exactly two subinstances in ${\mathcal{I}}$, or $e$ is a non-special edge of exactly one subinstance in ${\mathcal{I}}$;
\label{prop: edge appearance}
\end{properties}
and moreover, there exists a well-structured drawing $\hat \phi$ for $\hat I$, such that, with probability $1-O(1/\mu^{a})$,
\begin{properties}[2]{C}
\item for each instance $I'\in {\mathcal{I}}$, the drawing of $I'$ induced by $\hat \phi$ is a well-structured solution for $I'$;
\label{prop: well-structured solutions}
\item $|E(G)\setminus E(\hat G)|\le \mathsf{cr}(\phi)\cdot \operatorname{poly}(\mu)/|E(G)|$;
\label{prop: few deleted edges}
\item for each special cycle $R\in {\mathcal{R}}$, its images in drawings $\phi$ and $\hat \phi$ are identical, and the disc $D_R$ it defines in $\phi$ and $\hat \phi$ are the same;
for each edge $e\in E(\hat G)$ and each special cycle $R\in {\mathcal{R}}$: either in both $\phi$ and $\hat \phi$, the image of $e$ is disjoint from the interior of disc $D_R$, or the intersections of the image of $e$ with the interior of disc $D_R$ are identical in $\phi$ and $\hat \phi$;
and
\label{prop: forbidden discs}
\item if $\mathsf{cr}(\phi)\le m^2/\mu^{c}$ for some large enough constant $c>0$, then $\mathsf{cr}(\hat\phi)\le O(\mathsf{cr}(\phi))$.
\label{prop: number of crossings}
\end{properties}
Moreover, there is an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}})$, that given, for each instance $I'\in {\mathcal{I}}$, a feasible solution $\phi(\tilde I')$ the contracted instance $\tilde I'$ of $I'$, computes a well-structured solution $\phi(\hat I)$ for instance $\hat I$, of cost
$\mathsf{cr}(\phi(\hat I))\leq O\big(\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(\tilde I')\big)$.
\end{lemma}
The remainder of this subsection is dedicated to the proof of \Cref{lem: many path main}.
\iffalse
\begin{properties}{C}
\item each instance $I'\in {\mathcal{I}}$ is a well-structured subinstance;
\label{prop: well-structured instances}
\item for each contracted instance $\tilde I'=(\tilde G',\tilde\Sigma') \in \tilde {\mathcal{I}}$, either $\tilde I'$ is non-wide, or $|E(\tilde G')|\le m/\mu$;\label{prop: no longer wide}
\item the sets $\set{\hat E(I')\mid I'\in {\mathcal{I}}}$ of edges are mutually disjoint;\label{prop: private edges disjoint}
\item $|E^{\sf del}|\le \mathsf{cr}\cdot\mu^{O({50})}/m$, and edges of $E^{\sf del}$ does not belong to any instance of ${\mathcal{I}}$;\label{prop: few deleted edges in phase 1}
\item every edge of $E(G)\setminus \bigg(E^{\sf del}\cup \big(\bigcup_{I'\in {\mathcal{I}}}\hat E(I')\big)\bigg)$ is a special edge of exactly two instances in ${\mathcal{I}}$;\label{prop: few edges}
\item there exists, for every instance $I'\in {\mathcal{I}}$, a well-structured solution $\phi_{I'}$, such that\\
$\expect[]{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot \nu$; and \label{prop: small solution cost}
\item there is an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}})$, that, given, for each contracted instance $\tilde I' \in \tilde{\mathcal{I}}$, a feasible solution $\phi(\tilde I')$, computes a feasible solution $\phi$ for instance $I$, of cost $\mathsf{cr}(\phi^*)\leq O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$. \label{prop: alg to put together}
\end{properties}
\znote{to modify the properties?}
In the second phase, we will compute, for each instance $I'\in {\mathcal{I}}$, a new instance $I''$ obtained from $I'$ by deleting a set of edges from $I'$, such that
\znote{to add more properties}
\fi
\input{many_paths_main_decomposition}
\input{many_paths_main_decomposition_analysis}
\subsection{Phase 1: Decomposition into small or non-wide subinstances}
\ifhide
\iffalse
From the above discussion, in both cases eventually we obtain two distinct vertices $v',v''$ of $\hat G$ and a set ${\mathcal{P}}$ of $\floor{m/\mu^{{50}}}$ edge-disjoint paths connecting $v'$ to $v''$. Therefore, from now on we will work with graph $\hat G$, vertices $v',v''$ and the set ${\mathcal{P}}$ of paths in $\hat G$.
Note that the rotation system $\Sigma$ naturally defines a rotation system $\hat \Sigma$ on graph $\hat G$.
We first apply the algorithm from \Cref{lem: non_interfering_paths} to the instance $(\hat G,\hat \Sigma)$ and the set ${\mathcal{P}}$ of paths. We rename the set of paths that we obtain by ${\mathcal{P}}$. From \Cref{lem: non_interfering_paths}, the new set ${\mathcal{P}}$ contains $\floor{m/\mu^{{50}}}$ paths connecting $v'$ to $v''$, that are non-transversal with respect to $\hat \Sigma$.
Also note that the drawing $\phi^*$ of $G$ naturally induces a drawing of $\hat G$, which we denote by $\hat\phi$. It is easy to verify that $\hat\phi$ respects the rotation system $\hat \Sigma$ on $\hat G$, and $\mathsf{cr}(\hat\phi)=\mathsf{cr}$.
\fi
The algorithm for \Cref{lem: many path main} is recursive. Throughout, we maintain a well-structured subinstances of $\hat I$ together with its structure $(\hat K,\hat \rho,\hat {\mathcal{F}},\hat{\mathcal{G}})$, and a set $\hat E^{\sf del}$ of edges of $G$. Initially, instance $\hat I= I$, with structure $(\hat K,\hat \rho,\hat {\mathcal{F}},\hat{\mathcal{G}})=(K,\rho,{\mathcal{F}},{\mathcal{G}})$, and the set $\hat E^{\sf del}$ contains no edges. As the algorithm proceeds, the instance $\hat I$ and the set $\hat E^{\sf del}$ evolve.
\iffalse
\begin{itemize}
\item every instance that ever appears in collection $\hat{{\mathcal{I}}}$ is well-structured; and
\item over the course of the algorithm, the total number of cycles in $\set{{\mathcal{R}}(I')}_{I'\in \hat {\mathcal{I}}}$ may not exceed $m/\tau^s$ \znote{to fix}.
\end{itemize}
\fi
The algorithm continues to be executed as long as there exists some face $F\in \hat {\mathcal{F}}$, such that its contracted instance $\tilde I'_F$ is $\tau^i$-wide and not $\tau^s$-small.
We now describe an iteration of the algorithm. Let $\hat I$ be the current instance with structure $(\hat K,\hat \rho,\hat {\mathcal{F}},\hat{\mathcal{G}})$ and let $F$ be a face in $\hat {\mathcal{F}}$ such that the contracted subinstance $\tilde I_F$ is $\tau^i$-wide and not $\tau^s$-small.
Denote $m'=|E(\tilde G_F)|$, so $m'>\tau^s$.
Recall that there is a subgraph $G_F$ in the collection $\hat {\mathcal{G}}$ associated with face $F$.
Let ${\mathcal{R}}$ be the set of cycles in $\partial F$ and denote $V_{{\mathcal{R}}}=\set{v_R\mid R\in {\mathcal{R}}}$, so $V(\tilde G'_F)=\big( V(G'_F)\setminus V({\mathcal{R}}) \big)\cup V_{{\mathcal{R}}}$.
We use the following simple claim, whose proof uses standard max-flow techniques and is deferred to \Cref{apd: Proof of different cases}.
\iffalse\begin{claim}
\label{clm: find the witness}
There is an efficient algorithm, that, given an instance $I'=(G',\Sigma')$, checks whether or not instance $I'$ is wide, and if so, compute a set ${\mathcal{P}}$ of paths as a witness for $I'$.
\end{claim}
We first use the algorithm in \Cref{clm: find the witness} to compute a set ${\mathcal{P}}$ of paths as the witness for $\tilde I'$.
Consider now the subgraph $\tilde G''$ of graph $\tilde G'$ induced by all edges of $E({\mathcal{P}})$.
We use the following claim, whose proof uses standard techniques and is deferred to \Cref{apd: Proof of different cases}.\fi
\begin{claim}
\label{clm: different cases}
There is an efficient algorithm, that takes as input a well-structured instance $\hat I$ with structure $(\hat K,\hat \rho,\hat {\mathcal{F}},\hat{\mathcal{G}})$, a face $F\in \hat{\mathcal{F}}$ and a parameter $\tau^i>0$, checks whether or not the contracted instance $\tilde I_F$ is $\tau^i$-wide, and if so, compute a set ${\mathcal{P}}$ of edge-disjoint paths or cycles in graph $\tilde G_F$, such that $|{\mathcal{P}}|\ge m'/3\tau^i$, each path of ${\mathcal{P}}$ has length at most $10\tau^i$, and at least one of the following happens:
\begin{enumerate}
\item there are a pair $v',v''\in V(\tilde G')\setminus V_{{\mathcal{R}}}$ of vertices in $\tilde G'$, such that all elements of ${\mathcal{P}}$ are paths that connect $v'$ to $v''$ and do not contain any vertex of $V_{{\mathcal{R}}}$;
\label{case: clean paths}
\item there is a vertex $v_R\in V_{{\mathcal{R}}}$, such that all elements of ${\mathcal{P}}$ are cycles that contain vertex $v_R$ and do not contain any vertex from $V_{{\mathcal{R}}}\setminus \set{v_R}$;
\label{case: cycles}
\item there are a vertex $v_R\in V_{{\mathcal{R}}}$ and a vertex $v\in V(\tilde G')\setminus V_{{\mathcal{R}}}$, such that all elements of ${\mathcal{P}}$ are paths that connect $v_R$ to $v$ and do not contain any vertex from $V_{{\mathcal{R}}}\setminus \set{v_R}$;\label{case: paths with one node}
\item there are two vertices $v_R, v_{R'}\in V_{{\mathcal{R}}}$, such that all elements of ${\mathcal{P}}$ are paths that connect $v_R$ to $v_{R'}$ and do not contain any vertex from $V_{{\mathcal{R}}}\setminus \set{v_R, v_{R'}}$.
\label{case: paths with two nodes}
\end{enumerate}
\end{claim}
\znote{overview here}
\fi
\subsubsection{Step 1. Computing a new special cycle $K$}
\label{Step 1}
Let $v',v''$ be the endpoints of all paths of ${\mathcal{P}}$. Since no paths of ${\mathcal{P}}$ contains any vertex of $V_{{\mathcal{R}}}$, the paths of ${\mathcal{P}}$ are also paths in graph $G'$.
We view all paths of ${\mathcal{P}}$ as being directed from vertex $v'$ to vertex $v''$.
We denote $L=|{\mathcal{P}}|$ and denote by $E_1=\set{e_0,\ldots,e_{L-1}}$ the set of edges that serve as the first edge of paths of ${\mathcal{P}}$, where the edges are indexed according to the ordering ${{\mathcal{O}}}_{v'}\in \Sigma$.
We choose an index $z\in \set{0,\ldots,\floor{L/2}-1}$ uniformly at random, and let $z'= z+\floor{L/2}$. We denote by $\tilde P$ the path in ${\mathcal{P}}$ whose first edge is $e_z$, and denote by $\tilde P'$ the path in ${\mathcal{P}}$ whose first edge is $e_{z'}$. We define $\tilde K=\tilde P\cup \tilde P'$, so $\tilde K$ is a subgraph of $G'$. Since $I'=(G',\Sigma')$ is a subinstance of the input instance $I=(G,\Sigma)$, $G'$ is a subgraph of $G$, so $\tilde K$ is also a subgraph of $G$.
If paths $\tilde P$ and $\tilde P'$ do not share inner vertices, then we denote $P=\tilde P$, $P'=\tilde P'$, and $K=\tilde K$.
If paths $\tilde P$ and $\tilde P'$ share inner vertices, then we let $v^*$ be the first inner vertex of $\tilde P$ that is also contained in $\tilde P'$. We denote by $P$ the subpath of $\tilde P$ between $v'$ and $v^*$, and denote by $P'$ the subpath of $\tilde P'$ between $v'$ and $v^*$. Therefore, the paths $P, P'$ are internally vertex-disjoint.
We unname the previous vertex $v''$ and rename $v^*$ by $v''$.
We then define $K=P\cup P'$, so $K$ is a cycle, and $K\subseteq\tilde K$ is a subgraph of $G'$. Therefore, $P$ and $P'$ are paths of $G'$ and are also paths of $G$.
Since every path of ${\mathcal{P}}$ has length at most $\tau^i\cdot\mu^{10{50}}$, from the construction of $K$, $|E(K)|\le 2\tau^i\cdot\mu^{10{50}}$.
In the following steps, we will ``cut the instance $I'$ along $K$'' into two subinstances $I'_1, I'_2$, and cycle $K$ computed in this step will be a special cycle in both $I'_1$ and $I'_2$.
\iffalse
We prove the following observation.
\iffalse{whp version}
\begin{observation}
\label{obs: skeleton number of edges}
The probability that $K$ contains more than $\mu^{{50}+a}$ edges is at most $O(1/\mu^{a})$.
\end{observation}
\begin{proof}
For each $0\le i\le \floor{L/2}-1$, we denote by $K_i$ the subgraph formed by the union of the path in ${\mathcal{P}}$ whose first edge is $e_i$ and the path in ${\mathcal{P}}$ whose first edge is $e_{i+\floor{L/2}}$. Since the paths of ${\mathcal{P}}$ are edge-disjoint, the subgraphs $K_0,\ldots ,K_{\floor{L/2}-1}$ are edge-disjoint, and therefore the expected number of edges of a random subgraph of $\set{K_0,\ldots,K_{\floor{L/2}-1}}$ is at most $m/\floor{L/2}< O(\mu^{{50}})$. From Markov's bound, the probability that $K$ contains more than $\mu^{{50}+a}$ edges is at most $O(\mu^{{50}})/\mu^{{50}+a}=O(1/\mu^{a})$.
\end{proof}
\fi
\begin{observation}
\label{obs: skeleton number of edges}
$\expect[]{|E(K)|}=O(\tau^i)$.
\end{observation}
\begin{proof}
For each $0\le z\le \floor{L/2}-1$, we denote by $\tilde K_z$ the subgraph formed by the union of the path in ${\mathcal{P}}$ whose first edge is $e_z$ and the path in ${\mathcal{P}}$ whose first edge is $e_{z+\floor{L/2}}$. Since the paths of ${\mathcal{P}}$ are edge-disjoint, the subgraphs $\tilde K_0,\ldots, \tilde K_{\floor{L/2}-1}$ are edge-disjoint, and therefore the expected number of edges in $\tilde K$, a random element in set $\set{\tilde K_0,\ldots, \tilde K_{\floor{L/2}-1}}$, is at most $O(m'/L) \le O(\tau^i)$. Since $K\subseteq \tilde K$, we get that $\expect[]{|E(K)|}=O(\tau^i)$.
\end{proof}
\paragraph{Bad Event ${\cal{E}}_1$.}
We say that the bad event ${\cal{E}}_1$ happens if $K$ contains more than $\tau^i\cdot\mu^{a}$ edges.
From \Cref{obs: skeleton number of edges} and Markov's bound, $\Pr[{\cal{E}}_1]\le O(1/\mu^{a})$.
\fi
\subsubsection{Step 2. Computing the inner and outer sides of $K$}
\label{Step 2}
Recall that all paths of ${\mathcal{P}}$ are being directed from $v'$ to $v''$. Recall that we have denoted by $e_0,\ldots, e_{L-1}$ the set of first edges of paths in ${\mathcal{P}}$, where edges are indexed according to the ordering ${\mathcal{O}}_{v'}$. For each $0\le j\le L-1$, we denote by $P_j$ the path in ${\mathcal{P}}$ that has $e_{j}$ as its first edge, and we denote by $e'_{j}$ the last edge of $P_j$.
In order to ``cut instance $I'$ along $K$'' into subinstances $I'_1$ and $I'_2$, we will need to define the ``inner and outer sides'' of $K$, which we do in this step.
First, since the a drawing of any graph and its mirror image are viewed as identical drawings, we can fix the orientation of the ordering ${\mathcal{O}}_{v'}$ to be positive, without loss of generality.
This oriented ordering naturally induces an oriented ordering on the set $\set{P_0,\ldots, P_{L-1}}$ of paths, which we denote by $({\mathcal{O}}',b')$.
Next we consider the vertex $v''$. Note that edges $e'_{0},\ldots,e'_{L-1}$ belong to set $\delta(v'')$, and the circular ordering ${\mathcal{O}}_{v''}\in \Sigma$ is an circular ordering on the set $\set{e'_{0},\ldots,e'_{L-1}}$, which in turn induces an circular ordering on the set $\set{P_{0},\ldots,P_{L-1}}$ of paths, that we denote by ${\mathcal{O}}''$.
Clearly, computing an orientation of the unoriented ordering ${\mathcal{O}}_{v''}$ is equivalent to computing an orientation of the unoriented ordering ${\mathcal{O}}''$.
We compute an orientation $b''\in \set{-1,1}$ of ${\mathcal{O}}''$ as follows.
We set $b''=1$ iff $\mbox{\sf dist}(({\mathcal{O}}', b'),({\mathcal{O}}'',1))\le \mbox{\sf dist}(({\mathcal{O}}', b'),({\mathcal{O}}'',-1))$; otherwise we set $b''=-1$.
In other words,
\[b''=\arg\min_{b\in \set{-1,1}}\set{\mbox{\sf dist}(({\mathcal{O}}', b'),({\mathcal{O}}'',b))}.\]
Consider now a vertex $v\in V(K)$. Assume without loss of generality that $v$ belongs to path $P$. We denote by ${\mathcal{P}}_v=\set{P_{i_1},\ldots,P_{i_q}}$ the set of all paths ${\mathcal{P}}$ that contains vertex $v$.
For each $1\le j\le q$, we denote by $\tilde e_{i_j}$ the first edge of $P_{i_j}$ that is incident to $v$ (recall that paths of ${\mathcal{P}}$ are being directed from $v'$ to $v''$),
and recall that we have denoted by $e_{i_j}$ the first edge of $P_{i_j}$, so $e_{i_j}$ is incident to $v'$.
Note that we have already fixed an orientation of the ordering ${\mathcal{O}}_{v'}$ to be positive. Let $b(v)\in \set{-1,1}$ be an orientation of ordering ${\mathcal{O}}_v$.
Let $e$ be the first edge of $P$ and let $e'$ be the first edge of $P'$.
We say that an index $i_j$ is \emph{screwed under $b(v)$}, iff either edges $e,e_{i,j},e'$ appear clockwise in the oriented ordering $({\mathcal{O}}_{v'},+1)$ and edges $\tilde e, \tilde e_{i,j},\tilde e'$ appear clockwise in the oriented ordering $({\mathcal{O}}_{v},b(v))$ or edges $e',e_{i,j},e$ appear clockwise in the oriented ordering $({\mathcal{O}}_{v'},+1)$ and edges $\tilde e',\tilde e_{i,j},\tilde e$ appear clockwise in the oriented ordering $({\mathcal{O}}_{v},b(v))$.
We then set $b(v)$ as follows. If the number of indices in $\set{i_1,\ldots, i_q}$ that are screwed under $1$ is greater than the number of indices in $\set{i_1,\ldots, i_q}$ that are screwed under $-1$, then we set $b(v)=-1$, otherwise we set $b(v)=1$.
\iffalse
Note that the circular ordering ${\mathcal{O}}'$ on set ${\mathcal{P}}$ of paths naturally induces a circular ordering on the set ${\mathcal{P}}_v$ of paths, that we denote by ${\mathcal{O}}'_v$. Also note that the circular ordering ${\mathcal{O}}_v\in \Sigma$ naturally induces a circular ordering on the set $\set{\tilde e_{i_j}\mid 1\le j\le q}$ of edges, which in turn induces an ordering on the set ${\mathcal{P}}_v$ of paths, that we denote by ${\mathcal{O}}''_v$.
Clearly, computing an orientation of the unoriented ordering ${\mathcal{O}}_{v}$ is equivalent to computing an orientation of the unoriented ordering ${\mathcal{O}}''_v$.
We compute an orientation $b''_v\in \set{-1,1}$ of ${\mathcal{O}}''_v$ as follows.
We set $b(v)=1$ iff $\mbox{\sf dist}(({\mathcal{O}}'_v, b'),({\mathcal{O}}''_v,1))\le \mbox{\sf dist}(({\mathcal{O}}'_v, b'),({\mathcal{O}}''_v,-1))$; otherwise we set $b(v)=-1$.
In other words,
\[b(v)=\arg\min_{b\in \set{-1,1}}\set{\mbox{\sf dist}(({\mathcal{O}}'_v, b'),({\mathcal{O}}''_v,b))}.\]
\fi
So far we have computed, for each vertex of $K$, an orientation of its circular ordering in $\Sigma$. Let $v$ be an inner vertex of $P$, and denote $\delta(v)=\set{\tilde e, \hat e_1, \hat e_2,\ldots, \hat e_p,\tilde e', \hat e'_1, \hat e'_2, \ldots, \hat e'_q}$, where edges $\tilde e,\tilde e'$ belong to $P$, edge $\tilde e$ precedes edge $\tilde e'$ in $P$, and the edges appear clockwise in the oriented ordering on $v$.
We say that edges $\hat e_1, \hat e_2,\ldots, \hat e_p$ are on the \emph{inner side} of $K$, and say that edges $\hat e'_1, \hat e'_2, \ldots, \hat e'_q$ are on the \emph{outer side} of $K$.
Similarly, let $v$ be an inner vertex of $P'$, and denote $\delta(v)=\set{\tilde e, \hat e_1, \hat e_2,\ldots, \hat e_p,\tilde e', \hat e'_1, \hat e'_2, \ldots, \hat e'_q}$, where edges $\tilde e,\tilde e'$ belong to $P'$, edge $\tilde e$ precedes edge $\tilde e'$ in $P'$, and the edges appear clockwise in the oriented ordering on $v$.
We say that edges $\hat e_1, \hat e_2,\ldots, \hat e_p$ are on the \emph{outer side} of $K$, and say that edges $\hat e'_1, \hat e'_2, \ldots, \hat e'_q$ are on the \emph{inner side} of $K$.
For vertex $v''$, we denote assume that $\delta(v'')=\set{\tilde e, \hat e_1, \hat e_2,\ldots, \hat e_p,\tilde e', \hat e'_1, \hat e'_2, \ldots, \hat e'_q}$, where edges $\tilde e$ is the last edge of $P$ and $\tilde e'$ is the last edge of $P'$, and the edges appear clockwise in the oriented ordering at $v''$.
We say that edges $\hat e_1, \hat e_2,\ldots, \hat e_p$ are \emph{on the inner side} of $K$, and say that edges $\hat e'_1, \hat e'_2, \ldots, \hat e'_q$ are \emph{on the outer side} of $K$.
\subsubsection{Step 3. Decomposing instance $I'$ into two subinstances}
\label{Step 3}
Consider the contracted subinstance $\tilde I'=(\tilde G',\tilde \Sigma')$ of the well-structured instance $I'$. Since we have assumed that paths of ${\mathcal{P}}$ do not contain special vertices of graph $G'$, $K$ is also a cycle in $\tilde G'$, and for each vertex $v\in V(K)$, $\delta_{G'}(v)=\delta_{\tilde G'}(v)$.
We will decompose graph $\tilde G'$ into two subgraphs, and then construct two subinstances of $I'$ based on them.
We denote by $\hat E$ the set of all edges of $E(\tilde G')\setminus E(K)$ that is incident to a vertex of $K$, so each edge of $\hat E$ is incident to either vertex $v'$, or vertex $v''$, or an inner vertex of $P$ or $P'$.
We denote by $\hat E_{\mathsf{in}}$ the set of all edges of $\hat E$ that are on the inner side of $K$, and denote by $\hat E_{\mathsf{out}}$ the set of all edges of $\hat E$ that are on the outer side of $K$. Clearly, sets $\hat E_{\mathsf{in}},\hat E_{\mathsf{out}}$ partition $\hat E$. Note that, from the construction of cycle $K$, $|\hat E_{\mathsf{in}}|\ge |\hat E_{\mathsf{in}}\cap \delta_{G'}(v')| \ge \floor{L/2}$, and similarly $|\hat E_{\mathsf{out}}|\ge |\hat E_{\mathsf{out}}\cap \delta_{G'}(v')|\ge \floor{L/2}$.
In order to decompose graph $\tilde G'$ into two subgraphs, it will be convenient for us to slightly modify the graph $\tilde G'$, by splitting each vertex $v\in V(K)$ into three vertices: vertex $v_{\mathsf{in}}$, that is incident to the edges of $\hat E_{\mathsf{in}}\cap \delta(v)$, vertex $v_{\mathsf{out}}$, that is incident to the edges of $\hat E_{\mathsf{out}}\cap \delta(v)$, and vertex $v_{M}$, that is incident to two edges of $E(K)\cap \delta(v)$.
We denote by $\hat G$ the resulting graph.
We define $V_{\mathsf{in}}=\set{v_{\mathsf{in}}\mid v\in V(K)}$, and define sets $V_{\mathsf{out}}$ and $V_{M}$ similarly, so $V(\hat G)=V_{\mathsf{in}}\cup V_{\mathsf{out}}\cup V_{M}\cup (V(G')\setminus V(K))$.
Note that in graph $\hat G$, $K$ is a cycle on vertices of $V_M$ and is disconnected from the rest of the graph.
We then compute, using the standard max-flow techniques, a minimum-cardinality set $E^{*}$ of edges of $\hat G$, whose removal separates vertices of $V_{\mathsf{in}}$ from $V_{\mathsf{out}}$ in $\hat G$. We add the edges of $E^*$ into the set $\hat E^{\sf del}$.
We now construct two new instances as follows. First, let $\tilde G_{\mathsf{in}}$ be the graph obtained from subgraph of $\hat G$ consisting of all connected components of $\hat G\setminus E^{*}$ that contain vertices of $V_{\mathsf{in}}\cup V_{M}$, by identifying, for each vertex $v\in V(K)$, the corresponding vertex $v_{\mathsf{in}}\in V_{\mathsf{in}}$ with the corresponding vertex $v_M\in V_M$.
It is easy to verify that $\tilde G_{\mathsf{in}}$ is a subgraph of $\tilde G'$.
We now let $G_{\mathsf{in}}$ be the graph obtained from $\tilde G_{\mathsf{in}}$ by uncontracting each special vertex $v_R$ back to the cycle $R$. It is easy to verify that graph $G_{\mathsf{in}}$ is a subgraph of $G'$. We then let $\Sigma_{\mathsf{in}}$ be the rotation system on $G_{\mathsf{in}}$ induced by the rotation system $\Sigma'$ on $G'$, and define the instance $I_{\mathsf{in}}=(G_{\mathsf{in}}, \Sigma_{\mathsf{in}})$. For $I_{\mathsf{in}}$ to be a well-structured subinstance, we still need to define its set ${\mathcal{R}}(I_{\mathsf{in}})$ of special cycles.
Note that each special cycle of instance $I'$ now either entirely belongs to $G_{\mathsf{in}}$ or entirely belongs to $G_{\mathsf{out}}$.
We then let set ${\mathcal{R}}(I_{\mathsf{in}})$ contain all special cycles in ${\mathcal{R}}(I')$ that entirely belongs to $G_{\mathsf{in}}$ and the cycle $K$, namely ${\mathcal{R}}(I_{\mathsf{in}})=\set{R\mid v_R\in V(\tilde G_{\mathsf{in}})}\cup \set{K}$. Clearly, the cycles in ${\mathcal{R}}(I_{\mathsf{in}})$ are edge-disjoint (and even vertex-disjoint in this case).
This completes the construction of instance $I_{\mathsf{in}}=(G_{\mathsf{in}}, \Sigma_{\mathsf{in}})$.
It is then easy to verify that $I_{\mathsf{in}}$ is a well-structured subinstance of $I'$.
We construct the well-structured subinstance $I_{\mathsf{out}}=(G_{\mathsf{out}}, \Sigma_{\mathsf{out}})$ and its set ${\mathcal{R}}(I_{\mathsf{out}})$ of special cycles similarly.
We prove the following immediate observations, that will be later used in \Cref{subsec: analysis of decomposition lemma} for the analysis of the algorithm.
\begin{observation}
\label{obs: number of edges in base}
$|E(G_{\mathsf{in}})|+|E(G_{\mathsf{out}})|\le |E(G')|+|E(K)|$, and
$|E(G_{\mathsf{in}})|, |E(G_{\mathsf{out}})|\le |E(G')|-\tau^s/6\tau^i$.
\end{observation}
\begin{proof}
From the construction of graphs $G_{\mathsf{in}},G_{\mathsf{out}}$, $E(G_{\mathsf{in}}),E(G_{\mathsf{out}})\subseteq E(G')$, and the only edges of $G'$ that belong to both $E(G_{\mathsf{in}})$ and $E(G_{\mathsf{out}})$ are edges of $E(K)$, so $|E(G_{\mathsf{in}})|+|E(G_{\mathsf{out}})|\le |E(G')|+|E(K)|$.
On the other hand, observe that edges of $\hat E_{\mathsf{out}}$ may not belong to set $E(G_{\mathsf{in}})$, since otherwise some vertex of $V_{\mathsf{out}}$ (the endpoint of any edge of $\hat E_{\mathsf{out}}\cap E(G_{\mathsf{in}})$ in $V_{\mathsf{out}}$) must be reachable from some vertex of $V_{\mathsf{in}}$ in $G_{\mathsf{in}}$, contradicting the definition of $E^*$ and the construction of $\tilde G_{\mathsf{in}}$.
Therefore, $|E(G_{\mathsf{in}})|\le |E(G')|-\floor{L/2}$, and similarly $|E(G_{\mathsf{out}})|\le |E(G')|-\floor{L/2}$.
From \Cref{clm: different cases}, $L\ge m'/3\tau^i\ge \tau^s/3\tau^i$, and \Cref{obs: number of edges in base} now follows.
\end{proof}
\begin{observation}
\label{obs: 2 new special cycles}
$|{\mathcal{R}}(I_{\mathsf{in}})|+|{\mathcal{R}}(I_{\mathsf{out}})|= |{\mathcal{R}}(I')|+2$.
\end{observation}
\begin{proof}
Since the cycle $K$ in graph $\tilde G'$ does not contain any special vertex, from the construction of the graphs $G_{\mathsf{in}}, G_{\mathsf{out}}$, each special cycle in ${\mathcal{R}}(I')$ belongs to either $G_{\mathsf{in}}$ or $G_{\mathsf{out}}$ (but not both). Also note that the sets ${\mathcal{R}}(I_{\mathsf{in}})$ and ${\mathcal{R}}(I_{\mathsf{out}})$ contain one additional new cycle $K$. Therefore, $|{\mathcal{R}}(I_{\mathsf{in}})|+|{\mathcal{R}}(I_{\mathsf{out}})|= |{\mathcal{R}}(I')|+2$.
\end{proof}
This completes the description of the three steps in an iteration. Finally, we replace the subinstance $I'=(G',\Sigma')$ in the collection $\hat {\mathcal{I}}$ with two new subinstances $I_{\mathsf{in}}=(G_{\mathsf{in}}, \Sigma_{\mathsf{in}})$ and $I_{\mathsf{out}}=(G_{\mathsf{out}}, \Sigma_{\mathsf{out}})$, and proceed to the next iteration.
We denote by ${\mathcal{I}}$ the resulting collection $\hat{{\mathcal{I}}}$ at the end of the algorithm, and denote by $E^{\sf del}$ the resulting set $\hat E^{\sf del}$ at the end of the algorithm.
Finally, we define the subinstance $\hat I$ as follows. Its graph $\hat G=G\setminus E^{\sf del}$, its rotation system is $\Sigma$, and its set of special cycles is ${\mathcal{R}}$, the set of special cycles of $I$. This completes the description of the algorithm.
\subsubsection{Analysis of the algorithm}
\label{subsec: analysis of decomposition lemma}
In this subsection we provide the proof of \Cref{lem: many path main} with some details deferred to subsequent sections.
First, note that initially the collection $\hat{{\mathcal{I}}}$ contains one well-structured subinstance, and from the description of the algorithm, in each iteration we replace one well-structured subinstance in $\hat{{\mathcal{I}}}$ with two new well-structured subinstances. Therefore, all subinstances in the resulting collection ${\mathcal{I}}$ are well-structured subinstances. On the other hand, recall that the recursive algorithm continues to be executed as long as there exists some instance $I'\in \hat{\mathcal{I}}$, such that its contracted subinstance $\tilde I'$ is $\tau^i$-wide and not $\tau^s$-small. Therefore, when the algorithm terminates, property \ref{prop: well-structured and contracted} has to be satisfied.
Second, note that in each iteration, we start with an instance $I'$ in the current collection $\hat {\mathcal{I}}$, compute a cycle $K$ that is edge-disjoint from all special cycles in ${\mathcal{R}}(I')$ and then decompose instance $I'$ into two subinstances $I_{\mathsf{in}}$ and $I_{\mathsf{out}}$, such that only edges of $E(K)$ appear in both instances. Therefore, if an edge in the input instance $I'$ is not a special edge of any subinstance that ever appeared in $\hat {\mathcal{I}}$, then it either belongs to exactly one resulting instance or belongs to set $E^{\sf del}$; otherwise it eventually belongs to exactly two resulting instances. Therefore, property \ref{prop: edge appearance} holds.
In order to show that the properties \ref{prop: well-structured solutions}--\ref{prop: number of crossings} also hold and there exists an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}})$ that satisfies the properties required in \Cref{lem: many path main}, it is convenient for us to associate a tree $T$ with the algorithm, called the \emph{partitioning tree}. Each vertex of $T$ corresponds to a well-structured subinstance that once appeared in the collection $\hat{{\mathcal{I}}}$. We grow the tree $T$ along with the iterative algorithm as follows. Initially, $T$ contains a single root denoted by $v(I)$, where $I$ is the input well-structured subinstance to \Cref{lem: many path main}. Consider now some vertex $v(I')$ in $T$, where $I'$ is a subinstance that undergoes Steps 1, 2 and 3. From the algorithm, the instance $I'$ is replaced by two new subinstances $I_{\mathsf{in}}$ and $I_{\mathsf{out}}$. We then create two nodes, $v(I_{\mathsf{in}})$ and $v(I_{\mathsf{out}})$, that become the child vertices of $v(I')$ in $T$.
\subsubsection*{Number of Iterations}
We first analyze the number of the iterative algorithm using $T$. We use the following observation.
\begin{observation}
\label{obs: number of iterations}
If \fbox{$\tau^s\ge 100\cdot\tau^i\mu^{10{50}}$}, then
$|V(T)|\le 200\cdot m\tau^i/\tau^s$.
\end{observation}
\begin{proof}
We show that, if $\tau^s\ge 100\cdot\tau^i\mu^{10{50}}$, then the algorithm will terminate in $200\cdot m\tau^i/\tau^s$ iterations, which implies \Cref{obs: number of iterations}.
In fact, we will show that, for each subinstance $I'=(G',\Sigma')$ that ever appeared in $\hat {\mathcal{I}}$,
the number of vertices in the subtree of $T$ rooted at node $v(I')$, which we denote by $N_{T}(I')$, is at most $$f(I')=\max\set{\frac{|E(G')|-\tau^s}{\tau^s/100\tau^i},0}+3,$$
which, when applied to the initial instance $I=(G,\Sigma)$ where $|E(G)|=m$, immediately implies that the algorithm will terminate in $200\cdot m\tau^i/\tau^s$ iterations.
Consider now an instance $I'$ that once appeared in $\hat {\mathcal{I}}$. Assume first that $v(I')$ is a leaf of $T$, then the number of vertices in the subtree of $T$ rooted at node $v(I')$ is $1$, which is at most $f(I')$ (note that $f(I')\ge 3$ always holds).
Assume now that $v(I')$ is not a leaf of $T$, we let $I'_1=(G'_1,\Sigma'_1)$, $I'_2=(G'_2,\Sigma'_2)$ be two new instances constructed the iteration of processing $I'$. Since $|E(K)|\le 2\tau^i\cdot\mu^{10{50}}$, from \Cref{obs: number of edges in base} and the definition of ${\cal{E}}_1$, $|E(G'_1)|+|E(G'_2)|\le |E(G')|+2\tau^i\cdot\mu^{10{50}}$ and
$|E(G'_1)|, |E(G'_2)|\le |E(G')|-\tau^s/6\tau^i$. We consider the following cases.
\textbf{Case 1. Both nodes $v(I'_1), v(I'_2)$ are leaves of $T$.}
In this case, $N_T(I')\le 3 \le f(G')$.
\textbf{Case 2. One of the nodes $v(I'_1), v(I'_2)$ is a leaf of $T$.}
Assume without loss of generality that $v(I'_1)$ is a leaf and $v(I'_2)$ is not a leaf, so $|E(G')|-\tau^s/6\tau^i\ge |E(G'_2)|\ge \tau^s$, so
\[
\begin{split}
N_{T}(I')=2+N_{T}(I'_2)\le 2+f(I'_2)= & \text{ }2+ \frac{|E(G'_2)|-\tau^s}{\tau^s/10\tau^i}+3
\\
\le & \text{ } 2+ \frac{|E(G')|-(\tau^s/6\tau^i)-\tau^s}{\tau^s/100\tau^i}+3
\\
\le & \text{ } 2+ \frac{|E(G')|-\tau^s}{\tau^s/100\tau^i}-16+3
\\
\le & \text{ } \frac{|E(G')|-\tau^s}{\tau^s/100\tau^i}+3 =f(I').
\end{split}
\]
\textbf{Case 3. Both nodes $v(I'_1), v(I'_2)$ are not leaves of $T$.}
Then $|E(G'_1)|, |E(G'_1)|\ge \tau^s$, and so
\[
\begin{split}
N_{T}(I')\le 1+N_{T}(I'_1)+N_{T}(I'_2)\le & \text{ } 1+f(I'_1)+f(I'_2)
\\
= & \text{ } 7+ \frac{|E(G'_1)|-\tau^s}{\tau^s/100\tau^i}+\frac{|E(G'_2)|-\tau^s}{\tau^s/100\tau^i}
\\
\le & \text{ } 7+ \frac{|E(G')|+2\tau^i\cdot\mu^{10{50}}-2\tau^s}{\tau^s/100\tau^i}
\\
\le & \text{ } 7+ \frac{|E(G')|-\tau^s}{\tau^s/100\tau^i}
+
\frac{2\tau^i\cdot\mu^{10{50}}-\tau^s}{\tau^s/100\tau^i}
\\
\le & \text{ } \frac{|E(G')|-\tau^s}{\tau^s/100\tau^i}+3 =f(I'),
\end{split}
\]
where the last inequality uses the fact $\tau^s\ge 100\tau^i\mu^{10{50}}$.
\end{proof}
\iffalse{number of all special edges}
As an immediate corollary, we have the following bound the number of special edges in all resulting instances in ${\mathcal{I}}$.
\begin{corollary}
\label{cor: number of special edges}
The total number of special edges in all instances in ${\mathcal{I}}$ is at most
$800m(\tau^i)^2\mu^{10{50}}/\tau^s$.
\end{corollary}
\begin{proof}
From \Cref{obs: 2 new special cycles}, after each iteration, the number of special cycles in all subinstances in the collection $\hat{\mathcal{I}}$ increases by $2$. Since the number of iterations is bounded by $200\cdot m\tau^i/\tau^s$, and the length of each special cycle is at most $2\tau^i\cdot\mu^{10{50}}$, we get that the total number of special edges in all resulting instances in ${\mathcal{I}}$ is bounded by
$2\cdot \big(200\cdot m\tau^i/\tau^s\big)\cdot \big(2\tau^i\cdot\mu^{10{50}}\big)=800m(\tau^i)^2\mu^{10{50}}/\tau^s$.
\end{proof}
\fi
We now proceed to analyze the iterative algorithm using the partitioning tree $T$.
We will maintain a well-structured drawing $\tilde \phi$ of the input instance $I=(G,\Sigma)$, that is initialized to be the input well-structured drawing $\phi$ of $I$ and evolves as the iterative algorithm proceeds.
\znote{the correct notion and claim?}
We denote by $\bar I$ the well-structured instance obtained from $I$ by only deleting from it the edges of $\hat E^{\sf del}$. As more edges are added to set $\hat E^{\sf del}$, the instance $\bar I$ evolves, and it is easy to verify that eventually instance $\bar I$ becomes the output instance $\hat I$.
We say that a well-structured drawing $\tilde \phi$ of instance $\bar I$ is \emph{well-structured with respect to the current collection $\hat {\mathcal{I}}$}, iff for each subinstance $I'\in \hat{\mathcal{I}}$, the drawing of $I'$ induced by $\tilde \phi$ is a well-structured drawing of ${\mathcal{I}}$.
We first focus on an iteration, in which some instance $I'=(G',\Sigma')$ in collection ${\mathcal{I}}$ is replaced by two new subinstances $I_{\mathsf{in}}$ and $I_{\mathsf{out}}$.
\begin{observation}
\label{obs: not crossing your own special then not crossing any special}
Let $\phi$ be a drawing of instance $\bar I$ that is well-structured with respect to the current collection $\hat {\mathcal{I}}$. For an edge $e$ that belongs to a unique instance $I'\in \hat {\mathcal{I}}$, if $e$ does not cross any special edge of $I'$ in $\tilde \phi$, then $e$ does not cross any special edge of any instnace of $\hat {\mathcal{I}}$ in $\tilde \phi$.
\end{observation}
\begin{proof}
\znote{to complete}
\end{proof}
We use the following claim.
\begin{claim}
\label{lem: well-structured drawing splitting}
Consider an iteration of the algorithm in which a well-structured instance $I'=(G',\Sigma')$ is processed and eventually replaced by two new well-structured instances, that we denote by $I_{\mathsf{in}}, I_{\mathsf{out}}$.
Let $\tilde \phi$ be a well-structured drawing of $\bar I$ with respect to the collection $\hat{\mathcal{I}}$ before this iteration.
Then there exists a well-structured drawing $\tilde \phi'$ of $\bar I$ with respect to the collection $\hat{\mathcal{I}}$ after this iteration, such that
\begin{itemize}
\item for each special cycle $R$ in the collection $\hat{\mathcal{I}}$ before this iteration, its images in drawings $\tilde\phi$ and $\tilde \phi'$ are identical, and the disc $D_R$ it defines in $\phi$ and $\hat \phi$ are the same; for each edge $e\in E(\hat G)$ and each special cycle $R\in {\mathcal{R}}$: either in both $\phi$ and $\hat \phi$, the image of $e$ is disjoint from the interior of disc $D_R$, or the intersections of the image of $e$ with the interior of disc $D_R$ are identical in $\phi$ and $\hat \phi$; and
\item if \fbox{$\mathsf{cr}(\tilde\phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$}, then
$\mathsf{cr}(\tilde\phi')\le \mathsf{cr}(\tilde \phi)\cdot(1+1/\mu^{a})$.
\end{itemize}
\end{claim}
We provide the proof of \Cref{lem: well-structured drawing splitting} in \Cref{sec: Proof of well-structured drawing splitting}, after we complete the proof of \Cref{lem: many path main} using \Cref{lem: well-structured drawing splitting}.
\subsubsection{Proof of \Cref{lem: well-structured drawing splitting}}
\label{sec: Proof of well-structured drawing splitting}
In this subsection we provide the proof of \Cref{lem: well-structured drawing splitting}, thus completing the proof of \Cref{lem: many path main}.
We focus on an iteration, in which some instance $I'=(G',\Sigma')$ in collection ${\mathcal{I}}$ is replaced by two new subinstances $I_{\mathsf{in}}$ and $I_{\mathsf{out}}$.
We denote $m'=|E(G')|$.
Recall that currently we have a drawing $\tilde\phi$ of the instance $\bar I$ before this iteration, that is well-structured with respect to the collection $\hat{\mathcal{I}}$ before this iteration.
Recall that in Step 1 we computed a cycle $K$ that consists of two internally disjoint paths $P,P'$ and is a subgraph of the input graph $G$.
We first show several crucial properties of $K$.
\subsubsection*{Modifying the drawing $\tilde \phi$}
We now proceed to modify the drawing $\tilde \phi$ into a drawing that is well-structured with respect to the collection $\hat {\mathcal{I}}$ after this iteration. That is, we will modify the image of edges of $I'$ in $\tilde \phi$, such that the resulting drawing induces well-structured drawings for new instances $I_{\mathsf{in}}, I_{\mathsf{out}}$.
We now provide a summary of the stages in modifying the drawing $\tilde \phi$. For simplicity of notations we denote $I'_1=I_{\mathsf{in}}$ and $I'_2=I_{\mathsf{out}}$ from now on.
We first delete the edges that are added to set $\hat E^{\sf del}$ in this iteration, and rename the resulting drawing by $\tilde \phi$. Clearly, the new drawing $\tilde \phi$ contains at most the same number of crossings as the old one.
In the first stage, we modify the drawing $\tilde\phi$ into a new drawing $\hat\phi$ that induces almost well-structured drawings of instances $I'_1$ and $I'_2$ (recall that an almost well-structured drawing is a drawing that satisfies properties \ref{prop: simple curve}--\ref{prop: first segments}).
In the second stage, we locally modify the image of edges incident to vertices of $K$, such that the orientations of vertices of $K$ coincide the orientations computed in Step 2 (\Cref{Step 2}).
In the third and the fourth stages, we
further modify drawing $\hat\phi$ into a drawing $\tilde \phi'$ that induces well-structured drawings of instances $I'_1$ and $I'_2$, respectively.
In each of these stages, we will make sure not to increase the number of crossings in the drawing by too much.
We now describe each of the stages in turn.
\iffalse
From now on we denote $\phi=\tilde \phi$.
We now construct well-structured drawings of the subinstances $I'_1$ and $I'_2$ using the well-structured drawing $\phi$ of instance $I'$ in four stages. In the first stage, we modify the well-structured drawing $\phi$ of $I'$ into an almost well-structured drawing $\hat \phi$ of $I'$ (recall that an almost well-structured drawing is a drawing that satisfies properties \ref{prop: simple curve}--\ref{prop: discs interior not crossing}), such that the image of $K$ in $\hat\phi$ is a simple closed curve, and $\mathsf{cr}(\hat\phi)$ is not much larger than
$\mathsf{cr}(\phi)$. In the second stage, we locally modify the image of edges incident to vertices of $K$, such that the orientations of vertices of $K$ coincide the orientations computed in Step 2 (\Cref{Step 2}).
In the third and the fourth stages, we
split the drawing $\hat \phi$ into an almost well-structured drawing $\hat \phi_1$ of $I'_1$ and an almost well-structured drawing $\hat\phi_2$ of $I'_2$, and then we further modify drawings $\hat\phi_1, \hat\phi_2$ into well-structured drawings $\phi'_1,\phi'_2$ of instances $I'_1$ and $I'_2$, respectively.
We now describe each of the stages in turn.
\fi
\subsubsection*{Stage 1. Removing self-loops of the image of $K$ in $\tilde\phi$}
In the first stage, we modify the drawing $\tilde \phi$ into a new drawing $\hat \phi$ that induces almost well-structured drawings of instances $I'_1,I'_2$, via the following claim.
\begin{claim}
\label{clm: skeleton can be drawn in a natural way}
If event ${\cal{E}}$ does not happen, then there is a drawing $\hat\phi$ of instance $\bar I$ after this iteration, in which (i) for each edge not incident to any vertex of $K$, its image in $\hat \phi$ remains the same as in $\tilde \phi$; (ii) the image of $K$ in $\hat \phi$ is a simple closed curve that does not cross any special edge of $I'$, and (iii)
$\mathsf{cr}(\hat\phi)\le \mathsf{cr}(\phi)+O\big(h^2(\tau^i)^2\mu^{2a}\big).$
\end{claim}
\begin{proof}
We construct the drawing $\hat\phi$ by iteratively ``opening'' the self-loops of the curve representing the image of $P$ and the curve representing the image of $\tilde\phi$ until the image of $K$ becomes a simple closed curve. Recall that, from Property \ref{f_prop: P and P' do not cross}, the images of $P$ and $P'$ do not cross in $\tilde\phi$, and from Property \ref{f_prop: K does not cross special edges}, the image of $K$ does not cross any special edge of $I'$ in $\tilde\phi$.
We denote by $\gamma$ the image of $P$ in $\tilde\phi$. Note that $\gamma$ is not necessarily a simple curve. We first iteratively compute a set $Z$ of self-loops of $\gamma$ and another simple curve $\gamma'$ connecting $\tilde\phi(v')$ to $\tilde\phi(v'')$, as follows.
We view the curve $\gamma$ as being directed from $\tilde\phi(v')$ to $\tilde\phi(v'')$. Initially we set $Z=\emptyset$. We start from the first endpoint $\tilde\phi(v')$ of $\gamma$ and find the first point $z$ on $\gamma$ that appears twice on $\gamma$. Denote by $\zeta$ the subcurve of $\gamma$ between the first and the last (second) appearance of $z$, so $\zeta$ is a self-loop that starts and ends at $z$. We add the self-loop $\zeta$ to set $Z$ and delete $\zeta$ from $\gamma$. We continue this process until the curve $\gamma$ no longer contains self-loops. Denote by $\gamma'$ the resulting simple curve we obtain, and we also view curves $\gamma'$ as being directed from $\tilde\phi(v')$ to $\tilde\phi(v'')$. See \Cref{fig: remove_loop} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Curve $\gamma$ (with direction) is shown in blue, with vertices $v_1,v_2,v_3$ appearing on it sequentially.]{\scalebox{0.0855}{\includegraphics{figs/remove_loop_1.jpg}}}
\hspace{0.1cm}
\subfigure[Curve $\gamma'$ is shown in green. Self-loop $\zeta$ is shown in light blue. Thin strip around $\zeta$ is shown in grey.]{
\scalebox{0.085}{\includegraphics{figs/remove_loop_2.jpg}\label{fig: thin strip}}}
\caption{Illustration for constructing curve $\gamma'$. \label{fig: remove_loop}}
\end{figure}
The algorithm for constructing the drawing $\hat\phi$ is iterative. In each iteration, we process a self-loop in $Z$.
We now describe an iteration.
Let $\zeta$ be a loop in $Z$, and we denote by $z$ the unique point of $\zeta$ that lies on $\gamma'$.
%
We modify the drawing as follows.
Let $v_1,\ldots,v_r$ be the vertices whose images appear on the self-loop $\zeta$, where the vertices are indexed according to the ordering in which they appear on $P$ (see \Cref{fig: remove_loop}).
We denote $V_{\zeta}=\set{v_1,\ldots,v_r}$.
Let $\sigma$ be a tiny segment of $\gamma'$ that contains $z$.
We remove the image of vertices $v_1,\ldots,v_r$ and all their incident edges from the current drawing, and place them on $\sigma$, such that $v_1,\ldots,v_r$ appear sequentially in the direction of $P$.
See \Cref{fig: remove_loop_3} for an illustration.
\iffalse
\begin{figure}[h]
\centering
\subfigure[Before: the images of $v_1,v_2$ appear sequentially on the image of $P$, and they belong to the loop $\zeta$.]{\scalebox{0.09}{\includegraphics{figs/loop_before.jpg}\label{fig: loop_before}}}
\hspace{0.1cm}
\subfigure[After: the images of $v_1,v_2$ are moved out of $\zeta$ to the segment $\sigma$ (shown in red).]{
\scalebox{0.1}{\includegraphics{figs/loop_after.jpg}}\label{fig: loop_after}}
\caption{Illustration for the process of opening a loop $\zeta$. \label{fig: loop}}
\end{figure}
\fi
It remains to add back the edges incident to $v_1,\ldots,v_r$. We denote by $z_1,\ldots,z_r$ the points representing the new images of vertices $v_1,\ldots,v_r$ respectively.
%
%
Let $S$ be an arbitrarily thin strip around $\zeta$, such that the entire segment $\sigma$ is contained in $S$ (see \Cref{fig: thin strip}).
Let $E_{\zeta}$ be the set of all edges incident to a vertex of $V_{\zeta}$. Clearly, for each edge $e\in E_{\zeta}$, the current image of $e$ has to cross the boundary of $S$. For each $e\in E_{\zeta}$, we denote by $z_e$ the last crossing between the curve $\tilde\phi(e)$ and $S$, where the curve $\tilde\phi(e)$ is viewed as being directed from its endpoint in $V_{\zeta}$ to its endpoint outside of $V_{\zeta}$.
Now we have obtained two sets $\set{z_e\mid e\in E_{\zeta}}$ and $\set{z_1,\ldots,z_r}$ of points that all lie within $S$ or on the boundary of $S$.
We now compute, for each edge $e\in E_{\zeta}$ with endpoint $v_i\in V_{\zeta}$, a curve $\xi_e$ connecting $z_i$ to $z_e$ that lies within the thin strip $S$, such that the curves in $\Xi_{\zeta}=\set{\xi_e\mid e\in E_{\zeta}}$ are in general position.
Then for every $1\le i\le r$ and for every edge $e$ that is incident to vertex $v_i$, we define the curve $\eta_e$ be the concatenation of (i) the curve $\xi_e$; and (ii) the segment of its original image between the last crossing with the boundary of $S$ and the image of its other endpoint in $\tilde\phi$.
Note that the curve $\eta_e$ may still intersect $\gamma'$ times. Lastly we modify curve $\eta_e$ as follows. We view it as being directed from its endpoint in $V_{\zeta}$ to its endpoint outside $V_{\zeta}$. Let $p_e$ be a point of $\eta_e$ just before the first point on $\eta_e$ that also belongs to $\gamma'$, and let $p'_e$ be a point of $\eta_e$ just after the last point on $\eta_e$ that also belongs to $\gamma'$. We then replace the subcurve of $\eta_e$ between $p_e$ and $p'_e$ by a new curve that lies within the thin strip around $\gamma'$, and cross $\gamma'$ at most once.
See \Cref{fig: remove_loop_4} for an illustration.
Clearly, in this way we obtain a new drawing of the graph $G'$ that no longer has the loop $\zeta$.
%
\begin{figure}[h]
\centering
\subfigure[Segment $\sigma$ is shown in red, with the new images of $v_1,v_2,v_3$ placed on it. $e,e'$ are edges incident to $v_1, v_2$ respectively. Their corresponding curves $\eta_{e},\eta_{e'}$ are shown in pink.]{\scalebox{0.13}{\includegraphics{figs/remove_loop_3.jpg}\label{fig: remove_loop_3}}}
\hspace{0.1cm}
\subfigure[The adjusted image of $e'$: now it no longer intersect $\gamma$ more than once.]{
\scalebox{0.13}{\includegraphics{figs/remove_loop_4.jpg}}\label{fig: remove_loop_4}}
\caption{Illustration for the process of adding edges back. \label{fig: remove_loop_next}}
\end{figure}
We first show that, after this iteration, the drawing we obtain still induces a drawing of $I'$ that satisfies properties \ref{prop: simple curve}--\ref{prop: discs interior not crossing}.
Note that we have only modified the images of vertices of $V_{\zeta}$, and the special cycles in ${\mathcal{R}}(I')$ are node-disjoint from $K$.
Therefore, the image of all special cycles remain the same as in $\tilde\phi$, and so the properties \ref{prop: simple curve}--\ref{prop: discs interior not crossing} are still satisfied.
We now show that, after this iteration, the image of $K$ still does not cross any special edge in $I'$.
Recall that $K$ crosses no special cycle of ${\mathcal{R}}(I')$ in $\tilde\phi$. And note that the new images of edges in $K$ are entirely contained in the image of $K$ in $\tilde\phi$. So after this iteration, in the new drawing we obtained, the image of $K$ still does not cross any special edge in $I'$.
Consider the resulting drawing obtained by processing each self-loop in $Z$ in this way. Finally, we perform type-1 uncrossing to the set of curves representing the images of edges in $E_{\zeta}$. Denote by $\hat \phi$ the resulting drawing.
It is easy to verify that $\hat\phi$ induces a drawing for instance $I'$ that satisfies properties \ref{prop: simple curve}--\ref{prop: discs interior not crossing}, and moreover, in $\hat \phi$ the image of $K$ is a simple closed curve that does not cross any special edge of $I'$.
We now bound the number of new crossings in the drawing. Consider the iteration where some self-loop $\zeta$ is processed. Note that $|E_{\zeta}|\le \sum_{1\le i\le r}\deg_{G'}(v_i)$.
Note that, since we have performed type-1 uncrossing to the images of edges in $E_{\zeta}$, the resulting images of edges in $E_{\zeta}$ may create at most $|E_{\zeta}|^2$ new crossings. On the other hand, let $E^*_{\zeta}$ be the set of edges in $E(G')$ that cross the self-loop $\zeta$ in the original drawing $\tilde\phi$, so the new images of $E_{\zeta}$ may create $|E^*_{\zeta}|\cdot |E_{\zeta}|$ new crossings with images of $E^*_{\zeta}$.
Finally, since we modified the image of edges in $E_{\zeta}$ such that each of these edges cross $\gamma'$ at most once, the number of intersections between the final images of edges of $E_{\zeta}$ and curve $\gamma'$ is at most $|E_{\zeta}|$. Clearly, these are all new crossings that we may create in an iteration.
Therefore, summing over all iterations,
%
\[
\mathsf{cr}(\hat\phi)\le \text{ } \mathsf{cr}(\tilde\phi)+\sum_{\zeta \in Z}\bigg(|E_{\zeta}|+|E^*_{\zeta}|+1\bigg)\cdot |E_{\zeta}|
\le \text{ } \mathsf{cr}(\tilde\phi)+ \bigg(\sum_{\zeta \in Z}|E_{\zeta}|+|E^*_{\zeta}|\bigg)^2.
\]
Since event ${\cal{E}}_1$ does not happen, $\sum_{\zeta \in Z}|E^*_{\zeta}|\le (\mathsf{cr}(\tilde\phi)\cdot\tau^i\cdot\mu^{a})/m'$.
And from Property \ref{f_prop: no significant in loop}, no significant vertices may belong to any self-loop in $Z$, so the self-loops of $Z$ only contains images of type-1 high-degree vertices and low-degree vertices of $P$.
Then from Properties \ref{f_prop: K does not cross special edges} and \ref{f_prop: low-degree incident edges small}, $\sum_{\zeta \in Z}|E_{\zeta}|\le O(h\tau^i\mu^{a})$.
Altogether, we get that
$$\mathsf{cr}(\hat\phi)\le \text{ } \mathsf{cr}(\tilde\phi)+O\bigg( \frac{\mathsf{cr}(\tilde\phi)\cdot\tau^i\cdot\mu^{a}}{m'}+ h\cdot\tau^i\mu^a \bigg)^2=\mathsf{cr}(\tilde\phi)+O\bigg(h^2(\tau^i)^2\mu^{2a}\bigg).$$
\end{proof}
\iffalse
$\ $
We say that event ${\cal{E}}$ happens if any of events ${\cal{E}}_1,{\cal{E}}_1,{\cal{E}}_5,{\cal{E}}_3,{\cal{E}}_4$ happens, namely ${\cal{E}}={\cal{E}}_1\cup{\cal{E}}_1\cup{\cal{E}}_5\cup{\cal{E}}_3\cup{\cal{E}}_4$. From the above discussion, $\Pr[{\cal{E}}]\le O(1/\mu^{a})$.
Assuming that event ${\cal{E}}$ does not happen. The properties of the skeleton $K$ can be summarized as follows.
\begin{properties}{F}
\item $K$ contains at most $\mu^{{50}+a}$ edges and at most $\mu^{{50}+a}$ vertices; \label{fact: skeleton is short}
\item edges of $K$ participate in less than $(\mathsf{cr}\cdot\mu^{{50}+a})/m$ crossings in $\hat\phi$;\label{fact: skeleton is slightly crossed}
\item less than $2h'\mu^{{50}+a}$ edges are incident to a type-1 high-degree vertex of $K$;
\label{fact: few incident edges on type-1}
\item less than $2h\mu^{{50}+a}$ edges are incident to a low-degree vertex of $K$;
\label{fact: few incident edges on low-deg}
\item there is a drawing $\hat\phi'$ of $\hat G'$, such that $\hat \phi'$ obeys the rotation system $\hat \Sigma$, the image of $K$ in $\hat\phi'$ is a simple closed curve, and $\mathsf{cr}(\hat\phi')\le \mathsf{cr}+O(h\mu^{{50}+a})^2$.\label{fact: skeleton drawn as a cycle}
\end{properties}
\fi
\subsubsection*{Stage 2. Correcting the orientations of vertices in $V(K)$}
Recall that we have computed, in Step 2 (\Cref{Step 2}), an orientation $b(v)$ for each vertex $v\in V(K)$, and then used them to define inner and outer sides of $K$, which are then used to compute the instances $I'_1, I'_2$.
However, although the drawing $\hat\phi$ computed in the last stage obeys the rotation system $\Sigma'$, it may not yet match the orientations on vertices of $K$ computed in Step 2, which is required in Property~\ref{prop: first segments} if it induces well-structured solutions for instances $I'_1, I'_2$. In this stage, we further modify $\hat{\phi}$ by correcting the orientations of vertices of $K$.
First, we prove the following observations, which show that, for vertex $v''$ and all significant vertices of $K$, the orientation computed in Step 2 is correct (matched by the drawing $\tilde\phi$, and so also matched by $\hat\phi$ since we have not modified the orientations of vertices when constructing $\hat \phi$ from $\tilde\phi$ in the last stage).
\begin{observation}
The orientation $b''$ (computed in Step 2) on vertex $v''$ coincides the orientation of $v''$ in drawing $\tilde\phi$.
\end{observation}
\begin{proof}
Observe that the distances between oriented orderings satisfy the triangle inequality, namely for any oriented rotations $({\mathcal{O}}_1,b_1),({\mathcal{O}}_2,b_2),({\mathcal{O}}_3,b_3)$ on the same set of element, $\mbox{\sf dist}(({\mathcal{O}}_1,b_1),({\mathcal{O}}_3,b_3))\le \mbox{\sf dist}(({\mathcal{O}}_1,b_1),({\mathcal{O}}_2,b_2))+\mbox{\sf dist}(({\mathcal{O}}_2,b_2),({\mathcal{O}}_3,b_3))$. In fact, from the definition of the reordering curves, given a set $\Gamma_{12}$ of reordering curves for oriented orderings $({\mathcal{O}}_1,b_1),({\mathcal{O}}_2,b_2)$ and a set $\Gamma_{23}$ of reordering curves for oriented orderings $({\mathcal{O}}_2,b_2),({\mathcal{O}}_3,b_3)$, it is immediate to concatenate curves in $\Gamma_{12}$ with curves in $\Gamma_{23}$ to obtain a set $\Gamma_{13}$ of reordering curves for $({\mathcal{O}}_1,b_1),({\mathcal{O}}_3,b_3)$, such that $\chi(\Gamma_{13})\le \chi(\Gamma_{12})+\chi(\Gamma_{23})$.
On the one hand, since ${\mathcal{O}}''$ is an unoriented ordering on a set of $\Omega(m'/\tau^i)$ elements, it is easy to see that $\mbox{\sf dist}(({\mathcal{O}}'',-1),({\mathcal{O}}'',1))=\Omega\big((m'/\tau^i)^2\big)$. Therefore, since $m'\ge \tau^s$ and \fbox{$\mathsf{cr}(\tilde\phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$}, \[\mbox{\sf dist}(({\mathcal{O}}'',-1),({\mathcal{O}}',b'))+\mbox{\sf dist}(({\mathcal{O}}'',1),({\mathcal{O}}',b'))\ge \mbox{\sf dist}(({\mathcal{O}}'',-1),({\mathcal{O}}'',1))\ge \Omega\big((\tau^s/\tau^i)^2\big)>2\cdot\mathsf{cr}(\tilde\phi),\]
and so at most one of $\mbox{\sf dist}(({\mathcal{O}}'',-1),({\mathcal{O}}',b')),\mbox{\sf dist}(({\mathcal{O}}'',1),({\mathcal{O}}',b'))$ is less than $\mathsf{cr}(\tilde\phi)$.
On the other hand, since the paths of ${\mathcal{P}}$ are non-transversal with respect to $\Sigma'$, it is easy to show that paths of ${\mathcal{P}}$ participate in at most $\mathsf{cr}(\tilde\phi)$ crossings in drawing $\tilde\phi$, so the orientation $b''$ on $v''$ computed in Step 2 coincides the orientation of $v''$ in $\tilde\phi$.
\end{proof}
\begin{observation}
If event ${\cal{E}}$ does not happen, then for each significant vertex $u\ne v',v''$, the orientation $b(u)$ we computed in Step 2 coincides with the orientation of $u$ in drawing $\tilde\phi$.
\end{observation}
\begin{proof}
Let $u$ be a significant vertex and let ${\mathcal{P}}_u$ be the set of paths in ${\mathcal{P}}$ that contains vertex $u$. From the definition of a significant vertex, $|{\mathcal{P}}_u|\ge h$.
From Property \ref{f_prop: K crossed few times}, edges of $K$ participates in at most $(\mathsf{cr}(\tilde\phi)\cdot\tau^i\cdot\mu^{a})/m'$ crossings in $\tilde\phi$.
Denote ${\mathcal{P}}_u=\set{P_{i_1},\ldots, P_{i_q}}$, and for each $1\le j\le q$, we denote by $e_{i_j}$ the first edge of $P_{i_j}$ (the edge of $P_{i_j}$ incident to $v'$), and denote by $\tilde e_{i_j}$ the first edge of $P_{i_j}$ incident to $u$. Recall that in Step 2, we say that an index $i_j$ is screwed under $b(v)$, iff when the orientation of $v$ is $b(v)$, edges $e_{i_j}, \tilde e_{i_j}$ are on different sides of $K$. It is easy to show that, if index $i_j$ is screwed under $b(v)$, then the image of path $P_{i_j}$ has to cross $K$. It is also easy to see that, each index $i_j$ is screwed under either $-1$ or $1$ but not both.
Since event ${\cal{E}}$ does not happen, from Property \ref{f_prop: K crossed few times}, $K$ participate in at most $(\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m'<h/2$ crossings in $\tilde \phi$.
Since $u$ is a type-2 high-degree vertex, there is exactly one orientation in $\set{-1,1}$ under which the number of screwed indices is less than $h/2$. Therefore, the orientation $b(u)$ computed in Step 2 coincides with the orientation of $u$ in drawing $\tilde\phi$.
\end{proof}
Therefore, we only need to modify the orientation of low-degree vertices and type-1 high-degree vertices. We denote by $V'$ the set of all low-degree vertices and all type-1 high-degree vertices $v$ of $V(K)$, such that the orientation of $v$ in the drawing $\hat\phi$ does not coincide with the orientation $b(v)$ computed in Step 2.
Consider now a vertex $v\in V'$.
Assume without loss of generality that $v$ is a vertex of $P$, and denote $\delta(v)=\set{\tilde e, \hat e_1, \hat e_2,\ldots, \hat e_p,\tilde e', \hat e'_1, \hat e'_2, \ldots, \hat e'_q}$, where edges $\tilde e,\tilde e'$ belong to $P$, $\tilde e$ precedes $\tilde e'$ in $P$, and the edges appear clockwise in the oriented ordering $({\mathcal{O}}_v,b(v))$ of $v$.
Denote by $D=D_{\hat \phi}(v)$ the tiny $v$-disc in drawing $\hat \phi$.
Let $\tilde z$ be the intersection between $\hat\phi(\tilde e)$ with the boundary of disc $D$, and we define (i) a point $\tilde z'$ for $\tilde e'$; (ii) for each $1\le i\le p$, a point $ z_i$ for $\hat e_i$; and (iii) for each $1\le j\le q$, a point $ z'_j$ for $\hat e'_j$ similarly.
See \Cref{fig: flip} for an illustration.
We place a disc $D'$ around $\hat\phi(v)$, that lies entirely within disc $D$. We then place points $\delta(v)=\set{\tilde y, y'_1, y'_2,\ldots, y'_q,\tilde y', y_1, y_2, \ldots, y_p}$ on the boundary of $D'$, such that the points appear on the boundary of $D'$ clockwise in this order.
We then compute, a set $\Gamma$ of curves within the annulus $D\setminus D'$ that contains, for each $1\le i\le p$, a curve $\gamma_i$ connecting the point $ y_i$ to $ z_i$, and, for each $1\le j\le q$, a curve $\gamma'_j$ connecting the point $ y'_j$ to $ z'_j$, such that the curves of $\Gamma$ are in general position.
We modify the drawing $\hat\phi$ locally within disc $D$ as follows.
We erase the drawing of $\hat\phi$ inside disc $D$ except for the image of vertex $v$ and the images of edges $\tilde e,\tilde e'$. We now let, for each $1\le i\le p$, the new image of edge $\hat e_i$ be the concatenation of: (i) the line segment within $D'$ connecting $\hat\phi(v)$ to point $\hat y_i$; (ii) the curve in $\gamma_i\in \Gamma$ connecting $ y_i$ to $ z_i$; and (iii) the original image of edge $\hat e_i$ outside the disc $D$, namely the subcurve of $\hat\phi(\hat e_i)$ between $ z_i$ and the image of the other endpoint of $\hat e_i$.
Similarly, we let, for each $1\le j\le q$, the new image of edge $\hat e'_j$ be the concatenation of: (i) the line segment within $D'$ connecting $\hat\phi(v)$ to point $ y'_j$; (ii) the curve in $\gamma'_j\in \Gamma$ connecting $ y'_j$ to $ z'_j$; and (iii) the original image of edge $\hat e'_j$ outside the disc $D$, namely the subcurve of $\hat\phi(\hat e'_j)$ between $ z'_j$ and the image of the other endpoint of $\hat e'_j$.
See \Cref{fig: flip} for an illustration.
Clearly, the number of new crossings we have created in this local modification step is at most $\deg(v)^2$.
We modify the drawing $\hat \phi$ at all vertices of $V'$ in the same way, and denote by $\hat \phi'$ the resulting drawing. It is easy to see that drawing $\hat \phi'$ coincides with all oriented orderings on vertices of $K$ computed in Step 2.
Note that, from properties \ref{f_prop: low-degree incident edges small} and \ref{f_prop: type-1 incident edges small}, the total number of edges that are incident to a type-1 high degree vertex or a low-degree vertex is at most $O\big(h\cdot \tau^i\mu^{a}\big)$. Therefore,
\[\mathsf{cr}(\hat\phi')\le \mathsf{cr}(\hat\phi)+\sum_{v\in V(K)}
\deg(v)^2\le \mathsf{cr}(\hat\phi)+O\bigg(h^2(\tau^i)^2\mu^{2a}\bigg).\]
\begin{figure}[h]
\centering
\subfigure[A schematic view of the drawing $\hat\phi$ around $v$.]{\scalebox{0.1}{\includegraphics{figs/rotation_flip_before.jpg}}}
\hspace{1cm}
\subfigure[Curves in $\Gamma$ are shown in dash lines.]{
\scalebox{0.1}{\includegraphics{figs/rotation_flip_after.jpg}}}
\caption{An Illustration of orientation-flip at $v$ in Stage 2. \label{fig: flip}}
\end{figure}
Since we have only modified the drawing $\hat \phi$ only locally within tiny discs around vertices of $K$, in the resulting drawing $\hat \phi'$, it still holds that the image of $K$ is a simple closed curve that does not cross any special edge. Therefore, drawing $\hat \phi'$ induces almost well-structured solutions for instances $I'_1$ and $I'_2$.
\subsubsection*{Stage 3. Split the drawing and further modification on vertices}
So far we have computed a drawing $\hat \phi'$ that induces almost well-structured drawings of instances $I'_1$ and $I'_2$.
However, it does not induce well-structured drawings of $I'_1$ and $I'_2$ yet. In particular, vertices of $I'_1$ and $I'_2$ may be drawn on both sides of $K$ (so Property \ref{prop: discs interior no vertex} is not satisfied), and the number of edges crossing $K$ may be too large (so Property \ref{prop: niubi edge crossing special cycle} is not satisfied).
In this stage and the next stage, we further modify drawing $\hat \phi'$ so that it induces well-structured drawings of instances $I'_1$ and $I'_2$ respectively.
Recall that $I'_1$ corresponds to the inner side of $K$, namely $G'_1=G_{\mathsf{in}}$.
Also recall that, in drawing $\hat\phi_1$, the image of $K$ is a simple closed curve. We denote by $\eta_K$ this simple closed curve, so $\eta_K$ partitions the sphere into two faces. We denote by $F$ the face that contains the first segments of the images of edges in $E_{\mathsf{in}}$.
In this stage, we move the image of all vertices of $G'_{1}$ that are currently drawn out of $F$ inside $F$. In the next stage, we move the image of some edges inside $F$ to avoid crossing $K$.
\newcommand{\mathsf{Move}}{\mathsf{Move}}
We now start to describe the algorithm in this stage.
We first describe a subroutine called $\mathsf{Move}$ and then show how it is used in this stage.
\paragraph{Subroutine $\mathsf{Move}$.} The input to the subroutine $\mathsf{Move}$ consists of
\begin{itemize}
\item a drawing $\psi$ of $G'_1$;
\item a cluster $W$ of $G'_1$;
\item a disc $D$ such that all vertices of $W$ are drawn outside of $D$ in $\psi$;
\item an edge $e^*_W\in \delta(W)$ such that the endpoint of $e^*_W$ outside $W$ is drawn inside $D$;
\item a set ${\mathcal{Q}}$ of paths routing edges of $\delta(W)$ to $e^*_W$ inside $W$; and
\item a set ${\mathcal{P}}=\set{P(e)\mid e\in \delta(W)}$ of paths, such that for each $e\in \delta(W)$, the path $P(e)$ has $e$ as its first edge and has the last endpoint drawn inside disc $D$.
\end{itemize}
See \Cref{fig: move_layout} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Disc $D$ is shown on the left and cluster $W$ is shown on the right. Edges of $\delta(W)$ with the other endpoint drawn inside $D$ are shown in blue, and edges of $\delta(W)$ with the other endpoint drawn outside $D$ are shown in red.]{\scalebox{0.1}{\includegraphics{figs/move_layout_1.jpg}}}
\hspace{1cm}
\subfigure[$e^*_W=e_2$, and the paths of ${\mathcal{Q}}$ are shown in orange dash lines.]{
\scalebox{0.1}{\includegraphics{figs/move_layout_2.jpg}}}
\caption{An Illustration of the input to subroutine $\mathsf{Move}$. \label{fig: move_layout}}
\end{figure}
The subroutine $\mathsf{Move}(\psi, W, D, e^*_W, {\mathcal{Q}},{\mathcal{P}})$ outputs a new drawing $\psi'$ of graph $G'_1$ in which the cluster $W$ is drawn inside $D$. We now describe the steps in the subroutine. Denote by $u,u'$ the endpoints of edge $e^*_W$ where $u\notin W$ and $u '\in W$, so the image of $u$ in $\psi$ lies in disc $D$.
For each edge $e\in\delta(W)$, we denote by $Q(e)$ the path in ${\mathcal{Q}}$ that routes $e$ to $e^*_W$.
We denote by $p$ the first crossing between the images of $e^*_W$ and $K$ (we view the curve representing the image of $e^*$ as being directed from $u'$ to $u$).
Let $D'$ be a tiny disc that (i) is entirely contained in $D$; (ii) is arbitrarily close to $p$ but does not contain $p$; and (iii) intersects with no other edge than $e^*_W$. See \Cref{fig: move_1} for an illustration. We then construct, for each edge $e\in \delta(W)\setminus \set{e^*_W}$, a curve $\gamma_e$ that consists of (i) a curve that starts at the image of the endpoint of $e$ in $W$, travels through the thin strip of the image of path $Q(e)$, and ends at some point $x_e$ lying in the tiny $u'$-disc $D_{\psi}(u')$; and (ii) a curve that starts at $x_e$, travels through the thin strip of the image of $e^*_W$, and ends at a point $y_e$ on the boundary of $D'$; such that the curves $\Gamma=\set{\gamma_e}_{e\in \delta(W)\setminus \set{e^*_W}}$ are in general position (we will later specify how to choose the ordering in which points in $\set{y_e\mid e\in \delta(W)}$ appear on the boundary of $D'$).
We also denote by $y_{e^*_W}$ the last crossing of edge $e^*_W$ and the boundary of $D'$ (so the segment of image of $e^*_W$ between $u$ and $y_{e^*_W}$ is disjoint from the interior of $D'$).
See \Cref{fig: move_2} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Drawing $\psi$ and disc $D'$ (shown in light green).]{\scalebox{0.095}{\includegraphics{figs/move_1.jpg}}\label{fig: move_1}}
\hspace{1cm}
\subfigure[Curves in $\Gamma$ are shown in dash lines.]{
\scalebox{0.095}{\includegraphics{figs/move_2.jpg}}\label{fig: move_2}}
\caption{An Illustration of constructing curves of $\Gamma$.}
\end{figure}
We now erase the image of $W$ from $\psi$ and put it inside $D'$. Lastly, we compute a set $Z$ of curves connecting points in $\set{y_e\mid e\in \delta(W)}$ to the new image of the endpoint of edges in $\delta(W)$. See \Cref{fig: move_inside} for an illustration.
We now show in the following observation that we can indeed computing a circular ordering on points in $\set{y_e\mid e\in \delta(W)}$ and the set $Z$ of curves, such that the number of crossings between $Z$ and the new image of $W$ can be charged to the number of crossings in drawing $\psi$.
\begin{observation}
\label{obs: inside_disc drawing}
There exists a circular ordering ${\mathcal{O}}_Y$ on points in $Y=\set{y_e\mid e\in \delta(W)}$ and a set $Z$ of curves that contains, for each $e\in \delta(W)$, a curve $\zeta_e$ inside disc $D'$ connecting point $y_e$ to the new image of the endpoint of $e$ in $W$, such that the number of crossings between $Z$ and the new image of $W$ is at most the number of crossings between $W$ and paths in ${\mathcal{P}}$ in drawing $\psi$.
\end{observation}
\begin{proof}
Let $\hat W$ be the graph obtained from graph $W\cup \delta(W)$ by contracting all outside-$W$ endpoints of edges in $\delta(W)$ into a single vertex, that we denote by $\hat v$. It is easy to verify that, computing a circular ordering ${\mathcal{O}}_Y$ on points in $Y$ and a set $Z$ of curves as required in \Cref{obs: inside_disc drawing} is equivalent to computing a circular ordering $\hat{\mathcal{O}}$ on the incident edges of $\hat v$ and a drawing $\hat \phi$ of graph $\hat W$ that respect the orderings ${\mathcal{O}}_v$ for all $v\in V(W)$ and the new ordering $\hat{{\mathcal{O}}}$.
For each edge $e\in \delta(W)$, we consider the curve representing the image of path $P(e)$ in $\psi$, and denote by $y'_e$ the last crossing between this curve and the image of $K$ in $\psi$ (where we view the curve as being directed from its endpoint in $W$ to its endpoint that is drawn inside $D$), and we denote by $\gamma'_e$ the subcurve of this curve between its endpoint in $W$ and $y'_e$.
We now define $\hat {\mathcal{O}}'$ to be the circular ordering on set $Y'=\set{y'_e\mid e\in \delta(W)}$ in which points of $Y'$ appear on the image of $K$.
Note that the circular ordering $\hat {\mathcal{O}}'$ naturally induces a circular ordering on $\delta(W)$ and therefore on the set of incident edges of $\hat v$, that we denote by $\hat {\mathcal{O}}$.
We now a drawing $\hat \phi$ of $\hat W$ as follows. We consider the drawing of $W$ induced by $\psi$ and the curves in $\set{\gamma'_e\mid e\in \delta(W)}$. Note that each curve $\gamma'_e$ contains has an endpoint being the image of a vertex of $W$ and the other endpoint $y'_e$ lying on the boundary of $D$. Recall that all vertices of $W$ are drawn outside $D$. We now contract $D$ into a single point, which automatically identify the points in $Y'$. We view this point as the image of $\hat v$. It is easy to see that the drawing obtained in this way is a drawing of $\hat W$ that obeys all circular orderings in $\set{\hat{\mathcal{O}}}\cup \set{{\mathcal{O}}_v\mid v\in V(W)}$. Moreover, the number of crossings in $\hat \phi$ between the images of edges in $\delta(\hat v)$ and the image of $W$ is at most the number of crossings in $\psi$ between the curves in $\set{\gamma'_e\mid e\in \delta(W)}$ and the image of $W$. \Cref{obs: inside_disc drawing} now follows.
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[scale=0.1]{figs/move_3.jpg}
\caption{An illustration of curves in $Z$ (shown in blue and red solid curves).}\label{fig: move_inside}
\end{figure}
Finally, we set, for each edge $e\in \delta(W)$, the new image of $e$ to be the concatenation of (i) the original image of $e$ in $\psi$; (ii) the curve $\gamma_e$; and (iii) the curve $\zeta_e$. This completes the description of the subroutine $\mathsf{Move}(\psi, W, D, e^*_W, {\mathcal{Q}},{\mathcal{P}})$.
We denote by $\psi'$ the drawing produced by the subroutine.
Clearly, in $\psi'$, the cluster $W$ is completely drawn inside $D'$ and therefore inside $D$.
We now make some observations on drawing $\psi'$.
From the construction, the original crossings between edges of $W$ remain in the $\psi'$, the original crossings between edges of $\delta(W)$ and edges of $W$ remain in the $\psi'$ (within disc $D'$), since we copied the drawing of $W$ inside $D'$. For each $e\in \delta(W)\setminus \set{e^*_W}$, its new image contains its original image, together with the image of the path $Q(e)$ in $\psi$. Therefore, for each edge $e'\in E(W)\cup\delta(W)$, in the new drawing $\psi'$ there are $\cong_{{\mathcal{Q}}}(e')$ curves contained in the thin strip of $\psi(e)$.
\iffalse
For each edge $e\in E'$, we view its image $\hat \phi_1(e)$ as being directed from its endpoint in $V'$ to its other endpoint that are drawn inside $F$, and we denote by $z_e$ the first crossing between $e$ and $K$.
Define $\hat G$ as the graph obtained from $G_{\mathsf{in}}$ by splitting each edge $e\in E'$ with a new vertex $x_e$, and we denote $X=\set{x_e\mid e\in E'}$.
Clearly, if we view, for each vertex $x_e\in X$, the point $z_e$ as its image, then the drawing $\hat \phi_1$ naturally induces a drawing of $\hat G$.
For each edge $e\in E'$, we denote by $e^{a}$ the new edge split from $e$ that is incident to the endpoint of $e$ in $V'$, and we denote by $e^{b}$ the other edge split from $e$.
We define $E^a=\set{e^a\mid e\in E'}$ and $E^b=\set{e^b\mid e\in E'}$.
See \Cref{fig: well-structured_layout} for an illustration.
We now pick an arbitrary vertex $x_{e^*}\in X$. Let $z^*$ be a point in the intersection of the tiny-$x_{e^*}$ disc $D'=D_{\hat \phi_1}(x_{e^*})$ and $F$. Let $D^*$ be a tiny-$z^*$ disc that lies entirely in the intersection of the $D'\cap F$.
\begin{figure}[h]
\centering
\subfigure[Disc $D$ is shown on the left and cluster $W$ is shown on the right. Edges of $\delta(W)$ with the other endpoint drawn inside $D$ are shown in blue, and edges of $\delta(W)$ with the other endpoint drawn outside $D$ are shown in red.]{\scalebox{0.1}{\includegraphics{figs/well-structured_drawing_1.jpg}}}
\hspace{1cm}
\subfigure[]{
\scalebox{0.1}{\includegraphics{figs/well-structured_drawing_2.jpg}}}
\caption{An Illustration of the input to subroutine $\mathsf{Move}$. \label{fig: well-structured_layout}}
\end{figure}
Define $H'=H\cup E^a$.
We erase the drawing of $H'$ from drawing $\hat \phi_1$, and place the drawing $\hat \phi_1(H)$ of $H$ inside disc $D^*$, such that the new images of vertices of $X$ lie on the boundary of $D^*$. We now add the edges of $E^b$ back to the drawing as follows. Let $\zeta$ be the segment of $\partial F$ that lies in disc $D'$. We first designate, for each vertex $x_e\in X$, a distinct point $y_e$ on $\zeta$. We then compute a set $\gamma$ of curves that contains, for each vertex $x_e\in X$, a curve $\gamma_e$ that connects $z_e$ to $y_e$ and lies entirely outside of $F$, such that every pair of curves in $\Gamma$ crosses at most once. Finally, we let the new image of $e_2$ be the sequential concatenation of: (i) the previous image of $e_2$ between its endpoints originally drawn inside $F$ to the point $z_e$; and (ii) the curve $\gamma_e$ in $\Gamma$. In this way we obtain a new drawing of graph $\hat G$, and suppressing vertices of $X$, this drawing also induces a drawing of $G_1$, with all vertices lying inside $F$. We then perform type-1 uncrossings (the algorithm from \Cref{thm: type-1 uncrossing}) to the images of all edges in $E'$. We denote by $\phi'_1$ the resulting drawing. See \Cref{fig: well-structured_drawing} for an illustration.
It is easy to observe that, in the resulting drawing, the set of edges that cross $K$ is still $E'$.
\begin{figure}[h]
\centering
\subfigure[Before: disc $D^*$ is shown in ligh blue, and segment $\zeta$ is shown in black.]{\scalebox{0.1}{\includegraphics{figs/well-structured_drawing_3.jpg}}}
\hspace{0.2cm}
\subfigure[After: drawing of $H'$ is moved into $D^*$, and the new images of edges of $E^a, E^b$ are shown in pink and red, respectively.]{
\scalebox{0.1}{\includegraphics{figs/well-structured_drawing_4.jpg}}}
\caption{An Illustration of drawing modification in Stage 3. \label{fig: well-structured_drawing}}
\end{figure}
\fi
We now describe the algorithm in the third stage, that utilizes the subroutine $\mathsf{Move}$ and iteratively move the images of all vertices inside $F$.
We start from the drawing $\hat\phi_1$.
Let $V'$ be the set of vertices whose image in $\hat \phi_1$ lie outside $F$, and let $H$ be the subgraph of $G'_1$ induced by vertices of $V'$.
We first apply the algorithm from \Cref{thm: layered well linked decomposition} to graph $G'_1$ and its subgraph $G'_1\setminus H$ to compute a layered $\alpha$-well-linked decomposition $({\mathcal{W}}, {\mathcal{L}}_1,\ldots,{\mathcal{L}}_r)$ of graph $G'_1$ with respect to $G'_1\setminus H$, for $\alpha=O(1/\log^{2.5} m)$.
Now for each cluster $W\in {\mathcal{W}}$, we select uniformly at random an edge of $\delta^{\operatorname{down}}(W)$ as $e^*_W$ and then use the algorithm from
\Cref{cor: simple guiding paths}
to compute a collection ${\mathcal{Q}}_W$ of paths routing edges of $\delta(W)\setminus \set{e^*_W}$ to $e^*_W$ (note that, instead of selecting a path set in $\set{{\mathcal{Q}}^{(e)}\mid e\in \delta(W)}$ uniformly at random, we select a path set in $\set{{\mathcal{Q}}^{(e)}\mid e\in \delta^{\operatorname{down}}(W)}$ uniformly at random).
Then, we iteratively modify the current drawing $\hat \phi_1$ as follows.
Throughout, we maintain a drawing $\tilde \phi$ of $G'_1$, that is initialized to be $\hat \phi_1$.
The algorithm continues to be executed as long as there is still some cluster $W\in {\mathcal{W}}$ whose image in the current drawing $\tilde \phi$ lies outside $F$.
In each iteration, we select a cluster $W$ of ${\mathcal{W}}$, that, among all clusters of ${\mathcal{W}}$ that are drawn outside of $F$, minimizes the layer index of the layered $\alpha$-well-linked decomposition that it belongs to.
Clearly, at this moment all clusters from lower layers than $W$ are already moved into $F$ in the current drawing $\tilde \phi$. Since edge $e^*_W$ is chosen from $\delta^{\operatorname{down}}(W)$, the endpoint of $e^*_W$ outside $W$ lies inside $F$ in $\tilde \phi$.
We then run the subroutine $\mathsf{Move}(\tilde\phi, W, F, e^*_W, {\mathcal{Q}}_W)$, and update $\tilde{\phi}$ with the drawing output by the subroutine. We denote by $\phi_1$ the resulting drawing after we processed all clusters of ${\mathcal{W}}$ in this way.
\newcommand{\mathsf{load}}{\mathsf{load}}
In order to analyze the new crossings created in this stage and prepare for the next stage, we define \emph{load} on edges of $E(H)\cup\delta(H)$, based on the edges $\set{e^*_W}_{W\in {\mathcal{W}}}$ and paths sets of $\set{{\mathcal{Q}}_W}_{W\in {\mathcal{W}}}$, as follows.
First consider a cluster $W\in {\mathcal{L}}_r$. For each edge of $\delta(W)\setminus \set{e^*_W}$, we define its load to be $0$; for edge $e^*_W$, we define its load to be $|\delta(W)|$.
For each edge $e\in E(W)$, we define its load to be $\cong_W({\mathcal{Q}}_W,e)$.
Assume now that the load for each edge with at least one endpoint lying in $\bigcup_{j+1\le t\le r}V({\mathcal{L}}_t)$ are already defined. Consider now a cluster $W\in {\mathcal{L}}_j$. For each edge of $\delta(W)\setminus \set{e^*_W}$ whose load is not yet defined (the edges of $\delta(W)$ with the other endpoint lying in $\bigcup_{1\le t\le j}V({\mathcal{L}}_t)$), we define its load to be $0$; for edge $e^*_W$, we define its load to be the sum of load of all other edges of $\delta(W)$ (notice that the load of all such edges are already defined at this point). For each edge $e\in E(W)$, we define the load of $e$ to be the sum of load of all edges of $\delta(W)$ whose corresponding path in set ${\mathcal{Q}}_W$ contains edge $e$. This completes the definition of load on all edges of $E(H)\cup\delta(H)$.
Note that, since the edges $\set{e_W}_{W\in {\mathcal{W}}}$ and path sets $\set{{\mathcal{Q}}_W}_{W\in {\mathcal{W}}}$ are chosen randomly, the load on edges of $E(H)\cup\delta(H)$ are random variables.
We prove the following observation.
\begin{observation}
\label{obs: expected load}
For each edge $e\in E(H)\cup\delta(H)$, $\expect[]{\mathsf{load}(e)}=O(\log^{O(1)} m)$.
\end{observation}
\begin{proof}
Note that $E(H)=E^{\mathsf{out}}({\mathcal{W}})\cup (\bigcup_{W\in {\mathcal{W}}}E(H))$.
We first consider the edges of $E^{\mathsf{out}}({\mathcal{W}})\cup \delta(H)$.
Let $1\le j\le r$ be a layer index and denote $V_j=\bigcup_{j\le t\le r}V({\mathcal{L}}_t)$.
We show by induction on $j$ that, for each edge
$e\in E^{\mathsf{out}}({\mathcal{W}})$ with at least one endpoint lying in $V_j$, $\expect[]{\mathsf{load}(e)}\le (1+1/\log m)^{r+1-j}$. The base case is when $j=r$. Consider a cluster $W\in {\mathcal{L}}_r$. From Property \ref{condition: layered decomp edge ratio} in the definition of a layered well-linked decomposition, $|\delta^{\operatorname{up}}(W)|\le |\delta^{\operatorname{down}}(W)|/\log m$. Since the edge $e^*_W$ is picked uniformly at random from edges of $\delta^{\operatorname{down}}(W)$, the expected load on each edge of $\delta^{\operatorname{up}}(W)$ is $0$, and the expected load on each edge of $\delta^{\operatorname{down}}(W)$ is $|\delta(W)|/|\delta^{\operatorname{down}}(W)|=(|\delta^{\operatorname{up}}(W)|+|\delta^{\operatorname{down}}(W)|)/|\delta^{\operatorname{down}}(W)|\le (1+1/\log m)$.
Assume now that the claim is true for layers $r,\ldots,j+1$. Consider now a cluster $W\in {\mathcal{L}}_j$. From the inductive hypothesis, from the linearity of expectations, the expected load of every edge of $\delta(W)$ with the other endpoint lying in $V_{j+1}$ is at most $(1+1/\log m)^{r-j}$. From the definition of load, the load of every edge that connects two distinct clusters of ${\mathcal{L}}_j$ is $0$. On the other hand, $|\delta(W)|/|\delta^{\operatorname{down}}(W)|\le (1+1/\log m)$. Therefore, since the edge $e^*_W$ is picked uniformly at random from set $\delta^{\operatorname{down}}(W)$, the expected load on each edge of $\delta^{\operatorname{down}}(W)$ is at most $(1+1/\log m)^{r-j}\cdot (|\delta(W)|/|\delta^{\operatorname{down}}(W)|)\le (1+1/\log m)^{r+1-j}$. As a corollary, the expected load for every edge of $E^{\mathsf{out}}({\mathcal{W}})\cup \delta(H)$ is at most $(1+1/\log m)^{r}\le O(1)$ (as $r\le \log m$ from \Cref{thm: layered well linked decomposition}).
We now consider edges of $\bigcup_{W\in {\mathcal{W}}}E(H)$. Let $W$ be a cluster of ${\mathcal{W}}$. From \Cref{cor: simple guiding paths} and our construction, for each edge $e\in E(W)$, $\expect[]{\cong_W({\mathcal{Q}}_W,e)}=O(\log^{O(1)} m)$.
Note that the choice of path set ${\mathcal{Q}}_W$ is independent of the load on edges of connecting $W$ to a vertex in $V_{j+1}$.
Since the expected load for each edge of $\delta(W)$ is $O(1)$, from the definition on loads of edges of $E(H)$ and the linearity of expectations, $\expect[]{\mathsf{load}(e)}=O(\log^{O(1)} m)$. This completes the proof of \Cref{obs: expected load}.
\end{proof}
We now analyze the number of new crossings created in this stage. First, from the construction, in each iteration we move the drawing of some cluster $W\in {\mathcal{W}}$ into some tiny disc inside $F$, so their new image do not cross the image of $G'_1\setminus H$.
\znote{informal from here}
Then from the algorithm, the new image of each cluster $W\in {\mathcal{W}}$ participate in at most the same number of crossings as in the original drawing $I'$, with the additive term being $\mathsf{cr}(\tilde \phi)/\mu^{a}$ (allowing pairs of edges that cross $K$ to cross). Since we have not modified the image of $G'_1\setminus H$, the new crossings must each involve at least one edge of $E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)$. Note that each edge of $\delta(H)$ has to cross $K$ in drawing $\tilde \phi$, and $K$ participates in at most $(\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m'$ crossings in $\tilde \phi$. Therefore, $|\delta(H)|\le (\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m'$. Moreover, from the definition of a layered well-linked decomposition, $|E^{\mathsf{out}}({\mathcal{W}})|\le O(|\delta(H)|)$, so
\[|E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)|=O((\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m').\]
Therefore, the number of new crossings that involve a pair of edges in $E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)$ is at most
\[
O((\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m')^2\le O(\mathsf{cr}(\tilde \phi)/\mu^{a}),
\]
where we have used the fact that $m'\ge \tau^s$ and \fbox{$\mathsf{cr}(\tilde \phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$}.
Denote $E'=E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)$.
Lastly, for each edge $e\in E'$, in the new drawing there are $\mathsf{load}(e)$ images lying in the thin strip of the original image of $e$, so the number of crossings that involve $e$ is multiplied by a factor of $\mathsf{load}(e)$.
\subsubsection*{Stage 4. Further modification on edges}
We now describe the last stage, in which we move the image of some edges inside $F$.
We say that a vertex $v\in V(K)$ is \emph{heavy} iff $v$ is contained in at least $m/(\tau^i)^{5}$ paths of ${\mathcal{P}}$, otherwise we say it is \emph{light}.
We prove the following observations.
\begin{observation}
\label{obs: few edges incident to non-heavy edges}
The expected number of edges in $G'$ that is incident to a light vertex of $V(K)$ is $O(m'/(\tau^i)^{4})$.
\end{observation}
\begin{proof}
Let $v$ be a light vertex. Since it belongs to at most $m'/(\tau^i)^{5}$ paths in ${\mathcal{P}}$, the probability that $v$ belongs to path $P$ is $O((m'/(\tau^i)^{5})/(m'/\tau^i))=O(1/(\tau^i)^4)$. Therefore, for an edge $e\in E(G')$ that is incident to a type-1 high-degree vertex, the probability that its type-1 high-degree endpoint belongs to $P$ is $O(1/(\tau^i)^4)$. From the union bound on all edges in $E(G')$, the expected number of edges that are incident to some type-1 high-degree vertex on $P$ is $O(m'\cdot O(1/(\tau^i)^4))=O(m'/(\tau^i)^{4})$.
The proof for path $P'$ is symmetric. Therefore, the expected number of edges in $G'$ that is incident to a non-heavy vertex of $V(K)$ is $O(m'/(\tau^i)^{4})$.
\end{proof}
\paragraph{Bad Event ${\cal{E}}_6$.}
We say that the bad event ${\cal{E}}_6$ happens if the number of edges in $G'$ incident to a light vertex of $V(K)$ is at least $m'\cdot\mu^{a}/(\tau^i)^{4}$.
From \Cref{obs: few edges incident to non-heavy edges} and Markov's bound, $\Pr[{\cal{E}}_6]\le O(1/\mu^{a})$.
Consider now an edge $e\in E'$. Note that both its endpoints are drawn inside (or on) $C$.
\paragraph{Clean edges.} Consider now the resulting drawing $\phi_1$ in the last stage.
Recall that $E'=E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)$, and in $\phi_1$ only edges of $E'$ may cross $K$.
We say that an edge $e\in E'$ is clean, iff, when we denote by $x_1,x_2,\ldots,x_k$ the crossings between $e$ and $K$, where the crossings are indexed according to their appearance on the image of $e$, such that, for each even $1\le i\le k$, the subcurve of the image of $e$ between $x_i$ and $x_{i+1}$, which we denote by $\gamma_i(e)$, lies entirely outside $F$, and moreover, one of the two segments of the image of $K$ separated by points $x_i,x_{i+1}$ contains no heavy vertices.
We prove the following claim.
\begin{claim}
The number of non-clean edges is at most $O(\mu^{2a}(\tau^i)^{10})$.
\end{claim}
\subsubsection{Analysis of the algorithm}
\label{subsec: analysis of decomposition lemma}
In this subsection we provide the proof of \Cref{lem: many path main} with some details deferred to subsequent sections.
First, note that initially the collection $\hat{{\mathcal{I}}}$ contains one well-structured subinstance, and from the description of the algorithm, in each iteration we replace one well-structured subinstance in $\hat{{\mathcal{I}}}$ with two new well-structured subinstances. Therefore, all subinstances in the resulting collection ${\mathcal{I}}$ are well-structured subinstances. On the other hand, recall that the recursive algorithm continues to be executed as long as there exists some instance $I'\in \hat{\mathcal{I}}$, such that its contracted subinstance $\tilde I'$ is $\tau^i$-wide and not $\tau^s$-small. Therefore, when the algorithm terminates, property \ref{prop: well-structured and contracted} has to be satisfied.
Second, note that in each iteration, we start with an instance $I'$ in the current collection $\hat {\mathcal{I}}$, compute a cycle $K$ that is edge-disjoint from all special cycles in ${\mathcal{R}}(I')$ and then decompose instance $I'$ into two subinstances $I_{\mathsf{in}}$ and $I_{\mathsf{out}}$, such that only edges of $E(K)$ appear in both instances. Therefore, if an edge in the input instance $I'$ is not a special edge of any subinstance that ever appeared in $\hat {\mathcal{I}}$, then it either belongs to exactly one resulting instance or belongs to set $E^{\sf del}$; otherwise it eventually belongs to exactly two resulting instances. Therefore, property \ref{prop: edge appearance} holds.
In order to show that the properties \ref{prop: well-structured solutions}--\ref{prop: number of crossings} also hold and there exists an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}})$ that satisfies the properties required in \Cref{lem: many path main}, it is convenient for us to associate a tree $T$ with the algorithm, called the \emph{partitioning tree}. Each vertex of $T$ corresponds to a well-structured subinstance that once appeared in the collection $\hat{{\mathcal{I}}}$. We grow the tree $T$ along with the iterative algorithm as follows. Initially, $T$ contains a single root denoted by $v(I)$, where $I$ is the input well-structured subinstance to \Cref{lem: many path main}. Consider now some vertex $v(I')$ in $T$, where $I'$ is a subinstance that undergoes Steps 1, 2 and 3. From the algorithm, the instance $I'$ is replaced by two new subinstances $I_{\mathsf{in}}$ and $I_{\mathsf{out}}$. We then create two nodes, $v(I_{\mathsf{in}})$ and $v(I_{\mathsf{out}})$, that become the child vertices of $v(I')$ in $T$.
\subsubsection*{Number of Iterations}
We first analyze the number of the iterative algorithm using $T$. We use the following observation.
\begin{observation}
\label{obs: number of iterations}
If \fbox{$\tau^s\ge 100\cdot\tau^i\mu^{10{50}}$}, then
$|V(T)|\le 200\cdot m\tau^i/\tau^s$.
\end{observation}
\begin{proof}
We show that, if $\tau^s\ge 100\cdot\tau^i\mu^{10{50}}$, then the algorithm will terminate in $200\cdot m\tau^i/\tau^s$ iterations, which implies \Cref{obs: number of iterations}.
In fact, we will show that, for each subinstance $I'=(G',\Sigma')$ that ever appeared in $\hat {\mathcal{I}}$,
the number of vertices in the subtree of $T$ rooted at node $v(I')$, which we denote by $N_{T}(I')$, is at most $$f(I')=\max\set{\frac{|E(G')|-\tau^s}{\tau^s/100\tau^i},0}+3,$$
which, when applied to the initial instance $I=(G,\Sigma)$ where $|E(G)|=m$, immediately implies that the algorithm will terminate in $200\cdot m\tau^i/\tau^s$ iterations.
Consider now an instance $I'$ that once appeared in $\hat {\mathcal{I}}$. Assume first that $v(I')$ is a leaf of $T$, then the number of vertices in the subtree of $T$ rooted at node $v(I')$ is $1$, which is at most $f(I')$ (note that $f(I')\ge 3$ always holds).
Assume now that $v(I')$ is not a leaf of $T$, we let $I'_1=(G'_1,\Sigma'_1)$, $I'_2=(G'_2,\Sigma'_2)$ be two new instances constructed the iteration of processing $I'$. Since $|E(K)|\le 2\tau^i\cdot\mu^{10{50}}$, from \Cref{obs: number of edges in base} and the definition of ${\cal{E}}_1$, $|E(G'_1)|+|E(G'_2)|\le |E(G')|+2\tau^i\cdot\mu^{10{50}}$ and
$|E(G'_1)|, |E(G'_2)|\le |E(G')|-\tau^s/6\tau^i$. We consider the following cases.
\textbf{Case 1. Both nodes $v(I'_1), v(I'_2)$ are leaves of $T$.}
In this case, $N_T(I')\le 3 \le f(G')$.
\textbf{Case 2. One of the nodes $v(I'_1), v(I'_2)$ is a leaf of $T$.}
Assume without loss of generality that $v(I'_1)$ is a leaf and $v(I'_2)$ is not a leaf, so $|E(G')|-\tau^s/6\tau^i\ge |E(G'_2)|\ge \tau^s$, so
\[
\begin{split}
N_{T}(I')=2+N_{T}(I'_2)\le 2+f(I'_2)= & \text{ }2+ \frac{|E(G'_2)|-\tau^s}{\tau^s/10\tau^i}+3
\\
\le & \text{ } 2+ \frac{|E(G')|-(\tau^s/6\tau^i)-\tau^s}{\tau^s/100\tau^i}+3
\\
\le & \text{ } 2+ \frac{|E(G')|-\tau^s}{\tau^s/100\tau^i}-16+3
\\
\le & \text{ } \frac{|E(G')|-\tau^s}{\tau^s/100\tau^i}+3 =f(I').
\end{split}
\]
\textbf{Case 3. Both nodes $v(I'_1), v(I'_2)$ are not leaves of $T$.}
Then $|E(G'_1)|, |E(G'_1)|\ge \tau^s$, and so
\[
\begin{split}
N_{T}(I')\le 1+N_{T}(I'_1)+N_{T}(I'_2)\le & \text{ } 1+f(I'_1)+f(I'_2)
\\
= & \text{ } 7+ \frac{|E(G'_1)|-\tau^s}{\tau^s/100\tau^i}+\frac{|E(G'_2)|-\tau^s}{\tau^s/100\tau^i}
\\
\le & \text{ } 7+ \frac{|E(G')|+2\tau^i\cdot\mu^{10{50}}-2\tau^s}{\tau^s/100\tau^i}
\\
\le & \text{ } 7+ \frac{|E(G')|-\tau^s}{\tau^s/100\tau^i}
+
\frac{2\tau^i\cdot\mu^{10{50}}-\tau^s}{\tau^s/100\tau^i}
\\
\le & \text{ } \frac{|E(G')|-\tau^s}{\tau^s/100\tau^i}+3 =f(I'),
\end{split}
\]
where the last inequality uses the fact $\tau^s\ge 100\tau^i\mu^{10{50}}$.
\end{proof}
\iffalse{number of all special edges}
As an immediate corollary, we have the following bound the number of special edges in all resulting instances in ${\mathcal{I}}$.
\begin{corollary}
\label{cor: number of special edges}
The total number of special edges in all instances in ${\mathcal{I}}$ is at most
$800m(\tau^i)^2\mu^{10{50}}/\tau^s$.
\end{corollary}
\begin{proof}
From \Cref{obs: 2 new special cycles}, after each iteration, the number of special cycles in all subinstances in the collection $\hat{\mathcal{I}}$ increases by $2$. Since the number of iterations is bounded by $200\cdot m\tau^i/\tau^s$, and the length of each special cycle is at most $2\tau^i\cdot\mu^{10{50}}$, we get that the total number of special edges in all resulting instances in ${\mathcal{I}}$ is bounded by
$2\cdot \big(200\cdot m\tau^i/\tau^s\big)\cdot \big(2\tau^i\cdot\mu^{10{50}}\big)=800m(\tau^i)^2\mu^{10{50}}/\tau^s$.
\end{proof}
\fi
We now proceed to analyze the iterative algorithm using the partitioning tree $T$.
We will maintain a well-structured drawing $\tilde \phi$ of the input instance $I=(G,\Sigma)$, that is initialized to be the input well-structured drawing $\phi$ of $I$ and evolves as the iterative algorithm proceeds.
\znote{the correct notion and claim?}
We denote by $\bar I$ the well-structured instance obtained from $I$ by only deleting from it the edges of $\hat E^{\sf del}$. As more edges are added to set $\hat E^{\sf del}$, the instance $\bar I$ evolves, and it is easy to verify that eventually instance $\bar I$ becomes the output instance $\hat I$.
We say that a well-structured drawing $\tilde \phi$ of instance $\bar I$ is \emph{well-structured with respect to the current collection $\hat {\mathcal{I}}$}, iff for each subinstance $I'\in \hat{\mathcal{I}}$, the drawing of $I'$ induced by $\tilde \phi$ is a well-structured drawing of ${\mathcal{I}}$.
We first focus on an iteration, in which some instance $I'=(G',\Sigma')$ in collection ${\mathcal{I}}$ is replaced by two new subinstances $I_{\mathsf{in}}$ and $I_{\mathsf{out}}$.
\begin{observation}
\label{obs: not crossing your own special then not crossing any special}
Let $\phi$ be a drawing of instance $\bar I$ that is well-structured with respect to the current collection $\hat {\mathcal{I}}$. For an edge $e$ that belongs to a unique instance $I'\in \hat {\mathcal{I}}$, if $e$ does not cross any special edge of $I'$ in $\tilde \phi$, then $e$ does not cross any special edge of any instnace of $\hat {\mathcal{I}}$ in $\tilde \phi$.
\end{observation}
\begin{proof}
\znote{to complete}
\end{proof}
We use the following claim.
\begin{claim}
\label{lem: well-structured drawing splitting}
Consider an iteration of the algorithm in which a well-structured instance $I'=(G',\Sigma')$ is processed and eventually replaced by two new well-structured instances, that we denote by $I_{\mathsf{in}}, I_{\mathsf{out}}$.
Let $\tilde \phi$ be a well-structured drawing of $\bar I$ with respect to the collection $\hat{\mathcal{I}}$ before this iteration.
Then there exists a well-structured drawing $\tilde \phi'$ of $\bar I$ with respect to the collection $\hat{\mathcal{I}}$ after this iteration, such that
\begin{itemize}
\item for each special cycle $R$ in the collection $\hat{\mathcal{I}}$ before this iteration, its images in drawings $\tilde\phi$ and $\tilde \phi'$ are identical, and the disc $D_R$ it defines in $\phi$ and $\hat \phi$ are the same; for each edge $e\in E(\hat G)$ and each special cycle $R\in {\mathcal{R}}$: either in both $\phi$ and $\hat \phi$, the image of $e$ is disjoint from the interior of disc $D_R$, or the intersections of the image of $e$ with the interior of disc $D_R$ are identical in $\phi$ and $\hat \phi$; and
\item if \fbox{$\mathsf{cr}(\tilde\phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$}, then
$\mathsf{cr}(\tilde\phi')\le \mathsf{cr}(\tilde \phi)\cdot(1+1/\mu^{a})$.
\end{itemize}
\end{claim}
We provide the proof of \Cref{lem: well-structured drawing splitting} in \Cref{sec: Proof of well-structured drawing splitting}, after we complete the proof of \Cref{lem: many path main} using \Cref{lem: well-structured drawing splitting}.
\subsubsection{Proof of \Cref{lem: well-structured drawing splitting}}
\label{sec: Proof of well-structured drawing splitting}
In this subsection we provide the proof of \Cref{lem: well-structured drawing splitting}, thus completing the proof of \Cref{lem: many path main}.
We focus on an iteration, in which some instance $I'=(G',\Sigma')$ in collection ${\mathcal{I}}$ is replaced by two new subinstances $I_{\mathsf{in}}$ and $I_{\mathsf{out}}$.
We denote $m'=|E(G')|$.
Recall that currently we have a drawing $\tilde\phi$ of the instance $\bar I$ before this iteration, that is well-structured with respect to the collection $\hat{\mathcal{I}}$ before this iteration.
Recall that in Step 1 we computed a cycle $K$ that consists of two internally disjoint paths $P,P'$ and is a subgraph of the input graph $G$.
We first show several crucial properties of $K$.
\subsubsection*{Properties of $K$}
We prove the following observations on cycle $K$.
\begin{observation}
\label{obs: skeleton number of crossings}
The expected number of crossings in $\tilde\phi$ that involve edges of $K$ is $O(\mathsf{cr}(\tilde\phi)\cdot\tau^i/m')$.
\end{observation}
\begin{proof}
For each index $0\le z\le \floor{L/2}-1$, we denote by $\tilde K_z$ the subgraph formed by the union of the path in ${\mathcal{P}}$ whose first edge is $e_z$ and the path in ${\mathcal{P}}$ whose first edge is $e_{z+\floor{L/2}}$.
Since the subgraphs $\tilde K_0,\ldots, \tilde K_{\floor{L/2}-1}$ are edge-disjoint, the expected number of crossings in $ \tilde \phi$ that involve an edge in a random subgraph in $\set{\tilde K_0,\ldots, \tilde K_{\floor{L/2}-1}}$ is at most $O(\mathsf{cr}(\tilde \phi)/L) < O( \mathsf{cr}(\tilde \phi)\cdot\tau^i/m')$.
Since $K\subseteq \tilde K$, we get that the expected number of crossings in $\tilde\phi$ that involve edges of $K$ is at most $O(\mathsf{cr}(\tilde\phi)\cdot\tau^i/m')$.
\end{proof}
\paragraph{Bad Event ${\cal{E}}_1$.}
We say that the bad event ${\cal{E}}_1$ happens iff the edges of $K$ participate in more than $(\mathsf{cr}(\tilde \phi)\cdot\tau^i\cdot\mu^{a})/m'$ crossings in $\tilde \phi$.
From \Cref{obs: skeleton number of crossings} and Markov's bound, $\Pr[{\cal{E}}_1]\le O(1/\mu^{a})$.
\begin{observation}
\label{obs: skeleton paths not crossing each other}
If \fbox{$\mathsf{cr}(\tilde \phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$}, and event ${\cal{E}}_1$ does not happen, then the images of $P$ and $P'$ do not cross in $\tilde \phi$.
\end{observation}
\begin{proof}
The proof of \Cref{obs: skeleton paths not crossing each other} is similar to the proof of Claim 9.9 in \cite{chuzhoy2020towards}.
Assume the contrast that the images of $P$ and $P'$ cross in $\tilde\phi$, then there is a pair of edges $e\in E(P)$, $e'\in E(P')$ whose images cross in $\tilde\phi$. Consider the curve $\tilde\phi(P)$, that we view as being directed from $v'$ to $v''$. Let $p$ be the first point on this curve that belongs to the curve $\tilde\phi(P')$ and is not the image of a vertex. Let $\gamma$ be the simple curve connecting $\tilde\phi(v')$ to $p$, that is obtained from the segment of $\tilde\phi(P)$ from $\tilde\phi(v')$ to $p$ by deleting all its loops. We define the simple curve $\gamma'$ similarly. Note that the curves $\gamma$ and $\gamma'$ can contain other common points which are vertex-images in $\tilde\phi$, but they cannot cross at those points.
For every vertex $u\neq v'$ whose image is contained in both $\gamma$ and $\gamma'$, let $D_{\tilde\phi}(u)$ be the tiny-$u$ disc in $\tilde\phi$, and we modify the curves $\gamma$ and $\gamma'$ so that they are disjoint from the interior of $D_{\tilde\phi}(u)$, and instead follow its boundary on either side (see Figure~\ref{fig: curve_separation}).
%
\begin{figure}[h]
\centering
\subfigure[The curves $\gamma$ and $\gamma'$ touch at the image of $u$ in $\tilde\phi$.]{\scalebox{0.13}{\includegraphics{figs/curve_separation_before.jpg}}}
\hspace{0.1cm}
\subfigure[The curves $\hat \gamma,\hat \gamma'$ share only endpoints $p$ and $\tilde\phi(v')$.]{
\scalebox{0.13}{\includegraphics{figs/curve_separation_after.jpg}}}
\caption{Illustration for the proof of \Cref{obs: skeleton paths not crossing each other}. \label{fig: curve_separation}}
\end{figure}
%
Let $\hat \gamma$ and $\hat \gamma'$ denote the resulting two curves. Then the union of $\hat \gamma$ and $\hat \gamma'$ defines a closed simple curve that we denote by $\lambda$. Curve $\lambda$ partitions the plane into two faces, $F$ and $F'$. We assume w.l.o.g. that vertex $v''$ lies on the interior of face $F'$.
From the definition of $K$, there are at least $L/3$ paths in ${\mathcal{P}}$ whose image in $\tilde\phi$ intersects the interior of face $F$.
Note that, since the paths of ${\mathcal{P}}$ are non-transversal with respect to $\Sigma'$, the image of every path in ${\mathcal{P}}$ in $\tilde\phi$ cannot cross the boundary of face $F$ through the disc $D_{\tilde\phi}(u)$ of any vertex $u\in V(P)\cup V(P')$. Therefore, edges of $E(K)$ participate in at least $L/3$ crossings.
However, since we have assumed that \fbox{ $\mathsf{cr}(\tilde \phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$}, we get that
$$\frac{L}{3}\ge \frac{m'}{30\cdot\tau^i} \ge \frac{\tau^s}{30\cdot\tau^i}>\frac{\mathsf{cr}(\tilde \phi)\cdot\tau^i\cdot\mu^{a}}{\tau^s}>\frac{\mathsf{cr}(\tilde \phi)\cdot\tau^i\cdot\mu^{a}}{m'},$$
a contradiction to the assumption that the event ${\cal{E}}_1$ does not happen.
Therefore, the images of $P$ and $P'$ do not cross in $\tilde\phi$.
\end{proof}
\begin{observation}
\label{obs: new cycle not crossing old ones}
If \fbox{$\mathsf{cr}(\tilde \phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$}, then with probability $1-O(1/\mu^{a})$, in the drawing $\tilde \phi$ the edges of $K$ do not cross any special edge of any instance in the collection $\hat {\mathcal{I}}$ before this iteration.
\end{observation}
\begin{proof}
Since drawing $\tilde \phi$ is well-structured with respect to the collection $\hat{\mathcal{I}}$ before this iteration, and since instance $I'$ belongs to the collection ${\mathcal{I}}$ before this iteration, the drawing of $I'$ induced by $\tilde \phi$ is a well-structured drawing of $I'$.
We will show that with probability $1-O(1/\mu^{a})$, in the drawing $\tilde \phi$ the edges of $K$ do not cross any special edge of $I'$.
Then \Cref{obs: new cycle not crossing old ones} follows from \Cref{obs: not crossing your own special then not crossing any special}.
Denote by $E'$ the set of all edges in $E(G')$ that crosses an edge of $E({\mathcal{R}}(I'))$ in $\tilde \phi$.
From Property \ref{prop: niubi edge crossing special cycle} in the definition of a well-structured solution, $|E'|\le \mathsf{cr}(\tilde \phi)\cdot(\tau^i)^{10}/m'$.
We define the edge-disjoint subgraphs $\tilde K_0,\ldots, \tilde K_{\floor{L/2}-1}$ in the same way as \Cref{obs: skeleton number of crossings}.
Since the subgraphs $\tilde K_0,\ldots, \tilde K_{\floor{L/2}-1}$ are edge-disjoint, the probability that a random subgraph in $\set{\tilde K_0,\ldots, \tilde K_{\floor{L/2}-1}}$ contains an edge of $E'$ is at most
\[
\frac{|E'|}{\floor{L/2}}\le \frac{\mathsf{cr}(\tilde \phi)\cdot(\tau^i)^{10}/m'}{m'/3\tau^i}
\le
\frac{\mathsf{cr}(\tilde \phi)\cdot(\tau^i)^{9}}{(m')^2}
\le
\frac{\mathsf{cr}(\tilde \phi)\cdot(\tau^i)^{9}}{(\tau^s)^2}
<\frac{1}{\mu^{a}},
\]
where the last but one step uses the assumption that $\mathsf{cr}(\tilde \phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$.
\end{proof}
\paragraph{Bad Event ${\cal{E}}_2$.}
We say that the bad event ${\cal{E}}_2$ happens iff the drawing $\tilde \phi$ contains a crossing that involves an edge of $K$ and a special edge of some instance in the collection $\hat {\mathcal{I}}$ before this iteration. From \Cref{obs: new cycle not crossing old ones}, $\Pr[{\cal{E}}_2]=O(1/\mu^{a})$.
\paragraph{High-degree vertices.}
Denote $h=\mathsf{cr}(\tilde \phi)\cdot \mu^{11{50}+3a}\cdot(\tau^i)^4/m'$.
We say that a vertex $v\in V({\mathcal{P}})\setminus \set{v',v''}$ is a \emph{high-degree} vertex iff $\deg_{G'}(v)\ge h$, otherwise we say that it is a \emph{low-degree} vertex.
We say that a high-degree vertex $v$ is a \emph{type-1} high-degree vertex iff $v$ is contained in at most $h$ paths in ${\mathcal{P}}$, otherwise we say that $v$ is a \emph{type-2} high-degree vertex.
We denote by $V_{\sf{low}}(P)$ the set of all low-degree vertices of $P$, and we define the set $V_{\sf{low}}(P')$ similarly. We use the following observation.
\iffalse{whp version}
\begin{observation}
\label{obs: skeleton do not have many insignificant edges}
The probability that $\sum_{v\in V_{\sf{low}}(P)}\deg_{\hat G'}(v)\ge h\mu^{{50}+a}$ is $O(1/\mu^{a})$. Similarly, the probability that $\sum_{v\in V_{\sf{low}}(P')}\deg_{\hat G'}(v)\ge h\mu^{{50}+a}$ is $O(1/\mu^{a})$.
\end{observation}
\begin{proof}
Let $\hat P$ be a path of ${\mathcal{P}}$ and let $e$ be an edge that does not belong to set $E'$.
We say that $e$ is \emph{insignificant for $\hat P$}, iff $e$ is incident to a low-degree vertex of $\hat P$. Denote $e=(u,v)$. If $u$ is a low-degree vertex, then $e$ is insignificant for at most $h$ paths in ${\mathcal{P}}$ that contains $u$. Otherwise, $e$ is not insignificant for any path in ${\mathcal{P}}$ that contains $u$. Similarly, we get that $e$ is insignificant for at most $h$ paths in ${\mathcal{P}}$ that contains $v$. Therefore, an edge $e\notin E'$ is insignificant for at most $2h$ paths in ${\mathcal{P}}$.
Therefore, if we denote $\deg_{\sf{low}}(\hat P)=\sum_{v\in V_{\sf{low}}(P)}\deg_{\hat G}(v)$, then $\sum_{\hat P\in {\mathcal{P}}}\deg_{\sf{low}}(\hat P)\le 2hm$, and so the expected value $\deg_{\sf{low}}(\hat P)$ for a random path in ${\mathcal{P}}$ is at most $2hm/(m/\mu^{{50}})\le O(h\mu^{{50}})$. From Markov's bound, the probability that $\deg_{\sf{low}}(P)\ge h\mu^{{50}+a}$ is $O(1/\mu^{a})$, and similar the probability that $\deg_{\sf{low}}(P')\ge h\mu^{{50}+a}$ is $O(1/\mu^{a})$.
\Cref{obs: skeleton do not have many insignificant edges} now follows.
\end{proof}
\fi
\begin{observation}
\label{obs: skeleton do not have many insignificant edges}
$\expect[]{\sum_{v\in V_{\sf{low}}(P)}\deg_{G'}(v)}= O (h\tau^i)$. Similarly, $\expect[]{\sum_{v\in V_{\sf{low}}(P')}\deg_{G'}(v)}= O (h\tau^i)$.
\end{observation}
\begin{proof}
Let $\hat P$ be a path of ${\mathcal{P}}$ and let $e$ be an edge that does not belong to set $E({\mathcal{P}})$.
We say that $e$ is \emph{insignificant for $\hat P$}, iff $e$ is incident to a low-degree vertex of $\hat P$. Denote $e=(u,v)$. If $u$ is a low-degree vertex, then $e$ is insignificant for at most $h$ paths in ${\mathcal{P}}$ that contains $u$. Otherwise, $e$ is not insignificant for any path in ${\mathcal{P}}$ that contains $u$. Similarly, we get that $e$ is insignificant for at most $h$ paths in ${\mathcal{P}}$ that contains $v$. Therefore, an edge $e\notin E({\mathcal{P}})$ is insignificant for at most $2h$ paths in ${\mathcal{P}}$.
Therefore, if we denote $\deg_{\sf{low}}(\hat P)=\sum_{v\in V_{\sf{low}}(P)}\deg_{ G'}(v)$, then $\sum_{\hat P\in {\mathcal{P}}}\deg_{\sf{low}}(\hat P)\le 2hm'$, and so the expected value $\deg_{\sf{low}}(\hat P)$ for a random path in ${\mathcal{P}}$ is at most $O(2hm'/(m'/\tau^i))\le O(h\tau^i)$. The proof for path $P'$ is symmetric.
\end{proof}
We denote by $V_{\sf{high(1)}}(P)$ the set of all type-1 high-degree vertices of $P$, and we define set $V_{\sf{high(1)}}(P')$ similarly. We use the following observation.
\iffalse{whp version}
\begin{observation}
\label{obs: not many edges incident to a type-1 high-degree vertex in skeleton}
The probability that the total number of edges that are incident to a type-1 high-degree vertex on path $P$ is more than $h'\mu^{{50}+a}$ is $O(1/\mu^{a})$. Similarly, the probability that the total number of edges that are incident to a type-1 high-degree vertex on path $P'$ is more than $h'\mu^{{50}+a}$ is $O(1/\mu^{a})$.
\end{observation}
\begin{proof}
Let $v$ be a type-1 high-degree vertex. Since it belongs to at most $h'$ paths in ${\mathcal{P}}$, the probability that $v$ belongs to path $P$ is $O(h'/(m/\mu^{{50}}))$. Therefore, for an edge $e\in E(\hat G)$ that is incident to a type-1 high-degree vertices, the probability that its type-1 high-degree endpoint belongs to $P$ is $O(h'/(m/\mu^{{50}}))$. From the union bound on all edges in $E(\hat G)$, the expected number of edges that are incident to some type-1 high-degree vertex on path $P$ is $O(m\cdot h'/(m/\mu^{{50}}))=O(h'\mu^{{50}})$. Then from Markov's bound, the probability that number of edges that are incident to some type-1 high-degree vertex on path $P$ is greater than $h'\mu^{{50}+a}$ is $O(1/\mu^{a})$.
The proof for path $P'$ is symmetric.
\end{proof}
\fi
\begin{observation}
\label{obs: not many edges incident to a type-1 high-degree vertex in skeleton}
$\expect[]{\sum_{v\in V_{\sf{high(1)}}(P)}\deg_{G'}(v)}= O (h\tau^i)$, and $\expect[]{\sum_{v\in V_{\sf{high(1)}}(P')}\deg_{G'}(v)}= O (h\tau^i)$.
\end{observation}
\begin{proof}
Let $v$ be a type-1 high-degree vertex. Since it belongs to at most $h$ paths in ${\mathcal{P}}$, the probability that $v$ belongs to path $P$ is $O(h/(m'/\tau^i))$. Therefore, for an edge $e\in E(G')$ that is incident to a type-1 high-degree vertex, the probability that its type-1 high-degree endpoint belongs to $P$ is $O(h/(m'/\tau^i))$. From the union bound on all edges in $E(G')$, the expected number of edges that are incident to some type-1 high-degree vertex on $P$ is $O(m'\cdot h/(m'/\tau^i))=O(h\tau^i)$.
The proof for path $P'$ is symmetric.
\end{proof}
\paragraph{Bad Events ${\cal{E}}_3, {\cal{E}}_4$.}
We say that the bad event ${\cal{E}}_3$ happens if either $\sum_{v\in V_{\sf{low}}(P)}\deg_{G'}(v)\ge h\tau^i\mu^{a}$ holds or $\sum_{v\in V_{\sf{low}}(P')}\deg_{G'}(v)\ge h\tau^i\mu^{a}$ holds. From \Cref{obs: skeleton do not have many insignificant edges} and Markov's bound, $\Pr[{\cal{E}}_3]\le O(1/\mu^{a})$.
We say that the bad event ${\cal{E}}_4$ happens iff there are at least $h\tau^i\mu^{a}$ edges incident to a type-1 high-degree vertex of $K$. From \Cref{obs: not many edges incident to a type-1 high-degree vertex in skeleton} and Markov's bound, $\Pr[{\cal{E}}_4]\le O(1/\mu^{a})$.
\paragraph{Unbalanced vertices.}
Let $\hat P$ be a path of ${\mathcal{P}}$ and let $v$ be an inner vertex of $\hat P$. Let $e_v,e'_v$ be the two edges of $\hat P$ that are incident to $v$.
Consider now the edges of $\delta_{G'}(v)\cap E({\mathcal{P}})$, and assume that they appear in the ordering ${\mathcal{O}}_v\in \Sigma$ in the following order: $e_v,\hat e_1,\ldots,\hat e_s, e'_v,\hat e'_1,\ldots,\hat e'_t$.
We say that $v$ is \emph{unbalanced with respect to $\hat P$} iff $v$ is a type-2 high-degree vertex and $\min\set{s,t}\le \max\set{s,t}/\mu^{a}$.
\begin{observation}
\label{obs: skeleton loops do not contain significant nodes}
The probability that $P$ contains a type-2 high-degree vertex that is unbalanced with respect to $P$ is $O(1/\mu^{a})$. Similarly, the probability that $P'$ contains a type-2 high-degree vertex that is unbalanced with respect to $P'$ is $O(1/\mu^{a})$.
\end{observation}
\begin{proof}
For each vertex $u$, we denote by $d(u)$ the number of paths in ${\mathcal{P}}$ that contains $u$.
We say that a type-2 high-degree vertex $u$ is \emph{bad}, iff among all $d(u)$ paths of ${\mathcal{P}}$ that contains $u$, at least $d(u)/2\mu^{a}\tau^i$ of them participate in at least $d(u)/2\mu^{a}\tau^i$ crossings each; otherwise we say $u$ is \emph{good}.
We first show that the probability that $K$ contains a bad vertex is $O(1/\mu^a)$.
For each $\log h\le j\le \log n$, we say that a bad vertex $u$ is in \emph{class-$j$} iff $2^j\le d(u)<2^{j+1}$. Consider now an index $\log h\le j\le \log n$. On the one hand, from the construction of $K$, for each class-$j$ bad vertex, the probability that it belongs to $K$ is $O(2^{j}/(m'/\tau^i))$. On the other hand, we prove that the number of class-$j$ bad vertices is at most $O(\mathsf{cr}(\tilde \phi)\cdot \mu^{10{50}}/(2^j/2\mu^{a}\tau^i)^2)$.
We say that a crossing in drawing $\tilde \phi$ is \emph{witnessed by} a bad vertex $u$, iff one of the edges that participates in the crossing belongs to a path in ${\mathcal{P}}$ that contains $u$.
Since each path of ${\mathcal{P}}$ contains at most $\tau^i\mu^{10{50}}$ vertices, and the paths of ${\mathcal{P}}$ are edge-disjoint, each crossing is witnessed by at most $2\tau^i\mu^{10{50}}$ bad vertices. Note that, from the definition of bad vertices, each class-$j$ bad vertices witness at least $(2^j/2\mu^{a}\tau^i)^2$ crossings. It follows that the number of class-$j$ bad vertices is at most $O(\mathsf{cr}(\tilde \phi)\cdot 2\tau^i\mu^{10{50}}/(2^j/\mu^a)^2)$.
Therefore, the probability that $K$ contains a class-$j$ bad vertex is at most
\[
O\bigg(\frac{\mathsf{cr}(\tilde \phi)\cdot2\tau^i \mu^{10{50}}}{(2^j/2\mu^{a}\tau^i)^2}\cdot \frac{2^{j}}{(m'/\tau^i)}\bigg)\le O\bigg(\frac{\mathsf{cr}(\tilde \phi)\cdot \mu^{10{50}+2a}(\tau^i)^4}{2^j\cdot m'}\bigg)\le O\bigg(\frac{\mathsf{cr}(\tilde \phi)\cdot \mu^{10{50}+2a}(\tau^i)^4}{\mathsf{cr}(\tilde \phi)\cdot \mu^{11{50}+3a}(\tau^i)^4}\bigg)=O(1/\mu^{{50}+a}),
\]
where the last inequality uses the fact that $2^j\ge h$ and the definition of $h$. Then from the union bound on all indices $\log h\le j\le \log n$, the probability that $K$ contains a bad vertex is at most $O(1/\mu^a)$ (since $\mu\gg \log n$).
We next claim that, for each good type-2 high-degree vertex $u$, $u$ is unbalanced with respect to at most $O(d(u)/\mu^{a}\tau^i)$ paths in ${\mathcal{P}}$ that contains $u$. Before we prove the claim, we first show that this implies that the probability that $P$ or $P'$ contains a good type-2 high degree vertex that is unbalanced with respect to it is $O(1/\mu^{a})$, which, together with the arguments in the last paragraph, imply
\Cref{obs: skeleton loops do not contain significant nodes} immediately. Let $U$ be the set of all good type-2 high-degree vertices. Clearly, $\sum_{u\in U}d(u)\le m'$. For each vertex $u\in U$, we denote by $d'(u)$ the number of paths in ${\mathcal{P}}$ to which $u$ is unbalanced, then the number of paths such that there exists some good type-2 high-degree vertex being unbalanced with respect to it is at most
$\sum_{u\in U}d(u)\le O(m'/\mu^{a}\tau^i)\le O(L/\mu^{a})$. Therefore, the probability that $P$ or $P'$ is one of these paths is $O(1/\mu^{a})$.
We now provide the proof of the claim.
Consider now a good type-2 high-degree vertex $u\in U$.
Denote $M_u=d(u)/2\mu^{a}\tau^i$.
We denote by ${\mathcal{P}}_u$ the set of paths in ${\mathcal{P}}$ that contains $u$ and participates in at most $M_u$ crossings in $\tilde \phi$. Since $u$ is good, the number of paths in ${\mathcal{P}}$ that contains $u$ and participates in more than $M_u$ crossings in $\tilde \phi$ is less than $M_u$.
Let $P^*$ be an arbitrary path in ${\mathcal{P}}_u$, and let $e^*_1,e^*_2$ be the edges of $P^*$ that are incident to $u$, such that $e^*_1$ precedes $e^*_2$ in $P^*$. Recall that we have computed an orientation $b(u)$ for $u$ in Step 2 (\Cref{Step 2}). Denote $\delta(u)=\set{e^*_1,e^R_1,\ldots,e^R_x,e^*_2,e^L_1,\ldots,e^L_y}$, where edges are indexed according to their appearance (clockwise) in the oriented ordering at $u$. We define $E^R(u)=\set{e^R_1,\ldots,e^R_x}$ and $E^L(u)=\set{e^L_1,\ldots,e^L_y}$.
See \Cref{fig: unbal_1} for an illustration.
We denote the first edges of paths in ${\mathcal{P}}_u$ by $e^*, e_1,\ldots, e_r$, where $e^*$ is the first edge of path $P^*$, and other edges are indexed according to their appearance (clockwise) in the oriented ordering at $v$. For each index $1\le i\le r$, we denote by $P_i$ the path in ${\mathcal{P}}_u$ whose first edge is $e_i$.
Since the paths of ${\mathcal{P}}$ are non-transversal at $u$, for each path in ${\mathcal{P}}_u$, its two edges incident to $u$ have to either both belong to $E^L(u)$ or both belong to $E^R(u)$.
For each path $P_i$, we label it by $L$ iff the edges of $P_i$ incident to $u$ belong to set $E^L(u)$, and we label it by $R$ iff the edges of $P_i$ incident to $u$ belong to set $E^R(u)$.
Now we let ${\mathcal{P}}'_u$ be the union of (i) the set ${\mathcal{P}}^R_u$ that contains the $2M_u$ paths of ${\mathcal{P}}_u$ with smallest indices that are labelled by $R$ (if there are not that many, then take all of them); and (ii) the set ${\mathcal{P}}^L_u$ that contains the $2M_u$ paths of ${\mathcal{P}}_u$ with largest indices that are labelled by $L$ (if there are not that many, then take all of them).
See \Cref{fig: unbal_2} for an illustration.
\begin{figure}[h]
\centering
\subfigure[An illustration of edges of $\delta(u)$. Edges of $E^L(u)$ are shown in purple, and edges of $E^R(u)$ are shown in orange.]{\scalebox{0.14}{\includegraphics{figs/unbal_1.jpg}}\label{fig: unbal_1}}
\hspace{0.1cm}
\subfigure[An illustration of edges of $\delta(v')$ (assuming $M_u=1$). The first edges of paths laballed by $L$ are shown in purple, the first edges of paths laballed by $R$ are shown in orange, and the first edges of paths in set ${\mathcal{P}}'_u$ are with check marks.]{
\scalebox{0.12}{\includegraphics{figs/unbal_2.jpg}}\label{fig: unbal_2}}
\caption{Illustrations of edge sets $\delta(u)$ and $\delta(v')$.}
\end{figure}
Let $P^L$ be the path of ${\mathcal{P}}^L_u$ with the largest index, and let $P^R$ be the path of ${\mathcal{P}}^R_u$ with the smallest index. We first show that the index of $P^R$ has to be greater than the index of $P^R$. Assume otherwise, then consider subpaths of paths in set ${\mathcal{P}}^R$ between $v'$ and $u$ and the area surrounded by the images of (i) the subpath of $P^L$ between $u$ and $v'$; and (ii) the subpath of $P^*$ between $u$ and $v'$, that we denote by $F^L$. Clearly, the first segments of those subpaths of paths in set ${\mathcal{P}}^R$ lie inside the area $F_L$, while their last segments lie outside $F_L$. Therefore, each of these paths will either cross $P^*$ or cross $P^L$ (see \Cref{fig: unbal_3}), implying that either $P^*$ or $P^L$ participates in at least $M_u$ crossings in $\tilde \phi$, a contradiction.
\iffalse
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/unbal_3.jpg}
\caption{An illustration of paths $P^L$, $P^R$ and paths in ${\mathcal{P}}^R$ (shown in thin orange). The red circles mark the crossings between paths in ${\mathcal{P}}^R$ and path $P^L$ or $P^*$ (shown in blue).}\label{fig: unbal_3}
\end{figure}
\fi
Recall we have denoted by $F_L$ the area surrounded by the image of subpaths of $P^L$ and $P^*$ between $v'$ and $u$.
We now show that the image of $v''$ do not lie in $F_L$. In fact, if the image of $v''$ lies in $F_L$, then from similar arguments, all subpaths of paths labelled $R$ have to cross either $P^*$ or $P^L$, implying that either $P^*$ or $P^L$ participates in at least $M_u$ crossings in $\tilde \phi$, a contradiction.
Similarly, we can show that the image of $v''$ do not lie in $F_R$, the area surrounded by the image of subpaths of $P^R$ and $P^*$ between $v'$ and $u$.
\begin{figure}[h]
\centering
\subfigure[An illustration of paths $P^L,P^R$ and paths in ${\mathcal{P}}^R$ (in thin orange). The red circles are crossings between paths in ${\mathcal{P}}^R$ and path $P^L$ or $P^*$ (in blue).]{\scalebox{0.09}{\includegraphics{figs/unbal_3.jpg}}\label{fig: unbal_3}}
\hspace{0.1cm}
\subfigure[An illustration of areas $F^L$, $F^R$ (shown in grey) and the behavior of paths in $\hat{\mathcal{P}}_u$ at $u$.]{
\scalebox{0.09}{\includegraphics{figs/unbal_4.jpg}}\label{fig: unbal_4}}
\caption{Illustrations of crossing patterns of paths in ${\mathcal{P}}_u$.}
\end{figure}
We delete from ${\mathcal{P}}_u$ all paths of ${\mathcal{P}}'_u$ and all paths that cross $P^L$ or $P^R$ or $P^*$, and denote the resulting path set by $\hat{\mathcal{P}}_u$. Clearly, $|{\mathcal{P}}_u\setminus \hat{\mathcal{P}}_u|=O(M_u)$. It is now easy to show that, for all paths in $\hat{\mathcal{P}}_u$ labelled $L$, the last segment of the image of its first edge incident to $u$ lies in $F^L$, and the last segment of the image of its first edge incident to $u$ lies outside $F^L$. Similarly for all paths in $\hat{\mathcal{P}}_u$ labelled $R$, the last segment of the image of its first edge incident to $u$ lies in $F^R$, and the last segment of the image of its first edge incident to $u$ lies outside $F^L$ (see \Cref{fig: unbal_4}).
\iffalse
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/unbal_4.jpg}
\caption{An illustration of areas $F^L$, $F^R$ and paths in $\hat{\mathcal{P}}_u$.}\label{fig: unbal_4}
\end{figure}
\fi
Since the paths are non-transversal with respect to $\Sigma'$. It is now easy to see that, by definition, $u$ is unbalanced with respect to at most $O(M_u)$ paths in $\hat{\mathcal{P}}_u$.
Therefore, altogether, $u$ is unbalanced with respect to at most $O(M_u)$ paths in ${\mathcal{P}}$ that contains $u$. This completes the proof of the claim, and also completes the proof of \Cref{obs: skeleton loops do not contain significant nodes}.
\iffalse
\znote{To modify from here}
Consider now a path $\hat P$ in ${\mathcal{P}}_u\setminus {\mathcal{P}}'_u$ and assume without loss of generality that $\hat P$ is labelled $L$.
Let $P^L$ be the path of ${\mathcal{P}}^L_u$ with the $M_u$-th largest index.
Denote by $\hat e_a^L, \hat e_b^L$ the edges of $P^L$ incident to $u$, where $\hat e_a^L$ precedes $\hat e_b^L$ in $P_L$.
First it is easy to observe that edges $e^*_2,\hat e_b^L,\hat e_a^L, e^*_1$ must appear in the oriented rotation at $u$ in this order, since otherwise the paths in ${\mathcal{P}}^L_u$ with larger indices than $P^L$ will each participate in more than $M_u$ crossings in $\tilde \phi$, contradicting the definition of ${\mathcal{P}}_u$.
Then it is easy to see that the first edge of $\hat P$ that is incident to $u$ must appear clockwise between $\hat e^L_a$ and $e^*_1$, since otherwise path $\hat P$ has to cross every path in ${\mathcal{P}}^L_u$ with a smaller index than $P^L$, contradicting the fact that the path $P^L$ participates in less than $M_u$ crossings in $\tilde \phi$. Similarly, the last edge of $\hat P$ that is incident to $u$ must appear clockwise between $e^*_2$ and $\hat e^L_b$. Therefore, $u$ is unbalanced with respect to at most $M_u$ paths labelled $L$ in $P_u\setminus {\mathcal{P}}'_u$. Similarly, $u$ is unbalanced with respect to at most $M_u$ labelled $R$ in $P_u\setminus {\mathcal{P}}'_u$. Therefore, $u$ is unbalanced with respect to at most $2M_u$ path in ${\mathcal{P}}_u\setminus {\mathcal{P}}'_u$.\fi
\end{proof}
\paragraph{Bad Event ${\cal{E}}_5$.}
We say that bad event ${\cal{E}}_5$ happens if either $P$ contains an inner vertex that is a type-2 high-degree vertex and unbalanced with respect to $P$, or $P'$ contains an inner vertex that is a type-2 high-degree vertex and unbalanced with respect to $P'$. From \Cref{obs: skeleton loops do not contain significant nodes}, $\Pr[{\cal{E}}_5]\le O(1/\mu^{a})$.
From now on, we call type-2 high-degree vertices \emph{significant vertices}.
\begin{observation}
\label{obs: skeleton image loop does not contain significant vertices}
If events ${\cal{E}}_1$ and ${\cal{E}}_5$ do not happen, then no self-loop of the curve representing the image of $P$ in $\tilde \phi$ contains the image of any significant vertex of $V(P)$, and similarly no self-loop of the curve representing the image of $P$ in $\tilde \phi$ contains the image of any significant vertex of $V(P')$.
\end{observation}
\begin{proof}
We show that, if events ${\cal{E}}_1,{\cal{E}}_5$ do not happen, then no loop of curve $\tilde\phi(P)$ contains the image of any significant vertex of $V(P)$, and the proof for $P'$ is symmetric.
Let $\zeta$ be a self-loop of $\tilde\phi(P)$.
Clearly, $\zeta$ partitions the sphere into two faces, that we denote by $F$ and $F'$. Assume without loss of generality that face $F$ does not contain the image of $v'$.
Assume for the sake of contradiction that loop $\zeta$ does contain the image of a significant vertex $u\in V(P)$.
Let $e_u,e'_u$ be the two edges of $P$ that are incident to $u$.
Denote $\delta_{G'}(u)\cap E({\mathcal{P}})=\set{e_u,\hat e_1,\ldots,\hat e_s, e'_u,\hat e'_1,\ldots,\hat e'_t}$, where the edges are listed according to the ordering ${\mathcal{O}}_u\in \Sigma'$. We assume without loss of generality that the first segments of the curves representing the images of edges $\hat e_1,\ldots,\hat e_s$ lie in the face $F$. Since ${\cal{E}}_5$ does not happen, the significant vertex $u$ is not unbalanced with respect to $P$. Then since $\deg_{G'}(u)\ge h$ and $u$ is not unbalanced with respect to $P$, $s > h/2\mu^{a}$.
Note that, since the paths of ${\mathcal{P}}$ are non-transversal with respect to $\Sigma'$, edges $\hat e_1,\ldots,\hat e_s$ in fact come from a subset of $s/2$ paths of ${\mathcal{P}}$, which we denote by $\hat {\mathcal{P}}$.
%
We will show that, for each path $\hat P\in \hat{{\mathcal{P}}}$, the curve $\tilde\phi(\hat P)$ must cross $\zeta$. Note that this implies that the number of crossings that involve the curve $\zeta$ is at least $s/2 > h/4\mu^{a}>(\mathsf{cr}(\tilde \phi)\cdot\tau^i\cdot\mu^{a})/m'$, causing a contradiction to the assumption that event ${\cal{E}}_1$ does not happen, and therefore completes the proof of \Cref{obs: skeleton image loop does not contain significant vertices}.
We now show that the curve $\tilde\phi(\hat P)$ must cross $\zeta$. Note that the curve $\tilde\phi(\hat P)$ connects $\tilde\phi(v')$ to $\tilde\phi(v'')$, and the image of $v'$ lie outside the face $F$. Therefore, there must exist a point $z$ that belongs to both curve $\tilde\phi(\hat P)$ and $\zeta$, such that, if we denote by $\gamma_z, \gamma'_z$ the tiny segments of $\tilde\phi(\hat P)$ right before and after $z$, then one of $\gamma_z, \gamma'_z$ is contained in the interior of face $F$ and the other is contained in the interior of face $F'$.
Assume that $z$ is the image of a vertex. However, since the paths of ${\mathcal{P}}$ are non-transversal with respect to $\Sigma$, segments $\gamma_z$ and $\gamma'_z$ have to lie on the same side of $\zeta$, a contradiction.
Therefore $z$ is not the image of a vertex, and so $z$ is a crossing between $\tilde\phi(\hat P)$ and $\zeta$.
\end{proof}
\paragraph{Summary of properties of $K$.}
We define event ${\cal{E}}=\bigcup_{1\le i\le 5}{\cal{E}}_i$. Assume that event ${\cal{E}}$ does not happen, then the cycle $K$ and the well-structured drawing $\tilde \phi$ satisfies the following properties.
\begin{properties}{F}
\item $|E(K)|\le 2\tau^i\cdot\mu^{10{50}}$;
\item edges of $K$ participate in at most $(\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m'$ crossings in $\tilde \phi$;
\label{f_prop: K crossed few times}
\item in drawing $\tilde \phi$, the edges of $K$ do not cross any special edge of any instance in the collection $\hat {\mathcal{I}}$ before this iteration;\label{f_prop: K does not cross special edges}
\item the number of edges that are incident to a low-degree vertex (a vertex of degree at most $h$ in $G'$) of $K$ is at most $h\cdot \tau^i\cdot\mu^{a}$, where $h=\mathsf{cr}(\tilde \phi)\cdot \mu^{11{50}+3a}\cdot(\tau^i)^4/m'$;
\label{f_prop: low-degree incident edges small}
\item the number of edges that are incident to a type-1 high-degree vertex (a vertex that has degree at least $h$ and is contained in at most $h$ paths of ${\mathcal{P}}$) of $K$ is at most $h\cdot \tau^i\cdot\mu^{a}$;
\label{f_prop: type-1 incident edges small}
\item the images of $P$ and $P'$ do not cross in $\tilde \phi$;
\label{f_prop: P and P' do not cross}
\item no self-loop of the curve representing the image of $P$ in $\tilde \phi$ contains the image of any significant vertex, and similarly no self-loop of the curve representing the image of $P'$ in $\tilde \phi$ contains the image of any significant vertex.
\label{f_prop: no significant in loop}
\end{properties}
\subsubsection*{Modifying the drawing $\tilde \phi$}
We now proceed to modify the drawing $\tilde \phi$ into a drawing that is well-structured with respect to the collection $\hat {\mathcal{I}}$ after this iteration. That is, we will modify the image of edges of $I'$ in $\tilde \phi$, such that the resulting drawing induces well-structured drawings for new instances $I_{\mathsf{in}}, I_{\mathsf{out}}$.
We now provide a summary of the stages in modifying the drawing $\tilde \phi$. For simplicity of notations we denote $I'_1=I_{\mathsf{in}}$ and $I'_2=I_{\mathsf{out}}$ from now on.
We first delete the edges that are added to set $\hat E^{\sf del}$ in this iteration, and rename the resulting drawing by $\tilde \phi$. Clearly, the new drawing $\tilde \phi$ contains at most the same number of crossings as the old one.
In the first stage, we modify the drawing $\tilde\phi$ into a new drawing $\hat\phi$ that induces almost well-structured drawings of instances $I'_1$ and $I'_2$ (recall that an almost well-structured drawing is a drawing that satisfies properties \ref{prop: simple curve}--\ref{prop: first segments}).
In the second stage, we locally modify the image of edges incident to vertices of $K$, such that the orientations of vertices of $K$ coincide the orientations computed in Step 2 (\Cref{Step 2}).
In the third and the fourth stages, we
further modify drawing $\hat\phi$ into a drawing $\tilde \phi'$ that induces well-structured drawings of instances $I'_1$ and $I'_2$, respectively.
In each of these stages, we will make sure not to increase the number of crossings in the drawing by too much.
We now describe each of the stages in turn.
\iffalse
From now on we denote $\phi=\tilde \phi$.
We now construct well-structured drawings of the subinstances $I'_1$ and $I'_2$ using the well-structured drawing $\phi$ of instance $I'$ in four stages. In the first stage, we modify the well-structured drawing $\phi$ of $I'$ into an almost well-structured drawing $\hat \phi$ of $I'$ (recall that an almost well-structured drawing is a drawing that satisfies properties \ref{prop: simple curve}--\ref{prop: discs interior not crossing}), such that the image of $K$ in $\hat\phi$ is a simple closed curve, and $\mathsf{cr}(\hat\phi)$ is not much larger than
$\mathsf{cr}(\phi)$. In the second stage, we locally modify the image of edges incident to vertices of $K$, such that the orientations of vertices of $K$ coincide the orientations computed in Step 2 (\Cref{Step 2}).
In the third and the fourth stages, we
split the drawing $\hat \phi$ into an almost well-structured drawing $\hat \phi_1$ of $I'_1$ and an almost well-structured drawing $\hat\phi_2$ of $I'_2$, and then we further modify drawings $\hat\phi_1, \hat\phi_2$ into well-structured drawings $\phi'_1,\phi'_2$ of instances $I'_1$ and $I'_2$, respectively.
We now describe each of the stages in turn.
\fi
\subsubsection*{Stage 1. Removing self-loops of the image of $K$ in $\tilde\phi$}
In the first stage, we modify the drawing $\tilde \phi$ into a new drawing $\hat \phi$ that induces almost well-structured drawings of instances $I'_1,I'_2$, via the following claim.
\begin{claim}
\label{clm: skeleton can be drawn in a natural way}
If event ${\cal{E}}$ does not happen, then there is a drawing $\hat\phi$ of instance $\bar I$ after this iteration, in which (i) for each edge not incident to any vertex of $K$, its image in $\hat \phi$ remains the same as in $\tilde \phi$; (ii) the image of $K$ in $\hat \phi$ is a simple closed curve that does not cross any special edge of $I'$, and (iii)
$\mathsf{cr}(\hat\phi)\le \mathsf{cr}(\phi)+O\big(h^2(\tau^i)^2\mu^{2a}\big).$
\end{claim}
\begin{proof}
We construct the drawing $\hat\phi$ by iteratively ``opening'' the self-loops of the curve representing the image of $P$ and the curve representing the image of $\tilde\phi$ until the image of $K$ becomes a simple closed curve. Recall that, from Property \ref{f_prop: P and P' do not cross}, the images of $P$ and $P'$ do not cross in $\tilde\phi$, and from Property \ref{f_prop: K does not cross special edges}, the image of $K$ does not cross any special edge of $I'$ in $\tilde\phi$.
We denote by $\gamma$ the image of $P$ in $\tilde\phi$. Note that $\gamma$ is not necessarily a simple curve. We first iteratively compute a set $Z$ of self-loops of $\gamma$ and another simple curve $\gamma'$ connecting $\tilde\phi(v')$ to $\tilde\phi(v'')$, as follows.
We view the curve $\gamma$ as being directed from $\tilde\phi(v')$ to $\tilde\phi(v'')$. Initially we set $Z=\emptyset$. We start from the first endpoint $\tilde\phi(v')$ of $\gamma$ and find the first point $z$ on $\gamma$ that appears twice on $\gamma$. Denote by $\zeta$ the subcurve of $\gamma$ between the first and the last (second) appearance of $z$, so $\zeta$ is a self-loop that starts and ends at $z$. We add the self-loop $\zeta$ to set $Z$ and delete $\zeta$ from $\gamma$. We continue this process until the curve $\gamma$ no longer contains self-loops. Denote by $\gamma'$ the resulting simple curve we obtain, and we also view curves $\gamma'$ as being directed from $\tilde\phi(v')$ to $\tilde\phi(v'')$. See \Cref{fig: remove_loop} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Curve $\gamma$ (with direction) is shown in blue, with vertices $v_1,v_2,v_3$ appearing on it sequentially.]{\scalebox{0.0855}{\includegraphics{figs/remove_loop_1.jpg}}}
\hspace{0.1cm}
\subfigure[Curve $\gamma'$ is shown in green. Self-loop $\zeta$ is shown in light blue. Thin strip around $\zeta$ is shown in grey.]{
\scalebox{0.085}{\includegraphics{figs/remove_loop_2.jpg}\label{fig: thin strip}}}
\caption{Illustration for constructing curve $\gamma'$. \label{fig: remove_loop}}
\end{figure}
The algorithm for constructing the drawing $\hat\phi$ is iterative. In each iteration, we process a self-loop in $Z$.
We now describe an iteration.
Let $\zeta$ be a loop in $Z$, and we denote by $z$ the unique point of $\zeta$ that lies on $\gamma'$.
%
We modify the drawing as follows.
Let $v_1,\ldots,v_r$ be the vertices whose images appear on the self-loop $\zeta$, where the vertices are indexed according to the ordering in which they appear on $P$ (see \Cref{fig: remove_loop}).
We denote $V_{\zeta}=\set{v_1,\ldots,v_r}$.
Let $\sigma$ be a tiny segment of $\gamma'$ that contains $z$.
We remove the image of vertices $v_1,\ldots,v_r$ and all their incident edges from the current drawing, and place them on $\sigma$, such that $v_1,\ldots,v_r$ appear sequentially in the direction of $P$.
See \Cref{fig: remove_loop_3} for an illustration.
\iffalse
\begin{figure}[h]
\centering
\subfigure[Before: the images of $v_1,v_2$ appear sequentially on the image of $P$, and they belong to the loop $\zeta$.]{\scalebox{0.09}{\includegraphics{figs/loop_before.jpg}\label{fig: loop_before}}}
\hspace{0.1cm}
\subfigure[After: the images of $v_1,v_2$ are moved out of $\zeta$ to the segment $\sigma$ (shown in red).]{
\scalebox{0.1}{\includegraphics{figs/loop_after.jpg}}\label{fig: loop_after}}
\caption{Illustration for the process of opening a loop $\zeta$. \label{fig: loop}}
\end{figure}
\fi
It remains to add back the edges incident to $v_1,\ldots,v_r$. We denote by $z_1,\ldots,z_r$ the points representing the new images of vertices $v_1,\ldots,v_r$ respectively.
%
%
Let $S$ be an arbitrarily thin strip around $\zeta$, such that the entire segment $\sigma$ is contained in $S$ (see \Cref{fig: thin strip}).
Let $E_{\zeta}$ be the set of all edges incident to a vertex of $V_{\zeta}$. Clearly, for each edge $e\in E_{\zeta}$, the current image of $e$ has to cross the boundary of $S$. For each $e\in E_{\zeta}$, we denote by $z_e$ the last crossing between the curve $\tilde\phi(e)$ and $S$, where the curve $\tilde\phi(e)$ is viewed as being directed from its endpoint in $V_{\zeta}$ to its endpoint outside of $V_{\zeta}$.
Now we have obtained two sets $\set{z_e\mid e\in E_{\zeta}}$ and $\set{z_1,\ldots,z_r}$ of points that all lie within $S$ or on the boundary of $S$.
We now compute, for each edge $e\in E_{\zeta}$ with endpoint $v_i\in V_{\zeta}$, a curve $\xi_e$ connecting $z_i$ to $z_e$ that lies within the thin strip $S$, such that the curves in $\Xi_{\zeta}=\set{\xi_e\mid e\in E_{\zeta}}$ are in general position.
Then for every $1\le i\le r$ and for every edge $e$ that is incident to vertex $v_i$, we define the curve $\eta_e$ be the concatenation of (i) the curve $\xi_e$; and (ii) the segment of its original image between the last crossing with the boundary of $S$ and the image of its other endpoint in $\tilde\phi$.
Note that the curve $\eta_e$ may still intersect $\gamma'$ times. Lastly we modify curve $\eta_e$ as follows. We view it as being directed from its endpoint in $V_{\zeta}$ to its endpoint outside $V_{\zeta}$. Let $p_e$ be a point of $\eta_e$ just before the first point on $\eta_e$ that also belongs to $\gamma'$, and let $p'_e$ be a point of $\eta_e$ just after the last point on $\eta_e$ that also belongs to $\gamma'$. We then replace the subcurve of $\eta_e$ between $p_e$ and $p'_e$ by a new curve that lies within the thin strip around $\gamma'$, and cross $\gamma'$ at most once.
See \Cref{fig: remove_loop_4} for an illustration.
Clearly, in this way we obtain a new drawing of the graph $G'$ that no longer has the loop $\zeta$.
%
\begin{figure}[h]
\centering
\subfigure[Segment $\sigma$ is shown in red, with the new images of $v_1,v_2,v_3$ placed on it. $e,e'$ are edges incident to $v_1, v_2$ respectively. Their corresponding curves $\eta_{e},\eta_{e'}$ are shown in pink.]{\scalebox{0.13}{\includegraphics{figs/remove_loop_3.jpg}\label{fig: remove_loop_3}}}
\hspace{0.1cm}
\subfigure[The adjusted image of $e'$: now it no longer intersect $\gamma$ more than once.]{
\scalebox{0.13}{\includegraphics{figs/remove_loop_4.jpg}}\label{fig: remove_loop_4}}
\caption{Illustration for the process of adding edges back. \label{fig: remove_loop_next}}
\end{figure}
We first show that, after this iteration, the drawing we obtain still induces a drawing of $I'$ that satisfies properties \ref{prop: simple curve}--\ref{prop: discs interior not crossing}.
Note that we have only modified the images of vertices of $V_{\zeta}$, and the special cycles in ${\mathcal{R}}(I')$ are node-disjoint from $K$.
Therefore, the image of all special cycles remain the same as in $\tilde\phi$, and so the properties \ref{prop: simple curve}--\ref{prop: discs interior not crossing} are still satisfied.
We now show that, after this iteration, the image of $K$ still does not cross any special edge in $I'$.
Recall that $K$ crosses no special cycle of ${\mathcal{R}}(I')$ in $\tilde\phi$. And note that the new images of edges in $K$ are entirely contained in the image of $K$ in $\tilde\phi$. So after this iteration, in the new drawing we obtained, the image of $K$ still does not cross any special edge in $I'$.
Consider the resulting drawing obtained by processing each self-loop in $Z$ in this way. Finally, we perform type-1 uncrossing to the set of curves representing the images of edges in $E_{\zeta}$. Denote by $\hat \phi$ the resulting drawing.
It is easy to verify that $\hat\phi$ induces a drawing for instance $I'$ that satisfies properties \ref{prop: simple curve}--\ref{prop: discs interior not crossing}, and moreover, in $\hat \phi$ the image of $K$ is a simple closed curve that does not cross any special edge of $I'$.
We now bound the number of new crossings in the drawing. Consider the iteration where some self-loop $\zeta$ is processed. Note that $|E_{\zeta}|\le \sum_{1\le i\le r}\deg_{G'}(v_i)$.
Note that, since we have performed type-1 uncrossing to the images of edges in $E_{\zeta}$, the resulting images of edges in $E_{\zeta}$ may create at most $|E_{\zeta}|^2$ new crossings. On the other hand, let $E^*_{\zeta}$ be the set of edges in $E(G')$ that cross the self-loop $\zeta$ in the original drawing $\tilde\phi$, so the new images of $E_{\zeta}$ may create $|E^*_{\zeta}|\cdot |E_{\zeta}|$ new crossings with images of $E^*_{\zeta}$.
Finally, since we modified the image of edges in $E_{\zeta}$ such that each of these edges cross $\gamma'$ at most once, the number of intersections between the final images of edges of $E_{\zeta}$ and curve $\gamma'$ is at most $|E_{\zeta}|$. Clearly, these are all new crossings that we may create in an iteration.
Therefore, summing over all iterations,
%
\[
\mathsf{cr}(\hat\phi)\le \text{ } \mathsf{cr}(\tilde\phi)+\sum_{\zeta \in Z}\bigg(|E_{\zeta}|+|E^*_{\zeta}|+1\bigg)\cdot |E_{\zeta}|
\le \text{ } \mathsf{cr}(\tilde\phi)+ \bigg(\sum_{\zeta \in Z}|E_{\zeta}|+|E^*_{\zeta}|\bigg)^2.
\]
Since event ${\cal{E}}_1$ does not happen, $\sum_{\zeta \in Z}|E^*_{\zeta}|\le (\mathsf{cr}(\tilde\phi)\cdot\tau^i\cdot\mu^{a})/m'$.
And from Property \ref{f_prop: no significant in loop}, no significant vertices may belong to any self-loop in $Z$, so the self-loops of $Z$ only contains images of type-1 high-degree vertices and low-degree vertices of $P$.
Then from Properties \ref{f_prop: K does not cross special edges} and \ref{f_prop: low-degree incident edges small}, $\sum_{\zeta \in Z}|E_{\zeta}|\le O(h\tau^i\mu^{a})$.
Altogether, we get that
$$\mathsf{cr}(\hat\phi)\le \text{ } \mathsf{cr}(\tilde\phi)+O\bigg( \frac{\mathsf{cr}(\tilde\phi)\cdot\tau^i\cdot\mu^{a}}{m'}+ h\cdot\tau^i\mu^a \bigg)^2=\mathsf{cr}(\tilde\phi)+O\bigg(h^2(\tau^i)^2\mu^{2a}\bigg).$$
\end{proof}
\iffalse
$\ $
We say that event ${\cal{E}}$ happens if any of events ${\cal{E}}_1,{\cal{E}}_1,{\cal{E}}_5,{\cal{E}}_3,{\cal{E}}_4$ happens, namely ${\cal{E}}={\cal{E}}_1\cup{\cal{E}}_1\cup{\cal{E}}_5\cup{\cal{E}}_3\cup{\cal{E}}_4$. From the above discussion, $\Pr[{\cal{E}}]\le O(1/\mu^{a})$.
Assuming that event ${\cal{E}}$ does not happen. The properties of the skeleton $K$ can be summarized as follows.
\begin{properties}{F}
\item $K$ contains at most $\mu^{{50}+a}$ edges and at most $\mu^{{50}+a}$ vertices; \label{fact: skeleton is short}
\item edges of $K$ participate in less than $(\mathsf{cr}\cdot\mu^{{50}+a})/m$ crossings in $\hat\phi$;\label{fact: skeleton is slightly crossed}
\item less than $2h'\mu^{{50}+a}$ edges are incident to a type-1 high-degree vertex of $K$;
\label{fact: few incident edges on type-1}
\item less than $2h\mu^{{50}+a}$ edges are incident to a low-degree vertex of $K$;
\label{fact: few incident edges on low-deg}
\item there is a drawing $\hat\phi'$ of $\hat G'$, such that $\hat \phi'$ respects the rotation system $\hat \Sigma$, the image of $K$ in $\hat\phi'$ is a simple closed curve, and $\mathsf{cr}(\hat\phi')\le \mathsf{cr}+O(h\mu^{{50}+a})^2$.\label{fact: skeleton drawn as a cycle}
\end{properties}
\fi
\subsubsection*{Stage 2. Correcting the orientations of vertices in $V(K)$}
Recall that we have computed, in Step 2 (\Cref{Step 2}), an orientation $b(v)$ for each vertex $v\in V(K)$, and then used them to define inner and outer sides of $K$, which are then used to compute the instances $I'_1, I'_2$.
However, although the drawing $\hat\phi$ computed in the last stage respects the rotation system $\Sigma'$, it may not yet match the orientations on vertices of $K$ computed in Step 2, which is required in Property~\ref{prop: first segments} if it induces well-structured solutions for instances $I'_1, I'_2$. In this stage, we further modify $\hat{\phi}$ by correcting the orientations of vertices of $K$.
First, we prove the following observations, which show that, for vertex $v''$ and all significant vertices of $K$, the orientation computed in Step 2 is correct (matched by the drawing $\tilde\phi$, and so also matched by $\hat\phi$ since we have not modified the orientations of vertices when constructing $\hat \phi$ from $\tilde\phi$ in the last stage).
\begin{observation}
The orientation $b''$ (computed in Step 2) on vertex $v''$ coincides the orientation of $v''$ in drawing $\tilde\phi$.
\end{observation}
\begin{proof}
Observe that the distances between oriented orderings satisfy the triangle inequality, namely for any oriented rotations $({\mathcal{O}}_1,b_1),({\mathcal{O}}_2,b_2),({\mathcal{O}}_3,b_3)$ on the same set of element, $\mbox{\sf dist}(({\mathcal{O}}_1,b_1),({\mathcal{O}}_3,b_3))\le \mbox{\sf dist}(({\mathcal{O}}_1,b_1),({\mathcal{O}}_2,b_2))+\mbox{\sf dist}(({\mathcal{O}}_2,b_2),({\mathcal{O}}_3,b_3))$. In fact, from the definition of the reordering curves, given a set $\Gamma_{12}$ of reordering curves for oriented orderings $({\mathcal{O}}_1,b_1),({\mathcal{O}}_2,b_2)$ and a set $\Gamma_{23}$ of reordering curves for oriented orderings $({\mathcal{O}}_2,b_2),({\mathcal{O}}_3,b_3)$, it is immediate to concatenate curves in $\Gamma_{12}$ with curves in $\Gamma_{23}$ to obtain a set $\Gamma_{13}$ of reordering curves for $({\mathcal{O}}_1,b_1),({\mathcal{O}}_3,b_3)$, such that $\chi(\Gamma_{13})\le \chi(\Gamma_{12})+\chi(\Gamma_{23})$.
On the one hand, since ${\mathcal{O}}''$ is an unoriented ordering on a set of $\Omega(m'/\tau^i)$ elements, it is easy to see that $\mbox{\sf dist}(({\mathcal{O}}'',-1),({\mathcal{O}}'',1))=\Omega\big((m'/\tau^i)^2\big)$. Therefore, since $m'\ge \tau^s$ and \fbox{$\mathsf{cr}(\tilde\phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$}, \[\mbox{\sf dist}(({\mathcal{O}}'',-1),({\mathcal{O}}',b'))+\mbox{\sf dist}(({\mathcal{O}}'',1),({\mathcal{O}}',b'))\ge \mbox{\sf dist}(({\mathcal{O}}'',-1),({\mathcal{O}}'',1))\ge \Omega\big((\tau^s/\tau^i)^2\big)>2\cdot\mathsf{cr}(\tilde\phi),\]
and so at most one of $\mbox{\sf dist}(({\mathcal{O}}'',-1),({\mathcal{O}}',b')),\mbox{\sf dist}(({\mathcal{O}}'',1),({\mathcal{O}}',b'))$ is less than $\mathsf{cr}(\tilde\phi)$.
On the other hand, since the paths of ${\mathcal{P}}$ are non-transversal with respect to $\Sigma'$, it is easy to show that paths of ${\mathcal{P}}$ participate in at most $\mathsf{cr}(\tilde\phi)$ crossings in drawing $\tilde\phi$, so the orientation $b''$ on $v''$ computed in Step 2 coincides the orientation of $v''$ in $\tilde\phi$.
\end{proof}
\begin{observation}
If event ${\cal{E}}$ does not happen, then for each significant vertex $u\ne v',v''$, the orientation $b(u)$ we computed in Step 2 coincides with the orientation of $u$ in drawing $\tilde\phi$.
\end{observation}
\begin{proof}
Let $u$ be a significant vertex and let ${\mathcal{P}}_u$ be the set of paths in ${\mathcal{P}}$ that contains vertex $u$. From the definition of a significant vertex, $|{\mathcal{P}}_u|\ge h$.
From Property \ref{f_prop: K crossed few times}, edges of $K$ participates in at most $(\mathsf{cr}(\tilde\phi)\cdot\tau^i\cdot\mu^{a})/m'$ crossings in $\tilde\phi$.
Denote ${\mathcal{P}}_u=\set{P_{i_1},\ldots, P_{i_q}}$, and for each $1\le j\le q$, we denote by $e_{i_j}$ the first edge of $P_{i_j}$ (the edge of $P_{i_j}$ incident to $v'$), and denote by $\tilde e_{i_j}$ the first edge of $P_{i_j}$ incident to $u$. Recall that in Step 2, we say that an index $i_j$ is screwed under $b(v)$, iff when the orientation of $v$ is $b(v)$, edges $e_{i_j}, \tilde e_{i_j}$ are on different sides of $K$. It is easy to show that, if index $i_j$ is screwed under $b(v)$, then the image of path $P_{i_j}$ has to cross $K$. It is also easy to see that, each index $i_j$ is screwed under either $-1$ or $1$ but not both.
Since event ${\cal{E}}$ does not happen, from Property \ref{f_prop: K crossed few times}, $K$ participate in at most $(\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m'<h/2$ crossings in $\tilde \phi$.
Since $u$ is a type-2 high-degree vertex, there is exactly one orientation in $\set{-1,1}$ under which the number of screwed indices is less than $h/2$. Therefore, the orientation $b(u)$ computed in Step 2 coincides with the orientation of $u$ in drawing $\tilde\phi$.
\end{proof}
Therefore, we only need to modify the orientation of low-degree vertices and type-1 high-degree vertices. We denote by $V'$ the set of all low-degree vertices and all type-1 high-degree vertices $v$ of $V(K)$, such that the orientation of $v$ in the drawing $\hat\phi$ does not coincide with the orientation $b(v)$ computed in Step 2.
Consider now a vertex $v\in V'$.
Assume without loss of generality that $v$ is a vertex of $P$, and denote $\delta(v)=\set{\tilde e, \hat e_1, \hat e_2,\ldots, \hat e_p,\tilde e', \hat e'_1, \hat e'_2, \ldots, \hat e'_q}$, where edges $\tilde e,\tilde e'$ belong to $P$, $\tilde e$ precedes $\tilde e'$ in $P$, and the edges appear clockwise in the oriented ordering $({\mathcal{O}}_v,b(v))$ of $v$.
Denote by $D=D_{\hat \phi}(v)$ the tiny $v$-disc in drawing $\hat \phi$.
Let $\tilde z$ be the intersection between $\hat\phi(\tilde e)$ with the boundary of disc $D$, and we define (i) a point $\tilde z'$ for $\tilde e'$; (ii) for each $1\le i\le p$, a point $ z_i$ for $\hat e_i$; and (iii) for each $1\le j\le q$, a point $ z'_j$ for $\hat e'_j$ similarly.
See \Cref{fig: flip} for an illustration.
We place a disc $D'$ around $\hat\phi(v)$, that lies entirely within disc $D$. We then place points $\delta(v)=\set{\tilde y, y'_1, y'_2,\ldots, y'_q,\tilde y', y_1, y_2, \ldots, y_p}$ on the boundary of $D'$, such that the points appear on the boundary of $D'$ clockwise in this order.
We then compute, a set $\Gamma$ of curves within the annulus $D\setminus D'$ that contains, for each $1\le i\le p$, a curve $\gamma_i$ connecting the point $ y_i$ to $ z_i$, and, for each $1\le j\le q$, a curve $\gamma'_j$ connecting the point $ y'_j$ to $ z'_j$, such that the curves of $\Gamma$ are in general position.
We modify the drawing $\hat\phi$ locally within disc $D$ as follows.
We erase the drawing of $\hat\phi$ inside disc $D$ except for the image of vertex $v$ and the images of edges $\tilde e,\tilde e'$. We now let, for each $1\le i\le p$, the new image of edge $\hat e_i$ be the concatenation of: (i) the line segment within $D'$ connecting $\hat\phi(v)$ to point $\hat y_i$; (ii) the curve in $\gamma_i\in \Gamma$ connecting $ y_i$ to $ z_i$; and (iii) the original image of edge $\hat e_i$ outside the disc $D$, namely the subcurve of $\hat\phi(\hat e_i)$ between $ z_i$ and the image of the other endpoint of $\hat e_i$.
Similarly, we let, for each $1\le j\le q$, the new image of edge $\hat e'_j$ be the concatenation of: (i) the line segment within $D'$ connecting $\hat\phi(v)$ to point $ y'_j$; (ii) the curve in $\gamma'_j\in \Gamma$ connecting $ y'_j$ to $ z'_j$; and (iii) the original image of edge $\hat e'_j$ outside the disc $D$, namely the subcurve of $\hat\phi(\hat e'_j)$ between $ z'_j$ and the image of the other endpoint of $\hat e'_j$.
See \Cref{fig: flip} for an illustration.
Clearly, the number of new crossings we have created in this local modification step is at most $\deg(v)^2$.
We modify the drawing $\hat \phi$ at all vertices of $V'$ in the same way, and denote by $\hat \phi'$ the resulting drawing. It is easy to see that drawing $\hat \phi'$ coincides with all oriented orderings on vertices of $K$ computed in Step 2.
Note that, from properties \ref{f_prop: low-degree incident edges small} and \ref{f_prop: type-1 incident edges small}, the total number of edges that are incident to a type-1 high degree vertex or a low-degree vertex is at most $O\big(h\cdot \tau^i\mu^{a}\big)$. Therefore,
\[\mathsf{cr}(\hat\phi')\le \mathsf{cr}(\hat\phi)+\sum_{v\in V(K)}
\deg(v)^2\le \mathsf{cr}(\hat\phi)+O\bigg(h^2(\tau^i)^2\mu^{2a}\bigg).\]
\begin{figure}[h]
\centering
\subfigure[A schematic view of the drawing $\hat\phi$ around $v$.]{\scalebox{0.1}{\includegraphics{figs/rotation_flip_before.jpg}}}
\hspace{1cm}
\subfigure[Curves in $\Gamma$ are shown in dash lines.]{
\scalebox{0.1}{\includegraphics{figs/rotation_flip_after.jpg}}}
\caption{An Illustration of orientation-flip at $v$ in Stage 2. \label{fig: flip}}
\end{figure}
Since we have only modified the drawing $\hat \phi$ only locally within tiny discs around vertices of $K$, in the resulting drawing $\hat \phi'$, it still holds that the image of $K$ is a simple closed curve that does not cross any special edge. Therefore, drawing $\hat \phi'$ induces almost well-structured solutions for instances $I'_1$ and $I'_2$.
\subsubsection*{Stage 3. Split the drawing and further modification on vertices}
So far we have computed a drawing $\hat \phi'$ that induces almost well-structured drawings of instances $I'_1$ and $I'_2$.
However, it does not induce well-structured drawings of $I'_1$ and $I'_2$ yet. In particular, vertices of $I'_1$ and $I'_2$ may be drawn on both sides of $K$ (so Property \ref{prop: discs interior no vertex} is not satisfied), and the number of edges crossing $K$ may be too large (so Property \ref{prop: niubi edge crossing special cycle} is not satisfied).
In this stage and the next stage, we further modify drawing $\hat \phi'$ so that it induces well-structured drawings of instances $I'_1$ and $I'_2$ respectively.
Recall that $I'_1$ corresponds to the inner side of $K$, namely $G'_1=G_{\mathsf{in}}$.
Also recall that, in drawing $\hat\phi_1$, the image of $K$ is a simple closed curve. We denote by $\eta_K$ this simple closed curve, so $\eta_K$ partitions the sphere into two faces. We denote by $F$ the face that contains the first segments of the images of edges in $E_{\mathsf{in}}$.
In this stage, we move the image of all vertices of $G'_{1}$ that are currently drawn out of $F$ inside $F$. In the next stage, we move the image of some edges inside $F$ to avoid crossing $K$.
\newcommand{\mathsf{Move}}{\mathsf{Move}}
We now start to describe the algorithm in this stage.
We first describe a subroutine called $\mathsf{Move}$ and then show how it is used in this stage.
\paragraph{Subroutine $\mathsf{Move}$.} The input to the subroutine $\mathsf{Move}$ consists of
\begin{itemize}
\item a drawing $\psi$ of $G'_1$;
\item a cluster $W$ of $G'_1$;
\item a disc $D$ such that all vertices of $W$ are drawn outside of $D$ in $\psi$;
\item an edge $e^*_W\in \delta(W)$ such that the endpoint of $e^*_W$ outside $W$ is drawn inside $D$;
\item a set ${\mathcal{Q}}$ of paths routing edges of $\delta(W)$ to $e^*_W$ inside $W$; and
\item a set ${\mathcal{P}}=\set{P(e)\mid e\in \delta(W)}$ of paths, such that for each $e\in \delta(W)$, the path $P(e)$ has $e$ as its first edge and has the last endpoint drawn inside disc $D$.
\end{itemize}
See \Cref{fig: move_layout} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Disc $D$ is shown on the left and cluster $W$ is shown on the right. Edges of $\delta(W)$ with the other endpoint drawn inside $D$ are shown in blue, and edges of $\delta(W)$ with the other endpoint drawn outside $D$ are shown in red.]{\scalebox{0.1}{\includegraphics{figs/move_layout_1.jpg}}}
\hspace{1cm}
\subfigure[$e^*_W=e_2$, and the paths of ${\mathcal{Q}}$ are shown in orange dash lines.]{
\scalebox{0.1}{\includegraphics{figs/move_layout_2.jpg}}}
\caption{An Illustration of the input to subroutine $\mathsf{Move}$. \label{fig: move_layout}}
\end{figure}
The subroutine $\mathsf{Move}(\psi, W, D, e^*_W, {\mathcal{Q}},{\mathcal{P}})$ outputs a new drawing $\psi'$ of graph $G'_1$ in which the cluster $W$ is drawn inside $D$. We now describe the steps in the subroutine. Denote by $u,u'$ the endpoints of edge $e^*_W$ where $u\notin W$ and $u '\in W$, so the image of $u$ in $\psi$ lies in disc $D$.
For each edge $e\in\delta(W)$, we denote by $Q(e)$ the path in ${\mathcal{Q}}$ that routes $e$ to $e^*_W$.
We denote by $p$ the first crossing between the images of $e^*_W$ and $K$ (we view the curve representing the image of $e^*$ as being directed from $u'$ to $u$).
Let $D'$ be a tiny disc that (i) is entirely contained in $D$; (ii) is arbitrarily close to $p$ but does not contain $p$; and (iii) intersects with no other edge than $e^*_W$. See \Cref{fig: move_1} for an illustration. We then construct, for each edge $e\in \delta(W)\setminus \set{e^*_W}$, a curve $\gamma_e$ that consists of (i) a curve that starts at the image of the endpoint of $e$ in $W$, travels through the thin strip of the image of path $Q(e)$, and ends at some point $x_e$ lying in the tiny $u'$-disc $D_{\psi}(u')$; and (ii) a curve that starts at $x_e$, travels through the thin strip of the image of $e^*_W$, and ends at a point $y_e$ on the boundary of $D'$; such that the curves $\Gamma=\set{\gamma_e}_{e\in \delta(W)\setminus \set{e^*_W}}$ are in general position (we will later specify how to choose the ordering in which points in $\set{y_e\mid e\in \delta(W)}$ appear on the boundary of $D'$).
We also denote by $y_{e^*_W}$ the last crossing of edge $e^*_W$ and the boundary of $D'$ (so the segment of image of $e^*_W$ between $u$ and $y_{e^*_W}$ is disjoint from the interior of $D'$).
See \Cref{fig: move_2} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Drawing $\psi$ and disc $D'$ (shown in light green).]{\scalebox{0.095}{\includegraphics{figs/move_1.jpg}}\label{fig: move_1}}
\hspace{1cm}
\subfigure[Curves in $\Gamma$ are shown in dash lines.]{
\scalebox{0.095}{\includegraphics{figs/move_2.jpg}}\label{fig: move_2}}
\caption{An Illustration of constructing curves of $\Gamma$.}
\end{figure}
We now erase the image of $W$ from $\psi$ and put it inside $D'$. Lastly, we compute a set $Z$ of curves connecting points in $\set{y_e\mid e\in \delta(W)}$ to the new image of the endpoint of edges in $\delta(W)$. See \Cref{fig: move_inside} for an illustration.
We now show in the following observation that we can indeed computing a circular ordering on points in $\set{y_e\mid e\in \delta(W)}$ and the set $Z$ of curves, such that the number of crossings between $Z$ and the new image of $W$ can be charged to the number of crossings in drawing $\psi$.
\begin{observation}
\label{obs: inside_disc drawing}
There exists a circular ordering ${\mathcal{O}}_Y$ on points in $Y=\set{y_e\mid e\in \delta(W)}$ and a set $Z$ of curves that contains, for each $e\in \delta(W)$, a curve $\zeta_e$ inside disc $D'$ connecting point $y_e$ to the new image of the endpoint of $e$ in $W$, such that the number of crossings between $Z$ and the new image of $W$ is at most the number of crossings between $W$ and paths in ${\mathcal{P}}$ in drawing $\psi$.
\end{observation}
\begin{proof}
Let $\hat W$ be the graph obtained from graph $W\cup \delta(W)$ by contracting all outside-$W$ endpoints of edges in $\delta(W)$ into a single vertex, that we denote by $\hat v$. It is easy to verify that, computing a circular ordering ${\mathcal{O}}_Y$ on points in $Y$ and a set $Z$ of curves as required in \Cref{obs: inside_disc drawing} is equivalent to computing a circular ordering $\hat{\mathcal{O}}$ on the incident edges of $\hat v$ and a drawing $\hat \phi$ of graph $\hat W$ that respect the orderings ${\mathcal{O}}_v$ for all $v\in V(W)$ and the new ordering $\hat{{\mathcal{O}}}$.
For each edge $e\in \delta(W)$, we consider the curve representing the image of path $P(e)$ in $\psi$, and denote by $y'_e$ the last crossing between this curve and the image of $K$ in $\psi$ (where we view the curve as being directed from its endpoint in $W$ to its endpoint that is drawn inside $D$), and we denote by $\gamma'_e$ the subcurve of this curve between its endpoint in $W$ and $y'_e$.
We now define $\hat {\mathcal{O}}'$ to be the circular ordering on set $Y'=\set{y'_e\mid e\in \delta(W)}$ in which points of $Y'$ appear on the image of $K$.
Note that the circular ordering $\hat {\mathcal{O}}'$ naturally induces a circular ordering on $\delta(W)$ and therefore on the set of incident edges of $\hat v$, that we denote by $\hat {\mathcal{O}}$.
We now a drawing $\hat \phi$ of $\hat W$ as follows. We consider the drawing of $W$ induced by $\psi$ and the curves in $\set{\gamma'_e\mid e\in \delta(W)}$. Note that each curve $\gamma'_e$ contains has an endpoint being the image of a vertex of $W$ and the other endpoint $y'_e$ lying on the boundary of $D$. Recall that all vertices of $W$ are drawn outside $D$. We now contract $D$ into a single point, which automatically identify the points in $Y'$. We view this point as the image of $\hat v$. It is easy to see that the drawing obtained in this way is a drawing of $\hat W$ that respects all circular orderings in $\set{\hat{\mathcal{O}}}\cup \set{{\mathcal{O}}_v\mid v\in V(W)}$. Moreover, the number of crossings in $\hat \phi$ between the images of edges in $\delta(\hat v)$ and the image of $W$ is at most the number of crossings in $\psi$ between the curves in $\set{\gamma'_e\mid e\in \delta(W)}$ and the image of $W$. \Cref{obs: inside_disc drawing} now follows.
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[scale=0.1]{figs/move_3.jpg}
\caption{An illustration of curves in $Z$ (shown in blue and red solid curves).}\label{fig: move_inside}
\end{figure}
Finally, we set, for each edge $e\in \delta(W)$, the new image of $e$ to be the concatenation of (i) the original image of $e$ in $\psi$; (ii) the curve $\gamma_e$; and (iii) the curve $\zeta_e$. This completes the description of the subroutine $\mathsf{Move}(\psi, W, D, e^*_W, {\mathcal{Q}},{\mathcal{P}})$.
We denote by $\psi'$ the drawing produced by the subroutine.
Clearly, in $\psi'$, the cluster $W$ is completely drawn inside $D'$ and therefore inside $D$.
We now make some observations on drawing $\psi'$.
From the construction, the original crossings between edges of $W$ remain in the $\psi'$, the original crossings between edges of $\delta(W)$ and edges of $W$ remain in the $\psi'$ (within disc $D'$), since we copied the drawing of $W$ inside $D'$. For each $e\in \delta(W)\setminus \set{e^*_W}$, its new image contains its original image, together with the image of the path $Q(e)$ in $\psi$. Therefore, for each edge $e'\in E(W)\cup\delta(W)$, in the new drawing $\psi'$ there are $\cong_{{\mathcal{Q}}}(e')$ curves contained in the thin strip of $\psi(e)$.
\iffalse
For each edge $e\in E'$, we view its image $\hat \phi_1(e)$ as being directed from its endpoint in $V'$ to its other endpoint that are drawn inside $F$, and we denote by $z_e$ the first crossing between $e$ and $K$.
Define $\hat G$ as the graph obtained from $G_{\mathsf{in}}$ by splitting each edge $e\in E'$ with a new vertex $x_e$, and we denote $X=\set{x_e\mid e\in E'}$.
Clearly, if we view, for each vertex $x_e\in X$, the point $z_e$ as its image, then the drawing $\hat \phi_1$ naturally induces a drawing of $\hat G$.
For each edge $e\in E'$, we denote by $e^{a}$ the new edge split from $e$ that is incident to the endpoint of $e$ in $V'$, and we denote by $e^{b}$ the other edge split from $e$.
We define $E^a=\set{e^a\mid e\in E'}$ and $E^b=\set{e^b\mid e\in E'}$.
See \Cref{fig: well-structured_layout} for an illustration.
We now pick an arbitrary vertex $x_{e^*}\in X$. Let $z^*$ be a point in the intersection of the tiny-$x_{e^*}$ disc $D'=D_{\hat \phi_1}(x_{e^*})$ and $F$. Let $D^*$ be a tiny-$z^*$ disc that lies entirely in the intersection of the $D'\cap F$.
\begin{figure}[h]
\centering
\subfigure[Disc $D$ is shown on the left and cluster $W$ is shown on the right. Edges of $\delta(W)$ with the other endpoint drawn inside $D$ are shown in blue, and edges of $\delta(W)$ with the other endpoint drawn outside $D$ are shown in red.]{\scalebox{0.1}{\includegraphics{figs/well-structured_drawing_1.jpg}}}
\hspace{1cm}
\subfigure[]{
\scalebox{0.1}{\includegraphics{figs/well-structured_drawing_2.jpg}}}
\caption{An Illustration of the input to subroutine $\mathsf{Move}$. \label{fig: well-structured_layout}}
\end{figure}
Define $H'=H\cup E^a$.
We erase the drawing of $H'$ from drawing $\hat \phi_1$, and place the drawing $\hat \phi_1(H)$ of $H$ inside disc $D^*$, such that the new images of vertices of $X$ lie on the boundary of $D^*$. We now add the edges of $E^b$ back to the drawing as follows. Let $\zeta$ be the segment of $\partial F$ that lies in disc $D'$. We first designate, for each vertex $x_e\in X$, a distinct point $y_e$ on $\zeta$. We then compute a set $\gamma$ of curves that contains, for each vertex $x_e\in X$, a curve $\gamma_e$ that connects $z_e$ to $y_e$ and lies entirely outside of $F$, such that every pair of curves in $\Gamma$ crosses at most once. Finally, we let the new image of $e_2$ be the sequential concatenation of: (i) the previous image of $e_2$ between its endpoints originally drawn inside $F$ to the point $z_e$; and (ii) the curve $\gamma_e$ in $\Gamma$. In this way we obtain a new drawing of graph $\hat G$, and suppressing vertices of $X$, this drawing also induces a drawing of $G_1$, with all vertices lying inside $F$. We then perform type-1 uncrossings (the algorithm from \Cref{thm: type-1 uncrossing}) to the images of all edges in $E'$. We denote by $\phi'_1$ the resulting drawing. See \Cref{fig: well-structured_drawing} for an illustration.
It is easy to observe that, in the resulting drawing, the set of edges that cross $K$ is still $E'$.
\begin{figure}[h]
\centering
\subfigure[Before: disc $D^*$ is shown in ligh blue, and segment $\zeta$ is shown in black.]{\scalebox{0.1}{\includegraphics{figs/well-structured_drawing_3.jpg}}}
\hspace{0.2cm}
\subfigure[After: drawing of $H'$ is moved into $D^*$, and the new images of edges of $E^a, E^b$ are shown in pink and red, respectively.]{
\scalebox{0.1}{\includegraphics{figs/well-structured_drawing_4.jpg}}}
\caption{An Illustration of drawing modification in Stage 3. \label{fig: well-structured_drawing}}
\end{figure}
\fi
We now describe the algorithm in the third stage, that utilizes the subroutine $\mathsf{Move}$ and iteratively move the images of all vertices inside $F$.
We start from the drawing $\hat\phi_1$.
Let $V'$ be the set of vertices whose image in $\hat \phi_1$ lie outside $F$, and let $H$ be the subgraph of $G'_1$ induced by vertices of $V'$.
We first apply the algorithm from \Cref{thm: layered well linked decomposition} to graph $G'_1$ and its subgraph $G'_1\setminus H$ to compute a layered $\alpha$-well-linked decomposition $({\mathcal{W}}, {\mathcal{L}}_1,\ldots,{\mathcal{L}}_r)$ of graph $G'_1$ with respect to $G'_1\setminus H$, for $\alpha=O(1/\log^{2.5} m)$.
Now for each cluster $W\in {\mathcal{W}}$, we select uniformly at random an edge of $\delta^{\operatorname{down}}(W)$ as $e^*_W$ and then use the algorithm from
\Cref{cor: simple guiding paths}
to compute a collection ${\mathcal{Q}}_W$ of paths routing edges of $\delta(W)\setminus \set{e^*_W}$ to $e^*_W$ (note that, instead of selecting a path set in $\set{{\mathcal{Q}}^{(e)}\mid e\in \delta(W)}$ uniformly at random, we select a path set in $\set{{\mathcal{Q}}^{(e)}\mid e\in \delta^{\operatorname{down}}(W)}$ uniformly at random).
Then, we iteratively modify the current drawing $\hat \phi_1$ as follows.
Throughout, we maintain a drawing $\tilde \phi$ of $G'_1$, that is initialized to be $\hat \phi_1$.
The algorithm continues to be executed as long as there is still some cluster $W\in {\mathcal{W}}$ whose image in the current drawing $\tilde \phi$ lies outside $F$.
In each iteration, we select a cluster $W$ of ${\mathcal{W}}$, that, among all clusters of ${\mathcal{W}}$ that are drawn outside of $F$, minimizes the layer index of the layered $\alpha$-well-linked decomposition that it belongs to.
Clearly, at this moment all clusters from lower layers than $W$ are already moved into $F$ in the current drawing $\tilde \phi$. Since edge $e^*_W$ is chosen from $\delta^{\operatorname{down}}(W)$, the endpoint of $e^*_W$ outside $W$ lies inside $F$ in $\tilde \phi$.
We then run the subroutine $\mathsf{Move}(\tilde\phi, W, F, e^*_W, {\mathcal{Q}}_W)$, and update $\tilde{\phi}$ with the drawing output by the subroutine. We denote by $\phi_1$ the resulting drawing after we processed all clusters of ${\mathcal{W}}$ in this way.
\newcommand{\mathsf{load}}{\mathsf{load}}
In order to analyze the new crossings created in this stage and prepare for the next stage, we define \emph{load} on edges of $E(H)\cup\delta(H)$, based on the edges $\set{e^*_W}_{W\in {\mathcal{W}}}$ and paths sets of $\set{{\mathcal{Q}}_W}_{W\in {\mathcal{W}}}$, as follows.
First consider a cluster $W\in {\mathcal{L}}_r$. For each edge of $\delta(W)\setminus \set{e^*_W}$, we define its load to be $0$; for edge $e^*_W$, we define its load to be $|\delta(W)|$.
For each edge $e\in E(W)$, we define its load to be $\cong_W({\mathcal{Q}}_W,e)$.
Assume now that the load for each edge with at least one endpoint lying in $\bigcup_{j+1\le t\le r}V({\mathcal{L}}_t)$ are already defined. Consider now a cluster $W\in {\mathcal{L}}_j$. For each edge of $\delta(W)\setminus \set{e^*_W}$ whose load is not yet defined (the edges of $\delta(W)$ with the other endpoint lying in $\bigcup_{1\le t\le j}V({\mathcal{L}}_t)$), we define its load to be $0$; for edge $e^*_W$, we define its load to be the sum of load of all other edges of $\delta(W)$ (notice that the load of all such edges are already defined at this point). For each edge $e\in E(W)$, we define the load of $e$ to be the sum of load of all edges of $\delta(W)$ whose corresponding path in set ${\mathcal{Q}}_W$ contains edge $e$. This completes the definition of load on all edges of $E(H)\cup\delta(H)$.
Note that, since the edges $\set{e_W}_{W\in {\mathcal{W}}}$ and path sets $\set{{\mathcal{Q}}_W}_{W\in {\mathcal{W}}}$ are chosen randomly, the load on edges of $E(H)\cup\delta(H)$ are random variables.
We prove the following observation.
\begin{observation}
\label{obs: expected load}
For each edge $e\in E(H)\cup\delta(H)$, $\expect[]{\mathsf{load}(e)}=O(\log^{O(1)} m)$.
\end{observation}
\begin{proof}
Note that $E(H)=E^{\mathsf{out}}({\mathcal{W}})\cup (\bigcup_{W\in {\mathcal{W}}}E(H))$.
We first consider the edges of $E^{\mathsf{out}}({\mathcal{W}})\cup \delta(H)$.
Let $1\le j\le r$ be a layer index and denote $V_j=\bigcup_{j\le t\le r}V({\mathcal{L}}_t)$.
We show by induction on $j$ that, for each edge
$e\in E^{\mathsf{out}}({\mathcal{W}})$ with at least one endpoint lying in $V_j$, $\expect[]{\mathsf{load}(e)}\le (1+1/\log m)^{r+1-j}$. The base case is when $j=r$. Consider a cluster $W\in {\mathcal{L}}_r$. From Property \ref{condition: layered decomp edge ratio} in the definition of a layered well-linked decomposition, $|\delta^{\operatorname{up}}(W)|\le |\delta^{\operatorname{down}}(W)|/\log m$. Since the edge $e^*_W$ is picked uniformly at random from edges of $\delta^{\operatorname{down}}(W)$, the expected load on each edge of $\delta^{\operatorname{up}}(W)$ is $0$, and the expected load on each edge of $\delta^{\operatorname{down}}(W)$ is $|\delta(W)|/|\delta^{\operatorname{down}}(W)|=(|\delta^{\operatorname{up}}(W)|+|\delta^{\operatorname{down}}(W)|)/|\delta^{\operatorname{down}}(W)|\le (1+1/\log m)$.
Assume now that the claim is true for layers $r,\ldots,j+1$. Consider now a cluster $W\in {\mathcal{L}}_j$. From the inductive hypothesis, from the linearity of expectations, the expected load of every edge of $\delta(W)$ with the other endpoint lying in $V_{j+1}$ is at most $(1+1/\log m)^{r-j}$. From the definition of load, the load of every edge that connects two distinct clusters of ${\mathcal{L}}_j$ is $0$. On the other hand, $|\delta(W)|/|\delta^{\operatorname{down}}(W)|\le (1+1/\log m)$. Therefore, since the edge $e^*_W$ is picked uniformly at random from set $\delta^{\operatorname{down}}(W)$, the expected load on each edge of $\delta^{\operatorname{down}}(W)$ is at most $(1+1/\log m)^{r-j}\cdot (|\delta(W)|/|\delta^{\operatorname{down}}(W)|)\le (1+1/\log m)^{r+1-j}$. As a corollary, the expected load for every edge of $E^{\mathsf{out}}({\mathcal{W}})\cup \delta(H)$ is at most $(1+1/\log m)^{r}\le O(1)$ (as $r\le \log m$ from \Cref{thm: layered well linked decomposition}).
We now consider edges of $\bigcup_{W\in {\mathcal{W}}}E(H)$. Let $W$ be a cluster of ${\mathcal{W}}$. From \Cref{cor: simple guiding paths} and our construction, for each edge $e\in E(W)$, $\expect[]{\cong_W({\mathcal{Q}}_W,e)}=O(\log^{O(1)} m)$.
Note that the choice of path set ${\mathcal{Q}}_W$ is independent of the load on edges of connecting $W$ to a vertex in $V_{j+1}$.
Since the expected load for each edge of $\delta(W)$ is $O(1)$, from the definition on loads of edges of $E(H)$ and the linearity of expectations, $\expect[]{\mathsf{load}(e)}=O(\log^{O(1)} m)$. This completes the proof of \Cref{obs: expected load}.
\end{proof}
We now analyze the number of new crossings created in this stage. First, from the construction, in each iteration we move the drawing of some cluster $W\in {\mathcal{W}}$ into some tiny disc inside $F$, so their new image do not cross the image of $G'_1\setminus H$.
\znote{informal from here}
Then from the algorithm, the new image of each cluster $W\in {\mathcal{W}}$ participate in at most the same number of crossings as in the original drawing $I'$, with the additive term being $\mathsf{cr}(\tilde \phi)/\mu^{a}$ (allowing pairs of edges that cross $K$ to cross). Since we have not modified the image of $G'_1\setminus H$, the new crossings must each involve at least one edge of $E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)$. Note that each edge of $\delta(H)$ has to cross $K$ in drawing $\tilde \phi$, and $K$ participates in at most $(\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m'$ crossings in $\tilde \phi$. Therefore, $|\delta(H)|\le (\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m'$. Moreover, from the definition of a layered well-linked decomposition, $|E^{\mathsf{out}}({\mathcal{W}})|\le O(|\delta(H)|)$, so
\[|E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)|=O((\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m').\]
Therefore, the number of new crossings that involve a pair of edges in $E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)$ is at most
\[
O((\mathsf{cr}(\tilde \phi)\cdot \tau^i\cdot\mu^{a})/m')^2\le O(\mathsf{cr}(\tilde \phi)/\mu^{a}),
\]
where we have used the fact that $m'\ge \tau^s$ and \fbox{$\mathsf{cr}(\tilde \phi)<(\tau^s)^2/30\mu^{a}(\tau^i)^{20}$}.
Denote $E'=E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)$.
Lastly, for each edge $e\in E'$, in the new drawing there are $\mathsf{load}(e)$ images lying in the thin strip of the original image of $e$, so the number of crossings that involve $e$ is multiplied by a factor of $\mathsf{load}(e)$.
\subsubsection*{Stage 4. Further modification on edges}
We now describe the last stage, in which we move the image of some edges inside $F$.
We say that a vertex $v\in V(K)$ is \emph{heavy} iff $v$ is contained in at least $m/(\tau^i)^{5}$ paths of ${\mathcal{P}}$, otherwise we say it is \emph{light}.
We prove the following observations.
\begin{observation}
\label{obs: few edges incident to non-heavy edges}
The expected number of edges in $G'$ that is incident to a light vertex of $V(K)$ is $O(m'/(\tau^i)^{4})$.
\end{observation}
\begin{proof}
Let $v$ be a light vertex. Since it belongs to at most $m'/(\tau^i)^{5}$ paths in ${\mathcal{P}}$, the probability that $v$ belongs to path $P$ is $O((m'/(\tau^i)^{5})/(m'/\tau^i))=O(1/(\tau^i)^4)$. Therefore, for an edge $e\in E(G')$ that is incident to a type-1 high-degree vertex, the probability that its type-1 high-degree endpoint belongs to $P$ is $O(1/(\tau^i)^4)$. From the union bound on all edges in $E(G')$, the expected number of edges that are incident to some type-1 high-degree vertex on $P$ is $O(m'\cdot O(1/(\tau^i)^4))=O(m'/(\tau^i)^{4})$.
The proof for path $P'$ is symmetric. Therefore, the expected number of edges in $G'$ that is incident to a non-heavy vertex of $V(K)$ is $O(m'/(\tau^i)^{4})$.
\end{proof}
\paragraph{Bad Event ${\cal{E}}_6$.}
We say that the bad event ${\cal{E}}_6$ happens if the number of edges in $G'$ incident to a light vertex of $V(K)$ is at least $m'\cdot\mu^{a}/(\tau^i)^{4}$.
From \Cref{obs: few edges incident to non-heavy edges} and Markov's bound, $\Pr[{\cal{E}}_6]\le O(1/\mu^{a})$.
Consider now an edge $e\in E'$. Note that both its endpoints are drawn inside (or on) $C$.
\paragraph{Clean edges.} Consider now the resulting drawing $\phi_1$ in the last stage.
Recall that $E'=E^{\mathsf{out}}({\mathcal{W}})\cup\delta(H)$, and in $\phi_1$ only edges of $E'$ may cross $K$.
We say that an edge $e\in E'$ is clean, iff, when we denote by $x_1,x_2,\ldots,x_k$ the crossings between $e$ and $K$, where the crossings are indexed according to their appearance on the image of $e$, such that, for each even $1\le i\le k$, the subcurve of the image of $e$ between $x_i$ and $x_{i+1}$, which we denote by $\gamma_i(e)$, lies entirely outside $F$, and moreover, one of the two segments of the image of $K$ separated by points $x_i,x_{i+1}$ contains no heavy vertices.
We prove the following claim.
\begin{claim}
The number of non-clean edges is at most $O(\mu^{2a}(\tau^i)^{10})$.
\end{claim}
\begin{proof}
\end{proof}
We now describe the algorithm in this stage. Intuitively, we move the image of all clean edges inside $F$, as follows.
$\ $
\subsection{Phase 1}
\subsection{Phase 2}
\subsection{Completing the Proof of \Cref{lem: many paths}}
We will use the following lemma from \cite{chuzhoy2020towards}.
\begin{lemma}[Lemma 9.2 of \cite{chuzhoy2020towards}]
There is an efficient algorithm, that, given an instance $(H,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, a subset $E'\subseteq E(H)$ of edges of $H$, and a drawing $\phi$ of graph $H\setminus E'$ that respects $\Sigma$, compute a drawing $\phi'$ of $G$ that respects $\Sigma$, such that $\mathsf{cr}(\phi')\le \mathsf{cr}(\phi)+|E'|\cdot |E(H)|$.
\end{lemma}
\subsection{Phase 1: Decomposition into small or non-wide subinstances}
\iffalse
From the above discussion, in both cases eventually we obtain two distinct vertices $v',v''$ of $\hat G$ and a set ${\mathcal{P}}$ of $\floor{m/\mu^{{50}}}$ edge-disjoint paths connecting $v'$ to $v''$. Therefore, from now on we will work with graph $\hat G$, vertices $v',v''$ and the set ${\mathcal{P}}$ of paths in $\hat G$.
Note that the rotation system $\Sigma$ naturally defines a rotation system $\hat \Sigma$ on graph $\hat G$.
We first apply the algorithm from \Cref{lem: non_interfering_paths} to the instance $(\hat G,\hat \Sigma)$ and the set ${\mathcal{P}}$ of paths. We rename the set of paths that we obtain by ${\mathcal{P}}$. From \Cref{lem: non_interfering_paths}, the new set ${\mathcal{P}}$ contains $\floor{m/\mu^{{50}}}$ paths connecting $v'$ to $v''$, that are non-transversal with respect to $\hat \Sigma$.
Also note that the drawing $\phi^*$ of $G$ naturally induces a drawing of $\hat G$, which we denote by $\hat\phi$. It is easy to verify that $\hat\phi$ respects the rotation system $\hat \Sigma$ on $\hat G$, and $\mathsf{cr}(\hat\phi)=\mathsf{cr}$.
\fi
The algorithm for \Cref{lem: many path main} is recursive. Throughout, we maintain a collection $\hat{{\mathcal{I}}}$ of well-structured subinstances of $I$, that is initialized to be $\hat{{\mathcal{I}}}=\set{I}$, and a set $\hat E^{\sf del}$ of edges of $G$, that initially contains no edges.
As the algorithm proceeds, the collection $\hat{{\mathcal{I}}}$ and the set $\hat E^{\sf del}$
evolve, and we will ensure that every instance that ever appears in collection $\hat{{\mathcal{I}}}$ is well-structured.
\iffalse
\begin{itemize}
\item every instance that ever appears in collection $\hat{{\mathcal{I}}}$ is well-structured; and
\item over the course of the algorithm, the total number of cycles in $\set{{\mathcal{R}}(I')}_{I'\in \hat {\mathcal{I}}}$ may not exceed $m/\tau^s$ \znote{to fix}.
\end{itemize}
\fi
The algorithm continues to be executed as long as there exists some instance $I'\in \hat{\mathcal{I}}$, such that its contracted subinstance $\tilde I'$ is $\tau^i$-wide and not $\tau^s$-small.
We now describe an iteration of the algorithm. Let $I'=(G',\Sigma')$ be an instance in the current collection $\hat{\mathcal{I}}$ whose contracted subinstance is $\tau^i$-wide and not $\tau^s$-small.
We denote by $\tilde I'=(\tilde G',\tilde \Sigma')$ its contracted subinstance.
Denote $m'=|E(\tilde G')|$, so $m'>\tau^s$.
Let ${\mathcal{R}}$ be the set of special cycles for instance $I'$ and denote $V_{{\mathcal{R}}}=\set{v_R\mid R\in {\mathcal{R}}}$, so $V(\tilde G')=\big( V(G')\setminus V({\mathcal{R}}) \big)\cup V_{{\mathcal{R}}}$.
We use the following simple claim, whose proof uses standard max-flow techniques and is deferred to \Cref{apd: Proof of different cases}.
\iffalse\begin{claim}
\label{clm: find the witness}
There is an efficient algorithm, that, given an instance $I'=(G',\Sigma')$, checks whether or not instance $I'$ is wide, and if so, compute a set ${\mathcal{P}}$ of paths as a witness for $I'$.
\end{claim}
We first use the algorithm in \Cref{clm: find the witness} to compute a set ${\mathcal{P}}$ of paths as the witness for $\tilde I'$.
Consider now the subgraph $\tilde G''$ of graph $\tilde G'$ induced by all edges of $E({\mathcal{P}})$.
We use the following claim, whose proof uses standard techniques and is deferred to \Cref{apd: Proof of different cases}.\fi
\begin{claim}
\label{clm: different cases}
There is an efficient algorithm, that, takes as input a well-structured instance $I'=(G',\Sigma')$ and a parameter $\tau^i>0$, checks whether or not its contracted subinstance $\tilde I'=(\tilde G', \tilde\Sigma')$ is $\tau^i$-wide, and if so, compute a set ${\mathcal{P}}$ of paths or cycles in graph $\tilde G'$, such that $|{\mathcal{P}}|\ge m'/3\tau^i$, each path of ${\mathcal{P}}$ has length at most $\tau^i\cdot\mu^{10{50}}$, and at least one of the following happens:
\begin{enumerate}
\item there are a pair $v',v''\in V(\tilde G')\setminus V_{{\mathcal{R}}}$ of vertices in $\tilde G'$, such that all elements of ${\mathcal{P}}$ are paths that connect $v'$ to $v''$ and do not contain any vertex of $V_{{\mathcal{R}}}$;
\label{case: clean paths}
\item there is a vertex $v_R\in V_{{\mathcal{R}}}$, such that all elements of ${\mathcal{P}}$ are cycles that contain vertex $v_R$ and do not contain any vertex from $V_{{\mathcal{R}}}\setminus \set{v_R}$;
\label{case: cycles}
\item there are a vertex $v_R\in V_{{\mathcal{R}}}$ and a vertex $v\in V(\tilde G')\setminus V_{{\mathcal{R}}}$, such that all elements of ${\mathcal{P}}$ are paths that connect $v_R$ to $v$ and do not contain any vertex from $V_{{\mathcal{R}}}\setminus \set{v_R}$;\label{case: paths with one node}
\item there are two vertices $v_R, v_{R'}\in V_{{\mathcal{R}}}$, such that all elements of ${\mathcal{P}}$ are paths that connect $v_R$ to $v_{R'}$ and do not contain any vertex from $V_{{\mathcal{R}}}\setminus \set{v_R, v_{R'}}$.
\label{case: paths with two nodes}
\end{enumerate}
\end{claim}
We now consider each of the cases in \Cref{clm: different cases} in turn, as follows.
We first consider Case \ref{case: clean paths}.
\znote{to consider other cases}
\znote{overview here}
\subsubsection{Step 1. Computing a new special cycle $K$}
\label{Step 1}
Let $v',v''$ be the endpoints of all paths of ${\mathcal{P}}$. Since no paths of ${\mathcal{P}}$ contains any vertex of $V_{{\mathcal{R}}}$, the paths of ${\mathcal{P}}$ are also paths in graph $G'$.
We view all paths of ${\mathcal{P}}$ as being directed from vertex $v'$ to vertex $v''$.
We denote $L=|{\mathcal{P}}|$ and denote by $E_1=\set{e_0,\ldots,e_{L-1}}$ the set of edges that serve as the first edge of paths of ${\mathcal{P}}$, where the edges are indexed according to the ordering ${{\mathcal{O}}}_{v'}\in \Sigma$.
We choose an index $z\in \set{0,\ldots,\floor{L/2}-1}$ uniformly at random, and let $z'= z+\floor{L/2}$. We denote by $\tilde P$ the path in ${\mathcal{P}}$ whose first edge is $e_z$, and denote by $\tilde P'$ the path in ${\mathcal{P}}$ whose first edge is $e_{z'}$. We define $\tilde K=\tilde P\cup \tilde P'$, so $\tilde K$ is a subgraph of $G'$. Since $I'=(G',\Sigma')$ is a subinstance of the input instance $I=(G,\Sigma)$, $G'$ is a subgraph of $G$, so $\tilde K$ is also a subgraph of $G$.
If paths $\tilde P$ and $\tilde P'$ do not share inner vertices, then we denote $P=\tilde P$, $P'=\tilde P'$, and $K=\tilde K$.
If paths $\tilde P$ and $\tilde P'$ share inner vertices, then we let $v^*$ be the first inner vertex of $\tilde P$ that is also contained in $\tilde P'$. We denote by $P$ the subpath of $\tilde P$ between $v'$ and $v^*$, and denote by $P'$ the subpath of $\tilde P'$ between $v'$ and $v^*$. Therefore, the paths $P, P'$ are internally vertex-disjoint.
We unname the previous vertex $v''$ and rename $v^*$ by $v''$.
We then define $K=P\cup P'$, so $K$ is a cycle, and $K\subseteq\tilde K$ is a subgraph of $G'$. Therefore, $P$ and $P'$ are paths of $G'$ and are also paths of $G$.
Since every path of ${\mathcal{P}}$ has length at most $\tau^i\cdot\mu^{10{50}}$, from the construction of $K$, $|E(K)|\le 2\tau^i\cdot\mu^{10{50}}$.
In the following steps, we will ``cut the instance $I'$ along $K$'' into two subinstances $I'_1, I'_2$, and cycle $K$ computed in this step will be a special cycle in both $I'_1$ and $I'_2$.
\iffalse
We prove the following observation.
\iffalse{whp version}
\begin{observation}
\label{obs: skeleton number of edges}
The probability that $K$ contains more than $\mu^{{50}+a}$ edges is at most $O(1/\mu^{a})$.
\end{observation}
\begin{proof}
For each $0\le i\le \floor{L/2}-1$, we denote by $K_i$ the subgraph formed by the union of the path in ${\mathcal{P}}$ whose first edge is $e_i$ and the path in ${\mathcal{P}}$ whose first edge is $e_{i+\floor{L/2}}$. Since the paths of ${\mathcal{P}}$ are edge-disjoint, the subgraphs $K_0,\ldots ,K_{\floor{L/2}-1}$ are edge-disjoint, and therefore the expected number of edges of a random subgraph of $\set{K_0,\ldots,K_{\floor{L/2}-1}}$ is at most $m/\floor{L/2}< O(\mu^{{50}})$. From Markov's bound, the probability that $K$ contains more than $\mu^{{50}+a}$ edges is at most $O(\mu^{{50}})/\mu^{{50}+a}=O(1/\mu^{a})$.
\end{proof}
\fi
\begin{observation}
\label{obs: skeleton number of edges}
$\expect[]{|E(K)|}=O(\tau^i)$.
\end{observation}
\begin{proof}
For each $0\le z\le \floor{L/2}-1$, we denote by $\tilde K_z$ the subgraph formed by the union of the path in ${\mathcal{P}}$ whose first edge is $e_z$ and the path in ${\mathcal{P}}$ whose first edge is $e_{z+\floor{L/2}}$. Since the paths of ${\mathcal{P}}$ are edge-disjoint, the subgraphs $\tilde K_0,\ldots, \tilde K_{\floor{L/2}-1}$ are edge-disjoint, and therefore the expected number of edges in $\tilde K$, a random element in set $\set{\tilde K_0,\ldots, \tilde K_{\floor{L/2}-1}}$, is at most $O(m'/L) \le O(\tau^i)$. Since $K\subseteq \tilde K$, we get that $\expect[]{|E(K)|}=O(\tau^i)$.
\end{proof}
\paragraph{Bad Event ${\cal{E}}_1$.}
We say that the bad event ${\cal{E}}_1$ happens if $K$ contains more than $\tau^i\cdot\mu^{a}$ edges.
From \Cref{obs: skeleton number of edges} and Markov's bound, $\Pr[{\cal{E}}_1]\le O(1/\mu^{a})$.
\fi
\subsubsection{Step 2. Computing the inner and outer sides of $K$}
\label{Step 2}
Recall that all paths of ${\mathcal{P}}$ are being directed from $v'$ to $v''$. Recall that we have denoted by $e_0,\ldots, e_{L-1}$ the set of first edges of paths in ${\mathcal{P}}$, where edges are indexed according to the ordering ${\mathcal{O}}_{v'}$. For each $0\le j\le L-1$, we denote by $P_j$ the path in ${\mathcal{P}}$ that has $e_{j}$ as its first edge, and we denote by $e'_{j}$ the last edge of $P_j$.
In order to ``cut instance $I'$ along $K$'' into subinstances $I'_1$ and $I'_2$, we will need to define the ``inner and outer sides'' of $K$, which we do in this step.
First, since the a drawing of any graph and its mirror image are viewed as identical drawings, we can fix the orientation of the ordering ${\mathcal{O}}_{v'}$ to be positive, without loss of generality.
This oriented ordering naturally induces an oriented ordering on the set $\set{P_0,\ldots, P_{L-1}}$ of paths, which we denote by $({\mathcal{O}}',b')$.
Next we consider the vertex $v''$. Note that edges $e'_{0},\ldots,e'_{L-1}$ belong to set $\delta(v'')$, and the circular ordering ${\mathcal{O}}_{v''}\in \Sigma$ is an circular ordering on the set $\set{e'_{0},\ldots,e'_{L-1}}$, which in turn induces an circular ordering on the set $\set{P_{0},\ldots,P_{L-1}}$ of paths, that we denote by ${\mathcal{O}}''$.
Clearly, computing an orientation of the unoriented ordering ${\mathcal{O}}_{v''}$ is equivalent to computing an orientation of the unoriented ordering ${\mathcal{O}}''$.
We compute an orientation $b''\in \set{-1,1}$ of ${\mathcal{O}}''$ as follows.
We set $b''=1$ iff $\mbox{\sf dist}(({\mathcal{O}}', b'),({\mathcal{O}}'',1))\le \mbox{\sf dist}(({\mathcal{O}}', b'),({\mathcal{O}}'',-1))$; otherwise we set $b''=-1$.
In other words,
\[b''=\arg\min_{b\in \set{-1,1}}\set{\mbox{\sf dist}(({\mathcal{O}}', b'),({\mathcal{O}}'',b))}.\]
Consider now a vertex $v\in V(K)$. Assume without loss of generality that $v$ belongs to path $P$. We denote by ${\mathcal{P}}_v=\set{P_{i_1},\ldots,P_{i_q}}$ the set of all paths ${\mathcal{P}}$ that contains vertex $v$.
For each $1\le j\le q$, we denote by $\tilde e_{i_j}$ the first edge of $P_{i_j}$ that is incident to $v$ (recall that paths of ${\mathcal{P}}$ are being directed from $v'$ to $v''$),
and recall that we have denoted by $e_{i_j}$ the first edge of $P_{i_j}$, so $e_{i_j}$ is incident to $v'$.
Note that we have already fixed an orientation of the ordering ${\mathcal{O}}_{v'}$ to be positive. Let $b(v)\in \set{-1,1}$ be an orientation of ordering ${\mathcal{O}}_v$.
Let $e$ be the first edge of $P$ and let $e'$ be the first edge of $P'$.
We say that an index $i_j$ is \emph{screwed under $b(v)$}, iff either edges $e,e_{i,j},e'$ appear clockwise in the oriented ordering $({\mathcal{O}}_{v'},+1)$ and edges $\tilde e, \tilde e_{i,j},\tilde e'$ appear clockwise in the oriented ordering $({\mathcal{O}}_{v},b(v))$ or edges $e',e_{i,j},e$ appear clockwise in the oriented ordering $({\mathcal{O}}_{v'},+1)$ and edges $\tilde e',\tilde e_{i,j},\tilde e$ appear clockwise in the oriented ordering $({\mathcal{O}}_{v},b(v))$.
We then set $b(v)$ as follows. If the number of indices in $\set{i_1,\ldots, i_q}$ that are screwed under $1$ is greater than the number of indices in $\set{i_1,\ldots, i_q}$ that are screwed under $-1$, then we set $b(v)=-1$, otherwise we set $b(v)=1$.
\iffalse
Note that the circular ordering ${\mathcal{O}}'$ on set ${\mathcal{P}}$ of paths naturally induces a circular ordering on the set ${\mathcal{P}}_v$ of paths, that we denote by ${\mathcal{O}}'_v$. Also note that the circular ordering ${\mathcal{O}}_v\in \Sigma$ naturally induces a circular ordering on the set $\set{\tilde e_{i_j}\mid 1\le j\le q}$ of edges, which in turn induces an ordering on the set ${\mathcal{P}}_v$ of paths, that we denote by ${\mathcal{O}}''_v$.
Clearly, computing an orientation of the unoriented ordering ${\mathcal{O}}_{v}$ is equivalent to computing an orientation of the unoriented ordering ${\mathcal{O}}''_v$.
We compute an orientation $b''_v\in \set{-1,1}$ of ${\mathcal{O}}''_v$ as follows.
We set $b(v)=1$ iff $\mbox{\sf dist}(({\mathcal{O}}'_v, b'),({\mathcal{O}}''_v,1))\le \mbox{\sf dist}(({\mathcal{O}}'_v, b'),({\mathcal{O}}''_v,-1))$; otherwise we set $b(v)=-1$.
In other words,
\[b(v)=\arg\min_{b\in \set{-1,1}}\set{\mbox{\sf dist}(({\mathcal{O}}'_v, b'),({\mathcal{O}}''_v,b))}.\]
\fi
So far we have computed, for each vertex of $K$, an orientation of its circular ordering in $\Sigma$. Let $v$ be an inner vertex of $P$, and denote $\delta(v)=\set{\tilde e, \hat e_1, \hat e_2,\ldots, \hat e_p,\tilde e', \hat e'_1, \hat e'_2, \ldots, \hat e'_q}$, where edges $\tilde e,\tilde e'$ belong to $P$, edge $\tilde e$ precedes edge $\tilde e'$ in $P$, and the edges appear clockwise in the oriented ordering on $v$.
We say that edges $\hat e_1, \hat e_2,\ldots, \hat e_p$ are on the \emph{inner side} of $K$, and say that edges $\hat e'_1, \hat e'_2, \ldots, \hat e'_q$ are on the \emph{outer side} of $K$.
Similarly, let $v$ be an inner vertex of $P'$, and denote $\delta(v)=\set{\tilde e, \hat e_1, \hat e_2,\ldots, \hat e_p,\tilde e', \hat e'_1, \hat e'_2, \ldots, \hat e'_q}$, where edges $\tilde e,\tilde e'$ belong to $P'$, edge $\tilde e$ precedes edge $\tilde e'$ in $P'$, and the edges appear clockwise in the oriented ordering on $v$.
We say that edges $\hat e_1, \hat e_2,\ldots, \hat e_p$ are on the \emph{outer side} of $K$, and say that edges $\hat e'_1, \hat e'_2, \ldots, \hat e'_q$ are on the \emph{inner side} of $K$.
For vertex $v''$, we denote assume that $\delta(v'')=\set{\tilde e, \hat e_1, \hat e_2,\ldots, \hat e_p,\tilde e', \hat e'_1, \hat e'_2, \ldots, \hat e'_q}$, where edges $\tilde e$ is the last edge of $P$ and $\tilde e'$ is the last edge of $P'$, and the edges appear clockwise in the oriented ordering at $v''$.
We say that edges $\hat e_1, \hat e_2,\ldots, \hat e_p$ are \emph{on the inner side} of $K$, and say that edges $\hat e'_1, \hat e'_2, \ldots, \hat e'_q$ are \emph{on the outer side} of $K$.
\subsubsection{Step 3. Decomposing instance $I'$ into two subinstances}
\label{Step 3}
Consider the contracted subinstance $\tilde I'=(\tilde G',\tilde \Sigma')$ of the well-structured instance $I'$. Since we have assumed that paths of ${\mathcal{P}}$ do not contain special vertices of graph $G'$, $K$ is also a cycle in $\tilde G'$, and for each vertex $v\in V(K)$, $\delta_{G'}(v)=\delta_{\tilde G'}(v)$.
We will decompose graph $\tilde G'$ into two subgraphs, and then construct two subinstances of $I'$ based on them.
We denote by $\hat E$ the set of all edges of $E(\tilde G')\setminus E(K)$ that is incident to a vertex of $K$, so each edge of $\hat E$ is incident to either vertex $v'$, or vertex $v''$, or an inner vertex of $P$ or $P'$.
We denote by $\hat E_{\mathsf{in}}$ the set of all edges of $\hat E$ that are on the inner side of $K$, and denote by $\hat E_{\mathsf{out}}$ the set of all edges of $\hat E$ that are on the outer side of $K$. Clearly, sets $\hat E_{\mathsf{in}},\hat E_{\mathsf{out}}$ partition $\hat E$. Note that, from the construction of cycle $K$, $|\hat E_{\mathsf{in}}|\ge |\hat E_{\mathsf{in}}\cap \delta_{G'}(v')| \ge \floor{L/2}$, and similarly $|\hat E_{\mathsf{out}}|\ge |\hat E_{\mathsf{out}}\cap \delta_{G'}(v')|\ge \floor{L/2}$.
In order to decompose graph $\tilde G'$ into two subgraphs, it will be convenient for us to slightly modify the graph $\tilde G'$, by splitting each vertex $v\in V(K)$ into three vertices: vertex $v_{\mathsf{in}}$, that is incident to the edges of $\hat E_{\mathsf{in}}\cap \delta(v)$, vertex $v_{\mathsf{out}}$, that is incident to the edges of $\hat E_{\mathsf{out}}\cap \delta(v)$, and vertex $v_{M}$, that is incident to two edges of $E(K)\cap \delta(v)$.
We denote by $\hat G$ the resulting graph.
We define $V_{\mathsf{in}}=\set{v_{\mathsf{in}}\mid v\in V(K)}$, and define sets $V_{\mathsf{out}}$ and $V_{M}$ similarly, so $V(\hat G)=V_{\mathsf{in}}\cup V_{\mathsf{out}}\cup V_{M}\cup (V(G')\setminus V(K))$.
Note that in graph $\hat G$, $K$ is a cycle on vertices of $V_M$ and is disconnected from the rest of the graph.
We then compute, using the standard max-flow techniques, a minimum-cardinality set $E^{*}$ of edges of $\hat G$, whose removal separates vertices of $V_{\mathsf{in}}$ from $V_{\mathsf{out}}$ in $\hat G$. We add the edges of $E^*$ into the set $\hat E^{\sf del}$.
We now construct two new instances as follows. First, let $\tilde G_{\mathsf{in}}$ be the graph obtained from subgraph of $\hat G$ consisting of all connected components of $\hat G\setminus E^{*}$ that contain vertices of $V_{\mathsf{in}}\cup V_{M}$, by identifying, for each vertex $v\in V(K)$, the corresponding vertex $v_{\mathsf{in}}\in V_{\mathsf{in}}$ with the corresponding vertex $v_M\in V_M$.
It is easy to verify that $\tilde G_{\mathsf{in}}$ is a subgraph of $\tilde G'$.
We now let $G_{\mathsf{in}}$ be the graph obtained from $\tilde G_{\mathsf{in}}$ by uncontracting each special vertex $v_R$ back to the cycle $R$. It is easy to verify that graph $G_{\mathsf{in}}$ is a subgraph of $G'$. We then let $\Sigma_{\mathsf{in}}$ be the rotation system on $G_{\mathsf{in}}$ induced by the rotation system $\Sigma'$ on $G'$, and define the instance $I_{\mathsf{in}}=(G_{\mathsf{in}}, \Sigma_{\mathsf{in}})$. For $I_{\mathsf{in}}$ to be a well-structured subinstance, we still need to define its set ${\mathcal{R}}(I_{\mathsf{in}})$ of special cycles.
Note that each special cycle of instance $I'$ now either entirely belongs to $G_{\mathsf{in}}$ or entirely belongs to $G_{\mathsf{out}}$.
We then let set ${\mathcal{R}}(I_{\mathsf{in}})$ contain all special cycles in ${\mathcal{R}}(I')$ that entirely belongs to $G_{\mathsf{in}}$ and the cycle $K$, namely ${\mathcal{R}}(I_{\mathsf{in}})=\set{R\mid v_R\in V(\tilde G_{\mathsf{in}})}\cup \set{K}$. Clearly, the cycles in ${\mathcal{R}}(I_{\mathsf{in}})$ are edge-disjoint (and even vertex-disjoint in this case).
This completes the construction of instance $I_{\mathsf{in}}=(G_{\mathsf{in}}, \Sigma_{\mathsf{in}})$.
It is then easy to verify that $I_{\mathsf{in}}$ is a well-structured subinstance of $I'$.
We construct the well-structured subinstance $I_{\mathsf{out}}=(G_{\mathsf{out}}, \Sigma_{\mathsf{out}})$ and its set ${\mathcal{R}}(I_{\mathsf{out}})$ of special cycles similarly.
We prove the following immediate observations, that will be later used in \Cref{subsec: analysis of decomposition lemma} for the analysis of the algorithm.
\begin{observation}
\label{obs: number of edges in base}
$|E(G_{\mathsf{in}})|+|E(G_{\mathsf{out}})|\le |E(G')|+|E(K)|$, and
$|E(G_{\mathsf{in}})|, |E(G_{\mathsf{out}})|\le |E(G')|-\tau^s/6\tau^i$.
\end{observation}
\begin{proof}
From the construction of graphs $G_{\mathsf{in}},G_{\mathsf{out}}$, $E(G_{\mathsf{in}}),E(G_{\mathsf{out}})\subseteq E(G')$, and the only edges of $G'$ that belong to both $E(G_{\mathsf{in}})$ and $E(G_{\mathsf{out}})$ are edges of $E(K)$, so $|E(G_{\mathsf{in}})|+|E(G_{\mathsf{out}})|\le |E(G')|+|E(K)|$.
On the other hand, observe that edges of $\hat E_{\mathsf{out}}$ may not belong to set $E(G_{\mathsf{in}})$, since otherwise some vertex of $V_{\mathsf{out}}$ (the endpoint of any edge of $\hat E_{\mathsf{out}}\cap E(G_{\mathsf{in}})$ in $V_{\mathsf{out}}$) must be reachable from some vertex of $V_{\mathsf{in}}$ in $G_{\mathsf{in}}$, contradicting the definition of $E^*$ and the construction of $\tilde G_{\mathsf{in}}$.
Therefore, $|E(G_{\mathsf{in}})|\le |E(G')|-\floor{L/2}$, and similarly $|E(G_{\mathsf{out}})|\le |E(G')|-\floor{L/2}$.
From \Cref{clm: different cases}, $L\ge m'/3\tau^i\ge \tau^s/3\tau^i$, and \Cref{obs: number of edges in base} now follows.
\end{proof}
\begin{observation}
\label{obs: 2 new special cycles}
$|{\mathcal{R}}(I_{\mathsf{in}})|+|{\mathcal{R}}(I_{\mathsf{out}})|= |{\mathcal{R}}(I')|+2$.
\end{observation}
\begin{proof}
Since the cycle $K$ in graph $\tilde G'$ does not contain any special vertex, from the construction of the graphs $G_{\mathsf{in}}, G_{\mathsf{out}}$, each special cycle in ${\mathcal{R}}(I')$ belongs to either $G_{\mathsf{in}}$ or $G_{\mathsf{out}}$ (but not both). Also note that the sets ${\mathcal{R}}(I_{\mathsf{in}})$ and ${\mathcal{R}}(I_{\mathsf{out}})$ contain one additional new cycle $K$. Therefore, $|{\mathcal{R}}(I_{\mathsf{in}})|+|{\mathcal{R}}(I_{\mathsf{out}})|= |{\mathcal{R}}(I')|+2$.
\end{proof}
This completes the description of the three steps in an iteration. Finally, we replace the subinstance $I'=(G',\Sigma')$ in the collection $\hat {\mathcal{I}}$ with two new subinstances $I_{\mathsf{in}}=(G_{\mathsf{in}}, \Sigma_{\mathsf{in}})$ and $I_{\mathsf{out}}=(G_{\mathsf{out}}, \Sigma_{\mathsf{out}})$, and proceed to the next iteration.
We denote by ${\mathcal{I}}$ the resulting collection $\hat{{\mathcal{I}}}$ at the end of the algorithm, and denote by $E^{\sf del}$ the resulting set $\hat E^{\sf del}$ at the end of the algorithm.
Finally, we define the subinstance $\hat I$ as follows. Its graph $\hat G=G\setminus E^{\sf del}$, its rotation system is $\Sigma$, and its set of special cycles is ${\mathcal{R}}$, the set of special cycles of $I$. This completes the description of the algorithm.
\subsubsection*{Step 3. Segments Connecting Centers of Clustes}
In this step, we construct paths (that we call \emph{segments}), that connect center vertices of consecutive verterbrae.
Consider some edge $e\in \hat E$ and the corresponding mid-segment $P^2(e)\in {\mathcal{P}}^2$ of the nice guiding path $P(e)\in \hat {\mathcal{P}}$.
Assume that path $P^2(e)$ originates at a vertex of $S_{i'}$ and terminates at a vertex of $S_{j'}$, for some $i'<j'$. We define $\operatorname{span}'(e)=\set{i',i'+1,\ldots,j'-1}$. Clearly, from our definition, $\operatorname{span}(e)\subseteq \operatorname{span}'(e)$.
From the definition of nice guiding paths, there is a sequence $e_{i'},e_{i'+1},\ldots,e_{j'-1}$ of edges that appear on path $P^2(e)$ in this order, such that, for all $i'\leq z\leq j'-1$, edge $e_z$ connects a vertex of $S_z$ to a vertex of $S_{z+1}$. Moreover, if $\set{P_{i'}(e),P_{i'+1}(e),\ldots,P_{j'}(e)}$ is the sequence of paths obtained from $P^2$ after deleting the edges in set $\set{e_z}_{i'\leq z<j'}$ from it, then for all $i'\leq z\leq j'$, $P_z(e)\subseteq S_z$.
For each $i'\leq z\leq j'-1$, let $R_{z}(e)$ be a new path, that is obtained by concatenating two paths: path $Q_{z}(e_z)\in {\mathcal{Q}}_z$, that routes edge $e_z$ to the center vertex $u_z$ of $S_z$; and path $Q_{z+1}(e_z)\in {\mathcal{Q}}_{z+1}$, that routes edge $e_z$ to the center vertex $u_{z+1}$ of $S_{z+1}$. Clearly, path $R_z(e)$ connects $u_z$ to $u_{z+1}$, and we view it as being directed from $u_z$ to $u_{z+1}$.
Fix an index $1\leq t<r$. We define a path set ${\mathcal{R}}_t$, containing the segment $R_t(e)$ of every edge $e\in \hat E$, for which index $t$ lies in the set $\operatorname{span}'(e)$.
Clearly, all paths in ${\mathcal{R}}_{t}$ originate at vertex $u_t$ and terminate at vertex $u_{t+1}$. Let $N_t$ denote the number of all edges
$e\in \hat E$, with $t\in \operatorname{span}'(e)$. Then $|{\mathcal{R}}_t|=N_t$ must hold.
Consider again some edge $e\in \hat E$, and assume that $\operatorname{span}'(e)=\set{i',i'+1,\ldots,j'-1}$. Recall that we have also defined a path $P^{\mathsf{out}}(e)$, whose endpoints lie in $\set{u_1,\ldots,u_r}$. We denote the endpoints of $P^{\mathsf{out}}(e)$ by $u_{i''},u_{j''}$, where $i''<j''$, and we define $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$.
For an index $1\leq t<r$, we let $N'_t$ be the number of all edges
$e\in \hat E$, with $t\in \operatorname{span}''(e)$. We need the following claim, whose proof appears in \Cref{subsec: enough segments}.
\begin{claim}\label{claim: enough segments}
For all $1\leq t<r$, $N'_t=N_t$.
\end{claim}
We now provide some intuition. Since, from \Cref{claim: enough segments}, for all $1\leq t<r$, $N'_t=N_t=|{\mathcal{R}}_t|$, we could assign, to each edge $e\in \hat E$ with $t\in \operatorname{span}''(e)$, a distinct segment $R'_t(e)\in {\mathcal{R}}_t$. Consider now some edge $e\in \hat E$, and assume that $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$. Then for all $i''\leq t\leq j''-1$, we have assigned a segment $R_t'(e)$ to edge $e$, and we could let path $P^{\mathsf{in}}(e)$ be obtained by concatenating all these segments. This final path connects the endpoints of the path $P^{\mathsf{out}}(e)$, and we could then let the auxiliary cycle $W(e)$ be the union of the paths $P^{\mathsf{in}}(e)$ and $P^{\mathsf{out}}(e)$. However, if $W(e),W(e')$ are two cycles in the resulting collection $\set{W(e'')\mid e''\in \hat E}$ of auxiliary cycles, then there could be a large number of vertices $v\in V(W(e))\cap V(W(e'))$, such that the intersection of $W(e)$ and $W(e')$ at $v$ is transversal with respect to $\Sigma$. We will eventually use the
auxiliary cycles in order to construct solutions for subinstances of the input instance $I$ that we compute, and such transversal intersections of cycles may give rise to a large number of crossings in the resulting drawings, which we cannot afford.
In order to overcome this difficulty, for each for $1\leq t<r$, we assign the segments of ${\mathcal{R}}_t$ to the edges $e\in \hat E$ with $t\in \operatorname{span}''(e)$ more carefully, in the next step. The resulting segments will be combined into the paths $P^{\mathsf{in}}(e)$ for edges $e\in \hat E$ as before, and the final auxiliary cycle $W(e)$ will be obtained by combining $P^{\mathsf{out}}(e)$ with $P^{\mathsf{in}}(e)$ as before. However, we will now ensure that, for every pair $W(e),W(e')$ of the resulting auxiliary cycles, there is at most one vertex $v\in V(W(e))\cap V(W(e'))$, such that the intersection of $W(e)$ and $W(e')$ at $v$ is transversal. This will allow us to bound the number of crossings in the optimal solutions of the subinstances of $I$ that we construct. For convenience, for all $1\leq t<r$, we call the paths in set ${\mathcal{R}}_t$ \emph{segments}.
\iffalse
\mynote{this seems a bit problematic. Paths of ${\mathcal{R}}^*_{(t-1)}$ and of ${\mathcal{R}}^*_{(t)}$ both contain vertices of $S_t$. You want to also say that paths of ${\mathcal{R}}^*_{(t-1)}\cup {\mathcal{R}}^*_{(t)}$ are non-interfering inside $S_t$. Originally, the paths of ${\mathcal{Q}}_t$ were non-interfering inside $S_t$. But does this property still hold after this step? If yes, this needs a proof! Also: do we actually need this step? If the paths of ${\mathcal{Q}}_t$ are non-interfering, and the paths of ${\mathcal{Q}}_{t+1}$ are non-interfering, won't this mean that the paths of ${\mathcal{P}}^*_{(t)}$ are non-interfering?}
We then apply the algorithm in Lemma~\ref{lem: non_interfering_paths} to the instance $(G,\Sigma')$, the set ${\mathcal{P}}^*_{(t)}$ of paths, and the two multisets that contains $|{\mathcal{P}}^*_{(t)}|$ copies of $u_{t}$ and $|{\mathcal{P}}^*_{(t)}|$ copies of $u_{t+1}$ respectively, and obtain a set ${\mathcal{R}}^*_{(t)}$ of $|{\mathcal{R}}^*_{(t)}|=|{\mathcal{P}}^*_{(t)}|$ paths connecting $u_t$ to $u_{t+1}$, such that for every $e\in E(G')$, $\cong_{G'}({\mathcal{R}}^*_{(t)},e)\le \cong_{G'}({\mathcal{P}}^*_{(t)},e)$, and the set ${\mathcal{R}}^*_{(t)}$ of paths are non-transversal with respect to $\Sigma$.
\fi
\subsubsection*{Step 4. Constructing the Paths of ${\mathcal{P}}^{\mathsf{in}}=\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$}
Consider an edge $e\in \hat E$, and assume that $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$. We will construct a collection $\tilde {\mathcal{R}}(e)=\set{\tilde R_{i''}(e),\ldots,\tilde R_{j''-1}(e)}$ of paths, where for all $i''\leq t<j''$, $\tilde R_t(e)$ is a segment from the set ${\mathcal{R}}_t$. The final path $P^{\mathsf{in}}(e)$ will then be obtained by concatenating all segments in $\tilde {\mathcal{R}}(e)$.
We construct the collections $\tilde {\mathcal{R}}(e)$ of segments for all edges $e\in \hat E$ gradually, over the course of $r-1$ iterations.
Initially, we set $\tilde {\mathcal{R}}(e)=\emptyset$ for every edge $e\in \hat E$.
For all $1\leq t <r$, over the course of iteration $t$, we will define the segment $\tilde R_t(e)\in {\mathcal{R}}_t$ for every edge $e\in \hat E$ with $t\in \operatorname{span}''(e)$; the segment $\tilde R_t(e)$ is then added to set $\tilde {\mathcal{R}}(e)$.
We now fix an index $1\leq t<r$, and describe the execution of the $t$th iteration. We construct a multiset $A\subseteq \delta_G(u_t)$ of edges, as follows. Consider any edge $e\in \hat E$, with $t\in \operatorname{span}''(e)$. If $t$ is the first index in $\operatorname{span}''(e)$, then we let $e'$ be the first edge on path $P^{\mathsf{out}}(e)$, which must be incident to $u_t$. Otherwise, segment $\tilde R_{t-1}(e)$ is defined and belongs to $\tilde {\mathcal{R}}(e)$. We then let $e'$ be the last edge on segment $\tilde R_{t-1}(e)$, which must be incident to $u_t$. We then add edge $e'$ to set $A$. We say that this copy of edge $e'$ in multiset $A$ is owned by the edge $e$. Notice that
the number of edges in multiset $A$ is equal to the number of edges of $e\in \hat E$ with $t\in \operatorname{span}'(e)$, which is equal to $N_t$, from \Cref{claim: enough segments}. We also define another multiset $A'$ of edges incident to $u_t$, that contains the first edge on every path in set ${\mathcal{R}}_t$. We think of every edge $e'\in A'$ as representing a distinct path $R\in {\mathcal{R}}_t$, for which $e'$ serves as the first edge. From the definition, $|A'|=N_t$. For every edge $e'\in \delta_G(u_t)$, we now let $n^-(e')$ be the number of times that $e'$ appears in set $A$, and $n^+(e)$ the number of times that $e'$ appears in $A'$. From the above discussion, $\sum_{e\in \delta_G(u_t)}n^-(e)=\sum_{e\in \delta_G(u_t)}n^+(e)=N_t$.
We use the algorithm from \Cref{obs:rerouting_matching_cong}, with the ordering ${\mathcal{O}}_{u_t}\in \Sigma$ of the edges of $\delta_G(u_t)$, in order to compute a multiset $M \subseteq \delta_G(u_t)\times \delta_G(u_t)$ of $N_t$ ordered pairs of the edges of $\delta_G(v)$, such that each edge $e'\in \delta_G(u_t)$ participates in $n^-_e$ pairs in $M$ as the first edge, and in $n^+_e$ pairs in $M$ as the second edge. Recall that we are also guaranteed that, for every pair $(e^-_1,e^+_1),(e^-_2,e^+_2)\in M$, the intersection between path $P_1=(e^-_1,e^+_1)$ and path $P_2=(e^-_2,e^+_2)$ at vertex $u_t$ is non-transversal with respect to ${\mathcal{O}}_{u_t}$.
Consider now some edge $e\in \hat E$ with $t\in \operatorname{span}''(e)$, and let $e'\in A$ be the edge that $e$ owns. Assume that $e'$ is matched with an edge $e''\in A'$ by $M$, and that $e''$ represents a segment $R\in {\mathcal{R}}_t$. We then set $\tilde R_t(e)=R$, and we add the segment $R$ to set $\tilde {\mathcal{R}}_t$. In this way, every segment of ${\mathcal{R}}_t$ is assigned to a different edge $e\in \hat E$ with $t\in \operatorname{span}''(e)$.
The algorithm terminates once all indices $1\leq t<r$ are processed. Consider now some edge $e\in \hat E$, and assume that $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$. Set $\tilde {\mathcal{R}}(e)$ now contains, for each $i''\leq t<j''_1$, a path $\tilde R_t(e)$, that connects $u_t$ to $u_{t+1}$. We let $P^{\mathsf{in}}(e)$ be the path obtained by concatenating all paths in $\tilde {\mathcal{R}}(e)$. Observe that path $P^{\mathsf{in}}(e)$ originates at vertex $u_{i''}$, and terminates at vertex $u_{j''}$ -- the two endpoints of path $P^{\mathsf{out}}(e)$.
We then let $W(e)$ be the cycle obtained from the union of paths $P^{\mathsf{out}}(e)$ and $P^{\mathsf{in}}(e)$. We say that $W(e)$ is the \emph{auxiliary cycle} of edge $e$.
We denote the set of all auxiliary cycles by ${\mathcal{W}}=\set{W(e)\mid e\in \hat E}$.
Consider some edge $e\in \hat E$. We note that, while path $P^{\mathsf{out}}(e)$ must be a simple path, it is possible that path $P^{\mathsf{in}}(e)$ is not a simple path. Specifically, for an index $1\leq t<r$, let $P'=P^{\mathsf{in}}(e)\cap (S_t\cup \delta(S_t))$. Then either $P'=\emptyset$, or $P'$ is a contiguous path, that is obtained by concatenating two simple paths from ${\mathcal{Q}}_t$. In the latter case, it is possible that some vertex of $S_t$ lies on both these paths. Since it is important for us that, whenever $P^{\mathsf{in}}(e)\cap S_t\neq \emptyset$, path $P^{\mathsf{in}}(e)$ contains the vertex $u_t$, we do not turn $P^{\mathsf{in}}(e)$ into a simple path. Notice however that, from the above discussion, every vertex may appear at most twice on $P^{\mathsf{in}}(e)$.
We need the following simple observation.
\begin{observation}\label{obs: self-non-transversal}
Let $e\in \hat E$ be an edge, and let $v$ be any vertex that appears twice on path $P^{\mathsf{in}}(v)$. Let $e_1$, $e_2$ be the edges immediately preceding and immediately following the first appearence of $v$ on $W(e)$, respectively, and let $e'_1$, $e'_2$ are the edges immediately preceding and immediately following the second appearence of $v$ on $W(e)$, respectively. Then the circular ordering of the edges $e_1,e_2,e_1',e_2'$ in ${\mathcal{O}}_v\in \Sigma$ is either $(e_1,e_2,e_1',e_2')$, or $(e_1,e_2,e_2',e_1')$ (or the reverse of one of these orderings).
\end{observation}
\begin{proof}
Let $1\leq t\leq r$ be the index for which $v\in V(S_t)$. Recall that $P^{\mathsf{in}}(e)\cap S_t$ is a contiguious path, which is a concatenation of two paths in ${\mathcal{Q}}_t$. Since the paths in ${\mathcal{Q}}_t$ are non-transversal with respect to $\Sigma$, the observation follows.
\end{proof}
Assume now that we are given two edges $e,e'\in \hat E$, and some vertex $v$ that appears on both $W(e)$ and $W(e')$. We view the cycles $W(e)$ as being directed so that path $P^{\mathsf{in}}(e)$ and path $P^{\mathsf{in}}(e')$ visit the vertebrae in the increasing order of their indices. Let $v$ be any vertex that lies on both $W(e)$ and $W(e')$ (note that $v$ may appear up to twice on each of the cycles). We say that the intersection of $W(e)$ and $W(e')$ is non-transversal at $v$ iff the following hold: for every appearence of $v$ on $W(e)$, and for every appearence of $v$ on $W(e')$, if we denote by $e_1$ and $e_2$ the edges immediately preceding and immediately following this appearence of $v$ on $W(e)$, and by $e'_1$ and $e'_2$ the edges immediately preceding and immediately following this appearence of $v$ on $W(e')$, then the circular ordering of the edges $e_1,e_2,e_1',e_2'$ in ${\mathcal{O}}_v\in \Sigma$ is either $(e_1,e_2,e_1',e_2')$, or $(e_1,e_2,e_2',e_1')$ (or the reverse of these orderings).
Recall that we denoted, for each index $1\leq t<r$, by $\hat E_t\subseteq \hat E$ the set of all edges $e\in \hat E$, with $t\in \operatorname{span}(e)$.
We need the following observation.
\begin{observation}
\label{obs: auxiliary cycles non-transversal at at most one}
For every index $1\leq t<r$, for every pair $e, e'\in \hat E_t$ of distinct edges, there is at most one vertex $v\in V(W(e'))\cap V(W(e'))$, such that the intersection of cycles $W(e)$ and $W(e')$ is non-transversal at $v$. If such a vertex $v$ exists, then $v=u_t$ for some index $t$ that is the last index in either $\operatorname{span}''(e)$ or $\operatorname{span}''(e')$.
\end{observation}
\begin{proof}
Fix an index $1\leq t<r$ and a pair $e,e'\in \hat E_t$ of edges. Let $v$ be any vertex that lies on both $W(e)$ and $W(e')$. We consider three cases. The first case is when $v\in V(G)\setminus V'$, that is, there is some index $1\leq t'\leq r$, such that $v\in V(\tilde S_{t'})\setminus V(S_{t'})$. Since the inner vertices of paths $P^{\mathsf{in}}(e),P^{\mathsf{in}}(e')$ are contained in $V'$, vertex $v$ must be an inner vertex of both $P^{\mathsf{out}}(e)$ and $P^{\mathsf{out}}(e')$. From \Cref{claim: out-paths non-transversal}, the intersection of $P^{\mathsf{out}}(e)$ and $P^{\mathsf{out}}(e')$ at vertex $v$ is non-transversal. Therefore, the intersection of $W(e)$ and $W(e')$ at $v$ is non-transversal.
The second case is when there is some index $1\leq t'\leq r$, such that $v\in V(S_{t'})$, but $v\neq u_{t'}$.
Note that in this case, $v$ may appear up to twice on $P^{\mathsf{in}}(e)$, and up to twice on $P^{\mathsf{in}}(e')$. We fix one appearance of $v$ on $P^{\mathsf{in}}(e)$ and one appearance of $v$ on $P^{\mathsf{in}}(e')$. There must be a path $Q\in {\mathcal{Q}}_{t'}$, such that $Q\subseteq P^{\mathsf{in}}(e)$, and it contains the selected appearance of $v$ on $P^{\mathsf{in}}(e)$, and similarly, there must be a path $Q'\in {\mathcal{Q}}_{t'}$, with $Q'\subseteq P^{\mathsf{in}}(e')$, such that $Q'$ contains the selected appearance of $v$ on $P^{\mathsf{in}}(e')$. Recall that all paths in ${\mathcal{Q}}_{t'}$ are non-transversal with respect to $\Sigma$. Therefore, the intersection of cycles $W(e)$ and $W(e')$ is non-transversal at $v$.
The third case is when there is some index $t'$, such that $v=u_{t'}$. If $t'\in \operatorname{span}''(e)\cap \operatorname{span}''(e')$ holds, then, from the way the segments of ${\mathcal{R}}_{t'}$ were assigned to edges of $\hat E$, it is immediate that
the intersection of cycles $W(e)$ and $W(e')$ is non-transversal at $u_{t'}$.
Lastly, assume that $t'\not\in \operatorname{span}''(e)$ or $t'\not\in \operatorname{span}''(e)$. This may only happen if $u_{t'}$ is the last vertex on $P^{\mathsf{out}}(e)$ or $P^{\mathsf{out}}(e')$. If $u_{t'}$ is the last vertex on both these paths, then this is the unique vertex at which the intersection of the cycles $W(e)$ and $W(e')$ may be transversal.
Assume now that $u_{t'}$ is the last vertex on one of these two paths (say $P^{\mathsf{out}}(e)$), but it is not the last vertex on $P^{\mathsf{out}}(e')$. Assume that the last vertex on $P^{\mathsf{out}}(e')$ is $u_{t''}$. Since vertex $u_{t'}$ lies on path $P^{\mathsf{in}}(e')$, $t''>t'$ must hold, and so $u_{t''}$ may not lie on path $P^{\mathsf{in}}(e)$. Therefore, $u_{t'}$ is the unique vertex at which the intersection of the cycles $W(e)$ and $W(e')$ may be transversal.
\iffalse
Assume that $e_1$ is an edge of $\hat E_{i_1,j_1}$ and $e_2$ is an edge of $\hat E_{i_2,j_2}$, such that $j_1\ge j_2$, so $u_{j_2}$ is a common vertex of auxiliary cycles $W(e_1)$ and $W(e_2)$.
We will show that, the intersection between $W(e_1)$ and $W(e_2)$ is non-transversal at all their shared vertices except (possibly) for $u_{j_2}$, thus completing the proof of \Cref{obs: auxiliary cycles non-transversal at at most one}.
Recall that $W(e_1)=P^{\mathsf{out}}(e_1)\cup P^{\mathsf{in}}(e_1)$ and $W(e_2)=P^{\mathsf{out}}(e_2)\cup P^{\mathsf{in}}(e_2)$, and furthermore $P^{\mathsf{out}}(e_1)=P^{\mathsf{out}}_{\operatorname{left}}(e_1)\cup P^{\mathsf{out}}_{\operatorname{mid}}(e_1)\cup P^{\mathsf{out}}_{\operatorname{right}}(e_1)$ and $P^{\mathsf{out}}(e_2)=P^{\mathsf{out}}_{\operatorname{left}}(e_2)\cup P^{\mathsf{out}}_{\operatorname{mid}}(e_2)\cup P^{\mathsf{out}}_{\operatorname{right}}(e_2)$.
Note that the paths $P^{\mathsf{out}}_{\operatorname{mid}}(e_1),P^{\mathsf{out}}_{\operatorname{mid}}(e_2)$ are internally disjoint from $V'$, while the paths of $\set{P^{\mathsf{out}}_{\operatorname{left}}(e_k),P^{\mathsf{out}}_{\operatorname{right}}(e_k),P^{\mathsf{in}}(e_k)}_{k\in \set{1,2}}$ only use vertices of $V'$.
First, from Step 2, the paths $P^{\mathsf{out}}_{\operatorname{mid}}(e_1),P^{\mathsf{out}}_{\operatorname{mid}}(e_2)$ are non-transversal with respect to $\Sigma$.
Second, note that for each $1\le i\le r-1$, the set ${\mathcal{R}}^*_{(i)}$ of paths are non-transversal with respect to $\Sigma$.
Since the $i$-th segments of paths in ${\mathcal{P}}^{\mathsf{in}}$ are paths in ${\mathcal{R}}^*_{(i)}$, it follows that the paths $P^{\mathsf{in}}(e_1)$ and $P^{\mathsf{in}}(e_2)$ are non-transversal at all vertices of $V(G')\setminus\set{u_1,\ldots,u_r}$.
Third, from the algorithm and \Cref{obs:rerouting_matching_cong}, it is easy to verify that the paths of ${\mathcal{P}}^{\mathsf{in}}$ are also non-transversal at $u_1,\ldots,u_{j_2-1}$.
Lastly, since for each $1\le i\le r$, the set ${\mathcal{Q}}_{i}$ of paths are non-transversal with respect to $\Sigma$, and the intersection between paths of set ${\mathcal{R}}^*_{(i)}$ and cluster $S_i$ are paths of ${\mathcal{Q}}_i$, we get that the intersection between cycles $W(e_1)$ and $W(e_2)$ is non-transversal at all vertices of $V(P^{\mathsf{out}}_{\operatorname{left}}(e_1))\cup V(P^{\mathsf{out}}_{\operatorname{left}}(e_2))$.
Altogether, we get that the intersection between $W(e_1)$ and $W(e_2)$ is non-transversal at all their shared vertices except (possibly) for $u_{j_2}$.\fi
\end{proof}
\iffalse
----------------------
We now re-organize the paths in sets ${\mathcal{R}}^*_{(1)},\ldots, {\mathcal{R}}^*_{(r-1)}$ to obtain the paths in ${\mathcal{P}}^{\mathsf{in}}=\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$.
Note that, if we select, for each $i\le t\le j-1$, a path of ${\mathcal{R}}^*_{(t)}$, then, by taking the union of them, we can obtain a path $\hat P$ connecting $u_i$ to $u_j$.
We call the selected path of ${\mathcal{R}}^*_{(t)}$ the \emph{$t$-th segment} of $\hat P$. Therefore, a path connecting $u_i$ to $u_j$ obtained in this way is the union of its $i$-th segment, its $(i+1)$-th segment, $\ldots$ , and its $(j-1)$-th segment.
We now incrementally construct the set ${\mathcal{P}}^{\mathsf{in}}$ of paths.
Intuitively, we will simultaneously construct all paths of ${\mathcal{P}}^{\mathsf{in}}$ in a total of $r-1$ iterations, where in the $i$-th iteration, we will determine the $i$-th segment of all paths in ${\mathcal{P}}^{\mathsf{in}}$.
For each pair $1\le i'<j'\le r$, we denote by $\hat E_{i',j'}$ the set of all edges $e\in \hat E$ such that the path $P^{\mathsf{out}}(e)$ constructed in Step 2 has $u_{i'}$ as its last endpoint and $u_{j'}$ as its first endpoint.
Throughout, we will maintain a set $\hat{{\mathcal{P}}}=\set{\hat P_e\mid e\in \hat E}$ of paths, where each path $\hat P_e$ is indexed by an edge $e$ of $\hat E$.
For an edge $e\in \hat E_{i',j'}$, the path $\hat P_e$ is supposed to originate at $u_{i'}$ and terminate at $u_{j'}$.
We call $u_{i'}$ the \emph{destined origin} of $\hat P_e$ and call $u_{j'}$ the \emph{destined terminal} of $\hat P_e$.
Initially, $\hat{{\mathcal{P}}}$ contains $|\hat E|$ empty paths.
We will sequentially process vertices $u_1,\ldots,u_{r-1}$, and, for each $1\le i\le r-1$, upon processing vertex $u_i$, determine which path of ${\mathcal{R}}^*_{(i)}$ serves as the $i$-th segment of which path of $\hat{\mathcal{P}}$.
We now fix an index $1\le i\le r-1$ and describe the iteration of processing vertex $u_i$. The current set $\hat{\mathcal{P}}$ of paths can be partitioned into four sets: set $\hat{\mathcal{P}}^{o}_i$ contains all paths of $\hat{\mathcal{P}}$ whose destined origin is $u_i$; set $\hat{\mathcal{P}}^{t}_i$ contains all paths of $\hat{\mathcal{P}}$ whose destined terminal is $u_i$; set $\hat{\mathcal{P}}^{\textsf{thr}}_i$ contains all paths of $\hat {\mathcal{P}}$ whose destined origin is $u_{i'}$ for some index $i'<i$ and whose destined terminal is $u_{j'}$ for some index $j'>i$; and set $\hat{\mathcal{P}}\setminus (\hat{\mathcal{P}}^{o}_i\cup \hat{\mathcal{P}}^{t}_i\cup \hat{\mathcal{P}}^{\textsf{thr}}_i)$ contains all other paths.
Note that the paths in set $\hat{\mathcal{P}}^{t}_i$ and set $\hat{\mathcal{P}}\setminus (\hat{\mathcal{P}}^{o}_i\cup \hat{\mathcal{P}}^{t}_i\cup \hat{\mathcal{P}}^{\textsf{thr}}_i)$ do not contain an $i$-th segment, so in this iteration we will determine the $i$-th segment of paths in the sets $\hat{\mathcal{P}}^{o}_i$ and $\hat{\mathcal{P}}^{\textsf{thr}}_i$.
Note that the paths in $\hat{\mathcal{P}}^{o}_i$ currently contain no edges,
and the paths in $\hat{\mathcal{P}}^{\textsf{thr}}_i$ currently contain up to its $(i-1)$-th segment.
We then denote
\begin{itemize}
\item by $L^-_i$ the multi-set of the current last edges (the edges incident to $u_i$) of paths in $\hat{\mathcal{P}}^{\textsf{thr}}_i$;
\item by $L^+_i$ the multi-set of the first edges (the edges incident to $u_i$) of paths in ${\mathcal{R}}^*_{(i)}$ (note that these paths are currently not designated as the $i$-th segment of any path in $\hat{{\mathcal{P}}}$); and
\item by $L^{\mathsf{out}}_i$ the multi-set of last edges of paths in ${\mathcal{P}}^{\mathsf{out}}_{i,*}=\bigg\{P^{\mathsf{out}}_{\operatorname{left}}(e) \text{ }\bigg|\text{ } e\in \big(\bigcup_{j>i}\hat E_{i,j}\big)\bigg\}$.
\end{itemize}
Clearly, elements in sets $L^-_i, L^+_i,L^{\mathsf{out}}_i$ are edges of $\delta(u_i)$.
We then define, for each $e\in \delta(u_i)$, $n^-_e=n_{L^-_i}(e)+n_{L^{\mathsf{out}}_i}(e)$ and
$n^+_e=n_{L^+_i}(e)$.
We use the following simple observation.
\begin{observation}
For each edge $e\in \delta(u_i)$, $\sum_{e\in \delta(u_i)}n^-_e=\sum_{e\in \delta(u_i)}n^+_e$.
\end{observation}
\begin{proof}
On the one hand, $\sum_{e\in \delta(u_i)}n_{L^{\mathsf{out}}_i}(e)=|\bigcup_{j\ge i+2}\hat E_{i,j}|=|\hat{\mathcal{P}}^{o}_i|$, and $\sum_{e\in \delta(u_i)}n_{L^{-}_i}(e)=|L^-_i|=|\hat{\mathcal{P}}^{\textsf{thr}}_i|$.
On the other hand, recall that $|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|$, and the set ${\mathcal{P}}^*_{(i)}$ contains a path connecting $u_i$ to $u_{i+1}$ for each edge $e\in \big(\bigcup_{i'\le i-1,j'\ge i+1}\hat E_{i',j'}\big)\cup \big(\bigcup_{j'\ge i+2}\hat E_{i',j'}\big)$. Therefore,
$\sum_{e\in \delta(u_i)}n_{L^{+}_i}(e)=|L^+_i|=|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|=|\bigcup_{i'\le i-1,j'\ge i+1}\hat E_{i',j'}|+|\bigcup_{j'\ge i+2}\hat E_{i,j'}|=|\hat{\mathcal{P}}^{\textsf{thr}}_i|+|\hat{\mathcal{P}}^{o}_i|$.
\end{proof}
We apply the algorithm in \Cref{obs:rerouting_matching_cong} to graph $G$, vertex $u_i$, rotation ${\mathcal{O}}_{u_i}$ and integers $\set{n^-_e,n^+_{e}}_{e\in \delta(u_i)}$.
Let $M$ be the multi-set of ordered pairs of the edges of $\delta(u_i)$ that we obtain.
We then designate:
\begin{itemize}
\item for each path $\hat P_e\in \hat{\mathcal{P}}^{\textsf{thr}}_i$ with $e^-$ as its current last edge, a path of ${\mathcal{R}}^*_{(i)}$ that contains the edge $e^+$ as its first edge with $(e^-,e^+)\in M$, as the $i$-th segment of $\hat P_e$; and
\item for each path $P^{\mathsf{out}}_{\operatorname{left}}(e)\in{\mathcal{P}}^{\mathsf{out}}_{i,*}$ with $e^-$ as its last edge (recall that we view $u_i$ as the last endpoint of such a path), a path of ${\mathcal{R}}^*_{(i)}$ that contains the edge $e^+$ as its first edge with $(e^-,e^+)\in M$, as the $i$-th segment of $\hat P_e$;
\end{itemize}
such that each path of ${\mathcal{R}}^*_{(i)}$ is assigned to exactly one path of $\hat {\mathcal{P}}^{\textsf{thr}}_i\cup {\mathcal{P}}^{\mathsf{out}}_{i,*}$.
This completes the description of the $i$-th iteration.
See Figure~\ref{fig: inner_path} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Edges in $\delta(u_i)$: only the last/first edges of paths are shown. Here ${\mathcal{P}}^{\mathsf{out}}_{*,i}=\set{P^{\mathsf{out}}(e) \mid e\in\big( \bigcup_{i'< i}\hat E_{i',i}\big)}$.]{\scalebox{0.32}{\includegraphics{figs/inner_path_1.jpg}}}
\hspace{1pt}
\subfigure[Sets $L^-_i, L^+_i,L^{\mathsf{out}}_i$ and the pairing (shown in dash pink lines) given by the algorithm in \Cref{obs:rerouting_matching_cong}.]{
\scalebox{0.32}{\includegraphics{figs/inner_path_2.jpg}}}
\caption{An illustration of an iteration in Step 4 of constructing paths of $\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$.}\label{fig: inner_path}
\end{figure}
Let ${\mathcal{P}}^{\mathsf{in}}$ be the set of paths that we obtain after processing all vertices $u_1,\ldots,u_{r-1}$. Then for every $e\in \hat E$, we rename the path $\hat P_e$ at the end of the algorithm by $P^{\mathsf{in}}(e)$.
From the algorithm, it is easy to see that, for each pair $1\le i<j\le r$ and each edge $e\in \hat E_{i,j}$, the inner path $P^{\mathsf{in}}(e)$ starts at $u_i$ and ends at $u_j$ as it is supposed to, and path $P^{\mathsf{in}}(e)$ contains, for each $i\le t\le j-1$, a path of ${\mathcal{R}}^*_{(t)}$ as its $t$-th segment.
Therefore, path $P^{\mathsf{in}}(e)$ visits vertices $u_i,u_{i+1},\ldots,u_j$ sequentially.
\fi
\iffalse
\begin{observation}
\label{obs: non_transversal_1}
For each $1\le i\le r-1$, if we denote, for each $e\in E_i^{\operatorname{right}}$, by $R_e$ the path consisting of the last edge of path $P^{\mathsf{out}}_e$ and the first edge of path $P_e$, then the paths in $\set{P_e\mid e\in \bigcup_{ i'<i<j'}E(S'_{i'},S'_{j'})}$ and $\set{R_e\mid e\in E_i^{\operatorname{right}}}$ are non-transversal at $u_i$.
\end{observation}
\fi
\iffalse
\begin{observation}
\label{obs: non_transversal_2}
The inner paths in ${\mathcal{P}}$ are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
On one hand, note that for each $1\le i\le r-1$, the set ${\mathcal{R}}^*_{(i)}$ of paths are non-transversal with respect to $\Sigma$.
Since the $i$-th segments of paths in ${\mathcal{P}}$ are paths in ${\mathcal{R}}^*_{(i)}$, it follows that the paths of ${\mathcal{P}}$ are non-transversal at all vertices of $V(G)\setminus\set{u_1,\ldots,u_k}$.
On the other hand, from the algorithm and \Cref{obs:rerouting_matching_cong}, it is easy to verify that the paths of ${\mathcal{P}}$ are also non-transversal at $u_1,\ldots,u_k$.
\end{proof}
\fi
Lastly, we bound the congestion that is caused by the auxiliary cycles, in the following observation.
\begin{observation}
\label{obs: edge_occupation in outer and inner paths}
Consider any edge $e\in E(G)$. If $e\not \in \bigcup_{1\leq t\leq r}E(S_t)$, then $e$ belongs to at most $O(\log^{18}m)$ cycles of ${\mathcal{W}}$. Otherwise, if $e\in E(S_t)$ for some $1\leq t\leq r$, then the number of cycles of ${\mathcal{W}}$ that $e$ belongs to is at most $\cong_G({\mathcal{Q}}_t,e)\cdot O(\log^{18}m)$.
\end{observation}
\begin{proof}
Consider first an edge $e\in E(G)\setminus\textsf{left}(\bigcup_{1\leq t\leq r}E(S_t)\textsf{right} )$. If both endpoints of $e$ lie in set $V'$, then there is an index $1\leq t<r$, such that $e$ connects a vertex of $S_t$ to a vertex of $S_{t+1}$. In this case, the number of cycles in ${\mathcal{W}}$ containing edge $e$ is precisely the number of paths in ${\mathcal{R}}_t$ containing $e$, which is in turn bounded by $\cong_G({\mathcal{P}}^2,e)\leq O(\log^{18}m)$ from the definition of nice guiding paths.
If at least one endpoint of $e$ does not lie in $V'$, then the number of cycles in ${\mathcal{W}}$ that contain $e$ is $\cong_G({\mathcal{P}}^{\mathsf{out}}\leq O(\log^{18}m)$, from \Cref{claim: computing out-paths}.
Assume now that there is some index $1\leq t\leq r$, such that $e\in E(S_t)$. Let $A$ be the set of all edges $e'\in \delta_G(S_t)$, such that the path $Q_t(e')\in {\mathcal{Q}}_t$ contains the edge $e$. Clearly, $|A|=\cong_G({\mathcal{Q}}_t,e)$. Notice that a cycle $W\in {\mathcal{W}}$ may only contain the edge $e$ if it contains one of the edges in set $A$. Since, from the above discussion, every edge of $A$ may lie on at most $O(\log^{18}m)$ cycles of ${\mathcal{W}}$, the total number of cycles in ${\mathcal{W}}$ that contain $e$ is bounded by $|A|\cdot O(\log^{18}m)\leq \cong_G({\mathcal{Q}}_t,e)\cdot O(\log^{18}m)$.
\end{proof}
\subsection{Proof of \Cref{lem: find ordering of terminals}}
\label{sec:ordering of terminals}
We start by defining a new expanded graph $H'$, whose construction is similar to that of $H^*$, except that now we expand every vertex $v\in V(H)\setminus (\tilde T\cup \set{x})$ into a grid. Specifically,
we start with $H'=\emptyset$, and then process every vertex $u\in V(H)\setminus (\tilde T\cup \set{x})$ one by one. We denote by $d(u)$ the degree of the vertex $u$ in graph $H$. We now describe an iteration when a vertex $u\in V(H)\setminus (\tilde T\cup \set{x})$ is processed. Let $e_1(u),\ldots,e_{d(u)}(u)$ be the edges that are incident to $u$ in $H$, indexed according to their ordering in ${\mathcal{O}}_u\in \Sigma$. We let $\Pi(u)$ be the $(d(u)\times d(u))$ grid, and we denote the vertices on the first row of this grid by $s_1(u),\ldots,s_{d(u)}(u)$ indexed in their natural left-to-right order.
We add the vertices and the edges of the grid $\Pi(u)$ to graph $H'$. As before, we refer to the edges in the resulting grids $\Pi(u)$ as inner edges.
Once every vertex $u\in V(H)\setminus (\tilde T\cup \set{x})$ is processed, we add
the vertices of $\tilde T$ to the graph $H'$. Recall that every terminal $t\in \tilde T$ has degree $2$ in $H$. We denote $s_1(t)=s_2(t)=t$, and we arbitrarily denote the two edges incident to $t$ by $e_1(t)$ and $e_2(t)$.
We also add vertex $x$ to $H'$. We denote $s_1(x)=\cdots=s_{d(x)}(x)=x$, and we denote the edges incident to $x$ by $e_1(x),\ldots,e_{d(x)}(x)$, indexed consistently with the circular ordering ${\mathcal{O}}_x\in \Sigma$, where $\Sigma$ is the rotation system for graph $H$.
Next, we add
a collection of outer to graph $H'$, exactly as before. Consider any edge $e=(u,v)\in E(H)$. Assume that $e$ is the $i$th edge of $u$ and the $j$th edge of $v$, that is, $e=e_i(u)=e_j(v)$. Then we add an edge $e'=(s_i(u),s_j(v))$ to graph $H'$, and we view this edge as the copy of the edge $e\in E(H)$. This completes the definition of graph $H'$.
The partition $(X,Y)$ of the vertices of $V(H)\setminus \tilde T$ naturally defines a partition $(X',Y')$ of the vertices of $V(H')\setminus \tilde T$, as follows: $X'=\textsf{left} ( \bigcup_{u\in X\setminus\set{x}}V(\Pi(u))\textsf{right} )\cup \set{x}$ and $Y'= \bigcup_{u\in Y}V(\Pi(u))$.
We denote $H'_1=H'[X'\cup \tilde T]$ and $H'_2=H'[Y\cup \tilde T]$.
Let ${\mathcal{W}}_X=\set{\Pi(u)\mid u\in X\setminus\set{x}}$. Then ${\mathcal{W}}_X$ is a collection of disjoint clusters in graph $H'_1$. Moreover, since, for every vertex $u\in X\setminus\set{x}$, the set of vertices on the first row of a grid $\Pi(u)$ is $1$-well-linked in $\Pi(u)$ (from \Cref{obs: grid 1st row well-linked}), the corresponding cluster $\Pi(u)$ has the $1$-bandwidth property.
\begin{observation}\label{obs: routing terminals to $x$}
There is a collection ${\mathcal{J}}'$ of paths in graph $H_1'$, routing all terminals of $\tilde T$ to $x$, with $\cong_{H_1'}({\mathcal{J}}')\leq O\textsf{left}(\frac{\eta\log^{36}m}{\alpha^6(\alpha')^2}\textsf{right} )$.
\end{observation}
\begin{proof}
Recall that there exists a set ${\mathcal{J}}$ of edge-disjoint paths in graph $\hat H_1$, routing a subset $\tilde T_0\subseteq \tilde T$ of terminals to $x$, with $|{\mathcal{J}}|\geq \tilde k'$. Since every cluster in ${\mathcal{C}}$ has the $\alpha'$-bandwidth property, from
\Cref{claim: routing in contracted graph}, there is a collection ${\mathcal{J}}_0$ of edge-disjoint paths in graph $H_1$, routing a subset $\tilde T_0'\subseteq \tilde T_0$ to $x$, where $|\tilde T'_0|\geq \alpha'\tilde k'/2$. Using the $1$-bandwidth property of the clusters $\Pi(u)$ in graph $H'_1$, it is easy to verify that there is a collection ${\mathcal{J}}'_0$ of edge-disjoint paths in graph $H_1'$, routing the terminals of $\tilde T'_0$ to vertex $x$.
Recall that $\tilde k'=\tilde k\alpha^5/(c'\eta\log^{36}m)$, where $c'$ is a constant whose value was fixed in the proof of \Cref{lem: high opt or lots of paths}.
From Property \ref{prop after step 1: terminals in H1}, the set $\tilde T$ of terminals is $\tilde \alpha$-well-linked in $\hat H_1$, and so, from \Cref{clm: contracted_graph_well_linkedness}, it is $(\tilde \alpha\alpha')$-well-linked in $H_1$. Moreover, since $H_1=(H_1')_{|{\mathcal{W}}_X}$, and since each cluster in ${\mathcal{W}}_X$ has the $1$-bandwidth property, from \Cref{clm: contracted_graph_well_linkedness}, the set $\tilde T$ of terminals is $(\tilde \alpha\alpha')$-well-linked in $H_1'$.
From \Cref{lem: routing path extension}, there is a set ${\mathcal{J}}'$ of paths in graph $H'_1$, routing the terminals in $\tilde T$ to vertex $x$, with:
\[\cong_{H_1'}({\mathcal{J}}')\leq O\textsf{left}(\frac {|\tilde T|}{|\tilde T_0'|}\cdot\frac{1}{\alpha\alpha'}\textsf{right} )
\leq O\textsf{left}(\frac {\tilde k}{\tilde k'\alpha(\alpha')^2}\textsf{right} )
\leq O\textsf{left}(\frac{\eta\log^{36}m}{\alpha^6(\alpha')^2}\textsf{right} ).\]
\iffalse
\begin{lemma}
\label{lem: routing path extension}
Let $G$ be a graph, let $T$ be a set of vertices that are $\alpha$-well-linked in $G$, and let $T'$ be a subset of $T$. Let ${\mathcal{P}}'$ be a set of paths in $G$ routing the vertices of $T'$ to some vertex $x$ of $G$. Then there is a set ${\mathcal{P}}$ of paths routing vertices of $T$ to $x$, such that, for every edge $e\in E(G)$,
$\cong_G({\mathcal{P}},e)\le \ceil{\frac{|T|}{|T'|}}(\cong_G({\mathcal{P}}',e)+\ceil{1/\alpha})$.
\end{lemma}
Let $z=\ceil{\tilde k/\tilde k'}=O\textsf{left}(\frac{\eta\log^{36}n}{\alpha^5\alpha'}\textsf{right} )$. Next, we arbitrarily partition the terminals of $\tilde T\setminus \tilde T_0$ into $z$ subsets $\tilde T_1,\ldots,\tilde T_z$, of cardinality at most $\tilde k'$ each.
Consider now some index $1\leq i\leq z$. Using the algorithm from \Cref{thm: bandwidth_means_boundary_well_linked}, we can compute a collection ${\mathcal{J}}_i'$ of paths in graph $H_1'$, routing vertices of $\tilde T_i$ to vertices $\tilde T_0$, such that the paths in $\tilde T_i$ cause edge-congestion $O(1/(\tilde \alpha\alpha'))$, and each vertex of $\tilde T_0\cup\tilde T_i$ is the endpoint of at most one path in ${\mathcal{J}}_i'$. By concatenating the paths in ${\mathcal{J}}_i'$ with paths in ${\mathcal{J}}_0$, we obtain a collection ${\mathcal{J}}_i$ of paths in graph $H_1'$, connecting every terminal of $\tilde T_i$ to $x$, that cause edge-congestion $O(1/(\tilde \alpha\alpha'))$. Let ${\mathcal{J}}'=\bigcup_{i=0}^z{\mathcal{J}}_i$ be the resulting set of paths. Observe that set ${\mathcal{J}}'$ contains $\tilde k$ paths, routing the terminals in $\tilde T$ to the vertex $x$ in graph $H_1'$, with $\cong_{H_1'}({\mathcal{J}}')\leq O\textsf{left}(\frac z{\tilde \alpha\alpha'}\textsf{right} )\leq O\textsf{left}(\frac{\eta\log^{36}n}{\alpha^6(\alpha')^2}\textsf{right} )$.
\fi
\end{proof}
For convenience, we denote by $\rho=O\textsf{left}(\frac{\eta\log^{36}m}{\alpha^6(\alpha')^2}\textsf{right} )$ the bound on $\cong_{H_1'}({\mathcal{J}}')$. Note that we can compute such a collection ${\mathcal{J}}'$ of paths efficiently using standard maximum $s$-$t$ flow computation. In order to compute the ordering $\tilde {\mathcal{O}}$ of the terminals in $\tilde T$, we need one additional property from the paths in ${\mathcal{J}}'$: we need them to be \emph{confluent}. In order to define confluent paths, we need to assign each path a direction, so that one of its endpoint becomes the first vertex on the path.
If $P$ is a simple directed path, whose first endpoint is $s$ and last endpoint is $t$, a \emph{suffix} of $P$ is any subpath $P'\subseteq P$ that contains the vertex $t$.
\begin{definition}[Confluent Paths]
Let ${\mathcal{P}}$ be a collection of directed paths. We say that the paths in ${\mathcal{P}}$ are \emph{confluent} iff for every pair $P_1,P_2\in {\mathcal{P}}$ of paths, either $P_1\cap P_2=\emptyset$, or $P_1\cap P_2$ is a suffix of both $P_1$ and $P_2$.
\end{definition}
The following claim, that easily follows from the work of \cite{confluent},
allows us to transform the set ${\mathcal{J}}'$ of paths into a confluent one.
\begin{claim}\label{claim: confluent paths}
There is an efficient algorithm that computes a set ${\mathcal{J}}''$ of confluent paths in graph $H_1'$, routing the vertices of $\tilde T$ to $x$, (where every path is directed towards $x$), with $\cong_{H'_1}({\mathcal{J}}'')\leq O(\rho \log m)$.
\end{claim}
\begin{proof}
We use the notion of confluent flows from \cite{confluent}. The following definitions are from \cite{confluent}. Let $G=(V,E)$ be a directed graph, and let $\operatorname{Dem}:V\rightarrow {\mathbb R}^+$ be a demand function for vertices $v\in V$. Let $S\subseteq V$ denote a collection of sink vertices. We assume that every edge incident to a sink vertex $s\in S$ is directed towards $s$. A flow $f: E\rightarrow {\mathbb R}^+$ is a \emph{valid flow} if it satisfies, for every vertex $v\in V\setminus S$:
\[\sum_{e=(v,w)\in E}f(e)-\sum_{e=(u,v)\in E}f(e)=\operatorname{Dem}(v). \]
In other words, the total amount of flow leaving $v$ is equal to the demand on $v$ plus the total amount of flow entering $v$. The \emph{congestion} on vertex $v$ is defined to be as:
\[ \sum_{e=(u,v)\in E}f(e)+\operatorname{Dem}(v), \]
the total amount of flow entering $v$ plus the demand at $v$. The congestion of $f$ is the maximum, over all vertices $v\in V$, of the congestion of $f$ at $v$.
We say that a flow $f$ is \emph{confluent} if for every vertex $v\in V$, there is at most one edge $(v,u)$ with $f(v,u)>0$. A confluent flow therefore defines a subgraph of $G$ (induced by edges carrying non-zero flow), consisting of disjoint components $\set{T_1,\ldots,T_k}$, where each $T_i$ is an arborescence directed towards the root $s_i\in S$. In each such arborescence $T_i$, the maximum vertex congestion occurs at the sink $s_i$, and is equal to the total demand of all vertices of $T_i$.
The following result was proved in \cite{confluent}.
\begin{theorem}[Theorem 3 in \cite{confluent}]\label{thm: confluent flow}
There is an efficient algorithm, that, given a directed $n$-vertex graph $G$ with a collection $S\subseteq V(G)$ of sinks, and demands $\operatorname{Dem}(v)\geq 0$ for vertices $v\in V(G)$, such that there exists a (regular, splittable) flow $f$ with node congestion $1$ satisfying the demands in graph $G$, computes a confluent flow satisfying all demands, with congestion $O(\log n)$.
\end{theorem}
We construct a flow network from graph $H_1'$, as follows. First, we subdivide every edge $e$ that is incident to $x$ with a new sink vertex $s_e$, setting $S=\set{s_e\mid e\in \delta_{H_1'}(x)}$, and delete the vertex $x$ from the graph. Next, we bi-direct every edge of the resulting graph (by replacing it with two anti-parallel edges), except that for every sink vertex $s_e\in S$, the unique edge incident to $s_e$ is directed towards $s_e$. For every terminal $t\in \tilde T$, we set its demand $\operatorname{Dem}(t)=1/(4\rho)$, and we set the demands of all other vertices to $0$. Let $G$ be the resulting flow network. Denote $n=|V(G)|$, and observe that $|V(G)|=O\textsf{left}(\sum_{u\in V(H_1)}(\deg_{H_1}(u))^2\textsf{right} )\leq O(m^2)$. Note that the collection ${\mathcal{J}}'$ of paths in graph $H_1'$ immediately defines a collection $\tilde {\mathcal{J}}$ of paths, routing the set $\tilde T$ of terminals to the vertices of $S$, in graph $G$, with edge-congestion at most $\rho$. Since the degree of every vertex in $G$ is at most $4$, by sending $1/(4\rho)$ flow units on every path in ${\mathcal{J}}''$, we obtain a valid flow from vertices of $\tilde T$ to vertices of $S$, satisfying all demands, with vertex-congestion at most $1$. From \Cref{thm: confluent flow}, there is a confluent flow $f$ in graph $G$ with vertex-congestion $O(\log n)\leq O(\log m)$, satisfying all demands. We can use standard flow-decomposition of $f$ to obtain a collection $\tilde {\mathcal{J}}'$ of flow-paths, routing the terminals in $\tilde T$ to vertices of $S$, such that the paths in ${\mathcal{J}}''$ are confluent. Each such flow-paths carries $1/(4\rho)$ flow units in the flow $f$, so the total edge-congestion caused by paths in $\tilde {\mathcal{J}}'$ is at most $O(\rho\log m)$. Moreover, every sink vertex $s_e\in S$ is an endpoint of at most $O(\rho\log m)$ paths. The set $\tilde {\mathcal{J}}'$ of paths naturally define a set ${\mathcal{J}}''$ of confluent paths in graph $H_1'$, routing the set $\tilde T$ of terminals to vertex $x$, with edge-congestion $O(\rho \log m)$.
\end{proof}
We will use the set ${\mathcal{J}}''$ of confluent paths to both compute the desired ordering $\tilde {\mathcal{O}}$ of the terminals, and to show that there exists the desired drawing $\phi$ of graph $H^*$.
For convenence, let $d$ denote the degree of vertex $x$ in $H$ and the edges incident to $x$ by $e_1(x),\ldots,e_{d}(x)$, indexed consistently with the circular ordering ${\mathcal{O}}_x\in \Sigma$, where $\Sigma$ is the rotation system for graph $H$. For all $1\leq i\leq d$ let ${\mathcal{P}}_i\subseteq{\mathcal{J}}''$ be the subset of paths whose last edge is $e_i$, so $({\mathcal{P}}_1,\ldots,{\mathcal{P}}_d)$ is a partition of ${\mathcal{J}}''$. We define a circular ordering of the paths in ${\mathcal{J}}''$ as follows: for each $1\leq i\leq d$, the paths in ${\mathcal{P}}_i$ appear consecutively in this ordering, in an arbitrary order, and paths belonging to different subsets appear in the natural ordering ${\mathcal{P}}_1,\ldots,{\mathcal{P}}_d$ of these subsets. Denote ${\mathcal{J}}''=\set{P_1,\ldots,P_{\tilde k}}$, where the paths are indexed by the ordering that we have just defined. For all $1\leq j\leq |\tilde T|$, let $t_j$ be the terminal of $\tilde T$ that serves as an endpoint of path $P_j$. We have then defined an ordering $t_1,\ldots,t_{\tilde k}$ of the terminals in $\tilde T$, that we denote by $\tilde {\mathcal{O}}$. For all $1\leq i\leq \tilde k$, let $e_i$ denote the edge $(x,t_i)$ in graph $H^*$.
It is now enough to show that there is a drawing $\phi$ of graph $H^*$, in which the inner edges do not participate in crossings, and the images of edges $e_1,\ldots,e_{\tilde k}$ enter the image of $x$ in this order, such that $\mathsf{cr}(\phi)\leq O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\frac{\eta^2\log^{74}m}{\alpha^{12}(\alpha')^4}\textsf{right} ) + \textsf{left} (
\frac{\tilde k \eta\log^{37}m}{\alpha^6(\alpha')^2}\textsf{right} )$. The following observation will then finish the proof of \Cref{lem: find ordering of terminals}.
\begin{observation}
There is a drawing $\phi$ of graph $H^*$ with at most $O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\frac{\eta^2\log^{74}m}{\alpha^{12}(\alpha')^4}\textsf{right} ) + O\textsf{left} (
\frac{\tilde k \eta\log^{37}m}{\alpha^6(\alpha')^2}\textsf{right} )$ crossings, in which all crossings are between pairs of outer edges, and the images of edges $e_1,\ldots,e_{\tilde k}$ enter the image of $x$ in this circular order.
\end{observation}
\begin{proof}
Let $\phi_1$ be the optimal solution to instance $I$ of \textnormal{\textsf{MCNwRS}}\xspace, and denote by $\chi_1=\mathsf{OPT}_{\mathsf{cnwrs}}(I)$ its number of crossings. We can easily transform drawing $\phi_1$ to a drawing $\phi_2$ of graph $H'$, with the same number of crossings, such that every crossing in $\phi_2$ is between a pair of outer edges. In order to do so, we consider, for every vertex $v\in V(H)\setminus(\tilde T\cup \set{x})$, the tiny $v$-disc $D_{\phi}(v)$. We place a drawing of the grid $\Pi(v)$ inside the disc, using the natural layout of the grid (depending on the orientation of vertex $v$ in $\phi$, we may need to flip the image of $\Pi(v)$).
Next, we slightly modify the graph $H'$, as follows.
First, for every edge $e_i$ that is incident to $x$, we subdivide $e_i$ with a new vertex $s_i$, and then delete $x$ from the graph. Let $S=\set{s_1,\ldots,s_d}$ be the resulting set of new vertices. We modify the paths in ${\mathcal{J}}''$ so that each path now connects a distinct vertex of $\tilde T$ to some vertex of $S$, and the paths remain confluent. As before, we denote by ${\mathcal{P}}_i\subseteq {\mathcal{J}}''$ the set of paths that terminate at vertex $s_i$.
For every edge $e$ of the resulting graph, we let $N_e$ the number of paths in ${\mathcal{J}}''$ in which edge $e$ participates; recall that $N_e\leq O(\rho\log m)$ must hold. For every edge $e\in E(H_1')$, if $N_e=0$, then we delete edge $e$, and otherwise we replace $e$ with $N_e$ parallel copies.
We denote the resulting graph by $H''$, and we denote by $H_1''$ the subgraph of $H''$ corresponding to $H_1'$, that is, if we let $X''=V(H_1')\cap V(H'')$, then
$H_1''$ is the subgraph of $H''$ that is induced by vertices of $X''\cup S$. Observe that graph $H_1''$ can be viewed as consisting of disjoint trees $\tau_1,\ldots,\tau_d$ (though some of the trees may consist of a single vertex $s_i$), where for all $1\leq i\leq d$, the root of $\tau_i$ is the vertex $s_i$, but these trees may have parallel edges. If $v\in V(\tau_i)\setminus\set{s_i}$, and the subtree rooted at $v$ contains $n_v$ terminal vertices, then the edge connecting $v$ to its parent has exactly $n_v$ parallel copies, and $n_v\leq O(\rho\log m)$. We further modify the paths set ${\mathcal{J}}''$, so that the paths become edge-disjoint in graph $H_1''$, that is, we ensure that for every edge $e\in E(H_1')$, each path in ${\mathcal{J}}''$ containing $e$
uses a different copy of the edge $e$.
We can modify the drawing $\phi_2$ of graph $H'$ to obtain a drawing $\phi_3$ of graph $H''$, as follows. First, consider the tiny $x$-disc $D=D_{\phi_2}(x)$. Recall that, that for every edge $e_i$, the intersection of the image of $e_i$ in $\phi_2$ and the boundary of the disc $D$ is a single point, that we denote by $p_i$. We place the image of the new vertex $s_i$ on point $p_i$, and we erase the portion of the image of $e_i$ that lies in disc $D$. We also delete the image of $x$. We delete images of edges and vertices as needed, and then, for every edge $e\in E(H_1'')$ with $N_e>0$, we create $N_e$ copies of $e$, all of which are drawn in parallel to the original image of $e$, very close to it, so that the images of these copies of $e$ do not cross each other. Let $\phi_3$ denote this resulting drawing of the graph $H''$. Since, for every edge $e$, $N_e\leq O(\rho\log m)$, it is easy to verify that $\mathsf{cr}(\phi_3)\leq \mathsf{cr}(\phi_2)\cdot O(\rho^2\log^2m)\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \rho^2\log^2m)$.
For every vertex $v\in V(H'')$, this drawing $\phi_3$ of $H''$ naturally defines a circular ordering of the edges of $\delta_{H''}(v)$, that we denote by ${\mathcal{O}}^3_v$ -- the order in which the images of the edges of $\delta_{H''}(v)$ enter the image of $v$ in this drawing. (Note that the drawing $\phi_3$ depends on the optimal solution to instance $I$ of \textnormal{\textsf{MCNwRS}}\xspace, so we cannot compute the drawing or the resulting orderings ${\mathcal{O}}^3(v)$ for $v\in V(H'')$ efficiently; we only use their existence here).
Let $\Sigma^3=\set{{\mathcal{O}}_v^3}_{v\in V(H'')}$ be the resulting rotation system for graph $H''$.
Recall that for all $1\leq i\leq d$, ${\mathcal{P}}_i\subseteq {\mathcal{J}}''$ is a set of paths routing a subset of the terminals of $\tilde T$ to $s_i$, and for $1\leq i\neq j\leq d$, no vertex may belong to a path in ${\mathcal{P}}_i$ and to a path in ${\mathcal{P}}_j$.
For all $1\leq i\leq d$, let $\tilde T_i\subseteq \tilde T$ be the set of terminals that serve as endpoints of paths in ${\mathcal{P}}_i$.
We apply \Cref{lem: non_interfering_paths} to each such path set ${\mathcal{P}}_i$ separately, to obtain a path set ${\mathcal{P}}'_i$, that is non-transversal with respect to the rotation system $\Sigma^3$. The lemma ensures that the paths in ${\mathcal{P}}'_i$ route terminals of $\tilde T_i$ to vertex $s_i$ in $H''$; the paths are edge-disjoint, and moreover, an edge $e\in E(H'')$ may only belong to a path of ${\mathcal{P}}'_i$ if it belonged to a path of ${\mathcal{P}}_i$. Let ${\mathcal{J}}'''=\bigcup_{i=1}^d{\mathcal{P}}'_i$.
We are now ready to define a drawing $\phi$ of the graph $H^*$. Notice that graph $H''$ can be viewed as the union of graph $H^*\setminus\set{x}$, and the paths in ${\mathcal{J}}'''$. As discussed above, there is a drawing $\phi_3$ of $H''$ with at most $O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \rho^2\log^2m)$ crossings. We have defined a rotation system $\Sigma^3$ for graph $H''$, such that the drawing $\phi_3$ is consistent with this rotation system. Moreover, from the definition of the drawing $\phi_3$, there is a disc $D$ (that originally contained the image of the vertex $x$), whose interior is disjoint from the drawing $\phi_3$, with vertices $s_1,\ldots,s_d$ appearing on the boundary of $D$ in this circular order. Lastly, the paths in sets ${\mathcal{P}}_1',\ldots,{\mathcal{P}}_d'$ are all non-transversal with respect to $\Sigma_3$, and for all $1\leq i\neq j\leq d$, paths in ${\mathcal{P}}_i$ and paths in ${\mathcal{P}}_j$ cannot share vertices with each other.
In order to obtain the drawing $\phi$ of $H^*$, we slightly modify the drawing $\phi_3$, as follows. First, we place an image of vertex $x$ in the interior of the disc $D$.
For every edge $e_j=(t_j,x)$ of $H^*$, let $P_j\in {\mathcal{J}}'''$ be the path whose endpoint is $t_j$, and let $s_{i_j}$ be its other endpoint. Let $\gamma_j$ be the image of the path $P_j$ in the drawing $\phi_3$. Since all paths in set ${\mathcal{J}}'''$ are non-transversal with respect to $\Sigma_3$ we can apply the nudging algorithm from \Cref{claim: curves in a disc} to the image of every vertex of $H''$ that lies on some path in ${\mathcal{J}}'''$, in order to compute, for each $1\leq j\leq \tilde k$, a curve $\gamma_j'$ connecting the image of $t_j$ to the image of $s_{i_j}$, so that, if we delete from $\phi_3$ the images of all inner vertices and of all edges participating in the paths in ${\mathcal{J}}'''$, and add instead the curves $\gamma'_1,\ldots,\gamma'_{\tilde k}$, then the number of crossings does not increase. For all $1\leq j\leq d$, curve $\gamma'_j$ is obtained from $\gamma_j$ by ``nudging'' it in the vicinity of every vertex $v\in V(P_j)$ using the algorithm from \Cref{claim: curves in a disc}.
Note that for all $1\leq i\leq d$, for every terminal $t_j\in \tilde T_i$, the curve $\gamma'_j$ terminates at the image of the vertex $s_i$, and moreover, the images of the vertices $s_1,\ldots,s_d$ appear in this circular order on the boundary of the disc $D$. However, the order in which the curves in $\Gamma'_i=\set{\gamma'_j\mid t_j\in \tilde T_i}$ enter the image of $s_i$ may be different from the ordering of the corresponding terminals in $\tilde T$. We need to reorder the curves in $\Gamma'_i$ in the vicinity of $s_i$, so that they enter the image of $s_i$ in the order consistent with $\tilde {\mathcal{O}}$. Since for all $1\leq j\leq d$, $|\Gamma_j|\leq O(\rho\log m)$, we can perform these reorderings while introducing crossings whose number is bounded by:
\[\sum_{i=1}^d|\Gamma'_i|^2\leq \sum_{i=1}^d|\Gamma'_i|\cdot O(\rho\log m)\leq O(\tilde k\rho \log m). \]
For all $1\leq j\leq \tilde k$, let $\gamma''_j$ be the curve obtained form $\gamma'_j$ after the reordering, so that $\gamma''_j$ connects the image of $t_j$ to the image of $s_{i_j}$. By slighty extending this curve inside the disc $D$, we can ensure that it terminates at the image of $x$. This can be done for all $1\leq j\leq \tilde k$ without introducing any new crossings, while ensuring that the resulting curves $\gamma''_1,\ldots,\gamma''_{\tilde k}$ enter the image of $x$ in this order. We then let, for all $1\leq j\leq \tilde k$, $\gamma''_j$ be the image of the edge $(t_j,x)$, obtaining a drawing $\phi$ of $H^*$. It is immediate to verify that every crossing in $\phi$ is between a pair of outer edges, and from our construction, the edges $(t_1,x),\ldots,(t_{\tilde k},x)$ enter the image of $x$ in this circular order. From the above discussion, the total number of crossings in $\phi$ is at most:
\[O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \rho^2\log^2m)+O(\tilde k\rho \log m) \leq O\textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\frac{\eta^2\log^{74}m}{\alpha^{12}(\alpha')^4}\textsf{right} ) + \textsf{left} (
\frac{\tilde k \eta\log^{37}m}{\alpha^6(\alpha')^2}\textsf{right} ).
\]
\end{proof}
\section{An Algorithm for \ensuremath{\mathsf{MCNwRS}}\xspace -- Proof of \Cref{thm: main_rotation_system}}
\label{sec: high level}
In this section we provide a proof of Theorem \ref{thm: main_rotation_system}, with some of the details deferred to subsequent sections.
Throughout the paper, we denote by $I^*=(G^*,\Sigma^*)$ the input instance of the \textnormal{\textsf{MCNwRS}}\xspace problem, and we denote $m^*=|E(G^*)|$. We also use the following parameter that is central to our algorithm: ${\mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}}$,
where $c^*$ is a large enough constant.
As mentioned already, our algorithm for solving the \textnormal{\textsf{MCNwRS}}\xspace problem is recursive, and, over the course of the recursion, we will consider subinstances of instance $I^*$ (see \Cref{subsec: subinstances} for a definition).
The algorithm proceeds by recursively decomposing a given subinstance $I$ of $I^*$ into a collection of smaller subinstances.
The main technical ingredient of the proof is the following theorem.
\begin{theorem}
\label{thm: main}
There is a constant $c'$, and an efficient randomized algorithm, that, given a subinstance $I=(G,\Sigma)$ of $I^*$ with $m=|E(G)|\geq \mu^{c'}$, either returns FAIL, or produces a non-empty collection ${\mathcal{I}}$ of subinstances of $I$ with the following properties:
\begin{itemize}
\item for each subinstance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\leq m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\cdot (\log m)^{O(1)}$;
\item if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then with probability at least $15/16$:
$\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{O((\log m)^{3/4}\log\log m)} +m\cdot\mu^{O(1)}$; and
\item there is an efficient algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, that, given a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, of cost at most $O(\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{O(1)}$.
\end{itemize}
Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/16$.
\end{theorem}
\iffalse
We now define collections of subinstances that have some desirable properties. The following definition is similar in spirit to the definition of a $\nu$-decomposition of an instance, but, since it uses slightly different parameters, we distinguish between these two definitions.
\begin{definition}[Good and Perfect Collection of Subinstances]
Let $I=(G,\Sigma)$ be a subinstance of instance $I^*$, with $|E(G)|=m$, and let ${\mathcal{I}}$ be a collection of subinstances of $I$. We say that ${\mathcal{I}}$ is a \emph{good collection of subinstances} for $I$ if the following hold.
\begin{itemize}
\item for each subinstance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\leq m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\cdot (\log m)^{c_g}$, where $1000<c_g<c^*$ is some large enough universal constant whose value will be set later; and
\item there is an efficient algorithm, that we call \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, that, given a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, of cost at most $c_g\cdot(\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{c_g}$.
\end{itemize}
We say that ${\mathcal{I}}$ is a \emph{perfect} collection of subinstances for $I$ if, additionally:
$$\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{O((\log m)^{3/4}\log\log m)} +m\cdot\mu^{c_g}.$$
\end{definition}
The main technical ingredient of the proof is the following theorem.
\begin{theorem}
\label{thm: main}
There is an efficient randomized algorithm, whose input is a subinstance $I=(G,\Sigma)$ of $I^*$, with $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$ and $|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$. The algorithm outputs a good collection ${\mathcal{I}}$ of subinstances of $I$, and, with probability at least $7/8$, ${\mathcal{I}}$ is a perfect collection of subinstances for $I$.
\end{theorem}
\fi
The remainder of this paper is dedicated to the proof of \Cref{thm: main}. In the following subsection, we complete the proof of \Cref{thm: main_rotation_system} using \Cref{thm: main}.
\subsection{Proof of \Cref{thm: main_rotation_system}}
Throughout the proof, we assume that $m^*$ is larger than a sufficiently large constant, since otherwise we can return a trivial solution to instance $I^*$, from \Cref{thm: crwrs_uncrossing}.
We let $100<c_g<c^*/2$ be a large enough constant, so that, if the algorithm from \Cref{thm: main}, when applied to a subinstance $I=(G,\Sigma)$ of $I^*$ with $m=|E(G)|\geq \mu^{c'}$, returns a family ${\mathcal{I}}$ of subinstances of $I$, with $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\cdot (\log m)^{c_g}$. Additionally, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then with probability at least $15/16$: $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot 2^{c_g((\log m)^{3/4}\log\log m)} +m\cdot\mu^{c_g}$. We also assume that Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, given a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, of cost at most $c_g\cdot (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{c_g}$.
We use a simple recursive algorithm called \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace, that appears in Figure \ref{fig: algrec}.
\program{\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace}{fig: algrec}{
\noindent{\bf Input:} a subinstance $I=(G,\Sigma)$ of the input instance $I^*$ of the \textnormal{\textsf{MCNwRS}}\xspace problem.
\noindent{\bf Output:} a feasible solution to instance $I$.
\begin{enumerate}
\item Use the algorithm from \Cref{thm: crwrs_planar} to determine whether or not $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$. If so, use the algorithm from \Cref{thm: crwrs_planar} to compute a solution to $I$ of cost $0$. Return this solution, and terminate the algorithm.
\item Use the algorithm from Theorem~\ref{thm: crwrs_uncrossing} to compute a trivial solution $\phi'$ to instance $I$.
\item If $|E(G)|\leq \mu^{c'}$, return the trivial solution $\phi'$ and terminate the algorithm.
\item For $1\leq j\leq \ceil{\log m^*}$:
\begin{enumerate}
\item Apply the algorithm from \Cref{thm: main} to instance $I$.
\item If the algorithm returns FAIL, let $\phi_j=\phi'$ be the trivial solution to instance $I$, and set ${\mathcal{I}}_j(I)=\emptyset$.
\item Otherwise:
\begin{enumerate}
\item Let ${\mathcal{I}}_j(I)$ be the collection of subinstances of $I$ that the algorithm returns.
\item For every subinstance $I'\in {\mathcal{I}}_j(I)$, apply Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace to instance $I'$, to obtain a solution $\phi(I')$ to this instance.
\item Apply Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace from \Cref{thm: main} to solutions $\set{\phi(I')}_{I'\in {\mathcal{I}}_j(I)}$, to obtain a solution $\phi_j$ to instance $I$.
\end{enumerate}
\end{enumerate}
\item
Return the best solution of $\set{\phi',\phi_1,\ldots,\phi_{\ceil{\log m^*}}}$
\end{enumerate}
}
In order to analyze the algorithm, it is convenient to associate a \emph{partitioning tree} $T$ with it, whose vertices correspond to all subinstances considered over the course of the algorithm. Let $L=\ceil{\log m^*}$. We start with the tree $T$ containing a single root vertex $v(I^*)$, representing the input instance $I^*$. Consider now some vertex $v(I)$ of the tree, representing some subinstance $I=(G,\Sigma)$ of $I^*$.
When Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace was applied to instance $I$, if it did not terminate after the first three steps, it created $L$ collections ${\mathcal{I}}_1(I),\ldots,{\mathcal{I}}_L(I)$ of subinstances of $I$ (some of which may be empty, in case the algorithm from \Cref{thm: main} returned FAIL). For each such subinstance $I'\in \bigcup_{j=1}^L{\mathcal{I}}_j(I)$, we add a vertex $v(I')$ representing instance $I'$, that becomes a child vertex of $v(I)$. This concludes the description of the partitioning tree $T$.
We denote by ${\mathcal{I}}^*=\set{I\mid v(I)\in V(T)}$ the set of all subinstances of instance $I^*$ whose corresponding vertex appears in the tree $T$. For each such instance $I\in {\mathcal{I}}^*$, its \emph{recursive level} is the distance from vertex $v(I)$ to the root vertex $v(I^*)$ in the tree $T$ (so the recursive level of $v(I^*)$ is $0$).
For $j\geq 0$, we denote by $\hat {\mathcal{I}}_j\subseteq {\mathcal{I}}^*$ the set of all instances $I\in {\mathcal{I}}^*$, whose recursive level is $j$.
Lastly, the \emph{depth} of the tree $T$, denoted by $\mathsf{dep}(T)$, is the largest recursive level of any instance in ${\mathcal{I}}^*$.
In order to analyze the algorithm, we start with the following two simple observations.
\begin{observation}\label{obs: few recursive levels}
$\mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$.
\end{observation}
\begin{proof}
Consider any non-root vertex $v(I)$ in the tree $T$, and let $v(I')$ be the parent-vertex of $v(I)$. Denote $I=(G,\Sigma)$ and $I'=(G',\Sigma')$.
From the construction of tree $T$, instance $I$ belongs to some collection of subinstances obtained by applying the algorithm from \Cref{thm: main} to instance $I'$. Therefore, from \Cref{thm: main}, $|E(G)|\leq |E(G')|/\mu$ must hold. Therefore, for all $j\geq 0$, for every instance $I=(G,\Sigma)\in \hat {\mathcal{I}}_j$, $|E(G)|\leq m^*/\mu^j$. Since $\mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}$, we get that $\mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$.
\end{proof}
\begin{observation}\label{obs: num of edges}
$\sum_{I=(G,\Sigma)\in {\mathcal{I}}^*}|E(G)|\le m^*\cdot 2^{(\log m^*)^{1/8}}$.
\end{observation}
\begin{proof}
Consider any non-leaf vertex $v(I)$ of the tree $T$, and denote $I=(G,\Sigma)$.
Recall that, when Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace is applied to instance $I$, it applies the algorithm from \Cref{thm: main} to $I$ and compute $L$ collections ${\mathcal{I}}_1(I),\ldots,{\mathcal{I}}_L(I)$ of subinstances, such that, if we denote $|E(G)|=m$, then, for all $1\leq j\leq L$:
$$\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_j(I)}|E(G')|\leq m\cdot (\log m)^{c_g}\leq m\cdot (\log m^*)^{c_g}$$
%
(we have used the fact that, if $I$ is a subinstance of $I^*$, then $m\leq m^*$ must hold). Since $L\leq 2\log m^*$, and $m^*$ is sufficiently large, we get that:
%
$$\sum_{j=1}^L\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_j(I)}|E(G')|\leq m\cdot (\log m^*)^{c_g+2}.$$
%
For all $j\geq 0$, we denote by $N_j$ the total number of edges in all instances in set $\hat {\mathcal{I}}_j$, $N_j=\sum_{I=(G,\Sigma)\in \hat {\mathcal{I}}_j}|E(G)|$. Clearly, $N_0=m^*$, and, from the above discussion, for all $j>0$, $N_j\leq N_{j-1}\cdot (\log m^*)^{c_g+2}$.
Since $\mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$,
we conclude that:
\[
\sum_{I=(G,\Sigma)\in {\mathcal{I}}^*}|E(G)|\leq m^*\cdot (\log m^*)^{2c_g\cdot (\log m^*)^{1/8}/(c^*\log\log m^*)}\leq m^*\cdot 2^{(\log m^*)^{1/8}},
\]
since $c_g\leq c^*/2$.
\end{proof}
We use the following corollary, that follows immediately from \Cref{obs: num of edges}.
\begin{corollary}\label{cor: num of instances}
The number of instances $I=(G,\Sigma)\in {\mathcal{I}}^*$ with $|E(G)|\geq \mu^{c'}$ is at most $m^*$.
\end{corollary}
\iffalse{original corollary does not depend on the notion of leaf instance}
For an instance $I\in {\mathcal{I}}^*$, we say that it is a \emph{leaf instance}, if vertex $v(I)$ is a leaf vertex of the tree $T$, and we say that it is a non-leaf instance otherwise. We use the following corollary, that follows immediately from \Cref{obs: num of edges}, and the fact (from the description of Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace) that, if $I=(G,\Sigma)\in {\mathcal{I}}^*$ is a non-leaf instance, then $|E(G)|\geq \mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}$ must hold.
\begin{corollary}\label{cor: num of instances}
The number of instances $I=(G,\Sigma)\in {\mathcal{I}}^*$ with $|E(G)|\geq \mu^{c'}$ is at most $m^*$.
\end{corollary}
\fi
For an instance $I\in {\mathcal{I}}^*$, we say that it is a \emph{leaf instance}, if vertex $v(I)$ is a leaf vertex of the tree $T$, and we say that it is a non-leaf instance otherwise.
Consider now a non-leaf instance $I\in {\mathcal{I}}^*$. We say that a bad event ${\cal{E}}(I)$ happens, if all the following conditions hold:
\begin{itemize}
\item $0<\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$;
\item $|E(G)|\geq \mu^{c'}$; and
\item for all $1\leq j\leq L$, we have set $\phi_j=\phi'$ and ${\mathcal{I}}_j(I)=\emptyset$, since
the algorithm from \Cref{thm: main} returned FAIL in each of the $L$ iterations when it was applied to $I$.
\end{itemize}
Clearly, from \Cref{thm: main}, $\prob{{\cal{E}}(I)}\le (1/16)^L\leq 1/(m^*)^4$.
We say that a bad event ${\cal{E}}'(I)$ happens if the following conditions hold:
\begin{itemize}
\item $0<\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$;
\item $|E(G)|\geq \mu^{c'}$; and
\item for all $1\leq j\leq L$, $\sum_{I'\in {\mathcal{I}}_j(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')> \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{c_g\cdot (\log m)^{3/4}\log\log m} +m\cdot\mu^{c_g}$.
\end{itemize}
As before, from \Cref{thm: main}, $\prob{{\cal{E}}'(I)}\le (1/16)^L\leq 1/(m^*)^4$.
Let ${\cal{E}}$ be the bad event that event ${\cal{E}}(I)$ or event ${\cal{E}}'(I)$ happened for any instance $I\in {\mathcal{I}}^*$. From the Union Bound and \Cref{cor: num of instances}, we get that $\prob{{\cal{E}}}\leq 1/(m^*)^2$.
We use the following immediate observation.
\begin{observation}\label{obs: leaf}
If Event ${\cal{E}}$ does not happen, then for every leaf vertex $v(I)$ of $T$ with $I=(G,\Sigma)$, either $|E(G)|\leq \mu^{c'}$; or $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$; or $\mathsf{OPT}_{\mathsf{cnwrs}}(I)> |E(G)|^2/\mu^{c'}$.
\end{observation}
We use the following lemma to complete the proof of \Cref{thm: main_rotation_system}.
\begin{lemma}\label{lem: solution cost}
If Event ${\cal{E}}$ does not happen, then Algorithm $\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace$ computes a solution for instance $I^*=(G^*,\Sigma^*)$ of cost at most $2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right})$.
\end{lemma}
\begin{proof}
Consider a non-leaf instance $I=(G,\Sigma)$, and let ${\mathcal{I}}_1(I),\ldots,{\mathcal{I}}_L(I)$ be families of subinstances of $I$ that the Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace computes. Recall that, for each $1\le j\le L$ with ${\mathcal{I}}_j(I)\neq \emptyset$, the algorithm computes a solution $\phi_j$ to instance $I$, by first solving each of the instances in ${\mathcal{I}}_j(I)$ recursively, and then combining the resulting solutions using Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace. Eventually, the algorithm returns the best solution of $\set{\phi',\phi_1,\ldots,\phi_L}$, where $\phi'$ is the trivial solution, whose cost is at most $|E(G)|^2$. We fix an index $1\leq j\leq L$, for which solution $\phi_j$ has the lowest cost, breaking ties arbitrarily. Note that the cost of the solution to instance $I$ that the algorithm returns is at most $\mathsf{cr}(\phi_j)$. We then \emph{mark} the vertices of $\set{v(I')\mid I'\in {\mathcal{I}}_j(I)}$ in the tree $T$. We also mark the root vertex of the tree.
Let $T^*$ be the subgraph of $T$ induced by all marked vertices. It is easy to verify that $T^*$ is a tree, and moreover, every leaf vertex of $T^*$ is also a leaf vertex of $T$. For a vertex $v(I)\in V(T^*)$, we denote by $h(I)$ the length of the longest path in tree $T^*$, connecting vertex $v(I)$ to any of its descendants in the tree.
We use the following claim, whose proof is straightforward conceptually but somewhat technical; we defer the proof to \Cref{Appx: inductive bound proof}.
\begin{claim}\label{claim: bound by level}
Assume that Event ${\cal{E}}$ did not happen. Then there is a fixed constant $\tilde c>0$, such that, for every vertex $v(I)\in V(T^*)$, whose corresponding subinstance of $I^*$ is denoted by $I=(G,\Sigma)$, the cost of the solution that the algorithm computes for $I$ is at most:
$$2^{\tilde c\cdot h(I)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c'\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+(\log m^*)^{4c_g h(I)}\mu^{2c'\cdot \tilde c}\cdot|E(G)|.$$
\end{claim}
We are now ready to complete the proof of \Cref{lem: solution cost}. Recall that $h(I^*)=\mathsf{dep}(T^*)\leq \mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$ from \Cref{obs: few recursive levels}. Therefore, from \Cref{claim: bound by level}, the cost of the solution that the algorithm computes for instance $I^*$ is bounded by:
\[
\begin{split}
& \text{ } 2^{O(\mathsf{dep}(T))\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{O(1)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+(\log m^*)^{O(\mathsf{dep}(T))}\cdot\mu^{O(1)}\cdot m^*\\
\leq & \text{ } 2^{O((\log m^*)^{7/8})}\cdot \mu^{O(1)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+(\log m^*)^{O((\log m^*)^{1/8}/\log\log m^*) }\cdot\mu^{O(1)}\cdot m^*\\
\leq & \text{ } 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}),
\end{split}
\]
since $\mu=2^{O((\log m^*)^{7/8}\log\log m^*)}$.
\end{proof}
In order to complete the proof of \Cref{thm: main_rotation_system}, it is now enough to prove Theorem~\ref{thm: main}. The remainder of the paper is dedicated to the proof of \Cref{thm: main}.
\subsection{Proof of Theorem~\ref{thm: main} -- Main Definitions and Theorems}
We classify subinstances of the input instance $I^*$ into \emph{wide} and \emph{narrow}, and provide different algorithms for decomposing instances of each of the two kinds.
Throughout this subsection, we use the parameter ${50}=50$.
\begin{definition}[Wide and Narrow Instances]
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$. We say that $I$ is a \emph{wide} instance, iff there is a vertex $v\in V(G)$, a partition $(E_1,E_2)$ of the edges of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and a collection ${\mathcal{P}}$ of at least $\floor{m/\mu^{{50}}}$ simple edge-disjoint cycles in $G$, such that every cycle $P\in {\mathcal{P}}$ contains one edge of $E_1$ and one edge of $E_2$.
If no such cycle set ${\mathcal{P}}$ exists in $G$, then we say that $I$ is a \emph{narrow} instance.
\end{definition}
Note that there is an efficient algorithm to check whether a given instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace is wide, and, if so, to compute the corresponding cycle set ${\mathcal{P}}$, via standard algorithms for maximum flow. (For every vertex $v\in V(G)$, we can try all possible partitions $(E_1,E_2)$ of $\delta_G(v)$ with the required properties, as the number of such partitions is bounded by $|\delta_G(v)|^2$.)
We will use the following simple observation regarding narrow instances.
\begin{observation}\label{obs: narrow prop 2}
If an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace is narrow, then for every pair $u,v$ of distinct vertices of $G$, and any set ${\mathcal{P}}$ of edge-disjoint paths connecting $u$ to $v$ in $G$, $|{\mathcal{P}}|\leq 2\ceil{m/\mu^{{50}}}+2$ must hold.
\end{observation}
\begin{proof}
Assume for contradiction that $|{\mathcal{P}}|>2\ceil{m/\mu^{{50}}}+2$, and denote $|{\mathcal{P}}|=k$.
Let $E'\subseteq \delta_G(v)$ be the set of all edges $e\in \delta_G(v)$, such that $e$ is the first edge on some path in ${\mathcal{P}}$. We denote $E'=\set{e_1,\ldots,e_k}$, where the edges are indexed according to their ordering in the rotation ${\mathcal{O}}_v\in \Sigma$. We also denote ${\mathcal{P}}=\set{P(e_i)\mid 1\leq i\leq k}$, where path $P(e_i)$ contains the edge $e_i$ as its first edge. We can then compute a partition $(E_1,E_2)$ of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$. Additionally, we can ensure that $e_1,\ldots,e_{\ceil{k/2}}\in E_1$, while the remaining edges of $E'$ lie in $E_2$. For each $1\leq i\leq \ceil{m/\mu^{{50}}}$, we let $Q_i$ be the cycle obtained by concatenating the paths $P(e_i)$ and $P(e_{k-i+1})$. We can turn $Q_i$ into a simple cycle, by removing all cycles that are disjoint from vertex $v$ from it. It is then immediate to verify that cycle set $\set{Q_i\mid 1\leq i\leq \ceil{m/\mu^{{50}}}}$ has all the required properties to establish that instance $I$ is wide, a contradiction.
\end{proof}
\iffalse
\begin{definition}[Wide and Narrow Subinstances]
A subinstance $I=(G,\Sigma)$ of $I^*$ is an \emph{interesting} instance, iff there is a
collection ${\mathcal{P}}$ of at least $m/\mu^{{50}}$ edge-disjoint paths in $G$, that we refer to as the \emph{witness path set for $I$}, such that one of the following holds:
\begin{itemize}
\item either there are two distinct vertices $u,v\in V(G)$, such that every path in ${\mathcal{P}}$ originates at $v$ and terminates at $u$; or
\item there is a single vertex $v\in V(G)$, and a partition $(E_1,E_2)$ of the edges of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$. Every path in ${\mathcal{P}}$ must contain an edge of $E_1$ as its first edge and an edge of $E_2$ as its last edge.
\end{itemize}
%
If no such path set ${\mathcal{P}}$ exists in $G$, then we say that $I$ is an \emph{narrow} instance.
\end{definition}
\fi
The proof of \Cref{thm: main} relies on the following two theorems. The first theorem deals with wide subinstances of $I^*$, and its proof appears in \Cref{sec: many paths}.
\mynote{The theorem has changed. The old version appears below this new version for comparison. This section will need to be updated to work with the new theorem (but wait until I finish Sec 5 in case there are other changes).}
\begin{theorem}\label{lem: many paths}
There is an efficient randomized algorithm, that, given a wide subinstance $I=(G,\Sigma)$ of $I^*$ with $m=|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$, either returns FAIL, or computes a non-empty collection ${\mathcal{I}}$ of subinstances of $I$, such that $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\le |E(G)|$, and, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, either $|E(G')|\le m/\mu$, or instance $I'$ is not wide. Additionally, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$ then, with probability at least $1-1/\mu^2$, all of the following hold:
\begin{itemize}
\item the algorithm does not return FAIL;
\item $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log m)^{O(1)}$; and
\item there is an efficient algorithm, that, given a solution $\phi(I')$ for every instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq \sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) + \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\mu^{O(1)}$.
\end{itemize}
\end{theorem}
\mynote{old version of same theorem}
\begin{theorem}\label{lem: many paths}
There is an efficient randomized algorithm, that, given a wide subinstance $I=(G,\Sigma)$ of $I^*$ with $m=|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$, either returns FAIL, or computes a non-empty collection ${\mathcal{I}}$ of subinstances of $I$, such that:
\begin{itemize}
\item for each instance $I'=(G',\Sigma')\in {\mathcal{I}}$, if $I'$ is a wide instance, then $|E(G')|\le m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\le |E(G)|$; and
\item
there is an efficient algorithm, that, given a solution $\phi(I')$ for every instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq \sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) + (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}$.
\end{itemize}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/\mu^2$.
Additionally, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, then with probability at least $31/32$:
$$\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{O(1)}.$$
\end{theorem}
\iffalse
\begin{theorem}\label{lem: many paths}
There is a randomized efficient algorithm, that, given a wide subinstance $I=(G,\Sigma)$ of $I^*$, with $m=|E(G)|>\mu^{100{50}}$ and $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{100{50}}$,
computes a collection ${\mathcal{I}}$ of subinstances of $I$, such that:
\begin{itemize}
\item for each instance $I'=(G',\Sigma')\in {\mathcal{I}}$, if $I'$ is a wide instance, then $|E(G')|\le m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\le |E(G)|$;
and
\item $\expect[]{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{O(1)}$.
\end{itemize}
Moreover, there is an efficient algorithm, that, given, a solution $\phi(I')$ for every instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq \sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) + (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}$. \znote{constants subject to change}
\end{theorem}
\fi
\iffalse{original Theorem 3.6}
\begin{theorem}\label{lem: many paths}
There is a randomized efficient algorithm, that, given an interesting subinstance $I=(G,\Sigma)$, with $|E(G)|=m>\mu^{39}$ and $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{40}$,
computes two subinstances $I'=(G',\Sigma'),I''=(G'',\Sigma'')$ of $I$, such that $m/\mu^{10}\le |E(G')|,|E(G'')|\leq m\cdot (1-1/\mu^{10})$ and $|E(G')|+|E(G'')|\leq m$, and with probability at least $1-1/(40\mu^{11})$,
$$\mathsf{OPT}_{\mathsf{cnwrs}}(I')+\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I) \cdot(1+1/\mu^{12})+m \cdot \mu^{c},$$
where $c>0$ is some universal integer.
Moreover, there is an efficient algorithm, that, given solutions $\phi'$ to instance $I'$ and $\phi''$ to instance $I''$, computes a solution $\phi$ to instance $I'$ , with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\phi')+\mathsf{cr}(\phi'')+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\operatorname{poly}(\mu)$.
\end{theorem}
\fi
The second theorem deals with narrow instances.
\begin{theorem}\label{lem: not many paths}
There is an efficient randomized algorithm, that, given a narrow subinstance $I=(G,\Sigma)$ of $I^*$, with $m=|E(G)|\geq \mu^{20}$, either returns FAIL, or computes a $\nu$-decomposition ${\mathcal{I}}$ of $I$, for $\nu= 2^{O((\log m)^{3/4}\log\log m)}$, such that, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\le m/\mu$. Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{16}$, then the probability that the algorithm returns FAIL is at most $1/\mu^2$.
\end{theorem}
The majority of the remainder of this paper is dedicated to the proof of the above theorem: we develop the central technical tools used in the theorem's proof in \Cref{sec: guiding paths orderings basic disengagement} -- \Cref{sec: main disengagement}, and complete the proof in \Cref{sec: computing the decomposition}.
In the remainder of this section, we complete the proof of Theorem \ref{thm: main} using Theorems \ref{lem: many paths} and \ref{lem: not many paths}.
\iffalse
\mynote{restate the lemma in terms of $\alpha$-decomposition into subinstances}
\begin{lemma}\label{lem: not many paths}
There is an efficient randomized algorithm, that, given a non-interesting subinstance $I=(G,\Sigma)$ of $I^*$ with $|E(G)|=m\geq \mu^{10}$, either correctly certifies that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \Omega(m^2/\mu^5)$, or computes a collection ${\mathcal{I}}$ of subinstances of $I$, such that
the following hold:
\begin{itemize}
\item for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\le m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq O(m)$; and
\item $\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot 2^{O((\log m)^{3/4}\log\log m)}$.
\end{itemize}
Moreover, there is an efficient algorithm, that, given, for each instance $I'\in {\mathcal{I}}$, a feasible solution $\phi(I')$, computes a feasible solution $\phi$ for instance $I$, of cost $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$.
\end{lemma}
\fi
Recall that we are given an instance $I=(G,\Sigma)$, that is a subinstance of the input instance $I^*=(G^*,\Sigma^*)$, with $|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$. We denote $m=|E(G)|$. We compute the desired collection ${\mathcal{I}}$ of subinstances of $I$ in two steps.
If instance $I$ is a narrow instance, then we skip the first step. We let ${\mathcal{I}}'$ be a collection of subinstances of $I$, that consists of instance $I$ only. Assume now that $I$ is a wide instance. Then we execute Step 1, by applying the algorithm from \Cref{lem: many paths} to $I$.
If the algorithm returns FAIL, then we terminate the algorithm and return FAIL as well.
Otherwise, let ${\mathcal{I}}'$ be the resulting collection of subinstances of $I$. Recall that we are guaranteed that, for each instance $I'=(G',\Sigma')\in {\mathcal{I}}'$, either $I'$ is a narrow instance, or $|E(G')|\le m/\mu$. Additionally, we are guaranteed that $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\le |E(G)|$;
and, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{c'}$, then with probability at least $31/32$:
$$\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{c_g(\log m)^{3/4}\log\log m}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g},$$
for some constant $c_g$.
We say that a bad event ${\cal{E}}'_1$ happens if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, but: $$\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')> 2^{c_g(\log m)^{3/4}\log\log m}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g}.$$
From \Cref{lem: many paths}, $\prob{{\cal{E}}'_1}\leq 1/32$.
We say that a bad event ${\cal{E}}''_1$ happens if algorithm from \Cref{lem: many paths} returns FAIL when applied to $I$.
From \Cref{lem: many paths}, $\prob{{\cal{E}}''_1}\leq 1/\mu^2$.
Define the bad event ${\cal{E}}_1={\cal{E}}'_1\cup {\cal{E}}''_1$, so $\prob{{\cal{E}}_1}\leq 1/31$.
This completes the first step of the algorithm.
We now describe the second step.
We start by partitioning the set ${\mathcal{I}}'$ of instances into two subsets: set ${\mathcal{I}}_1'$ containing all instances $I'=(G',\Sigma')$ with $|E(G')|\leq m/\mu$, and set ${\mathcal{I}}_2'$ containing all remaining instances, that must be narrow.
We use the following simple observation.
\begin{observation}\label{obs: optbound for narrow}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and Event ${\cal{E}}_1$ does not happen, then for every instance $I'=(G',\Sigma')\in {\mathcal{I}}_2'$, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq |E(G')|^2/\mu^{16}$.
\end{observation}
\begin{proof}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and that Event ${\cal{E}}_1$ does not happen. Then:
$$\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{c_g(\log m)^{3/4}\log\log m}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g}.$$
Consider now some instance $I'=(G',\Sigma')\in {\mathcal{I}}_1'$. We then get that:
$$\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{c_g(\log m)^{3/4}\log\log m}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g}\leq \mu\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g},$$
since $\mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}$.
Since $m\geq \mu^{c'}$, and since we can assume that $c'$ is a large enough constant, $\mu^{c_g}\leq m/\mu^{c'/2}$. Further, since we have assumed that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and since $|E(G')|\geq m/\mu$ (since instance $I'=(G',\Sigma')$ belongs to set ${\mathcal{I}}_2'$), we get that:
$$\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq m^2/\mu^{c'-1}+m^2/\mu^{c'/2}\leq |E(G')|^2/\mu^{16},$$
assuming that $c'$ is large enough.
\end{proof}
Next, we process every instance $I'\in {\mathcal{I}}_2'$ one by one. Notice that for each such instance $I'=(G',\Sigma')$, $|E(G')|\geq m/\mu\geq \mu^{20}$ must hold, since $m\geq \mu^{c'}$. When instance $I'$ is processed, we apply the algorithm from \Cref{lem: not many paths} to it. If the algorithm returns FAIL, then we terminate the algorithm and return FAIL as well. Otherwise, we obtain a collection ${\mathcal{I}}(I')$ of subinstances of $I'$ (and hence of $I$). If the algorithm did not terminate with a FAIL, then we obtain the final collection ${\mathcal{I}}^*={\mathcal{I}}_1'\cup \textsf{left} (\bigcup_{I'\in {\mathcal{I}}_2'}{\mathcal{I}}(I')\textsf{right})$ of subinstances of $I$ that the algorithm returns.
This completes the description of the algorithm.
In order to analyze it, we first show that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/16$.
\begin{observation}\label{obs: fail prob}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/16$.
\end{observation}
\begin{proof}
For an instance $I'=(G',\Sigma')\in {\mathcal{I}}'$, we say that a bad event ${\cal{E}}(I')$ happens if the algorithm from \Cref{lem: not many paths}, when applied to $I'$, returns FAIL. From \Cref{lem: not many paths}, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(I')<|E(G')|^2/\mu^{16}$, then the probability that the algorithm returns FAIL is at most $1/\mu^2$. Therefore, from \Cref{obs: optbound for narrow}, if Event ${\cal{E}}_1$ does not happen, then $\prob{{\cal{E}}(I')}\leq 1/\mu^2$. Let ${\cal{E}}_2$ be the bad event that ${\cal{E}}(I')$ happens for any instance $I'\in {\mathcal{I}}'_2$. Since every instance $I'\in {\mathcal{I}}_2'$ contains at least $m/\mu$ edges, while, as we have established already, the total number of edges in all instances in ${\mathcal{I}}'$
is at most $m$, we get that $|{\mathcal{I}}'_2|\leq \mu$. From the Union Bound, assuming that the constant $c^*$ in the definition of the parameter $\mu$ is large enough, $\prob{{\cal{E}}_2\mid \neg {\cal{E}}_1}\leq 1/100$. Clearly, $\prob{{\cal{E}}_2}\leq \prob{{\cal{E}}_1}+\prob{{\cal{E}}_2\mid \neg {\cal{E}}_1}\leq 1/31+1/100< 1/20$. Note that, if events ${\cal{E}}_1$ and ${\cal{E}}_2$ do not happen, then algorithm may not return FAIL. We conclude that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/20<1/16$.
\end{proof}
We define the bad event ${\cal{E}}={\cal{E}}_1\cup {\cal{E}}_2$, where event ${\cal{E}}_2$ is defined in the proof of above observation. From the above discussion $\prob[]{{\cal{E}}}\le 1/20$.
Fron now on we assume that the algorithm did not return FAIL.
Recall that \Cref{lem: not many paths} gurantees that, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}_2'$, for every subinstance $I''=(G'',\Sigma'')\in {\mathcal{I}}(I')$, $|E(G'')|\leq |E(G')|/\mu\leq m/\mu$ holds. Therefore, for every instance $I''=(G'',\Sigma'')\in{\mathcal{I}}^*$, $|E(G'')|\leq m/\mu$ must hold.
Additionally, from the definition of a $\nu$-decomposition, we are guaranteed that for all $I'=(G',\Sigma')\in {\mathcal{I}}_2'$, $\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}(I')}|E(G'')|\leq |E(G')|\cdot (\log (|E(G')|)^{O(1)}\leq |E(G')|\cdot (\log m)^{O(1)}$.
Moreover, since $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\le |E(G)|$, we get that:
$$\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}^*}|E(G'')|\leq \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\cdot (\log m)^{O(1)}\leq m\cdot(\log m)^{O(1)}.$$
Next, we provide Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace.
\begin{claim}\label{claim: combine drawings}
There is an efficient algorithm, called \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, that, given a solution $\phi(I'')$ to each instance $I''\in {\mathcal{I}}^*$, computes a solution $\phi(I)$ to instance $I$, of cost at most $O(\sum_{I''\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I''))) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{O(1)}$.
\end{claim}
\begin{proof}
We construct the solution $\phi(I)$ to instance $I$ in two steps.
In the first step, we compute a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}'_2$, as follows. Consider any such instance $I'=(G',\Sigma')\in {\mathcal{I}}'_2$. Recall that we are given, for each instance $I''\in {\mathcal{I}}(I')$, a solution $\phi(I'')$. Recall also that ${\mathcal{I}}(I')$ is a $\nu$-decomposition of $I'$. We apply the efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}}(I'))$ from the definition of $\nu$-decompositions of instances to the drawings $\set{\phi(I'')}_{I''\in {\mathcal{I}}(I')}$, to obtain a feasible solution $\phi(I')$ for instance $I'$, of cost $\mathsf{cr}(\phi(I'))\leq O\textsf{left} (\sum_{I''\in {\mathcal{I}}(I')}\mathsf{cr}(\phi(I''))\textsf{right} )$.
We have now obtained a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}'$. In the second step, we apply the algorithm in the third bullet of the statement of \Cref{lem: many paths} to these solutions, to obtain a solution $\phi(I)$ to instance $I$.
The cost of the solution is bounded by:
\[\begin{split}
\mathsf{cr}(\phi(I)) &\leq \sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I')) + (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}\\
&\leq \sum_{I'\in {\mathcal{I}}'_1}\mathsf{cr}(\phi(I')) +\sum_{I'\in {\mathcal{I}}'_2}\mathsf{cr}(\phi(I'))+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}\\
&\leq \sum_{I'\in {\mathcal{I}}'_1}\mathsf{cr}(\phi(I')) +\sum_{I'\in {\mathcal{I}}'_2}O\textsf{left} (\sum_{I''\in {\mathcal{I}}(I')}\mathsf{cr}(\phi(I''))\textsf{right} )+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}\\
&\leq O\textsf{left} (\sum_{I''\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I'))\textsf{right} ) +(m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}.
\end{split}
\]
\end{proof}
In order to complete the proof of \Cref{thm: main}, it is now enough to prove the following observation.
\begin{observation}\label{obs: bound sum of opts}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then with probability at least $15/16$: $$\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{O((\log m)^{3/4}\log\log m)} +m\cdot\mu^{O(1)}.$$
\end{observation}
\begin{proof}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$. Recall that, if Event ${\cal{E}}$ does not happen, then:
$$\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{c_g(\log m)^{3/4}\log\log m)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g}.$$
From the definition of a $\nu$-decomposition, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}'_2$,
\[\expect{\sum_{I''\in {\mathcal{I}}(I')}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} )\cdot \nu \leq \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} ) \cdot 2^{O((\log m)^{3/4}\log\log m)},\]
Therefore, if ${\cal{E}}$ does not happen, then:
\[
\begin{split}
\expect{\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}&\leq \sum_{I'\in {\mathcal{I}}'_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I') +\sum_{I'\in {\mathcal{I}}'_2}\expect{\sum_{I''\in {\mathcal{I}}(I')}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}\\
&\leq \sum_{I'\in {\mathcal{I}}'_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'_2}\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} ) \cdot 2^{O((\log m)^{3/4}\log\log m)}\\
&\leq \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} )\cdot 2^{O((\log m)^{3/4}\log\log m)}\\
&\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{O(1)}.
\end{split} \]
We denote this expectation by $\eta'$. Let $\hat {\cal{E}}$ be the bad event that
$\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')>100\eta'$. From Markov's inequality, $\prob{\hat {\cal{E}}\mid \neg{\cal{E}}}<1/100$. Note that, if neither of the events $\hat {\cal{E}},{\cal{E}}$ happens, then we are guaranteed that $\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{O(1)}$. The probability that either one of these events happens is bounded by $\prob{{\cal{E}}}+\prob{\hat {\cal{E}}\mid \neg{\cal{E}}}\leq 1/20+ 1/100\leq 1/16$. We conclude that, If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then with probability at least $15/16$: $$\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{O((\log m)^{3/4}\log\log m)} +m\cdot\mu^{O(1)}.$$
\end{proof}
\iffalse
Moreover, we are guaranteed that,
\[\expect[]{\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}\leq \nu''\cdot \bigg( \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\big(\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\big)\bigg ). \]
Assume that event ${\cal{E}}$ did not happen. Then,
\[
\begin{split}
\expect[]{\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')} &\text{ }\leq \nu''\cdot \expect[]{ \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\big(\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\big)}\\
&\text{ }\leq \nu''\cdot \expect[]{\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}+\nu''\cdot\bigg(\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\bigg)\\
&\text{ }\leq \nu''\cdot \bigg(\nu'\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^c\bigg)+\nu''\cdot(\log|E(G)|)^{O(1)}\cdot|E(G)|\\
&\text{ }\leq (\nu'\nu'')\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+O(\nu''\cdot\mu^c)\cdot|E(G)|.
\end{split} \]
Let $\nu=\nu'\cdot\nu''$.
Since $\nu',\nu''=2^{O((\log m)^{3/4}\log\log m)}$, $\nu=\nu'\cdot\nu''=2^{O((\log m)^{3/4}\log\log m)}$.
Since $\mu^{c_g}>\nu''\cdot\mu^{c}$, with probability at least $9/10$, the resulting collection ${\mathcal{I}}^*$ of instances is perfect for $I$.
\fi
\iffalse
In order to do so,
@@@@ we compute a denote by ${\mathcal{I}}(I')=\set{I''_1,\ldots,I''_k}$ the set of subinstances obtained by applying the algorithm from \Cref{lem: not many paths} to it. From the definition of a $\nu''$-decomposition, we are also given an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}}(I'))$, that takes as input a solution of each instance in ${\mathcal{I}}(I')$ and returns a solution of $I'$. We apply this algorithm to the solutions $\set{\phi(I''_i)}_{1\le i\le k}$ to the subinstances in ${\mathcal{I}}(I')$ and let $\phi(I')$ be the solution we get. From the definition of a $\nu''$-decomposition, there exists some universal constant $c^*>0$, such that $\mathsf{cr}(\phi(I'))\le c^*\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I''_i))$.
Clearly, after the first step, we have obtained, for each subinstance $I'\in {\mathcal{I}}'$, a solution $\phi(I')$ to instance $I'$.
Recall that, in \Cref{lem: many paths} we are also given an efficient algorithm (that we denote by $\ensuremath{\mathsf{Alg}}\xspace'({\mathcal{I}}')$), that, given, for every instance $I'\in {\mathcal{I}}'$, a solution $\phi(I')$ to instance $I'$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq \big(\sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I'))\big)+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^c$.
In the second step, we simply apply the algorithm to the set $\set{\phi(I')}_{I'\in {\mathcal{I}}'}$ of solutions and let $\phi(I)$ be the solution to instance $I$ that we obtain. From \Cref{lem: many paths},
\[
\begin{split}
\mathsf{cr}(\phi)\leq & \text{ } \bigg(\sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I'))\bigg)+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^c\\
\leq & \text{ } c^*\cdot \bigg(\sum_{I''\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I''))\bigg)+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^c.
\end{split}
\]
Now \Cref{thm: main} follows by setting $c_g$ to be a contant such that $c_g> \max\set{c^*, c}$.
\fi
\iffalse{previous algorithm algcombine}
Consider the partitioning tree $T^*$ that is obtained from the partitioning tree $T$ in the first Stage, by adding, for each leaf $v(I')$ of $T$ with $I'\in {\mathcal{I}}'_2$ and for each instance $I''\in {\mathcal{I}}(I')$, a new vertex $v(I'')$ and an edge connecting it to $v(I')$. Clearly, the instances in ${\mathcal{I}}^*$ correspond to leaves in $T^*$. Assume now that we are given, for each instance $I'$ such that $v(I')$ is a leaf in $T^*$, a solution $\phi(I')$ to instance $I'$. We will construct a drawing of instance $I$ in two steps as follows.
In the first step, for each instance $I'\in {\mathcal{I}}'_2$, we denote by ${\mathcal{I}}(I')=\set{I''_1,\ldots,I''_k}$ the set of subinstances obtained by applying the algorithm from \Cref{lem: not many paths} to it. Recall that, from the definition of a $\nu$-decomposition in \Cref{lem: not many paths} we are also given an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}}(I'))$, that takes as input a solution of each instance in ${\mathcal{I}}(I')$ and returns a solution of $I'$. We apply this algorithm to the solutions $\set{\phi(I''_i)}{1\le i\le k}$, and let $\phi(I')$ be the solution we get. From the definition of a $\nu$-decomposition, $\mathsf{cr}(\phi')\le c^*\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I''_i))$ for some universal constant $c^*$.
In the second step, we look at the partitioning tree $T$ of Stage 1, and note that now we have, for each instance $I'$ such that $v(I')$ is a leaf in $T$, a solution $\phi(I')$ of $I'$. We now construct a solution for every instance that corresponds to a vertex in $T$ in a bottom-up fashion as follows. In each iteration, we take a vertex $v(I')$ in $T$ with children $v(I''_1),v(I''_2)$ in $T$, such that we have not yet computed a solution of instance $I'$, but we have already computed a solution $\phi(I''_1)$ for instance $I''_1$ and a solution $\phi(I''_2)$ for instance $I''_2$. Note that it is not hard to see that such a vertex has to exist in each iteration.
We not apply the solution-combining algorithm from \Cref{lem: many paths} to the solutions $\phi(I''_1), \phi(I''_2)$, and obtain a solution $\phi(I')$ to instance $I'$. This completes the description of an iteration. We keep performing iterations until we obtain a solution $\phi(I)$ to the input instance $I$.
It remains to upper bound $\mathsf{cr}(\phi(I))$. Recall that at the end of the first step, we have obtained, for each instance $I'\in {\mathcal{I}}'$ that corresponds to a leaf in $T$, a solution $\phi(I')$ to $I'$, such that $\sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I'))\le c^*\cdot\sum_{I'\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I'))$. Recall that the depth of tree $T$ is at most $2\mu^{11}$.
Moreover, for each non-leaf instance $I'=(G',\Sigma')$ of $T$, if we denote by $I''_1,I''_2$ the instances that correspond to its children in $T$, then $\mathsf{cr}(\phi(I'))\le \mathsf{cr}(\phi(I''_1))+\mathsf{cr}(\phi(I''_2))+(\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|)\cdot \mu^c$.
Via similar arguments as in the proof of \Cref{cor: stage 1} we can show that, for each $1\le i\le \floor{2\mu^{11}}$, if we denote by ${\mathcal{I}}'_i$ be the set of subinstances obtained after iteration $i$, then
\[
\sum_{I'\in {\mathcal{I}}'_{i+1}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot \mu^{c+13}\leq \bigg(\sum_{I'\in {\mathcal{I}}'_{i}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot\mu^{c+13}\bigg)\cdot (1+1/\mu^{12}),
\]
Therefore,
\[
\sum_{I': v(I')\in V(T)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot \mu^{c+13}\leq 4\cdot \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+ m\cdot \mu^{c+13}\bigg).
\]
It follows that
\[
\begin{split}
\mathsf{cr}(\phi(I))
& \le \sum_{I' \in {\mathcal{I}}'} \mathsf{cr}(\phi(I')) +\mathsf{dep}(T)\cdot O\bigg(\sum_{I \notin {\mathcal{I}}'} \big(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c+13}\big)\bigg)\\
& \le c^*\cdot \sum_{I' \in {\mathcal{I}}^*} \mathsf{cr}(\phi(I')) +8\cdot\bigg(\mu^{11}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+\mu^{11}\cdot \mu^{c+13}\cdot |E(G)|\big)\bigg).
\end{split}
\]
Now \Cref{thm: main} follows by setting $c_g$ to be a contant such that $c_g\ge \max\set{c^*, c+25}$.
\fi
\iffalse
It is easy to verify that $T^*$ is
We say that the instances in ${\mathcal{I}}_i(I)$ are \emph{critical subinstance} to instance $I$, iff $\mathsf{cr}(\phi_i)=\min\set{\mathsf{cr}(\phi_1),\ldots, \mathsf{cr}(\phi_L),|E(G)|^2}$.
Consider now the tree $T^*$, which is the subtree of $T$ induced by $v(I^*)$ and all vertices $v(I)$ such that the instance $I$ is critical.
It is easy to see that every leaf of $T^*$ is also a leaf of $T$.
Let $I$ be an instance with $v(I)\in V(T^*)$ and let $I_1,\ldots,I_k$ be its critical subinstances, where $I_i=(G_i,\Sigma_i)$ for each $i$.
From \Cref{thm: main} and since the event ${\cal{E}}$ does not happen,
$\sum_{1\le i\le k}\mathsf{OPT}_{\mathsf{cnwrs}}(I_i)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu +|E(G)|\cdot\mu^{c_g}$ and $\sum_{1\le i\le k}|E(G_i)|\leq |E(G)|\cdot (\log m^*)^{c_g}$ (where we have used the fact that $|E(G)|\le m^*$).
Therefore, since $(\log m^*)^{c_g}<\nu$,
\[
\sum_{1\le i\le k}\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I_i)+|E(G_i)|\cdot \mu^{c_g}\bigg)\leq \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c_g}\bigg)\cdot \nu.
\]
It is easy to see that, if we denote by ${\mathcal{I}}^{(j)}$ the set of all critical instances at recursive level $j$, then
\[
\sum_{I=(G,\Sigma)\in {\mathcal{I}}^{(j)}}\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c_g}\bigg)\leq \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\cdot \mu^{c_g}\bigg)\cdot \nu^{j}.
\]
It follows that, if we denote by ${\mathcal{I}}^*$ the set of all critical subinstances, then
\[
\begin{split}
\sum_{I=(G,\Sigma)\in {\mathcal{I}}^{^*}}\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c_g}\bigg) & \leq \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\cdot \mu^{c_g}\bigg)\cdot 2^{O((\log m^*)^{7/8}\log\log m^*)}\\
& = 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}).
\end{split}
\]
Consider now any instance $I=(G,\Sigma)$ with $v(I)\in V(T^*)$. We first denote $\operatorname{cost}^*(I)=\mu^{c_g}\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{2c_g}$. From the above discussion,
\[
\sum_{I=(G,\Sigma)\in {\mathcal{I}}^{^*}}\operatorname{cost}^*(I) \leq 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}).
\]
We then define the \emph{height} of $I$, denoted by $h(I)$, to be the largest distance from $v(I)$ to a descendant vertex of $v(I)$ in the tree $T^*$. In particular, if $v(I)$ is a leaf in $T^*$, then $h(I)=0$, and, from \Cref{obs: few recursive levels}, for every non-leaf subinstance $I$, $h(I)\le \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$ always holds.
For an instance $I'$ such that vertex $v(I')$ is a descendant of vertex $v(I)$ in $T^*$, we write $I'\prec I$.
We prove by induction on $h(I)$ that the value of the solution $\phi(I)$ that $\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace(I)$ computes satisfies that
$$\mathsf{cr}(\phi(I))\le \mu^{2c'}\cdot\operatorname{cost}^*(I)+\mu^{2c'}\cdot\sum_{I' \prec I} \operatorname{cost}^*(I')\cdot c_g^{h(I)-h(I')}.$$
The induction base is when $h(I)=0$, so $v(I)$ is a leaf vertex of $T$. We have shown that the claim holds in this case.
Let $I=(G,\Sigma)$ be a non-leaf subinstance with $v(I)\in V(T^*)$ and let $I_1,\ldots,I_k$ be its critical subinstances. If we denote by $\phi(I)$ the drawing of $I$ returned by the algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace, and similarly we define drawings $\set{\phi(I_i)}_{1\le i\le k}$, then from \Cref{thm: main},
\[
\mathsf{cr}(\phi(I)) \le c_g\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I_i))+ \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\bigg)\cdot \mu^{c_g} \le \operatorname{cost}^*(I)+c_g\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I_i)).
\]
Note that for each $1\le i\le k$, $h(I_i)\le h(I)-1$. Therefore, from the induction hypothesis,
\[
\begin{split}
\mathsf{cr}(\phi(I)) & \leq \operatorname{cost}^*(I)+c_g\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I_i))\\
& \le \operatorname{cost}^*(I)+c_g\cdot \sum_{1\le i\le k}\bigg(\mu^{2c'}\cdot\operatorname{cost}(I_i)+\mu^{2c'}\cdot\sum_{I' \prec I_i} \operatorname{cost}^*(I')\cdot c_g^{h(I_i)-h(I')}\bigg)\\
& \le \mu^{2c'}\cdot\operatorname{cost}^*(I)+ \mu^{2c'}\cdot\sum_{1\le i\le k}\bigg(c_g\cdot\operatorname{cost}(I_i)+\sum_{I' \prec I_i} \operatorname{cost}^*(I')\cdot c_g^{h(I_i)-h(I')+1}\bigg)\\
& \le \mu^{2c'}\cdot\operatorname{cost}^*(I)+ \mu^{2c'}\cdot\sum_{I'\prec I} \operatorname{cost}^*(I')\cdot c_g^{h(I)-h(I')}.
\end{split}
\]
Therefore, since $h(I^*)\le \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$, we get that for each $I'\in {\mathcal{I}}^*$, $c_g^{h(I^*)-h(I')}\le \mu$, and so
\[
\begin{split}
\mathsf{cr}(\phi(I^*))\le & \text{ } \mu^{2c'}\cdot\operatorname{cost}^*(I^*)+\mu^{2c'}\cdot\sum_{I' \prec I} \operatorname{cost}^*(I')\cdot c_g^{h(I^*)-h(I')}\\
\le & \text{ } 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}).
\end{split}
\]
\fi
\iffalse
Therefore, if we denote by ${\mathcal{I}}^{**}$ the set of all subinstances $I$ such that $v(I)$ is a leaf of tree $T^*$, then
\[
\begin{split}
\sum_{I\in {\mathcal{I}}^*} \mathsf{cr}(\phi(I))
& \le \mathsf{dep}(T)\cdot\bigg(\sum_{I\in {\mathcal{I}}^{**}} \mathsf{cr}(\phi(I)) +\sum_{I=(G,\Sigma)\in ({\mathcal{I}}^{*}\setminus {\mathcal{I}}^{**})} \big(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c_g}\big)\bigg)\cdot \mu^{c_g}\\
& =\mu^{O(c_g)}\cdot \bigg(\mu^{2c'}\cdot\sum_{I\in {\mathcal{I}}^{**}} \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right})
\bigg)\\
& = 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}).
\end{split}
\]
And since $\mathsf{cr}(\phi(I^*))\le \sum_{I\in {\mathcal{I}}^{^*}} \mathsf{cr}(\phi(I)) + \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\bigg)\cdot \mu^{c_g}$, it follows that $\mathsf{cr}(\phi(I^*))=2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right})$.
\fi
\iffalse
\znote{cannot prove by induction, need to re-write from here}
We prove by induction on $h(I)$ that the value of the solution $\phi(I)$ that $\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace(I)$ computes satisfies that $\mathsf{cr}(\phi(I))\le (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot \mu^{c_g}\cdot \nu^{h(I)}$. The induction base is when $h(I)=0$, so $v(I)$ is a leaf vertex of $T$. We have shown that the claim holds in this case.
Consider now some instance $I=(G,\Sigma)$ with $h(I)>0$. Assume first that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)> |E(G)|^2/\mu^{64}$. Recall that the algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace considers a trivial solution $\phi'$ to instance $I$, of cost at most $|E(G)|^2$. Since the algorithm eventually returns the best of the considered solutions, the cost of the solution that it returns may not be higher than $|E(G)|^2\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \mu^{64}$.
Therefore, we assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{64}$ from now on. Since we have assumed that Event ${\cal{E}}$ did not happen, neither did ${\cal{E}}(I)$. Therefore, there is an instance $1\leq i\leq L$, such that the family ${\mathcal{I}}_i(I)$ of instances is a perfect family for instance $I$. In particular, from the definition of a perfect family, we have that
$\sum_{I'\in {\mathcal{I}}_i(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu +|E(G)|\cdot\mu^{c_g}$.
Using the induction hypothesis, if we denote by $\phi(I')$ the solution that the algorithm computes for each instance $I'\in {\mathcal{I}}_i(I)$, then:
\[
\begin{split}
\sum_{I'=(G'\Sigma')\in {\mathcal{I}}_i(I)}\mathsf{cr}(\phi(I'))\leq & \sum_{I'=(G'\Sigma')\in {\mathcal{I}}_i(I)}\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot \nu^{h(I)-1}+|E(G')|\cdot \mu^{c_g}\cdot \nu^{h(I)-1}\bigg)\\
\leq & \sum_{I'=(G'\Sigma')\in {\mathcal{I}}_i(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot \nu^{h(I)-1}+\sum_{I'=(G'\Sigma')\in {\mathcal{I}}_i(I)}|E(G')|\cdot \mu^{c_g}\cdot \nu^{h(I)-1}\\
\leq & \text{ }\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\nu+|E(G)|\cdot \mu^{c_g} \bigg)\cdot \nu^{h(I)-1} + c_g\cdot |E(G)|\cdot \mu^{c_g}\cdot \nu^{h(I)-1}\\
\leq & \text{ }\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu^{h(I)} + (c_g+1)\cdot |E(G)|\cdot \mu^{c_g}\cdot \nu^{h(I)-1}\\
\leq & \text{ }\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu^{h(I)} + |E(G)|\cdot \mu^{c_g}\cdot \nu^{h(I)}.\\
\end{split}
\]
Let $\phi$ be the solution to instance $I$ obtained by applying Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace to solutions $\phi(I')$ to instances $I'\in {\mathcal{I}}_i(I)$. Then, from \Cref{thm: main}, and since we can assume that $c_g$ is a large enough constant, we are guaranteed that:
\[
\begin{split}
\mathsf{cr}(\phi) &\leq \bigg(\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) \bigg) + (\mathsf{OPT}(I)+|E(G)|)\cdot \mu^{c_g} \\
&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu^{h(I)} + |E(G)|\cdot \mu^{c_g}\cdot \nu^{h(I)}+ (\mathsf{OPT}(I)+|E(G)|)\cdot \mu^{c_g}\\
&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\nu^{h(I)}+\mu^{c_g}) + |E(G)|\cdot \mu^{c_g}\cdot (\nu^{h(I)}+1)
\end{split}
\]
Since $h(I^*)\leq \frac{1}{c^*}\sqrt{\frac{\log m^*}{\log\log m^*}}$, we conclude that the cost of the solution $\phi^*$ that the algorithm computes for instance $I^*$ is bounded by:
\[
\begin{split}
(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m^*)\cdot \mu^{O(1)}\cdot (\log m)^{\sqrt{\log m^*/\log\log m^*}}&
\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m^*)\cdot \mu^{O(1)}\cdot 2^{\sqrt{\log m^*\log\log m^*}}\\ &\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m^*)\cdot \mu^{O(1)}.
\end{split} \]
\f
\iffalse{backup: previous stage 1}
\paragraph{Stage 1.}
We initialize the set ${\mathcal{I}}$ to contain a single instance $I=(G,\Sigma)$. The first stage continues as long as there is any instance $\hat I=(\hat G,\hat \Sigma)\in {\mathcal{I}}$, that is an interesting subinstance of $I$, with $|E(\hat G)|>|E(G)|/\mu$. The first stage consists of iterations, where in every iteration we start with an interesting subinstance $\hat I= (\hat G,\hat \Sigma)\in {\mathcal{I}}$ with $|E(\hat G)|>m/\mu$.
We apply the algorithm from \Cref{lem: many paths} to instance $\hat I$,
obtaining two new instances $\hat I'=(\hat G',\hat \Sigma'),\hat I''=(\hat G'',\hat \Sigma'')$. We then replace $\hat I$ with $\hat I',\hat I''$ in ${\mathcal{I}}$, and continue to the next iteration. Let $i$ denote the index of the current iteration.
We say that the bad event ${\cal{E}}_i$ happens if (i) $|E(\hat G)|>\mu^{39}$; (ii) $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)<|E(\hat G)|^2/\mu^{40}$; and (iii) $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')+\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I'') > \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I) \cdot(1+1/\mu^{12})+m \cdot \mu^{c}$. From \Cref{lem: many paths}, the probability that ${\cal{E}}_i$ happens is bounded by $1/(40\mu^{11})$.
This completes the description of the algorithm for Stage 1. In order to analyze it, it is convenient to define a partitioning tree $T$ associated with it.
For every instance $\hat I$ that ever belonged to set ${\mathcal{I}}$ over the course of the Stage 1 algorithm, we add a vertex $v(\hat I)$ to the tree. If the algorithm from \Cref{lem: many paths} was applied to instance $\hat I$, producing two new instances $\hat I'$ and $\hat I''$, then vertices $v(\hat I'),v(\hat I'')$ become children of vertex $v(\hat I)$ in the tree. The root of the tree is the vertex $v(I)$.
We say that bad event ${\cal{E}}$ happens if bad event ${\cal{E}}_i$ happens for any iteration $i\leq 4\mu^{11}$ of Stage 1. From the union bound over all $1\le i\leq 4\mu^{11}$, we get that $\Pr[{\cal{E}}]\le 1/10$.
Let $T'$ be the subtree of $T$ that is induced by all vertices $v(\hat I)$, such that $\hat I$ belonged to ${\mathcal{I}}$ at some point before the completion of iteration $\ceil{2\mu^{11}}$.
Assume now that bad event $\hat {\cal{E}}$ did not happen. Since we only apply the algorithm from \Cref{lem: many paths} to instances $(\hat G,\hat \Sigma)$ with $|E(\hat G)|>|E(G)|/\mu$, every instance $\hat I'=(\hat G',\hat \Sigma')$ with $v(I')\in T'$ has $|E(\hat G')|\geq |E(G)|/\mu^{11}$. But then the number of the leaf vertices in tree $T'$ is bounded by $\mu^{11}$, and the total number of vertices in tree $T'$ must be bounded by $2\mu^{11}$, since every inner vertex of $T'$ has at least two children.
This can only happen if the algorithm terminates before the $\floor{2\mu^{11}}$-th iteration. Therefore, if Event ${\cal{E}}$ does not happen, the algorithm has fewer than $2\mu^{11}$ iterations.
Moreover, since every instance $\hat I=(\hat G,\hat \Sigma)$ that ever belonged to ${\mathcal{I}}$, $|E(\hat G)|\geq |E(G)|/\mu^{11}$ holds, and since we are guaranteed that $|E(G)|\geq \mu^{c'}\geq \mu^{50}$, we conclude that $|E(\hat G)|\geq \mu^{39}$ must hold.
We use the following claim.
\begin{claim}\label{claim: few crossings in each instance}
If event ${\cal{E}}$ did not happen, then for every instance $\hat I=(\hat G,\hat \Sigma)$ that ever belonged to the set ${\mathcal{I}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq |E(\hat G)|^2/\mu^{40}$.
\end{claim}
\begin{proof}
We say that a vertex $v(\hat I)$ lies at level $\ell$ of the tree $T$ iff the distance from $v(\hat I)$ to the root $v(I)$ of the tree is $\ell$. We will prove that, for all $0\leq \ell \leq \floor{2\mu^{11}}$, for every vertex $v(\hat I)$ lying at level $\ell$ of the tree $T$, with $\hat I=(\hat G,\hat\Sigma)$,
\begin{equation}
\label{eqn: induction}
\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell}+m\cdot\ell\cdot\mu^{c+13},
\end{equation}
The proof is by induction on $\ell$.
The base case is when $\ell=0$, and $\hat I$ is the input instance $I=(G,\Sigma)$. Recall that, from the statement of \Cref{thm: main}, $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\leq m^2/\mu^{63}\leq m^2/\mu^{40}$ as required, and it is immediate to verify that the inequality \ref{eqn: induction} holds.
Consider now some integer $0\leq \ell \leq \floor{2\mu^{11}}$. We assume that the claim holds for all integers below $\ell$, and prove it for integer $\ell$. Let $\hat I'=(\hat G',\hat \Sigma')$ be any instance, whose corresponding vertex $v(\hat I')$ lies at level $\ell$ of the tree $T$, and let $v(\hat I)$ be the parent vertex of $v(\hat I')$ in the tree, where $\hat I=(\hat G,\hat \Sigma)$. As observed already, if Event ${\cal{E}}$ does not happen, $|E(\hat G)|\geq \mu^{39}$ must hold, and from the induction hypothesis, $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell-1}+m\cdot(\ell-1)\cdot\mu^{c+13}$ and $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq |E(\hat G)|^2/\mu^{40}$ must hold. If we denote by $v(\hat I'')$ the second child vertex of $v(\hat I)$, then, since we have assumed that Event $\hat {\cal{E}}$ does not happen, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')+\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I) \cdot(1+1/\mu^{12})+m\cdot\mu^{c}$ must hold. Therefore,
\[
\begin{split}
\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq & \text{ } \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell-1}+m\cdot(\ell-1)\cdot\mu^{c+13}\bigg) \cdot(1+1/\mu^{12})+m\cdot\mu^{c}\\
\leq & \text{ } \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell}+\bigg(m\cdot(\ell-1)\cdot\mu^{c+13}\bigg) \cdot(1+1/\mu^{12})+m\cdot\mu^{c}\\
\leq & \text{ } \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell}+ m\cdot\ell\cdot\mu^{c+13}.
\end{split}
\]
Lastly, as observed already, $|E(\hat G')|\geq m/\mu^{11}$.
We then get that:
%
\[
\begin{split}
\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell}+ m\cdot\ell\cdot\mu^{c+13}\\
&\leq 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ 2 m\cdot\mu^{c+24}\\
&\leq \frac{4m^2}{\mu^{63}} \leq \frac{4\cdot |E(\hat G')|^2\cdot \mu^{22}}{\mu^{63}}\leq \frac{|E(\hat G')|^2}{\mu^{40}}.
\end{split}
\]
%
Here we have used the fact that $\ell\le \floor{2\mu^{11}}$ and the facts $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{63}$ and $|E(G)|\geq \mu^{c'}$ from the statement of \Cref{thm: main}, together with the assumption that $c'>c+50$.
\end{proof}
\begin{claim}\label{cor: stage 1}
If Event ${\cal{E}}$ did not happen, then $\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ 2\cdot m\cdot \mu^{c+13}$.
\end{claim}
\begin{proof}
For each $1\le i\le \floor{2\mu^{11}}$, we denote by ${\mathcal{I}}'_i$ be the set of subinstances obtained after iteration $i$. Therefore, from \Cref{lem: many paths},
\[
\sum_{I'\in {\mathcal{I}}'_{i+1}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \bigg(\sum_{I'\in {\mathcal{I}}'_{i}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\bigg)\cdot (1+1/\mu^{12})+m\cdot\mu^{c},
\]
and so
\[
\sum_{I'\in {\mathcal{I}}'_{i+1}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot \mu^{c+13}\leq \bigg(\sum_{I'\in {\mathcal{I}}'_{i}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot\mu^{c+13}\bigg)\cdot (1+1/\mu^{12}),
\]
Therefore,
\[
\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c+13}\bigg)\cdot (1+1/\mu^{12})^{\floor{2\mu^{11}}}\le 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ 2\cdot m\cdot \mu^{c+13}.
\]
\end{proof}
We partition the instances of ${\mathcal{I}}'$ into two subsets: set ${\mathcal{I}}_1'$ contains all instances $I'=(G',\Sigma')$ with $|E(G')|\leq m/\mu$, while set ${\mathcal{I}}_2'$ contains all remaining instances. Notice that by the definition of the Stage 1 algorithm, every instance in ${\mathcal{I}}_2'$ is a non-interesting subinstance of the input intance $I$.
\fi
\section{An Algorithm for \ensuremath{\mathsf{MCNwRS}}\xspace -- Proof of \Cref{thm: main_rotation_system}}
\label{sec: high level}
In this section we provide the proof of Theorem \ref{thm: main_rotation_system}, with some of the details deferred to subsequent sections.
Throughout the paper, we denote by $I^*=(G^*,\Sigma^*)$ the input instance of the \textnormal{\textsf{MCNwRS}}\xspace problem, and we denote $m^*=|E(G^*)|$. We also use the following parameter that is central to our algorithm: ${\mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}}$,
where $c^*$ is a large enough constant.
As mentioned already, our algorithm for solving the \textnormal{\textsf{MCNwRS}}\xspace problem is recursive, and, over the course of the recursion, we will consider various other instances $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace. Throughout the algorithm, parameters $m^*$ and $\mu$ remain unchanged, and are defined with respect to the original input instance $I^*$.
The main technical ingredient of the proof is the following theorem.
\begin{theorem}
\label{thm: main}
There is a constant $c''$, and an efficient randomized algorithm, that, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace with $m=|E(G)|$, such that $\mu^{c''}\leq m\leq m^*$, either returns FAIL, or computes a collection ${\mathcal{I}}$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace with the following properties:
\begin{itemize}
\item for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\leq m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\cdot (\log m)^{O(1)}$;
\item there is an efficient algorithm called \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, that, given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$; and
\item if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c''}$, then with probability at least $15/16$, all of the following hold:
\begin{itemize}
\item the algorithm does not return FAIL;
\item ${\mathcal{I}}\neq \emptyset$;
\item $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot 2^{O((\log m)^{3/4}\log\log m)}$; and
\item if algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace is given as input a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}$, then the resulting solution $\phi$ to instance $I$ that it computes has cost at most: $$O\bigg(\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\bigg) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{O(1)}.$$
\end{itemize}
\end{itemize}
\end{theorem}
\iffalse
We now define collections of subinstances that have some desirable properties. The following definition is similar in spirit to the definition of a $\nu$-decomposition of an instance, but, since it uses slightly different parameters, we distinguish between these two definitions.
\begin{definition}[Good and Perfect Collection of Subinstances]
Let $I=(G,\Sigma)$ be a subinstance of instance $I^*$, with $|E(G)|=m$, and let ${\mathcal{I}}$ be a collection of subinstances of $I$. We say that ${\mathcal{I}}$ is a \emph{good collection of subinstances} for $I$ if the following hold.
\begin{itemize}
\item for each subinstance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\leq m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\cdot (\log m)^{c_g}$, where $1000<c_g<c^*$ is some large enough universal constant whose value will be set later; and
\item there is an efficient algorithm, that we call \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, that, given a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, of cost at most $c_g\cdot(\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{c_g}$.
\end{itemize}
We say that ${\mathcal{I}}$ is a \emph{perfect} collection of subinstances for $I$ if, additionally:
$$\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{O((\log m)^{3/4}\log\log m)} +m\cdot\mu^{c_g}.$$
\end{definition}
The main technical ingredient of the proof is the following theorem.
\begin{theorem}
\label{thm: main}
There is an efficient randomized algorithm, whose input is a subinstance $I=(G,\Sigma)$ of $I^*$, with $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$ and $|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$. The algorithm outputs a good collection ${\mathcal{I}}$ of subinstances of $I$, and, with probability at least $7/8$, ${\mathcal{I}}$ is a perfect collection of subinstances for $I$.
\end{theorem}
\fi
The remainder of this paper is dedicated to the proof of \Cref{thm: main}. In the following subsection, we complete the proof of \Cref{thm: main_rotation_system} using \Cref{thm: main}.
\subsection{Proof of \Cref{thm: main_rotation_system}}
Throughout the proof, we assume that $m^*$ is larger than a sufficiently large constant, since otherwise we can return a trivial solution to instance $I^*$, from \Cref{thm: crwrs_uncrossing}.
We let $c_g>100$ be a large enough constant, so that, for example, when the algorithm from \Cref{thm: main} is applied to an instance $I=(G,\Sigma)$ with $m=|E(G)|$, such that $\mu^{c''}\leq m\leq m^*$ holds, it is guaranteed to return a family ${\mathcal{I}}$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace, with $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\cdot (\log m)^{c_g}$.
We say that the algorithm from \Cref{thm: main} is \emph{successful} if all of the following hold:
\begin{itemize}
\item the algorithm does not return FAIL;
\item if ${\mathcal{I}}$ is the collection of instances returned by the algorithm, then ${\mathcal{I}}\neq \emptyset$;
\item $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot 2^{c_g((\log m)^{3/4}\log\log m)}$; and
\item if algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace is given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}$, then it computes a solution $\phi$ to instance $I$, of cost at most $c_g\cdot (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{c_g}$.
\end{itemize}
By letting $c_g$ be a large enough constant, \Cref{thm: main} guarantees that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c''}$, then with probability at least $15/16$ the algorithm is successful.
We assume that the parameter $c^*$ in the definition of $\mu$ is sufficiently large, so that, e.g., $c^*>2c_g$.
We use a simple recursive algorithm called \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace, that appears in Figure \ref{fig: algrec}.
\program{\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace}{fig: algrec}{
\noindent{\bf Input:} an instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, with $|E(G)|\leq m^*$.
\noindent{\bf Output:} a feasible solution to instance $I$.
\begin{enumerate}
\item Use the algorithm from \Cref{thm: crwrs_planar} to determine whether $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$. If so, use the algorithm from \Cref{thm: crwrs_planar} to compute a solution to $I$ of cost $0$. Return this solution, and terminate the algorithm.
\item Use the algorithm from Theorem~\ref{thm: crwrs_uncrossing} to compute a trivial solution $\phi'$ to instance $I$.
\item If $|E(G)|\leq \mu^{c''}$, return the trivial solution $\phi'$ and terminate the algorithm.
\item For $1\leq j\leq \ceil{\log m^*}$:
\begin{enumerate}
\item Apply the algorithm from \Cref{thm: main} to instance $I$.
\item If the algorithm returns FAIL, let $\phi_j=\phi'$ be the trivial solution to instance $I$, and set ${\mathcal{I}}_j(I)=\emptyset$.
\item Otherwise:
\begin{enumerate}
\item Let ${\mathcal{I}}_j(I)$ be the collection of instances computed by the algorithm.
\item For every instance $I'\in {\mathcal{I}}_j(I)$, apply Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace to instance $I'$, to obtain a solution $\phi(I')$ to this instance.
\item Apply Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace from \Cref{thm: main} to solutions $\set{\phi(I')}_{I'\in {\mathcal{I}}_j(I)}$, to obtain a solution $\phi_j$ to instance $I$.
\end{enumerate}
\end{enumerate}
Return a solution to instance $I$ from among $\set{\phi',\phi_1,\ldots,\phi_{\ceil{\log m^*}}}$ that has fewest crossings.
\end{enumerate}
}
In order to analyze the algorithm, it is convenient to associate a \emph{partitioning tree} $T$ with it, whose vertices correspond to all instances of \ensuremath{\mathsf{MCNwRS}}\xspace considered over the course of the algorithm. Let $L=\ceil{\log m^*}$. We start with the tree $T$ containing a single root vertex $v(I^*)$, representing the input instance $I^*$. Consider now some vertex $v(I)$ of the tree, representing some instance $I=(G,\Sigma)$.
When Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace was applied to instance $I$, if it did not terminate after the first three steps, it constructed $L$ collections ${\mathcal{I}}_1(I),\ldots,{\mathcal{I}}_L(I)$ of instances (some of which may be empty, in case the algorithm from \Cref{thm: main} returned FAIL in the corresponding iteration). For each such instance $I'\in \bigcup_{j=1}^L{\mathcal{I}}_j(I)$, we add a vertex $v(I')$ representing instance $I'$ to $T$, that becomes a child vertex of $v(I)$. This concludes the description of the partitioning tree $T$.
We denote by ${\mathcal{I}}^*=\set{I\mid v(I)\in V(T)}$ the set of all instances of \ensuremath{\mathsf{MCNwRS}}\xspace, whose corresponding vertex appears in the tree $T$. For each such instance $I\in {\mathcal{I}}^*$, its \emph{recursive level} is the distance from vertex $v(I)$ to the root vertex $v(I^*)$ in the tree $T$ (so the recursive level of $v(I^*)$ is $0$).
For $j\geq 0$, we denote by $\hat {\mathcal{I}}_j\subseteq {\mathcal{I}}^*$ the set of all instances $I\in {\mathcal{I}}^*$, whose recursive level is $j$.
Lastly, the \emph{depth} of the tree $T$, denoted by $\mathsf{dep}(T)$, is the largest recursive level of any instance in ${\mathcal{I}}^*$.
In order to analyze the algorithm, we start with the following two simple observations.
\begin{observation}\label{obs: few recursive levels}
$\mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$.
\end{observation}
\begin{proof}
Consider any non-root vertex $v(I)$ in the tree $T$, and let $v(I')$ be the parent-vertex of $v(I)$. Denote $I=(G,\Sigma)$ and $I'=(G',\Sigma')$.
From the construction of tree $T$, instance $I$ belongs to some collection of instances obtained by applying the algorithm from \Cref{thm: main} to instance $I'$. Therefore, from \Cref{thm: main}, $|E(G)|\leq |E(G')|/\mu$ must hold. Therefore, for all $j\geq 0$, for every instance $I=(G,\Sigma)\in \hat {\mathcal{I}}_j$, $|E(G)|\leq m^*/\mu^j$. Since $\mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}$, we get that $\mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$.
\end{proof}
\begin{observation}\label{obs: num of edges}
$\sum_{I=(G,\Sigma)\in {\mathcal{I}}^*}|E(G)|\le m^*\cdot 2^{(\log m^*)^{1/8}}$.
\end{observation}
\begin{proof}
Consider any non-leaf vertex $v(I)$ of the tree $T$, and denote $I=(G,\Sigma)$.
Recall that, when Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace is applied to instance $I$, it uses the algorithm from \Cref{thm: main} to compute $L$ collections ${\mathcal{I}}_1(I),\ldots,{\mathcal{I}}_L(I)$ of instances, such that, if we denote $|E(G)|=m$, then, for all $1\leq j\leq L$:
%
$$\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_j(I)}|E(G')|\leq m\cdot (\log m)^{c_g}\leq m\cdot (\log m^*)^{c_g}$$
%
(since $m\leq m^*$ must hold). Since $L\leq 2\log m^*$, and $m^*$ is sufficiently large, we get that:
%
$$\sum_{j=1}^L\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_j(I)}|E(G')|\leq m\cdot (\log m^*)^{c_g+2}.$$
%
For all $j\geq 0$, we denote by $N_j$ the total number of edges in all instances in set $\hat {\mathcal{I}}_j$, $N_j=\sum_{I=(G,\Sigma)\in \hat {\mathcal{I}}_j}|E(G)|$. Clearly, $N_0=m^*$, and, from the above discussion, for all $j>0$, $N_j\leq N_{j-1}\cdot (\log m^*)^{c_g+2}$.
Since $\mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$,
we conclude that:
%
\[
\sum_{I=(G,\Sigma)\in {\mathcal{I}}^*}|E(G)|\leq m^*\cdot (\log m^*)^{2c_g\cdot (\log m^*)^{1/8}/(c^*\log\log m^*)}\leq m^*\cdot 2^{(\log m^*)^{1/8}},
\]
%
since $c_g\leq c^*/2$.
\end{proof}
We use the following corollary, that follows immediately from \Cref{obs: num of edges}.
\begin{corollary}\label{cor: num of instances}
The number of instances $I=(G,\Sigma)\in {\mathcal{I}}^*$ with $|E(G)|\geq \mu^{c''}$ is at most $m^*$.
\end{corollary}
We say that an instance $I\in {\mathcal{I}}^*$ is a \emph{leaf instance}, if vertex $v(I)$ is a leaf vertex of the tree $T$, and we say that it is a non-leaf instance otherwise.
Consider now a non-leaf instance $I=(G,\Sigma)\in {\mathcal{I}}^*$. We say that a bad event ${\cal{E}}(I)$ happens, if $0<\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c''}$, and, for all $1\leq j\leq L$, the $j$th application of the algorithm from \Cref{thm: main} to instance $I$ was unsuccessful.
Clearly, from \Cref{thm: main}, $\prob{{\cal{E}}(I)}\le (1/16)^L\leq 1/(m^*)^4$.
Let ${\cal{E}}$ be the bad event that event ${\cal{E}}(I)$ happened for any instance $I\in {\mathcal{I}}^*$. From the Union Bound and \Cref{cor: num of instances}, we get that $\prob{{\cal{E}}}\leq 1/(m^*)^2$.
We use the following immediate observation.
\begin{observation}\label{obs: leaf}
If Event ${\cal{E}}$ does not happen, then for every leaf vertex $v(I)$ of $T$ with $I=(G,\Sigma)$, either $|E(G)|\leq \mu^{c''}$; or $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$; or $\mathsf{OPT}_{\mathsf{cnwrs}}(I)> |E(G)|^2/\mu^{c''}$.
\end{observation}
We use the following lemma to complete the proof of \Cref{thm: main_rotation_system}.
\begin{lemma}\label{lem: solution cost}
If Event ${\cal{E}}$ does not happen, then Algorithm $\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace$ computes a solution for instance $I^*=(G^*,\Sigma^*)$ of cost at most $2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right})$.
\end{lemma}
\begin{proof}
Consider a non-leaf instance $I=(G,\Sigma)$, and let ${\mathcal{I}}_1(I),\ldots,{\mathcal{I}}_L(I)$ be families of instances of \ensuremath{\mathsf{MCNwRS}}\xspace that Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace computed, when applied to instance $I$. Recall that, for each $1\le j\le L$ with ${\mathcal{I}}_j(I)\neq \emptyset$, the algorithm computes a solution $\phi_j$ to instance $I$, by first solving each of the instances in ${\mathcal{I}}_j(I)$ recursively, and then combining the resulting solutions using Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace. Eventually, the algorithm returns the best solution of $\set{\phi',\phi_1,\ldots,\phi_L}$, where $\phi'$ is the trivial solution, whose cost is at most $|E(G)|^2$. We fix an arbitrary index $1\leq j\leq L$, such that the $j$th application of the algorithm from \Cref{thm: main} to instance $I$ was successful. Note that the cost of the solution to instance $I$ that the algorithm returns is at most $\mathsf{cr}(\phi_j)$. We then \emph{mark} the vertices of $\set{v(I')\mid I'\in {\mathcal{I}}_j(I)}$ in the tree $T$. We also mark the root vertex of the tree.
Let $T^*$ be the subgraph of $T$ induced by all marked vertices. It is easy to verify that $T^*$ is a tree, and moreover, if ${\cal{E}}$ did not happen, every leaf vertex of $T^*$ is also a leaf vertex of $T$. For a vertex $v(I)\in V(T^*)$, we denote by $h(I)$ the length of the longest path in tree $T^*$, connecting vertex $v(I)$ to any of its descendants in the tree.
We use the following claim, whose proof is straightforward conceptually but somewhat technical; we defer the proof to \Cref{Appx: inductive bound proof}.
\begin{claim}\label{claim: bound by level}
Assume that Event ${\cal{E}}$ did not happen. Then there is a fixed constant $\tilde c\geq \max\set {c'',c_g,c^*}$, such that, for every vertex $v(I)\in V(T^*)$, whose corresponding instance is denoted by $I=(G,\Sigma)$, the cost of the solution that the algorithm computes for $I$ is at most:
$$2^{\tilde c\cdot h(I)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c''\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+(\log m^*)^{4c_g h(I)}\mu^{2c''\cdot \tilde c}\cdot|E(G)|.$$
\end{claim}
We are now ready to complete the proof of \Cref{lem: solution cost}. Recall that $h(I^*)=\mathsf{dep}(T^*)\leq \mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$ from \Cref{obs: few recursive levels}. Therefore, from \Cref{claim: bound by level}, the cost of the solution that the algorithm computes for instance $I^*$ is bounded by:
%
\[
\begin{split}
& 2^{O(\mathsf{dep}(T))\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{O(1)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+(\log m^*)^{O(\mathsf{dep}(T))}\cdot\mu^{O(1)}\cdot m^*\\
& \quad \quad \quad \quad \quad \quad \leq2^{O((\log m^*)^{7/8})}\cdot \mu^{O(1)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+(\log m^*)^{O((\log m^*)^{1/8}/\log\log m^*) }\cdot\mu^{O(1)}\cdot m^*\\
&\quad \quad \quad \quad \quad \quad \leq 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}),
\end{split}
\]
%
since $\mu=2^{O((\log m^*)^{7/8}\log\log m^*)}$.
\end{proof}
In order to complete the proof of \Cref{thm: main_rotation_system}, it is now enough to prove Theorem~\ref{thm: main}. The remainder of the paper is dedicated to the proof of \Cref{thm: main}.
\subsection{Proof of Theorem~\ref{thm: main} -- Main Definitions and Theorems}
We classify instances of \ensuremath{\mathsf{MCNwRS}}\xspace into \emph{wide} and \emph{narrow}. Wide instances are, in turn, classified into \emph{well-connected} and not well-connected instances. We then provide different algorithms for decomposing instances of each of the resulting three kinds.
We use the following notion of a high-degree vertex.
\begin{definition}[High-degree vertex]
Let $G$ be any graph. A vertex $v\in V(G)$ is a \emph{high-degree} vertex, if $\deg_G(v)\geq |E(G)|/\mu^4$.
\end{definition}
We are now ready to define wide and narrow instances.
\begin{definition}[Wide and Narrow Instances]
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$. We say that $I$ is a \emph{wide} instance, if there is a high-degree vertex $v\in V(G)$, a partition $(E_1,E_2)$ of the edges of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and there is a collection ${\mathcal{P}}$ of at least $\floor{m/\mu^{{50}}}$ simple edge-disjoint cycles in $G$, such that every cycle $P\in {\mathcal{P}}$ contains one edge of $E_1$ and one edge of $E_2$.
An instance that is not wide is called \emph{narrow}.
\end{definition}
Note that there is an efficient algorithm to check whether a given instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace is wide, and, if so, to compute the corresponding set ${\mathcal{P}}$ of cycles, via standard algorithms for maximum flow. (For every vertex $v\in V(G)$, we try all possible partitions $(E_1,E_2)$ of $\delta_G(v)$ with the required properties, as the number of such partitions is bounded by $|\delta_G(v)|^2$.)
We will use the following simple observation regarding narrow instances.
\begin{observation}\label{obs: narrow prop 2}
If an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace is narrow, then for every pair $u,v$ of distinct high-degree vertices of $G$, and any set ${\mathcal{P}}$ of edge-disjoint paths connecting $u$ to $v$ in $G$, $|{\mathcal{P}}|\leq 2\ceil{|E(G)|/\mu^{{50}}}$ must hold.
\end{observation}
\begin{proof}
Assume for contradiction that $I=(G,\Sigma)$ is a narrow instance of \ensuremath{\mathsf{MCNwRS}}\xspace, with $|E(G)|=m$, and that there are two high-degree vertices $u,v$ of $G$, and a set ${\mathcal{P}}$ of more than $2\ceil{m/\mu^{{50}}}$ edge-disjoint paths in $G$ connecting $u$ to $v$. We denote $|{\mathcal{P}}|=k$.
Let $E'\subseteq \delta_G(v)$ be the set of all edges $e\in \delta_G(v)$, such that $e$ is the first edge on some path in ${\mathcal{P}}$. We denote $E'=\set{e_1,\ldots,e_k}$, where the edges are indexed according to their ordering in the rotation ${\mathcal{O}}_v\in \Sigma$. We also denote ${\mathcal{P}}=\set{P(e_i)\mid 1\leq i\leq k}$, where path $P(e_i)$ contains the edge $e_i$ as its first edge. We can then compute a partition $(E_1,E_2)$ of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$. Additionally, we can ensure that $e_1,\ldots,e_{\ceil{k/2}}\in E_1$, while the remaining edges of $E'$ lie in $E_2$. For each $1\leq i\leq \ceil{m/\mu^{{50}}}$, we let $Q_i$ be the cycle obtained by concatenating the paths $P(e_i)$ and $P(e_{k-i+1})$. We turn $Q_i$ into a simple cycle, by removing from it all cycles that are disjoint from vertex $v$. It is then immediate to verify that the set $\set{Q_i\mid 1\leq i\leq \ceil{m/\mu^{{50}}}}$ of cycles has all the required properties to establish that instance $I$ is wide, a contradiction.
\end{proof}
Next, we define well-connected wide instances.
\begin{definition}[Well-Connected Wide Instances]
Let $I=(G,\Sigma)$ be a wide instance of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$. We say that it is a \emph{well-connected} instance iff for every pair $u,v$ of distinct vertices of $G$ with $\deg_G(v),\deg_G(u)\geq m/\mu^5$, there is a collection of at least $\frac{8m}{\mu^{{50}}}$ edge-disjoint paths connecting $u$ to $v$ in $G$.
\end{definition}
The proof of \Cref{thm: main} relies on the following three theorems. The first theorem deals with wide instances that are not necessarily well-connected. Its proof is deferred to \Cref{sec: not well connected}.
\begin{theorem}\label{thm: not well connected}
There is an efficient randomized algorithm, whose input is a wide instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with $m=|E(G)|$, such that $ \mu^{20}\leq m\leq m^*$ holds. The algorithm computes a $\nu$-decomposition ${\mathcal{I}}$ of $I$, for $\nu= 2^{O((\log m)^{3/4}\log\log m)}$, such that every instance $I'=(G',\Sigma')\in {\mathcal{I}}$ is a subinstance of $I$, and one of the following holds for it:
\begin{itemize}
\item either $|E(G')|\le m/\mu$;
\item or $I'$ is a narrow instance;
\item or $I'$ is a wide and well-connected instance.
\end{itemize}
\end{theorem}
The second theorem deals with wide well-connected instances. Its proof appears in \Cref{sec: many paths}.
\begin{theorem}\label{lem: many paths}
There is an efficient randomized algorithm, whose input is a wide and well-connected instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with $m=|E(G)|$, such that $\mu^{c'}\leq m\leq m^*$ holds, for some large enough constant $c'$. The algorithm either returns FAIL, or computes a non-empty collection ${\mathcal{I}}$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace, such that the following hold:
\begin{itemize}
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\le 2|E(G)|$;
\item for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, either $|E(G')|\le m/\mu$, or instance $I'$ narrow;
\item
there is an efficient algorithm called $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace'$, that, given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$; and
\item if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$ then, with probability at least $1-1/\mu^2$, all of the following hold:
\begin{itemize}
\item the algorithm does not return FAIL;
\item $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log m)^{O(1)}$; and
\item if algorithm $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace'$ is given as input a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}$, then the resulting solution $\phi$ to instance $I$ that it computes has cost at most: $\mathsf{cr}(\phi)\leq \sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) + \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\mu^{O(1)}$.
\end{itemize}
\end{itemize}
\end{theorem}
The third theorem deals with narrow instances, and its proof appears in \Cref{sec: computing the decomposition}.
\begin{theorem}\label{lem: not many paths}
There is an efficient randomized algorithm, whose input is a narrow instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with $m=|E(G)|$, such that $ \mu^{50}\leq |E(G)|\leq 2m^*$. The algorithm either returns FAIL, or computes a $\nu$-decomposition ${\mathcal{I}}$ of $I$, for $\nu= 2^{O((\log m)^{3/4}\log\log m)}$, such that, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\le m/(2\mu)$. Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{21}$, then the probability that the algorithm returns FAIL is at most $O(1/\mu^2)$.
\end{theorem}
The majority of the remainder of this paper is dedicated to the proofs of the above three theorems. Before we provide these proofs, we develop central technical tools that they use, in Sections \ref{sec: guiding paths orderings basic disengagement} -- \ref{sec: main disengagement}.
In the remainder of this section, we complete the proof of Theorem \ref{thm: main} using Theorems \ref{thm: not well connected}, \ref{lem: many paths}, and \ref{lem: not many paths}.
\iffalse
\mynote{restate the lemma in terms of $\alpha$-decomposition into subinstances}
\begin{lemma}\label{lem: not many paths}
There is an efficient randomized algorithm, that, given a non-interesting subinstance $I=(G,\Sigma)$ of $I^*$ with $|E(G)|=m\geq \mu^{10}$, either correctly certifies that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \Omega(m^2/\mu^5)$, or computes a collection ${\mathcal{I}}$ of subinstances of $I$, such that
the following hold:
\begin{itemize}
\item for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\le m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq O(m)$; and
\item $\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot 2^{O((\log m)^{3/4}\log\log m)}$.
\end{itemize}
Moreover, there is an efficient algorithm, that, given, for each instance $I'\in {\mathcal{I}}$, a feasible solution $\phi(I')$, computes a feasible solution $\phi$ for instance $I$, of cost $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$.
\end{lemma}
\fi
Recall that we are given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, with $\mu^{c''}\leq |E(G)|\leq m^*$, for some large enough constant $c''$. We assume that $c''>100c'$, where $c'$ is the constant in \Cref{lem: many paths}. We use another large constant $c'_g$, and we assume that $c^*> c'_g >c''$, where $c^*$ is the constant in the definition of the parameter $\mu$. Throughout, we denote $m=|E(G)|$. We compute the desired collection ${\mathcal{I}}^*$ of instances in three steps.
\subsubsection*{Step 1}
Assume first that the input instance $I$ is a wide instance.
We apply the algorithm from \Cref{thm: not well connected} to $I$.
Let $\hat {\mathcal{I}}$ be the resulting collection of instances. We partition the set $\hat{\mathcal{I}}$ of instances into three subsets. The first set, denoted by $\hat {\mathcal{I}}_{\textsf {small}}$, contains all instances $I'=(G',\Sigma')\in \hat {\mathcal{I}}$ with $|E(G')|\le m/\mu$. The second set, denoted by $\hat {\mathcal{I}}^{(n)}_{\textsf {large}}$, contains all narrow instances in $\hat {\mathcal{I}}\setminus \hat {\mathcal{I}}_{\textsf {small}}$. The third set, denoted by $\hat {\mathcal{I}}^{(w)}_{\textsf {large}}$, contains all remaining instances of $\hat {\mathcal{I}}$. From \Cref{thm: not well connected}, every instance in $\hat {\mathcal{I}}^{(w)}_{\textsf {large}}$ is wide and well-connected. Since every instance $I'=(G',\Sigma')\in \hat {\mathcal{I}}$ is a subinstance of $I$, $|E(G')|\leq |E(G)|\leq m^*$ must hold.
Recall that, from \Cref{thm: not well connected}, $\hat {\mathcal{I}}$ is a $\nu_1$-decomposition for $I$, for
$\nu_1= 2^{O((\log m)^{3/4}\log\log m)}$. Therefore:
\begin{equation}
\sum_{I'=(G',\Sigma')\in \hat {\mathcal{I}}}|E(G')|\le m\cdot (\log m)^{c'_g},\label{eq: num of edges step 1}
\end{equation}
and
$$\expect{\sum_{I'\in \hat{\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \nu_1.$$
\paragraph{Bad Event ${\cal{E}}_1$.}
We say that a bad event ${\cal{E}}_1$ happens if $\sum_{I'\in \hat{\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')> 100\cdot\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \nu_1$. From the Markov Bound, $\prob{{\cal{E}}_1}\le 1/100$.
Note that, if event ${\cal{E}}_1$ did not happen, then for each instance $I'\in \hat{\mathcal{I}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')\le 100\cdot\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \nu_1$.
We need the following simple observation.
\begin{observation}\label{obs: optbound for narrow 1}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c''}$, and that Event ${\cal{E}}_1$ did not happen. Then for every instance $I'=(G', \Sigma')\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}\cup \hat {\mathcal{I}}^{(w)}_{\textsf {large}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq |E(G')|^2/\mu^{c'}$.
\end{observation}
\begin{proof}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\le m^2/\mu^{c''}$, and Event ${\cal{E}}_1$ did not happen, then for every instance $I'\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}\cup \hat {\mathcal{I}}^{(w)}_{\textsf {large}}$:
$$\mathsf{OPT}_{\mathsf{cnwrs}}(I')\le 100\cdot\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \nu_1\le \mu\cdot \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\le \mu\cdot \textsf{left} (\frac{m^2}{\mu^{c''}}+m\textsf{right} )\le \frac{m^2}{\mu^{c'}}$$
(since $c''>100c'$ is a large enough constant and $m\ge \mu^{c''}$).
\end{proof}
Assume now that instance $I$ is a narrow instance. Then we simply set $\hat{\mathcal{I}}=\hat {\mathcal{I}}^{(n)}_{\textsf {large}}=\set{I}$ and $\hat {\mathcal{I}}_{\textsf {small}}=\hat {\mathcal{I}}^{(w)}_{\textsf {large}}=\emptyset$.
This completes the description of the first step.
\subsubsection*{Step 2}
In the second step, we apply the algorithm from \Cref{lem: many paths} to every instance $I'\in\hat {\mathcal{I}}^{(w)}_{\textsf {large}}$.
If the algorithm returns FAIL, then we terminate our algorithm and return FAIL as well. Assume now that the algorithm from \Cref{lem: many paths}, when applied to instance $I'$, did not return FAIL. We let $\tilde{\mathcal{I}}(I')$ be the collection of instances that the algorithm computes.
Recall that we are guaranteed that, for each instance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde{\mathcal{I}}(I')$, either $\tilde I$ is a narrow instance, or $|E(\tilde G)|\le \frac{|E(G')|}{\mu}\le \frac m {\mu}$ (we have used the fact that $|E(G')|\le m$, since $I'=(G',\Sigma')$ is a subinstance of $I$).
Additionally, we are guaranteed that:
\begin{equation}
\sum_{\tilde I=(\tilde G, \tilde\Sigma)\in \tilde{\mathcal{I}}(I')}|E(\tilde G)|\le 2|E(G')|.\label{ineq: edges step 2}
\end{equation}
In particular, for every instance $\tilde I=(\tilde G, \tilde\Sigma)\in \tilde{\mathcal{I}}(I')$, $|E(\tilde G)|\leq 2|E(G')|\leq 2m\leq 2m^*$.
We say that the application of the algorithm from \Cref{lem: many paths} to an instance $I'=(G', \Sigma')\in \hat {\mathcal{I}}^{(w)}_{\textsf {large}}$ is \emph{successful}, if (i) the algorithm does not return FAIL; (ii)
$\sum_{\tilde I\in \tilde{\mathcal{I}}(I')}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot(\log m)^{c'_g}$; and (iii) there is an efficient algorithm $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace'$, that, given a solution $\phi(\tilde I)$ to every instance $\tilde I\in \tilde {\mathcal{I}}(I')$, computes a solution $\phi(I')$ to instance $I'$ with
$\mathsf{cr}(\phi(I'))\leq \sum_{\tilde I \in \tilde {\mathcal{I}}(I')}\mathsf{cr}(\phi(\tilde I)) + \mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot\mu^{c'_g}$.
\paragraph{Bad Event ${\cal{E}}_2$.}
For an instance $I'=(G', \Sigma')\in \hat {\mathcal{I}}^{(w)}_{\textsf {large}}$, we say that a bad event ${\cal{E}}_2(I')$ happens if the algorithm from \Cref{lem: many paths}, when applied to instance $I'$, was not successful. From \Cref{lem: many paths} and \Cref{obs: optbound for narrow 1}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c''}$, then $\prob{{\cal{E}}_2(I')\mid\neg{\cal{E}}_1}\leq 1/\mu^2$ (since we can assume that $c'_g$ is a large enough constant).
We let ${\cal{E}}_2$ be the bad event that at least of the events in $\set{{\cal{E}}_2(I')\mid I'\in \hat {\mathcal{I}}^{(w)}_{\textsf {large}}}$ happened.
Recall that, from the definition of the set ${\mathcal{I}}^{(w)}_{\textsf {large}}$ of instances, for every instance $I'=(G',\Sigma')\in \hat {\mathcal{I}}^{(w)}_{\textsf {large}}$, $|E(G')|\ge \frac{m}{\mu}$ holds. On the other hand, from Equation \ref{eq: num of edges step 1},
$$ \sum_{I'=(G',\Sigma')\in \hat {\mathcal{I}}^{(w)}_{\textsf {large}}}|E(G')|\le \sum_{I'=(G',\Sigma')\in \hat {\mathcal{I}}}|E(G')|\le m\cdot (\log m)^{c'_g}.$$
Therefore, $|\hat {\mathcal{I}}^{(w)}_{\textsf {large}}|\le \mu\cdot (\log m)^{c'_g}$. From the Union Bound, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq \frac{m^2}{\mu^{c''}}$, then $\prob{{\cal{E}}_2\mid\neg{\cal{E}}_1}\le \frac{\mu\cdot (\log m)^{c'_g}}{\mu^2}\le \frac 1 {100}$.
Let $\tilde {\mathcal{I}}=\bigcup_{I'\in \hat {\mathcal{I}}^{(w)}_{\textsf {large}}}\tilde{\mathcal{I}}(I')$.
Note that, from Inequalities \ref{eq: num of edges step 1} and \ref{ineq: edges step 2}, we get that:
\begin{equation}\label{ineq: total edges step 2}
\sum_{\tilde I=(\tilde G,\tilde \Sigma)\in \tilde{\mathcal{I}}}|E(\tilde G)|\leq 2m\cdot (\log m)^{c'_g}.
\end{equation}
We partition the instances in set $\tilde {\mathcal{I}}$ into two subsets: set $\tilde{{\mathcal{I}}}_{\textsf {small}}$, containing all instances $\tilde I=(\tilde G,\tilde \Sigma)$ in $\tilde{\mathcal{I}}$ with $|E(\tilde G)|\le m/\mu$, and set $\tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$, containing all remaining instances. From \Cref{lem: many paths}, every instance $\tilde I\in \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$ is narrow. This completes the description of the second step.
\subsection*{Step 3}
We focus on four sets of instances that we have constructed so far: $\hat{{\mathcal{I}}}_{\textsf {small}}, \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}, \tilde{{\mathcal{I}}}_{\textsf {small}}, \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$. Recall that, if instance $I'=(G',\Sigma')$ belongs to set $\hat{{\mathcal{I}}}_{\textsf {small}}\cup \tilde{{\mathcal{I}}}_{\textsf {small}}$, then $|E(G')|\leq m/\mu$. If instance $I'=(G',\Sigma')$ belongs to set $\hat{{\mathcal{I}}}^{(n)}_{\textsf {large}},\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$, then $m/\mu< |E(G')|\leq 2m$, and instance $I'$ is narrow.
We use the following simple observation.
\begin{observation}\label{obs: optbound for narrow}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c''}$, and neither of the events ${\cal{E}}_1,{\cal{E}}_2$ happened, then for every instance $I'=(G',\Sigma')\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')< |E(G')|^2/\mu^{21}$.
\end{observation}
\begin{proof}
From \Cref{obs: optbound for narrow 1}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c''}$, and the bad event ${\cal{E}}_1$ did not happen, then for every instance $I'=(G', \Sigma')\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}\cup \hat {\mathcal{I}}^{(w)}_{\textsf {large}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq |E(G')|^2/\mu^{c'}$.
Consider now some instance $I'=(G',\Sigma')\in \hat{{\mathcal{I}}}^{(w)}_{\textsf {large}}$. If, additionally, event ${\cal{E}}_2$ did not happen, then:
$$\sum_{\tilde I\in \tilde{\mathcal{I}}(I')}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)\le \mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot (\log m)^{c'_g}.$$
Therefore, for every instance $\tilde I\in \tilde{\mathcal{I}}(I')$:
$$\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)\le \mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot (\log m)^{c'_g}\le \frac{|E(G')|^2}{\mu^{c'}}\cdot (\log m)^{c'_g}\le \frac{m^2}{\mu^{c'-1}}.$$
We conclude that for every instance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)\leq \frac{m^2}{\mu^{c'-1}}$. Since, from the definition of the set $ \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$ of instances, $|E(\tilde G)|\geq \frac m{\mu}$, we get that:
$$\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)\leq \frac{m^2}{\mu^{c'-1}}< \frac{|E(\tilde G)|^2}{\mu^{21}},$$
assuming that $c'$ is a large enough constant.
\end{proof}
Next, we process every instance $I'\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$ one by one. Notice that for each such instance $I'=(G',\Sigma')$, $|E(G')|\geq m/\mu\geq \mu^{50}$ must hold, since $m\geq \mu^{c''}$. Additionally, as observed already, $|E(G')|\leq 2m\leq 2m^*$. When instance $I'=(G',\Sigma')$ is processed, we apply the algorithm from \Cref{lem: not many paths} to it.
If the algorithm returns FAIL, then we terminate the algorithm and return FAIL as well. Otherwise, we obtain a collection $\overline{\mathcal{I}}(I')$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace. From \Cref{lem: not many paths}, for every instance $I''=(G'',\Sigma'')\in \overline{\mathcal{I}}(I')$, $|E(G'')|\leq \frac{|E(G')|}{2\mu} \leq \frac m {\mu}$. Moreover, from the definition of a $\nu$-decomposition of an instance, and from the fact that $|E(G')|\leq 2m$, we get that:
\begin{equation}\label{eq: few edges 3}
\sum_{I''=(G'',\Sigma'')\in \overline {\mathcal{I}}(I')}|E(G'')|\leq |E(G')|\cdot (\log m)^{c'_g}.
\end{equation}
\paragraph{Bad Events ${\cal{E}}_3$ and ${\cal{E}}$.}
For an instance $I'=(G',\Sigma')\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$, we say that the bad event ${\cal{E}}_3(I')$ happens if the algorithm from \Cref{lem: not many paths}, when applied to instance $I'$, returns FAIL. From \Cref{lem: not many paths}, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(I')<|E(G')|^2/\mu^{21}$, then the probability that the algorithm returns FAIL is at most $O(1/\mu^2)$. Therefore, from \Cref{obs: optbound for narrow}, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c''}$, then $\prob{{\cal{E}}_3(I')\mid \neg{\cal{E}}_1\wedge\neg{\cal{E}}_2}\leq O(1/\mu^2)$. We let ${\cal{E}}_3$ to be the bad event that ${\cal{E}}_3(I')$ happened for any instance $I'\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$. Recall that, for every instance $I'=(G',\Sigma')\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}$, $|E(G')|\geq \frac m{\mu}$.
On the other hand, from Inequality \ref{eq: num of edges step 1},
$\sum_{I'=(G',\Sigma')\in \hat {\mathcal{I}}^{(n)}_{\textsf {large}}}|E(G')|\le m\cdot (\log m)^{c'_g}$,
and from Inequality \ref{ineq: total edges step 2},
$\sum_{I'=(G',\Sigma')\in \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}}|E(G')|\leq 2m\cdot (\log m)^{c'_g}$.
Therefore, $|\hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}|\leq 3\mu\cdot (\log m)^{c'_g}$. From the Union Bound, assuming that the constant $c^*$ in the definition of the parameter $\mu$ is large enough, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c''}$, then $\prob{{\cal{E}}_3\mid \neg{\cal{E}}_1\wedge\neg{\cal{E}}_2}\leq O\textsf{left} ( \frac{\mu\cdot(\log m)^{c'_g}}{\mu^2}\textsf{right} )\leq \frac 1 {100}$.
Lastly, we define bad event ${\cal{E}}$ to be the event that at least one of the events ${\cal{E}}_1,{\cal{E}}_2,{\cal{E}}_3$ happened.
Note that $\prob{{\cal{E}}}\leq \prob{{\cal{E}}_1}+\prob{{\cal{E}}_2\mid \neg{\cal{E}}_1}+\prob{{\cal{E}}_3\mid\neg {\cal{E}}_1\wedge\neg{\cal{E}}_2}$. Therefore, altogether, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c''}$, then
$\prob[]{{\cal{E}}}\le \frac{3}{100}\leq \frac 1{30}$. Note that, if bad event ${\cal{E}}$ does not happen, then the algorithm does not return FAIL.
If the third step of the algorithm did not terminate with a FAIL, we let $\overline{\mathcal{I}}_{\textsf{small}}= \bigcup_{I'\in \hat{{\mathcal{I}}}^{(n)}_{\textsf {large}}\cup \tilde{{\mathcal{I}}}^{(n)}_{\textsf {large}}} \overline{\mathcal{I}}(I')$.
By combining Equations \ref{eq: num of edges step 1}, \ref{ineq: total edges step 2} and \ref{eq: few edges 3}, we get that:
\begin{equation}\label{eq: few edges 4}
\sum_{I''=(G'',\Sigma'')\in \overline {\mathcal{I}}_{\textsf{small}}}|E(G'')|\leq 3m\cdot (\log m)^{2c'_g}.
\end{equation}
The output of the algorithm is the collection
${\mathcal{I}}^*=\hat{{\mathcal{I}}}_{\textsf {small}}\cup\tilde{{\mathcal{I}}}_{\textsf {small}}
\cup \overline{\mathcal{I}}_{\textsf{small}}$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace. From the above discussion, for every instance $I''=(G'',\Sigma'')\in {\mathcal{I}}^*$, $|E(G'')|\leq m/\mu$. As discussed already, if bad event ${\cal{E}}$ does not happen, then the algorithm does not return FAIL.
From now on we assume that the algorithm did not return FAIL.
From Inequalities \ref{eq: num of edges step 1}, \ref{ineq: total edges step 2} and \ref{eq: few edges 4}, we get that:
$$\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}^*}|E(G'')|\leq 6m\cdot(\log m)^{2c'_g}.$$
Next, we provide Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace in the following claim, whose proof is conceptually straightforward but somewhat technical, and is deferred to Section \ref{Appx: Proof of combine drawings} of Appendix.
\begin{claim}\label{claim: combine drawings}
There is an efficient algorithm, called \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, that, given a solution $\phi(I'')$ to every instance $I''\in {\mathcal{I}}^*$, computes a solution $\phi(I)$ to instance $I$.
Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\le m^2/\mu^{c''}$, and event ${\cal{E}}$ did not happen, then $\mathsf{cr}(\phi(I))\le O(\sum_{I''\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I''))) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{O(1)}$.
\end{claim}
The following observation, whose proof is deferred to Section \ref{Appx: Proof of bound sum of opts} of Appendix, will complete the proof of \Cref{thm: main}.
\begin{observation}\label{obs: bound sum of opts}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$ and bad event ${\cal{E}}$ did not happen, then for some constant $c$, with probability at least $99/100$: $$\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot2^{c(\log m)^{3/4}\log\log m}.$$
\end{observation}
Let ${\cal{E}}'$ be the bad event that $\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')> (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot2^{c(\log m)^{3/4}\log\log m}$. Clearly, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c''}$, then the probability that either of the events ${\cal{E}}$ or ${\cal{E}}'$ happens is at most $\prob{{\cal{E}}}+\prob{{\cal{E}}'\mid\neg{\cal{E}}}\leq 1/16$. Therefore, we conclude that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c''}$, then with probability at least $15/16$, all of the following hold: (i) the algorithm does not return FAIL; (ii) ${\mathcal{I}}^*\neq \emptyset$; (iii) $\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot 2^{O((\log m)^{3/4}\log\log m)}$; and (iv) if algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace is given as input a solution $\phi(I'')$ to every instance $I''\in {\mathcal{I}}^*$, then the resulting solution $\phi$ to instance $I$ that it computes has cost at most: $O\bigg(\sum_{I''\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I''))\bigg) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m)\cdot\mu^{O(1)}$. This concludes
the proof of Theorem \ref{thm: main} from Theorems \ref{thm: not well connected}, \ref{lem: many paths}, and \ref{lem: not many paths}.
\section{An Algorithm for \ensuremath{\mathsf{MCNwRS}}\xspace -- Proof of \Cref{thm: main_rotation_system}}
\label{sec: high level}
In this section we provide a proof of Theorem \ref{thm: main_rotation_system}, with some of the details deferred to subsequent sections.
Throughout the paper, we denote by $I^*=(G^*,\Sigma^*)$ the input instance of the \textnormal{\textsf{MCNwRS}}\xspace problem, and we denote $m^*=|E(G^*)|$. We also use the following parameter that is central to our algorithm: ${\mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}}$,
where $c^*$ is a large enough constant.
As mentioned already, our algorithm for solving the \textnormal{\textsf{MCNwRS}}\xspace problem is recursive, and, over the course of the recursion, we will consider subinstances of instance $I^*$ (see \Cref{subsec: subinstances} for a definition).
The algorithm proceeds by recursively decomposing a given subinstance $I$ of $I^*$ into a collection of smaller subinstances.
The main technical ingredient of the proof is the following theorem.
\mynote{this theorem has changed. Instead of producing one family ${\mathcal{I}}$ of instances, it now produces $\log m^*$ such families. I also increased probability of success. Please update Sec 3.1 to take this into account.}
\begin{theorem}
\label{thm: main}
There is a constant $c'$, and an efficient randomized algorithm, that, given a subinstance $I=(G,\Sigma)$ of $I^*$ with $m=|E(G)|\geq \mu^{c'}$, either returns FAIL, or produces $z=\ceil{\log m^*}$ collections ${\mathcal{I}}_1,\ldots,{\mathcal{I}}_z$ of subinstances of $I$ with the following properties:
\begin{itemize}
\item at least one of the collections ${\mathcal{I}}_1,\ldots,{\mathcal{I}}_z$ is non-empty;
\item for each subinstance $I'=(G',\Sigma')\in \bigcup_{j=1}^z{\mathcal{I}}_j$, $|E(G')|\leq m/\mu$;
\item for all $1\leq j\leq z$, $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_j}|E(G')|\leq m\cdot (\log m)^{O(1)}$;
\item there is an efficient algorithm called \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, that, given an index $1\leq j\leq z$, and a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}_j$, computes a solution $\phi$ to instance $I$;
\item if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then with probability at least $(1-1/\mu)$, the algorithm does not return FAIL, and
there is an index $1\leq j\leq z$ for which the following hold:
\begin{itemize}
\item ${\mathcal{I}}_j\neq \emptyset$;
\item $\sum_{I'\in {\mathcal{I}}_j}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{O((\log m)^{3/4}\log\log m)} +m\cdot\mu^{O(1)}$; and
\item if algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace is given as input a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}_j$, then the resulting solution $\phi$ to instance $I$ that it computes has cost at most: $$O\bigg(\sum_{I'\in {\mathcal{I}}_j}\mathsf{cr}(\phi(I'))\bigg) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{O(1)}.$$
\end{itemize}
\end{itemize}
\end{theorem}
\iffalse
We now define collections of subinstances that have some desirable properties. The following definition is similar in spirit to the definition of a $\nu$-decomposition of an instance, but, since it uses slightly different parameters, we distinguish between these two definitions.
\begin{definition}[Good and Perfect Collection of Subinstances]
Let $I=(G,\Sigma)$ be a subinstance of instance $I^*$, with $|E(G)|=m$, and let ${\mathcal{I}}$ be a collection of subinstances of $I$. We say that ${\mathcal{I}}$ is a \emph{good collection of subinstances} for $I$ if the following hold.
\begin{itemize}
\item for each subinstance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\leq m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\cdot (\log m)^{c_g}$, where $1000<c_g<c^*$ is some large enough universal constant whose value will be set later; and
\item there is an efficient algorithm, that we call \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, that, given a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, of cost at most $c_g\cdot(\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{c_g}$.
\end{itemize}
We say that ${\mathcal{I}}$ is a \emph{perfect} collection of subinstances for $I$ if, additionally:
$$\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{O((\log m)^{3/4}\log\log m)} +m\cdot\mu^{c_g}.$$
\end{definition}
The main technical ingredient of the proof is the following theorem.
\begin{theorem}
\label{thm: main}
There is an efficient randomized algorithm, whose input is a subinstance $I=(G,\Sigma)$ of $I^*$, with $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$ and $|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$. The algorithm outputs a good collection ${\mathcal{I}}$ of subinstances of $I$, and, with probability at least $7/8$, ${\mathcal{I}}$ is a perfect collection of subinstances for $I$.
\end{theorem}
\fi
The remainder of this paper is dedicated to the proof of \Cref{thm: main}. In the following subsection, we complete the proof of \Cref{thm: main_rotation_system} using \Cref{thm: main}.
\subsection{Proof of \Cref{thm: main_rotation_system}}
Throughout the proof, we assume that $m^*$ is larger than a sufficiently large constant, since otherwise we can return a trivial solution to instance $I^*$, from \Cref{thm: crwrs_uncrossing}.
We let $100<c_g<c^*/2$ be a large enough constant, so that, if the algorithm from \Cref{thm: main}, when applied to a subinstance $I=(G,\Sigma)$ of $I^*$ with $m=|E(G)|\geq \mu^{c'}$, returns a family ${\mathcal{I}}$ of subinstances of $I$, with $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\cdot (\log m)^{c_g}$. Additionally, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then with probability at least $15/16$: $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot 2^{c_g((\log m)^{3/4}\log\log m)} +m\cdot\mu^{c_g}$. We also assume that Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, given a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$, of cost at most $c_g\cdot (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{c_g}$.
We use a simple recursive algorithm called \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace, that appears in Figure \ref{fig: algrec}.
\program{\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace}{fig: algrec}{
\noindent{\bf Input:} a subinstance $I=(G,\Sigma)$ of the input instance $I^*$ of the \textnormal{\textsf{MCNwRS}}\xspace problem.
\noindent{\bf Output:} a feasible solution to instance $I$.
\begin{enumerate}
\item Use the algorithm from \Cref{thm: crwrs_planar} to determine whether or not $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$. If so, use the algorithm from \Cref{thm: crwrs_planar} to compute a solution to $I$ of cost $0$. Return this solution, and terminate the algorithm.
\item Use the algorithm from Theorem~\ref{thm: crwrs_uncrossing} to compute a trivial solution $\phi'$ to instance $I$.
\item If $|E(G)|\leq \mu^{c'}$, return the trivial solution $\phi'$ and terminate the algorithm.
\item For $1\leq j\leq \ceil{\log m^*}$:
\begin{enumerate}
\item Apply the algorithm from \Cref{thm: main} to instance $I$.
\item If the algorithm returns FAIL, let $\phi_j=\phi'$ be the trivial solution to instance $I$, and set ${\mathcal{I}}_j(I)=\emptyset$.
\item Otherwise:
\begin{enumerate}
\item Let ${\mathcal{I}}_j(I)$ be the collection of subinstances of $I$ that the algorithm returns.
\item For every subinstance $I'\in {\mathcal{I}}_j(I)$, apply Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace to instance $I'$, to obtain a solution $\phi(I')$ to this instance.
\item Apply Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace from \Cref{thm: main} to solutions $\set{\phi(I')}_{I'\in {\mathcal{I}}_j(I)}$, to obtain a solution $\phi_j$ to instance $I$.
\end{enumerate}
\end{enumerate}
\item
Return the best solution of $\set{\phi',\phi_1,\ldots,\phi_{\ceil{\log m^*}}}$
\end{enumerate}
}
In order to analyze the algorithm, it is convenient to associate a \emph{partitioning tree} $T$ with it, whose vertices correspond to all subinstances considered over the course of the algorithm. Let $L=\ceil{\log m^*}$. We start with the tree $T$ containing a single root vertex $v(I^*)$, representing the input instance $I^*$. Consider now some vertex $v(I)$ of the tree, representing some subinstance $I=(G,\Sigma)$ of $I^*$.
When Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace was applied to instance $I$, if it did not terminate after the first three steps, it created $L$ collections ${\mathcal{I}}_1(I),\ldots,{\mathcal{I}}_L(I)$ of subinstances of $I$ (some of which may be empty, in case the algorithm from \Cref{thm: main} returned FAIL). For each such subinstance $I'\in \bigcup_{j=1}^L{\mathcal{I}}_j(I)$, we add a vertex $v(I')$ representing instance $I'$, that becomes a child vertex of $v(I)$. This concludes the description of the partitioning tree $T$.
We denote by ${\mathcal{I}}^*=\set{I\mid v(I)\in V(T)}$ the set of all subinstances of instance $I^*$ whose corresponding vertex appears in the tree $T$. For each such instance $I\in {\mathcal{I}}^*$, its \emph{recursive level} is the distance from vertex $v(I)$ to the root vertex $v(I^*)$ in the tree $T$ (so the recursive level of $v(I^*)$ is $0$).
For $j\geq 0$, we denote by $\hat {\mathcal{I}}_j\subseteq {\mathcal{I}}^*$ the set of all instances $I\in {\mathcal{I}}^*$, whose recursive level is $j$.
Lastly, the \emph{depth} of the tree $T$, denoted by $\mathsf{dep}(T)$, is the largest recursive level of any instance in ${\mathcal{I}}^*$.
In order to analyze the algorithm, we start with the following two simple observations.
\begin{observation}\label{obs: few recursive levels}
$\mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$.
\end{observation}
\begin{proof}
Consider any non-root vertex $v(I)$ in the tree $T$, and let $v(I')$ be the parent-vertex of $v(I)$. Denote $I=(G,\Sigma)$ and $I'=(G',\Sigma')$.
From the construction of tree $T$, instance $I$ belongs to some collection of subinstances obtained by applying the algorithm from \Cref{thm: main} to instance $I'$. Therefore, from \Cref{thm: main}, $|E(G)|\leq |E(G')|/\mu$ must hold. Therefore, for all $j\geq 0$, for every instance $I=(G,\Sigma)\in \hat {\mathcal{I}}_j$, $|E(G)|\leq m^*/\mu^j$. Since $\mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}$, we get that $\mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$.
\end{proof}
\begin{observation}\label{obs: num of edges}
$\sum_{I=(G,\Sigma)\in {\mathcal{I}}^*}|E(G)|\le m^*\cdot 2^{(\log m^*)^{1/8}}$.
\end{observation}
\begin{proof}
Consider any non-leaf vertex $v(I)$ of the tree $T$, and denote $I=(G,\Sigma)$.
Recall that, when Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace is applied to instance $I$, it applies the algorithm from \Cref{thm: main} to $I$ and compute $L$ collections ${\mathcal{I}}_1(I),\ldots,{\mathcal{I}}_L(I)$ of subinstances, such that, if we denote $|E(G)|=m$, then, for all $1\leq j\leq L$:
$$\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_j(I)}|E(G')|\leq m\cdot (\log m)^{c_g}\leq m\cdot (\log m^*)^{c_g}$$
%
(we have used the fact that, if $I$ is a subinstance of $I^*$, then $m\leq m^*$ must hold). Since $L\leq 2\log m^*$, and $m^*$ is sufficiently large, we get that:
%
$$\sum_{j=1}^L\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_j(I)}|E(G')|\leq m\cdot (\log m^*)^{c_g+2}.$$
%
For all $j\geq 0$, we denote by $N_j$ the total number of edges in all instances in set $\hat {\mathcal{I}}_j$, $N_j=\sum_{I=(G,\Sigma)\in \hat {\mathcal{I}}_j}|E(G)|$. Clearly, $N_0=m^*$, and, from the above discussion, for all $j>0$, $N_j\leq N_{j-1}\cdot (\log m^*)^{c_g+2}$.
Since $\mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$,
we conclude that:
\[
\sum_{I=(G,\Sigma)\in {\mathcal{I}}^*}|E(G)|\leq m^*\cdot (\log m^*)^{2c_g\cdot (\log m^*)^{1/8}/(c^*\log\log m^*)}\leq m^*\cdot 2^{(\log m^*)^{1/8}},
\]
since $c_g\leq c^*/2$.
\end{proof}
We use the following corollary, that follows immediately from \Cref{obs: num of edges}.
\begin{corollary}\label{cor: num of instances}
The number of instances $I=(G,\Sigma)\in {\mathcal{I}}^*$ with $|E(G)|\geq \mu^{c'}$ is at most $m^*$.
\end{corollary}
\iffalse{original corollary does not depend on the notion of leaf instance}
For an instance $I\in {\mathcal{I}}^*$, we say that it is a \emph{leaf instance}, if vertex $v(I)$ is a leaf vertex of the tree $T$, and we say that it is a non-leaf instance otherwise. We use the following corollary, that follows immediately from \Cref{obs: num of edges}, and the fact (from the description of Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace) that, if $I=(G,\Sigma)\in {\mathcal{I}}^*$ is a non-leaf instance, then $|E(G)|\geq \mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}$ must hold.
\begin{corollary}\label{cor: num of instances}
The number of instances $I=(G,\Sigma)\in {\mathcal{I}}^*$ with $|E(G)|\geq \mu^{c'}$ is at most $m^*$.
\end{corollary}
\fi
For an instance $I\in {\mathcal{I}}^*$, we say that it is a \emph{leaf instance}, if vertex $v(I)$ is a leaf vertex of the tree $T$, and we say that it is a non-leaf instance otherwise.
Consider now a non-leaf instance $I\in {\mathcal{I}}^*$. We say that a bad event ${\cal{E}}(I)$ happens, if all the following conditions hold:
\begin{itemize}
\item $0<\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$;
\item $|E(G)|\geq \mu^{c'}$; and
\item for all $1\leq j\leq L$, we have set $\phi_j=\phi'$ and ${\mathcal{I}}_j(I)=\emptyset$, since
the algorithm from \Cref{thm: main} returned FAIL in each of the $L$ iterations when it was applied to $I$.
\end{itemize}
Clearly, from \Cref{thm: main}, $\prob{{\cal{E}}(I)}\le (1/16)^L\leq 1/(m^*)^4$.
We say that a bad event ${\cal{E}}'(I)$ happens if the following conditions hold:
\begin{itemize}
\item $0<\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$;
\item $|E(G)|\geq \mu^{c'}$; and
\item for all $1\leq j\leq L$, $\sum_{I'\in {\mathcal{I}}_j(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')> \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{c_g\cdot (\log m)^{3/4}\log\log m} +m\cdot\mu^{c_g}$.
\end{itemize}
As before, from \Cref{thm: main}, $\prob{{\cal{E}}'(I)}\le (1/16)^L\leq 1/(m^*)^4$.
Let ${\cal{E}}$ be the bad event that event ${\cal{E}}(I)$ or event ${\cal{E}}'(I)$ happened for any instance $I\in {\mathcal{I}}^*$. From the Union Bound and \Cref{cor: num of instances}, we get that $\prob{{\cal{E}}}\leq 1/(m^*)^2$.
We use the following immediate observation.
\begin{observation}\label{obs: leaf}
If Event ${\cal{E}}$ does not happen, then for every leaf vertex $v(I)$ of $T$ with $I=(G,\Sigma)$, either $|E(G)|\leq \mu^{c'}$; or $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$; or $\mathsf{OPT}_{\mathsf{cnwrs}}(I)> |E(G)|^2/\mu^{c'}$.
\end{observation}
We use the following lemma to complete the proof of \Cref{thm: main_rotation_system}.
\begin{lemma}\label{lem: solution cost}
If Event ${\cal{E}}$ does not happen, then Algorithm $\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace$ computes a solution for instance $I^*=(G^*,\Sigma^*)$ of cost at most $2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right})$.
\end{lemma}
\begin{proof}
Consider a non-leaf instance $I=(G,\Sigma)$, and let ${\mathcal{I}}_1(I),\ldots,{\mathcal{I}}_L(I)$ be families of subinstances of $I$ that the Algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace computes. Recall that, for each $1\le j\le L$ with ${\mathcal{I}}_j(I)\neq \emptyset$, the algorithm computes a solution $\phi_j$ to instance $I$, by first solving each of the instances in ${\mathcal{I}}_j(I)$ recursively, and then combining the resulting solutions using Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace. Eventually, the algorithm returns the best solution of $\set{\phi',\phi_1,\ldots,\phi_L}$, where $\phi'$ is the trivial solution, whose cost is at most $|E(G)|^2$. We fix an index $1\leq j\leq L$, for which solution $\phi_j$ has the lowest cost, breaking ties arbitrarily. Note that the cost of the solution to instance $I$ that the algorithm returns is at most $\mathsf{cr}(\phi_j)$. We then \emph{mark} the vertices of $\set{v(I')\mid I'\in {\mathcal{I}}_j(I)}$ in the tree $T$. We also mark the root vertex of the tree.
Let $T^*$ be the subgraph of $T$ induced by all marked vertices. It is easy to verify that $T^*$ is a tree, and moreover, every leaf vertex of $T^*$ is also a leaf vertex of $T$. For a vertex $v(I)\in V(T^*)$, we denote by $h(I)$ the length of the longest path in tree $T^*$, connecting vertex $v(I)$ to any of its descendants in the tree.
We use the following claim, whose proof is straightforward conceptually but somewhat technical; we defer the proof to \Cref{Appx: inductive bound proof}.
\begin{claim}\label{claim: bound by level}
Assume that Event ${\cal{E}}$ did not happen. Then there is a fixed constant $\tilde c>0$, such that, for every vertex $v(I)\in V(T^*)$, whose corresponding subinstance of $I^*$ is denoted by $I=(G,\Sigma)$, the cost of the solution that the algorithm computes for $I$ is at most:
$$2^{\tilde c\cdot h(I)\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{c'\cdot c_g}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+(\log m^*)^{4c_g h(I)}\mu^{2c'\cdot \tilde c}\cdot|E(G)|.$$
\end{claim}
We are now ready to complete the proof of \Cref{lem: solution cost}. Recall that $h(I^*)=\mathsf{dep}(T^*)\leq \mathsf{dep}(T)\leq \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$ from \Cref{obs: few recursive levels}. Therefore, from \Cref{claim: bound by level}, the cost of the solution that the algorithm computes for instance $I^*$ is bounded by:
\[
\begin{split}
& \text{ } 2^{O(\mathsf{dep}(T))\cdot (\log m^*)^{3/4}\log\log m^*}\cdot \mu^{O(1)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+(\log m^*)^{O(\mathsf{dep}(T))}\cdot\mu^{O(1)}\cdot m^*\\
\leq & \text{ } 2^{O((\log m^*)^{7/8})}\cdot \mu^{O(1)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+(\log m^*)^{O((\log m^*)^{1/8}/\log\log m^*) }\cdot\mu^{O(1)}\cdot m^*\\
\leq & \text{ } 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}),
\end{split}
\]
since $\mu=2^{O((\log m^*)^{7/8}\log\log m^*)}$.
\end{proof}
In order to complete the proof of \Cref{thm: main_rotation_system}, it is now enough to prove Theorem~\ref{thm: main}. The remainder of the paper is dedicated to the proof of \Cref{thm: main}.
\subsection{Proof of Theorem~\ref{thm: main} -- Main Definitions and Theorems}
We classify subinstances of the input instance $I^*$ into \emph{wide} and \emph{narrow}, and provide different algorithms for decomposing instances of each of the two kinds.
Throughout this subsection, we use the parameter ${50}=50$.
\begin{definition}[Wide and Narrow Instances]
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$. We say that $I$ is a \emph{wide} instance, iff there is a vertex $v\in V(G)$, a partition $(E_1,E_2)$ of the edges of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and a collection ${\mathcal{P}}$ of at least $\floor{m/\mu^{{50}}}$ simple edge-disjoint cycles in $G$, such that every cycle $P\in {\mathcal{P}}$ contains one edge of $E_1$ and one edge of $E_2$.
If no such cycle set ${\mathcal{P}}$ exists in $G$, then we say that $I$ is a \emph{narrow} instance.
\end{definition}
Note that there is an efficient algorithm to check whether a given instance $I$ of \ensuremath{\mathsf{MCNwRS}}\xspace is wide, and, if so, to compute the corresponding cycle set ${\mathcal{P}}$, via standard algorithms for maximum flow. (For every vertex $v\in V(G)$, we can try all possible partitions $(E_1,E_2)$ of $\delta_G(v)$ with the required properties, as the number of such partitions is bounded by $|\delta_G(v)|^2$.)
We will use the following simple observation regarding narrow instances.
\begin{observation}\label{obs: narrow prop 2}
If an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace is narrow, then for every pair $u,v$ of distinct vertices of $G$, and any set ${\mathcal{P}}$ of edge-disjoint paths connecting $u$ to $v$ in $G$, $|{\mathcal{P}}|\leq 2\ceil{m/\mu^{{50}}}+2$ must hold.
\end{observation}
\begin{proof}
Assume for contradiction that $|{\mathcal{P}}|>2\ceil{m/\mu^{{50}}}+2$, and denote $|{\mathcal{P}}|=k$.
Let $E'\subseteq \delta_G(v)$ be the set of all edges $e\in \delta_G(v)$, such that $e$ is the first edge on some path in ${\mathcal{P}}$. We denote $E'=\set{e_1,\ldots,e_k}$, where the edges are indexed according to their ordering in the rotation ${\mathcal{O}}_v\in \Sigma$. We also denote ${\mathcal{P}}=\set{P(e_i)\mid 1\leq i\leq k}$, where path $P(e_i)$ contains the edge $e_i$ as its first edge. We can then compute a partition $(E_1,E_2)$ of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$. Additionally, we can ensure that $e_1,\ldots,e_{\ceil{k/2}}\in E_1$, while the remaining edges of $E'$ lie in $E_2$. For each $1\leq i\leq \ceil{m/\mu^{{50}}}$, we let $Q_i$ be the cycle obtained by concatenating the paths $P(e_i)$ and $P(e_{k-i+1})$. We can turn $Q_i$ into a simple cycle, by removing all cycles that are disjoint from vertex $v$ from it. It is then immediate to verify that cycle set $\set{Q_i\mid 1\leq i\leq \ceil{m/\mu^{{50}}}}$ has all the required properties to establish that instance $I$ is wide, a contradiction.
\end{proof}
\iffalse
\begin{definition}[Wide and Narrow Subinstances]
A subinstance $I=(G,\Sigma)$ of $I^*$ is an \emph{interesting} instance, iff there is a
collection ${\mathcal{P}}$ of at least $m/\mu^{{50}}$ edge-disjoint paths in $G$, that we refer to as the \emph{witness path set for $I$}, such that one of the following holds:
\begin{itemize}
\item either there are two distinct vertices $u,v\in V(G)$, such that every path in ${\mathcal{P}}$ originates at $v$ and terminates at $u$; or
\item there is a single vertex $v\in V(G)$, and a partition $(E_1,E_2)$ of the edges of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$. Every path in ${\mathcal{P}}$ must contain an edge of $E_1$ as its first edge and an edge of $E_2$ as its last edge.
\end{itemize}
%
If no such path set ${\mathcal{P}}$ exists in $G$, then we say that $I$ is an \emph{narrow} instance.
\end{definition}
\fi
The proof of \Cref{thm: main} relies on the following two theorems. The first theorem deals with wide subinstances of $I^*$, and its proof appears in \Cref{sec: many paths}.
\mynote{The theorem has changed significantly. the rest of the proof needs to be updated.}
\begin{theorem}\label{lem: many paths}
There is an efficient randomized algorithm, that, given a wide subinstance $I=(G,\Sigma)$ of $I^*$ with $m=|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$, either returns FAIL, or computes $z=\ceil{\log m^*}$ collections ${\mathcal{I}}_1,\ldots,{\mathcal{I}}_z$ of subinstances of $I$, such that the following hold:
\begin{itemize}
\item for all $1\leq j\leq z$, $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_j}|E(G')|\le |E(G)|$;
\item for every instance $I'=(G',\Sigma')\in \bigcup_{j=1}^z{\mathcal{I}}_j$, either $|E(G')|\le m/\mu$, or instance $I'$ is not wide;
\item
there is an efficient algorithm called $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace'$, that, given an index $1\leq j\leq z$, and a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}_j$, computes a solution $\phi$ to instance $I$; and
\item if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$ then, with probability at least $1-1/\mu^2$, the algorithm does not return FAIL, and there is an index $1\leq j\leq z$ for which all of the following hold:
\begin{itemize}
\item ${\mathcal{I}}_j\neq \emptyset$;
\item $\sum_{I'\in {\mathcal{I}}_j}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log m)^{O(1)}$; and
\item if algorithm $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace'$ is given as input a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}_j$, then the resulting solution $\phi$ to instance $I$ that it computes has cost at most: $\mathsf{cr}(\phi)\leq \sum_{I'\in {\mathcal{I}}_j}\mathsf{cr}(\phi(I')) + \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\mu^{O(1)}$.
\end{itemize}
\end{itemize}
\end{theorem}
The second theorem deals with narrow instances.
\begin{theorem}\label{lem: not many paths}
There is an efficient randomized algorithm, that, given a narrow subinstance $I=(G,\Sigma)$ of $I^*$, with $m=|E(G)|\geq \mu^{20}$, either returns FAIL, or computes a $\nu$-decomposition ${\mathcal{I}}$ of $I$, for $\nu= 2^{O((\log m)^{3/4}\log\log m)}$, such that, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\le m/\mu$. Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{16}$, then the probability that the algorithm returns FAIL is at most $1/\mu^2$.
\end{theorem}
The majority of the remainder of this paper is dedicated to the proof of the above theorem: we develop the central technical tools used in the theorem's proof in \Cref{sec: guiding paths orderings basic disengagement} -- \Cref{sec: main disengagement}, and complete the proof in \Cref{sec: computing the decomposition}.
In the remainder of this section, we complete the proof of Theorem \ref{thm: main} using Theorems \ref{lem: many paths} and \ref{lem: not many paths}.
\iffalse
\mynote{restate the lemma in terms of $\alpha$-decomposition into subinstances}
\begin{lemma}\label{lem: not many paths}
There is an efficient randomized algorithm, that, given a non-interesting subinstance $I=(G,\Sigma)$ of $I^*$ with $|E(G)|=m\geq \mu^{10}$, either correctly certifies that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\geq \Omega(m^2/\mu^5)$, or computes a collection ${\mathcal{I}}$ of subinstances of $I$, such that
the following hold:
\begin{itemize}
\item for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, $|E(G')|\le m/\mu$;
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq O(m)$; and
\item $\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot 2^{O((\log m)^{3/4}\log\log m)}$.
\end{itemize}
Moreover, there is an efficient algorithm, that, given, for each instance $I'\in {\mathcal{I}}$, a feasible solution $\phi(I')$, computes a feasible solution $\phi$ for instance $I$, of cost $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$.
\end{lemma}
\fi
Recall that we are given an instance $I=(G,\Sigma)$, that is a subinstance of the input instance $I^*=(G^*,\Sigma^*)$, with $|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$. We denote $m=|E(G)|$. We compute the desired collection ${\mathcal{I}}$ of subinstances of $I$ in two steps.
If instance $I$ is a narrow instance, then we skip the first step. We let ${\mathcal{I}}'$ be a collection of subinstances of $I$, that consists of instance $I$ only. Assume now that $I$ is a wide instance. Then we execute Step 1, by applying the algorithm from \Cref{lem: many paths} to $I$.
If the algorithm returns FAIL, then we terminate the algorithm and return FAIL as well.
Otherwise, let ${\mathcal{I}}'$ be the resulting collection of subinstances of $I$. Recall that we are guaranteed that, for each instance $I'=(G',\Sigma')\in {\mathcal{I}}'$, either $I'$ is a narrow instance, or $|E(G')|\le m/\mu$. Additionally, we are guaranteed that $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\le |E(G)|$;
and, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\mu^{c'}$, then with probability at least $31/32$:
$$\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{c_g(\log m)^{3/4}\log\log m}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g},$$
for some constant $c_g$.
We say that a bad event ${\cal{E}}'_1$ happens if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, but: $$\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')> 2^{c_g(\log m)^{3/4}\log\log m}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g}.$$
From \Cref{lem: many paths}, $\prob{{\cal{E}}'_1}\leq 1/32$.
We say that a bad event ${\cal{E}}''_1$ happens if algorithm from \Cref{lem: many paths} returns FAIL when applied to $I$.
From \Cref{lem: many paths}, $\prob{{\cal{E}}''_1}\leq 1/\mu^2$.
Define the bad event ${\cal{E}}_1={\cal{E}}'_1\cup {\cal{E}}''_1$, so $\prob{{\cal{E}}_1}\leq 1/31$.
This completes the first step of the algorithm.
We now describe the second step.
We start by partitioning the set ${\mathcal{I}}'$ of instances into two subsets: set ${\mathcal{I}}_1'$ containing all instances $I'=(G',\Sigma')$ with $|E(G')|\leq m/\mu$, and set ${\mathcal{I}}_2'$ containing all remaining instances, that must be narrow.
We use the following simple observation.
\begin{observation}\label{obs: optbound for narrow}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and Event ${\cal{E}}_1$ does not happen, then for every instance $I'=(G',\Sigma')\in {\mathcal{I}}_2'$, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq |E(G')|^2/\mu^{16}$.
\end{observation}
\begin{proof}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and that Event ${\cal{E}}_1$ does not happen. Then:
$$\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{c_g(\log m)^{3/4}\log\log m}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g}.$$
Consider now some instance $I'=(G',\Sigma')\in {\mathcal{I}}_1'$. We then get that:
$$\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{c_g(\log m)^{3/4}\log\log m}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g}\leq \mu\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g},$$
since $\mu=2^{c^*(\log m^*)^{7/8}\log\log m^*}$.
Since $m\geq \mu^{c'}$, and since we can assume that $c'$ is a large enough constant, $\mu^{c_g}\leq m/\mu^{c'/2}$. Further, since we have assumed that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$, and since $|E(G')|\geq m/\mu$ (since instance $I'=(G',\Sigma')$ belongs to set ${\mathcal{I}}_2'$), we get that:
$$\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq m^2/\mu^{c'-1}+m^2/\mu^{c'/2}\leq |E(G')|^2/\mu^{16},$$
assuming that $c'$ is large enough.
\end{proof}
Next, we process every instance $I'\in {\mathcal{I}}_2'$ one by one. Notice that for each such instance $I'=(G',\Sigma')$, $|E(G')|\geq m/\mu\geq \mu^{20}$ must hold, since $m\geq \mu^{c'}$. When instance $I'$ is processed, we apply the algorithm from \Cref{lem: not many paths} to it. If the algorithm returns FAIL, then we terminate the algorithm and return FAIL as well. Otherwise, we obtain a collection ${\mathcal{I}}(I')$ of subinstances of $I'$ (and hence of $I$). If the algorithm did not terminate with a FAIL, then we obtain the final collection ${\mathcal{I}}^*={\mathcal{I}}_1'\cup \textsf{left} (\bigcup_{I'\in {\mathcal{I}}_2'}{\mathcal{I}}(I')\textsf{right})$ of subinstances of $I$ that the algorithm returns.
This completes the description of the algorithm.
In order to analyze it, we first show that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/16$.
\begin{observation}\label{obs: fail prob}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/16$.
\end{observation}
\begin{proof}
For an instance $I'=(G',\Sigma')\in {\mathcal{I}}'$, we say that a bad event ${\cal{E}}(I')$ happens if the algorithm from \Cref{lem: not many paths}, when applied to $I'$, returns FAIL. From \Cref{lem: not many paths}, if
$\mathsf{OPT}_{\mathsf{cnwrs}}(I')<|E(G')|^2/\mu^{16}$, then the probability that the algorithm returns FAIL is at most $1/\mu^2$. Therefore, from \Cref{obs: optbound for narrow}, if Event ${\cal{E}}_1$ does not happen, then $\prob{{\cal{E}}(I')}\leq 1/\mu^2$. Let ${\cal{E}}_2$ be the bad event that ${\cal{E}}(I')$ happens for any instance $I'\in {\mathcal{I}}'_2$. Since every instance $I'\in {\mathcal{I}}_2'$ contains at least $m/\mu$ edges, while, as we have established already, the total number of edges in all instances in ${\mathcal{I}}'$
is at most $m$, we get that $|{\mathcal{I}}'_2|\leq \mu$. From the Union Bound, assuming that the constant $c^*$ in the definition of the parameter $\mu$ is large enough, $\prob{{\cal{E}}_2\mid \neg {\cal{E}}_1}\leq 1/100$. Clearly, $\prob{{\cal{E}}_2}\leq \prob{{\cal{E}}_1}+\prob{{\cal{E}}_2\mid \neg {\cal{E}}_1}\leq 1/31+1/100< 1/20$. Note that, if events ${\cal{E}}_1$ and ${\cal{E}}_2$ do not happen, then algorithm may not return FAIL. We conclude that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then the algorithm returns FAIL with probability at most $1/20<1/16$.
\end{proof}
We define the bad event ${\cal{E}}={\cal{E}}_1\cup {\cal{E}}_2$, where event ${\cal{E}}_2$ is defined in the proof of above observation. From the above discussion $\prob[]{{\cal{E}}}\le 1/20$.
Fron now on we assume that the algorithm did not return FAIL.
Recall that \Cref{lem: not many paths} gurantees that, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}_2'$, for every subinstance $I''=(G'',\Sigma'')\in {\mathcal{I}}(I')$, $|E(G'')|\leq |E(G')|/\mu\leq m/\mu$ holds. Therefore, for every instance $I''=(G'',\Sigma'')\in{\mathcal{I}}^*$, $|E(G'')|\leq m/\mu$ must hold.
Additionally, from the definition of a $\nu$-decomposition, we are guaranteed that for all $I'=(G',\Sigma')\in {\mathcal{I}}_2'$, $\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}(I')}|E(G'')|\leq |E(G')|\cdot (\log (|E(G')|)^{O(1)}\leq |E(G')|\cdot (\log m)^{O(1)}$.
Moreover, since $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\le |E(G)|$, we get that:
$$\sum_{I''=(G'',\Sigma'')\in {\mathcal{I}}^*}|E(G'')|\leq \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\cdot (\log m)^{O(1)}\leq m\cdot(\log m)^{O(1)}.$$
Next, we provide Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace.
\begin{claim}\label{claim: combine drawings}
There is an efficient algorithm, called \ensuremath{\mathsf{AlgCombineDrawings}}\xspace, that, given a solution $\phi(I'')$ to each instance $I''\in {\mathcal{I}}^*$, computes a solution $\phi(I)$ to instance $I$, of cost at most $O(\sum_{I''\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I''))) +(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot\mu^{O(1)}$.
\end{claim}
\begin{proof}
We construct the solution $\phi(I)$ to instance $I$ in two steps.
In the first step, we compute a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}'_2$, as follows. Consider any such instance $I'=(G',\Sigma')\in {\mathcal{I}}'_2$. Recall that we are given, for each instance $I''\in {\mathcal{I}}(I')$, a solution $\phi(I'')$. Recall also that ${\mathcal{I}}(I')$ is a $\nu$-decomposition of $I'$. We apply the efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}}(I'))$ from the definition of $\nu$-decompositions of instances to the drawings $\set{\phi(I'')}_{I''\in {\mathcal{I}}(I')}$, to obtain a feasible solution $\phi(I')$ for instance $I'$, of cost $\mathsf{cr}(\phi(I'))\leq O\textsf{left} (\sum_{I''\in {\mathcal{I}}(I')}\mathsf{cr}(\phi(I''))\textsf{right} )$.
We have now obtained a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}'$. In the second step, we apply the algorithm in the third bullet of the statement of \Cref{lem: many paths} to these solutions, to obtain a solution $\phi(I)$ to instance $I$.
The cost of the solution is bounded by:
\[\begin{split}
\mathsf{cr}(\phi(I)) &\leq \sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I')) + (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}\\
&\leq \sum_{I'\in {\mathcal{I}}'_1}\mathsf{cr}(\phi(I')) +\sum_{I'\in {\mathcal{I}}'_2}\mathsf{cr}(\phi(I'))+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}\\
&\leq \sum_{I'\in {\mathcal{I}}'_1}\mathsf{cr}(\phi(I')) +\sum_{I'\in {\mathcal{I}}'_2}O\textsf{left} (\sum_{I''\in {\mathcal{I}}(I')}\mathsf{cr}(\phi(I''))\textsf{right} )+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}\\
&\leq O\textsf{left} (\sum_{I''\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I'))\textsf{right} ) +(m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^{O(1)}.
\end{split}
\]
\end{proof}
In order to complete the proof of \Cref{thm: main}, it is now enough to prove the following observation.
\begin{observation}\label{obs: bound sum of opts}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then with probability at least $15/16$: $$\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{O((\log m)^{3/4}\log\log m)} +m\cdot\mu^{O(1)}.$$
\end{observation}
\begin{proof}
Assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$. Recall that, if Event ${\cal{E}}$ does not happen, then:
$$\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2^{c_g(\log m)^{3/4}\log\log m)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c_g}.$$
From the definition of a $\nu$-decomposition, for every instance $I'=(G',\Sigma')\in {\mathcal{I}}'_2$,
\[\expect{\sum_{I''\in {\mathcal{I}}(I')}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} )\cdot \nu \leq \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} ) \cdot 2^{O((\log m)^{3/4}\log\log m)},\]
Therefore, if ${\cal{E}}$ does not happen, then:
\[
\begin{split}
\expect{\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}&\leq \sum_{I'\in {\mathcal{I}}'_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I') +\sum_{I'\in {\mathcal{I}}'_2}\expect{\sum_{I''\in {\mathcal{I}}(I')}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}\\
&\leq \sum_{I'\in {\mathcal{I}}'_1}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'_2}\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} ) \cdot 2^{O((\log m)^{3/4}\log\log m)}\\
&\leq \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\textsf{right} )\cdot 2^{O((\log m)^{3/4}\log\log m)}\\
&\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{O(1)}.
\end{split} \]
We denote this expectation by $\eta'$. Let $\hat {\cal{E}}$ be the bad event that
$\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')>100\eta'$. From Markov's inequality, $\prob{\hat {\cal{E}}\mid \neg{\cal{E}}}<1/100$. Note that, if neither of the events $\hat {\cal{E}},{\cal{E}}$ happens, then we are guaranteed that $\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{O(1)}$. The probability that either one of these events happens is bounded by $\prob{{\cal{E}}}+\prob{\hat {\cal{E}}\mid \neg{\cal{E}}}\leq 1/20+ 1/100\leq 1/16$. We conclude that, If $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{c'}$, then with probability at least $15/16$: $$\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot2^{O((\log m)^{3/4}\log\log m)} +m\cdot\mu^{O(1)}.$$
\end{proof}
\iffalse
Moreover, we are guaranteed that,
\[\expect[]{\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')}\leq \nu''\cdot \bigg( \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\big(\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\big)\bigg ). \]
Assume that event ${\cal{E}}$ did not happen. Then,
\[
\begin{split}
\expect[]{\sum_{I''\in {\mathcal{I}}^*}\mathsf{OPT}_{\mathsf{cnwrs}}(I'')} &\text{ }\leq \nu''\cdot \expect[]{ \sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\big(\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|\big)}\\
&\text{ }\leq \nu''\cdot \expect[]{\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}+\nu''\cdot\bigg(\sum_{I'=(G',\Sigma')\in {\mathcal{I}}'}|E(G')|\bigg)\\
&\text{ }\leq \nu''\cdot \bigg(\nu'\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^c\bigg)+\nu''\cdot(\log|E(G)|)^{O(1)}\cdot|E(G)|\\
&\text{ }\leq (\nu'\nu'')\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+O(\nu''\cdot\mu^c)\cdot|E(G)|.
\end{split} \]
Let $\nu=\nu'\cdot\nu''$.
Since $\nu',\nu''=2^{O((\log m)^{3/4}\log\log m)}$, $\nu=\nu'\cdot\nu''=2^{O((\log m)^{3/4}\log\log m)}$.
Since $\mu^{c_g}>\nu''\cdot\mu^{c}$, with probability at least $9/10$, the resulting collection ${\mathcal{I}}^*$ of instances is perfect for $I$.
\fi
\iffalse
In order to do so,
@@@@ we compute a denote by ${\mathcal{I}}(I')=\set{I''_1,\ldots,I''_k}$ the set of subinstances obtained by applying the algorithm from \Cref{lem: not many paths} to it. From the definition of a $\nu''$-decomposition, we are also given an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}}(I'))$, that takes as input a solution of each instance in ${\mathcal{I}}(I')$ and returns a solution of $I'$. We apply this algorithm to the solutions $\set{\phi(I''_i)}_{1\le i\le k}$ to the subinstances in ${\mathcal{I}}(I')$ and let $\phi(I')$ be the solution we get. From the definition of a $\nu''$-decomposition, there exists some universal constant $c^*>0$, such that $\mathsf{cr}(\phi(I'))\le c^*\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I''_i))$.
Clearly, after the first step, we have obtained, for each subinstance $I'\in {\mathcal{I}}'$, a solution $\phi(I')$ to instance $I'$.
Recall that, in \Cref{lem: many paths} we are also given an efficient algorithm (that we denote by $\ensuremath{\mathsf{Alg}}\xspace'({\mathcal{I}}')$), that, given, for every instance $I'\in {\mathcal{I}}'$, a solution $\phi(I')$ to instance $I'$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq \big(\sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I'))\big)+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^c$.
In the second step, we simply apply the algorithm to the set $\set{\phi(I')}_{I'\in {\mathcal{I}}'}$ of solutions and let $\phi(I)$ be the solution to instance $I$ that we obtain. From \Cref{lem: many paths},
\[
\begin{split}
\mathsf{cr}(\phi)\leq & \text{ } \bigg(\sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I'))\bigg)+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^c\\
\leq & \text{ } c^*\cdot \bigg(\sum_{I''\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I''))\bigg)+ (m+\mathsf{OPT}_{\mathsf{cnwrs}}(I))\cdot\mu^c.
\end{split}
\]
Now \Cref{thm: main} follows by setting $c_g$ to be a contant such that $c_g> \max\set{c^*, c}$.
\fi
\iffalse{previous algorithm algcombine}
Consider the partitioning tree $T^*$ that is obtained from the partitioning tree $T$ in the first Stage, by adding, for each leaf $v(I')$ of $T$ with $I'\in {\mathcal{I}}'_2$ and for each instance $I''\in {\mathcal{I}}(I')$, a new vertex $v(I'')$ and an edge connecting it to $v(I')$. Clearly, the instances in ${\mathcal{I}}^*$ correspond to leaves in $T^*$. Assume now that we are given, for each instance $I'$ such that $v(I')$ is a leaf in $T^*$, a solution $\phi(I')$ to instance $I'$. We will construct a drawing of instance $I$ in two steps as follows.
In the first step, for each instance $I'\in {\mathcal{I}}'_2$, we denote by ${\mathcal{I}}(I')=\set{I''_1,\ldots,I''_k}$ the set of subinstances obtained by applying the algorithm from \Cref{lem: not many paths} to it. Recall that, from the definition of a $\nu$-decomposition in \Cref{lem: not many paths} we are also given an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}}(I'))$, that takes as input a solution of each instance in ${\mathcal{I}}(I')$ and returns a solution of $I'$. We apply this algorithm to the solutions $\set{\phi(I''_i)}{1\le i\le k}$, and let $\phi(I')$ be the solution we get. From the definition of a $\nu$-decomposition, $\mathsf{cr}(\phi')\le c^*\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I''_i))$ for some universal constant $c^*$.
In the second step, we look at the partitioning tree $T$ of Stage 1, and note that now we have, for each instance $I'$ such that $v(I')$ is a leaf in $T$, a solution $\phi(I')$ of $I'$. We now construct a solution for every instance that corresponds to a vertex in $T$ in a bottom-up fashion as follows. In each iteration, we take a vertex $v(I')$ in $T$ with children $v(I''_1),v(I''_2)$ in $T$, such that we have not yet computed a solution of instance $I'$, but we have already computed a solution $\phi(I''_1)$ for instance $I''_1$ and a solution $\phi(I''_2)$ for instance $I''_2$. Note that it is not hard to see that such a vertex has to exist in each iteration.
We not apply the solution-combining algorithm from \Cref{lem: many paths} to the solutions $\phi(I''_1), \phi(I''_2)$, and obtain a solution $\phi(I')$ to instance $I'$. This completes the description of an iteration. We keep performing iterations until we obtain a solution $\phi(I)$ to the input instance $I$.
It remains to upper bound $\mathsf{cr}(\phi(I))$. Recall that at the end of the first step, we have obtained, for each instance $I'\in {\mathcal{I}}'$ that corresponds to a leaf in $T$, a solution $\phi(I')$ to $I'$, such that $\sum_{I'\in {\mathcal{I}}'}\mathsf{cr}(\phi(I'))\le c^*\cdot\sum_{I'\in {\mathcal{I}}^*}\mathsf{cr}(\phi(I'))$. Recall that the depth of tree $T$ is at most $2\mu^{11}$.
Moreover, for each non-leaf instance $I'=(G',\Sigma')$ of $T$, if we denote by $I''_1,I''_2$ the instances that correspond to its children in $T$, then $\mathsf{cr}(\phi(I'))\le \mathsf{cr}(\phi(I''_1))+\mathsf{cr}(\phi(I''_2))+(\mathsf{OPT}_{\mathsf{cnwrs}}(I')+|E(G')|)\cdot \mu^c$.
Via similar arguments as in the proof of \Cref{cor: stage 1} we can show that, for each $1\le i\le \floor{2\mu^{11}}$, if we denote by ${\mathcal{I}}'_i$ be the set of subinstances obtained after iteration $i$, then
\[
\sum_{I'\in {\mathcal{I}}'_{i+1}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot \mu^{c+13}\leq \bigg(\sum_{I'\in {\mathcal{I}}'_{i}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot\mu^{c+13}\bigg)\cdot (1+1/\mu^{12}),
\]
Therefore,
\[
\sum_{I': v(I')\in V(T)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot \mu^{c+13}\leq 4\cdot \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+ m\cdot \mu^{c+13}\bigg).
\]
It follows that
\[
\begin{split}
\mathsf{cr}(\phi(I))
& \le \sum_{I' \in {\mathcal{I}}'} \mathsf{cr}(\phi(I')) +\mathsf{dep}(T)\cdot O\bigg(\sum_{I \notin {\mathcal{I}}'} \big(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c+13}\big)\bigg)\\
& \le c^*\cdot \sum_{I' \in {\mathcal{I}}^*} \mathsf{cr}(\phi(I')) +8\cdot\bigg(\mu^{11}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+\mu^{11}\cdot \mu^{c+13}\cdot |E(G)|\big)\bigg).
\end{split}
\]
Now \Cref{thm: main} follows by setting $c_g$ to be a contant such that $c_g\ge \max\set{c^*, c+25}$.
\fi
\iffalse
It is easy to verify that $T^*$ is
We say that the instances in ${\mathcal{I}}_i(I)$ are \emph{critical subinstance} to instance $I$, iff $\mathsf{cr}(\phi_i)=\min\set{\mathsf{cr}(\phi_1),\ldots, \mathsf{cr}(\phi_L),|E(G)|^2}$.
Consider now the tree $T^*$, which is the subtree of $T$ induced by $v(I^*)$ and all vertices $v(I)$ such that the instance $I$ is critical.
It is easy to see that every leaf of $T^*$ is also a leaf of $T$.
Let $I$ be an instance with $v(I)\in V(T^*)$ and let $I_1,\ldots,I_k$ be its critical subinstances, where $I_i=(G_i,\Sigma_i)$ for each $i$.
From \Cref{thm: main} and since the event ${\cal{E}}$ does not happen,
$\sum_{1\le i\le k}\mathsf{OPT}_{\mathsf{cnwrs}}(I_i)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu +|E(G)|\cdot\mu^{c_g}$ and $\sum_{1\le i\le k}|E(G_i)|\leq |E(G)|\cdot (\log m^*)^{c_g}$ (where we have used the fact that $|E(G)|\le m^*$).
Therefore, since $(\log m^*)^{c_g}<\nu$,
\[
\sum_{1\le i\le k}\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I_i)+|E(G_i)|\cdot \mu^{c_g}\bigg)\leq \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c_g}\bigg)\cdot \nu.
\]
It is easy to see that, if we denote by ${\mathcal{I}}^{(j)}$ the set of all critical instances at recursive level $j$, then
\[
\sum_{I=(G,\Sigma)\in {\mathcal{I}}^{(j)}}\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c_g}\bigg)\leq \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\cdot \mu^{c_g}\bigg)\cdot \nu^{j}.
\]
It follows that, if we denote by ${\mathcal{I}}^*$ the set of all critical subinstances, then
\[
\begin{split}
\sum_{I=(G,\Sigma)\in {\mathcal{I}}^{^*}}\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c_g}\bigg) & \leq \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\cdot \mu^{c_g}\bigg)\cdot 2^{O((\log m^*)^{7/8}\log\log m^*)}\\
& = 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}).
\end{split}
\]
Consider now any instance $I=(G,\Sigma)$ with $v(I)\in V(T^*)$. We first denote $\operatorname{cost}^*(I)=\mu^{c_g}\cdot\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{2c_g}$. From the above discussion,
\[
\sum_{I=(G,\Sigma)\in {\mathcal{I}}^{^*}}\operatorname{cost}^*(I) \leq 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}).
\]
We then define the \emph{height} of $I$, denoted by $h(I)$, to be the largest distance from $v(I)$ to a descendant vertex of $v(I)$ in the tree $T^*$. In particular, if $v(I)$ is a leaf in $T^*$, then $h(I)=0$, and, from \Cref{obs: few recursive levels}, for every non-leaf subinstance $I$, $h(I)\le \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$ always holds.
For an instance $I'$ such that vertex $v(I')$ is a descendant of vertex $v(I)$ in $T^*$, we write $I'\prec I$.
We prove by induction on $h(I)$ that the value of the solution $\phi(I)$ that $\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace(I)$ computes satisfies that
$$\mathsf{cr}(\phi(I))\le \mu^{2c'}\cdot\operatorname{cost}^*(I)+\mu^{2c'}\cdot\sum_{I' \prec I} \operatorname{cost}^*(I')\cdot c_g^{h(I)-h(I')}.$$
The induction base is when $h(I)=0$, so $v(I)$ is a leaf vertex of $T$. We have shown that the claim holds in this case.
Let $I=(G,\Sigma)$ be a non-leaf subinstance with $v(I)\in V(T^*)$ and let $I_1,\ldots,I_k$ be its critical subinstances. If we denote by $\phi(I)$ the drawing of $I$ returned by the algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace, and similarly we define drawings $\set{\phi(I_i)}_{1\le i\le k}$, then from \Cref{thm: main},
\[
\mathsf{cr}(\phi(I)) \le c_g\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I_i))+ \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\bigg)\cdot \mu^{c_g} \le \operatorname{cost}^*(I)+c_g\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I_i)).
\]
Note that for each $1\le i\le k$, $h(I_i)\le h(I)-1$. Therefore, from the induction hypothesis,
\[
\begin{split}
\mathsf{cr}(\phi(I)) & \leq \operatorname{cost}^*(I)+c_g\cdot \sum_{1\le i\le k}\mathsf{cr}(\phi(I_i))\\
& \le \operatorname{cost}^*(I)+c_g\cdot \sum_{1\le i\le k}\bigg(\mu^{2c'}\cdot\operatorname{cost}(I_i)+\mu^{2c'}\cdot\sum_{I' \prec I_i} \operatorname{cost}^*(I')\cdot c_g^{h(I_i)-h(I')}\bigg)\\
& \le \mu^{2c'}\cdot\operatorname{cost}^*(I)+ \mu^{2c'}\cdot\sum_{1\le i\le k}\bigg(c_g\cdot\operatorname{cost}(I_i)+\sum_{I' \prec I_i} \operatorname{cost}^*(I')\cdot c_g^{h(I_i)-h(I')+1}\bigg)\\
& \le \mu^{2c'}\cdot\operatorname{cost}^*(I)+ \mu^{2c'}\cdot\sum_{I'\prec I} \operatorname{cost}^*(I')\cdot c_g^{h(I)-h(I')}.
\end{split}
\]
Therefore, since $h(I^*)\le \frac{(\log m^*)^{1/8}}{c^*\log\log m^*}$, we get that for each $I'\in {\mathcal{I}}^*$, $c_g^{h(I^*)-h(I')}\le \mu$, and so
\[
\begin{split}
\mathsf{cr}(\phi(I^*))\le & \text{ } \mu^{2c'}\cdot\operatorname{cost}^*(I^*)+\mu^{2c'}\cdot\sum_{I' \prec I} \operatorname{cost}^*(I')\cdot c_g^{h(I^*)-h(I')}\\
\le & \text{ } 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}).
\end{split}
\]
\fi
\iffalse
Therefore, if we denote by ${\mathcal{I}}^{**}$ the set of all subinstances $I$ such that $v(I)$ is a leaf of tree $T^*$, then
\[
\begin{split}
\sum_{I\in {\mathcal{I}}^*} \mathsf{cr}(\phi(I))
& \le \mathsf{dep}(T)\cdot\bigg(\sum_{I\in {\mathcal{I}}^{**}} \mathsf{cr}(\phi(I)) +\sum_{I=(G,\Sigma)\in ({\mathcal{I}}^{*}\setminus {\mathcal{I}}^{**})} \big(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\cdot \mu^{c_g}\big)\bigg)\cdot \mu^{c_g}\\
& =\mu^{O(c_g)}\cdot \bigg(\mu^{2c'}\cdot\sum_{I\in {\mathcal{I}}^{**}} \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right})
\bigg)\\
& = 2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right}).
\end{split}
\]
And since $\mathsf{cr}(\phi(I^*))\le \sum_{I\in {\mathcal{I}}^{^*}} \mathsf{cr}(\phi(I)) + \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\bigg)\cdot \mu^{c_g}$, it follows that $\mathsf{cr}(\phi(I^*))=2^{O((\log m^*)^{7/8}\log\log m^*)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I^*)+|E(G^*)|\textsf{right})$.
\fi
\iffalse
\znote{cannot prove by induction, need to re-write from here}
We prove by induction on $h(I)$ that the value of the solution $\phi(I)$ that $\ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace(I)$ computes satisfies that $\mathsf{cr}(\phi(I))\le (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)\cdot \mu^{c_g}\cdot \nu^{h(I)}$. The induction base is when $h(I)=0$, so $v(I)$ is a leaf vertex of $T$. We have shown that the claim holds in this case.
Consider now some instance $I=(G,\Sigma)$ with $h(I)>0$. Assume first that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)> |E(G)|^2/\mu^{64}$. Recall that the algorithm \ensuremath{\mathsf{AlgRecursiveCNwRS}}\xspace considers a trivial solution $\phi'$ to instance $I$, of cost at most $|E(G)|^2$. Since the algorithm eventually returns the best of the considered solutions, the cost of the solution that it returns may not be higher than $|E(G)|^2\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \mu^{64}$.
Therefore, we assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{64}$ from now on. Since we have assumed that Event ${\cal{E}}$ did not happen, neither did ${\cal{E}}(I)$. Therefore, there is an instance $1\leq i\leq L$, such that the family ${\mathcal{I}}_i(I)$ of instances is a perfect family for instance $I$. In particular, from the definition of a perfect family, we have that
$\sum_{I'\in {\mathcal{I}}_i(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu +|E(G)|\cdot\mu^{c_g}$.
Using the induction hypothesis, if we denote by $\phi(I')$ the solution that the algorithm computes for each instance $I'\in {\mathcal{I}}_i(I)$, then:
\[
\begin{split}
\sum_{I'=(G'\Sigma')\in {\mathcal{I}}_i(I)}\mathsf{cr}(\phi(I'))\leq & \sum_{I'=(G'\Sigma')\in {\mathcal{I}}_i(I)}\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot \nu^{h(I)-1}+|E(G')|\cdot \mu^{c_g}\cdot \nu^{h(I)-1}\bigg)\\
\leq & \sum_{I'=(G'\Sigma')\in {\mathcal{I}}_i(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\cdot \nu^{h(I)-1}+\sum_{I'=(G'\Sigma')\in {\mathcal{I}}_i(I)}|E(G')|\cdot \mu^{c_g}\cdot \nu^{h(I)-1}\\
\leq & \text{ }\bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\nu+|E(G)|\cdot \mu^{c_g} \bigg)\cdot \nu^{h(I)-1} + c_g\cdot |E(G)|\cdot \mu^{c_g}\cdot \nu^{h(I)-1}\\
\leq & \text{ }\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu^{h(I)} + (c_g+1)\cdot |E(G)|\cdot \mu^{c_g}\cdot \nu^{h(I)-1}\\
\leq & \text{ }\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu^{h(I)} + |E(G)|\cdot \mu^{c_g}\cdot \nu^{h(I)}.\\
\end{split}
\]
Let $\phi$ be the solution to instance $I$ obtained by applying Algorithm \ensuremath{\mathsf{AlgCombineDrawings}}\xspace to solutions $\phi(I')$ to instances $I'\in {\mathcal{I}}_i(I)$. Then, from \Cref{thm: main}, and since we can assume that $c_g$ is a large enough constant, we are guaranteed that:
\[
\begin{split}
\mathsf{cr}(\phi) &\leq \bigg(\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) \bigg) + (\mathsf{OPT}(I)+|E(G)|)\cdot \mu^{c_g} \\
&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \nu^{h(I)} + |E(G)|\cdot \mu^{c_g}\cdot \nu^{h(I)}+ (\mathsf{OPT}(I)+|E(G)|)\cdot \mu^{c_g}\\
&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\nu^{h(I)}+\mu^{c_g}) + |E(G)|\cdot \mu^{c_g}\cdot (\nu^{h(I)}+1)
\end{split}
\]
Since $h(I^*)\leq \frac{1}{c^*}\sqrt{\frac{\log m^*}{\log\log m^*}}$, we conclude that the cost of the solution $\phi^*$ that the algorithm computes for instance $I^*$ is bounded by:
\[
\begin{split}
(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m^*)\cdot \mu^{O(1)}\cdot (\log m)^{\sqrt{\log m^*/\log\log m^*}}&
\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m^*)\cdot \mu^{O(1)}\cdot 2^{\sqrt{\log m^*\log\log m^*}}\\ &\leq (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m^*)\cdot \mu^{O(1)}.
\end{split} \]
\f
\iffalse{backup: previous stage 1}
\paragraph{Stage 1.}
We initialize the set ${\mathcal{I}}$ to contain a single instance $I=(G,\Sigma)$. The first stage continues as long as there is any instance $\hat I=(\hat G,\hat \Sigma)\in {\mathcal{I}}$, that is an interesting subinstance of $I$, with $|E(\hat G)|>|E(G)|/\mu$. The first stage consists of iterations, where in every iteration we start with an interesting subinstance $\hat I= (\hat G,\hat \Sigma)\in {\mathcal{I}}$ with $|E(\hat G)|>m/\mu$.
We apply the algorithm from \Cref{lem: many paths} to instance $\hat I$,
obtaining two new instances $\hat I'=(\hat G',\hat \Sigma'),\hat I''=(\hat G'',\hat \Sigma'')$. We then replace $\hat I$ with $\hat I',\hat I''$ in ${\mathcal{I}}$, and continue to the next iteration. Let $i$ denote the index of the current iteration.
We say that the bad event ${\cal{E}}_i$ happens if (i) $|E(\hat G)|>\mu^{39}$; (ii) $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)<|E(\hat G)|^2/\mu^{40}$; and (iii) $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')+\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I'') > \mathsf{OPT}_{\mathsf{cnwrs}}(\hat I) \cdot(1+1/\mu^{12})+m \cdot \mu^{c}$. From \Cref{lem: many paths}, the probability that ${\cal{E}}_i$ happens is bounded by $1/(40\mu^{11})$.
This completes the description of the algorithm for Stage 1. In order to analyze it, it is convenient to define a partitioning tree $T$ associated with it.
For every instance $\hat I$ that ever belonged to set ${\mathcal{I}}$ over the course of the Stage 1 algorithm, we add a vertex $v(\hat I)$ to the tree. If the algorithm from \Cref{lem: many paths} was applied to instance $\hat I$, producing two new instances $\hat I'$ and $\hat I''$, then vertices $v(\hat I'),v(\hat I'')$ become children of vertex $v(\hat I)$ in the tree. The root of the tree is the vertex $v(I)$.
We say that bad event ${\cal{E}}$ happens if bad event ${\cal{E}}_i$ happens for any iteration $i\leq 4\mu^{11}$ of Stage 1. From the union bound over all $1\le i\leq 4\mu^{11}$, we get that $\Pr[{\cal{E}}]\le 1/10$.
Let $T'$ be the subtree of $T$ that is induced by all vertices $v(\hat I)$, such that $\hat I$ belonged to ${\mathcal{I}}$ at some point before the completion of iteration $\ceil{2\mu^{11}}$.
Assume now that bad event $\hat {\cal{E}}$ did not happen. Since we only apply the algorithm from \Cref{lem: many paths} to instances $(\hat G,\hat \Sigma)$ with $|E(\hat G)|>|E(G)|/\mu$, every instance $\hat I'=(\hat G',\hat \Sigma')$ with $v(I')\in T'$ has $|E(\hat G')|\geq |E(G)|/\mu^{11}$. But then the number of the leaf vertices in tree $T'$ is bounded by $\mu^{11}$, and the total number of vertices in tree $T'$ must be bounded by $2\mu^{11}$, since every inner vertex of $T'$ has at least two children.
This can only happen if the algorithm terminates before the $\floor{2\mu^{11}}$-th iteration. Therefore, if Event ${\cal{E}}$ does not happen, the algorithm has fewer than $2\mu^{11}$ iterations.
Moreover, since every instance $\hat I=(\hat G,\hat \Sigma)$ that ever belonged to ${\mathcal{I}}$, $|E(\hat G)|\geq |E(G)|/\mu^{11}$ holds, and since we are guaranteed that $|E(G)|\geq \mu^{c'}\geq \mu^{50}$, we conclude that $|E(\hat G)|\geq \mu^{39}$ must hold.
We use the following claim.
\begin{claim}\label{claim: few crossings in each instance}
If event ${\cal{E}}$ did not happen, then for every instance $\hat I=(\hat G,\hat \Sigma)$ that ever belonged to the set ${\mathcal{I}}$, $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq |E(\hat G)|^2/\mu^{40}$.
\end{claim}
\begin{proof}
We say that a vertex $v(\hat I)$ lies at level $\ell$ of the tree $T$ iff the distance from $v(\hat I)$ to the root $v(I)$ of the tree is $\ell$. We will prove that, for all $0\leq \ell \leq \floor{2\mu^{11}}$, for every vertex $v(\hat I)$ lying at level $\ell$ of the tree $T$, with $\hat I=(\hat G,\hat\Sigma)$,
\begin{equation}
\label{eqn: induction}
\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell}+m\cdot\ell\cdot\mu^{c+13},
\end{equation}
The proof is by induction on $\ell$.
The base case is when $\ell=0$, and $\hat I$ is the input instance $I=(G,\Sigma)$. Recall that, from the statement of \Cref{thm: main}, $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)\leq m^2/\mu^{63}\leq m^2/\mu^{40}$ as required, and it is immediate to verify that the inequality \ref{eqn: induction} holds.
Consider now some integer $0\leq \ell \leq \floor{2\mu^{11}}$. We assume that the claim holds for all integers below $\ell$, and prove it for integer $\ell$. Let $\hat I'=(\hat G',\hat \Sigma')$ be any instance, whose corresponding vertex $v(\hat I')$ lies at level $\ell$ of the tree $T$, and let $v(\hat I)$ be the parent vertex of $v(\hat I')$ in the tree, where $\hat I=(\hat G,\hat \Sigma)$. As observed already, if Event ${\cal{E}}$ does not happen, $|E(\hat G)|\geq \mu^{39}$ must hold, and from the induction hypothesis, $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell-1}+m\cdot(\ell-1)\cdot\mu^{c+13}$ and $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)\leq |E(\hat G)|^2/\mu^{40}$ must hold. If we denote by $v(\hat I'')$ the second child vertex of $v(\hat I)$, then, since we have assumed that Event $\hat {\cal{E}}$ does not happen, $\mathsf{OPT}_{\mathsf{cnwrs}}(I')+\mathsf{OPT}_{\mathsf{cnwrs}}(I'')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I) \cdot(1+1/\mu^{12})+m\cdot\mu^{c}$ must hold. Therefore,
\[
\begin{split}
\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq & \text{ } \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell-1}+m\cdot(\ell-1)\cdot\mu^{c+13}\bigg) \cdot(1+1/\mu^{12})+m\cdot\mu^{c}\\
\leq & \text{ } \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell}+\bigg(m\cdot(\ell-1)\cdot\mu^{c+13}\bigg) \cdot(1+1/\mu^{12})+m\cdot\mu^{c}\\
\leq & \text{ } \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell}+ m\cdot\ell\cdot\mu^{c+13}.
\end{split}
\]
Lastly, as observed already, $|E(\hat G')|\geq m/\mu^{11}$.
We then get that:
%
\[
\begin{split}
\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I')&\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot(1+1/\mu^{12})^{\ell}+ m\cdot\ell\cdot\mu^{c+13}\\
&\leq 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ 2 m\cdot\mu^{c+24}\\
&\leq \frac{4m^2}{\mu^{63}} \leq \frac{4\cdot |E(\hat G')|^2\cdot \mu^{22}}{\mu^{63}}\leq \frac{|E(\hat G')|^2}{\mu^{40}}.
\end{split}
\]
%
Here we have used the fact that $\ell\le \floor{2\mu^{11}}$ and the facts $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq |E(G)|^2/\mu^{63}$ and $|E(G)|\geq \mu^{c'}$ from the statement of \Cref{thm: main}, together with the assumption that $c'>c+50$.
\end{proof}
\begin{claim}\label{cor: stage 1}
If Event ${\cal{E}}$ did not happen, then $\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ 2\cdot m\cdot \mu^{c+13}$.
\end{claim}
\begin{proof}
For each $1\le i\le \floor{2\mu^{11}}$, we denote by ${\mathcal{I}}'_i$ be the set of subinstances obtained after iteration $i$. Therefore, from \Cref{lem: many paths},
\[
\sum_{I'\in {\mathcal{I}}'_{i+1}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \bigg(\sum_{I'\in {\mathcal{I}}'_{i}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\bigg)\cdot (1+1/\mu^{12})+m\cdot\mu^{c},
\]
and so
\[
\sum_{I'\in {\mathcal{I}}'_{i+1}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot \mu^{c+13}\leq \bigg(\sum_{I'\in {\mathcal{I}}'_{i}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')+m\cdot\mu^{c+13}\bigg)\cdot (1+1/\mu^{12}),
\]
Therefore,
\[
\sum_{I'\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\cdot\mu^{c+13}\bigg)\cdot (1+1/\mu^{12})^{\floor{2\mu^{11}}}\le 2\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(I)+ 2\cdot m\cdot \mu^{c+13}.
\]
\end{proof}
We partition the instances of ${\mathcal{I}}'$ into two subsets: set ${\mathcal{I}}_1'$ contains all instances $I'=(G',\Sigma')$ with $|E(G')|\leq m/\mu$, while set ${\mathcal{I}}_2'$ contains all remaining instances. Notice that by the definition of the Stage 1 algorithm, every instance in ${\mathcal{I}}_2'$ is a non-interesting subinstance of the input intance $I$.
\fi
\subsubsection{Step 2: Constructing the Paths of ${\mathcal{P}}^{\mathsf{in}}$ and the Auxiliary Cycles}
Consider some edge $e=(u,v)\in \hat E$, and assume that $u$ is the left endpoint of $e$. Assume that $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$, and denote by
$\hat e_{i''-1}$ the first edge on path $P^{\mathsf{out}}(e)$, and by $\hat e_{j''}$ the last edge on path $P^{\mathsf{out}}(e)$.
In this step we will construct another path $P^{\mathsf{in}}(e)$, whose first edge is $\hat e_{i''-1}$ and last edge is $\hat e_{j''}$. In order to do so, we will select, for all $i''\leq z<j''$, some edge $\hat e_z\in E_z$, that we assign to the edge $e$, and we will compute a path $R_z(e)$ (that we call a \emph{segment}), whose first edge is $\hat e_{z-1}$, last edge is $\hat e_z$, and all inner vertices are contained in $S_z$. The final path $P^{\mathsf{in}}(e)$ will be obtained by concatenating the segments $R_{i''}(e),\ldots,R_{j''}(e)$. Note that path $P^{\mathsf{in}}(e)$ has $\hat e_{i''-1}$ and $\hat e_{j''}$ as its first and last edges. By concatenating the paths $P^{\mathsf{in}}(e)$ and $P^{\mathsf{out}}(e)$, we will then obtain the auxiliary cycle $W(e)$. Our goal in constructing the set ${\mathcal{P}}^{\mathsf{in}}=\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$ of paths is to ensure that these paths cause low congestion, and that these paths are \emph{mostly} non-transversal with respect to $\Sigma$. In fact, we will ensure that, for every pair $P,P'\in {\mathcal{P}}^{\mathsf{in}}$ of such paths, there is at most one vertex $v$, such that the intersection of $P$ and $P'$ at $v$ is transversal. Intuitively, the resulting auxiliary cycles in $\set{W(e)\mid e\in \hat E}$ will be exploited in order to show the existence of cheap solutions to the subinstances of the input instance $I$ that we compute. Each transversal intersection between a pair of such cycles may give rise to a crossing in these solutions, which motivates the requirement that the paths in ${\mathcal{P}}^{\mathsf{in}}$ have few transversal intesrections is low.
Consider again an edge $e\in \hat E$, and assume that $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$. Recall that we have already defined edges $\hat e_{i''-1}\in \delta(S_{i''-1})$ and $\hat e_{j''}\in \delta(S_{j''-1})$. We construct a collection $\tilde {\mathcal{R}}(e)=\set{R_{i''}(e),\ldots, R_{j''}(e)}$ of paths, and define, for all $i''\leq z<j''$, edge $\hat e_z\in E_z$, such that, for all $i''\leq z\leq j''$, path $R_z(e)$ connects edge $\hat e_{z-1}$ to edge $\hat e_z$, and its inner vertices lie in $S_z$. In order to do so, we initially set $\tilde {\mathcal{R}}(e)=\emptyset$ for every edge $e\in \hat E$. We then process indices $1\leq z\leq r$ one by one. When index $z$ is processed, we will define, for every edge $e\in \hat E$ with $z\in \operatorname{span}''(e)$ or $z-1\in \operatorname{span}''(e)$, the segment $R_z(e)$; if $z\in \operatorname{span}''(e)$, we will also define the edge $\hat e_z\in E_z$, which is the last edge on path $R_z(e)$.
We will ensure that every edge $e'\in E_z$ is assigned to at most $O(\log^{18}m)$ edges of $\hat E$.
We now describe an interation where index $1\leq z\leq r$ is processed.
\paragraph{Iteration Description.}
We fix an index $1\leq z\leq r$, and describe an iteration for processing index $z$. Let $A_z\subseteq \hat E$ be the set of all edges $e\in \hat E$, with $z\in \operatorname{span}''(e)$. Note that for every edge $e\in A_z$, the corresponding edge $\hat e_{z-1}\in E_{z-1}$ is already fixed. Let $A'_z\subseteq \hat E\setminus A_z$ be the set of all edges $e\in \hat E$, such that $z-1\in \operatorname{span}''(e)$ but $z\not\in \operatorname{span}''(e)$. Notice that, if $e\in A'_z$, then both edges $\hat e_{z-1},\hat e_z\in \delta_G(S_z)$ are already fixed (in this case, edge $\hat e_z$ is the last edge on path $P^{\mathsf{out}}(e)$).
Consider the augmentation $S_z^+$ of the cluster $S_z$, that we denote for convenience by $H$. Recall that, in order to obtain graph $H$, we start by subdividing every edge $e'\in \delta_G(S_z)$ by vertex $t_{e'}$, and then denote by $T=\set{t_{e'}\mid e'\in \delta_G(S_z)}$ the set of newly added vertices, that we call \emph{terminals}. We then let $H$ be the subgraph of the resulting graph induced by $V(S_z)\cup T$.
Recall that, from the definition of nice witness structure, cluster $S_z$ has the $\alpha^*=\Omega(1/\log^{12}m)$-bandwidth property in $G$, and so, from \Cref{obs: wl-bw}, vertex set $T$ is $\alpha^*$-well-linked in $H$.
We now define a collection $M$ of pairs of terminals, that we call \emph{demand pairs}, that are associated with the edges of $A'_z$. Consider an edge $e\in A'_z$, and recall that edges $\hat e_{z-1},\hat e_z\in \delta_G(S_z)$ are already defined. The demand pair associated with edge $e$ is $(x_e,y_e)$, where $x_e=t_{\hat e_{z-1}}$ (the terminal vertex associated with edge $\hat e_{z-1}$), and $y_e=t_{\hat e_z}$ (the terminal vertex associated with edge $\hat e_z$). We then set $M=\set{(x_e,y_e)\mid e\in A'_z}$.
Recall that the edges of ${\mathcal{P}}^{\mathsf{out}}$ cause congestion at most $\eta=O(\log^{18}m)$, and every edge of $E_{z-1}$ is assigned to at most $\eta$ edges of $\hat E$. Therefore, a terminal $t_{e'}$ may participate in at most $\eta$ pairs in $M$. Using a standard greedy algorithm, we can now compute $2\eta$ sets of terminal pairs $M_1,\ldots,M_{2\eta}$, such that $M=\bigcup_{a=1}^{2\eta}M_a$, and, for all $1\leq a\leq 2\eta$, each terminal participates in at most one pair in $M_a$. For each $1\leq a\leq 2\eta$, we use the algorithm from \Cref{cor: routing well linked vertex set}, to compute a collection ${\mathcal{R}}(M_a)=\set{R(x,y)\mid (x,y)\in M_a}$ of paths in graph $H$, where for every pair $(x,y)\in M_a$, path $R(x,y)$ connects $x$ to $y$. Since the vertices of $T$ are $\alpha^*$-well-linked in $G$, with high probability, the paths in ${\mathcal{R}}(M_a)$ cause congestion $O(\log^{4}m/\alpha^*)=O(\log^{16}m)$. If the paths in ${\mathcal{R}}(M_a)$ cause a higher congestion, then we terminate the algorithm and return FAIL.
Note that the set $\bigcup_{a=1}^{2\eta}{\mathcal{R}}(M_a)$ of paths in graph $H$ naturally defines a set ${\mathcal{R}}''=\set{R(e)\mid e\in A'_z}$ of paths in graph $G$, where, for every edge $e\in A'_z$, path $R(e)$ has $\hat e_{z-1}$ as its first edge and $\hat e_z$ as its last edge, while all inner vertices of $R(e)$ lie in $S_z$. From the above discussion, the paths in ${\mathcal{R}}''$ cause congestion at most $\eta'=2\eta\cdot O(\log^{16}m)=O(\log^{34}m)$.
Next, we consider the set $A_z\subseteq E_{z-1}$ of edges. Let $X$ be a multiset of vertices of $T$ that contains, for every edge $e'\in A_z$, the corresponding vertex $t_{e'}$. Since every edge of $E_{z-1}$ may only be assigned to at most $\eta$ edges of $\hat E$, a vertex may appear in set $X$ at most $\eta$ times.
Recall that $|A_z|=N'_z$, and, from \Cref{claim: enough segments}, $N'_z\leq \eta\cdot |E_z|$. Therefore, we can define a multiset $Y$ that contains $|A|$ elements, each of which is a vertex from $\set{t_{e'}\mid e'\in E_z}$, such that at most $\eta$ copies of each such vertex $t_{e'}$ appear in set $Y$. We let $M'$ be an arbitrary matching between elements of $X$ and elements of $Y$. Using the same procedure as the one employed for the edges of $A'_z$, we construct a set ${\mathcal{R}}'=\set{R(e)\mid e\in A_z}$ of paths in graph $G$, where, for every edge $e\in A_z$, path $R(e)$ has $\hat e_{z-1}$ as its first edge and some edge of $E_{z}$ as its last edge, while all inner vertices of $R(e)$ lie in $S_z$. Additionally, the paths in ${\mathcal{R}}''$ cause congestion at most $\eta'$, and every edge of $E_z$ appears on at most $\eta$ edges of ${\mathcal{R}}$. (As before, if the paths in set ${\mathcal{R}}''$ cause a higher congestion, we terminate the algorithm and return FAIL.)
Note that we can assume without loss of generality that all paths in ${\mathcal{R}}'\cup {\mathcal{R}}''$ are simple paths.
To summarize, we have now constructed two sets ${\mathcal{R}}'=\set{R(e)\mid e\in A_z},{\mathcal{R}}''=\set{R(e)\mid e\in A'_z}$ of paths, with the following properties:
\begin{properties}{I}
\item Paths in each of the sets ${\mathcal{R}}',{\mathcal{R}}''$ cause congestion at most $\eta'=O(\log^{34}m)$; \label{inv: edge congestion}
\item For every edge $e\in A_z$, path $R(e)\in {\mathcal{R}}'$ has $\hat e_{z-1}$ as its first edge, some edge of $E_{z}$ as its last edge, and all inner vertices of $R(e)$ lie in $S_z$; \label{inv: first kind of paths}
\item Every edge of $E_z$ participates in at most $\eta$ paths of ${\mathcal{R}}'$;
\item For every edge $e\in A'_z$, path $R(e)\in {\mathcal{R}}''$ has $\hat e_{z-1}$ as its first edge, $\hat e_z$ as its last edge, and all inner vertices of $R(e)$ lie in $S_z$; and \label{inv: second kind of paths}
\item All paths in set ${\mathcal{R}}'\cup {\mathcal{R}}''$ are simple. \label{inv: simple paths}
\end{properties}
Next, we will iteratively modify the paths in ${\mathcal{R}}'\cup {\mathcal{R}}''$, while ensuring that Properties \ref{inv: edge congestion}--\ref{inv: simple paths} hold at the end of each iteration. In every iteration, we will attempt to reduce the number of transversal intersections between the paths of ${\mathcal{R}}'\cup {\mathcal{R}}''$. In fact we will guarantee that, after each iteration, either $\sum_{R\in {\mathcal{R}}'\cup {\mathcal{R}}''}|E(R)|$ decreases, or $\sum_{R\in {\mathcal{R}}'\cup {\mathcal{R}}''}|E(R)|$ remains unchanged, and the number of triples in set $\Pi^{T}({\mathcal{R}}'\cup {\mathcal{R}}'')$ strictly decreases (see definition immediately after \Cref{def: non-transversal paths})
We now describe a single iteration. Assume first that there are two paths $R(e),R(e')\in {\mathcal{R}}'$, and some vertex $v$ that is an inner vertex of both $R(e)$ and $R(e')$, such that the intersection of $R(e)$ with $R(e')$ at $v$ is transversal. In this case, we splice paths $R(e)$ and $R(e')$ at vertex $v$ (see \Cref{subsec: non-transversal paths and splicing}), obtaining two new paths. The first path, that replaces $R(e)$ in ${\mathcal{R}}'$, originates at edge $\hat e_{z-1}$ and terminates at the edge of $E_z$ that served as the last edge of $R(e')$. The second path, that replaces $R(e')$ in ${\mathcal{R}}'$, originates at edge $e'_{z-1}$, and terminates at the edge of $E_z$ that served as the last edge of the original path $R(e)$. Both paths only contain vertices of $S_z$ as inner vertices. From \Cref{obs: splicing}, either at least one of the two new paths $R(e), R(e')$ becomes a non-simple path; or both paths remain simple paths, but $|\Pi^T({\mathcal{R}}'\cup {\mathcal{R}}'')|$ decreases. Notice that, in the latter case, $\sum_{R\in {\mathcal{R}}'\cup {\mathcal{R}}''}|E(R)|$ remains unchanged. If the former case happens, then we remove cycles from paths $R(e),R(e')$, until they become simple paths. In this case, $\sum_{R\in {\mathcal{R}}'\cup {\mathcal{R}}''}|E(R)|$ decreases. In either case, it is easy to verify that Invariants \ref{inv: edge congestion}--\ref{inv: simple paths} continue to hold. We then proceed to the next iteration.
Assume now that there is a path $R(e)\in {\mathcal{R}}'\cup {\mathcal{R}}''$ and another path $R(e')\in{\mathcal{R}}''$, and two distinct vertices $v,v'$, both of which are inner vertices on both $R(e)$ and $R(e')$, such that $R(e)$ and $R(e')$ intersect transversally at both $v$ and $v'$. Let $Q$ be the subpath of $R(e)$ between $v$ and $v'$, and let $Q'$ be the subpath of $R(e')$ between $v$ and $v'$. We splice the paths $R(e)$ and $R(e')$ at both $v$ and $v'$. Equivalently, we modify path $R(e)$ by replacing its segment $Q$ with $Q'$, and we modify path $R(e')$ by replacing its segment $Q'$ with $Q$. Note that the first and the last edge on each path remains the same, and the congestion caused by the set ${\mathcal{R}}'\cup {\mathcal{R}}''$ of paths remains the same. If any of the resulting paths $R(e),R(e')$ becomes a non-simple path, then we delete cycles from it, until it becomes a simple path. In this case, $\sum_{R\in {\mathcal{R}}'\cup {\mathcal{R}}''}|E(R)|$ decreases. Otherwise, we can use \Cref{obs: splicing} to conclude that $|\Pi^T({\mathcal{R}}'\cup {\mathcal{R}}'')$ has decreased. Indeed, it is easy to verify that, for any vertex $v''\in V(S_z)\setminus\set{v,v'}$, the number of triples $(R_1,R_2,v'')\in \Pi^T({\mathcal{R}}'\cup {\mathcal{R}}'')$ did not grow. The number of triples $(R_1,R_2,v)\in \Pi^T({\mathcal{R}}'\cup {\mathcal{R}}'')$, and the number of triples $(R_1,R_2,v')\in \Pi^T({\mathcal{R}}'\cup {\mathcal{R}}'')$ have both decreased (as can be seen by applying \Cref{obs: splicing} to the set of paths that contains, for every path $R^*\in {\mathcal{R}}'\cup {\mathcal{R}}''$ with $v\in R^*$, a subpath of $R^*$ consisting of the two edges of $R^*$ incident to $v$, and doing the same for vertex $v'$). This completes the description of an iteration. It is easy to verify that Invariants \ref{inv: edge congestion}--\ref{inv: simple paths} continue to hold.
The algorithm for processing the index $z$ terminates when the path set ${\mathcal{R}}'$ becomes non-traversal with respect to $\Sigma$, and, for every pair $R\in {\mathcal{R}}'\cup{\mathcal{R}}''$, $R'\in {\mathcal{R}}''$ of paths, there is at most one vertex $v$ such that the intersection of $R$ and $R'$ at $v$ is transversal. We then denote ${\mathcal{R}}_z={\mathcal{R}}'\cup {\mathcal{R}}''$. For every edge $e\in A_z\cup A'_z$, we set $R_z(e)=R(e)$ (the unique path in ${\mathcal{R}}_z$ that originates at edge $\hat e_{z-1}$). If $e\in A'_z$, then path $R(e)$ is guaranteed to terminate at edge $\hat e_z$, from Invariant \ref{inv: second kind of paths}. If $e\in A_z$, then we let $\hat e_z$ be the last edge on path $R_z(e)$. We then add path $R_z(e)$ to set $\tilde {\mathcal{R}}(e)$.
Once all indices $1\leq z\leq r$ are processed, we obtain, for every edge $e\in \hat E$, the desired set $\tilde {\mathcal{R}}(e)$ of paths. If $\operatorname{span}(e)=\set{i'',\ldots,j''-1}$, then $\tilde {\mathcal{R}}(e)=\set{R_{i''}(e),\ldots,R_{j''}(e)}$. We then let $P^{\mathsf{in}}(e)$ be the path obtained by concatenating the paths in $\tilde {\mathcal{R}}(e)$. Recall that the first edge on $P^{\mathsf{in}}(e)$ is $\hat e_{i''-1}$, which is the first edge of $P^{\mathsf{out}}(e)$, and similary, the last edge on $P^{\mathsf{in}}(e)$ is $\hat e_{j''}$, the last edge of $P^{\mathsf{out}}(e)$. We obtain the auxiliary cycle $W(e)$ by concatenating the paths $P^{\mathsf{in}}(e)$ and $P^{\mathsf{out}}(e)$ (after deleting the extra copies of edges $\hat e_{i''-1},\hat e_{j''}$). It is immediate to verify that cycle $W(e)$ is a simple cycle. Lastly, we set ${\mathcal{P}}^{\mathsf{in}}=\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$ and ${\mathcal{W}}=\set{W(e)\mid e\in \hat E}$. We refer to ${\mathcal{W}}$ as the \emph{set of auxiliary cycles}.
From the above discussion and \Cref{claim: computing out-paths}, we obtain the following immediate observation:
\begin{observation}\label{obs: bound congestion of cycles}
Every edge $e\in \bigcup_{z=1}^rE(S_z)$ appears on at most $\eta'=O(\log^{34}m)$ cycles of ${\mathcal{W}}$. Every edge $e\in E(G)\setminus \textsf{left} ( \bigcup_{z=1}^rE(S_z) \textsf{right} )$ appears on at most $\eta=O(\log^{18}m)$ cycles of ${\mathcal{W}}$.
\end{observation}
Recall that we denoted, for each index $1\leq z<r$, by $\hat E_z\subseteq \hat E$ the set of all edges $e\in \hat E$, with $z\in \operatorname{span}(e)$.
We need the following observation.
\begin{observation}
\label{obs: auxiliary cycles non-transversal at at most one}
For every index $1\leq z<r$, for every pair $e, e'\in \hat E_z$ of distinct edges, there is at most one vertex $v\in V(W(e))\cap V(W(e'))$, such that the intersection of cycles $W(e)$ and $W(e')$ is transversal at $v$. If such a vertex $v$ exists, then $v\in S_j$ for some index $z< j<r$, and either $j-1$ is the last index in both $\operatorname{span}''(e),\operatorname{span}''(e')$, or $j-1$ is the last index in one of these sets, while $j$ belongs to another.
\end{observation}
\begin{proof}
Fix an index $1\leq z<r$ and a pair $e,e'\in \hat E_z$ of edges. Let $v$ be any vertex that lies on both $W(e)$ and $W(e')$. We consider two cases. The first case is when $v\in V''$, that is, there is some index $1\leq z'\leq r$, such that $v\in V(\tilde S_{z})\setminus V(S_{z})$. Since all inner vertices of paths $P^{\mathsf{in}}(e),P^{\mathsf{in}}(e')$ lie in $V'$, vertex $v$ must be an inner vertex of both $P^{\mathsf{out}}(e)$ and $P^{\mathsf{out}}(e')$. From \Cref{claim: out-paths non-transversal}, the intersection of $P^{\mathsf{out}}(e)$ and $P^{\mathsf{out}}(e')$ at vertex $v$ is non-transversal. Therefore, the intersection of $W(e)$ and $W(e')$ at $v$ is non-transversal.
The second case is when there is some index $1\leq z'\leq r$, such that $v\in V(S_{z'})$. Assume that the intersection of $W(e)$ and $W(e')$ at $v$ is transversal. From our construction of the path set ${\mathcal{R}}_{z'}$, it must be the case that at least one of the edges $e,e'$ lies in $A'_{z'}$.
Assume without loss of generality that this edge is $e$. Notice that, from the construction of set $A'_{z'}$, the last index of $\operatorname{span}''(e)$ must be $z'-1$. Since $e'\in A_{z'}\cup A'_{z'}$, either $z'\in \operatorname{span}''(e')$, or the last index in $\operatorname{span}''(e')$ is $z'-1$. Therefore, if we denote by $j$ the last index of $\operatorname{span}''(e')$, then $j\geq z'-1$ must hold. From our construction, $v$ is the only vertex of $S_{z'}$, such that the intersection of $P^{\mathsf{in}}(e)$ and $P^{\mathsf{in}}(e')$ at $v$ is transversal. Moreover, since $j\geq z'-1$, and $z'-1$ is the last index in $\operatorname{span}''(e)$, for every index $z''\neq z'$, for every vertex $v'\in S_{z''}$ that lies on both $P^{\mathsf{in}}(e)$ and $P^{\mathsf{in}}(e')$, the intersection of the two paths at $v'$ must be non-transversal.
\end{proof}
\iffalse
Nothing below here
Note that in this case, $v$ may appear up to twice on $P^{\mathsf{in}}(e)$, and up to twice on $P^{\mathsf{in}}(e')$. We fix one appearance of $v$ on $P^{\mathsf{in}}(e)$ and one appearance of $v$ on $P^{\mathsf{in}}(e')$. There must be a path $Q\in {\mathcal{Q}}_{t'}$, such that $Q\subseteq P^{\mathsf{in}}(e)$, and it contains the selected appearance of $v$ on $P^{\mathsf{in}}(e)$, and similarly, there must be a path $Q'\in {\mathcal{Q}}_{t'}$, with $Q'\subseteq P^{\mathsf{in}}(e')$, such that $Q'$ contains the selected appearance of $v$ on $P^{\mathsf{in}}(e')$. Recall that all paths in ${\mathcal{Q}}_{t'}$ are non-transversal with respect to $\Sigma$. Therefore, the intersection of cycles $W(e)$ and $W(e')$ is non-transversal at $v$.
The third case is when there is some index $z'$, such that $v=u_{t'}$. If $z'\in \operatorname{span}''(e)\cap \operatorname{span}''(e')$ holds, then, from the way the segments of ${\mathcal{R}}_{t'}$ were assigned to edges of $\hat E$, it is immediate that
the intersection of cycles $W(e)$ and $W(e')$ is non-transversal at $u_{t'}$.
Lastly, assume that $z'\not\in \operatorname{span}''(e)$ or $z'\not\in \operatorname{span}''(e)$. This may only happen if $u_{t'}$ is the last vertex on $P^{\mathsf{out}}(e)$ or $P^{\mathsf{out}}(e')$. If $u_{t'}$ is the last vertex on both these paths, then this is the unique vertex at which the intersection of the cycles $W(e)$ and $W(e')$ may be transversal.
Assume now that $u_{t'}$ is the last vertex on one of these two paths (say $P^{\mathsf{out}}(e)$), but it is not the last vertex on $P^{\mathsf{out}}(e')$. Assume that the last vertex on $P^{\mathsf{out}}(e')$ is $u_{t''}$. Since vertex $u_{t'}$ lies on path $P^{\mathsf{in}}(e')$, $z''>t'$ must hold, and so $u_{t''}$ may not lie on path $P^{\mathsf{in}}(e)$. Therefore, $u_{t'}$ is the unique vertex at which the intersection of the cycles $W(e)$ and $W(e')$ may be transversal.
\iffalse
Assume that $e_1$ is an edge of $\hat E_{i_1,j_1}$ and $e_2$ is an edge of $\hat E_{i_2,j_2}$, such that $j_1\ge j_2$, so $u_{j_2}$ is a common vertex of auxiliary cycles $W(e_1)$ and $W(e_2)$.
We will show that, the intersection between $W(e_1)$ and $W(e_2)$ is non-transversal at all their shared vertices except (possibly) for $u_{j_2}$, thus completing the proof of \Cref{obs: auxiliary cycles non-transversal at at most one}.
Recall that $W(e_1)=P^{\mathsf{out}}(e_1)\cup P^{\mathsf{in}}(e_1)$ and $W(e_2)=P^{\mathsf{out}}(e_2)\cup P^{\mathsf{in}}(e_2)$, and furthermore $P^{\mathsf{out}}(e_1)=P^{\mathsf{out}}_{\operatorname{left}}(e_1)\cup P^{\mathsf{out}}_{\operatorname{mid}}(e_1)\cup P^{\mathsf{out}}_{\operatorname{right}}(e_1)$ and $P^{\mathsf{out}}(e_2)=P^{\mathsf{out}}_{\operatorname{left}}(e_2)\cup P^{\mathsf{out}}_{\operatorname{mid}}(e_2)\cup P^{\mathsf{out}}_{\operatorname{right}}(e_2)$.
Note that the paths $P^{\mathsf{out}}_{\operatorname{mid}}(e_1),P^{\mathsf{out}}_{\operatorname{mid}}(e_2)$ are internally disjoint from $V'$, while the paths of $\set{P^{\mathsf{out}}_{\operatorname{left}}(e_k),P^{\mathsf{out}}_{\operatorname{right}}(e_k),P^{\mathsf{in}}(e_k)}_{k\in \set{1,2}}$ only use vertices of $V'$.
First, from Step 2, the paths $P^{\mathsf{out}}_{\operatorname{mid}}(e_1),P^{\mathsf{out}}_{\operatorname{mid}}(e_2)$ are non-transversal with respect to $\Sigma$.
Second, note that for each $1\le i\le r-1$, the set ${\mathcal{R}}^*_{(i)}$ of paths are non-transversal with respect to $\Sigma$.
Since the $i$-th segments of paths in ${\mathcal{P}}^{\mathsf{in}}$ are paths in ${\mathcal{R}}^*_{(i)}$, it follows that the paths $P^{\mathsf{in}}(e_1)$ and $P^{\mathsf{in}}(e_2)$ are non-transversal at all vertices of $V(G')\setminus\set{u_1,\ldots,u_r}$.
Third, from the algorithm and \Cref{obs:rerouting_matching_cong}, it is easy to verify that the paths of ${\mathcal{P}}^{\mathsf{in}}$ are also non-transversal at $u_1,\ldots,u_{j_2-1}$.
Lastly, since for each $1\le i\le r$, the set ${\mathcal{Q}}_{i}$ of paths are non-transversal with respect to $\Sigma$, and the intersection between paths of set ${\mathcal{R}}^*_{(i)}$ and cluster $S_i$ are paths of ${\mathcal{Q}}_i$, we get that the intersection between cycles $W(e_1)$ and $W(e_2)$ is non-transversal at all vertices of $V(P^{\mathsf{out}}_{\operatorname{left}}(e_1))\cup V(P^{\mathsf{out}}_{\operatorname{left}}(e_2))$.
Altogether, we get that the intersection between $W(e_1)$ and $W(e_2)$ is non-transversal at all their shared vertices except (possibly) for $u_{j_2}$.\fi
\end{proof}
where for all $i''\leq z<j''$, $\tilde R_z(e)$ is a segment from the set ${\mathcal{R}}_z$. The final path $P^{\mathsf{in}}(e)$ will then be obtained by concatenating all segments in $\tilde {\mathcal{R}}(e)$.
We construct the collections $\tilde {\mathcal{R}}(e)$ of segments for all edges $e\in \hat E$ gradually, over the course of $r-1$ iterations.
Initially, we set $\tilde {\mathcal{R}}(e)=\emptyset$ for every edge $e\in \hat E$.
For all $1\leq z <r$, over the course of iteration $z$, we will define the segment $\tilde R_z(e)\in {\mathcal{R}}_z$ for every edge $e\in \hat E$ with $z\in \operatorname{span}''(e)$; the segment $\tilde R_z(e)$ is then added to set $\tilde {\mathcal{R}}(e)$.
We now fix an index $1\leq z<r$, and describe the execution of the $z$zh iteration. We construct a multiset $A\subseteq \delta_G(u_z)$ of edges, as follows. Consider any edge $e\in \hat E$, with $z\in \operatorname{span}''(e)$. If $z$ is the first index in $\operatorname{span}''(e)$, then we let $e'$ be the first edge on path $P^{\mathsf{out}}(e)$, which must be incident to $u_z$. Otherwise, segment $\tilde R_{t-1}(e)$ is defined and belongs to $\tilde {\mathcal{R}}(e)$. We then let $e'$ be the last edge on segment $\tilde R_{t-1}(e)$, which must be incident to $u_z$. We then add edge $e'$ to set $A$. We say that this copy of edge $e'$ in multiset $A$ is owned by the edge $e$. Notice that
the number of edges in multiset $A$ is equal to the number of edges of $e\in \hat E$ with $z\in \operatorname{span}'(e)$, which is equal to $N_z$, from \Cref{claim: enough segments}. We also define another multiset $A'$ of edges incident to $u_z$, that contains the first edge on every path in set ${\mathcal{R}}_z$. We think of every edge $e'\in A'$ as representing a distinct path $R\in {\mathcal{R}}_z$, for which $e'$ serves as the first edge. From the definition, $|A'|=N_z$. For every edge $e'\in \delta_G(u_z)$, we now let $n^-(e')$ be the number of times that $e'$ appears in set $A$, and $n^+(e)$ the number of times that $e'$ appears in $A'$. From the above discussion, $\sum_{e\in \delta_G(u_z)}n^-(e)=\sum_{e\in \delta_G(u_z)}n^+(e)=N_z$.
We use the algorithm from \Cref{obs:rerouting_matching_cong}, with the ordering ${\mathcal{O}}_{u_z}\in \Sigma$ of the edges of $\delta_G(u_z)$, in order to compute a multiset $M \subseteq \delta_G(u_z)\times \delta_G(u_z)$ of $N_z$ ordered pairs of the edges of $\delta_G(v)$, such that each edge $e'\in \delta_G(u_z)$ participates in $n^-_e$ pairs in $M$ as the first edge, and in $n^+_e$ pairs in $M$ as the second edge. Recall that we are also guaranteed that, for every pair $(e^-_1,e^+_1),(e^-_2,e^+_2)\in M$, the intersection between path $P_1=(e^-_1,e^+_1)$ and path $P_2=(e^-_2,e^+_2)$ at vertex $u_z$ is non-transversal with respect to ${\mathcal{O}}_{u_z}$.
Consider now some edge $e\in \hat E$ with $z\in \operatorname{span}''(e)$, and let $e'\in A$ be the edge that $e$ owns. Assume that $e'$ is matched with an edge $e''\in A'$ by $M$, and that $e''$ represents a segment $R\in {\mathcal{R}}_z$. We then set $\tilde R_z(e)=R$, and we add the segment $R$ to set $\tilde {\mathcal{R}}_z$. In this way, every segment of ${\mathcal{R}}_z$ is assigned to a different edge $e\in \hat E$ with $z\in \operatorname{span}''(e)$.
The algorithm terminates once all indices $1\leq z<r$ are processed. Consider now some edge $e\in \hat E$, and assume that $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$. Set $\tilde {\mathcal{R}}(e)$ now contains, for each $i''\leq z<j''_1$, a path $\tilde R_z(e)$, that connects $u_z$ to $u_{t+1}$. We let $P^{\mathsf{in}}(e)$ be the path obtained by concatenating all paths in $\tilde {\mathcal{R}}(e)$. Observe that path $P^{\mathsf{in}}(e)$ originates at vertex $u_{i''}$, and terminates at vertex $u_{j''}$ -- the two endpoints of path $P^{\mathsf{out}}(e)$.
We then let $W(e)$ be the cycle obtained from the union of paths $P^{\mathsf{out}}(e)$ and $P^{\mathsf{in}}(e)$. We say that $W(e)$ is the \emph{auxiliary cycle} of edge $e$.
We denote the set of all auxiliary cycles by ${\mathcal{W}}=\set{W(e)\mid e\in \hat E}$.
Consider some edge $e\in \hat E$. We note that, while path $P^{\mathsf{out}}(e)$ must be a simple path, it is possible that path $P^{\mathsf{in}}(e)$ is not a simple path. Specifically, for an index $1\leq z<r$, let $P'=P^{\mathsf{in}}(e)\cap (S_z\cup \delta(S_z))$. Then either $P'=\emptyset$, or $P'$ is a contiguous path, that is obtained by concatenating two simple paths from ${\mathcal{Q}}_z$. In the latter case, it is possible that some vertex of $S_z$ lies on both these paths. Since it is important for us that, whenever $P^{\mathsf{in}}(e)\cap S_z\neq \emptyset$, path $P^{\mathsf{in}}(e)$ contains the vertex $u_z$, we do not turn $P^{\mathsf{in}}(e)$ into a simple path. Notice however that, from the above discussion, every vertex may appear at most twice on $P^{\mathsf{in}}(e)$.
We need the following simple observation.
\begin{observation}\label{obs: self-non-transversal}
Let $e\in \hat E$ be an edge, and let $v$ be any vertex that appears twice on path $P^{\mathsf{in}}(v)$. Let $e_1$, $e_2$ be the edges immediately preceding and immediately following the first appearence of $v$ on $W(e)$, respectively, and let $e'_1$, $e'_2$ are the edges immediately preceding and immediately following the second appearence of $v$ on $W(e)$, respectively. Then the circular ordering of the edges $e_1,e_2,e_1',e_2'$ in ${\mathcal{O}}_v\in \Sigma$ is either $(e_1,e_2,e_1',e_2')$, or $(e_1,e_2,e_2',e_1')$ (or the reverse of one of these orderings).
\end{observation}
\begin{proof}
Let $1\leq z\leq r$ be the index for which $v\in V(S_z)$. Recall that $P^{\mathsf{in}}(e)\cap S_z$ is a contiguious path, which is a concatenation of two paths in ${\mathcal{Q}}_z$. Since the paths in ${\mathcal{Q}}_z$ are non-transversal with respect to $\Sigma$, the observation follows.
\end{proof}
Assume now that we are given two edges $e,e'\in \hat E$, and some vertex $v$ that appears on both $W(e)$ and $W(e')$. We view the cycles $W(e)$ as being directed so that path $P^{\mathsf{in}}(e)$ and path $P^{\mathsf{in}}(e')$ visit the vertebrae in the increasing order of their indices. Let $v$ be any vertex that lies on both $W(e)$ and $W(e')$ (note that $v$ may appear up to twice on each of the cycles). We say that the intersection of $W(e)$ and $W(e')$ is non-transversal at $v$ iff the following hold: for every appearence of $v$ on $W(e)$, and for every appearence of $v$ on $W(e')$, if we denote by $e_1$ and $e_2$ the edges immediately preceding and immediately following this appearence of $v$ on $W(e)$, and by $e'_1$ and $e'_2$ the edges immediately preceding and immediately following this appearence of $v$ on $W(e')$, then the circular ordering of the edges $e_1,e_2,e_1',e_2'$ in ${\mathcal{O}}_v\in \Sigma$ is either $(e_1,e_2,e_1',e_2')$, or $(e_1,e_2,e_2',e_1')$ (or the reverse of these orderings).
\fi
\iffalse
In this step, we construct paths (that we call \emph{segments}), that connect center vertices of consecutive verterbrae.
Consider some edge $e\in \hat E$ and the corresponding mid-segment $P^2(e)\in {\mathcal{P}}^2$ of the nice guiding path $P(e)\in \hat {\mathcal{P}}$.
Assume that path $P^2(e)$ originates at a vertex of $S_{i'}$ and terminates at a vertex of $S_{j'}$, for some $i'<j'$. We define $\operatorname{span}'(e)=\set{i',i'+1,\ldots,j'-1}$. Clearly, from our definition, $\operatorname{span}(e)\subseteq \operatorname{span}'(e)$.
From the definition of nice guiding paths, there is a sequence $e_{i'},e_{i'+1},\ldots,e_{j'-1}$ of edges that appear on path $P^2(e)$ in this order, such that, for all $i'\leq z\leq j'-1$, edge $e_z$ connects a vertex of $S_z$ to a vertex of $S_{z+1}$. Moreover, if $\set{P_{i'}(e),P_{i'+1}(e),\ldots,P_{j'}(e)}$ is the sequence of paths obtained from $P^2$ after deleting the edges in set $\set{e_z}_{i'\leq z<j'}$ from it, then for all $i'\leq z\leq j'$, $P_z(e)\subseteq S_z$.
For each $i'\leq z\leq j'-1$, let $R_{z}(e)$ be a new path, that is obtained by concatenating two paths: path $Q_{z}(e_z)\in {\mathcal{Q}}_z$, that routes edge $e_z$ to the center vertex $u_z$ of $S_z$; and path $Q_{z+1}(e_z)\in {\mathcal{Q}}_{z+1}$, that routes edge $e_z$ to the center vertex $u_{z+1}$ of $S_{z+1}$. Clearly, path $R_z(e)$ connects $u_z$ to $u_{z+1}$, and we view it as being directed from $u_z$ to $u_{z+1}$.
Fix an index $1\leq z<r$. We define a path set ${\mathcal{R}}_z$, containing the segment $R_z(e)$ of every edge $e\in \hat E$, for which index $z$ lies in the set $\operatorname{span}'(e)$.
Clearly, all paths in ${\mathcal{R}}_{t}$ originate at vertex $u_z$ and terminate at vertex $u_{t+1}$. Let $N_z$ denote the number of all edges
$e\in \hat E$, with $z\in \operatorname{span}'(e)$. Then $|{\mathcal{R}}_z|=N_z$ must hold.
Consider again some edge $e\in \hat E$, and assume that $\operatorname{span}'(e)=\set{i',i'+1,\ldots,j'-1}$. Recall that we have also defined a path $P^{\mathsf{out}}(e)$, whose endpoints lie in $\set{u_1,\ldots,u_r}$. We denote the endpoints of $P^{\mathsf{out}}(e)$ by $u_{i''},u_{j''}$, where $i''<j''$, and we define $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$.
For an index $1\leq z<r$, we let $N'_z$ be the number of all edges
$e\in \hat E$, with $z\in \operatorname{span}''(e)$. We need the following claim, whose proof appears in \Cref{subsec: enough segments}.
\begin{claim}\label{claim: enough segments}
For all $1\leq z<r$, $N'_z=N_z$.
\end{claim}
We now provide some intuition. Since, from \Cref{claim: enough segments}, for all $1\leq z<r$, $N'_z=N_z=|{\mathcal{R}}_z|$, we could assign, to each edge $e\in \hat E$ with $z\in \operatorname{span}''(e)$, a distinct segment $R'_z(e)\in {\mathcal{R}}_z$. Consider now some edge $e\in \hat E$, and assume that $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$. Then for all $i''\leq z\leq j''-1$, we have assigned a segment $R_z'(e)$ to edge $e$, and we could let path $P^{\mathsf{in}}(e)$ be obtained by concatenating all these segments. This final path connects the endpoints of the path $P^{\mathsf{out}}(e)$, and we could then let the auxiliary cycle $W(e)$ be the union of the paths $P^{\mathsf{in}}(e)$ and $P^{\mathsf{out}}(e)$. However, if $W(e),W(e')$ are two cycles in the resulting collection $\set{W(e'')\mid e''\in \hat E}$ of auxiliary cycles, then there could be a large number of vertices $v\in V(W(e))\cap V(W(e'))$, such that the intersection of $W(e)$ and $W(e')$ at $v$ is transversal with respect to $\Sigma$. We will eventually use the
auxiliary cycles in order to construct solutions for subinstances of the input instance $I$ that we compute, and such transversal intersections of cycles may give rise to a large number of crossings in the resulting drawings, which we cannot afford.
In order to overcome this difficulty, for each for $1\leq z<r$, we assign the segments of ${\mathcal{R}}_z$ to the edges $e\in \hat E$ with $z\in \operatorname{span}''(e)$ more carefully, in the next step. The resulting segments will be combined into the paths $P^{\mathsf{in}}(e)$ for edges $e\in \hat E$ as before, and the final auxiliary cycle $W(e)$ will be obtained by combining $P^{\mathsf{out}}(e)$ with $P^{\mathsf{in}}(e)$ as before. However, we will now ensure that, for every pair $W(e),W(e')$ of the resulting auxiliary cycles, there is at most one vertex $v\in V(W(e))\cap V(W(e'))$, such that the intersection of $W(e)$ and $W(e')$ at $v$ is transversal. This will allow us to bound the number of crossings in the optimal solutions of the subinstances of $I$ that we construct. For convenience, for all $1\leq z<r$, we call the paths in set ${\mathcal{R}}_z$ \emph{segments}.
\fi
\iffalse
\mynote{this seems a bit problematic. Paths of ${\mathcal{R}}^*_{(t-1)}$ and of ${\mathcal{R}}^*_{(t)}$ both contain vertices of $S_z$. You want to also say that paths of ${\mathcal{R}}^*_{(t-1)}\cup {\mathcal{R}}^*_{(t)}$ are non-interfering inside $S_z$. Originally, the paths of ${\mathcal{Q}}_z$ were non-interfering inside $S_z$. But does this property still hold after this step? If yes, this needs a proof! Also: do we actually need this step? If the paths of ${\mathcal{Q}}_z$ are non-interfering, and the paths of ${\mathcal{Q}}_{t+1}$ are non-interfering, won't this mean that the paths of ${\mathcal{P}}^*_{(t)}$ are non-interfering?}
We then apply the algorithm in Lemma~\ref{lem: non_interfering_paths} to the instance $(G,\Sigma')$, the set ${\mathcal{P}}^*_{(t)}$ of paths, and the two multisets that contains $|{\mathcal{P}}^*_{(t)}|$ copies of $u_{t}$ and $|{\mathcal{P}}^*_{(t)}|$ copies of $u_{t+1}$ respectively, and obtain a set ${\mathcal{R}}^*_{(t)}$ of $|{\mathcal{R}}^*_{(t)}|=|{\mathcal{P}}^*_{(t)}|$ paths connecting $u_z$ to $u_{t+1}$, such that for every $e\in E(G')$, $\cong_{G'}({\mathcal{R}}^*_{(t)},e)\le \cong_{G'}({\mathcal{P}}^*_{(t)},e)$, and the set ${\mathcal{R}}^*_{(t)}$ of paths are non-transversal with respect to $\Sigma$.
\fi
\iffalse
----------------------
We now re-organize the paths in sets ${\mathcal{R}}^*_{(1)},\ldots, {\mathcal{R}}^*_{(r-1)}$ to obtain the paths in ${\mathcal{P}}^{\mathsf{in}}=\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$.
Note that, if we select, for each $i\le t\le j-1$, a path of ${\mathcal{R}}^*_{(t)}$, then, by taking the union of them, we can obtain a path $\hat P$ connecting $u_i$ to $u_j$.
We call the selected path of ${\mathcal{R}}^*_{(t)}$ the \emph{$z$-th segment} of $\hat P$. Therefore, a path connecting $u_i$ to $u_j$ obtained in this way is the union of its $i$-th segment, its $(i+1)$-th segment, $\ldots$ , and its $(j-1)$-th segment.
We now incrementally construct the set ${\mathcal{P}}^{\mathsf{in}}$ of paths.
Intuitively, we will simultaneously construct all paths of ${\mathcal{P}}^{\mathsf{in}}$ in a total of $r-1$ iterations, where in the $i$-th iteration, we will determine the $i$-th segment of all paths in ${\mathcal{P}}^{\mathsf{in}}$.
For each pair $1\le i'<j'\le r$, we denote by $\hat E_{i',j'}$ the set of all edges $e\in \hat E$ such that the path $P^{\mathsf{out}}(e)$ constructed in Step 2 has $u_{i'}$ as its last endpoint and $u_{j'}$ as its first endpoint.
Throughout, we will maintain a set $\hat{{\mathcal{P}}}=\set{\hat P_e\mid e\in \hat E}$ of paths, where each path $\hat P_e$ is indexed by an edge $e$ of $\hat E$.
For an edge $e\in \hat E_{i',j'}$, the path $\hat P_e$ is supposed to originate at $u_{i'}$ and terminate at $u_{j'}$.
We call $u_{i'}$ the \emph{destined origin} of $\hat P_e$ and call $u_{j'}$ the \emph{destined terminal} of $\hat P_e$.
Initially, $\hat{{\mathcal{P}}}$ contains $|\hat E|$ empty paths.
We will sequentially process vertices $u_1,\ldots,u_{r-1}$, and, for each $1\le i\le r-1$, upon processing vertex $u_i$, determine which path of ${\mathcal{R}}^*_{(i)}$ serves as the $i$-th segment of which path of $\hat{\mathcal{P}}$.
We now fix an index $1\le i\le r-1$ and describe the iteration of processing vertex $u_i$. The current set $\hat{\mathcal{P}}$ of paths can be partitioned into four sets: set $\hat{\mathcal{P}}^{o}_i$ contains all paths of $\hat{\mathcal{P}}$ whose destined origin is $u_i$; set $\hat{\mathcal{P}}^{t}_i$ contains all paths of $\hat{\mathcal{P}}$ whose destined terminal is $u_i$; set $\hat{\mathcal{P}}^{\textsf{thr}}_i$ contains all paths of $\hat {\mathcal{P}}$ whose destined origin is $u_{i'}$ for some index $i'<i$ and whose destined terminal is $u_{j'}$ for some index $j'>i$; and set $\hat{\mathcal{P}}\setminus (\hat{\mathcal{P}}^{o}_i\cup \hat{\mathcal{P}}^{t}_i\cup \hat{\mathcal{P}}^{\textsf{thr}}_i)$ contains all other paths.
Note that the paths in set $\hat{\mathcal{P}}^{t}_i$ and set $\hat{\mathcal{P}}\setminus (\hat{\mathcal{P}}^{o}_i\cup \hat{\mathcal{P}}^{t}_i\cup \hat{\mathcal{P}}^{\textsf{thr}}_i)$ do not contain an $i$-th segment, so in this iteration we will determine the $i$-th segment of paths in the sets $\hat{\mathcal{P}}^{o}_i$ and $\hat{\mathcal{P}}^{\textsf{thr}}_i$.
Note that the paths in $\hat{\mathcal{P}}^{o}_i$ currently contain no edges,
and the paths in $\hat{\mathcal{P}}^{\textsf{thr}}_i$ currently contain up to its $(i-1)$-th segment.
We then denote
\begin{itemize}
\item by $L^-_i$ the multi-set of the current last edges (the edges incident to $u_i$) of paths in $\hat{\mathcal{P}}^{\textsf{thr}}_i$;
\item by $L^+_i$ the multi-set of the first edges (the edges incident to $u_i$) of paths in ${\mathcal{R}}^*_{(i)}$ (note that these paths are currently not designated as the $i$-th segment of any path in $\hat{{\mathcal{P}}}$); and
\item by $L^{\mathsf{out}}_i$ the multi-set of last edges of paths in ${\mathcal{P}}^{\mathsf{out}}_{i,*}=\bigg\{P^{\mathsf{out}}_{\operatorname{left}}(e) \text{ }\bigg|\text{ } e\in \big(\bigcup_{j>i}\hat E_{i,j}\big)\bigg\}$.
\end{itemize}
Clearly, elements in sets $L^-_i, L^+_i,L^{\mathsf{out}}_i$ are edges of $\delta(u_i)$.
We then define, for each $e\in \delta(u_i)$, $n^-_e=n_{L^-_i}(e)+n_{L^{\mathsf{out}}_i}(e)$ and
$n^+_e=n_{L^+_i}(e)$.
We use the following simple observation.
\begin{observation}
For each edge $e\in \delta(u_i)$, $\sum_{e\in \delta(u_i)}n^-_e=\sum_{e\in \delta(u_i)}n^+_e$.
\end{observation}
\begin{proof}
On the one hand, $\sum_{e\in \delta(u_i)}n_{L^{\mathsf{out}}_i}(e)=|\bigcup_{j\ge i+2}\hat E_{i,j}|=|\hat{\mathcal{P}}^{o}_i|$, and $\sum_{e\in \delta(u_i)}n_{L^{-}_i}(e)=|L^-_i|=|\hat{\mathcal{P}}^{\textsf{thr}}_i|$.
On the other hand, recall that $|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|$, and the set ${\mathcal{P}}^*_{(i)}$ contains a path connecting $u_i$ to $u_{i+1}$ for each edge $e\in \big(\bigcup_{i'\le i-1,j'\ge i+1}\hat E_{i',j'}\big)\cup \big(\bigcup_{j'\ge i+2}\hat E_{i',j'}\big)$. Therefore,
$\sum_{e\in \delta(u_i)}n_{L^{+}_i}(e)=|L^+_i|=|{\mathcal{R}}^*_{(i)}|=|{\mathcal{P}}^*_{(i)}|=|\bigcup_{i'\le i-1,j'\ge i+1}\hat E_{i',j'}|+|\bigcup_{j'\ge i+2}\hat E_{i,j'}|=|\hat{\mathcal{P}}^{\textsf{thr}}_i|+|\hat{\mathcal{P}}^{o}_i|$.
\end{proof}
We apply the algorithm in \Cref{obs:rerouting_matching_cong} to graph $G$, vertex $u_i$, rotation ${\mathcal{O}}_{u_i}$ and integers $\set{n^-_e,n^+_{e}}_{e\in \delta(u_i)}$.
Let $M$ be the multi-set of ordered pairs of the edges of $\delta(u_i)$ that we obtain.
We then designate:
\begin{itemize}
\item for each path $\hat P_e\in \hat{\mathcal{P}}^{\textsf{thr}}_i$ with $e^-$ as its current last edge, a path of ${\mathcal{R}}^*_{(i)}$ that contains the edge $e^+$ as its first edge with $(e^-,e^+)\in M$, as the $i$-th segment of $\hat P_e$; and
\item for each path $P^{\mathsf{out}}_{\operatorname{left}}(e)\in{\mathcal{P}}^{\mathsf{out}}_{i,*}$ with $e^-$ as its last edge (recall that we view $u_i$ as the last endpoint of such a path), a path of ${\mathcal{R}}^*_{(i)}$ that contains the edge $e^+$ as its first edge with $(e^-,e^+)\in M$, as the $i$-th segment of $\hat P_e$;
\end{itemize}
such that each path of ${\mathcal{R}}^*_{(i)}$ is assigned to exactly one path of $\hat {\mathcal{P}}^{\textsf{thr}}_i\cup {\mathcal{P}}^{\mathsf{out}}_{i,*}$.
This completes the description of the $i$-th iteration.
See Figure~\ref{fig: inner_path} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Edges in $\delta(u_i)$: only the last/first edges of paths are shown. Here ${\mathcal{P}}^{\mathsf{out}}_{*,i}=\set{P^{\mathsf{out}}(e) \mid e\in\big( \bigcup_{i'< i}\hat E_{i',i}\big)}$.]{\scalebox{0.32}{\includegraphics{figs/inner_path_1.jpg}}}
\hspace{1pt}
\subfigure[Sets $L^-_i, L^+_i,L^{\mathsf{out}}_i$ and the pairing (shown in dash pink lines) given by the algorithm in \Cref{obs:rerouting_matching_cong}.]{
\scalebox{0.32}{\includegraphics{figs/inner_path_2.jpg}}}
\caption{An illustration of an iteration in Step 4 of constructing paths of $\set{P^{\mathsf{in}}(e)\mid e\in \hat E}$.}\label{fig: inner_path}
\end{figure}
Let ${\mathcal{P}}^{\mathsf{in}}$ be the set of paths that we obtain after processing all vertices $u_1,\ldots,u_{r-1}$. Then for every $e\in \hat E$, we rename the path $\hat P_e$ at the end of the algorithm by $P^{\mathsf{in}}(e)$.
From the algorithm, it is easy to see that, for each pair $1\le i<j\le r$ and each edge $e\in \hat E_{i,j}$, the inner path $P^{\mathsf{in}}(e)$ starts at $u_i$ and ends at $u_j$ as it is supposed to, and path $P^{\mathsf{in}}(e)$ contains, for each $i\le t\le j-1$, a path of ${\mathcal{R}}^*_{(t)}$ as its $z$-th segment.
Therefore, path $P^{\mathsf{in}}(e)$ visits vertices $u_i,u_{i+1},\ldots,u_j$ sequentially.
\fi
\iffalse
\begin{observation}
\label{obs: non_zransversal_1}
For each $1\le i\le r-1$, if we denote, for each $e\in E_i^{\operatorname{right}}$, by $R_e$ the path consisting of the last edge of path $P^{\mathsf{out}}_e$ and the first edge of path $P_e$, then the paths in $\set{P_e\mid e\in \bigcup_{ i'<i<j'}E(S'_{i'},S'_{j'})}$ and $\set{R_e\mid e\in E_i^{\operatorname{right}}}$ are non-transversal at $u_i$.
\end{observation}
\fi
\iffalse
\begin{observation}
\label{obs: non_zransversal_2}
The inner paths in ${\mathcal{P}}$ are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
On one hand, note that for each $1\le i\le r-1$, the set ${\mathcal{R}}^*_{(i)}$ of paths are non-transversal with respect to $\Sigma$.
Since the $i$-th segments of paths in ${\mathcal{P}}$ are paths in ${\mathcal{R}}^*_{(i)}$, it follows that the paths of ${\mathcal{P}}$ are non-transversal at all vertices of $V(G)\setminus\set{u_1,\ldots,u_k}$.
On the other hand, from the algorithm and \Cref{obs:rerouting_matching_cong}, it is easy to verify that the paths of ${\mathcal{P}}$ are also non-transversal at $u_1,\ldots,u_k$.
\end{proof}
\fi
\iffalse
Lastly, we bound the congestion that is caused by the auxiliary cycles, in the following observation.
\begin{observation}
\label{obs: edge_occupation in outer and inner paths}
Consider any edge $e\in E(G)$. If $e\not \in \bigcup_{1\leq z\leq r}E(S_z)$, then $e$ belongs to at most $O(\log^{18}m)$ cycles of ${\mathcal{W}}$. Otherwise, if $e\in E(S_z)$ for some $1\leq z\leq r$, then the number of cycles of ${\mathcal{W}}$ that $e$ belongs to is at most $\cong_G({\mathcal{Q}}_z,e)\cdot O(\log^{18}m)$.
\end{observation}
\begin{proof}
Consider first an edge $e\in E(G)\setminus\textsf{left}(\bigcup_{1\leq z\leq r}E(S_z)\textsf{right} )$. If both endpoints of $e$ lie in set $V'$, then there is an index $1\leq z<r$, such that $e$ connects a vertex of $S_z$ to a vertex of $S_{t+1}$. In this case, the number of cycles in ${\mathcal{W}}$ containing edge $e$ is precisely the number of paths in ${\mathcal{R}}_z$ containing $e$, which is in turn bounded by $\cong_G({\mathcal{P}}^2,e)\leq O(\log^{18}m)$ from the definition of nice guiding paths.
If at least one endpoint of $e$ does not lie in $V'$, then the number of cycles in ${\mathcal{W}}$ that contain $e$ is $\cong_G({\mathcal{P}}^{\mathsf{out}}\leq O(\log^{18}m)$, from \Cref{claim: computing out-paths}.
Assume now that there is some index $1\leq z\leq r$, such that $e\in E(S_z)$. Let $A$ be the set of all edges $e'\in \delta_G(S_z)$, such that the path $Q_z(e')\in {\mathcal{Q}}_z$ contains the edge $e$. Clearly, $|A|=\cong_G({\mathcal{Q}}_z,e)$. Notice that a cycle $W\in {\mathcal{W}}$ may only contain the edge $e$ if it contains one of the edges in set $A$. Since, from the above discussion, every edge of $A$ may lie on at most $O(\log^{18}m)$ cycles of ${\mathcal{W}}$, the total number of cycles in ${\mathcal{W}}$ that contain $e$ is bounded by $|A|\cdot O(\log^{18}m)\leq \cong_G({\mathcal{Q}}_z,e)\cdot O(\log^{18}m)$.
\end{proof}
\fi
\subsubsection{Step 3: Laminar Family ${\mathcal{L}}$ of Clusters, and Internal and External Routers for Clusters of ${\mathcal{L}}$}
We define a laminar family ${\mathcal{L}}=\set{U_1,\ldots,U_r}$ of clusters of $G$, that will be later used in order to compute a decomposition of the input instance $I$ into subinstances.
For each $1\le i\le r$, we define cluster $U_i$ to be the subgraph of $G$ induced by $\bigcup_{1\le z\le i}V(\tilde S_z)$. For convenience, we also denote by $\overline{U}_i$ the subgraph of $G$ induced by $\bigcup_{i< z\le r}V(\tilde S_z)$. We then define a laminar family ${\mathcal{L}}=\set{U_1,\ldots,U_r}$ of clusters of $G$. Notice that $U_r=G$, and $U_1\subseteq U_2\subseteq\cdots\subseteq U_r$.
We now fix an index $1\leq i\leq r$ and consider the set $\delta_G(U_i)=E(U_i,\overline{U}_i)$ of edges. We can partition this edge set into two subsets: set $E_i=E(S_i,S_{i+1})$, and set $\hat E_i$, containing all remaining edges. Notice that $\hat E_i$ is precisely the set of all edges $e\in \hat E$ with $i\in \operatorname{span}(e)$.
\iffalse
We need the following simple observation.
\begin{observation}\label{obs: cycles visit centers}
For every edge $e\in \hat E_i$, path $P^{\mathsf{in}}(e)$ contains an edge $\hat e_i\in E_i$.
\end{observation}
\begin{proof}
Fix an edge $e\in \hat E_i$, and
assume that $\operatorname{span}''(e)=(i'',j'')$.
Since $e\in \hat E_i$, $i\in \operatorname{span}(e)$, and, since $\operatorname{span}(e)\subseteq \operatorname{span}''(e)$, we get that $i\in \operatorname{span}''(e)$. From the construction of the path $P^{\mathsf{in}}(e)$, it now immediately follows that the path contains an edge $\hat e_i\in E_i$.
\end{proof}
\fi
For all $1\leq i\leq r$, we will define an internal router ${\mathcal{Q}}(U_i)$, and an external router ${\mathcal{Q}}'(U_i)$ for cluster $U_i$.
The internal routers ${\mathcal{Q}}(U_i)$ of the clusters $U_i\in {\mathcal{L}}$ will be used in order to compute the decomposition of $I$ into subinstances. Both the internal and the external routers ${\mathcal{Q}}(U_i),{\mathcal{Q}}'(U_i)$ will be used in order to argue that the resulting instances have a relatively cheap solution. In order to define these routers, we first need to define, for all $1\leq i\leq r$, an internal router ${\mathcal{Q}}(S_i)$ for cluster $S_i\in {\mathcal{S}}$. The algorithm for computing these routers is randomized, and is provided next.
\subsubsection*{Algorithm for Computing Internal Routers for the Vertebrae.}
Recall that we have defined a parameter $\hat \eta=2^{O((\log m)^{3/4}\log\log m)}$.
We also use a new parameter
$\beta^*=2^{O(\sqrt{\log m}\cdot \log\log m)}$.
We now provide a randomized algorithm that computes,
for each cluster $S_i\in {\mathcal{S}}$ an internal $S_i$-router ${\mathcal{Q}}(S_i)$. Additionally, we compute a partition $({\mathcal{S}}^{\operatorname{bad}},{\mathcal{S}}^{\operatorname{light}})$ of the clusters in ${\mathcal{S}}$. Initially, we set ${\mathcal{S}}^{\operatorname{bad}}={\mathcal{S}}^{\operatorname{light}}=\emptyset$.
Recall that, from the definition of the nice witness structure, every cluster $S_i\in {\mathcal{S}}$ has the $\alpha^*$-bandwidth property, for $\alpha^*=\Omega(1/\log^{12}m)$.
For each $1\leq i\leq r$ in turn, we apply Algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace from \Cref{thm:algclassifycluster} to instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace and cluster $J=S_i$, with parameter $p=1/m^{100}$.
If the algorithm returns a distribution ${\mathcal{D}}(S_i)$ over internal $S_i$-routers in $\Lambda(S_i)$, such that $S_i$ is $\beta^*$-light with respect to ${\mathcal{D}}(S_i)$, then we sample an internal $S_i$-router ${\mathcal{Q}}(S_i)$ from the distribution ${\mathcal{D}}(S_i)$, and we let $u_i$ be the vertex of $S_i$ that serves as the common endpoint of all paths in ${\mathcal{Q}}(S_i)$. We refer to vertex $u_i$ as the \emph{center vertex of $S_i$}. We also add cluster $S_i$ to set ${\mathcal{S}}^{\operatorname{light}}$ in this case.
Otherwise, Algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace returns FAIL. In this case, we add cluster $S_i$ to ${\mathcal{S}}^{\operatorname{bad}}$, and we apply the algorithm from \Cref{cor: simple guiding paths} to graph $G$ and cluster $S_i$. Let ${\mathcal{D}}(S_i)$ be the distribution over the set $\Lambda_G(S_i)$ of internal $S_i$-routers that the algorithm returns. We then let ${\mathcal{Q}}(S_i)$ be an internal $S_i$-router sampled from the distribution ${\mathcal{D}}(S_i)$. From \Cref{cor: simple guiding paths}, for each edge $e\in E(S_i)$, $\expect{\cong({\mathcal{Q}}(S_i),e)}\leq O(\log^4m/\alpha^*)=O(\log^{16}m)$.
\paragraph{Bad Event ${\cal{E}}$.} For an index $1\le i\le r$, we say that bad event ${\cal{E}}_i$ happens, if $S_i$ is not a $\hat \eta$-bad cluster, but Algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace returned FAIL when applied to it.
From \Cref{thm:algclassifycluster}, $\Pr[{\cal{E}}_i]\le 1/m^{100}$. We also let ${\cal{E}}$ be the bad event that event ${\cal{E}}_i$ happens for any index $1\le i\le r$. From the union bound, $\Pr[{\cal{E}}]\le 1/m^{99}$.
Consider again any index $1\leq i\leq r$. We will now slightly modify the paths in set ${\mathcal{Q}}(S_i)$, to ensure that they are non-traversal with respect to the rotation system $\Sigma$. In order to do so, we start by subdividing every edge $e\in \delta_G(S_i)$ with a vertex $t(e)$, and we let $X_i=\set{t(e)\mid e\in \delta_G(S_i)}$ be the resulting set of new vertices. We truncate the paths of ${\mathcal{Q}}(S_i)$, so that each such path originates at a distinct vertex of $X_i$ and terminates at vertex $u_i$. We then let $Y_i$ be the multiset of vertices containing the last vertex on every path of ${\mathcal{Q}}(S_i)$ (so $Y_i$ contains $|{\mathcal{Q}}(S_i)|$ copies of vertex $u_i$). We apply the algorithm from \Cref{lem: non_interfering_paths} to the resulting instance of \textnormal{\textsf{MCNwRS}}\xspace, set ${\mathcal{Q}}(S_i)$ of paths and vertex multisets $X_i,Y_i$, to obtain another set $\tilde {\mathcal{Q}}(S_i)$ of paths that are non-transversal with respect to $\Sigma$. Recall that for every edge $e\in E(G)$, $\cong_G(\tilde{\mathcal{Q}}(S_i),e)\leq \cong_G({\mathcal{Q}}(S_i),e)$, so in particular all inner vertices of the paths in $\tilde {\mathcal{Q}}(S_i)$ lie in $S_i$.
The path set $\tilde {\mathcal{Q}}(S_i)$ naturally defines a set of paths (internal $S_i$-router) in graph $G$,
that route the edges of $\delta_G(S_i)$ to vertex $u_i$, and are non-transversal with respect to $\Sigma$.
For convenience of notation, we denote this internal $S_i$-router by ${\mathcal{Q}}(S_i)$ from now on.
The following observation follows immediately from the above discussion, \Cref{thm:algclassifycluster} and the definition of $\beta^*$-light and $\hat \eta$-bad clusters (see \Cref{def: light cluster} and \Cref{def: bad cluster} in \Cref{subsec: guiding paths rotations}), and from the fact that $\beta^*\leq \hat \eta$.
\begin{observation}
\label{obs: congestion square of internal routers}
For every cluster $S_i\in {\mathcal{S}}^{\operatorname{light}}$, for every edge $e\in E(S_i)$:
$\expect{\textsf{left} (\cong_{G}({\mathcal{Q}}(S_i),e)\textsf{right} )^2}\le \hat \eta$.
Additionally, for every cluster $S_i\in {\mathcal{S}}^{\operatorname{bad}}$, for every edge $e\in E(S_i)$:
$\expect{\cong_G({\mathcal{Q}}(S_i),e)}\leq O(\log^{16}m)$. Moreover, if ${\cal{E}}$ did not happen, then every cluster $S_i\in {\mathcal{S}}^{\operatorname{bad}}$ is $\hat \eta$-bad, that is: $$\mathsf{OPT}_{\mathsf{cnwrs}}(S_i,\Sigma(S_i))+|E(S_i)|\geq \frac{|\delta_G(S_i)|^2}{\hat \eta},$$ where $\Sigma(S_i)$ is the rotation system for graph $S_i$ induced by $\Sigma$. Lastly, $\prob{{\cal{E}}}\leq 1/m^{99}$.
\end{observation}
\iffalse
\begin{observation}
For each $1\le i\le r$, if event ${\cal{E}}_i$ does not happen, then
$$\expect[]{\sum_{e\in E(S_i)}\bigg(\cong_{G'}({\mathcal{Q}}(S_i),e)\bigg)^2}\le \max\set{\beta^*,\mu^*}\cdot\bigg(\chi^2_{\phi^*}(S_i)+|E(S_i)|\bigg),$$
where $\phi^*$ is the optimal solution to instance $(G',\Sigma')$.
\end{observation}
\begin{proof}
Assume first that the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace does not return FAIL when it is applied to cluster $S_i$, then we obtain a distribution ${\mathcal{D}}(S_i)$ over internal routers in $\Lambda(S)$, such that $S_i$ is $\beta^*$-light with respect to ${\mathcal{D}}(S_i)$. Then from the construction of ${\mathcal{Q}}(S_i)$, we get that, for each $e\in E(S_i)$, $\expect[]{\cong_{{\mathcal{Q}}(S_i)}(e))^2}\le \beta^*$, and it follows that $\expect[]{\sum_{e\in E(S_i)}(\cong_{{\mathcal{Q}}_i}(e))^2}\le \beta^*\cdot|E(S_i)|$.
Assume now that the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace returns FAIL when it is applied to cluster $S_i$. Since event ${\cal{E}}_i$ does not happen, $S_i$ is a $\hat \eta$-bad cluster, meaning that $\mathsf{OPT}_{\mathsf{cnwrs}}(S_i,\Sigma_{S_i})+|E(S_i)|\geq \frac{|\delta_G(S_i)|^2}{\hat \eta}$. Therefore,
$$\expect[]{\sum_{e\in E(S_i)}(\cong_{{\mathcal{Q}}_i}(e))^2}\le \textsf{left}(\expect[]{\sum_{e\in E(S_i)}\cong_{{\mathcal{Q}}_i}(e)}\textsf{right})^2\le \bigg(|E(S_i)|\cdot O(\log^4m/\alpha_0)\bigg)^2\le \hat \eta\cdot |E(S_i)|.$$
\end{proof}
\fi
\subsubsection*{Internal and External Routers for Clusters of $\Lambda$}
Fix an index $1\leq i\leq r$. We now define an internal router ${\mathcal{Q}}(U_i)$ and an external router ${\mathcal{Q}}'(U_i)$ for cluster $U_i\in \Lambda$. We will ensure that all paths of ${\mathcal{Q}}(U_i)$ terminate at the center vertex $u_i$ of $S_i$, and all paths of $ {\mathcal{Q}}'(U_i)$ terminate at the center vertex $u_{i+1}$ of $S_{i+1}$.
In order to do so, we consider the edges $e\in \delta_G(U_i)$ one by one. For each such edge $e$, we define a path $Q(e)$, whose first edge is $e$, last vertex is $u_i$, and all inner vertices lie in $U_i$, and we define a path $Q'(e)$, whose first edge is $e$, last vertex is $u_{i+1}$, and all inner vertices lie in $U_{i+1}$. We will then set ${\mathcal{Q}}(U_i)=\set{Q(e)\mid e\in \delta(U_i)}$, and ${\mathcal{Q}}'(U_i)=\set{Q'(e)\mid e\in \delta(U_i)}$.
We now fix an edge $e\in \delta_G(U_i)$, and define the two paths $Q(e),Q'(e)$. Recall that $\delta_G(U_i)=E_i\cup \hat E_i$. Assume first that $e\in E_i$. In this case, $e\in \delta_G(S_i)$ and $e\in \delta_G(S_{i+1})$ must hold. We let $Q(e)$ be the unique path of the internal $S_i$-router ${\mathcal{Q}}(S_i)$ whose first edge is $e$, and we let $Q'(e)$ be the unique path of the internal $S_{i+1}$-router ${\mathcal{Q}}(S_{i+1})$, whose first edge is $e$. As required, path $Q(e)$ connects $e$ to $u_{i}$ and only contains vertices of $U_i$ as inner vertices, while path $Q'(e)$ connects $e$ to $u_{i+1}$, and only contains vertices of $\overline{U}_{i}$ as inner vertices.
Assume now that $e\in \hat E_i$. We denote $e=(u,v)$, and we assume that $u$ is the left endpoint of $e$.
Since $e\in \hat E_i$, $i\in \operatorname{span}(e)$, and, since $\operatorname{span}(e)\subseteq \operatorname{span}''(e)$, we get that $i\in \operatorname{span}''(e)$. From the construction of the path $P^{\mathsf{in}}(e)$, it must contain an edge $\hat e_i\in E_i$. Additionally, it must contain some edge $\hat e_{i-1}\in \delta_G(S_i)\setminus\set{\hat e_i}$ and some edge $\hat e_{i+1}\in \delta_G(S_{i+1})\setminus\set{\hat e_i}$ (if $e\in \delta_G(S_i)$, then $\hat e_{i-1}=e$, and if $e\in \delta_G(S_{i+1})$, then $\hat e_{i+1}=e$); see \Cref{fig: NN3a}.
Let $\rho(e)\subseteq P^{\mathsf{in}}(e)$ be the subpath of $P^{\mathsf{in}}(e)$ that starts at edge $\hat e_{i-1}$ and terminates at edge $\hat e_{i+1}$. Consider now the graph that is obtained from the auxiliary cycle $W(e)$ by deleting the edge $e$, and all edges of $\rho(e)$ excluding $\hat e_{i-1}$ and $\hat e_{i+1}$. Once we delete all isolated vertices in the resulting graph, we obtain two contiguous paths. The first path, that we denote by $P$, originates at $u$ and has edge $\hat e_{i-1}\in \delta_G(S_i)$ as its last edge; all edges and vertices of $P$ lie in $U_i$ (if $e=\hat e_{i-1}$, then $P=\set{u}$). The second path, that we denote by $P'$, originates at $v$ and has $\hat e_{i+1}\in \delta_G(S_{i+1})$ as its last edge; all edges and vertices of $P'$ lie in $\overline{U}_i$ (if $e=\hat e_{i+1}$, then $P'=\set{v}$). We let $Q(e)$ be the path obtained by concatenating the edge $e$ with the path $P$, and the unique path of the internal $S_i$-router ${\mathcal{Q}}(S_i)$ that originates at edge $\hat e_{i-1}$ (see \Cref{fig: NN3b}). Clearly, path $Q(e)$ has $e$ as its first edge, $u_i$ as its last vertex, and all its inner vertices lie in $U_i$. Similarly, we let $Q'(e)$ be the path obtained by concatenating the edge $e$ with the path $P'$, and the unique path of the internal $S_{i+1}$-router ${\mathcal{Q}}(S_{i+1})$ that originates at edge $\hat e_{i+1}$. Clearly, path $Q'(e)$ has $e$ as its first edge, $u_{i+1}$ as its last vertex, and all its inner vertices lie in $\overline{U}_i$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN3a.jpg}
\caption{Construction of paths $Q(e)$ and $Q'(e)$ for an edge $e\in \hat E_i$. Path $\rho(e)$ is shown in green, and the cut $(U_i,V\setminus U_i)$ is shown in a pink dashed line.
}\label{fig: NN3a}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.12]{figs/NN3b.jpg}
\caption{Paths $Q(e)$ is the union of edge $e$ and the green path; path $Q'(e)$ is the union of edge $e$ and the brown path.}\label{fig: NN3b}
\end{figure}
Once every edge of $\delta_G(U_i)$ is processed, we set ${\mathcal{Q}}(U_i)=\set{Q(e)\mid e\in \delta(U_i)}$ and ${\mathcal{Q}}'(U_i)=\set{Q'(e)\mid e\in \delta(U_i)}$. It is immediate to verify that ${\mathcal{Q}}(U_i)$ is an internal $U_i$-router, while ${\mathcal{Q}}'(U_i)$ is an external $U_i$-router. Note that the construction of both routers is randomized, and the only randomized component in the construction is the selection of the internal routers for the vertebrae.
We need the following simple observation.
\begin{observation}\label{obs: inner non-transversal}
For all $1\leq i<r$, the set ${\mathcal{Q}}(U_i)$ of paths is non-transversal with respect to $\Sigma$. Additionally, for every pair $Q'(e),Q'(e')\in {\mathcal{Q}}'(U_i)$ of paths, there is at most one vertex $v$, such that $Q'(e)$ and $Q'(e')$ have a transversal intersection at $v$. If such a vertex $v$ exists, then $v$ is the unique vertex, such that the auxiliary cycles $W(e),W(e')$ have a transversal intersection at $v$.
\end{observation}
\begin{proof}
We start by considering a pair of paths $Q(e),Q(e')\in {\mathcal{Q}}(U_i)$. Let $v$ be any vertex that serves as an inner vertex on both paths. We consider three cases. First, if $v\in V''$, then $e,e'\in \hat E_i$ must hold, and, from \Cref{obs: auxiliary cycles non-transversal at at most one}, the intersection of $Q(e)$ and $Q(e')$ at $v$ is non-transversal. The second case is when $v\in V(S_i)$. In this case, there must be two paths $\tilde Q_1,\tilde Q_2\in {\mathcal{Q}}(S_i)$, such that $\tilde Q_1\subseteq Q(e)$, $\tilde Q_2\subseteq Q(e')$, and $v$ is an inner vertex on both $\tilde Q_1$ and $\tilde Q_2$ (note that it is possible that $\tilde Q_1=\tilde Q_2$). In this case, from the construction of the internal $S_i$-router ${\mathcal{Q}}(S_i)$, the intersection of $Q(e)$ and $Q(e')$ at $v$ is non-transversal. The third case is when $v\in V'\setminus V(S_{i})$. In this case, $e,e'\in \hat E_i$ must hold. Assume that $v\in V(S_z)$ for some index $z<i$. Clearly, $z$ may not be the last index of $\operatorname{span}''(e)$ or of $\operatorname{span}''(e')$. Therefore, from \Cref{obs: auxiliary cycles non-transversal at at most one}, the intersection of $Q(e)$ and of $Q(e')$ at $v$ is non-transversal. We conclude that the set ${\mathcal{Q}}(U_i)$ of paths is non-transversal with respect to $\Sigma$.
Consider now some pair $Q'(e),Q'(e')\in {\mathcal{Q}}'(U_i)$ of paths. Let $v$ be any vertex that serves as an inner vertex on both paths. We again consider three cases. First, if $v\in V''$, then $e,e'\in \hat E_i$ must hold, and, from \Cref{obs: auxiliary cycles non-transversal at at most one}, the intersection of $Q'(e)$ and $Q'(e')$ at $v$ is non-transversal. The second case is when $v\in V(S_{i+1})$. In this case, there must be two paths $\tilde Q_1,\tilde Q_2\in {\mathcal{Q}}(S_{i+1})$, such that $\tilde Q_1\subseteq Q'(e)$, $\tilde Q_2\subseteq Q'(e')$, and $v$ is an inner vertex on both $\tilde Q_1$ and $\tilde Q_2$. In this case, from the construction of the internal $S_{i+1}$-router ${\mathcal{Q}}(S_{i+1})$, the intersection of $Q'(v)$ and $Q'(v')$ at $v$ is non-transversal. The third case is when $v\in V'\setminus V(S_i)$. In this case, $e,e'\in \hat E_i$ must hold, and $v$ also lies on cycles $W(e)$ and $W(e')$. Moreover, the intersection of $W(e)$ and $W(e')$ must be transversal at $v$.
From \Cref{obs: auxiliary cycles non-transversal at at most one}, there may be at most one vertex $v'$, such that the intersection of $W(e)$ and $W(e')$ at $v'$ is transversal. We conclude that there is at most one vertex $v\in V(Q'(e))\cap V(Q'(e'))$, such that the intersection of $Q'(e)$ and $Q'(e')$ at $v$ is transversal. If such a vertex $v$ exists, then $v$ is the unique vertex, such that the auxiliary cycles $W(e),W(e')$ have a transversal intersection at $v$.
\end{proof}
From \Cref{obs: bound congestion of cycles}, and the construction of the internal and external $U_i$-routers, we obtain the following immediate observation.
\begin{observation}\label{obs: bound congestion of routers}
For all $1\leq i< r$, an edge $e\in E(S_i)$ may appear on at most
$O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_i),e)$ paths of ${\mathcal{Q}}(U_i)$, and an edge $e\in E(U_i)\setminus E(S_i)$ may appear on at most $O(\log^{34}m)$ paths of ${\mathcal{Q}}(U_i)$. Similarly, an edge $e\in E(S_{i+1})$ may appear on at most $O(\log^{34}m)\cdot \cong_G({\mathcal{Q}}(S_{i+1}),e)$ paths of ${\mathcal{Q}}'(U_i)$, and an edge $e\in E(\overline{U}_i)\setminus E(S_{i+1})$ may appear on at most $O(\log^{34}m)$ paths of ${\mathcal{Q}}'(U_i)$.
\end{observation}
We will also use the following simple corollary of the observation.
\begin{corollary}\label{cor: few edges crossing cuts}
For all $1\leq z\leq r$, $|\delta_G(U_{z})|\leq |\delta_G(S_{z})|\cdot O(\log^{34}m)$, and $|\delta_G(U_{z})|\leq |\delta_G(S_{z+1})|\cdot O(\log^{34}m)$.
\end{corollary}
\begin{proof}
Recall that we have defined a collection ${\mathcal{Q}}(U_{z})$ of paths, routing the edges of $\delta_G(U_z)$ to vertex $u_{z}$. Each such path must contain an edge of $\delta_G(S_{z})$. Moreover, from \Cref{obs: bound congestion of routers}, an edge of $\delta_G(S_{z})$ may lie on at most $O(\log^{34}m)$ paths of ${\mathcal{Q}}(U_{z})$. Therefore, $|\delta_G(U_{z})|\leq |\delta_G(S_{z})|\cdot O(\log^{34}m)$.
Similarly, we have defined a collection ${\mathcal{Q}}'(U_{z})$ of paths, routing the edges of $\delta_G(U_z)$ to vertex $u_{z+1}$. Each such path must contain an edge of $\delta_G(S_{z+1})$. From \Cref{obs: bound congestion of routers}, an edge of $\delta_G(S_{z+1})$ may lie on at most $O(\log^{34}m)$ paths of ${\mathcal{Q}}'(U_{z})$. Therefore, $|\delta_G(U_{z})|\leq |\delta_G(S_{z+1})|\cdot O(\log^{34}m)$.
\end{proof}
\subsubsection{Step 1. Constructing the Paths of ${\mathcal{P}}^{\mathsf{out}}$}
In this step, we construct the set ${\mathcal{P}}^{\mathsf{out}}=\set{P^{\mathsf{out}}(e)\mid e\in \hat E}$ of paths, by slightly modifying the prefixes and the suffixes of the paths in $\hat {\mathcal{P}}$ to make them non-transversal.
Throughout, we denote $V'=\bigcup_{S\in {\mathcal{S}}}V(S)$ and $V''=V(G)\setminus V'$.
Consider an edge $e=(u,v)\in \hat E$, and assume that $u\in S_i,v\in S_j$, and $i<j$. For convenience, we will call $u$ the \emph{left endpoint of edge $e$}, and $v$ the \emph{right endpoint of edge $e$}.
We also define two sets of indices associated with edge $e$. The first set of indices is $\operatorname{span}(e)=\set{i,i+1,\ldots,j-1}$.
In order to define the second set of indices, assume that the last vertex on path $P^1(e)$ (vertex that lies in $V'$) belongs to cluster $S_{i'}$, while the last vertex on path $P^3(e)$ belongs to cluster $S_{j'}$. From the definition of nice guiding paths, $i'\leq i<j\leq j'$ must hold. We then let $\operatorname{span}'(e)=\set{i',i'+1,\ldots,j'-1}$.
It will be convenient for us to define the notion of left-monotone and right-monotone paths. The definition is similar to the one used in \Cref{subsec: getting nice structure better clustering}, but not identical.
\begin{definition}
Let $R$ be a (directed) path in graph $G$ that contains at least one edge, let $(v_1,\ldots,v_z)$ be the sequence of vertices appearing on $R$, and, for all $1\leq a\leq z$, assume that $v_a\in \tilde S_{i_a}$. We say that $R$ is a \emph{left-monotone path} if:
\begin{itemize}
\item $v_z\in V'$;
\item for $1<a<z$, $v_a\in V''$; and
\item $i_1\geq i_2\geq\cdots\geq i_z$.
\end{itemize}
Similarly, we say that $R$ is a \emph{right-monotone path}, if:
\begin{itemize}
\item $v_z\in V'$;
\item for $1<a<z$, $v_a\in V''$; and
\item $i_1\leq i_2\leq \cdots\leq i_z$.
\end{itemize}
\end{definition}
For each edge $e\in \hat E$, we view the paths $P^1(e)\in {\mathcal{P}}^1$, $P^3(e)\in {\mathcal{P}}^3$ as directed paths that originate at an endpoint of edge $e$ and terminate at a vertex of $V'$ (notice that it is possible that one or even both endpoints of $e$ lie in $V'$).
For every vertex $v\in V'$, we denote by $n_1(v)$ the total number of paths in $\hat {\mathcal{P}}^1$ that terminate at $v$, and by $n_3(v)$ the total number of paths in $\hat {\mathcal{P}}^3$ that terminate at $v$. Let $\eta=O(\log^{18}m)$ be such that the set $\hat {\mathcal{P}}$ of paths causes congestion at most $\eta$ in $G$.
We use the following claim in order to construct the paths of ${\mathcal{P}}^{\mathsf{out}}$; the proof of the claim uses standard techniques and is deferred to \Cref{subsec: computing out-paths}.
\begin{claim}\label{claim: computing out-paths}
There is an efficient algorithm to compute two sets ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}=\set{P^{\mathsf{out},\operatorname{left}}(e)\mid e\in \hat E}$ and ${\mathcal{P}}^{\mathsf{out},\operatorname{right}}=\set{P^{\mathsf{out},\operatorname{right}}(e)\mid e\in \hat E}$ of simple paths in graph $G$, each of which causes congestion at most $\eta$, such that the paths in set ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}$ are non-transversal with respect to $\Sigma$, and so are the paths in set ${\mathcal{P}}^{\mathsf{out},\operatorname{right}}$. Additionally, for every edge $e\in \hat E$, path $P^{\mathsf{out},\operatorname{left}}(e)$ has $e$ as its first edge, and it is left-monotone, while path $P^{\mathsf{out},\operatorname{right}}(e)$ has $e$ as its first edge, and is right-monotone. Moreover, for every vertex $v\in V'$, exactly $n_1(v)$ paths of ${\mathcal{P}}^{\mathsf{out},\operatorname{left}}$ terminate at $v$, and exactly $n_3(v)$ paths of ${\mathcal{P}}^{\mathsf{out},\operatorname{right}}$ terminate at $v$.
\end{claim}
Consider now some edge $e=(u,v)\in \hat E$, and assume that $u\in \tilde S_i$, $v\in \tilde S_j$, and $i<j$ holds. Consider the paths $P^{\mathsf{out},\operatorname{left}}(e),P^{\mathsf{out},\operatorname{right}}(e)$. Assume that the last vertex on path $P^{\mathsf{out},\operatorname{left}}$ is $u'$, and the last vertex on path $P^{\mathsf{out},\operatorname{right}}$ is $v'$. We let $P^{\mathsf{out}}(e)$ be the path obtained by concatenating path $P^{\mathsf{out},\operatorname{left}}(e)$ with the reversed path $P^{\mathsf{out},\operatorname{right}}(e)$, after deleting the extra copy of edge $e$. We view path $P^{\mathsf{out}}(e)$ as being directed rom $u'$ to $v'$. Therefore, path $P^{\mathsf{out}}(e)$ originates at vertex $u'$ and termiantes at vertex $v'$, and it contains the edge $e$. All inner vertices on $P^{\mathsf{out}}(e)$ belong to $V''$. We will sometimes refer to $u'$ and to $v'$ as the first and the last endpoints of path $P^{\mathsf{out}}(e)$. We will also refer to the edge of $P^{\mathsf{out}}(e)$ that is incident to $u'$ as the \emph{first edge} of path $P^{\mathsf{out}}(e)$, and to the edge of $P^{\mathsf{out}}(e)$ that is incident to $v'$ as the \emph{last edge} of path $P^{\mathsf{out}}(e)$.
Assume that $u'\in V(S_{i''})$ and $v'\in V(S_{j''})$. We define another set of indices associated with edge $e$: $\operatorname{span}''(e)=\set{i'',i''+1,\ldots,j''-1}$. Notice that $\operatorname{span}(e)\subseteq \operatorname{span}''(e)$ must hold by the definition of left-monotone and right-monotone paths. Lastly, we set ${\mathcal{P}}^{\mathsf{out}}=\set{P^{\mathsf{out}}(e)\mid e\in \hat E}$.
For an index $1\leq i<r$, let $\hat E_i\subseteq \hat E$ be the set of all edges $e\in \hat E$, with $i\in \operatorname{span}(e)$. We need the following simple claim.
\begin{claim}\label{claim: out-paths non-transversal}
For every index $1\leq i<r$, the paths in set $\set{P^{\mathsf{out}}(e)\mid e\in \hat E_i}$ are non-transversal with respect to $\Sigma$. Moreover, for each edge $e=(u,v)\in \hat E_i$ whose left endpoint is $u$, every vertex of $P^{\mathsf{out},\operatorname{left}}(e)\setminus\set{v}$ lies in $\bigcup_{z\leq i}V(\tilde S_z)$, and every vertex of $P^{\mathsf{out},\operatorname{right}}(e)\setminus\set{u}$ lies in $\bigcup_{z> i}V(\tilde S_z)$.
\end{claim}
\begin{proof}
The fact that, for every edge $e\in \hat E_i$, every vertex of $P^{\mathsf{out},\operatorname{left}}(e)\setminus\set{v}$ lies in $\bigcup_{z\leq i}V(\tilde S_z)$, and every vertex of $P^{\mathsf{out},\operatorname{right}}(e)\setminus\set{u}$ lies in $\bigcup_{z> i}V(\tilde S_z)$ follows immediately from the definition of left-monotone and right-monotone paths and edge set $\hat E_i$. Consider now any pair $e,e'\in \hat E_i$ of edges, and a vertex $v$ that is an inner vertex of both $P^{\mathsf{out}}(e)$ and $P^{\mathsf{out}}(e')$. Assume that $v\in V(\tilde S_j)$, for some index $1\leq j\leq r$.
From the definition of left-monotone and right-monotone paths, either $j\leq
i$ and $v$ is an inner vertex of both $P^{\mathsf{out},\operatorname{left}}(e)$ and $P^{\mathsf{out},\operatorname{left}}(e')$; or $j>i$, and $v$ is an inner vertex of both $P^{\mathsf{out},\operatorname{right}}(e)$ and $P^{\mathsf{out},\operatorname{right}}(e')$. In either case, \Cref{claim: computing out-paths} ensures that the intersection of $P^{\mathsf{out}}(e)$ and $P^{\mathsf{out}}(e')$ at $v$ is non-transversal.
\end{proof}
For every index $1\leq t<r$, we denote by $N_t$ the number of all edges
$e\in \hat E$, with $t\in \operatorname{span}'(e)$, and we denote by $N'_t$ be the number of all edges
$e\in \hat E$, with $t\in \operatorname{span}''(e)$. We also denote by $E_t\subseteq \tilde E'$ the set of all edges with one endpoint in $S_t$ and another in $S_{t+1}$.
We need the following claim, whose proof appears in \Cref{subsec: enough segments}.
\begin{claim}\label{claim: enough segments}
For all $1\leq t<r$, $N'_t=N_t$, and $N_t\leq O(\log^{18}m)\cdot |E_t|$.
\end{claim}
\subsubsection{Step 4: Constructing the Collection of Subinstances}
\label{subsec: instances}
Consider again an index $1\leq i<r$, and the internal $U_i$-router ${\mathcal{Q}}(U_i)=\set{Q(e)\mid e\in \delta_G(U_i)}$. Recall that all paths in ${\mathcal{Q}}(U_i)$ terminate at vertex $u_i$.
Denote $\delta_G(u_i)=\set{e_1^i,\ldots,e_{|\delta(u_i)|}^i}$, where the edges are indexed according to their order in ${\mathcal{O}}_{u_i}\in \Sigma$. For all $1\leq j\leq |\delta(u_i)|$, let $A_j^i\subseteq \delta_G(U_i)$ be the set of all edges $e\in \delta_G(U_i)$, such that the uniue path $Q(e)\in {\mathcal{Q}}(U_i)$ that has $e$ as its first edge contains edge $e^i_j$ as its last edge. We now define an ordering $\tilde {\mathcal{O}}_i$ of the edges of $\delta_G(U_i)$: the edges in sets $A_1^i,A_2^i,\ldots,A_{|\delta(u_i)|}^i$ appear in the order of the indices of their sets, and, for each $1\leq j\leq |\delta(u_i)|$, the ordering of the edges in set $A_j^i$ is arbitrary. Notice that the resulting ordering $\tilde {\mathcal{O}}_i$ of the edges of $\delta_G(U_i)$ is precisely the ordering ${\mathcal{O}}^{\operatorname{guided}}({\mathcal{Q}}(U_i),\Sigma)$, that is guided by the internal $U_i$-router ${\mathcal{Q}}(U_i)$ (see the definition in \Cref{subsec: guiding paths rotations}). For $i=r$, $\delta_G(U_i)=\emptyset$, and the ordering $\tilde {\mathcal{O}}_r$ of the edges of $\delta_G(U_i)$ is the trivial one.
We let ${\mathcal{I}}_2$ be a collection of subinstances of $I$ obtained by computing a laminar family-based decomposition of $I$ (defined in \Cref{subsec: laminar-based decomposition}) via the laminar family ${\mathcal{L}}=\set{U_i}_{1\leq i\leq r}$ of clusters, and the orderings $\tilde {\mathcal{O}}_i$ of the edge sets $\delta(U_i)$ for all $1\leq i\leq r$. We denote ${\mathcal{I}}_2=\set{I_1,\ldots,I_r}$, where, for $1\leq z\leq r$, instance $I_z=(G_z,\Sigma_z)$ is the instance associated with the cluster $U_z$. Recall that instance $I_z$ is defined as follows. Assume first that $1<z<r$. Then graph $G_z$ is obtained from graph $G$ by first contracting all vertices of $U_{z-1}$ into vertex $v^*_z$, and then contracting all vertices of $U_{z+1}$ into vertex $v^{**}_z$.
Notice that, equivalently, graph $G_z$ consists of the cluster $\tilde S_z\in \tilde {\mathcal{S}}$, the two special vertices $v^*_z,v^{**}_z$, and possibly some additional edges that are incident to these two special vertices.
The rotation system $\Sigma_z$ is defined as follows. Observe first that, for every vertex $v\in V(G_z)\setminus\set{v^*_z,v^{**}_z}$, $\delta_{G_z}(v)=\delta_G(v)$. The ordering ${\mathcal{O}}_v\in \Sigma_z$ of the edges incident to $v$ in $\Sigma_z$ remains the same as in $\Sigma$. Notice that $\delta_{G_z}(v^*_z)=\delta_G(U_{z-1})$. We let the ordering ${\mathcal{O}}_{v^*_z}\in \Sigma_z$ of the edges of $\delta_{G_z}(v^*_z)$ be the ordering $\tilde {\mathcal{O}}_{z-1}$ that we defined above. Lastly, observe that $\delta_{G_z}(v^{**}_z)=\delta_G(U_{z})$. We let the ordering ${\mathcal{O}}_{v^{**}_z}\in \Sigma_z$ of the edges of $\delta_{G_z}(v^{**}_z)$ be the ordering $\tilde {\mathcal{O}}_{z}$ that we defined above.
For $z=1$, instance $I_1=(G_1,\Sigma_1)$ is defined similarly, except that the instance does not contain vertex $v^*_1$. Instance $I_r=(G_r,\Sigma_r)$ is also defined similarly, except that it does not contain vertex $v^{**}_r$.
We now verify that the resulting collection ${\mathcal{I}}_2$ of instances has all required properties. Fix some index $1\leq z\leq r$. Recall that, from the definition of a nice witness structure, there is at most one cluster $C\in {\mathcal{C}}$ with $C\subseteq \tilde S_z$. Recall also that, for each cluster $C\in {\mathcal{C}}$, there is exactly one cluster $S_i\in {\mathcal{S}}$ that contains $C$. If some cluster $C\in {\mathcal{C}}$ is contained in $\tilde S_z$, then $E(\tilde S_z)\subseteq E(C)\cup E(G_{|{\mathcal{C}}})$ must hold, and so $E(G_z)\subseteq E(C)\cup E(G_{|{\mathcal{C}}})$ as well. Otherwise, $E(\tilde S_z)\subseteq E(G_{|{\mathcal{C}}})$, and so $E(G_z)\subseteq E(G_{|{\mathcal{C}}})$.
\iffalse
-----------------------
We now apply the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace to each cluster $U_i\in {\mathcal{L}}$ and its internal router ${\mathcal{W}}_i$, and let $I_1,\ldots, I_r$ be the subinstances we obtain. We recall here how the instances are constructed in \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace.
For each $1\le i\le r$, we denote $\tilde {\mathcal{O}}_i={\mathcal{O}}^{\operatorname{guided}}({\mathcal{W}}_i,\Sigma)$, the ordering on the set $E(U_i, \overline{U}_i)$ of edges, that is guided by ${\mathcal{W}}_i$ and $\Sigma$
First, the instance $I_1=(G_1,\Sigma_1)$ is the cluster-based instance associated with cluster $\tilde S_1$, namely $(G_1,\Sigma_1)=(G_{\tilde S_1},\tilde \Sigma_{\tilde S_1})$. Recall that $V(G_{\tilde S_1})=V(\tilde S_1)\cup \set{v^*_1}$, and the rotation on $v^*_1$, the vertex obtained by contracting $G\setminus \tilde S_1$, is $\tilde {\mathcal{O}}_1$ defined above.
Then for each index $2\le i\le r-1$, the instance $I_i=(G_i,\Sigma_i)$ is obtained from the cluster-based instance $(G_{U_i},\tilde \Sigma_{U_i})$, by taking its contracted instance with respect to cluster $U_{i-1}$.
So $V(G_i)=V(\tilde S_i)\cup \set{v^*_i, v'_i}$, where vertex $v^*_i$ is obtained by contracting $\overline{U}_i$ and vertex $v'_i$ is obtained by contracting $U_{i-1}$. The rotation on vertex $v^*_i$ is $\tilde {\mathcal{O}}_i$, and the rotation on vertex $v'_i$ is $\tilde {\mathcal{O}}_{i-1}$.
Finally, the instance $I_r=(G_r,\Sigma_r)$ is the contracted instance of $(G,\Sigma)$ with respect to cluster $U_{r-1}$. Recall that $V(G_r)=V(\tilde S_r)\cup \set{v'_r}$, and the rotation on $v'_r$, the vertex obtained from contracting the cluster $U_{r-1}$, is $\tilde {\mathcal{O}}_{r-1}$.
\fi
\iffalse
In order to construct the subinstances using the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace, we need to also specify distributions on internal routers for each cluster of $U_1,\ldots,U_{r-1}$, or equivalently construct a random internal router for each of these clusters, which we do next.
Consider the cluster $U_i$ for any $i\ge 1$.
We construct the internal router ${\mathcal{W}}_i$ as follows.
For each edge $e\in E(U_i, \overline{U}_i)$ (clearly this edge belongs to set $\hat E$ by definition), we define path $W_i(e)$ as the subpath of cycle $W(e)$ between (including) the edge $e$ and vertex $u_i\in S_i$, that entirely lies entirely in $U_i$ (see \Cref{obs: outer_path_monotone}). We then let ${\mathcal{W}}_i=\set{W_i(e)\mid e\in E(U_i, \overline{U}_i)}$.
\fi
\iffalse
\begin{figure}[h]
\centering
\subfigure[Set $E(U_i,\overline{U}_i)=\hat E_i\cup E^{\operatorname{right}}_{i}\cup E^{\operatorname{left}}_{i+1}\cup E^{\operatorname{over}}_{i}$ where $\hat E_{i}=\set{\hat e_1,\ldots,\hat e_4}$, $E^{\operatorname{right}}_{i}=\set{e^b}$, $E^{\operatorname{left}}_{i+1}=\set{e^g_1,e^g_2}$ and $E^{\operatorname{over}}_{i}=\set{e^r}$.
Paths of ${\mathcal{W}}_i$ excluding their first edges are shown in dash lines.]{\scalebox{0.13}{\includegraphics{figs/in_rotation_1.jpg}}}
\hspace{3pt}
\subfigure[The circular ordering $\tilde{\mathcal{O}}_{i}$ on the edges of $E(U_i,\overline{U}_i)$.]{
\scalebox{0.14}{\includegraphics{figs/in_rotation_2.jpg}}}
\caption{An illustration of path set ${\mathcal{W}}_i$ and the circular ordering $\tilde{\mathcal{O}}_{i}$.}\label{fig: in rotation good}
\end{figure}
\fi
\iffalse
\begin{figure}[h]
\centering
\subfigure[Layout of edge sets in $G$. Edges of $E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}$ are shown in green, and edges of $E^{\textsf{thr}}_{i}$ are shown in red. ]{\scalebox{0.14}{\includegraphics{figs/subinstance_1.jpg}
}
\hspace{5pt}
\subfigure[Graph $G_i$, where $\delta_{G_i}(v_i^{*})=\hat E_i \cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, and $\delta_{G_i}(v_i')=\hat E_{i-1} \cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_{i}$.]{
\scalebox{0.12}{\includegraphics{figs/subinstance_2.jpg}}}
\caption{An illustration of the construction of subinstance $(G_i,\Sigma_i)$.}\label{fig: subinstance}
\end{figure}
\fi
From \Cref{lem: basic disengagement combining solutions},
there is an efficient algorithm, that, given, for each instance $I_z\in {\mathcal{I}}$, a solution $\phi_z$, computes a solution for instance $I$ of value at most $\sum_{t=1}^r\mathsf{cr}(\phi_z)$.
Next, we bound the total number of edges in all resulting instances.
\iffalse
\subsubsection{Completing the Proof of \Cref{thm: advanced disengagement - disengage nice instances}}
\label{subsec: proof of disengagement with routing}
In this section we complete the proof of \Cref{thm: advanced disengagement - disengage nice instances}, by showing that the collection ${\mathcal{I}}_2$ of subinstances constructed in \Cref{subsec: instances} is indeed a $2^{O((\log m)^{3/4}\log\log m)}$-decomposition that satisfies the properties in \Cref{thm: advanced disengagement - disengage nice instances}.
Since the subinstances are constructed using the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace, \Cref{lem: basic disengagement combining solutions} holds. In particular, if we are given, for each $1\le i\le r$, a solution $\phi_i$ to subinstance $I_i$, then we can efficiently combine them together to obtain a solution $\phi$ for the input instance $I$ of cost at most $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
On the other hand, from the construction of the subinstances $I_1=(G_1,\Sigma_1),\ldots,I_r=(G_r,\Sigma_r)$, it is clear that, for each index $1\le i\le r$, $E(\tilde S_i)\subseteq E(G_i)$, and all other edges of $G_i$ lie in $E^{\textnormal{\textsf{out}}}(\tilde{\mathcal{S}})$. Note that each cluster in $\tilde {\mathcal{S}}$ contains at most one cluster of ${\mathcal{C}}'$, so all edges of $E^{\textnormal{\textsf{out}}}(\tilde{\mathcal{S}})$ belongs to $E(G_{\mid {\mathcal{C}}'})$.
We first assume that there exists a cluster $C_i\in {\mathcal{C}}'$ with $C_i\subseteq S_i$. Then from the definition of a nice witness structure, $E(\tilde S_i)\subseteq E(C_i)\cup E(G_{\mid {\mathcal{C}}'})$. Altogether, we get that $E(C_i)\subseteq E(G_i)$, and $E(G_i)\subseteq E(C_i)\cup E(G_{\mid {\mathcal{C}}'})$. Assume now that there does not exist a cluster $C_i\in {\mathcal{C}}'$ with $C_i\subseteq S_i$. Then from the definition of a nice witness structure, $E(\tilde S_i)\subseteq E(G_{\mid {\mathcal{C}}'})$, and therefore $E(G_i)\subseteq E(G_{\mid {\mathcal{C}}'})$.
For each $1\le i\le r$, we denote
$\hat E_i=E(\tilde S_i, \tilde S_{i+1})$,
$E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(\tilde S_i, \tilde S_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(\tilde S_i, \tilde S_{j'})$, and $E_i^{\textsf{thr}}=\bigcup_{i'<i<j'}E(\tilde S_i, \tilde S_{j'})$.
We prove the following observation.
\fi
\begin{observation}
\label{obs: disengaged instance size}
$\sum_{1\le z\le r}|E(G_z)|\le O(|E(G)|\cdot \log^{34}m)$.
\end{observation}
\begin{proof}
Fix an index $1\leq z\leq r$. From our construction, $|E(G_z)|\leq |E(\tilde S_z)|+|\delta_G(U_{z-1})|+|\delta_G(U_z)|$.
From \Cref{cor: few edges crossing cuts}, $|\delta_G(U_{z-1})|\leq |\delta_G(S_{z-1})|\cdot O(\log^{34}m)$, and $|\delta_G(U_z)|\leq |\delta_G(S_{z+1})|\cdot O(\log^{34}m)$.
Therefore, $|E(G_z)|\leq |E(\tilde S_z)|+(|\delta_G(S_{z-1})|+|\delta_G(S_{z+1})|)\cdot O(\log^{34}m)$. Summing up over all indices $z$, we get that:
\[\sum_{z=1}^r|E(G_z)|\leq \sum_{z=1}^r|E(\tilde S_z)|+ O(\log^{34}m)\cdot \sum_{z=1}^r|\delta_G(S_z)|\leq O(|E(G)|\cdot \log^{34}m). \]
\end{proof}
Recall that we have used a randomized algorithm to compute, for all $1\leq i\leq r$, an internal $U_i$-router ${\mathcal{Q}}(U_i)$ and an external $U_i$-router ${\mathcal{Q}}'(U_i)$.
In order to complete the proof of \Cref{thm: advanced disengagement - disengage nice instances}, it is now enough to show that the expected total optimal solution costs of all instances in ${\mathcal{I}}_2$ (over the random choices performed by the algorithm that computed the internal and the external $U_i$-routers) is bounded by $2^{O((\log m)^{3/4}\log\log m)}\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)$. The following claim, whose proof appears in \Cref{subsec: bound opt costs} will finish the proof of \Cref{thm: advanced disengagement - disengage nice instances}.
\begin{claim}
\label{claim: existence of good solutions special}
$\expect{\sum_{z=1}^r\mathsf{OPT}_{\mathsf{cnwrs}}(I_z)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|)$.
\end{claim}
\iffalse
Old stuff below
In the second stage, we exploit the auxiliary cycles that were constructed in the first stage, in order to compute a decomposition ${\mathcal{I}}_2(I)$ of the input instance $I$. In order to compute the decompmositin, we define a laminar family ${\mathcal{L}}=\set{U_1,\ldots,U_r}$ of clusters of $G$, and we exploit the set ${\mathcal{W}}$ of auxiliary cycles in order to define an internal router and an external router for each resulting cluster. We then use the algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace defined from \Cref{subsec: basic disengagement} in order to construct the family ${\mathcal{I}}_2(I)$ of subinstances of $I$. We will rely on \Cref{lem: basic disengagement combining solutions} to show that there is an algorithm that combines solutions to the subinstances in ${\mathcal{I}}_2(I)$ in order to obtain a cheap solution for instance $I$. However, since the depth of the laminar family ${\mathcal{L}}$ may be quite large, we will not rely on \Cref{lem: number of edges in all disengaged instances} and on \Cref{lem: disengagement final cost} in order to bound the total number of edges in all resulting instances and their total optimal solution cost. Instead, we will prove these bounds directly.
For each $1\le i\le r$, we define cluster $U_i$ to be the subgraph of $G$ induced by $\bigcup_{1\le t\le i}V(\tilde S_z)$. For convenience, we also denote by $\overline{U}_i$ the subgraph of $G$ induced by $\bigcup_{i+1\le t\le r}V(\tilde S_z)$. We then define a laminar family ${\mathcal{L}}=\set{U_1,\ldots,U_r}$ of clusters of $G$. Notice that $U_r=G$, and $U_1\subseteq U_2\subseteq\cdots\subseteq U_r$.
---------
We now fix an index $1\leq i\leq r$ and consider the set $\delta_G(U_i)=E(U_i,\overline{U}_i)$ of edges. We can partition the edge set $\delta_G(U_i)$ into two subsets: set $E_i=E(S_i,S_{i+1})$, and set $\hat E_i$, containing all remaining edges. Notice that $\hat E_i$ is precisely the set of all edges $e\in \hat E$ with $i\in \operatorname{span}(e)$. We need the following simple observation.
\begin{observation}\label{obs: cycles visit centers}
For every edge $e\in \hat E_i$, path $P^{\mathsf{in}}(e)$ contains a contiguious subpath $R_i(e)$ that starts at $u_i$, terminates at $u_{i+1}$, and contains exactly one edge of $E_i$.
\end{observation}
\begin{proof}
Fix an edge $e\in \hat E_i$, and
assume that $\operatorname{span}'(e)=(i'',j'')$. Then there is a subpath of $P^{\mathsf{out},\operatorname{left}}(e)$, that starts with edge $e$, terminates at a vertex of $\delta(S_{i''})$, and is left-monotone (the path constructed in \Cref{claim: computing out-paths}) From the definition of a left-monotone path, $i''\leq i$ must hold. Using a similar argument, $j''\geq j$ must hold. But then, from the definition of the path $P^{\mathsf{in}}(e)$, it must contain a segment $R_i(e)\in {\mathcal{R}}_i$, that starts at $u_i$, terminates at $u_{i+1}$, and contains exactly one edge of $E_i$.
\end{proof}
Next, we define an internal router $\tilde {\mathcal{Q}}_i$, and an external router $\tilde{\mathcal{Q}}'_i$ for cluster $U_i$. All paths in $\tilde {\mathcal{Q}}_i$ will terminate at $u_i$, and all paths in $\tilde {\mathcal{Q}}'_i$ will terminate at $u_{i_1}$. We start with $\tilde {\mathcal{Q}}_i=\tilde {\mathcal{Q}}'_i=\emptyset$, and then consider each edge $e\in \delta(U_i)$ one by one. If $e\in E_i$, then we add to $\tilde {\mathcal{Q}}_i$ the unique path $Q_i(e)\in {\mathcal{Q}}_i$ routing $e$ to $u_i$; recall that all vertices of $Q_i(e)$ are contained in $S_i$. We also add to $\tilde {\mathcal{Q}}'_i$ the unique path $Q_{i+1}(e)\in {\mathcal{Q}}_{i+1}$, routing $e$ to $u_{i+1}$; recall that all vertices of $Q_{i+1}(e)$ are contained in $S_i$.
Assume now that $e=(u,v)\in \hat E_i$, and that $u$ is the left endpoint of the edge. After we delete edge $e$ and path $R_i(e)$ (given by \Cref{obs: cycles visit centers}) from the auxiliary cycle $W(e)$, we obtain two paths: the first path, that we denote by $Q$, connects $u$ to $u_i$, and is contained in $U_i$; the second path, that we denote by $Q'$, connects $v$ to $u_{i+1}$, and is contained in $\overline{U}_i$. We add to $\tilde{\mathcal{Q}}_i$ the path obtained from $Q$ by appending edge $e$ to the beginning of the path, so it connects edge $e$ to $u_i$, and we add to $\tilde {\mathcal{Q}}_i'$ the path obtained from $Q'$ by appending edge $e$ to the beginning of the path, so it connects edge $e$ to $u_{i+1}$.
This completes the construction of the internal router $\tilde {\mathcal{Q}}_i$, and the external router $\tilde{\mathcal{Q}}'_i$ for cluster $U_i$. From the fact that the paths in each of the sets ${\mathcal{Q}}_i,{\mathcal{Q}}_{i+1}$ are non-transversal with respect to $\Sigma$, and from \Cref{obs: auxiliary cycles non-transversal at at most one}, we obtain the following immediate observation.
\begin{observation}\label{obs: inner non-transversal}
For all $1\leq i<r$, the set $\tilde {\mathcal{Q}}_i$ of paths is non-transversal with respect to $\Sigma$.
\end{observation}
-------------
\fi
\subsubsection{Constructing Sub-Instances}
\label{sec: compute advanced disengagement}
In this step we will construct, for each cluster $C_i\in {\mathcal{C}}$, an sub-instance $I_i=(G_i,\Sigma_i)$ of $(G,\Sigma)$, such that the instances ${\mathcal{I}}=\set{I_1,\ldots,I_r}$ satisfy the properties in \Cref{thm: disengagement - main}.
Recall that, for each $1\le i\le r$, $E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$, and $E^{\textsf{thr}}_i=\bigcup_{i'\le i-1,j'\ge i+1}E(C_{i'},C_{j'})$.
\paragraph{Instances $(G_2,\Sigma_2),\ldots,(G_{r-1},\Sigma_{r-1})$.}
We first fix some index $2\le i\le r-1$ and define the instance $(G_i,\Sigma_i)$ as follows.
The graph $G_i$ is obtained from $G$ by first contracting clusters $C_1,\ldots,C_{i-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_i$, and then contracting clusters $C_{i+1},\ldots,C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_i$, and finally deleting self-loops on the super-nodes $v^{\operatorname{left}}_i$ and $v^{\operatorname{right}}_i$. So $V(G_i)=V(C_i)\cup \set{ v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$. See \Cref{fig: disengaged instance} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edge sets in $G$. Edges of $E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}$ are shown in green, and edges of $E^{\textsf{thr}}_{i}$ are shown in red. ]{\scalebox{0.32}{\includegraphics{figs/disengaged_instance_1.jpg}
}
\hspace{5pt}
\subfigure[Graph $G_i$. Note that $\delta_{G_i}(v_i^{\operatorname{right}})=\hat E_i \cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$, and $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1} \cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_{i}$.]{
\scalebox{0.32}{\includegraphics{figs/disengaged_instance_2.jpg}}}
\caption{An illustration of the construction of sub-instance $(G_i,\Sigma_i)$.}\label{fig: disengaged instance}
\end{figure}
We now define the orderings in $\Sigma_i$. First, for each vertex $v\in V(C_i)$, the ordering on its incident edges is defined to be ${\mathcal{O}}_v$, the rotation on vertex $v$ in the given rotation system $\Sigma$.
It remains to define the rotations of super-nodes $v^{\operatorname{left}}_i,v^{\operatorname{right}}_i$.
We first consider $v^{\operatorname{left}}_i$.
Note that $\delta_{G_i}(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$.
For each edge $\hat e\in \hat E_{i-1}$, recall that $Q_{i-1}(\hat e)$ is the path in $C_{i-1}$ routing edge $\hat e$ to $u_{i-1}$.
For each edge $e\in E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i$, we denote by $W^{\operatorname{left}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i-1}$ and contains its entire left auxiliary path.
We then denote
$${\mathcal{W}}^{\operatorname{left}}_i=\set{W^{\operatorname{left}}_i(e)\mid e\in E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i}\cup \set{Q_{i-1}(\hat e)\mid \hat e\in \hat E_{i-1}}.$$
\znote{need to treat the red edges that enter $C_{i-1}$ specially}
Intuitively, the rotation on vertex $v^{\operatorname{left}}_i$ is defined to be the ordering in which the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ enter $u_{i-1}$.
Formally, for every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$, let $e^*_W$ be the unique edge of path $W$ that is incident to $u_{i-1}$. We first define a circular ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$, as follows: the paths are ordered according to the circular ordering of their corresponding edges $e^*_W$ in ${\mathcal{O}}_{u_{i-1}}\in \Sigma$, breaking ties arbitrarily. Since every path $W\in {\mathcal{W}}^{\operatorname{left}}_i$ is associated with a unique edge in $\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$, this ordering of the paths in ${\mathcal{W}}^{\operatorname{left}}_i$ immediately defines a circular ordering of the edges of $\delta_{G_i}(v_i^{\operatorname{left}})=\hat E_{i-1}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{left}}_i$, that we denote by ${\mathcal{O}}^{\operatorname{left}}_i$. See Figure~\ref{fig: v_left rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i-1}=\set{\hat e_1,\ldots,\hat e_4}$, $E^{\operatorname{left}}_{i}=\set{e^g_1}$ and $E^{\textsf{thr}}_{i}=\set{e^r_1}$.
Paths of ${\mathcal{W}}^{\operatorname{left}}_i$ excluding their first edges are shown in dash lines.]{\scalebox{0.13}{\includegraphics{figs/rotation_left_1.jpg}}}
\hspace{3pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{left}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. Set $\delta_{G_i}(v^{\operatorname{left}}_i)=\set{\hat e_1,\hat e_2,\hat e_3,\hat e_4,e^g_1, e^r_1}$. The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on them around $v^{\operatorname{left}}_i$ is shown above.]{
\scalebox{0.16}{\includegraphics{figs/rotation_left_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_left rotation}
\end{figure}
The rotation ${\mathcal{O}}^{\operatorname{right}}_{i}$ on vertex $v^{\operatorname{right}}_i$ is defined similarly.
Note that $\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$.
For each edge $\hat e'\in \hat E_{i}\cup E_i^{\operatorname{right}}$, recall that $Q_{i}(\hat e')$ is the path in $C_{i}$ routing edge $\hat e'$ to vertex $u_{i}$.
For each edge $e\in E^{\textsf{thr}}_i$, we denote by $W^{\operatorname{right}}_i(e)$ the subpath of the auxiliary cycle $R_e$ that connects $e$ to $u_{i}$ and contains its entire left auxiliary path.
We then denote
$${\mathcal{W}}^{\operatorname{right}}_i=\set{W^{\operatorname{right}}_i(e)\text{ }\big|\text{ } e\in E^{\textsf{thr}}_i}\cup \set{Q_{i}(\hat e')\text{ }\big|\text{ } \hat e'\in \hat E_{i}\cup E^{\operatorname{right}}_i}.$$
The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ is then defined in a similar way as the rotation ${\mathcal{O}}^{\operatorname{left}}_i$ on vertex $v^{\operatorname{left}}_i$, according to the paths of ${\mathcal{W}}^{\operatorname{right}}_i$ and the rotation ${\mathcal{O}}_{u_{i}}\in \Sigma$.
See Figure~\ref{fig: v_right rotation} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Layout of edges and paths, where $\hat E_{i}=\set{\hat e_1',\ldots,\hat e_4'}$, $E^{\operatorname{right}}_{i}=\set{\tilde e^g_1}$ and $E^{\textsf{thr}}_{i}=\set{\tilde e^r_1}$.
Paths of ${\mathcal{W}}^{\operatorname{right}}_i$ excluding their first edges are shown in dash lines. ]{\scalebox{0.13}{\includegraphics{figs/rotation_right_1.jpg}
}
\hspace{3pt}
\subfigure[The edges in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and the rotation ${\mathcal{O}}^{\operatorname{left}}_i$. Set $\delta_{G_i}(v^{\operatorname{right}}_i)=\set{\hat e_1',\hat e_2',\hat e_3',\hat e_4',\tilde e^g_1,\tilde e^r_1}$. The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on them around $v^{\operatorname{right}}_i$ is shown above.]{
\scalebox{0.16}{\includegraphics{figs/rotation_right_2.jpg}}}
\caption{The rotation ${\mathcal{O}}^{\operatorname{right}}_i$ on vertex $v^{\operatorname{right}}_i$ in the instance $(G_i,\Sigma_i)$.}\label{fig: v_right rotation}
\end{figure}
\paragraph{Instances $(G_1,\Sigma_1)$ and $(G_r,\Sigma_r)$.}
The instances $(G_1,\Sigma_1)$ and $(G_r,\Sigma_r)$ are defined similarly, but instead of two super-nodes, the graphs $G_1$ and $G_r$ contain one super-node each.
In particular, graph $G_1$ is obtained from $G$ by contracting clusters $C_2,\ldots, C_r$ into a super-node, that we denote by $v^{\operatorname{right}}_1$, and then deleting self-loops on it.
So $V(G_1)=V(C_1)\cup \set{v^{\operatorname{right}}_{1}}$ and $\delta_{G_1}(v^{\operatorname{right}}_{1})=\hat E_1\cup E^{\operatorname{right}}_1$.
The rotation of a vertex $v\in V(C_1)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{right}}_1$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{right}}_i$ for any index $2\le i\le r-1$.
Graph $G_r$ is obtained from $G$ by contracting clusters $C_1,\ldots, C_{r-1}$ into a super-node, that we denote by $v^{\operatorname{left}}_r$, and then deleting self-loops on it.
So $V(G_r)=V(C_r)\cup \set{v^{\operatorname{left}}_{r}}$ and $\delta_{G_r}(v^{\operatorname{left}}_{r})=\hat E_{r-1}\cup E^{\operatorname{left}}_r$.
The rotation of a vertex $v\in V(C_r)$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the given rotation system $\Sigma$, and the rotation ${\mathcal{O}}^{\operatorname{left}}_r$ is defined in a similar way as ${\mathcal{O}}^{\operatorname{left}}_i$ for any index $2\le i\le r-1$.
We will use the following claims later for completing the proof of \Cref{thm: disengagement - main} in the special case.
\begin{observation}
\label{obs: disengaged instance size}
$\sum_{1\le i\le r}|E(G_i)|\le O(|E(G)|)$, and for each $1\le i\le r$, $|E(G_i)|\le m/\mu$. \znote{parameter?}
\end{observation}
\begin{proof}
Note that, in the sub-instances $\set{(G_i,\Sigma_i)}_{1\le i\le r}$, each graph of $\set{G_i}_{1\le i\le r}$ is obtained from $G$ by contracting some sets of clusters of ${\mathcal{C}}$ into a single super-node, so each edge of $G_i$ corresponds to an edge in $E(G)$.
Therefore, for each $1\le i\le r$,
$$|E(G_i)|=|E_G(C_i)|+|\delta_G(C_i)|\le |E_G(C_i)|+|E^{\textsf{out}}({\mathcal{C}})|\le m/(100\mu)+m/(100\mu)\le m/\mu.$$
Note that $E(G_i)=E(C_i)\cup \hat E_i\cup \hat E_{i-1}\cup E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}\cup E_i^{\textsf{thr}}$.
First, each edge of $\bigcup_{1\le i\le r}E(C_i)$ appears in exactly one graphs of $\set{G_i}_{1\le i\le r}$.
Second, each edge of $\bigcup_{1\le i\le r}\hat E_i$ appears in exactly two graphs of $\set{G_i}_{1\le i\le r}$.
Consider now an edge $e\in E'$. If $e$ connects a vertex of $C_i$ to a vertex of $C_j$ for some $j\ge i+2$, then $e$ will appear as an edge in $E_i^{\operatorname{right}}\subseteq E(G_i)$ and an edge in $E_j^{\operatorname{left}}\subseteq E(G_j)$, and it will appear as an edge in $E_k^{\textsf{thr}}\subseteq E(G_k)$ for all $i<k<j$.
On one hand, we have $\sum_{1\le i\le r}|E_i^{\operatorname{left}} \cup E_i^{\operatorname{right}}|\le 2|E(G)|$.
On the other hand, note that $E_k^{\textsf{thr}}=E(C_{k-1},C_{k+1})\cup E^{\operatorname{over}}_{k-1}\cup E^{\operatorname{over}}_{k}$, and each edge of $e$ appears in at most two graphs of $\set{G_i}_{1\le i\le r}$ as an edge of $E(C_{k-1},C_{k+1})$.
Moreover, from \Cref{obs: bad inded structure}, $|E^{\operatorname{over}}_{k-1}|\le |\hat E_{k-1}|$ and $|E^{\operatorname{over}}_{k}|\le |\hat E_{k}|$.
Altogether, we have
\begin{equation}
\begin{split}
\sum_{1\le i\le r}|E(G_i)| & =
\sum_{1\le i\le r}\textsf{left}(
|E(C_i)|+ |\hat E_i|+ |\hat E_{i-1}|+ |E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|+|E_i^{\textsf{thr}}|
\textsf{right})\\
& = \sum_{1\le i\le r}
|E(C_i)|+ \sum_{1\le i\le r} \textsf{left}(|E_i^{\operatorname{left}}| + |E_i^{\operatorname{right}}|\textsf{right})+ \sum_{1\le i\le r}\textsf{left}(|E_i^{\textsf{thr}}|+ |\hat E_i|+ |\hat E_{i-1}|\textsf{right})\\
& \le |E(G)|+ 2\cdot |E(G)| + \sum_{1\le i\le r}\textsf{left}(|E(C_{i-1},C_{i+1})|+ 2|\hat E_i|+ 2|\hat E_{i-1}|\textsf{right})\\
& \le 8\cdot |E(G)|.
\end{split}
\end{equation}
This completes the proof of \Cref{obs: disengaged instance size}.
\end{proof}
\begin{observation}
\label{obs: rotation for stitching}
For each $1\le i\le r-1$, if we view the edge in $\delta_{G_i}(v^{\operatorname{right}}_i)$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$ as edges of $E(G)$, then $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$, and moreover, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{observation}
\begin{proof}
Recall that for each $1\le i\le r-1$,
$\delta_{G_i}(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\textsf{thr}}_i\cup E^{\operatorname{right}}_i$ and $\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})=\hat E_{i}\cup E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}$. From the definition of sets $E_i^{\textsf{thr}},E_{i+1}^{\textsf{thr}}, E^{\operatorname{right}}_i, E^{\operatorname{left}}_{i+1}$,
\[
\begin{split}
E_i^{\textsf{thr}}\cup E^{\operatorname{right}}_i
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i<j'\text{ or }i'=i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'\le i<j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'< i+1\le j'}\\
= & \set{e\in E(C_{i'},C_{j'})\mid i'<i+1<j'\text{ or }i'<i+1=j'}=E^{\textsf{thr}}_{i+1}\cup E^{\operatorname{left}}_{i+1}.
\end{split}
\]
Therefore, $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{G_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Moreover, from the definition of path sets ${\mathcal{W}}^{\operatorname{right}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_{i+1}$, it is not hard to verify that, for every edge $e\in \delta_{G_i}(v^{\operatorname{right}}_i)$, the path in ${\mathcal{W}}^{\operatorname{right}}_i$ that contains $e$ as its first edge is identical to the path in ${\mathcal{W}}^{\operatorname{left}}_{i+1}$ that contains $e$ as its first edge. According to the way that rotations ${\mathcal{O}}^{\operatorname{right}}_i,{\mathcal{O}}^{\operatorname{left}}_{i+1}$ are defined, ${\mathcal{O}}^{\operatorname{right}}_i={\mathcal{O}}^{\operatorname{left}}_{i+1}$.
\end{proof}
\subsubsection{Completing the Proof of \Cref{thm: disengagement - main} in the Special Case}
\label{sec: path case with no problematic index}
In this section we complete the proof of \Cref{thm: disengagement - main} in the special case where the Gomory-Hu tree of $H=G_{\mid {\mathcal{C}}}$ is a path and we are additionally given, for each edge $e\in E'$, an inner path $P_e$ in graph $H$, such that set $\set{P_e}_{e\in E'}$ of paths causes congestion at most $\eta$ in $H$, for some $\eta=2^{O((\log m)^{3/4}\log\log m)}$.
Specifically, we use the following two claims, whose proofs will be provided later.
\begin{claim}
\label{claim: existence of good solutions special}
$\expect{\sum_{1\le i\le r}\mathsf{OPT}_{\mathsf{cnwrs}}(G_i,\Sigma_i)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$.
\end{claim}
\begin{claim}
\label{claim: stitching the drawings together}
There is an efficient algorithm, that given, for each $1\le i\le r$, a feasible solution $\phi_i$ to the instance $(G_i,\Sigma_i)$, computes a solution to the instance $(G,\Sigma)$, such that $\mathsf{cr}(\phi)\le \sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
\end{claim}
We use the algorithm described in~\Cref{sec: guiding and auxiliary paths} and~\Cref{sec: compute advanced disengagement}, and return the disengaged instances $(G_1,\Sigma_1),\ldots,(G_r,\Sigma_r)$ as the collection of sub-instances of $(G,\Sigma)$.
From the previous subsections, the algorithm for producing the sub-instances is efficient.
On the other hand, it follows immediately from \Cref{obs: disengaged instance size}, \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together} that the sub-instances $(G_1,\Sigma_1),\ldots,(G_r,\Sigma_r)$ satisfy the properties in \Cref{thm: disengagement - main}.
This completes the proof of \Cref{thm: disengagement - main}.
We now provide the proofs of \Cref{claim: existence of good solutions special}, and~\Cref{claim: stitching the drawings together}.
$\ $
\begin{proofof}{Claim~\ref{claim: existence of good solutions special}}
Let $\phi^*$ be an optimal drawing of the instance $(G,\Sigma)$.
We will construct, for each $1\le i\le r$, a drawing $\phi_i$ of $G_i$ that respects the rotation system $\Sigma_i$, based on the drawing $\phi^*$, such that $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)\le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot 2^{O((\log m)^{3/4}\log\log m)})$, and \Cref{claim: existence of good solutions special} then follows.
\paragraph{Drawings $\phi_2,\ldots,\phi_{r-1}$.}
First we fix some index $2\le i\le r-1$, and describe the construction of the drawing $\phi_i$. We start with some definitions.
Recall that $E(G_i)=E_G(C_i)\cup (\hat E_{i-1}\cup \hat E_{i}) \cup (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})$.
We define the auxiliary path set ${\mathcal{W}}_i={\mathcal{W}}^{\operatorname{left}}_i\cup {\mathcal{W}}^{\operatorname{right}}_i$, so
$${\mathcal{W}}_i=\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}\cup \set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}},$$
where for each $e\in E^{\operatorname{left}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$; for each $e\in E^{\operatorname{right}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$ and the subpath of its inner path $P_e$ between $u_{i+1}$ and its last endpoint; and for each $e\in E^{\textsf{thr}}_{i}$, the path $W_e$ is the union of its outer path $P^{\mathsf{out}}_e$, the subpath of its inner path $P_e$ between its first endpoint and $u_{i-1}$, and the subpath of $P_e$ between $u_{i+1}$ and its last endpoint.
\iffalse
We use the following observation.
\begin{observation}
\label{obs: wset_i_non_interfering}
The set ${\mathcal{W}}_i$ of paths are non-transversal with respect to $\Sigma$.
\end{observation}
\begin{proof}
Recall that the paths in ${\mathcal{Q}}_{i-1}$ only uses edges of $E(C_{i-1})\cup \delta(C_{i-1})$, and they are non-transversal.
And similarly, the paths in ${\mathcal{Q}}_{i+1}$ only uses edges of $E(C_{i+1})\cup \delta(C_{i+1})$, and they are non-transversal.
Therefore, the paths in $\set{Q_{i+1}(\hat e)\text{ }\big|\text{ }\hat e\in \hat E_{i}}\cup \set{Q_{i-1}(\hat e)\text{ }\big|\text{ } \hat e\in \hat E_{i-1}}$ are non-transversal.
From \Cref{obs: non_transversal_1} and \Cref{obs: non_transversal_2}, the paths in $\set{W_e \text{ }\big|\text{ } e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i}\cup E^{\operatorname{right}}_{i})}$ are non-transversal. Therefore, it suffices to show that, the set ${\mathcal{W}}_i$ of paths are non-transversal at all vertices of $C_{i-1}$ and all vertices of $C_{i+1}$. Note that, for each edge $e\in (E^{\operatorname{left}}_{i}\cup E^{\textsf{thr}}_{i})$, $W_e\cap (C_{i-1}\cup \delta(C_{i-1}))$ is indeed a path of ${\mathcal{Q}}_{i-1}$. Therefore, the paths in ${\mathcal{W}}_i$ are non-transversal at all vertices of $C_{i-1}$. Similarly, they are also non-transversal at all vertices of $C_{i+1}$. Altogether, the paths of ${\mathcal{W}}_i$ are non-transversal with respect to $\Sigma$.
\end{proof}
\fi
For uniformity of notations, for each edge $\hat e\in \hat E_i$, we rename the path $Q_{i+1}(\hat e)$ by $W_{\hat e}$, and similarly for each edge $\hat e\in \hat E_{i-1}$, we rename the path $Q_{i-1}(\hat e)$ by $W_{\hat e}$. Therefore, ${\mathcal{W}}_i=\set{W_e\mid e\in E(G_i)\setminus E(C_i)}$.
Put in other words, the set ${\mathcal{W}}_i$ contains, for each edge $e$ in $G_i$ that is incident to $v^{\operatorname{left}}_i$ or $v^{\operatorname{right}}_i$, a path named $W_e$.
It is easy to see that all paths in ${\mathcal{W}}_i$ are internally disjoint from $C_i$.
We further partition the set ${\mathcal{W}}_i$ into two sets: ${\mathcal{W}}_i^{\textsf{thr}}=\set{W_e\mid e\in E^{\textsf{thr}}_i}$ and $\tilde {\mathcal{W}}_i={\mathcal{W}}_i\setminus {\mathcal{W}}_i^{\textsf{thr}}$.
We are now ready to construct the drawing $\phi_i$ for the instance $(G_i,\Sigma_i)$.
Recall that $\phi^*$ is an optimal drawing of the input instance $(G,\Sigma)$.
We start with the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$, that we denote by $\phi^*_i$. We will not modify the image of $C_i$ in $\phi^*_i$, but will focus on constructing the image of edges in $E(G_i)\setminus E(C_i)$, based on the image of edges in $E({\mathcal{W}}_i)$ in $\phi^*_i$.
Specifically, we proceed in the following four steps.
\paragraph{Step 1.}
For each edge $e\in E({\mathcal{W}}_i)$, we denote by $\pi_e$ the curve that represents the image of $e$ in $\phi^*_i$. We create a set of $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting the endpoints of $e$ in $\phi^*_i$, that lies in an arbitrarily thin strip around $\pi_e$.
We denote by $\Pi_e$ the set of these curves.
We then assign, for each edge $e\in E({\mathcal{W}}_i)$ and for each path in ${\mathcal{W}}_i$ that contains the edge $e$, a distinct curve in $\Pi_e$ to this path.
Therefore, each curve in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ is assigned to exactly one path of ${\mathcal{W}}_i$,
and each path $W\in {\mathcal{W}}_i$ is assigned with, for each edge $e\in E(W)$, a curve in $\Pi_e$. Let $\gamma_W$ be the curve obtained by concatenating all curves in $\bigcup_{e\in E({\mathcal{W}}_i)}\Pi_e$ that are assigned to path $W$, so $\gamma_W$ connects the endpoints of path $W$ in $\phi^*_i$.
In fact, when we assign curves in $\bigcup_{e\in \delta(u_{i-1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{left}}_i$ (recall that $\delta(v^{\operatorname{left}}_i)=\hat E_{i-1}\cup E^{\operatorname{left}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{left}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{left}}_i)}$), we additionally ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{left}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{left}}_i)$ enter $u_{i-1}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
And similarly, when we assign curves in $\bigcup_{e\in \delta(u_{i+1})}\Pi_e$ to paths in ${\mathcal{W}}^{\operatorname{right}}_i$ (recall that $\delta(v^{\operatorname{right}}_i)=\hat E_{i}\cup E^{\operatorname{right}}_i \cup E^{\textsf{thr}}_i$ and ${\mathcal{W}}^{\operatorname{right}}_i=\set{W_{e'}\mid e'\in \delta(v^{\operatorname{right}}_i)}$), we ensure that, if we view, for each edge $e'\in \delta(v^{\operatorname{right}}_i)$, the curve $\gamma_{W_{e'}}$ as the image of $e'$, then the image of edges in $\delta(v^{\operatorname{right}}_i)$ enter $u_{i+1}$ in the same order as ${\mathcal{O}}^{\operatorname{right}}_i$.
Note that this can be easily achieved according to the definition of ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$.
We denote $\Gamma_i=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$, and we further partition the set $\Gamma_i$ into two sets: $\Gamma_i^{\textsf{thr}}=\set{\gamma_{W}\mid W\in {\mathcal{W}}^{\textsf{thr}}_i}$ and $\tilde \Gamma_i=\Gamma_i\setminus \Gamma_i^{\textsf{thr}}$.
We denote by $\hat \phi_i$ the drawing obtained by taking the union of the image of $C_i$ in $\phi^*_i$ and all curves in $\Gamma_i$.
For every path $P$ in $G_i$, we denote by $\hat{\chi}_i(P)$ the number of crossings that involves the ``image of $P$'' in $\hat \phi_i$, which is defined as the union of, for each edge $e\in E(\tilde{\mathcal{W}}_i)$, an arbitrary curve in $\Pi_e$.
Clearly, for each edge $e\in E({\mathcal{W}}_i)$, all curves in $\Pi_e$ are crossed by other curves of $(\Gamma_i\setminus \Pi_e)\cup \phi^*_i(C_i)$ same number of times.
Therefore, $\hat{\chi}_i(P)$ is well-defined.
For a set ${\mathcal{P}}$ of paths in $G_i$, we define $\hat{\chi}_i({\mathcal{P}})=\sum_{P\in {\mathcal{P}}}\hat{\chi}_i(P)$.
\iffalse
We use the following observation.
\znote{maybe remove this observation?}
\begin{observation}
\label{obs: curves_crossings}
The number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$,
and the number of intersections between a curve in $\tilde\Gamma_i$ and the image of edges of $C_i$ in $\phi^*_i$, are both $O(\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W))$.
\end{observation}
\begin{proof}
We first show that the number of points that belongs to at least two curves in $\tilde\Gamma_i$ and is not the image of a vertex in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Note that, from the construction of curves in $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i}$, every crossing between a pair $W,W'$ of curves in $\tilde\Gamma_i$ must be the intersection between a curve in $\Pi_e$ for some $e\in E(W)$ and a curve in $\Pi_{e'}$ for some $e'\in E(W')$, such that the image $\pi_e$ for $e$ and the image $\pi_{e'}$ for $e'$ intersect in $\phi^*$.
Therefore, for each pair $W,W'$ of paths in $\tilde{\mathcal{W}}_i$, the number of points that belong to only curves $\gamma_W$ and $\gamma_{W'}$ is at most the number of crossings between the image of $W$ and the image of $W'$ in $\phi^*$.
It follows that the number of points that belong to exactly two curves of $\tilde\Gamma_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Altogether, the number of intersections between curves in $\tilde\Gamma_i$ is at most $|V(\tilde {\mathcal{W}}_i)|+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
We now show that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ that are not vertex-image is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
Let $W$ be a path of $\tilde {\mathcal{W}}_i$ and consider the curve $\gamma_W$. Note that $\gamma_W$ is the union of, for each edge $e\in E(W)$, a curve that lies in an arbitrarily thin strip around $\pi_e$.
Therefore, the number of crossings between $\gamma_W$ and the image of $C_i$ in $\phi^*_i$ is identical to the number of crossings the image of path $W$ and the image of $C_i$ in $\phi^*_i$, which is at most $\hat\mathsf{cr}(W)$.
It follows that the number of intersections between a curve in $\tilde\Gamma_i$ and the image of $C_i$ in $\phi^*_i$ is at most $\sum_{W\in \tilde{\mathcal{W}}_i}\hat\mathsf{cr}(W)$.
\end{proof}
\fi
\paragraph{Step 2.}
For each vertex $v\in V({\mathcal{W}}_i)$, we denote by $x_v$ the point that represents the image of $v$ in $\phi^*_i$, and we let $X$ contains all points of $\set{x_v\mid v\in V({\mathcal{W}}_i)}$ that are intersections between curves in $\Gamma_i$.
We now manipulate the curves in $\set{\gamma_W\mid W\in {\mathcal{W}}_i}$ at points of $X$, by processing points of $X$ one-by-one, as follows.
Consider a point $x_v$ that is an intersection between curves in $\Gamma_i$, where $v\in V({\mathcal{W}}_i)$, and let $D_v$ be an arbitrarily small disc around $x_v$.
We denote by ${\mathcal{W}}_i(v)$ the set of paths in ${\mathcal{W}}_i$ that contains $v$, and further partition it into two sets: ${\mathcal{W}}^{\textsf{thr}}_i(v)={\mathcal{W}}_i(v)\cap {\mathcal{W}}^{\textsf{thr}}_i$ and $\tilde{\mathcal{W}}_i(v)={\mathcal{W}}_i(v)\cap \tilde{\mathcal{W}}_i$. We apply the algorithm from \Cref{obs: curve_manipulation} to modify the curves of $\set{\gamma_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ within disc $D_v$. Let $\set{\gamma'_W\mid W\in\tilde{\mathcal{W}}_i(v)}$ be the set of new curves that we obtain. From \Cref{obs: curve_manipulation},
(i) for each path $W\in \tilde{\mathcal{W}}_i(v)$, the curve $\gamma'_W$ does not contain $x_v$, and is identical to the curve $\gamma_W$ outside the disc $D_v$;
(ii) the segments of curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc $D_v$ are in general position; and
(iii) the number of icrossings between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside $D_v$ is bounded by $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\iffalse{just for backup}
\begin{proof}
Denote $d=\deg_G(v)$ and $\delta_G(v)=\set{e_1,\ldots,e_d}$, where the edges are indexed according to the ordering ${\mathcal{O}}_v\in \Sigma$.
For each path $W\in \tilde{\mathcal{W}}_i(v)$, we denote by $p^{-}_W$ and $p^{+}_W$ the intersections between the curve $\gamma_W$ and the boundary of ${\mathcal{D}}_v$.
We now compute, for each $W\in W\in \tilde{\mathcal{W}}_i(v)$, a curve $\zeta_W$ in ${\mathcal{D}}_v$ connecting $p^{-}_W$ to $p^{+}_W$, such that (i) the curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ are in general position; and (ii) for each pair $W,W'$ of paths, the curves $\zeta_W$ and $\zeta_{W'}$ intersects iff the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{-}_{W'},p^{+}_{W},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_{W'},p^{-}_{W},p^{+}_{W'})$.
It is clear that this can be achieved by first setting, for each $W$, the curve $\zeta_W$ to be the line segment connecting $p^{-}_W$ to $p^{+}_W$, and then slightly perturb these curves so that no point belong to at least three curves in $\set{\zeta_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
We now define, for each $W$, the curve $\gamma'_W$ to be the union of the part of $\gamma_W$ outside ${\mathcal{D}}_v$ and the curve $\zeta_W$.
See Figure~\ref{fig: curve_con} for an illustration.
Clearly, the first and the second condition of \Cref{obs: curve_manipulation} are satisfied.
It remains to estimate the number of intersections between curves of $\set{\gamma'_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$, which equals the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$.
Since the paths in $\tilde{\mathcal{W}}_i(v)$ are non-transversal with respect to $\Sigma$ (from \Cref{obs: wset_i_non_interfering}), from the construction of curves $\set{\gamma_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$, if a pair $W,W'$ of paths in $\tilde {\mathcal{W}}_i(v)$ do not share edges of $\delta(v)$, then the order in which the points $p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'}$ appear on the boundary of ${\mathcal{D}}_v$ is either $(p^{-}_W,p^{+}_W,p^{-}_{W'},p^{+}_{W'})$ or $(p^{+}_W,p^{-}_W,p^{-}_{W'},p^{+}_{W'})$, and therefore the curves $\zeta_{W}$ and $\zeta_{W'}$ will not intersect in ${\mathcal{D}}_v$.
Therefore, only the curves $\zeta_W$ and $\zeta_{W'}$ intersect iff $W$ and $W'$ share an edge of $\delta(v)$.
Since every such pair of curves intersects at most once, the number of intersections between curves of $\set{\zeta_{W}\text{ }\big|\text{ }W\in \tilde{\mathcal{W}}_i(v)}$ inside disc ${\mathcal{D}}_v$ is at most $\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$.
\end{proof}
\begin{figure}[h]
\centering
\subfigure[Before: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are shown in black, and curves of $\tilde{\mathcal{W}}_i(v)$ are shown in blue, red, orange and green. ]{\scalebox{0.32}{\includegraphics{figs/curve_con_1.jpg}}}
\hspace{1pt}
\subfigure[After: Curves of ${\mathcal{W}}^{\textsf{thr}}_i(v)$ are not modified, while curves of $\tilde{\mathcal{W}}_i(v)$ are re-routed via dash lines within disc ${\mathcal{D}}_v$.]{
\scalebox{0.32}{\includegraphics{figs/curve_con_2.jpg}}}
\caption{An illustration of the step of processing $x_v$.}\label{fig: curve_con}
\end{figure}
\fi
We then replace the curves of $\set{\gamma_W\mid W\in \tilde{\mathcal{W}}_i(v)}$ in $\Gamma_i$ by the curves of $\set{\gamma'_W\mid W\in \tilde{\mathcal{W}}_i(v)}$.
This completes the description of the iteration of processing point the point $x_v\in X$. Let $\Gamma'_i$ be the set of curves that we obtain after processing all points in $X$ in this way. Note that we have never modified the curves of $\Gamma^{\textsf{thr}}_i$, so $\Gamma^{\textsf{thr}}_i\subseteq\Gamma'_i$, and we denote $\tilde\Gamma'_i=\Gamma'_i\setminus \Gamma^{\textsf{thr}}_i$.
We use the following observation.
\begin{observation}
\label{obs: general_position}
Curves in $\tilde\Gamma'_i$ are in general position, and if a point $p$ lies on more than two curves of $\Gamma'_i$, then either $p$ is an endpoint of all curves containing it, or all curves containing $p$ belong to $\Gamma^{\textsf{thr}}_i$.
\end{observation}
\begin{proof}
From the construction of curves in $\Gamma_i$, any point that belong to at least three curves of $\Gamma_i$ must be the image of some vertex in $\phi^*$. From~\Cref{obs: curve_manipulation}, curves in $\tilde\Gamma'_i$ are in general position; curves in $\tilde\Gamma'_i$ do not contain any vertex-image in $\phi^*$ except for their endpoints; and they do not contain any intersection of a pair of paths in $\Gamma_i^{\textsf{thr}}$. \Cref{obs: general_position} now follows.
\end{proof}
\paragraph{Step 3.}
So far we have obtained a set $\Gamma'_i$ of curves that are further partitioned into two sets $\Gamma'_i=\Gamma^{\textsf{thr}}_i\cup \tilde\Gamma'_i$,
where set $\tilde\Gamma'_i$ contains, for each path $W\in \tilde {\mathcal{W}}_i$, a curve $\gamma'_W$ connecting the endpoints of $W$, and the curves in $\tilde\Gamma'_i$ are in general position;
and set $\Gamma^{\textsf{thr}}_i$ contains, for each path $W\in {\mathcal{W}}^{\textsf{thr}}_i$, a curve $\gamma_W$ connecting the endpoints of $W$.
Recall that all paths in ${\mathcal{W}}^{\textsf{thr}}_i$ connects $u_{i-1}$ to $u_{i+1}$.
Let $z_{\operatorname{left}}$ be the point that represents the image of $u_{i-1}$ in $\phi_i^*$ and let $z_{\operatorname{right}}$ be the point that represents the image of $u_{i+1}$ in $\phi_i^*$.
Then, all curves in $\Gamma^{\textsf{thr}}_i$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
We view $z_{\operatorname{left}}$ as the first endpoint of curves in $\Gamma^{\textsf{thr}}_i$ and view $z_{\operatorname{right}}$ as their last endpoint.
We then apply the algorithm in \Cref{thm: type-2 uncrossing}, where we let $\Gamma=\Gamma^{\textsf{thr}}_i$ and let $\Gamma_0$ be the set of all other curves in the drawing $\phi^*_i$. Let $\Gamma^{\textsf{thr}'}_i$ be the set of curves we obtain. We then designate, for each edge $e\in E^{\textsf{thr}}_i$, a curve in $\Gamma^{\textsf{thr}'}_i$ as $\gamma'_{W_e}$, such that the curves of $\set{\gamma'_{W_e}\mid e\in \hat E_{i-1}\cup E^{\operatorname{left}}_i\cup E^{\textsf{thr}}_i}$ enters $z_{\operatorname{left}}$ in the same order as ${\mathcal{O}}^{\operatorname{left}}_i$.
Recall that ${\mathcal{W}}_i=\set{W_e\mid e\in (E_i^{\operatorname{left}}\cup E_i^{\textsf{thr}}\cup E_i^{\operatorname{right}}\cup \hat E_{i-1}\cup \hat E_i)}$, and, for each edge $e\in E_i^{\operatorname{left}}\cup \hat E_{i-1}$, the curve $\gamma'_{W_e}$ connects its endpoint in $C_i$ to $z_{\operatorname{left}}$; for each edge $e\in E_i^{\operatorname{right}}\cup \hat E_{i}$, the curve $\gamma'_{W_e}$ connects the endpoint of $e$ to $z_{\operatorname{right}}$; and for each edge $e\in E_i^{\textsf{thr}}$, the curve $\gamma'_{W_e}$ connects $z_{\operatorname{left}}$ to $z_{\operatorname{right}}$.
Therefore, if we view $z_{\operatorname{left}}$ as the image of $v^{\operatorname{left}}_i$, view $z_{\operatorname{right}}$ as the image of $v^{\operatorname{right}}_i$, and for each edge $e\in E(G_i)\setminus E(C_i)$, view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi'_i$.
It is clear from the construction of curves in $\set{\gamma'_{W_e}\mid e\in E(G_i)\setminus E(C_i)}$ that this drawing respects all rotations in $\Sigma_i$ on vertices of $V(C_i)$ and vertex $v^{\operatorname{left}}_i$.
However, the drawing $\phi'_i$ may not respect the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$. We further modify the drawing $\phi'_i$ at $z_{\operatorname{right}}$ in the last step.
\paragraph{Step 4.}
Let ${\mathcal{D}}$ be an arbitrarily small disc around $z_{\operatorname{right}}$ in the drawing $\phi'_i$, and let ${\mathcal{D}}'$ be another small disc around $z_{\operatorname{right}}$ that is strictly contained in ${\mathcal{D}}$.
We first erase the drawing of $\phi'_i$ inside the disc ${\mathcal{D}}$, and for each edge $e\in \delta(v^{\operatorname{right}}_i)$, we denote by $p_{e}$ the intersection between the curve representing the image of $e$ in $\phi'_i$ and the boundary of ${\mathcal{D}}$.
We then place, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, a point $p'_e$ on the boundary of ${\mathcal{D}}'$, such that the order in which the points in $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ appearing on the boundary of ${\mathcal{D}}'$ is precisely ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We then apply \Cref{lem: find reordering} to compute a set of reordering curves, connecting points of $\set{p_e\mid e\in \delta(v^{\operatorname{right}}_i)}$ to points $\set{p'_e\mid e\in \delta(v^{\operatorname{right}}_i)}$.
Finally, for each edge $e\in \delta(v^{\operatorname{right}}_i)$, let $\gamma_e$ be the concatenation of (i) the image of $e$ in $\phi'_i$ outside the disc ${\mathcal{D}}$; (ii) the reordering curve connecting $p_e$ to $p'_e$; and (iii) the straight line segment connecting $p'_e$ to $z_{\operatorname{right}}$ in ${\mathcal{D}}'$.
We view $\gamma_e$ as the image of edge $e$, for each $e\in \delta(v^{\operatorname{right}}_i)$.
We denote the resulting drawing of $G_i$ by $\phi_i$. It is clear that $\phi_i$ respects the rotation ${\mathcal{O}}^{\operatorname{right}}_i$ at $v^{\operatorname{right}}_i$, and therefore it respects the rotation system $\Sigma_i$.
We use the following claim.
\begin{claim}
\label{clm: rerouting_crossings}
The number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\end{claim}
\begin{proof}
Denote by ${\mathcal{O}}^*$ the ordering in which the curves $\set{\gamma'_{W_e}\mid e\in \delta_{G_i}(v_i^{\operatorname{right}})}$ enter $z_{\operatorname{right}}$, the image of $u_{i+1}$ in $\phi'_i$.
From~\Cref{lem: find reordering} and the algorithm in Step 4 of modifying the drawing within the disc ${\mathcal{D}}$, the number of crossings of $\phi_i$ inside the disc ${\mathcal{D}}$ is at most $O(\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}}))$. Therefore, it suffices to show that $\mbox{\sf dist}({\mathcal{O}}^*,{\mathcal{O}}_i^{\operatorname{right}})=O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
In fact, we will compute a set of curves connecting the image of $u_i$ and the image of $u_{i+1}$ in $\phi^*_i$, such that each curve is indexed by some edge $e\in\delta_{G_i}(v_i^{\operatorname{right}})$ these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$ and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$, and the number of crossings between curves of $Z$ is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
For each $e\in E^{\textsf{thr}}_i$, we denote by $\eta_e$ the curve obtained by taking the union of (i) the curve $\gamma'_{W_e}$ (that connects $u_{i+1}$ to $u_{i-1}$); and (ii) the curve representing the image of the subpath of $P_e$ in $\phi^*_i$ between $u_i$ and $u_{i-1}$.
Therefore, the curve $\eta_e$ connects $u_i$ to $u_{i+1}$.
We then modify the curves of $\set{\eta_e\mid e\in E^{\textsf{thr}}_i}$, by iteratively applying the algorithm from \Cref{obs: curve_manipulation} to these curves at the image of each vertex of $C_{i-1}\cup C_{i+1}$. Let $\set{\zeta_e\mid e\in E^{\textsf{thr}}_i}$ be the set of curves that we obtain. We call the obtained curves \emph{red curves}. From~\Cref{obs: curve_manipulation}, the red curves are in general position. Moreover, it is easy to verify that the number of intersections between the red curves is $O(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1}))$.
We call the curves in $\set{\gamma'_{W_e}\mid e\in \hat E_i}$ \emph{yellow curves}, call the curves in $\set{\gamma'_{W_e}\mid e\in E^{\operatorname{right}}_i}$ \emph{green curves}. See \Cref{fig: uncrossing_to_bound_crossings} for an illustration.
From the construction of red, yellow and green curves, we know that these curves enter $u_i$ in the order ${\mathcal{O}}^{\operatorname{right}}_i$, and enter $u_{i+1}$ in the order ${\mathcal{O}}^*$.
Moreover, we are guaranteed that the number of intersections between red, yellow and green curves is at most $\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W))$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{figs/uncross_to_bound_crossings.jpg}
\caption{An illustration of red, yellow and green curves.}\label{fig: uncrossing_to_bound_crossings}
\end{figure}
\end{proof}
From the above discussion and Claim~\ref{clm: rerouting_crossings}, for each $2\le i\le r-1$,
\[
\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right}).
\]
\iffalse
We now estimate the number of crossings in $\phi_i$ in the next claim.
\begin{claim}
\label{clm: number of crossings in good solutions}
$\mathsf{cr}(\phi_i)=\chi^2(C_i)+O\textsf{left}(\sum_{W\in \tilde{\mathcal{W}}_i}\mathsf{cr}(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})$.
\end{claim}
\begin{proof}
$2\cdot \sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2=\sum_{v\in V(G)}\sum_{e\in \delta_G(v)}\cong_G(\tilde {\mathcal{W}}_i,e)^2.$
\znote{need to redefine the orderings ${\mathcal{O}}^{\operatorname{left}}_i$ and ${\mathcal{O}}^{\operatorname{right}}_i$ to get rid of $\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2$ here, which we may not be able to afford.}
\end{proof}
\fi
\paragraph{Drawings $\phi_1$ and $\phi_{r}$.}
The drawings $\phi_1$ and $\phi_{r}$ are constructed similarly. We describe the construction of $\phi_1$, and the construction of $\phi_1$ is symmetric.
Recall that the graph $G_1$ contains only one super-node $v_1^{\operatorname{right}}$, and $\delta_{G_1}(v_1^{\operatorname{right}})=\hat E_1\cup E^{\operatorname{right}}_1$. We define ${\mathcal{W}}_1=\set{W_e\mid e\in E^{\operatorname{right}}_1}\cup \set{Q_2(\hat e)\mid \hat e\in \hat E_1}$. For each $\hat e\in \hat E_1$, we rename the path $Q_2(\hat e)$ by $W_e$, so ${\mathcal{W}}_1$ contains, for each edge $e\in \delta_{G_1}(v^1_{\operatorname{right}})$, a path named $W_e$ connecting its endpoints to $u_2$. Via similar analysis in \Cref{obs: wset_i_non_interfering}, it is easy to show that the paths in ${\mathcal{W}}_1$ are non-transversal with respect to $\Sigma$.
We start with the drawing of $C_1\cup E({\mathcal{W}}_1)$ induced by $\phi^*$, that we denote by $\phi^*_1$. We will not modify the image of $C_i$ in $\phi^*_i$ and will construct the image of edges in $\delta(v_1^{\operatorname{right}})$.
We perform similar steps as in the construction of drawings $\phi_2,\ldots,\phi_{r-1}$.
We first construct, for each path $W\in {\mathcal{W}}_1$, a curve $\gamma_W$ connecting its endpoint in $C_1$ to the image of $u_2$ in $\phi^*$, as in Step 1.
Let $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
We then process all intersections between curves of $\Gamma_1$ as in Step 2. Let $\Gamma'_1=\set{\gamma'_W\mid W\in {\mathcal{W}}_1}$ be the set of curves we obtain.
Since $\Gamma^{\textsf{thr}}_1=\emptyset$, we do not need to perform Steps 3 and 4. If we view the image of $u_2$ in $\phi^*_1$ as the image of $v^{\operatorname{right}}_1$, and for each edge $e\in \delta(v^{\operatorname{right}}_1)$, we view the curve $\gamma'_{W_e}$ as the image of $e$, then we obtain a drawing of $G_i$, that we denote by $\phi_i$. It is clear that this drawing respects the rotation system $\Sigma_1$. Moreover,
\[\mathsf{cr}(\phi_1)=\chi^2(C_1)+O\textsf{left}(\hat\chi_1({\mathcal{Q}}_2)+\sum_{W\in {\mathcal{W}}_1}\hat\chi_1(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_1,e)^2\textsf{right}).\]
Similarly, the drawing $\phi_k$ that we obtained in the similar way satisfies that
\[\mathsf{cr}(\phi_k)=\chi^2(C_k)+O\textsf{left}(\hat\chi_k({\mathcal{Q}}_{r-1})+\sum_{W\in {\mathcal{W}}_k}\hat\chi_k(W)+\sum_{e\in E(G)}\cong_G({\mathcal{W}}_k,e)^2\textsf{right}).\]
We now complete the proof of \Cref{claim: existence of good solutions special}, for which it suffices to estimate $\sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
Recall that, for each $1\le i\le r$, $\tilde {\mathcal{W}}_i=\set{W_e\mid e\in \hat E_{i-1}\cup \hat E_{i-1}\cup E^{\operatorname{left}}_{i}\cup E^{\operatorname{right}}_{i}}$, where $E^{\operatorname{left}}_{i}=E(C_i,\bigcup_{1\le j\le i-2}C_j)$, and
$E^{\operatorname{right}}_{i}=E(C_i,\bigcup_{i+2\le j\le r}C_j)$. Therefore, for each edge $e\in E'\cup (\bigcup_{1\le i\le r-1}\hat E_i)$, the path $W_e$ belongs to exactly $2$ sets of $\set{\tilde{\mathcal{W}}_i}_{1\le i\le r}$.
Recall that the path $W_e$ only uses edges of the inner path $P_e$ and the outer path $P^{\mathsf{out}}_e$.
Let $\tilde{\mathcal{W}}=\bigcup_{1\le i\le r}\tilde{\mathcal{W}}_i$, from \Cref{obs: edge_occupation in outer and inner paths},
for each edge $e\in E'\cup (\bigcup_{1\le i\le r-1}\hat E_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(1)$, and
for $1\le i\le r$ and for each edge $e\in E(C_i)$, $\cong_G(\tilde{\mathcal{W}},e)=O(\cong_G({\mathcal{Q}}_i,e))$.
Therefore, on one hand,
\[
\begin{split}
\sum_{1\le i\le r}\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} 2\cdot \cong_G(\tilde {\mathcal{W}},e)\cdot\cong_G(\tilde {\mathcal{W}},e')\\
& \le
\sum_{(e,e'): e,e'\text{ cross in }\phi^*} \textsf{left}(\cong_G(\tilde {\mathcal{W}},e)^2+\cong_G(\tilde {\mathcal{W}},e')^2\textsf{right})\\
& \le
\sum_{e\in E(G)} \chi(e)\cdot \cong_G(\tilde {\mathcal{W}},e)^2
= O(\mathsf{cr}(\phi^*)\cdot\beta),
\end{split}
\]
and on the other hand,
\[
\begin{split}
\sum_{1\le i\le r}\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2
& \le \sum_{e\in E(G)} \textsf{left}(\sum_{1\le i\le r} \cong_G(\tilde {\mathcal{W}}_i,e)\textsf{right})^2\\
& \le O\textsf{left}(\sum_{e\in E(G)} \cong_G(\tilde {\mathcal{W}},e)^2 \textsf{right})\\
& \le O\textsf{left}(|E(G)|+\sum_{1\le i\le r}\sum_{e\in E(C_i)} \cong_G({\mathcal{Q}}_i,e)^2\textsf{right})=O(|E(G)|\cdot\beta).
\end{split}
\]
Moreover, $\sum_{1\le i\le r}\chi^2(C_i)\le O(\mathsf{cr}(\phi^*))$, and $\sum_{1\le i\le r}\hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})\le O(\mathsf{cr}(\phi^*)\cdot\beta)$.
Altogether,
\[
\begin{split}
\sum_{1\le i\le r}\mathsf{cr}(\phi_i)
& \le
O\textsf{left}(\sum_{1\le i\le r}
\textsf{left}(
\chi^2(C_i)+ \hat\chi_i({\mathcal{Q}}_{i-1}\cup {\mathcal{Q}}_{i+1})+\sum_{W\in \tilde{\mathcal{W}}_i}\hat\chi_i(W)+\sum_{e\in E(G)}\cong_G(\tilde {\mathcal{W}}_i,e)^2\textsf{right})
\textsf{right})\\
& \le O(\mathsf{cr}(\phi^*))+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(\mathsf{cr}(\phi^*)\cdot\beta)+O(|E(G)|\cdot\beta)\\
& \le O((\mathsf{cr}(\phi^*)+|E(G)|)\cdot\beta).
\end{split}
\]
This completes the proof of \Cref{claim: existence of good solutions special}.
\end{proofof}
$\ $
\iffalse
We denote by $\hat{\phi_i}$ the resulting drawing, and it is clear that $\hat{\phi_i}$ is a drawing of the instance $(\hat G_i,\hat\Sigma_i)$, such that $\mathsf{cr}(\hat{\phi_i})\le \mathsf{cr}(C_i)+O(\operatorname{cost}({\mathcal{W}}_i)\cdot\operatorname{poly}\log n)$.
We first define an instance $(\hat G_i,\hat\Sigma_i)$ as follows. We start with the subgraph of $G$ induced by all edges of $E_G(C_i)$ and $E({\mathcal{W}}_i)$, and then create, for each edge $e$ in the subgraph, $\cong_{{\mathcal{W}}_i}(e)$ parallel copies of it. This finishes the description of graph $\hat G_i$.
For each vertex $v$ in $\hat G_i$, we define its rotation $\hat{\mathcal{O}}_v$ as follows. We start with the rotation ${\mathcal{O}}_v\in \Sigma$, and then replace, for each edge $e\in E({\mathcal{W}}_i)$, the edge $e$ in the ordering ${\mathcal{O}}_v$ by its $\cong_{{\mathcal{W}}_i}(e)$ copies that appears consecutively at the location of $e$ in ${\mathcal{O}}_v$, where the ordering among the copies is arbitrary.
On the one hand, in graph $\hat G_i$, we can make paths of ${\mathcal{W}}_i$ edge-disjoint by letting, for each edge $e\in E({\mathcal{W}}_i)$, each path of ${\mathcal{W}}_i$ that contains $e$ now take a distinct copy of $e$ in $\hat G_i$.
On the other hand, a drawing $\hat \phi_i$ of the instance $(\hat G_i,\hat\Sigma_i)$ can be easily computed from $\phi^*$, as follows.
We start with $\hat\phi'_i$, the drawing of $C_i\cup E({\mathcal{W}}_i)$ induced by $\phi^*$. Then for each edge of ${\mathcal{W}}_i$, let $\gamma_e$ be the curve that represents the image of $e$, and we create $\cong_{{\mathcal{W}}_i}(e)$ mutually internally disjoint curves connecting endpoints of $e$, that lies in an arbitrarily thin strip around $\gamma_e$. We denote by $\hat{\phi_i}$ the resulting drawing, and it is clear that $\hat{\phi_i}$ is a drawing of the instance $(\hat G_i,\hat\Sigma_i)$, such that $\mathsf{cr}(\hat{\phi_i})\le \mathsf{cr}(C_i)+O(\operatorname{cost}({\mathcal{W}}_i)\cdot\operatorname{poly}\log n)$.
For each edge $e\in E_G(C_i)$, we denote by $\gamma_e$ the curve in $\hat \phi_i$ that represents the image of $e$; and for each path $W\in {\mathcal{W}}_i$, we denote by $\gamma_W$ the curve in $\hat \phi_i$ that represents the image of $W$.
We define $\Gamma_0=\set{\gamma_e\mid e\in E_G(C_i)}$ and $\Gamma_1=\set{\gamma_W\mid W\in {\mathcal{W}}_i}$. It is immediate to verify that the sets $\Gamma_0,\Gamma_1$ of curves satisfy the condition of \Cref{thm: type-2 uncrossing}. We then apply the algorithm in \Cref{thm: type-2 uncrossing} to $\Gamma_0,\Gamma_1$, and let $\Gamma_1'$ be the set of curves that we obtain.
From \Cref{thm: type-2 uncrossing}, the curves in $\Gamma_1'$ do not intersect internally between each other, and have the same sets of first endpoints and last endpoints.
We now show that we can obtain a drawing $\phi_i$ of the instance $(G_i,\Sigma_i)$ using the curves $\Gamma_0,\Gamma_1'$.
For each edge $e\in E_G(C_i)$, we still let the curve $\gamma_e\in \Gamma_0$ be the image of $e$.
For each edge path $W\in {\mathcal{W}}$, from \Cref{thm: type-2 uncrossing}, there is a curve $\gamma'\in \Gamma_1'$ connecting the endpoint of $W$ to we still let the curve $\gamma_e\in \Gamma_0$ be the image of $e$.
\znote{maybe we need another type of uncrossing here.}
\fi
\begin{proofof}{Claim~\ref{claim: stitching the drawings together}}
We first define a set $\set{(U_i,\Sigma'_i)\mid 1\le i\le r}$ of instances of \textnormal{\textsf{MCNwRS}}\xspace as follows.
We define $U_1=G$, and for each $2\le i\le r$, we define $U_i$ to be the graph obtained from $G$ by contracting the clusters $C_1,\ldots, C_{i-1}$ into a vertex $v^{\operatorname{left}}_i$.
Note that each edge in $U_i$ is also an edge in $G$, and we do not distinguish between them.
Note that $U_1=G$, we define the rotation system $\Sigma'_1$ on $U_1$ to be $\Sigma$.
For each $2\le i\le r$, we define the rotation system $\Sigma'_i$ on $U_i$ as follows.
For each vertex $v\in \bigcup_{i\le t\le r}V(C_t)$, note that its incident edges in $U_i$ are the edges of $\delta_G(v)$, and its rotation in $\Sigma'_i$ is defined to be ${\mathcal{O}}_v$, the rotation on $v$ in the input rotation system $\Sigma$.
For vertex $v^{\operatorname{left}}_i$, note that its incident edges are the edges of $\delta_{G_i}(v^{\operatorname{left}}_i)$, and its rotation in $\Sigma'_i$ is defined to be ${\mathcal{O}}^{\operatorname{left}}_i$, the rotation on $v^{\operatorname{left}}_i$ of instance $(G_i,\Sigma_i)$.
Note that $U_k=G_k$ and $\Sigma'_k=\Sigma_k$, so the drawing $\phi_k$ of the instance $(G_k,\Sigma_k)$ is also a drawing of the instance $(U_k,\Sigma'_k)$. For clarity, when we view this drawing as a solution to the instance $(U_k,\Sigma'_k)$, we rename it by $\psi_k$.
We will sequentially, for $i=r-1,\ldots,1$, compute a drawing of the instance $(U_i,\Sigma'_i)$ using the drawing $\psi_{i+1}$ of $(U_{i+1},\Sigma'_{i+1})$ and the drawing $\phi_i$ of $(G_{i},\Sigma_{i})$, and eventually, we return the drawing $\psi_1$ of $(U_{1},\Sigma'_{1})$ as the solution to the instance $(G,\Sigma)$.
We now fix an index $1\le i< r-1$ and construct the drawing $\psi_i$ to the instance $(U_{i},\Sigma'_{i})$, assuming that we have computed a drawing $\psi_{i+1}$ to the instance $(U_{i+1},\Sigma'_{i+1})$, as follows.
Recall that $V(G_i)=V(C_i)\cup\set{v^{\operatorname{left}}_i,v^{\operatorname{right}}_i}$ if $i\ge 2$ and $V(G_i)=V(C_i)\cup\set{v^{\operatorname{right}}_i}$ if $i=1$, and $\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{U_{i+1}}(v^{\operatorname{left}}_{i+1})$. Moreover, from \Cref{obs: rotation for stitching} and the definition of instance $(U_{i+1},\Sigma'_{i+1})$, the rotation on $v^{\operatorname{right}}_i$ in $\Sigma_{i}$ is identical to the rotation on $v^{\operatorname{left}}_{i+1}$ in $\Sigma'_{i+1}$.
Denote $F=\delta_{G_i}(v^{\operatorname{right}}_i)=\delta_{U_{i+1}}(v^{\operatorname{left}}_{i+1})$.
Let ${\mathcal{D}}$ be an arbitrarily small disc around the image of $v_{i+1}^{\operatorname{left}}$ in $\psi_{i+1}$. For each edge $e\in F$, we denote by $p_e$ the intersection between the image of $e$ with the boundary of ${\mathcal{D}}$.
Therefore, the order in which the points $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}$ is ${\mathcal{O}}^{\operatorname{left}}_{i+1}$.
We erase the drawing of $\psi_{i+1}$ inside the disc ${\mathcal{D}}$, and view the area inside the disc ${\mathcal{D}}$ as the outer face of the drawing.
Similarly, let ${\mathcal{D}}'$ be an arbitrarily small disc around the image of $v_{i}^{\operatorname{right}}$ in $\phi_{i}$. For each edge $e\in F$, we denote by $p'_e$ the intersection between the image of $e$ with the boundary of ${\mathcal{D}}'$.
Therefore, the order in which the points $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}'$ is ${\mathcal{O}}^{\operatorname{right}}_{i}$.
We erase the drawing of $\phi_{i}$ inside the disc ${\mathcal{D}}'$, and let ${\mathcal{D}}''$ be another disc that is strictly contained in ${\mathcal{D}}'$. We now place the drawing of $\psi_{i+1}$ inside ${\mathcal{D}}'$ (after we erase part of it inside ${\mathcal{D}}$), so that the boundary of ${\mathcal{D}}$ in $\psi_{i+1}$ coincide with the boundary of ${\mathcal{D}}''$ in $\phi_{i}$, while the interior of ${\mathcal{D}}$ coincides with the exterior of ${\mathcal{D}}''$. We then compute a set $\set{\zeta_e\mid e\in F}$ of curves lying in ${\mathcal{D}}'\setminus {\mathcal{D}}''$, where for each $e\in F$, the curve $\zeta_e$ connects $p'_e$ to $p_e$, such that all curves of $\set{\zeta_e\mid e\in F}$ are mutually disjoint. Note that this can be done since the order in which $\set{p_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}'$ is identical to $\set{p'_e}_{e\in F}$ appear on the boundary of ${\mathcal{D}}$.
We denote by $\psi_i$ the resulting drawing we obtained.
See Figure~\ref{fig: stitching} for an illustration.
Clearly, $\psi_i$ is a drawing of $U_i$ that respects $\Sigma'_i$ if we view, for each edge $e\in F$, the union of (i) the image of $e$ in $\phi_i$ outside the disc ${\mathcal{D}}'$; (ii) the curve $\zeta_e$; and (iii) the image of $e$ in $\psi_{i+1}$ inside the disc ${\mathcal{D}}''$, as the image of $e$.
\begin{figure}[h]
\centering
\subfigure[The drawing $\psi_{i+1}$, where the boundary of ${\mathcal{D}}$ is shown in dash black.]{\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_1.jpg}
}
\hspace{0.45cm}
\subfigure[The drawing $\psi_{i+1}$ after we erase its part inside ${\mathcal{D}}$ and view the interior of ${\mathcal{D}}$ as the outer face.]{
\scalebox{0.35}{\includegraphics[scale=1.0]{figs/stitching_2.jpg}}}
\hspace{0.45cm}
\subfigure[The drawing $\psi_i$, where the curves of $\set{\zeta_e\mid e\in F}$ are shown in dash line segments.]{
\scalebox{0.36}{\includegraphics[scale=1.0]{figs/stitching_3.jpg}}}
\caption{An illustration of constructing the drawing $\psi_i$ from $\psi_{i+1}$.}\label{fig: stitching}
\end{figure}
Clearly, any crossing in the drawing $\psi_i$ is either a crossing of $\phi_i$ or a crossing of $\psi_{i+1}$, so $\mathsf{cr}(\psi_i)\le \mathsf{cr}(\psi_{i+1})+\mathsf{cr}(\phi_{i})$.
Therefore, if we rename the drawing $\psi_1$ of the instance $(U_1,\Sigma'_1)$ by $\phi$, then $\phi$ is a drawing of $G$ that respects $\Sigma$, and $\mathsf{cr}(\phi)\le \sum_{1\le i\le r}\mathsf{cr}(\phi_i)$.
\end{proofof}
\subsection{The Second Special Case: Path of Clusters}
In this section prove a special case of \Cref{thm: disengagement - main}. Specifically, we will assume that the Gomory-Hu tree of the contracted graph $G_{\mid{\mathcal{C}}}$ is a path.
Let $H=G_{\mid{\mathcal{C}}}$ be the contracted graph.
We denote the clusters in ${\mathcal{C}}$ by $C_1,C_2,\ldots,C_r$. For convenience, for each $1\leq i\leq r$, we denote by $x_i$ the vertex of graph $H$ that represents the cluster $C_i$.
\iffalse
Throughout this subsection, we assume that the Gomory-Hu tree $\tau$ of graph $H$ is a path, and we assume without loss of generality that the clusters are indexed according to their appearance on the path $\tau$. Note that each edge in $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$ corresponds to an edge in $H$, and we do not distinguish between them.
For each $1\le i\le r$, we define edge sets
$\hat E_i=E(C_i,C_{i+1})$,
$E_i^{\operatorname{right}}=\bigcup_{j>i+1}E(C_i,C_j)$, $E_i^{\operatorname{left}}=\bigcup_{j'<i-1}E(C_i,C_{j'})$, $E_i^{\textsf{thr}}=\bigcup_{i'<i<j'}E(C_{i'},C_{j'})$, $E_i^{\operatorname{over}}=\bigcup_{i'<i,j'>i+1}E(C_{i'},C_{j'})$.
\fi
For each pair $1\le j< j'\le r$, we define $C_{[j,j']}$ as the subgraph of $G$ induced by vertices of all clusters $C_t$ with index $t$ belonging to the interval $[j,j']$. We define clusters
$C_{(j,j']}$,
$C_{[j,j')}$, and
$C_{(j,j')}$ similarly.
We say that a partition ${\mathcal{Z}}$ of the cluster $C_{[j,j']}$ into a sequence ${\mathcal{Z}}=\set{Z_1,\ldots, Z_k}$ of clusters is \emph{${\mathcal{C}}$-respecting}, iff there exist indices $j<i_1<\cdots <i_{k-1}<j'$, such that for each $1\le s\le k$, $Z_s=C_{(i_{s-1}, i_s]}$ (where $j=i_0$ and $j'=i_k$).
In this section we will prove the following lemma.
\begin{lemma}
\label{lem: path case}
There is an efficient algorithm, that given an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace with $|E(G)|=m$, and a collection ${\mathcal{C}}=\set{C_1,\ldots,C_r}$ of vertex-disjoint clusters of $G$, such that
\begin{itemize}
\item vertex sets $\set{V(C)}_{C\in {\mathcal{C}}}$ partition $V(G)$;
\item each cluster $C\in {\mathcal{C}}$ has the $\alpha_0$-bandwidth property (where $\alpha_0=1/\log^3m$); and
\item the Gomory-Hu tree $\tau$ of the contracted graph $H=G_{\mid{\mathcal{C}}}$ is the path $(x_1,x_2,\ldots,x_r)$,
\end{itemize}
computes
\begin{itemize}
\item a ${\mathcal{C}}$-respecting partition of graph $G$ into a sequence ${\mathcal{Z}}=\set{Z_1,\ldots,Z_k}$ of vertex-disjoint clusters, such that each cluster $Z_j$ is either a {helpful}\xspace cluster or a {unhelpful}\xspace cluster;
\item for each {unhelpful}\xspace cluster $Z_j$, three sets ${\mathcal{Q}}^{\operatorname{left}}_j$, ${\mathcal{Q}}^{\operatorname{right}}_j$, ${\mathcal{Q}}^{\textsf{thr}}_j$ of paths. \znote{alternative name?}
\item a set ${\mathcal{P}}=\set{P_e\mid e\in E'}$ of paths in $H$, that contains, for each edge $e\in E'$, an auxiliary path $P_e$ of $e$, such that $\cong_H({\mathcal{P}})=O(1)$.
\end{itemize}
\end{lemma}
\iffalse
\paragraph{Inner Paths.} For an edge $e\in E'$ connecting $x_i$ to $x_j$ in $H$, we say that a path $P$ in $H$ is a \emph{left inner path} of $e$, iff the endpoints of $P$ are $x_i$ and $x_{i+1}$, and $P$ only contains vertices of $S_{i+1}$; we say that a path $P$ in $H$ is a \emph{right inner path} of $e$, iff the endpoints of $P$ are $x_{j-1}$ and $x_{j}$, and $P$ only contains vertices of $\overline{S}_{j-2}$; and we say that a path $P$ in $H$ is a \emph{middle inner path} of $e$, iff path $P$ sequentially visits vertices $x_{i+1}, x_{i+2},\dots,x_{j-1}$.
We say that a path $P$ is an \emph{inner path} of $e$, iff $P$ is the union of (i) a left inner path of $e$; (ii) a middle inner path of $e$; and (iii) a right inner path of $e$.
See \Cref{fig: LMR_inner} for an illustration.
\begin{figure}[h]
\centering
\includegraphics[scale=0.20]{figs/LMR_inner.jpg}
\caption{An illustration of left inner path (orange), middle inner path (purple) and right inner path (green) of edge $(x_i,x_j)\in E'$.}\label{fig: LMR_inner}
\end{figure}
In the remainder of this subsection, we will additionally assume that we are given, for each edge $e\in E'$, an inner path $P_e$ of $e$ in graph $H$, such that the set $\set{P_e}_{e\in E'}$ of paths causes congestion at most $\eta$, for some $\eta=2^{O((\log m)^{3/4}\log\log m)}$.
We now provide the proof of \Cref{thm: disengagement - main} with these additional assumptions.
\fi
Before we present the algorithm for proving \Cref{lem: path case} in several steps in the following subsections, we first introduce a crucial notion called \emph{bad index} in the following subsection.
\input{bad_index}
\subsubsection{Completing the proof of \Cref{lem: path case}}
In this subsection we present the algorithm for \Cref{lem: path case}.
We denote by ${\mathcal{B}}$ the set of all bad indices.
Let $i,i'$ be a pair of distinct bad indices in ${\mathcal{B}}$. We say that $i$ \emph{dominates} $i'$ iff $\mathsf{LM}(i)<i'<\mathsf{RM}(i)-1$. Note that \Cref{obs: bad index non-interleaving} simply says that no two bad indices can dominate each other.
\subsubsection*{Step 1. Selecting a subset of bad indices}
\begin{claim}
\label{clm: select bad indices}
We can efficiently compute a set ${\mathcal{B}}'\subseteq{\mathcal{B}}$ of bad indices, such that (i) for every pair $i,i'$ of bad indices of ${\mathcal{B}}'$, neither $i$ dominates $i'$ nor $i'$ dominates $i$; and (ii) for each $i\in {\mathcal{B}}$, either $i\in {\mathcal{B}}'$ or there exists some $i'\in {\mathcal{B}}$ such that $i'$ dominates $i$.
\end{claim}
\begin{proof}
We construct a directed graph $L$ as follows. Its vertex set $V(L)={\mathcal{B}}$. For every pair $i,i'\in {\mathcal{B}}$, the directed edge $(i,i')$ belongs to $L$ iff $i$ dominates $i'$. We first show that $L$ is acyclic. Note that from similar arguments in \Cref{obs: bad index non-interleaving}, we can show that if $i$ dominates $i'$, then $|\hat E_i|\le |\hat E_{i'}|/2000$. Therefore, if $L$ contains a cycle $(i_1,i_2,\ldots,i_k,i_1)$, then
\[
|\hat E_1|\le \frac{|\hat E_{2}|}{2000} \le \frac{|\hat E_{3}|}{2000^2}\le \cdots\le \frac{|\hat E_{k}|}{2000^{k-1}}\le \frac{|\hat E_{1}|}{2000^k}\text{ },
\]
a contradiction, so $L$ is acyclic. We call the vertices of $L$ with no out-edges \emph{sources}. Since $L$ is acyclic, it must contain at least one source. We then repeatedly perform the following operation until $L$ becomes an empty graph: select a source of $L$, add it to ${\mathcal{B}}'$, and then delete this source and all its out-neighbors from $L$. Clearly, after each operation, graph $L$ remains acyclic. Let ${\mathcal{B}}'$ be the resulting set of indices. It is easy to verify that ${\mathcal{B}}'$ satisfies the required properties in \Cref{clm: select bad indices}.
\end{proof}
\iffalse
For each $1\le j< j'\le r$, we define $C_{[j,j']}=\bigcup_{j\le t\le j'}C_t$,
$C_{(j,j']}=\bigcup_{j< t\le j'}C_t$,
$C_{[j,j')}=\bigcup_{j\le t< j'}C_t$, and
$C_{(j,j')}=\bigcup_{j< t< j'}C_t$.
And we define clusters $X_{[j,j']},X_{(j,j']},X_{[j,j')},X_{(j,j')}$ in $H$ similarly.
We say that a partition of $C_{[j,j']}$ into clusters $Y_1,\ldots, Y_k$ is \emph{${\mathcal{C}}$-respecting}, iff each $Z_s$ is the union of a set of consecutive clusters in $\set{C_j,\ldots,C_{j'}}$. In other words, there exist indices $j<i_1<\cdots <i_{k-1}<j'$, such that for each $1\le s\le k$, $Z_s=C_{[i_{s-1}+1, i_s]}$ (where $j=i_0$ and $j'=i_k$).
\fi
\subsubsection*{Step 2. Dealing with unbonded pairs}
We define the sequence $\sigma=(i_1,i_2,\ldots,i_k)$ where indices $i_1<i_2<\cdots<i_k$ are bad indices in set ${\mathcal{B}}'$ given by \Cref{clm: select bad indices}. We say that a pair $i_j,i_{j+1}$ of consecutive indices in $\sigma$ are \emph{bonded}, iff $i_j=\mathsf{LM}(i_{j+1})$ and $i_{j+1}=\mathsf{RM}(i_{j})-1$.
We say that a consecutive subsequence $\sigma_{[j,j']}=(i_{j},i_{j+1},\ldots,i_{j'})$ is \emph{bonded}, iff for each $j\le s<j'$, the pair $i_{s},i_{s+1}$ of indices are bonded.
We use the following claim.
\begin{claim}
\label{clm: dealing with non-bonding indices}
If a pair $i_j,i_{j+1}$ of consecutive indices in ${\mathcal{B}}'$ are not bonded, then we can efficiently find a ${\mathcal{C}}$-respecting partition of $C_{(i_j,i_{j+1}]}$ into {helpful}\xspace clusters ${\mathcal{Y}}=\set{Y_1,\ldots,Y_k}$ (with $k\ge 2$), such that for every $1\le s\le k-1$, the pair $Y_s,Y_{s+1}$ of clusters are nice.
\end{claim}
\begin{proof}
For convenience, we denote $a=i_j$ and $b=i_{j+1}$. We distinguish between the following cases.
\textbf{Case 1. $\mathsf{RM}(a)\le \mathsf{LM}(b)$.}
In this case we define $Y_1=C_{(a,\mathsf{RM}(a))}$, $Y_2=C_{\mathsf{RM}(a)}$, \ldots, $Y_{k-1}=C_{\mathsf{LM}(b)}$ and $Y_k=C_{(\mathsf{LM}(b),\mathsf{RM}(b)]}$. See \Cref{fig: unbonded case 1} for an illustration.
We first show that all cluster $Y_1,\ldots,Y_k$ are {helpful}\xspace. First, it is clear that clusters $Y_2,\ldots,Y_{k-1}$ are {helpful}\xspace, since each of them contains a single cluster of ${\mathcal{C}}$, and each cluster in ${\mathcal{C}}$ has the $\alpha_0$-bandwidth property. Consider now the cluster $Y_1$. From \Cref{claim: inner and outer paths}, there is a set of paths routing the edges of $\delta_H(X_{(a,\mathsf{RM}(a))})$ to $x_{\mathsf{RM}(a)-1}$ in $X_{(a,\mathsf{RM}(a))}$ with congestion $O(1)$, so cluster $X_{(a,\mathsf{RM}(a))}$ has the $O(1)$-bandwidth property in $H$. From \Cref{clm: contracted_graph_well_linkedness}, the cluster $C_{(a,\mathsf{RM}(a))}$ has the $O(\alpha_0)$-bandwidth property in $G$, and is therefore {helpful}\xspace by definition. Similarly, we can show that cluster $Y_k$ is also {helpful}\xspace.
We now show that every pair $Y_s,Y_{s+1}$ of consecutive clusters are nice.
From \Cref{clm: contraction does not create new bad indices}, it suffices to show that every index $i$ satisfying that $\mathsf{RM}(a)-1\le i\le \mathsf{LM}(b)$ is not a bad index. In fact, since $a,b$ are consecutive bad indices in ${\mathcal{B}}'$, if $i$ is a bad index, then it is not dominated by any index in ${\mathcal{B}}'$, contradicting the property guaranteed by
\Cref{clm: select bad indices}. Therefore, for each $1\le s\le k-1$, the pair $Y_s,Y_{s+1}$ of consecutive clusters are nice.
\textbf{Case 2. $\mathsf{RM}(a)> \mathsf{LM}(b)$.} In this case we consider all indices between $\mathsf{LM}(b)$ and $\mathsf{RM}(a)$. Assume first that there exists an index $\mathsf{LM}(b)\le i^*< \mathsf{RM}(a)$ that is not a bad index, we then set $Y_1=C_{(a,i^*]}$ and $Y_2=C_{(i^*,b]}$. From similar arguments in Case 1 (using \Cref{claim: inner and outer paths} and \Cref{clm: contracted_graph_well_linkedness}), we can show that clusters $Y_1,Y_2$ are {helpful}\xspace. Then from \Cref{clm: contraction does not create new bad indices}, since $i^*$ is not a bad index, the pair $Y_1,Y_2$ of clusters are nice.
Assume now that all indices $\mathsf{LM}(b)\le i< \mathsf{RM}(a)$ are bad indices. First observe that $\mathsf{RM}(a)\ge \mathsf{LM}(b)+2$, since if $\mathsf{RM}(a)=\mathsf{LM}(b)+1$ then the index $\mathsf{LM}(b)$ is not dominated by any index in ${\mathcal{B}}'$, a contradiction. Second, if there exists an index $\mathsf{LM}(b)\le i< \mathsf{RM}(a)$, such that at least one of $\mathsf{LM}(i)=a, \mathsf{RM}(i)=b+1$ does not hold, then we set $Y_1=C_{(a,i]}$ and $Y_2=C_{(i,b]}$. From similar arguments in Case 1, we can show that clusters $Y_1,Y_2$ are {helpful}\xspace. To see why the pair $Y_1,Y_2$ are nice, assume without loss of generality that $\mathsf{LM}(i)\ne a$, so $\mathsf{LM}(i)> a$ since otherwise the bad indices $i$ and $a$ dominate each other, a contradiction to \Cref{obs: bad index non-interleaving}. Then $C_{\mathsf{LM}(i)}\subseteq Y_1$. From the definition of $\mathsf{LM}(i)$, we get that $|E(Y_1,Y_2)|\ge |E^{\operatorname{left}}_i|/2$, so $Y_1,Y_2$ are nice by definition. If all indices $\mathsf{LM}(b)\le i< \mathsf{RM}(a)$ satisfy that $\mathsf{LM}(i)=a$ and $\mathsf{RM}(i)=b+1$, then clearly, the two indices $\mathsf{LM}(b), \mathsf{LM}(b)+1$ dominate each other, a contradiction to \Cref{obs: bad index non-interleaving}.
\begin{figure}[h]
\centering
\subfigure[The clusters $Y_1,\ldots,Y_k$ in Case 1.]{\scalebox{0.12}{\includegraphics{figs/unbonded_case1.jpg}}\label{fig: unbonded case 1}
}
\hspace{4pt}
\subfigure[The clusters $Y_1,Y_2$ in Case 2.]{\scalebox{0.12}{\includegraphics{figs/unbonded_case2.jpg}}\label{fig: unbonded case 2}}
\caption{An illustration of partition ${\mathcal{Y}}$ in \Cref{clm: dealing with non-bonding indices}.
\end{figure}
\end{proof}
\subsubsection*{Step 3. Construct the clusters and path sets}
We denote by $\Pi$ the set of all maximal bonded subsequences of $\sigma$. Here we also view a subsequence that contains an single index $i_j\in \sigma$ as a bonded subsequence.
Let $\Pi=\set{\sigma_{[j_s,j'_s]}\mid 1\le s\le p}$.
We now construct the set ${\mathcal{Z}}$ of clusters in \Cref{lem: path case} as follows.
For each $1\le s\le p-1$, we apply the algorithm from \Cref{clm: dealing with non-bonding indices} to the pair $j'_s,j_{s+1}$ of bad indices that are not bonded. Let $Y'_s, Y_{s+1}$ be the first and the last helpful cluster we get, and let ${\mathcal{Y}}_s$ be the set of all other helpful clusters we get.
Additionally, we define $Y_1=C_{(\mathsf{LM}(i_{j_1}),i_{j_1})}$ and $Y'_s=C_{(i_{j_p},\mathsf{RM}(i_{j_p})]}$, and we define ${\mathcal{Y}}_{0}=\set{C_1,\ldots,C_{\mathsf{LM}(i_{j_1})}}$ and
${\mathcal{Y}}_{p}=\set{C_{\mathsf{RM}(i_{j_p})},\ldots,C_{r}}$.
We let set ${\mathcal{Z}}$ contains: (i) for each $0\le s\le p$, all clusters in ${\mathcal{Y}}_s$; and (ii) for each $1\le s\le p$, a cluster $Z_s=Y_s\cup C_{(i_{j_s},i_{j'_s}]}\cup Y'_{s}$. Clearly, the clusters in $\bigcup_{0\le s\le p}{\mathcal{Y}}_s$ are all {helpful}\xspace clusters.
\paragraph{Constructing the set ${\mathcal{P}}=\set{P_e\mid e\in E'}$ of auxiliary paths.}
From the above discussion, since every pair of consecutive clusters in $\bigcup_{0\le s\le p}{\mathcal{Y}}_s$ are nice, from \Cref{cor: nice pairs imply no bad index}, the Gomory-Hu tree of the contracted graph $G_{\mid{\mathcal{Z}}}$ is a path with no bad index. We now compute the set of auuxiliary paths as follows.
We process edges in $E'$ one-by-one. Throughout, for each edge $\hat e\in \bigcup_{1\le i\le k-1}\hat E_i$, we maintain an integer $z_{\hat e}$ indicating how many times the edge $\hat e$ has been used, that is initialized to be $0$.
Consider an iteration of processing an edge $e\in E'$. Assume $e$ connects $C_i$ to $C_j$ in $G$, we then pick, for each $i\le t\le j-1$, an edge $\hat e$ of $\hat E_t$ with minimum $z_{\hat e}$ over all edges of $\hat E_t$, and let $P_e$ be the path obtained by taking the union of all picked edges.
Note that, in $H$, each edge of $\hat E_t$ connects $x_t$ to $x_{t+1}$, so the path $P_e$ sequentially visits nodes $x_i, x_{i+1},\dots,x_j$.
We then increase the value of $z_{\hat e}$ by $1$ for all picked edges $\hat e$, and proceed to the next iteration. After processing all edges of $E'$, we obtain a set ${\mathcal{P}}=\set{P_e}_{e\in E'}$ of paths in $H$.
We use the following observation.
\begin{observation}
\label{obs:central_congestion}
For each edge $\hat e\in \bigcup_{1\le i\le k-1}\hat E_i$, $\cong_{H}({\mathcal{P}},\hat e)\le 5000$.
\end{observation}
\begin{proof}
From the algorithm, for each $1\le i\le k-1$, the paths of ${\mathcal{P}}$ that contains an edge of $\hat E_i$ are $\set{P_e\mid e\in \big(E_i^{\operatorname{right}} \cup E_{i+1}^{\operatorname{left}} \cup E_i^{\operatorname{over}}\big)}$.
Since $|E_i^{\operatorname{right}}|+|E_{i+1}^{\operatorname{left}}|+|E_i^{\operatorname{over}}|\le 5000|\hat E_i|$, each edge of $\hat E_i$ is used at most $5000$ times. Observation~\ref{obs:central_congestion} then follows.
\end{proof}
\paragraph{Constructing the path sets for {unhelpful}\xspace clusters.}
We now show that, for each $1\le s\le p$, the cluster $Z_s$ is an {unhelpful}\xspace cluster, and construct the sets ${\mathcal{Q}}^{\operatorname{left}}_{s},{\mathcal{Q}}^{\operatorname{right}}_{s},{\mathcal{Q}}^{\textsf{thr}}_{s}$ of paths.
Recall that $Z_s=Y_s\cup C_{(i_{j_s},i_{j'_s}]}\cup Y'_{s}$.
For convenience, we rename $Z_s$ by $Z$, $Y_s$ by $Y$ and $Y'_s$ by $Y'$. We denote $Z=C_{[a,b]}$ and denote by $i_1,\ldots,i_k$ the bonded consecutive subsequence of $\sigma$ that belongs to $Z$, so $Y=C_{[a,i_1]}$ and $Y'=C_{(i_k,b]}$.
Moreover, for each $1\le t\le k-1$, $\mathsf{RM}(i_t)=i_{t+1}+1$ and $\mathsf{LM}(i_{t+1})=i_{t}$.
We first set the left and right centers of $Z$ as follows.
Exactly the same as in \Cref{sec: guiding and auxiliary paths}, for each $a\le t\le b$, we apply the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace from \Cref{thm:algclassifycluster} to the instance and (potentially) the algorithm from \Cref{lem: simple guiding paths} to get a set ${\mathcal{Q}}_t$ of paths routing edges of $\delta_G(C_t)$ to a vertex $u_t\in C_t$.
We then set the left center of $Z$ to be $u^{\operatorname{left}}=u_a$ and the right center of $Z$ to be $u^{\operatorname{right}}=u_b$.
We denote ${\mathcal{C}}'=\set{C_a,\ldots,C_b}$ and consider the contracted graph $H'=G_{\mid {\mathcal{C}}'}$. It is easy to verify that $H'$ is the subgraph of $H$ induced by vertices of $X_{[a,b]}$.
We construct the set ${\mathcal{Q}}^{\textsf{thr}}$ of paths as follows. From \Cref{obs: bad inded structure}, for each $a\le t\le b-1$, $|E(U_{a-1}, \overline{U}_b)|\le |\hat E_t|$. Therefore, we can compute a set ${\mathcal{P}}$ of edge-disjoint paths connecting to $x_a$ to $x_b$, that only use edges of $\bigcup_{a\le t\le b-1}\hat E_t$.
Then using the algorithm of \Cref{lem: convert path}, we convert the set ${\mathcal{P}}$ into a set ${\mathcal{Q}}^{\textsf{thr}}$ of paths.
Next, we denote $X^{\operatorname{left}}=X_{[a,i_1]}$, $X^{\operatorname{right}}=X_{(i_k,b]}$ and for each $1\le t\le k-1$, $X^t=X_{(i_{t},t_{t+1}]}$.
From \Cref{claim: inner and outer paths},
there exists a set ${\mathcal{P}}^{\operatorname{left}}$ routing edges of $\delta_{H'}(X^{\operatorname{left}})$ to $x_a$ inside $X^{\operatorname{left}}$, with congestion $O(1)$; and similarly, there exists a set ${\mathcal{P}}^{\operatorname{right}}$ routing edges of $\delta_{H'}(X^{\operatorname{right}})$ to $x_b$ inside $X^{\operatorname{right}}$, with congestion $O(1)$.
Moreover, for each $1\le t\le k-1$, there exists a set of paths routing its boundary to some vertex inside it, with congestion $O(1)$.
Consider now the contracted graph $H''$ obtained from $H'$ by contracting each cluster of $X^{\operatorname{left}}, X^{\operatorname{right}}, \set{X^t}_{1\le t\le k-1}, Y'_{s-1}, Y_{s+1}$ into a single vertex.
We denote by $y',y,w^{\operatorname{left}},w^{\operatorname{right}}$ the vertices of $H'$ that correspond to clusters $Y'_{s-1}, X^{\operatorname{left}},X^{\operatorname{right}},Y_{s+1}$, respectively. For each $1\le t\le k-1$, we denote by $w^t$ the vertex that corresponds to cluster $X^t$.
See \Cref{fig: unhelpful path} for an illustration.
We will construct sets of paths in $H''$ and then convert them into sets ${\mathcal{Q}}^{\operatorname{left}},{\mathcal{Q}}^{\operatorname{right}}$ of paths.
\begin{figure}[h]
\centering
\subfigure[Layout of part of $H'$.]{\scalebox{0.21}{\includegraphics{figs/unhelpful_path1.jpg}}\label{fig: unhelpful path 1}
}
\subfigure[Layout of part of $H''$.]{\scalebox{0.21}{\includegraphics{figs/unhelpful_path2.jpg}}\label{fig: unhelpful path 2}}
\caption{An illustration of part of $H'$ and $H''$.}
\label{fig: unhelpful path}
\end{figure}
Denote $W=\set{w^{\operatorname{left}},w^{\operatorname{right}}}\cup \set{w^t\mid 1\le t\le k-1}$ and $W'=W\cup \set{y,y'}$. We define $U^{\operatorname{left}}$ to the set of vertices that correspond to vertices of $U_{a-1}$ in original graph $G$, and we define $U^{\operatorname{right}}$ similarly.
Note that, from the construction of $H''$, for each $1\le t\le k-1$, $|E_{H''}(U^{\operatorname{left}},W)|\le 2\cdot (|E(w^t,w^{t-1})|+|E(w^t,w^{t-1})|)$ (where $w^{\operatorname{left}}=w^0$ and $y'=w^{-1}$). Additionally, since the pair $Y'_{s-1},Y_s$ of clusters are nice, $|E(y',w^{\operatorname{left}})|\ge |E_{H''}(U^{\operatorname{left}},W)|/10000$.
Denote $E^{\operatorname{left}}= E_{H''}(U^{\operatorname{left}},W)$.
We construct a set $\hat {\mathcal{P}}^{\operatorname{left}}$ of paths that contains, for each edge $e\in E^{\operatorname{left}}$, a path $\hat P^{\operatorname{left}}_e$ connecting $e$ to $w^{\operatorname{left}}$ within $W'$ as follows.
We first use similar arguments in the construction of ${\mathcal{Q}}^{\textsf{thr}}$ to compute, for each edge $e\in E^{\operatorname{left}}$ that is incident to $w^t$, a path $P'_e$ connecting $w^{t-1}$ to $w^{\operatorname{left}}$, such that all these paths are edge-disjoint in $H''$.
Next, for each edge $e\in E^{\operatorname{left}}$ that is incident to $w^{\operatorname{left}}$, we let $P^{\operatorname{left}}_e$ be the single-edge-path $e$.
For each $2\le t\le k-1$ and for each edge $e\in E^{\operatorname{left}}$ that is incident to $w^{1}$, we assign an edge $e^*$ of $E(w^t,w^{t-1})\cup E(w^t,w^{t-2})$, such that each edge of $E(w^t,w^{t-1})\cup E(w^t,w^{t-2})$ is assigned at most twice. Consider now an edge $e\in E^{\operatorname{left}}$. If $e^*\in E(w^t,w^{t-1})$, then we define $\hat P^{\operatorname{left}}_e$ to be the $e\cup e^*\cup P'_e$. If $e^*\in E(w^t,w^{t-2})$, then we define $\hat P^{\operatorname{left}}_e$ to be the concatenation of $e$, $e^*$ and the subpath of $P'_e$ between $w^{t-2}$ and $w^{\operatorname{left}}$.
\znote{below needs fixing, need to use the whole structure to treat edges of $E^{\operatorname{left}}$ that is incident to $w^{1}$.}
Lastly, consider the edges of $E^{\operatorname{left}}$ that is incident to $w^{1}$. For each $e\in E^{\operatorname{left}}$, we assign an edge $e^*\in E(y',w^{\operatorname{left}})$ to it, such that each edge of $E(y',w^{\operatorname{left}})$ is assigned at most $10000$ times, and then we define $\hat P^{\operatorname{left}}_e$ to be $e\cup e'$. This completes the definition of $\hat{\mathcal{P}}^{\operatorname{left}}$. It is easy to verify that $\cong_{H''}({\mathcal{P}}^{\operatorname{left}})=O(1)$.
\znote{Use $\hat{\mathcal{P}}^{\operatorname{left}}, {\mathcal{P}}^{\operatorname{left}}$ and ${\mathcal{Q}}_a$ to get ${\mathcal{Q}}^{\operatorname{left}}$, and similarly get ${\mathcal{Q}}^{\operatorname{right}}$.}
\subsection{Phase 1 of the Algorithm}
\label{sec: phase 1 interesting}
In Phase 1, we compute a collection ${\mathcal{I}}$ of subinstances of the subdivided instance $\check I$ that almost have all required properties, except that we will not be able to guarantee that the sum of the optimal solution costs of the resulting instances is suitably bounded. However, we will ensure that all resulting instances have a convenient structure, that will be utilized in Phase 2, in order to produce the final collection of subinstances of $\check I$. The algorithm that is used in Phase 1 is summarized in the following theorem.
\begin{theorem}\label{thm: phase 1}
There is an efficient randomized algorithm, whose input consists of a wide and well-connected instance $\check I^*=(\check G^*,\check \Sigma^*)$, with $\check m=|E(\check G^*)|\geq \mu^{c'}$, for some large enough constant $c'$. Let $\check I=(\check G,\check \Sigma)$ be the coresponding subdivided instance. The algorithm either returns FAIL, or computes a non-empty collection ${\mathcal{I}}$ of subinstances of $\check I$, such that, for every instance $I=(G,\Sigma)\in {\mathcal{I}}$, $G\subseteq \check G$, and $I$ is the subinstance of $\check I$ defined by $G$. Additionally, the algorithm computes, for every instance $I\in {\mathcal{I}}$, a core structure ${\mathcal{J}}(I)$ for $I$, such that, if we denote, for every instance $I\in {\mathcal{I}}$, the ${\mathcal{J}}(I)$-contracted subinstance of $I$ by $\hat I$, and let $\hat {\mathcal{I}}=\set{\hat I\mid I\in {\mathcal{I}}}$, then the following hold:
\begin{itemize}
\item for every instance $I\in {\mathcal{I}}$, if the corresponding contracted instance $\hat I=(\hat G,\hat \Sigma)$ is a wide instance, then $|E(\hat G)|\le \check m/\mu$;
\item $\sum_{\hat I=(\hat G,\hat \Sigma)\in \hat {\mathcal{I}}}|E(\hat G)|\le 2\check m$; and
\item there is an efficient algorithm, called $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace_1$, that, given a solution $\phi(\hat I)$ to every instance $\hat I\in \hat {\mathcal{I}}$, computes a solution $\check\phi$ to instance $\check I$.
\end{itemize}
Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then with probability at least $(1-1/\mu^{200})$, all of the following hold:
\begin{enumerate}
\item the algorithm does not return FAIL;\label{item: no fail}
\item for every instance $I\in {\mathcal{I}}$, there is a solution $\psi(I)$ to $I$, that is ${\mathcal{J}}(I)$-valid, with
$\sum_{I\in {\mathcal{I}}}\mathsf{cr}(\psi(I))\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}}{\check m}$; and
\item if algorithm $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace_1$ is given as input a solution $\phi(\hat I)$ to every instance $\hat I\in \hat {\mathcal{I}}$, then the solution $\check\phi$ to instance $\check I$ that it computes has cost at most: $\sum_{\hat I\in \hat {\mathcal{I}}}\mathsf{cr}(\phi(\hat I)) + \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot\mu^{8000}$.\label{item: combine}
\end{enumerate}
\end{theorem}
If the algorithm from \Cref{thm: phase 1} returns a collection ${\mathcal{I}}$ of subinstances of $\check I$ together with a core structure ${\mathcal{J}}(I)$ for each such subinstance, such that properties (\ref{item: no fail}) -- (\ref{item: combine}) hold, then we say that the algorithm is successful, and otherwise we say that it is unsuccessful. Notice that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then the probability that the algorithm is unsuccessful is bounded by $1/\mu^{200}$.
The remainder of this subsection is dedicated to the proof of \Cref{thm: phase 1}. The algorithm repeatedly applies Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace$ to subinstances of the input instance $\check I$. Throughout, we set the constant $b$ used in Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace$ to $b=52$.
We may not always be able to ensure that the input to Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace$ is valid. We will always ensure that the input consists of a graph $G\subseteq \check G$, a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for the subinstance $I$ of $\check I$ defined by $G$, and a promising set ${\mathcal{P}}$ of paths for $I$ and ${\mathcal{J}}$ of cardinality $\floor{\frac{|E(G)|}{\mu^{52}}}$.
But unfortunately we may not be able to ensure that there exists a solution $\phi$ to instance $I$ that is ${\mathcal{J}}$-valid, with $\mathsf{cr}(\phi)\leq \frac{|E(G)|^2}{\mu^{60b}}$ and $|\chi^{\mathsf{dirty}}(\phi)|\leq \frac{|E(G)|}{\mu^{60b}}$, since drawing $\phi$ is not given explicitly as part of input. If the input to Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace$ is not valid, then the procedure may fail during its execution. In this case, we will assume that the procedure returned FAIL (we will also say that the procedure fails). It is also possible that the output $({\mathcal{A}},I_1,I_2)$ of the procedure is not a valid output. We can verify efficiently that ${\mathcal{A}}$ is a valid enhancement structure for ${\mathcal{J}}$, and that $(I_1,I_2)$ is a valid split of $I$ along ${\mathcal{A}}$. We can also efficiently verify that Property \ref{prop: smaller graphs} holds for the resulting output. If we establish that either of these properties does not hold, then we will also assume that the procedure returned FAIL, or that it failed. However, it is possible that all above properties hold for the procedure's output, but properties \ref{prop output deleted edges} or \ref{prop output drawing} do not. As we are unable to efficiently verify these latter two properties, we will say in such a case that the procedure did not fail, but that it was unsuccessful. If the input to procedure \ensuremath{\mathsf{ProcSplit}}\xspace is valid, it is still possible that, with small probability (up to $2^{10}/\mu^{520}$), its output is not valid. As before, if ${\mathcal{A}}$ is not a valid enhancement structure for ${\mathcal{J}}$, or $(I_1,I_2)$ is not a valid split of $I$ along ${\mathcal{A}}$, or Property \ref{prop: smaller graphs} does not hold (which we can verify efficiently), we will say that the procedure returned FAIL, or that the procedure failed. Otherwise, if all these properties hold but the output of the procedure is not valid, we will say that the application of the procedure was unsuccessful. If the procedure returns a valid output, then we say that its application was successful.
As before, we denote $|E(\check G^*)|$ by $\check m$.
The algorithm for Phase 1 consists of a number of iterations. The input to iteration $j\geq 1$ consists of a collection ${\mathcal{I}}_j$ of subinstances of instance $\check I$, where for every instance $I=(G,\Sigma)\in {\mathcal{I}}_j$, $G\subseteq \check G$, and $I$ is the subinstance of $\check I$ defined by $G$. Additionally, for every instance $I\in {\mathcal{I}}_j$, we are given a core structure ${\mathcal{J}}(I)$ for $I$. We will ensure that, with high probability, the subinstances in ${\mathcal{I}}_j$ satisfy the following properties:
\begin{properties}{A}
\item for every instance $I=(G,\Sigma)\in {\mathcal{I}}_j$,
either the ${\mathcal{J}}(I)$-contracted subinstance $\hat I=(\hat G,\hat \Sigma)$ of $I$ is narrow, or
$|E(\hat G)|\leq \max\set{\frac{\check m}{\mu},2\check m-(j-1)\cdot \frac{\check m}{32\mu^{53}}}$; \label{invariant: fewer edgesx}
\item if we denote by $E^{\mathsf{del}}_j=E(\check G)\setminus \textsf{left}(\bigcup_{I=(G,\Sigma)\in {\mathcal{I}}_j}E(G)\textsf{right} )$, then $|E^{\mathsf{del}}_j|\leq \frac{j\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6000}}{\check m}$;\label{inv: few deleted edges}
\item for every instance $I\in {\mathcal{I}}_j$, there exists a solution $\psi(I)$ to instance $I$ that is ${\mathcal{J}}(I)$-valid, such that $\sum_{I\in {\mathcal{I}}_j}\mathsf{cr}(\psi(I)) \leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}_j}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{j\cdot \mu^{800}\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)}{\check m}$; \label{invariant: solutions to instances}
\item for every instance $I=(G,\Sigma)\in {\mathcal{I}}_j$, whose corresponding ${\mathcal{J}}(I)$-contracted instance $\hat I=(\hat G,\hat \Sigma)$ is wide with $|E(\hat G)|>\check m/\mu$, for every vertex $v\in V(G)$ with $\deg_G(v)\geq\frac{\check m}{\mu^5}$, there is a collection ${\mathcal{Q}}(v)$ of at least $\frac{8\check m}{\mu^{50}}-|E^{\mathsf{del}}_j|$ edge-disjoint paths in $G$ connecting $v$ to the vertices of the core $J(I)$ associated with the core structure ${\mathcal{J}}(I)$; \label{inv: route to core}
\item if we denote, for every instance $I\in {\mathcal{I}}_j$, by $\hat m(I)$ the number of edges in the corresponding ${\mathcal{J}}(I)$-contracted instance $\hat I$, then $\sum_{I\in {\mathcal{I}}_j}\hat m(I)\leq 2\check m$; \label{inv: disjoint edges} and
\item there is an efficient algorithm, that, given, for every instance $I\in {\mathcal{I}}_j$, a solution $\phi(I)$ that is clean with respect to ${\mathcal{J}}(I)$, constructs a solution $\phi(\check I)$ to instance $\check I$, of cost at most $\sum_{I\in {\mathcal{I}}_j}\mathsf{cr}(\phi(I))+ j\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6000}$.\label{invariant: putting solutions together}
\end{properties}
Specifically, for all $j\geq 1$, the input to iteration $j$ consists of a collection ${\mathcal{I}}_j$ of subinstances of $\check I$, where for every instance $I=(G,\Sigma)$, $G\subseteq \check G$, and $I$ is the subgraph of $\check I$ defined by $G$. Additionally, for every instance $I\in {\mathcal{I}}_j$, we are given a core structure ${\mathcal{J}}(I)$ for $I$. We denote, for each instance $I=(G,\Sigma)\in {\mathcal{I}}_j$, the corresponding ${\mathcal{J}}(I)$-contracted instance by $\hat I=(\hat G,\hat \Sigma)$, and we denote by $\hat m(I)=|E(\hat G)|$. We will guarantee that, if Properties \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold for input ${\mathcal{I}}_j$ to iteration $j$, then, with probability at least $1-\frac{1}{\mu^{400}}$, at the end of the iteration, we obtain a collection ${\mathcal{I}}_{j+1}$ of subinstances of $\check I$, each of which is defined by a subgraph of $\check G$, and, for every instance $I\in {\mathcal{I}}_{j+1}$, a core structure ${\mathcal{J}}(I)$, for which Properties \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold. With the remaining probability, the algorithm may either return FAIL, or produce an output for which some of the properties \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} do not hold. In each of these two cases, we say that the iteration was unsuccessful. If the input ${\mathcal{I}}_j$ to iteration $j$ does not have properties \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together}, then it is possible that the algorithm returns FAIL, or it returns output ${\mathcal{I}}_{j+1}$ for which some of the invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} do not hold. In both of these cases, we say that the iteration was unsuccessful. If the iteration produces output ${\mathcal{I}}_{j+1}$ for which properties \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold, then we say that it was successful. For all $j\geq 1$, we denote by $\tilde {\cal{E}}_j$ the bad event that iteration $j$ was unsuccessful.
The number of iterations in our algorithm is at most $z=128\ceil{\mu^{53}}$.
Note that, from Invariant \ref{invariant: fewer edgesx}, for every instance $I=(G,\Sigma)\in {\mathcal{I}}_z$, either the corresponding ${\mathcal{J}}(I)$-contracted instance $\hat I=(\hat G,\hat \Sigma)$ is narrow, or $|E(\hat G)|\leq \check m/\mu$. We will ensure that, for all $1\leq j\leq z$, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then $\prob{\tilde {\cal{E}}_j\mid \neg\tilde {\cal{E}}_1\wedge\cdots\wedge\neg\tilde {\cal{E}}_{j-1}}\leq \frac{1}{\mu^{400}}$. This will guarantee that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then with probability at least $1-1/\mu^{200}$, Properties \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold for ${\mathcal{I}}_z$.
The input to the first iteration is a set ${\mathcal{I}}_1$ of instances, consisting of a single instance $\check I$. Since instance $\check I^*$ is wide, there is at least one vertex $v^*$ with $\deg_{\check G}(v^*)\geq \check m/\mu^4$. We define
a core structure ${\mathcal{J}}(\check I)=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ associated with instance $\check I$ as follows. The core $J$ consists of a single vertex $v^*$, and its orientation $b_v$ is set to be arbitrary (say $1$). Drawing $\rho_J$ is the unique trivial drawing of $J$, and face $F^*(\rho_J)$ is the unique face of this drawing.
It is easy to verify that Invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold for ${\mathcal{I}}_1$. The only invariant that is not immediate is \ref{inv: route to core}. This invariant follows from the fact that the input instance $\check I^*$ is wide and well-connected. Therefore, for every vertex $u$ of $\check G$ with $\deg_{\check G}(u)\geq \check m/\mu^5$, $\deg_{\check G^*}(u)\geq \check m/\mu^5$ also holds, and there is a collection of at least $\frac{8\check m}{\mu^{50}}$ edge-disjoint paths connecting $u$ to $v^*$ in $\check G^*$ and hence in $\check G$.
We now describe the execution of iteration $j$. Consider an instance $I=(G,\Sigma)\in {\mathcal{I}}_j$, and its corresponding core structure ${\mathcal{J}}(I)$. We say that instance $I$ is \emph{inactive} if either the ${\mathcal{J}}(I)$-contracted subinstance $\hat I=(\hat G,\hat \Sigma)$ of $I$ is narrow, or $|E(\hat G)|\leq \check m/\mu$. Otherwise, we say that instance $I$ is \emph{active}. We denote by ${\mathcal{I}}^A_j$ the set of all active instances in ${\mathcal{I}}_j$, and by ${\mathcal{I}}^I_j$ the set of all inactive instances. We start with the set ${\mathcal{I}}_{j+1}$ containing every instance in ${\mathcal{I}}^{I}_j$. We also maintain the set $E^{\mathsf{del}}_{j+1}$ of deleted edges, that is initialized to $E^{\mathsf{del}}_j$. We then process every active instance $I\in {\mathcal{I}}^{A}_j$ one by one. We now describe the algorithm for processing one such instance $I=(G,\Sigma)$.
\paragraph{Processing instance $I=(G,\Sigma)\in {\mathcal{I}}^{A}_j$.}
Assume that Invariants \ref{invariant: fewer edgesx}, \ref{invariant: solutions to instances} and \ref{inv: route to core} hold for ${\mathcal{I}}_j$. Denote ${\mathcal{J}}(I)=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$, $|E(G)|=m$, and $|E(\hat G)|=\hat m(I)$. Since instance $I$ is active, $\hat m(I)>\check m/\mu$. Since $G\subseteq \check G$, $m\leq 2\check m$. Therefore, $\frac{m}{2\mu}\leq \hat m(I)\leq m$.
Consider any vertex $v\in V(G)$ with $\deg_{G}(v)\geq\frac{\hat m(I)}{\mu^4}$. Since $ \hat m(I)\geq \frac{\check m}{\mu}$, we get that $\deg_G(v)=\deg_{\hat G}(v)\geq \frac{\check m}{\mu^5}$. From Invariant \ref{inv: route to core},
there is a collection ${\mathcal{Q}}(v)$ of at least $\frac{8\check m}{\mu^{50}}-|E^{\mathsf{del}}_j|\geq \frac{8\check m}{\mu^{50}}- \frac{j\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6000}}{\check m}$ edge-disjoint paths in $G$ connecting $v$ to vertices of $J$. Since $j\leq z=128\ceil{\mu^{53}}$, if we assume that $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$ for a large enough constant $c'$, we get that $|{\mathcal{Q}}(v)|\geq \frac{4\check m}{\mu^{50}}\geq \frac{2\hat m(I)}{\mu^{50}}$.
We apply the algorithm from \Cref{claim: find potential augmentors}
to instance $I$ and core structure ${\mathcal{J}}(I)$, to obtain a promising set of paths ${\mathcal{P}}$, of cardinality $\floor{\frac{\hat m(I)}{\mu^{50}}}\geq \floor{\frac{m}{\mu^{52}}}$, since $\hat m(I)\geq \frac{m}{2\mu}$.
We use the following claim.
\begin{claim}\label{claim: invariants give valid input}
If $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$ for some large enough constant $c'$, and Invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold for ${\mathcal{I}}_j$, then $(G,{\mathcal{J}}(I),{\mathcal{P}})$ is a valid input to Procedure \ensuremath{\mathsf{ProcSplit}}\xspace.
\end{claim}
\begin{proof}
From the invariants it is immediate to verify that ${\mathcal{J}}(I)$ is a valid core structure for the subinstance $I$ of $\check I$ defined by $G$.
Consider the solution $\psi(I)$ to instance $I$, that is given by Invariant \ref{invariant: solutions to instances}. This solution is guaranteed to be ${\mathcal{J}}(I)$-valid. It is enough to verify that $\mathsf{cr}(\psi(I))\leq \frac{m^2}{\mu^{3120}}$ and $|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{m}{\mu^{3120}}$.
Recall that Invariant \ref{invariant: solutions to instances} guarantees that
$\sum_{I'\in {\mathcal{I}}_j}\mathsf{cr}(\psi(I')) \leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$.
In particular, $\mathsf{cr}(\psi(I))\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \frac{\check m^2}{\mu^{c'}}$ must hold for a large enough constant $c'$. Since $m\geq \hat m(I)\geq \frac{\check m}{\mu}$, we get that $\mathsf{cr}(\psi(I))\leq \frac{m^2}{\mu^{3120}}$ as required. Similarly, Invariant \ref{invariant: solutions to instances} guarantees that $\sum_{I'\in {\mathcal{I}}_j}|\chi^{\mathsf{dirty}}(\psi(I'))|\leq \frac{j\cdot \mu^{800}\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)}{\check m}$. In particular, $|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{j\cdot \mu^{800}\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)}{\check m}$. Since $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \frac{\check m^2}{\mu^{c'}}$ for a large enough constant $c'$, while $j\leq z=128\ceil{\mu^{53}}$, we get that $|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{\check m}{\mu^{c'-852}}$. Since
$m\geq \hat m(I)\geq \frac{\check m}{\mu}$, we get that $|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{m}{\mu^{3120}}$.
\end{proof}
In order to process instance $I\in {\mathcal{I}}_j^A$, we
apply Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace$ to input $(G,{\mathcal{J}}(I),{\mathcal{P}})$. If the procedure returns FAIL, then we terminate the algorithm and return FAIL as well. In this case we say that the current iteration failed. Otherwise, the procedure returns a ${\mathcal{J}}(I)$-enhancement structure ${\mathcal{A}}$, and
a split $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ of $I$ along ${\mathcal{A}}$. Let $P^*$ be the enhancement path of ${\mathcal{A}}$, and let $({\mathcal{J}}_1,{\mathcal{J}}_2)$ be the split of the core structure ${\mathcal{J}}$ along ${\mathcal{A}}$.
Denote $E^{\mathsf{del}}(I)=E(G)\setminus (E(G_1)\cup E(G_2))$. Let $G'=G\setminus E^{\mathsf{del}}(I)$, let $\Sigma'$ be the rotation system for $G'$ induced by $\Sigma$, and let $I'=(G',\Sigma')$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace.
We add the edges of $E^{\mathsf{del}}(I)$ to set $E^{\mathsf{del}}_{j+1}$, and we add instances $I_1,I_2$ to the collection ${\mathcal{I}}_{j+1}$ of instances, letting ${\mathcal{J}}(I_1)={\mathcal{J}}_1$ and ${\mathcal{J}}(I_2)={\mathcal{J}}_2$. From the definition of a split of an instance along an enhancement structure, $G_1,G_2\subseteq G$, ${\mathcal{J}}_1$ is a valid core structure for $I_1$, and ${\mathcal{J}}_2$ is a valid core structure for $I_2$. This completes the description of the algorithm for processing an instance $I\in {\mathcal{I}}_j^A$, and of the $j$th iteration. We now analyze its properties.
We say that iteration $j$ is \emph{good} if, for every instance $I\in {\mathcal{I}}_j^A$, the algorithm from \Cref{claim: find potential augmentors}, when applied to instance $I$ and core structure ${\mathcal{J}}(I)$ returned a promising set of paths ${\mathcal{P}}$ of cardinality $\floor{\frac{\hat m(I)}{\mu^{50}}}$, and additionally, the application of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace to input $(G,{\mathcal{J}}(I),{\mathcal{P}})$ was successful. We use the following claim to show that iteration $j$ is good with high probability.
\begin{claim}\label{claim: prob good iteration}
If Invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold for ${\mathcal{I}}_j$ and $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$,
then the probability that iteration $j$ is good is at least $1-1/\mu^{498}$.
\end{claim}
\begin{proof}
From the discussion above, if Invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold for ${\mathcal{I}}_j$, and $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then for every instance $I\in {\mathcal{I}}^A_j$, the algorithm from \Cref{claim: find potential augmentors}, when applied to instance $I$ and core structure ${\mathcal{J}}(I)$ returns a promising set of paths ${\mathcal{P}}(I)$ of cardinality $\floor{\frac{\hat m(I)}{\mu^{50}}}$. Additionally, from
\Cref{claim: invariants give valid input}, if Invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold for ${\mathcal{I}}_j$, and $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then for every instance $I\in {\mathcal{I}}_j^A$, $(G,{\mathcal{J}}(I),{\mathcal{P}}(I))$ is a valid input to Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. In this case, from \Cref{thm: procsplit}, the probability that Procedure \ensuremath{\mathsf{ProcSplit}}\xspace is either unsuccessful or fails, when applied to $(I,{\mathcal{J}}(I),{\mathcal{P}}(I))$, is at most $2^{20}/\mu^{520}$.
Since, from Invariant \ref{inv: disjoint edges}, $\sum_{I\in {\mathcal{I}}_j}\hat m(I)\leq 2\check m$, while for every active instance $I\in{\mathcal{I}}_j^A$, $\hat m(I)\geq \check m/\mu$, we get that $|{\mathcal{I}}^A_j|\leq 2\mu$. From the Union Bound, we conclude that, if Invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold, and $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then the probability that iteration $j$ is good is at least $1-1/\mu^{498}$.
\end{proof}
Lastly, the next claim allows us to bound the probability of the bad event $\tilde {\cal{E}}_z$.
\begin{claim}\label{claim: success prob}
Assume that
Invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold for ${\mathcal{I}}_j$, $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, and that iteration $j$ is good. Then the bad event $\tilde {\cal{E}}_j$ does not happen.
\end{claim}
\begin{proof}
Throughout the proof, we assume that Invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} hold for ${\mathcal{I}}_j$, $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, and iteration $j$ is good.
From the definition of a split of an instance (see \Cref{def: split}), for every instance $I=(G,\Sigma)\in {\mathcal{I}}_{j+1}$, $G\subseteq \check G$, and $I$ is the subinstance of $\check I$ defined by graph $G$.
It is now enough to show that Invariants \ref{invariant: fewer edgesx}--\ref{invariant: putting solutions together} continue to hold for the collection ${\mathcal{I}}_{j+1}$ of instances.
We first observe that, for each inactive instance $I\in {\mathcal{I}}^I_{j}$, invariant \ref{invariant: fewer edgesx} continue to hold for $I$, and $\hat m(I)$ does not change.
Consider now some active instance $I=(G,\Sigma)\in {\mathcal{I}}^A_j$, and let $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ be the split of $I$ that was computed by Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. We also let ${\mathcal{A}}$ be the core enhancement structure computed by the procedure, and we let $({\mathcal{J}}_1,{\mathcal{J}}_2)$ be the split of the core structure ${\mathcal{J}}(I)$ via ${\mathcal{A}}$.
Note that Property \ref{prop: smaller graphs} of a valid output for Procedure \ensuremath{\mathsf{ProcSplit}}\xspace ensures that
$|E(G_1)|,|E(G_2)|\leq |E(G)|-\frac{|E(G)|}{32\mu^{52}}$. Since $|E(G)|\geq \hat m(I)\geq \frac{\check m}\mu$, we get that $|E(G_1)|,|E(G_2)|\leq |E(G)|-\frac{\check m}{32\mu^{53}}\leq 2\check m-j\cdot \frac{\check m}{32\mu^{53}}$ (from the fact that Property \ref{invariant: fewer edgesx} holds for ${\mathcal{I}}_j$). This establishes Property \ref{invariant: fewer edgesx} for ${\mathcal{I}}_{j+1}$.
Recall that Property \ref{prop output deleted edges} of a valid output for Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace$ ensures that
$|E^{\mathsf{del}}(I)|\leq \frac{2\mathsf{cr}(\psi(I))\cdot \mu^{2000}}{|E(G)|}+|\chi^{\mathsf{dirty}}(\psi(I))|$. Therefore, we get that:
\[
\begin{split}
|E^{\mathsf{del}}_{j+1}|&\leq |E^{\mathsf{del}}_j|+\sum_{I=(G,\Sigma)\in {\mathcal{I}}^A_j}\textsf{left} (\frac{2\mathsf{cr}(\psi(I))\cdot \mu^{2000}}{|E(G)|}+|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} )\\
&\leq \frac{j\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6000}}{\check m}+\sum_{I\in {\mathcal{I}}^A_j}\frac{\mathsf{cr}(\psi(I))\cdot \mu^{2002}}{\check m}+\sum_{I\in {\mathcal{I}}^A_j}|\chi^{\mathsf{dirty}}(\psi(I))|.
\end{split}
\]
(we have used the fact that Invariant \ref{inv: few deleted edges} holds for ${\mathcal{I}}_j$, and that, for every instance $I=(G,\Sigma)\in {\mathcal{I}}^A_j$, $|E(G)|\geq \hat m(I)\geq \frac{\check m}{\mu}$). Recall that, from Invariant \ref{invariant: solutions to instances}, $\sum_{I\in {\mathcal{I}}_j}\mathsf{cr}(\psi(I)) \leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}_j}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{j\cdot \mu^{800}\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)}{\check m}\leq \frac{ \mu^{854}\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)}{\check m}$ (since $j\leq z=128\ceil{\mu^{53}}$). Altogether, we get that $|E^{\mathsf{del}}_{j+1}|\leq \frac{(j+1)\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6000}}{\check m}$, establising Invariant \ref{inv: few deleted edges} for ${\mathcal{I}}_{j+1}$.
Next, we establish Invariant \ref{invariant: solutions to instances} for ${\mathcal{I}}_{j+1}$. For every instance $I=(G,\Sigma)\in {\mathcal{I}}_j^I$, its solution $\psi(I)$ remains unchanged. Consider now some instance $I=(G,\Sigma)\in{\mathcal{I}}_j^A$, and the two subinstances $(I_1,I_2)$ of $I$ that Procedure \ensuremath{\mathsf{ProcSplit}}\xspace produced.
Let $G'=G\setminus E^{\mathsf{del}}(I)$, let $\Sigma'$ be the rotation system for $G'$ induced by $\Sigma$, and let $I'=(G',\Sigma')$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Consider the solution $\phi'$ for instance $I'$ that is guaranteed by Property \ref{prop output drawing} of valid output of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. Let $\phi_1$ be the solution to instance $I_1$ induced by $\phi'$, and let $\phi_2$ be the solution to instance $I_2$ induced by $\phi'$.
Property \ref{prop output drawing} guarantees that $\phi_1$ is a ${\mathcal{J}}_1$-valid solution to $I_1$, and $\phi_2$ is a ${\mathcal{J}}_2$-valid solution to $I_2$. We implicitly set $\psi(I_1)=\phi_1$ and $\psi(I_2)=\phi_2$.
From the definition of an instance split, (see \Cref{def: split}), the only edges that may be shared by graphs $G_1$ and $G_2$ are edges of $E(J)\cup E(P^*)$. Since no pair of edges in $E(J)\cup E(P^*)$ may cross each other in $\phi'$, we get that $\mathsf{cr}(\phi_1)+\mathsf{cr}(\phi_2)\leq \mathsf{cr}(\phi')$. Moreover, if $(e,e')_p\in \chi^{\mathsf{dirty}}(\phi_1)$, then either $(e,e')_p\in \chi^{\mathsf{dirty}}(\phi')$, or one of the edges $e,e'$ lies on $P^*$. Since the edges of $P^*$ participate in at most $\frac{\mathsf{cr}(\psi(I))\cdot \mu^{624}}{|E(G)|}\leq \frac{\mathsf{cr}(\psi(I))\cdot \mu^{625}}{\check m}$ crossings (as $|E(G)|\geq \hat m(I)\geq \check m/\mu$), we get that $| \chi^{\mathsf{dirty}}(\phi_1)|+| \chi^{\mathsf{dirty}}(\phi_2)|\leq |\chi^{\mathsf{dirty}}(\psi(I))|+\frac{\mathsf{cr}(\psi(I))\cdot \mu^{625}}{\check m}$.
Overall, we get that:
\[\sum_{I\in {\mathcal{I}}_{j+1}}\mathsf{cr}(\psi(I)) \leq \sum_{I\in {\mathcal{I}}_j}\mathsf{cr}(\psi(I)) \leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I);\]
and:
\[ \begin{split}
\sum_{I\in {\mathcal{I}}_{j+1}}|\chi^{\mathsf{dirty}}(\psi(I))|&\leq \sum_{I\in {\mathcal{I}}_{j}}|\chi^{\mathsf{dirty}}(\psi(I))|+\sum_{I\in {\mathcal{I}}_j^A}\frac{\mathsf{cr}(\psi(I))\cdot \mu^{625}}{\check m}\\
&\leq \frac{j\cdot \mu^{800}\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)}{\check m}+\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{625}}{\check m}\\
&\leq \frac{(j+1)\cdot \mu^{800}\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)}{\check m},
\end{split} \]
establishing Invariant \ref{invariant: solutions to instances} for ${\mathcal{I}}_{j+1}$.
Next, we establish Invariant \ref{inv: route to core}. Consider some instance $\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}_{j+1}$ with $\hat m(\tilde I)>\check m/\mu$, whose corresponding ${\mathcal{J}}(\tilde I)$-contracted graph is wide. If $\tilde I\in {\mathcal{I}}_j$, then $\tilde I$ is an inactive instance, and so Invariant \ref{inv: route to core} holds for it. Otherwise, there is some instance $I\in {\mathcal{I}}_j^A$, such that, if $(I_1,I_2)$ is the split of instance $I$ that we have computed, $\tilde I=I_1$ or $\tilde I=I_2$ holds. We assume w.l.o.g. that it is the former. We denote $I=(G,\Sigma)$, $I_1=(G_1,\Sigma_1)$, and we let $J$, $J_1$, and $J_2$ be the cores associated with the core structures ${\mathcal{J}}(I)$, ${\mathcal{J}}(I_1)$, and ${\mathcal{J}}(I_2)$, respectively.
Consider now any vertex $v\in V(G_1)$, whose degree in $G_1$ is at least $\frac{\check m}{\mu^5}$. Then $\deg_G(v)\geq \frac{\check m}{\mu^5}$ must hold as well. From Invariant \ref{inv: route to core}, there is a collection ${\mathcal{Q}}(v)$ of at least $\frac{8\check m}{\mu^{50}}-|E^{\mathsf{del}}_j|$ edge-disjoint paths in $G$ connecting $v$ to the vertices of $J$. We assume w.l.o.g. that the paths in ${\mathcal{Q}}(v)$ are internally disjoint from $V(J)$. Let ${\mathcal{Q}}'(v)\subseteq {\mathcal{Q}}(v)$ be the set of paths that do not contain edges of $E^{\mathsf{del}}(I)$. Clearly, $|{\mathcal{Q}}'(v)|\geq \frac{8\check m}{\mu^{50}}-|E^{\mathsf{del}}_j|-|E^{\mathsf{del}}(I)|\geq \frac{8\check m}{\mu^{50}}-|E^{\mathsf{del}}_{j+1}|$. We direct the paths in ${\mathcal{Q}}'(v)$ from $v$ to the vertices of $V(J)$. Notice that every path $Q\in {\mathcal{Q}}'(v)$ is contained in graph $G'$. Consider now any such path $Q\in {\mathcal{Q}}'(v)$. If path $Q$ contains a vertex of $P^*$ as an inner vertex, then we truncate it so it connects $v$ to a vertex of $P^*$, and is internally disjoint from $V(J)\cup V(P^*)$.
We claim that the resulting path $Q$ must be contained in graph $G_1$. This is since, from the definition of a split of an instance,
$V(G_1)\cup V(G_2)=V(G)$, and every vertex $u\in V(G_1)\cap V(G_2)$ belongs to $V(J_1)\cap V(J_2)$, while $J_1,J_2\subseteq J\cup P^*$. Since $E(G')=E(G_1)\cup E(G_2)$, we get that every path $Q$ in the resulting set ${\mathcal{Q}}'(v)$ is contained in graph $G_1$, and it connects $v$ to a vertex of $J_1$. This establishes Invariant \ref{inv: route to core} for ${\mathcal{I}}_{j+1}$.
Invariant \ref{inv: disjoint edges} follows from the fact that, for every instance $I\in {\mathcal{I}}_j^A$, if $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ is the split of instance $I$ that we have computed, then $E(G_1)\cap E(G_2)\subseteq E(J_1)\cap E(J_2)$ (since, from definition of a split, every vertex $u\in V(G_1)\cap V(G_2)$ belongs to $V(J_1)\cap V(J_2)$, and since a subdivided instance may not have parallel edges).
It now remains to establish Invariant \ref{invariant: putting solutions together}.
Assume we are given, for every instance $I'\in {\mathcal{I}}_{j+1}$, a solution $\phi(I')$ that is clean with respect to ${\mathcal{J}}(I')$.
Consider any active instance $I=(G,\Sigma)\in {\mathcal{I}}_j^A$, and let $(I_1=(G_1,\Sigma_2), I_2=(G_2,\Sigma_2))$ be the split of $I$ that we have constructed. We apply the algorithm from \Cref{obs: combine solutions for split} in order to obtain a solution $\phi(I)$ to instance $I$ that is clean with respect to ${\mathcal{J}}(I)$, and $\mathsf{cr}(\phi(I))\leq \mathsf{cr}(\phi(I_1))+\mathsf{cr}(\phi(I_2))+|E^{\mathsf{del}}(I)|\cdot |E(G)|$. From Property \ref{prop output deleted edges} of a valid output for \ensuremath{\mathsf{ProcSplit}}\xspace,
$|E^{\mathsf{del}}(I)|\leq \frac{2\mathsf{cr}(\phi)\cdot \mu^{2000}}{|E(G)|}+|\chi^{\mathsf{dirty}}(\psi(I))|$.
Overall, we have now obtained a solution $\phi(I)$ for every instance $I\in {\mathcal{I}}_j$, that is clean with respect to ${\mathcal{J}}(I)$, such that:
\[
\begin{split}
\sum_{I\in {\mathcal{I}}_j}\mathsf{cr}(\phi(I))&\leq \sum_{I'\in {\mathcal{I}}_{j+1}}\mathsf{cr}(\phi(I'))+\sum_{I=(G,\Sigma)\in {\mathcal{I}}^A_j}|E(G)|\cdot \textsf{left} ( \frac{\mathsf{cr}(\psi(I))\cdot \mu^{2000}}{|E(G)|}+|\chi^{\mathsf{dirty}}(\psi(I))| \textsf{right} ) \\
&\leq \sum_{I'\in {\mathcal{I}}_{j+1}}\mathsf{cr}(\phi(I'))+\sum_{I=(G,\Sigma)\in {\mathcal{I}}^A_j}\mathsf{cr}(\psi(I))\cdot \mu^{2000} + 2\check m\cdot \sum_{I=(G,\Sigma)\in {\mathcal{I}}^A_j} |\chi^{\mathsf{dirty}}(\psi(I))|.
\end{split}
\]
From Invariant \ref{invariant: solutions to instances}, $\sum_{I\in {\mathcal{I}}_j}\mathsf{cr}(\psi(I)) \leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}_j}|\chi^{\mathsf{dirty}}(\psi(I))|\leq \frac{j\cdot \mu^{800}\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)}{\check m}$.
Therefore, altogether:
\[
\begin{split}
\sum_{I\in {\mathcal{I}}_j}\mathsf{cr}(\phi(I))&\leq \sum_{I'\in {\mathcal{I}}_{j+1}}\mathsf{cr}(\phi(I'))+\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{2000} + 2z\cdot \mu^{800}\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\\
&\leq \sum_{I'\in {\mathcal{I}}_{j+1}}\mathsf{cr}(\phi(I'))+2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{2000},
\end{split}
\]
since $z=128\ceil{\mu^{53}}$.
We then apply the algorithm that is guaranteed by Invariant \ref{invariant: putting solutions together} to the collection ${\mathcal{I}}_j$ of instances, to compute a solution $\phi(\check I)$ to instance $\check I$, whose is at most:
\[
\begin{split}
\sum_{I\in {\mathcal{I}}_j}\mathsf{cr}(\phi(I))+j\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6000}&\leq \sum_{I'\in {\mathcal{I}}_{j+1}}\mathsf{cr}(\phi(I'))+j\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6000}+2\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{2000}\\
&\leq \sum_{I'\in {\mathcal{I}}_{j+1}}\mathsf{cr}(\phi(I'))+(j+1)\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6000}.
\end{split}\]
\end{proof}
From Claims \ref{claim: prob good iteration} and \ref{claim: success prob}, for all $1\leq j\leq z$, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\leq \check m^2/\mu^{c'}$, then $\prob{\tilde {\cal{E}}_j\mid \neg\tilde {\cal{E}}_1\wedge\cdots\wedge\neg\tilde {\cal{E}}_{j-1}}\leq 1/\mu^{498}$. Therefore,
$$\prob{\tilde {\cal{E}}_z}\leq \prob{\tilde {\cal{E}}_z\mid \neg\tilde {\cal{E}}_1\wedge\cdots\wedge\neg\tilde {\cal{E}}_{z-1}}+\prob{\tilde {\cal{E}}_{z-1}\mid \neg\tilde {\cal{E}}_1\wedge\cdots\wedge\neg\tilde {\cal{E}}_{z-2}}+\ldots+\prob{\tilde {\cal{E}}_1}\leq z/\mu^{498}\leq 1/\mu^{400},$$
since $z=128\ceil{\mu^{53}}$.
If the algorithm did not return FAIL, then we return the set ${\mathcal{I}}_z$ of subinstances of $\check I$, and, for every instance $I\in {\mathcal{I}}_z$, the corresponding core structure ${\mathcal{I}}(I)$.
Assume that Event $\tilde {\cal{E}}_z$ did not happen.
From Invariant \ref{invariant: fewer edgesx}, we are guaranteed that, for every instance $I\in {\mathcal{I}}$, if the corresponding contracted instance $\hat I=(\hat G,\hat \Sigma)$ is a wide instance, then $|E(\hat G)|\le \check m/\mu$. Invariant \ref{inv: disjoint edges} ensures that $\sum_{\hat I=(\hat G,\hat \Sigma)\in \hat {\mathcal{I}}}|E(\hat G)|\le 2\check m$, and Invariant \ref{invariant: putting solutions together} provides an efficient algorithm for combining clean solutions $\phi(I)$ to instances in $I\in{\mathcal{I}}_z$ to obtain a solution $\check \phi$ to instance $\check I$.
If Event $\tilde {\cal{E}}_z$ does not happen, then we are guaranteed that:
$$\mathsf{cr}(\check \phi)\leq \sum_{I\in {\mathcal{I}}_z}\mathsf{cr}(\phi(I))+ z\cdot \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6000}\leq \sum_{I\in {\mathcal{I}}_z}\mathsf{cr}(\phi(I))+ \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{6054},$$
(since $z=128\ceil{\mu^{53}}$). Since there is an efficient algorithm that, given, for every instance $I\in {\mathcal{I}}_z$, a solution $\hat \phi(I)$ to the corresponding ${\mathcal{J}}(I)$-contracted instance $\hat I$, computes a solution $\phi(I)$ to instance $I$ that is clean with respect to ${\mathcal{J}}(I)$ with $\mathsf{cr}(\phi(I))\leq \mathsf{cr}(\hat \phi(I))$, we obtain the desired efficient algorithm for combining solutions to instances in $\hat{\mathcal{I}}$ to obtain a solution to instance $\check I$.
Lastly, Invariant \ref{invariant: solutions to instances} ensures that, if Event $\tilde {\cal{E}}_z$ did not happen, then,
for every instance $I\in {\mathcal{I}}_z$, there exists a solution $\psi(I)$ to $I$ that is ${\mathcal{J}}(I)$-valid, such that $\sum_{I\in {\mathcal{I}}_z}\mathsf{cr}(\psi(I))\leq \mathsf{OPT}_{\mathsf{cnwrs}}(\check I)$, and $\sum_{I\in {\mathcal{I}}_z}|\chi^{\mathsf{dirty}}(\psi(I))|\leq
\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot z\mu^{800}}{\check m}\le
\frac{\mathsf{OPT}_{\mathsf{cnwrs}}(\check I)\cdot \mu^{900}}{\check m}$ (since $z=128\ceil{\mu^{53}}$).
Notice also that, if Event $\tilde {\cal{E}}_z$ does not happen, then the algorithm does not return FAIL.
This completes the proof of \Cref{thm: phase 1}.
\subsection{Phase 2 of the Algorithm}
\label{subsec: phase 2}
The goal of this phase is to ``repair'' each one of the instances $I\in {\mathcal{I}}$ computed in the first phase in order to ensure that each such instance has a cheap solution that is clean with respect to the core structure ${\mathcal{J}}(I)$. This, in turn, will ensure that the corresponding contracted graph has a cheap solution as well. In order to repair an instance $I=(G,\Sigma)\in {\mathcal{I}}$, we will compute a collection $E^{\mathsf{del}}(I)$ of edges of $G$. We will ensure that no edge of the core $J(I)$ corresponding to the core structure ${\mathcal{J}}(I)$ lies in $E^{\mathsf{del}}(I)$. We can then define a new instance $I'=(G',\Sigma')$, where $G'=G\setminus E^{\mathsf{del}}(I)$, and $\Sigma'$ is the rotation system for $G'$ that is induced by $\Sigma$. Note that ${\mathcal{J}}(I)$ remains a valid core structure for $I'$. Our goal is to ensure that, on the one hand, $|E^{\mathsf{del}}(I)|$ is not too large, and, on the other hand, there is a solution $\psi(I')$ to instance $I'$ that is clean with respect to ${\mathcal{J}}(I')$, and $\mathsf{cr}(\psi(I'))$ is not too large compared to $\mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(\psi(I))|^2$, where $\psi(I)$ is the ${\mathcal{J}}(I)$-valid solution for instance $I$ from \Cref{thm: phase 1}.
We now state the main result of this subsection, summarizing the algorithm for Phase 2.
\begin{theorem}\label{thm: phase 2}
There is an efficient randomized algorithm, whose input consists of a large enough constant $b$, a subinstance $I=(G,\Sigma)$ of $\check I$ with $|E(G)|=m$ and $G\subseteq \check G$, and a core structure ${\mathcal{J}}(I)$ for $I$, whose corresponding core is denoted by $J(I)$.
The algorithm computes a set $E^{\mathsf{del}}(I)\subseteq E(G)\setminus E(J(I))$ of edges, for which the following hold.
Let $G'=G\setminus E^{\mathsf{del}}(I)$, and let $I'=(G',\Sigma')$ be the subinstance of $\check I$ defined by $G'$.
The algorithm ensures that, if there is a solution $\psi(I)$ to instance $I$ that is ${\mathcal{J}}(I)$-valid, with $\mathsf{cr}(\psi(I))\leq m^2/\mu^{240b}$, and $|\chi^{\mathsf{dirty}}(\psi(I))|\leq m/\mu^{240b}$, then with probability at least $1-1/\mu^{2b}$,
$|E^{\mathsf{del}}(I)|\leq \textsf{left} (\frac{\mathsf{cr}(\psi(I))}{m}+|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} ) \cdot \mu^{O(b)}$, and there is a solution $\psi(I')$ to instance $I'$ that is clean with respect to ${\mathcal{J}}(I)$, with $\mathsf{cr}(\psi(I'))\leq \textsf{left} (\mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(G)|}{\mu^b}\textsf{right} )\cdot (\log m)^{O(1)}$.
\end{theorem}
If there is a solution $\psi(I)$ to instance $I$ that is ${\mathcal{J}}(I)$-valid, with $\mathsf{cr}(\psi(I))\leq m^2/\mu^{240b}$, and $|\chi^{\mathsf{dirty}}(\psi(I))|\leq m/\mu^{240b}$, then we let $\psi(I)$ be this solution, and we say that $\psi(I)$ is a good solution to instance $I$. Otherwise, we let $\psi(I)$ be any solution to instance $I$ that is ${\mathcal{J}}(I)$-valid, and we say that $\psi(I)$ is a bad solution to instance $I$.
We say that an application of the algorithm from \Cref{thm: phase 2} is \emph{successful}, if (i) $|E^{\mathsf{del}}(I)|\leq \textsf{left} (\frac{\mathsf{cr}(\psi(I))}{m}+|\chi^{\mathsf{dirty}}(\psi(I))|\textsf{right} ) \cdot \mu^{O(b)}$, and (ii)
there is a solution $\psi(I')$ to the resulting instance $I'=(G',\Sigma')$, that is clean with respect to ${\mathcal{J}}(I)$, with:
$$\mathsf{cr}(\psi(I'))\leq \textsf{left} (\mathsf{cr}(\psi(I))+|\chi^{\mathsf{dirty}}(\psi(I))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(I))|\cdot |E(G)|}{\mu^b}\textsf{right} )\cdot (\log m)^{O(1)}.$$
From \Cref{thm: phase 2}, if there is a good solution $\psi(I)$ to instance $I$, then the algorithm is successful with probability at least $1-1/\mu^{2b}$.
We provide the proof of the theorem below, after we complete the proof of \Cref{lem: many paths} using it.
\input{finish-proof-interesting}
\input{phase2-intuition}
\input{phase2-defs}
\input{phase2-the-alg}
\input{phase2-good-drawing}
\subsection{Phase 2: Layered Well-Linked Decomposition, Further Disengagement, and Fixing the Flower Cluster}
From now on we focus on the proof of \Cref{thm: decomposing problematic instances}. In order to simplify the notation, we denote the input problematic instance by $I=(G,\Sigma)$. Our goal is to design an efficient randomized algorithm that computes a collection ${\mathcal{I}}'$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace, such that, for each resulting instance $\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}'$, $|E(\tilde G)|\leq m/\mu$,
and additionally,
$\sum_{\tilde I=(\tilde G,\tilde \Sigma)\in {\mathcal{I}}'}|E(\tilde G)|\leq O(|E(G)|)$, and $\expect{\sum_{\tilde I\in {\mathcal{I}}'}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$. We also need to provide an efficient algorithm ${\mathcal{A}}(I)$, that, given a solution $\phi(\tilde I)$ to each instance $\tilde I\in {\mathcal{I}}'$, computes a solution $\phi$ to instance $I$, with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{\tilde I\in {\mathcal{I}}'}\mathsf{cr}(\phi(\tilde I))\textsf{right} )$.
We denote by $C$ the unique flower cluster of ${\mathcal{C}}^f$ contained in $G$, by $u^*$ its center vertex, and by ${\mathcal{X}}=\set{X_1,\ldots,X_k}$ its petals. This phase consists of two steps. In the first step, we compute a layered well-linked decomposition of the graph $G$ with respect to $C$, and perform disengagement of the resulting clusters. In the second step, we modify the flower cluster $C$ and its petals to ensure that every petal is routable in the resulting instance.
\subsubsection{Step 1: Layered Well-Linked Decomposition and Second Disengagement}
In this step, we apply the algorithm from \Cref{thm: layered well linked decomposition} to graph $G$ and cluster $C$, in order to compute
a valid layered $\alpha$-well-linked decomposition $({\mathcal{W}}, ({\mathcal{L}}_1,\ldots,{\mathcal{L}}_r))$ of $G$ with respect to $C$, for $\alpha=\frac{1}{c\log^{2.5}m}$, where $c$ is some large enough constant independent of $m$, and $r\leq \log m$. Note that, since we have assumed that $m$ is sufficiently large, every cluster $W\in {\mathcal{W}}$ has the $\alpha_0$-bandwidth property, for $\alpha_0=1/\log^3m$. Recall that we are additionally guaranteed that $\bigcup_{W\in {\mathcal{W}}}V(W)=V(G)\setminus V(C)$, and that, for every cluster $W\in {\mathcal{W}}$, $|\delta_G(W)|\leq |\delta_G(C)|$.
Recall that, for every cluster $W\in {\mathcal{W}}$ with $W\in {\mathcal{L}}_i$, we have partitioned the set $\delta_G(W)$ of edges into two subsets: set $\delta^{\operatorname{down}}(W)$ connecting vertices of $W$ to vertices that lie in the clusters of $\set{C}\cup {\mathcal{L}}_1\cup\cdots\cup {\mathcal{L}}_{i-1}$; and set $\delta^{\operatorname{up}}(W)$ containing all remaining edges, and we are guaranteed that $|\delta^{\operatorname{up}}(W)|<|\delta^{\operatorname{down}}(W)|/\log m$.
Lastly, recall that, for every cluster $W\in {\mathcal{W}}$, there is a collection ${\mathcal{P}}(W)$ of paths in $G$, routing the edges of $\delta_G(W)$ to edges of $\delta_G(C)$, such that the paths in ${\mathcal{P}}(W)$ avoid $W$, and cause congestion at most $200/\alpha$. Recall that, from Properties \ref{prop: flower cluster petal intersection} and \ref{prop: flower cluster routing} of the flower cluster, there is a collection ${\mathcal{Q}}\subseteq \bigcup_{i=1}^k{\mathcal{Q}}_i$ of edge-disjoint paths, routing the edges of $\delta_G(C)$ to the vertex $u^*$, such that all inner vertices on every path lie in $C$. By concatenating the paths in ${\mathcal{P}}(W)$ and the paths in ${\mathcal{Q}}$, we obtain a new collection ${\mathcal{Q}}'(W)$ of paths, that route the edges of $\delta_G(W)$ to vertex $u^*$, so that all inner vertices on every path lie outside $W$, and cause congestion at most $200/\alpha=O(\log^{2.5}m)$.
We partition the set ${\mathcal{W}}$ of clusters into two subsets ${\mathcal{W}}^{\operatorname{light}}$, and ${\mathcal{W}}^{\operatorname{bad}}$, as follows. We apply the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace from \Cref{thm:algclassifycluster} to each cluster cluster $W\in {\mathcal{W}}$ in turn, with parameter $p=1/(m^*)^4$. If the algorithm returns FAIL, then we add cluster $W$ to ${\mathcal{W}}^{\operatorname{bad}}$.
Recall that the probability that Algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace errs, that is, it returns FAIL when $W$ is not $\eta^*$-bad, for $\eta^*=2^{O((\log m)^{3/4}\log\log m)}$, is at most $1/(m^*)^4$. Otherwise, the algorithm returns a distribution ${\mathcal{D}}(W)$ over the set $\Lambda_G(W)$ of internal $W$-routers, such that cluster $W$ is $\beta^*$-light with respect to ${\mathcal{D}}(W)$, where $\beta^*=2^{O(\sqrt{\log m}\cdot \log\log m)}$. We add $W$ to ${\mathcal{W}}^{\operatorname{light}}$ in this case. This finishes the algorithm for partitioning the set ${\mathcal{W}}$ of clusters into ${\mathcal{W}}^{\operatorname{light}}$ and ${\mathcal{W}}^{\operatorname{bad}}$. We let $\beta=\max\set{\beta^*,\eta^*}$, so that $\beta\leq 2^{O((\log m)^{3/4}\log\log m)}$.
We say that a bad event ${\cal{E}}_{\operatorname{bad}}$ happens if set ${\mathcal{W}}^{\operatorname{bad}}$ contains a cluster that is not $\beta$-bad. From the above discussion, $\prob{\beta_{\operatorname{bad}}}\leq 1/(m^*)^3$, and every cluster $W\in {\mathcal{W}}^{\operatorname{light}}$ is $\beta$-good with respect to the distribution ${\mathcal{D}}(W)$ over the set $\Lambda_G(W)$ of internal $W$-routers.
Recall that we are also given, for every cluster $W\in {\mathcal{W}}$, a set ${\mathcal{Q}}'(W)$ of paths routing the edges of $\delta_G(W)$ to vertex $u^*$ (external $W$-router), with congestion at most $O(\log^{2.5}m)\leq \beta$.
For every cluster $W\in {\mathcal{W}}$, we define its distribution over the set $\Lambda'_G(W)$ of external $W$-routers to be the distribution that assign probability $1$ to the path set ${\mathcal{Q}}'(W)$.
We let ${\mathcal{L}}$ be a laminar family of clusters of $G$, containing cluster $G$ and every cluster of ${\mathcal{W}}$.
We now apply Algorithm \ensuremath{\mathsf{AlgBasicDisengagement}}\xspace to the instance $I=(G,\Sigma)$, using the laminar family ${\mathcal{L}}$ of clusters, its partition $({\mathcal{W}}^{\operatorname{bad}},{\mathcal{W}}^{\operatorname{light}}\cup \set{G})$, and distributions $\set{{\mathcal{D}}(W)}_{W\in {\mathcal{W}}^{\operatorname{light}}}$ and $\set{{\mathcal{D}}'(W)}_{W\in {\mathcal{L}}}$, that is given in \Cref{subsec: basic disengagement}.
We denote the resulting family of instances by ${\mathcal{I}}_2(I)$. Recall that family ${\mathcal{I}}_2(I)$ of instances contains a single global instance $\hat I$, and additionally, for every cluster $W\in {\mathcal{W}}$, an instance $ I_W$.
If we denote the global instance by $\hat I=(\hat G,\hat \Sigma)$, then graph $\hat G$ is obtained from graph $G$ by contracting every cluster $W\in {\mathcal{W}}$ into a supernode $v_W$. In particular, the flower cluster $C$ is contained in $\hat G$. For every cluster $W\in {\mathcal{W}}$, if we denote corresponding instance by $I_W=(G_W,\Sigma_W)$, then graph $G_W$ is obtained from graph $G$ by contracting all vertices of $V(G)\setminus V(W)$ into a single vertex $v^*$.
In particular, no edge of cluster $C$ may lie in $G_W$, and so every edge of $G_W$ is an edge of set $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$, where ${\mathcal{C}}$ is the set of clusters computed in Phase 1. Therefore, $|E(G_W)|\leq m/(2\mu)$.
Note that the depth of laminar family ${\mathcal{L}}$ is $1$. From \Cref{lem: disengagement final cost}, if event ${\cal{E}}_{\operatorname{bad}}$ did not happen,
\[
\begin{split}
\expect{\sum_{I'\in {\mathcal{I}}_2(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}& \leq O\textsf{left} (\beta^2\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\textsf{right} )\\
& \leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} ).
\end{split}
\]
If bad event ${\cal{E}}_{\operatorname{bad}}$ happened, then $\sum_{I'\in {\mathcal{I}}_2(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq m^3$. Since $\prob{{\cal{E}}_{\operatorname{bad}}}\leq 1/(m^*)^3$, we get that overall:
\begin{equation}\label{eq: small total opt}
\begin{split}
\expect{\sum_{I'\in {\mathcal{I}}_2(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}&
\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} ).
\end{split}
\end{equation}
Additionally, from \Cref{lem: number of edges in all disengaged instances}, we get that $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}_2(I)}|E(G')|\leq O(|E(G)|)$.
Lastly, from \Cref{lem: basic disengagement combining solutions},
there is an efficient algorithm, that, given, for each instance $I'\in {\mathcal{I}}_2(I)$, a solution $\phi(I')$, computes a solution for instance $I$ of value at most $\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))$.
Note that the set ${\mathcal{I}}_2(I)$ of instances has all properties required in \Cref{thm: decomposing problematic instances}, with one exception: it is possible that, in the global instance $\hat I=(\hat G,\hat \Sigma)$, $|E(\hat G)|>m/(2\mu)$.
In order to overcome this difficulty, we further decompose instance $\hat I$ into subinstances, proving the following lemma.
\begin{lemma}\label{lem: decomposing problematic instances by petal}
There is an efficient randomized algorithm, that either returns FAIL, or computes a $\nu_2$-decomposition $\tilde {\mathcal{I}}$ of instance $\hat I$, for $\nu_2=2^{O((\log m)^{3/4}\log\log m)}$, such that, for each instance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}$, $|E(\tilde G)|\leq m/(2\mu)$. Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)<\frac{ m^2}{c'' \mu^{13}}$ for some large enough constant $c''$, then the probability that the algorithm returns FAIL is at most $1/(8\mu^4)$.
\end{lemma}
We prove the lemma below, after we complete the proof of
\Cref{thm: decomposing problematic instances}
using it.
If the algorithm from \Cref{lem: decomposing problematic instances by petal} returns $\nu_2$-decomposition $\tilde {\mathcal{I}}$ of instance $\hat I$, then we return the collection $\tilde {\mathcal{I}}(I)=\tilde {\mathcal{I}}\cup \set{I_W\mid W\in {\mathcal{W}}}$ of instances, which is now guaranteed to be a $\nu_1$-decomposition of instance $I$, where $\nu_1=2^{O((\log m)^{3/4}\log\log m)}$. We are also guaranteed that, for each instance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}$, $|E(\tilde G)|\leq m/(2\mu)$.
Assume now that
$\mathsf{OPT}_{\mathsf{cnwrs}}(I)< m^2/\textsf{left} (\mu^{18}\cdot 2^{c'(\log m)^{3/4}\log\log m}\textsf{right} )$ for some large enough constant $c'$.
Recall that, from Equation \ref{eq: small total opt},
$\expect{\sum_{I'\in {\mathcal{I}}_2(I)}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$, and in particular $\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)}\leq \nu^{*}\cdot \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$, for some $\nu^{*}= 2^{O((\log m)^{3/4}\log\log m)}$. We say that a bad event ${\cal{E}}'$ happens if $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)>8\mu^4\cdot \nu^{*}\cdot \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )$. From Markov's inequality, $\prob{{\cal{E}}'}\leq 1/(8\mu^4)$.
By letting $c'$ be a large enough constant, we can assume that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)< m^2/\textsf{left} (\mu^{18}\cdot 2^{c'(\log m)^{3/4}\log\log m}\textsf{right} )$, then $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<\frac{m^2}{16c''\mu^{18}\cdot \nu^{*}}$, where $c''$ is the constant from \Cref{lem: decomposing problematic instances by petal}. Since we have assumed (in the statement of \Cref{lem: not many paths}) that $m>\mu^{50}$ and since $\mu>\nu^*$, we get that
$|E(G)|<\frac{m^2}{16c''\mu^{18}\cdot \nu^{*}}$.
To conclude, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)< m^2/\textsf{left} (\mu^{18}\cdot 2^{c'(\log m)^{3/4}\log\log m}\textsf{right} )$, then $(\mathsf{OPT}_{\mathsf{cnwrs}}(I))+|E(G)|)<\frac{m^2}{8c''\mu^{18}\cdot \nu^{*}}$. If, additionally, Event ${\cal{E}}'$ did not happen, then $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)<\frac{m^2}{c'' \mu^{11}}$. In this case, the algorithm from \Cref{lem: decomposing problematic instances by petal} may only return FAIL with probability at most $1/(8\mu^4)$.
To conclude, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)< m^2/\textsf{left} (\mu^{18}\cdot 2^{c'(\log m)^{3/4}\log\log m}\textsf{right} )$, then our algorithm may return FAIL in only two cases: either (i) event ${\cal{E}}'$ happened (which happens with probability at most $1/(8\mu^4)$); or (ii) $\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)<\frac{c'' m^2}{ \mu^{11}}$, and yet the algorithm from \Cref{lem: decomposing problematic instances by petal} returns FAIL (which happens with probability at most $1/(8\mu^3)$). Overall, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)<m^2/\textsf{left} (\mu^{18}\cdot 2^{c'(\log m)^{3/4}\log\log m}\textsf{right} )$, then the algorithm only returns FAIL with probability at most $1/(4\mu^4)$.
\iffalse
\mynote{please restate the lemma in terms of $\nu$-decomposition}
\begin{lemma}\label{lem: decomposing problematic instances by petal}
There is an efficient randomized algorithm, that computes a $\nu_2$-decomposition of instance $\hat I$, with the following properties:
\begin{itemize}
\item $\sum_{\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}(\hat I)}|E(\tilde G)|\leq O(|E(\hat G)|)$;
\item for each subinstance $\tilde I=(\tilde G,\tilde \Sigma)\in \tilde {\mathcal{I}}$, $|E(\tilde G)|\leq m/\mu$;
and
\item $\expect{\sum_{\tilde I\in \tilde {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde I)}\leq 2^{O((\log m)^{3/4}\log\log m)}\cdot \textsf{left}(\mathsf{OPT}_{\mathsf{cnwrs}}(\hat I)+|E(\hat G)|\textsf{right} )$.
\end{itemize}
Additionally, there is an efficient algorithm ${\mathcal{A}}(\hat I)$, that, given a solution $\phi(\tilde I)$ to each instance $\tilde I\in \tilde {\mathcal{I}}$, computes a solution $\phi$ to instance $\hat I$, with $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{\tilde I\in \tilde {\mathcal{I}}(I')}\mathsf{cr}(\phi(\tilde I))\textsf{right} )$.
\end{lemma}
\fi
From now on we focus on the proof of \Cref{lem: decomposing problematic instances by petal}. Recall that we have denoted $\hat I=(\hat G,\hat \Sigma)$, and that graph $\hat G$ is obtained from $G$ by contracting every cluster $W\in {\mathcal{W}}$ into a vertex $v_W$, that we refer to as a \emph{supernode}. We denote the resulting set of supernodes by $U=\set{v_W\mid W\in {\mathcal{W}}}$. Recall that the flower cluster $C\subseteq\hat G$, and the edges of $E(\hat G)\setminus E(C)$ lie in $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$, where ${\mathcal{C}}$ is the set of clusters computed in Phase 1. Therefore, $|E(\hat G)\setminus E(C)|\leq m/(160\mu)$.
Partition $({\mathcal{L}}_1,\ldots,{\mathcal{L}}_r)$ of the set ${\mathcal{W}}$ of clusters into $r\leq \log m$ layers naturally defines a partition $L_1,\ldots,L_r$ of the set $U$ of vertices into layers, where vertex $v_W$ lies in layer $L_i$ iff $W\in {\mathcal{L}}_i$. For convenience, we denote $L_0=V(C)$. For all $1\leq i \leq r$, for every vertex $v\in L_i$, we partition the set $\delta(v)$ of its edges into two subsets: set $\delta^{\operatorname{down}}(v)$ connecting $v$ to vertices of $L_0\cup \cdots \cup L_{i-1}$ and set $\delta^{\operatorname{up}}(v)$ containing all remaining edges, that connect $v$ to vertices of $L_i\cup \cdots \cup L_r$. In the following step, we may move some vertices of $U$ from their current layer to layer $L_0$. The definition of the sets $\delta^{\operatorname{down}}(v'),\delta^{\operatorname{up}}(v')$ of edges is always with respect to the current partition of vertices of $\hat G$ into layers. Observe that Property \ref{condition: layered decomp edge ratio} of layered well-linked decomposition ensures the following property:
\begin{properties}{P}
\item For every vertex $v\in U$,
$|\delta^{\operatorname{up}}(v)|<|\delta^{\operatorname{down}}(v)|/\log m$; \label{prop: layered decomp edge ratio for vertices}
\end{properties}
For convenience of notation, in the remainder of this proof we denote instance $\hat I=(\hat G,\hat \Sigma)$ by $I=(G,\Sigma)$. We use the parameter $m$ from before, so $|E(G)|\leq m$ holds.
\subsubsection{Step 2: Fixing Petals for Routability}
Recall that, as part of the definiton of the flower cluster $C$, we are given a collection ${\mathcal{X}}=\set{X_1,\ldots,X_k}$ of petals of $C$. Consider now some petal $X_i\in {\mathcal{X}}$. Let $\hat E_i=\delta_G(X_i)\setminus \delta_G(u^*)$, where $u^*$ is the center of the flower cluster $C$.
We will use the following definition.
\begin{definition}
Let $G$ be a graph, and let $C^f$ be a flower cluster in $G$, with center $u^*$ and a set ${\mathcal{X}}=\set{X_1,\ldots,X_k}$ of petals. For $1\leq i\leq k$, we say that petal $X_i$ is \emph{routable} in $G$ if there is a collection ${\mathcal{Q}}'_i=\set{Q'(e)\mid e\in \hat E_i}$ of paths in $G$, where for each edge $e\in \hat E_i$, path $Q'(e)$ has $e$ as its first edge, terminates at vertex $u^*$, and its inner vertices are disjoint from $X_i$, such that the paths in ${\mathcal{Q}}'_i$ cause congestion at most $3000$.
\end{definition}
As we show later, if every petal in ${\mathcal{X}}$ is routable, then we can decompose the current instance $I$ into smaller instances, each of which will correspond to a distinct petal in ${\mathcal{X}}$ (together with an additional ``global'' instance). Unfortunately, it is possible that some petals in ${\mathcal{X}}$ are not routable. We overcome this difficulty by ``fixing'' the flower cluster $C$. We do so iteratively, while ensuring that Property \ref{prop: layered decomp edge ratio for vertices} continues to hold after each iteration. In every iteration, we select some vertex of $U$ to be added to some petal $X_i$ of ${\mathcal{X}}$. The set
$\hat E_i=\delta_G(X_i)\setminus \delta_G(u^*)$ of edges is always defined with respect to the current petal $X_i$. In addition to maintaining Property \ref{prop: layered decomp edge ratio for vertices}, we will maintain the following important property:
\begin{properties}[1]{P}
\item For every petal $X_i\in {\mathcal{X}}$, there is a set ${\mathcal{Q}}_i=\set{Q(e)\mid e\in \hat E_i}$ of edge-disjoint paths, where for each edge $e\in \hat E_i$, path $Q(e)$ has $e$ as its first edge, vertex $u^*$ as its last vertex, and all inner vertices of $Q(e)$ lie in $X_i$. \label{prop: routing inside clusters}
\end{properties}
We now describe the algorithm for fixing the petals of $C$. While there is some petal $X_i\in {\mathcal{X}}$, and some vertex $v\in U$, such that at least $|\delta_G(v)|/2$ neighbors of $v$ in $G$ lie in $X_i$, we add $v$ to $X_i$, and remove it from $U$.
In other words, we update $X_i$ to be the subgraph of $G$ induced by vertex set $V(X_i)\cup \set{v}$, and we update $C$ to be the subgraph of $G$ induced by vertex set $V(C)\cup\set{v}$.
We also remove $v$ from its current layer $L_j$ and add it to $L_0$. It is immediate to verify that Property \ref{prop: layered decomp edge ratio for vertices} continues to hold after each iteration. We now show that the same is true for Property \ref{prop: routing inside clusters}.
Consider an iteration, when some vertex $v\in U$ was added to some petal $X_i\in {\mathcal{X}}$. Partition the edges of $\delta_G(v)$ into two subsets: set $\delta'(v)$ connecting vertex $v$ to vertices of $X_i$, and set $\delta''(v)$ containing all remaining edges. From our definitions, at the beginning of the current iteration, $\delta'(v)\subseteq \hat E_i$ held. Therefore, set ${\mathcal{Q}}_i$ contained, for each edge $e\in \delta'(v)$, a path $Q(e)$, connecting $e$ to $u^*$, such that all inner vertices of $Q(e)$ belong to $X_i$. At the end of the current iteration, the edges of $\delta'(v)$ no longer lie in $\hat E_i$, and the edges of $\delta''(v)$ are added to $\hat E_i$ instead. Since $|\delta''(v)|\leq |\delta'(v)|$, we can define a mapping $M$, that maps every edge of $\delta''(v)$ to a distinct edge of $\delta'(v)$. We update the set ${\mathcal{Q}}_i$ of paths as follows: first, we remove from it all paths whose first edge lies in $\delta'(v)$. Next, for each edge $e\in \delta''(v)$, we add a new path $Q(e)$ to ${\mathcal{Q}}_i$, that is obtained by appending $e$ to the original path $Q(e')$, where $e'=M(e)$ is the edge of $\delta'(v)$ to which edge $e$ is mapped. Therefore, Property \ref{prop: routing inside clusters} continues to hold after each iteration. Lastly, we consider the cluster $C$ and its corresponding set ${\mathcal{X}}$ of petals obtained at the end of the algorithm.
We slightly modify Property \ref{prop: flower cluster small boundary size} of the flower cluster, and replace it with the following property, that we refer to as Modified Property \ref{prop: flower cluster small boundary size}:
$$|\delta_G(C)|\leq 96m/\mu^{42} \mbox{ and\ \ }|\bigcup_{i=1}^k\delta(X_i)|\leq \frac{192m}{\mu^{42}}.$$
If Properties
\ref{prop: flower cluster vertex induced petals too} -- \ref{prop: flower cluster routing} hold for a cluster $C'$, with Property \ref{prop: flower cluster small boundary size} replaced with its modified counterpart, then we say that $C'$ is a \emph{modified flower cluster}. We now prove that cluster $C$ is a valid modified cluster.
\begin{claim}\label{claim: remains flower cluster}
Cluster $C$ is a valid modified flower cluster in the current graph $G$.
\end{claim}
\begin{proof}
It is immediate to verify that throughout the algorithm, Property \ref{prop: flower cluster vertex induced petals too} continues to hold.
Recall that, from Property \ref{prop: flower cluster small boundary size} in the definition of a flower cluster, at the beginning of the algorithm, $|\delta_G(C)|\leq 96m/\mu^{42}$ held. We claim that this property continues to hold throughout the algorithm. Indeed, when a vertex $v\in U$ is added to $C$, there is some petal $X_i\in {\mathcal{X}}$, such that $|\delta'(v)|\geq |\delta''(v)|$, where $\delta'(v)$ contains all edges connecting $v$ to vertices of $X_i$, and $\delta''(v)$ contains all remaining edges of $\delta(v)$. Notice that edges of $\delta'(v)$ are removed from $\delta_G(C)$ at the end of the iteration, while only the edges of $\delta''(v)$ may be added to $\delta_G(C)$ at the end of the current iteration. Therefore, $|\delta_G(C)|$ does not increase, and modified Property \ref{prop: flower cluster small boundary size} continues to hold throughout the algorithm. From the above discussion, whenever a vertex $v$ is added to cluster $C$, $\deg_G(v)\leq 2|\delta_G(C)|\leq 192m/\mu^{42}$ (from Property \ref{prop: flower cluster small boundary size}). Therefore, Property \ref{prop: flower center has large degree, everyone else no} holds throughout the algorithm.
It is immediate to verify that Properties \ref{prop: flower cluster petal intersection} and \ref{prop: flower cluster edges near center partition} continue to hold throughout the algorithm, and we have already established Property \ref{prop: flower cluster routing} for the final cluster $C$.
\end{proof}
Lastly, we show that, once the algorithm terminates, every petal in ${\mathcal{X}}$ is routable in $G$.
\begin{claim}\label{claim: routable petals}
At the end of the algorithm, every petal of ${\mathcal{X}}$ is routable in $G$.
\end{claim}
\begin{proof}
\iffalse {the following paragraph is moved to prelim}
Let ${\mathcal{P}}$ be some collection of paths in graph $G$, where each path is associated with a direction. A \emph{flow} defined over the set ${\mathcal{P}}$ of paths is an assignment of non-negative values $f(P)\geq 0$, called \emph{flow-values}, to every path $P\in {\mathcal{P}}$. We sometimes refer to paths in ${\mathcal{P}}$ as \emph{flow-paths for flow $f$}. For each edge $e\in E(G)$, let ${\mathcal{P}}(e)\subseteq {\mathcal{P}}$ be the set of all paths whose first edge is $e$, and let ${\mathcal{P}}'(e)\subseteq {\mathcal{P}}$ be the set of all paths whose last edge is $e$. We say that edge $e$ \emph{sends $z$ flow units} in $f$ iff $\sum_{P\in {\mathcal{P}}(e)}f(e)=z$. Similarly, we say that edge $e$ \emph{receives $z$ flow units} in $f$ iff $\sum_{P\in {\mathcal{P}}'(e)}f(P)=z$. The \emph{congestion} of the flow $f$ is the maximum, over all edges $e\in E(G)$ of $\sum_{\stackrel{P\in {\mathcal{P}}:}{e\in E(P)}}f(P)$.
\fi
Consider some petal $X_i\in {\mathcal{X}}$. Recall that we have defined the set $\hat E_i=\delta_G(X_i)\setminus \delta_G(u^*)$ of edges, where $u^*$ is the center of the flower cluster $C$. Recall that our goal is to show that there is a collection ${\mathcal{Q}}'_i=\set{Q'(e)\mid e\in \hat E_i}$ of paths, where for each edge $e\in \hat E_i$, path $Q'(e)$ has $e$ as its first edge, terminates at vertex $u^*$, and is internally disjoint from $X_i$, such that the paths in ${\mathcal{Q}}'_i$ cause congestion at most $3000$. Let $\hat {\mathcal{Q}}_i$ be the set of all paths in graph $G$, where each path $Q\in \hat{\mathcal{Q}}_i$ contains some edge of $\hat E_i$ as its first edge, terminates at vertex $u^*$, and is internally disjoint from $X_i$. From the integrality of flow, it is enough to show that there exists a flow $\hat f_i$, defined over the set $\hat {\mathcal{Q}}_i$ of paths, in which every edge of $\hat E_i$ sends one flow unit, such that flow $\hat f_i$ causes congestion at most $3000$. From now on we focus on proving that such a flow indeed exists.
For the sake of the proof we will define layer $L_0$ slightly differently than before: we let $L_0=V(C)\setminus V(X_i\setminus\set{u^*})$.
For each index $0\leq j\leq r$, we let $S_j=L_0\cup L_1\cup\cdots\cup L_j$. We then let the set $E^*_j$ of edges contain all edges of $\delta_G(S_j)$, except for those insident to vertex $u^*$. Notice that in particular, since $S_r=V(G)\setminus (V(X_i)\setminus\set{u^*})$, edge set $E^*_r$ is precisely the edge set $\hat E_i=E_G(X_i)\setminus \delta_G(u^*)$.
For all $0\leq j\leq r$, we let ${\mathcal{P}}^*_j$ be the set of all paths $P$, such that the first edge of $P$ lies in $E^*_j$, the last vertex of $P$ is $u^*$, and all inner vertices of $P$ lie in $S_j$.
We prove the following claim.
\begin{claim}\label{claim: layer by layer}
For all $0\leq j\leq r$, there is a flow $f^*_j$ defined over the set ${\mathcal{P}}^*_j$ of paths, in which every edge of $E^*_j$ sends one flow unit, such that the paths in ${\mathcal{P}}^*_j$ cause congestion at most $\textsf{left} (1+\frac 8 {\log m}\textsf{right} )^j$.
\end{claim}
Note that proof of \Cref{claim: layer by layer} will finish the proof of \Cref{claim: routable petals}. Indeed,
as observed already, $E^*_r=\hat E_i$, and it is easy to verify that ${\mathcal{P}}^*_r=\hat {\mathcal{Q}}_i$. In flow $f^*_r$, every edge of $\hat E_i$ sends one flow unit, as required, and the congestion of the flow is at most $\textsf{left} (1+\frac 8 {\log m}\textsf{right} )^r\leq 3000$, since $r\leq \log m$. Therefore, in order to complete the proof of \Cref{claim: routable petals}, it is now enough to prove \Cref{claim: layer by layer}, which we do next.
\begin{proofof}{\Cref{claim: layer by layer}}
The proof is by induction on $j$. The base is when $j=0$.
Recall that $S_0=L_0=V(C)\setminus \textsf{left} (V(X_i)\setminus u^*\textsf{right} )$. The set $E^*_0$ of edges is then a subset of $\bigcup_{i'\neq i}\hat E_{i'}$.
Recall that, from the definition of the flower
cluster, for all $1\leq i'\leq k$, there is a collection ${\mathcal{Q}}_{i'}$ of edge-disjoint paths routing the edges of $\hat E_{i'}$ to vertex $u^*$, with all inner vertices on every path contained in $X_{i'}$.
For each index $i\neq i'$, for each edge $e\in \hat E_{i'}$, let $Q(e)\in {\mathcal{Q}}_{i'}$ be the unique path whose first edge is $e$. Observe that $\bigcup_{i'\neq i}{\mathcal{Q}}_{i'}\subseteq {\mathcal{P}}^*_0$. By sending one flow unit on each path in $\set{Q(e)\mid e\in \hat E_0}$, we obtain the desired flow $f^*_0$, defined over the set ${\mathcal{P}}^*_0$ of paths, in which each edge of $E^*_0$ sends one flow unit. The congestion of the flow is $1$.
We now prove that the claim holds for an index $1\leq j\leq r$, provided that it holds for index $j-1$.
Consider some vertex $v\in L_j$. We partition the edges of $\delta^{\operatorname{down}}(v)$ into two subsets: set $\delta_1^{\operatorname{down}}(v)$ containing all edges connecting $v$ to vertices of $X_i$, and set $\delta_2^{\operatorname{down}}(v)$ containing all remaining edges of $\delta^{\operatorname{down}}(v)$.
Notice that the edges of $\delta_2^{\operatorname{down}}(v)$ lie in $E^*_{j-1}$ but not in $E^*_j$, while edges of $\delta^{\operatorname{up}}(v)$ lie in $E^*_j$ but not in $E^*_{j-1}$. In fact, since $S_j=S_{j-1}\cup L_j$, $E^*_j\subseteq \textsf{left} (E^*_{j-1}\setminus \textsf{left}(\bigcup_{v\in L_j}\delta_2^{\operatorname{down}}(v)\textsf{right} )\textsf{right} )\cup \textsf{left}(\bigcup_{v\in L_j}\delta^{\operatorname{up}}(v)\textsf{right} )$ (we use inclusion rather than equality since an edge of $\delta^{\operatorname{up}}(v)$ may connect $v$ to a vertex of $L_j$).
Consider again some vertex $v\in L_j$.
Recall that $|\delta^{\operatorname{down}}_1(v)|\leq |\deg_G(v)|/2$ (since the algorithm for fixing the flower cluster $C$ has terminated), while
$|\delta^{\operatorname{up}}(v)|\leq |\delta^{\operatorname{down}}(v)|/\log m\leq \deg_G(v)/\log m$. Therefore, $|\delta^{\operatorname{down}}_2(v)|\geq \textsf{left} (\frac 1 2 -\frac 1 {\log m}\textsf{right} )\deg_G(v)$, while $|E^*_j\cap \delta_G(v)|\subseteq |\delta^{\operatorname{up}}(v)|+|\delta^{\operatorname{down}}_1(v)|\leq \textsf{left} (\frac 1 2 +\frac 1 {\log m}\textsf{right} )\deg(v)$. Overall, we get that $|E_j^*\cap \delta_G(v)|\leq \textsf{left} (1+\frac 8 {\log m}\textsf{right} )|\delta_2^{\operatorname{down}}(v)|$.
Let ${\mathcal{R}}_j(v)$ denote the collection of all paths that can be obtained by combining two edges: an edge of $E^*_j\cap \delta_G(v)$ and an edge of $\delta_2^{\operatorname{down}}(v)$; the paths are directed towards edges of $\delta_2^{\operatorname{down}}(v)$. Clearly, there is a flow $f'_v$, defined over the paths in ${\mathcal{R}}_j(v)$, where every edge of $E^*_j\cap \delta_G(v)$ sends one flow unit, every edge of $\delta_2^{\operatorname{down}}(v)$ receives at most $\textsf{left} (1+\frac 8 {\log m}\textsf{right} )$ flow units, and the flow causes congestion at most $\textsf{left} (1+\frac 8 {\log m}\textsf{right} )$ (for example, we can obtain such a flow by spreading the flow originating at every edge of $E^*_j\cap \delta_G(v)$ evenly among the edges of $\delta_2^{\operatorname{down}}(v)$).
Next, we define a new flow $f_j(v)$, in which every edge of $E^*_j\cap \delta_G(v)$ sends one flow unit via a subset of paths of ${\mathcal{P}}^*_j$.
Consider any flow-path $R\in {\mathcal{R}}_j(v)$, and assume that $R$ consists of two edges: $e\in E^*_j\cap \delta_G(v)$ and $e'\in \delta_2^{\operatorname{down}}(v)$. Recall that edge $e'$ sends $1$ flow unit in flow $f^*_{j-1}$. For every path $P\in {\mathcal{P}}^*_{j-1}$ whose first edge is $e'$, we consider a path $R^P$ obtained by appending the edge $e$ at the beginning of the path, so that path $R^P$ now starts with edge $e$, and terminates at vertex $u^*$ as before. Observe that path $R^P$ lies in path set ${\mathcal{P}}^*_{j}$. Let $x=f_{j-1}^*(P)$ be the amount of flow sent via path $P$ in flow $f_{j-1}$, and let $x'=f'_v(R)$ be the amount of flow sent via path $R$ in flow $f'_v$. We then send $(x\cdot x')$ flow units via path $R^P$ in flow $f_j(v)$. Notice that, since every edge of $E^*_j\cap \delta_G(v)$ sends one flow unit in flow $f'_v$, and every edge of $\delta_2^{\operatorname{down}}(v)$ sends one flow unit in flow $f_{j-1}$, this ensures that every edge of $E^*_j\cap \delta_G(v)$ sends one flow unit in the new flow $f_j(v)$ that we just defined. Moreover, since every edge $e'\in \delta_2^{\operatorname{down}}(v)$ receives at most $\textsf{left} (1+\frac 8 {\log m}\textsf{right} )$ flow units in $f'_v$, for each flow-path $P\in {\mathcal{P}}^*_{j-1}$ whose first edge is $e'$,
the total amount of flow sent along path $P$ in the new flow $f_j(v)$ is at most $\textsf{left} (1+\frac 8 {\log m}\textsf{right} )$ times the amount of flow sent via path $P$ in $f^*_{j-1}$. In other words, we can think of flow $f_j(v)$ as obtained as follows: we start with flow $f^*_{j-1}$, and discard flow on all flow-paths except those whose first edge lies in $\delta_2^{\operatorname{down}}(v)$. Next, we scale the flow on each resulting flow-path by at most factor $\textsf{left} (1+\frac 8 {\log m}\textsf{right} )$. Lastly, we combine the resulting flow with flow $f'_v$.
We are now ready to define the final flow $f^*_j$.
Recall that the set $E^*_j$ of edges can be obtained from edge set $E^*_{j-1}$ by first deleting the edges of $\bigcup_{v\in L_j}\delta_2^{\operatorname{down}}(v)$ from it, and then adding a subset of the edges of $\bigcup_{v\in L_j}\delta^{\operatorname{up}}(v)$ to it. For every edge $e\in E^*_j\cap E^*_{j-1}$, for every path $P\in {\mathcal{P}}^*_{j-1}\cap {\mathcal{P}}^*_{j}$, whose first edge is $e$, the flow $f^*_j(P)$ remains the same as the flow $f^*_{j-1}(P)$. This ensures that each edge of $E^*_j\cap E^*_{j-1}$ sends one flow unit in the new flow, as ${\mathcal{P}}_{j-1}\subseteq {\mathcal{P}}_j$. For every vertex $v\in L_j$, we use the flow $f_j(v)$ in order to send flow from the edges of $\delta^{\operatorname{up}}(v)\cap E^*_j$. Specifically, for each edge $e\in \delta^{\operatorname{up}}(v)\cap E^*_j$, for every path $P\in {\mathcal{P}}^*_j$ whose first edge is $e$, we set the flow $f_j^*(P)$ to be equal to the flow sent via this path by $f_j(v)$. This ensures that every edge in $E^*_j\setminus E^*_{j-1}$ sends one flow unit in the new flow $f^*_j$. This finishes the description of the flow $f^*_j$. From the above discussion, every edge of $E^*_j$ sends one flow unit in $f^*_j$. It now remains to analyze the congestion of the flow.
Observe that flow $f^*_j$ can be obtained as follows. We start with the flow $f^*_{j-1}$, and we scale the flow on some of the flow-paths by at most factor $\textsf{left} (1+\frac{8}{\log m}\textsf{right})$ (this is since for every vertex $v\in L_j$, for each edge $e\in \delta_2^{\operatorname{down}}(v)$, edge $e$ receives at most $\textsf{left} (1+\frac{8}{\log m}\textsf{right} )$ flow units via flow $f'_v$, and this flow utilizes the flow that $e$ sends in $f^*_{j-1}$ in order to reach vertex $u^*$). Lastly, we combine the resulting flow with the flows $f'_v$ for all vertices $v\in L_j$. It is then easy to verify that the congestion caused by flow $f^*_j$ is bounded by the congestion caused by flow $f^*_{j-1}$ times $\textsf{left} (1+\frac{8}{\log m}\textsf{right} )$. Since, from the induction hypothesis, flow $f^*_{j-1}$ causes congestion at most
$\textsf{left} (1+\frac 8 {\log m}\textsf{right} )^{j-1}$, flow $f^*_j$ causes congestion at most $\textsf{left} (1+\frac 8 {\log m}\textsf{right} )^j$.
\end{proofof}
\end{proof}
\subsubsection{Proof of \Cref{thm: phase 2} -- Main Definitions and Notation}
\label{phase 2 defs}
In this subsection we define the main notions that we use in the proof of \Cref{thm: phase 2}, and also state the main lemmas from which the proof of \Cref{thm: phase 2} follows.
Suppose $G'$ is any subgraph of $\check G'$. Let $\Sigma'$ be the rotation system for $G'$ induced by $\check \Sigma'$, and let $I'=(G',\Sigma')$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. For brevity, we will say that $I'=(G',\Sigma')$ is the \emph{subinstance of $\check I'$ defined by $G'$}. Recal that $\check G'\subseteq \check G$, so $I'$ is also the subinstance of $\check I$ defined by $G'$.
\paragraph{Planar Drawings and Face Boundaries.}
Suppose we are given a planar graph $H$, and a drawing $\phi$ of $H$ on the sphere with no crossings. As before, we denote by ${\mathcal{F}}(\phi)$ the set of all faces of the drawing $\phi$. For a face $F\in {\mathcal{F}}(\phi)$, we denote by $\partial_{\phi}(H,F)$ the subgraph of $H$ containing all vertices and edges whose images are contained in the boundary of the face $F$. We omit the subscript $\phi$ when clear of context. We sometimes say that the vertices and edges of $\partial_{\phi}(H,F)$ serve as the boundary of face $F$ in $\phi$.
\paragraph{Skeleton and Skeleton Structure.}
We now define the central notions that the proof of \Cref{thm: phase 2} uses, namely a skeleton and a skeleton structure, that can be thought of as extending the notions of a core and a core structure.
\begin{definition}[Skeleton]
Let $K$ be a subgraph of $\check G'$. We say that $K$ is a \emph{skeleton graph}, if $\check J'\subseteq K$, and, for every edge $e\in E(K)$, graph $K\setminus\set{e}$ is connected.
\end{definition}
While the definition of the skeleton $K$ is quite general, we will construct the skeleton using a specific procedure. At the beginning of the algorithm, we let $K=\check J'$. In every iteration of the algorithm, we augment $K$ by adding to it a simple path (or a cycle) $P$, whose both endpoints belong to $K$, and all inner vertices are disjoint from $K$.
Next, we define a skeleton structure.
\begin{definition}[Skeleton Structure]
A skeleton structure ${\mathcal K}$ consists of the following three ingredients:
\begin{itemize}
\item a skeleton graph $K$;
\item for every vertex $u\in V(K)$, an orientation $b_u\in \set{-1,1}$, such that, for every vertex $u\in V(\check J')$, its orientation $b_u$ is identical to that given by the core structure $\check{\mathcal{J}}$; and
\item a drawing $\rho_K$ of graph $K$ on the sphere with no crossings, such that $\rho_K$ obeys the rotation system $\check \Sigma'$, the orientation of every vertex $u\in V(K)$ in $\rho_K$ is $b_u$, and drawing $\rho_K$ is clean with respect to $\check{\mathcal{J}}'$.
\end{itemize}
\end{definition}
Consider now some skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$. For brevity of notation, we denote by $\tilde{\mathcal{F}}({\mathcal K})={\mathcal{F}}(\rho_K)$ the set of all faces in the drawing $\rho_K$. Since drawing $\rho_K$ is clean with respect to $\check{\mathcal{J}}'$, the image of every vertex and every edge of $K$ appears in the region $F^*=F^*(\rho_{\check J'})$ of this drawing (including its boundary). Therefore, every forbidden face $F\in {\mathcal{F}}^{\operatorname{X}}(\rho_{\check J'})$ is also a face of $\tilde{\mathcal{F}}({\mathcal K})$. We denote by $\tilde{\mathcal{F}}^{\operatorname{X}}({\mathcal K})={\mathcal{F}}^{\operatorname{X}}(\rho_{\check J'})$ the set of all such faces, that we refer to as \emph{forbidden faces of drawing $\rho_K$}, and we denote by $\tilde{\mathcal{F}}'({\mathcal K})=\tilde{\mathcal{F}}({\mathcal K})\setminus\tilde{\mathcal{F}}^{\operatorname{X}}({\mathcal K})$ the set of all remaining faces.
Consider some face $F\in \tilde{\mathcal{F}}({\mathcal K})$, and let $J_F=\partial_{\rho_K}(K,F)$ be the graph consisting of all vertices and edges of $K$ whose images appear on the boundary of face $F$ in drawing $\rho_K$. From the definition of a skeleton graph $K$, graph $J_F$ is a core. We can then define a core structure ${\mathcal{J}}_F=(J_F,\set{b_u}_{u\in V(J_F)},\rho_{J_F}, F^*(\rho_{J_F}))$ as follows: for every vertex $u\in V(J_F)$, its orientation $b_u$ remains the same as in ${\mathcal K}$. Drawing $\rho_{J_F}$ of graph $J_F$ is the drawing induced by $\rho_K$. Notice that face $F\in \tilde{\mathcal{F}}({\mathcal K})$ remains a face in the drawing $\rho_{J_F}$. Face $F^*(\rho_{J_F})$ is then defined to be the face $F$. (We note that a core structure is generally defined for some specific graph $G'$, for which the properties specified in \Cref{def: valid core 2} must hold. Therefore, for now we view the core structure ${\mathcal{J}}_F$ as simply a tuple $(J_F,\set{b_u}_{u\in V(J_F)},\rho_{J_F}, F^*(\rho_{J_F}))$, and we will later define a subgraph $G_F$ of $\check G'$, for which ${\mathcal{J}}_F$ will be a valid core structure.)
\paragraph{${\mathcal K}$-Valid Drawings.}
Next, we consider a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$ and a subgraph $G'\subseteq G$ with $K\subseteq G'$. We then define ${\mathcal K}$-valid drawings of graph $G'$, in a natural way.
\begin{definition}[${\mathcal K}$-valid drawings]
Let ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$ be a skeleton structure, let $G'$ be a subgraph of $\check G'$ with $K\subseteq G'$, and let $I'=(G',\Sigma')$ be the subinstance of $\check I'$ defined by $G'$. We say that a solution $\phi$ to instance $I'$ is ${\mathcal K}$-valid, if the drawing of the skeleton $K$ induced by $\phi$ is identical to $\rho_K$, and the orientation of every vertex $u\in V(K)$ in drawing $\phi$ is identical to the orientation $b_u$ given by ${\mathcal K}$. We also say that a ${\mathcal K}$-valid solution $\phi$ to instance $I'$ is a \emph{${\mathcal K}$-valid drawing} of graph $G'$.
\end{definition}
Note that, if $\phi$ is a ${\mathcal K}$-valid solution to instance $I'$, for any skeleton structure ${\mathcal K}$, then it is also a $\check{\mathcal{J}}$-valid solution to $I'$.
Since we will be considering various core structures ${\mathcal{J}}_F$ associated with faces $F\in \tilde{\mathcal{F}}({\mathcal K})$ of a given skeleton structure ${\mathcal K}$, the definition of dirty edges and dirty crossings will change depending on which core structure we consider. As the algorithm pogresses, the drawing $\psi$ that we maintain for the current graph $G'=\check G'\setminus E^{\mathsf{del}}(\check I')$ will evolve. We need to keep track of the crossings in which the edges of the original core $\check J$ participate, and of the edges involved in these crossings.
Therefore, given a subgraph $G'\subseteq \check G'$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$, and a ${\mathcal K}$-valid solution to subinstance $I'=(G',\Sigma')$ defined by graph $G'$, we denote by $\chi^*(\phi)$ the set of all crossings $(e,e')_p$ in $\phi$ where $e$ or $e'$ belong to the core $\check J$.
\paragraph{A ${\mathcal K}$-Decomposition of $\check G'$ and Face-Based Instances.}
Over the course of the algorithm, we will maintain a skeleton structure ${\mathcal K}$, and an associated decomposition of graph $\check G'$ into subgraphs. Intuitively, for every face $F\in \tilde{\mathcal{F}}({\mathcal K})$, we will define a subgraph $G_F$ of $\check G'$ that we associate with face $F$. We now define a ${\mathcal K}$-decomposition of $\check G'$.
\begin{definition}[A ${\mathcal K}$-decomposition of $\check G'$]
Let ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$ be a skeleton structure, and, for every face $F\in \tilde {\mathcal{F}}({\mathcal K})$, let ${\mathcal{J}}_F=(J_F,\set{b_u}_{u\in V(J_F)},\rho_{J_F}, F^*(\rho_{J_F}))$ be the core structure that ${\mathcal K}$ defines for face $F$. A ${\mathcal K}$-decomposition of the input graph $\check G'$ is a collection ${\mathcal{G}}=\set{G_{ F}\mid F\in \tilde {\mathcal{F}}({\mathcal K})}$ of subgraphs of $\check G'$, for which the following hold:
\begin{itemize}
\item for every face $ F\in \tilde {\mathcal{F}}({\mathcal K})$, the core structure ${\mathcal{J}}_F$ associated with face $F$ is a valid core structure for instance $I_F=(G_F,\Sigma_F)$ defined by graph $G_F$;
\item for every forbidden face $F\in \tilde {\mathcal{F}}^{\operatorname{X}}({\mathcal K})$, $G_F=J_F$;
\item for every face $F\in \tilde {\mathcal{F}}({\mathcal K})$, $G_F\cap K=J_F$; and
\item for every pair $F, F'\in \tilde{\mathcal{F}}({\mathcal K})$ of distinct faces, $V(G_{F})\cap V(G_{F'})\subseteq V(J_F)\cap V(J_{F'})$, and $E(G_{F})\cap E(G_{F'})\subseteq E(J_F)\cap E(J_{F'})$.
\end{itemize}
We will sometimes refer to the subinstance $I_F=(G_F,\Sigma_F)$ of $\check I'$ defined by graph $G_F$ associated with a face $F\in \tilde {\mathcal{F}}({\mathcal K})$ as a \emph{face-based subinstance} associated with face $F$.
\end{definition}
In order to prove \Cref{thm: phase 2}, we will gradually construct a skeleton $K$ and its associated skeleton structure ${\mathcal K}$, starting with $K=\check J$. We will also maintain a ${\mathcal K}$-decomposition ${\mathcal{G}}$ of the graph $\check G'$. We will denote by $E^{\mathsf{del}}=E(\check G')\setminus\textsf{left}(\bigcup_{ F\in \tilde {\mathcal{F}}({\mathcal K})}E(G_{ F})\textsf{right} )$, and we will view $E^{\mathsf{del}}$ as the set of \emph{deleted edges}, that will eventually be added to set $E^{\mathsf{del}}(\check I')$. We will ensure that $|E^{\mathsf{del}}|$ will remain small over the course of the algorithm. Consider the graph $G'=\check G'\setminus E^{\mathsf{del}}$ and its associated subinstance $I'=(G',\Sigma')$ of $\check I'$. We will ensure that throughout the algorithm, there is a solution $\phi$ to instance $I'$, that is compatible with the solution $\psi(\check I')$ to instance $\check I'$ (with respect to the core structure $\check{\mathcal{J}}'$), with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\psi(\check I'))$ and $|\chi^{*}(\phi)|\leq |\chi^{*}(\psi(\check I'))|$. The algorithm terminates once every instance $G_{ F}$ in the resulting decomposition ${\mathcal{G}}$ becomes ``acceptable'' -- a notion that we define next.
Given a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$, a ${\mathcal K}$-decomposition ${\mathcal{G}}=\set{G_F\mid F\in \tilde {\mathcal{F}}({\mathcal K})}$ of graph $\check G'$, and a face $F\in \tilde {\mathcal{F}}({\mathcal K})$, we denote by $\tilde E_F$ the set of all edges $e\in E(G_F)$, such that exactly one endpoint of $e$ lies in $J_F$; in other words, $\tilde E_F=\delta_{G_F}(J_F)$ (recall that, since $G_F$ is a subgraph of the subdivided graph $\check G$, no edge of $E(G_F)\setminus E(J_F)$ may have both its endpoints in the core $J_F$). Recall that, when we defined a core structure, we have also defined an ordering ${\mathcal{O}}(J_F)$ of the edges of $\tilde E_F$. Intuitively, this is the order in which the edges of $\tilde E_F$ are encountered in any ${\mathcal{J}}_F$-valid solution to instance $I_F$, as we follow along the boudary of face $F^*(\rho_{J_F})$ inside the face, in the counter-clock-wise direction.
\begin{definition}[Acceptable Instances]\label{def: acceptable instance}
Let ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$ be skeleton structure, let ${\mathcal{G}}=\set{G_F\mid F\in \tilde {\mathcal{F}}({\mathcal K})}$ be a ${\mathcal K}$-decomposition of graph $\check G'$, and let $F\in \tilde {\mathcal{F}}({\mathcal K})$ be a face. Consider a graph $H_F$, that is obtained from graph $G_{F}$ by first subdividing every edge $e\in \tilde E_F$ with a vertex $t_e$, and then deleting all vertices and edges of $J_{F}$ from it. We say that instance $G_{F}$ is \emph{acceptable} if, for every partition $(E_1,E_2)$ of the edges of $\tilde E_F$, such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}(J_F)$, there is a cut $(X,Y)$ in graph $H_{F}$ with vertex set $\set{t_e\mid e\in E_1}$ contained in $X$, vertex set $\set{t_e\mid e\in E_2}$ contained in $Y$, and $|E_{H_{F}}(X,Y)|\leq \check m'/\mu^{2b}$.
\end{definition}
The main ingredients in the proof of \Cref{thm: phase 2} are the following two lemmas.
\begin{lemma}\label{lem: compute phase 2 decomposition}
There is an efficient randomized algorithm, whose input consists of a large enough constant $b\geq 1$, a subinstance $\check I'=(\check G',\check \Sigma')$ of $\check I$ with $|E(\check G')|=\check m'$ and $\check G'\subseteq \check G$, and a core structure $\check{\mathcal{J}}$ for $\check I'$, whose corresponding core is denoted by $\check J$.
The algorithm either returns FAIL, or computes a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$, and a ${\mathcal K}$-decomposition ${\mathcal{G}}$ of $\check G'$, such that, for every face $F\in \tilde {\mathcal{F}}({\mathcal K})$, the corresponding subinstance $I_F=(G_F,\Sigma_F)$ of $\check I'$ defined by the graph $G_F\in {\mathcal{G}}$ is acceptable.
Moreover, if there is a solution $\psi(\check I')$ to instance $\check I'$ that is $\check{\mathcal{J}}$-valid, with $\mathsf{cr}(\psi(\check I'))\leq (\check m')^2/\mu^{240b}$, and $|\chi^{*}(\psi(\check I'))|\leq \check m'/\mu^{240b}$, then with probability at least $1-1/\mu^{2b}$, the following hold:
\begin{itemize}
\item the algorithm does not return FAIL;
\item the cardinality of edge set $E^{\mathsf{del}}(\check I')=E(\check G')\setminus \textsf{left}(\bigcup_{F\in \tilde {\mathcal{F}}({\mathcal K})}E(G_F)\textsf{right} )$ is bounded by: $$\textsf{left} (\frac{\mathsf{cr}(\psi(\check I'))}{\check m'}+|\chi^*(\psi(\check I'))|\textsf{right} ) \cdot \mu^{O(b)};\mbox{ and }$$
\item if we denote $G'=\check G'\setminus E^{\mathsf{del}}(\check I')$, and let $I'=(G',\Sigma')$ be the subinstance of $\check I'$ defined by graph $G'$, then there is a ${\mathcal K}$-valid solution $\phi$ to instance $I'$, with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\psi(\check I'))$, $|\chi^*(\phi)|\leq |\chi^*(\psi(\check I'))|$, and the total number of crossings in which the edges of $E(K)\setminus E(\check J)$ participate is at most $\frac{\mathsf{cr}(\psi(\check I'))\cdot \mu^{50b}}{\check m'}$.
\end{itemize}
\end{lemma}
\begin{lemma}\label{lem: computed decomposition is good enough}
Suppose we are given a subinstance $\check I'=(\check G',\check \Sigma')$ of $\check I$ with $\check G'\subseteq \check G$ and $|E(\check G')|=\check m'$, a core structure $\check{\mathcal{J}}$ for $\check I'$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$, and a ${\mathcal K}$-decomposition ${\mathcal{G}}$ of $\check G'$, such that, for every face $F\in \tilde {\mathcal{F}}({\mathcal K})$, the corresponding subinstance $I_F=(G_F,\Sigma_F)$ of $\check I'$ defined by the graph $G_F\in {\mathcal{G}}$ is acceptable. Let $E^{\mathsf{del}}(\check I')=E(\check G')\setminus \textsf{left}(\bigcup_{F\in \tilde {\mathcal{F}}({\mathcal K})}E(G_F)\textsf{right} )$, $G'=\check G'\setminus E^{\mathsf{del}}(\check I')$, and let $I'=(G',\Sigma')$ be the subinstance of $\check I'$ defined by graph $G'$. Assume further that there is a ${\mathcal K}$-valid solution $\phi$ to instance $I'$, so that the total number of crossings in which the edges of $K$ participate is $N\leq \frac{\check m'}{\mu^{3b}}$ and $|\chi^{*}(\phi)|\leq \check m'/\mu^{240b}$. Then
there is a solution $\psi(I')$ to instance $I'$ that is clean with respect to $\check{\mathcal{J}}$, with $\mathsf{cr}(\psi(I'))\leq \textsf{left} (\mathsf{cr}(\phi)+N^2+|\chi^{*}(\phi)|^2+\frac{|\chi^{*}(\phi)|\cdot \check m'}{\mu^b}\textsf{right} )\cdot (\log \check m')^{O(1)}$.
\end{lemma}
We provide the proofs of \Cref{lem: compute phase 2 decomposition} and \Cref{lem: computed decomposition is good enough} in Sections \ref{subsec: compute phase 2 decomposition} and \ref{subsec: computed decomposition is good enough}, respectively. The proof of \Cref{thm: phase 2} easily follows from the above two lemmas. Indeed, consider the input instance
$\check I'=(\check G',\check \Sigma')$ of \ensuremath{\mathsf{MCNwRS}}\xspace that is a subinstance of $\check I$, with $\check G'\subseteq \check G$ and $|E(\check G')|=\check m'$, and a core structure $\check{\mathcal{J}}$ for $\check I'$, whose corresponding core is denoted by $\check J$. We apply the algorithm from \Cref{lem: compute phase 2 decomposition} to this input. If the algorithm returns FAIL, then we set $E^{\mathsf{del}}(\check I')=E(\check G')\setminus E(\check J)$, and return this edge set as the output. We say that the algorithm from \Cref{lem: compute phase 2 decomposition} \emph{fails} in this case. Otherwise, the algorithm from \Cref{lem: compute phase 2 decomposition} computes a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$, and a ${\mathcal K}$-decomposition ${\mathcal{G}}$ of $\check G'$, such that, for every face $F\in \tilde {\mathcal{F}}({\mathcal K})$, the corresponding subinstance $I_F=(G_F,\Sigma_F)$ of $\check I'$ defined by the graph $G_F\in {\mathcal{G}}$ is acceptable.
Let $E^{\mathsf{del}}(\check I')=E(\check G')\setminus \textsf{left}(\bigcup_{F\in \tilde {\mathcal{F}}({\mathcal K})}E(G_F)\textsf{right} )$, let $G'=\check G'\setminus E^{\mathsf{del}}(\check I')$, and let $I'=(G',\Sigma')$ be the subinstance of $\check I'$ defined by graph $G'$.
We say that the algorithm from \Cref{lem: compute phase 2 decomposition} is \emph{successful} if (i) it does not fail; (ii) $|E^{\mathsf{del}}(\check I')|\leq \textsf{left} (\frac{\mathsf{cr}(\psi(\check I'))}{\check m'}+|\chi^{*}(\psi(\check I'))|\textsf{right} ) \cdot \mu^{O(b)}$; and (iii) there is a ${\mathcal K}$-valid solution $\phi$ to instance $I'$, with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\psi(\check I'))$, $|\chi^*(\phi)|\leq |\chi^*(\psi(\check I'))|$, and the total number of crossings in which the edges of $E(K)\setminus E(\check J)$ participate is at most $\frac{\mathsf{cr}(\psi(\check I'))\cdot \mu^{50b}}{\check m'}$. If the algorithm is not successful, then we say that it is \emph{unsuccessful}. \Cref{thm: phase 2} guarantees that, if there is a solution $\psi(\check I')$ to instance $\check I'$ that is $\check{\mathcal{J}}$-valid, with $\mathsf{cr}(\psi(\check I'))\leq (\check m')^2/\mu^{240b}$, and $|\chi^{*}(\psi(\check I'))|\leq \check m'/\mu^{240b}$, then the algorithm from \Cref{lem: compute phase 2 decomposition} is successful with probability at least $1-1/\mu^{2b}$.
If the algorithm from \Cref{lem: compute phase 2 decomposition} does not fail, then we return edge set $E^{\mathsf{del}}(\check I')$ as the outcome of the algorithm.
In order to complete the proof of \Cref{thm: phase 2}, it is enough to show that, if there is a solution $\psi(\check I')$ to instance $\check I'$ that is $\check{\mathcal{J}}$-valid, with $\mathsf{cr}(\psi(\check I'))\leq (\check m')^2/\mu^{240b}$ and $|\chi^{*}(\psi(\check I'))|\leq \check m'/\mu^{240b}$, and the algorithm from \Cref{lem: compute phase 2 decomposition} is successful, then there is a solution $\psi(I')$ to instance $I'$ that is clean with respect to ${\mathcal{J}}(I)$, with $\mathsf{cr}(\psi(I'))\leq \textsf{left} (\mathsf{cr}(\psi(\check I'))+|\chi^{\mathsf{dirty}}(\psi(\check I'))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(\check I'))|\cdot \check m'}{\mu^b}\textsf{right} )\cdot (\log \check m')^{O(1)}$.
Assume that there is a solution $\psi(\check I')$ to instance $\check I'$ that is $\check{\mathcal{J}}$-valid, with $\mathsf{cr}(\psi(\check I'))\leq (\check m')^2/\mu^{240b}$ and $|\chi^{*}(\psi(\check I'))|\leq \check m'/\mu^{240b}$, and that the algorithm from \Cref{lem: compute phase 2 decomposition} is successful. Then
there is a ${\mathcal K}$-valid solution $\phi$ to instance $I'$, with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\psi(\check I'))$, $|\chi^*(\phi)|\leq |\chi^*(\psi(\check I'))|\leq \check m'/\mu^{240b}$, and the total number of crossings in which the edges of $K$ participate is at most $N=\frac{\mathsf{cr}(\psi(\check I'))\cdot \mu^{50b}}{\check m'}+|\chi^{*}(\psi(\check I'))|\leq \frac{\check m'}{\mu^{3b}}$ (since $\mathsf{cr}(\psi(\check I'))\leq \frac{(\check m')^2}{\mu^{240b}}$ and $|\chi^{*}(\psi(\check I'))|\leq \frac{\check m'}{\mu^{240b}}$). From \Cref{lem: computed decomposition is good enough},
there is a solution $\psi(I')$ to instance $I'$ that is clean with respect to $\check{\mathcal{J}}$, with $\mathsf{cr}(\psi(I'))\leq \textsf{left} (\mathsf{cr}(\phi)+N^2+|\chi^{*}(\phi)|^2+\frac{|\chi^{*}(\phi)|\cdot \check m'}{\mu^b}\textsf{right} )\cdot (\log \check m')^{O(1)}$.
Note that $\mathsf{cr}(\phi)\leq \mathsf{cr}(\psi(\check I'))$ and $|\chi^*(\phi)|\leq |\chi^*(\psi(\check I'))|=|\chi^{\mathsf{dirty}}(\psi(\check I'))|$.
Additionally,
$N\leq \frac{\mathsf{cr}(\psi(\check I'))\cdot \mu^{50b}}{\check m'}+|\chi^{\mathsf{dirty}}(\psi(\check I'))|$, so
$N^2\leq \frac{2(\mathsf{cr}(\psi(\check I')))^2\cdot \mu^{100b}}{(\check m')^2}+2|\chi^{\mathsf{dirty}}(\psi(\check I'))|^2\leq \mathsf{cr}(\psi(\check I'))$, since we have assumed that $\mathsf{cr}(\psi(\check I'))\leq (\check m')^2/\mu^{240b}$. Therefore, we get that:
$$\mathsf{cr}(\psi(I'))\leq \textsf{left} (\mathsf{cr}(\psi(\check I'))+|\chi^{\mathsf{dirty}}(\psi(\check I'))|^2+\frac{|\chi^{\mathsf{dirty}}(\psi(\check I'))|\cdot \check m'}{\mu^b}\textsf{right} )\cdot (\log \check m')^{O(1)}$$ as required.
In order to complete the proof of \Cref{thm: phase 2}, it is now enough to prove
\Cref{lem: compute phase 2 decomposition} and \Cref{lem: computed decomposition is good enough}, which we do next.
\subsubsection{Proof of \Cref{lem: computed decomposition is good enough}}
\label{subsec: computed decomposition is good enough}
We assume that we are given
a subinstance $\check I'=(\check G',\check \Sigma')$ of $\check I$ with $|E(\check G')|=\check m'$, a core structure $\check{\mathcal{J}}$ for $\check I'$, whose corresponding core graph is denoted by $\check J$, a skeleton structure ${\mathcal K}=(K,\set{b_u}_{u\in V(K)},\rho_K)$, and a ${\mathcal K}$-decomposition ${\mathcal{G}}$ of $\check G'$, such that, for every face $F\in \tilde {\mathcal{F}}({\mathcal K})$, the corresponding subinstance $I_F=(G_F,\Sigma_F)$ of $\check I'$ defined by the graph $G_F\in {\mathcal{G}}$ is acceptable. Let $E^{\mathsf{del}}(\check I')=E(\check G')\setminus \textsf{left}(\bigcup_{F\in \tilde {\mathcal{F}}({\mathcal K})}E(G_F)\textsf{right} )$, $G=\check G'\setminus E^{\mathsf{del}}(\check I')$, and let $I=(G,\Sigma)$ be the subinstance of $\check I'$ defined by graph $G$. We also assume that there is a ${\mathcal K}$-valid solution $\phi$ to instance $I$, so that the total number of crossings in which the edges of $K$ participate is at most $N\leq \frac{\check m'}{\mu^{3b}}$. Note that $\check{\mathcal{J}}$ remains a valid core structure, and ${\mathcal K}$ remains a valid skeleton structure for instance $I$. From this point onward we will only work with instance $I$, and we will not need the initial subinstance $\check I'$ of instance $\check I$, or the set $E^{\mathsf{del}}(\check I')$ of edges, but we will use the parameter $\check m'=|E(\check G')|$.
Our goal is to prove that
there is a solution $\psi$ to instance $I$ that is clean with respect to $\check{\mathcal{J}}$, with $\mathsf{cr}(\psi)\leq \textsf{left} (\mathsf{cr}(\phi)+N^2+|\chi^{*}(\phi)|^2+\frac{|\chi^{*}(\phi)|\cdot \check m'}{\mu^b}\textsf{right} )\cdot (\log \check m')^{O(1)}$.
Since the statement of the lemma is existential, it is sufficient to show an algorithm that transforms the solution $\phi$ to instance $I$ into another solution $\psi$ that is clean with respect to $\check{\mathcal{J}}$, with the number of crossings bounded as above.
In order to do so, we need to ``repair'' the drawing, so that for every edge $e\in E(G)$, the image of $e$ is disjoint from the interior of the forbidden faces in $\tilde{\mathcal{F}}^{\operatorname{X}}({\mathcal K})$. We do so in two steps. In the first step, we modify the drawing $\phi$ so that, for every non-forbidden face $F\in\tilde{\mathcal{F}}({\mathcal K})\setminus \tilde{\mathcal{F}}^{\operatorname{X}}({\mathcal K})$, the images of all vertices of graph $G_F\in {\mathcal{G}}$ associated with face $F$ lie in the region $F$ of the drawing. We say that an edge $e\in E(G)\setminus E(K)$ is \emph{bad} if the image of $e$ in the resulting drawing intersects the interior of at least one forbidden face in $\tilde{\mathcal{F}}^{\operatorname{X}}({\mathcal K})$. Notice that for each such bad edge $e$, there must be some face $F\in \tilde{\mathcal{F}}({\mathcal K})\setminus \tilde{\mathcal{F}}^{\operatorname{X}}({\mathcal K})$ with $e\in G_F$, so the images of the endpoints of $e$ lie in region $F$ of the current drawing. In the second step, we further modify the drawing to obtain a ${\mathcal K}$-clean solution to instance $I$. In order to do so, for every bad edge $e$, if $e\in E(G_F)$ for some face $F\in \tilde{\mathcal{F}}({\mathcal K})\setminus \tilde{\mathcal{F}}^{\operatorname{X}}({\mathcal K})$, we ``move'' the image of $e$ so it is drawn completely inside face $F$. In order to ensure that the new image of $e$ crosses few edges, we may need to rearrange the current drawing inside the face $F$. This step exploits the fact that instance $I_F$ corresponding to face $F$ is acceptable.
We now proceed to describe each of the steps in turn.
For convenience, we denote $\tilde {\mathcal{F}}'=\tilde {\mathcal{F}}({\mathcal K})\setminus \tilde {\mathcal{F}}^{\operatorname{X}}({\mathcal K})$.
\input{good-drawing-step1}
\input{good-drawing-step2}
\input{drawing-in-disc}
\subsubsection{Proof of \Cref{thm: phase 2} -- Intuition}
\label{phase 2 intuition}
For simplicity of exposition, assume that the core $\check J$ corresponding to the core structure $\check{\mathcal{J}}$ is a simple cycle.
Generally it is not difficult to modify the solution $\check \psi$ to instance $\check I'$ so that it becomes semi-clean, while only increasing the number of crossings by at most $|\chi^{\mathsf{dirty}}(\check \psi)|^2$. In order to do so, we let $E^{\mathsf{dirty}}$ be the set of all dirty edges -- that is, edges whose image in $\check \psi$ crosses the image of some edge of $\check J$. Let ${\mathcal{C}}$ be the set of all connected components of $\check G'\setminus E^{\mathsf{dirty}}$. It is easy to verify that for each component $C\in {\mathcal{C}}$, either the images of all vertices and edges of $C$ in $\check \psi$ lie in the region $F^*$; or the images of all vertices and edges of $C$ in $\check \psi$ are disjoint from $F^*$. In the latter case, we move the image of $C$ to lie in the interior of the face $F^*$, without changing the image itself. We then need to modify the images of the edges in set $E^{\mathsf{dirty}}$, so that they connect the new images of their endpoints. This can be easily done while introducing at most $|\chi^{\mathsf{dirty}}(\check \psi)|^2$ new crossings. We do not provide the details here, since we do not use this algorithm eventually.
Let $\check \psi'$ denote this semi-clean solution to instance $\check I'$ with respect to $\check{\mathcal{J}}$. Denote by $\gamma$ the image of the cycle $\check J$ in $\check \psi$, which must be a simple closed curve. For convenience, we will now denote by $E^{\mathsf{dirty}}$ the set of all dirty edges of $\check G'$ -- edges whose image in $\check \psi'$ crosses the image of some edge of $\check J$.
Consider now some dirty edge $e\in E^{\mathsf{dirty}}$. For simplicity of exposition, assume that $e$ is not incident to any vertex of $\check J$.
Since $\check \psi'$ is a semi-clean drawing of $\check G'$ with respect to $\check{\mathcal{J}}$, the images of the endpoints of $e$ must lie in region $F^*$. Therefore, there must be at least two points on $\check \psi'(e)$ that lie on $\gamma$. We assign the curve $\check \psi(e)$ an arbitrary direction, denote by $p$ the first point on $\check \psi'(e)$ that lies on $\gamma$, and by $p'$ the last point on $\check \psi'(e)$ that lies on $\gamma$. Points $p$ and $p'$ partition the curve $\gamma$ into two disjoint simple open curves, that we denote by $\gamma'$ and $\gamma''$, respectively. A simple way to ``repair'' the drawing of the edge $e$ so that it no longer crosses the edges of $\check J$ would be to replace the segment of $\check \psi(e)$ between $p$ and $p'$ with a new segment $\sigma(e)$, that follows the curve $\gamma'$ closely, in the interior of region $F^*$ (see \Cref{fig: fixdrawing}).
A problem with this approach is that this may greatly increase the number of crossings, as the segment $\sigma(e)$ may cross many edges in drawing $\check \psi'$. Intuitively, the requirements of \Cref{thm: phase 2} allow us to add up to $\frac{\check m'(\log \check m')^{O(1)}}{\mu^b}$ new crossings to the drawing $\check \psi$ for each dirty edge whose image we modify, but unfortunately it is possible that, after the modification, $\sigma(e)$ crosses the images of many more edges.
\begin{figure}[h]
\centering
\subfigure[Before: the image of $e$ (green) and its intersections $p,p'$ (red) with the image of $\check J$ (blue). Region $F^*$ is shown in gray.]{\scalebox{0.16}{\includegraphics{figs/fixdrawing_1.jpg}}\label{fig: fixdrawing_1}}
\hspace{0.5cm}
\subfigure[After: the new image of $e$ is shown in green
]{\scalebox{0.16}{\includegraphics{figs/fixdrawing_2}}\label{fig: fixdrawing_2}}
\caption{Repairing the image of an edge $e\in E^{\mathsf{dirty}}$.
}\label{fig: fixdrawing}
\end{figure}
Let $S$ denote the set of all vertices of $\check J$ whose images lie on $\gamma'$, and let $T$ be defined similary for $\gamma''$. Consider the minimun cut $(X,Y)$ separating vertices of $S$ from vertices of $T$ in graph $\check G'\setminus E^{\mathsf{dirty}}$, so $S\subseteq X$ and $T\subseteq Y$. Assume first that $|E_{\check G'}(X,Y)|<\check m'/\mu^b$. In this case, we can rearrange the drawing $\check \psi'$, so that all vertices and edges of $\check G[X]\setminus E^{\mathsf{dirty}}$ are drawn very close to the segment $\gamma'$ (but in the interior of region $F^*$), and similarly all vertices and edges of $\check G[Y]\setminus E^{\mathsf{dirty}}$ are drawn very close to the segment $\gamma''$. We can then define a curve $\sigma(e)$ connecting points $p$ and $p'$, so that $\sigma(e)$ is contained in $F^*$, and it only crosses the images of the edges in $E_{\check G'}(X,Y)$. Therefore, we can ensure that $\sigma(e)$ participates in few crossings. We can then modify the image of edge $e$ to follow the segment $\sigma(e)$ as before, without increasing the number of crossings by too much.
Note that each dirty edge $e\in E^{\mathsf{dirty}}$ may define a different partition $(S,T)$ of the vertices of $\check J$, and a different cut $(X,Y)$. However, if we can ensure that the number of edges crossing each such cut is sufficiently low, then we can still rearrange the drawing $\check \psi'$, and modify the drawings of all edges in $E^{\mathsf{dirty}}$, so that they become contained in region $F^*$, while ensuring that the total number of crossings only increases moderately.
It is however possible that, for some edge $e\in E^{\mathsf{dirty}}$, and its corresponding partition $(S,T)$ of $V(\check J)$, the minimum cut separating $S$ from $T$ in $\check G'$ contains more than $\check m'/\mu^b$ edges. In this case, there must be at least $\ceil{\check m'/\mu^b}$ edge-disjoint paths in $\check G'$ connecting vertices of $S$ to vertices of $T$. We can treat this set of paths as a promising set of paths, that can be used in order to define an enhancement $P$ of the core structure $\check{\mathcal{J}}$, using Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. We can also use the procedure in order to compute an enhancement structure ${\mathcal{A}}$, and a split $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ of instance $\check I'$ along ${\mathcal{A}}$. Unlike the algorithm from Phase 1, we will not view the resulting two instances $I_1,I_2$ as separate instances. Instead, we will initialize the set $E^{\mathsf{del}}(\check I')$ of deleted edges to edge set $E(\check G')\setminus (E(G_1)\cup E(G_2))$. We then consider the graph $K=\check J\cup P$, that we call a \emph{skeleton }, and fix a planar drawing $\rho_{K}$ of it (which is uniquely defined).
From Property \ref{prop output drawing} of valid output to Procedure \ensuremath{\mathsf{ProcSplit}}\xspace, there is a drawing $\psi$ of graph $\check G'\setminus E^{\mathsf{del}}(\check I')$, that obeys the rotation system $\check \Sigma'$, such that drawing $\psi$ is $\check{\mathcal{J}}$-valid, and the edges of $\check J\cup P$ do not cross each other in $\psi$, with $\mathsf{cr}(\psi)\leq \mathsf{cr}(\check \psi')$. If we consider the drawing $\rho_{\check J}$ of the core ${\check J}$, then the image of path $P$ partitions face $F^*$ into two new faces, that we denote by $F_1$ and $F_2$. Consider the split $({\mathcal{J}}_1,{\mathcal{J}}_2)$ of the core structure $\check{\mathcal{J}}$ along ${\mathcal{A}}$. The cores $J_1,J_2$ associated with the core structures ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$, respectively, serve as the boundaries of the faces $F_1$, $F_2$, respectively, in the drawing $\rho_K$ of graph $K$.
We note that the drawing of instance $I_1$ induced by $\check \psi$ is not necessarily semi-clean with respect to ${\mathcal{J}}_1$, and the same is true regarding instance $I_2$ and core structure ${\mathcal{J}}_2$. But we could modify $\check \psi$ to ensure this property, obtaining a new drawing $\psi'$ of graph $\check G'\setminus E^{\mathsf{del}}(\check I')$ (though this process would increase the number of crossings by factor $\operatorname{poly}\log \check m'$; we ignore this technicality for now). If we now consider some edge $e$, whose image in $\psi'$ crosses the image of some edge in ${\check J}$, then edge $e$ must either lie in graph $G_1$, or in graph $G_2$. Assume w.l.o.g. that it is the former.
In the modified drawing $\psi'$ of $\check G'\setminus E^{\mathsf{del}}(\check I')$, both endpoints of edge $e$ are drawn inside the region $F_1$. The image of $e$ must then cross the boundary of region $F_1$ in at least two points (recall that the boundary of region $F_1$ in $\psi'$ is the image of the core $J_1$). We can again use these two points to define a partition $(S,T)$ of the vertices of $J_1$, and compute a minimum cut $(X,Y)$ separating $S$ from $T$ in $G_1$. As before, if the value of this minimum cut is small, then we can modify the current drawing $\psi'$ locally inside region $F_1$ and modify the image of the edge $e$, so that it is contained in $F_1$, and no longer crosses the edges of ${\check J}$. If the value of this minimum cut is large, then we can again define a promising set of paths for instance $I_1$ and core structure ${\mathcal{J}}_1$, and then invoke Procedure \ensuremath{\mathsf{ProcSplit}}\xspace in order to further split core structure ${\mathcal{J}}_1$ and instance $I_1$, thereby adding new edges to set $E^{\mathsf{del}}(\check I')$.
At a high level, our algorithm can be thought of as maintaining a single \emph{skeleton} graph $K$ -- a planar subgraph of $\check G'$ with ${\check J}\subseteq K$, such that, for every edge $e\in E(K)$, graph $K\setminus\set{e}$ is connected.
We also maintain a \emph{skeleton structure} ${\mathcal K}$, that, in addition to the skeleton $K$, specifies the orientation $b_u$ of every vertex $u\in V(K)$, and a planar drawing $\rho_K$ of graph $K$ on the sphere. We require that, for every vertex $u\in V({\check J})$, its orientations in ${\mathcal K}$ and $\check{\mathcal{J}}$ are identical, and that drawing $\rho_K$ of $K$ is clean with respect to core structure $\check{\mathcal{J}}$, and is consistent with rotation system $\Sigma$ and orientations $b_u$ of the vertices $u\in V(K)$. Let ${\mathcal{F}}(\rho_K)$ be the set of all faces of the drawing $\rho_K$. Since drawing $\rho_K$ is clean with respect to $\check{\mathcal{J}}$, every forbidden face $F\in {\mathcal{F}}^{\operatorname{X}}(\rho_{\check J})$ is also a face of ${\mathcal{F}}(\rho_K)$. We denote by ${\mathcal{F}}^{\operatorname{X}}(\rho_K)={\mathcal{F}}^{\operatorname{X}}(\rho_{\check J})$ the set of all such faces of ${\mathcal{F}}(\rho_K)$, that we refer to as \emph{forbidden faces} of drawing $\rho_K$.
For every face $F\in {\mathcal{F}}(\rho_K)$, the set of vertices and edges of $K$ lying on its boundary define a core $J_F$. Using the skeleton structure ${\mathcal K}$, we can define a core structure ${\mathcal{J}}_F$ associated with the core $J_F$.
We also maintain, for every face $F\in {\mathcal{F}}(\rho_K)$, a subgraph $G_F$ of $\check G'$. We let $\Sigma_F$ be the rotation system for $G_F$ induced by $\Sigma$, and we let ${\mathcal{I}}_F=(G_F,\Sigma_F)$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. We require that ${\mathcal{J}}_F$ is a valid core structure for instance $I_F$, and, if face $F$ is a forbidden face, then $G_F=J_F$ holds. We will ensure that, for every pair $F,F'\in {\mathcal{F}}(\rho_K)$ of distinct faces, every vertex and every edge of $G_F\cap G_{F'}$ belong to $J_F\cap J_{F'}$, and that $E^{\mathsf{del}}(\check I')=E(\check G')\setminus \textsf{left}(\bigcup_{F\in {\mathcal{F}}(\rho_K)}E(G_F)\textsf{right} )$.
Consider now some face $F\in {\mathcal{F}}(\rho_K)\setminus {\mathcal{F}}^{{\operatorname{X}}}(\rho_K)$. Intuitively (though somewhat imprecisely), we say that the corresponding instance $I_F$ is \emph{acceptable} if, for every parition $(S,T)$ of the vertices of the core $J_F$, where the vertices of $S$ appear consecutively on the cycle $J_F$, there is a small cut in $G_F$ separating $S$ from $T$ (the definition is slightly more involved when $I_F$ is not a simple cycle). If instance $I_F$ is unacceptable, then we will use Procedure \ensuremath{\mathsf{ProcSplit}}\xspace in order to further augment the skeleton and partition instance $I_F$ into two subinstances. Once we reach a state where, for every face $F\in {\mathcal{F}}(\rho_K)$ of the current skeleton $K$, the corresponding instance $I_F$ is acceptable, we terminate the algorithm. We show that the resulting drawing $\psi$ of graph $\check G'\setminus E^{\mathsf{del}}(\check I')$ can be modified so that all crossings with the edges of the original core $\check J$ are eliminated, and the number of crossings only increases moderately.
\subsubsection{Proof of \Cref{lem: compute phase 2 decomposition}}
\label{subsec: compute phase 2 decomposition}
The proof of \Cref{lem: compute phase 2 decomposition} is somewhat similar to the proof of \Cref{thm: phase 1}, in that we repeatedly apply Procedure \ensuremath{\mathsf{ProcSplit}}\xspace to obtain the desired decomposition, though the details are different.
For simplicity of notation, we say that a solution $\psi$ to instance $\check I'$ is \emph{good}, if it is $\check{\mathcal{J}}$-valid, with $\mathsf{cr}(\psi)\leq (\check m')^2/\mu^{240b}$, and $|\chi^{*}(\psi)|\leq \check m'/\mu^{240b}$. If such a solution exists, then we denote it by $\psi$ throughout the algorithm (though we note that the solution is not known to the algorithm). If such a solution does not exist, then we let $\psi$ be any $\check{\mathcal{J}}$-valid solution to instance $\check I'$.
Given a skeleton structure ${\mathcal K}$, a ${\mathcal K}$-decomposition ${\mathcal{G}}$ of instance $\check I'$, and some face $F\in \tilde {\mathcal{F}}({\mathcal K})$, we denote by $m_F=|E(G_F)|$ and by $\hat m_F=|E(G_F)\setminus E(K)|$, where $G_F\in {\mathcal{G}}$ is the graph associated with face $F$.
Our algorithm consists of at most $\ceil{32\mu^{6b}}$ stages. For $1\leq i\leq \ceil{32\mu^{6b}}$, the input to stage $i$ is a skeleton structure ${\mathcal K}_i$, whose corresponding skeleton is denoted by $K_i$, and a ${\mathcal K}_i$-decomposition ${\mathcal{G}}_i$ of graph $\check G'$. We will ensure that, with high enough probability, the following properties hold.
\begin{properties}{A'}
\item for every face $F\in \tilde {\mathcal{F}}({\mathcal K}_i)$, either $m_F\leq \check m'\cdot \textsf{left} (1-\frac{i-1}{32\mu^{6b}}\textsf{right} )$, or the corresponding instance $I_F=(G_F,\Sigma_F)$ with $G_F\in {\mathcal{G}}_i$ is acceptable; \label{invariant: fewer edges2}
\item if we denote by $E^{\mathsf{del}}_i=E(\check G')\setminus\textsf{left}(\bigcup_{G_F\in {\mathcal{G}}_i}E(G_F)\textsf{right} )$, then $|E^{\mathsf{del}}_i|\leq (i-1)\cdot \textsf{left}( \frac{\mathsf{cr}(\psi)\mu^{100b}}{\check m'}+|\chi^{*}(\psi)|\cdot \mu^{3b}\textsf{right}) $; and \label{invariant: few deleted edges}
\item let $G_i=\check G'\setminus E^{\mathsf{del}}_i$, and let $\Sigma_i$ be the rotation system for $G_i$ induced by $\check \Sigma'$. Then there exists a solution $\phi_i$ to instance $(G_i,\Sigma_i)$, that is ${\mathcal K}_i$-valid, and has the following additional properties:\label{invariant: drawing}
\begin{itemize}
\item $\mathsf{cr}(\phi_i)\leq \mathsf{cr}(\psi)$;
\item $|\chi^*(\phi_i)|\leq |\chi^{*}(\psi)|$; and
\item if we denote by $N(\phi_i)$ the total number of crossings of $\phi_i$ in which the edges of $E(K_i)\setminus E(\check J)$ participate, then $N(\phi_i)\leq \frac{\mathsf{cr}(\psi)\cdot i\cdot \mu^{40b}}{\check m'}$.
\end{itemize}
\end{properties}
We say that the execution of stage $i$ is \emph{successful} if the output of the stage satisfies the invariants \ref{invariant: fewer edges2}--\ref{invariant: drawing}. We denote by $\tilde {\cal{E}}_i'$ the bad event that the execution of stage $i$ is unsuccessful.
We start by constructing the input to the first iteration. The skeleton structure ${\mathcal K}_1$ is defined by the core structure $\check{\mathcal{J}}=(\check J,\set{b_u}_{u\in V(\check J)},\rho_{\check J}, F^*(\rho_{\check J}))$ in a natural way: we let the skeleton $K_1$ be $\check J$, the orientations $b_u$ for vertices $u\in V(\check J)$ remain unchanged, and the drawing $\rho_{\check K_1}$ of skeleton $\check K_1$ is $\rho_{\check J}$. Notice that the collection $\tilde {\mathcal{F}}({\mathcal K}_1)$ of faces has a single non-forbidden face $F^*= F^*(\rho_{\check J})$, and we let $G_{F^*}=\check G'$ be the graph associated with that face. For every other face $F\in \tilde {\mathcal{F}}({\mathcal K}_1)$, skeleton structure ${\mathcal K}_1$ defines a corresponding core structure ${\mathcal{J}}_F$, whose core graph is denoted by $J_F$. We then let $G_F=J_F$ be the graph associated with face $F$. We let ${\mathcal{G}}_1=\set{G_F\mid F\in \tilde {\mathcal{F}}({\mathcal K})}$ be the resulting ${\mathcal K}_1$-decomposition of $\check G'$, so $E_1^{\mathsf{del}}=\emptyset$, and we let $\phi_1=\psi$ be the solution to instance $\check I'=(G_1,\Sigma_1)$ that we defined above. Note that $N(\phi_1)=0$.
We have now obtained an input to Stage $1$ of the algorithm. If $\psi$ is a good solution to $\check I'$, then Invariants \ref{invariant: fewer edges2}--\ref{invariant: drawing} hold for this input.
\paragraph{Stage Execution.}
We now describe an execution of Stage $i$, for $1\leq i\leq \ceil{32\mu^{6b}}$. At the beginning of Stage $i$, we set ${\mathcal K}_{i+1}={\mathcal K}_i$, denoting the corresponding skeleton by $K_{i+1}$, ${\mathcal{G}}_{i+1}={\mathcal{G}}_i$, and $E^{\mathsf{del}}_{i+1}=E^{\mathsf{del}}_i$. Let ${\mathcal{F}}'_i\subseteq \tilde {\mathcal{F}}({\mathcal K}_i)$ be the collection of all faces $F$, for which the corresponding instance $I_F=(G_F,\Sigma_F)$ is not acceptable, where $G_F\in {\mathcal{G}}_i$ is the graph associated with face $F$.
Denote ${\mathcal{F}}'_i=\set{F_1,\ldots,F_q}$. Consider any face $F_j\in {\mathcal{F}}'_i$. Since instance $I_{F_j}=(G_{F_j},\Sigma_{F_j})$ is not acceptable, from the definition of acceptable instances (see \Cref{def: acceptable instance}), $|E(G_{F_j})|\geq \check m'/\mu^{2b}$ must hold. From the definition of a ${\mathcal K}_i$-decomposition, for every pair $G_F,G_{F'}$ of graphs, if an edge $e$ lies in both graphs, then $e\in E(J_F)\cap E(J_{F'})$. Since an edge $e\in E(K_i)$ may lie on the boundary of at most two faces of the drawing $\rho_{K_i}$, we get that $\sum_{j=1}^q|E(G_{F_j})|\leq 2\check m'$. Therefore, $q\leq 2\mu^{2b}$ must hold.
The algorithm for Stage $i$ consists of $q$ iterations. In iteration $j$, we apply Procedure \ensuremath{\mathsf{ProcSplit}}\xspace from \Cref{thm: procsplit}
to instance $I_{F_j}$ that is defined by graph $G_{F_j}$, and the corresponding core structure ${\mathcal{J}}_{F_j}$. We will use parameter $b'=2b$ instead of parameter $b$ in Procedure \ensuremath{\mathsf{ProcSplit}}\xspace. The set ${\mathcal{P}}_j$ of promising paths of cardinality $\floor{\frac{|E(G_{F_j})|}{\mu^{2b}}}$ is computed as follows. From the definition of acceptable instances (see \Cref{def: acceptable instance}), if instance $I_{F_j}$ is not acceptable, then there is a partition $(E_1,E_2)$ of the edges of $\tilde E_{F_j}$ (the edges of $G_{F_j}$ with exactly one endpoint in the core $J_{F_j}$), such that the edges of $E_1$ appear consecutively in the ordering ${\mathcal{O}}(J_{F_j})$, and the minimum cut in the correpsonding graph $H_{F_j}$ separating separating these two sets of edges contains more than $\check m'/\mu^{2b}$ edges. From the Maximum Flow / Minimum Cut theorem, there is a collection ${\mathcal{P}}_j$ of $\floor{\frac{\check m'}{\mu^{2b}}}\geq \floor{\frac{|E(G_{F_j})|}{\mu^{2b}}}$
edge-disjoint paths in graph $G_{F_j}$, where each path $P\in {\mathcal{P}}_j$ has an edge of $E_1$ as its first edge, an edge of $E_2$ as its last edge, and is internally disjoint from $J_{F_j}$. Moreover, a set of paths with these properties can be computed efficiently.
Let ${\mathcal{A}}_j=\set{P_j,\set{b_u}_{u\in V(J_{F_j})\cup P_j},\rho'}$ be the enhancement structure computed by Procedure \ensuremath{\mathsf{ProcSplit}}\xspace, and let $(I_1^j=(G_1^j,\Sigma_1^j), I_2^j=(G_2^j,\Sigma_2^j))$ be the split of instance $I_{F_j}$ along ${\mathcal{A}}_j$ that the algorithm returns. We denote by $E^{\mathsf{del}}(I_{F_j})=E(G_j)\setminus (E(G^j_1)\cup E(G^j_2))$ the set of the deleted edges. We add the edges of $E^{\mathsf{del}}(I_{F_j})$ to $E^{\mathsf{del}}_{i+1}$, and we define a new skeleton $K'_{i+1}$ and skeleton structure ${\mathcal K}'_{i+1}$ using the enhancement ${\mathcal{A}}_j$ in a natural way: we let $K'_{i+1}=K_{i+1}\cup P_j$. The orientations of vertices $u$ that belong to $K_{i+1}$ remain unchanged in ${\mathcal K}'_{i+1}$, and the orientations of inner vertices on path $P_j$ are set to be identical to those given by ${\mathcal{A}}_j$. Consider now drawing $\rho'$ of graph $J_{F_j}\cup P_j$. In this drawing, the image of the core $J_{F_j}$ is identical to that in the drawing $\rho_{K_{i+1}}$ of skeleton $K_{i+1}$, and the image of path $P_j$ is drawn inside face $F_j$, partitioning it into two faces, $F_1^j$ and $F_2^j$. We obtain a drawing $\rho_{K_{i+1}'}$ of the new skeleton $K_{i+1}'$ by starting with the drawing $\rho_{K_{i+1}}$ of skeleton $K_{i+1}$, and then adding the drawing of path $P_j$ inside face $F_j$ exactly like in the drawing $\rho'$. This completes the definition of the new skeleton structure ${\mathcal K}_{i+1}'$. Notice that $\tilde {\mathcal{F}} ({\mathcal K}'_{i+1})=(\tilde {\mathcal{F}}({\mathcal K}_{i+1})\setminus\set{F_j})\cup \set{F_1^j\cup F_2^j}$. We modify the decomposition ${\mathcal{G}}_{i+1}$ by replacing graph $G_{F_j}$ with the graphs $G_{F_1^{j}}$ (that becomes associated with face $F_1^j$) and $G_{F_2^{j}}$ (that becomes associated with face $F_2^j$). We then replace skeleton structure ${\mathcal K}_{i+1}$ with the new skeleton structure ${\mathcal K}_{i+1}'$ and continue to the next iteration. We denote by ${\cal{E}}_j^i$ the bad event that the output of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace computed in iteration $j$ is not a valid output.
This completes the definition of the algorithm for stage $i$.
We start with the following observation.
\begin{observation}\label{obs: valid input to procsplit}
Assume that there is a good solution $\psi$ to instance $\check I'$, that the Invariants \ref{invariant: fewer edges2}--\ref{invariant: drawing} hold at the beginning of stage $i$.
Then for all $1\leq j\leq q$, the input to Procedure \ensuremath{\mathsf{ProcSplit}}\xspace in iteration $j$ of stage $i$ is a valid input.
\end{observation}
\begin{proof}
Since we have assumed that there is a good solution $\psi$ to instance $\check I'$, and that the Invariants \ref{invariant: fewer edges2}--\ref{invariant: drawing} hold at the beginning of stage $i$,
from Invariant \ref{invariant: drawing}, at the beginning of stage $i$, there exists a solution $\phi_i$ to instance $(G_i,\Sigma_i)$, that is ${\mathcal K}_i$-valid, with $\mathsf{cr}(\phi_i)\leq \mathsf{cr}(\psi)$,
$|\chi^*(\phi_i)|\leq |\chi^{*}(\psi)|$, and
$N(\phi_i)\leq \frac{\mathsf{cr}(\psi)\cdot i\cdot \mu^{40b}}{\check m'}$.
Consider now some index $1\leq j\leq q$. Let $\phi_{i,j}$ be the solution to instance $I_{F_j}$ defined by the graph $G_{F_j}\in {\mathcal{G}}_i$ that is induced by $\phi_i$.
This solution must be ${\mathcal{J}}_{F_j}$-valid.
From the above discussion,
$$\mathsf{cr}(\phi_{i,j})\leq \mathsf{cr}(\phi_i)\leq \mathsf{cr}(\psi)\leq \frac{(\check m')^2}{\mu^{240b}}\leq \frac{|E(G_{F_j})|^2}{\mu^{120b}},$$
since $|E(G_{F_j})|\geq \frac{\check m'}{\mu^{2b}}$. Next, we bound the number of dirty crossings of drawing $\phi_{i,j}$ with respect to the core structure ${\mathcal{J}}_{F_j}$. The set of such dirty crossings may include all crossings of $\chi^*(\phi_i)$ (whose number is bounded by $ |\chi^{*}(\psi)|\leq \frac{\check m'}{\mu^{240b}}\leq \frac{|E(G_{F_j})|}{\mu^{238b}}$), and the crossings in which the edges of $J_{F_j}\setminus \check J'$ participate, whose number is bounded by:
$$N(\phi_i)\leq \frac{\mathsf{cr}(\psi)\cdot i\cdot \mu^{40b}}{\check m'}\leq \frac{i\check m}{\mu^{200b}}\leq \frac{|E(G_{F_j})|}{\mu^{150b}},$$
since $\mathsf{cr}(\psi)\leq \frac{(\check m')^2}{\mu^{240b}}$, $i\leq \ceil{32\mu^{6b}}$ and $|E(G_{F_1})|\geq \check m'/\mu^{2b}$. Therefore, the total number of dirty crossings in drawing $\phi_{i,j}$ with respect to the core structure ${\mathcal{J}}_{F_j}$ is bounded by $\frac{|E(G_{F_1})|}{\mu^{120b}}$, as required. We conclude that there exists
there exists a solution $\phi_{i,j}$ to instance $I_{G_{F_j}}$ that is ${\mathcal{J}}_{F_j}$-valid, with $\mathsf{cr}(\phi_{i,j})\leq \frac{|E(G_{F_j})|^2}{\mu^{60b'}}$ and $|\chi^{\mathsf{dirty}}(\phi_{i,j})|\leq \frac{|E(G_{F_j})|}{\mu^{60b'}}$, and so the input to Procedure \ensuremath{\mathsf{ProcSplit}}\xspace in iteration $j$ of stage $i$ is valid.
\end{proof}
The following claim is central in the analysis of the algorithm.
\begin{claim}\label{claim: stage execution}
Assume that there is a good solution $\psi$ to instance $\check I'$, that Invariants \ref{invariant: fewer edges2}--\ref{invariant: drawing} hold at the beginning of stage $i$, and that neither of the bad events ${\cal{E}}_1^i,\ldots,{\cal{E}}_q^i$ happened over the course of the $i$th stage. Then Event $\tilde {\cal{E}}'_{i+1}$ does not happen either, and so Invariants \ref{invariant: fewer edges2}--\ref{invariant: drawing} hold at the end of stage $i+1$.
\end{claim}
\begin{proof}
Throughout the proof, we assume that there is a good solution $\psi$ to instance $\check I'$, that Invariants \ref{invariant: fewer edges2}--\ref{invariant: drawing} hold at the beginning of stage $i$, and that neither of the bad events ${\cal{E}}_1^i,\ldots,{\cal{E}}_q^i$ happens.
We start by establishing Invariant \ref{invariant: fewer edges2}. Consider some face $F\in \tilde {\mathcal{F}}({\mathcal K}_{i+1})$. If $F\in \tilde {\mathcal{F}}({\mathcal K}_i)$, then, since $F$ was not added to the set $\set{F_1,\ldots,F_q}$ of faces to be processed in stage $i$, the corresponding instance $I_F=(G_F,\Sigma_F)$ with $G_F\in {\mathcal{G}}_i$ is acceptable. Since graph $G_F$ remains unchagnged in ${\mathcal{G}}_{i+1}$, instance $I_F$ remains an acceptable instance. Assume now that $F\not\in \tilde {\mathcal{F}}({\mathcal K}_i)$. Then there must be an index $1\leq j\leq q$, for which $F=F^j_1$ or $F=F^j_2$. In other words, face $F$ was created in iteration $j$ of Stage $i$. Since Event ${\cal{E}}_j^i$ did not happen, the output produced by Procedure \ensuremath{\mathsf{ProcSplit}}\xspace in iteration $j$ is a valid output. From Property \ref{prop: smaller graphs}, $|E(G_{F^j_1})|, |E(G_{F^j_2})|\leq |E(G_{F})|-\frac{|E(G_{F_j})|}{32\mu^{b'}}\leq |E(G_{F_j})|-\frac{\check m'}{32\mu^{4b}}$, since $b'=2b$, and since $|E(G_{F_j})|\geq \check m/\mu^{2b}$ must hold, as instance $I_{F_j}$ is not acceptable. Since, from Invariant \ref{invariant: fewer edges2}, $|E(G_{F_j})|\leq \check m'\cdot \textsf{left} (1-\frac{i-1}{32\mu^{6b}}\textsf{right} )$, we get that $|E(G_{F^j_1})|, |E(G_{F^j_2})|\leq \check m'\cdot \textsf{left} (1-\frac{i}{32\mu^{6b}}\textsf{right} )$. Therefore, invariant \ref{invariant: fewer edges2} continues to hold.
Next, we establish Invariant \ref{invariant: few deleted edges}. Fix an index $1\leq j\leq q$. Using the arguments from the proof of \Cref{obs: valid input to procsplit},
there is a soluton $\phi_{i,j}$ to instance $I_{F_j}$ defined by the graph $G_{F_j}\in {\mathcal{G}}_i$, that is ${\mathcal{J}}_{F_j}$-valid, with
$\mathsf{cr}(\phi_{i,j})\leq \mathsf{cr}(\psi)\leq \frac{|E(G_{F_j})|^2}{\mu^{60b'}}$, and $|\chi^{\mathsf{dirty}}(\phi_{i,j})|\leq |\chi^{*}(\psi)|+N(\phi_i)\leq \frac{|E(G_{F_j})|}{\mu^{60b'}}$.
From Property \ref{prop output deleted edges}:
\[\begin{split}
|E^{\mathsf{del}}(I_{F_j})|&\leq \frac{2\mathsf{cr}(\phi_{i,j})\cdot \mu^{38b'}}{|E(G_{F_j})|}+|\chi^{\mathsf{dirty}}(\phi_{i,j})|\\
&\leq \frac{2\mathsf{cr}(\psi)\cdot \mu^{78b}}{\check m'}+|\chi^{*}(\psi)|+N(\phi_i)\\
&\leq \frac{2\mathsf{cr}(\psi)\cdot \mu^{78b}}{\check m'}+|\chi^{*}(\psi)| +\frac{\mathsf{cr}(\psi)\cdot i\cdot \mu^{40b}}{\check m'}\\
&\leq \frac{2\mathsf{cr}(\psi)\cdot \mu^{78b}}{\check m'}+|\chi^{*}(\psi)| +\frac{32\mathsf{cr}(\psi)\cdot \mu^{46b}}{\check m'}\\
&\leq \frac{\mathsf{cr}(\psi)\cdot \mu^{80b}}{\check m'}+|\chi^{*}(\psi)|.
\end{split}\]
(we have used the fact that $|E(G_{F_j})|\geq \check m/\mu^{2b}$, $b'=2b$, and $i\leq \ceil{32\mu^{6b}}$). Since $q\leq 2\mu^{2b}$, we get that $|E^{\mathsf{del}}_{i+1}|\leq |E^{\mathsf{del}}_i|+\frac{\mathsf{cr}(\psi)\cdot \mu^{100b}}{\check m'}+|\chi^{*}(\psi)|\cdot \mu^{3b}$, establishing invariant \ref{invariant: few deleted edges}.
It now remains to establish Invariant \ref{invariant: drawing}.
For convenience, we denote by $G^0=G_i$, and, for $1\leq j\leq q$, we denote by $G^j$ the graph obtained after the $j$th iteration of the $i$th stage, that is, $G^j=G^{j-1}\setminus E^{\mathsf{del}}(I_{F_j})$. We denote by $I^j$ the subinstance of $\check I'$ defined by graph $G^j$. We also denote by $K^0=K_i$ the initial skeleton at the beginning of phase $i$, and, for $1\leq j\leq q$, we denote by $K^j$ the skeleton obtained at the end of iteration $j$, so $K^j=K^{j-1}\cup P_j$. We denote by ${\mathcal K}^j$ the skeleton structure associated with skeleton $K^j$, that can be obtained from ${\mathcal K}^{j-1}$ and the enhancement structure ${\mathcal{A}}_j$ as described above. Lastly, we will define, for all $0\leq j\leq q$, a solution $\phi^j$ to instance $I^j$ that is ${\mathcal K}^j$-valid. We will ensure that for all $0\leq j<j'\leq q$, the drawing of graph $G_{F_{j'}}$ in $\phi^j$ is identical to that in $\phi^0$.
Additionally, we will ensure that $\mathsf{cr}(\phi^j)\leq \mathsf{cr}(\psi)$ and $|\chi^*(\phi^j)|\leq |\chi^*(\psi)|$.
Initially, we let $\phi^0=\phi_i$ be the solution for instance $I^0$ that is guaranteed to exist by Invariant \ref{invariant: drawing}. Recall that this solution is
${\mathcal K}^0$-valid; $\mathsf{cr}(\phi^0)\leq \mathsf{cr}(\psi)$;
$|\chi^*(\phi^0)|\leq |\chi^{*}(\psi)|$; and the total number of crossings of $\phi^0$ in which the edges of $E(K^0)\setminus E(\check J)$ participate is at most $\frac{\mathsf{cr}(\psi)\cdot i\cdot \mu^{40b}}{\check m'}$. For $0\leq j\leq q$, we denote by $N^j$ the total number of crossings in which the edges of $E(K^j)\setminus E(\check J)$ participate in $\phi^j$.
Consider now some index $1\leq j\leq q$, and assume that we are given a solution $\phi^{j-1}$ to instance $I^{j-1}$ that is ${\mathcal K}^{j-1}$-valid. Recall that the drawing of graph $G_{F_j}$ induced by $\phi^{j-1}$ is identical to that in $\phi^0$. Therefore, the drawing of $G_{F_j}$ induced by $\phi^{j-1}$ is precisely $\phi_{i,j}$. From Property \ref{prop output drawing} of the valid output to procedure \ensuremath{\mathsf{ProcSplit}}\xspace, there is a ${\mathcal{J}}_{F_j}$-valid solution $\phi'_j$ to the subinstance of $\check I'$ that is defined by graph $G'_{F_j}=G_{F_j}\setminus E^{\mathsf{del}}(I_{F_j})$, that is compatible with $\phi_{i,j}$, in which the edges of $E(J_{F_j})\cup E(P_j)$ do not cross each other, and the number of crosings in which the edges of $P_j$ participate is at most $\frac{\mathsf{cr}(\phi^{j-1})\cdot \mu^{12b'}}{|E(G_{F_j})|}\leq \frac{\mathsf{cr}(\psi)\cdot \mu^{26b}}{\check m'}$, since $\mathsf{cr}(\phi^{j-1})\leq \mathsf{cr}(\psi)$, $b'=2b$, and $|E(G_{F_j})|\geq \check m'/\mu^{2b}$.
Note that, from the definition of compatible drawings (see \Cref{def: compatible drawing}), the image of the core $J_{F_j}$ in $\phi'_j$ is identical to that in $\phi_{i,j}$. The only difference between drawing $\phi'_j$ and $\phi_{i,j}$ is that the images of the edges of $E^{\mathsf{del}}(I_{F_j})$ were deleted, and some additional local changes were made within the face $F_j$. We are guaranteed that, if a point $p$ is an inner point on the image of some edge $\phi_j'$, then is an inner point on the image of some edge in $\phi_{i,j}$.
Moreover, if $p$ is an image of some vertex $v$ in $\phi'_j$, then either (i) $p$ is the image of $v$ in $\phi_{i,j}$; or (ii) the degree of $p$ in $G_{F_j}\setminus E^{\mathsf{del}}(I_{F_j})$ is $2$, and $p$ is an inner point on the image of some edge in $\phi_{i,j}$.
In order to obtain drawing $\phi^{j}$ of graph $G^j$, we first delete, from drawing $\phi^{j-1}$, the images of all edges in $E^{\mathsf{del}}(I_{F_j})$. Next, we delete the image of the graph $G_{F_j}$ from the current drawing, and copy instead the image of the graph $G_{F_j}\setminus E^{\mathsf{del}}(I_{F_j})$ from drawing $\phi'_j$. Note that the two images of graph $G_{F_j}\setminus E^{\mathsf{del}}(I_{F_j})$ are identical except for some small local changes inside face $F_j$. While it is possible that edges and vertices of $G^j\setminus G_{F_j}$ are drawn inside face $F_j$ in $\phi^{j-1}$, it is easy to see that no new crossings between edges of $G_{F_j}$ and edges of $G^j\setminus G_{F_j}$ are introduced. Since $\mathsf{cr}(\phi'_j)\leq \mathsf{cr}(\phi_{i,j})$, we get that $\mathsf{cr}(\phi^j)\leq\mathsf{cr}(\phi^{j-1})\leq\mathsf{cr}(\psi)$. Since the only changes to the drawing outside face $F_j$ is the deletion of the segments of the images of some edges, $|\chi^*(\phi^j)|\leq |\chi^*(\phi^{j-1})|\leq |\chi^*(\psi)|$. Since
we are guaranteed from Property \ref{prop output drawing} that the number of crosings in which the edges of $P_j$ participate in $\phi'_j$ is at most $\frac{\mathsf{cr}(\psi)\cdot \mu^{26b}}{\check m'}$, we get that $N^j\leq N^{j-1}+\frac{\mathsf{cr}(\psi)\cdot \mu^{26b}}{\check m'}$.
We define the solution $\phi_{i+1}$ to instance $(G_{i+1},\Sigma_{i+1})$ to be $\phi^q$. From the above discussion, drawing $\phi^q$ is ${\mathcal K}_{i+1}$-valid, $\mathsf{cr}(\phi_{i+1})\leq \mathsf{cr}(\psi)$, and
$|\chi^*(\phi_i)|\leq |\chi^{*}(\psi)|$. Since the number of iterations $q\leq 2\mu^{2b}$, and in every iteration we introduce at most $\frac{\mathsf{cr}(\psi)\cdot \mu^{26b}}{\check m'}$ new crossings with the edges of the new skeleton $K_{i+1}$, we get that $N(\phi_{i+1})\leq N(\phi_i)+2\mu^{2b}\cdot \frac{\mathsf{cr}(\psi)\cdot \mu^{26b}}{\check m'}\leq \frac{\mathsf{cr}(\psi)\cdot i\cdot \mu^{40b}}{\check m'}+ \frac{2\mathsf{cr}(\psi)\cdot \mu^{28b}}{\check m'}\leq \frac{\mathsf{cr}(\psi)\cdot (i+1)\cdot \mu^{40b}}{\check m'}$.
\end{proof}
From \Cref{obs: valid input to procsplit} and \Cref{thm: procsplit}, for all $1\leq j\leq q$, $\prob{{\cal{E}}^i_j\mid \neg\tilde {\cal{E}}'_1\wedge\cdots\wedge\neg{\cal{E}}'_i}\leq \frac{2^{20}}{\mu^{10b'}}\leq \frac{2^{20}}{\mu^{20b}}$. Since $q\leq 2\mu^{2b}$, we get that $\prob{{\cal{E}}'_{i+1}\mid \neg\tilde {\cal{E}}'_1\wedge\cdots\wedge\neg{\cal{E}}'_i}\leq \frac{2^{21}}{\mu^{18b}}$.
Let $\tilde{\cal{E}}'$ be the bad event that either of the events $\tilde {\cal{E}}'_1,\ldots,\tilde {\cal{E}}'_z$ happened. Since $z=\ceil{32\mu^{6b}}$, we get that $\prob{\tilde {\cal{E}}'_z}\geq \frac{z}{\mu^{20b}}\leq \frac{1}{\mu^{10b}}$.
We return the skeleton structure ${\mathcal K}_z$, and the ${\mathcal K}_z$-decomposition ${\mathcal{G}}_z$ of $\check G'$. From Invariant \ref{invariant: fewer edges2}, for every face $F\in \tilde {\mathcal{F}}({\mathcal K}_i)$, the corresponding instance $I_F=(G_F,\Sigma_F)$ with $G_F\in {\mathcal{G}}_i$ must be acceptable (this is since the graph associated with an unacceptable instance must have at least $\check m'/\mu^{2b}$ edges). Assume now that there is a good solution $\psi$ to instance $\check I'$, and that bad event $\tilde {\cal{E}}'$ did not happen. Then we are guaranteed that the algorithm does not return FAIL, and, from Invariant \ref{invariant: few deleted edges}, $|E^{\mathsf{del}}_z|\leq z\cdot \textsf{left}( \frac{\mathsf{cr}(\psi)\mu^{100b}}{\check m'}+|\chi^{*}(\psi)|\cdot \mu^{3b}\textsf{right}) \leq \frac{\mathsf{cr}(\psi)\mu^{108b}}{\check m'}+|\chi^{\mathsf{dirty}}(\psi)|\cdot \mu^{10b}$, since $z=\ceil{32\mu^{6b}}$. Lastly, Invariant \ref{invariant: drawing} guarantees that there is a solution $\phi$ to instance $(G_z,\Sigma_z)$, where $G_z=\check G'\setminus E^{\mathsf{del}}_z$, and $\Sigma_z$ is the rotation system for $G_z$ induced by $\check \Sigma'$, with the following properties. First, drawing $\phi$ is ${\mathcal K}_z$-valid. Additionally, $\mathsf{cr}(\phi)\leq \mathsf{cr}(\psi)$, $|\chi^*(\phi)|\leq |\chi^{*}(\psi)|$, and the total number of crossings in which the edges of $E(K_z)\setminus E(\check J)$ participate is at most $\frac{\mathsf{cr}(\psi)\cdot z\cdot
\mu^{40b}}{\check m'}\leq\frac{\mathsf{cr}(\psi)\cdot \mu^{47b}}{\check m'} $.
Since $\prob{\tilde {\cal{E}}'}\leq 1/\mu^{10b}$, this completes the proof of \Cref{lem: compute phase 2 decomposition}.
\section{Definitions, Notation, Known Results, and their Easy Extensions}
\label{sec:long prelim}
In this section we provide additional definitions and notation, together with known results and their easy extensions that we use throughout the paper.
\input{prelims-clusters}
\input{prelims-cuts}
\input{prelims-expanders-embedding}
\input{prelims-curves}
\input{prelims-contracted}
\section{Definitions, Notation, Known Results, and their Easy Extensions}
\label{sec:long prelim}
In this section we provide additional definitions and notation, together with known results and their easy extensions that we use throughout the paper.
All graphs that we consider in this paper are undirected. However, it will sometimes be convenient for us to assign direction to paths in such graphs. We do so by designating one endpoint of the path as its first endpoint, and another endpoint as its last endpoint. We will then view the path as being directed from its first endpoint towards its last endpoint. We will sometimes refer to a path with an assigned direction as a directed path, even though the underlying graph is an undirected graph.
\subsection{Clusters and Augmentations of Clusters}
We will refer to vertex-induced subgraphs of $G$ as
\emph{clusters}. For a set ${\mathcal{C}}$ of disjoint clusters of $G$, we denote by $E^{\textsf{out}}_G({\mathcal{C}})$ the set of edges connecting distinct clusters of ${\mathcal{C}}$.
We sometimes omit the subscript $G$ when clear from the context.
Given a cluster $C$ in a graph $G$, and a path $P$ in $G$, whose endpoints are $u$ and $v$ we say that $P$ is \emph{internally disjoint} from $C$, iff $V(C)\cap V(P)\subseteq \set{u,v}$.
Next, we define the notaion of augmentation of a cluster.
\begin{definition}[Augmentation of Clusters]
\label{def: Graph C^+}
Let $C$ be a cluster of $G$. The graph $C^+$, that is called the \emph{augmentation} of cluster $C$, is defined as follows. First, we subdivide every edge $e\in \delta_G(C)$ with a vertex $t_e$, and let $T(C)=\set{t_e\mid e\in \delta_G(C)}$ be the set of newly added vertices, that we sometimes refer to as \emph{terminals of $C$}. We then let $C^+$ be the subgraph of the resulting graph induced by $V(C)\cup T(C)$.
\end{definition}
\iffalse
Throughout the paper, we use the following definition.
\begin{definition}[Curves in general position]
Let $\Gamma$ be a set of of simple curves in the plane.
We say that the curves of $\Gamma$ are \emph{in general position}, iff the following four conditions hold:
\begin{itemize}
\item for every pair $\gamma,\gamma'\in \Gamma$ of distinct curves, there is a finite number of points $p$ with $p\in \gamma\cap \gamma'$;
\item if a point $p$ is an endpoint of a curve in $\Gamma$, then it may not be an inner point on any curve in $\Gamma$; and
\item if a point $p'$ is contained in more than two curves in $\Gamma$, then it may not be an inner point on any curve in $\Gamma$.
\end{itemize}
\end{definition}
Given a set $\Gamma$ of curves in general position, a \emph{crossing} between a pair $\gamma,\gamma'\in \Gamma$ of curves is a point $p$ that is an inner point on both $\gamma$ and $\gamma'$.
We also say that curves $\gamma,\gamma'$ cross at point $p$.
\iffalse
We will sometimes distinguish between two types of crossings of curves: \emph{transversal} and \emph{non-transversal}.
\begin{definition}[Transversal crossing of curves]
Let $\Gamma$ be a set of curves in general position, and let $\gamma,\gamma'\in \Gamma$ be two distinct curves that cross at point $p$. Let $D$ be an arbitrarily small disc that contains $p$ in its interior. Let $q_1,q_2$ be the points of $\gamma$ lying on the boundary of $D$, and let $q_1',q_2'$ be the points of $\gamma'$ lying on the boundary of $D$. We say that the crossing $p$ between $\gamma$ and $\gamma'$ is \emph{transversal} iff the points $q_1,q_1',q_2,q_2'$ appear in this circular ordering on the boundary of $D$; otherwise, we say that the crossing is \emph{non-transversal}. If the crossing $p$ between $\gamma$ and $\gamma'$ is non-transversal, then we sometimes say that $\gamma$ and $\gamma'$ \emph{touch} at point $p$.
\end{definition}
\fi
We are now ready to define a graph drawing.
\begin{definition}[Graph Drawing and Crossings]
A \emph{drawing} $\phi$ of a graph $G$ is an embedding of $G$ into the plane, that maps every vertex $v$ of $G$ to a point $\phi(v)$ (called the \emph{image of $v$}), and every edge $e=(u,v)$ of $G$ to a simple curve $\phi(e)$ (called the \emph{image of $e$}), whose endpoints are $\phi(u)$ and $\phi(v)$, such that the set $\set{\phi(e)\mid e\in E(G)}$ of curves is in general position. If $p$ is an inner point on images $\phi(e),\phi(e')$ of distinct edges $e,e'\in E(G)$,
then we say that edges $e$ and $e'$ \emph{cross} at $p$, and that $p$ is a \emph{crossing} in the drawing $\phi$, that we also sometimes denote by $(e,e')_p$, where $(e,e')$ is an unordered pair (omitting the subscript $p$ if the crossing is uniquely defined). We denote by $\mathsf{cr}(\phi)$ the total number of crossings in the drawing $\phi$. If $\mathsf{cr}(\phi)=0$, then we say that the drawing $\phi$ is \emph{planar}.
\end{definition}
\iffalse
that connects the images of its endpoints. We require that the interiors of the curves representing the edges do not contain the images of any of the vertices. We say that two edges \emph{cross} at a point $p$, if the images of both edges contain $p$, and $p$ is not the image of a shared endpoint of these edges. We require that no three edges cross at the same point in a drawing of $\phi$.
We say that $\phi$ is a \emph{planar drawing} of $G$ iff no pair of edges of $G$ crosses in $\phi$.
For a vertex $v\in V(G)$, we denote by $\phi(v)$ the image of $v$, and for an edge $e\in E(G)$, we denote by $\phi(e)$ the image of $e$ in $\phi$.
\fi
Note that a drawing of $G$ in the plane naturally defines a drawing of $G$ on the sphere and vice versa; we use both types of drawings.
\fi
\subsection{Curves in a Plane or on a Sphere}
\subsubsection{Reordering Curves}
Assume that we are given two oriented orderings $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$ of elements in set $U=\set{u_1,\ldots,u_r}$. Assume for simplicity that $b=b'=0$ (otherwise the corresponding ordering can be flipped). Consider a disc $D$, with a collection $\set{p_1,\ldots,p_r}$ of distinct points appearing on the boundary of $D$ (where we view each point $p_i$ as representing element $u_i$ of $U$), whose clock-wise order on the boundary of $D$ is precisely ${\mathcal{O}}$. Similarly, let $D'\subseteq D$ be any other disc that is strictly contained in $D$, with a collection $\set{p'_1,\ldots,p'_r}$ of points appearing on the boundary of $D'$, where for $1\leq i\leq r$, point $p'_i$ represents element $u_i\in U$, such that the clock-wise order of the points in $\set{p'_1,\ldots,p'_r}$ on the boundary of $D$ is precisely ${\mathcal{O}}'$.
For two unoriented orderings ${\mathcal{O}},{\mathcal{O}}'$ on $U$, we define $\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')=\min_{b,b'\in \set{0,1}}\set{\mbox{\sf dist}(({\mathcal{O}},0),({\mathcal{O}}',b'))}$.
\begin{definition}[Reordering curves]
We say that a collection $\Gamma=\set{\gamma_1,\ldots,\gamma_r}$ of curves is a \emph{set of reordering curves for $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$} iff the curves in $\Gamma$ are simple curves in general position, the interior of every curve $\gamma_i\in \Gamma$ is contained in $D\setminus D'$, and, for all $1\leq i\leq r$, curve $\gamma_i$ has $p_i,p'_i$ as its endpoints. The \emph{cost} of the collection $\Gamma$ of curves is defined to be the total number of crossings between these curves.
\end{definition}
\begin{definition}[Distance between orderings]
Let $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$ be two oriented orderings of a collection $U$ of elements. The \emph{distance} between the two orderings, denoted by $\mbox{\sf dist}(({\mathcal{O}},b),({\mathcal{O}}',b'))$, is the smallest cost of any set $\Gamma$ of reordering curves for $({\mathcal{O}},b)$ and $({\mathcal{O}},b')$.
\end{definition}
We use the following lemma, whose proof is provided Appendix~\ref{apd: Proof of find reordering}.
\begin{lemma}
\label{lem: ordering modification}
There is an algorithm, that, given a set $U$ of elements, a set $\Gamma=\set{\gamma_u\mid u\in U}$ of curves that shares an endpoint $z$, and two circular orderings ${\mathcal{O}},{\mathcal{O}}'$ on elements of $U$, such that the curves of $\Gamma$ enter $z$ in the order ${\mathcal{O}}$, computes a new set $\Gamma'=\set{\gamma'_u\mid u\in U}$ of curves, such that:
\begin{itemize}
\item the curves of $\Gamma'$ enter $z$ in the order ${\mathcal{O}}'$;
\item for each $u\in U$, the curve $\gamma_u$ differs from the curve $\gamma'_u$ only within some tiny disc $D$ that contains $z$; and
\item the number of crossings between curves of $\Gamma'$ within $D$ is $O(\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}'))$.
\end{itemize}
\end{lemma}
\iffalse
\subsection{Problem Definitions}
In the \textsf{Minimum Crossing Number}\xspace~problem, the input is an $n$-vertex graph $G$, and the goal is to compute a drawing of $G$ in the plane with minimum number of crossings.
The value of the optimal solution, also called the \emph{crossing number} of $G$, is denoted by $\mathsf{OPT}_{\mathsf{cr}}(G)$.
We also consider a closely related problem called Crossing Number with Rotation System (\textnormal{\textsf{MCNwRS}}\xspace). In this problem, the input is a graph $G$, and, for every vertex $v\in V(G)$, a circular ordering ${\mathcal{O}}_v$ of the set $\delta_G(v)$ of edges. The collection $\Sigma=\set{{\mathcal{O}}_v}_{v\in V(G)}$ of all these orderings is called a \emph{rotation system} of $G$. We say that a drawing $\phi$ of $G$ \emph{respects} the rotation system $\Sigma$ if the following holds. For a vertex $v\in V(G)$, let $D(v)$ be an arbitrarily small disc around $v$ in $\phi$. Then the images of the edges of $\delta_G(v)$ in $\phi$ must intersect the boundary of $\eta(v)$ in a circular order that is identical to ${\mathcal{O}}_v$ (but the orientation of this ordering may be arbitrary).
Given an instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, we say that a drawing $\phi$ of $G$ is a \emph{feasible solution} for $I$ iff $\phi$ respects the rotation system $\Sigma$. The \emph{cost} of the solution is the number of crossings in $\phi$. The goal in the \textnormal{\textsf{MCNwRS}}\xspace problem is to compute a feasible solution to the given instance $I$ of the problem of smallest possible cost. We denote the cost of the optimal solution of the \textnormal{\textsf{MCNwRS}}\xspace instance $I$ by $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$.
We use the following simple theorems about the \textnormal{\textsf{MCNwRS}}\xspace problem, whose proofs are deferred to
Appendix~\ref{apd: Proof of crwrs_planar} and Appendix~\ref{apd: Proof of crwrs_uncrossing}, respectively.
\begin{theorem}
\label{thm: crwrs_planar}
There is an efficient algorithm, that, given an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, determines if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$, and, if so, computes a feasible solution of cost $0$.
\end{theorem}
\begin{theorem}
\label{thm: crwrs_uncrossing}
There is an efficient algorithm, that given an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, computes a feasible solution to instance $I$, of cost at most $|E(G)|^2$.
\end{theorem}
We refer to the solution computed by the algorithm from Theorem~\ref{thm: crwrs_uncrossing} as a \emph{trivial solution} to instance $I=(G,\Sigma)$.
\iffalse
\subsection{$\alpha$-Decompositions of an \textnormal{\textsf{MCNwRS}}\xspace Instance}
Let $I=(G,\Sigma)$ be an instance of \textnormal{\textsf{MCNwRS}}\xspace, let ${\mathcal{C}}$ be a set of vertex-disjoint clusters of $G$ such that the sets $\set{V(C)}_{C\in {\mathcal{C}}}$ partition $V(G)$, and let $\alpha>0$ be a real number. An \emph{$\alpha$-decomposition of $I$ with respect to ${\mathcal{C}}$} is a collection ${\mathcal{I}}$ of instances of \textnormal{\textsf{MCNwRS}}\xspace with the following properties:
\begin{enumerate}
\item \label{prop: alpha deco 1} for each subinstance $I'=(G',\Sigma')\in {\mathcal{I}}$, there is at most one cluster $C\in {\mathcal{C}}$ with $E(C)\subseteq E(G')$; all other edges of $G'$ lie in set $E^{\textnormal{\textsf{out}}}({\mathcal{C}})$;
\item \label{prop: alpha deco 2} $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq O(|E(G)|)$; and
\item \label{prop: alpha deco 3} $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \alpha\cdot \big(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)| \big)$.
\end{enumerate}
We say that a randomized algorithm ${\mathcal{A}}$ computes an $\alpha$-decomposition of instance $I=(G,\Sigma)$ with respect to ${\mathcal{C}}$, if it computes a random family ${\mathcal{I}}$ of subinstances that satisfies Property~\ref{prop: alpha deco 1} and Property~\ref{prop: alpha deco 2}, and the following property in replace of Property~\ref{prop: alpha deco 3}:
\[\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\leq \alpha\cdot \bigg(\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\bigg).\]
\fi
\subsection{Subinstances}\label{subsec: subinstance}
Our algorithm for solving the \textnormal{\textsf{MCNwRS}}\xspace problem is recursive. Given the input instance $I^*=(G^*,\Sigma^*)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, it will consider various instances $(G,\Sigma)$ of the problem, that we call \emph{subinstances} of $I^*$. We use the following definition of subinstances.
\begin{definition}[Subinstances]
Let $I=(G,\Sigma)$ and $I'=(G',\Sigma')$ be two instances of \ensuremath{\mathsf{MCNwRS}}\xspace. We say that instance $I'$ is a \emph{subinstance} of instance $I$, if there is a subgraph $\tilde G\subseteq G$, and a collection $S_1,\ldots,S_r$ of mutually disjoint subsets of vertices of $\tilde G$, such that graph $G'$ can be obtained from $\tilde G\subseteq G$ by contracting, for all $1\leq i\leq r$, every vertex set $S_i$ into a supernode $u_i$; we keep parallel edges but remove self-loops. We do not distiguish between the edges incident to the supernodes in graph $G'$ and their counterparts in graph $G$. Additionally, for every vertex $v\in V(G')\cap V(G)$, its rotation ${\mathcal{O}}'_v$ in $\Sigma'$ must remain the same as in $\Sigma$, while for every supernode $u_i$, its rotation ${\mathcal{O}}_{u_i}$ can be defined arbitrarily.
\end{definition}
Notice that subinstance relation is transitive: if $(G_1,\Sigma_1)$ is a subinstance of $(G_0,\Sigma_0)$, and $(G_2,\Sigma_2)$ is a subinstance of $(G_1,\Sigma_1)$, then $(G_2,\Sigma_2)$ is a subinstance of $(G_0,\Sigma_0)$.
\fi
\subsubsection{Curves in a Disc}
\label{sec: curves in a disc}
Suppose we are given a disc $D$, and a collection $\set{s_1,t_1,\ldots,s_k,t_k}$ of distinct points on the boundary of the disc $D$. For all $1\leq i< j\leq k$, we say that the two pairs $(s_i,t_i),(s_j,t_j)$ of points \emph{cross} iff points $s_i,s_j,t_i,t_j$ are encountered in one of the following orderings when traversing the boundary of the disc in any direction: either $(s_i,s_j,t_i,t_j)$, or $(s_i,t_j,t_i,s_j)$.
We use the following simple claim, whose proof is deferred to \Cref{apd: Proof of curves in a disc}.
\begin{claim}\label{claim: curves in a disc}
There is an efficient algorithm that, given a disc $D$, and a collection $\set{s_1,t_1,\ldots,s_k,t_k}$ of distinct points on the boundary of $D$, computes a collection $\Gamma=\set{\gamma_1,\ldots,\gamma_k}$ of curves, such that, for all $1\leq i\leq k$, curve $\gamma_i$ has $s_i$ and $t_i$ as its endpoints, and its interior is contained in the interior of $D$. Moreover, for every pair $1\leq i<j\leq k$ of indices, if the two pairs $(s_i,t_i),(s_j,t_j)$ of points cross then curves $\gamma_i,\gamma_j$ intersect at exactly one point; otherwise, curves $\gamma_i,\gamma_j$ do not intersect.
\end{claim}
\subsubsection{Uncrossing of Curves}
In this subsection, we consider a set $\Gamma$ of curves in the plane that are in general position, and provide two different ways to ``simplify'' this collection of curves by \emph{uncrossing} them. The two types of uncrossings are referred to as \emph{type-$1$ uncrossing} and \emph{type-$2$ uncrossing}, respectively. With the first type of uncrossing, we ensure that, after the modification, each curve still has exactly the same endpoints as before, while we eliminate some of the crossings. With the second type of uncrossing we only ensure that the multiset containing the first endpoint of each curve and the multiset containing the last endpoint of each curve are preserved (but they may be matched to each other differently by the new curves).
\subsubsection*{Type-1 Uncrossing}
Assume that we are given a set $\Gamma$ of curves that are in the general position, and let $\gamma,\gamma'\in \Gamma$ be a pair of distinct simple curves. Recall that we say that $\gamma,\gamma'$ \emph{cross} at a point $p$, iff $p$ belongs to both curves and $p$ is not an endpoint of either of these curves; such a point $p$ is also called an \emph{crossing} between the curves $\gamma,\gamma'$.
We denote by $\chi(\gamma, \gamma')$ the number of crossings between curves $\gamma,\gamma'$.
For a set $\Gamma$ of simple curves and another simple curve $\gamma\notin \Gamma$, such that the curves of $\set{\gamma}\cup \Gamma$ are in general position, we denote by $\chi(\gamma,\Gamma)$ the number of crossings between $\gamma$ and curves in $\Gamma$, that is, $\chi(\gamma,\Gamma)=\sum_{\gamma'\in \Gamma}\chi(\gamma,\gamma')$. We also denote by $\chi(\Gamma)$ the total number of crossings between curves in $\Gamma$, namely $\chi(\Gamma)=\sum_{\gamma,\gamma'\in \Gamma, \gamma\ne \gamma'}\chi(\gamma,\gamma')$.
Intuitively, when two distinct curves $\gamma,\gamma'\in \Gamma$ cross more than once, we can locally modify them in the way shown in Figure~\ref{fig:type_1_uncrossing}, so that each of the two curves still has the same endpoints, at least one crossing between the two curves is eliminated, and no new crossings between $\gamma,\gamma'$ and other curves are created.
We call this process, that is formally stated in the following theorem, \emph{type-1 uncrossing}. The proof is standard and is provided in Appendix~\ref{apd: type-1 uncrossing} for completeness.
\begin{figure}[h]
\centering
\subfigure[Before: Curves $\gamma$ and $\gamma'$ cross twice, at points $p$ and $q$, and they have two crossings in total with the pink curve twice.]{\scalebox{0.48}{\includegraphics{figs/type_1_uncross_1.jpg}
}
\hspace{0.8cm}
\subfigure[After: Each of the new curves $\gamma$ and $\gamma'$ has same endpoints as before. The two curves no longer cross each other, and they still have two crossings in total with the pink curve.]{
\scalebox{0.48}{\includegraphics{figs/type_1_uncross_2.jpg}}}
\caption{An illustration of a type-1 uncrossing.}\label{fig:type_1_uncrossing}
\end{figure}
\begin{theorem}[Type-1 Uncrossing]
\label{thm: type-1 uncrossing}
There is an efficient algorithm, that, given a set $\Gamma$ of simple curves in general position, that are partitioned into two disjoint sets $\Gamma_1,\Gamma_2$, computes, for each curve $\gamma\in \Gamma_1$, a simple curve $\gamma'$ that has same endpoints as $\gamma$, such that, if we denote by $\Gamma_1'=\set{\gamma'\mid \gamma\in \Gamma_1}$, then the following hold:
\begin{itemize}
\item curves of $\Gamma'_1\cup \Gamma_2$ are in general position;
\item every pair of curves in $\Gamma_1'$ cross at most once;
\item for every curve $\gamma\in \Gamma_2$, $\chi(\gamma,\Gamma'_1)\le \chi(\gamma,\Gamma_1)$; and
\item $\chi(\Gamma'_1\cup\Gamma_2)\le \chi(\Gamma)$.
\end{itemize}
\end{theorem}
\subsubsection*{Type-2 Uncrossing}
\znote{maybe modify the name of subsections?}
We will sometimes give a curve $\gamma$ a \emph{direction}, by designating one of its endpoints as its first endpoint, which we denote by $s(\gamma)$, and the other as its last endpoint, which we denote by $t(\gamma)$.
Given a set $\Gamma$ of directed curves, we denote $S(\Gamma)=\set{s(\gamma)\mid \gamma\in \Gamma}$ and $T(\Gamma)=\set{t(\gamma)\mid \gamma\in \Gamma}$, where both $S(\Gamma),T(\Gamma)$ are multisets.
Consider some graph $G$ and a drawing $\phi$ of $G$ in the plane. Recall that we have defined, for every vertex $v\in V(G)$, a disc $D_{\phi}(v)$, that we referred to as a \emph{tiny $v$-disc}.
\begin{definition}
Let $G$ be a graph and $\phi$ a drawing of $G$ in the plane. We say that a curve $\gamma$ is \emph{aligned} with the drawing $\phi$ of $G$ iff there is a sequence $(e_1,e_2,\ldots,e_{r-1})$ of edges in $G$, and a partition $(\sigma_1,\sigma_1',\sigma_2,\sigma_2',\ldots,\sigma'_{r-1},\sigma_r)$ of $\gamma$ into consecutive segments, such that, if we denote, for all $1\leq i\leq r$, $e_i=(v_i,v_{i+1})$, then the following holds:
\begin{itemize}
\item for all $1\leq i\leq r-1$, segment $\sigma'_i$ is a contiguous segment of non-zero length of the image of the edge $e_i$ in $\phi$, and it is disjoint from all discs in $\set{D_{\phi}(u)}_{u\in V(G)}$, except that its first endpoint may lie on the boundary of of disc $D_{\phi}(v_i)$, and its last endpoint may lie on the boundary of disc $D_{\phi}(v_{i+1})$;
\item for all $1\leq i\leq r$, segment $\sigma_i$ is either contained in disc $D_{\phi}(v_i)$, or it is empty. In the latter case, $1<i<r$ must hold, and, if we denote by $p$ the last endpoint of $\sigma'_{i-1}$, then $p$ is also the first endpoint of $\sigma'_{i+1}$, and it must be a point at which images of edge $e_{i-1}$ and edge $e_{i}$ cross;
\item segment $\sigma_1$ is precisely the part of the image of edge $e_1$ that is contained in $D_{\phi}(v_1)$; and
\item segment $\sigma_r$ is precisely the part of the image of edge $e_{r-1}$ that is contained in $D_{\phi}(v_r)$.
\end{itemize}
\end{definition}
We use the following theorem, whose proof is deferred to \Cref{apd: new type 2 uncrossing}.
\begin{theorem}
\label{thm: new type 2 uncrossing}
There is an efficient algorithm, whose input contains a graph $G$, a drawing $\phi$ of $G$ on the sphere, a subgraph $C$ of $G$, and a collection ${\mathcal{Q}}$ of edge-disjoint paths in $G$ that are internally disjoint from $C$, such that, for each path $Q\in {\mathcal{Q}}$, one of its endpoints is designated as its first endpoint and is denoted by $s_Q$, the other endpoint, denoted by $t_Q$, is designated as its last endpoint. The algorithm computes a set $\Gamma=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$ of directed simple curves on the sphere with the following properties:
\begin{enumerate}
\item every curve $\gamma(Q)\in \Gamma$ is aligned with the graph $\bigcup_{Q'\in {\mathcal{Q}}}Q'$;
\item for each path $Q\in {\mathcal{Q}}$, $s(\gamma(Q))=\phi(s_Q)$;
\item if the first endpoint of every path in ${\mathcal{Q}}$ is the same vertex $u^*$, then for each path $Q\in {\mathcal{Q}}$ whose first edge (the edge incident to its first endpoint) is denoted by $e_1(Q)$, curve $\gamma(Q)$ contains the segment of the image of $e_1(Q)$ that is lies in the disc $D_{\phi}(u^*)$;
\item the multiset $T(\Gamma)$ (containing the last endpoint of every curve in $\Gamma$) is precisely the multiset $\set{\phi(t_Q)\mid Q\in {\mathcal{Q}}}$; and
\item the curves in $\Gamma$ do not cross each other.
\end{enumerate}
\end{theorem}
Note that, from the definition of graph-aligned curves, and from the fact that the curves in $\Gamma$ do not cross each other, for every edge $e\in E(C)$, the number of crossings between the image of edge $e$ in $\phi$ and the curves in $\Gamma$ is bounded by the number of crossings between $e$ and edges of $G\setminus C$ in drawing $\phi$, namely $\chi(\phi(e),\Gamma)\le \chi_{\phi}(e, G\setminus C)$. We emphasize that the curves in $\Gamma$ may match the mutisets $\set{s(\gamma(Q))\mid Q\in {\mathcal{Q}}}$ and $\set{t(\gamma(Q))\mid Q\in {\mathcal{Q}}}$ differently from the paths in ${\mathcal{Q}}$.
We use the following corollary, whose proof is deferred to \Cref{apd: cor new type 2 uncrossing}.
\begin{corollary}
\label{cor: new type 2 uncrossing}
There is an efficient algorithm, whose input contains a graph $G$, a drawing $\phi$ of $G$ on the sphere, a subgraph $C$ of $G$, and a collection ${\mathcal{Q}}$ of paths in $G$ that are internally disjoint from $C$, such that, for each path $Q\in {\mathcal{Q}}$, one of its endpoints is designated as its first endpoint and is denoted by $s_Q$, the other endpoint, denoted by $t_Q$, is designated as its last endpoint. The algorithm computes a set $\Gamma=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$ of directed curves on the sphere with the following properties:
\begin{itemize}
\item for each path $Q\in {\mathcal{Q}}$, $s(\gamma(Q))=\phi(s_Q)$;
\item the multiset $T(\Gamma)$ (containing the last endpoint of every curve in $\Gamma$) is precisely the multiset $\set{\phi(t_Q)\mid Q\in {\mathcal{Q}}}$;
\item the curves in $\Gamma$ do not cross each other; and
\item for each edge $e\in E(C)$, the number of crossings between the image of $e$ in $\phi$ and the curves in $\Gamma$ is bounded by the number of crossings between the image of $e$ and images of the edges of $G\setminus C$ times $\cong_G({\mathcal{Q}})$, namely $\chi(\phi(e),\Gamma)\le \chi_{\phi}(e, G\setminus C)\cdot \cong_G({\mathcal{Q}})$.
\end{itemize}
\end{corollary}
\iffalse{the original type-2 uncrossing}
\znote{To check if the original type-2 uncrossing can be removed}
We will sometimes give a curve a \emph{direction}, by designating one of its endpoints as its first endpoint, and the other as its last endpoint.
Given a set $\Gamma$ of directed curves,
for every curve $\gamma\in \Gamma$, let $s(\gamma)$ be the first endpoint of $\gamma$, and let $t(\gamma)$ be its last endpoint. Let $S(\Gamma)=\set{s(\gamma)\mid \gamma\in \Gamma}$, and let $T(\Gamma)=\set{t(\gamma)\mid \gamma\in \Gamma}$, where both $S(\Gamma),T(\Gamma)$ are multisets, so, for example, for a point $p$, $n_{S(\Gamma)}(p)$ is the number of curves $\gamma\in \Gamma$ with $p=s(\gamma)$.
\begin{figure}[h]
\centering
\subfigure[Before: curve set $\Gamma$ contains all curves except for the yellow curve. Note that there are four crossing points between the yellow curve and the curves in $\Gamma$.]{\scalebox{0.5}{\includegraphics{figs/type_2_uncross_1.jpg}
}
\hspace{0.5cm}
\subfigure[After: Curves of $\Gamma$ are uncrossed and they do not cross each other anymore. The multisets $S(\Gamma)$, $T(\Gamma)$ remain the same, but they are matched to each other differently. There are still four crossing points between the yellow curve and the remaining curves.]{
\scalebox{0.51}{\includegraphics{figs/type_2_uncross_2.jpg}}}
\caption{An illustration of type-2 uncrossing.}
\label{fig:type_2_uncrossing}
\end{figure}
Intuitively, given a set $\Gamma$ of curves (not necessarily in general position), we can compute another set $\Gamma'$ of curves as shown in Figure~\ref{fig:type_2_uncrossing}, such that the curves of $\Gamma'$ do not cross with each other, $S(\Gamma)=S(\Gamma')$ and $T(\Gamma)=T(\Gamma')$, and and no additional crossings are created.
We call this process, \emph{type-2 uncrossing}.
In fact we describe this process in a more general setting, where we are given a number of collections $\Gamma_0,\Gamma_1,\ldots,\Gamma_r$ of curves, and we apply the uncrossing described above to each of the sets $\Gamma_1,\ldots,\Gamma_r$ separately. We show that we can do so without increasing the total number of crossings.
We now formally define type-2 uncrossing in the following theorem, whose proof is provided in Appendix~\ref{apd: type-2 uncrossing}.
\begin{theorem}[Type-2 Uncrossing]
\label{thm: type-2 uncrossing}
There is an efficient algorithm, that receives as input collections $\Gamma_0,\Gamma_1,\ldots,\Gamma_r$ of simple directed curves, such that: (i) the number of points that lie in at least two curves of $\bigcup_{i=0}^r\Gamma_i$ is finite; (ii) for each curve in of $\bigcup_{i=0}^r\Gamma_i$, neither of its endpoint is an inner point of any other curve; and (iii) if a point $p$ lies on more than two curves of $\bigcup_{i=0}^r\Gamma_i$, then either it is an endpoint of every curve that contains $p$, or there is some index $1\leq i\leq r$, such that all curves containing $p$ belong to $\Gamma_i$, and $p$ is an inner point of each of these curves.
The algorithm computes, for each $1\le i\le r$, a set $\Gamma'_i$ of new simple directed curves, such that, if we denote by $\Gamma=\Gamma_0\cup (\bigcup_{i=1}^r\Gamma_i)$ and by $\Gamma'=\Gamma_0\cup(\bigcup_{i=1}^r\Gamma_i')$, the following hold:
\begin{itemize}
\item the curves in $\Gamma'$ are in general position;
\item for each $1\le i \le r$, $S(\Gamma_i)=S(\Gamma_i')$ and $T(\Gamma_i)=T(\Gamma_i')$;
\item for each $1\le i \le r$, the curves in $\Gamma'_i$ do not cross;
\item for every curve $\gamma\in \Gamma_0$, $\chi(\gamma,\Gamma'\setminus \Gamma_0)\le \chi(\gamma,\Gamma\setminus \Gamma_0)$; and
\item $\chi(\Gamma')\le \chi(\Gamma)$.
\end{itemize}
\end{theorem}
As mentioned already, although the algorithm from Theorem~\ref{thm: type-2 uncrossing} ensures that for each $1\le i\le r$, $S(\Gamma_i)=S(\Gamma_i')$ and $T(\Gamma_i)=T(\Gamma_i')$, the pairing between the points in $S(\Gamma_i)$ and the points in $T(\Gamma_i')$ that is defined by the curves in the new set $\Gamma_i'$ may be completely different from that provided by the curves in $\Gamma_i$.
The algorithm from Theorem~\ref{thm: type-2 uncrossing} processes each index $1\leq i\leq r$ one by one. In order to process index $i$, we iterate, as long as there is some point $p$ in the current drawing that serves as an inner point for at least two curves in the current set $\Gamma_i$. We then consider a very small disc $\mu(p)$ containing the point $p$. Let $\Gamma_i(p)\subseteq\Gamma_i$ be the set of curves of $\Gamma_i$ containing the point $p$ as an inner point. For each such curve $\gamma$, we delete the segment of $\gamma$ lying in disc $\mu(p)$, thereby obtaining two segments of $\gamma$, denote by $\gamma_1$ and $\gamma_2$, where $\gamma_1$ is the segment whose endpoint lies in $S(\Gamma_i)$. We then consider two resulting sets of curves: $\Gamma'_i(p)=\set{\gamma_1\mid \gamma\in \Gamma_i(p)}$ and $\Gamma''_i(p)=\set{\gamma_2\mid \gamma\in \Gamma_i(p)}$. We ``stitch'' the two sets of curves together by defining a new collection $\Gamma'''(p)$ of curves, that are mutually disjoint and contained in the disc $\mu(p)$, such that for each curve $\gamma_1\in \Gamma'_i(p)$ there is exactly one curve in $\Gamma'''_i(p)$ originating at the endpoint of $\gamma_1$ lying on the boundary of $\mu(p)$, and similarly, for each curve $\gamma_2\in \Gamma''_i(p)$ there is exactly one curve in $\Gamma'''_i(p)$ terminating at the endpoint of $\gamma_2$ lying on the boundary of $\mu(p)$. We also ensure that $|\Gamma_i'''(p)|=|\Gamma_i'(p)|=|\Gamma_i''(p)|$. By combining the curves in $\Gamma'(p),\Gamma''_i(p)$ and $\Gamma'''_i(p)$, we obtain a new set of curves that replace the curves of $\Gamma_i(p)$ in set $\Gamma_i$, and continue to the next iteration. Therefore, the above algorithm ensures one additional property that will be useful for us later:
\begin{properties}{C}
\item For all $1\leq i\leq r$, if $p$ is a point that lies on some curve of $\Gamma'_i$, but $p$ does not lie on any curve of $\Gamma_i$, then there is some other point $p^*$, that serves as an inner point of at least two curves of $\Gamma_i$, and point $p^*$ lies in the small disc $\mu(p^*)$ containing $p^*$. \label{prop: curve rerouting}
\end{properties}
In other words, the property ensures that the sets of points on the sphere that lie on the curves of $\Gamma'_i$ is contained in the set of points on the sphere that lie on the curves of $\Gamma_i$, and the discs $\mu(p^*)$ of points $p^*$ that serve as inner points on at least two curves of $\Gamma_i$.
{the original type-2 uncrossing}
\fi
\iffalse{for edge-disjoint paths}
Let $G$ be a graph and $\Sigma$ be a rotation system on $G$.
Let $P_1,P_2$ be edge-disjoint paths in graph $G$, and some vertex $u$ that is an inner vertex of both $P_1$ and $P_2$. Denote by $e_1,e_1'$ the two edges of $P_1$ that are incident to $u$, and similarly denote by $e_2,e_2'$ the two edges of $P_2$ that are incident to $u$. We say that the intersection of the paths $P_1,P_2$ at vertex $u$ is \emph{non-transversal} iff the edges $e_1,e_1',e_2,e_2'$ appear in the ordering ${\mathcal{O}}_u\in \Sigma$ in one of the following two orders: either (i) $(e_1,e_1',e_2,e_2')$ or (ii) $(e_1',e_1,e_2,e_2')$.
We say that a set ${\mathcal{P}}$ of edge-disjoint paths is \emph{locally non-interfering with respect to $\Sigma$}, iff for every pair $P_1,P_2$ of distinct paths in ${\mathcal{P}}$ and for every vertex $u$ that is an inner vertex on both paths, the intersection of $P_1$ and $P_2$ at $u$ is non-transversal.
\fi
\iffalse
Throughout this subsection, we assume that we are given an instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem. We start by defining the notion of non-transversal intersection of paths, which we then use to define non-transversal paths.
\begin{definition}[Non-transversal intersection of paths]
Let $P_1,P_2$ be simple paths in graph $G$, and let $u$ be a vertex that is an inner vertex of both $P_1$ and $P_2$. Denote by $e_1,e_1'$ the two edges of $P_1$ that are incident to $u$, and similarly denote by $e_2,e_2'$ the two edges of $P_2$ that are incident to $u$. We say that the intersection of the paths $P_1,P_2$ at vertex $u$ is \emph{non-transversal} iff one of the following hold:
\begin{itemize}
\item either the set $\set{e_1,e_1',e_2,e_2'}$ contains fewer than $4$ distinct edges; or
\item all edges in set $\set{e_1,e_1',e_2,e_2'}$ are distinct,
and they appear in the ordering ${\mathcal{O}}_u\in \Sigma$ in one of the following two circular orders: either (i) $(e_1,e_1',e_2,e_2')$ or (ii) $(e_1',e_1,e_2,e_2')$.
\end{itemize}
Otherwise, we say that the intersection of the paths $P_1,P_2$ is \emph{transversal} at vertex $u$.
\end{definition}
Similarly, for simple cycles $R_1,R_2$ in $G$ and a vertex $u\in V(R_1)\cap V(R_2)$, we define the ``non-transversal intersection of $R_1$ and $R_2$ at vertex $u$ in the same way.
\iffalse
Let $G$ be a graph, let $P$ be a path in $G$, and let $v\in V(P)$ be a vertex of $P$. We denote by $P[v]$ the subpath of $P$ consisting of the edges in $P$ that are incident to $v$. Namely, if $P=(v_1,v_2,\ldots,v_k)$, then $P[v_1]=(v_1,v_2)$ and $P[v_k]=(v_{k-1},v_k)$, and for each $2\le i\le k-1$, $P[v_i]=(v_{i-1},v_i,v_{i+1})$.
For a set ${\mathcal{P}}$ of paths in $G$ and a vertex $v$, we denote by ${\mathcal{P}}_v\subseteq {\mathcal{P}}$ the subset of paths that contain the node $v$, and we define ${\mathcal{P}}[v]=\set{P[v]\mid P\in {\mathcal{P}}_v}$, and define $G[{\mathcal{P}},v]$ be the subgraph of $G$ that contains all paths in ${\mathcal{P}}[v]$. Therefore, $G[{\mathcal{P}},v]$ is a star graph with $v$ serving as the center of the star.
Given a rotation ${\mathcal{O}}_v$ on the edges of $\delta_G(v)$, we say that the set ${\mathcal{P}}$ of paths is \emph{locally non-interfering at $v$ with respect to ${\mathcal{O}}_v$}, if the paths of ${\mathcal{P}}[v]$ are non-interfering with respect to the unique planar drawing of $G[{\mathcal{P}},v]$ that respects ${\mathcal{O}}_v$.
\fi
\iffalse
We use the following lemma, whose proof appears in Appendix~\ref{apd:locally_non_interfering}.
\begin{lemma}
\label{lem:local_non_int}
There exists an efficient algorithm that, given a graph $G$, a rotation system $\Sigma$ on $G$ and a set $\tilde{\mathcal{P}}$ of edge-disjoint paths connecting a pair $v,v'$ of distinct vertices of $G$, computes another set ${\mathcal{P}}$ of $|{\mathcal{P}}|=|\tilde{{\mathcal{P}}}|$ edge-disjoint paths in $G$ connecting $v$ to $v'$, such that $E({\mathcal{P}})\subseteq E(\tilde{\mathcal{P}})$, and ${\mathcal{P}}$ is locally non-interfering at all vertices of $V(G)\setminus\set{v,v'}$ with respect to their rotations/semi-rotations in $\Sigma$.
\end{lemma}
\fi
We will repeatedly use the following lemma for modifying a set of paths that share a common vertex. Its proof appears in \Cref{apd: Proof of rerouting_matching_cong}.
\begin{lemma}
\label{obs:rerouting_matching_cong}
There is an efficient algorithm, that, given a graph $G$, a vertex $v$ of $G$, an ordering ${\mathcal{O}}_v$ on edges of $\delta_G(v)$ and for each edge $e\in \delta_G(v)$, two integers $n^-_e, n^+_e\ge 0$, such that $\sum_{e\in \delta_G(v)}n^-_e=\sum_{e\in \delta_G(v)}n^+_e$, computes a multiset $M\subseteq \delta_G(v)\times \delta_G(v)$ of $|M|=\sum_{e\in \delta_G(v)}n^+_e$ ordered pairs of the edges of $\delta_G(v)$, such that
\begin{itemize}
\item for each $e\in \delta_G(v)$, $M$ contains $n^-_e$ pairs $(e^-,e^+)$ with $e^-=e$;
\item for each $e\in \delta_G(v)$, $M$ contains $n^+_e$ pairs $(e^-,e^+)$ with $e^+=e$; and
\item for each pair $(e^-_1,e^+_1),(e^-_2,e^+_2)$ of ordered pairs in $M$, the intersection between path $P_1=(e^-_1,e^+_1)$ and path $P_2=(e^-_2,e^+_2)$ at vertex $v$ is non-transversal with respect to ${\mathcal{O}}_v$.
\end{itemize}
\end{lemma}
\begin{definition}[Non-transversal paths]
Let $\Sigma$ be a rotation system on graph $G$.
We say that a set ${\mathcal{P}}$ of simple paths in graph $G$ is \emph{non-transversal with respect to $\Sigma$}, iff for every pair $P_1,P_2$ of distinct paths in ${\mathcal{P}}$, for every vertex $u$ that is an inner vertex on both $P_1$ and $P_2$, the intersection between $P_1$ and $P_2$ at vertex $u$ is non-transversal.
\end{definition}
Using \Cref{obs:rerouting_matching_cong}, we can prove the following lemma that allows us to transform an arbitrary set ${\mathcal{R}}$ of paths into a set ${\mathcal{R}}'$ of non-transversal paths that have the same endpoints as paths in ${\mathcal{R}}$, without increasing the congestion on any edge.
The proof of the lemma below is similar to the proof of Lemma 9.5 in~\cite{chuzhoy2020towards}, and is provided in Appendix~\ref{apd: Proof of non_interfering_paths} for completeness.
\znote{maybe we need to introduce congestion before this subsection? (for the following lemmas and theorems)}
\fi
\iffalse
\begin{lemma}
\label{lem: non transversal cost of cycles bounded by cr}
Let $I=(G,\Sigma)$ be an instance of \textnormal{\textsf{MCNwRS}}\xspace and let $\phi$ be any solution to $I$ on the sphere. Let ${\mathcal{R}}$ be a set of simple cycles in $G$, such that, for every pair $R,R'$ of edge-disjoint cycles in ${\mathcal{R}}$, the intersection of $R$ and $R'$ are transversal at at most one of their common vertices. Then
\[\mathsf{cost}_{\mathsf {NT}}({\mathcal{R}},\Sigma) \le \sum_{e\in E(G)}\chi_{\phi}(e)\cdot(\cong_G({\mathcal{R}},e))^2.\]
\end{lemma}
\fi
\iffalse
\paragraph{Non-Transversal Cost.} Given an instance $(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace and a set ${\mathcal{Q}}$ of paths in $G$, we define the \emph{non-transversal cost of ${\mathcal{Q}}$ with respect to $\Sigma$}, which is denoted by $\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}};\Sigma)$, as the number of unordered pairs $Q,Q'$ of paths in ${\mathcal{Q}}$, such that paths $Q$ and $Q'$ are edge-disjoint, and the intersection of $Q,Q'$ are non-transversal at some vertex that is an inner vertex of both $Q$ and $Q'$.
Note that, if the set ${\mathcal{Q}}$ of paths is non-transversal with respect to $\Sigma$, then $\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}};\Sigma)=0$. Similarly, given a set ${\mathcal{R}}$ of cycles in $G$, the non-transversal cost $\mathsf{cost}_{\mathsf {NT}}({\mathcal{R}},\Sigma)$ of ${\mathcal{R}}$ with respect to $\Sigma$ is defined similarly.
Additionally, for two sets ${\mathcal{Q}},{\mathcal{Q}}'$ of paths in $G$, we define $\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}},{\mathcal{Q}}';\Sigma)$ as the number of pairs $(Q,Q')$ of paths with $Q\in {\mathcal{Q}}$ and $Q'\in {\mathcal{Q}}'$, such that paths $Q$ and $Q'$ are edge-disjoint, and the intersection of $Q,Q'$ are non-transversal at some vertex that is an inner vertex of both $Q$ and $Q'$.
By definition, $\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}};\Sigma)= \mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}},{\mathcal{Q}};\Sigma)$.
We will use the following lemma. Its proof appears in \Cref{apd: Proof of non transversal cost of cycles bounded by cr}.
\begin{lemma}
\label{lem: non transversal cost of cycles bounded by cr}
Let $I=(G,\Sigma)$ be an instance of \textnormal{\textsf{MCNwRS}}\xspace and let $\phi$ be any solution to $I$ on the sphere. Let ${\mathcal{R}},{\mathcal{R}}'$ be two sets of simple cycles in graph $G$, such that, for every pair $R,R'$ of edge-disjoint cycles with $R\in {\mathcal{R}}$ and $R'\in {\mathcal{R}}'$, the intersection of $R$ and $R'$ are transversal with respect to $\Sigma$ at at most one of their common vertices. Then
\[\mathsf{cost}_{\mathsf {NT}}({\mathcal{R}},{\mathcal{R}}';\Sigma) \le \sum_{e,e'\in E(G)}\chi_{\phi}(e,e')\cdot\bigg(\cong_G({\mathcal{R}},e)\cdot \cong_G({\mathcal{R}}',e')+\cong_G({\mathcal{R}},e')\cdot \cong_G({\mathcal{R}}',e)\bigg).\]
\end{lemma}
We will use the following theorem. Its proof appears in \Cref{apd: Proof of curve_manipulation}.
\begin{theorem}
\label{thm: new nudging}
There is an efficient algorithm, whose input contains a graph $G$, a drawing $\phi$ of $G$ on the sphere, a subgraph $C$ of $G$, two sets ${\mathcal{Q}},{\mathcal{Q}}'$ of paths in $G$ that share an endpoint $t^*$, that is refered to as their common last endpoint, and an circular ordering ${\mathcal{O}}_{t^*}$ on its incident edges, such that (i) all paths of ${\mathcal{Q}}\cup {\mathcal{Q}}'$ are internally disjoint from $C$; (ii) for each $Q\in {\mathcal{Q}}\cup {\mathcal{Q}}'$, the other endpoint of $Q$, denoted by $s_Q$, is designated as its first endpoint; and (iii) for every pair $Q,Q'$ of edge-disjoint paths in ${\mathcal{Q}}\cup {\mathcal{Q}}'$, the intersection of $Q$ and $Q'$ is transversal at at most one of their common vertices. The algorithm computes a set $\Gamma=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}\cup {\mathcal{Q}}'}$ of directed curves on the sphere with the following properties:
\begin{itemize}
\item the curves in $\Gamma$ are in general position;
\item for each path $Q\in {\mathcal{Q}}\cup {\mathcal{Q}}'$, $s(\gamma(Q))=\phi(s_Q)$ and $t(\gamma(Q))=\phi(t^*)$;
\item $\chi(\Gamma)\le |{\mathcal{Q}}'|^2+\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}};\Sigma)+\mathsf{cost}_{\mathsf {NT}}({\mathcal{Q}},{\mathcal{Q}}';\Sigma)+\sum_{e\in E(G)}(\chi_{\phi}(e)+1)\cdot (\cong_G({\mathcal{Q}},e))^2\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }
+\sum_{e\in E(G)} \cong_G({\mathcal{Q}},e)\cdot\cong_G({\mathcal{Q}}',e)+\sum_{e,e'\in E(G)}
\chi_{\phi}(e,e')\cdot \cong_G({\mathcal{Q}},e)\cdot\cong_G({\mathcal{Q}}',e')$;
\item the circular ordering in which curves of $\Gamma$ enter $\phi(t^*)$ is identical to ${\mathcal{O}}^{\operatorname{guided}}({\mathcal{Q}}\cup {\mathcal{Q}}',{\mathcal{O}}_{t^*})$, the circular ordering in which paths of ${\mathcal{Q}}\cup {\mathcal{Q}}'$ enter $t^*$; and
\item for each edge $e\in E(C)$, the number of crossings between the image of $e$ in $\phi$ and the curves in $\Gamma$ is bounded by the number of crossings between the image of $e$ and images of the edges of $G\setminus C$ times $\cong_G({\mathcal{Q}}\cup {\mathcal{Q}}')$, namely $\chi(\phi(e),\Gamma)\le \chi_{\phi}(e, G\setminus C)\cdot \cong_G({\mathcal{Q}}\cup {\mathcal{Q}}')$.
\end{itemize}
\end{theorem}
Given an instance $(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace and a simple path $P$ with endpoints $s,t$ in $G$, we define the \emph{$P$-extraction} as the following operations on $G$ and $\Sigma$:
(i) delete all edges of $E(P)$ from $G$;
(ii) for each internal vertex $u\in V(P)$, delete the two edges of $P$ that are incident to $u$ from the ordering ${\mathcal{O}}_u\in \Sigma$;
(ii) add an edge $e_P=(s,t)$ to $G$; and
(iv) replace the edge of $P$ that is incident to $s$ with the edge $e_P$ in the ordering ${\mathcal{O}}_s\in \Sigma$, and replace the edge of $P$ that is incident to $t$ with the edge $e_P$ in the ordering ${\mathcal{O}}_t\in \Sigma$.
Similarly, if we are given a set ${\mathcal{P}}$ of edge-disjoint paths in $G$, then \emph{${\mathcal{P}}$-extraction} is defined to be a sequence of operations that contains a $P$-extraction for each path $P\in {\mathcal{P}}$. It is easy to see that the order in which these $P$-extractions appear in this sequence does not matter.
We use the following lemma that allows us to modify a solution $\phi$ of instance $(G,\Sigma)$ to a solution of the instance $(G',\Sigma')$ that is obtained from applying ${\mathcal{P}}$-contraction to $(G,\Sigma)$. Its proof appears in \Cref{apd: Proof of curves from non-transversal paths}.
\begin{lemma}
\label{lem: curves from non-transversal paths}
There is an algorithm, that, given an instance $(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, a feasible solution $\phi$ for this instance, and a set ${\mathcal{P}}$ of edge-disjoint paths that are non-transversal with respect to $\Sigma$, efficiently computes a feasible solution $\phi'$ for the instance $(G',\Sigma')$ that is obtained by applying ${\mathcal{P}}$-contraction to $(G,\Sigma)$, such that $\mathsf{cr}(\phi')\le \mathsf{cr}(\phi)$.
\end{lemma}
\fi
\subsection{Routing Paths}
\label{sec: routing paths}
\paragraph{Routing Paths.} Given a graph $G$ and two sets $S,T\subseteq V(G)$,
a \emph{routing of vertices of $S$ to vertices of $T$} is a collection ${\mathcal{Q}}=\set{Q_v\mid v\in S}$ of paths, where for each vertex $v\in S$, path $Q_v$ originates at a vertex of $v$ and terminates at a vertex of $T$.
If, additionally, for every vertex of $T$, exactly one path in ${\mathcal{Q}}$ terminates at $T$, then we say that ${\mathcal{Q}}$ is an \emph{one-to-one routing} of vertices of $S$ to vertices of $T$.
Similarly, given two sets $E_1,E_2$ of edges of $G$, we say that a set ${\mathcal{Q}}=\set{Q_e\mid e\in E_1}$ of paths is a routing of edges of $E_1$ to edges of $E_2$, or that ${\mathcal{Q}}$ routes edges of $E_1$ to edges of $E_2$, iff for every edge $e\in E_1$, path $Q_e$ has $e$ as its first edge, and some edge of $E_2$ as its last edge. If, additionally, every edge of $E_2$ serves as the last edge on exactly one path in ${\mathcal{Q}}$, then we say that ${\mathcal{Q}}$ is an one-to-one routing of edges of $E_1$ to edges of $E_2$.
\begin{definition}[Routing Paths inside and outside a Cluster]
Let $G$ be a graph and let $C$ be a cluster of $G$. We say that a set ${\mathcal{Q}}(C)=\set{Q_e\mid e\in \delta_G(C)}$ of paths \emph{route the edges of $\delta_G(C)$ inside $C$ to a vertex $v\in V(C)$}, iff for all $e\in \delta_G(C)$, path $Q_e$ has $e$ as its first edge, $v$ as its last vertex, and $Q_e\setminus\set{e}$ is contained in $C$.
We say that a set ${\mathcal{Q}}'(C)=\set{Q'_e\mid e\in \delta_G(C)}$ of paths \emph{route the edges of $\delta_G(C)$ outside $C$ to a vertex $v'\in V(G)\setminus V(C)$}, iff for all $e\in \delta_G(C)$, path $Q'_e$ has $e$ as its first edge, $v'$ as its last vertex, and $Q'_e\setminus\set{e}$ is disjoint from $C$.
\end{definition}
We repeatedly use the following simple claim, whose proof appears in \Cref{apd: Proof of remove congestion}.
\begin{claim}\label{claim: remove congestion}
Let $G$ be a graph, let $S$, $T$ be two disjoint multisets of its vertices, and let ${\mathcal{P}}$ be a collection of paths in $G$, where every path in ${\mathcal{P}}$ originates at a vertex of $S$ and terminates at a vertex of $T$,
such that for every vertex $v$, at most $n_S(v)$ paths in ${\mathcal{P}}$ originate at $v$ and at most $n_T(v)$ paths in ${\mathcal{P}}$ terminate at $v$. Assume further that the paths in ${\mathcal{P}}$ cause congestion $\rho$. Then there is a collection ${\mathcal{P}}'$ of edge-disjoint paths in $G$, with $|{\mathcal{P}}'|\geq |{\mathcal{P}}|/\rho$, such that for every vertex $v$, at most $n_S(v)$ paths in ${\mathcal{P}}'$ originate at $v$ and at most $n_T(v)$ paths in ${\mathcal{P}}'$ terminate at $v$.
\end{claim}
\subsection{Non-Transversal Paths and Path Splicing}
\label{subsec: non-transversal paths and splicing}
We start by defining the notion of non-transversal intersection of paths, which we then use to define non-transversal paths.
\begin{definition}[Non-transversal intersection of paths and cycles]
Let $I=(G,\Sigma)$ be an intance of \ensuremath{\mathsf{MCNwRS}}\xspace, let $P_1,P_2$ be two simple paths in $G$, and let $u$ be a vertex in $V(P_1)\cap V(P_2)$. Denote by $E_1$ the set of (one or two) edges of $P_1$ that are incident to $u$, and similarly denote by $E_2$ the set of (one or two) edges of $P_2$ that are incident to $u$. We say that the intersection of the paths $P_1,P_2$ at vertex $u$ is \emph{non-transversal with respect to $\Sigma$} iff one of the following hold:
\begin{itemize}
\item either the set $E_1\cup E_2$ contains fewer than $4$ distinct edges; or
\item $E_1=\set{e_1,e_1'}$ and $E_2=\set{e_2,e_2'}$, all edges in set $\set{e_1,e_1',e_2,e_2'}$ are distinct,
and they appear in the ordering ${\mathcal{O}}_u\in \Sigma$ in one of the following two circular orders: $(e_1,e_1',e_2,e_2')$ or $(e_1',e_1,e_2,e_2')$.
\end{itemize}
Otherwise, we say that the intersection of the paths $P_1,P_2$ at vertex $u$ is \emph{transversal}.
If $R_1,R_2$ are simple cycles in $G$, and $u$ is a vertex in $V(R_1)\cap V(R_2)$, then we classify the intersection of $R_1$ and $R_2$ and $u$ as transversal or non-transversal with respect to $\Sigma$ similarly.
\end{definition}
\begin{definition}[Non-transversal set of paths]\label{def: non-transversal paths}
Let $I=(G,\Sigma)$ be an intance of \ensuremath{\mathsf{MCNwRS}}\xspace, and let ${\mathcal{P}}$ be a collection of simple paths in $G$. We say that path set ${\mathcal{P}}$ is \emph{non-transversal with respect to $\Sigma$} iff for every pair $P_1,P_2\in {\mathcal{P}}$ of paths, for every vertex $u\in V(P_1)\cap V(P_2)$,
the intersection of $P_1$ and $P_2$ at $u$ is non-transversal with respect to $\Sigma$.
\end{definition}
Assume now that we are given some instance $I=(G,\Sigma)$, and a collection ${\mathcal{Q}}$ of simple paths in $G$.
We let $\Pi^T({\mathcal{Q}})$ denote the set of all triples $(Q,Q',v)$, such that $Q,Q'\in {\mathcal{Q}}$, $v$ is an inner vertex of both $Q$ and $Q'$, and the intersection of $Q$ and $Q'$ at $v$ is transversive with respect to $\Sigma$.
In some cases, we will be given some set ${\mathcal{P}}$ of simple paths in a graph $G$, and we will need to transform it into a set ${\mathcal{P}}'$ of paths that is non-transversal with respect to the given rotation system $\Sigma$ for $G$, while ensuring that $S({\mathcal{P}}')=S({\mathcal{P}})$ and $T({\mathcal{P}}')=T({\mathcal{P}})$. Below we provide a procedure for doing so. The procedure uses a simple subroutine that we call \emph{path splicing} and describe next.
\paragraph{Path Splicing.}
Suppose we are given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, two simple paths $P,P'$ in $G$, and a vertex $v$, that serves as an inner vertex of both $P$ and $P'$, such that the intersection of $P$ and $P'$ at vertex $v$ is transversal with respect to $\Sigma$. We assume that each of the paths $P,P'$ is assigned a direction, and we denote by $s$ and $t$ the first and the last endpoints of $P$, respectively, and by $s'$ and $t'$ the first and the last endpoints of $P'$, respectively. The \emph{splicing} of $P$ and $P'$ at vertex $v$ produces two new paths: path $\tilde P$ that is a concatenation of the subpath of $P$ from $s$ to $v$, and the subpath of $P'$ from $v$ to $t'$; and path $\tilde P'$, that is a concatenation of the subpath of $P'$ from $s'$ to $v$, and the subpath of $P$ from $v$ to $t$. We use the following simple observation regarding the splicing procedure, whose proof is deferred to \Cref{apd: Proof of splicing}.
\begin{observation}\label{obs: splicing}
Let $I=(G,\Sigma)$ be an instance of $\ensuremath{\mathsf{MCNwRS}}\xspace$, let ${\mathcal{P}}$ be a set of simple (directed) paths in $G$,
and let $(P,P',v)$ be a triple in $\Pi^T({\mathcal{P}})$. Let $\tilde P,\tilde P'$ be the pair of paths obtained by splicing $P$ and $P'$ at $v$, and let ${\mathcal{P}}'=\big({\mathcal{P}}\setminus\set{P,P'}\big)\cup \set{\tilde P,\tilde P'}$.
Then $S({\mathcal{P}}')=S({\mathcal{P}})$ and $T({\mathcal{P}}')=T({\mathcal{P}})$. Additionally, either (i) at least one of the paths $\tilde P,\tilde P'$ is a non-simple path; or (ii) $|\Pi^T({\mathcal{P}}')|<|\Pi^T({\mathcal{P}})|$.
\end{observation}
\iffalse
Let $G$ be a graph, let $P$ be a path in $G$, and let $v\in V(P)$ be a vertex of $P$. We denote by $P[v]$ the subpath of $P$ consisting of the edges in $P$ that are incident to $v$. Namely, if $P=(v_1,v_2,\ldots,v_k)$, then $P[v_1]=(v_1,v_2)$ and $P[v_k]=(v_{k-1},v_k)$, and for each $2\le i\le k-1$, $P[v_i]=(v_{i-1},v_i,v_{i+1})$.
For a set ${\mathcal{P}}$ of paths in $G$ and a vertex $v$, we denote by ${\mathcal{P}}_v\subseteq {\mathcal{P}}$ the subset of paths that contain the node $v$, and we define ${\mathcal{P}}[v]=\set{P[v]\mid P\in {\mathcal{P}}_v}$, and define $G[{\mathcal{P}},v]$ be the subgraph of $G$ that contains all paths in ${\mathcal{P}}[v]$. Therefore, $G[{\mathcal{P}},v]$ is a star graph with $v$ serving as the center of the star.
Given a rotation ${\mathcal{O}}_v$ on the edges of $\delta_G(v)$, we say that the set ${\mathcal{P}}$ of paths is \emph{locally non-interfering at $v$ with respect to ${\mathcal{O}}_v$}, if the paths of ${\mathcal{P}}[v]$ are non-interfering with respect to the unique planar drawing of $G[{\mathcal{P}},v]$ that respects ${\mathcal{O}}_v$.
\fi
\iffalse
We use the following lemma, whose proof appears in Appendix~\ref{apd:locally_non_interfering}.
\begin{lemma}
\label{lem:local_non_int}
There exists an efficient algorithm that, given a graph $G$, a rotation system $\Sigma$ on $G$ and a set $\tilde{\mathcal{P}}$ of edge-disjoint paths connecting a pair $v,v'$ of distinct vertices of $G$, computes another set ${\mathcal{P}}$ of $|{\mathcal{P}}|=|\tilde{{\mathcal{P}}}|$ edge-disjoint paths in $G$ connecting $v$ to $v'$, such that $E({\mathcal{P}})\subseteq E(\tilde{\mathcal{P}})$, and ${\mathcal{P}}$ is locally non-interfering at all vertices of $V(G)\setminus\set{v,v'}$ with respect to their rotations/semi-rotations in $\Sigma$.
\end{lemma}
\fi
\iffalse
\mynote{I suspect that this lemma is not needed anymore, but we can decide whether to keep it later.}
We will repeatedly use the following lemma for modifying a set of paths that share a common vertex. Its proof appears in \Cref{apd: Proof of rerouting_matching_cong}.
\begin{lemma}
\label{obs:rerouting_matching_cong}
There is an efficient algorithm, that, given a graph $G$, a vertex $v$ of $G$, an ordering ${\mathcal{O}}_v$ on edges of $\delta_G(v)$ and for each edge $e\in \delta_G(v)$, two integers $n^-_e, n^+_e\ge 0$, such that $\sum_{e\in \delta_G(v)}n^-_e=\sum_{e\in \delta_G(v)}n^+_e$, computes a multiset $M\subseteq \delta_G(v)\times \delta_G(v)$ of $|M|=\sum_{e\in \delta_G(v)}n^+_e$ ordered pairs of the edges of $\delta_G(v)$, such that
\begin{itemize}
\item for each $e\in \delta_G(v)$, $M$ contains $n^-_e$ pairs $(e^-,e^+)$ with $e^-=e$;
\item for each $e\in \delta_G(v)$, $M$ contains $n^+_e$ pairs $(e^-,e^+)$ with $e^+=e$; and
\item for each pair $(e^-_1,e^+_1),(e^-_2,e^+_2)$ of ordered pairs in $M$, the intersection between path $P_1=(e^-_1,e^+_1)$ and path $P_2=(e^-_2,e^+_2)$ at vertex $v$ is non-transversal with respect to ${\mathcal{O}}_v$.
\end{itemize}
\end{lemma}
\fi
\iffalse
\begin{definition}[Non-transversal paths]
Let $\Sigma$ be a rotation system on graph $G$.
We say that a set ${\mathcal{P}}$ of simple paths in graph $G$ is \emph{non-transversal with respect to $\Sigma$}, iff for every pair $P_1,P_2$ of distinct paths in ${\mathcal{P}}$, for every vertex $u$ that is an inner vertex on both $P_1$ and $P_2$, the intersection between $P_1$ and $P_2$ at vertex $u$ is non-transversal.
\end{definition}
\fi
\iffalse
\mynote{for the lemma below: 1. do we need to require that the paths are edge-disjoint? 2. The lemma does not say that the routing is one-to-one. I think the lemma should be resetated like this:
}
\fi
Using \Cref{obs: splicing}, we can prove the following lemma that allows us to transform an arbitrary set ${\mathcal{R}}$ of paths into a set ${\mathcal{R}}'$ of non-transversal paths that have the same endpoints as paths in ${\mathcal{R}}$, without increasing the congestion on any edge.
The proof of the lemma below is similar to the proof of Lemma 9.5 in~\cite{chuzhoy2020towards}, and is provided in Appendix~\ref{apd: Proof of non_interfering_paths} for completeness.
\iffalse
\begin{lemma}
\label{lem: non_interfering_paths}
There is an efficient algorithm, that, given an instance $I=(G, \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, two disjoint multisets $S,T$ of vertices of $G$, and a set ${\mathcal{R}}$ of simple edge-disjoint paths in $G$ routing vertices of $S$ to vertices of $T$, such that the vertices of $S\cup T$ may not serve as inner vertices on any path in ${\mathcal{R}}$, computes a set ${\mathcal{R}}'$ of $|{\mathcal{R}}'|=|{\mathcal{R}}|$ edge-disjoint paths in $G$ vertices of $S$ to vertices of $T$, such that vertices of $S\cup T$ may not be an inner vertex on any path in ${\mathcal{R}}$, and (i) the paths in ${\mathcal{R}}'$ are non-transversal with respect to $\Sigma$; and (ii) $E({\mathcal{R}}')\subseteq E({\mathcal{R}})$.
\end{lemma}
\fi
\begin{lemma}
\label{lem: non_interfering_paths}
There is an efficient algorithm, that, given an instance $(G, \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace and a set ${\mathcal{R}}$ of (directed) paths in $G$, computes another set ${\mathcal{R}}'$ of simple (directed) paths in $G$, such that $S({\mathcal{R}}')=S({\mathcal{R}})$, $T({\mathcal{R}}')=T({\mathcal{R}})$, and the paths in ${\mathcal{R}}'$ are non-transversal with respect to $\Sigma$. Moreover, for every edge $e\in E(G)$, $\cong_G({\mathcal{R}}',e)\leq \cong_G({\mathcal{R}},e)$.
\end{lemma}
\subsection{Balanced Cut and Sparsest Cut}
Suppose we are given a graph $G=(V,E)$, and a subset $T\subseteq V$ of its vertices. We say that a cut $(X,Y)$ in $G$ is a valid $T$-cut iff $X\cap T,Y\cap T\neq \emptyset$. The \emph{sparsity} of a valid $T$-cut $(X,Y)$ is $\frac{|E(X,Y)|}{\min\set{|X\cap T|, |Y\cap T|}}$.
In the Sparsest Cut problem, given a graph $G$ and a subset $T$ of its vertices, the goal is to compute a valid $T$-cut of minimum sparsity. Arora, Rao and Vazirani~\cite{ARV} have shown an $O(\sqrt {\log n})$-approximation algorithm for the sparsest cut problem\footnote{The algorithm was originally designed for finding approximate sparsest cut on simple graphs, but it can be easily generalized to graphs with parallel edges by using edge costs.}.
We denote this algorithm by \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace, and its approximation factor by $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)=O(\sqrt{\log n})$.
Let $\theta$ be some parameter such that $0<\theta<1/2$. We say that a cut $(A,B)$ of $G$ is \emph{$\theta$-vertex-balanced} iff $|A|,|B|\ge \theta\cdot |V(G)|$.
We say that a cut $(A,B)$ of $G$ is \emph{$\theta$-edge-balanced} iff $|E(A)|,|E(B)|\ge \theta\cdot|E(G)|$.
In this paper, when we refer to a $\theta$-balanced cut, we mean a $\theta$-edge-balanced cut.
We will use the following theorem and its corollary for computing an approximate minimum balanced cut.
\begin{theorem}[Corollary 2 in~\cite{ARV}]
\label{thm: ARV}
For any constant $0<c<1/2$, there exists another constant $0<c'<c$, and an efficient algorithm, that, given any \textbf{simple} connected graph $G$ with $n$ vertices and $m$ edges, computes a $c'$-vertex-balanced cut of $G$, whose size is at most $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)$ times the size of a minimum $c$-vertex-balanced cut of $G$.
\end{theorem}
The proof of the following corollary of \Cref{thm: ARV} is provided in Appendix~\ref{apd: Proof of approx_balanced_cut}.
\begin{corollary}
\label{cor: approx_balanced_cut}
For any constant $0<\hat c<1/2$, there exists another constant $0<\hat c'<\hat c$, and an efficient algorithm, that, given any connected (not necessarily simple) graph $G$ with $n$ vertices and $m$ edges, computes a $\hat c'$-edge-balanced cut of $G$, whose size is at most $O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))$ times the size of a minimum $\hat c$-edge-balanced cut of $G$.
\end{corollary}
We will use the following lemma, that is a simple consequence of the Lipton-Tarjan separator theorem for planar graphs \cite{lipton1979separator}.
We note that the version for vertex-balanced cuts of the lemma below are proved in~\cite{pach1996applications}.
For completeness, we provide its proof appears in Appendix~\ref{apd: Proof of min_bal_cut}.
\begin{lemma}
\label{lem:min_bal_cut}
Let $G=(V,E)$ be a connected graph with maximum degree $\Delta\le |E|/2^{100}$. If $\mathsf{OPT}_{\mathsf{cr}}(G)\le |E|^2/2^{100}$, then the minimum $(1/4)$-edge-balanced cut in $G$ has size $O(\sqrt{\mathsf{OPT}_{\mathsf{cr}}(G)+\Delta\cdot|E|})$.
\end{lemma}
\subsection{Well-Linkedness, Bandwidth Property and Well-linked Decomposition}
The notion of well-linkedness has played a central role in graph theory and graph algorithms in the past decade (e.g. \cite{racke2002minimizing,chekuri2004edge,andrews2010approximation,chuzhoy2012polylogarithmic,chuzhoy2012routing,chekuri2016polynomial,chuzhoy2016improved,chuzhoy2019towards}).
We use the following standard definitions, which are equivalent to
those used in \cite{chuzhoy2012polylogarithmic,chuzhoy2012routing,chekuri2016polynomial,chuzhoy2016improved,chuzhoy2019towards}.
\begin{definition}[Well-Linkedness]
We say that a set $T$ of vertices of $G$ is \emph{$\alpha$-well-linked} in $G$, iff the value of the sparsest cut in $G$ with respect to $T$ is at least $\alpha$; equivalently, for every partition $(A,B)$ of $V(G)$ with $A,B\neq \emptyset$, $|E_G(A,B)|\geq \alpha\cdot \min\set{|A\cap T|,|B\cap T|}$.
\end{definition}
The next simple observation, that has been used extensively in previous work, shows that the set of vertices lying in the first row of the $(r\times r)$-grid is $1$-well-linked. For completeness, we provide a proof in \Cref{apd: Proof grid 1st row well-linked}.
\begin{observation}
\label{obs: grid 1st row well-linked}
Let $r\geq 1$ be an integer, and let $H$ be the $(r\times r)$-grid. Let $S$ be the set of vertices lying in the first row of the grid. Then vertex set $S$ is $1$-well-linked in $H$.
\end{observation}
\iffalse
Assume now that we are given a graph $G$, and some subgraph $C$ of $G$ that we call cluster. Throughout this paper, we will sometimes use a graph $C^+$ that is associated with $C$, whose definition is given below.
\begin{definition}[Graph $C^+$]
Let $G$ be a graph and $C\subseteq G$ be a cluster. Graph $C^+$ associated with $C$ is constructed as follows. First, we subdivide every edge $e\in \delta_G(C)$ with a vertex $t_e$, and let $T(C)=\set{t_e\mid e\in \delta_G(C)}$ be the set of newly added vertices, that we sometimes refer to as \emph{terminals of $C$}. We then let $C^+$ be the subgraph of the resulting graph induced by $V(C)\cup T(C)$.
\end{definition}
\fi
Next, we define the notion of bandwidth property, that was also used extensively in graph algorithms.
\begin{definition}[$\alpha$-Bandwidth Property]
We say that a cluster $C$ of graph $G$ has the \emph{$\alpha$-bandwidth property}, for some parameter $\alpha>0$, iff for every partition $(A,B)$ of vertices of $C$,
$|E(A,B)|\ge \alpha\cdot\min\set{|\delta_G(A)\cap \delta_G(C)|, |\delta_G(B)\cap \delta_G(C)|}$.
\end{definition}
The following immediate observation provides an equivalent definition of the bandwidth property that is helpful to keep in mind.
\begin{observation}\label{obs: wl-bw}
Let $G$ be a graph and $C\subseteq G$ a subgraph of $G$, and let $0<\alpha<1$ be a parameter. Cluster $C$ has the $\alpha$-bandwidth property iff the set $T(C)$ of vertices is $\alpha$-well-linked in graph $C^+$ (see \Cref{def: Graph C^+}).
\end{observation}
The next observation, that is quite standard, shows that a cluster with a bandwidth property must contain many edges.
\begin{observation}\label{obs: bw many edges}
Let $G$ be a graph and $C\subseteq G$ a subgraph of $G$ that has the $\alpha$-bandwidth property, for some $0<\alpha<1$. Denote $|\delta_G(C)|=k$, and let $v$ be the vertex maximizing $k_v=|\delta_G(C)\cap \delta_G(v)|$ among all vertices of $C$. Then $|E(C)|\geq \alpha\cdot (k-k_v)/3$.
\end{observation}
\begin{proof}
We rank all vertices $u$ of $C$ in decreasing order of $k_u$: $u_1,u_2,\ldots,u_r$ where $u_1=v$.
Assume first that $k_v<2k/3$. For each $1\le i\le r$, we denote $s_i=\sum_{1\le t\le i}k_{u_i}$. We first show that there exists an integer $1\le i<r$, such that $k/3\le s_i\le 2k/3$. In fact, if $k_{u_1}\ge k/3$, then $s_1$ satisfies the above property. Otherwise we have $k_{u_1}<k/3$, and we know that the sequence $s_1,\ldots,s_r$ is increasing and $s_r=k$, and the difference between every pair $s_i,s_{i+1}$ is less than $k/3$. It is easy to verify that there exists an integer $1\le i<r$, such that $k/3\le s_i\le 2k/3$. Let $i^*$ be such an integer, and consider the cut $(\set{u_t\mid 1\le t\le i^*},\set{u_t\mid i^*< t\le r})$ of $C$. From the $\alpha$-bandwidth property of $C$, the number of edges in this cut is at least $\alpha\cdot (k/3)\ge \alpha(k-k_v)/3$. Assume now that $k_v\ge 2k/3$. Then consider the the cut $(\set{u_1},\set{u_t\mid 2\le t\le r})$ of $C$. From the $\alpha$-bandwidth property of $C$, the number of edges in this cut is at least $\alpha(k-k_v)\ge \alpha(k-k_v)/3$.
\end{proof}
We use the following theorem and its immediate corollary, which are easy consequences of the max-flow min-cut theorem. For completeness, we provide its proof in Appendix~\ref{apd: Proof of bandwidth_means_boundary_well_linked}.
\begin{theorem}
\label{thm: bandwidth_means_boundary_well_linked}
There is an efficient algorithm, that, given a graph $G$, a set $T$ of vertices of $G$ that is $\alpha$-well-linked, and a pair $T_1,T_2$ of disjoint equal-cardinality subsets of $T$, computes an one-to-one routing ${\mathcal{Q}}$ of vertices of $T_1$ to vertices of $T_2$ with $\cong_G({\mathcal{Q}})\leq \ceil{1/\alpha}$.
\end{theorem}
\begin{corollary}
\label{cor: bandwidth_means_boundary_well_linked}
There is an efficient algorithm, that, given a graph $G$, a cluster $S$ of $G$ that has $\alpha$-bandwidth property for some $0<\alpha<1$, and a pair $E_1,E_2$ of disjoint equal-cardinality subsets of $\delta(S)$, computes an one-to-one routing ${\mathcal{Q}}$ of edges of $E_1$ to edges of $E_2$, with $\cong_G({\mathcal{Q}})=\ceil{1/\alpha}$, such that for each path $Q\in{\mathcal{Q}}$, all its edges except for the first and the last edge of $Q$ belong to $E(S)$.
\end{corollary}
The technique of well-linked decompositions has been used extensively in graph algorithms (see e.g. \cite{racke2002minimizing,chekuri2004edge,andrews2010approximation,chuzhoy2012polylogarithmic,chuzhoy2012routing,chekuri2016polynomial,chuzhoy2016improved,chuzhoy2019towards}). We provide a well-known variant, that generalizes Theorem 2.8 in \cite{chuzhoy2012routing}, that shows the existence of an algorithm for decomposing a cluster of a graph into subclusters that have $\alpha$-bandwidth property.
We denote this algorithm by $\ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{WLD}}}}}\xspace^{(\alpha)}$, and its proof is provided in Appendix~\ref{apd: Proof of well_linked_decomposition}.
\begin{theorem}
\label{thm:well_linked_decomposition}
There is an efficient algorithm, that, given a graph $G$ with $n$ vertices and $m$ edges, a vertex-induced subgraph $S$ of $G$, and a parameter $\alpha\le 1/(48\log^2 m)$, computes a collection ${\mathcal{R}}$ of vertex-induced subgraphs (called clusters) of $S$, such that:
\begin{itemize}
\item the vertex sets $\set{V(R)}_{R\in {\mathcal{R}}}$ partition $V(S)$;
\item for every cluster $R\subseteq{\mathcal{R}}$, $|\delta(R)|\le |\delta(S)|$;
\item every cluster $R\subseteq{\mathcal{R}}$ has the $\alpha$-bandwidth property; and
\item $\sum_{R\in {\mathcal{R}}}|\delta(R)|\le |\delta(S)|\cdot\textsf{left}(1+O(\alpha\cdot \log m)\textsf{right})$.
\end{itemize}
Additionally, the algorithm computes,
for every cluster $R\in {\mathcal{R}}$, a set ${\mathcal{P}}(R)=\set{P(e)\mid e\in \delta_G(R)}$ of paths, such that, for every edge $e\in \delta_G(R)$, path $P(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, and all inner vertices of $P(e)$ lie in $V(S)\setminus V(R)$. Moreover, for every cluster $R\in {\mathcal{R}}$, the set ${\mathcal{P}}(R)$ of paths causes edge-congestion at most $100$ in $G$.
\end{theorem}
We will also repeatedly use the following lemma, whose proof is provided in \Cref{apd: Proof of routing path extension}.
\begin{lemma}
\label{lem: routing path extension}
Let $G$ be a graph, let $T$ be a set of vertices that are $\alpha$-well-linked in $G$, and let $T'$ be a subset of $T$. Let ${\mathcal{P}}'$ be a set of paths in $G$ routing the vertices of $T'$ to some vertex $x$ of $G$. Then there is a set ${\mathcal{P}}$ of paths routing vertices of $T$ to $x$, such that, for every edge $e\in E(G)$,
$\cong_G({\mathcal{P}},e)\le \ceil{\frac{|T|}{|T'|}}(\cong_G({\mathcal{P}}',e)+\ceil{1/\alpha})$.
\end{lemma}
\input{for-prelims}
\iffalse
\subsection{Laminar Family of Clusters and the Decomposition Tree}
Let $G$ be any graph, and let ${\mathcal{L}}$ be a family of connected subgraphs of $G$, that we call {clusters}. We say that ${\mathcal{L}}$ is a \emph{laminar family}, iff for all $S,S'\in {\mathcal{L}}$, either $S\cap S'=\emptyset$, or $S\subseteq S'$, or $S\subseteq S'$ holds.
We will also always assume that $G\in {\mathcal{L}}$ for any laminar family of subgraphs of $G$.
Given a laminar family ${\mathcal{L}}$ of clusters of $G$, we associate a \emph{paritioning tree} $\tau({\mathcal{L}})$ with it, which is a rooted tree defined as follows. The vertex set of the tree is $\set{v(S)\mid S\in {\mathcal{L}}}$. The root of the tree is $v(G)$ -- the vertex associated with the graph $G$ itself.
In order to define the edge set, consider a pair $S,S'\in {\mathcal{L}}$ of clusters. If $S\subsetneq S'$, and there is no other cluster $S''\in {\mathcal{L}}$ with $S\subsetneq S''\subsetneq S'$, then we add an edge $(v(S),v(S'))$ to the tree $\tau({\mathcal{L}})$; vertex $v(S)$ becomes a child vertex of $v(S')$ in the tree. In this case, we say that cluster $S$ is a \emph{child cluster} of cluster $S'$, and cluster $S'$ is the \emph{parent cluster} of $S$. We denote by ${\mathcal{W}}_{{\mathcal{L}}}(S)$ the set of all child clusters of a cluster $S\in {\mathcal{L}}$.
\fi
\subsection{Minimum Cuts and Gomory-Hu Tree}
\label{subsec: GH tree}
Let $G$ be a graph and let $s,t$ be two vertices of $G$. We denote by $\text{min-cut}_G(s,t)$ the size of a minimum cut $(S,T)$ in $G$, where $s\in S$ and $t\in T$.
Gomory and Hu~\cite{gomory1961multi} proved the following theorem.
\begin{theorem}[\cite{gomory1961multi}]\label{thm: GH tree properties}
There is an efficient algorithm, that, given a weighted graph $G=(V,E,w)$, computes a weighted tree $\tau=(V,E',w')$, such that
\begin{itemize}
\item for every pair $s,t$ of distinct vertices of $V$, if we denote by $P_{s,t}$ the unique path in $\tau$ connecting $s$ to $t$, then $\text{min-cut}_G(s,t)=\min_{e\in P_{s,t}}\set{w'(e)}$; and
\item for every pair $s,t$ of distinct vertices of $V$ and a min-cut $(S,T)$ in $G$ separating $s$ from $t$, then the same cut $(S,T)$ is also a min-cut in $\tau$ separating $s$ from $t$, and vice versa.
\end{itemize}
\end{theorem}
Let $G$ be a weighted graph, then the tree $\tau$ obtained from applying the algorithm in the above theorem to $G$ is called the \emph{Gomory-Hu tree} of $G$.
In this paper, we will use the following immediate corollary of the above theorem.
\begin{corollary}
\label{cor: G-H tree_edge_cut}
Let $G$ be a graph and let $\tau$ be a Gomory-Hu tree of graph $G$. Then for each edge $e=(u,u')\in E(\tau)$, if we denote by $U,U'$ the two subtrees obtained from $\tau$ by removing the edge $e$, with $u\in U$ and $u'\in U'$, then the cut $(V(U),V(U'))$ is a min-cut in $G$ separating $u$ from $u'$.
\end{corollary}
We will use the following lemma to handle graphs whose Gomory-Hu tree is a path. Its proof appears in Appendix~\ref{apd: Proof of GH tree path vs contraction}.
\begin{lemma}
\label{lem: GH tree path vs contraction}
Let $G$ be a graph whose Gomory-Hu tree $\tau$ is a path.
Let ${\mathcal{C}}$ be a set of vertex-disjoint clusters of $G$, such that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$, and the vertices in each cluster of ${\mathcal{C}}$ induce a subpath in $\tau$ (equivalently, if we denote $\tau=(v_1,\ldots,v_n)$, where vertices appear on path $\tau$ in this order, then for each cluster $C\in {\mathcal{C}}$, there exist two integers $i_C,j_C$, such that $V(C)=\set{v_{t}\mid i_C\le t\le j_C}$).
Then $\tau_{|{\mathcal{C}}}$, the contracted path, is a Gomory-Hu tree of the contracted graph $G_{|{\mathcal{C}}}$.
\end{lemma}
We will also use the following lemma on min-cuts in a graph. Its proof is provided in Appendix~\ref{apd: Proof of multiway cut with paths sets}.
\begin{lemma}\label{lem: multiway cut with paths sets}
There is an efficient algorithm, that, given a graph $G$ and a collection $S=\set{s_1,\ldots,s_k}$ of its vertices, computes, for all $1\leq i\leq k$, a set $A_i$ of vertices of $G$, and a collection ${\mathcal{Q}}_i$ of paths in $G$, such that the following hold:
\begin{itemize}
\item for all $1\leq i\leq k$, $S\cap A_i=\set{s_i}$, and moreover, $(A_i,V(G)\setminus A_i)$ is a minimum cut separating $s_i$ from the vertices of $S\setminus\set{s_i}$ in $G$;
\item for all $1\leq i<i'\leq k$, $A_i\cap A_{i'}= \emptyset$; and
\item for all $1\leq i\leq k$, ${\mathcal{Q}}_i=\set{Q_i(e)\mid e\in \delta_G(A_i)}$, where for each $e\in \delta_G(A_i)$, path $Q_i(e)$ has $e$ as its first edge and $s_i$ as its last vertex, with all internal vertices of $Q_i(e)$ lying in $A_i$. Moreover, the paths in ${\mathcal{Q}}_i$ are mutually edge-disjoint.
\end{itemize}
\end{lemma}
\iffalse
\mynote{is this needed?}
If a crossing is caused by the intersection of the images of a pair of edges $e,e'$ (namely $\phi(e)$ and $\phi(e')$), then we say that the edges $e$ and $e'$ \emph{participate} in the crossing, or the pair of edges $e$ and $e'$ \emph{cross}.
We also say that an edge $e$ is \emph{crossed} in a drawing $\phi$ if the edge $e$ participates in at least one crossing of $\phi$.
For any pair $E_1, E_2 \subseteq E$ of subsets of edges, we denote by $\mathsf{cr}_{\phi}(E_1, E_2)$ the number of crossings in $\phi$ in which the images of edges of $E_1$ intersect the images of edges of $E_2$.
Clearly, for each subset $E' \subseteq E$, $\mathsf{cr}(\phi)=\mathsf{cr}_{\phi}(E',E')+\mathsf{cr}_{\phi}(E',E\setminus E')+\mathsf{cr}_{\phi}(E\setminus E',E\setminus E')$.
Since at most two edges participate in a crossing, for disjoint subsets $E_1,\ldots,E_r$ of $E$, $\sum_{1\le i\le r}\mathsf{cr}_{\phi}(E,E_i)\le 2\cdot\mathsf{cr}(\phi)$.
For a subgraph $C$ of $G$, we denote $\mathsf{cr}_{\phi}(C)=\mathsf{cr}_{\phi}(E(C),E(C))$, or equivalently $\mathsf{cr}_{\phi}(C)=\mathsf{cr}(\phi(C))$. Similarly, for two subgraphs $C,C'$, we denote $\mathsf{cr}_{\phi}(C,C')=\mathsf{cr}_{\phi}(E(C),E(C'))$.
\fi
\iffalse
\textbf{Bridges and Extensions of subgraphs.}
Let $G$ be a graph, and let $C\subseteq G$ be a subgraph of $G$. A \emph{bridge} for $C$ in graph $G$ is either (i) an edge $e=(u,v)\in E(G)$ with $u,v\in V(C)$ and $e\not \in E(C)$; or (ii) a connected component of $G\setminus V(C)$.
We denote by ${\mathcal{R}}_G(C)$ the set of all bridges for $C$ in graph $G$. For each bridge $R\in {\mathcal{R}}_G(C)$, we define the set of vertices $L(R)\subseteq V(C)$, called the \emph{legs of $R$},
as follows. If $R$ consists of a single edge $e$, then $L(R)$ contains the endpoints of $e$. Otherwise, $L(R)$ contains all vertices $v\in V(C) $, such that $v$ has a neighbor that belongs to $R$.
Next, we define an \emph{extension} of the subgraph $C\subseteq G$, denoted by $X_G(C)$. The extension contains, for every bridge $R\in {\mathcal{R}}_G(C)$, a tree $T_R$, that is defined as follows. If $R$ is a bridge consisting of a single edge $e$, then the corresponding tree $T_R$ only contains the edge $e$. Otherwise,
let $R'$ be the subgraph of $G$ consisting of the graph $R$, the vertices of $L(R)$, and all edges of $G$ connecting vertices of $R$ to vertices of $L(R)$.
We let $T_R\subseteq R'$ be a tree, whose leaves are precisely the vertices of $L(R)$, that contains smallest number of edges among all such trees. Note that such a tree exists because graph $R$ is connected.
We let the extension of $C$ in $G$ be $X_G(C)=\set{T_R\mid R\in {\mathcal{R}}_G(C)}$.
\fi
\iffalse
\textbf{Sparsest Cut.}
Suppose we are given a graph $G=(V,E)$, and a subset ${\mathcal T}\subseteq V$ of $k$ vertices, called terminals.
The sparsity of a cut $(S,\overline S)$ in $G$ is $\Phi(S)=\frac{|E(S,\overline S)|}{\min\set{|S\cap {\mathcal T}|, |\overline S\cap {\mathcal T}|}}$, and the value of the sparsest cut in $G$ is defined to be:
$\Phi(G)=\min_{S\subset V}\set{\Phi(S)}$.
The goal of the sparsest cut problem is to find a cut of minimum sparsity. Arora, Rao and Vazirani~\cite{ARV} have shown an $O(\sqrt {\log n})$-approximation algorithm for the sparsest cut problem. We denote this algorithm by \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace, and its approximation factor by $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)=O(\sqrt{\log n})$.
\fi
\iffalse
Let $G$ be a graph, and let $S$ be a subset of vertices of $G$, and let $0<\alpha<1$, such that $S$ is $\alpha$-well-linked.
We now define the sparsest cut and the concurrent flow instances corresponding to $S$ as follows. For each edge in $\mathsf{out}(S)$, we subdivide the edge by adding a new vertex $t_e$ to it. Let $G'$ denote the resulting graph and let $T$ denote the set of all vertices $t_e$ for $e\in \mathsf{out}_G(S)$. Consider the graph $H=G'[S]\cup \mathsf{out}_{G'}(S)$. We can naturally define an instance of the non-uniform sparsest cut problem on $H$ where the set of terminals is $T$. We then have the following well-known observation.
\begin{observation}
Let $G$, $S$, $H$ and $T$ be defined as above, and let $0<\alpha <1$, such that $S$ is $\alpha$-well-linked. Then every pair of vertices in $T$ can send one flow unit to each other in $H$, such that the maximum congestion on any edge is at most $\beta(n)|T|/\alpha$. Moreover if $M$ is any partial matching on the vertices of $T$, then we can send one flow unit between every pair $(u,v)\in M$ in graph $H$, with maximum congestion at most $2\beta(n)/\alpha$.
\end{observation}
\iffalse
The following theorem shows the technique of \emph{well-linked decomposition}, which has been used extensively in graph algorithms (e.g., see~\cite{racke2002minimizing,chekuri2005multicommodity}).
\begin{theorem}[Terminal Well-Linked Decomposition]
\label{thm:well_linked-decomposition}
There is an efficient algorithm, that, given any connected graph $G$ and a subset $T\subseteq V(G)$ of its vertices with $|T|=k$, computes a collection $\set{G_1,\ldots,G_r}$ of connected subgraphs of $G$, such that:
\begin{enumerate}
\item for each $1\le i\le r$, denote $V_i=V(G_i)$, then $(V_1,\ldots,V_r)$ forms a partition of $V(G)$;
\item for each $1\le i\le r$, denote $T_i=T\cap V_i$, then the subset $T_i\cup \Gamma(V_i)$ of vertices is $\Omega(\frac{1}{\log^{3/2}k\cdot\log\log k})$-well-linked in $G_i$; and
\item $\sum_{1\le i\le r}|\mathsf{out}(V_i)|\le 0.01 k$, and $\sum_{1\le i\le r}|T_i\cup \Gamma(V_i)|\le 1.01 k$.
\end{enumerate}
\end{theorem}
\fi
\fi
\iffalse
\paragraph{Non-interfering paths.}
To be used later for locally non-interfering/non-interfering paths (for this the definition should be adapted). We may also keep it in prelims as a definition of crossing paths. Maybe we should also define a drawing of a path as a concatenation of images of its edges. We use it quite a bit.
\begin{definition}
Let $\gamma,\gamma'$ be two curves in the plane or on the sphere. We say that $\gamma$ and $\gamma'$ \emph{cross}, iff there is a disc $D$, whose boundary is a simple closed curve that we denote by $\beta$, such that:
\begin{itemize}
\item $\gamma\cap D$ is a simple open curve, whose endpoints we denote by $a$ and $b$;
\item $\gamma'\cap D$ is a simple open curve, whose endpoints we denote by $a'$ and $b'$; and
\item $a,a',b,b'\in \beta$, and they appear on $\beta$ in this circular order.
\end{itemize}
\end{definition}
Given a graph $G$ embedded in the plane or on the sphere, we say that two paths $P,P'$ in $G$ cross iff their images cross. Similarly, we say that a path $P$ crosses a curve $\gamma$ iff the image of $P$ crosses $\gamma$.
\fi
\subsection{Clusters, Paths, Flows, and Routers}
\label{subsec: clusters and paths}
\subsubsection{Clusters and Augmentations of Clusters}
Let $G$ be a graph. A \emph{cluster} of $G$ is a vertex-induced connected subgraph of $G$. For a set ${\mathcal{C}}$ of mutually disjoint clusters of $G$, we denote by $E^{\textsf{out}}_G({\mathcal{C}})$ the set of all edges $e=(u,v)$ of $G$, with endpoints $u$ and $v$ lying in distinct clusters of ${\mathcal{C}}$.
We sometimes omit the subscript $G$ when clear from the context.
Next, we define the notion of augmentation of a cluster.
\begin{definition}[Augmentation of Clusters]
\label{def: Graph C^+}
Let $C$ be a cluster of a graph $G$. The \emph{augmentation} of cluster $C$, denoted by $C^+$, is a graph that is obtained from $G$ as follows. First, we subdivide every edge $e\in \delta_G(C)$ with a vertex $t_e$, and let $T(C)=\set{t_e\mid e\in \delta_G(C)}$ be the resulting set of newly added vertices. We then let $C^+$ be the subgraph of the resulting graph induced by the set $V(C)\cup T(C)$ of vertices.
\end{definition}
\subsubsection{Paths and Flows}
\label{sec: routing paths}
As mentioned already, all graphs that we consider in this paper are undirected. However, sometimes it will be convenient for us to assign direction to paths in such graphs. We do so by designating one endpoint of the path as its first endpoint, and another endpoint as its last endpoint. We will then view the path as being directed from its first endpoint towards its last endpoint. We will sometimes refer to a path with an assigned direction as a directed path, even though the underlying graph is an undirected graph.
Let $G$ be a graph, and let ${\mathcal{P}}$ be a collection of paths in $G$. We say that the paths of ${\mathcal{P}}$ are \emph{edge-disjoint} if every edge of $G$ belongs to at most one path of ${\mathcal{P}}$. We say that the paths in ${\mathcal{P}}$ are \emph{vertex-disjoint} if every vertex of $G$ belongs to at most one path of ${\mathcal{P}}$. We say that the paths in ${\mathcal{P}}$ are \emph{internally disjoint} if every vertex $v\in V(G)$ that serves as an inner vertex of some path in ${\mathcal{P}}$ only belongs to one path of ${\mathcal{P}}$. Given a subset $S$ of vertices of $G$, we say that the paths in ${\mathcal{P}}$ are \emph{internally disjoint from $S$} if no vertex of $S$ serves as an inner vertex of any path in ${\mathcal{P}}$. Abusing the notation, for a subgraph $C$ of $G$, we sometimes say that a set ${\mathcal{P}}$ of paths is internally disjoint from $C$ to indicate that it is internally disjoint from $V(C)$.
\paragraph{Flows.}
Let $G$ be a graph, and let ${\mathcal{P}}$ be a collection of directed paths in graph $G$. A \emph{flow} over the set ${\mathcal{P}}$ of paths is an assignment of non-negative values $f(P)\geq 0$, called \emph{flow-values}, to every path $P\in {\mathcal{P}}$. We sometimes refer to paths in ${\mathcal{P}}$ as \emph{flow-paths for flow $f$}. For each edge $e\in E(G)$, let ${\mathcal{P}}(e)\subseteq {\mathcal{P}}$ be the set of all paths whose first edge is $e$, and let ${\mathcal{P}}'(e)\subseteq {\mathcal{P}}$ be the set of all paths whose last edge is $e$. We say that edge $e$ \emph{sends $z$ flow units} in $f$ if $\sum_{P\in {\mathcal{P}}(e)}f(e)=z$, and we say that edge $e$ \emph{receives $z$ flow units} in $f$ if $\sum_{P\in {\mathcal{P}}'(e)}f(P)=z$. Similarly, for a vertex $v\in V(G)$, we say that $v$ sends $z$ flow units in $f$ if the sum of flow-values of all paths $P\in {\mathcal{P}}$ that originate at $v$ is $z$. We say that $v$ receives $z$ flow units in $f$ if the sum of the flow-values of all paths $P\in {\mathcal{P}}$ that terminate at $v$ is $z$. The \emph{congestion} that flow $f$ causes on an edge $e$ is $\sum_{\stackrel{P\in {\mathcal{P}}:}{e\in E(P)}}f(P)$, and the \emph{total congestion} of the flow $f$ is the maximum congestion that it causes on any edge $e\in E(G)$.
An \emph{$s$-$t$ flow network} consists of a graph $G$, non-negative capacities $c(e)\geq 0$ for each edge $e\in E(G)$, and two special vertices: source $s$ and destination $t$. Let ${\mathcal{P}}$ be the set of all paths in graph $G$ originating at $s$ and terminating at $t$. An $s$-$t$ flow in $G$ is a flow $f$ that is defined over the set ${\mathcal{P}}$ of paths, such that for every edge $e\in E(G)$, the congestion that $f$ causes on edge $e$ is at most $c(e)$. The \emph{value} of the flow is $\sum_{P\in {\mathcal{P}}}f(P)$. Maximum $s$-$t$ flow is an $s$-$t$ flow of largest possible value. We say that a flow $f$ is \emph{integral} if, for every path $P$, value $f(P)$ is an integer. It is a well known fact (called \emph{integrality of flow}) that, if all edge capacities in a flow network are integral, then there is a maximum $s$-$t$ flow that is integral, and such a flow can be found efficiently. In case where the capacity of every edge is unit, such a flow defines a maximum-cardinality collection of edge-disjoint $s$-$t$ paths.
\paragraph{Congestion Reduction.}
We repeatedly use the following simple claim, whose proof follows from integrality of flow, and appears in \Cref{apd: Proof of remove congestion}.
\begin{claim}\label{claim: remove congestion}
Let $G$ be a graph and let ${\mathcal{P}}$ be a set of directed paths in $G$. For each vertex $v\in V(G)$, let $n_S(v)$ and $n_T(v)$ denote the numbers of paths in ${\mathcal{P}}$ originating and terminating at $v$, respectively. Then there is a set ${\mathcal{P}}'$ of at least $|{\mathcal{P}}|/\cong_G({\mathcal{P}})$
edge-disjoint directed paths in $G$, such that, for every vertex $v$, at most $n_S(v)$ paths of ${\mathcal{P}}'$ originate at $v$, and at most $n_T(v)$ paths of ${\mathcal{P}}'$ terminate at $v$. Moreover, there is an efficient algorithm, that, given $G$ and ${\mathcal{P}}$, computes a set ${\mathcal{P}}'$ of paths with these properties.
\end{claim}
\subsubsection{Routing Paths, Internal Routers and External Routers}
\label{subsubsection: routing paths}
\paragraph{Routing Paths.} Suppose we are given a graph $G$, two sets $S,T\subseteq V(G)$ of its vertices, and a set ${\mathcal{Q}}$ of paths. We say that ${\mathcal{Q}}$ is
a \emph{routing of vertices of $S$ to vertices of $T$}, or that ${\mathcal{Q}}$ \emph{routes vertices of $S$ to vertices of $T$} if ${\mathcal{Q}}=\set{Q_v\mid v\in S}$, and, for every vertex $v\in S$, path $Q_v$ originates at $v$ and terminates at a vertex of $T$.
If, additionally, for every vertex $t\in T$, exactly one path in ${\mathcal{Q}}$ terminates at $t$, then we say that ${\mathcal{Q}}$ is a \emph{one-to-one routing} of vertices of $S$ to vertices of $T$.
Similarly, given two sets $E_1,E_2$ of edges of $G$, we say that a set ${\mathcal{Q}}=\set{Q_e\mid e\in E_1}$ of paths is a \emph{routing of edges of $E_1$ to edges of $E_2$}, or that ${\mathcal{Q}}$ \emph{routes edges of $E_1$ to edges of $E_2$}, if, for every edge $e\in E_1$, path $Q_e$ has $e$ as its first edge, and some edge of $E_2$ as its last edge. If, additionally, every edge of $E_2$ serves as the last edge of exactly one path in ${\mathcal{Q}}$, then we say that ${\mathcal{Q}}$ is a \emph{one-to-one routing of edges of $E_1$ to edges of $E_2$}.
\iffalse
Next, we consider a cluster $C$ in a graph $G$, and we define routing paths in the interior and in the exterior of cluster $C$.
\begin{definition}[Routing Paths for a Cluster]
Let $G$ be a graph and let $C$ be a cluster of $G$. We say that paths in a set ${\mathcal{Q}}(C)=\set{Q_e\mid e\in \delta_G(C)}$ \emph{route the edges of $\delta_G(C)$ in $C$ to a vertex $v\in V(C)$}, iff for all $e\in \delta_G(C)$, path $Q_e$ has $e$ as its first edge, $v$ as its last vertex, and $Q_e\setminus\set{e}$ is contained in $C$.
We say that a set ${\mathcal{Q}}'(C)=\set{Q'_e\mid e\in \delta_G(C)}$ of paths \emph{route the edges of $\delta_G(C)$ outside $C$ to a vertex $v'\in V(G)\setminus V(C)$}, iff for all $e\in \delta_G(C)$, path $Q'_e$ has $e$ as its first edge, $v'$ as its last vertex, and $Q'_e\setminus\set{e}$ is disjoint from $C$.
\end{definition}
\fi
Next, we define the notions of internal and external routers for clusters, which are central notions that are used throughout our algorithms.
\begin{definition}[Internal and External Routers for Clusters]\label{def: routers}
Let $G$ be a graph, let $C$ be a cluster of $G$, and let ${\mathcal{Q}}(C)$ be a set of paths in $G$. We say that ${\mathcal{Q}}(C)$ is an \emph{internal router} for $C$, or an \emph{internal $C$-router}, if there is some vertex $u\in V(C)$, such that ${\mathcal{Q}}(C)=\set{Q_e\mid e\in \delta_G(C)}$, and, for each edge $e\in \delta_G(C)$, path $Q_e$ has $e$ as its first edge, $u$ as its last vertex, and all edges of $E(Q_e)\setminus\set{e}$ lie in $C$. We refer to vertex $u$ as the \emph{center of the router}.
Similarly, we say that a set ${\mathcal{Q}}'(C)$ of paths in $G$ is an \emph{external router} for $C$, or an \emph{external $C$-router}, if there is some vertex $u'\in V(G)\setminus V(C)$, such that ${\mathcal{Q}}'(C)=\set{Q'_e\mid e\in \delta_G(C)}$, and, for each edge $e\in \delta_G(C)$, path $Q_e$ has $e$ as its first edge, $u'$ as its last vertex, is internally disjoint from $C$. We refer to $u'$ as the \emph{center of the router}.
We denote by $\Lambda_G(C)$ the set of all internal $C$-routers, and by $\Lambda'_G(C)$ the set of all external $C$-routers in $G$. We may omit the subscript $G$ when clear from the context.
\end{definition}
Throughout the paper, we will be working with distributions over the set $\Lambda_G(C)$ of internal $C$-routers and distributions over the set $\Lambda'_G(C)$ of external $C$-routers for various clusters $C$ of a given graph $G$. We say that a distribution ${\mathcal{D}}$ over a set $U$ of elements is given \emph{explicitly}, if we are given a list $U'\subseteq U$ of elements, whose probability in ${\mathcal{D}}$ is non-zero, together with their associated probability values. We say that distribution ${\mathcal{D}}$ is given \emph{implicitly} if we are given an efficient randomized algorithm that draws an element from $U$ according to the distribution. When the distribution ${\mathcal{D}}$ is over a set of routers in a graph $G$, the running time of the algorithm should be bounded by $\operatorname{poly}(|E(G)|)$.
\subsubsection{Non-Transversal Paths and Path Splicing}
\label{subsec: non-transversal paths and splicing}
We start by defining the notions of transversal and non-transversal intersections of paths and cycles, which we then use to define non-transversal paths.
\begin{definition}[Non-transversal Intersection of Paths and Cycles]
Let $I=(G,\Sigma)$ be an intance of \ensuremath{\mathsf{MCNwRS}}\xspace, let $P_1,P_2$ be two simple paths in $G$, and let $u$ be a vertex in $V(P_1)\cap V(P_2)$. Denote by $E_1$ the set of (one or two) edges of $P_1$ that are incident to $u$, and similarly denote by $E_2$ the set of (one or two) edges of $P_2$ that are incident to $u$. We say that the intersection of the paths $P_1,P_2$ at vertex $u$ is \emph{non-transversal with respect to $\Sigma$} if one of the following holds:
\begin{itemize}
\item either the set $E_1\cup E_2$ contains fewer than $4$ distinct edges; or
\item $E_1=\set{e_1,e_1'}$ and $E_2=\set{e_2,e_2'}$, all edges in set $\set{e_1,e_1',e_2,e_2'}$ are distinct,
and they appear in the ordering ${\mathcal{O}}_u\in \Sigma$ in one of the following circular orders: $(e_1,e_1',e_2,e_2')$, or $(e_1,e_1',e_2',e_2)$ (recall that the orderings are unoriented, so the reversals of the above two orderings are also included in this definition).
\end{itemize}
Otherwise, we say that the intersection of the paths $P_1,P_2$ at vertex $u$ is \emph{transversal} (see \Cref{fig:non_trans}).
If $R_1,R_2$ are simple cycles in $G$, and $u$ is a vertex in $V(R_1)\cap V(R_2)$, then we classify the intersection of $R_1$ and $R_2$ and $u$ as transversal or non-transversal with respect to $\Sigma$ similarly.
\end{definition}
\begin{figure}[h]
\centering
\subfigure[The intersection of paths $P_1$ (red) and $P_2$ (purple) is transversal at $v$.]{\scalebox{0.14}{\includegraphics{figs/non_trans_1.jpg}}}
\hspace{0.2cm}
\subfigure[The intersection of path $P_1$ (red) and path $P_2$ (purple) is non-transversal at $v$.]{
\scalebox{0.14}{\includegraphics{figs/non_trans_2.jpg}}}
\caption{Transversal and non-transversal intersections of paths.
}\label{fig:non_trans}
\end{figure}
\begin{definition}[Non-transversal Set of Paths]\label{def: non-transversal paths}
Let $I=(G,\Sigma)$ be an intance of \ensuremath{\mathsf{MCNwRS}}\xspace, and let ${\mathcal{P}}$ be a collection of simple paths in $G$. We say that the set ${\mathcal{P}}$ of paths is \emph{non-transversal with respect to $\Sigma$} if, for every pair $P_1,P_2\in {\mathcal{P}}$ of paths, for every vertex $u\in V(P_1)\cap V(P_2)$, the intersection of $P_1$ and $P_2$ at $u$ is non-transversal with respect to $\Sigma$.
\end{definition}
Assume now that we are given some instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, and a collection ${\mathcal{Q}}$ of simple paths in $G$.
We let $\Pi^T({\mathcal{Q}})$ denote the set of all triples $(Q,Q',v)$, such that $Q,Q'\in {\mathcal{Q}}$, $v$ is an inner vertex of both $Q$ and $Q'$, and the intersection of $Q$ and $Q'$ at $v$ is transversal with respect to $\Sigma$.
We need to design a subroutine, that, given a set ${\mathcal{Q}}$ of simple directed paths in a graph $G$, transforms it into a set ${\mathcal{Q}}'$ of paths that is non-transversal with respect to the given rotation system $\Sigma$ for $G$. We need to ensure that the multisets containing the first vertex of every path in ${\mathcal{Q}}$ and in ${\mathcal{Q}}'$, respectively, remain unchanged, and the same holds for multisets containing the last vertex of every path in both path sets. We also need to ensure that for each edge $e\in E(G)$, $\cong_G({\mathcal{Q}}')\leq \cong_G({\mathcal{Q}})$.
Below we provide a procedure for performing such a transformation. The procedure uses a simple subroutine that we call \emph{path splicing} and describe next.
\paragraph{Path Splicing.}
Suppose we are given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, two simple paths $P,P'$ in $G$, and a vertex $v$, that serves as an inner vertex of both $P$ and $P'$, such that the intersection of $P$ and $P'$ at vertex $v$ is transversal with respect to $\Sigma$. We assume that each of the paths $P,P'$ is assigned a direction, and we denote by $s$ and $t$ the first and the last endpoints of $P$, respectively, and by $s'$ and $t'$ the first and the last endpoints of $P'$, respectively. The \emph{splicing} of $P$ and $P'$ at vertex $v$ produces two new paths: path $\tilde P$, that is a concatenation of the subpath of $P$ from $s$ to $v$, and the subpath of $P'$ from $v$ to $t'$; and path $\tilde P'$, that is a concatenation of the subpath of $P'$ from $s'$ to $v$, and the subpath of $P$ from $v$ to $t$.
See \Cref{fig: path_splicing} for an illustration.
\begin{figure}[h]
\centering
\subfigure[Before: Path $P$ is shown in red and path $P'$ is shown in purple.]{\scalebox{0.12}{\includegraphics{figs/pathsplicing_1.jpg}}}
\hspace{0.7cm}
\subfigure[After: Path $\tilde P$ is shown in red and path $\tilde P'$ is shown in purple.]{
\scalebox{0.12}{\includegraphics{figs/pathsplicing_2.jpg}}}
\caption{An illustration of path splicing at vertex $v$.}\label{fig: path_splicing}
\end{figure}
For a set ${\mathcal{P}}$ of directed paths in a graph $G$, we denote by $S({\mathcal{P}})$ and $T({\mathcal{P}})$ the multisets containing the first vertex on every path in ${\mathcal{P}}$, and the last vertex on every path in ${\mathcal{P}}$, respectively.
We use the following simple observation regarding the splicing procedure, whose proof is deferred to Section \ref{apd: Proof of splicing} of Appendix.
\begin{observation}\label{obs: splicing}
Let $I=(G,\Sigma)$ be an instance of $\ensuremath{\mathsf{MCNwRS}}\xspace$, let ${\mathcal{P}}$ be a set of simple directed paths in $G$,
and let $(P,P',v)$ be a triple in $\Pi^T({\mathcal{P}})$. Let $\tilde P,\tilde P'$ be the pair of paths obtained by splicing $P$ and $P'$ at $v$, and let ${\mathcal{P}}'=\big({\mathcal{P}}\setminus\set{P,P'}\big)\cup \set{\tilde P,\tilde P'}$.
Then $S({\mathcal{P}}')=S({\mathcal{P}})$ and $T({\mathcal{P}}')=T({\mathcal{P}})$. Additionally, either (i) at least one of the paths $\tilde P,\tilde P'$ is a non-simple path; or (ii) $|\Pi^T({\mathcal{P}}')|<|\Pi^T({\mathcal{P}})|$.
\end{observation}
Using \Cref{obs: splicing}, we can prove the following lemma that allows us to transform an arbitrary set ${\mathcal{R}}$ of paths into a set ${\mathcal{R}}'$ of non-transversal paths, while preserving the multisets containing the first endpoint and the last endpoint of every path, and without increasing the congestion on any edge.
The proof of the lemma below is similar to the proof of Lemma 9.5 in~\cite{chuzhoy2020towards}, and is provided in Appendix~\ref{apd: Proof of non_interfering_paths} for completeness.
\begin{lemma}
\label{lem: non_interfering_paths}
There is an efficient algorithm, that, given an instance $(G, \Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace and a set ${\mathcal{R}}$ of directed paths in $G$, computes another set ${\mathcal{R}}'$ of simple directed paths in $G$, such that $S({\mathcal{R}}')=S({\mathcal{R}})$, $T({\mathcal{R}}')=T({\mathcal{R}})$, and the paths in ${\mathcal{R}}'$ are non-transversal with respect to $\Sigma$. Moreover, for every edge $e\in E(G)$, $\cong_G({\mathcal{R}}',e)\leq \cong_G({\mathcal{R}},e)$.
\end{lemma}
\iffalse
\noindent{\bf Flows and cuts.}
Let $G$ be a graph and let ${\mathcal{P}}$ be the set of all paths in $G$. A \emph{flow} $f$ in graph $G$ is a function $f: {\mathcal{P}}\to \mathbb{R}^{\ge 0}$ that assigns a non-negative value to each path $P\in {\mathcal{P}}$.
The \emph{value} of the flow $f$ is $\sum_{P\in {\mathcal{P}}}f(P)$. Let $P$ be path in ${\mathcal{P}}$ that originates at a vertex $u\in V(G)$ and terminates at some other vertex $u'\in V(G)$. We say that the node $u$ \emph{sends} $f(P)$ units of flow to $u'$ along the path $P$. For each edge $e\in E(G)$, we define the congestion of the flow $f$ on the edge $e$ to be $\sum_{P\in {\mathcal{P}}: e\in P}f(P)$, namely the total amount of flow of $f$ sent through $e$.
The total congestion of flow $f$ is the maximum congestion of $f$ on any edge of $G$.
A \emph{cut} in a graph $G$ is a bipartition of its vertex set $V$ into non-empty subsets. The \emph{value} of a cut $(S,V\setminus S)$ is $|E_G(S,V\setminus S)|$.
\fi
\subsection{Contracted Graphs}
Let $G$ be a graph and let ${\mathcal{C}}$ be a collection of disjoint clusters of $G$. We define the \emph{contracted graph} $G_{|{\mathcal{C}}}$ to be the graph obtained from $G$ by contracting each cluster $C\in {\mathcal{C}}$ into a supernode $v_C$; we remove self-loops but keep parallel edges. Note that every edge of graph $G_{|{\mathcal{C}}}$ corresponds to some edge of graph $G$. We do not distinguish between such edges, so $E(G_{|{\mathcal{C}}})\subseteq E(G)$. We refer to vertices of $G_{|{\mathcal{C}}}$ that are not supernodes as \emph{regular} vertices.
In the following claim, we derive well-linkedness properties of a set $T$ of vertices in a graph $G$ from well-linkedness of $T$ in a contracted graph $G_{|{\mathcal{C}}}$ and bandwidth properties of the clusters of ${\mathcal{C}}$ in $G$. The proof is deferred to Section \ref{apd: Proof of contracted_graph_well_linkedness} of Appendix.
\begin{claim}
\label{clm: contracted_graph_well_linkedness}
Let $G=(V,E)$ be a graph, $T\subseteq V$ a subset of its vertices, and ${\mathcal{C}}$ a collection of disjoint clusters of $G$, such that $T\cap (\bigcup_{C\in {\mathcal{C}}}V(C))=\emptyset$. Assume that each cluster $C\in {\mathcal{C}}$ has the $\alpha_1$-bandwidth property in $G$, and that the set $T$ of vertices is $\alpha_2$-well-linked in the contracted graph $G_{|{\mathcal{C}}}$, for some parameters $0<\alpha_1,\alpha_2<1$. Then $T$ is $(\alpha_1\cdot \alpha_2)$-well-linked in $G$.
\end{claim}
The following corollary of \Cref{clm: contracted_graph_well_linkedness} essentially replaces the well-linkedness property of the set $T$ of vertices with the equivalent bandwidth property of a cluster of a given graph $G$.
The proof is deferred to Section \ref{apd: Proof of cor contracted_graph_well_linkedness} of Appendix.
\begin{corollary}
\label{cor: contracted_graph_well_linkedness}
Let $G$ be a graph, and let $R$ be a cluster of $G$. Let ${\mathcal{C}}$ be a collection of disjoint clusters of $R$, such that every cluster $C\in {\mathcal{C}}$ has the $\alpha_1$-bandwidth property in graph $G$, for some parameter $0<\alpha_1<1$. Denote $\hat R=R_{\mid{\mathcal{C}}}$ and $\hat G=G_{\mid{\mathcal{C}}}$, and assume further that $\hat R$ has the $\alpha_2$-bandwidth property in graph $\hat G$, for some $0<\alpha_2<1$. Then cluster $R$ has the $(\alpha_1\cdot\alpha_2)$-bandwidth property in graph $G$.
\end{corollary}
The following simple claim allows us to transform a routing in a contracted graph $G_{|{\mathcal{C}}}$ into a routing in the original graph $G$. The proof appears in Section \ref{apx: contracted graph routing} of Appendix.
\begin{claim}
\label{claim: routing in contracted graph}
There is an efficient algorithm, that takes as input a graph $G$, a set ${\mathcal{C}}$ of disjoint clusters of $G$, such that each cluster $C\in {\mathcal{C}}$ has the $\alpha$-bandwidth property in $G$ for some $0<\alpha<1$, and a collection ${\mathcal{P}}$ of edge-disjoint paths in the contracted graph $G_{|{\mathcal{C}}}$, routing some set $T\subseteq V(G)\cap V(G_{|{\mathcal{C}}})$ of vertices to some vertex $x\in V(G)\cap V(G_{|{\mathcal{C}}})$. The algorithm produces a collection ${\mathcal{P}}'$ of paths in graph $G$, routing the vertices of $T$ to vertex $x$, such that, for each edge $e\in E(G)\setminus \textsf{left}( \bigcup_{C\in {\mathcal{C}}}E(C)\textsf{right} )$, $\cong_G({\mathcal{P}}',e)\le 1$, and for each edge $e\in \bigcup_{C\in {\mathcal{C}}}E(C)$, $\cong_{G}({\mathcal{P}}',e)\leq \ceil{1/\alpha}$. Additionally, the algorithm produces another set ${\mathcal{P}}''$ of edge-disjoint paths in graph $G$, of cardinality at least $\alpha \cdot |T|/2$, routing a subset $T'\subseteq T$ of vertices to $x$.
\end{claim}
The following claim allows us to bound the crossing number of a contracted graph. The proof is provided in Section \ref{apd: Proof of crossings in contr graph} of the Appendix.
\begin{claim}\label{lem: crossings in contr graph}
Let $I=(G,\Sigma)$ be an instance of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, and let ${\mathcal{C}}$ be a collection of disjoint clusters of $G$, such that each cluster in ${\mathcal{C}}$ has the $\alpha$-bandwidth property, for some $0<\alpha<1$. Then there is a drawing $\phi$ of the contracted graph $G_{|{\mathcal{C}}}$, with $\mathsf{cr}(\phi)\leq O(\mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot \log^8m/\alpha^2)$, where $m=|E(G)|$. Moreover, for every regular vertex $x\in V(G_{|{\mathcal{C}}})\cap V(G)$, the ordering of the edges of $\delta_G(x)$ as they enter $x$ in $\phi$ is consistent with the rotation ${\mathcal{O}}_x\in \Sigma$.
\end{claim}
\subsection{Curves in the Plane or on a Sphere}
\subsubsection{Reordering Curves}
Assume that we are given two oriented orderings $({\mathcal{O}},b), ({\mathcal{O}}',b')$ on a set $U=\set{u_1,\ldots,u_r}$ of elements. Assume for simplicity that $b=b'=1$ (otherwise the corresponding ordering can be flipped). Consider a disc $D$, with a collection $\set{p_1,\ldots,p_r}$ of distinct points appearing on the boundary of $D$ (we will view each point $p_i$ as representing element $u_i$ of $U$), such that the order in which these points are encountered, as we traverse the boundary of $D$ in the counter-clock-wise direction, is precisely ${\mathcal{O}}$. Let $D'\subseteq D$ be another disc that is contained in $D$, whose boundary is disjont from the boundary of $D$. Assume that a collection $\set{p'_1,\ldots,p'_r}$ of points appear on the boundary of $D'$, and that the order in which these points are encountered as we traverse the boundary of $D'$ in the counter-clock-wise direction is precisely ${\mathcal{O}}'$. As before, for each $1\leq i\leq r$, we view point $p'_i$ as representing element $u_i\in U$.
We now define reordering curves between the oriented orderings $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$, which are then used in order to define the distance between the two orderings.
\begin{definition}[Reordering curves]
We say that a collection $\Gamma=\set{\gamma_1,\ldots,\gamma_r}$ of curves is a \emph{set of reordering curves for oriented orderings $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$} iff (i) the curves in $\Gamma$ are in general position; (ii) each curve $\gamma_i\in \Gamma$ is simple and its interior is contained in $D\setminus D'$; and (iii) for all $1\leq i\leq r$, curve $\gamma_i$ has $p_i,p'_i$ as its endpoints. The \emph{cost} of the collection $\Gamma$ is the total number of crossings between its curves.
\end{definition}
\begin{definition}[Distance between orderings]
Let $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$ be two oriented orderings on a set $U$ of elements. The \emph{distance} between the two oriented orderings, denoted by $\mbox{\sf dist}(({\mathcal{O}},b),({\mathcal{O}}',b'))$, is the smallest cost of any collection $\Gamma$ of reordering curves for $({\mathcal{O}},b)$ and $({\mathcal{O}},b')$. For two unoriented orderings ${\mathcal{O}},{\mathcal{O}}'$ on $U$, we define $\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}')=\min_{b,b'\in \set{-1,1}}\set{\mbox{\sf dist}(({\mathcal{O}},b),({\mathcal{O}}',b'))}$.
\end{definition}
The following lemma, that follows from Section 4 of~\cite{pelsmajer2009odd} and Section 5.2 of~\cite{pelsmajer2011crossing}, provides an efficient algorithm to compute a collection of reordering curves of near-optimal cost for a given pair of oriented orderings. The proof is deferred to Section \ref{subsec: compute reordering} of Appendix.
\begin{lemma}\label{lem: find reordering}
There is an efficient algorithm, that, given a pair $({\mathcal{O}},b)$, $({\mathcal{O}}',b')$ of oriented orderings on a set $U$ of elements, computes a collection $\Gamma$ of reordering curves for $({\mathcal{O}},b)$ and $({\mathcal{O}}',b')$, of cost at most $2\cdot\mbox{\sf dist}(({\mathcal{O}},b),({\mathcal{O}}',b'))$.
\end{lemma}
We will use the following simple corollary of the lemma, whose proof is provided Appendix~\ref{apd: Proof of find reordering}.
\begin{corollary}
\label{lem: ordering modification}
There is an efficient algorithm, whose input is a graph $G$, a drawing $\phi$ of $G$ in the plane, a vertex $v\in V(G)$, and a circular ordering ${\mathcal{O}}_v$ of the edges of $\delta_G(v)$. Let ${\mathcal{O}}'_v$ be the circular order in which the edges of $\delta_G(v)$ enter the image of $v$ in $\phi$, and let $D=D_{\phi}(v)$ be a tiny $v$-disc. The algorithm produces a new drawing $\phi'$ of $G$, with $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)+2\cdot\mbox{\sf dist}({\mathcal{O}}_v,{\mathcal{O}}'_v)$, such that the following hold:
\begin{itemize}
\item the images of the edges of $\delta_G(v)$ enter the image of $v$ in the order ${\mathcal{O}}_v$ in $\phi'$; and
\item the drawings $\phi$ and $\phi'$ are identical, except that, for each edge $e\in \delta_G(v)$, the segment of the image of $e$ lying inside the disc $D$ may be different in the two drawings.
\end{itemize}
\end{corollary}
\iffalse
We will also need the following simple claim about orderings.
\mynote{please add proof to appendix and link to it. It's OK to replace $k^2/2$ by $k^2/4$ or some other $\Theta(k^2)$.}
\begin{claim}\label{claim: correct orientation}
Let $U$ be a set of $k$ elements, and let ${\mathcal{O}},{\mathcal{O}}'$ be two unoriented orderings on the elements of $U$. Then $\mbox{\sf dist}(({\mathcal{O}},1),({\mathcal{O}}',1))+\mbox{\sf dist}(({\mathcal{O}},1),({\mathcal{O}}',-1))\geq k^2/2$.
\end{claim}
\fi
\iffalse
The following lemma, whose proof is provided Appendix~\ref{apd: Proof of find reordering}.
\mynote{problem with this lemma: we don't define a tiny disc around a point, the curves should be in general position, we didn't define ordering of curves entering a point. Replace all this with a drawing of a graph and ordering of edges entering a vertex?}
\begin{lemma}
\label{lem: ordering modification}
There is an algorithm, that, given a set $U$ of elements, a set $\Gamma=\set{\gamma_u\mid u\in U}$ of curves that shares an endpoint $z$, and two circular orderings ${\mathcal{O}},{\mathcal{O}}'$ on elements of $U$, such that the curves of $\Gamma$ enter $z$ in the order ${\mathcal{O}}$, computes a new set $\Gamma'=\set{\gamma'_u\mid u\in U}$ of curves, such that:
\begin{itemize}
\item the curves of $\Gamma'$ enter $z$ in the order ${\mathcal{O}}'$;
\item for each $u\in U$, the curve $\gamma_u$ differs from the curve $\gamma'_u$ only within some tiny disc $D$ that contains $z$; and
\item the number of crossings between curves of $\Gamma'$ within $D$ is $O(\mbox{\sf dist}({\mathcal{O}},{\mathcal{O}}'))$.
\end{itemize}
\end{lemma}
\fi
\subsubsection{Type-1 Uncrossing of Curves}
\label{subsec: uncrossing type 1}
In this subsection we consider a set $\Gamma$ of curves in the plane (or on a sphere) that are in general position, and provide a simple operation, called \emph{type-$1$ uncrossing}, whose goal is to ``simplify'' this collection of curves by eliminating some of the crossings between them. Specifically, we modify the curves in $\Gamma$, without changing their endpoints, to ensure that every pair of curves cross at most once. We now describe the type-1 uncrossing operation formally.
Let $\Gamma$ be a set of simple curves in the plane that are in general position.
For a pair $\Gamma_1,\Gamma_2$ of disjoint subsets of $\Gamma$, we denote by $\chi(\Gamma_1,\Gamma_2)$ the total number of crossings between the curves in $\Gamma_1$ and the curves in $\Gamma_2$. In other words, $\chi(\Gamma_1,\Gamma_2)$ is the number of points $p$, such that $p$ lies on a curve in $\Gamma_1$ and on a curve in $\Gamma_2$, and $p$ is not an endpoint of these curves. If $\Gamma_1=\set{\gamma}$, then we use the shorthand $\chi(\gamma,\Gamma_2)$ instead of $\chi(\set{\gamma},\Gamma_2)$.
The type-1 uncrossing operation iteratively considers pairs $\gamma,\gamma'\in \Gamma$ of distinct curves that cross more than once, and then locally modifies them, as shown in Figure~\ref{fig:type_1_uncrossing}, to eliminate two crossings. This operation ensures that no new crossings are created, and preserves the endpoints of both curves. The following theorem summarizes this operation. The proof of the theorem is standard and is deferred to Section \ref{apd: type-1 uncrossing} of Appendix for completeness.
\begin{figure}[h]
\centering
\subfigure[Before: Curves $\gamma$ and $\gamma'$ cross twice, at points $p$ and $q$. The crossing points of both curves with the third curve are circled as well.]{\scalebox{0.12}{\includegraphics{figs/type_1_uncross_1.jpg}}}
\hspace{0.8cm}
\subfigure[After: Each of the new curves $\gamma$ and $\gamma'$ has same endpoints as before. The two curves no longer cross each other, and the pink curve still participates in two crossings with $\gamma$ and $\gamma'$.]{
\scalebox{0.12}{\includegraphics{figs/type_1_uncross_2.jpg}}}
\caption{Type-1 uncrossing operation.}\label{fig:type_1_uncrossing}
\end{figure}
\begin{theorem}[Type-1 Uncrossing]
\label{thm: type-1 uncrossing}
There is an algorithm, that, given a set $\Gamma$ of simple curves in general position, that are partitioned into two disjoint subsets $\Gamma_1,\Gamma_2$, computes, for each curve $\gamma\in \Gamma_1$, a simple curve $\gamma'$ that has same endpoints as $\gamma$, such that, if we denote by $\Gamma_1'=\set{\gamma'\mid \gamma\in \Gamma_1}$, then the following hold:
\begin{itemize}
\item the curves in set $\Gamma'_1\cup \Gamma_2$ are in general position;
\item every pair of distinct curves in $\Gamma_1'$ cross at most once;
\item for every curve $\gamma\in \Gamma_2$, $\chi(\gamma,\Gamma'_1)\le \chi(\gamma,\Gamma_1)$; and
\item the total number of crossings between the curves of $\Gamma_1'\cup \Gamma_2$ is bounded by the total number of crossings between the curves of $\Gamma$.
\end{itemize}
The running time of the algorithm is bounded by $\operatorname{poly}(n\cdot N)$, where $n$ is the number of bits in the representation of the set $\Gamma$ of curves, and $N$ is the number of crossing points between the curves of $\Gamma$.
\end{theorem}
\subsubsection{Curves in a Disc and Nudging of Curves}
\label{sec: curves in a disc}
Suppose we are given a disc $D$, and a collection $\set{s_1,t_1,\ldots,s_k,t_k}$ of distinct points on its boundary. For all $1\leq i< j\leq k$, we say that the two pairs $(s_i,t_i),(s_j,t_j)$ of points \emph{cross} iff the unoriented circular ordering of the points $s_i,s_j,t_i,t_j$ on the boundary of $D$ is $(s_i,s_j,t_i,t_j)$.
We use the following simple claim, whose proof is deferred to \Cref{apd: Proof of curves in a disc}.
\begin{claim}\label{claim: curves in a disc}
There is an efficient algorithm that, given a disc $D$, and a collection $\set{s_1,t_1,\ldots,s_k,t_k}$ of distinct points on the boundary of $D$, computes a collection $\Gamma=\set{\gamma_1,\ldots,\gamma_k}$ of curves, such that, for all $1\leq i\leq k$, curve $\gamma_i$ has $s_i$ and $t_i$ as its endpoints, and its interior is contained in the interior of $D$. Moreover, for every pair $1\leq i<j\leq k$ of indices, if the two pairs $(s_i,t_i),(s_j,t_j)$ of points cross then curves $\gamma_i,\gamma_j$ intersect at exactly one point; otherwise, curves $\gamma_i,\gamma_j$ do not intersect. Lastly, every point in the interior of $D$ may be contained in at most two curves of $\Gamma$.
\end{claim}
\paragraph{Nudging Procedure.}
In a nudging procedure, we are given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a subset $U$ of vertices of $G$, and a collection ${\mathcal{P}}$ of edge-disjoint paths, such that, for every path $P\in {\mathcal{P}}$, all inner vertices of $P$ lie in $U$, and the endpoints of $P$ do not lie in $U$. Additionally, we are given some solution $\phi$ to instance $I$. For every path $P\in {\mathcal{P}}$, we denote by $\gamma(P)$ the image of path $P$ in $\phi$, that is, $\gamma(P)$ is the concatenation of the images of all edges of $P$. Notice that the resulting collection $\Gamma=\set{\gamma(P)\mid P\in {\mathcal{P}}}$ may not be in general position. This is since some vertices $u\in U$ may lie on more than $2$ paths in ${\mathcal{P}}$, and in such a case more than $2$ curves in $\Gamma$ contain the point $\phi(u)$. The purpose of the nudging procedure is to slightly modify the curves in $\Gamma$ in the viccinity of the images of such vertices to ensure that the resulting collection of curves $\Gamma'=\set{\gamma'(P)\mid P\in {\mathcal{P}}}$ is in general position, while introducing relatively few crossings. Additionally, the procedure ensures that, for every path $P\in {\mathcal{P}}$, the endpoints of the new curve $\gamma'(P)$ are identical to those of the original curve $\gamma(P)$.
We start by letting, for every path $P$, curve $\gamma'(P)$ be the original curve $\gamma(P)$, and we set $\Gamma'=\set{\gamma'(P)\mid P\in {\mathcal{P}}}$. We then process every vertex $u\in U$ one by one. Consider an iteration when any such vertex $u$ is processed. Let ${\mathcal{P}}(u)\subseteq {\mathcal{P}}$ be a set of all paths $P\in {\mathcal{P}}$ with $u\in V(P)$. We denote ${\mathcal{P}}(u)=\set{P_1,\ldots,P_k}$. Consider the tiny $u$-disc $D(u)=D_{\phi}(u)$ in the drawing $\phi$ of graph $G$. For all $1\leq i\le k$, we let $s_i,t_i$ be the two points at which curve $\gamma'(P_i)$ intersects the boundary of the disc $D(u)$. Note that all points $s_1,t_1,\ldots,s_k,t_k$ must be distinct, as the paths in ${\mathcal{P}}$ are edge-disjoint. We use the algorithm from \Cref{claim: curves in a disc} in order to construct a collection $\set{\sigma_1,\ldots,\sigma_k}$ of curves, such that, for all $1\leq i\leq k$, curve $\sigma_i$ has $s_i$ and $t_i$ as its endpoints, and is completely contained in $D(u)$. Recall that the claim ensures that, for every pair $1\leq i<j\leq k$ of indices, if the two pairs $(s_i,t_i),(s_j,t_j)$ of points cross, then curves $\sigma_i,\sigma_j$ intersect at exactly one point; otherwise, curves $\sigma_i,\sigma_j$ do not intersect. The former may only happen if paths $P_i,P_j$ have a transversal intersection at vertex $u$.
For all $1\leq i\leq k$, we modify the curve $\gamma'(P_i)$ as follows: we replace the segment of the curve between points $s_i,t_i$ with the curve $\sigma_i$.
Once every vertex of $U$ is processed, we obtain the final collection $\Gamma'=\set{\gamma'(P)\mid P\in {\mathcal{P}}}$ of curves. From the above discussion, we get the following observation.
\begin{observation}\label{obs: nudging summary}
The set $\Gamma'=\set{\gamma'(P)\mid P\in {\mathcal{P}}}$ of curves is in general position, and, for every path $P\in {\mathcal{P}}$, the endpoints of curve $\gamma'(P)$ are identical to the endpoints of curve $\gamma(P)$. Moreover, if $\chi$ denotes the set of all crossings $(e,e')_p$ in $\phi$, where $e$ and $e'$ are edges of $\bigcup_{P\in {\mathcal{P}}}E(P)$, then the number of crossings between the curves of $\Gamma'$ is bounded by $|\chi|+|\Pi^T({\mathcal{P}})|$. Lastly, if the paths in ${\mathcal{P}}$ are non-transversal with respect to $\Sigma$, then for every path $P\in {\mathcal{P}}$, the number of crossings between $\gamma'(P)$ and $\Gamma'\setminus\set{\gamma'(P)}$ is bounded by the number of crossings $(e,e')_p$ in $\phi$ where exactly one of the edges $e,e'$ belongs to $P$.
\end{observation}
\iffalse
One common scenario in which we will use \Cref{claim: curves in a disc} is \emph{nudging} of curves. Consider some graph $G$, its drawing $\phi$, and a collection ${\mathcal{Q}}$ of edge-disjoint paths in $G$. Assume that, if a vertex $v$ is an endpoint of a path in ${\mathcal{Q}}$, then it may not serve as an inner vertex of any path in ${\mathcal{Q}}$. We can then define, for every path $Q\in {\mathcal{Q}}$, a curve $\gamma(Q)$, that corresponds to the image of the path $Q$ in $\phi$. Specifically, $\gamma(Q)$ is the concatenation of the images of all edges of $Q$. Unfortunately, the resulting set $\Gamma=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$ of curves is not necessarily in general position. This is since, if a vertex $v$ serves as an inner vertex on more than two paths in ${\mathcal{Q}}$, then point $\phi(v)$ serves as an inner point on more than two curves in $\Gamma$. We will use the algorithm from \Cref{claim: curves in a disc} to slightly adjust the segments of the curves in $\Gamma$ that lie inside tiny $v$-discs $D_{\phi}(v)$ for such vertices $v\in V(G)$, to ensure that the resulting set of curves is in general position. We informally refer to this adjustment operation as \emph{nudging}.
\fi
\subsubsection{Type-2 Uncrossing of Curves}
\label{subsec: type-2 uncrossing}
In this subsection we provide another subroutine, called \emph{type-2 uncrossing of curves}, that allows us to simplify a given set $\Gamma$ of curves by removing some of the crossings between them. Unlike the type-1 uncrossing operation, we no longer preserve the endpoints of every curve, but we ensure that the multisets containing the endpoints of the curves are preserved under this operation.
It will sometimes be useful for us to assign a \emph{direction} to a curve $\gamma$, by designating one of its endpoints, that we denote by $s(\gamma)$, as its first endpoint, and the other endpoint, denoted by $t(\gamma)$, as its last endpoint. If $\Gamma$ is a collection of curves, and each curve in $\Gamma$ is assigned a direction, then we say that $\Gamma$ is a collection of \emph{directed curves}. In such a case, we let $S(\Gamma)$ be the multiset of points containing the first endpoint of every curve in $\Gamma$, and we let $T(\Gamma)$ be the multiset of points containing the last endpoint of every curve in $\Gamma$.
For the type-2 uncrossing operation, we will consider curves that arise from some drawing $\phi$ of a graph $G$. We first need to define curves that are aligned with a graph drawing. For intuition, consider first some planar graph $G$, and its planar drawing $\phi$. In this case, curve $\gamma$ is aligned with the drawing $\phi$ of $G$, if there is some path $P$ in $G$, such that $\gamma$ can be obtained by first concatenating the images of all edges if $P$, and then possibly modifying the resulting curve within tiny discs $D_{\phi}(v)$ for vertices $v\in V(P)$ (typically via a nudging operation). If $\phi$ is a non-planar drawing of some graph $G$, then the definition of a curve $\gamma$ being aligned with the drawing is similar, but now we allow the curve $\gamma$ to ``switch'' from the image of one edge to another, at a crossing point between the two edges. Therefore, we can define a sequence $e_1,e_2,\ldots,e_{r-1}$ of edges of $G$, such that the curve ``follows'' segments of these edges. The curve $\gamma$ itself can then be partitioned into segments $\sigma_1,\sigma_1',\sigma_2,\sigma_2',\ldots,\sigma'_{r-1},\sigma_r$, where for all $1\leq i\leq r-1$, $\sigma_i'$ is a contiguous segment of the image of edge $e_i$. For a pair $\sigma'_i,\sigma'_{i+1}$ of such segments, either the last endpoint of $\sigma'_i$ and the first endpoint of $\sigma_{i+1}'$ are identical (and it is a crossing point between the images of $e_i$ and $e_{i+1}$); or segment $\sigma_{i+1}$ is contained in disc $D_{\phi}(v_{i+1})$, where $v_{i+1}$ is a common endpoint of $e_i$ and $e_{i+1}$. With this intuition in mind, we now define the notion of alignment of a curve with a drawing of a graph.
\begin{definition}[Curve aligned with a drawing of a graph]
Let $G$ be a graph and $\phi$ a drawing of $G$ in the plane. We say that a curve $\gamma$ is \emph{aligned} with the drawing $\phi$ of $G$ if there is a sequence $(e_1,e_2,\ldots,e_{r-1})$ of edges of $G$, and a partition $(\sigma_1,\sigma_1',\sigma_2,\sigma_2',\ldots,\sigma'_{r-1},\sigma_r)$ of $\gamma$ into consecutive segments, such that, if we denote, for all $1\leq i< r$, $e_i=(v_i,v_{i+1})$, then the following hold:
\begin{itemize}
\item for all $1\leq i\leq r-1$, $\sigma'_i$ is a contiguous segment of non-zero length of $\phi(e_i)$, and it is disjoint from all discs in $\set{D_{\phi}(u)}_{u\in V(G)}$, except that its first endpoint may lie on the boundary of $D_{\phi}(v_i)$, and its last endpoint may lie on the boundary of $D_{\phi}(v_{i+1})$;
\item for all $1\leq i\leq r$, segment $\sigma_i$ is either contained in disc $D_{\phi}(v_i)$, or it is contained in a tiny $p$-disc $D(p)$, where $p$ is a crossing point of $\phi(e_{i-1})$ and $\phi(e_{i})$;
\item $\sigma_1=\phi(e_1)\cap D_{\phi}(v_1)$; and
\item $\sigma_r=\phi(e_{r-1})\cap D_{\phi}(v_r)$.
\end{itemize}
\end{definition}
In order to perform a type-2 uncrossing operation, we consider a graph $G$, a drawing $\phi$ of $G$, and a set ${\mathcal{Q}}$ of simple directed paths in $G$.
We assume that no vertex of $G$ may serve simultaneously as an endpoint of a path of ${\mathcal{Q}}$ and an inner vertex of some other path of ${\mathcal{Q}}$.
We can then define a set $\Gamma=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$ of curves, where, for every path $Q\in {\mathcal{Q}}$, curve $\gamma(Q)$ is obtained by concatenating the images of the edges of $Q$. Note however that the curves in the resulting set $\Gamma$ are not necessarily in general position. Type-2 uncrossing allows us to fix this, and moreover to eliminate all crossings between the resulting set $\Gamma'$ of curves. Unlike type-1 uncrossing, we only guarantee that the multisets containing the first and last endpoints of the curves in $\Gamma'$ remain identical to those corresponding to $\Gamma$, but we no longer guarantee that they are matched in the same way to each other. For technical reasons, we need to consider two different settings for the type-2 uncrossing: one where the paths in set ${\mathcal{Q}}$ are edge-disjoint, in which case we can provide somewhat stronger guarantees, and another where this is not the case. These two settings for type-2 uncrossing are provided in the following theorem and its corollary, whose proofs are simple and are deferred to Sections \ref{apd: new type 2 uncrossing} and \ref{apd: cor new type 2 uncrossing} of Appendix, respectively. We start with the setting where the paths in set ${\mathcal{Q}}$ are edge-disjoint.
\begin{theorem}
\label{thm: new type 2 uncrossing}
There is an efficient algorithm, whose input consists of a graph $G$, a drawing $\phi$ of $G$ on the sphere, and a collection ${\mathcal{Q}}$ of edge-disjoint paths in $G$, such that no vertex of $G$ may simultaneously serve as an endpoint of some path in ${\mathcal{Q}}$ and an inner vertex of some path in ${\mathcal{Q}}$. Additionally, for each path $Q\in {\mathcal{Q}}$, one of its endpoints is designated as its first endpoint and is denoted by $s(Q)$, and the other endpoint is designated as its last endpoint and denoted by $t(Q)$.
The algorithm computes a set $\Gamma=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$ of directed simple curves on the sphere with the following properties:
\begin{itemize}
\item every curve $\gamma(Q)\in \Gamma$ is aligned with the drawing of the graph $\bigcup_{Q'\in {\mathcal{Q}}}Q'$ induced by $\phi$;
\item for each path $Q\in {\mathcal{Q}}$, $s(\gamma(Q))=\phi(s(Q))$; moreover, if $e_1(Q)$ is the first edge of $Q$, then curve $\gamma(Q)$ contains the segment $\phi(e_1(Q))\cap D_{\phi}(s(Q))$;
\item the multiset $T(\Gamma)$, containing the last endpoint of every curve in $\Gamma$, is precisely the multiset $\set{\phi(t(Q))\mid Q\in {\mathcal{Q}}}$, containing the image of the last vertex on every path of ${\mathcal{Q}}$ in $\phi$; and
\item the curves in $\Gamma$ do not cross each other.
\end{itemize}
\end{theorem}
We emphasize that the curves in $\Gamma$ may match the mutisets $\set{\phi(s(Q))\mid Q\in {\mathcal{Q}}}$ and $\set{\phi(t(Q))\mid Q\in {\mathcal{Q}}}$ differently from the paths in ${\mathcal{Q}}$.
We will sometimes use \Cref{thm: new type 2 uncrossing} in a setting where we are additionally given a subgraph $C\subseteq G$, and the paths of ${\mathcal{Q}}$ are internally disjoint from $C$. In such a case, from the definition of aligned curves, and from the fact that the curves of $\Gamma$ do not cross each other, for every edge $e\in E(C)$, the number of crossings between $\phi(e)$ and the curves in $\Gamma$ is bounded by the number of crossings between $\phi(e)$ and the curves of $\set{\phi(e')\mid e'\in \bigcup_{Q\in {\mathcal{Q}}}E(Q)}$.
We use the following corollary of \Cref{thm: new type 2 uncrossing}, that deals with the setting where paths in set ${\mathcal{Q}}$ may share edges. The proof is deferred to Section \ref{apd: cor new type 2 uncrossing} of Appendix.
\begin{corollary}
\label{cor: new type 2 uncrossing}
There is an efficient algorithm, whose input consists of a graph $G$, a drawing $\phi$ of $G$ on the sphere, a subgraph $C$ of $G$, and a collection ${\mathcal{Q}}$ of paths in $G$, that are internally disjoint from $C$, such that no vertex of $G$ may simultaneously serve as an endpoint of some path in ${\mathcal{Q}}$ and an inner vertex of some path in ${\mathcal{Q}}$. Additionally, for each path $Q\in {\mathcal{Q}}$, one of its endpoints is designated as its first endpoint and is denoted by $s(Q)$, and the other endpoint is designated as its last endpoint and is denoted by $t(Q)$.
The algorithm computes a set $\Gamma=\set{\gamma(Q)\mid Q\in {\mathcal{Q}}}$ of directed simple curves on the sphere with the following properties:
\begin{itemize}
\item for every path $Q\in {\mathcal{Q}}$, $s(\gamma(Q))=\phi(s(Q))$;
\item the multiset $T(\Gamma)$, containing the last endpoint of every curve in $\Gamma$, is precisely the multiset $\set{\phi(t(Q))\mid Q\in {\mathcal{Q}}}$, containing the image of the last vertex on every path of ${\mathcal{Q}}$;
\item the curves in $\Gamma$ do not cross each other; and
\item for each edge $e\in E(C)$, the number of crossings between $\phi(e)$ and the curves in $\Gamma$ is bounded by
$\sum_{e'\in E(G)\setminus E(C)}\chi(e,e')\cdot \cong_G({\mathcal{Q}},e')$, where $\chi(e,e')$ is the number of crossings between $\phi(e)$ and $\phi(e')$.
\end{itemize}
\end{corollary}
\subsection{Cuts, Well-Linkedness, and Related Notions}
\label{subsec: cuts, wl}
\subsubsection{Minimum Cuts}
A \emph{cut} in a graph $G$ is a bipartition $(A,B)$ of its vertices into non-empty subsets. The \emph{value} of the cut is $|E(A,B)|$.
We sometimes consider cuts in edge-capacitated graphs. Given a graph $G$ with capacities $c(e)\geq 0$ on edges $e\in E(G)$ and a cut $(A,B)$ in $G$, the value of the cut is $\sum_{e\in E_G(A,B)}c(e)$.
When edge capacities are not specified, we assume that they are unit.
Given two disjoint subsets $S,T$ of vertices of $G$, an \emph{$S$-$T$} cut, or a \emph{cut separating $S$ from $T$} is a cut $(A,B)$ with $S\subseteq A$, $T\subseteq B$.
A \emph{minimum $S$-$T$ cut} is an $S$-$T$ cut $(A,B)$ of minimum value. When $S=\set{s}$ and $T=\set{t}$, we refer to $S$-$T$ cuts as $s$-$t$ cuts.
We will use the following lemma, whose proof is provided in Section~\ref{apd: Proof of multiway cut with paths sets} of Appendix.
\begin{lemma}\label{lem: multiway cut with paths sets}
There is an efficient algorithm, that, given a graph $G$ and a collection $S=\set{s_1,\ldots,s_k}$ of its vertices, computes, for all $1\leq i\leq k$, a set $A_i$ of vertices of $G$, and a collection ${\mathcal{Q}}_i$ of paths in $G$, such that the following hold:
\begin{itemize}
\item for all $1\leq i\leq k$, $S\cap A_i=\set{s_i}$, and moreover, $(A_i,V(G)\setminus A_i)$ is a minimum cut separating $s_i$ from the vertices of $S\setminus\set{s_i}$ in $G$;
\item for all $1\leq i<i'\leq k$, $A_i\cap A_{i'}= \emptyset$; and
\item for all $1\leq i\leq k$, ${\mathcal{Q}}_i=\set{Q_i(e)\mid e\in \delta_G(A_i)}$, where for each $e\in \delta_G(A_i)$, path $Q_i(e)$ has $e$ as its first edge, $s_i$ as its last vertex, and all internal vertices of $Q_i(e)$ lie in $A_i$. Moreover, the paths in set ${\mathcal{Q}}_i$ are edge-disjoint.
\end{itemize}
\end{lemma}
\subsubsection{Gomory-Hu Trees}
\label{subsec: GH tree}
Gomory-Hu tree is a convenient structure that represents all minimum $s$-$t$ cuts in a given graph $G$. We summarize its properties in the following theorem.
\begin{theorem}[\cite{gomory1961multi}]\label{thm: GH tree properties}
There is an efficient algorithm, that, given a graph $G=(V,E)$ with capacities $c(e)\geq 0$ on its edges $e\in E$, computes a tree $\tau=(V,E')$ with capacities $c'(e)\geq 0$ on its edges $e\in E'$, such that the following hold:
\begin{itemize}
\item for every pair $s,t$ of distinct vertices of $V$, the value of the minimum $s$-$t$ cut in $G$ is equal to $\min_{e\in E(P_{s,t})}\set{c'(e)}$, where $P_{s,t}$ is the unique path connecting $s$ to $t$ in $\tau$; and
\item for every pair $s,t$ of distinct vertices of $V$, if $(A,B)$ is a minimum $s$-$t$ cut in graph $G$, then $(A,B)$ is a minimum $s$-$t$ cut in graph $\tau$, and vice versa.
\end{itemize}
\end{theorem}
We obtain the following immediate corollary of \Cref{thm: GH tree properties}.
\begin{corollary}
\label{cor: G-H tree_edge_cut}
Let $G$ be an edge-capacitated graph, and let $\tau$ be a Gomory-Hu tree of graph $G$. Then for every edge $e=(u,u')\in E(\tau)$, if we denote by $U,U'$ the vertex sets of the two connected components of $\tau\setminus\set{e}$, with $u\in U$, then $(U,U')$ is a minimum $u$-$u'$ cut in graph $G$.
\end{corollary}
\iffalse
We will use the following lemma to handle graphs whose Gomory-Hu tree is a path. Its proof appears in Appendix~\ref{apd: Proof of GH tree path vs contraction}.
\begin{lemma}
\label{lem: GH tree path vs contraction}
Let $G$ be a graph whose Gomory-Hu tree $\tau$ is a path.
Let ${\mathcal{C}}$ be a set of vertex-disjoint clusters of $G$, such that $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)$, and the vertices in each cluster of ${\mathcal{C}}$ induce a subpath in $\tau$ (equivalently, if we denote $\tau=(v_1,\ldots,v_n)$, where vertices appear on path $\tau$ in this order, then for each cluster $C\in {\mathcal{C}}$, there exist two integers $i_C,j_C$, such that $V(C)=\set{v_{t}\mid i_C\le t\le j_C}$).
Then $\tau_{|{\mathcal{C}}}$, the contracted path, is a Gomory-Hu tree of the contracted graph $G_{|{\mathcal{C}}}$.
\end{lemma}
\fi
\subsubsection{Balanced Cut and Sparsest Cut}
Suppose we are given a graph $G=(V,E)$, and a subset $T\subseteq V$ of its vertices. We say that a cut $(X,Y)$ in $G$ is a valid $T$-cut iff $X\cap T,Y\cap T\neq \emptyset$. The \emph{sparsity} of a valid $T$-cut $(X,Y)$ with respect to $T$ is $\frac{|E(X,Y)|}{\min\set{|X\cap T|, |Y\cap T|}}$.
In the Sparsest Cut problem, given a graph $G$ and a subset $T$ of its vertices, the goal is to compute a valid $T$-cut of minimum sparsity. Arora, Rao and Vazirani~\cite{ARV} designed an $O(\sqrt {\log n})$-approximation algorithm for the sparsest cut problem\footnote{The algorithm was originally designed for simple graphs, but it can be easily generalized to graphs with parallel edges by exploiting edge capacities.}, where $n=|V(G)|$.
We denote this algorithm by \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace, and its approximation factor by $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)=O(\sqrt{\log n})$.
We say that a cut $(A,B)$ in a graph $G$ is \emph{$\eta$-edge-balanced}, or just \emph{$\eta$-balanced}, for a parameter $0<\eta<1$, if $|E(A)|,|E(B)|\leq \eta\cdot|E(G)|$. We say that a cut $(A,B)$ is a \emph{minimum $\eta$-balanced cut in $G$} if $(A,B)$ is an $\eta$-balanced cut of minimum value $|E(A,B)|$.
We will use the following theorem that follows from the work of \cite{ARV}.
The proof is provided in Section~\ref{apd: Proof of approx_balanced_cut} of Appendix.
\begin{theorem}
\label{cor: approx_balanced_cut}
For every constant $1/2<\hat \eta <1$, there is another constant $\hat \eta<\hat \eta'<1$ and an efficient algorithm, that, given a connected (not necessarily simple) graph $G$ with $m$ edges, computes a $\hat \eta'$-balanced cut $(A,B)$ in $G$, whose value is at most $O(\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m))$ times the value of the minimum $\hat \eta$-balanced cut in $G$.
\end{theorem}
The following lemma is a simple consequence of the Planar Separator Theorem of Lipton and Tarjan \cite{lipton1979separator}.
A version of the lemma for vertex-balanced cuts was proved in~\cite{pach1996applications}.
For completeness, we provide the proof of the lemma in Section~\ref{apd: Proof of min_bal_cut} of Appendix.
\begin{lemma}
\label{lem:min_bal_cut}
Let $G$ be a connected (not necessarily simple) graph with $m$ edges and maximum vertex degree $\Delta\le m/2^{40}$. If $\mathsf{OPT}_{\mathsf{cr}}(G)\le m^2/2^{40}$, then the value of a minimum $(3/4)$-edge-balanced cut in $G$ is at most $O(\sqrt{\mathsf{OPT}_{\mathsf{cr}}(G)+\Delta\cdot m})$.
\end{lemma}
\subsubsection{Well-Linkedness, Bandwidth Property, and Routing Well-Linked Vertex Sets}
The notion of well-linkedness plays a central role in graph theory and graph algorithms (see e.g. \cite{racke2002minimizing,chekuri2004edge,andrews2010approximation,chuzhoy2012polylogarithmic,chuzhoy2012routing,chekuri2016polynomial,chuzhoy2016improved,chuzhoy2019towards}).
We use the following standard definitions, which are equivalent to
those used in much of previous work.
\begin{definition}[Well-Linkedness]
We say that a set $T$ of vertices in a graph $G$ is \emph{$\alpha$-well-linked}, for a parameter $0<\alpha<1$, if the sparsity of every valid $T$-cut in graph $G$ is at least $\alpha$. Equivalently, for every partition $(A,B)$ of $V(G)$ with $A\cap T,B\cap T\neq \emptyset$, $|E_G(A,B)|\geq \alpha\cdot \min\set{|A\cap T|,|B\cap T|}$ must hold.
\end{definition}
The next simple observation, that has been used extensively in previous work, shows that the set of vertices lying on the first row of the $(r\times r)$-grid is $1$-well-linked. For completeness, we provide its proof in Section \ref{apd: Proof grid 1st row well-linked} of Appendix.
\begin{observation}
\label{obs: grid 1st row well-linked}
Let $r\geq 1$ be an integer, and let $H$ be the $(r\times r)$-grid graph. Let $S$ be the set of vertices lying on the first row of the grid. Then vertex set $S$ is $1$-well-linked in $H$.
\end{observation}
Next, we define the notion of bandwidth property, that was also used extensively in graph algorithms.
\begin{definition}[$\alpha$-Bandwidth Property]
We say that a cluster $C$ of a graph $G$ has the \emph{$\alpha$-bandwidth property} in $G$, for some parameter $0<\alpha<1$, if, for every partition $(A,B)$ of vertices of $C$,
$|E_G(A,B)|\ge \alpha\cdot\min\set{|\delta_G(A)\cap \delta_G(C)|, |\delta_G(B)\cap \delta_G(C)|}$.
\end{definition}
The following immediate observation provides an equivalent definition of the bandwidth property that is helpful to keep in mind. Recall that, for a cluster $C$ of a graph $G$, its augmentation $C^+$ is a graph that is obtained from graph $G$ as follows. We subdivide every edge $e\in \delta_G(C)$ with a vertex $t_e$, and let $T(C)=\set{t_e\mid e\in \delta_G(C)}$ be the resulting set of newly added vertices. We then let $C^+$ be the subgraph of the resulting graph induced by vertex set $V(C)\cup T(C)$.
\begin{observation}\label{obs: wl-bw}
Let $G$ be a graph, let $C\subseteq G$ a cluster of $G$, and let $0<\alpha<1$ be a parameter. Cluster $C$ has the $\alpha$-bandwidth property iff the set $T(C)=\set{t_e\mid e\in \delta_G(C)}$ of vertices is $\alpha$-well-linked in graph $C^+$, which is the augmentation of cluster $C$ in $G$.
\end{observation}
One useful property of well-linked sets of vertices is that routing is easy between vertices of such sets. We summarize this property, that has been used extensively in past work, in the following theorem, and we provide its proof in Appendix~\ref{apd: Proof of bandwidth_means_boundary_well_linked} for completeness. The theorem uses the notion of one-to-one routing that was defined in \Cref{subsubsection: routing paths}.
\begin{theorem}
\label{thm: bandwidth_means_boundary_well_linked}
There is an efficient algorithm, that, given a graph $G$, a set $T$ of vertices of $G$ that is $\alpha$-well-linked, and a pair $T_1,T_2$ of disjoint equal-cardinality subsets of $T$, computes a one-to-one routing ${\mathcal{Q}}$ of vertices of $T_1$ to vertices of $T_2$, with $\cong_G({\mathcal{Q}})\leq \ceil{1/\alpha}$.
\end{theorem}
The next corollary follows immediately from \Cref{obs: wl-bw} and \Cref{thm: bandwidth_means_boundary_well_linked}.
\begin{corollary}
\label{cor: bandwidth_means_boundary_well_linked}
There is an efficient algorithm, that, given a graph $G$, a cluster $S$ of $G$ that has the $\alpha$-bandwidth property for some $0<\alpha<1$, and a pair $E_1,E_2$ of disjoint equal-cardinality subsets of the edge set $\delta_G(S)$, computes a one-to-one routing ${\mathcal{Q}}$ of edges of $E_1$ to edges of $E_2$, with $\cong_G({\mathcal{Q}})\leq \ceil{1/\alpha}$, such that, for every path $Q\in{\mathcal{Q}}$, all inner vertices of $Q$ lie in $S$.
\end{corollary}
\subsubsection{Basic Well-Linked Decomposition}
Typically, in a well-linked decomposition, we are given a graph $G$ together with a cluster $S$ of $G$, and our goal is to compute a partition of $S$ into clusters, each of which has the $\alpha$-bandwidth property in graph $G$, for some given parameter $0<\alpha<1$. Algorithms for computing well-linked decompositions were used extensively in prior work on graph-based problems (see e.g. \cite{racke2002minimizing,chekuri2004edge,andrews2010approximation,chuzhoy2012polylogarithmic,chuzhoy2012routing,chekuri2016polynomial,chuzhoy2016improved,chuzhoy2019towards}). We use a variation of this technique, that, in addition to ensuring that each cluster $R$ in the decomposition has the $\alpha$-bandwidth property, provides a collection ${\mathcal{P}}(R)$ of paths routing the edges of $\delta_G(R)$ to edges of $\delta_G(S)$, such that the paths in ${\mathcal{P}}(R)$ are internally disjoint from $R$ and cause low congestion. The proof uses standard techniques and is deferred to Section~\ref{apd: Proof of well_linked_decomposition} of Appendix.
\begin{theorem}
\label{thm:well_linked_decomposition}
There is an efficient algorithm, whose input is a graph $G$, a connected cluster $S$ of $G$, and parameters $m$ and $\alpha$, for which $|E(G)|\leq m$ and $0<\alpha< \min\set{\frac 1 {64\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m},\frac 1 {48\log^2 m}}$ hold.
The algorithm computes a collection ${\mathcal{R}}$ of vertex-disjoint clusters of $S$, such that:
\begin{itemize}
\item $\bigcup_{R\in {\mathcal{R}}}V(R)=V(S)$;
\item for every cluster $R\in{\mathcal{R}}$, $|\delta_G(R)|\le |\delta_G(S)|$;
\item every cluster $R\in{\mathcal{R}}$ has the $\alpha$-bandwidth property in graph $G$; and
\item $\sum_{R\in {\mathcal{R}}}|\delta_G(R)|\le |\delta_G(S)|\cdot\textsf{left}(1+O(\alpha\cdot \log^{1.5} m)\textsf{right})$.
\end{itemize}
Additionally, the algorithm computes,
for every cluster $R\in {\mathcal{R}}$, a set ${\mathcal{P}}(R)=\set{P(e)\mid e\in \delta_G(R)}$ of paths in graph $G$ with $\cong_G({\mathcal{P}}(R))\leq 100$, such that, for every edge $e\in \delta_G(R)$, path $P(e)$ has $e$ as its first edge and some edge of $\delta_G(S)$ as its last edge, and all inner vertices of $P(e)$ lie in $V(S)\setminus V(R)$.
\end{theorem}
We note that, while the above theorem requires that cluster $S$ is connected, it can also be used when this is not the case, by simply applying the algorithm to every connected component of $S$ and then taking the union of all resulting sets of clusters; all properties that the theorem guarantees will continue to hold.
\subsubsection{Layered Well-Linked Decomposition}
\label{subsec: layered wld}
To the best of our knowledge, layered well-linked decomposition was first introduced by Andrews \cite{andrews2010approximation}. It is similar to the basic well-linked decomposition, except that it has some additional useful properties. We start by defining a layered well-linked decomposition formally. Our definition is very similar to that of \cite{andrews2010approximation}, except that we require some additional properties.
Let $H$ be a graph with $|E(H)|=m$ and $C\subseteq H$ a cluster of $H$. Let ${\mathcal{W}}$ be a collection of disjoint clusters of $H\setminus C$ with $\bigcup_{W\in {\mathcal{W}}}V(W)=V(H\setminus C)$, and let $({\mathcal{L}}_1,{\mathcal{L}}_2,\ldots,{\mathcal{L}}_r)$ be a partition of ${\mathcal{W}}$ into subsets that we call \emph{layers}. We denote ${\mathcal{L}}_0=\set{C}$, and, for all $1\leq i\leq r$, for every cluster $W\in {\mathcal{L}}_i$, we partition the set $\delta_H(W)$ of edges into two subsets: set $\delta^{\operatorname{down}}(W)$ containing all edges $(u,v)$ with $u\in V(W)$ and $v$ lying in a cluster of ${\mathcal{L}}_0\cup\cdots\cup{\mathcal{L}}_{i-1}$, and set $\delta^{\operatorname{up}}(W)$ containing all remaining edges of $\delta(W)$, namely: all edges $(u,v)$ with $u\in V(W)$ and $v$ lying in a cluster of ${\mathcal{L}}_i\cup\cdots\cup{\mathcal{L}}_{r}$ (see \Cref{fig: LWLD}). We say that the collection ${\mathcal{W}}$ of clusters, together with its partition $({\mathcal{L}}_1,{\mathcal{L}}_2,\ldots,{\mathcal{L}}_r)$ into layers is a \emph{valid layered $\alpha$-well-linked decomposition of $H$ with respect to $C$}, for some parameter $0<\alpha<1$, iff the following conditions hold:
\begin{properties}{L}
\item For every pair $W,W'$ of distinct clusters in ${\mathcal{W}}$, $V(W)\cap V(W')=\emptyset$, and $\bigcup_{W\in {\mathcal{W}}}V(W)=V(H)\setminus V(C)$; \label{condition: layered decomposition is partition}
\item each cluster $W\in {\mathcal{W}}$ has the $\alpha$-bandwidth property in $H$; \label{condition: layered well linked}
\item for every cluster $W\in {\mathcal{W}}$, $|\delta_H(W)|\leq |\delta_H(C)|$, and $|E_H(W)|\geq |\delta_H(W)|/(64\log m)$; \label{condition: layered decomp each cluster prop}
\item for every cluster $W\in {\mathcal{W}}$, $|\delta^{\operatorname{up}}(W)|<|\delta^{\operatorname{down}}(W)|/\log m$; \label{condition: layered decomp edge ratio}
\item $\sum_{W\in {\mathcal{W}}}|\delta_H(W)|\leq 4|\delta_H(C)|$; \label{condition: layered decomposition few edges}
and
\item for every cluster $W\in {\mathcal{W}}$, there is a collection ${\mathcal{P}}(W)=\set{P(e)\mid e\in \delta_H(W)}$ of paths in $H$, that cause congestion at most $200/\alpha$, and for all $e\in \delta_H(W)$, path $P(e)$ contains $e$ as its first edge, some edge $e'\in \delta_H(C)$ as its last edge, and all inner vertices of $P(e)$ are disjoint from $W$. \label{condition: layered decomposition routing}
\end{properties}
\begin{figure}[h]
\centering
\includegraphics[scale=0.08]{figs/LWLD.jpg}
\caption{An illustration of a layered well-linked decomposition of $H$ with respect to $C$. For cluster $W\in {\mathcal{L}}_2$, the edges of $\delta^{\operatorname{up}}(W)$ are shown in red, and the edges of $\delta^{\operatorname{down}}(W)$ are shown in blue.}\label{fig: LWLD}
\end{figure}
Recall that, given a graph $H$ and two sets $E',E''$ of its edges, we say that a set ${\mathcal{P}}$ of paths in $H$ routes edges of $E'$ to edges of $E''$ if ${\mathcal{P}}=\set{P(e)\mid e\in E'}$, and, for each edge $e\in E'$, path $P(e)$ has $e$ as its first edge and some edge of $E''$ as its last edge. Given a cluster $W$ of $H$, we say that the set ${\mathcal{P}}$ of paths \emph{avoids} $W$ if, for every path $P\in {\mathcal{P}}$, no inner vertex of $P$ lies in $W$. Therefore, Condition \ref{condition: layered decomposition routing} equivalently requires that for every cluster $W\in {\mathcal{W}}$, there is a collection ${\mathcal{P}}(W)$ of paths in $H$
routing the edges of $\delta_H(W)$ to the edges of $\delta_H(C)$, such that the paths in ${\mathcal{P}}(W)$ avoid $W$. This property is the main difference between our definition of a layered well-linked decomposition and that of \cite{andrews2010approximation}, which did not require this property.
The following theorem allows us to compute a layered well-linked decomposition in any graph. Its proof is practically identical to the algorithm of \cite{andrews2010approximation}. The main difference is that we need to prove that the resulting decomposition has property \ref{condition: layered decomposition routing}.
The proof of the theorem is deferred to Section \ref{sec: layered well linked} of Appendix.
\begin{theorem}\label{thm: layered well linked decomposition}
There is a large enough constant $c$, and an efficient algorithm, that given a connected graph $H$ with $|E(H)|=m\geq c$ and a cluster $C$ of $H$,
computes a valid layered $\alpha$-well-linked decomposition $({\mathcal{W}}, ({\mathcal{L}}_1,\ldots,{\mathcal{L}}_r))$ of $H$ with respect to $C$, for $\alpha=\frac{1}{c\log^{2.5}m}$. The number of layers in the decomposition is $r\leq \log m$.
\end{theorem}
\subsection{Expanders, Graph Embeddings, and Routing Well-Linked Sets}
We will use the notion of expanders, that we define next.
\begin{definition}[Expanders]
We say that a graph $W$ is an $\alpha$-expander, for some $0<\alpha<1$, if, for every partition $(A,B)$ of $V(W)$ into non-empty subsets, $|E_W(A,B)|\geq \alpha\cdot\min\set{|A|,|B|}$; equivalently, the set $V(W)$ of vertices is $\alpha$-well-linked in $W$.
\end{definition}
We will also use a standard notion of graph embeddings.
\begin{definition}[Embedding of Graphs]
Let $H$, $G$ be a pair of graphs with $V(H)\subseteq V(G)$. An \emph{embedding} of $H$ into $G$ is a collection ${\mathcal{P}}=\set{P(e)\mid e\in E(H)}$ of paths in graph $G$, where for each edge $e=(u,v)\in E(H)$, path $P(e)$ has endpoints $u$ and $v$. The \emph{congestion of the embedding} is $\cong_{G}({\mathcal{P}})$.
\end{definition}
The following well known claim shows a connection between well-linked sets of vertices and embeddings of expanders. The proof is standard and deferred to Section~\ref{apd: Proof of embed expander} of Appendix.
\begin{claim}\label{claim: embed expander}
There is a universal constant $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}$, and an efficient randomized algorithm that, given a graph $G$ together with a subset $T$ of its vertices of cardinality $k$, such that $T$ is $\alpha$-well-linked in $G$, for some $0<\alpha<1$, constructs another graph $W$ with $V(W)=T$ and maximum vertex degree at most $\ensuremath{c_{\mbox{\tiny{\sc CMG}}}}\log^2k$, together with an embedding ${\mathcal{P}}$ of $W$ into $G$ with congestion at most $\frac{\ensuremath{c_{\mbox{\tiny{\sc CMG}}}} \log^2k}{\alpha}$, such that with high probability graph $W$ is an $(1/4)$-expander.
\end{claim}
We show in the following observation that, if $W$ is the outcome expander of the algorithm from \Cref{claim: embed expander}, then it has a high crossing number. The proof is provided in Section~\ref{apd: Proof of cr of exp} of Appendix.
\begin{observation}\label{obs: cr of exp}
There is some constant $c$, such that, if $W$ is an $(1/4)$-expander, with $|V(W)|=k>c$ and maximum vertex degree $O(\log^2k)$, then $\mathsf{OPT}_{\mathsf{cr}}(W)\geq k^2/(c\log^8k)$.
\end{observation}
We obtain the following useful corollary of \Cref{claim: embed expander}, that allows us to route specific pairs of vertices of a well-linked vertex set $T$. We provide its proof in Appendix~\ref{apd: Proof of routing well linked vertex set}.
\begin{corollary}\label{cor: routing well linked vertex set}
There is an efficient randomized algorithm that, given a graph $G$, a subset $T$ of its vertices of cardinality $k$, that is $\alpha$-well-linked in $G$, for some $0<\alpha<1$, together with a partial matching $M$ over the vertices of $T$, computes a set ${\mathcal{R}}(M)=\set{R(u,v)\mid (u,v)\in M}$ of paths in graph $G$, such that for every pair $(u,v)\in M$ of vertices, path $R(u,v)$ connects $u$ to $v$. Moreover, with high probability, the congestion caused by the paths in ${\mathcal{R}}(M)$ in $G$ is $O((\log^4 k)/\alpha)$.
\end{corollary}
Let $K_z$ be a complete graph, whose vertex set has cardinality $z$. We obtain the following immediate corollary of \Cref{cor: routing well linked vertex set}, whose proof appears in Section~\ref{apd: Proof of embed complete graph} of Appendix.
\begin{corollary}\label{cor: embed complete graph}
There is an efficient randomized algorithm that, given a graph $G$ and a subset $T$ of its vertices of cardinality $z$, such that $T$ is $\alpha$-well-linked in $G$, for some $0<\alpha<1$, computes an embedding $\tilde {\mathcal{P}}$ of the complete graph $K_z$ with $V(K_z)=T$ into $G$, such that, with high probability, the congestion of the embedding is $O((z\log^4z)/\alpha)$.
\end{corollary}
\subsubsection{Constructing Internal Routers}
\iffalse
We now give some intuition for internal routers. One of the main subroutines that our algorithm uses is an \emph{disengagement procedure}. Intuitively, suppose we are given some drawing $\phi$ of a graph $G$, and assume additionally that we are given a collection ${\mathcal{C}}$ of clusters of $G$. Assume further that each cluster $C\in {\mathcal{C}}$ has the $\alpha$-bandwidth property for some $0<\alpha<1$. In a disengagement procedure (the analogues of which were used in \cite{chuzhoy2011algorithm,chuzhoy2020towards}), we slightly modify the drawing $\phi$ in order to ensure that every cluster $C$ is drawn ``separately''. In other words, we will define, for every cluster $C\in {\mathcal{C}}$, a disc $D(C)$, such that for every pair $C,C'\in {\mathcal{C}}$ of distinct clusters, $D(C)\cap D(C')=\emptyset$, and we will ``move'' the drawing of $C$ to appear inside $D(C)$. We then need to ``connect'' this new drawing of $C$ to the remainder of the graph, by adding drawings of edges of $\delta_G(C)$ to the current drawing. A central tool in this disengagement procedure is an internal router for cluster $C$: intuitively, it is a collection ${\mathcal{Q}}(C)$ of paths, routing edges of $\delta_G(C)$ to some fixed vertex $u_C$ of $C$. Ideally, we would like all paths in ${\mathcal{Q}}(C)$ to cause small congestion, but this may be impossible to achieve, for example, if all vertex degrees in $C$ are low. Instead, it is sufficient for us to ensure that the paths in ${\mathcal{Q}}(C)$ are chosen by a randomized algorithm such that, for every edge $e\in E(C)$, $\expect{(\cong({\mathcal{Q}}(C),e))^2}$ is small. Even this weaker property is quite challenging to achieve. Instead, we now show a simple result that allows us to ensure that for every edge $e\in E(C)$, $\expect{\cong({\mathcal{Q}}(C),e)}$ is small. This bound will already be sufficient for us to obtain some useful results, that are described later in this section. To motivate the statement of the next lemma, recall that $\alpha$-bandwidth property of cluster $C$ is equivalent to stating that, if we let $G'$ be obtained from graph $G$ by subdividing every edge $e\in \delta_G(C)$ with a vertex $t_e$, letting $T=\set{t_e\mid e\in \delta_G(e)}$, and we let $C'=G'[V(C)\cup T]$, then the set $T$ of vertices is $\alpha$-well-linked in $C'$. In the following lemma, it may be convenient to think of graph $G$ as being the graph $C'$ that we just defined.
\fi
We now provide an efficient algorithm, that, given a graph $G$ and a cluster $C$ of $G$ that has the $\alpha$-bandwidth property, constructs a distribution ${\mathcal{D}}(C)$ over the internal $C$-routers, such that the expected congestion on every edge of $C$ is small. We start with the following lemma, that provides a similar result for a graph $G$ and a set $T$ of vertices of $G$ that is well-linked.
\begin{lemma}\label{lem: simple guiding paths}
There is an efficient randomized algorithm, whose input is a graph $G$ and set $T$ of its vertices called terminals, such that $|T|=z$, and $T$ is $\alpha$-well-linked in $G$, for some $0<\alpha<1$. The algorithm computes, for every terminal $t\in T$, a set ${\mathcal{Q}}_t=\set{Q_t(t')\mid t'\in T\setminus\set{t}}$ of paths, where, for all $t'\in T\setminus\set{t}$, path $Q_t(t')$ connects $t'$ to $t$. Moreover, if we select a vertex $t\in T$ uniformly at random, then, for every edge $e\in E(G)$, $\expect{\cong({\mathcal{Q}}_t,e)}\leq O(\log^4z/\alpha)$.
\end{lemma}
\begin{proof}
We use the algorithm from \Cref{cor: embed complete graph}, in order to compute an embedding $\tilde{\mathcal{P}}$ the complete graph $K_z$ with $V(K_z)=T$ into $G$. Recall that the algorithm ensures that, with high probability, the congestion of the embedding is at most $(cz\log^4z)/\alpha$, for some constant $c$. If the congestion caused by the paths in $\tilde {\mathcal{P}}$ is greater than this bound, then we
say that the algorithm from \Cref{cor: embed complete graph} failed.
We repeat the algorithm from \Cref{cor: embed complete graph} $O(\log |E(G)|)$ times. Let ${\cal{E}}_1$ be the event that the algorithm failed in each of these applications. Then $\prob{{\cal{E}}_1}\leq 1/\operatorname{poly}(z)$. In this case, for every terminal $t\in T$, we return a set ${\mathcal{Q}}_t=\set{Q_t(t')\mid t'\in T\setminus\set{t}}$ of paths, where for every terminal $t'\in T\setminus\set{t}$, $Q_t(t')$ is an arbitrary path connecting $t$ to $t'$ in $G$. Clearly, for all $t\in T$, for every edge $e\in E(G)$, $\cong_G({\mathcal{Q}}_t,e)\leq z$.
We assume from now on that, in some application of the algorithm from \Cref{cor: embed complete graph}, it returned a set $\tilde {\mathcal{P}}$ of paths with $\cong_G(\tilde {\mathcal{P}})\leq O((z\log^4z)/\alpha)$.
We now fix a terminal $t\in T$, and define the corresponding set ${\mathcal{Q}}_t=\set{Q_t(t')\mid t'\in T\setminus\set{t}}$ of paths. For every terminal $ t'\in T\setminus\set{t}$, we let $Q_t(t')$ be the unique path in set $\tilde {\mathcal{P}}$ that serves as the embedding of the edge $(t,t')\in E(K_z)$. Clearly, path $Q_t(t')$ connects $t'$ to $t$ as required.
Consider now an edge $e$, and let $\eta_e=\cong_G(\tilde {\mathcal{P}},e)\leq O((z\log^4z)/\alpha)$. Since every path of $\tilde {\mathcal{P}}$ may lie in at most two path sets of $\set{{\mathcal{Q}}_t}_{t\in T}$, we get that $\sum_{t\in T}\cong_G({\mathcal{Q}}_t,e)\leq 2\eta_e $. Therefore, if Event ${\cal{E}}_1$ did not happen, and a terminal $t\in T$ is selected uniformly at random, then
$\expect{\cong({\mathcal{Q}}_t,e)}\leq 2\eta_e/z\leq O(\log^4z/\alpha)$. Overall, for every edge $e\in E(G)$, $\expect{\cong({\mathcal{Q}}_t,e)}\leq \expect{\cong({\mathcal{Q}}_t,e)\mid \neg{\cal{E}}_1}+\expect{\cong({\mathcal{Q}}_t,e)\mid {\cal{E}}_1}\cdot \prob{{\cal{E}}_1}\leq O(\log^4z/\alpha)+O(1/z)\leq O(\log^4z/\alpha)$.
\end{proof}
The following corollary allows us to compute a distribution over internal $C$-routers for a cluster $C$ of a graph $G$, such that the expected congestion on every edge of $C$ is small. The corollary follows immediately by applying the algorithm from \Cref{lem: simple guiding paths} to the augmentation $C^+$ of the cluster $C$ in graph $G$. The proof of the corollary is omitted.
\begin{corollary}\label{cor: simple guiding paths}
There is an efficient randomized algorithm, whose input is a graph $G$ and a cluster $C$ of $G$ that has the $\alpha$-bandwidth property for some $0<\alpha<1$. The algorithm returns (explicitly) a distribution ${\mathcal{D}}$ over the set $\Lambda(C)$ of internal $C$-routers, such that, for every edge $e\in E(C)$, $\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{\cong({\mathcal{Q}},e)}\leq O((\log |\delta_G(C)|)^4/\alpha)$.
\end{corollary}
\section{Useful things from FOCS paper}
By default, all logarithms are to the base of $2$. All graphs are finite, simple and undirected. Graphs with parallel edges are explicitly referred to as multi-graphs.
We follow standard graph-theoretic notation. Assume that we are given a graph $G=(V,E)$.
For a vertex $v\in V$, we denote by $\delta_G(v)$ the set of all edges of $G$ that are incident to $v$.
For two disjoint subsets $A,B$ of vertices of $G$, we denote by $E_G(A,B)$ the set of all edges with one endpoint in $A$ and another in $B$.
For a subset $S\subseteq V$ of vertices, we denote by $E_G(S)$ the set of all edges with both endpoints in $S$, and we denote by $\mathsf{out}_G(S)$ the subset of edges of $E$ with exactly one endpoint in $S$, namely $\mathsf{out}_G(S)=E_G(S, V\!\setminus\! S)$.
We denote by $G[S]$ the subgraph of $G$ induced by $S$.
We sometimes omit the subscript $G$ if it is clear from the context.
We say that a graph $G$ is $\ell$-connected for some integer $\ell>0$, if there are $\ell$ vertex-disjoint paths between every pair of vertices in $G$.
Given a graph $G=(V,E)$, a \emph{drawing} $\phi$ of $G$ is an embedding of the graph into the plane, that maps every vertex to a point and every edge to a continuous curve that connects the images of its endpoints. We require that the interiors of the curves representing the edges do not contain the images of any of the vertices. We say that two edges \emph{cross} at a point $p$, if the images of both edges contain $p$, and $p$ is not the image of a shared endpoint of these edges. We require that no three edges cross at the same point in a drawing of $\phi$.
We say that $\phi$ is a \emph{planar drawing} of $G$ iff no pair of edges of $G$ crosses in $\phi$.
For a vertex $v\in V(G)$, we denote by $\phi(v)$ the image of $v$, and for an edge $e\in E(G)$, we denote by $\phi(e)$ the image of $e$ in $\phi$. For any subgraph $C$ of $G$, we denote by $\phi(C)$ the union of images of all vertices and edges of $C$ in $\phi$. For a path $P\subseteq G$, we sometimes refer to $\phi(P)$ as the \emph{image of $P$ in $\phi$}. Note that a drawing of $G$ in the plane naturally defines a drawing of $G$ on the sphere and vice versa; we use both types of drawings.
Given a graph $G$ and a drawing $\phi$ of $G$ in the plane, we use $\mathsf{cr}(\phi)$ to denote the number of crossings in $\phi$. Let $\phi'$ be the drawing of $G$ that is a mirror image of $\phi$. We say that $\phi$ and $\phi'$ are \emph{identical} drawings of $G$, and that their \emph{orientations} are different. We sometime say that $\phi'$ is obtained by \emph{flipping} the drawing $\phi$. If $\gamma$ is a simple closed curve in $\phi$ that intersects $G$ at vertices only, and $S$ is the set of vertices of $G$ whose images lie on $\gamma$, with $|S|\geq 3$, then we say that the circular orderings of the vertices of $S$ along $\gamma$ in $\phi$ and $\phi'$ are identical, but the orientations of the two orderings are different, or opposite.
Whitney \cite{whitney1992congruent} proved that every $3$-connected planar graph has a unique planar drawing.
Throughout, for a $3$-connected planar graph $G$, we denote by $\rho_G$ the unique planar drawing of $G$.
\textbf{Problem Definitions.}
The goal of the \textsf{Minimum Crossing Number}\xspace~problem is to compute a drawing the input graph $G$ in the plane with smallest number of crossings.
The value of the optimal solution, also called the \emph{crossing number} of $G$, is denoted by $\mathsf{OPT}_{\mathsf{cr}}(G)$.
We also consider a closely related problem called Crossing Number with Rotation System (\textnormal{\textsf{MCNwRS}}\xspace). In this problem, we are given a multi-graph $G$, and, for every vertex $v\in V(G)$, a circular ordering ${\mathcal{O}}_v$ of its incident edges. We denote $\Sigma=\set{{\mathcal{O}}_v}_{v\in V(G)}$, and we refer to $\Sigma$ as a \emph{rotation system}. We say that a drawing $\phi$ of $G$ \emph{respects} the rotation system $\Sigma$ if the following holds. For every vertex $v\in V(G)$, let $\eta(v)$ be an arbitrarily small disc around $v$ in $\phi$. Then the images of the edges of $\delta_G(v)$ in $\phi$ must intersect the boundary of $\eta(v)$ in a circular order that is identical to ${\mathcal{O}}_v$ (but we can choose the orientation of this ordering, that may be either clock-wise or counter-clock-wise). In the \textnormal{\textsf{MCNwRS}}\xspace problem, the input is a {\bf multi-graph} $G$ with a rotation system $\Sigma$, and the goal is to compute a drawing of $G$ in the plane that respects $\Sigma$ and minimizes the number of crossings.
\iffalse
\mynote{is this needed?}
If a crossing is caused by the intersection of the images of a pair of edges $e,e'$ (namely $\phi(e)$ and $\phi(e')$), then we say that the edges $e$ and $e'$ \emph{participate} in the crossing, or the pair of edges $e$ and $e'$ \emph{cross}.
We also say that an edge $e$ is \emph{crossed} in a drawing $\phi$ if the edge $e$ participates in at least one crossing of $\phi$.
For any pair $E_1, E_2 \subseteq E$ of subsets of edges, we denote by $\mathsf{cr}_{\phi}(E_1, E_2)$ the number of crossings in $\phi$ in which the images of edges of $E_1$ intersect the images of edges of $E_2$.
Clearly, for each subset $E' \subseteq E$, $\mathsf{cr}(\phi)=\mathsf{cr}_{\phi}(E',E')+\mathsf{cr}_{\phi}(E',E\setminus E')+\mathsf{cr}_{\phi}(E\setminus E',E\setminus E')$.
Since at most two edges participate in a crossing, for disjoint subsets $E_1,\ldots,E_r$ of $E$, $\sum_{1\le i\le r}\mathsf{cr}_{\phi}(E,E_i)\le 2\cdot\mathsf{cr}(\phi)$.
For a subgraph $C$ of $G$, we denote $\mathsf{cr}_{\phi}(C)=\mathsf{cr}_{\phi}(E(C),E(C))$, or equivalently $\mathsf{cr}_{\phi}(C)=\mathsf{cr}(\phi(C))$. Similarly, for two subgraphs $C,C'$, we denote $\mathsf{cr}_{\phi}(C,C')=\mathsf{cr}_{\phi}(E(C),E(C'))$.
\fi
{\bf Faces and Face Boundaries.}
Suppose we are given a planar graph $G$ and a drawing $\phi$ of $G$ in the plane. The set of faces of $\phi$ is the set of all connected regions of $\mathbb{R}^2\setminus \phi(G)$. We designate a single face of $\phi$ as the ``outer'', or the ``infinite'' face.
The \emph{boundary} $\delta(F)$ of a face $F$ is a sub-graph of $G$ consisting of all vertices and edges of $G$ whose image is incident to $F$. Notice that, if graph $G$ is not connected, then boundary of a face may also be not connected. Unless $\phi$ has a single face, the boundary $\delta(F)$ of every face $F$ of $\phi$ must contain a simple cycle $\delta'(F)$ that separates $F$ from all other faces. We sometimes refer to graph $\delta(F)\setminus\delta'(F)$ as the \emph{inner boundary} of $F$. Lastly, observe that, if $G$ is $2$-connected, then the boundary of every face of $\phi$ is a simple cycle.
\iffalse
\begin{definition}[face, outer face]
Given a graph $G$ and a planar drawing $\phi$ of $G$, the set of faces of $\phi$ are defined to be the set of connected regions in $\mathbb{R}^2\setminus \phi(G)$, and we denote it by ${\mathcal{F}}(\phi)$. Moreover, exactly one of these faces is infinite which we refer to as the \emph{outer face}.
\end{definition}
\begin{definition}[boundary, inner boundary, outer boundary, and point of contact]
For each face $F\in {\mathcal{F}}(\phi)$, the boundary of the face $F$ is defined as the set of vertices and edges that are incident to the face $F$, and we denote it by $\delta(F)$. Note that if the graph is connected, the boundary of $F$ also forms a connected graph. We may also call it the full boundary to emphasize the contrast to the inner/outer boundary defined next.
For a face that is not the outer face, we define the \emph{outer boundary} $\delta^{out}(F)\subseteq \delta(F)$ to be the simple cycle that separates the face $F$ from the outer face. In other words, if $D(F)$ denotes the closed disc whose boundary is $\delta^{out}(F)$, then $D(F)$ should contain $F$.
Note that if $G$ is $2$-connected, the boundary of each face $F\in {\mathcal{F}}(\phi)$ is a simple cycle, i.e., $\delta(F)=\delta^{out}(F)$.
Moreover, we define the \emph{inner boundary} as $\delta^{in}(F)$ to be the set of edges in $\delta(F)\setminus \delta^{out}(F)$, together with their endpoint vertices.
Finally, for a connected graph, we can define for each vertex $v$ on the boundary $\delta(F)$, the \emph{point of contact} of $v$ to be the vertex on the outer boundary with minimum distance to $v$.
\end{definition}
\fi
\textbf{Bridges and Extensions of Sub-Graphs.}
Let $G$ be a graph, and let $C\subseteq G$ be a sub-graph of $G$. A \emph{bridge} for $C$ in graph $G$ is either (i) an edge $e=(u,v)\in E(G)$ with $u,v\in V(C)$ and $e\not \in E(C)$; or (ii) a connected component of $G\setminus V(C)$.
We denote by ${\mathcal{R}}_G(C)$ the set of all bridges for $C$ in graph $G$. For each bridge $R\in {\mathcal{R}}_G(C)$, we define the set of vertices $L(R)\subseteq V(C)$, called the \emph{legs of $R$},
as follows. If $R$ consists of a single edge $e$, then $L(R)$ contains the endpoints of $e$. Otherwise, $L(R)$ contains all vertices $v\in V(C) $, such that $v$ has a neighbor that belongs to $R$.
Next, we define an \emph{extension} of the subgraph $C\subseteq G$, denoted by $X_G(C)$. The extension contains, for every bridge $R\in {\mathcal{R}}_G(C)$, a tree $T_R$, that is defined as follows. If $R$ is a bridge consisting of a single edge $e$, then the corresponding tree $T_R$ only contains the edge $e$. Otherwise,
let $R'$ be the sub-graph of $G$ consisting of the graph $R$, the vertices of $L(R)$, and all edges of $G$ connecting vertices of $R$ to vertices of $L(R)$.
We let $T_R\subseteq R'$ be a tree, whose leaves are precisely the vertices of $L(R)$, that contains smallest number of edges among all such trees. Note that such a tree exists because graph $R$ is connected.
We let the extension of $C$ in $G$ be $X_G(C)=\set{T_R\mid R\in {\mathcal{R}}_G(C)}$.
\iffalse
\textbf{Sparsest Cut.}
Suppose we are given a graph $G=(V,E)$, and a subset ${\mathcal T}\subseteq V$ of $k$ vertices, called terminals.
The sparsity of a cut $(S,\overline S)$ in $G$ is $\Phi(S)=\frac{|E(S,\overline S)|}{\min\set{|S\cap {\mathcal T}|, |\overline S\cap {\mathcal T}|}}$, and the value of the sparsest cut in $G$ is defined to be:
$\Phi(G)=\min_{S\subset V}\set{\Phi(S)}$.
The goal of the sparsest cut problem is to find a cut of minimum sparsity. Arora, Rao and Vazirani~\cite{ARV} have shown an $O(\sqrt {\log n})$-approximation algorithm for the sparsest cut problem. We denote this algorithm by \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace, and its approximation factor by $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)=O(\sqrt{\log n})$.
\fi
\textbf{Sparsest Cut and Well-Linkedness.}
Suppose we are given a graph $G=(V,E)$, and a subset $\Gamma\subseteq V$ of its vertices. We say that a cut $(X,Y)$ in $G$ is a valid $\Gamma$-cut iff $X\cap \Gamma,Y\cap \Gamma\neq \emptyset$. The \emph{sparsity} of a valid $\Gamma$-cut $(X,Y)$ is $\frac{|E(X,Y)|}{\min\set{|X\cap \Gamma|, |Y\cap \Gamma|}}$.
In the Sparsest Cut problem, given a graph $G$ and a subset $\Gamma$ of its vertices, the goal is to compute a valid $\Gamma$-cut of minimum sparsity. Arora, Rao and Vazirani~\cite{ARV} have shown an $O(\sqrt {\log n})$-approximation algorithm for the sparsest cut problem. We denote this algorithm by \ensuremath{{\mathcal{A}}_{\mbox{\textup{\scriptsize{ARV}}}}}\xspace, and its approximation factor by $\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(n)=O(\sqrt{\log n})$.
We say that a set $\Gamma$ of vertices of $G$ is \emph{$\alpha$-well-linked} in $G$, iff the value of the sparsest cut in $G$ with respect to $\Gamma$ is at least $\alpha$.
\iffalse
Let $G$ be a graph, and let $S$ be a subset of vertices of $G$, and let $0<\alpha<1$, such that $S$ is $\alpha$-well-linked.
We now define the sparsest cut and the concurrent flow instances corresponding to $S$ as follows. For each edge in $\mathsf{out}(S)$, we sub-divide the edge by adding a new vertex $t_e$ to it. Let $G'$ denote the resulting graph and let $T$ denote the set of all vertices $t_e$ for $e\in \mathsf{out}_G(S)$. Consider the graph $H=G'[S]\cup \mathsf{out}_{G'}(S)$. We can naturally define an instance of the non-uniform sparsest cut problem on $H$ where the set of terminals is $T$. We then have the following well-known observation.
\begin{observation}
Let $G$, $S$, $H$ and $T$ be defined as above, and let $0<\alpha <1$, such that $S$ is $\alpha$-well-linked. Then every pair of vertices in $T$ can send one flow unit to each other in $H$, such that the maximum congestion on any edge is at most $\beta(n)|T|/\alpha$. Moreover if $M$ is any partial matching on the vertices of $T$, then we can send one flow unit between every pair $(u,v)\in M$ in graph $H$, with maximum congestion at most $2\beta(n)/\alpha$.
\end{observation}
\iffalse
The following theorem shows the technique of \emph{well-linked decomposition}, which has been used extensively in graph algorithms (e.g., see~\cite{racke2002minimizing,chekuri2005multicommodity}).
\begin{theorem}[Terminal Well-Linked Decomposition]
\label{thm:well_linked-decomposition}
There is an efficient algorithm, that, given any connected graph $G$ and a subset $T\subseteq V(G)$ of its vertices with $|T|=k$, computes a collection $\set{G_1,\ldots,G_r}$ of connected subgraphs of $G$, such that:
\begin{enumerate}
\item for each $1\le i\le r$, denote $V_i=V(G_i)$, then $(V_1,\ldots,V_r)$ forms a partition of $V(G)$;
\item for each $1\le i\le r$, denote $T_i=T\cap V_i$, then the subset $T_i\cup \Gamma(V_i)$ of vertices is $\Omega(\frac{1}{\log^{3/2}k\cdot\log\log k})$-well-linked in $G_i$; and
\item $\sum_{1\le i\le r}|\mathsf{out}(V_i)|\le 0.01 k$, and $\sum_{1\le i\le r}|T_i\cup \Gamma(V_i)|\le 1.01 k$.
\end{enumerate}
\end{theorem}
\fi
\fi
\paragraph{Non-interfering paths.}
To be used later for locally non-interfering/non-interfering paths (for this the definition should be adapted). We may also keep it in prelims as a definition of crossing paths. Maybe we should also define a drawing of a path as a concatenation of images of its edges. We use it quite a bit.
\begin{definition}
Let $\gamma,\gamma'$ be two curves in the plane or on the sphere. We say that $\gamma$ and $\gamma'$ \emph{cross}, iff there is a disc $D$, whose boundary is a simple closed curve that we denote by $\beta$, such that:
\begin{itemize}
\item $\gamma\cap D$ is a simple open curve, whose endpoints we denote by $a$ and $b$;
\item $\gamma'\cap D$ is a simple open curve, whose endpoints we denote by $a'$ and $b'$; and
\item $a,a',b,b'\in \beta$, and they appear on $\beta$ in this circular order.
\end{itemize}
\end{definition}
Given a graph $G$ embedded in the plane or on the sphere, we say that two paths $P,P'$ in $G$ cross iff their images cross. Similarly, we say that a path $P$ crosses a curve $\gamma$ iff the image of $P$ crosses $\gamma$.
\section{Second Main Tool: Cluster Classification}
\label{sec: routing within a cluster}
In this section we introduce our second main tool, the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace, that is summarized in the following theorem.
\begin{theorem}\label{thm:algclassifycluster}
There is a randomized algorithm, that, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace problem with $|E(G)|=m$, a cluster $J\subseteq G$ that has the $\alpha_0$-bandwidth property in $G$, for $\alpha_0=1/\log^{50}m$, and a parameter $0<p<1$, either returns FAIL, or computes a distribution ${\mathcal{D}}(J)$ over the set $\Lambda(J)$ of internal $J$-routers, such that cluster $J$ is $\beta^*$-light with respect to ${\mathcal{D}}(J)$, where $\beta^*=2^{O(\sqrt{\log m}\cdot \log\log m)}$. Moreover, if cluster $J$ is not $\eta^*$-bad, for $\eta^*=2^{O((\log m)^{3/4}\log\log m)}$, then the probability that the algorithm returns FAIL is at most $p$. The running time of the algorithm is $\operatorname{poly}(m\cdot \log(1/p))$.
\end{theorem}
We will sometimes say that the algorithm \ensuremath{\mathsf{AlgClassifyCluster}}\xspace \emph{errs} if it returns FAIL and yet cluster $J$ is not $\eta^*$-bad. Clearly, the probability that the algorithm errs is at most $p$. We note that the distribution ${\mathcal{D}}(J)$ over the set $\Lambda(J)$ of internal $J$-routers that the algorithm computes may be returned by the algorithm implicitly, by providing another efficient algorithm to draw a router from the distribution.
In order to prove \Cref{thm:algclassifycluster}, it is sufficient to prove the following theorem.
\begin{theorem}\label{thm:algclassifycluster easier}
There is an efficient randomized algorithm, that, given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$, a
cluster $J\subseteq G$ that has the $\alpha_0$-bandwidth property in $G$, for $\alpha_0=\Omega(1/\log^{50}m)$, either returns FAIL, or computes a distribution ${\mathcal{D}}(J)$ over the set of internal $J$-routers, such that cluster $J$ is $\beta^*$-light with respect to ${\mathcal{D}}(J)$, where $\beta^*=2^{O(\sqrt{\log m}\cdot \log\log m)}$. Moreover, if cluster $J$ is not $\eta^*$-bad, for $\eta^*=2^{O((\log m)^{3/4}\log\log m)}$, then the probability that the algorithm returns FAIL is at most $1/2$.
\end{theorem}
Indeed, given a graph $G$, a cluster $J$ of $G$ and a parameter $0<p<1$, as in the statement of \Cref{thm:algclassifycluster}, we simply run the algorithm from \Cref{thm:algclassifycluster easier} $\ceil{\log(1/p)}$ times on the input instance $(G,\Sigma)$ and cluster $J$ of $G$. If the algorithm returns FAIL in every of these iterations, then we also return FAIL. Otherwise, in at least one of the iterations, the algorithm from \Cref{thm:algclassifycluster easier} returns a distribution ${\mathcal{D}}(J)$ over the set $\Lambda(J)$ of internal $J$-routers, such that cluster $J$ is $\beta^*$-light with respect to ${\mathcal{D}}(J)$. We then return the distribution ${\mathcal{D}}(J)$ as the algorithm's outcome. It is immediate to verify that the probability that the algorithm errs is at most $p$, and that its running time is $\operatorname{poly}(m\cdot \log(1/p))$, as required.
In the remainder of this section, we focus on the proof of \Cref{thm:algclassifycluster easier}.
It will be convenient for us to consider the augmentation $J^+$ of cluster $J$. Recall that this is the graph that is obtained from $G$ by subdiving every edge $e\in \delta_G(J)$ with a vertex $t_e$, letting $T=\set{t_e\mid e\in \delta_G(J)}$ be the set of the new vertices, and then letting $J^+$ be the subgraph of the resulting graph induced by $T\cup V(J)$. We refer to vertices of $T$ as \emph{terminals}, and we denote $|T|=k$. Recall that, from the $\alpha_0$-bandwidth property of cluster $J$, the set $T$ of terminals is $\alpha_0$-well-linked in $J^+$. Since the degree of every terminal in $J^+$ is $1$, the rotation system $\Sigma$ for graph $G$ naturally defines a unique rotation system $\Sigma(J^+)$ for $J^+$. Moreover, cluster $J$ is $\eta^*$-bad iff $\mathsf{OPT}_{\mathsf{cnwrs}}(J^+,\Sigma(J^+))+|E(J^+\setminus T)|\geq k^2/\eta^*$.
Let $\Lambda(J^+,T)$ denote the collection of all sets ${\mathcal{Q}}$ of paths, such that paths in ${\mathcal{Q}}$ route all vertices of $T$ to some vertex $x\in V(J^+)\setminus T$, in graph $J^+$.
We sometimes also call ${\mathcal{Q}}$ a router, and refer to $x$ as the \emph{center vertex} of the router.
Notice that, if we are given a distribution ${\mathcal{D}}$ over sets of paths in $\Lambda(J^+,T)$, such that, for every edge $e\in E(J^+)$, $\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_{J^+}({\mathcal{Q}},e))^2}\leq \beta^*$, then we can immediately obtain a distribution ${\mathcal{D}}(J)$ over the set $\Lambda(J)$ of internal $J$-routers, such that cluster $J$ is $\beta^*$-light with respect to ${\mathcal{D}}(J)$.
From now on we will only focus on graph $J^+$ and the corresponding rotation system $\Sigma(J^+)$, so it will be convenient for us to denote graph $J^+$ by $G$ and $\Sigma(J^+)$ by $\Sigma$. We denote by $I=(G,\Sigma)$ the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace.
From now on our goal is to design a randomized algorithm, that either computes a distribution ${\mathcal{D}}$ over the set $\Lambda(G,T)$ of internal routers, such that, for every edge $e\in E(G)$, $\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_{G}({\mathcal{Q}},e))^2}\leq \beta^*$, or returns FAIL. We need to ensure that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G\setminus T)|< k^2/\eta^*$, then the probability that the algorithm returns FAIL is at most $1/2$.
We now provide some intuition.
We first show below an algorithm called \ensuremath{\mathsf{AlgFindGuiding}}\xspace\xspace that ``almost'' provides the required guarantees.
Specifically, if we are guaranteed that $|E(G)|\leq k\cdot \eta$ for some small parameter $\eta$, then the algorithm either computes a distribution ${\mathcal{D}}$ over sets of paths in $\Lambda(G,T)$, such that, for every edge $e\in E(G)$, $\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_G({\mathcal{Q}},e))^2}\leq \operatorname{poly}(\log m)/\operatorname{poly}(\alpha_0)$, or it returns FAIL, with the guarantee that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+|E(G\setminus T)|<\frac{k^2\operatorname{poly}(\alpha_0)}{\operatorname{poly}(\eta\log m)}$, then the probability that the algorithm returns FAIL is at most $1/2$. We could use this theorem directly if $|E(G)|\leq k\cdot \eta$ holds for some $\eta\leq (\eta^*)^{\epsilon}$ where $\epsilon$ is a constant, but unfortunately this is not guaranteed in the statement of \Cref{thm:algclassifycluster}, and $|E(G)|$ may be arbitrarily large compared to $k$. In order to overcome this difficulty, we use another algorithm, that, given a graph $G$ and a set $T$ of its terminals as above, computes a collection ${\mathcal{C}}$ of disjoint clusters of $G\setminus T$, such that, for each cluster $C\in {\mathcal{C}}$, either (i) there is an internal $C$-router ${\mathcal{Q}}(C)\in \Lambda(C)$ such that the paths in ${\mathcal{Q}}(C)$ are edge-disjoint; or (ii) $C$ is $\eta$-bad for some parameter $\eta\ll \eta^*$; or (iii) $|E(C)|\leq |\delta_G(C)|\cdot O(\operatorname{poly}(\eta\log m))$. In the latter case, we say that $C$ is a \emph{concise} cluster.
We then apply the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace to each concise cluster. As a result, for each such concise cluster $C\in {\mathcal{C}}$, we will either establish, with high probability, that it is a $\eta'$-bad cluster, for some parameter $\eta'$, or we will compute a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers, such that cluster $C$ is $\beta'$-light with respect to ${\mathcal{D}}(C)$, for some parameter $\beta'$. The algorithm for computing the collection ${\mathcal{C}}$ of clusters of $G$ also guarantees that each cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property, for $\alpha'=\Omega(1/\log^{1.5}m)$, and that the corresponding contracted graph $G_{|{\mathcal{C}}}$ contains significantly fewer edges: $|E(G_{|{\mathcal{C}}})|\leq |E(G)|/\eta$. Intuitively, we would then like to continue with the contracted graph $G_{|{\mathcal{C}}}$, applying exactly the same algorithm to this graph. We could continue this process, obtaining a clustering ${\mathcal{C}}'$ of this new contracted graph, and so on, until we reach a final contracted graph $\hat G$, with $|E(\hat G)|\leq O(k\eta)$. At this point we can apply the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace to graph $\hat G$ directly, and as a result, we either obtain the desired distribution ${\mathcal{D}}$ over path sets in $\Lambda(G,T)$, or establish, with high probability, that $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+|E(G\setminus T)|$ is sufficiently high. A problem with this approach is that the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace requires a rotation system $\Sigma'$ for its input graph $H$. Recall that the algorithm guarantees that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma')+|E(H)|$ is sufficiently low, then it only returns FAIL with probability at most $1/2$. The difficulty is that, if $H$ is the contracted graph $G_{|{\mathcal{C}}}$, then it is not immediately clear how to define the rotation system $\Sigma'$ for $H$, such that $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\Sigma')$ is not much higher than $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)$.
In order to overcome this difficulty, we design the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace for a more general setting. In this setting, the input is a graph $H$, a rotation system $\Sigma'$ for $H$, and a set $T'$ of terminals of $H$, such that the terminals of $T'$ are $\alpha$-well-linked in $H$. Additionally, we are given some collection ${\mathcal{C}}'$ of disjoint $\eta'$-bad clusters in $H$. We also require that $|E(H_{|{\mathcal{C}}'})|\leq |T'|\eta'$, for some parameter $\eta'$.
The algorithm either returns FAIL, or computes a distribution ${\mathcal{D}}'$ over the set $\Lambda(H,T')$ of routers, such that, for every edge $e\in E(H)$,
$\expect[{\mathcal{Q}}\sim {\mathcal{D}}']{(\cong({\mathcal{Q}},e))^2}$ is sufficiently low. We are also guaranteed that, if $|\mathsf{OPT}_{\mathsf{cnwrs}}(H,T')|+|E(H\setminus T')|$ is sufficiently small compared to $|T'|^2$, then the probability that the algorithm returns FAIL is at most $1/2$.
This stronger version of algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace will allow us to carry out the algorithm outlined above.
We now provide formal descriptions of the two main tools that our algorithm uses. The first tool allows us to compute a decomposition of an input graph $G$ into a collection of clusters, each of which is either light, bad, or concise.
The proof of the theorem uses rather standard techniques and is deferred to Section \ref{sec: appx-decomposition-good-bad-other} of Appendix.
\begin{theorem}\label{thm: basic decomposition of a graph}
There is an efficient algorithm, that we refer to as \ensuremath{\mathsf{AlgInitPartition}}\xspace, whose input consists of a connected $m$-edge graph $G$, a set $T\subseteq V(G)$ of $k$ vertices called \emph{terminals}, such that each vertex of $T$ has degree $1$ in $G$, and a parameter $\eta>\log m$, such that $k\leq \frac{m}{16\eta\log m}$.
The algorithm computes a collection ${\mathcal{C}}$ of vertex-disjoint clusters of $G\setminus T$, a partition $({\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3)$ of ${\mathcal{C}}$ into three subsets, and, for every cluster $C\in {\mathcal{C}}_3$, an internal $C$-router ${\mathcal{Q}}(C)\in \Lambda(C)$, where the paths of ${\mathcal{Q}}(C)$ are edge-disjoint, such that the following additional properties hold:
\begin{itemize}
\item every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property, where $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}=\Omega\textsf{left}(\frac{1}{\log^{1.5}m}\textsf{right} )$;
\item for every cluster $C\in {\mathcal{C}}_1$, $|E(C)|\leq O(\eta^4\log^8m)\cdot |\delta_G(C)|$;
\item for every cluster $C\in {\mathcal{C}}_2$, $\mathsf{OPT}_{\mathsf{cr}}(C)\geq \Omega(|E(C)|^2/(\eta^2\operatorname{poly}\log m))$, and $|E(C)|\ge \Omega(\eta^4 |\delta_G(C)|\log^8m)$;
\item $\bigcup_{C\in {\mathcal{C}}}V(C)=V(G)\setminus T$; and
\item $|E(G_{|{\mathcal{C}}})|\leq |E(G)|/\eta$.
\end{itemize}
\end{theorem}
Note that, from the theorem statement, for every cluster $C\in {\mathcal{C}}_2$, $\mathsf{OPT}_{\mathsf{cr}}(C)\geq \Omega(|\delta_G(C)|^2\eta^6/\operatorname{poly}\log m)$. We will informally refer to clusters in ${\mathcal{C}}_1$ as concise clusters.
\iffalse
We start with intuition. Consider an instance $I=(G,\Sigma)$ of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, and let $C\subseteq G$ be a subgraph of $G$ that we refer to as a cluster. Fix an optimal solution $\phi^*$ to instance $I$; we define $\mathsf{cr}(C)$ as usual with respect to $\phi^*$ -- the number of crossings in $\phi^*$ in which the edges of $C$ participate. For a set ${\mathcal{Q}}$ of paths in $G$, the cost $\operatorname{cost}({\mathcal{Q}})$ is also defined as before, with respect to $\phi^*$.
Assume that the boundary of $C$ is $\alpha$-well-linked in $C$, and denote $|E(C)|/|\delta(C)|=\eta$. We show below a randomized algorithm that either computes a vertex $u\in V(C)$ and a set ${\mathcal{Q}}$ of paths routing $\delta(C)$ to $u$, whose expected cost is comparable to $\mathsf{cr}(C)$; or with high probability correctly establishes that $\mathsf{cr}(C)>|\delta(C)|^2/\operatorname{poly}(\eta)$.
We say that cluster $C$ is \emph{good} if the former outcome happens, and we say that it is \emph{bad} otherwise.
Intuitively, given our input instance $I=(G,\Sigma)$, we can compute a decomposition ${\mathcal{C}}$ of $G$ into clusters, such that for each cluster $C\in {\mathcal{C}}$, the boundary of $C$ is $1/\operatorname{poly}\log m$-well-linked in $C$, and moreover $|\delta(C)|\geq |E(C)|/\operatorname{poly}(\mu)$. In order to compute the decomposition, we start with ${\mathcal{C}}$ containing a single cluster $G$, and the interate. In every iteration, we select a cluster $C\in {\mathcal{C}}$, and then either compute a sparse balanced cut $(A,B)$ in $C$, or compute a cut $(A,B)$ that is sparse with respect to the edges of $\delta(C)$ (that is, $|E(A,B)|<\min\set{|\delta(C)\cap \delta(A)|,|\delta(C)\cap \delta(B)|}/\operatorname{poly}\log n$; or return FAIL. In the former two cases, we replace $C$ with $C[A]$ and $C[B]$ in ${\mathcal{C}}$, while in the latter case we declare the cluster $C$ \emph{settled}; we will show that in this case the boundary of $C$ is $1/\operatorname{poly}\log n$-well-linked in $C$ and $|\delta(C)|\geq |E(C)|/\operatorname{poly}(\mu)$. The algorithm terminates once every cluster in ${\mathcal{C}}$ is settled. We show that each of the resulting clusters $C\in {\mathcal{C}}$ is sufficiently small (that is, $|E(C)|\leq m/\mu$), and the total number of edges connecting different clusters is also sufficiently small (at most $m/\mu$). If each resulting cluster is a light cluster, then we obtain the desired decomposition ${\mathcal{C}}$, together with the desired collections $\set{{\mathcal{Q}}(C)}_{C\in {\mathcal{C}}}$ of paths. Unfortunately, some of the clusters $C\in {\mathcal{C}}$ may be bad. For each such cluster we are guaranteed with high probability that $\mathsf{cr}(C)>|\delta(C)|^2/\operatorname{poly}(\mu)$, but unfortunately the sum of values of $|\delta(C)|^2$ over all bad clusters $C$ may be quite small, significantly smaller than say $|E(G)|^2$, so we cannot conclude that $\mathsf{OPT}(I)\geq |E(G)|^2/\operatorname{poly}(\mu)$.
In order to overcome this difficulty, we partition the set ${\mathcal{C}}$ of clusters into a set ${\mathcal{C}}'$ of light clusters, and a set ${\mathcal{C}}''$ of bad clusters. Let $G'=G\setminus (\bigcup_{C\in {\mathcal{C}}'}C)$, and let $G''=G'_{|{\mathcal{C}}''}$ be obtained from $G'$ by contracting each bad cluster into a super-node. We then apply the same decomposition algorithm to graph $G''$. We continue this process until no bad clusters remain, so we have obtained a decomposition of $G$ into a collection of light clusters, or we compute a bad cluster $C$ with $|\delta(C)|\geq |E(G)|/\operatorname{poly}(\mu)$, so we can conclude that $\mathsf{OPT}(I)\geq|E(G)|^2/\operatorname{poly}(\mu)$.
In order to be able to carry out this process, we need an algorithm that, given a cluster $C$, either computes the desired set ${\mathcal{Q}}$ of paths routing the edges of $\delta(C)$ to some vertex $u$ of $C$, such that the cost of ${\mathcal{Q}}$ is low; or establishes that $\mathsf{cr}(C)>|\delta(C)|^2/\operatorname{poly}(\mu)$. But because we will need to apply this algorithm to graphs obtained from sub-graphs of $G$ by contracting some bad clusters, we need a more general algorithm, that is summarized in the following theorem.
\fi
Let $H$ be a graph and let $T$ be a set of vertices of $H$ called terminals. We say that a set ${\mathcal{Q}}=\set{Q(t)\mid t\in T}$ of paths in graph $H$ is a \emph{router for $H$ and $T$} if there is a vertex $x\in V(H)$, such that, for every terminal $t\in T$, path $Q(t)$ originates at $t$ and temrinates at $x$. We denote by $\Lambda(H,T)$ the set of all routers for $H$ and $T$.
Our second tool is algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace, summarized in the following theorem.
\begin{theorem}\label{thm: find guiding paths}
There are universal constants $c_0$ and $c^*$, and an efficient randomized algorithm, called \ensuremath{\mathsf{AlgFindGuiding}}\xspace, that receives as input an instance $I=(H,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, where $|E(H)|=m$, a set $T\subseteq V(H)$ of $k$ vertices of $H$ called terminals, and a collection ${\mathcal{C}}$ of disjoint clusters of $H\setminus T$. Additionally, the algorithm receives as input parameters $0\leq \alpha,\alpha'\leq 1$ and $\eta,\eta'\geq 1$, such that the following conditions hold:
\begin{itemize}
\item $\eta\geq \frac{c^*\log^{46}m}{\alpha^{10}(\alpha')^2}$ and $\eta'\geq \eta^{13}$;
\item $k\geq |E(H_{|{\mathcal{C}}})|/\eta$;
\item every terminal $t\in T$ has degree $1$ in $H$;
\item the set $T$ of terminals is $\alpha$-well-linked in the contracted graph $H_{|{\mathcal{C}}}$; and
\item every cluster $C\in {\mathcal{C}}$ is $\eta'$-bad and has the $\alpha'$-bandwidth property in $H$.
\end{itemize}
The algorithm either returns FAIL or (explicitly) returns a distribution ${\mathcal{D}}$ over the routers in $\Lambda(H,T)$, such that,
for every edge $e\in E(H)$,
$\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{32}m}{\alpha^{12}(\alpha')^8}\textsf{right} )$. Moreover, if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(H\setminus T)|\leq \frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}m}$, then the probability that the algorithm returns FAIL is at most $1/2$.
\end{theorem}
The proof of \Cref{thm: find guiding paths} is quite technical, and is deferred to Section \ref{sec: guiding paths}. We note that the algorithm returns the distribution ${\mathcal{D}}$ explicitly, that is, it lists all routers ${\mathcal{Q}} \in \Lambda(H,T)$ that have a non-zero probability, together with their probability values in ${\mathcal{D}}$.
In the remainder of this section, we complete the proof of \Cref{thm:algclassifycluster}, using the algorithms \ensuremath{\mathsf{AlgInitPartition}}\xspace and \ensuremath{\mathsf{AlgFindGuiding}}\xspace.
Note that we can assume, throughout the proof, that $m$ is sufficiently large (larger than some large enough constant). Otherwise, since the vertices of $T$ are $\alpha$-well-linked in $G$, we get that $k\leq m\leq O(1)$. We can then let ${\mathcal{D}}$ be a distribution that gives a probability $1$ to an internal router ${\mathcal{Q}}$ with target vertex $u$, where $u$ is an arbitrary vertex of $V(G)\setminus T$, and ${\mathcal{Q}}$ is an arbitrary collection of simple paths routing the vertices of $T$ to $u$. Clearly, for every edge $e\in E(G)$, $(\cong_G({\mathcal{Q}},e))^2\leq k^2\leq O(1)$.
\subsection{Main Parameters}
We now introduce some parameters that our algorithm uses. The main parameter is $\eta=2^{(\log m)^{3/4}}$. Our algorithm will consist of $(\ell-1)$ phases, for $\ell\leq O\textsf{left}(\frac{\log m}{\log \eta}\textsf{right})=O((\log m)^{1/4})$.
At the beginning of the $i$-th phase, we will be given a collection ${\mathcal{C}}_i$ of disjoint clusters of $G\setminus T$.
\paragraph{Parameters for bandwidth property.}
We will use the following parameters for bandwidth property of the clusters. Recall that $\alpha_0=\frac{1}{(\log m)^{50}}$ is the parameter from the statement of \Cref{thm:algclassifycluster}. For $1\leq i\leq \ell$, $\alpha_i=(\alpha_0)^i=\frac{1}{(\log m)^{50\cdot i}}$. We will ensure that for all $1\leq i\leq \ell$, every cluster in ${\mathcal{C}}_i$ has the $\alpha_{i}$-bandwidth property. Note that $\alpha_{\ell}=1/(\log m)^{50\cdot\ell}=1/2^{O((\log m)^{1/4}\log\log m)}$.
\paragraph{Parameters for light clusters.}
We set $\beta_0=1$, and for $1\leq i\leq \ell$, $\beta_i=\frac{(\log m)^{56}}{(\alpha_0)^{12}\cdot (\alpha_{i})^8}\cdot \beta_{i-1}$. It is easy to verify that $\beta_{\ell}\leq (\log m)^{O(\ell^2)}\leq 2^{O((\log m)^{1/2}\log\log m)}\leq \beta^*$.
\iffalse
\begin{definition}[Good clusters]
Given a cluster $C\subseteq G\setminus T$, and a distribution ${\mathcal{D}}(C)$ over pairs in $\Lambda(C)$, we say that $C$ is \emph{level-$i$ good} with respect to ${\mathcal{D}}(C)$, iff for every edge $e\in E(C)$: $$\expect[(u(C),{\mathcal{Q}}(C)) \in_{{\mathcal{D}}(C)}\Lambda(C)]{(\cong_G({\mathcal{Q}}(C),e))^2}\leq \beta_i.$$
\end{definition}
\fi
\paragraph{Parameters for bad clusters.}
We define the following parameters: $\eta_0=\eta^4\cdot \log^9m=2^{O ((\log m)^{3/4} )}$; $\eta_1=\max\set{(2\eta_0)^{13},\eta_0\cdot (\log m)^{80}}=2^{O ((\log m)^{3/4} )}$, and for each $1\leq i\leq \ell$,
$\eta_i=\eta_{i-1}\cdot \max\set{(\log m)^{80+50\cdot i},\beta^3_{i-1}}$.
Clearly,
$\eta_i\leq (\beta_{\ell})^3\cdot\eta_{i-1}$, and
$\eta_{\ell}\leq \eta_1\cdot (\beta_{\ell})^{3\ell}\leq 2^{O ((\log m)^{3/4}\log\log m )}\leq \eta^*$.
\iffalse
\begin{definition}[Bad clusters]
We say that a cluster $C\subseteq G\setminus T$ is \emph{level-$i$ bad} iff $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq \frac{|\delta_G(C)|^2}{\eta_i}$.
\end{definition}
\fi
\iffalse
\subsection{Main Parameters}
We start by introducing some parameters that our algorithm uses. The main parameter is $\eta=2^{(\log m)^{2/3}}$. Our algorithm will consist of at most $\ell=O\textsf{left}(\frac{\log m}{\log \eta}\textsf{right})=O((\log m)^{1/3})$ phases.
At the beginning of the $i$th phase, we will be given a collection ${\mathcal{C}}_i$ of disjoint vertex-induced subgraphs (clusters) of $G\setminus T$.
\paragraph{Parameters for bandwidth property.}
We will use the following parameters for bandwidth property of the clusters: $\alpha_0=\frac{1}{\log^{50}m}$, and for $1\leq i\leq \ell$, $\alpha_i=(\alpha_0)^i=\frac{1}{(\log m)^{50\cdot i}}$. We will ensure that for all $1\leq i\leq \ell$, every cluster in ${\mathcal{C}}_i$ has the $\alpha_{i-1}$-bandwidth property. Note that $\alpha_{\ell}=1/(\log m)^{3\ell}=1/2^{O((\log m)^{1/3}\log\log m)}$.
\paragraph{Parameters for bad clusters.}
We also define the following parameters: $\eta_0=\eta^4\cdot \log^9m=2^{O ((\log m)^{2/3} )}$; $\eta_1=\max\set{\eta_0^{13},\eta_0\cdot (\log m)^{80}}=2^{O ((\log m)^{2/3} )}$, and for $1\leq i\leq \ell$, $\eta_i=\eta_{i-1}\cdot (\log m)^{80+3i}$. Clearly, $\eta_{\ell}\leq \eta_1\cdot (\log m)^{O(\ell^2)}\leq 2^{O ((\log m)^{2/3}\log\log m )}$.
\begin{definition}[Bad clusters]
We say that a cluster $C\subseteq G\setminus T$ is \emph{level-$i$ bad} iff $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\Sigma_C)\geq \frac{|\delta_G(C)|^2}{\eta_i}$.
\end{definition}
\paragraph{Parameters for light clusters.}
Our last set of parameters is defined as follows. We set $\beta_0=1$, and for $1\leq i\leq \ell$, $\beta_i=\frac{(\log m)^{33}}{(\alpha_0)^{12}\cdot (\alpha_{i-1})^8}$. It is easy to verify that $\beta_{\ell}\leq (\log m)^{O(\ell^2)}\leq 2^{O((\log m)^{2/3}\log\log m)}$
\begin{definition}[light clusters]
Given a cluster $C\subseteq G\setminus T$, and a distribution ${\mathcal{D}}(C)$ over pairs in $\Lambda(C)$, we say that $C$ is \emph{level-$i$ light} with respect to ${\mathcal{D}}(C)$, iff for every edge $e\in E(C)$: $$\expect[(u(C),{\mathcal{Q}}(C)) \in_{{\mathcal{D}}(C)}\Lambda(C)]{(\cong_G({\mathcal{Q}}(C),e))^2}\leq \beta_i.$$
\end{definition}
\fi
\subsection{Algorithm Execution}
As already mentioned, the algorithm consists of $(\ell-1)$ phases, where $\ell= O((\log m)^{1/4})$. At the beginning of the $i$th phase, we will be given a collection ${\mathcal{C}}_{i}$ of disjoint clusters of $G\setminus T$, which is partitioned into two subsets: ${\mathcal{C}}_{i}^{\operatorname{light}}$ and ${\mathcal{C}}_{i}^{\operatorname{bad}}$. For each cluster $C\in {\mathcal{C}}_{i}^{\operatorname{light}}$, we will also be given a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers.
For all $1\leq i\leq \ell$, we will also define a bad event ${\cal{E}}_i$, and we will ensure that it happens with probability at most $i/m^{10}$.
We will ensure that the following properties hold for all $1\leq i\leq \ell$:
\begin{properties}{R}
\item each cluster $C\in {\mathcal{C}}_{i}$ has the $\alpha_{i}$-bandwidth property; \label{prop 1 of clusters bw}
\item the number of edges in the contracted graph $G_{|{\mathcal{C}}_{i}}$ is at most $m/\eta^{i-1}$; \label{prop 2 of clusters small contracted graph}
\item each cluster $C\in {\mathcal{C}}_{i}^{\operatorname{light}}$ is $\beta_{i}$-light with respect to the distribution ${\mathcal{D}}(C)$; \label{prop 3 of clusters good} and
\item if the bad event ${\cal{E}}_i$ does not happen, then each cluster $C\in {\mathcal{C}}_{i}^{\operatorname{bad}}$ is $\eta_{i}$-bad.\label{prop 4 last of clusters bad}
\end{properties}
The input to the first phase, ${\mathcal{C}}_1=\emptyset$. Clearly, all properties \ref{prop 1 of clusters bw}--\ref{prop 4 last of clusters bad} hold for this set of clusters.
We now describe the execution of the $i$th phase.
We assume that we are given as input a collection ${\mathcal{C}}_{i}$ of disjoint clusters of $G\setminus T$, which is partitioned into two subsets, ${\mathcal{C}}_{i}^{\operatorname{light}}$ and ${\mathcal{C}}_{i}^{\operatorname{bad}}$. We are also given, for each cluster $C\in {\mathcal{C}}_{i}^{\operatorname{light}}$, a distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers, and we are guaranteed that Properties \ref{prop 1 of clusters bw}--\ref{prop 4 last of clusters bad} hold.
We consider the contracted graph $G'=G_{|{\mathcal{C}}_{i}}$. The execution of the $i$th phase consists of two steps: in the first step, we apply the algorithm \ensuremath{\mathsf{AlgInitPartition}}\xspace to the contracted graph $G'$, obtaining a collection ${\mathcal{C}}$ of clusters of $G'$, which we then convert into clusters of $G$. The set ${\mathcal{C}}$ of clusters is partitioned into three subsets. Informally, the clusters in the first subset are concise, the clusters in the second subset are $\eta_{i+1}$-bad if event ${\cal{E}}_{i}$ did not happen, and the clusters in the third set are $\beta_{i+1}$-light with respect to a distribution over the internal routers that we construct. In the second step we further process each concise cluster, using the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace, in order to determine whether it is a $\beta_{i+1}$-light or an $\eta_{i+1}$-bad cluster, and in the former case, to compute the corresponding distribution ${\mathcal{D}}(C)$ over the set $\Lambda(C)$ of internal $C$-routers.
\subsection{Step 1: Partition}
We assume first that $|E(G')|>16\eta k\log m $, where $\eta=2^{(\log m)^{3/4}}$ is the parameter that we have defined above.
If the inequality does not hold, then the current phase is the last phase of the algorithm, and we show how to execute this phase at the end of this subsection.
We apply the algorithm \ensuremath{\mathsf{AlgInitPartition}}\xspace from \Cref{thm: basic decomposition of a graph} to graph $G'$, the set $T$ of terminals, and the parameter $\eta$ that we have defined. Notice that, since $\eta=2^{(\log m)^{3/4}}$, and since we have assumed that $m$ is greater than some large enough constant, $\eta>\log m\geq \log (|E(G')|)$ must hold.
We now consider the output of the algorithm, that consists of a collection ${\mathcal{C}}$ of disjoint clusters of $G'\setminus T$, a partition $({\mathcal{C}}_1',{\mathcal{C}}_2',{\mathcal{C}}_3')$ of ${\mathcal{C}}$ into three subsets, and, for every cluster $C\in {\mathcal{C}}_3'$, a vertex $u(C)\in V(C)$, and an internal $C$-router ${\mathcal{Q}}(C)$, consisting of edge-disjoint paths routing the edges of $\delta_G(C)$ to $u(C)$. Recall that we are also guaranteed that every cluster $C\in {\mathcal{C}}$ has the $\alpha'$-bandwidth property, where $\alpha'=\frac{1}{16\ensuremath{\beta_{\mbox{\tiny{\sc ARV}}}}(m)\cdot \log m}\geq \alpha_0$.
Consider any cluster $C\in {\mathcal{C}}$. Recall that $C$ is a cluster of the contracted graph $G'$, and it has the $\alpha_0$-bandwidth property. Let ${\mathcal{W}}(C)$ be the set of all clusters $W\in {\mathcal{C}}_{i}$, whose corresponding supernode $v_W\in V(C)$. Recall that every cluster of ${\mathcal{C}}_{i}$ has the $\alpha_{i}$-bandwidth property from Property \ref{prop 1 of clusters bw}. Let $U_C$ be the set of vertices of $G$, that contains every regular (non-supernode) vertex of $C$, and every vertex lying in clusters of ${\mathcal{W}}(C)$. In other words, $U_C=(V(G)\cap V(C))\cup \textsf{left}(\bigcup_{W\in {\mathcal{W}}(C)}V(W)\textsf{right} )$. We then let $\tilde C=G[U_C]$.
Since cluster $C$ has the $\alpha_0$-bandwidth property, and every cluster in ${\mathcal{W}}(C)$ has the $\alpha_{i}$-bandwidth property, from \Cref{clm: contracted_graph_well_linkedness} and \Cref{obs: wl-bw}, cluster $\tilde C$ has the $\alpha_{i}\cdot \alpha_0=\alpha_{i+1}$-bandwidth property. We let ${\mathcal{C}}_{i+1}=\set{\tilde C\mid C\in {\mathcal{C}}}$.
Notice that we have just established Property \ref{prop 1 of clusters bw} for clusters in ${\mathcal{C}}_{i+1}$. It is immediate to verify that $G_{|{\mathcal{C}}_{i+1}}=G'_{|{\mathcal{C}}}$. Since \Cref{thm: basic decomposition of a graph} guarantees that $|E(G'_{|{\mathcal{C}}})|\leq |E(G')|/\eta$, and, from Property \ref{prop 2 of clusters small contracted graph}, $|E(G')|=|E(G_{|{\mathcal{C}}_{i}})|\leq m/\eta^{i-1}$, we get that $|E(G_{|{\mathcal{C}}_{i+1}})|\leq |E(G')|/\eta\leq m/\eta^i$, establishing Property \ref{prop 2 of clusters small contracted graph} for the set ${\mathcal{C}}_{i+1}$ of clusters.
We now construct the partition $({\mathcal{C}}_{i+1}^{\operatorname{bad}}, {\mathcal{C}}_{i+1}^{\operatorname{light}})$ of the set ${\mathcal{C}}_{i+1}$ of clusters.
We start by letting ${\mathcal{C}}_{i+1}^{\operatorname{bad}}=\set{\tilde C\mid C\in {\mathcal{C}}_2'}$ and ${\mathcal{C}}_{i+1}^{\operatorname{light}}=\emptyset$. We then consider every cluster $C\in {\mathcal{C}}_3'$ one by one. Recall that for each such cluster $C$, the algorithm from \Cref{thm: basic decomposition of a graph} provides an internal router ${\mathcal{Q}}(C)$, routing the edges of $\delta_{G'}(C)$ to $u(C)$, such that the paths in ${\mathcal{Q}}(C)$ are edge-disjoint. If vertex $u(C)$ is a supernode, whose corresponding cluster $W\in {\mathcal{C}}_{i}$ lies in set ${\mathcal{C}}_{i}^{\operatorname{bad}}$, then we add $\tilde C$ to ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$; otherwise, we add $\tilde C$ to ${\mathcal{C}}_{i+1}^{\operatorname{light}}$.
Lastly, we set ${\mathcal{C}}_{i+1}^{\operatorname{concise}}=\set{\tilde C\mid C\in {\mathcal{C}}_1'}$, and we refer to clusters in ${\mathcal{C}}_{i+1}^{\operatorname{concise}}$ as \emph{concise clusters}. In Step 2, we will further process clusters in ${\mathcal{C}}_{i+1}^{\operatorname{concise}}$, and we will eventually add each such cluster to either ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$ or to ${\mathcal{C}}_{i+1}^{\operatorname{light}}$. Before we do so, we establish Property \ref{prop 4 last of clusters bad} for clusters that are currently in ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$, and we define a distribution ${\mathcal{D}}(\tilde C)$ over the set $\Lambda_G(\tilde C)$ of internal $\tilde C$-routers for every cluster $\tilde C$ that is currently in ${\mathcal{C}}_{i+1}^{\operatorname{light}}$, such that $\tilde C$ is $\beta_{i+1}$-light with respect to ${\mathcal{D}}(\tilde C)$ (that is, we establish Property \ref{prop 3 of clusters good} for clusters that are currently in ${\mathcal{C}}_{i+1}^{\operatorname{light}}$).
\paragraph{Bad Clusters.}
Recall that \Cref{thm: basic decomposition of a graph} guaranteed that, for every cluster $C\in {\mathcal{C}}_2'$, $\mathsf{OPT}_{\mathsf{cr}}(C)\geq \Omega(|E(C)|^2/(\eta^2\operatorname{poly}\log m))$, and $|E(C)|> \Omega(\eta^4 |\delta_G(C)|\log^8m)$. Therefore:
\[ \mathsf{OPT}_{\mathsf{cr}}(C)\geq \Omega\textsf{left} (\frac{|E(C)|^2}{\eta^2\operatorname{poly}\log m}\textsf{right} )\geq \Omega\textsf{left} (\frac{|\delta_G(C)|^2\eta^6}{\operatorname{poly}\log m}\textsf{right} )\geq |\delta_G(C)|^2.\]
From the definition of cluster $\tilde C$, graph $C$ is a contracted graph of $\tilde C$ with respect to clusters in ${\mathcal{W}}(C)$, that is, $C=\tilde C_{|{\mathcal{W}}(C)}$. As each cluster in ${\mathcal{W}}(C)\subseteq {\mathcal{C}}_i$ has the $\alpha_{i}$-bandwidth property (from Property \ref{prop 1 of clusters bw}), from \Cref{lem: crossings in contr graph}, there is a drawing of $C$ containing at most
$O(\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde C, \Sigma_{\tilde C})\cdot \log^8m/(\alpha_{i})^2)$ crossings, where $\Sigma_{\tilde C}$ is the rotation system for $\tilde C$ induced by $\Sigma$. Since we have established that $\mathsf{OPT}_{\mathsf{cr}}(C)\geq |\delta_G(C)|^2$, we get that:
\[\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde C, \Sigma_{\tilde C})\geq \Omega \textsf{left} (\frac{|\delta_G(C)|^2\cdot (\alpha_{i})^2}{\log^8m} \textsf{right} )\geq \frac{|\delta_G(C)|^2}{\eta_{i+1}}, \]
since, by the definition, $\eta_{i+1}> \eta \geq 2^{(\log m)^{3/4}}$, while
$\alpha_{i}\geq \alpha_{\ell}\geq 1/2^{O((\log m)^{1/4}\log\log m)}$, and since we have assume that $m$ is large enough. We conclude that every cluster in
$\set{\tilde C\mid C\in {\mathcal{C}}_2'}$ is $\eta_{i+1}$-bad.
Consider now some
cluster $C\in {\mathcal{C}}_3'$, such that vertex $u(C)$ that serves as the center of the router ${\mathcal{Q}}(C)$ provided by the algorithm from \Cref{thm: basic decomposition of a graph} is a supernode, whose corresponding cluster $W\in {\mathcal{C}}_{i}^{\operatorname{bad}}$. From Property \ref{prop 4 last of clusters bad}, if Event ${\cal{E}}_{i}$ did not happen, cluster $W$ is an $\eta_{i}$-bad cluster, that is, $\mathsf{OPT}_{\mathsf{cnwrs}}(W,\Sigma_W)+|E(W)|\geq \frac{|\delta_G(W)|^2}{\eta_{i}}$, where $\Sigma_W$ is the rotation system for $W$ induced by $\Sigma$. Since $W\subseteq \tilde C$, we get that $\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde C,\Sigma_{\tilde C})+|E(\tilde C)|\geq \frac{|\delta_G(W)|^2}{\eta_{i}}$. Lastly, since there is a set ${\mathcal{Q}}(C)$ of edge-disjoint paths routing the edges of $\delta_{G'}(C)$ to vertex $u(C)$ inside $C$, we conclude that $|\delta_G(\tilde C)|\leq |\delta_G(W)|$. Altogether, from the fact that $\eta_{i+1}>\eta_{i}$, we get that
$\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde C,\Sigma_{\tilde C})+|E(\tilde C)|\geq \frac{|\delta_G(\tilde C)|^2}{\eta_{i+1}}$.
We conclude that, if event ${\cal{E}}_{i}$ did not happen, then every cluster that we have added to set ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$ so far is an $\eta_{i+1}$-bad cluster.
\paragraph{Light Clusters.}
Consider now some cluster $\tilde C\in {\mathcal{C}}_{i+1}^{\operatorname{light}}$, and let $C\in {\mathcal{C}}$ be its corresponding cluster in graph $G'$. Recall that the algorithm from \Cref{thm: basic decomposition of a graph} provides a collection ${\mathcal{Q}}(C)$ of edge-disjoint paths routing the edges of $\delta_{G'}(C)$ to $u(C)$, such that, for every path in ${\mathcal{Q}}(C)$, all inner vertices of the path lie in $C$. We will now define a distribution ${\mathcal{D}}(\tilde C)$ over the set $\Lambda_G(\tilde C)$ of internal $\tilde C$-routers, so that $\tilde C$ is $\beta_{i+1}$-light with respect to ${\mathcal{D}}(\tilde C)$.
Assume first that vertex $u(C)$ is a regular vertex in cluster $C$, that is, it is not a supernode. Since every cluster $W\in {\mathcal{W}}(C)$ has the $\alpha_{i}$-bandwidth property, we can use the algorithm from
\Cref{claim: routing in contracted graph} to compute a collection ${\mathcal{Q}}(\tilde C)$ of paths, routing the edges of $\delta_G(\tilde C)$ to $u(C)$, such that, for every path of ${\mathcal{Q}}(\tilde C)$, all its inner vertices lie in $\tilde C$, and the largest congestion on an edge of $\tilde C$ is bounded by $\ceil{1/\alpha_{i}}$.
The resulting distribution ${\mathcal{D}}(\tilde C)$ then consists of a single internal $\tilde C$-router ${\mathcal{Q}}(\tilde C)$, that is chosen with probability $1$.
Clearly, for every edge $e\in E(\tilde C)$:
$$\expect[{\mathcal{Q}}(\tilde C) \sim {\mathcal{D}}(\tilde C)]{(\cong_G({\mathcal{Q}}(\tilde C),e))^2}\leq \ceil{1/\alpha_{i}}^2\leq \beta_{i+1},$$
since by definition $\beta_{i+1}=\frac{(\log m)^{56}}{(\alpha_0)^{12}\cdot (\alpha_{i+1})^8}\cdot \beta_{i}$.
Assume now that $u(C)$ is a supernode, corresponding to some cluster $W^*\in {\mathcal{W}}(C)$, such that $W^*\in {\mathcal{C}}_{i}^{\operatorname{light}}$. Let $\tilde C'$ be the cluster obtained from $\tilde C$, after we contract cluster $W^*$ into a supernode $v_{W^*}=u(C)$. Using the same reasoning as in the previous case, we can compute a set ${\mathcal{Q}}(\tilde C')$ of paths in graph $\tilde C'$, routing the edges of $\delta_G(\tilde C)$ to $u(C)$, such that, for every path in ${\mathcal{Q}}(\tilde C')$, every inner vertex on the path lies in $\tilde C'$, and the largest congestion on an edge of $\tilde C'$ is bounded by $\ceil{1/\alpha_{i}}$. Moreover, the algorithm from \Cref{claim: routing in contracted graph} guarantees that every edge in $\delta_{\tilde C'}(u(C))$ belongs to at most one path in ${\mathcal{Q}}(\tilde C')$ (and it is the last edge on that path).
Recall that cluster $W^*$ is $\beta_{i}$-light with respect to the distribution ${\mathcal{D}}(W^*)$ over the set $\Lambda_G(W^*)$ of internal $W^*$-routers.
We choose an internal $W^*$-router ${\mathcal{Q}}(W^*)\in \Lambda(W^*)$ from the distribution ${\mathcal{D}}(W^*)$, routing the edges of $\delta_G(W^*)$ to a vertex $u(W^*)$ of $W^*$. We now consider every path $Q\in {\mathcal{Q}}(\tilde C')$ one by one. Let $e\in \delta_G(\tilde C)$ be the first edge on $Q$, and let $e'\in \delta_G(W^*)$ be the last edge on $Q$. Let $Q^*$ be the unique path in ${\mathcal{Q}}(W^*)$ whose first edge is $e'$, and let $Q'$ be obtained by first deleting the edge $e'$ from $Q$, and then concatenating the resulting path with path $Q^*$. Notice that path $Q'$ connects the edge $e$ to the vertex $u(W^*)$, in graph $\tilde C\cup \delta_G(\tilde C)$. We then set ${\mathcal{Q}}(\tilde C)=\set{Q'\mid Q\in {\mathcal{Q}}(\tilde C')}$, so that ${\mathcal{Q}}(\tilde C)$ is an internal $\tilde C$-router in $\Lambda_G(\tilde C)$.
This finishes the definition of the distribution ${\mathcal{D}}(\tilde C)$ over the set $\Lambda(\tilde C)$ of internal $\tilde C$-routers. Note that the distribution is given implicitly, that is, we provide an efficient algorithm to draw a router from the distribution.
Consider now some edge $e\in E(\tilde C)$. If $e\not\in E(W^*)$, then with probability $1$ (over the choices of internal $\tilde C$-routers ${\mathcal{Q}}(\tilde C)$ from ${\mathcal{D}}(\tilde C)$), $\cong_{G}({\mathcal{Q}}(\tilde C),e)\le \ceil{1/\alpha_{i}}^2$, and so $(\cong_{G}({\mathcal{Q}}(\tilde C),e))^2\leq \beta_{i+1}$ as argued before. If $e\in E(W^*)$, then $\cong_G({\mathcal{Q}}(\tilde C),e)=\cong_{G}({\mathcal{Q}}(W^*),e)$, and so:
$$\expect[{\mathcal{Q}}(\tilde C) \sim {\mathcal{D}}(\tilde C)]{(\cong_G({\mathcal{Q}}(\tilde C),e)^2}=\expect[{\mathcal{Q}}(W^*) \sim {\mathcal{D}}(W^*)]{(\cong_G({\mathcal{Q}}(W^*),e)^2}
\leq \beta_{i}\leq \beta_{i+1}.$$
We conclude that cluster $\tilde C$ is $\beta_{i+1}$-light with respect to the distribution ${\mathcal{D}}(\tilde C)$ over the set $\Lambda_G(\tilde C)$ of intenral $\tilde C$-routers that we have computed.
Recall that so far we assumed that $|E(G')|>16\eta k\log m$ held, where $G'=G_{|{\mathcal{C}}_i}$. Assume now that $|E(G')|\leq 16\eta k\log m $. In this case, we let ${\mathcal{C}}_{i+1}^{\operatorname{bad}}={\mathcal{C}}_{i+1}^{\operatorname{light}}=\emptyset$ and we let ${\mathcal{C}}_{i+1}^{\operatorname{concise}}$ contain a single cluster, $\tilde C=G\setminus T$. We also let ${\mathcal{C}}_1'$ contain a single cluster, $C=G'\setminus T$, and we set ${\mathcal{C}}_2'={\mathcal{C}}_3'=\emptyset$. We denote ${\mathcal{W}}(C)={\mathcal{C}}_i$. The current phase will become the final phase of the algorithm.
\subsection{Step 2: Concise Clusters}
Observe that every cluster $C\in {\mathcal{C}}_1'$ has the $\alpha_0$-bandwidth property in $G'$, and $|E(C)|\leq \eta_0\cdot |\delta_{G'}(C)|$ holds. Indeed, if $|E(G')|\leq 16\eta k\log m $ holds, then set ${\mathcal{C}}_1'$ contains a single cluster $C=G'\setminus T$, and $|E(C)|\leq 16\eta k\log m\leq \eta_0|\delta_{G'}(C)|$ holds.
Since the set $T$ of terminals is $\alpha_0$-well-linked in $G$ (from the statement of \Cref{thm:algclassifycluster}), cluster $\tilde C=G\setminus T$ has the $\alpha_0$-bandwidth property in $G$, and cluster $C$ has the $\alpha_0$-bandwidth property in $G'$.
Otherwise, \Cref{thm: basic decomposition of a graph} guarantees that every cluster $C\in {\mathcal{C}}_1'$ has the $\alpha_0$-bandwidth property, and moreover,
$|E(C)|\leq O(\eta^4\log^8m)\cdot |\delta_{G'}(C)|\leq \eta_0\cdot |\delta_G(C)|$
(since $\eta_0=\eta^4\cdot \log^9m$, and we have assumed that $m$ is large enough).
Recall that for every cluster $C\in {\mathcal{C}}_1'$, we have defined a collection ${\mathcal{W}}(C)\subseteq {\mathcal{C}}_{i}$ of clusters, such that for each cluster $W\in {\mathcal{W}}(C)$, its corresponding supernode $v_W$ lies in $C$. We have also defined a cluster $\tilde C\in {\mathcal{C}}^{\operatorname{concise}}_{i+1}$, that is a subgraph of $G$ correpsonding to $C$. In other words, we can think of $\tilde C$ as being obtained from $C$ by un-contracting every cluster $W\in {\mathcal{W}}(C)$.
We would now like to apply the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace to each such cluster $ C\in {\mathcal{C}}_1'$, in order to classify the corresponding cluster $\tilde C$ of $G$ as either an $\eta_{i+1}$-bad or a $\beta_{i+1}$-light cluster. Notice however that the algorithm requires that we define a rotation system for $C$, and, if the algorithm classifies $C$ as an $\eta_{i+1}$-bad cluster (by returning ``FAIL''), then we are only guaranteed that it is likely that the value of the optimal solution of the resulting instance is high. Therefore, ideally we would like to define a rotation system $\hat \Sigma(C)$ for cluster $C$ of $G'$, such that $\mathsf{OPT}_{\mathsf{cnwrs}}(C,\hat \Sigma(C))$ is not much higher than $\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde C,\Sigma_{\tilde C})$.
Unfortunately, it is not immediately clear how to define such a rotation system, mainly because it is unclear how to define the orderings on edges incident to supernodes. In order to overcome this difficulty, we consider a different graph, that can be thought of as an intermediate graph between $C$ and $\tilde C$. This new graph, that we denote by $H(\tilde C)$, is obtained as follows.
We start
from graph $\tilde C^+$.
Recall that $\tilde C^+$ is obtained from graph $G$ by first subdividing every edge $e\in \delta_G(\tilde C)$ with a vertex $t_e$, and then letting $\tilde C^+$ be the subgraph of the resulting graph induced by $V(\tilde C)\cup \set{t_e\mid e\in \delta_G(\tilde C)}$. We denote $T'=\set{t_e\mid e\in \delta_G(\tilde C)}$. We partition the set ${\mathcal{W}}(C)$ of clusters into two subsets: set ${\mathcal{W}}^{\operatorname{light}}(C)={\mathcal{W}}(C)\cap {\mathcal{C}}_{i}^{\operatorname{light}}$ and ${\mathcal{W}}^{\operatorname{bad}}(C)={\mathcal{W}}(C)\cap {\mathcal{C}}_{i}^{\operatorname{bad}}$.
Graph $H(\tilde C)$ is then obtained from graph $\tilde C^+$, by contracting every cluster $W\in {\mathcal{W}}^{\operatorname{light}}(C)$ into a supernode $v_W$.
Additionally, we denote by $H'(\tilde C)$ the graph obtained from $\tilde C^+$ by contracting every cluster $W\in {\mathcal{W}}(C)$ into a supernode $v_W$.
Notice that graph $H'(\tilde C)$ is precisely the augmentation $C^+$ of the cluster $C$ in $G'$.
In the remainder of this step, we focus on one specific cluster $C\in {\mathcal{C}}_1'$, so for convenience, we will denote $H(\tilde C)$ by $H$, $H'(\tilde C)$ by $H'$, ${\mathcal{W}}(C)$ by ${\mathcal{W}}$, and ${\mathcal{W}}^{\operatorname{light}}(C),{\mathcal{W}}^{\operatorname{bad}}(C)$ by ${\mathcal{W}}^{\operatorname{light}}$ and ${\mathcal{W}}^{\operatorname{bad}}$, respectively. From our construction, $H'=H_{|{\mathcal{W}}^{\operatorname{bad}}}$.
Recall that we have already established that cluster $C$ has the $\alpha_0$-bandwidth property in $G'$. Therefore, the set $T'$ of vertices is $\alpha_0$-well-linked in graph $H'$.
Additionally, from Property \ref{prop 1 of clusters bw}, every cluster $W\in {\mathcal{W}}^{\operatorname{bad}}$ has the $\alpha_{i}$-bandwidth property, and, if event ${\cal{E}}_{i}$ did not happen, each such cluster is $\eta_{i}$-bad.
Recall also that we are guaranteed that $|E(C)|\leq \eta_0\cdot |\delta_{G'}(C)|=\eta_0\cdot |T'|$. Therefore, $|E(H')|=|E(C)|+|T'|\leq 2\eta_0|T'|$.
Intuitively, we would now like to apply the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace from \Cref{thm: find guiding paths} to graph $H$, the set $T'$ of terminals, and the correpsonding collection ${\mathcal{W}}^{\operatorname{bad}}$ of clusters. In order to do so, we need to define a rotation system $\hat\Sigma$ for graph $H$. We do so using a randomized algorithm that exploits the distributions ${\mathcal{D}}(W)$ over the set $\Lambda_G(W)$ of internal $W$-routers for clusters $W\in {\mathcal{W}}^{\operatorname{light}}$. We would like to use the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace in order to decide whether to add cluster $\tilde C$ to the set ${\mathcal{C}}_{i+1}^{\operatorname{light}}$ of light clusters or to the set ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$ of bad clusters. Specifically, if the algorithm returns FAIL, we would like to add it to ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$, and otherwise we would like to add it to set ${\mathcal{C}}_{i+1}^{\operatorname{light}}$, together with the distribution ${\mathcal{D}}(\tilde C)$ over the set $\Lambda_G(\tilde C)$ of internal $\tilde C$-routers that we can compute using the distribution over internal routers in $\Lambda(H,T')$ that the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace computes.
Notice however, that, even if $\mathsf{OPT}_{\mathsf{cnwrs}}(H,\hat \Sigma)$ is small, the algorithm may return FAIL with a constant probability. Additionally, the random choices that we make in defining the rotation system $\hat \Sigma$ for graph $H$ may also result in an instance whose solution value is too high (though this can only happen with relatively small probability). In order to ensure that our algorithm classifies cluster $\tilde C$ as a light or a bad cluster correctly with high probability, we will perform $m$ identical iterations (but in each iteration we construct the rotation system $\hat \Sigma$ for $H$ from scratch). We now describe a single iteration.
\paragraph{Execution of a single iteration.}
In order to perform a single iteration of the algorithm, we construct a rotation system $\hat \Sigma$ for graph $H$, as follows. Consider any vertex $v\in V(H)$. If $v\in T'$, then the degree of $v$ in $H$ is $1$, and the corresponding ordering of its incident edges is trivial. Assume now that $v\in V(H)\setminus T'$, and that $v$ is not a supernode. In this case, there is a one-to-one correpsondence between the edges in set $\delta_{H}(v)$ and the edges in set $\delta_G(v)$. We use the ordering ${\mathcal{O}}_v\in \Sigma$ of the edges in $\delta_G(v)$ in order to define an ordering of the edges in $\delta_{H}(v)$, for the rotation system $\hat \Sigma$. Lastly, we assume that vertex $v\in V(H)\setminus T'$ is a supernode. In other words, $v=v_W$, where $W\in {\mathcal{W}}^{\operatorname{light}}$ is a light cluster. Recall that we are given a distribution ${\mathcal{D}}(W)$ over the set $\Lambda_G(W)$ of internal $W$-routers, such that $W$ is $\beta_i$-light with respect to this distribution. We randomly select an internal $W$-router ${\mathcal{Q}}(W)\in \Lambda_G(W)$ from the distribution ${\mathcal{D}}(W)$. Let $u(W)$ be the center vertex of ${\mathcal{Q}}(W)$, so that ${\mathcal{Q}}(W)$ is a set of paths routing the edges of $\delta_{G}(W)$ to vertex $u(W)$. Also recall that, from the definition of light clusters, for every edge $e\in E(W)$, $\expect[{\mathcal{Q}}(W)\sim{\mathcal{D}}(W)]{(\cong_{G}({\mathcal{Q}}(W),e))^2}\leq \beta_{i}$.
Observe that the edges of $\delta_H(v_W)$ are precisely the edges of $\delta_G(W)$. Next, we transform the set ${\mathcal{Q}}(W)$ of paths into a set of non-transversal paths, by applying the algorithm from \Cref{lem: non_interfering_paths} to the set ${\mathcal{Q}}(W)$ of paths. We denote the resulting set of paths by $\hat{\mathcal{Q}}(W)$; note that $\hat {\mathcal{Q}}(W)$ is an internal $W$-router.
Recall that we have defined an ordering of edges of $\delta_G(W)$ guided by the internal $W$-router $\hat{\mathcal{Q}}(W)$ (see \Cref{subsec: guiding paths rotations}). We let the ordering $\hat {\mathcal{O}}_{v_W}\in \hat \Sigma$ be the ordering of the edges of $\delta_G(W)$ guided by the paths in $\hat {\mathcal{Q}}(W)$.
We need the following observation, whose proof follows arguments that are similar to those used in the proof of \Cref{lem: disengagement final cost}, and is deferred to Section \ref{subsec: proof of obs opt is small} of Appendix.
\begin{observation}\label{obs: opt is small}
$\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(H, \hat \Sigma)}\leq O\textsf{left}(\beta_{i}\cdot \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde C,\Sigma_{\tilde C})+|E(\tilde C)|\textsf{right} )\textsf{right} )$.
\end{observation}
A single iteration of our algorithm consists of computing a rotation system $\hat \Sigma$ for graph $H$ from scratch, and then applying the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace from \Cref{thm: find guiding paths} to instance $I=(H,\hat \Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace,
with the set $T'$ of terminals, and the collection ${\mathcal{C}}={\mathcal{W}}^{\operatorname{bad}}$ of clusters. We set the parameters for the algorithm from \Cref{thm: find guiding paths} as follows: $\alpha=\alpha_0$, $\alpha'=\alpha_{i}$, $\eta=2\eta_0$, and $\eta'=\eta_{i}$. Recall that the set $T'$ of terminals is $\alpha_0$-well-linked in the contracted graph $H'=H_{|{\mathcal{W}}^{\operatorname{bad}}}$, and $|T'|\geq |E(H'(\tilde C))|/(2\eta_0)$. Moreover, each cluster in ${\mathcal{W}}^{\operatorname{bad}}(C)$ has the $\alpha_{i}$-bandwidth property, and, if event ${\cal{E}}_{i}$ did not happen, then each such cluster is $\eta_{i}$-bad (note that, if $i=1$ then ${\mathcal{W}}^{\operatorname{bad}}(C)=\emptyset$).
Notice that $\eta'\geq \eta_1\geq (2\eta_0)^{13}\geq \eta^{13}$, from the definition of the parameter $\eta_i$.
It remains to verify that $\eta\geq \frac{c^*\log^{46}m}{\alpha^{10}(\alpha')^2}$, or, equivalently, that $\eta_0\geq \frac{c^*\log^{46}m}{2\alpha_0^{10}(\alpha_{i})^2}$.
Recall that we set $\eta_0=\eta^4\cdot \log^9m=2^{O ((\log m)^{3/4} )}$, and we ensure that $\alpha_{i},\alpha_0\geq \alpha_{\ell}=1/(\log m)^{50\cdot\ell}=1/2^{O((\log m)^{1/4}\log\log m)}$. Since we assume that $m$ is large enough, the inequality clearly holds.
Therefore, all conditions of
\Cref{thm: find guiding paths} hold, and we can apply the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace to instance $I=(H,\hat \Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace,
with the set $T'$ of terminals, the collection ${\mathcal{C}}={\mathcal{W}}^{\operatorname{bad}}$ of clusters, and the parameters defined above.
Recall that we perform $m$ such iterations. If, in every iteration, algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace returns FAIL, then we add cluster $\tilde C$ to set ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$. We next show that, if Event ${\cal{E}}_{i}$ did not happen,
and $\tilde C$ is not an $\eta_{i+1}$-bad cluster in $G$, then the probability that $\tilde C$ is added to ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$ is small.
\begin{claim}\label{claim: small probability of mistake for bad cluster}
Let ${\cal{E}}_{i+1}(\tilde C)$ denote the bad event that $\tilde C$ is not an $\eta_{i+1}$-bad cluster, but our algorithm adds $\tilde C$ to ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$. Then $\prob{{\cal{E}}_{i+1}(\tilde C)\mid \neg {\cal{E}}_i}\leq (3/4)^m$.
\end{claim}
\begin{proof}
Consider a single iteration of the algorithm. Recall that, from \Cref{obs: opt is small},
$$\expect{\mathsf{OPT}_{\mathsf{cnwrs}}(H, \hat \Sigma)}\leq O\textsf{left}(\beta_{i}^2\cdot \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde C,\Sigma_{\tilde C})+|E(\tilde C)|\textsf{right} )\textsf{right} ).$$
If cluster $\tilde C$ is not $\eta_{i+1}$-bad, then $|E(\tilde C)|+\mathsf{OPT}_{\mathsf{cnwrs}}(\tilde C,\Sigma_{\tilde C})<|T'|^2/\eta_{i+1}$. So for some constant $c'$:
$$\expect{|E(H\setminus T')|+\mathsf{OPT}_{\mathsf{cnwrs}}(H(\tilde C), \hat \Sigma)}\leq c'\beta_{i}^2\cdot|T'|^2/\eta_{i+1}.$$
Let ${\cal{E}}'$ denote the event that $|E(H\setminus T')|+\mathsf{OPT}_{\mathsf{cnwrs}}(H, \hat \Sigma)>4c'\beta_{i}^2\cdot|T'|^2/\eta_{i+1}$. From Markov's bound, $\prob{{\cal{E}}'}\leq 1/4$.
Denote $k=|T'|$, and recall that we have set $\eta'=\eta_{i}$, $\alpha=\alpha_0$, and $\alpha'=\alpha_{i}$.
Since $\eta_{i+1}\geq \eta_{i}\cdot \beta^3_{i}$, and
$\beta_{i}=\frac{(\log m)^{56}}{(\alpha_0)^{12}\cdot (\alpha_{i})^8}\cdot \beta_{i-1}$,
we get that:
\[\frac{4c'\beta_{i}^2}{\eta_{i+1}}\leq \frac{4c'}{\beta_{i}\eta_{i}}\leq \frac{(\alpha_0)^{12}\cdot (\alpha_{i})^8}{c_0\eta_{i}\log^{50}m}.\]
We conclude that, if event ${\cal{E}}'$ did not happen, and $\tilde C$ is not an $\eta_{i+1}$-bad cluster, then $|E(H\setminus T')|+\mathsf{OPT}_{\mathsf{cnwrs}}(H, \hat \Sigma)\leq \frac{(k\alpha^4 \alpha')^2}{c_0\eta'\log^{50}m}$.
Let ${\cal{E}}''$ be the bad event that the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace returned FAIL. From \Cref{thm: find guiding paths}, if cluster $\tilde C$ is not $\eta_{i+1}$-bad, then $\prob{{\cal{E}}''\mid \neg {\cal{E}}'\wedge \neg{\cal{E}}_{i}}\leq 1/2$.
Overall, assuming that the event ${\cal{E}}_{i}$ did not happen and cluster $\tilde C$ is not $\eta_i$-bad, then the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace may only return FAIL if either ${\cal{E}}'$ or ${\cal{E}}''$ happen, which, from the above discussion, happens with probability at most $(3/4)$. Overall, since we repeat the above algorithm $m$ times, the probability that in every iteration the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace returns FAIL is at most $(3/4)^m$.
\end{proof}
Assume now that, in any one of the iterations, the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace did not return FAIL, and instead returned
a distribution ${\mathcal{D}}$ over the routers of $\Lambda(H,T')$, such that for every edge $e\in E(H)$, $\expect[{\mathcal{Q}}\sim{\mathcal{D}}]{(\cong({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{32}m}{\alpha_0^{12}\alpha_{i}^8}\textsf{right} )$.
We now provide a distribution ${\mathcal{D}}(\tilde C)$ over the set $\Lambda_G(\tilde C)$ of internal $\tilde C$-routers, such that $\tilde C$ is $\beta_{i+1}$-light with respect to ${\mathcal{D}}(\tilde C)$. The distribution is provided implicitly: that is, we provide an efficient algorithm for drawing an internal $\tilde C$-router from the distribution.
In order to draw an internal $\tilde C$-router from distribution ${\mathcal{D}}(\tilde C)$, we start by choosing a router ${\mathcal{Q}}\in \Lambda(H,T')$ from the distribution ${\mathcal{D}}$. Let $x$ be the center vertex of ${\mathcal{Q}}$, so $x$ is a vertex of $V(H\setminus T')$, and ${\mathcal{Q}}$ is a collection of paths in $H$, routing all terminals in $T'$ to $x$. Equivalently, we can view ${\mathcal{Q}}$ as a collection of paths that route the edges of $\delta_G(\tilde C)$ to the vertex $x$, in the contracted graph $\tilde C_{|{\mathcal{W}}^{\operatorname{light}}(C)}\cup \delta_G(\tilde C)$. Additionally, for every cluster $W\in {\mathcal{W}}^{\operatorname{light}}(C)$, we select an internal $W$-router ${\mathcal{Q}}(W)\in \Lambda_G(W)$ from the distribution ${\mathcal{D}}(W)$, and we denote by $u(W)\in V(W)$ its center vertex.
Assume first that $x$ is a regular vertex in graph $H$, that is, it is not a supernode representing a cluster of ${\mathcal{W}}^{\operatorname{light}}(C)$. In this case, we set $u(\tilde C)=x$, and we will use $u(\tilde C)$ as the center vertex for internal router ${\mathcal{Q}}(\tilde C)\in \Lambda_G(\tilde C)$ that we construct. Otherwise, if $x=v_W$ for some cluster $W\in {\mathcal{W}}^{\operatorname{light}}(C)$, then we set $u(\tilde C)=u(W)$, where $u(W)$ is the center vertex of the internal router ${\mathcal{Q}}(W)\in \Lambda_G(W)$ that we have selected for cluster $W$.
Next, we consider every path $Q\in {\mathcal{Q}}$ one by one. Let $Q$ be any such path, and assume that the first edge on $Q$ is $e\in \delta_G(\tilde C)$. We transform $Q$ into a path $Q'$ connecting $e$ to $u(\tilde C)$ in $G$, as follows. We consider supernodes $v_{W'}$ that lie on $Q$ one by one. For any such supernode $v_{W'}$ that is an inner vertex of $Q$, we let $e',e''$ be the two edges that appear immediately before and immediately after $v_{W'}$ on $Q$. Observe that $e',e''\in\delta_G(W')$. Therefore, there is a path $P(e')\in {\mathcal{Q}}(W')$ connecting $e'$ to $u(W')$, whose inner vertices lie in $W'$, and a path $P(e'')\in {\mathcal{Q}}(W')$ connecting $e''$ to $u(W')$, whose inner vertices lie in $W'$. By concatenating these two paths, we obtain a path $P^*(Q,W')$, whose first edge is $e'$, last edge is $e''$, and all remaining edges lie in $W'$. We then replace the segment of path $Q$ consisting of the edges $e',e''$ with the path $P^*(Q,W')$. Lastly, if $v_W$ is the last vertex on path $Q$ (in which case $x=v_W$), then we let $e'$ be the last edge on $Q$. Notice that $e'\in \delta_G(W)$ must hold. Then there must be a path $P(e')\in {\mathcal{Q}}(W)$, whose first edge is $e'$ and last vertex is $u(W)=u(\tilde C)$. We then replace the edge $e'$ on path $Q$ with the path $P(e')$. Let $Q'$ be the final path that is otbained from $Q$ after this transformation. Then $Q'$ is a path in graph $G$, whose first edge is $e$, last vertex is $u(\tilde C)$, and all inner edges and vertices are contained in $\tilde C$.
Lastly, we let ${\mathcal{Q}}(\tilde C)=\set{Q'\mid Q\in {\mathcal{Q}}}$ be the resulting router in $\Lambda_G(\tilde C)$.
This finishes the definition of the distribution ${\mathcal{D}}(\tilde C)$ over the set $\Lambda(\tilde C)$ of internal $\tilde C$-routers. Notice that we do not provide the distribution explicitly, and instead we have described an algorithm that, given access to distribution ${\mathcal{D}}$ computed by the algorithm \ensuremath{\mathsf{AlgFindGuiding}}\xspace, and distributions $\set{{\mathcal{D}}(W)}$ for clusters $W\in {\mathcal{W}}^{\operatorname{light}}(C)$ (that may also be given implicitly), samples an internal $\tilde C$-router from the distribution ${\mathcal{D}}(\tilde C)$.
We add cluster $\tilde C$ to set ${\mathcal{C}}_{i+1}^{\operatorname{light}}$, together with the distribution ${\mathcal{D}}(\tilde C)$.
It now remains to show that cluster $\tilde C$ is $\beta_{i+1}$-light with respect to the distribution ${\mathcal{D}}(\tilde C)$, which we do in the following claim.
\begin{claim}\label{claim: good cluster is good}
Cluster $\tilde C$ is $\beta_{i+1}$-light with respect to the distribution ${\mathcal{D}}(\tilde C)$.
\end{claim}
\begin{proof}
Consider some edge $e\in E(\tilde C)$. Assume first that edge $e$ does not lie in any cluster $W\in {\mathcal{W}}^{\operatorname{light}}(C)$. In this case:
\[\expect[{\mathcal{Q}}(\tilde C)\sim {\mathcal{D}}(\tilde C)]{(\cong_G({\mathcal{Q}}(\tilde C),e))^2}=\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_{H}({\mathcal{Q}},e))^2}\leq O\textsf{left} (\frac{\log^{32}m}{\alpha_0^{12}(\alpha_{i})^8}\textsf{right} ), \]
from \Cref{thm: find guiding paths}. Since $\beta_{i+1}=\frac{(\log m)^{56}}{(\alpha_0)^{12}\cdot (\alpha_{i+1})^8}\cdot \beta_{i}$, and $\alpha_{i+1}\leq \alpha_i$, we get that:
$$\expect[{\mathcal{Q}}(\tilde C)\sim {\mathcal{D}}(\tilde C)]{(\cong_G({\mathcal{Q}}(\tilde C),e))^2}\leq \beta_{i+1}.$$
Next, we assume that $e$ lies in some cluster $W\in {\mathcal{W}}^{\operatorname{light}}(C)$. In order to analyze $\expect{(\cong_G({\mathcal{Q}}(\tilde C),e))^2}$, we consider the following two-step process. In the first step, we select an internal $W$-router ${\mathcal{Q}}(W)\in \Lambda(W)$ from the distribution ${\mathcal{D}}(W)$, and denote its center vertex by $u(W)$. Then, in the second step, we select a router ${\mathcal{Q}}\in \Lambda(H,T')$ from distribution ${\mathcal{D}}$. Lastly, composing the paths in ${\mathcal{Q}}$ with the paths in ${\mathcal{Q}}(W)$, similarly to our construction of the final set ${\mathcal{Q}}(\tilde C)$ of paths, will establish the final congestion on edge $e$.
Let ${\mathcal{Q}}(W)\in \Lambda(W)$ be the internal $W$-router that was chosen from distribution ${\mathcal{D}}(W)$, and assume that the paths in ${\mathcal{Q}}(W)$ cause congestion $z$ on edge $e$. We denote ${\mathcal{Q}}(W)=\set{Q(e')\mid e'\in \delta_G(W)}$, where for every edge $e'\in \delta_G(W)$, path $Q(e')$ originates at edge $e'$ and terminates at vertex $u(W)$. Let $E'\subseteq \delta_G(W)$ be the set of edges $e'$ whose corresponding path $Q(e')$ contains the edge $e$, so $|E'|=z$. Denoting $E'=\set{e_1,\ldots,e_z}$, and assuming that the set ${\mathcal{Q}}(W)$ of paths is fixed, we can now write:
\[
\begin{split}
\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{(\cong_{G}({\mathcal{Q}}(\tilde C),e))^2}&=\expect[{\mathcal{Q}}\sim {\mathcal{D}}]{\textsf{left}(\sum_{i=1}^z\cong_{H}({\mathcal{Q}},e_i)\textsf{right} )^2 } \\
&\leq \expect[{\mathcal{Q}}\sim {\mathcal{D}}]{\sum_{i=1}^z2z\cdot(\cong_{H}({\mathcal{Q}},e_i))^2 }\\
&\leq z^2\cdot O\textsf{left} (\frac{\log^{32}m}{\alpha_0^{12}\alpha_i^8}\textsf{right} ).
\end{split} \]
Recall that $z$ is the congestion caused by the set ${\mathcal{Q}}(W)$ of paths on edge $e$. Therefore, overall:
\[\begin{split}
\expect[{\mathcal{Q}}(\tilde C) \sim {\mathcal{D}}(\tilde C)]{(\cong_G({\mathcal{Q}}(\tilde C),e))^2}
&\leq \expect[{\mathcal{Q}}(W)\sim {\mathcal{D}}(W)]{(\cong_G({\mathcal{Q}}(W),e))^2}\cdot O\textsf{left} (\frac{\log^{32}m}{\alpha_0^{12}\alpha_i^8}\textsf{right} )\\
&\leq \beta_{i}\cdot O\textsf{left} (\frac{\log^{32}m}{\alpha_0^{12}\alpha_i^8}\textsf{right} ) \leq \beta_{i+1}.
\end{split}
\]
%
(here we have used the fact that cluster $W$ is $\beta_{i}$-light with respect to distribution ${\mathcal{D}}(W)$, that $\beta_{i+1}=\frac{(\log m)^{56}}{(\alpha_0)^{12}\cdot (\alpha_{i+1})^8}\cdot \beta_{i}$), and that $\alpha_{i+1}\leq \alpha_i$.
\end{proof}
We now summarize the second step of the algorithm. First, from \Cref{claim: good cluster is good}, every cluster $\tilde C$ that we have added to set ${\mathcal{C}}_{i+1}^{\operatorname{light}}$ over the course of Step 2 is a $\beta_{i+1}$-light cluster with respect to the distribution ${\mathcal{D}}(\tilde C)$ over the set $\Lambda_G(\tilde C)$ of internal $\tilde C$-routers that we have defined. Let ${\cal{E}}_{i+1}$ be the bad event that any cluster $\tilde C$ that was added to set ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$ over the course of the current phase is not $\eta_{i+1}$-bad.
From the discussion above, event ${\cal{E}}_{i+1}$ may only happen if either event ${\cal{E}}_i$ happened, or there is some cluster $C\in {\mathcal{C}}_1'$, for which event ${\cal{E}}_{i+1}(\tilde C)$ happened. From \Cref{claim: small probability of mistake for bad cluster}, and Property \ref{prop 4 last of clusters bad}, the probability of ${\cal{E}}_{i+1}$ is bounded by $\prob{{\cal{E}}_i}+m\cdot (3/4)^m\leq i/m^{10}+m\cdot (3/4)^m\leq (i+1)/m^{10}$, since we have assumed that $m$ is large enough.
Lastly, if the current phase is the last phase, that is, $|E(G')|\leq 16\eta k\log m$ holds, then set ${\mathcal{C}}_1'$ contains a single cluster $C=G'\setminus T$.
If our algorithm added the corresponding cluster $\tilde C=G\setminus T$ to set ${\mathcal{C}}_{i+1}^{\operatorname{bad}}$, then we return FAIL. Notice that, if $\mathsf{OPT}_{\mathsf{cnwrs}}(G,\Sigma)+|E(G)\setminus T|< k^2/\eta^*\leq k^2/\eta_{\ell}\leq k^2/\eta_{i+1}$, then the algorithm may only return FAIL if event ${\cal{E}}_{i+1}$ happened, with may only happen with probability at most $\frac{i+1}{m^{10}}<\frac 1 2$. Otherwise, our algorithm added cluster $\tilde C$ to set ${\mathcal{C}}_{i+1}^{\operatorname{light}}$, and constructed a distribution ${\mathcal{D}}(\tilde C)$ over the set $\Lambda(\tilde C)$ of internal $\tilde C$-routers, such that cluster $\tilde C$ is $\beta_{i+1}$-light with respect to ${\mathcal{D}}(\tilde C)$. Notice that ${\mathcal{D}}(\tilde C)$ can also be viewed as a distribution over the routers of $\Lambda(G,T)$, and we are guaranteed that, for every edge $e\in E(G)$, $\expect[{\mathcal{Q}}\sim {\mathcal{D}}(\tilde C)]{(\cong_{G}({\mathcal{Q}},e))^2}\leq \beta_{\ell}\leq \beta^*$. From Property \ref{prop 2 of clusters small contracted graph}, after $(\ell-1)$ phases the algorithm terminates, for $\ell=O\textsf{left}(\frac{\log m}{\log \eta}\textsf{right} )$.
\subsection{Well-Structured Subinstances, Well-Structured Solutions and a Decomposition Lemma}
\newif\ifhide
\hidefalse
\ifhide
\znote{maybe we need the notion "boundary of a face" $\partial F$ back in the prelim}
\paragraph{Well-Structured subinstances and contracted subinstances.}
We say that a subinstance $I=(G,\Sigma)$ of the input instance $\check I$ is \emph{well-structured}, iff $G$ is a subgraph of $\check G$, and we are given
\begin{itemize}
\item a $2$-connected subgraph $K$ of $G$, called the \emph{skeleton} of $I$;
\item a planar drawing $\rho$ of $K$ in the sphere (we denote by ${\mathcal{F}}$ the set of faces in $\rho$).
\item a collection ${\mathcal{G}}=\set{G_F}_{F\in {\mathcal{F}}}$ of subgraphs of $G$, where each subgraph $G_F\in {\mathcal{G}}$ is indexed by a face $F\in{\mathcal{F}}$, such that
\begin{itemize}
\item each vertex $v\notin V(K)$ belongs to exactly one subgraphs in ${\mathcal{G}}$;
\item for each vertex $v\in V(K)$ and each face $F\in {\mathcal{F}}$, subgraph $G_F$ contains vertex $v$ iff $v$ lies on the boundary of face $F$ (namely $v\in \partial F$);
\item for each face $F\in {\mathcal{F}}$ and each vertex $v\in \partial F$, if we denote by $e,e'$ the edges of $\partial F$ incident to $v$, then edges of $\delta_{G_F}(v)$ must be consecutive in the circular ordering ${\mathcal{O}}_v\in \Sigma$ and must lie between $e$ and $e'$. Specifically, if $\delta_G(v)=\set{e,e_1,\ldots,e_s,e',e'_1,\ldots,e'_t}$, where the other edges are indexed according to ${\mathcal{O}}_v$, then either $\delta_{G_F}(v)=\set{e_1,\ldots,e_s}$ or $\delta_{G_F}(v)=\set{e'_1,\ldots,e'_t}$.
\end{itemize}
\end{itemize}
We call the tuple $(K,\rho,{\mathcal{F}},{\mathcal{G}})$ the \emph{structure} of instance $I=(G,\Sigma)$.
Additionally, we call edges of $K$ \emph{special edges} of instance $I$.
Let $F$ be a face in ${\mathcal{F}}$. We define the subinstance $I_F=(G_F,\Sigma_F)$, where the rotation system $\Sigma_F$ is induced by $\Sigma$.
We define the subinstance $\tilde I_F=(\tilde G_F,\tilde \Sigma_F)$ as follows. Since skeleton $K$ is $2$-connected and $F$ is a face in a planar drawing of $K$, graph $\partial F$ is the union of a set of vertex-disjoint cycles.
We denote by ${\mathcal{R}}(F)$ the set of vertex-disjoint cycles in $\partial F$.
Graph $\tilde G_F$ is obtained from $G_F$ by contracting, for each cycle $R$ in ${\mathcal{R}}(F)$, all vertices of $R$ into a supernode, which we denote by $v_R$, so $V(\tilde G)=(V(G)\setminus V(\partial F))\cup \set{v_R\mid R\in {\mathcal{R}}(F)}$.
We now define the rotation system $\tilde \Sigma_F$.
Note that, for every vertex $v\in V(G)\setminus V(\partial F)$, $\delta_{\tilde G_F}(v)=\delta_{\tilde G}(v)$, and we define the ordering $\tilde \oset_v$ in $\tilde \Sigma$ to be ${\mathcal{O}}_v$, the ordering on vertex $v$ in $\Sigma$. It remains to define the orderings $\set{\tilde \oset_{v_{R}}\mid R\in {\mathcal{R}}(F)}$. Consider a cycle $R\in {\mathcal{R}}(F)$ with $R=(v_0,v_1,\ldots,v_{s-1},v_0=v_s)$. For each $0\le i\le s-1$, we denote by $e_i$ the edge in $R$ connecting $v_i$ to $v_{i+1}$, and we denote $\delta_{G_F}(v_i)=\set{e_{i-1}, e^i_1,\ldots,e^i_{d_i},e_i}$, where the edges are indexed according to the ordering ${\mathcal{O}}_{v_i}$ in $\Sigma$. From the definition of graph $\tilde G_F$,
$$\delta_{\tilde G}(v_R)=\bigg(\bigcup_{0\le i\le s-1}\delta_{G_F}(v_i)\bigg)\setminus E(R)=\set{e^0_1,\ldots,e^0_{d_0},e^1_1,\ldots,e^1_{d_1},\ldots,e^{s-1}_1,\ldots,e^{s-1}_{d_{s-1}}}.$$ We then define the ordering $\tilde \oset_{v_{R}}$ to be
$e^0_1,\ldots,e^0_{d_0},e^1_1,\ldots,e^1_{d_1},\ldots,e^{s-1}_1,\ldots,e^{s-1}_{d_{s-1}}$. This completes the definition of instance $\tilde I_F$. We call $\tilde I_F$ the
\emph{contracted subinstance} of $I_F$.
\paragraph{Well-Structured solutions.}
Let $I=(G,\Sigma)$ be a well-structured instance with structure $(K,\rho,{\mathcal{F}},{\mathcal{G}})$. We say that a solution $\phi$ of instance $I$ is a \emph{well-structured solution} (or a \emph{well-structured drawing}) for $I$, iff
\begin{properties}{P}
\item the drawing of $K$ induced by drawing $\phi$ is identical to $\rho$;
\item for each face $F\in {\mathcal{F}}$,
\begin{itemize}
\item all vertices of $G_F\setminus \partial F$ are drawn in the interior of face $F$;
\item for each edge $e\in (G_F\setminus F)$ that is incident to some vertex $v\in F$, the intersection of image of $e$ and the tiny $v$-disc $D_{\phi}(v)$ is entirely contained in face $F$;
and
\end{itemize}
\item the number of edges that crosses a special edge of $I$ in $\phi$ is at most $\mathsf{cr}(\phi)\cdot (\tau^i)^{10}/|E(G)|$.
\label{prop: niubi edge crossing special cycle}
\end{properties}
The main result in this subsection is the following lemma.
\znote{parameters to be determined.}
\fi
\newcommand{E^{\sf del}}{E^{\sf del}}
\newcommand{well-structured\xspace}{well-structured\xspace}
\begin{lemma}
\label{lem: many path main}
There is an efficient randomized algorithm, that given a well-structured subinstance $I=(G,\Sigma)$ of $\check I$, with structure $(K,\rho,{\mathcal{F}},{\mathcal{G}})$, for which there exists a well-structured\xspace drawing $\phi$, computes a well-structured subinstance $I'=(G',\Sigma)$ of $\check I$ with structure $(K',\rho',{\mathcal{F}}',{\mathcal{G}}')$, such that
\begin{properties}{C}
\item $G'\subseteq G$, $K\subseteq K'$, and the drawing of $K$ induced by $\rho'$ is identical to $\rho$;
\item for each face $F'\in {\mathcal{F}}'$, the contracted subinstance $\tilde I'_F$ is either $\tau^s$-small or $\tau^i$-narrow;
\label{prop: well-structured and contracted}
\end{properties}
and moreover, with probability $1-O(1/\mu^{a})$,
\begin{properties}[2]{C}
\item $|E(G)\setminus E(G')|\le \mathsf{cr}(\phi)\cdot \operatorname{poly}(\mu)/|E(G)|$;
\label{prop: few deleted edges}
\item for each face $F$ and each edge $e\in G_F$, the segment(s) of curve $\phi(e)$ that lies outside face $F$ is identical to the segment(s) of curve $\phi'(e)$ that lies outside face $F$.
\label{prop: forbidden discs}
\item if $\mathsf{cr}(\phi)\le m^2/\mu^{c}$ for some large enough constant $c>0$, then there exists a well-structured solution $\phi'$ for $I'$, such that $\mathsf{cr}(\phi')\le O(\mathsf{cr}(\phi))$.
\label{prop: number of crossings}
\end{properties}
Moreover, there is an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace(I')$, that given, for each face $F'\in {\mathcal{F}}'$, a feasible solution $\phi(\tilde I'_F)$ to the contracted instance $\tilde I'_F$, computes a well-structured solution $\phi(I')$ for instance $I'$, of cost
$\mathsf{cr}(\phi(I'))\leq O\big(\sum_{F'\in {\mathcal{F}}'}\mathsf{cr}(\phi(\tilde I'_F)\big)$.
\end{lemma}
The remainder of this subsection is dedicated to the proof of \Cref{lem: many path main}.
\iffalse
\begin{properties}{C}
\item each instance $I'\in {\mathcal{I}}$ is a well-structured subinstance;
\label{prop: well-structured instances}
\item for each contracted instance $\tilde I'=(\tilde G',\tilde\Sigma') \in \tilde {\mathcal{I}}$, either $\tilde I'$ is non-wide, or $|E(\tilde G')|\le m/\mu$;\label{prop: no longer wide}
\item the sets $\set{\hat E(I')\mid I'\in {\mathcal{I}}}$ of edges are mutually disjoint;\label{prop: private edges disjoint}
\item $|E^{\sf del}|\le \mathsf{cr}\cdot\mu^{O({50})}/m$, and edges of $E^{\sf del}$ does not belong to any instance of ${\mathcal{I}}$;\label{prop: few deleted edges in phase 1}
\item every edge of $E(G)\setminus \bigg(E^{\sf del}\cup \big(\bigcup_{I'\in {\mathcal{I}}}\hat E(I')\big)\bigg)$ is a special edge of exactly two instances in ${\mathcal{I}}$;\label{prop: few edges}
\item there exists, for every instance $I'\in {\mathcal{I}}$, a well-structured solution $\phi_{I'}$, such that\\
$\expect[]{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot \nu$; and \label{prop: small solution cost}
\item there is an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}})$, that, given, for each contracted instance $\tilde I' \in \tilde{\mathcal{I}}$, a feasible solution $\phi(\tilde I')$, computes a feasible solution $\phi$ for instance $I$, of cost $\mathsf{cr}(\phi^*)\leq O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$. \label{prop: alg to put together}
\end{properties}
\znote{to modify the properties?}
In the second phase, we will compute, for each instance $I'\in {\mathcal{I}}$, a new instance $I''$ obtained from $I'$ by deleting a set of edges from $I'$, such that
\znote{to add more properties}
\fi
\input{many_paths_main_decomposition}
\input{many_paths_main_decomposition_analysis}
\section{Preliminaries}
\label{sec: short_prelim}
By default, all logarithms in this paper are to the base of $2$. All graphs are undirected and finite. Graphs may contain parallel edges but they may not contain self loops.
Graphs without parallel edges are explicitly referred to as simple graphs.
\subsection{Graph-Theoretic Notation}
We follow standard graph-theoretic notation. Let $G=(V,E)$ be a graph.
For a vertex $v\in V$, we denote by $\delta_G(v)$ the set of all edges of $G$ that are incident to $v$, and we denote $\deg_G(v)=|\delta_G(v)|$.
For two disjoint subsets $A,B$ of vertices of $G$, we denote by $E_G(A,B)$ the set of all edges with one endpoint in $A$ and the other in $B$.
For a subset $S\subseteq V$ of vertices, we denote by $G[S]$ the subgraph of $G$ induced by $S$, by $E_G(S)$ the set of all edges with both endpoints in $S$, and by $\delta_G(S)$ the set of all edges with exactly one endpoint in $S$.
Abusing the notation, for a subgraph $C$ of $G$, we use $\delta_G(C)$ to denote $\delta_G(V(C))$.
\iffalse
Throughout, given a graph $G=(V,E)$, we denote by $\Delta(G)$ the maximum vertex degree in $G$.
We follow standard graph-theoretic notation.
For a vertex $v\in V$, we denote by $\delta_G(v)$ the set of all edges of $G$ that are incident to $v$, and we denote $\deg_G(v)=|\delta(v)|$.
For two disjoint subsets $A,B$ of vertices of $G$, we denote by $E_G(A,B)$ the set of all edges with one endpoint in $A$ and another in $B$.
For a subset $S\subseteq V$ of vertices, we denote by $G[S]$ the subgraph of $G$ induced by $S$, by $E_G(S)$ the set of all edges with both endpoints in $S$, and we denote $\overline{S}=V\setminus S$.
We also denote by $\delta_G(S)$ the subset of edges of $E$ with exactly one endpoint in $S$, namely $\delta_G(S)=E_G(S, V\!\setminus\! S)$.
Similarly, for a subgraph $C$ of $G$, we denote by $\delta_G(C)=\delta_G(V(C))$ the set of edges of $G$ with exactly one endpoint in $C$.
We will refer to vertex-induced subgraphs of $G$ as
\emph{clusters}.
\fi
\begin{definition}[Congestion]
Let $G$ be a graph, let $e$ be an edge of $G$, and let ${\mathcal{Q}}$ be a set of paths in $G$. The \emph{congestion that the set ${\mathcal{Q}}$ of paths causes on edge $e$}, denoted by $\cong_G({\mathcal{Q}},e)$, is the number of paths in ${\mathcal{Q}}$ that contain $e$. The \emph{total congestion of ${\mathcal{Q}}$ in $G$} is $\cong_G({\mathcal{Q}})=\max_{e\in E(G)}\set{\cong_G({\mathcal{Q}}, e)}$.
\end{definition}
\subsection{Curves in General Position, Graph Drawings, Faces, and Crossings}
Let $\gamma$ be an open curve in the plane, and let $P$ be a set of points in the plane. We say that $\gamma$ is \emph{internally disjoint} from $P$ if no inner point of $\gamma$ lies in $P$. In other words, $P\cap \gamma$ may only contain the endpoints of $\gamma$.
Given a set $\Gamma$ of open curves in the plane, we say that the curves in $\Gamma$ are \emph{internally disjoint} if, for every pair $\gamma,\gamma'\in \Gamma$ of distinct curves, every point $p\in \gamma\cap\gamma'$ is an endpoint of both curves.
We use the following definition of curves in general position.
\begin{definition}[Curves in general position]
Let $\Gamma$ be a finite set of open curves in the plane.
We say that the curves of $\Gamma$ are \emph{in general position}, if the following conditions hold:
\begin{itemize}
\item for every pair $\gamma,\gamma'\in \Gamma$ of distinct curves, there is a finite number of points $p$ with $p\in \gamma\cap \gamma'$;
\item for every pair $\gamma,\gamma'\in \Gamma$ of distinct curves, an endpoint of $\gamma$ may not serve as an inner point of $\gamma'$ or of $\gamma$; and
\item for every triple $\gamma,\gamma',\gamma''\in \Gamma$ of distinct curves, if some point $p$ lies on all three curves, then it must be an endpoint of each of these three curves.
\end{itemize}
\end{definition}
Let $\Gamma$ be a set of curves in general position,
and let $\gamma,\gamma'\in \Gamma$ be a pair of curves. Let $p$ be any point that lies on both $\gamma$ and $\gamma'$, but is not an endpoint of either curve. We then say that point $p$ is
a \emph{crossing} between $\gamma$ and $\gamma'$, or that curves $\gamma$ and $\gamma'$ \emph{cross} at point $p$.
We are now ready to formally define graph drawings.
\begin{definition}[Graph Drawings]
A \emph{drawing} $\phi$ of a graph $G$ in the plane is a map $\phi$, that maps every vertex $v$ of $G$ to a point $\phi(v)$ in the plane (called the \emph{image of $v$}), and every edge $e=(u,v)$ of $G$ to a simple curve $\phi(e)$ in the plane whose endpoints are $\phi(u)$ and $\phi(v)$ (called the \emph{image of $e$}), such that all points in set $\set{\phi(v)\mid v\in V(G)}$ are distinct, and the set $\set{\phi(e)\mid e\in E(G)}$ of curves is in general position. Additionally, for every vertex $v\in V(G)$ and edge $e\in E(G)$, $\phi(v)\in \phi(e)$ only if $v$ is an endpoint of $e$.
\end{definition}
Assume now that we are given some drawing $\phi$ of graph $G$ in the plane, and assume that for some pair $e,e'$ of edges, their images $\phi(e),\phi(e')$ cross at point $p$. Then we say that $(e,e')_p$ is a \emph{crossing} in the drawing $\phi$ (we may sometimes omit the subscript $p$ if the images of the two edges only cross at one point).
We also say that $p$ is a \emph{crossing point} of drawing $\phi$.
We denote by $\mathsf{cr}(\phi)$ the total number of crossings in the drawing $\phi$.
Note that a drawing of a graph $G$ in the plane naturally defines a drawing of $G$ on the sphere and vice versa; we use both types of drawings.
For convenience, given a drawing $\phi$ of a graph $G$, we sometimes will not distinguish between the edges of $G$ and their images. For example, we may say that edges $e,e'$ cross in drawing $\phi$ to indicate that their images cross. Similarly, we may not distinguish between vertices and their images. For example, we may talk about the order in which edges of $\delta_G(v)$ enter vertex $v$ in drawing $\phi$, to mean the order in which the images of the edges of $\delta_G(v)$ enter the image of $v$. We denote by $\phi(G)=\textsf{left} (\bigcup_{e\in E(G)}\phi(e)\textsf{right} )\cup \set{\phi(v)\mid v\in V(G)}$.
\paragraph{Images of Paths.}
Assume that we are given a graph $G$, its drawing $\phi$, and a path $P$ in $G$. The \emph{image of path $P$ in $\phi$}, denoted by $\phi(P)$, is the curve that is obtained by concatenating the images of all edges $e\in E(P)$. Equivalently, $\phi(P)=\bigcup_{e\in E(P)}\phi(e)$. If $P=\set{v}$ for some vertex $v$, then $\phi(P)=\phi(v)$.
\paragraph{Planar Graphs and Planar Drawings.}
A graph $G$ is \emph{planar} if there is a drawing of $G$ in the plane with no crossings. A drawing $\phi$ of a graph $G$ in the plane with $\mathsf{cr}(\phi)=0$ is called a \emph{planar drawing} of $G$. We use the following result by Hopcroft and Tarjan.
\begin{theorem}[\cite{hopcroft1974efficient}]\label{thm: testing planarity}
There is an algorithm, that, given a graph $G$, correctly establishes whether $G$ is planar, and if so, computes a planar drawing of $G$. The running time of the algorithm in $O(|V(G)|)$.
\end{theorem}
\paragraph{Faces of a Drawing.}
Suppose we are given a graph $G$ and a drawing $\phi$ of $G$ in the plane or on the sphere. The set of faces of $\phi$ is the set of all connected regions of $\mathbb{R}^2\setminus \phi(G)$. If $G$ is drawn in the plane, then we designate a single face of $\phi$ as the ``outer'', or the ``infinite'' face.
\iffalse
\mynote{we have a problem here. Boundary of a face is well-defined because a face is a contiguous region in the plane that has a boundary. Here we associate a subgraph of $G$ with that boundary. 1. do we need it? 2. would it be better to say that this is a subgraph corresponding to the boundary (and not that the subgraph is the boundary?). }
If the drawing $\phi$ of $G$ is planar, then for every face $F$ of the drawing, the boundary of $F$ corresponds to a subgraph $\partial(F)$ of $G$, consisting of all vertices and edges of $G$ whose images are incident to $F$. Abusing the notation, we will sometimes refer to $\partial(F)$ as the boundary of $F$. Notice that, if graph $G$ is not connected, then graph $\partial(F)$ may also be not connected.
Lastly, observe that, if $G$ is $2$-connected, then for every face $F$ of the drawing $\phi$, graph $\partial(F)$ is a simple cycle.
\fi
\paragraph{Identical Drawings and Orientations.}
Assume that we are given some planar drawing $\phi$ of a graph $G$.
We can associate, with every face $F$ of this drawing, a subgraph $\partial(F)$ of $G$, containing all vertices and edges of $G$ whose images are contained in the boundary of $F$.
Drawing $\phi$ of $G$ can be uniquely defined by the list ${\mathcal{F}}$ of all its faces, and, for every face $F\in {\mathcal{F}}$, the corresponding subgraph $\partial(F)$ of $G$. In particular, if $\phi,\phi'$ are two planar drawings of the graph $G$, and there is an one-to-one mapping between the set ${\mathcal{F}}$ of the faces of $\phi$ and the set ${\mathcal{F}}'$ of the faces of $\phi'$, and, for every face $F$, the graph $\partial(F)$ is identical in both drawings, then we say that drawings $\phi$ and $\phi'$ are \emph{identical}.
Assume now that we are given a (possibly non-planar) drawing $\phi$ of a graph $G$. Let $G'$ be the graph obtained from $G$ by placing a vertex on every crossing of $\phi$. We then obtain a planar drawing $\psi$ of the resulting graph $G'$, where every vertex $v\in V(G')\setminus V(G)$ corresponds to a unique crossing point of $\phi$. For every edge $e\in E(G)$, let $L(e)$ be the list of all vertices of $G'$ that correspond to crossings in which edge $e$ participates in $\phi$, ordered in the order in which these crossings appear on the image of edge $e$ in $\phi$, as we traverse it from one endpoint to another. Graph $G'$, its planar drawing $\psi$, and the lists $\set{L(e)}_{e\in E(G)}$ uniquely define the drawing $\phi$ of $G$. In other words, if $\phi,\phi'$ are two drawings of the graph $G$, for which (i) the corresponding graphs $G'$ are the same (up to renaming the vertices of $V(G')\setminus V(G)$); (ii) the induced planar drawings $\psi$ of $G'$ are identical; and (iii) the vertex lists $\set{L(e)}_{e\in E(G)}$ are identical, then $\phi$ and $\phi'$ are \emph{identical drawings} of $G$.
Assume now that $\phi$ is a drawing of a graph $G$ in the plane, and let $\phi'$ be the drawing of $G$ that is the mirror image of $\phi$. We say that $\phi$ and $\phi'$ are \emph{identical} drawings of $G$, and that their \emph{orientations} are \emph{different}, or \emph{opposite}. We sometime say that $\phi'$ is obtained by \emph{flipping} the drawing $\phi$.
We say that a graph $G$ is $3$-connected, if for every pair $u,v\in V(G)$ of its vertices, $G\setminus\set{u,v}$ is a connected graph. We use the following well known result.
\begin{theorem}[\cite{whitney1992congruent}]
Every $3$-connected planar graph has a unique planar drawing.
\end{theorem}
\iffalse
\begin{definition}[face, outer face]
Given a graph $G$ and a planar drawing $\phi$ of $G$, the set of faces of $\phi$ are defined to be the set of connected regions in $\mathbb{R}^2\setminus \phi(G)$, and we denote it by ${\mathcal{F}}(\phi)$. Moreover, exactly one of these faces is infinite which we refer to as the \emph{outer face}.
\end{definition}
\begin{definition}[boundary, inner boundary, outer boundary, and point of contact]
For each face $F\in {\mathcal{F}}(\phi)$, the boundary of the face $F$ is defined as the set of vertices and edges that are incident to the face $F$, and we denote it by $\delta(F)$. Note that if the graph is connected, the boundary of $F$ also forms a connected graph. We may also call it the full boundary to emphasize the contrast to the inner/outer boundary defined next.
For a face that is not the outer face, we define the \emph{outer boundary} $\delta^{out}(F)\subseteq \delta(F)$ to be the simple cycle that separates the face $F$ from the outer face. In other words, if $D(F)$ denotes the closed disc whose boundary is $\delta^{out}(F)$, then $D(F)$ should contain $F$.
Note that if $G$ is $2$-connected, the boundary of each face $F\in {\mathcal{F}}(\phi)$ is a simple cycle, i.e., $\delta(F)=\delta^{out}(F)$.
Moreover, we define the \emph{inner boundary} as $\delta^{in}(F)$ to be the set of edges in $\delta(F)\setminus \delta^{out}(F)$, together with their endpoint vertices.
Finally, for a connected graph, we can define for each vertex $v$ on the boundary $\delta(F)$, the \emph{point of contact} of $v$ to be the vertex on the outer boundary with minimum distance to $v$.
\end{definition}
\fi
\subsection{Grids and Their Standard Drawings}
The $(r\times r)$-grid is a graph whose vertex set is $\set{v_{i,j}\mid 1\le i,j\le r}$, and edge set is the union of the set $\set{(v_{i,j},v_{i,{j+1}})\mid 1\le i\le r, 1\le j< r}$ of \emph{horizontal edges}, and the set $\set{(v_{i,j},v_{i+1,{j}})\mid 1\le i< r, 1\le j\le r}$ of \emph{vertical edges}.
For $1\le i\le r$, the \emph{$i$th row} of the grid is the subgraph of the grid graph induced by vertex set $\set{v_{i,j}\mid 1\le j\le r}$. Similarly, for $1\leq j\leq r$, the \emph{$j$th column} of the grid is the subgraph of the grid graph induced by vertex set $\set{v_{i,j}\mid 1\le i\le r}$.
Given an $(r\times r)$-grid, we refer to vertices $v_{1,1},v_{1,r},v_{r,1}$, and $v_{r,r}$ as the \emph{corners} of the grid.
We also refer to the graph that is obtained from the union of row $1$, row $r$, column $1$, and column $r$, as the \emph{boundary} of the grid.
It is not hard to see that the $(r\times r)$-grid has a unique planar drawing (this is since the $(1\!\times \!1)$-grid and the $(2\!\times \!2)$-grid have unique planar drawings, and for all $r\ge 3$, if we suppress the corner vertices of the grid, we obtain a planar $3$-connected graph, that has a unique planar drawing). We refer to this unique planar drawing of the grid as its \emph{standard drawing}
(see \Cref{fig: grid}).
For all $1\le i,j\le r-1$, we let $\mathsf{Cell}_{i,j}$ be the face of the standard drawing, that contains the images of the vertices $v_{i,j},v_{i,j+1},v_{i+1,j},v_{i+1,j+1}$ on its boundary.
\begin{figure}[h]
\centering
\includegraphics[scale=0.1]{figs/grid.jpg}
\caption{The standard drawing of the $(r\times r)$-grid with $\mathsf{Cell}_{2,2}$ shown in green
}\label{fig: grid}
\end{figure}
\subsection{Circular Orderings, Orientations, and Rotation Systems}
Suppose we are given a collection $U=\set{u_1,\ldots,u_r}$ of elements. Let $D$ be any disc in the plane.
Assume further that we are given, for every element $u_i\in U$, a point $p_i$ on the boundary of $D$, so that all resulting points in $\set{p_1,\ldots,p_r}$ are distinct. As we traverse the boundary of the disc $D$ in the clock-wise direction, the order in which we encounter the points $p_1,\ldots,p_r$ defines a \emph{circular ordering ${\mathcal{O}}$ of the elements of $U$}. If we traverse the boundary of the disc $D$ in the counter-clock-wise direction, we obtain a circular ordering ${\mathcal{O}}'$ of the elements of $U$, which is the mirror image of the ordering ${\mathcal{O}}$. We say that the orderings ${\mathcal{O}}$ and ${\mathcal{O}}'$ are \emph{identical} but their \emph{orientations} are different, or opposite: ${\mathcal{O}}$ has a negative and ${\mathcal{O}}'$ has a positive orientation. Whenever we refer to an ordering ${\mathcal{O}}$ of elements, we view it as \emph{unoriented} (that is, the orientation can be chosen arbitrarily). When the orientation of the ordering is fixed, we call it an \emph{oriented ordering}, and denote it by $({\mathcal{O}},b)$, where ${\mathcal{O}}$ is the associated (unoriented) ordering of elements of $U$, and $b\in \set{-1,1}$ is the orientation, with $b=-1$ indicating a negative (that is, clock-wise), orientation.
We will also consider graph drawings on the sphere.
In this case, when we say we traverse the boundary of a disc $D$ in the clock-wise direction, we mean that we traverse the boundary of $D$ so that the interior of $D$ lies to our right. Similarly, we traverse the boundary of $D$ in the counter-clock-wise direction, if the interior of $D$ lies to our left.
Circular orderings and orientations are then defined similarly.
Given a graph $G$ and a vertex $v\in V(G)$, a circular ordering ${\mathcal{O}}_v$ of the edges of $\delta_G(v)$ is called a \emph{rotation}. A collection of circular orderings ${\mathcal{O}}_v$ for all vertices $v\in V(G)$ is called a \emph{rotation system} for graph $G$.
\subsection{Tiny $v$-Discs and Drawings that Obey Rotations}
Given a graph $G$, its drawing $\phi$, and a vertex $v\in V(G)$, we will sometimes utilize the notion of a \emph{tiny $v$-disc}, that we define next.
\begin{definition}[Tiny $v$-Disc]\label{def: tiny v-disc}
Let $G$ be a graph and let $\phi$ be a drawing of $G$ on the sphere or in the plane. For each vertex $v\in V(G)$, we denote by $D_{\phi}(v)$ a very small disc containing the image of $v$ in its interior, and we refer to $D_{\phi}(v)$ as \emph{tiny $v$-disc}. Disc $D_{\phi}(v)$ must be small enough to ensure that, for every edge $e\in \delta_G(v)$, the image $\phi(e)$ of $e$ intersects the boundary of $D_{\phi}(v)$ at a single point, and $\phi(e)\cap D_{\phi}(v)$ is a contiguous curve. Additionally, we require that for every vertex $u\in V(G)\setminus\set{v}$, $\phi(u)\not\in D_{\phi}(v)$; for every edge $e'\in E(G)\setminus\delta_G(v)$, $\phi(e')\cap D_{\phi}(v)=\emptyset$; and that no crossing point of drawing $\phi$ is contained in $D_{\phi}(v)$. Lastly, we require that all discs in $\set{D_{\phi}(v)\mid v\in V(G)}$ are mutually disjoint.
\end{definition}
Consider now a graph $G$, a vertex $v\in V(G)$, and a drawing $\phi$ of $G$. Consider the tiny $v$-disc $D=D_{\phi}(v)$. For every edge $e\in \delta_G(v)$, let $p_e$ be the (unique) intersection of the image $\phi(e)$ of $e$ and the boundary of the disc $D$. Let ${\mathcal{O}}$ be the (unoriented) circular ordering in which the points of $\set{p_e}_{e\in \delta_G(v)}$ appear on the boundary of $D$. Then ${\mathcal{O}}$ naturally defines a circular ordering ${\mathcal{O}}^*_v$ of the edges of $\delta_G(v)$: ordering ${\mathcal{O}}^*_v$ is obtained from ${\mathcal{O}}$ by replacing, for each edge $e\in \delta_G(v)$, point $p_e$ with the edge $e$. We say that \emph{the images of the edges of $\delta_G(v)$ enter the image of $v$ in the order ${\mathcal{O}}^*_v$} in the drawing $\phi$. For brevity, we may sometimes say that the edges of $\delta_G(v)$ enter $v$ in the order ${\mathcal{O}}^*_v$ in $\phi$. While we view the ordering ${\mathcal{O}}^*_v$ as unoriented, drawing $\phi$ also defines an orientation for this ordering. If the points in set $\set{p_e\mid e\in \delta_G(v)}$ are encountered in the order ${\mathcal{O}}^*_v$ when traversing the boundary of $D$ in the counter-clock-wise direction, then the orientation is $1$, and otherwise it is $-1$.
Assume now that we are given a graph $G$ and a rotation system $\Sigma$ for $G$. Let $\phi$ be a drawing of $G$. Consider any vertex $v\in V(G)$, and its rotation ${\mathcal{O}}_v\in \Sigma$. We say that the drawing $\phi$ \emph{obeys the rotation ${\mathcal{O}}_v\in \Sigma$}, if the order in which the edges of $\delta_G(v)$ enter $v$ in $\phi$ is precisely ${\mathcal{O}}_v$ (note that both orderings are unoriented).
We say that the \emph{orientation of $v$ is $-1$}, or \emph{negative}, in the drawing $\phi$ if the orientation of the ordering ${\mathcal{O}}_v$ of the edges of $\delta_G(v)$ as they enter $v$ is $-1$, and otherwise, the orientation of $v$ in $\phi$ is $1$, or positive.
We say that drawing $\phi$ of $G$ \emph{obeys the rotation system $\Sigma$}, if it obeys the rotation ${\mathcal{O}}_v\in \Sigma$ for every vertex $v\in V(G)$.
Assume now that we are given a set $\Gamma$ of curves in general position, where each curve $\gamma\in \Gamma$ is an open curve. Let $p$ be any point that serves as an endpoint of at least one curve in $\Gamma$, and let $\Gamma'\subseteq \Gamma$ be the set of curves for which $p$ serves as an endpoint. We then define a \emph{tiny $p$-disc} $D(p)$ to be a small disc that contains the point $p$ in its interior; does not contain any other point that serves as an endpoint of a curve in $\Gamma$; and does not contain any crossing point of curves in $\Gamma$. Additionally, we ensure that, for every curve $\gamma\in \Gamma$, if $\gamma\in \Gamma'$, then $\gamma\cap D(p)$ is a simple curve, and otherwise $\gamma\cap D(p)=\emptyset$. For every curve $\gamma\in \Gamma'$, let $q(\gamma)$ be the unique point of $\gamma$ lying on the boundary of the disc $D(p)$. Note that all points in $\set{q(\gamma)\mid \gamma\in \Gamma'}$ are distinct. Let ${\mathcal{O}}$ be the circular order in which these points are encountered when we traverse the boundary of $D(p)$. As before, this ordering naturally defines a circular ordering ${\mathcal{O}}'$ of the curves in $\Gamma'$. We then say that the curves of $\Gamma'$ \emph{enter the point $p$ in the order ${\mathcal{O}}'$}.
\subsection{Problem Definitions and Trivial Algorithms}
\label{subsec: prelim problem definitions}
In the \textsf{Minimum Crossing Number}\xspace~problem, the input is an $n$-vertex graph $G$, and the goal is to compute a drawing of $G$ in the plane with minimum number of crossings.
The value of the optimal solution, also called the \emph{crossing number} of $G$, is denoted by $\mathsf{OPT}_{\mathsf{cr}}(G)$.
We also consider a closely related problem called Minimum Crossing Number with Rotation System (\textnormal{\textsf{MCNwRS}}\xspace). In this problem, the input is a graph $G$, and a rotation system $\Sigma$ for $G$.
Given an instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, we say that a drawing $\phi$ of $G$ is a \emph{feasible solution} for $I$ if $\phi$ obeys the rotation system $\Sigma$. The \emph{cost} of the solution is the number of crossings in $\phi$. The goal in the \textnormal{\textsf{MCNwRS}}\xspace problem is to compute a feasible solution to the given input instance $I$ of smallest possible cost. We denote the cost of the optimal solution of the \textnormal{\textsf{MCNwRS}}\xspace instance $I$ by $\mathsf{OPT}_{\mathsf{cnwrs}}(I)$.
We use the following two simple theorems about the \textnormal{\textsf{MCNwRS}}\xspace problem, whose proofs are deferred to
Appendix~\ref{apd: Proof of crwrs_planar} and Appendix~\ref{apd: Proof of crwrs_uncrossing}, respectively.
\begin{theorem}
\label{thm: crwrs_planar}
There is an efficient algorithm, that, given an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, correctly determines whether $\mathsf{OPT}_{\mathsf{cnwrs}}(I)=0$, and, if so, computes a feasible solution to instance $I$ of cost $0$.
\end{theorem}
\begin{theorem}
\label{thm: crwrs_uncrossing}
There is an efficient algorithm, that given an instance $I=(G,\Sigma)$ of \textnormal{\textsf{MCNwRS}}\xspace, computes a feasible solution to $I$, of cost at most $|E(G)|^2$.
\end{theorem}
We refer to the solution computed by the algorithm from Theorem~\ref{thm: crwrs_uncrossing} as a \emph{trivial solution}.
We will also use the following lemma from \cite{chuzhoy2020towards}, that allows us to insert edges into a partial solution to \ensuremath{\mathsf{MCNwRS}}\xspace problem instance.
\begin{lemma}[Lemma 9.2 of \cite{chuzhoy2020towards}]
\label{lem: edge insertion}
There is an efficient algorithm, that, given an instance $I=(G,\Sigma)$ of the \textnormal{\textsf{MCNwRS}}\xspace problem, a subset $E'\subseteq E(G)$ of edges of $G$, and a drawing $\phi$ of graph $G\setminus E'$ that obeys $\Sigma$, computes a solution $\phi'$ to instance $I$, with $\mathsf{cr}(\phi')\le \mathsf{cr}(\phi)+|E'|\cdot |E(G)|$.
\end{lemma}
\subsection{A $\nu$-Decomposition of an Instance}
\label{subsec: subinstances}
A central tool that we use in our divide-and-conquer algorithm is a $\nu$-decomposition of instances.
\begin{definition}[$\nu$-Decomposition of Instances]
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$, and let $\nu\geq 1$ be a parameter. We say that a collection ${\mathcal{I}}$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace is a \emph{$\nu$-decomposition of $I$}, if the following hold:
\begin{properties}{D}
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\leq m\cdot (\log m)^{O(1)}$;\label{prop: few edges}
\item $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+m\textsf{right} )\cdot \nu$; and \label{prop: small solution cost}
\item there is an efficient algorithm $\ensuremath{\mathsf{Alg}}\xspace({\mathcal{I}})$, that, given, a feasible solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}$, computes a feasible solution $\phi$ to instance $I$, of cost $\mathsf{cr}(\phi)\leq O\textsf{left} (\sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I'))\textsf{right} )$. \label{prop: alg to put together}
\end{properties}
We say that a randomized algorithm $\ensuremath{\mathsf{Alg}}\xspace$ is a \emph{$\nu$-decomposition algorithm for a family ${\mathcal{I}}^*$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace} if $\ensuremath{\mathsf{Alg}}\xspace$ is an efficient algorithm, that, given an instance $I=(G,\Sigma)\in {\mathcal{I}}^*$, produces a collection ${\mathcal{I}}$ of instances that has properties \ref{prop: few edges} and \ref{prop: alg to put together}, and ensures the following additional property (that replaces Property \ref{prop: small solution cost}):
\begin{properties}[1]{D'}
\item $\expect{\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')}\le \textsf{left} (\mathsf{OPT}_{\mathsf{cnwrs}}(I)+|E(G)|\textsf{right} )\cdot \nu$.\label{prop: modified expectation}
\end{properties}
\end{definition}
In the following claim, whose proof appears in \Cref{apd: Proof of compose algs}, we show that algorithms for computing $\nu$-decompositions can be naturally composed together.
\begin{claim}\label{claim: compose algs}
Let $\ensuremath{\mathsf{Alg}}\xspace_1$ be a randomized $\nu'$-decomposition algorithm for some family ${\mathcal{I}}^*$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace. Assume that, given an instance $I\in {\mathcal{I}}^*$, algorithm $\ensuremath{\mathsf{Alg}}\xspace_1$ produces a collection ${\mathcal{I}}'$ of instances, all of which belong to some family ${\mathcal{I}}^{**}$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace. Let $\ensuremath{\mathsf{Alg}}\xspace_2$ be a randomized $\nu''$-decomposition algorithm for family ${\mathcal{I}}^{**}$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace. Lastly, let $\ensuremath{\mathsf{Alg}}\xspace$ be a randomized algorithm, that, given an instance $I\in {\mathcal{I}}^*$ of \ensuremath{\mathsf{MCNwRS}}\xspace, applies Algorithm $\ensuremath{\mathsf{Alg}}\xspace_1$ to $I$, to obtain a collection ${\mathcal{I}}'$ of instances, and then, for every instance $I'\in {\mathcal{I}}'$, applies Algorithm $\ensuremath{\mathsf{Alg}}\xspace_2$ to $I'$, obtaining a collection ${\mathcal{I}}''(I')$ of instances. The output of algorithm $\ensuremath{\mathsf{Alg}}\xspace$ is the collection ${\mathcal{I}}=\bigcup_{I'\in {\mathcal{I}}'}{\mathcal{I}}''(I')$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace. Then $\ensuremath{\mathsf{Alg}}\xspace$ is a randomized $\nu$-decomposition algorithm for family ${\mathcal{I}}^*$ of instances of $\ensuremath{\mathsf{MCNwRS}}\xspace$, for $\nu=\nu''\cdot\max\set{2\nu',(\log m)^{O(1)}}$, where $m$ is the number of edges in instance $I$.
\end{claim}
\subsection{Subinstances}
We use the following definition of subinstances.
\begin{definition}[Subinstances]\label{def: subinstance}
Let $I=(G,\Sigma)$ and $I'=(G',\Sigma')$ be two instances of \ensuremath{\mathsf{MCNwRS}}\xspace. We say that instance $I'$ is a \emph{subinstance} of instance $I$, if there is a subgraph $\tilde G\subseteq G$, and a collection $S_1,\ldots,S_r$ of mutually disjoint subsets of vertices of $\tilde G$, such that graph $G'$ can be obtained from $\tilde G$ by contracting, for all $1\leq i\leq r$, every vertex set $S_i$ into a supernode $u_i$; we keep parallel edges but remove self-loops\footnote{Note that this definition is similar to the definition of a minor, except that we do not require that the induced subgraphs $G[S_i]$ of $G$ are connected.}. We do not distinguish between the edges incident to the supernodes in graph $G'$ and their counterparts in graph $G$. For every vertex $v\in V(G')\cap V(G)$, its rotation ${\mathcal{O}}'_v$ in $\Sigma'$ must be consistent with the rotation ${\mathcal{O}}_v\in \Sigma$, while for every supernode $u_i$, its rotation ${\mathcal{O}}'_{u_i}$ in $\Sigma'$ can be defined arbitrarily.
\end{definition}
Observe that, if instance $I'=(G',\Sigma')$ is a subinstance of $I=(G,\Sigma)$, then $|E(G')|\le |E(G)|$.
Also notice that the subinstance relation is transitive: if instance $I_1$ is a subinstance of instance $I_0$, and instance $I_2$ is a subinstance of $I_1$, then $I_2$ is a subinstance of $I_0$.
\subsubsection*{Case 1: ${\mathcal{P}}$ was a type-1 promising path set}
\paragraph{Computing the Split.}
Note that in order to complete the enhancement structure ${\mathcal{A}}$, we need to define a drawing $\rho'$ of the graph $K'=K\cup P$ with no crossings, so that $\rho'$ is consistent with the rotation system $\Sigma$ and the orientations $\set{b_u}_{u\in V(K')}$, which must be a clean drawing with respect to ${\mathcal K}$. If we consider any such drawing $\rho'$, then the image of the cycle $P$ in $\rho'$ partitions the sphere into two discs, that we denote again by $D$ and $D'$. For every core structure ${\mathcal{J}}_i\in {\mathcal K}$, the image of $J_i$ in $\rho'$ is essentially fixed -- it must be drawn inside disc $D_i$, and the drawing must be identical to $\rho_i$. But it is possible that $J_i$ is drawn either inside disc $D$ or inside disc $D'$. Once we determine, for all $1\leq i\leq r$, whether $J_i$ is drawn inside $D$ or inside $D'$, drawing $\rho'$ of $K'$ becomes fixed.
We now describe a procedure that cuts the graph $G'$ into two subgraphs, one to be drawn inside $D$, and another to be drawn inside $D'$. This partition of $G'$ will also determine, for all $1\leq i\leq r$, whether $J_i$ is drawn inside $D$ or inside $D'$. Therefore, this procedure will allow us to both complete the computation of the enhancement structure ${\mathcal{A}}$ (by finalizing the drawing $\rho'$ of $K'$), and to compute the split of instance $I$ along ${\mathcal{A}}$. The procedure itself is very simple conceptually: we simply compute a minimum cut in $G'$ separating the edges of $\tilde E'_1$ from edges of $\tilde E'_2$. In order to ensure that the cores are not separated by this cut, we contract each core into a single vertex before computing the cut. We now describe this cutting procedure.
We construct a flow network $H$ as follows. We start with $H=G'$, and then, for all $1\leq i\leq r$, we contract all vertices of the core $J_i$ into a supernode $v_{J_i}$. Next, we subdivide every edge $e\in \tilde E$ with a vertex $t_e$, and denote $T_1=\set{t_e\mid e\in \tilde E_1'}$, $T_2=\set{t_e\mid e\in \tilde E_2'}$. We delete all vertices of $P$ and their adjacent edges from the resulting graph, contract all vertices of $T_1$ into a source vertex $s$, and contract all vertices of $T_2$ into a destination vertex $t$. We then compute a minimum $s$-$t$ cut $(A,B)$ in the resulting flow network $H$, and we denote by $E''=E_H(A,B)$. Eventually, we will delete the edges of $E''$ from $G'$, so these edges will not belong to any of the two instances that we construct. The following claim bounds the cardinality of $E''$. The proof is deferred to Section \ref{subsec: small cut set in case 1} of Appendix.
\begin{claim}\label{claim: cut set small}
If Events ${\cal{E}}_1$ and ${\cal{E}}_3$ did not happen, then $|E''|\leq \frac{64\mu^{700}\mathsf{cr}(\phi)}{m}$.
\end{claim}
We are now ready to complete the construction of the enhancement structure ${\mathcal{A}}$. We have already defined a type-1 enhancement $\Pi$, and an orientation $b_u$ for every vertex $u\in (V(P^*_1)\cup V(P^*_2))\setminus V(K)$. It now remains to define a drawing $\rho'$ of the graph $K'$. In order to define the drawing $\rho'$, we start by drawing the cycle $P^*_1$ on the plane in the natural way without any crossings, and we fix the orientations of the vertices $u\in P^*_1$ to be consistent with the orientations $b_u$ that we have computed for them.
We denote by $F$ the unique face in this drawing whose boundary is the image of $P^*_1$ and by $F'$ the infinite face. We can view this current drawing of $P^*_1$ as the drawing induced by $\phi'$ (since both drawings are identical). Note that in the drawing $\phi'$ of $G'$, either for every edge $e\in \tilde E_1$, $\sigma(e)\subseteq F$, or for every edge $e\in \tilde E_2$, $\sigma(e)\subseteq F$.
We can ensure that it is the former (by flipping the image of the path $P^*_1$, thereby reversing the orientations of all its vertices, if necessary).
For all $1\leq i\leq r$, if $v_{J_i}\in A$, then we place the disc $D_i$ inside face $F$, and otherwise we place it inside face $F'$. The discs are placed so that all of them remain disjoint from each other and from the image of $P^*_1$. For all $1\leq i\leq r$, we then plant the image $\rho_i$ of the core $J_i$ given by the core structure ${\mathcal{J}}_i$ inside the disc $D_i$.
It is immediate to verify that $\rho'$ is a drawing of graph $K'$ with no crossings, and that it is a clean drawing of $K'$ with respect to ${\mathcal K}$. This completes the construction of the enhancement structure ${\mathcal{A}}$. We let $({\mathcal K}_1,{\mathcal K}_2)$ denote the split of ${\mathcal K}$ along the enhancement structure ${\mathcal{A}}$, where ${\mathcal K}_1$ contains all core structures ${\mathcal{J}}_i$ with $v_{J_i}\in A$.
Next, we define a split $(I_1,I_2)$ of instance $I$ along the enhacement structure ${\mathcal{A}}$. In order to do so, we define two vertex sets $A',B'$ in graph $G'$, as follows. We start with $A'=A\setminus\set{s}$ and $B'=B\setminus\set{t}$. For every index $1\leq i\leq r$, if $v_{J_i}\in A$, then we replace $v_{J_i}$ with vertex set $V(J_i)$ in $A'$, and otherwise we replace $v_{J_i}$ with vertex set $V(J_i)$ in $B'$. We then let $G_1=G'[A'\cup V(P)]$ and $G_2=G'[B'\cup V(P)]$. The rotation system $\Sigma_1$ for graph $G_1$ and the rotation system $\Sigma_2$ for graph $G_2$ are induced by $\Sigma$.
Let $I_1=(G_1,\Sigma_1)$ and $I_2=(G_2,\Sigma_2)$ be the resulting two instances of \ensuremath{\mathsf{MCNwRS}}\xspace. We now prove that $(I_1,I_2)$ is a split of $I$ along ${\mathcal{A}}$.
Notice that $V(G_1)\cap V(G_2)=V(P)$ and $E(G_1)\cap E(G_2)=E(P)$. Additionally, $E(G_1)\cup E(G_2)\subseteq E(G)$. Let $E^{\mathsf{del}}=E(G)\setminus (E(G_1)\cup E(G_2))$. Then $E^{\mathsf{del}}=E'\cup E''$. From Claims \ref{claim: third bad event bound} and
\ref{claim: cut set small}, if Event ${\cal{E}}$ did not happen, then $|E'|+|E''|\leq 2\cdot\mathsf{cr}(\phi)\cdot \mu^{2150}/m$.
Recall that skeleton structure ${\mathcal K}_1$ contains, for every supernode $v_{J_i}\in A$, the corresponding core structure ${\mathcal{J}}_i$ of graph $G$. Since, from our construction, $J_i\subseteq G_1$, it is also a valid core structure for instance $I_1$. Additionally, ${\mathcal K}_1$ contains a core structure ${\mathcal{J}}$, whose associated core $J$ is the cycle $P$. The drawing $\rho_{J}$ associated with $J$ is the natural drawing described above, with face $F$ serving as an infinite face of the drawing. It is easy to verify that ${\mathcal{J}}$ is a valid core structure for instance $I_1$.
Similarly, skeleton stucture ${\mathcal K}_2$ contains, for every supernode $v_{J_i}\in B$, the corresponding core structure ${\mathcal{J}}_i$ of graph $G$. Since, from our construction, $J_i\subseteq G_2$, it is also a valid core structure for instance $I_2$. Additionally, ${\mathcal K}_2$ contains a core structure ${\mathcal{J}}'$, whose associated core $J'$ is the cycle $P$, and whose associated drawing $\rho_{J'}$ is the natural drawing of $P$ described above, with face $F'$ serving as an infinite face of the drawing. As before, ${\mathcal{J}}'$ is a valid core structure for instance $I_2$.
Therefore, ${\mathcal K}_1$ is a valid skeleton structure for $I_1$ and ${\mathcal K}_2$ is a valid skeleton structure for $I_2$. We conclude that $(I_1,I_2)$ is a valid split of instance $I$ along the enhancement structure ${\mathcal{A}}$.
In the following two observations we show that, if the bad event ${\cal{E}}$ did not happen, then the enhancement structure ${\mathcal{A}}$, together with the split $(I_1,I_2)$ of $I$ along ${\mathcal{A}}$, is a valid output for \ensuremath{\mathsf{ProcSplit}}\xspace.
The proofs are deferred to Sections \ref{subsec: few edges in split Case 1} and \ref{subsec: getting semi-clean solution for case 1} of Appendix, respectively.
\begin{observation}\label{obs: few edges in split graphs}
If Event ${\cal{E}}$ did not happen, then $|E(G_1)|,|E(G_2)|\leq m-\frac{m}{100\mu^{50}}$.
\end{observation}
\begin{observation}\label{obs: semi-clean case 1}
If bad event ${\cal{E}}$ does not happen, then
there is a semi-clean solution $\phi_1$ for instance $I_1$ with respect to ${\mathcal K}_1$, and a semi-clean solution $\phi_2$ for instance $I_2$ with respect to ${\mathcal K}_2$, such that $|\chi^*(\phi_1)|+|\chi^*(\phi_2)|\leq |\chi^*(\phi)|$, and $|\chi^{\mathsf{dirty}}(\phi_1)|+ |\chi^{\mathsf{dirty}}(\phi_2)|\leq |\chi^{\mathsf{dirty}}(\phi)|+\frac{64\mu^{700}\mathsf{cr}(\phi)}{m}$.
\end{observation}
Recall that we have already showed that, if Event ${\cal{E}}$ did not happen, then $|E(G)\setminus (E(G_1)\cup E(G_2))|=|E^{\mathsf{del}}|\leq 2\cdot\mathsf{cr}(\phi)\cdot \mu^{2150}/m< \mathsf{cr}(\phi)\cdot \mu^{2200}/m+|\chi^{\mathsf{dirty}}(\phi)|$. Altogether, if Event ${\cal{E}}$ did not happen, then the algorithm produces a valid output for \ensuremath{\mathsf{ProcSplit}}\xspace. Since the probability of Event ${\cal{E}}$ is at most $1/\mu^{399}$, this completes the proof of \Cref{thm: procsplit} for Case 1.
\subsubsection*{Case 2: ${\mathcal{P}}$ was a type-2 promising path set}
\paragraph{Computing the Split.}
The algorithm for computing the drawing $\tilde \rho$ of the graph $K'=K\cup P$ and of the split $(I_1,I_2)$ of the instance $I$ is almost identical to that in Case 1. The slight difference is that in this case, the endpoints of $P$ belong to the core $J$.
We construct a flow network $H$ as follows. We start with $H=G'$, and then, for all $1\leq i\leq r$ with $i\neq i^*$, we contract all vertices of the core $J_i$ into a supernode $v_{J_i}$. Next, we subdivide every edge $e\in \tilde E$ with a vertex $t_e$, and denote $T_1=\set{t_e\mid e\in \tilde E_1'}$, $T_2=\set{t_e\mid e\in \tilde E_2'}$. We delete all vertices of $P\cup J$ and their adjacent edges from the resulting graph, contract all vertices of $T_1$ into a source vertex $s$, and contract all vertices of $T_2$ into a destination vertex $t$. We then compute a minimum $s$-$t$ cut $(A,B)$ in the resulting flow network $H$, and we denote by $E''=E_H(A,B)$. The following claim, that is an analogue of \Cref{claim: cut set small} for Case 2, bounds the cardinality of $E''$. The proof is almost identical to that of \Cref{claim: cut set small}. We provide its sketch in Section \ref{subsec: small cut set in case 2} of Appendix.
\begin{claim}\label{claim: cut set small case2}
If Events ${\cal{E}}$ did not happen, then $|E''|\leq \frac{64\mu^{700}\mathsf{cr}(\phi)}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
\end{claim}
In order to complete the construction of the enhancement structure ${\mathcal{A}}$, we now only need to define a drawing $\rho'$ of the graph $K'$. In order to define the drawing $\rho'$, we start
with the unique planar drawing $\tilde \rho$ of graph $J\cup P$, in which the orientation of every vertex $u\in V(J\cup P)$ is $b_u$, and the drawing of $J$ induced by $\tilde \rho$ is $\rho_{J}$.
Recall that this drawing is precisely the drawing of $J\cup P$ induced by $\phi'$, though its orientation may be different. The faces $F_1$ and $F_2$ in drawing $\tilde \rho$ are well defined and can be computed efficiently.
For all $1\leq i\leq r$ with $i\neq i^*$, if $v_{J_i}\in A$, then we place the disc $D_i$ with the drawing $\rho_{J_i}$ of graph $J_i$ inside face $F_1$, and otherwise we place it inside face $F_2$. The discs are placed so that all of them remain disjoint from each other and from the image of $J\cup P$. It is immediate to verify that $\rho'$ is a drawing of graph $K'$ with no crossings, and that it is a clean drawing of $K'$ with respect to ${\mathcal K}$. This completes the construction of the enhancement structure ${\mathcal{A}}$. We let $({\mathcal K}_1,{\mathcal K}_2)$ denote the split of ${\mathcal K}$ along the enhancement structure ${\mathcal{A}}$, where ${\mathcal K}_1$ contains all core structures ${\mathcal{J}}_i$ with $v_{J_i}\in A$.
Next, we define a split $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ of instance $I$ along the enhacement structure ${\mathcal{A}}$. In order to do so, we define two vertex sets $A',B'$ in graph $G'$, as before. We start with $A'=A\setminus\set{s}$ and $B'=B\setminus\set{t}$. For every index $1\leq i\leq r$ with $i\neq i^*$, if $v_{J_i}\in A$, then we replace $v_{J_i}$ with vertex set $V(J_i)$ in $A'$, and otherwise we replace $v_{J_i}$ with vertex set $V(J_i)$ in $B'$.
Additionally, we let $J\subseteq J\cup P$ be the graph containing all vertices and edges of $J\cup P$, whose images in $\tilde \rho$ lie on the boundary of $F$, and we let $\rho_J$ denote the drawing of $J$ induced by $\tilde \rho$, where we view $F_1$ as the infinite face of the drawing. Similarly, we let $J'\subseteq J\cup P$ be the graph containing all vertices and edges of $J\cup P$, whose images in $\tilde \rho$ lie on the boundary of $F_2$, and we let $\rho_{J'}$ denote the drawing of $J'$ induced by $\tilde \rho$, where we view $F_2$ as the infinite face of the drawing. Note that $V(J')\cup V(J'')=V(J)$, and edge sets $E(J'),E(J'')$ define a partition of $E(J)$.
We then let $G_1$ be the subgraph of $G'$, whose vertex set is $V(A')\cup V(J)$, and edge set contains all edges of $E_{G'}(A')$, $E_{G'}(A',V(J))$, and all edges of $J$. Similarly, we let $G_2$ be the subgraph of $G'$, whose vertex set is $V(B')\cup V(J')$, and edge set contains all edges of $E_{G'}(B')$, $E_{G'}(B',V(J'))$, and all edges of $J'$.
The rotation system $\Sigma_1$ for graph $G_1$ and the rotation system $\Sigma_2$ for graph $G_2$ are induced by $\Sigma$.
Let $I_1=(G_1,\Sigma_1)$ and $I_2=(G_2,\Sigma_2)$ be the resulting two instances of \ensuremath{\mathsf{MCNwRS}}\xspace. We now prove that $(I_1,I_2)$ is a split of $I$ along ${\mathcal{A}}$.
Notice that $V(G_1)\cap V(G_2)\subseteq V(P\cup J)$ and $E(G_1)\cap E(G_2)\subseteq E(P\cup J)$. Additionally, $E(G_1)\cup E(G_2)\subseteq E(G)$. Let $E^{\mathsf{del}}=E(G)\setminus (E(G_1)\cup E(G_2))$. Then $E^{\mathsf{del}}=E'\cup E''$. From Claims \ref{claim: third bad event bound} and
\ref{claim: cut set small case2}, if Event ${\cal{E}}$ did not happen, then $|E'|+|E''|\leq \frac{2\mathsf{cr}(\phi)\cdot \mu^{2150}}m +|\chi^{\mathsf{dirty}}(\phi)|$.
Recall that skeleton structure ${\mathcal K}_1$ contains, for every supernode $v_{J_i}\in A$, the corresponding core structure ${\mathcal{J}}_i$ of graph $G$. Since, from our construction, $J_i\subseteq G_1$, it is also a valid core structure for instance $I_1$. Additionally, ${\mathcal K}_1$ contains a core structure ${\mathcal{J}}$, whose associated core is $J$. It is easy to verify that ${\mathcal{J}}$ is a valid core structure for instance $I_1$. Therefore, ${\mathcal K}_1$ is a valid skeleton structure for $I_1$.
From similar arguments, ${\mathcal K}_2$ is a valid skeleton structure for instance $I_2$. We conclude that $(I_1,I_2)$ is a valid split of instance $I$ along the enhancement structure ${\mathcal{A}}$. Notice that $|{\mathcal K}_1|,|{\mathcal K}_2|\leq r$ must hold (this is unlike Case 1, where it is possible that one of the skeleton structures contains $r+1$ core structures).
In the following two observations we show that, if the bad event ${\cal{E}}$ did not happen, then the enhancement structure ${\mathcal{A}}$, together with the split $(I_1,I_2)$ of $I$ along ${\mathcal{A}}$, is a valid output for \ensuremath{\mathsf{ProcSplit}}\xspace.
The first observation is an analogue of \Cref{obs: few edges in split graphs} for Case 2. Since the parameters for the two cases are slightly different, we
provide a proof in Section \ref{subsec: few edges in split Case 2} of appendix. The second observation is an analogue of \Cref{obs: semi-clean case 1} for Case 2. Its proof is almost identical to that of \Cref{obs: semi-clean case 1}. We provide a proof sketch in Section \ref{subsec: getting semi-clean solution for case 2} of Appendix.
\begin{observation}\label{obs: few edges in split graphs case2}
If Event ${\cal{E}}$ did not happen, then $|E(G_1)|,|E(G_2)|\leq m-\frac{m}{32\mu^{300}}$.
\end{observation}
\begin{observation}\label{obs: semi-clean case 2}
If bad event ${\cal{E}}$ does not happen, then
there is a semi-clean solution $\phi_1$ for instance $I_1$ with respect to ${\mathcal K}_1$, and a semi-clean solution $\phi_2$ for instance $I_2$ with respect to ${\mathcal K}_2$, such that $|\chi^*(\phi_1)|+|\chi^*(\phi_2)|\leq |\chi^*(\phi)|$, and $|\chi^{\mathsf{dirty}}(\phi_1)|+ |\chi^{\mathsf{dirty}}(\phi_2)|\leq |\chi^{\mathsf{dirty}}(\phi)|+\frac{64\mu^{700}\mathsf{cr}(\phi)}{m}$.
\end{observation}
Recall that we have already showed that, if Event ${\cal{E}}$ did not happen, then $|E(G)\setminus (E(G_1)\cup E(G_2))|=|E^{\mathsf{del}}|\leq 2\cdot\mathsf{cr}(\phi)\cdot \frac{\mu^{2150}}m+|\chi^{\mathsf{dirty}}(\phi)| < \mathsf{cr}(\phi)\cdot \mu^{2200}/m+|\chi^{\mathsf{dirty}}(\phi)|$. Altogether, if Event ${\cal{E}}$ did not happen, then the algorithm produces a valid output for \ensuremath{\mathsf{ProcSplit}}\xspace. Since the probability of Event ${\cal{E}}$ is at most $1/\mu^{399}$, this completes the proof of \Cref{thm: procsplit} for Case 2.
\subsubsection*{Case 3: ${\mathcal{P}}$ was a type-3 promising path set}
In the third case, ${\mathcal{P}}$ was a type-3 promising path set. We let $1\leq i^*<i^{**}\leq r$ be the indices for which all paths in ${\mathcal{P}}$ originate at vertices of $J$ and terminate at vertices of $J_{i^{**}}$. We denote $E_1=\delta_G(J)$ and $E_2=\delta_G(J)$.
If the two paths chosen from ${\mathcal{P}}^*$ share at least one vertex of $V(G)\setminus V(K)$, then we have constructed a type-2 enhancement whose both endpoints lie in $J$. In this case, the algorithm and its analysis are practically identical to those from Case 2 and are omitted here. We now assume that the two chosen paths are disjoint from each other. For convenience of notation, we denote $P=P^*_1$ and $P'=P^*_2$. Throughout, we denote by $\tilde J$ the graph obtained from the union of $J,J_{i^{**}},P$ and $P'$.
Recall that, from \Cref{claim: new drawing}, if neither of the events ${\cal{E}}_1,{\cal{E}}_3$ happened, drawing $\phi'$ contains no crossings between the edges of $E(K')$.
Throughout, we denote by $\gamma$ the image of path $P$ and by $\gamma'$ the image of path $P'$ in $\phi'$.
We recall how the paths $P$ and $P'$ were constructed.
We started with a set ${\mathcal{P}}^*$ of $k'=\ceil{\frac{m}{\mu^{300}}}$ simple edge-disjoint paths in graph $G$, with every path originating at a vertex of $J_{i^*}$ and terminating at a vertex of $J_{i^{**}}$. Additionally, the paths in ${\mathcal{P}}^*$ are internally disjoint from $K$, and they are non-transversal with respect to $\Sigma$. We denoted by $E^*_1\subseteq \delta(J_{i^{*}})$ the subset of edges that belong to the paths of ${\mathcal{P}}^*$, letting $E^*_1=\set{e_1,\ldots,e_{k'}}$, where the edges are indexed in the order of their appearence in the ordering ${\mathcal{O}}(J_{i^{*}})$. For all $1\leq j\leq k'$, we denoted by $P_j\in {\mathcal{P}}^*$ the unique path originating at the edge $e_j\in E^*_1$. We then selected an index $\floor{k'/8}<j^*<\ceil{k'/4}$ uniformly at random, that determined the chosen paths $P_{j^*}$ and $P_{j^*+\floor{k'/2}}$. Since we assumed that the chosen paths do not share any vertices of $V(G)\setminus V(K)$, we eventually set $P=P_{j^*}$ and $P'=P_{j^*+\floor{k'/2}}$.
Let $\tilde \rho$ be the drawing of graph $\tilde J$ induced by $\phi'$. This drawing has no crossings, and it has two faces, that we denote by $F_1$ and $F_2$, respectively, whose boundaries contain both curves $\gamma$ and $\gamma'$. The drawing of $J$ induced by $\tilde \rho$ is $\rho_{J}$, and the drawing of $J_{i^{**}}$ induced by $\tilde \rho$ is $\rho_{J_{i^{**}}}$, which are known to our algorithm. However, in order to construct the drawing $\tilde \rho$ of graph $\tilde J$ efficiently, we also need to know the orientations of these two drawings with respect to each other in $\tilde \rho$.
In order to overcome this difficulty, we choose another index $j^*< j^{**}<j^*+\floor{k'/2}$ uniformly at random, and we let $P''=P_{j^{**}}$. We say that a bad event ${\cal{E}}_5$ happens if $P''$ is a bad path, or some edge $e\in E(P'')$ crosses the image of an edge of $P\cup P'$ in the drawing $\phi$.
From \Cref{obs: number of bad paths}, the number of bad paths in ${\mathcal{P}}^*$ is at most $\frac{m}{8\mu^{700}}$. Moreover, if bad event ${\cal{E}}$ did not happen, then paths $P$ and $P'$ are both good paths, and so the number of crossings in which the edges of $E(P)\cup E(P')$ participate is at most $\frac{32\mu^{700}\cdot \mathsf{cr}(\phi)}{m}\leq \frac{m}{8\mu^{700}}$ (from Property \ref{prop valid input drawing} of valid input to Procedure \ensuremath{\mathsf{ProcSplit}}\xspace). Therefore, if Event ${\cal{E}}$ did not happen, there are at most $\frac{m}{8\mu^{700}}$ indices $1\leq i\leq k$, such that path $P_i$ is bad, or some edge of $P_i$ crosses an edge of $E(P)\cup E(P')$ in $\phi$.
Since $k'=\ceil{\frac{m}{\mu^{300}}}$, $\prob{{\cal{E}}_5\mid \neg{\cal{E}}}\leq \frac{1}{\mu^{400}}$.
Consider now the graph $\tilde J'=\tilde J\cup P''=J_{i^*}\cup J_{i^{**}}\cup P\cup P'\cup P''$, and let $\tilde \rho'$ be the drawing of $\tilde J'$ induced by $\phi'$. If neither of the events ${\cal{E}},{\cal{E}}_5$ happened, then $\tilde \rho'$ is a unique planar drawing of graph $\tilde J'$, such that the drawing of $J$ induced by $\tilde \rho'$ is $\rho_{J}$, and the drawing of $J_{i^{**}}$ induced by $\tilde \rho$ is $\rho_{J_{i^{**}}}$. Therefore, we can compute the drawing $\tilde \rho'$ of graph $\tilde J'$ efficiently. The drawing $\tilde \rho$ of graph $\tilde J$ is then identical to the drawing of $\tilde J$ induced by $\tilde \rho'$ (if bad events ${\cal{E}},{\cal{E}}_5$ did not happen). Therefore, if bad events ${\cal{E}},{\cal{E}}_5$ did not happen, then we can compute the drawing $\tilde \rho$ of $\tilde J$ efficiently, though we do not know its orientation in $\phi'$.
As in previous two cases, for every vertex $u\in V(\tilde J)$, we consider the tiny $u$-disc $D_{\phi'}(u)$. For every edge $e\in \delta_{G'}(u)$, we denote by $\sigma(e)$ the segment of $\phi'(e)$ that is drawn inside the disc $D_{\phi'}(u)$. Let $\tilde E=\textsf{left} (\bigcup_{u\in V(\tilde J)}\delta_{G'}(u)\textsf{right} )\setminus E(\tilde J)$. We partition edge set $\tilde E$ into a set $\tilde E^{\mathsf{in}}$ of \emph{inner edges} and the set $\tilde E^{\mathsf{out}}$ of outer edges exactly as before: Edge set $\tilde E^{\mathsf{in}}$ contains all edges $e\in \tilde E$ with $\sigma(e)\subseteq F_1$, and $\tilde E^{\mathsf{out}}$ contains all remaining edges. Let $e_1$ be an arbitrary fixed edge of $E_1$ that does not lie on $P$ or $P'$. As before, we assume without loss of generality that $e_1\in \tilde E^{\mathsf{in}}$.
The algorithm for computing the orientation of the inner vertices of $P$ and $P'$ is practically identical to the one used in Case 2, with a minor difference in the analysis that we highlight below.
We define an orientation of the ordering ${\mathcal{O}}(J)$ in $\phi'$ exactly as in Case 2. As in Case 1, we can now efficiently determine, for every edge $e\in E_1\cup E_2$, whether it belongs to $E^{\mathsf{in}}$ or $E^{\mathsf{out}}$. In the former case, we add the edge to set $\tilde E_1'$, and in the latter case, we add it to $\tilde E_2'$.
The algorithm for computing the orientation of every inner vertex $u$ of $P$ and $P'$ is identical to that used in Case 2.
The proof that the orientations of the vertices are computed correctly with respect to $\phi'$ provided that neither of the events ${\cal{E}},{\cal{E}}_5$ happenned (that is, an analogue of \Cref{claim: orientations and edge split is computed correctly Case2}) is also almost identical. The only difference is that now the images of the paths in sets ${\mathcal{P}}_1(u),{\mathcal{P}}_2(u),{\mathcal{P}}'_1(u),{\mathcal{P}}'_2(u)$ may cross the images of the edges of $J_{i^{**}}$. However, the total number of such paths, whose images cross the edges of $E(J)\cup E(J_{i^{**}})$, remains bounded by $N^{\operatorname{bad}}(u)$, so the same calculations work in Case 3 as well. We note however that the orientation of the vertices $u\in V(J_{i^{**}})$ are computed differently, and they are determined by the orientation of the drawing $\rho_{J_{i^{**}}}$ of $J_{i^{**}}$ induced by $\tilde \rho$: we either keep the orientations of all such vertices unchanged from ${\mathcal{J}}_{i^{**}}$, or we may flip all of them, depending on the orientation of drawing $\rho_{J_{i^{**}}}$ in $\tilde \rho$.
\paragraph{Computing the Split.}
The algorithm for computing the drawing $\tilde \rho$ of the graph $K'=K\cup P$ and of the split $(I_1,I_2)$ of the instance $I$ is almost identical to that in Cases 1 and 2. We provide a sketch below, highlighting the differences.
We construct a flow network $H$ as follows. We start with $H=G'$, and then, for all $1\leq i\leq r$ with $i\not\in\set{i^*,i^{**}}$, we contract all vertices of the core $J_i$ into a supernode $v_{J_i}$. Next, we subdivide every edge $e\in \tilde E$ with a vertex $t_e$, and denote $T_1=\set{t_e\mid e\in \tilde E_1'}$, $T_2=\set{t_e\mid e\in \tilde E_2'}$. We delete all vertices of $\tilde J$ and their adjacent edges from the resulting graph, contract all vertices of $T_1$ into a source vertex $s$, and contract all vertices of $T_2$ into a destination vertex $t$. We then compute a minimum $s$-$t$ cut $(A,B)$ in the resulting flow network $H$, and we denote by $E''=E_H(A,B)$. The following claim, that is an analogue of \Cref{claim: cut set small case2} for Case 2. The proof is virtually identical and is omitted here.
\begin{claim}\label{claim: cut set small case3}
If neither of the events ${\cal{E}},{\cal{E}}_5$ happen, then $|E''|\leq \frac{64\mu^{700}\mathsf{cr}(\phi)}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
\end{claim}
The drawing $\rho'$ of the graph $K'$ is constructed exactly as before: we start with the drawing $\tilde \rho$ of $\tilde J$, and then plant the discs $D_i$ corresponding to the cores $J_i\in {\mathcal K}\setminus\set{J,J_{i^{**}}}$ inside face $F$ or inside face $F'$, depending on whether the corresponding supernode $v_{J_i}$ lies in $A$ or $B$.
Consider now the split $({\mathcal K}_1,{\mathcal K}_2)$ of ${\mathcal K}$ along the resulting enhancement structure ${\mathcal{A}}$.
Then ${\mathcal K}_1$ contains all core structures ${\mathcal{J}}_i$ with $v_{J_i}\in A$, while ${\mathcal K}_2$ contains all core structures ${\mathcal{J}}_i$ with $v_{J_i}\in B$. Additionally, ${\mathcal K}_1$ contains a core structure ${\mathcal{J}}$, whose corresponding core $J$ contains all vertices and edges of $\tilde J$ that lie on the boundary of face $F$ in drawing $\tilde \rho$. The drawing $\rho_J$ associated with $J$ is its induced drawing in $\tilde \rho$, with face $F$ serving as the infinite face. Similarly, ${\mathcal K}_2$ contains a core structure ${\mathcal{J}}'$, whose corresponding core $J'$ contains all vertices and edges of $\tilde J$ that lie on the boundary of face $F'$ in drawing $\tilde \rho$. The drawing $\rho_{J'}$ associated with $J'$ is its induced drawing in $\tilde \rho$, with face $F'$ serving as the infinite face. Notice that $V(J)\cup V(J')=V(\tilde J)$, and $E(J)\cup E(J')=E(\tilde J)$.
The split $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ of instance $I$ along the enhacement structure ${\mathcal{A}}$ is defined exactly as before. We start with $A'=A\setminus\set{s}$ and $B'=B\setminus\set{t}$. For every core structure ${\mathcal{J}}_i\in{\mathcal K}\setminus\set{J,J_{i^{**}}}$, if $v_{J_i}\in A$, then we replace $v_{J_i}$ with vertex set $V(J_i)$ in $A'$, and otherwise we replace $v_{J_i}$ with vertex set $V(J_i)$ in $B'$.
We then let $G_1$ be the subgraph of $G'$, whose vertex set is $V(A')\cup V(J)$, and edge set contains all edges of $E_{G'}(A')$, $E_{G'}(A',V(J))$, and all edges of $J$. Similarly, we let $G_2$ be the subgraph of $G'$, whose vertex set is $V(B')\cup V(J')$, and edge set contains all edges of $E_{G'}(B')$, $E_{G'}(B',V(J'))$, and all edges of $J'$.
The rotation system $\Sigma_1$ for graph $G_1$ and the rotation system $\Sigma_2$ for graph $G_2$ are induced by $\Sigma$.
Let $I_1=(G_1,\Sigma_1)$ and $I_2=(G_2,\Sigma_2)$ be the resulting two instances of \ensuremath{\mathsf{MCNwRS}}\xspace. The proof that $(I_1,I_2)$ is a split of $I$ along ${\mathcal{A}}$ is the same as in Case 2, and is omitted here.
We denote $E^{\mathsf{del}}=E(G)\setminus (E(G_1)\cup E(G_2))$. As before, $E^{\mathsf{del}}=E'\cup E''$, and, if Event ${\cal{E}}$ did not happen, then $|E^{\mathsf{del}}|\leq \frac{2\mathsf{cr}(\phi)\cdot \mu^{2150}}m +|\chi^{\mathsf{dirty}}(\phi)| < \frac{\mathsf{cr}(\phi)\cdot \mu^{2200}}{m}+|\chi^{\mathsf{dirty}}(\phi)|$.
As in Case 2, it is easy to verify that $|{\mathcal K}_1|,|{\mathcal K}_2|\leq r$ must hold.
The following observation is an analogue of \Cref{obs: few edges in split graphs case2} for Case 3. The proof is virtually identical (with a very slight change in parameters) and is omitted here.
\begin{observation}\label{obs: few edges in split graphs case3}
If Event ${\cal{E}}$ did not happen, then $|E(G_1)|,|E(G_2)|\leq m-\frac{m}{32\mu^{300}}$.
\end{observation}
The next observation is an analogue of \Cref{obs: semi-clean case 2} for Case 3. Its proof is almost identical to that of \Cref{obs: semi-clean case 2}. We provide a proof sketch in Section \ref{subsec: getting semi-clean solution for case 3} of Appendix.
\begin{observation}\label{obs: semi-clean case 3}
If neither of the bad events ${\cal{E}},{\cal{E}}_5$ happen, then
there is a semi-clean solution $\phi_1$ for instance $I_1$ with respect to ${\mathcal K}_1$, and a semi-clean solution $\phi_2$ for instance $I_2$ with respect to ${\mathcal K}_2$, such that $|\chi^*(\phi_1)|+|\chi^*(\phi_2)|\leq |\chi^*(\phi)|$, and $|\chi^{\mathsf{dirty}}(\phi_1)|+ |\chi^{\mathsf{dirty}}(\phi_2)|\leq |\chi^{\mathsf{dirty}}(\phi)|+\frac{64\mu^{700}\mathsf{cr}(\phi)}{m}$.
\end{observation}
Altogether, if neither of the events ${\cal{E}}$, ${\cal{E}}_5$ happens, then the algorithm produces a valid output for \ensuremath{\mathsf{ProcSplit}}\xspace. Since $\prob{{\cal{E}}}\leq \frac{1}{2\mu^{399}}$, while $\prob{{\cal{E}}_5\mid \neg{\cal{E}}}\leq \frac{1}{\mu^{400}}$, this completes the proof of \Cref{thm: procsplit}.
\section{From Sec 3}
\begin{definition}[High-degree vertex]
Let $G$ be any graph. A vertex $v\in V(G)$ is a \emph{high-degree} vertex, if $\deg_G(v)\geq |E(G)|/\mu^4$.
\end{definition}
\begin{definition}[Wide and Narrow Instances]
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$. We say that $I$ is a \emph{wide} instance, iff there is a high-degree vertex $v\in V(G)$, a partition $(E_1,E_2)$ of the edges of $\delta_G(v)$, such that the edges of $E_1$ appear consecutively in the rotation ${\mathcal{O}}_v\in \Sigma$, and so do the edges of $E_2$, and there is a collection ${\mathcal{P}}$ of at least $\floor{m/\mu^{{50}}}$ simple edge-disjoint simple cycles in $G$, such that every cycle $P\in {\mathcal{P}}$ contains one edge of $E_1$ and one edge of $E_2$.
If no such cycle set ${\mathcal{P}}$ exists in $G$, then we say that $I$ is a \emph{narrow} instance.
\end{definition}
\begin{definition}[Well-Connected Wide Instances]
Let $I=(G,\Sigma)$ be a wide instance of \ensuremath{\mathsf{MCNwRS}}\xspace with $|E(G)|=m$. We say that instance $I$ is \emph{well-connected} iff for every pair $u,v$ of distinct vertices of $G$ with $\deg_G(v),\deg_G(u)\geq m/\mu^5$, there is a collection of at least $\frac{8m}{\mu^b}$ edge-disjoint paths connecting $u$ to $v$ in $G$.
\end{definition}
\begin{theorem}\label{lem: many paths}
There is an efficient randomized algorithm, that, given a wide and well-connected subinstance $I=(G,\Sigma)$ of $I^*$ with $m=|E(G)|\geq \mu^{c'}$, for some large enough constant $c'$, either returns FAIL, or computes a non-empty collection ${\mathcal{I}}$ of subinstances of $I$, such that the following hold:
\begin{itemize}
\item $\sum_{I'=(G',\Sigma')\in {\mathcal{I}}}|E(G')|\le |E(G)|$;
\item for every instance $I'=(G',\Sigma')\in {\mathcal{I}}$, either $|E(G')|\le m/\mu$, or instance $I'$ narrow;
\item
there is an efficient algorithm called $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace'$, that, given a solution $\phi(I')$ to every instance $I'\in {\mathcal{I}}$, computes a solution $\phi$ to instance $I$; and
\item if $\mathsf{OPT}_{\mathsf{cnwrs}}(I)\leq m^2/\mu^{c'}$ then, with probability at least $1-1/\mu^2$, all of the following hold:
\begin{itemize}
\item the algorithm does not return FAIL;
\item $\sum_{I'\in {\mathcal{I}}}\mathsf{OPT}_{\mathsf{cnwrs}}(I')\leq \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot (\log m)^{O(1)}$; and
\item if algorithm $\ensuremath{\mathsf{AlgCombineDrawings}}\xspace'$ is given as input a solution $\phi(I')$ to each instance $I'\in {\mathcal{I}}$, then the resulting solution $\phi$ to instance $I$ that it computes has cost at most: $\mathsf{cr}(\phi)\leq \sum_{I'\in {\mathcal{I}}}\mathsf{cr}(\phi(I')) + \mathsf{OPT}_{\mathsf{cnwrs}}(I)\cdot\mu^{O(1)}$.
\end{itemize}
\end{itemize}
\end{theorem}
=========================================================
\section{Core and Core structure}
=======================
${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$
\begin{definition}[Core and Core Structure]\label{def: valid core 1}
Given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a \emph{core structure} for $I$ consists of the following:
\begin{itemize}
\item a connected subgraph $J$ of $G$ called a \emph{core}, such that, for every edge $e\in E(J)$, graph $J\setminus \set{e}$ is connected;
\item an orientation $b_u\in \set{-1,1}$ for every vertex $u\in V(J)$;
\item a drawing $\rho_J$ of $J$ on the sphere with no crossings, that is consistent with the rotation system $\Sigma$ and the orientations in $\set{b_u}_{u\in V(J)}$. In other words, for every vertex $u\in V(J)$, the images of the edges in $\delta_J(u)$ enter the image of $u$ in $\rho_J$ according to their order in the rotation ${\mathcal{O}}_u\in \Sigma$ and orientation $b_u$ (so, e.g. if $b_u=1$ then the orientation is counter-clock-wise); and
\item a distinguished face $F^*(\rho_J)\in {\mathcal{F}}(\rho_J)$, such that the image of every vertex $u\in V(J)$, and the image of every edge $e\in E(J)$ is contained in the boundary of face $F^*(\rho_J)$ in drawing $\rho_J$.
\end{itemize}
\end{definition}
We denote a core structure by ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$, and we refer to graph $J$ as the \emph{core associated with ${\mathcal{J}}$}. We denote ${\mathcal{F}}^X(\rho_J)={\mathcal{F}}(\rho_J)\setminus\set{F^*(\rho_J)}$, and we refer to the faces in ${\mathcal{F}}^X(\rho_J)$ as the \emph{forbidden faces} of the drawing $\rho_J$.
\begin{definition}[Valid Core Structure]\label{def: valid core 2}
We say that a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J,F^*_{\rho_C})$ is \emph{valid} iff for every vertex $u\in V(J)$, for every edge $e_i^u\in \delta_G(u)\setminus E(J)$, the corresponding point $p_i^u$ lies in the interior of face $F^*(\rho_J)$ (and hence on the boundary of the disc $D(J)$).
\end{definition}
=================================
\paragraph{Ordering ${\mathcal{O}}(J)$ of the Edges of $\delta_G(J)$.}
Consider a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J,F^*(\rho_J))$, the drawing $\rho_J$ of $J$, and its corresponding disc $D(J)$. Recall that, for every vertex $u\in V(J)$ and every edge $e_i^u\in \delta_G(J)$, we have defined a point $p_i^u$ on the boundary of the disc $D(J)$ representing the edge $e_i^u$. We define a circular ordering ${\mathcal{O}}(J)$ of the edges of $\delta_G(J)$ to be the circular order in which the points $p_i^u$ corresponding to the edges of $\delta_G(J)$ are encountered, as we traverse the boundary of the disc $D(J)$ in counter-clock-wise direction.
================================================================
\section{Drawings of Graphs}
Next, we define a valid drawing of a graph $G$ with respect to a core structure ${\mathcal{J}}$.
\begin{definition}[A ${\mathcal{J}}$-Valid Drawing of a Graph]\label{def: valid drawing}
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace, and let ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J,F^*(\rho_J))$ be a core structure for $I$. A drawing $\phi$ of $G$ on the sphere is a \emph{${\mathcal{J}}$-valid drawing} iff we can define a disc $D'(J)$ that contains the images of all vertices and edges of the core $J$ in its interior, and the image of the core $J$ in $\phi$ is identical to $\rho_{J}$ (though its orientation may be arbitrary), with disc $D'(J)$ in $\phi$ playing the role of the disc $D(J)$in $\rho_{J}$.
\end{definition}
Abusing the notation, we will not distinguish between disc $D(J)$ in $\rho_J$ and disc $D'(J)$ in $\phi$, denoting both discs by $D(J)$.
Consider now some solution $\phi$ to instance $I$, that is ${\mathcal{J}}$-valid, with respect to some core structure ${\mathcal{J}}$. The image of graph $J$ in $\phi$ partitions the sphere into regions, each of which corresponds to a unique face of ${\mathcal{F}}(\rho_J)$. We do not distinguish between these regions and faces of ${\mathcal{F}}(\rho_J)$, so we view ${\mathcal{F}}(\rho_J)$ as a collection of regions in the drawing $\phi$ of $G$.
Next, we define special types of ${\mathcal{J}}$-valid drawings, called clean and semi-clean drawings.
\begin{definition}[Clean and Semi-Clean Drawings]\label{def: semiclean drawing}
Let $I=(G,\Sigma)$ be an instance of the \ensuremath{\mathsf{MCNwRS}}\xspace problem, let ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J,F^*(\rho_J))$ be a core structure for $I$, and let $\phi$ be a solution to instance $I$. We say that $\phi$ is a \emph{semi-clean solution to instance $I$}, or a \emph{semi-clean drawing of $G$}, with respect to ${\mathcal{J}}$ if it is a ${\mathcal{J}}$-valid drawing, and, additionally, the image of every vertex of $V(G)\setminus V(J)$ lies outside of the disc $D(J)$ (so in particular it must lie in the interior of the region $F^*(\rho_J)\in {\mathcal{F}}(\rho_J)$).
If, additionally, the drawing of every edge of $G$
is contained in region $F^*(\rho_J)$, then we say that $\phi$ is a \emph{clean} solution to $I$ with respect to ${\mathcal{J}}$, or that it is a \emph{${\mathcal{J}}$-clean} solution.
\end{definition}
Notice that, from the definition, if $\phi$ is a clean solution to instance $I$ with respect to core structure ${\mathcal{J}}$, then the edges of $J$ may not participate in any crossings in $\phi$.
If $\phi$ is a semi-clean solution to $I$ with respect to ${\mathcal{J}}$, then the edges of $J$ may participate in crossings, but no two edges of $J$ may cross each other.
Consider now a crossing $(e,e')_p$ in drawing $\phi$ of $G$, that is semi-clean with respect to ${\mathcal{J}}$. We say that crossing $(e,e')_p$ is \emph{dirty} if exactly one of the two edges $e,e'$ lies in $E(J)$. We denote by $\chi^{\mathsf{dirty}}(\phi)$ the set of all dirty crossings of drawing $\phi$. We denote by $\chi^*(\phi)$ the set of all crossings $(e,e')_{p}$ of $\phi$, for which the crossing point $p$ lies in the inerior of the region $F^*(\rho_J)$. We call assume w.l.o.g. that each such point $p$ lies outside the disc $D(J)$ (by making the disc smaller if necessary).
\paragraph{A ${\mathcal{J}}$-Contracted Instance.}
\begin{observation}\label{obs: clean solution to contracted}
There is an efficient algorithm, whose input consists of an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for $I$, and a ${\mathcal{J}}$-clean solution $\phi$ to instance $I$. The output of the algorithm is a solution $\hat \phi$ to the corresponding ${\mathcal{J}}$-contracted instance $\hat I=(\hat G,\hat \Sigma)$, with $\mathsf{cr}(\hat \phi)=\mathsf{cr}(\phi)$.
\end{observation}
====================================================
\section{Core Enhancement Structure}
=====================================================
\begin{definition}[Core Enhancement]
Given an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$ and a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for $I$, an \emph{enhancement} of the core structure ${\mathcal{J}}$ is a simple path $P\subseteq G$ (that may be a simple cycle), whose both endpoints belong to $J$, such that $P$ is internally disjoint from $J$ (See \Cref{fig: enhance} for an illustration.).
\end{definition}
For simplicity of notation, we will sometimes refer to an enhancement of a core structure ${\mathcal{J}}$ as an enhancement of the corresponding core $J$, or as a ${\mathcal{J}}$-enhancement. Next, we define a core enhancement structure.
\begin{definition}[Core Enhancement Structure]
Given an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$ and a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$,
a \emph{${\mathcal{J}}$-enhancement structure} consists of:
\begin{itemize}
\item a ${\mathcal{J}}$-enhancement $P$;
\item an orientation $b_u\in \set{-1,1}$ for every vertex $u\in V(P)\setminus V(J)$; and
\item a drawing $\rho'$ of the graph $J'=J\cup P$ with no crossings, such that $\rho'$ is consistent with the rotation system $\Sigma$ and the orientations $b_u$ for all vertices $u\in V(J')$ (the orientations of vertices of $J$ are determined by ${\mathcal{J}}$), and moreover, $\rho'$ is a clean drawing of $J'$ with respect to ${\mathcal{J}}$.
\end{itemize}
\end{definition}
================================
The split
================================
Suppose we are given an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$, and a ${\mathcal{J}}$-enhancement ${\mathcal{A}}=\set{P,\set{b_u}_{u\in V(J')},\rho'}$, where $J'=J\cup P$.
We now show an efficient algorithm that, given ${\mathcal{J}}$ and ${\mathcal{A}}$, splits the core structure ${\mathcal{J}}$ into two new core structures, ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$, using the enhancement structure ${\mathcal{A}}$.
We refer to $({\mathcal{J}}_1,{\mathcal{J}}_2)$ as a \emph{split} of the core structure ${\mathcal{J}}$ via the enhancement structure ${\mathcal{A}}$.
We let $\rho'$ be the drawing of the graph $J'$ on the sphere given by the enhancement structure ${\mathcal{A}}$. Recall that there is a disc $D(J)$ that contains the image of $J$ in $\rho'$, whose drawing in $D(J)$ is identical to $\rho_J$ (up to orientation). Additionally, all vertices and edges of $P$ must be drawn in region $F^*(\rho_J)$ of $\rho'$. Therefore, in drawing $\rho'$ of $J'$, there are two faces incident to the image of the path $P$, that we denote by $F_1$ and $F_2$, respectively, and $F_1\cup F_2=F^*(\rho_J)$ holds.
We let $J_1\subseteq J'$ be the graph containing all vertices and edges, whose images lie on the boundary of face $F_1$ in $\rho'$, and we define graph $J_2\subseteq J'$ similarly for face $F_2$.
We now define the core structure ${\mathcal{J}}_1$, whose corresponding core graph is $J_1$. For every vertex $u\in V(J_1)$, its orientation $b_u$ is the same as in ${\mathcal{A}}$. The drawing $\rho_{J_1}$ of $J_1$ is defined to be the drawing of $J_1$ induced by the drawing $\rho'$ of $J'$. Note that $F_1$ remains a face in the drawing $\rho_{J_1}$. We then let $F^*(\rho_{J_1})=F_1$.
The definition of the core structure ${\mathcal{J}}_2$ is symmetric, except that we use core $J_2$ instead of $J_1$ and face $F_2$ instead of $F_1$ (see \Cref{fig: type_2_enhance} for an illustration). This completes the description of the algorithm for computing a split $({\mathcal{J}}_1,{\mathcal{J}}_2)$ of the core structure ${\mathcal{J}}$ via the enhancement structure ${\mathcal{A}}$.
\begin{definition}[Splitting an Instance along an Enhancement Structure]\label{def: split}
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace, let ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ be a core structure for $I$, and let ${\mathcal{A}}=\set{P,\set{b_u}_{u\in V(J')},\rho'}$ be an enhancement structure for ${\mathcal{J}}$, where $J'=J\cup P$. Let $({\mathcal{J}}_1,{\mathcal{J}}_2)$ be the split of ${\mathcal{J}}$ via the enhancement structure ${\mathcal{A}}$, and denote by $J_1,J_2$ the cores of ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$, respectively.
A \emph{split} of instance $I$ along ${\mathcal{A}}$ is a pair $I_1=(G_1,\Sigma_1), I_2=(G_2,\Sigma_2)$ of instances of \ensuremath{\mathsf{MCNwRS}}\xspace, for which the following hold.
\begin{itemize}
\item $V(G_1)\cup V(G_2)=V(G)$ and $E(G_1)\cup E(G_2)\subseteq E(G)$;
\item every vertex $v\in V(G_1)\cap V(G_2)$ belongs to $V(J_1)\cap V(J_2)$;
\item ${\mathcal{J}}_1$ is a valid core structure for $I_1$, and ${\mathcal{J}}_2$ is a valid core structure for $I_2$.
\end{itemize}
\end{definition}
\begin{observation}\label{obs: combine solutions for split}
There is an efficient algorithm, whose input consists of
an instance $I=(G,\Sigma)$ of \ensuremath{\mathsf{MCNwRS}}\xspace, a core structure ${\mathcal{J}}$ for $I$, a ${\mathcal{J}}$-enhancement structure ${\mathcal{A}}$, and a split $(I_1,I_2)$ of $I$ along ${\mathcal{A}}$, together with a clean solution $\phi_1$ to instance $I_1$ with respect to ${\mathcal{J}}_1$, and a clean solution $\phi_2$ to instance $I_{2}$ with respect to ${\mathcal{J}}_2$, where $({\mathcal{J}}_1,{\mathcal{J}}_2)$ is the split of ${\mathcal{J}}$ along ${\mathcal{A}}$. The algorithm computes a clean solution $\phi$ to instance $I$ with respect to ${\mathcal{J}}$, with $\mathsf{cr}(\phi)\leq \mathsf{cr}(\phi_1)+\mathsf{cr}(\phi_2)+|E^{\mathsf{del}}|\cdot |E(G)|$, where $E^{\mathsf{del}}=E(G)\setminus (E(G_{1})\cup E(G_{2}))$.
\end{observation}
\subsubsection{Auxiliary Claim}
\begin{claim}\label{claim: curves orderings crossings}
Let $I=(G,\Sigma)$ be an instance of \ensuremath{\mathsf{MCNwRS}}\xspace, and let ${\mathcal{P}}=\set{P_1,\ldots,P_{4k+2}}$ be a collection of directed edge-disjoint simple paths in $G$, that are non-transversal with respect to $\Sigma$. For all $1\leq i\leq 4k+2$, let $e_i$ be the first edge on path $P_i$. Assume that there are two distinct vertices $u,v\in V(G)$, such that all paths in ${\mathcal{P}}$ originate at $u$ and terminate at $v$, and assume further that
edges $e_1,\ldots,e_{4k+2}$ appear in this order in the rotation ${\mathcal{O}}_u\in \Sigma$. Lastly, let $\phi$ be any solution to instance $I$, such that the number of crossings $(e,e')_p$ in $\phi$ with $e$ or $e'$ lying in $E(P_1)$ is at most $k$, and assume that the same is true for $E(P_{2k+1})$. Then $\phi$ does not contain a crossing between an edge of $P_1$ and an edge of $P_{2k+1}$.
\end{claim}
======================================
Procedure \ensuremath{\mathsf{ProcSplit}}\xspace
=======================================
\paragraph{Valid Input for \ensuremath{\mathsf{ProcSplit}}\xspace.}
A valid input for \ensuremath{\mathsf{ProcSplit}}\xspace consists of an instance $I=(G,\Sigma)$ of $\ensuremath{\mathsf{MCNwRS}}\xspace$, and a a core structure ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ for $I$, for which the following hold:
\begin{properties}{C}
\item graph $G$ is a subgraph of $\check G$, and $\Sigma$ is the rotation system for $G$ induced by $\check \Sigma$; \label{prop valid input size}
\item the ${\mathcal{J}}$-contracted instance $\hat I$ corresponding to instance $I$ is wide; \label{prop valid input wide}
\item $|E(G)|\geq \check m/\mu$; \label{prop many edges}
\item for every vertex $x\in V(G)$ with $\deg_G(x)\geq \check m/\mu^5$, there is a collection of at least $\frac{4\check m}{\mu^{50}}$ edge-disjoint paths in $G$ connecting $x$ to vertices of $J$; and \label{prop: connect high degree to core}
\item there exists a solution $\phi$ to instance $I$ that is semi-clean with respect to ${\mathcal{J}}$, such that $\mathsf{cr}(\phi)\leq \frac{|E(G)|^2}{\mu^{1500}}$ and $|\chi^{\mathsf{dirty}}(\phi)|\leq \frac{|E(G)|}{ \mu^{1500}}$. \label{prop valid input drawing}
\end{properties}
We emphasize that the semi-clean solution $\phi$ is not known to the algorithm.
\paragraph{Valid Output for \ensuremath{\mathsf{ProcSplit}}\xspace.}
A valid output for \ensuremath{\mathsf{ProcSplit}}\xspace consists of a
${\mathcal{J}}$-enhancement structure ${\mathcal{A}}$, and
a split $(I_1=(G_1,\Sigma_1),I_2=(G_2,\Sigma_2))$ of $I$ along ${\mathcal{A}}$, for which the following hold. Let $({\mathcal{J}}_1,{\mathcal{J}}_2)$ be the split of the core structure ${\mathcal{J}}$ along ${\mathcal{A}}$. Then:
\begin{properties}{P}
\item $|E(G)\setminus (E(G_1)\cup E(G_2))|\leq \frac{\mathsf{cr}(\phi)\cdot \mu^{2200}}{|E(G)|}+|\chi^{\mathsf{dirty}}(\phi)|$; \label{prop output deleted edges}
\item each of the graphs $G_{1},G_{2}$ contains at most $|E(G)|-\frac{|E(G)|}{32\mu^{300}}$ edges; and \label{prop: smaller graphs}
\item there is a semi-clean solution $\phi_1$ to instance $I_1$ with respect to ${\mathcal{J}}_1$, and a semi-clean solution $\phi_2$ to instance $I_2$ with respect to ${\mathcal{J}}_2$, such that $|\chi^{\mathsf{dirty}}(\phi_1)|+|\chi^{\mathsf{dirty}}(\phi_2)|\leq |\chi^{\mathsf{dirty}}(\phi)|+\frac{64\mu^{700}\mathsf{cr}(\phi)}{|E(G)|}$, and $|\chi^*(\phi_1)|+|\chi^*(\phi_2)|\leq |\chi^*(\phi)|$.\label{prop output drawing}
\end{properties}
The main theorem of this subsection summarizes the properties of Procedure \ensuremath{\mathsf{ProcSplit}}\xspace.
\begin{theorem}\label{thm: procsplit}
There is an efficient randomized algorithm, that, given a valid input $I=(G,\Sigma)$, ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho_J, F^*(\rho_J))$ to procedure \ensuremath{\mathsf{ProcSplit}}\xspace, with probability at least $1-1/\mu^{399}$ computes a valid output for the procedure.
\end{theorem}
\begin{claim}\label{claim: find potential augmentors}
There is an efficient algorithm that, given a valid input $I=(G,\Sigma)$, ${\mathcal{J}}=(J,\set{b_u}_{u\in V(J)},\rho, F^*)$ to procedure \ensuremath{\mathsf{ProcSplit}}\xspace, computes a promising path set of cardinality $\ceil{\frac{m}{6\mu^{50}}}$.
\end{claim}
\paragraph{Valid Input for $\ensuremath{\mathsf{ProcSplit}}\xspace'$.}
A valid input to Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$ consists of a skeleton augmenting structure ${\mathcal{W}}$ for $\check{\mathcal{K}}$ (whose corresponding skeleton augmentation we denote by $W$), parameters $g_1,g_2$, a ${\mathcal{W}}$-decomposition ${\mathcal{G}}$ of $\check G'$, and a face $F\in \tilde {\mathcal{F}}({\mathcal{W}})$, for which the following hold. Denote $E^{\mathsf{del}}=E(\check G')\setminus\textsf{left} (\bigcup_{G_{F'}\in {\mathcal{G}}}E(G_{F'})\textsf{right})$, $\tilde G=\check G'\setminus E^{\mathsf{del}}$, and let $\tilde \Sigma$ be the rotation system for graph $\tilde G$ induced by $\check \Sigma'$. Let $\tilde I=(\tilde G,\tilde \Sigma)$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Then:
\begin{properties}{C}
\item Instance $I_F=(G_F,\Sigma_F)$ associated with the graph $G_F\in {\mathcal{G}}$ is not acceptable;
\label{prop: instance unacceptable}
\item for every subset $E'\subseteq E(\tilde G')\setminus E(W)$ of edges, with $|E'|\leq \textsf{left} (\frac{g_1}{\check m'}+g_2\textsf{right} )\cdot \mu^{100b}$, there is a connected component $C$ in graph $\tilde G'\setminus E'$ with $W\subseteq C$; \label{prop: new}
\item there exists a solution $\phi$ to instance $\tilde I$, that is semi-clean with respect to $\check{\mathcal{K}}$, and has the following additional properties: \label{prop: drawing}
\begin{itemize}
\item drawing $\phi$ is ${\mathcal{W}}$-compatible;
\item $\mathsf{cr}(\phi)\leq \min\set{g_1,(\check m')^2/\mu^{60b}}$;
\item $|\chi^{\mathsf{dirty}}(\phi)|\leq \set{g_2,\check m'/\mu^{60b}}$ (as before, a crossing of $\phi$ is dirty if it involves an edge of the skeleton $\check K$); and
\item the total number of crossings of $\phi$ in which the edges of $E(W)\setminus E(\check K)$ participate is at most $\mathsf{cr}(\phi)\mu^{34b}/\check m'$.
\end{itemize}
\end{properties}
As before, the semi-clean solution $\phi$ is not known to the algorithm.
\paragraph{Valid Output for $\ensuremath{\mathsf{ProcSplit}}\xspace'$.}
A valid output for Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$ consists of
a skeleton augmenting structure ${\mathcal{W}}'$ for $\check{\mathcal{K}}$ (whose corresponding skeleton augmentation we denote by $W'$), an edge set $E^{\mathsf{del}}(F)\subseteq E(G_F)\setminus E(W')$ of cardinality at most $\textsf{left}(\frac{g_1}{|E(G_F)|}+g_2\textsf{right})\cdot \mu^{50b}$, and a ${\mathcal{W}}'$-decomposition ${\mathcal{G}}'$ of graph $\check G'$, for which the following hold.
Denote $E^{\mathsf{del}}=E(\check G')\setminus\textsf{left} (\bigcup_{G_{F'}\in {\mathcal{G}}}E(G_{F'})\textsf{right})$, $\tilde G'=\check G'\setminus (E^{\mathsf{del}}\cup E^{\mathsf{del}}(F))$, and let $\tilde \Sigma'$ be the rotation system for graph $\tilde G'$ induced by $\check \Sigma'$. Let $\tilde I'=(\tilde G',\tilde \Sigma')$ be the resulting instance of \ensuremath{\mathsf{MCNwRS}}\xspace. Then:
\begin{properties}{P}
\item $W\subseteq W'$, and $\tilde {\mathcal{F}}({\mathcal{W}}')=\textsf{left}(\tilde{\mathcal{F}}({\mathcal{W}})\setminus\set{F}\textsf{right})\bigcup\set{F_1,F_2}$;\label{prop: same faces}
\item for every face $F'\in \tilde {\mathcal{F}}({\mathcal{W}})\setminus\set{F}$, the graph $G_{F'}$ associated with face $F'$ in the decomposition ${\mathcal{G}}$ is identical to the graph $G'_{F'}$ associated with face $F'$ in the decomposition ${\mathcal{G}}'$; \label{prop: same graphs}
\item if $G_F$ is the graph associated with face $F$ in ${\mathcal{G}}$, and $G_{F_1},G_{F_2}$ are the graphs associated with faces $F_1$ and $F_2$, respectively, in ${\mathcal{G}}'$, then $E(G_F)\setminus (E(G_{F_1})\cup E(G_{F_2}))= E^{\mathsf{del}}(F)$. Moreover, $|E(G_{F_1})\setminus E(W')|, |E(G_{F_2})\setminus E(W')|\leq |E(G_{F})\setminus E(W)|-\check m'/\mu^{5b}$; \label{prop: small instances} and
\item there exists a solution $\phi'$ to instance $\tilde I'$, that is semi-clean with respect to $\check{\mathcal{K}}$, and has the following additional properties: \label{prop: drawing after}
\begin{itemize}
\item drawing $\phi'$ is ${\mathcal{W}}'$-compatible;
\item $\mathsf{cr}(\phi')\leq \mathsf{cr}(\phi)$;
\item $|\chi^{\mathsf{dirty}}(\phi)|\leq |\chi^{\mathsf{dirty}}(\phi')|$; and
\item if we denote by $N(\phi)$ the total number of crossings of $\phi$ in which the edges of $E(W)\setminus E(\check K)$ participate, and define $N(\phi')$ similarly for $\phi'$, then $N(\phi')\leq N(\phi)+\mathsf{cr}(\phi)\mu^{26b}/\check m'$.
\end{itemize}
\end{properties}
We summarize our algorithm for Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$ in the following lemma.
\begin{lemma}\label{lem: alg procsplit'}
There is an efficient randomized algorithm, that, given a valid input to Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$, with probability at least $1-1/\mu^{10b}$ produces a valid output for Procedure $\ensuremath{\mathsf{ProcSplit}}\xspace'$.
\end{lemma}
======================
Parmeters for \ensuremath{\mathsf{ProcSplit}}\xspace and bad events
========================
$k=\floor{\frac{m}{\mu^{50}}}$
Probability to select path into enhancement: $\frac{16}{k}\leq \frac{32\mu^{50}}{m}$.
\begin{definition}[Good path]
We say that a path $P'\in {\mathcal{P}}$ is \emph{good} if the following happen:
\begin{itemize}
\item the number of crossings in which the edges of $P'$ participate in $\phi$ is at most $\frac{64\mu^{450}\mathsf{cr}(\phi)}{m}$; and
\item there are no crossings in $\phi$ between edges of $P'$ and edges of $J$.
\end{itemize} A path that is not good is called a \emph{bad path}.
\end{definition}
We now bound the number of bad paths in ${\mathcal{P}}$.
\begin{observation}\label{obs: number of bad paths}
The number of bad paths in ${\mathcal{P}}$ is at most $\frac{m}{8\mu^{450}}$.
\end{observation}
We say that bad event ${\cal{E}}_1$ happens if the enhancement path $P$ is bad.
===================================
\subsubsection{Trimming clusters in ${\mathcal{C}}_2$ and well-linked decomposition}
We say that a cluster $C$ is a \emph{flower cluster}, iff
In this step, we further modify the sets ${\mathcal{C}}_1, {\mathcal{C}}_2, {\mathcal{C}}_3$ of clusters of $G'$ that we get from Section~\ref{sec:size_reduction}, to obtain new sets ${\mathcal{C}}'_1, {\mathcal{C}}'_2, {\mathcal{C}}'_3$ of clusters of $G'$ (we denote ${\mathcal{C}}'={\mathcal{C}}'_1\cup {\mathcal{C}}'_2\cup {\mathcal{C}}'_3$), such that:
\begin{itemize}
\item the vertex sets $\set{V(C)}_{C\in {\mathcal{C}}'}$ partitions $V(G')$;
\item $|E^{\textsf{out}}({\mathcal{C}}')|= O(m/\mu)$;
\item for each cluster $C\in {\mathcal{C}}_1$, $|E(C)|\le m/\mu^2$, and $C$ is $\alpha$-well-linked;
\item for each cluster $C\in {\mathcal{C}}_2$, $|E(C)|> m/\mu^2$, and $C$ is a flower cluster;
\item for each cluster $C\in {\mathcal{C}}_3$, $|E(C)|> m/\mu^2$ and $C$ is a good cluster.
\end{itemize}
Recall that in Section~\ref{sec:size_reduction}, we are given a valid sub-instance $(G',\Sigma')$ (with $|E(G')|=m$) of the input instance $(G,\Sigma)$, and we compute a set ${\mathcal{C}}={\mathcal{C}}_1\cup {\mathcal{C}}_2\cup {\mathcal{C}}_3$ of disjoint clusters of $G'$, where ${\mathcal{C}}_1$ contains clusters of at most $m/\mu^{2}$ edges, ${\mathcal{C}}_3$ contains good clusters with at most $m/\mu^{2}$ edges, and for each cluster $C\in {\mathcal{C}}_2$, $|E(C)|> m/\mu^2$, $C$ is not a good cluster and $C$ contains exactly one high-degree vertex.
We directly set ${\mathcal{C}}'_3={\mathcal{C}}_3$. All steps in this section are applied to clusters in ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$.
\paragraph{Step 1. Trim clusters in ${\mathcal{C}}_2$.}
We consider each cluster in ${\mathcal{C}}_2$ one-by-one, and further decompose it as follows. Let $C$ be a type-2 cluster, and let $u$ be the unique high-degree vertex in $C$ (so $\deg_{C}(u)\ge m/\mu^5$).
Let $\tilde C$ be the graph obtained from $C\cup\delta(C)$ by contracting all vertices of $\Gamma_G(C)$ into a single vertex, that we denote by $\tilde u$.
We compute a min-cut $(A,B)$ of $C'$ separating $u$ from $\tilde u$, with $u\in A$ and $\tilde u\in B$. We then decompose $C$ into two clusters: the subgraph of $C$ induced by vertices of $A$, that we denote by $C'$; and the subgraph of $C$ induced by vertices of $B\setminus \set{\tilde u}$, that we denote by $C''$.
Notice that there is a set of edge-disjoint paths in $C'$ connecting $\delta(C)$ to $u$.
If $|E(C')|>m/\mu^2$, then we replace the cluster $C$ in ${\mathcal{C}}_2$ and with cluster $C'$. Otherwise, we delete $C$ from ${\mathcal{C}}_2$ and add $C'$ to ${\mathcal{C}}_1$.
Note that the cluster $C''$ does not contain any high-degree vertex, and the graph $C''$ may be not connected.
We apply the algorithm in Lemma~\ref{lem:main_decomposition} to each connected component of $C''$ as a separate cluster to ${\mathcal{C}}_1$.
Clearly, we will obtain a set $\tilde{\mathcal{C}}=\set{C''_1,\ldots,C''_k}$ of clusters, such that $|E(C''_i)|\le |E(C'')|/\mu^2\le m/\mu^2$ for each $1\le i\le k$, and $|E^{\textsf{out}}(\tilde{\mathcal{C}})|\le O(|E(C'')|/\mu)$. We add all clusters of $\tilde{\mathcal{C}}$ to ${\mathcal{C}}_1$.
\paragraph{Step 2. Process clusters in ${\mathcal{C}}_2$ to get flower clusters.}
In this step, we further process the clusters in ${\mathcal{C}}_2$ one-by-one as follows. Throughout, we set the parameter $d=m/\mu^{5}$.
Let $C$ be a cluster of ${\mathcal{C}}_2$ and let $u$ be the unique high-degree vertex. Denote $\delta(u)=\set{e_1,\ldots,e_{\deg(u)}}$, where the edges are indexed according to the rotation at $u$ in $\Sigma$.
Denote $r=\lfloor\frac{\deg(u)}{d}\rfloor+1$.
For each $1\le i\le r-1$, let $E_i=\set{e_{(i-1)d+1},\ldots,e_{id}}$, and let $E_{r}=\set{e_{(r-1)d+1},\ldots,e_{\deg(u)}}$.
Clearly, sets $E_1,E_2,\ldots,E_r$ are consecutive subsets of edges in $\delta(u)$, each containing at most $d$ edges, and $r<O(\mu^5)$.
Let the cluster $C'$ be obtained from $C$ by replacing the vertex $u$ with $r$ distinct new vertices $u_1,\ldots,u_r$, such that the edges of $E_i$ are incident to $u_i$.
Denote $T=\set{u_1,\ldots,u_r}$.
For each $1\le i\le r$, let $(A_i,B_i)$ be the min-cut in $C'$, that separates $u_i$ from all vertices in $T\setminus \set{u_i}$, where $u_i\in A_i$. We denote by $X_i$ the subgraph of $C'$ induced by vertices of $A_i$,
and we denote ${\mathcal{X}}=\set{X_1,\ldots,X_r}$.
For each $1\le i\le r$, we denote $\hat E_i=\delta(X_i)$, then $\hat E_i$ can be partitioned into two subsets: set $\hat E'_i$ connecting a vertex of $X_i$ to a vertex of some cluster of ${\mathcal{X}}\setminus \set{X_{i}}$, and set $\hat E''_i$, connecting $X_i$ to vertices of $C'$ that do not belong to any cluster of ${\mathcal{X}}$ (such an edge may also belong to $\delta(C)$).
See Figure~\ref{fig:flower_edge} for an illustration.
\begin{figure}[h]
\includegraphics[scale=0.3]{figs/flower_edge.jpg}
\caption{An illustration of the edge sets in a flower cluster.\label{fig:flower_edge}}
\end{figure}
We use the following observation.
\begin{observation}
For each $1\le i\le r$, there is a set ${\mathcal{Q}}_i$ of paths in $X_i$ connecting edges of $\hat E_i$ to $u_i$, and $\cong({\mathcal{Q}}_i)\le 2$.
\end{observation}
\begin{proof}
edges of $\hat E'_i$ are routed because $X_i$ is a min cut separating $u_i$ from other terminals; edges of $\hat E''_i$ are routed because of the min-cut $(A,B)$, and possibly using the paths connecting $\hat E'_i$ to $u_i$.
\end{proof}
Let $C''$ be the cluster obtained by taking the union of $X_1,\ldots,X_r$, and unifying $v_1,\ldots,v_r$ back into the vertex $v$, so $C''$ is the subgraph of $C$ induced by vertices of $\bigcup_{1\le i\le r}V(X_i)$. We decompose the cluster $C$ into $C''$ and $C\setminus C''$.
Clearly, the subgraph $C\setminus C''$ does not contain any high-degree vertex. We apply the algorithm of Lemma~\ref{lem:main_decomposition} to $C\setminus C''$, and add all the obtained clusters to ${\mathcal{C}}_1$.
On the other hand, if $|E(C'')|>m/\mu^2$, then $C''$ is a flower cluster, and we replace the $C$ in ${\mathcal{C}}_2$ with $C''$. Otherwise we delete $C$ from ${\mathcal{C}}_2$ and add $C''$ to ${\mathcal{C}}_1$.
\paragraph{Step 3. Well-linked decomposition on clusters of ${\mathcal{C}}_1$.}
In this step, we process the clusters in ${\mathcal{C}}_1$ one-by-one as follows. Let $C$ be a cluster of ${\mathcal{C}}_1$. We apply Theorem~\ref{thm:well_linked_decomposition} to $C$ and obtain a set $\set{C_1,\ldots,C_k}$ of clusters. We then replace the cluster $C$ in ${\mathcal{C}}_1$ with clusters $C_1,\ldots,C_k$.
This completes the description of the algorithm. We return the sets ${\mathcal{C}}_1, {\mathcal{C}}_2, {\mathcal{C}}_3$ at the end of the algorithm as the desired sets ${\mathcal{C}}'_1, {\mathcal{C}}'_2,{\mathcal{C}}'_3$, and let ${\mathcal{C}}'={\mathcal{C}}'_1\cup{\mathcal{C}}'_2\cup{\mathcal{C}}'_3$. It is clear from the algorithm that the vertex sets $\set{V(C)}_{C\in {\mathcal{C}}}$ partitions $V(G')$; and the sets ${\mathcal{C}}'_1, {\mathcal{C}}'_2,{\mathcal{C}}'_3$ contains clusters with desired properties, respectively.
It remains to show that $|E^{\textsf{out}}({\mathcal{C}}')|\le m/\mu$.
\subsubsection{Tool Lemmas}
\iffalse
\begin{theorem}[Theorem 3 of \cite{chuzhoy2012vertex}]
\label{thm:weak_well_linked_decomposition}
There is an efficient algorithm, that given any graph $G=(V,E)$ and any cluster $S\subseteq V$ of vertices with $|\delta(S)|=z$, computes a set ${\mathcal{R}}$ of clusters of $S$, such that,
\begin{itemize}
\item the vertex sets $\set{V(R)}_{R\in {\mathcal{R}}}$ partitions $V(S)$;
\item for each cluster $R\subseteq{\mathcal{R}}$, $|\delta(R)|\le |\delta(S)|$;
\item for each cluster $R\subseteq{\mathcal{R}}$, $R$ is $\Omega(\log^{-3/2}z)$-boundary-well-linked;
\item $\sum_{R\in {\mathcal{R}}}|\delta(R)|\le 1.2|\delta(S)|$.
\end{itemize}
\fi
\begin{theorem}[Corollary 1 of \cite{chuzhoy2012routing}]
\label{thm:well_linked_decomposition}
There is an efficient algorithm, that given any graph $G=(V,E)$ and any cluster $S$ of $G$ with $|\delta(S)|=z$, computes a set ${\mathcal{R}}$ of clusters of $S$, such that,
\begin{itemize}
\item the vertex sets $\set{V(R)}_{R\in {\mathcal{R}}}$ partitions $V(S)$;
\item for each cluster $R\subseteq{\mathcal{R}}$, $|\delta(R)|\le |\delta(S)|$;
\item for each cluster $R\subseteq{\mathcal{R}}$, $R$ is $\Omega(\log^{-7/2}z)$-boundary-well-linked;
\item $\sum_{R\in {\mathcal{R}}}|\delta(R)|\le |\delta(S)|(1+O(\log^{-1/2}n))$.
\end{itemize}
\end{theorem}
|
1,116,691,499,391 | arxiv |
\section{Proof of Theorem~\ref{thm:stronger-mdc-qq3}}
Our proof in this section involves a refinement of modules, that is called {\em $p$-connected components}.
The notion of $p$-connectedness also generalizes connectivity in graphs.
A graph $G=(V,E)$ is $p$-connected if and only if, for every bipartition $(V_1,V_2)$ of $V$, there exists a path of length four with vertices in both $V_1$ and $V_2$.
The $p$-connected components of a graph are its maximal induced subgraphs which are $p$-connected.
Furthermore, a $p$-connected graph is termed {\em separable} if there exists a bipartition $(V_1,V_2)$ of its vertex-set such that, for every crossing $P_4$, its two ends are in $V_2$ and its two internal vertices are in $V_1$.
The latter bipartition $(V_1,V_2)$ is called a separation, and if it exists then it is unique.
We need a strengthening of Theorem~\ref{thm:modular-dec}:
\begin{theorem}[~\cite{JaO95}]\label{thm:primeval-dec}
For an arbitrary graph $G$ exactly one of the following conditions is satisfied.
\begin{enumerate}
\item $G$ is disconnected;
\item $\overline{G}$ is disconnected;
\item There is a unique proper separable $p$-connected component of $G$, with its separation being $(V_1,V_2)$ such that every vertex not in this component is adjacent to every vertex of $V_1$ and nonadjacent to every vertex of $V_2$;
\item $G$ is $p$-connected.
\end{enumerate}
\end{theorem}
If $G$ or $\overline{G}$ is disconnected then it corresponds to a degenerate node in the modular decomposition tree. So we know how to handle with the two first cases.
It remains to study the $p$-connected components of $(q,q-3)$-graphs.
For that, we need to introduce the class of $p$-trees:
\begin{definition}[~\cite{Bab00}]\label{def:p-trees}
A graph $G=(V,E)$ is a $p$-tree if one of the following conditions hold:
\begin{itemize}
\item the quotient graph $G'$ of $G$ is a $P_4$.
Furthermore, $G$ is obtained from $G'$ by replacing one vertex by a cograph.
\item the quotient graph $G'$ of $G$ is a spiked $p$-chain $P_k$, or its complement.
Furthermore, $G$ is obtained from $G'$ by replacing any of $x,y,v_1,v_k$ by a module inducing a cograph.
\item the quotient graph $G'$ of $G$ is a spiked $p$-chain $Q_k$, or its complement.
Furthermore, $G$ is obtained from $G'$ by replacing any of $v_1,v_k, z_2, z_3, \ldots, z_{k-5}$ by a module inducing a cograph.
\end{itemize}
\end{definition}
We stress that the case where the quotient graph $G'$ is a $P_4$, and so, of order $4 \leq 7 \leq q$ can be ignored in our analysis.
Other characterizations for $p$-trees can be found in~\cite{Bab98}.
The above Definition~\ref{def:p-trees} is more suitable to our needs.
\begin{theorem}[~\cite{BaO99}]\label{thm:p-comp}
A $p$-connected component of a $(q,q-3)$-graph either contains less than $q$ vertices, or is isomorphic to a prime spider, to a disc or to a $p$-tree.
\end{theorem}
Finally, before we can prove Theorem~\ref{thm:stronger-mdc-qq3}, we need to further characterize the {\em separable} $p$-connected components.
We use the following characterization of separable $p$-connected components.
\begin{theorem}[~\cite{JaO95}]\label{thm:separable-p-comp}
A $p$-connected graph $G=(V,E)$ is separable if and only if its quotient graph is a split graph.
Furthermore, its unique separation $(V_1,V_2)$ is given by the union $V_1$ of the strong modules inducing the clique and the union $V_2$ of the strong modules inducing the stable set.
\end{theorem}
We are now ready to prove Theorem~\ref{thm:stronger-mdc-qq3}.
\begin{proofof}{Theorem~\ref{thm:stronger-mdc-qq3}}
Suppose $G$ and $\overline{G}$ are connected (otherwise we are done).
By Theorem~\ref{thm:primeval-dec} there are two cases.
First we assume $G$ to be $p$-connected.
By Theorem~\ref{thm:p-comp}, $G$ either contains less than $q$ vertices, or is isomorphic to a prime spider, to a disc or to a $p$-tree.
Furthermore, if $G$ is a $p$-tree then according to Definition~\ref{def:p-trees}, the nontrivial modules can be characterized.
So, we are done in this case.
Otherwise, $G$ is not $p$-connected.
Let $V = V_1 \cup V_2 \cup V_3$ such that: $H = G[V_1 \cup V_2]$ is a separable $p$-component with separation $(V_1,V_2)$, every vertex of $V_3$ is adjacent to every vertex of $V_1$ and nonadjacent to every vertex of $V_2$.
Note that $G'$ is obtained from the quotient graph $H'$ of $H$ by possibly adding a vertex adjacent to all the strong modules in $V_1$.
In particular, by Theorem~\ref{thm:separable-p-comp} $H'$ is a split graph, and so, $G'$ is also a split graph.
By Lemma~\ref{lem:reduce-qq3}, it implies that $G'$ is either a prime spider, a spiked $p$-chain $Q_k$, a spiked $p$-chain $\overline{Q_k}$, or a graph with at most $q$ vertices.
Furthermore, if $G'$ is a prime spider then by Theorem~\ref{thm:p-comp} so is $H$, hence $G$ is a spider.
Otherwise, $G'$ is either a spiked $p$-chain $Q_k$ or a spiked $p$-chain $\overline{Q_k}$.
It implies that $H$ is a $p$-tree.
In particular, the nontrivial modules in $H$ can be characterized according to Definition~\ref{def:p-trees}.
The only nontrivial module of $G$ that is not a nontrivial module of $H$ (if any) contains $V_3$.
Finally, since the module that contains $V_3$ has no neighbour among the modules in $V_2$, the corresponding vertex in the quotient can only be a $z_i$, for some $i$.
So, we are also done in this case.
\end{proofof}
\section{Applications to other graph classes}
\label{sec:applications}
Our algorithmic schemes in Sections~\ref{sec:dist} and~\ref{sec:maxmatching} are all based on preprocessing methods with either split decomposition or modular decomposition.
If the prime subgraphs of the decomposition have constant-size then the input graph has bounded clique-width.
However, when the prime subgraphs are ``simple'' enough w.r.t. the problem considered, we may well be able to generalize our techniques in order to apply to some graph classes with unbounded clique-width.
In what follows, we present such examples.
\smallskip
A graph is {\em weak bipolarizable} if every prime subgraph in its modular decomposition is a chordal graph~\cite{Ola89}.
Some cycle problems such as {\sc Girth} (trivially) and {\sc Triangle Counting} (by using a clique-tree) can be easily solved in linear-time for chordal graphs.
The latter extends to the larger class of weak bipolarizable graphs by using our techniques.
\smallskip
Another instructive example is the class of graphs with small prime subgraphs for {\em c-decomposition}.
The c-decomposition consists in successively decomposing a graph by the modular decomposition and the split decomposition until all the subgraphs obtained are either degenerate (complete, edgeless or star) or prime for both the modular decomposition and the split decomposition~\cite{Lan01}.
Let us call c-width the minimum $k \geq 2$ such that any prime subgraph in the c-decomposition has order at most $k$.
The following was proved in~\cite{Rao08}.
\begin{theorem}[~\cite{Rao08}]
The class of graphs with c-width $2$ ({\it i.e.}, completely decomposable by the c-decomposition) has unbounded clique-width.
\end{theorem}
It is not clear how to compute the c-decomposition in linear-time.
However, both the modular decomposition and the split decomposition of graphs with small c-width already have some interesting properties which can be exploited for algorithmic purposes.
Before concluding this section we illustrate this fact with {\sc Eccentricities}.
\begin{lemma}\label{lem:split-dec-cdec}
Let $G=(V,E)$ be a graph with c-width at most $k$ that is prime for modular decomposition.
Every split component of $G$ that is not degenerate either has order at most $k$ or contains a universal vertex.
\end{lemma}
\begin{proof}
Since $G$ has c-width at most $k$, every non degenerate split component of $G$ with order at least $k+1$ can be modularly decomposed.
We show in the proof of Lemma~\ref{lem:mw-to-sw} that if a non degenerate graph can be modularly decomposed and it does not contain a universal vertex then it has a split.
Therefore, every non degenerate split component of size at least $k+1$ contains a universal vertex since it is prime for split decomposition.
\end{proof}
We now revisit the algorithmic scheme of Theorem~\ref{thm:sw-ecc}.
\begin{proposition}\label{prop:diam-unbounded-cw}
For every $G=(V,E)$ with c-width at most $k$, {\sc Eccentricities} can be solved in ${\cal O}(k^2 \cdot n + m)$-time.
In particular, {\sc Diameter} can also be solved in ${\cal O}(k^2 \cdot n + m)$-time.
\end{proposition}
\begin{proof}
Let $G'=(V',E')$ be the quotient graph of $G$.
Note that $G'$ has c-width at most $k$.
Furthermore, by Theorem~\ref{thm:mw-ecc} the problem reduces in linear-time to solve {\sc Eccentricities} for $G'$.
We compute the split-decomposition of $G'$.
It takes linear-time~\cite{CDR12}.
By Lemma~\ref{lem:split-dec-cdec} every split component of $G'$ either has order at most $k$ or it has diameter at most $2$.
Let us consider the following subproblem for every split component $C_i$.
Given a weight function $e : V(C_i) \to \mathbb{N}$, compute $\max_{u \in V(C_i) \setminus \{v\}} dist_{C_i}(u,v) + e(u)$ for every $v \in C_i$.
Indeed, the algorithm for Theorem~\ref{thm:sw-ecc} consists in solving the above subproblem a constant-number of times for every split component, with different weight functions $e$ that are computed by tree traversal on the split decomposition tree.
In particular, if the above subproblem can be solved in ${\cal O}(k^2 \cdot |V(C_i)| + |E(C_i)|)$-time for every split component $C_i$ then we can solve {\sc Eccentricities} for $G'$ in ${\cal O}(k^2 \cdot |V(G')| + |E(G')|)$-time.
There are two cases.
If $C_i$ has order at most $k$ then the above subproblem can be solved in ${\cal O}(|V(C_i)||E(C_i)|)$-time, that is in ${\cal O}(k^2 \cdot |V(C_i)|)$.
Otherwise, by Lemma~\ref{lem:split-dec-cdec} $C_i$ contains a universal vertex, that can be detected in ${\cal O}(|V(C_i)| + |E(C_i)|)$-time.
In particular, $C_i$ has diameter at most two.
Let $V(C_i) = (v_1,v_2,\ldots,v_{|V(C_i)|})$ be totally ordered such that, for every $j < j'$ we have $e(v_j) \geq e(v_{j'})$.
An ordering as above can be computed in ${\cal O}(|V(C_i)|)$-time, for instance using a bucket-sort algorithm.
Then, for every $v \in V(C_i)$ we proceed as follows.
We compute $D_v = 1 + \max_{u \in N_{C_i}(v)} e(u)$.
It takes ${\cal O}(deg_{C_i}(v))$-time.
Then, we compute the smallest $j$ such that $v_j$ and $v$ are nonadjacent (if any).
Starting from $v_1$ and following the ordering, it takes ${\cal O}(deg_{C_i}(v))$-time.
Finally, we are left to compare, in constant-time, $D_v$ with $2 + e(v_i)$.
Overall, the subproblem is solved in ${\cal O}(|V(C_i)|+|E(C_i)|)$-time in this case.
Therefore, {\sc Eccentricities} can be solved in ${\cal O}(k^2 \cdot n + m)$-time for $G$.
\end{proof}
\section{Cycle problems on bounded clique-width graphs}
\label{sec:cycle}
Clique-width is the smallest parameter that is considered in this work.
We start studying the possibility for $k^{{\cal O}(1)} \cdot (n+m)$-time algorithms on graphs with clique-width at most $k$.
Positive results are obtained for two variations of {\sc Triangle Detection}, namely \textsc{Triangle Counting} and \textsc{Girth}.
We define the problems studied in Section~\ref{sec:pbs-cw}, then we describe the algorithms in order to solve these problems in Section~\ref{sec:algos-cw}.
\subsection{Problems considered}\label{sec:pbs-cw}
We start introducing our basic cycle problem.
\begin{center}
\fbox{
\begin{minipage}{.95\linewidth}
\begin{problem}[\textsc{Triangle Detection}]\
\label{prob:triangle-detect}
\begin{description}
\item[Input:] A graph $G=(V,E)$.
\item[Question:] Does there exist a triangle in $G$?
\end{description}
\end{problem}
\end{minipage}
}
\end{center}
Note that for general graphs, {\sc Triangle Detection} is conjectured not to be solvable in ${\cal O}(n^{3-\varepsilon})$-time, for any $\varepsilon > 0 $, with a combinatorial algorithm~\cite{VaW10}.
It is also conjectured not to be solvable in ${\cal O}(n^{\omega - \varepsilon})$-time for any $\varepsilon > 0$, with $\omega$ being the exponent for fast matrix multiplication~\cite{AbV14}.
Our results in this section show that such assumptions do not hold when restricted to bounded clique-width graphs.
More precisely, we next describe fully polynomial parameterized algorithms for the two following generalizations of {\sc Triangle Detection}.
\begin{center}
\fbox{
\begin{minipage}{.95\linewidth}
\begin{problem}[\textsc{Triangle Counting}]\
\label{prob:triangle-count}
\begin{description}
\item[Input:] A graph $G=(V,E)$.
\item[Output:] The number of triangles in $G$.
\end{description}
\end{problem}
\end{minipage}
}
\end{center}
\begin{center}
\fbox{
\begin{minipage}{.95\linewidth}
\begin{problem}[\textsc{Girth}]\
\label{prob:girth}
\begin{description}
\item[Input:] A graph $G=(V,E)$.
\item[Output:] The girth of $G$, that is the minimum size of a cycle in $G$.
\end{description}
\end{problem}
\end{minipage}
}
\end{center}
In~\cite{VaW10}, the three of {\sc Triangle Detection}, {\sc Triangle Counting} and {\sc Girth} are proved to be subcubic equivalent when restricted to combinatorial algorithms.
\subsection{Algorithms}\label{sec:algos-cw}
Roughly, our algorithms in what follows are based on the following observation.
Given a labeled graph $\langle G, \ell \rangle$ (obtained from a $k$-expression), in order to detect a triangle in $G$, resp. a minimum-length cycle in $G$, we only need to store the adjacencies, resp. the distances, between every two label classes.
Hence, if a $k$-expression of length $L$ is given as part of the input we obtain algorithms running in time ${\cal O}(k^2L)$ and space ${\cal O}(k^2)$.
\smallskip
Our first result is for {\sc Triangle Counting} (Theorem~\ref{thm:cw-triangle}). It shares some similarities with a recent algorithm for listing all triangles in a graph~\cite{BFNN17}.
However, unlike the authors in~\cite{BFNN17}, we needn't use the notion of $k$-modules in our algorithms.
Furthermore, since we only ask for {\em counting} triangles, and not to list them, we obtain a better time complexity than in~\cite{BFNN17}.
\begin{theorem}\label{thm:cw-triangle}
For every $G=(V,E)$, \textsc{Triangle Counting} can be solved in ${\cal O}(k^2 \cdot (n+m))$-time if a $k$-expression of $G$ is given.
\end{theorem}
\begin{proof}
We need to assume the $k$-expression is {\em irredundant}, that is, when we add a complete join between the vertices labeled $i$ and the verticed labeled $j$, there was no edge before between these two subsets.
Given a $k$-expression of $G$, an irredundant $k$-expression can be computed in linear-time~\cite{CoB00}.
Then, we proceed by dynamic programming on the irredundant $k$-expression.
More precisely, let $\langle G, \ell \rangle$ be a labeled graph, $\ell : V(G) \to \{1, \ldots, k\}$.
We denote by $T(\langle G, \ell \rangle)$ the number of triangles in $G$.
In particular, $T(\langle G, \ell \rangle) = 0$ if $G$ is empty.
Furthermore, $T(\langle G, \ell \rangle) = T(\langle G', \ell' \rangle)$ if $\langle G, \ell \rangle$ is obtained from $\langle G', \ell' \rangle$ by: the addition of a new vertex with any label, or the identification of two labels.
If $\langle G, \ell \rangle$ is the disjoint union of $\langle G_1, \ell_1 \rangle$ and $\langle G_2, \ell_2 \rangle$ then $T(\langle G, \ell \rangle) = T(\langle G_1, \ell_1 \rangle) + T(\langle G_2, \ell_2 \rangle)$.
Finally, suppose that $\langle G, \ell \rangle$ is obtained from $\langle G', \ell' \rangle$ by adding a complete join between the set $V_i$ of vertices labeled $i$ and the set $V_j$ of vertices labeled $j$.
For every $p,q \in \{1, \ldots, k\}$, we denote by $m_{p,q}$ the number of edges in $\langle G', \ell' \rangle$ with one end in $V_p$ and the other end in $V_q$.
Let $n_{p,q}$ be the number of (non necessarily induced) $P_3$'s with an end in $V_p$ and the other end in $V_q$.
Note that we are only interested in the number of {\em induced} $P_3$'s for our algorithm, but this looks more challenging to compute.
Nevertheless, since the $k$-expression is irredundant, $n_{i,j}$ is exactly the number of induced $P_3$'s with one end in $V_i$ and the other in $V_j$.
Furthermore after the join is added we get: $|V_i|$ new triangles per edge in $G'[V_j]$, $|V_j|$ new triangles per edge in $G'[V_i]$, and one triangle for every $P_3$ with one end in $V_i$ and the other in $V_j$.
Summarizing:
$$T(\langle G, \ell \rangle) = T(\langle G', \ell' \rangle) + |V_j| \cdot m_{i,i} + |V_i| \cdot m_{j,j} + n_{i,j}.$$
In order to derive the claimed time bound, we are now left to prove that, after any operation, we can update the values $m_{p,q} \ \mbox{and} \ n_{p,q}, \ p,q \in \{1, \ldots, k\}$, in ${\cal O}(k^2)$-time.
Clearly, these values cannot change when we add a new (isolated) vertex, with any label, and they can be updated by simple summation when we take the disjoint union of two labeled graphs.
We now need to distinguish between the two remaining cases.
In what follows, let $m_{p,q}'$ and $n_{p,q}'$ represent the former values.
\begin{itemize}
\item
Suppose that label $i$ is identified with label $j$.
Then:
$$m_{p,q} = \begin{cases}
0 & \mbox{if} \ i \in \{p,q\} \\
m_{i,i}' + m_{i,j}' + m_{j,j}' & \mbox{if} \ p=q=j \\
m_{p,j}' + m_{p,i}' & \mbox{if} \ q = j, \ p \notin \{i,j\} \\
m_{j,q}' + m_{i,q}' & \mbox{if} \ p = j, \ q \notin \{i,j\} \\
m_{p,q}' & \mbox{else}
\end{cases},$$
$$n_{p,q} = \begin{cases} 0 & \mbox{if} \ i \in \{p,q\} \\
n_{j,j}' + n_{j,i}' + n_{i,i}' & \mbox{if} \ p=q=j \\
n_{p,j}' + n_{p,i}' & \mbox{if} \ q = j, \ p \notin \{i,j\} \\
n_{j,q}' + n_{i,q}' & \mbox{if} \ p = j, \ q \notin \{i,j\} \\
n_{p,q}' & \mbox{else}\end{cases}.$$
\item Otherwise, suppose that we add a complete join between the set $V_i$ of vertices labeled $i$ and the set $V_j$ of vertices labeled $j$.
Then, since the $k$-expression is irredundant: $$m_{p,q} = \begin{cases} |V_i| \cdot |V_j| & \mbox{if} \ \{i,j\} = \{p,q\} \\ m_{p,q}' & \mbox{else}\end{cases}.$$
For every $u_i,v_i \in V_i$ and $w_j \in V_j$ we create a new $P_3$ $(u_i,w_j,v_i)$.
Similarly, for every $u_j,v_j \in V_j$ and $w_i \in V_i$ we create a new $P_3$ $(u_j,w_i,v_j)$.
These are the only new $P_3$'s with two edges from the complete join.
Furthermore, for every edge $\{u_i,v_i\}$ in $V_i$ and for every $w_j \in V_j$ we can create the two new $P_3$'s $(u_i,v_i,w_j)$ and $(v_i,u_i,w_j)$.
Similarly, for every edge $\{u_j,v_j\}$ in $V_j$ and for every $w_i \in V_i$ we can create the two new $P_3$'s $(u_j,v_j,w_i)$ and $(v_j,u_j,w_i)$.
Finally, for every edge $\{v,u_j\}$ with $u_j \in V_j, \ v \notin V_i \cup V_j$, we create $|V_i|$ new $P_3$'s, and for every edge $\{v,u_i\}$ with $u_i \in V_i, \ v \notin V_i \cup V_j$, we create $|V_j|$ new $P_3$'s.
Altogether combined, we deduce the following update rules:
$$n_{p,q} = \begin{cases} n_{i,i}' + |V_i| \cdot |V_j| \cdot (|V_i|-1)/2 & \mbox{if} \ p = q = i \\
n_{j,j}' + |V_j| \cdot |V_i| \cdot (|V_j|-1)/2 & \mbox{if} \ p = q = j \\
n_{i,j}' + 2 \cdot |V_j| \cdot m_{i,i}' + 2 \cdot |V_i| \cdot m_{j,j}' & \mbox{if} \ \{p,q\} = \{i,j\} \\
n_{i,q}' + |V_i| \cdot m_{j,q}' & \mbox{if} \ p = i, q \notin \{i,j\} \\
n_{p,i}' + |V_i| \cdot m_{p,j}' & \mbox{if} \ q = i, p \notin \{i,j\} \\
n_{j,q}' + |V_j| \cdot m_{i,q}' & \mbox{if} \ p = j, q \notin \{i,j\} \\
n_{p,j}' + |V_j| \cdot m_{p,i}' & \mbox{if} \ q = j, p \notin \{i,j\} \\
n_{p,q}' & \mbox{else}\end{cases}.$$
\end{itemize}
\end{proof}
Our next result is about computing the {\em girth} of a graph (size of a smallest cycle).
To the best of our knowledge, the following Theorem~\ref{thm:cw-girth} gives the first known polynomial parameterized algorithm for {\sc Girth}.
\begin{theorem}\label{thm:cw-girth}
For every $G=(V,E)$, \textsc{Girth} can be solved in ${\cal O}(k^2 \cdot (n+m))$-time if a $k$-expression of $G$ is given.
\end{theorem}
\begin{proof}
The same as for Theorem~\ref{thm:cw-triangle}, we assume the $k$-expression to be irredundant.
It can be enforced up to linear-time preprocessing~\cite{CoB00}.
We proceed by dynamic programming on the $k$-expression.
More precisely, let $\langle G, \ell \rangle$ be a labeled graph, $\ell : V(G) \to \{1, \ldots, k\}$.
We denote by $\mu(\langle G, \ell \rangle)$ the girth of $G$.
By convention, $\mu(\langle G, \ell \rangle) = +\infty$ if $G$ is empty, or more generally if $G$ is a forest.
Furthermore, $\mu(\langle G, \ell \rangle) = \mu(\langle G', \ell' \rangle)$ if $\langle G, \ell \rangle$ is obtained from $\langle G', \ell' \rangle$ by: the addition of a new vertex with any label, or the identification of two labels.
If $\langle G, \ell \rangle$ is the disjoint union of $\langle G_1, \ell_1 \rangle$ and $\langle G_2, \ell_2 \rangle$ then $\mu(\langle G, \ell \rangle) = \min \{\mu(\langle G_1, \ell_1 \rangle), \mu(\langle G_2, \ell_2 \rangle) \}$.
Suppose that $\langle G, \ell \rangle$ is obtained from $\langle G', \ell' \rangle$ by adding a complete join between the set $V_i$ of vertices labeled $i$ and the set $V_j$ of vertices labeled $j$.
For every $p,q \in \{1, \ldots, k\}$, we are interested in the minimum length of a {\em nonempty} path with an end in $V_p$ and an end in $V_q$.
However, for making easier our computation, we consider a slightly more complicated definition.
If $p \neq q$ then we define $d_{p,q}$ as the minimum length of a $V_pV_q$-path of $G'$.
Otherwise, $p=q$, we define $d_{p,q}$ as the minimum length taken over all the paths with two distinct ends in $V_p$, and all the nontrivial closed walks that intersect $V_p$ ({\it i.e.}, there is at least one edge in the walk, we allow repeated vertices or edges, however a same edge does not appear twice consecutively).
Intuitively, $d_{p,p}$ may not represent the length of a path only in some cases where a cycle of length at most $d_{p,p}$ is already ensured to exist in the graph (in which case we needn't consider this value).
Furthermore note that such paths or closed walks as defined above may not exist.
So, we may have $d_{p,q} = + \infty$.
Then, let us consider a minimum-size cycle $C$ of $G$.
We distinguish between four cases.
\begin{itemize}
\item
If $C$ does not contain an edge of the join, then it is a cycle of $G'$.
\item
Else, suppose that $C$ contains exactly one edge of the join.
Then removing this edge leaves a $V_iV_j$-path in $G'$; this path has length at least $d_{i,j}$.
Conversely, if $d_{i,j} \neq +\infty$ then there exists a cycle of length $1 + d_{i,j}$ in $G$, and so, $\mu(\langle G, \ell \rangle) \leq 1 + d_{i,j}$.
\item
Else, suppose that $C$ contains exactly two edges of the join.
In particular, since $C$ is of minimum-size, and so, it is an induced cycle, the two edges of the join in $C$ must have a common end in the cycle.
It implies that removing the two edges from $C$ leaves a path of $G'$ with either its two ends in $V_i$ or its two ends in $V_j$.
Such paths have respective length at least $d_{i,i}$ and $d_{j,j}$.
Conversely, there exist closed walks of respective length $2 + d_{i,i}$ and $2+d_{j,j}$ in $G$.
Hence, $\mu(\langle G, \ell \rangle) \leq 2 + \min \{ d_{i,i}, d_{j,j} \}$.
\item Otherwise, $C$ contains at least three edges of the join. Since $C$ is induced, it implies that $C$ is a cycle of length four with two vertices in $V_i$ and two vertices in $V_j$.
Such a (non necessarily induced) cycle exists if and only if $\min \{|V_i|,|V_j|\} \geq 2$.
\end{itemize}
Summarizing: $$\mu(\langle G, \ell \rangle) = \begin{cases} \min \{ \mu(\langle G', \ell' \rangle), 1 + d_{i,j}, 2 + d_{i,i}, 2 +d_{j,j} \} & \mbox{if} \ \min\{|V_i|,|V_j|\} = 1 \\ \min \{ \mu(\langle G', \ell' \rangle), 1 + d_{i,j}, 2 + d_{i,i}, 2 +d_{j,j}, 4 \} & \mbox{otherwise.}\end{cases}$$
In order to derive the claimed time bound, we are now left to prove that, after any operation, we can update the values $d_{p,q}, \ p,q \in \{1, \ldots, k\}$, in ${\cal O}(k^2)$-time.
Clearly, these values cannot change when we add a new (isolated) vertex, with any label, and they can be updated by taking the minimum values when we take the disjoint union of two labeled graphs.
We now need to distinguish between the two remaining cases.
In what follows, let $d_{p,q}'$ represent the former values.
\begin{itemize}
\item
Suppose that label $i$ is identified with label $j$.
Then: $$ d_{p,q} = \begin{cases} + \infty & \mbox{if} \ i \in \{p,q\} \\ \min\{d_{i,i}', d_{i,j}', d_{j,j}'\} & \mbox{if} \ p=q=j \\ \min\{ d_{p,i}', d_{p,j}' \} & \mbox{if} \ q = j \\ \min\{ d_{i,q}', d_{j,q}' \} &\mbox{if} \ p = j \\ d_{p,q}' & \mbox{else}.\end{cases}$$
\item Otherwise, suppose that we add a complete join between the set $V_i$ of vertices labeled $i$ and the set $V_j$ of vertices labeled $j$.
The values $d_{p,q}'$ can only be decreased by using the edges of the join.
In particular, using the fact that the $k$-expression is irredundant, we obtain: $$ d_{p,q} = \begin{cases}
1 & \mbox{if} \ \{p,q\} = \{i,j\} \\
\min\{ 2, d_{p,q}' \} & \mbox{if} \ p=q=i, \ |V_i| \geq 2 \ \mbox{or} \ p=q=j, \ |V_j| \geq 2 \\
\min\{ d_{i,i}', 1 + d_{i,j}', 2 + d_{j,j}' \} & \mbox{if} \ p=q=i, \ |V_i| = 1 \\
\min\{ d_{j,j}', 1 + d_{i,j}', 2 + d_{i,i}' \} & \mbox{if} \ p=q=j, \ |V_j| = 1 \\
\min\{ d_{i,q}', 1 + d_{j,q}' \} & \mbox{if} \ p = i, \ q \notin \{i,j\} \\
\min\{ d_{p,i}', d_{p,j}' + 1 \} & \mbox{if} \ q = i, \ p \notin \{i,j\} \\
\min\{ d_{j,q}', 1 + d_{i,q}' \} & \mbox{if} \ p = j, \ q \notin \{i,j\} \\
\min\{ d_{p,j}', d_{p,i}' + 1 \} & \mbox{if} \ q = j, \ p \notin \{i,j\}
\end{cases}.$$
\end{itemize}
For all the remaining values of $p$ and $q$, the difficulty is to account for the cases where two consecutive edges of the join must be used in order to decrease the value $d_{p,q}'$.
We do so by using the {\em updated} values $d_{p,i},d_{i,q},d_{p,j},d_{j,q}$ instead of the former values $d_{p,i}',d_{i,q}',d_{j,p}',d_{q,j}'$.
More precisely,
$$d_{p,q} = \min \{ d_{p,q}', d_{p,i} + 1 + d_{j,q}, d_{p,j} + 1 + d_{i,q} \} \ \mbox{else}.$$
\end{proof}
The bottleneck of the above algorithms is that they require a $k$-expression as part of the input.
So far, the best-known approximation algorithms for clique-width run in ${\cal O}(n^3)$-time, that dominates the total running time of our algorithms~\cite{OuS06}.
However, on a more positive side a $k$-expression can be computed in linear time for many classes of bounded clique-width graphs.
In particular, combining Theorems~\ref{thm:cw-triangle} and~\ref{thm:cw-girth} with Lemmas~\ref{lem:mw-cw},~\ref{lem:sw-cw} and~\ref{lem:qq3-cw} we obtain the following result.
\begin{corollary}
For every $G=(V,E)$, {\sc Triangle Counting} and {\sc Girth} can be solved in ${\cal O}(k^2 \cdot (n+m))$-time, for every $k \in \{mw(G),sw(G),q(G)\}$.
\end{corollary}
\subsection{Hardness results for clique-width}\label{sec:hardness-cw}
The goal in this section is to prove that we cannot solve the problems of Section~\ref{sec:dist-pbs} in time $2^{o(cw)} n^{2 - \varepsilon}$, for any $\varepsilon > 0$ (Theorems~\ref{thm:cw-diam}---\ref{thm:cw-hyp}).
These are the first known hardness results for clique-width in the field of ``Hardness in P''.
Our results are conditioned on the Strong Exponential Time Hypothesis ({\sc SETH}): {\sc SAT} cannot be solved in ${\cal O}^*(2^{c \cdot n})$-time, for any $c < 1$~\cite{IPZ98}.
Furthermore, they are derived from similar hardness results obtained for {\em treewidth}.
Precisely, a {\it tree decomposition} $(T,{\cal X})$ of $G=(V,E)$ is a pair consisting of a tree $T$ and of a family ${\cal X}=(X_t)_{t \in V(T)}$ of subsets of $V$ indexed by the nodes of $T$ and satisfying:
\begin{itemize}
\item $\bigcup_{t \in V(T)}X_t=V$;
\item for any edge $e=\{u,v\} \in E$, there exists $t\in V(T)$ such that $u,v \in X_t$;
\item for any $v \in V$, the set of nodes $\{ t \in V(T) \mid v \in X_t\}$ induces a subtree, denoted by $T_v$, of $T$.
\end{itemize}
The sets $X_t$ are called {\it the bags} of the decomposition.
The {\it width} of a tree decomposition is the size of a largest bag minus one.
Finally, the {\it treewidth} of a graph $G$, denoted by $tw(G)$, is the least possible width over its tree decompositions.
\smallskip
Several hardness results have already been obtained for treewidth~\cite{AVW16}.
However, $cw(G) \leq 2^{tw(G)}$ for general graphs~\cite{CoR05}, that does not help us to derive our lower-bounds.
Roughly, we use relationships between treewidth and clique-width in some graph classes ({\it i.e.}, bounded-degree graphs~\cite{Cou12,GuW00}) in order to transpose the hardness results for treewidth into hardness results for clique-width.
Namely:
\begin{lemma}[\cite{Cou12,GuW00}]\label{lem:rel-tw-cw}
If $G$ has maximum degree at most $d$ (with $d \geq 1$), we have:
\begin{itemize}
\item $tw(G) \leq 3d \cdot cw(G) - 1$;
\item $cw(G) \leq 20d \cdot tw(G) + 22$.
\end{itemize}
\end{lemma}
Our reductions in what follows are based on Lemma~\ref{lem:rel-tw-cw}, and on previous hardness results for bounded treewidth graphs and bounded-degree graphs~\cite{AVW16,EvD16}.
\begin{theorem}\label{thm:cw-diam}
Under {\sc SETH}, we cannot solve {\sc Diameter} in $2^{o(k)} \cdot n^{2-\varepsilon}$-time on graphs with maximum degree $4$ and treewidth at most $k$, for any $\varepsilon > 0$.
In particular, we cannot solve {\sc Diameter} in $2^{o(k)} \cdot n^{2-\varepsilon}$-time on graphs with clique-width at most $k$, for any $\varepsilon > 0$.
\end{theorem}
\begin{proof}
In~\cite{AVW16}, they proved that under {\sc SETH}, we cannot solve {\sc Diameter} in ${\cal O}(n^{2-\varepsilon})$-time, for any $\varepsilon > 0$, in the class of tripartite graphs $G = (A \cup C \cup B, E)$ such that: $|A| = |B| = n$, $|C| = {\cal O}(\log n)$, and all the edges in $E$ are between $C$ and $A \cup B$.
Note that there exists a tree decomposition $(T,{\cal X})$ of $G$ such that $T$ is a path and the bags are the sets $\{a\} \cup C, \ a \in A$ and $\{b\} \cup C, \ b \in B$.
Hence, $tw(G) = {\cal O}(|C|) = {\cal O}(\log n)$~\cite{AVW16}.
Then, we use the generic construction of~\cite{EvD16} in order to transform $G$ into a bounded-degree graph.
We prove that graphs with treewidth ${\cal O}(\log n)$ can be generated with this construction\footnote{Our construction has less degrees of freedom than the construction presented in~\cite{EvD16}.}.
More precisely, let $T_{big}$ and $T_{small}$ be rooted balanced binary trees with respective number of leaves $|A| = |B| = n$ and $|C| = {\cal O}(\log n)$.
There is a bijective correspondance between the leaves of $T_{big}$ and the vertices in $A$, resp. between the leaves of $T_{big}$ and the vertices in $B$.
Similarly, there is a bijective correspondance between the leaves of $T_{small}$ and the vertices of $C$.
In order to construct $G'$ from $G$, we proceed as follows:
\begin{itemize}
\item We replace every vertex $u \in A \cup B$ with a disjoint copy $T_{small}^u$ of $T_{small}$.
We also replace every vertex $c \in C$ with two disjoint copies $T_{big}^{c,A}, T_{big}^{c,B}$ of $T_{big}$ with a common root.
\item For every $a \in A, c \in C$ adjacent in $G$, we add a path of length $p$ (fixed by the construction) between the leaf of $T_{small}^a$ corresponding to $c$ and the leaf of $T_{big}^{c,A}$ corresponding to $a$.
In the same way, for every $b \in B, c \in C$ adjacent in $G$, we add a path of length $p$ between the leaf of $T_{small}^b$ corresponding to $c$ and the leaf of $T_{big}^{c,B}$ corresponding to $b$.
\item Let $T_{big}^{A}$ and $T_{big}^{B}$ be two other disjoint copies of $T_{big}$.
For every $a \in A$ we add a path of length $p$ between the leaf corresponding to $a$ in $T_{big}^{A}$ and the root of $T_{small}^a$.
In the same way, for every $b \in B$ we add a path of length $p$ between the leaf corresponding to $b$ in $T_{big}^{B}$ and the root of $T_{small}^b$.
\item Finally, for every $u \in A \cup B$, we add a path of length $p$ with one end being the root of $T_{small}^u$.
\end{itemize}
The resulting graph $G'$ has maximum degree $4$.
In~\cite{EvD16}, they prove that, under {\sc SETH}, we cannot compute $diam(G')$ in ${\cal O}(n^{2-\varepsilon})$-time, for any $\varepsilon > 0$.
\smallskip
We now claim that $tw(G') = {\cal O}(\log n)$.
Since by Lemma~\ref{lem:rel-tw-cw} we have $tw(G') = \Theta(cw(G'))$ for bounded-degree graphs, it will imply $cw(G') = {\cal O}(\log n)$.
In order to prove the claim, we assume w.l.o.g. that the paths added by the above construction have length $p=1$\footnote{The hardness result of~\cite{EvD16} holds for $p=\omega(\log n)$. We reduce to the case $p=1$ {\em only} for computing the treewidth.}.
Indeed, subdividing an edge does not change the treewidth~\cite{BoK06}.
Note that in this situation, we can also ignore the pending vertices added for the last step of the construction.
Indeed, removing the pending vertices does not change the treewidth either~\cite{Bod06}.
Hence, from now on we consider the graph $G'$ resulting from the three first steps of the construction by taking $p=1$.
Let $(T',{\cal X}')$ be a tree decomposition of $T_{big}$ of unit width.
There is a one-to-one mapping between the nodes $t \in V(T')$ and the edges $e_t \in E(T_{big})$.
Furthermore, let $e_t^A, \ e_t^B, e_t^{c,A} \ \mbox{and} \ e_t^{c,B}$ be the copies of edge $e_t$ in the trees $T_{big}^A, \ T_{big}^B, \ T_{big}^{c,A} \ \mbox{and} \ T_{big}^{c,B}, \ c \in C$, respectively.
For every node $t \in V(T')$, we define a new bag $Y_t$ as follows.
If $e_t$ is not incident to a leaf-node then we set $Y_t = e_t^A \cup e_t^B \cup \left[ \bigcup_{c \in C} \left( e_t^{c,A} \cup e_t^{c,B} \right) \right]$.
Otherwise, $e_t$ is incident to some leaf-node.
Let $a_t \in A, \ b_t \in B$ correspond to the leaf.
We set $Y_t = V\left(T_{small}^{a_t}\right) \cup V\left(T_{small}^{b_t}\right) \cup e_t^A \cup e_t^B \cup \left[ \bigcup_{c \in C} \left( e_t^{c,A} \cup e_t^{c,B} \right) \right]$.
By construction, $(T', (Y_t)_{t \in V(T')})$ is a tree decomposition of $G'$.
In particular, $tw(G') \leq \max_{t \in V(T')} |Y_t| = {\cal O}(\log n)$, that finally proves the claim.
\smallskip
Finally, suppose by contradiction that $diam(G')$ can be computed in $2^{o(tw(G'))} \cdot n^{2-\varepsilon}$-time, for some $\varepsilon > 0$.
Since $tw(G') = {\cal O}(\log n)$, it implies that $diam(G')$ can be computed in ${\cal O}(n^{2-\varepsilon})$-time, for some $\varepsilon > 0$.
The latter refutes {\sc SETH}.
Hence, under {\sc SETH} we cannot solve {\sc Diameter} in $2^{o(k)} \cdot n^{2-\varepsilon}$-time on graphs with maximum degree $4$ and treewidth at most $k$, for any $\varepsilon > 0$.
This negative result also holds for clique-width since $cw(G') = \Theta(tw(G'))$.
\end{proof}
The following reduction to {\sc Betweenness Centrality} is from~\cite{EvD16}.
Our main contribution is to upper-bound the clique-width and the treewidth of their construction.
\begin{theorem}\label{thm:cw-bc}
Under {\sc SETH}, we cannot solve {\sc Betweenness Centrality} in $2^{o(k)} \cdot n^{2-\varepsilon}$-time on graphs with maximum degree $4$ and treewidth at most $k$, for any $\varepsilon > 0$.
In particular, we cannot solve {\sc Betweenness Centrality} in $2^{o(k)} \cdot n^{2-\varepsilon}$-time on graphs with clique-width at most $k$, for any $\varepsilon > 0$.
\end{theorem}
\begin{proof}
Let $G'$ be the graph from the reduction of Theorem~\ref{thm:cw-diam}.
In~\cite{EvD16}, the authors propose a reduction from $G'$ to $H$ such that, under {\sc SETH}, we cannot solve {\sc Betweenness Centrality} for $H$ in ${\cal O}(n^{2-\varepsilon})$-time, for any $\varepsilon > 0$.
In order to prove the theorem, it suffices to prove $tw(H) = \Theta(tw(G'))$.
Indeed, the construction of $H$ from $G'$ is as follows.
\begin{itemize}
\item For every $u \in A \cup B$, we remove the path of length $p$ with one end being the root of $T_{small}^u$, added at the last step of the construction of $G'$.
This operation can only decrease the treewidth.
\item Then, we add a path of length $p$ between the respective roots of $T_{big}^A$ and $T_{big}^B$.
Recall that we can assume $p=1$ since subdividing an edge does not modify the treewidth~\cite{BoK06}.
Adding an edge to a graph increases its treewidth by at most one.
\item Finally, let $H_1$ be the graph so far constructed.
We make a disjoint copy $H_2$ of $H_1$.
This operation does not modify the treewidth.
Then, for every $b \in B$, let ${T_{small}^b}'$ be the copy of $T_{small}^b$ in $H_2$.
We add a new vertex $b'$ that is uniquely adjacent to the root of ${T_{small}^b}'$.
The addition of pending vertices does not modify the treewidth either~\cite{Bod06}.
\end{itemize}
Overall, $tw(H) \leq tw(G') + 1 = {\cal O}(\log n)$.
Furthermore, since $H$ has maximum degree at most $4$, by Lemma~\ref{lem:rel-tw-cw} we have $cw(H) = \Theta(tw(H)) = {\cal O}(\log n)$.
\end{proof}
Our next reduction for {\sc Hyperbolicity} is inspired from the one presented in~\cite{BCH16}.
However, the authors in~\cite{BCH16} reduce from a special case of {\sc Diameter} where we need to distinguish between graphs with diameter either $2$ or $3$.
In order to reduce from a more general case of {\sc Diameter} we need to carefully refine their construction.
\begin{theorem}\label{thm:cw-hyp}
Under {\sc SETH}, we cannot solve {\sc Hyperbolicity} in $2^{o(k)} \cdot n^{2-\varepsilon}$-time on graphs with clique-width and treewidth at most $k$, for any $\varepsilon > 0$.
\end{theorem}
\begin{proof}
We use the graph $G'$ from the reduction of Theorem~\ref{thm:cw-diam}.
More precisely, let us take $p = \omega(\log n)$ for the size of the paths in the construction.
It has been proved in~\cite{EvD16} that either $diam(G') = (4+o(1)) p$ or $diam(G') = (6+o(1)) p$.
Furthermore, under {\sc SETH} we cannot decide in which case we are in truly subquadratic time.
Our reduction is inspired from~\cite{BCH16}.
Let $H$ be constructed from $G'$ as follows (see also Fig.~\ref{fig:reduction-hyp}).
\begin{itemize}
\item We add two disjoint copies $V_x,V_y$ of $V(G')$ and the three vertices $x,y,z \notin V(G')$.
We stress that $V_x$ and $V_y$ are independent sets.
Furthermore, for every $v \in V$, we denote by $v_x$ and $v_y$ the copies of $v$ in $V_x$ and $V_y$, respectively.
\item For every $v \in V(G')$, we add a $vv_x$-path $P_v^x$ of length $(3/2+o(1))p$, and similarly we add a $vv_y$-path $P_v^y$ of length $(3/2+o(1))p$.
\item Furthermore, for every $v \in V(G')$ we also add a $xv_x$-path $Q_v^x$ of length $(3/2+o(1))p$; a $yv_y$-path $Q_v^y$ of length $(3/2+o(1))p$; a $zv_x$-path $Q_v^{z,x}$ of length $(3/2+o(1))p$ and a $zv_y$-path $Q_v^{z,y}$ of length $(3/2+o(1))p$.
\end{itemize}
\begin{figure}[h!]
\centering
\includegraphics[width=.45\textwidth]{Fig/reduction-hyp}
\caption{The graph $H$ from the reduction of Theorem~\ref{thm:cw-hyp}.}
\label{fig:reduction-hyp}
\end{figure}
\smallskip
We claim that the resulting graph $H$ is such that $tw(H) = tw(G') + {\cal O}(1)$ and $cw(H) = cw(G') + {\cal O}(1)$.
Indeed, let us first consider $H' = H \setminus \{x,y,z\}$.
The graph $H'$ is obtained from $G'$ by adding some disjoint trees rooted at the vertices of $V(G')$.
In particular, it implies $tw(H') = tw(G')$, hence (by adding $x,y,z$ in every bag) $tw(H) \leq tw(H') + 3 \leq tw(G') + 3$.
Furthermore, let us fix a $k$-expression for $G'$. We transform it to a $(k+16)$-expression for $H$ as follows.
We start adding $x,y,z$ with three distinct new labels.
Then, we follow the $k$-expression for $G'$.
Suppose a new vertex $v \in V(G')$, with label $i$ is introduced.
It corresponds to some tree $T_v$ in $H'$, that is rooted at $v$.
Every such a tree has clique-width at most $3$~\cite{GoR00}.
So, as an intermediate step, let us fix a $3$-expression for $T_v$.
We transform it to a $12$-expression for $T_v$: with each new label encoding the former label in the $3$ expression ($3$ possibilities), and whether the node is either the root $v$ or adjacent to one of $x,y,z$ ($4$ possibilities).
This way, we can make $x,y,z$ adjacent to their neighbours in $T_v$, using the join operation.
Then, since the root $v$ has a distinguished label, we can ``freeze'' all the other nodes in $T_v \setminus v$ using an additional new label and relabeling operations.
Finally, we relabel $v$ with its original label $i$ in the $k$-expression of $G'$, and then we continue following this $k$-expression.
Summarizing, $cw(H) \leq cw(G') + 16$.
\smallskip
Next, we claim that $\delta(H) \geq (3+o(1))p$ if $diam(G') = (6+o(1))p$, while $\delta(H) \leq (11/4+o(1))p$ if $diam(G') = (4+o(1))p$.
Recall that by Theorem~\ref{thm:cw-diam}, under {\sc SETH} we cannot decide in which case we are in time $2^{o(tw(G'))} n^{2-\varepsilon} = 2^{o(cw(G'))} n^{2-\varepsilon}$, for any $\varepsilon > 0$.
Therefore, proving the claim will prove the theorem.
First suppose that $diam(G') = (6+o(1))p$.
Let $u,v \in V(G')$ satisfy $dist_{G'}(u,v) = (6+o(1))p$.
Observe that $diam(G') \leq (6+o(1))p = 4 \cdot (3/2 + o(1))p$, therefore $G'$ is an isometric subgraph of $H$ by construction.
Then, $S_1 = dist_H(u,v) + dist_H(x,y) = (12+o(1))p$;
$S_2 = dist_H(u,x) + dist_H(v,y) = (6+o(1))p$;
$S_3 = dist_H(u,y) + dist_H(v,x) = S_2$.
As a result, we obtain $\delta(H) \geq (S_1 - \max\{S_2,S_3\})/2 = (3+o(1))p$.
Second, suppose that $diam(G') = (4+o(1))p$.
We want to prove $\delta(H) \leq (11/4+o(1))p$.
By contradiction, let $a,b,c,d \in V(H)$ satisfy:
$$S_1 = dist_H(a,b) + dist_H(c,d) \geq S_2 = dist_H(a,c) + dist_H(b,d) \geq S_3 = dist_H(a,d) + dist_H(b,c),$$
$$S_1 - S_2 > (11/2+o(1))p.$$
The hyperbolicity of a given $4$-tuple is upper-bounded by the minimum distance between two vertices of the $4$-tuple~\cite{BCCM15,CCL15,Sot11}.
So, let us consider the distances in $H$.
\begin{itemize}
\item
Let $v \in V(G')$.
For every $u \in V(G'), \ dist_H(u,v) \leq dist_G(u,v) \leq (4+o(1))p$.
Furthermore for every $u' \in P_u^x$, $dist_H(v,u') \leq dist_H(v,u) + dist_H(u,u') \leq (11/2 + o(1))p$.
Similarly for every $u' \in P_u^y$, $dist_H(v,u') \leq dist_H(v,u) + dist_H(u,u') \leq (11/2 + o(1))p$.
For every $u' \in Q_u^x$, $dist_H(v,u') \leq dist_H(v,x) + dist_H(x,u') \leq (9/2+o(1))p$.
We prove in the same way that for every $u' \in Q_u^y \cup Q_u^{z,x} \cup Q_u^{z,y}, \ dist_H(v,u') \leq (9/2+o(1))p$.
Summarizing, $ecc_H(v) \leq (11/2 + o(1))p$.
\item
Let $v' \in P_v^x$, for some $v \in V(G')$.
For every $u \in V(G')$ and $u' \in P_u^x$ there are two cases.
Suppose that $dist_H(u',u_x) \leq p+o(1)$ or $dist_H(v',v_x) \leq p+o(1)$.
Then, $dist_H(v',u') \leq dist_H(v',v_x) + dist_H(v_x,u_x) + dist_H(u_x,u') \leq (1 + 3 + 3/2 + o(1))p = (11/2+o(1))p$.
Otherwise, $\max\{dist_H(u',u),dist_H(v',v)\} \leq (1/2+o(1))p$, and so, $dist_H(u',v') \leq dist_H(u',u) + dist_H(u,v) + dist_H(v',v) \leq (5+o(1))p$.
Similarly (replacing $u_x$ with $u_y$), for every $u' \in P_u^y$ we have $dist(v',u') \leq (11/2+o(1))p$.
For every $u' \in Q_u^x, \ dist_H(v',u') \leq dist_H(v',v_x) + dist_H(x,v_x) + dist_H(x,u') \leq (9/2+o(1))p$.
In the same way for every $u' \in Q_u^{z,x} \cup Q_u^{z,y}, \ dist_H(v',u') \leq dist_H(v',v_x) + dist_H(z,v_x) + dist_H(z,u') \leq (9/2+o(1))p$.
For every $u' \in Q_u^y$, we first need to observe that $dist_H(v_x,u_y) = (3+o(1))p\ \mbox{and} \ dist_H(v,y) = (3+o(1))p$.
In particular if $dist_H(v,v') \leq p +o(1)$ then, $dist_H(v',u') \leq dist_H(v,v') + dist_H(v,y) + dist_H(y,u') \leq (11/2+o(1))p$.
Otherwise, $dist_H(v',u') \leq dist_H(v',v_x) + dist_H(v_x,u_y) + dist_H(u_y,u') \leq (1/2 + 3 + 3/2 + o(1))p = (5+o(1))p$.
Summarizing, $ecc_H(v') \leq (11/2 + o(1))p$.
\item
Let $v' \in P_v^y$, for some $v \in V(G')$.
In the same way as above, we prove $ecc_H(v') \leq (11/2 + o(1))p$.
\item
Let $v' \in Q_v^{z,x} \cup Q_v^{z,y}$, for some $v \in V(G')$.
For every $u \in V(G')$ and for every $u' \in Q_u^{z,x} \cup Q_u^{z,y}$ we have $dist_H(v',u') \leq dist_H(v',z) + dist_H(z,u') \leq (3+o(1))p$.
For every $u' \in Q_u^x$ we have $dist_H(v',u') \leq dist_H(v',z) + dist_H(z,u_x) + dist_H(u_x,u') \leq (9/2+o(1))p$.
Similarly for every $u' \in Q_u^y$ we have $dist_H(v',u') \leq (9/2+o(1))p$.
Summarizing, $ecc_H(v') \leq (11/2 + o(1))p$.
\end{itemize}
In particular, every vertex in $H$ has eccentricity at most $(11/2 + o(1))p$, except maybe those in $\bigcup_{v \in V(G')} Q_v^x = X$ and those in $\bigcup_{v \in V(G')} Q_v^y = Y$.
However, $S_1 - S_2 \leq \min \{ dist_H(a,b), dist_H(c,d) \}$~\cite{CCL15}.
So, we can assume w.l.o.g. $a,c \in X$ and $b,d \in Y$.
Furthermore, $S_1 - S_2 \leq 2 \cdot dist_H(a,c)$~\cite{BCCM15,Sot11}.
Hence, $(11/2+o(1))p < S_1 - S_2 \leq 2 \cdot dist_H(a,c) \leq 2 \cdot ( dist_H(a,x) + dist_H(c,x) )$.
It implies $\max\{dist_H(a,x),dist_H(c,x)\} > (11/8 + o(1))p = (3/2 - 1/8 + o(1))p$.
Assume by symmetry that $dist_H(a,x) > (3/2 - 1/8 + o(1))p$.
Then, $dist_H(a,V_x) < (1/8+o(1))p$.
However, $dist_H(a,c) \leq dist_H(a,V_x) + (3+o(1))p + dist_H(c,V_y) < (1/8+3+3/2+o(1))p < (11/2+o(1))p$.
A contradiction.
Therefore, we obtain as claimed that $\delta(H) \leq (11/4 + o(1))p$.
\end{proof}
It is open whether any of these above problems can be solved in time $2^{{\cal O}(k)} \cdot n$ on graphs with clique-width at most $k$ (resp., on graphs with treewidth at most $k$, see~\cite{AVW16,Hus16}).
\subsection{Kernelization methods with modular decomposition}\label{sec:algos-modular-dec}
The purpose of the subsection is to show how to apply the previous results, obtained with split decomposition, to modular decomposition.
On the way, improvements are obtained for the running time.
Indeed, it is often the case that only the quotient graph $G'$ needs to be considered.
We thus obtain algorithms that run in ${\cal O}(mw(G)^{{\cal O}(1)} + n + m)$-time.
See~\cite{MNN16} for an extended discussion on the use of {\em Kernelization} for graph problems in P.
We start with the following lemma:
\begin{lemma}[folklore]\label{lem:mw-to-sw}
For every $G=(V,E)$ we have $sw(G) \leq mw(G)+1$.
\end{lemma}
\begin{proof}
First we claim that $mw(H) \leq mw(G)$ for every {\em induced} subgraph $H$ of $G$.
Indeed, for every module $M$ of $G$ we have that $M \cap V(H)$ is a module of $H$, thereby proving the claim.
We show in what follows that a ``split decomposition'' can be computed from the modular decomposition of $G$ such that all the non degenerate split components have size at most $mw(G)+1$\footnote{Formally this is only a partial split decomposition, since there are subgraphs that could be further decomposed.}.
Applying this result to every prime split component of $G$ in its canonical split decomposition proves the lemma.
W.l.o.g., $G$ is connected (otherwise, we consider each connected component separately).
Let ${\cal M}(G) = \{ M_1,M_2, \ldots, M_k \}$ ordered by decreasing size.
\begin{enumerate}
\item If $|M_1| =1$ ($G$ is either complete or prime for modular decomposition) then we output $G$.
\item Otherwise, suppose $|M_1| < n-1$.
We consider all the maximal strong modules $M_1, M_2, \ldots, M_t$ such that $|M_i| \geq 2$ sequentially.
For every $1 \leq i \leq t$, we have that $(M_i, V \setminus M_i)$ is a split.
Furthermore if we apply the corresponding simple decomposition then we obtain two subgraphs, one being the subgraph $G_i$ obtained from $G[M_i]$ by adding a universal vertex $b_i$, and the other being obtained from $G$ by replacing $M_i$ by a unique vertex $a_i$ with neighbourhood $N_G(M_i)$.
Then, there are two subcases.
\begin{itemize}
\item Subcase ${\cal M}(G) = \{M_1,M_2\}$.
In particular, $|M_2| \geq 2$.
We perform a simple decomposition for $M_1$.
The two resulting subgraphs are exactly $G_1$ and $G_2$.
\item Subcase $\{M_1,M_2\} \subsetneq {\cal M}(G)$.
We apply simple decompositions for $M_1,M_2, \ldots, M_t$ sequentially.
Indeed, let $i \in \{1, \ldots, t\}$ and suppose we have already applied simple decompositions for $M_1, M_2, \ldots, M_{i-1}$.
Then, since there are at least three modules in ${\cal M}(G)$ we have that $(M_i, \{a_1,a_2, \ldots a_{i-1}\} \cup \bigcup_{j > i}M_j)$ remains a split, and so, we can apply a simple decomposition.
The resulting components are exactly: the quotient graph $G'$ and, for every $1 \leq i \leq t$, the subgraph $G_i$ obtained from $G[M_i]$.
\end{itemize}
Furthermore, in both subcases we claim that the modular decomposition of $G_i$ can be updated from the modular decomposition of $G[M_i]$ in constant-time.
Indeed, the set of all universal vertices in a graph is a clique and a maximal strong module.
We output $G'$ (only if $\{M_1,M_2\} \subsetneq {\cal M}(G)$) and, for every $1 \leq i \leq t$, we apply the procedure recursively for $G_i$.
\item Finally, suppose $|M_1| = n-1$.
In particular, ${\cal M}(G) = \{ M_1, M_2 \}$ and $M_2$ is trivial.
Let ${\cal M}(G[M_1]) = \{ M_1', M_2', \ldots, M_p' \}$ ordered by decreasing size.
If $|M_1'| = 1$ ({\it i.e.}, $G[M_1]$ is either edgeless, complete or prime for modular decomposition) then we output $G$.
Otherwise we apply the previous Step 2 to the modular partition $M_1', M_2', \ldots, M_p', M_2$.
\end{enumerate}
The procedure takes linear-time if the modular decomposition of $G$ is given.
Furthermore, the subgraphs obtained are either: the quotient graph $G'$; a prime subgraph for modular decomposition with an additional universal vertex; or a degenerate graph (that is obtained from either a complete subgraph or an edgeless subgraph by adding a universal vertex).
\end{proof}
\begin{corollary}
For every $G=(V,E)$ we can solve:
\begin{itemize}
\item {\sc Eccentricities} and {\sc Diameter} in ${\cal O}(mw(G)^2 \cdot n + m)$-time;
\item {\sc Hyperbolicity} in ${\cal O}(mw(G)^3 \cdot n + m)$-time;
\item {\sc Betweenness Centrality} in ${\cal O}(mw(G)^2 \cdot n + m)$-time.
\end{itemize}
\end{corollary}
In what follows, we explain how to improve the above running times in some cases.
\begin{theorem}\label{thm:mw-ecc}
For every $G=(V,E)$, {\sc Eccentricities} can be solved in ${\cal O}(mw(G)^3 + n+m)$-time.
In particular, {\sc Diameter} can be solved in ${\cal O}(mw(G)^3 + n+m)$-time.
\end{theorem}
\begin{proof}
W.l.o.g., $G$ is connected.
Consider the (partial) split decomposition obtained from the modular decomposition of $G$ (Lemma~\ref{lem:mw-to-sw}).
Let $T$ be the corresponding split decomposition tree.
By construction, there exists a modular partition $M_1,M_2, \ldots, M_k$ of $G$ with the two following properties:
\begin{itemize}
\item All but at most one split components of $G$ are split components of some $G_i, \ 1 \leq i \leq k$, where the graph $G_i$ is obtained from $G[M_i]$ by adding a universal vertex $b_i$.
\item Furthermore, the only remaining split component (if any) is the graph $G'$ obtained by replacing every module $M_i$ with a single vertex $a_i$.
Either $G'$ is degenerate (and so, $diam(G') \leq 2$) or $k \leq mw(G)+1$.
We can also observe in this situation that if we root $T$ in $G'$ then the subtrees of $T \setminus \{G'\}$ are split decomposition trees of the graphs $G_i, \ 1 \leq i \leq k$.
\end{itemize}
We can solve {\sc Eccentricities} for $G$ as follows.
First for every $1 \leq i \leq k$ we solve {\sc Eccentricities} for $G_i$.
In particular, $diam(G_i) \leq 2$, and so, for every $v \in V(G_i)$ we have: $ecc_{G_i}(v) = 0$ if and only if $V(G_i) = \{v\}$; $ecc_{G_i}(v) = 1$ if and only if $v$ is universal in $G_i$; otherwise, $ecc_{G_i}(v) = 2$.
Therefore, we can solve {\sc Eccentricities} for $G_i$ in ${\cal O}(|V(G_i)|+|E(G_i)|)$-time.
Overall, this step takes ${\cal O}(\sum_{i=1}^k |V_i| + |E_i|) = {\cal O}(n+m)$-time.
Then there are two subcases.
Suppose $G'$ is not a split component.
We deduce from Lemma~\ref{lem:mw-to-sw} $G=G[M_1] \oplus G[M_2]$.
In this situation, for every $i \in \{1,2\}$, for every $v \in V(G_i)$ we have $ecc_{G}(v) = \max\{ ecc_{G_i}(v), 1 \}$.
Otherwise, let us compute {\sc Eccentrities} for $G'$.
It takes ${\cal O}(|V(G')|) = {\cal O}(n)$-time if $G'$ is degenerate, and ${\cal O}(mw(G)^3)$-time otherwise.
Applying the algorithmic scheme of Theorem~\ref{thm:sw-ecc}, one obtains $ecc_G(v) = \max\{ecc_{G_i}(v), dist_{G_i}(v,a_i) + ecc_{G'}(b_i) - 1\} = \max \{ecc_{G_i}(v), ecc_{G'}(b_i) \}$ for every $v \in M_i$.
Hence, we can compute $ecc_G(v)$ for every $v \in V$ in ${\cal O}(n)$-time.
\end{proof}
\begin{corollary}
For every connected $G=(V,E)$, $diam(G) \leq \max \{ mw(G), 2 \}$.
\end{corollary}
Next, we consider {\sc Hyperbolicity}.
It is proved in~\cite{Sot11} that, for every $G=(V,E)$ with quotient graph $G'$, $\delta(G') \leq \delta(G) \leq \max\{\delta(G'),1\}$.
The latter immediately implies the following result:
\begin{theorem}\label{thm:mw-hyp}
For every $G=(V,E)$, we can decide whether $\delta(G) >1$, and if so, compute $\delta(G)$, in ${\cal O}(mw(G)^4 + n + m)$-time.
\end{theorem}
However, we did not find a way to preprocess $G$ in linear-time so that we can compute $\delta(G)$ from $\delta(G')$.
Indeed, let $G_M$ be a graph of diameter at most $2$.
Solving {\sc Eccentricities} for $G_M$ can be easily done in linear-time.
However, the following shows that it is not that simple to do so for {\sc Hyperbolicity}.
\begin{lemma}[~\cite{CoDu14}]\label{lem:hyp-diam1}
For every $G=(V,E)$ we have $\delta(G) \leq \left\lfloor diam(G) /2 \right\rfloor$.
Furthermore, if $diam(G) \leq 2$ then $\delta(G) < 1$ if and only if $G$ is $C_4$-free.
\end{lemma}
The detection of an induced $C_4$ in ${\cal O}(mw(G)^{{\cal O}(1)} + n +m)$-time remains an open problem.
\subsubsection*{Short digression: using neighbourhood diversity}
We show that by imposing more constraints on the modular partition, some more kernels can be computed for the problems in Section~\ref{sec:dist-pbs}.
Two vertices $u,v$ are {\em twins} in $G$ if $N_G(u) \setminus v = N_G(v) \setminus u$.
Being twins induce an equivalence relationship over $V(G)$.
The number of equivalence classes is called the {\em neighbourhood diversity} of $G$, sometimes denoted by $nd(G)$~\cite{Lam12}.
Observe that every set of pairwise twins is a module of $G$.
Hence, $mw(G) \leq nd(G)$.
\begin{theorem}\label{thm:nd-hyp}
For every $G=(V,E)$, {\sc Hyperbolicity} can be solved in ${\cal O}(nd(G)^4 + n+m)$-time.
\end{theorem}
\begin{proof}
Let $V_1,V_2,\ldots,V_k, \ k = nd(G)$, partition the vertex-set $V$ in twin classes.
The partition can be computed in linear-time~\cite{Lam12}.
Furthermore, since it is a modular partition, we can compute a (partial) split decomposition as described in Lemma~\ref{lem:mw-to-sw}.
Let $G'=(V',E')$ such that $V' = \{v_1,v_2,\ldots,v_k\}$ and $E' = \{ \{v_i,v_j\} \mid V_i \times V_j \subseteq E \}$.
Then, the split components are either: $G'$, stars $S^i$ (if the vertices of $V_i$ are pairwise nonadjacent, {\it i.e.}, false twins) or complete graphs $K^i$ (if the vertices of $V_i$ are pairwise adjacent, {\it i.e.}, true twins).
Applying the algorithmic scheme of Theorem~\ref{thm:sw-hyp}, in order to solve {\sc Hyperbolicity} for $G$ it suffices to compute, for every split component $C_j$, the hyperbolicity value $\delta(C_j)$ {\em and} all the simplicial vertices in $C_j$.
This can be done in ${\cal O}(|V(C_j)|)$-time if $C_j$ is a star or a complete graph, and in ${\cal O}(nd(G)^4)$-time if $C_j = G'$.
Therefore, we can solve {\sc Hyperbolicity} for $G$ in total ${\cal O}(nd(G)^4 + n+m)$-time.
\end{proof}
In~\cite{FKMN+17}, the authors propose an ${\cal O}(2^{{\cal O}(k)} + n +m)$-time algorithm for computing {\sc Hyperbolicity} with $k$ being the vertex-cover of the graph.
Their algorithm is pretty similar to Theorem~\ref{thm:nd-hyp}.
This is no coincidence since every graph with vertex-cover at most $k$ has neighbourhood diversity at most $2^{{\cal O}(k)}$~\cite{Lam12}.
Finally, the following was proved implicitly in~\cite{PEZB14}.
\begin{theorem}[~\cite{PEZB14}]\label{thm:nd-bc}
For every $G=(V,E)$, {\sc Betweenness Centrality} can be solved in ${\cal O}(nd(G)^3 + n+m)$-time.
\end{theorem}
\subsection{Distance problems considered}\label{sec:dist-pbs}
\subsubsection*{Eccentricity-based problems}
The first problem considered is computing the diameter of a graph (maximum length of a shortest-path).
\begin{center}
\fbox{
\begin{minipage}{.95\linewidth}
\begin{problem}[\textsc{Diameter}]\
\label{prob:diam}
\begin{description}
\item[Input:] A graph $G=(V,E)$.
\item[Output:] The diameter of $G$, that is $\max_{u,v \in V} dist_G(u,v)$.
\end{description}
\end{problem}
\end{minipage}
}
\end{center}
Hardness results for {\sc Diameter} have been proved, {\it e.g.}, in~\cite{RoV13,AGV15,BCH16,AVW16,EvD16}.
Our new hardness results are proved for {\sc Diameter}, while our fully polynomial parameterized algorithms apply to the following more general version of the problem.
The {\em eccentricity} of a given vertex $v$ is defined as $ecc_G(v) = \max_{u \in V} dist_G(u,v)$.
Observe that $diam(G) = \max_v ecc_G(v)$.
\begin{center}
\fbox{
\begin{minipage}{.95\linewidth}
\begin{problem}[\textsc{Eccentricities}]\
\label{prob:ecc}
\begin{description}
\item[Input:] A graph $G=(V,E)$.
\item[Output:] The eccentricities of the vertices in $G$, that is $\max_{u \in V} dist_G(u,v)$ for every $v \in V$.
\end{description}
\end{problem}
\end{minipage}
}
\end{center}
\subsubsection*{Gromov hyperbolicity}
Then, we consider the parameterized complexity of computing the Gromov hyperbolicity of a given graph.
Gromov hyperbolicity is a measure of how close (locally) the shortest-path metric of a graph is to a tree metric~\cite{Gro87}.
We refer to~\cite{ducoffe:tel-01485328} for a survey on the applications of Gromov hyperbolicity in computer science.
\begin{boxedproblem}{Hyperbolicity}{prob:hyp}
\begin{description}
\item[Input:] A graph $G=(V,E)$.
\item[Output:] The hyperbolicity $\delta$ of $G$, that is:
\begin{align*}
\max_{u,v,x,y \in V} \frac {dist_G(u,v) + dist_G(x,y) - \max \{ dist_G(u,x) + dist_G(v,y), dist_G(u,y) + dist_G(v,y) \}} 2.
\end{align*}
\end{description}
\end{boxedproblem}
Hardness results for {\sc Hyperbolicity} have been proved in~\cite{BCH16,CoDu14,FIV15}.
Some fully polynomial parameterized algorithms, with a different range of parameters than the one considered in this work, have been designed in~\cite{FKMN+17}.
\subsubsection*{Centrality problems}
There are different notions of centrality in graphs.
For clarity, we choose to keep the focus on one centrality measurement, sometimes called the Betweenness Centrality~\cite{Free77}.
More precisely, let $G=(V,E)$ be a graph and let $s,t \in V$.
We denote by $\sigma_G(s,t)$ the number of shortest $st$-paths in $G$.
In particular, for every $v \in V$, $\sigma_G(s,t,v)$ is defined as the number of shortest $st$-paths passing by $v$ in $G$.
\begin{boxedproblem}{Betweenness Centrality}{prob:bc}
\begin{description}
\item[Input:] A graph $G=(V,E)$.
\item[Output:] The betweenness centrality of every vertex $v \in V$, defined as:
\begin{align*}
BC_G(v) = \sum_{s,t \in V \setminus v} \sigma_G(s,t,v)/\sigma_G(s,t).
\end{align*}
\end{description}
\end{boxedproblem}
See~\cite{AGV15,BCH16,EvD16} for hardness results on {\sc Betweenness Centrality}.
\subsection{Applications to graphs with few $P_4$'s}
\label{sec:dist-qq3}
Before ending Section~\ref{sec:dist}, we apply the results of the previous subsections to the case of $(q,q-3)$-graphs.
For that we need to consider all the cases where the quotient graph has super-constant size $\Omega(q)$ (see Lemma~\ref{lem:reduce-qq3}).
\subsubsection*{Eccentricities}
\begin{theorem}\label{thm:qq3-ecc}
For every $G=(V,E)$, {\sc Eccentricities} can be solved in ${\cal O}(q(G)^3 + n+m)$-time.
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:mw-to-sw}, there exists a partial split decomposition of $G$ such that the only split component with diameter possibly larger than $2$ is its quotient graph $G'$.
Furthermore, as shown in the proof of Theorem~\ref{thm:mw-ecc}, solving {\sc Eccentricities} for $G$ can be reduced in ${\cal O}(n+m)$-time to the solving of {\sc Eccentricities} for $G'$.
By Lemma~\ref{lem:reduce-qq3} we only need to consider the following cases.
We can check in which case we are in linear-time~\cite{Bab00}.
\begin{itemize}
\item Suppose $G'=(S'\cup K'\cup R',E')$ is a prime spider.
There are two subcases.
\begin{enumerate}
\item
If $G'$ is a thick spider then it has diameter two.
Since in addition, there is no universal vertex in $G'$, therefore every vertex of $G'$ has eccentricity exactly two.
\item
Otherwise, $G'$ is a thin spider.
Since there is no universal vertex in $G'$, every vertex has eccentricity at least two.
In particular, since $K'$ is a clique dominating set of $G'$, $ecc_{G'}(v) = 2$ for every $v \in K'$.
Furthermore, since there is a join between $K'$ and $R'$, $ecc_{G'}(v) = 2$ for any $v \in R'$.
Finally, since every two vertices of $S'$ are pairwise at distance three, $ecc_{G'}(v) = 3$ for every $v \in S$.
\end{enumerate}
\item Suppose $G'$ is isomorphic either to a cycle $C_{n'}$, or to a co-cycle $\overline{C_{n'}}$, for some $n' \geq 5$.
\begin{enumerate}
\item
If $G'$ is isomorphic to a cycle $C_{n'}$ then every vertex of $G'$ has eccentricity $\left\lfloor n' / 2 \right\rfloor$.
\item
Otherwise, $G'$ is isomorphic to a co-cycle $\overline{C_{n'}}$.
We claim that every vertex of $G'$ has eccentricity $2$.
Indeed, let $v \in \overline{C_{n'}}$ be arbitrary and let $u,w \in \overline{C_{n'}}$ be the only two vertices nonadjacent to $v$.
Furthermore, let $u',w'$ be the unique vertices of $\overline{C_{n'}} \setminus v$ that are respectively nonadjacent to $u$ and to $w$.
Since $n' \geq 5$, we have $u' \neq w'$.
In particular, $(v,u',w)$ and $(v,w',u)$ are, respectively, a shortest $vw$-path and a shortest $vu$-path.
Hence, $ecc_{G'}(v) = 2$.
\end{enumerate}
\item Suppose $G'$ is a spiked $p$-chain $P_k$, or its complement.
\begin{itemize}
\item Subcase $G'$ is a spiked $p$-chain $P_k$.
In particular, $G'$ contains the $k$-node path $P_k = (v_1,v_2,\ldots,v_k)$ as an isometric subgraph.
Furthermore, if $x \in V(G')$ then $dist_{G'}(v_1,x) = 2$, and $x$ and $v_2$ are twins in $G' \setminus v_1$.
Similarly, if $y \in V(G')$ then $dist_{G'}(v_k,y) = 2$, and $y$ and $v_{k-1}$ are twins in $G' \setminus v_k$.
As a result: for every $1 \leq i \leq k, \ ecc_{G'}(v_i) = ecc_{P_k}(v_i) = \max\{i-1,k-i\}$; if $x \in V(G')$ then $ecc_{G'}(x) = ecc_{P_k}(v_2) = k-2$; if $y \in V(G')$ then $ecc_{G'}(y) = ecc_{P_k}(v_{k-1}) = k-2$.
\item Subcase $G'$ is a spiked $p$-chain $\overline{P_k}$.
In particular, $\overline{G'}$ is a spiked $p$-chain $P_k$.
Since $k \geq 6$, every spiked $p$-chain $P_k$ has diameter more than four.
Hence, $diam(G') \leq 2$, that implies {\sc Eccentricities} can be solved for $G'$ in linear-time.
\end{itemize}
\item Suppose $G'$ is a spiked $p$-chain $Q_k$, or its complement.
\begin{itemize}
\item Subcase $G'$ is a spiked $p$-chain $Q_k$.
There is a clique-dominating set $K' = \{v_2,v_4,\ldots,v_{2j},\ldots\}$ of $G'$.
In particular, every vertex of $K'$ has eccentricity two.
Furthermore, any $z_i$ is adjacent to both $v_2,v_4$.
Every vertex $v_{2i-1}$, except $v_3$, is adjacent to $v_2$.
Finally, every vertex $v_{2i-1}$, except $v_1$ and $v_5$, is adjacent to $v_4$.
As a result, any vertex $z_i$ has eccentricity two; any vertex $v_{2i-1}, \ i \notin \{1,2,3\}$, also has eccentricity two.
However, since $v_2$ and $v_4$ are, respectively, the only neighbours of $v_1$ and $v_3$, we get $dist_{G'}(v_1,v_3) = dist_{G'}(v_3,v_5) = 3$.
Hence $ecc_{G'}(v_1) = ecc_{G'}(v_3) = ecc_{G'}(v_5) = 3$.
\item Subcase $G'$ is a spiked $p$-chain $\overline{Q_k}$.
Roughly, we reverse the roles of vertices $v_{2i}$ with even index with the roles of vertices $v_{2i-1}$ with odd index.
More precisely, there is a clique-dominating set $K' = \{v_1,v_3,\ldots,v_{2j-1},\ldots\}$ of $G'$.
In particular, every vertex of $K'$ has eccentricity two.
Furthermore, any $z_i$ is adjacent to both $v_1,v_3$.
Every vertex $v_{2i}$, except $v_2$, is adjacent to $v_1$.
Finally, every vertex $v_{2i}$, except $v_4$, is adjacent to $v_3$.
As a result, any vertex $z_i$ has eccentricity two; any vertex $v_{2i}, \ i \notin \{1,2\}$, also has eccentricity two.
However, since $v_3$ is the only neighbour of $v_2$, we get $dist_{G'}(v_2,v_4) = 3$, hence $ecc_{G'}(v_2) = ecc_{G'}(v_4) = 3$.
\end{itemize}
\item Otherwise, $|V(G')| \leq q(G)$.
Then, solving {\sc Eccentricities} for $G'$ can be done in ${\cal O}(q(G)^3)$-time.
\end{itemize}
Therefore, in all the above cases, {\sc Eccentricities} can be solved for $G'$ in ${\cal O}(\min\{q(G)^3,n+m\})$-time.
\end{proof}
\begin{corollary}\label{cor:qq4-diam}
For every connected $(q,q-4)$-graph $G=(V,E)$, $diam(G) \leq q$.
\end{corollary}
Corollary~\ref{cor:qq4-diam} does not hold for $(q,q-3)$-graphs because of cycles and spiked $p$-chains $P_k$.
\subsubsection*{Gromov hyperbolicity}
\begin{theorem}\label{thm:qq4-hyp}
For every $G=(V,E)$, {\sc Hyperbolicity} can be solved in ${\cal O}(q(G)^3 \cdot n+m)$-time.
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:mw-to-sw}, we can compute a partial split decomposition from the modular decomposition of $G$.
It takes ${\cal O}(n+m)$-time.
Let $C_1, C_2, \ldots, C_k$ be the split components.
By using the algorithmic scheme of Theorem~\ref{thm:sw-hyp}, solving {\sc Hyperbolicity} can be reduced in ${\cal O}(\sum_i |V(C_i)|+|E(C_i)|)$-time to the computation, for every $1 \leq i \leq k$, of the hyperbolicity value $\delta(C_i)$ {\em and} of all the simplicial vertices in $C_i$.
We claim that it can be done in ${\cal O}(q(G)^3 \cdot |V(C_i)| + |E(C_i)|)$-time.
Since $\sum_i |V(C_i)| = {\cal O}(n)$ and $\sum_i |E(C_i)| = {\cal O}(n+m)$~\cite{Rao08b}, the latter claim will prove the desired time complexity.
If $C_i$ is degenerate then the above can be done in ${\cal O}(|V(C_i)|)$-time.
Otherwise, $C_i$ is obtained from a prime subgraph $G'$ in the modular decomposition of $G$ by possibly adding a universal vertex.
In particular, we have: $\delta(G') = \delta(C_i)$ if $G'=C_i$; $\delta(C_i) = 0$ can be decided in ${\cal O}(|V(C_i)|+|E(C_i)|)$-time~\cite{How79}; otherwise, $diam(C_i) \leq 2$, and so, by Lemma~\ref{lem:hyp-diam1} we have $\delta(C_i) = 1$ if and only if $C_i$ contains an induced cycle of length four (otherwise, $\delta(C_i) = 1/2$).
Therefore, we are left to compute the following for every prime subgraph $G'$ in the modular decomposition of $G$:
\begin{itemize}
\item Compute $\delta(G')$;
\item Decide whether $G'$ contains an induced cycle of length four;
\item Compute the simplicial vertices in $G'$.
\end{itemize}
In particular, if $|V(G')| \leq q(G)$ then it can be done in ${\cal O}(|V(G')|^4)={\cal O}(q(G)^3 \cdot |V(G')|)$-time.
Otherwise, by Lemma~\ref{lem:reduce-qq3} we only need to consider the following cases.
We can check in which case we are in linear-time~\cite{Bab00}.
\begin{itemize}
\item Suppose $G'$ is a prime spider.
In particular it is a split graph, and so, it does not contain an induced cycle of length more than three.
Furthermore the simplicial vertices of any chordal graph, and so, of $G'$, can be computed in linear-time.
If $G'$ is a thin spider then it is a block-graph, and so, $\delta(G') = 0$~\cite{How79}.
Otherwise, $G'$ is a thick spider, and so, it contains an induced diamond.
The latter implies $\delta(G') \geq 1/2$.
Since $diam(G') \leq 2$ and $G'$ is $C_4$-free, by Lemma~\ref{lem:hyp-diam1} $\delta(G') < 1$, hence we have $\delta(G') = 1/2$.
\item Suppose $G'$ is a cycle or a co-cycle of order at least five.
Since cycles and co-cycles are non complete regular graphs they do not contain any simplicial vertex~\cite{AiF84}.
Furthermore, a cycle of length at least five of course does not contain an induced cycle of length four; a co-cycle of order five is a $C_5$, and a co-cycle or order at least six always contains an induced cycle of length at least four since there is an induced $2K_2 = \overline{C_4}$ in its complement.
Finally, the hyperbolicity of a given cycle can be computed in linear-time~\cite{CCDL17+}; for every co-cycle of order at least six, since it has diameter at most two and it contains an induced cycle of length four, by Lemma~\ref{lem:hyp-diam1} it has hyperbolicity equal to $1$.
\item Suppose $G'$ is a spiked $p$-chain $P_k$, or its complement.
In particular, if $G'$ is a spiked $p$-chain $P_k$ then it is a block-graph, and so, a chordal graph.
It implies $\delta(G') = 0$~\cite{How79}, $G'$ does not contain any induced cycle of length four, furthermore all the simplicial vertices of $G'$ can be computed in linear-time.
Else, $G'$ is a spiked $p$-chain $\overline{P_k}$.
Since, in $\overline{G'}$, every vertex is nonadjacent to at least one edge of $P_k$, it implies that $G'$ has no simplicial vertex.
Furthermore, since $P_k$, and so, $\overline{G'}$, contains an induced $2K_2$, the graph $G'$ contains an induced cycle of length four.
Since $diam(G')=2$, it implies by Lemma~\ref{lem:hyp-diam1} $\delta(G') = 1$.
\item Otherwise, $G'$ is a spiked $p$-chain $Q_k$, or its complement.
In both cases, $G'$ is a split graph, and so, a chordal graph.
It implies that $G'$ does not contain an induced cycle of length four, and that all the simplicial vertices of $G'$ can be computed in linear-time.
Furthermore, we can decide in linear-time whether $\delta(G') = 0$~\cite{How79}.
Otherwise, it directly follows from the characterization in~\cite{BKM01} that a necessary condition for a chordal graph to have hyperbolicity at least one is to contain two disjoint pairs of vertices at distance $3$.
Since there are no such pairs in $G'$, $\delta(G') = 1/2$.
\end{itemize}
\end{proof}
The solving of {\sc Betweenness Centrality} for $(q,q-3)$-graphs is left for future work.
We think it is doable with the techniques of Theorem~\ref{thm:sw-bc}.
However, this would require to find ad-hoc methods for every graph family in Lemma~\ref{lem:reduce-qq3}.
The main difficulty is that we need to consider weighted variants of these graph families, and the possibility to add a universal vertex.
\subsection{Parameterized algorithms with split decomposition}\label{sec:algos-split}
We show how to use split decomposition as an efficient preprocessing method for {\sc Diameter}, {\sc Eccentricities}, {\sc Hyperbolicity} and {\sc Betweenness Centrality}.
Improvements obtained with modular decomposition will be discussed in Section~\ref{sec:algos-modular-dec}.
Roughly, we show that in order to solve the problems considered, it suffices to solve some {\em weighted} variant of the original problem for every split component (subgraphs of the split decomposition) separately.
However, weights intuitively represent the remaining of the graph, so, we need to account for some dependencies between the split components in order to define the weights properly.
In order to overcome this difficulty, we use in what follows a tree-like structure over the split components in order to design our algorithms.
A {\em split decomposition tree} of $G$ is a tree $T$ where the nodes are in bijective correspondance with the subgraphs of the split decomposition of $G$, and the edges of $T$ are in bijective correspondance with the simple decompositions used for their computation.
More precisely:
\begin{itemize}
\item If $G$ is either degenerate, or prime for split decomposition, then $T$ is reduced to a single node;
\item Otherwise, let $(A,B)$ be a split of $G$ and let $G_A = (A \cup \{b\}, E_A), \ G_B = (B \cup \{a\}, E_B)$ be the corresponding subgraphs of $G$.
We construct the split decomposition trees $T_A, T_B$ for $G_A$ and $G_B$, respectively.
Furthermore, the split marker vertices $a$ and $b$ are contained in a unique split component of $G_A$ and $G_B$, respectively.
We obtain $T$ from $T_A$ and $T_B$ by adding an edge between the two nodes that correspond to these subgraphs.
\end{itemize}
A split decomposition tree can be constructed in linear-time~\cite{CDR12}.
\subsubsection*{Diameter and Eccentricities}
\begin{lemma}\label{lem:diam-split}
Let $(A,B)$ be a split of $G=(V,E)$ and let $G_A = (A \cup \{b\}, E_A), \ G_B = (B \cup \{a\}, E_B)$ be the corresponding subgraphs of $G$.
Then, for every $u \in A$ we have: $$ecc_G(u) = \max \{ ecc_{G_A}(u), dist_{G_A}(u,b) + ecc_{G_B}(a) - 1\}.$$
\end{lemma}
\begin{proof}
Let $C = N_G(B) \subseteq A$ and $D = N_G(A) \subseteq B$.
In order to prove the claim, we first need to observe that, since $(A,B)$ is a split of $G$, we have, for every $v \in V$:
$$dist_G(u,v) = \begin{cases} dist_{G_A}(u,v) \ \mbox{if} \ v \in A \\
dist_G(u,C) + 1 + dist_G(v,D) \ \mbox{if} \ v \in B.\end{cases}$$
Furthermore, $dist_G(u,C) = dist_{G_A}(u,b) - 1$, and similarly $dist_G(v,D) = dist_{G_B}(v,a) - 1 \leq ecc_{G_B}(a) - 1$.
Hence, $ecc_G(u) \leq \max \{ ecc_{G_A}(u), dist_{G_A}(u,b) + ecc_{G_B}(a) - 1\}$.
Conversely, $ecc_{G_A}(u) = \max \{dist_{G_A}(u,b)\} \cup \{dist_{G_A}(u,v) \mid v \in A\} = \max \{dist_{G}(u,D)\} \cup \{dist_{G}(u,v) \mid v \in A\} \leq ecc_G(u)$.
In the same way, let $v \in B$ maximize $dist_G(v,C)$.
We have: $dist_G(u,v) = dist_{G_A}(u,b) + ecc_{G_B}(a) - 1 \leq ecc_G(u)$.
\end{proof}
\begin{theorem}\label{thm:sw-ecc}
For every $G=(V,E)$, {\sc Eccentricities} can be solved in ${\cal O}(sw(G)^2 \cdot n +m)$-time.
In particular, {\sc Diameter} can be solved in ${\cal O}(sw(G)^2 \cdot n +m)$-time.
\end{theorem}
\begin{proof}
Let $T$ be a split decomposition tree of $G$, with its nodes being in bijective correspondance with the split components $C_1, C_2, \ldots, C_k$.
It can be computed in linear-time~\cite{CDR12}.
We root $T$ in $C_1$.
For every $1 \leq i \leq k$, let $T_i$ be the subtree of $T$ that is rooted in $C_i$.
If $i > 1$ then let $C_{p(i)}$ be its parent in $T$.
By construction of $T$, the edge $\{C_{p(i)}, C_i\} \in E(T)$ corresponds to a split $(A_i,B_i)$ of $G$, where $V(C_i) \subseteq A_i$.
Let $G_{A_i} = (A_i \cup \{b_i\}, E_{A_i}), \ G_{B_i} = (B_i \cup \{a_i\}, E_{B_i})$ be the corresponding subgraphs of $G$.
We observe that $T_i$ is a split decomposition tree of $G_{A_i}$, $T \setminus T_i$ is a split decomposition tree of $G_{B_i}$.
Our algorithm proceeds in two main steps, with each step corresponding to a different traversal of the tree $T$.
First, let $G_1 = G$ and let $G_i = G_{A_i}$ for every $i > 1$.
We first compute, for every $1 \leq i \leq k$ and for every $v_i \in V(C_i)$, its eccentricity in $G_i$.
In order to do so, we proceed by dynamic programming on the tree $T$:
\begin{itemize}
\item If $C_i$ is a leaf of $T$ then {\sc Eccentricities} can be solved: in ${\cal O}(|V(C_i)|)$-time if $C_i$ induces a star or a complete graph; and in ${\cal O}(|V(C_i)|^3) = {\cal O}(sw(G)^2 \cdot |V(C_i)|)$-time else.
\item Otherwise $C_i$ is an internal node of $T$.
Let $C_{i_1}, C_{i_2}, \ldots, C_{i_l}$ be the children of $C_i$ in $T$.
Every edge $\{C_i, C_{i_t}\} \in E(T), \ 1 \leq t \leq l$ corresponds to a split $(A_{i_t},B_{i_t})$ of $G_i$, where $V(C_{i_t}) \subseteq A_{i_t}$.
We name $b_{i_t} \in V(C_{i_t}), \ a_{i_t} \in V(C_i)$ the vertices added after the simple decomposition.
Furthermore, let us define $e(a_{i_t}) = ecc_{G_{i_t}}(b_{i_t}) - 1$.
For every other vertex $u \in V(C_i) \setminus \{a_{i_1}, a_{i_2}, \ldots, a_{i_k}\}$, we define $e(u) = 0$.
Then, applying Lemma~\ref{lem:diam-split} for every split $(A_{i_t},B_{i_t})$ we get: $$\forall u \in V(C_i), \ ecc_{G_i}(u) = \max\limits_{v \in V(C_i)} dist_{C_i}(u,v) + e(v).$$
We distinguish between three cases.
\begin{enumerate}
\item
If $C_i$ is complete, then we need to compute $x_i \in V(C_i)$ maximizing $e(x_i)$, and $y_i \in V(C_i) \setminus \{x_i\}$ maximizing $e(y_i)$.
It can be done in ${\cal O}(|V(C_i)|)$-time.
Furthermore, for every $u \in V(C_i)$, we have $ecc_{G_i}(u) = 1 + e(x_i)$ if $u \neq x_i$, and $ecc_{G_i}(x_i) = \max \{e(x_i), 1 + e(y_i)\}$.
\item
If $C_i$ is a star with center node $r$, then we need to compute a leaf $x_i \in V(C_i) \setminus \{r\}$ maximizing $e(x_i)$, and another leaf $y_i \in V(C_i) \setminus \{x_i,r\}$ maximizing $e(y_i)$.
It can be done in ${\cal O}(|V(C_i)|)$-time.
Furthermore, $ecc_{G_i}(r) = \max \{e(r), 1 + e(x_i)\}, \ ecc_{G_i}(x_i) = \max\{ e(x_i), 1 + e(r), 2 + e(y_i) \}$, and for every other $u \in V(C_i) \setminus \{x_i,r\}$ we have $ecc_{G_i}(u) = \max \{1 + e(r), 2 + e(x_i)\}$.
\item Otherwise, $|V(C_i)| \leq sw(G)$, and so, all the eccentricities can be computed in ${\cal O}(|V(C_i)||E(C_i)|) = {\cal O}(sw(G)^2 \cdot |V(C_i)|)$-time.
\end{enumerate}
\end{itemize}
Overall, this step takes total time ${\cal O}(sw(G)^2 \cdot \sum_i |V(C_i)|) = {\cal O}(sw(G)^2 \cdot n)$.
Furthermore, since $G_1 = G$, we have computed $ecc_G(v_1)$ for every $v_1 \in V(C_1)$.
\medskip
Second, for every $2 \leq i \leq k$, we recall that by Lemma~\ref{lem:diam-split}: $$\forall v_i \in V(G_i), \ ecc_{G}(v_i) = \max \{ecc_{G_i}(v_i), dist_{G_i}(v_i,b_i) + ecc_{G_{B_i}}(a_i) - 1\}.$$
In particular, since we have already computed $ecc_{G_i}(v_i)$ for every $v_i \in V(C_i)$ (and as a byproduct, $dist_{G_i}(v_i,b_i)$), we can compute $ecc_G(v_i)$ from $ecc_{G_{B_i}}(a_i)$.
So, we are left to compute $ecc_{G_{B_i}}(a_i)$ for every $2 \leq i \leq k$.
In order to do so, we proceed by reverse dynamic programming on the tree $T$.
More precisely, let $C_{p(i)}$ be the parent node of $C_i$ in $T$, and let $C_{j_0} = C_i, C_{j_1}, C_{j_2}, \ldots, C_{j_k}$ denote the children of $C_{p(i)}$ in $T$.
For every $0 \leq t \leq k$, the edge $\{C_{p(i)}, C_{j_t}\}$ represents a split $(A_{j_t},B_{j_t})$, where $V(C_{j_t}) \subseteq A_{j_t}$.
So, there has been vertices $b_{j_t} \in V(C_{j_t}), \ a_{j_t} \in V(C_{p(i)})$ added by the corresponding simple decomposition.
We define $e'(a_{j_t}) = ecc_{G_{j_t}}(b_{j_t}) - 1$.
Furthermore, if $p(i) > 1$, let $C_{p^2(i)}$ be the parent of $C_{p(i)}$ in $T$.
Again, the edge $\{C_{p^2(i)},C_{p(i)}\}$ represents a split $(A_{p(i)},B_{p(i)})$, where $V(C_{p(i)}) \subseteq A_{p(i)}$.
So, there has been vertices $b_{p(i)} \in V(C_{p(i)}), \ a_{p(i)} \in V(C_{p^2(i)})$ added by the corresponding simple decomposition.
Let us define $e'(b_{p(i)}) = ecc_{G_{B_{p(i)}}}(a_{p(i)}) - 1$ (obtained by reverse dynamic programming on $T$).
Finally, for any other vertex $u \in V(C_{p(i)})$, let us define $e'(u) = 0$.
Then, by applying Lemma~\ref{lem:diam-split} it comes: $$\forall 0 \leq t \leq k, \ ecc_{G_{B_{i_t}}}(a_{i_t}) = \max\limits_{v \in V(C_{p(i)}) \setminus \{a_{i_t}\}} dist_{C_{p(i)}}(a_{i_t},v) + e'(v).$$
We can adapt the techniques of the first step in order to compute all the above values in ${\cal O}(sw(G)^2 \cdot |V(C_{p(i)})|)$-time.
Overall, the time complexity of the second step is also ${\cal O}(sw(G)^2 \cdot n)$.
Finally, since a split decomposition can be computed in ${\cal O}(n+m)$-time, and all of the subsequent steps take ${\cal O}(sw(G)^2 \cdot n)$-time, the total running time of our algorithm is an ${\cal O}(sw(G)^2 \cdot n + m)$.
\end{proof}
\subsubsection*{Gromov hyperbolicity}
It has been proved in~\cite{Sot11} that for every graph $G$, if every split component of $G$ is $\delta$-hyperbolic then $\delta(G) \leq \max\{1,\delta\}$.
We give a self-contained proof of this result, where we characterize the gap between $\delta(G)$ and the maximum hyperbolicity of its split components.
\begin{lemma}\label{lem:hyp-split}
Let $(A,B)$ be a split of $G=(V,E)$ and let $C = N_G(B) \subseteq A, \ D = N_G(A) \subseteq B$.
Furthermore, let $G_A = (A \cup \{b\}, E_A), \ G_B = (B \cup \{a\}, E_B)$ be the corresponding subgraphs of $G$.
Then, $\delta(G) = \max \{\delta(G_A), \delta(G_B), \delta^*\}$ where:
$$\delta^* =
\begin{cases}
1 & \mbox{if neither} \ C \ \mbox{nor} \ D \ \mbox{is a clique};\\
1/2 & \mbox{if} \ \min\{|C|,|D|\} \geq 2 \ \mbox{and exactly one of} \ C \ \mbox{or} \ D \ \mbox{is a clique}; \\
0 & \mbox{otherwise}.
\end{cases}$$
\end{lemma}
\begin{proof}
Since $G_A,G_B$ are isometric subgraphs of $G$, we have $\delta(G) \geq \max \{\delta(G_A), \delta(G_B)\}$.
Conversely, for every $u,v,x,y \in V$ define $L$ and $M$ to be the two largest sums amongst $\{ dist_G(u,v) + dist_G(x,y), dist_G(u,x) + dist_G(v,y), dist_G(u,y) + dist_G(v,x) \}$.
Write $\delta(u,v,x,y) = (L-M)/2$.
Furthermore, assume that $\delta(u,v,x,y) = \delta(G)$.
W.l.o.g., $|\{u,v,x,y\} \cap A| \geq |\{u,v,x,y\} \cap B|$.
In particular, if $u,v,x,y \in A$ then $\delta(u,v,x,y) \leq \delta(G_A)$.
Otherwise, there are two cases.
\begin{itemize}
\item Suppose $|\{u,v,x,y\} \cap A| = 3$.
W.l.o.g., $y \in B$.
Then, for every $w \in \{u,v,x\}$ we have $dist_G(w,y) = dist_{G_A}(w,b) + dist_{G_B}(a,y) - 1$.
Hence, $\delta(u,v,x,y) = \delta(u,v,x,b) \leq \delta(G_A)$.
\item Otherwise, $|\{u,v,x,y\} \cap A| = 2$.
W.l.o.g. $x,y \in B$.
Observe that $M = dist_G(u,x) + dist_G(v,y) = dist_G(u,y) + dist_G(v,x) = dist_{G_A}(u,b) + dist_{G_A}(v,b) + dist_{G_B}(a,x) + dist_{G_B}(a,y) - 2$.
Furthermore, $L = dist_G(u,v) + dist_G(x,y) \leq dist_{G_A}(u,b) + dist_{G_A}(v,b) + dist_{G_B}(a,x) + dist_{G_B}(a,y)$.
Hence, $\delta(u,v,x,y) = \max \{0, L - M\}/2 \leq 1$.
In particular:
\begin{itemize}
\item Suppose $\min \{|C|,|D|\} = 1$.
Then, the $4$-tuple $u,v,x,y$ is disconnected by some cut-vertex $c$.
In particular, $M = dist_G(u,c) + dist_G(v,c) + dist_G(c,x) + dist_G(c,y) \geq L$, and so, $\delta(u,v,x,y) = 0$.
Thus we assume from now on that $\min \{|C|,|D|\} \geq 2$.
\item Suppose $L - M = 2$.
It implies both $a$ is on a shortest $xy$-path (in $G_B$) and $b$ is on a shortest $uv$-path (in $G_A$).
Since there can be no simplicial vertices on a shortest path, we obtain that neither $a$ nor $b$ can be simplicial.
Thus, $C$ and $D$ are not cliques.
Conversely, if $C$ and $D$ are not cliques then there exists an induced $C_4$ with two ends in $C$ and two ends in $D$.
As a result, $\delta(G) \geq 1$.
\item Suppose $L - M = 1$.
Either $C$ or $D$ is not a clique.
Conversely, if either $C$ or $D$ is not a clique then, since we also assume $\min\{|C|,|D|\} \geq 2$, there exists either an induced $C_4$ or an induced diamond with two vertices in $C$ and two vertices in $D$.
As a result, $\delta(G) \geq 1/2$.
\end{itemize}
\end{itemize}
\end{proof}
\begin{theorem}\label{thm:sw-hyp}
For every $G=(V,E)$, {\sc Hyperbolicity} can be solved in ${\cal O}(sw(G)^3 \cdot n+m )$-time.
\end{theorem}
\begin{proof}
First we compute in linear-time the split components $C_1, C_2, \ldots, C_k$ of $G$.
By Lemma~\ref{lem:hyp-split}, we have $\delta(G) \geq \max_i\delta(C_i)$.
Furthermore, for every $1 \leq i \leq k$ we have: if $C_i$ induces a star or a complete graph, then $\delta(C_i) = 0$; otherwise, $|V(C_i)| \leq sw(G)$, and so, $\delta(C_i)$ can be computed in ${\cal O}(|V(C_i)|^4) = {\cal O}(sw(G)^3 \cdot |V(C_i)|)$-time, simply by iterating over all possible $4$-tuples.
Summarizing, we can compute $\max_i\delta(C_i)$ in ${\cal O}(sw(G)^3 \cdot \sum_i |V(C_i)|) = {\cal O}(sw(G)^3 \cdot n)$-time.
By Lemma~\ref{lem:hyp-split} we have $\delta(G) \leq \max \{1, \max_i\delta(C_i) \}$.
Therefore, if $\max_i\delta(C_i) \geq 1$ then we are done.
Otherwise, in order to compute $\delta(G)$, by Lemma~\ref{lem:hyp-split} it suffices to check whether the sides of every split used for the split decomposition induce a complete subgraph.
For that, we use a split decomposition tree $T$ of $G$.
Indeed, recall that the edges of $T$ are in bijective correspondance with the splits.
Let us root $T$ in $C_1$.
Notations are from the proof of Theorem~\ref{thm:sw-ecc}.
In particular, for every $1 \leq i \leq k$ let $T_i$ be the subtree of $T$ that is rooted in $C_i$.
If $i > 1$ then let $C_{p(i)}$ be its parent in $T$.
By construction of $T$, the edge $\{C_{p(i)}, C_i\} \in E(T)$ corresponds to a split $(A_i,B_i)$ of $G$, where $V(C_i) \subseteq A_i$.
Let $G_{A_i} = (A_i \cup \{b_i\}, E_{A_i}), \ G_{B_i} = (B_i \cup \{a_i\}, E_{B_i})$ be the corresponding subgraphs of $G$.
Vertex $a_i$ is simplicial in $G_{B_i}$ if and only if the side $N_G(A_i)$ is a clique.
Similarly, vertex $b_i$ is simplicial in $G_{A_i}$ if and only if the side $N_G(B_i)$ is a clique.
So, we perform tree traversals of $T$ in order to decide whether $a_i$ and $b_i$ are simplicial.
More precisely, we recall that $T_i$ and $T \setminus T_i$ are split decomposition trees of $G_{A_i}$ and $G_{B_i}$, respectively.
We now proceed in two main steps.
\begin{itemize}
\item
First, we decide whether $b_i$ is simplicial in $G_{A_i}$ by dynamic programming.
More precisely, let $C_{i_1}, C_{i_2}, \ldots, C_{i_k}$ be the children of $C_i$ in $T$. (possibly, $k=0$ if $C_i$ is a leaf).
Then, $b_i$ is simplicial in $G_{A_i}$ if and only if: it is simplicial in $C_i$; and for every $1 \leq t \leq k$ such that $\{b_i,a_{i_t}\} \in E(C_i)$, we have that $b_{i_t}$ is simplicial in $G_{A_{i_t}}$.
In particular, testing whether $b_{i}$ is simplicial in $C_i$ takes time: ${\cal O}(1)$ if $C_i$ induces a star or a complete graph; and ${\cal O}(|V(C_i)|^2) = {\cal O}(sw(G) \cdot |V(C_i)|)$ otherwise.
Since a vertex can have at most $|V(C_i)| - 1$ neighbours in $C_i$, testing whether $b_i$ is simplicial in $G_{A_i}$ can be done in ${\cal O}(|V(C_i)|)$ additional time.
So, overall, the first step takes ${\cal O}(sw(G) \cdot \sum_i|V(C_i)|) = {\cal O}(sw(G) \cdot n)$-time.
\item
Second, we decide whether $a_i$ is simplicial in $G_{B_i}$ by reverse dynamic programming.
Let $C_{j_0} = C_i, C_{j_1}, C_{j_2}, \ldots, C_{j_k}$ denote the children of $C_{p(i)}$ in $T$.
Furthermore, if $p(i) \neq 1$ then let $C_{p^2(i)}$ be the parent of $C_{p(i)}$ in $T$.
Then, $a_i$ is simplicial in $G_{B_i}$ if and only if: it is simplicial in $C_{p(i)}$; for every $1 \leq t \leq k$ such that $\{a_i,a_{j_t}\} \in E(C_{p(i)})$, we have that $b_{j_t}$ is simplicial in $G_{A_{j_t}}$; if $p(i) \neq 1$ and $\{a_i,b_{p(i)}\} \in E(C_{p(i)})$, we also have that $a_{p(i)}$ is simplicial in $G_{B_{p(i)}}$.
Testing, for every $0 \leq t \leq k$, whether $a_{j_t}$ is simplicial in $C_{p(i)}$ takes total time: ${\cal O}(|V(C_{p(i)})|)$ if $C_{p(i)}$ induces a star or a complete graph; and ${\cal O}(|V(C_{p(i)})|^3) = {\cal O}(sw(G)^2 \cdot |V(C_{p(i)})|)$ otherwise.
Then, for stars and prime components, we can test, for every $0 \leq t \leq k$, whether $a_{j_t}$ is simplicial in $G_{B_{j_t}}$ in total ${\cal O}(|E(C_{p(i)})|)$-time, that is ${\cal O}(|V(C_{p(i)})|)$ for stars and ${\cal O}(|V(C_{p(i)})|^2) = {\cal O}(sw(G) \cdot |V(C_{p(i)})|)$ for prime components.
For the case where $C_{p(i)}$ is a complete graph then, since all the vertices in $C_{p(i)}$ are pairwise adjacent, we only need to check whether there is at least one vertex $a_{j_t}$ such that $b_{j_t}$ is non simplicial in $G_{A_{j_t}}$, and also if $p(i) > 1$ whether $a_{p(i)}$ is non simplicial in $G_{B_{p(i)}}$.
It takes ${\cal O}(|V(C_{p(i)})|)$-time.
So, overall, the second step takes ${\cal O}(sw(G)^2 \cdot n)$-time.
\end{itemize}
\end{proof}
\begin{corollary}[~\cite{Sot11}]
For every connected $G=(V,E)$ we have $\delta(G) \leq \max\{1, \left\lfloor (sw(G)-1) /2 \right\rfloor \}$.
\end{corollary}
\subsubsection*{Betweenness Centrality}
The following subsection can be seen as a broad generalization of the preprocessing method presented in~\cite{PEZB14}.
We start introducing a generalization of {\sc Betweenness Centrality} for {\em vertex-weighted} graphs.
Admittedly, the proposed generalization is somewhat technical.
However, it will make easier the dynamic programming of Theorem~\ref{thm:sw-bc}.
Precisely, let $G=(V,E,\alpha,\beta)$ with $\alpha,\beta : V \to \mathbb{N}$ be weight functions.
Intuitively, for a split marker vertex $v$, $\alpha(v)$ represents the side of the split replaced by $v$, while $\beta(v)$ represents the total number of vertices removed by the simple decomposition.
For every path $P=(v_1,v_2,\ldots,v_{\ell})$ of $G$, the {\em length} of $P$ is equal to the number $\ell$ of edges in the path, while the {\em cost} of $P$ is equal to $\prod_{i=1}^{\ell} \alpha(v_i)$.
Furthermore, for every $s,t \in V$, the value $\sigma_G(s,t)$ is obtained by summing the cost over all the shortest $st$-paths in $G$.
Similarly, for every $s,t,v \in V$, the value $\sigma_G(s,t,v)$ is obtained by summing the cost over all the shortest $st$-paths in $G$ that contain $v$.
The betweenness centrality of vertex $v$ is defined as: $$\frac 1 {\alpha(v)} \sum_{s,t \in V \setminus v} \beta(s)\beta(t) \frac {\sigma_G(s,t,v)} {\sigma_G(s,t)}.$$
Note that if all weights are equal to $1$ then this is exactly the definition of Betweenness Centrality for unweighted graphs.
\begin{lemma}\label{lem:bc-split}
Let $(A,B)$ be a split of $G=(V,E,\alpha,\beta)$ and let $C = N_G(B) \subseteq A, \ D = N_G(A) \subseteq B$.
Furthermore, let $G_A = (A \cup \{b\}, E_A, \alpha_A, \beta_A), \ G_B = (B \cup \{a\}, E_B, \alpha_B, \beta_B)$ be the corresponding subgraphs of $G$, where:
$$\begin{cases}
\alpha_A(v) = \alpha(v), \ \beta_A(v) = \beta(v) \ \mbox{if} \ v \in A \\
\alpha_B(u) = \alpha(u), \ \beta_B(u) = \beta(u) \ \mbox{if} \ u \in B \\
\alpha_A(b) = \sum_{u \in D} \alpha(u), \ \beta_A(b) = \sum_{u \in B} \beta(u) \\
\alpha_B(a) = \sum_{v \in C} \alpha(v), \ \beta_B(a) = \sum_{v \in A} \beta(v).
\end{cases}$$
Then for every $v \in A$ we have:
$$BC_G(v) = BC_{G_A}(v) + [v \in C] BC_{G_B}(a).$$
\end{lemma}
\begin{proof}
Let $v \in A$ be fixed.
We consider all possible pairs $s,t \in V \setminus v$ such that $dist_G(s,t) = dist_G(s,v) + dist_G(v,t)$.
\smallskip
Suppose that $s,t \in A \setminus v$.
Since $(A,B)$ is a split, the shortest $st$-paths in $G$ are contained in $N_G[A] = A \cup D$.
In particular, the shortest $st$-paths in $G_A$ are obtained from the shortest $st$-paths in $G$ by replacing any vertex $d \in D$ by the split marker vertex $b$.
Conversely, the shortest $st$-paths in $G$ are obtained from the shortest $st$-paths in $G_A$ by replacing $b$ with any vertex $d \in D$.
Hence, $\sigma_{G_A}(s,t,b) = \sum_{d \in D} \sigma_G(s,t,d)$, that implies $\sigma_G(s,t) = \sigma_{G_A}(s,t)$.
Furthermore, $\sigma_G(s,t,v) = \sigma_{G}(s,v) \sigma_G(v,t) = \sigma_{G_A}(s,v)\sigma_{G_A}(v,t) = \sigma_{G_A}(s,t,v)$.
As a result, $\sigma_G(s,t,v)/\sigma_G(s,t) = \sigma_{G_A}(s,t,v)/\sigma_{G_A}(s,t)$.
\smallskip
Suppose that $s \in B, \ t \in A \setminus v$.
Every shortest $st$-path in $G$ is the concatenation of a shortest $sD$-path with a shortest $tC$-path.
Therefore, $\sigma_G(s,t) = \frac {\sigma_{G_B}(s,a) \cdot \sigma_{G_A}(b,t)}{\alpha_B(a)\cdot \alpha_A(b)}$.
We can furthermore observe $v$ is on a shortest $st$-path in $G$ if, and only if, $v$ is on a shortest $bt$-path in $G_A$.
Then, $\sigma_G(s,t,v) = \sigma_{G}(s,v) \sigma_G(v,t) = \frac {\sigma_{G_B}(s,a) \cdot \sigma_{G_A}(b,v)}{\alpha_B(a)\cdot \alpha_A(b)} \sigma_{G_A}(v,t)$.
As a result, $\sigma_G(s,t,v)/\sigma_G(s,t) = \sigma_{G_A}(b,t,v)/\sigma_{G_A}(b,t)$.
\smallskip
Finally, suppose that $s,t \in B$.
Again, since $(A,B)$ is a split the shortest $st$-paths in $G$ are contained in $N_G[B] = B \cup C$.
In particular, $\sigma_G(s,t,v) \neq 0$ if, and only if, we have $v \in C$ and $\sigma_{G_B}(s,t,a) \neq 0$.
More generally, if $v \in C$ then $\sigma_G(s,t,v) = \frac {\alpha_A(v)} {\alpha_B(a)} \sigma_{G_B}(s,t,a)$.
As a result, if $v \in C$ then $\sigma_G(s,t,v)/\sigma_G(s,t) = \frac {\alpha_A(v)} {\alpha_B(a)} \cdot \sigma_{G_B}(s,t,a)/\sigma_{G_B}(s,t)$.
\medskip
Overall, we have:
\begin{align*}
BC_G(v) &= \frac 1 {\alpha(v)} \sum_{s,t \in V \setminus v} \beta(s)\beta(t) \frac {\sigma_G(s,t,v)} {\sigma_G(s,t)} \\
&= \frac 1 {\alpha(v)} \sum_{s,t \in A \setminus v} \beta(s)\beta(t) \frac {\sigma_G(s,t,v)} {\sigma_G(s,t)} + \frac 1 {\alpha(v)} \sum_{s \in B, \ t \in A \setminus v} \beta(s)\beta(t) \frac {\sigma_G(s,t,v)} {\sigma_G(s,t)} + \frac 1 {\alpha(v)} \sum_{s,t \in B} \beta(s)\beta(t) \frac {\sigma_G(s,t,v)} {\sigma_G(s,t)}\\
&= \frac 1 {\alpha_A(v)} \sum_{s,t \in A \setminus v} \beta_A(s)\beta_A(t) \frac {\sigma_{G_A}(s,t,v)}{\sigma_{G_A}(s,t)} + \frac 1 {\alpha_A(v)} \sum_{s \in B, \ t \in A \setminus v} \beta_B(s)\beta_A(t)\frac {\sigma_{G_A}(b,t,v)}{\sigma_{G_A}(b,t)} \\ &\ + \frac 1 {\alpha_A(v)} [v \in C] \sum_{s,t \in B} \beta_B(s)\beta_B(t)\frac {\alpha_A(v)} {\alpha_B(a)} \cdot \frac {\sigma_{G_B}(s,t,a)}{\sigma_{G_B}(s,t)}\\
&= \left( BC_{G_A}(v) - \frac {\beta_A(b)} {\alpha_A(v)} \sum_{t \in A \setminus v} \beta_A(t) \frac{\sigma_{G_A}(b,t,v)}{\sigma_{G_A}(b,t)} \right) + \frac {\sum_{s \in B} \beta(s)} {\alpha_A(v)} \sum_{t \in A \setminus v} \beta_A(t) \frac{\sigma_{G_A}(b,t,v)}{\sigma_{G_A}(b,t)} + [v \in C] BC_{G_B}(a) \\
&= BC_{G_A}(v) + [v \in C] BC_{G_B}(a),
\end{align*}
that finally proves the lemma.
\end{proof}
\begin{theorem}\label{thm:sw-bc}
For every $G=(V,E)$, {\sc Betweenness Centrality} can be solved in ${\cal O}(sw(G)^2 \cdot n + m)$-time.
\end{theorem}
\begin{proof}
Let $T$ be a split decomposition tree of $G$, with its nodes being in bijective correspondance with the split components $C_1, C_2, \ldots, C_k$.
It can be computed in linear-time~\cite{CDR12}.
As for Theorem~\ref{thm:sw-ecc}, we root $T$ in $C_1$.
For every $1 \leq i \leq k$, let $T_i$ be the subtree of $T$ that is rooted in $C_i$.
If $i > 1$ then let $C_{p(i)}$ be its parent in $T$.
We recall that by construction of $T$, the edge $\{C_{p(i)}, C_i\} \in E(T)$ corresponds to a split $(A_i,B_i)$ of $G$, where $V(C_i) \subseteq A_i$.
Furthermore, let $G_{A_i} = (A_i \cup \{b_i\}, E_{A_i}), \ G_{B_i} = (B_i \cup \{a_i\}, E_{B_i})$ be the corresponding subgraphs of $G$.
We observe that $T_i$ is a split decomposition tree of $G_{A_i}$, while $T \setminus T_i$ is a split decomposition tree of $G_{B_i}$.
Let us assume $G=(V,E,\alpha,\beta)$ to be vertex-weighted, with initially $\alpha(v) = \beta(v) = 1$ for every $v \in V$.
For every $i > 1$, let $G_{A_i} = (A_i \cup \{b_i\}, E_{A_i}, \alpha_{A_i}, \beta_{A_i}), \ G_{B_i} = (B_i \cup \{a_i\}, E_{B_i}, \alpha_{B_i}, \beta_{B_i})$ be as described in Lemma~\ref{lem:bc-split}.
In particular, for every $i > 1$:
$$\begin{cases}
\alpha_{A_i}(v) = \alpha(v) = 1, \ \beta_{A_i}(v) = \beta(v) = 1 \ \mbox{if} \ v \in A_i \\
\alpha_{B_i}(u) = \alpha(u) = 1, \ \beta_{B_i}(u) = \beta(u) = 1 \ \mbox{if} \ u \in B_i \\
\alpha_{A_i}(b_i) = |N_G(A_i)|, \ \beta_{A_i}(b_i) = |B_i| \\
\alpha_{B_i}(a_i) = |N_G(B_i)|, \ \beta_{B_i}(a_i) = |A_i|.
\end{cases}$$
Hence, all the weights can be computed in linear-time by dynamic programming over $T$.
We set $G_1 = G$ while $G_i = G_{A_i}$ for every $i >1$.
Furthermore, we first aim at computing $BC_{G_i}(v)$ for every $v \in V(C_i)$.
If $C_i$ is a leaf of $T$ then there are three cases to be considered.
\begin{enumerate}
\item Suppose $G_i$ is a complete graph.
Then, for every $v \in V(C_i)$ we have $BC_{G_i}(v) = 0$.
\item Suppose $G_i$ is a star, with center node $r$.
In particular, $BC_{G_i}(v) = 0$ for every $v \in V(C_i) \setminus \{r\}$.
Furthermore, since $r$ is onto the unique shortest path between every two leaves $s,t \in V(C_i) \setminus \{r\}$, we have $\sigma_{G_i}(s,t,r) = \sigma_{G_i}(s,t)$.
Let us write $\beta(G_i) = \sum_{v \in V(C_i) \setminus \{r\}} \beta_{G_i}(v)$.
We have:
\begin{align*}
BC_{G_i}(r) &= \frac 1 {\alpha_{G_i}(r)} \sum_{s,t \in V(C_i) \setminus \{r\}} \beta_{G_i}(s)\beta_{G_i}(t) \\
&= \frac 1 {2\alpha_{G_i}(r)} \sum_{s \in V(C_i) \setminus \{r\}} \beta_{G_i}(s) \left(\sum_{t \in V(C_i) \setminus \{r,s\}} \beta_{G_i}(t)\right) \\
&= \frac 1 {2\alpha_{G_i}(r)} \sum_{s \in V(C_i) \setminus \{r\}} \beta_{G_i}(s) \left(\beta(G_i) - \beta_{G_i}(s)\right).
\end{align*}
It can be computed in ${\cal O}(|V(C_i)|)$-time.
\item Finally, suppose $G_i$ is prime for split decomposition.
Brandes algorithm~\cite{Bra01} can be generalized to that case.
For every $v \in V(C_i)$, we first compute a BFS ordering from $v$.
It takes ${\cal O}(|E(C_i)|)$-time.
Furthermore for every $u \in V(C_i) \setminus \{v\}$, let $N^+(u)$ be the neighbours $w \in N_{C_i}(u)$ such that $w$ is on a shortest $uv$-path.
We compute $\sigma_{G_i}(u,v)$ by dynamic programming.
Precisely, $\sigma_{G_i}(v,v) = \alpha_{G_i}(v)$, and for every $u \neq v, \ \sigma_{G_i}(u,v) = \alpha_{G_i}(u) \cdot \left(\sum_{w \in N^+(u)}\sigma_{G_i}(w,v)\right)$.
It takes ${\cal O}(|E(C_i)|)$-time.
Overall in ${\cal O}(|V(C_i)||E(C_i)|)$-time, we have computed $\sigma_{G_i}(u,v)$ and $dist_{G_i}(u,v)$ for every $u,v \in V(C_i)$.
Then, for every $v \in V(C_i)$, we can compute $BC_{G_i}(v)$ in ${\cal O}(|V(C_i)|^2)$-time by enumerating all the pairs $s,t \in V(C_i) \setminus \{v\}$.
Since $G_i$ is prime, the total running time is in ${\cal O}(|V(C_i)|^3) = {\cal O}(sw(G)^3)$, and so, in ${\cal O}(sw(G)^2 \cdot |V(C_i)|)$.
\end{enumerate}
Otherwise, $C_i$ is an internal node of $T$.
Let $C_{i_1}, C_{i_2}, \ldots, C_{i_k}$ be the children of $C_i$ in $T$.
Assume that, for every $1 \leq t \leq k$, $BC_{G_{i_t}}(b_{i_t})$ has been computed (by dynamic programming over $T$).
Let us define the following weight functions for $C_i$:
$$\begin{cases}
\alpha_i(a_{i_t}) = \alpha_{B_{i_t}}(a_{i_t}), \ \beta_i(a_{i_t}) = \beta_{B_{i_t}}(a_{i_t}) \\
\alpha_i(v) = \alpha_{A_{i}}(v), \ \beta_i(v) = \beta_{A_{i}}(v) \ \mbox{otherwise.}
\end{cases}$$
Observe that every edge $\{C_i,C_{i_t}\}$ also corresponds to a split $(A_{i_t}',B_{i_t}')$ of $G_i$, where $V(C_{i_t}) \subseteq A_{i_t}' = A_{i_t}$.
By applying all the corresponding simple decompositions, one finally obtains $H_i = (V(C_i),E(C_i),\alpha_i,\beta_i)$.
Then, let us define $\ell_i(a_{i_t}) = BC_{G_{i_t}}(b_{i_t})$ and $\ell_i(v) = 0$ else.
Intuitively, the function $\ell_i$ is a corrective term updated after each simple decomposition.
More precisely, we obtain by multiple applications of Lemma~\ref{lem:bc-split}, for every $v \in V(C_i)$: $$BC_{G_i}(v) = BC_{H_i}(v) + \sum_{u \in N_{H_i}(v)} \ell_i(u) $$
Clearly, this can be reduced in ${\cal O}(|E(H_i)|)$-time, resp. in ${\cal O}(|V(H_i)|)$-time when $H_i$ is complete, to the computation of $BC_{H_i}(v)$.
So, it can be done in ${\cal O}(sw(G)^2 \cdot |V(C_i)|)$-time ({\it i.e.}, as explained for the case of leaf nodes).
Overall, this first part of the algorithm takes time ${\cal O}(sw(G)^2 \cdot \sum_i |V(C_i)|) = {\cal O}(sw(G)^2 \cdot n)$.
Furthermore, since $G_1 = G$, we have computed $BC_G(v)$ for every $v \in V(C_1)$.
Then, using the same techniques as above, we can compute $BC_{G_{B_i}}(a_i)$ for every $i > 1$ by reverse dynamic programming over $T$.
It takes ${\cal O}(sw(G)^2 \cdot n)$-time.
Finally, by Lemma~\ref{lem:bc-split} we can compute $BC_G(v)$ from $BC_{G_i}(v)$ and $BC_{G_{B_i}}(a_i)$, for every $v \in V(C_i)$.
It takes linear-time.
\end{proof}
\section{Parameterization, Hardness and Kernelization for some distance problems on graphs}\label{sec:dist}
We prove separability results between clique-width and the upper-bounds for clique-width presented in Section~\ref{sec:prelim}.
More precisely, we consider the problems {\sc Diameter}, {\sc Eccentricities}, {\sc Hyperbolicity} and {\sc Betweenness Centrality} (defined in Section~\ref{sec:dist-pbs}), that have already been well studied in the field of ``Hardness in P''.
On the negative side, we show in Section~\ref{sec:hardness-cw} that we cannot solve these three above problems with a {\em fully polynomial} parameterized algorithm, when parameterized by clique-width.
However, on a more positive side, we prove the existence of such algorithms in Sections~\ref{sec:algos-split},~\ref{sec:algos-modular-dec} and~\ref{sec:dist-qq3}, when parameterized by either the modular-width, the split-width or the $P_4$-sparseness.
\input{dist-pbs}
\input{dist-hardness}
\input{dist-split}
\input{dist-modular}
\input{dist-qq-3}
\section{Introduction}
The classification of problems according to their complexity is one of the main goals in computer science.
This goal was partly achieved by the theory of NP-completeness which helps to identify the problems that are unlikely to have polynomial-time algorithms.
However, there are still many problems in P for which it is not known if the running time of the best current algorithms can be improved.
Such problems arise in various domains such as computational geometry, string matching or graphs.
Here we focus on the existence and the design of {\em linear-time} algorithms, for solving several graph problems when restricted to classes of bounded {\em clique-width}.
The problems considered comprise the detection of short cycles ({\it e.g.}, {\sc Girth} and {\sc Triangle Counting}), some distance problems ({\it e.g.}, {\sc Diameter}, {\sc Hyperbolicity}, {\sc Betweenness Centrality}) and the computation of maximum matchings in graphs.
We refer to Sections~\ref{sec:pbs-cw},~\ref{sec:dist-pbs} and~\ref{sec:maxmatching}, respectively, for a recall of their definitions.
Clique-width is an important graph parameter in structural graph theory, that intuitively represents the closeness of a graph to a cograph --- {\it a.k.a.}, $P_4$-free graphs~\cite{CPS85,CoB00}.
Some classes of perfect graphs, including distance-hereditary graphs, and so, trees, have bounded clique-width~\cite{GoR00}.
Furthermore, clique-width has many algorithmic applications.
Many algorithmic schemes and metatheorems have been proposed for classes of bounded clique-width~\cite{CMR00,Cou12,EGW01}.
Perhaps the most famous one is Courcelle's theorem, that states that every graph problem expressible in Monadic Second Order logic ($MSO_1$) can be solved in $f(k) \cdot n$-time when restricted to graphs with clique-width at most $k$, for some computable function $f$ that only depends on $k$~\cite{CMR00}.
Some of the problems considered in this work can be expressed as an $MSO_1$ formula.
However, the dependency on the clique-width in Courcelle's theorem is super-polynomial, that makes it less interesting for the study of graphs problems in P.
Our goal is to derive a {\em finer-grained} complexity of polynomial graph problems when restricted to classes of bounded clique-width, that requires different tools than Courcelle's theorem.
\medskip
Our starting point is the recent theory of ``Hardness in P'' that aims at better hierarchizing the complexity of polynomial-time solvable problems~\cite{VaW15}.
This approach mimics the theory of NP-completeness.
Precisely, since it is difficult to obtain unconditional hardness results, it is natural to obtain hardness results assuming some complexity theoretic conjectures.
In other words, there are key problems that are widely believed not to admit better algorithms such as 3-SAT (k-SAT), 3SUM and All-Pairs Shortest Paths (APSP).
Roughly, a problem in P is hard if the existence of a faster algorithm for this problem implies the existence of a faster algorithm for one of these fundamental problems mentioned above.
In their seminal work, Williams and Williams~\cite{VaW10} prove that many important problems in graph theory are all equivalent under subcubic reductions.
That is, if one of these problems admits a truly sub-cubic algorithms, then all of them do.
Their results have extended and formalized prior work from, {\it e.g.},~\cite{GaO95,KrS06}.
The list of such problems was further extended in~\cite{AGV15,BCH16}.
Besides purely negative results ({\it i.e.}, conditional lower-bounds) the theory of ``Hardness in P'' also comes with renewed algorithmic tools in order to leverage the existence, or the nonexistence, of improved algorithms for some graph classes.
The tools used to improve the running time of the above mentioned problems are similar to the ones used to tackle NP-hard problems, namely approximation and FPT algorithms.
Our work is an example of the latter, of which we first survey the most recent results.
\paragraph{Related work: Fully polynomial parameterized algorithms.}
FPT algorithms for polynomial-time solvable problems were first considered by Giannopoulou et al.~\cite{GMN17}.
Such a parameterized approach makes sense for any problem in P for which a conditional hardness result is proved, or simply no linear-time algorithms are known.
Interestingly, the authors of~\cite{GMN17} proved that a matching of cardinality at least $k$ in a graph can be computed in ${\cal O}(kn + k^3)$-time.
We stress that {\sc Maximum Matching} is a classical and intensively studied problem in computer science~\cite{DuP14,FGV99,FPT97,GaT83,KaS81,MiV80,MNN16,YuY93}.
The well known ${\cal O} (m\sqrt{n})$-time algorithm in~\cite{MiV80} is essentially the best so far for {\sc Maximum Matching}.
Approximate solutions were proposed by Duan and Pettie~\cite{DuP14}.
More related to our work is the seminal paper of Abboud, Williams and Wang~\cite{AVW16}.
They obtained rather surprising results when using {\em treewidth}: another important graph parameter that intuitively measures the closeness of a graph to a tree~\cite{Bod06}.
Treewidth has tremendous applications in pure graph theory~\cite{RoS86} and parameterized complexity~\cite{Cou90}.
Furthermore, improved algorithms have long been known for ''hard'' graph problems in P, such as {\sc Diameter} and {\sc Maximum Matching}, when restricted to trees~\cite{Jor69}.
However, it has been shown in~\cite{AVW16} that under the Strong Exponential Time Hypothesis, for any $\varepsilon >0$ there can be no $2^{o(k)} \cdot n^{2-\varepsilon}$-time algorithm for computing the diameter of graphs with treewidth at most $k$.
This hardness result even holds for {\em pathwidth}, that leaves little chance to find an improved algorithm for any interesting subclass of bounded-treewidth graphs while avoiding an exponential blow-up in the parameter.
We show that the situation is different for clique-width than for treewidth, in the sense that the hardness results for clique-width do not hold for important subclasses.
We want to stress that a familiar reader could ask why the hardness results above do not apply to clique-width directly since it is upper-bounded by a function of treewidth~\cite{CoR05}.
However, clique-width cannot be {\em polynomially} upper-bounded by the treewidth~\cite{CoR05}.
Thus, the hardness results from~\cite{AVW16} do not preclude the existence of, say, an ${\cal O}(k n)$-time algorithm for computing the diameter of graphs with clique-width at most $k$.
\medskip
On a more positive side, the authors in~\cite{AVW16} show that {\sc Radius} and {\sc Diameter} can be solved in $2^{{\cal O}(k\log{k})} \cdot n^{1+{\cal O}(1)}$-time, where $k$ is treewidth.
Husfeldt~\cite{Hus16} shows that the eccentricity of every vertex in an undirected graph on $n$ vertices can be computed in time $n \cdot \text{exp}\left[{\cal O}(k \log d) \right]$, where $k$ and $d$ are the treewidth and the diameter of the graph, respectively.
More recently, a tour de force was achieved by Fomin et al.~\cite{FLPS+17} who were the first to design parameterized algorithms with {\em polynomial dependency} on the treewidth, for {\sc Maximum Matching} and {\sc Maximum Flow}.
Furthermore they proved that for graphs with treewidth at most $k$, a tree decomposition of width ${\cal O}(k^2)$ can be computed in ${\cal O}(k^7 \cdot n\log n)$-time.
We observe that their algorithm for {\sc Maximum Matching} is {\em randomized}, whereas ours are deterministic.
We are not aware of the study of another parameter than treewidth for polynomial graph problems.
However, some authors choose a different approach where they study the parameterization of a fixed graph problem for a broad range of graph invariants~\cite{BFNN17,FKMN+17,MNN16}.
As examples, {\em clique-width} is part of the graph invariants used in the parameterized study of {\sc Triangle Listing}~\cite{BFNN17}.
Nonetheless, clique-width is not the main focus in~\cite{BFNN17}.
Recently, Mertzios, Nichterlein and Niedermeier~\cite{MNN16} propose algorithms for {\sc Maximum Matching} that run in time ${\cal O} (k^{{ \cal O} (1)} \cdot (n+m))$, for several parameters such as feedback vertex set or feedback edge set.
Moreover, the authors in~\cite{MNN16} suggest that {\sc Maximum Matching} may become the ``drosophila'' of the study of the FPT algorithms in P.
We advance in this research direction.
\subsection{Our results}
In this paper we study the parameterized complexity of several classical graph problems under a wide range of parameters such as clique-width and its upper-bounds {\em modular-width}~\cite{CoB00}, {\em split-width}~\cite{Rao08b}, {\em neighbourhood diversity}~\cite{Lam12} and {\em $P_4$-sparseness}~\cite{BaO99}.
The results are summarized in Table~\ref{tab:summary}.
\smallskip
Roughly, it turns out that some hardness assumptions for general graphs do not hold anymore for graph classes of bounded clique-width.
This is the case in particular for {\sc Triangle Detection} and other cycle problems that are subcubic equivalent to it such as, {\it e.g.}, {\sc Girth}, that all can be solved in linear-time, with quadratic dependency on the clique-width, with the help of dynamic programming (Theorems~\ref{thm:cw-triangle} and~\ref{thm:cw-girth}).
The latter complements the results obtained for {\sc Triangle Listing} in~\cite{BFNN17}.
However many hardness results for {\em distance problems} when using treewidth are proved to also hold when using clique-width (Theorems~\ref{thm:cw-diam},~\ref{thm:cw-bc} and~\ref{thm:cw-hyp}).
These negative results have motivated us to consider some upper-bounds for clique-width as parameters, for which better results can be obtained than for clique-width.
Another motivation stems from the fact that the existence of a parameterized algorithm for computing the clique-width of a graph remains a challenging open problem~\cite{CHLB+00}.
We consider some upper-bounds for clique-width that are defined via {\em linear-time} computable graph decompositions.
Thus if these parameters are small enough, say, in ${\cal O}(n^{1-\varepsilon})$ for some $\varepsilon > 0$, we get truly subcubic or even truly subquadratic algorithms for a wide range of problems.
\input{summary-results}
\subsubsection*{Graph parameters and decompositions considered}
Let us describe the parameters considered in this work as follows.
The following is only an informal high level description (formal definitions are postponed to Section~\ref{sec:prelim}).
\paragraph*{\sc Split Decomposition.}
A {\em join} is a set of edges inducing a complete bipartite subgraph.
Roughly, clique-width can be seen as a measure of how easy it is to reconstruct a graph by adding joins between some vertex-subsets.
A {\em split} is a join that is also an edge-cut.
By using pairwise non crossing splits, termed ``strong splits'', we can decompose any graph into degenerate and prime subgraphs, that can be organized in a treelike manner.
The latter is termed {\em split decomposition}~\cite{GiP12}.
We take advantage of the treelike structure of split decomposition in order to design dynamic programming algorithms for distance problems such as {\sc Diameter}, {\sc Gromov Hyperbolicity} and {\sc Betweenness Centrality} (Theorems~\ref{thm:sw-ecc},~\ref{thm:sw-hyp} and~\ref{thm:sw-bc}, respectively).
Although clique-width is also related to some treelike representations of graphs~\cite{CHMPR15}, the same cannot be done for clique-width as for split decomposition because the edges in the treelike representations for clique-width may not represent a join.
\paragraph*{\sc Modular Decomposition.}
Then, we can improve the results obtained with split decomposition by further restricting the type of splits considered.
As an example, let $(A,B)$ be a bipartition of the vertex-set that is obtained by removing a split.
If every vertex of $A$ is incident to some edges of the split then $A$ is called a {\em module} of $G$.
That is, for every vertex $v \in B$, $v$ is either adjacent or nonadjacent to every vertex of $A$.
The well-known {\em modular decomposition} of a graph is a hierarchical decomposition that partitions the vertices of the graph with respect to the modules~\cite{HaP10}.
Split decomposition is often presented as a refinement of modular decomposition~\cite{GiP12}.
We formalize the relationship between the two in Lemma~\ref{lem:mw-to-sw}, that allows us to also apply our methods for split decomposition to modular decomposition.
However, we can often do better with modular decomposition than with split decomposition.
In particular, suppose we partition the vertex-set of a graph $G$ into modules, and then we keep exactly one vertex per module.
The resulting {\em quotient graph} $G'$ keeps most of the distance properties of $G$.
Therefore, in order to solve a distance problem for $G$, it is often the case that we only need to solve it for $G'$.
We so believe that modular decomposition can be a powerful {\em Kernelization} tool in order to solve graph problems in P.
As an application, we improve the running time for some of our algorithms, from time ${\cal O}(k^{{\cal O}(1)} \cdot n + m)$ when parameterized by the {\em split-width} (maximum order of a prime subgraph in the split decomposition), to ${\cal O}(k^{{\cal O}(1)} + n + m)$-time when parameterized by the {\em modular-width} (maximum order of a prime subgraph in the modular decomposition).
See Theorem~\ref{thm:mw-ecc}.
Furthermore, for some more graph problems, it may also be useful to further restrict the internal structures of modules.
We briefly explore this possibility through a case study for {\em neighbourhood diversity}.
Roughly, in this latter case we only consider modules that are either independent sets (false twins) or cliques (true twins).
New kernelization results are obtained for {\sc Hyperbolicity} and {\sc Betweenness Centrality} when parameterized by the neighbourhood diversity (Theorems~\ref{thm:nd-hyp} and~\ref{thm:nd-bc}, respectively).
It is worth pointing out that so far, we have been unable to obtain kernelization results for {\sc Hyperbolicity} and {\sc Betweenness Centrality} when only parameterized by the modular-width.
It would be very interesting to prove separability results between split-width, modular-width and neighbourhood diversity in the field of fully polynomial parameterized complexity.
\paragraph*{\sc Graphs with few $P_4$'s.}
We finally use modular decomposition as our main tool for the design of new linear-time algorithms when restricted to graphs with few induced $P_4$'s.
The $(q,t)$-graphs have been introduced by Babel and Olariu in~\cite{BaO98}.
They are the graphs in which no set of at most $q$ vertices can induce more than $t$ paths of length four.
Every graph is a $(q,t)$-graph for some large enough values of $q$ and $t$.
Furthermore when $q$ and $t$ are fixed constants, $t \leq q-3$, the class of $(q,t)$-graphs has bounded clique-width~\cite{MaR99}.
We so define the $P_4$-sparseness of a given graph $G$, denoted by $q(G)$, as the minimum $q \geq 7$ such that $G$ is a $(q,q-3)$-graph.
The structure of the quotient graph of a $(q,q-3)$-graph, $q$ being a constant, has been extensively studied and characterized in the literature~\cite{Bab98,BaO98,BaO99,Bab00,JaO95}.
We take advantage of these existing characterizations in order to generalize our algorithms with modular decomposition to ${\cal O}(q(G)^{{\cal O}(1)} \cdot n + m)$-time algorithms (Theorems~\ref{thm:qq3-ecc} and~\ref{thm:qq4-hyp}).
Let us give some intuition on how the $P_4$-sparseness can help in the design of improved algorithms for hard graph problems in P.
We consider the class of {\em split graphs} ({\it i.e.}, graphs that can be bipartitioned into a clique and an independent set).
Deciding whether a given split graph has diameter $2$ or $3$ is hard~\cite{BCH16}.
However, suppose now that the split graph is a $(q,q-3)$-graph $G$, for some fixed $q$.
An induced $P_4$ in $G$ has its two ends $u,v$ in the independent set, and its two middle vertices are, respectively, in $N_G(u) \setminus N_G(v)$ and $N_G(v) \setminus N_G(u)$.
Furthermore, when $G$ is a $(q,q-3)$-graph, it follows from the characterization of~\cite{Bab98,BaO98,BaO99,Bab00,JaO95} either it has a quotient graph of bounded order ${\cal O}(q)$ or it is part of a well-structured subclass where the vertices of all neighbourhoods in the independent set follow a rather nice pattern (namely, spiders and a subclass of $p$-trees, see Section~\ref{sec:prelim}).
As a result, the diameter of $G$ can be computed in ${\cal O}(\max\{q^3,n+m\})$-time when $G$ is a $(q,q-3)$ split graph.
We generalize this result to every $(q,q-3)$-graph by using modular decomposition.
\medskip
All the parameters considered in this work have already received some attention in the literature, especially in the design of FPT algorithms for NP-hard problems~\cite{Bab00,GaP03,GiP12,GLO13,Rao08b}.
However, we think we are the first to study clique-width and its upper-bounds for polynomial problems.
There do exist linear-time algorithms for {\sc Diameter}, {\sc Maximum Matching} and some other problems we study when restricted to some graph classes where the split-width or the $P_4$-sparseness is bounded ({\it e.g.}, cographs~\cite{YuY93}, distance-hereditary graphs~\cite{Dra97,DrN00}, $P_4$-tidy graphs~\cite{FPT97}, etc.).
Nevertheless, we find the techniques used for these specific subclasses hardly generalize to the case where the graph has split-width or $P_4$-sparseness at most $k$, $k$ being any fixed constant.
For instance, the algorithm that is proposed in~\cite{DrN00} for computing the diameter of a given distance-hereditary graph is based on some properties of LexBFS orderings.
Distance-hereditary graphs are exactly the graphs with split-width at most two~\cite{GiP12}.
However it does not look that simple to extend the properties found for their LexBFS orderings to bounded split-width graphs in general.
As a byproduct of our approach, we also obtain new linear-time algorithms when restricted to well-known graph families such as cographs and distance-hereditary graphs.
\subsubsection*{Highlight of our {\sc Maximum Matching} algorithms}
Finally we emphasize our algorithms for {\sc Maximum Matching}.
Here we follow the suggestion of Mertzios, Nichterlein and Niedermeier~\cite{MNN16} that {\sc Maximum Matching} may become the ``drosophila'' of the study of the FPT algorithms in P.
Precisely, we propose ${\cal O} (k^4 \cdot n + m)$-time algorithms for {\sc Maximum Matching} when parameterized either by modular-width or by the $P_4$-sparseness of the graph (Theorems~\ref{thm:mw-maxmatching} and~\ref{thm:qq3-maxmatching}).
The latter subsumes many algorithms that have been obtained for specific subclasses~\cite{FPT97,YuY93}.
\smallskip
Let us sketch the main lines of our approach.
Our algorithms for {\sc Maximum Matching} are recursive.
Given a partition of the vertex-set into modules, first we compute a maximum matching for the subgraph induced by every module separately.
Taking the union of all the outputted matchings gives a matching for the whole graph, but this matching is not necessarily maximum.
So, we aim at increasing its cardinality by using augmenting paths~\cite{Ber57}.
In an unpublished paper~\cite{Nov89}, Novick followed a similar approach and, based on an integer programming formulation, he obtained an ${\cal O}(k^{{\cal O}(k^3)}n + m)$-time algorithm for {\sc Maximum Matching} when parameterized by the modular-width.
Our approach is more combinatorial than his.
Our contribution in this part is twofold.
First we carefully study the possible ways an augmenting path can cross a module.
Our analysis reveals that in order to compute a maximum matching in a graph of modular-width at most $k$ we only need to consider augmenting paths of length ${\cal O}(k)$.
Then, our second contribution is an efficient way to compute such paths.
For that, we design a new type of characteristic graph of size ${\cal O}(k^4)$.
The same as the classical quotient graph keeps most distance properties of the original graph, our new type of characteristic graph is tailored to enclose the main properties of the current matching in the graph.
We believe that the design of new types of characteristic graphs can be a crucial tool in the design of improved algorithms for graph classes of bounded modular-width.
\medskip
We have been able to extend our approach with modular decomposition to an ${\cal O}(q^4 \cdot n + m)$-time algorithm for computing a maximum matching in a given $(q,q-3)$-graph.
However, a characterization of the quotient graph is not enough to do that.
Indeed, we need to go deeper in the $p$-connectedness theory of~\cite{BaO99} in order to better characterize the nontrivial modules in the graphs (Theorem~\ref{thm:stronger-mdc-qq3}).
Furthermore our algorithm for $(q,q-3)$-graph not only makes use of the algorithm with modular decomposition.
On our way to solve this case we have generalized different methods and reduction rules from the literature~\cite{KaS81,YuY93}, that is of independent interest.
We suspect that our algorithm with modular decomposition can be used as a subroutine in order to solve {\sc Maximum Matching} in linear-time for bounded split-width graphs.
However, this is left for future work.
\subsection{Organization of the paper}
In Section~\ref{sec:prelim} we introduce definitions and basic notations.
\medskip
Then, in Section~\ref{sec:cycle} we show FPT algorithms when parameterized by the clique-width.
The problems considered are \textsc{Triangle Counting} and \textsc{Girth}.
To the best of our knowledge, we present the first known polynomial parameterized algorithm for {\sc Girth} (Theorem~\ref{thm:cw-girth}).
Roughly, the main idea behind our algorithms is that given a labeled graph $G$ obtained from a $k$-expression, we can compute a minimum-length cycle for $G$ by keeping up to date the pairwise distances between every two label classes.
Hence, if a $k$-expression of length $L$ is given as part of the input we obtain algorithms running in time ${\cal O}(k^2L)$ and space ${\cal O}(k^2)$.
\medskip
In Section~\ref{sec:dist} we consider distance related problems, namely: {\sc Diameter}, {\sc Eccentricities}, {\sc Hyperbolicity} and {\sc Betweenness Centrality}.
\smallskip
We start proving, in Section~\ref{sec:hardness-cw}, none of these problems above can be solved in time $2^{o(k)} n^{2 - \varepsilon}$, for any $\varepsilon > 0$, when parameterized by the clique-width (Theorems~\ref{thm:cw-diam}---\ref{thm:cw-hyp}).
These are the first known hardness results for clique-width in the field of ``Hardness in P''.
Furthermore, as it is often the case in this field, our results are conditioned on the Strong Exponential Time Hypothesis~\cite{IPZ98}.
In summary, we take advantage of recent hardness results obtained for {\em bounded-degree} graphs~\cite{EvD16}.
Clique-width and treewidth can only differ by a constant-factor in the class of bounded-degree graphs~\cite{Cou12,GuW00}.
Therefore, by combining the hardness constructions for bounded-treewidth graphs and for bounded-degree graphs, we manage to derive hardness results for graph classes of bounded clique-width.
\smallskip
In Section~\ref{sec:algos-split} we describe fully polynomial FPT algorithms for {\sc Diameter}, {\sc Eccentricity}, {\sc Hyperbolicity} and {\sc Betweenness centrality} parameterized by the split-width.
Our algorithms use split-decomposition as an efficient preprocessing method.
Roughly, we define weighted versions for every problem considered (some of them admittedly technical).
In every case, we prove that solving the original distance problem can be reduced in linear-time to the solving of its weighted version for every subgraph of the split decomposition separately.
\smallskip
Then, in Section~\ref{sec:algos-modular-dec} we apply the results from Section~\ref{sec:algos-split} to modular-width.
First, since $sw(G) \leq mw(G)+1$ for any graph $G$, all our algorithms parameterized by split-width are also algorithms parameterized by modular-width.
Moreover for {\sc Eccentricities}, and for {\sc Hyperbolicity} and {\sc Betweenness Centrality} when parameterized by the neighbourhood diversity, we show that it is sufficient only to process the quotient graph of $G$.
We thus obtain algorithms that run in ${\cal O}(mw(G)^{{\cal O}(1)} + n + m)$-time, or ${\cal O}(nd(G)^{{\cal O}(1)} + n + m)$-time, for all these problems.
\smallskip
In Section~\ref{sec:dist-qq3} we generalize our previous algorithms to be applied to the $(q,q-3)$-graphs.
We obtain our results by carefully analyzing the cases where the quotient graph has size $\Omega(q)$.
These cases are given by Lemma~\ref{lem:reduce-qq3}.
\medskip
Section~\ref{sec:maxmatching} is dedicated to our main result, linear-time algorithms for {\sc Maximum Matching}.
First in Section~\ref{sec:matching-mw} we propose an algorithm parameterized by the modular-width that runs in ${\cal O}(mw(G)^{4} \cdot n + m)$-time.
In Section~\ref{sec:matching-qq3} we generalize this algorithm to $(q,q-3)$-graphs.
\medskip
Finally, in Section~\ref{sec:applications} we discuss applications to other graph classes.
\section{New Parameterized algorithms for {\sc Maximum Matching}}\label{sec:maxmatching}
A matching in a graph is a set of edges with pairwise disjoint end vertices.
We consider the problem of computing a matching of maximum size.
\begin{center}
\fbox{
\begin{minipage}{.95\linewidth}
\begin{problem}[\textsc{Maximum Matching}]\
\label{prob:matching}
\begin{description}
\item[Input:] A graph $G=(V,E)$.
\item[Output:] A matching of $G$ with maximum cardinality.
\end{description}
\end{problem}
\end{minipage}
}
\end{center}
{\sc Maximum Matching} can be solved in polynomial time with Edmond's algorithm~\cite{Edm65}. A naive implementation of the algorithm runs in ${\cal O}(n^4)$ time. Nevertheless, Micali and Vazirani~\cite{MiV80} show how to implement Edmond's algorithm in time ${\cal O}(m\sqrt{n})$.
In~\cite{MNN16}, Mertzios, Nichterlein and Niedermeier design some new algorithms to solve {\sc Maximum Matching}, that run in ${\cal O}(k^{{\cal O}(1)} \cdot (n+m))$-time for various graph parameters $k$.
They also suggest to use {\sc Maximum Matching} as the ``drosophilia'' of the study of fully polynomial parameterized algorithms.
In this section, we present ${\cal O}(k^4 \cdot n + m)$-time algorithms for solving {\sc Maximum Matching}, when parameterized by either the modular-width or the $P_4$-sparseness of the graph.
The latter subsumes many algorithms that have been obtained for specific subclasses~\cite{FPT97,YuY93}.
\subsection{Computing short augmenting paths using modular decomposition}
\label{sec:matching-mw}
Let $G=(V,E)$ be a graph and $F \subseteq E$ be a matching of $G$.
A vertex is termed matched if it is incident to an edge of $F$, and unmatched otherwise.
An $F$-augmenting path is a path where the two ends are unmatched, all edges $\{x_{2i},x_{2i+1}\}$ are in $F$ and all edges $\{x_{2j-1}, x_{2j}\}$ are not in $F$.
We can observe that, given an $F$-augmenting path $P = (x_1,x_2, \ldots, x_{2k})$, the matching $E(P)\Delta F$ (obtained by replacing the edges $\{x_{2i},x_{2i+1}\}$ with the edges $\{x_{2j-1}, x_{2j}\}$) has larger size than $F$.
\begin{theorem}[Berge,~\cite{Ber57}]\label{thm:berge}
A matching $F$ in $G=(V,E)$ is maximum if and only if there is no $F$-augmenting path.
\end{theorem}
We now sketch our approach.
Suppose that, for every module $M_i \in {\cal M}(G)$, a maximum matching $F_i$ of $G[M_i]$ has been computed.
Then, $F = \bigcup_i F_i$ is a matching of $G$, but it is not necessarily maximum.
Our approach consists in computing short augmenting paths (of length ${\cal O}(mw(G))$) using the quotient graph $G'$, until we obtain a maximum matching.
For that, we need to introduce several reduction rules.
\smallskip
The first rule (proved below) consists in removing, from every module $M_i$, the edges that are not part of its maximum matching $F_i$.
\begin{lemma}\label{lem:mw-matching-reduction}
Let $M$ be a module of $G=(V,E)$, let $G[M] = (M,E_M)$ and let $F_M \subseteq E_M$ be a maximum matching of $G[M]$.
Then, every maximum matching of $G_M' = (V, (E \setminus E_M) \cup F_M)$ is a maximum matching of $G$.
\end{lemma}
\begin{proof}
Let us consider an arbitrary maximum matching of $G$.
We totally order $M = \{v_1,v_2, \ldots, v_l\}$, in such a way that unmatched vertices appear first, and for every edge in the matching $F_M$ the two ends of it are consecutive.
Let $S \subseteq M, \ |S| = k$ be the vertices of $M$ that are matched with a vertex of $V \setminus M$.
We observe that with the remaining $|M| - k$ vertices of $M \setminus S$, we can only obtain a matching of size at most $\mu_M = \min \{|F_M|, \left\lfloor (|M| - k)/2 \right\rfloor\}$.
Conversely, if $S=\{v_1,v_2,\ldots,v_k\}$ then we can always create a matching of size exactly $\mu_M$ with the vertices of $M \setminus S$ and the edges of $F_M$.
Since $M$ is a module of $G$, this choice can always be made without any loss of generality.
\end{proof}
From now on we shall assume each module induces a matching.
In particular, for every $M \in {\cal M}(G)$, the set $V(E(G[M]))$ stands for the non isolated vertices in the subgraph $G[M]$.
Then, we need to upper-bound the number of edges in an augmenting path that are incident to a same module.
\begin{lemma}\label{lem:bound-edges-augmenting-path}
Let $G=(V,E)$ be a graph such that every module $M \in {\cal M}(G)$ induces a matching.
Furthermore let $G'=({\cal M}(G),E')$ be the quotient graph of $G$, and let $F \subseteq E$ be a non maximum matching of $G$.
There exists an $F$-augmenting path $P=(x_1,x_2,\ldots,x_{2\ell})$ such that the following hold for every $M \in {\cal M}(G)$:
\begin{itemize}
\item $|\{ i \mid x_{2i-1}, x_{2i} \in M\}| \leq 1$;
furthermore if $|\{ i \mid x_{2i-1}, x_{2i} \in M\}| = 1$ then, for every $M' \in N_{G'}(M)$ we have $\{ i \mid x_{2i-1}, x_{2i} \in M'\} = \emptyset$;
\item $|\{ i \mid x_{2i}, x_{2i+1} \in M\}| \leq 1$;
\item $|\{ i \mid x_{2i-1} \notin M, \ x_{2i} \in M\}| \leq 1$;
\item $|\{ i \mid x_{2i} \notin M, \ x_{2i+1} \in M\}| \leq 2$;
furthermore if $|\{ i \mid x_{2i} \notin M, \ x_{2i+1} \in M\}| = 2$ then there exist $x_{2i_0+1},x_{2i_0+3},x_{2i_0+4} \in M$;
\item $|\{ i \mid x_{2i-1} \in M, \ x_{2i} \notin M\}| \leq 1$;
\item $|\{ i \mid x_{2i} \in M, \ x_{2i+1} \notin M\}| \leq 2$;
furthermore if $|\{ i \mid x_{2i} \in M, \ x_{2i+1} \notin M\}| = 2$ then there exist $x_{2i_0-1},x_{2i_0},x_{2i_0+2} \in M$.
\end{itemize}
In particular, $P$ has length ${\cal O}(|{\cal M}(G)|)$.
\end{lemma}
\begin{proof}
Let $P$ be a {\em shortest} $F$-augmenting path that minimizes $i(P) = \left|E(P) \cap \left(\bigcup_{M \in {\cal M}(G)}E(G[M])\right)\right|$.
Equivalently, $P$ is a shortest augmenting path with the minimum number of edges $i(P)$ with their two ends in a same module.
There are four cases.
\begin{enumerate}
\item
Suppose by contradiction there exist $i_1 < i_2$ such that $x_{2i_1-1}, x_{2i_1}, x_{2i_2-1}, x_{2i_2} \in M$.
See Fig.~\ref{fig-matching-1}-\ref{fig-matching-2}.
In particular, $i_2 - i_1 \geq 2$ since $M$ induces a matching.
Furthermore, $x_{2i_1+1}, x_{2i_2-2} \in N_G(M)$.
Then, $(x_1,\ldots, x_{2i_1-1}, x_{2i_1+1}, x_{2i_1}, x_{2i_2-2}, x_{2i_2-1}, x_{2i_2}, \ldots x_{2\ell})$ is an $F$-augmenting path, thereby contradicting the minimality of $i(P)$.
\begin{figure}[h!]
\hfill\begin{minipage}{.48\textwidth}
\centering
\includegraphics[width=.45\textwidth]{Fig/case1-1}
\caption{Case $x_{2i_1-1}, x_{2i_1}, x_{2i_2-1}, x_{2i_2} \in M$.}
\label{fig-matching-1}
\end{minipage}\hfil
\begin{minipage}{.48\textwidth}
\centering
\includegraphics[width=.45\textwidth]{Fig/case1-2}
\caption{Local replacement of $P$.}
\label{fig-matching-2}
\end{minipage}
\end{figure}
Similarly, suppose by contradiction there exist $x_{2i_1-1},x_{2i_1} \in M$ and there exist $x_{2i_2-1}, x_{2i_2} \in M', \ M' \in N_{G'}(M)$.
See Fig~\ref{fig-matching-3}.
We assume by symmetry $i_1 < i_2$.
In this situation, either $i_1 = 1$, and so, $x_{2i_1-1} = x_1$ is unmatched, or $i_1 > 1$ and so, $x_{2i_1-1}$ is matched to $x_{2i_1-2} \neq x_{2i_2}$.
Then, $(x_1,\ldots,x_{2i_1-1},x_{2i_2},\ldots,x_{2\ell})$ is an $F$-augmenting path, thereby contradicting the minimality of $|V(P)|$.
\begin{figure}[h!]
\centering
\includegraphics[width=.25\textwidth]{Fig/case2-1}
\caption{Case $x_{2i_1-1},x_{2i_1} \in M$ and $x_{2i_2-1}, x_{2i_2} \in M'$.}
\label{fig-matching-3}
\end{figure}
\item
Suppose by contradiction there exist $i_1 < i_2$ such that $x_{2i_1}, x_{2i_1+1}, x_{2i_2}, x_{2i_2+1} \in M$.
See Fig~\ref{fig-matching-4}.
In particular, $x_{2i_1-1} \in N_G(M)$ since $M$ induces a matching.
Then, $(x_1,\ldots, x_{2i_1-1}, x_{2i_2}, x_{2i_2+1}, \ldots x_{2\ell})$ is an $F$-augmenting path, thereby contradicting the minimality of $|V(P)|$.
\begin{figure}[h!]
\centering
\includegraphics[width=.25\textwidth]{Fig/case3-1}
\caption{Case $x_{2i_1}, x_{2i_1+1}, x_{2i_2}, x_{2i_2+1} \in M$.}
\label{fig-matching-4}
\end{figure}
\item
Suppose by contradiction there exist $i_1 < i_2$ such that $x_{2i_1-1},x_{2i_2-1} \notin M, \ x_{2i_1},x_{2i_2} \in M$.
See Fig~\ref{fig-matching-5}.
Either $i_1 = 1$, and so, $x_{2i_1-1} = x_1$ is unmatched, or $i_1 > 1$ and so, $x_{2i_1-1}$ is matched to $x_{2i_1-2} \neq x_{2i_2}$.
Then, $(x_1,\ldots,x_{2i_1-1},x_{2i_2},\ldots,x_{2\ell})$ is an $F$-augmenting path, thereby contradicting the minimality of $|V(P)|$.
\begin{figure}[h!]
\centering
\includegraphics[width=.16\textwidth]{Fig/case4-1}
\caption{Case $x_{2i_1-1},x_{2i_2-1} \notin M, \ x_{2i_1},x_{2i_2} \in M$.}
\label{fig-matching-5}
\end{figure}
By symmetry, the latter also proves that $|\{ i \mid x_{2i-1} \in M, \ x_{2i} \notin M\}| \leq 1$.
\item
Finally suppose by contradiction there exist $i_1 < i_2 < i_3$ such that $x_{2i_1},x_{2i_2},x_{2i_3} \notin M$, $x_{2i_1+1},x_{2i_2+1},x_{2i_3+1} \in M$.
See Fig~\ref{fig-matching-6}.
Then, $(x_1,\ldots,x_{2i_1},x_{2i_1+1},x_{2i_3},x_{2i_3+1}\ldots,x_{2\ell})$ is an $F$-augmenting path, thereby contradicting the minimality of $|V(P)|$.
\begin{figure}[h!]
\centering
\includegraphics[width=.25\textwidth]{Fig/case5-1}
\caption{Case $x_{2i_1},x_{2i_2},x_{2i_3} \notin M$, $x_{2i_1+1},x_{2i_2+1},x_{2i_3+1} \in M$.}
\label{fig-matching-6}
\end{figure}
We prove in the same way that if there exist $i_1 < i_2$ such that $x_{2i_1},x_{2i_2}\notin M$ and $x_{2i_1+1},x_{2i_2+1} \in M$ then $i_2 = i_1+1$.
See Fig~\ref{fig-matching-7}.
Furthermore, if $x_{2i_2+2} = x_{2i_1+4} \notin M$ then $(x_1,\ldots,x_{2i_1},x_{2i_1+1},x_{2i_2+2},x_{2i_2+3}\ldots,x_{2\ell})$ is an $F$-augmenting path, thereby contradicting the minimality of $|V(P)|$.
\begin{figure}[h!]
\centering
\includegraphics[width=.2\textwidth]{Fig/case6-1}
\caption{Case $x_{2i_2+2} = x_{2i_1+4} \notin M$.}
\label{fig-matching-7}
\end{figure}
By symmetry, the same proof as above applies to $\{ i \mid x_{2i} \in M, \ x_{2i+1} \notin M\}$.
\end{enumerate}
Overall, every $M \in {\cal M}(G)$ is incident to at most $8$ edges of $P$, and so, $P$ has length ${\cal O}(|{\cal M}(G)|)$.
\end{proof}
Based on Lemmas~\ref{lem:mw-matching-reduction} and~\ref{lem:bound-edges-augmenting-path}, we introduce in what follows a {\em witness subgraph} in order to find a matching.
We think the construction could be improved but we chose to keep is as simple as possible.
\begin{definition}\label{def:witness-subgraph}
Let $G=(V,E)$ be a graph, $G'=({\cal M}(G),E')$ be its quotient graph and $F \subseteq E$ be a matching of $G$.
The witness matching $F'$ is obtained from $F$ by keeping a representative for every possible type of edge in an augmenting path.
Precisely:
\begin{itemize}
\item Let $M \in {\cal M}(G)$.
If $E(G[M]) \cap F \neq \emptyset$ then there is exactly one edge $\{u_M,v_M\} \in E(G[M]) \cap F$ such that $\{u_M,v_M\} \in F'$.
Furthermore if $E(G[M]) \setminus F \neq \emptyset$ then we pick an edge $\{x_M,y_M\} \in E(G[M]) \setminus F$ and we add in $F'$ every edge in $F$ that is incident to either $x_M$ or $y_M$.
\item Let $M,M' \in {\cal M}(G)$ be adjacent in $G'$.
There are exactly $\min \{ 4, |F \cap (M \times M')| \}$ edges $\{v_M,v_{M'}\}$ added in $F'$ such that $v_M \in M, \ v_{M'} \in M'$ and $\{v_M,v_{M'}\} \in F$.
\end{itemize}
The witness subgraph $G_F'$ is the subgraph induced by $V(F')$ with at most two unmatched vertices added for every strong module.
Formally, let $M \in {\cal M}(G)$.
The submodule $M_F \subseteq M$ contains exactly $\min \{ 2, |M \setminus V(F)| \}$ vertices of $M \setminus V(F)$.
Then, $$G_F' = G\left[ V(F') \cup \left( \bigcup_{M \in {\cal M}(G)} M_F \right) \right]. $$
\end{definition}
As an example, suppose that every edge of $F$ has its two ends in a same module and every module induces a matching.
Then, $G_F'$ is obtained from $G'$ by substituting every $M \in {\cal M}(G)$ with at most one edge (if $F \cap E(G[M]) \neq \emptyset$) and at most two isolated vertices (representing unmatched vertices).
From the algorithmic point of view, we need to upper-bound the size of the witness subgraph, as follows.
\begin{lemma}\label{lem:witness-size}
Let $G=(V,E)$ be a graph, $G'=({\cal M}(G),E')$ be its quotient graph and $F \subseteq E$ be a matching of $G$.
The witness subgraph $G_F'$ has order ${\cal O}(|E(G')|)$.
\end{lemma}
\begin{proof}
By construction for every $M \in {\cal M}(G)$ we have $|M \cap V(G_F')| = {\cal O}(deg_{G'}(M))$.
Therefore, $|V(G_F')| = \sum_{M \in {\cal M}(G)} |M \cap V(G_F')| = {\cal O}(|E(G')|)$.
\end{proof}
Our algorithm is based on the correspondance between $F$-augmenting paths in $G$ and $F'$-augmenting paths in $G'_F$, that we prove next.
The following Lemma~\ref{lem:witness-maxmatching} is the key technical step of the algorithm.
\begin{lemma}\label{lem:witness-maxmatching}
Let $G=(V,E)$ be a graph such that every module $M \in {\cal M}(G)$ induces a matching.
Let $F \subseteq E$ be a matching of $G$ such that $\bigcup_{M \in {\cal M}(G)} V(E(G[M])) \subseteq V(F)$.
There exists an $F$-augmenting path in $G$ if and only if there exists an $F'$-augmenting path in $G_F'$.
\end{lemma}
\begin{proof}
In one direction, $G_F'$ is an induced subgraph of $G$.
Furthermore, according to Definition~\ref{def:witness-subgraph}, $F' \subseteq F$ and $V(F') = V(F) \cap V(G_F')$.
Thus, every $F'$-augmenting path in $G_F'$ is also an $F$-augmenting path in $G$.
Conversely, suppose there exists an $F$-augmenting path in $G$.
Let $P=(v_1,v_2,\ldots,v_{2\ell})$ be an $F$-augmenting path in $G$ that satisfies the conditions of Lemma~\ref{lem:bound-edges-augmenting-path}.
We transform $P$ into an $F'$-augmenting path in $G_F'$ as follows.
For every $1 \leq i \leq 2\ell$ let $M_i \in {\cal M}(G)$ such that $v_i \in M_i$.
\begin{itemize}
\item We choose $u_1 \in M_1 \cap V(G_F'), \ u_{2\ell} \in M_{2\ell} \cap V(G_F')$ unmatched.
Furthermore, if $M_1 = M_{2\ell}$ then we choose $u_1 \neq u_{2\ell}$.
The two of $u_1,u_{2\ell}$ exist according to Definition~\ref{def:witness-subgraph}.
\item Then, for every $1 \leq i \leq \ell-1$, we choose $u_{2i} \in M_{2i} \cap V(G_F'), \ u_{2i+1} \in M_{2i+1} \cap V(G_F')$ such that $\{u_{2i},u_{2i+1}\} \in F'$.
Note that if $M_{2i} = M_{2i+1}$ then $\{u_{2i},u_{2i+1}\}$ is the unique edge of $F' \cap E(G[M_{2i}])$.
By Lemma~\ref{lem:bound-edges-augmenting-path} we also have that $\{v_{2i},v_{2i+1}\}$ is the unique edge of $E(P) \cap F$ such that $v_{2i},v_{2i+1} \in M_{2i}$.
Otherwise, $M_{2i} \neq M_{2i+1}$.
If there are $p$ edges $e \in F$ with one end in $M_{2i}$ and the other end in $M_{2i+1}$ then there are at least $\min\{p,4\}$ such edges in $F'$.
By Lemma~\ref{lem:bound-edges-augmenting-path} there are at most $\min\{p,4\}$ edges $e \in E(P) \cap F$ with one end in $M_{2i}$ and the other end in $M_{2i+1}$.
Hence, we can always ensure the $u_j$'s, $1 \leq j \leq 2\ell$, to be pairwise different.
\end{itemize}
The resulting sequence ${\cal S}_P = (u_1,u_2,\ldots,u_{2\ell})$ is not necessarily a path, since two consecutive vertices $u_{2i-1},u_{2i}$ need not be adjacent in $G_F'$.
Roughly, we insert alternating subpaths in the sequence in order to make it a path.
However, we have to be careful not to use twice a same vertex for otherwise we would only obtain a walk.
Let $I_P = \{ i \mid \{u_{2i-1},u_{2i}\} \notin E \}$.
Observe that for every $i \in I_P$ we have $M_{2i-1} = M_{2i}$.
In particular, since we assume $\bigcup_{M \in {\cal M}(G)} V(E(G[M])) \subseteq V(F)$, it implies $i \notin \{1,\ell\}$.
Furthermore, $M_{2i-2} \neq M_{2i}$ and $M_{2i} \neq M_{2i+1}$ since otherwise $v_{2i-2},v_{2i-1},v_{2i} \in M_{2i}$ or $v_{2i-1},v_{2i},v_{2i+1} \in M_{2i}$ thereby contradicting that $M_{2i}$ induces a matching.
According to Definition~\ref{def:witness-subgraph} there exist $x_i,y_i \in M_{2i}$ such that $\{x_i,y_i\} \in E(G[M_{2i}]) \setminus F$ and every edge of $F$ that is incident to either $x_i$ or $y_i$ is in $F'$.
Such two edges always exist since we assume $\bigcup_{M \in {\cal M}(G)} V(E(G[M])) \subseteq V(F)$, hence there exist $w_i,z_i$ such that $\{w_i,x_i\}, \{y_i,z_i\} \in F'$.
Note that $w_i,z_i \notin M_{2i}$ since $\{x_i,y_i\} \in E(G[M_{2i}])$ and $M_{2i}$ induces a matching.
Since by Lemma~\ref{lem:bound-edges-augmenting-path} $\{v_{2i-1},v_{2i}\}$ is the unique edge $e \in E(P) \setminus F$ such that $e \subseteq M_{2i}$, the vertices $x_i,y_i, \ i \in I_P$ are pairwise different.
Furthermore, we claim that there can be no $i_1, i_2 \in I_P$ such that $s_{i_1} \in \{x_{i_1},y_{i_1}\}$ and $t_{i_2} \in \{x_{i_2},y_{i_2}\}$ are adjacent in $G$.
Indeed otherwise, $M_{2i_1} \in N_{G'}(M_{2i_2})$, there exist $v_{2i_1-1},v_{2i_1} \in M_{2i_1}$ and $v_{2i_2-1},v_{2i_2} \in M_{2i_2}$, thereby contradicting Lemma~\ref{lem:bound-edges-augmenting-path}.
As a result, all the vertices $w_i,x_i,y_i,z_i, \ i \in I_P$ are pairwise different.
However, we may have $\{w_i,x_i\} = \{u_{2j},u_{2j+1}\}$ or $\{y_i,z_i\} = \{u_{2j},u_{2j+1}\}$ for some $j$.
We consider the indices $i \in I_P$ sequentially, by increasing value.
By Lemma~\ref{lem:bound-edges-augmenting-path}, $v_{2j} \notin M_{2i}, \ v_{2j+1} \in M_{2i}$ for some $j \neq i-1$ implies $j=i-2$.
Similarly (obtained by reverting the indices, from $v_1'=v_{2\ell}$ to $v_{2\ell}'=v_1$), $v_{2j} \in M_{2i}, \ v_{2j+1} \notin M_{2i}$ for some $j \neq i$ implies $j=i+1$.
Therefore, if $(w_i,x_i) \in {\cal S}_p$ then $(w_i,x_i) \in \{ (u_{2i-4},u_{2i-3}), (u_{2i-2},u_{2i-1}), (u_{2i+1},u_{2i}), (u_{2i+3},u_{2i+2}) \}$, and the same holds for $(y_i,z_i)$.
Note also that the pairs $(w_i,x_i)$ and $(z_i,y_i)$ play a symmetric role.
Thus we can reduce by symmetries (on the sequence and on the two of $(w_i,x_i)$ and $(z_i,y_i)$) to the six following cases:
\begin{itemize}
\item Case $x_i,y_i \notin {\cal S}_P$.
See Fig~\ref{fig-replacement-1}.
In particular, $w_i,z_i \notin {\cal S}_P$.
\begin{figure}[h!]
\centering
\includegraphics[width=.15\textwidth]{Fig/replacement-1}
\caption{Case $x_i,y_i \notin {\cal S}_P$.}
\label{fig-replacement-1}
\end{figure}
We insert the $F'$-alternating subpath $(u_{2i-1},w_i,x_i,y_i,z_i,u_{2i})$.
\item Case $x_i \notin {\cal S}_P$, $y_i \in \{u_{2i-1},u_{2i}\}$.
We assume by symmetry $y_i = u_{2i}$.
See Fig~\ref{fig-replacement-2}.
In particular, we have $w_i \notin {\cal S}_P$.
\begin{figure}[h!]
\centering
\includegraphics[width=.15\textwidth]{Fig/replacement-2}
\caption{Case $x_i \notin {\cal S}_P$, $y_i = u_{2i}$.}
\label{fig-replacement-2}
\end{figure}
We insert the $F'$-alternating subpath $(u_{2i-1},w_i,x_i,y_i = u_{2i})$.
Note that the case $x_i \in \{u_{2i-1},u_{2i}\}$, $y_i \notin {\cal S}_P$ is symmetrical to this one.
\item Case $x_i \notin {\cal S}_P$, $y_i \in {\cal S}_p \setminus \{u_{2i-1},u_{2i}\}$.
We assume by symmetry $(y_i,z_i) = (u_{2i+2},u_{2i+3})$ (the case $(z_i,y_i) = (u_{2i-4},u_{2i-3})$ is obtained by reverting the indices along the sequence).
See Fig~\ref{fig-replacement-4}.
\begin{figure}[h!]
\centering
\includegraphics[width=.2\textwidth]{Fig/replacement-4}
\caption{Case $x_i \notin {\cal S}_P$, $(y_i,z_i) = (u_{2i+2},u_{2i+3})$.}
\label{fig-replacement-4}
\end{figure}
We replace $(u_{2i-1},u_{2i},u_{2i+1},y_i=u_{2i+2})$ by the $F'$-alternating subpath $(u_{2i-1},w_i,x_i,y_i)$.
Note that the case $x_i \in {\cal S}_p \setminus \{u_{2i-1},u_{2i}\}$, $y_i \notin {\cal S}_P$ is symmetrical to this one.
\item Case $(w_i,x_i) = (u_{2i-4},u_{2i-3})$, $y_i = u_{2i}$.
See Fig~\ref{fig-replacement-6}.
\begin{figure}[h!]
\centering
\includegraphics[width=.16\textwidth]{Fig/replacement-6}
\caption{Case $(w_i,x_i) = (u_{2i-4},u_{2i-3})$, $y_i = u_{2i}$.}
\label{fig-replacement-6}
\end{figure}
We replace $(u_{2i-3}=x_i,u_{2i-2},u_{2i-1},u_{2i}=y_i)$ by the $F'$-alternating subpath $(x_i,y_i)$.
Note that the case $x_i = u_{2i-1}$, $(y_i,z_i) = (u_{2i+2},u_{2i+3})$, and the two more cases obtained by switching the respective roles of $(x_i,w_i)$ and $(y_i,z_i)$, are symmetrical to this one.
\item Case $(w_i,x_i) = (u_{2i-4},u_{2i-3})$, $y_i = u_{2i-1}$.
See Fig~\ref{fig-replacement-6-b}.
\begin{figure}[h!]
\centering
\includegraphics[width=.16\textwidth]{Fig/replacement-6-bis}
\caption{Case $(w_i,x_i) = (u_{2i-4},u_{2i-3})$, $y_i = u_{2i-1}$.}
\label{fig-replacement-6-b}
\end{figure}
We replace $(u_{2i-3}=x_i,u_{2i-2}=z_i,u_{2i-1}=y_i,u_{2i})$ by the $F'$-alternating subpath $(x_i,y_i,z_i,u_{2i})$.
Note that the case $x_i = u_{2i}$, $(y_i,z_i) = (u_{2i+2},u_{2i+3})$, and the two more cases obtained by switching the respective roles of $(x_i,w_i)$ and $(y_i,z_i)$, are symmetrical to this one.
\item Case $(w_i,x_i) = (u_{2i-4},u_{2i-3})$, $(y_i,z_i) = (u_{2i+2},u_{2i+3})$.
See Fig~\ref{fig-replacement-8}.
\begin{figure}[h!]
\centering
\includegraphics[width=.16\textwidth]{Fig/replacement-8}
\caption{Case $(w_i,x_i) = (u_{2i-4},u_{2i-3})$, $(y_i,z_i) = (u_{2i+2},u_{2i+3})$.}
\label{fig-replacement-8}
\end{figure}
Since $x_i = u_{2i-3}, \ y_i = u_{2i+2}$ are adjacent we can remove $(u_{2i-2},u_{2i-1},u_{2i},u_{2i+1})$ from ${\cal S}_P$.
\end{itemize}
Overall, in every case the procedure only depends on the subsequence between $u_{2i-4}$ and $u_{2i+3}$.
In order to prove correctness of the procedure, it suffices to prove that this subsequence has not been modified for a smaller $i' \in I_P, \ i' < i$.
Equivalently, we prove that the above procedure does not modify the subsequence between $u_{2j-4}$ and $u_{2j+3}$ for any $j \in I_P, \ j > i$.
First we claim $j \geq i+2$.
Indeed, $M_{2i+1} \in N_{G'}(M_{2i})$.
Hence, by Lemma~\ref{lem:bound-edges-augmenting-path}, $i \in I_P$ implies $i+1 \notin I_P$, that proves the claim.
In this situation, $2j-4 \geq 2i$.
Furthermore, the subsequence $(u_{2i},\ldots,u_{2\ell})$ is modified only if $u_{2i+2} \in \{x_i,y_i\}$.
However in the latter case we have $u_{2i+3} \notin M_{2i}$, hence $M_{2i+3} \in N_{G'}(M_{2i})$, and so, by Lemma~\ref{lem:bound-edges-augmenting-path} $j \geq i+3$.
In particular, $2j-4 \geq 2i+2$ and the subsequence $(u_{2i+2},\ldots,u_{2\ell})$ is not modified by the procedure.
Altogether, it proves that the above procedure is correct.
Finally, applying the above procedure for all $i \in I_P$ leads to an $F'$-alternating path in $G_F'$.
\end{proof}
We can now state the main result in this subsection.
\begin{theorem}\label{thm:mw-maxmatching}
For every $G=(V,E)$, {\sc Maximum Matching} can be solved in ${\cal O}(mw(G)^4 \cdot n + m)$-time.
\end{theorem}
\begin{proof}
The algorithm is recursive.
If $G$ is trivial (reduced to a single node) then we output an empty matching.
Otherwise, let $G' = ({\cal M}(G),E')$ be the quotient graph of $G$.
For every module $M \in {\cal M}(G)$, we call the algorithm recursively on $G[M]$ in order to compute a maximum matching $F_M$ of $G[M]$.
Let $F^* = \bigcup_{M \in {\cal M}(G)} F_M$.
By Lemma~\ref{lem:mw-matching-reduction} (applied to every $M \in {\cal M}(G)$ sequentially), we are left to compute a maximum matching for $G^* = (V, (E \setminus \bigcup_{M \in {\cal M}(G)} E(G[M])) \cup F^*)$.
Therefore from now on assume $G = G^*$.
\smallskip
If $G'$ is edgeless then we can output $F^*$.
Otherwise, by Theorem~\ref{thm:modular-dec} $G'$ is either prime for modular decomposition or a complete graph.
Suppose $G'$ to be prime.
We start from $F_0 = F^*$.
Furthermore, we ensure that the two following hold at every step $t \geq 0$:
\begin{itemize}
\item All the vertices that are matched in $F^*$ are also matched in the current matching $F_t$.
For instance, it is the case if $F_t$ is obtained from $F_0$ by only using augmenting paths in order to increase the cardinality of the matching.
\item For every $M \in {\cal M}(G)$ we store $|F_M \cap F_t|$.
For every $M,M' \in {\cal M}(G)$ adjacent in $G'$ we store $|(M \times M') \cap F_t|$.
In particular, $|F_M \cap F_0| = |F_M|$ and $|(M \times M') \cap F_0| = 0$.
So, it takes time ${\cal O}(\sum_{M \in {\cal M}(G)}deg_{G'}(M))$ to initialize this information, that is in ${\cal O}(|E(G')|) = {\cal O}(mw(G)^2)$.
Furthermore, it takes ${\cal O}(\ell)$-time to update this information if we increase the size of the matching with an augmenting path of length $2\ell$.
\end{itemize}
We construct the graph $G'_{F_t}$ according to Definition~\ref{def:witness-subgraph}.
By using the information we store for the algorithm, it can be done in ${\cal O}(|E(G'_{F_t})|)$-time, that is in ${\cal O}(|E(G')|^2) = {\cal O}(mw(G)^4)$ by Lemma~\ref{lem:witness-size}.
Furthermore by Theorem~\ref{thm:berge} there exists an $F_t$-augmenting path if and only if $F_t$ is not maximum.
Since we can assume all the modules in ${\cal M}(G)$ induce a matching, by Lemma~\ref{lem:witness-maxmatching} there exists an $F_t$-augmenting path in $G$ if and only if there exists an $F_t'$-augmenting path in $G'_{F_t}$.
So, we are left to compute an $F_t'$-augmenting path in $G'_{F_t}$ if any.
It can be done in ${\cal O}(|E(G_{F_t}')|)$-time~\cite{GaT83}, that is in ${\cal O}(mw(G)^4)$.
Furthermore, by construction of $G_{F_t}'$, an $F_t'$-augmenting path $P'$ in $G'_{F_t}$ is also an $F_t$-augmenting path in $G$.
Thus, we can obtain a larger matching $F_{t+1}$ from $F_t$ and $P$.
We repeat the procedure above for $F_{t+1}$ until we reach a maximum matching $F_{t_{\max}}$.
The total running time is in ${\cal O}(mw(G)^4 \cdot t_{\max})$.
\smallskip
Finally, assume $G'$ to be complete.
Let ${\cal M}(G) = \{M_1,M_2, \ldots,M_k\}$ be linearly ordered.
For every $1 \leq i \leq k$, write $G_i = G[\bigcup_{j \leq i}M_j]$.
We compute a maximum matching $F^i$ for $G_i$, from a maximum matching $F^{i-1}$ of $G_{i-1}$ and a maximum matching $F_{M_i}$ of $G[M_i]$, sequentially.
For that, we apply the same techniques as for the prime case, to some ``pseudo-quotient graph'' $G_i'$ isomorphic to $K_2$ ({\it i.e.}, the two vertices of $G_i'$ respectively represent $V(G_{i-1})$ and $M_i$).
Since the pseudo-quotient graphs have size two, this step takes total time ${\cal O}(|V(G')| + (|F^k| - |F^*|))$.
\medskip
Overall, summing the order of all the subgraphs in the modular decomposition of $G$ amounts to ${\cal O}(n)$~\cite{Rao08b}.
Furthermore, a maximum matching of $G$ also has cardinality ${\cal O}(n)$.
Therefore, the total running time is in ${\cal O}(mw(G)^4 \cdot n)$ if the modular decomposition of $G$ is given.
The latter decomposition can be precomputed in ${\cal O}(n+m)$-time~\cite{TCHP08}.
\end{proof}
\subsection{More structure: $(q,q-3)$-graphs}
\label{sec:matching-qq3}
The second main result in Section~\ref{sec:maxmatching} is an ${\cal O}(q(G)^4 \cdot n + m)$-time algorithm for {\sc Maximum Matching} (Theorem~\ref{thm:qq3-maxmatching}).
Our algorithm for $(q,q-3)$-graphs reuses the algorithm described in Theorem~\ref{thm:mw-maxmatching} as a subroutine.
However, applying the same techniques to a case where the quotient graph has super-constant size $\Omega(q)$ happens to be more challenging.
Thus we need to introduce new techniques in order to handle with all the cases presented in Lemma~\ref{lem:reduce-qq3}.
Computing a maximum matching for the {\em quotient graph} is easy.
However, we also need to account for the edges present inside the modules.
For that, we need the following stronger variant of Lemma~\ref{lem:reduce-qq3}.
The latter generalizes similar structure theorems that have been obtained for some specific subclasses~\cite{GRT97}.
\begin{theorem}\label{thm:stronger-mdc-qq3}
For an arbitrary $(q,q-3)$-graph $G$, $q \geq 7$, and its quotient graph $G'$, exactly one of the following conditions is satisfied.
\begin{enumerate}
\item $G$ is disconnected;
\item $\overline{G}$ is disconnected;
\item $G$ is a disc (and so, $G=G'$ is prime for modular decomposition);
\item $G$ is a spider (and so, $G'$ is a prime spider);
\item $G'$ is a spiked $p$-chain $P_k$, or a spiked $p$-chain $\overline{P_k}$.
Furthermore, for every $v \in V(G')$, if the corresponding module $M_v \in {\cal M}(G)$ is such that $|M_v| \geq 2$ then we have $v \in \{v_1,v_k,x,y\}$;
\item $G'$ is a spiked $p$-chain $Q_k$, or a spiked $p$-chain $\overline{Q_k}$.
Furthermore, for every $v \in V(G')$, if the corresponding module $M_v \in {\cal M}(G)$ is such that $|M_v| \geq 2$ then we have either $v \in \{v_1,v_k\}$ or $v = z_i$ for some $i$;
\item $|V(G')| \leq q$.
\end{enumerate}
\end{theorem}
The proof of Theorem~\ref{thm:stronger-mdc-qq3} is postponed to the appendix.
It is based on a refinement of modular decomposition called {\em primeval decomposition}.
In what follows, we introduce our techniques for the cases where the quotient graph $G'$ is neither degenerate nor of constant size.
\subsubsection*{Simple cases}
\begin{lemma}\label{lem:disc}
For every disc $G=(V,E)$, a maximum matching can be computed in linear-time.
\end{lemma}
\begin{proof}
If $G=C_n, \ n \geq 5$ is a cycle then the set of edges $\{ \{2i,2i+1\} \mid 0 \leq i \leq \left\lfloor n/2 \right\rfloor - 1 \}$ is a maximum matching.
Otherwise, $G=\overline{C_n}$ is a co-cycle.
Let $F_n$ contain all the edges $\{4i,4i+2\}, \ \{4i+1,4i+3\}, \ 0 \leq i \leq \left\lfloor n/4 \right\rfloor - 1$.
There are three cases.
If $n = 0 \pmod 4$ or $n = 1 \pmod 4$ then there is at most one vertex unmatched by $F_n$, and so, $F_n$ is a maximum matching of $G$.
Otherwise, if $n = 3 \pmod 4$ then a maximum matching of $G$ is obtained by adding the edge $\{n-3,n-1\}$ to $F_n$.
Finally, assume $n = 2 \pmod 4$.
By construction, $F_n$ leaves unmatched the two of $n-2,n-1$.
We obtain a perfect matching of $G$ from $F_n$ by replacing $\{0,2\}$ with $\{n-2,0\}, \ \{n-1,2\}$.
Note that it is possible to do that since $n \geq 5$.
\end{proof}
\begin{lemma}\label{lem:spider}
If $G=(S \cup K \cup R, E)$ is a spider then there exists a maximum matching of $G$ composed of: a perfect matching between $K$ and $S$; and a maximum matching of $G[R]$.
\end{lemma}
\begin{proof}
We start from a perfect matching $F_0$ between $K$ and $S$.
We increase the size of $F_0$ using augmenting paths until it is no more possible to do so.
By Theorem~\ref{thm:berge}, the obtained matching $F_{\max}$ is maximum.
Furthermore, either there is a perfect matching between $K$ and $S$ or there is at least one vertex of $S$ that is unmatched.
Since $V(F_0) \subseteq V(F_{\max})$ the latter proves the lemma.
\end{proof}
\subsubsection*{The case of prime $p$-trees}
Roughly, when the quotient graph $G'$ is a prime $p$-tree, our strategy consists in applying the following reduction rules until the graph is empty.
\begin{enumerate}
\item Find an isolated module $M$ (with no neighbour).
Compute a maximum matching for $G[M]$ and for $G[V \setminus M]$ separately.
\item Find a pending module $M$ (with one neighbour $v$).
Compute a maximum matching for $G[M]$.
If it is not a perfect matching then add an edge between $v$ and any unmatched vertex in $M$, then discard $M \cup \{v\}$.
Otherwise, discard $M$ (Lemma~\ref{lem:pending-module}).
\item Apply a technique known as ``{\sc SPLIT} and {\sc MATCH}''~\cite{YuY93} to some module $M$ and its neighbourhood $N_G(M)$.
We do so only if $M$ satisfies some properties.
In particular, we apply this rule when $M$ is a universal module (with a complete join between $M$ and $V \setminus M$).
See Definition~\ref{def:split-and-match} and Lemma~\ref{lem:join}.
\end{enumerate}
We introduce the reduction rules below and we prove their correctness.
\paragraph{Reduction rules.}
The following lemma generalizes a well-known reduction rule for {\sc Maximum Matching}: add a pending vertex and its unique neighbour to the matching then remove this edge~\cite{KaS81}.
\begin{lemma}\label{lem:pending-module}
Let $M$ be a module in a graph $G=(V,E)$ such that $N_G(M) = \{v\}$, $F_M$ is a maximum matching of $G[M]$ and $F_M^*$ is obtained from $F_M$ by adding an edge between $v$ and any unmatched vertex of $M$ (possibly, $F_M^* = F_M$ if it is a perfect matching).
There exists a maximum matching $F$ of $G$ such that $F_M^* \subseteq F$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:mw-matching-reduction}, every maximum matching for $G_M' = (V, (E \setminus E(G[M]) \cup F_M)$ is also a maximum matching for $G$.
There are two cases.
Suppose there exists $u \in M \setminus V(F_M)$.
Then, $u$ is a pending vertex of $G_M'$.
There exists a maximum matching of $G_M'$ that contains the edge $\{u,v\}$~\cite{KaS81}.
Furthermore, removing $u$ and $v$ disconnects the vertices of $M \setminus u$ from $V \setminus N_G[M]$.
It implies that a maximum matching $F'$ of $G \setminus (u,v)$ is the union of any maximum matching of $G[M \setminus u]$ with any maximum matching of $G[V \setminus N_G[M]]$.
In particular, $F_M$ is contained in some maximum matching $F'$ of $G \setminus (u,v)$.
Since $\{u,v\}$ is contained in a maximum matching of $G$, therefore $F = F' \cup \{\{u,v\}\}$ is a maximum matching of $G$.
We are done since $F_M^* = F_M \cup \{\{u,v\}\} \subseteq F$ by construction.
Otherwise, $F_M$ is a perfect matching of $G[M]$.
For every edge $\{x,y\} \in F_M$, we have that $x,y$ have degree two in $G_M'$.
The following reduction rule has been proved to be correct in~\cite{KaS81}: remove any $x$ of degree two, merge its two neighbours and increase the size of the solution by one unit.
In our case, since $N_{G_M'}[y] \subseteq N_{G_M'}[v]$ the latter is equivalent to put the edge $\{x,y\}$ in the matching.
Overall, applying the reduction rule to all edges $\{x,y\} \in F_M$ indeed proves the existence of some maximum matching $F$ such that $F_M = F_M^* \subseteq F$.
\end{proof}
Then, we introduce a technique known as ``{\sc SPLIT} and {\sc MATCH}'' in the literature~\cite{YuY93}.
\begin{definition}\label{def:split-and-match}
Let $G=(V,E)$ be a graph, $F \subseteq E$ be a matching of $G$.
Given some module $M \in {\cal M}(G)$ we try to apply the following two operations until none of them is possible:
\begin{itemize}
\item Suppose there exist $u \in M, \ v \in N_G(M)$ unmatched.
We add an edge $\{u,v\}$ to the matching ({\sc MATCH}).
\item Otherwise, suppose there exist $u,u' \in M, \ v,v' \in N_G(M)$ such that $u$ and $u'$ are unmatched, and $\{v,v'\}$ is an edge of the matching.
We replace the edge $\{v,v'\}$ in the matching by the two new edges $\{u,v\}, \ \{u',v'\}$ ({\sc SPLIT}).
\end{itemize}
\end{definition}
The ``{\sc SPLIT} and {\sc MATCH}'' has been applied to compute a maximum matching in linear-time for cographs and some of its generalizations~\cite{FGV99,FPT97,YuY93}.
Our Theorem~\ref{thm:mw-maxmatching} can be seen as a broad generalization of this technique.
In what follows, we introduce more cases where the ``{\sc SPLIT} and {\sc MATCH}'' technique can be used in order to compute a maximum matching directly.
\begin{lemma}\label{lem:join}
Let $G = G_1 \oplus G_2$ be the join of two graphs $G_1,G_2$ and let $F_1,F_2$ be maximum matchings for $G_1,G_2$, respectively.
For $F = F_1 \cup F_2$, applying the `{\sc SPLIT} and {\sc MATCH}'' technique to $V(G_1)$, then to $V(G_2)$ leads to a maximum matching of $G$.
\end{lemma}
\begin{proof}
The lemma is proved in~\cite{YuY93} when $G$ is a cograph.
In particular, let $G^* = (V, (V(G_1) \times V(G_2)) \cup F_1 \cup F_2)$.
Since it ignores the edges from $(E(G_1) \setminus F_1) \cup (E(G_2) \setminus F_2)$, the procedure outputs the same matching for $G$ and $G^*$.
Furthermore, $G^*$ is a cograph, and so, the outputted matching is maximum for $G^*$.
By Lemma~\ref{lem:mw-matching-reduction}, a maximum matching for $G^*$ is a maximum matching for $G$.
\end{proof}
\paragraph{Applications.}
We can now combine our reductions rules as follows.
\begin{proposition}\label{prop:maxmatching-prime-ptree}
Let $G=(V,E)$ be a $(q,q-3)$-graph, $q \geq 7$, such that its quotient graph $G'$ is isomorphic to a prime $p$-tree.
For every $M \in {\cal M}(G)$ let $F_M$ be a maximum matching of $G[M]$ and let $F^* = \bigcup_{M \in {\cal M}(G)}F_M$.
A maximum matching $F_{\max}$ for $G$ can be computed in ${\cal O}(|V(G')|+|E(G')|+|F_{\max}|-|F^*|)$-time if $F^*$ is given as part of the input.
\end{proposition}
\begin{proof}
There are five cases.
If $G'$ has order at most $7$ then we can apply the same techniques as for Theorem~\ref{thm:mw-maxmatching}.
Otherwise, $G'$ is either a spiked $p$-chain $P_k$, a spiked $p$-chain $\overline{P_k}$, a spiked $p$-chain $Q_k$ or a spiked $p$-chain $\overline{Q_k}$.
\paragraph{Case $G$ is a spiked $p$-chain $P_k$.}
By Theorem~\ref{thm:stronger-mdc-qq3} we have that $(v_2,v_3,\ldots,v_{k-1})$ are vertices of $G$.
In this situation, since $N_{G'}(v_1) = v_2$, $M_{v_1}$ is a pending module.
We can apply the reduction rule of Lemma~\ref{lem:pending-module} to $M_{v_1}$.
Doing so, we discard $M_{v_1}$ and possibly $v_2$.
Let $S = M_x$ if $v_2$ has already been discarded and let $S = M_x \cup \{v_2\}$ otherwise.
We have that $S$ is a pending module in the resulting subgraph, with $v_3$ being its unique neighbour.
Furthermore, by Lemma~\ref{lem:pending-module} we can compute a maximum matching of $G[S]$ from $F_{M_{v_x}}$, by adding an edge between $v_2$ (if it is present) and an unmatched vertex in $M_x$ (if any).
So, we again apply the reduction rule of Lemma~\ref{lem:pending-module}, this time to $S$.
Doing so, we discard $S$, and possibly $v_3$.
Then, by a symmetrical argument we can also discard $M_{v_k}, \ M_y, \ v_{k-1}$ and possibly $v_{k-2}$.
We are left with computing a maximum matching for some subpath of $(v_3,v_4,\ldots,v_{k-2})$, that can be done in linear-time by taking half of the edges.
\paragraph{Case $G$ is a spiked $p$-chain $\overline{P_k}$.}
By Theorem~\ref{thm:stronger-mdc-qq3}, the nontrivial modules of ${\cal M}(G)$ can only be $M_{v_1},M_{v_k},M_x,M_y$.
In particular, $F^* = F_{M_{v_1}} \cup F_{M_{v_k}} \cup F_{M_x} \cup F_{M_y}$.
Let $U = M_{v_1} \cup M_{v_k} \cup M_x \cup M_y$.
The graph $G \setminus U$ is isomorphic to $\overline{P_{k-2}}, \ k \geq 6$.
Furthermore, let $F_{k-2}$ contain the edges $\{v_2,v_{\left\lceil k/2 \right\rceil +1}\}, \ \{v_{\left\lfloor k /2 \right\rfloor}, v_{k-1}\}$ plus all the edges $\{v_i,v_{k+1-i}\}, \ 3 \leq i \leq \left\lfloor k/2 \right\rfloor -1$.
Observe that $F_{k-2}$ is a maximum matching of $\overline{P_{k-2}}$.
In particular it is a perfect matching of $\overline{P_{k-2}}$ if $k$ is even, and if $k$ is odd then it only leaves vertex $v_{\left\lceil k/2 \right\rceil}$ unmatched.
We set $F_0 = F^* \cup F_{k-2}$ to be the initial matching.
Then, we repeat the procedure below until we cannot increase the matching anymore.
We consider the modules $M \in \{ M_{v_1},M_{v_k},M_x,M_y \}$ sequentially.
For every $M$ we try to apply the {\sc SPLIT} and {\sc MATCH} technique of Definition~\ref{def:split-and-match}.
Overall, we claim that the above procedure can be implemented to run in constant-time per loop.
Indeed, assume that the matched vertices (resp., the unmatched vertices) are stored in a list in such a way that all the vertices in a same module $M_v, \ v \in V(G')$ are consecutive.
For every matched vertex $u$, we can access to the vertex that is matched with $u$ in constant-time.
Furthermore for every $v \in V(G')$, we keep a pointer to the first and last vertices of $M_v$ in the list of matched vertices (resp., in the list of unmatched vertices).
For any loop of the procedure, we iterate over four modules $M$, that is a constant.
Furthermore, since $|N_G(M)| \geq |V(G) \setminus M|-2$ then we only need to check three unmatched vertices of $V \setminus M$ in order to decide whether we can perform a {\sc MATCH} operation.
Note that we can skip scanning the unmatched vertices in $M$ using our pointer structure, so, it takes constant-time.
In the same way, we only need to consider three matched vertices of $V \setminus M$ in order to decide whether we can perform a {\sc SPLIT} operation.
Again, it takes constant-time.
Therefore, the claim is proved.
\smallskip
Let $F_{\max}$ be the matching so obtained.
By the above claim it takes ${\cal O}(|F_{\max}| - |F_0|)$-time to compute it with the above procedure.
Furthermore, we claim that $F_{\max}$ is maximum.
Suppose for the sake of contradiction that $F_{\max}$ is not a maximum matching.
By Lemma~\ref{lem:mw-matching-reduction}, $F_{\max}$ cannot be a maximum matching of $G^*$, obtained from $G$ by removing the edges in $(E(G[M_{v_1}]) \cup E(G[M_{v_x}])\cup E(G[M_{v_y}]) \cup E(G[M_{v_k}]))\setminus F^*$.
Let $P=(u_1,u_2,\ldots,u_{2\ell})$ be a {\em shortest} $F_{\max}$-augmenting path in $G^*$, that exists by Theorem~\ref{thm:berge}.
We prove as an intermediate subclaim that both $u_1,u_{2\ell}$ must be part of a same module amongst $M_{v_1},M_{v_k},M_x,M_y$.
Indeed, for every distinct $M,M' \in \{ M_{v_1},M_{v_k},M_x,M_y \}$, every vertex of $M$ is adjacent to every vertex of $M'$.
Furthermore, $V(F_{k-2}) \subseteq V(F_{\max})$ by construction and $v_{\left\lceil k/2 \right\rceil}$ (the only vertex of $\overline{P_{k-2}}$ possibly unmatched) is adjacent to every vertex of $U$.
Therefore, if the subclaim were false then $u_1,u_{2\ell}$ should be adjacent, hence they should have been matched together with a {\sc MATCH} operation.
A contradiction.
So, the subclaim is proved.
Let $M \in \{ M_{v_1},M_{v_k},M_x,M_y \}$ so that $u_1,u_{2\ell} \in M$.
Since $E(G^*[M])=F_M$, and $V(F_M) \subseteq V(F^*) \subseteq V(F_{\max})$ by construction, we have $u_2 \in N_G(M)$.
Furthermore, $u_3 \notin N_G(M)$ since otherwise, by considering $u_1,u_{2\ell} \in M$ and $u_2,u_3 \in N_G(M)$, we should have increased the matching with a {\sc SPLIT} operation.
In this situation, either $u_3 \in M$ or $u_3 \in V \setminus N_G[M]$.
We prove as another subclaim that $u_3,u_4 \in M$.
Indeed, suppose by contradiction $u_4 \in N_G(M)$.
In particular, $(u_1,u_4,u_5,\ldots,u_{2\ell})$ is a shorter augmenting path than $P$, thereby contradicting the minimality of $P$.
Therefore, $u_4 \notin N_G(M)$.
Moreover, if $u_3 \in V \setminus N_G[M]$ then, since the set $V \setminus N_G[M]$ induces a stable, we should have $u_4 \in N_G(M)$.
A contradiction.
So, $u_3 \in M$, and $u_4 \in N_G[M] \setminus N_G(M) = M$, that proves the subclaim.
The above subclaim implies $\{u_3,u_4\} \in F_M$.
Since $\{u_3,u_4\} \notin F_{\max}$, there exists a module $M'$ such that $u_2,u_5 \in M'$, and the edges $\{u_2,u_3\}, \ \{u_4,u_5\}$ have been obtained with a {\sc SPLIT} operation.
However, since $u_1,u_{2\ell} \in M$ are unmatched, and $M \subseteq N_G(M')$, we should have performed two {\sc MATCH} operations intead of performing a {\sc SPLIT} operation.
A contradiction.
Therefore, as claimed, $F_{\max}$ is a maximum matching of $G$.
\paragraph{Case $G$ is a spiked $p$-chain $Q_k$.}
For every $1 \leq i \leq \left\lceil k/2 \right\rceil$, let $V_i = \bigcup_{j \geq i} (M_{v_{2j-1}}\cup M_{v_{2j}} \cup M_{z_{2j-1}} \cup M_{z_{2j}})$ (by convention $M_{v} = \emptyset$ if vertex $v$ is not present).
Roughly, our algorithm tries to compute recursively a maximum matching for $G_i = G[V_i \cup U_{i-1}]$, where $U_{i-1}$ is a union of modules in $\{M_{v_{2i-2}}, \ M_{z_{2i-2}}\}$.
Initially, we set $i=1$ and $U_0 = \emptyset$.
See Fig.~\ref{fig-p-tree-matching}.
\begin{figure}[h!]
\centering
\includegraphics[width=.5\textwidth]{Fig/matching-qq3}
\caption{Schematic view of graph $G_i$.}
\label{fig-p-tree-matching}
\end{figure}
If $i= \left\lceil k/2 \right\rceil$ then the quotient subgraph $G_i'$ has order at most six.
We can reuse the same techniques as for Theorem~\ref{thm:mw-maxmatching} in order to solve this case.
Thus from now on assume $i < \left\lceil k/2 \right\rceil$.
We need to observe that $v_{2i-1}$ is a pending vertex in the quotient subgraph $G_i'$, with $v_{2i}$ being its unique neighbour.
By Theorem~\ref{thm:stronger-mdc-qq3}, $v_{2i} \in V(G)$, hence $M_{v_{2i-1}}$ is a pending module of $G_i$.
Thus, we can apply the reduction rule of Lemma~\ref{lem:pending-module}.
Doing so, we can discard the set $S_i$, where $S_i = M_{v_{2i-1}} \cup \{v_{2i}\}$ if $F_{M_{v_{2i-1}}}$ is not a perfect matching of $G[M_{v_{2i-1}}]$, and $S_i = M_{v_{2i-1}}$ otherwise.
Furthermore, in the case where $U_{i-1} \neq \emptyset$, there is now a complete join between $U_{i-1}$ and $V_i \setminus S_i$.
By Lemma~\ref{lem:join} we can compute a maximum matching of $G_i \setminus S_i$ from a maximum matching of $G[U_{i-1}]$ and a maximum matching of $G[V_i \setminus S_i]$.
In particular, since $U_{i-1}$ is a union of modules in $\{M_{v_{2i-2}}, \ M_{z_{2i-2}}\}$ and there is a complete join between $M_{v_{2i-2}}$ and $M_{z_{2i-2}}$, by Lemma~\ref{lem:join} a maximum matching of $G[U_{i-1}]$ can be computed from $F_{M_{v_{2i-2}}}$ and $F_{M_{z_{2i-2}}}$.
So, we are left to compute a maximum matching of $G[V_i \setminus S_i]$.
Then, there are two subcases.
If $v_{2i} \in S_i$ then $M_{z_{2i-1}}$ is disconnected in $G[V_i \setminus S_i]$.
Let $U_i = M_{z_{2i}}$.
The union of $F_{M_{z_{2i-1}}}$ with a maximum matching of $G_{i+1} = G[V_{i+1} \cup U_i]$ is a maximum matching of $G[V_i \setminus S_i]$.
Otherwise, $M_{z_{2i-1}}$ is a pending module of $G[V_i \setminus S_i]$ with $v_{2i}$ being its unique neighbour.
We apply the reduction rule of Lemma~\ref{lem:pending-module}.
Doing so, we can discard the set $T_i$, where $T_i = M_{z_{2i-1}} \cup \{v_{2i}\}$ if $F_{M_{z_{2i-1}}}$ is not a perfect matching of $G[M_{z_{2i-1}}]$, and $T_i = M_{z_{2i-1}}$ otherwise.
Let $U_i = M_{z_{2i}}$ if $v_{2i} \in T_i$ and $U_i = M_{z_{2i}} \cup M_{v_{2i}}$ otherwise.
We are left to compute a maximum matching of $G_{i+1} = G[V_{i+1} \cup U_i]$.
Overall, the procedure stops after we reach an empty subgraph, that takes ${\cal O}(|V(G')|)$ recursive calls.
\paragraph{Case $G$ is a spiked $p$-chain $\overline{Q_k}$.}
Roughly, the case where $G'$ is isomorphic to a spiked $p$-chain $\overline{Q_k}$ is obtained by reverting the role of vertices with even index and vertices with odd index.
For every $1 \leq i \leq \left\lfloor k/2 \right\rfloor$, let $V_i = \bigcup_{j \geq i} (M_{v_{2j}} \cup M_{z_{2j}} \cup M_{v_{2j+1}} \cup M_{z_{2j+1}})$.
Our algorithm tries to compute recursively a maximum matching for $G_i = G[V_i \cup U_{i-1}]$, where $U_{i-1}$ is a union of modules in $\{M_{v_{2i-1}}, \ M_{z_{2i-1}}\}$.
Initially, we set $i=1$ and $U_0 = M_{v_1}$.
If $i= \left\lfloor k/2 \right\rfloor$ then the quotient subgraph $G_i'$ has order at most six.
We can reuse the same techniques as for Theorem~\ref{thm:mw-maxmatching} in order to solve this case.
Thus from now on assume $i < \left\lfloor k/2 \right\rfloor$.
We need to observe that $v_{2i}$ is a pending vertex in the quotient subgraph $G_i'$, with $v_{2i+1}$ being its unique neighbour.
By Theorem~\ref{thm:stronger-mdc-qq3}, $v_{2i+1} \in V(G)$, hence $M_{v_{2i}}$ is a pending module of $G_i$.
Thus, we can apply the reduction rule of Lemma~\ref{lem:pending-module}.
Doing so, we can discard the set $S_i$, where $S_i = M_{v_{2i}} \cup \{v_{2i+1}\}$ if $F_{M_{v_{2i}}}$ is not a perfect matching of $G[M_{v_{2i}}]$, and $S_i = M_{v_{2i}}$ otherwise.
Furthermore, in the case where $U_{i-1} \neq \emptyset$, there is now a complete join between $U_{i-1}$ and $V_i \setminus S_i$.
By Lemma~\ref{lem:join} we can compute a maximum matching of $G_i \setminus S_i$ from a maximum matching of $G[U_{i-1}]$ and a maximum matching of $G[V_i \setminus S_i]$.
In particular, since $U_{i-1}$ is a union of modules in $\{M_{v_{2i-1}}, \ M_{z_{2i-1}}\}$ and there is a complete join between $M_{v_{2i-1}}$ and $M_{z_{2i-1}}$, by Lemma~\ref{lem:join} a maximum matching of $G[U_{i-1}]$ can be computed from $F_{M_{v_{2i-2}}}$ and $F_{M_{z_{2i-2}}}$.
So, we are left to compute a maximum matching of $G[V_i \setminus S_i]$.
Then, there are two subcases.
If $v_{2i+1} \in S_i$ then $M_{z_{2i}}$ is disconnected in $G[V_i \setminus S_i]$.
Let $U_i = M_{z_{2i+1}}$.
The union of $F_{M_{z_{2i}}}$ with a maximum matching of $G_{i+1} = G[V_{i+1} \cup U_i]$ is a maximum matching of $G[V_i \setminus S_i]$.
Otherwise, $M_{z_{2i}}$ is a pending module of $G[V_i \setminus S_i]$ with $v_{2i+1}$ being its unique neighbour.
We apply the reduction rule of Lemma~\ref{lem:pending-module}.
Doing so, we can discard the set $T_i$, where $T_i = M_{z_{2i}} \cup \{v_{2i+1}\}$ if $F_{M_{z_{2i}}}$ is not a perfect matching of $G[M_{z_{2i}}]$, and $T_i = M_{z_{2i}}$ otherwise.
Let $U_i = M_{z_{2i+1}}$ if $v_{2i+1} \in T_i$ and $U_i = M_{z_{2i+1}} \cup M_{v_{2i+1}}$ otherwise.
We are left to compute a maximum matching of $G_{i+1} = G[V_{i+1} \cup U_i]$.
Overall, the procedure stops after ${\cal O}(|V(G')|)$ recursive calls.
\end{proof}
\subsubsection*{Main result}
\begin{theorem}\label{thm:qq3-maxmatching}
For every $G=(V,E)$, {\sc Maximum Matching} can be solved in ${\cal O}(q(G)^4 \cdot n + m)$-time.
\end{theorem}
\begin{proof}
We generalize the algorithm for Theorem~\ref{thm:mw-maxmatching}.
In particular the algorithm is recursive.
If $G$ is trivial (reduced to a single node) then we output an empty matching.
Otherwise, let $G' = ({\cal M}(G),E')$ be the quotient graph of $G$.
For every module $M \in {\cal M}(G)$, we call the algorithm recursively on $G[M]$ in order to compute a maximum matching $F_M$ of $G[M]$.
Let $F^* = \bigcup_{M \in {\cal M}(G)} F_M$.
If $G'$ is either edgeless, complete or a prime graph with no more than $q(G)$ vertices then we apply the same techniques as for Theorem~\ref{thm:mw-maxmatching} in order to compute a maximum matching $F_{\max}$ for $G$.
It takes constant-time if $G'$ is a stable, ${\cal O}(q(G)^4\cdot(|F_{\max}|-|F^*|))$-time if $G'$ is prime and ${\cal O}(|V(G')|+ (|F_{\max}|-|F^*|))$-time if $G'$ is a complete graph.
Otherwise by Theorem~\ref{thm:stronger-mdc-qq3} the following cases need to be considered.
\begin{itemize}
\item Suppose $G$ is a disc.
In particular, $G = G'$.
By Lemma~\ref{lem:disc}, we can compute a maximum matching for $G$ in ${\cal O}(|V(G')|+|E(G')|)$-time.
\item Suppose $G=(S \cup K \cup R, E)$ is a spider.
In particular, $G'=(S \cup K \cup R',E')$ is a prime spider.
By Lemma~\ref{lem:spider}, the union of $F_{R} = F^*$ with a perfect matching between $S$ and $K$ is a maximum matching of $G$.
It can be computed in ${\cal O}(|V(G')|+|E(G')|)$-time.
\item Otherwise $G'$ is a prime $p$-tree.
By Proposition~\ref{prop:maxmatching-prime-ptree}, a maximum matching $F_{\max}$ for $G$ can be computed in ${\cal O}(|V(G')|+|E(G')|+|F_{\max}|-|F^*|)$-time.
\end{itemize}
Overall, summing the order of all the subgraphs in the modular decomposition of $G$ amounts to ${\cal O}(n)$~\cite{Rao08b}.
Summing the size of all the subgraphs in the modular decomposition of $G$ amounts to ${\cal O}(n+m)$~\cite{Rao08b}.
Furthermore, a maximum matching of $G$ also has cardinality ${\cal O}(n)$.
Therefore, the total running time is in ${\cal O}(q(G)^4 \cdot n+m)$ if the modular decomposition of $G$ is given.
The latter decomposition can be precomputed in ${\cal O}(n+m)$-time~\cite{TCHP08}.
\end{proof}
\section{Preliminaries}\label{sec:prelim}
We use standard graph terminology from~\cite{BoM08,Die10}.
Graphs in this study are finite, simple (hence without loops or multiple edges) and unweighted -- unless stated otherwise.
Furthermore we make the standard assumption that graphs are encoded as adjacency lists.
We want to prove the existence, or the nonexistence, of graph algorithms with running time of the form $k^{{\cal O}(1)} \cdot (n+m)$, $k$ being some fixed graph parameter.
In what follows, we introduce the graph parameters considered in this work.
\subsubsection*{Clique-width}
A labeled graph is given by a pair $\langle G, \ell \rangle$ where $G=(V,E)$ is a graph and $\ell : V \to \mathbb{N}$ is called a labeling function.
A {\em k-expression} can be seen as a sequence of operations for constructing a labeled graph $\langle G, \ell \rangle$, where the allowed four operations are:
\begin{enumerate}
\item Addition of a new vertex $v$ with label $i$ (the labels are taken in $\{1, 2, \ldots, k\}$), denoted $i(v)$;
\item Disjoint union of two labeled graphs $\langle G_1, \ell_1 \rangle$ and $\langle G_2, \ell_2 \rangle$, denoted $\langle G_1, \ell_1 \rangle \oplus \langle G_2, \ell_2 \rangle$;
\item Addition of a join between the set of vertices labeled $i$ and the set of vertices labeled $j$, where $i \neq j$, denoted $\eta(i,j)$;
\item Renaming label $i$ to label $j$, denoted $\rho(i,j)$.
\end{enumerate}
See Fig.~\ref{fig:cw-example} for examples.
The {\em clique-width} of $G$, denoted by $cw(G)$, is the minimum $k$ such that, for some labeling $\ell$, the labeled graph $\langle G, \ell \rangle$ admits a $k$-expression~\cite{CER93}.
We refer to~\cite{CMR00} and the references cited therein for a survey of the many applications of clique-width in the field of parameterized complexity.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{.15\textwidth}\centering
\includegraphics[width=.25\textwidth]{Fig/cw-example.pdf}
\caption{$1(v_a)$}
\end{subfigure}\hfill
\begin{subfigure}[b]{.15\textwidth}\centering
\includegraphics[width=.5\textwidth]{Fig/cw-example-2}
\caption{$2(v_b)$}
\end{subfigure}\hfill
\begin{subfigure}[b]{.15\textwidth}\centering
\includegraphics[width=.5\textwidth]{Fig/cw-example-3}
\caption{$\eta(1,2)$}
\end{subfigure}\hfill
\begin{subfigure}[b]{.15\textwidth}\centering
\includegraphics[width=.5\textwidth]{Fig/cw-example-4}
\caption{$\rho(1,3)$}
\end{subfigure}\hfill
\begin{subfigure}[b]{.15\textwidth}\centering
\includegraphics[width=.75\textwidth]{Fig/cw-example-5}
\caption{$1(v_c)$}
\end{subfigure}\hfill
\begin{subfigure}[b]{.15\textwidth}\centering
\includegraphics[width=.75\textwidth]{Fig/cw-example-6}
\caption{$\eta(2,1)$}
\end{subfigure}\vspace{15pt}
\begin{subfigure}[b]{.15\textwidth}\centering
\includegraphics[width=.75\textwidth]{Fig/cw-example-7}
\caption{$\rho(2,3)$}
\end{subfigure}\hfill
\begin{subfigure}[b]{.15\textwidth}\centering
\includegraphics[width=\textwidth]{Fig/cw-example-8}
\caption{$2(v_d)$}
\end{subfigure}\hfill
\begin{subfigure}[b]{.15\textwidth}\centering
\includegraphics[width=\textwidth]{Fig/cw-example-9}
\caption{$\eta(1,2)$}
\end{subfigure}
\caption{A $3$-expression for the path $P_4$.}
\label{fig:cw-example}
\end{figure}
Computing the clique-width of a given graph is NP-hard~\cite{FRRS09}.
However, on a more positive side the graphs with clique-width two are exactly the cographs and they can be recognized in linear-time~\cite{CPS85,CoB00}.
Clique-width three graphs can also be recognized in polynomial-time~\cite{CHLB+00}.
The parameterized complexity of computing the clique-width is open.
In what follows, we focus on upper-bounds on clique-width that are derived from some graph decompositions.
\subsubsection*{Modular-width}
A {\em module} in a graph $G=(V,E)$ is any subset $M \subseteq V(G)$ such that for any $v \in V \setminus M$, either $M \subseteq N_G(v)$ or $M \cap N_G(v) = \emptyset$.
Note that $\emptyset, \ V, \ \mbox{and} \ \{v\}$ for every $v \in V$ are trivial modules of $G$.
A graph is called {\em prime} for modular decomposition if it only has trivial modules.
A module $M$ is {\em strong} if it does not overlap any other module, {\it i.e.}, for any module $M'$ of $G$, either one of $M$ or $M'$ is contained in the other or $M$ and $M'$ do not intersect.
Furthermore, let ${\cal M}(G)$ be the family of all inclusion wise maximal strong modules of $G$ that are proper subsets of $V$.
The {\em quotient graph} of $G$ is the graph $G'$ with vertex-set ${\cal M}(G)$ and an edge between every two $M,M' \in {\cal M}(G)$ such that every vertex of $M$ is adjacent to every vertex of $M'$.
Modular decomposition is based on the following structure theorem from Gallai.
\begin{theorem}[~\cite{Gal67}]\label{thm:modular-dec}
For an arbitrary graph $G$ exactly one of the following conditions is satisfied.
\begin{enumerate}
\item $G$ is disconnected;
\item its complement $\overline{G}$ is disconnected;
\item or its quotient graph $G'$ is prime for modular decomposition.
\end{enumerate}
\end{theorem}
Theorem~\ref{thm:modular-dec} suggests the following recursive procedure in order to decompose a graph, that is sometimes called modular decomposition.
If $G = G'$ ({\it i.e.}, $G$ is complete, edgeless or prime for modular decomposition) then we output $G$.
Otherwise, we output the quotient graph $G'$ of $G$ and, for every strong module $M$ of $G$, the modular decomposition of $G[M]$.
The modular decomposition of a given graph $G=(V,E)$ can be computed in linear-time~\cite{TCHP08}.
See Fig.~\ref{fig:modular-dec} for an example.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{.46\textwidth}\centering
\includegraphics[width=.35\textwidth]{Fig/modular-graph}
\end{subfigure}\hfill
\begin{subfigure}[b]{.52\textwidth}\centering
\includegraphics[width=.65\textwidth]{Fig/modular-graph-dec}
\end{subfigure}
\caption{A graph and its modular decomposition.}
\label{fig:modular-dec}
\end{figure}
Furthermore, by Theorem~\ref{thm:modular-dec} the subgraphs from the modular decomposition are either edgeless, complete, or prime for modular decomposition.
The {\em modular-width} of $G$, denoted by $mw(G)$, is the minimum $k \geq 2$ such that any prime subgraph in the modular decomposition has order (number of vertices) at most $k$~\footnote{This term has another meaning in~\cite{Rao08}. We rather follow the terminology from~\cite{CoB00}.}.
The relationship between clique-width and modular-width is as follows.
\begin{lemma}[~\cite{CMR00}]\label{lem:mw-cw}
For every $G=(V,E)$, we have $cw(G) \leq mw(G)$, and a $mw(G)$-expression defining $G$ can be constructed in linear-time.
\end{lemma}
We refer to~\cite{HaP10} for a survey on modular decomposition.
In particular, graphs with modular-width two are exactly the cographs, that follows from the existence of a cotree~\cite{Sum73}.
Cographs enjoy many algorithmic properties, including a linear-time algorithm for {\sc Maximum Matching}~\cite{YuY93}.
Furthermore, in~\cite{GLO13} Gajarsk{\`y}, Lampis and Ordyniak prove that for some $W$-hard problems when parameterized by clique-width there exist FPT algorithms when parameterized by modular-width.
\subsubsection*{Split-width}
A {\em split} $(A,B)$ in a {\em connected} graph $G=(V,E)$ is a partition $V = A \cup B$ such that: $\min \{ |A|, |B| \} \geq 2$; and there is a complete join between the vertices of $N_G(A)$ and $N_G(B)$.
For every split $(A,B)$ of $G$, let $a \in N_G(B), \ b \in N_G(A)$ be arbitrary.
The vertices $a,b$ are termed {\em split marker vertices}.
We can compute a ``simple decomposition'' of $G$ into the subgraphs $G_A = G[A \cup \{b\}]$ and $G_B = G[B \cup \{a\}]$.
There are two cases of ``indecomposable'' graphs.
Degenerate graphs are such that every bipartition of their vertex-set is a split.
They are exactly the complete graphs and the stars~\cite{Cun82}.
A graph is prime for split decomposition if it has no split.
A split decomposition of a connected graph $G$ is obtained by applying recursively a simple decomposition, until all the subgraphs obtained are either degenerate or prime.
A split decomposition of an {\em arbitrary} graph $G$ is the union of a split decomposition for each of its connected components.
Every graph has a canonical split decomposition, with minimum number of subgraphs, that can be computed in linear-time~\cite{CDR12}.
The {\em split-width}
of $G$, denoted by $sw(G)$, is the minimum $k \geq 2$ such that any prime subgraph in the canonical split decomposition of $G$ has order at most $k$.
See Fig.~\ref{fig:split-dec} for an illustration.
\begin{lemma}[~\cite{Rao08b}]\label{lem:sw-cw}
For every $G=(V,E)$, we have $cw(G) \leq 2 \cdot sw(G) + 1$, and a $(2 \cdot sw(G) + 1)$-expression defining $G$ can be constructed in linear-time.
\end{lemma}
We refer to~\cite{GaP03,GiP12,Rao08b} for some algorithmic applications of split decomposition.
In particular, graphs with split-width at most two are exactly the distance-hereditary graphs~\cite{BaM86}.
Linear-time algorithms for solving {\sc Diameter} and {\sc Maximum Matching} for distance-hereditary graphs are presented in~\cite{DrN00,Dra97}.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{.46\textwidth}\centering
\includegraphics[width=.45\textwidth]{Fig/split-graph}
\end{subfigure}\hfill
\begin{subfigure}[b]{.52\textwidth}\centering
\includegraphics[width=.75\textwidth]{Fig/split-graph-dec}
\end{subfigure}
\caption{A graph and its split decomposition.}
\label{fig:split-dec}
\end{figure}
We stress that split decomposition can be seen as a refinement of modular decomposition.
Indeed, if $M$ is a module of $G$ and $\min\{|M|,|V\setminus M|\} \geq 2$ then $(M, V \setminus M)$ is a split.
In what follows, we prove most of our results with the more general split decomposition.
\subsubsection*{Graphs with few $P_4$'s}
A $(q,t)$-graph $G = (V,E)$ is such that for any $S \subseteq V$, $|S| \leq q$, $S$ induces at most $t$ paths on four vertices~\cite{BaO98}.
The {\em $P_4$-sparseness} of $G$, denoted by $q(G)$, is the minimum $q \geq 7$ such that $G$ is a $(q,q-3)$-graph.
\begin{lemma}[~\cite{MaR99}]\label{lem:qq3-cw}
For every $q \geq 7$, every $(q,q-3)$-graph has clique-width at most $q$, and a $q$-expression defining it can be computed in linear-time.
\end{lemma}
The algorithmic properties of several subclasses of $(q,q-3)$-graphs have been considered in the literature.
We refer to~\cite{BaO99} for a survey.
Furthermore, there exists a canonical decomposition of $(q,q-3)$-graphs, sometimes called the {\em primeval decomposition}, that can be computed in linear-time~\cite{Bau96}.
Primeval decomposition can be seen as an intermediate between modular and split decomposition.
We postpone the presentation of primeval decomposition until Section~\ref{sec:maxmatching}.
Until then, we state the results in terms of modular decomposition.
\smallskip
More precisely, given a $(q,q-3)$-graphs $G$, the prime subgraphs in its modular decomposition may be of super-constant size $\Omega(q)$.
However, if they are then they are part of one of the well-structured graph classes that we detail next.
\smallskip
A {\em disc} is either a cycle $C_n$, or a co-cycle $\overline{C_n}$, for some $n \geq 5$.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{.48\textwidth}\centering
\includegraphics[width=.45\textwidth]{Fig/thin-spider}
\caption{Thin spider.}
\end{subfigure}\hfill
\begin{subfigure}[b]{.48\textwidth}\centering
\includegraphics[width=.5\textwidth]{Fig/thick-spider}
\caption{Thick spider.}
\end{subfigure}
\caption{Spiders.}
\label{fig:splider}
\end{figure}
\smallskip
A \emph{spider} $G=(S \cup K \cup R,E)$ is a graph with vertex set $V=S \cup K \cup R$ and edge set $E$ such that:
\begin{enumerate}
\item $(S,K,R)$ is a partition of $V$ and $R$ may be empty;
\item the subgraph $G[K\cup R]$ induced by $K$ and $R$ is the complete join $K \oplus R$, and $K$ separates $S$ and $R$, i.e. any path from a vertex in $S$ and a vertex in $R$ contains a vertex in $K$;
\item
$S$ is a stable set, $K$ is a clique, $|S| = |K|\geq 2$, and
there exists a bijection $f: S \longrightarrow K$ such that, either for all
vertices $s\in S$, $N(s)\cap K = K - \{f(s)\}$ or $N(s)\cap K = \{f(s)\}$. Roughly speaking, the edges between $S$ and $K$ are either a matching or an anti-matching. In the former case or if $|S| = |K| \leq 2$, $G$ is called \emph{thin}, otherwise $G$ is \emph{thick}.
See Fig.~\ref{fig:splider}.
\end{enumerate}
If furthermore $|R| \leq 1$ then we call $G$ a {\em prime spider}.
\begin{figure}[h!]
\centering
\includegraphics[width=.45\textwidth]{Fig/spiked-pk}
\caption{Spiked $p$-chain $P_k$.}
\label{fig:spiked-pk}
\end{figure}
\smallskip
Let $P_k = (v_1,v_2,v_3, \ldots, v_k), \ k \geq 6$ be a path of length at least five.
A {\em spiked $p$-chain $P_k$} is a supergraph of $P_k$, possibly with the additional vertices $x,y$ such that: $N(x) = \{v_2,v_3\}$ and $N(y) = \{v_{k-2},v_{k-1}\}$.
See Fig.~\ref{fig:spiked-pk}.
Note that one or both of $x$ and $y$ may be missing.
In particular, $P_k$ is a spiked $p$-chain $P_k$.
A {\em spiked $p$-chain $\overline{P_k}$} is the complement of a spiked $p$-chain $P_k$.
Let $Q_k$ be the graph with vertex-set $\{v_1,v_2, \ldots, v_k\}, \ k \geq 6$ such that, for every $i \geq 1$,
$N_{Q_k}(v_{2i-1}) = \{ v_{2j} \mid j \leq i, \ j \neq i-1 \}$
and $N_{Q_k}(v_{2i}) = \{v_{2j} \mid j \neq i \} \cup \{ v_{2j-1} \mid j \geq i, \ j \neq i+1 \}$.
A {\em spiked $p$-chain $Q_k$} is a supergraph of $Q_k$, possibly with the additional vertices $z_2, z_3, \ldots, z_{k-5}$ such that:
\begin{itemize}
\item $N(z_{2i-1}) = \{v_{2j} \mid j \in [1;i]\} \cup \{z_{2j} \mid j \in [1;i-1]\}$;
\item $\overline{N(z_{2i})} = \{v_{2j-1} \mid j \in [1;i+1]\} \cup \{z_{2j-1} \mid j \in [2;i]\}$
\end{itemize}
Any of the vertices $z_i$ can be missing, so, in particular, $Q_k$ is a spiked $p$-chain $Q_k$.
See Fig.~\ref{fig:spiked-qk}.
A {\em spiked $p$-chain $\overline{Q_k}$} is the complement of a spiked $p$-chain $Q_k$.
\begin{figure}[h!]
\centering
\includegraphics[width=.25\textwidth]{Fig/spiked-qk}
\caption{Spiked $p$-chain $Q_k$.}
\label{fig:spiked-qk}
\end{figure}
Finally, we say that a graph is a {\em prime $p$-tree} if it is either: a spiked $p$-chain $P_k$, a spiked $p$-chain $\overline{P_k}$, a spiked $p$-chain $Q_k$, a spiked $p$-chain $\overline{Q_k}$, or part of the seven graphs of order at most $7$ that are listed in~\cite{MaR99}.
\begin{lemma}[~\cite{Bab00,MaR99}]\label{lem:reduce-qq3}
Let $G=(V,E)$, $q \geq 7$, be a connected $(q,q-3)$-graph such that $G$ and $\overline{G}$ are connected.
Then, one of the following must hold for its quotient graph $G'$:
\begin{itemize}
\item either $G'$ is a prime spider;
\item or $G'$ is a disc;
\item or $G'$ is a prime $p$-tree;
\item or $|V(G')| \leq q$.
\end{itemize}
\end{lemma}
A simpler version of Lemma~\ref{lem:reduce-qq3} holds for the subclass of $(q,q-4)$-graphs:
\begin{lemma}[~\cite{Bab00}]\label{lem:reduce-qq4}
Let $G=(V,E)$, $q \geq 4$, be a connected $(q,q-4)$-graph such that $G$ and $\overline{G}$ are connected.
Then, one of the following must hold for its quotient graph $G'$:
\begin{itemize}
\item $G'$ is a prime spider;
\item or $|V(G')| \leq q$.
\end{itemize}
\end{lemma}
The subclass of $(q,q-4)$-graphs has received more attention in the literature than $(q,q-3)$-graphs.
Our results hold for the more general case of $(q,q-3)$-graphs.
|
1,116,691,499,392 | arxiv |
\section*{Appendix}
\section{Proof of Theorem 1}\label{app:Th:compute_g}
\begin{proof}(Theorem 1)
Formally, denote $g_Z$ to estimate $\nabla_{\theta}\log Z_{\theta}$ as follows using importance sampling:
\begin{align}
g_Z &=\frac{1}{M_2} \sum_{\tau\in\mathcal{T}^{Q}_{M_2,1}}\frac{P(\tau|\theta,T)}{Q(\tau|\hat{\theta})}\nabla_{\theta}R_{\theta}(\tau)\\
&=\frac{1}{M_2}\sum_{\tau\in\mathcal{T}^{Q}_{M_2,1}}\frac{Z_{\hat{\theta}}}{Z_{\theta}}\frac{D(\tau)e^{-R_{\theta}(\tau)}} {e^{-\hat{\theta}^Tf(\tau)}}\nabla_{\theta}R_{\theta}(\tau)
\end{align}
where $M_2$ is the sample size of importance distribution $Q$. $\mathcal{T}$ denotes the set of all feasible trajectories, and $\mathcal{T}_M^Q$ denotes the set of $M$ trajectories sampled from distribution $Q$.
Then, we can write $Z_{\theta}$ out and further use importance sampling
\begin{align}
g_Z&=\frac{\frac{Z_{\hat{\theta}}}{M_2}\sum_{\tau\in\mathcal{T}^{Q}_{M_2,1}}\frac{D(\tau)e^{-R_{\theta}(\tau)}} {e^{-\hat{\theta}^Tf(\tau)}}\nabla_{\theta}R_{\theta}(\tau)}{\sum_{\tau\in\mathcal{T}} e^{-R_{\theta}(\tau)}D(\tau)I_C(\tau) }\\
&=\frac{\frac{Z_{\hat{\theta}}}{M_2}\sum_{\tau\in\mathcal{T}^{Q}_{M_2,1}}\frac{D(\tau)e^{-R_{\theta}(\tau)}} {e^{-\hat{\theta}^Tf(\tau)}}\nabla_{\theta}R_{\theta}(\tau)}{\frac{1}{M_2}\sum_{\tau\in\mathcal{T}^Q_{M_2,2}} \frac{e^{-R_{\theta}(\tau)}D(\tau)I_C(\tau)}{I_C(\tau)e^{-\hat{\theta}^Tf(\tau)}}Z_{\hat{\theta}} }
\end{align}
where we can see the term $Z_{\hat{\theta}}/M_2$ in both the nominator and the denominator are cancelled out. Therefore, by simplifying this equation we get
\begin{align}
g_Z=\frac{\sum_{\tau\in\mathcal{T}^{Q}_{M_2,1}}\frac{D(\tau)e^{-R_{\theta}(\tau)}} {e^{-\hat{\theta}^Tf(\tau)}}\nabla_{\theta}R_{\theta}(\tau)}{\sum_{\tau\in\mathcal{T}^Q_{M_2,2}} \frac{e^{-R_{\theta}(\tau)}D(\tau)}{e^{-\hat{\theta}^Tf(\tau)}}}
\end{align}
Especially, if we represent $R_{\theta}(\tau)$ as linear combination of features and let $\hat{\theta}=\theta$, this equation further simplifies to
\begin{align}
g_Z =\frac{\sum_{\tau\in\mathcal{T}^{Q}_{M_2,1}}D(\tau)f(\tau)}{\sum_{\tau\in\mathcal{T}^Q_{M_2,2}}D(\tau)}
\end{align}
where $f(\tau)$ is the feature of trajectory $\tau$. Furthermore, we also have the expectation of $g_Z$ as
\begin{align}
\mathbb{E}[g_Z]&=\mathbb{E}_{Q}[\frac{P(\tau|\theta,T)}{Q(\tau|\theta)}\nabla_{\theta}R_{\theta}(\tau)]\\
&=\int P(\tau|\theta,T)f(\tau)\\
&=\mathbb{E}_{P}[f(\tau)]
\end{align}
Therefore, because of
\begin{align*
g_{\theta}&=\frac{1}{M_1}\sum_{\tau\in \mathcal{D}_{M_1}} f(\tau)-g_Z\\
&=\frac{1}{M_1}\sum_{\tau\in \mathcal{D}_{M_1}} f(\tau)-\frac{\sum_{\tau\in\mathcal{T}^{Q}_{M_2,1}}D(\tau)f(\tau)}{\sum_{\tau\in\mathcal{T}^Q_{M_2,2}}D(\tau)}
\end{align*}
we can see the expectation of $g_{\theta}$ is
\begin{align}
\mathbb{E}[g_{\theta}]=\mathbb{E_{\mathcal{D}}}[f(\tau)]-\mathbb{E}_{P}[f(\tau)]=\nabla_{\theta}\log Z_{\theta}
\end{align}
This completes the proof.
\end{proof}
\section{Proof of Theorem 3}\label{app:Th:main}
To prove Theorem 3, we first introduce a Lemma as follows:
\begin{Lm}\label{Lm:L-smooth}
If the total variation $\max_{\theta}Var_{P}(f(\tau))\leq \sigma_2^2$, then $L(\theta)$ is $\sigma_2^2$-smooth w.r.t. $\theta$.
\end{Lm}
\begin{proof}
Since $L(\theta)=-\frac{1}{N}\sum_{\tau\in\mathcal{D}}\log P(\tau|\theta,T)$, $L$-smoothness requires that
\begin{align*}
||\nabla L(\theta_1)-\nabla l(\theta_2)||_2\leq L||\theta_1-\theta_2||_2
\end{align*}
where $L$ is a constant. Because of the mean value theorem, there exists a point $\tilde{\theta}\in(\theta_1, \theta_2)$ such that
\begin{align*}
\nabla L(\theta_1)-\nabla L(\theta_2)=\nabla(\nabla L(\tilde{\theta}))(\theta_1-\theta_2).
\end{align*}
Taking the $L_2$ norm for both sides, we have
\begin{align}
||\nabla L(\theta_1)-\nabla L(\theta_2)||_2=&
||\nabla(\nabla L(\tilde{\theta}))(\theta_1-\theta_2)||_2\nonumber\\
\leq&||\nabla(\nabla L(\tilde{\theta}))||_2~||\theta_1-\theta_2||_2\label{eq:lem11}
\end{align}
Then, the problem is to bound the matrix 2-norm $||\nabla(\nabla L(\tilde{\theta}))||_2$. Since we know the explicit form of $L(\theta)$, we know
\begin{align}
\nabla L(\theta)&=\nabla\log Z_{\theta}-\frac{1}{N}\sum_{\tau\in\mathcal{D}}f(\tau), \nonumber\\
\nabla(\nabla L(\theta))&= \sum_{\tau\in\mathcal{T}}[f(\tau)-\nabla\log Z_{\theta}][f(\tau)-\nabla\log Z_{\theta}]^T P(\tau|\theta,T)\label{eq:lem12},
\end{align}
where $\nabla(\nabla L(\theta))$ is the co-variance matrix. Denote Cov$_{\theta}[f(\tau)]=\nabla(\nabla L(\theta))$, which is both symmetric and positive semi-definite. We have
\begin{align*}
||\nabla(\nabla L(\tilde{\theta}))||_2=||\text{Cov}_{\theta}[f(\tau)]||_2=\lambda_{max},
\end{align*}
where $\lambda_{max}$ is the maximum eigenvalue of the matrix Cov$_{\theta}[f(\tau)]$. Then, because of the positive semi-definiteness of the co-variance matrix, all the eigenvalues are non-negative, and we can bound $\lambda_{max}$ as
\begin{align*}
\lambda_{max}\leq\sum_{i}\lambda_i=Tr(\text{Cov}_{\theta}[f(\tau)]),
\end{align*}
where $Tr($Cov$_{\theta}[\phi(X)])$ is the trace of matrix Cov$_{\theta}[f(\tau)]$. Using the definition in Equation \ref{eq:lem12}, $Tr($Cov$_{\theta}[f(\tau)])$ can be further derived as:
\begin{align*}
Tr(\text{Cov}_{\theta}[f(\tau)])=\mathbb{E}_{P}[||f(\tau)||_2^2]-||\mathbb{E}_{P}[f(\tau)]||_2^2,
\end{align*}
which is equal to the total variation $Var_{P}(f(\tau))$. Therefore, we have
\begin{align*}
||\nabla(\nabla L(\tilde{\theta}))||_2\leq Var_{P}(f(\tau))\leq \sigma_2^2.
\end{align*}
Combining this with Equation \ref{eq:lem11}, we know
\begin{align*}
||\nabla L(\theta_1)-\nabla L(\theta_2)||_2\leq \sigma_2^2~||\theta_1-\theta_2||_2.
\end{align*}
This completes the proof.
\end{proof}
We give the full proof of Theorem 3 as follows:
\begin{proof}(Theorem 3)
Since we use $M_1$ samples from the training set $\{\tau_i\}_{i=1}^{M_1}$ and $2M_2$ samples $\tau'_1, \dots, \tau'_{M_2}$ from $Q(\tau|\theta)$ using XOR-Sampling at each iteration, we have
$$g_k = \frac{1}{M_1}\sum_{j=1}^{M_1}f(\tau_j)-\frac{\sum_{j=1}^{M_2}D(\tau'_j)f(\tau'_j)}{\sum_{j=M_2+1}^{2M_2}D(\tau'_j)}$$
Assume these samples $\tau'$ form distribution $Q'$, and denote $g_k^i=\frac{1}{M_1}\sum_{j=1}^{M_1}f(\tau_j)-\frac{D(\tau'_i)f(\tau'_i)}{\mathbb{E}_{Q'}[D(\tau)]}$, we have the expectation of $g_k$ as
\begin{align*}
\mathbb{E}_{\mathcal{D},Q'}[g_k]&=\frac{\mathbb{E}_{Q'}[\mathbb{E}_{\mathcal{D}}[f(\tau)]\mathbb{E}_{Q'}[D(\tau)]-D(\tau')f(\tau')]}{\mathbb{E}_{Q'}[D(\tau)]}\\
&=\mathbb{E}_{\mathcal{D},Q'}[g_k^i].
\end{align*}
In each iteration $k$ we can adjust the parameters in XOR-Sampling to give the constant factor approximation of both the denominator and the nominator, then for each $g_k^i$ we can obtain from Theorem 2 that
\begin{align}
\frac{1}{\delta^2} [\nabla L(\theta_k)]^+ &\leq \mathbb{E}_{\mathcal{D},Q'}[g_k^{i+}]\leq \delta^2 [\nabla L(\theta_k)]^+,\label{eq:pos2}\\
\delta^2 [\nabla L(\theta_k)]^- &\leq \mathbb{E}_{\mathcal{D},Q'}[g_k^{i-}]\leq \frac{1}{\delta^2}[\nabla L(\theta_k)]^-.\label{eq:neg2}
\end{align}
where $\nabla L(\theta_k)$ is the true gradient at $k$-th iteration. Denote $g_k^+=\max\{g_k,\textbf{0}\}$ and $g_k^-=\min\{g_k,\textbf{0}\}$.
Clearly, $g_k^{i+} \geq 0$ and $g_k^{i-} \leq 0$. Moreover,
for a given dimension, either $g_k^{i+}=0$ for that dimension or $g_k^{i-}=0$.
Evaluating $g_k$ dimension by dimension, we can see that
$g_k^+=\frac{1}{M_2}\sum_{i=1}^{M_2} g_k^{i+}$ and $g_k^-=\frac{1}{M_2}\sum_{i=1}^{M_2} g_k^{i-}$.
Combined with Equation~\ref{eq:pos2} and \ref{eq:neg2}, we know
\begin{align*}
\frac{1}{\delta^2} [\nabla L(\theta_k)]^+ \leq \mathbb{E}[g_k^+]
\leq \delta^2 [\nabla L(\theta_k)]^+,\\
\delta^2 [\nabla L(\theta_k)]^- \leq \mathbb{E}[g_k^-]
\leq \frac{1}{\delta^2} [\nabla L(\theta_k)]^-.
\end{align*}
In terms of variance, assume the variance of each $f(\tau'_j)$ can be bounded by $Var_{P}(f(\tau'_j))\leq \sigma_2^2$ resulted from both importance sampling and XOR-Sampling, and because $\mathbb{E}_{\mathcal{D},P}[g_k]=\mathbb{E}_{\mathcal{D},P}[g_k^i]$, the variance of $g_k$, denoted as $Var_{\mathcal{D},P}(g_k)$, can then be bounded as
\begin{align*}
Var_{\mathcal{D},P}(g_k)&= Var_D(\frac{1}{M_1}\sum_{j=1}^{M_1}f(\tau_j)) +\\
& Var_{P}(\frac{1}{M_2}\sum_{i=1}^{M_2}f(\tau'_i))\\
& =\frac{1}{M_1}Var_D(f(\tau_j)) + \frac{1}{M_2}Var_{P}(f(\tau'_i))\\
& \leq \frac{\sigma_1^2}{M_1} + \frac{\sigma_2^2}{M_2}
\end{align*}
Therefore, since $L(\theta)$ is convex and $\sigma_2^2-$smooth from Lemma \ref{Lm:L-smooth}, we can then apply Theorem 4 to get the result in Theorem 3.
\begin{align*}
\mathbb{E}[L(\overline{\theta_K})]-OPT &\leq \frac{\delta^2||\theta_0-\theta^*||_2^2}{2\eta K}+\frac{\eta\max_{\theta_k}\{Var_{\mathcal{D},P}(g_k)\}}{\delta^2}\\
&\leq\frac{\delta^2||\theta_0-\theta^*||_2^2}{2\eta K}+\frac{\eta\sigma_1^2}{\delta^2 M_1}+\frac{\eta\sigma_2^2}{\delta^2M_2}.
\end{align*}
This completes the proof.
\end{proof}
\section{Proof of Theorem 5}
\begin{proof} (Theorem 5)
Since we use flow constraints to ensure valid trajectories, the number of binary variables in XOR-Sampling in then $O(|\mathcal{S}||\mathcal{A}|)$.
From Theorem 2 we know that in each iteration of X-MEN, we need to access $O(|\mathcal{S}||\mathcal{A}|\ln\frac{|\mathcal{S}||\mathcal{A}|}{\gamma})$ queries of NP oracles in order to generate one sample. However, as specified also in \cite{ermon2013embed}, only the first sample needs those many queries. Once we have the first sample, the number of XOR constraints to add can be known in generating future samples for this SGD iteration.
Therefore, we fix the number of XOR constraints added starting the generation of the second sample.
As a result, we only need one NP oracle query in generating each of the following $2(M_2-1)$ samples.
Therefore, total queries in each iteration will be $O(|\mathcal{S}||\mathcal{A}|\ln\frac{|\mathcal{S}||\mathcal{A}|}{\gamma}+M_2)$. To complete all $K$ SGD iterations, X-MEN needs $O(K|\mathcal{S}||\mathcal{A}|\ln\frac{|\mathcal{S}||\mathcal{A}|}{\gamma}+KM_2)$ NP oracle queries in total.
\end{proof}
\section{ADAPT SAMPLE DISTRIBUTION VIA VARIANCE REDUCTION}
We have shown that one can solve the constrained inverse reinforcement learning problem by estimating the expectation term in maximum likelihood learning by importance sampling, where a proposal distribution is used to get valid samples that satisfy additional constraints via XOR-Sampling. In this section we show that parameter $\hat{\theta}$ of the proposal distribution can be adaptively updated based on variance reduction. Denote the importance weight $w(\tau)=\frac{P(\tau|\theta,T)}{Q(\tau|\hat{\theta})}$. Since we use importance sampling, the expectation $\mu_q=\mathbb{E}[g_Z]$ is an unbiased estimator of the true value $\mu=\nabla_{\theta}\log Z_{\theta}$, and the variance is $\sigma_Q^2/n$ where
\begin{align}
\sigma_Q^2 &= \sum_{\tau} \frac{\nabla_{\theta}R(\theta)^2P(\tau|\theta,T)^2}{Q(\tau|\hat{\theta})}-\mu^2 \nonumber\\
&= \mathbb{E}_{Q}[\nabla_{\theta}R(\theta)^2w(\tau)^2]-\mu^2
\end{align}
When the reward function $R_{\theta}(\tau)$ is represented by a linear combination of hand-crafted features, i.e., $R_{\theta}(\tau)=\theta^Tf(\tau)$, we let the sample distribution $Q$ share weights with the nominal distribution $P$, i.e., $\hat{\theta}=\theta$, which means each iteration we update parameters $\theta$ in $P$, we update parameters as $\hat{\theta}=\theta$ at the same time. However, in a more general case when reward function $R_{\theta}(\tau)$ is represented not in a linear form, such as a neural network presentation, we can not simply let $\hat{\theta}=\theta$. In this case, we need to find the optimal $\hat{\theta}$ by optimization on some loss functions. One possible way is to minimize the variance via stochastic gradient descent, namely,
\begin{align}
\hat{\theta}=\text{argmin}_{\hat{\theta}}\sigma_Q^2=\text{argmin}_{\hat{\theta}}\mathbb{E}_{Q}[\nabla_{\theta}R(\theta)^2w(\tau)^2]
\end{align}
In this paper we only let $R_{\theta}(\tau)=\theta^Tf(\tau)$ where $f(\tau)$ is either hand-crafted features or neural networks. However, even in this case we can still use the variance reduction method to update $\hat{\theta}$. We leave this version of updating $\hat{\theta}$ for future work where we can see it might not still hold the theoretical guarantee of convergence rate, yet might work better in practice due to the strong representation power of deep neural network.
\section{Human Obstacle Avoidance}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{figs/obstacle.pdf}
\caption{Overlaid trajectories generated by X-MEN to learn the human preferences. The goal is to move from $S_0$ to $S_G$ and the action space contains only going up and right. The shaded regions represent obstacles in the human’s environment, and the red circle represent a "must pass" point. Additional constraints are that human cannot take the same action consecutively for 3 times. We can see the generated trajectories from X-MEN satisfy all the constraints and follow the shortest possible paths, similar to what human demonstrators' actions.}\label{fig:obstacle}
\vspace*{-0.3cm}
\end{figure}
The human agents are attempting to reach a fixed goal state $S_G$ from a given initial state $S_0$, as shown in Figure \ref{fig:obstacle} in the appendix. The agent has only two actions for moving up or moving right. The shaded regions represent obstacles in the human’s environment that cannot be passed through, and the red circle represent a "must pass" choke point that every person has to walk through. Additional hard constraints are that human cannot take the same action consecutively for more than 3 times.
We only show the generated trajectories of X-MEN in Figure \ref{fig:obstacle}. We can see X-MEN is able to successfully avoid obstacles and pass the "must go" choke point, and the 10 generated trajectories shown in the figure are indeed the shortest paths from the start state to the goal (matching human demonstrations). What is worth noting is that X-MEN learns to go up first before passing through the gap between two obstacles, because otherwise the trajectory has to violate the constraint of taking the same action consecutively for no more than 3 times.
\section{CONCLUSION}
We proposed X-MEN, a novel XOR maximum entropy framework for constrained Inverse Reinforcement Learning.
We showed theoretically that X-MEN converges in linear speed towards the global optimum of the likelihood function for solving IRL problems. Empirically, we demonstrated the superior performance of X-MEN on two navigation tasks with additional hard combinatorial constraints.
In all tasks, X-MEN generates 100\% valid samples and the generated trajectories closely match the distribution of the training set.
For future work, we would like to extend X-MEN to model-free reinforcement learning while preserving the theoretical guarantees. We also intend to test richer representations of the reward function in form of deep networks on real-world, large-scale constrained IRL tasks.
\subsection{Valid trajectories generation via XOR-Sampling}
In X-MEN, we use XOR-Sampling \cite{ermon2013embed} to get samples from $Q(\tau|\hat{\theta})$,
which leverages recent advancements in sampling via hashing and randomization. In particular, this sampling scheme guarantees that the probability of drawing a sample is sandwiched between a constant bound of the true probability.
We only present the general idea of XOR-Sampling on unweighted functions
here and refer the readers to the paper \cite{ermon2013embed}
for the weighted case. Notice that XOR-Sampling can directly draw samples from unnormalized distributions without the knowledge of the partition function $Z_{\hat{\theta}}$.
For the unweighted case, assuming $w(\tau)=I_C(\tau)e^{-\hat{\theta}^Tf(\tau)}$ takes binary values, we need to draw samples from the set $\mathcal{W}=\{\tau: w(\tau)=1\}$ uniformly at random; i.e., suppose $|\mathcal{W}|=2^l$, then each member in $\mathcal{W}$ should have $2^{-l}$ probability to be sampled.
Following notations from the SAT community, we call one assignment $\tau_0$ which makes $w(\tau_0)=1$ a ``satisfying assignment''. XOR-Sampling obtains near-uniform samples by querying a NP oracle to find one satisfying assignment subject to additional randomly generated XOR constraints.
Initially, the NP oracle will find one satisfying assignment subject to zero XOR constraints, albeit not at random. We then keep adding XOR constraints.
We can prove that in expectation, each newly added XOR constraint rules out approximately half of the satisfying assignments at random. Therefore, if we start with $2^l$ satisfying assignments in $\mathcal{W}$, after adding $l$ XOR constraints, we will be left with only one satisfying assignment in expectation.
We return this assignment as our first sample. Because we can prove that the assignments are ruled out randomly, we can guarantee that the returned assignment must be a randomly chosen one from $\mathcal{W}$ \cite{Gomes2006NearUniformSampling,Gomes2007XORCounting}. Interestingly, since $I_C(\tau)$ here is a indicator function of whether trajectory $\tau$ satisfies a set of combinatorial constraints $C(\tau)$, we can add these constraints together with those XOR constraints and solve them by Mix Integer Programming solvers like Cplex simultaneously. In addition, we also add flow constraints to ensure each samples is a valid trajectory.
For the weighted case, one needs to draw samples from an unnormalized function $w(\tau)$, i.e., the probability of getting a sample $\tau_0$, $Q(\tau_0|\hat{\theta})$ is proportional to $w(\tau)$.
The idea is to discretize $w(\tau)$ and transform
the weighted problem into an unweighted one with additional variables.
Our paper uses the constant approximation bounds of XOR sampling on weighted functions through the following theorem.
we refer the readers to \cite{ermon2013embed,fan2021xorcd,fan2021xorsgd} for the details on the discretization scheme and the choice of the parameters of the original algorithm to reach the bound in Theorem \ref{Th:bound3}.
\begin{Th}\label{Th:bound3}\cite{ermon2013embed}~
Let $\delta>1$, $0 < \gamma < 1$, $w: \{0,1\}^n \rightarrow \mathbb{R}^+$ be an unnormalized
probability density function. $Q(\tau|\hat{\theta}) \propto w(\tau)$ is the normalized distribution. Then, with probability at least $1-\gamma$, XOR-Sampling$(w, \delta, \gamma)$ succeeds and outputs a sample $\tau_0$ by querying $O(n\ln(\frac{n}{\gamma}))$ NP oracles. Upon success, each $\tau_0$ is produced
with probability $Q'(\tau_0)$.
Then, let $\phi:\{0,1\}^n\rightarrow\mathbb{R}^+$ be one non-negative function, then the expectation of
one sampled $\phi(\tau)$ satisfies,
\begin{align}
\frac{1}{\delta}\mathbb{E}_{Q}[\phi(\tau)]\leq\mathbb{E}_{Q'}[\phi(\tau)] \leq\delta\mathbb{E}_{Q}[\phi(\tau)].\label{eq:bound_eq2}
\end{align}
\end{Th}
\section{EXPERIMENTS}
\begin{figure*}[t]
\centering
\subfigure[reward map]{\label{fig:reward}
\includegraphics[width=0.3\linewidth]{figs/grid_reward.pdf}}
\subfigure[ground truth]{\label{fig:gt}
\includegraphics[width=0.3\linewidth]{figs/grid_gt.pdf}}
\subfigure[X-MEN]{\label{fig:x-men}
\includegraphics[width=0.35\linewidth]{figs/grid_recover.pdf}}
\subfigure[Maxent]{\label{fig:maxent}
\includegraphics[width=0.3\linewidth]{figs/grid_maxent.pdf}}
\subfigure[RE-IRL]{\label{fig:re}
\includegraphics[width=0.3\linewidth]{figs/grid_re.pdf}}
\subfigure[MLCI]{\label{fig:mlci}
\includegraphics[width=0.35\linewidth]{figs/grid_mlci.pdf}}
\caption{The superior performance of X-MEN against baselines in the grid world environment.\textbf{(a)} The ground truth reward map of the $9\times9$ gridworld. The reward of each state is 0, except for $S_G$ which is 1. Red symbols denotes constraints, where the red triangle denotes the state that must be passed through first among all the symbols, red crosses denote the states that can never be passed through, and the agent must pass through only one red square and one red circle. \textbf{(b)-(e)} The marginal probability of passing through each state of the ground truth demonstration and the distribution generated by different learning algorithms. We can see distribution of trajectories from X-MEN matches with the demonstration the most. Neither Maxent IRL nor RE-IRL can handle constraints. While MLCI knows "where not to go", it has difficulty in knowing "where must go" and we show in Figure \ref{fig:structure} that it can not generate $100\%$ trajectories satisfying constraints.}
\label{fig:gridworld}
\end{figure*}
\begin{figure*}[t]
\subfigure[]{\label{fig:sample2sample}
\includegraphics[width=0.32\linewidth]{figs/valid_sample2sample.pdf}}
\subfigure[]{\label{fig:sample2time}
\includegraphics[width=0.32\linewidth]{figs/valid_sample2time.pdf}}
\subfigure[]{\label{fig:distribution}
\includegraphics[width=0.32\linewidth]{figs/distribution.pdf}}
\caption{X-MEN outperforms competing approaches by producing 100\% valid trajectories while capturing the inductive bias in demonstration on a $9\times 9$ gridworld benchmark shown in Figure \ref{fig:reward}. (\textbf{Left}) The percentage of valid trajectories generated by different algorithms, varying the number of demonstration trajectories. (\textbf{Middle}) The percentage of valid trajectories generated by different algorithms varying training time. (\textbf{Right}) The dashed line shows the percentage of valid trajectories generated from different algorithms. The bars show the distributions of these valid trajectories grouped by different types of paths (upper paths or lower paths). X-MEN generates 100$\%$ valid trajectories, and has the minimal KL divergence $0.005$ towards that of demonstration.}
\label{fig:structure}
\end{figure*}
We conduct the experiments similar to those in \cite{scobee2019maximum}, where we first show the superior performance of X-MEN in a synthetic grid world benchmark and then analyze trajectories from humans as they navigate around obstacles and follow certain constraints on the floor. To obtain final trajectories, X-MEN first draws trajectories from the proposal distribution, and then re-samples from this trajectory pool according to the importance weights.
For comparison, we compare with classic Max-Ent IRL \citep{ziebart2008maximum}, RE-IRL \citep{boularias2011relative} and recently proposed maximum likelihood constraint inference (MLCI) \cite{scobee2019maximum} which can mask out the "not to go" states in the transition distribution. We implement X-MEN using IBM ILOG CPLEX Optimizer 12.63 for queries to NP oracles and XOR-Sampling parameters are same as \cite{fan2021xorcd}. Experiments are carried out on a cluster, where each node has 24 cores and 96GB memory.
\subsection{Grid World}
We consider a 9×9 grid world. The state corresponds to the location of the agent on the grid. The agent has three actions for moving up, right, or diagonally to the upper right by one cell. The objective is to move from the starting state in the bottom-left corner $s_0$ to the goal state in the up-right corner $s_G$. ). Every state-action pair produces a distance feature, and the cumulative reward is inverse proportional to distance, which encourages short trajectories. There are additionally three more types of constraints, denoted as red symbols shown in Figure \ref{fig:reward}. The red triangle denotes the state that must be passed through first among all the symbols, red crosses denote the states that can never be passed through, and the agent must pass through only one red square and one red circle. The demonstration trajectories satisfies all the constraints and have an inductive bias: $70\%$ trajectories move along the upper paths and $30\%$ move along the lower path.
Due to the presence of hard constraints, recovering the reward map cannot be considered as the sole performance metric for a learning algorithm.
In fact, an IRL agent with the groundtruth reward map may produce sub-optimal actions if he violates constraints.
Therefore, we show in Figure \ref{fig:gt}-\ref{fig:mlci} the marginal distributions of passing each grid cell generated by aggregating 100 trajectories produced by different learning algorithms and the groundtruth demonstrations. We can see distribution of trajectories from X-MEN matches the demonstrations the most. Neither Maxent IRL nor RE-IRL can handle constraints. While MLCI knows "where not to go", it has difficulty in knowing "where must go" as the probability of the state marked as triangle is not 1 (we constrain that the agent must go through the triangle). Figure \ref{fig:structure} further computes the percentage of valid trajectories generated by different algorithms varying the number of demonstration trajectories (\ref{fig:sample2sample}) and training time (\ref{fig:sample2time}). X-MEN always generates $100\%$ valid trajectories while the competing methods satisfy no more than $50\%$. Moreover, we can see from the trend that even we keep increasing the number of demonstrations and the training time, the increase in baseline performance is minimal.
Figure \ref{fig:distribution} compares the recovered distribution of the trajectories, where we can see X-MEN has the minimal KL divergence 0.005 towards the ground truth distribution of demonstration.
\subsection{Human Obstacle Avoidance}
In our second example, we analyze trajectories from humans as they navigate around obstacles on the floor and follow certain constraints. We map these continuous trajectories into trajectories through a grid world where each cell represents a a 1ft-by-1ft area on the ground. The state corresponds to the location of the agent on the grid. The human agents are attempting to reach a fixed goal state $S_G$ from a given initial state $S_0$, as shown in Figure 1 in the appendix. The agent has only two actions for moving up or moving right. The shaded regions represent obstacles in the human’s environment that cannot be passed through, and the red circle represent a "must pass" choke point that every person has to walk through. Additional hard constraints are that human cannot take the same action consecutively for more than 3 times.
Demonstrations were collected from 10 volunteers, who want to move from the start state to the goal state without violating any constraints.
Empirical observations reveal that volunteers tend to follow the shortest paths given these constraints. We train both our model and the competing approaches using these demonstrations within the same training time of $4$ hours and use 16 trajectory samples in each SGD iteration. Generated trajectories are shown in Figure 1 in the appendix, where we can see X-MEN is able to successfully avoid obstacles and pass the "must go" choke point, and the 10 generated trajectories shown in the figure are indeed the shortest paths from the start state to the goal (matching human demonstrations). Competing approaches do not generate trajectories that satisfy constraints, while the trajectories generated by X-MEN are $100\%$ valid.
\section{INTRODUCTION}
\begin{figure*}[t]
\subfigure[Add no constraint]{\label{fig:intuition1}
\includegraphics[width=0.32\linewidth]{figs/intuition1.pdf}}
\subfigure[Add single-state constraints $\mathcal{C}_1$]{\label{fig:intuition2}
\includegraphics[width=0.32\linewidth]{figs/intuition2.pdf}}
\subfigure[Add multi-state constraint $\mathcal{C}_2$]{\label{fig:intuition3}
\includegraphics[width=0.32\linewidth]{figs/intuition3.pdf}}
\caption{Examples of constrained IRL problems. The agent wants to move from the start state $S_0$ (blue grid) to the goal state $S_G$ (green grid). Ground truth demonstration is shown in the red line. The same initial reward function before learning is used for all 3 situations, with one-step reward listed in each grid. Most likely trajectories under the initial reward function (e.g.,those maximizing rewards and subject to constraints) are shown using blue dashed arrows. (\textbf{a}) When no constraint is added to the MDP, the agent finds the shortest path directly upward from $S_0$ to $S_G$. (\textbf{b}) When single-state constraint $\mathcal{C}_1$, which forbids the agent to to pass through the red grids, is imposed, the agent can detour from either the left or the right side. (\textbf{c}) When there are an additional multi-state constraint $\mathcal{C}_2$ imposed, which constrains at least half of all passing states in the shaded area, the optimal trajectory is to detour from the right side. Notice that this behavior aligns with the demonstration.}
\label{fig:intuition}
\end{figure*}
Inverse Reinforcement Learning (IRL) \citep{ng2000algorithms,abbeel2004apprenticeship,ziebart2008maximum,arora2021survey,li2017deep}
provides an important way to learn from demonstrations.
IRL assumes that the demonstrator implicitly maximizes the cumulative reward of a Markov Decision Process (MDP).
The goal of IRL is to recover the unknown reward function from the observed demonstrations.
Various IRL algorithms have been proposed, including Linear IRL \citep{ng2000algorithms,abbeel2004apprenticeship} and Large-Margin Q-Learning \citep{ratliff2006maximum}.
To differentiate among multiple reward functions which lead to similar behaviors, Maximum Entropy Inverse Reinforcement Learning (MaxEnt IRL) \citep{ziebart2008maximum,wulfmeier2015maximum,finn2016guided,ho2016generative} assumes that the demonstrator samples trajectories from a maximum entropy distribution parameterized by the cumulative reward.
In this paper, we focus on IRL problems where certain constraints are known beforehand and hence do not need to be rediscovered by the learning algorithm.
The trajectories from the demonstrator are known to satisfy these constraints and we require the IRL agent to follow trajectories satisfying these constraints as well.
Indeed, standard IRL algorithms \citep{abbeel2007application,vasquez2014inverse,scobee2018haptic} can be applied to this scenario without modifications and they eventually discover the optimal policy.
Nevertheless, it may require a large amount of demonstrations to learn these constraints.
Worse still, it is still possible for the IRL agent to produce trajectories which occasionally violate constraints even after many training epochs.
This is especially problematic in safety critical domains, such as autonomous driving, robotic surgery, etc.
Recent work has attempted to embed constraints into IRL. For example, the work of \citep{vazquez2017learning,kalweit2020deep} uses demonstrations to learn a rich class of possible specifications that can represent a task. Others have focused specifically on learning constraints, that is, behaviors that are expressly forbidden or infeasible \citep{chou2018learning,subramani2018inferring,mcpherson2018modeling,scobee2019maximum,anwar2020inverse,mcpherson2021maximum}.
Nevertheless, so far the attempts have been focused on \textit{single-state} constraints, where a handful of actions are forbidden in certain states and these forbidden actions have little impact for future state-action transitions. Their approaches cannot address \textit{multi-state combinatorial} constraints, which limits a chain of actions spanning multiple time stamps.
For example, Figure~\ref{fig:intuition} (c) demonstrates a navigation task where constraints require at least half of the states in each trajectory is located in the shaded area.
With this constraint imposed, only trajectories passing the right-hand side are possible.
Such constraints cannot be addressed with previous approaches which mask out actions from certain states.
In this work, we propose \textbf{X}OR-\textbf{M}aximum \textbf{EN}tropy (X-MEN) Constrained Inverse Reinforcement Learning, which \textbf{\textit{provably converges to the optimal reward function for MaxEnt IRL in linear number of training steps}}, even in the presence of hard combinatorial constraints.
X-MEN also guarantees to produce trajectories which satisfy multi-state combinatorial constraints.
X-MEN is based on the Maximum Entropy IRL learning \citep{ziebart2008maximum,boularias2011relative}.
Distinctively, X-MEN harnesses XOR-sampling to estimate the gradient of the expected reward from the current model distribution.
The recently proposed XOR-Sampling \citep{Gomes2006NearUniformSampling,Ermon13Wish,ermon2013embed} reduces the sampling problem into queries of NP oracles via hashing and projection, and guarantees a constant factor approximation for the expectation estimation.
After obtaining samples, X-MEN uses Stochastic Gradient Descent (SGD) to maximize the difference between expected reward from the demonstration and from the trajectories sampled from the current model distribution, a procedure closely resembling contrastive divergence learning, to maximize the likelihood of the demonstrated behavior.
Theoretic analysis reveals that X-MEN provably converges to the \textit{global optimum} of the likelihood function in linear number of SGD iterations.
In addition, X-MEN can handle rewards parameterized either in a linear form or in the representation of a neural network.
During testing, the policy learned by X-MEN can also be adapted to satisfying additional constraints without retraining.
In experiments, we compare the performance of X-MEN against MaxEnt IRL \citep{ziebart2008maximum} and additional baselines such as Reletive Entropy IRL (RE-IRL) \citep{boularias2011relative} and recently proposed maximum likelihood constraint inference (MLCI) \citep{scobee2019maximum} on several grid world environments and in an imitation learning environment with human data of navigating around obstacles. All these environments require the agent to follow constraints.
Our experiment shows the learned trajectories of X-MEN 100\% satisfy constraints, while a majority of trajectories produced by competing approaches do not ($\geq 60\%$ violate constraints).
Also X-MEN produces trajectories that closely imitate those of the demonstrations.
In summary, our contributions are as follows:
\begin{itemize}
\item We propose X-MEN, an algorithm that provably converges to the optimal reward function for MaxEnt IRL in linear number of training steps, even in the presence of multi-state combinatorial constraints.
\item X-MEN is guaranteed to produce trajectories which satisfy combinatorial constraints, beyond the capability of previous approaches.
\item Experimental results reveal that X-MEN produces trajectories that closely resemble demonstration while satisfying constraints, outperforming a series of constrained IRL baselines.
\end{itemize}
\section{XOR Maximum Entropy IRL}
In this section we propose \textbf{X}OR-\textbf{M}aximum \textbf{EN}tropy Constrained Inverse Reinforcement Learning (X-MEN), to solve the inverse reinforcement learning problem with multi-state combinatorial constraints.
We develop X-MEN based on maximum entropy inverse reinforcement learning \citep{ziebart2008maximum,boularias2011relative,finn2016guided}. Specifically, the model assumes that the expert samples the demonstrated trajectories $\{\tau_i\}$ from the distribution $P(\tau|\theta,T)$ in Equation \ref{eq:constarined_p},
where $R_{\theta}(s_t, a_t)=\theta^Tf(s_t,a_t)$ is represented by a linear combination of feature vector $f(s_t,a_t)$. $f(s_t,a_t)$ can be hand-crafted or generated by a deep neural network.
Forward-backward dynamic programming can hardly solve this problem even if the dynamics is given due to the presence of the hard combinatorial constraints $I_C(\tau)$.
Our X-MEN has the ability to solve this problem by leveraging XOR and importance sampling to estimate $P(\tau|\theta,T)$.
After learning with X-MEN, we can always generate valid trajectories. We use Stochastic Gradient Descent (SGD) to optimize the objective, where in each iteration we compute the gradient of the negative log likelihood:
\begin{align}\label{eq:grad_ll}
&\nabla_{\theta}L(\theta)=\frac{1}{|\mathcal{D}|}\sum_{\tau\in \mathcal{D}}\nabla_{\theta} R_{\theta}(\tau) + \nabla_{\theta}\log Z_{\theta}\notag\\
&=\frac{1}{|\mathcal{D}|}\sum_{\tau\in \mathcal{D}}\nabla_{\theta} R_{\theta}(\tau) -\sum_{\tau}P(\tau|\theta,T) \nabla_{\theta}R_{\theta}(\tau).
\end{align}
The first term in Equation~\ref{eq:grad_ll} is $\mathbb{E}_D[\nabla_{\theta} R_{\theta}(\tau)]$ and the second term is $\mathbb{E}_P[\nabla_{\theta} R_{\theta}(\tau)]$. To compute the gradient, we estimate the second term $\mathbb{E}_P[\nabla_{\theta} R_{\theta}(\tau)]$ using importance sampling, where the following proposal distribution $Q(\tau|\hat{\theta})$ is used:
\begin{align}\label{eq:Q}
Q(\tau|\hat{\theta})=\frac{1}{Z_{\hat{\theta}}}I_C(\tau)e^{-\hat{\theta}^Tf(\tau)},
\end{align}
where $f(\tau)=\sum_{t=1}^L\gamma^t f(s_t,a_t)$ is the combined feature vector of the whole trajectory.
$\hat{\theta}$ are the parameters to adjust for the proposal $Q$. While we can set other parameters, in this paper we let $\hat{\theta}=\theta$. In other words, we use the current learned parameters $\theta$ to replace $\hat{\theta}$.
Importance sampling is needed because the probabilities of state transitions $D(\tau)$ may not have an exponential family form. Currently, the implementation of XOR-sampling samples from exponential family distributions. Nevertheless, we notice this importance sampling step can be avoided if $D(\tau)$ has an exponential family distribution.
To approximate $\nabla_{\theta}L(\theta)$ in Equation~\ref{eq:grad_ll}, we sample $M_1$ trajectories from the dataset of demonstrations to form the set $\mathcal{D}_{M_1}$.
Then we sample $2M_2$ trajectories from the proposal distribution $Q(\tau | \theta)$ to form two sets $\mathcal{T}_{M_2,1}^Q$, $\mathcal{T}_{M_2,2}^Q$ (each set contains $M_2$ trajectories; details on how to obtain samples from $Q$ are discussed later).
Then, we can use $g_{\theta}$ in the following Theorem \ref{Th:compute_g} to approximate $\nabla_{\theta}L(\theta)$:
\begin{Th}\label{Th:compute_g}
Let the model distribution $P(\tau|\theta,T)$ defined in Equation \ref{eq:constarined_p} and the gradient of the likelihood function defined in Equation \ref{eq:grad_ll}. Let $g_{\theta}$ be
\begin{align}\label{eq:g_theta}
g_{\theta}=\frac{1}{M_1}\sum_{\tau\in \mathcal{D}_{M_1}} f(\tau)-\frac{\sum_{\tau\in\mathcal{T}^{Q}_{M_2,1}}D(\tau)f(\tau)}{\sum_{\tau\in\mathcal{T}^Q_{M_2,2}}D(\tau)},
\end{align}
where $\mathcal{D}_{M_1}$, $\mathcal{T}_{M_2,1}^Q$, $\mathcal{T}_{M_2,2}^Q$ are defined above. We must have $g_{\theta}$ is an unbiased estimation of $\nabla_{\theta}L(\theta)$, ie., $\mathbb{E}[g_{\theta}] = \nabla_{\theta}L(\theta)$.
\end{Th}
We leave the proof of Theorem \ref{Th:compute_g} to the supplementary materials.
However, in practice sampling from distribution $Q$ is intractable due to the existence of the partition function $Z_{\theta}$ and the unbiased estimation is hard to obtain. In this paper we incorporate the recently proposed XOR-Sampling to get a constant approximation of the true gradient $\nabla_{\theta}L(\theta)$.
XOR-Sampling is used to obtain samples from the proposal distribution $Q$ such that the probability of drawing a sample is sandwiched between a constant multiplicative bound of the true probability.
XOR-Sampling is the result of a rich line of research \citep{ermon2013embed,Gomes06XORCounting,Gomes2007XORCounting}, which translates the \#-P complete sampling problem into queries to NP oracles with provable guarantees.
The high level idea of XOR sampling is as follows.
Suppose one would like to draw one ball uniformly at random from an urn, with access to an oracle that returns one ball from the urn once queried (implemented as an NP-oracle when sampling in a combinatorial space). Notice that the oracle will not return the balls uniformly at random; i.e., it may return the same ball every time.
XOR-sampling removes the balls from the urn by introducing additional XOR constraints. One can prove that half of the balls are removed at random, each time when one XOR constraint is introduced.
Hence, one keeps adding XOR constraints until there are only one ball remaining. Then the last ball is returned.
Since the balls are removed at random, the last left must be a random one drawn from the original set of balls.
In practice, XOR-sampling also works with weighted probability distributions.
Our paper uses the probabilistic bound of XOR-sampling via Theorem~\ref{Th:bound2}.
We refer the readers to \cite{ermon2013embed,fan2021xorcd,fan2021xorsgd} for the details on the discretization scheme and the choice of the parameters of XOR-sampling to obtain the bound in Theorem \ref{Th:bound2}.
\begin{Th}\label{Th:bound2}\citep{ermon2013embed}~
Let $\delta>1$, $0 < \gamma < 1$, $w: \{0,1\}^n \rightarrow \mathbb{R}^+$ be an unnormalized
probability density function where $n=|\mathcal{S}||\mathcal{A}|$. $Q(\tau|{\theta}) \propto w(\tau)$ is the normalized distribution and $C(\tau)$ is the set of hard combinatorial constraints. Then, with probability at least $1-\gamma$, XOR-Sampling$(w, C(\tau), \delta, \gamma)$ succeeds and outputs a sample $\tau_0$ by querying $O(n\ln(\frac{n}{\gamma}))$ NP oracles. Upon success, each $\tau_0$ is produced with probability $Q'(\tau_0)$.
Then, let $\phi:\{0,1\}^n\rightarrow\mathbb{R}^+$ be one non-negative function, then the expectation of
one sampled $\phi(\tau)$ satisfies,
\begin{align}
\frac{1}{\delta}\mathbb{E}_{Q}[\phi(\tau)]\leq\mathbb{E}_{Q'}[\phi(\tau)] \leq\delta\mathbb{E}_{Q}[\phi(\tau)].\label{eq:bound_eq2}
\end{align}
\end{Th}
The detailed procedure of X-MEN is shown in Algorithm \ref{alg:X-MEN}. Here we demonstrate the version of X-MEN, where the only parameter to optimize is $\theta$.
A variant of this algorithm can be developed which back-propagate the gradient over the feature vector $f(s,a)$ as well, if $f(s,a)$ is represented as a neural network and also can be updated during learning.
X-MEN takes as inputs the feature vector $f(s,a)$, transition probability $D(\tau)$, constraint set $C(\tau)$, training data $\{\tau_i\}_{i=1}^N$, initial model parameter $\theta_0$,
the learning rate $\eta$, the number of SGD iterations $K$,
XOR-Sampling parameters $(\delta, \gamma)$, and batch sizes $M_1$, $M_2$, and outputs the averaged learned parameter $\overline{\theta_{K}}$.
To approximate $\mathbb{E}_{P}[\nabla_{\theta}R_{\theta}(\tau)]$ at the $k$-th iteration, X-MEN draws $2M_2$ samples $\tau'_1, \dots, \tau'_{2M_2}$ from the proposal distribution $Q(\tau|{\theta})$ using XOR-Sampling, where $M_2$ is a user-determined sample size.
Because XOR-Sampling has a failure rate, X-MEN repeatedly call XOR-Sampling until all $2M_2$ samples are obtained successfully (line 3 -- 8). Then, X-MEN also draws $M_1$ samples from the training set $\{\tau_i\}_{i=1}^N$ uniformly at random to approximate $\mathbb{E}_{\mathcal{D}}[\nabla_{\theta}R(\theta)]$.
Once all the samples are obtained, X-MEN uses
$g_k = \frac{1}{M_1}\sum_{\tau\in \mathcal{D}_{M_1}} f(\tau)-\frac{\sum_{j=1}^{M_2}D(\tau'_j)f(\tau'_j)}{\sum_{j=M_2+1}^{2M_2}D(\tau'_j)}$ as an approximation for the gradient of the negative log likelihood.
$\theta$ is updated following the rule $\theta_{k+1} = \theta_{k}-\eta g_k$ for $K$ steps, where $\eta$ is the learning rate. Finally, the average of $\theta_1, \ldots, \theta_K$, namely $\overline{\theta_{K}}=\frac{1}{K}\sum_{k=1}^{K}\theta_k$ is the output of the algorithm.
We show in the next sections that X-MEN enjoys the property of convergence to the global optimum of the objective in linear number of iterations, and illustrate how to incorporate XOR-Sampling into our framework for sample generation with strict constraint satisfaction.
\begin{algorithm}[t!]
\caption{XOR Maximum Entropy Constrained Inverse Reinforcement Learning (X-MEN)}
\label{alg:X-MEN}
\LinesNumbered
\KwIn{$\theta_0, f(s, a), K, \eta, \delta, \gamma, D(\tau), C(\tau), M_1,M_2,\mathcal{D}$.}
\For{$k=0$ {\bfseries to} $K$}{
$j\gets 1$ \tcp*[f]{\text{$M_1$ and $M_2$ are batch size}}\\
\While{$j\leq 2M_2$}{
$\tau'\gets$ XOR-Sampling$\left(e^{-{\theta_k}^T f(\tau)}, C(\tau), \delta, \gamma\right)$
\If{$\tau' \neq Failure$} {
$\tau'_j\gets \tau'$; $j\gets j+1$
}
}
Get samples~~ $\mathcal{D}_{M_1}=\{\tau_j\}_{j=1}^{M_1}$ from $\mathcal{D}$.\\
$g_k=\frac{1}{M_1}\sum_{\tau\in \mathcal{D}_{M_1}} f(\tau)-\frac{\sum_{j=1}^{M_2}D(\tau'_j)f(\tau'_j)}{\sum_{j=M_2+1}^{2M_2}D(\tau'_j)}$\\
Update the parameters~~ $\theta_{k+1} = \theta_{k}-\eta g_k$
}
\textbf{return}$~~\overline{\theta_{K}}=\frac{1}{K}\sum_{k=1}^{K}\theta_k$
\end{algorithm}
\subsection{Linearly Converge to the Global Optimum}
Suppose the only parameter to learn is $\theta$, in other words, $f(x,a)$ are fixed,
the reward function $R_{\theta}(\tau)$ is represented by a linear combination of hand-crafted features),
we can easily see that the objective is convex with regard to $\theta$. Under this circumstance, we show that X-MEN converges to the global optimum of the log likelihood function in addition to a vanishing term.
Moreover, the speed of the convergence is linear
with respect to the number of stochastic gradient descent steps.
Denote $Var_{\mathcal{D}}(f(\tau)) = \mathbb{E}_{\mathcal{D}}[||f(\tau)||_2^2] - ||\mathbb{E}_{\mathcal{D}}[f(\tau)]||_2^2$ and
$Var_{P}(f(\tau)) = \mathbb{E}_{{P}}[||f(\tau)||_2^2] - ||\mathbb{E}_{{P}}[f(\tau)]||_2^2$ as the total variations of $f(\tau)$ w.r.t. the data distribution $P_{\mathcal{D}}$ and model distribution $P(\tau|\theta,T)$.
The precise mathematical theorem states
\begin{Th}\label{Th:main}
(main)~ Let $P(\tau|\theta,T)$ and $Q(\tau|{\theta})$ as defined in Equation \ref{eq:constarined_p} and \ref{eq:Q}, $R_{\theta}(\tau)=\theta^Tf(\tau)$.
Given trajectories $\mathcal{D}=\{\tau_i\}_{i=1}^N$ and the objective function $L(\theta)$, denote $OPT=\min_{\theta} L(\theta)$ and $\theta^*=\text{argmin}_{\theta}L(\theta)$.
Let $Var_{\mathcal{D}}(f(\tau))\leq\sigma_1^2$ and $\max_{\theta}Var_{P}(f(\tau))\leq \sigma_2^2$.
Suppose $1\leq\delta\leq\sqrt{2}$ is used in XOR-sampling, the learning rate $\eta\leq \frac{2-\delta^2}{\sigma_2^2\delta}$, and $\overline{\theta_K}$ is the output of X-MEN. We have:
\begin{align*}
\mathbb{E}[L(\overline{\theta_K})]-OPT \leq\frac{\delta^2||\theta_0-\theta^*||_2^2}{2\eta K}+\frac{\eta\sigma_1^2}{\delta^2 M_1}+\frac{\eta\sigma_2^2}{\delta^2M_2}.
\end{align*}
\end{Th}
X-MEN is the first provable algorithm which converges to the global optimum of the likelihood function and a tail term for constrained inverse reinforcement learning problems. Moreover, the rate of the convergence is linear in the number of SGD iterations $K$. Previous approaches for IRL problems with hard combinatorial constraints do not have such tight bounds.
The main challenge to prove Theorem~\ref{Th:main}
lies in the fact that we cannot ensure the unbiasedness of the
gradient estimator.
Because the objective is convex with respect to $\theta$ and smooth, a gradient descent algorithm can be proven
to be linearly convergent towards the optimal value if the
expectation of the estimated gradient is unbiased, ie, $\mathbb{E}[g_k] = \nabla_{\theta} L(\theta_k)$.
However, even though we apply XOR-sampling, which has
a constant approximation bound in generating
samples from the model distribution,
we still cannot guarantee the unbiasedness of $g_k$.
Instead, using the constant factor approximation of XOR-Sampling, which is formally stated in Theorem~\ref{Th:bound2}, the bound
for $g_{k}$ is in the following form:
\begin{align}
\frac{1}{\delta^2} [\nabla L(\theta_k)]^+ \leq \mathbb{E}[g_k^+]
\leq \delta^2 [\nabla L(\theta_k)]^+,\label{eq:gb1}\\
\delta^2 [\nabla L(\theta_k)]^- \leq \mathbb{E}[g_k^-]
\leq \frac{1}{\delta^2} [\nabla L(\theta_k)]^-.\label{eq:gb2}
\end{align}
Here, $\delta>1$ is a constant factor, $[f]^+$ means the positive part of $f$, ie,
$[f]^+ = \max\{f, \mathbf{0}\}$, and $[f]^-$ means the negative part of $f$, ie,
$[f]^- = \min\{f, \mathbf{0}\}$.
The bound in Equation~\ref{eq:gb1} and \ref{eq:gb2}
can be proven by bounding the nominator and the denominator of Equation \ref{eq:g_theta}, and we leave the proof in the supplementary materials.
The proof of Theorem \ref{Th:main} relies mainly on the following Theorem \ref{Th:XOR_SGD_bound} which bounds the errors of Stochastic Gradient
Descent (SGD) algorithms which only have
access to constant approximate gradient vectors.
Theorem~\ref{Th:XOR_SGD_bound} was proved in \cite{fan2021xorcd}, to help bound the errors of learning an exponential family model.
Theorem \ref{Th:XOR_SGD_bound} requires function
$f$ to be $L$-smooth. $f(\theta)$ is $L$-smooth if and only if $||f(\theta_1) - f(\theta_2)||_2 \leq L ||\theta_1 - \theta_2||_2$.
Notice that the conditions of Theorem \ref{Th:main}
automatically guarantee the $L$-smoothness of the objective and we leave the proof in the appendix.
\begin{Th}\label{Th:XOR_SGD_bound}\citep{fan2021xorcd}~
Let $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be a $L$-smooth convex function and $\theta^*=\text{argmin}_{\theta} f(\theta)$. In iteration $k$ of SGD, $g_k$ is the estimated gradient, i.e., $\theta_{k+1}=\theta_{k}-\eta g_k$. If $Var(g_k)\leq \sigma^2$, and there exists $1\leq c\leq\sqrt{2}$ s.t. $\frac{1}{c}[\nabla f(\theta_k)]^+ \leq \mathbb{E}[g_k^+]\leq c[\nabla f(\theta_k)]^+$ and $c[\nabla f(\theta_k)]^- \leq \mathbb{E}[g_k^-]\leq \frac{1}{c}[\nabla f(\theta_k)]^-$, then for any $K>1$ and step size $\eta\leq \frac{2-c^2}{Lc}$, let $\overline{\theta_K}=\frac{1}{K}\sum_{k=1}^K \theta_k$, we have
\begin{align}\label{eq:XOR_SGD_bound}
\mathbb{E}[f(\overline{\theta_K})]-f(\theta^*)\leq \frac{c||\theta_0-\theta^*||_2^2}{2\eta K}+\frac{\eta\sigma^2}{c}.
\end{align}
\end{Th}
The proof of Theorem \ref{Th:main} is to apply
Theorem \ref{Th:XOR_SGD_bound} on the objective $L(\theta)$ and noticing that $L(\theta)$ is $L$-smooth when the total variation $Var_P(f(\tau))$ is bounded \citep{fan2021xorcd}.
Theorem~\ref{Th:main} states that in expectation, the difference between the output of X-MEN and the true optimum $OPT$ is bounded by a term that is inversely proportional to the number of iterations $K$ and a tail term $\frac{\eta\sigma_1^2}{\delta^2 M_1}+\frac{\eta\sigma_2^2}{\delta^2M_2}$.
To reduce the tail term with fixed steps $\eta$, we can generate more samples at each iteration to reduce the variance (increase $M_1$ and $M_2$).
In addition, to quantify the computational complexity of X-MEN,
we prove the following theorem in the supplementary materials detailing the number of queries to NP oracles needed for X-MEN.
\begin{Th}\label{Th:num_queries}
Let $|\mathcal{S}|$ and $|\mathcal{A}$ be the number of all possible states and all possible actions, respectively, then X-MEN in Algorithm \ref{alg:X-MEN} uses $O\left(K|\mathcal{S}||\mathcal{A}|\ln\frac{|\mathcal{S}||\mathcal{A}|}{\gamma}+KM_2\right)$ queries to NP oracles.
\end{Th}
\section{Preliminary}
\section{INVERSE REINFORCEMENT LEARNING}
Here we present a brief overview of IRL. $\mathcal{M}=\{\mathcal{S}, \mathcal{A}, T, R, \gamma\}$ is a Markov Decision Process (MDP), where $\mathcal{S}$ denotes the state space of all states $s$, $\mathcal{A}$ denotes the set of possible actions $a$, $T$ denotes the transition probability function, $R$ denotes the reward function, and $\gamma \in [0, 1]$ is the discount factor. Given an MDP, an optimal policy $\pi^*$ is the one to maximize the expected cumulative reward.
IRL considers the case where the reward function is unknown. Instead, a set of expert demonstrations $\mathcal{D}=\{\tau_1,\ldots, \tau_N\}$ is provided which are sampled from a user policy $\pi$, i.e. provided by a demonstrator. Each demonstration
consists of a series of state-action pairs $\tau_i=\{(s_{i0},a_{i0}), (s_{i1},a_{i1}),\ldots,(s_{iL},a_{iL_{i}})\}$, where $L_i$ denotes the length of the trajectory. The goal of IRL is to uncover the hidden reward $r$ from the demonstrations.
\subsection{Maximum Entropy IRL}
A number of approaches have been proposed to tackle the IRL problem \citep{ng2000algorithms,abbeel2004apprenticeship,ratliff2006maximum}. One crucial problem to address for IRL is to differentiate among multiple reward functions that lead to the same demonstrations.
An influential formulation is Maximum Entropy IRL \citep{ziebart2008maximum}, which can also be viewed as a special case of Relative Entropy IRL (RE-IRL) \citep{boularias2011relative,snoswell2020revisiting}.
In this formulation, the probability that the demonstrator chooses a given trajectory is proportional to the exponent of the reward along the path. Denote $R_{\theta}(\tau)=\sum_{t=1}^{L}\gamma^t R_{\theta}(s_t,a_t)$ as the discounted cumulative reward parameterized by $\theta$. The probability of choosing trajectory $\tau$ is:
\begin{align}
P_{choice}(\tau|\theta,T) = \frac{1}{Z_{\theta}}e^{R_{\theta}(\tau)}.\label{eq:zchoice}
\end{align}
Here $Z_\theta$ is a normalization constant to ensure $P_{choice}(\tau|\theta, T)$ is a probability distribution.
Let $d_{0}$ as the probability distribution of the initial state. %
$D(\tau)=d_0(s_1)\prod_{t=1}^{L}T(s_{t+1}|s_t,a_t)$ is the probability of state transitions which leads to the trajectory $\tau$.
The overall probability of observing trajectory $\tau$ from demonstrations hence is the product of the choice probability times the state transition probability:
\begin{align}
P(\tau|\theta,T) = \frac{1}{Z_{\theta}}e^{R_{\theta}(\tau)}D(\tau).
\end{align}
\subsection{IRL with Multi-state Combinatorial Constraints}
Despite the success of many IRL models, many real world tasks require additional constraints to be satisfied when learning from demonstrations.
In this work, we restrict ourselves to dealing with hard combinatorial constraints, as shown in Figure \ref{fig:intuition}. Note that this is not particularly restrictive since, for example, safety constraints are often hard constraints as well are constraints imposed by physical laws. Different from previous work that only defines constraints as a set of forbidden state-action pairs, which we call single-state constraints, here we consider more general cases of multi-state combinatorial constraints. Denote $C(\tau)=\{c_i(\tau)\}$ as the set of constraints that each trajectory must satisfy, and $I_C(\tau)$ the indicator function of whether constraints $C(\tau)$ are satisfied. Formally,
\begin{align*}
I_C(\tau)=\begin{cases}
1, ~~~~ \text{if}~ \tau~ \text{satisfies the constraints set }~ C(\tau)\\
0, ~~~~ \text{otherwise}
\end{cases}
\end{align*}
We augment the MDP into the constrained MDP: $\mathcal{M}^C=\{\mathcal{S}, \mathcal{A}, T, R, C\}$. The probability of observing a trajectory $\tau$ now becomes\footnote{Notice $Z_\theta$ in Equation \ref{eq:constarined_p} is different from $Z_\theta$ in Equation \ref{eq:zchoice} because of the introduction of $I_C$. $Z_\theta$ still normalizes the probability in Equation \ref{eq:constarined_p}. Without too much cluttering, we use the same symbol $Z_\theta$ in both equations.}
\begin{align}\label{eq:constarined_p}
P(\tau|\theta,T)=\frac{1}{Z_{\theta}}e^{R_{\theta}(\tau)}D(\tau)I_C(\tau),
\end{align}
Given the set of expert demonstrations $\mathcal{D}$, we want to find the best reward function by minimizing the negative log likelihood function $L(\theta)$.
\begin{align*}
\text{argmin}_{\theta}L(\theta)= \text{argmin}_{\theta}\frac{1}{|\mathcal{D}|}\sum_{\tau\in \mathcal{D}}-R_{\theta}(\tau) + \log Z_{\theta}.
\end{align*}
Notice only the terms related to the optimization variable $\theta$ are included in the rightmost equation.
\section{RELATED WORK}
Max-Ent IRL models were first proposed \citep{ziebart2008maximum} to addresses the inherent ambiguity of possible reward functions and induced policies for an observed behavior, during the training of which a forward-backward dynamic programming algorithm were used to exactly compute the partition function and marginal probability \citep{snoswell2020revisiting}, assuming the knowledge of the transition probability. Relative Entropy IRL \citep{boularias2011relative} extends this work by leveraging an importance sampling approach to estimate the partition function unbiasedly without knowing the dynamics. Guided Cost Learning \citep{finn2016guided} further learns a Max-Ent model with policy optimization. Later work accommodates arbitrary nonlinear reward functions such as neural networks \citep{finn2016guided,kalweit2020deep,wulfmeier2015maximum}, instead of a linear combination of features. Recently proposed Generative Adversarial Imitation Learning (GAIL) \citep{ho2016generative} is an imitation learning method that does not require estimating likelihoods.
However, while Markovian rewards do often provide a succinct and expressive way to specify the objectives of a task, they cannot capture all possible task specifications, especially additional constraints \citep{vazquez2017learning}. Recent work on constrained IRL only focuses on local constraints of states, actions and features \citep{chou2018learning,subramani2018inferring,mcpherson2018modeling}, which can hardly represent all the real world scenarios as most constraints are trajectory long. Other methods focus on learning constraints from the demonstrations, such as maximum likelihood constraint inference \citep{scobee2019maximum,kalweit2020deep,anwar2020inverse,mcpherson2021maximum}. Our approach differs from all the existing methods and addresses the open question of learning with hard combinatorial constraints. We adapt the Max-Ent framework to allow us to reason about all the trajectories that satisfy the constraints during the contrastive learning process. Here we only consider pre-defined constraints. One should notice that even with the full knowledge of transition probability, dynamic programming cannot work well under trajectory-long constraints since it has no knowledge of any hard combinatorial information.
X-MEN was motivated by the recent proposed probabilistic inference via hashing and randomization technique for both sampling \citep{ermon2013embed,ivrii2015computing}, counting \citep{Gomes2006NearUniformSampling,ding2019towards}, and marginal inference problems \citep{Ermon13Wish,kuck2019adaptive,Chakraborty2014DistributionAwareSA,Chakraborty2015WeightedCounting,belle2015hashing} with constant approximation guarantees. Latest work also show the success of XOR-Sampling \citep{ermon2013embed} to boost stochastic optimization algorithms \citep{fan2021xorsgd} and improve machine learning tasks on structure generation \citep{fan2021xorcd}.
\subsubsection*{References}}
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\title{X-MEN: Guaranteed XOR-Maximum Entropy \\
Constrained Inverse Reinforcement Learning}
\author[1]{\href{mailto:<[email protected]>?Subject=Your UAI 2022 paper}{Fan Ding}{}}
\author[1]{Yexiang Xue}
\affil[1]{%
Computer Science Dept.\\
Purdue University\\
West Lafayette, Indiana, USA
}
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|
1,116,691,499,393 | arxiv | \section{Introduction}
Dynamical spin systems have played a central role in non-equilibrium
statistical models. The Ising model is widely studied in
statistical mechanics, as it is simple and allows one to understand
many features of phase transitions. The non-equilibrium properties of
the Ising model follow from the spin dynamics. In his article \cite{RG},
Glauber introduced a dynamical model formulating the dynamics
of spins, based on the rates coming from a detailed balance analysis.
It is a simple non-equilibrium model of interacting spins
with spin flip dynamics. An extension of the kinetic Ising model with
nonuniform coupling constants on a one-dimensional lattice was introduced
in \cite{DKSM}. In \cite{TV}, a damage spreading method was used to study
the sensitivity of the time evolution of a kinetic Ising model with
Glauber dynamics against the initial conditions. The full time dependence
of the space-dependent magnetization and of the equal time
spin-spin correlation functions were studied in \cite{SSG}.
Non-equilibrium two-time correlation and response functions for
the ferromagnetic Ising chain with Glauber dynamics have been studied
in \cite{GL,DRS}. The dynamics of a left-right asymmetric Ising
chain has been studied in \cite{God}. The response function to
an infinitesimal magnetic field for the Ising-Glauber model with
arbitrary exchange couplings was addressed in \cite{chat2003}.
In \cite{KA2002}, a Glauber model on a one-dimensional lattice
with boundaries was studied, for both
ferromagnetic and anti-ferromagnetic couplings.
The large-time behavior of the one-point function was studied.
It was shown that the system exhibits a dynamical phase transition,
which is controlled by the rate of spin flip at the boundaries.
It was shown in \cite{KA2008,KA2011}
that for a nonuniform extension of the kinetic Ising model,
there are cases where the system exhibits static and dynamical phase transitions.
Using a transfer matrix method, it was shown that there are
cases where the system exhibits a static phase transition,
which is a change of behavior of the static profile
of the expectation values of the spins near end points \cite{KA2008}.
Using the same method, it was shown in \cite{KA2011} that a dynamic
phase transition could occur as well: there is a fast phase
where the relaxation time is independent of the reaction rates
at the boundaries, and a slow phase where the relaxation time
does depend on the reaction rates at the boundaries.
Most of the studies on reaction diffusion models have
been on cases where the boundary conditions are constant
in time. Among the few models with time dependent
boundary conditions, is the asymmetric simple exclusion
process on a semi-infinite chain coupled at the end to
a reservoir with a particle density that changes periodically
in time \cite{PSS2008}. The situation is similar regarding
the case of the kinetic Ising model as well. Among the exceptions
are the study of the dynamical response of a two-dimensional
Ising model subject to a square-wave external field \cite{BR2008},
and the study of a harmonic oscillator linearly coupled with
a linear chain of Ising spins \cite{PBC2010,PBC2010-2}.
In this article a one dimensional Ising model at temperature $T$
on a semi-infinite lattice with time varying boundary spin is investigated.
The paper is organized as follows. In section 2 a brief review of
the formalism is presented, mainly to introduce the notation. In
section 3, a semi-infinite lattice with oscillating boundary spin is studied.
The exact solution for the expectation values of the spin at any site is obtained.
It is shown that there the boundary produces an evanescent wave in the lattice.
The low and high frequency limits are studied in greater detail.
The total magnetization of the lattice, $M(t)$, is also obtained.
It is shown that for rapidly changing boundary conditions,
the total magnetization is equal to the the boundary spin itself,
plus something proportional to the time integral of the boundary spin.
A nonuniform model in also investigated. It is shown that its evolution
operator eigenvalues are real. For the specific case of a
two-part lattice with each part being homogeneous, the reflection and
transmission coefficients corresponding to a harmonic source
at the end of the lattice are calculated. Finally, section 4 is
devoted to the concluding remarks.
\section{One-dimensional Ising model with nonuniform coupling constants}
Consider an Ising model on a one-dimensional lattice
with $L$ sites, labeled from $1$ to $L$. At each site
of the lattice there is a spin interacting with
its nearest neighboring sites according to the Ising Hamiltonian.
At the boundaries there are fixed magnetic fields. Denoting
the spin at the site $j$ by $s_j$, and the magnetic field at
the sites 1 and $L$ by $\mathfrak{B}_1$ and $\mathfrak{B}_L$,
one has for the Ising Hamiltonian
\begin{equation}\label{bo.1}
{\mathcal H} =-\sum_{\alpha=1+\mu}^{L-\mu}J_\alpha\,s_{\alpha-\mu}\,
s_{\alpha+\mu}-\mathfrak{B}_1\,s_1-\mathfrak{B}_L\,s_L.
\end{equation}
where $J_\alpha$ is the coupling constant in the link $\alpha$, and
\begin{equation}\label{bo.2}
\mu=\frac{1}{2}.
\end{equation}
The link $\alpha$ links the sites $\alpha-\mu$ and $\alpha+\mu$,
so that $\alpha\pm \mu$ are integers, and $\alpha$
runs from $\mu$ up to $(L-\mu)$. Throughout this paper, sites
are denoted by Latin letters which represent integers, while links
are denoted by Greek letters which represent integers plus one half
($\mu$). The spin variable $s_j$ takes the values $+1$ for spin up
($\uparrow$), or $-1$ for spin down ($\downarrow$). Define
\begin{equation}\label{bo.3}
K_\alpha:=\begin{cases}
\beta\,J_\alpha,& 1<\alpha<L\\
\beta\,\mathfrak{B}_1,& \alpha=\mu\\
\beta\,\mathfrak{B}_L,& \alpha=L+\mu
\end{cases}
\end{equation}
where
\begin{equation}\label{bo.4}
\beta:=\frac{1}{k_\mathrm{B}\,T},
\end{equation}
and $k_\mathrm{B}$ is the Boltzmann's constant, and
$T$ is the temperature. Denoting the reaction rate
from the configuration $A$ to the configuration
$B$ by $\omega(A\to B)$,
and assuming that in each step only one spin flips,
detailed balance demands the following for the reaction rates.
\begin{align}\label{bo.5}
\omega[(S', s_j)\to(S',-s_j)]&=\Gamma_j\,
[1-s_j\,\tanh(K_{j-\mu}\,s_{j-1}+K_{j+\mu}\,s_{j+1})],\nonumber\\
&\qquad 1<j<L,\\ \label{bo.6}
\omega[(S', s_1)\to(S',-s_1)]&=\Gamma_1\,
[1-s_1\,\tanh(K_\mu+K_{1+\mu}\,s_2)],\\ \label{bo.7}
\omega[(S', s_L)\to(S',-s_L)]&=\Gamma_L\,
[1-s_L\,\tanh(K_{L-\mu}\,s_{L-1}+K_{L+\mu})].
\end{align}
$\Gamma_j$'s are independent of the configurations. For simplicity,
we take them to be independent of the site. Then, rescaling
the time they are set equal to one.
So the evolution equation for the expectation value of the spin in
the site $j$ is
\begin{align}\label{bo.8}
\frac{\mathrm{d}}{\mathrm{d} t}\langle s_j\rangle&=-2\,\langle s_j\rangle+
[\tanh(K_{j-\mu}+K_{j+\mu})+\tanh(K_{j-\mu}-K_{j+\mu})]\,\langle
s_{j-1}\rangle\nonumber\\ &\quad+
[\tanh(K_{j-\mu}+K_{j+\mu})-\tanh(K_{j-\mu}-K_{j+\mu})]\, \langle
s_{j+1}\rangle, \quad 1<j<L,\nonumber\\
\frac{\mathrm{d}}{\mathrm{d} t}\langle s_1\rangle&=-2\,\langle s_1\rangle+
[\tanh(K_\mu+K_{1+\mu})+\tanh(K_\mu-K_{1+\mu})]\nonumber\\ &\quad+
[\tanh(K_\mu+K_{1+\mu})-\tanh(K_\mu-K_{1+\mu})]\,\langle s_2\rangle,\nonumber\\
\frac{\mathrm{d}}{\mathrm{d} t}\langle s_L\rangle&=-2\,\langle s_L\rangle+
[\tanh(K_{L-\mu}+K_{L+\mu})+\tanh(K_{L-\mu}-K_{L+\mu})]\,\langle
s_{L-1}\rangle\nonumber\\ &\quad+
[\tanh(K_{L-\mu}+K_{L+\mu})-\tanh(K_{L-\mu}-K_{L+\mu})].
\end{align}
These can be written in the form
\begin{align}\label{bo.9}
\frac{\mathrm{d}}{\mathrm{d} t}\langle s_j\rangle&=-2\,\langle s_j\rangle+
[\tanh(K_{j-\mu}+K_{j+\mu})+\tanh(K_{j-\mu}-K_{j+\mu})]\,\langle
s_{j-1}\rangle\nonumber\\ &\quad+
[\tanh(K_{j-\mu}+K_{j+\mu})-\tanh(K_{j-\mu}-K_{j+\mu})]\,\langle
s_{j+1}\rangle,\quad 1\leq j\leq L,\\ \label{bo.10}
\langle s_0\rangle&=1,\\ \label{bo.11}
\langle s_{L+1}\rangle&=1.
\end{align}
\section{Time varying boundary conditions on a semi-infinite lattice}
Consider a lattice for which the boundary spins ($s_0$ and
$s_{L+1}$) are externally controlled, but the reactions at
the internal sites satisfy detailed balance. The evolution
equation is then the same as (\ref{bo.9}), but combined
with boundary conditions different from (\ref{bo.10}) and
(\ref{bo.11}). A semi-infinite lattice the boundary of
which is externally controlled, is obtained by letting
$L$ tend to infinity, and using the following boundary
conditions
\begin{align}\label{bo.12}
&\langle s_0\rangle=f(t),\\ \label{bo.13}
&\mbox{$\langle s_j\rangle$ does not blow up as $j$ tends to infinity},
\end{align}
instead of (\ref{bo.10}) and (\ref{bo.11}).
A general solution of (\ref{bo.9}), combined with
(\ref{bo.12}) and (\ref{bo.13}), can be written
as the sum of a particular solution plus a general
solution of (\ref{bo.9}), combined with the homogeneous
boundary conditions.
\subsection{Semi-infinite lattice with uniform couplings: the homogeneous solution}
For a lattice with uniform couplings, $K_\alpha$'s are denoted by
$K$. The solution to the homogenous equation (vanishing $f$) is
denoted by $\langle s_j\rangle_\mathrm{h}$. One arrives at
\begin{equation}\label{bo.14}
\frac{\mathrm{d}}{\mathrm{d} t}\langle s_j\rangle_\mathrm{h}=-2\,\langle s_j\rangle_\mathrm{h}+
[\tanh(2\,K)]\,(\langle s_{j-1}\rangle_\mathrm{h}+\langle s_{j+1}\rangle_\mathrm{h}),\quad 0<j.
\end{equation}
Defining
\begin{equation}\label{bo.15}
\langle s_j\rangle_\mathrm{h}:=-\langle s_{-j}\rangle_\mathrm{h},\quad j<0,
\end{equation}
one arrives at
\begin{equation}\label{bo.16}
\frac{\mathrm{d}}{\mathrm{d} t}\langle s_j\rangle_\mathrm{h}=-2\,\langle s_j\rangle_\mathrm{h}+
[\tanh(2\,K)]\,(\langle s_{j-1}\rangle_\mathrm{h}+\langle s_{j+1}\rangle_\mathrm{h}),
\end{equation}
which holds for all integers $j$. Denoting the linear operator
acting on $\langle s_l\rangle$'s in the right-hand side of
(\ref{bo.16}) by $h$, the above equation is of the form
\begin{equation}\label{bo.17}
\frac{\mathrm{d}}{\mathrm{d} t}\langle s_j\rangle_\mathrm{h}=h^l_j\,\langle s_l\rangle_\mathrm{h},
\end{equation}
where $h^l_j$'s are the matrix elements of $h$.
Defining the generating function $G$ through
\begin{equation}\label{bo.18}
G(z,t):=\sum_{j=-\infty}^\infty\,z^j\,\langle s_j\rangle_\mathrm{h}(t),
\end{equation}
one arrives at
\begin{equation}\label{bo.19}
\frac{\partial G}{\partial t}=[-2+(z+z^{-1})\,\tanh(2\,K)]\,G,
\end{equation}
resulting in
\begin{align}\label{bo.20}
G(z,t)&=\exp\{[-2+(z+z^{-1})\,\tanh(2\,K)]\,t\}\,G(z,0),\nonumber\\
&=\exp(-2\,t)\,\sum_{k=-\infty}^\infty z^k\,\mathrm{I}_k[2\,t\,\tanh(2\,K)]\,G(z,0),\nonumber\\
&=\exp(-2\,t)\,\sum_{j=-\infty}^\infty z^j\,\sum_{l=-\infty}^\infty
\mathrm{I}_{j-l}[2\,t\,\tanh(2\,K)]\,\langle s_l\rangle_\mathrm{h}(0),
\end{align}
where $\mathrm{I}_k$ is the modified Bessel function of first kind of order $k$.
(\ref{bo.20}) results in
\begin{align}\label{bo.21}
\langle s_j\rangle_\mathrm{h}(t)&=\exp(-2\,t)\,\sum_{l=-\infty}^\infty
\mathrm{I}_{j-l}[2\,t\,\tanh(2\,K)]\,\langle s_l\rangle_\mathrm{h}(0),\nonumber\\
&=\exp(-2\,t)\,\sum_{l=1}^\infty\{\mathrm{I}_{j-l}[2\,t\,\tanh(2\,K)]-\mathrm{I}_{j+l}[2\,t\,\tanh(2\,K)]\}
\,\langle s_l\rangle_\mathrm{h}(0).
\end{align}
Using the large argument behavior of the modified Bessel functions,
it is seen that
\begin{equation}\label{bo.22}
\langle s_j\rangle_\mathrm{h}(t)\sim\exp\{-2\,[1-\tanh(2\,K)]\,t\},\quad j>0,
\end{equation}
showing that the homogeneous solution tends to zero at large times.
\subsection{Semi-infinite lattice with uniform couplings: the particular solution
corresponding to harmonic boundary conditions}
The harmonic boundary condition is
\begin{equation}\label{bo.23}
\langle s_0\rangle=\mathrm{Re}[\sigma_0\,\exp(-\mathrm{i}\,\omega\,t)]
\end{equation}
The following ansatz for a particular solution $\langle s_j\rangle_\mathrm{p}$
to equations (\ref{bo.9}) and (\ref{bo.12})
\begin{equation}\label{bo.24}
\langle s_j\rangle_\mathrm{p}=\mathrm{Re}[\sigma_j\,\exp(-\mathrm{i}\,\omega\,t)]
\end{equation}
results in
\begin{equation}\label{bo.25}
(\mathrm{i}\,\omega-2)\,\sigma_j+[\tanh(2\,K)](\sigma_{j+1}+\sigma_{j-1})=0.
\end{equation}
This has a solution of the form
\begin{equation}\label{bo.26}
\sigma_j=c\,z^j,
\end{equation}
where $z$ satisfies
\begin{equation}\label{bo.27}
z+z^{-1}=\frac{-i\omega +2}{\tanh(2K)}.
\end{equation}
It is obvious that changing the sign of $K$ results
in changing the sign of $z$, while changing the sign of
$\omega$ results in changing $z$ to its complex conjugate.
So it is sufficient to consider only nonegative values of
$K$ and $\omega$. From now on, it is assumed that
$K$ and $\omega$ are nonnegative. (\ref{bo.27}) has
two solution for $z$, which are inverse of each other,
and none are unimodular. The boundary condition at infinity
imposes that of the two solutions of type (\ref{bo.26}),
only that solution is acceptable which corresponds to
the root of (\ref{bo.27}) with modulus less than
one. From now, only this root is denoted by $z$:
\begin{equation}\label{bo.28}
z:=r\,\exp(\mathrm{i}\,\theta)
\end{equation}
where $r$ and $\theta$ are real and $r$ is positive
and less that one. The solution to (\ref{bo.25}) is then
\begin{equation}\label{bo.29}
\sigma_j=\sigma_0\,z^j.
\end{equation}
As $|z|$ is less than one, the particular solution (\ref{bo.24})
describes an evanescent wave. Obviously, the rate of decay
length and the phase speed, $\ell$ and $v$ respectively,
satisfy
\begin{align}\label{bo.30}
\ell&=-\frac{1}{\ln r},\nonumber\\
v&=\frac{\theta}{\omega}.
\end{align}
As the homogenous solution (\ref{bo.21}) tends to zero for large times,
the particular solution (\ref{bo.24}) is in fact the large times solution
to the problem of harmonic boundary condition.
Defining
\begin{align}\label{bo.31}
a&:=\frac{\omega}{2},\nonumber\\
b&:=\tanh 2\,K,\nonumber\\
u&:=\frac{r+r^{-1}}{2},
\end{align}
the real and imaginary parts of (\ref{bo.27}) read
\begin{align}\label{bo.32}
u\,\cos\theta&=\frac{1}{b},\nonumber\\
\sqrt{u^2-1}\,\sin\theta&=\frac{a}{b}.
\end{align}
So $u$ satisfies
\begin{equation}\label{bo.33}
b^2\,u^4-(a^2+b^2+1)\,u^2+1=0,
\end{equation}
from which one arrives, for the solution which is
larger than one, at
\begin{equation}\label{bo.34}
u=\left(\frac{1+a^2+b^2+\sqrt{(1+a^2+b^2)^2-4 b^2}}{2\,b^2}\right)^{1/2}.
\end{equation}
This is increasing with respect to $a$, and decreasing
with respect to $b$. Noting that
\begin{equation}\label{bo.35}
\frac{\mathrm{d} u}{\mathrm{d} r}=\frac{1}{2}\,\left(1-\frac{1}{r^2}\right),
\end{equation}
which shows that $u$ is decreasing with respect to $r$,
it is seen that $r$ is decreasing with respect to $\omega$,
and increasing with respect to $K$. One also has
\begin{equation}\label{bo.36}
r=u-\sqrt{u^2-1}.
\end{equation}
Regarding $\theta$, differentiating the first equation in
(\ref{bo.32})with respect to $a$, one has
\begin{equation}\label{bo.37}
\cos\theta\,\frac{\partial u}{\partial a}-u\,\sin\theta\,\frac{\partial\theta}{\partial a}=0,
\end{equation}
resulting in
\begin{align}\label{bo.38}
\frac{\partial\theta}{\partial a}&=\frac{\cos\theta}{u\,\sin\theta}\,
\frac{\partial u}{\partial a},\nonumber\\
&=\frac{b^2\,u^2-b^2}{b\,u\,\sqrt{(1+a^2+b^2)^2-4\,b^2}}\,\frac{\sin\theta}{a},\nonumber\\
&=\frac{b^2\,u^2-b^2}{b\,u\,(b^2\,u^2-u^{-2})}\,\frac{\sin\theta}{a}.
\end{align}
Equation (\ref{bo.34}) shows that
\begin{equation}\label{bo.39}
b\,u\geq 1,
\end{equation}
from which it is seen that
\begin{equation}\label{bo.40}
0<\frac{\partial\theta}{\partial a}\leq\frac{\sin\theta}{a}.
\end{equation}
The first inequality shows $\theta$ is an increasing function of $a$,
so it is an increasing function of $\omega$.
The second inequality results in
\begin{equation}\label{bo.41}
\frac{\partial\theta}{\partial a}\leq\frac{\theta}{a},
\end{equation}
which shows that $(\theta/a)$ is a decreasing function
of $a$. So $(a/\theta)$ is an increasing function of $a$, or
$(\omega/\theta)$ is an increasing function of $\omega$.
One also has
\begin{equation}\label{bo.42}
\frac{\partial(2\,b^2\,u^2)}{\partial b^2}=1+\frac{a^2+b^2-1}{\sqrt{(1+a^2+b^2)^2-4\,b^2}},
\end{equation}
and as
\begin{equation}\label{bo.43}
\sqrt{(1+a^2+b^2)^2-4\,b^2}\geq 1-b^2,
\end{equation}
it turns out that $(b\,u)$ is increasing with $b$, so that $\theta$
is increasing with $b$. Hence $(\omega/\theta)$ is decreasing with
$K$.
The asymptotic behavior of $r$ and $\theta$ is summarized as
\begin{equation}\label{bo.44}
r=\begin{cases}
\displaystyle{\tanh K},&
\omega\ll 1,\\ \\
\displaystyle{\frac{\tanh(2\,K)}{\omega}},& 1\ll\omega,\\ \\
\displaystyle{\frac{\tanh(2\,K)}{\sqrt{4+\omega^2}}},&
K\ll 1,\\ \\
\displaystyle{\frac{\sqrt{8+\omega^2+\omega\,\sqrt{16+\omega^2}}-
\sqrt{\omega^2+\omega\,\sqrt{16+\omega^2}}}{\sqrt{8}}},&1\ll K
\end{cases},
\end{equation}
and
\begin{equation}\label{bo.45}
\theta=\begin{cases}
\displaystyle{\frac{\omega\,\cosh(2\,K)}{2}},&
\omega\ll 1,\\ \\
\displaystyle{\frac{\pi}{2}},& 1\ll\omega,\\ \\
\displaystyle{\tan^{-1}\frac{\omega}{2}},&
K\ll 1,\\ \\
\displaystyle{\cos^{-1}\sqrt{\frac{8+\omega^2-\omega\,\sqrt{16+\omega^2}}{8}}},&1\ll K
\end{cases},
\end{equation}
Among other things, it is seen that the phase speed, at
low frequencies approaches the constant value
$2/[\cosh(2\,K)]$, while at high frequencies varies
like $(2\,\omega/\pi)$.
Figure \ref{fig1} is a plot of $r$ versus $\omega$ for different values of
$\tanh(2\,K)$ from $0.1$ to $0.9$. Figure \ref{fig2} is a plot of
the phase speed $(\omega/\theta)$ versus $\omega$ for
different values of $\tanh(2\,K)$ from $0.1$ to $0.9$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.3]{boundoscr.eps}
\setlength{\unitlength}{1pt}
\put(-163,160){$r$}
\put(-7,4){$\omega$}
\put(-122,1){\scriptsize{1}}
\put(-172,31){\scriptsize{0.1}}
\put(-5,13){\scriptsize{0.1}}
\put(-5,53){\scriptsize{0.9}}
\put(-8,13){\vector(0,1){50}}
\put(-5,33){$\tanh(2\,K)$}
\caption{\label{fig1}}{The plot of $r$ versus $\omega$ for different values of $\tanh(2\,K)$}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.3]{boundoscc.eps}
\setlength{\unitlength}{1pt}
\put(-186,160){$(\omega/\theta)$}
\put(-7,4){$\omega$}
\put(-122,1){\scriptsize{1}}
\put(-166,51){\scriptsize{1}}
\put(-170,95){\vector(0,-1){55}}
\put(-182,92){\scriptsize{0.1}}
\put(-182,47){\scriptsize{0.9}}
\put(-216,69){$\tanh(2\,K)$}
\caption{\label{fig2}}{The plot of the phase speed $(\omega/\theta)$ versus $\omega$
for different values of $\tanh(2\,K)$}
\end{center}
\end{figure}
The total magnetization, defined as the sum of the expectation
values of the spins, is denoted by $M$. At large times only
the particular solution contributes to the magnetization. So,
\begin{equation}\label{bo.46}
M=\mathrm{Re}\left[\frac{\sigma_0}{1-z}\,\exp(-\mathrm{i}\,\omega\,t)\right],\quad t\to\infty.
\end{equation}
For the time-independent boundary condition, this leads to
\begin{equation}\label{bo.47}
M=\frac{\sigma_0}{1-\tanh K},\quad (t\to\infty,\;\omega=0).
\end{equation}
For high frequencies,
\begin{equation}\label{bo.48}
M=\mathrm{Re}\left\{\sigma_0\,\left[1-\frac{\tanh(2\,K)}{-\mathrm{i}\,\omega}\right]^{-1}\,\exp(-\mathrm{i}\,\omega\,t)\right\},
\quad (t\to\infty,\;\omega\to\infty).
\end{equation}
This can be simplified to
\begin{align}\label{bo.49}
M&=\mathrm{Re}\left\{\sigma_0\,\left[1+\frac{\tanh(2\,K)}{-\mathrm{i}\,\omega}\right]\,\exp(-\mathrm{i}\,\omega\,t)\right\},
\nonumber\\ &=\langle s_0\rangle(t)+[\tanh(2\,K)]\,[S_0(t)-\bar S_0],
\quad (t\to\infty,\;\omega\to\infty),
\end{align}
where
\begin{align}\label{bo.50}
S_0(t)&:=\int_0^t\mathrm{d} t'\,\langle s_0\rangle(t'),\nonumber\\
\bar S_0&:=\lim_{T\to\infty}\left[\frac{1}{T}\,\int_T\mathrm{d} t\;S_0(t)\right].
\end{align}
One then arrives at a similar result for the magnetization
when the boundary condition is any arbitrary rapidly varying function
of time (so that its low frequency components are negligible):
\begin{align}\label{bo.51}
&M=\langle s_0\rangle(t)+[\tanh(2\,K)]\,[S_0(t)-\bar S_0],\nonumber\\
&\quad (t\to\infty,\mbox{ rapidly varying boundary conditions}).
\end{align}
\subsection{Semi-infinite lattice with two parts of uniform couplings: the particular solution
corresponding to harmonic boundary conditions}
Consider a semi-infinite lattice consisting of two parts, so that
\begin{equation}\label{bo.52}
K_\alpha=\begin{cases}K_1,&\alpha<N\\
K_2,&\alpha>N
\end{cases},
\end{equation}
The time evolution equations for the expectation values of the spins are
\begin{align}\label{bo.53}
\langle\dot s_j\rangle&=-2\,\langle s_j\rangle+
[\tanh(2\,K_1)]\,(\langle s_{j-1}\rangle+\langle s_{j+1}\rangle), \quad 0<j<N,\\ \label{bo.54}
\langle\dot s_N\rangle&=-2\,\langle s_N\rangle+\kappa_-\,\langle s_{N-1}\rangle+
\kappa_+\,\langle s_{N+1}\rangle,\\ \label{bo.55}
\langle\dot s_j\rangle&=-2\,\langle s_j\rangle+
[\tanh(2\,K_2)]\,(\langle s_{j-1}\rangle+\langle s_{j+1}\rangle), \quad N<j.
\end{align}
where
\begin{align}\label{bo.56}
\kappa_-&:=\tanh(K_1+K_2)+\tanh(K_1-K_2),\nonumber\\
\kappa_+&:=\tanh(K_1+K_2)-\tanh(K_1-K_2).
\end{align}
Applying a harmonic boundary condition (\ref{bo.23}), one has for
the particular solution of the kind (\ref{bo.24}),
\begin{equation}\label{bo.57}
\sigma_j=\begin{cases}(A_1\,z_1^j+B_1\,z_1^{-j}),& 0\leq j\leq N\\
A_2\,z_2^j,& N\leq j\end{cases},
\end{equation}
where
\begin{equation}\label{bo.58}
z_l+ z_l^{-1}=\frac{-\mathrm{i}\,\omega+2}{\tanh(2\,K_l)},\quad l=1,2
\end{equation}
and $|z_l|$ is smaller than one. The boundary condition results in
\begin{equation}\label{bo.59}
A_1+B_1=\sigma_0.
\end{equation}
From (\ref{bo.57}) for $j=N$, one arrives at
\begin{equation}\label{bo.60}
A_1\,z_1^N+B_1\,z_1^{-N}=A_2\,z_2^N.
\end{equation}
Finally, (\ref{bo.54}) results in
\begin{equation}\label{bo.61}
\kappa_-\,(A_1\,z_1^{N-1}+B_1\,z_1^{-N+1})+\kappa_+\,A_2\,z_2^{N+1}=(-\mathrm{i}\,\omega +2)\,A_2\,z_2^N.
\end{equation}
Equations (\ref{bo.59}) through (\ref{bo.61}) give
\begin{align}\label{bo.62}
A_1&=\frac{(\kappa_-\,z_1+\kappa_+\,z_2+\mathrm{i}\,\omega-2)\,z_1^{-N}\,\sigma_0}
{\kappa_-\,(z_1^{-N+1}-z_1^{N-1})+(\kappa_+\,z_2+\mathrm{i}\,\omega-2)\,(z_1^{-N}-z_1^N)},\nonumber\\
B_1&=\frac{-(\kappa_-\,z_1^{-1}+\kappa_+\,z_2+\mathrm{i}\,\omega-2)\,z_1^N\,\sigma_0}
{\kappa_-\,(z_1^{-N+1}-z_1^{N-1})+(\kappa_+\,z_2+\mathrm{i}\,\omega-2)\,(z_1^{-N}-z_1^N)},\nonumber\\
A_2&=\frac{\kappa_-\,(z_1-z_1^{-1})\,z_2^{-N}\,\sigma_0}
{\kappa_-\,(z_1^{-N+1}-z_1^{N-1})+(\kappa_+\,z_2+\mathrm{i}\,\omega-2)\,(z_1^{-N}-z_1^N)}.
\end{align}
For large $N$, these simplify to
\begin{align}\label{bo.63}
A_1&=\sigma_0,\nonumber\\
B_1&=\frac{-(\kappa_-\,z_1^{-1}+\kappa_+\,z_2+\mathrm{i}\,\omega-2)\,z_1^{2\,N}\,\sigma_0}
{\kappa_-\,z_1+\kappa_+\,z_2+\mathrm{i}\,\omega-2},\nonumber\\
A_2&=\frac{\kappa_-\,(z_1-z_1^{-1})\,z_2^{-N}\,z_1^N\,\sigma_0}
{\kappa_-\,z_1+\kappa_+\,z_2+\mathrm{i}\,\omega-2}.
\end{align}
It can be easily shown that for the nonuniform lattice and at
high frequencies, up to first term in $\omega^{-1}$ the magnetization
is similar to the case of the uniform lattice.
\subsection{Semi-infinite lattice with nonuniform couplings:
the relaxation times}
The general solution of (\ref{bo.9}) with (\ref{bo.12}) and (\ref{bo.13})
is the sum of a particular solution and the general solution to
(\ref{bo.9}) and (\ref{bo.12}) and (\ref{bo.13}) with vanishing $f$.
The latter (the homogeneous solution) satisfies
\begin{align}\label{bo.64}
\frac{\mathrm{d}}{\mathrm{d} t}\langle s_j\rangle_\mathrm{h}&=-2\,\langle s_j\rangle_\mathrm{h}+
[\tanh(K_{j-\mu}+K_{j+\mu})+\tanh(K_{j-\mu}-K_{j+\mu})]\,\langle
s_{j-1}\rangle_\mathrm{h}\nonumber\\ &\quad+
[\tanh(K_{j-\mu}+K_{j+\mu})-\tanh(K_{j-\mu}-K_{j+\mu})]\,\langle
s_{j+1}\rangle_\mathrm{h},\quad 1\leq j\leq L,\nonumber\\
\langle s_0\rangle_\mathrm{h}&=0,\nonumber\\
\langle s_{L+1}\rangle_\mathrm{h}&=0,
\end{align}
which can be written as (\ref{bo.17}). Denoting an eigenvalue
of $h$ by $E$, and the corresponding eigenvector by $\psi_E$,
it is seen that there are solutions to (\ref{bo.64}) of the form
\begin{equation}\label{bo.65}
\langle s_j\rangle_\mathrm{h}(t)=\psi_{E\,j}\,\exp(E\,t).
\end{equation}
These solutions decay with a relaxation time $\tau$ satisfying
\begin{equation}\label{bo.66}
\tau=-\frac{1}{\mathrm{Re}(E)}.
\end{equation}
One can see that the eigenvalues of the operator $h$ are real.
To see this, one notices that equations (\ref{bo.64}) are
the same as the equations corresponding to the homogeneous
solution of (\ref{bo.8}). So the homogeneous solution to
the Ising chain externally driven at ends, is the same as
the homogenous solution to the Ising chain with magnetic
fields at boundaries. The evolution equation for the latter
satisfies the detailed balance. For any evolution satisfying
detailed balance, the eigenvalues of the evolution operator
are real. To see this, one notices that the criterion of
the detailed balance is
\begin{equation}\label{bo.67}
\omega(B\to A)=Y^A_B\,\exp[\beta(\mathcal{E}_B-\mathcal{E}_A)],
\end{equation}
where $A$ and $B$ are two different state, $Y^A_B$'s are real nonnegative
numbers (for $B\ne A$) satisfying
\begin{equation}\label{bo.68}
Y^B_A=Y^A_B,
\end{equation}
and $\mathcal{E}$ is the energy of the system in the state $A$.
So the matrix $Y$ is Hermitian. Equation (\ref{bo.67})
means that the evolution matrix $H$, the components of
which are $\omega(B\to A)$'s, is a similarity-transformed of
$Y$. As $Y$ is Hermitian, the eigenvalues of $Y$ are real.
As $H$ is a similarity-transformed of $Y$, the eigenvalues of
$H$ are the same as the eigenvalues of $Y$. So
the eigenvalues of $H$ are real (\cite{db} for example).
The eigenvalues of $h$ are eigenvalues of $H$ as well. So
the eigenvalues of $h$ are real.
\section{Concluding remarks}
A one dimensional kinetic Ising model at temperature $T$,
with time varying boundary conditions was studied, for the case
the lattice is semi-infinite. The evolution equation for
the expectation values of the spins was investigated. For
the case of harmonic boundary conditions, with uniform couplings,
exact particular solutions were obtained for the expectation values
of the spins, as well as the total magnetization. The low- and
high-frequency behaviors were studied in more detail.
Models for which the coupling constant is nonuniform were
also studied. Physically, such nonuniform couplings
could arise when either the interaction between spins or
the temperature depends on the position.
As a specific example, the harmonic solution on
a semi-infinite lattice consisting of two homogeneous
parts studied.
Finally, it was shown that for a general (nonuniform) lattice,
the eigenvalues corresponding to the evolution operator are real.
\\[\baselineskip]
\textbf{Acknowledgement}: This work was supported by
the research council of the Alzahra University.
\newpage
|
1,116,691,499,394 | arxiv | \section{Introduction}
The discovery of the Higgs boson on July 2012 provided a well established
explanation of the mass origin of the fermions and gauge bosons of
the standard model (SM). However, despite this great breakthrough,
many questions are still open, such as the origin of neutrino mass
and its smallness, the dark matter (DM) nature, that appears necessary
at the galactic scale. In addition to these unanswered questions,
it is so puzzling that most of the mass-dimension parameters in the
SM are of the order of $\mathcal{O}(100)$~$\mathrm{GeV}$. Therefore, we still
have too much to learn about the mechanism(s) of masses at the Universe.
In regards to this open issue, recent works have studied several extensions
of the SM that possess a scale-invariance (SI) symmetry, where the
electroweak symmetry breaking (EWSB) occurs as a quantum effect via
the radiative symmetry breaking a la Coleman-Weinberg~\cite{Coleman:1973jx}.
These models are phenomenologically rich, and generally predict new
$\mathrm{TeV}$ mass particles, that could be within the reach of the LHC.
A popular explanation of the smallness of the neutrino mass is provided by
the so-called seesaw mechanism~\cite{Gell}, where a hierarchy between
the charged leptons and neutrinos masses emerges due to the hierarchy
between the electroweak (EW) scale and the heavy singlet Majorana
fermion mass, that are added to the SM. The new high scale (Majorana
fermion mass) makes this scenario impossible to be probed at the current
and near future high energy colliders. Another attractive approach
is that the radiative neutrino mass is massless at tree level and
acquire a naturally small Majorana mass term at the loop level: one-loop~\cite{Zee:1985rj,Ma:1998dn},
two loops~\cite{Zee:1985id,Babu:1988ki,Aoki:2010ib,Guo:2012ne,Kajiyama:2013zla},
three loops~\cite{Krauss:2002px,Aoki:2008av,Gustafsson:2012vj,Ahriche:2014cda,Hatanaka:2014tba,Nishiwaki:2015iqa,Ahriche:2015wha,Ahriche:2015loa,Okada:2015hia,Nomura:2016ezz,Gu:2016xno,Cheung:2017efc,Dutta:2018qei},
and four loops~\cite{Nomura:2016fzs}. Some of these models address
the DM problem where a massive Majorana DM with a mass range from
$\mathrm{GeV}$ to $\mathrm{TeV}$ can play an important role as a
DM candidate~\cite{Ma:1998dn,Krauss:2002px,Aoki:2008av,Okada:2015hia},
in addition to the neutrino oscillation data. Furthermore, these models
predict interesting signatures at current/future collider experiments~\cite{Aoki:2016wyl,Ahriche:2014xra,Toma:2013zsa}
(for a review, see~\cite{Cai:2017jrq}).
Extending some of the $\nu$-DM motivated models by incorporating
the SI symmetry, makes the model phenomenology modified and richer~\cite{Foot:2007ay}.
Indeed, one can emphasize that by considering the SI symmetry the
issues of ESWB, neutrino mass and DM are solved all together at the
EW scale. For example, the models in~\cite{Ahriche:2016cio} and
\cite{Ahriche:2015loa} are SI generalizations of a one-loop (scotogenic~\cite{Ma:2006km})
and a three-loop (KNT~\cite{Krauss:2002px}) model, where the lightest
Majorana fermion plays the DM role. In addition to the DM annihilation
channel $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$
in the scotogenic model (and similarly with the SI-KNT model~\cite{Ahriche:2015loa}),
other channels such as $N_{1}N_{1}\rightarrow h_{i}h_{k},\,VV,q\bar{q}$,
are possible to take place due to the DM coupling with the Higgs/dilaton
in the SI extensions. Here, $V=W,Z$ are the SM gauge bosons, and
$h_{i}$ could be the Higgs $H$ and/or the the lighter CP-even eigenstate
$D$, that is called the dilaton in the SI models. It is strictly massless
at tree-level and acquires its mass via the loop corrections. In the
non SI models (for example~\cite{Krauss:2002px} and~\cite{Ma:2006km}),
for a given DM mass $M_{1}$, the relic density dictates the values
of the new Yukawa couplings ($g_{i\alpha}$) that couple the left-handed
lepton doublets (or the charged right-handed leptons in the case of
KNT model) and the inert doublet (charged singlet) scalar. This makes
the values of these couplings ($g_{i\alpha}$) too constrained since
they should be also in agreement with the neutrino oscillation data~\cite{Fukuda:2001nj},
and the lepton flavor violating (LFV) constraints~\cite{PDG2021}.
In the SI framework, the non negligible contributions of the new DM
annihilation channels $N_{1}N_{1}\rightarrow SS,\,VV,q\bar{q}$ could
relax the bounds on these new couplings and make the parameters space
(especially the range of the couplings $g_{i\alpha}$) much larger.
The new DM-Higgs/dilaton interactions that allow the new DM annihilation
channels will lead to a tree-level DM-nucleon scattering cross section,
contrary to the non SI models where it is a pure one-loop effect.
Consequently, this makes the effect of the requirement of both the
direct detection (DD) constraints and the relic density, very important
on the scalar sector within the SI models. For this reason, investigating
these effects is the motivation of our work by considering the model~\cite{Ahriche:2016cio}
as an example. In the non-SI scotogenic model with fermionic DM~\cite{Ahriche:2017iar},
the neutrino mass smallness is achieved by imposing a mass degeneracy
between the neutral inert CP-even and CP-odd scalars, since relatively
large values for the $g_{i\alpha}$ couplings are dictated by the
DM relic density. This implies that one of the scalar quartic couplings
must be suppressed $\lambda_{5}=\mathcal{O}(10^{-10})$. Nonetheless,
this fine-tuning could be avoided by extending the model with a
real singlet that is charged under a global $Z_{4}/Z_{2}$ symmetry~\cite{Ahriche:2020pwq}.
Here, we will investigate the possibility of non-suppressed $\lambda_{5}$
value within the SI framework due to the existence of new DM annihilation
channels ($N_{1}N_{1}\rightarrow SS,\,VV,q\bar{q}$).
The paper is organized as follows. Section~\ref{sec:The-Model} is
devoted to describe the SI scotogenic $\nu$-DM model, in addition
to the EWSB description within the SI framework. In section~\ref{sec:DM},
the DM relic density is estimated via an explicit and exact calculation
of the cross section of all channels. In addition, to impose the DD
bounds, the DM nucleon cross section formula is given. In section~\ref{sec:NR},
we perform a random scan for the model parameters space to probe all
the possible correlation aspects between the scalar sector and the
DM requirements. Our conclusions are given in section~\ref{sec:Conc}.
\section{The Scale Invariant Scotogenic Model\label{sec:The-Model}}
Here, the SM is extended by one inert doublet scalar, $S\sim(1,2,1)$,
three singlet Majorana fermions $N_{i}\sim(1,1,0)$, and one real
neutral singlet scalar $\phi\sim(1,1,0)$ to assist the radiative
EWSB. The model is assigned by a global $Z_{2}$ symmetry $\{S,\,N_{i}\}\rightarrow\{-S,\,-N_{i}\}$,
where all other fields being $Z_{2}$-even, in order to make the lightest
$Z_{2}$-odd field $N_{1}\equiv N_{\text{DM}}$ stable, hence it
plays the DM candidate role. The Lagrangian contains the following
terms
\begin{equation}
\mathcal{L}\supset-\;\{g_{i\alpha}\overline{N_{i}^{c}}S^{\dagger}L_{\alpha}+\mathrm{h.c}\}-\frac{1}{2}y_{i}\phi\overline{N_{i}^{c}}\,N_{i}-V(H,S,\phi),\label{L:Ma}
\end{equation}
where the inert doublet can be presented as $S^{T}=\Big(S^{+}\,,\,(S^{0}+i\,A^{0})/\sqrt{2}\Big)\sim(1,2,1)$,
$L_{\beta}$ and $\ell_{\alpha R}$ the left-handed lepton doublet
and right-handed leptons; the Greek letters label the SM flavors,
$\alpha,\,\beta\in\{e,\,\mu,\,\tau\}$, $g_{i\alpha}$ and $y_{i}$
are new Yukawa couplings. The most general SI scalar potential that
obeys the $Z_{2}$ symmetry is given by
\begin{align}
V_{0}(H,\,S,\,\phi) & =\frac{1}{6}\lambda_{H}(\left|H\right|^{2})^{2}+\frac{\lambda_{\phi}}{24}\phi^{2}+\frac{\lambda_{S}}{2}|S|^{4}+\frac{\omega_{1}}{2}|H|^{2}\phi^{2}+\frac{\omega_{2}}{2}\,\phi^{2}|S|^{2}+\lambda_{3}\,|H|^{2}|S|^{2}\nonumber \\
& +\lambda_{4}\,|H^{\dagger}S|^{2}+\{\frac{\lambda_{5}}{2}(H^{\dagger}S)^{2}+h.c.\}\label{V:Ma}
\end{align}
The first term in (\ref{L:Ma}) and the last term in (\ref{V:Ma})
are responsible for generating neutrino mass via the one-loop diagrams
as illustrated in Fig.~\ref{fig:SI3l}.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.4\textwidth]{SI_one_loop}
\par\end{centering}
\caption{The neutrino mass is generated in the SI-scotogenic model at one-loop
level.}
\label{fig:SI3l}
\end{figure}
The neutrino mass matrix elements are given by~\cite{Ma:2006km}
\begin{equation}
m_{\alpha\beta}^{(\nu)}=\sum_{i}\frac{g_{i\alpha}g_{i\beta}M_{i}}{16\pi^{2}}\left[\frac{m_{S^{0}}^{2}}{m_{S^{0}}^{2}-M_{i}^{2}}\log\frac{m_{S^{0}}^{2}}{M_{i}^{2}}-\frac{m_{A^{0}}^{2}}{m_{A^{0}}^{2}-M_{i}^{2}}\log\frac{m_{A^{0}}^{2}}{M_{i}^{2}}\right].\label{eq:MaNU}
\end{equation}
The neutrino mass matrix elements in (\ref{eq:MaNU}) can be written
as $m_{\alpha\beta}^{(\nu)}=\sum_{i}g_{i\alpha}g_{i\beta}\Lambda_{i}=\left(g^{T}.\varLambda.g\right)_{\alpha\beta}$,
which permits us to write the new Yukawa couplings according to the
Casas-Ibarra parameterization as~\cite{Casas:2001sr}
\begin{equation}
g=D_{\sqrt{\varLambda^{-1}}}RD_{\sqrt{m_{\nu}}}U_{\nu}^{T},
\end{equation}
with $D_{\sqrt{\varLambda^{-1}}}=\textrm{diag\{\ensuremath{\varLambda_{1}^{-1/2}},\ensuremath{\varLambda_{2}^{-1/2}},\ensuremath{\varLambda_{3}^{-1/2}}\}},\,D_{\sqrt{m_{\nu}}}=\textrm{diag}\{m_{1}^{1/2},m_{2}^{1/2},m_{3}^{1/2}\}$,
$R$ is an arbitrary $3\times3$ orthogonal matrix, $m_{i}$ represents
the neutrino mass eigenstates and $U_{\nu}$ is the Pontecorvo-Maki-Nakawaga-Sakata
(PMNS) mixing matrix. These couplings are subject of the LFV bounds
on the branching ratios of $\ell_{\alpha}\rightarrow\ell_{\beta}\gamma$
and $\ell_{\alpha}\rightarrow\ell_{\beta}\ell_{\beta}\ell_{\beta}$~\cite{Toma:2013zsa}.
In this model, the DM candidate could be fermionic (the lightest Majorana
fermion, $N_{1}$) or a scalar (the lightest among $S^{0}$ and $A^{0}$).
In the case of a scalar DM, the situation matches the inert doublet
model case~\cite{Banerjee:2021oxc}, where the co-annihilation effect
should be considered in order to have viable parameters space. For
Majorana DM case, the Yukawa couplings $g_{i\alpha}$ values are constrained
by the relic density, and therefore the neutrino mass smallness should
be achieved by the $S^{0}-A^{0}$ mass degeneracy, i.e., imposing
a very small values for\textbf{ $\lambda_{5}=\mathcal{O}(10^{-10})$}.
After the EWSB, the CP-even neutral scalars acquire VEV's as
\begin{equation}
H\rightarrow\frac{\upsilon+h}{\sqrt{2}}\begin{pmatrix}0\\
1
\end{pmatrix},\,\,\,\phi\rightarrow\frac{x+\phi}{\sqrt{2}},\label{vev}
\end{equation}
and hence give masses to all model fields, where we get two CP-even
eigenstates
\begin{equation}
\begin{pmatrix}H\\
D
\end{pmatrix}=\begin{pmatrix}c_{\alpha}~-s_{\alpha}\\
s_{\alpha}~c_{\alpha}
\end{pmatrix}\begin{pmatrix}h\\
\phi
\end{pmatrix},\label{scMix}
\end{equation}
where $H$ denotes the 125 $\mathrm{GeV}$ Higgs, $D$ is the dilaton scalar and
$\alpha$ is the Higgs-dilaton mixing angle, which is defined at tree-level
as $s_{\alpha}=\sin\alpha=\upsilon/\sqrt{\upsilon^{2}+x^{2}}$ and
$c_{\alpha}=\cos\alpha=x/\sqrt{\upsilon^{2}+x^{2}}$. In the SI approach,
the EWSB is triggered by the the radiative corrections. Then, the
one-loop effective potential in (\ref{V:Ma}) can be expressed in
the $\overline{DR}$ scheme as
\begin{eqnarray}
V^{1-\ell}(h,\phi) & = & \frac{\lambda_{H}+\delta\lambda_{H}}{24}h^{4}+\frac{\lambda_{\phi}+\delta\lambda_{\phi}}{24}\phi^{4}+\frac{\omega+\delta\omega}{4}h^{2}\,\phi^{2}\nonumber\\
& & + \frac{1}{64\pi^{2}}\sum_{i}n_{i}m_{i}^{4}(h,\phi)\left[\log\frac{m_{i}^{2}(h,\phi)}{\Lambda^{2}}-\frac{3}{2}\right],\label{eq:V}
\end{eqnarray}
where $\delta\lambda_{H},\,\delta\lambda_{\phi},\,\delta\omega$ are
counter-terms, $n_{i}$ and $m_{i}^{2}(h,\phi)$ are the field multiplicity
and field dependant masses, respectively. Here $\Lambda=m_{h}=125.18\,\mathrm{GeV}$
is the renormalization scale and $m_{h}$ is the measured Higgs mass. Here,
we choose the counter-terms $\delta\lambda_{H},\,\delta\lambda_{\phi}$
and $\delta\omega$, where the minimum $\{h=\upsilon,\phi=x\}$ is
still the vacuum at one-loop; and the Higgs mass at one-loop must
correspond to the measured value. The field dependent masses for the
gauge bosons and fermions are the same as in the SM. The field dependent
masses of the Goldstone and the Majorana singlets are $m_{\chi}^{2}=\frac{1}{6}\lambda_{H}h^{2}+\frac{1}{2}\omega\phi^{2}$
and $M_{i}^{2}=y_{i}^{2}\phi^{2}$, respectively. The Higgs-dilation
eigenmasses are the eigenvalues of the matrix whose elements are $m_{hh}^{2}=\frac{1}{2}\lambda_{H}h^{2}+\frac{1}{2}\omega\phi^{2}$,
$m_{\phi\phi}^{2}=\frac{1}{2}\lambda_{\phi}\phi^{2}+\frac{1}{2}\omega h^{2}$
and $m_{h\phi}^{2}=\omega h\phi$, and the inert field dependant masses
are given by $m_{S^{\pm}}^{2}=\frac{1}{2}\omega_{2}\phi^{2}+\frac{1}{2}\lambda_{3}h^{2},~m_{S^{0},A^{0}}^{2}=m_{S^{\pm}}^{2}+\frac{1}{2}(\lambda_{4}\pm\lambda_{5})h^{2}$.
Since the dilaton acquires mass via radiative effects, then its mass
squared should be positive $m_{D}^{2}>0$, which is basically guaranteed
by the vacuum stability conditions. The vacuum stability could be
ensured by imposing the coefficients of the term $\varphi^{4}\log\varphi$
to be positive rather than the coefficients of the term $\varphi^{4}$, where $\varphi$ refers
to any direction in the $h-\phi$ plan. These conditions are translated
into
\begin{equation}
\begin{array}{c}
-12y_{t}^{4}+\tfrac{9}{4}g_{2}^{4}+\tfrac{3}{4}g_{1}^{4}+\frac{4}{3}\lambda_{H}^{2}+\frac{4}{3}\omega_{1}^{2}+2\lambda_{3}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})^{2}+(\lambda_{3}+\lambda_{4}-\lambda_{5})^{2}>0,\\
\lambda_{\phi}^{2}+4\omega_{1}^{2}+4\omega_{2}^{2}-8(y_{1}^{4}+y_{2}^{4}+y_{3}^{4})>0.
\end{array}\label{eq:Ma-VSt}
\end{equation}
As mentioned earlier, the EWSB is triggered by the radiative corrections,
where the Higgs/dilaton sector has a similar structure for all the
SI $\nu$-DM models. However, there may be some few differences due
to the nature and multiplicity of the new fields. Indeed, the new
model couplings are not fully free and are severely constrained by
many conditions such as the measured Higgs mass and the vacuum stability.
Since the dilaton squared mass is purely radiative, we try to quantify
the radiative effects by the values of both of the dilation squared
mass and the one-loop correction to Higgs/dilaton mixing
\begin{equation}
\Delta_{s_{\alpha}}=\frac{(s_{\alpha})_{1-loop}-(s_{\alpha})_{tree-level}}{(s_{\alpha})_{tree-level}}.\label{eq:DS}
\end{equation}
These two quantities, in addition to the singlet VEV controls the
new contribution of the DM annihilation via the interactions with the
Higgs/dilaton, i.e., via the channels $N_{1}N_{1}\rightarrow h_{i}h_{k},\,VV,q\bar{q}$
with $h_{i}=H,D$.
One has to mention that for other regions of space parameters, the light CP-even could match the SM-like Higgs, where the Higgs mass
is generated fully radiative~\cite{Ahriche:2021frb}. In this setup, the radiative effects push simultaneously the light CP-even mass to match the measured Higgs mass; and the scalar mixing to be in agreement with the Higgs signal strength measurements at the LHC~\cite{ATLAS:2016neq}.
In addition, this scenario is in agreement with all the previously mentioned constraints~\cite{Ahriche:2021frb}.
\section{Dark Matter\label{sec:DM}}
In this section, we discuss the relic density estimation approach
and give the exact cross section formulas for the different annihilation
channels. Then, we give the DM-nucleon scattering cross section that
is subject of DD experiments.
\textbf{DM Relic Density}: the DM candidate is the lightest Majorana
fermion ($N_{1}$), where there are many possible annihilation channels,
that can be classify into two categories according to the DM interactions:
(1) the annihilation via the new Yukawa interactions ($g_{i\alpha}$),
i.e., into $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$;
and (2) the annihilation via the interactions with the Higgs/dilaton,
i.e., $N_{1}N_{1}\rightarrow h_{i}h_{k},\,VV,q\bar{q}$, where $S_{i}=H,\,D$
and $V=W,Z$. Clearly, for suppressed Higgs-dilation mixing $s_{\alpha}$,
the situation matches the non SI models, i.e., the DM annihilation
must fully achieved via the new Yukawa interactions. In the opposite
case, the DM annihilation could be dominated by all channels (or simply
by just one channel) that are mediated by the Higgs/dilaton. This
depends on the DM mass $M_{1}$, dilaton mass and the Higgs-dilation
mixing $s_{\alpha}$.
In what follows, we show how could the DM relic density be estimated
at the freeze-out temperature~\cite{Srednicki:1988ce}. When the
temperature of the Universe drops below the DM mass, the DM decouples
from the thermal bath and then preserves the DM particles number.
In other words, after the freeze-out the ratio $n_{DM}/n_{s}$ remains
constant during the Universe expansion, where $n_{DM}$ and $n_{s}$
are the DM number and entropy densities, respectively. The estimated
relic density must match the Planck observation~\cite{Aghanim:2018eyx}
\begin{equation}
\Omega_{{\rm DM}}h^{2}=0.120\pm0.001\,,\label{eq:omegah}
\end{equation}
where $h$ is the reduced Hubble constant and $\Omega_{{\rm DM}}$
denotes the DM energy density scaled by the critical density. Up to
a very good approximation, the cold DM relic
abundance of the WIMP scenario is given by~\cite{Srednicki:1988ce}
\begin{equation}
\Omega_{N_{1}}h^{2}\simeq\frac{(1.07\times10^{9})x_{F}}{\sqrt{g_{\ast}}M_{pl}(\mathrm{GeV})\left\langle \sigma(N_{\text{1}}\ N_{\text{1}})\upsilon_{r}\right\rangle },
\end{equation}
where $x_{F}=M_{1}/T_{F}$ represents the freeze-out temperature,
that can be determined iteratively from the equation
\begin{equation}
x_{F}=\log\left(\sqrt{\frac{45}{8}}\frac{M_{1}M_{pl}\left\langle \sigma(N_{1}N_{1})\upsilon_{r}\right\rangle }{\pi^{3}\sqrt{g_{\ast}x_{F}}}\right).
\end{equation}
Here, $\upsilon_{r}$ denotes the relative velocity, $M_{pl}$ is
the Plank mass, $g_{\ast}$ counts the effective degrees of freedom
of the relativistic fields in equilibrium, and
\begin{align}
\left\langle \sigma(N_{1}N_{1})\upsilon_{r}\right\rangle & =\sum_{X}\left\langle \sigma(N_{1}N_{1}\rightarrow X)\upsilon_{r}\right\rangle =\sum_{X}\int_{4M_{1}^{2}}^{\infty}ds~\sigma_{N_{1}N_{1}\rightarrow X}(s)\frac{\left(s-4M_{1}^{2}\right)}{8TM_{1}^{4}K_{2}^{2}\left(\frac{M_{1}}{T}\right)}\sqrt{s}K_{1}\left(\frac{\sqrt{s}}{T}\right),
\end{align}
represents the total thermally averaged annihilation cross section.
We have $s$ is the Mandelstam variable, $K_{1,2}$ are the modified
Bessel functions and $\sigma_{N_{1}N_{1}\rightarrow X}(s)$ is the
partial annihilation cross due to the channel $N_{1}N_{1}\rightarrow X$,
at the CM energy $\sqrt{s}$, where the possible channels are shown
in Fig.~\ref{DM-ahn}.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=1.0\textwidth]{Feyn}
\par\end{centering}
\caption{Different DM annihilation channels in the SI-scotogenic model. The cross section in (\ref{eq:XS1}) is estimated based on the diagrams (a) and (b), while diagram (c) yields the cross section of the SM channels in (\ref{csSM}). The Higgs/dilaton annihilation cross section in (\ref{HHDD}) is computed using the diagrams (d), (e) and (f).}
\label{DM-ahn}
\end{figure}
We provide here the exact formulas for the partial annihilation cross
sections. For the channels $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$,
we have~\cite{Lalili,Ahriche:2020pwq}
\begin{align}
\sigma\upsilon_{r} & =\frac{1}{32\pi s^{2}}\sum_{\alpha,\beta}\sum_{X,Y}\left|\eta_{X}\eta_{Y}\,g_{1\alpha}g_{1\beta}^{*}\right|^{2}\,\lambda(s,m_{\alpha}^{2},m_{\beta}^{2})\left\{ \mathcal{R}(Q_{X},Q_{Y},T_{+},T_{-},B)+\right.\nonumber \\
& \left.-\frac{M_{1}^{2}(s-m_{\alpha}^{2}-m_{\beta}^{2})}{B(Q_{X}+Q_{Y})}\log\left|\frac{(Q_{X}+B)(Q_{Y}+B)}{(Q_{X}-B)(Q_{Y}-B)}\right|\right\},\label{eq:XS1}\\
Q_{X} & =\frac{1}{2}(s+2m_{X}^{2}-2M_{1}^{2}-m_{\alpha}^{2}-m_{\beta}^{2}),\,T_{\pm}=\frac{1}{2}(s\pm m_{\alpha}^{2}\mp m_{\beta}^{2}),\nonumber \\
B & =\frac{1}{2s}\lambda(s,m_{\alpha}^{2},m_{\beta}^{2})\lambda(s,M_{1}^{2},M_{1}^{2}),\,\lambda(x,y,z)=\sqrt{(x-y-z)^{2}-4yz},\nonumber\\
\mathcal{R}(r,t,w,q,\eta) & =2-\frac{(t-w)(t-q)}{\eta\,(t-r)}\log\left|\frac{t+\eta}{t-\eta}\right|-\frac{(r-w)(r-q)}{\eta\,(r-t)}\log\left|\frac{r+\eta}{r-\eta}\right|.\nonumber
\end{align}
For the channel $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta}$,
we have: $m_{\alpha}=m_{\ell_{\alpha}},\,m_{\beta}=m_{\ell_{\beta}}$,
$\eta_{S^{\pm}}=i$ and $\{X,Y\}=\{S^{\pm},S^{\pm}\}$. While, for
the channel $N_{1}N_{1}\rightarrow\nu_{\alpha}\bar{\nu}_{\beta}$,
we have: $m_{\alpha}=m_{\beta}=0,\,\{X,Y\}=\{S^{0},S^{0}\},\{S^{0},A^{0}\},\{A^{0},A^{0}\}$
and $\eta_{S^{0}}=\frac{i}{_{\sqrt{2}}},\,\eta_{A^{0}}=\frac{1}{_{\sqrt{2}}}$.
For the channels that are mediated by the Higgs/dilaton ($N_{1}N_{1}\rightarrow X\bar{X},\,X=W,Z,b,t$),
one can write the cross section as~\cite{Ahriche:2016cio}
\begin{equation}
\sigma(N_{1}N_{1}\rightarrow X\bar{X})\upsilon_{r}=8\sqrt{s}s_{\alpha}^{2}c_{\alpha}^{2}y_{1}^{2}\left\vert \frac{1}{s-m_{H}^{2}+im_{H}\Gamma_{H}}-\frac{1}{s-m_{D}^{2}+im_{D}\Gamma_{D}}\right\vert ^{2}\Gamma_{H\rightarrow X\bar{X}}(m_{H}\rightarrow\sqrt{s}),\label{csSM}
\end{equation}
where $\Gamma_{H\rightarrow X\bar{X}}(m_{H}\rightarrow\sqrt{s})$\ is
the total SM Higgs decay width with the Higgs mass replaced by $m_{H}\rightarrow\sqrt{s}$.
Here, $\Gamma_{H,D}$ are the Higgs and dilaton total decay widths,
respectively. For simplicity, we take in our numerical
scan $\Gamma_{H}\simeq\Gamma_{h}^{SM}$ and $\Gamma_{D}\simeq0$.
For the annihilation into Higgs/dilaton, we have the exact cross section
formula at CM energy $\sqrt{s}$ given by
\begin{align}
\sigma(N_{1}N_{1}\rightarrow h_{i}h_{j})\upsilon_{r} & =\frac{1}{8\pi s}\left\{ R+\frac{W}{2B}\log\left|\frac{A-B}{A+B}\right|+\frac{Z}{A^{2}-B^{2}}\right\} ,\label{HHDD}
\end{align}
with $h_{i}=H,D$ and
\begin{align*}
R & =\frac{1}{4}y_{1}^{2}(s-M_{1}^{2})|Q|^{2},\,W=\frac{1+\delta_{ij}}{2}q_{i}q_{j}y_{1}^{3}(s-M_{1}^{2})M_{1}\left(\Re(Q)+\frac{\delta_{ij}q_{i}q_{j}y_{1}M_{1}}{(m_{h_{i}}^{2}+m_{h_{j}}^{2}-s)}\right),\\
Z & =\frac{1+\delta_{ij}}{4}q_{i}^{2}q_{j}^{2}y_{1}^{4}M_{1}^{2}(s-M_{1}^{2}),\,Q=\frac{-s_{h}\lambda_{Hij}}{s-m_{H}^{2}+im_{H}\Gamma_{H}}+\frac{c_{h}\lambda_{ijD}}{s-m_{D}^{2}+im_{D}\Gamma_{D}},\\
A & =\frac{1}{2}(s-m_{h_{i}}^{2}-m_{h_{j}}^{2}),\,B=\frac{1}{2s}\lambda(s,M_{1}^{2},M_{1}^{2})\lambda(s,m_{h_{i}}^{2},m_{h_{j}}^{2}),
\end{align*}
where $\lambda_{ijk}$ are the Higgs/dilaton triple couplings and
\{$q_{H}=-s_{\alpha},\,q_{D}=c_{\alpha}$\}.
\vspace{1cm}
\textbf{DM Direct Detection}: in the SI $\nu-$DM models, the sensitivity
to the direct-detection experiments could be due to the interactions
between the DM and quarks via the mediation of the Higgs/dilaton.
The effective low-energy Lagrangian of this interaction can be written
as
\begin{equation}
\mathcal{L}_{N_{1}-q}^{(eff)}=-\frac{1}{2}s_{\alpha}c_{\alpha}y_{q}y_{1}\left[\frac{1}{m_{H}^{2}}-\frac{1}{m_{D}^{2}}\right]\,\bar{q}q\,\overline{N}_{\text{1}}^{c}N_{1},
\end{equation}
where $y_{q}$ is the light quark Yukawa coupling. Consequently,
the effective nucleon-DM interaction can be written as
\begin{equation}
\mathcal{L}_{N_{1}-\mathcal{N}}^{(eff)}=\frac{s_{\alpha}c_{\alpha}(m_{\mathcal{N}}-\frac{7}{9}m_{\mathcal{B}})M_{1}}{x\,\upsilon}\left[\frac{1}{m_{h}^{2}}-\frac{1}{m_{D}^{2}}\right]\mathcal{\bar{N}N}\overline{N}_{\text{1}}^{c}N_{\text{1}},
\end{equation}
where $m_{\mathcal{N}}$ is the nucleon mass and $m_{\mathcal{B}}$
the baryon mass in the chiral limit~\cite{He:2008qm}. This leads
to the nucleon-DM spin-independent elastic cross section in the chiral
limit~\cite{Ahriche:2016cio}
\begin{equation}
\sigma_{\det}=\frac{c_{\alpha}^{2}s_{\alpha}^{2}m_{\mathcal{N}}^{2}(m_{\mathcal{N}}-\frac{7}{9}m_{\mathcal{B}})^{2}M_{1}^{4}}{\pi\upsilon^{2}x^{2}(M_{1}+m_{\mathcal{B}})^{2}}\left[\frac{1}{m_{h}^{2}}-\frac{1}{m_{D}^{2}}\right]^{2}.\label{eq:DD}
\end{equation}
In what follows, we will consider the recent upper bound reported
by Xenon 1T experiment~\cite{XENON:2018voc}. In addition, we compare
our results with the projected sensitivities for the future proposed
experiments: PandaX-4t~\cite{PandaX:2018wtu}, LUX-Zeplin~\cite{LUX-ZEPLIN:2018poe},
XENONnT with 20 ton-yr exposure~\cite{XENON:2020kmp} and DARWIN~\cite{DARWIN:2016hyl}.
\section{Numerical Results and Discussion\label{sec:NR}}
This model is subject to many theoretical and experimental constraints.
Here, we will be interested in all phenomenological and experimental
aspects of the Higgs/dilaton interactions and their correlation with
DM relic density and DD, which may imply some constraints on the neutrino
mass and the LFV processes. Then, in our numerical scan, we consider
the following constraints: perturbativity, perturbative unitarity,
the different Higgs decay channels (di-photon, invisible and undetermined),
the electroweak precision tests, LEP negative searches for light scalar
(in our case, it applies to the cross section of $e^{-}e^{+}\rightarrow ZD$),
and the Higgs signal strength at the LHC $\mu_{{\rm tot}}\geq0.89$
at 95\%~CL~\cite{ATLAS:2016neq}. The latter constraint (Higgs signal strength at the LHC~\cite{ATLAS:2016neq})
can be expressed as $\mu_{{\rm tot}}=c_{\alpha}^{2}\times(1-\mathcal{B}_{BSM})\geq0.89$,
with $c_{\alpha}$ being the Higgs/dilaton mixing angle and $\mathcal{B}_{BSM}$
is the branching ratio of any non-SM Higgs decay channel (could be
the invisible channel like $H\rightarrow N_{1}N_{1}$ or/and an undermined
channel like $H\rightarrow DD$), which are constrained as $\mathcal{B}_{BSM}\leq0.47$~\cite{Aad:2019mbh}.
For a given value of $\mathcal{B}_{BSM}$, the condition $\mu_{{\rm tot}}\geq0.89$
can be translated at tree-level into a direct constraint on the singlet
VEV value as: $x>700.36\,\mathrm{GeV},\,600\,\mathrm{GeV},\,480\,\mathrm{GeV},\,335\,\mathrm{GeV},\,233\,\mathrm{GeV}$
for $\mathcal{B}_{BSM}=0,\,0.038,\,0.11,\,0.27,\,0.469$, respectively.
However, one has to mention that in the SI models $\mathcal{B}_{BSM}$
is likely to take values much smaller than the experimental bound
due to the facts that the Higgs invisible decay branching ratio is
$s_{\alpha}^{2}$ proportional, and the triple coupling $\lambda_{HDD}$
is suppressed since it has only a pure one-loop contribution. Here,
we consider the input parameters ranges
\begin{equation}
\begin{array}{c}
113.5\,\mathrm{GeV}<m_{S^{\pm}}<1\,\mathrm{TeV},\,10\,\mathrm{GeV}<m_{S^{0},A^{0}}<1\,\mathrm{TeV},\,6\,\mathrm{GeV}<M_{1}<1\,\mathrm{TeV},\\
1\,\mathrm{GeV}<m_{D}<100\,\mathrm{GeV},\,y_{i}^{2},\,\left|\lambda_{i}\right|\,\left|g_{i\alpha}\right|^{2}<4\pi,
\end{array}\label{eq:free}
\end{equation}
where $\lambda_{i}$ denotes all the couplings in (\ref{V:Ma}). In
the absence of BSM decay channels for the Higgs, the constraints $\mu_{{\rm tot}}\geq0.89$
implies that the singlet VEV is larger than $x>700.36\,\mathrm{GeV}$,
i.e., $x^{2}/(\upsilon^{2}+x^{2})\geq0.89$. However, by considering
the radiative correction in the mixing estimation, the singlet VEV
could smaller as $x\geq\,\mathrm{GeV}$. This is understood from
the fact that the radiative corrections could be large so that the
dilaton mass, which is a pure radiative effect, could reach values
as $m_{D}\sim100\,\,\mathrm{GeV}$, and the ratio $\Delta_{s_{\alpha}}$
in (\ref{eq:DS}) lies between $-1500\%<\Delta_{s_{\alpha}}<1500\%$.
This means that the scalar mixing could be dominated by the radiative corrections, and hence in our analysis, we will consider the one-loop value of the mixing $s_{\alpha}$.
The existence of the Higgs/dilaton coupling to the Majorana DM candidate,
$N_{1}$ in the SI models, leads to an addition contribution to the
DM annihilation cross section via the channels $N_{1}N_{1}\rightarrow h_{i}h_{k},\,VV,q\bar{q}$,
with $h_{i}$ denotes the Higgs/dilaton and $V$ denotes the gauge
bosons. This new contribution could be either small or dominant depending
on the parameters $m_{D},~M_{1}$ and the mixing $\sin\alpha$. Therefore,
such a non-negligible new contribution makes the contribution of the
channels $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$
smaller with respect to the non SI extended models. In other words,
larger (smaller) contribution of the Higgs/dilaton mediated DM annihilation
channels leads to smaller (larger) contribution of the channels $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$,
and therefore smaller (larger) values of the new Yukawa couplings
$g_{i\alpha}$, i.e., more precisely the combination $\sum_{\alpha,\beta}|g_{1\alpha}g_{1\beta}^{*}|^{2}$.
Then, it would useful to define the ratio
\begin{equation}
\mathcal{R}_{f}=\frac{\left\langle \sigma(N_{1}N_{1}\rightarrow f)\upsilon_{r}\right\rangle }{\left\langle \sigma(N_{1}N_{1})\upsilon_{r}\right\rangle },\label{eq:RL}
\end{equation}
that represents the contribution of the channel $N_{1}N_{1}\rightarrow f$
to the total thermally averaged cross section at the freeze-out. Clearly,
we have $\sum_{X}\mathcal{R}_{f}=1$, however, we will be interested
in the two ratios: $\mathcal{R}_{LL}=\sum_{\alpha.\beta}(\mathcal{R}_{\ell_{\alpha}\ell_{\beta}}+\mathcal{R}_{\nu_{\alpha}\bar{\nu}_{\beta}})$
and $\mathcal{R}_{hh}=\sum_{h_{i,k}=H,D}\mathcal{R}_{h_{i}h_{k}}$ that
represent the relative contributions of the channels $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$
and $N_{1}N_{1}\rightarrow h_{i}h_{k}$ to the annihilation cross
section, respectively. Clearly the ratio $\mathcal{R}_{LL}$ is proportional
to the combination $\sum_{\alpha,\beta}|g_{1\alpha}g_{1\beta}^{*}|^{2}$
in the limit of degenerate charged lepton masses. For a given DM mass
value $M_{1}$, let's call $\left\langle \sigma_{0}\upsilon_{r}\right\rangle =\left\langle \sigma(N_{1}N_{1})\upsilon_{r}\right\rangle $
the correct cross section value at the freeze-out that matches the
observed relic density (\ref{eq:omegah}). Then, the cross section
of all Higgs/dilaton mediated channels $\sum_{f}\left\langle \sigma(N_{1}N_{1}\rightarrow f)\upsilon_{r}\right\rangle $
($f=h_{i}h_{k},\,VV,q\bar{q}$) at the freeze-out should be smaller than
$\left\langle \sigma_{0}\upsilon_{r}\right\rangle $. Therefore, the
condition $\sum_{f}\left\langle \sigma(N_{1}N_{1}\rightarrow f)\upsilon_{r}\right\rangle \geq\left\langle \sigma_{0}\upsilon_{r}\right\rangle $
leads to a value for the relic density that is smaller than the measured
value and can not be compensated by the channels $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$.
This condition, together with the spin-independent DM DD cross section
(\ref{eq:DD}) would eliminate a significant part of the parameters
space of any SI model that addresses the neutrino mass and Majorana
DM together.
In order to investigate the impact of these constraints, we perform
a random scan over the input parameters ranges in (\ref{eq:free}),
while taking into account the theoretical and experimental
constraints mentioned earlier. Since the case where the DM annihilation does fully occur
via the channel $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$,
is phenomenologically identical the minimal scotogenic model~\cite{Ahriche:2017iar},
we will focus on the regions of the parameters space where the contributions
of the other channels $N_{1}N_{1}\rightarrow h_{i}h_{k},\,VV,q\bar{q}$
are not vanishing. We consider 4000 benchmark points (BPs) that fulfill
all the conditions mentioned above and shown in Fig.~\ref{SP}-top
and Fig.~\ref{SP}-bottom, the possible values of \{$m_{D}-s_{\alpha}^{2}$\}
for different values of the DM mass $M_{1}$, singlet VEV $x$, and
the ratios $\mathcal{R}_{LL}$ and $\mathcal{R}_{hh}$.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.49\textwidth]{SP11}
\includegraphics[width=0.49\textwidth]{SP22}\\
\includegraphics[width=0.49\textwidth]{SP33}\includegraphics[width=0.49\textwidth]{SP44}\\
\par\end{centering}
\caption{The dilaton mass versus the Higgs/dilaton mixing, where the palette
shows the DM mass (up-right), the singlet VEV (up-left), the ratio
$\mathcal{R}_{LL}$ (bottom-right) and the ratio $\mathcal{R}_{hh}$
(bottom-left). The DM requirements of relic density and direct
detection bounds, in addition to the LEP (OPAL) bound on $e^{-}e^{+}\rightarrow Z^{*}D$
are already considered.}
\label{SP}
\end{figure}
One has to mention that for a significant part the BPs (not considered
in Fig.~\ref{SP}) that pass all the requirements, the DM annihilation
is almost fully achieved via $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$.
From Fig.~\ref{SP}, one can learn many conclusions, for instance,
light dilaton ($m_{D}<25~\mathrm{GeV}$) and large mixing ($s_{\alpha}>0.1$)
correspond to light DM and large singlet VEV. In this region the DM
annihilation is dominated by light quarks, charged leptons and/or
neutrinos. Most of the BPs with large dilaton mass ($m_{D}>m_{H}/2$)
and small mixing ($s_{\alpha}<0.1$) correspond to light singlet VEV
($x<900~\mathrm{GeV}$) and heavy DM ($M_{1}\sim O(100~\mathrm{GeV})$). In this region,
the DM annihilation could be dominated either by $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$
or by the channel $N_{1}N_{1}\rightarrow S_{i}S_{k}$. For the region
of large mixing ($s_{\alpha}>0.3$) the DM annihilation channel $N_{1}N_{1}\rightarrow S_{i}S_{k}$
is always suppressed despite the dilaton mass, while the channel $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$
is dominated for part of the BPs. It should be stated that for the
region of small mixing ($s_{\alpha}<0.01$) and small dilaton mass
($m_{D}<m_{H}/2$), there exist possible viable BPs but with less
population. For these BPs, the DM annihilation is fully achieved via
the channel dominated either by $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$,
that matches phenomenologically the non-SI (minimal) scotogenic model.
For a complete picture, we show in Fig\@.~\ref{DM} the DM-nucleon spin-independent
cross section versus the DM mass, while the palette represents the
dilaton mass (left) and the scalar mixing (right).
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.49\textwidth]{sdetmd1}~\includegraphics[width=0.49\textwidth]{sdetsa1}
\par\end{centering}
\caption{The DM direct detection cross section versus the DM mass is presented, where the recent bound of Xenon 1T experiment is shown~\cite{XENON:2018voc}. the palette illustrates the dilaton mass (left) and the scalar mixing (right).
In addition, we show the projected sensitivities for the future experiments:
PandaX-4t~\cite{PandaX:2018wtu}, LUX-Zeplin~\cite{LUX-ZEPLIN:2018poe},
XENONnT with 20 ton-yr exposure~\cite{XENON:2020kmp} and DARWIN~\cite{DARWIN:2016hyl}.
The black line represents the neutrino floor, where any of the DM detection
via nucleon scattering is not possible.}
\label{DM}
\end{figure}
Clearly, from Fig.~\ref{DM} a significant part of the BPs
are within the reach of future DD experiments such as PandaX-4t~\cite{PandaX:2018wtu},
LUX-Zeplin~\cite{LUX-ZEPLIN:2018poe} and XENONnT~\cite{XENON:2020kmp},
while few of them (either with suppressed mixing $|s_{\alpha}|<10^{-3}$
or with light DM $M_{1}<10~\mathrm{GeV}$) could not before detected since
they are below the neutrino floor. For viable DM with mass values
larger than $M_{1}>100~\mathrm{GeV}$, the scalar mixing should be $|s_{\alpha}|\lesssim 0.03$
and the dilaton mass $m_{D}\gtrsim70\,\mathrm{GeV}$. However, for $20\,\mathrm{GeV}\leq M_{1}\leq60\,\mathrm{GeV}$,
the scalar mixing is almost maximal $|s_{\alpha}|\sim0.1$. The population
distribution around $M_{1}\sim m_{H}/2$ implies that the analysis
in this region requires a careful treatment.
The contribution of the channel $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$
to the total DM annihilation may take any value from almost 0\% to
almost 100\%, for the whole range of both dilation and DM masses.
This implies a relative freedom in the new Yukawa couplings range
($g_{i\alpha}$) contrary to the case of non SI version where the
new Yukawa coupling values are strictly dictated by the relic density.
In order to illustrate this point, we show in Fig.~\ref{lm5}-left
the correlation between the ratios $\mathcal{R}_{LL}$ and $\mathcal{R}_{hh}$
and the quartic coupling $\lambda_{5}$ for the 4000 BPs used previously
in Fig.~\ref{SP}. In order to show the explicit dependence of the
scalar coupling $\lambda_{5}$ on the ratio$\mathcal{R}_{LL}$, we
consider the BPs shown in Table.~\ref{Tab} among the 4000 points,
which all correspond to a maximal $\mathcal{R}_{LL}\sim1$.
\begin{table}[!h]
\begin{centering}
\begin{tabular}{|c|c|c|c|c|}
\hline
& BP1 & BP2 & BP3 & BP4\tabularnewline
\hline
\hline
$m_{D}$ (GeV) & 16.25 & 89.47 & 60.27 & 93.42\tabularnewline
\hline
$s_{\alpha}$ & 0.22685 & 0.01467 & -0.01905 & -0.07311\tabularnewline
\hline
x (GeV) & 856.34 & 393.7 & 1115.3 & 1657.4\tabularnewline
\hline
$m_{S^{\pm}}$ (GeV) & 257.6 & 308.12 & 445.60 & 667.25\tabularnewline
\hline
$\overline{m}$ (GeV) & 250.6 & 409.37 & 428.21 & 750.46\tabularnewline
\hline
$M_{1}$ (GeV) & 6.31 & 81.94 & 175.20 & 467.81\tabularnewline
\hline
$M_{2}$ (GeV) & 6.67 & 88.69 & 186.59 & 532.69\tabularnewline
\hline
$M_{3}$ (GeV) & 7.34 & 96.14 & 220.03 & 602.23\tabularnewline
\hline
$x_{f}$ & 21.94 & 24.27 & 24.89 & 25.89\tabularnewline
\hline
$\sum_{i,k}|g_{1,i}g_{1k}^{*}|^{2}$ & $2.48\times10^{-5}$ & $3.51\times10^{-3}$ & $1.31\times10^{-2}$ & $7.02\times10^{-2}$\tabularnewline
\hline
$\mathcal{R}_{LL}$ & 0.9918 & 0.98063 & 0.96442 & 0.95498\tabularnewline
\hline
$\mathcal{R}_{hh}$ & $1.21\times10^{-33}$ & 0.01857 & 0.0355 & 0.0450\tabularnewline
\hline
$\lambda_{5}$ & $5.6196\times10^{-7}$ & $1.1265\times10^{-8}$ & $2.7543\times10^{-9}$ & $-1.8715\times10^{-9}$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{The benchmark points used in Fig.~\ref{lm5}-bottom. Here, we have
$\overline{m}^{2}=\frac{1}{2}(m_{S^{0}}^{2}+m_{A^{0}}^{2})$, $x_{f}=M_{1}/T_{f}$
is freeze-out parameter, and the couplings combination $\sum_{i,k}|g_{1,i}g_{1k}^{*}|^{2}$
is proportional to the cross section of $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$ in the limit of massless charged leptons.}
\label{Tab}
\end{table}
Next, we assume that one can make the ratio $\mathcal{R}_{LL}$ smaller
by re-scaling the new Yukawa couplings $g_{i\alpha}$ to smaller values,
while keeping the inert parameters ($M_{i},\,x,~\omega_{2},~\lambda_{3}$
and $\lambda_{4}$) constant. In this case, both neutrino mass matrix
elements in (\ref{eq:MaNU}) and the DM relic density can be kept
in agreement with the data, only by allowing $\lambda_{5}$ to have
larger values, by and tuning the radiative effects (dilaton mass and
scalar mixing), respectively. Therefore, the dependence of the ratio
$\mathcal{R}_{LL}$ on $\lambda_{5}$ is shown in Fig.~\ref{lm5}-right
where the palette shows the couplings combination $\sum_{i,k}|g_{1,i}g_{1k}^{*}|^{2}$
that is proportional to the cross section of $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$
in the massless charged leptons limit.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.49\textwidth]{lm55}~\includegraphics[width=0.49\textwidth]{lm5BP}
\par\end{centering}
\caption{Left: the ratio $\mathcal{R}_{LL}$ versus the scalar coupling $\lambda_{5}$
for the 4000 BPs used previously in Fig.~\ref{SP}, where the palette
shows the ratio $\mathcal{R}_{hh}$. Right: the dependence of the
ratio $\mathcal{R}_{LL}$ on the scalar coupling $\lambda_{5}$ for
the BPs shown in Table.~\ref{Tab}, where both DM relic density and
neutrino masses are kept in agreement with the data. The palette shows
the couplings combination $\sum_{i,k}|g_{1,i}g_{1k}^{*}|^{2}$ that
is proportional to the cross section of $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$
in the limit of massless charged leptons.}
\label{lm5}
\end{figure}
For the 4000 BPs considered in Fig.~\ref{lm5}-left, the
scalar coupling can be relaxed up to $\lambda_{5}\sim10^{-5}$ contrary
to the non-SI case where $\lambda_{5}\sim10^{-10}.$ From Fig.~\ref{lm5}-right,
one notices that by tuning the radiative effects (dilaton mass and
scalar mixing), one can push the scalar coupling up to larger values
as $\lambda_{5}\sim1$. Consequently, the new Yukawa couplings can
be much small as shown by the palette in Fig.~\ref{lm5}-right, i.e.,
$\lambda_{5}\sim1\Longrightarrow\sum_{i,k}|g_{1,i}g_{1k}^{*}|^{2}\leq10^{-4}$.
This conclusion can be extrapolated into other models like the SI-KNT
model~\cite{Ahriche:2015loa}, however, getting larger values for
$\lambda_{5}\sim0.1-1$ in any SI model is not possible except for
the case where the model new couplings and masses can be tuned without
being in conflict with the above mentioned theoretical and experimental
constraints. \\
The SI-scotogenic model with Majorana DM candidate has almost the
same LHC predictions as the minimal scotogenic model~\cite{Ahriche:2017iar}.
For instance, a pair of charged scalars that are produced at the LHC ($pp\rightarrow S^{\pm}S^{\mp}$)
leads to many distinguishable signatures such as $4\textrm{jets}+\slashed{E}_{T},~1\ell+2\textrm{jets}+\slashed{E}_{T}$
and $2\ell+\slashed{E}_{T}$. Moreover, the process $pp\rightarrow S^{\pm}S^{0},\,S^{\pm}A^{0}$ may lead to the
signatures $1\ell+2\textrm{ jets}+\slashed{E}_{T}$ and $1\ell+\slashed{E}_{T}$. However, the process $pp\rightarrow S^{0}S^{0},\,A^{0}A^{0}$
results the final states $\slashed{E}_{T}+\textrm{ ISR}$ (mono-jet, mono-$\gamma$).
In order to distinguish the SI-scotogeneic model among its minimal version at colliders, one has to look for signatures that exit only in the
SI-version like $pp\to N_{1}N_{1}\gamma(H)$, $pp\to H\to DD\to4b,4\tau,2b2\tau,2b2\gamma$.
As mentioned previously, the new Yukawa couplings $g_{i\alpha}$
can take small values, especially in the case where ${\cal R}_{LL} << 1$, which basically makes the life-time of the charged
scalar ($S^{\pm}$) longer and then, displaced vertices can be seen in $pp\rightarrow S^{\pm}S^{\pm},\,S^{\pm}S^{0},\,S^{\pm}A^{0}$ due to $S^{\pm}\to\ell^{\pm}N_{1}$.
\section{Conclusion\label{sec:Conc}}
In this work, we have considered the SI-scotogenic model that addresses
neutrino oscillation data, DM nature and the EWSB, all together at
the weak scale. In this model, the EWSB is triggered by radiative
effects due to new fields and interactions in addition to the SM one.
The DM candidate could be the lightest among the CP-even, CP-odd scalars
or lightest Majorana singlet fermion, here we adopted a Majorana DM
candidate. After discussing the EWSB we imposed different theoretical
and experimental constraints like the vacuum stability, the electroweak
precision tests, LEP negative searches for light scalars, the Higgs
decays ($h\rightarrow\gamma\gamma$, invisible, undertermined), DM
relic density and DM DD experiments. The DM relic density was estimated
following~\cite{Srednicki:1988ce}, where the cross section of different
annihilation channels was estimated exactly, including those of $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$.
In the non SI (minimal scotogenic) version of the model, the DM annihilation
occurs via the channel channel $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$,
which dictates the values of the new Yukawa couplings $g_{i\alpha}$
that relatively large. In addition, the neutrino mass smallness can
be achieved by imposing the a mass degeneracy between the CP-even
and the CP-odd scalars, which means a suppressed value for the coupling
$\lambda_{5}=(m_{S^{0}}^{2}-m_{A^{0}}^{2})/\upsilon^{2}\sim10^{-10}.$
Due to non negligible coupling for DM with the Higgs/dilaton, other
DM annihilation channels such as $N_{1}N_{1}\rightarrow h_{i}h_{k},\,VV,q\bar{q}$
are possible, and therefore the new Yukawa couplings $g_{i\alpha}$
could take smaller values with respect to the non-SI case. This means
that Majorana DM is possible without a mass degeneracy between the
CP-even and the CP-odd scalars.
By considering all the previous listed theoretical constraints, and
focusing on benchmark points (BPs) that are different from non SI
version, i.e., BPs with non negligible contributions of the channels
$N_{1}N_{1}\rightarrow h_{i}h_{k},\,VV,q\bar{q}$ to the total DM
annihilation, we performed a numerical scan over the parameters space.
We have found that the DM annihilation could be dominated by light
quarks, charged leptons and/or neutrinos channels for light dilaton
($m_{D}<25~\mathrm{GeV}$), large mixing ($s_{\alpha}>0.1$), light DM ($M_{1}<10\,\mathrm{GeV}$)
and large singlet VEV ($x>2\,\mathrm{TeV}$). Another interesting region for
large dilaton mass ($m_{D}>m_{H}/2$), small mixing ($s_{\alpha}<0.1$),
small singlet VEV ($x<900~\mathrm{GeV}$) and heavy DM ($M_{1}\sim O(100~\mathrm{GeV})$),
where the DM annihilation could be dominated either by $N_{1}N_{1}\rightarrow\ell_{\alpha}\ell_{\beta},\,\nu_{\alpha}\bar{\nu}_{\beta}$
or by the channel $N_{1}N_{1}\rightarrow HH,HD,DD$. We found also
that the values of the scalar quartic coupling can be relaxed up to
$\lambda_{5}\sim10^{-5}$ contrary to the non-SI case where $\lambda_{5}\sim10^{-10}.$
By possible tuning of the radiative effects (in our model due to $M_{i},\,x,~\omega_{2},~\lambda_{3}$
and $\lambda_{4}$), the scalar coupling could be $\lambda_{5}\sim1$,
which implies very small values for the new Yukawa couplings $\sum_{i,k}|g_{1,i}g_{1k}^{*}|^{2}\leq10^{-4}$.
These regions are the most important regions where the SI extension
of the scotogenic model (as well other models like the SI-KNT), in
which the Majorana DM phenomenology is different than the non SI version.
These regions are with a great interest and their phenomenology studies at
colliders is ongoing in a future work~\cite{future}.
\subsection*{Acknowledgements}
This work is supported by the University of Sharjah via the following grants: \textit{Probing the Majorana Neutrino Nature in Radiative Neutrino Mass Models at Current and Future Colliders} (No. 1802143057-P), \textit{Extended Higgs Sectors at Colliders: Constraints \& Predictions} (No. 21021430100) and \textit{Hunting for New Physics
at Colliders} (No. 21021430107). The authors thank S. Nasri for his valuable comments.
|
1,116,691,499,395 | arxiv | \section{I. Introduction}
\hspace*{3.35ex}Hole-doped perovskite manganites \textit{Re}$_{1-x}$\textit{A}$_{x}$MnO$_{3}$ (\textit{Re}: rare earth metal, \textit{A}: alkaline earth metal) have rich electronic and magnetic phase diagrams that depend on the hole concentration and the average ionic radius of the \textit{A}-site (\textit{Re}/\textit{A}) ions \cite{Ref1}. Antiferromagnetic insulating, ferromagnetic metallic, as well as charge- and orbital-ordered phases emerge for specific combinations of \textit{A}-site ionic radii and hole concentrations (\textit{e.g.}, Nd$_{0.5}$Sr$_{0.5}$MnO$_{3}$ and Pr$_{0.5}$Sr$_{0.5}$MnO$_{3}$) \cite{Ref1, Ref2}. Because dramatic phase transitions are induced in hole-doped perovskite manganites by applying external magnetic fields \cite{Ref2}, they have been intensively studied in order to utilize their colossal magnetoresistance and tunneling magnetoresistance (TMR) effects in spintronic devices \cite{Ref3}. For TMR devices, two ferromagnetic layers with different coercivities are necessary for independent reversal of magnetization directions under an applied magnetic field. However, perovskite manganites have the disadvantage of having quite small coercivities ($\sim$ 50 Oe at 5 K in La$_{1-x}$Sr$_{x}$MnO$_{3}$ (LSMO)) \cite{Ref3, Ref4}. Nevertheless, using a ferromagnetic layer of Co-doped Pr$_{0.8}$Ca$_{0.2}$MnO$_3$ (PCMO), which has an enhanced coercivity, manganite-based spintronic devices with a TMR ratio greater than 120 \% were successfully fabricated recently \cite{Ref5}. \\
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=6cm]{figure1.eps}
\caption{(color online) Phase diagram of PCMCO (\textit{y} = 0--0.6) determined from the transport and SQUID measurements in Ref. 6. F. I. and P. I. denote the ferromagnetic insulating and paramagnetic insulating phases, respectively.}
\label{FIG1}
\end{center}
\end{figure}
\hspace*{3.35ex}Figure 1 shows the phase diagram of Co-doped PCMO, Pr$_{0.8}$Ca$_{0.2}$Mn$_{1-y}$Co$_y$O$_3$ (PCMCO). PCMCO is a ferromagnetic insulator in the Co concentration range of \textit{y} = 0--0.3 \cite{Ref5, Ref6}. Its Curie temperature (\textit{T}$_{\rm C}$) increases with Co substitution. While Co doping in PCMO is useful for improving the magnetic properties, the origin is not clear because of the lack of information about the electronic and magnetic states of Mn and Co. To solve this problem, we have employed x-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD) measurements, both of which offer element-selective capabilities. From the XAS measurements, the valence states of the two transition metal ions could be determined and from the XMCD measurements, the element-selective magnetic information could be obtained. In addition to these two spectroscopic methods, hard x-ray photoemission spectroscopy (HAXPES) measurements were performed to obtain information about the electronic structure near the Fermi level (\textit{E}$_{\rm F}$) that dominates the transport properties. \\
\section{II. Experimental}
\hspace*{3.35ex}Epitaxial PCMCO (\textit{y} = 0--0.3) thin films were grown onto Nb-doped SrTiO$_3$ (001) substrates by pulsed laser deposition. Detailed growth conditions of PCMCO films (grown on LSAT substrates) were described elsewhere. \cite{Ref6} In the present study, use of the Nb-SrTiO$_3$ substrates was necessary for preventing charging effects in the spectroscopic experiments. All PCMCO films were coherently grown and were therefore under compressive strain from the substrates, as shown by four-circle x-ray diffraction measurements (not shown). The \textit{T}$_{\rm C}$, determined from the \textit{M}-\textit{T} curves (not shown), ranged from 75 (\textit{y} = 0) to 120 K (\textit{y} = 0.3) \cite{Ref6, Ref7}. The saturation magnetizations were determined from the \textit{M}-\textit{H} curves.\cite{Ref8}\\
\hspace*{3.35ex}The XMCD measurements were performed at the undulator beamline BL-16A of the Photon Factory, using an XMCD system equipped with a vector-type superconducting magnet \cite{Ref9}. A magnetic field of \textit{B} = 1 T was applied parallel to the photon direction, and the angle between the sample surface and the magnetic field was set to 30$^\circ$. The photon helicity was reversed and the magnetic field direction was fixed for the XMCD measurements. The degree of circular polarization was $\pm$95$\pm$4 \% \cite{Ref10}. All measurements were performed at 20 K, which is well below the \textit{T}$_{\rm C}$ of PCMCO. The XAS spectra are defined as the average of the absorption spectra taken with positive and negative photon helicities.\\
\hspace*{3.35ex}The HAXPES measurements were carried out at the undulator beamline BL47XU of SPring-8. Synchrotron radiation from the undulator was monochromatized to 7.94 keV with a Si 111 double crystal monochromator and a Si 444 channel-cut monochromator, which reduced the energy bandwidth to 80 meV. All HAXPES spectra were recorded at room temperature using a Scienta R-4000 electron energy analyzer with a total energy resolution of 250 meV. The \textit{E}$_{\rm F}$ of the samples was referenced to that of a gold foil that was in electrical contact with the samples.\\
\section{III. Results and Discussion}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7cm]{figure2.eps}
\caption{(color online) XAS spectra of PCMCO (\textit{y} = 0--0.3). (a) Co 2\textit{p} XAS spectra. The XAS spectrum of Co$^{2+}$ for CoO is also shown as a reference. (b) Mn 2\textit{p} XAS spectra. The XAS spectra of Mn$^{3+}$ (LaMnO$_3$) and Mn$^{4+}$ (SrMnO$_3$) are also shown as references.}
\label{FIG2}
\end{center}
\end{figure}
\hspace*{3.35ex}Figure 2(a) shows the Co 2\textit{p} XAS spectra of PCMCO (\textit{y} = 0.1--0.3). For comparison, the XAS spectrum of CoO is also included \cite{Ref11}. Each Co 2\textit{p} XAS spectrum shows two features of the \textit{L}$_3$ (h$\nu$ = 775--785 eV) and \textit{L}$_2$ (h$\nu$ = 790--800 eV) edges. The Co \textit{L}$_3$ edge consists of three sharp peak structures located at 777, 778, and 779.5 eV. The \textit{L}$_3$ edge also has shoulder structures on the higher photon energy side ($\sim$782 eV). On the other hand, the \textit{L}$_2$ edge has a broad structure centered at 793.5 eV. Because the XAS spectral shapes strongly depend on the valence state, the valence of the Co ions can be evaluated in comparison with reference XAS spectra: The Co 2\textit{p} XAS spectra of PCMCO are quite similar to the spectrum of CoO, suggesting that the Co ions in PCMCO and CoO have an identical valence state, \textit{i.e.}, the divalent Co ions in CoO$_6$ octahedra have the high-spin configuration ($t_{2g}^5$, $e_{g}^2$) \cite{Ref11}. Note that the octahedrally-coordinated trivalent Co ions in LaCoO$_3$ show a completely different XAS spectrum (not shown) \cite{Ref11}. These results indicate that the Co ions in PCMCO are in the divalent state with a high-spin configuration in PCMCO, regardless of the Co concentration (\textit{y} = 0.1--0.3).\\
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7cm]{figure3.eps}
\caption{(color online)HAXPES spectra of PCMCO (\textit{y} = 0--0.3). (a) Valence-band spectra. (b) Mn 2\textit{p} core level spectra. (c) Plot of the core-level shifts as a function of the Co concentration.}
\label{FIG3}
\end{center}
\end{figure}
\hspace*{3.35ex}Because the divalent Co ions substitute for the Mn sites with valences between 3+ and 4+, holes should be doped into the remaining Mn atoms. To evaluate the valence of the Mn ions, the Mn 2\textit{p} XAS spectra were measured as shown in Fig. 2(b). For reference, the Mn 2\textit{p} XAS spectra of Mn$^{3+}$ (LaMnO$_3$) \cite{Ref11} and Mn$^{4+}$ (SrMnO$_3$) \cite{Ref11} are also shown in the same figure. For the parent PCMO compound, the nominal valence of the Mn ions is +3.2 and the XAS spectral shape is similar to that of LaMnO$_3$. The XAS spectrum of PCMCO (\textit{y} = 0.3) is somewhat different from that of LaMnO$_3$, especially in the region of the characteristic peak structure of the \textit{L}$_3$ edge. This peak is also observed in the XAS spectrum of SrMnO$_3$, suggesting that the valence of the Mn ion increases with Co substitution. In addition, the center of gravity of the Mn \textit{L}$_3$ peak is shifted toward higher photon energies with increasing Co concentration. Because the same shift has been reported in the hole-doped perovskite manganite LSMO \cite{Ref12}, the present result can be taken as evidence for hole doping of the Mn atoms induced by the Co substitution. \\
\hspace*{3.35ex}The Co substitution in PCMO also influences the valence band and core levels measured by HAXPES as shown in Fig. 3. Figure 3 (a) shows the valence-band spectra of PCMCO (\textit{y} = 0--0.3). The valence band mainly consists of two prominent O 2\textit{p}-derived structures centered at 6 and 3.5 eV below \textit{E}$_{\rm F}$. The deeper structure consists of Mn 3\textit{d}-O 2\textit{p} bonding states and the shallower one of O 2\textit{p} non-bonding states \cite{Ref13}. These states are shifted toward lower binding energies with Co substitution, suggesting that a downward chemical potential shift occurs due to the hole doping. Pr 4\textit{d}, Mn 3\textit{d} \textit{t}$_{2g}$, and Co 3\textit{d} \textit{t}$_{2g}$ states are expected to exist around the shoulder of the O 2\textit{p} non-bonding states \cite{Ref13} although they cannot be isolated in the spectra. The density of states (DOS) closest to \textit{E}$_{\rm F}$ does not reach \textit{E}$_{\rm F}$, in agreement with the insulating nature of PCMCO. With increasing Co concentration, the DOS near \textit{E}$_{\rm F}$ increases. When the hole doping on Mn atoms occurs, the Mn 3\textit{d} \textit{e}$_{g}$ DOS should be reduced because the doped holes primarily enter the Mn 3\textit{d} \textit{e}$_{g}$ band. Therefore, we conclude that the DOS is derived from both Mn 3\textit{d} \textit{e}$_{g}$ and Co 3\textit{d} \textit{e}$_{g}$ states and that they appear nearly at the same energies.\\
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7cm]{figure4.eps}
\caption{(color online) Proportions of Mn ions with different valences as a function of the Co concentration in PCMCO. The inset shows the nominal Mn valence as a function of Co substitution. }
\label{FIG4}
\end{center}
\end{figure}
\hspace*{3.35ex}The Mn 2\textit{p} core-level photoemission spectra show evolution of the spectral line shape with Co substitution. Figure 3(b) shows the Mn 2\textit{p} core-level spectra of PCMCO (\textit{y} = 0--0.3), the Mn 2\textit{p}$_{3/2}$ peak shows higher intensity on the lower binding energy side for lower Co concentrations (\textit{y} = 0 and 0.1) while for higher Co concentrations (\textit{y} = 0.2 and 0.3), Mn 2\textit{p}$_{3/2}$ peak shows higher intensity on the higher binding energy side. The chemical shifts of the Mn core levels also support the Mn hole doping scenario. In contrast to the Mn 2\textit{p}$_{3/2}$ peaks, the Mn 2\textit{p}$_{1/2}$ peak structure remained unchanged, but the peak position shifted toward higher binding energies with increasing Co concentration. This energy shift is opposite to that of the O 2\textit{p} states as shown in Fig. 3(a). To investigate this behavior, we have plotted the energy shifts of the other core levels as a function of Co concentration in Fig. 3(c). All core levels except for the Mn-derived states are shifted toward lower binding energies by about 0.2 eV. Similar core-level shifts have been reported in the hole doped PCMO \cite{Ref13}. The Mn 2\textit{p} and Mn 2s core levels show opposite energy shifts to the O, Pr, and Ca core levels. \\
\hspace*{3.35ex}The shift of a core level with varying chemical composition is given by,
\begin{eqnarray}
\Delta E &=& \Delta \mu +K\Delta Q +\Delta V_M-\Delta E_R
\end{eqnarray}
where $\Delta \mu$ is the change in the chemical potential, $\Delta Q$ is the change in the number of valence electrons on the atom considered, $\Delta V_M$ is the change in the Madelung potential, and $\Delta E_R$ is the change in the extra-atomic relaxation energy derived from changes in the screening of the core-hole potential by metallic conduction electrons and surrounding ions \cite{Ref14}. For the Pr, Ca and O atoms, $\Delta \mu$ dominates the core-level shifts because the valence of Pr, Ca and O ions is independent of Co doping ($\Delta Q$ = 0). $\Delta E_R$ is negligible because PCMCO is an insulator. A change of the Madelung potential ($\Delta V_M$) can also be excluded because it would cause shifts of the core levels of the O anion and the Pr and Ca cations in different directions. Meanwhile, the Mn core-level shifts are mainly influenced by a change in the number of valence electrons, which is called the chemical shift, although a change in the chemical potential is also included \cite{Ref15}. The Co substitution increases the Mn valence due to hole doping, which corresponds to a smaller number of electrons ($\Delta Q <$ 0). The direction of the core-level shift for the transition metal atoms is determined by a competition between the chemical shift and the chemical potential shift \cite{Ref16}. Such opposite core-level shifts between elements have been observed in many complex transition-metal oxides, including the high-\textit{T}$_{\rm C}$ cuprate La$_{2-x}$Sr$_{x}$CuO$_{4}$ \cite{Ref17}. The present core-level shifts are fully consistent with hole-doping at the Mn sites, justifying our basic picture.\\
\hspace*{3.35ex}Assuming that all Co ions in PCMCO (\textit{y} $\leq$ 0.3) are divalent, the valence of Mn ions can be estimated to be (3.2-2\textit{y})/(1-\textit{y}). The resultant Mn valence as a function of Co concentration is plotted in the inset of Fig 4. The nominal valence of Mn monotonically increases with increasing Co concentration. From the nominal Mn valence, the concentrations of Mn$^{3+}$ and Mn$^{4+}$ ions can be deduced and are plotted in Fig. 4. The ratio of Mn$^{4+}$ linearly increases and that of Mn$^{3+}$ linearly decreases as a function of Co concentration. These results are utilized to estimate the total magnetization and the orbital magnetic moment of each element using the XMCD sum rules \cite{Ref18}. \\
\hspace*{3.35ex}Figure 5(a) shows the Co 2\textit{p} XMCD spectra of PCMCO. The XMCD spectrum of La$_2$MnCoO$_6$ (LMCO) is also plotted as a reference \cite{Ref11}. All the XMCD spectra of PCMCO are normalized to the integrated XAS intensity as shown in Fig. 2(a). The spectral line shapes of PCMCO are quite similar to those of LMCO. In addition, the XMCD spectra of PCMCO (\textit{y} = 0.1--0.3) are similar to each other in terms of line shapes and intensities, indicating that the same degree of ferromagnetic Co$^{2+}$ states are present in PCMCO, regardless of the Co concentration. The stronger negative and weaker positive XMCD signals at the \textit{L}$_3$ and \textit{L}$_2$ edges represent a finite orbital magnetic moment parallel to the spin magnetic moment. A quantitative analysis is presented below.\\
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7cm]{figure5.eps}
\caption{(color online) XMCD spectra of PCMCO (\textit{y} = 0.0--0.3). (a) Co 2\textit{p} XMCD spectra. The XMCD spectrum of Co$^{2+}$ for La$_2$MnCoO$_6$ (LMCO) is also shown for reference. (b) Mn 2\textit{p} XMCD spectra. The XMCD spectra of Mn$^{3+}$ (LaMnO$_{3+\delta}$) and Mn$^{4+}$(LMCO) are also shown as references.}
\label{FIG5}
\end{center}
\end{figure}
\hspace*{3.35ex}Mn 2\textit{p} XMCD spectra were also recorded as shown in Fig. 5(b). The spectra have also been normalized to the integrated XAS intensity shown in Fig. 2(b). The reference XMCD spectra of Mn$^{3+}$ (LaMnO$_{3+\delta}$) and Mn$^{4+}$ (LMCO) are shown in the same figure. The XMCD signal is negative at the \textit{L}$_3$ edge and positive at the \textit{L}$_2$ edge, which indicates that the spin magnetic moment of the Mn ions is aligned parallel to that of the Co ions. This result indicates that ferromagnetic coupling exists between the Mn and Co ions. \\
\hspace*{3.35ex}In contrast to the Co 2\textit{p} XMCD spectra, the Mn 2\textit{p} XMCD spectral line shape changes systematically with Co concentration. The XMCD spectrum of the parent compound PCMO (with the nominal valence of Mn$^{3.2+}$) is similar to that of the oxygen-excess LMO (Mn$^{(3+2\delta )+})$ \cite{Ref19}. With increasing Co concentration, a kink structure appears at about 645 eV of the \textit{L}$_3$ edge in the XMCD spectra of PCMCO (\textit{y} = 0.1 and 0.2). It finally becomes a peak in the \textit{y} = 0.3 sample. From comparison with the reference XMCD spectrum of Mn$^{4+}$ (LMCO), we conclude that the peak comes from the ferromagnetic Mn$^{4+}$ ion. \\
\hspace*{3.35ex}For element-selective quantitative analysis of the ferromagnetic moment in PCMCO, the XMCD spin and orbital sum rules are applied to the Mn and Co 2\textit{p} XMCD spectra. The spin and orbital magnetic moments are obtained using the following equations,
\begin{eqnarray}
\nonumber m_{\rm spin}+7m_{T_{z}} &=& \frac{-2(\Delta I_{L3}-2\Delta I_{L2})}{I_{L3}+I_{L2}}n_{h}\mu_{\rm B},\\
m_{\rm orbital}&=&\frac{-4(\Delta I_{L3}+\Delta I_{L2})}{3(I_{L3}+I_{L2})}n_{h}\mu_{\rm B}
\end{eqnarray}
where $I_{Li}$ and $\Delta I_{Li}$ represent the integrals of the XAS and XMCD spectra in the \textit{L}$_i$ edge region, \textit{n}$_{h}$ is the number of holes per atom and $\mu _{\rm B}$ is the Bohr magneton. Here, $m_{T_{z}}$ denotes the magnetic dipole moment, which is negligible compared to \textit{m}$_{\rm spin}$ for ions in the cubic symmetry \cite{Ref20}. For XMCD sum rule analysis, the number of holes on each atom needs to be known. In this analysis, the number of holes on the Co atoms is assumed from the nominal valence: the Co$^{2+}$ (\textit{d}$^7$) ion has 3 holes in the whole Co concentration range because the Co 2\textit{p} XAS and XMCD spectra of Co$^{2+}$ are well reproduced by the cluster model calculation of Co$^{2+}$ ion without charge transfer from the oxygen ligands. The number of holes on the Mn atoms is assumed using the following equation with linear interpolation in order to take the charge transfer into account.\\
\begin{eqnarray}
\nonumber n_{h, \rm Mn 3\textit{d}}=(4+\Delta n_{\rm Mn 3\textit{d}})\times (4-z)+\\
(3+\Delta n^{'}_{\rm Mn 3\textit{d}})\times (3-z)
\end{eqnarray}
where the parameters are set to $\Delta$n$_{Mn 3d}$ = 0.5 for LaMnO$_3$ and $\Delta$n$^{'}_{Mn 3d}$ = 0.8 for SrMnO$_3$ \cite{Ref19} and \textit{z} is the nominal valence of the Mn ions, estimated from Fig. 4. The spin and orbital magnetic moments estimated in this way for Co and Mn are shown in Fig. 6. \\
\hspace*{3.35ex}Figure 6 shows the magnetic moments calculated using the XMCD sum rules as a function of Co concentration. The ferromagnetic moment of the Co ion is roughly constant with Co substitution. The spin and orbital magnetic moments of Co atoms (\textit{m}$_{\rm Co, spin}$ and \textit{m}$_{\rm Co, orbital}$) are about 1 and 0.25 $\mu _{\rm B}$/Co, respectively. The sum ($\sim$1.25 $\mu _{\rm B}$/Co) is smaller than the total magnetization obtained from SQUID measurements (\textit{M}$_{\rm total, SQUID}$ = 1.5$\sim$2.0 $\mu _{\rm B}$/unit cell) \cite{Ref8}, suggesting that the magnetization of Mn atoms increases with Co doping. In fact, the spin magnetic moment of the Mn atoms (\textit{m}$_{\rm Mn, spin}$) increases with Co doping. On the other hand, the orbital magnetic moment of Mn atoms (\textit{m}$_{\rm Mn, orbital}$) remains almost zero in the whole Co concentration range. A very small orbital magnetic moment has been reported for several perovskite Mn oxides, (\textit{e.g.} LSMO) \cite{Ref19}. The total magnetization calculated using the XMCD sum rules (\textit{M}$_{\rm total, XMCD}$) was obtained from the following equation:
\begin{eqnarray}
\nonumber M_{\rm total, XMCD}=(1-y)\times(m_{\rm Mn, spin}+m_{\rm Mn, orbital})+\\
y\times (m_{\rm Co, spin}+m_{\rm Co, orbital})
\end{eqnarray}
which yielded an \textit{M}$_{\rm total, XMCD}$ of about 1.5 $\mu _{\rm B}$/unit cell for the whole Co concentration range, consistent with \textit{M}$_{\rm total, SQUID}$. \\
\hspace*{3.35ex}Finally, we briefly discuss the origin of the interesting magnetic behavior of PCMCO. The parent compound, PCMO has a nominal Mn valence of 3.2+ with high spin configuration, suggesting a maximum spin magnetic moment of 3.8 $\mu _{\rm B}$/unit cell. The experimental magnetic moment obtained from the XMCD and SQUID measurements (1.5$\sim$2.0 $\mu _{\rm B}$/unit cell) are much smaller than the ideal value. This suggests that antiferromagnetic coupling, \textit{e.g.}, due to a superexchange interaction between neighboring Mn$^{3+}$ ions via oxygen ions affects the magnetic properties of PCMO but is weakened by hole doping. \\
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8cm]{figure6.eps}
\caption{(color online) Ferromagnetic moments in PCMCO as a function of Co concentration. The total magnetization deduced from XMCD (\textit{M}$_{\rm total, XMCD}$) has been obtained using the \textit{m}$_{\rm spin}$ and \textit{m}$_{\rm orbital}$ of Mn and Co atoms and the B-atom concentration shown in Fig. 4. The total magnetization deduced from SQUID measurements (\textit{M}$_{\rm total, SQUID}$) has been determined from the \textit{M}-\textit{H} curves \cite{Ref8}.}
\label{FIG6}
\end{center}
\end{figure}
\hspace*{3.35ex}Considering the phase diagram of bulk PCMO \cite{Ref21}, the ferromagnetic ground states appears only in a Ca concentration range from \textit{x} = 0.15 (Mn$^{3.15+}$) to 0.3 (Mn$^{3.3+}$). However, the Mn atoms have a nominal valence of 3.5+ and 3.7+ in PCMCO with \textit{y} = 0.2 and \textit{y} = 0.3, respectively, suggesting that Co ions play other roles besides hole doping, such as including a ferromagnetic superexchange interaction between Co$^{2+}$ and Mn$^{4+}$, as expected from Goodenough-Kanamori rule \cite{Ref22, Ref23}. Because of electrostatic interactions, Mn atoms neighboring the Co$^{2+}$ ions tend to be tetravalent, and these two ions are expected to be ferromagnetically coupled with each other in the whole Co concentration range. As a result, the magnetization of Co is almost independent of the Co concentration while the magnetization of Mn increases with Co concentration through a double-exchange interaction enhanced by the hole doping. The combination of ferromagnetic superexchange and double-exchange interactions aligns the spins of Co$^{2+}$ and Mn$^{3+/4+}$ ions, stabilizing the ferromagnetic ground state in this system and leading to an increase of the magnetization of Mn and \textit{T}$_{\rm C}$. \\
\hspace*{3.35ex}Another characteristic magnetic behavior of PCMCO is the enhanced coercivity compared to PCMO \cite{Ref6}. The substituted Co ions have a larger orbital magnetic moment than Mn, which induces the larger coercivity for the higher Co concentrations. \\
\section{IV. Conclusion}
\hspace*{3.35ex}In summary, we have investigated the electronic and magnetic properties of Co-doped PCMO using HAXPES, XAS, and XMCD. The HAXPES and XAS results reveal the valence states of the Co and Mn ions. The Co ions are in a high-spin divalent state in the whole Co concentration range, resulting in hole doping at the Mn sites. The XMCD results indicate that the Co and Mn spins are ferromagnetically coupled, which we attribute to the ferromagnetic superexchange interaction between the Co$^{2+}$ and Mn$^{4+}$ ions in addition to the double-exchange interaction between the Mn$^{3+}$ and Mn$^{4+}$ ions. The Co$^{2+}$ ions with large orbital magnetic moments are expected to induce the large coercivity in PCMCO. \\
\section{Acknowledgement}
This work was supported by a Grant-in-Aid for Scientific Research (S22224005) andResearch Activity Start-up (25887021) from the Japan Society for the Promotion of Science (JSPS) program and the Quantum Beam Technology Development Program from the Japan Science and Technology. This research is granted by JSPS through the ÅgFunding Program for World-Leading Innovative R\&D on Science and Technology (FIRST Program),Åh initiated by the Council for Science and Technology Policy (CSTP). K.Y. acknowledges the financial support from JSPS. The synchrotron radiation experiment at SPring-8 was done under the approvals of the Japan Synchrotron Radiation Research Institute (2011A1624). The work at KEK-PF was done under the approval of the Program Advisory Committee (proposals 2010S2-001 and 2012G667) at the Institute of Materials Structures Science, KEK.
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1,116,691,499,396 | arxiv | \section{#1}\setcounter{equation}{0}}
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\begin{document}
\begin{flushright}
\texttt{\today}
\end{flushright}
\begin{centering}
\vspace{2cm}
\textbf{\Large{
Flat-space Holography and Correlators of \\ Robinson-Trautman Stress tensor }}
\vspace{0.8cm}
{\large Reza Fareghbal, Isa Mohammadi }
\vspace{0.5cm}
\begin{minipage}{.9\textwidth}\small
\begin{center}
{\it Department of Physics,
Shahid Beheshti University,
G.C., Evin, Tehran 19839, Iran. }\\
\vspace{0.5cm}
{\tt r$\[email protected], [email protected]}
\\ $ \, $ \\
\end{center}
\end{minipage}
\begin{abstract}
We propose a quasi-local stress tensor for the four-dimensional asymptotically flat Robinson-Trautman geometries by taking the flat-space limit from the corresponding asymptotically AdS solutions. This stress tensor results in the correct charges of the generators of BMS symmetry if we define conformal infinity by an anisotropic scaling of the metric components. Using flat-space holography this stress tensor is related to expectation values of the stress tensor in a dual field theory called BMS-invariant field theory (BMSFT). We also calculate the two and three point functions of the proposed stress tensor.
\end{abstract}
\end{centering}
\newpage
\section{Introduction}
According to the proposal of \cite{Bagchi:2010zz, Bagchi:2012cy}, the holographic dual of asymptotically flat spacetimes in (d+1) dimensions are $d$-dimensional field theories which are invariant under BMS symmetry (BMSFT). On the gravity side, BMS symmetry is the asymptotic symmetry of the asymptotically flat spacetimes at null infinity \cite{BMS, Ashtekar:1996cd, Barnich:2006av, Barnich:4dBMS, aspects}. Similar to the AdS/CFT correspondence, the asymptotic symmetry of the gravity theory in the bulk is the same as the exact symmetry of the filed theory living on the boundary. BMS symmetry in three and four dimensions is infinite-dimensional. Hence, similar to the two-dimensional conformal field theories (CFT$_2$), one would expect some universal properties for the dual BMSFT$_2$ and BMSFT$_3$. These properties can be studied by holographic methods in the context of Flat/BMSFT correspondence.
Taking the flat-space limit (zero cosmological constant limit) from the asymptotically AdS spacetimes, written in the suitable coordinates, results in the asymptotically flat spacetimes. It is plausible to find some aspects of BMSFTs by starting from AdS/CFT correspondance and taking the flat space limit. Most of the previous works have been concentrated on BMSFT$_2$ (see \cite{Prohazka:2017equ,Prohazka:2017lqb} for a complete list of related works). However BMSFT$_3$ is also interesting due to its holographic connection with four-dimensional spacetimes. In this paper we focus on the latter case.
Our approach in this paper is similar to \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O}. We start from some asymptotically flat solutions and find the asymptotically AdS spacetimes whose flat-space limit leads to the original asymptotically flat metric. Then, we use the dictionary of AdS/CFT to find the quasi-local stress tensor of the corresponding asymptotically AdS spacetimes \cite{Brown.AdS}. We take the flat-space limit of the stress tensor components and propose the quasi-local stress tensor of the asymptotically flat spacetimes. Using stress tensor components and applying Brown and York's method \cite{Brown.qe}, we calculate the corresponding charges of symmetry generators. Our results are exactly the same as the charges derived by covariant phase space method \cite{Barnich:2001jy, Wald.L, Wald.Z}.
The stress tensor components are given in terms of the parameters of the gravity solution. Variation of these parameters with respect to the symmetry generators is used to derive variation of the stress tensor components. We can use them on the boundary side and impose invariance of the stress-tensor correlators under the action of the global part of the BMS symmetry to derive the universal forms of BMSFT$_3$ stress tensor one, two and three point functions. The final correlators are consistent with those of \cite{Bagchi:2016bcd}, where the results are obtained by using the highest-weight representation of the BMS$ _4 $ algebra.
We have used the above approach for BMSFT$_2$ in papers \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni, Fareghbal.A.O} and for the quasi-local stress tensor of the Kerr black hole in \cite{Fareghbal:2016hqr}. In this paper we apply our method for a class of four-dimensional asymptotically flat spacetimes known as Robinson-Trautman (RT) solutions \cite{Robinson:1960zzb} (see also \cite{RT} ). Schwarzschild black hole and non-rotating time-dependent solutions are in this category. RT solutions with the negative cosmological constant have been studied in the context of AdS/CFT correspondence \cite{deFreitas:2014lia}-\cite{Ciambelli:2017wou}. Although they are not invariant under the action of a generic BMS transformation, but it is not difficult to check that these types of solutions, with the zero cosmological constant, satisfy the BMS boundary conditions introduced in \cite{Barnich:4dBMS, aspects}. It is straightforward to write down the corresponding asymptotically AdS geometry of an asymptotically flat RT type solution. This property makes it easy to holographically study the asymptotically flat RT solution by starting from the AdS/CFT correspondence and taking the flat-space limit.
The quasi-local stress tensor of the asymptotically AdS RT spacetimes are given by the dictionary of AdS/CFT correspondence. Although, taking the flat-space limit from the metric components is straightforward but stress tensor components are either zero or singular after taking the naive flat-space limit. Hence, similar to \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O}, we first multiply appropriate powers of AdS radius and then take the flat-space limit. In order to show that our proposed stress tensor is correct, we shall use it to compute the conserved charges of symmetry generators. To do so, we use the Brown and York's method \cite{Brown.qe} which needs an integration over a surface. The well-known definition of conformal infinity for the four-dimensional asymptotically flat spacetimes yields a two-dimensional surface. Apparently, BMSFT$_3$ cannot live on these surfaces. To solve the problem we define spacetimes where BMSFT coulde live on it by using an anisotropic scaling of the asymptotically flat metric. Currently we do not know a clear justification of this proposal but it should be similar to the non-relativistic holography discussed in \cite{Horava:2009vy}. In our previous works \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O}, we have successfully used this idea to compute charges by using proposed flat-space stress tensor. Our observation shows that if we find an asymptotically flat spacetimes by taking flat-space limit from an asymptotically AdS spacetimes, the dual BMSFT lives on a spacetime which has the same metric as the spacetime which CFT lives on it but the AdS radius should be replaced by proper powers of Newton's constant.
This paper is organized as follows: in section two, we briefly review Flat/BMSFT correspondence and Robinson-Trautman geometries. In section three we propose the quasi-local stress tensor of asymptotically flat Robinson-Trautman geometries and use it to compute the corresponding charges of the BMS symmetry. In section four we calculate the correlators of BMSFT$_4$ stress tensor. Last section is devoted to conclusions.
\section{Flat-space holography and Robinson-Trautman geometry}
Asymptotic symmetry of the asymptotically flat spacetimes at null infinity in three and four dimensions is infinite dimensional \cite{Barnich:2006av,Barnich:4dBMS, aspects}. These symmetries which are the extensions of Poincare symmetry are known as the BMS symmetries. Similar to the two dimensional conformal symmetry, the generators of BMS$_3$ and BMS$_4$ are not globally well-defined and divided into two parts: super-translation and super-rotation. They are the infinite extensions of translation and rotation of the Poincare symmetry.
In this paper we focus on Einstein gravity in four dimensions. The BMS$_4$ algebra is given by
\begin{eqnarray}\label{BMS4-algebra}
\nonumber [L_m,L_n]&=&(m-n)L_{m+n},\qquad [\bar L_m,\bar L_n]=(m-n)\bar L_{m+n},\qquad [L_m,\bar L_n]=0,\\
{[}L_l,M_{m,n}{]}&=&\({l+1\over2}-m\)M_{m+l,n},\qquad {[}\bar L_l,M_{m,n}{]}= \({l+1\over2}-n\)M_{m,n+l}.
\end{eqnarray}
where $l$, $m$ and $n$ are integers and Poincare sub-algebra is given by $\{L_0, L_{\pm 1}, \bar L_0, \bar L_{\pm 1}, M_{00}, M_{01}, M_{10},$ $M_{11}\}$. The first six generators are the generators of the Lorentz symmetry and the last four generate translation. Thus, $L_n$ and $\bar L_n$ for all integers $n$ are known as super-rotations and $M_{nm}$ are super-translations.
A useful representation for the generators of algebra \eqref{BMS4-algebra} is achieved if one starts from a generic solution of Einstein gravity, in an appropriate gauge (known as BMS gauge), and imposes particular boundary conditions \cite{Barnich:4dBMS, aspects}. In the coordinate $x^0=u$, $x^1=r$, $x^2=\theta$ and $x^3=\phi$, metric of the four-dimensional asymptotically flat spacetime is given by
\begin{equation}\label{4d metric}
ds^2=e^{2\beta}{V\over r} du^2-2e^{2\beta} du dr+g_{AB}\left(dx^A-U^A du\right)\left(dx^B-U^B du\right)
\end{equation}
where $A,B=2,3$ and $V,\beta,U^A, g_{AB}$ are functions of coordinates. The boundary conditions which result in \eqref{BMS4-algebra} are given by \cite{Barnich:4dBMS, aspects}
\begin{align}\label{boundary condition}
\nonumber g_{uu}&=-2r\partial_u\varphi-e^{-2\varphi}+ \bar\nabla\varphi +\mathcal{O}(r^{-1}),\qquad g_{ur}=-1+\mathcal{O}(r^{-2}),\\
g_{uA}&=\mathcal{O}(1),\qquad g_{rr}=g_{rA}=0,\qquad g_{AB}=r^2 \bar\gamma_{AB}+\mathcal{O}(r),
\end{align}
where $\varphi=\varphi(u,\theta,\phi)$, $\bar\gamma_{AB}dx^Adx^B= e^{2\varphi}\left(d\theta^2+\sin^2\theta d\phi^2\right)$ and $\bar\nabla$ denotes the Laplacian with respect to $\bar \gamma_{AB}$.
According to the AdS/CFT correspondence, the symmetry of the dual field theory is the same as the asymptotic symmetry of the gravity solutions. Thus, it makes sense to think of BMS symmetry as the symmetry of the dual theory of the asymptotically flat spacetimes. More precisely, algebra \eqref{BMS4-algebra} denotes the symmetries of the dual three-dimensional theory of the four-dimensional asymptotically flat spacetimes \eqref{4d metric}. The observation of \cite{Bagchi:2012cy} is that the whole BMS$_3$ and the global part of BMS$_4$ algebra \eqref{BMS4-algebra} are given by an ultra-relativistic contraction of the conformal symmetry in two and three-dimensions, respectively. Thus BMSFTs are ultra-relativistic field theories.
Asymptotically flat spacetimes given by \eqref{4d metric} are dual to some states in the dual BMSFT. In the gravity side, distinct solutions are characterized by functions $V,\beta,U^A$ and $g_{AB}$ satisfying some equations which are resulted in from the Einstein field equations. Although explicit forms of these functions are not known but it is possible to find their expansions in terms of the $r$ coordinate \cite{Barnich:4dBMS, aspects}. In this paper we restrict our analysis to a specific class of \eqref{4d metric} with $U^A=0$. More explicitly, we consider asymptotically flat spacetimes with line-element
\begin{equation}\label{RT metric}
ds^2=-qdu^2-2 du dr+2 r^2 e^{P(u,\zeta,\bar \zeta)}d\zeta d\bar\zeta,
\end{equation}
where $\zeta$ and $\bar \zeta$ are complex coordinates given by $\zeta=e^{i\phi}\cot {\theta\over 2}$ and $\bar \zeta=e^{-i\phi}\cot {\theta\over 2}$ and $q$ is given by
\begin{equation}
q=-{2 M(u)\over r}+r \partial_u P(u,\zeta,\bar\zeta)-e^{-P(u,\zeta,\bar \zeta)}\partial_\zeta\partial_{\bar\zeta} P.
\end{equation}
The functions $ M $ and $P$ must satisfy the following equation:
\begin{equation}
3M\partial_u P+2\partial_u M+e^{-P}\partial_\zeta\partial_{\bar\zeta}\left(e^{-p}\partial_\zeta\partial_{\bar\zeta}P\right)=0.
\end{equation}
The line-element \eqref{RT metric} is known as the Robinson-Trautman geometry (RT). The Schwarzschild black hole and asymptotically flat time dependent solutions can be put in this form.
Starting from \eqref{4d metric} , one can find the generic form of the asymptotic killing vectors $\xi^\mu$ which respect the boundary conditions \eqref{boundary condition} \cite{Barnich:4dBMS, aspects}:
\begin{align}\label{explicit form of AKV}
\nonumber \xi^u&= f(u,\zeta,\bar \zeta),\\
\nonumber \xi^\zeta&= F(\zeta)-{1\over r}e^{-P}\partial_{\bar\zeta}f,\\
\nonumber \xi^{\bar\zeta}&= \bar F(\bar\zeta)-{1\over r}e^{-P}\partial_{\zeta}f,\\
\xi^r&=-{1\over 2}r\left(f\partial_u P+k\right)+e^{-P}\partial_\zeta\partial_{\bar\zeta} f,
\end{align}
where
\begin{equation}
k=e^{-P}\left(\partial_\zeta\left(e^P F\right)+\partial_{\bar\zeta}\left(e^P \bar F\right)\right),
\end{equation}
\begin{equation}
f=e^{P\over2}\left(T(\zeta,\bar\zeta)+{1\over 2}\int du\,e^{-{P\over 2}}k\right).
\end{equation}
$F(\zeta)$, $\bar F(\bar\zeta)$ and $T(\zeta,\bar\zeta)$ are arbitrary functions of the complex coordinates $(\zeta,\bar\zeta)$. Generators $L_n$, $\bar L_n$ and $M_{nm}$ of BMS$_4$ algebra \eqref{BMS4-algebra} are given by
\begin{align}\label{basic definition}
\nonumber L_n&=\xi\left(F(\zeta)=-\zeta^{n+1},\, \bar F(\bar\zeta)=0,\, T(\zeta,\bar \zeta)=0\right),\\
\nonumber \bar L_n&=\xi\left(F(\zeta)=0,\, \bar F(\bar\zeta)=-\bar\zeta^{n+1},\, T(\zeta,\bar \zeta)=0\right),\\
M_{mn}&=\xi\left(F(\zeta)=0,\, \bar F(\bar\zeta)=0,\, T(\zeta,\bar \zeta)=\zeta^{m}\bar\zeta^n\right).
\end{align}
The RT spacetimes \eqref{RT metric} are characterized by two functions $M$ and $P$. Variation of these functions under the action of the asymptotic killing vectors \eqref{explicit form of AKV} reads
\begin{align}\label{variation of functions}
\nonumber \delta_\xi P&=0, \\
\delta_\xi M&=2f\partial_u M+3M\left(k+f\partial_u P\right)-e^{-P}\left[\partial_\zeta f\,\partial_{\bar\zeta}\left(e^{-P}\partial_\zeta\partial_{\bar\zeta}P\right)+\partial_{\bar\zeta} f\,\partial_{\zeta}\left(e^{-P}\partial_\zeta\partial_{\bar\zeta}P\right)\right].
\end{align}
It is clear from the variation of $M$ that the transformation \eqref{explicit form of AKV} do not respect the form of the RT solutions and after the BMS transformation, the final spacetime is in the form of \eqref{4d metric} with $U^A\neq 0$.
\section{Anisotropic conformal boundary and Robinson-Trautman stress tensor}
A possible way to study the flat-space holography is taking the flat-space limit from the AdS/CFT calculations.
In this method, it is necessary to find a corresponding asymptotically AdS metric for each of the asymptotically flat spacetimes which are related by taking the flat-space limit. An interesting property of the RT solutions is that for all of the possible values of the cosmological constant (including zero value), metric is given by \eqref{RT metric}. The functions $q$ and $P$ of the asymptotically flat solutions are given by taking the flat-space limit from their asymptotically AdS counterparts. For the asymptotically AdS solutions, $q$ is given by
\begin{equation}\label{q for AdS}
q=-{2 {M}(u)\over r}+r \partial_u P(u,\zeta,\bar\zeta)-e^{-P(u,\zeta,\bar \zeta)}\partial_\zeta\partial_{\bar\zeta} P+ {r^2\over\ell^2}
\end{equation}
where $\ell$ is the AdS radius.
In this section, starting from the asymptotically AdS RT solutions, we take appropriate flat-space limit from the components of the stress tensor to propose asymptotically flat RT stress tensors. Then we justify our proposal.
We consider Einstein gravity with negative cosmological constant in four-dimensions,
\begin{equation}\label{action0}
S_0={1\over 16 \pi G}\int dx^4\sqrt{-g}\left(R+{6\over\ell^2}\right).
\end{equation}
RT spacetimes, with $q$ given by \eqref{q for AdS} and appropriate functions $P$ satisfying the equations of motion, are solutions of this theory. The quasi-local stress tensor of these solutions is given by the Brown and York method \cite{Brown.AdS, Brown.qe},
\begin{equation}\label{def.BY}
T^{\mu\nu}={2\over\sqrt{-\gamma}}\dfrac{\delta S}{\delta \gamma_{\mu\nu}},
\end{equation}
where $\gamma_{\mu\nu}=g_{\mu\nu}-n_\mu n_\nu$ is the boundary metric and $n_\mu$ is the outward pointing normal vector
to the boundary. $S$ is given by
\begin{equation}
S=S_0-{1\over 8\pi G}\int_{\partial M} d^3x {\mathcal{K}}+S_{ct},
\end{equation}
where $\mathcal{K}$ is the extrinsic curvature of the boundary and $S_{ct}$ has the following form
\begin{equation}\label{CTerm}
S_{ct}=-{1\over 4 \pi G \ell}\int_{\partial M} d^3 x \sqrt{-\gamma}\left(1-{\ell^2\over 4}R_{(3)}\right),
\end{equation}
where $R_{(3)}$ is the Ricci scalar of $\gamma_{\mu\nu}$. $S_{ct}$ has to be added to remove the divergent terms at the boundary \cite{Brown.AdS}. Using \eqref{def.BY}-\eqref{CTerm} we find the stress-tensor components as
\begin{align}\label{stress tensor AdS}
\nonumber T_{uu}&={M\over4\pi G r \ell },\\
\nonumber T_{u\zeta}&=T_{\zeta u}=-{\ell\over16\pi G r}\partial_{\zeta}\left(e^{-P}\partial_\zeta\partial_{\bar\zeta}P\right),\\
\nonumber T_{u\bar\zeta}&=T_{\bar\zeta u}=-{\ell\over16\pi G r}\partial_{\bar\zeta}\left(e^{-P}\partial_\zeta\partial_{\bar\zeta}P\right),\\
\nonumber T_{\zeta\zeta}&=-{\ell^3 e^{P}\over16\pi G r}\partial_{\zeta}\left(e^{-P}\partial_u\partial_{\zeta}P\right),\\
\nonumber T_{\bar\zeta\bar\zeta}&=-{\ell^3 e^{P}\over16\pi G r}\partial_{\bar\zeta}\left(e^{-P}\partial_u\partial_{\bar\zeta}P\right),\\
T_{\zeta\bar\zeta}&=T_{\bar\zeta\zeta}={\ell Me^{P}\over 8\pi G r}
\end{align}
We would like to take the flat-space limit from these components and propose a stress tensor for the asymptotically flat RT solution. The flat-space limit is given by $G/\ell^2\to 0$ while keeping $G$ fixed\footnote{$G/\ell^2$ is dimensionless in four-dimensions. }. It is clear from \eqref{stress tensor AdS} that in this limit the stress tensor components are either zero or singular. Thus the direct limit is not well-defined. However, it is possible to define the stress tensors of the asymptotically flat cases by taking limit from the multiplication of corresponding components in the asymptotically AdS cases to some powers of $G/\ell^2\to 0$. This method has been used previously in \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O}, for several cases. We expect that this method results in correct components of RT stress tensor. However, in all of the previous examples \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O}, the stress tensor components were just a function of $\ell$ or $1/\ell$. $\ell^3$ factors in $T_{\zeta\zeta}$ and $T_{\bar\zeta\bar\zeta}$ are new. Apparently, the method of \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O} needs to be refined in this case. In this paper we do not intend to study this problem but we would like to restrict our calculation to the case that the method of \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O} is applicable. Thus we consider the RT solutions whose $P$ function is u-independent. Although this choice of $P$ reduces the generality of our results but it still consists a large class of asymptotically flat solutions including Schwarzschild black hole.
Another point is that the conservation of stress tensor of the asymptotically flat case requires that the components with one $u$ index, become non-symmetric. The fact that BMSFTs are not relativistic theories may justify this non-symmetric stress tensors. From the bulk point of view although BMS symmetry is asymptotic symmetry which includes four-dimensional Poincare sub-group but boundary BMSFT$_3$ does not contain three-dimensional Poincare symmetry.
Putting all together and setting $G=1$, we propose the following components for the asymptotically flat RT solutions with $u$-independent $P$:
\begin{align}\label{stress tensor flat}
\nonumber T_{uu}&={M\over4\pi r },\\
\nonumber T_{u\zeta}&=-{1\over16\pi r}\partial_{\zeta}\left(e^{-P}\partial_\zeta\partial_{\bar\zeta}P\right),\qquad \,\,\,\,\,T_{\zeta u}=0,\\
\nonumber T_{u\bar\zeta}&=-{1\over16\pi r}\partial_{\bar\zeta}\left(e^{-P}\partial_\zeta\partial_{\bar\zeta}P\right),\,\,\,\,\,\qquad T_{\bar\zeta u}=0,\\
\nonumber T_{\zeta\zeta}&=0,\qquad\qquad\qquad\qquad\qquad\qquad T_{\bar\zeta\bar\zeta}=0,\\
T_{\zeta\bar\zeta}&=T_{\bar\zeta\zeta}={ Me^{P}\over 8\pi r},
\end{align}
where $M=M(u)$ and $P=P(\zeta,\bar\zeta)$, using equation of motion, satisfy
\begin{equation}\label{EOM flat and u-independent p}
2\partial_u M+e^{-P}\partial_\zeta\partial_{\bar\zeta}\left(e^{-p}\partial_\zeta\partial_{\bar\zeta}P\right)=0.
\end{equation}
From the above equation we conclude that $M$ is a linear function of $u$ coordinate. In the rest of this section we try to justify our proposal \eqref{stress tensor flat}.
Now we want to show that the proposed RT stress tensor \eqref{stress tensor flat} results in the correct charges of the symmetry generators. To do so, we use the Flat/BMSFT correspondence which proposes that the stress tensor \eqref{stress tensor flat}, calculated in the gravity side, is also the stress tensor of BMSFT. $u$, $\zeta$ and $\bar\zeta$ are coordinates of the spacetime on which BMSFT lives and $r$ is just a constant in this view. The main challenging problem is to determine the spacetime metric on which BMSFT lives. For the asymptotically AdS RT metric, given by \eqref{RT metric} and \eqref{q for AdS}, the dual CFT lives on the spacetime whose metric is simply given by the standard definition of the conformal boundary. For the current case, metric of the boundary reads
\begin{equation}\label{metric of boundary AdS}
ds^2={r^2\over\ell^2}\left(-du^2+2\ell^2 e^{P} d\zeta d\bar\zeta\right).
\end{equation}
Definition of the conformal infinity for the asymptotically flat spacetimes leads to a two-dimensional metric which is not suitable for our analysis. The proposal to deal with this problem is to use anisotropic conformal infinity as the spacetime where BMSFT lives on it. This proposal has been succefully used in \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O} to compute the charges using flat-space stress tensor. The main idea goes back to \cite{Horava:2009vy} in which the authors have studied anisotropic conformal infinity in the context of holography for the non-relativistic theories. The fact that BMSFTs have ultra-relativistic conformal symmetry is a hint that the methods of relativistic theories does not work. Currently we do not know a well-established procedure to define the anisotropic conformal infinity in the context of the flat-space holography. However, in all of the previous works \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O} the geometry of spacetime on which the BMSFT lives, was the same as the parent CFT whose contraction resulted in BMSFT. The only difference is that the AdS radius $\ell$ must be replaced by an appropriate power of Newton's constant $G$ which has the same dimension as $\ell$. If we accept the lessons of \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O}, the dual BMSFT of RT solution lives on a spacetime with the following metric,
\begin{equation}\label{metric of boundary flat}
ds^2={r^2\over G}\left(-du^2+2 G e^{P} d\zeta d\bar\zeta\right).
\end{equation}
For $G=1$ and assuming $P$ as an $u$-independent function, we can use \eqref{stress tensor flat} and \eqref{metric of boundary flat} to calculate the symmetry charges. Before doing this calculation and using the fact that $M$ depends on $u$, it is not difficult to show that the conservation equation of stress tensor, $\nabla_\mu T^{\mu\nu}=0$, leads to \eqref{EOM flat and u-independent p} which is a result of equations of motion in the gravity side. This is a good check for the correctness of our assumptions.
To calculate the charges $Q_\xi$, associated to a symmetry generator $\xi$, we use the Brown and York formula \cite{Brown.qe},
\begin{equation}\label{def-of-charge}
Q_\xi=\int_\Sigma d\sigma \sqrt{\sigma}v^\mu\xi^\nu T_{\mu\nu},
\end{equation}
where $\Sigma$ is a $u$-constant surface in the anisotropic conformal boundary, $\sigma_{ab}$ is the metric of $\Sigma$ and $v^\mu$ is the unit timelike vector normal to $\Sigma$. Using \eqref{explicit form of AKV}, \eqref{stress tensor flat} and \eqref{metric of boundary flat} we find,
\begin{equation}\label{final charge}
Q_\xi=\frac{1}{16 \pi}\int d\zeta d\bar\zeta e^{P}\left[4fM-\left(F\partial_\zeta+\bar F\partial_{\bar\zeta}\right)\left(e^{-P}\partial_\zeta\partial_{\bar\zeta}P\right)\right].
\end{equation}
This is exactly the same as the results of the covariant phase space method in \cite{Barnich:2011mi}.
\section{Correlators of stress tensor}
We can use \eqref{stress tensor flat} and \eqref{variation of functions} to find the correlation functions of stress tensor. The procedure is similar to the CFT case. We assume that the correlators of BMSFT stress tensor are invariant under the action of the global part of the BMSFT symmetry. Using the asymptotic symmetry generators \eqref{explicit form of AKV} and the definition of BMS symmetry generators \eqref{basic definition}, the global part is given by $\{L_0, L_{\pm 1}, \bar L_0, \bar L_{\pm 1}, M_{00}, M_{01}, M_{10},$ $M_{11}\}$ for fixed and large values of $r$ coordinate.
According to \eqref{stress tensor flat}, the stress tensor components are given in terms of $M$ and $P$ functions. Thus, in order to vary correlators we need to know the variation of $M$ and $P$ under the action of the global part of the BMS symmetry. This is given by \eqref{variation of functions}. Since $\delta_\xi P=0$, all of the correlators consisting just $P$ functions are zero. This shows that all of the correlators which do not have at least one $T_{uu}$ or $T_{\zeta\bar\zeta}$ are zero. Moreover, $\delta_\xi P=0$ expresses that to study asymptotically flat spacetimes we can restrict ourselves to the class of solutions with particular $P$ functions. The simplest case is $P=0$. As a result, using \eqref{stress tensor flat} we see that all of the correlators can be determined by using, $\{\langle M^1\rangle, \langle M^1 M^2\rangle, \langle M^1 M^2 M^3\rangle,\cdots \}$ where the index in $M^i$ denotes the $i$-th insertion corresponding to the point $(u_i,\zeta_i,\bar\zeta_i)$. Invariance of the correlators under the action of the global part results in
\begin{equation}\label{1point M}
\Phi_1:= \langle M^1\rangle=0
\end{equation}
\begin{equation}\label{2pointM}
\Phi_2:=\langle M^1 M^2\rangle={C\ \over (\zeta_1-\zeta_2)^3(\bar\zeta_1-\bar\zeta_2)^3}
\end{equation}
\begin{equation}\label{3point function}
\Phi_3:= \langle M^1 M^2 M^3\rangle= {\bar C\over (\zeta_1-\zeta_2)^{3\over 2}(\zeta_1-\zeta_3)^{3\over 2} (\zeta_2-\zeta_3)^{3\over 2} (\bar\zeta_1-\bar\zeta_2)^\frac32 (\bar\zeta_1-\bar\zeta_3)^\frac32 (\bar\zeta_2-\bar\zeta_3)^\frac32}
\end{equation}
where $C$ and $\bar C$ are constants.
Using \eqref{1point M}-\eqref{3point function} we find that the non-zero two and three point functions are $\langle T_{uu}^1 T_{uu}^2 \rangle, \langle T_{uu}^1 T_{\zeta\bar\zeta}^2 \rangle, \langle T_{\zeta\bar\zeta}^1 T_{\zeta\bar\zeta}^2 \rangle, \langle T_{uu}^1 T_{uu}^2 T_{uu}^3\rangle, \langle T_{uu}^1 T_{uu}^2 T_{\zeta\bar\zeta}^3 \rangle, \langle T_{uu}^1 T_{\zeta\bar\zeta}^2 T_{\zeta\bar\zeta}^3\rangle, \langle T_{\zeta\bar\zeta}^1 T_{\zeta\bar\zeta}^2 T_{\zeta\bar\zeta}^3 \rangle $. The two point functions are proportional to $\Phi_2$ and the three point functions are proportional to $\Phi_3$. Furthermore, the correlators are independent of u coordinate. Our results are in complete agreement with the results of \cite{Bagchi:2016bcd} where the correlators of the primary operators of BMSFT have been calculated by considering highest-weight representation of the BMS$_4$ algebra. Comparing our results with \cite{Bagchi:2016bcd} shows that in the case of the stress tensor operators, the eigenvalues of $L_0$ and $\bar L_0$ are $3/2$. $L_0+\bar L_0$ is identified as the dilatation operator and $L_0-\bar L_0$ is the rotation operator. Thus we find that the weight under the dilatation operator is $3$. This may be an interesting properties of the three dimensional BMSFTs and needs more studies.
\section{Conclusion}
In this paper we generalized the method of \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O} to four-dimensional asymptotically flat RT solutions. On the gravity side the quasi-local stress tensor yields the correct charges of the BMS symmetry generators. On the field theory side, the structure of correlation functions are consistent with the previous work \cite{Bagchi:2016bcd}. This shows that when an asymptotically flat metric is given by taking the flat-space limit from an asymptotically AdS spacetime, one can use the dictionary of AdS/CFT to gain some insights into flat space holography.
This paper is the first step to study BMSFT$_3$ by using the method of \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O}. The final goal is to consider a generic four-dimensional asymptotically flat metric which satisfies the BMS boundary conditions and find quasi-local stress tensor correlators.
In all of our calculations in this paper and \cite{ Fareghbal:2013ifa,Fareghbal.Hosseni,Fareghbal:2016hqr, Fareghbal.A.O} it is assumed that BMSFTs live on the anisotropic conformal infinity of the asymptotically flat spacetimes. Its metric can be found when the asymptotically flat geometry is given by taking the flat-space limit of an asymptotically AdS spacetime. However, it is still an open question to develop a systematic procedure to find the anisotropic conformal infinity in the context of flat-space holography.
\subsubsection*{Acknowledgements}
The authors would like to thank Mohammad Asadi, Seyed Morteza Hosseini and Pedram Karimi for useful comments. This work is supported by Iran National Science Foundation (INSF), project No. 9526713.
|
1,116,691,499,397 | arxiv | \section{I. introduction}
The observations of the cosmic microwave background (CMB) strongly
support inflation as the paradigm of early universe. To discover the
nature of inflation, intensive analysis of the CMB has been performed.
The latest results by the Planck
collaboration~\cite{Ade:2015lrj,Ade:2015xua} provide the bounds on the
scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ of the
primordial density fluctuations,
\begin{align}
&n_s=0.9655\pm0.0062\,\,
(68\%\,{\rm CL}) \,,
\nonumber \\
&r<0.10\,\, (95\%\,{\rm CL})\,.
\label{eq:nsr_obs}
\end{align}
In fact, some inflation models,
such as canonical chaotic inflation~\cite{Linde:1983gd} and hybrid
inflation~\cite{Linde:1993cn}, are already disfavored due to the
bounds. Although they are not supported by the current observations,
the models are simple and still attractive in theoretical point of
view.
Recently Refs.\,\cite{Buchmuller:2012ex,Buchmuller:2013zfa} studied
the hybrid inflation in the framework of superconformal
supergravity~\cite{Einhorn:2009bh,Kallosh:2010ug,Ferrara:2010yw,Ferrara:2010in}.
It was found that the Starobinsky model~\cite{Starobinsky:1980te}
emerges in the supersymmetric D-term hybrid
inflation~\cite{Binetruy:1996xj,Halyo:1996pp,Kallosh:2003ux}, to give
a good accordance with the Planck observations. On the other hand,
the D-term hybrid inflation was considered in a different context. In
a shift symmetric K\"{a}hler potential~\cite{Kawasaki:2000yn}, a
`chaotic regime' was found in the subcritical value of the inflaton
field~\cite{Buchmuller:2014rfa}. In the framework, inflation lasts
even after the critical point of the hybrid inflation to give rise to
different predictions from chaotic inflation. The following
study~\cite{Buchmuller:2014dda} showed that the energy scale of
inflation coincides with the Grand Unification (GUT) scale using the
Planck 2013 data~\cite{Planck:2013jfk}. However, there is a tension
between the predictions and the observations, especially the Planck
2015 data~\cite{Ade:2015lrj,Ade:2015xua}.
In this letter we revisit D-term hybrid inflation in superconformal
framework. It will be shown that there exists a single slow-rolling
field in the subcritical value of the inflaton field. Since inflation
continues for sufficiently long period, cosmic strings are
unobservable as in
Refs.\,\cite{Buchmuller:2014rfa,Buchmuller:2014dda}. The potential in
the subcritical region turns out to be in a general class of
superconformal $\alpha$
attractors~\cite{Kallosh:2013yoa,Kallosh:2013xya}, especially similar
to the simplest version of the model. Consequently, non-trivial
behavior and different predictions from the simplest ones are
discovered.
\section{II. subcritical regime in superconformal D-term inflation}
We consider D-term hybrid inflation in supergravity with
superconformal matter~\cite{Buchmuller:2012ex,Buchmuller:2013zfa}. In
the model three chiral superfields $S_\pm$ and $\Phi$, which have
local U(1) charge $\pm q$ ($q>0$) and $0$, respectively, are
introduced. The superpotential and K\"{a}hler potential after fixing
a gauge for the local conformal symmetry are respectively given by,
\begin{align}
& W=\lambda S_+ S_- \Phi \,, \\
& K=-3\log \Omega^{-2}\,,
\label{eq:Kahler}
\end{align}
with,
\begin{eqnarray}
\Omega^{-2}=1-\frac{1}{3}\left(|S_+|^2+|S_-|^2+|\Phi|^2\right) -
\frac{\chi}{6}\left(\Phi^2+\bar{\Phi}^2\right)\,,
\nonumber \\
\end{eqnarray}
where $\lambda$ and $\chi$ are constants.\footnote{Throughout this
letter we use the same notation for chiral superfields and scalar
fields and take the reduced Planck mass $M_{\rm pl}=1$ unit. } The
term proportional to $\chi$ in the K\"{a}hler potential breaks
superconformal symmetry explicitly. In the model the Fayet-Iliopoulos (FI)
term can be accommodated. Then, the D-term potential in the Einstein
frame is~\cite{Buchmuller:2012ex},
\begin{eqnarray}
V_D=\frac{1}{2}g^2\left(q\Omega^2(|S_+|^2-|S_-|^2)-\xi\right)^2\,,
\end{eqnarray}
where $g$ is the gauge coupling and $\xi$ is the FI term, which is
taken as a constant. (See
Refs.\,\cite{Binetruy:2004hh,Komargodski:2010rb,Dienes:2009td,
Catino:2011mu,Wieck:2014xxa,Domcke:2014zqa} for the subtleties of
this issue in supergravity.) The F-term potential in the Einstein
frame, on the other hand, is given in a simple form without
exponentially growing terms~\cite{Buchmuller:2012ex,Einhorn:2012ih},
\begin{align}
V_F &=\Omega^{4} \lambda^2\biggl[
|\Phi|^2\left(|S_+|^2+|S_-|^2\right)+|S_+S_-|^2 \nonumber \\
&
-\frac{\chi^2|S_+S_-\Phi|^2}
{3+\frac{\chi}{2}\left(\Phi^2+\bar{\Phi}^2\right)+\chi^2|\Phi|^2} \biggr]\,.
\end{align}
As in the canonical hybrid inflation, $S_-$ is stabilized to its
origin meanwhile $S_+$ suffers from the tachyonic instability
depending on the field value of $\Phi$. The nature of $\Phi$ depends
on the value of $\chi$. In the K\"{a}hler potential there is a shift
symmetry under ${\rm Re}\,\Phi \,({\rm Im}\,\Phi) \to {\rm Re}\,\Phi\,
({\rm Im}\,\Phi)\,+$\,const. for $\chi=-1\,(+1)$, and ${\rm Re}\,\Phi
\,({\rm Im}\,\Phi)$ can play a role of inflaton, as mentioned in
Ref.\,\cite{Buchmuller:2012ex}. We consider $\chi \le -1$ in the later
discussion without loss of generality. Then, the total potential is
given by the waterfall field $s\equiv \sqrt{2}|S_+|$ and the inflaton
field $\phi\equiv \sqrt{2}{\rm Re}\,\Phi$,
\begin{align}
V_{\rm tot}(\phi,s)&=V_F+V_D \nonumber \\
&=\frac{\Omega^4(\phi,s) \lambda^2}{4} s^2 \phi^2
+ \frac{g^2}{8}\left(q \Omega^2(\phi,s)s^2-2\xi\right)^2\,, \\
\Omega^{-2}(\phi,s)&=1-\frac{1}{6}\left(s^2+(1+\chi)\phi^2\right)\,.
\end{align}
The waterfall field becomes tachyonic below the critical value
$\phi_c$ of the inflaton field,
\begin{align}
\phi_c^2=\frac{6qg^2\xi}{3\lambda^2+(1+\chi)qg^2\xi}\,.
\end{align}
After the tachyonic growth, the waterfall field is expected to reach
its local minimum, which is obtained by $\partial V_{\rm
tot}(\phi,s)/\partial s =0$,
\begin{align}
s_{\rm min}^2
&=\frac{2\xi\Omega^{-2}(\phi,0)}{q(1+\tilde{\xi})}
\frac{1-\Psi^2}{1+\frac{\tilde{\xi}}{1+\tilde{\xi}}\Psi^2}\,,
\label{eq:smin}
\end{align}
where $\tilde{\xi}\equiv \xi/3q$ and,
\begin{align}
\Psi\equiv \frac{\Omega(\phi,0)\phi}{\Omega(\phi_c,0)\phi_c}
=\frac{\Omega(\phi,0)\phi}{\sqrt{2qg^2\xi/\lambda^2}}\,.
\end{align}
The expression for the local minimum given in
Refs.\,\cite{Buchmuller:2014rfa,Buchmuller:2014dda} corresponds to the
case for $\chi=-1$ (and $q=1$) from the facts that
$\Omega(\phi,0)|_{\chi=-1}=1$ and $\tilde{\xi}\sim {\cal O}(10^{-4})$
in our targeted parameter space. Following
Refs.\,\cite{Buchmuller:2014rfa,Asaka:2001ez} (see also Appendix), we
have confirmed numerically that the waterfall field reaches to the
local minimum after ${\cal O}(1/H_c)$ where $H_c(=g\xi/\sqrt{6})$ is
the Hubble parameter at the critical point, and then it becomes a
single field inflation. Since the inflation lasts well over ${\cal
O}(10^{2}/H_c)$, cosmic strings, which are produced during the
tachyonic growth, are unobservable. After the waterfall field relaxed
to the local minimum, the dynamics of the inflaton is described by the
potential,
\begin{align}
V&\equiv V_{\rm tot}(\phi,s_{\rm min}) \nonumber \\
&= g^2\xi^2(1+\tilde{\xi})\Psi^2
\frac{1-\frac{\Psi^2}{2(1+\tilde{\xi})}}
{1+2\tilde{\xi}\Psi^2} \,.
\label{eq:V}
\end{align}
As in Eq.\,\eqref{eq:smin}, it is easily to see that the potential $V$
with $\chi=-1$ agrees with one given in
Refs.\,\cite{Buchmuller:2014rfa,Buchmuller:2014dda} up to ${\cal
O}(\xi)$.\footnote{$q$ and $g$ can be absorbed by the redefinition
of $\lambda$ and $\xi$, $\bar{\lambda}\equiv \lambda /\sqrt{qg}$ and
$\bar{\xi}\equiv g \xi$ if we ignore terms proportional to
$\tilde{\xi}$ that are irrelevant numerically. Although we will use
$\lambda$ and $\xi$ in the following discussion, the results in
terms of $\bar{\lambda}$ and $\bar{\xi}$ can be obtained by $q\to 1$,
$g\to 1$, $\lambda \to \bar{\lambda}$, and $\xi \to \bar{\xi}$.}
We note that non-zero $\lambda$ explicitly breaks the shift symmetry
for ${\rm Re}\,\Phi$ as well as $\chi$ that deviates from $-1$
does. Thus, a parameter $\lambda \ll 1$ and $\chi\simeq-1$ is
consistent with each other under the approximate shift symmetry. In
addition, $\chi\simeq -1$ is required for $\lambda \ll 1$ otherwise
$\phi^2_c$ gets negative. As it will be seen, the observational data
indeed implies such a parameter space.
\section{III. cosmological consequences}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.5]{nsr.pdf}
\end{center}
\caption{Scalar spectral index and tensor-to-scalar ratio for
various values of $\delta\chi$ ($0<\delta\chi<1$) and $N_{*}=50$
and $60$ as `superconf.'. Result in the shift symmetric K\"{a}hler
potential in Ref.\,\cite{Buchmuller:2014dda} is also given as
`shift sym.' (updated using the Planck 2015 results). Here $q=g=1$
and appropriate values of $\lambda$ and $\xi$ to satisfy the
observed scalar amplitude is taken. }
\label{fig:nsr}
\end{figure}
The slow roll parameters for the inflaton dynamics are given as,
\begin{align}
\epsilon(\phi) = \frac{1}{2}\left(\frac{V'}{V}\right)^2\,,
\quad \quad
\eta(\phi) =\frac{V''}{V}\,,
\end{align}
where $V'=dV/d\hat{\phi}$ and $V''=d^2V/d\hat{\phi}^2$. Here
$\hat{\phi}$ is canonically-normalized inflaton field that is related
to $\phi$ as,
\begin{align}
\frac{d \phi}{d\hat{\phi}}=K_{\Phi \bar{\Phi}}^{-1/2}\,,
\label{eq:dphidphihat}
\end{align}
where $K_{\alpha\bar{\alpha}}\equiv\partial^2
K/\partial\alpha\partial{\bar{\alpha}}$. $|S_-|=0$,
$\Phi=\bar{\Phi}=\phi/\sqrt{2}$, and $|S_+|=s_{\rm min}/\sqrt{2}$ are
implicit here. (Parametrically $s_{\rm min}\simeq 0$ is a good
approximation as discussed later.) Inflation ends at $\phi=\phi_f\equiv{\rm
Max}\{\phi_{\epsilon}$,\,$\phi_{\eta}$\} where
$\epsilon(\phi_\epsilon)=1$ and $|\eta(\phi_\eta)|=1$, and the last
$e$-folds $N_*$ before the end of inflation is obtained by,
\begin{align}
N_*=\int_{\phi_f}^{\phi_*}d\phi \frac{V}{dV/d\phi}K_{\Phi\bar{\Phi}}\,.
\end{align}
The cosmological observables, {\it i.e.,} the scalar amplitude $A_s$,
the spectral index, and the tensor-to-scalar ratio, are then
determined by,
\begin{align}
A_s=&\frac{V(\phi_*)}{24\pi^2 \epsilon(\phi_*)}\,,
\\
n_s=1+2\eta(\phi_*)-6\epsilon&(\phi_*)\,, \quad \quad
r=16 \epsilon(\phi_*)\,.
\end{align}
We normalize the scalar amplitude by using the Planck 2015
data\,\cite{Ade:2015xua} $A_s=2.198^{+0.076}_{-0.085}\times 10^{-9}$
and compute $n_s$ and $r$ for a given $N_*$.
As we have stated before, our target is the parameter space $\lambda
\ll 1$. To search such a region, it is convenient to parametrize
$\chi$ as,
\begin{align}
\chi=-1-\frac{3\lambda^2}{qg^2\xi}\delta \chi \quad \quad
(0<\delta\chi<1)\,,
\end{align}
to satisfy $\phi_c^2=2qg^2\xi/\lambda^2(1-\delta\chi)>0$.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.65]{Alwdxi.pdf}
\end{center}
\caption{Allowed region for $N_*=55$ to 60 from the bounds on $n_s$
(68\% CL) and $r$ (95\% CL). Line contents are the same as
Fig.\,\ref{fig:nsr}. Here we have updated the result for the shift
symmetric K\"{a}hler case by using the Planck 2015 data.}
\label{fig:lamxi}
\end{figure}
Now we are ready to discuss the cosmological
consequences. Fig.\,\ref{fig:nsr} shows the predictions of $n_s$ and
$r$ in our model. Here $q=g=1$ is taken (see footnote 2), and
$\lambda$ and $\xi$ are determined for a $\delta \chi$ and $N_*$ by
using the scalar amplitude observed by the Planck collaboration. In
Fig.\,\ref{fig:lamxi}, the allowed regions due to the bounds on $n_s$
and $r$ are shown for $N_*=55$--60.\footnote{There is no allowed
region for $N_*=50$ except for $\delta \chi=0.9$. } The upper and
lower bounds on $\xi$ corresponds to the upper limit on $r$ and lower
limit on $n_s$, respectively. In the $n_s$-$r$ plane, smaller values
of $n_s$ and $r$ are obtained for larger $\lambda$ (and smaller
$\xi$). In Fig.\,\ref{fig:nsr} the result in the previous work
\cite{Buchmuller:2014dda}, {\it i.e.}, the shift symmetric K\"{a}hler
potential case, is also given as `shift sym.'.\footnote{ Do not
confuse with the shift symmetric K\"{a}hler case with the present
superconformal case where the shift symmetry is (weakly) broken in
the K\"{a}hler potential.} We have checked that the result for
$\delta \chi=0$ agrees with it numerically and the similar behavior is
seen around $\delta \chi\simeq 0$. When $\delta \chi$ gets close to
unity, on the contrary, a different behavior is observed. It is seen
that $r$ gets smaller meanwhile $n_s$ tends to stay in the same value,
which is within the Planck bounds. As a result, a wider allowed
parameter space is obtained, which is seen in Fig.\,\ref{fig:lamxi}.
It is seen $\lambda\sim 10^{-4}$--$10^{-3}$ and $\sqrt{\xi}\sim
10^{16}$\,GeV are consistent region with the Planck observation.
Although the allowed region becomes larger, $\sqrt{\xi}$ tends to sit
around the GUT scale even for $\delta \chi=0.9$. As a consequence, the
predicted $r$ is not extremely small. For example, $r>0.0020$ (0.075)
for $N_*=60$ (50) for $\delta \chi=0.9$. The value of $\chi$ in the
allowed region, on the other hand, is found as $-1.41$
$(-1.016)<\chi<-1.0046$ ($-1.0092$) for $N_*=60$ (50). Therefore, the
parameter space $\lambda \ll 1$ and $\chi\sim -1$ is indeed favored by
the observations.
In order to interpret the results, it is instructive to consider a
canonically-normalized inflaton field $\hat{\phi}$. Although the r.h.s
of Eq.\,\eqref{eq:dphidphihat} is complicated, it can be approximated
in the parameter space we are considering as,
\begin{align}
\frac{d \phi}{d\hat{\phi}}
\simeq \sqrt{1-\frac{1}{6}(1+\chi)\phi^2}\,.
\label{eq:dphidphihat_app}
\end{align}
Then it is solved analytically,
\begin{align}
\phi=\frac{1}{\sqrt{\beta}}\sinh\sqrt{\beta}(\hat{\phi}+C)\,,
\label{eq:phiinphihat}
\end{align}
where $C$ is a constant and,
\begin{align}
\beta\equiv -\frac{1+\chi}{6}=\frac{\lambda^2}{2qg^2\xi}\delta\chi
=\frac{\delta \chi}{\phi^2_c(1-\delta\chi)}\,.
\label{eq:beta}
\end{align}
We have found that $C=0$ is appropriate choice. Then $\Psi$ is simply
given as,
\begin{align}
\Psi\simeq \delta \chi^{-1/2} \tanh\sqrt{\beta} \hat{\phi}\,,
\end{align}
to express the potential in terms of $\hat{\phi}$,
\begin{align}
V\simeq g^2\xi^2 \delta \chi^{-1} \tanh^2\sqrt{\beta} \hat{\phi}
\biggr[1-\frac{\delta \chi^{-1}}{2} \tanh^2\sqrt{\beta}
\hat{\phi}\biggl]\,.
\label{eq:Vcano}
\end{align}
This potential is valid in $\hat{\phi}\le \hat{\phi}_c=
\frac{1}{\sqrt{\beta}}\sinh^{-1}\sqrt{\beta}\phi_c$. It is
straightforward to check that the r.h.s is equal to $g^2 \xi^2/2$ for
$\hat{\phi}=\hat{\phi}_c$, and $\hat{\phi}_c\to \infty$ for $\delta
\chi\to 1$. We note that the potential coincides with a general class
of superconformal $\alpha$ attractors~\cite{Kallosh:2013yoa}. It
especially resembles to the simplest class of the model,
\begin{align}
V_{\alpha \mathchar`-attr}=
\Lambda^4 \tanh^{2m}\frac{\hat{\phi}}{\sqrt{6\alpha}}\,.
\label{eq:Valpha}
\end{align}
Due to the additional term, however, it has a different asymptotic
behavior as we will see below.
In the small $\lambda$ (and large $\xi$) region, $\beta$ gets small,
then the potential reduces to,
\begin{align}
V\simeq
g^2\xi^2(1-\delta\chi) \frac{\hat{\phi}^2}{\phi_c^2}
\left[1-\frac{1+(4/3)\delta\chi}{2(1-\delta\chi)}
\frac{\hat{\phi}^2}{\phi^2_c}\right]\,.
\end{align}
This is nothing but the potential for the shift symmetric K\"{a}hler
case given in Refs.\,\cite{Buchmuller:2014rfa,Buchmuller:2014dda} in
the limit $\delta \chi \to 0$, which leads to $\hat{\phi}\to \phi$.
This feature is clearly seen in Fig.\,\ref{fig:nsr}. We note that the
quadratic term is rewritten as $(\lambda^2\xi/2q)\hat{\phi}^2$, which
is independent of $\delta\chi$. Therefore, $n_s$ and $r$ approach to
those in quadratic chaotic inflation in small $\lambda$ limit (while
$\lambda^2 \xi\simeq$ constant), independent of $\delta \chi$.
(Such a region is excluded, thus it is not shown in
Fig.\,\ref{fig:lamxi}.) The potential $V_{\alpha \mathchar`-attr}$,
on the other hand, has a similar structure,
\begin{align}
V_{\alpha \mathchar`-attr}\simeq \frac{\Lambda^4}{6\alpha}\hat{\phi}^{2m}
\left[1-\frac{m\hat{\phi}^2}{9\alpha}\right]\,.
\end{align}
Although it coincides with $V$ in the limit $\hat{\phi}\to 0$ for
$m=1$ and $\Lambda^4/6\alpha=\lambda^2\xi/2q$, it is not possible
to get the same factor for the quartic term.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.65]{potentials.pdf}
\end{center}
\caption{Potential as function of canonically-normalized inflaton
field $\hat{\phi}$. $\delta\chi=0.9$ and $1$ cases (`superconf.')
are shown, which are compared with superconformal $\alpha$
attractors (`$\alpha$ attr.'), $R^2$ inflation (`$R^2$'), and
the shift symmetric K\"{a}hler case (`shift sym.'). The field
values $\hat{\phi}_f$ and $\hat{\phi}_*$ at the end of inflation
and the last 60 $e$-folds, respectively, are also indicated for
$\delta\chi=0.9$ and $1$ cases. }
\label{fig:V}
\end{figure}
In large $\lambda$ (and small $\xi$) region, on the contrary, $\beta$
increases, which leads us to expand $\Psi$ in large
$\sqrt{\beta}\hat{\phi}$ limit to obtain,
\begin{align}
V\simeq \frac{1}{2}g^2\xi^2(2-\delta\chi^{-1})
\left[1+a_1e^{-2\sqrt{\beta}\hat{\phi}}
-a_2e^{-4\sqrt{\beta}\hat{\phi}}\right]\,,
\label{eq:Vlargefield}
\end{align}
with $a_1=8(1-\delta\chi)/(2\delta\chi-1)$ and
$a_2=16(2-\delta\chi)/(2\delta\chi-1)$. This expression should be
compared with Eq.\,\eqref{eq:Valpha} in the $\alpha\ll 1$ limit. As
shown in Ref.\,\cite{Kallosh:2013yoa}, it reduces to the potential in
$R^2$ inflation~\cite{Whitt:1984pd} at large field value\footnote{To
be precise, $\alpha=1$ gives the original $R^2$ inflation. The
factor $4m (>0)$ is quantitatively irrelevant for the slow-roll
predictions.},
\begin{align}
V_{\alpha \mathchar`-attr}\simeq \Lambda^4\left[
1-4me^{-\frac{2\hat{\phi}}{\sqrt{6\alpha}}}\right]\,.
\end{align}
Now it is clear that the form of the potential with $\delta\chi=1$ (in
large $\lambda$ region) reduces to $R^2$ inflation, or the simplest
class of superconformal $\alpha$ attractors in $\alpha\ll 1$ limit. To
be specific, a choice of $\Lambda^4=g^2\xi^2/2$ and $\alpha=1/24\beta$
leads to the same asymptotic form. Then, we get $n_s\simeq
1-2/(N_*+1)-3qg^2\xi/8\lambda^2(N_*+1)^2$, $r\simeq
qg^2\xi/\lambda^2(N_*+1)^2$, while satisfying $\lambda^2 \xi\simeq$
constant. Namely, when $\lambda$ increases $n_s$ approaches to
$1-2/(N_*+1)$ and $r$ gets smaller and smaller. We have confirmed
this behavior using Eq.\,\eqref{eq:Vcano} with $\delta\chi=1$. Recall
that, however, the critical value becomes infinity, which is
unphysical.
Such a behavior, on the contrary, can not be seen for $\delta \chi\neq
1$ case shown in Fig.\,\ref{fig:nsr}. This arises from non-zero $a_1$
in Eq.\,\eqref{eq:Vlargefield}. This is why we have seen the different
cosmological consequences.
In Fig.\,\ref{fig:V}, the potential as function of
canonically-normalized inflaton field is plotted for $\delta \chi=0.9$
and $1$. Here $\lambda=9.4 \times 10^{-4}$ ($1.9\times 10^{-3}$),
$\sqrt{\xi}=1.3\times 10^{16}$ ($5.7 \times 10^{15}$)\,GeV for $\delta
\chi=0.9$ (1) to give $n_s=0.966$ and $r=0.051$ (0.00052) for
$N_*=60$. In the plot the potentials in superconformal $\alpha$
attractors $V_{\alpha\mathchar`-attr}$, $R^2$ inflation, and the shift
symmetric K\"{a}hler case, are also shown for comparison. Each
potential is normalized to unity when $\hat{\phi}$ reaches to the
critical point ($\delta \chi=0.9$ case and the shift symmetric
K\"{a}hler case) or infinity ($\delta \chi=1$ case, $\alpha$
attractors, and $R^2$ inflation). For the shift symmetric K\"{a}hler
case, the critical point is taken to the same value as
$\delta\chi=0.9$ case. We take the parameters for $\alpha$ attractors
and $R^2$ inflation to have the same asymptotic form as $\delta
\chi=1$ case in large field limit. It is seen that $\delta \chi=1$
case is similar to $\alpha$ attractors, but not exactly the same. As
a result, the predictions for the slow-roll quantities are different,
{\it i.e.,} $n_s=0.967$ and $r=0.00044$. It is clear, on the other
hand, that $\delta \chi=0.9$ case shows a different behavior from the
others. To summarize, the model has a nature of both the shift
symmetric K\"{a}hler case and the simplest superconformal $\alpha$
attractors, and the slow-roll predictions change accordingly.
\section{IV. conclusion}
We have revisited superconformal D-term hybrid inflation. After
reaching its critical value, the inflaton field is slowly rolling thus
inflation continues for a small coupling $\lambda$ of inflaton to the
other fields. Because of a sufficiently long period of slow-roll
regime, cosmic strings, which are formed during the tachyonic growth
of the waterfall field, are unobservable. The potential which
determines the dynamics of the canonically-normalized inflaton in the
subcritical regime has been found to resemble to the simplest version
of superconformal $\alpha$ attractors but with an additional term.
Consequently, different predictions for the slow-roll parameters are
obtained. For $\lambda\sim 10^{-4}$--$10^{-3}$ and $\sqrt{\xi}\sim
10^{16}\,{\rm GeV}$, $n_s$ and $r$ are consistent with the Planck
data.
The predictions depend on a parameter $\chi$ that explicitly breaks
superconformal symmetry in the K\"{a}hler potential. In addition, the
K\"{a}hler potential with $|\chi|=1$ has a shift symmetry for the
inflaton field, which is explicitly broken by non-zero $\lambda$ in
the superpotential. On the other hand, $|\chi|\simeq 1$ is required
from the consistency of the model setting, thus $\lambda\ll 1$ is
parametrically natural. It has been found that the observational
bounds indeed prefer such a parameter space.
\vspace{0.2cm}
\noindent
{\it Acknowledgments}\\
\noindent
We are grateful to Wilfried Buchm\"{u}ller for valuable discussions
and helpful comments on the manuscript. This work was supported by
JSPS KAKENHI Grant Numbers JP17H05402, JP17K14278 and JP17H02875.
|
1,116,691,499,398 | arxiv | \section{INTRODUCTION}
\label{sec:intro}
The Wide Field Infrared Survey Telescope (WFIRST) Coronagraph Instrument (CGI) is a high-contrast imager, polarimeter, and integral field spectrograph (IFS) that will enable the study of exoplanets and circumstellar disks at visible wavelengths ($\sim 550-850$~nm).\footnote{Instrument parameters and simulations are available at \url{https://wfirst.ipac.caltech.edu/}}
CGI aims to achieve detection limits of $\sim 10^{-9}$ the flux of the host star at separations of $\sim 0.15''-1.5''$ (Figure \ref{fig:theplot}), in order to serve as a pathfinder for future terrestrial planet finding missions that require $\sim 10^{-10}$ flux ratio capability. If CGI achieves its predicted performance, it will be capable of imaging and low-resolution spectroscopy of gas giant planets in reflected light and of detection of exozodiacal dust disks at unprecedented sensitivity. These observations would begin to constrain cloud properties of mature Jupiter analogues, to shed light on the planet formation process in protoplanetary disks, and to identify the ``cleanest'' (least dusty) systems for future exo-Earth searches. To achieve these goals, WFIRST CGI will build not only on a legacy of previous space observatories, but also on a legacy of ground-based high-contrast instrumentation. We refer the reader to Debes et al., 2015 \cite{Debes2015} for an excellent discussion of lessons learned from the Hubble and James Webb Space Telescopes. In these proceedings, we discuss relevant lessons learned from ground-based instruments in the areas of wavefront sensing and control, observing strategies, calibration, and data post-processing.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=9cm]{flux_ratio_spie_vb.png}
\end{tabular}
\end{center}
\caption[example] { \label{fig:theplot}
Predicted CGI performance on a V=5 star, in the context of current high-contrast instrumentation\cite{ThePlot}. Curves and planets are color-coded by bandpass central wavelength. Curves are $5\sigma$ post-processed point source detection limits; bold lines are predictions for the three official CGI observing modes. CGI integration times are noted in the plot, while other instruments' performances are typically based on $\sim1$~hr of integration time. Known self-luminous imaged planets (colored points) are shown at their observed IR flux ratios as well as at their predicted flux ratios at visible wavelengths. Gray triangles are predicted flux ratios for known giant planets detected by the radial velocity technique, when viewed at quadrature with assumed albedoes of 0.5.}
\end{figure}
\section{WAVEFRONT SENSING AND CONTROL}
\label{sec:wavefront control}
At small working angles, contrast is governed by star centering behind the coronagraph occulting spot and by low order aberrations.
Control of both fast tip/tilt jitter and of slower drift in low order modes is necessary in order to achieve the best performance.
Several ground-based instruments have dedicated low order wavefront sensing and control (LOWFS/C) systems, including P1640\cite{Cady2013}, SPHERE\cite{NDiaye2016}, SCExAO\cite{Singh2015}, and GPI\cite{Hartung2014}.
One common lesson learned from all of these systems is that for optimal performance, LOWFS/C must be operated continuously, in parallel with science observations.
The LOWFS should not take any light from the science camera, and so designs utilizing light rejected by the coronagraph are preferred.
Furthermore, the LOWFS/C loop speed must be matched to the power spectrum of the appropriate modes (both fast tip/tilt jitter and slower drifts in higher modes).
In CGI, tip/tilt errors will originate from both sub-Hz observatory pointing drift and from structural vibrations excited by the telescope reaction wheels (1-100Hz).
Longer timescale thermal drifts in the spacecraft will be the primary contributors to errors in other low order modes.
To compensate, CGI will have a dedicated LOWFS/C system for Zernike modes 2-11\cite {Shi2016}.
The Zernike phase contrast wavefront sensor will use the starlight reflected by the coronagraph occulting masks and will operate during science observations.
A fast steering mirror will correct tip/tilt jitter at frequencies $\lesssim 20$~Hz; other modes will be corrected at 5~mHz with a combination of a dedicated focus corrector and deformable mirrors (DMs).
At larger working angles in the so-called ``dark hole,'' a High Order Wavefront Sensing and Control (HOWFS/C) system is needed.
All ground-based high-contrast instruments have dedicated high order wavefront sensors to sense atmospheric turbulence; CGI does not have a need for an analogous system.
CGI does, however, require HOWFS/C to counteract optical aberrations in the telescope and instrument\footnote{In ground-based systems, such internal aberrations are often referred to as ``non-common path aberrations,'' because they are not in the optical path of the Adaptive Optics system's dedicated wavefront sensor.}.
These aberrations are best sensed using the images from the science camera itself.
Several focal plane wavefront sensing techniques have been tested on P1640, Keck, SCExAO, and GPI\cite{Martinache2014, Matthews2017, Savransky2012a, Bottom2016a, Bottom2017}.
However, they have achieved varying degrees of success on sky, because precise knowledge of the instrument model and deformable mirror calibration is required.
Furthermore, drifts in the instrument cause evolution of the point spread function (PSF), and so the quality of the dark hole degrades without continuous HOWFS/C.
CGI has elected to use pairwise probing and electric field conjugation; instrument model calibration is the subject of ongoing efforts\cite{Giveon2011a, Seo2017, Cady2017}.
CGI will be photon-starved on science targets; hence, the current baseline is to dig the dark hole on a bright reference star and to freeze the high-order DM correction throughout the science sequence (Figure \ref{fig:os6}).
Ongoing work is assessing the requirements on instrument thermal stability and on DM calibration for this ``optimize and freeze'' operational scenario.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.7\textwidth]{OS6-2.jpg}
\end{tabular}
\end{center}
\caption[example]{ \label{fig:os6} CGI baseline observing scenario. CGI will alternate between one or more bright reference stars and the science target, typically separated by $15^\circ - 20^\circ$ on the sky. The initial dark hole will be optimized on the first visit to the reference star. The high-order wavefront correction will be frozen throughout the science target observation, and will be re-optimized on each subsequent reference star visit. CGI will alternate between two roll angles on the science target. Simulated images and IFS data for this observing scenario are \href{https://wfirst.ipac.caltech.edu/sims/Coronagraph_public_images.html}{publicly available.}}
\end{figure}
\section{OBSERVING STRATEGY AND PSF SUBTRACTION}
For any high-contrast imager, the observing and post-processing strategies are interdependent.
To recover faint sources, one must first subtract the PSF of the primary star.
Several different strategies for PSF synthesis and subtraction have been developed on ground-based instruments, and each has a corresponding observing strategy.
The differential imaging techniques are then combined with PSF synthesis algorithms such as a median combination of images, LOCI\cite{Lafreniere2007}, or KLIP\cite{Soummer2007}.
\subsection{Angular Differential Imaging}
In ADI\cite{Marois2006}, the telescope pupil rotates with respect to the sky. The stellar PSF is fixed in detector coordinates, while planets or disks appear rotate around the central star.
A minimum amount of rotation, $\theta$, is required to avoid self-subtraction of planet light, typically $r\theta \sim 1$~FWHM (Full Width at Half Max), where $r$ is the star-planet separation.
This observing strategy is common practice for high-contrast ground-based instruments.
CGI can roll a maximum of $26^\circ$ due to Sun-to-spacecraft angle constraints, limiting the effectiveness of ADI at small working angles.
Unlike ground-based instruments, CGI will not have continuous roll angle coverage.
Instead, it will roll between the two extreme possible angles\footnote{Ongoing work is investigating the utility of adding a small number of intermediate roll angles.} (Figure \ref{fig:os6}), in a strategy more typical of space-based observatories.
The PSF of the coronagraph used in IFS mode has extended sidelobes in the azimuthal direction, so CGI's limited roll angle means that ADI cannot be used near the inner edge of the field of view of the IFS.
Additionally, regardless of the coronagraph design, ADI acts as a high-pass filter on extended sources, and so it is not optimal for disks.
For these reasons, CGI must employ Reference Differential Imaging as well.
\subsection{Reference Differential Imaging}
An RDI observing sequence alternates between the science target and one or more reference targets, and the reference target images are used for PSF synthesis.
RDI can be used in concert with ADI (RDI+ADI) if the science target images are included in the reference library used for PSF synthesis.
Even in pure RDI, observing the science target at multiple roll angles helps to average over PSF residuals.
RDI is preferred over ADI in situations where self-subtraction is significant: when there is little sky rotation or when targeting extended emission in circumstellar disks.
RDI requires PSF stability between the science and reference stars.
Therefore, it is commonly used in space-based observations\cite{Choquet2015} and will be the default observing strategy for CGI (Figure \ref{fig:os6}).
Observations of bright reference stars are already required for CGI wavefront control, as described in Section \ref{sec:wavefront control}, and so the additional images needed for RDI can be added with minimal overhead.
RDI is less commonly used by ground-based facilities, where changing seeing conditions introduce PSF variability, and where target and reference star brightness must be closely matched in order to achieve similar AO correction quality.
Nevertheless, ground-based RDI has been successfully used to improve sensitivity at small working angles in ``snapshot'' surveys where each target has little sky rotation\cite{Xuan2018} and in studies of disks\cite{Mawet2009, Rameau2012}.
Several factors must be considered when choosing a reference star.
The star should not have any close companions or a circumstellar disk.
The stars should be close enough on the sky (typically $<20^\circ$ for CGI) that the telescope and instrument conditions do not change appreciably between reference and science pointings.
Both imaging and IFS modes are insensitive to stellar spectral type mismatch; the chromatic and time variations of the speckles dominate over object color differences.
\subsection{Spectral Differential Imaging}
SDI relies on the deterministic evolution of PSF speckles with wavelength.
When speckles are due only to phase aberrations in the wavefront, their location varies radially as $\lambda$, while their flux varies as the stellar spectrum modulated by $\lambda^{-2}$.
Planets, on the other hand, remain at a fixed location and can have distinct spectral features (e.g.: CH$_4$ absorption).
These relationships can be used to synthesize the PSF at one wavelength from the PSF at another wavelength.
In its simplest form, SDI consists of subtraction of scaled images from two adjacent filters (e.g.: in and out of a CH$_4$ or $H\alpha$ features\cite{Racine1999, Marois2000a, Artigau2008, Dohlen2008, Close2014b}).
The IFS implementation can take a more sophisticated approach, utilizing information from multiple spectral channels, but following the same underlying methodology\cite{Sparks2002}.
Current ground-based instruments operate in a regime where phase aberrations dominate over amplitude aberrations.
However, when speckles are induced by both phase and amplitude aberrations, the same wavelength scaling does not apply, and SDI breaks down.
When phase and amplitude effects are comparable, as in CGI, the resulting speckle field is chromatic (Figure \ref{fig:psfs}, right panel).
CGI cannot use SDI unless future algorithms are developed that can disentangle the contributions from the two types of speckles.
Even though SDI cannot be used for PSF synthesis, the spectral information still adds value.
Matched filtering, which compares the spectrum of a candidate to that of a planet model\cite{Ruffio2017}, can still be used to differentiate planets and residual speckles.
\subsection{Polarimetric Differential Imaging}
PDI\cite{Langlois2014, Perrin2015} can be used to extract polarized signals from companions or disks, under the assumption of polarization-independent speckles.
In this method, the target is observed at multiple (linear) polarization angles, and the polarization-invariant signal (the unpolarized stellar speckle field and the unpolarized astrophysical signal) is removed.
Unlike ADI, PDI has the benefit of preserving extended structures, so long as these structures are polarized.
PDI has enabled the ground-based characterization of a number of circumstellar disks and has achieved working angles smaller than those possible with ADI\cite{Perrin2015, 2017Msngr.169...32G}.
In ground-based facilities, speckle fields are dominated by polarization-independent effects, but again, this is not the case for CGI.
The fast telescope primary mirror and non-optimal mirror coatings induce polarization-dependent speckles, or ``polarization aberrations''\cite{Breckinridge2015, Krist2016, Krist2018} (Figure \ref{fig:psfs}, left panel).
Future trade studies will investigate the possibility of observing in only a single polarization for improved dark hole optimization at the expense of throughput.
Note that although PDI is not expected to improve CGI post-processed sensitivity, CGI will still be capable of polarimetry of bright sources (Section \ref{sec:pol cal}).
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=\textwidth]{CGIpsfs.jpg}
\end{tabular}
\end{center}
\caption[example]{ \label{fig:psfs} Simulated CGI speckle fields for imaging mode with the Hybrid Lyot Coronagraph, all with the same log scale display. Speckle fields vary with wavelength and are polarization-dependent.
\textit{Left:} Raw image in X polarization and the absolute difference of X and Y polarization images.
\textit{Right:} Raw images in the same polarization at 546nm and 604nm. Please refer to companion proceedings for additional information about CGI modeling efforts\cite{Krist2018,Rizzo2018,Zhou2018}.}
\end{figure}
\section{CALIBRATION}
\subsection{Polarimetry}
\label{sec:pol cal}
Although CGI will not be capable of PDI, it will still be capable of (linear) polarimetry on bright sources with fluxes significantly above the polarization-dependent speckle noise floor.
CGI is required to measure linear polarization fraction to better than 3\% systematic accuracy in the high signal to noise limit.
CGI will have a set of four interchangeable linear analyzers (0$^\circ$, 45$^\circ$, 90$^\circ$, 135$^\circ$).
Because CGI will not have a modulator, it will not be possible to self-calibrate the instrumental polarization from the science sequences, as is common practice in ground-based instruments with modulators\cite{Millar-Blanchaer2016, vanHolstein2017, Murakawa2004}.
Routine observations of calibrators and telescope and instrument modeling will be critical.
Figure \ref{fig:pol cal} shows the effect of throughput, crosstalk, and instrumental polarization on the astrophysical polarization signal.
The instrumental polarization ($IP$) will have both constant terms and terms that vary with each target (i.e.: polarization-dependent speckles).
Therefore, $IP$ should be calibrated on each science target, perhaps even after each optimization of the dark hole during a science sequence.
Efficiency in each polarization ($\eta$) and crosstalk between polarizations (eg: $Q\rightarrow U$) are likely to evolve more slowly, requiring less frequent calibration on polarized standard stars (cadence to be determined).
Catalogs of unpolarized and polarized standard stars\footnote{eg: \url{http://www.ukirt.hawaii.edu/instruments/irpol/irpol_stds.html}} may be supplemented with observations of pre-determined CGI PSF reference stars.
Note that CGI will not measure circular polarization (Stokes V), nor are circumstellar disks expected to be significantly circularly polarized.
The effect of all crosstalk, including Stokes V, will have to be modeled to determine polarimetric efficiency and accuracy; this is the subject of ongoing efforts\cite{Breckinridge2015, Krist2017, Schmid2018, vanHolstein2018}.
\begin{figure} [ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=.6\textwidth]{polarimetry.jpg}
\end{tabular}
\end{center}
\caption[]{ \label{fig:pol cal} Effect of instrument response (Mueller Matrix) on the source intrinsic polarization. The quantities in the purple box should be calibrated on each target, while the quantities in the orange box should require less frequent calibration. CGI will not measure circular polarization and source-intrinsic circular polarization is expected to be small in most cases.}
\end{figure}
\subsection{Astrometry and Photometry}
The high dynamic range and small field of view (FOV) of CGI images pose a significant challenge for both relative and absolute astrometric and photometric calibration.
CGI will achieve single-frame flux ratios of $10^{-8}$ or better; this dynamic range exceeds detector capabilities, and so all science frames will be taken with the primary star occulted by the coronagraph.
Without coronagraph masks, the FOVs of the imaging and IFS channels are 4.5'' and 1.1'' in radius, respectively; with coronagraph masks and corresponding field stops, the FOVs correspond to the detection limit curves shown in Figure \ref{fig:theplot}.
There are three primary challenges: determining the location of the star behind the coronagraph mask, flux calibrating coronagraphic images, and determining the plate scale, rotation, and distortion map.
The latter requires periodic calibration, while the former two must be done for every science image and sequence.
Many ground-based systems use ``satellite spots'' for relative astrometric and photometric calibration (i.e.: relative to the host star flux and location).
Satellite spots are copies of the primary star PSF, injected at known flux levels and known offsets from the central star.
There are two ways to inject satellite spots: pupil plane amplitude modulation (grids on pupil plane optics\cite{Sivaramakrishnan2010a}) or pupil plane phase modulation (sinewaves on deformable mirrors\cite{Jovanovic2015}).
The former has the advantage of stability, but is also therefore inflexible; the location and amplitude cannot be adjusted.
Satellite spots can also suffer from interactions with residual stellar speckles; the four spots in the GPI image in Figure \ref{fig:sat spots} show slightly different morphologies.
If deformable mirror sinewaves are used, the amplitude and location of the spots can be adjusted on each target.
Furthermore, the phase of the sinewaves can be modulated to average over quasistatic speckle interactions\cite{Jovanovic2015}. SCExAO uses this approach, and CGI would likely follow suit.
One drawback of the sinewave method is that flux calibration relies on precise knowledge of the amplitude of the applied sinewaves; actuator calibration errors that would be removed in a closed-loop operation are not removed in this open-loop operation.
DM calibration for CGI is the subject of ongoing efforts; the actuator accuracy will determine CGI's flux calibration accuracy.
\begin{figure}[ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=.4\textwidth]{sat_spots.jpg}
\end{tabular}
\end{center}
\caption[]{ \label{fig:sat spots}
A coronagraphic image with ``satellite spots'' (enclosed by blue dashed circles). Satellite spots provide both relative astrometry and photometry in each frame. CGI will inject satellite spots by placing one or more sinewaves on its deformable mirrors.}
\end{figure}
Absolute photometric calibration of the satellite spots can be achieved in one of two ways: observations of an astrophysical binary system or self-calibration with a series of tiered satellite spots.
The former requires a well-calibrated astrophysical point source with projected separation of several tenths of an arcsecond and a flux $\lesssim 100$ times the flux of the satellite spots ($10^{-5}-10^{-6}$).
Such an object is beyond reach of all current visible-light high-contrast instruments, and so this method may be infeasible for CGI.
An alternative approach would employ a series images with satellite spots of varying flux ratios.
A detailed CGI calibration plan is under development and will also investigate whether neutral density filters and/or variable detector gains are necessary.
Absolute astrometric calibration requires both an initial calibration, including a distortion map, and periodic checks of plate scale and orientation\cite{Maire2016,Wang2016a}.
The initial calibration will be conducted pre-launch with calibration light sources.
In-flight checks of the orientation and separations of the satellite spots could use relatively bright astrophysical calibrators (such as globular cluster fields), as the fluxes of the satellite spots could be significantly increased relative to their fluxes in science frames.
The astrometric fields would be calibrated by instruments on well-characterized observatories such as HST and Keck.
CGI could also be tied to the WFIRST Wide Field Instrument (WFI); a WFI snapshot taken at each pointing could be used for absolute astrometry, provided that the WFI-to-CGI mapping is calibrated periodically.
\section{INTEGRAL FIELD SPECTROGRAPH DATA PROCESSING}
The CGI IFS will follow in the footsteps of several high-contrast ground-based IFSs, including OSIRIS\cite{Larkin2006}, P1640\cite{Hinkley2011}, SPHERE\cite{Claudi2008}, GPI\cite{Larkin2014}, and CHARIS\cite{Peters2012}.
IFSs optically subdivide the focal plane into spatial pixels (``spaxels'') with mirrors or lenslets before passing the light through a disperser.
The resulting array of ``microspectra'' (one low resolution spectrum per spaxel) are imaged onto the detector.
The CGI IFS will use a lenslet array and an $R\sim50$ prism, with a maximum bandpass of 20\%. The current baseline includes two 18\% bandpass filters that will span roughly 600--800~nm.
Post-processing is required to reconstruct the 3D data cube (sky x, sky y, and wavelength) from the array of microspectra.
The most straightforward approach is boxcar (aperture) extraction; all pixels in the extraction region are given equal weight\cite{Mesa2015, Wang2017}.
A more sophisticated approach, Horne extraction, uses a wavelength-dependent axi-symmetric Gaussian PSF profile\cite{Horne1986}\footnote{Horne originally referred to this as ``optimal extraction,'' and this terminology is sometimes still used, although this method is not optimal in all cases.}.
These approaches do not inherently account for wavelength channel covariance caused by the extended PSF wings\cite{Greco2016}.
A further refinement, least squares ($\chi^2$), uses high-fidelity lenslet PSFs to fit the spectrum with a linear combination of tophats of varying intensity and central wavelength, convolved with the lenslet PSFs\cite{Brandt2017}.
This approach decreases the effects of channel-to-channel covariance and lenselet PSF variance across the FOV.
Figure \ref{fig:CHARIS redux} shows a single wavelength channel of CHARIS IFS data, extracted using each of these different techniques: boxcar, Horne, and $\chi^2$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{CHARIS_redux.jpg}
\end{center}
\caption[]{ \label{fig:CHARIS redux}
Single wavelength channel of CHARIS broadband IFS data, reduced with three different spectral extraction techniques. From left to right: boxcar, Horne, and $\chi^2$. Image stretch is logarithmic and is the same for all three images.}
\end{figure}
Wavelength calibration requires knowledge of the detector x/y location and dispersion profile of each microspectrum.
For both ground-based instruments and CGI, calibration proceeds in two stages\cite{Wolff2016, Brandt2017, Zimmerman2011}.
The initial calibration of lenslet locations and dispersion profiles is conducted off-sky/pre-launch with bright lab sources such as arclamps, tunable filters, or lasers.
The lenslet arrays may shift globally over the course of operations, and so the global translation and low-order distortion should be checked frequently, with cadence determined by the stability of the instrument.
Ground-based instruments have the option to take snapshots of arclamps or other iternal sources for this purpose.
However, the CGI baseline design does not include an internal light source, and so wavelength calibration must be conducted on-sky; both 1\% narrowband observations of reference stars and observations of absorption or emission line targets are under consideration.
\section{SUMMARY}
The Coronagraph Instrument on the Wide Field Infrared Survey Telescope will demonstrate visible-light high-contrast imaging, polarimetry, and integral field spectroscopy at unprecedented sensitivity.
CGI will enable the study of exoplanets and circumstellar disks in reflected visible light, at fluxes as low as $10^{-9}$ the flux of the primary star.
This will be an improvement of roughly three orders of magnitude beyond current ground-based capabilities, and will constitute a shift from primarily near infrared wavelengths to wavelengths as low as 550~nm.
CGI can apply many of the lessons learned from ground-based high-contrast instruments in the areas of wavefront control, observing strategy, calibration, and integral field spectrograph data processing.
Continuous control of low-order aberrations with a dedicated LOWFS/C system is needed for optimal sensitivity at small working angles.
Sensing of non-common path low order aberrations and of high order aberrations is best achieved with focal plane WFS using science camera images themselves.
Successful WFS/C requires precise knowledge of the instrument model and deformable mirror calibration.
Spectral and polarimetric differential imaging are not likely to be applicable due to chromatic and polarization-dependent speckle behavior, and so CGI observing strategies will be designed around angular and reference differential imaging.
Relative astrometric calibration will use ``satellite spots'' injected by sinewaves on the deformable mirror; absolute astrometry will then be calibrated against known systems and perhaps against the WFIRST Wide Field Instrument.
Relative photometry will also use satellite spots; absolute calibration of the flux of the spots requires precise knowledge of the deformable mirror shape.
A combination of on-sky calibrators and instrument models must account for instrumental polarization as well as crosstalk between polarization states and transmission efficiency of each linear state.
Spectral extraction requires both wavelength calibration and data processing techniques that account for covariances between wavelength channels.
\section*{ACKNOWLEDGEMENTS}
\copyright 2018.
This research was carried out in part at at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration; government sponsorship acknowledged.
|
1,116,691,499,399 | arxiv | \section{Introduction}
The detection of single atoms with high fidelity is an important requirement for many modern atomic physics experiments, and in particular for quantum information science. Single particle imaging has been achieved using fluorescence imaging of ions~\cite{Neuhauser1980} and neutral atoms~\cite{Schlosser2001, Nelson2007, Wilzbach2009, Bucker2009, Heine2010}, but is difficult in atom-chip experiments due to a scattering background from the chip~\cite{Wilzbach2009}. A different approach is to enhance the atom-photon interaction using resonant cavities~\cite{Goldwin2011, Bochmann2010, Gehr2010}. However, free-space absorption imaging of single particles~\cite{Wineland1987, Streed2012, Tey2009} is still a daunting task.
In this paper we propose absorption imaging in a standing wave as a simple and effective way to improve the signal-to-noise ratio (SNR) of atom detection. This is particularly useful in the context of atom-chip experiments~\cite{Smith2011} where a standing wave naturally forms if the imaging is done perpendicular to the chip surface. We will take the setup currently in use in our group as a reference~\cite{Leung2011, Gerritsma2007, Whitlock2009} for realistic experimental parameters.
In the first section we will calculate the absorption signal of a single atom, both for traditional one-pass imaging and for reflection imaging as employed in our experiment.
In the second half of the paper we present numerical simulations of the expected absorption images and signal-to-noise ratio (SNR) under realistic conditions, taking into account all common sources of noise, for in-trap atoms as well as for untrapped atoms. To do so we use a hybrid approach, first calculating the movement of the atom in the imaging beam based on scattering individual photons, and subsequently investigating the absorption signal due to this atom, now taking the imaging beam as a classical wave. We show that using reflection imaging drastically increases the expected SNR and thus the single-atom detection accuracy.
\section{Single-Atom Absorption Imaging}
Our starting point is the absorption signal of a single atom at rest, illuminated by a resonant laser beam with homogeneous intensity distribution (see Fig. \ref{fig:setup}). As is shown in the appendix the intensity in the imaging plane is then given by
\begin{equation}
\frac{I_\mathrm{abs}(\rho)}{I_\mathrm{in}(\rho)}=\frac{\sigma}{1+s}\,\left[\Re{p(\rho)}-a\,\chi\,|p(\rho)|^2\right]
\end{equation}
where $I_\mathrm{abs}(\rho)$ is the absorbed intensity, i.e. the missing intensity in the image plane with image-plane coordinates $\rho$, $I_\mathrm{in}$ is the incoming intensity, $\sigma$ is the absorption cross section of the atom, $s$ the saturation parameter in the object plane, $p(\rho)$ the point-spread function of the imaging system (see below), $a=\left[\int{|p(\rho)|^2\,d^2\rho}\right]^{-1}$ the effective area over which the signal is distributed in the detection plane and $\chi$ is the fraction of the scattered light collected by the imaging system.
In an experiment the above quantities $a$ and $I_\mathrm{abs}(0)$ are not directly accessible due to the finite size of any detector and the finite time resolution. Instead one would observe the quantities
\begin{equation}
\begin{split}
N_\mathrm{det}(x) &= \int_Ad\rho\int_0^\tau dt\left(I_\mathrm{in}-I_\mathrm{abs}\right)\\
N_\mathrm{ref}(x) &= \int_Ad\rho\int_0^\tau dt I_\mathrm{in}
\label{eq:Nphot}
\end{split}
\end{equation}
where $x$ is an index over camera pixels, $A$ is the size of a pixel and $\tau$ is the exposure time, i.e. the duration of the imaging pulse. Both $\sigma$ and $s$ now depend on time, as the atom acquires a finite velocity during the imaging pulse, leading to Doppler shifts. In addition the local saturation parameter in the object plane varies in the case of standing wave imaging, as will be discussed below. We can then define the apparent column density of atoms per pixel in the image plane as
\begin{equation}
n(x) = \frac{1+s_0}{\sigma_0}\frac{N_\mathrm{det}(x)}{N_\mathrm{ref}(x)}
\label{eq:atomdensity}
\end{equation}
where $s_0$ is the saturation parameter for the intensity of the incoming beam and $\sigma_0$ is the absorption cross section of an atom at rest for the incoming intensity. The apparent atom number per pixel would be given by $N_\mathrm{app}(x) = A\,n(x)$.
As the atoms will move out of the focal plane during the imaging pulse, simulating this signal will require a point-spread function that is a function not only of the image plane coordinates, but also takes defocussing into account. This PSF can be numerically determined as
\begin{equation}
p\left(\rho, f; t\right) = \frac{2\pi}{\rho_0^2}\int_{0}^{1}r\,\mathrm{exp}(i f r^2)\times J_0\left(\nicefrac{2\pi\rho\,r}{\rho_0}\right)dr
\label{eq:psf}
\end{equation}
for any given defocus $f=\frac{2\pi}{\lambda}Z(1-\sqrt{1-\mathrm{NA}^2})$ where Z is a real-space coordinate in the imaging direction~\cite{Janssen2002}, NA is the numerical aperture of the imaging system and both $\rho$ and $Z$ might depend on time. $\rho_0=\lambda/\mathrm{NA}$ is proportional to the resolution of the imaging system, determined by the wavelength and the numerical aperture. Aberrations in the imaging system can be taken into account in a similar manner. We assume these to be negligible near the optical axis. In the simulations the numerical point-spread function given above is used to calculate the intensity in the image plane.
\subsection{Analytical Estimates}
\label{sec:analytics}
For a better intuition of the above results we consider some simple analytic estimates. Assuming a stationary atom in perfect focus of a two-lens imaging system with unit magnification ($f$-$2f$-$f$) we can use a simple point-spread function of the form $p(\rho) = 1/(\rho\rho_0)J_1(2\pi\rho/\rho_0)$ where $p(0)=\pi/\rho_0^2=a^{-1}$. Using $\chi=(\nicefrac{3}{8})\mathrm{NA}^2$ and assuming a 2-level system where $\sigma_0=3\lambda^2/2\pi$ the maximum signal at the center of the image simplifies to
\begin{align}
\frac{I_\mathrm{abs}(0)}{I_\mathrm{in}(0)} &= \frac{1}{1+s_0}\left(\frac{3}{2}\mathrm{NA}^2-\frac{9}{16}\mathrm{NA}^4\right)
\label{eq:intensity_estimate}
\end{align}
which is thus determined purely by the numerical aperture of the imaging system and the saturation parameter $s_0$. For a numerical aperture $\mathrm{NA}=0.4$ as is the case in our experiment we then find an absorption in the centre of the image of $I_\mathrm{abs}(0)/I_\mathrm{in}(0)=0.23$ in the low saturation limit $s_0\ll1$ for a single atom.
We can further consider the total apparent number of atoms one would extract from this signal as the peak signal times the area over which the signal is distributed,
\begin{equation}
N_\mathrm{app} = a\frac{1+s_0}{\sigma_0}\frac{I_\mathrm{abs}(0)}{I_\mathrm{in}(0)} = 1-\frac{3}{8}\mathrm{NA}^2
\label{eq:napp}
\end{equation}
This shows that the expected number of atoms is reduced by the second, NA-dependent term. This term is due to light which is scattered by the atom, but scattered into the solid angle of the lens, thus reducing the apparent amount of light that is absorbed. For a numerical aperture of 0.4 we therefore expect to find an apparent atom number $N_\mathrm{app}$ of 0.94 rather than one.
The commonly used quantity \emph{optical density} (OD) can thus be defined for a single particle by the point-spread function of the imaging system, i.e. $\mathrm{OD} = -\ln\left(1-\nicefrac{I_\mathrm{abs}}{I_\mathrm{in}}\right)$. At the center of the image (for vanishing pixel size) we find for the peak optical density a value of $\mathrm{OD}(0) \approx \nicefrac{3}{2}\mathrm{NA}^2+\nicefrac{9}{16}\mathrm{NA}^4$ in the low-saturation limit (where the sign change is due to a series expansion of the logarithm). For our numerical aperture this equals a peak optical density of 0.26.
It is worth noting here that the atom number extracted in the usual way from optical density, in this case
\begin{equation}
N_\mathrm{OD}=-a\frac{1+s_0}{\sigma_0}\,\ln\left(1-\frac{I_\mathrm{abs}\left(0\right)}{I_\mathrm{in}\left(0\right)}\right)
\label{eq:nod}
\end{equation} would yield 1.07 atoms rather than 0.94. The atom number is slightly overestimated, as Lambert-Beers law is valid only for a continuously absorbing medium, not for a single absorber. In the following we always use the spatially dependent equivalent of \eqref{eq:napp} rather than \eqref{eq:nod} for the atom number.
\subsection{Reflection Imaging}
\begin{figure}[htb]
\centering
\includegraphics[width=7cm]{setup}%
\caption{Sketch of the imaging system considered for this paper. On the left the situation of a travelling wave without reflection is depicted. On the right the situation of reflective imaging (with the atom chip on top) is shown.}%
\label{fig:setup}%
\end{figure}
The above discussion is for an atom imaged by a travelling wave, as depicted in Fig.~\ref{fig:setup}(a). In reflection imaging, such as in Fig.~\ref{fig:setup}(b) the situation is slightly different. In this case the probe light forms a standing wave at the position of the atoms. For Rb atoms in our magnetic lattice~\cite{Leung2011, Gerritsma2007, Whitlock2009} we expect a ground-state size of the atomic wavefunction of about \unit{40}{\nano\meter}, about a factor of $10$ smaller than the period of the standing wave. We therefore approximate the atom as a point-particle which can initially be positioned in an anti-node of the standing wave. In our atom chip experiment the trap position can be adjusted by means of an externally controlled magnetic field to achieve this.
The intensity of the probe beam then varies sinusoidally along the probe direction and the maximum intensity in the anti-nodes is a factor four greater than the intensity of the incoming travelling wave. The coherent scattering amplitude $\mathcal{A}$ scales with the the local field amplitude (at low saturation). Thus the scattered wave is two times as strong as for a travelling wave, leading to an accordingly higher observed signal. More precisely, the saturation parameter will also change.
It should be noted here that while in the limit of low saturation and short exposure times the observed signal will indeed be two times larger than for a travelling wave, for realistic imaging with finite exposure time the atom starts to probe the spatial varation of the light field, leading to an observed signal which depends on the exact imaging parameters. The expected signal can then only be predicted by simulations such as described below.
A further important difference to travelling wave imaging is that the atom is not pushed out of the focus of the imaging system by the probe beam, but performs a random walk in all dimensions. This point will also be further discussed in the simulations below.
Another factor of two could in principle be gained because the scattered wave is emitted towards and reflected by the mirror. This again doubles the amplitude of the scattered wave in the detection plane. A practical consideration here is that for single-atom imaging one will typically use high-NA imaging, with a Rayleigh length $z_R=\lambda/\left(\pi\mathrm{NA}^2\right) \lesssim \unit{10}{\micro\meter}$. For the typical situation in our experiment the atom is at about \unit{10}{\micro\meter} from the surface. This implies that the atom and its mirror image are not simultaneously in focus in the image plane. In practice this reflected wave hardly contributes to the optical density for our experimental parameters. If the imaging is done extremely close to a reflecting surface, or if the Rayleigh length of the imaging system is significantly larger this can however be a significant effect.
\section{Simulations}
\label{sec:simulations}
After having calculated the absorption signal of a single atom at rest in the previous section, we would now like to determine the accuracy with which we can detect a single atom in a more realistic setup. To do so we have to take two important effects into account: recoil blurring and imaging noise. The first describes the process where the atom starts to move due to scattering photons, leading to a blurring of the image. The second describes the uncertainty with which one can determine the local intensity of the imaging beam, both for fundamental reasons, i.e. shot-noise, and technical reasons such as read-out noise of the camera. These two cases impose counteracting constraints: recoil blurring is minimal for short exposure times and weak probe intensities, shot-noise is reduced for large photon counts, i.e. long exposure times and high probe intensities. In addition, at high intensities the signal saturates. Therefore there exists an optimum signal-to-noise ratio (SNR) for a certain exposure time and probe intensity. We use a hybrid approach in treating these effects: recoil blurring is calculated based on scattering individual photons off the atom (see below), while the absorption signal is calculated based on a classical-wave approach as described above.
In our simulations we distinguish four cases:
\begin{description}
\item[a)] free atom in travelling wave
\item[b)] trapped atom in travelling wave
\item[c)] free atom in standing wave
\item[d)] trapped atom in standing wave
\end{description}
where the cases a) and b) are for traditional one-pass imaging, and the cases c) and d) are for our reflective atom-chip setup (see Fig. \ref{fig:setup}). While case a) and c) assume a freely moving atom, case b) and d) assume the atom to be held in a magnetic trap. For this trap we assume the potential calculated from the magnetic pattern of microtraps present on our atom chip~\cite{Leung2011} which is approximately equivalent to trap frequencies of $(\omega_x, \omega_y, \omega_z) = 2\pi\times$\unit{(38.1, 36.5, 14.0)}{\kilo\hertz} and a trap depth of \unit{8.8}{G}. In the simulations the calculated potential is used without harmonic approximations. The atom is treated as free as soon as it acquires sufficient energy to escape the trap. We assume optical pumping to non-trapped states is negligible for an appropriate choice of polarisation for the imaging beam.
\begin{table}
\centering
\begin{tabular}{cccccccc}
case & $\tau$ ($\mu$s) & $I/I_\mathrm{sat}$ & SNR$_\mathrm{px}$ & SNR$_\mathrm{CRB}$ & N$_\mathrm{app}$ & $p_{1, 1}$ & $p_{1, 0}$\\\hline
a) & 12.4 & 0.59 & 1.55 & 1.57 & 0.94 & 57\% & 21\%\\
b) & 12.0 & 0.65 & 1.57 & 1.59 & 0.95 & 57\% & 20\%\\
c) & 17.0 & 0.56 & 1.73 & 1.76 & 0.74 & 62\% & 19\%\\
d) & 42.5 & 0.56 & 2.63 & 2.68 & 0.83 & 82\% & 9.5\%\\
\end{tabular}
\caption{Optimum exposure parameters, and resulting SNR for the four cases described in the text. SNR$_\mathrm{px}$ is the SNR obtainable from evaluating a single pixel, while SNR$_\mathrm{CRB}$ is the SNR for an estimator achieving the Cramér-Rao bound (without taking fringe-removal into account). The normalization factor N$_\mathrm{app}$ for case a) and b) is as expected from the analytic treatment of section \ref{sec:analytics}. The normalization factor in cases c) and d) can only be obtained from the simulations. Finally we give the probability of true positive measurements $p_{1, 1}$ and false positive measurements due to the presence of zero atoms $p_{1, 0}$ (see \ref{sec:noise}).}
\label{tab:parms}
\end{table}
\subsection{Recoil Blurring}
As the atom is illuminated by the imaging beam it scatters photons at a rate determined by the local intensity at the position of the atom as well as any detuning due to e.g. Doppler shifts as the atom acquires momentum with each of these scattering events. In the case of a travelling wave the photon scattering drives the atom strongly along the direction of the imaging beam, as the absorption of a photon always happens in this direction, performing a biased random walk. This quickly leads to the atom moving out of the focal plane of the imaging lens. The atom further performs an unbiased random walk in momentum space in the plane perpendicular to the imaging direction due to the spontaneous emission of a scattered photon, leading to a blurring of the image. In the case of a standing wave the atom can absorb a photon from both directions along the imaging axis, giving rise to an unbiased random walk in the imaging direction as well as the plane perpendicular to it. The atom therefore leaves the focus of the lens much more slowly than in the case of a travelling wave.
We simulate the movement of the atoms by calculating stochastic trajectories where random scattering events occur at the local scattering rate, taking the effect of the trap into account as a classical potential. In the case of an untrapped travelling wave we recover the expected behaviour, i.e. the root-mean-square (RMS) position along the imaging beam growing as $t^2$ and the RMS position in a plane perpendicular to the imaging direction growing as $t^{3/2}$, as is expected for uniform acceleration and a random walk in momentum space respectively. The effect of the trap is almost negligible in the case of a travelling wave, as the force of the light field is much greater than the restoring force of the trap potential. In the case of reflective imaging the behaviour proportional to $t^{3/2}$ is also expected in the probe direction. This is however modified by the periodic intensity of the standing wave. Since the net force on the atoms is much lower than in the case of a travelling wave the influence of the trapping field is also more important, particularly at low intensities.
These calculated RMS trajectories are then used in the integrations of Eq. \eqref{eq:Nphot} to determine the signal after a given exposure time for a certain probe intensity.
\subsection{Detection Noise}
\label{sec:noise}
To determine the feasibility of absorption imaging for single atoms one needs to consider various sources of noise in addition to the atomic trajectories to determine the signal-to-noise ratio. In the following we assume near shot-noise limited imaging, with further contributions from camera read-out noise and dark counts, as well as a finite quantum efficiency of the imaging system. Equations \eqref{eq:Nphot} determine the number of photons per pixel in the signal and light fields, and can be trivially modified to take a finite quantum-efficiency of the imaging system into account. With these values we can calculate the expected signal $n(x)$ as atom density per pixel (Eq. \ref{eq:atomdensity}) for any combination of exposure time and intensity. The variance of this signal is given by
\begin{equation}
\sigma_n(x)^2 = \left(\frac{1+s_0}{\sigma_0}\right)^2\left(\frac{\sigma_\mathrm{det}^2}{N_\mathrm{ref}^2} + \frac{N_\mathrm{det}^2\sigma_\mathrm{ref}^2}{N_\mathrm{ref}^4}\right)
\end{equation}
where $N_\mathrm{det}$ and $N_\mathrm{ref}$ are the number of electrons per pixel in the signal and reference images respectively, also integrated over pixel size and exposure time. $\sigma_\mathrm{det}^2$ and $\sigma_\mathrm{ref}^2$ are the variances of these quantities, mostly determined by photon shot noise ($\propto N$) and camera readout noise.
The signal to noise ratio per pixel is then given by $\nicefrac{n(x)}{\sigma_n(x)}$; one can further improve the signal to noise ratio by not only evaluating the central pixel, but using an optimal estimator achieving the Cramér-Rao bound, such as
\begin{equation}
N = \frac{\sum{n(x) q(x)}}{\sum q(x)^2}
\label{eq:estimator}
\end{equation}
where $q(x)$ is the spatial mode function of the signal~\cite{Ockeloen2010}. In our results we list the SNR both for evaluating only the central pixel and for an estimator achieving the Cramér-Rao bound.
We define the accuracy of the measurement as the probability of finding true positives $p_{1, 1}$, i.e. finding one atom if there is one atom present. Here we assume all measurement outcomes for which $0.5 < \mathrm{N} < 1.5$ to indicate the presence of exactly one atom. We have determined this value by simulations of $10^6$ individual absorption images for each case. The results are in excellent agreement with those expected for a normal distribution, for which the accuracy is given by $F=\mbox{Erf}(\nicefrac{z}{\sqrt{2}})$ with $z=\nicefrac{N}{2\sigma}$.
We can also determine the probability of finding false positives in the absence of any atoms $p_{1, 0}$, i.e. finding one atom if zero atoms are present. False positives due to the presence of more than one atom can not easily be determined, as an accurate calculating of the absorption signal of two atoms is non-trivial. We expect these to be of similar magnitude as those due to the absence of any atoms.
\subsection{Simulation Results}
\begin{figure}[ht]
\centering
\includegraphics[width=7cm]{SNR}
\caption{Simulated SNR as function of intensity of the incoming beam (in units of saturation intensity) and exposure time. The four subplots correspond to the four cases described in the text as indicated near the top and right axes, the color scale is normalized to the maximum SNR of case d). The black dot marks the position of the optimum.}
\label{fig:SNR}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=7cm]{OD}
\caption{Simulated single-shot absorption images for optimum exposure parameters, including fringe removal. The four subplots correspond to the four cases described in section \ref{sec:simulations}. Bright colors correspond to high optical density. Each subplot shows $5\times5$ pixels, with the atom initially located at the center. At larger distances from the initial position the PSF becomes negligibly small. Using Eq. \ref{eq:estimator} we extract N=(1.32, 1.39, 0.56, 1.26) atoms for subplots a)-d) respectively. These atom numbers are normalized by $N_\mathrm{app}$ for each imaging setup.}
\label{fig:od}
\end{figure}
We simulate the expected absorption image and SNR assuming a numerical aperture of the objective lens of 0.4 (Edmund Optics NT47-727 with custom AR coating), a total magnification of the imaging system of 10, a pixel size of the camera of $\unit{(13 \times 13)}{\micro\meter\squared}$ (Andor iKon-M 934), a quantum efficiency of 0.9 and a readout noise of 13 counts per pixel~\cite{Ockeloen2010}. We consider resonant $\sigma^+$-transitions on the D2-line of Rubidium 87, where $\sigma_0=\unit{2.907\times10^{-9}}{\centi\meter\squared}$ and the saturation intensity $I_\mathrm{sat}=\unit{1.669}{\milli\watt\per\centi\meter\squared}$~\cite{steck2010}. We have confirmed that in the limit of low saturation our simulations yield the expected number of atoms in all four cases.
We find optimum imaging parameters by numerically maximizing the signal-to-noise ratio of the central pixel as a function of exposure time and intensity, i.e. maximizing SNR(0) where
\begin{equation}
\mathrm{SNR}(x) = \nicefrac{n(x)}{\sigma_n(x)}
\end{equation}
using the precalculated trajectories of an atom in the integrations of Eqs. \eqref{eq:Nphot}.
To avoid confusion in the case of reflective imaging where the intensity is spatially varying, the intensity is here always given in units of saturation intensity for the incoming beam (cf. Fig. \ref{fig:setup}). The local intensity at the initial position of the atoms is therefore higher by a factor of four for the standing wave.
We also determine the SNR of an estimator achieving the Cramér-Rao bound by weighting the signal by the mode function of the atomic distribution as described above. SNR as a function of exposure time and intensity is depicted in Fig. \ref{fig:SNR}. Table \ref{tab:parms} lists the signal-to noise ratio for optimum imaging parameters. Note that this can be further improved by a factor $\approx \sqrt{2}$ by using the \emph{fringe-removal algorithm}~\cite{Ockeloen2010} which reduces the shot noise in the light field by optimal averaging of reference images even in the absence of any fringes. Fig. \ref{fig:od} finally shows simulated single-shot absorption images of a single atom for the parameters given above, including the use of fringe removal. The number of atoms extracted from these images using the optimum estimator is given in the figure caption.
Depending on the imaging method the effect of recoil blurring on the optical resolution of the imaging system can be significant. For our optimum exposure parameters we find the strongest recoil blurring in case c), where the RMS position of the atoms in the object plane increases by \unit{0.9}{\micro\meter} during the imaging pulse. For all other cases considered here recoil blurring is smaller by at least a factor of 3. This blurring should be compared to the base resolution of the imaging system of \unit{1.2}{\micro\meter} (Rayleigh criterion). Furthermore the images are discretized by the effective pixel size of \unit{1.3}{\micro\meter} in object space.
\section{Discussion \& Conclusion}
In this paper we have outlined the theoretical prospects of detecting single rubidium atoms using absorption imaging. We find the optical density of a single atom in single-pass imaging to be ultimately determined only by the numerical aperture of the imaging system with a peak optical density of $OD = \nicefrac{3}{2}\mathrm{NA}^2\left(1+\nicefrac{3}{8}\mathrm{NA}^2\right)$ in the low saturation limit. We have calculated optimum imaging parameters for a number of different cases and could show that the use of a reflective imaging setup can significantly improve the signal-to-noise ratio of absorption imaging. We expect an accuracy of about $82\%$ for the detection of single atoms for the optimum parameters given above in case d), which can be improved to $94\%$ (with a false-positive rate due to zero atoms of $4\%$) using the reduction in shot noise achieved by the fringe removal algorithm~\cite{Whitlock2010}. This compares to a detection accuracy of only $74\%$ (with a false-positive rate due to zero atoms of $15\%$) for single-pass imaging with the same imaging system. For measuring statistical averages in a regular array of traps further improvements can be made using correlation analysis~\cite{Whitlock2010}.
We would like to thank Jasper Reinders for help with the simulations and N.J. van Druten for helpful discussions of the manuscript.
This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).
|
1,116,691,499,400 | arxiv | \section{Introduction}
The GRism ACS Program for Extra-galactic Science (GRAPES)
project is a slitless spectroscopic survey of
the Hubble Ultra Deep Field (UDF) region (Beckwith 2006).
GRAPES exploits the high spatial
resolution and low background of the G800L grism on the
Advanced Camera for Surveys (ACS). The resulting spectra
can detect continuum levels down to $z_{AB} > 27$ magnitude,
making them the deepest ever obtained for continuum spectroscopy.
Additionally, they can detect emission lines to $\sim 5\times
10^{-18} \ergcm2s$, more sensitive than any previous slitless survey
and comparable to the sensitivities of typical slit
spectra on $6$--$10$ meter class telescopes.
GRAPES thereby provides a rich
data set with a wide range of potential applications, from
Lyman breaks in distant galaxies (Malhotra et al. 2005),
to 4000 \AA\ breaks in galaxies with older stellar populations (Daddi
et al. 2005; Pasquali et al. 2005), to spectral classification of
faint Galactic stars (Pirzkal et al. 2005).
The use of slitless spectrographic surveys to search for emission-line
galaxies dates back more than three decades. The most famous surveys
of this type used objective prisms on Schmidt telescopes. Early surveys
utilized photographic plates in order to maximize the field-of-view.
Surveys of this type include the Tololo (Smith 1975; Smith et al. 1976)
and Michigan (MacAlpine et al. 1977, MacAlpine \& Williams 1981) surveys.
More recently the Universidad Complutense de Madrid (Zamorano et al.
1994, 1996) survey defined a well-studied sample of H$\alpha$ selected
galaxies. The advent of large CCD detectors available on Schmidt
telescopes has made possible large-scale digital objective-prism surveys.
The first of this type is the KPNO International Spectroscopic Survey
(KISS; Salzer et al. 2000). Objective prism surveys from the ground
have in general been most effective for relatively nearby galaxies.
Efforts to search for high redshift Lyman-$\alpha$ emitting galaxies
with slitless spectrographic surveys also be dated back to as early
as 1980s (e.g., Koo \& Kron 1980; for a review, see Pritchet 1994).
However, only the advent of slitless spectroscopic capabilities
on the Hubble Space Telescope on both the STIS and NICMOS instruments,
enabled such surveys to achieve high sensitivity. Examples include
H$\alpha$ from redshift
0.75 to 1.9 with NICMOS (McCarthy et al. 1999; Yan et al. 1999)
and [\ion{O}{2}] $\lambda$3727\ from 0.43 to 1.7 with STIS (Teplitz et al.
2003a,b). These surveys were done in parallel observing mode to maximize
the total area observed. These were followed by parallel mode
surveys using ACS during the first year of ACS operations: The APPLES
survey, led by Rhoads (see Pasquali et al 2005),
and a similar effort led by L. Yan (see Drozdovsky et al 2005).
GRAPES is a natural successor to these efforts, and represents
a major step forward in sensitivity and robustness, thanks to the
improved experimental design allowed by pointed observations.
In the present paper, we present a redshift catalog for strong emission
lines galaxies identified in the GRAPES spectra for both nearby and
distant galaxies. We combine our line wavelengths
with photometric redshift estimates from broad band photometry to
obtain accurate redshifts for most of the emission line sources in the
UDF. Such redshifts are a starting point in studies of cosmological
evolution. Emission line galaxies are of particular physical interest
for several reasons. $H\alpha$, [\ion{O}{2}] $\lambda$3727\ and [\ion{O}{3}] $\lambda \lambda$4959,5007\ can be
used to study the evolution of star formation rate (e.g., Gallego et
al. 1995, 2002). We will present such analysis from the
GRAPES emission lines, including the completeness analysis,
in forthcoming paper (Gronwall et al., in preparation).
Ly$\alpha$ can be a prominent signpost of actively star forming
galaxies at the highest
presently available redshifts, and can be used to probe physical
conditions in these galaxies (Malhotra \& Rhoads 2002) and to study
the ionization state of the intergalactic medium (Malhotra \& Rhoads
2004, 2006).
The paper is organized as follows. We describe the data reduction
and emission line search procedure in section 2, then we present
our results in section 3, followed by a discussion
section (section 4). We present our redshift catalog for
emission line galaxies in table~1.
\section{Observations and Reductions}
Slitless spectroscopy using the ACS G800L grism
on the Hubble Space Telescope has its own unique characteristics,
which we discuss briefly here as useful background to understanding
the strengths and limitations of our data set.
First, the high spatial resolution and low sky background
afforded by HST result in highly sensitive spectra. Second,
the dispersion is low, at $40$\AA\ per pixel.
Third, the slitless nature of the observations imply that spectra may
overlap, and that the effective spectral resolution scales inversely
with the angular size of the source. Thus, to detect an emission
line, its equivalent width (in \AA) should not be far below
the object size times spectrograph dispersion. Our
selection effects for a range of continuum magnitude, line flux,
and equivalent width are discussed below. Fourth, continuum subtraction
is nontrivial in grism spectra. The basic
difficulty is in identifying object-free regions of the dispersed
grism data to scale and subtract a sky model. This can be done very well,
but not perfectly, with residuals at the level of a few $\times 10^{-4}$
counts/s/pixel, corresponding to $\sim 10^{-3}$ of the sky count rate
(see Pirzkal et al 2004 for further discussion).
The implication for emission line
studies is that equivalent width measurements at the faintest continuum
fluxes are subject to potential error induced by continuum subtraction
residuals. Fifth, the grism signal-to-noise ratio and the contamination
can both be helped by using a narrow extraction window, but this
comes at the cost of aperture losses in the extracted spectrum.
Because the ACS point spread function does not vary strongly
with wavelength, the aperture losses should be largely wavelength
independent, and will have little effect on line ratio measurements.
The GRAPES observations consisted of a total of 5 epochs of observations
with comparable exposure time at 5 different orientations
(also quoted as Position Angles or PAs). By splitting the
observations into different orientations, we are able
to mitigate the effects of overlapping spectra: most objects
are uncontaminated in at least some roll angles.
Due to slight offsets in sky coverage for different roll angles,
we have varied depth in the exposures. About 10.5 square arcmin
are covered by at least 4 roll angles, and more than 12.5 square arcmin
are covered by at least one roll angle.
Because the spectra are $\sim 100$ pixels long, while the
field size is $4096$ pixels, the fraction of spectra truncated
by the field edge (and thereby lost) is only $\sim 2\%$.
A more detailed
description of GRAPES project, especially the observations and
the data reduction, can be found in Pirzkal et al. (2004).
We note that the spectral resolution, line sensitivity,
and equivalent width threshold for GRAPES emission lines are all
comparable to typical modern narrowband surveys (e.g.,
Rhoads et al 2000, 2004; Rhoads \& Malhotra 2001; Malhotra \& Rhoads
2002; Cowie \& Hu 1998; Hu et al 1998, 2002, 2004; Kudritzki et
al 2000; Fynbo, Moller, \& Thomsen 2001; Pentericci et al 2000;
Stiavelli et al 2001; Ouchi et al 2001, 2003, 2004; Fujita et al
2003; Shimasaku et al 2003, 2006; Kodaira et al 2003; Ajiki et
al 2004; Taniguchi et al 2005; Venemans et al 2002, 2004).
However, the HST grism survey gives much broader redshift coverage
and higher spatial resolution over a smaller solid angle. It
yields immediate spectroscopic redshifts in many cases, and is
immune to the strong redshift selection effects introduced in
ground-based data by the forest of night sky OH emission lines.
After the data are reduced the spectra are extracted with
the software package aXe\footnote{
http://www.stecf.org/software/aXe}.
We search for the emission lines on all extracted spectra
of the objects in the UDF field whose z-band (ACS F850LP filter)
magnitudes reach as faint as $z_{AB} = 29 \mag.$ While the
detection limit for continuum emission in the GRAPES
spectra is brighter than this ($z_{AB} \approx 27.2 \mag$;
see Pirzkal et al.~2004),
a fraction of such faint objects can have detectable emission lines.
The emission line search for this paper was performed on the 1D extracted
spectra, searching both in the spectra from individual PAs and on the
combined spectra of all PAs.
To detect lines automatically, we wrote an IDL script (``emlinecull'')
that identifies and fits lines as follows.
We first remove the continuum by subtracting a median filtered version
of the spectrum from the original data. We then determine whether the
resulting high-pass filtered spectrum shows evidence for emission
lines by sorting its pixels on signal to noise ratio and determining
the maximum {\it net} signal to noise ratio in a running total of (1,
2, ..., N) pixels. (See Pirzkal et al. 2004 for further discussion of
this ``net significance'' parameter applied to unfiltered 1D spectra.)
In addition to ``net significance,'' we also calculate a cumulative
continuum flux (based on the median spectrum that we subtracted). An
object is retained as a likely emission line source provided that (a)
its net significance exceeds $2.5$, and (b) the ratio of ``continuum''
(i.e., low-pass filtered flux) to ``line'' (i.e., high-pass filtered
flux) is $\le 10$.
The second criterion eliminates broad features of low equivalent
width, thus avoiding a large catalog contamination by cool dwarf stars
with ``bumpy'' continuum spectra. The precise line profile and other
details affect the precise correspondence between this cutoff and a
true equivalent width. This criterion will of course introduce some
catalog incompleteness at low equivalent width, but this is inevitable
anyway, given the limited spectral resolution of the grism.
Once an object is identified as a likely emission line source, the
most significant peak in the filtered spectrum is identified, fitted
with a Gaussian profile, and subtracted. This peak finding and
subtraction is iterated until no pixel remaining in the residual
spectrum exceeds the noise by a factor $> 2.5/\sqrt{2} = 1.77$.
(This implies a net line significance $\ga 2.5\sigma$ for any real
selected line, since the line spread function is never narrower
than two ACS pixels.) The
output line list gives for each line the central wavelength, flux,
line width, continuum level, and equivalent width, along with
estimated uncertainties in each parameter. The fitting code is based
on the IDL ``MPFIT'' package\footnote{
http://cow.physics.wisc.edu/$\sim$craigm/idl/idl.html} written by
Craig Markwardt.
The fitting code then checks the measured significance of each line,
because line locations are initially based on the significance of
single pixels, while the final significance is based on the full
Gaussian fit to the line. At this stage we discard any candidate line whose
final significance is $<1.8\sigma$ Because this $1.8\sigma$ cutoff is
applied in individual position angles, the final significance of a
line after combining data from multiple GRAPES position angles (see
below) will usually be $\ga 3$. This step also discards any candidate lines
with a fitted FWHM $<10$\AA, because such narrow fitted widths are far
below the instrumental resolution and therefore indicate a noise spike
rather than a real spectral line.
The observed line width is usually determined by the angular size of
the object rather than its velocity width. Formally it is a
convolution of the true emission line profile and the (projected)
spatial profile of the object multiplied by the $40$\AA\ per pixel
dispersion. In general, this is dominated by the spatial term, unless
the line width exceeds $30,000 \kms \times (\theta / \hbox{arcsec})$,
where $\theta$ is the angular size of the object in the dispersion
direction. Realistically, such
extreme linewidth to size ratios are only expected in broad-lined
active galactic nuclei (AGN). Here the observed spatial scale will be
the PSF size ($0.1''$) regardless of distance, and the minimum
velocity width for a resolved line becomes $3000 \kms$.
After the emission line lists from individual PAs
are generated through EmlineCull, we merge the lists according
to following criteria: 1) A line is retained
in one individual PA if the estimated contamination from
overlapping spectra is less than 25\% of the (line plus
continuum) flux integrated over the line profile.
Otherwise, if the estimated contamination exceeds 25\%,
the line is simply discarded.
2) Two lines from two different PAs are considered
to be the same line if their wavelengths agree within their
combined line widths (FWHM), after possible offsets between
the cataloged object position and the location of strong
line emitting regions are taken into account.
3) A line is considered as a detection if it is detected
in at least 2 PAs. After this merged emission line object list
is created, we visually examine all the individual spectra on
both 1-dimensional extraction and 2-dimensional cut as a
sanity check. This resulted in removal of some objects
from the list. With the help of the emission line object
list generated from the PA combined spectra, we also visually
inspected all the combined spectra of the UDF objects down
to and sometimes fainter than $i_{AB} = 28.0 \mag$ as a completeness check.
This resulted in the addition of a few emission line objects.
Most of these were bright objects with broad emission lines of low
equivalent width, which were missed in the automated line finding
because they exceeded the threshold for continuum to line ratio.
Among the parameters returned by ``emlinecull,'' the central
wavelength (and its associated error) are the most reliable. The
other parameters are potentially affected by systematic effects. Line
flux can be suppressed to some degree by the continuum subtraction
procedure, and is furthermore underestimated due to aperture losses in
the 1D extraction. The equivalent width is vulnerable to sky
subtraction uncertainties for the faintest objects, and may also
suffer if the spatial distribution of line emitting regions does not
match that of continuum emitting regions, so that the equivalent width
within the extracted aperture is not the correct spatially averaged
equivalent width for the object. As discussed above, the line width
is essentially a measure of object angular size for most objects.
We have therefore chosen to report only the line wavelengths and
approximate fluxes.
To improve the line flux estimates, we re-calculated them as follows.
First we fit the continuum to a pair of baseline regions, one on each
side of the line, and subtract the resulting linear continuum fit
from the spectrum. The typical width for the continuum
fitting is about $2 \hbox{FWHM}$ on each side of the line.
We then do a direct integral of the the continuum-subtracted line to
determine its final flux estimate.
Because we do not know the spatial distribution of line emitting
regions in our sources {\it a priori}, we have chosen not to apply
aperture corrections to the line fluxes. However, we can calculate
what the aperture correction would be if the line flux
were distributed like the continuum flux, using the direct
images from the HUDF. To do so, we convolved the ACS i-band
PSF with a gaussian of full width half maximum equal to the size of
the object perpendicular to the grism direction, and then computed
the fraction of the total flux falling within the grism extraction
width. The resulting aperture corrections were factors of
$\sim 1.5$ to $\sim 3$, with a majority falling near the upper
end of this range.
The final emission line object list is presented in Table 1. A total
of 115 objects are listed in Table 1, of which 101 objects have UDF ID
numbers from the released UDF catalogue. Those without UDF ID are
either near a brighter object (with which they are blended in the UDF
catalog) or else lie just outside the UDF field. Note that the GRAPES
field of view covers a slightly larger area than that of UDF field due
to the combination of exposures at different orientations. The total
number of lines listed in Table 1 is 147, as multiple emission lines
are detected in some galaxies.
The detected emission lines are then identified to
the following template line list: a) Lyman-$\alpha$ ($\lambda$1216);
b) [\ion{O}{2}] $\lambda$3727; c) [O III] $\lambda$4959 and [O III] $\lambda$5007,
which are not resolved at the ACS grism resolution ($\sim 40$ \AA/pixel)
below redshift 1.0 or for extended sources, so are treated as a single feature
at [\ion{O}{3}] $\lambda$4995, their line ratio averaged wavelength;
and d) H$\alpha$ ($\lambda$6563).
These lines are chosen because they are usually the strongest
optical / near UV emission lines found in star forming galaxies.
Some AGN emission lines are also used as a template in identifying AGNs.
But the AGNs are identified primarily based on their point-like morphologies
(Pirzkal et al. 2005).
The general line identification strategy is summarized
as follows. 1) First we measure the photometric redshifts for
all the objects in the list using the photo-z code (Mobasher et al.~2004).
Only 4 objects outside the UDF field of view do not have photo-z
measurements. Images from 7 filters are used to determine the redshift.
These are: 4 ACS bands (BViz), 2 NICMOS bands (JH)
and 1 ISAAC band (K). 2) We then use the photometric
redshifts as the input to identify observed emission lines to the template
lines and recalculate the redshift. If the recomputed redshift falls
within the 95\% confidence region of the photo-z redshift, we take
it as measured redshift. 3) Visually examine all the
line identifications. In last step, we have several different
approaches: a) If an object is found to have 2 or more lines,
we calculate the wavelength ratio of different lines to
re-identify the lines and recalculate the redshift if necessary.
b) In both single and double emission line cases, if a relatively
smooth break feature is found across the emission line region,
it is very likely that this feature is the 4000 \AA\ break,
and the corresponding line can be identified as [\ion{O}{2}] $\lambda$3727.
c) In the single emission line case, if a Lyman break feature is
found near the emission line, and this Lyman break feature
can be further confirmed with the broad band fluxes from the
direct images, then the corresponding
emission line is identified as Lyman-$\alpha$.
d) If a line cannot be identified with
any method mentioned above, we simply present the measurements
of the line without deriving its redshift. In the final list,
87\% of the sample has line redshifts that are consistent
with their photometric redshifts.
Note that the GRAPES redshifts presented here do not in general
provide an independent check on photometric redshift estimates,
since a photometric redshift is used to help identify the emission
lines in the GRAPES spectra. An exception can be made for
objects with two GRAPES emission lines, in which case the wavelength
ratio of the lines is usually measured with sufficient precision to
identify them with no further information.
To assess the selection effects in our sample, we performed extensive
Monte Carlo simulations of our line identification and measurement
procedures. These simulations were performed in one dimension, using
Bruzual \& Charlot (2003) models for the continuum, with added
emission lines. These lines were taken to be gaussians with various
widths and fluxes. We selected 100~Myr, 5~Myr, and 1.4~Gyr templates
from the BC03 library, with metallicities of 0.08, 0.2 and 0.5 solar,
thus yielding 9 different templates with a range of 4000\AA\ break
strengths and stellar absorption line strengths. In total, we
simulated 1.5 million emission lines, spanning a wide range of
continuum levels (e.g. scaled BC03 template), line fluxes, redshifts,
and object sizes (which determine the effective resolution of each
grism observation and the width of the observed lines).
For each simulation, we stored the line flux,
EW, i and z band broad band AB magnitude, redshift, and object size.
Each of the simulated spectrum was multiplied by the ACS grism
response function and resampled to the grism resolution of 40\AA\ per
pixel, prior to adding noise. Our noise calculation included both
count noise from the input spectrum and the contribution of the sky
background ($\approx 0.1$ DN/pixel/s), and was scaled to match the
noise levels observed in the final, combined, GRAPES spectra. We
identified spectral lines in each simulated spectrum using the same
``emlinecull'' script used to analyze the GRAPES data, and we
determined a simulated line to be successfully detected if its
measured line center was within one ACS grism resolution element
(i.e. 40\AA) of the simulated line center. Using these simulations,
we were able to study the effect that wavelength, EW, line flux,
object size, and line blending (in the case of 4959/5007\AA) has on
the fraction of successfully detected lines.
Figure~\ref{f1} shows the recovery fraction in the simulations for
several input parameters. In each case, the remaining parameters
are fixed at values that allow for easy line identification.
Lyman-$\alpha$\ presents a special case here because it is invariably located
atop a significant spectral break when observed at the relevant
redshifts. This means that the line not only needs to be significantly
detected with respect to the photon noise ($s/n > 2.5$ as usual),
but also must pass an equivalent width threshold that depends on the
spectral resolution, in order to appear as a distinct emission line that
the algorithm will identify.
We characterized this effect
by convolving a model spectrum of a Lyman-$\alpha$\ emission line
atop the corresponding Lyman-$\alpha$\ break with line spread functions
of width $60 \hbox{\AA} \la \Delta \lambda \la 250$\AA,
adding random noise, and running our detection algorithm.
In the noise-free case, $\hbox{EW} \ga 0.4 \Delta \lambda$ is
required for the line to be recovered. If we lower the continuum
signal-to-noise ratio, the EW rises. The required equivalent
widths for 80\% completeness, given $s/n \approx 100$, $10$,
$4.5$, $2$, and $1$ per pixel in the continuum,
become $\approx 0.5 \Delta \lambda$, $ 0.75 \Delta \lambda$,
$\Delta \lambda$, $1.5 \Delta \lambda$, and
$3 \Delta \lambda$, respectively. In the limit of vanishingly
small continuum, the problem reverts to detecting an isolated line,
and for $s/n < 1$ per pixel,
the threshold equivalent width scales as $1/(s/n)$ as expected.
\section{Results}
Table 1 lists a total of 115 galaxies that have detectable emission lines.
Among these, 9 are high redshift Lyman-$\alpha$ emitters; at least
3 (and possibly up to 6) are active galactic nuclei (AGN);
and the remainder are star forming galaxies detected in some
combination of their [\ion{O}{2}] $\lambda$3727, [\ion{O}{3}] $\lambda \lambda$4959,5007, and
H$\alpha$ lines. In this latter category, 11 of them are
detected with both [\ion{O}{2}] $\lambda$3727\ and [\ion{O}{3}] $\lambda \lambda$4959,5007\, and 16 are detected with both
[\ion{O}{3}] $\lambda \lambda$4959,5007\ and H$_{\alpha}$.
The first column of this table lists the UDF ID of the objects.
An ID of ``$\le -100$'' means that the object is outside or near the edge
of the UDF field so it does not have a UDF ID. An ID between 0 and
-100 means that the object is too close to an extended galaxy so it
might be taken as part of that galaxy thus not assigned a UDF ID.
The objects added into the
line list after visual inspection of their spectra are marked with \dag.
Column ``i-mag'' is the i-band AB magnitude of the object.
Column ``wavelength'' is the central wavelength of the detected line,
along with its $1\sigma$ statistical error estimate.
Column ``Line Flux'' is line flux in the observed frame, as measured
directly from the 1D spectra {\it without aperture correction}. The
true line flux is in general larger by a factor of $1.5$ to $3$.
Column ``Redshift'' is redshift determined from the emission
lines, with a redshift of ``-1'' indicating that this line is not
identified.
Column ``Line'' is the physical identification of the observed
line.
Some examples of the emission line galaxy spectra are presented
in figure~\ref{f3}.
Figure~\ref{f4} shows the redshift distributions of the
emission line galaxies according to the detected lines. The redshift
range covered by each emission line is effectively set by the
wavelength coverage of the grism and the
rest wavelengths of the lines. The grism response extends
from 5500\AA\ to 10500\AA.
In practice, we find we can detect emission lines usefully over
the wavelength range 5700 \AA\ $\la \lambda \la$ 9700\AA,
which corresponds to a grism throughput $\ga 25\%$ of the
peak throughput. In this figure, a curve of
the number (per redshift bin) of emission line galaxies that would be expected
in the corresponding redshift (bin) are also overplotted.
To derive these curves, we first generated a modified line
luminosity function directly from the GRAPES data,
using the $1/V$ method and the empirical sample, but disregarding
both selection effects and evolution. We then used this function
to estimate the number of objects that ought to have been detected
in each redshift bin.
The minimum luminosity for line detection was empirically calibrated
to the faintest detected GRAPES emission lines, with the redshift
dependence based on the grism sensitivity and the luminosity distance
calculated in the current concordance cosmology
($\Omega_m = 0.3, \Omega_\Lambda = 0.7, H_0 = 70 \kmsMpc$;
see Spergel et al. 2003, 2006). The dotted curves are
in some way an interpolation of the observed number-redshift
counts, but they improve on a direct interpolation by
incorporating our knowledge of the cosmology and instrument
properties. Because the histograms are plotted based on detected
lines, galaxies can
contribute to more than one histogram if they have multiple
detected emission lines.
\section{Discussion}
The GRAPES spectroscopy goes to much fainter continuum flux levels
than is typical for ground-based followup spectroscopy. The
median broad band magnitude for the emission line objects in table~1
is $i_{AB}=24.67 \mag$ (see also figure~\ref{f2}).
Over 40\% of the sample has $i_{AB}>25 \mag$, which is
about the faintest magnitude level routinely targeted for ground-based
followup spectroscopy, and 15\% is fainter than $i_{AB}=26 \mag$. It is
thus likely that 30--40\% of our emission line sample would be missed
by ground-based followup efforts. The GRAPES sample thus provides a
unique resource for studying emission line luminosity functions and
the star formation rate density (SFRD) at moderate redshifts.
The results of such studies will be presented in the
companion paper (Gronwall et al. 2007).
The morphologies of the emission line galaxies are presented in
Pirzkal et al. (2006).
The 115 redshifts from this emission line catalog represent a
fraction $\sim 5\%$ of the sources in the Hubble Ultra Deep Field
brighter than 28th magnitude (AB). The primary factors leading
to this comparatively small fraction are the selection effects
demonstrated in figure~1. For an object to be selected in our
emission line catalog with a reasonable probability, it must
have $i_{AB} \la 26.5$, $\hbox{EW} \ga 50$\AA, size $< 0.75''$,
and an emission line with observed wavelength $5800{\rm \AA} \la
\lambda \la 9600{\rm \AA}$. Indeed, if we only consider objects
with $i_{AB} < 26.5$, the redshift completeness rises to $9.5\%$.
The GRAPES survey has also published continuum break redshifts of
Lyman break galaxies (Malhotra et al 2005) and distant elliptical
galaxies (Daddi et al 2005; Pasquali et al 2005), and there are
additional redshift samples in progress for later type galaxies
with $4000$\AA\ breaks (Hathi et al 2007, in preparation) and
for an overarching sample of galaxies with significant spectroscopic
continuum information in the GRAPES spectra (Ryan et al
2007, see below). So the spectroscopic success rate is reasonably
high for sources where sufficient information can be expected
given the properties of the instrument and data set. In particular,
the fraction of redshifts lost to crowding and overlap is
modest, because of our multiple roll angle observing strategy.
As a sanity check, we compared our measured redshifts with those
available in the Chandra Deep Field South and found good agreement.
We use redshifts from five available references (Vanzella
et al.~2005; Le Fevre et al.~2005, Szokoly et al.~2004,
Vanzella et al.~2006, and Grazian et al.~2006).
The results of this comparison are given in table~2.
Both the overall agreement and the
agreement with individual references was generally good.
For example, we found that there are 7 objects in common
in our emission line catalog and the VLT/FORS2
catalog of Vanzella et al.~2005. Among these, redshifts
are in agreement for 6 objects. Moreover, the one
that does not agree, UDF -100, also shows up in Le Fevre et al.~2004,
where its redshift {\it does} agree with our measurement.
Among the 9 objects in Le Fevre, only one (UDF3484) disagrees.
We also found 4 out of 5 in agreement with Szokoly et al.~2004
(here the object that does not agree is UDF4445, a 2-line object).
The total sample is 23 objects with both GRAPES and ground-based
redshifts available. Among these, $2.5$ show ``catastrophic'' redshift
mismatches (where the ``0.5'' object is our ID -100,
with two inconsistent ground-based redshifts). This 11\% failure rate
is essentially the same as the catastrophic failure rate for photometric
redshifts, and for essentially the same reason: Most of our redshifts
are constrained to match a photometric redshift at the outset.
The catastrophic failures in our list correspond to mis-identified
emission lines.
In a future paper (Ryan et al 2007) we will combine the
continuum shape information from the grism with multi-band optical
and near-IR photometry to derive spectro-photometric redshifts for
the GRAPES data set. This effort may reduce the catastrophic
failure rate in line identifications, since the extra information
from the grism continuum will help distinguish among the redshift
likelihood peaks in photometric redshift fitting.
To estimate our redshift accuracy, we study the cleanest subset
of these object with ground-based and GRAPES redshifts.
We first exclude the catastrophic failures from consideration (excluding
ID -100 along with the other two). Our catalog
also has a few lines from the manually identified emission line
galaxies where we lack an automated line wavelength uncertainty
estimate, and we exclude these also for the present.
Among the remaining 15 overlap objects, the RMS redshift difference
between GRAPES and ground-based data is $\hbox{RMS}(\Delta z) = 0.008$.
For comparison, the median estimated redshift uncertainty from
our line centroiding algorithm is $\hbox{median}(\delta \lambda /
\lambda_{rest}) = 0.009$.
Thus, we infer a typical redshift uncertainty just below $0.01$
for our emission line catalog.
Examining the redshift residuals $z_{GRAPES} - z_{VLT}$
as a function of redshift shows weak evidence for a systematic
offset at $z \la 0.3$, with $\langle z_{GRAPES} - z_{VLT} \rangle
\sim 0.01$, but this is based on only 3 or 4 data points and
may be a coincidence.
If we examine our subsample with two emission lines and compare the
redshifts derived separately from the two lines, we reach a similar
conclusion-- the offset between these measurements is again fully
consistent with characteristic redshift errors $\delta z \la 0.01$.
The comparison with ground-based spectra also allows us to
estimate the systematic error floor in our line wavelength measurements:
A systematic error component of $\sim 12$\AA\ is quite sufficient
to account for the observed redshift offset between
ground and GRAPES data for those lines whose formal wavelength
uncertainty $\delta \lambda < 5$\AA. A larger comparison
sample might refine this estimate, but is unlikely to change
it dramatically.
\section{Conclusion}
We find that a deep spectroscopic survey
like GRAPES offers a unique opportunity to identify emission lines
and determine redshifts for faint galaxies.
In this paper, we present a catalog of emission lines
identified in GRAPES, including wavelength measurements, flux
estimates, line identifications, and redshifts.
Over 40\% of the sample comes from objects fainter than
the typical continuum magnitude limit for ground-based multiobject
spectroscopic followup programs. These
objects might never have been identified as emission line galaxies
without a space based observation as ours. Based on comparison
with ground-based spectra for a subset of our objects, we infer
a typical redshift accuracy of $\delta z = 0.009$ for our
catalog.
\acknowledgments
This work was supported by grant GO-09793.01-A from the Space
Telescope Science Institute, which is operated by AURA under
NASA contract NAS5-26555. ED acknowledge support from
NASA through the Spitzer Fellowship Program, under award 1268429.
This project has made use of the aXe
extraction software, produced by ST-ECF, Garching, Germany.
We also made use of the "mpfit" IDL library, and we thank Craig
Markwardt for making this package public.
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1,116,691,499,401 | arxiv | \section{Introduction}
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Space-time dependent or `local' couplings $\lambda^i = \lambda^i(x)$ are a
standard tool in quantum field theory. They may be viewed as sources for
composite operators. Well-defined operator
insertions are obtained by functionally varying the generating
functional with respect to the local
couplings,
\begin{equation} \label{introins}
\frac{\delta}{\delta \lambda^i (x)} W = \, \langle {\cal O}_i (x)
\rangle
\, .
\end{equation}
The concept of local couplings is particularly appealing
in supersymmetric theories
where holomorphy is at the origin of non-renormalisation
theorems~\cite{SV1,LK,Seiberg:1993vc,LK2}.
As we will see below, new results for
supersymmetric gauge theories are obtained by promoting the couplings, both
the gauge couplings as well as the superpotential couplings, to full chiral
and antichiral space-time dependent superfields $\lambda (z) $ and $\bar
\lambda (\bar z) $, respectively, satisfying $\bar D_{\dot \alpha} \lambda = 0$ and
$D_\alpha \bar \lambda = 0$.
The consequences of allowing for local supercouplings in ${\cal N}=1$ theories have
recently been investigated in a series of papers
\cite{Kraus:2001tg}-\cite{Kraus:2001kn} within a perturbative approach.
Both a component \cite{Kraus:2001id} and a superspace \cite{Kraus:2002nu}
approach were taken to study pure ${\cal N}=1$
super Yang-Mills theory. The Wess-Zumino model was considered in
\cite{KrausWZ}.
Most importantly, for pure ${\cal N}=1$ gauge theory it was shown
in these publications that in the presence of
local couplings there is an additional new anomaly. This anomaly appears in
the Ward identity for the topological symmetry associated
with the theta angle and
manifests itself in an anomalous divergence of the topological current.
At one loop, this anomaly is given by
\cite{Kraus:2002nu}
\begin{gather} \label{pontanomintro} \left(
\int \! d^6z \, \frac{\delta}{\delta \lambda} -
\int\! d^6\bar z \, \frac{\delta}{\delta \bar \lambda} \right)
\Gamma
= \frac{\mathcal{A}_1}{4}\left(\int\! d^6z \,
\frac{1}{\lambda^{\prime}} \, \mathrm{tr}(W^{\alpha}W_{\alpha})-\int\! d^6\bar{z}
\, \frac{1}{
\bar{\lambda}^{\prime} } \, \mathrm{tr}(\bar{W}_{\dot{\alpha}}
\bar{W}^{\dot{\alpha}}) \right),
\end{gather}
where $\lambda^{\prime}(z)=\lambda(z)+1/2g^2$ and
$\bar{\lambda}^{\prime}(\bar{z})=\bar{\lambda}(\bar{z})+1/2g^2$.
$\Gamma$ is the vertex functional of pure ${\cal N}=1$ Yang-Mills
theory.\footnote{$\Gamma$ is the vertex functional
of the BPHZ approach. Within perturbation theory,
by virtue of the so-called `action
principle', $\Gamma$ is equivalent to the `quantum effective action'
$\Gamma_{\rm eff}$. This in turn is
a local function of the fields and of
the couplings, constructed order by order in perturbation
theory. In this sense $\Gamma_{\rm eff}$
corresponds to the bare action in the standard perturbative
approach. - A review of the BPHZ approach
may be found in \cite{BPHZ}.}
Moreover the one-loop coefficient ${\cal A}_1$ is given by
${\cal A}_1 = C_2(G)/8 \pi^2$.
The l.h.s. of (\ref{pontanomintro})
is the symmetry transformation of the vertex
functional under the topological symmetry. The r.h.s. is the new anomaly which
vanishes in the limit of constant couplings where the integrand becomes the
Pontryagin density.
The one-loop coefficient of the anomaly in (\ref{pontanomintro}) has been
calculated in \cite{Kraus:2001id} in a component approach. In fact,
an alternative way of obtaining an equivalent expression for
the lowest component of (\ref{pontanomintro}) is to vary the component
action with respect to the space-time dependent theta angle.
This gives rise to the one-loop result \cite{Kraus:2001id,Bos}
\begin{gather} \label{Jtop1} \,
\frac{\delta}{\delta \tilde \theta (x)} \, \Gamma \, = \, - \,
(1 \, + \, {{\cal A}}_1 \,
\, g^2 ) \left( {\textstyle \frac{1}{4}} F^a_{\mu \nu}
\tilde F^a{}^{\mu \nu} + \partial^\mu (
\bar \lambda^a \sigma_\mu \lambda^a ) \right) (x) \, , \qquad \tilde \theta
= \frac{\theta} {8 \pi^2} \, .
\end{gather}
Here $\lambda_\alpha{}^a$
are the gauginos. (\ref{Jtop1}) implies that the divergence
of the topological current $J_\mu$,
defined by
\begin{gather}
J_\mu = \, {\ts \frac{1}{8}} \, \varepsilon_{\mu \sigma \nu \rho} ( A^{\sigma a} \partial^\nu
A^{\rho a} + {\textstyle \frac{1}{3}}
A^{\sigma a} A^{\nu b} A^{\rho c} f_{abc} ) +
\bar \lambda^a \sigma_\mu \lambda^a \, , \label{Jtop}
\end{gather}
is anomalous. In fact, classically we have $\partial^\mu J_\mu =
{\textstyle \frac{1}{4}} F^{a}_{\mu \nu} \tilde F^a{}^{\mu \nu} + \partial^\mu (
\bar \lambda^a \sigma_\mu \lambda^a )$ and therefore (\ref{Jtop1}) may
equivalently be written as
\begin{gather} \,
\frac{\delta}{\delta \tilde \theta } \, \Gamma = \, - \,
(1 + {{\cal A}_1} \, g^2) \,
\partial^\mu J_\mu \,,
\qquad {\cal A}_1 = \frac{C_2(G)}{8\pi^2} \, , \label{introcpnts}
\end{gather}
where the anomalous contribution is ${\cal A}_1 g^2 \, \partial^\mu J_\mu$.
In~\cite{Kraus:2002nu}, the anomaly in (\ref{pontanomintro})
is used to obtain a general scheme independent result for the
gauge $\beta$ function to all orders.
The main point in this derivation is to add an appropriate counterterm
to the action which shifts the anomaly from the topological Ward identity
(\ref{pontanomintro}) to the Callan-Symanzik equation.
There is an interesting analogy between this anomaly shift and the approach
of~\cite{Arkani-Hamed:1997mj}, where anomalous transformations of the path
integral functional measure are considered.
- Note in particular that the one-loop coefficient
${\cal A}_1 = C_2(G)/8 \pi^2$ is the coefficient which appears in
the denominator of the NSVZ $\beta$-function~\cite{Novikov:ic}.
A further use of local couplings, so far used for non-supersymmetric field
theories or for SUSY theories in components, is that they allow for an elegant
formulation of {\it local} renormalisation group (RG) or Callan-Symanzik (CS)
equations. These determine how conformal symmetry is broken in a quantised
field theory. This is in contrast to the standard (or {\it global}) RG
equations, which express the breakdown of scale invariance upon quantisation.
For the formulation of local RG equations it is necessary to couple the
quantum field theory to a classical curved space background, with the metric
acting as source for the energy-momentum
tensor~\cite{Osborn:gm,Kraus:1992ru} (see also \cite{Jack}).
The significance of this approach is that it
may be of relevance for a proof of an analogue of
the Zamolodchikov C-theorem~\cite{Zamolodchikov} in
more than two dimensions. In particular in~\cite{Osborn:gm} an alternative
derivation of the two-dimensional C-theorem was given using local
couplings. It is essential for this analysis that in the presence of local
couplings, there are new conformal anomalies in addition to the familiar
anomalies involving the curvature of the background metric.
These new conformal anomalies involve derivatives of the
couplings. Furthermore in \cite{Osborn:gm} within the local coupling approach,
the flow of a candidate {\it a}-function (related to the coefficient of the
Euler anomaly) for a possible four-dimensional C-theorem~\cite{Cardy:cw} is
related to a quadratic form, which however so far has not been shown to be positive definite.
A related approach to supersymmetric theories was used in~\cite{Osborn:2003vk}
and in particular in~\cite{FreedmanOsborn} where a four-loop expression for
the candidate {\it a}-function was given and shown to coincide with
non-perturbative results for $a_{\rm UV} - a_{\rm IR}$ found
in~\cite{Anselmi:1997am}. These non-perturbative results were obtained using
't Hooft anomaly matching and by performing explicit one-loop computations of
three-point correlators involving the R current, the energy-momentum
tensor and a particular anomaly-free current.
A different approach to a possible four-dimensional C-theorem for
supersymmetric theories is the principle called
`$a$ maximization' which has recently been proposed
and investigated
in
\cite{Intriligator:2003jj}-\cite{Kutasov:2004xu}. The local coupling approach taken here is
complementary to $a$ maximization, though relations between the two approaches
exist \cite{Barnes:2004jj}.
The purpose of this paper is twofold. First we consider
$\mathcal{N}=1$ SUSY gauge
theories with local chiral supercouplings both without and with chiral matter.
We show how the NSVZ $\beta$-function may easily be derived from the
topological anomaly present for local couplings.
This $\beta$-function is naturally
associated to a particular renormalisation scheme which we describe in
detail.
Secondly we consider ${\cal N}=1$ SUSY gauge theories with local couplings which in
addition are coupled to a classical supergravity background in
superspace. The aim of this analysis is to find new results for the
coefficients of the gravitational anomalies. -
For the superspace formulation of
local CS equations, the supercurrent associated to
superconformal transformations
has to be coupled to the appropriate superspace supergravity
field~\cite{Buchbinder:qv,Gates:nr}. An analysis of local CS equations for
supersymmetric theories with constant coupling has been given
in~\cite{Erdmenger:1998tu,Erdmenger:1998xv,Erdmenger:1999uw}.
Here we extend this approach to the local coupling case.
The resulting superconformal Ward identities may be viewed as a
generalisation of previous results for the anomalous divergence of the
supercurrent~\cite{Clark:1980dw,Shifman:1986zi,Kogan:1995mr,Leigh:1995ep}
to the off-shell
case with curved superspace background. With this approach we are able to give
an all-order derivation of an expression for the central charge $c$, the
coefficient of the Weyl tensor squared contribution to the conformal anomaly,
in terms of the beta and gamma functions of the theory, of the form
\begin{gather} \label{cc}
c = c_1 + \frac{1}{24}\left( N_V \frac{\beta_g}{g} -
{\gamma_i} {N_\chi{}^i} \right) \, , \;\; c_1 = {
\frac{1}{24}}\,
(3 N_V +
N_\chi) \, , \qquad N_\chi =\sum\limits_{i} N_\chi{}^i \, .
\end{gather}
Here $N_\chi{}^i$ denotes the number of chiral fields with anomalous
dimension $\gamma_i$ (We take the anomalous dimension matrix to be
diagonal). $c_1$ is the one-loop result for the central charge.
The expression (\ref{cc}) was
first presented in~\cite{Anselmi:1996dd}, and is
based on the two-loop calculations
of~\cite{Jack:wd}. Our all-order
derivation presented here relies on the fact that on curved
superspace, the topological Ward identity has an additional one-loop
anomaly of the form
\begin{gather} \label{pontanomintro2} \left(
\int\! d^6z \,\frac{\delta}{\delta \lambda} -
\int\! d^6\bar z \,\frac{\delta}{\delta \bar \lambda} \right)
\Gamma^\prime
= \frac{\mathcal{C}_{1}}{4}\left(\int\! d^6z \,
\frac{1}{\lambda^{\prime}} \mathrm{tr}(W^{\alpha\beta\gamma}W_{\alpha\beta \gamma})
-\int\! d^6\bar{z}
\, \frac{1}{
\bar{\lambda}^{\prime} } \mathrm{tr}(\bar{W}_{\dot{\alpha}{\dot \beta}{\dot \gamma}}
\bar{W}^{\dot{\alpha}{\dot \beta}{\dot \gamma}}) \right),
\end{gather}
where $W_{\alpha \beta\gamma}$ is the superspace Weyl density. $\Gamma'$ is the
vertex functional in which a suitable local counterterm has been added to
$\Gamma$ in
(\ref{pontanomintro}) such as to cancel the r.h.s.~of (\ref{pontanomintro}).
Note again that
the anomaly (\ref{pontanomintro2}) vanishes in the constant coupling limit.
We calculate the coefficient
${\mathcal C}_1$ to one loop in the component decomposition of
\cite{Anselmi:1997am}, and
show how it gives rise to the desired result for $c$ in the same
renormalisation scheme as used for
the derivation of the NSVZ $\beta$ function.
Moreover we extend our result to theories with matter by
considering an off-shell version of the Konishi anomaly on curved
space background.
The essential point of our derivation is again - as in \cite{Kraus:2002nu} -
that with a suitable local
counterterm, the anomaly in (\ref{pontanomintro2}) may be
shifted from the topological Ward identity to the superconformal Ward identity
and thus to the Callan-Symanzik equation.
The coefficient of the Euler central charge $a$ is inherently more difficult
to derive in the approach presented here,
since it contains terms non-linear in the
anomalous dimension, of the form ${\rm tr}(\gamma \gamma)$ and ${\rm tr}(\gamma \gamma
\gamma)$. We expect to return to a derivation of $a$
in the future. At least already
at the present stage we are able to show that there cannot be any
terms of the form ${\rm tr} (\gamma) $ contributing to $a$.
Moreover our results are consistent with the expected factor of the
$\beta(g)/g$ contribution to $a$.
The outline of this paper is as follows.
We begin in section~\ref{internal} with a brief summary of the results of \cite{Kraus:2002nu}
relevant for our analysis, the
implications of local chiral couplings in supersymmetric Yang-Mills theory
and the associated internal anomalies. Then we identify the constraints
leading to the particular renormalisation scheme which gives rise to the
NSVZ $\beta$-function. Moreover we extend the analysis to
gauge theories with matter.
In section~\ref{external} we find expressions for the central
charge $c$ by considering the external
anomalies. Firstly, the gauge field contribution to the new topological
anomaly is calculated and is shown to give rise to
the $\beta (g)/g$ contribution to $c$. Next we include the matter
contributions by
considering the Konishi anomaly in an off-shell approach for a curved
superspace background.
Finally in section~\ref{conclusion} we
conclude with comments on the implications for the Euler central charge $a$
and an outlook on future directions.
The appendix contains the necessary one-loop triangle diagram computations.
\section{Local Chiral Couplings - Internal anomalies}\label{internal}
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We begin by giving a summary of the results in~\cite{Kraus:2002nu,Kraus:2002se} which show that in a pure gauge theory, a new
topological anomaly appears when the couplings are allowed to be space-time
dependent. We then show how the NSVZ beta function arises in this approach for
a particular renormalisation scheme.
\subsection{Topological anomaly in pure gauge theory}
The starting point is pure $ \mathcal{N}=1 $ SYM with gauge group $G$, whose
classical action is, in the conventions of~\cite{Kraus:2002nu},
\begin{equation}
S_{{\rm constant \, coupling}}[V]=\, - \frac{1}{4g^2} \int\! d^6z \,
\mathrm{tr}(W^{\alpha}W_{\alpha}) - \frac{1}{4g^2}\int\! d^6\bar{z} \,
\mathrm{tr}(\bar{W}_{\dot{\alpha}} \bar{W}^{\dot{\alpha}}),
\end{equation}
with
\begin{gather} \label{ws}
W_\alpha=\,\frac{1}{8}\bar{D}^2(e^{-2V}D_{\alpha}e^{2V}), \qquad
\bar{W}_{\dot \alpha}=-\frac{1}{8}D^2(e^{2V}\bar{D}_{\dot{\alpha}}e^{-2V}).
\end{gather}
Next the gauge coupling $ g $ is promoted to
a local chiral and antichiral superfield, $ \lambda(z) $ and $
\bar{\lambda}(\bar{z}) $ respectively, such that the action becomes
\begin{equation}\label{action1}
{\cal S} [V]=\, -
\frac{1}{2}\int \!d^6z\, \lambda(z) \mathrm{tr}(W^{\alpha}W_{\alpha})
- \frac{1}{2}\int \! d^6\bar{z}\,
\bar{\lambda}(\bar{z}) \mathrm{tr}(\bar{W}_{\dot{\alpha}} \bar{W}^{\dot{\alpha}}).
\end{equation}
As discussed in \cite{Kraus:2001id},
the vector superfield $ V $ may \emph{not} be fixed to the Wess-Zumino gauge if
manifest supersymmetry is to be preserved in the presence of local couplings. In the case where the
Wess-Zumino gauge is fixed, the supersymmetry algebra closes
only up to gauge transformations and
hence is not linearly realised.
In order to be able to make recourse to perturbation theory, there has to be a
well-defined free field limit. For the action (\ref{action1}),
there are two ways to proceed:
If the lowest components of $\lambda(z)$ and $\bar{\lambda}(\bar{z})$ are
taken to coincide with the local coupling and local theta angle by virtue of
\begin{gather} \label{coupling2}
\lambda(z)|_{\theta}+\bar{\lambda}(\bar{z})|_{\bar{\theta}}
=\frac{1}{2g^2(x)}, \quad
\lambda(z)|_{\theta}-\bar{\lambda}(\bar{z})|_{\bar{\theta}}
=\, - \frac{i}{16\pi^2}\Theta(x)
\, ,
\end{gather}
then a rescaling of $V$ by
\begin{gather}
V\rightarrow (\lambda+\bar{\lambda})^{-1/2}V \, ,
\end{gather}
leads to a well-defined free field limit of the action (\ref{action1}).
It would be very interesting to study the renormalisation behaviour of the
action with this normalization.
Here, however, we follow another approach for
calculational simplicity, first used in \cite{Kraus:2002nu}.
In this approach a well-defined free field limit is obtained by shifting
the lowest components of the superfields by a constant such that
\begin{gather} \label{coupling}
\lambda^{\prime}(z) = \lambda(z)+\frac{1}{2g^2} \, , \qquad
\bar{\lambda}^{\prime}(\bar{z}) = \bar{\lambda}(\bar{z})+\frac{1}{2g^2}.
\end{gather}
The new action with these couplings reads
\begin{gather} \label{gauge}
S[V]=\, -
\frac{1}{2}\int\! d^6z\,\lambda^{\prime}(z) \mathrm{tr}(W^{\alpha}W_{\alpha})
- \frac{1}{2}\int\! d^6\bar{z}\,\bar{\lambda}^{\prime}(\bar{z})
\mathrm{tr}(\bar{W}_{\dot{\alpha}} \bar{W}^{\dot{\alpha}}) \, .
\end{gather}
From this classical action a perturbative expansion for the associated
quantum theory is obtained as follows.
Varying the vertex functional corresponding to (\ref{gauge})
with respect to $\lambda$ or $\bar \lambda$ gives rise to well-defined
operator insertions, which in turn correspond to the vertices of the
perturbation expansion.
Moreover the action (\ref{gauge})
has a well-defined free field limit, which is obtained by
setting $ \lambda=0, \bar{\lambda}=0 $.
This allows for an unambiguous definition of the free propagator
but with an additional factor of $g^2$ in its definition as compared to the
standard constant coupling perturbative approach.
Kraus and collaborators show that the perturbative expansion obtained
in this way - vertices from functional differentiation and modified
propagators - satisfies
\begin{equation} \label{ng}
N_g=2(N_{\lambda}+N_{\bar{\lambda}})+2(l-1)+N_{V},
\end{equation}
where $N_{g}$ denotes the power of the constant coupling $g$
in a Feynman graph,
$N_{\lambda}$ and $N_{\bar{\lambda}}$ count the
operator insertions obtained by varying with respect to $\lambda$ or
$\bar \lambda$, respectively, and $N_{V}$ the number of external
superfield legs. $l$ is the number of loops.
The relation (\ref{ng}) ensures that
the loop expansion is a power series.
For the action (\ref{gauge}) there are
two important classical Ward identities which become
anomalous upon quantisation. These arise due to a new global symmetry
associated with the local couplings. The relevant
symmetry transformation consists of shifting the
local couplings by a complex constant $\omega$,
\begin{gather} \label{coupling3}
\lambda(z)\rightarrow \lambda(z)+\omega, \qquad
\bar{\lambda}(\bar{z})\rightarrow\bar{\lambda}(\bar{z})+\bar{\omega}.
\end{gather}
If we define the following operators
\begin{equation} \label{plusminus}
\Delta^{\pm}\equiv \int\!
d^6z\,\frac{\delta}{\delta\lambda(z)}\pm\int\! d^6\bar{z}\,
\frac{\delta}{\delta\bar{\lambda}(\bar{z})},
\end{equation}
then it is easily verified that the classical action $ S[V] $
satisfies the {\it shift
equation}, induced by the real part of $\omega$,
\begin{equation} \label{shift}
\Delta^{+}S=-g^3\partial_gS \, .
\end{equation}
This identity is crucial for establishing the equivalence between the
perturbation expansion described above and the standard one.
In addition, there is also the {\it Pontryagin identity} arising from
the imaginary part
of $\omega$,
\begin{equation}
\Delta^{-}S=\, - \frac{1}{2}\int\! d^6z \,
\mathrm{tr}(W^{\alpha}W_{\alpha}) + \frac{1}{2}\int\!
d^6\bar{z}\, \mathrm{tr}(\bar{W}_{\dot{\alpha}} \bar{W}^{\dot{\alpha}})=0 .
\end{equation}
The r.h.s. vanishes on flat space since it is an integral over a topological density.
The crucial result of \cite{Kraus:2002nu} is that in the quantised theory, the
Pontryagin identity becomes anomalous. At one loop, the anomaly is given by
\begin{equation} \label{pontanom}
\Delta^{-}\Gamma= \frac{\mathcal{A}_{1}}{4}\left(\int\! d^6z \,
\frac{1}{\lambda^{\prime} (z)} \mathrm{tr}(W^{\alpha}W_{\alpha})-\int\! d^6\bar{z}
\, \frac{1}{
\bar{\lambda}^{\prime}(\bar{z})} \mathrm{tr}(\bar{W}_{\dot{\alpha}}
\bar{W}^{\dot{\alpha}}) \right) \, ,
\end{equation}
with
$\Gamma$ the BPHZ vertex functional.
(Capital calligraphic letters denote anomaly coefficients and the number 1
denotes 1-loop.)
The anomaly coefficient $ \mathcal{A}_1 $ was calculated in~\cite{Kraus:2001id}
and is given by
\begin{equation}
\mathcal{A}_1=\frac{1}{8\pi^{2}}C_2(G),
\end{equation}
where the group theory factor is $ (T^AT^A)_{R}=C(R) {\bf 1} $ for a representation $R$ of the gauge group, and $R=G$ is the adjoint representation. For $G= SU(N_c) $ we have
\begin{equation}
\mathcal{A}_1=\frac{N_c}{8\pi^{2}}.
\end{equation}
It is essential to note that the local couplings enter
the anomaly in (\ref{pontanom}) in the form
$1/\lambda'$, $1/{\bar \lambda}'$.
This is consistent with the fact that when taking the constant
coupling limit, the integrand of (\ref{pontanom}) reduces
to a component form that contains a factor of $g^2$ as in (\ref{introcpnts})
(see \cite{Kraus:2001id}).
\subsection{Shifting the anomaly}
We have seen that the Pontryagin equation has an anomaly given by
(\ref{pontanom}). However by using the freedom of adding local
counterterms,
$ \Gamma^{ct}(V) $, to the quantum action, it is possible to move
the anomaly
to the shift equation (\ref{shift}). This is
analogous to the situation for the chiral anomaly, where
one has the freedom to shift the anomaly between the axial and
vector Ward identities. The specific counterterm chosen
in~\cite{Kraus:2002nu}, which shifts the anomaly from (\ref{pontanom}) to
(\ref{shift}), is
\begin{gather}
\Gamma^{ct}(V)= \, - \frac{\mathcal{A}_1}{4}\left(\int\!
d^6z\, [\ln(2\lambda^{\prime}(z))+\ln g^2] \mathrm{tr}(W^{\alpha}W_{\alpha})+\int\!
d^6\bar{z}\, [\ln(2\bar{\lambda}^{\prime}(\bar{z}))+\ln g^2]
\mathrm{tr}(\bar{W}_{\dot{\alpha}} \bar{W}^{\dot{\alpha}}) \right). \label{counterterm1}
\end{gather}
The $ \ln g^2 $ terms in this expression are
necessary to ensure that this counterterm vanishes in the constant coupling limit
$ \lambda=0, \bar{\lambda}=0 $, such that the constant coupling perturbation
expansion remains a power series.
The action
\begin{gather} \label{Gammaprime}
\Gamma^{\prime}=\Gamma+\Gamma^{ct}(V)
\end{gather}
satisfies the Pontryagin identity
\begin{gather}\label{ponteq}
\Delta^{-}\Gamma^{\prime}=0 \, .
\end{gather}
However the shift equation now becomes anomalous,
\begin{equation} \label{qshift0}
\Delta^{+}\Gamma^{\prime}=-g^3\partial_{g}
\Gamma^{\prime} - \frac{\mathcal{A}_1g^2}{2}\left(\int\!
d^6z\,\mathrm{tr}(W^{\alpha}W_{\alpha})
+\int\! d^6\bar{z} \, \mathrm{tr}(\bar{W}_{\dot{\alpha}} \bar{W}^{\dot{\alpha}}) \right).
\end{equation}
The anomaly term is just the classical SYM action being acted upon by the
operator $ \Delta^{+} $, such that at one loop, (\ref{qshift0}) may be written as
\begin{gather}\label{qshift}
\Delta^{+}\Gamma^{\prime}=-g^3\partial_{g}\Gamma^{\prime}+\mathcal{A}_1g^2\Delta^{+}S.
\end{gather}
This result is scheme independent.
We now proceed by chosing a particular scheme, which we show to
coincide with the NSVZ scheme.
A particular scheme is obtained by assuming that (\ref{ponteq}) and
(\ref{qshift})
are valid to {\it all} orders in perturbation theory, which
corresponds to replacing (\ref{qshift}) by the full quantum equation
\begin{gather} \label{qshift2}
\Delta^{+}\Gamma^{\prime}=-g^3\partial_{g}
\Gamma^{\prime}+\mathcal{A}_1g^2\Delta^{+}\Gamma^{\prime}.
\end{gather}
This scheme choice requires in particular that
the Pontryagin anomaly we discussed above is exhausted at one-loop in this
particular scheme.\footnote{It appears feasible to prove one-loop
exactness of the Pontryagin anomaly by an argument based on the
results of \cite{Bos} together with supersymmetry. However finite
renormalisations at higher order are always possible.}
It is straightforward to see that the scheme
defined by passing from (\ref{qshift}) to (\ref{qshift2}) is indeed the NSVZ
scheme in which the beta function takes the form
derived by NSVZ in~\cite{Novikov:ic}.
Indeed, a rearrangement of~(\ref{qshift2}) gives
\begin{gather} \label{shifta}
\Delta^{+}\Gamma^{\prime}=-\frac{g^3}{1-\mathcal{A}_1g^2}
\partial_{g}\Gamma^{\prime}.
\end{gather}
This has already a form reminiscent of the NSVZ $ \beta
$-function. For an exact identification we now
establish the connection to scale transformations
of the vertex functional as expressed by a renormalisation group
flow.
As discussed in \cite{Kraus:2002nu}, when (\ref{ponteq}) holds to all
orders, the local coupling $\lambda$ (or $\bar \lambda$) is not
renormalised beyond one loop and remains holomorphic (or antiholomophic)
in the quantised theory. This is due to the consistency condition
\begin{gather} \label{consistency1}
\left[ \, \Delta^- \, , \; \mu \frac{\partial}{\partial \mu} \; \right]
\Gamma^\prime = 0 \, ,
\end{gather}
with $\Delta^-$ defined in (\ref{plusminus}) and $[ \, , \, ]$ the
commutator.
Therefore the Callan-Symanzik equation reads
\begin{gather} \label{CS1}
\left( \mu \frac{\partial}{\partial \mu} \, + \, {\cal B}_1 \Delta^+ \right)
\, \Gamma^\prime \, = 0 \, , \qquad {\cal B}_1 = \frac{3}{16 \pi^2}
C_2(G) \, ,
\end{gather}
with $\Delta^+$ defined in (\ref{plusminus}) and ${\cal B}_1$ the
standard one-loop coefficient of the gauge beta function.
Inserting (\ref{shifta}) into (\ref{CS1}) then gives rise to
to the Callan-Symanzik equation
\begin{gather} \label{CS}
\left(\mu\frac{\partial}{\partial \mu}+\beta(g)\partial_{g}
\right)\Gamma^{\prime}=0 \, ,
\end{gather}
with
\begin{equation}
\beta(g) = -\frac{\mathcal{B}_1g^3}{1-\mathcal{A}_1g^2}=-\frac{g^3}{16\pi^{2}}\left(\frac{3C_2(G)}{1-C_2(G)g^2/(8\pi^2)}\right),
\end{equation}
which is precisely the NSVZ $\beta$-function.
A further important point is that if (\ref{ponteq}) is valid to all
orders in perturbation theory, then a well-defined
local operator insertion is obtained by virtue of
\begin{gather} \label{localins}
\frac{\delta}{\delta \lambda(z)} \, \Gamma^\prime \, = \, - \,
\frac{1}{2} \,
{\rm tr} \, W^\alpha W_\alpha (z) \, .
\end{gather}
In standard notation, local insertions of composite operators are often
denoted by square brackets, ie.~$[W^\alpha W_\alpha]$, but we omit this here
and in the remainder of this paper to simplify the notation.
The Callan-Symanzik equation (\ref{CS1}) may be interpreted as an
anomalous Ward identity for scale transformations.
Scale transformation are a subgroup of superconformal transformations,
and together with the local equation (\ref{localins}), (\ref{CS1})
implies that the superconformal Ward identity consistent with
(\ref{CS1}) is given by
\begin{gather} \label{oneloop}
\bar D^{\dot \alpha} T_{\alpha {\dot \alpha}}\, = \, D_\alpha T \, , \qquad T = - \frac{1}{6} {\cal
B}_1 \, {\rm tr} \, W^\beta W_\beta \, .
\end{gather}
Here $T_{\alpha {\dot \alpha}}$ is the supercurrent from which all currents of the
superconformal group may be obtained. Note that for the theory with
local couplings considered here, the supercurrent has contributions
which involve derivatives of the couplings. Moreover the
superconformal anomaly $T$ is one-loop.
\subsection{Curved superspace background}
We proceed by deriving a local version of the Callan-Symanzik equation
(\ref{CS}).
For this purpose we couple the quantised local coupling theory to a
classical curved superspace background. As in \cite{Buchbinder:qv,Gates:nr},
the curved superspace involves a chiral compensator $\phi$ and a real
Weyl invariant superfield $H^{\alpha {\dot \alpha}}$, such that local quantum insertions of
the supercurrent $T_{\alpha {\dot \alpha}}$ and of the superconformal anomaly $T$
are given by
\begin{gather}
\label{scano}
\bar {\cal D}^{\dot \alpha} T_{\alpha {\dot \alpha}} \, = \, {\cal D}_\alpha T \; ,
\qquad T_{\alpha {\dot \alpha}} =
\frac{\delta}{\delta H^{\alpha {\dot \alpha}}} \Gamma^\prime \, , \quad T \, = \, \frac{1}{3}
\, \phi
\frac{\delta}{\delta \phi} \Gamma^\prime \, .
\end{gather}
It is important to note that in the case of constant couplings, $T$ as
given
by (\ref{scano}) has higher-order quantum corrections. It is possible to
redefine the supercurrent $T_{\alpha {\dot \alpha}}$ such that $T$ is one-loop, but then
the coupling to curved superspace as given by (\ref{scano})
is inconsistent (see
\cite{supercurrentPS,supercurrentGZ,Piguet:1986ug,Gates:nr}).
However for the local coupling theory considered here, the curved superspace
background given by (\ref{scano}) is consistent with $T$ being
one-loop, as we now show. This is possible essentially since the
supercurrent is modified by additional terms involving derivatives of
the couplings.
When
coupled to a curved superspace background as in
\cite{Buchbinder:qv,Gates:nr},
the classical action (\ref{gauge}) becomes
\begin{gather} \label{gaugecurved}
S[V]=\, -
\frac{1}{2}\int\! d^6z\, \phi^3 \,
\lambda^{\prime}(z) \mathrm{tr}(W^{\alpha}W_{\alpha})
- \frac{1}{2}\int\! d^6\bar{z} \, \bar \phi^3 \, \bar{\lambda}^{\prime}(\bar{z})
\mathrm{tr}(\bar{W}_{\dot{\alpha}} \bar{W}^{\dot{\alpha}}) \, .
\end{gather}
The topological anomaly discussed in section 2.1 is present also on the curved
superspace background. A quantum action $\Gamma'$ satisfying the topological
Ward identity $\Delta^- \Gamma^\prime \, = \, 0$ as in (\ref{ponteq})
is obtained from the quantum action $\Gamma$ corresponding to the classical
action (\ref{gaugecurved}) by adding a suitable local counterterm,
$ \Gamma^\prime \, = \, \Gamma \, + \, \Gamma^{ct}(V)$ given by
\begin{align} \label{counterterm2}
\Gamma^{ct}(V) = -& \frac{\mathcal{A}_1}{4} \int\!
d^6z \,\phi^3 [\ln(2\lambda^{\prime}(z))+\ln g^2]
\mathrm{tr}(W^{\alpha}W_{\alpha}) \nonumber \\
-& \frac{\mathcal{A}_1}{4} \int\! d^6\bar{z}\, \bar \phi^3
[\ln(2\bar{\lambda}^{\prime}
(\bar{z}))+\ln g^2]
\mathrm{tr}(\bar{W}_{\dot{\alpha}} \bar{W}^{\dot{\alpha}}) \, ,
\end{align}
which is the curved superspace analogue of (\ref{counterterm1}).
This counterterm is Weyl invariant.
The Ward identity $\Delta^- \Gamma^\prime \, = \, 0 $
is crucial for our construction. It implies that
a local operator insertion is obtained by virtue of
\begin{gather}
\frac{\delta}{\delta \lambda} \, \Gamma^\prime \, = \, \phi^3 \,
{\rm tr} \, W^\alpha W_\alpha \, .
\end{gather}
Then
the consistency condition
\begin{gather} \label{consistency2}
\left[ \, \phi \frac{\delta}{\delta \phi} \, , \, \frac{\delta}{\delta
\lambda} \, \right] \, \Gamma^\prime \, = \, 0
\end{gather}
implies that
\begin{gather} \label{TTT}
T \, = \, \frac{1}{3} \, \phi \frac{\delta}{\delta \phi} \, \Gamma^\prime
\end{gather}
is one-loop.
Note that $T_{\alpha {\dot \alpha}}$, the
insertion of the supercurrent given by (\ref{scano}),
contains derivatives of the local couplings and
is therefore different from the supercurrent in the standard constant
coupling theory.
By dimensional analysis, the
quantum vertex functional $\Gamma^\prime$ satisfies
the scale relation
\begin{gather} \label{cseqn}
\mu\frac{\partial}{\partial\mu}\Gamma^{\prime}\, + \, \left(\int\!
d^6z\, \phi\frac{\delta}{\delta\phi}\, + \int\!
d^6\bar{z}\, \bar{\phi}\frac{\delta}{\delta\bar{\phi}}\right)\Gamma^{\prime}
\, = \, 0 \, .
\end{gather}
The consistency condition (\ref{consistency2}) implies that
the Callan-Symanzik equation of the flat space case (\ref{CS1})
is also valid for the theory coupled to curved superspace,
\begin{gather} \label{CS3}
\left( \mu \frac{\partial}{\partial \mu} \, + \, {\cal B}_1 \Delta^+ \right)
\, \Gamma^\prime \, = 0 \, , \qquad {\cal B}_1 = \frac{3}{16 \pi^2}
C_2(G) \, .
\end{gather}
A local Callan-Symanzik equation consistent with
(\ref{CS3}) and (\ref{cseqn}) is given by
\begin{gather} \label{susylocalRG}
\phi \frac{\delta}{\delta \phi} \Gamma^\prime \, = \,
{\cal B}_1 \,
\frac{\delta}{\delta \lambda } \Gamma^\prime
\, .
\end{gather}
The superconformal anomaly reads
\begin{gather} \label{wti}
\bar {\cal D}^{\dot \alpha} T_{\alpha {\dot \alpha}} \, = \, {\cal D}_\alpha T \; , \qquad
T \, = \, - \frac{{\cal B}_1}{6} \,
\phi^3 \, {\rm tr} W^\alpha W_\alpha \, .
\end{gather}
This is the curved superspace analogue of (\ref{oneloop}).
\subsection{Theories with matter}
Lets us now consider the action of a gauge theory with matter,
\begin{align}
S = S_{\rm gauge} + S_{\rm matter} \, , \label{ac}
\end{align}
where the gauge part of the action is as in (\ref{gauge}) above,
and the matter part contains
$n$ chiral fields $\Phi^i$ transforming
in the representations $R_i$ of the gauge group $G$.
On a curved space background, the matter action is given by
(see~\cite{Buchbinder:qv,Gates:nr} for
superspace notation)
\begin{equation}\label{matteract}
S_{\rm matter}
= \, \frac{1}{4} \, \int \! d^8 z \, \tilde E\bar{\Phi}_i e^{2V} \Phi^i \, .
\end{equation}
$\tilde E$ is the appropriate curved superspace integration measure and
$i$ is the flavour index. We suppress colour indices for notational
simplicity. We first restrict to the case where there is no
superpotential, and include it in a second step below.
The new vertex functional corresponding to (\ref{ac})
still satisfies the Pontryagin
equation~(\ref{ponteq}) with the same anomaly as before. Therefore
the shift equation~(\ref{qshift2}) is preserved. This implies in particular
that the denominator for the NSVZ $\beta$-function is unchanged as expected.
We now generalise the local Callan-Symanzik equation
(\ref{susylocalRG}) to the action (\ref{ac}) which includes chiral
matter. Our starting point is the result
of~\cite{Shifman:1986zi,Kogan:1995mr,Leigh:1995ep},
based on an earlier result in~\cite{Clark:1980dw},
according to which on flat space,
the supercurrent anomaly for the action (\ref{ac})
may be written as
\begin{gather} \label{LS}
\bar D^{\dot \alpha} T_{\alpha {\dot \alpha}} = \, - \frac{1} {3} \, D_\alpha \left(
\frac{{\mathcal B}_1{}'}{2} {\rm tr} (W^\beta W_\beta)
+ \, \frac{1}{4} \bar D^2
\sum\limits_{i=1}^{n} \gamma_i
\bar \Phi_i e^{2V} \Phi^i \right) \, ,
\end{gather}
where the coefficient of the gauge anomaly is one-loop and $\gamma_i$ is the
anomalous dimension\footnote{As in~\cite{Leigh:1995ep}, we assume that
the anomalous dimension
matrix is diagonal,
$\gamma^i{}_j = \gamma_{(i)} \delta^i{}_j$. Note also that the
anomalous dimension of $\Phi^i$ used here is half the value of the mass
anomalous dimension used
in~\cite{Kogan:1995mr,Leigh:1995ep,Anselmi:1997am}. We use the superspace
conventions of \cite{Leigh:1995ep}.} of the field $\Phi^i$.
The coefficient $\mathcal{B}_1^{\prime} $ is given by
\begin{gather} \label{Bstrich}
\mathcal{B}_1^{\prime}=\frac{1}{16\pi^2}
\left(3C_2(G)- \sum\limits_{i=1}^{n} T(R_i)\right),
\end{gather}
where $T(R_i)$ is the Dynkin index in the representation $R_i$ of $G$,
$\mathrm{tr}(T^AT^B)_{R_i}=T(R_i)\delta^{AB}$.
We now generalise (\ref{LS}) in two ways: We consider the off-shell case
and we couple the quantised theory to a classical supergravity
background as in section 2.4. Then the supercurrent anomaly is given by
\begin{gather} \label{scm}
\bar{{\cal D}}^{{\dot \alpha}} \frac{\delta}{\delta H^{\alpha {\dot \alpha}}} \Gamma' = \, \frac{1}{3} \, {\cal D}_\alpha \left[
\left( \phi \frac{ \delta}{\delta \phi} - \Phi^i \frac{\delta}{\delta \Phi^i}
\right) \Gamma' \right] \, ,
\end{gather}
where the r.h.s.~is the transformation of the action under the super Weyl
transformation given by $\delta \phi = \sigma \phi$, $\delta \Phi^i = - \sigma
\Phi^i$ with $\sigma(z)$ the (chiral) Weyl transformation parameter.\footnote{For a
detailed analysis of the transformation properties of quantum actions and
of the quantum supercurrent under superconformal transformations,
see~\cite{Erdmenger:1998tu,Erdmenger:1998xv,Erdmenger:1999uw}.}
(\ref{scm}) generalises (\ref{scano}) to the matter case.
$\Gamma^\prime$ is the vertex functional obtained from the classical action
(\ref{ac}), together with the necessary counterterm
(\ref{counterterm2}) to guarantee $\Delta^-
\Gamma^\prime =0$ with $\Delta^-$ as in (\ref{plusminus}).
As discussed in section 2.4, the supercurrent
in the local coupling theory on
curved superspace has a one-loop gauge anomaly. Therefore instead of the flat
space equation (\ref{LS}) we may now write
\begin{gather} \label{csp}
\left( \phi \frac{ \delta}{\delta \phi} - \Phi^i \frac{\delta}{\delta \Phi^i}
\right) \Gamma' \, = \, - \,
\frac{{\mathcal B}_1{}'}{2} \phi^3 \,
{\rm tr} \, (W^\alpha W_\alpha) + \, \frac{1}{4} \phi^3 (\bar {\cal D}^2 +R) \,
\sum\limits_{i=1}^{n} \gamma_i
\bar \Phi_i e^{2V} \Phi^i \, ,
\end{gather}
which generalises (\ref{TTT}) and (\ref{wti}).
The anomalous Weyl transformation (\ref{csp})
is the starting point for deriving the
local Callan-Symanzik equation in the presence of matter fields.
It should be kept in mind that there are also contributions from the
gravitational anomalies to (\ref{csp}). These are discussed separately
in section \ref{external}.
Next we combine (\ref{csp}) with the Konishi anomaly. The Konishi anomaly
corresponds to an anomaly in an axial symmetry under which the matter fields
transform as $\delta \Phi^i = i v \Phi^i$, $ v \in {\mathbb R}$ .
The classical action
is invariant under this symmetry,
\begin{gather} \label{konishi} \left[
\int\! d^6z \, \Phi^i \frac{\delta}{\delta \Phi^i} \, - \,
\int\! d^6\bar z \, \bar \Phi^i \frac{\delta}{\delta \bar \Phi^i} \right]
\, S \, = \, 0 \, .
\end{gather}
The current associated with this symmetry
is precisely the Konishi current: The local version of (\ref{konishi}) is
\begin{gather}
\Phi^i\frac{\delta}{\delta\Phi^i}S=
- \frac{1}{4}
\phi^3 (\bar{\mathcal{D}}^2+R)(\bar{\Phi}_i e^{2V} \Phi^i) \, .
\end{gather}
In the quantised theory, the Konishi current has
an anomaly~\cite{Konishi:1983hf} which takes the off-shell form\footnote{again
in the conventions used in \cite{Leigh:1995ep} for the Konishi anomaly.}
\begin{gather}
\Phi^i\frac{\delta}{\delta\Phi^i}\Gamma^{\prime}
= \frac{1}{4} \phi^3 (\bar{\mathcal{D}}^2+R)(\bar{\Phi}_ie^{2V}\Phi^i)
\, - \,
\frac{1}{16\pi^2} \sum\limits_{i=1}^{n}
T(R_i)\, \phi^3 \mathrm{tr}(W^{\alpha}W_{\alpha}) \, ,
\end{gather}
or
\begin{gather} \label{kon}
\gamma_i
\Phi^i\frac{\delta}{\delta\Phi^i}\Gamma^{\prime}
= \gamma_i \frac{1}{4} \phi^3 (\bar{\mathcal{D}}^2+R)(
\bar{\Phi}_i e^{2V}\Phi^i) \, - \,
\frac{1}{16\pi^2} \sum\limits_{i=1}^{n} \gamma_i
T(R_i) \, \phi^3 \mathrm{tr}(W^{\alpha}W_{\alpha}) \, .
\end{gather}
Combining (\ref{kon}) and (\ref{csp}) we obtain the local Callan-Symanzik
equation
\begin{gather} \left[
\phi \frac{\delta}{\delta \phi} - (1- \gamma_i)
\Phi^i\frac{\delta}{\delta\Phi^i} \right] \Gamma^{\prime}
\, = \,
\left(\mathcal{B}'_1 + \frac{2}{16 \pi^2} \sum\limits_{i=1}^n
\gamma_i T(R_i)\right) \frac{\delta}
{\delta \lambda } \Gamma^{\prime} \, .
\end{gather}
Using
\begin{gather} \label{cseqnPhi}
\mu\frac{\partial}{\partial\mu}\Gamma^{\prime}\, + \, \left(\int\!
d^6z \,\left( \phi\frac{\delta}{\delta\phi} - \Phi^i\frac{\delta}{\delta\Phi^i}
\right) \, + c.c. \right) \, \Gamma^{\prime}
\, = \, 0 \, ,
\end{gather}
which generalises (\ref{cseqn}), and the definition of $\Delta^+$ given in
(\ref{plusminus}), we have
\begin{equation}
\left[\mu\frac{\partial}{\partial\mu} +
\left({\cal B}^{\prime}_1+ \frac{2}{16\pi^2} \sum\limits_{i=1}^n \gamma_i \right)
\Delta^+ - \gamma_i {\cal N}_i
\right]\Gamma^{\prime}=0,
\,
\end{equation}
with
\begin{equation}
{\cal N}_i \equiv \int\!
d^6z \,\left( \Phi^i\frac{\delta}{\delta\Phi^i}+c.c.\right) \, .
\end{equation}
By virtue of the shift equation (\ref{shifta}) we obtain the standard
Callan-Symanzik equation
\begin{gather}
\left[\mu\frac{\partial}{\partial\mu} +
\left({\cal B}^{\prime}_1+ \frac{2}{16\pi^2} \sum\limits_{i=1}^n \gamma_i \right)
\left(\frac{-g^3}{1-\mathcal{A}_1g^2}\right)\partial_g - \gamma_i {\cal N}_i
\right]\Gamma^{\prime}=0 \, .
\end{gather}
Thus with ${\cal B}'_1$ given by (\ref{Bstrich})
we find for the $\beta$-function that
\begin{gather}
\beta(g)=-\frac{g^3}{16\pi^2}\left(\frac{3C_2(G)-\sum_{i=1}^{n}(T(R_i)
-2\gamma_iT(R_i))}{1-C_2(G)g^2/8\pi^2}\right),
\end{gather}
which coincides with the NSVZ beta function.
It is straightforward to include a superpotential with local chiral
supercoupling into this discussion. In this case the matter part of the action
is given by
\begin{equation}\label{matteract2}
S_{\rm matter}
= \frac{1}{4}
\int\! d^8z \, \tilde E \,
\bar{\Phi}_i e^{2V} \Phi^i +\frac{1}{3!}\int\! d^6z \, \phi^3 Y_{ijk}
\Phi^i\Phi^j\Phi^k+\frac{1}{3!}\int\! d^6\bar{z} \, \bar \phi^3
\bar{Y}^{ijk}\bar{\Phi}_i\bar{\Phi}_j\bar{\Phi}_k.
\end{equation}
In this theory the matter beta function satisfies
\begin{gather}
\beta^{ijk}(Y)=3\gamma^{(i}_{\;m}Y^{jk)m}.\label{betamatt}
\end{gather}
As before the classical action is invariant under the Konishi symmetry, under
which the matter fields and the local matter couplings transform as
\begin{align} \label{kos}
&\Phi^i \rightarrow e^{i v}\Phi^i, \;\;\; Y_{ijk}\rightarrow
e^{-3i v}Y_{ijk},\\
&\bar{\Phi}_i\rightarrow e^{-i v}\bar{\Phi}_i,\; \bar{Y}^{ijk}
\rightarrow e^{3i v}\bar{Y}^{ijk} \, , \qquad v \in {\mathbb R} \, .
\nonumber
\end{align}
The superpotential remains invariant under this transformation, as well as the
matter kinetic term. Note also that this symmetry differs from R
symmetry since it leaves the chiral compensator invariant.
The symmetry (\ref{kos}) leads to the Ward identity
\begin{gather}\label{konishi2}
\left(\int\! d^6z \, \left[3Y^{ ijk}\frac{\delta}{\delta
Y^{ijk}}-\Phi^i\frac{\delta}{\delta\Phi^i}\right]-c.c.\right)S = 0 \, .
\end{gather}
In the quantised theory the Konishi anomaly gives rise to
\begin{gather} \label{kon3}
\left( \Phi^i\frac{\delta}{\delta\Phi^j} - 3 Y^{ijk} \frac{\delta}
{\delta Y^{ijk} } \right) \Gamma^{\prime}
= \frac{1}{4} \phi^3 (\bar{\mathcal{D}}^2+R)(\bar{\Phi}_i e^{2V}\Phi^i) \, - \,
\frac{1}{16\pi^2} \sum\limits_{i=1}^{n}T(R_i)
\phi^3\mathrm{tr}(W^{\alpha}W_{\alpha}) \, .
\end{gather}
Note that due to the variation with respect to the supercoupling on the
l.h.s., there is no contribution from the superpotential on the r.h.s. of
(\ref{kon3}). The absence of an anomaly involving the superpotential from this
equation may be seen as follows: As discussed for instance in
\cite{Piguet:1986ug}, for renormalisable massless
SUSY field theories there is
no superpotential contribution to the conformal Ward identity
(\ref{LS}). This is a consequence of R symmetry and
leads in particular to the relation
(\ref{betamatt}) between the matter beta and gamma
functions. Therefore there is no superpotential contribution to the r.h.s.~of
(\ref{csp}). The same argument implies that there is no superpotential
contribution to the r.h.s.~of (\ref{kon3}).~\footnote{We are
grateful to E.~Sokatchev for a discussion on this point. See \cite{Sokatchev}
for cases where mixing occurs.}
By adding (\ref{kon3}) to the integrated off-shell super Weyl identity
(\ref{csp}) we obtain the Callan-Symanzik equation
\begin{gather}
\left[\mu\frac{\partial}{\partial\mu} +
\beta(g) \partial_g + \left( \int\! d^6z \, \beta^{ijk} \frac{\delta}
{\delta Y^{ijk} } + \int\! d^6\bar z \, \bar \beta_{ijk} \frac{\delta}
{\delta \bar Y_{ijk} } \right) - \gamma_i {\cal N}_i
\right]\Gamma^{\prime}=0 \, . \label{ccs}
\end{gather}
with the same NSVZ gauge $\beta$-function as before. There are
additional contributions from the gravitational anomalies
to the r.h.s.~of (\ref{ccs}) which we discuss next.
\section{External Anomalies - Central Functions}\label{external}
\setcounter{equation}{0}
In addition to the internal anomalies discussed
above, there will also be external anomalies involving the super Euler density
and the square of the supergravity Weyl tensor. These external anomalies
appear both in the topological and in the conformal
Ward identities. We calculate the coefficient of the gravitational anomalies
in the topological Ward identity (\ref{ponteq})
to one-loop order and show how an anomaly
shift similar to the one performed in section 2 above allows us to derive an
all-order expression for the central charge $c$ (the coefficient of the Weyl
tensor squared) in a particular well-defined renormalisation scheme. This
expression coincides with the one found in
\cite{Anselmi:1997am} based on a two-loop result of
Jack \cite{Jack:wd}.
\subsection{Gauge field contribution}
As before we first consider pure gauge theory.
In analogy to the gauge anomaly in the topological Ward identity we
expect a gravitational anomaly of the form
\begin{gather}\label{extanom1}
\Delta^{-}\Gamma^{\prime}=\frac{\mathcal{F} N_V}{24\pi^2} \, W^2
-\frac{\mathcal{G} N_V}{24\pi^2} \, E^2,
\end{gather}
where
\begin{align}
W^2 =&\left(\int\! d^6z\,\phi^{3}\lambda^{\prime
-1}(z)W^{\alpha\beta\gamma}W_{\alpha\beta\gamma}-\int\!
d^6\bar{z}\, \bar{\phi}^{3}\bar{\lambda}^{\prime
-1}(\bar{z})\bar{W}_{\dot{\alpha}\dot{\beta}\dot{\gamma}}
\bar{W}^{\dot{\alpha}\dot{\beta}\dot{\gamma}} \right) \, , \nonumber \\
E^2 =& \int\! d^6z\, \phi^{3}\lambda^{\prime
-1}(z)(W^{\alpha\beta\gamma}W_{\alpha\beta\gamma}+(\mathcal{\bar{D}}^2+R)
[G^aG_a+2R\bar{R}])\nonumber \\
&-\int\! d^6\bar{z}\,\bar{\phi}^{3}\bar{\lambda}^{\prime -1}(\bar{z})
(\bar{W}_{\dot{\alpha}\dot{\beta}\dot{\gamma}} \bar{W}^{\dot{\alpha}
\dot{\beta}\dot{\gamma}}+(\mathcal{D}^2+\bar{R})[G^aG_a+2R\bar{R}]).
\end{align}
Here $ W_{\alpha\beta\gamma} $ is the super-Weyl tensor and
$(W^{\alpha\beta\gamma}W_{\alpha\beta\gamma}+(\mathcal{\bar{D}}^2+R)[G^aG_a+2R\bar{R}])$
is the chiral projection of the the super Euler
density~\cite{Buchbinder:qv,Gates:nr}.
By performing the one-loop calculation of appendix A.2,
we find for the coefficients ${\cal F}$, ${\cal G}$ to one-loop order:
\begin{gather}
\mathcal{F}_1 = \, - \frac{1}{32}\, , \; \;
\mathcal{G}_1 = \, - \frac{3}{16}\, . \label{FG}
\end{gather}
A summary of the one-loop calculation of appendix A.2 is as follows:
We decompose (\ref{extanom1})
into components and perform the required triangle diagram computations
for the topological current, in analogy to the gauge anomaly computation
outlined in (\ref{Jtop}), (\ref{introcpnts}). In particular it is necessary to
calculate one-loop three point functions which contain the the
topological current (\ref{Jtop}) and two copies of either the R current or the
energy-momentum tensor. -
In the subsequent we assume that an appropriate scheme may be chosen
such that there are no higher order contributions to ${\cal F}$. This
gives a result for $c$ consistent with expectations. As far as ${\cal
G}$ is concerned, we work with its lowest-order value here and leave
an investigation of its higher order contributions to future
investigations.
In analogy to the discussion of section 2.2, the anomaly (\ref{extanom1})
may be shifted to the `shift
equation' (\ref{qshift2})
by adding an appropriate counterterm to the action.
This counterterm reads
\begin{align}
\Gamma^{ct}(W,E)=&-\frac{\mathcal{F}_1 N_V}{24\pi^2}
\int\! d^6z\, \phi^{3}[\ln(\lambda^{\prime}(z))+\ln 2 g^2]
W^{\alpha\beta\gamma}W_{\alpha\beta\gamma}\nonumber \\
&-\frac{\mathcal{F}_1 N_V}{24\pi^2}
\int\! d^6\bar{z}\, \bar{\phi}^{3}[\ln(\bar{\lambda}^{\prime}(\bar{z}))+\ln
2 g^2]\bar{W}_{\dot{\alpha}\dot{\beta}\dot{\gamma}}
\bar{W}^{\dot{\alpha}\dot{\beta}\dot{\gamma}} \nonumber \\
& + \frac{\mathcal{G}_1N_V}{24\pi^2}\int\! d^6z \,\phi^{3}
\ln(\lambda^{\prime}(z))(W^{\alpha\beta\gamma}W_{\alpha\beta\gamma}
+ (\mathcal{\bar{D}}^2+R)[G^aG_a+2R\bar{R}])
\nonumber \\
&+\frac{\mathcal{G}_1N_V}{24\pi^2}\int\!
d^6\bar{z}\, \bar{\phi}^{3}\ln(\bar{\lambda}^{\prime}(\bar{z}))
(\bar{W}_{\dot{\alpha}\dot{\beta}\dot{\gamma}}
\bar{W}^{\dot{\alpha}\dot{\beta}\dot{\gamma}} +
(\mathcal{D}^2 +\bar{R})[G^aG_a+2R\bar{R}]) .
\end{align}
The $\ln 2 g^2$ is necessary as before to obtain a well-defined constant
coupling limit.
Note since the Euler density is topological, the terms in which it is
multiplied by the constant $\ln 2 g^2$ actually vanish, and such terms are
absent from the counterterm above.
For
\begin{equation} \label{Gamma2strich}
\Gamma^{\prime \prime} = \Gamma' + \Gamma^{ct}(W,E)
\end{equation}
the quantum Pontryagin
equation is anomaly free,
\begin{gather}
\Delta^{-}\Gamma^{\prime\prime}= 0 \, ,
\end{gather}
but the shift equation (\ref{qshift2}) becomes anomalous as in the case of the
internal anomaly discussed in section 2. Let us derive the exact form
of this external anomaly.
For this purpose we first
determine how the topological gravitational anomaly contributes to the
conformal anomaly. We begin by considering the contributions
of the gravitational anomaly to the supercurrent divergence,
\begin{align}
\bar{{\cal D}}^{\dot{\alpha}}T_{\alpha\dot{\alpha}}=& \,
{\cal D}_{\alpha}(T_{internal}
+T_{external}) \, , \\
T_{internal} =-&\frac{1}{6} \mathcal{B}_1 \, \phi^3W^{\beta}W_{\beta} \,
,\, \qquad \mathcal{B}_1=\frac{3}{16\pi^2}C_2(G), \nonumber \\
T_{external}=&-\frac{c_1-a_1 }{48\pi^2}\, \phi^3
W^{\alpha\beta\gamma}W_{\alpha\beta\gamma} \nonumber \\
&+\frac{a_1}{48\pi^2}\,
\phi^3 (\mathcal{\bar{D}}^2+R)(G^aG_a+2R\bar{R}) \label{T}
\end{align}
Just as the internal anomaly contributes with the one-loop coefficient
${\cal B}_1 $ to the superconformal anomaly (see (\ref{consistency2})
and (\ref{wti})),
the gravitational anomaly
contributes with one-loop coefficients $c_1$ and $a_1$ given
by\footnote{For the pure gauge theory considered here,
$N_\chi = 0$ but we include
the matter contribution for reference in the next section.}
\begin{gather} \label{ca}
c_1 = {\textstyle \frac{1}{24}} ( 3 N_V + N_\chi) \, , \quad
a_1 = {\textstyle \frac{1}{24}} (9 N_V + N_\chi) \, .
\end{gather}
(\ref{T}) implies that
in the presence of the supergravity background, the local
Callan-Symanzik equation (\ref{susylocalRG}) reads
\begin{align}
\phi\frac{\delta}{\delta\phi}\Gamma^{\prime\prime}=\,
\mathcal{B}_1\frac{\delta}{\delta\lambda}\Gamma^{\prime\prime}
-&\frac{c_{1}-a_1}{16\pi^2} \, \phi^3 W^{\alpha\beta\gamma}
W_{\alpha\beta\gamma} \nonumber \\
+&\frac{a_1}{16\pi^2}\, \phi^3 (\mathcal{\bar{D}}^2+R)[G^aG_a+2R\bar{R}] \, .
\end{align}
The shift equation (\ref{shifta}) now becomes
\begin{gather}
\mathcal{B}_1\Delta^{+}\Gamma^{\prime\prime}=\beta(g)\partial_{g}\Gamma^{\prime\prime}+\frac{2\mathcal{F}_1N_V}{24\pi^2}\left(\frac{\beta(g)}{g}\right)\left(\int\! d^6z\,
\phi^{3}W^{\alpha\beta\gamma}W_{\alpha\beta\gamma}+c.c \right), \label{sb}
\end{gather}
where we use the scheme, given by (\ref{shifta}),
in which $\beta(g)$ is the NSVZ $\beta$-function.
Note again the absence of the Euler density from the shift
equation (\ref{sb}),
since this is a topological invariant and its integral vanishes.
The Callan-Symanzik equation now reads
\begin{gather}
\left(\mu\frac{\partial}{\partial
\mu}+\beta(g)\partial_g\right)\Gamma^{\prime\prime}=
\frac{1}{16\pi^2}\left(c_1- \frac{4}{3} \mathcal{F}_1 N_V
\frac{\beta(g)}{g} \right) \left(\int\! d^6z \,
\phi^{3}W^{\alpha\beta\gamma}W_{\alpha\beta\gamma}+c.c \right).
\end{gather}
From this expression we read off the result for the central charge
$c$ to all orders,
\begin{gather}
c(g)= c_1 - \frac{4}{3} N_V \mathcal{F}_1\frac{\beta(g)}{g} \, .
\end{gather}
Using the one-loop result (\ref{FG}) for
$\mathcal{F}_1$ we have
\begin{gather}
c(g)= c_1 + \frac{N_V}{24} \frac{\beta(g)}{g}.
\end{gather}
This coincides with the result of \cite{Anselmi:1997am}.
\subsection{Matter contribution}
Let us no include matter fields into the discussion of the section 3.1
in view of deriving an expression for the central charge $c$ in the
presence of matter.
In generalisation of
(\ref{csp}) we have, including the gravitational anomaly,
\begin{align} \label{cspc}
\left( \phi \frac{ \delta}{\delta \phi} - \Phi^i \frac{\delta}{\delta \Phi^i}
\right) \Gamma' \, = & \, - \,
\frac{{\mathcal B}_1{}'}{2} \phi^3 \,
{\rm tr} ( W^\alpha W_\alpha) + \, \frac{1}{4} \phi^3 (\bar {\cal D}^2 +R) \,
\sum\limits_{i=1}^{n} \gamma_i
\bar \Phi_i e^{2V} \Phi_i \, \nonumber\\ & - \frac{c_1 - a_1}{16 \pi^2}
\phi^3 W^{\alpha \beta \gamma}W_{\alpha \beta \gamma} + \frac{a_1}{16 \pi^2} \phi^3
(\bar {\cal D}^2 +R) ( G^{\alpha {\dot \alpha}}G_{\alpha {\dot \alpha}} +2 R \bar R) \, ,
\end{align}
where $c_1$, $a_1$ are the one-loop coefficients (\ref{ca})
of the gravitational Weyl
anomaly. Similarly there are now gravitational anomalies contributing to the
Konishi identity as well,
\begin{align} \label{Konishicurved}
\Phi^i\frac{\delta}{\delta\Phi^i}\Gamma^{\prime}
= &
\frac{1}{4} \phi^3 (\bar{\mathcal{D}}^2+R)(\bar{\Phi}_ie^{2V}\Phi^i) \, - \,
\frac{1}{16\pi^2}\sum\limits_{i}
T(R_i) \, \phi^3 \mathrm{tr}(W^{\alpha}W_{\alpha}) \nonumber\\
& - \frac{1}{24 \pi^2} ({\cal H} - {\cal I})
\phi^3 W^{\alpha \beta \gamma}W_{\alpha \beta \gamma} + \frac{{\cal I}}{24 \pi^2} \phi^3
(\bar {\cal D}^2 +R) ( G^{\alpha {\dot \alpha}}G_{\alpha {\dot \alpha}} +2 R \bar R) \, .
\end{align}
We calculate ${\cal H}$, ${\cal I}$ to one loop in appendix \ref{p calculation} and find
\begin{gather}\label{Hres}
{\cal H}_1 = - \frac{1}{16} \, \sum\limits_{i} N_\chi{}^i \, , \quad {\cal I}_1 = 0 \, .
\end{gather}
$N_\chi{}^i$ is the number of chiral fields with anomalous dimension
$\gamma_i$.
For example, in SQCD with $N_f$
flavours, there is only one $\gamma$ and $N_\chi= 2 N_f N_c$.
The fact the ${\cal I}$ vanishes at one loop agrees with the fact that there is no
contribution linear in $\gamma$ to the central charge $a$, as discuss
in section 4 below.
By combining the Weyl identity (\ref{cspc}) and the Konishi identity
(\ref{Konishicurved}) as before, and using $\Gamma^{\prime \prime}$ defined in
(\ref{Gamma2strich}), we obtain
\begin{align}
\left(\mu \frac{\partial}{\partial \mu} - \gamma_i {\cal N}_i \right) \Gamma^{\prime \prime}
= & - ({\cal B}_1' + \frac{2}{16\pi^2} \sum\limits_{i=1}^n \gamma_i) \Delta^+
\Gamma^{\prime \prime} \nonumber\\ & + \frac{1}{16 \pi^2} \left( c_1 -
\frac{\gamma_i}{24} {N_\chi{}^i} \right) \left( \int\! d^6z \, \phi^3\,
W^{\alpha \beta \gamma}W_{\alpha \beta \gamma} \; + c.c. \right) \, .
\end{align}
Therefore we have
\begin{gather} \label{finalCS}
\left(\mu \frac{\partial}{\partial \mu} + \beta(g) \partial_g
- \gamma_i {\cal N}_i \right) \Gamma^{\prime \prime} = \, \frac{1}{16 \pi^2}
\, c \, \left( \int\! d^6z \, \phi^3\,
W^{\alpha \beta \gamma}W_{\alpha \beta \gamma} \; + c.c. \right) \, ,
\end{gather}
with the central charge
\begin{gather}
c = c_1 + \frac{1}{24}\left( N_V \frac{\beta_g}{g} -
{\gamma_i} {N_\chi{}^i} \right) \, , \qquad c_1 = {
\frac{1}{24}} (3 N_V + \sum\limits_i
N_\chi{}^i) \, . \label{cresult}
\end{gather}
This all-order expression
coincides with the result of~\cite{Anselmi:1997am}.\footnote{
Note that the anomalous dimension used in \cite{Anselmi:1997am}
is twice the size of the one used here. At the same time $N_\chi=N_f
N_c$ in \cite{Anselmi:1997am}, whereas our conventions give $N_\chi=2 N_f
N_c$ for the theories considered in
\cite{Anselmi:1997am}.}
\section{Conclusions}\label{conclusion}
\setcounter{equation}{0}
In this paper we have given a derivation of the NSVZ beta function and of the
central charge $c$ in a well-determined renormalization scheme. Our derivation
is based on the topological anomaly which is present when the couplings are
allowed to be space-time dependent.
As an outlook we comment on the implications of our results on the central
charge $a$, the coefficient of the Euler density anomaly which is expected to
be of relevance for generalizations of the C theorem to four dimensions. In
\cite{FreedmanOsborn}, a four-loop expression for a candidate
C function $\tilde a$ was obtained. Here we chose a particular renormalisation
scheme in which the result of \cite{FreedmanOsborn} reads
\begin{gather}
\tilde a \equiv a + \beta^i w_i \, , \quad \tilde a = a_1 - \frac{1}{8} {\rm {\rm tr}}
(\gamma \gamma) + \frac{1}{12} {\rm tr} (\gamma \gamma \gamma) + \frac{1}{4}
N_V \frac{\beta(g)}{g} + \frac{1}{3} Y^{ijk}\beta_{ijk} \label{FO} \, .
\end{gather}
Here $w_i$ is a one-form in coupling space which may be identified with the
coefficient of a local conformal anomaly involving derivatives of the
couplings. For a derivation of (\ref{FO}) valid to all orders using the approach presented in this
paper, it would be necessary to calculate the coefficients of the new
anomalies - for instance of ${\cal H}$ and ${\cal I}$ in \ref{Konishicurved} -
to higher order, which we have not undertaken here. So far our statements are
limited to expressions linear in the beta and gamma functions.
However the results of this paper are consistent with (\ref{FO}) in at least
two respects. First, the fact that ${\cal I} = 0$ at
one loop in (\ref{Konishicurved}) is
in agreement with the fact that there is no contribution of the form ${\rm tr}
(\gamma)$ (linear in $\gamma$ without further
contributions from the couplings)
to $\tilde a$ as given by (\ref{FO}):
A term of the form ${\cal I} {\rm tr}(\gamma)$ would enter $a$ when
adding (\ref{Konishicurved}) multiplied by $\gamma$ to (\ref{cspc}).
Secondly, the contribution $\frac{1}{4} N_V \frac{\beta(g)}{g}$ to (\ref{FO})
is consistent with ${\cal G} = - \frac{3}{16}$ in (\ref{FG}), just as ${\cal F}
= - \frac{1}{32}$ leads to the $\frac{1}{24} N_V \frac{\beta(g)}{g}$
contribution to $c$ as given by (\ref{cresult}).
The relative numerical factor is $4/3$ in both cases. However the derivation
of $c$ from ${\cal F}$ presented here does not apply to $a$ and ${\cal G}$
since the Euler density contribution is absent from the Callan-Symanzik
equation (\ref{finalCS}). - We intend to return to the question of
deriving an all-order expression for $a$ within the local coupling
approach in the future.
Finally let us note that local couplings appear naturally within the AdS/CFT
correspondence and its generalisations since the couplings appear as boundary
values of the supergravity fields. In particular the holographic C theorem of
\cite{FGPW}, valid for field theories in arbitrary dimensions, may be
interpreted within field theory within the local coupling approach
\cite{Skenderis,Porrati,Erdmenger:2001ja}. We expect that the results
of the present paper will allow for a further understanding of the
field-theoretical implications of the holographic C theorem.
\bigskip \bigskip \bigskip\bigskip \bigskip \bigskip
{\bf Acknowledgements}
We are very grateful to Hugh Osborn for numerous discussions and helpful
comments. Moreover we thank E.~Sokatchev and B.~Wecht for useful discussions.
The research of J.E.~is supported by DFG (Deutsche
Forschungsgemeinschaft) within the `Emmy Noether' programme, grant
ER301/1-4. J.B.~acknowledges support through a Research Fellowship of
the Alexander von Humboldt Foundation while in Berlin.
Part of the research of J.E. for this work was carried out at the KITP, Santa
Barbara, USA.
\bigskip \bigskip \bigskip\bigskip \bigskip \bigskip
|
1,116,691,499,402 | arxiv | \section{Introduction}
A \emph{hole} is a chordless (or an induced) cycle in a graph. The \emph{chordality} of a graph $G$, denoted by $\mathcal{C}(G)$, is defined to be the size of a largest hole in $G$, if there exists a cycle in $G$. If $G$ is acyclic, then its chordality is taken as $0$. A graph $G$ is \emph{$k$-chordal} if $\mathcal{C}(G) \leq k$. In other words, a graph is $k$-chordal if it has no holes with more than $k$ vertices in it. Chordal graphs are exactly the class of $3$-chordal graphs and chordal bipartite graphs are bipartite, $4$-chordal graphs.
$k$-chordal graphs
have been studied in the literature in \cite{BodThil}, \cite{Sun5}, \cite{CSL}, \cite{YonDour},
\cite{Dragan20051} and \cite{Spinrad1991227}. For example, Chandran and Ram \cite{Sun5}
proved that the number of minimum cuts in a $k$-chordal graph is at most $\frac{(k+1)n}{2}-k$.
Spinrad\cite{Spinrad1991227} showed that $(k-1)$-chordal graphs can be recognized in
$O(n^{k-3}M)$ time, where $M$ is the time required to multiply two $n$ by $n$ matrices.
Powering and its effects on the chordality of a graph has been a topic of interest.
The $m$-th power of a graph $G$, denoted by $G^m$, is a graph with vertex set
$V(G^m) = V(G)$ and edge set $E(G^m)=\{(u,v)~|~u \neq v, d_G(u,v) \leq m\}$, where
$d_G(u,v)$ represents the distance between $u$ and $v$ in $G$.
Balakrishnan and Paulraja \cite{BalPaul1} proved that odd powers of chordal graphs are chordal.
Chang and Nemhauser \cite{changNemhauser} showed that if $G$ and $G^2$ are chordal then so are
all powers of $G$. Duchet \cite{Duchet} proved a stronger result which says that if $G^m$ is
chordal then so is $G^{m+2}$. Brandst\"adt et al. in \cite{BrandPower} showed that if $G^m$ is $k$-chordal then so is $G^{m+2}$, where $k\geq 3$ is an integer. Studies on families of graphs that are closed under powering can also be seen in the literature. For instance, it is known that interval graphs,
proper interval graphs \cite{Raychauduri2}, strongly chordal graphs \cite{Lub1}, circular-arc graphs \cite{Raychauduri}\cite{Flotow3}, cocomparability
graphs \cite{Flotow2} etc. are closed under taking powers.
Subclasses of bipartite graphs, like chordal bipartite graphs, are not closed under powering since the $m$-th power of a bipartite graph need not be even bipartite. Chandran et al. in \cite{SunMatRog} introduced the notion of \emph{bipartite powering} to retain the
bipartitedness of a bipartite graph while taking power.
Given a bipartite graph $G$ and an odd positive integer $m$, $G^{[m]}$ is a bipartite graph with $V\left(G^{[m]}\right)=V(G)$ and
$E\left(G^{[m]}\right)=\{(u,v) ~|~ u,v\in V(G), d_G(u,v)\mbox{ is odd, and }
d_G(u,v)\leq m\}$. The graph $G^{[m]}$ is called the \emph{$m$-th bipartite power} of $G$. It was shown in \cite{SunMatRog} that, for every odd positive integer $m$, the $m$-th bipartite power of a tree is chordal bipartite. The intention there was to construct chordal bipartite graphs of high boxicity. The fact that the chordal bipartite graph under consideration was obtained as a bipartite power of a tree was crucial for proving that its boxicity was high. Since trees are a subclass of chordal bipartite graphs, a natural question that came up was the following: is it true that the $m$-th bipartite power of every chordal bipartite graph is chordal bipartite? In this paper we answer this question in the affirmative. In fact, we prove a more general result.
\subsubsection{Our Result}
Let $m$, $k$ be positive integers such that $m$ is odd and $k \geq 4$. Let $G$ be a bipartite graph.
If $G$ is $k$-chordal, then so is $G^{[m]}$. Note that the special case when $k=4$ gives us the following result: chordal bipartite graphs are closed under bipartite powering.
\section{Graph Preliminaries}
Throughout this paper we consider only finite, simple, undirected graphs.
For a graph $G$, we use $V(G)$ to denote the set of vertices of $G$. Let $E(G)$ denote its edge set. For every $x,y \in V(G)$, $d_G(x,y)$ represents the distance between $x$ and $y$ in $G$.
For every $u \in V(G)$, $N_G(u)$ denotes its \emph{open neighborhood} in $G$, i.e. $N_G(u) = \{v~|~(u,v)\in E(G)\}$.
A path $P$ on the vertex set $V(P) = \{v_1,v_2,\ldots,v_n\}$ (where $n\geq 2$) has its edge set $E(P) = \{(v_i,v_{i+1})~|~1 \leq i \leq n-1\}$.
Such a path is denoted by $v_1 v_2 \ldots v_n$. If $v_i,v_j\in V(P)$, $v_iPv_j$ is the path $v_i v_{i+1}\ldots v_{j}$.
The length of a path $P$ is the number of edges in it and is denoted by $||P||$.
A cycle $C$ with vertex set $V(C)=\{v_1,v_2,\ldots,v_n\}$, and edge set $E(C)=\{(v_i,v_{i+1}) ~|~1\leq i\leq n-1\}\cup \{(v_n,v_1)\}$ is denoted as $C = v_1 v_2 \ldots v_n v_1$. We use $||C||$ to denote the length of cycle $C$.
\section{Holes in Bipartite Powers}
Let $H$ be a bipartite graph. Let $\mathcal{B}(H)$ be a family of graphs constructed from $H$ in the following manner: $H' \in \mathcal{B}(H)$ if corresponding to each vertex $v \in V(H)$ there exists a nonempty bag of vertices, say $B_v$, in $H'$ such that (a) for every $x \in B_u$, $y \in B_v$, $(x,y) \in E(H')$ if and only if $(u,v) \in E(H)$, and (b) vertices within each bag in $H'$ are pairwise non-adjacent. Below we list a few observations about $H$ and every $H'$ (, where $H' \in \mathcal{B}(H)$):
\begin{remark}
\label{observation1}
$H'$ is bipartite.
\end{remark}
\begin{remark}
\label{observation2}
$H$ is an induced subgraph of $H'$.
\end{remark}
\begin{remark}
\label{observation3}
Let $k$ be an integer such that $k \geq 4$. If $H$ is $k$-chordal, then so is $H'$.
\end{remark}
\begin{proof}
Any hole of size greater than $4$ in $H'$ cannot have more than one vertex from the same bag, say $B_v$, as such vertices have the same neighborhood. Hence, the vertices of a hole (of size greater than $4$) in $H'$ belong to different bags and thus there is a corresponding hole of the same size in $H$.
\end{proof}
\begin{theorem}\label{bippowtheorem}
Let $m$, $k$ be positive integers such that $m$ is odd and $k \geq 4$. Let $G$ be a bipartite graph.
If $G$ is $k$-chordal, then so is $G^{[m]}$.
\end{theorem}
\begin{proof}
We prove this by contradiction. Let $p$ denote the size of a largest induced cycle, say $C = u_0 u_1 \ldots u_{p-1} u_0$, in $G^{[m]}$.
Assume $p > k$. Then, $p \geq 6$ (since $k \geq 4$ and $G^{[m]}$ is bipartite).
Between each
$u_{i-1}$ and $u_i$, where $i \in \{0, \ldots, p-1\}$, there exists a shortest path of length not more than $m$ in
$G$ \footnotemark[1].
Let $P_i$ be one such shortest path between $u_{i-1}$ and $u_i$ in $G$.
Let $H$ be the subgraph induced on the vertex set $\bigcup_{i=0}^{p-1}V(P_i)$ in $G$. As mentioned in the beginning of this section, construct a graph $H'$ from $H$, where $H' \in \mathcal{B}(H)$, in the following manner: for each $v \in V(H)$, let $|B_v| = |\{P_i~|~0\leq i \leq p-1, v \in V(P_i)\}|$ i.e., let $B_v$ have as many vertices as the number of paths in $\{P_0 \ldots P_{p-1}\}$ that share vertex $v$ in $H$.
For each $i \in \{0, \ldots , p-1\}$, let $Q_i' = u_{i-1}Q_i$ be a shortest path between $u_{i-1}$ and $u_i$ in $H'$ such that no two paths $Q_i$ and $Q_j$ (where $i \neq j$) share a vertex \footnotemark[1]
\footnotetext[1]{throughout this proof expressions involving subscripts of $u$, $P$, $Q$, and $Q'$ are to be taken modulo $p$. Every such expression should be evaluated to a value in $\{0, \ldots , p-1 \}$. For example, consider a vertex $u_i$, where $i < p$
Then, $p + i = i$.}.
From our construction of $H'$ from $H$ it is easy to see that such paths exist. Let $Q_i = v_{i,1} v_{i,2} \ldots v_{i,r_i}u_i$, where $r_i = ||Q_i|| \geq 0$. Thus, $Q_i' = u_{i-1}v_{i,1} v_{i,2} \ldots v_{i,r_i}u_i$.
Clearly, $||Q_i'|| = ||P_i|| \leq m$. The reader may also note that the cycle $C$ ($ = u_0 u_1 \ldots u_{p-1} u_0$) which is present in $G^{[m]}$ will be present in $H^{[m]}$ and thereby in $H'^{[m]}$ too.
In order to prove the theorem, it is enough to show that there exists an induced cycle of size at least $p$ in $H'$. Then by combining Observation \ref{observation3} and the fact that $H$ is an induced subgraph of $G$, we get $k \geq \mathcal{C}(G) \geq \mathcal{C}(H) \geq \mathcal{C}(H') \geq p$ contradicting our assumption that $p > k$. Hence, in the rest of the proof we show that $\mathcal{C}(H') \geq p$.
Consider the following drawing of the
graph $H'$. Arrange the vertices $u_0, u_1,$ $\ldots, u_{p-1}$ in that order
on a circle in clockwise order.
Between each $u_{i-1}$ and $u_{i}$ on the circle arrange the vertices $v_{i,1}, v_{i,2}, \ldots , v_{i,r_i}$ in that order in clockwise order. Recall that these vertices are the internal vertices of path $Q_i'$.
\begin{remark}
\label{neighborObv}
In this circular arrangement of vertices of $H'$, each vertex has an edge (in $H'$) with both its left neighbor and right neighbor in the arrangement.
\end{remark}
Let $x_1, x_2 \in V(H')$, where $x_1 \in V(Q_i)$, $x_2 \in V(Q_j)$. We define the \emph{clockwise distance from $x_1$ to $x_2$}, denoted by $clock\_dist(x_1,x_2)$, as the minimum non-negative integer $s$ such that $j = i + s$. Similarly, the \emph{clockwise distance from $x_2$ to $x_1$}, denoted by $clock\_dist(x_2,x_1)$, is the minimum non-negative integer $s'$ such that $i = j+s'$. Let $x,y,z \in V(H')$.
We say $y<_x z$ if scanning the vertices of $H'$ in
clockwise direction along the circle starting from $x$, vertex $y$ is encountered before $z$.
Let $x \in V(Q_i)$. Vertex $y$ is called the \emph{farthest neighbor of $x$ before $z$} if $y \in N_{H'}(x)$, $y \in V(Q_{i}) \cup V(Q_{i+1}) \cup V(Q_{i+2})$, $y <_x z$, and for every other $w \in N_{H'}(x)$ either $z <_x w$ or $w \notin V(Q_{i}) \cup V(Q_{i+1}) \cup V(Q_{i+2})$ or both.
\begin{remark}
\label{farthestNeighborObv}
There always exists a vertex which is the farthest neighbor of $x$ before $z$, unless $(x,z) \in E(H')$ and $z \in V(Q_{i}) \cup V(Q_{i+1}) \cup V(Q_{i+2})$.
\end{remark}
Let $\{A,B\}$ be the bipartition of the bipartite graph $H'$. We categorize the edges of $H'$ as follows: an edge $(x,y) \in E(H')$
is called an \emph{$l$-edge}, if $l = \min(clock\_dist(x,y),clock\_dist(y,x) )$.
\begin{figure}[H]
\centerline{\input{lEdge.pstex_t}}
\caption{$x \in V(Q_i), y \in V(Q_{i+l})$ and let $(x,y) \in E(H')$ be an $l$-edge, where $l > 2$. The dotted line between $u_{i-1}$ and $u_i$ indicate the path $Q_i$. Similarly, the dotted line between $u_{i+l-1}$ and $u_{i+l}$ indicate the path $Q_{i+l}$.}
\label{l-edgeFig}
\end{figure}
\begin{claim}
\label{Ledgeclaim}
$H'$ cannot have an $l$-edge, where $l>2$.
\end{claim}
\begin{proof}
Suppose $H'$ has an $l$-edge, where $l > 2$, between $x \in Q_i$ and $y \in Q_{i+l}$ (see Fig. \ref{l-edgeFig}). Let $a_1 = ||u_{i-1}Q_i'x||,~ b_1 = ||xQ_i'u_i||,~ a_2 = ||u_{i+l-1}Q_{i+l}'y||$ and $ b_2 = ||yQ_{i+l}'u_{i+l}||$. We consider the following two cases:
\begin{case}
$l$ is even.
\end{case}
In this case $u_{i-1}$ and $u_{i+l-1}$ will be on the same side of the bipartite graph $H'$. Without loss of generality, let $u_{i-1}, u_{i+l-1} \in A$. Then, $u_{i}, u_{i+l} \in B$.
We know that, for every $w_1, w_2 \in V(H'^{[m]})$ with $w_1 \in A$ and $w_2 \in B$, if $(w_1,w_2) \notin E(H'^{[m]})$ then $d_{H'}(w_1,w_2) \geq m+2$ (recalling $m$ and $d_{H'}(w_1,w_2)$ are odd integers).
Therefore, we have
$a_1 + 1 + b_2 \geq d_{H'}(u_{i-1},u_{i+l}) \geq m+2$.
Similarly, $b_1 + 1 + a_2 \geq d_{H'}(u_{i},u_{i+l-1}) \geq m+2$. Summing up the two inequalities we get, $(a_1 + b_1) + (a_2 + b_2) \geq 2m + 2$. This implies that either $||Q_i'||$ or $||Q_{i+l}'||$ is greater than $m$ which is a contradiction.
\begin{case}
$l$ is odd (proof is similar to the above case and hence omitted).
\end{case}
Hence we prove the claim.
\end{proof}
We find a cycle $C' = z_0z_1\ldots z_qz_0$ in $H'$ using Algorithm $3.1$ \footnotemark[1]. Please read the algorithm before proceeding further.
\footnotetext[1]{throughout this proof expressions involving subscripts of $z$ are to be taken modulo $q+1$. Every such expression should be evaluated to a value in $\{0, \ldots , q \}$. For example, consider a vertex $z_a$, where $a < q+1$. Then, $q+1+a = a$.}.
\begin{algorithm}[H]
\begin{algorithmic}
\label{cycle-construction}
\caption{Finding Cycle $C'$ in $H'$ such that $||C'|| \geq ||C||$}
\setcounter{line}{0}
\MYSTATE{$l \leftarrow \max_{l'}(H' \mbox{has an }l'\mbox{-edge})$. Without loss of generality assume that this $l$-edge is between a vertex in $Q_0$ and a vertex in $Q_{l}$}
\MYSTATE{Scan the vertices of $Q_0$ in clockwise direction to find the first vertex $z_0$, where $z_0 \in V(Q_0)$, which has an $l$-edge to a vertex in $Q_{l}$.}
\MYSTATE{Scan the vertices of $Q_{l}$ in clockwise direction to find the last vertex in $Q_l$ which is a neighbor of $z_0$ in $H'$. Call it $z_1$.}
\MYSTATE{Find the farthest neighbor of $z_1$ before $z_0$. Call it $z_2$.} /* refer proof of Claim \ref{farthestNeighborClaim} for a proof of existence of such a $z_2$*/
\MYSTATE{$s \leftarrow 2$.}
\WHILE{$(z_s,z_0) \notin E(H')$}
\MYSTATE{Find the farthest neighbor of $z_s$ before $z_0$. Call it $z_{s+1}$.} /* such a neighbor exists by Observation \ref{farthestNeighborObv}*/
\MYSTATE{ $s \leftarrow s + 1$.}
\ENDWHILE
\MYSTATE{ $q \leftarrow s$.}
\MYSTATE{ Return cycle $C' = z_0 z_1\ldots z_{q} z_0$.}
\end{algorithmic}
\end{algorithm}
\begin{claim}
\label{farthestNeighborClaim}
There always exists a farthest neighbor of $z_1$ before $z_0$.
\end{claim}
\begin{proof}
Note that $z_0 \in Q_0$ and $z_1 \in Q_l$, where $l \leq 2$ (by Claim \ref{Ledgeclaim}). Recalling that $||C|| = p \geq 6$, we have $z_0 \notin V(Q_{l}) \cup V(Q_{l+1}) \cup V(Q_{l+2})$. Hence by Observation \ref{farthestNeighborObv}, the claim is true.
\end{proof}
\begin{claim}
\label{whileClaim}
The while loop in Algorithm $3.1$ terminates after a finite number of iterations.
\end{claim}
\begin{proof}
From Observation \ref{neighborObv}, we know that each vertex has an edge (in $H'$) with both its left neighbor and right neighbor in the circular arrangement. Each time when Step $6$ of Algorithm $3.1$ is executed, a vertex $z_{s+1}$ is chosen such that $z_{s+1}$ is the farthest neighbor of $z_s$ before $z_0$.
Since $H'$ is a finite graph, there will be a point of time in the execution of the algorithm when in Step $6$ it picks a $z_{s+1}$ such that $(z_{s+1}, z_0) \in E(H')$ .
\end{proof}
From Claim \ref{whileClaim}, we can infer that $C'$ is a cycle.
\begin{claim}\label{inducedcycleclaim}
$C'$ is an induced cycle in $H'$.
\end{claim}
\begin{proof}
Suppose $C'$ is not an induced cycle. Then there exists a chord $(z_a,z_b)$ in $C'$. Since $(z_a, z_b)$ is a chord, we have $b \neq a-1$ or $b \neq a+1$.
Let $l =\max_{l'}(H'$ has an $l'\mbox{-edge})$.
Let $z_a \in V(Q_i)$, $z_b \in V(Q_j)$.
We know that $\min (clock\_dist(z_a, z_b),$ $clock\_dist(z_b, z_a)) \leq l$. Without loss of generality, assume $clock\_dist(z_a, z_b) \leq l \leq 2$ (from Claim \ref{Ledgeclaim}). That is, $j-i \leq l \leq 2$ and $(z_a,z_b)$ is a $(j-i)$-edge. If $z_a = z_0$, then $z_b \neq z_1$ and the algorithm exits from the while loop, when $q=b$, thus returning a cycle $z_0 \ldots z_b z_0$. But in such a cycle $(z_b, z_0)$ is not a chord. Therefore, $z_a \neq z_0$. Similarly, $z_b \neq z_0$.
We know that $z_{a+1} \neq z_b$, $z_{a+1} <_{z_a} z_b$, and $z_{a+1} \in V(Q_{i}) \cup V(Q_{i+1}) \cup V(Q_{i+2})$. Since $j-i \leq 2$, $z_b \in V(Q_{i}) \cup V(Q_{i+1}) \cup V(Q_{i+2})$. If $z_b <_{z_a} z_0$, then it contradicts the fact that $z_{a+1}$ is the farthest neighbor of $z_a$ before $z_0$. Therefore, $z_0 <_{z_a} z_b$. Then, either $z_b = z_1$ or $z_1 <_{z_a} z_b$. Recall that $l = \max_{l'}(H' \mbox{has an }l'\mbox{-edge})$, and $(z_0,z_1)$ is an $l$-edge with $z_0 \in V(Q_0)$ and $z_1 \in V(Q_{l})$. Since (i) $(z_a,z_b)$ is a $(j-i)$-edge, where $j-i \leq l$, (ii) $z_0 <_{z_a} z_b$, and (iii) $z_b =z_1$ or $z_1 <_{z_a} z_b$, we have $l \geq j-i = clock\_dist(z_a, z_b) \geq clock\_dist(z_0, z_b) \geq clock\_dist(z_0, z_1) = l$. Hence, $j-i = l$ and $(z_a,z_b)$ is an $l$-edge. We know that $(z_0,z_1)$ is also an $l$-edge with $z_0 \in V(Q_0)$ and $z_1 \in V(Q_l)$. Since $z_0 <_{z_a} z_b$ and $z_b = z_1$ or $z_1 <_{z_0} z_b$, we get $z_a \in V(Q_0)$ and $z_b \in V(Q_l)$.
From Step 2 of the algorithm we know that $z_0$ is the first vertex (in a clockwise scan) in $Q_0$ which has an $l$-edge to a vertex in $Q_l$. This implies that, since $z_0 <_{z_a} z_b$, $z_a = z_0$ which is a contradiction. Hence we prove the claim.
\end{proof}
What is left now is to show that $q+1 \geq p$, i.e., $||C'|| \geq ||C||$, where $C' = z_0 \ldots z_qz_0$ and $C = u_0 \ldots u_{p-1}u_0$. In order to show this, we state and prove the following claims.
\begin{claim}
\label{cycleSizeClaim}
For every $j \in \{0, \ldots , p-1\}$, $(V(Q_j) \cup V(Q_{j+1}))\cap V(C') \neq \emptyset$.
\end{claim}
\begin{proof}
Suppose the claim is not true. Find the minimum $j$ that violates the claim. Clearly, $j \neq 0$ as $z_0 \in V(Q_0)$. We claim that $z_q \in V(Q_{j-1})$. Suppose $z_q \notin V(Q_{j-1})$. Let $a = \max \{i~|~z_i \in V(Q_{j-1})\}$ (note that, since $j \neq 0$, by the minimality of $j$, $(V(Q_{j-1}) \cup V(Q_j)) \cap V(C') \neq \emptyset$ and therefore $V(Q_{j-1}) \cap V(C') \neq \emptyset$). Since $z_a \neq z_q$, by the maximality of $a$, we have $z_{a+1} \notin V(Q_{j-1})$. From our assumption, $(V(Q_j) \cup V(Q_{j+1}))\cap V(C') = \emptyset$ and therefore $z_{a+1} \notin V(Q_{j-1}) \cup V(Q_j) \cup V(Q_{j+1})$. Thus $z_a \neq z_q$ and $z_{a+1}$ is not the farthest neighbor of $z_a$ before $z_0$. This is a contradiction to the way $z_{a+1}$ is chosen by Algorithm $3.1$.
Hence, $z_q \in V(Q_{j-1})$. We know that $(z_q, z_0) \in E(H')$ with $z_q \in V(Q_{j-1})$ and $z_0 \in V(Q_0)$. Since $l = \max_{l'}(H'\mbox{ has an }l'\mbox{-edge})$, we have $\min(clock\_dist(z_q,z_0),$ $clock\_dist(z_0,z_q)) \leq l$ . That is, $j \geq p + 1-l$ or $j \leq 1+l$. As $l \leq 2$ (by Claim \ref{Ledgeclaim}), we have $j=p-1$ or $j \leq 1+l$. Since $z_0 \in V(Q_0)$, $(V(Q_{p-1}) \cup V(Q_{0}))\cap V(C') \neq \emptyset$ and hence $j \neq p-1$. Therefore, $j \leq 1+l$. Since $z_0 \in V(Q_0)$ and $z_1 \in V(Q_{l})$ (recall $l \leq 2$), we get $j = 1 + l$.
We know that, for every $z_a, z_b \in V(C')$, if $a<b$ then $z_a <_{z_0} z_b$. Therefore, $z_1 <_{z_0} z_q$. We have $z_1 \in V(Q_l)$. Since $j = 1+l$, we also have $z_q \in V(Q_l)$. Thus, we have $z_1, z_q \in V(Q_l)$ and $z_1 <_{z_0} z_q$. But this contradicts the fact that $z_1$ is the last vertex in $Q_l$ encountered in a clockwise scan that has $z_0$ as its neighbor.
\end{proof}
\begin{figure}
\centerline{\input{0Edge1.pstex_t}}
\caption{Figure illustrates the case when path $P$ defined in Claim \ref{0-edgeClaim} is a trivial path. The dotted lines between each $u_{i-1}$ and $u_i$ indicate the path $Q_i'$. Each continuous arc corresponds to an edge in the cycle $C' = z_0\ldots z_qz_0$.}
\label{0-edgeFig1}
\end{figure}
\begin{figure}
\centerline{\input{0Edge2.pstex_t}}
\caption{Figure illustrates the case when path $P$ defined in Claim \ref{0-edgeClaim} is
$P = z_{a+1} z_{a+2}\ldots z_{a+1+s}$, where $s \geq 1$ and $z_{a+1+s} = z_b$. The dotted lines between each $u_{i-1}$ and $u_i$ indicate the path $Q_i'$. Each continuous arc corresponds to an edge in the cycle $C' = z_0\ldots z_qz_0$.}
\label{0-edgeFig2}
\end{figure}
\begin{claim}
\label{0-edgeClaim}
Let $(z_a,z_{a+1}), (z_b,z_{b+1}) \in E(C')$ be two $2$-edges, where $a < b$. Let $P$, $P'$ denote the clockwise $z_{a+1}-z_b$, $z_{b+1}-z_a$ paths respectively in $C'$.
Both $P$ and $P'$ contain at least one $0$-edge.
\end{claim}
\begin{proof}
Consider the path $P$ (proof is similar in the case of path $P'$). Path $P$ is a non-trivial path only if $z_{a+1} \neq z_b$. Suppose $z_{a+1} = z_b$ (see Fig. \ref{0-edgeFig1}). Let $z_a \in V(Q_f)$.
For the sake of ease of notation, assume $f=1$ (the same proof works for any value of $f$). Let $a_1 = ||u_0Q_1'z_a||$, $b_1 = ||z_aQ_1'u_1||$, $a_2 = ||u_2Q_3'z_b||$, $b_2 = ||z_bQ_3'u_3||$, $a_3 = ||u_4Q_5'z_{b+1}||$, and $b_3 = ||z_{b+1}Q_5'u_5||$.
We know that, for every $w_1, w_2 \in V(H'^{[m]})$ with $w_1 \in A$ and $w_2 \in B$, if $(w_1,w_2) \notin E(H'^{[m]})$ then $d_{H'}(w_1,w_2) \geq m+2$.
Since $(u_0,u_3) \notin E(H'^{[m]}),~(u_1,u_4) \notin E(H'^{[m]})$ and $(u_2,u_5) \notin E(H'^{[m]})$, we have $a_1 + b_2 \geq m+1$, $b_1 + a_3 \geq m$, and $a_2 + b_3 \geq m+1$.
Adding the three inequalities and by applying an easy averaging argument we can infer that either $a_1 + b_1 = ||Q_1|| > m$, $a_2 + b_2 = ||Q_{3}|| > m$, or $a_3 + b_3 =||Q_{5}|| > m$ which is a contradiction. Therefore $P$ is a non-trivial path i.e., $z_{a+1} \neq z_b$. Assume $P$ does not contain any $0$-edge. Let $P = z_{a+1} z_{a+2}\ldots z_{a+1+s}$, where $s \geq 1$, $a+1+s = b$, and $(z_{a+1},z_{a+2}) \ldots (z_{a+s},z_{a+1+s})$ are $1$-edges (see Fig. \ref{0-edgeFig2}). Since $(u_0,u_3) \notin E(H'^{[m]}),~(u_1,u_4) \notin E(H'^{[m]})$, we have $c_a+d_{a+1} \geq m+1$ and $d_{a} + d_{a+2} \geq m$ (please refer Fig. \ref{0-edgeFig2} for knowing what $c_a, d_a, \ldots ,c_{b+1},d_{b+1}$ are). Summing up the two inequalities, we get $d_{a+1} + d_{a+2} \geq 2m+1 - (c_{a} + d_a)$. We know that, for each $i \in \{0, \ldots p-1\}$, $||Q_i'|| \leq m$. Therefore, we have $c_a + d_a \leq m$. Hence, $d_{a+1} + d_{a+2} \geq m+1$. Since $(c_{a+1} + d_{a+1}) + (c_{a+2} + d_{a+2}) \leq 2m$, we get
\begin{eqnarray}
\label{c1c2ineq}
c_{a+1} + c_{a+2} & \leq & m-1
\end{eqnarray}
Since $(u_{s+2},u_{s+5}) \notin E(H'^{[m]}),~(u_{s+1},u_{s+4}) \notin E(H'^{[m]})$, we have,
\begin{eqnarray*}
c_b + d_{b+1} & \geq & m+1 \\
c_{a+s} + c_{b+1} & \geq & m
\end{eqnarray*}
Summing up the two inequalities, we get
\begin{eqnarray*}
c_b + c_{a+s} & \geq & 2m+1 - (c_{b+1} + d_{b+1})
\end{eqnarray*}
Since $b=a+s+1$ and $c_{b+1} + d_{b+1} \leq m$, we get
\begin{eqnarray}
\label{cscs+1ineq}
c_{a+s+1} + c_{a+s} & \geq & m+1
\end{eqnarray}
Substituting for $s=1$ in Inequality \ref{cscs+1ineq}, we get $c_{a+2} + c_{a+1} \geq m+1$. But this contradicts Inequality \ref{c1c2ineq}. Hence $s>1$. Suppose $s=2$. Since $(u_{2},u_{5}) \notin E(H'^{[m]}))$, we have $c_{a+1} + d_{a+3} \geq m$. Adding this with Inequality \ref{cscs+1ineq}, we get $c_{a+1} + c_{a+2} \geq (2m+1) - (c_{a+3} + d_{a+3}) \geq m+1$. But this contradicts Inequality \ref{c1c2ineq}. Hence $s>2$. Since $(u_{s},u_{s+3}) \notin E(H'^{[m]})), \ldots ,(u_{2},u_{5}) \notin E(H'^{[m]}))$, we have the following inequalities:-
\begin{eqnarray*}
c_{a+s-1} + d_{a+s+1} & \geq & m \\
\vdots & \vdots & \vdots \\
c_{a+1} + d_{a+3} & \geq & m
\end{eqnarray*}
Adding the above set of inequalities and applying the fact that $c_i + d_i \leq m $, $\forall i \in \{0, \ldots q\}$, we get $c_{a+1} + c_{a+2} + d_{a+s} + d_{a+s+1} \geq 2m$. Adding this with Inequality \ref{cscs+1ineq}, we get $c_{a+1} + c_{a+2} \geq (3m+1) - (c_{a+s+1} + d_{a+s+1}) - (c_{a+s} + d_{a+s}) \geq m+1$. But this contradicts Inequality \ref{c1c2ineq}. Hence we prove the claim.
\end{proof}
\begin{claim}
\label{pathContribClaim}
For every $j,j' \in \{0, \ldots, p-1\}$, where $j < j'$ and $(V(Q_j) \cup V(Q_{j'}))\cap V(C') = \emptyset$, there exist $i, i' \in \{0, \ldots, p-1\}$, where only $i$ satisfies $j < i < j'$, such that $|V(Q_{i})\cap V(C')| \geq 2$ and $|V(Q_{i'})\cap V(C')| \geq 2$.
\end{claim}
\begin{proof}
By Claim \ref{cycleSizeClaim}, (i) $j' \neq j+1$ or $j' \neq j-1$, and (ii) there exist $r,r' \in \{0, \ldots , q\}$ such that $(z_{r}, z_{r+1})$ is a $2$-edge with its endpoints on $Q_{j-1}$ and $Q_{j+1}$ and $(z_{r'}, z_{r'+1})$ is a $2$-edge with its endpoints on $Q_{j'-1}$ and $Q_{j'+1}$. By Claim \ref{0-edgeClaim}, we know that if $P$, $P'$ denote the clockwise $z_{r+1}- z_{r'}$, $z_{r'+1}-z_r$ paths respectively in $C'$, then both $P$ and $P'$ contains at least one $0$-edge. This proves the claim.
\end{proof}
In order to show that the size of cycle $C'$ ($= z_0 \ldots z_qz_0$) is at least $p$, we consider the following three cases:- \vspace{0.05in} \\
\emph{case $|\{Q_j \in \{Q_0 \ldots Q_{p-1}\}~|~V(Q_j) \cap V(C') = \emptyset \}| = 0$}: In this case, for every $j \in \{0, \ldots p-1\}$, $Q_j$ contributes to $V(C')$ and therefore $||C'|| \geq p = ||C||$. \vspace{0.05in} \\
\emph{case $|\{Q_j \in \{Q_0 \ldots Q_{p-1}\}~|~V(Q_j) \cap V(C') = \emptyset \}| = 1$}: Let $Q_j$ be that only path (among $Q_0 \ldots Q_{p-1}$) that does not contribute to $V(C')$. Then we claim that there exists a $Q_{j'}$, where $j' \neq j$, such that $V(C') \cap V(Q_{j'}) \geq 2$. Suppose the claim is not true then it is easy to see that $||C'|| = p-1$ which is an odd number thus contradicting the bipartitedness of $H'$. Hence the claim is true. Now, by applying the claim it is easy to see that $||C'|| = \sum_{j}|V(C') \cap V(Q_j)| \geq p = ||C||$. \vspace{0.05in} \\
\emph{case $|\{Q_j \in \{Q_0 \ldots Q_{p-1}\}~|~V(Q_j) \cap V(C') = \emptyset \}| > 1$}: Scan vertices of $H'$ starting from any vertex in clockwise direction. Claim \ref{pathContribClaim} ensures that between every $Q_j$ and $Q_{j'}$, which do not contribute to $V(C')$, encountered there exists a $Q_i$ which compensates by contributing at least two vertices to $V(C')$. Therefore, $||C'|| \geq p = ||C||$.
\hfill \rule{0.6em}{0.6em}
\end{proof}
|
1,116,691,499,403 | arxiv | \section{Introduction}
\label{S:1}
\subsection*{Federated Learning Overview}
The ever-growing use of big data systems, industrial-scale IoT platforms, and smart devices contribute to the exponential growth in data dimensions~\cite{s19204354}. This exponential growth of data has accelerated the adoption of machine learning in many areas, such as natural language processing and computer vision. However, many machine learning systems suffer from insufficient training data. The reason is mainly due to the increasing concerns on data privacy, e.g., restrictions on data sharing with external systems for machine learning purposes. For instance, the General Data Protection Regulation (GDPR)~\cite{(gdpr)_2019} stipulate a range of data protection measures, and data privacy is now one of the most important ethical principles expected of machine learning systems \cite{jobin2019global}. Furthermore, raw data collected are unable to be used directly for model training for most circumstances. The raw data needs to be studied and pre-processed before being used for model training and data sharing restrictions increase the difficulty to obtain training data. Moreover, concept drift~\cite{9084352} also occurs when new data is constantly collected, replacing the outdated data. This makes the model trained on previous data degrades at a much faster rate. Hence, a new technique that can swiftly produce models that adapt to the concept drift when different data is discovered in clients is essential.
\begin{figure}[tbh!]
\centering\includegraphics[width=0.9\linewidth]{FL_overview.pdf}
\caption{Federated Learning Overview.}
\label{Fig:FLOps}
\end{figure}
To effectively address the lack of training data limitations, concept drift, and the data-sharing restriction while still enabling effective data inferences by the data-hungry machine learning models, Google introduced the concept of federated learning~\cite{mcmahan2017communicationefficient} in 2016. Fig.~\ref{Fig:FLOps} illustrates the overview of federated learning. There are three stakeholders in a federated learning system: (1) learning coordinator (i.e., system owner), (2) contributor client - data contributor \& local model trainer, and (3) user client - model user. Note that a contributor client can also be a user client. There are two types of system nodes (i.e., hardware components): (1) central server, (2) client device.
\subsection*{A Motivation Example}
Imagine we use federated learning to train the next-word prediction model in a mobile phone keyboard application. The learning coordinator is the provider of the keyboard application, while contributor clients are the mobile phone users. The user clients will be the new or existing mobile phone users of the keyboard application. The differences in ownership, geolocation, and usage pattern cause the local data to possess non-IID\footnote{Non-Identically and Independently Distribution: Highly-skewed and personalised data distribution that vary heavily between different clients and affects the model performance and generalisation~\cite{8889996}.} characteristics, which is a design challenge of federated learning systems. The federated learning process starts when a training task is created by the learning coordinator. For instance, the keyboard application provider produces and embeds an initial global model into the keyboard applications. The initial global model (includes task scripts \& training hyperparameters) is broadcast to the participating client devices. After receiving the initial model, the model training is performed locally across the client devices, without the centralised collection of raw client data. Here, the smartphones that have the keyboard application installed receive the model to be trained. The client devices typically run on different operating systems and have diverse communication and computation resources, which is defined as the system heterogeneity challenges.
Each training round takes one step of gradient descent on the current model using each client's local data. In this case, the smartphones optimise the model using the keyboard typing data. After each round, the model update is submitted by each participating client device to the central server. The central server collects all the model updates and performs model aggregation to form a new version of the global model. The new global model is re-distributed to the client devices for the next training round. This entire process iterates until the global model converges. As a result, communication and computation efficiency are crucial as many local computation and communication rounds are required for the global model to converge. Moreover, due to the limited resources available on each device, the device owners might be reluctant to participate in the federated learning process. Therefore, client motivatability becomes a design challenge. Furthermore, the central server communicates with multiple devices for the model exchange which makes it vulnerable to the single-point-of-failure. The trustworthiness of the central server and the possibility of adversarial nodes participating in the training process also creates system reliability and security challenges. After the completion of training, the learning coordinator stores the converged model and deploys it to the user clients. For instance, the keyboard application provider stores the converged model and embeds it to the latest version of the application for existing or new application users. Here, the different versions of the local models associated with the global model created need to be maintained.
\subsection*{Design Challenges}
Compared to centralised machine learning, federated learning is more advantageous from the data privacy perspective and dealing with the lack of training data. However, a federated learning system, as a large-scale distributed system, presents more architectural design challenges~\cite{10.1145/3450288}, especially when dealing with the interactions between the central server and client devices and managing trade-offs of software quality attributes. The main design challenges are summarised as follows.
\begin{itemize}[leftmargin=*]
\item
Global models might have low accuracy, and lack generality when client devices generate non-IID data. Centralising and randomising the data is the approach adopted by conventional machine learning to deal with data heterogeneity but the inherent privacy-preserving nature of federated learning render such techniques inappropriate.
\item To generate high-quality global models that are adaptive to concept drift, multiple rounds of communication are required to exchange local model updates, which could incur high communication costs.
\item Client devices have limited resources to perform the multiple rounds of model training and communications required by the system, which may affect the model quality.
\item As numerous client devices participate in federated learning, it is challenging to coordinate the learning process and ensure model provenance, system reliability and security.
\end{itemize}
How to select appropriate designs to fulfill different software quality requirements and design constraints is non-trivial for such a complex distributed system. Although much effort has been put into federated learning from the machine learning techniques side, there is still a gap on the architectural design considerations of the federated learning systems. A systematic guidance on architecture design of federated learning systems is required to better leverage the existing solutions and promote federated learning to enterprise-level adoption.
\begin{figure*}
\centering
\includegraphics[width=0.75\linewidth]{Protocol.pdf}
\caption{Pattern Collection Process.}
\label{Fig:CollectionProtocol}
\end{figure*}
\subsection*{Research Contribution}
In this paper, we present a collection of patterns for the design of federated learning systems. In software engineering, an architectural pattern is a reusable solution to a problem that occurs commonly within a given context in software design~\cite{Beck1987}. Our pattern collection includes three client management patterns, four model management patterns, three model training patterns, and four model aggregation patterns. We define the lifecycle of a model in a federated learning system and associate each identified pattern to a particular state transition in the lifecycle.
The main contribution of this paper includes:
\begin{itemize}[leftmargin=*]
\item The collection of architectural patterns provides a design solution pool for practitioners to select from for real-world federated learning system development.
\item The federated learning model lifecycle with architectural pattern annotations, which serves as a systematic guide for practitioners during the design and development of a federated learning system.
\end{itemize}
\subsection*{Paper Structure}
The remainder of the study is organised as follows. Section~\ref{S:methodology} introduces the research methodology. Section~\ref{S:patterns} provides an overview of the patterns in the federated learning lifecycle, followed by the detailed discussions on each pattern. Section~\ref{S:discussion} summarises and discusses some repeating challenges of federated learning systems and Section~\ref{S:related} presents the related work. Finally, Section~\ref{S:conclusion} concludes the paper.
\begin{table*}[!h]
\scriptsize
\linespread{1.3}
\centering
\caption{Sources of Patterns.}
\begin{tabular}{l l c c c}
\toprule
\textbf{\footnotesize{Category}} & \textbf{\makecell[c]{\footnotesize{Pattern}}} & \textbf{\makecell[l]{\footnotesize{SLR papers}}} & \textbf{\makecell[l]{\footnotesize{ML/FL papers}}} & \textbf{\makecell[l]{\footnotesize{Real-world applications}}}\\
\midrule
\multirow{3}{0.2\columnwidth}{Client management patterns} & Pattern 1: Client registry & 0 & 2 & 3\\
& Pattern 2: Client selector & 4 & 2 & 1\\
& Pattern 3: Client cluster & 2 & 2 & 2\\
\cmidrule(l){1-5}
\multirow{4}{0.2\columnwidth}{Model management patterns} & Pattern 4: Message compressor & 8 & 6 & 0\\
& Pattern 5: Model co-versioning registry & 0 & 0 & 4\\
& Pattern 6: Model replacement trigger & 0 & 1 & 3\\
& Pattern 7: Deployment selector & 0 & 0 & 3\\
\cmidrule(l){1-5}
\multirow{3}{0.2\columnwidth}{Model training patterns} & Pattern 8: Multi-task model trainer & 2 & 1 & 3\\
& Pattern 9: Heterogeneous data handler & 1 & 2 & 0\\
& Pattern 10: Incentive registry & 18 & 1 & 0\\
\cmidrule(l){1-5}
\multirow{4}{0.2\columnwidth}{Model aggregation patterns} & Pattern 11: Asynchronous aggregator & 4 & 1 & 0\\
& Pattern 12: Decentralised aggregator & 5 & 2 & 0\\
& Pattern 13: Hierarchical aggregator & 4 & 2 & 0\\
& Pattern 14: Secure aggregator & 31 & 0 & 3\\
\bottomrule
\label{tab:mapping}
\end{tabular}
\end{table*}
\section{Methodology}
\label{S:methodology}
Fig.~\ref{Fig:CollectionProtocol} illustrates the federated learning pattern extraction and collection process. Firstly, the patterns are collected based on the results of our previous systematic literature review (SLR) on federated learning~\cite{10.1145/3450288}. SLR and situational method engineering (SME)~\cite{BRINKKEMPER1996275} are some of the renowned systematic methodologies for derivation of pattern languages. For instance, several pattern derivations on cloud migration and software architecture have used SLR (e.g., Zdun et al.~\cite{zdun2007systematic}, Aakash Ahmad et al.~\cite{ahmad2014pattern}, and Jamshidi et al.~\cite{jamshidi2017pattern}). Moreover, Balalaie et al.~\cite{balalaie2018microservices} have derived the pattern languages in the context of cloud-native and microservices using situational method engineering.
For this work, we have adopted the SLR method as the currently available materials and research works on federated learning are still highly academic-based. Secondly, we intend to propose design patterns for software architectural design aspects of building federated learning systems rather than for their development/engineering processes. This is because, during the SLR work, we have identified many architectural design challenges and lack of systematic design approaches to federated learning. Furthermore, while SME has the benefit of offering a systematic methodology for selecting appropriate method components from a repository of reusable method components, it is more suitable for pattern extraction of an information system development (ISD) process~\cite{mirbel2006situational}.
The SLR was performed according to Kitchenham’s SLR guideline~\cite{Kitchenham07guidelinesfor}, and the number of final studied papers is 231. During the SLR, we developed a comprehensive mapping between federated learning challenges and approaches. Additionally, we reviewed 22 machine learning and federated learning papers published after the SLR search cut-off date (31st Jan 2020) and 22 real-world applications. The additional literature review on machine learning and federated learning, and the review of the real-world applications are conducted based on our past real-world project implementation experience. Table~\ref{tab:mapping} shows a mapping between each pattern with its respective number of source papers and real-world applications. Twelve patterns were initially collected from SLR or additional literature review, whereas the remaining two patterns were identified through real-world applications.
We discussed the proposed patterns according to the pattern form presented in \cite{10.5555/273448.273487}. The form comprehensively describes the patterns by discussing the \textbf{Context}, \textbf{Problem}, \textbf{Forces}, \textbf{Solution}, \textbf{Consequences}, \textbf{Related patterns}, and \textbf{Known-uses} of the pattern.
The \textbf{Context} is the description of the situation where a problem occurs, in which the solution proposed is applicable, or the problem is solvable by the pattern. \textbf{Problem} comprehensively elicits the challenges and limitations that occur under the defined context. \textbf{Forces} describe the reasons and causes for a specific design or pattern decision to be made to resolve the problem.
\textbf{Solution} describes how the problem and the conflict of forces can be resolved by a specific pattern. \textbf{Consequences} reason about the impact of applying a solution, specifically on the contradictions among the benefits, costs, drawbacks, tradeoffs, and liabilities of the solution. \textbf{Related patterns} record the other patterns from this paper that are related to the current pattern. \textbf{Known-uses} refer to empirical evidence that the solution has been used in the real world.
\section{Federated Learning Patterns}
\label{S:patterns}
Fig.~\ref{Fig:FLLifeCycle} illustrates the lifecycle of a model in a federated learning system. The lifecycle covers the deployment of the completely trained global model to the client devices (i.e., model users) for data inference. The deployment process involves the communication between the central server and the client devices. We categorise the federated learning patterns as shown in Table \ref{tab:overview} to provide an overview. There are four main groups: (1) client management patterns, (2) model management patterns, (3) model training patterns, and (4) model aggregation patterns.
\begin{table*}[tbp]
\scriptsize
\centering
\caption{Overview of architectural patterns for federated learning}
\label{tab:overview}
\begin{tabular}{p{0.3\columnwidth}p{0.4\columnwidth}p{1.2\columnwidth}}
\toprule
\textbf{\makecell[l]{\footnotesize{Category}}} &
\textbf{\makecell[c]{\footnotesize{Name}}} &
\textbf{\makecell[c]{\footnotesize{Summary}}}\\
\midrule
\multirow{5}{0.3\columnwidth}{\textbf{Client management patterns}} & \multirow{1}{0.32\columnwidth}{Client registry} & Maintains the information of all the participating client devices for client management.\\
\cmidrule(l){2-3}
& \multirow{1}{0.32\columnwidth}{Client selector} & Actively selects the client devices for a certain round of training according to the predefined criteria to increase model performance and system efficiency.\\
\cmidrule(l){2-3}
& \multirow{1}{0.32\columnwidth}{Client cluster} & Groups the client devices (i.e., model trainers) based on their similarity of certain characteristics (e.g., available resources, data distribution, features, geolocation) to increase the model performance and training efficiency.\\
\cmidrule(l){1-3}
\cmidrule(l){1-3}
\multirow{8}{0.3\columnwidth}{\textbf{Model management patterns}} & \multirow{2}{0.32\columnwidth}{Message compressor} & Compresses and reduces the message data size before every round of model exchange to increase the communication efficiency.\\
\cmidrule(l){2-3}
& \multirow{1}{0.4\columnwidth}{Model co-versioning registry} & Stores and aligns the local models from each client with the corresponding global model versions for model provenance and model performance tracking.\\
\cmidrule(l){2-3}
& \multirow{1}{0.4\columnwidth}{Model replacement trigger} & Triggers model replacement when the degradation in model performance is detected.\\
\cmidrule(l){2-3}
& \multirow{1}{0.32\columnwidth}{Deployment selector} & Selects and matches the converged global models to suitable client devices to maximise the global models' performance for different applications and tasks.\\
\cmidrule(l){1-3}
\cmidrule(l){1-3}
\multirow{8}{0.3\columnwidth}{\textbf{Model training patterns}} & \multirow{1}{0.4\columnwidth}{Multi-task model trainer} & Utilises data from separate but related models on local client devices to improve learning efficiency and model performance.\\
\cmidrule(l){2-3}
& \multirow{1}{0.4\columnwidth}{Heterogeneous data handler} & Solves the non-IID and skewed data distribution issues through data volume and data class addition while maintaining the local data privacy.\\
\cmidrule(l){2-3}
& \multirow{1}{0.32\columnwidth}{Incentive registry} & Measures and records the performance and contributions of each client and provides incentives to motivate clients' participation. \\
\cmidrule(l){1-3}
\multirow{8}{0.3\columnwidth}{\textbf{Model aggregation patterns}} & \multirow{1}{0.4\columnwidth}{Asynchronous aggregator} & Performs aggregation asynchronously whenever a model update arrives without waiting for all the model updates every round to reduce aggregation latency.\\
\cmidrule(l){2-3}
& \multirow{1}{0.4\columnwidth}{Decentralised aggregator} & Removes the central server from the system and decentralizes its role to prevent single-point-of-failure and increase system reliability.\\
\cmidrule(l){2-3}
& \multirow{1}{0.32\columnwidth}{Hierarchical aggregator} & Adds an edge layer to perform partial aggregation of local models from closely-related client devices to improve model quality and system efficiency. \\
\cmidrule(l){2-3}
& \multirow{1}{0.32\columnwidth}{Secure aggregator}
& The adoption of secure multiparty computation (MPC) protocols that manages the model exchange and aggregation security to protect model security. \\
\bottomrule
\end{tabular}
\end{table*}
\begin{figure*}
\centering\includegraphics[width=0.8\linewidth]{FL_lifecycle.pdf}
\caption{A Model's Lifecycle in Federated Learning.}
\label{Fig:FLLifeCycle}
\end{figure*}
\subsection{\textbf{Client Management Patterns}}\label{ClientManagementPatterns}
Client management patterns describe the patterns that manage the client devices' information and their interaction with the central server. A \textit{client registry} manages the information of all the participating client devices. \textit{Client selector} selects client devices for a specific training task, while \textit{client cluster} increases the model performance and training efficiency through grouping client devices based on the similarity of certain characteristics (e.g., available resources, data distribution).
\subsubsection{\textbf{Pattern 1: Client Registry}}\label{ClientRegistry}
\begin{figure}[h]
\centering\includegraphics[width=0.7\linewidth]{Client_registry.pdf}
\caption{Client Registry.}
\label{Fig:ClientRegistry}
\end{figure}
\textbf{Summary:} A client registry maintains the information of all the participating client devices for client management. According to Fig.~\ref{Fig:ClientRegistry}, the client registry is maintained in the central server. The central server sends the request for information to the client devices. The client devices then send the requested information together with the first local model updates.
\textbf{Context:} Client management is centralised, and global and local models are exchanged between the central server the massive number of distributed client devices with dynamic connectivity and diverse resources.
\textbf{Problem:} It is challenging for a federated learning system to track any dishonest, failed, or dropout node. This is crucial to secure the central server and client devices from adversarial threats. Moreover, to effectively align the model training process of each client device for each aggregation round, a record of the connection and training information of each client device that has interacted with the central server is required.
\textbf{Forces:} The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Updatability.} The ability to keep track of the participating devices is necessary to ensure the information recorded is up-to-date.
\item \textit{Data privacy.} The records of client information expose the clients to data privacy issues. For instance, the device usage pattern of users may be inferred from the device connection up-time, device information, resources, etc.
\end{itemize}
\textbf{Solution:} A client registry records all the information of client devices that are connected to the system from the first time. The information includes device ID, connection up \& downtime, device resource information (computation, communication, power \& storage). The access to the client registry could be restricted according to the agreement between the central server and participating client devices.
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Maintainability.} The client registry enables the system to effectively manage the dynamically connecting and disconnecting clients.
\item \textit{Reliability.} The client registry provides status tracking for all the devices, which is essential for problematic node identification.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Data privacy.} The recording of the device information on the central server leads to client data privacy issues.
\item \textit{Cost.} The maintenance of client device information requires extra communication cost and storage cost, which further surges when the number of client devices increases.
\end{itemize}
\textbf{Related patterns:} \textit{Model Co-Versioning Registry, Client Selector, Client Cluster, Asynchronous Aggregator, Hierarchical Aggregator}
\textbf{Known uses:}
\begin{itemize}[leftmargin=*]
\item \textit{IBM Federated Learning\footnote{\url{https://github.com/IBM/federated-learning-lib}}}: \textit{Party Stack} component manages the client parties of IBM federated learning framework, that contains sub-components such as protocol handler, connection, model, local training, and data handler for client devices registration and management.
\item doc.ai\footnote{\url{https://doc.ai/}}: Client registry is designed for medical research applications to ensure that updates received apply to a current version of the global model, and not a deprecated global model.
\item \textit{SIEMENS Mindsphere Asset Manager \footnote{\url{https://documentation.mindsphere.io/resources/html/asset-manager/en-US/index.html}}}:To support the collaboration of federated learning clients in industrial IoT environment,~\textit{Industrial Metadata Management} is introduced as a device metadata and asset data manager.
\end{itemize}
\subsubsection{\textbf{Pattern 2: Client Selector}}\label{ClientSelector}
\begin{figure}[h]
\centering\includegraphics[width=0.7\linewidth]{Client_selector.pdf}
\caption{Client Selector.}
\label{Fig:ClientSelector}
\end{figure}
\textbf{Summary:} A client selector actively selects the client devices for a certain round of training according to the predefined criteria to increase model performance and system efficiency. As shown in Fig.~\ref{Fig:ClientSelector}, the central server assesses the performance of each client according to the information received. Based on the assessment results, the second client is excluded from receiving the global model.
\textbf{Context:} Multiple rounds of model exchanges are performed and communication cost becomes a bottleneck. Furthermore, multiple iterations of aggregations are performed and consume high computation resources.
\textbf{Problem:} The central server is burdensome to accommodate the communication with massive number of widely-distributed client devices every round.
\textbf{Forces:} The problem requires the following forces to be balanced:
\begin{itemize}[leftmargin=*]
\item \textit{Latency.} Client devices have system heterogeneity (difference in computation, communication, \& energy resources) that affect the local model training and global model aggregation time.
\item \textit{Model quality.} Local data are statistically heterogeneous (different data distribution/quality) which produce local models that overfit the local data.
\end{itemize}
\textbf{Solution:} Selecting client devices with predefined criteria can optimise the formation of the global model. The client selector on the central server performs client selection every round to include the best fitting client devices for global model aggregation. The selection criteria can be configured as follows:
\begin{itemize}[leftmargin=*]
\item Resource-based: The central server assesses the resources available on each client devices every training round and selects the client devices with the satisfied resource status (e.g., WiFi connection, pending status, sleep time)
\item Data-based: The central server examines the information of the data collected by each client, specifically on the number of data classes, distribution of data sample volume per class, and data quality. Based on these assessments, the model training process includes devices with high-quality data, higher data volume per class, and excludes the devices with low-quality data, or data that are highly heterogeneous in comparison with other devices.
\item Performance-based: Performance-based client selection can be conducted through local model performance assessment (e.g., performance of the latest local model or the historical records of local model performance).
\end{itemize}
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Resource optimisation.} The client selection optimises the resource usage of the central server to compute and communicate with suitable client devices for each aggregation round, without wasting resources to aggregate the low-quality models.
\item \textit{System performance.} Selecting clients with sufficient power and network bandwidth greatly reduces the chances of clients dropping out and lowers the communication latency.
\item \textit{Model performance.} Selecting clients with the higher local model performance or lower data heterogeneity increases the global model quality.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Model generality.} The exclusion of models from certain client devices may lead to the missing of essential data features and the loss of the global model generality.
\item \textit{Data privacy.} The central server needs to acquire the system and resource information (processor's capacity, network availability, bandwidth, online status, etc.) every round to perform devices ranking and selection. Access to client devices' information creates data privacy issues.
\item \textit{Computation cost.} Extra resources are spent on transferring of the required information for selection decision-making.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Deployment Selector}
\textbf{Known uses:}
\begin{itemize}[leftmargin=*]
\item \textit{Google's FedAvg}: \textit{FedAvg} \cite{mcmahan2017communicationefficient} algorithm includes client selection that randomly selects a subset of clients for each round based on predefined environment conditions and device specification of the client devices.
\item In \textit{IBM's Helios} \cite{xu2019helios}, there is a training consumption profiling function that fully profiles the resource consumption for model training on client devices. Based on that profiling, a resource-aware soft-training scheme is designed to accelerate local model training on heterogeneous devices and prevent stragglers from delaying the collaborative learning process.
\item \textit{FedCS} suggested by OMRON SINIC X Corporation\footnote{\url{https://www.omron.com/sinicx/}} sets a certain deadline for clients to upload the model updates.
\item \textit{Communication-Mitigated Federated Learning (CMFL)} \cite{8885054} excludes the irrelevant local updates by identifying the relevance of a client update by comparing its global tendency of model updating with all the other clients.
\item \textit{CDW\_FedAvg}~\cite{9233457} takes the centroid distance between the positive and negative classes of each client dataset into account for aggregation.
\end{itemize}
\subsubsection{\textbf{Pattern 3: Client Cluster}}\label{ClientCluster}
\begin{figure}[h]
\centering\includegraphics[width=0.6\linewidth]{Client_cluster.pdf}
\caption{Client Cluster.}
\label{Fig:ClientCluster}
\end{figure}
\textbf{Summary:} A client cluster groups the client devices (i.e., model trainers) based on their similarity of certain characteristics (e.g., available resources, data distribution, features, geolocation) to increase the model performance and training efficiency. In Fig.~\ref{Fig:ClientCluster}, the client devices are clustered into 2 groups by the central server, and the central server will broadcast the global model that is more related to the clusters accordingly.
\textbf{Context:} The system trains models over client devices which have diverse communication and computation resources, resulted in statistical and system heterogeneity challenges.
\textbf{Problem:} Federated learning models generated under non-IID data properties are deemed to be less generalised. This is due to the lack of significantly representative data labels from the client devices. Furthermore, local models may drift significantly from each other.
\textbf{Forces:} The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Computation cost and training time.} More computation costs and longer training time are required to overcome the non-IID issue of client devices.
\item \textit{Data privacy.} Data privacy contradicts with the Access to the entire or parts of the client's raw data is needed by the learning coordinator to resolve the non-IID issue which creates data privacy risks.
\end{itemize}
\textbf{Solution:} Client devices are clustered into different groups according to their properties (e.g., data distribution, features similarities, gradient loss). By creating clusters of clients with similar data patterns, the global model generated will have better performance for the non-IID-severe client network, without accessing the local data.
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Model quality.} The global model created by client clusters can have a higher model performance for highly personalised prediction tasks.
\item \textit{Convergence speed.} The consequent deployed global models can have faster convergence speed as the models of the same cluster can identify the gradient's minima much faster when the clients' data distribution and IIDness are similar.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Computation cost.} The central server consumes extra computation cost and time for client clustering and relationship quantification.
\item \textit{Data privacy.} The learning coordinator (i.e., central server) requires extra client device information (e.g., data distribution, feature similarities, gradient loss) to perform clustering. This exposes the client devices to the possible risk of private data leakage.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Client Selector, Deployment Selector}
\textbf{Known uses:}
\begin{itemize}[leftmargin=*]
\item Iterative Federated Clustering Algorithm (\textit{IFCA})\footnote{\url{https://github.com/jichan3751/ifca}} is a \\framework introduced by UC Berkley and Google to cluster client devices based on the loss values of the client's gradient.
\item Clustered Federated Learning (\textit{CFL})\footnote{\url{https://github.com/felisat/clustered-federated-learning}}
uses a cosine simila\\rity-based clustering method that creates a bi-partitioning to group client devices with the same data generating distribution into the same cluster.
\item \textit{TiFL} \cite{10.1145/3369583.3392686} is a tier-based federated learning system that adaptively groups client devices with similar training time per round to mitigate the heterogeneity problem without affecting the model accuracy.
\item Patient clustering in a federated learning system is implemented by Massachusetts General Hospital to improve efficiency in predicting mortality and hospital stay time \cite{HUANG2019103291}.
\end{itemize}
\subsection{\textbf{Model Management Patterns}}\label{ModelManagementPatterns}
Model management patterns include patterns that handle mo\\-del transmission, deployment, and governance. A \textit{message compressor} reduces the transmitted message size. A \textit{model co-versioning registry} records all local model versions from each client and aligns them with their corresponding global model. A \textit{model replacement trigger} initiates a new model training task when the converged global model's performance degrades. A \textit{deployment selector} deploys the global model to the selected clients to improve the model quality for personalised tasks.
\subsubsection{\textbf{Pattern 4: Message Compressor}}\label{MessageCompressor}
\begin{figure}[h]
\centering\includegraphics[width=0.85\linewidth]{Message_compressor.pdf}
\caption{Message Compressor.}
\label{Fig:MessageCompressor}
\end{figure}
\textbf{Summary:} A message compressor reduces the message data size before every round of model exchange to increase the communication efficiency. Fig.~\ref{Fig:MessageCompressor} illustrates the operation of the pattern on both ends of the system (client device and central server).
\textbf{Context:} Multiple rounds of model exchanges occurs between a central server and many client devices to complete the model training.
\textbf{Problem:} Communication cost for model communication (e.g., transferring model parameters or gradients) is often a critical bottleneck when the system scales up, especially for bandwidth-limited client devices.
\textbf{Forces:} The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Computation cost.} High computation costs are required by the central server to aggregate all the bulky model parameters collected every round.
\item \textit{Communication cost.} Communication costs are scarce to communicate the model parameters and gradients between resource-limited client devices and the central server.
\end{itemize}
\textbf{Solution:} The model parameters and the training task script as one message package is compressed before being transferred between the central server and client devices.
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Communication efficiency.} The compression of model parameters reduces the communication cost and network\\ throughput for model exchanges.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Computation cost.} Extra computation is required for message compression and decompression every round.
\item \textit{Loss of information.} The downsizing of the model parameters might cause the loss of essential information.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Model Co-Versioning Registry}
\textbf{Known uses:}
\begin{itemize}[leftmargin=*]
\item \textit{Google Sketched Update}~\cite{konecny2017federated}: Google proposes two communication efficient update approaches: structured update and sketched update. Structured update directly learns an update from a restricted space that can be parametrised using a smaller number of variables, whereas sketched update compresses the model before sending it to the central server.
\item \textit{IBM PruneFL}~\cite{jiang2020model} adaptively prunes the distributed parameters of the models, including initial pruning at a selected client and further pruning as part of the federated learning process.
\item \textit{FedCom}~\cite{haddadpour2020federated} compresses messages for uplink communication from the client device to the central server. The central server produces a convex combination of the previous global model and the average of updated local models to retain the essential information of the compressed model parameters.
\end{itemize}
\subsubsection{\textbf{Pattern 5: Model Co-versioning Registry}}\label{ModelCo-versioningRegistry}
\begin{figure}[h]
\centering\includegraphics[width=0.75\linewidth]{Model_co-versioning_registry.pdf}
\caption{Model Co-versioning Registry.}
\label{Fig:ModelCo-versioningRegistry}
\end{figure}
\textbf{Summary:} A model co-versioning registry records all local model versions from each client and aligns them with their corresponding global model. This enables the tracing of model quality and adversarial client devices at any specific point of the training to improve system accountability. Fig.~\ref{Fig:ModelCo-versioningRegistry} shows that the registry collects and maps the local model updates to the associated global model versions.
\textbf{Context:} Multiple new versions of local models are generated from different client devices and one global model aggregated each round. For instance, a federated learning task that runs for 100 rounds on 100 devices will create 10,000 local models and 100 global models in total.
\textbf{Problem:} With high numbers of local models created each round, it is challenging to keep track of which local models contributed to the global model of a specific round. Furthermore, the system needs to handle the challenges of asynchronous updates, client dropouts, model selection, etc.
\textbf{Forces:} The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Updatability.} The system needs to keep track of the local and global models concerning each client device's updates (application's version or device OS/firmware updates) and ensure that the information recorded is up-to-date.
\item \textit{Immutability.} The records and storage of the models co-versions and client IDs needs to be immutable.
\end{itemize}
\textbf{Solution:} A model co-versioning registry records the local model version of each client device and the global model it corresponds to. This enables seamless synchronous and asynchronous model updates and aggregation. Furthermore, the model co-versioning registry enables the early-stopping of complex model training (stop training when the local model overfits and retrieve the best performing model previously). This can be done by observing the performance of the aggregated global model. Moreover, to provide accountable model provenance and co-versioning, blockchain is considered as one alternative to the central server due to immutability and decentralisation properties.
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Model quality.} The mapping of local models with their corresponding version of the global model allows the study of the effect of each local model quality on the global model.
\item \textit{Accountability.} System accountability improves as stakeholders can trace the local models that correspond to the current or previous global model versions.
\item \textit{System security.} It enables the detection of adversarial or dishonest clients and tracks the local models that poisons the global model or causes system failure.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Storage cost.} Extra storage cost is incurred to store all the local and global models and their mutual relationships. The record also needs to be easily retrievable and it is challenging if the central server host more task.
\end{itemize}
\textbf{Related patterns:}~\textit{Client Registry}
\textbf{Known uses:}
\begin{itemize}[leftmargin=*]
\item \textit{DVC}\footnote{\url{https://dvc.org/}} is an online machine learning version control platform built to make models shareable and reproducible.
\item \textit{MLflow Model Registry}\footnote{\url{https://docs.databricks.com/applications/mlflow/model-registry.html}} on Databricks is a centralized model store that provides chronological model lineage, model versioning, stage transitions, and descriptions.
\item \textit{Replicate.ai}\footnote{\url{https://replicate.ai/}} is an open-source version control platform for machine learning that automatically tracks code, hyperparameters, training data, weights, metrics, and system dependencies.
\item \textit{Pachyderm}\footnote{\url{https://www.pachyderm.com/}} is an online machine learning pipeline platform that uses containers to execute the different steps of the pipeline and also solves the data versioning provenance issues.
\end{itemize}
\subsubsection{\textbf{Pattern 6: Model Replacement Trigger}}\label{ModelReplacementTrigger}
\begin{figure}[h]
\centering\includegraphics[width=0.7\linewidth]{Model_replacement_trigger.pdf}
\caption{Model Replacement Trigger.}
\label{Fig:ModelReplacementTrigger}
\end{figure}
\textbf{Summary:} Fig.~\ref{Fig:ModelReplacementTrigger} illustrates a model replacement trigger that initiates a new model training task when the current global model's performance drops below the threshold value or when a degrade on model prediction accuracy is detected.
\textbf{Context:} The client devices use converged global models for inference or prediction.
\textbf{Problem:} As new data is introduced to the system, the global model accuracy might reduce gradually. Eventually, with the degrading performance, the model is no longer be suitable for the application.
\textbf{Forces:} The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Model quality.} The global model deployed might experience a performance drop when new data are used for inference and prediction.
\item \textit{Performance degradation detection.} The system needs to effectively determine the reason for the global model's performance degradation before deciding whether to activate a new global model generation.
\end{itemize}
\textbf{Solution:} A model replacement trigger initiates a new model training task when the acting global model's performance drops below the threshold value. It will compare the performance of the deployed global model on a certain number of client devices to determine if the degradation is a global event. When the global model performance is lower than the preset threshold value for more than a fixed number of consecutive times, given that performance degradation is a global event, a new model training task is triggered.
\textbf{Consequences: }
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Updatability.} The consistent updatability of the global model helps to maintain system performance and reduces the non-IID data effect. It is especially effective for clients that generate highly personalised data that causes the effectiveness of the global model to reduce much faster as new data is generated.
\item \textit{Model quality.} The ongoing model performance monitoring is effective to maintain the high quality of the global model used by the clients.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Computation cost.} The client devices will need to perform model evaluation periodically that imposes extra computational costs.
\item \textit{Communication cost.} The sharing of the evaluation results among clients to know if performance degradation is a global event is communication costly.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Client Selector, Model Co-versioning Registry}
\textbf{Known uses: }
\begin{itemize}[leftmargin=*]
\item \textit{Microsoft Azure Machine Learning Designer}\footnote{\url{https://azure.microsoft.com/en-au/services/machine-learning/designer/}} provides a platform for machine learning pipeline creation that enables models to be retrained on new data.
\item \textit{Amazon SageMaker}\footnote{\url{https://aws.amazon.com/sagemaker/}} provides model deployment and monitoring services to maintain the accuracy of the deployed models.
\item \textit{Alibaba Machine Learning Platform}\footnote{\url{https://www.alibabacloud.com/product/machine-learning}} provides end-to-end machine learning services, including data processing, feature engineering, model training, model prediction, and model evaluation.
\end{itemize}
\subsubsection{\textbf{Pattern 7: Deployment Selector}}\label{DeploymentSelector}
\begin{figure}[h]
\centering\includegraphics[width=0.68\linewidth]{Deployment_selector.pdf}
\caption{Deployment Selector.}
\label{Fig:DeploymentSelector}
\end{figure}
\textbf{Summary:} A deployment selector deploys the converged global model to the selected model users to improve the prediction quality for different applications and tasks. As shown in Fig.~\ref{Fig:DeploymentSelector}, different versions of converged models are deployed to different groups of clients after evaluation.
\textbf{Context:} Client devices train local models using multitask federated learning settings (a model is trained using data from multiple applications to perform similar and related tasks). These models need to be deployed to suitable groups of client devices.
\textbf{Problem:} Due to the inherent diversity and non-IID distribution among local data, the globally trained model may not be accurate enough for all clients or tasks.
\textbf{Forces:} The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Identification of clients.} The central server needs to match and deploy the global models to the different groups of client devices.
\item \textit{Training and storage of different models.} The central server needs to train and store different global models for diverse clients or applications.
\end{itemize}
\textbf{Solution:} A deployment selector examines and selects clients (i.e., model users) to receive the trained global model specified for them based on their data characteristics or applications. The deployment selector deploys the model to the client devices once the global model is completely trained.
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Model performance.} Deploying converged global models to suitable groups of client devices enhances the model performance to the specific groups of clients or applications.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Cost.} There are extra costs for training of multiple personalised global models, deployment selection, storage of multiple global models.
\item \textit{Model performance.} The statistical heterogeneity of model trainers produces personalised local models, which is then generalised through~\textit{FedAvg} aggregation. We need to consider the performance trade-off of the generalised global model deployed for different model users and applications.
\item \textit{Data privacy.} Data privacy challenges exist when the central server collects the client information to identify suitable models for different clients. Moreover, the global model might be deployed to model users that have never joined the model training process.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Client Selector}
\textbf{Known uses: }
\begin{itemize}[leftmargin=*]
\item \textit{Azure Machine Learning}\footnote{\url{https://docs.microsoft.com/en-us/azure/machine-learning/concept-model-management-and-deployment}} supports mass deployment with a step of compute target selection.
\item \textit{Amazon SageMaker}\footnote{\url{https://docs.aws.amazon.com/sagemaker/latest/dg/multi-model-endpoints.html}} can host multiple models with multi-model endpoints.
\item \textit{Google Cloud}\footnote{\url{https://cloud.google.com/ai-platform/prediction/docs/deploying-models}} uses model resources to manage different versions of models.
\end{itemize}
\subsection{\textbf{Model Training Patterns}}\label{ModelTrainingPatterns}
Patterns about the model training and data preprocessing are group together as model training patterns, including \textit{multi-task model trainer} that tackles non-IID data characteristics, \textit{heterogeneous data handler} that deals with data heterogeneity in training datasets, and \textit{incentive registry} that increases the client's motivatability through rewards.
\subsubsection{\textbf{Pattern 8: Multi-Task Model Trainer}}\label{Multi-taskModelTrainer}
\begin{figure}[h]
\centering\includegraphics[width=0.63\linewidth]{Multi-task_model_trainer.pdf}
\caption{Multi-Task Model Trainer.}
\label{Fig:Multi-taskModelTrainer}
\end{figure}
\textbf{Summary:} In federated learning, a multi-task model trainer trains separated but related models on local client devices to improve learning efficiency and model performance. As shown in Fig.~\ref{Fig:Multi-taskModelTrainer}, there are two groups of applications: (i) text-related applications (e.g., messaging, email, etc.), and (ii) image-related applications (camera, video, etc.). The related models are trained on client devices using data of related tasks.
\textbf{Context:} Local data has statistical heterogeneity property where the data distribution among different clients is skewed and a global model cannot capture the data pattern of each client.
\textbf{Problem:} Federated learning models trained with non-IID data suffer from low accuracy and are less generalised to the entire dataset. Furthermore, the local data that is highly personalised to the device users' usage pattern creates local models that diverge in different directions. Hence, the global model may have relatively low averaged accuracy.
\textbf{Forces:} The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Computation cost.} The complex model that solves the non-IID issue consumes more computation and energy resources every round compared to general federated model training. It also takes longer to compute all the training results from the different tasks before submitting them to the central server.
\item \textit{Data privacy.} To address the non-IID issue, more information from the local data needs to be explored to understand the data distribution pattern. This ultimately exposes client devices to local data privacy threats.
\end{itemize}
\textbf{Solution:} The multi-task model trainer performs similar-but-related machine learning tasks on client devices. This enables the local model to learn from more local data that fit naturally to the related local models for different tasks. For instance, a multi-task model for the next-word prediction task is trained using the on-device text messages, web browser search strings, and emails with similar mobile keyboard usage patterns. \textit{MOCHA}~\cite{smith2018federated} is a state-of-the-art federated multi-task learning algorithm that realises distributed multi-task learning on federated settings.
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Model quality.} Multi-task learning improves the model performance by considering local data and loss in optimization and obtaining a local weight matrix through this process. The local model fits for non-IID data in each node better than a global model.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Model quality.} Multi-task training often works only with convex loss functions and performs weak on non-convex loss functions.
\item \textit{Model portability.} As each client has a different model, the model's portability is a problem that makes it hard to apply multi-task training on cross-device FL.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Model Co-versioning Registry, Client Cluster, Deployment Selector}
\textbf{Known uses: }
\begin{itemize}[leftmargin=*]
\item \textit{MultiModel}\footnote{\url{https://ai.googleblog.com/2017/06/multimodel-multi-task-machine-learning.html}} is a neural network architecture by Google that simultaneously solves several problems spanning multiple domains, including image recognition, translation, and speech recognition.
\item \textit{MT-DNN}\footnote{\url{https://github.com/microsoft/MT-DNN}} is an open-source natural language understanding toolkit by Microsoft to train customized deep learning models.
\item \textit{Yahoo Multi-Task Learning for Web Ranking} is a multi-task learning framework developed by Yahoo! Labs to rank in web search.
\item \textit{VIRTUAL}~\cite{corinzia2019variational} is an algorithm for federated multi-task learning with non-convex models. The server and devices are treated as a star-shaped bayesian network, and model learning is performed on the network using approximated variational inference.
\end{itemize}
\subsubsection{\textbf{Pattern 9: Heterogeneous Data Handler}}\label{HeterogeneousDataHandler}
\begin{figure}[h]
\centering\includegraphics[width=0.75\linewidth]{Heterogeneous_data_handler.pdf}
\caption{Heterogeneous Data Handler.}
\label{Fig:HeterogeneousDataHandler}
\end{figure}
\textbf{Summary:} Heterogeneous data handler solves the non-IID and skewed data distribution issues through data volume and data class addition (e.g., data augmentation or generative adversarial network) while maintaining the local data privacy. The pattern is illustrated in Fig.~\ref{Fig:HeterogeneousDataHandler}, where the heterogeneous data handler operates at both ends of the system.
\textbf{Context:} Client devices possess heterogeneous data characteristics due to the highly personalized data generation pattern. Furthermore, the raw local data cannot be shared so the data balancing task becomes extremely challenging.
\textbf{Problem:} The imbalanced and skewed data distribution of client devices produces local models that are not generalised to the entire client network. The aggregation of these local models reduces global model accuracy.
\textbf{Forces:} The problem requires the following forces to be balanced:
\begin{itemize}[leftmargin=*]
\item \textit{Data efficiency.} It is challenging to articulate the suitable data volume and classes to be augmented to solve data heterogeneity on local client devices.
\item \textit{Data accessibility.} The heterogeneous data issue that exists within the client device can be solved by collecting all the data under a centralized location. However, this violates the data privacy of client devices.
\end{itemize}
\textbf{Solution:} A heterogeneous data handler balances the data distribution and solves the data heterogeneity issue in the client devices through data augmentation and federated distillation. Data augmentation solves data heterogeneity by generating augmented data locally until the data volume is the same across all client devices. Furthermore, the classes in the datasets are also populated equally across all client devices. Federated distillation enables the client devices to obtain knowledge from other devices periodically without directly accessing the data of other client devices. Other methods includes taking the quantified data heterogeneity weightage (e.g, Pearson's correlation, centroid averaging-distance, etc.) into account for model aggregation.
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Model quality.} By solving the non-IID issue of local datasets, the performance and generality of the global model are improved.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Computation cost.} It is computationally costly to deal with data heterogeneity together with the local model training.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Client Selector, Client Cluster}
\textbf{Known uses:}
\begin{itemize}[leftmargin=*]
\item \textit{Astreae}~\cite{8988732} is a self-balancing federated learning framework that alleviates the imbalances by performing global data distribution-based data augmentation.
\item Federated Augmentation (\textit{FAug})~\cite{jeong2018communicationefficient} is a data augmentation scheme that utilises a generative adversarial network (GAN) which is collectively trained under the trade-off between privacy leakage and communication overhead.
\item Federated Distillation (\textit{FD})~\cite{8904164} is a method that adopted knowledge distillation approaches to tackle the non-IID issue by obtaining the knowledge from other devices during the distributed training process, without accessing the raw data.
\end{itemize}
\subsubsection{\textbf{Pattern 10: Incentive Registry}}\label{IncentiveRegistry}
\begin{figure}[h]
\centering\includegraphics[width=0.6\linewidth]{Incentive_registry.pdf}
\caption{Incentive Registry.}
\label{Fig:IncentiveRegistry}
\end{figure}
\textbf{Summary:} An incentive registry maintains the list of participating clients and their rewards that correspond to clients' contributions (e.g., data volume, model performance, computation resources, etc.) to motivate clients' participation. Fig.~\ref{Fig:IncentiveRegistry} illustrates a blockchain \& smart contract-based incentive mechanism.
\textbf{Context:} The model training participation rate of client devices is low while the high participation rate is crucial for the global model performance.
\textbf{Problem:} Although the system relies greatly on the participation of clients to produce high-quality global models, clients are not mandatory to join the model training and contribute their data/resources.
\textbf{Forces:} The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Incentive scheme.} It is challenging to formulate the form of rewards to attract different clients with different participation motives. Furthermore, the incentive scheme needs to be agreed upon by both the learning coordinator and the clients, e.g., performance-based, data-contribution-based, resource-contribution-based, and provision-frequency-\\based.
\item \textit{Data privacy.} To identify the contribution of each client device, the local data and client information is required to be studied and analysed by the central server. This exposes the client devices' local data to privacy threats.
\end{itemize}
\textbf{Solution:} An incentive registry records all client's contributions and incentives based on the rate agreed. There are various ways to formulate the incentive scheme, e.g., deep reinforcement learning, blockchain/smart contracts, and the Stackelberg game model.
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Client motivatability.} The incentive mechanism is effective in attracting clients to contribute data and resources to the training process.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{System security.} There might be dishonest clients that submit fraudulent results to earn rewards illegally and harm the training process.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Client Selector}
\textbf{Known uses:}
\begin{itemize}[leftmargin=*]
\item \textit{FLChain}~\cite{8905038} is a federated learning blockchain providing an incentive scheme for collaborative training and a market place for model trading.
\item \textit{DeepChain}~\cite{8894364} is a blockchain-based collaborative training framework with an incentive mechanism that encourages parties to participate in the deep learning model training and share the obtained local gradients.
\item \textit{FedCoin}~\cite{Liu2020} is a blockchain-based peer-to-peer payment system for federated learning to enable Shapley Value (SV) based reward distribution.
\end{itemize}
\subsection{\textbf{Model Aggregation Patterns}}\label{ModelAggregationPatterns}
Model aggregation patterns are design solutions of model aggregation used for different purposes. \textit{Asynchronous aggregator} aims to reduce aggregation latency and increase system efficiency, whereas \textit{decentralised aggregator} targets to increase system reliability and accountability. \textit{Hierarchical aggregator} is adopted to improve model quality and optimises resources. \textit{Secure aggregator} is designed to protect the models' security.
\subsubsection{\textbf{Pattern 11: Asynchronous Aggregator}}\label{AsynchronousAggregator}
\begin{figure}[h]
\centering
\includegraphics[width=0.7\linewidth]{Asynchronous_aggregator.pdf}
\caption{Asynchronous Aggregator.}
\label{Fig:AsynchronousAggregator}
\end{figure}
\textbf{Summary:} To increase the global model aggregation speed every round, the central server can perform model aggregation asynchronously whenever a model update arrives without waiting for all the model updates every round. In Fig.~\ref{Fig:AsynchronousAggregator}, the asynchronous aggregator pattern is demonstrated as the first client device asynchronously uploads its local model during the second aggregation round while skipping the first aggregation round.
\textbf{Context:} In conventional federated learning, the central server receives all local model updates from the distributed client devices synchronously and performs model aggregation every round. The central server needs to wait for every model to arrive before performing the model aggregation for that round. Hence, the aggregation time depends on the last model update that reaches the central server.
\textbf{Problem:} Due to the difference in computation resources, the model training lead time is different per device. Furthermore, the difference in bandwidth availability, communication efficiency affects the model's transfer rate. Therefore, the delay in model training and transfer increases the latency in global model aggregation.
\textbf{Forces:} The problem requires the following forces to be balanced:
\begin{itemize}[leftmargin=*]
\item \textit{Model quality.} There will be possible bias in the global model if not all local model updates are aggregated in every iteration as important information or features might be left out.
\item \textit{Aggregation latency.} The aggregation of local models can only be performed when all the model updates are collected. Therefore, the latency of the aggregation process is affected by the slowest model update that arrives at the central server.
\end{itemize}
\textbf{Solution:} The global model aggregation is conducted whenever a model update is received, without being delayed by other clients. Then the server starts the next iteration and distributes the new central model to the clients that are ready for training. The delayed model updates that are not included in the current aggregation round will be added in the next round with some reduction in the weightage, proportioned to their respective delayed time.
\textbf{Consequences:}
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Low aggregation latency.} Faster aggregation time per round is achievable as there is no need to wait for the model updates from other clients for the aggregation round. The bandwidth usage per iteration is reduced as fewer local model updates are transferred and receive simultaneously every round.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Communication cost.} The number of iteration to collect all local mode updates increases for the asynchronous approach. More iterations are required for the entire training process to train the model till convergence compares to synchronous global model aggregation.
\item \textit{Model bias.} The global model of each round does not contain all the features and information of every local model update. Hence the global model might have a certain level of bias in prediction.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Client Selector, Model Co-versioning Registry, Client Update Scheduler}
\textbf{Known uses:}
\begin{itemize}[leftmargin=*]
\item Asynchronous Online Federated Learning (\textit{ASO-fed})~\cite{chen2020asynchronous} is a framework for federated learning that adopted asynchronous aggregation. The central server update the global model whenever it receives a local update from one client device (or several client devices if the local updates are received simultaneously). On the client device side, online-learning is performed as data continue to arrive during the global iterations.
\item Asynchronous federated SGD-Vertical Partitioned (\textit{AFSGD-VP})~\cite{gu2020privacypreserving} algorithm uses a tree-structured communication scheme to perform asynchronous aggregation. The algorithm does not need to align the iteration number of the model aggregation from different client devices to compute the global model.
\item Asynchronous Federated Optimization (\textit{FedAsync})~\cite{xie2020asynchronous} is an approach that leverages asynchronous updating technique and avoids server-side timeouts and abandoned rounds while requires no synchronous model broadcast to all the selected client devices.
\end{itemize}
\subsubsection{\textbf{Pattern 12: Decentralised Aggregator}}\label{DecentralisedAggregator}
\begin{figure}[h]
\centering\includegraphics[width=0.65\linewidth]{Decentralised_aggregator.pdf}
\caption{Decentralised Aggregator.}
\label{Fig:DecentralisedAggregator}
\end{figure}
\textbf{Summary:} A decentralised aggregator improves system reliability and accountability by removing the central server that is a possible single-point-of-failure. Fig.~\ref{Fig:DecentralisedAggregator} illustrates the decentralised federated learning system built using blockchain and smart contract, while the model updates are performed through the exchange between neighbour devices.
\textbf{Context:} The model training and aggregation are coordinated by a central server and both the central server and the owner may not be trusted by all the client devices that join the training process.
\textbf{Problem:} In~\textit{FedAvg}, all the chosen devices have to submit the model parameters to one central server every round. This is extremely burdensome to the central server and network congestion may occur. Furthermore, centralised federated learning possesses a single-point-of-failure. Data privacy threats may also occur if the central server is compromised by any unauthorised entity. The mutual trust between the client devices and the central server may not be specifically established.
\textbf{Forces: }The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Decentralised model management.} The federated learning systems face challenges to collect, store, examine, and aggregate the local models due to the removal of the central server.
\item \textit{System ownership.} Currently, the central server is own by the learning coordinator that creates the federated learning jobs. The removal of the central server requires the re-definition of system ownership. It includes the authority and accessibility of learning coordinator in the federated learning systems.
\end{itemize}
\textbf{Solution:} A decentralised aggregator replaces the central server's role in a federated learning system. The aggregation and update of the models can be performed through peer-to-peer exchanges between client devices. First, a random client from the system can be an aggregator by requesting the model updates from the other clients that are close to it. Simultaneously, the client devices conduct local model training in parallel and send the trained local models to the aggregator. The aggregator then produces a new global model and sends it to the client network. Blockchain is the alternative to the central server for model storage that prevents single-point-of-failure. The ownership of the blockchain belongs to the learning coordinator that creates the new training tasks and maintains the blockchain. Furthermore, the record of models on a blockchain is immutable that increases the reliability of the system. It also increases the trust of the system as the record is transparent and accessible by all the client devices.
\textbf{Consequences: }
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{System reliability.} The removal of single-point-of-failure increases the system reliability by reducing the security risk of the central server from any adversarial attack or the failure of the entire training process due to the malfunction of the central server.
\item \textit{System accountability.} The adoption of blockchain promotes accountability as the records on a blockchain is immutable and transparent to all the stakeholders.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{Latency.} Client device as a replacement of the central server for model aggregation is not ideal for direct communication with multiple devices (star-topology). This may cause latency in the model aggregation process due to blockchain consensus protocols.
\item \textit{Computation cost.} Client devices have limited computation power and resource to perform model training and aggregation parallel. Even if the training process and the aggregation is performed sequentially, the energy consumption to perform multiple rounds of aggregation is very high.
\item \textit{Storage cost.} High storage cost is required to store all the local and global models on storage-limited client devices or blockchain.
\item \textit{Data privacy.} Client devices can access the record of all the models under decentralised aggregation and blockchain settings. This might expose the privacy-sensitive information of the client devices to other parties.
\end{itemize}
\textbf{Related patterns:} \textit{Model Co-versioning Registry, Incentive Registry}
\textbf{Known uses:}
\begin{itemize}[leftmargin=*]
\item \textit{BrainTorrent}~\cite{roy2019braintorrent} is a server-less, peer-to-peer approach to perform federated learning where clients communicate directly among themselves, specifically for federated learning in medical centers.
\item \textit{FedPGA}~\cite{9205506} is a decentralised aggregation algorithm developed from~\textit{FedAvg}. The devices in~\textit{FedPGA} exchange partial gradients rather than full model weights. The partial gradient exchange pulls and merges the different slice of the updates from different devices and rebuild a mixed update for aggregation.
\item A fully decentralised framework~\cite{lalitha2018fully} is an algorithm in which users update their beliefs by aggregate information from their one-hop neighbors to learn a model that best fits the observations over the entire network.
\item A Segmented gossip approach~\cite{hu2019decentralized} splits a model into segmentation that contains the same number of non-overlapping model parameters. Then, the gossip protocol is adopted where each client stochastically selects a few other clients to exchange the model segmentation for each training iteration without the orchestration of a central server.
\end{itemize}
\subsubsection{\textbf{Pattern 13: Hierarchical Aggregator}}\label{HierarchicalAggregator}
\begin{figure}[h]
\centering\includegraphics[width=0.8\linewidth]{Hierarchical_aggregator.pdf}
\caption{Hierarchical Aggregator.}
\label{Fig:HierarchicalAggregator}
\end{figure}
\textbf{Summary:} To reduce non-IID effects on the global model and increase system efficiency, a hierarchical aggregator adds an intermediate layer (e.g., edge server) to perform partial aggregations using the local model parameters from closely-related client devices before the global aggregation. In Fig.~\ref{Fig:HierarchicalAggregator}, edge servers are added as an intermediate layer between the central server and client devices to serve the client devices that are closer to them.
\textbf{Context:} The communication between the central server and the client devices is slowed down or frequently disrupted due to being physically distant from each other and are wirelessly connected.
\textbf{Problem:} The central server can access and store more data but requires high communication overhead and suffers from latency due to being physically distant from the client devices. Moreover, client devices possess non-IID characteristics that affect global model performance.
\textbf{Forces:} The problem requires the following forces to be balanced:
\begin{itemize}[leftmargin=*]
\item \textit{System efficiency.} The system efficiency of the server-client setting to perform federated learning is low, as the central server is burdensome to accommodate the communication and the model aggregations of the widely-distributed client devices.
\item \textit{Data heterogeneity.} In the server-client setting of a federated learning system, the data heterogeneity characteristics of client devices become influential and dominant to the global model production, as the central server deals with all the client devices that generate non-IID data.
\end{itemize}
\textbf{Solution:} A hierarchical aggregator adds edge servers between the central server and client devices. The combination of server-edge-client architecture can improve both computation and communication efficiency of the federated model training process. Edge servers collect local models from the nearest client devices, a subset of all the client devices. After every k1 round of local training on each client, each edge server aggregates its clients’ models. After every k2 edge model aggregations, the cloud server aggregates all the edge servers’ models, which means the communication with the central server happens every k1k2 local updates~\cite{9148862}.
\textbf{Consequences: }
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Communication efficiency.} The hierarchical aggregators speed up the global model aggregation and improve communication efficiency.
\item \textit{Scalability.} Adding an edge layer helps to scale the system by improving the system's ability to handling more client devices.
\item \textit{Data heterogeneity and non-IID reduction.} The partial aggregation in a hierarchical manner aggregates local models that have similar data heterogeneity and non-IIDness before the global aggregation on the central server. This greatly reduces the effect of data heterogeneity and non-IIDness on global models.
\item \textit{Computation and storage efficiency.} The edge devices are rich with computation and storage resources to perform partial model aggregation. Furthermore, edge devices are nearer to the client devices which increase the model aggregation and computation efficiency.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{System reliability.} The failure of edge devices may cause the disconnection of all the client devices under those edge servers and affect the model training process, model performance, and system reliability.
\item \textit{System security.} Edge servers could become security breach points as they have lower security setups than the central server and the client devices. Hence, they are more prone to network security threats or becoming possible points-of-failure of the system.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Client Cluster, Model Co-versioning Registry}
\textbf{Known uses: }
\begin{itemize}[leftmargin=*]
\item \textit{HierFAVG}
is an algorithm that allows multiple edge servers to perform partial model aggregation incrementally from the collected updates from the client devices.
\item Hierarchical Federated Learning (\textit{HFL}) enables hierarchical model aggregation in large scale networks of client devices where communication latency is prohibitively large due to limited bandwidth. The \textit{HFL} seeks a consensus on the model and uses edge servers to aggregate model updates from client devices that are geographically near.
\item Federated Learning + Hierarchical Clustering (\textit{FL+HC}) is the addition of a hierarchical clustering algorithm to the federated learning system. The cluster is formed according to the data distributions similarity based on the following distance metrics: (1) Manhattan, (2) Euclidean, (3) Cosine distance metrics.
\item \textit{Astraea} is a federated learning framework that tackles non-IID characteristics of federated clients. The framework introduces a mediator to the central server and the client devices to balance the skewed client data distributions. The mediator performs the z-score-based data augmentation and downsampling to relieve the global imbalanced of training data.
\end{itemize}
\subsubsection{\textbf{Pattern 14: Secure Aggregator}}\label{SecureAggregator}
\begin{figure}[h]
\centering\includegraphics[width=0.68\linewidth]{Secure_Aggregator.pdf}
\caption{Secure Aggregator.}
\label{Fig:SecureAggregator}
\end{figure}
\textbf{Summary:} A security aggregator manages the model exchange and aggregation security protocols to protect model security. Fig.~\ref{Fig:SecureAggregator} illustrates the security aggregator on each components with the security protocols embedded in them.
\textbf{Context:} The central server sends global models to any existing or unknown device every round with no data privacy and security protocols that protect the communication from unauthorised access. Furthermore, model parameters contain pieces of private user information that can be inferred by the data-hungry machine learning algorithms.
\textbf{Problem:} There are no security measures to tackle the honest-but-curious and active adversary security threats which exist in federated learning systems.
\textbf{Forces:} The problem requires to balance the following forces:
\begin{itemize}[leftmargin=*]
\item \textit{Client device security.} Client device security issues exist when dishonest and malicious client devices join the training process and poison the overall model performance by disrupting the training process or providing false updates to the central server.
\item \textit{Data security.} Data security of the client devices is challenged when the gradients or model parameters are inferred by unauthorised parties through the data-hungry machine learning algorithms.
\end{itemize}
\textbf{Solution:} A security aggregator handles the secure multiparty computation (SMC) protocols for model exchanges and aggregations. The protocols provide security proof to guarantee that each party knows only its input and output. For instance, homomorphic encryption is a method to encrypt the models and only allow authorised client devices and the central server to decrypt and access the models. Pairwise masking and differential privacy (DP) methods are applied to reduce the interpretability of the model by unauthorised party~\cite{10.1145/3298981}. The technique involves adding noise to the parameters or gradient or uses a generalised method.
\textbf{Consequences: }
Benefits:
\begin{itemize}[leftmargin=*]
\item \textit{Data security.} The secure aggregator protects the model from being access by adversarial and unauthorised parties through homomorphic encryptions and prevents information leakage due to the data-hungry property of machine learning models.
\end{itemize}
Drawbacks:
\begin{itemize}[leftmargin=*]
\item \textit{System efficiency.} The extra security processes affect the system efficiency if excessive security steps are required every round for every device. It also lowers the training and aggregation speed due to encryption and decryption time.
\item \textit{Model performance-privacy trade-off.} The model performance is affected if the model privacy methods aggressively interfere with the model's interpretability due to being excessively obscure.
\item \textit{Compromised key.} For encryption and decryption functions, the possible compromise of the security keys increases the privacy threat.
\end{itemize}
\textbf{Related patterns:} \textit{Client Registry, Model Co-versioning Registry}
\textbf{Known uses: }
\begin{itemize}[leftmargin=*]
\item \textit{SecAgg}\cite{10.1145/3133956.3133982} a practical protocol by Google for secure aggregation in the federated learning settings.
\item \textit{HybridAlpha}~\cite{10.1145/3338501.3357371} is a framework that manages the client devices that join the federated learning process. The security operation includes functional encryption, DP, and SMC.
\item \textit{TensorFlow Privacy Library}\footnote{\url{https://github.com/tensorflow/privacy/}} provides an implementation of DP-SGD machine learning.
\item \textit{ZamaAI}\footnote{\url{https://zama.ai/}} is an AI service platform that provides encryption technology that uses a homomorphic compiler to convert the model into an end-to-end encrypted parameters.
\item Simple Encrypted Arithmetic Library (\textit{SEAL}\footnote{\url{https://www.microsoft.com/en-us/research/project/microsoft-seal/}}) is a homomorphic encryption API introduced by Microsoft AI to allow computations to be performed directly on encrypted data.
\end{itemize}
\section{Discussion}
\label{S:discussion}
Various patterns are proposed to improve the architectural design challenges of a federated learning system. The main challenges include communication \& computation efficiency, data privacy, model performance, system security, and reliability. First, \textit{client registry}, \textit{client selector}, and \textit{client cluster} are proposed for client device management in the job creation stage. These patterns manage client devices to improve model performance, system, and training efficiency.
During the model training stage, the performance trade-off often occurs due to the non-IID nature of the local data. The patterns proposed to address this issue are \textit{heterogeneous data handler} and \textit{incentive registry}. Furthermore, the non-IID data that enhances the local personalisation of the model also hurts the generalisation of the global model produced. The patterns proposed to address this issue are \textit{client cluster}, \textit{hierarchical aggregator}, \textit{multi-task model trainer}, and \textit{deployment selector}. Specifically, \textit{multi-task model trainer} adopts multi-task or transfer learning techniques to learn different models or personalise a global model on local data to optimise the model performance for clients with different local data characteristics, whereas the \textit{deployment selector} effectively selects the user clients to receive the personalised models that fit their local data.
For the model exchange and aggregation stages, communication and computation efficiency become a system bottleneck. To effectively tackle these issues, \textit{client selector}, \textit{client cluster}, \textit{deployment selector}, \textit{asynchronous aggregator}, and \textit{hierarchical aggregator} are embedded in the system to optimise resource consumption. However, these patterns require extra client information (i.e., resource or performance) to perform the selection or scheduling of updates. Intuitively, the collection and analysis of the client information on the central server may lead to another form of data privacy violation. Furthermore, extra computation and communication resources are consumed to collect and analyse the client information, in addition to the model training task and the fundamental tasks of the client devices. Hence, \textit{message compressor} and \textit{hierarchical aggregator} are proposed to tackle these issues. Moreover, \textit{incentive registry} is proposed to encourage more client devices to join the training to improve the model performance.
The system security issue is present due to the distributed ownership of federated learning system components. The client nodes are mostly owned by different parties which are not governed by the system owner. Therefore, unauthorised clients may join the system and obtain model parameters from the system. Furthermore, adversarial clients may harm the model or system performance by uploading dishonest updates. \textit{Secure aggregator}, \textit{model co-versioning registry}, and \textit{client registry} aim to solve these challenges. Lastly, the trustworthiness between the client devices and the central server is also a challenge to gain the participation of clients. The central server may also be a single-point-of-failure that may affect the reliability of the system. Hence, \textit{decentralised aggregator} is proposed to solve the issue.
\section{Related Work}
\label{S:related}
In many real-world scenarios, machine learning applications are usually embedded as a software component to a larger software system at enterprise level. Hence, to promote enterprise level adoption of machine learning-based applications, many researchers view machine learning models as a component of a software system so that the challenges in building machine learning models can be tackled through systematic software engineering approaches.
Wan et al. ~\cite{8812912} studied how the adoption of machine learning changes software development practices. The work characterises the differences in various aspects of software engineering and task involved for machine learning system development and traditional software development. Lwakatare et al.
~\cite{10.1007/978-3-030-19034-7_14} propose a taxonomy that depicts maturity stages of use of machine learning components in the industrial software system and mapped the challenges to the machine learning pipeline stages.
Building machine learning models is becoming an engineering discipline where practitioners take advantage of tried-and-proven methods to address recurring problems~\cite{49406}.\\Washizaki et al.~\cite{8945075} studies the machine learning design patterns and architectural patterns. The authors also proposed an architectural pattern for machine learning for improving operational stability~\cite{8712157}. The work separates machine learning systems' components into business logic and machine learning components and focuses on the machine learning pipeline management, data management, and machine learning model versioning operations.
The research on federated learning system design was first done by Bonawitz et. al~\cite{47976}, focusing on the high-level design of a basic federated learning system and its protocol definition. However, there is no study on the definition of architecture patterns or reusable solutions to address federated learning design challenges currently. Our work addresses this particular gap with respect to software architecture designs for federated learning as a distributed software system. To the best of our knowledge, this is the first comprehensive and systematic collection of federated learning architectural patterns. The outcomes are intended to provide architectural guidance for practitioners to better design and develop federated learning systems.
\section{Conclusion}
\label{S:conclusion}
Federated learning is a data privacy-preserving, distributed machine learning approach to fully utilise the data and resources on IoT and smart mobile devices. Being a distributed system with multiple components and different stakeholders, architectural challenges need to be solved before federated learning can be effectively adopted in the real-world. In this paper, we present 14 federated learning architectural patterns associated with the lifecycle of a model in federated learning. The pattern collection is provided as architectural guidance for architects to better design and develop federated learning systems. In our future work, we will explore the architectural designs that help improve the trust in federated learning.
\bibliographystyle{cas-model2-names}
|
1,116,691,499,404 | arxiv | \section{Introduction}\label{intro}
About thirty ultra-compact X-ray binaries (UCXBs) candidates have been identified to date \citep{Zand2007}. These systems are a subclass of low mass X-ray binaries (LMXBs) with very short orbital period, typically $\lesssim $ 80 min. The compact nature of UCXBs implies that the companion star's density is higher than typical main sequence stars, and so can be excluded (\citealt{Nelson1986}; \citealt{Savonije1986}). Because their sizes are comparable to the donor Roche lobe dimension, the most likely companion star candidates for UCXBs are white dwarfs or helium burning stars \citep{Nelson1986}. This inference was later confirmed through the observation of strong helium and carbon-oxygen lines in their spectra (\citealt{Schulz2001}; \citealt{Nelemans2004}; \citealt{Nelemans2006}).
The UCXB candidate 4U 0614+091 is a persistent LMXB \citep{Forman1978} that has been identified in the optical with a faint ($\sim18 \rm \, mag$), blue, variable star in the galactic plane \citep{Paradijs1994}. After the detection of Type I bursts \citep{Kuulkers2009}, the compact object of the system was identified as a neutron star. Assuming the Eddington luminosity for the bursts, a first estimate of the system distance was obtained ($< 3 \, \rm kpc$; \citealt{Brandt1992}), that was recently revised to 3.2 kpc \citep{Kuulkers2010}.
As expected for an UCXB, optical spectroscopy revealed significant carbon and oxygen lines, but no hydrogen or helium, leading to the conclusion that the system possesses a carbon-oxygen accretion disc (\citealt{Nelemans2004}; \citealt{Nelemans2006}). The observation of a broad emission line associated to O VIII Ly$\alpha$ emission in the X-ray band led to the identification of the companion star with an oxygen-rich donor star \citep{Nelemans2004}. However, the observation of type I bursts remained unexplained, since the presence of high quantities of helium is needed to account for these thermonuclear reactions. To solve this puzzle \citet{Juett2003} supposed that spallation reactions at the neutron star surface could split C and O nuclei into H and He which can then recombine again into heavier elements, explaining both the absence of H and He in the spectra and the occurrence of type I bursts.
Several attempts have been made to determine the orbital period of the system. Modulations in the optical/NIR band can be due to different phenomena, like the heating of one side of the companion by X-ray irradiation from the neutron star, but also to the presence of superhumps or hot spots from the impact point of the accretion stream with the disc. Firstly \citet{O'Brien2005} reported a $\sim$50 min orbital period from high-speed optical data obtained with ULTRACAM. A $\sim$50 min period was also suggested from \citet{Zhang2012} thanks to the observation of a quasi-periodic oscillation. \citet{Shahbaz20084U} found evidence for three different periods based on optical photometry (42, 51.3 and 64 min) with the 51.3 min being the clearest modulation.
\citet{Hakala2011} was not able to find any periodicity for the system despite their extensive observations. \citet{Madej2013} finally noted a weak periodical signal modulated at a period of $ \sim$30 min in the red-wing/blue-wing flux ratio of the most prominent emission feature at $ \sim 4650\,\AA $. They proposed that this modulation could be due to Doppler shifts of the CIII and OII lines as well as variable flux ratio of the CIII with respect to the OII lines forming the feature. They hypothesized that this periodic signal could represent the orbital period of the source.
A recent work \citep{Migliari10} presented the most complete energy spectrum of 4U 0614+091, from radio to X-rays, suggesting the presence of optically thick and thin synchrotron emission from a jet of relativistic particles. Such a phenomenon is thought to be strictly bound to accretion \citep{Fender01} and in the LMXB scenario it is mostly expected to happen in persistent systems or in transient systems during outburst. The emission of a jet makes a polarimetric study of 4U 0614+091 particularly intriguing, since synchrotron emission is known to be intrinsically linearly polarised (\citealt{Ribicky79}).
The paper is organised as follows: in Section 2 we present the results of the first optical polarimetric study (\textit{r}-band) performed on the persistent LMXB 4U 0614+091; in Section 3 we report our attempts to determine the orbital period of the system, through a photometric and spectroscopic analysis. All the errors are indicated at the $ 68 \% $ confidence level (c.l.), unless differently stated.
\section{TNG and NOT optical polarimetry}
The system 4U 0614+091 was observed on January 27, 2013 with the 3.6 m FGG TNG telescope at La Palma, equipped with the PAOLO polarimeter. A set of 28 images of 240 s integration each was taken, with the optical \textit{r} filter (6200 \AA). The night was clear, with seeing degrading with time (from $1.0''$ to $1.6''$). 4U 0614+091 was then observed a year later (March 11, 2014) in the same optical band with the ALFOSC instrument in polarimetric mode (using the Wedged Double Wollaston configuration, WeDoWo) mounted at the 2.5 m NOT telescope at La Palma, for a totality of 2 images of 900 seconds integration each.
Image reduction was carried out following standard procedures: subtraction of an averaged bias frame, division by a normalized flat frame. Flux measurements have been performed through aperture photometry with \textit{\tt daophot} \citep{Stetson1987} for all the objects in the field.
Both the polarimeter PAOLO \citep{Covino_PAOLO} and the WeDoWo device \citep{Oliva1997} consist of a double Wollaston prism (DW) mounted in the filter wheel, which produces four simultaneous polarisation states of the field of view. The images are then separated by a special wedge, producing four image slices on the CCD that correspond to four different position angles with respect to the telescope axis (0$ ^{\circ} $, 45$ ^{\circ} $, 90$ ^{\circ} $ and 135$ ^{\circ} $). Such images possess all the information needed in order to provide linear polarisation measurements. This is a fundamental requirement for PAOLO, since the instrument is mounted on the Nasmyth focus of the TNG, introducing varying instrumental polarization of the order of 2-3 $ \% $ \citep{Giro2003}.
The normalised Stokes parameters for linear polarisation, $ Q $ and $ U $, are defined as follows:
\begin{equation}
Q=\frac{f(0^{\circ})-f(90^{\circ})}{f(0^{\circ})+f(90^{\circ})} ; \,\,\,\,\, U=\frac{f(45^{\circ})-f(135^{\circ})}{f(45^{\circ})+f(135^{\circ})},
\end{equation}
where $ f $ corresponds to the measured flux of the source.
An estimate of the observed linear polarisation degree of the incoming radiation can then be achieved from:
\begin{equation}\label{Peq}
P_{\rm obs}= \sqrt{Q^{2}+U^{2}},
\end{equation}
which should be corrected for a bias factor (\citealt{Wardle1974}; \citealt{Serego}) in order to account for the non-gaussianity of the statistic describing the polarisation distribution. In particular:
\begin{equation}\label{bias_corr}
P=P_{\rm obs}\sqrt{1-\left( \frac{\sigma_{\rm P}}{P_{\rm obs}}\right)^{2} },
\end{equation}
where $ \sigma_{\rm P} $ is the r.m.s. error on the polarisation degree.
The polarisation angle $ \theta $ can be obtained from the relation:
\begin{equation}\label{pol_angle}
\theta = 0.5 \tan^{-1}(U/Q).
\end{equation}
\subsection{Average degree of linear polarisation}\label{ave_degree_sec}
We considered first of all the set of images obtained in 2013 with the PAOLO instrument. In order to reach a high S/N ratio, that is fundamental in polarimetry, we tried to estimate the linear polarisation degree in the $ r $-band starting from the average of all the images with good seeing. In particular, we defined a limit to the seeing of $ \lesssim 1.3'' $, which allowed us to use 12 images (48 min).
The Stokes parameters $ Q $ and $ U $ of all the objects in the field have been evaluated by means of the instrumental polarization model extensively described in \citet{Covino_PAOLO}. Since the field stars we chose as references cluster rather well around a common value in the $ Q-U $ plane (Fig. \ref{U_Q_media}), we could safely assume them to be intrinsecally unpolarised and that the interstellar polarisation for these bright stars in the field is probably low. This hypothesis is supported from the evaluation of the instrumental contribution only to the polarisation of the field stars, that was possible thanks to the tools described in \citet{Covino_PAOLO} and that was consistent with the results obtained for the Stokes parameters of the chosen reference stars. In particular, the modelling of the instrumental polarization by means of observations of a suitable set of polarimetric (polarized and/or unpolarized) standard stars has shown routinely a rms residual at about 0.2\% or even better if the observations cover a rather limited range in Hour Angles, as this is the case.
This fact allowed us to correct the values of $ Q $ and $ U $ of the target for the average $ Q $ and $ U $ obtained for the field stars, in order to account for the not negligible effects of instrumental polarisation. The amount of this correction is reported in Tab. \ref{tab_Q_U} as the weighted mean.
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.27]{Q_U_average.png}
\caption{$ U $ vs. $ Q $ for the averaged image of the optical $ r $ filter, for 4U0614+091 (blue triangle) and for five reference field stars (black squares), Tab. \ref{tab_Q_U}. With a red dot we indicated the weighted mean of the reference stars Stokes parameters. The parameters reported are not corrected for interstellar or instrumental effects.}
\label{U_Q_media}
\end{center}
\end{figure}
\begin{table}
\caption{Values of the Q and U Stokes parameters not corrected for interstellar or instrumental effects represented in Fig. \ref{U_Q_media} for 4U0614+091 and for five reference field stars. The weighted mean of the reference stars Stokes parameters has been reported in the last raw and corresponds to the amount of correction that we applied to $ Q $ and $ U $ of the target.}
\label{tab_Q_U}
\centering
\begin{tabular}{|c c c|}
\hline\hline
Object & Q ($ \% $) & U ($ \% $) \\
\hline
4U0614+091 & $-2.24 \pm 0.93$ & $0.32 \pm 0.66$ \\
Star 1 & $-0.32 \pm 0.17$ & $-1.47 \pm 0.13$\\
Star 2 & $0.46 \pm 0.29$ &$-1.24 \pm 0.26$ \\
Star 3 & $1.29 \pm 0.33$ & $-2.90 \pm 0.26$\\
Star 4 & $-0.97 \pm 0.65$ & $-2.50 \pm 0.49$\\
Star 5 & $0.32 \pm 0.36$ & $-2.35 \pm 0.26$\\
Weighted mean & $0.08 \pm 0.12$ & $-1.80 \pm 0.10$\\
\hline
\end{tabular}
\end{table}
Starting from these Stokes parameters (that can be supposed to be normally distributed), we used a Monte Carlo simulation to obtain the probability distribution that describes the polarisation degree $ P $ of the radiation (Rice distribution, \citealt{Wardle1974}). Passing from a Cartesian coordinate system with axis \textit{Q} and \textit{U} to a polar system, where $ P $ is the radius (eq. \ref{Peq}), we had to correct the distribution for the geometrical factor 1/$ P$, obtained evaluating the Jacobian determinant of the coordinate change. The probability density function obtained after this correction can be demonstrated to be at first approximation a Gaussian, simply supposing the errors of \textit{Q} and \textit{U} to be similar between each others in the Rice distribution.
From the fit of this new distribution with a Gaussian function, we evaluated the most probable value of $ P $ and its uncertainty to be $ P=2.85 \pm 0.96 \%$ that is significant at a $ 3\sigma $ level. This value does not need any bias correction (eq. \ref{bias_corr}), since it has been evaluated without supposing $ P $ to be normally distributed.
Furthermore we obtained an estimate of the polarisation angle $ \theta $ (eq. \ref{pol_angle}) of $ 85.9^{\circ} \pm 8.4^{\circ} $.
Following \citet{Serkowski75} we could evaluate the maximum expected interstellar contribution to the linear polarisation $ P_{\rm max} $ of 4U 0614+091. In particular we made use of the empirical formula $ P_{\rm max} \leq 3A_{\rm V}$, where $ A_{\rm V}=1.4 $ is the total $ V $-band Galactic extinction expected along the source line of sight\footnote{\url{http://ned.ipac.caltech.edu/forms/calculator.html}}. According to this relation, the maximum contribution to the linear polarisation for 4U 0614+091 due to interstellar effects should remain under $ 4\% $. For this reason it is not possible to fully rule out the possible involvement of the interstellar dust in the observed polarisation degree of the target.
Future multi-wavelength observations will allow us to obtain a polarimetric spectral energy distribution, that, if fitted with the Serkowski law \citep{Serkowski75}, will permit us to verify whether the observed polarisation degree has an interstellar origin or is intrinsic to the target.
We then performed a similar analysis on the two images taken with the WeDoWo device in 2014, summing them together in order to achieve a higher significance. The Stokes parameters have been extracted thanks to another custom set of command-line tools developed for the WeDoWo instrument, and were corrected through the observation of the standard non-polarised star G191B2B\footnote{\url{http://www.not.iac.es/instruments/turpol/std/zpstd.html}}. The polarisation degree $ P $ was obtained through the same analysis described above. Unfortunately a combination between the faintness of the source and the bad seeing of the night ($ >1.5'' $) did not permit us to obtain a significant polarisation detection for the source. A $ 3\sigma $ upper limit to $ P $ of the 3.4$ \% $ was obtained, that is consistent with the result achieved for the 2013 dataset.
\subsection{Time-dependent linear polarisation}\label{pol_light curve_sec}
With the aim of detecting some kind of variability in the polarisation degree of the source, we analised the 28 images obtained with the PAOLO polarimeter, extracting the normalised Stokes parameters $ Q $ and $ U $ as above. The polarisation degree trend as function of time for the target and for one of the reference stars is shown in Fig. \ref{upper}. For $ \sim $ the $ 50 \% $ of the images the target did not possess a polarisation degree different from 0 at a $ \geqslant \, 1 \sigma $ level; in that cases we decided to report in Fig. \ref{upper} the $ 3\sigma $ upper limits on $ P $.
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{pol_campo.png}
\caption{\textit{Top} panel: $ r $-band polarisation curve of 4U0614+091. With black squares we reported the polarisation detection ($ P/\sigma_{\rm P}\geqslant 1 $) with their error bars, whereas with red arrows we represented all the upper limits that we evaluated when a polarisation measure was not possible, due for example to lower S/N ratios. The superimposed horizontal line represents the value $ P_{\rm mean} $ obtained by fitting the polarisation detections and upper limits with a constant. The gray band represents the 1$ \sigma $ level. Note that a better determination of the polarisation level is obtained from the probability density function (as discussed in the text), resulting in a similar central value but a much smaller error ($0.96\%$). \textit{Bottom} panel: $ r $-band polarisation curve of one field star chosen as reference (black squares: polarisation detections; red arrows: upper limits).}
\label{upper}
\end{center}
\end{figure}
The polarisation degree of 4U 0614+091 does not show any particular trend with time, remaining almost constant around a common value $ P_{\rm mean} $. In order to take into account both the polarisation detections and upper limits in Fig. \ref{upper} (\textit{top} panel), and maintaining the same cut to seeing $ \lesssim 1.3 $ used in Sec. \ref{ave_degree_sec}, we obtained $ P_{\rm mean} $ by summing together the distributions of polarisation of the selected images and by fitting the obtained distribution with a Gaussian function. With this method, $ P_{\rm mean}= 3.1 \% \pm 1.8 \%$, that is consistent with the value of $ P $ obtained in Sec. \ref{ave_degree_sec}.
\section{The orbital period of 4U 0614+091}
\subsection{$ r $-band light curve of 4U0614+091}
As stated in Sec. \ref{intro}, no decisive measure of the orbital period $ P_{\rm orb} $ of 4U 0614+091 has been obtained to date. A possibility for measuring $ P_{\rm orb} $ is to observe a periodic modulation in the optical flux emitted from the source. In fact in the case of LMXBs, where the dominant optical/IR emission from a system in quiescence is the companion star, they are often subject to ellipsoidal modulations. This effect derives from the fact that the companion star suffers for a tidal distortion due to the large gravitational field of the compact object, and for this reason the projected surface area of the distorted star is different at quadratures than at conjunctions, resulting in maxima and minima of the observed flux, respectively. When the systems possess shorter orbital periods and smaller orbital separations (as in case of UCXBs), supposing the companion star emission to be the most relevant component, the effects of irradiation should dominate the observed fluxes, causing the light curves to have a single maximum and a single minimum around the phases of superior and inferior conjunction, respectively. Observing these patterns would allow us to measure precisely the orbital period of the source. However, such methods are usually effective with quiescent LMXBs, where the companion star is expected to dominate the optical emission.
The photo-polarimetric images of 4U 0614+091 taken with PAOLO allowed us to compute the system $ r $-band light curve. Infact, not considering a negligible loss in flux, the sum of the intensities measured in the four slices for each image results in the total flux coming from our target of observation.
We therefore summed the fluxes corresponding to the four different position angles for the system 4U 0614+091 and for 5 isolated stars in the field of view. We performed differential photometry with respect to this selection of reference stars, in order to minimize any systematic effect.
The calibration of the magnitudes was performed using the R2 magnitudes of the USNO B1.0 catalogue\footnote{\url{http://www.nofs.navy.mil/data/fchpix/}}, whose magnitudes were transformed into $ r $-band magnitudes using the transformation equations of \citet{Jordi2006}. However, a systematic error of a few tenths of a magnitude should be taken into account due to the inaccuracy of the USNO magnitudes.
\begin{figure}
\begin{center}
\includegraphics[scale=0.42]{lightcurves.png}
\caption{\textit{Top} panel: $ r $-band light curve obtained for the system 4U0614+091. Superimposed, the fit of the dataset with a straight line. \textit{Bottom} panel: residual light curve after the subtraction of the straight line obtained with the fit of the light curve in the \textit{top} panel from the dataset itself. Superimposed, the fit of the curve with a sinusoidal function + a constant. With a dashed line we indicated in both panels the fraction of the day that corresponds to the possible polarisation flare (Fig. \ref{upper}.}
\label{light curve}
\end{center}
\end{figure}
The light curve (Fig. \ref{light curve}, \textit{top} panel) shows a general decreasing trend with time, probably due to accretion variations in the disc, with superimposed a significant short term variability. The fit with a straight line $y=mx+q$ produces a $ reduced $ $ \chi ^{2}$ of $ \sim 1.04 $, with $ q= 18.19 \pm 0.01$ and $ m=0.61\pm 0.13 $. We could then subtract from the light curve the straight line obtained from the fit, ending up with the residuals in Fig. \ref{light curve} (\textit{bottom} panel) that suggest a sinusoidal modulation (with significance probability of $ \sim 98\% $ given by an f-test) with a periodicity of $ 44.9 \pm 2.3 $ min and a semi-amplitude of $ (2.15\pm 0.73) \times10^{-2} $ mag (reduced $ \chi ^{2}$ of $ \sim 0.67 $).
\subsection{Spectroscopy with FORS1}\label{spectroscopy}
A set of 16 low resolution ($ \sim 160\,\, \rm km\, s^{-1} $) spectra of 300 s integration each was taken of 4U 0614+091 on 3 September 2007 with the ESO VLT (Very Large Telescope), equipped with the FORS1 spectrograph. All the spectra were taken with the 300V grism with a 1-arcsec slit, using 2 $ \times $ 2 on-chip binning and covering the wavelength range 4300-8000 \AA. The night was clear, with seeing variable around a mean value of $ \sim 1.2 '' $. These observations cover $ \sim 1.5 $ times the possible $ \sim 50 $ minutes orbital period of the system. The extraction of the spectrum was performed with the ESO-MIDAS\footnote{\url{http://www.eso.org/projects/esomidas/}} software
package. Wavelength and flux calibration of the spectra were achieved using helium-argon lamp and observing spectrophotometric stars.
In Fig. \ref{spectrum} we report the average spectrum of the source. The spectrum shows both absorption and emission lines, the latter possibly due to the presence of partially ionised carbon and oxygen, that possess a lot of lines in the investigated wavelength range.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.42]{4u0614_spectrum.jpeg}
\caption{Average spectrum of 4U 0614+091 with indicated the most prominent identified emission lines (Tab. \ref{tab_spectrum}).}
\label{spectrum}
\end{figure}
We could not find any trace of hydrogen and helium lines in our average spectrum, leading to the conclusion that the companion star of the system must be a hydrogen-poor star. The identification of the emission lines was made by the comparison with the accurate results of \citet{Nelemans2004}, that in turn used the local thermodynamic equilibrium (LTE) model proposed by \citet{Marsh1991}. In particular, our results are in agreement with the companion star being a C/O-rich white dwarf as stated in \citet{Nelemans2004}. The most prominent identified features in our spectrum are reported in Tab. \ref{tab_spectrum}.
\begin{table}
\caption{Strongest features identified in the spectrum of 4U 0614+091 (Fig. \ref{spectrum}). For details about the transitions, see Tab. 3 in the work of \citet{Nelemans2004}.}
\label{tab_spectrum}
\centering
\begin{tabular}{|c c|}
\hline\hline
Feature (\AA) & Ion \\
\hline
4650 & CIII\\
& OII \\
4700 & OII \\
4935 & OII \\
5700 & CIII\\
5810 & CIII\\
& CIV?\\
6580 & CII \\
&(OII)\\
6790 & CII \\
7240 & CII \\
\hline
\end{tabular}
\end{table}
In order to determine the orbital period of the source, we studied the variations of the equivalent width (EW) of the emission lines CII/OII 6580 \AA $\,$ and CII 7240 \AA, indicated in Fig. \ref{spectrum}. We performed sinusoidal fitting of the EW variations, averaging the spectra two by two with the aim of enhancing the signal to noise ratio. The results of the fits are reported in Tab. \ref{tab_fit_sin}: the most significant periodicity is the obtained with the 7240 \AA $\,$line (Fig. \ref{EWfit}), for which $ P=40.9 \pm 6.8 $ min, that is consistent with the period of the oscillation obtained from the $ r $-band light curve (Fig. \ref{light curve}, \textit{bottom} panel). We also tried to measure the radial velocity of the CII/OII 6580 $\AA$ and CII 7240 $\AA$ emission lines. We found no evidence for periodical Doppler motions, likely due to the poor resolution of our spectra.
\begin{table}
\caption{Results of the sinusoidal fit of the EW variations for the two selected lines. In the last column we reported the confidence level of the fit, obtained with an F-test with respect to a fit with a constant. All the uncertainties are reported at the 90$ \% $ confidence level. }
\label{tab_fit_sin}
\centering
\begin{tabular}{|c c c c|}
\hline\hline
$\lambda$ (\AA) & Period (min) & Reduced $ \chi ^{2} $ & CL \\
\hline
6580 & $37.9 \pm 2.6$ & 5.98 & 45.7 $\%$\\
7240 & $40.9 \pm 6.8$ & 0.29 & 1.5 $ \% $\\
\hline
\end{tabular}
\end{table}
\begin{figure}[!h]
\centering
\includegraphics[scale=0.3]{EW_cri_1.png}
\caption{EW variation curve of CII 7240 $\AA$ line, obtained by averaging the spectra two by two, with superimposed a sinusoidal fit with period 41 min.}
\label{EWfit}
\end{figure}
\section{Discussion}
\subsection{Polarimetric signatures of a jet}
Relativistic particle jets have been observed to be emitted from different kinds of systems, like Active Galactic Nuclei (AGN), Super Soft Sources and X-ray binaries (XRBs), all subject to the phenomenon of accretion (disc-jet coupling, \citealt{Fender01}).
The existence of jets in XRBs in particular does not seem to depend on the type of compact object hosted in the system, that can be both a neutron star and a black hole (recently evidence for a transient jet in a binary
system containing a white dwarf has been found, \citealt{Kording08}).
However, XRBs hosting black holes are among the most studied jet emitters sources, also because they are brighter then, e.g, neutron stars systems in the wavelength range where the jet contributes the most to the emission, i.e. the radio band. Generally, XRBs that emit jets are characterised by an optically thick, flat synchrotron radio spectrum, interpreted as a signature of the presence of the compact jet (for some galactic black hole XRBs this has been confirmed by radio imaging of the jet itself; \citealt{Stirling01}; \citealt{Dhawan2000}). Because the size scale of the emitting region in the jet is expected to scale inversely with frequency (\citealt{Nowak2005}; \citealt{Blandford79}), the jet break frequency marks the start of the particle acceleration in the jet \citep{Polko2010}.
A break from an optically thick to optically thin synchrotron spectrum is then expected in the IR/optical range \citep{Falcke2004}.
The break is detected in 4U0614+09 \citep{Migliari10} and also seen in the black hole candidates GX 339--4 (\citealt{Corbel02}; \citealt{Gandhi11}; \citealt{Corbel2013}), XTE J1118+480 \citep{Hynes2006}, XTE J1550-564 \citep{Chaty2011}, V404 Cyg \citep{Russell2013a} and MAXI J1836-194 \citep{Russell2013b}.
Synchrotron emission is expected to be intrinsically linearly polarised at a high level, up to tens of per cent, especially in case of ordered magnetic fields. A polarimetric signature of synchrotron-emitting jets can be observed both in the NIR and in the optical in LMXBs; however, due to tangled and turbulent magnetic fields at the base of the jets, polarisation degrees in these sources have never been found to exceed a few per cent, as stated in \citealt{Russell11}. The only evidence of ordered magnetic fields in LMXBs jets has been observed for the persistent system Cyg X-1 \citep{Russell14}. Only a few LMXBs have been observed with polarimetric techniques to date, both persistent and transient ones (\citealt{Charles80}; \citealt{Dolan89}; \citealt{Gliozzi98}; \citealt{Hannikainen00}; \citealt{Schultz04}; \citealt{Brocksopp07}; \citealt{Shahbaz08}; \citealt{Russell08}; \citealt{Russell11}; \citealt{Baglio_cen2014}).
The aim of our work was to obtain additional evidence for the emission of a compact jet from the UCXB 4U 0614+091. Since the presence of a jet was stated from \citet{Migliari10}, we expected to observe a degree of linear polarisation of at least a few $ \% $, according to previous observations of polarised XRBs, due to the jet synchrotron radiation for a tangled magnetic field. With our analysis, we measured an average polarisation degree of $ 2.85 \% \pm 0.96 \% $, as expected.
We should also consider that in systems where an accretion disc is present, not only synchrotron radiation from a relativistic particles jet could be the cause of emission of polarised light. In particular, hydrogen in the accretion disc is expected to be almost totally ionised, due to the high temperatures caused by viscosity. In this state, Thomson scattering of the emitted radiation with these free electrons is expected, causing a polarisation degree of at most a few per cent in the optical (\citealt{Brown78}; \citealt{Dolan84}; \citealt{Cheng88}). However, no hydrogen is present in the accretion disc of 4U 0614+091, since the companion star of the system has been found to be an oxygen-rich mass donor star, and hydrogen and helium lines have never been observed.
We deduce that the significant non-zero degree of polarisation of 4U0614+091 can be due to synchrotron radiation from a relativistic particle jet emitted from its central regions, which is inferred from the SED of the system \citep{Migliari10}.
Following \citet{Migliari10}, we built the infrared SED of 4U 0614+091 adding to the infrared and radio points reported in that paper and the archival data in the ALLWISE catalog (Tab. \ref{tab_sed}; Fig. \ref{sed}). These data are not contemporary to that of \citet{Migliari10}, but they fit well with the previous dataset, meaning that the average spectrum of the target does not vary significantly with time. From the fit with a linear function we obtained a spectral index $ \alpha $ of $ \sim $0.03 in the optically thick part of the spectrum, whereas in the optically thin part $ \alpha \sim -0.43$, consistent with \cite{Migliari10}. The break frequency $ \nu_{\rm break} $ between optically thin and optically thick synchrotron emission remains in the same range of \citealt{Migliari10} ($ 1.25\times10^{13} \rm Hz < \nu_{\rm break}< 3.71\times10^{13} \rm Hz$).
\begin{table}
\caption{RADIO and IR fluxes (uncorrected for the negligible Galactic reddening)} obtained from the ALLWISE catalogue and from \citet{Migliari10} and $ r $-band de-reddened flux ($ A_{\rm V}=1.4 $) from the TNG data used to build the SED in Fig. \ref{sed}.
\label{tab_sed}
\centering
\begin{tabular}{|c c |}
\hline\hline
Band & Flux (mJy) \\
\hline
4.86 GHz \citep{Migliari10} & $0.281 \pm 0.014$ \\
8.46 GHz \citep{Migliari10} & $0.276 \pm 0.010$ \\
\hline
$1.250 \times 10^{13}$ Hz (24 $\mu m$, \citealt{Migliari10}) & $0.35 \pm 0.06$ \\
\hline
$3.75 \times 10^{13}$ Hz (8 $\mu m$, \citealt{Migliari10}) & $0.26 \pm 0.02$ \\
$5.17 \times 10^{13}$ Hz (5.8 $\mu m$, \citealt{Migliari10}) & $0.20 \pm 0.02$ \\
$6.67 \times 10^{13}$ Hz (4.5 $\mu m$, \citealt{Migliari10}) & $0.20 \pm 0.02$ \\
$8.33 \times 10^{13}$ Hz (3.6 $\mu m$, \citealt{Migliari10}) & $0.17 \pm 0.01$ \\
\hline
$8.8 \times 10^{13}$ Hz (ALLWISE W1) & $0.18 \pm 0.01$ \\
$6.5 \times 10^{13}$ Hz (ALLWISE W2) & $0.20 \pm 0.02$ \\
$2.5 \times 10^{13}$ Hz (ALLWISE W3) & $<$ 0.36\\
\hline
$4.82 \times 10^{14}$ Hz ($ r $-band, this work)& $0.49 \pm 0.01$\\
\hline
\end{tabular}
\end{table}
\begin{figure}[!h]
\centering
\includegraphics[scale=0.38]{fit_sed_allwise.png}
\caption{Spectral energy distribution of the system 4U 0614+091 built starting from the radio (green triangles) and infrared points (blue squares) of \citet{Migliari10}, from the ALL WISE catalog magnitudes (red circles) and from the $ r $-flux obtained in this work (black x). With an orange arrow we indicated the W3-band WISE upper limit obtained from the ALLWise catalog. Superimposed, the two linear fits of the SED. All the points are normalized to the fluxes of \citet{Migliari10}.}
\label{sed}
\end{figure}
Under the hypothesis of jet emission, starting from the linear fit to the infrared part of the SED in Fig. \ref{sed} we could estimate the expected $ r $-band flux due to the jet emission ($F_{\rm jet}\sim 0.1$ mJy). Since we know that the total de-reddened $ r $-band flux of the source at the time of the TNG polarimetric observation was $ F_{\rm r}\sim $ 0.5 mJy, we could obtain from the ratio between $F_{\rm jet}$ and $ F_{\rm r}$ an estimate of the intrinsic linear polarisation degree of the jet only of $\sim 15 \% $ .
The polarisation light curve (Fig. \ref{upper}, \textit{top} panel) shows an almost constant trend with time, as stated in Sec. \ref{pol_light curve_sec}. Nevertheless it has to be noted that for higher times a hint of a small polarisation flare seems to be present; if we suppose the emission of the jet to be constant with time, the decrease of the flux that is observed in Fig. \ref{light curve} contemporary to the possible polarisation flare (indicated with a dashed line) makes the percentile flux of the jet increase, and this could explain the possible higher polarisation degree observed.
\subsection{Attempts in determining 4U0614+091 orbital period}\label{attempts}
\citet{Shahbaz20084U} found evidence of three different orbital periods for 4U0614+091, with the clearest modulation at 51.3 minutes likely due to a superhump rather than the orbital period of the system. With the aim of eliminating any ambiguity, we tried to obtain an estimate of the system orbital period by means of photometric and spectroscopic observations. We obtained the $ r $-band light curve of the system starting from the polarimetric measurements, and we could observe a slight decrease of the flux with time (Fig. \ref{light curve}, \textit{top} panel), with a possible superimposed sinusoidal modulation with a periodicity of $ \sim 45 $ min (Fig. \ref{light curve}, \textit{bottom} panel). This modulation could arise from the X-ray irradiation of the inner face of the companion star from the compact object (in this case, giving direct information about the orbital period of the binary) or from the presence of hot spots or superhumps in the accretion disc, that can create a modulation in the light curve due to the rotation of the system.
We then tried to identify any periodicity in the system through the analysis of the EW variation curves of the CII 7240 $\AA$ line. From a sinusoidal fit (Fig. \ref{EWfit}) we obtained a periodical modulation at a $40.9 \pm 6.8$ minutes period, that corresponds to the orbital period of the line-emitting outer regions of the disc, and is consistent with the one measured from the optical light curve (Fig. \ref{light curve}, \textit{bottom} panel). This suggests that both the modulations could be caused by the same phenomenon; specifically, we propose that the EW variation of the CII 7240 $\AA$ emission line could be due to the modulation of the continuum emission, that we observe indeed in our light curve, and that should be linked to the accretion disc, as our spectroscopic analysis pointed out. Following this hypothesis, we could rule out the X-ray irradiation as the origin of the observed modulation in the light curve, that can be thus explained e.g. referring to the presence of hot spots or superhumps in the accretion disc.
Unfortunately the calibration precision of our spectra is too low in order to detect a modulation in the continuum emission as small as the one that we observed in the light curve (i.e. $ \sim 2.15\times10^{-2} $ mag), whose amplitude of oscillation does not exceed $ 10\% $.
Since the periodicity that we measured from the EW variation is linked to the outer regions of the disc, in order to obtain an estimate of the orbital period of the source we approximated the outer radius of the disc with its tidal radius $ R_{\rm tid} $ \citep{King96}:
\begin{equation}
R_{\rm tid}=0.9R_{\rm L},
\end{equation}
where $ R_{\rm L} $ is the Roche lobe radius of the compact object of the system, and is given by:
\begin{equation}\label{roche_lobe_eq}
\frac{R_{\rm L}}{a}=\frac{0.49\, q^{2/3}}{0.6\, q^{2/3} + \ln \left(1+q^{1/3}\right)},
\end{equation}
where $ q=M_{\rm NS}/M_{*} $ is the ratio between the neutron star and the companion star mass ($ M_{\rm NS} $ and $M_{*}$, respectively) and $ a $ is the binary separation, that is an upper limit to the radius of the companion star orbit.
From the Kepler law, we know that:
\begin{equation}\label{orb_period}
\frac{P(R_{\rm tid})}{P_{\rm orb}}\propto \left( \frac{R_{\rm out}}{a}\right)^{3/2} ,
\end{equation}
where $ P(r_{\rm tid}) $ and $ P_{\rm orb} $ are, respectively, the orbital period of the outer region of the disc and of the companion star of the system.
Considering a mass ratio of at most $\sim 0.1 $, that is typical of a UCXB system, we obtain $ P_{\rm orb} \gtrsim 2P(r_{\rm tid})$, depending on the value of $ q $ that we consider, and meaning that the orbital period of our system should exceed 1 hour.
\section{Conclusions}
In this work, we presented the results of a $ r $-band polarimetric and a spectroscopic analysis of the persistent UCXB 4U0614+091, based on observations obtained in 2007, 2013 and 2014 with the VLT FORS1 spectrograph, the TNG PAOLO and the NOT WeDoWo polarimeters, respectively.
In \citet{Migliari10} the authors stated the presence of an infrared excess in the spectral energy distribution of the system, that they interpreted as the signature of synchrotron emission from a relativistic particles jet. We were for this reason interested in searching for an optical linear polarisation degree of the order of some per cent, since synchrotron emission is expected to be intrinsically linearly polarised at this level. We obtained a $ r $-band polarisation degree of $ 2.85 \% \pm 0.96 \% $ from the TNG dataset, and a 3$ \sigma $ upper limit of 3.4$ \% $ from the NOT data. A polarisation degree of a few per cent in the optical is exactly what is expected in case of a jet emission; for this reason we can confirm the presence of this further component in the system emitted radiation. Under this hypothesis, we could moreover estimate an intrinsic linear polarisation degree of the jet only in the $ r $-band of $ \sim 15\% $.
We then tried to determine the system orbital period. We built the light curve of 4U 0614+091 starting from the polarimetric images obtained with the PAOLO polarimeter, observing a decreasing trend of the flux with time, possibly due to accretion variations in the disc, with superimposed a sinusoidal modulation at a $ \sim 45 $ min period. From the spectroscopic FORS1 images we could then extract the EW variation curve of the CII 7240 $ \AA $ line, and from its sinusoidal fit we could obtain a periodicity of $ 40.9 \pm 6.8 $ min, that refers to the outer regions of the disc and is consistent with the period of the modulation obtained with the optical light curve, suggesting for a common origin of the two periodicities (probably a hot spot or a superhump in the accretion disc). From the Kepler law, using the tidal radius of the accretion disc as an approximation for its outer radius, we could estimate for the system 4U0614+091 an orbital period $ \gtrsim $ 1 hour.
\begin{acknowledgements}
MCB acknowledges S. Crespi for helpful discussions and T. Pursimo and I. Andreoni for their support during her observing run in La Palma. DM acknowledges Prof. M. Colpi (Universit\`{a} di Milano-Bicocca) for supportive discussions and the INAF-Osservatorio Astronomico di Brera for kind hospitality during her bachelor thesis. TS was supported by the Spanish Ministry of Economy and Competitiveness (MINECO) under the grant AYA2010-18080.
\end{acknowledgements}
\addcontentsline{toc}{chapter}{Bibliografia}
|
1,116,691,499,405 | arxiv | \section{Introduction}\label{sec:intro}
Let $t$ be a rooted tree and $n_i(t)$ the number of nodes in $t$ having $i$ children. The sequence $(n_i(t),i\geq 0)$ is called the degree sequence of $t$, and satisfies $ \sum_{i\ge 0} n_i(t)=1+\sum_{i\ge 0} in_i(t)=|t|$, the number of nodes in $t$.
The aim of this paper is to study trees chosen under $`P_{{\bf s}}$, the uniform distribution on the set of { plane} trees with specified degree sequence ${\bf s}=(n_i,i\geq 0)$, and then size $|{\bf s}|:=\sum_{i\ge 0} n_i$. More precisely, a sequence of degree sequences $({\bf s}(\kappa),\kappa \geq 0)$ with ${\bf s}(\kappa)=(n_i(\kappa),i\geq 0)$, corresponding to trees with size ${\sf n}_\kappa:=|{\bf s}(\kappa)|\to +\infty$ is given, and the investigations concern the limiting behaviour of tree under $`P_{{\bf s}(\kappa)}$.
\begin{figure}[htb]
\centerline{\includegraphics[height=1.6cm]{exa}}
\caption{The 10 trees of $\mathbb T_{\bf s}$ for the degree sequence ${\bf s}=(3,1,2,0,0,\dots)$.}
\end{figure}
We now introduce some notation valid { in the entire paper}. We denote by ${\bf p}(\kappa)=(p_i(\kappa),i\geq 0)$ the degree distribution under $`P_{{\bf s}(\kappa)}$:
\begin{equation}\label{eq:pik}
p_i(\kappa)=\frac{n_i(\kappa)}{{\sf n}_\kappa}.
\end{equation}
{ Let also
\begin{equation}\label{eq:sigma}
\sigma_\kappa^2:=\sum_{i\ge 1} \frac{n_i(\kappa)}{{\sf n}_\kappa-1} i^2 - 1;
\end{equation}
$\sigma_\kappa^2$ is ``almost'' the associated variance, this choice of definition yields shorter formulae in the following.} The maximum degree of any tree with degree sequence ${\bf s}(\kappa)$ is
\[\Delta_\kappa=\max\{i:n_i(\kappa)>0\}.\]
{ Throughout} the paper ${\bf p}=(p_i,i\geq 0)$ is a distribution with mean 1, and variance $\sigma^2_{\bf p}\in(0,+\infty)=\sum_{i\ge 0} i^2p_i -1\in (0,\infty).$ In the following theorem, which is the main result of the present paper, ${\bf p}(\kappa)\Rightarrow {\bf p}$ means equivalence in distribution, which here means that for any $i\geq 0$, $p_i(\kappa)\to p_i$, as $\kappa\to \infty$.
\begin{theo}\label{thm:main_gh}
Let $({\bf s}(\kappa),\kappa\geq 0)$ be a sequence of degree sequences such that ${\sf n}_\kappa\to +\infty$, $\Delta_\kappa=o({\sf n}_\kappa^{1/2})$, ${\bf p}(\kappa)\Rightarrow {\bf p}$ with $\sigma_\kappa^2\to \sigma^2_{\bf p}$, that is convergence of second moment. Let ${\bf t}$ be a plane tree chosen under $`P_{{\bf s}(\kappa)}$ and let $d_{{\bf t}}$ be the graph distance in ${\bf t}$. Under $`P_{{\bf s}(\kappa)}$, when $\kappa\to +\infty$, $({\bf t},\sigma_\kappa {\sf n}_\kappa^{-1/2} d_{{\bf t}})$ converges in distribution to Aldous' continuum random tree (encoded by twice a Brownian excursion), in the Gromov--Hausdorff sense.
\end{theo}
First observe that the very strong result of \citet{HaMi2010} about the asymptotics of Markov branching trees that has been used to give asymptotics for random trees in a wide variety of settings does not apply in the present case of trees with a prescribed degree sequence. Indeed, the subtrees of a given node are not independent given their sizes when one fixes the degree sequence. Our approach uses instead the observation done by \citet{MA-MO} that all natural encodings of the trees are asymptotically proportional in the case of Galton-Watson trees conditioned by the size. The same property will also hold here. In particular, the \emph{height process} or the \emph{contour process} both encoding the metric structure of the tree resemble the \emph{depth-first queue process} encoding the sequence of degrees observed when performing a depth-first traversal. This fact was used by \citet{MA-MO} to give an alternative proof of Aldous' result in the case of Galton--Watson trees conditioned on the total progeny under some moment condition (\citet{BK} also observed this phenomenon).
\medskip
One of the crucial questions underlying our work is that of the universality of the convergence of random trees to the continuum random tree (CRT).
We are motivated by the metric structure of graphs with a prescribed degree sequence. Introduced by \citet{BeCa1978a} and by \citet{Bo1980a} in the form of the configuration model, these graphs have received a lot of attention since the first tight analysis of the size of connected components by \citet{MoRe1995,MoRe1998}. This is mainly because the model allows for a lot of flexibility in the degree sequence. In particular, the model provides a construction of random graphs with degree sequences that may match the observations in large real-world networks.
Of course, random graphs with a prescribed degree sequence are much more complex than trees with a prescribed degree sequence, but there is no doubt that the analysis of trees is a first step towards the identification of the metric structure of the corresponding graphs. Indeed, recent results of \citet{Joseph2010a} show that under some moment condition, the sizes of the connected components of random graphs with a prescribed \emph{critical} degree sequence are similar to those of Erd\H{o}s--R\'enyi $G(n,p)$ random graphs \cite{ErRe1960,Bollobas2001,JaLuRu2000}: they may be asymptotically described in terms of the lengths of the excursions of a Brownian motion with parabolic drift above its current minimum, as demonstrated by \citet{Aldous1997}. (See also \cite{Riordan2011a}, where it is supposed that the maximum degree is bounded.) On the other hand, the metric structure of $G(n,p)$ inside the critical window has recently been identified in terms of modifications of Brownian CRT by \citet*{AdBrGo2010,AdBrGo2010a}. In other words, the present analysis is one more building block towards an invariance principle for scaling limits of random graphs, i.e., that critical random graphs with a prescribed degree sequence have (under a suitable moment condition on the degree distribution) the same scaling limit (as sequence of compact metric spaces) as classical random graphs \cite{AdBrGo2010a}. This is at least what is suggested by the results of \citet*{Hofstad2009,BhHoLe2009}, \citet{Joseph2010a} and \citet{Riordan2011a}. \par
Moreover, in the same way that uniform random trees or forests may be seen as the results of coagulation/fragmentation processes involving particles \cite{Pitman1999b,Pitman2006}, trees with a prescribed degree sequence appear naturally in similar aggregation processes. The model where particles have constrained valence may appear more ``physically'' grounded. The relevant underlying coalescing procedure is the additive coalescent \cite{AlPi1998a, Bertoin2000a}, a Markov process whose dynamics are such that particles merge at a rate proportional to the sum of their masses/sizes. The additive coalescent is the aggregation process appearing in Knuth's modification of R\'enyi's parking problem \cite{Renyi1958,Hemmer1989} or the hashing with linear probing \cite{ChLo2002,BeMi2006}. The reader may find more information about coagulation/fragmentation processes in the monograph by \citet{Bertoin2006} or the recent survey by \citet{Berestycki2009}. \medskip
The model $`P_{{\bf s}}$ is related to Galton--Watson trees \cite{AtNe1972, Harris1963}, also called simply generated trees in the combinatorial literature, by a simple conditioning: the distribution $`P_{{\bf s}}$ coincides with the distribution of the family tree ${\bf t}$ of a Galton--Watson process with offspring distribution $(\nu_i,i\geq 0)$ (which must satisfies $\nu_i>0$ if $n_i>0$) conditioned on $\{n_i({\bf t})=n_i,i\geq 0\}$. { Indeed, $`P_{{\bf s}}$ assigns the same probability to all trees with the same degree sequence.}
In this sense, the distribution $\nu$ plays a role of secondary importance, and $`P_{{\bf s}}$ appears to be a model of combinatorial nature, { far from the world} of Galton--Watson processes.
Nevertheless, we will see that Theorem \ref{thm:main_gh} implies the following result of Aldous (stated in a slightly different form in \cite{Aldous1991}) (see also \cite{Aldous1991,Aldous1991b,Aldous1993a, MA-MO, Legall1993}), { where $H_{\bf t}$ is the height process of ${\bf t}$ (the definition is recalled in the next section)}.
\begin{pro}[Aldous \cite{Aldous1991}]\label{pro:Ald} Let $\mu=(\mu_i,i\geq 0)$ be a distribution with mean $m_\mu=1$ and variance $\sigma^2_\mu\in(0,+\infty)$, and let $`P_{\mu}$ be the distribution of a Galton--Watson tree with offspring distribution $\mu$. Along the subsequence $\{n~: `P_\mu(|{\bf t}|=n)>0\}$, under $`P_\mu(~\cdot~|\,|{\bf t}|=n)$
\[\left(\frac{H_{\bf t}(nx)}{\sqrt{n}}\right)_{x\in[0,1]}\xrightarrow[n\to\infty]{(law)} \frac{2}{\sigma_\mu} {\sf e}\]
where ${\sf e}$ denotes a standard Brownian excursion, the convergence holding in the space ${\mathcal C}[0,1]$ equipped with the topology of uniform convergence.
\end{pro}
We will see that this theorem may be seen indeed as a consequence of Theorem \ref{thm:main_gh}; the argument morally relies on the fact that under $`P_\mu(~.~|\,|{\bf t}|=n)$, the empirical degree sequence satisfies the { hypotheses} of Theorem \ref{thm:main_gh} with probability going to 1 (this is stated in Lemma \ref{lem:GW}). The proof of this theorem is postponed { until} Section \ref{sec:pt}.
{ Note also results of \citet{Rizzolo2011a} and \citet{Kort} that have a flavor similar to our Theorem~\ref{thm:main_gh} (although neither implies the other): they proved that Galton--Watson trees conditioned on the number of nodes having their degrees in a subset $A$ of the support of the measure $\mu$ has a limiting behaviour depending on $A$. For instance, they consider trees conditioned on the number of leaves, the number of nodes with other out-degrees being left free. The proofs in \citet{Rizzolo2011a} rely ultimately on the approach based on Markov branching trees developed by \citet{HaMi2010}.
}
\medskip
\noindent\textsc{Plan of the paper.}\ In Section~\ref{sec:tree} we introduce precisely the model of trees we consider. Section~\ref{sec:backbone} is devoted to a useful backbone decomposition for these trees. We then prove our main result, the convergence of rescaled trees to the continuum random trees, in Section~\ref{sec:proof}. Finally, the application to coagulation processes with particles with constrained valence is developed in Section~\ref{sec:coagulation}.
\section{Trees with prescribed degree sequence}
\label{sec:tree}
We here define formally the combinatorial object discussed in this paper.
For convenience we write $\mathbb{N}=\{1,2,\dots\}$ for the set of positive natural numbers. First recall some definitions related to standard rooted plane trees. Let $\mathcal U=\bigcup_{n\geq 0}\mathbb{N}^n$
be the set of finite words on the alphabet $\mathbb{N}$, where $\mathbb N^0=\{\varnothing\}$, and $\varnothing$ denotes the empty word. Denote by $uv$ the concatenation of $u$ and $v$; by convention ${\varnothing} u=u{\varnothing} =u$.
A subset $T$ of $\mathcal U$ is a \it plane tree \rm (see Figure \ref{fig:pt}) if
{ \begin{itemize}
\item it contains $\varnothing$ (called the root),
\item it is stable by prefix (if $uv\in T$ for $u$ and $v$ in $\mathcal U$, then $u\in T$), and
\item if ($uk\in T$ for some $k>1$ and $u\in U$) then $uj\in T$ for $j$ in $\{1,\dots,k\}$.
\end{itemize} }
This last condition appears necessary to get a unique tree with a given genealogical structure.
The set of plane trees will be denoted by $\mathbb T$.
\begin{figure}[ht]
\psfrag{v}{$\varnothing$}
\psfrag{1}{1}
\psfrag{11}{11}
\psfrag{12}{12}
\psfrag{13}{13}
\psfrag{14}{14}
\psfrag{15}{15}
\psfrag{131}{131}
\psfrag{151}{151}
\psfrag{152}{152}
\centerline{\includegraphics[height=3cm]{plan_tree}}
\caption{\label{fig:pt} Usual representation of the plane tree $\{\varnothing,1,11,12,13,14,15,131,151,152\}$}
\end{figure}
Notice that the lexicographical order $<$ on $\mathcal U$, also named the depth-first order, induces a total order on any tree $t$; this is of prime importance for the encodings of $t$ we will present.
For $t\in \mathbb T$, and $u\in t$, let $c_t(u)=\max\{i~: ui \in t\}$ be the number of children of $u$ in $t$. The depth of $u$ in $t$, its number of letters as a word in $\mathcal U$, is denoted $|u|$. The notation $|t|$ refers to the cardinality of $t$, its number of nodes including the root $\varnothing$.
With a tree $t\in \mathbb T$, one can associate its degree sequence ${\bf s}(t)=(n_i(t), i\ge 0)$, where $n_i(t)=\#\{u\in t: c_t(u)=i\}$ is the number of nodes with degree $i$ in $t$. For a fixed degree sequence ${\bf s}$, write $\mathbb T_{\bf s}$ for the set of trees $t\in \mathbb T$ such that ${\bf s}(t)={\bf s}$, and let $`P_{\bf s}$ be the uniform distribution on $\mathbb T_{\bf s}$. To investigate the shape of random trees under $`P_{\bf s}$, we will use the usual encodings: \emph{height process} $H$ and \emph{depth-first walk} $S$ (or \L ukasiewicz path) { and \emph{contour process} ${\bf{C}}$}. These encodings are defined by first fixing their values at the integral points, and then linear interpolation in between (See Figure~\ref{fig:thw}). For a tree $t\in \mathbb T$, let $\tilde u_1=\varnothing< \tilde u_2<\dots<\tilde u_{|t|}$ denote the nodes of $t$ sorted according to the lexicographic order. Then we define $H=H_t$ by $H(i)=|\tilde u_{i+1}|$, $S=S_t$ by $S_t(i)=\sum_{j=1}^i (c_t(\tilde u_j)-1)$; the process $H_t$ is defined on $[0,|t|-1]$ and $S_t$ on $[0,|t|]$.
{ For the contour process ${\bf{C}}_t$ of $t$, we need to define first a function $f_t:\{0,\dots,2(|t|-1)\}\mapsto t$ which can be regarded as a walk around $t$; first set $f_t(0)= \varnothing$, the root. For $i<2(|t|-1)$, given $f_t(i)=v$, $f_t(i+1) $ is $u$, the smallest child of $v$ (for the lexicographical order) absent from the list $\{f_t(0),\dots,f_t(i)\}$ , and the father of $v$ if no such $u$ exists. The contour process has the following values on integer positions
\[{\bf{C}}_t(i)=|f_t(i)|,~~i\in\{0,\dots,2(|t|-1)\}.\]}
\begin{figure}[ht]
\centerline{\includegraphics[height=3cm]{luka-height}}
\caption{\label{fig:thw} A plane tree $t\in \mathbb T$, its height process $H_t$, \L ukasiewicz walk $S_t$ and its contour process ${\bf{C}}_t$.}
\end{figure}
\begin{theo}\label{thm:main_encodings}Under the hypothesis of Theorem \ref{thm:main_gh}, under $`P_{{\bf s}(\kappa)}$,
\begin{equation}\label{eq:all-conv}
\left(\frac{H_{\bf t}( x ({\sf n}_\kappa-1))}{{\sf n}_\kappa^{1/2}},{ \frac{C_{\bf t}( x 2({\sf n}_\kappa-1))}{{\sf n}_\kappa^{1/2}}}, \frac{S_{\bf t}( x {\sf n}_\kappa )}{{\sf n}_\kappa^{1/2}}\right)_{x\in [0,1]}\xrightarrow[\kappa\to\infty]{} \left(\frac 2 {\sigma_{\bf p}}\,{\sf e},\frac 2 {\sigma_{\bf p}}\,{\sf e},\sigma_{\bf p}{\sf e}\right)
\end{equation}
in distribution in the space $\mathcal C([0,1],`R^3)$ of continuous functions from $[0,1]$ with values in $`R^3$, equipped with the supremum distance.
\end{theo}{ The contour process is a kind of interpolation of the height process. The fact that both these processes have the same asymptotic behaviour is well understood in some general settings : it is shown in Marckert and Mokkadem (Lemma 3.19 \cite{MAMO2}) that, if under any model of random trees, the height process has a continuous limit after a non trivial normalisation, then the contour process has the same limit with the same space normalisation (and time normalisation multiplied by 2 to take into account the relative durations of these processes). This property has been noticed before in the case of Galton--Watson trees conditioned by the size \cite{BK,MA-MO}.\par
As a consequence (of Lemma 3.19 \cite{MAMO2}), to establish
\begin{equation}\label{eq:all-conv2}
\left(\frac{H_{\bf t}( x ({\sf n}_\kappa-1))}{{\sf n}_\kappa^{1/2}}, \frac{S_{\bf t}( x {\sf n}_\kappa )}{{\sf n}_\kappa^{1/2}}\right)_{x\in [0,1]}\xrightarrow[\kappa\to\infty]{} \left(\frac 2 {\sigma_{\bf p}}\,{\sf e},\sigma_{\bf p}{\sf e}\right)
\end{equation}
is sufficient to deduce \eref{eq:all-conv}. }
Note now that the condition $\sigma^2_{\bf p}>0$ is necessary in Theorem \ref{thm:main_encodings}: it ensures that $p_0=\lim_{\kappa\to\infty} n_0(\kappa)/{\sf n}_\kappa>0$ and that large trees are not close to a linear tree, where most of the nodes have degree one.\par
A tree $t\in \mathbb T$ can also be seen as a metric space when equipped with the graph distance $d_t$. A consequence of Theorem~\ref{thm:main_encodings} is that, under $`P_{{\bf s}(\kappa)}$, the metric space $$\left({\bf t},\frac{\sigma_\kappa}{\sqrt{{\sf n}_\kappa}}d_{\bf t}\right)$$ converges to the continuum random tree encoded by $2{\sf e}$ in the sense of Gromov--Hausdorff distance between equivalence classes of compact metric spaces. The fact that the convergence { of the contour process (or the height process)} implies the convergence of the trees for the Gromov--Hausdorff topology is well known, see for example Lemma 2.3 in Le Gall \cite{LGRRT}. So, in particular, to prove Theorem~\ref{thm:main_gh} it suffices to prove Theorem~\ref{thm:main_encodings} and for this, it is sufficient to prove \eref{eq:all-conv2}.
\medskip
\noindent\textbf{Remark.}\ One can define other models of random trees with a prescribed degree sequence: for example, \emph{rooted labelled trees}. Let $`Q_{{\bf s}(k)}$ be the uniform distribution on those with degree sequence ${\bf s}(k)$. Since labelled trees have a canonical ordering (using an order on the labels to order the children of each node), forgetting the labels, they can be seen as plane trees with the same degree sequence, inducing a distribution $`P'_{{\bf s}(k)}$ on the set of plane trees. By a simple counting argument, it turns out that $`P'_{{\bf s}(k)}=`P_{{\bf s}(k)}$. This situation is drastically different from the general case, since the projection of uniform labelled trees on plane tree (that is without fixing the degree sequence) does not induce the uniform distribution on plane trees. As a consequence, Theorem~\ref{thm:main_gh} is also valid for the model of labelled trees with a prescribed degree sequence.
\section{Combinatorial considerations: a backbone decomposition}\label{sec:backbone}
In this section we develop a decomposition of trees under $`P_{{\bf s}(k)}$ along a branch. It is essentially the usual \emph{backbone decomposition} for Galton--Watson trees due to \citet*[see, e.g.,][]{LyPePe95a} transposed under $`P_{{\bf s}(k)}$. The decomposition amounts to describing the structure of the branch from the root to a distinguished node $u$, together with the (ordered) forest formed by the trees rooted at the neighbours of that branch. \medskip
\noindent\textsc{Forest with a given degree sequence.}\
A forest ${\sf f}=(t_1,\dots,t_k)$ is a finite sequence of trees; its degree sequence ${\bf s}({\sf f})=\sum_{i=1}^k {\bf s}(t_i)$ is the (component-wise) sum of the degree sequences of the trees which compose it.
If ${\bf s}=(n_i,i\geq 0)$ is the degree sequence of a forest ${\sf f}$, then the number of roots of ${\sf f}$ is given by $r=|{\bf s}|-\sum_{i\ge 0} i n_i$. Let ${\mathbb F}_{\bf s}$ be the set of forests of ($r$ ordered) plane trees having degree sequence ${\bf s}$.
We have (see, e.g., \cite{Pitman1999b}, p. 128)
\begin{equation}\label{eq:nbf}
\#{\mathbb F}_{\bf s}=\frac{r}{|{\bf s}|}\binom{|{\bf s}|}{(n_i,i\geq 0)}=\frac r {|{\bf s}|} \cdot \frac{|{\bf s}| !}{\prod_{i\ge 0} n_i!}.
\end{equation}
\medskip
\noindent\textsc{The content of a branch.}\ Let $t$ be a plane tree, and let $u=i_1\dots i_{|u|}$ be one of its nodes, where $i_j\in \mathbb{N}$ for any $j$.
For $j\leq |u|$, write $u_j=i_1\dots i_j$, the ancestor of $u$ having depth $j$ (with the convention $u_0=\varnothing$, the root of $t$). The set $\cro{\varnothing,u}=\{u_j~: j <|u|\}$ is called the branch of $u$ (notice that $u$ is excluded).
For any $i\geq 0$, the number of ancestors of $u$ having $i$ children is written
\[M_i(u,t)=\#\{v~:v\textrm{ strict ancestors of }u,c_t(v)=i\}.\]
We refer to ${\bf M}(u,t)=(M_i(u,t), i\ge 0)$ as the composition of the branch. Note that we necessarily have $M_0(u,t)=0$.
Clearly if $u\in t$, then
\begin{equation} |u|=\sum_{i\geq 1}M_i(u,t)=|{\bf M}(u,t)|.\end{equation}
Further let ${\sf LR}(u,t)$ (for left or right) be { the set of nodes that are children of some node in $\cro{\varnothing, u}$ without being themselves in $\cro{\varnothing, u}$; note that because of our convention for $\cro{\varnothing,u}$, $u$ belongs to ${\sf LR}(u,t)$ (see Figure~\ref{fig:RandLR})}.
\begin{figure}[h]\psfrag{u}{$u$}
\centerline{\includegraphics[height=4.5cm]{dec-u}}
\caption{\label{fig:RandLR} { A tree $t$ with a marked node $u$; the sets in the two right-hand side pictures show the sets ${\sf R}(u,t)$ and ${\sf LR}(u,t)$.}}
\end{figure}
Let also ${\sf R}(u,t)$ be the subset of ${\sf LR}(u,t)$, of nodes lying to the right of the path $\cro{\varnothing,u}$ { (therefore $u\notin {\sf R}(u,t)$)}.
A node $v$ is in ${\sf R}(u,t)$ if it is a child of some $u_i$, for $i\in \{0,\dots, |u|-1\}$, and satisfies $v>u_{i+1}$ in the lexicographic order on $\mathcal U$. Therefore{
\begin{align*}
|{\sf LR}(u,t)|&=\sum_{j=0}^{|u|-1} \left(c_t(u_j)-1\right)+1 =\sum_{i\ge 0} M_i(u,t)(i-1)+1\\
|{\sf R}(u,t)|&= \sum_{j=0}^{|u|-1} (c_t(u_j)-i_{j+1}).
\end{align*}}
Let $\tilde u_1=\varnothing<\tilde u_2<\dots< \tilde u_{|t|}$ be the nodes of $t$, in increasing lexicographic order. Then {
\begin{equation}\label{eq:H_and_S}
H_t(k)=|\tilde u_{k+1}| \textrm{~~and~~} S_t(k)=|{\sf R}(\tilde u_k,t)|+c_t(\tilde u_k)-1,
\end{equation}} so that the discrepancy between $H_t$ and $S_t$ can be accessed using the number of nodes to the right of the paths to $\tilde u_i$, $i= 1, \dots, |t|$. This observation lies at the heart of our approach.
The set of plane trees with degree sequence ${\bf s}$ and a distinguished node (marked plane trees) is denoted by $\mathbb T^{\bullet}_{\bf s}=\left\{(t,u)~: t\in \mathbb T_{\bf s}, u\in t\right\}$, and
the uniform distribution on this set is denoted $`P^{\bullet}_{\bf s}$. Under $`P^\bullet_{\bf s}$, a marked tree $(t,u)$ is distributed as $(t',u')$ where $t'$ is a tree sampled under $`P_{\bf s}$ and $u'$ is a uniformly random node in $t'$.
We now decompose a marked tree $(t,u)$ along the branch $\cro{\varnothing,u}$.
First, consider the structure of this branch, that we call the contents:
\[{\sf Cont}(t,u):=\big((c_t(u_0),i_1),\dots,(c_t(u_{|u|-1}),i_{|u|})\big).\]
We write $J^{\bf m}$ for the set of potential vectors ${\sf Cont}(t,u)$ when the composition of the branch $\cro{\varnothing, u}$ is ${\bf M}(u,t)={\bf m}$.
Besides, notice that
\begin{equation}\label{eq:cjm}
|J^{{\bf m}}|=\binom{|{\bf m}|}{(m_i,i\geq 1)}\prod_{i\geq 1} i^{m_i}.
\end{equation}
Since, if ${\sf Cont}(u,t)\in J^{{\bf m}}$ { then $|{\sf LR}(u,t)|=1+\sum_{i\ge 0}(i-1)m_i$}, we will use the following notation:
\[|{\sf LR}({\bf m})|:=1+\sum_{i\ge 0}(i-1)m_i.\]
\medskip
\noindent\textsc{The forest off a distinguished path.}\
For a tree $t$ and any node $v\in t$, let $t_v=\{w~:vw \in t\}$ be the subtree of $t$ rooted at $v$. The sequence of trees ${\sf F}(t,u) = (t_v, v \in {\sf LR}(u,t))$
is the forest constituted by the subtrees of $t$ rooted at the vertices belonging to ${\sf LR}(u,t)$, and sorted according to the rank of their root for the lexicographic order.
The decomposition which associates $({\sf Cont}(t,u),{\sf F}(t,u))$ to a marked tree $(t,u)$ is clearly one-to-one. The following proposition characterises the distributions of ${\bf M}({\bf u},{\bf t})$, ${\sf Cont}({\bf u},{\bf t})$, and $|{\sf R}({\bf u},{\bf t})|$ when $({\bf t},{\bf u})$ is sampled under $`P^\bullet_{\bf s}$. { In the following, for two sequences of integers ${\bf s}=(n_0,n_1,\dots)$ and ${\bf m}=(m_0,m_1,\dots)$ we write ${\bf s}-{\bf m}=(n_0-m_0,n_1-m_1,\dots)$.}
\begin{pro}\label{pro:fond-comb} Let ${\bf s}=(n_0,n_1,\dots)$ be a degree sequence and let ${{\bf m}}=(m_0,m_1,\dots)$ be such that $m_0=0$, and $m_i\leq n_i$ for any $i\geq 1$. Let $({\bf t},{\bf u})$ be chosen according to $`P^{\bullet}_{\bf s}$.\\
(a) We have
\[`P^{\bullet}_{\bf s}\left({\bf M}({\bf u},{\bf t})={\bf m}\right)=\frac{|{\sf LR}({\bf m})|\,|{\bf m}|! \,|{\bf s}-{\bf m}|!}{|{\bf s}|!\,|{\bf s}-{\bf m}|}\cdot \prod_{i\geq 1}\binom{n_i}{m_i}{i^{m_i}}.\]
(b) Moreover, for any vector $C\in J^{\bf m}$,
\[`P^{\bullet}_{\bf s}\left({\sf Cont}({\bf u},{\bf t})=C~|~{\bf M}({\bf u},{\bf t})={\bf m}\right)= 1/{\#J^{\bf m}}.\]
(c) For any $x\geq 0$, and ${\bf m}$ such that $`P_{\bf s}^\bullet({\bf M}({\bf u},{\bf t})={\bf m})>0$,
\begin{equation}\label{eq:rep}
`P^{\bullet}_{\bf s}\left(\left||{\sf R}({\bf u},{\bf t})|-\frac{\sigma_{\bf s}^2}{2}|{\bf u}|\right|\geq x~\Bigg|~{\bf M}({\bf u},{\bf t})={\bf m}\right)=`P\left(\left|\sum_{j\geq 1}\sum_{k=1}^{m_j} U^{(k)}_j-\frac{\sigma_{\bf s}^2}{2}|{\bf m}|\right|\geq x\right)
\end{equation}
where the $U^{(k)}_j$ are independent random variables, $U^{(k)}_j$ is uniform in $\{0,\dots,j-1\}$ and where $\sigma_{\bf s}^2$ is the variance associated with $(p_i=n_i/|{\bf s}|$, $i\ge 0$) (as done on \eref{eq:sigma}).
\end{pro}
\begin{proof}Since the backbone decomposition is a bijection, we have for any
vector $C\in J^{\bf m}$, we have
\begin{eqnarray*}
`P^{\bullet}_{\bf s}\left({\sf Cont}({\bf u},{\bf t})=C\right)&=&\frac{\#{\mathbb F}_{{\bf s}-{\bf m}}}{|{\bf s}|\cdot \#{\mathbb F}{\bf s}}\\
&=& \frac{|{\sf LR}({\bf m})|}{|{\bf s}-{\bf m}|}\binom{|{\bf s}-{\bf m}|}{(n_i-m_i,i\geq 0)} \bigg/\binom{|{\bf s}|}{(n_i,i\geq 0)},
\end{eqnarray*}
by the expression for the number of forests in \eref{eq:nbf}. As $`P^\bullet_{\bf s}\left({\sf Cont}({\bf u},{\bf t})=C\right)$ is independent of $C\in J^{\bf m}$, it suffices to multiply by $\#J^{{\bf m}}$ in order to get $`P^{\bullet}_{\bf s}\left({\bf M}({\bf u},{\bf t})={\bf m}\right)$. After simplification, this yields the first statement in (a), and then (b). Now, (b) implies that
for any $R\geq 0$, and any composition ${\bf m}$ for which $`P^{\bullet}_{\bf s}({\bf M}({\bf u},{\bf t})={\bf m})>0$, we have
\[`P^{\bullet}_{\bf s}\left(|{\sf R}({\bf u},{\bf t})|=R~|~{\bf M}({\bf u},{\bf t})={\bf m}\right)=`P\left(\sum_{j\geq 1}\sum_{k=1}^{m_j} U^{(k)}_j=R\right),\]
where the $U^{(k)}_j$ are independent random variables, and $U^{(k)}_j$ is uniform in $\{0,\dots,j-1\}$. This implies assertion (c) and completes the proof.
\end{proof}
\section{Convergence of uniform trees to the CRT: Proof of Theorem~\ref{thm:main_encodings}}
\label{sec:proof}
\subsection{The general approach}\label{sec:conv_approach}
Our approach uses the phenomenon observed in Marckert \& Mokkadem \cite{MA-MO} in the case of critical Galton--Watson tree (having a variance): under some mild assumptions the \L ukasiewicz path $S_{\bf t}$ and the height process $H_{\bf t}$ are asymptotically proportional, that is, up to a scalar normalisation, the difference between these processes converge to the zero function. It turns out that a similar phenomenon occurs when the degree sequence is prescribed, and this is the basis of our proof.
In order to prove Theorem~\ref{thm:main_encodings} we proceed in two steps: the first one consists in showing that the depth-first walk $S_{\bf t}$ associated to a tree sampled under $`P_{{\bf s}(\kappa)}$ converges to a Brownian excursion. The process $S_{\bf t}$ is much easier to deal with than $H_{\bf t}$, { since $S_{\bf t}$ is \emph{essentially} a random walk conditioned to stay non-negative, and forced to end up at the origin (precisely at $-1$).} We provide the details in Section~\ref{sec:luka} below. The core of the work lies in the second step, which consists in proving that { rescaled versions of } $S_{\bf t}$ and $H_{\bf t}$ are indeed close, uniformly on $[0,1]$. More precisely, by Theorem 3.1 p.~27 of \cite{BIL}, the following proposition is sufficient to show that ${\sf n}_\kappa^{-1/2} 2 S_{\bf t}({\sf n}_\kappa\cdot)$ and ${\sf n}_\kappa^{-1/2} \sigma_{\kappa}^2 H_{\bf t}(({\sf n}_\kappa-1)\cdot )$ have the same limit in $({\mathcal C}[0,1], \|_\infty)$.
\begin{pro}\label{pro:compar_unif}Under the hypothesis of Theorem \ref{thm:main_gh}, there exists $c_\kappa=o({\sf n}_\kappa^{1/2})$ such that, as $\kappa\to\infty$,
\[`P_{{\bf s}(\kappa)}\left( \sup_{x\in[0,1]}\left|S_{\bf t}(x{\sf n}_\kappa)- \frac{\sigma_{\kappa}^2}2 H_{\bf t}(x({\sf n}_\kappa-1))\right|\geq c_\kappa \right)\xrightarrow[\kappa\to\infty]{} 0.\]
\end{pro}
In order to prove Proposition~\ref{pro:compar_unif}, { recall the representations of $S_t$ and $H_t$ in terms of $|{\sf R}(u,t)|$ and $|u|$ given in \eref{eq:H_and_S}. }
A non-uniform version of the claim in Proposition~\ref{pro:compar_unif} is the following:
\begin{pro}\label{pro:compar}Assume the hypothesis of Theorem \ref{thm:main_gh}. Let $({\bf t},{\bf u})$ chosen under $`P_{{\bf s}(\kappa)}^\bullet$. There exists $c_\kappa=o({\sf n}_\kappa^{1/2})$ such that,
\[`P_{{\bf s}(\kappa)}^\bullet\left( \left||{\sf R}({\bf u},{\bf t})|- \frac{\sigma_{\kappa}^2}2|{\bf u}|\right|\geq c_\kappa \right)\xrightarrow[\kappa\to\infty]{} 0.\]
\end{pro}
{ Again, by \eref{eq:H_and_S}, one sees that
{ $$\left|\left||{\sf R}({\bf u},{\bf t})|- \frac{\sigma_{\kappa}^2}2|{\bf u}|\right|-\left|S_{\bf t}({\bf u})- \frac{\sigma_{\kappa}^2}2 H_{\bf t}({\bf u}-1)\right|\right|\le \Delta_\kappa,$$
and $\Delta_\kappa=o(\sqrt {\sf n}_\kappa)$, by assumption.} Therefore, Proposition~\ref{pro:compar} implies then that the proportion of indexes $m\in\cro{0,{\sf n}_\kappa}$ for which $$\left|S_{\bf t}(m+1)- \frac{\sigma_{\kappa}^2}2 H_{\bf t}(m)\right|\geq c_\kappa{ -}\Delta_k$$ goes to 0 { (we will choose $c_\kappa$ such that $\Delta_\kappa=o(c_\kappa)$)}. In this case, if the sequence of processes $(D_\kappa:={\sf n}_\kappa^{-1/2}( S_{\bf t}(x{\sf n}_\kappa)- \frac{\sigma_{\kappa}^2}2 H_{\bf t}(x({\sf n}_\kappa-1))),\kappa\geq 1)$ is tight, we can deduce the convergence of the finite distributions of $(D_\kappa,\kappa\geq 1)$ to those of the null process on $[0,1]$. Hence, to show Proposition~\ref{pro:compar_unif}, it suffices to show Proposition~\ref{pro:compar} together with the tightness of $(D_\kappa,\kappa\geq 1)${ ; the tightness is actually also} needed to show the convergence of $D_\kappa$ in distribution in $C[0,1]$ (see \cite[see, e.g.][]{BIL}). Since under the sequence of distributions $`P_{{\bf s}(\kappa)}$,
the family of rescaled versions of $S_{\bf t}$ (see Section~\ref{sec:luka}) is tight, it suffices to prove that the family of rescaled versions of $H_{\bf t}$ is tight as well.\par
We need also to say a word about the fact that both processes $S_t$ and $H_t$ have a small difference in their time rescaling. Again, this is not a problem since the process $S_t$ has its increments bounded by $\Delta_\kappa=o(\sqrt({\sf n}_\kappa))$.}
\medskip
\noindent\textbf{Remark.}\ Under slightly stronger assumptions on the degree sequences, it is possible to control the discrepancy between the height process and the \L ukasiewicz path at \emph{every point} in $\{0,1,\dots, {\sf n}_\kappa-1\}$. More precisely it would be possible to show that
\begin{equation}\label{eq:control_everywhere}
`P_{{\bf s}(\kappa)}^{\bullet}\left( \left||{\sf R}({\bf u},{\bf t})|- \frac{\sigma_{\kappa}^2}2|{\bf u}|\right|\geq c_{\kappa} \right) =o(1/{\sf n}_\kappa).
\end{equation}
Using the union bound, this yields the convergence of the rescaled height process to a Brownian excursion, as a random function in ${\mathcal C}[0,1]$. One is easily convinced that with the optimal assumptions for Theorem~\ref{thm:main_encodings}, the bound in \eqref{eq:control_everywhere} might just not hold.
\medskip
We now move on to the ingredients of the proof: we first give the details of the convergence of ${\sf n}_\kappa^{-1/2}S_{\bf t}(~\cdot~{\sf n}_\kappa)$ to a Brownian excursion in Section~\ref{sec:luka}, then we prove tightness for ${\sf n}_\kappa^{-1/2}H_{\bf t}~(\cdot~({\sf n}_\kappa-1))$ in Section~\ref{sec:tightness}. The longer proof of Proposition~\ref{pro:compar} is delayed until Section~\ref{sec:fdd}.
\subsection{Convergence of the \L ukasiewicz walk}\label{sec:luka}
In this section, we give the details of the proof of the convergence of the depth-first walk under $`P_{{\bf s}(\kappa)}$ towards the Brownian excursion.
\begin{lem}\label{lem:exchan}Assume the hypothesis of Theorem \ref{thm:main_gh}. Under $`P_{{\bf s}(\kappa)}$,
$$\left(\frac{S_{\bf t}(x {\sf n}_\kappa)}{\sigma_\kappa {\sf n}_\kappa^{1/2}}\right)_{x\in[0,1]} \xrightarrow[\kappa\to+\infty]{(law)} {\sf e}$$
as random functions in $\mathcal C[0,1]$.
\end{lem}
\begin{proof}{
Let ${\mathbf c}=\{c_1,c_2,\dots, c_{{\sf n}_\kappa}\}$ be a multiset of ${\sf n}_\kappa$ integers whose distribution is given by ${\bf s}(\kappa)$. Let $\pi=(\pi_1,\pi_2,\dots, \pi_{{\sf n}_\kappa})$ be a uniform random permutation of $\{1,2,\dots, {\sf n}_\kappa\}$, and for $j\in\{1,\dots, {\sf n}_\kappa\}$, define
$$W_{\pi}(j)=\sum_{i=1}^j (c_{\pi_i}-1).$$
Theorem 20.7 of Aldous \cite{Aldous1983} (see also Theorem~24.1 in \cite{BIL}) ensures that, when $\Delta_\kappa =o(\sqrt{{\sf n}_\kappa})$,
$$\left(\frac{W_\pi(s {\sf n}_\kappa)}{\sigma_\kappa {\sf n}_\kappa^{1/2}}\right)_{s\in [0,1]}\xrightarrow[\kappa\to+\infty]{(law)} {\sf b},$$
in $\mathcal C[0,1]$, where ${\sf b}=({\sf b}(s), s\in [0,1])$ is a standard Brownian bridge. \par
The increments of the walk $(W_\pi(j), 0\le j\le {\sf n}_\kappa)$ satisfy $c_{\pi_i}-1\ge -1$ for every $i$ (such walks are sometimes called \emph{left-continuous}), and furthermore, $W_\pi({\sf n}_\kappa)=-1$. The cycle lemma \cite{DvMo1947a} ensures that there is a unique way to turn the process $W_\pi$ into an excursion by shifting the increments cyclically (in each rotation class there is a unique excursion) : to see this, first extend the definition of the permutation, setting $\pi_j:=\pi_{j-{\sf n}_\kappa}$ for any $j\in\{{\sf n}_\kappa+1,\dots, 2{\sf n}_\kappa\}$. For $j_\pi$ the location of the first minimum of the walk $W_\pi$ in $\{1,\dots, {\sf n}_\kappa\}$, we have that $W_\pi(j+j_\pi)-W_{\pi}(j_\pi)$ is an excursion in the following sense:
$$ \tilde S_\pi(j):=W_\pi(j+j_\pi)-W_\pi(j_\pi)\ge 0 \quad \text{for } j<{\sf n}_\kappa\qquad \text{and }\tilde S_\pi({\sf n}_\kappa)=-1.$$
Since in each rotation class there is exactly one excursion, and since the set of excursions hence obtained is exactly the set of depth-first walk of the trees in $\mathbb T_{{\bf s}(\kappa)}$, it is then easy to conclude that for ${\bf t}$ uniformly chosen in $\mathbb T_{{\bf s}(\kappa)}$,
$$(S_{\bf t}(j), 0\le j\le {\sf n}_\kappa) \ensuremath{\stackrel{d}{=}} (\tilde S_\pi(j), 0\le j\le {\sf n}_\kappa),$$
for $\pi$ a random permutation of $\{1,\dots, {\sf n}_\kappa\}$. }
Since the Brownian bridge ${\sf b}$ has almost surely a unique minimum, the claim follows by the mapping theorem \cite{BIL}.
\end{proof}
\subsection{Tightness for the height process}\label{sec:tightness}
The rescaled height process under $`P_{\bf s}(\kappa)$ is the process in $\mathcal C[0,1]$, $h_\kappa={\sf n}_\kappa^{-1/2} H(~\cdot~({\sf n}_\kappa-1))$.
In this section, we prove that the family $(h_\kappa,\kappa >0)$ is tight (we will omit the $\kappa$ when unnecessary). Since $h_\kappa(0)=0$, the following lemma is sufficient to prove tightness \cite[see, e.g.,][]{BIL}. \par
Let $\omega_h$ be the modulus of continuity of the rescaled height process $h$: for $\delta>0$
$$\omega_h(\delta)=\sup_{|t-s|\le \delta} |h(s)-h(t)|.$$
\begin{lem}\label{lem:tightness}Under the hypothesis of Theorem \ref{thm:main_gh}, for any $\epsilon>0$ and $\eta>0$, there exists $\delta>0$ such that, for all $\kappa$ large enough,
$$`P_{{\bf s}(\kappa)}(\omega_h(\delta)>\epsilon)<\eta.$$
\end{lem}
The bound we provide consists in reducing the bounds on the variations of $h$ to bounds on the variations of the \L ukasiewicz path $S$, which is known to be tight since it converges in distribution (Lemma~\ref{lem:exchan}). The underlying ideas are due to \citet{AdDeJa2010a} and \citet{Addario2011a} to prove Gaussian tail bounds for the height and width of Galton--Watson trees and random trees with a prescribed degree sequence, respectively.
For a plane tree $t\in \mathbb T$, let $t^-$ be the mirror image of $t$, or in other words, the tree obtained by flipping the order of the children of every node. Then, we let $S^-_t:=S_{t^-}$ be the \emph{reverse depth-first walk}. Observe that the mirror flip is a bijection, so that $S_t$ and $S^-_t$ have the same distribution under $`P_{{\bf s}(\kappa)}$.
\begin{proof}[Proof of Lemma~\ref{lem:tightness}]In this proof, we identify the nodes of a tree $t$ and their index in the lexicographic order; so in particular, we write $H_t(u)$ for the height of a node $u$ in $t$, and we write $|u-v|\le \delta {\sf n}_\kappa$ to mean that $u$ and $v$ are within $\delta {\sf n}_\kappa$ in the lexicographic order (that is, $u=\tilde u_i$ and $v=\tilde u_j$ for some $i$ and $j$ satisfying $|i-j|\le \delta {\sf n}_\kappa$).
Consider a tree $t$ and two nodes $u$ and $v$. Write $u\wedge v$ for the (deepest) first common ancestor of $u$ and $v$ in $t$. In the following we write $u\preceq v$ to mean that $u$ is an ancestor of $v$ in $t$ ($u=v$ is allowed). Then,
\begin{align}
|H_t(u)-H_t(v)|
&\le |H_t(u)-H_t(u\wedge v)| + |H_t(v)-H_t(u\wedge v)|,
\end{align}
so that it suffices to bound variations of $H_t$ between two nodes on the same path to the root:
\begin{align*}
\sup_{|u-v|\le \delta {\sf n}_\kappa} |H_t(u)-H_t(v)| \le { 2 + }2\sup_{w\preceq u,|u-w|\le \delta {\sf n}_\kappa} |H_t(u)-H_t(w)|.
\end{align*}
{ (The extra two in the previous bound is needed because of the following reason: the closest common ancestor $u\wedge v$ might not be within distance $\delta {\sf n}_\kappa$ of either $u$ and $v$; however, there is certainly a node $w$ lying within distance one of $u\wedge v$ that is visited between $u$ and $v$.)}
Now, observe that, for $w\preceq u$, every node $v$ on the path between $w$ and $u$ which has degree more than one contributes at least one to the number of nodes off the path between $w$ and $u$:
$$1+\sum_{w\preceq v\preceq u} (c_t(v)-1)\ge H_t(u)-H_t(w)-\sum_{w\preceq v\preceq u} {\bf 1}_{\{c_t(v)=1\}}$$
However, one may also bound this same number of nodes in terms of the depth-first walk $S_t$, and the reverse depth-first walk $S^-_t$:
\begin{equation}\label{eq:bo1}
1+\sum_{w\preceq v\preceq u} (c_t(v)-1)\le S_t(u)-S_t(w)+S_t^-(u)-S_t^-(w)+{ 2}c_t(w).
\end{equation}
In other words, we have
\begin{align}\label{eq:bound_modulus}
\sup_{|u-v|\le \delta {\sf n}_\kappa} |H_t(v)-H_t(u)|
&\le { 2 + } 2\sup_{|u-w|\le \delta {\sf n}_\kappa, w\preceq u} |S_t(u)-S_t(w)| + 2\sup_{|u-w|\le \delta {\sf n}_\kappa, w\preceq u} |S_t^-(u)-S_t^-(w)|\nonumber\\
&\quad+{ 2}\max_{w} c_t(w) + \sup_{|u-w|\le \delta {\sf n}_\kappa} \sum_{w\preceq v\preceq u} {\bf 1}_{\{c_t(v)=1\}}\nonumber\\
&\le { 2 + } 2\sup_{|u-w|\le \delta n} |S_t(u)-S_t(w)| + 2\sup_{|u-w|\le \delta {\sf n}_\kappa} |S_t^-(u)-S_t^-(w)|\nonumber\\
&\quad + { 2}\Delta_\kappa + \sup_{|u-w|\le \delta {\sf n}_\kappa} \sum_{w\preceq v\preceq u} {\bf 1}_{\{c_t(v)=1\}}\nonumber\\
&\le { 2 + }2 {\sf n}_\kappa^{1/2} \omega_s(\delta) +2 {\sf n}_\kappa^{1/2} \omega_{s-}(\delta)+ { 2}\Delta_\kappa + \sup_{|u-w|\le \delta {\sf n}_\kappa} \sum_{w\preceq v\preceq u} {\bf 1}_{\{c_t(v)=1\}},
\end{align}
where $\omega_s$ and $\omega_{s^-}$ denote the moduli of continuity of the rescaled \L ukasiewicz path $ {\sf n}_\kappa^{-1/2} S_t$ and $ {\sf n}_\kappa^{-1/2} S^-_t$, respectively.
The first four terms in \eqref{eq:bound_modulus} are easy to bound since $\Delta_\kappa=o(\sqrt {\sf n}_\kappa)$ and, { after renormalisation}, $S_t$ and $S^-_t$ are tight under $`P_{{\bf s}(\kappa)}$. The only term remaining to control is the one concerning the number of nodes of degree one:
$$Y_t(\delta):=\sup_{|u-w|\le \delta {\sf n}_\kappa} \sum_{w\preceq v\preceq u} {\bf 1}_{\{c_t(v)=1\}}.$$
To bound $Y_t(\delta)$ we relate the distribution of trees under $`P_{{\bf s}(\kappa)}$ to those under $`P_{{\bf s}(\kappa)^\star}$, where ${\bf s}(\kappa)^\star=(n_0^\star, n_1^\star, \dots)$ is obtained from ${\bf s}(\kappa)$ by removing all nodes of degree one, i.e., $n_1^\star=0$ and $n_i^\star=n_i$ for every $i\neq 1$. Then, in a tree $t^\star$ sampled under $`P_{{{\bf s}(\kappa)}^\star}$, one has $Y_{t^\star}(\delta)=0$. Recall also that $\Delta_\kappa=o(\sqrt {\sf n}_\kappa)$.
Now, for a sum of three terms to be at least $\epsilon$, at least one term must exceed $\epsilon/3$. So for every $\epsilon,\delta>0$, there exists a $\delta>0$ such that, for all $\kappa$ large enough,
\begin{align*}
`P_{{\bf s}(\kappa)^\star}(\omega_h(\delta)\ge \epsilon)
&\le `P_{{\bf s}(\kappa)^\star}(2\omega_s(\delta)>\epsilon/3)+`P_{{\bf s}(\kappa)^\star}(2\omega_{s^-}(\delta)>\epsilon/3)\\
&= 2 `P_{{\bf s}(\kappa)^\star}( 6\omega_s(\delta)\ge \epsilon )<\eta,
\end{align*}
since, under $`P_{{\bf s}(\kappa)^\star}$, $S_t$ and $S^-_t$ have the same distribution and ${\sf n}_\kappa^{-1/2}S_t$ is tight, and since $`P_{{\bf s}(\kappa)^\star}(\Delta_\kappa{\sf n}_\kappa^{-1/2}\geq `e/3)$ is zero for $\kappa$ large enough.
This proves that ${\sf n}_\kappa^{-1/2}H_t$ is tight under $`P_{{\bf s}(\kappa)^\star}$.
Now, we can couple the trees sampled under $`P_{{\bf s}(\kappa)^\star}$ and $`P_{\bf s}(\kappa)$. Since the nodes of degree one do not modify the tree structure, a tree $t$ under $`P_{{\bf s}(\kappa)}$ may be obtained by first sampling $t^\star$ using $`P_{{\bf s}(\kappa)^\star}$, and then placing the nodes of degree one uniformly at random : precisely, this insertion of nodes is done inside the edges of $t^\star$ (plus a phantom edge below the root). Given any ordering of the edges of $t^\star$ (plus the one below the root), the vector $(X_1^\star,\dots, X_{{\sf n}_\kappa-n_1(\kappa)}^\star)$ of numbers of nodes of degree one falling in these edges
is such that
$$(X_1^\star,\dots, X_{{\sf n}_\kappa-n_1(\kappa)}^\star) \ensuremath{\stackrel{d}{=}} \text{Multinomial}\left(n_1(\kappa); \frac{1}{{\sf n}_\kappa-n_1(\kappa)},\dots,\frac{1}{{\sf n}_\kappa-n_1(\kappa)}\right).$$
Conversely, $t^\star$ is obtained from $t$ by removing the nodes of degree one, so that $t$ and $t^\star$ can be thought as random variables in the same probability space under $`P_{{\bf s}(\kappa)}$. To bound $Y_t(\delta)$, observe that it is unlikely that adding the nodes of degree one in this way creates too long paths.\par
In fact, ``the length of paths'' is expected to be multiplied by $1+q_\kappa$ for
$q_\kappa=n_1(\kappa)/({\sf n}_\kappa-n_1(\kappa))$. Let $\alpha=2+q_\kappa$, and fix $\delta>0$ such that $`P_{{\bf s}(\kappa)^\star}(\omega_h(\delta)\ge \epsilon/\alpha)<\eta/2$; such a $\delta>0$ exists since the height process is tight under $`P_{{\bf s}(\kappa)^\star}$. Note that since we add nodes in the construction of $t$ under $`P_{{\bf s}(\kappa)}$ from $t^\star$ under $`P_{{\bf s}(\kappa)^\star}$, nodes that are within $\delta {\sf n}_\kappa$ in $t$ are also within $\delta {\sf n}_\kappa$ in $t^\star$. Write $h^\star$ for the rescaled height process obtained from $t^\star$, the tree associated with $t$ by deletion of all nodes of degree one (the rescaling stays $\sqrt {\sf n}_\kappa$). We have,
\begin{align*}
`P_{{\bf s}(\kappa)}(\omega_h(\delta)\ge \epsilon)
&\le `P_{{\bf s}(\kappa)}(\omega_{h^\star}(\delta)\ge \epsilon/\alpha) + `P_{{\bf s}(\kappa)}(\omega_h(\delta)\ge \epsilon~,~\omega_{h^\star}(\delta)\le\epsilon/\alpha)\\
&\leq `P_{{\bf s}(\kappa)^\star}(\omega_h(\delta)\ge \epsilon/\alpha) + `P_{{\bf s}(\kappa)}(\omega_h(\delta)\ge \epsilon~|
~\omega_{h^\star}(\delta)\le\epsilon/\alpha)\\
&\le `P_{{\bf s}(\kappa)^\star}(\omega_h(\delta)\ge \epsilon/\alpha)+\delta {\sf n}_\kappa^2 `P\left(\sum_{i=1}^{\epsilon \sqrt{{\sf n}_\kappa}/\alpha} (1+X_i^\star) \ge `e \sqrt{{\sf n}_\kappa}\right)\\
&\le \eta/2 + \delta {\sf n}_\kappa^2 `P\left(\sum_{i=1}^{\epsilon\sqrt{{\sf n}_\kappa}/\alpha} X_i \ge `e\sqrt{{\sf n}_\kappa}(1-1/\alpha)\right),
\end{align*}
where the $X_i$ are i.i.d.\ Binomial$(n_1, 1/({\sf n}_\kappa-n_1))$ random variables. The last line follows from the standard fact that the numbers $(X_i^\star)$ obtained from a sampling\emph{ without replacement} (of the $n_1(\kappa)$ nodes of degree one) are more concentrated than their counterpart $(X_i)$ coming from a sampling \emph{with replacement} \cite{Aldous1983}.
Now, the sum in the right-hand side is itself a binomial random variable:
$$\sum_{i=1}^{\epsilon\sqrt{{\sf n}_\kappa}/\alpha} X_i \ensuremath{\stackrel{d}{=}}\text{Binomial}\left(\epsilon\sqrt{{\sf n}_\kappa} n_1/\alpha , \frac 1{{\sf n}_\kappa-n_1}\right)$$
whose mean is $`e\sqrt{n_k}q_\kappa/(2+q_\kappa)$ when $`e\sqrt{{\sf n}_\kappa}(1-1/\alpha)=`e\sqrt{{\sf n}_\kappa}(1+q_\kappa)/(2+q_\kappa)$. By Chernoff's bound, using that $q_\kappa$ converges, it follows that for some constant $c>0$ valid for $\kappa$ large enough,
$$`P_{{\bf s}(\kappa)}(\omega_h(\delta)\ge \epsilon)\le \eta/2 + \delta {\sf n}_\kappa^2 e^{-c \sqrt{{\sf n}_\kappa}/\epsilon}.$$
Finally, for all $\kappa$ large enough, with this value for $\delta$, we have $`P_{{\bf s}(\kappa)}(\omega_h(\delta)\ge \epsilon)<\eta$, which completes the proof.
\end{proof}
\section{Finite dimensional distributions: Proof of Proposition~\ref{pro:compar}}\label{sec:fdd}
\subsection{A roadmap to Proposition~\ref{pro:compar}: identifying the bad events}
Our approach consists in showing that if the event in Proposition~\ref{pro:compar} occurs, then one of the following three events must occur: (1) either the depth $|{\bf u}|$ of node ${\bf u}$ is unusually large, (2) or the content of the branch $\llbracket \varnothing, {\bf u}\rrbracket$ is atypical, (3) or the number of nodes to the right of the path is not what it should be, despite of the length $|{\bf u}|$ and content ${\bf M}({\bf u},{\bf t})$ being typical.
We will then prove that those simpler events are unlikely. For $h\geq 0$, and two sequences $a=(a_\kappa,\kappa\ge 0)$, and $b=(b_\kappa,\kappa\ge 0)$ we define families of sets $A_{h,a,b}$ as follows. Given a sequence of degree distribution $({\bf s}(\kappa),\kappa \geq 0)$,
\begin{align*}
A_{h,a,b}(\kappa):=\left\{{\bf m}~ :
|{\bf m}|=h, \left|\left(\sum_{i\ge 0} m_i\frac{i-1}2\right)-\frac{h\sigma^2_{\kappa}}{2}\right|\leq a_{\kappa},
\sum_{i\ge 1} m_i i^2 \leq b_{\kappa}\right\}.\nonumber
\end{align*}
If ${\bf m}\in A_{h,a,b}(\kappa)$ then $|{\bf m}|=h$, and ${\bf m}$ corresponds to the content of a branch $\cro{\varnothing,u}$ such that $|u|=h$. The set $A_{h,a,b}(\kappa)$ are designed to contain most typical contents of a branch of length $h$ under $`P_{{\bf s}(\kappa)}$, provided the choices for the sequences $a$ and $b$ are suitable. The decomposition of the bad event we have outlined above is then expressed formally by
\begin{align}\label{eq:a-control}
`P_{{\bf s}(\kappa)}^{\bullet}\left(\left||{\sf R}({\bf u},{\bf t})|-\frac{\sigma_\kappa^2}{2}|{\bf u}|\right|\geq c_\kappa\right)&\leq
`P_{{\bf s}(\kappa)}^{\bullet}(|{\bf u}|\geq x \sqrt{{\sf n}_\kappa})\nonumber\\
&\quad+~`P_{{\bf s}(\kappa)}^\bullet\left(|{\sf LR}({\bf u},{\bf t})|\ge x \sqrt{{\sf n}_\kappa}\right)\nonumber\\
&\quad+~`P_{{\bf s}(\kappa)}^{\bullet}\left(|{\bf u}|\vee |{\sf LR}({\bf u},{\bf t})|\le x \sqrt{{\sf n}_\kappa}, {\bf M}({\bf u},{\bf t})\notin \bigcup_{h\leq x\sqrt{{\sf n}_\kappa}}A_{h,a,b}(\kappa)\right)\nonumber\\
&\quad+\sum_{h\leq x\sqrt{{\sf n}_\kappa} \atop{{\bf m} \in A_{h,a,b}(\kappa)}}`P_{{\bf s}(\kappa)}^{\bullet}\left(\left||{\sf R}({\bf u},{\bf t})|-\frac{\sigma_\kappa^2}{2}|{\bf u}|\right|\geq c_\kappa, {\bf M}({\bf u},{\bf t})={\bf m}\right).
\end{align}
{ Proving Proposition~\ref{pro:compar} reduces to proving that every term in the right-hand side above can be made arbitrarily small for large $\kappa$ by a judicious choice of $a_\kappa$, $b_\kappa$, $c_\kappa$ and $x$.} The bound on the first term is a direct consequence of the Gaussian tail bounds for the height of trees recently proved by \citet{Addario2011a} in the very setting we use:
\begin{equation}\label{eq:bound_height}
`P_{{\bf s}(\kappa)}^{\bullet}(|{\bf u}|\geq x \sqrt{{\sf n}_\kappa})\le `P_{{\bf s}(\kappa)}\left(\max_{u\in t} |u|\geq x\sqrt{{\sf n}_\kappa}\right)\le \exp(-c x^2 /\sigma_\kappa^2),
\end{equation}
for a universal constant $c>0$ and all sufficiently large $\kappa$. The second term is bounded using the depth-first walk $S$ and the reverse depth-first walk $S^-$, as in the proof of Lemma~\ref{lem:tightness}:
\begin{align*}
`P_{{\bf s}(\kappa)}^\bullet(|{\sf LR}({\bf u},{\bf t})|\ge x\sqrt{{\sf n}_\kappa})
&\le `P_{{\bf s}(\kappa)}\left(\max_{0\le k\le {\sf n}_\kappa}\{S(k)+S^-(k)\}+\Delta_\kappa \ge x\sqrt{{\sf n}_\kappa}\right)\\
&\le 2 `P_{{\bf s}(\kappa)}\left(\max_{0\le k \le {\sf n}_\kappa} S(k)\ge \frac x 3\sqrt{{\sf n}_\kappa}\right),
\end{align*}
for all $\kappa$ large enough, since $\Delta_\kappa=o({\sf n}_\kappa)$ and $S$ and $S^-$ have the same distribution under $`P_{{\bf s}(\kappa)}$. We finish using the tightness of ${\sf n}_\kappa^{-1/2}S({\sf n}_\kappa.)$ under $`P_{{\bf s}(\kappa)}$; more precisely, we have
\begin{align}\label{eq:bound_LR}
`P_{{\bf s}(\kappa)}^\bullet(|{\sf LR}({\bf u},{\bf t})|\ge x\sqrt{{\sf n}_\kappa}) \le 16 \cdot 9 \cdot \frac{\sigma_\kappa^2}{x^2},
\end{align}
by Lemma~20.5 of \cite{Aldous1983}. The bounds on the two remaining terms are stated in Lemmas~\ref{lem:content} and~\ref{lem:discrepancy}, the proof of which appear in Sections~\ref{sec:content} and~\ref{sec:discrepancy}, respectively.
\begin{lem}\label{lem:content}Since { $\Delta_k=o(\sqrt {\sf n}_\kappa)$} there exists $`e_\kappa$ such that $\Delta_\kappa\le\varepsilon_\kappa \sqrt{{\sf n}_\kappa}$, with $0<\varepsilon_\kappa\to 0$. Let $a_\kappa=\varepsilon_\kappa^{1/4} \sqrt{{\sf n}_\kappa}$ and $b_\kappa=\varepsilon_\kappa^{1/2} {\sf n}_\kappa $. Then, for every $x>0$, and all $\kappa$ large enough,
\begin{align*}
`P_{{\bf s}(\kappa)}^{\bullet}\left(|{\bf u}|\vee |{\sf LR}({\bf u},{\bf t})|\leq x \sqrt{{\sf n}_\kappa}, {\bf M}({\bf u},{\bf t})\not\in \bigcup_{h\leq x \sqrt{{\sf n}_\kappa}} A_{h,a,b}(\kappa)\right)
&\le 6x^2 e^{x^2}
\exp\left(-\frac {\varepsilon_\kappa^{-1/2}}{2x(\sigma^2_\kappa+1)+2}\right).
\end{align*}
\end{lem}
\begin{lem}\label{lem:discrepancy}Since { $\Delta_k=o(\sqrt {\sf n}_\kappa)$} there exists $`e_\kappa$ such that $\Delta_\kappa\le\varepsilon_\kappa \sqrt{{\sf n}_\kappa}$, with $0<\varepsilon_\kappa\to 0$ and $\varepsilon_\kappa^{-3/4}=o({\sf n}_\kappa)$ as $\kappa\to\infty$. Let $a_\kappa=\varepsilon_\kappa^{1/4} \sqrt{{\sf n}_\kappa}$, $b_\kappa=\varepsilon_\kappa^{1/2} {\sf n}_\kappa$, and $c_\kappa=\varepsilon_\kappa^{1/8}\sqrt{{\sf n}_\kappa}$. Then, for all $\kappa$ large enough,
\begin{equation}\label{eq:devi}
\sum_{h\leq x\sqrt{{\sf n}_\kappa}\atop{{\bf m} \in A_{h,a,b}(\kappa)}}
`P_{{\bf s}(\kappa)}^{\bullet}\left(\left||{\sf R}({\bf u},{\bf t})|-\frac{\sigma_\kappa^2}{2}|{\bf u}|\right|\geq c_\kappa, {\bf M}({\bf u},{\bf t})={\bf m}\right)\le 2 e^{-\varepsilon_\kappa^{-1/2}}.
\end{equation}
\end{lem}
Before proceeding with the proofs of these two lemmas, we indicate how to use them in order to complete the proof of Proposition~\ref{pro:compar}. Let $\varepsilon_\kappa$ be such that $\Delta_\kappa\le \varepsilon_\kappa \sqrt{{\sf n}_\kappa}$, with $\varepsilon_\kappa\to0$ as $\kappa\to\infty$. Then, set $a_\kappa=\varepsilon_\kappa^{1/4} \sqrt{{\sf n}_\kappa}$, $b_\kappa=\varepsilon_\kappa^{1/2} {\sf n}_\kappa$ and $c_\kappa=\varepsilon_\kappa^{1/8} \sqrt{{\sf n}_\kappa}$. Let now $\epsilon>0$ be arbitrary. Pick $x>0$ large enough such that, for all $\kappa$ large enough,
\begin{align*}
`P_{{\bf s}(\kappa)}^{\bullet}(|{\bf u}|\geq x \sqrt{{\sf n}_\kappa}) + `P_{{\bf s}(\kappa)}^\bullet(|{\sf LR}({\bf u},{\bf t})|\ge x\sqrt{{\sf n}_\kappa}) < \epsilon/2.
\end{align*}
The bounds in \eqref{eq:bound_height} and \eqref{eq:bound_LR}, and the fact that $\sigma^2_\kappa \to \sigma^2_{\bf p}$ ensure that this is possible. The value for $x$ being fixed, Lemmas~\ref{lem:content} and~\ref{lem:discrepancy} now make it possible to choose $\kappa_0$ large enough such that, for all $\kappa\ge \kappa_0$, the two remaining terms in the right-hand side of \eqref{eq:a-control} also sum to at most $\epsilon/2$. Thus, for all $\kappa\ge \kappa_0$, we have
$$`P_{{\bf s}(\kappa)}^{\bullet}\left(\left||{\sf R}({\bf u},{\bf t})|-\frac{\sigma_\kappa^2}{2}|{\bf u}|\right|\geq c_\kappa\right)<\epsilon,$$
which completes the proof, since $\epsilon$ was arbitrary.
\subsection{The content of a branch is very likely typical: Proof of Lemma~\ref{lem:content}}\label{sec:content}
We now prove that, on the event that $|{\bf u}|$ and $|{\sf LR}({\bf u},{\bf t})|$ are not too large, the content of the branch $\cro{\varnothing, {\bf u}}$ is typical with high probability.
We start by rewriting the probability of interest using Proposition \ref{pro:fond-comb}:
\begin{align}\label{eq:content1}
&`P_{{\bf s}(\kappa)}^{\bullet}\left(|{\bf u}|\vee |{\sf LR}({\bf u},{\bf t})|\leq x\sqrt{{\sf n}_\kappa}, {\bf M}({\bf u},{\bf t})\not\in \bigcup_{h\leq x\sqrt{{\sf n}_\kappa}} A_{h,a,b}(\kappa)\right)\nonumber\\
&=\sum_{h\leq x\sqrt{{\sf n}_\kappa}} `P_{{\bf s}(\kappa)}^{\bullet}\left(|{\bf u}|=h, |{\sf LR}({\bf u},{\bf t})|\le x\sqrt{{\sf n}_\kappa},{\bf M}({\bf u},{\bf t})\not\in A_{h,a,b}(\kappa)\right)\nonumber\\
&=\sum_{h\le x\sqrt{{\sf n}_\kappa}} \sum_{|{\bf m}|=h\atop{{\bf m}\not\in A_{h,a,b}(\kappa)}, |{\sf LR}({\bf m})|\le x\sqrt{{\sf n}_\kappa}}`P_{{\bf s}(\kappa)}^{\bullet}\left(|{\bf u}|=h,{\bf M}({\bf u},{\bf t})={\bf m}\right)\nonumber\\
&=\sum_{h\le x\sqrt{{\sf n}_\kappa}} \sum_{|{\bf m}|=h\atop{{\bf m}\not\in A_{h,a,b}(\kappa)},|{\sf LR}({\bf m})|\le x\sqrt{{\sf n}_\kappa}}\frac{|{\sf LR}({\bf m})|\,h! \,({\sf n}_\kappa-h)!}{{\sf n}_\kappa!({\sf n}_\kappa-h)}\prod_{i\geq 1}\binom{n_i}{m_i}{i^{m_i}}.
\end{align}
where, for short, we have written $n_i$ instead of $n_i(\kappa)$.
{ We now reduce the right-hand side to an expected value with respect to multinomial random variables. Let $(P_i, i\ge 1)$ be multinomial with parameters $h$ and $(in_i/({\sf n}_\kappa-1), i\ge 1)$.} Then, for any ${\bf m}=(0, m_1, m_2, \dots)$ such that $|{\bf m}|=h$, we have
$$ `P\left((P_i, i\ge 1)=(m_i, i\ge 1)\right)= \frac{h!}{\prod_{i\ge 1} m_i!} \cdot \prod_{i\ge 1} \left(\frac {in_i}{{\sf n}_\kappa-1}\right)^{m_i}.$$
Now, since $(1-x)^{-1}\leq \exp(2x)$ for $|x|\le 1/2$, we have for all $h\le x\sqrt{{\sf n}_\kappa}$, and all $\kappa$ large enough,
\begin{align*}
\frac{({\sf n}_\kappa-h)!{\sf n}_\kappa^h}{{\sf n}_\kappa!}\le \prod_{i=0}^{h-1} \frac1{1-i/{{\sf n}_\kappa}}\le \prod_{i=0}^{h-1} e^{2i/{{\sf n}_\kappa}} \le e^{x^2}.
\end{align*}
Note also that, for every $i\ge 1$, we have $n_i!\le n_i^{m_i} (n_i-m_i)!$, so that, rewriting \eqref{eq:content1} in terms of events with respect to $(P_i, i\ge 1)$, we obtain
\begin{align*}
&`P_{{\bf s}(\kappa)}^{\bullet}\left(|{\bf u}|\vee |{\sf LR}({\bf u},{\bf t})|\leq x\sqrt{{\sf n}_\kappa}, {\bf M}({\bf u},{\bf t})\not\in \bigcup_{h\leq x\sqrt{{\sf n}_\kappa}} A_{h,a,b}(\kappa)\right)\\
& = \sum_{h\le x\sqrt{{\sf n}_\kappa}} \sum_{|{\bf m}|=h\atop{{\bf m}\not\in A_{h,a,b}(\kappa)}, |{\sf LR}({\bf m})|\le x \sqrt{{\sf n}_\kappa}} \frac{|{\sf LR}({\bf m})|}{{\sf n}_\kappa-h}\cdot \frac{({\sf n}_\kappa-h)! { ({\sf n}_\kappa-1)^h}}{{\sf n}_\kappa!} \prod_{i\ge 1} \frac{n_i!}{n_i^{m_i} (n_i-m_i)!} \cdot `P\left((P_i, i\ge 1)=(m_i, i\ge 1)\right)\\
&\le \sum_{h\le x\sqrt{{\sf n}_\kappa}}\frac {2x}{\sqrt{{\sf n}_\kappa}} e^{x^2} \sum_{|{\bf m}|=h\atop{{\bf m}\not\in A_{h,a,b}(\kappa)}} `P\left((P_i, i\ge 1)=(m_i, i\ge 1)\right)\\
& \le 2 x^2 e^{x^2} \sup_{h\le x \sqrt{{\sf n}_\kappa}} `P((P_i, i\ge 1) \not\in A_{h,a,b}(\kappa)).
\end{align*}
Now, we decompose the set of ${\bf m}$ in the right-hand side so as to obtain bad events that are individually simpler to deal wit
\begin{align*}
`P_{{\bf s}(\kappa)}^{\bullet}\left(|{\bf u}|\vee |{\sf LR}({\bf u},{\bf t})|\le x\sqrt{{\sf n}_\kappa},{\bf M}({\bf u},{\bf t})\not\in A_{h,a,b}(\kappa)\right)\le 2 x^2 e^{x^2} \sup_{h\le x\sqrt{{\sf n}_\kappa}}(\zeta_1+\zeta_2)
\end{align*}
where
\begin{align*}
\zeta_1=`P\left(\left|\left(\sum_{i\ge 1} P_i\frac{i-1}2\right)-\frac{h\sigma^2_\kappa}{2}\right|\geq a_\kappa \right)\qquad \text{and}\qquad
\zeta_2= `P\left(\sum_{i\ge 1} i^2 P_i >b_\kappa\right).
\end{align*}
We now bound the terms $\zeta_1$ and $\zeta_2$ individually.
\medskip
\noindent\textsc{The first term $\zeta_1$.}
Observe first, that
$$`E\left[\sum_{i\ge 1} P_i \frac{i-1}2\right]=\frac{h\sigma^2_\kappa}{2},$$
so that bounding $\zeta_1$ consists in bounding the deviations of (a function of) a multinomial vector. However, one can write
$$\sum_{i\ge 1} P_i \cdot \frac{i-1}{2} - h \frac{\sigma_\kappa^2}2 \stackrel{d}{=} \sum_{j=1}^h (B_j -`E B_j),$$
where $B_j$, $j=1,\dots, h$, are i.i.d.\ random variables taking value $(i-1)/2$ with probability $ i n_i/({\sf n}_\kappa-1)$, for $i\ge 1$. Now, the sums $\sum_{j=1}^\ell (B_i-\Ec{B_j})$, $\ell=0, 1, \dots, h$, form a martingale. We bound their deviations using a concentration inequality from \cite{McDiarmid1998a} (Theorem 3.15), which says that if $S$ is a sum of independent random variable $X_1+\dots+X_n$ such that $`E(S)=\mu$, $\var(S)=V$, and if for all k $X_k-`E(X_k)\leq b$, then $`P(S-\mu\geq t)\leq e^{-t^2/(2V(1+bt/(3V))}$. The variance of $B_j$ may be bounded as follows:
$$\var(B_j) \le \Ec{B_j^2} = \sum_{i\ge 1} \frac{(i-1)^2}4 \frac{i n_i}{ {\sf n}_\kappa-1} \le \Delta_\kappa \sum_{i\ge 1} \frac{i-1}4 \frac{i n_i}{ {\sf n}_\kappa-1}{ =} \Delta_\kappa \sigma_\kappa^2/4,$$
for all $\kappa$ large enough.
Now, since $\max\{|B_j-`E(B_j)|: j=0,\dots, h \}\le \Delta_\kappa$, one has, for $h\le x\sqrt{{\sf n}_\kappa}$,
\begin{align*}
`P\left(\left|\sum_{j=1}^h (B_j-`E B_j)\right|\ge a_\kappa\right)
& \le 2 \exp\left(- \frac{a_\kappa^2}{2 h \Delta_\kappa \sigma^2_\kappa/4 +2\Delta_\kappa a_\kappa/3}\right)\\
& \le 2 \exp\left(- \frac {a_\kappa^2}{ x \sqrt{{\sf n}_\kappa} \Delta_\kappa \sigma_\kappa^2}\right),
\end{align*}
for all $\kappa$ large enough, since $a_\kappa=\varepsilon_\kappa^{1/4}\sqrt{{\sf n}_\kappa}=o(\sqrt{{\sf n}_\kappa})$. It follows that, for every $h\le x\sqrt{{\sf n}_\kappa}$, we have
\begin{equation}\label{eq:zeta1}
\zeta_1=\sup_{h\leq x\sqrt{{\sf n}_\kappa}}`P\left(\left|\left(\sum_{i\ge 1} P_i\frac{i-1}2\right)-\frac{h\sigma^2_\kappa}{2}\right|\geq a_\kappa\right) \le 2 \exp\left(- \frac {\varepsilon_\kappa^{-1/2}}{ x \sigma_\kappa^2}\right).
\end{equation}
\medskip
\noindent\textsc{The second term $\zeta_2$}.\
We bound $\zeta_2$ using the idea we used when bounding $\zeta_1$: one can express the event in terms of independent random variables $B_j$, $j=1,\dots, h$, where $B_j$ takes value $i^2$ with probability $ i n_i/({\sf n}_\kappa-1)$.
Observe first that
$$\E{\sum_{i\ge 1} i^2 P_i }=\E{\sum_{j=1}^h B_j} = h \sum_{i\ge 1} i^2 \cdot \frac{i n_i}{ {\sf n}_\kappa-1} \le h \Delta_\kappa (\sigma_\kappa^2+1).$$
So, we have
\begin{align*}
`P\left(\sum_{i\ge 1} i^2P_i >b_\kappa\right)
&=`P\left(\sum_{j=1}^h B_j > b_\kappa\right)\\
&\le `P\left(\sum_{j=1}^h (B_j - \Ec{B_j})> \frac{b_\kappa}2\right),
\end{align*}
for all $\kappa$ large enough, since $h \Delta_\kappa \le x \varepsilon_\kappa {\sf n}_\kappa=o(\varepsilon_\kappa^{1/2} {\sf n}_\kappa)=o(b_\kappa)$. The right-hand side above can be bounded using the martingale inequality in \cite{McDiarmid1998a} (Theorem~3.15). We note that the variance of $B_j$ satisfies
$$\var(B_j)\le \Ec{B_j^2}=\sum_{i\ge 1} i^4 \cdot \frac {in_i}{ {\sf n}_\kappa-1} \le \Delta_\kappa^3 (\sigma_\kappa^2 +1).$$
Since $\max\{|B_i|: i=1,\dots, h \}\le \Delta_\kappa^2$, it follows by McDiarmid's inequality that
\begin{align}\label{eq:zeta3}
\zeta_2\le`P\left(\sum_{j=1}^h (B_j - \Ec{B_j})> \frac{b_\kappa}2\right)
& \le \exp\left(-\frac{b_\kappa^2/4}{2 x\sqrt{{\sf n}_\kappa} \Delta_\kappa^3 (\sigma_\kappa^2+1)+2\Delta_\kappa^2 b_\kappa/3}\right)\nonumber\\
& \le \exp\left(-\frac{b_\kappa}{2(x(\sigma_\kappa^2+1)+1/3) \Delta_\kappa^2}\right)\nonumber\\
& = \exp\left(-\frac {\varepsilon_\kappa^{-3/2}}{2x(\sigma^2_\kappa+1)+2/3}\right),
\end{align}
for all $\kappa$ large enough, since $ \Delta_\kappa \sqrt{{\sf n}_\kappa} =o(b_\kappa)$.
To complete the proof, it suffices to combine the bounds in \eqref{eq:zeta1}--\eqref{eq:zeta3}, and observe that they imply the claim for $\kappa$ large enough, since the { upper bound in \eqref{eq:zeta3} is much smaller than the one in \eqref{eq:zeta1}.}
\subsection{The structure of a branch with typical content: Proof of Lemma~\ref{lem:discrepancy}}\label{sec:discrepancy}
Finally, we consider the probability that the structure of a branch is not what one expects, in spite of the length and content being close to the typical values. The left hand side in \eref{eq:devi} is bounded by
\[\sup_{h\leq x\sqrt{{\sf n}_\kappa}\atop{{\bf m} \in A_{h,a,b}(\kappa)}}\!\!\!\!`P_{{\bf s}(\kappa)}^{\bullet}\left(\left||{\sf R}({\bf u},{\bf t})|-\frac{\sigma_\kappa^2}{2}|{\bf u}|\right|\geq c_\kappa ~\Bigg|~ {\bf M}({\bf u},{\bf t})={\bf m}\right)
= \sup_{h\leq x\sqrt{{\sf n}_\kappa}\atop{{\bf m} \in A_{h,a,b}(\kappa)}}\!\!\!\!`P\left(\left|\frac{\sigma_\kappa^2}{2}h-\sum_{j\geq 1}\sum_{k=1}^{m_j} U^{(k)}_j\right|\geq c_\kappa \right),
\]
by Proposition \ref{pro:fond-comb} (3), where $U_j^{(k)}$ are independent random variables with $U_j^{(k)}$ uniform on $\{0,1,\dots, j-1\}$.
By the triangle inequality, the quantity in the right-hand side above is at most
\begin{equation} \label{eq:11-24}
\sup_{h\leq x\sqrt{{\sf n}_\kappa}\atop{{\bf m} \in A_{h,a,b}(\kappa)}}
`P\left(\left|\sum_{j\ge 1} m_j\frac{j-1}2-\sum_{j\geq 1}\sum_{k=1}^{m_j} U^{(k)}(j)\right|\geq c_\kappa-\left|\frac{\sigma_\kappa^2h}{2}-\sum_{j\ge 1} m_j\frac{j-1}2\right|\right).
\end{equation}
By definition of $A_{h,a,b}(\kappa)$, and since $c_\kappa>2 a_\kappa$ for all $\kappa$ large enough,
the quantity in \eref{eq:11-24} is bounded by
\[
\sup_{h\leq x\sqrt{{\sf n}_\kappa} \atop{{\bf m} \in A_{h,a,b}(\kappa)}}
`P\left(\left|\sum_{j\ge 1} m_j\frac{j-1}2-\sum_{j\geq 1}\sum_{k=1}^{m_j} U^{(k)}_j\right|\geq \frac{c_\kappa}2\right).
\]
Now, since all the random variables $U_j^{(k)}$, $j\ge 1$, $k=1,\dots, m_j$ are symmetric about their respective mean $(j-1)/2$, one obtains using Chernoff's bounding method
\begin{align}\label{eq:bigbound}
`P\left(\left|\sum_{j\ge 1} m_j\frac{j-1}2-\sum_{j\geq 1}\sum_{k=1}^{m_j} U^{(k)}_j\right|\geq \frac{c_\kappa}2\right)\nonumber
&\leq2\inf_{t\geq 0}e^{-tc_\kappa/2}\E{e^{t\sum_{j\geq 1}\sum_{k=1}^{m_j}\left(U^{(k)}_j-\frac{(j-1)}{2}\right)}}\\
&=2\inf_{t\geq 0}e^{-tc_\kappa/2}\prod_{j\geq 1}\left(\frac{\sinh(tj/2)}{j\sinh(t/2)}\right)^{m_j}\nonumber\\
&\le 2\inf_{t\geq 0}\exp\left(-t\frac{c_\kappa}2+\sum_{j\geq 1}m_j\left(\frac{j^2t^2}{24}-\frac{t^2}{24}+\frac{t^4}{2880}\right)\right)\nonumber\\
&\le 2\inf_{t\in (0,1)}\exp\left(-t\frac{c_\kappa}2+\sum_{j\geq 1}m_j\left(\frac{j^2t^2}{24}-\frac{t^2}{48}\right)\right)\\
&\le 2\inf_{t\in (0,1)}\exp\left(-t\frac{c_\kappa}2+\frac{t^2}{24}\sum_{j\geq 1}m_j j^2\right).\nonumber
\end{align}
Here the third line follows from the bounds $\log(\sinh(s))\leq \log(s)+s^2/6$ and $\log(\sinh(s))\geq \log(s)+s^2/6-s^4/180$ valid for $s\geq 0$. Finally, we obtain
\begin{align*}
\sup_{h\leq x\sqrt{{\sf n}_\kappa} \atop{{\bf m} \in A_{h,a,b}(\kappa)}}
`P\left(\left|\sum_{j\ge 1} m_j\frac{j-1}2-\sum_{j\geq 1}\sum_{k=1}^{m_j} U^{(k)}_j\right|\geq \frac{c_\kappa}2\right)
&\le 2\inf_{t\in(0,1)}\exp\left(-t\frac{c_\kappa}2+ \frac{t^2 b_\kappa}{24}\right)\\
&\le 2 e^{-3c_\kappa^2/(2b_\kappa)},
\end{align*}
upon choosing $t=6c_\kappa/b_\kappa$, which is indeed in $(0,1)$ for $\kappa$ large enough (we restricted the range of $t$ in \eqref{eq:bigbound}). This completes the proof since $3c_\kappa^2/(2b_\kappa)=3\varepsilon_\kappa^{-3/4}/2\ge \varepsilon_\kappa^{-1/2}$, for all $\kappa$ large enough.
\section{The limit of rescaled Galton--Watson trees: Proof of Proposition~\ref{pro:Ald}}
\label{sec:pt}
{ Consider the family tree of a Galton-Watson tree ${\bf t}$ with offspring distribution $\mu=(\mu_i,i\geq 0)$ starting with one individual. Let $`P_\mu$ be the probability distribution of ${\bf t}$.}
Denote by $\wh{{\bf s}}_{\bf t}:=(\wh{n}_i({\bf t}),i\geq 0)$ the empirical degree sequence of ${\bf t}$, let {
\begin{align*}
\wh{\mu}_i &= \wh{n}_i({\bf t})/|{\bf t}|,\\
\wh{\sigma}^2&=\sum_{i\ge 0} i^2\frac{\wh{n}_{i}({\bf t})}{|{\bf t}|-1}-1\\
\wh{\Delta} &=\max\{i:\wh{n}_i>0\}.
\end{align*}
Note that $\wh{\sigma}^2$ is not the variance of the empirical distribution $(\wh{\mu}_i, i\ge 0)$ but has been chosen to be consistent with the definition of $\sigma_{{\bf s}(\kappa)}^2$ in \eqref{eq:sigma}. Write $`P_\mu^n(\,\cdot\,)=`P_\mu(\,\cdot\,|\,|{\bf t}|=n)$.} In what follows, all the assertions containing `` $`P_\mu^n$'' are to be understood ``for $n$ such that $`P_\mu(|{\bf t}|=n)>0$''; similarly, the limit with respect to $`P_\mu^n$ are to be understood in the same manner, along subsequences included in $\{n: `P_\mu(|{\bf t}|=n)>0\}$.
\begin{lem}\label{lem:GW}
Assume that $\mu$ has mean 1 and variance $\sigma^2_\mu\in(0,+\infty)$. Then
under $`P_\mu^n$,
\begin{equation}\label{eq:cv-tr}
(\hat{\mu},\wh{\sigma}^2,{\wh{\Delta}}/{\sqrt{n}})\xrightarrow[n]{(d)} (\mu,\sigma^2_\mu, 0),
\end{equation}
where the convergence holds in the space ${\cal M}(`N)\times `R\times `R$ equipped with the product topology.
\end{lem}
{ In this lemma,} ${\cal M}(`N)$ is the set of probability measures on $`N$. The topology on ${\cal M}(`N)$ is metrizable, for example, by the distance
\[D(\nu,\nu')=\sum_{i\geq 0} { \frac{1}{2^i}}d_{\textup{TV}}(\nu[i],\nu'[i])\]
where $\nu[i]$ is the distribution of the $i$th first marginals under $\nu$ and $d_{\textup{TV}}$ is the distance in total variation. Since here the limit is the deterministic measure $\mu$, it suffices to show that, for all $i$, $\hat{\mu}_i\to \mu_i$ in probability as $n\to\infty$. With $D$ it is easy to construct a metric on ${\cal M}(`N)\times `R\times `R$ making of this space a Polish space. Hence, by the Skohorod theorem there exists a probability space where versions of $(\hat{\mu},\wh{\sigma}^2,{\wh{\Delta}}/{\sqrt{n}})$ under $`P_\mu^n$ converges almost surely to $(\mu,\sigma^2_\mu, 0)$. So on the conditional space, the hypotheses of Theorem~\ref{thm:main_gh} hold almost surely, and then its conclusion, which is a limit in distribution, also holds. { Of course, we do not mean that any sequence of trees for which the degree distribution satisfies the conditions of Theorem~\ref{thm:main_gh} converges to the continuum random tree; one also needs that for any fixed $\kappa$, conditional on the degree sequence ${\bf s}$ the trees are distributed according to $`P_{\bf s}$. This fact certainly holds for conditioned Galton--Watson trees: under $`P_\mu$ all trees with the same degree sequence occur with the same probability, and conditional on its degree sequence ${\bf s}$, a Galton--Watson tree is precisely distributed according to $`P_{{\bf s}}$. To summarise, to prove Proposition~\ref{pro:Ald} it suffices to prove Lemma~\ref{lem:GW}.}
\begin{proof}[Proof of Lemma \ref{lem:GW}]
The claim is about properties of the degree sequence of Galton--Watson trees conditioned on their total progeny. We first provide a way to construct the degree sequence. Consider the \L ukasiewicz walk $S_n$ associated with a tree ${\bf t}$ under $`P_\mu^n$; the degree sequence of the tree ${\bf t}$ is essentially (just shift by one) the empirical distribution of the increments of $S_n$. More precisely, consider first a random walk $W=(W_k,k=0,\dots,n)$, with i.i.d.\ increments $X_k=W_k-W_{k-1},k=1,\dots,n$ with distribution \[\nu_i=`P(X_k=i)=\mu_{i+1}\quad i\geq -1;\]
then $S=(S_0,\dots,S_n)$ is distributed as $W$ conditioned on
$W\in A^+_{-1}(n)$ where
$$A^+_{-1}(n)=\{w=(w_0,\dots,w_n)~:~ w_0=0, w_k\geq 0, 1\le k <n,w_n=-1\}$$ is the set of discrete excursions of length $n$.
Write $K_i=\#\{ k : X_k=i-1 \}$, and $K=(K_i,i\geq 0)$. Then, if $W\in A^+_{-1}(n)$, the sequence $K=(K_i, i\ge 0)$ is distributed as the degree sequence of a tree under $`P_\mu^n$. In other words, we have
\[`P(K\in B~|~W\in A^+_{-1}(n))=`P_{\mu}^n((\wh{n_i}({\bf t}),i\geq 0)\in B ).\]
By the rotation principle, we may remove the positivity condition :
\[`P(K\in B~|~W_n=-1)=`P_{\mu}^n((\wh{n_i}({\bf t}),i\geq 0)\in B ).\]
Our aim is now to show that the condition that $W$ is a bridge imposed by $W_n=-1$ does not completely wreck the properties of $W$ in the following sense: let ${\cal F}_k=\sigma(W_0,\dots,W_k)$ be the $\sigma$-field generated by the $k$ first $W_i$; then there exists a constant $c\in (0,\infty)$ such that for any $n$ large enough, and for any event $B\in {\cal F}_{\floor{n/2}}$ one has
\begin{equation}\label{eq:absolute_cont}`P(B ~|~ W_n=-1) \leq c\, `P(B).\end{equation}
That is: any event $B$ in ${\cal F}_{\floor{n/2}}$ with a very small probability for a standard (unconditioned) random walk also has a small probability in the bridge case (conditional on $W_n=-1$).
The argument proving this claim is given in \citet{JM}, page 662 and goes as follows:
\begin{align*}
`P(B\,|\,W_{n}=-1)
&=\sum_{x} `P(B\,|\,W_{\floor{n/2}}=x,W_{n}=-1) \cdot \frac{`P(W_{\floor{n/2}}=x,W_{n}=-1)}{`P(W_n=-1)}\\
&=\sum_{x} `P(B\, |\, W_{\floor{n/2}}=x)`P(W_{\floor{n/2}}=x)\cdot \frac{`P(W_{n-\floor{n/2}}=-x-1)}{`P(W_n=-1)}.
\end{align*}
It then suffices to (a) observe that $\sup_x `P(W_{n-\floor{n/2}}=-x-1)\le c/\sqrt{n}$ for some constant $c_1\in (0,\infty)$ \cite[][Theorem 2.2 p. 76]{PET1995}, and (b) use a local limit theorem to show that $`P(W_n=-1)\ge c_2 \sqrt n$, for some constant $c_2\in (0,\infty)$ and all $n$ large enough \cite[page 233]{GK-54}. This gives the result in \eqref{eq:absolute_cont} with $c=c_1/c_2$.
Now using that the increments $(X_1,\dots,X_n)$ under $`P(\,\cdot\,| \, W_n=-1)$ are exchangeable, any concentration principle for the first half of them easily extends to the second half (the easy details are omitted). Consider the degree sequence induced by the first half of the walk:
let $K^{1/2}_i=\#\{ k : X_k=i-1, k \leq \floor{n/2} \}$, and note that the $K_i^{1/2}$ are ${\cal F}_{\floor{n/2}}$-measurable.
For $W$ (that is, with no conditioning), we have
\begin{equation}\label{ec:cvsigma_mu}
\frac 1 {\floor{n/2}} \sum_{i\ge 0} K^{1/2}_i(i-1)^2=\frac{1}{\floor{n/2}} \sum_{j=1}^{\floor{n/2}} X_j^2\xrightarrow[n\to\infty]{} \Ec{X_1^2}=\sigma_\mu^2
\end{equation}
by the law of large number, since $X_i$ owns a (finite) moment of order 2.
Hence, for any $`e>0$, writing
\[Ev(`e)= \left\{ \left|\frac 1{\floor{n/2}}\sum_{i\ge 0} K^{1/2}_i(i-1)^2-\sigma_\mu^2 \right|\geq `e \right\},\]
we have
$`P(Ev(`e))\to 0$ and thus, according to the bound in \eqref{eq:absolute_cont}, $`P(Ev(`e) |W_n=-1)\to 0$, as $n\to\infty$. Using the argument twice (one for each half of the walk) yields convergence $\wh{\sigma}^2\to \sigma_\mu^2$ in probability as $n\to\infty$.
The same argument also proves that
$$`P\left( \left|\frac{K^{1/2}_i}{\floor{n/2}}-\mu_i\right|\geq `e~\Bigg|\,W_n=-1\right)\to 0,$$
which yields $\wh{\mu_i}\to \mu_i$ in probability.
The fact that $\wh{\Delta}=o(\sqrt{n})$ (in probability) under $`P_\mu^n$ is also a consequence of the convergence of the sum given in \eref{ec:cvsigma_mu}. To see this, let $C(`a)= \{k ~:~`P(X_1^2\geq k)\geq `a/k\}$. Since $\Ec{X_1^2}=\sum_{k\ge 0} `P(X_1^2\geq k)<+\infty$, then $k`P(X_1^2\geq k)\to 0$ , entailing $\#C(`a)<+\infty$ for any $\alpha>0$. In particular, for any $`e>0$,
$$\#\{n : n`P(X_1^2\geq `e n)\geq `a/`e\}<+\infty.$$ Taking $`a=`e`e'$, one obtains that $\#\{n : n`P(X_1^2\geq `e n)\geq `e'\}<+\infty$, which implies that
$$`P(\max \{X_i:i\le n/2\}\geq `e \sqrt n)\le n `P(X_1^2\ge `e n)\xrightarrow [n\to\infty]{} 0.$$ So under the unconditioned law one has $\wh\Delta = o(\sqrt n)$; we complete the proof using the bound in \eqref{eq:absolute_cont}.
\end{proof}
\section{Application to constrained coalescing processes}\label{sec:coagulation}
In this final section, we discuss an application of Theorem~\ref{thm:main_gh} to a coalescence process with particles having constrained valences.
The famous additive coalescent \cite{AlPi1998a,Bertoin2000a, Bertoin2006, Pitman1999b,Pitman2006} can be seen as arising from the following natural microscopic description. Consider a set of $n$ distinct particles $\{1,2,\dots, n\}$. The particles are initially free, and form $n$ clusters; the clusters are organised as rooted trees. The clusters merge according to the following dynamics. At each step, choose a particle $u$ uniformly at random; it belongs to some cluster $T$ rooted at $r$. Choose uniformly a second cluster $T'\neq T$, with root $r'$. Add an edge between $r'$ and $u$ to obtain a new cluster rooted at $r$. At each step, the system consists of a forest of general rooted labelled trees (an acyclic graph on $\{1,2,\dots, n\}$ with a distinguished node per connected component). The process stops after $n-1$ steps, when the system consists of a single rooted labelled tree. The final tree is then uniform among all rooted labelled trees.
One can similarly define a system of coalescing particles where the degrees would be constrained. Different algorithms might be used, depending on the precise way the uniform choices are made, that yield \emph{a priori} different trees.
\medskip
\noindent\textsc{Labelled particles.} Consider the set of particles $\{1,2,\dots, n\}$, and a set of degrees $c_1\le c_2\le\dots\le c_n$. Write ${\bf s}=(n_i, i\ge 0)$ for the associated degree sequence, $n_i=\#\{j: c_j=i\}$. Assign randomly the particles a degree. For instance, this can be done using a random permutation $\sigma=(\sigma(1),\dots,\sigma(n))$ of $\{1,2,\dots, n\}$ and assigning degree $c_{\sigma(i)}$ to particle $i$. Think now of the particle $i$ as initially having edges to $c_{\sigma(i)}$ free slots that can each contain a single particle. The particles will now merge to form clusters. Each cluster is represented by a tree with a distinguished vertex (the root). Initially, each particle sits in a tree containing a single node (which is then also the root). Proceed with the following algorithm to merge the particles, as long as there are free slots left:
\begin{itemize}
\item Pick a free slot $s$ uniformly at random; say it is bound to particle $p$ lying in the cluster rooted at $r$.
\item Pick \emph{another} cluster, uniformly at random, rooted at some node $r'$.
\item Merge the two clusters by assigning $r'$ to the free slot $s$; this creates an edge between the particles $p$ and $r'$, and removes the slot $s$ from the set of free slots. The new cluster is rooted at $r$.
\end{itemize}
At every iteration, precisely one slot is filled and the process stops after $n-1$ steps. The process yields a random tree \emph{labelled} tree $T_n^L$.
The labelled tree $T_n^L$ is uniform in the set of labelled trees having the same specified degree sequence. To see this, just consider the encoding of the process by the final labelled tree, together with a labelling of the edge indicating their order of appearance. At iteration $i\in \{1,\dots, n-1\}$, there are $n-i$ free slots left and $n-i+1$ connected components, so that the probability that any couple free slot/other connected component is precisely
$$\frac 1 {(n-i)^2}.$$
Overall, the probability to obtain any particular pairing free slots/particles together with a history is
$$\prod_{i=1}^n \frac 1{(n-i)^2}=\frac 1 {(n-1)!^2}.$$
The same particle adjacency ---hence the same labelled tree--- is obtained by the $\prod_{j=1}^n c_j!$ ways to pair the free slots with particles; and for any labelled tree there are exactly $(n-1)!$ distinct histories. Finally, among the $n!$ ways to assign the labels to particles in the first place, $\prod_{i\ge 0}n_i!$ correspond to the degree/label pattern of the tree, it follows that the probability of seeing any labelled tree after $n-1$ iterations is precisely
\begin{equation}\label{eq:coalescing_labelled}
\frac{\prod_{i\ge 0}n_i!}{n!} \times \frac 1 {(n-1)!^2} \times (n-1)! \times \prod_{i=1}^n c_i! = \frac{\prod_{i\ge 0} i!^{n_i}}{(n-1)!} \times \binom{n}{(n_i, i\ge 0)}^{-1},
\end{equation}
which depends only on the degree sequence, so that trees with the same degree sequence are chosen uniformly. (This is also, as it should, the inverse of the number of labelled trees with degree sequence given by ${\bf s}=(n_i, i\ge 0)$ \cite[][Example 6.2.2]{Pitman2006}.)
\medskip
\noindent\textsc{Unlabelled particles.} Consider a degree sequence in the form of ${\bf s}=(n_i, i\ge 0)$ where $n_i$ denotes the number of nodes of degree $i$. For $c_1\le c_2\le \dots \le c_n$ of size $n$. So $\sum_{i\ge 0} c_i=n-1$. As before, we think of the particles as having empty slots, but since there are no labels, we impose that the slots of any given particle be ordered. The particles then merge according to the same algorithm, in order to distinguish particles use the canonical labelling giving label $i$ to the particle with degree $c_i$. After forgetting the canonical labelling, the process yields a plane tree $T_n$.
Again, the plane tree $T_n$ is uniform among all plane trees with the correct degree sequence. The arguments are similar, only simpler, to those we used in the labelled case. Since, for a given plane tree, there are $\prod_{i\ge 0} n_i!$ ways to assign the canonical labels to the nodes, the probability to obtain any given plane tree is
$$\prod_{i\ge 0} n_i! \times \frac 1 {(n-1)!^2} \times (n-1)! = n \binom{n}{(n_i, i\ge 0)}^{-1}$$
\medskip
In these coalescing particle systems, one of the parameters of interest is the metric structure of the cluster (structure of the ``molecule'') eventually obtained after all particles have coalesced into a single component. In the unrestricted case, the metric structure is described by the CRT of Aldous. Our result shows that the quenched version, conditional on the degree sequence, is also valid under reasonable conditions on the degree sequence imposed. Results for Galton--Watson trees conditioned on the size only are recovered by sampling the degree sequence.
For instance, to recover the unrestricted version of the merging process, one can sample $n$ independent Poisson(1) random variables, and keep them if their sum equals $n-1$; the $n$ exchangeable values obtained are then the degrees $C_1,C_2,\dots, C_n$ of the $n$ particles.
\subsection*{Acknowledgements }
{ We are grateful to the referees for the many relevant remarks they made on the paper.}
{
\setlength{\bibsep}{.2em}
\small
\bibliographystyle{abbrvnat}
|
1,116,691,499,406 | arxiv | \section{Introduction}
Higher-spin (HS) gauge theory is a
theory of an infinite set of gauge fields of all spins. Since
gauge invariant HS interaction vertices contain higher derivatives
of degrees increasing with spin
\cite{Bengtsson:1983pd,Berends:1984wp,Fradkin:1987ks,Fradkin:1991iy},
HS gauge theory is not a local
field theory in the usual sense. Some arguments that HS gauge
theory has to be essentially non-local were given in
\cite{Sleight:2017pcz} based on the holographic correspondence
with the boundary (critical) sigma-model, conjectured by Klebanov
and Polyakov \cite{Klebanov:2002ja} (see also \cite{Sezgin:2002rt}).
To be free from the assumptions of holographic correspondence it
is important to analyze the issue of (non)locality of HS gauge
theory directly in the bulk. Based on the nonlinear HS equations of
\cite{Vasiliev:1990en,more} such analysis was performed in
\cite{Vasiliev:2016xui,Gelfond:2017wrh,Vas,4a1, 4a2}
in different
sectors of the theory at some lowest orders.
All vertices derived in these papers turned out to
be spin-local\footnote{Roughly speaking, spin-locality implies that the vertices are local for
any finite subset of fields of different spins. More precisely, this is literally the
case at the lowest interaction order but may need some further elaboration at higher orders.
For more detail on these issues we refer the reader to \cite{2a2}. } including some of the quintic vertices in the
Lagrangian counting. Also somewhat different arguments pointing out at
locality of the HS gauge theory were presented in a recent paper
\cite{David:2020ptn}.
The obtained vertices agree with holographic prediction at cubic
order \cite{Sezgin:2017jgm, Didenko:2017lsn}. However, the bulk
vertices derived from nonlinear HS equations so far did not
contain the $\phi^4$ vertex for the spin-zero field $\phi$. On the
other hand, it is this vertex \cite{Bekaert:2015tva} that was
argued to be highly non-local \cite{Sleight:2017pcz} in the HS
theory holographically dual to the boundary sigma-model
\cite{Klebanov:2002ja}. The degree of non-locality prescribed by
the analysis of \cite{Sleight:2017pcz} led the authors to a
conclusion of a fundamental failure for HS holographic
reconstruction programme beyond cubic order. Still, the same
vertex was also analyzed in \cite{Ponomarev:2017qab} concluding
that the non-locality if present is of a very special form.
The aim of this paper is to carry out holographically independent
approach to the locality problem. We do it by extending the
analysis of locality of \cite{Vas, 2a1, 4a1, 4a2, 2a2} to the
vertices of order $C^3$ in the sector of equations on the
zero-forms $C$ that contain in particular the $\phi^4$ vertex of
interest in the form of $\phi^3$ contribution to the field
equations. Note that the vertices studied in this paper include an
$AdS_4$ extension of those obtained by Metsaev in
\cite{Metsaev:1991mt}.
In this paper we give general arguments based on the so-called
$Z$-dominance Lemma of \cite{2a1} that the holomorphic, \ie
$\eta^2$, and antiholomorphic $\bar\eta^2$ vertices in HS gauge
theory must be spin-local, where $\eta$ is a complex parameter in
the HS equations. We explicitly demonstrate by direct calculation
that every individual contribution to the (anti)holomorphic part
of a quartic vertex acquires a form that results in a complete
spin-locality of the entire piece. While our analysis is sufficient to
see that the result is local it does not give directly the manifestly
local form of the remaining local vertex.
The derivation of the latter, which uses partial integrations and
Schouten identity, we leave for the future. The analysis of the
mixed $\eta\bar \eta$ vertex that is the only remaining sector in
the analysis of locality of the $\phi^4$ sector, needs somewhat
different tools and is beyond the scope of this paper.
The paper is organized as follows. In Section \ref{Section2}, the
necessary background on HS equations
is presented. Section \ref{Section3} contains brief
recollection on the so called limiting shifted homotopy and the
interpretation of the $Z$-dominance lemma via space $\Hp$
of star-product functions. In
Section \ref{RES}, we collect expressions for the holomorphic
vertices obtained from the generating equations.
Discussion of the obtained results and problems yet to be solved
is placed in Section \ref{Conclussion}. Some useful formulas are collected in Appendix A.
Appendices B and C
contain the detailed derivation of the third-order
contribution
$B_3^{\eta\eta}$ to the zero-form and second-order contribution
$W_2^{\eta\eta}$ to the one-form fields, respectively.
\section{Recollection of higher-spin equations}\label{Section2}
In the frame-like formalism \cite{Vasiliev:1980as}, unfolded equations for interacting HS fields in $AdS_4$
can be schematically put into the form \cite{Vasiliev:1988sa}
\begin{equation}\label{HSsketch1}
\dr_x \omega+\go\ast \go=\Upsilon(\go,\go,C)+\Upsilon(\go,\go,C,C)+\ldots,
\end{equation}
\begin{equation}\label{HSsketch2}
\dr_x C+\omega \ast C-C\ast \omega=\Upsilon(\go,C,C)+\Upsilon(\go,C,C,C)+\ldots\,,
\end{equation}
where HS fields are encoded in two generating functions, the one-form
\begin{equation}
\label{goc}
\omega(Y,x)=\dr x^\mu \omega_\mu(Y,x)=\sum_{n,m} \dr x^\mu\omega_{\mu\;\alpha_1 \dots \alpha_n,\dot{\alpha}_1\ \dots \dot{\alpha}_m}(x)y^{\alpha_1}\dots y^{\alpha_n}\bar{y}^{\dot{\alpha}_1}\dots \bar{y}^{\dot{\alpha}_m}\; ;\; m+n=2(s-1),
\end{equation}
and zero-form
\begin{equation}\label{Cfield}
C(Y,x)=\sum_{n,m}C_{\alpha_1 \dots \alpha_n,\dot{\alpha}_1\ \dots \dot{\alpha}_m}(x)y^{\alpha_1}\dots y^{\alpha_n}\bar{y}^{\dot{\alpha}_1}\dots \bar{y}^{\dot{\alpha}_m}\; ;\; |m-n|=2s\,
\end{equation}
with two-component indices $\ga\,, \dot \ga=1,2$.
$Y^A=(y^\alpha,\bar{y}^{\dot{\alpha}})$ is $sp(4)$ spinor. Field
components of definite $s$ are associated with spin-$s$ massless
fields, encoding the original Fronsdal field along with all its
on-shell nontrivial space-time derivatives. In (\ref{HSsketch1}),
(\ref{HSsketch2}) and in the sequel all products of the fields are
wedge products which is implicit. $\dr_x=\dr x^\mu\frac{\p}{\p
x^\mu}$ is space-time De Rham differential.
Star product is defined as follows
\begin{equation}
f(y,\bar{y})\ast g(y,\bar{y})=\int\frac{d^2u d^2v}{(2\pi)^2}\frac{d^2 \bar{u} d^2\bar{v}}{(2\pi)^2}e^{iu_\alpha v^\alpha+i\bar{u}_{\dot{\alpha}} \bar{v}^{\dot{\alpha}}}f(y+u,\bar{y}+\bar{u})g(y+v,\bar{y}+\bar{v}).
\end{equation}
The form of the vertices on \rhs of \eqref{HSsketch1} and \eqref{HSsketch2}
is determined by the consistency condition with $\dr_x^2=0$. This determines
the vertices up to field redefinitions
\be
\label{fr}
\go' = F(\go,C)\q C' = G(C)\,,
\ee
where $F(\go,C)$ is linear in the one-form $\go$, while both
$F(\go,C)$ and $G(C)$ can be nonlinear in $C$. Indeed, a field
redefinition in the consistent system produces another consistent
system. Since $\go$ and $C$ contain all on-shell nontrivial
derivatives of the Fronsdal fields, non-linear field redefinitions
(\ref{fr}) may contain infinite tails of higher derivatives thus
being non-local though having particular quasi-local form
expandable in power series in terms of components of (\ref{goc})
and (\ref{Cfield}) So, if system (\ref{HSsketch1}),
(\ref{HSsketch2}) is local or spin-local (for more detail see
\cite{2a2}) in some specific choice
of variables it may lose this property in other variables. Other
way around, if system (\ref{HSsketch1}), (\ref{HSsketch2}) is
non-local in some set of variables, this does not necessarily
imply that there is no other set of variables making the system
spin-local.
Direct computation of vertices consistent with a given locality
requirement from compatibility conditions is technically involved.
Indeed, deriving HS equations from formal consistency may
result in non-localities due to an inherent natural field
redefinition freedom (see \cite{Vasiliev89} for an earlier
account). To avoid such non-localities one has to impose extra
conditions on field variables that are {\it a priori} unknown. An
alternative scheme making the derivation of vertices much easier
and setting control on field variables is based on the
generating system of \cite{more}, that has the form
\be
\dr_x W+W*W=0\,,\label{HS1}
\ee
\be
\dr_x S+W*S+S*W=0\,,\label{HS2}
\ee
\be\dr_x B+[W,B]_*=0\,,\label{HS3}
\ee
\be S*S=i(\theta^{A} \theta_{A}+ B*\Gamma) \q \Gamma =\eta \gga +\bar \eta \bar\gga\,,
\label{HS4}\ee
\be
[S,B]_*=0\,,\label{HS5}
\ee
where master fields $W$, $S$ and $B$ depend on space-time
coordinates $x$ and commuting spinor coordinates $Y_A$ and
$Z_A=(z_{\al},\bar z_{\dal})$. In what follows the $x$--dependence
is implicit. In addition there is also a dependence on discrete
involutive Klein elements $K=(k, \bar k)$ such that
(to simplify formulae, in the sequel the star-product symbol $*$
is implicit in the products with Klein elements, \ie
$AK\equiv A*K $)
\be\label{hcom}
\{k,y_{\al}\}=\{k,z_{\al}\}=0\,,\qquad [k,\bar y_{\dal}]=[k,\bar
z_{\dal}]=0\,,\qquad k^2=1\q [k\,,\bar k]=0\,
\ee
and analogously for $\bar k$.
Star product $*$ acts on functions of $Y$ and $Z$ according to
\be\label{star}
(f*g)(Z, Y)=\ff{1}{(2\pi)^4}\int \dr^4 U \dr^4 V f(Z+U; Y+U)g(Z-V;
Y+V)\exp(iU_{A}V^{A})\,,
\ee
where $sp(4)$ indices $A,B,\dots$ are raised and lowered by the
antisymmetric form $\gep_{AB}=-\gep_{BA}$ as follows
$X^{A}=\gep^{AB}X_{B}$ and $X_A=X^{B}\gep_{BA}$. Master fields are
differential forms with respect to space-time differential
$\dr x^{\nu}$ and auxiliary spinor differential $\theta_A=(\theta_\al,
\bar\theta_{\dal})$ satisfying
\be
\{\theta_{A}\,,\theta_{B}\}=0\q
\{\theta_{\al},k\}=\{\bar\theta_{\dal},\bar k\}=0\,,\qquad
[\theta_{\al},\bar k]=[\bar\theta_{\dal},k]=0\,.
\ee
$B(Z,Y;K|x)$ is a zero-form, while $W(Z,Y;K|x)=\mathrm{d}x^{\mu} W_{\mu}(Z,Y;K|x)$ is a one-form in space-time differential and
$S(Z,Y;K|x)=\theta^{\al} S_{\al}(Z,Y;K|x)+\bar{\theta}^{\dal} \bar{S}_{\dal}(Z,Y;K|x)$ is a one-form in auxiliary spinor differentials. Finally,
$\gga$ and $\bar\gga$ are central two-forms attributed to the Klein operators
\be\label{klein}
\gga=\exp({iz_{\al}y^{\al}})k\theta^{\al} \theta_{\al}\,,\qquad
\bar\gga=\exp({i\bar{z}_{\dal}\bar{y}^{\dal}})\bar
k\bar\theta^{\dal}\bar\theta_{\dal}\,.
\ee
To see that they are central \cite{more}, one should use that $\theta^3 =\bar\theta^3=0$
and that the star-product elements
$\kappa:= \exp({iz_{\al}y^{\al}})$ and $\bar\kappa:=\exp({i\bar{z}_{\dal}\bar{y}^{\dal}})$
have the properties analogous to (\ref{hcom}) with respect to star products with
$f(Y,Z)$ but commute with the Klein operators and differentials $\theta^\ga$ and
$\bar{\theta}^{\dal}$.
\subsection{Perturbation theory}
A proper HS vacuum is the following exact solution of
\eqref{HS1}-\eqref{HS5}
\begin{align}
&B_0=0\,,\label{B0}\\
&S_0=\theta^\al z_{\al}+\bar{\theta}^{\dal}\bar
z_{\dal}\,,\label{S0}\\
&W_0=\go(Y|x)\,,\quad \dr_x\go+\go*\go=0\,.
\end{align}
The flat connection $\go$ can be chosen to describe $AdS_4$.
Since vacuum value of $S_0$ is non-trivial it is going to generate
via star (anti)commutators PDEs in $Z$ for master fields. Indeed, consider equation \eqref{HS5} in the first order
\begin{equation}
[S_0,B_1]_\ast+[S_1,B_0]_\ast=0\,.
\end{equation}
Using star product \eqref{star} one can check that
\begin{equation}
[Z_A,f(Z,Y;\theta)]_\ast=-2i\frac{\partial}{\partial Z^A}f(Z,Y;\theta).
\end{equation}
It means that $B_1$ field is $Z$ - independent
\begin{equation}
B_1(Z,Y)=C(Y)\,,
\end{equation}
which is the generating function for HS curvatures \eqref{Cfield}. Lower index in $B_1$, shows the order of the expression in the $C$-field. Correspondingly, $B_2$ is of second order in $C$ and so on
\begin{equation}\label{expansionB}
B=C+B_2(C,C)+B_3(C,C,C)+\ldots
\end{equation}
The same rule applies to $S$ and $W$, \ie
\begin{equation}\label{expansionSW}
S=S_0+S_1(C)+S_2(C,C)+\ldots\q W=\omega+W_1(\omega,C)+W_2(\go,C,C)+\ldots
\end{equation}
To solve for $Z$-dependence
of master fields at each perturbation order one has to solve equation of the form
\be\label{steq}
\mathrm{d}_{Z} f(Z;Y;\theta)=J(Z;Y;\theta)\q \dr_Z:=\theta^A \frac{\partial}{\partial Z^A}\,,
\ee
where $J$ originates from the lower-order terms and $f$ is either
$B$, $S$ or $W$. Let us note that it is vacuum solution of the
auxiliary field $S_0$ \eqref{S0} manifesting itself in operator
$\dr_Z$ that allows one to capture the field redefinition
ambiguity as its kernel. Different solutions can be obtained by
one or another contracting homotopy operator $\triangle$
\be
f=\triangle J\,
\ee
resulting from the standard homotopy trick. For an operator $\partial$, that should be also nilpotent,
$\p^2=0$, one considers operator
\begin{equation}
N=\dr_Z \partial+\partial \dr_Z.
\end{equation}
If $N$ is diagonalizable one can introduce the almost inverse
operator $N^\ast$. Since all the $\dr_Z$ - cohomologies are in the
kernel of $N$ one rewrites the solution to \eqref{steq} as
\begin{equation}
f=\partial N^\ast J=\triangle J.
\end{equation}
The simplest choice for $\partial$ is
\begin{equation}
\partial=Z^A\frac{\partial}{\partial \theta^A}\,.
\end{equation}
Then $N$ turns out to be an Euler operator and thus easily invertible
\begin{equation}
N=Z^A\frac{\partial}{\partial Z^A}+\theta^A \frac{\partial}{\partial \theta^A}\q N^\ast J(Z;Y;\theta)=\int_0^1\frac{dt}{t}J(tZ;Y;t\theta).
\end{equation}
This choice of $\partial$ leads to contracting homotopy operator
\bee\label{oldres}
\hmt_{0}J(Z;Y;\theta)&=&Z^{A}\ff{\p}{\p\theta^{A}}\int_{0}^{1}dt\ff1t
J(t Z; Y; t\theta)\,,
\eee
referred to as the {\it conventional} homotopy operator in
\cite{4a1}.
Using \eqref{HS4} and \eqref{S0} one finds in particular
\begin{equation}
-2i \dr_z S_1^\eta=i\eta C\ast \gamma=i\eta \theta^\alpha \theta_\alpha e^{iz_\alpha y^\alpha}
C(-z\,,\bar y)k\,.
\end{equation}
Using now \eqref{oldres} we find solution for $S_1^\eta$ in the
form
\begin{equation}\label{S1vvedenie}
S_1^\eta=\eta \theta^\alpha z_\alpha\int_0^1 dt\, t \exp\big\{itz_\alpha y^\alpha\big\}C(-tz,\bar y)k.
\end{equation}
Different choices of homotopy operators represent gauge and field
redefinition ambiguity. A particular class of the so-called
shifted homotopies can be defined by considering $Z^A-Q^A$ instead
of $Z^A$ with some $Z$-independent $Q^A$. Local properties of HS
vertices crucially depend on properties of the chosen homotopy
operators. A class of homotopy operators consistent with locality
requirement based on shifted homotopies at non-trivial interaction
level was proposed in \cite{4a2}. Let us also note that
another way of fixing the $Z$-dependence based on the so-called
gauge function method is reviewed in \cite{Iazeolla:2020jee}
(see also \cite{Aros:2019pgj} for applications to various backgrounds).
\subsubsection{Notation}
Let us set up our notation. Derivative with respect to holomorphic
argument of the $C$-fields is denoted as $\partial_{i\,\alpha}$
where index $i$ indicates position of the $C$-field in expression
that contains several $C$'s as seen from left to right. Derivative
with respect to holomorphic argument of $\omega$ - field is
denoted as $\partial_{\go\, \alpha}$.
Whenever arguments of $C$'s or $\go$ are not written explicitly we
assume the {\it exponential} form. This means the following:
suppose one has $\go CCC$. Then it should be understood as
\begin{equation}
\omega(\mathsf{y}_\go,\bar{y})\ust C(\mathsf{y}_1,\bar{y})\ust C(\mathsf{y}_2,\bar{y})\ust C(\mathsf{y}_3,\bar{y})\,,
\end{equation}
{ where $\ust$ denotes the star product with respect to the
barred variables.}
Derivatives $\partial_\omega$ and $\partial_i$ act as
\begin{equation}
\partial_{\go \, \alpha}=\frac{\partial}{\partial \mathsf{y}_\go^{\alpha}}\q \partial_{i\,\alpha}=\frac{\partial}{\partial \mathsf{y}_i^\alpha}
\end{equation}
followed by all the auxiliary variables set to zero, \ie
\begin{equation}
\mathsf{y}_\go=\mathsf{y}_i=0.
\end{equation}
To make contact with the $p,t$ notation of \cite{4a1} and
\cite{4a2} note that
\begin{equation}
t_\alpha=-i\partial_{\go \, \alpha}\q p_{i\, \alpha}=-i\partial_{i\,\alpha}.
\end{equation}
\section{ Limiting homotopy procedure, subspace $\Hp$ \\and $Z$-dominance lemma}\label{Section3}
The limiting shifted contracting homotopy was introduced in
\cite{4a2} as the generalization of the shifted homotopy
introduced in \cite{4a1}
\begin{multline}
\hmt_{q,\beta}f(z,y\vert \theta)=\\
=\int \frac{d^2u\, d^2 v}{(2\pi)^2}e^{iu_\alpha v^\alpha}\left(z^\alpha+q^\alpha +u^\alpha\right)
\frac{\partial}{\partial \theta^\alpha}\int_0^1 \frac{dt}{t}f(t z -(1-t)(q+u),y+\beta v\vert t \theta)\,,
\end{multline}
where $q^\ga$ is a $z$-independent spinorial shift parameter while
$\gb\in (-\infty , 1)$ is a free parameter. For simplicity we
confine ourselves to the holomorphic sector of undotted spinors
which is of most interest in this paper. (Antiholomorphic sector
of dotted spinors is analysed analogously.)
This operator satisfies the following resolution of identity
\begin{equation}
\label{1}
\dr_z \hmt_{q,\beta}+\hmt_{q,\beta}\dr_z=1-h_{q,\beta}\,,
\end{equation}
where
\begin{equation}\label{projector}
h_{q,\beta}f(z,y\vert \theta)=\int \frac{d^2 u d^2 v}{(2\pi)^2}\,
e^{iu_\alpha v^\alpha}f(-q-u,y+\beta v\vert 0)
\end{equation}
is the projector on $\dr_z$-cohomology.
We say that function $f(z,y)$ of the form
\begin{equation}
\label{class}
f(z,y\vert \theta)=\int_0^1 d\mathcal{T}\, e^{i\mathcal{T}z_\alpha y^\alpha}\phi
\left(\mathcal{T}z,y\vert \mathcal{T} \theta,\mathcal{T}\right)\,
\end{equation}
belongs to the space $\Hp$ if there exists
such real $\varepsilon>0$, that
\begin{equation}\label{limit}
\lim_{\mathcal{T}\rightarrow 0}\mathcal{T}^{1-\varepsilon}\phi(w,u\vert
\theta,\mathcal{T})=0\,.
\end{equation}
Note that the definition of space $\Hp$ is relaxed compared to
that of space $\Sp^{+0}$ of \cite{2a2} because it does not require
any specific behaviour of the $\phi$ at $\mathcal{T}\to1$.
Nevertheless in our calculations sometimes it is convenient to use
specific form degree relations of \cite{2a2} that describe star products
for the forms of specific degrees $p\,,p'$, belonged to spaces $\Sp_p^{0+}$ and $\Sp_{p'}^{+0}$.
There are two main options that appear in the computations below to satisfy \eqref{limit}:
\begin{equation}\label{kernels}
\phi_1(\mathcal{T}z,y\vert \mathcal{T} \theta,
\mathcal{T})=\frac{\mathcal{T}^{\delta_1}}{\mathcal{T}}\widetilde{\phi}_1(\mathcal{T}z,y\vert
\mathcal{T} \theta)\q \phi_2(\mathcal{T} z,y\vert
\mathcal{T}\theta,
\mathcal{T})=\theta(\mathcal{T}-\delta_2)\frac{1}{\mathcal{T}}\widetilde{\phi}_2(\mathcal{T}z,y\vert
\mathcal{T} \theta)
\end{equation}
with some $\delta_{1,2}>0$ and step-function $\theta(x)$. (Note that according to \cite{2a2} the
poles in $\mathcal{T}$ in (\ref{kernels}) are fictitious being
cancelled by the $\mathcal{T}$-dependence of $z$- and
$\theta$-dependent terms in \eq{class}). Functions $\widetilde{\phi}_i(\T
z,y\vert \T \theta)$ are formal power series in variables $\T z$
and $y$ as they are realized as star products of various powers of
$C$'s and $\omega$'s. Functions of the form $\phi_2$ result from the
decomposition $1=\theta(a-\epsilon) +\theta(\epsilon -a)$.
Space $\Hp$ can be represented as the
direct sum
\begin{equation}
\Hp=\Hp_0 \oplus \Hp_1 \oplus \Hp_2\,,
\end{equation}
where $\Hp_p$ are spanned by the degree-$p$ functions in $\theta$
with kernels that satisfy \eqref{limit}.
Equations of motion (\ref{HSsketch1}), (\ref{HSsketch2}) resulting
from nonlinear system (\ref{HS1})-(\ref{HS5}) have \rhss
independent of $Z^A$ and $\theta^A$ since they belong to the
sector of zero-forms in $\theta$ and are $\dr_Z$-closed as a
consequence of equations (\ref{HS1})-(\ref{HS5}) resolved at the
previous stages. On the other hand, various terms contributing to
the \rhss of equations (\ref{HSsketch1}), (\ref{HSsketch2}) as a
result of solution of equations (\ref{HS1})-(\ref{HS5}) are of the
form (\ref{class}). In particular, each of these terms is usually
$Z$-dependent. While \rhss of (\ref{HSsketch1}),
(\ref{HSsketch2}) are $Z$-independent as a consequence of
equations (\ref{HS1})-(\ref{HS5}), the fact that
the sum of all of them is $Z$-independent is not obvious,
demanding an appropriate partial integrations over homotopy
parameters that appear at various stages of the order-by-order
analysis of nonlinear HS equations. After all, functions
(\ref{class}) can be $Z$-independent only if they have a
distributional measure supported at $\mathcal{T}=0$, \ie after
appropriate partial integrations the measure contains a factor of
$\delta(\mathcal{T})$. Such a measure has dimension $-1$ in
$\mathcal{T}$. If a function contains an additional factor of
$\mathcal{T}^\gvep$, it cannot contribute to the $Z$-independent
answer. This just means that functions of the class $\Hp_0$ cannot
contribute to the $Z$-independent equations (\ref{HSsketch1}),
(\ref{HSsketch2}). This is the content of $Z$-dominance Lemma of
\cite{2a1}: any terms in $\phi(w,u\vert \theta, \mathcal{T}) $
dominated by a positive power of $\mathcal T$ do not contribute to
the dynamical equations (\ref{HSsketch1}), (\ref{HSsketch2}).
Application of this fact to locality is straightforward once it is
shown that all terms containing
infinite towers of higher derivatives in the vertices of interest
belong to $\Hp_0$ and, therefore, do not contribute to HS
equations (\ref{HSsketch2}). This is what is shown in this paper.
A related fact is that, as shown in \cite{4a2}, the space $\Hp$ exhibits special properties under the action of the
limiting shifted homotopy $\hmt_{q,\beta}$ at $\gb\to-\infty$ leading to
local HS interactions. Namely, it maps $\Hp_1$ to $\Hp_0$ \cite{2a2},
\begin{equation}\label{classSave}
\lim_{\beta \rightarrow -\infty} \hmt_{q,\beta} f_1(z,y\vert \theta)=f_0(z,y\vert 0)
\q \forall f_1\in \Hp_1\q f_0 \in \Hp_0\,.
\end{equation}
Since elements of $\Hp_0$ do not contribute to $Z$-independent
physical vertex by $Z$-dominance lemma, this property allows us to discard all terms from
$\Hp$ in the analysis of the $\go C^3$ vertex in equation
(\ref{HSsketch2}).
More in detail,
to prove that the vertex is spin-local we find
it most efficient to represent the \rhs
of equations in the $\dr_Z$--exact spin-local form modulo terms from $\Hp$ that do not contribute to
the final vertex by $Z$-dominance lemma.
In other words, to solve equation (\ref{steq}) we represent $J$ on its \rhs in the form
\be
\label{ex}
J=\dr_Z \widehat{f} +J^+\q J^+\in \Hp\,,
\ee
where $\widehat{f}$ is spin-local. Since $\dr_Z J=0$ and $\dr_Z^2=0$, we conclude that
\be
\dr_Z J^+=0\,.
\ee
We now note that for the sum of two $\dr_Z$-closed terms one can
use independent contracting homotopies for each of the summands.
Hence, we can write for $f$ on the \lhs of (\ref{steq})
\be
\label{fex}
f=\widehat{f} + \lim_{\gb\to-\infty}\hmt_{q,\gb} J^+
\ee
with some $q$. Since $J^+\in \Hp$, with this definition the contribution of $J^+$
will give zero to the dynamical field equations for any $q$ while $\widehat{f}$ will give a
spin-local contribution. Details of the derivation of the decomposition (\ref{ex}) in the $B$ and $W$ sectors are given in Appendices B and C, respectively.
To put it differently, to eliminate $Z$-dependence of a
seemingly $Z$-dependent expression we manage to show that the
vertices $\Upsilon^{\eta\eta}(\go,C,C,C)$ and
$\Upsilon^{\bar{\eta}\bar{\eta}}(\go,C,C,C)$ are spin-local modulo
terms from $\Hp$ that vanish by $Z$-dominance lemma, \ie
\be\label{schemUps}\qquad \Upsilon(Y)= \widehat{\Upsilon}(Z,Y)+ \Upsilon_+(Z,Y)\q \Upsilon_+\in \Hp\,,\ee
where $\widehat{\Upsilon}$ is spin-local but $Z$-dependent expression.
Now we observe that since the physical vertex $ \Upsilon(Y)$ is $\theta$, $Z$-independent it can be written in
the form
\be
\Upsilon(Y)= \widehat{\Upsilon}(Z,Y)+ \Upsilon_+(Z,Y) = h_{q,\gb} (\widehat{\Upsilon}(Z,Y)+ \Upsilon_+(Z,Y))
\ee
where $ h_{q,\gb}$ is the cohomology projector with any shift parameters $q$ and $\gb$ (recall that,
as is obvious from (\ref{projector}), cohomology
projectors leave $\theta, Z$-independent functions invariant \cite{4a2}). Taking the limit
$\gb\to-\infty$ we find that, for any $q$, components of
$\widehat{\Upsilon}(Z,Y)$ can contribute to the resulting vertex $\Upsilon(Y)$ while $\Upsilon_+(Z,Y)$ cannot
since the limiting projector $h_{q,-\infty}$ \eqref{projector} acts trivially on
$\Hp_0$ \cite{2a2},
\begin{equation}
\label{coh}
\lim_{\beta \rightarrow -\infty} h_{q,\beta} f_0(Z,Y\vert 0)=0\q f_0 \in \Hp_0\,.
\end{equation}
This implies that
\be h_{q,-\infty}(\widehat{\Upsilon}(Z,Y)+ \Upsilon_+(Z,Y))=h_{q,-\infty}(\widehat{\Upsilon}(Z,Y) )\,.
\ee
This formula provides an alternative interpretation of $Z$-dominance Lemma stating that
elements of $\Hp_0$ do not contribute to the physical $Z$-independent vertices though
practically, its application with any $q$ may not be useful since
in most cases the result has a seemingly non-local form.
On the other hand, by $Z$-dominance lemma, a spin-local vertex $\widehat{\Upsilon}(Z,Y)$ must be
decomposable into a sum
\be
\label{ll}
\widehat{\Upsilon}(Z,Y)=\Upsilon^{loc}(Y) + \Upsilon^{loc}_+(Z,Y) \q \Upsilon^{loc}_+\in \Hp
\ee
with a spin-local $Z$-independent $\Upsilon^{loc}(Y)$.
Hence one can conclude that $\Upsilon^{loc}_+(Z,Y)+\Upsilon
_+(Z,Y)=0$ which fact is not at all manifest being a consequence
of Schouten identity and
various relations between integrals over homotopy parameters.
Of course, once $\Upsilon^{loc}(Y)$ is found, the application of
$h_{q,-\infty}$ with any $q$ to the \rhs of \eq{schemUps} gives a
spin-local vertex $\Upsilon (Y)=\Upsilon^{loc}(Y)$. It should be
stressed again however that, though one can formally obtain the final
result via application of any cohomology projector, it will be
spin-local, but not manifestly spin-local, containing non-local
contributions that will all cancel by virtue of partial
integrations and Schouten identity used in the derivation of the
decomposition (\ref{ll}) which is practically easier to find. As
will be demonstrated in the
forthcoming paper \cite{GelKor}, being technically involved,
this approach makes it possible to compute explicit form of the
physical spin-local vertices.
\section{Final results}\label{RES}
\subsection{General structure of equations}
Dynamical equations up to the third order in the zero-forms $C$ can be schematically
put into the form
\begin{equation}\label{EOMsch}
\dr_x C+[\go,C]_\ast=\Upsilon^\eta(\go,C,C)+\Upsilon^{\bar{\eta}}(\go,C,C)+\Upsilon^{\eta\eta}(\go,C,C,C)+\Upsilon^{\bar{\eta}\bar{\eta}}(\go,C,C,C)+\Upsilon^{\eta\bar{\eta}}(\go,C,C,C)+\ldots
\end{equation}
The vertex $\Upsilon^{\eta\eta}(\go,C,C,C)$ resulting
from system \eqref{HS1}-\eqref{HS5} has the form
\begin{equation}\label{rhs}
\Upsilon^{\eta\eta}(\go,C,C,C)=-\dr_x B_3^{\eta\eta}-\dr_x B_2^{\eta}-
[\go,B_3^{\eta\eta}]_\ast-[W_1^\eta,B_2^\eta]_\ast-[W_2^{\eta\eta},C]_\ast\,.
\end{equation}
Here $W^\eta_1,W_2^{\eta\eta}$ and $B_2^\eta, B_3^{\eta\eta}$ are master fields of the corresponding orders from expansions \eqref{expansionB}, \eqref{expansionSW} which are to be obtained from the generating system via solving equation of the type \eqref{steq}. Each term on the \rhs of this equation depends both on $Y$ and on $Z$. These vertices can be decomposed into two parts
\begin{equation}\label{upsz}
\Upsilon^{\eta\eta}(\go,C,C,C)=\widehat{\Upsilon}^{\eta\eta}(\go,C,C,C)+
{\Upsilon}_+^{\eta\eta}(\go,C,C,C)\q
{\Upsilon}_+^{\eta\eta}(\go,C,C,C)\in \Hp_0\,,
\end{equation}
where unlike the whole $\Upsilon^{\eta\eta}(\go,C,C,C)$ its two
contributions on the right of \eqref{upsz} can be $z$--dependent.
In this paper we compute the
$\widehat{\Upsilon}^{\eta\eta}(\go,C,C,C)$ part of the vertices.
This part turns out to be free from
contractions between holomorphic variables of the $C$-fields because such terms
belong to $\Hp_0$. Consistency of
equations \eqref{HS1}-\eqref{HS5}
guarantees that $\Upsilon^{\eta\eta}(\go,C,C,C)$ is $Z$-independent
and, according to $Z$-dominance Lemma, it can be realized only as $\delta(\T)$ in the kernel.
Hence $Z$-independent expression for $\Upsilon^{\eta\eta}(\go,C,C,C)$ must be free from infinite tower of contractions
between holomorphic variables which implies spin-locality of the resulting HS equations.
In this section we present final expression for $\widehat{\Upsilon}^{\eta\eta}(\go,C,C,C)$
\begin{equation}
\label{V}
\widehat{\Upsilon}^{\eta\eta}(\go,C,C,C)=\widehat{\Upsilon}^{\eta\eta}_{\go CCC}
+\widehat{\Upsilon}_{C\go CC}^{\eta\eta}+\widehat{\Upsilon}_{CC\go C}^{\eta\eta}
+\widehat{\Upsilon}_{CCC\go}^{\eta\eta}
\end{equation}
obtained from the generating system \eqref{HS1}-\eqref{HS5} using the perturbation
scheme up to the third order in $C$-field. Details of their derivation are presented
in Appendices B and C.
The vertices in (\ref{V}) are composed from the following terms
\begin{equation}\label{goCCC}
\widehat{\Upsilon}^{\eta\eta}_{\go CCC}\approx-\dr_x \widehat{B}_3^{\eta\eta}\Big|_{\go CCC}
- \omega\ast \widehat{B}_3^{\eta\eta}-\dr_x B_2^{\eta\, loc}\Big|_{\go CCC}-W^{\eta}_{1\, \go C}\ast
B_2^{\eta\, loc}-\widehat{W}_{2\, \go CC}^{\eta\eta}\ast C,
\end{equation}
\begin{equation}
\widehat{\Upsilon}_{C\go CC}^{\eta\eta}\approx-\dr_x \widehat{B}_3^{\eta\eta}\Big|_{C\go CC}
-\dr_x B_2^{\eta\, loc}\Big|_{C\go CC}-W^{\eta}_{1\, C\go}\ast B_2^{\eta\, loc}
-\widehat{W}_{2\, C\go C}^{\eta\eta}\ast C+C\ast \widehat{W}_{2\, \go CC}^{\eta\eta},
\end{equation}
\begin{equation}
\widehat{\Upsilon}^{\eta\eta}_{CC\go C}\approx-\dr_x \widehat{B}_3^{\eta\eta}\Big|_{CC\go C}-\dr_x
B_2^{\eta\, loc}\Big|_{CC\go C}+B_2^{\eta\, loc}\ast W^\eta_{1\, \go C}
-W^{\eta\eta}_{2\, CC\go}\ast C+C\ast W^{\eta\eta}_{2\, C\go C},
\end{equation}
\begin{equation}\label{CCCgo}
\widehat{\Upsilon}_{CCC\go}^{\eta\eta}\approx-\dr_x
\widehat{B}_3^{\eta\eta}\Big|_{CCC\go}+\widehat{B}_3^{\eta\eta}\ast
\go-\dr_x B_2^{\eta\, loc}\Big|_{CCC\go}+B_2^{\eta\, loc}\ast
W_{1\, C\go}^{\eta}+C\ast \widehat{W}_{2\, CC\go}^{\eta\eta}.
\end{equation}
The expression for $B_2^{\eta\, loc}$
has the form \cite{Vas}
\begin{multline}\label{B2loc}
B_2^{\eta\, loc}=\frac{\eta}{2}\int d^3\tau_+ \big[\delta^\prime (1-\sum_{i=1}^{3}\tau_i)
-iz_\alpha y^\alpha \delta(1-\sum_{i=1}^{3}\tau_i)\big]\exp({i\tau_1\,z_\alpha y^\alpha
+i\tau_1 \partial_{1\alpha}\partial_2 {}^\alpha})\times\\
C(-\tau_1 z+\tau_2 y,\bar{y})\ust C(-\tau_1 z-\tau_3 y,\bar{y})k\,,
\end{multline}
where we use a short-hand notation
\be
d^3\tau_+:= d\tau_1 d\tau_2 d\tau_3 \theta(\tau_1) \theta(\tau_2) \theta(\tau_3)\,.
\ee
Whenever this notation is used there is always a delta-function in a corresponding expression $\delta(1-\tau_1-\tau_2-\tau_3)$ that bounds the $\tau_i$ variables from above.
Note that
$B_2^{\eta loc}$ is a sum of $B_2^\eta$ obtained in \cite{4a1} and
local cohomology (\ie $Z$-independent) shift $\delta B_2^\eta$
\bee\label{redef}
B_2^{\eta loc}&=&B_2^\eta+\delta B_2^\eta\q \\ \nn
\delta B_2^\eta&=&\frac{\eta}{2}\int d^2\tau_+
\delta(1-\tau_1-\tau_2)C(\tau_1 y,\bar{y})\ust C(-\tau_2y,\bar{y})k.
\eee
The expressions $W_{1\, \go C}^{\eta}$ and $W_{1\, C\go}^\eta$ were obtained in \cite{4a1}, having the form
\begin{multline}\label{W1goCeta}
W_{1\, \omega C}^\eta=-\frac{\eta}{2}\int_0^1 d\tau_1 \int_0^1 d\sigma\, (1-\tau_1)
\left(z^\alpha \partial_{\omega \alpha}\right)\exp\Big\{ i\tau_1\, z_\alpha y^\alpha
+i(1-(1-\tau_1)\sigma)\partial_{\omega\alpha}\partial_1 {}^\alpha\Big\}\times \\
\omega(-\tau_1 z+(1-\tau_1)\sigma y,\bar{y})\ust C(-\tau_1 z,\bar{y})k\,,
\end{multline}
\begin{multline}\label{W1Cgoeta}
W_{1\,C\omega}^\eta=-\frac{\eta}{2}\int_0^1 d\tau_1 \int_0^1 d\sigma\,
(1-\tau_1)\left(z^\alpha \partial_{\omega \alpha}\right)\exp\Big\{i\tau_1 z_\alpha y^\alpha
+i(1-(1-\tau_1)\sigma)\partial_{1\alpha}\partial_{\go}{}^\alpha\Big\}\times\\
C(-\tau_1 z,\bar{y})\ust \go(-\tau_1 z-(1-\tau_1)\sigma y,\bar{y})k\,.
\end{multline}
Note that the terms with $\dr_x$ contribute to the third order via the second-order
contribution to $\dr_x C$
\be\label{C2EOM}
\dr_x C = C\ast \omega -\omega \ast C+\Upsilon^\eta_{\go CC}+\delta \Upsilon^\eta_{\go CC}
+\Upsilon^{\eta}_{C\go C}+\delta\Upsilon^{\eta}_{C\go C}+\Upsilon^{\eta}_{CC\go}
+\delta \Upsilon^{\eta}_{CC\go}\,.
\ee
Here $\Upsilon^\eta_{\go CC}, \Upsilon^{\eta}_{C\go C}$ and
$\Upsilon^{\eta}_{CC\go}$ are vertices obtained in \cite{4a1} and
$\delta\Upsilon^\eta_{\go CC}, \delta\Upsilon^{\eta}_{C\go
C},\delta\Upsilon^{\eta}_{CC\go}$ result from the local field
redefinition of $B_2^\eta$
\eq{redef} giving
\begin{multline}\label{VgoCC}
\Upsilon_{\go CC}^\eta +\delta \Upsilon_{\go CC}^\eta=-\frac{\eta}{2}\int d^3\tau_+\,
\delta\left(1-\sum_{i=1}^3 \tau_i\right)\Big(y^\alpha \partial_{\omega \alpha}\Big)
\exp\Big\{i(1-\tau_2)\partial_{\omega \alpha}\partial_1^\alpha
-i\tau_2 \partial_{\omega \alpha}\partial_2^\alpha\Big\}\times\\
\times \omega\big((1-\tau_3)y,\bar{y}\big)\ust C\big(\tau_1 y,\bar{y}\big)\ust C\big((\tau_1-1)y,\bar{y}\big)k,
\end{multline}
\begin{multline}\label{VCCgo}
\Upsilon_{CC\go}^\eta+\delta \Upsilon_{CC\go}^\eta=-\frac{\eta}{2}\int d^3\tau_+
\,\delta\left(1-\sum_{i=1}^3 \tau_i\right)\Big(y^\alpha \partial_{\omega \alpha}\Big)
\exp\Big\{i(1-\tau_2)\partial_{2 \alpha}\partial_{\go}^\alpha
-i\tau_2\partial_{1\alpha} \partial_{\go}^\alpha\Big\}
\times\\\times C\big((1-\tau_1)y,\bar{y}\big)\ust C\big(-\tau_1 y,\bar{y}\big)
\ust \omega\big((\tau_3-1)y,\bar{y}\big)k,
\end{multline}
\begin{multline}\label{VCgoC}
\Upsilon^\eta_{C\go C}+\delta \Upsilon^\eta_{C\go C}=-\frac{\eta}{2}\int d^3 \tau_+\,
\delta\left(1-\sum_{i=1}^3 \tau_i\right)\Big(y^\alpha \partial_{\omega \alpha}\Big)
\exp\Big\{i\tau_2 \partial_{1\alpha}\partial_{\go}^\alpha+i(1-\tau_2)\partial_{\omega \alpha}\partial_2^\alpha
\Big\}
\times\\
\times C\big(\tau_3 y,\bar{y}\big)\ust \omega\big(-\tau_1 y,\bar{y}\big)\ust C\big((\tau_3-1)y,\bar{y}\big)k\\
-\frac{\eta}{2}\int d^3 \tau_+\, \delta\left(1-\sum_{i=1}^3 \tau_i\right)
\Big(y^\alpha \partial_{\omega \alpha}\Big)\exp\Big\{i(1-\tau_2)\partial_{1\alpha}\partial_{\go}^\alpha
+i\tau_2 \partial_{\omega \alpha}\partial_2^\alpha\Big\}\times\\
\times C\big((1-\tau_3)y,\bar{y}\big)\ust \omega\big(\tau_1 y,\bar{y}\big)\ust C\big(-\tau_3 y,\bar{y}\big)k.
\end{multline}
\subsection{The fields}
The expressions for $\widehat{B}^{\eta\eta}_3$ and $\widehat{W}^{\eta\eta}_2$ derived in Appendices B and С are
\begin{multline}\label{B3final}
\widehat{B}_{3}^{\eta\eta}=-\frac{\eta^2}{4} \int_0^1 d\mathcal{T}\, \mathcal{T} \int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right) \int_0^1 d\xi\, \frac{\rho_1\, (z_\alpha y^\alpha)^2 }{(\rho_1+\rho_2)(\rho_1+\rho_3)}\times\\
\exp\Big\{i\mathcal{T}\, z_\alpha y^\alpha+\mathcal{T} z^\alpha\Big(-(\rho_1+\rho_3)\partial_{1\alpha}+(\rho_2-\rho_3)\partial_{2\alpha}+(\rho_1+\rho_2)\partial_{3\alpha}\Big)\\
+(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1\alpha}-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}\right)+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}\partial_{3\alpha}-\frac{\rho_3}{\rho_1+\rho_3}\partial_{2\alpha}\right)\Big\}CCC\,,
\end{multline}
\begin{multline}\label{W2CCgofinal}
\widehat{W}_{2\, CC\go}^{\eta\eta}=\frac{\eta^2}{4}\int_0^1 d\mathcal{T}\, \mathcal{T}
\int d^4 \rho_+\, \delta\left(1-\sum_{i=1}^4 \rho_i\right)
\frac{\rho_1\left(z^\gamma \partial_{\go \gamma}\right)^2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\times\\
\times\exp\Big\{i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha
\Big(-(\rho_1+\rho_2)\partial_{1 \alpha}+(\rho_3+\rho_4)\partial_{2 \alpha}+(1-\rho_2)\partial_{\go \alpha}\Big)\\
+\frac{\rho_1\rho_3}{(\rho_1+\rho_2)(\rho_3+\rho_4)}y^\alpha \partial_{\go \alpha}
+i\left(\frac{(1-\rho_4)\rho_2}{\rho_1+\rho_2}+\rho_4\right)\partial_{2 \alpha}\partial_{\go} {}^\alpha
-i\frac{\rho_1\rho_4}{\rho_3+\rho_4}\partial_{1 \alpha}\partial_{\go} {}^\alpha\Big\}CC\go,
\end{multline}
\begin{multline}\label{W2goCCfinal}
\widehat{W}_{2\, \go CC}^{\eta\eta}=\frac{\eta^2}{4}\int_0^1 d\mathcal{T}\,\T
\int d^4\rho_+\, \delta\left(1-\sum_{i=1}^4 \rho_i\right)
\frac{\rho_1 \left(z^\gamma \partial_{\go \gamma}\right)^2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\times\\
\times \exp\Big\{i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big((1-\rho_2)\partial_{\go\alpha}-(\rho_3+\rho_4)\partial_{1\alpha}+(\rho_1+\rho_2)\partial_{2 \alpha}\Big)\\
+\frac{\rho_1\rho_3}{(\rho_1+\rho_2)(\rho_3+\rho_4)}y^\alpha \partial_{\go \alpha}
+i\left(\frac{(1-\rho_4)\rho_2}{\rho_1+\rho_2}+\rho_4\right)\partial_{\go \alpha}\partial_1 {}^\alpha
-i\frac{\rho_4\rho_1}{\rho_3+\rho_4}\partial_{\go \alpha}\partial_2 {}^\alpha\Big\}\go CC,
\end{multline}
\begin{multline}\label{W2CgoCfinal}
\widehat{W}_{2\, C\go C}^{\eta \eta}=-\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}
\int d^4\rho_+\, \delta\left(1-\sum_{i=1}^4 \rho_i\right)
\frac{(\rho_1+\rho_3)\left(z^\gamma\partial_{\go \gamma}\right)^2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\times\\
\times \exp\Big\{i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha
\Big(-(\rho_3+\rho_4)\partial_{1 \alpha}+(\rho_1-\rho_3)\partial_{\go \alpha}
+(\rho_1+\rho_2)\partial_{2 \alpha}\Big)
\\-\frac{\rho_3\rho_1}{(\rho_1+\rho_2)(\rho_3+\rho_4)}y^\alpha \partial_{\go\alpha}
+i\frac{\rho_4(1-\rho_2)}{\rho_3+\rho_4}\partial_{\go \alpha}\partial_{2} {}^\alpha
+i\frac{\rho_2(1-\rho_4)}{\rho_1+\rho_2}\partial_{1 \alpha}\partial_\go {}^\alpha\Big\}C\go C.
\end{multline}
From now on we skip antiholomorphic
(barred) variables for brevity. More precisely,
$CCC$ on the \rhs of \eq{B3final}
is to be understood as $C(\mathsf{y}_1, \by)\ust C (\mathsf{y}_2, \by)\ust C(\mathsf{y}_3, \by)
\big|_{\mathsf{y}_i=0}$,
$\go CC$ on the \rhs of \eq{W2goCCfinal}
as $\go(\mathsf{y}_\go, \by)\ust C(\mathsf{y}_1, \by)\ust C (\mathsf{y}_2, \by)
\big|_{\mathsf{y}_\go=\mathsf{y}_i=0}$ {\it etc}.
Expressions \eqref{B3final}-\eqref{W2CgoCfinal} are spin-local
because the exponential factors in all of them are free from terms
$\p_{i\ga}\p_j^\ga$ describing contractions between higher
components of the zero-forms $C(Y)$ bringing higher-derivative
vertices for fields of particular spins. So are the terms induced
by these expressions in vertices \eqref{goCCC}-\eqref{CCCgo}.
Indeed, differentiating $\widehat{B}_3^{\eta\eta}$ one should use
only first-order part from \rhs of \eqref{C2EOM} which does not
bring contractions between $C$-fields, similarly star product with
$\omega$ does not bring contractions due to \eqref{YL},
\eqref{YR}. On the other hand, though star product of
$\widehat{W}_2^{\eta\eta}$ with $C$ does bring contractions
between the fields $C$, all of them result from the $z$-dependent
terms in the exponentials (\ref{W2CCgofinal})-(\ref{W2CgoCfinal})
that carry at least one power of $\mathcal{T}$. Such terms contain
an additional factor of $\T$ in front of the contraction terms
$\p_{i\ga}\p_j^\ga$ thus belonging to $\Hp$. Hence all the
contributions to the vertex (\ref{rhs}) induced from
$B_3^{\eta\eta}$ and $W_2^{\eta\eta}$ are spin-local modulo terms
in $\Hp$.
\subsection{Equations}
\subsubsection{$B_3$ driven terms}
Direct computation of the $B_3$ induced terms using \eqref{C2EOM},
\eqref{B3final} and \eqref{YL}, \eqref{YR} gives
\begin{multline}
\dr_x \widehat{B}_3^{\eta\eta}\big|_{\go
CCC}\approx\frac{\eta^2}{4} \int_0^1 d\mathcal{T}\, \mathcal{T}
\int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right) \int_0^1
d\xi\, \frac{\rho_1\, (z_\alpha y^\alpha)^2
e^{i\mathcal{T}\, z_\alpha y^\alpha} }{(\rho_1+\rho_2)(\rho_1+\rho_3)}\times\\
\times\exp\Big\{\mathcal{T} z^\alpha
\Big(-(\rho_1+\rho_3)(\partial_{\go\alpha}+\partial_{1\alpha})+(\rho_2-\rho_3)\partial_{2\alpha}
+(\rho_1+\rho_2)\partial_{3\alpha}\Big)+i\partial_{\go \alpha}\partial_1 {}^\alpha\\
+(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}(\partial_{\go \alpha}+\partial_{1\alpha})-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}\right)+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}\partial_{3\alpha}-\frac{\rho_3}{\rho_1+\rho_3}\partial_{2\alpha}\right)\Big\}\go CCC,
\end{multline}
\begin{multline}
\dr_x \widehat{B}_3^{\eta\eta}\big|_{C\go CC}\approx-\frac{\eta^2}{4} \int_0^1 d\mathcal{T}\, \mathcal{T} \int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right) \int_0^1 d\xi\, \frac{\rho_1\, (z_\alpha y^\alpha)^2 e^{i\mathcal{T}\, z_\alpha y^\alpha} }{(\rho_1+\rho_2)(\rho_1+\rho_3)}\times\\
\times\exp\Big\{\mathcal{T} z^\alpha\Big(-(\rho_1+\rho_3)(\partial_{\go\alpha}+\partial_{1\alpha})+(\rho_2-\rho_3)\partial_{2\alpha}+(\rho_1+\rho_2)\partial_{3\alpha}\Big)+i\partial_{1 \alpha}\partial_\go {}^\alpha\\
+(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}(\partial_{\go \alpha}+\partial_{1\alpha})-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}\right)+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}\partial_{3\alpha}-\frac{\rho_3}{\rho_1+\rho_3}\partial_{2\alpha}\right)\Big\} C\go CC\\
+\frac{\eta^2}{4} \int_0^1 d\mathcal{T}\, \mathcal{T} \int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right) \int_0^1 d\xi\, \frac{\rho_1\, (z_\alpha y^\alpha)^2 e^{i\mathcal{T}\, z_\alpha y^\alpha} }{(\rho_1+\rho_2)(\rho_1+\rho_3)}\times\\
\times\exp\Big\{\mathcal{T} z^\alpha\Big(-(\rho_1+\rho_3)\partial_{1\alpha}+(\rho_2-\rho_3)(\partial_{\go\alpha}+\partial_{2\alpha})+(\rho_1+\rho_2)\partial_{3\alpha}\Big)+i\partial_{\go \alpha}\partial_2 {}^\alpha\\
+(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1\alpha}-\frac{\rho_2}{\rho_1+\rho_2}(\partial_{\go\alpha}+\partial_{2\alpha})\right)\\
+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}\partial_{3\alpha}-\frac{\rho_3}{\rho_1+\rho_3}(\partial_{\go \alpha}+\partial_{2\alpha})\right)\Big\} C\go CC,
\end{multline}
\begin{multline}
\dr_x \widehat{B}_3^{\eta\eta}\big|_{CC\go C}\approx-\frac{\eta^2}{4} \int_0^1 d\mathcal{T}\, \mathcal{T} \int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right) \int_0^1 d\xi\, \frac{\rho_1\, (z_\alpha y^\alpha)^2 e^{i\mathcal{T}\, z_\alpha y^\alpha} }{(\rho_1+\rho_2)(\rho_1+\rho_3)}\times\\
\times\exp\Big\{\mathcal{T} z^\alpha\Big(-(\rho_1+\rho_3)\partial_{1\alpha})+(\rho_2-\rho_3)(\partial_{\go \alpha}+\partial_{2\alpha})+(\rho_1+\rho_2)\partial_{3\alpha}\Big)+i\partial_{2 \alpha}\partial_\go {}^\alpha\\
+(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1\alpha}-\frac{\rho_2}{\rho_1+\rho_2}(\partial_{\go \alpha}+\partial_{2\alpha})\right)+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}\partial_{3\alpha}-\frac{\rho_3}{\rho_1+\rho_3}(\partial_{\go \alpha}+\partial_{2\alpha})\right)\Big\} CC\go C\\
+\frac{\eta^2}{4} \int_0^1 d\mathcal{T}\, \mathcal{T} \int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right) \int_0^1 d\xi\, \frac{\rho_1\, (z_\alpha y^\alpha)^2 e^{i\mathcal{T}\, z_\alpha y^\alpha} }{(\rho_1+\rho_2)(\rho_1+\rho_3)}\times\\
\times\exp\Big\{\mathcal{T} z^\alpha\Big(-(\rho_1+\rho_3)\partial_{1\alpha}+(\rho_2-\rho_3)\partial_{2\alpha}+(\rho_1+\rho_2)(\partial_{\go\alpha}+\partial_{3\alpha})\Big)+i\partial_{\go \alpha}\partial_3 {}^\alpha\\
+(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1\alpha}-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}\right)
+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}(\partial_{\go \alpha}+\partial_{3\alpha})-\frac{\rho_3}{\rho_1+\rho_3}\partial_{2\alpha}\right)\Big\} CC\go C,
\end{multline}
\begin{multline}
\dr_x \widehat{B}_3^{\eta\eta}\big|_{ CCC\go}\approx-\frac{\eta^2}{4} \int_0^1 d\mathcal{T}\, \mathcal{T} \int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right) \int_0^1 d\xi\, \frac{\rho_1\, (z_\alpha y^\alpha)^2 e^{i\mathcal{T}\, z_\alpha y^\alpha} }{(\rho_1+\rho_2)(\rho_1+\rho_3)}\times\\
\times\exp\Big\{\mathcal{T} z^\alpha\Big(-(\rho_1+\rho_3)\partial_{1\alpha}+(\rho_2-\rho_3)\partial_{2\alpha}+(\rho_1+\rho_2)(\partial_{\go \alpha}+\partial_{3\alpha})\Big)+i\partial_{3 \alpha}\partial_\go {}^\alpha\\
+(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1\alpha}-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}\right)+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}(\partial_{\go \alpha}+\partial_{3\alpha})-\frac{\rho_3}{\rho_1+\rho_3}\partial_{2\alpha}\right)\Big\}CCC\go.
\end{multline}
\begin{multline}
\omega\ast \widehat{B}_3^{\eta\eta}\approx-\frac{\eta^2}{4} \int_0^1 d\mathcal{T}\, \mathcal{T} \int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right) \int_0^1 d\xi\, \frac{\rho_1\, \left[z_\alpha\left(y^\alpha-i\partial_{\go} {}^\alpha\right)\right]^2 e^{i\mathcal{T}\, z_\alpha y^\alpha} }{(\rho_1+\rho_2)(\rho_1+\rho_3)}\times\\
\times\exp\Big\{\mathcal{T} z^\alpha\Big(-\partial_{\go \alpha}-(\rho_1+\rho_3)\partial_{1\alpha}+(\rho_2-\rho_3)\partial_{2\alpha}+(\rho_1+\rho_2)\partial_{3\alpha}\Big)+y^\alpha\partial_{\go \alpha}\\
+(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1\alpha}-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}\right)+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}\partial_{3\alpha}-\frac{\rho_3}{\rho_1+\rho_3}\partial_{2\alpha}\right)\\
+i\frac{(1-\xi)\rho_1}{\rho_1+\rho_2}\partial_{\go \alpha}\partial_1 {}^\alpha-i\left(\frac{(1-\xi)\rho_2}{\rho_1+\rho_2}+\frac{\xi\rho_3}{\rho_1+\rho_3}\right) \partial_{\go \alpha}\partial_2 {}^\alpha+i\frac{\xi\rho_1}{\rho_1+\rho_3}\partial_{\go \alpha}\partial_3 {}^\alpha\Big\} \go CCC,
\end{multline}
\begin{multline}
\widehat{B}_3^{\eta\eta}\ast \go \approx-\frac{\eta^2}{4} \int_0^1 d\mathcal{T}\, \mathcal{T} \int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right) \int_0^1 d\xi\, \frac{\rho_1\, \left[z_\alpha\left(y^\alpha+i\partial_{\go} {}^\alpha\right)\right]^2 e^{i\mathcal{T}\, z_\alpha y^\alpha} }{(\rho_1+\rho_2)(\rho_1+\rho_3)}\times\\
\times\exp\Big\{\mathcal{T} z^\alpha\Big(\partial_{\go \alpha}-(\rho_1+\rho_3)\partial_{1\alpha}+(\rho_2-\rho_3)\partial_{2\alpha}+(\rho_1+\rho_2)\partial_{3\alpha}\Big)+y^\alpha\partial_{\go \alpha}\\
+(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1\alpha}-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}\right)+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}\partial_{3\alpha}-\frac{\rho_3}{\rho_1+\rho_3}\partial_{2\alpha}\right)\\
+i\frac{(1-\xi)\rho_1}{\rho_1+\rho_2}\partial_{1 \alpha}\partial_\go {}^\alpha-i\left(\frac{(1-\xi)\rho_2}{\rho_1+\rho_2}+\frac{\xi\rho_3}{\rho_1+\rho_3}\right) \partial_{2 \alpha}\partial_\go {}^\alpha+i\frac{\xi\rho_1}{\rho_1+\rho_3}\partial_{3 \alpha}\partial_\go {}^\alpha\Big\} CCC\go.
\end{multline}
\subsubsection{$B_2$ driven terms}
The terms resulting from $\dr_x$ differentiation of $B_2$ and its multiplication with $W_1$
by virtue of \eqref{W1goCeta}, \eqref{W1Cgoeta}, \eqref{C2EOM} and \eqref{starPr} give
\begin{multline}
\dr_x B^{\eta\, loc}_2\big|_{\go CCC}\approx-\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}
\int_0^1 d\xi\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3\rho_i\right)\left(z_\alpha y^\alpha\right)
\Big[\left(\mathcal{T}z^\alpha-\xi y^\alpha\right)\partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{i\mathcal{T}z_\alpha y^\alpha+i(1-\rho_2)\partial_{\go \alpha}\partial_1 {}^\alpha
-i\rho_2 \partial_{\go \alpha}\partial_2 {}^\alpha+\mathcal{T}z^\alpha
\Big(-(\rho_1+\rho_2)\partial_{\go \alpha}
-\rho_1\partial_{1 \alpha}+(\rho_2+\rho_3)\partial_{2 \alpha}+\partial_{3\alpha}\Big)\\
+y^\alpha\Big(\xi(\rho_1+\rho_2)\partial_{\go \alpha}
+\xi\rho_1\partial_{1 \alpha}-\xi(\rho_2+\rho_3)\partial_{2 \alpha}+(1-\xi)\partial_{3\alpha}\Big) \Big\}\go CCC\,,
\end{multline}
\begin{multline}
\dr_x B^{\eta\, loc}_2 \big|_{C\go CC}\approx
-\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}\int_0^1 d\xi\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3\rho_i\right)
\left(z_\alpha y^\alpha\right)\Big[(\mathcal{T}z^\alpha-\xi y^\alpha)\partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i\rho_2\partial_{1 \alpha}\partial_{\go} {}^\alpha
+i(1-\rho_2)\partial_{\go \alpha}\partial_{2} {}^\alpha
+\mathcal{T}z^\alpha\Big(-\rho_3 \partial_{1 \alpha}+\rho_1 \partial_{\go \alpha}
+(\rho_1+\rho_2)\partial_{2 \alpha}+\partial_{3\alpha}\Big)\\
+y^\alpha\Big(\xi \rho_3 \partial_{1 \alpha}-\xi\rho_1 \partial_{\go \alpha}
-\xi(\rho_1+\rho_2)\partial_{2 \alpha}+(1-\xi)\partial_{3\alpha}\Big)\Big\}C \go CC\\
-\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}\int_0^1 d\xi
\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(z_\alpha y^\alpha\right)
\Big[(\mathcal{T}z^\alpha-\xi y^\alpha)\partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{ i\mathcal{T}z_\alpha y^\alpha+i(1-\rho_2)\partial_{1 \alpha}\partial_{\go} {}^\alpha
+i\rho_2 \partial_{\go \alpha}\partial_{\go} {}^\alpha
+\mathcal{T}z^\alpha\Big(-(\rho_1+\rho_2)\partial_{1 \alpha}-\rho_1\partial_{\go\alpha}
+\rho_3\partial_{2 \alpha}+\partial_{3\alpha}\Big)\\
+y^\alpha\Big(\xi(\rho_1+\rho_2)\partial_{1 \alpha}
+\xi\rho_1 \partial_{\go \alpha}-\xi\rho_3 \partial_{2 \alpha}+(1-\xi)\partial_{3\alpha}\Big)\Big\}C\go CC\\
-\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}\int_0^1 d\xi\int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)
\left(z_\alpha y^\alpha\right)\Big[ \left(\mathcal{T}z^\alpha+ (1-\xi)y^\alpha\right)
\partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha +i(1- \rho_2)
\partial_{\go \alpha}\partial_{2} {}^\alpha
-i\rho_2 \partial_{\go \alpha} \partial_3 {}^\alpha
+\mathcal{T}z^\alpha\Big(-\partial_{1 \alpha}-(\rho_1+\rho_2)\partial_{\go \alpha}
-\rho_1 \partial_{2 \alpha}+(\rho_2+\rho_3)\partial_{3\alpha}\Big)\\
+y^\alpha\Big(\xi\partial_{1\alpha}-(1-\xi)(\rho_1+\rho_2)\partial_{\go \alpha}
-(1-\xi)\rho_1 \partial_{2 \alpha}+(1-\xi)(\rho_2+\rho_3)\partial_{3\alpha}\Big)\Big\}C\go CC\,,
\end{multline
\begin{multline}
\dr_x B^{\eta\, loc}_2\big|_{CC\go C}\approx-\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}
\int_0^1 d\xi \int d^3\rho_+ \, \delta\left(1-\sum_{i=1}^3\rho_i\right)\left(z_\alpha y^\alpha\right)
\Big[\left(\mathcal{T}z^\alpha-\xi y^\alpha\right)\partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i(1-\rho_2)\partial_{2\alpha}\partial_{\go} {}^\alpha
-i\rho_2 \partial_{1 \alpha}\partial_{\go} {}^\alpha+\mathcal{T}z^\alpha
\Big(-(\rho_1+\rho_2)\partial_{1 \alpha}+\rho_3 \partial_{2 \alpha}+(\rho_2+\rho_3)\partial_{\go \alpha}
+\partial_{3\alpha}\Big)\\
+y^\alpha\Big(\xi(\rho_1+\rho_2)\partial_{1 \alpha}-\xi\rho_3 \partial_{2 \alpha}
-\xi(\rho_2+\rho_3)\partial_{\go \alpha}+(1-\xi)\partial_{3\alpha}\Big)\Big\}CC\go C\\
-\frac{i\eta^2}{4}\int_0^1 d\mathcal{T} \int_0^1 d\xi \int d^3\rho_+ \,
\delta\left(1-\sum_{i=1}^{3}\rho_i\right)\left(z_\alpha y^\alpha\right)
\Big[\left(\mathcal{T}z^\alpha +(1-\xi)y^\alpha\right)\partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i\rho_2\partial_{2\alpha}\partial_{\go} {}^\alpha
+i(1-\rho_2)\partial_{\go \alpha}\partial_3 {}^\alpha+\mathcal{T}z^\alpha
\Big(-\partial_{1 \alpha}-\rho_3 \partial_{2 \alpha}+\rho_1\partial_{\go \alpha}
+(\rho_1+\rho_2)\partial_{3\alpha}\Big)\\
+y^\alpha \Big(\xi \partial_{1 \alpha}-(1-\xi)\rho_3\partial_{2 \alpha}
+(1-\xi)\rho_1\partial_{\go \alpha}+(1-\xi)(\rho_1+\rho_2)\partial_{3\alpha}\Big)\Big\}CC\go C\\
-\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}\int_0^1 d\xi\int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)
\left(z_\alpha y^\alpha\right)\Big[\left(\mathcal{T}z^\alpha+(1-\xi)y^\alpha\right)\partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i(1-\rho_2)\partial_{2 \alpha}\partial_{\go} {}^\alpha
+i\rho_2 \partial_{\go \alpha}\partial_3 {}^\alpha+\mathcal{T}z^\alpha
\Big(-\partial_{1 \alpha}-(\rho_1+\rho_2)\partial_{2 \alpha}
-\rho_1\partial_{\go \alpha}+\rho_3 \partial_{3\alpha}\Big)\\
+y^\alpha\Big(\xi\partial_{1 \alpha}-(1-\xi)(\rho_1+\rho_2)\partial_{2 \alpha}
-(1-\xi)\rho_1\partial_{\go \alpha}+(1-\xi)\rho_3\partial_{3\alpha}\Big)\Big\}CC\go C\,,
\end{multline}
\begin{multline}
\dr_x B_2^{\eta\, loc}\big|_{CCC\go}\approx-\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}\int_0^1 d\xi
\int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3\rho_i\right)\left(z_\alpha y^\alpha\right)
\Big[\left(\mathcal{T}z^\alpha+(1-\xi)y^\alpha\right)\partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha +i(1-\rho_2)\partial_{3\alpha}\partial_{\go} {}^\alpha
-i\rho_2 \partial_{2 \alpha}\partial_{\go} {}^\alpha
+\mathcal{T}z^\alpha\Big(-\partial_{1 \alpha}-(\rho_2+\rho_3)\partial_{2 \alpha}
+\rho_1 \partial_{3\alpha}+(\rho_1+\rho_2)\partial_{\go \alpha}\Big)\\
+y^\alpha\Big(\xi\partial_{1 \alpha}-(1-\xi)(\rho_2+\rho_3)\partial_{2 \alpha}
+(1-\xi)\rho_1 \partial_{3\alpha}+(1-\xi)(\rho_1+\rho_2)\partial_{\go \alpha}\Big)\Big\}CCC\go\,,
\end{multline}
\begin{multline}
W_{1\, \go C}^\eta \ast B_2^{\eta\, loc}\approx\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}\T \int_0^1 d\sigma
\int d^3\rho_+\, \frac{\delta\left(1-\sum_{i=1}^3 \rho_i\right)}{\rho_1+\rho_2}\left(z^\gamma \partial_{\go \gamma}\right)\Big[z_\alpha y^\alpha+i\sigma z^\alpha \partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i(1-\sigma)\partial_{\go \alpha}\partial_1 {}^\alpha
-i\frac{\rho_1\sigma}{\rho_1+\rho_2} \partial_{\go \alpha}\partial_{2}{}^\alpha+i\frac{\rho_2\sigma}{\rho_1+\rho_2} \partial_{\go\alpha}\partial_3 {}^\alpha\\
+\mathcal{T}z^\alpha\Big(-(\rho_1+\rho_2+\sigma \rho_3)\partial_{\go \alpha}-(\rho_1+\rho_2)\partial_{1 \alpha}+(\rho_3-\rho_1)\partial_{2 \alpha}+(\rho_3+\rho_2)\partial_{3\alpha}\Big)\\
+y^\alpha\Big(\sigma\partial_{\go \alpha}-\frac{\rho_1}{\rho_1+\rho_2}\partial_{2 \alpha}+\frac{\rho_2}{\rho_1+\rho_2}\partial_{3\alpha}\Big)\Big\}\go CCC\,,
\end{multline}
\begin{multline}
W_{1\, C\go}^\eta \ast B_2^{\eta \, loc}\approx\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}\T
\int_0^1 d\sigma \int d^3\rho_+\, \frac{\delta\left(1-\sum_{i=1}^3 \rho_i\right)}{\rho_1+\rho_2}
\left(z^\gamma \partial_{\go \gamma}\right)\Big[z_\alpha y^\alpha+i\sigma z^\alpha \partial_{\go \alpha}\Big]\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha
+i(1-\sigma)\partial_{1 \alpha}\partial_\go {}^\alpha
-i\frac{\rho_1\sigma}{\rho_1+\rho_2} \partial_{\go \alpha}\partial_{2}{}^\alpha
+i\frac{\rho_2\sigma}{\rho_1+\rho_2} \partial_{\go\alpha}\partial_3 {}^\alpha\\
+\mathcal{T}z^\alpha\Big(-(\rho_1+\rho_2+\sigma \rho_3)\partial_{\go \alpha}
-(\rho_1+\rho_2)\partial_{1 \alpha}+(\rho_3-\rho_1)\partial_{2 \alpha}+(\rho_3+\rho_2)\partial_{3\alpha}\Big)\\
+y^\alpha\Big(\sigma\partial_{\go \alpha}-\frac{\rho_1}{\rho_1+\rho_2}\partial_{2 \alpha}
+\frac{\rho_2}{\rho_1+\rho_2}\partial_{3\alpha}\Big)\Big\}C \go CC\,,
\end{multline}
\begin{multline}
B_2^{\eta\, loc}\ast W_{1\, \go C}^\eta\approx\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}\T
\int_0^1 d\sigma \int d^3\rho_+\, \frac{\delta\left(1-\sum_{i=1}^3 \rho_i\right)}{\rho_1+\rho_2}
\Big[z_\alpha y^\ga-i\sigma z^\alpha \partial_{\go\alpha}\Big]\left(z^\gamma \partial_{\go \gamma}\right)\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i(1-\sigma)\partial_{\go \alpha}\partial_3 {}^\alpha
-i\frac{\rho_1 \sigma}{\rho_1+\rho_2}\partial_{1 \alpha}\partial_{\go} {}^\alpha
+i\frac{\rho_2 \sigma}{\rho_1+\rho_2}\partial_{2 \alpha}\partial_{\go} {}^\alpha\\
+\mathcal{T}z^\alpha\Big(-(\rho_3+\rho_1)\partial_{1 \alpha}-(\rho_3-\rho_2)\partial_{2 \alpha}
+(\rho_1+\rho_2-\sigma\rho_3)\partial_{\go \alpha}+(\rho_1+\rho_2)\partial_{3\alpha}\Big)\\
+y^\alpha\Big(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1 \alpha}-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}
-\sigma\partial_{\go\alpha}\Big)\Big\}CC\go C\,,
\end{multline}
\begin{multline}
B_2^{\eta\, loc}\ast W_{1\, C\go}^\eta\approx\frac{i\eta^2}{4}\int_0^1 d\mathcal{T}\T
\int_0^1 d\sigma \int d^3\rho_+\, \frac{\delta\left(1-\sum_{i=1}^3 \rho_i\right)}{\rho_1+\rho_2}
\Big[z_\alpha y^\ga-i\sigma z^\alpha \partial_{\go\alpha}\Big]\left(z^\gamma \partial_{\go \gamma}\right)\times\\
\times \exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i(1-\sigma)\partial_{3 \alpha}\partial_\go {}^\alpha
-i\frac{\rho_1 \sigma}{\rho_1+\rho_2}\partial_{1 \alpha}\partial_{\go} {}^\alpha
+i\frac{\rho_2 \sigma}{\rho_1+\rho_2}\partial_{2 \alpha}\partial_{\go} {}^\alpha\\
+\mathcal{T}z^\alpha\Big(-(\rho_3+\rho_1)\partial_{1 \alpha}-(\rho_3-\rho_2)\partial_{2 \alpha}+(\rho_1+\rho_2
-\sigma\rho_3)\partial_{\go \alpha}+(\rho_1+\rho_2)\partial_{3\alpha}\Big)\\
+y^\alpha\Big(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1 \alpha}-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}
-\sigma\partial_{\go\alpha}\Big)\Big\}CCC\go.
\end{multline}
\subsubsection{$W_2$ driven terms}
Terms resulting from star product with $W_2^{\eta\eta}$ are
\begin{multline}
C\ast\widehat{W}_{2\, \go CC}^{\eta\eta}\approx\frac{\eta^2}{4}
\int_0^1 d\mathcal{T}\,\T \int d^4\rho_+\, \delta\left(1-\sum_{i=1}^4 \rho_i\right)
\frac{\rho_1\left(z^\gamma \partial_{\go \gamma}\right)^2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\times\\
\times \exp\Big\{i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha
\Big(-\partial_{1 \alpha}+(1-\rho_2)\partial_{\go\alpha}-(\rho_3+\rho_4)\partial_{2\alpha}
+(\rho_1+\rho_2)\partial_{3 \alpha}\Big)+y^\alpha\partial_{1 \alpha}\\
+\frac{\rho_1\rho_3}{(\rho_1+\rho_2)(\rho_3+\rho_4)}
\left(y^\alpha \partial_{\go \alpha}+i\partial_{1 \alpha}\partial_{\go}{}^\alpha\right)
+i\left(\frac{(1-\rho_4)\rho_2}{\rho_1+\rho_2}+\rho_4\right)\partial_{\go \alpha}\partial_1 {}^\alpha
-i\frac{\rho_4\rho_1}{\rho_3+\rho_4}\partial_{\go \alpha}\partial_2 {}^\alpha\Big\}C \go CC,
\end{multline}
\begin{multline}
\widehat{W}_{2\, \go CC}^{\eta\eta}\ast C\approx\frac{\eta^2}{4}
\int_0^1 d\mathcal{T}\,\T \int d^4\rho_+\, \delta\left(1-\sum_{i=1}^4 \rho_i\right)
\frac{\rho_1 \left(z^\gamma \partial_{\go \gamma}\right)^2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\times\\
\times \exp\Big\{i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big((1-\rho_2)\partial_{\go\alpha}
-(\rho_3+\rho_4)\partial_{1\alpha}+(\rho_1+\rho_2)\partial_{2 \alpha}+\partial_{3 \alpha}\Big)
+y^\alpha\partial_{3 \alpha}\\
+\frac{\rho_1\rho_3}{(\rho_1+\rho_2)(\rho_3+\rho_4)}
\left(y^\alpha \partial_{\go \alpha}+i\partial_{\go \alpha}\partial_{3}{}^\alpha\right)
+i\left(\frac{(1-\rho_4)\rho_2}{\rho_1+\rho_2}+\rho_4\right)\partial_{\go \alpha}\partial_1 {}^\alpha
-i\frac{\rho_4\rho_1}{\rho_3+\rho_4}\partial_{\go \alpha}\partial_2 {}^\alpha\Big\} \go CC C,
\end{multline}
\begin{multline}
C\ast \widehat{W}_{2\, CC\go}^{\eta\eta}\approx\frac{\eta^2}{4}\int_0^1 d\mathcal{T}\, \mathcal{T}
\int d^4 \rho_+\, \delta\left(1-\sum_{i=1}^4 \rho_i\right)
\frac{\rho_1\left(z^\gamma \partial_{\go \gamma}\right)^2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\times\\
\times\exp\Big\{i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big(-\partial_{1 \alpha}
-(\rho_1+\rho_2)\partial_{2 \alpha}+(\rho_3+\rho_4)\partial_{3 \alpha}+(1-\rho_2)\partial_{\go \alpha}\Big)
+y^\alpha\partial_{1 \alpha}\\
+\frac{\rho_1\rho_3}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\left(y^\alpha \partial_{\go \alpha}
+i\partial_{1 \alpha}\partial_{\go} {}^\alpha\right)
+i\left(\frac{(1-\rho_4)\rho_2}{\rho_1+\rho_2}+\rho_4\right)\partial_{2 \alpha}\partial_{\go} {}^\alpha
-i\frac{\rho_1\rho_4}{\rho_3+\rho_4}\partial_{1 \alpha}\partial_{\go} {}^\alpha\Big\}CCC\go,
\end{multline}
\begin{multline}
\widehat{W}_{2\, CC\go}^{\eta\eta}\ast C\approx\frac{\eta^2}{4}\int_0^1 d\mathcal{T}\, \mathcal{T}\int d^4 \rho_+\, \delta\left(1-\sum_{i=1}^4 \rho_i\right)\frac{\rho_1\left(z^\gamma \partial_{\go \gamma}\right)^2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\times\\
\times\exp\Big\{i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big(-(\rho_1+\rho_2)\partial_{1 \alpha}+(\rho_3+\rho_4)\partial_{2 \alpha}+(1-\rho_2)\partial_{\go \alpha}+\partial_{3 \alpha}\Big)+y^\alpha\partial_{3 \alpha}\\
+\frac{\rho_1\rho_3}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\left(y^\alpha \partial_{\go \alpha}+i\partial_{\go \alpha}\partial_{3} {}^\alpha\right)+i\left(\frac{(1-\rho_4)\rho_2}{\rho_1+\rho_2}+\rho_4\right)\partial_{2 \alpha}\partial_{\go} {}^\alpha-i\frac{\rho_1\rho_4}{\rho_3+\rho_4}\partial_{1 \alpha}\partial_{\go} {}^\alpha\Big\}CC\go C,
\end{multline}
\begin{multline}
C\ast \widehat{W}_{2\, C\go C}^{\eta \eta}\approx-\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}
\int d^4\rho_+\, \delta\left(1-\sum_{i=1}^4 \rho_i\right)
\frac{(\rho_1+\rho_3)\left(z^\gamma\partial_{\go \gamma}\right)^2 }{(\rho_1+\rho_2)(\rho_3+\rho_4)}\times\\
\times \exp\Big\{i\mathcal{T}z_\alpha y^\alpha
+\mathcal{T}z^\alpha\Big(-\partial_{1 \alpha}-(\rho_3+\rho_4)\partial_{2 \alpha}
+(\rho_1-\rho_3)\partial_{\go \alpha}+(\rho_1+\rho_2)\partial_{3 \alpha}\Big)
+y^\alpha \partial_{1 \alpha}\\-\frac{\rho_3\rho_1}{(\rho_1+\rho_2)(\rho_3+\rho_4)}
\left(y^\alpha \partial_{\go\alpha}+i\partial_{1 \alpha}\partial_{\go} {}^\alpha\right)
+i\frac{\rho_4(1-\rho_2)}{\rho_3+\rho_4}\partial_{\go \alpha}\partial_{2} {}^\alpha
+i\frac{\rho_2(1-\rho_4)}{\rho_1+\rho_2}\partial_{1 \alpha}\partial_\go {}^\alpha\Big\}CC\go C,
\end{multline}
\begin{multline}
\widehat{W}_{2\, C\go C}^{\eta \eta}\ast C\approx-\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}
\int d^4\rho_+\, \delta\left(1-\sum_{i=1}^4 \rho_i\right)
\frac{(\rho_1+\rho_3)\left(z^\gamma\partial_{\go \gamma}\right)^2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\times\\
\times \exp\Big\{i\mathcal{T}z_\alpha y^\alpha
+\mathcal{T}z^\alpha\Big(-(\rho_3+\rho_4)\partial_{1 \alpha}+(\rho_1-\rho_3)\partial_{\go \alpha}
+(\rho_1+\rho_2)\partial_{2 \alpha}+\partial_{3 \alpha}\Big)+y^\alpha \partial_{3 \alpha}
\\-\frac{\rho_3\rho_1}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\left(y^\alpha \partial_{\go\alpha}
+i\partial_{\go \alpha}\partial_3 {}^\alpha\right)
+i\frac{\rho_4(1-\rho_2)}{\rho_3+\rho_4}\partial_{\go \alpha}\partial_{2} {}^\alpha
+i\frac{\rho_2(1-\rho_4)}{\rho_1+\rho_2}\partial_{1 \alpha}\partial_\go {}^\alpha\Big\}C\go CC.
\end{multline}
In the end of this section let us stress again that all terms on
the \rhs of vertex (\ref{rhs}) are free from $C$-field
contractions $\p_{i\ga}\p_j^\ga$ in the exponentials, hence being
spin-local. This is the central result of this paper.
\section{Conclusion}\label{Conclussion}
In this paper we have analyzed the $\go C^3$ vertices in the
equation for the zero-form $C$ (\ref{HSsketch2}) in the
holomorphic $\eta^2$ sector, showing that these vertices are
spin-local in the terminology of \cite{2a2}. In particular, they
contain the holomorphic part of the $\phi^4$ vertex in the
Lagrangian nomenclature for a spin-zero scalar field $\phi$. This
is another step in the analysis of locality of HS gauge theory
performed in \cite{4a1,4a2}. To complete the analysis of
spin-locality of the HS gauge theory at quartic order it remains
to extend these results to the mixed $\eta\bar \eta$ sector. This
problem differs in some respects from the (anti)holomorphic one
and will be analyzed elsewhere.
On the other hand, there are remaining problems even in the
holomorphic sector left unsolved. The most important one is to
find explicit $Z$-independent local form of the holomorphic vertex
$\go C^3$. The naive attempt to set $Z=0$ in the vertex obtained
in this paper does not necessarily lead to correct result since
the omitted terms in $\Hp_0$ are needed for consistency of the
equations and may contribute to the sector of equations. Indeed,
setting $Z=0$ corresponds to the application of the conventional
homotopy projector which does not eliminate the part of the vertex
in $\Hp$. Let us stress again in this regard that the elaborated
technique based on dropping off terms from $\Hp_0$ turns out to be
highly efficient for checking out spin-locality. To obtain
explicit form of these vertices there are two alternative ways of
the analysis.
One is to eliminate the $Z$-dependence from the vertex by direct partial integration.
Being technically involved and not at all obvious due to the
need of using Schouten identity and partial integrations, this program is realised
at least for a particular vertex in the forthcoming paper \cite{GelKor}.
Another is to apply the limiting shifted homotopy procedure with
appropriately chosen shift when solving for HS fields. Note that
the $Z$-dependence for HS master fields has been found using no
shifted homotopies in our paper. It would be interesting to
understand if the locality is reached within well elaborated contracting
homotopy approach. Since the choice of homotopy shift and hence cohomology
projector via resolution of identity (\ref{1}), (\ref{projector}) affects field redefinitions that can
themselves be non-local the art is to find a shift that makes the
result manifestly spin-local. This is an interesting problem for the
future.
To summarize, the results of this paper indicate that equations of motion of
HS gauge theory have a tendency of being spin-local. At this stage it is
crucially important to see whether this property extends to the mixed
$\eta\bar\eta \go C^3$ sector of equation (\ref{HSsketch2}) which is the most
urgent problem on the agenda.
\newcounter{appendix}
\setcounter{appendix}{1}
\renewcommand{\theequation}{\Alph{appendix}.\arabic{equation}}
\addtocounter{section}{1} \setcounter{equation}{0}
\renewcommand{\thesection}{\Alph{appendix}.}
\addcontentsline{toc}{section}{\,\,\,\,\,\,\,Appendix A. Useful formulas}
\section*{Acknowledgments}
This work was supported by
the Russian Science Foundation grant 18-12-00507.
\section*{Appendix A. Useful formulas}\label{Appendix}
Useful multiplication formula for the star product of functions of the form
\begin{equation}
f_j(z,y)=\int_0^1 d\tau_j \exp {i(\tau_j\, z_\alpha y^\alpha)} \phi_j(\tau_j z,(1-\tau_j)y\vert \tau_j \theta,\tau_j)
\end{equation}
is \cite{Vasiliev:2015wma}
\begin{multline}\label{starPr}
f_1\ast f_2 (z,y)=\int_0^1 d\tau_1 \int_0^1 d\tau_2 \int e^{iu_\alpha v^\alpha}\,
\exp {i( \tau_1 \circ \tau_2 z_\alpha y^\alpha)}\times\\
\phi_1\Big(\tau_1\big[(1-\tau_2)z-\tau_2 y+u\big],(1-\tau_1)\big[(1-\tau_2)y-\tau_2 z+u\big]\big| \tau_1 \theta,\tau_1\Big)\times\\
\phi_2\Big(\tau_2\big[(1-\tau_1)z+\tau_1 y-v\big]\big],(1-\tau_2)\big[(1-\tau_1)y+\tau_1 z+v\big]\big| \tau_2 \theta,\tau_2\Big)\,.
\end{multline}
For instance, if one function is $z$-independent the following formulas are handy in star-product computation
\begin{equation}\label{YL}
f(y)\ast \Gamma(z,y)=f(y)\Gamma(z+i\overleftarrow{\partial}_f,y-i\overleftarrow{\partial}_f)\, ,
\end{equation}
\begin{equation}\label{YR}
\Gamma(z,y)\ast f(y)=\Gamma(z+i\partial_f,y+i\partial_f)f(y)\,.
\end{equation}
\bigskip
Second-order zero-form vertices are \cite{4a1}
\begin{multline}
\Upsilon^\eta_{\go CC}=-\frac{i\eta}{2}\int d^3\tau_+\, \delta\left(1-\sum_{i=1}^3 \tau_i\right)
\partial_{\omega\alpha}\left(\partial_1^\alpha+\partial_2^\alpha\right)
\exp\Big\{i(1-\tau_2)\partial_{\omega \alpha}\partial_1^\alpha
-i\tau_2 \partial_{\omega \alpha}\partial_2^\alpha\Big\}\\
\times \omega\big((1-\tau_3)y,\bar{y}\big)\ust C\big(\tau_1 y,\bar{y}\big)\ust C\big((\tau_1-1)y,\bar{y}\big)k,
\end{multline}
\begin{multline}
\Upsilon_{CC\go}^\eta=-\frac{i\eta}{2}\int d^3\tau_+ \,\delta\left(1-\sum_{i=1}^3 \tau_i\right)
\partial_{\omega\alpha}\left(\partial_1^\alpha+\partial_2^\alpha\right)
\exp\Big\{i(1-\tau_1)\partial_{2 \alpha}\partial_{\go}^\alpha
-i\tau_1\partial_{1\alpha} \partial_{\go}^\alpha\Big\}\\
\times C\big((1-\tau_2)y,\bar{y}\big)\ust C\big(-\tau_2 y,\bar{y}\big)\ust \omega\big((\tau_3-1)y,\bar{y}\big)k,
\end{multline}
\begin{multline}
\Upsilon^\eta_{C\go C}=-\frac{i\eta}{2}\int d^3 \tau_+\, \delta\left(1-\sum_{i=1}^3 \tau_i\right)
\partial_{\omega\alpha}\left(\partial_1^\alpha+\partial_2^\alpha\right)
\exp\Big\{i\tau_3 \partial_{1\alpha}\partial_{\go}^\alpha
+i(1-\tau_3)\partial_{\omega \alpha}\partial_2^\alpha\Big\}\\
\times C\big(\tau_2 y,\bar{y}\big)\ust \omega\big(-\tau_1 y,\bar{y}\big)\ust C\big((\tau_2-1)y,\bar{y}\big)k\\
-\frac{i\eta}{2}\int d^3 \tau_+\, \delta\left(1-\sum_{i=1}^3 \tau_i\right)\partial_{\omega\alpha}
\left(\partial_1^\alpha+\partial_2^\alpha\right)\exp\Big\{i(1-\tau_2)\partial_{1\alpha}\partial_{\go}^\alpha
+i\tau_2 \partial_{\omega \alpha}\partial_2^\alpha\Big\}\\
\times C\big((1-\tau_3)y,\bar{y}\big)\ust \omega\big(\tau_1 y,\bar{y}\big)\ust C\big(-\tau_3 y,\bar{y}\big)k\,.
\end{multline}
\addtocounter{appendix}{1}
\renewcommand{\theequation}{\Alph{appendix}.\arabic{equation}}
\addtocounter{section}{1} \setcounter{equation}{0}
\addcontentsline{toc}{section}{\,\,\,\,\,\,\,Appendix B. $B_3^{\eta\eta}$}
\section*{Appendix B. $B_3^{\eta\eta}$}
\label{SecB3}
Computation of $B^{\eta\eta}_3$ goes as follows.
Equation for $B_3^{\eta\eta}$ from \eqref{HS5} is
\begin{equation}
2i\dr_z B_3^{\eta\eta}=[S_1^\eta,B_2]_\ast+[S_2^{\eta\eta},C]_\ast\,.
\end{equation}
An important observation of Section 6.2 of \cite{4a2} based on the
technique of re-ordering operators $O_{\gb}f(z,y)$ was that
if $S_2^{\eta\eta}$
is computed using $B_2^{\eta\, loc}$ \eqref{B2loc}, then the
contribution to the vertices $ \Upsilon^{\eta\eta}(\go,\go,C,C)$
from $S^{\eta\eta}_2$ vanishes at $\gb\to-\infty$. Proceeding analogously one can see that
contribution to the vertices $ \Upsilon^{\eta\eta}(\go,C,C,C)$ from such $S_2^{\eta\eta}$ also
vanishes at $\gb\to-\infty$.
Hence to find the part of $B_3^{\eta\eta}$ that contributes to
$\widehat{\Upsilon}^{\eta\eta}$ one has to solve the equation
\begin{equation}
\dr_z \widehat{B}_3^{\eta\eta}=\frac{i}{2}[B_2^{\eta\, loc},S_1^\eta]_\ast.
\end{equation}
$S_1^\eta$ is given by \eqref{S1vvedenie}. Replacing the
integration over simplex in \eqref{B2loc} by the integration over
unit square
\begin{equation}\label{simp_to_square1}
\int d^3\tau_+ \delta(1-\sum_{i=1}^3\tau_i)=\int_0^1 d\tau_1\int_0^1 d\sigma (1-\tau_1),
\end{equation}
by changing the variables as follows
\begin{equation}\label{simp_to_square2}
\tau_2=(1-\tau_1)\sigma,\;\; \tau_3=(1-\tau_1)(1-\sigma)
\end{equation} and then performing partial integration with respect to $\gt_1$
using star-product formula \eq{starPr} and
dropping the terms from $\Hp_1$ we obtain
\begin{multline}\label{commut1}
\left[B_2^{\eta\, loc} ,S_1^\eta\right]_\ast\approx-\frac{i\eta^2}{2}\theta^\beta
\int_0^1 d\tau_1 \int_0^1 dt\int_0^1 d\sigma \, t(1-t)(1-\tau_1)e^{i\tau_1\circ t \,z_\alpha y^\alpha}
(z_\alpha y^\alpha) z_\beta\times\\
\Bigg\{C\Big(\big[-\tau_1(1-t)-\sigma t(1-\tau_1)\big]z+\sigma y\Big)C
\Big(\big[-\tau_1(1-t)+t(1-\tau_1)(1-\sigma)\big]z-(1-\sigma)y\Big)C\Big(t(1-\tau_1)z\Big)\\
-C\Big(-t(1-\tau_1)z\Big)C\Big(\big[\tau_1(1-t)-\sigma t(1-\tau_1)\big]z-\sigma y\Big)C
\Big(\big[\tau_1(1-t)+(1-\sigma)(1-\tau_1)t\big]z+(1-\sigma)y\Big)
\Bigg\}.
\end{multline}
Recall that we use notation with hidden $\bar y$ variables:
\bee\label{ustbybz}&& C(-\tau_1 z+\sigma(1-\tau_1) y) C(-\tau_1 z-(1-\sigma)(1-\tau_1)y)\equiv
\\ \nn &&\equiv
C(-\tau_1 z+\sigma(1-\tau_1) y,\by)\ust C(-\tau_1 z-(1-\sigma)(1-\tau_1)y,\by)\,.
\eee
Since only small values of \be \label{circ}\mathcal{T}:=\tau_1\circ t=\tau_1(1-t)+t(1-\tau_1)\ee
contribute to $\widehat{\Upsilon}^{\eta\eta}$ one needs to consider two triangle regions of the init
square in $(\tau_1,t)$ coordinates.
Only the lower triangle with small $t$ and $\tau_1$ contributes because the upper-one gives $\mathcal{T}^3$ in the
pre-exponential thus belonging to $\Hp_1$. The following change of variables is handy
in the further analysis
\begin{multline}\label{lowTr1}
\int d\mathcal{T} \int d\tau_1\, dt\, \theta(\tau_1)\theta(t)\theta(\varepsilon-\tau_1-t) \delta(\mathcal{T}-\tau_1-t)f(\tau_1,t)=\\
=\int_0^\varepsilon d\mathcal{T} \int_0^\mathcal{T} dt\,
f(\mathcal{T}-t,t) =\int_0^\varepsilon d \mathcal{T} \int_0^1 dt^\prime \,
\mathcal{T}\, f(\mathcal{T}(1-t^\prime),\mathcal{T} t^\prime).
\end{multline}
Adding the terms from $\Hp$, which do not affect the HS field equations,
one can reach further simplifications.
For instance, one can add $\int_{\varepsilon}^{1}d\T\int_0^1 dt^\prime f(\T(1-t^\prime),\T t^\prime)$ to \eqref{lowTr1}, \ie
\begin{multline}\label{lowTr}
\int d\mathcal{T} \int d\tau_1\, dt\, \theta(\tau_1)\theta(t)\theta(\varepsilon-\tau_1-t) \delta(\mathcal{T}-\tau_1-t)f(\tau_1,t)\approx\\
\approx\int_0^1 d \mathcal{T} \int_0^1 dt^\prime \,
\mathcal{T}\, f(\mathcal{T}(1-t^\prime),\mathcal{T} t^\prime)\,.
\end{multline}
(Recall that sign $\approx$ means that equality is up to terms from $\Hp$.)
In \eqref{commut1} it is convenient to introduce new variables
\begin{equation}\label{simplex1}
\rho_1=t^\prime\sigma\, ,\;\;\; \rho_2=t^\prime(1-\sigma)\,,\;\;\; \rho_3=1-t^\prime.
\end{equation}
They form a simplex since $\rho_1+\rho_2+\rho_3=1$. The inverse formulas are
\begin{equation}\label{simplex2}
\sigma=\frac{\rho_1}{\rho_1+\rho_2}\, ,\;\;\; (1-\sigma)=\frac{\rho_2}{\rho_1+\rho_2}\, ,\;\;\; t^\prime=1-\rho_3=\rho_1+\rho_2.
\end{equation}
In these new variables, the commutator takes the form
\begin{multline}\label{commutunif}
\left[ B_2^{\eta\, loc} ,S_1^\eta
\right]_\ast\approx-\frac{i\eta^2}{2}\theta^\beta z_\beta
(z_\alpha y^\alpha) \int_0^1 d\mathcal{T}\, \mathcal{T}^2 \int d^3
\rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right)
e^{i\mathcal{T}\, z_\alpha y^\alpha}\times\\
\Big\{C\Big(-\mathcal{T}(\rho_1+\rho_3)z+\frac{\rho_1}{\rho_1+\rho_2}y\Big)C
\Big(\mathcal{T}(\rho_2-\rho_3)z-\frac{\rho_2}{\rho_1+\rho_2} y\Big)C\Big(\mathcal{T}(\rho_1+\rho_2)z\Big)\\
-C\Big(-\mathcal{T}(\rho_1+\rho_3)z\Big)C\Big(\mathcal{T}(\rho_2-\rho_3)z-\frac{\rho_3}{\rho_1+\rho_3}y\Big)
C\Big(\mathcal{T}(\rho_1+\rho_2)z+\frac{\rho_1}{\rho_1+\rho_3}y\Big)\Big\}.
\end{multline}
Note that $z$-dependence is the same in the both terms.
Introducing an additional integration parameter $\xi$, the
$y$-dependence can be uniformized as follows
\begin{multline}
\left[ B_2^{\eta\, loc} ,S_1^\eta
\right]_*\approx\frac{i\eta^2}{2}\theta^\beta z_\beta(z_\alpha y^\alpha)
\int_0^1 d\mathcal{T}\, \mathcal{T}^2\int d^3 \rho_+
\delta\left(1-\sum_{i=1}^3 \rho_i\right)
\int_0^1 d\xi\, \frac{\partial}{\partial \xi}
\exp \big\{ \mathcal{Z}\big\}CCC\,,
\end{multline}
where the following notations are used
\begin{equation}
D_\alpha=-(\rho_1+\rho_3)\partial_{1\alpha}+(\rho_2-\rho_3)\partial_{2\alpha}+(\rho_1+\rho_2)\partial_{3\alpha},
\end{equation}
\begin{equation}
\mathcal{Z}=i\T z_\alpha y^\alpha + \mathcal{T} z^\alpha D_\ga +(1-\xi)y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_2}\partial_{1\alpha}-\frac{\rho_2}{\rho_1+\rho_2}\partial_{2\alpha}\right)+\xi\, y^\alpha\left(\frac{\rho_1}{\rho_1+\rho_3}\partial_{3\alpha}-\frac{\rho_3}{\rho_1+\rho_3}
\partial_{2\alpha}\right)\,.
\end{equation}
Evaluating the derivative with respect to $\xi$ taking into account that
\begin{equation}
\frac{\partial \mathcal{Z} }{\partial \xi}=\frac{\rho_1}{\T(\rho_1+\rho_2)(\rho_1+\rho_3)}y^\alpha \frac{\partial
\mathcal{Z}}{\partial z^\alpha}
\end{equation}
along with the Schouten identity
\begin{multline}
(\theta^\beta z_\beta)\left(y^\alpha \frac{\partial \mathcal{Z}}{\partial z^\alpha}\right)=
(\theta^\beta y_\beta)\left(z^\alpha \frac{\partial \mathcal{Z}}{\partial z^\alpha}\right)
+(z_\alpha y^\alpha)\left(\theta^\beta \frac{\partial \mathcal{Z}}{\partial z^\beta}\right)=\\
=(\theta^\beta y_\beta)\left(\T \frac{\partial \mathcal{Z}}{\partial \T}\right)
+(z_\alpha y^\alpha)\dr_z \mathcal{Z},
\end{multline}
the expression for the commutator can be rewritten in the form
\begin{multline}
\left[ B_2^{\eta\, loc} ,S_1^\eta
\right]_\ast\approx\mathrm{d}_z\left[ -\frac{\eta^2(z_\alpha
y^\alpha)^2}{2} \int_0^1 d\mathcal{T}\, \mathcal{T}\int d^3 \rho_+
\delta\left(1-\sum_{i=1}^3 \rho_i\right)
\int_0^1 d\xi\, \frac{\rho_1\, \exp \big\{ \mathcal{Z}\big\}}{(\rho_1+\rho_2)(\rho_1+\rho_3)}
CCC\rule{0pt}{18pt}\right]\\
-\frac{\eta^2}{2} \theta^\beta y_\beta \int d\mathcal{T}\, \delta\left(1-\mathcal{T}\right)
\mathcal{T}^2 \int d^3 \rho_+ \delta\left(1-\sum_{i=1}^3 \rho_i\right)
\int_0^1 d\xi\, \frac{\rho_1\, (z_\alpha y^\alpha) \exp \big\{ \mathcal{Z}\big\} }{(\rho_1+\rho_2)(\rho_1+\rho_3)}CCC.
\end{multline}
Since the second (boundary) term belongs to $\Hp_1$ and thus contributes to
${\Upsilon}^{\eta\eta}_+$ the part of $B_3^{\eta\eta}$ that
contributes to $\widehat{\Upsilon}^{\eta\eta}$ can be chosen in the form \eqref{B3final}.
\addtocounter{appendix}{1}
\renewcommand{\theequation}{\Alph{appendix}.\arabic{equation}}
\addtocounter{section}{1} \setcounter{equation}{0}
\addcontentsline{toc}{section}{\,\,\,\,\,\,\,Appendix C. $W_2^{\eta\eta}$ }
\renewcommand{\thesubsection}{\Alph{appendix}.\arabic{subsection}}
\section*{Appendix C. $W_2^{\eta\eta}$}
\label{W}
A particular solution for $W^{\eta\eta}_2$ was found in
\cite{4a2} where it was used in the computation of vertices
$\Upsilon^{\eta\eta}(\go,\go, C,C)$. However, this solution turns
out to be technically inconvenient for the analysis modulo $\Hp$
subspace. In this section we apply the approach proposed in the
previous section that allows us to single out the $\Hp_0$ part
from $W_2^{\eta\eta}$. Vertices
$\Upsilon^{\eta\eta}(\go,\go,C,C)$ computed with $W_2^{\eta\eta}$
given by \eqref{W2CCgofinal}-\eqref{W2CgoCfinal} may differ from
those of \cite{4a2} by a local field redefinition.
To compute $W_2^{\eta\eta}$ consider the equation
\begin{multline}
\dr_x S_1^\eta+W_1^\eta\ast S_1^\eta+S_1^\eta\ast W_1^\eta+\dr_x S_2^{\eta\eta}+\omega\ast S_2^{\eta\eta}+S_2^{\eta\eta}\ast \omega+S_0\ast W_2^{\eta\eta}+W_2^{\eta\eta}\ast S_0=0.
\end{multline}
{As mentioned in Appendix B
contribution to the vertices $ \Upsilon^{\eta\eta}(\go,C,C,C)$ from $S_2^{\eta\eta}$
vanishes at $\gb\to-\infty$. The remaining equation to be solved is
\begin{equation}
2i\dr_z \widehat{W}_2\approx\dr_x S_1^\eta+W_1^\eta\ast S_1^\eta+S_1^\eta\ast W_1^\eta.
\end{equation}
Here $S_1^\eta$ is given by \eqref{S1vvedenie} while $W_1^\eta$ consists of two parts \eqref{W1goCeta} and \eqref{W1Cgoeta}.
To calculate $\dr_x S_1^\eta$ one needs second-order zero-form vertices $\Upsilon^\eta(\go, C,C)$
obtained in \cite{4a1} with
additional shifts $\delta\Upsilon^\eta(\go, C,C)$ generated by the local shift $\delta B_2^\eta$
\eq{redef}. These are given by \eqref{VgoCC}-\eqref{VCgoC}.
\subsection{$W_{2\, CC\go}^{\eta\eta}$}
\label{SecW2CCgo}
Equation for $W_{2\, CC\go}^{\eta\eta}$ has the form
\begin{equation}\label{dzW2=}
2i\dr_z \widehat{W}_{2\, CC\go}^{\eta\eta}\approx\dr_x S_1^\eta\Big|_{CC\go}+S_1^\eta \ast W_{1\, C\go}^\eta.
\end{equation}
Computing $S_1^\eta\ast W^\eta_{1\, C\go}$ and dropping terms from $\Hp_1$ one obtains
discarding barred variables as in \eqref{ustbybz}
\begin{multline}
S_1^\eta \ast W^\eta_{1\, C\go}\approx \frac{\eta^2}{2}\int_0^1 dt \int_0^1 d\tau_1 \int_0^1 d\sigma\,t(1-t)(1-\tau_1)^2 \left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\omega \alpha}\right)\exp\Big\{i\tau_1 \circ t z_\alpha y^\alpha+i(1-\sigma)\partial_{2 \alpha}\partial_{\go}{}^\alpha\Big\}\times\\
\times C\Big(-t(1-\tau_1)z\Big)C\Big(\tau_1(1-t)z\Big)\omega\Big(\big[\tau_1(1-t)+
\sigma t(1-\tau_1)\big]z+\sigma y\Big)\,.
\end{multline}
Taking into account that only small values of $\tau_1 \circ t$ contribute to $\widehat{\Upsilon}^{\eta\eta}$ one can change integration variables as in \eqref{lowTr} to obtain
\begin{multline}
S_1^\eta \ast W_{1\, C\go}^\eta\approx\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2 \int_0^1 dt^\prime \int_0^1 d\sigma\, t^\prime\, \left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\omega \alpha}\right)\exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i(1-\sigma)\partial_{2 \alpha}\partial_{\go} {}^\alpha\Big\}\times\\
\times C\Big(-\mathcal{T} t^\prime z\Big)C\Big(\mathcal{T}(1-t^\prime)z\Big)\omega\Big(\mathcal{T}\big[(1-t^\prime)+t^\prime \sigma\big]z+\sigma y\Big)\,.
\end{multline}
In the simplex variables \eqref{simplex1}, \eqref{simplex2} this expression
can be rewritten as
\begin{multline}\label{S1W1cgo}
S_1^\eta \ast W_{1\, C\go}^\eta\approx\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2 \int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\omega \alpha}\right)\exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i\frac{\rho_2}{\rho_1+\rho_2}\partial_{2 \alpha}\partial_{\go} {}^\alpha\Big\}\times\\
\times C\Big(-\mathcal{T} (\rho_1+\rho_2) z\Big)C\Big(\mathcal{T}\rho_3 z\Big)\omega\Big(\mathcal{T}(\rho_1+\rho_3)z+\frac{\rho_1}{\rho_1+\rho_2} y\Big)\,.
\end{multline}
To compute $\dr_x S_1^\eta$ one has to use vertex \eqref{VCCgo}
\begin{multline}\label{dxS1}
\dr_x S_1^\eta \big|_{CC\go}\approx\\
\approx-\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2 \int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\omega \alpha}\right)\exp\Big\{i\mathcal{T} z_\alpha y^\alpha+i(1-\rho_1)\partial_{2\alpha}\partial_{\go}{}^\alpha-i\rho_1 \partial_{1\alpha}\partial_{\go} {}^\alpha\Big\}\times\\
\times C\Big(-\mathcal{T}(\rho_1+\rho_2)z\Big)C\Big(\mathcal{T}\rho_3 z\Big)\go\Big(\mathcal{T}(\rho_1+\rho_3)z\Big).
\end{multline}
As in Appendix B, $y$-dependence in \eqref{dzW2=} can be uniformized with the help of the new integration parameter $\xi$.
Using new notation for brevity
\begin{multline}
\mathsf{Z}=i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big(-(\rho_1+\rho_2)\partial_{1\alpha}+\rho_3 \partial_{2 \alpha}+(\rho_1+\rho_3)\partial_{\omega \alpha}\Big)\\
+\xi\left(i\frac{\rho_2}{\rho_1+\rho_2}\partial_{2 \alpha}\partial_{\go}{}^\alpha+\frac{\rho_1}{\rho_1+\rho_2}y^\alpha \partial_{\omega \alpha}\right)+(1-\xi)\Big(i(1-\rho_1)\partial_{2 \alpha}\partial_{\go}{}^\alpha-i\rho_1 \partial_{1\alpha}\partial_{\go}{}^\alpha\Big)
\end{multline}
one has
from \eqref{dzW2=} taking into account \eqref{S1W1cgo} and \eqref{dxS1}
\begin{equation}
2i\dr_z \widehat{W}_{2\,
CC\go}^{\eta\eta}=\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\,
\mathcal{T}^2 \int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3
\rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha
\partial_{\omega \alpha}\right)\int_0^1 d\xi \,
\frac{\partial}{\partial \xi} e^{\mathsf{Z}} \, CC\go.
\end{equation}
Evaluating the derivative over $\xi$
\begin{equation}
\frac{\partial \mathsf{Z}}{\partial \xi} =\frac{\rho_1}{\rho_1+\rho_2}\Big(-i\rho_3 \partial_{2 \alpha}\partial_{\go}{}^\alpha+i(\rho_1+\rho_2)\partial_{1\alpha}\partial_\go {}^\alpha+y^\alpha\partial_{\omega \alpha}\Big)
\end{equation}
and taking into account that
\bee\label{w2dz} \frac{\partial \mathsf{Z}}{\partial \xi}
= \frac{-i\rho_1\, }{\T (\rho_1+\rho_2) } \partial_{\go}{}^\alpha \frac{\partial \mathsf{Z}}{\partial z^\ga}
\eee
along with the Schouten identity
\begin{equation}\label{SchoutenW}
\left(\theta^\beta z_\beta\right)\left(\partial_\go{}^\alpha \frac{\partial \mathsf{Z}}{\partial z^\alpha}\right)=\left(\theta^\beta \partial_{\omega \beta}\right)\left(z^\alpha \frac{\partial \mathsf{Z}}{\partial z^\alpha}\right)-\left(\theta^\beta \frac{\partial \mathsf{Z}}{\partial z^\beta}\right)\left(z^\alpha \partial_{\go \alpha}\right)
\q\end{equation}
the pre-exponential part can be written in the form
\begin{equation}
\left(\theta^\beta z_\beta\right)\left(\partial_\go{}^\alpha \frac{\partial \mathsf{Z}}{\partial z^\alpha}\right)=\left(\theta^\beta \partial_{\omega \beta}\right)\left(\T\frac{\partial \mathsf{Z}}{\partial \T} \right)-\left(z^\alpha \partial_{\go \alpha}\right)\dr_z \mathsf{Z}
\end{equation}
and thus \rhs of \eqref{dzW2=} can be put into the form
\begin{multline}
2i\dr_z \widehat{W}_{2\, CC\go}^{\eta\eta}=-\frac{i\eta^2}{2}\int_0^1 d\mathcal{T}\, \frac{\rho_1\mathcal{T}^2}{\rho_1+\rho_2} \int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\int_0^1 d\xi\, \left(\theta^\beta \partial_{\omega \beta}\right)\left(z^\gamma \partial_{\go \gamma}\right)\frac{\partial}{\partial \mathcal{T}}e^{\mathsf{Z}}CC\go\\
+\frac{i\eta^2}{2}\int_0^1 d\mathcal{T} \, \frac{\rho_1
\mathcal{T}^2}{\rho_1+\rho_2}\int d^3\rho_+\,
\delta\left(1-\sum_{i=1}^3 \rho_i\right)\int_0^1 d\xi
\left(z^\gamma \partial_{\go \gamma}\right)^2\dr_z e^{\mathsf{Z}}
CC\go.
\end{multline}
After integrating by parts with respect to $\T$ in the first term, the resulting boundary term belongs to $\Hp$ (cf. the second case of \eqref{kernels}) and hence can be discarded. This brings equation for $\widehat{W}_{2\, CC\go}^{\eta\eta}$ to the form
\begin{equation}
2i\dr_z \widehat{W}_{2\, CC\go}^{\eta \eta} \approx\dr_z
\left\{\frac{i\eta^2}{2}\int_0^1 d\mathcal{T}\int_0^1 d\xi \int
d^3\rho_+ \, \delta\left(1-\sum_{i=1}^3
\rho_i\right)\frac{\mathcal{T}\rho_1 \left(z^\gamma \partial_{\go
\gamma}\right)^2}{\rho_1+\rho_2}e^{\mathsf{Z}}CC\go \right\}.
\end{equation}
This allows us to choose the part of $W_{2\, CC\go}^{\eta\eta}$,
that contributes to $\widehat{\Upsilon}^{\eta\eta}$, in the form
\begin{multline}
\widehat{W}_{2\, CC\go}^{\eta \eta}=\frac{\eta^2}{4}\int_0^1 d\mathcal{T} \int_0^1 d\xi\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3\rho_i\right)\frac{\mathcal{T}\rho_1 \left(z^\gamma \partial_{\go \gamma}\right)^2}{\rho_1+\rho_2}\times\\
\times\exp\Big\{i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big(-(\rho_1+\rho_2)\partial_{1 \alpha}+\rho_3 \partial_{2 \alpha}+(\rho_1+\rho_3)\partial_{\omega \alpha}\Big)\\
+\xi\left(\frac{i\rho_2}{\rho_1+\rho_2}\partial_{2 \alpha}\partial_{\go}{}^\alpha+\frac{\rho_1}{\rho_1+\rho_2}y^\alpha\partial_{\omega \alpha}\right)+(1-\xi)\Big(i(1-\rho_1)\partial_{2 \alpha}\partial_{\go}{}^\alpha-\rho_1 \partial_{1 \alpha}\partial_{\go}{}^\alpha\Big)\Big\}CC\go.
\end{multline}
Finally, one can change the integration variables to rewrite
$\widehat{W}_{2\, CC\go}^{\eta \eta}$ in the form of the integral over
a four-dimensional simplex
according to
\begin{multline}\label{4simplex}
\int_0^1 d\xi \int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\, f(\xi,1-\xi;\rho_1,\rho_2,\rho_3)=\\
=\int d^2\xi_+\, \delta(1-\xi_1 -\xi_2)\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\, f(\xi_1,\xi_2;\rho_1,\rho_2,\rho_3)=\\
=\int d\zeta_1 \int d\zeta_2 \int d^2\xi_+\, \delta(1-\xi_1 -\xi_2)\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\,\delta(\zeta_1-\rho_3 \xi_1)\,\delta(\zeta_2-\rho_3 \xi_2) f(\xi_1,\xi_2;\rho_1,\rho_2,\rho_3)=\\
=\int d^2 \zeta_+ \int d^3 \rho_+\, \frac{1}{\rho_3^2} \delta\Big(1-\frac{\zeta_1}{\rho_3}-\frac{\zeta_2}{\rho_3}\Big)\delta\left(1-\sum_{i=1}^3 \rho_i\right)f\left(\frac{\zeta_1}{\rho_3},\frac{\zeta_2}{\rho_3};\rho_1,\rho_2,\rho_3\right)=\\
=\int d^2\zeta_+ \int d^2 \rho_+\, \frac{\delta\left(1-\rho_1-\rho_2-\zeta_1-\zeta_2\right)}{1-\rho_1-\rho_2}f\left(\frac{\zeta_1}{1-\rho_1-\rho_2},\frac{\zeta_2}{1-\rho_1-\rho_2};\rho_1,\rho_2,1-\rho_1-\rho_2\right)=\\
=\int d^4\rho_+\, \delta \left(1-\sum_{i=1}^4 \rho_i\right)\frac{1}{1-\rho_1-\rho_2}f\left(\frac{\rho_3}{1-\rho_1-\rho_2},
\frac{\rho_4}{1-\rho_1-\rho_2};\rho_1,\rho_2,1-\rho_1-\rho_2\right)\,.
\end{multline}
In these variables $\widehat{W}^{\eta\eta}_{2\, CC\go}$ acquires the form \eqref{W2CCgofinal}.
\subsection{$W_{2\, C\go C}^{\eta\eta}$}
\label{SecW2CgoC}Equation for this part of the connection is
\begin{equation}\label{W2CgoCEq}
2i \dr_z \widehat{W}_{2\, C\go C}^{\eta \eta}\approx\dr_x S_1^\eta\Big|_{C\go C}+S_1^\eta \ast W_{1\, \go C}^\eta +W_{1\, C\go}^\eta \ast S_1^\eta.
\end{equation}
Star product $S_1^\eta \ast W_{1\, \go C}^\eta$ can be computed by
\eqref{starPr}. Discarding terms in ${\Upsilon}_+^{\eta\eta}$ and
omitting the barred variables, $S_1^\eta \ast W_{1\, \go C}^\eta$
takes the form
\begin{multline}
S_1^\eta \ast W_{1\, \go C}^\eta\approx\frac{\eta^2}{2}\int_0^1 dt\int_0^1 d\tau_1 \int_0^1 d\sigma\,t(1-t)(1-\tau_1)^2 \left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\omega \alpha}\right)\exp\Big\{i\tau_1 \circ t\, z_\alpha y^\alpha+i(1-\sigma)\partial_{\go \alpha}\partial_{2}{}^\alpha\Big\}\times\\
\times C\Big(-t(1-\tau_1)z\Big)\omega\Big(\big[\tau_1(1-t)-\sigma t(1-\tau_1)\big]z-\sigma y\Big)C\Big(\tau_1(1-t)z\Big).
\end{multline}
Since only small values of $\T$ contribute to $\widehat{\Upsilon}^{\eta\eta}$ we consider only
lower triangle of the init square in $(\tau_1,t)$ and perform the same change of variables as in
\eqref{lowTr}. Using the simplex variables \eqref{simplex1}, \eqref{simplex2} the result
can be re-written as
\begin{multline}
S_1^\eta \ast W_{1\, \go C}^\eta\approx \frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\go \alpha}\right)\exp\Big\{i\mathcal{T}z_\alpha y^\alpha+i\frac{\rho_2}{\rho_1+\rho_2}\partial_{\go \alpha}\partial_2 {}^\alpha\Big\}\times\\
\times C\Big(-\mathcal{T}(1-\rho_3)z\Big)\go
\Big(-\mathcal{T}\rho_1 z+\mathcal{T}\rho_3z-\frac{\rho_1}{\rho_1+\rho_2}y\Big)C
\Big(\mathcal{T}\rho_3 z\Big)\,.
\end{multline}
Analogously, for star product $W_{1\, C\go}^\eta \ast S_1^\eta$
\begin{multline}
W_{1\, C\go}^\eta\ast S_1^\eta\approx\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\go \alpha}\right)\exp\Big\{i\mathcal{T}z_\alpha y^\alpha+i\frac{\rho_2}{\rho_1+\rho_2}\partial_{\go \alpha}\partial_2 {}^\alpha\Big\}\times\\
\times C\Big(-\mathcal{T}\rho_3 z\Big)\go\Big(\mathcal{T}\rho_1 z-\mathcal{T}\rho_3z-\frac{\rho_1}{\rho_1+\rho_2}y\Big)C\Big(\mathcal{T}(1-\rho_3) z\Big)\,.
\end{multline}
The $\dr_x S_1^\eta$ part computed using vertex \eqref{VCgoC} is
\begin{multline}
\dr_x S_1^\eta \big|_{C\go C}\approx\\
\approx-\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\go \alpha}\right)\exp\Big\{i\mathcal{T}z_\alpha y^\alpha+i\rho_2\partial_{1 \alpha}\partial_{\go}{}^\alpha+i(1-\rho_2)\partial_{\go \alpha}\partial_2 {}^\alpha\Big\}\times\\
\times C\Big(-\mathcal{T}\rho_3 z\Big)\go\Big(\mathcal{T}\rho_1 z\Big)
C\Big(\mathcal{T}(1-\rho_3) z\Big)\\
-\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\go \alpha}\right)\exp\Big\{i\mathcal{T}z_\alpha y^\alpha+i(1-\rho_2)\partial_{1 \alpha}\partial_{\go}{}^\alpha+i\rho_2\partial_{\go \alpha}\partial_2 {}^\alpha\Big\}\times\\
\times C\Big(-\mathcal{T}(1-\rho_3) z\Big)\go\Big(-\mathcal{T}\rho_1 z\Big)C\Big(\mathcal{T}\rho_3 z\Big)\,.
\end{multline}
It is natural to group the \rhs of (\ref{W2CgoCEq}) in the following way
\bee\nn&&
\dr_x S_1^\eta \big|_{C\go C}+S_1^\eta \ast W_{1\, \go C}^\eta+W_{1\, C\go}^\eta \ast S_1^\eta
\approx\\ \nn&&
\approx
\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)
\left(z^\alpha \partial_{\go \alpha}\right)\exp\big\{i\mathcal{T}z_\alpha y^\alpha\big\}\times
\\\nn&&\times
\Big[-\exp\big\{ i\rho_2\partial_{1 \alpha}\partial_{\go}{}^\alpha
+i(1-\rho_2)\partial_{\go \alpha}\partial_2 {}^\alpha\big\} \,\, C
\Big(-\mathcal{T}\rho_3 z\Big)\go\Big(\mathcal{T}\rho_1 z\Big)
C\Big(\mathcal{T}(1-\rho_3) z\Big)\\\nn&&
+ \exp\big\{ i\frac{\rho_2}{\rho_1+\rho_2}\partial_{\go \alpha}\partial_2 {}^\alpha\big\}\,
C\Big(-\mathcal{T}\rho_3 z\Big)\go\Big(\mathcal{T}\rho_1 z-\mathcal{T}\rho_3z-\frac{\rho_1}{\rho_1+\rho_2}y\Big)
C\Big(\mathcal{T}(1-\rho_3) z\Big)\\\nn&&
- \exp\big\{ i(1-\rho_2)\partial_{1 \alpha}\partial_{\go}{}^\alpha
+i\rho_2\partial_{\go \alpha}\partial_2 {}^\alpha\big\}\,\, C\big(-\mathcal{T}(1-\rho_3) z\Big)
\go\Big(-\mathcal{T}\rho_1 z\Big)C\Big(\mathcal{T}\rho_3 z\Big)\\\nn&&
+ \exp\Big\{ i\frac{\rho_2}{\rho_1+\rho_2}\partial_{\go \alpha}\partial_2 {}^\alpha\big\}\,\, C\Big(-\mathcal{T}\rho_3 z\Big)\go\Big(\mathcal{T}\rho_1 z-\mathcal{T}\rho_3z-\frac{\rho_1}{\rho_1+\rho_2}y\Big)C\Big(\mathcal{T}(1-\rho_3) z\Big)
\Big]\,.\eee
Introducing new notations for brevity
\bee &&
\mathsf{Z}_1=i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big(-\rho_3 \partial_{1 \alpha}
+\rho_1 \partial_{\go \alpha}+(1-\rho_3)\partial_{2 \alpha}\Big)\\\nn&&
+\xi\left(-\mathcal{T}\rho_3 z^\alpha \partial_{\go \alpha}-\frac{\rho_1}{\rho_1+\rho_2}
y^\alpha \partial_{\go \alpha}+i\frac{\rho_2}{\rho_1+\rho_2}\partial_{1 \alpha}\partial_{\go}{}^\alpha\right)+(1-\xi)\Big(i\rho_2\partial_{1 \alpha}\partial_{\go} {}^\alpha+i(1-\rho_2)\partial_{\go \alpha}\partial_2 {}^\alpha\Big),
\\ \nn &&\mathsf{Z}_2=i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big(-(1-\rho_3)
\partial_{1 \alpha}-\rho_1 \partial_{\go \alpha}+\rho_3\partial_{2 \alpha}\Big)
\\\nn&&
+\xi\left(\mathcal{T}\rho_3 z^\alpha \partial_{\go \alpha}-\frac{\rho_1}{\rho_1+\rho_2}y^\alpha
\partial_{\go \alpha}+i\frac{\rho_2}{\rho_1+\rho_2}\partial_{\go \alpha}\partial_{2}{}^\alpha\right)+(1-\xi)
\Big(i(1-\rho_2)\partial_{1 \alpha}\partial_{\go} {}^\alpha+i\rho_2\partial_{\go \alpha}\partial_2 {}^\alpha\Big)
\,,
\eee
the \rhs of \eqref{W2CgoCEq} can be written as an integral of a total derivative
\begin{multline}\label{Z1Z2}
2i\dr_z \widehat{W}_{2\, C\go
C}^{\eta\eta}\approx\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\,
\mathcal{T}^2\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3
\rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha
\partial_{\go \alpha}\right)\int_0^1 d\xi \,
\frac{\partial}{\partial
\xi}\Big(e^{\mathsf{Z}_1}+e^{\mathsf{Z}_2}\Big)C\go C.
\end{multline}
Since \begin{multline}
\frac{\partial \mathsf{Z}_1}{\partial \xi}
=-\rho_3\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}-\frac{\rho_1}{\rho_1+\rho_2}\Big[y^\alpha \partial_{\go \alpha}+i\rho_3 \partial_{1 \alpha}\partial_{\go} {}^\alpha+i(1-\rho_3)\partial_{\go \alpha}\partial_2 {}^\alpha\Big]=\\
=-\rho_3\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}+i\frac{\rho_1}{\T(\rho_1+\rho_2)}\partial_{\go}{}^\alpha
\frac{\partial\mathsf{Z}_1}{\partial z^\alpha}\,,
\end{multline}
analogously to \eqref{SchoutenW} by virtue of Schouten identity one has
\begin{multline}
\left(\theta^\beta z_\beta\right)\frac{\partial \mathsf{Z}_1}{\partial \xi}
=-\left(\theta^\beta z_\beta \right)\rho_3\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}-i\frac{\rho_1
\left(z^\gamma \partial_{\go \gamma}\right)}{\mathcal{T}(\rho_1+\rho_2)}\dr_z \mathsf{Z}_1+
i\frac{\rho_1 \left(\theta^\beta \partial_{\omega \beta}\right)}
{\rho_1+\rho_2}\frac{\partial \mathsf{Z}_1}{\partial \T}\,.
\end{multline}
Therefore the $\mathsf{Z}_1$-dependent part from the \rhs of \eqref{Z1Z2} can be rewritten in the form
\begin{multline}
\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\go \alpha}\right)\int_0^1 d\xi \, \frac{\partial}{\partial \xi} e^{\mathsf{Z}_1}C\go C\approx\\
\approx-\frac{\eta^2}{2}\left(\theta^\beta z_\beta\right)\left(z^\gamma \partial_{\go \gamma}\right)
\int_0^1 d\mathcal{T}\, \mathcal{T}^2 \int_0^1 d\xi \int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)
\rho_3\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}\,e^{\mathsf{Z}_1}C \go C\\
-\frac{i\eta^2}{2}\left(z^\gamma \partial_{\go \gamma}\right)^2 \int_0^1 d\mathcal{T}\, \mathcal{T} \int_0^1 d\xi\int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\frac{\rho_1}{\rho_1+\rho_2}\dr_z e^{\mathsf{Z}_1}C\go C\\
+\frac{i\eta^2}{2}\left(\theta^\beta \partial_{\omega
\beta}\right)\left(z^\gamma \partial_{\go \gamma}\right) \int_0^1
d\mathcal{T}\, \mathcal{T}^2 \int_0^1 d\xi\int d^3 \rho_+\,
\delta\left(1-\sum_{i=1}^3
\rho_i\right)\frac{\rho_1}{\rho_1+\rho_2}\frac{\partial}{\partial
\mathcal{T}} e^{\mathsf{Z}_1}C\go C.
\end{multline}
Integrating by parts in the last term, the resulting boundary term contributes to ${\Upsilon}_+^{\eta\eta}$. Hence
\begin{multline}
\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\go \alpha}\right)\int_0^1 d\xi \, \frac{\partial}{\partial \xi} e^{\mathsf{Z}_1}C\go C\approx\\
\approx-\frac{\eta^2}{2}\left(\theta^\beta z_\beta\right)
\left(z^\gamma \partial_{\go \gamma}\right)\int_0^1 d\mathcal{T}\, \mathcal{T}^2 \int_0^1 d\xi
\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\rho_3
\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}\,e^{\mathsf{Z}_1}C \go C\\
+\dr_z\left\{-\frac{i\eta^2}{2}\left(z^\gamma \partial_{\go
\gamma}\right)^2 \int_0^1 d\mathcal{T}\, \mathcal{T} \int_0^1
d\xi\int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3
\rho_i\right)\frac{\rho_1}{\rho_1+\rho_2} e^{\mathsf{Z}_1}C\go C
\right\}.
\end{multline}
Analogously, since
\be\nn
\frac{\partial \mathsf{Z}_2}{\partial \xi}
=\rho_3\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}
+i\frac{\rho_1}{\T(\rho_1+\rho_2)}\partial_{\go} {}^\alpha \frac{\partial \mathsf{Z}_2}{\partial z^\alpha}\,,
\ee
\begin{multline}
\left(\theta^\beta z_\beta\right)\frac{\partial \mathsf{Z}_2}{\partial \xi}=
\left(\theta^\beta z_\beta\right)\rho_3
\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}-i\frac{\rho_1
\left(z^\gamma \partial_{\go\gamma}\right)}{\T(\rho_1+\rho_2)}\dr_z \mathsf{Z}_2+
i\frac{\rho_1\left(\theta^\beta \partial_{\omega \beta}\right)}{\rho_1
+\rho_2}\frac{\partial \mathsf{Z}_2}{\partial \T}
\end{multline} and,
therefore, the $\mathsf{Z}_2$-dependent part of the r.h.s. of \eqref{Z1Z2} yields
\begin{multline}
\frac{\eta^2}{2}\int_0^1 d\mathcal{T}\, \mathcal{T}^2\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\left(\theta^\beta z_\beta\right)\left(z^\alpha \partial_{\go \alpha}\right)\int_0^1 d\xi \, \frac{\partial}{\partial \xi} e^{\mathsf{Z}_2}C\go C\approx\\
\approx\frac{\eta^2}{2}\left(\theta^\beta z_\beta\right)\left(z^\gamma \partial_{\go \gamma}\right)
\int_0^1 d\mathcal{T}\, \mathcal{T}^2 \int_0^1 d\xi \int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)
\rho_3\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}\,e^{\mathsf{Z}_2}C \go C\\
-\frac{i\eta^2}{2}\left(z^\gamma \partial_{\go \gamma}\right)^2 \int_0^1 d\mathcal{T}\, \mathcal{T}
\int_0^1 d\xi\int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\frac{\rho_1}{\rho_1+\rho_2}
\dr_z e^{\mathsf{Z}_2}C\go C\\
+\frac{i\eta^2}{2}\left(\theta^\beta \partial_{\omega
\beta}\right)\left(z^\gamma \partial_{\go \gamma}\right) \int_0^1
d\mathcal{T}\, \mathcal{T}^2 \int_0^1 d\xi\int d^3 \rho_+\,
\delta\left(1-\sum_{i=1}^3
\rho_i\right)\frac{\rho_1}{\rho_1+\rho_2}\frac{\partial}{\partial
\mathcal{T}} e^{\mathsf{Z}_2}C\go C\,.
\end{multline}
Modulo terms from $\mathcal{H}^+$ this equals to
\begin{multline}
\frac{\eta^2}{2}\left(\theta^\beta z_\beta\right)\left(z^\gamma \partial_{\go \gamma}\right)
\int_0^1 d\mathcal{T}\, \mathcal{T}^2 \int_0^1 d\xi \int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)
\rho_3\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}\,e^{\mathsf{Z}_2}C \go C\\
+\dr_z\left\{-\frac{i\eta^2}{2}\left(z^\gamma \partial_{\go
\gamma}\right)^2 \int_0^1 d\mathcal{T}\, \mathcal{T} \int_0^1
d\xi\int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3
\rho_i\right)\frac{\rho_1}{\rho_1+\rho_2} e^{\mathsf{Z}_2}C\go C
\right\}.
\end{multline}
As a result, the \rhs of \eqref{Z1Z2} acquires the form
\begin{multline}
2i\dr_z \widehat{W}_{2\, C\go C}^{\eta \eta}\approx\dr_z\left\{-\frac{i\eta^2}{2}\left(z^\gamma \partial_{\go \gamma}\right)^2 \int_0^1 d\mathcal{T}\, \mathcal{T} \int_0^1 d\xi\int d^3 \rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\frac{\rho_1}{\rho_1+\rho_2}\Big[e^{\mathsf{Z}_1}+ e^{\mathsf{Z}_2}\Big]C\go C \right\}\\
+\frac{\eta^2}{2}\left(\theta^\beta z_\beta\right)\left(z^\gamma \partial_{\go \gamma}\right)
\int_0^1 d\mathcal{T}\, \mathcal{T}^2 \int_0^1 d\xi \int d^3\rho_+\,
\delta\left(1-\sum_{i=1}^3 \rho_i\right)\rho_3
\big(\mathcal{T}z +i\partial_{1 }
+i \partial_2 {} \big){}^\alpha\partial_{\go \alpha}\left[e^{\mathsf{Z}_2}-e^{\mathsf{Z}_1}\right]C \go C .
\end{multline}
To see that the last term vanishes it is convenient to change
integration variables to those of the four-dimensional simplex
\eqref{4simplex}. In these four-dimensional simplicial variables
\begin{multline}\label{Z1=}
\mathsf{Z}_1=i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big(-(\rho_3+\rho_4)\partial_{1 \alpha}+(\rho_1-\rho_3)\partial_{\go \alpha}+(\rho_1+\rho_2)\partial_{2 \alpha}\Big)
-\frac{\rho_3\rho_1}{(\rho_1+\rho_2)(\rho_3+\rho_4)}y^\alpha \partial_{\go\alpha}\\+i\frac{\rho_3\rho_2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\partial_{1 \alpha}\partial_{\go} {}^\alpha+i\frac{\rho_4 \rho_2}{\rho_3+\rho_4}\partial_{1 \alpha}\partial_{\go} {}^\alpha+i\frac{\rho_4(1-\rho_2)}{\rho_3+\rho_4}\partial_{\go \alpha}\partial_2 {}^\alpha\,,
\end{multline}
\begin{multline}\label{Z2=}
\mathsf{Z}_2=i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big(-(\rho_1+\rho_2)\partial_{1 \alpha}+(\rho_3-\rho_1)\partial_{\go \alpha}+(\rho_3+\rho_4)\partial_{2 \alpha}\Big)
-\frac{\rho_3\rho_1}{(\rho_1+\rho_2)(\rho_3+\rho_4)}y^\alpha \partial_{\go\alpha}\\
+i\frac{\rho_3\rho_2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}\partial_{\go \alpha}\partial_{2} {}^\alpha+i\frac{\rho_4 \rho_2}{\rho_3+\rho_4}\partial_{\go \alpha}\partial_{2} {}^\alpha+i\frac{\rho_4(1-\rho_2)}{\rho_3+\rho_4}\partial_{1 \alpha}\partial_\go {}^\alpha\,.
\end{multline}
Shuffling the $\rho$-variables in $\mathsf{Z}_2$
\begin{equation}
\rho_1\longrightarrow\rho_3,\;\; \rho_3\longrightarrow \rho_1,\;\; \rho_2\longrightarrow\rho_4,\;\;
\rho_4\longrightarrow\rho_2\q
\end{equation}
taking into account that
\bee\nn
&&\gd(1-\rho_1-\rho_2-\rho_3-\rho_4)\left(\frac{\rho_1\rho_4}{(\rho_1+\rho_2)(\rho_3+\rho_4)}
+\frac{\rho_4\rho_2}{\rho_1+\rho_2}\right) =
\gd(1-\rho_1-\rho_2-\rho_3-\rho_4) \frac{\rho_4(1-\rho_2)}{\rho_3+\rho_4},
\\\nn&& \gd(1-\rho_1-\rho_2-\rho_3-\rho_4)\left(\frac{\rho_3\rho_2}{(\rho_1+\rho_2)(\rho_3+\rho_4)}
+\frac{\rho_4\rho_2}{\rho_3+\rho_4}\right)
=\gd(1-\rho_1-\rho_2-\rho_3-\rho_4)\frac{\rho_2(1-\rho_4)}{\rho_1+\rho_2}\q
\eee
one can see that
\begin{multline}
\mathsf{Z}_1=\mathsf{Z}_2=i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big(-(\rho_3+\rho_4)\partial_{1 \alpha}+(\rho_1-\rho_3)\partial_{\go \alpha}+(\rho_1+\rho_2)\partial_{2 \alpha}\Big)
\\-\frac{\rho_3\rho_1}{(\rho_1+\rho_2)(\rho_3+\rho_4)}y^\alpha \partial_{\go\alpha}
+i\frac{\rho_4(1-\rho_2)}{\rho_3+\rho_4}\partial_{\go \alpha}\partial_{2} {}^\alpha+i\frac{\rho_2(1-\rho_4)}{\rho_1+\rho_2}\partial_{1 \alpha}\partial_\go {}^\alpha.
\end{multline}
The part of $W_{2\, C\go C}^{\eta\eta}$ that contributes to $\widehat{\Upsilon}^{\eta\eta}$
thus has the form \eqref{W2CgoCfinal}.
\subsection{$W_{2\, \go CC}^{\eta\eta}$}
Equation for the part of $W_{2\, \go CC}^{\eta\eta}$ that contributes to $\widehat{\Upsilon}^{\eta\eta}$ is
\begin{equation}
2i\dr_z \widehat{W}_{2\, \go CC}^{\eta\eta}\approx\dr_x S_1^\eta \big|_{\go CC}+W_{1\, \go C}^\eta\ast S_1^\eta.
\end{equation}
Since computation is analogous to that for $\widehat{W}_{2\ CC\go}^{\eta\eta}$ we present only the final result
\begin{multline}
\widehat{W}_{2\, \omega CC}^{\eta\eta}=\frac{\eta^2}{4}\int_0^1 d\mathcal{T} \, \mathcal{T}\int_0^1 d\xi\int d^3\rho_+\, \delta\left(1-\sum_{i=1}^3 \rho_i\right)\frac{\rho_1 (z^\alpha \partial_{\omega\alpha})^2}{\rho_1+\rho_2}\times\\
\times\exp\bigg\{i\mathcal{T}z_\alpha y^\alpha+\mathcal{T}z^\alpha\Big((\rho_1+\rho_3) \partial_{\omega\alpha}-\rho_3 \partial_{1\alpha}+(\rho_1+\rho_2) \partial_{2\alpha}\Big)\\
+\xi\Big(i(1-\rho_1)\partial_{\omega\alpha}\partial_1 {}^\alpha-i\rho_1\partial_{\omega\alpha}\partial_2 {}^\alpha\Big)+(1-\xi)\left(i\frac{\rho_2}{\rho_1+\rho_2}\partial_{\omega\alpha}\partial_1 {}^\alpha+\frac{\rho_1}{\rho_1+\rho_2}y^\alpha \partial_{\omega \alpha}\right)\bigg\}\omega CC.
\end{multline}
Equivalently, as an integral over a four-dimensional simplex it is given in \eqref{W2goCCfinal}.
\addcontentsline{toc}{section}{\,\,\,\,\,\,\,References}
\section*{}
|
1,116,691,499,407 | arxiv | \section{Introduction}
Consider the optimal stopping problem
\begin{equation}\label{eq.intro.optimalstopping}
\sup_{\tau\in \mathcal{T}} \mathbb{E}_x[\delta(\tau)f(X_\tau)],
\end{equation}
where $X=(X_t)_{t\in[0,\infty)}$ is a time-homogeneous Markov process taking values in some state space $\mathbb{X}$, $\delta$ is a discount function and $f$ is a reward function. It is well known that when $\delta$ is not exponential, the problem \eqref{eq.intro.optimalstopping} may be time-inconsistent. That is, a stopping strategy that is optimal from today's point of view may no longer be optimal from a future's perspective. A popular approach to address this time-inconsistency is to look for a subgame perfect Nash equilibrium instead of solving \eqref{eq.intro.optimalstopping}: a strategy such that once it is imposed over the planning horizon, the current self has no incentive to deviate from the strategy, given all future selves will follow it.
There have been a lot of papers on equilibrium strategies for time-inconsistent control problems, and we refer to \cite{MR4288523, MR4328502, MR3626618} and the references therein. The development for theory of time-inconsistent stopping is more recent, and we refer to \cite{MR4250561,huang2018time,MR4067078,MR4273542,MR4116459,MR4080735,MR3880244,MR4205889,MR4332966,MR3980261,2022arXiv220107659B,MR3911711}.
Let us also mention the work \cite{MR4397932} which analyzes a time-inconsistent Dynkin game, and \cite{liang2021weak} which considers a time-inconsistent controller-stopper problem. It is worth to mention that most of the papers on time-inconsistent control and stopping focus on the characterization of equilibria. A few exceptions include \cite{MR3911711,MR4116459,MR4250561,MR4121091} where the optimality and selection of equilibria are first analyzed in the presence of multiple equilibria. In particular, it is shown in settings of these papers that there exists an optimal equilibrium which pointwisely dominates all other equilibria in terms of the associated value functions; moreover, this optimal equilibrium is given by the intersection of all equilibria and thus the smallest equilibria.
The focus of this paper differs from those in the existing literature on time-inconsistent problems: we consider the stability of (smallest optimal) equilibria as well as the optimal values induced by these equilibria (or by the optimal equilibria). More specifically, we investigate the continuity of the optimal equilibrium and optimal value with respect to (w.r.t.) the reward function $f$ and the transition kernel $Q$ of the Markov process $X$. Our first main result, Theorem \ref{t1}, states that, with the local convergence of $f$ and $Q$ which is equipped with the total variation distance, the optimal equilibria (in terms of inclusion) is lower semicontinuous, and the optimal value function is upper semicontinuous w.r.t. $(f,Q)$. We provide examples showing that the exact continuity w.r.t. $(f,Q)$ for either the optimal equilibrium or the optimal value function may fail. Moreover, we also construct an example in which the semi-continuity fails if the convergence of $Q$ in total variation is changed to weak convergence. Let us emphasize that our first main result contrasts with the stability of the optimal value w.r.t. $(f,Q)$ under time-consistent stopping (i.e., with exponential discounting): the continuity indeed holds for time-consistent stopping in our setup, as indicated in Remark \ref{rm.exponential}.
In our second main result, Theorem \ref{thm.Qf.continuity}, we recover the continuity (under a relaxation) of the optimal value function w.r.t. $(f,Q)$ by relaxing the equilibrium concept and including $\varepsilon$-equilibria: Specifically, we show that as $(f^n,Q^n)$ uniformly converges to $(f,Q)$, it holds that $\lim_{\varepsilon\searrow 0}\lim_{n\to\infty}V_\varepsilon^{Q^n}(\cdot,f^n)=V_0^{Q}(\cdot,f)$, where $V_\varepsilon^{Q^n}(\cdot,f^n)$ is the optimal value induced by all $\varepsilon$-equilibria w.r.t. $(f^n,Q^n)$. The two limits in $\varepsilon$ and $n$ cannot be changed due to the first main result; see Remark \ref{rm.exchange.application}. To prove the second main result, we introduce the notion of pseudo $\varepsilon$-equilibrium which captures the idea of penalizing the possible deviation in the continuation region but not in the stopping region; see Definition \ref{def.equi.pseudoepsi}. It turns out that pseudo $\varepsilon$-equilibria have better properties than $\varepsilon$-equilibria: One can embed the set of pseudo-$\varepsilon$-equilibria to pseudo equilibria corresponding to a perturbed reward function; see Lemma~\ref{lm.equi.epsilon}.
A remarkable observation is that the smallest optimal pseudo equilibrium is actually the smallest optimal equilibrium; see Proposition~\ref{prop.optimalequi.pseudo}. In Example~\ref{ex:countersecond}, we demonstrate that the continuity in our second main result may fail if we replace the uniform convergence of $(f,Q)$ with locally uniform convergence. In Proposition~\ref{prop.continue.pseudo}, however, we show that if the relaxation is over the pseudo $\varepsilon$ equilibria, then the uniform convergence can be loosened.
Stability analysis is an important topic in control and optimization problems. For the stability of equilibria, let us mention the very recent works \cite{feinstein2020continuity} and \cite{feinstein2022dynamic} on Nash games. To the best of our knowledge, there is no literature so far studying the stability of equilibria for time-inconsistent (stopping) problems. In this regards, our paper provides very novel and conceptual contributions to the stability analysis in the topic of time-inconsistent problems. Our results also give a theoretical guidance for the numerical computation of optimal equilibrium values for time-inconsistent stopping: with good estimation of the reward function $f$ and transition kernel $Q$, one needs to use $\varepsilon$-equilibria instead of perfect equilibria to estimate the optimal value induced by perfect equilibria.
The rest of the paper is organized as follows. The setup and main assumptions are introduced in Section \ref{sec:set.up}, together with several preliminary lemmata. In Section \ref{sec:upper.continue}, we present our first main result, the proof of which is given in Section \ref{subsec:thm1.proof}. In Section \ref{sec:continue.epsilon}, we provide the second main result by introducing (pseudo) $\varepsilon$-equilibria. The proof of this result is collected in Section \ref{subsec:proof.continue}. Appendix gathers the proofs of lemmata in Section \ref{sec:set.up}.
\section{Setup and preliminaries}\label{sec:set.up}
Consider a measurable space $(\Omega,\mathcal{F})$ and let $X=(X_t)_{t=0,1,\dotso}$ be a time-homogeneous Markov process in discrete time, taking values in some polish space $\mathbb{X}$. Let $\mathbb{F}$ be the filtration generated by $X$. Denote $\mathcal{B}$ the class of Borel sets of $\mathbb{X}$, and $\mathbb{N}:=\{0,1,2,\dotso\}$, $\overline\mathbb{N}:=\mathbb{N}\cup\{\infty\}$, $\mathbb{R}_+:=[0,\infty)$. Let $f:\mathbb{X}\rightarrow \mathbb{R}_+$ be a reward function that may be discontinuous. Denote $||f||_\infty:=\sup_{x\in\mathbb{X}}|f(x)|$. Let $\delta:\mathbb{N}\mapsto[0,1]$ be a discount function that is decreasing with $\delta(0)=1$, $\delta(1)<1$ and $\lim_{t\to\infty}\delta(t)=0$. We further make the following assumption on the discount function $\delta(\cdot)$.
\begin{Assumption}\label{assume.delta}
$\delta(\cdot)$ is log sub-additive, i.e.,
\begin{equation}\label{eq.assume.logsubadd}
\delta(t+s)\geq \delta(t)\delta(s),\quad \forall s,t\geq 0.
\end{equation}
\end{Assumption}
\begin{Remark}
Typical discount functions, including exponential, hyperbolic, generalized hyperbolic and pseudo-exponential discounting, satisfy Assumption \ref{assume.delta}.
\end{Remark}
Given the transition kernel $Q(x,dy)$ for $X$ and a stopping time $\tau$, define
$$
v^Q(x,\tau,f):=\mathbb{E}_x^Q[\delta(\tau)f(X_\tau)],
$$
where $\mathbb{E}_x^Q$ is the expectation w.r.t. $Q$ given $X_0=x$. For $S\in{\mathcal B}$, denote
$$\rho(S):= \inf\{t\geq 1, X_t\in S\},$$
and
$$J^Q(x,S,f):=\mathbb{E}^Q_x[\delta(\rho(S))f(X_{\rho(S)})]\cdot 1_{\{x\notin S\}}+f(x)\cdot 1_{\{x\in S\}},\quad \forall x\in \mathbb{X}.$$
We provide the definition of equilibria and optimal equilibria in the following.
\begin{Definition}[Equilibria and optimal equilibria]\label{def.equilibrium}
Fix a reward function $f$ and a transition kernel $Q$. A Borel set $S\subset\mathbb{X}$ is called an equilibrium (w.r.t. $f$ and $Q$) if
\begin{equation}\label{eq.def.equilibrium}
\begin{cases}
f(x)\leq \mathbb{E}^Q_x[\delta(\rho(S))f(X_{\rho(S)})],\quad \forall x\notin S,\\
f(x)\geq \mathbb{E}^Q_x[\delta(\rho(S))f(X_{\rho(S)})],\quad \forall x\in S.
\end{cases}
\end{equation}
Denote ${\mathcal E}^Q(f)$ the set of equilibria w.r.t. $f$ and $Q$.
$S\in{\mathcal E}^Q(f)$ is called an optimal equilibrium (w.r.t. $f$ and $Q$), if for any $T\in {\mathcal E}^Q(f)$,
$$
J^Q(x, S,f)\geq J^Q(x,T,f),\quad \forall x\in \mathbb{X}.
$$
\end{Definition}
Let
\begin{equation}\label{eq.value.optima}
V^Q(x,f):=\sup_{S\in {\mathcal E}^Q(f)} J^Q(x,S,f),\quad x\in \mathbb{X},
\end{equation}
which represents the optimal value generated over all equilibria. As indicated by results in \cite{MR3911711} (also see Lemma \ref{lm.iteration.sstar}) there exists an optimal equilibria and thus the supremum for $V^Q(x,f)$ is attained universally at the optimal equilibria for all $x\in\mathbb{X}$.
In this paper, we investigate the stability of $V^Q(x,f)$ w.r.t. the transition kernel $Q$ and reward function $f$. To begin with, recall the total variation distance between two measures $\mu$ and $\nu$,
$$ ||\mu- \nu||_\text{TV} := \sup_{g\in B(\mathbb{X};[0,1])} \left \{ \int_\mathbb{X} g \, d\mu - \int_\mathbb{X} g \, d\nu \right \},$$
where $B(\mathbb{X};[0,1])$ is the set of Borel measurable functions on $\mathbb{X}$ taking values in $[-1,1]$. We will use the following notions of convergence for $f$ and $Q$ for the stability analysis of $V^Q(x,f)$.
\begin{Definition}\label{def.f}
Let $(f^n)_{n\in \overline \mathbb{N}}$ be a sequence of functions on $\mathbb{X}$. We say $f^n$ converges to $f^\infty$ locally uniformly if for any compact set $K\subset \mathbb{X}$,
$$\lim_{n\to\infty}\sup_{x\in K} |f^n(x)-f^\infty(x)|=0.$$
Recall that $f^n$ converges to $f^\infty$ uniformly if $\|f^n-f^\infty\|_{\infty}\to 0$ as $n\to\infty$.
\end{Definition}
\begin{Definition}\label{def.variation}
Let $(Q^n)_{n\in \overline \mathbb{N}}$ be a sequence of transition kernels. We say $Q^n$ converges to $Q^\infty$ locally uniformly in total variation, if for any compact set $K\subset\mathbb{X}$,
$$\lim_{n\to\infty} \sup_{x\in K}||Q^n(x,\cdot)-Q^\infty(x,\cdot)||_\text{TV}= 0.$$
We say $Q^n$ converges to $Q^\infty$ uniformly in total variation, if
$$\lim_{n\to\infty}\sup_{x\in \mathbb{X}}||Q^n(x,\cdot)-Q^\infty(x,\cdot)||_\text{TV}=0.$$
\end{Definition}
\begin{Remark}
When $\mathbb{X}$ is countable and under the discrete topology, locally uniform convergence of $(Q^n(x,y))_{n\in\overline \mathbb{N}}$ in total variation is the same as the pointwise weak convergence.
When $\mathbb{X}$ is uncountable (e.g., the process under $Q^n$ is a time-discretized diffusion), then the locally uniform convergence of $(Q^n(x,y))_{n\in\overline \mathbb{N}}$ in total variation can be implied by the following condition: There exist a reference measure $\mu$ such that any $Q^n(x,\cdot)$ has a probability density $q^n(x,\cdot)$ w.r.t. $\mu$, i.e., $Q^n(x,dy)=q^n(x,y)\mu(dy)$, and for any compact set $K\subset\mathbb{X}$,
$$
\lim_{n\to\infty}\sup_{x\in K}\int_\mathbb{X}|q^n(x,y)-q^\infty(x,y)|d\mu(y)=0.
$$
\end{Remark}
Now we present three lemmata that will be used in later sections, and their proofs are collected in Appendix \ref{appendix}.
The first lemma is an analogue of Theorem 2.2 in \cite{MR4205889} for discrete time setting, which provides the existence of an optimal equilibrium, as well as an iterative approach for its construction. To this end, define
\begin{equation}\label{e001}
\quad S^*(f,Q):= \cap_{S\in {\mathcal E}^Q(f)} S.
\end{equation}
We have the following.
\begin{Lemma}\label{lm.iteration.sstar}
Let Assumption \ref{assume.delta} hold. Suppose $f$ is bounded and non-negative, and Q is a transition kernel. Define $S_0=\emptyset$ and for $k=1,2,\dotso$,
$$S_{k+1}:=S_k\cup\left\{x\in\mathbb{X}\setminus S_k:\ f(x)>\sup_{1\leq\tau\leq\rho(S_k)}v^Q(x,\tau,f)\right\}.$$
Then $\cup_{k\in \mathbb{N}} S_k=S^*(f,Q)$. Moreover, $S^*(f,Q)$ is an optimal equilibrium, and thus
$$
V^Q(x,f)=J^Q(x, S^*(f,Q),f),\quad \forall x\in \mathbb{X}.
$$
\end{Lemma}
\begin{Remark}
Lemma \ref{lm.iteration.sstar} indicates that there exists a ``smallest'' equilibrium, which is also an optimal one. The supremum for $V^Q(x,f)$ is achieved by the same equilibrium $S^*(f,Q)$. Moreover, If the discount function is exponential, i.e., when the stopping problem \eqref{eq.intro.optimalstopping} is time-consistent, a similar discussion as that in \cite{MR4205889} would show that $S^*(f,Q)$ and $V^Q(x,f)$ would coincide with the optimal stopping region and value respectively in the classical sense.
\end{Remark}
\begin{Lemma}\label{l2}
Let $(Q^n)_{n\in \overline \mathbb{N}}$ be transition kernels.
\begin{itemize}
\item[(a)] Suppose $Q^n$ converges to $Q^\infty$ locally uniformly in total variation. Then for any $x\in\mathbb{X}$ and $T\in\mathbb{N}$,
$$\lim_{n\to\infty}\sup_{g\in B(\mathbb{X}^T;[0,1])} \left |\mathbb{E}_x^{Q^n}g(X_1,X_2,\dotso,X_T)-\mathbb{E}_x^{Q^\infty} g(X_1,X_2,\dotso,X_T)\right |=0.$$
\item[(b)]
In addition to the condition in part (a), assume that, for any compact set $K$ and $\varepsilon>0$, there exists a compact set $K'$ such that $\sup_{x\in K} Q^\infty(x, K')\geq 1-\varepsilon$.
Then for any compact set $K$ and $T\in\mathbb{N}$,
$$\lim_{n\to\infty}\sup_{x\in K, g\in B(\mathbb{X}^T;[0,1])} \left |\mathbb{E}_x^{Q^n}g(X_1,X_2,\dotso,X_T)-\mathbb{E}_x^{Q^\infty} g(X_1,X_2,\dotso,X_T)\right |=0.$$
\item[(c)] Suppose $Q^n$ converges to $Q^\infty$ uniformly in total variation.
Then for any $T\in\mathbb{N}$,
$$\lim_{n\to\infty}\sup_{x\in \mathbb{X}, g\in B(\mathbb{X}^T;[0,1])} \left |\mathbb{E}_x^{Q^n}g(X_1,X_2,\dotso,X_T)-\mathbb{E}_x^{Q^\infty} g(X_1,X_2,\dotso,X_T)\right |=0.$$
\end{itemize}
\end{Lemma}
\begin{Remark}
Suppose under $Q^\infty$,
$$X_{t+1}=h(X_t,\xi_t),$$
where $\xi_0,\xi_1,\dotso$ are i.i.d. random variables and $h:\mathbb{X}\times\mathbb{R}^d\mapsto\mathbb{X}$ is continuous. Then the additional assumption in Lemma \ref{l2}(b) is satisfied. Indeed, fix compact set $K\subset\mathbb{X}$ and $\varepsilon>0$. There exists constant $C>0$ such that $\mathbb{P}(|\xi_0|\leq C)\geq 1-\varepsilon$. Let $C':=\sup_{(x,y)\in K\times \overline{B_{C}}}|h(x,y)|<\infty$ and $K':=\overline{B_{C'}}\subset\mathbb{X}$, where $B_r$ is the ball centered at zero with radius $r$. Then $\sup_{x\in K} Q^\infty(x,K')\geq\mathbb{P}(|\xi_0|\leq C)\geq 1-\varepsilon$.
\end{Remark}
\begin{Lemma}\label{lm.tau.uniform}
Let $(Q^n)_{n\in \overline \mathbb{N}}$ be transition kernels, and $(f^n)_{m\in \overline \mathbb{N}}$ be non-negative reward functions such that $\sup_{n\in \overline \mathbb{N}} \|f^n\|_{\infty}<\infty$. Suppose Assumption \ref{assume.delta} holds.
\begin{itemize}
\item[(a)] Suppose $Q^n$ converges to $Q^\infty$ locally uniformly in total variation and $f^n$ converges to $f^\infty$ locally uniformly. Then
$$
\lim_{n\rightarrow\infty} \sup_{\tau\in \mathcal{T}} |v^{Q^n}(x,\tau,f^n)-v^{Q^\infty}(x,\tau,f^\infty)|=0,\quad \forall x\in \mathbb{X}.
$$
\item[(b)] In addition to the conditions in part (a), assume that
for any compact set $K$ and $\varepsilon>0$, there exists a compact set $K'$ such that $\sup_{x\in K} Q^\infty(x, K')\geq 1-\varepsilon$.
Then for any compact set $K$,
$$
\lim_{n\rightarrow\infty} \sup_{x\in K, \tau\in \mathcal{T}} |v^{Q^n}(x,\tau,f^n)-v^{Q^\infty}(x,\tau,f^\infty)|=0.
$$
\end{itemize}
\item[(c)]
Suppose $Q^n$ converges to $Q^\infty$ uniformly in total variation
and $\|f^n-f^\infty\|_{\infty}\rightarrow0$.
Then
$$
\lim_{n\rightarrow\infty} \sup_{x\in \mathbb{X},\tau\in \mathcal{T}} |v^{Q^n}(x,\tau,f^n)-v^{Q^\infty}(x,\tau,f^\infty)|=0.
$$
\end{Lemma}
\section{Semi-contintuity of the smallest optimal equilibrium and its associated value}\label{sec:upper.continue}
In this section, we present the first main result: the semi-continuity of $V^Q(x,f)$ and $S^*(f^\infty, Q^\infty)$ w.r.t. $f$ and $Q$. The proof is collected in Section \ref{subsec:thm1.proof}. Examples for discontinuity are also provided.
\begin{Theorem}\label{t1}
Suppose Assumption \ref{assume.delta} holds. Let $(Q^n)_{n\in \overline \mathbb{N}}$ be transition kernels, and $(f^n)_{n\in \overline \mathbb{N}}$ be non-negative reward functions with $\sup_{n\in \overline \mathbb{N}} \|f^n\|_{\infty}<\infty$. Suppose $Q^n$ converges to $Q^\infty$ locally uniformly in total variation, and $f^n$ converges to $f^\infty$ locally uniformly. Then
\begin{equation}\label{e5}
S^*(f^\infty, Q^\infty) \subset \liminf_{n\rightarrow\infty}S^*(f^n,Q^n),
\end{equation}
and
\begin{equation}\label{e6}
V^{Q^\infty}(x,f^\infty)\geq\limsup_{n\rightarrow\infty}V^{Q^n}(x,f^n),\quad \forall x\in \mathbb{X}.
\end{equation}
\end{Theorem}
\begin{Remark}\label{rm.sets.semicontinue}
We also have the semi-continuity in terms of the equilibria sets: under the conditions in Theorem \ref{t1},
$$
\limsup_{n\to \infty} {\mathcal E}^{Q^n}(f^n)\subset {\mathcal E}^{Q^\infty}(f^\infty).
$$
Indeed, for $S\in \limsup_{n\to \infty} {\mathcal E}^{Q^n}(f^n)$, there exists a subsequence $(n_k)_k$ such that $S\in {\mathcal E}^{Q^{n_k}}(f^{n_k})$, and thus
$$
\begin{cases}
f^{n_k}(x)\leq \mathbb{E}^{Q^{n_k}}[\delta(\rho(S)f^{n_k}(X_{\rho(S)}))],\quad \forall x\notin S;\\
f^{n_k}(x)\geq \mathbb{E}^{Q^{n_k}}[\delta(\rho(S)f^{n_k}(X_{\rho(S)}))],\quad \forall x\in S.
\end{cases}
$$
By Lemma \ref{lm.tau.uniform}(a), letting $k\to\infty$ we can conclude that $S\in {\mathcal E}^{Q^\infty}(f^\infty)$.
\end{Remark}
\begin{Remark}\label{rm.exponential}
If $\delta$ is exponential, i.e., $\delta(t+s)=\delta(t)\delta(s)$ for any $s,t\geq 0$, then by a similar discussion as that in \cite{MR4205889}, we have that
\begin{equation}\label{eq.expon0}
V^{Q^n}(x,f^n)= \sup_{\tau\in \mathcal{T}} \mathbb{E}^{Q^n}_x[\delta(\tau) f^n(X_\tau)].
\end{equation}
By Lemma \ref{lm.tau.uniform}(a), \eqref{eq.expon0} implies that
\begin{equation}\label{eq.expon}
\lim_{n\to\infty} V^{Q^n}(x,f^n)=V^{Q^\infty}(x,f^\infty),\quad \forall x\in \mathbb{X},
\end{equation}
which is the continuity of the optimal value function.
However, we still only have the semi-continuity for the ``smallest" optimal stopping region $S^*(f^n,Q^n)$.
\end{Remark}
We now present three examples of discontinuity. The first two examples shows that the strict inequalities in \eqref{e5} and \eqref{e6} can happen. Example \ref{eg.upper.finite} is for discontinuity w.r.t. the transition kernel, and Example \ref{eg.upper.finite1} is for discontinuity w.r.t. the reward function. Then we provide a discontinuity example under weak convergence of transition kernels, which indicates that the locally uniform convergence in total variation for transition kernels is the right assumption.
\begin{Example}\label{eg.upper.finite}
Let $\mathbb{X}=\{a,b,c\}\subset \mathbb{R}$ with $c<b<a$, $\delta(1)=1/2$ and $\delta(2)=1/3$. Define
\begin{align*}
\begin{cases}
Q^n: Q^n(c,b)=1, \quad Q^n(b,a)=p_n=1-\frac{1}{n}, \quad Q^n(b,b)=\frac{1}{n}, \quad \forall n\in \overline \mathbb{N},\\
f(a)=2, \quad f(b)=1, \quad f(c)=\frac{1}{2},
\end{cases}
\end{align*}
where $\frac{1}{\infty}:=0$, and with a bit of abuse of notation $Q(x,y):=\mathbb{P}(X_1=y|X_0=x)$. It is easy to check that $Q^n$ converges to $Q^\infty$ uniformly in total variation.
Note that any equilibrium must contain the global maximum of the reward function. By computation
$$
J^{Q^\infty} (b, \{a\},f)=1=f(b)\quad\text{and}\quad J^{Q^\infty}(c,\{a\},f)=2/3>f(c),
$$
which imply that $S^*(f,Q^\infty)=\{a\}$. Moreover, since for $n<\infty$,
$$f(b)>J^{Q^n}(b,\{a\},f)=J^{Q^n}(b,\{a,c\},f),$$
any equilibrium w.r.t. $Q^n$ for $n<\infty$ must contain $\{a,b\}$. As $f(c)>\delta(1)f(b)$, $\{a,b\}$ is not equilibrium w.r.t. $Q^n$ for $n<\infty$. Consequently, ${\mathcal E}^{Q^n}(f)=\{\mathbb{X}\}$ for $n<\infty$. Hence,
$$
S^*(f,Q^\infty)=\{c\}\subsetneqq \mathbb{X}=S^*(f,Q^n),\quad \forall n<\infty,$$
and
$$V^{Q^n}(c,f)=f(c)<J^{Q^\infty}(c,\{a\})=V^{Q^\infty}(c,f).$$
\end{Example}
\begin{Example}\label{eg.upper.finite1}
Let $\mathbb{X}=\{a,b,c\}\subset \mathbb{R}$ with $c<b<a$, $\delta(1)=1/2$ and $\delta(2)=1/3$. Define
\begin{align*}
\begin{cases}
Q(c,b)=1, \quad Q(b,a)=1, \quad Q(a,a)=1,\\
f^n(a)=2, \quad f^n(b)=1+\frac{1}{n},\quad f^n(c)=\frac{1}{2}+(1+\delta(1))\frac{1}{n},\quad \forall n\in \overline \mathbb{N}.
\end{cases}
\end{align*}
Obviously, $\|f^n-f^\infty\|_{\infty}\rightarrow\infty$.
We can compute that
$$
J^Q (b, \{a\}, f^\infty)=1=f^\infty(b), \quad J^Q(c,\{a\},f^\infty)=2/3>f^\infty(c),$$
and thus $\hat{S}^\infty=\{a\}$. Meanwhile,
$$J^Q(b,\{a\},f^n)=J^Q(b,\{a,c\},f^n)=1<f^n(b),$$
so neither $\{a\}$ nor $\{a,c\}$ belongs to ${\mathcal E}^Q(f^n)$ for $n<\infty$. By $$
f^n(c)=\frac{1}{2}+(1+\delta(1))\frac{1}{n}>\frac{1}{2}+\delta(1)\frac{1}{n}=\delta(1)f^n(b),
$$
$\{a,b\}$ is not equilibrium for all $f^n$ for $n<\infty$. Therefore, $\mathbb{X}$ is the only equilibrium w.r.t. $f^n$ for $n<\infty$. Hence,
$$S^*(f^\infty,Q)=\{c\}\subsetneqq \mathbb{X}=S^*(f,Q^n),\quad\forall\,n<\infty,$$
and
$$
\limsup_{n\rightarrow \infty}V^Q(c,f^n)=\limsup_{n\rightarrow \infty}f^n(c)=\frac{1}{2}<\frac{2}{3}=V^Q(c,f^\infty).
$$
\end{Example}
When $\mathbb{X}$ is finite, convergence locally uniformly in total variation is equivalent to weak convergence. When $\mathbb{X}$ is not finite, we provide below an example showing that the semi-continuity in Theorem \ref{t1} fails when only weak convergence is assumed. Hence, weak convergence is too weak to establish the semi-continuity in Theorem \ref{t1}.
\begin{Example}\label{eg.weakconvergence}
Let $\mathbb{X}=\{y, x_\infty,x_1,x_2,...\}\subset \mathbb{R}$, where $0\leq x_n\nearrow x_\infty$ and $y=\dfrac{x_\infty}{\delta(2)}+1$. Let $f(x)=x$. Define for $n<\infty$,
\begin{align*}
Q^n:&
\begin{cases}
Q^n(x_i, x_n)=1, \quad \text{for} \quad i\neq n,\\
Q^n(x_\infty,x_n)=1, Q^n(x_n,y)=1, Q^n(y,y)=1,
\end{cases}
\text{and} \quad
Q^\infty:&
\begin{cases} Q^\infty(x_i, x_\infty)=1, \quad \text{for} \quad \forall i,\\
Q^\infty(x_\infty, x_\infty)=1, Q^\infty(y, y)=1.
\end{cases}
\end{align*}
It can be shown that $Q^n(z,\cdot)$ weakly converges to $Q^\infty(z,\cdot)$ for any $z\in\mathbb{X}$.
However, since $Q^n(x_1,\{x_\infty\})=0$ for $n<\infty$ while $Q^\infty(x_1,\{x_\infty\})=1$, the locally uniform convergence in total variation fails.
For $n<\infty$, since $y>\dfrac{x_\infty}{\delta(2)}$, we have that
$$
\mathbb{E}^{Q^n}_{x_i}[\delta(\rho(\{y\})f(X_{\rho(\{y\}}))]=\begin{cases}
\delta(2)y, & i\in \overline \mathbb{N}\setminus\{n\}\\
\delta(1)y,&i=n
\end{cases}\ >x_\infty\geq x_i.
$$
This implies $S^*(f,Q^n)=\{y\}$ for $n<\infty$. On the other hand, denote
$$S_1:=\left\{x\in\mathbb{X}:\ f(x)>\sup_{1\leq\tau}v^{Q^\infty}(x,\tau,f)\right\}.$$
Obviously, $\{x,y\}\subset S_1$. By Lemma \ref{lm.iteration.sstar}, we have that $\{x,y\}\subset S^*(f,Q^\infty)$. Hence,
$$
\limsup_{n\rightarrow \infty} S^*(f,Q^n)\subsetneqq S^*(f,Q^\infty) \quad \text{and}\quad V^{Q^\infty}(x_\infty,f)=x_\infty<\liminf_{n\to \infty}V^{Q^\infty}(x_\infty,f)=\delta(2)y.
$$
\end{Example}
\subsection{Proof of Theorem \ref{t1}}\label{subsec:thm1.proof}
\begin{proof}[{\bf \textit{Proof of Theorem \ref{t1}}}]
For $n\in\overline\mathbb{N}$, define $S_0^n=\emptyset$ and
\begin{equation}\label{eq.thm.0}
S_{k+1}^{n}:=S_k^n\cup\left\{x\in\mathbb{X}\setminus S_k^{n}:\ f(x)>\sup_{1\leq\tau\leq\rho(S_k^{n})}v^{Q^n}(x,\tau,f^n)\right\}.
\end{equation}
By Lemma \ref{lm.iteration.sstar}, $S^*(f^n,Q^n)=\cup_k S_k^{n}=\lim_{k\to\infty}S_k^{n}$, $\forall n\in \overline \mathbb{N}$. We show by induction that
\begin{equation}\label{e3}
S_k^{\infty}\subset \liminf_{n\rightarrow\infty} S_k^{n},\quad k=0,1,\dotso,
\end{equation}
which in particular implies that $S^*(f^\infty,Q^\infty)\subset\liminf_{n\to\infty}S^*(f^n,Q^n)$.
Obviously, \eqref{e3} holds for $k=0$. Suppose it holds for $k=i$ and consider the case $k=i+1$. Take $x\in S_{i+1}^{\infty}$. If $x\in S_i^{\infty}$, then by induction hypothesis $$
x\in\liminf_{n\to\infty} S_{i}^{n}\subset \liminf_{n\to\infty} S_{i+1}^{n}.
$$
Now assume $x\notin S_i^{\infty}$. Then
\begin{equation}\label{e4}
\alpha:=f^\infty(x)-\sup_{1\leq\tau\leq\rho(S_{i}^{\infty})}v^{Q^\infty}(x,\tau,f^\infty)>0.
\end{equation}
Denote the probability measure $\mathbb{P}^n$ induced by $Q^n$. By induction hypothesis,
$$\rho({S_i^{\infty}})\geq\rho\left(\underset{1\leq n<\infty}{\cup}\left(\underset{n\leq j<\infty,}{\cap}S_i^{j}\right)\right)=\lim_{n\to\infty}\rho\left(\underset{n\leq j<\infty}{\cap}S_i^{j}\right),\quad\mathbb{P}_x^\infty-\text{a.s.}.$$
Therefore, there exists $N\in\mathbb{N}$ such that for any $n\geq N$,
\begin{equation}\label{e7}
\mathbb{P}_x^{Q^\infty}\left[\rho(S_i^{n})>\rho({S_i^{\infty}})\right]\leq\mathbb{P}_x^{Q^\infty}\left[\rho\left(\underset{n\leq j<\infty}{\cap}S_i^{j}\right)>\rho({S_i^{\infty}})\right]<\frac{\alpha}{2M},
\end{equation}
where $M:=\sup_{n\in \overline\mathbb{N}} \|f^n\|_{\infty}<\infty$.
Then for any $\tau'$ with $1\leq\tau'\leq\rho(S_i^{n})$, we have that
$$v^{Q^\infty}(x,\tau',f^\infty)\leq v^{Q^\infty}(x,\tau'\wedge \rho(S_i^{\infty}),f^\infty)+\frac{\alpha}{2}\leq \sup_{1\leq\tau\leq\rho(S_i^{\infty})}v^{Q^\infty}(x,\tau,f^\infty)+\frac{\alpha}{2},$$
and thus
$$\sup_{1\leq\tau\leq\rho(S_i^{n})}v^{Q^\infty}(x,\tau,f^\infty)\leq \sup_{1\leq\tau\leq\rho(S_i^{\infty})}v^{Q^\infty}(x,\tau,f^\infty)+\frac{\alpha}{2},\quad\forall\,n\geq N.$$
This together with \eqref{e4} implies that
\begin{equation}\label{eq.1.thm}
f^\infty(x)-\sup_{1\leq\tau\leq\rho(S_{i}^{n})}v^{Q^\infty}(x,\tau,f^\infty)\geq\frac{\alpha}{2}>0.\end{equation}
By Lemma \ref{lm.tau.uniform} part (a), for $n$ large enough, we have that
\begin{equation}\label{eq.2.thm}
\begin{aligned}
\left|\sup_{1\leq\tau\leq\rho(S_{i}^{n})}v^{Q^\infty}(x,\tau,f^\infty)-\sup_{1\leq\tau\leq\rho(S_{i}^{n})}v^{Q^n}(x,\tau,f^n)\right|\leq & \sup_{1\leq\tau\leq\rho(S_{i}^{n})}\left|v^{Q^\infty}(x,\tau,f^\infty)-v^{Q^n}(x,\tau,f^n)\right| \\
\leq & \sup_{\tau\in\mathcal{T}}\left|v^{Q^\infty}(x,\tau,f^\infty)-v^{Q^n}(x,\tau,f^n)\right|<\frac{\alpha}{3}.
\end{aligned}
\end{equation}
Meanwhile, we can choose $N'$ such that for all $n\geq N'$ \eqref{eq.2.thm} holds and
\begin{equation}\label{eq.3.thm}
|f^n(x)-f^\infty(x)|\leq \frac{\alpha}{12}.
\end{equation}
Thus, for all $n\geq \max\{N, N'\}$, combine \eqref{eq.1.thm}, \eqref{eq.2.thm} and \eqref{eq.3.thm},
\begin{align*}
f^n(x)- \sup_{1\leq\tau\leq\rho(S_{i}^{n})}v^{Q^n}(x,\tau,f^n)=& f^n(x)-f^\infty(x)+f^\infty(x)-\sup_{1\leq\tau\leq\rho(S_{i}^{n})}v^{Q^\infty}(x,\tau,f^\infty)\\
&+\sup_{1\leq\tau\leq\rho(S_{i}^{n})}v^{Q^\infty}(x,\tau,f^\infty)-\sup_{1\leq\tau\leq\rho(S_{i}^{n})}v^{Q^n}(x,\tau,f^n)\\
\geq&-\frac{\alpha}{12}+\frac{\alpha}{2}-\frac{\alpha}{3}>0.
\end{align*}
Consequently, for $n$ large enough, no matter $x$ is in $S_i^{n}$ or not, we always have $x\in S_{i+1}^{n}$, and thus $x\in\liminf_{n\to\infty} S_{i+1}^{n}$. By the arbitrariness of $x$, \eqref{e3} holds for $k=i+1$. We have proved \eqref{e5}.
Now let $\varepsilon>0$ and $x\notin S^*(f^\infty,Q^\infty)$. Following the argument in \eqref{e7}, we can show that there exists $N\in\mathbb{N}$ such that for any $n>N$,
\begin{equation}\label{e8}
\mathbb{P}_x^{Q^\infty}\left[\rho(S^*(f^n,Q^n))>\rho({S^*(f^\infty,Q^\infty)})\right]<\frac{\varepsilon}{2M}.
\end{equation}
Then there exists $N'>N$ such that for any $n>N'$,
\begin{align*}
v^{Q^\infty}(x,\rho(S^*(f^\infty,Q^\infty)))\geq &v^{Q^\infty}(x,\rho(S^*(f^\infty,Q^\infty)\cup S^*(f^n,Q^n)))\geq v^{Q^\infty}(x,\rho(S^*(f^n,Q^n)))-\frac{\varepsilon}{2}\\
\geq & v^{Q^n}(x,\rho(S^*(f^n,Q^n)))-\varepsilon,
\end{align*}
where the first inequality follows from \cite[Lemma 3.1]{MR4116459} (or Lemma \ref{lm.equi.pseudo}), the second inequality follows from \eqref{e8}, the third inequality follows from Lemma \ref{lm.tau.uniform} part (a). As a result,
$$v^{Q^\infty}(x,\rho(S^*(f^\infty,Q^\infty)))\geq \limsup_{n\to\infty}v^{Q^\infty}(x,\rho(S^*(f^n,Q^n)))-\varepsilon.$$
By the arbitrariness of $\varepsilon$, we have \eqref{e6} holds.
\end{proof}
\section{Continuity under a relaxed limit}\label{sec:continue.epsilon}
As shown in the previous section, $V^Q(x,f)$ is not continuous w.r.t. $Q$ or $f$ in general. To achieve the stability, we need to relax the equilibrium set over which we take supremum.
\begin{Definition}\label{def.equi.epsi}
Fix a reward function $f$ and a transition kernel $Q$. Take $\varepsilon\geq 0$. A Borel set $S$ is called an $\varepsilon$-equilibrium (w.r.t. $f$ and $Q$), if
\begin{equation} \label{eq.def.epsequi}
\begin{cases}
f(x)\leq \mathbb{E}^Q_x[\delta(\rho(S))f(X_{\rho(S)})]+\varepsilon,\quad \forall x\notin S,\\
f(x)+\varepsilon\geq \mathbb{E}^Q_x[\delta(\rho(S))f(X_{\rho(S)})],\quad \forall x\in S.
\end{cases}
\end{equation}
Define
$$
{\mathcal E}^Q(f,\varepsilon):=\{\text{$S$ is an $\varepsilon$-equilibrium w.r.t. }f \text{ and }Q\}.
$$
When $\varepsilon=0$, we still call $S$ an equilibrium and may use the notation ${\mathcal E}^Q(f)$ instead of ${\mathcal E}^Q(f,0)$.
\end{Definition}
We also need the following notion of pseudo $\varepsilon$-equilibria, which loosens the criterion of $\varepsilon$-equilibrium by giving up the condition in \eqref{eq.def.epsequi} when $x\in S$.
\begin{Definition}\label{def.equi.pseudoepsi}
Fix a reward function $f$ and a transition kernel $Q$. Take $\varepsilon\geq 0$. A Borel set $S\subset\mathbb{X}$ is called a pseudo $\varepsilon$-equilibrium (w.r.t. $f$ and $Q$), if
\begin{equation}\label{eq.defem.outS}
f(x)\leq \mathbb{E}^Q_x[\delta(\rho(S))f(X_{\rho(S)})]+\varepsilon,\quad \forall x\notin S.
\end{equation}
Define
$$
{\mathcal G}^Q(f,\varepsilon):=\{\text{$S$ is a pseudo $\varepsilon$-equilibrium w.r.t. $f$ and $Q$}\}.
$$
When $\varepsilon=0$, we simply call $S$ is a pseudo equilibrium, and write ${\mathcal G}^Q(f)$ short for ${\mathcal G}^Q(f,0)$. We say $S\in {\mathcal G}^Q(f)$ is an optimal pseudo equilibrium (w.r.t. $f$ and $Q$), if for any $T\in {\mathcal G}^Q(f)$,
$$
J(x,S,f)\geq J(x,T,f),\quad \forall x\in \mathbb{X}.
$$
\end{Definition}
Now define
\begin{equation}\label{eq.def.WVepsilon}
W^Q_\varepsilon(x,f):=\sup_{S\in {\mathcal G}^Q(f,\varepsilon)} J^Q(x,S,f);\quad V^Q_\varepsilon(x,f):=\sup_{S\in {\mathcal E}^Q(f,\varepsilon)} J^Q(x,S,f).
\end{equation}
When $\varepsilon=0$ we write $W^Q(x,f)$ instead of $W^Q_0(x,f)$, and we keep using the notation $V^Q(x,f)$ in \eqref{eq.value.optima} instead of $V^Q_0(x,f)$.
Pseudo $\varepsilon$-equilibria have better properties than $\varepsilon$-equilibria. As we will see in Lemma~\ref{lm.equi.epsilon} below one can embed the set of pseudo-$\varepsilon$-equilibria to pseudo equilibria corresponding to a perturbed reward function. We will also observe that the smallest optimal pseudo equilibrium is actually the smallest optimal equilibrium in Proposition~\ref{prop.optimalequi.pseudo}. These two results form the backbone of the proof of the second main result which we state below. The proof of this result is provided in Section \ref{subsec:proof.continue}.
\begin{Theorem}\label{thm.Qf.continuity}
Suppose Assumption \ref{assume.delta} holds. Let $(Q^n)_{n\in \overline \mathbb{N}}$ be transition kernels, and $(f^n)_{n\in \overline \mathbb{N}}$ be bounded and non-negative reward functions.
Suppose $Q^n$ converges to $Q^\infty$ uniformly in total variation, and $\|f^n-f^\infty\|_{\infty}\rightarrow0$. Then
\begin{align*}
&\lim\limits_{\varepsilon\searrow 0}\Big( \liminf_{n\rightarrow \infty} V^{Q^n}_\varepsilon(x,f^n)\Big)= \lim\limits_{\varepsilon\searrow 0}\Big( \liminf_{n\rightarrow \infty} W^{Q^n}_\varepsilon(x,f^n)\Big)\\
=&\lim\limits_{\varepsilon\searrow 0}\Big( \limsup_{n\rightarrow \infty}V^{Q^n}_\varepsilon(x,f^n)\Big)= \lim\limits_{\varepsilon\searrow 0}\Big( \limsup_{n\rightarrow \infty} W^{Q^n}_\varepsilon(x,f^n)\Big)\\
=&V^{Q^\infty}(x,f^\infty),\quad \forall x\in \mathbb{X}.
\end{align*}
\end{Theorem}
Letting $f^n=f$ and $Q^n=Q$ for $n\in \overline \mathbb{N}$ in Theorem \ref{thm.Qf.continuity}, we achieve the following corollary, which shows that $V^Q(x,f)$ is indeed the limit of the supremum value over all $\varepsilon$-equilibria as $\varepsilon\searrow 0$.
\begin{Corollary}\label{cor}
Suppose Assumption \ref{assume.delta} holds. Given a bounded reward function $f\geq 0$ and a transition kernel $Q$, we have that
$$
\lim_{\varepsilon\searrow 0}V^Q_\varepsilon(x,f)=\lim_{\varepsilon\searrow 0}W^Q_\varepsilon(x,f)=V^Q(x,f),\quad \forall x\in \mathbb{X}.
$$
\end{Corollary}
\begin{Remark}\label{rm.exchange.application}
Combining Theorem \ref{t1} and Corollary \ref{cor}, we have
$$
\limsup\limits_{n\rightarrow \infty}\Big( \lim_{\varepsilon\searrow 0}V^{Q^n}_\varepsilon(x,f^n)\Big)=\limsup\limits_{n\rightarrow \infty} V^{Q^n}(x,f^n)\leq V^{Q^\infty}(x,f^\infty),\quad \forall x\in\mathbb{X}.
$$
Recall that the strict inequality above can be achieved as shown in Examples \ref{eg.upper.finite} and \ref{eg.upper.finite1}. Hence, together with Theorem \ref{thm.Qf.continuity}, we see that
the order of taking $\varepsilon\searrow 0$ and taking $n\rightarrow \infty$ cannot be exchanged.
Moreover, the main results in this paper provide a guideline for numerical approximation for $V^{Q^\infty}(x,f^\infty)$: With good approximations of the transition kernel $Q^\infty$ and reward function $f^\infty$, taking supremum only over equilibria may not provide good estimation for the target optimal value. Instead, one should take supremum over all $\varepsilon$-equilibria.
\end{Remark}
\begin{Remark}
Analogous to Remark \ref{rm.sets.semicontinue}, if the same conditions in Theorem \ref{thm.Qf.continuity} hold, then
$$
\lim_{\varepsilon\searrow 0}\left(\liminf_{n\to \infty} {\mathcal E}^{Q^n}_\varepsilon (f^n)\right)=\lim_{\varepsilon\searrow 0}\left(\limsup_{n\to \infty} {\mathcal E}^{Q^n}_\varepsilon (f^n)\right)={\mathcal E}^{Q^\infty} (f^\infty).
$$
\begin{proof}
By a similar argument as in Remark \ref{rm.sets.semicontinue}, we can show that
$$
\lim_{\varepsilon\searrow 0}\left(\limsup_{n\to \infty} {\mathcal E}^{Q^n}_\varepsilon (f^n)\right)\subset{\mathcal E}^{Q^\infty} (f^\infty).
$$
It remains to show that
\begin{equation}\label{eq.sets}
{\mathcal E}^{Q^\infty} (f^\infty)\subset \lim_{\varepsilon\searrow 0}\left(\liminf_{n\to \infty} {\mathcal E}^{Q^n}_\varepsilon (f^n)\right).
\end{equation}
For $S\in {\mathcal E}^{Q^\infty} (f^\infty)$, we have
$$
\begin{cases}
f^{\infty}(x)\leq \mathbb{E}^{Q^{\infty}}[\delta(\rho(S)f^{\infty}(X_{\rho(S)}))],\quad \forall x\notin S;\\
f^{\infty}(x)\geq \mathbb{E}^{Q^{\infty}}[\delta(\rho(S)f^{\infty}(X_{\rho(S)}))],\quad \forall x\in S.
\end{cases}
$$
Then for any $\varepsilon>0$, Lemma \ref{lm.tau.uniform} implies that, for $n$ big enough,
$$
\begin{cases}
f^{n}(x)-\varepsilon\leq \mathbb{E}^{Q^{n}}[\delta(\rho(S)f^{n}(X_{\rho(S)}))],\quad \forall x\notin S;\\
f^{n}(x)+\varepsilon\geq \mathbb{E}^{Q^{n}}[\delta(\rho(S)f^{n}(X_{\rho(S)}))],\quad \forall x\in S.
\end{cases}.
$$
Consequently, $S\in\liminf_{n\to \infty} {\mathcal E}^{Q^n}_\varepsilon (f^n)$ for any $\varepsilon>0$, which implies \eqref{eq.sets}.
\end{proof}
\end{Remark}
The following example shows that the continuity result in Theorem \ref{thm.Qf.continuity} may fail if the convergence of $(Q_n)_{n\in \mathbb{N}}$ in total variation is only assumed to be locally uniform instead of uniform.
\begin{Example}\label{ex:countersecond}
Let $\mathbb{X}=\{y, x_0, x_1,x_2,...\}\subset \mathbb{R}$. Define
\begin{align*}
&Q^n:
\begin{cases}
Q^n(x_i, x_{i+1})=\frac{1}{2}, Q^n(x_i,y)=\frac{1}{2},\quad & 0\leq i< n,\\
Q^n(x_i,y)=1,\quad & i> n\\
Q^n(x_n,x_n)=1, Q^n(y,y)=1.
\end{cases};\\
&Q^\infty:
\begin{cases}
Q^\infty(x_i, x_{i+1})=\frac{1}{2}, Q^\infty(x_i, y)=\frac{1}{2}, \quad \forall i\geq 0,\\
Q^\infty(y, y)=1.
\end{cases}
\end{align*}
One can easily see that $Q^n$ converges to $Q^\infty$ locally uniformly, but not uniformly. Let $f(x_i)=1$ for $i\in\mathbb{N}$, $f(y)=2.99$, and $\delta(k)=\frac{1}{1+k}$ for $k\in\mathbb{N}$.
We have
$\frac{1}{2}\delta(1)(1+f(y))=\frac{3.99}{4}<1$, and
$$
\sum_{k=1}^\infty \delta(k)\left(\frac{1}{2}\right)^k f(y)>\sum_{k=1}^3 \delta(k)\left(\frac{1}{2}\right)^k f(y)=2.99\left(\frac{1}{4}+\frac{1}{12}+\frac{1}{32}\right)>1.
$$
That is,
\begin{equation}\label{eq.eg.M}
\frac{1}{2}\delta(1)(1+f(y))<1<\sum_{k=1}^\infty \delta(k)\left(\frac{1}{2}\right)^k f(y).
\end{equation}
Take $\varepsilon$ with $0<\varepsilon<1-\frac{1}{2}\delta(1)(1+f(y))$. For any $n<\infty$ and $S\in {\mathcal E}^{Q^n}(f,\varepsilon)$, it is easy to check that $y,x_n\in S$. For any $i\leq n$, if $x_i\in S$, then by the first inequality in \eqref{eq.eg.M}, $x_{i-1}\in S$. Hence, for any $n<\infty$,
$$
\{x_0,x_1,...,x_n\}\subset S,\quad \forall S\in {\mathcal E}^{Q^n}(f,\varepsilon).
$$
As the above holds for any $\varepsilon$ with $0<\varepsilon<1-\frac{1}{2}\delta(1)(1+f(y))$, we have that
$$
\limsup_{n\rightarrow \infty}V_\varepsilon^n(x_0)=f(x_0),\quad \forall n<\infty,
$$
which leads to
\begin{equation}\label{eq.eg.x0}
\limsup_{\varepsilon\searrow 0}\limsup_{n\rightarrow \infty}V_\varepsilon^n(x_0)=f(x_0).
\end{equation}
On the other hand, the second inequality in \eqref{eq.eg.M} indicates $J^\infty(x_i, \{y\})>f(x_i)$ for any $i\in\mathbb{N}$. This together with $J^\infty(y, \{y\})<f(y)$ implies that
$$\hat{S}^\infty=\{y\}\quad\text{and}\quad V^\infty (x_0)=J^\infty(x_0,\{y\})=\sum_{k=1}^\infty \delta(k)\left(\frac{1}{2}\right)^k f(y).$$
Then by \eqref{eq.eg.x0} and the second inequality in \eqref{eq.eg.M},
$$
\limsup_{\varepsilon\searrow 0}\limsup_{n\rightarrow \infty}V_\varepsilon^n(x_0)<V^\infty(x_0).
$$
\end{Example}
However, if we use $W^{Q^n}_\varepsilon(.,f^n)$ (instead of $V^{Q^n}_\varepsilon(.,f^n)$) to approximate $V^{Q^\infty}(.,f^\infty)$, then we can weaken the uniform convergence in total variation condition to locally uniform convergence as shown in the following proposition.
\begin{Proposition}\label{prop.continue.pseudo}
Suppose the conditions for $(f^n)_{n\in \overline \mathbb{N}}$ and $\delta$ in Theorem \ref{thm.Qf.continuity} hold, and $Q^n$ converges to $Q^\infty$ locally uniformly in total variation. Assume that for any compact set $K$ and $\varepsilon>0$, there exists a compact set $K'$ such that $\sup_{x\in K} Q^\infty(x, K')\geq 1-\varepsilon$.
Then
\begin{align*}
& \lim\limits_{\varepsilon\searrow 0}\Big( \liminf_{n\rightarrow \infty} W^{Q^n}_\varepsilon(x,f^n)\Big)
= \lim\limits_{\varepsilon\searrow 0}\Big( \limsup_{n\rightarrow \infty} W^{Q^n}_\varepsilon(x,f^n)\Big)
=V^{Q^\infty}(x,f^\infty),\quad \forall x\in \mathbb{X}.
\end{align*}
\end{Proposition}
The proof of Proposition \ref{prop.continue.pseudo} is presented in Section \ref{subsec:proof.continue}
\subsection{Proofs of Theorem \ref{thm.Qf.continuity} and Proposition \ref{prop.continue.pseudo}}\label{subsec:proof.continue}
To prepare for the proofs of Theorem \ref{thm.Qf.continuity} and Proposition \ref{prop.continue.pseudo}, we first provide some auxiliary results for (pseudo) $\varepsilon$-equilibria.
\begin{Lemma}\label{lm.equi.pseduvs}
Fix a bounded reward function $f$ and a transition kernel $Q$. We have that
$$
{\mathcal E}^Q(f,\varepsilon)\subset {\mathcal G}^Q(f,\varepsilon),\quad \forall \varepsilon\geq 0,
$$
and
$$
V^Q_\varepsilon(x,f)\leq W^Q_\varepsilon(x,f),\quad \forall x\in \mathbb{X},\forall \varepsilon\geq 0.
$$
\end{Lemma}
\begin{proof}
The result directly follows from Definitions \ref{def.equi.epsi} and \ref{def.equi.pseudoepsi}.
\end{proof}
\begin{Lemma}\label{lm.equi.pseudo}
Let Assumption \ref{assume.delta} hold. Let $f\geq 0$ be a bounded reward function and $Q$ be a transition kernel.
\begin{itemize}
\item[(a)]
Given $S,T\in {\mathcal G}^Q(f)$, we have that
$
S\cap T\in {\mathcal G}^Q(f).
$
\item[(b)]
Let $S,R\in{\mathcal B}$ such that $S\in {\mathcal G}^Q(f)$ and $R\supset S$. Then
$$
J^Q(x,S,f)\geq J^Q(x,R,f),\quad \forall x\in \mathbb{X}.
$$
\end{itemize}
\end{Lemma}
\begin{proof}
Part (a):
We can use the same argument as that in the proof of \cite[lemma 4.1]{MR3911711} to get that
$$
J(x,S\cap T)\geq J(x, S)\vee J(x,T)\geq f(x), \quad\forall\, x\notin S\cap T,
$$
which implies $S\cap T\in {\mathcal G}^Q(f)$.
Part (b): Notice that $J^Q(x,S,f)=f(x)=J^Q(x,R,f)$, for all $x\in S$. For $x\notin S$, same discussion in the proof of \cite[Lemma 3.1]{MR4116459} (or \cite[Lemma 4.1]{MR4250561} ) can be applied to reach that
$$
J^Q(x, S,f)\geq J^Q(x,R,f).
$$
\end{proof}
Define
$$S_*(f,Q):=\cap_{s\in {\mathcal G}^Q(f)} S.$$
Recall the smallest optimal equilibrium, $S^*(f,Q)=\cap_{s\in {\mathcal E}^Q(f)} S$ defined in \eqref{e001}. The following proposition shows that $S_*(f,Q)$ is optimal among all pseudo equilibria and also coincides with $S^*(f,Q)$.
\begin{Proposition}\label{prop.optimalequi.pseudo}
Let Assumption \ref{assume.delta} hold. Given a bounded reward function $f\geq 0$ and a transition kernel $Q$, we have that
$$S_*(f,Q)=S^*(f,Q)\quad\text{and}\quad W^Q(x,f)=J^Q(x,S_*(f,Q),f)=V^Q(x,f),\ \forall x\in \mathbb{X}.
$$
\end{Proposition}
\begin{proof
By Lemma \ref{lm.equi.pseduvs}, ${\mathcal E}^Q(f)\subset {\mathcal G}^Q(f)$ and thus $S_*(f,Q)\subset S^*(f,Q)$. We show $S^*(f,Q)\subset S_*(f,Q)$ by the iterative construction for $S^*(f,Q)$. Recall $S^*(f,Q)=\cup_{n\in \mathbb{N}} S_n$ in Lemma \ref{lm.iteration.sstar}, where $(S_n)_{n\in \mathbb{N}}$ is an increasing sequence defined as $S_0=\emptyset$, and
$$
S_{n+1}:=\{x\in \mathbb{X}\setminus S_n: f(x)>\sup_{S:S_n\subset S\subset \mathbb{X}\setminus\{x\}} J^Q(x,S,f) \}, \quad n\in\mathbb{N}.
$$
For any $R\in {\mathcal G}^Q(f)$, we prove by induction that
\begin{equation}\label{e002}
S_n\subset R,\quad\forall\,n\in \mathbb{N}.
\end{equation}
We have $S_0=\emptyset\subset R$. Suppose $S_n\subset R$, then for any $x\notin R$,
$$
f(x)\leq J^Q(x,R,f)\leq \sup_{S:S_n\subset S\subset \mathbb{X}\setminus\{x\}} J^Q(x,S,f),
$$
and thus $x\notin S_{n+1}$. Therefore, $S_{n+1}\subset R$.
By \eqref{e002}, $S^*(f,Q)=\cup_{n\geq 0}S_n\subset R$ for any $R\in {\mathcal G}^Q(f)$, which implies $S^*(f,Q)\subset S_*(f,Q)$. Hence, $S_*(f,Q)=S^*(f,Q)$. Moroever, for any $S\in {\mathcal G}^Q(f)$, by Lemma \ref{lm.equi.pseudo} part (b),
$$
J^Q(x,S_*(f,Q),f)\geq J^Q(x,S,f),\quad \forall x\in \mathbb{X},
$$
so $J^Q(.,S_*(f,Q),f)=W^Q(.,f)$. Together with Lemma \ref{lm.iteration.sstar}, we have that
$$
W^Q(x,f)=J^Q(x,S_*(f,Q),f)=J^Q(x,S^*(f,Q),f)=V^Q(x,f),\quad \forall x\in \mathbb{X}.
$$
\end{proof}
\begin{Lemma}\label{lm.f.epsilon}
Suppose Assumption \ref{assume.delta} holds. For any $0\leq \varepsilon_1\leq \varepsilon_2$, we have that
\begin{equation}\label{eq.fepsilon.equiset}
{\mathcal G}^Q((f-\varepsilon_1)\vee 0)\subset {\mathcal G}^Q((f-\varepsilon_2)\vee 0).
\end{equation}
Therefore,
\begin{equation}\label{eq.fepsilon.optimalequi}
S_*((f-\varepsilon_1)\vee 0,Q)\supseteq S_*((f-\varepsilon_2)\vee 0,Q).
\end{equation}
\end{Lemma}
\begin{proof}
Let $S\in {\mathcal G}^Q(f-\varepsilon_1)$. For any $x\notin S$,
\begin{align*}
&|J^Q(x,S,(f-\varepsilon_1)\vee 0)-J^Q(x,S,(f-\varepsilon_2)\vee 0)| \\
= & \mathbb{E}^x\left[\delta(\rho(S)) \left(\left(\left(f\left(X_{\rho(S)}\right)-\varepsilon_1\right)\vee 0\right)-\left(\left(f\left(X_{\rho(S)}\right)-\varepsilon_2\right)\vee 0\right)\right)\right]\\
\leq & \mathbb{E}^x[\delta(\rho(S))(\varepsilon_2-\varepsilon_1)]
\leq \varepsilon_2-\varepsilon_1.
\end{align*}
If $f(x)\geq \varepsilon_2$, then
\begin{align*}
&J^Q(x,S,(f-\varepsilon_2)\vee 0)\geq J^Q(x,S,(f-\varepsilon_1)\vee 0)-(\varepsilon_2-\varepsilon_1)\\
\geq &f(x)-\varepsilon_1-(\varepsilon_2-\varepsilon_1)=f(x)-\varepsilon_2,
\end{align*}
where the second inequality follows that $S\in {\mathcal G}^Q(f-\varepsilon_1)$.
If $f(x)<\varepsilon_2$, then $J^Q(x,S,(f-\varepsilon_2)\vee 0)\geq 0=(f(x)-\varepsilon_2)\vee 0$. Hence, $S\in {\mathcal G}^Q(f-\varepsilon_2)$.
\end{proof}
\begin{Lemma}\label{lm.optimal.pseudo}
Suppose Assumption \ref{assume.delta} holds. Given a bounded reward function $f\geq 0$ and a transition kernel $Q$, we have that
\begin{equation}\label{eq.continuef.optimalequi}
S^*((f-\varepsilon)\vee 0,Q)=S_*((f-\varepsilon)\vee 0,Q) \uparrow S_*(f,Q)=S^*(f,Q), \quad \text{as}\; \varepsilon \searrow 0,
\end{equation}
and
\begin{equation}\label{eq.continuef.value}
\lim_{\varepsilon \searrow 0}V^Q(x,(f-\varepsilon)\vee 0)=V^Q(x,f),\quad \forall x\in \mathbb{X}.
\end{equation}
\end{Lemma}
\begin{proof}
As for \eqref{eq.continuef.optimalequi}, by Lemma \ref{lm.f.epsilon}, $S_*((f-\varepsilon)\vee 0,Q)$ increases as $\varepsilon\searrow0$, so
$$
S':=\cup_{\varepsilon>0 } S_*((f-\varepsilon)\vee 0,Q)\subset S_*(f,Q).
$$
Given $x\notin S'$,
\begin{align*}
\mathbb{E}^x[\delta(\rho(S'))f(X_{\rho(S')})]=&\lim\limits_{\varepsilon\searrow 0}E^x[\delta(\rho(S_*((f-\varepsilon)\vee 0,Q)))((f(X_{\rho(S_*((f-\varepsilon)\vee 0,Q))})-\varepsilon)\vee 0)]\\
=&\lim\limits_{\varepsilon\searrow 0}J^Q(x,S_*((f-\varepsilon)\vee 0,Q),(f-\varepsilon)\vee 0)
\geq \lim\limits_{\varepsilon\searrow 0} (f(x)-\varepsilon)\vee 0\\
=&f(x),
\end{align*}
where the second line follows that $x\notin S_*((f-\varepsilon)\vee 0,Q)$. Hence, $S'\in {\mathcal G}^Q(f)$ and $S_*(f,Q)\subset S'$, which implies $S'=S_*(Q,f)$. Then by Proposition \ref{prop.optimalequi.pseudo},
$$
S^*((f-\varepsilon)\vee 0,Q) =S_*((f-\varepsilon)\vee 0,Q) \uparrow S_*(f,Q)=S^*(f,Q) ,\quad \text{as}\; \varepsilon \searrow 0.
$$
Now we prove \eqref{eq.continuef.value}. By \eqref{eq.continuef.optimalequi}, for $x\in S^*(f,Q)$, $x\in S^*((f-\varepsilon)\vee 0,Q)$ for $\varepsilon$ small enough, and thus
$$\lim_{\varepsilon \searrow 0}V^Q(x,(f-\varepsilon)\vee 0)=\lim_{\varepsilon \searrow 0}(f(x)-\varepsilon)\vee 0=f(x)=V^Q(x,f),\quad \forall x\in S_*(f,Q).$$
For $x\notin S_*(f,Q)$, by \eqref{eq.continuef.optimalequi}, $\rho(S^*(f-\varepsilon)\vee 0,Q)\rightarrow \rho(S^*(f,Q))$ a.s. and $(f-\varepsilon)\vee 0\rightarrow f$ as $\varepsilon\searrow 0$. Then by Dominated Convergence Theorem,
\begin{align*}
\lim\limits_{\varepsilon\searrow 0}V^Q(x,(f-\varepsilon)\vee 0)=&\lim\limits_{\varepsilon\searrow 0} E^x[\delta(\rho(S^*((f-\varepsilon)\vee 0,Q)))((f(X_{\rho(S^*((f-\varepsilon)\vee 0,Q))})-\varepsilon)\vee 0)]\\
=& \mathbb{E}^x[\delta(\rho(S^*(f,Q)))f(X_{\rho(S^*(f,Q))})]=V^Q(x,f),\quad x\notin S_*(f,Q).
\end{align*}
which completes the proof of \eqref{eq.continuef.value}.
\end{proof}
\begin{Lemma}\label{lm.equi.epsilon}
Suppose Assumption \ref{assume.delta} holds. Let $f\geq 0$ be a bounded reward function and $Q$ be a transition kernel. Then for any $\varepsilon>0$, we have that
$$
{\mathcal G}^Q(f)\subset {\mathcal G}^Q((f-\varepsilon)\vee 0)\subset {\mathcal G}^Q(f,\varepsilon)\subset {\mathcal G}^Q\left(\left(f-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right).
$$
\end{Lemma}
\begin{proof}
${\mathcal G}^Q(f)\subset {\mathcal G}^Q((f-\varepsilon)\vee 0)$ follows Lemma \ref{lm.f.epsilon}.
Let $S\in {\mathcal G}^Q((f-\varepsilon)\vee 0)$. For any $x\notin S$, if $f(x)\geq \varepsilon$, then
$$
\mathbb{E}_x^Q[\delta(\rho(S)) f(X_{\rho(S)})]\geq \mathbb{E}_x^Q[\delta(\rho(S)) ((f(X_{\rho(S)})-\varepsilon)\vee 0)]\geq (f(x)-\varepsilon)\vee 0=f(x)-\varepsilon.
$$
If $f(x)<\varepsilon$, obviously, $ \mathbb{E}_x^Q[\delta(\rho(S)) f(X_{\rho(S)})]\geq 0>f(x)-\varepsilon$. So $S\in {\mathcal G}^Q(f,\varepsilon)$.
Let $S\in {\mathcal G}^Q(f,\varepsilon)$. Take $x\notin S$. If $f(x)\geq \frac{\varepsilon}{1-\delta(1)}$, then by $\rho(S)\geq 1$ we have that
\begin{align*}
\mathbb{E}_x^Q\left[\delta(\rho(S)) \left(\left(f(X_{\rho(S)})-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right)\right]\geq &\mathbb{E}_x^Q[\delta(\rho(S)) f(X_{\rho(S)})]-\delta(1)\cdot \frac{\varepsilon}{1-\delta(1)}\\
\geq & f(x)-\varepsilon-\frac{\delta(1)\varepsilon}{1-\delta(1)}
= f(x)-\frac{\varepsilon}{1-\delta(1)},
\end{align*}
where the second line follows from $S\in {\mathcal G}^Q(f,\varepsilon)$. If $f(x)<\frac{\varepsilon}{1-\delta(1)}$, then $$
\mathbb{E}_x^Q\left[\delta(\rho(S)) \left(\left(f(X_{\rho(S)})-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right)\right]\geq0=\left(f(x)-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0.
$$
Hence, $S\in {\mathcal G}^Q((f-\frac{\varepsilon}{1-\delta(1)})\vee 0)$.
\end{proof}
\begin{proof}[{\bf \textit{Proof of Theorem \ref{thm.Qf.continuity}}}]
The proof is a combination of the following two steps.
Step 1.
We first prove, under assumptions in Theorem \ref{thm.Qf.continuity}, that
\begin{equation}\label{eq.prop.liminf}
V^{Q^\infty}(x,f^\infty)\leq \liminf_{n\rightarrow \infty} V^{Q^n}_\varepsilon(x,f^n)\leq \liminf_{n\rightarrow \infty} W^{Q^n}_\varepsilon(x,f^n), \quad \forall \varepsilon>0.
\end{equation}
Let $\varepsilon>0$. Applying Lemma \ref{lm.tau.uniform}(c) with $\tau=\rho(S^*(f^\infty,Q^\infty))$, there exists $N\in \mathbb{N}$ such that
$$
\sup_{x\in \mathbb{X}} |v^{Q^n}(x, \rho(S^*(f^\infty,Q^\infty)), f^n)-v^{Q^\infty}(x, \rho(S^*(f^\infty,Q^\infty)), f^\infty)|\leq\varepsilon.
$$
Then
\begin{align*}
v^{Q^n}(x, \rho(S^*(f^\infty,Q^\infty)), f^n)&\geq v^{Q^\infty}(x, \rho(S^*(f^\infty,Q^\infty)), f^\infty)-\varepsilon\geq f(x)-\varepsilon,\quad \forall x\notin S^*(f^\infty,Q^\infty),\\
v^{Q^n}(x, \rho(S^*(f^\infty,Q^\infty)), f^n)&\leq v^{Q^\infty}(x, \rho(S^*(f^\infty,Q^\infty)), f^\infty)+\varepsilon\leq f(x)+\varepsilon,\quad \forall x\in S^*(f^\infty,Q^\infty).
\end{align*}
Hence, $S^*(f^\infty,Q^\infty)\in {\mathcal E}^{Q^n}_\varepsilon(f^n)$ for all $n\geq N$.
Now take $x\in \mathbb{X}$. For $n\geq N$, by Definition \ref{def.equi.pseudoepsi} and \eqref{eq.def.WVepsilon},
$$
V^{Q^n}_\varepsilon(x,f^n)\geq J^{Q^n}(x,S^*(f^\infty,Q^\infty),f^n)
$$
which leads to
$$
\liminf_{n\rightarrow \infty} V^{Q^n}_\varepsilon(x,f^n)\geq \liminf_{n\rightarrow \infty} J^{Q_n}(x,S^*(f^\infty,Q^\infty),f^n) =V^{Q^\infty}(x,f^\infty)
$$
where the second (in)equality follows from Lemma \ref{lm.tau.uniform}(a).
By Lemma \ref{lm.equi.pseduvs}, $W^{Q^n}_\varepsilon(x,f^n)\geq V^{Q^n}_\varepsilon(x,f^n)$, and Step 1 is completed.
Step 2. Now we show, under the same assumptions in Theorem \ref{t1} (which are weaker than the assumptions in Theorem \ref{thm.Qf.continuity}), that
\begin{equation}\label{eq.continue.upper}
\lim\limits_{\varepsilon\searrow 0} \left(\limsup_{n\rightarrow \infty} V^{Q^n}_\varepsilon(x,f^n)\right) \leq \lim\limits_{\varepsilon\searrow 0} \left(\limsup_{n\rightarrow \infty} W^{Q^n}_\varepsilon(x,f^n)\right)\leq V^{Q^\infty}(x,f^\infty),\quad \forall x\in \mathbb{X}.
\end{equation}
By Theorem \ref{t1} and Proposition \ref{prop.optimalequi.pseudo}, for any $\varepsilon\geq 0$,
\begin{equation}\label{eq0}
\begin{aligned}
\limsup_{n\rightarrow \infty}V^{Q^n}\left(x,\left(f^n-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right)=& \limsup_{n\rightarrow \infty}W^{Q^n}\left(x,\left(f^n-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right)\\
\leq & V^{Q^\infty}\left(x,\left(f^\infty-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right),\quad \forall x\in \mathbb{X}.
\end{aligned}
\end{equation}
Meanwhile, for $n\in\overline \mathbb{N}$,
\begin{equation}\label{eq1}
\begin{aligned}
V^{Q^n}_\varepsilon(x,f^n)\leq & W^{Q^n}_\varepsilon(x,f^n)\\
\leq & W^{Q^n}\left(x,\left(f^n-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right)+\frac{\varepsilon}{1-\delta(1)}\\
=&V^{Q^n}\left(x,\left(f^n-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right)+\frac{\varepsilon}{1-\delta(1)},\quad \forall x\in \mathbb{X}.
\end{aligned}
\end{equation}
where the first line follows from Lemma \ref{lm.equi.pseduvs}, the second line follows from ${\mathcal G}^{Q^n}(f^n,\varepsilon)\subset {\mathcal G}^{Q_n}((f^n-\frac{\varepsilon}{1-\delta(1)})\vee 0)$ implied by Lemma \ref{lm.equi.epsilon}, and the last line follows from Proposition \ref{prop.optimalequi.pseudo}.
By \eqref{eq0} and \eqref{eq1}, for any $\varepsilon\geq 0$ and $x\in\mathbb{X}$,
\begin{align*}
\limsup_{n\rightarrow \infty} V^{Q^n}_\varepsilon(x,f^n)\leq\limsup_{n\rightarrow \infty} W^{Q^n}_\varepsilon(x,f^n)\leq & \limsup\limits_{n\rightarrow \infty} V^{Q^n}\left(x,\left(f^n-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right)+\frac{\varepsilon}{1-\delta(1)}\\
\leq & V^{Q^\infty}\left(x,\left(f^\infty-\frac{\varepsilon}{1-\delta(1)}\right)\vee 0\right)+\frac{\varepsilon}{1-\delta(1)},
\end{align*}
Then \eqref{eq.continue.upper} follows by setting $Q=Q^\infty$ in \eqref{eq.continuef.value}.
\end{proof}
\begin{proof}[{\bf \textit{Proof of Proposition \ref{prop.continue.pseudo}}}]
Step 1. Let $\varepsilon>0$. We first prove that for any $x\in \mathbb{X}\setminus S^*(f^\infty,Q^\infty)$, there exists a set $S_x$ and $N\in\mathbb{N}$ such that $$
S_x\in {\mathcal G}^{Q^n}(f^n,\varepsilon),\quad
J^{Q^n}(x,S^*(f^\infty,Q^\infty),f^n)\leq J^{Q^n}(x,S_x,f^n)+\varepsilon\quad\text{and} \quad \forall n\geq N.
$$
Fix $x\notin S^*(f^\infty,Q^\infty)$. As $\sup_{n\in \overline \mathbb{N}} \|f^n\|_{\infty}=:M<\infty$, we can take $T\in \mathbb{N}$ such that $\delta(T)M<\varepsilon/2$. Then we apply the same discussion as \eqref{eq.n.T} to find a compact set $K$ and $N_1\in\mathbb{N}$ (that may depend on $x$) such that
\begin{equation}\label{eq.local.thm}
2M\left(1-\mathbb{P}^{Q^n}_x(X_t\in K,\ t=0,\dotso,T)\right)=2M\cdot \mathbb{P}^{Q^n}_x(\rho(\mathbb{X}\setminus K)\leq T)<\varepsilon/2, \quad \forall N_1\leq n\leq \infty.
\end{equation}
By Lemma \ref{lm.tau.uniform}(b),
$$
\lim_{n\rightarrow\infty } \sup_{y\in (K\setminus S^*(f^\infty, Q^\infty))}|J^{Q^\infty}(y,S^*(f^\infty, Q^\infty),f^\infty)-J^{Q^n}(y,S^*(f^\infty, Q^\infty),f^n)|=0.
$$
This together with the locally uniform convergence of $(f^n)_{n\in \mathbb{N}}$, we can find $N_2\in \mathbb{N}$ (that may depend on $x$) such that, for all $n\geq N_2$, $\sup_{y\in K}|f^n(y)-f^\infty(y)|<\frac{\varepsilon}{2}$ and
$$
J^{Q^\infty}(y,S^*(f^\infty, Q^\infty))-\frac{\varepsilon}{2}\leq J^{Q^n}(y,S^*(f^\infty,Q^\infty),f^\infty),\quad \forall y\in (K\setminus S^*(f^\infty, Q^\infty)).
$$
This imply that for all $n\geq N_2$,
\begin{align}
\notag f^n(y)-\varepsilon \leq & f^\infty(y)-\frac{\varepsilon}{2}\leq J^{Q^\infty}(y,S^*(f^\infty,Q^\infty),f^\infty)-\frac{\varepsilon}{2}\\
\label{e003} \leq &J^{Q^n}(y,S^*(f^\infty,Q^\infty),f^n),\quad \forall y\in (K\setminus S^*(f^\infty, Q^\infty)).
\end{align}
Let
$$S_x:=S^*(f^\infty,Q^\infty)\cup (\mathbb{X}\setminus K).$$
By \eqref{e003}, $S_x\in {\mathcal G}^{Q^n}_\varepsilon(f^n)$ for $n\geq N_2$. Moreover, for any $n\geq N:=N_1\vee N_2$,
\begin{align*}
&|J^{Q^n}(x,S^*(f^\infty,Q^\infty),f^n)-J^{Q^n}(x,S_x,f^n)|\\
\leq & \mathbb{E}^{Q^n}_x[|\delta(\rho(S^*(f^\infty,Q^\infty)))f^n(X_{\rho(S^*(f^\infty,Q^\infty))})-\delta(\rho(S_x))f^n(X_{\rho(S_x)})|\cdot 1_{\{X_{\rho(S_x)}\notin S^*(f^\infty,Q^\infty), \rho(S_x)\geq T\}}]\\
&+\mathbb{E}^{Q^n}_x[|\delta(\rho(S^*(f^\infty,Q^\infty)))f^n(X_{\rho(S^*(f^\infty,Q^\infty))})-\delta(\rho(S_x))f^n(X_{\rho(S_x)})|\cdot 1_{\{X_{\rho(S_x)}\notin S^*(f^\infty,Q^\infty), \rho(S_x)< T\}}]\\
\leq & 2M \delta(T)+2M\cdot \mathbb{P}^{Q^n}_x(\rho(\mathbb{X}\setminus K)\leq T)\\
<&\varepsilon,
\end{align*}
where the last line follows from \eqref{eq.local.thm} and $\delta(T)M<\varepsilon/2$. Step 1 is completed.
Step 2. For any $x\notin S^*(f^\infty,Q^\infty)$, we can find $N'\in\mathbb{N}$ (which may depend on $x$) such that $$
| J^{Q^\infty}(x, S^*(f^\infty,Q^\infty), f^\infty)- J^{Q^n}(x, S^*(f^\infty,Q^\infty), f^\infty)|<\frac{\varepsilon}{2},\quad \forall n\geq N'.
$$
Then from Step 1,
\begin{align*}
V^{Q^\infty}(x,f^\infty)= & J^{Q^\infty}(x, S^*(f^\infty,Q^\infty), f^\infty)\leq J^{Q^n}(x, S^*(f^\infty,Q^\infty), f^n)+\varepsilon\\
\leq &J^{Q^n}(x, S_x,f^n)+2\varepsilon\leq W^{Q^n}_\varepsilon(f^n)+2\varepsilon,\quad \forall n\geq N\vee N'.
\end{align*}
Letting $n\to \infty$ then $\varepsilon\searrow 0$, we have that
$$
V^{Q^\infty}(x,f^\infty)\leq \lim_{\varepsilon\searrow 0}\left(\liminf_{n\to \infty} W^{Q^n}_\varepsilon (x,f^n)\right),\quad \forall x\in \mathbb{X}.
$$
Then the rest follows from Step 2 in the proof of Theorem \ref{thm.Qf.continuity}.
\end{proof}
|
1,116,691,499,408 | arxiv | \section{Introduction}
The derivation of four-dimensional
low-energy effective actions arising from string theory requires
a detailed understanding of the geometries used as compactification
spaces. Since the early days of string theory much research has focused on the study of
Calabi-Yau manifolds of complex dimension three. These threefolds
were identified as valid compactification backgrounds to four space-time
dimensions and can yield to, for example when used in
the heterotic string theories, potentially
phenomenologically interesting four-dimensional effective theories
with the minimal amount of supersymmetry. In contrast, there is
significantly less known about the geometry of
Calabi-Yau manifolds of complex dimension four.
With the advent of F-theory \cite{Vafa:1996xn,Denef:2008wq,Weigand:2010wm} it became clear that
these fourfolds are relevant in obtaining
four-dimensional effective theories
with the minimal amount of supersymmetry from Type IIB
string theory. It is therefore crucial to further our understanding
of the geometry of Calabi-Yau fourfolds and investigate
the relation to couplings in the effective theories.
In contrast to Calabi-Yau threeforlds one finds that
Calabi-Yau fourfolds admit three non-trivial independent
Hodge numbers that count the number of
harmonic forms of different degree. In full analogy to threefolds,
two of them encode the number of complex structure and
K\"ahler structure deformations of the geometry.
When compactifying string theory or M-theory
on a Calabi-Yau fourfold, these deformations will appear
as massless fields, so-called moduli, in the effective action.
The moduli space geometry has been studied in various
works \cite{Greene:1993vm,Mayr:1996sh,Klemm:1996ts,Grimm:2009ef,
Honma:2013hma,Halverson:2013qca,Braun:2014xka,Bizet:2014uua}. The additional Hodge number on
Calabi-Yau fourfolds is associated to the number
of harmonic three-forms. In this work we will discuss in detail
how the presence of these three-forms affects the effective theory
when compactifying Type IIA string theory and
M-theory on Calabi-Yau fourfolds. The effective theory
will then admit new scalars $N_l$ with couplings non-trivially
varying over both the complex structure and K\"ahler structure
moduli spaces. This was already observed for M-theory compactifications
in \cite{Haack:1999zv,Haack:2001jz}, for Type IIA compactifications in \cite{Haack:2000di},
and for F-theory compactifications in \cite{Grimm:2010ks}.
We will show in this work that the moduli dependence at certain
points in the moduli space can actually be computed explicitly by
using mirror symmetry for Calabi-Yau fourfolds.
Our first focus is on a refined understanding
of the moduli variations of three-forms. Therefore, we begin
by introducing a basis
of $(2,1)$-forms on the fourfold that is convenient when performing
the dimensional reduction. The so-defined set of forms is adapted to the
underlying complex structure and it was pointed out in \cite{Grimm:2010ks}
that their variation with the complex structure moduli can
be captured by a holomorphic function $f_{kl}$, with indices
ranging over the number of harmonic $(2,1)$-forms.
This holomorphic function can be used in the compactification
of Type IIA string theory, accessed via its low-energy supergravity theory,
on a Calabi-Yau fourfold. Such dimensional reductions
of the Type IIA theory have already been investigated in \cite{Gukov:1999ya,Haack:2000di,Gates:2000fj}.
They are expected to yield two-dimensional effective theories with $\mathcal{N}=(2,2)$ supersymmetry
describing the dynamics of chiral and twisted-chiral multiplets.
Without including three-forms the supersymmetry properties of such Type IIA effective theories
were already discussed in \cite{Gates:2000fj} by extending earlier
results \cite{Gates:1994gx,deWit:1992xr,Grisaru:1995dr}. We will consider the generalization
of this result including the three-form scalars $N_l$
and suggest that it leads to a more general class of supersymmetric
dilaton supergravities.
The two-dimensional $\mathcal{N}=(2,2)$ effective action is expected to be
invariant under the action of mirror symmetry. More precisely, considering
Type IIA string theory on two Calabi-Yau fourfolds that are mirror manifolds to each other,
the resulting two effective actions should admit an identification under
an appropriate mirror map. This map identifies complex coordinates and couplings
at special points in moduli space.
Mirror symmetry exchanges complex structure and K\"ahler structure deformations, but
preserves the number of non-trivial three-forms and thus the number of three-form scalars $N_l$.
It also maps chiral to twisted-chiral
multiplets. Therefore, we are forced to perform an appropriate duality transformation
for the three-form scalars appearing in pairs of effective actions arising
from mirror manifolds. In both
effective actions the dynamics of the three-forms are described by two
holomorphic functions $f_{kl}$ and $h_l^k$. The former is holomorphic
in the complex structure moduli, while the latter is holomorphic in
the complexified
K\"ahler moduli. These functions are exchanged by mirror symmetry
and we are able to derive the complex structure dependence of $f_{kl}$ in the
large complex structure limit by using the results of a large volume compactification
on the mirror geometry.
Our results have several interesting applications, in particular, when using
the Calabi-Yau fourfolds with non-trivial three-forms
as F-theory backgrounds. To determine the four-dimensional
F-theory effective actions for such configurations one uses the duality with
M-theory \cite{Vafa:1996xn,Denef:2008wq,Grimm:2010ks}.
It was shown in \cite{Grimm:2010ks}
that the function $f_{lm}$ can either lift to a gauge coupling function
of R-R vector fields or to the metric of a special set of complex scalars.
In both cases it is desirable to explicitly
compute the moduli dependence of their coupling function.
For example, the three-form scalars lifting to four-dimensional
scalars naturally admit real shift symmetries or even a generalized
Heisenberg symmetry and might be of profound phenomenological interest
(see, for example, \cite{Grimm:2011tb,Grimm:2014vva,Grimm:2015ona}). Furthermore,
considering the weak string coupling limit of the F-theory setting following \cite{Sen:1996vd,Sen:1997gv}
the resulting effective theory should match with the orientifold effective
actions \cite{Grimm:2004uq,Grimm:2004ua}. In the case that such a limit exists one can associate
a Calabi-Yau threefold to the F-theory Calabi-Yau fourfold. We
are then able to show that our result for $f_{kl}$ obtained by
fourfold mirror symmetry is consistent with the weak coupling
analog obtained from threefold mirror symmetry.
This paper is organized as follows. In \autoref{three-form_facts} we recall some
basics about Calabi-Yau fourfolds and discuss a convenient basis of $(2,1)$-forms
parametrized by a holomorphic function $f_{lk}$. We dimensionally reduce
Type IIA supergravity on a Calabi-Yau fourfold in \autoref{IIAreduction}.
This allows us to investigate the $\mathcal{N}=(2,2)$ supersymmetric structure
of the effective theory and perform a set of important dualizations to
interchange chiral multiplets and twisted-chiral multiplets. In \autoref{mirror_section}
we discuss mirror symmetry with a focus on the $(2,1)$-form sector.
This allows us to determine $f_{lm}$ in the large complex structure limit.
We use these results in an F-theory compactification on
an elliptically fibered Calabi-Yau fourfold in \autoref{F-theoryapp}.
Moving to the weak string coupling limit, we find compatibility of
our result for $f_{lm}$ with the answers predicted by mirror symmetry
of Calabi-Yau threefolds. This work has two appendices with useful
computational results. In \autoref{3d-2dreduction} we perform the
circle reduction of a general three-dimensional un-gauged $\mathcal{N}=2$ supergravity theory
with focus on the bosonic action. We find interesting
conditions on the kinetic potential to match the proposed $\mathcal{N}=(2,2)$ action in two dimensions.
The dualization of chiral to twisted-chiral multiplets in the bosonic sector is
performed in \autoref{detailed_dual}. We again find conditions on the kinetic potential
in order that this dualization can be performed. The results of both
appendices are immediately applicable to Calabi-Yau fourfold effective actions
of Type IIA string theory and M-theory.
\section{On the geometry of Calabi-Yau fourfolds with three-form cohomology} \label{three-form_facts}
In this section we introduce important facts about the geometry of Calabi-Yau fourfolds $Y_4$. A brief summary
of some basics about their differential structure and topology will be given in \autoref{CY4basics}.
The focus of \autoref{3formbasics} will be to introduce relevant properties of the three-form
cohomology of $Y_4$. We argue that an appropriate definition of three-forms of Hodge-type
(2,1) can be given in terms of a function $f_{mn}$ holomorphic in the complex structure moduli.
This function will be of key interest throughout this work.
\subsection{Some basic properties of Calabi-Yau fourfolds} \label{CY4basics}
We define a compact real eight-dimensional manifold $Y_4$ to be a Calabi-Yau fourfold if its holonomy group is
exactly $SU(4)$. Such manifolds are K\"ahler, i.e.~admit a closed K\"ahler two-form $J$, and possess a unique Ricci-flat metric within the class of $J$. Furthermore, one can introduce a non-trivial closed $(4,0)$-form $\Omega$ on $Y_4$ that is unique up to constant rescalings. Note that $J$ and $\Omega$ can be used to form a
top-form on $Y_4$ and one has
\begin{equation} \label{def-Omega2}
\frac{1}{4!} J \wedge J \wedge J \wedge J = |\Omega|^{-2}\, \Omega \wedge \bar \Omega \ , \qquad \qquad |\Omega|^2= \frac{1}{\mathcal{V}} \int_{Y_4} \Omega \wedge \bar \Omega \ ,
\end{equation}
where $\mathcal{V}$ is the total volume of $Y_4$. The $SU(4)$ holonomy also allows one to introduce
one complex covariantly constant and no-where vanishing spinor of definite chirality.
The forms $J$ and $\Omega$ are obtained as bilinear contractions using this spinor.
With our definition of a Calabi-Yau fourfold, we can also constrain the Hodge numbers
$ h^{p,q} (Y_4) = \text{dim}(H^{p,q}(Y_4,\mathbb{C}) )$.
There are three independent Hodge numbers on $Y_4$: $h^{1,1}(Y_4) $, $h^{3,1}(Y_4) $, and $h^{2,1}(Y_4) $.
The significance of $h^{1,1}(Y_4) $ and $h^{3,1}(Y_4) $ in the dimensional reduction are very similar to
the case of a Calabi-Yau threefold (see e.g.~\cite{Candelas:1990pi}). On the one hand, the
number $ h^{1,1} (Y_4)$ counts the allowed K\"ahler structure deformations, which we will
denote by $v^A$. On the other hand, the number
$ h^{3,1} (Y_4)$ counts the complex structure deformations denoted by $z^K$. Both
turn out to become moduli fields in the effective theory obtained by dimensional reduction on $Y_4$
and will be discussed in more detail in \autoref{IIAreduction_details}.
The Hodge number $h^{2,1}$ has no threefold analog and understanding the geometries
with $h^{2,1}(Y_4)>0$ will be the main focus of this work.
Having three independent Hodge numbers, the Hodge diamond takes the form
\vspace{.5cm}
\arraycolsep=2,0pt\def1.4{1.4}
\setlength{\unitlength}{0.6cm}
\begin{picture}(21,8)
\put(5,8){$ h^{0,0} $}
\put(4,7){$ h^{1,0} $} \put(6,7){$ h^{0,1} $}
\put(3,6){$ h^{2,0} $} \put(5,6){$ h^{1,1} $} \put(7,6){$ h^{0,2} $}
\put(2,5){$ h^{3,0} $} \put(4,5){$ h^{2,1} $} \put(6,5){$ h^{1,2} $} \put(8,5){$ h^{0,3} $}
\put(1,4){$ h^{4,0} $} \put(3,4){$ h^{3,1} $} \put(5,4){$ h^{2,2} $} \put(7,4){$ h^{1,3} $} \put(9,4){$ h^{0,4} $}
\put(2,3){$ h^{4,1} $} \put(4,3){$ h^{3,2} $} \put(6,3){$ h^{2,3} $} \put(8,3){$ h^{1,4} $}
\put(3,2){$ h^{4,2} $} \put(5,2){$ h^{3,3} $} \put(7,2){$ h^{2,4} $}
\put(4,1){$ h^{4,3} $} \put(6,1){$ h^{3,4} $}
\put(5,0){$ h^{4,4} $}
\put(11,4){$ = $}
\put(17,8){$ 1 $}
\put(16,7){$ 0 $} \put(18,7){$ 0 $}
\put(15,6){$ 0 $} \put(17,6){$ h^{1,1} $} \put(19,6){$ 0 $}
\put(14,5){$ 0 $} \put(16,5){$ h^{2,1} $} \put(18,5){$ h^{2,1} $} \put(20,5){$ 0 $}
\put(13,4){$ 1 $} \put(15,4){$ h^{3,1} $} \put(17,4){$ h^{2,2} $} \put(19,4){$ h^{3,1} $} \put(21,4){$ 1\ \ , $}
\put(14,3){$ 0 $} \put(16,3){$ h^{2,1} $} \put(18,3){$ h^{2,1} $} \put(20,3){$ 0 $}
\put(15,2){$ 0 $} \put(17,2){$ h^{1,1} $} \put(19,2){$ 0 $}
\put(16,1){$ 0 $} \put(18,1){$ 0 $}
\put(17,0){$ 1 $}
\setlength{\unitlength}{0.06cm}
\multiput(20,73)(10,-10){7}{\line(1,-1){6}}
\end{picture}
\noindent
where we have indicated for later use the action of mirror symmetry on the Hodge numbers. More
precisely, mirror symmetry identifies two Calabi-Yau geometries with Hodge numbers
mirrored along the dashed line.
A more detailed discussion of mirror symmetry will be presented in \autoref{mirror_section}.
In addition, one finds the formulas \cite{Klemm:1996ts}
\begin{equation}
h^{2,2}(Y_4) = 2(22 + 2h^{1,1} + 2h^{3,1} - h^{2,1})\ , \qquad \quad \chi(Y_4)
= 6(8 + h^{1,1} + h^{3,1} - h^{2,1})
\end{equation}
where $ \chi = \sum_{p,q} (-1)^{p+q} h^{p,q}$ is the Euler characteristic of $Y_4$.
\subsection{Non-trivial three-forms on Calabi-Yau fourfolds} \label{3formbasics}
Of key importance in this work is the inclusion of non-trivial three-forms in the dimensional reduction and discussion
of mirror symmetry. In this subsection we summarize some basic properties of such three-forms that
will be useful throughout the later sections.
To begin with, we comment on the moduli dependence of three-forms when
choosing them to represent elements of $H^{2,1}(Y_4)$.
In order to do that, recall that the Hodge filtration of the three-cohomology
$ H^3 (Y_4, \mathbb{C} ) $ is given by the holomorphic bundles
$ F^p (Y_4) = \bigoplus_{j=p} ^3 H^{j, 3-j} $ over the complex structure moduli space.
Since $ H^{3,0}(Y_4) $ is trivial, this enables us to find a basis $ \psi_l $ of
$F^2(Y_4) = H^{2,1}(Y_4) $, which varies holomorphically with the complex structure moduli $z^K$, i.e.~one has
\begin{equation}
\frac{\partial}{ \partial {\bar{z}^K} }\psi_l = 0, \qquad l=1, \ldots, h^{2,1}(Y_4)\ ,
\end{equation}
where $K = 1, \ldots, h^{3,1}(Y_4)$ labels the complex structure moduli.
Note that in the dimensional reduction we can think of $\psi_l$ to be
the harmonic representative in each class of $H^{2,1}(Y_4) $.
At a given point in the complex structure space we can write this basis in the form
\begin{equation} \label{def-f1}
\psi_l = \alpha_l + i f_{lm} (z) \beta^m \qquad \in H^{2,1}(Y_4)\ ,
\end{equation}
where $ (\alpha_l ,\beta^m )$ comprise a real moduli-independent basis of $ H^3(Y_4, \mathbb{R}) $.\footnote{
It might be natural to chose $( \alpha_l , \beta^m)$ to be a basis of $ H^3(Y_4, \mathbb{Z})$, but quantization
of the coefficients will not be important in this work.}
Holomorphicity of the forms $\psi_l$ translates to the fact that
$ f_{lm}(z) $ is a holomorphic function of the complex structure moduli $ z^K$.
Furthermore, we assume that $(\alpha_l,\beta^m)$ is chosen such
that $\text{Re} f_{lm}$ is a positive definite and invertible matrix.\footnote{While we have no
complete proof that this is always possible, we note that $H^{2,1}(Y_4)/H^{3}(Y_4,\mathbb{Z})$ is
actually a complex torus. $f_{lm}$ sets the complex structure on this torus.}
After performing the dimensional reduction on $Y_4$ in \autoref{IIAreduction}, we
aim to find the proper complex fields that are compatible with two-dimensional supersymmetry.
For the zero-modes arising from the $\psi_l$ it turns out that a further
normalization is useful, i.e.~we introduce the (1,2)-forms
\begin{equation} \label{def-Psil}
\Psi^l = \frac{1}{2} (\text{Re} f)^{lm} \bar {\psi}_m = \frac{1}{2} (\text{Re} f)^{lm} ( \alpha_m - i \bar f_{mn} (z) \beta^n)
\qquad \in H^{1,2}(Y_4)\ .
\end{equation}
In this expression we have multiplied by the \textit{inverse} $(\text{Re} f)^{lm}$ of the real part of $ f_{lm}$, i.e.
$ (\text{Re} f)^{lm} \, (\text{Re} f)_{mk}= \delta^l _k $.
This definition allows to give a simple expressions for $\text{Im}\, \Psi^l$ and the derivative
of $ \Psi^l$ with respect to the complex structure moduli:
\begin{equation} \label{useful}
\bar \Psi^l - \Psi^l = i \beta^l\ ,\qquad \partial_{z^K} \Psi^l = - \Psi^k \, (\text{Re} f)^{lm}\, \partial_{z^K} (\text{Re} f)_{mk}\ ,
\end{equation}
and accordingly $ \partial_{z^K} \Psi^l = \partial_{z^K} \bar \Psi^l$.
Note that $\Psi^l$ is a $(1,2)$-form and therefore satisfies
\begin{equation} \label{starPsi}
* \Psi^l =- i J \wedge \Psi^l\ ,
\end{equation}
where $*$ is the Hodge-star for the Calabi-Yau metric on $Y_4$.
To evaluate the integrals appearing in the dimensional reduction we
impose one further condition on the basis $(\alpha_l,\beta^m)$.
More precisely, we demand
\begin{equation}
\beta^l \wedge \beta^m = 0\ , \qquad \forall \ l,m = 1, \ldots, h^{2,1}(Y_4)\ ,
\end{equation}
which is supposed to hold in cohomology.\footnote{
Considering hypersurfaces in toric varieties, this condition can be satisfied for non-trivial three-forms
arising from singular Riemann surfaces. This allows us to choose a symplectic basis
with respect to a certain divisor for the three-forms.}
Introducing a basis $\omega_A$ of $H^{1,1}(Y_4)$ we thus find that
\begin{equation} \label{betabeta_condition}
\int_{Y_4} \omega_A \wedge \beta^l \wedge \beta^m = 0 \ ,\qquad \forall \ A = 1, \ldots, h^{1,1}(Y_4) \ .
\end{equation}
The remaining integrals are in general non-trivial and denoted
by
\begin{equation} \label{Cdef}
C_{Am}{}^k = \int_{Y_4} \omega_A \wedge \alpha_m \wedge \beta^k\ ,\qquad
C_{Amk} = \int_{Y_4} \omega_A \wedge \alpha_m \wedge \alpha_k\ .
\end{equation}
Using a basis $(\alpha_m, \beta^k)$ satisfying \eqref{betabeta_condition}
one checks that the metric $\int \Psi^l \wedge * \bar \Psi^k$ is symmetric in the
indices $l,k$ and real. This property will be crucial in determining a
kinetic potential for this metric.
\section{Dimensional reduction of Type IIA supergravity} \label{IIAreduction}
In this section we perform the dimensional reduction of Type IIA supergravity on
a Calabi-Yau fourfold $Y_4$. Such reductions have already been
performed in \cite{Haack:1999zv,Haack:2001jz,Haack:2000di,Gates:2000fj}. Our analysis follows
\cite{Haack:1999zv,Haack:2001jz,Haack:2000di}, but we will apply in addition the improved understanding about
the three-form cohomology of \autoref{three-form_facts}.
\subsection{The effective action from a Calabi-Yau fourfold reduction} \label{IIAreduction_details}
The Kaluza-Klein reduction of Type IIA supergravity can be trusted in the
limit in which the typical length scale of the physical volumes of submanifolds of $Y_4$ are sufficiently
large compared to the string scale. This limit is referred to
as the large volume limit. Furthermore, these typical length scales
set the Kaluza-Klein scale which we assume to be sufficiently above the
energy scale of the effective action. We therefore keep only the massless Kaluza-Klein modes
in the following reduction.
Our starting point will be the bosonic part of the ten-dimensional Type IIA action in string-frame given by \footnote{
Note that for convenience we have set $\kappa^2 = 1$.}
\begin{align} \label{10Daction}
S^{(10)}_{\rm IIA} &= \int e^{-2\check \phi_{\rm IIA}} \left( \frac{1}{2}\check R \, \check \ast 1 + 2 d\check \phi_{\rm IIA} \wedge \check \ast d\check \phi_{\rm IIA} - \frac{1}{4} \check H_3 \wedge \check \ast \check H_3 \right) \nonumber \\
&\quad - \frac{1}{4} \int \Big( \check F_2 \wedge \check \ast \check F_2 + \check {\mathbf F}_4 \wedge \check \ast \check {\mathbf{F}}_4+ \check B_2 \wedge \check F_4 \wedge \check F_4\Big) \ .
\end{align}
where $\check \phi_{\rm IIA}$ is the ten-dimensional dilaton, $ \check H_3 = d \check B_2 $ is the field strength of the NS-NS two-form
$ \check B_2 $, and $\check F_p = d\check C_p$ are the field strengths of the R-R $p$-forms $\check C_1$ and $\check C_3$.
We also have used the modified field strength $ \check {\mathbf{F}}_4 = \check F_4 - \check C_1 \wedge \check H_3 $.
Here and in the following we will use a check to indicate ten-dimensional fields.
The background solution around which we want to consider the effective theory is
taken to be of the form $\mathbb{M}_{1,1} \times Y_4$, where $\mathbb{M}_{1,1}$
is the two-dimensional Minkowski space-time, and $Y_4$ is a Calabi-Yau fourfold
with properties introduced in \autoref{three-form_facts}. As pointed out there
such a manifold admits one complex covariantly constant spinor of definite chirality.
This spinor can be used to dimensionally reduce the $\mathcal{N}=(1,1)$ supersymmetry
of Type IIA supergravity to obtain a two-dimensional $\mathcal{N}=(2,2)$ supergravity theory.
In particular, the two ten-dimensional gravitinos of opposite chirality reduce to two
pairs of two-dimensional Majorana-Weyl gravitinos with opposite chirality. We will have more to
say about the supersymmetry properties of the two-dimensional action in \autoref{22dilatonSugra}.
Furthermore, recall that $Y_4$ admits a
Ricci-flat metric $g^{(8)}_{mn}$ and one can thus check that a metric of the form
\begin{equation} \label{metric-ansatz}
d\check s^2 = \eta_{\mu \nu} dx^\mu dx^\nu + g^{(8)}_{mn} dy^m dy^n\ ,
\end{equation}
solves the ten-dimensional equations of motion in the absence of background fluxes.\footnote{The
inclusion of background fluxes complicates the reduction further. In particular, it
requires to introduce a warp-factor. The M-theory reduction with warp-factor was recently
performed in \cite{GPW}.}
Note that in \eqref{metric-ansatz} we denote
by $ x^\mu $ the two-dimensional coordinates of the space-time $\mathbb{M}_{1,1} $,
whereas the eight-dimensional real coordinates on the Calabi-Yau fourfold $ Y_4 $ are denoted
by $ y^m $.
The massless perturbations around this background both consist of fluctuations
of the internal metric $g^{(8)}_{mn}$ that preserve the Calabi-Yau condition as
well as the fluctuations of the form fields $\check B_2$, $\check C_1,\check C_3$ and the dilaton $\check \phi_{\rm IIA}$.
The metric fluctuations give rise to the real K\"ahler structure moduli $v^A$, $A=1,\ldots, h^{1,1}(Y_4)$ that preserve the
complex structure and are given by
\begin{equation} \label{Kaehlermoduli}
g_{i\bar \jmath } + \delta g_{i \bar \jmath} = -i J_{i \bar \jmath} = -i v^A\, (\omega_{A})_{ i \bar \jmath},
\end{equation}
where $J$ is the K\"ahler form on $Y_4$ and $ \omega_A$ comprises a real basis of
harmonic $(1,1)$-forms spanning $H^{1,1}(Y_4) $. The K\"ahler structure moduli
appear also in the expression of the total string-frame volume $\mathcal{V}$ of $Y_4$ given by
\begin{equation}
\mathcal{V} \equiv \int_{Y_4} *1 = \frac{1}{4!} \int_{Y_4} J \wedge J \wedge J \wedge J\ .
\end{equation}
In addition to the K\"ahler structure moduli one finds a set of complex structure moduli $z^K$, $K = 1,\ldots, h^{3,1}(Y_4)$.
These fields parameterize the change in the complex structure of $Y_4$ preserving the class
of its K\"ahler form $J$. Infinitesimally they are given by the fluctuations $\delta z^K$ as
\begin{equation} \label{CSmoduli}
\delta g_{\bar \imath \bar \jmath} = - \frac{1}{3 | \Omega| ^2 } \overline{\Omega}_{\bar \imath}^{\ lmn} (\chi_{K})_{lmn \bar \jmath}\, \delta z^{K} \ ,\qquad \qquad
\end{equation}
where $\Omega$ is the $(4,0)$-form, the $\chi_{K}$ form a
basis of harmonic $(3,1)$-forms spanning $H^{3,1}(Y_4)$, and $|\Omega|^2$
was already given in \eqref{def-Omega2}.
The Kaluza-Klein ansatz for the remaining fields takes the form
\begin{align} \label{BC-expand}
&\check B_2 = b^A \omega_A \ , \qquad \check C_1 = A \ , \\
&\check C_3 = V^A \wedge \omega_A + N_l \Psi^l + \bar N_l \bar \Psi^l \ , \nonumber
\end{align}
where $\Psi^l$ is a basis of harmonic
$(1,2)$-forms spanning $H^{1,2}(Y_4)$ as introduced in \eqref{def-Psil}.
A discussion of the properties of $\Psi^l$ was already given in \autoref{three-form_facts}.
Finally, we dimensionally reduce the Type IIA dilaton by dropping its
dependence on the internal manifold $Y_4$. It turns out to be convenient
to define a two-dimensional dilaton $\phi_{\rm IIA}$ in
terms of the ten-dimensional dilaton $\check \phi_{\rm IIA}$ as
\begin{equation} \label{def-2dilaton}
e^{2 \phi_{\rm IIA}} \equiv \frac{e^{2 \check \phi_{\rm IIA}}}{\mathcal{V}}\ .
\end{equation}
In summary, we find in the two-dimensional $\mathcal{N}=(2,2)$ supergravity theory
the $2 h^{1,1}(Y_4) + 1$ real scalar fields $v^A(x)$, $b^A(x)$, $\phi_{\rm IIA}(x)$ as
well as the $h^{3,1}(Y_4) + h^{2,1}(Y_4)$ complex scalar fields $z^K$, $N_l$.
In addition there are $h^{1,1}(Y_4) + 1$ vectors $A$, $V^A$, which carry, however,
no physical degrees of freedom in a two-dimensional theory if they are not involved
in any gauging. Since the effective action considered here contains
no gaugings, we will drop these in the following analysis.
To perform the dimensional reduction one inserts the expansions \eqref{Kaehlermoduli},
\eqref{CSmoduli}, \eqref{BC-expand}, and \eqref{def-2dilaton} into
the Type IIA action \eqref{10Daction}.
It reduces
to the two-dimensional action
\begin{align} \label{2Daction}
S^{(2)} = \int e^{-2 \phi_{\rm IIA}}& \Big( \frac{1}{2} R\, *1 + 2 d \phi_{\rm IIA} \wedge * \phi_{\rm IIA}
- G_{K\bar L}\, dz^K \wedge * d\bar z^L - G_{A B} \, d t^A \wedge * d\bar t^B \Big) \nonumber\\
& - \frac{1}{2} v^A d_{A}{}^{l k} D N_l \wedge * \overline{D N}_k
- \frac{i}{4} d_{A}{}^{l k} db^A \wedge (N_l \overline{DN}_k - DN_{l} \bar N_k) \ .
\end{align}
We note that the NS-NS part, which
is summarized in the first line of \eqref{10Daction}, reduces to the first line of \eqref{2Daction},
while the R-R part, i.e.~the second line of \eqref{10Daction}, reduces to the second line of \eqref{2Daction}.
Let us introduce the various objects appearing in the action \eqref{2Daction}.
First, we have defined the complex coordinates
\begin{equation}
t^A \equiv b^A + i v^A\ ,
\end{equation}
which combine the K\"ahler structure moduli with the B-field moduli.
Furthermore, we have introduced the metric \footnote{The second
equality follows from the cohomological identity $* \omega_A = -\frac{1}{2} \omega_A \wedge J \wedge J + \frac{1}{36} \mathcal{V}^{-1}\mathcal{K}_A J \wedge J \wedge J$.}
\begin{equation} \label{GABmetric}
G_{A B} = \frac{1}{4 \mathcal{V} } \int_{Y_4} \omega_A \wedge * \omega_B
= - \frac{1}{8 \mathcal{V}} \Big( \mathcal{K}_{AB} - \frac{1}{18 \mathcal{V}} \mathcal{K}_A \mathcal{K}_B \Big)\ ,
\end{equation}
where $\mathcal{V}$, $\mathcal{K}_A$ and $\mathcal{K}_{AB}$ are given in terms of the
quadruple intersection numbers $\mathcal{K}_{ABCD}$ as
\begin{align}
&\mathcal{K}_{ABCD} = \int_{Y_4} \omega_A \wedge \omega_B\wedge \omega_C \wedge \omega_D\ , \\
&\mathcal{V} =\frac{1}{4!} \mathcal{K}_{ABCD} v^A v^B v^C v^D\ ,\quad \mathcal{K}_A = \mathcal{K}_{ABCD} v^B v^C v^D\ , \quad
\mathcal{K}_{AB} = \mathcal{K}_{ABCD} v^C v^D\ . \nonumber
\end{align}
With these definitions at hand, we can further evaluate the metric
$G_{A B}$ and show that it can be obtained from a K\"ahler potential as
\begin{equation}
G_{A B} =- \partial_{t^A} \partial_{\bar t^B} \log \, \mathcal{V} \ .
\end{equation}
Also the metric $G_{K\bar L}$ is actually a K\"ahler metric. It only depends on the complex structure
moduli $z^K$ and takes the form
\begin{equation}
G_{K\bar L} = - \frac{\int_{Y_4} \chi_K \wedge \bar \chi_L}{\int_{Y_4} \Omega \wedge \bar \Omega}
= - \partial_{z^K} \partial_{\bar z^L} \log \int_{Y_4} \Omega \wedge \bar \Omega\ .
\end{equation}
Note that both $G_{A B}$ and $G_{K\bar L}$ are actually positive definite and therefore
define physical kinetic terms in \eqref{2Daction}. Both terms scale
with the dilaton $\phi_{\rm IIA}$ and it is easy to check that this dependence cannot be
removed using a Weyl-rescaling of the two-dimensional metric. We will show in
\autoref{22dilatonSugra} that this is consistent with the form of the $\mathcal{N}=(2,2)$ dilaton
supergravity.
Let us now turn to the R-R part of the action \eqref{10Daction} and
discuss the couplings appearing in the second line of \eqref{2Daction}.
First, we introduce the coupling function
\begin{equation} \label{evaluate_d}
d_A{}^{lm} \equiv i \int_{Y_4} \omega_A \wedge \Psi^{l} \wedge \bar \Psi^m
= - \int_{Y_4} \omega_A \wedge \Psi^{l} \wedge \beta^m = - \frac{1}{2} (\text{Re} f)^{ln}C_{An}{}^m \ ,
\end{equation}
where we have used \eqref{useful} to evaluate the second equality, and \eqref{betabeta_condition}, \eqref{Cdef} to
show the third equality.
One also checks the relation
\begin{equation} \label{def-H}
H^{l m} \equiv \int_{Y_4} \Psi^{l} \wedge * \bar \Psi^m
= i \int_{Y_4} J \wedge \Psi^{l} \wedge \bar \Psi^m = v^A d_A{}^{l m} \ ,
\end{equation}
where we have used \eqref{starPsi} for the (1,2)-forms $\Psi^l$. This contraction gives precisely the
positive definite metric of the complex scalars $N_l$ in \eqref{2Daction}.
It turns out to be convenient to write
\begin{equation} \label{splitH}
H^{l m} = v^A d_A{}^{l m} = - \frac{1}{2} (\text{Re} f)^{ln} v^A C_{An}{}^m \equiv - \frac{1}{2} (\text{Re} f)^{ln}\ \text{Re}\, h_{n}^m\ ,
\end{equation}
where $h_{n}^m = - i t^A C_{An}{}^m$.
Note that $H^{lm}$ thus depends non-trivially on the complex structure moduli $z^K$
through the holomorphic functions $f_{kl}$ and on the K\"ahler structure moduli $t^A$
through the holomorphic function $h_{n}^m $.
Second, we note that the modified derivative $DN_l$ appearing in \eqref{2Daction} is a shorthand for
\begin{equation}
D N_l = d N_l - 2 \text{Re} N_m (\text{Re} f)^{mn} \partial_{z^K} (\text{Re} f_{nl} ) dz^K\ .
\end{equation}
Using this expression one easily reads off the coefficient function in front of $dN_l \wedge * dz^K$
and checks that it can be obtained by taking derivatives of a real function. In the next subsection
we show that this is true for all terms in \eqref{2Daction} and discuss the connection with two-dimensional
supersymmetry.
\subsection{Comments on two-dimensional $\mathcal{N}=(2,2)$ supergravity} \label{22dilatonSugra}
Having performed the dimensional reduction we next want to comment
on the supersymmetry properties of the
action \eqref{2Daction}. As pointed out already
in the previous subsection the counting of covariantly constant spinors on
the Calabi-Yau fourfold suggests that the two-dimensional effective theory admits
$\mathcal{N}=(2,2)$ supersymmetry. It was pointed out in \cite{Gates:2000fj}
that, at least in the case of $h^{2,1}(Y_4)=0$ one expects to be
able to bring the action \eqref{2Daction} into the standard form of
an two-dimensional $\mathcal{N}=(2,2)$ dilaton supergravity. In
this work the dilaton supergravity action was constructed using
superspace techniques. Earlier works in this direction include \cite{Gates:1994gx,deWit:1992xr,Grisaru:1995dr}.
In the following we comment on this matching for $h^{2,1}(Y_4)=0$
and then discuss the general case in which $h^{2,1}(Y_4)>0$.
In order to display the supergravity actions we first have to introduce two sets
of multiplets containing scalars in two-dimensions: (1) a set of chiral multiplets
with complex scalars $\phi^\kappa$,
(2) a set of twisted-chiral multiplets with complex scalar $\sigma^A$.
In a superspace description these multiplets obey the two
inequivalent linear spinor derivative constraints leading to
irreducible representations.
To discuss the actions we first focus on the
case $h^{2,1}(Y_4)=0$ and follow the constructions of \cite{Gates:2000fj}.
For simplicity we will not include gaugings or a scalar potential.
The superspace action used in \cite{Gates:2000fj} is
given by
\begin{equation} \label{suspace1}
S^{(2)}_{\rm dil} = \int d^2 x d^4 \theta E^{-1} e^{-2V - \mathcal{K}}\ .
\end{equation}
Here $E^{-1}$ is the superspace measure, $V$ is a real superfield with $V| = \varphi$ as lowest component,
and $\mathcal{K}$ is a function of the chiral and twisted-chiral multiplets with lowest components $\phi^\kappa$
and $\sigma^A$, respectively. To display the bosonic part of the action \eqref{suspace1}
we first set
\begin{equation}
e^{-2 \tilde \varphi} = e^{-2 \varphi - \mathcal{K}}\ ,
\end{equation}
where $\mathcal{K} (\phi^\kappa,\bar \phi^\kappa,\sigma^A,\bar \sigma^A)$ is evaluated as a function of
the bosonic scalars. With this definition at hand one finds the bosonic action
\begin{align} \label{bosonic_dilatonsugra}
S^{(2)}_{\rm dil} = \int e^{-2\tilde{\varphi}} & \left( \frac{1}{2}R *1 + 2d \tilde{\varphi} \wedge * d\tilde{\varphi}
- \mathcal{K}_{\phi^\kappa \bar \phi^\lambda} \, d\phi^\kappa \wedge * d \bar {\phi}^\lambda
+ \mathcal{K}_{ \sigma^A \bar \sigma^B }\, d \sigma^A \wedge * d \bar {\sigma}^B \right.\nonumber \\
& \ \ - \left. \mathcal{K}_{\phi^\kappa \bar \sigma^B} \, d \phi^\kappa \wedge d \bar \sigma^B
- \mathcal{K}_{\sigma^A \bar \phi^\lambda} \, d \bar {\phi}^\lambda \wedge d\sigma^A \right)\ ,
\end{align}
where $\mathcal{K}_{\phi^\kappa \bar \phi^\lambda } = \partial_{\phi^\kappa} \partial_{\bar \phi^\lambda} \mathcal{K} $,
$\mathcal{K}_{ \phi^\kappa \bar \sigma^A } = \partial_{\phi^\kappa} \partial_{\bar \sigma^A} \mathcal{K} $
with a similar notation for the other coefficients.
It is now straightforward to compare \eqref{bosonic_dilatonsugra} with the action
\eqref{2Daction} for the case $h^{2,1}(Y_4)=0$, i.e.~in the absence of any complex scalars $N_l$.
One first identifies
\begin{equation}
\tilde \varphi = \phi_{\rm IIA} \ ,\qquad \phi^K = z^K\ , \qquad \sigma^A = t^A\ ,
\end{equation}
and then infers that
\begin{equation} \label{hatKsimple}
\mathcal{K} = - \log \int_{Y_4} \Omega \wedge \bar \Omega + \log \mathcal{V} \ .
\end{equation}
Note that we find here a positive sign in front of the logarithm of $\mathcal{V}$. This is related
to the fact that there is an extra minus sign in the kinetic terms of the twisted-chiral
fields $\sigma^A$ in \eqref{bosonic_dilatonsugra}. Clearly, the kinetic terms of the
complex structure deformations $z^K$ and complexified
K\"ahler structure deformations $t^A$ in the action \eqref{2Daction} have both positive definite kinetic
terms.\footnote{Our discussion differs here from the one in \cite{Gates:2000fj}, where the
sign in front of $\log \mathcal{V}$ was claimed to be negative.}
Let us now include the complex scalars $N_l$. It is important to note that
the action \eqref{2Daction} cannot be brought into the form \eqref{bosonic_dilatonsugra}.
In fact, we see in \eqref{2Daction} that the terms independent of the two-dimensional
metric do not contain an $\phi_{\rm IIA}$-dependent pre-factor, while the terms of this type
in \eqref{bosonic_dilatonsugra} do admit an $\tilde \varphi$-dependence. Any field
redefinition in \eqref{2Daction} involving the dilaton seems to introduce new
undesired mixed terms that cannot be matched with \eqref{bosonic_dilatonsugra} either.
However, we note that the action \eqref{2Daction} actually can be
brought to the form
\begin{align} \label{bosonic_dilatonsugra_extended}
S^{(2)} = \int e^{-2\tilde{\varphi}} & \left( \frac{1}{2}R *1 + 2d \tilde{\varphi} \wedge * d\tilde{\varphi}
- \tilde K_{\phi^\kappa \bar \phi^\lambda} \, d\phi^\kappa \wedge * d \bar {\phi}^\lambda
+ \tilde K_{ \sigma^A \bar \sigma^B}\, d \sigma^A \wedge * d \bar {\sigma}^B \right.\nonumber \\
& - \left. \tilde K_{ \phi^\kappa \bar \sigma^B} \, d \phi^\kappa \wedge d \bar \sigma^B
- \tilde K_{ \sigma^A \bar \phi^\lambda} \, d \bar {\phi}^\kappa \wedge d\sigma^A \right)\ ,
\end{align}
where $\tilde K$ is now allowed to be dependent on $\tilde \varphi$ and given by
\begin{equation} \label{tildeK}
\tilde K = \mathcal{K} + e^{2 \tilde \varphi} \mathcal{S}\ ,
\end{equation}
Similar to $\mathcal{K}$, the new function $\mathcal{S}$ is allowed to depend on the chiral and twisted-chiral scalars $\phi^\kappa, \sigma^A$,
but is taken to be independent of $\tilde \varphi$. The action \eqref{bosonic_dilatonsugra_extended}
trivially reduces to \eqref{bosonic_dilatonsugra} for $\mathcal{S}=0$. Note that the new terms
induced by $\mathcal{S}$ do not scale with $e^{-2\tilde \varphi}$.
Comparison with \eqref{2Daction} reveals that one can identify
\begin{equation} \label{id_fields}
\tilde \varphi = \phi_{\rm IIA} \ ,\qquad \phi^\kappa = (z^K,N_l)\ , \qquad \sigma^A = t^A\ ,
\end{equation}
and introduce the generating functions
\begin{align} \label{idhatKS}
&\mathcal{K} = - \log \int_{Y_4} \Omega \wedge \bar \Omega + \log \mathcal{V} \ , \\
& \mathcal{S} = H^{l k} \ \text{Re}\, N_l\ \text{Re} N_k\ ,\qquad H^{l k} \equiv v^A d_A{}^{l k} \ . \nonumber
\end{align}
To show this, it is useful to note that $d_A{}^{l k}$ can be evaluated
as in \eqref{evaluate_d} and depends on the complex structure moduli through
the holomorphic function $f_{mn}(z)$ only.
Let us close this subsection with two remarks. First, note
that \eqref{bosonic_dilatonsugra_extended} is
expected to be compatible with $\mathcal{N}=(2,2)$ supersymmetry and gives an extension
of the two-dimensional dilaton supergravity action \eqref{suspace1}.
A suggestive form of the extended superspace action is
\begin{equation} \label{suspace2}
S^{(2)} = \int d^2 x d^4 \theta E^{-1} \big(e^{-2V - \mathcal{K}} + \mathcal{S}\big)\ ,
\end{equation}
where $\mathcal{S}$ is now evaluated as a function of the chiral and twisted-chiral superfields.
It would be interesting to check that \eqref{suspace2} indeed correctly reproduces the
bosonic action \eqref{bosonic_dilatonsugra_extended} with $\tilde K$ as in \eqref{tildeK}.
Second, the action \eqref{bosonic_dilatonsugra_extended} with the identification \eqref{id_fields}
can also be straightforwardly obtained by dimensionally reducing M-theory, or rather
eleven-dimensional supergravity, first on $Y_4$ and then on an extra circle of radius $r$.
The reduction of M-theory on $Y_4$ was carried out in \cite{Haack:1999zv,Haack:2001jz}. We give the resulting
three-dimensional action in \eqref{3daction} and briefly recall
this reduction in \autoref{Mreduction_details} when considering applications to F-theory.
Using the standard relation of eleven-dimensional supergravity on a circle and Type IIA supergravity,
one straightforwardly identifies
\begin{equation} \label{id_vA}
r = e^{-2\phi_{\rm IIA}}\ , \qquad e^{2 \phi_{\rm IIA}} v^A = \frac{v^A_{\rm M}}{\mathcal{V}_{\rm M}} \equiv L^A\ ,
\end{equation}
where $v^A_{\rm M}$ and $\mathcal{V}_{\rm M}$ are the analogs of $v^A$ and $\mathcal{V}$
used in the M-theory reduction. Note that the scalars $L^A$ are the appropriate fields to appear in three-dimensional
vector multiplets. Inserting the identification \eqref{id_vA}
into \eqref{tildeK} together with \eqref{id_fields}, \eqref{idhatKS} one finds
\begin{equation} \label{def-tildeKM}
\tilde K^{\rm M} = - \log \int_{Y_4} \Omega \wedge \bar \Omega + \log \Big(\frac{1}{4!} \mathcal{K}_{ABCD}L^A L^B L^C L^D \Big) + L^A d_A{}^{l k}\ \text{Re} N_l\ \text{Re} N_k\ ,
\end{equation}
where we have dropped the logarithm containing the circle radius. Indeed $\tilde K^{\rm M}$ agrees
precisely with the result found in \cite{Haack:1999zv,Haack:2001jz,Grimm:2010ks} from the M-theory reduction.
The general discussion of the circle reduction of a three-dimensional un-gauged $\mathcal{N}=2$ supergravity theory
to a two-dimensional $\mathcal{N}=(2,2)$ supergravity theory can be found in \autoref{3d-2dreduction}.
\subsection{Legendre transforms from chiral and twisted-chiral scalars} \label{sec_Legendre}
In this subsection we want to introduce an operation that allows to
translate the dynamics of certain chiral multiplets to twisted-chiral multiplets and
vice versa. More precisely, we will assume that some of the scalars, say the scalars
$\lambda_l$, in the $\mathcal{N}=(2,2)$ supergravity action have continuous shift
symmetries, i.e.~$\lambda_l \rightarrow \lambda_l + c_l$ for constant $c_l$.
These scalars therefore only appear with derivatives $d\lambda^l$ in the action.
By the standard duality of massless $p$-forms to $(D-p-2)$-forms in $D$ dimensions,
one can then replace the scalars $\lambda_l$ by
dual scalars $ \lambda'^{\, l}$. Accordingly, one has to adjust
the complex structure on the scalar field space by performing a
Legendre transform.
In the following we will give representative examples
of how this works in detail. We will see that this duality, in particular
as described in the first example, becomes crucial
in the discussion of mirror symmetry of \autoref{mirror_section}.
As a first example, let us consider the above theory with complex scalars
$z^K,N_l$ in chiral multiplets and complex scalars $t^A$ in twisted-chiral
multiplets. The kinetic potential for these fields $\tilde K$
was given in \eqref{tildeK} with \eqref{idhatKS}.
Two facts about this example are crucial for the following discussion.
First, the fields $N_l$ admit a shift symmetry $N_l \rightarrow N_l + i c_l$ in the
action, i.e.~the kinetic potential $\tilde K$ given in \eqref{idhatKS} is independent of $N_l - \bar N_l$.
Second, the $N_l$ only appear in the term $\mathcal{S}$ of
the kinetic potential and thus carry no dilaton pre-factor in the
action. One can thus straightforwardly
dualize $N_l - \bar N_l$ into real scalars $\lambda'^{\, l}$.
The new complex scalars $N'^{\, l}$ are then given by
\begin{equation} \label{N'}
N'^{\, l} = \frac{1}{2}\frac{\partial \mathcal{S}}{\partial \text{Re} \, N_l}+ i \lambda'^{\, l}\ ,
\end{equation}
where we have included a factor of $1/2$ for later convenience.
Furthermore, the new kinetic potential $\tilde K'$ is now a function
of $z^K,\ t^A,\ N'^{\, l}$ and given by the Legendre transform
\begin{equation} \label{tildeK'}
\tilde K' = \tilde K - 2\, e^{2\tilde \varphi} \text{Re} \, N'^{\, l} \text{Re} \, N_l \ ,
\end{equation}
where $ \text{Re} \, N_l $ has to be evaluated as a function of $\text{Re} \, N'^{\, l}$
and the other complex fields by solving \eqref{N'} for $\text{Re} \, N_l$.
One now checks that the scalars $N'^{\, l}$ actually reside in
twisted-chiral multiplets. Using the transformation \eqref{N'} and
\eqref{tildeK'} in the action \eqref{bosonic_dilatonsugra_extended} simply yields a dual description in which
certain chiral multiplets are consistently replaced by twisted-chiral multiplets.
It is simple to evaluate \eqref{N'}, \eqref{tildeK'} for $\mathcal{S}$ given in
\eqref{idhatKS} to find
\begin{eqnarray}
&N'^{\, l} &= H^{lm}\ \text{Re}\, N_m + i \lambda'^{\, l}\ , \label{N'ex} \\
&\tilde K' &= \mathcal{K} - e^{2 \phi_{\rm IIA}} H_{kl} \ \text{Re}\, N'^{\, k} \ \text{Re}\, N'^{\, l}\ ,\label{tildeK'ex}
\end{eqnarray}
where $H^{lm}$ is the inverse of the matrix $H_{lm}$ introduced in \eqref{def-H}, \eqref{splitH}.
It is interesting to realize that upon inserting \eqref{N'ex} into \eqref{tildeK'ex}
one finds that $\tilde K'$ evaluated as a function
of $N_k$ only differs by a minus sign in front of the term linear in $e^{2 \phi_{\rm IIA}}$
from the original $\tilde K$. This simple transformation arises from the
fact that $\tilde K$ is only quadratic in the $N_k$. This observation
will be crucial again in the discussion of mirror symmetry in \autoref{mirror_section}.
As a second example, we briefly want to discuss a dualization that
transforms all multiplets containing scalars to become chiral.
The detailed computation for a general $\mathcal{N}=(2,2)$ setting can be
found in \autoref{detailed_dual}.
For the example of \autoref{22dilatonSugra} we focus on the twisted-chiral
multiplets with complex scalars
$t^A$. These admit a shift symmetry $t^A \rightarrow t^A + c^A$ for
constant $c^A$, such that $\text{Re}\, t^A$ only appears with derivatives in
the action. Accordingly, the kinetic potential $\tilde K$
is independent of $t^A + \bar t^A$ as seen in \eqref{tildeK} with \eqref{idhatKS}.
Due to the shift symmetry we can dualize the scalars $t^A + \bar t^A$
to scalars $\rho_A$. However, note that by using the kinetic potential \eqref{tildeK}, \eqref{idhatKS}
there are couplings of $t^A$
in \eqref{bosonic_dilatonsugra_extended} that have a dilaton
factor $e^{\tilde \varphi}$, and others that are independent
of $e^{\tilde \varphi}$.
This seemingly prevents us from performing a straightforward Legendre transform
to bring the resulting action to the form \eqref{bosonic_dilatonsugra_extended}
with only chiral multiplets. Remarkably, the special properties of the
kinetic potential \eqref{tildeK}, \eqref{idhatKS}, however, allow us to nevertheless achieve this goal
as we will see in the following.
The action \eqref{bosonic_dilatonsugra_extended} for a setting with only chiral multiplets with
complex scalars $M^I$ takes the form
\begin{equation} \label{S2Kaehler}
S^{(2)} = \int e^{-2\tilde{\varphi}} \left( \frac{1}{2}R *1 + 2d \tilde{\varphi} \wedge * d\tilde{\varphi}
- \mathbf{K}_{M^I \bar M^J} \, dM^I\wedge * d \bar M^J \right)\ ,
\end{equation}
where $ \mathbf{K}_{M^I \bar M^J} = \partial_{M^I} \partial_{\bar M^J} \mathbf{K}$.
In other words, the potential $\mathbf{K}$ is in this case actually a K\"ahler potential
on the field space spanned by the complex coordinates $M^I$.
For our example \eqref{tildeK}, \eqref{idhatKS} the scalars $M^I$
consist of $z^K$, $N_l$, and $T_A$, where $T_A$ are the duals of the
complex fields $t^A$.
We make the following Ansatz for the dual coordinates
$T_A$
\begin{equation} \label{TA_Ansatz1}
T_A = e^{-2\tilde \varphi} \frac{\partial \tilde K}{\partial \text{Im}\, t^A} + i \rho_A=
e^{-2\tilde \varphi} \frac{\partial \mathcal{K} }{\partial \text{Im}\, t^A}+ \frac{\partial \mathcal{S}}{\partial \text{Im}\, t^A} + i \rho_A \ ,
\end{equation}
and the dual potential $\mathbf{K}$
\begin{equation} \label{TA_Ansatz2}
\mathbf{K} = \tilde K - e^{2\tilde \varphi} \text{Re} \, T_A \text{Im} \, t^A\ .
\end{equation}
These expressions describe the standard Legendre transform for $\text{Im} \,t^A$, but
crucially contain dilaton factors $e^{2\tilde \varphi}$. This latter fact
allows to factor out $e^{-2\tilde \varphi}$ as required in \eqref{S2Kaehler},
but requires to perform a two-dimensional Weyl rescaling as we will discuss
below. Using \eqref{tildeK} with \eqref{idhatKS} one
straightforwardly evaluates
\begin{eqnarray} \label{TAbfK}
&T_A &= e^{-2\phi_{\rm IIA}} \frac{1}{3!} \frac{\mathcal{K}_{A}}{\mathcal{V}} + d_{A}{}^{kl}\ \text{Re}\, N_l \ \text{Re}\, N_k + i \rho_A \ , \\
& \mathbf{K} &= - \log \int_{Y_4} \Omega \wedge \bar \Omega + \log \mathcal{V} \ .
\end{eqnarray}
Clearly, upon using the map \eqref{id_vA}
this result is familiar from the study of M-theory compactifications
on Calabi-Yau fourfolds \cite{Haack:1999zv,Haack:2001jz,Grimm:2010ks}. Also note that the contribution
$\mathcal{S}$ present in the kinetic potential \eqref{tildeK} is removed by the Legendre transform
in $\mathbf{K} $ and reappears in a more involved definition of the coordinates $T_A$.
At first it appears that \eqref{TA_Ansatz1} induces new mixed terms involving
one $d\tilde \varphi$ due to the dilaton dependence in front of the derivatives of $\mathcal{K}$.
Interestingly, these can be removed by a two-dimensional Weyl rescaling
if $\mathcal{K}$ satisfies the conditions
\begin{equation} \label{hatKcond}
\mathcal{K}_{t^A} \, \mathcal{K}^{t^A \bar t^B}\, \mathcal{K}_{\bar t^B} = k\ ,
\qquad \mathcal{K}_{v^A} \ d \text{Im} t^A = d f \ ,
\end{equation}
for some constant $k$ and some real field dependent function $f$.
In this expression $\mathcal{K}^{t^A \bar t^B}$ is the inverse of $\mathcal{K}_{t^A \bar t^B}$
and $\mathcal{K}_{v^A} \equiv \partial_{\text{Im}\, t^A} \mathcal{K}$.
In fact, one can perform the rescaling $\tilde g_{\mu \nu} = e^{2 \omega} g_{\mu \nu}$,
which transforms the Einstein-Hilbert action as
\begin{equation} \label{Weyl-EH}
\int e^{-2 \tilde \varphi} \frac{1}{2} \tilde R \ \tilde *1 =
\int e^{-2 \tilde \varphi} \left( \frac{1}{2} R \ *1 - 2 d\omega \wedge * d\tilde \varphi \right) \ ,
\end{equation}
while leaving all other terms invariant.
Using \eqref{Weyl-EH} to absorb the mixed terms one needs to chose
\begin{equation}
\omega = - \frac{k}{2} \tilde \varphi - \frac{f}{2} \ .
\end{equation}
The details of this computation can be found in \autoref{detailed_dual}.
Indeed, for the example \eqref{idhatKS} one finds $f= \text{log}\ \mathcal{V}$
and $k=-4$. Remarkably, the condition \eqref{hatKcond}
essentially states that $\mathcal{K}$ has to satisfy a no-scale like condition.
A recent discussion and further references on the subject of
studying four-dimensional supergravities satisfying
such conditions can be found in \cite{Ciupke:2015ora}.
\section{Mirror symmetry at large volume/large complex structure} \label{mirror_section}
In \autoref{IIAreduction} we have determined the two-dimensional action obtained from
Type IIA supergravity compactified on a Calabi-Yau fourfold. We commented
on its $\mathcal{N}=(2,2)$ supersymmetry structure which relies on the proper identification
of chiral and twisted-chiral multiplets in two dimensions. In this section
we are exploring the action of mirror symmetry. More precisely, we consider
pairs of geometries $Y_4$ and $\hat Y_4$ that are mirror manifolds \cite{Greene:1993vm,Mayr:1996sh,Klemm:1996ts}.
From a string theory world-sheet perspective one expects the two theories
obtained from string theory on $Y_4$ and $\hat Y_4$ to be dual. This implies
that after finding the appropriate identification of coordinates the two-dimensional effective
theories should be identical when considered at dual points in moduli space.
We will make this more precise for the large volume and large complex structure
point in this section. Note that in contrast to mirror symmetry for Calabi-Yau threefolds the mirror
theories encountered here are both arising in Type IIA string theory.\footnote{This can be seen
immediately when employing the SYZ-understanding of mirror symmetry as T-duality \cite{Strominger:1996it}.
Mirror symmetry is thereby understood as T-duality along half of the compactified
dimensions, i.e.~$Y_4$ is argued to contain real four-dimensional tori along which T-duality can be
performed. Clearly, this inverts an even number of dimensions for Calabi-Yau fourfolds.}
\subsection{Mirror symmetry for complex and K\"ahler structure}
Mirror symmetry arises from the observation that the conformal field theories associated with
$Y_4$ and $\hat Y_4$ are equivalent.
It describes the identification of
Calabi-Yau fourfolds $Y_4$, $\hat Y_4$
with Hodge numbers
\begin{equation}
h^{p,q}(Y_4) = h^{4-p,q}(\hat {Y}_4)\ .
\end{equation}
Note that this particularly includes the non-trivial conditions
\begin{eqnarray}
&h^{1,1}(Y_4) = h^{3,1} (\hat Y_4) \ , \qquad h^{3,1}(Y_4) = h^{1,1} (\hat Y_4)\ ,& \label{h11=h31} \\
&h^{2,1}(Y_4) = h^{2,1}(\hat Y_4)\ . \label{h21=h21} &
\end{eqnarray}
The first identification \eqref{h11=h31} together with the observations made in
\autoref{IIAreduction} implies that mirror symmetry exchanges K\"ahler structure deformations
of $Y_4$ ($\hat Y_4$) with complex structure deformations of $\hat Y_4$ ($Y_4$).
Accordingly one needs to
exchange chiral multiplets and twisted-chiral multiplets in the effective $\mathcal{N}=(2,2)$ supergravity
theory. The second identification \eqref{h21=h21} seems to suggest that for the fields $N_l$
the mirror map is trivial. However, as we will see in \autoref{3form_mirror} this is not the case
and one has to equally change from a chiral to a twisted-chiral description.
To present a more in-depth discussion of mirror symmetry we first
need to introduce some notation. All fields and couplings obtained
by compactification on $Y_4$ are denoted as in \autoref{IIAreduction}.
To destinguish them from the quantities obtained in the $\hat Y_4$
reduction we will dress the latter with a hat. In particular for the fields we write
\begin{eqnarray}
Y_4 :& \quad & \phi_{\rm IIA}\, ,\ t^A\, ,\ z^K \, ,\ N_l \ , \\
\hat Y_4 :& \quad & \hat \phi_{\rm IIA}\, ,\ \hat t^K\, ,\ \hat z^A \, ,\ \hat N_l \ . \nonumber
\end{eqnarray}
Note that we have exchanged the indices on $\hat t^K$ and $\hat z^A$
in accordance with the fact that complex structure and K\"ahler structure
deformations are interchanged by mirror symmetry. In other words,
$K= 1, \ldots, h^{1,1}(\hat Y_4)$ and $A = 1, \ldots, h^{3,1}(\hat Y_4)$
is compatible with the previous notation due to \eqref{h11=h31}.
Similarly we will adjust the notation for the couplings. For example,
the functions introduced in \eqref{idhatKS} and \eqref{def-f1}, \eqref{def-Psil} are
\begin{eqnarray}
Y_4 :& \quad & f_{mn}(z) \, ,\ H^{mn}(v,z) \, , \\
\hat Y_4 :& \quad &\hat f_{mn}(\hat z) \, , \ \hat H^{mn}(\hat v,\hat z)\, .
\end{eqnarray}
The functional form of the various couplings will in general differ for
$Y_4$ and $\hat Y_4$. A match of the two mirror-symmetric
effective theories should, however, be possible when identifying
the mirror map, which we denote formally by $\mathcal M[\cdot]$.
We want to focus on the sector of the theory independent of the three-forms.
Recall that in the two-dimensional effective theory obtained from
$Y_4$ the kinetic terms of the complex structure moduli $z^K$ and
K\"ahler structure moduli $t^A$ are obtained from the kinetic potential \eqref{hatKsimple}, \eqref{idhatKS} as
\begin{align} \label{hatKY4}
\mathcal{K}(Y_4) = \log \Big( \frac1{4!} \mathcal{K}_{ABCD}\, \text{Im} \,t^A \, \text{Im}\, t^B \, \text{Im} \, t^C\, \text{Im} \, t^D \Big)- \log \int_{Y_4} \Omega \wedge \overline{\Omega}
\end{align}
when used in the action \eqref{bosonic_dilatonsugra}.
Mirror symmetry exchanges the K\"ahler moduli $t^K$ of $Y_4$ with the
complex structure moduli $\hat z^K$ of $\hat Y_4$. The expression
\eqref{hatKY4} was computed at the large volume
point in K\"ahler moduli space, i.e.~with the assumption that $\text{Im} \, t^A \gg 1$ in string units.
Accordingly one has to evaluate $\mathcal{K}(\hat Y_4)$ at the large complex structure
point as
\begin{equation}
\int_{\hat Y_4} \hat \Omega \wedge \overline{ \hat \Omega} =
\frac1{4!} \mathcal{K}_{ABCD}\, \text{Im} \,\hat z^A \, \text{Im} \,\hat z^B \, \text{Im} \,\hat z^C\, \text{Im} \,\hat z^D\ ,
\end{equation}
where now $\text{Im} \, \hat z^A \gg 1$.
Similarly, one has to proceed for the K\"ahler moduli part of the kinetic potential $\mathcal{K}(\hat Y_4)$
and evaluate $\mathcal{K}(Y_4)$ at the large complex structure point
\begin{equation}
\int_{Y_4} \Omega \wedge \overline{ \Omega} = \frac1{4!} \hat \mathcal{K}_{KLMN}\, \text{Im}\, z^K \, \text{Im} \, z^L \, \text{Im} \, z^M\, \text{Im} \, z^N\ ,
\end{equation}
where $\hat \mathcal{K}_{KLMN}$ are now the quadruple intersection numbers on the geometry $\hat Y_4$.
Therefore, at the large volume and large complex structure point the two effective theories obtained
from $Y_4$ and $\hat Y_4$ are identified under the mirror map
\begin{equation} \label{mirror_tz}
\mathcal{M} \big[t^A \big]= \hat{z}^A\ , \quad \mathcal{M}\big[z^K\big]= \hat{t}^K\ ,
\end{equation}
and
\begin{equation} \label{mirror_phiIIA}
\mathcal{M} \big[ \mathcal{K}(Y_4) \big] = - \mathcal{K}(\hat Y_4)\ , \qquad \mathcal M \big[ \phi_{\rm IIA}\big] = \hat \phi_{\rm IIA} \ .
\end{equation}
It is important to stress that a sign change occurs when applying the mirror map to $\mathcal{K}$.
This can be traced back to the fact that scalars in chiral and twisted-chiral multiplets have different
sign kinetic terms in the actions \eqref{bosonic_dilatonsugra}, \eqref{bosonic_dilatonsugra_extended}.
The quantum corrections to $\mathcal{K}$ were discussed using mirror symmetry in \cite{Greene:1993vm,Mayr:1996sh,Klemm:1996ts,Grimm:2009ef}
and localization in \cite{Honma:2013hma,Halverson:2013qca} (using and extending
the results of \cite{Benini:2012ui,Doroud:2012xw,Jockers:2012dk}).
\subsection{Mirror symmetry for non-trivial three-forms} \label{3form_mirror}
Let us next include the moduli $N_l$ arising for Calabi-Yau fourfolds $Y_4$ with non-vanishing
$h^{2,1}(Y_4)$. In \autoref{IIAreduction} we have seen that these complex scalars
are part of chiral multiplets. Their dynamics was described by
the real function $\mathcal{S}$ in the kinetic potential $\tilde K$ given in \eqref{tildeK} and \eqref{idhatKS}.
For completeness we recall that
\begin{equation} \label{SY4}
\mathcal{S}(Y_4) = H^{l k}\ \text{Re}\, N_l\ \text{Re}\,N_k\ ,\qquad H^{l k} \equiv v^A d_A{}^{l k} \ ,
\end{equation}
where $d_A{}^{l k} $ is a function of the complex structure moduli of $Y_4$.
Mirror symmetry should map the fields $N_l$ to scalars $\hat N_l$ arising
in the reduction on the mirror Calabi-Yau fourfold $\hat Y_4$, i.e.~one
should have
\begin{equation} \label{cMN}
\mathcal M \big[ N_l \big] = Q_{l}( \hat N, \hat z, \hat t)\ ,
\end{equation}
where we have allowed the image of $N_l$ to be a non-trivial
function that will be determined in the following.
In fact, note that the map cannot be as simple as $\mathcal M(N_l) = \hat N_l$.
As already pointed out in \cite{Gates:2000fj}
the mirror duals $\mathcal M(N_l)$ need to be, in contrast to the $N_l$, parts
of \textit{twisted}-chiral multiplets. To
achieve this we need to use the
results of \autoref{sec_Legendre}.
Let us therefore consider the reduction on $\hat Y_4$ using the same notation as
in \autoref{IIAreduction} but with hatted symbols. The two-dimensional theory
will contain a set of complex scalars $\hat N_l$ that reside in chiral multiplets.
We can transform them to scalars in twisted-chiral multiplets using \eqref{N'ex}
and \eqref{tildeK'ex}. In other words, we find a dual description with
scalars $\hat N'^l$ defined as
\begin{equation} \label{hatN'}
\hat N'^{\, l} = \hat H^{lm}\, \text{Re}\, \hat N_m + i \hat \lambda'^{\, l}\ ,
\end{equation}
where $ \hat H^{lm}$ is a function of the mirror complex structure moduli $\hat z^A$
and K\"ahler moduli $\hat v^K$. The dual kinetic potential takes the form
\begin{equation} \label{tildeK'hatY4}
\tilde K'(\hat Y_4) = \mathcal{K}(\hat Y_4) - e^{2 \hat \phi_{\rm IIA}} \hat H_{kl}\ \text{Re}\, \hat N'^{\, k} \ \text{Re}\, \hat N'^{\, l}\ .
\end{equation}
The mirror map \eqref{mirror_tz}, \eqref{mirror_phiIIA} and \eqref{cMN}
exchanges chiral and twisted-chiral states and therefore has to take the form
\begin{eqnarray}
& \mathcal M \big[ N_l \big]& = \hat N'^{\, l}(\hat N,\hat z,\hat t)\ , \qquad \mathcal{M} \big[ t^A\big]= \hat{z}^A\ ,
\quad \mathcal{M}\big[ z^K \big] = \hat{t}^K\ , \label{full_mirror1} \\
& \mathcal{M}\big[ \tilde K(Y_4)\big] &= - \tilde {K}'(\hat Y_4)\ , \qquad \mathcal M\big[ \phi_{\rm IIA}\big] = \hat \phi_{\rm IIA}\ .
\label{full_mirror2}
\end{eqnarray}
and is evaluated as a function of $\hat N_l$, $\hat z^A$ and $\hat t^K$ by using \eqref{hatN'}.
Using these insights we are now able to infer the mirror image of the function $H_{mn}$ appearing
in $\tilde K(Y_4)$.
To do that, we apply the mirror map to the kinetic potential $\tilde K$. Note
that
\begin{equation}
\mathcal M \big[ \tilde K(Y_4) \big] = - \mathcal{K}(\hat Y_4) + e^{2 \hat \phi_{\rm IIA}} \mathcal M \big[ \mathcal{S}(Y_4) \big]\ ,
\end{equation}
where we have used \eqref{mirror_phiIIA}. Furthermore, we insert \eqref{full_mirror2} into \eqref{SY4}
to find
\begin{equation}
\mathcal M \big[ \mathcal{S}(Y_4) \big] = \sum_{k,l} \mathcal M \big[ H^{kl} \big] \text{Re}\, \hat N'^k \, \text{Re}\, \hat N'^l\ .
\end{equation}
We next apply \eqref{full_mirror2} together with \eqref{tildeK'hatY4} which requires
\begin{equation}
\sum_{k,l} \mathcal M \big[ H^{kl}\big] \text{Re}\, \hat N'^k \, \text{Re}\, \hat N'^l \overset{!}{=} \hat H_{kl}\ \text{Re}\, \hat N'^{\, k} \ \text{Re}\, \hat N'^{\, l}\ ,
\end{equation}
and thus enforces
\begin{equation} \label{cMH}
\mathcal M \big[ H^{kl} \big] \overset{!}{=} \hat H_{kl}\ .
\end{equation}
We therefore find that the mirror map actually identifies $H^{kl}$ with the \textit{inverse} $\hat H_{kl}$ of $\hat H^{kl}$.
This inversion is crucial and stems from the exchange of chiral an twisted-chiral multiplets under mirror symmetry.
In the final part of this section we evaluate the condition \eqref{cMH} at the large complex structure
point, since $H^{kl}$ given in \eqref{SY4} was computed at large volume.
Using the mirror map we are now able to determine the holomorphic function $f_{kl}$ appearing
in the definition of $H_{kl}$ at the large complex structure point.
Note that \eqref{splitH} translates on $Y_4$ and $\hat Y_4$ to
\begin{eqnarray} \label{recall_HhatH}
&H^{lm} &= - \frac{1}{2} (\text{Re} f)^{ln} \ \text{Re}\, h_n^m \ , \qquad h_n^m = - i t^A C_{An}{}^m \ , \qquad \\
&\hat H^{lm} &= -\frac{1}{2} (\text{Re} \hat f)^{ln} \ \text{Re}\, \hat h_n^m \ ,\qquad \hat h_n^m =-i\hat t^K \hat C_{Kn}{}^m\ , \nonumber
\end{eqnarray}
where on the mirror geometry we introduced the intersection numbers
\begin{equation}
\hat C_{Kn}{}^m = \int_{\hat Y_4} \hat{\omega}_K \wedge \hat{\alpha}_n \wedge \hat{\beta}^m\ .
\end{equation}
Using \eqref{full_mirror1}, \eqref{full_mirror2}, \eqref{cMH}, and \eqref{recall_HhatH} in the
mirror map one infers that a possible identification is \footnote{Note that
in general the basis $(\alpha_l,\beta^k)$ might not directly map to $(\hat \alpha_l,\hat \beta^k)$
on the mirror geometry $\hat Y_4$. In this expression we have assumed that there is no non-trivial
base change under mirror symmetry. }
\begin{equation}
\text{Re} f_{nm}= \text{Im} z^K \hat C_{Kn}{}^m\ .
\end{equation}
By holomorphicity of $f_{nm}$ we finally conclude
\begin{equation} \label{lin_fresults}
f_{nm}= -i z^K \hat C_{Kn}{}^m
\end{equation}
Having determined the function $f_{mn}$ at the large complex
structure point we have established a complete match of the
two two-dimensional effective theories obtained from $Y_4$ and
$\hat Y_4$ under the mirror map $\mathcal M[\cdot]$. The result \eqref{lin_fresults}
is not unexpected. In fact, from the variation of Hodge-structures one
could have expected a leading linear dependence on $z^K$. Furthermore,
we will find agreement with a dual Calabi-Yau threefold result when using
the geometry $Y_4$ as F-theory background and performing the orientifold limit.
This will be the task of the final section of this work.
\section{Applications for F-theory and Type IIB orientifolds} \label{F-theoryapp}
In this section we want to apply the result obtained by using
mirror symmetry to compactifications of F-theory and their orientifold limit.
The F-theory effective action is studied via the M-theory to
F-theory limit. Therefore, we will briefly review in \autoref{Mreduction_details}
the dimensional reduction of M-theory on a smooth Calabi-Yau fourfold including three-form moduli.
This reduction was already performed in \cite{Haack:2001jz}, but we will use the insights we have gained in the previous sections to include the three-form moduli more conveniently.
In \autoref{MFlimit} we will then restrict to a certain class
of elliptically fibered Calabi-Yau fourfolds and perform the M-theory to F-theory limit.
This allows us to identify the characteristic data determining the
four-dimensional $ \mathcal{N} = 1 $ F-theory effective action in terms of the geometric
quantities of the internal space \cite{Grimm:2010ks}. We note that for certain fourfolds the holomorphic
function $f_{kl}$ lifts to a four-dimensional gauge coupling function.
Starting from these F-theroy settings we will then perform the weak string coupling limit in \autoref{orientifold_limit}.
In this limit $f_{kl}$ can be partially computed by using mirror symmetry for Calabi-Yau threefolds
and we show compatibility with the fourfold result of \autoref{mirror_section}.
\subsection{M-theory on Calabi-Yau fourfolds} \label{Mreduction_details}
In this subsection we review the dimensional reduction of M-theory on a Calabi-Yau fourfold $ Y_4 $ in the large volume limit without fluxes. The ansatz here is similar to the one used for Type IIA supergravity in \autoref{IIAreduction_details}.
We start with eleven-dimensional supergravity as the low-energy limit of M-theory.
Its bosonic two-derivative action is given by
\begin{equation} \label{11Daction}
S^{(11)} = \int \frac{1}{2} \check R\ \check\ast 1 - \frac{1}{4} \check F_4 \wedge\check \ast \check F_4 - \frac{1}{12}\check C_3 \wedge \check F_4 \wedge \check F_4\ ,
\end{equation}
with $\check F_4 = d \check C_3 $ the eleven-dimensional three-form field strength. This will be dimensionally reduced on the background
\begin{equation}
d\check s ^2 = \eta^{(3)}_{\mu \nu} dx^\mu dx^\nu + g^{(8)}_{mn} dy^m dy^n\ ,
\end{equation}
where $ \eta^{(3)} $ is the metric of three-dimensional Minkowski space-time $ \mathbb{M}_{2,1} $ and $ g^{(8)} $ the metric of the Calabi-Yau fourfold $ Y_4 $. This is the analog to \eqref{metric-ansatz}
and, as we briefly discussed at the end of \autoref{22dilatonSugra}, the Type IIA supergravity vacuum can be obtained by
a circle-reduction of this Ansatz.
To perform the dimensional reduction one inserts similar expansions of \eqref{Kaehlermoduli},
\eqref{CSmoduli} and \eqref{BC-expand} into the eleven-dimensional action \eqref{11Daction}. For the metric deformations consisting of K\"ahler and complex structure deformations, this is exactly the same as \eqref{Kaehlermoduli} and
\eqref{CSmoduli}, hence we obtain $ h^{1,1}(Y_4) $ real scalars $ v^A_{\rm M}$ by expanding the M-theory K\"ahler form $J_{\rm M}$ as
\begin{equation} \label{JM}
J_{\rm M} = v^A _{\rm M}\omega_A
\end{equation}
and $ h^{3,1}(Y_4) $ complex scalars $ z^K $ in three dimensions. Since the eleven-dimensional three-form $\check C_3 $ is the common origin of the Type IIA fields
$\check {B}_2,\ \check{C}_3$, we expand
\begin{equation} \label{11Dthree-form}
\check C_3 = V^A \wedge \omega_A + N_l \Psi^l + \bar{N}_l \bar{\Psi}^l \, .
\end{equation}
This yields $ h^{2,1}(Y_4) $ three-dimensional complex scalars $ N_l $ and $ h^{1,1}(Y_4) $ vectors $ V^A $.
The latter combine with the real scalars $ v^A_{\rm M} $ into three-dimensional vector multiplets, whereas $ z^K, N_l $ give
rise to three-dimensional chiral multiplets. Combining the expansions \eqref{Kaehlermoduli},
\eqref{CSmoduli} and \eqref{11Dthree-form} with the action \eqref{11Daction} by using the notation of \autoref{IIAreduction_details} and \autoref{22dilatonSugra} we thus obtain the three-dimensional effective action
\footnote{The action has been Weyl-rescaled to the three-dimensional Einstein
frame by introducing ${g}^{\rm new}_{\mu \nu} = \mathcal{V}^{-2} g^{\rm old}_{\mu \nu} $}
\begin{align} \label{3daction}
S^{(3)} &= \int \frac{1}{2} R \ast 1 - G_{K\overline{L}} dz^K \wedge \ast d\overline{z}^L
- \frac{1}{2} d \log \mathcal{V}_{\rm M} \wedge \ast d \log \mathcal{V}_{\rm M} - G_{A B}^{\rm M} dv^A_{\rm M} \wedge \ast dv^B_{\rm M} \nonumber \\
&\quad - \frac{1}{2}v^A_{\rm M}\, d_{A}{}^{l k} DN_l \wedge \ast D\overline{N}_k
- \mathcal{V}_{\rm M}^2 G^{\rm M}_{A B} dV^A \wedge \ast dV^B \nonumber \\
&\quad + \frac{i}{4} d_A{} ^{l k} dV^A \wedge \big(N_l D\overline{N}_k - \overline{N}_k DN_l \big) \ .
\end{align}
Note the $G_{AB}^{\rm M}$ takes the same functional form as \eqref{GABmetric}, but uses the
M-theory K\"ahler structure deformations $v^A_{\rm M}$.
The three-dimensional action given in \eqref{3daction} is an $ \mathcal{N} = 2 $ supergravity theory.
The proper scalars in the vector multiplets are denoted by $L^A$ and
are expressed in terms of the $v^A_{\rm M}$ as $L^A = \frac{v^A_{\rm M}}{\mathcal{V}_{\rm M}}$,
as already given in \eqref{id_vA}. The complex scalars in the chiral multiplets
are collectively denoted by $\phi^\kappa = (z^K, N_l)$.
The action \eqref{3daction} can then be written using a kinetic potential $\tilde K^{\rm M}$ as
\begin{align} \label{chirallinear3D}
S^{(3)} = \int & \frac{1}{2} R^{(3)} \ast 1 + \frac{1}{4} \tilde{K}_{L^A L^B}^{\rm M} dL^A \wedge \ast dL^B
+ \frac{1}{4} \tilde{K}^{\rm M}_{L^A L^B} dV^A \wedge \ast dV^B \nonumber \\
&
- \tilde{K}^{\rm M}_{\phi^\kappa \bar \phi^\lambda}\, d\phi^\kappa \wedge \ast d\bar \phi^\lambda + dV^A \wedge \text{Im} (\tilde K^{\rm M}_{L^A \phi^\kappa} d \phi^\kappa )\ ,
\end{align}
where $\tilde{K}^{\rm M}_{L^A L^B} = \partial_{L^A} \partial_{L^B} \tilde K$,
$\tilde{K}^{\rm M}_{\phi^\kappa \bar \phi^\lambda} = \partial_{\phi^\kappa} \partial_{\bar \phi^\lambda} \tilde K^{\rm M}$,
and $\tilde K^{\rm M}_{L^A \phi^\kappa} = \partial_{L^A} \partial_{\phi^\kappa} \tilde K^{\rm M} $.
Comparing \eqref{3daction} with \eqref{chirallinear3D} the
kinetic potential obtained for this M-theory reduction therefore reads
\begin{equation} \label{def-tildeKM_Recall}
\tilde K^{\rm M} = - \log \int_{Y_4} \Omega \wedge \bar \Omega + \log \Big(\frac{1}{4!} \mathcal{K}_{ABCD}L^A L^B L^C L^D \Big) + \, L^A d_A{}^{l k} \ \text{Re}\, N_l \ \text{Re}\, N_k \ ,
\end{equation}
and was already given in \eqref{def-tildeKM}.
Recalling the discussion at the end of \autoref{22dilatonSugra} it is not hard to check
that \eqref{3daction} reduces to the Type IIA result found
in \autoref{IIAreduction_details} upon a circle compactification.
The detailed circle reduction is performed for a general three-dimensional
un-gauged $\mathcal{N}=2$ theory in \autoref{3d-2dreduction}.
\subsection{M-theory to F-theory lift} \label{MFlimit}
Let us now lift the result \eqref{chirallinear3D} of the M-theory reduction on a general smooth Calabi-Yau fourfold $ Y_4 $ to a four-dimensional effective F-theory compactification. To do so, we need to restrict $ Y_4 $ to be an elliptic fibration $ \pi: Y_4 \rightarrow B_3 $ over a base manifold $ B_3 $ which is a three-dimensional complex K\"ahler manifold. This four-dimensional theory exhibits $ \mathcal{N}=1 $ supersymmetry.
In the following we will not need to consider the full four-dimensional theory, but
will rather focus on the kinetic terms of the complex scalars and vectors without
including gaugings or a scalar potential.
Supersymmetry ensures that these can be written in the form \cite{Wess:1992cp}
\begin{equation} \label{S4gen}
S^{(4)} = \int \frac{1}{2} R \ast 1
- K^{\rm F}_{M^I \bar M^J} \, d M^I \wedge \ast d\bar{M}^J - \frac{1}{2} \text{Re} \, \mathbf{f}_{\Lambda \Sigma}
F^\Lambda \wedge \ast F^\Sigma - \frac{1}{2} \text{Im} \,\mathbf{f}_{\Lambda \Sigma} F^\Lambda \wedge F^\Sigma\ .
\end{equation}
In this expression we denoted by $M^I$ the bosonic degrees of freedom in chiral multiplets,
and by $F^\Lambda$ the field strengths of vectors $A^\Lambda$.
The metric $K^{\rm F}_{M^I \bar M^J} $ is K\"ahler and thus can be obtained from
a K\"ahler potential $K^{\rm F}$ via $K^{\rm F}_{M^I \bar M^J} = \partial_{M^I} \partial_{\bar M^J} K^{\rm F}$.
The gauge-kinetic coupling function $\mathbf{f}_{\Lambda \Sigma} $ is holomorphic
in the complex scalars $M^I$.
In order to determine the K\"ahler potential $K^{\rm F}$ and the gauge coupling
function $\mathbf{f}_{\Lambda \Sigma} $ via M-theory one next would have
to compactify \eqref{S4gen} on a circle. The resulting
three-dimensional theory then has to be pushed to the Coulomb
branch and all massive modes, including the excited
Kaluza-Klein modes of all four-dimensional fields, have to
be integrated out. The resulting three-dimensional effective theory
can then, after a number of dualizations, be compared with
the M-theory effective action \eqref{3daction}.
Performing all these steps is in general complicated.
However, a relevant special case
has been considered in \cite{Grimm:2010ks} and
will be the focus in the following discussion.
Despite the fact that we could refer
to \cite{Grimm:2010ks} we will try to keep the derivation of
$K^{\rm F}$ and $\mathbf{f}_{\Lambda \Sigma} $
in this subsection self-contained.
Let us therefore assume that $Y_4$ is an elliptically fibered
Calabi-Yau fourfold that satisfies the conditions
\begin{equation} \label{special_geom}
h^{2,1}(Y_4)= h^{2,1}(B_3)\ , \qquad \quad h^{1,1}(Y_4) = h^{1,1}(B_3) + 1\ .
\end{equation}
It is not hard to use toric geometry to construct examples
that satisfy these conditions (see, for example, refs.~\cite{Grimm:2009yu}).
From the point of view of F-theory, or Type IIB string theory, the first
condition in \eqref{special_geom} implies
that all scalars $N_l$ in \eqref{3daction} lift to R-R vectors $A^l$ in
four dimensions. In other words, one can compactify
Type IIB on the base $B_3$ and obtain vectors $A^l$
by expanding the R-R four-form as
\begin{equation} \label{C4expand}
C_4 = A^l \wedge \alpha_I - \tilde A_l \wedge \beta^l + \ldots\ .
\end{equation}
The vectors $\tilde A_l$ are the magnetic duals of the $A^l$
and can be eliminated by using the self-duality of the field-strength
of $C_4$.
The second condition in \eqref{special_geom}
implies that there are no further vectors in the four-dimensional
theory, i.e.~there are no massless vector degrees of
freedom arising from seven-branes.
The two-forms used in \eqref{JM} and \eqref{11Dthree-form} split simply as
\begin{equation}
\omega_A = (\omega_0, \omega_\alpha)\ ,
\end{equation}
where $ \omega_0$ is the Poincar\'e-dual of the base divisor $B_3$
and $\omega_\alpha$ is the Poincar\'e-dual of
the vertical divisors $ D^\alpha = \pi^{-1} (D^\alpha_{\rm b}) $ stemming from
divisors $D^{\alpha}_{\rm b}$ of $B_3$.
Accordingly one
splits the three-dimensional vector multiplets in \eqref{chirallinear3D}
as
\begin{equation}
L^A = (R, L^\alpha )\ , \qquad V^A = (A^0, A^\alpha) \ .
\end{equation}
One can now evaluate the kinetic potential \eqref{def-tildeKM_Recall}
for the special case \eqref{special_geom}.
The only relevant non-vanishing quadruple
intersection numbers are given by
\begin{equation} \label{base-triple}
\mathcal{K}_{0\alpha \beta \gamma} = \int_{Y_4} \omega_0 \wedge \omega_\alpha \wedge \omega_\beta \wedge \omega_\gamma \equiv \mathcal{K}_{\alpha \beta \gamma}\ ,
\end{equation}
which are simply the triple intersections $\mathcal{K}_{\alpha \beta \gamma}$ of the base $B_3$. Crucially,
for an elliptic fibration one has $\mathcal{K}_{\alpha \beta \gamma \delta} = 0$.
Furthermore, note that due to \eqref{special_geom} all non-trivial
three-forms come from the base $B_3$ and we can chose the
basis $(\alpha_I , \beta^l)$ such that
\begin{equation} \label{special_C}
C_{0m}{}^k = \int_{Y_4} \omega_0 \wedge \alpha_l \wedge \beta^k = \delta_l^k\ , \qquad C_{\alpha m}{}^{k} = C_{Amk} = 0\ ,
\end{equation}
with $C_{Am}{}^k$ and $C_{Amk}$ introduced in \eqref{Cdef}.
Inserting \eqref{base-triple} and \eqref{special_C} into \eqref{def-tildeKM_Recall} one finds
\begin{equation} \label{tildeKM_Special}
\tilde K^{\rm M} = - \log \int_{Y_4} \Omega \wedge \bar \Omega
+ \log \Big(\frac{1}{3!} \mathcal{K}_{\alpha \beta \gamma} L^\alpha L^\beta L^\gamma \Big) + \log(R) - \frac{1}{2} R\, \text{Re} f^{l k} \ \text{Re}\, N_l \ \text{Re}\, N_k \ ,
\end{equation}
where we have used that $L^A d_A{}^{l k} =-\frac{1}{2} L^A C_{A m}^{l} \text{Re} f^{m k} =-\frac{1}{2} R\, \text{Re} f^{l k}$,
and we have dropped terms in the logarithm that are higher order in $R$.
In order to compare this kinetic potential with the result of the
circle reduction of \eqref{S4gen} we next have to dualize $(L^\alpha, A^\alpha)$
into three-dimensional complex scalars $T_{\alpha}$, and $N_k$ into three-dimensional
vectors $(\xi^k, A^k)$. Due to our assumption \eqref{special_geom}
leading to \eqref{special_C} we can perform these dualizations independently.
The change from $(L^\alpha,A^\alpha)$ to $\text{Re} T_\alpha = \partial_{L^\alpha} \tilde K^{\rm M}$
is similar to \eqref{TAbfK}. It is conveniently
parameterized by the base K\"ahler deformations $v^\alpha_{\rm b}$
and the base volume $\mathcal{V}_{\rm b}$ defined as \cite{Grimm:2004uq,Grimm:2010ks}
\begin{equation} \label{def-vb}
L^\alpha = \frac{v^\alpha_{\rm b}}{\mathcal{V}_{\rm b}} \ , \qquad
\mathcal{V}_{\rm b} = \frac{1}{3!} \mathcal{K}_{\alpha \beta \gamma} v^{\alpha}_{\rm b} v^{\beta}_{\rm b} v^{\gamma}_{\rm b}\ .
\end{equation}
The dualization of the complex scalars $N_k$ into three-dimensional vectors
is similar to the dualization yielding \eqref{N'}, \eqref{tildeK'} and \eqref{N'ex}, \eqref{tildeK'ex}. First, one introduces
\begin{equation}
\xi^k = \partial_{\text{Re} N_k} \tilde K^{\rm M} \ , \qquad \tilde K^{\rm M \rightarrow F} = \tilde K^{\rm M} - \xi^k \text{Re} \, N_l \ ,
\end{equation}
and then dualizes the
field $\text{Im} N_k$ with a shift symmetry into the vector $A^k$.
Together both Legendre transforms yield
\begin{equation} \label{tildeKFM}
\tilde K^{\rm M \rightarrow F} = - \log \int_{Y_4} \Omega \wedge \bar \Omega
- 2 \log \, \mathcal{V}_{\rm b}\, + \log \, R + \frac{1}{2 R}\, \text{Re} f_{l k} \ \xi^l \xi^k \ ,
\end{equation}
which has to be evaluated as a function of $z^K$, $\xi^k$ and
\begin{equation} \label{Talphabase}
T_{\alpha}
= \partial_{L^\alpha} \tilde K^{\rm M} + i \rho_\alpha
= \frac{1}{2!} \mathcal{K}_{\alpha \beta \gamma} v^\beta_{\rm b} v^\gamma_{\rm b} + i \rho_\alpha\ .
\end{equation}
The kinetic potential \eqref{tildeKFM} is now in the correct frame to be lifted to
four space-time dimensions.
To derive $K^{\rm F}$, $\mathbf{f}_{kl} $
one reduces \eqref{S4gen} on a circle of radius $r$
with the usual Kaluza-Klein ansatz
the four-dimensional metric and vectors as
\begin{equation}
g^{(4)}_{\mu \nu} =
\begin{pmatrix}
g^{(3)} _{pq} + r^2 A^0 _p A^0 _q & r^2 A^0 _q \\
r^2 A^0 _p & r^2
\end{pmatrix},
\qquad A^k _\mu = (A^k_p + A^0 _p \zeta^k, \zeta^k)\ ,
\end{equation}
where we introduced the three-dimensional indices $ p,q = 0,1,2 $ and the Kaluza-Klein vector $ A^0 $. Note that we use for three-dimensional vectors the same symbol $ A^k $ as in four dimensions.
Furthermore, we introduced the new three-dimensional real scalars $ r, \zeta^k $ into the theory.
We next define
\begin{equation}
R = r^{-2}\ , \quad \xi^{\hat k} = ( R, R \zeta^k)\ , \quad A^{\hat k} = (A^0, A^k) \ .
\end{equation}
The three-dimensional theory obtained by reducing \eqref{S4gen} has thus
the field content: chiral multiplets with complex scalars $M^I$ and vector multiplets
$(\xi^{\hat k},A^{\hat k} )$.
Its action can be written in the form \eqref{chirallinear3D}
with a kinetic potential
\begin{equation} \label{redK}
\tilde K(M,\bar{M}, \xi) = K^F (M,\bar{M}) + \log (R) - \frac{1}{R} \text{Re}\, \mathbf{f}_{k l }(M) \xi^k \xi^l \ ,
\end{equation}
when replacing $L^A \rightarrow \xi^{\hat k}$, $V^A \rightarrow A^{\hat k}$, and $\phi^\kappa \rightarrow M^I$.
Finally, comparing \eqref{redK} with \eqref{tildeKFM} implies that one finds $M^I = \{T_\alpha, z^K \} $
\begin{eqnarray}
K^{\rm F} &=& -\log (\int_{Y_4} \Omega \wedge \overline{\Omega}) - 2 \log{\mathcal{V}_b}\ , \label{F-theoryKpot}\\
\mathbf{f}_{kl}&=& \frac{1}{2} f_{kl}\, .
\end{eqnarray}
In the next section, we want to derive the orientifold limit of this result relating the data of F-theory on $ Y_4 $ to Type IIB supergravity with $ O7/O3 $-planes on the closely related Calabi-Yau three-fold $ Y_3 $, a double cover of $ B_3 $.
\subsection{Orientifold limit of F-theory and mirror symmetry} \label{orientifold_limit}
In this final subsection we investigate the orientifold limit of the F-theory
effective action introduced above.
More precisely, we assume that the F-theory compactification on the
elliptically fibered geometry $Y_4$ admits a weak string coupling limit as
introduced by Sen \cite{Sen:1996vd,Sen:1997gv}. This limit takes one to a
special region in the complex structure moduli space of $Y_4$
in which the dilaton-axion $\tau = C_0 + ie^{-\phi_{\rm IIB}}$, given by the complex
structure of the two-torus fiber of $Y_4$, is almost everywhere
constant along the base $B_3$. The locations where $\tau$
is not constant are precisely the orientifold seven-planes (O7-planes).
In the weak string coupling limit the geometry $Y_4$ can
be approximated by
\begin{equation} \label{orientifold_projection}
Y_4 \cong (Y_3 \times T^2)/\tilde {\sigma}
\end{equation}
where we introduced the involution $ \tilde {\sigma} = (\sigma, -1,-1) $ with $ \sigma $ being a holomorphic and isometric
orientifold involution such that $ Y_3 / \sigma = B_3 $. The two one-cycles of the torus are both odd under the involution, but its volume form is even.
It was shown in \cite{Sen:1996vd,Sen:1997gv} that the double cover $Y_3$ of $B_3$ is actually a Calabi-Yau threefold. The location of the
O7-planes in $Y_3$ is simply the fixed-point set of $\sigma$.
In the limit \eqref{orientifold_projection} we can check compatibility of the mirror symmetry results of
\autoref{mirror_section} with the mirror symmetry of the Calabi-Yau threefold $Y_3$.
By using the mirror fourfold $\hat Y_4$ of $Y_4$ we have found that the function $f_{lk}$ is linear in the large
complex structure limit of $Y_4$. Here we recall that the weak string coupling expression gives a compatible result. Using
the mirror $\hat Y_3$ of $Y_3$ one shows that the function $f_{lk}$ is linear in the large
complex structure limit of $Y_3$. This can be depicted as
\begin{equation}
\begin{array}{ccc}
\text{F-theory on}\ Y_4 &\quad \xrightarrow{\quad \text{weak coupling} \quad } & \text{Type IIB orientifolds}\ Y_3 / \sigma \qquad \\
&& \qquad\qquad\qquad \updownarrow \quad \text{physical mirror duality}\\
&&\qquad \text{Type IIA orientifolds}\ \hat Y_3 / \hat \sigma \qquad
\end{array}
\end{equation}
Note that mirror symmetry of $Y_3$ and $\hat Y_3$ gives a physical map between Type IIB and Type IIA orientifolds.
The mirror map between $Y_4$ and $\hat Y_4$ has no apparent physical meaning in F-theory. Nevertheless,
using the geometry $Y_4$ in Type IIA compactifications it can be used to calculate
$f_{lk}$ as we explained in \autoref{mirror_section}.
Let us now introduce the function $f_{lm}$ for the geometry \eqref{orientifold_projection}.
In the orientifold setting one splits the cohomologies of $Y_3$ as $H^{p,q} (Y_3)= H^{p,q}_+ (Y_3) \oplus H^{p,q}_- (Y_3)$,
which are the two eigenspaces of $\sigma^*$. We denote their dimensions as $h^{p,q}_\pm (Y_3)$.
As reviewed, for example, in \cite{Denef:2008wq}
the complex structure moduli $z^K$ of $Y_4$ split into three sets of fields at weak string coupling.
First, there is the dilaton-axion $\tau$, which is now a modulus of the effective theory.
Second, there are $h^{2,1}_-$ complex structure moduli $z^{\kappa}$ of the quotient $Y_3/\sigma$.
Third, the remaining number of complex structure deformations of $Y_4$ correspond to
D7-brane position moduli. The last set are open string degrees of freedom
and are not captured by the geometry of $Y_3$. For simplicity, we will not include
them in the following discussion.
With this simplifying assumption one finds that the pure complex structure part of the F-theory
K\"ahler potential \eqref{F-theoryKpot} splits as
\begin{equation} \label{Omegasplit}
-\log (\int_{Y_4} \Omega \wedge \overline{\Omega} ) = - \log \big[ -i(\tau - \bar{\tau})\big] - \log \Big[ i \int_{Y_3} \Omega_3 \wedge \bar{\Omega}_3 \Big] +\ldots \, ,
\end{equation}
where $\Omega_3$ is the $(3,0)$-form on $Y_3$ that varies holomorphically
in the complex structure moduli $z^{\kappa}$.
The dots indicate that further corrections arise that are suppressed at weak string coupling $-i(\tau - \bar \tau) \gg 1$.
Taking the weak coupling limit for the K\"ahler potential \eqref{F-theoryKpot} of the
K\"ahler structure deformations is more straightforward. The deformations are counted by
$h^{1,1}_+(Y_3)$ and identified with the K\"ahler structure deformations $v_{\rm b}^\alpha$ of the
base $B_3$ introduced in \eqref{def-vb}. The orientifold K\"ahler potential for this
set of deformations is then simply the second term in \eqref{F-theoryKpot} and
the K\"ahler coordinates are given by \eqref{Talphabase}.
Turning to the gauge theory sector, we note that the number of R-R vectors $A^l$ arising
from $C_4$ as in \eqref{C4expand} are counted by $h^{2,1}_+(Y_3)$ in the orientifold setting.
The gauge coupling function for these vectors is determined as function of
the complex structure moduli $z^{\kappa}$ of $Y_3$ in \cite{Grimm:2004uq}.\footnote{Note that we
have slightly changed the index conventions with respect to \cite{Grimm:2004uq} in order to
match the F-theory discussion. }
It is given by
\begin{equation} \label{f-ori}
f_{kl}(z^\kappa) = -i \mathcal{F}_{k l}|(z^{\kappa}) \equiv \partial_{z^k} \partial_{z^l} \mathcal{F} | (z^{\kappa})\ ,
\end{equation}
where $\mathcal{F}$ is the pre-potential determining the moduli-dependence of the
$\Omega_3$ of the geometry $Y_3$. To evaluate \eqref{f-ori} one first
splits the complex structure moduli of $Y_3$ into $h^{2,1}_-(Y_3)$ fields
$z^{\kappa}$ and $h^{2,1}_+(Y_3)$ fields $z^k$. The pre-potential $\mathcal{F}(z^{\kappa},z^k)$
of $Y_3$ at first depends on both sets of fields.
Then one has to take derivatives of $\mathcal{F}$ with respect to $z^k$ and
afterwards set these fields to constant background values compatible
with the orientifold involution $\sigma$. This freezing of the $z^k$ is indicated
by the symbol $|$ in \eqref{f-ori}.
Using mirror symmetry for Calabi-Yau threefolds it is well-known
that the pre-potential at the large complex structure point of $Y_3$
is a cubic function of the complex structure moduli $z^{\kappa}$ and $z^{k}$.
Taking derivatives and evaluating the expression on the orientifold moduli
space one thus finds
\begin{equation}
f_{k l}(z^\kappa) = -i z^{\kappa} \hat \mathcal{K}_{\kappa k l } \ ,
\end{equation}
where $ \hat \mathcal{K}_{\kappa k l } = \int_{\hat{Y}_3} \hat{\omega}_{\kappa} \wedge \hat{\omega}_k \wedge \hat{\omega}_l $
are the triple intersection numbers of the mirror threefold $\hat Y_3$.
This result agrees with the one for Type IIA orientifolds, which have
been studied at large volume in \cite{Grimm:2004ua}.
Hence, we find consistency with the F-theory result \eqref{lin_fresults} obtained by using
mirror symmetry for $Y_4$ at the large complex structure point. To obtain a
complete match of the results the
intersection matrix $\hat C_{\kappa k}{}^l$ of $\hat Y_4$ is identified with the triple intersection $\hat \mathcal{K}_{\kappa k l } $
of $\hat Y_3$.
To close this section we stress again that we have only discussed the matching with
the orientifold limit for special geometries satisfying \eqref{special_geom}.
Furthermore, we have not included the open string degrees of freedom on the
orientifold side. Clearly, our result for $f_{lk}$ obtained in \autoref{mirror_section}
can be more generally applied. For example, a simple generalization
is the inclusion of $h^{1,1}_-(Y_3)$ moduli $G^a$ into the orientifold setting,
which arise in the expansion of the complex two-form $C_2-\tau B_2$.
In F-theory the same degrees of freedom appear
from the expansion \eqref{11Dthree-form} into non-trivial three-forms $\Psi_a$ that have two legs
in the base $B_3$ and one leg in the torus fiber, i.e.~are not present in
the geometries satisfying \eqref{special_geom}.
In the orientifold setting one finds that the fields $G^a$ correct the complex
coordinates \eqref{Talphabase}. We read off the result from \cite{Grimm:2004uq} to find \footnote{Note that
compared with \cite{Grimm:2004uq} we have redefined $\rho_\alpha$ to make the terms
in $T_\alpha$ involving the $G^a$ real.}
\begin{equation}
T_\alpha =\frac{1}{2!} \mathcal{K}_{\alpha \beta \gamma} v^\beta_{\rm b} v^\gamma_{\rm b}
+ \frac{1}{2 \, \text{Im} \tau}\ \mathcal{K}_{\alpha a b}\ \text{Im} \, G^a \text{Im} \, G^b + i \rho_\alpha \, .
\end{equation}
Comparing this expression with \eqref{TAbfK} we read off that
\begin{equation}
N^a = i G^a\ , \qquad d_{\alpha ab} = \frac{1}{2} \frac{1}{ \text{Im} \tau} \mathcal{K}_{\alpha a b}\ , \qquad f_{ab} (\tau)= i \tau \delta_{ab}\ ,
\end{equation}
in order to match the F-theory result as already done in \cite{Grimm:2005fa}.
Again we find that the result is linear in one of the complex structure moduli, namely the field $\tau$,
of the Calabi-Yau fourfold $Y_4$ in the orientifold limit \eqref{orientifold_projection}.
It would be interesting to generalize these results even further and also include the open
string moduli into the orientifold setting.
\section{Conclusions}
In this paper we first studied
the two-dimensional low-energy effective action obtained from Type IIA string theory
on a Calabi-Yau fourfold with non-trivial
three-form cohomology.
The couplings of the three-forms were shown to
be encoded by two holomorphic functions $f_{kl}$ and
$h_k^l$, where the former depends on the complex structure moduli
and the latter on the complexified K\"ahler structure moduli.
Performing a large volume dimensional reduction of Type IIA supergravity, we
were able to derive $h_k^l$ explicitly as a linear function.
We argued that $f_{kl}$ and $h_k^l$ computed
on mirror pairs of Calabi-Yau manifolds will be exchanged, at least,
if one considers the theories at large volume and large complex structure.
In order to show this, we investigated the non-trivial map
between the three-form moduli arising from mirror geometries
and argued that it involves a scalar field dualization together
with a Legendre transformation.
This can be also motivated by the fact that chiral and
twisted-chiral multiplets are expected to be exchanged
by mirror symmetry. We thus established a linear dependence of
the function $f_{lk}$ on the complex structure moduli near
the large complex structure point and determined the constant topological
pre-factor.
In this work we also included a discussion of
the superymmetry properties of the two-dimensional
low-energy effective action. This action is expected to be an
$\mathcal{N}=(2,2)$ supergravity theory, which we showed to
extend the dilaton supergravity action of \cite{Gates:2000fj}.
The bosonic action was brought to an elegant
form with all kinetic and topological terms determined
by derivatives of a single function $\tilde K = \mathcal{K} + e^{2 \tilde \varphi} \mathcal{S}$,
where $\mathcal{K}$ and $\mathcal{S}$ can depend on the scalars in chiral and
twisted-chiral multiplets, but are independent of the two-dimensional dilaton $\tilde \varphi$.
In the Type IIA supergravity reduction the three-form scalars only
appeared in the function $\mathcal{S}$ and are thus suppressed by
$e^{2 \tilde \varphi} = e^{2 \phi_{\rm IIA}}$.
In this analysis the complex structure moduli and the
three-form moduli were argued to fall into chiral multiplets, while the
complexified K\"ahler moduli are in twisted-chiral multiplets.
However, due to apparent shift symmetries of the three-form moduli
and complexified K\"ahler moduli a scalar dualization accompanied by
a Legendre transformation can be performed in
two dimensions. This lead to dual descriptions in which certain
chiral multiplets are replaced by twisted-chiral multiplets and vice versa.
Remarkably, if one dualizes a subset of scalars appearing in $\mathcal{K}$,
we found that the requirement
to bring the dual action back to the standard $\mathcal{N}=(2,2)$ dilaton supergravity
form imposes conditions on viable $\mathcal{K}$.
These constraints include a no-scale type condition on $\mathcal{K}$.
The emergence of such restrictions arose from general arguments
about two-dimensional theories coupled to an overall
$e^{-2\tilde \varphi}$ factor. For Calabi-Yau fourfold reductions we checked that these conditions are
indeed satisfied.
It would be interesting to investigate this further and to get a
deeper understanding of this result.
Having shown that in the large complex structure limit
the function $f_{kl}$ is linear in the complex structure
moduli, we discussed the
application of this result in an F-theory compactification.
By assuming that the Calabi-Yau fourfold is elliptically fibered and that
the three-forms exclusively arise from the base of this fibration,
we recalled that $f_{kl}$ is actually the gauge-coupling
function of four-dimensional R-R vector fields.
This gauge-coupling function was already evaluated
in the weak string coupling limit in the orientifold literature.
In this orientifold limit
one can double-cover the base with a Calabi-Yau threefold.
We found compatibility of the fourfold result with the expectation
from mirror symmetry for Calabi-Yau threefold orientifolds.
In this analysis we only included closed string moduli in
the orientifold setting. Clearly, the results obtained from the
Calabi-Yau fourfold analysis are more powerful and it would be
interesting to further investigate the open string dependence in
orientifolds using our results. Additionally we commented briefly
on the case in which the three-forms have legs in the fiber
of the elliptic fibration. In this situation the inverse of $\text{Re} f_{lk}$
sets the value of decay constants of four-dimensional axions \cite{Grimm:2014vva}.
Again we found compatibility in the closed string sector at weak string
coupling in which $f_{lk} \propto i \tau$. It would be interesting
to include the open string moduli in the orientifold
setting and derive corrections to $f_{lk}$ without restricting to the
weak string coupling limit. The latter task requires to compute $f_{lk}$ away from the
large complex structure limit for elliptically fibered Calabi-Yau fourfolds.
In order to derive the complete moduli dependence of $f_{lk}$
at various points in complex structure moduli space it would
be desirable to obtain differential equations obeyed by
the $(2,1)$-forms. This should be possible by investigating
the variations of Hodge structures and is expected to yield
equations of second order in derivatives. The linear solutions
found in this work can then provide the boundary
conditions for the complete solutions. It would be important to
develop the necessary tools for such an analysis and
we hope to return to this issue in a future publication.
\subsubsection*{Acknowledgments}
We would like to thank Hans Jockers, Andreas Kapfer, Denis Klevers,
Diego Regalado, and Matthias Weissenbacher for illuminating discussions.
This work was supported by a grant of the Max Planck Society.
|
1,116,691,499,409 | arxiv |
\section{Introduction}
Quarks and gluons, the fundamental degrees of freedom of Quantum Chromodynamics (QCD)
never appear in Nature as asymptotic one-particle states. This phenomena is called confinement.
Instead, only their bound states, i.e., mesons and baryons, are observed in experiment
and these hadronic states are the asymptotic states whose interactions can be parametrized by the S-matrix.
An ultimate goal for theoretical studies of the strong interaction, therefore, is to extract the properties of the hadronic S-matrix from QCD.
For a description of hadronic interactions, the nuclear force is one of the most fundamental quantities in nuclear physics. The origin of the nuclear force, however, is still one of the major unsolved problems in strong interaction physics even after the establishment of QCD.
Although the nuclear force is not well understood theoretically, a large number of proton-proton and neutron-proton scattering data as well as deuteron properties have been experimentally accumulated and summarized as the nucleon-nucleon ($NN$) potential, which is designed to reproduce these experimental properties through the Schr\"odinger equation for the $NN$ wave function. Once the potential is determined, it can be used to study systems with more than 2 nucleons by using various many-body techniques.
Phenomenological $NN$ potentials which can fit the $NN$ data precisely (e.g. more than 2000 data with $\chi^2/{\rm dof}\simeq 1$ ) at $T_{\rm lab} < 300 $ MeV are called the high-precision $NN$ potentials. Those in coordinate space, some of which are shown in Fig.~\ref{fig:NNpotential}, reflect characteristic features of the $NN$ interaction at different length scales\cite{Taketani1967,Hoshizaki1968,Brown1976,Machleidt1989,Machleidt2001}:
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.5\textwidth]{Figs/phen-pot_new.eps}
\caption{Three examples of the modern $NN$ potential in $^1S_0$ (spin-singlet and $S$-wave) channel: CD-Bonn\protect\cite{Machleidt2001a}, Reid93\protect\cite{Stoks:1994wp} and Argonne $v_{18}$\protect\cite{Wiringa:1994wb}. Taken from Ref.~\protect\cite{Ishii:2006ec}.}
\label{fig:NNpotential}
\end{center}
\end{figure}
\begin{enumerate}
\item[(i)]
The long range part of the nuclear force (at relative distance
$r > 2$ fm)
is dominated by one-pion exchange introduced by Yukawa\cite{Yukawa1935}.
Because of the pion's Nambu-Goldstone character, it
couples to the spin-isospin density of the nucleon and hence
leads to a strong spin-isospin dependent force, namely the tensor force.
\item[(ii)] The medium range part ($1\ {\rm fm} < r < 2$ fm) receives
significant contributions from the exchange of
two-pions ($\pi\pi$) and/or heavy mesons ($\rho$, $\omega$, and $\sigma$).
In particular, the spin-isospin independent attraction
of about 50 -- 100 MeV in this region plays an essential role
for the binding of atomic nuclei.
\item[(iii)] The short range part ($r < 1$ fm) is best described by
a strong repulsive core as originally introduced by Jastrow \cite{Jastrow1951}.
Such a short range repulsion is important for
the stability of atomic nuclei against collapse,
for determining the maximum mass of neutron stars, and for
igniting the Type II supernova explosions \cite{Tamagaki1993,Heiselberg2000,Lattimer2000}.
\end{enumerate}
It is then a challenge for the theoretical particle and nuclear physics communities to extract these properties of the $NN$ interaction from first principle non-perturbative QCD calculations, in particular lattice QCD simulations.
A theoretical framework suitable for such a purpose was first proposed by L\"{u}scher\cite{Luscher:1990ux}. For two hadrons in a finite box with the size $L \times L \times L$ with periodic boundary conditions, an exact relation between the energy spectra in the box
and the elastic scattering phase shift at these energies was derived. If the range of the hadron interaction $R$ is sufficiently smaller than the size of the box $R<L/2$, the behavior of the
two-particle Nambu-Bethe-Salpeter (NBS) wave function $\psi ({\bf r})$ in the interval $R < \vert {\bf r} \vert < L/2 $ has sufficient information to relate the phase shift and the two-particle spectrum. L\"{u}scher's method bypasses the difficulty to treat the real-time scattering process on the Euclidean lattice. Furthermore, it utilizes the finiteness of the lattice box effectively to extract the information of the on-shell scattering matrix and the phase shift.
A closely related but an alternative approach to the $NN$ interactions from lattice QCD has been proposed recently\cite{Ishii:2006ec,Aoki:2008hh,Aoki:2009ji}.
The starting point is the same NBS wave function $\psi (\bf r)$ as discussed in Ref.~\cite{Luscher:1990ux}. Instead of looking at the wave function outside the range of the interaction, the authors consider the internal region $ |\bf r | < R$ and define an energy-independent non-local potential $U(\bf r, \bf r')$ from $\psi (\bf r)$ so that it obeys the Schr\"{o}dinger type equation in a finite box.
Since $U(\bf r, \bf r')$ for strong interactions is localized in its spatial coordinates due to confinement of quarks and gluons, the potential receives only weak finite volume effect in a large box. Therefore, once $U$ is determined and is appropriately extrapolated to $L \rightarrow \infty$, one may simply use the Schr\"{o}dinger equation in infinite space to calculate the scattering phase shifts and bound state spectra to compare with experimental data. A further advantage of utilizing the potential is that it is a smooth function of the quark masses so that it is relatively easy to handle on the lattice. This is in sharp contrast to the scattering length which shows a singular behavior in the quark mass corresponding to the formation of the $NN$ bound state.
Since the recent progress for the study of the $NN$ interaction by the first method has already been reviewed in Ref.\cite{Beane:2010em}, the recent progress for the second method is mainly considered in this review.
In Sec.~\ref{sec:strategy}, the strategy of Ref.~\cite{Ishii:2006ec,Aoki:2008hh,Aoki:2009ji} to define the $NN$ potential in QCD is explained in detail, and the lattice formulation is introduced
in Sec.~\ref{sec:lattice}.
Results of lattice QCD calculations for $NN$ potentials are given in both quenched and full QCD
in Sec.~\ref{sec:NNpotential}. Central potentials at the leading order of the velocity expansion is shown to reproduce qualitative features of the $NN$ potential such as the repulsion at short distance and the attraction at medium to long distances.
The tensor potential, which exists also at leading order, is extracted. Contrary to the case of the central potentials, it does not have a repulsive core. Higher order contributions in the velocity expansion are also investigated and shown to be small at low energy and low orbital angular momentum $L$.
In Sec.~\ref{sec:hyperon}, the method to extract the potential is applied to the hyperon-nucleon interactions such as $N\Xi$ and $N\Lambda$ systems. Interactions between octet baryons in general are also investigated in the flavor SU(3) limit, where up, down and strange quark masses are all equal. In Sec.~\ref{sec:OPE},
we also consider a recent attempt to understand the origin of the repulsive core in the $NN$ potential. Using the operator product expansion and renormalization group analysis in QCD, the potential derived from the NBS wave function in Sec.~\ref{sec:NNpotential} is shown to have a repulsive core, whose functional form is also theoretically predicted.
In Sec.~\ref{sec:extension}, two extensions of the potential method are considered, together with explicit applications of these extensions to hadron interactions. One is the extension of the potential method to inelastic scattering, in order to investigate the $\Lambda\Lambda$ system,
while the other is the extraction of the potential from the time dependent NBS wave function in lattice QCD. With the latter method, the existence of the $H$-dibaryon is investigated in the flavor SU(3) limit.
In Sec.~\ref{sec:others}, applications of the method to the three nucleon force, meson-baryon potentials and the potential in 2-color QCD are considered.
Brief concluding remarks are given in Sec.~\ref{sec:conclusion}.
\section{Strategy to extract potential in QCD}
\label{sec:strategy}
\subsection{Nambu-Bethe-Salpeter (NBS) wave function and its asymptotic behavior}
A key quantity to extract the potential from QCD is the equal time Nambu-Bethe-Salpeter wave function, defined by
\begin{eqnarray}
\varphi^W({\bf x}) e^{-Wt}&=& \langle 0 \vert T\{N({\bf r} + {\bf x},t) N({\bf r}, t)\} \vert 2N,W, s_1 s_2\rangle,
\end{eqnarray}
where $\vert 2N, W, s_1s_2\rangle$ is an eigen-state of QCD for two nucleons with total energy $W=2\sqrt{{\bf k}^2+m_N^2}$ and the total three-momentum ${\bf P}=0$, whose helicities are denoted by $s_1$, $s_2$. Here the local nucleon operator is given by
\begin{eqnarray}
N_{\alpha}(x) &\equiv& \left(\begin{array}{c}
p_\alpha (x) \\
n_\alpha (x) \\
\end{array}\right) =
\varepsilon^{abc} \left( u_a (x) C\gamma_5 d_b(x) \right) q_{\alpha}(x), \quad
q(x)=\left(\begin{array}{c}
u (x) \\
d (x) \\
\end{array}\right),
\end{eqnarray}
where $x=({\bf x},t)$, the color indices are denoted by $a,b,c$, and $\alpha$ is the spinor index.
The charge conjugation matrix in the spinor space is given by $C=\gamma_2\gamma_4$, and
$p,n$ denotes proton and neutron operators while $u,d$ denote up and down quark operators.
Note that $\varphi^W$ implicitly has two pairs of spinor-flavor indices which come from $N_\alpha ({\bf r} + {\bf x},t) N_\beta({\bf r}, t)$ and two helicity indices $s_1$ and $s_2$.
The most remarkable property of the above NBS wave function is explained as follows.
If the $W$ is smaller than the threshold energy for one-pion production ({\it i.e.} $W < 2 m_N + m_\pi$ ), then its asymptotic behavior for large $\vert{\bf x} \vert$ can be
evaluated\cite{Ishizuka2009a,Aoki:2009ji}. The helicity component in the spin singlet channel ($S=0$) is given by $
\vert s_1 s_2 \rangle = \frac{1}{\sqrt{2}}\left(\vert +\frac{1}{2},+\frac{1}{2}\rangle +\vert -\frac{1}{2},-\frac{1}{2}\rangle\right)
$, where the relative $+$ sign is our convention.
For this case, we have
\begin{eqnarray}
\varphi^W({\bf r})_{S=0} &\simeq & \sum_{l,l_z} Z^{l,l_z} (S=0)Y_{l l_z}(\Omega_{\bf r}) \frac{\sin( k r - l\pi/2 + \delta_{l0}(k))}{k r} e^{i\delta_{l0}(k)}
\label{eq:asympt_singlet}
\end{eqnarray}
where $r=\vert{\bf r}\vert$, $k=\vert{\bf k}\vert$, and
$\delta_{lS}(k)$ is the $NN$ scattering phase shift in QCD with the total angular momentum $l$ and the total spin $S$, which is determined by the unitarity of S-matrix in QCD below the inelastic threshold\cite{Aoki:2009ji}. Here $Y_{lm}(\Omega_{\bf r})$ is the spherical harmonic function with the solid angle $\Omega_{\bf r}$ of ${\bf r}$. The coefficient $Z^{ll_z}(0)$ has spinor components $\alpha,\beta$ and is given by
\begin{eqnarray}
Z^{ll_z}_{\alpha\beta}(0) &=& Z D^l_{l_z0}(\Omega_{\bf k}) U_{\alpha \hat\alpha}(\nabla) U_{\beta \hat\beta}(-\nabla) \chi_{\hat\alpha \hat\beta} (0,0)
\end{eqnarray}
where $Z$ is the wave function renormalization for the nucleon operator $N(x)$, $D^l_{m\lambda}$ is the Wigner $D$-matrix, and $U_{\alpha \hat\alpha}(\nabla)$ and $U_{\beta \hat\beta}(-\nabla)$ are the $4\times 2$ matrices acting on the $2\times 2$ matrix $\chi_{\hat\alpha\hat\beta}(S,S_z)$. Explicitly we have
\begin{eqnarray}
U(\nabla) &=& \sqrt{W+m_N}\left(
\begin{array}{cc}
I_{2\times 2}, & \displaystyle\frac{-i\sigma\cdot \nabla}{W+ m_N} \\
\end{array}\right) \\
\chi (0,0)&=&\frac{1}{\sqrt{2}} i\sigma_2, \quad \chi(1,0) =\frac{1}{\sqrt{2}}\sigma_1,\quad
\chi(1,\pm 1) = \frac{1}{2}(I_{2\times 2}\pm \sigma_3).
\end{eqnarray}
For the spin triplet channel ($S=1$), the asymptotic behavior of $\varphi^W$ is more involved but schematically written as
\begin{equation}
\varphi^W({\bf r})_{S=1} \propto \sum Y_{l l_z}(\Omega_{\bf r}) \frac{\sin( k r - l\pi/2 + \delta_{l1}(k))}{k r} e^{i\delta_{l1}(k)}
\label{eq:asympt_triplet} .
\end{equation}
An explicit form of the asymptotic behavior for the NBS wave function in the triplet channel
is given in appendix~\ref{app:NBS}.
The asymptotic behaviors in eqs.~(\ref{eq:asympt_singlet}) and (\ref{eq:asympt_triplet}) tell us that the NBS wave function at large separation $r$ describes the scattering wave of the quantum mechanics whose phase shift agrees with the phase of the S-matrix in QCD.
Therefore
the NBS wave function satisfies the free Schr\"odinger equation at large $r$ as
\begin{equation}
\left[\frac{k^2}{2\mu} - H_0\right] \varphi^W({\bf r} )\simeq 0, \qquad H_0 =\frac{-\nabla^2}{2\mu}
\end{equation}
at $W < 2m_N + m_\pi$, where $ \mu = m_N/2$ is the reduced mass of the $NN$ system.
Note that these properties hold
without using the non-relativistic approximation/expansion. In particular, only the upper components of the spinor indices for the NBS wave function ($\alpha=1,2$ and $\beta =1,2$) are enough to reproduce all $NN$ scattering phase shifts $\delta_{lS}(k)$ with $l=0,1,2,3,\cdots$ and $S=0,1$. From these properties, (the upper spinor components of) the NBS wave function can be regarded as the "wave function" of the $NN$ system at $W < 2m_N + m_\pi$.
It is also noted that the equal-time constraint for the NBS wave function here is not a restriction to the extraction of physical observables such as the scattering phase shift, as evident from the fact that all informations on the scattering phase shifts are encoded in the asymptotic behavior of the equal-time NBS wave function. Moreover, the equal-time NBS wave function with non-zero total momentum (${\bf P}\not=0$) is equivalent to the NBS wave function in the center of mass frame with non-zero time separation. In this case the 4-dimensional distance between two nucleon operators is always space-like.
\subsection{Non-local potential from the NBS wave function}
Since the NBS wave function satisfies the free Schr\"odinger equation at large $r$, one can define short-ranged non-local potential by
\begin{eqnarray}
\left[E_k- H_0\right] \varphi^W_{\alpha\beta}({\bf x} ) &=&
\int U_{\alpha\beta;\gamma\delta}({\bf x}, {\bf y}) \varphi^W_{\gamma\delta}({\bf y} )d^3y, \qquad
E_k=\frac{k^2}{2\mu}.
\label{eq:schroedinger}
\end{eqnarray}
It is noted that the spinor indices $\alpha,\beta,\gamma,\delta$ here run from 1 to 2, since
all $NN$ scattering phase shifts can be reproduced from the NBS wave function with $\alpha,\beta\in \{1,2\}$ as discussed in the previous subsection.
Therefore $U_{\alpha\beta;\gamma\delta}$ has $4\times 4$ components, which can be determined from $4$ components of $\varphi_{\alpha\beta}^W$ for 4 different combinations of $(s_1, s_2)$.
Note that since the NBS wave function $\varphi^W$ is multiplicatively renormalized, the potential $U({\bf x},{\bf y})$ is finite and does not depend on the particular renormalization scheme .
The non-local function $U({\bf x},{\bf y})$ is shown to be energy-independent as follows.
Let $V_{\rm th}$ be the space spanned by the wave function with $W\le W_{\rm th} \equiv 2m_N+m_\pi$: $V_{\rm th}=\{\varphi^W\vert W\le W_{\rm th}\}$, and
the projection operator to $V_{\rm th}$ is defined as
\begin{eqnarray}
P^{W_{\rm th}}({\bf x},{\bf y}) &=&\int_{W_{1,2}\le W_{\rm th} }\rho(W_1) dW_1\, \rho(W_2) dW_2\, \varphi^{W_1}({\bf x}) N^{-1}(W_1,W_2)\varphi^{W_2}({\bf y})^\dagger \nonumber \\
&\equiv& \int_{W_1\le W_{\rm th} }\rho(W_1) dW_1\, P(W_1;{\bf x},{\bf y})
\end{eqnarray}
where $\rho(W)$ is the density of states at energy $W$, and $N^{-1}(W_1,W_2)$ is the inverse
of the hermitian operator $N(W_1,W_2)$ defined by
\begin{equation}
N(W_1,W_2) = \int \varphi^{W_1}({\bf r})^\dagger \varphi^{W_2}({\bf r})\, d^3 r,
\end{equation}
so that
\begin{equation}
\int \rho(W) dW\, N(W_1,W) N^{-1}(W,W_2) =\frac{1}{\rho(W_1)}\delta(W_1-W_2).
\end{equation}
The non-local potential is then defined by
\begin{eqnarray}
U^{W_{\rm th}}({\bf x},{\bf y}) &=& \int_{W_{1,2}\le W_{\rm th} }\rho(W_1) dW_1\, \rho(W_2) dW_2\, \left[E_k - H_0\right]\varphi^{W_1}({\bf x}) N^{-1}(W_1,W_2)\varphi^{W_2}({\bf y})^\dagger \nonumber \\
&=& \int_{W_1\le W_{\rm th} }\rho(W_1) dW_1\, \left[E_k - H_0\right] P(W_1;{\bf x},{\bf y}).
\label{eq:non-local}
\end{eqnarray}
It is easy to see that the above non-local potential satisfies eq.(\ref{eq:schroedinger}) at
$W\le W_{\rm th}$ as follows.
\begin{eqnarray}
\int U({\bf x},{\bf y})^{W_{\rm th}}\varphi^W({\bf y})\, d^3y &=&
\int_{W_1\le W_{\rm th} }\rho(W_1) dW_1\, \left[E_k - H_0\right]\varphi^{W_1}({\bf x})
\frac{1}{\rho(W)}\delta(W_1-W)\nonumber \\
&=& \theta(W_{\rm th}-W)\, \left[E_k - H_0\right]\varphi^{W }({\bf x}) .
\end{eqnarray}
It should be noted that the non-local potential $U({\bf x},{\bf y})$ which satisfied
eq.(~\ref{eq:schroedinger}) at $W\le W_{\rm th}$ is not unique: For example, we can add the following term
\begin{equation}
\int_{W>W_{\rm th}} \rho(W)\, dW\, f_W({\bf x}) P(W;{\bf x},{\bf y})
\end{equation}
with arbitrary functions $f_W({\bf x})$ to the non-local potential $U({\bf x},{\bf y})$ without affecting eq.(\ref{eq:schroedinger}) at $W\le W_{\rm th}$.
The non-local potential $U({\bf x},{\bf y})$ in eq. (\ref{eq:non-local}) is energy independent by construction.
Alternatively we can define a different non-local potential by
\begin{equation}
U^{\infty}({\bf x},{\bf y}) = \int_0^\infty \rho(W) dW\, \left[E_k - H_0\right] P(W;{\bf x},{\bf y}),
\end{equation}
which satisfies eq. (\ref{eq:schroedinger}) for all $W$. This potential, however, becomes long-ranged, due to the presence of inelastic contributions above $W_{\rm th}$.
The extension of this method to non-elastic cases will be discussed in Sec.~\ref{sec:extension}.
In Ref.~\cite{Beane:2010em}, it is claimed that the NBS wave function satisfies the Schr\"odinger equation with a non-local and energy-dependent potential. In general this is true but, as shown here, there is a scheme which makes the non-local potential energy-independent. This is sufficient for the strategy considered in this report.
\subsection{Velocity expansion of the non-local potential}
In principle, if one knows all NBS wave functions $\varphi^W$, the non-local potential $U$ can be constructed according to eq. (\ref{eq:non-local}). In practice, however, one can obtain only a few of them corresponding to the ground state as well as a few low lying excited states in lattice QCD simulations. Therefore, for practical applications, it is convenient to expand the non-local potential in terms of the velocity(derivative) with local functions as
\begin{equation}
U({\bf x},{\bf y}) = V({\bf x},\nabla) \delta^3({\bf x}-{\bf y}) .
\end{equation}
At the lowest few orders we have
\begin{eqnarray}
V({\bf r},\nabla) &=&\underbrace{V_0(r) + V_\sigma(r) \vec\sigma_1\cdot\vec\sigma_2 + V_T(r) S_{12}}_{\rm LO} + \underbrace{V_{\rm LS} (r){\bf L}\cdot{\bf S}}_{\rm NLO} + O(\nabla^2),
\label{eq:velocity_exp}
\end{eqnarray}
where $r=\vert{\bf r}\vert$, $\vec\sigma_i$ is the Pauli-matrix acting on the spin index of the $i$-th nucleon, ${\bf S}=(\vec\sigma_1+\vec\sigma_2)/2$ is the total spin, ${\bf L} = {\bf r}\times {\bf p}$ is the angular momentum, and
\begin{equation}
S_{12} = 3 \frac{({\bf r}\cdot \vec\sigma_1) ({\bf r}\cdot \vec\sigma_2) }{r^2} -\vec\sigma_1\cdot\vec\sigma_2
\end{equation}
is the tensor operator. Each coefficient function is further decomposed into its flavor components as
\begin{equation}
V_X(r) = V_X^0(r) + V_X^\tau(r) \vec\tau_1\cdot \vec\tau_2, \quad
X=0, \sigma, T, {\rm LS},\cdots,
\end{equation}
where $\vec \tau_i$ is the Pauli-matrix acting on the flavor index of the $i$-th nucleon.
The form of the velocity expansion (\ref{eq:velocity_exp}) agrees with the form determined by symmetries\cite{okubo1958}.
For the leading order of the velocity expansion, the local potential is given by
\begin{equation}
V^{\rm LO}({\bf r}) = V_0(r) + V_\sigma(r) \vec\sigma_1\cdot\vec\sigma_2 + V_T(r) S_{12},
\end{equation}
which can be obtained from the NBS wave function at one value of $W$. Since $S_{12}=0$ for the spin single state, for example, we have
\begin{equation}
V_c(r, S=0)\equiv V_c(r) - 3V_\sigma(r) = \frac{\left[E_k -H_0\right]\varphi^W({\bf r})}{\varphi^W({\bf r})} .
\end{equation}
\subsection{Remarks}
There are several remarks on the extraction of the potential from QCD described in the previous sections.
First of all, it should be noted that the potential itself is not a physical observable, and it is therefore not unique. In particular the potential depends on the choice of the nucleon operator to define the NBS wave function. The local nucleon operator is one choice here but other definitions are equally possible, though the local operator is a convenient choice for the reduction formula of composite particles such as the nucleon\cite{Nishijima,Zimmermann,Haag}. A choice of the nucleon operator to define the NBS wave function is considered to be a "scheme" to define the potential. The potential is therefore a scheme dependent quantity, while
physical observables such as the scattering phase shift and the binding energy of the deuteron are of course scheme independent.
Is such a scheme-dependent quantity useful ? The answer to this question is probably "yes", since the potential is useful to "understand" or "describe" the phenomena. For example, the repulsive core best summarizes the behavior of the NN scattering phase shift at larger energy in terms of the short distance behavior of the potential.
A well-known example for a scheme-dependent but useful quantity is the running coupling in quantum field theory. Although the running coupling is of course scheme dependent, it is useful to understand the deep inelastic scattering data at high energy ({\it i.e.} asymptotic freedom).
Among different schemes, there exist of course good schemes. A good convergence of the perturbative expansion for a certain class of observables is a reasonable criteria for a good scheme in the case of the running coupling. In the case of the potential, a good convergence of the velocity expansion is important. In this respect, a completely local and energy-independent potential is the best one, and moreover the inverse scattering method tells us that it must be unique if it exists. We think that such a local and energy-independent potential exists only if no inelastic scattering appears at all values of energy ({\it i.e.} $W_{\rm th}=\infty$).
There exists a criticism that the repulsive core in the phenomenological potentials is meaningless since the low energy scattering data can be described equally well by other potentials without the repulsive core. In other words, the repulsive core can be removed by a unitary transformation of the wave function without changing the scattering phase shift at low energy. Even though this may be true, the resulting "potential" is highly non-local, and therefore is beyond the concept of a potential.
This criticism thus corresponds to claiming that the asymptotic freedom is meaningless since it can be removed by some non-perturbative scheme of the "coupling", which is however beyond the concept of a coupling.
For the definition of the potential in QCD from the NBS wave function, the convergence of the velocity expansion can be checked by examining the energy ($W$) dependence of the lower order potentials. For example, if we have $\varphi^{W_n}$ for $n=1,2,\cdots N$, we can determine the $N-1$ unknown local functions of the velocity expansion in $N$ different ways.
The variation among $N$ different determinations gives an estimate of the size of the higher order terms neglected. Furthermore one of these higher order terms can be determined from $\varphi^{W_n}$ for $n=1,2,\cdots N$.
The convergence of the velocity expansion will be investigated explicitly in Sec.~\ref{sec:NNpotential}.
Let us consider what we have shown so far from a slightly different point of view.
Instead of adopting the point of view that the "potential" can be defined in QCD,
the analysis in this section shows that
the use of quantum mechanics with potentials to describe the NN scattering can be justified in a more fundamental quantum field theory (QCD) through the NBS wave function, whose asymptotic behavior encodes phases of the S-matrix for the NN scattering.
If the local approximation for the potential defined from the NBS wave function is good in the low energy region, one can use the local potential combined with quantum mechanics to investigate nuclear physics\footnote{It is more legitimated to show if three or more body potentials remain subdominant in the same framework in QCD before using the potential in nuclear physics. There exists another approach to investigate nuclei, by combining the chiral effective field theory with lattice calculations. (See Ref.\cite{Epelbaum:2010xt} and references therein.) The chiral effective can easily incorporate may-body interactions order by order in the chiral expansion, though there are many free parameters to be determined from experimental data. On the other hand, the potential from lattice QCD does not contain such free parameters. Therefore it is interesting and important to determine some parameters of the chiral effective theory from the potential in lattice QCD. } .
It is now clear that there is no unique definition for the NN potential. Ref.~\cite{Beane:2010em,Detmold:2007wk,Beane:2008ia}, however, criticized that the NBS wave function is not "the correct wave function for two nucleons" and that its relation to the correct wave function is given by
\begin{equation}
\varphi^W({\bf r}) = Z_{NN}(\vert {\bf r}\vert) \langle 0 \vert T\{N_0({\bf x}+{\bf r},0) N_0({\bf x},0)\}\vert 2N, W,s_1,s_2\rangle + \cdots
\end{equation}
where $N_0({\bf x},t)$ is "a free-field nucleon operator" and the ellipses denote "additional contributions from the tower of states of the same global quantum numbers".
Thus $\langle 0 \vert T\{N_0({\bf x}+{\bf r},0) N_0({\bf x},0)\}\vert 2N, W,s_1,s_2\rangle$ is considered to be "the correct wave function".
In this claim it is not clear what is "a free-field nucleon operator" in the interacting quantum field theory. An asymptotic {\it in} or {\it out} field operator may be a candidate. If the asymptotic field is used for $N_0$, however, the potential defined from the wave function identically vanishes for all ${\bf r}$ by construction.
To be more fundamental, a concept of "the correct wave function" is doubtful. If some wave function were "correct", the potential would be uniquely defined from it. This clearly contradicts the fact discussed above that the potential is not an observable and therefore is not unique.
This argument shows that the criticism of Ref.~\cite{Beane:2010em,Detmold:2007wk,Beane:2008ia} is flawed.
\section{Lattice formulation}
\label{sec:lattice}
In this section, we discuss the procedure to extract the NBS wave function from lattice QCD simulations.
For this purpose, let us consider the correlation function on the lattice defined by
\begin{equation}
F({\bf r},t-t_0)=\langle 0\vert T\{N({\bf x}+{\bf r},t ) N({\bf x},t)\} \overline{\cal J}(t_0)\vert 0 \rangle
\label{eq:4-pt}
\end{equation}
where $\overline{\cal J}(t_0)$ is a source operator which creates a two nucleon state of states and its explicit form will be considered later. By inserting a complete set and considering baryon number conservation, we have
\begin{eqnarray}
F({\bf r},t-t_0) &=&\langle 0\vert T\{N({\bf x}+{\bf r},t ) N({\bf x},t)\} \sum_{n,s_1,s_2} \vert 2N, W_n, s_1,s_2\rangle \langle 2N, W_n,s_1,s_2 \vert \overline{\cal J}(t_0)\vert 0 \rangle \nonumber \\
&=& \sum_{n, s_1,s_2} A_{n, s_1,s_2} \varphi^{W_n}({\bf r}) e^{-W_n (t-t_0)}, \quad
A_{n,s_1,s_2} =\langle 2N, W_n,s_1,s_2 \vert \overline{\cal J}(0)\vert 0 \rangle .
\end{eqnarray}
For a large time separation $(t-t_0)\rightarrow \infty$, we have
\begin{equation}
\lim_{(t-t_0)\rightarrow\infty} F({\bf r},t-t_0) = A_0 \varphi^{W_0}({\bf r}) e^{-W_0 (t-t_0)}
+ O(e^{- W_{n\not=0} (t-t_0)})
\label{eq:ground}
\end{equation}
where $W_0$ is assumed to be the lowest energy of NN states.
Since the source dependent term $A_0$ is just a multiplicative constant to the NBS wave function $\varphi^{W_0}({\bf r})$, the potential defined from $\varphi^{W_0}({\bf r})$ in the previous section is manifestly source-independent. Therefore the statement that the potential in this scheme is "source-dependent" in Ref.~\cite{Beane:2008dv} is clearly wrong.
In this extraction of the wave function, the ground state saturation for the correlation function $F$ in eq. (\ref{eq:ground}) is important. In principle, one can achieve this by taking a large $t-t_0$.
In practice, however, $F$ becomes very noisy at large $t-t_0$, so that the extraction of $\varphi^{W_0}$ becomes difficult at large $t-t_0$. Therefore it is crucial to find the region of $t$ where the ground state saturation is approximately satisfied while the signal is still reasonably good.
The choice of the source operator becomes important in order to have such a good $t$-region.
\subsection{Choice of source operator}
We can choose the source operator $\bar{\cal J}$ to fix quantum numbers of the state $\vert 2N, W, s_1,s_2\rangle $ such as $(J, J_z)$. Since a lattice QCD simulation is usually performed on the (finite) hyper-cubic lattice, we consider the cubic transformation group $SO(3,{\bf Z})$ instead of the $SO(3,{\bf R})$ as the symmetry of 3-dimensional space. Therefore the quantum number is classified in terms of the irreducible representation of $SO(3,{\bf Z})$, denoted by $A_1$, $A_2$, $E$, $T_1$, $T_2$ whose dimensions are $1,1,2,3,3$.
A relation of irreducible representations between $SO(3,{\bf Z})$ and $SO(3,{\bf R})$ is given in table~\ref{tab:cubic} for $L\le 6$, where $L$ represents the angular momentum for the irreducible representation in $SO(3,{\bf R})$. For example, the source operator $\bar{\cal J}(t_0)$ in the $A_1$ representation with positive parity generates states with $L=0,4,6,\cdots$ at $t=t_0$, while
the operator in the $T_1$ representation with negative parity produces states with $L=1,3,5,\cdots$. For two nucleons, the total spin $S$ becomes $1/2\otimes 1/2 = 1 \oplus 0$, which corresponds to $T_1$($S=1$) and $A_1$($S=0$) of the $SO(3,{\bf Z})$.
Therefore, the total representation $J$ for a two nucleon system is determined by the product $R_1\otimes R_2$, where $R_1=A_1,A_2,E,T_1,T_2$ for the orbital "angular momentum" while $R_2=A_1, T_1$ for the total spin. In table~\ref{tab:product}, the product $R_1\otimes R_2$ is decomposed into the direct sum of irreducible representations.
\begin{table}
\begin{center}
\caption{The number of each representation of $SO(3,{\bf Z})$ which appears in the angular momentum $L$ representation of $SO(3,{\bf R})$. $P=(-1)^L$ represents an eigenvalue under parity transformation.}
\label{tab:cubic}
\vspace{0.3cm}
\begin{tabular}{|cc|ccccc|}
\hline
$L$ & $P$ & $A_1$ & $A_2$ & $E$ & $T_1$ & $T_2$ \\
\hline
0 (S)& $+$ & 1 & 0 &0 &0 & 0 \\
1 (P)& $-$ & 0 & 0 &0 & 1 & 0\\
2 (D)& $+$ & 0 & 0 &1 & 0 & 1 \\
3 (F)& $-$ & 0 & 1 &0 & 1 & 1\\
4 (G)& $+$ & 1 & 0 &1 & 1 & 1 \\
5 (H)& $-$ & 0 & 0 &1 & 2 & 1\\
6 (I) & $+$ & 1 & 1 &1 & 1 & 2 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{The decomposition for a product of two irreducible representations, $R_1\otimes R_2$, into irreducible representations in $SO(3,{\bf Z})$. Note that $R_1\otimes R_2= R_2\otimes R_1$ by definition. }
\label{tab:product}
\vspace{0.3cm}
\begin{tabular}{|c||ccccc|}
\hline
& $A_1$ & $A_2$ & $E$ & $T_1$ & $T_2$ \\
\hline
\hline
$A_1$ & $A_1$ & $A_2$ & $E$ & $T_1$ & $T_2$ \\
$A_2$ & $A_2$ & $A_1$ & $E$ & $T_2$ & $T_1$\\
$E$ & $E$ & $E$ &$A_1\oplus A_2\oplus E$ & $T_1\oplus T_2$ & $T_1\oplus T_2$ \\
$T_1$ & $T_1$ & $T_2$& $T_1\oplus T_2$ & $A_1\oplus E \oplus T_1\oplus T_2$ & $A_2\oplus E \oplus T_1\oplus T_2$\\
$T_2$ & $T_2$ & $T_1$ &$T_1\oplus T_2$& $A_2\oplus E \oplus T_1\oplus T_2$ &$A_1\oplus E \oplus T_1\oplus T_2$ \\
\hline
\end{tabular}
\end{center}
\end{table}
Most of the results in this report use the wall source at $t=t_0$ defined by
\begin{equation}
{\cal J}^{\rm wall}(t_0)_{\alpha\beta, fg} = N^{\rm wall}_{\alpha,f} (t_0) N^{\rm wall}_{\beta,g} (t_0)
\end{equation}
where $\alpha,\beta=1,2$ are upper component spinor indices while $f,g$ are flavor indices.
Here $N^{\rm wall}(t_0)$ is obtained by replacing the local quark field $q(x)$ of $N(x)$ by the wall quark field,
\begin{equation}
q^{\rm wall}(t_0) \equiv \sum_{\bf x} q({\bf x},t_0)
\end{equation}
with the Coulomb gauge fixing only at $t=t_0$. Note that this gauge-dependence of the source operator disappears for the potential.
All states created by the wall source have zero total momentum. Among them
the state with zero relative momentum has the largest magnitude.
The most important reason to employ the wall source is that the ground state saturation for the potential at long distance is better achieved for the wall source than other sources such as the smeared source.
By construction, the source operator $\bar{\cal J}^{\rm wall}(t_0)$ has zero orbital angular momentum at $t=t_0$, which corresponds to the $A_1$ representation with positive parity.
By the spin projection operator $P^{(S)}$, e.g. $P^{(S=0)}=\sigma_2$ and $P^{(S=1,S_z=0)}=\sigma_1$, we fix the $J$ of the source as
\begin{equation}
{\cal J}(t_0;J^{P=+},I) = P^{(S)}_{\beta\alpha} {\cal J}^{\rm wall}(t_0)_{\alpha\beta,fg}
\end{equation}
where $P=\pm $ is the parity and $I=1,0$ is the total isospin of the system. Since the nucleon is a fermion, an exchange of two nucleon operators in the source should give a minus sign. This fact fixes the total isospin once the total spin is given: $(S,I)=(0,1)$ or $(1,0)$.
(Note that $S,I=0$ are antisymmetric while $S,I=1$ are symmetric under the exchange.)
Since $A_1^+\otimes A_1(S=0) = A_1^+$ and $A_1^+\otimes T_1(S=1) = T_1^+$, the state with either $(J^P,I) =(A_1^+, 1)$ for the spin-singlet or $(J^P,I) =(T_1^+, 0)$ for the spin-triplet is created at $t=t_0$ by the corresponding source operator. The NBS wave function extracted at $t > t_0$ has the same quantum numbers $(J^P,I)$ as they are conserved under QCD interactions.
In addition the total spin $S$ is conserved at $t > t_0$ for the two-nucleon system with equal up and down quark masses: Under the exchange of the two particles, the constraint $(-1)^{S+1+I+1}P=-1$ must be satisfied due to the fermionic nature of nucleon, while
the parity $P$ and the isospin $I$ are conserved in this system. Therefore $S$ is conserved.
It is noted, however, that $L$ is not conserved in general. While the state with $(J^P,I) =(A_1^+, 1)$ always has $L=A_1^+$ even at $t > t_0$, the one with $(J^P,I) =(T_1^+, 0)$ has both $L=A_1^+$ and $L=E^+$ components\footnote{This can be seen from Table~\ref{tab:product} for $R_2=T_1$(spin-triplet), which also tells us existences of $L= T_1^+$ and $L=T_2^+$ components in addition. These extra components are expected to be small since they appear as a consequence of the violation of $SO(3,{\bf R})$ on the hyper-cubic lattice.} at $t> t_0$, which corresponds to $L=0$ and $L=2$ in $SO(3,{\bf R})$, respectively.
Note that $J$ or $L$ in this report is used to represent the total or orbital quantum number of $SO(3,{\bf Z})$ as well as $SO(3,{\bf R})$, depending on the context.
The orbital angular momentum $L$ of the NBS wave function can be fixed to a particular value by the projection operator $P^{(L)}$ as
\begin{equation}
\varphi^W({\bf r}; J^P,I, L,S) = P^{(L)} P^{(S)} \varphi^W({\bf r}; J^P,I)
\end{equation}
where $\varphi^W({\bf r}; J^P,I) $ is extracted from
\begin{equation}
F({\bf r},t-t_0;J^P,I) \simeq A(J^P,I) \varphi^{W}({\bf r}; J^P,I)e^{-W(t-t_0)} ,\quad
A(J^P,I) =\langle 2N,W \vert \bar{\cal J}(0;J^P,I)\vert 0\rangle
\end{equation}
for large $t-t_0$. Here we also apply the total spin projection operator $P^{(S)}$ but this is redundant since the total spin $S$, already fixed by the source, is conserved as mentioned above.
The projection operator $P^{(L)}$ to an arbitrary function $\phi({\bf r})$ is defined in general by
\begin{equation}
P^{(L)} \phi({\bf r}) \equiv \frac{d_L}{24}\sum_{g\in SO(3,{\bf Z})} \chi^L(g) \phi(g^{-1}\cdot{\bf r})
\end{equation}
for $L=A_1,A_2,E,T_1,T_2$, where $\chi^L$ denotes the character for the representation $L$ of the cubic group $SO(3,{\bf Z})$, $g$ is one of 24 elements in $SO(3,{\bf Z})$ and $d_L$ is the dimension of $L$.
\subsection{Leading order potential: spin-singlet case}
We present the procedure to determine potentials at the reading order(LO):
\begin{equation}
V^{\rm LO}({\bf r} ) = V_0(r) +V_\sigma(r)(\vec\sigma_1\cdot\vec\sigma_2)+ V_T(r) S_{12}.
\end{equation}
Since $S_{12}=0$ and $\vec\sigma_1\cdot\vec\sigma_2= - 3$ for the spin-singlet case,
the LO central potential in the case of $(J^P,I)=(A_1^+,1)$ state is extracted as
\begin{equation}
V_C(r)^{(S,I)=(0,1)} \equiv V^{I=1}_0 (r)-3 V^{I=1}_\sigma (r)
= \frac{\left[E_k- H_0\right]\varphi^W({\bf r}; A_1^+,I=1,A_1,S=0)}{\varphi^W({\bf r}; A_1^+,I=1,A_1,S=0)},
\end{equation}
where $V_X^{I=1} = V_X^0+V_X^\tau$ in isospin space.
The potential $V_C({\bf r})^{(S,I)=(0,1)}$ in the above
is often referred to as the central potential for the $^1S_0$ state, where the notation $^{2S+1}L_J$ represents the orbital angular momentum $L$ (see table~\ref{tab:cubic}), the total spin $S$ and the total angular momentum $J$ of ${\bf J}={\bf L}+{\bf S}$. It is noted, however, that in the leading order of the velocity expansion, the potential does not depend on the quantum number of the state $J=L=A_1$. Moreover the $A_1$ state may contain $L=4,6,\cdots$ components other than $L=0$, though the $L=0$ component may dominate in the state. Therefore it is more precise to refer to $V_C({\bf r})^{(S,I)=(0,1)}$ as the spin-singlet (isospin-triplet) central potential determined from the state with $J=L=A_1$. A possible difference of spin-singlet central potentials between this determination and others such as the one determined from $J=L=E$ gives an estimate for contributions from higher order terms in the velocity expansion.
\subsection{Leading order potential: spin-triplet case}
Both the tensor potential $V_T$ and central potential $V_C$ appear at the LO
for the spin-triplet case. Let us consider the determination from the $(J^P,I)=(T_1^+,0)$ state. The
Schr\"odinger equation for this state becomes
\begin{equation}
\left[H_0+V_C(r)^{(S,I)=(1,0)} + V_T(r) S_{12}\right]\varphi^W({\bf r};J^P=T_1^+,I=0) = E_k \varphi^W({\bf r};J^P=T_1^+,I=0)
\end{equation}
where the spin-triplet central potential is given by
\begin{equation}
V_C(r)^{(S,I)=(1,0)} \equiv V_0^{I=0}(r) + V_\sigma^{I=0}(r), \qquad
V_X^{I=0} = V_X^0-3V_X^\tau .
\end{equation}
The projections to $A_1$ and $E$ components read
\begin{eqnarray}
{\cal P}\varphi^W_{\alpha\beta} &\equiv& P^{(A_1)}\varphi^W_{\alpha\beta} ({\bf r}; J^P=T_1^+,I=0)
\label{eq:projA1}
\\
{\cal Q}\varphi^W_{\alpha\beta} &\equiv& P^{(E)}\varphi^W_{\alpha\beta} ({\bf r}; J^P=T_1^+,I=0)
\simeq (1-P^{(A_1)}) \varphi^W_{\alpha\beta} ({\bf r}; J^P=T_1^+,I=0).
\label{eq:projE}
\end{eqnarray}
The last quantity in eq. (\ref{eq:projE}) is an approximation of the first line and a difference comes from $T_1$ and $T_2$ components, which are expected to be small.
This approximate representation for ${\cal Q}$ is often employed in numerical simulations.
Using these projections, $V_C$ and $V_T$ can be extracted as
\begin{eqnarray}
V_C(r)^{(1,0)} &=& E_k -\frac{1}{\Delta({\bf r})}\left( [{\cal Q} S_{12}\varphi^W]_{\alpha\beta}({\bf r})
H_0[{\cal P}\varphi^W]_{\alpha\beta}({\bf r}) - [{\cal P} S_{12}\varphi^W]_{\alpha\beta}({\bf r})
H_0[{\cal Q}\varphi^W]_{\alpha\beta}({\bf r})
\right) \label{eq:central}\\
V_T(r) &=& \frac{1}{\Delta({\bf r})}\left( [{\cal Q} \varphi^W]_{\alpha\beta}({\bf r})
H_0[{\cal P}\varphi^W]_{\alpha\beta}({\bf r}) - [{\cal P} \varphi^W]_{\alpha\beta}({\bf r})
H_0[{\cal Q}\varphi^W]_{\alpha\beta}({\bf r})
\right) \label{eq:tensor}
\\
\Delta({\bf r}) &\equiv& [{\cal Q} S_{12}\varphi^W]_{\alpha\beta}({\bf r})
[{\cal P}\varphi^W]_{\alpha\beta}({\bf r}) - [{\cal P} S_{12}\varphi^W]_{\alpha\beta}({\bf r})
[{\cal Q}\varphi^W]_{\alpha\beta}({\bf r}) .
\end{eqnarray}
In numerical simulations, $(\alpha,\beta)=(2,1)$ is mainly employed.
If one neglects $V_T$ by putting $V_T=0$ in the above, one obtains the effective central potential for the spin-triplet (isospin-singlet) state as
\begin{equation}
V_C^{\rm eff}(r)^{(1,0)} = \frac{\left[E_k- H_0\right] {\cal P}\varphi^W_{\alpha\beta}({\bf r}) }{{\cal P}\varphi^W_{\alpha\beta}({\bf r}) } .
\label{eq:effective_central}
\end{equation}
The difference between $V_C$ and $V_C^{\rm eff}$ is $O(V_T^2)$ in the second order perturbation for small $V_T$.
\subsection{A comparison with the finite volume method in lattice QCD}
In this subsection we briefly compare the potential method with the direct extraction of the phase shift via the finite volume method in lattice QCD.
First of all, by construction, the potential method gives the correct phase shift at $k =\sqrt{W^2/4-m_N^2}$ where $W$ is the total energy of the state from which the NBS wave function is defined, while phase shifts at other values of $k$ are approximated ones obtained in the velocity expansion of the non-local potential.
Secondly, the finite size correction to the potential is expected to be small. Indeed the finite volume method for the extraction of the phase shift in lattice QCD assumes that there is no finite size correction to the potential as long as the volume is large enough so that the interaction range of the potential is smaller than half of the lattice extension, $L/2$. Under this condition, there exists an asymptotic region in the periodic box where the scattering wave satisfies the free Schr\"odinger equation with a specific value of the energy, from which one can determine the phase shift at certain values of $k$ in the infinite volume. This is L\"uscher's finite volume formula for the phase shift\cite{Luscher:1990ux}.
Thirdly, we also expect that the quark mass dependence of the potential is much milder than that of physical observables such as the scattering length. While the scattering length is almost zero at the heavy quark mass region, it diverges when the bound state is formed at the lighter quark mass region.
In this situation, the scattering length varies from zero to infinity as the quark mass changes\cite{Kuramashi:1995sc}.
Such a drastic change of the scattering length can easily be realized by a small change to the shape of the potential as a function of the quark mass.
Let us assume that higher order terms in the velocity expansion give negligible contributions at low energy so that the leading order local potential well reproduces the scattering phase shift.
In this situation, some problems of the finite size method can be avoided by using
the potential method. To extract the phase shift in the finite size method in lattice QCD, one has to assume that one particular angular momentum gives a dominant contribution among possible angular momenta in a given representation of the cubic group. For example, although a state in the $A_1$ representation contains not only an $L=0$ contribution but also $L=4,6,\cdots$ contributions, one usually assumes that the $L=0$ contribution dominates so that the energy shift in the finite volume is related to the scattering phase for the $L=0$ state. In the case of the potential, on the other hand, such an assumption is unnecessary. One can determine the local potential in the velocity expansion from the $A_1$ state without specifying the dominant angular momentum. Once the potential is obtained, one can calculate the scattering phase shift for an arbitrary $L$ by solving the Schr\"odinger equation in the infinite volume with the extracted potential.
Furthermore, one can check the assumption made for the finite size method by comparing sizes of the scattering phases among different $L$'s.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.33\textwidth,angle=270]{Figs/wave_529.eps}
\includegraphics[width=0.33\textwidth,angle=270]{Figs/pot_529_1S0-3S1.eps}
\caption{(Left)The NN wave function for the spin-singlet and spin-triplet channels in the orbital $A_1^+$ representation at $m_\pi\simeq 529$ MeV and $a\simeq 0.137$ fm in quenched QCD. The insert is a three-dimensional plot of the spin-singlet wave function $\varphi^W(x,y,z=0)$.
(Right) The NN (effective) central potential for the spin-singlet (spin-triplet) channel determined from the orbital $A_1^+$ wave function. Both figures are taken from Ref.~\protect\cite{Aoki:2009ji}. }
\label{fig:wave_potential}
\end{center}
\end{figure}
\section{Results for nuclear potentials from lattice QCD}
\label{sec:NNpotential}.
\subsection{Quenched QCD results for (effective) central potentials}
Let us show results in the quenched QCD, where creations and annihilations of virtual quark-antiquark pairs are all neglected. For the simulations, the standard plaquette gauge action is employed on a 32$^4$ lattice at the bare gauge coupling constant $\beta=6/g^2=5.7$, which corresponds to the lattice spacing $a\simeq 0.137$ fm ($1/a=1.44(2)$ GeV), determined from the $\rho$ meson mass in the chiral limit, and the physical size of the lattice $L\simeq 4.4$ fm\cite{Ishii:2006ec}. As for the quark action, the standard Wilson fermion action is employed at three different values of the quark mass corresponding to the pion mass $m_\pi \simeq 731, 529, 380$ MeV and the nucleon mass $m_N \simeq 1560,1330,1200$ MeV, respectively.
Fig.~\ref{fig:wave_potential}(Left) shows the NBS wave functions for the spin-singlet and the spin-triplet channels in the orbital $A_1$ representation at $m_\pi \simeq 529$ MeV. These wave functions are normalized to be 1 at the largest spatial point $r\simeq 2.2$ fm.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=6.5cm,angle=270]{Figs/wave_fit_1S0.eps}
\includegraphics[width=6.5cm,angle=270]{Figs/wave_fit_3S1.eps}
\caption{(Left) The fit of the $NN$ wave functions at $m_\pi\simeq 529$ MeV for the spin-singlet channel in the orbital $A_1^+$ representation using the Green's function in the fit range $11\le r/a \le 15$. (Right) A similar fit for the spin-triplet channel. Taken from Ref.~\protect\cite{Aoki:2009ji}.}
\label{fig:wave_fit}
\end{center}
\end{figure}
The central potential in the spin-singlet channel and the effective central potential in the spin-triplet channel reconstructed from the wave functions at $m_\pi\simeq 529$ MeV are shown in Fig.~\ref{fig:wave_potential}(Right). These potentials reproduce the qualitative features of the phenomenological $NN$ potentials, namely the repulsive core at short distance surrounded by the attractive well at medium and long distances. From this figure one observes that the interaction range of the potential is smaller than 1.5 fm. Therefore the box size $L\simeq 4.4$ fm is large enough to extract the phase shift by the finite size method, and furthermore the finite size corrections to the potentials themselves are expected to be small. Labels $^1S_0$ and $^3S_1$
of the potentials in the figure represent the fact that potentials are determined from $A_1$ wave functions, which are dominated by $S$ wave components.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.35\textwidth,angle=270]{Figs/pot_1S0_mass.eps}
\caption{The central potentials for the spin-singlet channel from the orbital $A_1^+$ representation at three different pion masses in quenched QCD. Taken from Ref.~\protect\cite{Aoki:2009ji}.}
\label{fig:mass-dep}
\end{center}
\end{figure}
Instead of calculating the energy shift due to the finite size, one can extract the asymptotic momentum $k$, by fitting the NBS wave function $\varphi({\bf r})$ at large distance with the Green's function $G({\bf r};k^2)$ in a finite and periodic box for
the Helmholtz equation $(\nabla^2 + k^2)G({\bf r};k^2) = -\delta_{\rm lat}({\bf r})$ with $\delta_{\rm lat}({\bf r})$ being the periodic delta-function. Explicitly it is given by
\begin{equation}
G({\bf r};k^2) =\frac{1}{L^3}\sum_{{\bf n}\in{\bf Z}^3}\frac{e^{i(2\pi/L){\bf n\cdot r}}}{(2\pi/L)^2{\bf n^2}-k^2}.
\end{equation}
The asymptotic momentum $k$ is related to the scattering phase shift $\delta_0(k)$ or the scattering length $a_0$ for the S-states\footnote{ We here assume that the dominant component of the scattering wave in $A_1$ representation has $L=0$.}
as
\begin{equation}
k \cot \delta_0(k) = \frac{2}{\sqrt{\pi} L}Z_{00}(1;q^2) =\frac{1}{a_0} + O(k^2),
\label{eq:SL}
\end{equation}
where $Z_{00}(1,q^2)$ with $q=\frac{kL}{2\pi}$ is (the analytic continuation of) the generalized zeta-function $Z_{00}(s,q^2) = \displaystyle\frac{1}{\sqrt{4\pi}}\sum_{{\bf n}\in {\bf Z}^3}({\bf n}^2 -q^2)^{-s}$. Fig.~\ref{fig:wave_fit} shows the fits of the wave functions in the interval $11 a\simeq 1.5$ fm $\le r \le 16 a\simeq 2.2$ fm using the above form at $m_\pi = 529$ MeV. This leads to the values of the effective energy $E\equiv k^2/m_N$, which can be translated to the scattering length $a_0$ by the L\"uscher's formula (\ref{eq:SL}).
In Fig.\ref{fig:mass-dep}, we compare the $NN$ central potentials in the spin-singlet channel for three different pion masses. As the pion mass decreases, the repulsion at short distance and the attraction at medium distance are enhanced simultaneously.
In table~\ref{tab:EandSL}, we give values of $E$ and the S-wave scattering length $a_0$, which show a net attraction of the NN interactions in both channels at these pion masses, though the absolute magnitudes of the scattering length $a_0$ are much smaller than the experimental values at the physical pion mass $m_\pi \simeq 140$ MeV: $a_0^{(\rm exp)}=(^1S_0)\sim 20$ fm and $a_0^{(\rm exp)}=(^3S_1)\sim -5$ fm.
\begin{table}[bt]
\begin{center}
\caption{Effective center of mass energies $E$ obtained from the asymptotic momenta and the scattering length $a_0$ at different pion masses. Taken from Ref.~\protect\cite{Aoki:2009ji}.}
\label{tab:EandSL}
\vspace{0.3cm}
\begin{tabular}{|l || ll | ll |}
\hline
& \multicolumn{2}{c|}{$E$[MeV] } & \multicolumn{2}{c|}{$a_0$[fm] } \\
\hline
$m_\pi$[MeV] & spin-singlet & spin-triplet & $^1S_0$ & $^3S_1$ \\
\hline
\hline
731.1(4) & -0.40(8) & -0.48(10) & 0.12(3) & 0.14(3) \\
529.0(4) & -0.51(9) & -0.56(11) & 0.13(3) & 0.14(3) \\
379.7(9) & -0.68(26) & -0.97(37) & 0.15(7) & 0.23(10) \\
\hline
\end{tabular}
\end{center}
\end{table}
The above discrepancy is partly caused by the heavier pion masses and the absence of the dynamical quarks in quenched simulations.
If we get closer to the physical pion mass in full QCD simulations, there should appear the "unitary region" where the $NN$ scattering length shows the singularity associated with the formation of the di-nucleon bound state, so that $a_0$ changes sign\cite{Kuramashi:1995sc}. Therefore the $NN$ scattering length must become a non-linear function of the pion mass in this region.
Unlike the scattering length, on the other hand, the $NN$ potential does not necessarily have singular behavior in the unitary region, as demonstrated in the well-known quantum mechanical examples such as the low-energy scattering between ultracold atoms. To check this in QCD, it is of course important to study the $NN$ potential in the full QCD simulations at lighter pion masses.
In addition to the above reasoning, there is a possibility that extracted values of $k^2$ have large systematic uncertainties caused by the contamination of the excited states at large distance for the wave functions.
These $NN$ scattering lengths extracted from the NBS wave function agree in sign but are much smaller in magnitude than the previous quenched results from the finite size method in smaller volume\cite{Fukugita:1994na}, while they disagree even in sign with the recent full QCD results form the finite size method (See Ref.\cite{Beane:2010em} and references therein.).
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/wave.1665.eps}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/wave2.1665.eps}
\caption{(Left) $(\alpha,\beta)=(2,1)$ components of the orbital $A_1^+$ and non-$A_1^+$ wave functions from $J^P=T_1^+$ (and $J_z=S_z=0$) states at $m_\pi\simeq 529$ MeV. (Right) The same wave functions but the spherical harmonics components are removed from the non-$A_1^+$ part. Taken from Ref.~\protect\cite{Aoki:2009ji}.}
\label{fig:d-wave}
\end{center}
\end{figure}
\subsection{Tensor potential}
In Fig.~\ref{fig:d-wave}(Left), we show the $A_1$ and non-$A_1$ components of the NBS wave function obtained from the $J^P=T_1^+$ (and $J_z=S_z=0$) states at $m_\pi\simeq 529$ MeV, according to eqs. (\ref{eq:projA1}) and (\ref{eq:projE}). The $A_1$ wave function is multivalued as a function of $r$ due to its angular dependence. For example, the $(\alpha,\beta) = (2,1)$ component of the
$L=2$ part of the non-$A_1$ wave function is proportional to the spherical harmonics $Y_{20}(\theta,\phi) \propto 3\cos^2\theta -1$. Fig.~\ref{fig:d-wave}(Right) shows the non-$A_1$ component divided by $Y_{20}(\theta,\phi)$. It is clear that the multivaluedness is mostly removed, showing that the non-$A_1$ component is dominated by the $D$ ($L=2$) state.
\begin{figure}[bt]
\begin{center}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/VTVC.1665.eps}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/pot_tensor-fit.eps}
\caption{(Left) The central potential $V_C(r)^{(1,0)}$ and the tensor potential $V_T(r)$ obtained from the $J^P=T_1^+$ NBS wave function, together with the effective central potential $V_C^{\rm eff} (r)^{(1,0)}$, at $m_\pi\simeq 529$ MeV. (Right) Pion mass dependence of the tensor potential. The lines are the four-parameter fit using one-pion-exchange $+$ one-rho-exchange with Gaussian form factor. Taken from Ref.~\protect\cite{Aoki:2009ji}.}
\label{fig:tensor}
\end{center}
\end{figure}
Shown in Fig.~\ref{fig:tensor} (Left) are the central potential $V_C(r)^{(1,0)}$ and tensor potential $V_T(r)$ together with the effective central potential $V_C^{\rm eff}(r)^{(1,0)}$, at the leading order of the velocity expansion as given in eqs. (\ref{eq:central}), (\ref{eq:tensor}) and (\ref{eq:effective_central}), respectively.
Note that $V_C^{\rm eff}(r)$ contains the effect of $V_T(r)$ implicitly as higher order effects through processes such as ${}^3S_1\rightarrow {}^3D_1\rightarrow {}^3S_1$. At the physical pion mass,
$V_C^{\rm eff}(r)$ is expected to obtain sufficient attraction from the tensor potential, which causes the appearance of a bound deuteron in the spin-triplet (and flavor-singlet) channel while an absence of the bound dineutron in the spin-singlet (and flavor-triplet) channel. The difference between $V_C(r)^{(1,0)}$ and $V_C^{\rm eff}(r)$ in Fig.~\ref{fig:tensor} (Left) is still small in this quenched simulation due to relatively large pion mass. This is also consistent with the small scattering length
in the previous subsection.
The tensor potential in Fig.~\ref{fig:tensor} (Left) is negative for the whole range of $r$ within statistical errors and has a minimum around 0.4 fm. If the tensor potential receives a significant contribution from one-pion exchange as expected from the meson theory, $V_T(r)$ is rather sensitive to the change of the pion mass. As shown in Fig.~\ref{fig:tensor} (Right), it is indeed the case: Attraction of $V_T(r)$ is substantially enhanced as the pion mass decreases.
The central and tensor potentials obtained from lattice QCD are given at discrete data points. For practical applications to nuclear physics, however, it is more convenient to parameterize the lattice results by known functions. Such a fit for $V_T(r)$ is given by the form of one-pion-exchange $+$ one-rho-exchange with Gaussian form factors as
\begin{eqnarray}
V_T(r) &=& b_1(1-e^{-b_2r^2})^2\left(1+\frac{3}{m_\rho r}+\frac{3}{(m_\rho r)^2}\right)\frac{e^{-m_\rho r}}{r}
+
b_3(1-e^{-b_4r^2})^2\left(1+\frac{3}{m_\pi r}+\frac{3}{(m_\pi r)^2}\right)\frac{e^{-m_\pi r}}{r} ,
\nonumber\\
\end{eqnarray}
where $b_{1,2,3,4}$ are the fitting parameters while $m_\pi$ ($m_\rho$) is taken to be the pion mass (the rho meson mass) calculated at each pion mass.
The fit line for each pion mass is drawn in Fig.~\ref{fig:tensor} (Right).
It may be worth mentioning that the pion-nucleon coupling constant extracted from the parameter $b_3$ in the case of the lightest pion mass ($m_\pi = 380$ MeV) gives $g_{\pi N}^2/(4\pi) = 12.1
(2.7)$, which is encouragingly close to the empirical value.
\subsection{Convergence of the velocity expansion}
\begin{figure}[bt]
\begin{center}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/VC.1S0.+009.2.2fm.eps}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/VCeff_so3.1D2.+009.01.eps}
\caption{(Left) The spin-singlet central potential $V_C(r)^{(0,1)}$ obtained from the orbital $A_1^+$ channel at $E\simeq 45$ MeV (red solid circles) and at $E\simeq 0$ MeV (blue open circles)
in quenched QCD at $m_\pi\simeq 529$ MeV. (Right) The same potentials at $E\simeq 45$ MeV, obtained from the orbital $A_1^+$ representation (red open circles) and from the $T_2^+$ representation (cray solid circles). Taken from Ref.~\protect\cite{Murano:2010hh}.}
\label{fig:E-depA}
\end{center}
\end{figure}
The potentials are derived so far at the leading order of the velocity expansion. It is therefore important to investigate the convergence of the velocity expansion: How good is the leading order approximation ? How small are higher order contributions ? If the non-locality of the $NN$ potentials were absent, the leading order approximation for the potentials would give exact results at all energies. The non-locality of the potentials therefore becomes manifest in the energy dependence of the potentials.
So far the LO potentials are extracted with periodic boundary conditions in the spatial directions for quark fields. This leads to the lowest values of the effective center of mass energy $E$ almost zero.
To study the energy dependence, the leading order local potentials at $E\simeq 45$ MeV, realized by anti-periodic boundary conditions in the spatial directions, are calculated
in quenched QCD at $m_\pi\simeq 529$ MeV and $L\simeq 4.4$ fm\cite{Aoki:2008wy,Murano:2010tc,Murano:2010hh}.
In this case, 4 types of "momentum-wall" sources, defined by
\begin{equation}
q^{\rm wall}_f(t_0) \equiv \sum_{\bf x} q({\bf x},t_0) f({\bf x})
\end{equation}
are employed,
where $f({\bf x}) =\cos( (\pm x \pm y + z)\pi/L)$. Note that $f({\bf x})=1$ corresponds to the wall source used in the periodic boundary condition. These momentum-wall sources induce $L = T_2^+$ as well as $L=A_1^+$ states.
\begin{figure}[bt]
\begin{center}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/VC.+009.2.2fm.eps}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/VT.+009.2.2fm.eps}
\caption{(Left) The spin-triplet central potential $V_C(r)^{(1,0)}$ obtained from the orbital $A_1^+-T_2^+$ coupled channel in quenched QCD at $m_\pi\simeq 529$ MeV. (Right) The tensor potential $V_T(r)$ from the orbital $A_1^+-T_2^+$ coupled channel. For these two figures, symbols are same as in Fig.~\ref{fig:E-depA}(Left). Taken from Ref.~\protect\cite{Murano:2010hh}.
}
\label{fig:E-depB}
\end{center}
\end{figure}
In Fig.~\ref{fig:E-depA}(Left), the spin-singlet potential $V_C(r)^{(S,I)=(0,1)}$ obtained from the $L=A_1^+$ state at $E\simeq 45$ MeV (red circles) is compared with that at $E\simeq 0$ MeV (blue circles), while a comparison is made for the spin-triplet potentials in Fig.~\ref{fig:E-depB}, $V_C(r)^{(S,I)=(1,0)}$(left) and $V_T(r)$ (right).
Good agreements between results at two energies seen in these figures indicate that higher order contributions are rather small in this energy interval. In other words, these local potentials obtained at $E\simeq 0$ MeV can be safely used to describe the $NN$ scattering phase shift in both spin-singlet and -triplet channels between $E=0$ MeV and $E=45$ MeV at this pion mass in quenched QCD.
Non-locality of the potential may become manifest also in its angular momentum dependence, since the orbital angular momentum $L={\bf r}\times{\bf p}$ contains one derivative.
In Fig.~\ref{fig:E-depA} (Right), the spin-singlet potential $V_C(r)^{(S,I)=(0,1)}$ obtained from the $L=T_2^+$ state, whose main component has $L=2$, is compared to the one from the $L=A_1^+$ state, whose main component has $L=0$.
In this case local potentials are determined at the same energy, $E\simeq 45$ MeV, but different orbital angular momentum.
Although the statistical errors are rather large in the case of $L=T_2$, a good agreement between the two is again observed, suggesting that the $L$ dependence of the potential is small for the spin-singlet state.
By these comparisons, it is observed that both energy and orbital angular momentum dependences for local potentials are very weak within statistical errors. We therefore conclude that contributions from higher order terms in the velocity expansion are small and that the LO local potentials in the expansion obtained at $E\simeq 0$ MeV and $L= 0$ are good approximations for the non-local potentials at least up to the energy $E\simeq 45$ MeV and orbital angular momentum $L=2$.
Hereafter "potential" in this report means the local potential at the leading order, unless otherwise stated.
\subsection{Full QCD results}
\begin{figure}[bt]
\begin{center}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/potentials-full.eps}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/potentials-quench.eps}
\caption{(Left) 2+1 flavor QCD results for the central potential and tensor potentials at $m_\pi\simeq 701$ MeV. (Right) Quenched results for the same potentials at $m_\pi\simeq 731$ MeV.
Taken from Ref.~\protect\cite{Ishii:2009zr}.}
\label{fig:full}
\end{center}
\end{figure}
Needless to say, it is important to repeat calculations of $NN$ potentials in full QCD on larger volumes at lighter pion masses. The PACS-CS collaboration is performing $2+1$ flavor QD simulations, which cover the physical pion mass\cite{Aoki:2008sm,Aoki:2009ix}.
Gauge configurations are generated with the Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson quark action on a $32^3\times 64$ lattice.
The lattice spacing $a$ is determined from $m_\pi$, $m_K$ and $m_\Omega$ as $a\simeq 0.091$ fm, leading to $L\simeq 2.9$ fm.
Three ensembles of gauge configurations are used
to calculate $NN$ potentials at $(m_\pi, m_N)\simeq $(701 MeV, 1583 MeV), (570 MeV, 1412 MeV) and (411 MeV,1215 MeV )\cite{Ishii:2009zr} .
Fig.~\ref{fig:full}(Left) shows the $NN$ local potentials obtained from the PACS-CS configurations at $E\simeq 0$ and $m_\pi=701$ MeV, which is compared with the previous quenched results at comparable pion mass $m_\pi \simeq 731$ MeV but at $a\simeq 0.137$ fm, given in Fig.~\ref{fig:full}(Right). Both the repulsive core at short distance and the tensor potential become significantly enhanced in full QCD. The attraction at medium distances tends to be shifted to the outer region, while its magnitude remains almost unchanged. These differences may be caused by dynamical quark effects. For a definite conclusion on this point, however, a more controlled comparison at the similar lattice spacing is needed.
\begin{figure}[bt]
\begin{center}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/VC1S0.eps}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/PS_1S0_hikaku.eps}
\caption{(Left) 2+1 flavor QCD results for the spin-singlet central potential from the orbital $A_1^+$ channel at three values of the pion mass. (Right) Scattering phase shifts in $^1S_0$ channel from the corresponding lattice potential given in (Left), together with the empirical one.
Taken from Ref.~\protect\cite{Ishii:2009zr}. }
\label{fig:full_massA}
\end{center}
\end{figure}
In Fig.~\ref{fig:full_massA}(Left), the spin-singlet central potential $V_C(r)^{(0,1)}$ determined from the orbital $A_1$ channel is plotted at three pion masses, while the spin-triplet central potential $V_C(r)^{(1,0)}$ and the tensor potential $V_T(r)$ from the orbital $A_1^+-T_ 2^+$ couple channel are given in Fig.~\ref{fig:full_massB}.
As in the quenched QCD,
the repulsive cores at short distance, the attractive pocket at medium distance and the strength of the tensor potential are all enhanced as pion mass decreases.
\begin{figure}[bt]
\begin{center}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/VC3S1.eps}
\includegraphics[width=0.32\textwidth,angle=270]{Figs/VT.eps}
\caption{(Left) 2+1 flavor QCD results for the spin-triplet central potential $V_C(r)^{(1,0)}$ from the orbital $A_1^+-T_2^+$ coupled channel at three values of the pion mass. (Right) The tensor potential $V_T(r)$ at three values of the pion mass. Taken from Ref.~\protect\cite{Ishii:2009zr}.}
\label{fig:full_massB}
\end{center}
\end{figure}
The phase shifts of the $NN$ scattering for ${}^1S_0$ obtained from the above $V_C(r)^{(0,1)}$ are given in Fig.~\ref{fig:full_massA}(Right).
At low energy, the phase shift increases due to the attraction at medium distance, while at high energy it decreases as a consequence of the repulsive core at short distance.
The shape of the scattering phase shift as a function of energy is qualitatively similar to but is much smaller than in magnitude the experimental one, plotted by the black solid line in Fig.~\ref{fig:full_massA}(Right). As discussed before, the pion mass dependence is so large for the scattering phase shift that full QCD simulations at physical pion mass are needed to reproduce the experimental behavior.
It is noted here that ground state saturation has to be achieved to an accuracy of around 1 MeV, which is about 0.05 \% of the total mass of the two-nucleon system,
in order to determine the shift of the potential ($E=k^2/m_N$) at the same accuracy. The value of $E$ has a strong influence on the value of the scattering length.
Such a high precision is not yet attained in full QCD calculations, since significantly larger $t$ and, accordingly, larger statistics are required. An alternative method to overcome this difficulty will be discussed in Sec.~\ref{sec:extension} .
\section{Hyperon Interactions}
\label{sec:hyperon}
Hyperon($Y$) potentials (hyperon-nucleon and hyperon-hyperon) serve as the starting point in studying
the hyper-nuclei physics. Properties of these potentials can also determine structures in the core of neutron stars. In spite of their importance, only a limited knowledge for hyperon potentials is available so far, since experimental data such as scattering phase shifts are difficult to obtain, due to the short lifetime of hyperons. Therefore it is important to calculate hyperon potentials in lattice QCD by using the potential method.
\subsection{Quenched result for $N\Xi$ potentials}
\begin{figure}[bt]
\begin{center}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/NXi_left.eps}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/NXi_right.eps}
\caption{ (Left)
The spin-singlet central potential for $p\Xi^0$ obtained from the orbital $A_1^+$ channel
at $m_\pi\simeq 368$ MeV
(circle) and $m_\pi\simeq 511$ MeV (box).
The central part of the OPEP ($F/(F+D)=0.36)$ in
Eq.~(\ref{eq:opep}) is also given by solid line.
(Right) The spin-triplet effective central potential from the orbital $A_1^+$ channel
at $m_\pi\simeq 368$ MeV (triangle) and $m_\pi\simeq 511$ MeV (diamond), together with the OPEP (solid line). Taken from Ref.~\cite{Nemura:2008sp}.}
\label{fig:NXi}
\end{center}
\end{figure}
Since all octet-baryons and decuplet $\Omega$ are stable in the strong interaction,
there are many hyperon potentials in $2+1$ flavor QCD.
The method for the $NN$ potentials can be straightforwardly applied to the $I=1$ $N\Xi$ channel,
since $p\Xi$ is simply obtained from $ p n$ by replacing a $d$-quark in the neutron with the $s$-quark and it does not have strong decay into other channels.
Unstable channels such as the $I=0$ $N\Xi$, which can decay into $\Lambda\Lambda$ via the strong interaction, will be discussed later. In addition, experimentally, not much information has been available on the $N\Xi$ interaction except for a few studies: a recent report gives the upper limit of elastic and inelastic cross sections\cite{XN_Ahn} while earlier publications suggest weak attractions of $\Xi-$ nuclear interactions\cite{Nakazawa,Fukuda,Khaustov}. The $\Xi-$nucleus interactions will be soon studied as one of the day-one experiments at J-PARC\cite{JPARC} via $(K^-,K^+)$ reaction with a nuclear target.
Ref.~\cite{Nemura:2008sp} gives the first quenched result for $I=1$ $N\Xi$ potentials.
Lattice parameters are the same as for the quenched $NN$ potential. In addition to two values of the light quark mass, the quenched strange quark is introduced and is fixed to one value.
The potential is calculated at $(m_\pi, m_N, m_\Xi) = (511(1){\rm MeV}, 1300(4){\rm MeV}, 1419(4){\rm MeV})$ and $(368(1){\rm MeV}, 1167(7){\rm MeV}, 1383(6){\rm MeV})$, using the NBS wave function with the interpolation operators defined by
\begin{equation}
p_\alpha (x) =\varepsilon_{abc} (u^a(x)C\gamma_5 d^b(x)) u_\alpha^c(x), \quad
\Xi_\alpha^0(x) = \varepsilon_{abc} (u^a(x)C\gamma_5 s^b(x)) s_\alpha^c(x) .
\end{equation}
Since both $p$ and $\Xi^0$ have $(I, I_z)=1/2,1/2$, the $p\Xi^0$ system has $I=1$ with the strangeness $S=-2$.
The left (right) of Fig.~\ref{fig:NXi} gives the (effective) central potential of the $p\Xi^0$ system obtained from the $L=A_1$ representation for the spin-singlet (triplet) at $m_\pi = 511$ MeV and 368 MeV. Potentials in the $I=1$ $N\Xi$ system for both channels show a repulsive core at $r\le 0.5$ fm surrounded by an attractive well, similar to the $NN$ systems. In contrast to the $NN$ case, however, the repulsive core of the $p\Xi^0$ potential in the spin-singlet channel is substantially stronger than in the triplet channel. The attraction in the medium to long distance region( 0.6 fm $ \le r \le 1.2$ fm ) is similar in both channels.
The height of the repulsive core increases as the light quark mass decreases, while a significant difference is not seen for the attraction in the medium to long distance within statistical errors.
Potentials in Fig.~\ref{fig:NXi} are weakly attractive
on the whole in both spin channels at both pion masses, in spite of the repulsive core at short distance, though the attraction in the triplet is a little stronger than that in the singlet.
The solid lines in Fig.~\ref{fig:NXi} are the one-pion exchange potential (OPEP), given by
\begin{equation}
V_C^\pi = -(1-2\alpha)\frac{g_{\pi NN}^2}{4\pi} \frac{(\vec\tau_N\cdot\vec\tau_\Xi)(\vec\sigma_N\cdot\vec\sigma_\Xi)}{3}\left(\frac{m_\pi}{2m_N}\right)^2 \frac{e^{-m_\pi r}}{r}
\label{eq:opep}
\end{equation}
with $(m_\pi, m_N) =(368 {\rm MeV}, 1167 {\rm MeV})$,
where the pseudo-vector $\pi\Xi\Xi$ coupling $f_{\pi\Xi\Xi}$ is related to the $\pi NN$ coupling as $f_{\pi\Xi\Xi}= -f_{\pi NN} (1-2\alpha)$ with the parameter $\alpha = F/(F+D)$, and $g_{\pi NN} = f_{\pi NN}\frac{m_\pi}{2m_N}$. The empirical vales, $\alpha \simeq 0.36$ and $g_{\pi NN}/(4\pi) \simeq 14.0$, are used for the plot.
Unlike the $NN$ potential, the OPEP in the present case has opposite sign between the spin-singlet channel and spin-triplet channel. In addition, the absolute magnitude is smaller due to the factor $1-2\alpha$. No clear signature of the OPEP at long distance ($r\ge 1.2$ fm) is observed in Fig.~\ref{fig:NXi} within statistical errors. Furthermore, there is clear departure from the OPEP at medium distance (0.6 fm $\le r \le 1.2$ fm) in both channels. These observations may suggest an existence of state-independent attraction.
\subsection{Full and quenched QCD results for $N \Lambda$ potentials}
\begin{figure}[bt]
\begin{center}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/VC_1S0_full.eps}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/VCVT_3E1_full.eps}
\caption{ (Left)
The spin-singlet central potential for $N\Lambda$ obtained from the orbital $A_1^+$ channel
in 2+1 flavor QCD at $m_\pi\simeq 414$ MeV (red) and 699 MeV (green).
(Right) The spin-triplet central potential and the tensor potential for $N\Lambda$ obtained from the orbital $A_1^+-T_2^+$ coupled channel in 2+1 flavor QCD
at $m_\pi\simeq 414$ MeV (red and blue) and 699 MeV (green and magenta). Taken from Ref.~\protect\cite{Nemura:2009kc}. }
\label{fig:NL_full}
\end{center}
\end{figure}
Spectroscopic studies of the $\Lambda$ and $\Sigma$ hypernuclei, carried out both experimentally and theoretically, suggest that the $\Lambda$-nucleus interaction is attractive while the $\Sigma$-nucleus interaction is repulsive. If this is the case, the $\Lambda$ particle would be the first strange baryon instead of $\Sigma^-$ to appear in the core of the neutron stars\cite{SchaffnerBielich:2008kb}.
It is therefore interesting and important to investigate the nature of the $N\Lambda$ interaction in lattice QCD, by calculating the $N\Lambda$ potential using the method of this report.
Since the $\Lambda$ is the lightest hyperon, the $N\Lambda$ potential can be calculated as in the case of $NN$ potentials.
In Ref.~\cite{Nemura:2009kc}, the $N\Lambda$ potentials are calculated in both full and quenched QCD. The 2+1 flavor full QCD gauge configurations generated by the PACS-CS collaboration are employed for the calculations of the potentials on a $32^3\times 64$ lattice at $a= 0.091(1)$ fm, while in the quenched calculation, the potentials are obtained on a $32^3\times 48$ lattice at $a=0142(1)$ fm. Numerical values for some hadron masses for these calculations are given in Table~\ref{tab:NL_mass}, together with some lattice parameters.
Fig.~\ref{fig:NL_full} shows the $N\Lambda$ potentials obtained from 2+1 flavor QCD calculations as a function of $r$ at $m_\pi\simeq 699$ MeV and 414 MeV. The spin-singlet central potential obtained from the $J=A_1$ channel is plotted on the left, while the spin-triplet central potential and the tensor potential obtained from the $J=T_1^+$ channel are given on the right. The central potential multiplied by a volume factor ($r^2 V_C(r)$) is also shown in the left panel in addition to the $V_C(r)$ itself in the right panel, in order to compare the strength of the repulsion between two quark masses.
As can be seen in Fig.~\ref{fig:NL_full}, the attractive well of the central potentials moves to the outer region as the light quark mass decreases, while the depth of these attractive pockets does not change so much. The present results show that the tensor force is weaker than the $NN$ case in Fig.\ref{fig:full}. Moreover the quark mass dependence of the tensor force seems small.
Both repulsive and attractive parts of the central potentials increase in magnitude as the light quark mass decreases.
\begin{figure}[bt]
\begin{center}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/VC_1S0_quench.eps}
\includegraphics[width=0.45\textwidth,angle=0]{Figs/VCVT_3E1_quench.eps}
\caption{ (Left)
The spin-singlet central potential for $N\Lambda$ obtained from the orbital $A_1^+$ channel
in quenched QCD at $m_\pi\simeq 407$ MeV (red) and 512 MeV (green).
(Right) The spin-triplet central potential and the tensor potential for $N\Lambda$ obtained from the orbital $A_1^+-T_2^+$ coupled channel in quenched QCD
at $m_\pi\simeq 407$ MeV (red and blue) and 512 MeV (green and magenta).
Taken from Ref.~\protect\cite{Nemura:2009kc}. }
\label{fig:NL_quench}
\end{center}
\end{figure}
Fig.~\ref{fig:NL_quench} shows the $N\Lambda$ potentials in quenched QCD calculations at $m_\pi\simeq 512$ MeV and 407 MeV. The central potential in the spin-singlet channel from $J=A_1$ is on the left, while the central and the tensor potential in the triplet channel from $J=T_1$ are on the right. Qualitative features of these potentials are more or less similar to those in full QCD: the attractive pocket of the central potentials moves to the longer distance region as the quark mass decreases while the quark mass dependence of the tensor potential is small.
\begin{table}[tb]
\centering \leavevmode
\begin{tabular}{|cccccccc|}
\hline \hline
$m_\pi$ & $m_\rho$ & $m_K$ & $m_{K^\ast}$ & $m_N$ & $m_\Lambda$ & $m_{\Sigma}$ & $m_{\Xi}$ \\
\hline
\multicolumn{8}{|l|}{\underline{2+1 flavor QCD on a $32^3\times 64$ lattice at $a= 0.091(1)$ fm
}} \\
699.4(4) &
1108(3) &
786.8(4) &
1159(2) &
1572(6) &
1632(4) &
1650(5) &
1701(4) \\
%
567.9(6) &
1000(4) &
723.7(7) &
1081(3) &
1396(6) &
1491(4) &
1519(5) &
1599(4) \\
%
413.6(6) &
902(3) &
636.6(4) &
1026(3) &
1221(7) &
1349(4) &
1406(8) &
1505(4) \\
%
301(3) &
845(10) &
592(1) &
980(6) &
1079(12) &
1248(15) &
1308(13) &
1432(7) \\
%
\hline
%
\multicolumn{8}{|l|}{ \underline{quenched QCD on a $32^3\times 48$ lattice at $a=0.142(1)$ fm
}} \\
%
511.8(5) & 862(3) & 605.8(5) & 898(1) &
1297(6) & 1344(6) & 1375(5) & 1416(3) \\
%
463.6(6) & 842(4) & 586.3(5) & 895(2) &
1250(9) & 1314(9) & 1351(6) & 1404(4) \\
%
407(1) & 820(3) & 564.9(5) & 886(3) &
1205(13) & 1269(9) & 1326(9) & 1383(5) \\
\hline
\hline
135 & 770 & 494 & 892 &
940 & 1116 & 1190 & 1320 \\
%
\hline \hline
\end{tabular}
\caption{
Hadron masses in units of MeV in lattice QCD simulations. The last line shows experimental values. Taken from Ref.~\protect\cite{Nemura:2009kc}.
}
\label{tab:NL_mass}
\end{table}
\subsection{Flavor SU(3) limit}
In order to unravel the nature of the various channels in the hyperon interactions, it is more convenient to consider an idealized flavor SU(3) symmetric world, where $u,d$ and $s$ quarks are all degenerate with a common finite mass. In the flavor SU(3) limit, one may capture essential features of the interaction, in particular, the short range force without contamination from the quark mass difference. Calculations in the SU(3) limit allow us to extract potentials for irreducible flavor multiplets: Potentials between asymptotic baryon states are obtained by the recombination of the multiplets with suitable Clebsh-Gordan coefficients.
In the flavor SU(3) limit, the ground state baryon belongs to the flavor-octet with spin $1/2$, and two baryon states with a given angular momentum are labeled by the irreducible representation of SU(3) as
\begin{equation}
{\bf 8}\otimes {\bf 8} = \underbrace{{\bf 27}\oplus {\bf 8}\oplus {\bf 1}}_{\rm symmetric}\oplus
\underbrace{\overline{\bf 10}\oplus {\bf 10}\oplus {\bf 8}}_{\rm anti-symmetric},
\end{equation}
where "symmetric" and "anti-symmetric" stand for the symmetry under the exchange of the flavor for two baryons. For the system with orbital S-wave, the Pauli principle imposes {\bf 27}, {\bf 8} and {\bf 1} to be spin-singlet ($^1S_0$), while $\overline{\bf 10}$, {\bf 10} and {\bf 8} to be spin-triplet ($^3S_1$).
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figs/v27_kappa.eps
\includegraphics[width=0.45\textwidth]{Figs/v10s_kappa.eps}
\end{center}
\caption{
The BB potentials in {\bf 27} (Left) and $\overline{\bf 10}$(Right) representations from the orbital $A_1^+$ channel in the flavor SU(3) limit,
extracted from the lattice QCD simulation at
$m_{\pi}=1014$ MeV (red bars) and $m_{\pi}=835$ MeV (green crosses).
Taken from Ref.~\protect\cite{Inoue:2010hs}.
}
\label{fig:su3limitA}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figs/v8s_kappa.eps
\includegraphics[width=0.45\textwidth]{Figs/v10_kappa.eps}
\includegraphics[width=0.45\textwidth]{Figs/v1_kappa.eps
\includegraphics[width=0.45\textwidth]{Figs/v8a_kappa.eps}
\caption{ The BB potentials in ${\bf 8}_s$(Upper-Left), {\bf 10}(Upper-Right), {\bf 1}(Lower-Left) and ${\bf 8}_a$(Lower-Right) from the orbital $A_1^+$ channel in the flavor SU(3) limit,
extracted from the lattice QCD simulation at $m_{\pi}=1014$ MeV (red bars) and $m_{\pi}=835$ MeV (green crosses).
Taken from Ref.~\protect\cite{Inoue:2010hs}.
}
\label{fig:su3limitB}
\end{center}
\end{figure}
The NBS wave function is defined by
\begin{equation}
\varphi^{W, (X)}({\bf r}) = \langle 0 \vert (BB)^{(X)}({\bf r}) \vert W, B=2, X \rangle,
\end{equation}
from which the corresponding (effective) central potential is given as
\begin{equation}
V^{(X)}(r) = \frac{1}{2\mu}\left[\frac{\nabla^2 \varphi^{W,(X)}({\bf r})}{\varphi^{W,(X)}({\bf r})}+k^2\right]
\end{equation}
with $k^2 =W^2/4-m_B^2$ and $\mu = m_B/2$, where $m_B$ is the common octet baryon mass and
$(BB)^{(X)}({\bf r}) =\sum_{ij} C_{ij}B_i({\bf x}+{\bf r},0) B_j({\bf x},0)$ with $X={\bf 27}, {\bf 8}_s,{\bf 1},\overline{\bf 10}, {\bf 10}, {\bf 8}_a$.
Two baryon operators $BB^{(X)}$ in the flavor basis are given in terms of the particle basis in Appendix~\ref{app:octet}.
Potentials among octet baryons, both the diagonal part ($B_1B_2\rightarrow B_1B_2$) and the off-diagonal part ($B_1B_2\rightarrow B_3B_4$) are obtained by suitable combination of $V^{(X)}(r)$.
Using the 3 flavor full QCD gauge configuration generated by CP-PACS/JLQCD Collaborations on a $16^3\times 32$ lattice at $a\simeq 0.12$ fm where light and strange quark masses have same values\cite{CPPACS-JLQCD}, the (effective) central potentials are calculated\cite{Inoue:2010hs} at $(m_{PS}, m_B) = (1014(1) {\rm MeV}, 2026(3) {\rm MeV})$ and $(845(1) {\rm Mev}, 1752(3) {\rm MeV})$, where $m_{PS}$ and $m_B$ denote the octet pseudo-scalar (PS) meson mass and the octet baryon mass, respectively.
Figs.~\ref{fig:su3limitA} and \ref{fig:su3limitB} give the six baryon-baryon ($BB$) potentials in the flavor basis, where red (green) data correspond to the $m_{PS} = 1014$ MeV (835 MeV). The left panels show central potentials for the spin-singlet channel from the $J=A_1$ state, while the right panels give effective central potentials for the spin-triplet channel from the $J=T_1$ state.
\begin{table}[tb]
\begin{center}
\begin{tabular}{| c|l |}
\hline \hline
flavor multiplet & baryon pair (isospin) \\
\hline
{\bf 27} & \{NN\}(I=1), \{N$\Sigma$\}(I=3/2), \{$\Sigma\Sigma$\}(I=2), \\
& \{$\Sigma\Xi$\}(I=3/2), \{$\Xi\Xi$\}(I=1) \\
{\bf 8}$_s$ & none \\
~{\bf 1} & none \\
\hline
{\bf 10}$^*$ & [NN](I=0), [$\Sigma\Xi$](I=3/2) \\
{\bf 10} & [N$\Sigma$](I=3/2), [$\Xi\Xi$](I=0)\\
{\bf 8}$_a$ & [N$\Xi$](I=0) \\
\hline \hline
\end{tabular}
\caption{\label{tab:pairs}Baryon pairs in an irreducible flavor SU(3) representation,
where $\{BB'\}$ and $[BB']$ denotes $BB' + B'B$ and $BB' - B'B$, respectively.}
\end{center}
\end{table}
Note that some of octet-baryon pairs exclusively belong to an irreducible representation of SU(3) as shown in Table~\ref{tab:pairs}. For example, symmetric $NN$ belongs to ${\bf 27}$, which therefore can be considered as the $NN$ spin-singlet potential in the flavor SU(3) symmetric limit. Similarly $V^{(\overline{\bf 10})}$, $V^{({\bf 10})}$ and $V^{({\bf 8}_a)}$ can be considered as some $BB$ potentials of the particle basis in the SU(3) symmetric limit, while $V^{({\bf 1})}$ and $V^{({\bf 8}_s)}$ are always mixtures of different $BB$ potentials in the particle basis.
Fig.~\ref{fig:su3limitA} shows $V^{\bf (27)}(r)$ and $V^{\bf (\overline{10})}(r)$, which correspond to spin-singlet and spin-triplet NN potentials, respectively. Both have a repulsive core at short distance with an attractive pocket around 0.6 fm. These qualtative features are consistent with the previous results found for the NN potential in both quenched and full QCD.
The upper-right panel of Fig.~\ref{fig:su3limitB} shows that $V^{\bf (10)}(r)$ has a stronger repulsive core and a weaker attractive pocket than $V^{\bf (27,\overline{10})}(r)$. Furthermore $V^{({\bf 8}_s)}(r)$ in the upper-left panel of Fig.~\ref{fig:su3limitB} has a very strong repulsive core among all 6 channels, while $V^{({\bf 8}_a)}(r)$ in the lower-right panel has a very weak repulsive core.
In contrast to all other cases, $V^{\bf (1)}(r)$ shows attraction instead of repulsion at all distances,
as shown in the lower-left panel.
Above features are consistent with what has been observed in a phenomenological quark model\cite{Oka:2000wj}.
In particular, the potential in the ${\bf 8}_s$ channel in this quark model becomes strongly repulsive at short distance since the six quarks cannot occupy the same orbital state due to the Pauli exclusion for quarks. On the other hand, the potential in the {\bf 1} channel does not suffer from the quark Pauli exclusion and can become attractive due to the short-range gluon exchange. Such agreements between the lattice data and the phenomenological model suggest that the quark Pauli exclusion plays an essential role for the repulsive core in BB systems.
The BB potentials in the baryon basis can be obtained from those in the SU(3) basis by a unitary rotation as
\begin{equation}
V_{ij}(r) = \sum_X U_{i X} V^{(X)}(r)U_{X j}^\dagger
\label{eq:rotation}
\end{equation}
where $U$ is a unitary matrix which rotates the flavor basis $\ketv{X}>$
to the baryon basis $\ketv i>$, {\it i.e.} $\ketv i> = U_{i X} \ketv{X}>$.
The explicit forms of the unitary matrix $U$ in terms of the
CG coefficients are found in Appendix \ref{app:octet}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figs/pot_dia30.eps}\hfill
\includegraphics[width=0.45\textwidth]{Figs/pot_off30.eps}
\caption{\label{fig:pot_lamlam}
BB potentials in baryon basis for the S=$-$2, I=0, $^1S_0$ sector.
Three diagonal(off-diagonal) potentials are shown in left(right) panel.
Taken from Ref.~\protect\cite{Inoue:2010hs}.
}
\end{center}
\end{figure}
In Fig.~\ref{fig:pot_lamlam}, as characteristic examples,
let us show the spin-singlet potentials for S=$-$2, I=0 channel determined from the orbital $A_1^+$ representation at $m_\pi = 835$ MeV.
To obtain $V_{ij}(r)$, the potentials in the SU(3) basis are fitted
by the following form with five parameters $b_{1,2,3,4,5} $,
\begin{equation}
V(r) = b_1 e^{-b_2\,r^2} + b_3(1 - e^{-b_4\,r^2})\left( \frac{e^{-b_5\,r}}{r} \right)^2 .
\end{equation}
Then the right hand side of Eq. (\ref{eq:rotation}) is used
to obtain the potentials in the baryon basis.
The left panel of Fig.~\ref{fig:pot_lamlam} shows the diagonal part of the potentials.
The strong repulsion in the ${\bf 8}_s$ channel is reflected most in the $\Sigma\Sigma$(I=0) potential
due to its largest CG coefficient among three channels.
The strong attraction in the ${\bf 1}$ channel is reflected most in the $N\Xi$(I=0) potential
due to its largest CG coefficient.
Nevertheless, all three diagonal potentials have a repulsive core originating from the ${\bf 8}_s$ component.
The right panel of Fig.~\ref{fig:pot_lamlam} shows the off-diagonal parts of the potentials
which are comparable in magnitude to the diagonal ones.
Since the off-diagonal parts are not negligible in the baryon basis,
fully coupled channel analysis is necessary to study observables.
A similar situation holds even in (2+1)-flavors where the strange quark is heavier than up and down quarks: The SU(3) basis with approximately
diagonal potentials is useful for obtaining essential features of the BB interactions, while
the baryon basis with substantial magnitude of the off-diagonal potentials is
necessary for practical applications.
\if0
\begin{figure}[tpbh]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figs/pot_exclusive_1S0.eps}\hfill
\includegraphics[width=0.45\textwidth]{Figs/pot_exclusive_3S1.eps}
\includegraphics[width=0.45\textwidth]{Figs/pot_nsignlam_1S0.eps}\hfill
\includegraphics[width=0.45\textwidth]{Figs/pot_nsignlam_3S1.eps}
\includegraphics[width=0.45\textwidth]{Figs/pot_nxisiglam_1S0.eps}\hfill
\includegraphics[width=0.45\textwidth]{Figs/pot_nxisigsigsiglam_3S1.eps}
\includegraphics[width=0.45\textwidth]{Figs/pot_sigxilamxi_1S0.eps}\hfill
\includegraphics[width=0.45\textwidth]{Figs/pot_sigxilamxi_3S1.eps}
\caption{\label{fig:pot_particle}
BB potentials in baryon basis other than those shown in Fig.~\ref{fig:pot_lamlam}.
See the caption of Fig.~\ref{fig:pot_lamlam}.
}
\end{center}
\end{figure}
\fi
Other potentials in the baryon basis are given in Ref.~\cite{Inoue:2010hs}.
Since the ${\bf 8}_s$ state does not couple to the spin-triplet channel,
the repulsive cores in the spin-triplet channel
are relatively small.
The off-diagonal potentials are not generally small:
For example,
the $N\Lambda$-$N\Sigma$ potential in the spin-triplet channel
is comparable in magnitude at short distances with the
diagonal $N\Lambda$-$N\Lambda$ and $N\Sigma$-$N\Sigma$ potentials.
Although all quark masses of 3 flavors are degenerate and rather heavy in these simulations,
the coupled channel potentials in the baryon basis may give
useful hints for the behavior of hyperons ($\Lambda$, $\Sigma$ and $\Xi$)
in hyper-nuclei and in neutron stars \cite{Hashimoto:2006aw,SchaffnerBielich:2010am}.
The flavor singlet channel has attraction for all distances, which might produce the bound state, the $H$-dibaryon, in this channel. The present data, however, are not sufficient to make a definite
conclusion on the $H$-dibaryon, since a single lattice with small extension $L\simeq 2$ fm is employed. In order to investigate whether the $H$-dibaryon exists or not in the flavor SU(3) limit, data on several different volumes are needed. Such a study on the $H$-dibaryon will be discussed in Sec.~\ref{sec:extension}.
In order to extend the study in the flavor SU(3) limit to the real world where the strange quark is much heavier than light quarks, the potential method used so far has to be extended to more general cases, which will also be considered in Sec.~\ref{sec:extension}.
\section{Origin of repulsive core}
\label{sec:OPE}
As seen in the previous sections, lattice QCD calculations show that the $NN$ potential defined through the NBS wave function has not only the attraction at medium to long distance
that has long been well understood in terms of pion and other heavier meson exchanges, but also a characteristic repulsive core at short distance, whose origin is still theoretically unclear.
Furthermore, the $BB$ potentials in the flavor SU(3) limit show several different behaviors at short distance: some have a stronger/weaker repulsive core than $NN$ while the singlet has an attractive core. In this section,
recent attempts \cite{Aoki:2010kx, Aoki:2009pi,Aoki:2010uz} to theoretically understand the short distance behavior of the potential in terms of the operator product expansion (OPE) is explained.
\subsection{OPE and repulsive core}
Let us first explain the basic idea. We consider the equal time NBS wave function defined by
\begin{eqnarray}
\varphi^E_{AB}(\bf r) &=& \langle 0\vert T\{ O_A(\bf r/2,0) O_B(-\bf r/2,0)\} \vert E\rangle,
\end{eqnarray}
where $\vert E \rangle $ is some eigenstate of a certain system with total energy $E$, and $O_A$, $O_B$ are some operators of this system. (We suppress other quantum numbers of the state $\vert E\rangle$ for simplicity.) The OPE reads
\begin{eqnarray}
O_A(\bf r/2,0) O_B(-\bf r/2,0) \simeq \sum_C D_{AB}^C(\bf r) O_C({\bf 0},0),
\end{eqnarray}
Suppose that the coefficient function of the OPE behaves
in the small $r(=\vert \bf r\vert)$ limit as
\begin{eqnarray}
D_{AB}^C(\bf r ) &\simeq & r^{\alpha_C}( - \log r)^{\beta_C} f_C(\theta,\phi),
\end{eqnarray}
where $\theta,\phi$ are the angles of $\bf r$, the NBS wave function becomes
\begin{eqnarray}
\varphi_{AB}^E(\bf r) &\simeq& \sum_C r^{\alpha_C} (-\log r)^{\beta_C} f_C(\theta,\phi) D_C(E),
\end{eqnarray}
where
\begin{eqnarray}
D_C(E) &=& \langle 0 \vert O_C({\bf 0},0)\vert E\rangle.
\end{eqnarray}
The potential at short distances can be calculated from this expression.
For example, in the case of the Ising field theory in two dimensions, the OPE for the spin field $\sigma$ is given by
\begin{eqnarray}
\sigma(x,0) \sigma(0,0) \simeq G(r) {\bf 1} + c\, r^{3/4} O_1(0) + \cdots, \quad r=\vert x\vert,
\end{eqnarray}
where $O_1(x)$ ($= :\bar\psi\psi(x):$ in terms of free fermion fields) is an operator of dimension one. This leads to
\begin{eqnarray}
\varphi (r,E) \simeq r^{3/4} D(E) + O(r^{7/4}),\quad D(E)= c \langle 0 \vert O_1(0) \vert E \rangle ,
\end{eqnarray}
where $ \vert E \rangle $ is a two-particle state with energy $E=2\sqrt{k^2+ m^2}$.
From this expression the potential becomes
\begin{eqnarray}
V(r) &=& \frac{\varphi^{\prime\prime}(r,E) + k^2\varphi(r,E)}{m \varphi(r,E) }
\simeq -\frac{3}{16}\frac{1}{m r^2}
\end{eqnarray}
in the $r\rightarrow 0$ limit. The OPE predicts not only the $r^{-2}$ behavior of the potential at short distance but also its coefficient $-3/16$. Furthermore the potential at short distance does not depend on the energy of the state in this example\cite{Aoki:2008wy,Aoki:2008yw}.
In QCD the dominant terms at short distance have $\alpha_C=0$. Among these terms, we assume that $C$ has the largest contribution such that
$\beta_C > \beta_{C^\prime}$ for $\forall C^\prime\not= C$. Since such dominant operators with $\alpha_C=0$ mainly couple to the zero angular momentum ($L=0$) state,
let us consider the NBS wave function with $L=0$.
Applying $\nabla^2$ to this wave function,
we obtain the following classification of the short distance behavior of the potential.
\begin{enumerate}
\item[(1)] $\beta_C\not=0$: The potential at short distance is energy independent and becomes
\begin{eqnarray}
V(r) \simeq -\frac{\beta_C}{m r^2(-\log r)} \,,
\end{eqnarray}
which is attractive for $\beta_C > 0$ and repulsive for $\beta_C<0$.
\item[(2)] $\beta_C=0$: In this case the potential becomes
\begin{eqnarray}
V(r) \simeq \frac{ D_{C^\prime}(E)}{D_C(E)}\frac{-\beta_{C^\prime}}{m r^2}(-\log r)^{\beta_{C^\prime}-1} \,,
\label{eq:pot_ope}
\end{eqnarray}
where $\beta_{C^\prime} < 0$ is the second largest exponent. The sign of the potential at short distance depends on the sign of $ D_{C^\prime}(E)/D_{C}(E)$.
\end{enumerate}
On the lattice, we do not expect divergence at $r=0$ due to lattice artifacts at short distance. The above classification holds at $a \ll r \ll 1/\Lambda_{\rm QCD}$, while the potential becomes finite even at $r=0$ on the lattice.
Since QCD is an asymptotic free theory, the 1-loop calculation for anomalous dimensions becomes
exact at short distance.
The OPE in QCD is written as
\begin{eqnarray}
O_A(y/2) O_B(-y/2)&=&\sum_C D_{AB}^C(r,g,m,\mu)O_C(0)
\end{eqnarray}
where $g$ ($m$) is the renormalized coupling constant (quark mass) at scale $\mu$.
In the limit that $r=\vert y\vert = e^{-t} R \rightarrow 0 $ ( $t\rightarrow\infty$ with fixed $R$),
the renormalization group analysis leads to
\begin{eqnarray}
\lim_{r\rightarrow 0} D_{AB}^C(r,g,m,\mu) = (-2\beta^{(1)} g^2\log r)^{\gamma_{AB}^{C,(1)}/(2\beta^{(1)})} D_{AB}^C(R, 0,0,\mu),
\end{eqnarray}
where
$
\beta^{(1)} = \displaystyle \frac{1}{16\pi^2}\left(11-\frac{2N_f}{3}\right)
$
is the QCD beta-function at 1-loop, and
\begin{eqnarray}
\gamma_{AB}^{C,(1)} = \gamma_C^{(1)}-\gamma_A^{(1)} - \gamma_B^{(1)} \equiv \frac{1}{48\pi^2}\gamma .
\end{eqnarray}
Here $\gamma_X^{(1)}$ is the 1-loop anomalous dimension of the operator $O_X$.
An appearance of $D_{AB}^C(R,0,0,\mu)$ on the right-hand side tells us that
it is enough to know the OPE only at tree level.
From the above expression, $\beta_C$ is given by
$
\beta_C = \displaystyle \frac{\gamma_{AB}^{C,(1)}}{2\beta^{(1)}}.
$
\subsection{Two flavor case}
We first consider the OPE for $N_f=2$ QCD\cite{Aoki:2010kx}. The 1-loop calculations show that the largest value of $\beta_C$ is always zero for both spin-singlet and spin-triplet channels for $N_f=2$ QCD and that
the second largest value of $\beta_{C^\prime}$ is given by
\begin{equation}
\beta_{C^\prime}^{S=0} = -\frac{6}{33-2N_f}, \quad \beta_{C^\prime}^{S=1} = -\frac{2}{33-2N_f},
\end{equation}
where $S=0,1$ denotes the total spin.
This corresponds to the case (2) in the previous subsection. Therefore
the OPE and renormalization group analysis in QCD predicts the universal functional form of the $NN$ central potential at short distance as
\begin{equation}
V_C^S(r) \simeq \frac{D_{C^\prime}(E)}{D_C(E)} \frac{-\beta_{C^\prime}^S (-\log r)^{\beta_{C^\prime}^S -1}}{m_N r^2}, \qquad r\rightarrow 0,
\end{equation}
which is a little weaker than a $1/r^2$ singularity, while for the tensor potential we have
\begin{equation}
V_T(r)\simeq 0
\end{equation}
at the 1-loop order.
The OPE, however, can not tell whether the potential at short distance is repulsive or attractive, which is determined by the sign of the coefficient. If $D_X(E)$ and $D_Y(E)$ are evaluated by the non-relativistic quark model at the leading order, we obtain
\begin{equation}
\frac{D_{C^\prime}(E)}{D_C(E)}(S=0) \simeq \frac{D_{C^\prime}(E)}{D_C(E)}(S=1)\simeq 2 .
\end{equation}
For both cases, the ratio has positive sign, which gives repulsion at short distance, the repulsive core.
\subsection{Extension to three flavors}
The above calculation has been extended to $N_f=3$ QCD\cite{Aoki:2010uz}.
In the 3-flavor case, some channel may become attractive at short distance since the Pauli exclusion principle is less significant than in the 2-flavor case. Indeed the lattice QCD calculations in the flavor SU(3) limit shows the attractive potential for the singlet channel, as seen in the previous section.
The largest value of $\beta_C$ is given for each $X$ representation of the flavor SU(3) in unit of $1/(33-2N_f)$ in the table~\ref{tab:ope}, where we define
\begin{equation}
\beta_C =\frac{\gamma^{(X)}}{33- 2 N_f} .
\end{equation}
\begin{table}[t]
\begin{center}
\begin{tabular}{| c | llllll |}
\hline \hline
$X$ & ${\bf 27}$ & ${\bf 8}_s$ & ${\bf 1}$ & $\overline{\bf 10}$ & ${\bf 10}$ & ${\bf 8}_a$ \\
\hline
$\gamma^{(X)}$ & 0 & 6 & 12 & 0 & 0 & 4 \\
Non-relativistic op. & yes & no & yes & yes & yes & yes \\
\hline \hline
\end{tabular}
\caption{\label{tab:ope} The largest value of $\beta_C$ in unite of $1/(33-2N_f)$ of 3-flavor QCD for each representation. The last line indicates that the operator corresponding to the largest value of $\beta_C$ exists or not in the non-relativistic limit.}
\end{center}
\end{table}
From the table, we observe that the largest value of $\beta_C$ is zero in the ${\bf 27}$, $\overline{\bf 10}$ and ${\bf 10}$ channels. This is consistent with the nucleon case in the previous subsection, which belong to ${\bf 27}$ (spin-singlet) and $\overline{\bf 10}$(spin-triplet). These three channels correspond to the case (2), so that the potentials are given by eq. (\ref{eq:pot_ope}).
On the other hand, the largest value of $\beta_C$ becomes positive in the ${\bf 8}_s$, $\overline{\bf 8}_a$ and ${\bf 1}$ channels, which correspond to the case (1). Therefore the (effective) central potential becomes attractive at short distance as
\begin{equation}
V_C^{(X)}(r) \simeq -\frac{\gamma^{(X)}}{(33-2N_f)}\frac{1}{ m_B r^2(-\log r)} ,
\end{equation}
where $m_B$ is the octet baryon mass.
The attractive core of the potential in the flavor singlet channel agrees with the behavior of the potential found for the numerical simulation of lattice QCD in the previous section,
while for other two channels, ${\bf 8}_s$ and ${\bf 8}_a$, the prediction by the OPE disagrees with
the lattice QCD results: The potential in the ${\bf 8}_s$ channel is most repulsive among 6 channels and the potential in the ${\bf 8}_a$ channel still has a repulsive core, which is however weaker than others.
The disagreement between the OPE and the lattice QCD result for the ${\bf 8}_s$ channel may be understood by the fact that no local 6 quark operator exists for this channel in the non-relativistic limit, as shown in the table~\ref{tab:ope}: the ${\bf 8}_s$ operator with the largest positive $\beta_C$ has a very small coefficient at low energy, so that the other operators with zero or negative $\beta_C$ may still dominate at distance scales comparable to the lattice spacing $a= 0.1-0.2$ fm. For the ${\bf 8}_a$ case, the weakest repulsive core in the lattice QCD simulation suggests that
the attraction from the leading operator with the positive $\beta_C$ may be cancelled by other contributions from the sub-leading operators with zero or negative $\beta_C$ at the distance scale comparable to the lattice spacing of the simulations.
It is therefore important to confirm the prediction from the OPE, by investigating the behavior of the repulsive core for each channel in the flavor SU(3) limit at finer lattice spacings and hopefully in the continuum limit.
\section{Extensions}
\label{sec:extension}
In this section, recent extensions of the potential method are considered.
\subsection{Inelastic scattering}
The potential method discussed so far is shown to be quite successful in order to describe elastic hadron interactions. Hadron interactions in general, however, lead to inelastic scatterings as the total energy of the system increases. In order to extract hadron interactions which describe such inelastic scatterings from lattice QCD, an extension of the potential method is considered in this subsection.
Let us first discuss the case of $A+B\rightarrow C+D$ scattering where $A,B,C,D$ represent some 1-particle states. This is a simplified version of the octet baryon scattering in the strangeness $S=-2$ and isospin $I=0$ channel, where $\Lambda\Lambda$, $N\Xi$ and $\Sigma\Sigma$ appear as asymptotic states of the strong interaction if the total energy is larger than $2m_\Sigma$.
We here assume $m_A + m_B < m_C + m_D < W $, where $W=E_k^A + E_k^B$ is the total energy of the system, and $E_k^X =\sqrt{m_X^2 +\mbox{\boldmath $k$}^2}$. In this situation, the QCD eigen-state with the quantum numbers of the $AB$ state and center of mass energy $W$ is expressed in general as
\begin{eqnarray}
\vert W \rangle &=& c_{AB} \vert AB,W\rangle + c_{CD} \vert CD, W\rangle +\cdots\\
\vert AB, W\rangle &=& \vert A, \mbox{\boldmath $k$} \rangle_{\rm in}\otimes \vert B, -\mbox{\boldmath $k$} \rangle_{\rm in}, \quad
\vert CD, W\rangle = \vert C, \mbox{\boldmath $q$} \rangle_{\rm in}\otimes \vert D, -\mbox{\boldmath $q$} \rangle_{\rm in},
\end{eqnarray}
where $W=E_k^A+E_k^B = E_q^C+E_q^D$.
We define the following NBS wave functions,
\begin{eqnarray}
\varphi_{AB}(\bf r,\mbox{\boldmath $k$} )e^{-Wt} &=& \langle 0 \vert T\{ \varphi_A(\mbox{\boldmath $x$}+\bf r,t) \varphi_B(\mbox{\boldmath $x$},t) \}\vert W \rangle, \\
\varphi_{CD}(\bf r,\mbox{\boldmath $q$} )e^{-Wt} &=& \langle 0 \vert T\{ \varphi_C(\mbox{\boldmath $x$}+\bf r,t) \varphi_D(\mbox{\boldmath $x$},t) \}\vert W \rangle .
\end{eqnarray}
Using the partial wave decomposition such that\footnote{Here we ignore spins for simplicity.}
\begin{eqnarray}
\varphi_{XY}(\bf r,\mbox{\boldmath $k$} ) &=& 4\pi\sum_{l,m} i^l \varphi^{\ell}_{XY}(r,k)Y_{lm}(\Omega_{\mbox{\scriptsize \boldmath $r$}}) \overline{Y_{lm}(\Omega_{\mbox{\scriptsize \boldmath $k$}})} ,
\end{eqnarray}
the NBS wave function of the 2-channel system behaves for large $r$ as
\begin{eqnarray}
\left(
\begin{array}{l}
\varphi_{AB}^{\ell}(r,k) \\
\varphi_{CD}^{\ell}(r,q) \\
\end{array}
\right)
&\simeq&
\left(
\begin{array}{ll}
j_l(k r) & 0 \\
0 & j_l(q r) \\
\end{array}
\right)
\left(
\begin{array}{l}
c_{AB} \\
c_{CD}\\
\end{array}
\right)
+
\left(
\begin{array}{ll}
n_l(kr)+i j_l(k r) & 0 \\
0 & n_l(qr)+i j_l(q r) \\
\end{array}
\right)
\nonumber \\
\nonumber \\
&\times&
O(W)
\left(
\begin{array}{cc}
e^{i\delta_l^1(W)}\sin \delta_l^1(W) & 0 \\
0 & e^{i\delta_l^2(W)}\sin \delta_l^2(W) \\
\end{array}
\right)
O^{-1}(W)
\left(
\begin{array}{l}
c_{AB} \\
c_{CD}\\
\end{array}
\right) , \\
O(W) &=& \left(
\begin{array}{cc}
\cos\theta(W) & -\sin\theta(W) \\
\sin\theta(W) & \cos\theta(W) \\
\end{array}
\right) ,
\end{eqnarray}
where $\delta_l^i(W)$ is the scattering phase shift , whereas $\theta(W)$ is the mixing angle.
This expression shows that the NBS wave functions for large $r$ agree with scattering waves described by two scattering phases $\delta_l^i(W)$ ($i=1,2$) and one mixing angle $\theta(W)$.
Because of this property, these wave functions satisfy
\begin{equation}
(\nabla^2 + \mbox{\boldmath $k$}^2 )\varphi_{AB}(\bf r,\mbox{\boldmath $k$}) =0, \quad
(\nabla^2 + \mbox{\boldmath $q$}^2 )\varphi_{CD}(\bf r,\mbox{\boldmath $q$}) =0
\end{equation}
for $r\rightarrow \infty$.
Let us now consider QCD in the finite volume $V$.
In the finite volume, $ \vert AB, W \rangle $ and $ \vert CD, W \rangle $ are no longer eigen-states of the hamiltonian.
True eigenvalues are shifted from $W$ to $W_i = W + O(V^{-1})$ ($i=1,2$).
By diagonalization method in lattice QCD simulations,
it is relatively easy to determine $W_1$ and $W_2$. With these values L\"uscher's finite volume formula gives two conditions, which,
however, are insufficient to determine three observables,
$\delta_l^1$, $\delta_l^2$ and $\theta$. (See \cite{Liu:2005kr, Lage:2009zv,Bernard:2010fp} for recent proposals to overcome this difficulty.)
An alternative approach to extract three observables, $\delta_l^1$, $\delta_l^2$ and $\theta$, has been proposed in lattice QCD through the above NBS wave functions\cite{Ishii:2011,HALQCD:2011}.
We consider the NBS wave functions at two different values of energy, $W_1$ and $W_2$,
in the finite volume:
\begin{eqnarray}
\varphi_{AB}(\bf r,\mbox{\boldmath $k$}_i )e^{-W_i t} &=& \langle 0 \vert T\{ \varphi_A(\mbox{\boldmath $x$}+\bf r,t) \varphi_B(\mbox{\boldmath $x$},t) \}\vert W_i \rangle \\
\varphi_{CD}(\bf r,\mbox{\boldmath $q$}_i )e^{-W_i t} &=& \langle 0 \vert T\{ \varphi_C(\mbox{\boldmath $x$}+\bf r,t) \varphi_D(\mbox{\boldmath $x$},t) \}\vert W_i \rangle, \quad i=1,2 .
\end{eqnarray}
We then define the coupled channel non-local potentials from the coupled channel Schr\"odinger equation as
\begin{eqnarray}
\left[\frac{k_i^2}{2\mu_{AB}} - H_0\right] \varphi_{AB}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_i) &=&
\int d^3 y\ U_{AB,AB}(\mbox{\boldmath $x$};\mbox{\boldmath $y$})\ \varphi_{AB}(\mbox{\boldmath $y$},\mbox{\boldmath $k$}_i)+ \int d^3 y\ U_{AB,CD}(\mbox{\boldmath $x$};\mbox{\boldmath $y$})\ \varphi_{CD}(\mbox{\boldmath $y$},\mbox{\boldmath $q$}_i)\nonumber \\
\\
\left[\frac{q_i^2}{2\mu_{CD}} - H_0\right] \varphi_{CD}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_i) &=&
\int d^3 y\ U_{CD,AB}(\mbox{\boldmath $x$};\mbox{\boldmath $y$})\ \varphi_{AB}(\mbox{\boldmath $y$},\mbox{\boldmath $k$}_i)+ \int d^3 y\ U_{CD,CD}(\mbox{\boldmath $x$};\mbox{\boldmath $y$})\ \varphi_{CD}(\mbox{\boldmath $y$},\mbox{\boldmath $q$}_i) \nonumber \\
\end{eqnarray}
for $i=1,2$.
As before the velocity expansion is introduced as
\begin{eqnarray}
U_{XY,VZ} (\mbox{\boldmath $x$};\mbox{\boldmath $y$}) &=& V_{XY,VZ}(\mbox{\boldmath $x$},\nabla)\delta^3(\mbox{\boldmath $x$}-\mbox{\boldmath $y$})
= \left[V_{XY,VZ}(\mbox{\boldmath $x$}) + O(\nabla)\right]\delta^3(\mbox{\boldmath $x$}-\mbox{\boldmath $y$})
\end{eqnarray}
and at the leading order of the expansion, we have
\begin{eqnarray}
K_{AB}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_i)\equiv \left[\frac{k_i^2}{2\mu_{AB}} - H_0\right] \varphi_{AB}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_i) &=&
V_{AB,AB}(\mbox{\boldmath $x$})\ \varphi_{AB}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_i)+ V_{AB,CD}(\mbox{\boldmath $x$})\ \varphi_{CD}(\mbox{\boldmath $x$},\mbox{\boldmath $q$}_i)\nonumber \\
\\
K_{CD}(\mbox{\boldmath $x$},\mbox{\boldmath $q$}_i)\equiv \left[\frac{q_i^2}{2\mu_{CD}} - H_0\right] \varphi_{CD}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_i) &=&
V_{CD,AB}(\mbox{\boldmath $x$})\ \varphi_{AB}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_i)+ V_{CD,CD}(\mbox{\boldmath $x$})\ \varphi_{CD}(\mbox{\boldmath $x$},\mbox{\boldmath $q$}_i). \nonumber \\
\end{eqnarray}
These equations for $i=1,2$ can be solved as
\begin{eqnarray}
\left(
\begin{array}{ll}
V_{AB,AB}(\mbox{\boldmath $x$}) & V_{AB,CD}(\mbox{\boldmath $x$}) \\
V_{CD,AB}(\mbox{\boldmath $x$}) & V_{CD,CD}(\mbox{\boldmath $x$})\\
\end{array}
\right) &=&
\left(
\begin{array}{ll}
K_{AB}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_1) & K_{AB}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_2)\\
K_{CD}(\mbox{\boldmath $x$},\mbox{\boldmath $q$}_1) & K_{CD}(\mbox{\boldmath $x$},\mbox{\boldmath $q$}_2)\\
\end{array}
\right) \nonumber \\
&\times &
\left(
\begin{array}{ll}
\varphi_{AB}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_1) & \varphi_{AB}(\mbox{\boldmath $x$},\mbox{\boldmath $k$}_2) \\
\varphi_{CD}(\mbox{\boldmath $x$},\mbox{\boldmath $q$}_1) & \varphi_{CD}(\mbox{\boldmath $x$},\mbox{\boldmath $q$}_2) \\
\end{array}
\right)^{-1}.
\end{eqnarray}
Once we obtain the coupled channel local potentials $V_{XY, VZ}(\mbox{\boldmath $x$})$, we solve the coupled channel Schr\"odinger equation in {\it infinite} volume with some appropriate boundary condition such that the incoming wave has a definite $\ell$ and consists of the $AB$ state only , in order to extract
three observables for each $\ell$ ($\delta_l^1(W)$, $\delta_l^2(W)$ and $\theta(W)$)
at all values of $W$.
Of course, since $V_{XY,VZ}$ is the leading order approximation in the velocity expansion of $U_{XY,VZ}(\mbox{\boldmath $x$};\mbox{\boldmath $y$})$, results for three observables $\delta_l^1(W)$, $\delta_l^2(W)$ and $\theta(W)$ at $W\not=W_1, W_2$ are also approximate ones and might be different from the exact values. By performing an additional extraction of $V_{XY, VZ}(\mbox{\boldmath $x$})$ at $(W_3,W_4)\not=( W_1,W_2)$, we can test how good the leading order approximation is.
The method considered above can be generalized to inelastic scattering where a number of particles is not conserved. For illustration, let us consider the scattering $A+B\rightarrow A+B$ and $A+B\rightarrow A+B+C$ where the total energy $W$ satisfies
$ m_A+m_B+m_C < W < m_A+m_B+2m_C$.
The following NBS wave functions at the center of mass system are used:
\begin{eqnarray}
\varphi_{AB}^W(\mbox{\boldmath $x$})e^{-Wt} &=& \langle 0 \vert T\{\varphi_A(\bf r+\mbox{\boldmath $x$},t)\varphi_B(\bf r,t)\}\vert W \rangle \\
\varphi_{ABC}^W(\mbox{\boldmath $x$},\mbox{\boldmath $y$})e^{-Wt} &=& \langle 0 \vert T\{\varphi_A\left(\bf r+\mbox{\boldmath $x$}+\frac{\mbox{\boldmath $y$}\,\mu_{BC}}{m_C},t\right)\varphi_B(\bf r+\mbox{\boldmath $y$},t)\varphi_C(\bf r,t)\}\vert W \rangle ,
\end{eqnarray}
where
\begin{eqnarray}
\vert W \rangle &=& c_1\ \vert \mbox{\boldmath $k$} \rangle_{\rm in}\otimes \vert -\mbox{\boldmath $k$} \rangle_{\rm in} +
c_2\ \vert \mbox{\boldmath $q$}_x \rangle_{\rm in}\otimes \left\vert \mbox{\boldmath $q$}_y-\frac{\mbox{\boldmath $q$}_x\mu_{BC}}{m_C} \right\rangle_{\rm in}\otimes \left\vert -\mbox{\boldmath $q$}_y- \frac{\mbox{\boldmath $q$}_x\mu_{BC}}{m_B}\right\rangle_{\rm in}
\end{eqnarray}
with
\begin{eqnarray}
W &=& \sqrt{\mbox{\boldmath $k$}^2+m_A^2} +\sqrt{\mbox{\boldmath $k$}^2+m_B^2} \nonumber \\
&=& \sqrt{\mbox{\boldmath $q$}_x^2+m_A^2}+\sqrt{(\mbox{\boldmath $q$}_y-\frac{\mbox{\boldmath $q$}_x\mu_{BC}}{m_C})^2+m_B^2} +\sqrt{(\mbox{\boldmath $q$}_y+\frac{\mbox{\boldmath $q$}_x\mu_{BC}}{m_B})^2+m_C^2}
\end{eqnarray}
and $1/\mu_{AB} =1/m_B + 1/m_C$. Here $\mbox{\boldmath $y$} = \bf r_B-\bf r_C$ is a relative coordinate between
$B$ and $C$ with the reduced mass $\mu_{BC}$, while $\mbox{\boldmath $x$} =\bf r_A-{\bf R}_{BC}$ is the one between $A$ and the center of mass of $B$ and $C$ with ${\bf R}_{BC} = (m_B\bf r_B + m_C \bf r_C)/(m_B+ m_C)$.
We define the non-local potential from the coupled channel equations as
\begin{eqnarray}
K_{AB}^W(\mbox{\boldmath $x$})&\equiv& \left[\frac{\mbox{\boldmath $k$}^2}{2\mu_{AB}}-H_0^{AB}\right] \varphi_{AB}^W(\mbox{\boldmath $x$})=
\int d^3\,z\ U_{AB,AB}(\mbox{\boldmath $x$};\mbox{\boldmath $z$})\, \varphi_{AB}^W(\mbox{\boldmath $z$}) \nonumber \\
&+& \int d^3\,z\ d^3\,w\ U_{AB,ABC}(\mbox{\boldmath $x$};\mbox{\boldmath $z$},\mbox{\boldmath $w$})\, \varphi_{ABC}^W(\mbox{\boldmath $z$},\mbox{\boldmath $w$})\\
K_{ABC}^W(\mbox{\boldmath $x$},\mbox{\boldmath $y$})&\equiv&\left[\frac{\mbox{\boldmath $q$}_x^2}{2\mu_{A,BC}} + \frac{\mbox{\boldmath $q$}_y^2}{2\mu_{BC}} -H_0^{A,BC} - H_0^{BC}\right]
\varphi_{ABC}^W(\mbox{\boldmath $x$},\mbox{\boldmath $y$})=
\int d^3\,z\ U_{ABC,AB}(\mbox{\boldmath $x$},\mbox{\boldmath $y$};\mbox{\boldmath $z$})\nonumber \\
&\times& \varphi_{AB}^W(\mbox{\boldmath $z$})\nonumber
+ \int d^3\,z\ d^3\,w\ U_{ABC,ABC}(\mbox{\boldmath $x$},\mbox{\boldmath $y$};\mbox{\boldmath $z$},\mbox{\boldmath $w$})\, \varphi_{ABC}^W(\mbox{\boldmath $z$},\mbox{\boldmath $w$})
\end{eqnarray}
where
\begin{eqnarray}
H_0^{AB}&=&-\frac{\nabla_{\mbox{\scriptsize \boldmath $x$}}^2}{2\mu_{AB}}, \quad
H_0^{A,BC}=-\frac{\nabla_{\mbox{\scriptsize \boldmath $x$}}^2}{2\mu_{A,BC}}, \quad
H_0^{BC}=-\frac{\nabla_{\mbox{\scriptsize \boldmath $y$}}^2}{2\mu_{BC}}
\end{eqnarray}
with another reduced mass defined by $1/\mu_{A,BC}=1/m_A+1/(m_B+m_C)$.
We consider the following velocity expansions
\begin{eqnarray}
U_{AB,AB} (\mbox{\boldmath $x$}; \mbox{\boldmath $z$}) &=& \left[V_{AB,AB}(\mbox{\boldmath $x$}) + O(\nabla^x)\right]\delta^3(\mbox{\boldmath $x$}-\mbox{\boldmath $z$}) \\
U_{AB,ABC} (\mbox{\boldmath $x$}; \mbox{\boldmath $z$},\mbox{\boldmath $w$}) &=& \left[V_{AB,ABC}(\mbox{\boldmath $x$},\mbox{\boldmath $w$}) + O(\nabla^x)\right]\delta^3(\mbox{\boldmath $x$}-\mbox{\boldmath $z$}) \\
U_{ABC,AB} (\mbox{\boldmath $x$},\mbox{\boldmath $y$}; \mbox{\boldmath $z$}) &=& \left[V_{AB,ABC}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) + O(\nabla^x)\right]\delta^3(\mbox{\boldmath $x$}-\mbox{\boldmath $z$})\\
U_{ABC,ABC} (\mbox{\boldmath $x$},\mbox{\boldmath $y$}; \mbox{\boldmath $z$},\mbox{\boldmath $w$}) &=& \left[V_{ABC,ABC}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) + O(\nabla^x,\nabla^y)\right]\delta^3(\mbox{\boldmath $x$}-\mbox{\boldmath $z$}) \delta^3(\mbox{\boldmath $y$}-\mbox{\boldmath $w$}) ,
\end{eqnarray}
where the hermiticity of the non-local potentials gives $V_{AB,ABC}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) = V_{ABC,AB}(\mbox{\boldmath $x$},\mbox{\boldmath $y$})$.
At the leading order of the velocity expansions, the coupled channel equations become
\begin{eqnarray}
K_{AB}^W(\mbox{\boldmath $x$}) &=& V_{AB,AB}(\mbox{\boldmath $x$}) \varphi_{AB}^W(\mbox{\boldmath $x$}) + \int d^3\,w\ V_{AB,ABC}(\mbox{\boldmath $x$},\mbox{\boldmath $w$})\varphi_{ABC}^W(\mbox{\boldmath $x$},\mbox{\boldmath $w$} ) \\
K_{ABC}^W(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) &=& V_{ABC,AB}(\mbox{\boldmath $x$},\mbox{\boldmath $y$})\varphi_{AB}^W(\mbox{\boldmath $x$}) + V_{ABC,ABC}(\mbox{\boldmath $x$},\mbox{\boldmath $y$})\varphi_{ABC}^W(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) .
\end{eqnarray}
By considering two values of energy such that $W=W_1, W_2$,
we can determine $V_{ABC,AB}$ and $V_{ABC,ABC}$ from the second equation as
\begin{eqnarray}
\left(
\begin{array}{ll}
V_{ABC,AB}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) & V_{ABC,ABC}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) \\
\end{array}
\right) &=&
\left(
\begin{array}{ll}
K_{ABC}^{W_1}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) & K_{ABC}^{W_2}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) \\
\end{array}
\right) \nonumber \\
&\times&
\left(
\begin{array}{ll}
\Psi_{AB}^{W_1}(\mbox{\boldmath $x$}) & \Psi_{AB}^{W_2}(\mbox{\boldmath $x$}) \\
\Psi_{ABC}^{W_1}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) & \Psi_{ABC}^{W_2}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) \\
\end{array}
\right)^{-1} .
\end{eqnarray}
Using the hermiticity relation $V_{AB,ABC}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) = V_{ABC,AB}(\mbox{\boldmath $x$},\mbox{\boldmath $y$})$, we can extract $V_{AB,AB}$ from the first equation as
\begin{eqnarray}
V_{AB,AB}(\mbox{\boldmath $x$}) &=&\frac{1}{\Psi_{AB}^W(\mbox{\boldmath $x$})}\left[
K_{AB}^{W}(\mbox{\boldmath $x$}) -\int d^3\,w\ V_{ABC,AB}(\mbox{\boldmath $x$},\mbox{\boldmath $w$})\Psi_{ABC}^W(\mbox{\boldmath $x$},\mbox{\boldmath $w$} )
\right]
\end{eqnarray}
for $W=W_1, W_2$. A difference of $V_{AB,AB}(\mbox{\boldmath $x$})$ between two estimates at $W_1$ and $W_2$ gives an estimate for higher order contributions in the velocity expansions.
Once we obtain $V_{AB,AB}$, $V_{AB,ABC}=V_{ABC,AB}$ and $V_{ABC,ABC}$, we can solve the coupled channel Schr\"odinger equations in the {\it infinite} volume, in order to extract physical observables.
As $W$ increases and becomes larger than $m_A+m_B + n m_C$, the inelastic scattering $A+B\rightarrow A+B+n C $ becomes possible. As in the case of $A+B\rightarrow A+B+C$ in the above, we can define the coupled channel potentials including this channel, though calculations of the NBS wave functions for multi-hadron operators become more and more difficult in practice.
\subsection{Coupled channels with $S=-2$ and $I=0$}
\label{sec:coupled_channel}
\begin{table}[tb]
\begin{center}
\begin{tabular}{|c|c|cccccc|}
\hline \hline
& $N_{conf}$ & $m_\pi$ & $m_K$ & $m_N$ & $m_\Lambda$ & $m_\Sigma$ & $m_\Xi$ \\
\hline
Set 1 & $700$ & $875(1)$ & $916(1)$ & $1806(3)$ & $1835(3)$ & $1841(3)$ & $1867(2)$ \\
Set 2 & $800$ & $749(1)$ & $828(1)$ & $1616(3)$ & $1671(2)$ & $1685(2)$ & $1734(2)$ \\
Set 3 & $800$ & $661(1)$ & $768(1)$ & $1482(3)$ & $1557(3)$ & $1576(3)$ & $1640(3)$ \\
\hline \hline
\end{tabular}
\caption{Hadron masses in units of [MeV] and number of configurations for each set.}
\label{tab:hadron_2+1}
\end{center}
\end{table}
As an application of the method in the previous subsection, let us consider $BB$ potentials for the $S=-2$ and $I=0$ channel, which consist of the $\Lambda\Lambda$, $N\Xi$ and $\Sigma\Sigma$ components in terms of low-lying octet baryons. Mass differences of these components are quite small such that $2m_\Lambda = 2232$ Mev, $m_N+m_\Sigma = 2257$ MeV and $2m_\Sigma = 2386$ MeV. Using diagonalized source operators, the NBS wave functions at three different values of energy,
\begin{equation}
\varphi_{AB}^{W_i}(\bf r,\mbox{\boldmath $k$}_{AB}^i)e^{-W_i t} = \langle 0 \vert T\{ \varphi_A(\bf r+\mbox{\boldmath $x$},t)\varphi_B(\bf r,t)\}\vert W_i \rangle
\end{equation}
for $i=0,1,2$, are extracted in lattice QCD simulations,
where $ AB = \Lambda\Lambda$, $N\Xi$ and $\Sigma\Sigma$, and $\mbox{\boldmath $k$}_{AB}^i$ satisfies
\begin{equation}
W_i =\sqrt{(\mbox{\boldmath $k$}_{AB}^i)^2+ m_A^2} + \sqrt{(\mbox{\boldmath $k$}_{AB}^i)^2+ m_B^2} .
\end{equation}
Using the notation
\begin{equation}
K_{AB}^i ( \bf r) = \frac{1}{2\mu_{AB}}\left(\nabla^2+(\mbox{\boldmath $k$}_{AB}^i)^2\right)\varphi_{AB}^{W_i}(\bf r,\mbox{\boldmath $k$}_{AB}^i)
\end{equation}
where $1/\mu_{AB}=1/m_A + 1/m_B$,
the coupled channel $3\times 3$ potential matrix is given by
\begin{equation}
V_{AB,CD}(\bf r) = \sum_i K_{AB}^i (\bf r) \left[\varphi_{CD}^{W_i}(\bf r,\mbox{\boldmath $k$}_{CD}^i)\right]^{-1} .
\end{equation}
Here the last factor is the inverse of the $3\times 3$ matrix $\varphi_{CD}^{W_i}(\bf r,\mbox{\boldmath $k$}_{CD}^i)$
with indices $i$ and $CD$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\textwidth]{Figs/all_d_pb.eps}
\includegraphics[width=\textwidth]{Figs/all_nd_pb.eps}
\end{center}
\caption{The coupled channel potential matrix from the NBS wave function for Set 1. The vertical axis is the potential strength in units of [MeV], while the horizontal axis is the relative distance between two baryons in units of [fm]. Taken from Ref.~\protect\cite{Sasaki:2010bh}. }
\label{FIG:PTpbSet1}
\end{figure}
Gauge configurations generated on a $16^3\times 32$ lattice at $a\simeq 0.12$ fm ( therefore $L\simeq 1.9$ fm) in 2+1-flavor full QCD simulation are employed to calculate the coupled channel potentials at three different values of the light quark mass with the fixed bare strange quark mass\cite{Sasaki:2010bh}. Quark propagators are calculated with the spatial wall source at $t_0$ with the Dirichlet boundary condition in time at $t=t_0+16$. The wall source is placed at 16 different time slices on each gauge configuration, in order to enhance signals, together with the average over forward and backward propagations in time.
Corresponding hadron masses and number of gauge configurations are given in table~\ref{tab:hadron_2+1}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\textwidth]{Figs/potall.eps}
\end{center}
\caption{Transition potentials in the flavor SU(3) IR basis. Red, blue and green symbols correspond to results of Set1, Set2 and Set3, respectively. The result of the flavor SU(3) symmetric limit at the same strange quark mass is also plotted with brown symbols~\protect\cite{Inoue:2010hs}. Taken from Ref.~\protect\cite{Sasaki:2010bh}.}
\label{FIG:potall}
\end{figure}
The coupled channel potential matrix $V_{AB,CD}$ from the NBS wave function for Set 1 is shown in Figure~\ref{FIG:PTpbSet1}.
The flavor dependence of the height of the repulsive core at short distance region is observed. In particular,
the $\Sigma \Sigma$ potential has the strongest repulsive core of these three channels.
It is interesting to see that off-diagonal parts of the potential matrix roughly satisfy the hermiticity relation $V_{AB,CD} = V_{CD,AB}$ within statistical errors.
In addition the off-diagonal parts are similar in magnitude for
$V_{\Lambda \Lambda,\Sigma \Sigma}$ and $V_{N \Xi,\Sigma \Sigma}$ with the diagonal parts, but $V_{\Lambda \Lambda,N \Xi}$ is much smaller.
In order to compare the results of the potential matrix calculated in three configuration sets, the potentials from the particle basis are transformed to those in the flavor SU(3) irreducible representation (IR) basis as
\begin{eqnarray}
V^{IR} = U^\dagger V U =
\left( \begin{array}{ccc}
V_{1,1} & V_{1,8} & V_{1,27} \\
V_{8,1} & V_{8,8} &V_{8,27} \\
V_{27,1} & V_{27,8} & V_{27,27} \\
\end{array} \right)
\end{eqnarray}
where $U$ is a unitary transformation matrix whose explicit form is given in Appendix~\ref{app:octet}. The potential matrix in the IR basis is convenient and a good measure of SU(3) breaking effects by comparing three configuration sets since it is diagonal in the SU(3) symmetric limit.
In Figure~\ref{FIG:potall}, the results of the potential matrix in the IR basis are compared between different configuration sets, together with the one in the flavor SU(3) symmetric limit.
As the pion mass decreases, the repulsive core in the $V_{27,27}$ potential increases.
The $V_{1,27}$ and $V_{8,27}$ transition potentials are consistent with zero within statistical errors.
On the other hand, it is noteworthy that the flavor SU(3) symmetry breaking effect becomes manifest in the $V_{1,8}$ transition potential.
\subsection{Time dependent method}
One of the practical difficulties to extract the NBS wave function and the potential from
the correlation function Eq.(\ref{eq:4-pt}) is to achieve the ground state saturation in numerical simulations at large but finite $t-t_0$ with reasonably small statistical errors.
While the stability of the potential against $t-t_0$ has been confirmed within statistical errors in numerical simulations reviewed in the report, the determination of $W$ for the ground state suffers from systematic errors due to contaminations of possible excited states, which can be seen as follows. There exist 3 different methods to determine $W$. The most well-known method is to determine $W$ from the $t-t_0$ dependence of the correlation function eq.(\ref{eq:4-pt}) summed over $\bf r$ to pick up the zero momentum state. On the other hand, one can determine $\mbox{\boldmath $k$}^2$ of $W$ by fitting the $\bf r$ dependence of the NBS wave function with its expected asymptotic behavior at large $r$ or by reading off the constant shift of the Laplacian part of the potential from zero at large $r$.
Although the latter two methods usually give consistent results within statistical errors, the first method (the $t$ dependence method) sometimes leads to a result different from those determined by the latter two (called the $\bf r$ dependent method together) at the value of $t-t_0$ usually employed in numerical simulations. Although, in principle, the increase of $t-t_0$ is needed in order to see an agreement between $t$ and $\bf r$ dependence methods, it is difficult in practice due to larger statistical errors at larger $t-t_0$ for the two-baryon system.
In order to overcome this practical difficulty, the method to extract the potential from the NBS wave function has been modified as follows. Let us consider the correlation function Eq.(\ref{eq:4-pt}) again:
\begin{equation}
F(\bf r,t) = \sum_{W\le W_{\rm th} } A_W \phi^W(\bf r ) e^{-W t} + O(e^{-W_{\rm th}t}) .
\end{equation}
If $t$ is large enough so that contributions from $O(e^{-W_{\rm th}t})$ terms can be neglected\footnote{This limitation for $t$ can be loosened if the coupled channel potentials are introduced as in the previous subsections.}, we have
\begin{equation}
H_0 F(\bf r, t) \simeq \sum_W A_W\int d^3\bf r^\prime [E_W\delta^{(3)}(\bf r-\bf r^\prime) - U(\bf r,\bf r^\prime)]\varphi^W(\bf r^\prime) e^{-W t}
\end{equation}
where $E_W=k_W^2/(2\mu) = (W^2-4m_N^2)/(4m_N)$ with $\mu=m_N/2$. By using the non-relativistic approximation that $W=2\sqrt{k_W^2+m_N^2} =2m_N + k_W^2/m_N + O(k_W^4/m_N^3)$,
\begin{equation}
\left[ H_0 +\frac{d}{d t} +2m_N\right] F(\bf r, t) =- \int d^3\bf r^\prime U(\bf r,\bf r^\prime) F(\bf r, t)
\simeq -V^{\rm LO}(\bf r) F(\bf r, t)
\end{equation}
where the velocity expansion is introduced in the last line and higher other than the leading order terms are then omitted. The leading order potential is therefore given by
\begin{equation}
V^{\rm LO}(\bf r) = -\frac{\left[ H_0 +\frac{d}{d t} +2m_N\right] F(\bf r, t) }{F (\bf r, t)}
\end{equation}
or
\begin{equation}
V^{\rm LO}(\bf r) = -\frac{\left[ H_0 +\frac{d}{d t} \right] R(\bf r, t) }{R (\bf r, t)}
\end{equation}
where $R(\bf r, t) = F (\bf r, t)/e^{-2 m_N t}$. Here it is assumed that $O(e^{-W_{\rm th} t})$ contributions can be neglected at large $t$.
The non-relativistic formula for $V^{\rm LO}(r)$ above can be easily generalized to the case that masses of two particles are different, by the replacement that $R(\bf r, t) = F (\bf r, t)/e^{- (m_A+m_B) t}$.
Note also that the potential extracted in this method automatically satisfies that $V^{\rm LO}(r) \rightarrow 0$ as $r\rightarrow 0$ without the constant shift.
This property may be used to check whether this extraction works correctly or not.
On the lattice, the $t$ derivative should be approximated by the $t$ difference. In practice, one may adopt a particular method for the $t$ difference,
in order to reduce statistical as well as systematic errors for $V^{\rm LO}(\bf r)$.
The non-relativistic approximation can be removed by using the second order derivative in $t$ as
\begin{equation}
V^{\rm LO}(\bf r) =\frac{ \displaystyle \left[ -H_0 +\frac{1}{4m_N}\frac{d^2}{d t^2} -m_N\right] F(\bf r, t)}{ F (\bf r, t)} ,
\end{equation}
as long as $O(e^{-W_{\rm th} t})$ contributions are negligibly small. For this method to apply, two particles should have the same mass.
Statistical errors of the second order difference on the lattice must be kept small in numerical simulations.
One may introduce a more general correlation function as
\begin{equation}
F(\mbox{\boldmath $x$} , \mbox{\boldmath $y$}, t) = \int d^3\mbox{\boldmath $x$}_1d^3\mbox{\boldmath $y$}_1\langle 0 \vert T\{N(\mbox{\boldmath $x$}_1+\mbox{\boldmath $x$},t) N(\mbox{\boldmath $x$}_1,t)\} T\{\overline{N}(\mbox{\boldmath $y$}_1+\mbox{\boldmath $y$},0) \overline{N}(\mbox{\boldmath $y$}_1,0)\} \vert 0\rangle .
\end{equation}
Using this new quantity, we have
\begin{equation}
\left[ H_0 -\frac{1}{4m_N}\frac{d^2}{d t^2} +m_N\right] R(\mbox{\boldmath $x$},\mbox{\boldmath $y$}, t) = -\int d^3\,\mbox{\boldmath $z$} U(\mbox{\boldmath $x$},\mbox{\boldmath $z$}) F(\mbox{\boldmath $z$}, \mbox{\boldmath $y$},t),
\end{equation}
from which the non-local potential is extracted as
\begin{equation}
U(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) = \int d^3\mbox{\boldmath $z$} \left[- H_0 +\frac{1}{4m_N}\frac{d^2}{d t^2} -m_N\right] F(\mbox{\boldmath $x$},\mbox{\boldmath $z$}, t) \cdot
\tilde F^{-1}(\mbox{\boldmath $z$},\mbox{\boldmath $y$}, t).
\end{equation}
Here $\tilde F^{-1}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}, t)$ is the approximated inverse of the hermitian operator $F (\mbox{\boldmath $x$},\mbox{\boldmath $y$}, t)$, defined by
\begin{equation}
\tilde F^{-1}(\mbox{\boldmath $x$},\mbox{\boldmath $y$}, t) =\sum_{\lambda_n\not= 0 } \frac{1}{\lambda_n(t)} v_n(\mbox{\boldmath $x$}, t) v_n^\dagger(\mbox{\boldmath $y$},t)
\end{equation}
where $\lambda_n(t)$ and $v(\mbox{\boldmath $x$},t)$ are an eigenvalue and a corresponding eigenvector of $F (\mbox{\boldmath $x$},\mbox{\boldmath $y$}, t)$, respectively, and zero eigenvalues are removed in the summation.
Since the modified potential
\begin{equation}
\hat U(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) = U(\mbox{\boldmath $x$},\mbox{\boldmath $y$}) + \sum_{\lambda_n=0} c_n v_n(\mbox{\boldmath $x$}, t) v_n^\dagger(\mbox{\boldmath $y$},t)
\end{equation}
also satisfies the same Schr\"odinger equation $\forall\{c_n \}$, the non-local potential is NOT unique, and $U(\mbox{\boldmath $x$},\mbox{\boldmath $y$})$ is scheme dependent, as discussed before.
\subsection{Bound $H$ dibaryon in flavor SU(3) limit}
As an application of the method in the previous subsection, let us consider the singlet potential in the flavor SU(3) limit in order to investigate whether the bound $H$ dibaryon exists or not in this case.
At the leading order of the velocity expansion, the central potential is defined in this method by
\begin{equation}
V_C^{(X)} (r) = -\frac{ \left[H_0 +\displaystyle \frac{d}{d t}\right] R(\bf r,t-t_0)}{ R(\bf r,t-t_0)} ,
\end{equation}
which is calculated on $16^3\times 32$, $24^3\times 32$ and $32^3\times 32$ lattices at $a=0.121(2)$ fm and three values of the quark mass, where the PS meson mass and the octet baryon mass are given by $(m_{\rm PS}, m_B) = (1015(1){\rm MeV}, 2030(2){\rm MeV} )$, $(837(1){\rm MeV}, 1748(1){\rm MeV} )$ and $(673(1){\rm MeV}, 1485(2){\rm MeV} )$ on a $32^3\times 32$ lattice\cite{Inoue:2010es}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figs/volume_vr_27_K13710_dark.eps}
\end{center}
\vspace{-0.5cm}
\caption{The flavor 27-plet potential $V_C^{(27)}(r)$ obtained for $L=1.94,\, 2.90,\, 3.87$ fm
at $m_{\rm ps}=1015$ MeV and $(t-t_0)/a = 10$. Taken from Ref.~\protect\cite{Inoue:2010es}.
}
\label{fig:27-plet}
\end{figure}
To check the qualitative consistency with previous results, the central potential in the 27-plet channel is plotted in Fig.~\ref{fig:27-plet} obtained in three different lattice volumes with $L=1.94,\,2.90,\,3.87$ fm at $m_{\rm ps}=1015$ MeV and $(t-t_0)/a=10$.
This is the case corresponding to the spin-singlet NN potential.
Compared with statistical errors, the $L$ dependence is found to be negligible.
The $t$ dependence is also small as long as $(t-t_0)/a \geq 9$.
As expected, the potential approaches zero automatically for large $r$. The figure shows a repulsive core at short distance surrounded by an attractive well at medium and long distances, which is qualitatively consistent with previous results in quenched and full QCD simulations.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figs/volume_vr_1_K13710_wfit_dark.eps}
\includegraphics[width=0.45\textwidth]{Figs/mass_vr_1_wide_dark.eps}
\end{center}
\vspace{-0.5cm}
\caption{ The flavor-singlet potential $V_C^{(1)}(r)$ at $(t-t_0)/a = 10$.
(Left) Results for $L=1.94,\, 2.90,\, 3.87$ fm at $m_{\rm ps}=1015$ MeV.
(Right) Results for $L=3.87$ fm at $m_{\rm ps}=1015,\, 837,\, 673$ MeV.
Taken from Ref.~\protect\cite{Inoue:2010es}. }
\label{fig:singlet}
\end{figure}
Shown in Fig.~\ref{fig:singlet}(Left) and Fig.~\ref{fig:singlet}(Right)
are the volume and the quark mass dependences
of the central potential in the flavor-singlet channel $V_C^{(1)}(r)$, respectively, at $(t-t_0)/a=10$
where the potentials do not have appreciable change with respect to the choice of $t$.
The flavor-singlet potential is shown to have an ``attractive core" and to be well localized in space.
Because of the latter property, no significant volume dependence of the potential is observed within the statistical errors, as seen in Fig.~\ref{fig:singlet}(Left).
As the quark mass decreases in Fig.~\ref{fig:singlet}(Right) , the long range part of the attraction tends to increase.
The resultant potential is fitted by the following analytic function composed of
an attractive Gaussian core plus a long range (Yukawa)$^2$ attraction:
$
V(r) = b_1 e^{-b_2\,r^2} + b_3(1 - e^{-b_4\,r^2})\left( {e^{-b_5\,r}}/{r} \right)^2 .
$
With the five parameters, $b_{1,2,3,4,5}$, the lattice results can be fitted
reasonably well with $\chi^2/{\rm dof} \simeq 1$.
The fitted result for $L=3.87$ fm is shown by the dashed line in Fig.~\ref{fig:singlet}(Left).
Solving the Schr\"{o}dinger equation with the fitted potential in infinite volume,
the energies and the wave functions are obtained at the present quark masses in the flavor SU(3) limit.
It turns out that, at each quark mass, there is only one bound state
with binding energy 30--40 MeV.
In Fig.~\ref{fig:H-dibaryon}(Left),
the energy and the root-mean-squared (rms) distance
of the bound state are plotted in the case of
$(t-t_0)/a=9, 10, 11$ at $m_{\rm ps}=673$ MeV and $L=3.87$ fm, where
errors are estimated by the jackknife method.
Although the statistical error increases as $t$ increases,
we observe small changes of central values, which are considered as the systematic errors.
Fig.~\ref{fig:H-dibaryon}(Right) shows the energy and the rms distance of the bound state at each quark mass obtained from the potential at $L=3.87$ fm and $(t-t_0)/a=10$.
Despite the fact that the potential has quark mass dependence,
the resultant binding energies of the $H$-dibaryon are insensitive in the present range of the quark masses.
This is due to the fact that the increase of the attraction toward the lighter quark
mass is partially compensated by the increase of the kinetic energy for the lighter baryon mass.
It is noted that there appears no bound state for the potential of the 27-plet channel
in the present range of the quark masses.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.45\textwidth]{Figs/hdibaryon_K13800_L32_new_dark.eps}
\vspace{0.3cm}
\includegraphics[width=0.45\textwidth]{Figs/hdibaryon_mass_dark.eps}
\end{center}
\vspace{-0.5cm}
\caption{The bound state energy $E_0\equiv -B_H$ and the rms distance $\sqrt{\langle r^2\rangle}$ of the $H$-dibaryon
obtained from the potential at $L=3.87$ fm.
(Left) $(t-t_0)/a$ dependence at $m_{\rm ps}=673$ MeV.
(Right) Quark mass dependence at $(t-t_0)/a=10$. Taken from Ref.~\protect\cite{Inoue:2010es}.}
\label{fig:H-dibaryon}
\end{figure}
The final results of the binding energy $B_H$ and the rms distance $\sqrt{\langle r^2\rangle}$
are summarized below, where the 1st and 2nd parentheses correspond to statistical errors
and systematic errors from the $t$-dependence, respectively.
\begin{eqnarray}
m_{ps}=1015 ~\mbox{MeV} :~ \ B_H &=& 32.9 (4.5)(6.6) ~\mbox{MeV} \nonumber \\
~~ \sqrt{\langle r^2 \rangle} &=& 0.823(33)(40) ~\mbox{fm} \nonumber \\
m_{ps}=~837 ~\mbox{MeV} :~ \ B_H &=& 37.4 (4.4)(7.3) ~\mbox{MeV} \nonumber \\
~~ \sqrt{\langle r^2 \rangle} &=& 0.855(29)(61) ~\mbox{fm} \nonumber \\
m_{ps}=~673 ~\mbox{MeV} :~ \ B_H &=& 35.6 (7.4)(4.0) ~\mbox{MeV}
\nonumber \\
~~ \sqrt{\langle r^2 \rangle} &=& 1.011(63)(68) ~\mbox{fm}
\nonumber
\end{eqnarray}
Since the binding energy is insensitive to the quark masses,
there may be a possibility of weakly bound or resonant $H$-dibaryon even in the real world with lighter quark masses and the flavor SU(3) breaking. To make a definite conclusion on this point,
the $\Lambda\Lambda-N\Xi-\Sigma\Sigma$ coupled channel analysis is necessary
for $H$ in the (2+1)-flavor lattice QCD simulations, as discussed in Sec.~\ref{sec:coupled_channel}.
Recently the existence of $H$-dibaryon is also suggested by a direct calculation of its binding energy in 2+1 full QCD simulations\cite{Beane:2010hg}, where $B_H=16.6(2.1)(4.6)$ MeV is reported in the $L\rightarrow \infty$ extrapolation at $m_\pi \simeq 389$ MeV .
\section{Other applications}
\label{sec:others}
In the last section, some other applications of the potential method are reviewed.
\subsection{Three nucleon force}
Recent precise calculations of few-nucleon systems clearly indicate that the 2 nucleon force alone is insufficient to understand the nuclei, which calls for three (and/or more) nucleon forces.
Actually, the three nucleon force (TNF) is supposed to play an important and nontrivial role in various phenomena in nuclear and astrophysics.
For the binding energies of light nuclei, the attractive TNF is required to reproduce the experimental data. On the other hand, the repulsive TNF is necessary to reproduce the empirical saturation density of symmetric nuclear matter. For the equation of state(EoS) of asymmetric nuclear matter, a repulsive TNF is required
to explain the observed maximum neutron star mass.
Pioneered by Fujita-Miyazawa~\cite{Fujita:1957zz},
the TNF has been mainly studied from the two-pion exchange picture
with the $\Delta$-excitation.
In addition, the repulsive TNF is often introduced
phenomenologically~\cite{Pudliner:1995wk}.
Recently, the TNF based on chiral effective field theory is developing~\cite{vanKolck:1994yi},
but the unknown low-energy constants can be obtained
only by the fitting to the experimental data.
Since the TNF originates from the fact that the nucleon is not a fundamental particle,
it is essential to study the TNF from fundamental degrees of freedom(DoF) , i.e., quarks and gluons.
In Ref.~\cite{Doi:2010yh,Doi:2011gq}
such first-principle calculations of the TNF in lattice QCD have been reported.
If the potential method is applied to the three nucleon (3N) system,
a straightforward calculation is impossible due to the significantly enlarged DoF. In Ref.~\cite{Doi:2010yh,Doi:2011gq}, two different approaches have been considered.
The NBS wave function for 3N is defined by
\begin{equation}
\varphi^W(\bf r_{12},\bf r_{123})e^{-Wt} = \langle 0 \vert N(\mbox{\boldmath $x$}_1,t) N(\mbox{\boldmath $x$}_2,t) N(\mbox{\boldmath $x$}_3,t) \vert W \rangle
\end{equation}
where $\bf r_{12}\equiv \mbox{\boldmath $x$}_1-\mbox{\boldmath $x$}_2$, $\bf r_{123}\equiv \mbox{\boldmath $x$}_3-(\mbox{\boldmath $x$}_1+\mbox{\boldmath $x$}_2)/2$ are the Jacobi coordinates, and
$\vert W\rangle $ is the 3N state with energy $W$.
At the leading order of the velocity expansion, the NBS wave function satisfies
\begin{equation}
\left[ -\frac{1}{2\mu_{12}}\nabla^2_{r_{12}} -\frac{1}{2\mu_{123}}\nabla^2_{r_{123}} + \sum_{i<j}
V_{{\rm 2N},ij}(\mbox{\boldmath $x$}_i-\mbox{\boldmath $x$}_j) + V_{\rm TNF}(\bf r_{12},\bf r_{123})
\right] \varphi^W(\bf r_{12},\bf r_{123}) = E \varphi^W(\bf r_{12},\bf r_{123})
\end{equation}
where $V_{{\rm 2N},ij}(\mbox{\boldmath $x$}_i-\mbox{\boldmath $x$}_j) $ denotes the potential between $(i,j)$-pair, $V_{\rm TNF}(\bf r_{12},\bf r_{123})$ the TNF, $\mu_{12}=m_N/2$, $\mu_{123}=2m_N/3$ the reduced masses.
If $\varphi^W(\bf r_{12},\bf r_{123})$ is calculated for all $\bf r_{12},\bf r_{123}$ and all $V_{{\rm 2N},ij}(\mbox{\boldmath $x$}_i-\mbox{\boldmath $x$}_j) $ are available by lattice calculations, $V_{\rm TNF}(\bf r_{12},\bf r_{123})$ can be extracted.
Unfortunately, this is not the case:
Since both $\bf r_{12}$ and $\bf r_{123}$ have $L^3$ DoF,
the calculation cost is more expensive by a factor of $L^3$
compared to the 2N system.
Furthermore, the number of diagrams
to be calculated in the Wick contraction
tends to diverge with a factor of $N_u ! \times N_d !$
($N_{u,d}$ are numbers of u,d quarks in the system).
It is also noted that not all 2N potentials are available in lattice QCD at this moment:
Only parity-even 2N potentials have been obtained so far.
The first method in Ref.~\cite{Doi:2010yh} to avoid these problems is to consider the effective 2N potential in the 3N system by taking the summation over the location of the spectator nucleon $N(\mbox{\boldmath $x$}_3)$,
\begin{equation}
\varphi^W(\bf r_{12}) = \sum_{\mbox{\boldmath $x$}_3}\varphi^W(\bf r_{12},\bf r_{123})= \sum_{\bf r_{123}}\varphi^W(\bf r_{12},\bf r_{123}) .
\end{equation}
The effective potential between $N(\mbox{\boldmath $x$}_1)$ and $N(\mbox{\boldmath $x$}_2)$ is then defined by
\begin{equation}
\left[ -\frac{1}{2\mu_{12}}\nabla^2_{r_{12}} +V_{\rm eff}(\bf r_{12})
\right] \varphi^W(\bf r_{12}) = E \varphi^W(\bf r_{12}) .
\end{equation}
In this calculation, the DoF of $\bf r_{123}$ is integrated out beforehand, and thus
the calculation cost is reduced by a factor of $\sim 1/L^3$, compared to the straightforward calculation. The difference $V_{\rm eff} (\vec{r}) - V_{2N}(\vec{r})$ can be considered to be the ``finite density effect'' in the 3N system. Part of this effect is attributed
to the genuine 2N potential with the nontrivial 3N correlation,
while another originates from the genuine TNF.
The triton channel( $I=1/2$, $J^P= 1/2^+$) is studied as the 3N system.
Since the spectator nucleon is projected to the S-wave, the possible quantum numbers between the (effective) 2N are only $^{2S+1}L_J =$ $^1S_0$, $^3S_1$, $^3D_1$.
Gauge configurations in 2-flavor QCD on a $16^3\times 32$ lattice
at $a\simeq 0.16$ fm\cite{Aoki:2002uc} are employed for the calculation at $m_\pi \simeq 1.13$ GeV and $m_N\simeq 2.15$ GeV.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.32\columnwidth,angle=270]{Figs/pot_3b.all_av.NR__.t_009.r_av.eps}
\includegraphics[width=0.32\columnwidth,angle=270]{Figs/pot_diff.ten_aud.3b_NR__.t_009.2b_NR__.t_009.r_av.eps}
\end{center}
\caption{(Left) Effective 2N potentials,
where red, blue, brown points correspond to
$V_{C,{\rm eff}}^{I=1,S=0}$,
$V_{C,{\rm eff}}^{I=0,S=1}$,
$V_{T,{\rm eff}}^{I=0,S=1}$
potential, respectively.
(Right) The difference between the effective 2N and the genuine 2N
for $V_{T}^{I=0,S=1}$ potential.
Taken from Ref.~\protect\cite{Doi:2010yh}. }
\label{fig:pot_eff2N}
\end{figure}
Fig.~\ref{fig:pot_eff2N}(Left) show results for $V_{\rm eff}(r)$
in the triton channel at $t-t_0=8$, where the constant shift by energy is not included
for the central potentials. It is noteworthy that $V_{\rm eff}(r)$ is obtained with good precision even though the signal to noise ratio is expected to be worse for more quarks in the system.
Fig.~\ref{fig:pot_eff2N}(Right) gives $V_{\rm eff}(r) - V_{\rm 2N}(r)$ for the tensor potential, which is free from the constant shift. The difference is consistent with zero within a few MeV statistical errors. Similar results are reported for central potentials as well in Ref.\cite{Doi:2010yh}.
There is no indication of a TNF effect. A possible explanation is that the TNF effect is suppressed at heavy quark mass. Basically similar results are obtained, however, for lighter pion masses($m_\pi \simeq 0.7$ GeV and 0.57 GeV)\cite{Doi:2010yh}. Another possibility is that the TNF effect is suppressed by the summation over the location of the spectator nucleon.
While the TNF effect is expected to be enhanced when all three nucleons are close to each other, such 3-dimensional spacial configurations have small contributions in the spectator summations.
In order to assess this possibility, a second method has been investigated in Ref.\cite{Doi:2010yh,Doi:2011gq}, where the linear setup with $\bf r_{123}=0$ is used for the 3N wave function.
In this case, the third nucleon is attached to $(1,2)$-nucleon pair with only S-wave.
Considering the total 3N quantum numbers of $I=1/2, J^P=1/2^+$,
the wave function can be completely spanned by only three bases, which can be labeled
by the quantum numbers of $(1,2)$-pair as $^1S_0$, $^3S_1$, $^3D_1$.
Therefore, the Schr\"odinger equation is simplified to
the $3\times 3$ coupled channel equations with the bases of
$\varphi_{^1S_0}$, $\varphi_{^3S_1}$, $\varphi_{^3D_1}$.
Even in this case the subtraction of $V_{\rm 2N}$ remains nontrivial:
the parity-odd potentials, which must be subtracted, are not available in lattice QCD at this moment.
The subtraction problem of parity-odd potentials can be avoided for triton by using the symmetric wave function,
\begin{equation}
\varphi_S \equiv
\frac{1}{\sqrt{6}}
\Big[
- p_{\uparrow} n_{\uparrow} n_{\downarrow} + p_{\uparrow} n_{\downarrow} n_{\uparrow}
- n_{\uparrow} n_{\downarrow} p_{\uparrow} + n_{\downarrow} n_{\uparrow} p_{\uparrow}
+ n_{\uparrow} p_{\uparrow} n_{\downarrow} - n_{\downarrow} p_{\uparrow} n_{\uparrow}
\Big] .
\label{eq:psi_S}
\end{equation}
Combined with the Pauli principle,
it is automatically guaranteed that any 2N-pairs couple with even parity only,
since this wave function is anti-symmetric in spin/isospin spaces for any 2N-pairs.
Therefore the TNF can be extracted unambiguously in this channel,
without the information of parity-odd 2N potentials.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.32\columnwidth,angle=270]{Figs/wave_3b.all_______av.NR__.t_009.r_av.eps}
\includegraphics[width=0.32\columnwidth,angle=270]{Figs/pot_3b.TNR_scalar__av.3b_NR__.t_009.2b_NR__.t_009.r_av.eps}
\end{center}
\caption{(Left) The wave function with linear setup in the triton channel.
Red, blue, brown points correspond to
$\varphi_S$, $\varphi_M$, $\varphi_{^3D_1}$, respectively.
(Right) The scalar/isoscalar TNF in the triton channel,
plotted against the distance $r=\vert \bf r_{12}/2\vert$ in the linear setup.
Taken from Ref.~\protect\cite{Doi:2010yh}. }
\label{fig:TNR}
\end{figure}
The same gauge configurations used for the effective 2N potential study are employed in the numerical simulations.
Fig.~\ref{fig:TNR}(Left) gives each wave function of
$\varphi_S= \frac{1}{\sqrt{2}} ( - \psi_{^1S_0} + \psi_{^3S_1} )$, $\varphi_M\equiv \frac{1}{\sqrt{2}} ( + \psi_{^1S_0} + \psi_{^3S_1} )$, $\psi_{^3D_1}$ as a function of $r=\vert \bf r_{12}/2\vert$ in the triton channel at $t-t_0 = 8$.
Among these three
$\varphi_S$ dominates the wave function, since
$\varphi_S$ contains the component for which
all three nucleons are in S-wave.
By subtracting the $V_{2N}$ from the total potentials in the 3N system,
the TNF is detemined.
Fig.~\ref{fig:TNR} (Right) shows results
for the scalar/isoscalar TNF, where the $r$-independent shift of $V_{2N}$ is not included,
and thus about ${\cal O}(10)$ MeV systematic error is understood.
There are various physical implications in Fig.~\ref{fig:TNR} (Right).
At the long distance region of $r$, the TNF is small as is
expected.
At the short distance region,
the indication of a repulsive TNF is observed.
Recalling that the repulsive short-range TNF is phenomenologically required
to explain the saturation density of nuclear matter, etc.,
this is a very encouraging result.
Of course, further study is necessary to confirm this result,
e.g., the study of the ground state saturation,
the evaluation of the constant shift by energies,
the examination of the discretization error.
\subsection{Meson-baryon interactions}
The potential method can be naturally extended to the meson-baryon systems and the meson-meson systems. In this subsection, two applications of the potential method to the meson-baryon system are discussed.
The first application is the study of the $K N$ interaction in the $I(J^P) =0(1/2^-)$ and $1(1/2^-)$ channels by the potential method. These channels may be relevant for the possible exotic state $\Theta^+$, whose existence is still controversial.
The $KN$ potentials in isospin $I=0$ and $I=1$ channels have been calculated in $2+1$ full QCD simulations, employing 700 gauge configurations on a $16^3\times 32$ lattice at $a=0.121(1)$ fm and $(m_\pi, m_K, m_N)=(871(1),912(2),1796(7))$ in unit of MeV\cite{Ikeda:2010sg}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.45\columnwidth]{Figs/SWV_NK_0.eps}
\includegraphics[width=0.45\columnwidth]{Figs/SWV_pK.eps}
\end{center}
\caption{The NBS wave function of the $KN$ scattering in the $I=0$ (left) and the $I=1$ (right) channels. Taken from Ref.~\protect\cite{Ikeda:2010sg}. }
\label{fig:NBS_KN}
\end{figure}
Fig.~\ref{fig:NBS_KN} shows the NBS wave functions of the $KN$ scatterings
in the $I=0$ (left) and $I=1$ (right) channels.
The large $r$ behavior of the NBS wave functions in both channels do not show a sign of
a bound state, though more detailed analysis is needed with larger volumes for a definite conclusion. On the other hand, the small $r$ behavior of the NBS wave functions
suggests a repulsive interaction at short distance ($r<0.3$ fm).
The repulsion in the $I=1$ channel seems to be stronger than that in the $I=0$ channel.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.45\columnwidth]{Figs/SVR_NK_0_multiplot.eps}
\includegraphics[width=0.45\columnwidth]{Figs/SVR_pK_multiplot.eps}
\end{center}
\caption{The LO potential for the $KN$ state without the energy shift $E$ in the $I=0$ (left)
and the $I=1$ channels (right). }
\label{fig:pot_NK}
\end{figure}
The LO potential $V(r)$ for the $KN$ state without the constant energy shift $E$ is shown
in Fig.~\ref{fig:pot_NK} in the $I=0$ (left) and $I=1$ (right).
As expected from the NBS wave functions in Fig.~\ref{fig:NBS_KN},
the repulsive interactions are observed at short distance in both channels,
while the attractive well appears at the medium distance ($0.4<r<0.8$ fm) in the $I=0$ channel.
These results indicate that there are no bound states
in $I(J^{\pi})=0(1/2^-)$ and $1(1/2^-)$ states at $m_{\pi} \simeq 870$ MeV.
The second application is to study the charmonium-nucleon interaction using the potential method. Since charmonium does not share the same quark flavor with the nucleon,
the charmonium-nucleon interaction is mainly induced by the genuine QCD effect of multi-gluon exchanges. Theoretical studies based on QCD suggest that the $c\bar{c}$-$N$ interaction is weakly attractive. It is argued that the $c\bar{c}$-nucleus
($A$) bound system may be realized for the mass number $A\ge 3$ if the
attraction between the charmonium and the nucleon is sufficiently
strong~\cite{Brodsky:1989jd,Wasson:1991fb}. Precise information on the $c\bar{c}$-$N$ potential $V_{c\bar{c}N}(r)$ is therefore indispensable for exploring nuclear-bound charmonium states such as the $\eta_c$-${}^{3}{\rm He}$ or $J/\psi$-${}^{3}{\rm He}$ bound state in few body calculations~\cite{Belyaev:2006vn}.
In Ref.~\cite{Takahashi:2009ef}, the charmonium-nucleon potentials are calculated in quenched QCD on $16^3\times 48$ and $32^3\times 48$ lattices at $a\simeq 0.94$ fm at three different values of the light quark mass corresponding to $(m_\pi, m_N)\simeq (640,1430), (720,1520), (870,1700)$ in unit of MeV and one fixed value of the charm quark mass corresponding to $m_{\eta_c} \simeq 2920$ MeV and $m_{J/\Psi} \simeq 3000$ MeV.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.32\columnwidth,angle=270]{Figs/potential.eps}
\includegraphics[width=0.32\columnwidth,angle=270]{Figs/vol_dep.eps}
\end{center}
\caption{(Left) The effective central potential in the $s$-wave $\eta_c$-$N$ system at $m_\pi= 640$ MeV. The solid line is the fit by the Yukawa potential while the dashed line is the one by the phenomenological potential adopted in Ref.~\protect\cite{Brodsky:1989jd}.
(Right) The volume dependence of the $\eta_c$-$N$ potential. Taken from Ref.~\protect\cite{Kawanai:2010ev}.
}
\label{fig:pot_ccN}
\end{figure}
The effective central $\eta_c$-$N$ potential, evaluated from the NBS wave function with measured values of $E$ and $\mu$, is shown in Fig.~\ref{fig:pot_ccN}(Left).
The $\eta_c$-$N$ potential clearly exhibits an entirely
attractive interaction between charmonium and the nucleon
without any repulsion at all distances.
Absence of the short range repulsion (the repulsive core) may be related to absence of
the Pauli exclusion between the heavy quarkonium and the light hadron.
The interaction is exponentially screened in the long distance region
$r\ge 1$ fm.
The fit of the potential with the Yukawa form $-\gamma e^{-\alpha r}/r$ gives $\gamma \sim 0.1$ and $\alpha \sim 0.6$ GeV (solid line in Fig.~\ref{fig:pot_ccN}), which are compared with the phenomenological values $\gamma=0.6$ and $\alpha=0.6$ GeV in Ref.~\cite{{Brodsky:1989jd}} (dashed line). The strength of the Yukawa potential $\gamma$
is six times smaller than the phenomenological value, while the Yukawa
screening parameter $\alpha$ obtained from lattice QCD is
comparable with the phenomenological one. The $c\bar{c}$-$N$ potential is found to be
rather weak in lattice QCD.
As shown in Fig.~\ref{fig:pot_ccN}(Right),
there is no significant difference between potentials at two different spatial sizes ($La\approx 3.0$ and 1.5 fm). The size dependence of the $\eta_c$-$N$ potential seems small since
the $\eta_c$-$N$ potential is quickly screened to zero and turns out to be short
ranged.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.32\columnwidth,angle=270]{Figs/mass_dep.eps}
\includegraphics[width=0.32\columnwidth,angle=270]{Figs/potential_jpsi.eps}
\end{center}
\caption{(Left) The quark-mass dependence of the $\eta_c$-$N$ potential.
(Right) The spin-independent part of the central $J/\psi$-$N$
potential at $m_\pi = 640$ MeV, together with the $\eta_c$-$N$ potential for comparison.
Taken from Ref.~\protect\cite{Kawanai:2010ev}.
}
\label{fig:pot_JPsiN}
\end{figure}
No large quark-mass dependence is observed in Fig.~\ref{fig:pot_JPsiN}(Left).
This may be understood by the argument that
the $c\bar{c}$-$N$ interaction is mainly governed by multi-gluon exchanges,
which do not depend explicitly on the quark mass.
Taking a closer look at the inset of Fig.~\ref{fig:pot_JPsiN}(Left), however, one finds
that the attractive interaction in the $\eta_c$-$N$ system tends to get slightly weaker as the light
quark mass decreases.
The spin-independent part of the central $J/\psi$-$N$ potential is shown at $m_\pi = 640$ MeV in Fig.~\ref{fig:pot_JPsiN}(Right), together with the $\eta_c$-$N$ potential for comparison.
While no qualitative difference between the $\eta_c$-$N$ and $J/\psi$-$N$ potentials is observed, the attractive interaction in the $J/\psi$-$N$ potential is a little
stronger than the $\eta_c$-$N$ potential.
The attraction, however, is still not strong enough to form a bound state in the
$J/\psi$-$N$ channel.
\subsection{Two-color QCD and potentials}
In Ref.~\cite{Takahashi:2009ef}, potentials between baryons in two-color (SU(2)) QCD are investigated on the lattice.
In two-color QCD, two quarks (diquark) form a "baryon", whose local interpolating operator is given by
\begin{equation}
D_{ij,\Gamma}(\mbox{\boldmath $x$},t)\equiv \varepsilon^{ab}q_i(\mbox{\boldmath $x$},t)\Gamma q_j(\mbox{\boldmath $x$},t),
\end{equation}
where $\varepsilon^{ab}$ is the $2\times 2$ anti-symmetric tensor, $i,j$ are flavor indices and
$\Gamma=C,C\gamma_5, C\gamma_\mu, C\gamma_\mu\gamma_5$ with the charge conjugation matrix $C$. The NBS wave function is then defined by
\begin{equation}
\varphi^{W,\Gamma}_{ij,kl}(\bf r )e^{-Wt} = \langle 0\vert T\{ D_{ij,\Gamma}(\mbox{\boldmath $x$}+\bf r,t) D_{kl,\Gamma}
(\mbox{\boldmath $x$},t)\} \vert W\rangle
\end{equation}
where $\vert W \rangle $ is an eigenstate of two color QCD with total energy $W$ and "baryon" number 2. The potential derived from this wave function is denoted as
$V_{ij,kl}^\Gamma(\bf r)$.
The potentials have been calculated in SU(2) quenched QCD on a $24^3\times 48$ lattice at $a\simeq 0.1$ fm determined from the string tension with the assumption that $\sqrt{\sigma}=440$ MeV. The lightest "baryon" corresponds to the scalar diquark state ($\Gamma=C$). Therefore the NBS wave function is evaluated for $\Gamma=C$ at
four values of the quark mass, which give $m_S=1.044(2),0.836(2),0.618(2), 0.377(3)$ in lattice unit with $m_S$ being the scalar diquark mass. The potential is then extracted from the $A_1$ state with $\Gamma = C$, and is therefore denoted by $V_{ij,kl}(r)$.
Fig.~\ref{fig:potential_2color} shows the LO central potentials plotted as functions of $r$ for
$(i,j,k,l)=(1,2,3,4)$(left) and $(1,2,1,2)$(right), where quark-exchange diagrams are absent for the former while they exist for the latter .
The potential $V_{12,34}(r)$ has an attractive interaction at all length scales, which becomes stronger as $m_S$ decreases. The $r$-dependence is monotonic at all values of $m_S$.
At large $r$ ($r\ge 4$ in lattice units), the potential has small $m_S$ dependence, while it has stronger $m_S$ dependence at small $r$ ($r\le 4$). The magnitude of the potential decreases as $m_S$ increases and finally the potentials at $m_S=0.836$ and $1.044$ coincide with each other.
For $V_{12,12}$, on the other hand, strong repulsions appear at short distance.
The $m_S$ dependence is not monotonic: The potential is a smooth function of $r$ at small $m_S$, while it has a pocket at intermediate distance for $m_S=1.044$.
As $m_S$ decreases, the repulsive core rapidly increases and the attractive pocket disappears.
The qualitative difference between the two cases suggests that contributions from quark exchange diagrams, which more or less represent the Pauli exclusion effect, are responsible for the repulsive core. On the other hand, the attraction may be explained by gluon exchanges,
which are the main part of the potential $V_{12,34}(r)$.
More details analysis can be found in Ref.~\cite{Takahashi:2009ef}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.45\columnwidth]{Figs/potentials_diag1.eps}
\includegraphics[width=0.45\columnwidth]{Figs/potentials_diag12.eps}
\end{center}
\caption{
\label{fig:potential_2color}
The LO central potentials $V_{12,34}(r)$ (Left) and $V_{12,12}(r)$ (Right)
as functions of the relative distance $r$ in lattice unit. Here $\kappa = 0.135,14,145$ and $0.15$ correspond to $m_S=1.044,0.836, 0.618$ and $0.377$ in lattice unit, respectively.
Taken from Ref.~\protect\cite{Takahashi:2009ef}.
}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this report, we have reviewed the progress on the study of hadron interactions via the potential method in lattice QCD. The key quantity of the method is the NBS wave function of the two particle state. Not only is the asymptotic behavior of the NBS wave related to the scattering phase shift (the phase of the S-matrix) in QCD, but also the non-local but energy-independent potential (or the interaction kernel) can be extracted from the NBS wave function in the non-asymptotic region.
By construction the potential correctly reproduces the scattering phase shift at all energies below the inelastic threshold. In practice the non-local potential is approximated by the velocity expansion in terms of local functions, so that physical observables such as the scattering phase shift are approximately calculated. It is also possible to check the accuracy of the approximation.
Using the local potential approximation at the leading order in the velocity expansion, nucleon-nucleon, hyperon-nucleon and hyperon-hyperon interactions have already been investigated successfully. The BB potential in the flavor SU(3) limit shows the attraction in the flavor singlet channel, which is strong enough to form a bound state, $H$-dibaryon in this limit.
The repulsive core is analyzed in terms of the operator product expansion and the renormalization group.
The potential method is extended to the case that the total energy is above the inelastic threshold.
This method seems to work for the BB interaction with $S=-2$ and $I=0$, where
the coupled channel potentials among $\Lambda\Lambda$, $N\Xi$ and $\Sigma\Sigma$ are obtained.
The potential method is also applied to the three nucleon force, the meson-baryon interactions and two color QCD.
Of course, more careful studies of systematic errors such as the finite volume effect, dynamical quark effect, quark mass dependence and the lattice spacing effect are needed, before applying
potentials obtained in lattice QCD in order to investigations of nuclei and hyper-nuclei.
Finally I stress here that the potential derived from the NBS wave function adds a new tool to investigate hadron interactions in lattice QCD, in addition to the standard finite volume method proposed by L\"uscher~\cite{Luscher:1990ux}.
In particular, the hadron scattering above the inelastic threshold can be treated in lattice QCD by the potential method. Other extensions of this method will also be looked for.
\section*{Acknowledgement}
I would like to thank all members of HAL QCD collaboration (Drs. T.~Doi, T.~Hatsuda, Y.~Ikeda, T.~Inoue, N.~Ishii, K.~Murano, H.~Nemura, K.~Sasaki), Dr. J.~Balog and Dr. P.~Weisz for useful discussions. In particular I would like to thank Dr. P.~Weisz for careful reading of the manuscript and valuable comments.
I also thank T.~Kawanai for providing me with figures used in this report.
This work is supported in part by the Grant-in-Aid of the Ministry of Education, Science and Technology, Sports and Culture(MEXT) (No. 20340047) and by the MEXT Grant-in-Aid for Scientific Research on Innovative Areas (No. 2004: 20105001, 20105003).
|
1,116,691,499,410 | arxiv | \section*{Abstract}
{\bf
Following the many contributions KLOE-2 has done to Dark Matter (DM) searches, an alternative model, where the Dark Force mediator is an hypothetical leptophobic B boson, in contra-position to the U boson or "dark photon", is investigated. The B boson couples mainly to quarks and it can be searched in the Phi decay to $\eta$-B where B will decay in $\pi^{0}$-$\gamma$. So far, investigation of the $\pi^{0}$-$\gamma$ invariant mass shows no clear structure belonging to the signal of the DM mediator, hence, an upper limit in the number of events at 90\% with CLs the technique will be established for the decay.
}
\section{Introduction}
\label{sec:intro}
The Universe consists, in its majority, of Dark Matter (DM) and Dark Energy, being the contribution of visible matter as little as 4\% of the total.
The existence of DM is nowadays well established and supported by many cosmological observations.
However, several models try to describe the nature of the same from very different approaches. If DM is a new type of matter then, depending on how it interacts with the Standard Model (SM) particles,
it could explain some of its inconsistencies and open the possibility of
new hidden forces Beyond the Standard Model.
Of particular interest it is its involvement in the g-2 anomaly~\cite{AOYAMA20201}, a long standing discrepancy between theory and experimental values of the anomalous momentum of the muon, which has been confirmed at the level of 4.2$\sigma$~\cite{PhysRevLett.126.141801}. Several extensions of the SM have proposed models~\cite{ext1, ext2, ext3, ext4,ext5} with a Weakly Interacting Massive Particle
(WIMP) belonging to a secluded gauge sector. In the most popular model, the new gauge interaction would be mediated by a new vector gauge boson, the U boson or {\it dark photon}, which could interact with the photon via a kinetic-mixing term. A U boson, with mass of $\mathcal{O}(1\text{GeV})$ and in the range of $10^{−2} − 10^{−7}$, could be observed in e+e− colliders via different processes.
After several years of sterile searches of the U boson, specially in the $g-2$ preferred region, new models are gaining popularity. Among them, a proposal of a leptophobic {\it dark photon}~\cite{sean}, a new gauge field coupling to baryon number. This new dark matter mediator, or B boson, would mainly decay to baryons, opening new possibilities of discovery.
\section{KLOE at DA\texorpdfstring{$\Phi$}{Lg}NE}
\label{sec:kloesetup}
The KLOE (K LOng Experiment) and KLOE-2 experiments, at the DA$\Phi$NE $\phi$-factory at the Laboratori Nazionali di Frascati in Italy, were the two stages of a multi-purpose experiment able to measure mesons in the range of energies from 5 MeV to 1 GeV. In the KLOE experiment, the experimental apparatus consisted of a central detector made up of a large cylindrical drift chamber (DC)~\cite{DC} and a lead-scintillating fiber electromagnetic calorimeter~\cite{EMC} surrounded by a superconductive coil providing a magnetic field of $0.5~\text{T}$. The EMC energy and time resolutions are $\sigma_E/E = 5.7\%/\sqrt{E\text{[GeV]}}$ and $\sigma_t(E)=57\text{ps}/\sqrt{E\text{[GeV]}}\oplus100 \text{ps}$, respectively. The DC position resolutions are $\sigma_{xy} \sim 150 \mu\text{m}$ and $\sigma_z \sim 2\text{mm}$ and its momentum resolution, $\sigma_{p\perp}/p_{\perp}$, is better than $0.4\%$ for large angle tracks.
The KLOE experiment collected $2.5~\mathrm{fb^{-1}}$ at the $\phi$-peak during its 2002 to 2005 campaign and some more data off-peak. Its continuation, KLOE-2, started to acquire data on November 2014 and finalized in March 2018, collecting more than $5~\mathrm{fb^{-1}}$, thanks to an upgraded beam crossing scheme of the DA$\Phi$NE collider.
For the KLOE-2 run the original detector setup was upgraded with the installation of a inner tracker~\cite{cgem-it} and two calorimeters~\cite{CCALT,QCALT} close to the interaction region (IP), in order to improve the vertex reconstruction near the interaction point (IP) and increase tightness of the detector. Moreover, two couples of energy taggers~\cite{HET-LET} were installed along the machine layout to study $\gamma\gamma$ fusion. The full sample of KLOE and KLOE-2 data presents an invaluable and unique collection to carry out a broad program involving, kaon-, hadron-physics and dark matter searches among others~\cite{KLOE2_proposal,doi:10.1142/S0217751X19300126}.
\section{Leptophobic B boson}
\label{sec:another}
As mentioned in the introduction~\ref{sec:intro}, WIMPS have been proposed as plausible components of the DM component of our Universe. The interaction between these candidates and the SM particles is commonly proposed to be mediated by a new vector gauge boson called {\it{dark photon}}, U or A', which couples to the photon via a kinematic mixing term, $\epsilon^2$, which can be searched for in $e^+e^-$ colliders via processes $e^+e^- \rightarrow U \gamma$, $V \rightarrow P\gamma$ decays, where V and P are vector and pseudoscalar mesons, and $e^+e^- \rightarrow h'U$, where $h'$ is a Higgs-like particle responsible for the breaking of the hidden symmetry. The KLOE collaboration has profusely contributed to these searches studying different processes~\cite{combined_limit,enrico, KLOE_UL1, KLOE_UL2, ppg}, covering a large part of the expected mass region.
Additionally, a new model where the interaction is mediated by new gauge boson that couples preferably to quarks over leptons, as proposed in the U photon models, thus being quoted as leptophobic. In the simplest model, the new baryonic-force mediator, called B boson, would couple to baryon number and arise from a new $U(1)_{B}$ gauge symmetry~\cite{sean}. In this case, the interaction the lagrangian is described by:
\begin{equation}
\mathcal{L} = \frac{1}{3}g_{B}\Bar{q}\gamma^{\mu}qB_{\mu}
\label{eq:lagrangian}
\end{equation}
where $B_{\mu}$ is the new gauge field coupling to baryon number. The gauge coupling $g_{B}$ is universal for all quarks q. From this, a baryonic fine structure constant $\alpha_B \equiv g^2_B/(4\pi)$, analogous to the electromagnetic constant $\alpha_{em}$, is defined.
Since eq.~\ref{eq:lagrangian} preserves all low-energy symmetries of QCD and the gauge coupling is universal for all flavors, B does not transform under the flavor symmetry and is a singlet under isospin. This means that B has assigned the same quantum numbers as the $\omega$ meson: $I^G(J^{PC}) = 0^-(1^{--})$. Thus, representing a good guide of what we can expect for the decays of the B boson. From $\omega$ we know that the three main decay modes Branching Fractions are: $BR(\omega \rightarrow \pi^+ \pi^- \pi^0) \simeq 89\%$, $BR(\omega \rightarrow \pi^0 \gamma) \simeq 8\%$ and $BR(\omega \rightarrow \pi^+ \pi^- \simeq 1.5\%$. This scheme is expected be followed by the B boson in the range $m_{\pi} \le m_B \le \text{GeV}$. The different branching fractions can be seen in the fig.~\ref{fig:brBboson}. Above 1 GeV, the kaon channel $B \rightarrow K \bar{K}$ opens and the B decays become similar to those of the $\Phi$ meson.
\begin{figure}
\centering
\includegraphics[width=10cm]{brBboson.png}
\caption{Figure from~\cite{sean}. branching ratios for B decay. Thick lines have $\epsilon = eg_B/(4\pi)2$; thin dotted lines have $\epsilon = 0.1 \times eg_B/(4\pi)2$.}
\label{fig:brBboson}
\end{figure}
\subsection{Search of the B boson with KLOE detector}
As seen in fig.~\ref{fig:brBboson}, in the mass range between $m_{\pi}$ and $m_{3\pi} \approx 620 \text{MeV}$, the dominant channel is the decay $B \rightarrow \pi^0 \gamma$. With this in mind, we perform the search of the B boson with the KLOE data by looking for an enhancement in the invariant mass of $m_{\pi^0\gamma}$ from the decay chain $\Phi \rightarrow B \eta \rightarrow \pi^0 \gamma$, with $eta \rightarrow \gamma\gamma$ and $\pi^0 \rightarrow \gamma\gamma$.
A data sample of $1.7 fb^{-1}$ has been analyzed looking for $5-\gamma$ final states. For this, exactly five neutral clusters with a deposited energy in the calorimeter above $10 \text{MeV}$ and scattering angle $23\circ \le \theta \le 157\circ$, this is in the barrel section of the KLOE calorimeter, are selected. To this selection, a kinematic fit is applied to improve the resolution and allow to properly select the $\eta-\pi^0$ containing events.
After a refine selection and proper background discrimination, the mass of the $\pi0\gamma$ system can be reconstructed, as presented in fig.~\ref{fig:Bmass} in blue full circles. The expected background according to Monte Carlo simulations corresponds to the irreducible SM backgrounds coming from the decays $\Phi \rightarrow a_0 \gamma \rightarrow \eta \pi^0 \gamma$ and $\Phi \rightarrow \eta \gamma$ with $\eta \rightarrow 3\pi^0$, where final state photons can be undetected, therefore lost, or merged to another hits by the reconstruction algorithms. To avoid model dependency, the background in the invariant mass of $\pi^0\gamma$ is extracted by a side-band fit, and the interpolated background with this method is shown in fig.~\ref{fig:Bmass} as magenta full circles.
\begin{figure}
\centering
\includegraphics[width=8cm]{Bmass.png}
\caption{Preliminary spectra of the invariant mass of the $\pi^0\gamma$ system. The B boson signature would be a pronounce enhancement in the spectra. Full blue circles represent the data and full magenta circles the extrapolated background using a side-band fit.}
\label{fig:Bmass}
\end{figure}
Since no excess is observe in the spectra, we proceed to extract the upper limit in the B boson by using the CLs method~\cite{cls}. The preliminary result of the upper limit on the number of expected signal events at 90\% CLs is presented in fig.~\ref{fig:Nsig}. From this we can expect to set limits in the coupling constant of the B boson ($\alpha_B$) at the level of $\mathcal{O}(10^{-7})$ at 90\% CLs.
\begin{figure}
\centering
\includegraphics[width=12cm]{Nsig2.png}
\caption{Preliminary Upper Limit at 90\% CLs in the number of signal events using CLs method.}
\label{fig:Nsig}
\end{figure}
\section{Conclusion}
The KLOE-2 Collaboration continues contributing to the searches of DM and Physics Beyond the Standard Model by exploring the reaction $\Phi \rightarrow B \eta$ with B being a new gauge boson coupling mainly to quarks. The interested decay channel of this study is the process $B \rightarrow \pi^0 \gamma$, which can be mimicked by the SM process $\Phi \rightarrow a_0 \gamma$ and $\Phi \rightarrow \eta \gamma \rightarrow 3\pi^0 \gamma$ when photons are not detected or merged to another hits during the reconstruction. In the scenario of the discovery of the new mediator, the evidence would be an enhancement in the invariant mass of the $\pi^0\gamma$ pairs. Since no evidence of the B boson is found, we proceed to extract the upper limit on the coupling of the B boson to the SM particles, $\alpha_B$. In this work we present the preliminary upper limit in the number of B boson signal events extracted with the CLs method at 90\%, from which we expect to set limits of the oder of $\mathcal{O}(10^{-7})$ at 90\% CLs in the coupling constant.
\section*{Acknowledgements}
We warmly thank our former KLOE colleagues for the access to the data collected during the KLOE data-taking campaign. We thank the DA$\Phi$NE team for their efforts in maintaining low background running conditions and their collaboration during all data taking. We want to thank our technical staff: G.F. Fortugno and F. Sborzacchi for their dedication in ensuring efficient operation of the KLOE computing facilities; M. Anelli for his continuous attention to the gas system and detector safety; A. Balla, M. Gatta, G. Corradi and G. Papalino for electronics maintenance; C. Piscitelli for his help during major maintenance periods. This work was supported in part by the Polish National Science Centre through the Grants No. 2013/11/B/ST2/04245, 2014/14/E/ST2/00262, 2014/12/S/ST2/00459,\\
2016/21/N/ST2/01727,2016/23/N/ST2/01293, 2017/26/M/ST2/00697.
|
1,116,691,499,411 | arxiv | \section{Introduction}
Understanding the chemical diversity of materials in star- and planet-forming regions is one of the key topics of present-day astronomy.
Formation of stars and planets can occur in various kinds of galaxies, which differ in size, shape, age, and environment.
It is thus important to understand how galactic characteristics affect properties of interstellar and circumstellar medium.
It is known that infrared spectra of embedded young stellar objects (YSOs) show absorption bands due to ices in which a large amount of heavy elements and complex molecules are preserved \citep[e.g., ][]{vDB98,Boo04}.
A variety of ice species (H$_2$O, CO$_2$, CO, CH$_3$OH, CH$_4$) have been detected toward molecular clouds and YSOs \citep[e.g.,][]{Gib04,Dar05,Pon08,Boo08,Boo11,Obe11}.
These kinds of molecular species are also detected toward solar system objects \citep[e.g., comets,][]{Oot12}, and ices are believed to deliver important volatile molecules to planets \citep{Ehr00b}.
Furthermore, ices are also detected toward the central region of nearby external galaxies \citep[e.g.,][]{Spo03,Yam11,Yam13}.
Ice chemistry plays an essential role in the overall chemical evolution of molecular clouds since it efficiently proceeds in dense and cold regions (n$_H$ $\geq$ 10$^4$ cm$^{-3}$, T $\sim$10K).
Grain surface reactions as well as radiolysis and photolysis can produce complex molecules that are different from those produced by gas-phase reactions \citep[e.g.,][]{Tie82,Has92}.
Therefore, the chemical properties of ices should not be neglected to understand diversities of materials found in star- and planet-forming regions.
Observations of ices around extragalactic YSOs aim to understand which environment parameters of galaxies are relevant to ice chemistry.
It is highly probable that different galactic environments (e.g., metallicity, radiation field) could affect the properties of circumstellar materials because circumstellar materials are closely related to interstellar medium.
In particular, it is important to understand the impact of low metallicity environments on the interstellar and circumstellar chemistry.
These studies are essential to constrain chemical processes occurring in the past universe, since cosmic metallicity is believed to be increasing with the evolution of the universe \citep[e.g., ][]{Raf12}.
The Large Magellanic Cloud (LMC) is the nearest external star-forming galaxy \citep[d = 49.97 $\pm$ 1.11 kpc, ][]{Pie13} and a prime target for this study.
The LMC has been an excellent target for studies of molecular cloud evolution and star formation \citep[e.g.,][]{Mei06,FK10}.
It is well known that the metallicity of the interstellar medium (ISM) within the LMC is about half of the solar neighborhood \citep [e.g.,][]{Duf82,Wes90,And01,Rol02}.
Far-infrared and sub-millimeter observations show that dust temperatures in the LMC are, on average, higher than those in our Galaxy \citep[e.g., ][]{Agu03,Sak06}.
Further, gamma-ray observations of the LMC indicate that the cosmic-ray density in the LMC is 20$\%$ to 30$\%$ lower than typical Galactic values \citep{Abd10}.
Previous studies have reported properties of ices around embedded high-mass YSOs in the LMC \citep{ST,ST10,ST12,ST13,thesis,vanL05,vanL10, Oli06,Oli09,Oli11,Sea09,Sea10}.
\citet{ST10}, for example, report the results of 2.5--5.0 $\mu$m spectroscopic observations of embedded high-mass YSOs in the LMC with the Infrared Camera (IRC) on board the \textit{AKARI} satellite \citep{TON07,Mur07}.
They find that the CO$_2$/H$_2$O ice ratio of LMC YSOs is systematically higher than those measured toward Galactic counterparts.
This suggests that the chemical properties of circumstellar materials of YSOs may depend on the environment of the host galaxy.
The authors suggest that intense radiation field and/or warm dust temperature in the LMC could be responsible for the different molecular abundance of ices in the LMC.
Observations of the 15.2 $\mu$m CO$_2$ ice band toward LMC high-mass YSOs with \textit{Spitzer}/IRS suggest that a majority of CO$_2$ is locked in a water-rich (polar) ice mixture, but a fraction of CO$_2$ in a polar ice component is lower in the LMC than in our Galaxy \citep{Oli09,Sea10}.
In addition, these studies argued that a larger number of LMC high-mass YSOs show a CO$_2$ ice profile characteristic to a pure or annealed CO$_2$ ice component compared to Galactic similar sources, suggesting a higher degree of thermal processing of ices in the LMC.
The above previous studies suggest different physical and chemical properties of ices in the low metallicity environment, however, current studies of ices around LMC YSOs are mostly dedicated to major ice species such as H$_2$O and CO$_2$.
Detailed information of chemically important minor ice species is therefore needed to provide a comprehensive view of ice chemistry in the low metallicity environment of the LMC.
The 3.2--3.7 $\mu$m spectral region is one of the important wavelengths for studies of solids since various C--H stretching vibrations of carbon bearing species are observed in this region.
Solid methanol (CH$_3$OH), which has an absorption band at 3.53 $\mu$m, is an important ice mantle component and sometimes the second most abundant after water ice.
The CH$_3$OH ice has been detected toward several YSOs in our Galaxy with an abundance ranging from 5$\%$ to 30 $\%$ relative to the water ice \citep[e.g.,][]{Gri91,Dar99,Whi11}.
The abundance of CH$_3$OH in the solid phase is of great importance for molecular chemistry since it is believed to be a starting point for the formation of complex organic molecules in circumstellar environments of YSOs \citep[e.g., ][]{NM04,Her09}.
However, abundances of solid CH$_3$OH in extragalactic objects remain to be investigated.
The 3.47 $\mu$m absorption band is another interesting band in the C--H stretching region of embedded sources.
The band is widely detected toward a variety of embedded sources such as high- to intermediate-mass YSOs \citep[e.g., ][]{All92,Bro96,Bro99,Dar01,Dar02,Ish02}, low-mass YSOs \citep[e.g., ][]{Pon03,Thi06}, and quiescent dense molecular clouds \citep[e.g., ][]{Chi96,Chi11}.
The 3.47 $\mu$m band strength is known to correlate with the H$_2$O ice absorption depth.
The band carrier is proposed to be interstellar nano-diamonds \citep{All92,Pir07,Men08} or ammonia hydrates formed in H$_2$O:NH$_3$ mixture ice \citep{Dar01,Dar02,Boo15}.
However, the exact identification of the 3.47 $\mu$m band carrier is still under debate.
Furthermore, properties of the 3.47 $\mu$m band in low metallicity environments are poorly understood.
In this paper, we present the results of L-band spectroscopic observations toward embedded high-mass YSOs in the LMC with the Infrared Spectrometer And Array Camera (ISAAC) at the Very Large Telescope (VLT) of the European Southern Observatory (ESO).
Details of target selection, observations and data reduction are summarized in $\S$2.
Results of observations and spectral analysis are described in $\S$3.
Properties of solid methanol and the 3.47 $\mu$m absorption band in the LMC are discussed in $\S$4.
Implications for the formation of organic molecules in low metallicity galaxies are also discussed in this section.
Conclusions of this work are summarized in $\S$5.
\begin{table*}
\caption{Summary of the observations}
\label{target}
\centering
\begin{tabular}{ l c c c c c c}
\hline\hline
Source & Other ID &RA & Dec & [3.55]\tablefootmark{a} & t$_{int}$\tablefootmark{b} & Standard \\
& &(J2000) & (J2000) & (mag) & (min) & Star\tablefootmark{c} \\
\hline
ST1 & MSX LMC 940 & 05:39:31.15 & -70:12:16.8 & 10.52 & 70 & H29 \\
ST2 & MSX LMC 501 & 05:22:12.56 & -67:58:32.2 & 9.75 & 70 & H29 \\
ST3 & J052546.51-661411.5 & 05:25:46.69 & -66:14:11.3 & 9.98 & 60 & H16 \\
ST4 & J051449.43-671221.4 & 05:14:49.41 & -67:12:21.5 & 10.45 & 80 & H16 \\
ST5 & J053054.27-683428.2 & 05:30:54.27 & -68:34:28.2 & 10.03 & 30 & H29 \\
ST6 & J053941.12-692916.8 & 05:39:41.08 & -69:29:16.8 & 12.12 & 220 & H16, H29 \\
ST7 & J052351.13-680712.2 & 05:23:51.15 & -68:07:12.2 & 10.28 & 55 & H29 \\
ST10 & MSX LMC 1229 & 04:56:40.80 & -66:32:30.4 & 10.67 & 195 & H16, H29 \\
ST16 & MSX LMC 318 & 05:19:12.30 & -69:09:06.8 & 9.89 & 30 & H16 \\
\hline
\textit{Archival sources} & & & & & \\
ST14 & MSX LMC 1275 & 04:58:54.33 & -66:07:18.8 & 10.67 & 65 & H37 \\
ST17 & MSX LMC 94 & 05:10:24.15 & -70:14:06.7 & 11.51 & 65 & H37 \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{a}{\textit{Spitzer}/IRAC band 1 (3.55 $\mu$m) magnitude after color correction.}
\tablefoottext{b}{Total on-source integration time.}
\tablefoottext{c}{H16, H29 and H37 refer to HIP16368, HIP29134 and HIP37623, respectively.}
}
\end{table*}
\begin{table*}
\caption{Spectroscopic standard stars used for calibration in this study}
\label{std}
\centering
\begin{tabular}{ l c c c c c }
\hline\hline
Name & RA & Dec & Spectral & V\tablefootmark{a} & L'\tablefootmark{b} \\
& (J2000) & (J2000) & Type\tablefootmark{a} & (mag) & (mag) \\
\hline
HIP16368 & 03:30:51.71 & -66:29:23.0 & B8V & 5.04 & 5.20 \\
HIP29134 & 06:08:44.26 & -68:50:36.3 & B8V & 5.81 & 5.80 \\
HIP37623 & 07:43:11.98 & -36:03:00.3 & B5V & 5.59 & 5.99\tablefootmark{b} \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{a}{Data taken from the SIMBAD database. }
\tablefoottext{b}{Estimated assuming [K] -- [L'] = -0.05 (ref. ISAAC web page for spectroscopic standards). }
}
\end{table*}
\section{Observations and data reduction}
\subsection{Selection of targets}
Nine high-mass YSOs are selected based on our previous near-infrared spectroscopic observations with \textit{AKARI} presented in \citet{ST10} and \citet{thesis}.
In addition, we obtained ISAAC spectroscopic data from the ESO archive for two high-mass YSOs (089.C-0882(C); PI: J. M. Oliveira) whose \textit{AKARI} spectra are also presented in \citet{thesis}.
All of these YSOs show H$_2$O and CO$_2$ ice absorption bands in their near-infrared spectra, suggesting that they are appropriate targets to investigate minor ice species.
Tentative detections of solid CH$_3$OH, CO, and XCN bands were reported for several \textit{AKARI} samples in our previous study, but these were less conclusive due to the low spectral resolution (R $\sim$ 80) and low spatial resolution.
We selected sources with relatively large ice column densities and high L-band fluxes for the present observations.
Details of the targets and observations are summarized in Table \ref{target}.
\subsection{VLT/ISAAC L-band spectroscopy}
Spectra of nine high-mass YSOs were obtained between January 21 and 28, 2013, at the VLT UT3 using the infrared spectrograph ISAAC.
Observations were carried out as a part of the ESO normal program 090.C-0497(A) (PI: E. Dartois).
The VLT, located on Cerro Paranal, Chile, has a primary mirror of 8.2 m in diameter.
The ISAAC was installed at the Nasmyth A focus of VLT/UT3 at the time of our observations.
We conducted L-band slit spectroscopy with a low-resolution grating, which covers 2.8 $\mu$m to 4.2 $\mu$m with the 120$\arcsec$ length slit.
The central wavelength of the grating was set to 3.55 $\mu$m.
The pixel scale of the detector is 0.1484$\arcsec$ and the size is 1024 $\times$ 1024.
Target acquisition was carried out with the ESO L-band filter centered at 3.78 $\mu$m.
The atmospheric seeing was typically 0.6$\arcsec$--0.7$\arcsec$ during our observation nights.
The slit width was set to 0.6$\arcsec$ for every target except ST5, whose slit width was set to 0.3$\arcsec$ considering its brightness and atmospheric conditions of the night.
The slit width for archival sources (ST14 and ST17) is 1$\arcsec$.
The achieved spatial resolution in actual physical scale at the distance of the LMC is 0.15 pc and 0.07 pc for the 0.6$\arcsec$ and 0.3$\arcsec$ slit, respectively.
The resultant spectral resolution is R $\sim$600 for the 0.6$\arcsec$ slit on raw data.
The actual spectral resolution of fully calibrated data is slightly worse due to spectral binning (see $\S$2.3).
Sky background emission was canceled out by chopping and nodding with ABBA sequences, in which each target was observed at two different positions along the slit, and the sky was removed by subtracting one frame from the other.
The chopping throw and direction were carefully determined by inspecting the \textit{AKARI}/IRC 3.2 $\mu$m images \citep{Kat12} or the \textit{Spitzer}/IRAC 3.55 $\mu$m images \citep{Mei06} at each target position.
The adopted chopping throw ranges from 5$\arcsec$ to 15$\arcsec$ depending on the targets.
Standard stars were observed for calibration purpose before and after observations of each target.
Properties of the standard stars are summarized in Table \ref{std}.
Airmass differences between targets and standard stars were typically smaller than 0.1.
\begin{figure*}[!]
\begin{center}
\includegraphics[width=15.5cm, angle=0]{f1.eps}
\caption{VLT/ISAAC and {\it AKARI}/IRC 2.5--5 $\mu$m spectra of embedded high-mass YSOs in the LMC.
The blue circles connected by thick lines represent the VLT L-band spectra obtained in this work.
The red circles connected by thin lines represent the {\it AKARI} spectra (the original data taken from \citet{ST10} and \citet{thesis}).
The green open squares represent photometric fluxes of {\it Spitzer}/IRAC band 1 (3.55 $\mu$m) and band 2 (4.5 $\mu$m) measured in the SAGE survey.
Detected ice absorption bands are labeled in each panel and the wavelengths of hydrogen recombination lines are indicated in the upper panels.
A color version of this figure is available in the online journal.
}
\label{Spec}
\end{center}
\end{figure*}
\begin{figure*}[!]
\begin{center}
\includegraphics[width=15.5cm, angle=0]{f2.eps}
\caption{Same as Fig. \ref{Spec}, for archival sources.
}
\label{Spec_add}
\end{center}
\end{figure*}
\subsection{Data reduction}
The raw data are reduced using our own IDL-based programs developed for VLT/ISAAC slit spectroscopy data.
Data of two archival sources are reduced in the same manner as our target sources.
First, individual exposure data are coadded to cancel bad pixels and to increase the signal-to-noise ratio.
Since the present spectroscopic observations were performed with the ABBA chopping sequences, spectra of objects are detected at three positions along the chopping direction on coadded data (one positive spectrum at the ON position and two negative spectra at the OFF positions).
To extract each spectrum, we measure the central position and FWHM of signals along the spatial direction at multiple wavelength points.
We then extract the three spectra using the extraction width and position determined above and combine them into one spectrum.
Uncertainties of data are estimated based on the signals at sky positions, which are typically located at 15 pixels apart from the source position.
We extract spectra of targets and standard stars in the same manner.
The extracted spectra are smoothed by a Savitzky-Golay filter with an appropriate polynomial degree and filter width.
To obtain flux calibrated spectra, the target spectra are divided by the spectrum of the associated standard star.
The wavelength calibration is carried out using telluric absorption lines before this division, and slight wavelength shifts between target's and standard star's spectra are corrected.
We also correct airmass differences between targets and standard stars.
A theoretical spectrum of the standard stars is applied to the spectra for flux calibration.
We here assume blackbody spectra for the standard stars.
The temperature of the blackbody is estimated by fitting the Planck function to photometric fluxes in BVIJHK bands, which are taken from the SIMBAD database \footnote{http://simbad.u-strasbg.fr/simbad/}.
In addition, we add photospheric absorption lines of hydrogen (Pf $\zeta$ at 2.873 $\mu$m, Pf $\epsilon$ at 3.039 $\mu$m, Pf $\delta$ at 3.297 $\mu$m, Pf $\gamma$ at 3.741 $\mu$m, Br $\alpha$ at 4.0523 $\mu$m), which intrinsically exist in the spectrum of standard stars.
The strength and width of the absorption lines are measured by fitting a Gaussian to the Br $\alpha$ line, which is usually the most prominent and less contaminated with telluric lines.
Then, the strengths of other absorption lines relative to Br $\alpha$ are determined based on a Kurucz stellar atmosphere model (T$_{eff}$ = 11900 K, log $g$ = +4.04, [Fe/H] = 0), which is taken from The 1993 Kurucz Stellar Atmospheres Atlas.
The reduced spectra are fine-tuned with further wavelength calibration, absolute flux calibration, and appropriate smoothing as necessary.
For sources that show emission lines of Pf $\gamma$ and Br $\alpha$, the wavelength solutions are slightly modified by interpolating the peak positions of these two emission lines.
A geocentric radial velocity of 265 km/s is assumed to estimate the peak wavelength of hydrogen recombination lines on the sky \citep[the velocity is estimated based on Fig. 7 in][]{Fuk08}.
The adopted velocity may have uncertainty by several tens of km/s depending on the location of sources in the LMC, but such a small velocity difference does not significantly change the wavelength solution because the present spectra are medium resolution.
The overall wavelength accuracy of the calibrated spectra is estimated to be 0.004 $\mu$m.
The absolute fluxes are calibrated by scaling each spectrum to the corresponding \textit{Spitzer}/IRAC band 1 photometric fluxes at 3.55 $\mu$m, which are taken from the SAGE catalog \citep{Mei06}.
The color-correction is applied to the photometric data to accurately determine the in-band fluxes.
The appropriate color correction factors are estimated based on the gradient of each spectrum between 3.2 and 3.9 $\mu$m and the tables of color correction factors given in the IRAC instrument handbook.
Finally, the spectra are binned to achieve the S/N ratio that is necessary for the subsequent spectral analysis.
Noisy spectral regions due to telluric lines and bad pixels are carefully inspected and masked before data binning.
The resultant spectral resolution of the present VLT/ISAAC data is $\lambda$/$\Delta$$\lambda$ $\sim$500 for the 3.2--4.0 $\mu$m region.
The spectral resolution shorter than 3.2 $\mu$m is worse by a factor of two to four since more data points are binned in this wavelength region because of increased noise.
Calibrated VLT/ISAAC spectra are shown in Figs. \ref{Spec}--\ref{Spec_add}.
Since broad wavelength coverage is crucial for reliable continuum determination in the subsequent spectral analysis, \textit{AKARI}/IRC grism spectra ($\lambda$/$\Delta$$\lambda$ $\sim$100, data taken from \citet{ST10} and \citet{thesis}) are concatenated to the short and long wavelength sides of the ISAAC spectra.
The IRC spectra are scaled to match the ISAAC spectra around 3 $\mu$m and 4 $\mu$m (see also Fig. \ref{App_Spec} for comparison of ISAAC and IRC spectra).
The concatenated \textit{AKARI} spectra are also shown in the figures.
\begin{figure*}[!]
\begin{center}
\includegraphics[width=18cm, angle=0]{f3.eps}
\caption{
Results of the spectral analysis for the 3.47 $\mu$m and 3.53 $\mu$m (CH$_3$OH) absorption bands.
Left: derived continua (dashed lines, green) are shown with the observed spectra.
Middle: results of the spectral fitting.
The dashed lines (blue) represent the laboratory CH$_3$OH ice spectrum and the dot-dashed lines (red) represent the 3.47 $\mu$m absorption band.
The solid lines (gray) show the sum of those two components.
Right: residual spectra after subtraction of the 3.47 $\mu$m band. The spectra are only shown for the sources in which the 3.47 $\mu$m band is detected.
}
\label{Fit1}
\end{center}
\end{figure*}
\begin{figure*}[!]
\begin{center}
\includegraphics[width=18cm, angle=0]{f4.eps}
\caption{{\it Continued}
}
\label{Fit2}
\end{center}
\end{figure*}
\begin{figure*}[!]
\begin{center}
\includegraphics[width=15.5cm, angle=0]{f5.eps}
\caption{Results of the spectral analysis for the 3.05 $\mu$m H$_2$O ice band.
The dashed lines (blue) represent the laboratory H$_2$O ice spectrum fitted to the observed spectrum (solid lines, black).
Derived continua are shown in the lower right side of each panel with dashed lines (green).
}
\label{WFit1}
\end{center}
\end{figure*}
\section{Results}
\subsection{Observed spectra}
Figures \ref{Spec}--\ref{Spec_add} show the VLT/ISAAC spectra of high-mass embedded YSOs in the LMC together with their \textit{AKARI}/IRC spectra.
The 3.05 $\mu$m H$_2$O ice absorption band (O--H stretching mode) is detected toward all of the eleven sources.
Since the short-wavelength side of the water absorption is not detected owing to low atmospheric transmission, the present L-band spectroscopy mainly detects the long-wavelength side of the water ice absorption; i.e., the red wing of the water ice absorption \citep[e.g.,][]{Dar01,Dar02}.
The 3.53 $\mu$m CH$_3$OH ice absorption band (C--H stretching mode) is detected toward ST6 and ST10, and the strength of the absorption is found to be weak.
The band is absent toward all the other sources, but meaningful upper limits are obtained.
The 3.47 $\mu$m absorption band, which is often detected toward embedded sources in our Galaxy, is detected toward six LMC sources (ST3, ST5, ST6, ST7, ST10, ST17).
For ST17, however, the S/N ratio is poor.
The absorption band around 3.4 $\mu$m, which is due to aliphatic hydrocarbon and detected toward diffuse interstellar clouds, is not present in our observations.
Details of the absorption bands in the 3.4--3.5 $\mu$m region are discussed in $\S$4 after careful baseline correction of continuum.
Hydrogen recombination lines (Pf $\gamma$ at 3.7406 $\mu$m and Br $\alpha$ at 4.0523 $\mu$m) are also detected in several sources.
The VLT and \textit{AKARI} spectra are compared in Fig. \ref{App_Spec}.
Each panel in the figure shows three spectra including, a VLT spectrum before spectral smoothing, a VLT spectrum after spectral smoothing and concatenation with \textit{AKARI} data, and an original \textit{AKARI} spectrum after flux level adjustment.
Overall shapes are consistent between VLT and \textit{AKARI} data except for the region of PAH emission bands around 3.3 $\mu$m.
The PAH emission is significantly weaker in our VLT spectra.
Considering a different spatial resolution of VLT ($\sim$0.6$\arcsec$ = 0.15 pc at the LMC) and \textit{AKARI} ($\sim$6$\arcsec$ = 1.5 pc), this suggests that the PAH emission mainly arises from an extended region ($>$ 1 pc) surrounding our target YSOs, while continuum emission and ice absorption arise from a compact region ($\sim$0.1 pc).
\begin{table}[b]
\centering
\caption{Optical depths of the observed absorption bands for high-mass YSOs in the LMC}
\label{Tab_tau}
\begin{tabular}{ l c c c }
\hline\hline
Source & $\tau$$_{3.05 \mu m}$ & $\tau$$_{3.47 \mu m}$ & $\tau$$_{3.53 \mu m}$ \\
\hline
ST1 & 1.44 $\pm$ 0.19 & $<$0.04 & $<$0.02 \\
ST2 & 1.08 $\pm$ 0.09 & $<$0.02 & $<$0.02 \\
ST3 & 2.19 $\pm$ 0.04 & 0.07 $\pm$ 0.01 & $<$0.01 \\
ST4 & 1.39 $\pm$ 0.11 & $<$0.02 & $<$0.02 \\
ST5 & 2.57 $\pm$ 0.29 & 0.08 $\pm$ 0.03 & $<$0.04 \\
ST6 & 4.18 $\pm$ 0.58 & 0.15 $\pm$ 0.02 & 0.06 $\pm$ 0.02 \\
ST7 & 2.83 $\pm$ 0.12 & 0.10 $\pm$ 0.02 & $<$0.04 \\
ST10 & 3.54 $\pm$ 0.60 & 0.11 $\pm$ 0.01 & 0.04 $\pm$ 0.01 \\
ST14 & 1.38 $\pm$ 0.07 & $<$0.03 & $<$0.03 \\
ST16 & 1.14 $\pm$ 0.19 & $<$0.02 & $<$0.02 \\
ST17 & ... & 0.2 $\pm$ 0.1 & ... \\
\hline
\end{tabular}
\tablefoot{Uncertainties and upper limits are of 3 $\sigma$ level and do not include systematic errors due to continuum determination. }
\end{table}
\subsection{Spectral analysis of absorption bands}
The observed spectra are used to derive column densities of detected molecular species.
We derive column densities of the H$_2$O and CH$_3$OH ices by fitting laboratory ice spectra to the observed spectra after careful subtraction of continuum.
The optical depth of the 3.47 $\mu$m absorption band is also estimated in conjunction with spectral fitting of the CH$_3$OH ice band.
For ST17, we analyze only the 3.2--3.7 $\mu$m region because of lack of spectral data shorter than 3.2 $\mu$m.
Details of continuum subtraction and spectral fitting for the 3.2--3.7 $\mu$m region and the water ice band are described below.
\subsubsection{Continuum determination: 3.2--3.7 $\mu$m region}
Continuum baselines are carefully estimated to accurately derive the optical depth of the 3.47 $\mu$m and 3.53 $\mu$m absorption bands, since it is located in the red wing of the deep water ice absorption.
In our analysis, local continuum is derived by fitting a three- to fourth-order polynomial function to the selected continuum regions.
The wavelength regions used for the local continuum are typically 3.09--3.15 $\mu$m, 3.35--3.39 $\mu$m, 3.59--3.72 $\mu$m, 3.78--3.81 $\mu$m, 3.98--4.01 $\mu$m, and 4.07--4.15 $\mu$m.
The spectral region of PAH emission is not used for the continuum determination.
Each adopted continuum is shown in the left panel of Figs. \ref{Fit1}--\ref{Fit2}.
These local continua mimic the smooth profile of the water ice wing and enable us to extract the buried absorption component.
Observed spectra are divided by these continua and converted to optical depth (right panel of Fig. \ref{Fit1}--\ref{Fit2}).
A similar definition of continuum is adopted in \citet{Bro99} and \citet{Dar99}, which investigate the 3.47 $\mu$m and 3.53 $\mu$m absorption band for Galactic high-mass YSOs.
Since continuum subtraction significantly affects the absorption depth of the 3.53 $\mu$m band, we employ the similar continuum definition and compare absorption depths of LMC sources with those of the Galactic sources that are reported in the above-mentioned papers.
\subsubsection{Spectral fitting: 3.2--3.7 $\mu$m region}
We fit a laboratory ice spectrum to the observed spectra to derive ice column densities.
Since the 3.53 $\mu$m CH$_3$OH band partially overlaps with the 3.47 $\mu$m absorption band, these two absorption components are simultaneously fitted with a $\chi^2$ minimization method.
Wavelength regions between 3.3 $\mu$m and 3.7 $\mu$m are used for the fit.
For the CH$_3$OH band, we fit a laboratory profile of the pure CH$_3$OH ice at 10 K, which is taken from the Leiden Molecular Astrophysics database\footnote{http://www.strw.leidenuniv.nl/lab/databases/} \citep{Ger95,Ger96}.
We tried to fit various CH$_3$OH ice mixtures in this analysis and we confirm through visual inspection that the pure CH$_3$OH ice results in the best fit among the investigated ice mixtures.
It is reported that the profile of the pure CH$_3$OH ice also fits well to the 3.53 $\mu$m band of Galactic high-mass YSOs \citep{All92,Bro99}.
For the 3.47 $\mu$m band, we fit a Gaussian profile ($\lambda$$_c$ = 3.469 $\mu$m, FWHM = 0.105 $\mu$m) except for ST6.
The assumed profile is consistent with the typical profile of the 3.47 $\mu$m band observed toward Galactic high-mass YSOs \citep{Bro99}.
For ST6, we use another Gaussian profile ($\lambda$$_c$ = 3.460 $\mu$m, FWHM = 0.103 $\mu$m) for the fitting since the former profile results in a poor fit.
This kind of blueshifted profile is relatively rare but observed toward a Galactic high-mass YSO Mon R2 IRS 2 \citep{Bro99}.
Based on the above fitting, we derive the peak optical depth of the 3.53 $\mu$m CH$_3$OH band and then calculate ice column densities using the following equation:
\begin{equation}
N(CH_3OH) = \tau(3.53) \Delta\nu / A, \label{Eq_column}
\end{equation}
where $N(CH$$_3$$OH)$ is the column density of the CH$_3$OH ice in units of cm$^{-2}$, $\tau(3.53)$ is the peak optical depth of the 3.53 $\mu$m absorption band as derived by our spectral fitting, $\Delta$$\nu$ is the FWHM of the absorption band, and $A$ is the band strength as measured in the laboratory.
We adopt the band strengths of the pure CH$_3$OH ice as 5.3$\times$10$^{-18}$ cm molecule$^{-1}$ \citep{Ker99}, and $\Delta$$\nu$ is assumed to be 31 cm$^{-1}$ \citep{Sch96}.
The result of the fitting is shown in the right panel of Figs. \ref{Fit1}--\ref{Fit2}.
Derived optical depths and column densities are summarized in Tables \ref{Tab_tau} and \ref{Tab_column}.
\begin{table*}
\centering
\caption{Derived ice column densities and abundances for high-mass YSOs in the LMC}
\label{Tab_column}
\begin{tabular}{ l c c c c c c c}
\hline\hline
Source & \textit{N}(H$_2$O) & \textit{N}(CH$_3$OH) & \textit{N}(CO) & \textit{N}(CO$_2$) & \textit{N}(CH$_3$OH) & \textit{N}(CO) & \textit{N}(CO$_2$) \\
& & & & & /\textit{N}(H$_2$O) & /\textit{N}(H$_2$O) & /\textit{N}(H$_2$O) \\
& (10$^{17}$ cm$^{-2}$)& (10$^{17}$ cm$^{-2}$) & (10$^{17}$ cm$^{-2}$) & (10$^{17}$ cm$^{-2}$) & ($\%$) & ($\%$ ) & ($\%$ ) \\
\hline
ST1 & 24.88 $\pm$ 3.29 & $<$1.2 & 4.1 $\pm$ 1.1 & 5.9 $\pm$ 0.5 & $<$4.8 & 16.5 & 23.7 \\
ST2 & 18.65 $\pm$ 1.58 & $<$1.2 & $<$0.5 & 3.8 $\pm$ 0.4 & $<$6.4 & <3 & 20.4 \\
ST3 & 34.43 $\pm$ 0.66 & $<$0.6 & 1.5 $\pm$ 0.2\tablefootmark{a} & 10.1 $\pm$ 1.3 & $<$1.7 & 4.5 & 29.3 \\
ST4 & 23.22 $\pm$ 1.80 & $<$1.2 & 1.6 $\pm$ 0.1\tablefootmark{a} & 8.4 $\pm$ 1.1 & $<$5.2 & 6.9 & 36.2 \\
ST5 & 40.60 $\pm$ 4.64 & $<$2.3 & 1.9 $\pm$ 0.8 & 11.4 $\pm$ 0.9 & $<$5.7 & 4.7 & 28.1 \\
ST6 & 62.10 $\pm$ 8.62 & 3.5 $\pm$ 1.2 & $<$15 & 21.5 $\pm$ 6.2 & 5.6 & <24 & 34.6 \\
ST7 & 48.02 $\pm$ 2.10 & $<$2.3 & 2.6 $\pm$ 0.2\tablefootmark{a} & 26.0 $\pm$ 2.9 & $<$4.8 & 5.4 & 54.1 \\
ST10 & 61.42 $\pm$ 10.36 & 2.3 $\pm$ 0.6 & 9.8 $\pm$ 3.3 & 17.9 $\pm$ 3.4 & 3.7 & 16.0 & 29.1 \\
ST14 & 23.82 $\pm$ 1.24 & $<$1.8 & 2.8 $\pm$ 0.9 & 3.8 $\pm$ 0.3 & $<$7.6 & 11.8 & 16.0 \\
ST16 & 19.64 $\pm$ 3.21 & $<$1.2 & $<$2 & 2.7 $\pm$ 0.2 & $<$6.1 & <10 & 13.8 \\
ST17 & ... & ... & 5.5 $\pm$ 1.7 & 14.3 $\pm$ 1.8 & ... & ... & ... \\
\hline
\end{tabular}
\tablefoot{Uncertainties and upper limits are of 3 $\sigma$ level and do not include systematic errors due to continuum determination and adopted band strengths.
Ref. \tablefoottext{a}{\citet{Oli11}}
}
\end{table*}
\subsubsection{Continuum determination: H$_2$O, CO, and CO$_2$ ices}
For H$_2$O , CO, and CO$_2$ ices, we use global continuum that is derived by fitting a polynomial of the third or fourth order to the observed spectra.
The wavelength regions used for the global continuum are typically 2.6--2.7 $\mu$m, 4.1--4.15 $\mu$m, and 4.8--4.9 $\mu$m, which are set to avoid spectral regions that show prominent absorption or emission features.
The estimated continua are shown in Fig. \ref{WFit1} together with the optical depth spectra derived based on these global continuum.
\subsubsection{Spectral fitting: H$_2$O, CO, and CO$_2$ ices}
Although H$_2$O, CO, and CO$_2$ ice column densities of the present targets are derived in our previous studies \citep{ST10,thesis}, we re-evaluate their column densities using the combined VLT + \textit{AKARI} spectra obtained in this work.
The contamination by PAH emission bands and diffuse warm dust emission is significantly reduced in the present VLT spectra, thanks to the higher spatial resolution, which enables us to derive more reliable ice column densities.
For water ice, we fit laboratory ice spectra to the observed spectra with a $\chi^2$ minimization method and derived the column density by the following equation:
\begin{equation}
N(H_2O) = \int \tau d\nu / A, \label{Eq_column}
\end{equation}
where $N(H$$_2$$O)$ is the column density of the H$_2$O ice in units of cm$^{-2}$, $\tau$ is the optical depth, $\nu$ the wavenumber in units of cm$^{-1}$, $A$ is the band strength based on laboratory measurements.
The integration is performed over the wavelength region of the H$_2$O ice absorption band.
Wavelength regions used for the fit are set to avoid the bottom and the long-wavelength wing of the H$_2$O absorption band.
The fitted wavelengths are typically 2.75--2.95 $\mu$m and 3.1--3.15 $\mu$m.
Laboratory ice profiles of the H$_2$O and CO$_2$ ice mixture at 10 K and 80 K (H$_2$O:CO$_2$ = 100:14), which are similar to the typical chemical compositions of circumstellar ices around high-mass YSOs, are simultaneously fitted to the observed spectra.
The laboratory spectrum is taken from the Leiden database.
We adopt the band strength of the H$_2$O ice band as 2.0$\times$10$^{-16}$ cm molecule$^{-1}$ \citep{Ger95}.
The result of the fitting is shown in Fig. \ref{WFit1} and derived column densities are summarized in Tables \ref{Tab_tau} and \ref{Tab_column}.
The resultant H$_2$O ice column densities are somewhat larger than those derived by \textit{AKARI} data in \citet{ST10}, particularly for ST2, ST5, ST7, and ST10.
This is because ISAAC spectra are less contaminated by surrounding emissions by PAH and warm dust.
For CO and CO$_2$ ices, we use the same method as presented in \citet{ST10} for spectral fitting and for calculation of ice column densities.
The results of spectral fitting are shown in Fig. \ref{App_Spec2}.
The derived column densities are summarized in Tables \ref{Tab_column}.
The resultant ice column densities are generally larger than the previous estimates due to a decrease in the overall flux level of \textit{AKARI} spectra.
\subsubsection{Notes on individual sources}
\textbf{ST6:} The source shows the deepest absorptions of the H$_2$O, CH$_3$OH, and 3.47 $\mu$m band among the sources examined in this study.
This suggests the deeply-embedded and chemically-rich nature of the source.
We additionally analyzed \textit{Spitzer}/IRS 5--33 $\mu$m spectrum of the source to investigate properties of ice absorption bands in the mid-infrared region.
The IRS spectral data were taken from the SAGE-Spec database \citep{Kem10,Woo11}.
The result of the analysis is presented in $\S$ 3.3.
\textbf{ST10:} The source shows the second deepest absorptions of the H$_2$O, CH$_3$OH, and 3.47 $\mu$m band after ST6.
We examined the IRS spectrum of the source, but it was severely contaminated by PAH emissions.
\textbf{ST17:} The source shows absorptions of the H$_2$O and 3.47 $\mu$m band, although the S/N ratio of the spectrum is poor.
The optical depth of the 3.47 $\mu$m band is estimated by visual inspection, but the uncertainty is large since the bottom of the absorption band is noisy.
The source shows a hint of the CH$_3$OH ice band, but it is difficult to claim the detection with this S/N.
The \textit{AKARI} spectrum of the source is contaminated by a nearby source particularly in the wavelength region shorter than 4 $\mu$m.
Thus the short-wavelength side of the H$_2$O absorption band is not available for this source, which makes it difficult to estimate the column density of the H$_2$O ice.
Data of ST17 are not used in the following discussion section because of the low spectral quality.
\begin{figure*}[!]
\begin{center}
\includegraphics[width=18cm, angle=0]{f6.eps}
\caption{
(a) \textit{AKARI}/IRC 2.5--5 $\mu$m and \textit{Spitzer}/IRS 5--33 $\mu$m spectrum of ST6.
The derived continuum is shown with dashed lines (green).
Detected absorption bands of solids are labeled.
(b) Upper panel: optical depth spectrum of ST6 in the 8--10.5 $\mu$m region.
The smooth solid line (brown) indicates the fitted profile of the silicate absorption band.
Lower panel: residual spectrum after the subtraction of the silicate absorption band.
Absorption bands of the NH$_3$ ice at 9.0 $\mu$m and CH$_3$OH ice at 9.75 $\mu$m are not seen in the spectrum.
}
\label{ST6_IRCIRS}
\end{center}
\end{figure*}
\subsection{Ice absorption bands in the mid-infrared spectrum of ST6}
Mid-infrared spectral regions provide us with another piece of important information about ice mantle compositions that is complementary to the near-infrared spectral data.
Figure \ref{ST6_IRCIRS}a shows 2.5--33 $\mu$m spectrum of ST6, which shows the richest ice absorption bands in the infrared region among the sources we investigated.
We examined IRS data of the other sources, but those are severely contaminated by surrounding emissions by PAH and warm dust.
Detailed analysis of minor ice absorption bands is thus difficult except for those of 15.2 $\mu$m CO$_2$ ice band, which were already presented in previous studies \citep{Oli09,Sea10}.
The IRS spectrum of ST6 is presented in \citet{Mat14}, but we carried out the analysis of ice absorption bands.
We estimate the continuum emission by fitting a spline function to the wavelength regions that are free from absorption/emission features (2.7 $\mu$m, 4 $\mu$m, 5--5.5 $\mu$m, 7.5 $\mu$m, and 30 $\mu$m), and derive optical depth spectrum.
Figure \ref{ST6_IRCIRS}b shows the optical depth spectrum of ST6 in the 8--10.5 $\mu$m range.
To estimate the contribution of the NH$_3$ ice absorption band at 9.0 $\mu$m (umbrella mode), we fit and subtract the silicate absorption, and a residual spectrum is shown in the lower panel of the figure.
Following the method described in \citet{Bot10}, the profile of the silicate absorption band is approximated by a polynomial function.
The NH$_3$ ice 9.0 $\mu$m absorption band is not detected prominently in the residual spectrum.
However, it seems that the absorption band is partially contaminated by the PAH emission at 8.61 $\mu$m.
We place an upper limit of 0.1 for the peak optical depth of the NH$_3$ ice absorption band in ST6.
This optical depth corresponds to the NH$_3$ column density of $\sim$3$\times$10$^{-17}$ cm$^{-2}$, if we assume the band strength of A = 1.3$\times$10$^{-17}$ cm molecule$^{-1}$ and the FWHM of $\Delta$$\lambda$ = 0.3 $\mu$m \citep{Bot10}.
Thus, the upper limit for the abundance of solid NH$_3$ relative to water ice is $<$5 $\%$ for ST6.
The CH$_3$OH ice shows an absorption band at 9.75 $\mu$m (C--O stretching mode), which is not seen in the spectrum.
The IRS spectrum shows weak emission lines of molecular hydrogen at 6.91 $\mu$m (0--0 S(5)), 9.67 $\mu$m (0--0 S(3)), 12.28 $\mu$m (0--0 S(2)), 17.04 $\mu$m (0--0 S(1)), 28.22 $\mu$m (0--0 S(0)).
The 9.75 $\mu$m CH$_3$OH band is blended with a H$_2$ 0--0 S(3) line at 9.67 $\mu$m.
Thus it is difficult to place an upper limit for the optical depth of the CH$_3$OH ice band.
Figure \ref{ST6_IRS2} shows the IRS spectrum in the 5--8 $\mu$m region.
Several absorptions are seen in addition to the water ice absorption band at 6.0 $\mu$m (bending mode).
To estimate the contribution of these additional absorption components, we estimate the strength of the 6.0 $\mu$m water ice band based on the column density derived from the 3.05 $\mu$m band and subtracted from the spectrum.
We assume the laboratory spectrum of pure water ice at 10 K (data taken from the Leiden database) as a profile of the 6.0 $\mu$m band.
The residual spectrum is shown in the lower panel of the figure together with possible molecules that contribute to the absorption bands \citep{Boo08,Boo15}.
The figure hints at the possibility that various complex species could exist in the low metallicity environment of the LMC.
Higher spectral- and spatial-resolution observations will enable us a more detailed analysis of these minor solids bands for a larger number of LMC and SMC YSOs.
\begin{figure}[!]
\begin{center}
\includegraphics[width=9cm, angle=0]{f7.eps}
\caption{
Upper panel: optical depth spectrum of ST6 in the 5.5--8 $\mu$m region.
The smooth solid line (blue) indicates the expected spectrum of the 6.0 $\mu$m water ice band, whose strength is estimated based on the 3.05 $\mu$m band.
Lower panel: spectrum after subtraction of the 6.0 $\mu$m water ice band.
Absorption components that cannot be explained by the water ice are seen in the spectrum.
Possible identifications of these absorption bands are indicated.
}
\label{ST6_IRS2}
\end{center}
\end{figure}
\section{Discussions}
The present data enable us to discuss spectral properties of the C--H stretching vibration region of embedded high-mass YSOs in the LMC.
In this section, we discuss methanol ice chemistry and possible carriers of the 3.47 $\mu$m band in the LMC.
Implications for the formation of organic molecules in low metallicity environments are also discussed.
\subsection{Methanol ice in the LMC}
\subsubsection{Comparison of methanol ice abundance with Galactic sources}
Figure \ref{histo_CH3OH} compares the abundance of the CH$_3$OH ice between LMC and Galactic high-mass YSOs.
An abundance of ice is defined as a ratio of a column density relative to the water ice column density, since water is the most abundant solid species in dense clouds.
The plotted data in the figure are summarized in Tables \ref{Tab_column} and \ref{Tab_MW2}.
For all the lines of sight of the present ten LMC YSOs, the CH$_3$OH ice abundance is less than 5--8 $\%$.
Even for the sources with CH$_3$OH ice detection (ST6 and ST10), the derived abundances are as small as 5.6 and 3.7 $\%$, respectively.
On the other hand, four out of 13 (about one-third of the total) Galactic samples show CH$_3$OH ice abundances between 10 $\%$ and 40 $\%$.
Although statistical uncertainties still remain as a result of the small number of samples, the present results suggest that solid CH$_3$OH is less abundant in the LMC than in our Galaxy within the present investigations.
\citet{Sea10} also suggested low CH$_3$OH ice abundance in LMC YSOs on the basis of the spectral analysis of the 15.2 $\mu$m CO$_2$ ice band.
However, determination of CH$_3$OH ice abundances based on the profile analysis of the CO$_2$ ice band entails various uncertainties, as the authors cautioned in their paper.
The present analysis provides direct evidence of the low CH$_3$OH ice abundance in LMC YSOs based on the measurement of the C--H stretching vibration band of the CH$_3$OH ice.
Figure \ref{histo_H2O} shows a histogram of H$_2$O ice column densities for the LMC and Galactic samples.
Since water is the most abundant ice species except molecular hydrogen, its column densities dominate the total ice column density along the line of sight.
Although \citet{Oli11} suggested that the H$_2$O ice is selectively depleted in LMC YSOs, the present figure shows that the distribution of H$_2$O ice column densities is similar between the observed LMC YSOs and Galactic samples except for the extremely embedded sources, which are discussed below.
Further, luminosities and appearances of infrared spectra are also similar between the LMC and Galactic samples \citep[][see also Fig. \ref{Spec}--\ref{Spec} in this work]{ST10,Gib04}.
These facts suggest that the present LMC samples share similar properties (e.g., evolutionary stages) with the Galactic samples.
Nevertheless, we did not detect any source with a high CH$_3$OH ice abundance, which indicates that the solid methanol is less abundant in the LMC.
However, in the present LMC samples, we are still missing extremely embedded sources (N(H$_2$O) $>$ 10$^{19}$ cm$^{-2}$), such as W33A or AFGL7009S in our Galaxy.
Although such extreme sources are not yet detected in the LMC, their infrared spectral information will be crucial to improve our understanding of ice chemistry in low metallicity environments.
Further systematic observations of embedded YSOs in the LMC are obviously important.
\begin{figure}[!]
\begin{center}
\includegraphics[width=9.1cm, angle=0]{f8.eps}
\caption{
Histogram of the CH$_3$OH ice abundances for the LMC high-mass YSOs (red) and Galactic high-mass YSOs (green).
All the LMC samples show the CH$_3$OH ice abundance less than 10 $\%$ relative to water ice, while several Galactic samples show higher CH$_3$OH ice abundances.
}
\label{histo_CH3OH}
\end{center}
\end{figure}
\begin{figure}[!]
\begin{center}
\includegraphics[width=8.5cm, angle=0]{f9.eps}
\caption{
Histogram of the H$_2$O ice column densities for the LMC (red) and Galactic (green) high-mass YSOs.
The number of samples for which the CH$_3$OH ice absorption is detected is indicated with transverse lines.
}
\label{histo_H2O}
\end{center}
\end{figure}
\begin{table*}
\centering
\caption{Ice column densities and abundances for Galactic high-mass YSOs}
\label{Tab_MW2}
\begin{tabular}{ l c c c c c c c}
\hline\hline
Object & \textit{N}(H$_2$O) & \textit{N}(CH$_3$OH)\tablefootmark{*} & \textit{N}(CO) & \textit{N}(CO$_2$) & \textit{N}(CH$_3$OH) & \textit{N}(CO) & \textit{N}(CO$_2$) \\
& & & & & /\textit{N}(H$_2$O) & /\textit{N}(H$_2$O) & /\textit{N}(H$_2$O) \\
& (10$^{17}$ cm$^{-2}$) & (10$^{17}$ cm$^{-2}$) & (10$^{17}$ cm$^{-2}$) & (10$^{17}$ cm$^{-2}$) & ($\%$) & ($\%$) & ($\%$) \\
\hline
S140 IRS 1 & 19\tablefootmark{a} & $<$0.9\tablefootmark{b} & ... & 4.2\tablefootmark{c} & $<$4.5 & ... & 22.1 \\
Mon R2 IRS 2 & 35.6\tablefootmark{a} & 1.9\tablefootmark{d} & 2.7\tablefootmark{a} & 6.0\tablefootmark{c} & 5.4 & 7.6 & 16.9 \\
Mon R2 IRS 3 & 19\tablefootmark{a} & $<$1.3\tablefootmark{b} & ... & 1.6\tablefootmark{c} & $<$6.7 & ... & 8.4 \\
RAFGL989 & 22.4\tablefootmark{e} & 0.7\tablefootmark{e} & 4.5\tablefootmark{a} & 8.1\tablefootmark{c} & 3.3 & 20 & 36.2 \\
RAFGL2136 & 45.7\tablefootmark{e} & 3.5\tablefootmark{d} & 2.7\tablefootmark{a} & 7.8\tablefootmark{c} & 7.6 & 5.9 & 17.1 \\
RAFGL2591 & 12\tablefootmark{a} & 2.4\tablefootmark{a} & ... & 1.6\tablefootmark{c} & 20.3 & ... & 13.3 \\
RAFGL7009S & 113.1\tablefootmark{e} & 35.5\tablefootmark{b} & 18\tablefootmark{a} & 25\tablefootmark{c,+} & 31.4 & 16 & 22.1 \\
W33 A & 125.7\tablefootmark{e} & 19.9\tablefootmark{d} & 8.9\tablefootmark{a} & 14.5\tablefootmark{c} & 15.8 & 7.1 & 11.5 \\
NGC 7538 IRS1 & 22\tablefootmark{a} & $<$1.1\tablefootmark{b} & 1.8\tablefootmark{a} & 5.1\tablefootmark{c} & $<$5.1 & 8.2 & 23.2 \\
NGC 7538 IRS9 & 64.1\tablefootmark{e} & 4.1\tablefootmark{d} & 12\tablefootmark{a} & 16.3\tablefootmark{c} & 6.4 & 18.7 & 25.4 \\
Orion BN & 25\tablefootmark{a} & $<$1.1\tablefootmark{b} & ... & 2.9\tablefootmark{c} & $<$4.5 & ... & 11.6 \\
Orion IRC2 & 24.5\tablefootmark{a} & 3.6\tablefootmark{a} & ... & 2.6\tablefootmark{c} & 14.6 & ... & 10.6 \\
W3 IRS 5 & 56.5\tablefootmark{e} & $<$2.3\tablefootmark{b} & 1.6\tablefootmark{a} & 7.1\tablefootmark{c} & $<$4.1 & 3.1 & 12.6 \\
\hline
\end{tabular}
\tablefoot{All the values are taken from the literature.
\tablefoottext{*}{Methanol column densities are recalculated using the band strength of 5.3$\times$10$^{-18}$ cm molecule$^{-1}$ and $\Delta$$\nu$ of 31 cm$^{-1}$, except for RAGL7009S, whose column density is estimated from the combination modes at 3.9 $\mu$m. }
\tablefoottext{+}{Estimated from the 15.2 $\mu$m absorption band. } \\
Ref.
\tablefoottext{a}{\citet{Gib04}};
\tablefoottext{b}{\citet{Dar99}};
\tablefoottext{c}{\citet{Gib04}};
\tablefoottext{d}{\citet{Bro99}};
\tablefoottext{e}{\citet{Boo08}
}}
\end{table*}
\subsubsection{Warm ice chemistry}
We propose the warm ice chemistry hypothesis to explain the characteristic chemical properties of ices in the LMC revealed by the present and previous observations.
It is widely accepted that CH$_3$OH is mainly formed by solid-phase reactions.
One possible reaction mechanism is caused by diffusive grain surface chemistry.
Laboratory experiments suggest that hydrogenation of CO plays a key role in the formation of CH$_3$OH \citep[e.g.,][]{Wat02,Wat07}.
The temperature dependance of the formation of H$_2$CO and CH$_3$OH by hydrogenation is investigated by experiments, which have reported that their formation is significantly suppressed when the surface temperature is higher than 20 K.
Numerical simulations of grain surface chemistry also suggest that the formation efficiency of CH$_3$OH decreases as the temperature of dust grains increase \citep[e.g., ][]{Ruf01,Cup09,Cha12}.
These studies suggest that hydrogenation of CO, which leads to the formation of CH$_3$OH, becomes less efficient at high dust temperatures.
These behaviors can be explained qualitatively by very rapid diffusion or evaporation of hydrogen atoms at increased surface temperature.
The binding energy of a hydrogen atom on the ice surface is known to be much smaller than those of other species \citep[][and references therein]{HW13}.
When dust temperature is sufficiently low ($\sim$10 K), hydrogen atoms can diffuse on the surface at a moderate speed to find and to react with CO.
At increased temperatures ($\sim$20 K), hydrogen starts to diffuse very rapidly on the surface, which suppresses hydrogenation because hydrogen atoms move to other surface sites before reacting with CO even though they could meet on the surface.
At even higher temperatures, hydrogen atoms start to evaporate rapidly from the surface, which further suppresses hydrogenation.
A detailed study of solid CH$_3$OH for Galactic YSOs and quiescent clouds suggest that the efficiency of CH$_3$OH production in dense cores and protostellar envelopes is mediated by the degree of prior CO depletion \citep{Whi11}.
Abundances of solid CO in the lines of sight of the present LMC high-mass YSOs are summarized in Table \ref{Tab_column}, while those for Galactic sources are in Table \ref{Tab_MW2}.
Mean CO ice abundances and standard deviations excluding sources with upper limit are 9.4 $\pm$ 5.3 for the LMC samples and 10.8 $\pm$ 6.5 for the Galactic samples.
The similar CO ice abundance for LMC and Galactic sources suggests that freeze out of CO gas onto grain surfaces occurs even in the LMC.
Sufficient freeze out of CO is also suggested previously for several LMC YSOs \citep{Oli11}.
Therefore, we suggest that less efficient hydrogenation of CO due to warm dust temperatures suppress production of solid CH$_3$OH around high-mass YSOs in the LMC rather than inefficient depletion of CO.
An alternative formation mechanism for solid CH$_3$OH involves energetic processes of ice mantles, such as the formation through UV photon irradiation (photolysis) or proton irradiation (radiolysis), which are proposed by experimental studies \citep[e.g., ][]{Hud99,Ger01,Bar02,Wat07}.
The UV radiation field in dense shielded regions of molecular clouds is dominated by cosmic-ray induced UV photons.
The strength of cosmic-ray induced UV radiation field is closely related to the cosmic-ray density, which is reported to be lower in the LMC than in our Galaxy \citep{Abd10}.
This indicates that energetic processes induced by cosmic rays are expected to be less efficient in dense clouds in the LMC.
Thus, if the formation of solid CH$_3$OH is dominated by energetic processes, the low cosmic-ray density in the LMC can account for the observed low CH$_3$OH ice abundance.
However, this does not reject the hypothesis of warm ice chemistry because the reduction of energetic processes cannot explain the increased abundance of the CO$_2$ ice, which is discussed in Section 4.1.3.
With the current observational data, it is difficult to separate the relative contribution of warm ice chemistry and the low cosmic-ray density on the suppressed formation of CH$_3$OH in the LMC.
On the other hand, the strong interstellar UV radiation field in the LMC does not contribute to the energetic formation processes of CH$_3$OH.
This is because the formation of the CH$_3$OH ice requires much denser and more shielded environments (Av $\sim$17) than those required for the H$_2$O ice formation (Av $\sim$6), as discussed in \citet{Whi11}.
The interstellar UV photons are heavily attenuated in such very dense clouds, and the CH$_3$OH ice formation is not affected by the strength of the external UV radiation field.
Water ice, which is abundantly detected in the LMC, is also mainly formed by hydrogenation of surface species in dense clouds.
One possible reaction pathway for water ice formation is by $OH + H \to H_2O$ \citep[e.g., ][and references therein]{vanD13}.
Under the warm ice chemistry hypothesis, it should be explained why hydrogenation of OH can occur in the LMC, while hydrogenation of CO is suppressed.
We speculate that this is caused by the difference of an activation barrier in each hydrogenation pathway.
It is known that the surface reaction of $CO + H \to HCO$ and $H_2CO + H \to H_3CO$ possess a high activation barrier \citep[$>$ 400 K, ][]{Woo02,Fuc09}.
This indicates that CO and H have to stay on the same or adjacent surface site for a certain amount of time to react.
In this case, hydrogenation is suppressed at high temperatures because of very rapid diffusion or evaporation as mentioned above.
On the other hand, the hydrogenation of OH to H$_2$O is known to be barrierless.
Very rapid diffusion of hydrogen does not suppress hydrogenation of OH because their reaction proceeds immediately after the encounter of reactants.
In addition, the Eley-Rideal mechanism, which is caused by direct collisions of the gas-phase species to the surface species, may also contribute to the formation.
Thus, we speculate that formation of water ice is still possible even at moderately warm dust temperatures where CO hydrogenation becomes less efficient.
In the LMC, it is highly possible that the typical temperature of molecular clouds is higher than in our Galaxy due to stronger interstellar radiation field.
The temperature of dust in molecular clouds should be closely related to metallicity of the galaxy since the interstellar radiation field is attenuated by dust grains.
Several studies actually argue that the typical dust temperature of diffuse clouds in the LMC is higher than in our Galaxy based on far-infrared to sub-millimeter observations \citep{Agu03,Sak06}.
\citet{vanL10_b} reported that dust around high-mass YSOs and compact HII regions in the SMC is warmer (37--51 K) than those in comparable objects in the LMC (32-44 K).
For further extragalaxies, it is reported that the color temperature of galaxies measured by far-infrared observations is negatively correlated with their metallicities \citep[e.g.,][]{Eng08}.
Although dust temperatures of {dense} clouds in the LMC, where significant amount of ices are formed, are still an open question, the above studies imply the higher dust temperatures of dust in lower metallicity environments.
Spectral profiles of the 15.2 $\mu$m CO$_2$ ice band for high-mass YSOs in the LMC are well studied with \textit{Spitzer} observations \citep{Oli09,Sea10}.
\citet{Sea10} suggest that the CO$_2$ ices around high-mass YSOs in the LMC are more thermally processed than those in our Galaxy.
This also supports the idea that the increased dust temperature in the LMC affects the properties of ices around embedded YSOs.
The warm ice chemistry scenario suggests that the low CH$_3$OH ice abundance in the LMC favor warmer dust temperatures in dense shielded regions where ices are formed.
Future measurements of dust and gas temperatures of compact dense clouds in the LMC will be crucial for testing the present hypothesis and for understanding of ice chemistry in low metallicity environments.
\subsubsection{Decrease of CH$_3$OH and increase of CO$_2$ as a consequence of warm ice chemistry}
The relatively high abundance of the CO$_2$ ice in the LMC, which was reported in previous infrared observations \citep{ST,ST10,Oli09,Sea10}, is also consistent with the above warm ice chemistry scenario.
The CO$_2$/H$_2$O ice ratio for the present LMC high-mass YSOs are compared with those of Galactic high-mass YSOs in Fig. \ref{histo_CO2}.
The mean value and standard deviation of the CO$_2$/H$_2$O ice ratio is 28.5 $\pm$ 11.6 $\%$ for the ten LMC samples and 17.8 $\pm$ 7.8 $\%$ for the 13 Galactic samples, respectively (see Tables \ref{Tab_column} and \ref{Tab_MW2}).
The median value of the CO$_2$/H$_2$O ice ratio is 29.1 $\%$ for the LMC samples and 16.9 $\%$ for the Galactic samples, respectively.
The high-mass YSOs in the LMC show the higher CO$_2$ ice abundance compared to Galactic high-mass YSOs as suggested in previous studies.
It is known that the reaction of $CO + OH \to CO_2 + H$ is one of the dominant pathways for the formation of the CO$_2$ ice \citep[e.g.,][]{Oba10,Iop11}.
As suggested by numerical simulations \citep[e.g.,][]{Ruf01,Cha12}, this formation pathway becomes efficient as the dust temperature increases because the mobility of CO on the surface increases accordingly.
Since the binding energy of CO on the ice surface is much larger than that of hydrogen, CO can still remain and moderately diffuse on the surface even at high dust temperatures at which hydrogen atoms evaporate.
Therefore, the high dust temperature in the LMC can enhance the formation of CO$_2$ by increasing the mobility of CO, while it can reduce the formation of CH$_3$OH by suppressing the hydrogenation of CO.
The lower elemental C/O ratio in the LMC than in our Galaxy (see $\S$4.2.2) can also contribute to the low abundance of CH$_3$OH relative to H$_2$O.
However, we emphasize that the CO$_2$ ice abundance is enhanced in the LMC despite the low C/O ratio.
Warm ice chemistry can simultaneously account for the enhanced formation of CO$_2$ and the suppressed formation of CH$_3$OH.
This would suggest that warm ice chemistry is responsible for the characteristic chemical compositions of ices in the LMC.
\citet{Oli11} argued that the high CO$_2$/H$_2$O ice ratio in the LMC is caused by the depletion of the H$_2$O ice rather than the enhanced formation of the CO$_2$ ice.
However, the enhanced formation of CO$_2$ is more consistent with the observed low abundance of the CH$_3$OH ice in the LMC, since both are expected in grain surface chemistry at a relatively high dust temperature, as discussed above.
The H$_2$O ice column densities are similar between the present LMC samples and Galactic samples except for two extremely embedded Galactic sources, as discussed in $\S$4.1.1.
Therefore, we suggest that enhanced formation of CO$_2$ is responsible for the high CO$_2$/H$_2$O ice ratio in the LMC.
Based on the above discussions, we conclude that warm ice chemistry is responsible for the characteristic chemical compositions of ices around high-mass YSOs in the LMC.
The present results suggest that conditions to form CH$_3$OH are less often encountered in the LMC's environment presumably due to warm ice chemistry.
A key factor for warm ice chemistry is less efficient hydrogenation of surface species, which is triggered by the elevated temperature of dust grains in dense molecular clouds.
Dust temperatures of ice-forming dense clouds in metal-poor environments are expected to be higher than those in metal-rich environments, as discussed in the previous section.
We therefore suggest that warm ice chemistry is one of the characteristics of interstellar and circumstellar chemistry of dense ISM in low metallicity galaxies.
\begin{figure}[!]
\begin{center}
\includegraphics[width=9.2cm, angle=0]{f10.eps}
\caption{
Histogram of the CO$_2$ ice abundances for the LMC high-mass YSOs (red) and Galactic high-mass YSOs (green).
The LMC samples show the relatively high CO$_2$/H$_2$O ice ratio compared to the Galactic samples.
}
\label{histo_CO2}
\end{center}
\end{figure}
\subsubsection{Gas-phase methanol in the LMC}
Previous radio observations have reported the underabundance of methanol masers in the LMC in comparison to our Galaxy \citep{Bea96,Gre08,Ell10}.
Thermal emission lines of methanol are detected toward a limited number of HII regions in the LMC \citep{Hei99,Wan09}.
Recent unbiased spectral line surveys toward the LMC's molecular clouds suggest that thermal emission lines due to gas-phase methanol is significantly weaker in the LMC than in our Galaxy \citep[Shimonishi et al. in prep.;][]{Nis15}.
We suggest that the low production rate of solid methanol due to warm ice chemistry contributes to few detections of methanol gas in the LMC.
\subsection{The 3.47 $\mu$m band in the LMC}
In contrast to the very weak or absent CH$_3$OH ice absorption in the LMC, six out of eleven sources show the 3.47 $\mu$m absorption band.
We here discuss the difference and similarity of the band between the LMC and our Galaxy to understand properties of the band carrier in low metallicity environment.
\subsubsection{Possible carriers of the 3.47 $\mu$m band}
The 3.47 $\mu$m absorption band is widely detected toward a variety of embedded sources such as high- to intermediate-mass YSOs \citep[e.g., ][]{All92,Bro96,Bro99,Dar01,Dar02,Ish02}, low-mass YSOs \citep[e.g., ][]{Pon03,Thi06}, and quiescent dense molecular clouds \citep[e.g., ][]{Chi96,Chi11}.
Despite many spectroscopic detections, the carrier of the 3.47 $\mu$m band is still under debate and various candidates have been proposed in the literature.
\citet{All92} argued that the C--H vibration of hydrogen atoms bonded to tertiary carbon atoms could be responsible for the 3.47 $\mu$m band
\footnote{There are three classes of carbon atoms related to C--H vibrations of aliphatic hydrocarbons; primary, secondary, and tertiary aliphatic carbons.
In the primary aliphatic carbon, three of its four single bonds are directed to other carbons.
Similarly, in the secondary and the tertiary carbon, one or two of its four bonds are directed to other carbons \citep{All92}. }.
Although the primary and secondary hydrocarbons also show characteristic bands in 3.4--3.5 $\mu$m regions \citep[e.g., ][]{Men10}, the predominance of the 3.47 $\mu$m band in various embedded sources in our Galaxy puts constraints on the structure of hydrocarbons in dense clouds.
The authors suggest that the only way in which tertiary carbon atoms can dominate the spectrum is if the carbon atoms are arranged in a diamond structure.
These interstellar diamonds may have a connection with nano-sized diamonds in meteorites \citep[e.g., ][]{Lew87,Pir07}.
Alternatively, \citet{Dar01} and \citet{Dar02} argued that the 3.47 $\mu$m band is related to an ammonia hydrate formed in NH$_3$:H$_2$O mixture ice.
The authors show that a stretching vibration mode arising from the interaction between a nitrogen atom of NH$_3$ and an O--H bond of H$_2$O can account for a part of the 3.47 $\mu$m absorption.
It was previously reported for Galactic sources that optical depths of the 3.47 $\mu$m band correlate well with those of the H$_2$O ice band \citep{Bro99}.
An ammonia hydrate, as a carrier of the 3.47 $\mu$m band, is thus consistent with the correlation of the 3.47 $\mu$m band and the H$_2$O ice.
\begin{figure*}[!]
\begin{center}
\includegraphics[width=18cm, angle=0]{f11.eps}
\caption{
(a) Optical depths of the 3.05 $\mu$m H$_2$O ice band vs. the 3.47 $\mu$m band.
Filled (black) and open (green) squares represent data points of the LMC high-mass YSOs (this work) and Galactic high-mass YSOs (see Table \ref{Tab_MW} for references).
The upward and downward triangles represents the lower and upper limit.
Solid (black) and dashed (green) lines represent the results of a straight-line fit to LMC and Galactic data points, respectively.
The best-fit line is $\tau_{3.47} = 0.044\tau_{3.05} - 0.030$ for the LMC, and $\tau_{3.47} = 0.043\tau_{3.05} - 0.008$ for our Galaxy.
The straight-line fit without upper limit points is also shown with black dashed line for the LMC data.
The figure shows that the slope of the $\tau_{3.47}$ and $\tau_{3.05}$ correlation is similar between the LMC and Galactic sources.
(b) Optical depth ratio of $\tau_{3.47}$/$\tau_{3.05}$ vs. $\tau_{3.05}$.
The symbols are same as in the left panel.
For the LMC sources with $\tau_{3.05}$ $<$ 2, the $\tau_{3.47}$/$\tau_{3.05}$ ratio is significantly lower than Galactic sources, which suggests the presence of threshold $\tau_{3.05}$ for the appearance of the 3.47 $\mu$m band in the LMC.
}
\label{t31_t347}
\end{center}
\end{figure*}
\begin{table}[b]
\centering
\caption{Optical depths of the water ice and the 3.47 $\mu$m absorption bands for Galactic high-mass YSOs}
\label{Tab_MW}
\begin{tabular}{ l c c}
\hline\hline
Object & $\tau$$_{3.05 \mu m}$ & $\tau$$_{3.47 \mu m}$ \\
\hline
S140 IRS 1 & 1.12\tablefootmark{a} & 0.027\tablefootmark{b} \\
Mon R2 IRS 2 & 1.77\tablefootmark{a} & 0.083\tablefootmark{c} \\
Mon R2 IRS 3 & 1.12\tablefootmark{a} & 0.036\tablefootmark{c} \\
RAFGL989 & 1.34\tablefootmark{+} & ... \\
RAFGL2136 & 2.73\tablefootmark{+} & 0.14\tablefootmark{c} \\
RAFGL2591 & 0.74\tablefootmark{a} & 0.045\tablefootmark{c} \\
RAFGL7009S & 6.75\tablefootmark{+} & ... \\
W33 A & 7.50\tablefootmark{+} & $>$0.29\tablefootmark{c} \\
NGC 7538 IRS1 & 1.27\tablefootmark{a} & 0.052\tablefootmark{b} \\
NGC 7538 IRS9 & 3.83\tablefootmark{+} & 0.13\tablefootmark{c} \\
Orion BN & 1.44\tablefootmark{a} & 0.034\tablefootmark{b} \\
Orion IRC2 & 1.48\tablefootmark{a} & ... \\
W3 IRS 5 & 3.37\tablefootmark{+} & 0.13\tablefootmark{b} \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{+}{Optical depths of the H$_2$O ice are calculated based on the corresponding column densities in Table \ref{Tab_MW2} assuming $\Delta$$\nu$ = 335 cm$^{-1}$ and A = 2.0$\times$10$^{-16}$ cm molecule$^{-1}$. }
\\
Ref.
\tablefoottext{a}{\citet{Gib04}};
\tablefoottext{b}{\citet{Bro96}};
\tablefoottext{c}{\citet{Bro99}};
}
\end{table}
\subsubsection{Comparison of the band properties with Galactic sources}
Figure \ref{t31_t347}a compares the optical depths of the 3.47 $\mu$m band and the 3.05 $\mu$m H$_2$O ice band for the LMC and Galactic samples.
The plotted data are summarized in Tables \ref{Tab_tau} and \ref{Tab_MW}.
The above-mentioned correlation of the 3.47 $\mu$m band and the H$_2$O ice observed for Galactic sources is also seen in the LMC sources, as shown in the figure.
A possible explanation for the observed correlation is that the carrier of the 3.47 $\mu$m band is formed by a process similar to that of water ice formation.
This process is presumably related to hydrogenation of carbon or nitrogen atoms on the dust surface since water ice is also formed by hydrogenation.
The figure suggests that the 3.47 $\mu$m band and the H$_2$O ice band correlate similarly between the LMC and Galactic samples, but the LMC sources seem to require a higher H$_2$O ice threshold for the appearance of the 3.47 $\mu$m band.
A least-squares fit of a straight line to the LMC data points gives
\begin{equation}
\tau_{3.47} = (0.044 \pm 0.004)\tau_{3.05} - (0.030 \pm 0.007),
\end{equation}
where the fitting is not weighted and upper limits are included as data point.
When the upper limits are excluded in the fit, the slope is 0.038 $\pm$ 0.011 and the intercept is 0.015 $\pm$ 0.031.
This line, however, does not account for the upper limit points as seen in the figure.
The same fitting for the Galactic data points gives
\begin{equation}
\tau_{3.47} = (0.043 \pm 0.004)\tau_{3.05} - (0.008 \pm 0.005)
.\end{equation}
An intercept value of $\tau_{3.05}$ at $\tau_{3.47}$ = 0 axis (i.e., threshold $\tau_{3.05}$) is slightly different between the LMC and our Galaxy.
The threshold value is estimated to be $\tau_{3.05}$ = 0.69 $\pm$ 0.23 for the LMC and $\tau_{3.05}$ = 0.19 $\pm$ 0.14 for our Galaxy, respectively.
The difference may suggest that a more shielded environment is necessary for the formation of the 3.47 $\mu$m band carrier in the LMC.
Figure \ref{t31_t347}b compares the $\tau_{3.47}$/$\tau_{3.05}$ ratio against $\tau_{3.05}$.
The LMC sources with $\tau_{3.05}$ $<$ 2 show significantly lower $\tau_{3.47}$/$\tau_{3.05}$ ratio than those of Galactic sources.
This again suggests the non-zero value of threshold $\tau_{3.05}$ for the presence of the 3.47 micron absorption in the LMC.
However, the ratio may not be the best representation once it is seen that there is a threshold.
Follow-up observations toward a larger number of embedded sources in the LMC are highly required to provide solid evidence on the observed difference of the threshold $\tau_{3.05}$ in the LMC.
On the other hand, the comparison of the samples in which the 3.47 $\mu$m band is detected suggests that the ratio of $\tau_{3.47}$ and $\tau_{3.05}$ is not significantly different between the LMC and our Galaxy.
The mean value and standard deviation of $\tau_{3.47}$/$\tau_{3.05}$ ratio are 0.0331 $\pm$ 0.0024 for the five LMC samples and 0.0392 $\pm$ 0.0124 for the nine Galactic samples, respectively.
The median value of $\tau_{3.47}$/$\tau_{3.05}$ ratio is 0.0320 for the LMC samples and 0.0386 for the Galactic samples, respectively.
The LMC samples show only slightly lower $\tau_{3.47}$/$\tau_{3.05}$ ratios compared to Galactic samples (see also Figure \ref{t31_t347}b).
This would suggest that, in the well-shielded regions where the 3.47 $\mu$m band is detected, the lower metallicity as well as the different elemental abundances and different interstellar environment of the LMC have little effect on the abundance ratio of the 3.47 $\mu$m band carrier and water ice.
In the LMC, the elemental abundance ratio of both C/O and N/O are lower than those in the sun: [C/O]$_{LMC}$/[C/O]$_{Sun}$ $\sim$0.55, [N/O]$_{LMC}$/[N/O]$_{Sun}$ $\sim$0.30 \citep{Duf82}.
An underabundance of the gas-phase atomic nitrogen in the ISM of the LMC is also confirmed by \textit{Spitzer}/MIPS far-infrared observations \citep{vanL10}.
Hence, abundances of both carbon bearing and nitrogen bearing species relative to oxygen bearing species decrease if their ratios simply depend on the initial elemental abundances.
However, formation efficiencies of interstellar molecules are not that straightforward, and previous studies actually reported that the CO$_2$/H$_2$O ice ratio is higher in the LMC than in our Galaxy despite the lower C/O ratio in the LMC \citep[e.g.,][]{ST10}.
This fact suggests that the simple argument from the low C/O ratio cannot rule out C- or N-bearing species as a potential candidate of the 3.47 $\mu$m band.
In the next section, we discuss about possible carriers of the 3.47 $\mu$m band in the environment of the LMC.
\subsubsection{The carrier of the 3.47 $\mu$m band in the LMC}
The better correlation between $\tau_{3.47}$ and $\tau_{3.05}$ than the correlation of $\tau_{3.47}$ and $\tau_{9.7}$ (silicate band) reported in previous studies \citep{Bro96,Bro99} suggests that the formation of the 3.47 $\mu$m band carrier is related to dense and cold regions where ices are formed.
As mentioned earlier, there are two hypotheses for the carrier of the 3.47 $\mu$m band, i.e., C--H bonds on diamonds and an ammonia hydrate.
We discuss which hypothesis is more consistent with the interstellar environment and the observed properties of the 3.47 $\mu$m band in the LMC.
Under the hypothesis that interstellar diamonds are responsible for the 3.47 $\mu$m band, the formation and destruction of C--H bonds on carbon grains should be considered.
\citet{Men08,Men10} confirms the formation of tertiary C--H bonds after irradiation of hydrogen atoms to carbon particle samples coated by water ice.
In this case, hydrogenation of carbon atoms occurs at the interface of dust and ice by penetration of hydrogen atoms from the ice surface to the grain surface.
Owing to warm ice chemistry in the LMC, as discussed in Section 4.1.2, hydrogenation of surface species is less efficient if there is an activation barrier because hydrogen atoms rapidly diffuse or evaporate on/from the surface before a reaction.
This is caused by a relatively high dust temperature in LMC molecular clouds.
These characteristics, however, should not be simply applied to the hydrogenation that proceeds at the interface of dust and ice.
Currently, it is difficult to quantitatively discuss the temperature dependence of such an internal hydrogenation reaction because of a lack of relevant laboratory information, and thus the formation efficiency of tertiary C--H bonds in the LMC remains to be explained.
On the other hand, C--H bonds in dense clouds are mainly destroyed by cosmic-ray hits and internal UV induced by interaction of cosmic rays and molecular hydrogen \citep{Men10}.
The flux of cosmic-ray induced UV photons is proportional to cosmic-ray fluxes \citep[e.g., ][]{Pra83,She04}.
It is reported based on gamma-ray observations that the cosmic-ray density in the LMC is 20--30$\%$ lower than in our Galaxy \citep{Abd10}.
Such a low cosmic-ray density can decrease the destruction efficiency of C--H bonds in the LMC.
If the 3.47 $\mu$m band dominantly arises from the C--H vibration of hydrogen atoms bonded to tertiary carbons, this could be a possible cause for the similar $\tau_{3.47}$/$\tau_{3.05}$ ratio between the LMC and our Galaxy despite the low C/O ratio in the LMC.
Under the hypothesis that an ammonia hydrate is responsible for the 3.47 $\mu$m band, molecular abundance of ammonia in the LMC should be considered.
Gas-phase ammonia is reported to be deficient in star-forming regions in the LMC; the fractional abundance is estimated to be 1.5--5 orders of magnitude lower than those in Galactic star-forming regions \citep{Ott10}.
Recent spectral line surveys toward molecular clouds in the LMC suggest that nitrogen-containing gas-phase molecules, such as HCN, HNC, N$_2$H$^+$, are also underabundant in the LMC \citep{Nis15}.
These studies suggest that the low elemental abundance of nitrogen (see $\S$4.2.2) as well the intense interstellar UV radiation field in the LMC contribute to the low abundances of nitrogen-bearing molecules.
An upper limit is estimated for the abundance of solid ammonia in ST6, which shows the deepest 3.47 $\mu$m band among the sources we investigated.
The estimated upper limit of the solid ammonia abundance (NH$_3$/H$_2$O) is $<$5 $\%$ for ST6 based on the spectral analysis of the \textit{Spitzer}/IRS data (see $\S$3.3).
The abundance of solid ammonia, estimated on the basis of the ammonia hydrate hypothesis of the 3.47 $\mu$m band, is reported to be less than or equal to 7 $\%$ for the two Galactic high-mass YSOs RAFGL 989 and RAFGL 2136 \citep{Dar02} and about 5 $\%$ for a larger Galactic sample \citep{Dar01}.
The solid ammonia is also observed toward Galactic low-mass YSOs through the umbrella mode at 9.0 $\mu$m \citep{Bot10}.
The typical abundance is around 5 $\%$ over 23 sources in their Table 2, excluding the two sources mentioned in the article as likely upper limits and the source EC82, whose silicate dust continuum is clearly observed in emission.
Hence, the upper limit of the solid ammonia abundance for the LMC source ST6 is not significantly different from Galactic values.
Numerical simulations of grain surface chemistry suggest that the abundance of solid ammonia decreases as the temperature of dust grains increases \citep[e.g., ][]{Sta04,Cha14}.
This behavior is interpreted as a decrease in hydrogenation efficiency of nitrogen on the dust surface.
This would suggest that warm ice chemistry discussed in $\S$4.1.2 can lower the formation efficiency of solid ammonia in the LMC.
Although a current observational constraint on solid ammonia abundance in the LMC is still less conclusive because of a sample, we speculate that the interstellar environment of the LMC favors the low abundance of solid ammonia, which is consistent with radio observations of gas-phase ammonia and numerical simulation of grain surface chemistry mentioned above.
These discussions indicate that the contribution from ammonia hydrate may be less significant in the LMC.
Our interpretation cannot exclude the contribution of an ammonia hydrate to the 3.47 $\mu$m band observed in our Galaxy.
Further observational constraints on abundances of ammonia for larger number of LMC samples are key to improving our understanding of the 3.47 $\mu$m band carrier in the LMC.
\subsection{Implications for the formation of organic molecules in low metallicity galaxies}
Complex organic molecules are widely detected toward Galactic chemically-rich sources such as hot cores and hot corinos \citep[e.g.,][]{vDB98,Her09}.
Understanding of the formation and evolution of these molecules in space is of interest in exploring the chemical complexity of the interstellar medium.
Particularly, properties of complex organic molecules in low metallicity environments are of great interest in order to discuss the building blocks of prebiotic molecules in the past metal-poor universe.
Formation processes of complex organic molecules in interstellar and circumstellar environments are not well understood yet, but grain surface chemistry is believed to play an important role.
Theoretical models suggest that CH$_3$OH is a key to the formation of complex organic molecules \citep[e.g.,][]{Mil97,NM04,Gar08a,Cha14}.
The sublimation of solid CH$_3$OH into gas-phase and subsequent reactions in warm and dense circumstellar regions can enhance the formation of complex molecules \citep[e.g.,][]{NM04}.
Alternatively, CH$_3$OH can produce heavy radicals like CH$_3$O or CH$_2$OH by photodissociation, which subsequently evolve into complex molecules by grain surface reactions \citep[e.g.,][]{Gar08a}.
Thus, the low CH$_3$OH abundance in the LMC implies that formation of complex organic molecules from methanol-derived species is less efficient in the LMC.
However, it is obvious that we should take diverse mechanisms into account to discuss complex molecular chemistry.
A variety of carbon-/oxygen-/nitrogen-bearing species (e.g., hydrocarbon, PAH) are possible building blocks of complex organic molecules.
Further laboratory and theoretical studies on the formation of complex organic molecules are needed in conjunction with observational efforts to detect complex species in low metallicity galaxies.
\section{Summary}
We present the results of near-infrared spectroscopic observations toward embedded high-mass YSOs in the LMC using VLT/ISAAC.
The medium-resolution ($R$ $\sim$500) spectra in the 3--4 $\mu$m region are presented for eleven sources in the LMC.
The achieved spatial resolution is about 0.15 pc at the distance of the LMC, which is higher than those of previous satellite observations with \textit{AKARI} or \textit{Spitzer} by a factor of $\sim$10.
The properties of detected ice absorption bands (water, methanol, 3.47 $\mu$m band) are investigated and we obtained the following conclusions.
\begin{enumerate}
\item
The H$_2$O ice absorption band at 3.05 $\mu$m is detected for all of the YSO samples.
Thanks to the high spatial resolution achieved by the VLT, contamination by PAH emission bands is significantly reduced, which enables us to better constrain the water ice column densities than those estimated from our previous \textit{AKARI} observations.
\item
The 3.53 $\mu$m CH$_3$OH ice absorption band for the LMC YSOs is found to be absent or weak compared to those seen toward Galactic counterparts.
The absorption band is marginally detected for two out of eleven objects.
We estimate the abundance of CH$_3$OH ice relative to water ice, which suggests that solid CH$_3$OH is less abundant in the LMC high-mass YSOs than in Galactic sources.
\item
We propose that warm ice chemistry in the LMC is responsible for the low abundance of solid CH$_3$OH presented in this work as well as the relatively high abundance of solid CO$_2$ reported in previous observations \citep[e.g., ][]{ST10,Oli11}.
When the dust temperature in a molecular cloud is high due to strong interstellar radiation field, hydrogenation of CO on the grain surface to form CH$_3$OH would become less efficient because of the decrease in available hydrogen atoms on the surface.
On the other hand, the formation of CO$_2$ through the $CO + OH \to CO_2 + H$ reaction could be enhanced as a result of the increased mobility of parent species.
This suppresses the CH$_3$OH production, whereas enhances the CO$_2$ production.
Since dust temperature of molecular clouds is believed to be related to metallicity of the host galaxy, such warm ice chemistry is presumably one of the important characteristics of interstellar and circumstellar chemistry in low metallicity galaxies.
\item
The 3.47 $\mu$m absorption band, which is generally seen in embedded sources, is detected toward six out of eleven objects in the LMC.
The 3.47 $\mu$m band and the H$_2$O ice band correlate similarly between the LMC and Galactic samples, but the LMC sources seem to require a slightly higher H$_2$O ice threshold for the presence of the 3.47 $\mu$m band.
The LMC sources with small H$_2$O ice optical depths show significantly lower $\tau_{3.47}$/$\tau_{3.05}$ ratio than those of Galactic sources.
The difference in the threshold may suggest that more shielded environment is necessary for the formation of the 3.47 $\mu$m band carrier in the LMC.
For the LMC sources with relatively large H$_2$O ice optical depths, we found that the $\tau_{3.47}$/$\tau_{3.05}$ ratio is only marginally low compared to Galactic sources.
This would suggest that, in well-shielded regions, the lower metallicity as well as the different elemental abundances and different interstellar environment of the LMC have little effect on the abundance ratio of the 3.47 $\mu$m band carrier and water ice.
\item
The formation of complex organic molecules in low metallicity environment of the LMC is discussed based on the above results.
CH$_3$OH is believed to be a starting point for formation of more complex carbonaceous molecules.
The low CH$_3$OH ice abundance in the LMC implies that formation of complex organic molecules from methanol-derived species is less efficient in the LMC.
However, formation processes of complex organic molecules in space are not well understood, and thus further observational, theoretical, and laboratory studies are needed to constrain the complex molecular chemistry in the past metal-poor universe.
\end{enumerate}
We stress that the above conclusions are based on observations of a small number of extragalactic YSO samples.
Further spectroscopic observations with a larger number of samples with higher sensitivity and higher spectral resolution are critically needed for comprehensive understanding of ice chemistry in low metallicity environments.
\begin{acknowledgements}
This research is based on observations with the Very Large Telescope at the European Southern Observatory, Paranal, Chile (programme ID 090.C-0497).
We thank the VLT and ISAAC operation team members.
We are especially grateful to the support astronomers for many helpful comments on site during our observations.
This work uses data obtained by the \textit{AKARI} satellite, a JAXA project with the participation of ESA, and those obtained by NASA's \textit{Spitzer} Space Telescope.
We are grateful to all the members who contributed to these projects.
This work has made extensive use of the Leiden Ice Database.
This work is supported by the Japan Society for the Promotion of Science (JSPS) Research Fellow (24006664) and a Grant-in-Aid from the JSPS (15K17612).
Finally, we would like to thank our referee, Jacco Th. van Loon, whose suggestions greatly improved this paper.
\end{acknowledgements}
|
1,116,691,499,412 | arxiv | \section{Introduction}\label{sec: intro}
Laser cavitation is widely used for medical applications~\cite{Peng2008}, including ablation of biological tissue~\cite{Vogel2003} and lithotripsy~\cite{Fried2018a,Marks2007}. Lately, lasers have also been investigated for their use in laser-actuated needle-free jet injectors (NFJIs)~\cite{Schoppink2022}. These NFJIs present many advantages over conventional hypodermic needles, such as improved patient compliance~\cite{Daly2020}, reduction in needle waste~\cite{Mitragotri2005}, improved safety for healthcare workers~\cite{Mitragotri2006}, injection of high-viscous liquids~\cite{Williams2016} and better control over injection parameters~\cite{Schoppink2022,McKeage2018a}.
The working principle of these laser-actuated NFJIs is similar to that of laser-induced forward transfer, where a laser is heating a liquid resulting in the formation of a vapor bubble~\cite{Sopena2017,Santos2020}. For NFJIs, the liquid is contained in an open-ended microfluidic channel, such that the explosively growing bubble pushes the liquid through a small opening, which results in the formation of a fast microfluidic jet with the ability to penetrate the skin.
Jet velocities are controlled by the delivered energy~\cite{Tagawa2012}, optical beam size~\cite{Krizek2020}, channel size~\cite{Tagawa2012,Berrospe-Rodriguez2016} or the inclusion of a tapered nozzle~\cite{OyarteGalvez2020}. The injection depth into the tissue is proportional to this jet velocity~\cite{Krizek2020}, which can be used to target specific skin layers~\cite{Schoppink2022}.
Although the liquid is heated by the laser, temperature measurements show moderate and temporal temperature increase~\cite{Quinto-Su2014, Banks2019}. Furthermore, degradation studies showed no damage to the drug molecules~\cite{Krizek2020,Krizek2021}. To avoid any risks of degrading the liquid to be injected, some injectors include a membrane to split two chambers~\cite{Han2010,Ham2017}. For these reasons, laser heating of the liquid is not hindering the applicability of jet injection.
Besides the standard concept of laser-actuated jet injection, several additions have been introduced. Instead of using lenses to guide and focus the light, the use of optical fibers allows for smaller and more flexible devices and results in a reproducible beam size~\cite{Krizek2020,Krizek2021}. Furthermore, repetitive jetting results in the desired injection volume and can increase the injection depth~\cite{Jang2014,Ham2017,Krizek2020a,OyarteGalvez2019,Arora2007,Romgens2016}. Finally, the addition of a spacer between the injector fixes the stand-off distance and applies tensile stress on the skin to improve the injection reproducibility. All of these additions show improvements for the jet injector.
NFJIs can be actuated by a pulsed laser~\cite{Krizek2020,Ham2017,Robles2020,Rohilla2020,Tagawa2012,Tagawa2013}, or a continuous-wave (CW) laser~\cite{Berrospe-Rodriguez2016,Berrospe-Rodriguez2017, OyarteGalvez2020,Quetzeri-Santiago2021}. Although both laser types can create jets able to penetrate skin-like substrates, a comparison between the two methods is difficult, as all studies use different injector geometries, laser configurations, ejected liquid volumes and disparate substrates for injection~\cite{Schoppink2022}. The difference between the two laser types is in the time of irradiation, as shown in Figure \ref{fig: Laser types}. For the pulsed laser, the bubble forms shortly after the ns pulse with very high peak power ($\approx 0.5-2 GW/cm^2$). The CW laser irradiates continuously and therefore heats the liquid slower (ms). At a certain time after the beginning of radiation ($\tau_C \sim$~ms), there is enough energy for nucleation and subsequently, the bubble forms. The difference in the timescales (ns and ms) between these two lasers has an effect on the absorption of energy, heat and pressure confinement, which will be further discussed in Section \ref{sec: Theory}.
In this manuscript, we investigate laser-actuated cavitation at a fiber tip in an open capillary, and we compare two types of lasers, the ns-pulsed and the CW laser. This comparison provides a better understanding of the energy transfer from the optical energy into the kinetic energy, which is of great importance for the use of laser-actuated cavitation in needle-free jet injectors.
\begin{figure}[b!]
\includegraphics[width=0.6\linewidth]{Figures/LaserTypes.pdf}
\centering
\captionsetup{width=\linewidth}
\caption{Schematics indicating the main difference between the timescale of nucleation and power for the pulsed and continuous-wave laser. For the pulsed laser (green), all energy is delivered with a high peak power within 5 ns, after which nucleation occurs and the bubble forms. For the continuous-wave laser (red), the power is switched on and remains working continuously at low power. After some time during this laser illumination ($\tau\sim\textnormal{ms}$, depending on the power), bubble nucleation occurs. Although the timescales are different, the bubbles show similar dynamics and lifetimes ($\tau_\textnormal{P,B}\sim$~$\tau_\textnormal{C,B}\sim$~200~µs)}
\label{fig: Laser types}
\end{figure}
\section{Theory}\label{sec: Theory}
Figure \ref{fig: Laser types} shows the difference in timescales between nucleation by the pulsed (ns) and CW lasers (ms). The difference in timescale has an effect on three parameters: the absorption of the optical energy, thermal diffusion and potential pressure confinement. These will be explained below.
First of all, the energy absorption by the liquid can be split up into two categories: linear and non-linear. The former depends on the absorption coefficient $\alpha$ of the liquid at the laser wavelength. As linear absorption does not depend on timescale, it is resent in both pulsed and CW laser exposition. In the case of only linear absorption, the optical penetration can be calculated by~\cite{Jacques1993}
\begin{equation}
\Psi(z) = \Psi_0 \exp(-\alpha z),
\end{equation}
where $\Psi$ is the laser irradiance, $\Psi_0$ is the incident irradiance, and z is the position along the laser beam. Therefore, a typical penetration depth $d = \frac{1}{\alpha}$ can be defined, at which the irradiance has dropped to $\Psi(d) = \exp(-1) \Psi_0 \approx 0.37 \Psi_0$, which means that approximately 63\% has been absorbed. As linear absorption of water is negligible in the visible and near-infrared ($\lambda <$ 1300~nm)~\cite{Ruru2012}, a dye is typically added to increase the absorption~\cite{Padilla-Martinez2014}. Non-linear absorption may occur at sufficient power densities, which is a result of optical or thermal breakdown \cite{Vogel2003}.
The threshold for this optical breakdown is found to be around $10^{-11}$~W/m$^2$ or $\sim$~500~J/cm$^2$ for laser pulses of 6~ns at a wavelength of 1064~nm~\cite{Vogel1999}. However, the threshold is reduced up to three orders of magnitude when the target has a very high linear absorption coefficient~\cite{Vogel2003}.
Second of all, thermal diffusion plays a significant role when the timescale of nucleation is comparable to the thermal diffusion timescale. The thermal diffusion time $t_d$ is given as~\cite{Paltauf2003}
\begin{equation}\label{eq: thermal diffusion}
t_d = \frac{\delta^2}{4\kappa},
\end{equation}
where $\kappa$ is the thermal diffusivity of the liquid, and $\delta$ is the typical length scale, which is either the beam diameter or the typical absorption length, whichever is smaller. In the case of heating water ($\kappa~\approx~0.14~\textnormal{mm}^2/\textnormal{s}$) with a beam diameter of 50~µm and absorption length of 100~µm, the thermal diffusion time (over the length of the beam diameter) is approximately 4 ms. Therefore, thermal diffusion does not play a role in the pulsed laser actuation ($\tau_{pulsed} \sim $ ns), but does influence the CW actuation ($\tau_{cw} \sim $ ms).
However, even for pulsed lasers, heat transfer during the bubble's lifetime should be taken into account. Sun et al. found that the inclusion of heat transfer is required for a numerical model of the growth and collapse of the bubble~\cite{Sun2009}. They also found that the bubble collapse is slower than the growth in microchannels, and such an effect increases with smaller channel sizes.
Third of all, the rapid heating of the liquid results in thermoelastic stresses in the irradiated volume, as the system tends to reconfigure to a new equilibrium~\cite{Vogel2003}. Pressure confinement may play an important role in defining the threshold for liquid-gas transition \cite{Paltauf1998}. In the case of irradiation from an optical fiber, the finite size of the fiber enhances this effect~\cite{Paltauf1998}. When this heating is sufficiently fast, the pressure is confined within the volume of irradiation near the fiber tip. The time for a pressure wave to travel across the irradiated volume is~\cite{Frenz1996}
\begin{equation}
t_p = \frac{\delta}{c_\textnormal{water}},
\end{equation}
where $c_\textnormal{water}$ is the speed of sound in water (1480~m/s). For a fiber diameter of 50~µm and absorption length of 100~µm, $t_p~\approx$~34~ns. This means that for pulsed laser actuation, the pressure is mostly confined to the region close to the fiber. In the absence of pressure confinement (when the laser pulse duration is much longer than $t_p$), there is significant thermal expansion of the heated volume during the irradiation. Therefore, the thermoelastic stresses are reduced, and the built-up pressure at bubble formation is reduced~\cite{Vogel2003}.
\section{Experimental methods}\label{sec: methods}
\begin{figure}[b!]
\includegraphics[width=\linewidth]{Figures/Experimental_setup.pdf}
\centering
\captionsetup{width=\linewidth}
\caption{Experimental setups consisting of a laser, either pulsed (A) or continuous-wave (B), coupled into a multimode optical fiber, which is inserted into a capillary system (C). A high-speed camera (Photron NOVA S6), together with a light source (Schott CV-LS) is used for imaging. The inner diameter of the smaller capillary is 300~µm, and the distance between the fiber tip and the liquid-air interface is 950~µm. Due to the smaller inner diameter of the lower capillary compared to the upper capillary (ID = 1200~µm), surface tension ensures that the lower capillary will constantly be filled after each jet generation.}
\label{fig: ExperimentalSetup}
\end{figure}
Figure~\ref{fig: ExperimentalSetup} shows the experimental setups, which consist of a fiber-coupled laser and a microfluidic capillary system. Two laser types are used, a pulsed and a CW laser. For all the experiments, the laser is coupled into a multimode optical fiber with core diameter varying from 50, 105 and 200~µm (Thorlabs FG050LGA, FG105LCA, FG200LEA). This fiber is inserted into a capillary system filled with water. Upon laser illumination, a bubble forms at the laser tip.
The pulsed laser (Continuum, ML-II) has a pulse duration of 5~ns and a wavelength of 532~nm. The laser light is coupled into the optical fiber using a two-mirror system and a focusing objective. Before and after each experiment, the pulse energy at the output of the optical fiber is measured using an energy sensor (Thorlabs ES111C). The light energy coupled into the fiber varied between 50 and 700~µJ. For the 50 and 105~µm fiber the upper pulse energy limit was approximately 130 and 480~µJ due to the laser-induced damage threshold for the fiber tip.
The CW laser (BKTel Photonics, HPFL-2-350-FCAPC) has a wavelength of 1950~nm, which was deliberately chosen to match the absorption peak of water ($\alpha~\approx$ 12000~m$^{-1}$~\cite{Ruru2012}). The output power can be varied from 0.2 to 3W. The laser is initially coupled into a single-mode fiber (Corning SMF-28e), which is then connected through a mating sleeve to the respective multimode fiber. The laser also has a secondary fiber output at 1\% of the nominal power, which is connected to a photodetector (Thorlabs DET05D2) to monitor the output power using an oscilloscope (Tektronix MSO2014B).
The multimode fiber is inserted into a capillary system, which consists of two concentric connected capillaries with inner diameters of 1500 (top) and 300~µm (bottom). Initially, the capillary system is completely filled with water, and capillary flow from the larger to the smaller capillary ensures that the 300~µm capillary will be completely filled. The fiber is inserted partially into the 300~µm (bottom) capillary, and has a distance of 950~µm to the end of the capillary (the liquid-air interface).
For the experiments with the 532~nm laser and 105~µm fiber, we increased the absorption with aqueous solutions of Allura Red AC (ARAC, red food dye) with varying concentrations between 5 and 52~mM. The absorption coefficients were measured and compared with values reported in the literature (see Appendix A). For the 50 and 200~µm fiber, only the 10~mM ($\approx~12000~\textnormal{m}^{-1}$) and 52~mM ($\approx~19000~\textnormal{m}^{-1}$) were used. The 10~mM was chosen as the absorption coefficient matches the absorption coefficient of the CW laser experiment in this study, and the 52~mM was chosen as it was used in previous studies~\cite{Krizek2020,Krizek2020a}.
A Photron NOVA S6 high-speed camera was used in combination with Navitar 12x zoom lens system and a Schott CV-LS light source for visualization of the bubble dynamics. The camera is used at a framerate of 192k fps, a resolution of 256*80 and a pixel size of 5~µm. Figure~\ref{fig: BubbleVolumeTime} shows a few typical images during the bubble lifetime. The images are analyzed with a custom-made MATLAB algorithm, which tracks the bubble over time, and calculates the volume assuming cylindrical symmetry. The average growth rate is calculated as the maximum bubble size divided by the growth time.
\begin{figure}
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/BubbleVolumeTime.pdf}
\captionsetup{width=.9\linewidth}
\caption{Bubble volume over time for an experiment with the pulsed laser (d$_{\textrm{fiber}}$ = 105~µm, dye concentration C$_{\textrm{ARAC}}$ = 10~mM, E$_{\textrm{pulse}}$ = 233~µJ). \textbf{Left:} Five images during the experiment, with the tracked bubble in blue. Respectively: just before the laser pulse, during the growth phase, at maximum volume, during the collapse, and after the experiment showing the expelled volume. The white scale bar in the top figure measures 200 µm. \textbf{Right:} Bubble volume plotted versus time dependence, the red dots corresponds to the 5 images on the left (I-V).}
\label{fig: BubbleVolumeTime}
\end{minipage}
\vline
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/Pulsed105_Growth_and_Volume_vs_energy-eps-converted-to.pdf}
\captionsetup{width=.9\linewidth}
\caption{Maximum bubble volume (blue, right axis) and average volumetric growth rate (red, left axis) for a range of pulse energies with the 105~µm fiber and 52~mM dye concentration. Each data point is an average of at least 6 individual measurements with the error bars indicating standard deviation. Solid lines indicate a linear fit for both variables.}
\label{fig: Pulsed105 Volume Growth rate}
\end{minipage}
\end{figure}
\section{Results and discussion}\label{sec: results and discussion}
\subsection{Pulsed laser}\label{sec: pulsed laser}
\subsubsection*{Bubble volume and volumetric growth rate}
Figure~\ref{fig: BubbleVolumeTime}(a) shows typical experimental images of the bubble generated by the pulsed laser. The bubble appears at the fiber tip and grows towards the free surface until it reaches a maximum size and collapses. Occasionally, cavitation bubbles inside the bulk liquid appear due to the negative peak pressure of the shock wave within a few frames after the laser pulse ($\tau~<$~20~µs). As these cavitation bubbles remain small (R~$<10$~µm) and do not affect the main bubble or the jet, we will not discuss them further. These bubbles are further discussed in Ref.~\cite{Hayasaka2017}.
The calculated bubble volume over time is shown in Figure~\ref{fig: BubbleVolumeTime}(b). We can observe parabolic-like behavior, although the growth is typically slightly faster than the collapse ($\approx5-20\%$), which was also observed by Sun et al., both numerically as well as experimentally~\cite{Sun2009}. From the bubble volume curve, the average growth rate is calculated by the maximum size divided by the growth time.
Figure~\ref{fig: Pulsed105 Volume Growth rate} shows the maximum bubble volume and average volumetric growth rate for a range of laser pulse energies. Both the volume and the growth rate show a linear behaviour with the pulse energy, meaning that it is a good indication of the jet velocity, which also increases linearly with pulse energy~\cite{Krizek2020}. In this previous study, it was confirmed that the jet velocity grows linearly with laser energy using the same range of energies, resulting in velocities 0~-~125~m/s~\cite{Krizek2020} and using a dye concentration of (52~mM ARAC).
Interestingly, the offset of the fit, which can be seen as the threshold energy for bubble formation, is not equal for both fits. Therefore, extrapolating this linear relation would indicate that bubbles with zero volume still have a positive growth rate, which wouldn't be possible. We attribute this to the fact that for these very small bubbles, there are errors in bubble detection due to their small size and their short lifetime. These smallest bubbles typically only span 10 by 20 pixels in the image and have a bubble lifetime of approximately 40~ms. Thus, the growth time is only $\sim$~20~ms, which is 4 frames. This means that the actual maximum bubble size may actually be reached between two frames and not captured by the camera. Therefore, we think that the average growth rate is a better parameter to show the bubble dynamics as a function of pulse energy.
\subsubsection*{Dye concentration effect}
The bubble growth rate as a function of the pulse energy is plotted in Figure~\ref{fig: Growth_rate_vs_energy_concentrations}, for a range of dye concentrations and the 105~µm fiber tip. For all pulse energies, an increase in dye concentration results in a larger growth rate. This is explained by the fact that an increasing dye concentration increases the absorption coefficient and thus a more localized energy absorption. This results in higher liquid temperatures and more vaporization.
These results indicate that the bubble and resulting jet dynamics can be controlled by the absorption coefficient of the liquid. This means that a jet injector relying on a pulsed laser is not limited by the range of pulse energies, as an increase or decrease in dye concentration will result in a change in jet velocity and injection depth. Furthermore, by increasing the dye concentration, the required pulse energies will decrease, which would allow the use of lasers to be more affordable and possibly smaller in size.
\begin{figure}[t!]
\centering
\includegraphics[width=.85\linewidth]{Figures/Growth_rate_vs_energy_concentrations-eps-converted-to.pdf}
\captionsetup{width=.9\linewidth}
\caption{Average growth rate of the bubbles generated by the pulsed laser with the 105~µm fiber for a range of pulse energies and 5 different concentrations of ARAC. Error bars indicate standard deviation in pulse energies and bubble growth rate for at least 6 individual bubbles with identical initial conditions. For each concentration, the data is fitted with a linear fit, with corresponding slope (units of 10$^{-9}$~m$^3$/(s*µJ)).}
\label{fig: Growth_rate_vs_energy_concentrations}
\end{figure}
More specifically, the absorption coefficients we used of 0.7~-~2~$\times$~10$^4$~m$^{-1}$ are similar to the absorption coefficients of water around its peak at 2~µm~\cite{Ruru2012}. Holmium and Thulium lasers operate near this absorption peak and are often used for the irradiation of water and biological tissues~\cite{Wang2020}. Larger absorption coefficients have been used, either by employing a different dye~\cite{Tagawa2012}, or by using an Er:YAG laser~\cite{Park2012,Jang2013}, where the absorption coefficient of water is approximately two orders of magnitude larger~\cite{Hale1973}. In the first case, even lower pulse energies (E~=~19~µJ) than the ones in this study would result in a bubble and jet (although in a different setup). However, for the second case, the Er:YAG laser has a much longer pulse duration of 250~µs~\cite{Jacques1993,Jang2013}, for which reason there is no thermal or pressure confinement, and it operates similarly to the CW laser. Therefore this laser typically requires pulse energies 400-1000~mJ for bubble and jet formation~\cite{Jang2013,Jang2016}, which is three orders of magnitude larger than used in our study. Besides that, Er:YAG lasers are not compatible with fiber delivery~\cite{Fried2018a}. Near 1450~nm, there is another peak in the absorption coefficient of water, although twice as small ($\alpha$~=~3150~m$^{-1}$~\cite{Ruru2012}) compared to the lower limit in our study. A previous study~\cite{Krizek2021} with a 1574~nm laser ($\alpha\approx$~900~m$^{-1}$) showed that it would require pulse energies of 2-4~mJ, approximately one order of magnitude larger compared to the energies probed here. For even smaller wavelengths, absorption of water is negligible, and thus studies without the use of a dye rely fully on non-linear absorption and require more energy~\cite{Rohilla2020}.
\subsubsection*{Fiber core diameter}
Figure~\ref{fig: Fiber radius 10 and 52 mM} shows the volumetric bubble growth rates for the three different fiber core diameters: 50, 105 and 200~µm. Figure~\ref{fig: Fiber radius 10 and 52 mM}A is for the ARAC concentration of 10 mM and Figure~\ref{fig: Fiber radius 10 and 52 mM}B is for the ARAC concentration of 52 mM. The measurement data per fiber are fitted with the best linear fit. For both concentrations, larger fibers require larger pulse energy to obtain the same growth rate. This trend was also observed for jet velocity and fiber sizes~\cite{Krizek2020}, and explained by a larger energy density for the smaller fibers, and thus a larger pressure. This means that a smaller fiber with a less powerful laser can create the same bubble and resulting jet.
However, for the 50~µm fiber, we could not reach very large growth rates for two reasons: First, for the 52~mM dye concentration, we observed a plateau in the growth rate for pulse energies larger than 60~µJ. Second, the tip of the 50~µm fiber is very prone to damage - because of high light power densities, and the laser-induced damage threshold was found at approximately 120~µJ. This means that the larger fibers are more flexible in their use, although they require larger pulse energies.
\begin{figure}
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/FiberRad10mM-eps-converted-to.pdf}
\captionsetup{width=.9\linewidth}
\end{minipage}
\vline
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/FiberRad52mM-eps-converted-to.pdf}
\captionsetup{width=.9\linewidth}
\end{minipage}
\caption{Average volumetric bubble growth rate for three optical fiber sizes. Error bars indicate standard deviation of at least 6 individual measurements. Solid lines indicate the best linear fit, with corresponding slope (units of 10$^{-9}$~m$^3$/(s*µJ)). (A): Dye concentration of C$_\textnormal{dye}$~=~10~mM. (B): Dye concentrations of C$_\textnormal{dye}$~=~52~mM. }
\label{fig: Fiber radius 10 and 52 mM}
\end{figure}
\subsubsection*{Energy threshold for bubble nucleation}
The intersection with the x-axis of the linear fits in Figures~\ref{fig: Growth_rate_vs_energy_concentrations}~and~\ref{fig: Fiber radius 10 and 52 mM}, indicates an energy threshold for bubble formation. These energy thresholds are plotted against the irradiated volumes in Figure~\ref{fig: threshold_vs_volume_v2}. The irradiated volume is calculated until the typical absorption length $\delta$, assuming a conical shape (due to divergence) starting at the fiber tip with a diameter equal to the core diameter (see Appendix B). The linear trend of the energy threshold with the irradiated volume indicates a constant energy density required for bubble nucleation. Therefore, the energy threshold scales quadratic with the fiber radius and linear with the absorption length. However, in the case where the absorption length is much larger compared to the fiber radius, the beam divergence should be taken into account. In practical applications where less powerful lasers are preferred, the fiber should have a small core radius, small NA (small divergence) and the liquid should have a large absorption coefficient.
\begin{figure}
\centering
\includegraphics[width = 0.7\linewidth]{Figures/threshold_per_volume_new-eps-converted-to.pdf}
\caption{Energy threshold taken from offset in the linear fits from Figures~\ref{fig: Growth_rate_vs_energy_concentrations}~and~\ref{fig: Fiber radius 10 and 52 mM}, plotted against the irradiated volume. Colors indicate the different fiber diameters, symbols indicate dye concentrations and error bars indicate 95\% confidence interval for the linear fits. The black line indicates the best linear fit.}
\label{fig: threshold_vs_volume_v2}
\end{figure}
\subsection{CW laser}\label{sec: CW laser}
\subsubsection*{Nucleation time and delivered energy}
In contrast to the pulsed laser, for the CW experiment, only the power can be directly controlled, and not the energy. Here, the delivered energy depends on the power and the moment of nucleation, which depends on the laser power~\cite{Padilla-Martinez2014}. Figure~\ref{fig: CW power nucl time energy}A shows the influence of the laser power on the nucleation time for the three fibers. First, the nucleation time increases with decreasing laser power, as it takes longer to reach the nucleation temperature. Second, for fixed laser power, the nucleation time increases with increasing fiber diameter. This is explained by an increase in beam size and thus a decrease in intensity, for which reason it takes longer to reach the nucleation temperature. Due to a limited range of laser power, sub-ms nucleation times could not be reached for the 200~µm fiber.
Besides the influence of the power and fiber size, there are fluctuations in nucleation time for each individual data point. This fluctuation in nucleation time is called \textit{jitter}~\cite{Padilla-Martinez2014} and is explained by the stochastic nature of nucleation. These fluctuations linearly affect the delivered energy, and thus reduce the reproducibility. We find that the typical fluctuation (standard deviation) is approximately 8\% of the nucleation time, which is smaller compared to earlier findings up to approximately 60\%~\cite{Padilla-Martinez2014,Zhang2022}. We attribute this to the fact that bubbles develop directly at the optical fiber, instead of heating a liquid with a non-collimated laser beam, where the beam size is more difficult to control and reproduce. Furthermore, in our case we do only create individual bubbles, after which the liquid cools down again, which may result in a more reproducible nucleation time.
The delivered energies, which are calculated by multiplying the nucleation time and the power, are shown in Figure~\ref{fig: CW power nucl time energy}B. Similar to the nucleation times, a reduction in power or an increase in fiber size results in an increase of the energy delivered. This is because an increase in nucleation time results in an increase in heat diffusion. This heat diffusion will result in an even longer nucleation time as the local temperature increases at the fiber tip is slower, which then results in larger delivered energy.
For the 50 and 105~µm fibers, the CW laser results in comparable energies as the ns-laser.
However, for CW the energies required for bubble formation by the 200~µm fiber are an order of magnitude larger due to the longer nucleation times, which is caused by the larger fiber area and thus lower intensities.
Previous work of Ref.~\cite{OyarteGalvez2020} refers to a free-space laser diode with free-space light focusing with a 6*33~µm$^2$ laser beam. For a power of 0.5~W, they found a nucleation time of 600~µs, resulting in a delivered energy of 300~µJ, for a liquid with a similar absorption coefficient as ours~($\alpha$~=~1.0~and~1.2~*~10$^4$~m$^{-1}$ respectively). The beam size for the smallest fiber in our study is already 10 times larger and still results in twice as fast nucleation time for 0.5~W ($\sim$~300~µs). Therefore, the previous study with free-space optics and a glass microdevice showed a significant optical loss between the laser diode and the microfluidic channel. Here, due to the direct contact of the fiber and the liquid, the number of interfaces is minimized and can be no losses due to misalignment, resulting in higher energy transfer efficiency.
\begin{figure}[t!]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/CW_power_nucl_time-eps-converted-to.pdf}
\end{minipage}
\vline
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/CW_power_energy-eps-converted-to.pdf}
\end{minipage}
\caption{Nucleation time (A) and delivered energy (B) as a function of CW laser power for three optical fibers with different diameter. Each data point is an average over at least 6 individual measurements, with the error bars indicating the standard deviation.}
\label{fig: CW power nucl time energy}
\end{figure}
\subsubsection*{Bubble growth rate}
The bubble growth rates for the CW laser source are shown in Figure~\ref{fig: CW growth rates fibers}A. It is clear that the growth rates of the bubbles generated with the 200~µm fiber are much larger compared to the 50 and 105~µm fiber. The increased growth rate is explained by the delivered energy, which is an order of magnitude larger compared to the other fibers. As discussed above, smaller energies could not be delivered with the used CW laser for this fiber as it would require much larger powers. The bubbles generated at the 200~µm fiber are also much larger compared to the other two fibers, and all generated bubbles grow beyond the edge of the capillary, for which reason they coalesce with the surrounding air before reaching their maximum volume. An example of such a bubble generated at the 200~µm fiber is shown in Appendix C.).
Figure~\ref{fig: CW growth rates fibers}B shows a close-up of the growth rates of the bubbles generated at the two smaller fibers. It shows that for the CW laser, the bubble growth rate also increases linearly with increasing delivered energy. For most energies, the 50~µm fiber has a larger growth rate compared to the 105~µm fiber, which can be explained by the larger energy density. However, for delivered energies larger than 500~µJ, the 105~µm fiber results in faster-growing bubbles. We attribute it to the longer nucleation time for the 50~µm fiber to reach those levels of energy. For the 50~µm fiber, the nucleation times to reach E~$>$~500~µJ is t$_\textnormal{n}~>$~2ms, whereas for the 105~µm fiber, the nucleation time is only half. Therefore there is increased heat dissipation for the 50~µm fiber, and thus a smaller efficiency. Therefore, the growth rate no longer increases linearly with the delivered energy, and larger growth rates require much more energy for the 50~µm fiber.
Interestingly, extrapolating the linear fit of the 50~µm fiber in Figure~\ref{fig: CW growth rates fibers}B shows a positive growth rate for zero energy, which is physically not possible. However, due to thermal dissipation, it is questionable whether the growth rate for the CW-laser would actually be linear with the energy, as thermal dissipation may reduce the energy efficiency. Therefore, even though the initial behaviour for both fibers seems linear, extrapolating this to find an energy threshold does not seem to have a physical meaning.
\begin{figure}[t!]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/CW_growth_rate_incl_200-eps-converted-to.pdf}
\end{minipage}
\vline
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/CW_growth_rate_excl_200-eps-converted-to.pdf}
\end{minipage}
\caption{Average volumetric bubble growth rate for the CW laser. Figure (A) includes all three fiber sizes. As the energy and growth rates for the 200~µm fiber are much larger compared to the 50 and 105~µm fiber, figure (B) shows a close-up of the left bottom of (A), indicated by the dotted lines. Figure (B) includes the best linear fit for the 50 and 105~µm fiber.}
\label{fig: CW growth rates fibers}
\end{figure}
\subsubsection*{Secondary bubbles}
For small bubbles generated by the CW laser (R~$<$~150~µm), we observe the formation of a secondary bubble at the vapor-liquid interface, see~Figure~\ref{fig: Secondary bubble}. These events are different from twin cavitation bubbles reported in Ref.~\cite{Zhang2022}, where the initial bubble first partially collapses before forming a secondary bubble. In our case, the secondary bubble typically forms when the initial bubble reaches its maximum. Furthermore, in our case the volume of the secondary bubble is smaller compared to the first one, for which reason its appearance does not affect the parabolic shape of the bubble volume versus time.
These secondary bubbles can be explained by the fact that the laser remains irradiating during the bubble formation and, therefore, further heats the liquid during the bubble growth. In the case of large laser powers and short nucleation times, this heating during the bubble lifetime could create a secondary bubble. In the case of Figure~\ref{fig: Secondary bubble}, the growth time of the bubble takes 40~µs, whereas the initial nucleation time was 135~µs, which means that after nucleation, another 30\% of the initial energy is still delivered, which results in secondary nucleation at the vapor-liquid interface of the bubble.
The occurrence of these secondary bubbles depends on the laser power and nucleation time of the initial bubble. The secondary bubble forms during the lifetime of the bubble (typically 50-200~µs), during which enough additional energy has to be delivered to form a secondary bubble. We find that these secondary bubbles only appear when the intensities are sufficiently larger such that the initial nucleation time is smaller than approximately 450~µs, both for the 50 and 105~µm fiber. For the 200~µm fiber the intensities are smaller and thus the nucleation times are much larger we did not observe any secondary bubbles. However, we hypothesize that the use of a more powerful laser can also form these secondary bubbles with the 200~µm fiber. However, as mentioned, these secondary bubbles only happen for small nucleation times and large powers, which result in small bubbles overall. Therefore, we conclude that the larger and faster-growing primary bubbles are of more interest.
\begin{figure}[t!]
\centering
\begin{minipage}{.45\textwidth}
\centering
\hspace{-1cm}
\includegraphics[width=1.05\linewidth]{Figures/Secondary_Bubble_50_5_3-eps-converted-to.pdf}
\end{minipage}
\vline
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/Secondary_bubble_volume_curve_50_5_3-eps-converted-to.pdf}
\end{minipage}
\caption{\textbf{Left:} 16 consecutive frames showing the typical dynamics of a small bubble generated by the CW laser. As the laser is still running during the bubble growth, more liquid will be heated, which results in a secondary (frame 7-12) and even a third bubble (frame 13-14). Snapshots are consecutive, from top to bottom and left to right, with an inter-frame time of 5~ms. The initial nucleation time (top left frame) was 135~µs, and the bubble reached its maximum size (frame 8) at 175~µs. Scalebar on the right bottom measures 100~µm. \textbf{Right:} Calculated bubble volume for each frame, showing that the initial maximum is reached at frame 6, but due to secondary bubble formation, another maximum is reached at frame 8, although the difference is small.}
\label{fig: Secondary bubble}
\end{figure}
\subsection{Comparison between pulsed and CW}
A direct comparison between the bubbles generated by the pulsed and CW laser requires a matching absorption coefficient such that the volumes over which the energy is absorbed are identical. Therefore, we compare the results of the 10~mM ARAC solution, as its absorption coefficient at 532~nm is nearly identical compared to the absorption coefficient of water at 1950~nm. In this subsection, we compare the bubbles generated by the 50 and 105~µm fiber, as they show similar results in the growth rate (1~-~$7*10^{-7} \textnormal{m}^3/s$). The 200~µm fiber shows very different results for the CW laser, as the delivered energies are much larger (see Figure \ref{fig: CW growth rates fibers}).
\subsubsection*{Energy efficiency}
Budgeting of energy has practical implications on the right choice of laser source.
The growth rates for the CW and pulsed laser are shown in Figure~\ref{fig: GR_pulsed_CW_50_105}, for the 50 and 105~µm fiber. For the same delivered optical energy, the pulsed laser results in faster growth rates, although the required energies are in the same order of magnitude. Typically, the CW laser requires two to three times more optical energy than the pulsed laser to generate a bubble with the same growth rate. The reduced efficiency is explained by three reasons. First of all, for the bubbles generated by the CW laser heat diffusion cannot be neglected, as the nucleation time is 0.1~-~5~ms, which is in the same order as the thermal diffusion timescale ($\sim$~4~ms, see Equation~\ref{eq: thermal diffusion}). Most of this heat will be lost, as it will heat up the liquid around the irradiated volume. Though, this heat increase in the proximate area would be limited, and most of it does not result in the phase transition. Second, the absorption coefficient of water around 1950~nm decreases with increasing temperature, and the absorption coefficient at 100$\degree$C is only half the initial value at room temperature~\cite{Jansen1994,Lange2002}. Therefore, for the heating phase with the CW laser, the average absorption length is much larger compared to the absorption length of the pulsed laser experiments.
Third, for the pulsed laser, there might be non-linear absorption due to the high peak power. The measured absorption coefficient of the 10~mM ARAC is the same as the absorption coefficient of water for low optical intensities, but the absorption coefficient for the high-power pulses may be larger due to additional non-linear absorption by the liquid. This increased absorption would result in a smaller irradiated volume and thus a higher energy density, which will increase the bubble growth rate.
\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{Figures/Growth_rate_pulsed_CW_50_105-eps-converted-to.pdf}
\captionsetup{width=0.5\linewidth}
\caption{Growth rate of bubbles generated by the pulsed (red) and CW laser (blue), for fibers with core diameters of 50 (dark colors) and 105~µm (light colors). The absorption coefficients for both laser are the same ($\alpha$~$\approx$~12000~m$^{-1}$).}
\label{fig: GR_pulsed_CW_50_105}
\end{figure}
\subsubsection*{Bubble dynamics}
Bubble dynamics govern the subsequent microfluidic jet parameters.
As shown above, the typical growth rates are similar for both lasers. However, identical average growth rates may not result in the exact same bubble characteristics. An increase in growth time, results in a larger bubble, assuming the average growth rate is the same. The maximum volume against the average growth rate is shown in Figure~\ref{fig: GR_MV_pulsed_CW} for the bubbles generated by the pulsed (red) and CW (blue) laser. For identical average growth rates, the bubbles created by CW laser grow to a $\sim 15\%$ larger volume (due to the increased growth time). Alternatively, for bubbles with the same maximum volume, the bubbles created by the CW laser take longer to reach that volume. This reduced growth rate could be explained by the larger volume over which the delivered energy is dissipated, resulting in a less explosive phase transition. However, this also means that for the same average growth rate, the bubbles generated by the CW laser will push out the remaining liquid over a longer length and time. This increased time of energy transfer will mostly affect the jet tail, which is typically slower and more dispersed~\cite{Krizek2020,AndrewUnpublished}. We hypothesize that this longer growth time results in a faster and more reproducible jet tail.
A comparison between the growth and collapse can be made by normalizing the volume and time, as shown in Figure~\ref{fig: Volume_time_normalized}. For each individual bubble, the volume is normalized by its maximum volume and the time by the growth time. The lines show an average value over all bubbles, and the shaded region is the standard deviation. This figure shows that the initial normalized growth rate of the bubbles generated by the pulsed laser is larger compared to bubbles generated by the CW laser, as indicated by the steeper curve for small times.
Furthermore, it shows that for the pulsed laser, the collapse is on average $\sim 20\%$ slower compared to the growth, whereas for the CW they take equal time. Both of these findings are in agreement with previous studies on the bubble dynamics for pulsed laser~\cite{Zwaan2007,Sun2009} and CW~\cite{OyarteGalvez2020}. However, this difference is mainly caused by the $\sim 15\%$ smaller growth time for the pulsed laser, see Figure~\ref{fig: GR_MV_pulsed_CW}. If both curves were normalized by the same time, the time of collapse would only vary by $<5\%$.
\begin{figure}
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{Figures/Growth_rate_Max_size_shade-eps-converted-to.pdf}
\captionsetup{width=0.9\linewidth}
\caption{Maximum bubble volume vs average growth rate for individual bubbles generated by the pulsed laser (red) and the CW laser (blue). The shaded area encapsulates all data.}
\label{fig: GR_MV_pulsed_CW}
\end{minipage}
\vline
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{Figures/Growth_collapse_norm_pulsed_CW-eps-converted-to.pdf}
\captionsetup{width=0.9\linewidth}
\caption{Normalized bubble size vs normalized time, for CW (blue) and pulsed (red). Lines indicate averages over all bubbles, shaded regions indicate standard deviation. For each individual bubble, the volume is normalized by the maximum volume, and the time is normalized by the growth time.}
\label{fig: Volume_time_normalized}
\end{minipage}
\end{figure}
\subsubsection*{Reproducibility}
Reproducibility is an important factor considering application in health care. The error bars in Figure~\ref{fig: GR_pulsed_CW_50_105} show that the deviation in the bubble growth rate is larger for the CW laser (blue colors) compared to the pulsed laser (red colors). This indicates that even with identical initial conditions for the CW laser, the bubble dynamics can vary each time slightly. As discussed in section~\ref{sec: CW laser}, the energy delivered by the CW laser is not controlled directly but depends on the laser power and nucleation time. As nucleation is a stochastic event, there is a variance in the nucleation time, resulting in less or more delivered energy for early or late nucleation, respectively.
Figure~\ref{fig: CW growth individual} shows the growth rate of all individual bubbles generated by the CW laser together with the mean and standard deviation for a group of 6 or more with identical initial conditions (laser power). Each color indicates a different laser power, which influences the delivered energy. However, even for measurements with identical laser power, the delivered energy can vary up to $\pm~50$~µJ. For each of these laser powers, the individual results are typically in the bottom-left or top-right quadrant of the error bars. This indicates that there is a correlation between the additional energy for delayed nucleation and the bubble growth rate. Furthermore, per laser power, the individual data points are fitted with a linear fit, which is shown by the colored lines. The slope of these colored lines is very similar to the slope of the black line, which is the best fit when comprising all data. Therefore, the variance in nucleation time directly affects the bubble growth rate, which means that delayed nucleation results in more energy and a faster-growing bubble, and early nucleation results in less energy and a slower-growing bubble. However, as nucleation is a stochastic process, this results in a random deviation and reduces reproducibility.
For the pulsed laser, the standard deviation in delivered energy is only affected by the laser specifications. On average, it is much smaller with 2\% of the delivered energy, compared to 8\% for CW. As this deviation also directly affects the bubble growth rate, the bubble growth rate for the pulsed laser is thus more reproducible.
\begin{figure}[t!]
\centering
\includegraphics[width=1\textwidth]{Figures/CW_growth_individual-eps-converted-to.pdf}
\captionsetup{width=1\linewidth}
\caption{Growth rate for individual bubble events for the 105~µm fiber with the CW laser. Each color indicates a different laser power. The delivered energy mainly depends on the laser power but also on the moment of nucleation (e.g. late nucleation results in larger delivered energy). The circles are individual measurements, and the error bars indicate the mean and standard deviation of those individual measurements with the same laser power (at least 6). The colored lines indicate the best linear fit per laser power. The black line indicates the best linear fit for all data points. The slope of the colored lines is similar to the black line, indicating a correlation between the time of nucleation and bubble growth rate.}
\label{fig: CW growth individual}
\end{figure}
\subsection{Practical considerations of laser choice for applications}
Engineering a medical device based on laser-induced cavitation is limited by the available offer of laser sources on the market. Yet, choosing the right laser emitter is vital for the success of any potential product or prototype, where not only the best physical and optical parameters shall be considered. Other aspects like cost, the robustness of operation, size or compliance with safety hazards play a key role. Intuitively, the cheaper, smaller and safer source is favourable, but these are usually contradictory parameters. The results of this study bring a better understanding of what kind of laser parameters are critical to generate fast traveling liquid droplets of relevance for its use in eventual devices using laser-induced cavitation, e.g. needle-free jet injectors.
Although an in-depth techno-economical discussion on the selection of laser sources for specific applications is beyond the scope of this study, we will briefly discuss the impact of the different laser and device parameters here. The main differences are shown in Table \ref{Table: CW vs Pulsed}.
For applications where reproducibility and reliability of the bubble dynamics are most important, the pulsed laser is preferred over the CW laser. Due to the reduced control over the delivered energy by the CW laser, there is a larger deviation in individual bubble growth rates. In the case of a jet injection device, this affects the jet velocity and thus injection depth. However, the exact influence of the deviation in bubble dynamics on the injection depth should be investigated in a further study.
\begin{table}[t]
\caption{Overview of the difference of the laser and bubble characteristics of the CW and pulsed laser.}
\label{Table: CW vs Pulsed}
\begin{tabular}{|l|l|l|}
\hline
& \textbf{Pulsed laser} & \textbf{CW laser} \\ \hline
\textbf{Timescale} & 5~ns & 1~ms \\ \hline
\textbf{Peak power} & 10$^4$~W & 1~W \\ \hline
\textbf{Typical price$^*$} & $\sim$~1000~-~50000~\$ & $\sim$~100~-~1000~\$ \\ \hline
\textbf{Typical size} & Table top device & Handheld device \\ \hline
\textbf{Control over delivered energy} & Direct (input of laser) & Indirect (depends on P \& t$_\textnormal{n}$) \\ \hline
\textbf{Reproducibility (variation)} & 2\% & 8\% \\ \hline
\textbf{Ratio growth to collapse time} & 1:1.2 & 1:1 \\ \hline
\textbf{\begin{tabular}[c]{@{}l@{}}Influence of increase in fiber \\ size (laser input parameters \\ constant)\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Reduction in energy density\\ Increase in energy threshold\\ Smaller/slower growing bubbles\end{tabular} & \begin{tabular}[c]{@{}l@{}}Delayed nucleation\\ Increase in delivered energy\\ Larger/faster growing bubbles\end{tabular} \\ \hline
\textbf{\begin{tabular}[c]{@{}l@{}}Influence of increase in \\ absorption coefficient/\\ dye concentration\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Larger energy density\\ Decrease in energy threshold\\ Increase of bubble size\end{tabular} & Not examined in this study$^{**}$. \\ \hline
\end{tabular}
$^{*}$Laser prices are highly dependent on wavelength and exact characteristics. Given prices are just an indication for typical consumer price of a single device. \\
$^{**}$Refs. \cite{AfanadorDelgado2019,Zhang2022} show that an increase in dye concentration results in shorter nucleation times and smaller bubbles.
\end{table}
On the other hand, if the price and/or size of the device is of more importance, a CW laser would be preferred over a pulsed laser. Due to their lower power, they are smaller and more affordable. Although, portable pulsed lasers with moderate pulse energies, as the ones used in our studies, have become more widely available over the past decade.
Furthermore, the use of optical fiber allows for a small handheld device, even in the case of a large laser source. The laser source, including electronics, can then be placed in a bench-top part.
The choice of optical fiber has a large influence on bubble dynamics. For the pulsed laser, a smaller fiber reduces the required pulse energy. However, the smallest fiber (50~µm diameter) is limited in the range of bubble growth rate due to laser-induced damage of the fiber tip. For the CW laser, an increase in fiber size results in faster-growing bubbles, although it requires a more powerful laser (P~$>$~2W) to create a larger range of growth rates.
Changing the dye concentration has a large influence on the bubble dynamics. For the pulsed laser, increase of dye concentration and thus absorption coefficient results reduces the required pulse energies, as discussion in Section~\ref{sec: pulsed laser}. This allows for the use of smaller and more affordable pulsed lasers. Simultaneously, a single device could create a large range of bubbles by changing the absorption coefficient of the liquid. The influence of the absorption coefficient on the bubbles generated by the CW laser have not been investigated in this study. However, previous studies showed an influence on the nucleation time and bubble size~\cite{AfanadorDelgado2019,Zhang2022}. Moreover, adding the dye into drug formulation might bear further toxicity and regulatory issues. Future studies should focus on the exact influence of absorption coefficient on bubble dynamics.
\section{Conclusion}\label{sec: conclusion}
We compared the dynamics of bubbles generated by two lasers with different timescales (ns and ms) inside an open-ended capillary. Our comparative study is the first to show the resulting bubble dynamics in the same fluidic confinement with two different laser types. We have shown that these lasers create bubbles of comparable growth rates proportional to the delivered energy. This linear increase is in agreement with previously found linear increase between the energy and the jet velocity.
For the pulsed laser set-up, we found that the energy threshold for nucleation is proportional to the irradiated volume. This volume depends on the fiber core radius, beam divergence and absorption coefficient. Therefore, a decrease in irradiated volume, either by changing the fiber or increasing the absorption coefficient, resulting in a reduction of the energy threshold. This allows for the use of more affordable and/or less powerful pulsed laser sources while creating the same bubble.
For the continuous-wave laser, we found an efficiency increase and better reproducibility compared to previous experiments with free-space optics. This is explained by a reduction in the number of interfaces and easier alignment. Furthermore, we show that the delivered energy can be controlled by the laser power and the fiber size. For each fiber, a decrease in laser power results in an increase in delivered energy, allowing the creation of faster-growing bubbles. Furthermore, a larger fiber results in an increase in delivered energy for fixed laser power due to a longer nucleation time. Therefore, the 200~µm fiber results in much larger and faster-growing bubbles than the 50 or 105~µm fiber, when the same laser power is used.
A comparison between the two laser sources shows that the pulsed laser requires slightly less optical energy to create the same bubble growth rate, which we attribute to heat dissipation and a reduction in absorption coefficient during the CW laser heating. However, for the same average growth rate, bubbles generated by the CW laser grow for a longer time and are larger.
Finally, we made a comparison between both methods in terms of practical usage. Since the delivered energy by the CW laser cannot be controlled directly, there is a larger deviation in the delivered energy compared to the pulsed lasers (8\% and 2\% respectively), which also results in a larger deviation in bubble growth rates. This would be unfavourable for laser-based jet injection, as it decreases the control over jet velocity and, therefore, injection depth. However, if this variation is within the allowable error industry standards, then the CW laser could be advantageous due to its smaller size and lower price compared to pulsed lasers.
\section*{Acknowledgements}
J.J.S. would like to thank dr. Dani\"el J\'auregui-V\'azquez and dr. Jose Alvarez Chavez for their help with the optical set-up of the CW laser. J.J.S and D.F.R. acknowledge the funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 851630), and NWO Take-off phase 1 program funded by the Ministry of Education, Culture and Science of the Government of the Netherlands (No. 18844). J.K. and Ch.M. acknowledge the Innosuisse BRIDGE Proof of Concept grant funding. The authors are thankful for the insightful discussions with Dr. M. A. Quetzeri Santiago, D.L. van der Ven, K. Mohan and D. de Boer.
\section*{Competing interest}
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence the work reported in this paper.
\section*{CRediT authorship contribution statement}
\textbf{Jelle Schoppink:} Conceptualization, Methodology, Formal analysis, Investigation, Data Curation, Writing - Original Draft, Visualization
\textbf{Jan Krizek:} Conceptualization, Methodology, Writing - Review \& Editing
\textbf{Christophe Moser:} Conceptualization, Funding acquisition, Writing - Review \& Editing
\textbf{David Fernandez Rivas:} Conceptualization, Supervision, Project administration, Funding acquisition, Writing - Review \& Editing.
\printbibliography
\begin{figure}[b!]
\centering
\includegraphics[width=0.7\textwidth]{Figures/TOC_Only_Graphical_abstract.pdf}
\captionsetup{width=1\linewidth}
\caption{For Table of Contents Only}
\label{fig: Graphical abstract}
\end{figure}
\end{document}
\section*{Appendix A: Allura Red AC absorption coefficient}\label{sec: Appendix}
The measured absorption coefficients for the Allura Red AC dye (ARAC, 80\%, Sigma Aldrich) are shown in Figure~\ref{fig: alpha_vs_concentration}. The values are calculated from the transmission of the 532~nm laser through a rectangular capillary with a 100~µm path length. The values are compared with reported values in literature~\cite{Bevziuk2017,Garcia2017}, which closely matches our measured values for concentrations up to 5~mM. For larger concentrations, we note that the absorption coefficient does no longer increase linearly with the dye concentration.
\begin{figure}[h!]
\centering
\includegraphics[width=.7\linewidth]{Figures/Absorption_coefficient_concentration-eps-converted-to.pdf}
\caption{Measured absorption coefficients (blue symbols) for Allura Red AC (ARAC). The values are compared with an extrapolated value from literature (red curve)~\cite{Bevziuk2017,Garcia2017}. It can clearly be observed that the absorption coefficient does not behave linearly at these concentrations.}
\label{fig: alpha_vs_concentration}
\end{figure}
\newpage
\section*{Appendix B: Calculation of irradiated volume}\label{sec: Appendix B}
As the fiber core radii R$_\textnormal{f}$ ($25\textnormal{~µm}<\textnormal{R}_\textnormal{f}<100\textnormal{~µm}$) are of similar size of the absorption length d ($52\textnormal{~µm}<\textnormal{d}<131\textnormal{~µm}$), the beam divergence $\beta$ cannot be neglected when calculating the irradiated volume. The beam divergence is calculated by the NA of the fiber and the refractive index of the liquid $n$ (see Figure \ref{fig: BeamRadDiv})
\begin{equation*}
\beta = \arcsin{\frac{\textnormal{NA}}{n}} = \arcsin{\frac{\textnormal{0.22}}{1.33}} \approx 9.5\degree.
\end{equation*}
Then, the beam radius R along the propagation axis $x$ is calculated as
\begin{equation*}
\textnormal{R}(x) = \textnormal{R}_\textnormal{f}+x\tan{\beta}.
\end{equation*}
The irradiated volume $V$ can be calculated by integrating the beam area ($\pi$R$(x)^2$) from the fiber tip ($x=0$) to the absorption length ($x=\textnormal{d}$)
\begin{align*}
V = \pi\int_{0}^{\textnormal{d}}\textnormal{R}^2 dx = \pi\int_{0}^{\textnormal{d}}(\textnormal{R}_\textnormal{f}+x\tan{\beta})^2 dx = \pi\int_{0}^{\textnormal{d}}(\textnormal{R}_\textnormal{f}^2+2\textnormal{R}_\textnormal{f}x\tan{\beta}+x^2\tan^2{\beta})dx \\
= \pi |\textnormal{R}_\textnormal{f}^2x + \textnormal{R}_\textnormal{f}x^2\tan{\beta}+\frac{1}{3}x^3\tan^2{\beta}|_0^\textnormal{d} = \pi (\textnormal{R}_\textnormal{f}^2\textnormal{d} + \textnormal{R}_\textnormal{f}\textnormal{d}^2\tan{\beta}+\frac{1}{3}\textnormal{d}^3\tan^2{\beta)}
\end{align*}
\begin{figure}[h!]
\centering
\includegraphics[width = \linewidth]{Figures/BeamRadiusDivergence-eps-converted-to.pdf}
\caption{Schematic indicating the irradiated volume (green) of the liquid (red) near the fiber tip. Beam radius increases along the propagation axis due to the divergence. At the fiber tip, the radius is equal to the core radius. The irradiated volume is calculated by integrating the beam area from the fiber tip to the absorption length \textnormal{d}.}
\label{fig: BeamRadDiv}
\end{figure}
\newpage
\section*{Appendix C: CW bubble growth}\label{sec: Appendix C}
Figure~\ref{Appendix fig: CW 105 and 200} shows a few snapshots of the bubbles generated by the CW laser at the 105 and 200~µm fiber. For the 200~µm fiber (right snapshots), the bubble has a much larger energy, due to the longer nucleation time. Therefore, the bubble grows more explosively and always grows beyond the capillary edge. Therefore, no maximum bubble size or average growth rate could be obtained for the 200~µm fiber.
\begin{figure}[ht]
\centering
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/Appendix_Bubble105-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{.49\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figures/Appendix_Bubble200-eps-converted-to.pdf}
\end{minipage}
\caption{Snapshots of the bubble generated at a 105~µm (left) and 200~µm fiber (right) by the CW laser with a power of approximately 0.7~W. The bubbles generated at the 200~µm fiber have a much larger energy and grow beyond the capillary edge and expel all the liquid. Therefore, there is no maximum bubble size or average growth rate observed.}
\label{Appendix fig: CW 105 and 200}
\end{figure}
\printbibliography
\end{document} |
1,116,691,499,413 | arxiv | \section{Introduction} \label{sec:Introduction}
Only the most violent objects in the universe are capable of producing
high-energy particles in non-thermal acceleration processes and emit
photons with energies in the X-ray band and above. Spectral and
morphological studies in the X-ray and gamma-ray bands have become
established tools to study the non-thermal emission processes of various
astrophysical sources \cite{Frederick2010}. However, many of the regions
of interest (black hole vicinities, formation zones of relativistic
jets, etc.) are too small to be spatially resolved with current and
future instruments. Spectro-polarimetric X-ray observations are capable
of providing additional information~-- namely (i) the energy-resolved
fraction of linear polarization (e.g. what fraction of emission is
polarized), and (ii) the projected orientation of the polarization plane
(defined by the electric field vector of the photon) with respect to the
emitting source. Various emission mechanisms of compact objects lead to
comparable spectral signatures, but would differ in the polarization
characteristics. The measurements of polarization properties would
therefore help to constrain the geometry of the inner regions of
relativistic plasma jets, mass-accreting black holes (BHs) and neutron
stars \cite{Lei1997, Krawcz2011}.
So far, only a few space-borne missions have successfully measured
polarization in the X-ray regime. The Crab nebula is the only source for
which X-ray polarization has been established with a high level of
confidence. In measurements with the OSO-8 satellite, the Crab exhibits
a polarization fraction of $20 \%$ at energies of $2.6 - 5.2 \,
\rm{keV}$ and a direction angle of $30^{\circ}$ with respect to the
X-ray jet observed in the nebula \cite{Weisskopf1978}. At energies above
$100 \, \rm{keV}$, measurements resulted in a polarization fraction of
$46 \% \pm 10 \%$ with the direction aligned with the jet
\cite{Dean2008, Forot2008}. A second astrophysical source emitting
polarized X-rays was identified recently. {\it INTEGRAL}\xspace observations of the
X-ray binary Cygnus\,X-1 indicate a high fraction of polarization of
$(67 \pm 30)\%$ in the $400 \, \rm{keV} - 2 \, \rm{MeV}$ band, whereas a
$20\%$ upper limit was derived for the $250-400 \, \rm{keV}$ band
\cite{Laurent2011}. Various authors have reported tentative evidence for
polarized hard X-ray/soft gamma-ray emission from different gamma-ray
bursts \cite{Coburn2003, Kalemci2007, Yonetoku2011, Kostelecky2013}.
However, all of these detections have somewhat marginal significance,
possibly being impacted by unknown systematic effects in their
respective instrumentation.
Model predictions of polarized emission for various source types lie
slightly below the sensitivity of the past OSO-8 mission, making future,
more sensitive polarimetry missions particularly interesting. However,
there are currently no dedicated missions in orbit that are capable of
measuring X-ray polarization fractions in the $<10\%$ regime, a
requirement to study the corresponding emission mechanisms for a variety
of astrophysical source classes (see Sec.~\ref{sec:Science}). More
recently, various experiments have been proposed that could change the
situation as they combine broadband sensitivity with a high detection
efficiency. The proposed polarimeters use focusing mirrors to collect
photons from the source. Photoelectric effect polarimeters, like the
{\it Gravity and Extreme Magnetism SMEX} (GEMS) mission \cite{GEMS} and
XIPE \cite{XIPE}, track the direction of photo electrons ejected in
photoelectric effect interactions of the X-rays. The {\it ASTRO-H}
mission (to be launched in 2015) will carry the {\it Soft Gamma-Ray
Imager}. The detector combines an X-ray and gamma-ray collimator with a
Si scatterer and CdTe absorber. The mission will be able to do
scattering polarimetry at $E > 50 \, \rm{keV}$ energies
\cite{Tajima2010}. Our group is working on a proposal of the space borne
scattering polarimeter PolSTAR which uses a lower-Z LiH stick as a
scatterer to enable the detection of $3 - 80 \, \rm{keV}$ X-rays. The
PolSTAR concept will be described in a forthcoming paper. Missions like
GEMS, PolSTAR and XIPE aim at obtaining $1 \%$ minimum detectable
polarization fractions, see Eq.~(\ref{eqn:MDP}), for mCrab sources.
These missions would allow very high-signal-to-noise detections of
bright galactic sources (e.g. Cyg\,X-1, GRS\,1915+105, Her\,X-1) and
could detect $\sim1 \%$ polarization fractions for extra-galactic
sources (e.g. NGC\,4151).
Scattering polarimeters detect the direction into which photons scatter
when interacting in the detector. Although different scattering
processes dominate at different energies (coherent scattering below a
few keV, Thomson scattering at intermediate energies, and Compton
scattering at energies $> 20 \, \rm{keV}$), all scattering processes
share the property that the photons scatter preferentially perpendicular
to the polarization plane (electric field vector) of the incoming
photon. For example, the angular dependence of Compton scattering
processes is given by the Klein-Nishina cross section \cite{Evans1955}:
\begin{equation}
\label{eq:KleinNishinaCrossSection}
\frac{\rm{d}\sigma}{\rm{d}\Omega} = \frac{r_{0}^{2}}{2}
\frac{k_{1}^{2}}{k_{0}^{2}} \left[ \frac{k_{0}}{k_{1}} +
\frac{k_{1}}{k_{0}} -2 \sin^{2} \theta \cos^{2} \eta \right],
\end{equation}
where $\eta$ is the angle between the electric vector of the incident
photon and the scattering plane, $r_{0}$ is the classical electron
radius, $\bf{k}_{0}$ and $\bf{k}_{1}$ are the wave-vectors before and
after scattering, and $\theta$ is the scattering angle. The azimuthal
distribution of scattered events shows a sinusoidal modulation with a
$180^{\circ}$ periodicity and a maximum at $\pm 90^{\circ}$ to the
preferred electric field direction of a polarized X-ray signal.
In this paper we describe the design and performance of a scattering
polarimeter, X-Calibur. The polarimeter utilizes a plastic scintillator
as scatterer which can scatter $E > 20 \, \rm{keV}$ X-rays efficiently.
This is ideal for the operation on a balloon where the energy threshold
is well matched to the low energy cutoff of the transmissivity caused by
the residual atmosphere above the balloon altitude of 125,000 feet. An
assembly of Cadmium Zinc Telluride (CZT) detectors surrounds the
scintillator in order to record the azimuthal distribution of the
scattered photons, allowing one to reconstruct the polarization
properties of the incoming X-ray beam. The background of charged
particles and high-energy photons is suppressed by an active CsI shield.
The paper is structured as follows. Section~\ref{sec:Science} gives a
brief overview over the scientific potential of hard X-ray polarimetry.
The design of the X-Calibur polarimeter is described in
Sec.~\ref{sec:XCLB}. The data analysis methods are described in
Sec.~\ref{sec:Analysis}, followed by an outline of the simulation
procedure in Sec.~\ref{sec:Simulations}.
Section~\ref{sec:DetectorCharacterization} describes the calibration and
characterization studies of the CZT detectors. Measurements with the
assembled X-Calibur polarimeter to characterize the scattering
scintillator and the shield are described in
Sec.~\ref{sec:XCLB_Characterization}. Polarization measurements with
X-Calibur are described in Sec.~\ref{sec:XCLB_PolarimetryMeasurements}.
The paper ends with a summary and outlook in Sec.~\ref{sec:Conclusion}.
The X-Calibur data presented in this paper were taken (i) in a
laboratory environment at Washington University, (ii) at the Cornell
High Energy Synchrotron Source (CHESS), and (iii) in Ft.~Sumner, NM,
during a preparation campaign for an upcoming balloon flight.
\section{Scientific Potential} \label{sec:Science}
This section discusses the scientific potential for a scattering
polarimeter such as X-Calibur from a balloon platform. X-rays from
cosmic sources can be polarized owing to the anisotropy in the source
geometry and/or the emission characteristics of various processes
\cite{Lei1997}. Non-thermal emission, like synchrotron radiation,
results in a large polarization fraction $r$. Synchrotron emission will
result in linearly polarized photons with their electric fields oriented
perpendicular to the magnetic field lines (projected); the observed
polarization map can therefore be used to trace the magnetic field
structure of the source, a common practice in radio and optical
polarimetry. An electron population with a spectral energy distribution
of $\rm{d}N/\rm{d}E \propto E^{-p}$ emitting in a uniform magnetic field
will lead to an observable fraction of polarization of
\cite{Korchakov1962}:
\begin{equation}
r_{\rm{sync}} = \frac{p+1}{p+7/3}, \quad \rm{with} \, (p + 3) =
\frac{\alpha + 1} {\alpha + 5/3}.
\end{equation}
Here, $\alpha$ is the index of the X-ray power law spectrum. An observed
polarization fraction close to this limit can therefore be interpreted
as an indication of a highly ordered magnetic field since
non-uniformities in the magnetic field will reduce $r_{\rm{sync}}$. The
polarized synchrotron photons can in turn be inverse-Compton scattered
by relativistic electrons~-- weakening the fraction of polarization (but
not erasing it) and imprinting a scattering angle dependence to the
observed fraction of polarization \cite{Krawczynski2012b}. Such
inverse-Compton signals will usually (but not always) appear in hard
gamma-rays, where polarimetry is difficult, due to multiple scattering
in pair production detectors. Another important mechanism for polarizing
photons is Thomson scattering which creates a polarization perpendicular
to the scattering plane \cite{Rybicki1991}. Curvature radiation is
polarized, as well. The scientific potentials of spectro-polarimetric
observations over the broadest possible energy range are summarized
below; more detailed discussions can be found in \citet{Krawcz2011} and
\citet{Lei1997}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{Schnittman_Fig3}
\end{center}
\caption{Ray-traced image of direct radiation from a thermal disk around
a BH including returning radiation (observer located at an inclination
angle of $75^{\circ}$, gas on the left side of the disk moving toward
the observer causing the characteristic increase in intensity due to
relativistic beaming). The image is adopted from \citet{Schnittman2009}.
The observed intensity is color-coded on a logarithmic scale and the
energy-integrated polarization vectors are projected onto the image
plane with lengths proportional to the fraction of polarization.}
\label{fig:BH_Polarization}
\end{figure}
{\it Binary black hole systems.} Particle scattering in a Newtonian
accretion disk surrounding a BH will lead to the emission of polarized
X-rays. Relativistic aberration and beaming, gravitational lensing, and
gravito-magnetic frame-dragging will result in an energy-dependent
fraction of polarization since photons with higher energies originate
closer to the BH than the lower-energy photons \cite{Connors1977}.
\citet{Schnittman2009} calculate the expected polarization signature
including (i) the effects of deflection of photons emitted in the disk
by the strong gravitational forces in the regions surrounding the BH and
(ii) re-scattering these photons by the accretion disk
\cite{Schnittman2009, Schnittman2010}. The resulting effect is a swing
in the polarization direction from being horizontal at low energies to
vertical at high energies, i.e., parallel to the spin axis of the BH.
Spectro-polarimetric observations can therefore be used to constrain the
mass and spin of the BH \cite{Schnittman2009}, as well as the
inclination of the inner accretion disk and the shape of the corona
\cite{Schnittman2010}, see Fig.~\ref{fig:BH_Polarization}. In principle,
X-ray polarization can also be used to test General Relativity in the
strong gravity regime \cite{Krawczynski2012c}.
{\it Pulsars.} High-energy particles in pulsar magneto-spheres are
expected to emit synchrotron and/or curvature radiation which are
difficult to distinguish from one another, solely based on the observed
photon energy spectrum. However, since the orbital planes for
accelerating charges that govern these two radiation processes are
orthogonal to each other, their polarized emission will exhibit
different behavior in position angle and polarization fraction as
functions of energy and the rotation phase of the pulsar
\cite{Dean2008}. An illustration of the models of phase dependence of
the X-ray/gamma-ray polarization signatures in pulsars can be found in
\citet{Dyks2004}. In magnetars, the highly-magnetized cousins of
pulsars, polarization-dependent resonant Compton up-scattering is a
leading candidate for generating the observed hard X-ray tails
\cite{BaringHarding2007}. In both these classes, phase-dependent
spectro-polarimetry can probe the emission mechanism, and provide
insights into the magnetospheric locale of the emission region.
Cyclotron lines arising from transitions between Landau levels in
intense magnetic fields that occur in the polar regions of neutron stars
and magnetars are polarized. Also, the absorption scattering
cross-sections from the Landau levels are dependent on the polarization
state of the X-rays, so that the radiative transfer through such plasma
will lead to a polarization of the emergent radiation. The first such
cyclotron feature was observed by \citet{Truemper1978} in Her\,X-1 and
is interpreted as due to an absorption line around $40 \, \rm{keV}$
\cite{Staubert2007}. Such features allow an estimate of the magnetic
field strength \cite{Coburn2002}. Observations of polarized X-rays will
strongly confirm that these features are indeed due to cyclotron lines
in magnetic fields of $3-5 \times 10^{12} \, \rm{G}$. A detailed
discussion of the theoretical aspects of cyclotron radiation is given in
\citet{Leahy2010}.
{\it Pulsar wind nebulae.} The compact object (e.g. pulsar) resulting
from a previous supernova explosion can be surrounded by a synchrotron
emitting nebula. The nebula extends far beyond the magneto-sphere of the
central pulsar, but its emission is believed to be driven by the pulsar
which injects relativistic electrons/positrons that are further shock
accelerated in the nebula. Spectro-polarimetric observations can be used
to constrain the magnetic field and particle populations in such pulsar
wind nebulae~-- such as the Crab nebula, the leading driver for this
field of X-ray polarimetry. Given the more compact emission regions at
high energies, these objects potentially show a higher polarization
fraction at hard X-rays as compared to soft X-rays, reflecting the
contrast between jet and more diffuse nebular contributions
\cite{Forot2008}.
{\it Supernova remnants.} Supernova remnants (SNRs) present an
opportunity to perform X-ray polarimetry, as well. The remnants possess
tangled magnetic fields on large scales in their interiors, as is
evidenced in the classic radio polarization map of the Crab nebula
\cite{Velusamy1985}. Both, radio and X-ray signals, are believed to be
due to synchrotron emission, and so it is reasonably presumed that X-ray
emission from SNRs should be significantly polarized. The X-ray spectra
of such remnants are typically steeper than spectra in the radio, which
leads to the expectation of higher polarization fractions in the X-ray
band. Yet, since the electrons generating X-rays will diffuse on larger
spatial scales than their radio-emitting counterparts do, the X-ray
signals should capture the field morphology on larger scales. It may or
may not be more coherent than the field structure on smaller (radio)
scales. Due to instrumental limitations in angular resolution at X-ray
energies, however, it will not be possible to resolve individual
regions~-- depending on the angular size of the remnant. The
observational challenge will therefore be to overcome the competition
between compact regions causing highly polarized emission on one hand,
and on the other hand an averaging effect of different emission regions
with different orientations of their magnetic fields. An example of
expectations for the SNR synchrotron polarization properties can be
found in \citet{Bykov2009}.
{\it Relativistic jets in active galactic nuclei.} Relativistic
electrons in jets of active galactic nuclei (AGN) emit polarized
synchrotron radiation at radio/optical wavelengths. The same electron
population is believed to produce hard X-rays by inverse-Compton
scattering off a photon field. Simultaneous measurements of the
polarization angle and the fraction of polarization in the radio to hard
X-ray band could help to disentangle the following scenarios: (i) If the
electrons mainly up-scatter the co-spatial synchrotron photon field
(synchrotron self Compton), the polarization of the hard X-rays is
expected to track the polarization at radio/optical wavelengths
\cite{Poutanen1994}. The fraction of X-ray polarization could be close
to the fraction of polarization of the synchrotron emission measured in
the radio/optical bands and the polarization directions between radio,
optical and X-rays should be identical. (ii) If the electrons dominantly
up-scatter an external photon field (external Compton, e.g. photons of
the cosmic microwave background or from the accretion disk surrounding
the super-massive black hole) the hard X-rays will have a relatively
small ($<$10\%) fraction of polarization \cite{McNamara2009}. Hadronic
jet emission models for low-synchrotron-peaked AGN, on the other hand,
predict an even higher fraction of polarization at high energies,
compared to the leptonic SSC models \cite{Zhang2013}.
Polarization also allows one to test the structure of the magnetic field
of the jet. Particles accelerated in a helical field which are moving
through a standing shock can cause an X-ray synchrotron flare with a
continuous (in time) swing in polarization direction. Such an event was
observed from BL\,Lacertae at optical wavelengths \cite{Marscher2008}.
{\it Gamma-ray bursts.} Gamma-ray bursts are believed to be connected to
hyper-nova explosions and the formation/launch of relativistic jets
\cite{Woosley1993}. As in the case of the jets in AGN, the structure of
their jets and the particle distribution responsible for gamma-ray
bursts can be revealed by X-ray polarization measurements
\cite{Kostelecky2013}. The X-ray emission of a gamma-ray burst, however,
usually lasts for only a few minutes at most, so that rapid follow-up
observations in the X-ray band below $30 \, \rm{keV}$ would be the main
challenge for studying their polarization properties.
On a one-day balloon flight, we would achieve $5-15 \%$ MDPs, see
Eq.~(\ref{eqn:MDP}), for between 1 and 4 sources. For the Crab, the
phase resolved polarimetry would allow us to decide between emission
models. For Cyg\,X-1 and GRS\,1915+105 we could test corona models, and
for Her\,X-1, we could get a first estimate of the polarization fraction
of the X-ray emission.
\section{Design of X-Calibur} \label{sec:XCLB}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.99\textwidth]{X-Calibur_InFocus}
\end{center}
\caption{Schematic view of the functionality of X-Calibur (not to
scale). X-rays from an astrophysical source are focused with a grazing
incidence mirror onto the scattering rod of the polarimeter. The
scattered X-ray is recorded in one of the surrounding CZT detectors. The
polarimeter is embedded by a shield to suppress background of particles
not originating from the mirror.}
\label{fig:XCLB_Functionality}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[height=0.38\textheight]{X-Calibur_Schematic}
\hfill
\includegraphics[height=0.38\textheight]{P1080730_LabVersion4}
\hfill
\includegraphics[height=0.38\textheight]{P1090499_X-Calibur}
\end{center}
\caption{Design of the X-Calibur polarimeter. {\bf Left:} Schematic
illustration. Side view: incoming X-rays Compton-scatter in the
scintillator rod (aligned with the optical axis, read out by a PMT) and
are subsequently photo-absorbed in one of the surrounding CZT detectors
($8\times8$ pixels each, see Fig.~\ref{fig:DetectorRing}). A group of
four detectors surrounding the scintillator at a given depth are
referred to as ring ({\it R1} to {\it R8}). Each ring can be further
divided into top and bottom sub rings, as illustrated for ring {\it R8}.
Top view: Detector ring viewed along the optical axis, looking into the
X-ray beam. The polarization signature is imprinted in the azimuthal
scattering distribution (see Fig.~\ref{fig:X-Calibur_Configurations},
top, for more details). The four sides of detectors are referred to as
boards ({\it Bd0} to {\it Bd3}), each board comprising all eight
detectors per side. 3D view (`exploded'): four sides of detector columns
({\it Bd0} to {\it Bd3}) surround the central scintillator rod~--
covering the whole range of azimuthal scattering angles. {\bf Middle:}
Partly assembled polarimeter with two detector sides removed for better
visibility. The red wires provide the high voltage to the detectors. The
read-out electronics is stacked at the backside of the detectors. {\bf
Right:} Fully assembled polarimeter (flipped upside-down: X-ray entering
from the bottom).}
\label{fig:Design}
\end{figure*}
\begin{table}[t!]
\begin{tabular}{rrrrr}
ID & Serial-No. & Date & $C_{\rm{ch}}$ & $C_{\rm{wu/ft}}$ \\
\hline \hline
\noalign{\smallskip}
\multicolumn{5}{l}{{\bf Endicott 5\,mm}} \\
\hline
EN$_{5}$1 & 672992-01 & 01/11 & 0/5 & 0/5 \\
EN$_{5}$2 & 672992-02 & 01/11 & 1/5 & 1/5 \\
EN$_{5}$3 & 672992-03 & 01/11 & 2/5 & 2/5 \\
EN$_{5}$4 & 672992-04 & 01/11 & 3/5 & 3/5 \\
EN$_{5}$5 & 672994-04 & 02/11 & 0/4 & 0/4 \\
EN$_{5}$6 & 672994-03 & 02/11 & 1/4 & 1/4 \\
EN$_{5}$7 & 672994-02 & 02/11 & 2/4 & 2/4 \\
EN$_{5}$8 & 672994-01 & 02/11 & 3/4 & --- \\
\noalign{\smallskip}
\multicolumn{5}{l}{{\bf Creative Electron 5\,mm}} \\
\hline
CE$_{5}$1 & 721613 & 03/11 \\
\noalign{\smallskip}
\multicolumn{5}{l}{{\bf QuikPak 5\,mm}} \\
\hline
QP$_{5}$1 & 3627 & 03/11 & --- & 3/4 \\
QP$_{5}$3 & 3611 & 03/11 & --- & --- \\
QP$_{5}$4 & 3834 & 03/11 & 3/1 & 3/1 \\
QP$_{5}$6 & 6292 & 03/11 & 0/1 & 0/1 \\
QP$_{5}$7 & 6345 & 03/11 & 2/1 & 2/1 \\
QP$_{5}$8 & 721c602 & 03/11 & 1/1 & 1/1 \\
QP$_{5}$13 & 6886 & 04/12 & 0/3 & 0/3 \\
QP$_{5}$14 & 6977 & 04/12 & 2/3 & 2/3 \\
QP$_{5}$16 & 10814 & 04/12 & 2/2 & 2/2 \\
QP$_{5}$17 & 10819 & 04/12 & 0/2 & 0/2 \\
QP$_{5}$18 & 10829 & 04/12 & 1/2 & 1/2 \\
QP$_{5}$19 & 10847 & 04/12 & 3/3 & 3/3 \\
QP$_{5}$20 & 10848 & 04/12 & 3/2 & 3/2 \\
QP$_{5}$21 & 10860 & 04/12 & 1/3 & 1/3 \\
\noalign{\smallskip}
\multicolumn{5}{l}{{\bf Endicott 2\,mm}} \\
\hline
EN$_{2}$1 & 674326-01 & 02/11 & 0/6 & 0/8 \\
EN$_{2}$2 & 674327-01 & 02/11 & 2/8 & 2/8 \\
EN$_{2}$3 & 674328-01 & 02/11 & 2/6 & 2/6 \\
EN$_{2}$4 & 674328-02 & 02/11 & 0/8 & 0/6 \\
EN$_{2}$5 & 674329-01 & 02/11 & --- & --- \\
EN$_{2}$6 & 674330-01 & 02/11 & 1/6 & 1/6 \\
EN$_{2}$7 & 674331-01 & 02/11 & 3/6 & --- \\
EN$_{2}$8 & 674332-01 & 02/11 & --- & 3/8 \\
\noalign{\smallskip}
\multicolumn{5}{l}{{\bf Creative Electron 2\,mm}} \\
\hline
CE$_{2}$1 & 2180 & 03/11 & 0/7 & 0/7 \\
CE$_{2}$2 & 720612 & 03/12 & 3/8 & 3/6 \\
CE$_{2}$3 & 720511 & 03/12 & 1/8 & 1/8 \\
CE$_{2}$4 & 721541i & 03/12 & 1/7 & 1/7 \\
CE$_{2}$5 & 06172A & 03/12 & 2/7 & 2/7 \\
CE$_{2}$6 & 726712 & 03/12 & 3/7 & 3/7 \\
\end{tabular}
\caption{CZT detector sample. ID as in this paper (two letters for the
company, index for the detector thickness $[\rm{mm}]$, and a number),
serial number, date of delivery [MM/YY], and position in polarimeter
[Bd/ring] for the configurations $C_{\rm{ch}}$ and $C_{\rm{wu/ft}}$ (see
Fig.~\ref{fig:X-Calibur_Configurations}).}
\label{tab:Detectors}
\end{table}
\begin{figure*}[t]
\begin{center}
\includegraphics[height=0.255\textheight]{DetectorGold_Rotated}
\hfill
\includegraphics[height=0.255\textheight]{Detector_BondedAndASICs}
\hfill
\includegraphics[height=0.255\textheight]{p1060223_zoom}
\end{center}
\caption{Definition of a detector `ring'. {\bf Left:} $2 \times 2 \times
0.2 \, \rm{cm}^{3}$ CZT detector with 64 pixels (anode side). {\bf
Middle:} $2 \times 2 \times 0.2 \, \rm{cm}^{3}$ CZT detector bonded to a
ceramic chip carrier which is plugged into 2 ASIC readout boards. The
high-voltage cable is glued to the detector cathode (red wire). {\bf
Right:} Four CZT detectors surrounding the scintillator (blueish glow)
form a unit referred to as `ring'~-- covering the whole $360^{\circ}$
azimuthal scattering range with $4 \times 8 = 32$ pixels. The whole
polarimeter is equipped with eight rings.}
\label{fig:DetectorRing}
\end{figure*}
The balloon-borne version of X-Calibur is a low-Z Compton scattering
polarimeter that will allow one to measure polarization fractions in the
$20 - 80 \, \rm{keV}$ band down to the percentage level. X-Calibur will
be used in the focal plane of the X-ray mirror of the {\it InFOC$\mu$S}\xspace telescope
with a field-of-view of $10 \, \rm{arc min}$; X-Calibur does not provide
imaging capabilities. Owing to the fact that a grazing incidence mirror
reflects only under very shallow angles, it changes the polarization
properties of X-rays by less than $1\%$ \cite{Katsuta2009}. The
advantages of the X-Calibur design can be described as follows. (i) A
high detection efficiency is achieved, using roughly $80 \%$ of photons
impinging on the polarimeter. (ii) The use of a focusing optics instead
of a large detector volume results in a compact instrument design that
can be shielded efficiently~-- strongly reducing the background level.
(iii) The continuous rotation of the polarimeter strongly reduces
possible systematic effects that can hamper non-rotating polarimeters
due to asymmetric azimuthal detector responses. These characteristics,
as outlined in the following sections, make it a well-suited experiment
to study several sources mentioned in Sec.~\ref{sec:Science} in a
one-day balloon flight. The energy-dependent detection efficiency of the
polarimeter depends on (i) the effective area and point-spread function
of the X-ray mirror, (ii) the fraction of scatterings compared to
competing interactions such as photo-absorption, and (iii) the
geometrical detector coverage to record a high fraction of scattered
X-rays (minimization of possible escape paths) \cite{Guo2010}. This
section describes the overall design of the polarimeter, as well as the
characteristics of its individual components.
\subsection{Design}
The conceptual design of the X-Calibur polarimeter is illustrated in
Figures~\ref{fig:XCLB_Functionality} and \ref{fig:Design}. A low-Z
scintillator rod aligned with the optical axis of the telescope is used
as Compton-scatterer~-- leading to a polarization-dependent azimuthal
scattering distribution that is recorded by the surrounding assembly of
CZT detectors. The azimuthal distribution is resolved by $4\times8 = 32$
pixels for each of the $64$ depth bins along the optical axis. Detailed
information on the simulations to optimize the X-Calibur design (as
presented in this paper) can be found in \citet{Krawcz2011} and
\citet{Guo2013}.
Throughout the paper, we refer to a detector ring as a set of four CZT
detectors surrounding the scintillator rod on four sides at a given
depth along the optical axis (see Fig.~\ref{fig:DetectorRing}, right).
Ring {\it R1} is situated at the polarimeter entrance (top in
Fig.~\ref{fig:Design}, left), and ring {\it R8} is situated at its rear
end. Each ring covers the whole $360^{\circ}$ azimuthal scattering
range. A ring can further be subdivided into sub rings, eg. {\it
R8$_{\rm{t}}$} and {\it R8$_{\rm{b}}$} for the top and bottom half,
respectively (see Fig.~\ref{fig:Design}, left). The smallest possible
subdivision is a ring consisting of only one pixel row, referred to as
{\it single-pixel ring}\xspace. All eight detectors situated on one of the four sides are
referred to as detector board {\it Bd0} to {\it Bd3} (see
Fig.~\ref{fig:Design}, left).
In the scintillator, photoelectric effect interactions dominate below
$\sim$$15 \, \rm{keV}$. At $20 \, \rm{keV}$, however, the cross section
of the photoelectric absorption already drops to $0.1 \,
\rm{cm}^{2}/\rm{g}$ and can be neglected as compared to the cross
section of Compton scattering for which X-Calibur is designed. This
energy regime therefore defines the detection threshold of the
polarimeter. The mean free path for Compton scattering in the
scintillator material is $\approx$$4 \, \rm{cm}$, so that the length of
the scattering rod of $14 \, \rm{cm}$ covers $\simeq 3.5$ path lengths.
This translates into a $\simeq$$90 \%$ probability for
Compton-scattering in the energy regime of $20 - 80 \, \rm{keV}$. For
sufficiently energetic photons, the Compton interaction produces enough
scintillation light to trigger a photo-multiplier tube (PMT) attached to
the end of the scintillator. The trigger efficiency of the
scintillator/PMT unit is discussed in
Sec.~\ref{subsec:ScintillatorEfficiency}. The scattered X-rays are in
turn photo-absorbed in the surrounding rings of high-Z CZT detectors.
This combination of scatterer/absorber leads to a high fraction of
unambiguously detected Compton events~-- in contrast to background
events not entering the polarimeter along the optical axis for which the
scintillator/CZT coincidences are strongly reduced (see
Sec.~\ref{subsec:BG_Data}). Linearly polarized X-rays will preferably
Compton-scatter perpendicular to their electric field vector, see
Eq.~(\ref{eq:KleinNishinaCrossSection}). This will result in a
modulation of the measured azimuthal scattering distribution that is
used to determine the polarization properties of the X-rays (see
Sec.~\ref{subsec:AnalysisPolarization}).
\subsection{The CZT Detectors}
X-ray photons that penetrate a semi-conductor X-ray detector deposit
their energy in the detector volume, usually through photo-electric
effect interactions. A charge cloud proportional to the X-ray energy is
created and accelerated by a high-voltage electric field that is applied
between the detector cathode and the pixels on the anode side. The
moving charge is measured and digitized and the grid position of the
corresponding pixel reflects the position of the interaction. CZT is the
semi-conductor material of choice for X-ray detectors operating in the
$E>5 \, \rm{keV}$ to MeV energy band with a high probability for
photo-electric effect interactions, see for example \citet{VarPitch}.
The CZT detectors used in X-Calibur were ordered from different
companies\footnote{Endicott Interconnect: http://www.evproducts.com,
Quikpak/Redlen: http://redlen.ca, Creative Electron:
http://creativeelectron.com}. The sample of detectors is listed in
Tab.~\ref{tab:Detectors}. Each detector ($2\times2 \, \rm{cm}^{2}$) is
contacted with a 64-pixel anode grid ($2.5 \, \rm{mm}$ pixel pitch) and
a monolithic cathode facing the scintillator rod. Two different detector
thicknesses ($2 \, \rm{mm}$ and $5 \, \rm{mm}$) are used in the
polarimeter (Fig.~\ref{fig:Design}, left). Historically, our group had
been working with detector thicknesses of $5 \, \rm{mm}$ and higher.
Therefore, five of the polarimeter rings are equipped with $5 \,
\rm{mm}$ detectors. The remaining three rings are equipped with $2 \,
\rm{mm}$ detectors which are still sufficient to absorb more than $99\%$
of X-rays in the energy regime relevant for X-Calibur, and at the same
time measure a lower level of background (which scales roughly with the
detector volume). The cathodes of the detectors are biased at
$V_{\rm{bi,5}} = -500 \, \rm{V}$ ($5 \, \rm{mm}$ detectors) and at
$V_{\rm{bi,2}} = -150 \, \rm{V}$ ($2 \, \rm{mm}$ detectors),
respectively.
Each CZT detector is permanently bonded (anode side) to a ceramic chip
carrier which is plugged into the electronic readout board.
Figure~\ref{fig:DetectorRing} (left) shows a single CZT detector unit
with an $8 \times 8$ pixel matrix on the anode side as well as the
readout electronics. Each CZT detector is read out by two digitizer
boards, each consisting of a 32 channel ASIC and a 12-bit
analog-to-digital converter. The ASIC was developed by G.~De~Geronimo
(BNL) and E.~Wulf (NRL) \cite{Wulf2007}. The ASICs are operated at a
medium amplification (gain) of $28.5 \, \rm{mV/fC}$ and a signal peaking
time of $0.5 \, \mu \rm{s}$. These settings are a result of a previous
optimization to achieve an optimal energy resolution and low noise. Each
ASIC has a built-in capacitor that allows one to directly inject a
programmable amount of charge into the individual readout channels for
testing purposes. The readout noise of the ASIC is as low as $2.5 \,
\rm{keV}$ FWHM (see Fig.~\ref{fig:TempStudiesNoise} in
Sec.~\ref{subsec:TempStudies}). All 16 digitizer boards (reading eight
CZT detectors) are read out by one harvester board ({\it Bd0}-{\it Bd3},
see Fig.~\ref{fig:Design}) transmitting the data to a PC-104 computer
with a rate of 6.25~Mbits/s. X-Calibur comprises 2048 data channels. The
time to read and process a triggered event is about $130 \, \mu\rm{s}$
(ASIC dead time). However, only the ASIC involved in the triggered event
will be dead during the read-out. All other ASICs will still be
sensitive and can store events that will be read out once the previous
read-out cycle is completed.
\subsection{The Scintillator}
A plastic scintillator rod is used as Compton-scatterer. The advantage,
compared to other scattering materials, is the scintillation light
produced in the scattering interaction. The light is read by a PMT and
can be used (optional) in the analysis. The EJ-200 scintillator
(Hydrogen:Carbon ratio of $5.17:4.69$, $\left< Z \right> = 3.4$, $\rho
\approx 1 \, \rm{g} / \rm{cm}^{3}$, decay time $2.1 \, \rm{nsec}$) is
used, read by a Hamamatsu R7600U-200 PMT with a high quantum efficiency
super-bi-alkali photo cathode. To increase the optical yield, the
scintillator is wrapped in white {\it tyvek$^{\tiny \circledR}$} paper.
The PMT signal is amplified and digitized. A discriminator tests whether
the digitized PMT pulse exceeds a programmable trigger threshold and
activates a corresponding flag ($f_{\rm{sci}}$). The flag is kept high
for $6 \, \mu\rm{s}$ and is merged into the data stream of triggered CZT
detector events. The trigger efficiency of the scintillator is studied
in Sec.~\ref{subsec:ScintillatorEfficiency}. The $f_{\rm{sci}}$ flag
allows one to select scintillator/CZT events from the data, which
represent likely Compton-scattering candidates~-- strongly suppressing
other backgrounds (see Sec.~\ref{subsec:BG_Data}). However, the
polarimeter can be operated without the PMT trigger information, with
the scintillator acting only as a passive scatterer.
\begin{figure*}[t]
\begin{center}
\includegraphics[height=0.31\textheight]{beilicke_FRAWS_2013_02_fig03}
\includegraphics[height=0.31\textheight]{P1090353_MotorAndBearing_Plus_TungstenCap}
\hfill
\includegraphics[height=0.31\textheight]{P1090507_X-CaliburInVessel_Top}
\hfill
\includegraphics[height=0.31\textheight]{InfocusTelescope}
\end{center}
\caption{Active/passive shield, rotation design, and {\it InFOC$\mu$S}\xspace X-ray
telescope. {\bf Left:} X-Calibur polarimeter, embedded by the CsI active
shield, as well as the electronic readout and azimuthal rotation
bearing. {\bf 2$^{\rm{nd}}$ from left, top:} Tungsten cap with
collimator, only allowing X-rays from the mirror to enter the shield.
{\bf 2$^{\rm{nd}}$ from left, bottom:} Motor and ring bearing to rotate
polarimeter/shield assembly. {\bf 2$^{\rm{nd}}$ from right:} Opened
pressure vessel with shield/X-Calibur installed (tungsten cap removed).
The X-ray beam enters the polarimeter through the hole in the top plate.
{\bf Right:} The {\it InFOC$\mu$S}\xspace balloon gondola/truss with active pointing
control and the X-ray mirror installed. The pressure vessel is installed
in the back.}
\label{fig:ActiveShieldAndTelescope}
\end{figure*}
\subsection{The Shield}
During the balloon flight, the polarimeter will be hit by charged and
neutral particle backgrounds with different spectral signatures and
intensities. These backgrounds reduce the signal-to-noise ratio and, in
the case of non-isotropic fluxes, can even lead to a fake polarization
signature. In order to suppress these backgrounds, the polarimeter and
the front-end readout electronics are operated inside an active CsI(Na)
anti-coincidence shield. The $2.7 \, \rm{cm}$ thick CsI crystal of the
shield covers the sides and the bottom of the polarimeter and produces
scintillation light when particles interact. The top is protected by a
passive tungsten plate/collimator
(Fig.~\ref{fig:ActiveShieldAndTelescope}, left), blocking X-rays and
particles that do not come from the X-ray mirror. The (active) CsI
scintillator of the shield is read out by four Hamamtsu PMTs R\,6233
which are biased at $V_{\rm{bi}} = +800 \, \rm{V}$. The analog signal of
all four PMTs is merged and in turn digitized. A programmable, digital
discriminator decides on whether a shield flag $f_{\rm{shld}}$ is set on
the CZT readout board (kept up for $6 \, \mu\rm{s}$) and is merged into
the data stream. The values of the discriminator and the width of the
flag were optimized using a radio-active source to maximize the shield
efficiency and minimize chance coincidences (see
Sec.~\ref{subsec:BG_Data}).
In order to reduce the systematic uncertainties of the polarization
measurements, the polarimeter and the active shield will be rotated
around the optical axis with $\sim 2 \, \rm{rpm}$ using a ring bearing
(see Fig.~\ref{fig:ActiveShieldAndTelescope}). The angle between the
polarimeter/shield and the mounting fixture is read out by a code wheel
with the accuracy of $1^{\circ}$. A counter-rotating mass can be used to
cancel the net angular momentum of the rotating polarimeter assembly
during the balloon flight. The computer reading the PMT and CZT events
is part of the rotating assembly, and referred to as polarimeter CPU.
\subsection{The {\it InFOC$\mu$S}\xspace X-ray Telescope}
The X-Calibur polarimeter will be flown in a pressurized vessel located
in the focal plane of the {\it InFOC$\mu$S}\xspace X-ray telescope
\cite{InFocus_FirstFlight}. The telescope is shown in the right panel of
Fig.~\ref{fig:ActiveShieldAndTelescope}. A Wolter grazing incidence
mirror focuses the X-rays onto the polarimeter. The X-Calibur
scintillator rod will be aligned with the optical axis of the {\it InFOC$\mu$S}\xspace
X-ray telescope (see Sec.~\ref{subsec:FtSumnerData}). The focal length
of the mirror is $8 \, \rm{m}$ and the field of view is $\rm{FWHM} = 10
\, \rm{arc min}$. The telescope truss of {\it InFOC$\mu$S}\xspace is only coupled to the
gondola by a ball joint in a support cup with floating oil, allowing for
full inertial pointing of the telescope with an accuracy of $7 ''$ and
$15 ''$ RMS in altitude and azimuth, respectively. To maintain the
decoupling between truss and gondola, any communication between the two
systems is done by wireless connections. In addition to the rotating
polarimeter CPU (see above), a second CPU as part of the X-Calibur
subsystem (motor CPU) is installed in the pressure vessel (non-rotating)
and controls the motors, and a temperature system. Power and data
communication between the polarimeter CPU and the motor CPU is achieved
by a {\it Mercotac$^{\tiny \circledR}$} 830-SS rotating ring of mercury
sliding contacts. Communication between the pressure vessel and the
telescope gondola will be done via a wireless network. The data will be
stored on solid state drives and will in parallel be down-linked to the
ground.
\begin{figure}[th!]
\begin{center}
\includegraphics[width=0.44\textwidth]{X-Calibur_ConfigurationsCZT} \\
\includegraphics[width=0.46\textwidth]{X-Calibur_Configurations}
\end{center}
\caption{Different X-Calibur configurations referred to via
$C_{\rm{env}}^{\rm{stp}}$. {\bf Top row:} CZT detector geometries (see
Fig.~\ref{fig:Design}, left) viewed along the optical axis looking into
the X-Ray beam. Each detector is tangentially translated by $\Delta = 1
\, \rm{mm}$. $C^{1}$: CZT detector cathodes being $c = 16 \, \rm{mm}$
away from the optical axis (regular-sized sealed radioactive source fits
in between). $C^{2}$: Reduced distance of $c =11 \, \rm{mm}$ (only
compact sources). $C^{3}$: same as $C^{2}$ but with the scintillator rod
installed. The definitions of $\Phi$ and $\Delta \Phi$ as used in
Sec.~\ref{sec:XCLB_Characterization} are illustrated. {\bf Bottom:}
Schematic sketches (not to scale) of X-Calibur installed in the
different environments. $C_{\rm{wu}}$: Washington University; detector
calibration in copper box ($C_{\rm{wu1a}}$) or in passive CsI shield
($C_{\rm{wu1b}}$); performance measurements in the active shield with
different source positions ($C_{\rm{wu2a}}$, $C_{\rm{wu2b}}$, and
$C_{\rm{wu2c}}$). $C_{\rm{ch}}$: X-Calibur/CHESS setup in hutch {\it
C1}. The beam intensity is reduced by absorption foils; platinum
($C_{\rm{ch1}}$), or platinum and lead ($C_{\rm{ch3}}$). The X-Calibur
polarimeter can be rotated around its optical axis (angle $\alpha$)
which is aligned with the CHESS beam. $C_{\rm{ft}}$: X-Calibur installed
in the {\it InFOC$\mu$S}\xspace X-ray telescope during a flight preparation campaign in
Ft.~Sumner.}
\label{fig:X-Calibur_Configurations}
\end{figure}
\subsection{X-Calibur Configurations and Data Sets}
In order to characterize the different components and aspects of the
X-Calibur polarimeter, different types of measurements were performed
with different geometrical configurations of the instrument (e.g. with
and without the shield, measurements without the scattering rod to
calibrate the CZT detector response itself, illumination of the
instrument with different X-ray sources from different angles, etc.).
The measurements were performed at different facilities/locations (which
we refer to as environments). The energy calibration and
characterization of the CZT detectors was performed in the laboratory at
Washington University (Sec.~\ref{sec:DetectorCharacterization}).
Measurements of a polarized X-ray beam to study the performance of the
polarimeter were conducted at the CHESS synchrotron facility at Cornell
University (Sec.~\ref{subsec:CHESS}). Data were also taken with the
fully integrated X-Calibur/{\it InFOC$\mu$S}\xspace telescope in a field campaign in
Ft.Sumner, NM (Sec.~\ref{subsec:FtSumnerData}). Measurements of the
background (Sec.~\ref{subsec:BG_Data}) were performed at Washington
University, CHESS, and in Ft.Sumner. Throughout the paper, each
configuration and location is referred to as $C_{\rm{env}}^{\rm{stp}}$
(`env' referring to the environment/location and `stp' referring to the
experimental setup). These different configurations are shown in
Fig.~\ref{fig:X-Calibur_Configurations} and will be referred to in the
sections to follow in which the corresponding results are presented.
It should be noted, that some of the data presented in this paper were
taken without the CsI shield (e.g. the data taken at the CHESS
facility), not allowing one to use the shield veto for background
suppression. Since separate background runs were taken and subtracted
from the data, and the majority of measurements is signal-dominated,
this does not affect any result or conclusion presented in this paper.
\section{Reconstruction of Polarization Properties} \label{sec:Analysis}
The events recorded by X-Calibur consist of the digitized pulse height
of one (or more) detector pixel(s), a time stamp, and flags describing
whether the shield and/or the scintillator triggered. These raw events
are first transformed into measured energies using the detector
calibration. The reconstructed events are further processes in order to
derive energy spectra and the polarization properties of the measured
X-ray beam. This section outlines the corresponding procedures.
\subsection{Definitions}
The modulation in the measured azimuthal scattering distribution is the
defining signature from which the polarization properties are derived.
The modulation factor $\mu$ describes the polarimeter response to a
polarized beam and is used to reconstruct the polarization properties.
Assuming a $100 \%$ linearly polarized X-ray beam, the minimum
($C_{\rm{min}}$) and maximum ($C_{\rm{max}}$) number of counts of the
azimuthal scattering distribution define:
\begin{equation}
\label{eq:ModulationFactor}
\mu = \frac{C_{\rm{max}} - C_{\rm{min}}}{C_{\rm{max}} + C_{\rm{min}}}.
\end{equation}
It represents the modulation amplitude of a $100 \%$ polarized beam and
depends on the polarimeter design and the physics of Compton-scattering.
The performance of a polarimeter can be characterized by the minimum
detectable polarization (MDP) as the minimum fraction of polarization
that can be detected at the $99 \%$ confidence level for a given time of
observation $T$. Assuming a polarimeter that detects all
Compton-scattered photons with an ideal angular resolution~-- in this
case $\mu$ becomes the modulation amplitude averaged over all solid
angles and the Klein-Nishina cross section~-- one can estimate the MDP
by integrating the scattering probability distribution
\cite{Weisskopf2011, StokesAnalysis} ($R_{\rm{src}}$ and $R_{\rm{bg}}$
are the source and background count rates, respectively):
\begin{equation}
\label{eqn:MDP}
\rm{MDP} \simeq \frac{4.29}{\mu R_{\rm{src}}} \sqrt{\frac{R_{\rm{src}} +
R_{\rm{bg}}}{T}}.
\end{equation}
\subsection{X-Calibur Event Reconstruction and Selection}
Each recorded X-Calibur event contains an event number, a GPS time stamp
$T_{\rm{gps}}$, the orientation angle $\theta_{\rm{w}}$ of the
shield/polarimeter with respect to the mounting plate (read by a code
wheel), and a list of CZT detector pixels that were hit (up to nine)
including their digitized pulse heights. Furthermore, two flags are
merged into the data stream: (i) a flag $f_{\rm{shld}}$ that indicates
whether the PMTs reading the active CsI shield got a signal exceeding
the defined discriminator threshold, and (ii) a flag $f_{\rm{sci}}$ that
indicates if the PMT reading the central scintillator rod of the
polarimeter (see Fig.~\ref{fig:Design}, left) exceeded its discriminator
threshold. The average analog rise/fall times
$\tau_{\rm{r}}$/$\tau_{\rm{f}}$ (time for a signal rise from $10-90\%$
of the amplitude) of the folded scintillator/PMT response were measured
for the shield and for the scintillator. For the shield we find
$\tau_{\rm{r}}^{\rm{shld}} \approx 70 \, \rm{ns}$ and
$\tau_{\rm{f}}^{\rm{shld}} \approx 2.6 \, \mu\rm{s}$, respectively. The
response of the scintillator rod of the polarimeter is much faster:
$\tau_{\rm{r}}^{\rm{sci}} \approx 3.0 \pm 1.3 \, \rm{ns}$ and
$\tau_{\rm{f}}^{\rm{sci}} \approx 13.7 \pm 5.1 \, \rm{ns}$ for cosmic
rays, and $\tau_{\rm{r}}^{\rm{sci}} \approx 4.7 \pm 2.7 \, \rm{ns}$ and
$\tau_{\rm{f}}^{\rm{sci}} \approx 13.5 \pm 5.0 \, \rm{ns}$ for a
Cs$^{137}$ source placed at configuration $C_{\rm{wu2c}}^{3}$,
respectively. Upon a CZT trigger, the event readout is delayed by
$2.5-3.3 \, \mu\rm{s}$ (jitter) to allow the shield and scintillator PMT
signals to built-up and being converted into the corresponding flags.
The flags $f_{\rm{shld}}$ and $f_{\rm{sci}}$ are kept up for a duration
of $6 \, \mu\rm{s}$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{MultiplicityDistribution}
\end{center}
\caption{Distribution of pixel multiplicities $m$ for events taken with
different sources (recorded in detector ring {\it R3}): Cosmic ray (CR)
background in the laboratory, energy calibration using Eu$^{152}$,
Eu$^{152}$ calibration restricting event energies to $E < 80 \,
\rm{keV}$, and data taken at the $40 \, \rm{keV}$ CHESS beam (see
Sec.~\ref{subsec:CHESS}).}
\label{fig:MultiplicityDistribution}
\end{figure}
For each event, the pulse heights of all contributing channels/pixels
$i=1...m$ are transformed into energies $E_{i}$ using the channel
calibration, see Eq.~(\ref{eq:EnergyCalibration}) in
Sec.~\ref{subsec:CZT_Calibration}. The total energy of the CZT event is
$E = \sum_{i} E_{i}$. The number of pixels $m$ participating in the
event is referred to as the pixel multiplicity. Selection cuts can be
applied to the data based on the event properties mentioned above:
$T_{\rm{gps}}$, $\theta_{\rm{w}}$, $E$, $m$, $f_{\rm{shld}}$, and
$f_{\rm{sci}}$. Figure~\ref{fig:MultiplicityDistribution} shows the
distribution of $m$ for different source types (measured in detector
ring {\it R3}). It can be seen that sources with high energy
contributions (such as the cosmic ray background or Eu$^{152}$ with its
$E> 100 \, \rm{keV}$ energy lines) have $m=1$ pixel event contributions
of $\simeq 70 \%$. Sources with energies concentrated in the $20-80 \,
\rm{keV}$ interval, relevant for X-Calibur, have $m=1$ contributions of
$> 90 \%$ (e.g. the $40 \, \rm{keV}$ CHESS beam, see
Sec.~\ref{subsec:CHESS}). This is explained by the fact that low energy
X-rays deposit smaller and more concentrated charge clouds in the CZT
with a reduced chance of charge sharing between pixels (that would cause
$m \geq 2$ events). Therefore, an event selection cut on $m=1$ is a
reasonable way for background subtraction without loosing signal events
at low energies. Cuts on the energy $E$ and the shield flag
$f_{\rm{shld}}$ will further reduce the background (see
Sec.~\ref{subsec:BG_Data}). A selection cut on the scintillator flag
$f_{\rm{sci}}$ selects a very clean sample of events that
Compton-scattered in the scintillator. This has the potential to further
reduce the background~-- however, with a loss in efficiency at low
energies (see Sec.~\ref{subsec:ScintillatorEfficiency}). However, a cut
on $f_{\rm{sci}}$ is optional and not a requirement for sensitive
polarization measurements with X-Calibur.
\subsection{Energy Spectra}
Energy spectra are used to study the scattering properties of the
polarimeter and mark an intermediate step to derive the energy-dependent
polarization properties. Energy spectra can be derived for individual
pixels or for groups of pixels (e.g. a CZT detector or a detector ring
{\it R}). The energy spectra shown in this paper are normalized to the
acquisition time (dead time corrected), the anode detector surface
covered by the corresponding pixel group and the width of the energy
bins. Note, the limited dynamical range of the charge digitization of
individual pixel channels will lead to discrete energies. Given the
different energy calibrations of the channels, these will differ from
pixel to pixel. This difference can lead to binning artifacts in energy
spectra that are obtained from a small group of pixels.
\subsection{Polarization Properties} \label{subsec:AnalysisPolarization}
The signature of a polarized beam will be imprinted in the azimuthal
scattering distribution of recorded events (see for example
Figs.~\ref{fig:AzimuthDistribution} or \ref{fig:CHESS_2DAzimuthData} in
Sec.~\ref{subsec:CHESS}). Either the scattering distribution itself, or
a method involving the Stokes parameters, can be used to extract the
polarization fraction $r$ and polarization direction $\Omega$. First,
the details of the detector geometry have to be carefully considered in
the analysis, as outlined below.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.5\textwidth]{X-Calibur_CZT_BeamOffset}
\hfill
\includegraphics[width=0.48\textwidth]{StokesInterpol}
\end{center}
\caption{{\bf Left:} Angular coverage of two selected CZT pixels within
a {\it single-pixel ring}\xspace for different beam positions $P$. For a non-rotating detector
(left) the angular coverage for each pixel only needs to be calculated
once. In the rotating system (right) the beam offset in the horizon
system (solid arrow) leads to varying angular coverage per pixel
depending on the current rotation $\alpha$ that have to be updated on an
event-by-event basis. Note, the pixel contacts are physically deposited
on the backside of the detectors but are shown on the front side (facing
the scintillator) for a better illustration of the angular measurement.
The cyan dashed arrows indicate the $x/z$ detector coordinate system.
{\bf Right:} Stokes parameters $Q$, $U$, and $I$ as measured within a
{\it single-pixel ring}\xspace for a polarized beam with $\Omega = 90^{\circ}$. The pixels $j$
have varying angular coverage $\Delta \Phi_{j}$ (indicated by the
horizontal error bars). The gray bands indicate the gaps between
detector boards (see left panel). The filled data points show the
measured values. The open data points show the interpolated Stokes
parameters to recover the contributions missed by the gaps. Note, the
entries in the figure are normalized to the solid angle $\Delta \Phi$
covered by each pixel. Therefore, the normalization leads to apparent
jumps from the pixels to the values shown for the gaps. The open symbol
at $\Phi \simeq 360^{\circ}$ corresponds to a dead pixel.}
\label{fig:SchematicBeamOffset_AndStokes}
\end{figure*}
{\it Azimuthal pixel coverage.} The $360^{\circ}$ range in azimuth is
covered by the $4 \times 8 = 32$ pixels per {\it single-pixel ring}\xspace (see
Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, left). Different pixels
cover different azimuthal ranges $\Delta \Phi$ with respect to the
center of the scattering rod which defines the optical axis. Therefore,
the four sides of detector boards lead to a 4-fold symmetry in the
scattering distribution (Fig.~\ref{fig:AzimuthDistribution}). This
purely geometrical effect can be corrected for by dividing the counts
$C_{j}$ in each pixel $j$ by its azimuthal coverage $\Delta \Phi_{j}$.
{\it Azimuthal pixel coverage for beam offsets.} A special situation
arises if the optical axis of the X-ray beam is not aligned with the
geometrical axis of the polarimeter (see $P_{1}$ in
Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, left). Such an offset, if
not corrected, will introduce asymmetries in the azimuthal scattering
distributions and can mimic a wrong polarization signature (see
Sec.~\ref{subsec:Systematics} for a detailed study). Assuming a known
offset vector $\vec{P}1$, the azimuthal coverage $\Delta \Phi_{p1, j}$
can be recalculated for each pixel $j$. This correction re-aligns the
origin of the detector system with the beam axis and strongly reduces
the systematic effect introduced by the offset. In the case of
sufficient event statistics, first moments can be used to estimate the
beam offset from the data itself (see Sec.~\ref{subsec:Systematics}). If
the polarimeter is rotating, the beam offset in the horizon system will
rotate in the detector system (see
Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, left). In this case
$\Delta \Phi_{p1,j}$ has to be updated on an event-by-event basis,
taking into account the current polarimeter orientation $\alpha$
measured by the code wheel.
{\it Pixel acceptance and flat fielding.} Individual pixels have
different trigger efficiencies, energy thresholds and energy resolutions
affecting the number of counts derived from a given energy interval. To
correct for these differences, the pixels are flat fielded using
Compton-scattered events recorded from a non-polarized X-ray beam that
results in a flat azimuthal scattering distribution. For a {\it single-pixel ring}\xspace the
event counts per azimuthal coverage $C_{j} / \Delta \Phi_{j}$ are
averaged for all $j=1...32$ pixels and are used to determine a relative
azimuthal acceptance $a_{j}$:
\begin{equation}
w_{j} = \frac{1}{a_{j}} = \frac{\Delta \Phi_{j}}{C_{j}} \frac{1}{N}\sum_{i=1}^{N}\frac{C_{i}}{\Delta \Phi_{i}}.
\label{eqn:Acceptance}
\end{equation}
The pixel acceptance $a_{j}$ can in turn be used to weight individual
events with $w_{j}=1/a_{j}$. Implicitly, $a_{j}$ depends on the event
selection cuts~-- so that it has to be computed for the particular set
of cuts applied to the data. Dead pixels cannot be recovered by a
corresponding weight since they contribute zero events. However, once
the polarimeter/shield assembly is rotating with respect to the
polarization plane, the pixel acceptances can be ignored since they
average out throughout the measurement~-- this includes the treatment of
dead pixels.
{\it Polarization properties derived from the azimuthal scattering
distribution.} Integrating $C_{j} / \Delta \Phi_{j}$ for each pixel $j$
in a certain energy range and weighting the individual counts with
$w_{j}$ will result in the azimuthal scattering distribution. Here, the
angular coverage $\Delta \Phi_{j}$ determines the horizontal error bar
of the data point. The scattering distribution can be derived for
individual detector rings. Fitting a sinusoidal function to the
distribution (e.g. left panel in Fig.~\ref{fig:AzimuthDistribution},
Sec.~\ref{subsec:CHESS}) allows one to reconstruct (i) the orientation
of the polarization plane (minimum), as well as (ii) the modulation of
the data $\mu_{\rm{data}}$ following Eq.~(\ref{eq:ModulationFactor}).
The corresponding modulation factor $\mu_{\rm{sim}}$ is derived from
simulations of a $100 \%$ polarized beam (see
Sec.~\ref{sec:Simulations}) that are analyzed using the same set of
event selection cuts. The polarization fraction $r$ of the measured beam
is in turn calculated to be:
\begin{equation}
r = \frac{\mu_{\rm{data}}}{\mu_{\rm{sim}}}.
\label{eqn:PolFracPhiDistri}
\end{equation}
The effect of dead detector pixels and gaps between the detector boards
(see Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, left) is accounted
for automatically, since the corresponding points do not show up in the
distribution and will not affect the fit.
{\it Polarization properties derived from Stokes parameters.} An
alternative approach to reconstruct the polarization properties is based
on the Stokes parameters \cite{Chandrasekhar1960, StokesAnalysis} that
are calculated for each event $k$:
\begin{equation}
q_{k} = \cos(2 \Phi_{k}), \quad u_{k} = \sin(2 \Phi_{k}), \quad i_{k} = 1.
\label{eqn:StokesParams}
\end{equation}
The Stokes parameters can be summed for a subset of the data consisting
of $N$ events (e.g. over a specific energy interval and detector ring):
\begin{equation}
Q = \sum_{k=1}^{N} w_{j(k)} q_{k}, \quad U = \sum_{k=1}^{N} w_{j(k)} u_{k}, \quad I = \sum_{k=1}^{N} w_{j(k)}.
\label{eqn:StokesSum}
\end{equation}
Here, each event $k$ (originating from pixel $j$) is weighted with
$w_{j(k)} = 1/a_{j(k)}$. Since each pixel $j$ covers a range in azimuth
$\Delta \Phi_{j} = \Phi_{j,\rm{max}} - \Phi_{j,\rm{min}}$, the values
$q_{k}$ and $u_{k}$ derived from the mean angle $\Phi_{j(k)}$ in a
non-rotating system may lead to inaccurate and/or biased results.
Therefore, the mean Stokes parameters $\left< q \right>_{j(k)}$ and
$\left< u \right>_{j(k)}$ are calculated based on the covered range in
azimuth of the corresponding pixel $j$:
\begin{eqnarray}
\begin{split}
\left< q \right>_{j(k)} & = \frac{1}{\Delta \Phi_{j(k)}} \int_{\Phi_{j(k),\rm{min}}}^{\Phi_{j(k),\rm{max}}} \cos(2 \Phi) \, \rm{d}\Phi, \\
\left< u \right>_{j(k)} & = \frac{1}{\Delta \Phi_{j(k)}} \int_{\Phi_{j(k),\rm{min}}}^{\Phi_{j(k),\rm{max}}} \sin(2 \Phi) \, \rm{d}\Phi.
\end{split}
\label{eqn:MeanStokes}
\end{eqnarray}
The values of $\left< q \right>_{j(k)}$ and $\left< u \right>_{j(k)}$
are in turn used in Eq.~(\ref{eqn:StokesSum}). The proper treatment of
dead pixels and gaps between the detector boards {\it Bd0}-{\it Bd3}
(see Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, left) is crucial when
working with the Stokes parameters in a non-rotating coordinate system,
since a lack of events from a particular azimuthal direction
$\tilde{\Phi}$ will lead to an increase/decrease in $\sum_{k} \left< q
\right>_{j(k)}$ and/or $\sum_{k} \left< u \right>_{j(k)}$,
systematically affecting the reconstructed polarization properties (see
Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, right). The amplitude and
sign of the resulting effect depends on the orientation between the
polarization plane and the detector plane (assuming the gaps are located
at angles of $45^{\circ}$, $135^{\circ}$, $225^{\circ}$, and
$315^{\circ}$, see Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, left).
If the planes are exactly parallel or exactly perpendicular, the effect
leads to a maximal underestimation of $r$. An angle of $45^{\circ}$
(polarization plane being aligned with two of the four gaps) leads to
the maximal overestimation of $r$. At angles of $45/2 = 27.5^{\circ}$
(and multiples thereof) the effect of detector gaps cancels. A
simulation for the X-Calibur detector layout was performed and resulted
in a systematic effect of $\pm6 \%$ for a $100\%$ polarized beam.
Therefore, a correction has to be applied: (i) the $m=1,2,...$ azimuthal
ranges $\delta \tilde{\Phi}_{m}$ covered by the gaps have to be
identified in which the detector is not sensitive (ii) The distributions
of $\delta Q / \delta \Phi$, $\delta U / \delta \Phi$, and $\delta I /
\delta \Phi$ have to be collected as a function of $\Phi$
(Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, right). (iii) These
distributions are in turn used to interpolate the data gaps $\delta
\tilde{\Phi}_{m}$ from neighboring pixel to recover the `missing'
contributions in Eq.~(\ref{eqn:StokesSum}):
\begin{eqnarray}
\begin{split}
Q & \rightarrow Q + \sum_{m} \Delta \tilde{Q}_{m} \delta
\tilde{\Phi}_{m}, \\ U & \rightarrow U + \sum_{m} \Delta \tilde{U}_{m}
\delta \tilde{\Phi}_{m}, \\ I & \rightarrow I + \sum_{m} \Delta
\tilde{I}_{m} \delta \tilde{\Phi}_{m}.
\end{split}
\label{eqn:StokesGapCorrection}
\end{eqnarray}
The errors on the added sums are calculated using error propagation
during the interpolation. Given the scalar nature of the Stokes
analysis, there is no simple way of representing the azimuthal
modulation of the scattering distribution. Therefore, it is useful to
compare the Stokes results with the results obtained from the azimuthal
scattering distribution (previous paragraph) in order to identify
possible systematic effects. Note again, that a rotating polarimeter
will not require a correction for dead pixels or detector gaps.
The Stokes sums in (\ref{eqn:StokesSum}) or
(\ref{eqn:StokesGapCorrection}) can be used to reconstruct the
polarization fraction $r$ and polarization angle~$\Omega$:
\begin{eqnarray}
\begin{split}
r & = \frac{2}{\mu_{\rm{sim}}} \frac{\sqrt{Q^{2} + U^{2}}}{I}, \\
\Omega & = \frac{1}{2} \rm{atan} (U / Q).
\end{split}
\label{eqn:PolarizationFromStokes}
\end{eqnarray}
Results from $l$ independent measurements (e.g. from different detector
rings) can be combined in a weighted average. Since the polarization
fraction is always positive it can lead to an overestimation if the true
polarization fraction is in the MDP regime of the data set, see
Eq.~(\ref{eqn:MDP}), where the error bars are highly asymmetric. This
systematic effect can be avoided by averaging a set of modified Stokes
parameters:
\begin{eqnarray}
\begin{split}
Q' & = \sum_{l} \frac{2 w_{l}}{\mu_{\rm{sim,l}}} \frac{Q_{l}}{I_{l}}, \quad U' = \sum_{l} \frac{2 w_{l}}{\mu_{\rm{sim,l}}} \frac{U_{l}}{I_{l}}, \\
r' & = \frac{1}{L'} \sqrt{Q'^{2} + U'^{2}} \quad {\rm with} \quad L' = \sum_{l} w_{l}.
\end{split}
\label{eqn:Stokes_Q_U_Average}
\end{eqnarray}
The weights $w_{l} = 1/\sigma_{l}^{2}$ account for the statistical
uncertainties of the individual measurements $l$.
{\it Unfolding analysis.} An unfolding analysis that takes into account
the energy-dependent detector response and photon detection efficiencies
and that can be used to reconstruct polarization fraction and angle as a
function of true photon energy will be described in a separate paper
\cite{XCLB_LogLikelihoodAnalysis}.
{\it Forward folding.} Recording the azimuthal scattering distributions
with planar detectors will lead to projection effects that depend on the
polar angle of the scattering in the scintillator. For most parts of the
polarimeter these effects are negligible, since (i) a particular
detector ring sees the superposition of different polar scattering
angles canceling the effect, and (ii) the same effect is present in the
simulations that are used to determine the polarization fraction
following Eq.~(\ref{eqn:PolFracPhiDistri}). Only for polarization
fractions $r \ll 1$ measured in detector ring {\it R1}, which sees only
back-scatter events, a second order correction may be
needed\footnote{Here, the $r \propto \mu_{\rm{data}}$ relation in
Eq.~(\ref{eqn:PolFracPhiDistri}) will no longer be exactly linear.}. To
fully take into account these effects, the data can be analyzed with a
forward folding method (which is beyond the scope of this paper). The
modeling of the pixel acceptance in {\it R1}, only affecting
measurements with the non-rotating polarimeter, will also be slightly
affected by the effect.
\section{Simulations} \label{sec:Simulations}
Simulations of the energy-dependent response of the X-Calibur
polarimeter are needed in order to reconstruct the polarization
properties from measured data (see
Sec.~\ref{subsec:AnalysisPolarization}). The detector geometry is
modeled and the X-ray flux/spectrum was simulated for the different
experimental setups in which the data presented in this paper were
taken. The simulations were performed in the following steps.
\begin{enumerate}
\item Physics interactions, scatterings, and energy depositions in the
scintillator and the CZT detectors were simulated using {\it
GEANT4}\footnote{{\tt http://geant4.cern.ch/}} with the Livermore
low-energy electromagnetic model list.
\item The charge collection efficiency as a function of
depth-of-interaction in the CZT detectors was determined using an
in-house developed software to (i) calculate the 2D electric potential
inside the detector crystals followed by (ii) the integration of the
weighting potential \cite{Jung2007} along the charge transport tracks,
resulting in the collected charge for each energy deposition (using a
dielectric constant for CZT of $10$). The simulations are valid for
detectors with strip anode contacts but will roughly resemble the
response for pixelated detectors, as well. A mobility of electrons/holes
of $\mu_{\rm{e}} = 1000 \, \rm{cm}^{2} / \rm{V} / \rm{s}$, and
$\mu_{\rm{h}} = -120 \, \rm{cm}^{2} / \rm{V} / \rm{s}$ is assumed, as
well as life times of $\tau_{\rm{e}} = 10^{-6} \, \rm{s}$ and
$\tau_{\rm{h}} = 4 \cdot 10^{-6} \, \rm{s}$, respectively. This
corresponds to a mobility-lifetime product of $\mu_{\rm{e}}\tau_{\rm{e}}
= 10^{-3}$ which is in reasonable agreement with the values measured for
a selection of the detectors used in X-Calibur (see
Fig.~\ref{fig:TempStudies}, bottom).
\item The energy resolution (asymmetric Gaussian function, implicitly
including the electronic readout noise) and energy threshold were
measured from real data (Sec.~\ref{subsec:ThreshAndResolution}) and were
folded into the simulations on a pixel-by-pixel basis. However,
differences in channel trigger efficiencies were not simulated. Using
the reversed energy calibration in Eq.~(\ref{eq:EnergyCalibration}), the
simulated events were in turn converted into the X-Calibur data format
(ASIC/channel ID and digitized raw pulse height) and can be analyzed in
the same way as the measured data.
\item Since we find that the simulations do not properly account for low
energy tails in the detector response (see next paragraph), an empirical
model was used to scatter an exponential tail $T(E) = A \cdot
\exp(E/E_{\rm{t}})$ into the simulations with a relative fraction of
$f=0.45$ per energy deposit. The parameter $E_{\rm{t}} = 15 \, \rm{keV}$
(at $E=40 \, \rm{keV}$) and $E_{\rm{t}} = 30 \, \rm{keV}$ (at $E = 120
\, \rm{keV}$) was interpolated in $\log E$ for the energy range covered.
\item The scintillator trigger flag $f_{\rm{sci}}$ was generated for
each event based on the scintillator trigger efficiency determined from
real data (see Fig.~\ref{fig:Scintillator_TriggerEfficiency} in
Sec.~\ref{subsec:ScintillatorEfficiency}).
\end{enumerate}
Different scenarios/setups were simulated, reflecting the measurements
and studies presented in this paper.
{\it CZT detector response.} To test the validity of the simulation
chain, the direct illumination of a single CZT detector with a
Eu$^{152}$ point source was simulated\footnote{All lines with
intensities above $2 \%$ were generated according to their relative
emission intensities.}~-- corresponding to the experimental setup used
for the detector calibration measurements presented in
Sec.~\ref{subsec:CZT_Calibration}. The comparison between the
simulations and data (Fig.~\ref{fig:EnergyCalibration}, right) shows
reasonable agreement in terms of line positions, widths, and threshold
effects. However, if ignoring step (4) of the simulation chain (`Sim' in
the legend of Fig.~\ref{fig:EnergyCalibration}), a lack in continuum
emission can be seen in the simulations. This motivated the introduction
of step (4) which leads to a reasonable agreement between data and
simulations over the whole energy band relevant for X-Calibur (`Sim+' in
the figure legend). Possible reasons for the continuum in the data may
be related to details in the detector response or back-reflection of
emitted X-rays from the source off the surrounding fixture that is not
simulated\footnote{For similar CZT detectors we find a photo-peak
detection efficiency of order unity \cite{VarPitch}.}. It should be
noted that the goal of the CZT simulations is not to find an accurate
model for the detector response~-- but rather a suited parameterization
that reproduces the integral spectral response for the different energy
bins.
{\it CHESS beam.} X-Calibur performance measurements were performed at
the highly polarized synchrotron X-ray beam at the CHESS facility~--
providing a strong, mono-energetic X-ray beam (Sec.~\ref{subsec:CHESS}).
A corresponding set of simulations was performed using steps (1)-(5),
resembling the CHESS setup of pencil-beam X-rays (polarized and
non-polarized) at $40 \, \rm{keV}$, $80 \, \rm{keV}$, and $120 \,
\rm{keV}$ entering the polarimeter along the optical axis of the
scintillator. Detector pixels that were excluded during the CHESS data
runs were also excluded in the simulations to resemble a configuration
close to the one used for the measurements. The trigger efficiency of
the scintillator as a function of energy deposition was derived from the
CHESS data (see Fig.~\ref{fig:Scintillator_TriggerEfficiency} in
Sec.~\ref{subsec:ScintillatorEfficiency}) and was fed into the
simulations in step (5) to generate the trigger flag $f_{\rm{sci}}$ on
an event-by-event basis. No backgrounds were simulated in the case of
the CHESS measurements since the measurements were completely
signal-dominated.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{CrabSimulations}
\end{center}
\caption{Simulated X-Calibur observations of the Crab nebula ($5.6 \,
\rm{h}$), taken from \citet{Guo2010}. The data points show the
reconstructed flux (top), the fraction of polarization (middle), and
polarization angle (bottom). The lines show the assumed flux,
polarization fraction and polarization direction.}
\label{fig:XCLB_Sims}
\end{figure}
{\it Balloon flight.} A balloon flight in the focal plane of the
{\it InFOC$\mu$S}\xspace mirror assembly was assumed in an earlier simulation
\cite{Guo2010} that only involved step (1) in the above chain. The
effective detection areas of the X-ray mirror are $95/60/40 \,
\rm{cm}^{2}$ at $20/30/40 \, \rm{keV}$, respectively. We accounted for
atmospheric absorption at a floating altitude of $130,000$ feet using
the NIST XCOM attenuation coefficients\footnote{{\tt
http://www.nist.gov/pml/data/xcom/index.cfm}} and an atmospheric depth
of $2.9 \, \rm{g/cm}^{2}$ (observations performed at zenith); the
atmospheric transmissivity rapidly increases from $0$ to $0.6$ in the
$20 - 80 \, \rm{keV}$ range. The trigger efficiency of the scintillator
scatterer was assumed to be $f_{\rm{sci}} = 1$ above an energy
deposition of $2 \, \rm{keV}$ and $f_{\rm{sci}} = 0$ below.
We simulated the most important backgrounds such as the cosmic X-ray
background \cite{Ajello2008}, albedo photons and cosmic ray protons and
electrons \cite{Mizuno2004}. The neutron background was not modeled
since a detailed study of \citet{Parsons2004} showed that the
contribution in CZT can be neglected. Different shield configurations
and shield thicknesses were simulated. The configuration shown in
Fig.~\ref{fig:ActiveShieldAndTelescope} (left) represents an optimized
compromise balancing the background rejection power and the
mass/complexity of the shield. A Crab-like source was simulated for a
$5.6 \, \rm{hr}$ balloon flight. We assumed a power law energy spectrum,
and a continuous change of the polarization fraction and angle between
the values measured at $5.2 \, \rm{keV}$ with OSO-8 \cite{Weisskopf1978}
and at $E>100 \, \rm{keV}$ with {\it INTEGRAL}\xspace \cite{Dean2008} by modeling a
transition following a Fermi distribution.
For a Crab-like source the simulations predict an event rate of $1.1
(3.2) \, \rm{Hz}$ with (without) requiring a triggered scintillator
coincidence ($f_{\rm{sci}} = 1$). Figure~\ref{fig:XCLB_Sims} compares
the simulation results with the assumed model curves; the errors were
computed in a similar way as described by \citet{Weisskopf2010}.
Simulations performed at different zenith angles $\theta$ show that the
source rate scales with $(\cos \theta)^{1.3}$ which is taken into
account for simulating astrophysical observations. More details about
these simulations are discussed in \citet{Guo2010} and \citet{Guo2013}.
\section{CZT Detector Characterization} \label{sec:DetectorCharacterization}
The performance of the individual CZT detectors used in X-Calibur is
coupled to the performance of the polarimeter as a whole~-- including
its energy threshold and its energy resolution. This section describes
the energy calibration of the individual CZT detectors
(Sec.~\ref{subsec:CZT_Calibration}), as well as measurements of the
energy threshold and energy resolution
(Sec.~\ref{subsec:ThreshAndResolution}). The CZT performance serves as
important input for the simulations described in
Sec.~\ref{sec:Simulations}. In contrast to the laboratory, the
polarimeter will be operated in an environment of varying temperature
conditions during the balloon flight which motivates the study of the
temperature dependence of the CZT detector performance which will be
described in Sec.~\ref{subsec:TempStudies}.
In order to quantify the characteristics of individual pixels, the
emission lines in the calibrated energy spectra are fitted with a
Gaussian function. The peak position is described by the mean
$E_{\rm{p}}$. To account for the asymmetric shape of the peaks (see for
example Fig.~\ref{fig:EnergyCalibration}), the fitted function allows
for asymmetric spectral continua levels ($c_{1}$, $c_{2}$) and
asymmetric peak widths ($\sigma_{1}$, $\sigma_{2}$) for the $E <
E_{\rm{p}}$ (1) and $E > E_{\rm{p}}$ (2) regimes, respectively. The fit
parameters are used to characterize the measured peak. The energy
resolution is calculated as the full width half maximum, $\Delta E =
\rm{FWHM} = 2 \sqrt{2 \ln 2} \cdot \frac{1}{2} (\sigma_{1} +
\sigma_{2})$. The peak rate is determined by counting the events in the
interval $\pm 2 \, \rm{FWHM}$ centered around $E_{\rm{p}}$, normalized
by the observation time.
A Eu$^{152}$~source is used as calibration/test source in a variety of
studies presented in this paper. Eu$^{152}$~emits X-ray lines at
$39.5\,\rm{keV}$ ($\rm{K}_{\alpha2}$), $40.12\,\rm{keV}$
($\rm{K}_{\alpha1}$), $45.7 \, \rm{keV}$, $122.78 \, \rm{keV}$, $244.7
\, \rm{keV}$, $344.28 \, \rm{keV}$, and at higher energies. The line at
$40.12\,\rm{keV}$ is the strongest in the low energy triplet; relative
to the $40.12\,\rm{keV}$ line, the $39.5 \, \rm{keV}$ line is emitted at
$55 \%$ intensity and the $45.5 \, \rm{keV}$ line at $31 \%$ intensity.
The $40.12\,\rm{keV}$, used as low-energy performance marker, is
therefore fitted jointly with the two close-by lines that are set at
fixed distance $\Delta E$ and fixed intensity relative to the
$40.12\,\rm{keV}$ peak (with the free fit parameter $\sigma$ being the
same for all three lines since the energy resolution of a detector pixel
is not expected to change within a few keV). Including the two
neighboring lines avoids systematic shifts and artificial broadening of
the fitted peak (see Fig.~\ref{fig:EnergyCalibration}, left).
\subsection{Energy Calibration} \label{subsec:CZT_Calibration}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{ZoomChannelEu152_Asic19_Channel_10}
\hfill
\includegraphics[width=0.50\textwidth]{EnergyCalibrationSpectraEu152_Mosaic}
\end{center}
\caption{Eu$^{152}$ energy spectra (direct CZT illumination). The
vertical lines indicate the nominal X-ray line energies; relative
heights indicate the emission intensity folded with the absorption
probability in $5 \, \rm{mm}$ CZT and $2 \, \rm{mm}$ CZT (small tick),
respectively. Data were taken with X-Calibur configuration
$C_{\rm{wu1}}$ (see Fig.~\ref{fig:X-Calibur_Configurations}). {\bf
Left:} Spectrum of an individual detector pixel (linear energy axis,
$C_{\rm{wu1a}}^{1}$). The dashed vertical line indicates the trigger
energy threshold. The fitted functions ($40.1 \, \rm{keV}$ and $121.8 \,
\rm{keV}$) are used to determine the peak position and energy resolution
(FWHM). {\bf Right:} Energy spectra summed over all pixels of the
detectors in ring {\it R3} ($5 \, \rm{mm}$ thickness) and ring {\it R7}
($2 \, \rm{mm}$ thickness), respectively. Also shown are the simulated
spectra with/without (Sim+/Sim) additional continuum (see
Sec.~\ref{sec:Simulations}). A CR background spectrum (scaled by a
factor of $10$) is shown for reference. The horizontal dotted line
indicates the energy range relevant for X-Calibur.}
\label{fig:EnergyCalibration}
\end{figure*}
The energy calibration of the individual CZT detector pixels is done
with a compact Eu$^{152}$ source (cylindrical emitting volume with a
diameter of $\simeq$$3 \, \rm{mm}$) in the X-Calibur configuration
$C_{\rm{wu1b}}^{2}$ in which the the scintillator rod is not installed
(see Fig.~\ref{fig:X-Calibur_Configurations}). The source was
successively placed at the centers of the detector rings~-- allowing to
calibrate rings {\it R1} to {\it R8}, one at a time. In a first step, an
automatic routine is used to optimize the pixel trigger thresholds: data
are taken in a special acquisition mode that adjusts ASIC/channel
discriminators based on measured event rates for each pixel, such that
the trigger threshold is as low as possible (maximizing the integral
trigger rate), but at the same time is safely above the electronic noise
regime. The noise regime leads to very high (artificial) trigger rates
and differs from channel to channel. The routine works reliably for most
channels. However, a visual inspection of all 2048 recorded energy
spectra was performed to assure that channels with trigger thresholds
set too high or too low (failed automatic detection of the noise regime)
were adjusted manually.
About 5 million events were taken for each CZT detector (20 million
events per ring {\it R}), and the known energy lines at $40.1 \,
\rm{keV}$ and $122 \, \rm{keV}$ were used to determine the pedestal
$p_{0}$ and amplification slope $a$ for each channel\footnote{The
linearity of the channels was confirmed using the internal test pulse
generator of the ASIC.}. The energy of a measured pulse height $p$ is in
turn calculated using
\begin{equation}
\label{eq:EnergyCalibration}
E(p) = (p-p_{0})/a.
\end{equation}
The left panel of Figure~\ref{fig:EnergyCalibration} shows the
calibrated energy spectrum of a single pixel. The right panel shows the
averaged calibration spectra of two chosen detector rings ($4 \times 64$
pixels each). Also shown are the energy spectra obtained from the
simulations of the corresponding setup (see Sec.~\ref{sec:Simulations}).
It can be seen, that the $2 \, \rm{mm}$ detectors (ring {\it R7}) loose
performance at energies $E> 100 \, \rm{keV}$, as compared to the
detectors with $5 \, \rm{mm}$ thickness (ring {\it R3}). However, in the
$20-80 \, \rm{keV}$ energy band, relevant for X-Calibur, both types of
detector thickness perform at a similar level.
In addition to the temperature-dependence of the detector performance
discussed in Sec.~\ref{subsec:TempStudies}, another consideration has to
be made when applying the calibration to the data. The CZT detectors
used in X-Calibur are not setup for measuring the depth position of the
X-ray interaction/absorption between the detector cathode and anode~--
referred to as the depth-of-interaction (DOI). The energy calibration in
Eq.~(\ref{eq:EnergyCalibration}), however, depends on the average DOI of
a given energy. The mean DOI, however, changes with the cosine of the
inclination angle measured between the absorbed X-ray and the detector
plane. The calibration was determined with the X-ray source located $c
\simeq 11 \, \rm{mm}$ above the center of the detector cathode (see
Fig.~\ref{fig:X-Calibur_Configurations}, top). Depending on their
geometrical locations, the pixels are hit under angles between
$0-45^{\circ}$. X-rays Compton-scattering in the X-Calibur scintillator,
on the other hand, can hit the CZT detectors at angles between
$0-90^{\circ}$, which adds a systematic error/uncertainty to the
reconstructed energy for small incident angles. However, in the $20-80
\, \rm{keV}$ band the DOI distribution is very narrow and localized
close to the cathode. Therefore, the angle dependence is negligible~--
the systematic shift of the reconstructed $40.1 \, \rm{keV}$ line was
experimentally constrained to be less than $1\%$ for shallow inclination
angles.
A $< 3\%$ fraction of ASIC channels were found to be dead, too noisy, or
did not make contact to the detector pixel. These channels were excluded
from the analysis and are marked with `x' in the corresponding 2D plots
shown in this paper (e.g. Fig.~\ref{fig:PixelThreshold}).
\subsection{Energy Resolution and Threshold}
\label{subsec:ThreshAndResolution}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{Results_CFG1_Threshold}
\end{center}
\caption{2D distribution of energy thresholds of all X-Calibur pixels.
In this representation, the four detector sides surrounding the
scintillator ({\it Bd0} to {\it Bd3}, see Fig.~\ref{fig:Design}, left)
are unfolded into a plane. The segments enclosed by dashed and dotted
lines indicate the CZT detectors ($8 \times 8$ pixels each). The
elongated box (solid line in {\it Bd0} to {\it Bd3}, each) indicates the
position of the scintillator, even though it was not installed for this
particular data set (configuration $C_{\rm{wu1b}}^{2}$, see
Fig.~\ref{fig:X-Calibur_Configurations}, top). Pixels marked with `x'
were not included in the analysis. The left panels show the threshold
distributions per detector ring {\it R} (the gray distribution shows the
scaled average of all pixels).}
\label{fig:PixelThreshold}
\end{figure}
To study the performance of the X-Calibur CZT detectors, the calibration
data (Sec.~\ref{subsec:CZT_Calibration}) were used to characterize the
energy spectra of individual pixels (see left panel of
Fig.~\ref{fig:EnergyCalibration} for reference). The relevant properties
studied in this section are the energy threshold, the fitted line/peak
position, and the energy resolution (FWHM). As for the energy
calibration, the data were taken with the configuration
$C_{\rm{wu1b}}^{2}$ (see Fig.~\ref{fig:X-Calibur_Configurations}).
The energy threshold of a pixel is defined as the reconstructed energy
$E_{\rm{thr}}$ above which the corresponding ASIC channel starts to
trigger on events (see Fig.~\ref{fig:EnergyCalibration}, left). The
distribution of thresholds for all X-Calibur pixels is shown in
Fig.~\ref{fig:PixelThreshold}. The thresholds vary from pixel to pixel,
but no significant geometrical trends can be identified if comparing
edge pixels versus central pixels, or pixels of detectors with different
thickness ($5 \, \rm{mm}$, rings {\it R1-R5} versus $2 \, \rm{mm}$,
rings {\it R6-R8}). About $50\%$ of all pixels have an energy threshold
of $E<20 \, \rm{keV}$. It should be noted that one has to account for
the energy resolution of a pixel in order to determine the analysis
threshold that is about $3-4 \, \rm{keV}$ higher. The thresholds shown
in Fig.~\ref{fig:PixelThreshold} reflect the status of the compact
X-Calibur configuration. The thresholds of the detectors operated as a
single unit are up to $5 \, \rm{keV}$ lower (see
Fig.~\ref{fig:TempStudies}).
\begin{figure*}[t!]
\begin{center}
\includegraphics[height=0.42\textheight]{Results_CFG1_40keV_Resolution}
\hfill
\includegraphics[height=0.42\textheight]{Results_CFG1_Noise_Resolution_Zoom}
\hfill
\includegraphics[height=0.42\textheight]{Results_CFG1_122keV_Resolution}
\end{center}
\caption{2D distribution of the pixel-by-pixel energy resolution
(compare with Fig.~\ref{fig:PixelThreshold}). {\bf Left:} Energy
resolution at $40.1 \, \rm{keV}$. {\bf Middle:} Electronic readout noise
at $40 \, \rm{keV}$ as determined using the ASICs internal test pulse
generator (only shown for {\it Bd2}). The same range of the color scale
as in the left panel is used. {\bf Right:} Energy resolution at $121.8
\, \rm{keV}$. This energy line is not well-defined in the energy spectra
measured with the $2 \, \rm{mm}$ detectors (see
Fig.~\ref{fig:EnergyCalibration}) so that results are only shown for the
$5 \, \rm{mm}$ rings {\it R1} to {\it R5}.}
\label{fig:PixelResolution}
\end{figure*}
Figure~\ref{fig:PixelResolution} shows the energy resolutions at $40.1
\, \rm{keV}$ and $121.8 \, \rm{keV}$, respectively. Detector rings {\it
R2} and {\it R3} are the most sensitive ones when it comes to detecting
the Compton-scattered X-rays in the polarization measurements (see
Sec.~\ref{subsec:CHESS}). Therefore, the best performing $5 \, \rm{mm}$
detectors were positioned in these rings accordingly. Some of the lower
detector rings ({\it R4} to {\it R8}) show regions with clearly
poorer-than-average energy resolution. However, any asymmetry in
azimuthal detector performance will cancel out due to the rotation of
the polarimeter in the final mode of operation. The energy line at
$121.8 \, \rm{keV}$ is not well-defined in the spectra measured with the
$2 \, \rm{mm}$ detectors (see Fig.~\ref{fig:EnergyCalibration}, right).
Therefore, Fig.~\ref{fig:PixelResolution} only shows the $121.8 \,
\rm{keV}$ energy resolutions for the $5 \, \rm{mm}$ detectors. The
average energy resolution in rings {\it R1} to {\it R3} at $40.1 \,
\rm{keV}$ amounts to $4.0 \, \rm{keV}$ ($10.0\%$); the corresponding
value at $121.8 \, \rm{keV}$ is $4.9 \, \rm{keV}$ ($\simeq 4\%$). The
average performance of rings {\it R4} to {\it R7} at $40.1 \, \rm{keV}$
amounts to $4.6 \, \rm{keV}$ ($\simeq 11\%$). The performance of the
detectors in ring {\it R8} is modest.
The energy resolution of a detector pixel is mainly determined by two
factors~-- the quality of the CZT crystal and the noise of the read-out
electronics. The electronic readout noise was determined for all
channels using the ASICs internal pulse generator (see
Sec.~\ref{sec:XCLB}). The generator injects charge into the amplifier of
the corresponding channel and allows one to test the
trigger/digitization chain on a chennel-by-channel basis. 1000 events
were taken per channel with the detectors connected and biased at
nominal operation voltage (to also account for noise introduced by dark
currents in the CZT). The results are shown in the middle panel of
Fig.~\ref{fig:PixelResolution} for one of the four boards. More details
on the electronic noise studies are presented in
Sec.~\ref{subsec:TempStudies}. Note, that the internal ASIC capacitor
does not allow to inject charges that correspond to energies lower than
$\approx 200 \, \rm{keV}$. However, the noise versus energy trend seems
to level off for energies lower than $\simeq 500 \, \rm{keV}$ (see
Fig.~\ref{fig:TempStudiesNoise}). Therefore, the absolute noise
resolution measured at $\simeq 200 \, \rm{keV}$ was used to estimate the
relative noise contribution at $40 \, \rm{keV}$, as shown in the middle
panel of Fig.~\ref{fig:PixelResolution}.
A comparison between the electronic noise and the energy resolution
determined from the spectral lines shows that the low-energy resolution
is dominated by the electronic noise. This kind of comparison can in
general assist in localizing the cause for modestly performing
detectors, e.g. by disentangling the contributions of electronic noise
versus CZT crystal quality. The detector located in ring {\it R5} of
{\it Bd2}, for example, shows noisy regions in both, the Eu$^{152}$
data, as well as in the electronic noise measurement~-- indicating a
high leakage current as the reason for the sub-optimal performance. The
ring {\it R8} detector in the same board, on the other hand, does not
exhibit a poorer energy resolution than others in terms of noise~--
therefore, the poor performance visible in the Eu$^{152}$ data is
probably related to a low CZT crystal quality.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.48\textwidth]{PolarimeterTemperature}
\end{center}
\caption{Temperature-dependent performance of individual CZT detectors.
Each data point represents the average of $64$ pixels of a detector.
Vertical arrows indicate the shift of the curve if considering only edge
pixels of a detector (dashed direction), or only central pixels of a
detector (solid direction). {\bf Top:} Energy resolution at $122 \,
\rm{keV}$ (only shown for the $5 \, \rm{mm}$ detectors). {\bf
2$^{\rm{\bf{nd}}}$ from top:} Energy resolution at $59 \, \rm{keV}$.
{\bf 3$^{\rm{\bf{rd}}}$ from top:} Peak detection rate of the $59 \,
\rm{keV}$ line normalized to the rate measured at $T=15^{\circ} \,
\rm{C}$. {\bf 4$^{\rm{\bf{th}}}$ from top:} Energy threshold as
determined after threshold optimization performed at the corresponding
temperature. {\bf Bottom:} Mobility-lifetime product
$\mu_{\rm{e}}\tau_{\rm{e}}$ measured based on the $59 \, \rm{keV}$
line.}
\label{fig:TempStudies}
\end{figure}
On average, the side pixels of individual detectors exhibit a poorer
than average energy resolution. At low energies, this shows up as a
periodic structure on the left and right side of each detector. The same
pattern can be seen in the electronic noise measurement
(Fig.~\ref{fig:PixelResolution}, left versus middle), and can therefore
be attributed to the readout noise. This hypothesis is supported by the
fact that the leads between the corresponding edge pixels and the ASIC
channels are located on the `outside' region of the printed circuit
board~-- being more susceptible to noise pick up from the surrounding
electronics. Subtracting the readout noise, these edge pixels do no
longer show poorer energy resolution at low energies as compared to the
other pixels. For energies $E > 100 \, \rm{keV}$ (not relevant for
X-Calibur), the detector edge pixels show a reduced resolution in
addition to the electronic noise component (`frame-like' structure
surrounding each detector in Fig.~\ref{fig:PixelResolution}, right).
This is a known issue with CZT detectors and can be explained by a less
homogeneous electric field in the edge regions of a detector, affecting
the charge collection. The effect is less prominent (if visible at all)
for the horizontal edge pixels, for which the field is stabilized by the
neighboring detectors located in the same plane.
\subsection{CZT Performance at different Temperatures}
\label{subsec:TempStudies}
During a balloon flight the pressure vessel housing the polarimeter will
undergo several changes in temperature that can potentially affect its
performance. At float altitude, the vessel will be in an outside
temperature environment of around $-20^{\circ} \rm{C}$. During daytime,
the thermal radiation fields of the sun and the earth will provide
additional sources of energy. During the night, the thermal radiation of
the earth is the only external source of heat flow. The electronics
inside the vessel generate heat at a rate of less than $100 \, \rm{W}$.
The outside of the vessel will be insulated using a layer of aluminized
mylar (reflection of sun light and high emittance of thermal radiation)
that will be contact-separated from the surface of the vessel with a
layer of Dacron mesh. Furthermore, heater bands with a total power of
$175 \, \rm{W}$ are installed inside the vessel to guarantee a
controllable temperature in the range of $0$ to $25^{\circ} \rm{C}$.
This thermal design will prevent overheating during day-time and will
avoid cold temperatures during the night. Nonetheless, variations in
temperature of the polarimeter during flight are expected to some
extent. Therefore, it is important to understand the
temperature-dependent performance of the CZT detectors.
Individual CZT detectors were used to quantify the temperature
dependence of the energy resolution, the energy threshold and the
detection rate. Data were taken in a temperature chamber in the range of
$T \in [-25; +35]^{\circ} \, \rm{C}$ with the detectors being
illuminated with an Am$^{241}$ source ($59 \, \rm{keV}$) and a Co$^{57}$
source ($122 \, \rm{keV}$). The sources were located at a distance of
$\simeq 2 \, \rm{cm}$ above the detector cathode. The data were used to
re-calibrate each detector (pixel-by-pixel) for each environment
temperature in order to cancel temperature-dependent calibration
effects. Since these measurements were time-intensive and could only be
done for one detector at a time, the study was limited to a
representative subset of the detectors shown in
Tab.~\ref{tab:Detectors}.
{\it Temperature-dependent detector characteristics.} The results of the
measurements are shown in Fig.~\ref{fig:TempStudies}, averaged over all
64 pixels per detector. The $122 \, \rm{keV}$ line is not well defined
in the $2 \, \rm{mm}$ detectors, so that the high-energy results are
only shown for the $5 \, \rm{mm}$ detectors. Given the limited sample of
detectors, we cannot expect to attribute observed trends to a specific
detector class (such as the brand); however, some general findings can
be identified and are described in the following.
The energy resolution generally improves if reducing the temperature
from $T = 35^{\circ} \, \rm{C}$ to $T = 0^{\circ} \, \rm{C}$. This
effect is most prominent for the two tested Quikpak detectors and
amounts to an improvement of up to $50\%$/$30\%$ at $59/122 \,
\rm{keV}$, respectively. The effect is much weaker for the Creative
Electron detectors. For temperatures $T < 0^{\circ} \, \rm{C}$, the
resolution levels out. For some of the Endicott detectors ($2 \,
\rm{mm}$) it even gets poorer again; however, the two detectors showing
this effect (EN$_{2}$4 and EN$_{2}$5) show very poor performance in
general. All three $5 \, \rm{mm}$ detectors show a much better energy
resolution at $122 \, \rm{keV}$ in the central pixels (indicated by the
downward arrow in the top panel of Fig.~\ref{fig:TempStudies}) as
compared to the edge pixels (upward arrow). Here, the aspect ratio
allows for E-field lines to bulge out of the sides of the detector (see
also Fig.~\ref{fig:PixelResolution}). This difference is clearly less
pronounces at $59 \, \rm{keV}$ where the charge collection is much more
concentrated in the cathode region of the detector crystal.
The peak detection rate at $59 \, \rm{keV}$ (3$^{\rm{rd}}$ panel from
the top in Fig.~\ref{fig:TempStudies}) shows a clear trend for all
detectors: a reduced detection efficiency with decreasing temperature.
This effect is not understood. Although the measurements were carefully
set up, it cannot be excluded that a temperature-dependent contraction
of the casing/fixture could have lead to a slight change in distance
between the source and the detector as a function of environment
temperature.
For each temperature, the energy thresholds were re-optimized. The
results are shown in the 4$^{\rm{th}}$ panel of
Fig.~\ref{fig:TempStudies}. No clear trends can be identified~-- the
average energy threshold of the studied detectors lies between $15-20 \,
\rm{keV}$.
Measurements at different bias voltages $V_{\rm{bi}}$ were used to
determine the mobility lifetime product $\mu_{\rm{e}}\tau_{\rm{e}}$.
Data were taken at $V_{\rm{bi}} = -200 \, \rm{V}$, $-500 \, \rm{V}$, and
$-700 \, \rm{V}$ ($5 \, \rm{mm}$ detectors) and at $V_{\rm{bi}} = -100
\, \rm{V}$, $-150 \, \rm{V}$, and $-200 \, \rm{V}$ ($2 \, \rm{mm}$
detectors), respectively. The shift of the line position with increasing
$V_{\rm{bi}}$ was used to calculate $\mu_{\rm{e}}\tau_{\rm{e}}$
\cite{MuTauImproved}. The results are shown in the bottom panel of
Fig.~\ref{fig:TempStudies}. No significant change in
$\mu_{\rm{e}}\tau_{\rm{e}}$ can be identified for the temperature range
studied. \citet{Jung2007} discuss the temperature dependence of Imarad
High-Pressure Bridgman CZT detectors and find a decreasing trend of
$\mu_{\rm{e}} \tau_{\rm{e}}$ for temperatures $T > 10^{\circ} \, \rm{C}$
and for $T < -25^{\circ} \, \rm{C}$.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.99\textwidth]{PeakPosVsTemp}
\end{center}
\caption{Reconstructed line energy for different temperatures using the
calibration obtained at $T_{\rm{cal}} = 25^{\circ} \, \rm{C}$ (dashed
vertical line). The data points indicate the mean of the peak positions
of all 64 pixels and the error bars represent the standard deviation of
that distribution (separately derived for each temperature). The dashed
horizontal line indicates the nominal line energy.}
\label{fig:PeakPosVsTemp}
\end{figure*}
{\it Temperature-stability of the calibration.} In the studies shown in
Fig.~\ref{fig:TempStudies} all detector pixels were re-calibrated at
each temperature. It is important to understand how the calibration
itself changes with temperature. To study this effect, a reference
calibration at $T_{\rm{cal}} = 25^{\circ} \, \rm{C}$ was applied to the
measurements taken at the different temperatures. The line positions at
$59.5 \, \rm{keV}$ and $122.1 \, \rm{keV}$ were determined for the
individual pixels. Each distribution (one per temperature) of
reconstructed line positions was in turn characterized by its mean and
its standard deviation. The results are illustrated in
Fig.~\ref{fig:PeakPosVsTemp}, where the standard deviation (spread of
the corresponding distribution) is represented as error bar. For
reference, the spectra of detector QP$_{5}$1 were corrected with the
calibration obtained at $T_{\rm{cal}} = 5^{\circ} \, \rm{C}$. While the
mean reconstructed line energy does not change significantly, the
widening of the error band indicates that individual channels show a
temperature-dependent upward/downward drift of the line position. This
makes it difficult to globally correct for temperature-dependent changes
in the calibration~-- unless calibration data are taken for all 2048
X-Calibur channels at a variety of temperatures. The calibration changes
by up to $\simeq 2\%$ for temperatures varying around $\pm 10^{\circ} \,
\rm{C}$ relative to the reference calibration $T_{\rm{cal}}$.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{TempNoise}
\end{center}
\caption{Electronic readout noise measured with the ASIC's internal test
pulser for different temperatures and different energies (detector
CE$_{5}$1 and EN$_{5}$8). For reference, the gray asterisk marker
indicates the 1~std.dev. range of all 2048 channels as operated in the
X-Calibur assembly at room temperature of $T \approx 25^{\circ} \,
\rm{C}$ with/without biased detectors (solid/dashed line). {\bf Top:}
Channel-by-channel noise (vertical dotted lines) and detector average
with the detector being biased at $V_{\rm{bi}}=-500 \, \rm{V}$. {\bf
Middle:} Detector averages for three different temperatures with and
without the cathode bias $V_{\rm{bi}}$. {\bf Bottom:} A series of $T
\approx 25^{\circ} \, \rm{C}$ measurements with different configurations
as described in the text.}
\label{fig:TempStudiesNoise}
\end{figure}
{\it Electronic noise.} The read-out electronic contributes a certain
amount of jitter to the measured energy resolution of a detector pixel.
In order to quantify this contribution, a series of measurements was
taken with detectors CE$_{5}$1 and EN$_{5}$8 (see
Tab.~\ref{tab:Detectors}) at different temperatures using the internal
pulse generator of the ASIC (see Sec.~\ref{sec:XCLB}). The detectors
were operated in an electrically shielded copper box. With the detector
being plugged into the ASIC, the measurements actually reflect the
readout noise of the ASIC/detector assembly, rather than the noise of
the ASIC alone. Different amounts of charge were injected (corresponding
to different energies)\footnote{Note, that given the differences in
pixel acceptance and pixel calibration, a fixed amount of charge
injected into the ASIC translates to slightly varying reconstructed
energies (if comparing different channels).}. The measured pulse heights
were transformed to energies using the corresponding energy calibration
determined for each temperature. For each configuration and channel, a
total of 1000 events were injected. The calibrated distribution was
fitted in order to determine the corresponding mean energy and energy
resolution.
The results are shown in Fig.~\ref{fig:TempStudiesNoise}. At energies of
around $200 \, \rm{keV}$ the electronic noise of the ASIC/detector unit
increases by $(21 \pm 6)\%$ in the studied temperature range of
$-25^{\circ} \, \rm{C}$ to $+35^{\circ} \, \rm{C}$. The shape of the
energy-dependent noise curve depends on the temperature. At room
temperature, the noise increases by $(20 \pm 4)\%$ if going from $200 \,
\rm{keV}$ to $1.7 \, \rm{MeV}$ and levels out around $3 \, \rm{keV}$ at
low energies. This suggests, that the energy resolution in the X-Calibur
range ($\approx 4 \, \rm{keV}$ at $40 \, \rm{keV}$, see
Fig.~\ref{fig:PixelResolution}) is dominated by the electronic readout
noise, rather than charge transport properties in the CZT crystal. The
gray asterisk marker in Fig.~\ref{fig:TempStudiesNoise} shows the
1~std.dev.~range of the noise distribution of all 2048 data channels as
measured at room temperature ($\approx 25^{\circ} \, \rm{C}$) in the
final X-Calibur configuration, compare with
Fig.~\ref{fig:PixelResolution} (middle).
The middle panel of Fig.~\ref{fig:TempStudiesNoise} illustrates the
effect of the bias voltage of the detector. A biased cathode at $T =
25^{\circ} \, \rm{C}$ increases the low-energy noise by $(8 \pm 5)\%$,
whereas no noticeable change can be measured for temperatures lower than
that. Therefore, the cathode bias only seems to systematically affect
the readout noise for temperatures higher than $\simeq 15^{\circ} \,
\rm{C}$.
The bottom panel in Fig.~\ref{fig:TempStudiesNoise} shows the electronic
noise measured at room temperature for different configurations: (i) the
detector/ASIC unit with biased cathode, (ii) the detector/ASIC unit with
unbiased cathode, (iii) the ASIC with a ceramic chip carrier but no
detector bonded to it, and (iv) only the plain ASIC. It can be seen that
steps (i)--(iii) each add $\approx 0.2 \, \rm{keV}$ readout noise to the
single-detector system. The average readout noise of the whole X-Calibur
assembly (gray asterisk in Fig.~\ref{fig:TempStudiesNoise}) is shown for
reference. Another series of measurements was performed with the plain
ASIC at different temperatures (not shown). No significant noise trend
could be identified in the $T = -20^{\circ} \, \rm{C}$ to $+25^{\circ}
\, \rm{C}$ temperature range which leads to the conclusion that the
temperature dependence of the readout noise of the ASIC/detector unit
(top panel of Fig.~\ref{fig:TempStudiesNoise}) is mostly a result of the
temperature dependence of the dark currents in the CZT crystal.
{\it Caveats:} The electronic readout noise varies from ASIC to ASIC and
depends on the electronic shielding environment in which the ASIC is
operated. Therefore, the comparison between the absolute noise levels of
the single ASIC system shown in Fig.~\ref{fig:TempStudiesNoise} and the
average readout noise in the X-Calibur assembly should be treated with
care; the relative noise trends found, however, can likely be applied to
whole X-Calibur assembly. It should also be mentioned, that the cooling
aggregates of the temperature chamber (increased activity at low
temperatures) can potentially introduce external noise pick-up in the
ASIC.
\subsection{Summary of the Detector Calibration and Tests}
Each detector pixel has been energy calibrated according to
Eq.~(\ref{eq:EnergyCalibration}). With the current readout electronics
and the compact X-Calibur configuration, the CZT detectors achieve a
mean trigger threshold of $\simeq 21 \, \rm{keV}$. The mean energy
resolution at $40 \, \rm{keV}$ in the three front-side detector rings
{\it R1}-{\it R3} (detecting most of the scattered events in the
polarization measurements) is found to be $\Delta E_{\rm{czt}} \simeq 4
\, \rm{keV}$ FWHM, when operated at room temperature. The energy
resolution of the polarimeter as a whole is determined by the energy
resolution of the individual detectors and by the energy deposited/lost
in the scintillator ($\Delta E_{\rm{sci}} = 0 - 5.4 \, \rm{keV}$ at $40
\, \rm{keV}$). The energy resolution of the detectors is thus not
entirely negligible and X-Calibur would benefit from an optimized
readout ASIC. We are currently working on modifying and adopting the
HD-3 ASIC \cite{DeGeronimo2003, Vernon2010}. Using a pre-amplifier chain
optimized for the $2-100 \, \rm{keV}$ energy range, we expect a trigger
threshold of $1.7 \, \rm{keV}$ and electronic readout noise of $\simeq
550 \, \rm{eV}$ RMS. The noise contribution of the new ASIC to the
energy resolution of the polarimeter would be negligible for all
energies above $20 \, \rm{keV}$. The low energy threshold can
potentially be used on a satellite-borne version of the polarimeter.
The effective energy threshold of the polarimeter is slightly higher
than the energy threshold of the individual CZT detectors, as a $25 \,
\rm{keV}$ photon looses up to $2.2 \, \rm{keV}$ in the scatterer.
However, the polarimeter will detect a large fraction of the X-rays at
125,000 feet flight altitude, as the residual atmosphere only transmits
photons above $\simeq 25 \, \rm{keV}$.
Even though the temperature-dependent trends in energy resolution would
favor operating the detectors at $T \leq 0^{\circ} \, \rm{C}$
(Fig.~\ref{fig:TempStudies}, top), the thermal design of the X-Calibur
assembly and local internal heat built-up of the polarimeter during the
balloon flight makes an operation at $T \simeq (15 \pm 10)^{\circ} \,
\rm{C}$ a more likely scenario~-- still guaranteeing a reasonable energy
resolution for most of the detectors. A change in temperature, which
will be monitored during flight, leads to a shift in reconstructed
energy~-- which can go either way (Fig.~\ref{fig:PeakPosVsTemp}).
Ideally, one should use a data base of temperature-dependent calibration
values on a pixel-by-pixel basis to correct for the temperature trends.
If ignoring the temperature-dependence of the calibration, a systematic
error on the reconstructed energy of a few percent has to be accounted
for in the temperature interval of $\pm 10^{\circ} \, \rm{C}$ around the
calibration temperature. For the first X-Calibur flight, we will choose
the second option.
\section{X-Calibur: Instrument Characterization} \label{sec:XCLB_Characterization}
This section describes measurements of the fully assembled polarimeter
installed in the CsI shield. The goal of the measurements is to
characterize the efficiency of the shield, and to estimate the
background levels in the different (ground-based) environments the
polarimeter was operated in (Sec.~\ref{subsec:BG_Data}). The reduction
of the background is crucial in order to perform sensitive measurements
of the polarization properties of astrophysical sources, see
Eq.~(\ref{eqn:MDP}).
The X-rays that enter the polarimeter along the optical axis will
produce a certain amount of scintillation light when scattering in the
scintillator rod which is read out by a PMT. The efficiency curve,
describing the trigger probability for different energy depositions in
the scatterer, is discussed in Sec.~\ref{subsec:ScintillatorEfficiency}.
As will be shown in Sec.~\ref{subsec:BG_Data}, a high trigger efficiency
of the scintillator will allow a further reduction of the background.
The efficiency curve is fed into the simulations that were discussed in
Sec.~\ref{sec:Simulations}. The measurements described in this chapter
were done with disk-like radioactive sources with a diameter of $\simeq
0.5 \, \rm{cm}$.
\subsection{Shield Performance and Cosmic Ray Background} \label{subsec:BG_Data}
The shielding and suppression of backgrounds is a crucial task for
sensitive polarimetry measurements. The background in the ground-based
measurements presented in this paper (non-flight) results mostly from
secondary particles produced in air showers in the earth's atmosphere,
induced by cosmic rays (CRs). The flux of the secondary particles
depends on the geographical location at which the measurement is
performed, and on the structure/materials of the building in which the
polarimeter is operated (partly shielding the secondary particles). As
can be seen in Fig.~\ref{fig:MultiplicityDistribution}, the CR
background has a higher fraction of multiplicity $m \geq 2$ pixel events
as compared to Compton-scattered X-rays in the $E < 100 \, \rm{keV}$
regime, relevant for the X-Calibur polarimetry measurements. In general,
the distribution of $m$ depends on the energy and the kind of
interaction; high-energy muons (ionization), for example, trigger events
along a row of pixels (unless they cross the detector perpendicular to
the pixel plane). Low energy X-rays, on the other hand, are
photo-absorbed with a charge deposition usually contained well within
one pixel. Therefore, a requirement of $m = 1$ pixel events already
suppresses the background for the polarization measurements by a certain
amount.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{ShieldScalerVsDiscriminator}
\end{center}
\caption{Optimization of the shield discriminator using three different
data runs (a)-(c), as described in the text. The background contribution
is subtracted for the runs with Cs$^{137}$ outside shield (b), and for
Am$^{241}$/Eu$^{152}$ on the optical axis (c). The dashed vertical line
indicates the optimal discriminator setting. {\bf Top:} Shield trigger
rate for different threshold settings. The horizontal lines indicate the
rates corresponding to $10/5/2\%$ (from top to bottom) dead time
produced by the shield vetoes. The noise regime starts to dominate
around DAC $<20$; this in turn leads to an underestimation of the
Cs$^{137}$/Eu$^{152}$ trigger contribution since the CR background is
subtracted. In the case of the Am$^{241}$ source, no significant trigger
rate above background was measured. {\bf Bottom:} Fraction of detected
CZT events that are vetoed with $f_{\rm{shld}} = 1$.}
\label{fig:ShieldOptimization}
\end{figure}
{\it Optimization of the shield trigger threshold.} A particle crossing
the active shield (see Fig.~\ref{fig:ActiveShieldAndTelescope, left)
produces scintillation light in the CsI crystal. The crystal is read by
four PMTs which signals are summed and digitized.} A programmable
discriminator decides whether the shield veto flag $f_{\rm{shld}}$ is
activated ($f_{\rm{shld}}$ is kept active for $6 \, \mu\rm{s}$ per
trigger) and merged into the data stream. Events with the corresponding
flag can in turn be filtered out in the data analysis. In order to
optimize the shield trigger efficiency, a series of measurements was
performed with different settings of the discriminator. The measurements
comprise: (a) cosmic ray background only, (b) a collimated Cs$^{137}$
source aimed from outside the shield at the X-Calibur CZT detector
assembly (configuration $C_{\rm{wu2b}}^{3}$ in
Fig.~\ref{fig:X-Calibur_Configurations}), and (c) collimated
Am$^{241}$/Eu$^{152}$ sources placed on the optical axis of the
polarimeter (configuration $C_{\rm{wu2a}}^{3}$) to simulate X-rays from
the X-ray mirror entering the polarimeter without interaction in the
shield\footnote{Obviously, the lead used to collimate the sources leads
to indirect scatterings and shield contamination, in particular in the
case of the high energy lines of Eu$^{152}$.}. Two shield
characteristics were measured: (i) the raw trigger rate of the shield,
and (ii) the fraction of triggered CZT events with $f_{\rm{shld}} = 1$.
The results are shown in Fig.~\ref{fig:ShieldOptimization}. Since the
cosmic ray background was present in all three runs, the X-ray data runs
(b) and (c) were corrected for the rates measured in run (a). The
setting of the discriminator was optimized according to the following
criteria.
\begin{enumerate}
\item The dead time produced by noise triggers should not exceed a few
percent (horizontal lines in Fig.~\ref{fig:ShieldOptimization}). The
Am$^{241}$/Eu$^{152}$ sources located on the optical axis (c) should
result in CZT events with a $f_{\rm{shld}} = 1$ contribution as low as
possible (no signal suppression).
\item For the CR (a) and Cs$^{137}$ (b) runs, the fraction of CZT
triggers with $f_{\rm{shld}} = 1$ should be as high as possible,
reflecting a high rejection power.
\end{enumerate}
We determined an optimal shield discriminator setting of $\rm{DACQ} =
30$ which is indicated by the vertical line in
Fig.~\ref{fig:ShieldOptimization}. Note, the fraction of Eu$^{152}$
events with $f_{\rm{shld}} = 1$ is around $25 \%$, which is probably due
to X-rays that Compton scatter and interact with the shield and CZT~-- a
result of not having a well collimated X-ray beam entering the
polarimeter in this measurement.
\begin{table}[t!]
\begin{tabular}{lrr}
Data set & $\left< T_{5 \, \rm{mm}} \right>$ & $\left< T_{2 \, \rm{mm}} \right>$ \\
& [mHz] & [mHz] \\
\hline \hline
\noalign{\smallskip}
\multicolumn{3}{l}{CZT initial calibration (Washington Univ.): $C_{\rm{wu1a}}^{1}$} \\
\hline
Eu$^{152}$ illumination & $7100$ & $6200$ \\
Background & $59$ & $35$ \\
\noalign{\smallskip}
\multicolumn{3}{l}{Scintillator characterization (Wash. Univ.): $C_{\rm{wu2a}}^{3}$} \\
\hline
Eu$^{152}$ on optical axis$^{\rm{*}}$ & $31$ & $8.7$ \\
Background in shield (active) & $9.9(5.4)$ & $4.9(2.0)$ \\
\noalign{\smallskip}
\multicolumn{3}{l}{CHESS synchrotron beam (Cornell Univ.): $C_{\rm{ch}}^{3}$} \\
\hline
$40 \, \rm{keV}$ X-ray beam & $1100$ & $300$ \\
Background & $13$ & $8.4$ \\
\noalign{\smallskip}
\multicolumn{3}{l}{X-Calibur/shield/InFocus (Ft.~Sumner): $C_{\rm{ft}}^{3}$} \\
\hline
X-ray source$^{\rm{*}}$ & $853$ & $190$ \\
Background in shield (active) & $8.5(3.9)$ & $4.7(1.6)$ \\
\noalign{\smallskip}
\multicolumn{3}{l}{X-Calibur flight (simulations)} \\
\hline
Crab$^{\rm{*}}$ & $1.25$ & $0.3$ \\
\end{tabular}
\caption{Average event trigger rates per CZT detector pixel for $5 \,
\rm{mm}$ detectors $\left< T_{5 \, \rm{mm}} \right>$ and for $2 \,
\rm{mm}$ detectors $\left< T_{2 \, \rm{mm}} \right>$. Rates are shown
for the different measurements/environments presented in this paper (see
Fig.~\ref{fig:X-Calibur_Configurations}). No event selection cuts are
applied. Rates marked with a `*' are background subtracted. The
corresponding background spectra for the different data sets are shown
in Fig.~\ref{fig:BG_Spectra}.}
\label{tab:PixelRates}
\end{table}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{BackgroundSpectra_Mosaic}
\hfill
\includegraphics[width=0.49\textwidth]{BackgroundSpectra2_Mosaic}
\end{center}
\caption{Cosmic ray background spectra (events with pixel multiplicity
$m=1$, if not mentioned otherwise). The dotted vertical lines indicate
the ranges of the $\rm{KL}_{1-3}$ transition energies of tungsten (W)
and lead (Pb) which show up in some of the spectra. The
X-Calibur/{\it InFOC$\mu$S}\xspace sensitive energy range is indicated, as well. {\bf
Left:} Spectra measured in different environments (see
Fig.~\ref{fig:X-Calibur_Configurations}). Date and duration of the runs
is given in square brackets. The (passive/active) CsI shield strongly
reduces the background. The three dashed vertical lines on the
$C_{\rm{wu1a}}^{1}$ spectrum represent the high-energy lines of
Eu$^{152}$ ($244.7 \, \rm{keV}$ and $344.3 \, \rm{keV}$) and Cs$^{137}$
($661.7 \, \rm{keV}$) for which an onset indication can be seen; the
sources were stored in a lead bunker $\sim$$2 \, \rm{m}$ away during the
background measurements and were probably contributing to the spectrum
at a low level. {\bf Right:} Energy spectra for individual detector
rings measured at Washington University in the active shield (after
shield veto, $f_{\rm{shld}} = 0$). We also show energy spectra (scaled
by a factor of 0.1) from different pixels of the detectors on the
readout board {\it Bd2} to illustrate the difference between central
pixels and pixels located on the edge of detectors, see also
Fig.~\ref{fig:BG_Maps2D} for reference.}
\label{fig:BG_Spectra}
\end{figure*}
{\it Cosmic ray background levels.} Figure~\ref{fig:BG_Spectra} shows
the CR energy spectra ($m=1$ pixel multiplicity) measured in the CZT
detectors at the different locations: the laboratory at Washington
University ($C_{\rm{wu}}$), the CHESS X-ray beam facility
($C_{\rm{ch}}$, Sec.~\ref{subsec:CHESS}), and in Ft.~Sumner
($C_{\rm{ft}}$, Sec.~\ref{subsec:FtSumnerData}). For reference, the
corresponding event trigger rates ($m \geq 1$) per pixel are summarized
in Tab.~\ref{tab:PixelRates}. Note, that the actual pixel rates can vary
quite substantial, since different pixels see different income fluxes
depending on the type of measurement (e.g. an X-ray beam Compton
scattered in the scintillator leads to a strongly depth-dependent
illumination of CZT detectors, compare with
Fig.~\ref{fig:CHESS_2DAzimuthData}). However, it can be seen that for
most measurements presented in this paper the background can be
neglected. During the balloon flight, however, the expected
source-to-background ratio will be much lower and a proper understanding
and the background suppression will be crucial \cite{Guo2010, Guo2013}.
Even though the background at flight altitude will be different as
compared to the ground-based CR background, some characteristics in the
detector response can be discussed qualitatively based on the spectra
shown in Fig.~\ref{fig:BG_Spectra}. While $\alpha$-particles interact
close to the surface of the detector, muons as well as primary and
secondary high-energy gamma rays will penetrate deeper and their energy
deposition is proportional to the detector volume. This volume-dependent
background rate can be seen by comparing the spectra (and trigger rates)
measured with $5 \, \rm{mm}$ versus $2 \, \rm{mm}$ detectors. Note, that
the spectra shown in Fig.~\ref{fig:BG_Spectra} represent particle fluxes
folded with the energy-dependent response of the CZT detectors (with an
energy calibration derived from, and valid for, X-rays). The drop in
event rate below $30 \, \rm{keV}$, for example, is an effect of the
superposition of the different trigger thresholds of the pixels
contributing to the spectrum. The dynamical energy range covered by a
single pixel saturates around $2000 \, \rm{keV}$ (differing from pixel
to pixel), so that the combined $m=1$ pixel spectra shown in
Fig.~\ref{fig:BG_Spectra} drop off around this energy. Allowing events
with $m \geq 1$ multiplicities will extend the energy range (see
Fig.~\ref{fig:BG_Spectra}, right) which is, however, not relevant for
the operation of X-Calibur.
{\it The CsI shield efficiency.} Figure~\ref{fig:BG_Maps2D} shows the 2D
distribution of event count rates from background data taken with
X-Calibur installed in the CsI shield at Washington University. The
rates are shown for two different energy bands. The left panels show the
raw rates and the right panels show the rates after rejecting events
that triggered the active shield, only allowing non-vetoed events
($f_{\rm{shld}} = 0$). It again becomes obvious that the $5 \, \rm{mm}$
detectors ({\it R1}-{\it R5}) collect more background compared to the $2
\, \rm{mm}$ detectors ({\it R6}-{\it R8}). Furthermore, the edge pixels
of the detector rows on the individual boards {\it Bd0-Bd3} see a higher
background rate as compared to central pixels (likely because of the
higher exposed surface area detecting charged particles and low energy
X-rays). The energy-resolved difference between edge and central pixels
can be seen in Fig.~\ref{fig:BG_Spectra}, right ($5 \, \rm{mm}$
detectors of boards {\it Bd2}): the central pixels show an almost two
times lower background compared to edge pixels in the energy regime
relevant for X-Calibur.
The background distribution after shield veto (Fig.~\ref{fig:BG_Maps2D},
right) exhibits a spatial gradient with more background events being
detected closer to the front side of the experiment, as that side is
only shielded by the passive tungsten cap (see
Fig.~\ref{fig:ActiveShieldAndTelescope}, left). In particular, the
front-side (top) pixel row of ring {\it R1}, with its exposed detector
side walls, suffers strongly from primary radiation leaking through the
tungsten, as well as secondary particles being produced in the tungsten.
Here, spectral signatures of the tungsten KL transitions can be
identified in the measured spectra shown in Fig.~\ref{fig:BG_Spectra}
(right). Since this single pixel row is not crucial for the polarimetry
sensitivity it can be excluded from the analysis.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{BG_Maps2D_35-150keV} \\
\includegraphics[width=0.49\textwidth]{BG_Maps2D_200-1000keV}
\end{center}
\caption{2D maps of the CR count rates (pixel-by-pixel, representation
same as in Fig.~\ref{fig:PixelThreshold}) of a $192 \, \rm{h}$
background run with X-Calibur being installed in the active CsI shield
($C_{\rm{wu2c}}^{3}$ in Fig.~\ref{fig:X-Calibur_Configurations}). The
tungsten cap is located at the top of each panel. The count rate is
derived by integrating the corresponding energy spectra ($m=1$ events)
in the given energy range: $35-150\, \rm{keV}$ (top) and $200-1000 \,
\rm{keV}$ (bottom). The left panel shows the measured rate (passive
shield rejection only) and the right panel shows the non-vetoed events
with $f_{\rm{shld}} = 0$ (active and passive rejection). Left and right
panels are shown with the same axis/color ranges, each.}
\label{fig:BG_Maps2D}
\end{figure}
The passive/active rejection efficiency of the CsI shield becomes
obvious if comparing the different background spectra in
Fig.~\ref{fig:BG_Spectra}, left. Moving the polarimeter from its copper
housing (used in the initial test measurements, $C_{\rm{wu1}}^{1}$) into
the CsI shield ($C_{\rm{wu2c}}^{3}$) reduces the background by roughly
one order of magnitude due to passive shielding. Applying the
$f_{\rm{shld}} = 0$ veto from the active shield leads to an additional
background rejection by a factor of $\sim$$2$. Note, that the
recombination of the tungsten and lead $\rm{KL}_{1,2,3}$ transition
energies\footnote{http://www.nist.gov/pml/data/xraytrans/} (probably
activated by CRs) can be identified in the background spectra. This
feature is most prominent in the spectrum of ring {\it R1}
(Fig.~\ref{fig:BG_Spectra}, right) which is located closest to the
tungsten cap.
It should be noted that an additional cut on the scintillator rod
coincidence flag $f_{\rm{sci}} = 1$ reduces the background by another
$1.5$ orders of magnitude (Fig.~\ref{fig:BG_Spectra}, left)~-- strongly
rejecting events that did not enter the polarimeter along the optical
axis and interacted in the scintillator, see
Sec.~\ref{subsec:ScintillatorEfficiency}.
\subsection{Compton-scattering and Scintillator Trigger Efficiency} \label{subsec:ScintillatorEfficiency}
\begin{figure}[th!]
\begin{center}
\includegraphics[width=0.47\textwidth]{SpectraScintillatorVsDAC_R1b_Mosaic} \\
\includegraphics[width=0.47\textwidth]{SpectraScintillatorVsDAC_R2t_Mosaic} \\
\includegraphics[width=0.47\textwidth]{SpectraScintillatorVsDAC_R3_Mosaic}
\end{center}
\caption{A lead-collimated Eu$^{152}$ source enters the polarimeter
along the optical axis and Compton-scatterers in the scintillator rod
($C_{\rm{wu2a}}^{3}$ in Fig.~\ref{fig:X-Calibur_Configurations}). The
resulting CZT spectra ($m=1$, background subtracted) are shown for
different rings. The horizontal lines indicate the complete range of
possible energies $E_{\rm{czt}}$ after Compton-scattering (dotted:
$180-90^{\circ}$, solid: $90-0^{\circ}$). The attached vertical dotted
lines indicate the sub range for the particular ring~-- given its
geometrical coverage of the scintillator. Spectra are measured with
different scintillator trigger thresholds settings (Digital to Analog
Converter, DAC) and are compared to the spectra without coincidence
requirement (no cuts). The lead collimator prevents direct source hits
(rings {\it R1-R3}), but scattered X-rays and/or activation of the
$\rm{KL}_{1-3}$ transitions in tungsten (W) and lead (Pb) partly
contaminate the spectra.}
\label{fig:Scintillator_Spectra}
\end{figure}
The central scintillator rod of the polarimeter (Fig.~\ref{fig:Design})
acts as a Compton-scatterer for the incoming X-ray beam. At the same
time, it produces scintillation light that is read out by a PMT. In
order to characterize the Compton-scattering and the trigger response of
the scintillator, a collimated Eu$^{152}$ source was placed at the
polarimeter entrance, illuminating the scintillator along the optical
axis (see setup $C_{\rm{wu2a}}^{3}$ in
Fig.~\ref{fig:X-Calibur_Configurations}). The Compton-scattered X-rays
are recorded by the surrounding CZT detector rings. An X-ray enters the
polarimeter with a certain energy $E_{\rm{line}}$, deposits the energy
$\Delta E_{\rm{sci}}$ in the scintillator and is absorbed in a CZT
detector with its post-scattering energy of $E_{\rm{czt}} =
E_{\rm{line}} - \Delta E_{\rm{sci}}$. The amount of scintillation light
can be assumed to be proportional to $\Delta E_{\rm{sci}}$.
Data were taken with different settings of the scintillator/PMT
discriminator threshold (DAC). As can be seen in
Tab.~\ref{tab:PixelRates}, the background level is not negligible in
this measurement, so that background spectra without a X-ray source were
recorded and subtracted. It should be noted, however, that X-rays
scattered in the lead collimator and activation/recombination of the
$\rm{KL}_{1,2,3}$ electron transitions in the lead collimator and the
tungsten cap lead to a slight contamination of the measured spectra.
These corresponding spectral features do not cancel out after CR
background subtraction since they are (indirectly) introduced by the
Eu$^{152}$ source itself. Since those features will over-proportionally
affect the data without the $f_{\rm{sci}}=1$ coincidence (X-ray paths
that do not necessarily cross the scintillator), this may lead to an
underestimation of the trigger efficiency (which in the ideal case would
only compare Compton-scattered events in the scintillator with and
without the $f_{\rm{sci}} = 1$ trigger coincidence). The scintillator
trigger efficiency for different energies $\Delta E_{\rm{sci}}$ is
derived as follows.
{\it Compton spectra.} The resulting Compton-scattered energy spectra
$E_{\rm{czt}}$ are shown in Fig.~\ref{fig:Scintillator_Spectra} for
different CZT detector rings. Data are shown with and without the
scintillator coincidence requirement. The energy $\Delta E_{\rm{sci}}$
deposited in the scintillator depends on the scattering kinematics,
including the primary energy of the X-ray $E_{\rm{line}}$ and its
scattering angle. The theoretical range of energies after
$0-180^{\circ}$ Compton scattering is indicated by horizontal lines for
the four main Eu$^{152}$ energy lines. Due to geometrical reasons, some
of the detector rings can only detect X-rays originating from a limited
range of Compton-scattering angles. With the scintillator starting at
ring {\it R2} (Fig.~\ref{fig:Design}, left), the sub rings {\it
R1$_{\rm{t}}$}, and {\it R1$_{\rm{b}}$}, for example, can only detect
back-scattered X-rays which deposit/loose the highest amount of energy
$\Delta E_{\rm{sci}}$ in the scintillator~-- leading to a higher trigger
probability. For a given incoming energy $E_{\rm{line}}$, the range of
possible scattering angles translates into a range of possible
Compton-scattered energies $E_{\rm{czt}}$. These ring-dependent sub
ranges in scattered energy are indicated by the vertical lines in
Fig.~\ref{fig:Scintillator_Spectra}. Note, however, that the scattering
angles/energies will not be equally distributed within these boundaries.
One can qualitatively discuss the X-Calibur response to
Compton-scattering for the different scattered energy lines.
\begin{itemlist}
\item $E_{\rm{line}} = 40 \, \rm{keV}$: The Compton-scattered energy
range of the Eu$^{152}$ line doublet ($39.52\,\rm{keV}$ and
$40.12\,\rm{keV}$) only covers $E_{\rm{czt}} = 34.2 - 40.1\,\rm{keV}$ at
an energy resolution of $\Delta E_{40} \simeq 4 \, \rm{keV}$. This makes
it difficult to resolve structures in the scattered energy spectrum.
However, the geometrical coverage of ring {\it R1$_{\rm{b}}$}
constraints the energy deposition in the scintillator to $\Delta
E_{\rm{sci}} = 2.4-5.4 \, \rm{keV}$ since only back-scattering angles of
$80-180^{\circ}$ are possible (top panel in
Fig.~\ref{fig:Scintillator_Spectra}). A clear improvement in trigger
efficiency is obvious when lowering the scintillator DAC threshold from
170 to 10.
\item $E_{\rm{line}} = 121.8 \, \rm{keV}$: This well-defined line at
higher energy leads to a broader Compton-continuum which can be tested
at different energies $\Delta E_{\rm{sci}}$. While ring {\it
R1$_{\rm{b}}$} again only sees the back-scattered X-rays, the rings
further down cover a broad continuum range. The trigger efficiency for
the back-scattering of the $121.8 \, \rm{keV}$ line ($\Delta
E_{\rm{sci}} = 23-39 \, \rm{keV}$) does not seem to depend on the
scintillator DAC setting and is therefore already at its maximum. In the
close-to forward scattering regime ($\Delta E_{\rm{sci}}$ a few keV) in
ring {\it R3} (bottom panel in Fig.~\ref{fig:Scintillator_Spectra}) a
lower DAC threshold leads to a broader Compton continuum in the
$f_{\rm{sci}} = 1$ data, reflecting the increase in trigger efficiency.
Starting at ring {\it R3}, however, the effective thickness of the lead
collimator at the polarimeter entrance is not high enough to fully
absorb all X-rays; this leads to some direct detector hits in the raw
energy spectrum~-- and therefore an underestimation of the trigger
efficiency at the corresponding value of $\Delta E_{\rm{sci}}$.
\item $E_{\rm{line}} = 244.7 \, \rm{keV}$ and $344.3 \, \rm{keV}$: These
two high-energy (but lower intensity) lines lead to partially
overlapping Compton-continua. This makes it difficult to study the
response in more detail. It can also be seen, that $344.3 \, \rm{keV}$
direct CZT hits are not fully prevented by the lead collimator (peak in
raw spectra).
\end{itemlist}
{\it Scintillator efficiency.} The scintillator trigger efficiency as a
function of $\Delta E_{\rm{sci}} = E_{\rm{line}} - E_{\rm{czt}}$ was
estimated from the flux ratio at energy $E_{\rm{czt}}$ with and without
the $f_{\rm{sci}} = 1$ coincidence requirement (after continuum
subtraction). This can be done independently for the different line
energies and for different detector rings. Although the different
Eu$^{152}$ energies are ideal to test different values of $\Delta
E_{\rm{sci}}$, the reflection and re-processing of the high energy lines
contaminates the raw spectrum hampering the quantitative determination
of the trigger efficiency (underestimation). Therefore, additional data
were taken with an Am$^{241}$ source which has a line at $59.5 \,
\rm{keV}$, but no significant emission above. The trigger efficiencies
determined from the different source lines and detector rings are
summarized in Fig.~\ref{fig:Scintillator_TriggerEfficiency} for the
optimized discriminator threshold of DAC=10. The trigger distribution
was fitted by a function
\begin{equation}
f(\Delta E_{\rm{sci}}) = \frac{a}{1 + e^{-b (\Delta E_{\rm{sci}}-E_{0})} + e^{-c
(\Delta E_{\rm{sci}}-E_{0})}}.
\label{eqn:ScintTriggEffFit}
\end{equation}
For the optimized setting of DAC=10, the trigger efficiency reaches
$50\%$ at $\Delta E_{\rm{sci}} \simeq 4.6 \, \rm{keV}$. Note, that the
trigger fraction does not level out at $a = 1$ ($100 \%$). This is
potentially a result of the contamination of the spectrum (direct
detector hits)~-- caused by the non-ideal measurement setup that leads
to a systematic underestimation of the efficiency. For the simulations
in Sec.~\ref{sec:Simulations}, we therefore assumed the same trigger
function, however with a value of $a=1$. Note, that we currently do not
model any depth-dependence of $f(\Delta E_{\rm{sci}})$ which becomes
important when looking at the different detector rings. In a detailed
{\it Monte Carlo} study, \citet{PogoScintSims} simulate the trigger
efficiency of a similar scintillator material (EJ-204) and find a
threshold of $2-3 \, \rm{keV}$. The authors also find a $13 \%$ drop in
light yield along a $20 \, \rm{cm}$ long scintillator rod (from close to
the PMT towards the distant end).
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{ScintillatorTriggerFraction}
\end{center}
\caption{Fraction of CZT events with a scintillator coincidence trigger
($f_{\rm{sci}}=1$) as a function of energy deposition $\Delta
E_{\rm{sci}}$. Data were taken with Am$^{241}$, Eu$^{152}$, and at the
CHESS beam (Sec.~\ref{subsec:CHESS}, however with a much higher
scintillator threshold). The Compton-scattered continua (above
background, see for example Fig.~\ref{fig:Scintillator_Spectra}) were
used to derive the trigger efficiency for different $\Delta
E_{\rm{sci}}$. The distribution was fitted by a function shown in
Eq.~(\ref{eqn:ScintTriggEffFit}). The dashed horizontal line indicates
the range of scintillator energy depositions for incoming X-rays in the
$20-80 \, \rm{keV}$ band.}
\label{fig:Scintillator_TriggerEfficiency}
\end{figure}
The raw scintillator trigger rate (independent of CZT event triggers) of
the optimized threshold setting of DAC=10 was measured to be only a few
hundred Hz~-- a further reduction in trigger threshold by updating the
signal amplifier seems possible and is work in progress.
A cleaner setup to measure the scintillator response is the
well-collimated X-ray beam at CHESS (see Sec.~\ref{subsec:CHESS}). The
results are also shown in Fig.~\ref{fig:Scintillator_TriggerEfficiency}
for a non-polarized beam at $40 \, \rm{keV}$ and $120 \, \rm{keV}$,
respectively. However, the amplifier for the PMT reading the
scintillator was not optimized at the time of the CHESS measurements, so
that the $50\%$ trigger probability is only reached around $\Delta
E_{\rm{sci}} \simeq 15 \, \rm{keV}$. The corresponding fitted trigger
efficiency (again with the $a=1$ assumption)\footnote{Also the CHESS
measurements suffered from indirect contamination due to beam absorber
foils, see Sec.~\ref{subsec:CHESS}.} was fed into the simulations that
were done for the CHESS measurements discussed in
Sec.~\ref{subsec:CHESS}.
\subsection{Summary of the Instrument Characteristics}
The rejection power of the active shield was studied in the laboratory.
It reduces the background by more than one order of magnitude in the
energy range relevant for X-Calibur.
When we require a scintillator trigger ($f_{\rm{sci}} = 1$), the
effective energy threshold of the polarimeter increases to $\simeq 25 \,
\rm{keV}$ (with the current scintillator threshold of $\sim$$5 \,
\rm{keV}$). We are working on a further reduction of the scintillator
threshold \cite{Fabiani2013}. Above an energy of $30 \, \rm{keV}$, we
can detect a high fraction of events with a coincident scintillator
signal. For ground-based background measurements, the scintillator
coincidence reduces the background by another factor of $\sim$$30$. On
upcoming balloon flights, we will use events with (for best high-energy
signal to noise ratio) and without (for best low-energy response)
scintillator coincidences.
\section{X-Calibur: Polarization Measurements} \label{sec:XCLB_PolarimetryMeasurements}
This section discusses measurements performed with the fully assembled
and calibrated X-Calibur polarimeter. Measurements of non-polarized and
polarized X-ray beams at the CHESS facility at Cornell University are
described in Sec.~\ref{subsec:CHESS} and are compared to simulations.
The results illustrate the functionality of X-Calibur as an X-ray
polarimeter. Section~\ref{subsec:FtSumnerData} describes X-Calibur
measurements that were taken with the fully integrated
X-Calibur/{\it InFOC$\mu$S}\xspace X-ray telescope in Ft.~Sumner during a test
integration of a flight-ready balloon setup.
Systematic effects and asymmetries in the detector response can, under
certain circumstances, cause measurements of aparent polatization
fractions, even if the measured X-ray beam itself is non-polarized. It
is therefore crucial to understand and control these kinds of systematic
effects in order to correctly study the polarization properties of
astrophysical sources. The systematic effects (and corrections thereof)
caused by a X-ray beam hitting the polarimeter off-center are discussed
in Sec.~\ref{subsec:Systematics}.
\subsection{Polarized X-rays from the CHESS Beam} \label{subsec:CHESS}
In order to measure the response to a polarized X-ray beam, the
X-Calibur polarimeter was operated at the Cornell High Energy
Synchrotron Source (CHESS)\footnote{http://www.chess.cornell.edu/} for
one week in March 2013. CHESS provides a highly collimated and highly
polarized beam of synchrotron X-rays. Using Bragg reflection from a
2-bounce silicon (220) monochromator, a $40 \, \rm{keV}$ beam was
generated with the 2$^{\rm{nd}}$/3$^{\rm{rd}}$ harmonics at $80 / 120 \,
\rm{keV}$, respectively. The measurements were performed in hutch {\it
C1}. The polarimeter was mounted on an adjustable X/Y/Z stage table with
the scintillator being aligned with the X-ray beam (using X-ray
fluorescence paper). The setup, referred to as $C_{\rm{ch}}$, is shown
in Fig.~\ref{fig:X-Calibur_Configurations}. X-Calibur was mounted in a
fixture that allowed us to rotate the polarimeter around the
optical/beam axis, in order to test the response at different
orientations between the polarization plane and the detector. The
rotation angle is referred to as $\alpha$ with detector board {\it Bd0}
located at the top position at $\alpha = 0^{\circ}$ (see $C^{3}$ in the
top panel of Fig.~\ref{fig:X-Calibur_Configurations}). Looking into the
beam, a positive angle $\alpha$ corresponds to a counter clock-wise
rotation. The accuracy of setting the angle was estimated to be $\Delta
\alpha \simeq 2^{\circ}$. It should be noted that we did not use a high
precision rotation mechanism (as will be used during the balloon
flight). Therefore, an $\alpha$ dependent mis-alignment/tilt of the
scintillator during the measurements cannot be excluded.
{\it The CHESS X-ray flux.} A total of 30 synchrotron-emitting electron
bunches cycle the CHESS accelerator ring. The relative intensity of the
X-ray beam entering the {\it C1} hutch is controlled by a system of
slits, as well as the tunable orientation of the Bragg reflection
crystals. The intensity, however, varies with (i) decaying electron
population in the ring during a run (re-charged every $\sim$$2 \,
\rm{h}$), (ii) local heat built-up on the Bragg crystals, and (iii) the
position stability of the electron beam in the ring. The intensity of
the monochromatic beam entering the hutch was monitored using an argon
ion chamber ($6 \, \rm{cm}$ in length along the beam, set to a counter
range of $10^{-8} \, \rm{A}/\rm{V}$). The absolute flux calibration of
X-Calibur was not a major objective of the measurements; therefore, we
did not setup a system to automatically stabilize the beam intensity
entering the hutch~-- but rather kept it manually in the ballpark of
$1-2 \cdot 10^{8} \, \rm{cts}/\rm{s}$ (ion chamber) throughout the
measurements and logged the average intensity for each data run.
Absorber foils placed behind the ion chamber further reduced the beam
intensity by several orders of magnitude. This brought the X-ray flux
hitting the polarimeter down to the level of $1-2 \, \rm{kHz}$, a regime
that can be well handled by the readout electronics of the polarimeter.
{\it Configurations.} An aluminum collimator plate of thickness $0.5''$
was placed directly in front of the X-Calibur polarimeter with an
entrance hole of $0.25''$ in diameter. The CZT detector configuration
used for the CHESS measurements is listed in Tab.~\ref{tab:Detectors}.
CHESS data presented in this paper were taken with three configurations
(variations of $C_{\rm{ch}}$ as depicted in
Fig.~\ref{fig:X-Calibur_Configurations}).
\begin{itemlist}
\item {\bf $C_{\rm{ch1}}$}: the absorber foils consist of platinum (Pt),
a $125 \, \mu\rm{m}$ layer placed directly after the ion chamber and a
stack of four $90 \, \mu\rm{m}$ layers placed between the ion chamber
and X-Calibur. This setup is optimized for a band-pass at $40 \,
\rm{keV}$.
\item{\bf $C_{\rm{ch3}}$}: One layer of $125 \, \mu\rm{m}$ Pt absorber
foil is placed directly behind (downstream) the ion chamber, and a $1.27
\, \rm{mm}$ lead foil (Pb) is placed between the ion chamber and
X-Calibur. This setup is optimized for a band-pass at $80 \, \rm{keV}$.
\item{\bf $C_{\rm{ch4}}$}: Same as $C_{\rm{ch3}}$, but the ion chamber
was removed from the beam path (in order to extend the range of beam
offsets without interfering with the housing of the ion chamber).
\end{itemlist}
X-Calibur detects individual events, but only the ASIC that triggered
the event (corresponding to $1/64$ of the polarimeter) is dead during
the read-out which takes $\sim$$130 \,\mu\rm{s}$. The dead-time during
data runs is therefore usually not higher than $5\%$. In order to
guarantee a homogeneous data set, some additional pixels (as compared to
the calibration runs) were excluded from the analysis which had high
thresholds or were too noisy~-- even though the azimuthal acceptance
(see Sec.~\ref{sec:Analysis}) is in general capable of correcting for
most of these effects. Only events with multiplicity $m=1$ pixels were
selected. X-Calibur was not operated in its active CsI shield, and the
scintillator PMT discriminator was not yet optimized at the time of the
CHESS measurements.
{\it Data runs.} A series of measurements was taken with configuration
$C_{\rm{ch1}}$ (optimized for $40 \, \rm{keV}$) and $C_{\rm{ch3,4}}$
(optimized for $80 \, \rm{keV}$). Data were taken with different
polarimeter orientations $\alpha$. The $C_{\rm{ch1}}$ data runs span a
range of $\alpha \in [-90; +90]^{\circ}$ in steps of $10^{\circ}$ with
$4 \, \rm{Mio}$ events per orientation. A non-polarized beam was
`generated' by the superposition of data from two perpendicular
orientations. A higher number of events were taken for these sets: $20
\, \rm{Mio}$ events for the $\alpha = 0/-90^{\circ}$ pair, and $10 \,
\rm{Mio}$ events for the $\alpha = \pm 45^{\circ}$ reference pair. The
$C_{\rm{ch3,4}}$ series scanned a range of $\alpha \in
[-90;+10]^{\circ}$ with a non-polarized beam composed from $10 \,
\rm{Mio}$ events derived from the $\alpha = 0/-90^{\circ}$ orientations.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{CHESS_AllDetectorUnPolarized_Mosaic}
\end{center}
\caption{Energy spectra of the Compton-scattered non-polarized CHESS
X-ray beam (superposition of perpendicular polarization planes). The
measurements were performed with the two configurations $C_{\rm{ch1}}$
and $C_{\rm{ch3}}$ (see text and
Fig.~\ref{fig:X-Calibur_Configurations}). Shown are the background
subtracted integral spectra of all detectors~-- only ring R1 was
excluded due to flux contamination (see text and
Fig.~\ref{fig:CHESS_ComtponSpectra_UnPol}, right, for the reason). The
background level is shown for reference. The horizontal lines represent
the nominal energies of the three harmonics for (left to right): $180 -
90^{\circ}$ Compton-scattering (dotted), and $90 - 0^{\circ}$
Compton-scattering (solid). Spectra are shown with and without the
$f_{\rm{sci}} = 1$ scintillator coincidence requirement. Simulated
spectra are shown, as well.}
\label{fig:CHESS_ComtponSpectra_AllDetector_UnPol}
\end{figure}
Several background runs were taken without the X-ray beam entering the
hutch. The over-all background spectrum is shown in the left panel of
Fig.~\ref{fig:BG_Spectra} (red data points). The {\it C1} hutch is
shielded by (partly) lead-enforced walls which likely contribute to the
lower background level as compared to the laboratory at Washington
University. In fact, a signature at the lead $\rm{KL}_{1-3}$ transition
energies can be identified in the CHESS background spectrum. Although
the background is negligible compared to the strong X-ray signal (see
Figs.~\ref{fig:CHESS_ComtponSpectra_UnPol} and
\ref{fig:CHESS_ComtponSpectra_Pol} for reference), it was subtracted
from the spectra shown in this section. It should be noted that the beam
in the hutch may introduce an additional implicit/diffuse background by
scattering off several components (slits, absorber foils, etc.). This
kind of background was not determined in a dedicated measurement, but
can potentially be higher than the CR background.
{\it Synchronization between simulations and data.} The Bragg
monochromator only allows the $40/80/120 \, \rm{keV}$ harmonics of the
CHESS white beam to enter the C1 hutch. The spectral intensities of the
white beam, as well as the Bragg-reflected intensities of the
monochromator, were calculated using the {\it X-ray Oriented Programs}
(XOP) software
package\footnote{http://www.esrf.eu/computing/scientific/xop2.1/}. The
absorber foils further change the relative intensities. Given the
density of platinum of $\rho_{\rm{Pt}} = 21.45 \, \rm{g}/\rm{cm}^{3}$,
its mass attenuation coefficient at $40 \, \rm{keV}$ of
$(\mu/\rho)_{\rm{Pt}} = 12.45 \rm{cm}^{2}/\rm{g}$, and the thickness of
the Pt foils $d$, one can calculate the beam intensity $I$ entering the
polarimeter by $I/I_{0} = \exp(-\mu/\rho \cdot \rho \cdot d)$. The mass
attenuation coefficients $\mu/\rho$ were taken from the NIST X-ray
database\footnote{http://www.nist.gov/pml/data/xraycoef/index.cfm}.
However, an accurate prediction of the relative flux intensities
entering the polarimeter requires a detailed modeling of all
energy-dependent X-ray absorption/transmissions on the beam path
(including the entrance windows, the Bragg monochromator, and the
absorber foils).
Therefore, we instead used the overall spectrum measured by the whole
polarimeter to synchronize the relative $40/80/120 \, \rm{keV}$ flux
intensities with our Monte Carlo simulations.
Figure~\ref{fig:CHESS_ComtponSpectra_AllDetector_UnPol} shows the
overall X-Calibur responses (all detector rings except for R1, see
below) to the non-polarized CHESS beam, measured with configurations
$C_{\rm{ch1}}$ and $C_{\rm{ch3}}$. The corresponding relative
intensities of the three mono-energetic energies in the simulations were
matched accordingly and are in turn applied for all data versus
simulation comparisons that follow. Note, however, that the CHESS white
beam intensity varies. Therefore, the simulations were scaled according
to the relative difference in the integral X-Calibur trigger rate with
respect to the data run used to normalize the simulations. We estimate
the systematic error on the absolute flux normalization of the
simulations to be around $20 \%$.
Figure~\ref{fig:CHESS_ComtponSpectra_AllDetector_UnPol} also shows the
energy spectra with a coincident scintillator trigger ($f_{\rm{sci}} =
1$), although obtained with the non-optimal scintillator trigger
threshold.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{CHESS_ComptonSpectra_Mosaic}
\hfill
\includegraphics[width=0.49\textwidth]{CHESS_ComptonSpectra_Ring1t_Mosaic}
\end{center}
\caption{Energy spectra of the Compton scattered, non-polarized $40/80/120
\, \rm{keV}$ CHESS X-ray beam (averaged over the complete azimuthal
scattering range, $m=1$ events). The horizontal lines represent the
nominal energies after Compton-scattering. For reference, the CR
spectrum is shown for one of the $5 \, \rm{mm}$ rings. {\bf Left:}
Spectra from selected sub rings. In the case of ring {\it
R1$_{\rm{t}}$}, the first pixel row was removed (see text and right
panel). {\bf Right:} Special case of ring {\it R1$_{\rm{t}}$} which is
located at the polarimeter entrance; individual pixel rows are shown. It
can be seen that the first pixel row has significantly higher flux at
$E>40 \, \rm{keV}$ as compared to the other rows~-- likely due to
external X-ray contamination. The same spectra are shown with the
scintillator trigger condition ($f_{\rm{sci}} = 1$). Simulated spectra
are shown for the pixel rows 1 and 4. The spectrum obtained from the
first pixel row in Ring {\it R2$_{\rm{t}}$} is shown for reference.}
\label{fig:CHESS_ComtponSpectra_UnPol}
\end{figure*}
{\it Compton spectra of a non-polarized beam.} The energy spectra of the
non-polarized Compton scattered X-ray beam are shown for individual
detector rings in Fig.~\ref{fig:CHESS_ComtponSpectra_UnPol} (left). The
spectra are averaged over all azimuthal scattering angles $\Phi$ within
each ring. Ring {\it R1$_{\rm{t}}$} mostly detects back-scattered events
(the scintillator geometrically covering rings {\it R2}-{\it R8}, see
Fig.~\ref{fig:Design}, left) which deposit the maximal amount of energy
in the scintillator: $\Delta E_{\rm{sci}}$ up to $5.4 \, \rm{keV}$ for
the $40 \, \rm{keV}$ beam, depositing $E_{\rm{czt}} = 34.6 \, \rm{keV}$
in the CZT detector. The rings further downstream detect a superposition
of X-rays that underwent Compton-scattering under different angles. The
rear-side rings ({\it R7} and {\it R8}), to a larger extent, detect
X-rays which Compton scattered nearly in the forward direction~--
depositing only a small amount of energy in the scintillator, with an
energy deposition in the CZT ($E_{\rm{czt}}$) close to the energy of the
primary X-ray ($E_{\rm{line}}$). This explains the shift of the peak
positions in the spectra towards higher energies with increasing ring
number. The theoretical range expectations of $E_{\rm{czt}}$ are
indicated by the horizontal lines in
Fig.~\ref{fig:CHESS_ComtponSpectra_UnPol} (assuming an idealized energy
resolution).
The right panel of Fig.~\ref{fig:CHESS_ComtponSpectra_UnPol} shows the
energy spectra of individual pixel rows of ring {\it R1$_{\rm{t}}$}. It
can be seen that the first pixel row shows a strong anomaly at energies
$E_{\rm{czt}} > 50 \, \rm{keV}$ as compared to the pixel rows 2-4. The
comparison with simulations further underlines the difference. A likely
explanation of the additional continuum would be the following. The
first row of pixels has a rather high effective area (pixel sides) in
the plane perpendicular to the X-ray beam, as compared to other pixel
rows. These `side' pixels will detect X-rays that underwent diffuse
scattering in the Pt/Pb absorber foils or the polarimeter entrance plate
(see Fig.~\ref{fig:X-Calibur_Configurations}), and succeeded in entering
the polarimeter along a path not along the optical axis. Since this
involves X-ray transmission through some of the material of the
polarimeter fixture, it is more likely to happen for higher energies,
which is where the anomaly in the spectrum becomes more prominent. The
first pixel row was therefore excluded from the analysis of the data
presented in this section. It should be noted, that the contamination is
also seen in the pixel rows further downstream~-- however, with strongly
decreasing strength.
The right panel of Fig.~\ref{fig:CHESS_ComtponSpectra_UnPol} also shows
the spectra obtained from the same data with the additional requirement
of a scintillator trigger ($f_{\rm{sci}} = 1$). This favors high-energy
and/or back-scattered events which deposit a high amount of energy
$\Delta E_{\rm{sci}}$ in the scintillator. With the $f_{\rm{sci}} = 1$
cut, the first pixel rows no longer show the anomaly discussed above~--
underlining the hypothesis of an external contamination not interacting
with the scintillator. A more detailed study of the trigger efficiency
of the scintillator is presented in
Sec.~\ref{subsec:ScintillatorEfficiency}. Despite the contamination
issue seen in the first front-side pixel rows, the Compton-spectra
measured with X-Calibur are in good agreement with the simulations
(although not explicitly shown for all rings in
Fig.~\ref{fig:CHESS_ComtponSpectra_UnPol}, right, for reasons of
visibility.).
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{CHESS_ComptonSpectraPol_Mosaic}
\end{center}
\caption{Compton spectra of the polarized $40/80/120 \, \rm{keV}$ CHESS
beam. Spectra are shown for two of the four detector sides ({\it Bd0}
and {\it Bd1}, see Fig.~\ref{fig:X-Calibur_Configurations}, top) in ring
{\it R2$_{\rm{b}}$}, with {\it Bd0} aligned with the plane of
polarization ($\alpha = 0^{\circ}$). The {\it Bd1} spectrum is also
shown with the requirement of a scintillator trigger (the discriminator
threshold not being optimized at this point). The horizontal lines
represent the nominal energy ranges after Compton-scattering. The
simulated spectra are shown, as well.}
\label{fig:CHESS_ComtponSpectra_Pol}
\end{figure}
{\it Compton spectra of a polarized beam.} A polarized X-ray beam
introduces an azimuthal modulation with a $180^{\circ}$ periodicity to
the measured spectra. Figure~\ref{fig:CHESS_ComtponSpectra_Pol} shows
the Compton spectra measured on two of the four detector sides in ring
{\it R2$_{\rm{b}}$} (with the polarimeter being oriented at an angle of
$\alpha = 0^{\circ}$). As expected, and essential for the functionality
of the polarimeter, the detector side located perpendicular to the
polarization plane ({\it Bd0}) detects a higher number of
Compton-scattered X-rays. The energy spectra obtained from simulations
are shown in Fig.~\ref{fig:CHESS_ComtponSpectra_Pol}, as well, and are
found to be in reasonable agreement with the data. For the data versus
simulation comparison one has to keep in mind that the simulations were
performed for a $100 \%$ polarized beam, corresponding to the maximal
modulation and therefore a maximal scattering concentration in {\it
Bd0}. The polarization fraction of the CHESS beam, on the other hand, is
$r < 100 \%$. Therefore, the azimuthal asymmetry ({\it Bd0} versus {\it
Bd1}) in the data is expected to be smaller compared to the simulations.
Dividing the spectra into distinct bins in azimuth, and integrating
counts in a given energy interval, will lead to the azimuthal scattering
distributions that are discussed in the next paragraph.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{AzimuthDistribution_40keV}
\hfill
\includegraphics[width=0.49\textwidth]{AzimuthDistribution_SystematicExamples}
\end{center}
\caption{{\bf Left:} Azimuthal scattering distribution (CHESS at $40 \,
\rm{keV}$, ring R2$_{\rm{b}}$). The horizontal error bars reflect the
azimuthal ranges $\Delta \Phi_{j}$ covered by the corresponding pixels
$j$ (see Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, left). The raw
non-polarized beam shows the 4-fold symmetry of the four detectors boards
{\it Bd0-Bd3} (see Fig.~\ref{fig:X-Calibur_Configurations}). The
polarized beam has the additional $180^{\circ}$ amplification imprinted
which is clearly extracted once corrected for the azimuthal coverage
$\Delta \Phi_{j}$ and the pixel acceptance $a_{j}$. A sine function is
fitted to extract $C_{\rm{min}}$, $C_{\rm{max}}$, the modulation factor
$\mu$, and the angle of the polarization plane $\Omega_{\rm{p}}$ (dotted
vertical lines, indicating the error range). The dashed vertical line
indicates the nominal polarization plane. For reference, the
distribution of a simulated $40 \, \rm{keV}$ beam is shown ($100 \%$
polarized, reconstructed in the same way). {\bf Right:} Normalized
azimuthal scattering distributions derived from different data sets: a
non-polarized beam, a polarized beam measured at $\alpha = 70^{\circ}$, as
well as the response to a cosmic-ray background run.}
\label{fig:AzimuthDistribution}
\end{figure*}
{\it Azimuthal scattering distribution.} The azimuthal scattering
distributions were generated as outlined in
Sec.~\ref{subsec:AnalysisPolarization} by integrating the energy range
of Compton-scattered photons for the three harmonics of the X-ray beam
($34.6 - 40 \, \rm{keV}$, $60.9 - 80 \, \rm{keV}$, and $81.7 - 120 \,
\rm{keV}$, respectively). The energy intervals include an additional $2
\, \rm{keV}$ cushion to account for the energy resolution of the CZT
detector pixels. All measured event rates were in turn normalized on a
pixel-by-pixel basis using the azimuthal coverage $\Delta \Phi_{j}$ and
acceptances $a_{j}$. Following Eq.~(\ref{eqn:Acceptance}) described in
Sec.~\ref{subsec:AnalysisPolarization}, the pixel acceptances $a_{j}$
were determined from the non-polarized beam for each of the above energy
intervals.
An example of an azimuthal scattering distribution measured in the
$34.6-40 \, \rm{keV}$ band is shown in
Fig.~\ref{fig:AzimuthDistribution} (left). The normalized distribution
is used to derive the plane of polarization $\Omega_{\rm{p}}$ (nominal
value of $\Omega_{\rm{nom}} = 90^{\circ}$), as well as the relative
scattering amplitude $\mu_{\rm{ch}}$. The distribution is compared to
the one obtained from the simulation of a $100 \%$ polarized beam at $40
\, \rm{keV}$. Both are found to be in good agreement~-- except for the
slightly lower amplitude of the data which is a result of the $r < 100
\%$ polarization fraction of the CHESS beam (see below).
Figure~\ref{fig:CHESS_2DAzimuthData} illustrates the reconstruction of
the azimuthal scattering distributions for the complete polarimeter
(subrings {\it R1$_{\rm{t}}$}-{\it R8$_{\rm{b}}$}), based on simulations
(top panel) and based on measured CHESS data (bottom panel). The
response to a non-polarized $40 \, \rm{keV}$ beam is shown on the left
side of the figure. The second panel shows the polarimeter response to a
polarized beam at $40 \, \rm{keV}$, not yet corrected for pixel
acceptance $a_{j}$ and azimuthal coverage $\Delta \Phi_{j}$. Applying
the corrections leads to the smooth distribution shown in the third
panel, which is in good agreement if comparing the simulations to the
data. The pixel acceptances $a_{j}$ depend on (i) the energy threshold,
(ii) the energy resolution, and (iii) the trigger efficiency of the
pixel $j$. The simulations take into account (i) and (ii), derived from
the calibration measurements presented in
Sec.~\ref{subsec:CZT_Calibration}, but not (iii). The projected count
distributions lead to the azimuthal scattering distributions shown in
the right panel of Fig.~\ref{fig:CHESS_2DAzimuthData}. In contrast to
the 2D distributions shown in planar pixel coordinates, the $\Phi$
positions and error bars now represent the proper angular coverage of
each pixel (compare with Fig.~\ref{fig:SchematicBeamOffset_AndStokes},
left). The expected $180^{\circ}$ modulation is clearly revealed, and
the reconstructed orientation of the polarization plane agrees with the
direction of the CHESS beam setup~-- confirming the functionality of the
X-Calibur polarimeter.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.99\textwidth]{CHESS_Example_AzimuthDistribution_CHESS1_E_32_6_42_0keV_Angle-1111deg} \\
\includegraphics[width=0.99\textwidth]{CHESS_Example_AzimuthDistribution_CHESS1_E_32_6_42_0keV_Angle0deg} \\
\end{center}
\caption{X-Calibur 2D scattering distributions of the $40 \, \rm{keV}$
beam at CHESS ($C_{\rm{ch}}^{3}$, see
Fig.~\ref{fig:X-Calibur_Configurations}, the beam enters from the top).
Events are shown for reconstructed energies of $36-40 \, \rm{keV}$. The
top row shows the results of simulations of a $100\%$ polarized beam.
The bottom row shows the results of the CHESS measurement. The (blue)
boxes in the 2D maps indicate the projected outline of the scintillator
(Compton-scatterer). {\bf Left:} Raw count map (pixel-by-pixel) of a
non-polarized beam (neither corrected for $\Delta \Phi_{j}$ nor for pixel
acceptance $a_{j}$). All four detector sides, {\it Bd0}-{\it Bd3}, are
unfolded into a plane. The detector rings {\it R1} to {\it R8} are
indicated. {\bf Second:} Count map of a raw measurement of a polarized
beam. {\bf Third:} Count map of the polarized beam, corrected for
$\Delta \Phi_{j}$ and $a_{j}$. {\bf Right:} Normalized azimuthal
scattering distribution (corrected for $\Delta \Phi_{j}$ and $a_{j}$)
for different detector rings. The vertical lines indicate the nominal
plane $\Omega_{\rm{n}}$ of the electric field vector of the polarized
beam.}
\label{fig:CHESS_2DAzimuthData}
\end{figure*}
{\it Simulated modulation factors.} Each detector ring (see
Fig.~\ref{fig:CHESS_2DAzimuthData}), or a sub ring thereof, can be seen
as an independent detector that allows to reconstruct energy-dependent
polarization properties of the incoming X-ray beam.
Figure~\ref{fig:MC_ModulationFactor} shows the simulated modulation
factors $\mu_{\rm{sim}}$ for the different energies and X-Calibur
configurations as a function of detector ring/depth, measured along the
optical axis. The values for $\mu_{\rm{sim}}$ were obtained using the
Stokes analysis, and are in agreement with the corresponding results
obtained from the analysis of the azimuthal scattering distribution (not
shown). The simulations shown reflect the detector status of the CHESS
measurements (dead pixels, energy resolutions, etc.), not an idealized
detector. Ring {\it R1} detects only back-scattered X-rays with a
correspondingly lower $\mu_{\rm{sim}}$~-- but with a better energy
resolution since the scattering kinematics are better defined. The
step-like structure in $\mu_{\rm{sim}}^{120\,\rm{keV}}$ between rings
{\it R5} and {\it R6} can be explained by the strong degradation in
energy resolution at high energies in the $2 \, \rm{mm}$ detectors
located at rings {\it R6}-{\it R8}. The highest modulation is achieved
in ring {\it R2}.
The simulations of the modulation factor were also done involving the
scintillator flag $f_{\rm{sci}}$, based on the two scintillator trigger
efficiencies as measured in
Fig.~\ref{fig:Scintillator_TriggerEfficiency}. This reflects the
efficiency during the CHESS measurements (not optimized), as well as the
efficiency of the optimized scintillator trigger. The scintillator
improves the modulation factor for most rings~-- by requiring a minimum
energy deposition $\Delta E_{\rm{sci}}$ which effectively limits the
allowed range of polar scattering angles. This, however, also reduces
the event statistics (not imprinted in $\mu_{\rm{sim}}$). A scintillator
trigger efficiency of $100 \%$ would again lead to the same distribution
of modulation factors as no cut on $f_{\rm{sci}}$ does. The main benefit
of the scintillator trigger lies in its ability to suppress background
during a measurement (see for example Fig.~\ref{fig:BG_Spectra}, left).
The modulation factor $\mu$, together with the detection rate after
background subtraction, are the crucial characteristics that determine
the MDP detection sensitivity of the polarimeter following
Eq.~(\ref{eqn:MDP}).
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{MC_ModulationFactors}
\end{center}
\caption{Simulated modulation factors $\mu_{\rm{sim}}$ as a function of
detector ring {\it R}, shown for different energies. The geometrical
positions (along the optical axis $y$) are indicated for the
scintillator and the CZT ring assembly. The simulated data were analyzed
with the Stokes method. Results are also shown for the scintillator
trigger requirement $f_{\rm{sci}}=1$, assuming the PMT trigger
efficiency during the CHESS measurements (PMT$_{\rm{ch}}$) and the
efficiency of the optimized PMT threshold (PMT+), see
Fig.~\ref{fig:Scintillator_TriggerEfficiency}.}
\label{fig:MC_ModulationFactor}
\end{figure}
{\it Polarization fraction of the CHESS beam.} The azimuthal scattering
distributions for each sub ring and each data set were fitted to
determine $\mu_{\rm{ch}}$. The simulated modulation factors
$\mu_{\rm{sim}}$ (Fig.~\ref{fig:MC_ModulationFactor}) were in turn used
to determine the polarization fraction $r$ of the CHESS beam using
Eq.~(\ref{eqn:PolFracPhiDistri}). This is shown in
Fig.~\ref{fig:AzDistri_RecoMu} for the $40 \, \rm{keV}$ beam for
different orientations $\alpha$ of the polarimeter relative to the
polarization plane. Also shown are the residuals between the
reconstructed polarization plane and the nominal polarization plane.
Note, that for large values of $\alpha$ the azimuthal scattering
distribution shows an additional global slope, possibly caused by an
$\alpha$-dependent tilt of the rotation fixture (see
Fig.~\ref{fig:AzimuthDistribution}, right, for an example orientation of
$\alpha = 70^{\circ}$). This will affect the reconstructed polarization
properties and is further discussed in Sec.~\ref{subsec:Systematics}. To
reduce this systematic effect, the CHESS azimuthal distributions were
folded back into the $[0;180]^{\circ}$ interval before the sinusoidal
function was fitted to the data points.
\begin{table}[t!]
\begin{tabular}{lrr}
Setup & $\Delta \Omega \, [\rm{deg}]$ & $r_{\rm{rec}} \, [\%]$ \\
\hline \hline
\noalign{\smallskip}
\multicolumn{3}{l}{{beam energy: $40 \, \rm{keV}$}} \\
\hline
$C_{\rm{ch1}}$ & $-4.7 \pm 1.9$ & $91.0 \pm 0.6$ \\
& $-4.6 \pm 1.7$ & $88.6 \pm 0.4$ \\
\noalign{\smallskip}
\multicolumn{3}{l}{{beam energy: $80 \, \rm{keV}$}} \\
\hline
$C_{\rm{ch1}}$ & $-2.6 \pm 1.7$ & $83.5 \pm 0.8$ \\
& $-2.0 \pm 2.1$ & $78.3 \pm 0.8$ \\
$C_{\rm{ch1}}$, $f_{\rm{sci}} = 1$ & $-2.1 \pm 2.5$ & $86.5 \pm 1.1$ \\
& $-2.1 \pm 2.7$ & $82.8 \pm 1.0$ \\
$C_{\rm{ch4}}$ & $-2.5 \pm 1.8$ & $90.4 \pm 0.5$ \\
& $-2.0 \pm 2.2$ & $85.5 \pm 0.6$ \\
\noalign{\smallskip}
\multicolumn{3}{l}{{beam energy: $120 \, \rm{keV}$}} \\
\hline
$C_{\rm{ch1}}$ & $-0.5 \pm 2.9$ & $89.2 \pm 1.5$ \\
& $-0.2 \pm 3.3$ & $71.5 \pm 1.3$ \\
$C_{\rm{ch1}}$, $f_{\rm{sci}} = 1$ & $-0.7 \pm 4.6$ & $86.3 \pm 1.2$ \\
& $-0.8 \pm 3.7$ & $80.5 \pm 1.5$ \\
\end{tabular}
\caption{Residuals between the reconstructed polarization plane
$\Omega_{\rm{p}}$ and the nominal plane at $\Omega_{\rm{n}} =
90^{\circ}$: $\Delta \Omega = (\Omega_{\rm{n}}-\alpha) -
\Omega_{\rm{p}}$. The second column shows the reconstructed polarization
fractions $r_{\rm{rec}}$ of the CHESS beam at different energies and
different configurations/setups (partly with the $f_{\rm{sci}} = 1$
requirement). See Figs.~\ref{fig:AzDistri_RecoMu} and
\ref{fig:CHESS_MorePolFractions} for an illustration. The first row per
data set shows the results obtained from the analysis of the azimuthal
$\Phi$-scattering distribution following
Eq.~(\ref{eqn:PolFracPhiDistri}). The second row represents the results
obtained from the Stokes analysis following
Eq.~(\ref{eqn:PolarizationFromStokes}). The results are averaged over
all orientations $\alpha$, each.}
\label{tab:CHESS_PolFrac}
\end{table}
Figure~\ref{fig:CHESS_MorePolFractions} shows the corresponding
polarization fractions reconstructed from the $80 \, \rm{keV}$ harmonic.
The reconstructed polarization fractions obtained from all measurements
are summarized in Tab.~\ref{tab:CHESS_PolFrac}~-- in all cases averaged
over all measured orientations $\alpha$. Table~\ref{tab:CHESS_PolFrac}
also shows the results derived from the Stokes analysis following
Eq.~(\ref{eqn:PolarizationFromStokes}). Both methods are in reasonable
agreement, whereas the Stokes analysis seems to slightly underestimate
$r$ compared to the results obtained with the azimuthal scattering
distribution. The systematic error on the reconstructed polarization
fraction was estimated as described in Sec.~\ref{subsec:Systematics}.
The polarization of the CHESS beam as reconstructed using the
$\Phi$-distribution method is measured to be $r_{\rm{ch}}^{\Phi} = (87.8
\pm 0.4_{\rm{stat}} \pm 4.6_{\rm{sys}}) \, \%$ with no indication of an
energy dependence in the $40-120 \, \rm{keV}$ range. The corresponding
value reconstructed from the Stokes analysis is $r_{\rm{ch}}^{\rm{st}} =
(81.2 \pm 0.4_{\rm{stat}} \pm 10_{\rm{sys}}) \, \%$. The difference in
the systematic error is explained by the fact that the Stokes analysis
is more sensitive to the detector configuration (which was not perfectly
constrained during the CHESS measurements). A better defined geometry
and the rotation of the polarimeter will reduce the systematic error for
measurements performed during the balloon flight.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{PhiDistriFitResult_CHESS1_E_32_6_42_0keV}
\end{center}
\caption{Reconstructed polarization properties of the CHESS beam at $40
\, \rm{keV}$ (configuration $C_{\rm{ch1}}$). {\bf Top:} Modulation
factor $\mu$ versus polarimeter sub rings {\it Ri$_{\rm{t,b}}$}. The
results were derived from simulations (MC), as well as from data taken
under different X-Calibur orientations $\alpha$. For each $\alpha$ and
each ring, $\mu_{\rm{ch}}$ is derived from fits to acceptance corrected
$\Phi$-distributions (see Fig.~\ref{fig:AzimuthDistribution}, left). The
scintillator starts covering detector rings at $y \geq 1.8 \, \rm{cm}$.
The green error band reflects the $1 \, \rm{std.dev}$ range of measured
points in the particular $y$ slice. {\bf Middle:} Reconstructed
polarization fraction following Eq.~(\ref{eqn:PolFracPhiDistri}). The
green band reflects the $1 \, \rm{std.dev}$ range of all reconstructed
fractions in the corresponding $y$ slice. {\bf Bottom:} Residual between
reconstructed polarization plane $\Omega_{\rm{p}}$ and true polarization
plane $\Omega_{\rm{n}} = 90^{\circ}$, corrected for the X-Calibur
orientation $\alpha$: $\Delta \Omega = (\Omega_{\rm{n}} - \alpha)
-\Omega_{\rm{p}}$.}
\label{fig:AzDistri_RecoMu}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{PhiDistriFitResult_PolFrac_CHESS4_E_58_9_82_0keV} \\
\end{center}
\caption{Reconstructed polarization fractions of the CHESS beam (compare
with Fig.~\ref{fig:AzDistri_RecoMu}) derived from the $80 \, \rm{keV}$
harmonic (configuration $C_{\rm{ch3}}$).}
\label{fig:CHESS_MorePolFractions}
\end{figure}
{\it Polarization plane of the CHESS beam.}
Table~\ref{tab:CHESS_PolFrac} also summarizes the average reconstructed
polarization planes $\Omega_{\rm{p}}$ (corrected for the X-Calibur
orientation $\alpha$). Note, that the uncertainty in setting the
X-Calibur orientation was estimated to be $\Delta \alpha = \pm
2^{\circ}$. Within this systematic error, as well as the statistical
errors, the average of reconstructed polarization planes (all rings and
all X-Calibur orientations $\alpha$) is compatible with the nominal
plane $\Omega_{\rm{n}}$ of the CHESS beam. However, the measurements
consistently reconstruct $\Delta \Omega < 0^{\circ}$, which may indicate
a slight offset in the way the rotation mechanism was
installed/calibrated. The reconstructed polarization planes shown in
Fig.~\ref{fig:AzDistri_RecoMu} indicate a slight dependence on $\alpha$
and $y$. The left panel in Fig.~\ref{fig:AzimuthDistribution} shows a
corresponding azimuthal distribution measured with the X-Calibur
orientation of $\alpha = 70^{\circ}$, revealing a slight asymmetry. This
result can be seen as another indication of a slight mis-alignment
(offset and/or tilt) between the rotation axis of the scintillator and
the optical axis of the X-ray beam. A more detailed discussion of this
kind of systematic effect can be found in Sec.~\ref{subsec:Systematics}.
\subsection{Measurements with the X-Calibur/{\it InFOC$\mu$S}\xspace Assembly}
\label{subsec:FtSumnerData}
The full X-Calibur/{\it InFOC$\mu$S}\xspace experiment was assembled and tested during a
flight preparation campaign at the NASA {\it Columbia Scientific Balloon
Facility} (CSBF) site in Ft.~Sumner, NM, in the fall of 2014. The
polarimeter was installed in the rotating CsI shield assembly, which in
turn was installed in the pressure vessel as part of the {\it InFOC$\mu$S}\xspace X-ray
telescope (see Fig.~\ref{fig:ActiveShieldAndTelescope}). This setup,
$C_{\rm{ft}}$, is illustrated in
Fig.~\ref{fig:X-Calibur_Configurations}. The CZT detector configuration
of these measurements is listed in Tab.~\ref{tab:Detectors} and differs
slightly from the one used in the CHESS measurements (described in
Sec.~\ref{subsec:CHESS}). The X-ray mirror was installed on the optical
bench located at the front part of the {\it InFOC$\mu$S}\xspace telescope. The pressure
vessel is located at a focal distance of $8 \, \rm{m}$.
Before the balloon flight, a beryllium (Be) window is installed in the
top dome of the pressure vessel, to assure a pressure-sealed entrance to
the polarimeter with high transmissivity to hard X-rays. For the
ground-based measurements, presented in this section, the Be window was
not installed to allow to visually align the polarimeter with the
optical axis of the X-ray mirror. Throughout the measurements presented
in this section, the polarimeter was rotated at $4 \, \rm{rpm}$. Due to
the rotation, the application of the pixel acceptances $a_{j}$, defined
in Eq.~(\ref{eqn:Acceptance}), is no longer required, since each pixel
tests the complete, time-averaged azimuthal scattering range with
respect to the polarization plane. As another consequence, the azimuthal
binning can be chosen finer than in the non-rotating system in which it
was limited to the number of $32$ CZT pixels per row. For each event,
the hit pixel is de-rotated into the laboratory/horizon coordinate frame
(see Fig.~\ref{fig:SchematicBeamOffset_AndStokes}, left) to determine
the azimuthal scattering angle $\Phi$.
\begin{figure*}[t!]
\begin{center}
\includegraphics[height=0.32\textheight]{FtSumner_MirrorScanSetup}
\hfill
\includegraphics[height=0.32\textheight]{FtSumnerMirrorScan2D_Rotating}
\hfill
\includegraphics[height=0.32\textheight]{FtSumnerPolFraction}
\end{center}
\caption{{\it InFOC$\mu$S}\xspace/X-Calibur X-ray mirror scan, configuration
$C_{\rm{ft}}$ in Fig.~\ref{fig:X-Calibur_Configurations}. {\bf Left:}
The collimated X-ray source is aligned with the optical axis of the
mirror and the polarimeter which is situated in the focal plane at a
distance of $8 \, \rm{m}$ (not visible in this photograph). The X-ray
source can be automatically moved in the $X$/$Z$ plane to scan the
mirror with the X-rays traveling along $Y$. {\bf Middle:} The
Compton-scattered event distribution measured with X-Calibur (horizon
system, de-rotated) during the scan in the energy range of $30-50 \,
\rm{keV}$. {\bf Right:} Background subtracted measurements of the
collimated X-ray beam. Three data runs where taken with (i) the beam
hitting the center of the scintillator ($\Delta d = 0 \, \rm{mm}$), (ii)
the beam hitting at an offset of $\Delta d = -4 \, \rm{mm}$, and (iii)
the beam scanning the X-ray mirror. The top panel shows the
Compton-scattered energy spectra (all-detector average, each). The
bottom panel shows the energy-dependent polarization fraction
reconstructed from the azimuthal $\Phi$-scattering distribution, as well
as using the Stokes parameters. The given errors are statistical only.}
\label{fig:FtSumner_MirrorScan}
\end{figure*}
{\it X-ray mirror/X-Calibur alignment.} The alignment of the polarimeter
and the mirror are crucial in order to gain the maximal sensitivity for
polarization measurements and to reduce systematic effects. The optical
axis and the on-axis image location of the X-ray mirror were measured by
a CCD camera with an optical parallel beam. The X-ray mirror was placed
in the optical beam, and its tip and tilt were adjusted such that the
optical axis is parallel to the optical beam. Then, the CCD camera was
placed at the center of the mirror, looking along the optical axis
facing the polarimeter, and taking a picture of the on-axis image plane.
The image location was recorded in pixel coordinates, which determines
the optical axis as well as the on-axis image position. After the X-ray
mirror was installed onto the optical bench (see
Fig.~\ref{fig:FtSumner_MirrorScan}, left), a picture of the front
surface of the polarimeter's scintillator was taken by the camera. The
mirror tip and tilt were adjusted by shimming at the interface to the
optical bench, such that the center of the scintillator is at the
recorded location of the on-axis image of the X-ray mirror. An alignment
between the scintillator and mirror of better than $1\, \rm{mm}$ was
achieved which guarantees a systematic error on the reconstructed
polarization fraction of less than $2 \%$ (see
Sec.~\ref{subsec:Systematics}).
{\it Mirror scan.} The proper alignment of the telescope is
tested/confirmed in a mirror scan. A movable X-ray source scans the
surface of the mirror while the polarimeter response is measured. The
X-ray scanning system consists of an X-ray source with a $60\, \rm{cm}$
long collimator, tip and tilt stages, and $X$/$Z$ travel stages to which
the X-ray source is mounted. The X-ray source was positioned in front of
the X-ray mirror and could be used to either directly illuminate the
polarimeter (through the central hole in the X-ray mirror), or to scan
the whole mirror aperture. The setup is shown in
Fig.~\ref{fig:FtSumner_MirrorScan}, left. An Oxford 5011 electron impact
X-ray tube is used to generate the X-rays, with an active source spot
size of $0.05 \, \rm{mm}$ in diameter (molybdenum target, Mo). The
produced X-ray spectrum is made of emission line (Mo-K) as well as
bremsstrahlung. The X-ray source provides a continuum spectrum up to $50
\, \rm{keV}$. The current was adjusted to give a reasonable event rate
at the polarimeter located at $8 \, \rm{m}$ distance (see
Tab.~\ref{tab:PixelRates} for reference). The collimator can be equipped
with two interchangeable pin holes with diameters of $0.1 \, \rm{mm}$
and $1 \, \rm{mm}$, respectively. The beam size at $8 \, \rm{m}$
distance is around $2 \, \rm{mm}$ in diameter for the $0.1 \, \rm{mm}$
pin hole. The $1 \, \rm{mm}$ pin hole produces count rates in the
polarimeter above one kHz at $8 \, \rm{m}$ distance ($20-50 \,
\rm{keV}$, after air absorption). The tip and tilt stages change the
direction of the collimated X-ray beam. The $X$/$Z$ translation stages
allow the X-ray beam to scan over the entire aperture of the mirror
(with the central hole in the mirror blocked by a lead absorber). The
X-ray beam was aligned with the optical axis of the mirror. The $1 \,
\rm{mm}$ pin hole was used to scan the mirror within $51$ horizontal
rows along $X$. The response measured with X-Calibur is shown in
Fig.~\ref{fig:FtSumner_MirrorScan} (middle) in the de-rotated coordinate
system (the horizon intersects at $90^{\circ}$ and $270^{\circ}$ in this
representation). The corresponding energy spectrum is shown in
Fig.~\ref{fig:FtSumner_MirrorScan}, right. The results illustrate that
the mirror was successfully aligned with the polarimeter.
{\it Direct beam illumination.} As a reference measurement to the mirror
scan, data were taken with the polarimeter being directly illuminated by
the collimated X-ray source ($0.1 \, \rm{mm}$ pin hole). The X-rays
enter the scintillator along its optical axis, but do not pass the X-ray
mirror in this measurement. The spectrum of the mirror scan shown in
Fig.~\ref{fig:FtSumner_MirrorScan} (right, top) drops off faster
compared to the spectrum measured from the direct beam illumination.
This is a result of the energy-dependent effective area of the mirror.
An additional run was taken with the X-ray beam hitting the scintillator
off center by $\Delta d = -4 \, \rm{mm}$, which is discussed in
Sec.~\ref{subsec:Systematics}.
The polarization parameters were reconstructed in the same way as
described in Sec.~\ref{subsec:CHESS} (except for the acceptance
correction $a_{j}$). The depth-dependent modulation factors (see
Fig.~\ref{fig:MC_ModulationFactor}) obtained from the $40 \, \rm{keV}$
CHESS simulation were used to reconstruct the polarization fraction of
the X-ray beam in different energy bands (assuming $\mu_{\rm{sim}}$
being independent of energy in the $20-50 \, \rm{keV}$ band). The
reconstruction was done with both methods described in
Sec.~\ref{subsec:AnalysisPolarization}, namely using the analysis of the
azimuthal scattering distribution following
Eq.~(\ref{eqn:PolFracPhiDistri}), as well as using the Stokes parameters
following Eq.~(\ref{eqn:PolarizationFromStokes}). Both methods yield
comparable results which are shown in Fig.~\ref{fig:FtSumner_MirrorScan}
(right, bottom). An energy dependent polarization fraction is found in
the data. However, in contrast to the mono-energetic CHESS beam
(Sec.~\ref{subsec:CHESS}), the energy distribution of the incoming
X-rays (before Compton-scattering in the scintillator) follows a
continuum. To properly reconstruct the polarization fraction as a
function of incoming X-ray energy, one would have to utilize
reconstruction methods of forward folding or unfolding
\cite{XCLB_LogLikelihoodAnalysis}, which is beyond the scope of this
paper. However, a significant increase in polarization fraction can be
observed with increasing energy. The data of the mirror scan was also
used to reconstruct the energy-dependent polarization fraction, and is
found to be in reasonable agreement with the direct beam measurement.
This confirms the predictions by \citet{Katsuta2009} that an X-ray
mirror does not substantially affect the polarization properties.
\subsection{Systematic Effects in Polarization Measurements}
\label{subsec:Systematics}
\begin{figure*}[t!]
\begin{center}
\includegraphics[height=0.285\textheight]{AzimuthDistribution_OffsetsUnpolarized}
\hfill
\includegraphics[height=0.285\textheight]{PolFracSystError}
\end{center}
\caption{{\bf Left:} Azimuthal scattering distributions of a
non-polarized X-ray beam (CHESS data and simulations, both in a
non-rotating system) hitting the scintillator at different offsets $d$
relative to the optical axis. X-Calibur was oriented at $\alpha =
0^{\circ}$, with the beam `walking' from {\it Bd1} to {\it Bd3} with
increasing $\Delta X$ (see Fig.~\ref{fig:SchematicBeamOffset_AndStokes},
left, for a coordinate system). Scattering distributions are shown with
and without the offset correction (see
Sec.~\ref{subsec:AnalysisPolarization}). {\bf Right:} Apparent
polarization fraction $r$ of the non-polarized beam. The light blue region
indicates the diameter of the scintillator rod. The data were analyzed
using the Stokes parameters as described by
Eq.~(\ref{eqn:Stokes_Q_U_Average}). Results are shown without and with
the corrected beam offset.}
\label{fig:PolFracSysError}
\end{figure*}
The understanding and control of systematic effects of different nature
is crucial for the correct reconstruction of the polarization properties
from measured data. A series of measurements was performed at CHESS (see
Sec.~\ref{subsec:CHESS}) to study the effects of a mis-alignment
(offset) between the X-ray beam and the optical axis of the polarimeter.
The $X$/$Z$ stage of the table in hutch {\it C1} was used to
systematically scan the polarimeter response for offsets ranging from
$-5 \, \rm{mm}$ to $+5 \, \rm{mm}$ in steps of $1 \, \rm{mm}$ (the beam
comes in along $Y$, see Figs.~\ref{fig:X-Calibur_Configurations} and
\ref{fig:SchematicBeamOffset_AndStokes} (left) for the definition of the
coordinate system). The scan along $X$ was performed with a polarimeter
orientation of $\alpha = 0^{\circ}$, whereas the scan along $Z$ was
performed with $\alpha = -90^{\circ}$. This allowed us to pairwise
superimpose the data runs of perpendicular polarization planes that hit
the scintillator at the same position~-- being equivalent to a
non-polarized beam hitting at that particular offset position. The data
runs were taken with configuration $C_{\rm{ch4}}$, testing the response
of the $80 \, \rm{keV}$ harmonic of the CHESS X-ray beam. A set of
simulations with an $80 \, \rm{keV}$ beam (polarized and non-polarized)
were performed resembling the same offsets as measured at CHESS, as well
as offset simulations at $40 \, \rm{keV}$.
{\it Non-polarized beam.} The left panel of
Figure~\ref{fig:PolFracSysError} shows examples of azimuthal scattering
distributions measured with different beam offsets for a non-polarized
beam. The distributions were corrected for $\Delta \Phi_{j}$ (assuming a
central beam) and $a_{j}$, as before. It can be seen that beam offsets
of $\simeq 2 \, \rm{mm}$ already introduce systematic asymmetries.
Although the offset distributions are not flat, as expected for a
non-polarized beam, they also do not resemble the shape expected from a
polarized beam. Given the high event statistics of the data used in the
study, the sinusoidal fits used to determine the polarization properties
will therefore certainly fail~-- allowing one to detect/filter the
systematic effect. However, in the case of data with higher statistical
uncertainty, the observed asymmetries could potentially lead to the
reconstruction of an artificial polarization fraction of $r_{\rm{rec}} >
0$. Note, that folding the distributions into the $[0; 180]^{\circ}$
interval (not shown), will substantially reduce the asymmetry. The
Stokes analysis is per definition only considering the $[0;
180]^{\circ}$ interval. However, it is not sensitive to deviations from
the sinusoidal scattering distribution, and therefore does not provide a
goodness-of-fit measure that can be used to detect imprints of a beam
offset. Figure~\ref{fig:PolFracSysError} also shows examples of
distributions that were offset corrected following the description in
Sec.~\ref{subsec:AnalysisPolarization}. This, however, assumes that the
beam offset is known.
\begin{figure*}[t!]
\begin{center}
\includegraphics[height=0.285\textheight]{AzimuthDistribution_OffsetsPolarized}
\hfill
\includegraphics[height=0.285\textheight]{PolFracSystErrorPolarized}
\end{center}
\caption{{\bf Left:} Azimuthal scattering distributions of polarized
X-ray beams hitting the scintillator at different offsets $d$. Results
are shown for the CHESS data ($r_{\rm{ch}} \simeq 85\%$ polarized) and
simulations ($r = 100\%$ polarized), both in a non-rotating system.
Distributions corrected for the beam offset (see
Sec.~\ref{subsec:AnalysisPolarization}) are shown, as well. Results are
shown for offsets along $X$ ($\alpha = 0^{\circ}$) and $Z$ ($\alpha =
-90^{\circ}$, corresponding to a scan along $X$ with perpendicular
polarization). {\bf Right:} Reconstructed polarization fraction $r$ for
the off-center beams, analyzed using the Stokes parameters as in
Eq.~(\ref{eqn:Stokes_Q_U_Average}). The outline of the scintillator is
indicated. Also shown are data derived from the Ft.~Sumner measurement
shown in Fig.~\ref{fig:FtSumner_MirrorScan}, right. Results are shown
without and with the offset correction.}
\label{fig:PolFracSysErrorPolarized}
\end{figure*}
To quantify the artificial polarization $r_{\rm rec}$ inferred when
neglecting the beam offset, the data were analyzed using the Stokes
parameters following Eq.~(\ref{eqn:Stokes_Q_U_Average}).
Figure~\ref{fig:PolFracSysError} (right) shows $r_{\rm{rec}}$ as a
function of beam offset. The results are shown for the simulations as
well as for the CHESS data~-- both being in reasonable agreement. Also
shown is an analytic model that simply integrates the events per
azimuth angle interval in a plane perpendicular to the optical axis. The
right panel of Fig.~\ref{fig:PolFracSysError} shows the same
distribution after the beam offset correction described in
Sec.~\ref{subsec:AnalysisPolarization} was applied. The discrepancy
between the data and simulations after correction can be explained by an
uncertainty in the absolute beam alignment in the data and the fact
that, in contrast to the simulations, two separate measurements had to
be superimposed to generate the non-polarized beam, amplifying the effect
of unaccounted offsets or mis-alignments (any uncertainty in the offset
not only moves the data points along the beam offset axis, but also
along the $r$ axis).
In general, however, the first-order correction procedure greatly
reduces the systematic effect to less than a few percent for offsets $d
\leq 3 \, \rm{mm}$. The simulated data shown in the right panel of
Fig.~\ref{fig:PolFracSysError} suggest that the correction works better
for beam energies of $80 \, \rm{keV}$ as compared to $40 \, \rm{keV}$.
This can be explained by the fact that only geometrical offsets are
corrected for. Differences in absorption lengths in the scintillator
material, originating from positions other than $P_{0}$ (see left panel
in Fig.~\ref{fig:SchematicBeamOffset_AndStokes}), will cause
second-order effects that depend on the energy of the scattered X-ray.
However, addressing these additional correction terms is beyond the
scope of this paper.
{\it Polarized beam.} For a polarized beam (left panel in
Fig.~\ref{fig:PolFracSysErrorPolarized}), the offsets lead to either a
reduction or amplification of the reconstructed polarization fraction,
depending on the angle between the offset vector $\bf{d}$ and the plane
of polarization $\Omega_{\rm{p}}$. The contribution of the beam offset
to the reconstructed polarization is illustrated in the right panel of
Fig.~\ref{fig:PolFracSysErrorPolarized} for different energies,
including the results for the offset corrected analysis. The correction
substantially reduces the systematic effect on $r$. A diverging trend
can be identified in the corrected CHESS data for $d > 0 \, \rm{mm}$
(which is also visible in the case of the `non-polarized' beam shown in
the right panel of Fig.~\ref{fig:PolFracSysError}). This indicates an
inaccuracy in the experimental setup, e.g. the asymmetric/fractional
scattering off a fixture along the beam path before entering the
polarimeter.
An additional offset measurement was performed during the flight
preparation in Ft.~Sumner (see Sec.~\ref{subsec:FtSumnerData}) using the
partly polarized beam of the collimated X-ray source, hitting the
scintillator at an offset of $d = -4 \, \rm{mm}$ (horizon system). The
polarimeter/shield assembly was continuously rotating during the
measurements. The corresponding fraction of apparent polarization $r$ is
indicated in Fig.~\ref{fig:PolFracSysErrorPolarized} (right). Due to the
rotation of the polarimeter, the offset correction (see
Sec.~\ref{subsec:AnalysisPolarization}) was calculated on an
event-by-event basis. The results illustrate that the correction also
works in a rotated system, canceling the artificial fraction of
polarization introduced by the offset. The offset-corrected polarization
spectrum is shown in Fig.~\ref{fig:FtSumner_MirrorScan} (right),
reproducing the measured results obtained with no beam offset.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.49\textwidth]{Reco_MeanX}
\end{center}
\caption{Correlation between true beam offset and beam offset measured
following Eq.~(\ref{eqn:FirstMoments}) for data and simulations.}
\label{fig:Offset_RecoMeanX}
\end{figure}
{\it Determination of the beam offset.} The geometrical offset
correction described above assumes that the offset vector $\bf{d}$ is
known. For sufficient event statistics and the assumption of a
time-independent beam offset, the offset can be estimated from the data
itself using first moments, essentially substituting $2 \Phi_{k}
\rightarrow \Phi_{k}$ in Eq.~(\ref{eqn:StokesParams}):
\begin{eqnarray} \begin{split} \left< x \right> & = - \frac{c}{W}
\sum_{k=1}^{N} w_{j(k)} \sin(\Phi_{k}), \\ \left< z \right> & =
\frac{c}{W} \sum_{k=1}^{N} w_{j(k)} \cos(\Phi_{k}), \\ W & =
\sum_{k=1}^{N} w_{j(k)}. \end{split} \label{eqn:FirstMoments}
\end{eqnarray} Here, the constant $c=11 \, \rm{mm}$ reflects the
distance between the scintillator center and the detector plane (see
Fig.~\ref{fig:X-Calibur_Configurations}, top). The weights $w_{j(k)}$
are the same as in Eq.~(\ref{eqn:StokesSum}). As shown in
Fig.~\ref{fig:Offset_RecoMeanX}, the measured offsets are linearly
correlated with the true beam offsets. However, the slope of the
correlation depends to some extent on the true polarization fraction $r$
of the beam and the angle between $\Omega_{\rm{p}}$ and {\bf d}. Since
the offset of a non-polarized beam itself mimics a polarization fraction,
there will be a residual ambiguity that prevents to completely
disentangle $\bf{d}$ and $r$ from the data alone. Therefore, a
time-resolved external monitoring of the beam position during the
balloon flight is preferable.
Figure~\ref{fig:Offset_RecoMeanX} reveals a slight shift/translation
between the slope of the reconstructed offset measured in the CHESS data
versus the slope obtained from the simulations. This can be used to
estimate an accuracy of the alignment achieved during the CHESS
measurements to be $\Delta X_{\rm{sys}} \simeq 0.3 \, \rm{mm}$ for the
X-Calibur orientation of $\alpha = 0^{\circ}$. Note, with a rotation
axis possibly not exactly aligned with the axis of the scintillator,
this may translate into larger offsets $\Delta X$ for different
X-Calibur orientations $\alpha$.
{\it Systematic error on the polarization fraction.} The systematic
error on the reconstructed polarization fraction $\Delta r_{\rm{sys}}$
can be estimated as follows. For the polarization measurements presented
in this paper (Sec.~\ref{subsec:CHESS} and \ref{subsec:FtSumnerData}),
we assume an unaccounted beam/scintillator mis-alignment of $\Delta d = 1
\, \rm{mm}$. For a highly-polarized beam of $r = O(100\%)$ this leads to
an error of $\Delta r_{\rm{sys,\Delta d}} \simeq 2\%$ relative to the
reconstructed on-axis beam (see simulated curve in
Fig.~\ref{fig:PolFracSysErrorPolarized}). For a non-polarized beam
(simulations in Fig.~\ref{fig:PolFracSysError}) the offset leads to an
overestimation of $\simeq 2\%$. For simplicity, we describe the
systematic error introduced by the beam offset as independent of the
true polarization fraction: $\Delta r_{\rm{sys,\Delta d}} = 2\%$.
The X-Calibur version used during the CHESS and Ft.~Sumner measurements
(see Secs.~\ref{subsec:CHESS} and \ref{subsec:FtSumnerData}) allowed for
some flexibility in adjusting the distance between the plane of the CZT
detectors and the optical axis when assembling the polarimeter (see
Fig.~\ref{fig:X-Calibur_Configurations}, top: $c = 11 \, \rm{mm}$). We
measured $c_{\rm{ch}} = (11.3 \pm 0.5) \, \rm{mm}$ for the CHESS setup
and $c_{\rm{ft}} = (10.5 \pm 0.5) \, \rm{mm}$ for the Ft.~Sumner setup.
Conservatively, we assume $c = (11 \pm 1) \, \rm{mm}$. To study how an
increased/decreased value of $c$ affects $r_{\rm{rec}}$, we
correspondingly shifted pixel coordinates when analyzing a data set, but
determined $r_{\rm{rec}}$ with the modulation factor $\mu_{\rm{sim}}$
obtained from simulations with the nominal value of $c = 11 \, \rm{mm}$.
For $\Delta c = \pm 1 \, \rm{mm}$, a $100\%$ polarized beam is
reconstructed to be $r_{\rm{rec}} = r_{-0.5\%}^{+1\%}$ (overestimation
for a reduced distance $c$). For a non-polarized beam the effect is
estimated to be $<0.1\%$. We therefore assume that the distance related
systematic error\footnote{Note, that asymmetric distances of the
different detector sides will mimic beam offsets with potentially
stronger effects.} is proportional to $r$ with $\Delta r_{\rm{sys,c}} =
0.01 \, r_{\rm{rec}}$. In the case of the Stokes analysis, the
uncertainty in $c$ will introduce another systematic error: the angular
coverage of the detector gaps, corrected for by
Eq.~(\ref{eqn:StokesGapCorrection}), will be over/underestimated. For a
$100\%$ polarized beam and $\Delta c = -1 \, \rm{mm}$, we find maximal
deviation of $\Delta r_{\rm{sys,c}}^{\rm{Stk}} = \pm 6\%$, and for
$\Delta c = +1 \, \rm{mm}$ we find $\Delta r_{\rm{sys,c}}^{\rm{Stk}} =
\pm 4.5\%$, with the sign and strength depending on the orientation of
the detector planes relative to the polarization vector (see
Sec.~\ref{subsec:AnalysisPolarization}).
An error introduced by the analysis procedure can be estimated by
comparing the results obtained with the two analysis methods presented
in Sec.~\ref{subsec:AnalysisPolarization}. No significant differences
were found when analyzing simulated data of a non-polarized beam. Based
on the modulation factors of a $100\%$ polarized beam, see
Eq.~(\ref{eq:ModulationFactor}), we estimate the systematic error to be
on the order of $2\%$. Therefore, a relative error of $\Delta
r_{\rm{sys,a}} = 0.02 \, r_{\rm{rec}}$ is assumed. Reasons for the
difference can be the different treatment of dead pixels, detector gaps,
etc. (see Sec.~\ref{subsec:AnalysisPolarization}). The total systematic
error on the polarization fraction is:
\begin{eqnarray}
\begin{split}
\Delta r_{\rm{sys}} & = \Delta r_{\rm{sys,\Delta d}} + \Delta
r_{\rm{sys,c}} + \left( \Delta r_{\rm{sys,c}}^{\rm{Stk}} \right) +
\Delta r_{\rm{sys,a}} \\ & = 2\% + 0.01 \, r_{\rm{rec}} + \left(\Delta
r_{\rm{sys,c}}^{\rm{Stk}} \right)+ 0.02 \, r_{\rm{rec}}.
\end{split}
\label{eqn:SystErrorPolFrac}
\end{eqnarray}
This error is estimated and valid for the particular measurements
presented in this paper. A better alignment in future measurements, a
rotating polarimeter, and a more detailed study of the analysis methods
will allow one to further reduce the error.
{\it Second-order systematic effects.} The tilt of the scintillator axis
with respect to the X-ray beam will introduce another (depth-dependent)
systematic effect. It should be mentioned that the simulations presented
in this section assume a point-like X-ray beam, whereas the CHESS X-ray
beam had a square-shaped footprint with side lengths of the order of $1
\, \rm{mm}$. The systematic effects introduced by the offset/tilt might
be weakened by properly taking into account the point spread function of
the X-ray mirror that will be used for the astrophysical observations.
The temperature dependence of the energy calibration of the CZT
detectors (Sec.~\ref{subsec:TempStudies}), if not corrected for, can
also modestly affect the reconstructed modulation factors of the data if
the measurements are performed at temperatures other than the
calibration temperature. However, the study of the effects discussed in
this paragraph are beyond the scope of this paper.
\subsection{Summary of the Polarimeter Performance}
We used measurements at the CHESS X-ray beam facility to calibrate the
polarimeter. CHESS gives a highly polarized and well collimated beam
with well defined photon energies. We were able to demonstrate the full
functionality of X-Calibur and measured the beam polarization to be
$r_{\rm{ch}} = (88 \pm 5)\%$. In addition, we used a collimated X-ray
beam from an X-ray source to make end-to-end tests of the full
{\it InFOC$\mu$S}\xspace/X-Calibur assembly in the field.
When uncorrected for, the $\sim$$3\%$ of defect pixels and gaps between
detector boards produce an apparent polarization of a non-polarized beam
of up to $5\%$ (Stokes analysis only). After correction, the apparent
polarization goes down to $< 2 \%$. Once we rotate the polarimeter, we
expect that the systematic effect is reduced by a factor $\sim
\sqrt{n}$, with $n$ being the number of detected events. The systematic
error will then be much smaller than the statistical error.
The detection principle of X-Calibur requires to focus the X-rays onto
the center of the scatterer. If uncorrected for, an offset of $1 \,
\rm{mm}$ leads to an apparent polarization of $\simeq 2\%$ for a
non-polarized beam. For the upcoming balloon flight, our goal is to limit
the offset of the focal point from the center of the scatterer to $< 1
\, \rm{mm}$, and to monitor the offset with a backward looking camera
located close to the X-ray mirror. The camera will monitor a LED cross
hair. We can in addition use X-ray data to constrain the location of the
focal spot. The systematic error on the polarization fraction after
correcting for the offset of the focal point is $< 1 \%$. Thus, for the
upcoming balloon flight, we estimate that X-Calibur can detect $\geq
3~\%$ polarization fractions in the $20 - 80 \, \rm{keV}$ band.
\section{Summary and Conclusions} \label{sec:Conclusion}
We designed, optimized, and built an X-ray polarimeter, X-Calibur, and
studied its performance and sensitivity. The fully assembled X-Calibur
polarimeter was tested (i) in the laboratory at Washington University,
(ii) with a polarized X-ray beam at the Cornell High Energy Synchrotron
Source (CHESS), and (iii) during a X-Calibur/{\it InFOC$\mu$S}\xspace flight-integration
test in Ft.~Sumner, NM. X-Calibur makes use of the fact that polarized
photons Compton scatter preferentially perpendicular to their electric
field orientation. It combines a detection efficiency on the order of
$80 \%$, with a high modulation factor of $\mu \approx 0.5$ averaged
over the whole detector assembly, and with values up to $\mu \approx
0.7$ for select subsections of the polarimeter. Operated in a mode of
continuous rotation, X-Calibur allows for a good control over systematic
effects. Scattering polarimetry has the strength that it can operate
over a wide energy range. The low energy threshold is given by the
competition between photoelectric absorption and scattering processes,
the mirror reflectivity limits the sensitivity at high energies.
We calibrated all 2048 CZT detector pixels of the polarimeter and
studied their performance with respect to their energy threshold, energy
resolution (including the contribution of electronic readout noise), and
trigger efficiency. The CZT detectors achieve a mean energy threshold of
$21 \, \rm{keV}$ and a mean $40 \, \rm{keV}$ energy resolution of
$\Delta E_{\rm{czt}} \approx 4 \, \rm{keV}$ FWHM. The
temperature-dependencies of the pixel responses were studied, as well,
and we found that the effects can be sufficiently controlled for the
temperature range expected during a balloon flight.
We characterized the performance of the active CsI shield and find a
background suppression by more than one order of magnitude in the energy
range relevant for X-Calibur. We also measured the trigger efficiency of
the scintillator that is used as Compton scatterer and find a trigger
threshold around $5 \, \rm{keV}$ with the potential for further
reduction (using an improved amplification circuit). We used the CHESS
X-ray beam to test the polarimeter. Detailed comparisons of experimental
and simulated data allowed us to demonstrate the full functionality of
the polarimeter and to measure the polarization of the CHESS beam. We
studied different systematic effects that potentially affect the
reconstructed polarization properties and estimated the systematic error
to be smaller than $2\%$ for an upcoming balloon flight.
Our tentative observation program for an upcoming X-Calibur/{\it InFOC$\mu$S}\xspace
balloon flight includes galactic sources (Crab nebula, Her\,X-1,
Cyg\,X-1, GRS\,1915, EXO\,0331) and one extragalactic source (Mrk\,421)
for which sensitive polarization measurements will be carried through.
In principle, a similar space-borne scattering polarimeter could operate
over the broader $3-80 \, \rm{keV}$ energy band. Here, a LiH rod would
be used as passive scatterer. In contrast to the plastic scintillator
used in the balloon-borne polarimeter, the LiH scatterer does not yield
a coincidence signal. However, the lower atomic number of LiH results in
the possibility of using the polarimeter down to energies of a few keV.
\section*{Acknowledgments}
We are grateful for NASA funding from grants NNX10AJ56G, NNX12AD51G and
NNX14AD19G, as well as discretionary funding from the McDonnell Center
for the Space Sciences to build the X-Calibur polarimeter. Polarization
measurements: This work is based upon research conducted at the Cornell
High Energy Synchrotron Source (CHESS) which is supported by the
National Science Foundation and the National Institutes of
Health/National Institute of General Medical Sciences under NSF award
DMR-0936384. We would like to thank Ken Finkelstein for the excellent
support in setting up the experiment at CHESS and for the continuous
discussions thereafter.
|
1,116,691,499,414 | arxiv | \section{Introduction}
A region $\Omega \subset \mathbb C$, the boundary of which is a polygonal curve with angles multiple of $\pi/2$, is considered in this article. This region models the shape of a channel under a dam. The calculation of fluid flow in such channel boils down to a conformal mapping problem of $\Omega$ onto the upper half-plane. The solution of such problems is given by Christoffel-Schwartz integral (see, e.g., \cite{ShabatEng} or \cite{SchwartzChristoffel}), which in this case is naturally defined on an elliptic Riemann surface. In this paper the simple formula, which writes integral in terms of Weierstrass sigma function (see e.g., \cite{AkhiezerEng} or \cite{Chandra}), was found. This approach allows to avoid numerical integration and the mapping parameters can be found from a simple nonlinear system of equations; therefore, the computation is significantly simplified.
Similar problems have been considered in \cite{SigBog}, \cite{DamBogEng}, \cite{BogHept}, \cite{BogGrig}, and \cite{BogCond}, where Christoffel-Schwartz integral was efficiently represented by theta functions (see, e.g., \cite{FarkasKra}). In the paper \cite{BogCond} the application of Lauricella functions to these problems was studied, and also different approaches were compared.
The main advantage of Weierstrass functions over theta functions is that they have limiting values when the surface degenerates. The paper analyzes behavior of the constructed conformal mapping under the condition that the dam width tends to zero. It turns out that conformal mappings have a limit, which is a solution to the limiting problem. Thus, it is shown that the solution is stable under the considered degeneration.
The described property of the Weierstrass sigma function loses its value if the standard method is used for calculations, which expresses the sigma function in terms of the theta function (since it does not withstand degeneration). Thus, it is necessary to use an independent method for calculating sigma functions. In this paper, we use the expression for the coefficients of its expansion into Taylor series obtained by Weierstrass (see \cite{Weier}). Since the proof presented there is apparently not complete (at one point, the analyticity of the sigma function in three variables in a neighborhood of zero is used, which is not obvious), we give a more detailed proof in the appendix. The above formula, however, is not sufficient for the final numerical solution, since Taylor series are not suitable for calculations with large arguments (namely, such a need arises with degeneration). Thus, the problem of constructing an efficient computational method for the sigma function independent of theta functions still remains unsolved. If such a method is available, it will be possible to construct formulas that are stable under various degenerations and use them in calculations. The results of this work illustrate the need in such methods.
Problems in which hyperelliptic Riemann surfaces of higher genus arise can also be solved using the theory of sigma functions developed by Klein and Baker in \cite{Klein} and \cite{Baker} respectively (more detailed exposition can be found in \cite{Buch}). There is hope that it will be possible to prove the stability of formulas expressing the solution of the above problems in terms of high-order sigma functions. Thus, the construction of Weierstrass-type recurrent formulas (which are known for genus 1 and 2; see \cite{Buch}) and calculation methods for sigma functions can be extremely useful in applied problems.
\section{The statement and the origin of the problem}
Consider region $\Omega$ in the complex plane pictured on Figure 1.
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\draw[black,thick] (-5,-1) -- (5,-1);
\draw[black,thick] (-5,2) -- (0,2) node [pos=1,below left] {$w_4$};
\draw[black,thick] (0,2) -- (0,0) node [pos=1,left] {$w_3$};
\draw[black,thick] (0,0) -- (1,0) node [pos=1,below right] {$w_2$};
\draw[black,thick] (1,0) -- (1,1) node [pos=1,above] {$w_1$};
\draw[black,thick] (1,1) -- (5,1);
\draw[<->,black,dashed,thick] (0,-0.2) -- (1,-0.2) node [pos=0.5,below] {$\delta$};
\draw[<->,black,dashed,thick] (1.2,0) -- (1.2,1) node [pos=0.5,right] {$h$};
\draw[<->,black,dashed,thick] (-4,-1) -- (-4,2) node [pos=0.5,right] {$h^-$};
\draw[<->,black,dashed,thick] (4,-1) -- (4,1) node [pos=0.5,right] {$h^+$};
\end{tikzpicture}
\caption{Region $\Omega$.}
\end{figure}
It is bounded from below by a line and from above by a polygonal curve with four vertices $w_1,w_2,w_3,w_4$ (it is convenient to think that this region also has two vertices at $\pm \infty$). Let the line, which bounds the region from below be parallel to the real axis, and let the vertex $w_4$ be at the origin. Then, this region is determined by four real parameters $h^-,h^+,h,\delta$, where $h$ is equal to the length of segment $[w_1,w_2]$, $\delta$ is the length of $[w_2,w_3]$, and $h^-$ and $h^+$ are equal to the distance from the line that bounds the region from below to $w_4$ and $w_1$ respectively. These parameters are positive and satisfy inequalities $h^- - h^+ + h > 0$, which corresponds to positivity of the length of $[w_3,w_4]$, and $h < h^+$. The region is determined uniquely by these parameters.
Regions similar to $\Omega$ arise in problems connected with the computation of fluid flow through the porous material under a dam. Since the flow is continuous and satisfies the Darcy's law, the pressure $p$ is a harmonic function in $\Omega$. Assuming that segments $[w_1, w_2]$, $[w_2,w_3]$, $[w_3, w_4]$, and channel's bottom are impenetrable, we obtain natural boundary conditions: the normal derivative $\partial p/\partial n$ vanishes on impenetrable segments of the boundary, while on the remaining segments (i.e. on the half-lines starting from $w_1$ and $w_4$) $p$ is locally constant.
Consider a real-valued function $q$ in the region $\Omega$ such that $f = p + iq$ is holomorphic (such function exists because $\Omega$ is simply connected). The normal derivative of $p$ vanishing condition is easily equivalent to constancy of $q$ on the corresponding boundary segment. It follows, that, if $f$ is a function that conformally maps $\Omega$ onto a rectangle in a way, such that $w_1$, $w_4$, and vertices at infinity are mapped to the vertices of a rectangle, then $p = \RE f$ is a solution to the original problem. $q = \IM f$ is called the current function. Its level lines are the streamlines of the fluid under the dam. It is clear, that it is enough to solve the problem of conformal mapping of $\Omega$ onto upper half-plane $\mathbb C_+$ in order to solve the specified problem.
In what follows, the conformal mapping problem will be solved explicitly using the tools of Weierstrass elliptic functions. Below we show the calculation of streamlines in $\Omega$ obtained using the method constructed in this work.
\begin{figure}[h!]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\textwidth]{LinesColor2.pdf}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\textwidth]{LinesColorCloser2.pdf}
\end{subfigure}
\caption{Streamlines in the region $\Omega$.}
\end{figure}
\FloatBarrier
\section{The solution of the conformal mapping problem}
\subsection{The general form of the solution and parameter determination}
Since $\Omega$ is simply connected, there is a conformal mapping $W:\mathbb C_+ \rightarrow \Omega$, where $\mathbb C_+ = \{z \in \mathbb C: \IM z > 0\}$ is the upper half-plane (see, e.g., \cite{KartanEng} or \cite{ShabatEng}). Using, if necessary, a suitable automorphism of $\mathbb C_+$, one can make it such that the point $w_4$ is the preimage under $W$ (more precisely, under its continuation to the boundary) of $\infty$. Then, by Christoffel-Schwartz theorem (see \cite{SchwartzChristoffel}), there exist $x^-<x^+<x_1<x_2<x_3 \in \mathbb R$ and $C \in \mathbb C$ such that
\begin{equation}
dW = \phi = C \frac{\sqrt{(x - x_2)(x-x_3)}}{(x-x^-)(x-x^+)\sqrt{x-x_1}}dx.
\end{equation}
\begin{remark}
Here $x_i$ is the preimage of $w_i$ under $W$, and $x^-$ and $x^+$ are the points on the boundary of the upper half-plane in which $W$ goes to infinity (preimages of the vertices at infinity).
\end{remark}
The differential form $\phi$ can be considered on the hyperelliptic Riemann surface $V$ of genus $1$, defined by equation $y^2 = F(x) = 4(x - x_1)(x-x_2)(x-x_3)$. Using the shift of the upper half-plane we can set $x_1 + x_2 + x_3 = 0$ without loss of generality. Thus, $F(x) = 4x^3 - g_2x - g_3$ for some real $g_2,g_3$ (that are determined by $x_1,x_2,x_3$). We can rewrite $\phi$ on this surface in the form
\begin{equation}
\phi = 2C \frac{(x-x_2)(x-x_3)}{y(x-x^-)(x-x^+)}dx.
\end{equation}
Let us fix the branch of $\sqrt{F(x)}$ in the region that is made from $\mathbb C$ by throwing off segment $[x_1,x_2]$ and half-line $[x_3,\infty]$. Let this branch have positive values as the argument tends to half-line $(x_3, \infty)$ from the upper half-plane. Recalling that $dx/y$ is a holomorphic (everywhere non-zero) form on $V$, we obtain that $\phi$ has two zeros of multiplicity $2$ at $(x_2,0)$ and $(x_3,0)$ and also four simple poles at $(x^-,\pm \sqrt{F(x^-)})$ and $(x^+,\pm\sqrt{F(x^+)})$. Note that the residues of this form at these poles are equal to $\pm h^-/\pi$ and $\mp h^+/\pi$ respectively.
Now we shall use Abel map (see, e.g., \cite{ForsterEng}) which identifies $V$ with $\Jac(V)$ (as usual, we set the infinity as the initial point and $dx/y$ as the basis of holomorphic forms). Let us introduce the half-periods $$
\omega = \int_{x_1}^{x_2} \frac{dx}{y},\;\; \omega' = -\int_{x_2}^{x_3} \frac{dx}{y},
$$ and quantities $\eta = \zeta(\omega)$ and $\eta' = \zeta(\omega')$, where $\zeta$ is the Weierstrass zeta function (see \cite{AkhiezerEng}).
It is easy to see that $\omega,\eta \in \mathbb R$ and $\omega', \eta' \in i\mathbb R$. The set of points $(x,\sqrt{F(x)})$, where $x \in \mathbb C_+$, is mapped by this map onto the rectangle with vertices $0,\omega', \omega' - \omega, -\omega$. Let us denote the images of the points $(x^-, \sqrt{F(x^-)})$ and $(x^+,\sqrt{F(x^+)})$ by $z^-$ and $z^+$ respectively (see Figure 3, where the preimages of points are indicated in the brackets).
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale = 2]
\draw[black,thick] (0,0) -- (0,2)node [pos=0,right] {$0\;(\infty)$};
\draw[black,thick] (0,2) -- (-3,2)node [pos=0,right] {$\omega'\;(x_1)$};
\draw[black,thick] (-3,2) -- (-3,0)node [pos=0,left] {$\omega' - \omega\;(x_2)$};
\draw[black,thick] (-3,0) -- (0,0) node [pos=0,left] {$-\omega\;(x_3)$};
\filldraw [black] (0,0.7) circle (0.5pt) node[right] {$z^-$};
\filldraw [black] (0,1.2) circle (0.5pt) node[right] {$z^+$};
\end{tikzpicture}
\caption{Image of the upper half-plane under the Abel map.}
\end{figure}
The images of $(x^-, -\sqrt{F(x^-)})$ and $(x^+,-\sqrt{F(x^+)})$ in this case are equal to $-z^-$ and $-z^+$. Consider the differential form $\psi$ on the torus that corresponds to $\phi$ under this identification of $V$ with $\Jac(V)$. This form has $4$ simple poles in the points $\pm z^-$ and $\pm z^+$ and its residues are equal to $\pm h^-/\pi$ and $\mp h^+/\pi$ respectively.
Now we use the method of representing elliptic functions by Weierstrass functions that is described in \cite{AkhiezerEng}. Consider meromorphic function
\begin{equation}\label{eq413}
g(z) = \frac{h^-}{\pi}(\zeta(z - z^-) - \zeta(z + z^-)) - \frac{h^+}{\pi}(\zeta(z-z^+) - \zeta(z+z^+)).
\end{equation}
Using the quasiperiodic properties of $\zeta$ (see \cite{AkhiezerEng}) it is easy to conclude that $g$ is elliptic. Form $g(z)dz$ has the same simple poles as $\psi$ with the same residues. Therefore, $\psi - g(z)dz$ is a holomorphic form on the torus. Since the space of holomorphic $1$-forms on the torus is one-dimensional it follows that $\psi - g(z)dz = Ddz$, where $D$ is a constant (note that $D \in i\mathbb R$).
Now we return to the map $W$. It is clear that $$
W(x) = -\int_{x}^\infty \phi.
$$ In view of that, let \begin{equation}
Q(z) = \int_0^z \psi.
\end{equation}
Obviously, $W(x)$ is equal to $Q(z)$, where $z$ is the image of $(x,\sqrt{F(x)})$ under the Abel map. Thus, $Q$ conformally maps the rectangle with vertices $0,\omega', \omega' - \omega, -\omega$ onto $\Omega$, and $\omega'$ is mapped to $w_1$, $\omega'-\omega$ is mapped to $w_2$, and $-\omega$ to $w_3$ (also $0$ is mapped to $w_4$). Now we can derive the system of equations from the previously obtained relations:
\begin{equation}\label{eq415}
g(-\omega) + D = 0,\;\;\; g(\omega' - \omega) + D = 0,
\end{equation}
\begin{equation}\label{eq416}
Q(\omega'-\omega) - Q(\omega') = -ih,\;\;\; Q(-\omega) - Q(\omega'-\omega) = -\delta.
\end{equation}
\begin{remark}
The first pair of equations follows from the fact that $\phi$ has zeros in the points $(x_2,0)$ and $(x_3,0)$, and the second pair is a consequence of relations $w_3-w_2 = -\delta$, $w_2 - w_1 = -ih$.
\end{remark}
It remains to derive a reasonable formula for $Q$. Recall that $\zeta$ is a logarithmic derivative of $\sigma$. It easily follows that
\begin{equation}\label{eq417}
Q(z) = Dz + \frac{h^-}{\pi}\ln\left(\frac{\sigma(z-z^-)}{\sigma(z+z^-)}\right) - \frac{h^+}{\pi}\ln\left(\frac{\sigma(z-z^+)}{\sigma(z+z^+)}\right) - i(h^- - h^+),
\end{equation}
where $\ln$ denotes the branch of logarithm in the plane cut by negative imaginary half-line such that $\ln(1) = 0$. Substituting in \eqref{eq415} the formula for $g$ from \eqref{eq413} and using quasiperiodicity of $\sigma$ (see, e.g., \cite{AkhiezerEng}), we obtain the system of equations:
\begin{equation}\label{eq418}
\begin{dcases}
-D\omega - \frac{2h^+}{\pi}\eta z^+ + \frac{2h^-}{\pi}\eta z^- = -ih, \\
-D\omega' - \frac{2h^+}{\pi}\eta'z^+ + \frac{2h^-}{\pi}\eta'z^- = -\delta, \\
D + \frac{h^-}{\pi}(\zeta(\omega - z^-) - \zeta(\omega + z^-)) - \frac{h^+}{\pi}(\zeta(\omega-z^+) - \zeta(\omega+z^+)) = 0, \\
D + \frac{h^-}{\pi}(\zeta(\omega' + \omega - z^-) - \zeta(\omega' + \omega + z^-)) \\ \qquad-\frac{h^+}{\pi}(\zeta(\omega' + \omega-z^+) - \zeta(\omega' + \omega+z^+)) = 0.
\end{dcases}
\end{equation}
In this system of equations there are five variables (since the quantities $\omega,\omega',\eta,\eta'$ are determined by $g_2$ and $g_3$) $g_2,g_3,D,z^+,z^-$ (the first two are real and the other are imaginary) and four equations \eqref{eq418} (the first, third and fourth equations are imaginary and the second one is real). Thus, it is natural to consider a single parameter family of curves that necessarily contains a suitable one, i.e. to consider functions $g_2 = g_2(\gamma)$ and $g_3 = g_3(\gamma)$ and use the system \eqref{eq418} to determine the parameters $\gamma, D, z^+, z^-$.
In what follows we shall use the family of curves that is defined by the roots of the polynomial $F$: $x_1 = \gamma - 1/2$, $x_2 = -2\gamma$, $x_3 = \gamma+1/2$, $\gamma \in (-1/6,1/6)$ (more detailed analysis of this family is given during the study of the degeneration $\delta \rightarrow 0$). This family corresponds to the normalization condition $x_3 - x_1 = 1$ in addition to the already given relation $x_1 + x_2 + x_3 = 0$.
\begin{figure}[ht!]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figure_2.pdf}
\caption{ The rectangle $P$ and contours in it.}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figure_1.pdf}
\caption{The image of the rectangle and the contours.}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Figure_1alt.pdf}
\caption{The behaviour near $[w_2,w_3]$.}
\end{subfigure}%
\caption{The conformal mapping $Q$.}
\end{figure}
\subsection{On the numerical implementation}
It was decided to use for numerical implementation the explicit computation of the sigma function depending on parameters $g_2,g_3$ through its Taylor series (see \cite{Weier} or Theorem~\ref{tWeierSeries}). It is clear that for the effective solution of the system \eqref{eq418} it is necessary to compute all the quantities in it and their derivatives with respect to parameters. In the end it reduces to the computation of $\omega$ and $\omega'$ and their derivatives with respect to $g_2$ and $g_3$ and, also, $\zeta$ and its derivatives with respect to $z, g_2,g_3$. Since $$
\zeta = \frac{1}{\sigma} \frac{\partial \sigma}{\partial z} ,
$$
the problem of computation of $\zeta$ and its derivatives can be solved easily. In order to compute $\omega$ we note that $\sigma$ has zeros exactly in the points of the lattice $\{2m\omega + 2n\omega':n,m \in \mathbb Z\}$ and these zeros are simple. An effective way to localize a simple zero $z_0$ of a holomorphic function $f$ is to compute integral of $zf'(z)/2\pi if(z)$ on a contour enclosing $z_0$. To find a suitable contour it is possible to apply a variant of binary search using that $\omega \ge \pi/2$. Using the specified method either directly, or for an approximate calculation of zero and subsequent application of equation solving methods, it is easy to construct an effective and precise algorithm of computation of $\omega$ (and $\omega'$). In order to compute their derivatives it is possible to differentiate the integral of $z\sigma'(z)/\sigma(z)$ by $g_2$ or $g_3$, and to compute it explicitly by determining the residue in the zero of $\sigma$. Thus, the solution of the system \eqref{eq418} can be completely reduced to the computation of the sigma function and its derivatives with respect to $z$, $g_2$, and $g_3$.
We demonstrate the solution of a specific problem by this method. Let $h^+ = \pi$, $h^- = \pi + 0.5$, $h = 0.5$, $\delta = 0.2$. We shall search for the solution in the one parameter family of curves defined by $x_1 = \gamma - 1/2$, $x_2 = -2\gamma$, $x_3 = \gamma + 1/2$, $\gamma \in (-1/6, 1/6)$. The solution of the system \eqref{eq418} is $$(\gamma,D,z^+,z^-) = (0.1051616134,0.0203152915i,1.3043479103i,0.7195735824i).$$ Given this $\gamma$ we obtain $\omega = 1.6518996331$, $\omega' = 2.2939120295i$. On Figure 4 the image of the rectangle $P$ with vertices $0,\omega',\omega'-\omega,-\omega$ under the map $Q$ is shown.
\section{Stability of the solution under the degeneration of the region}
Here we shall consider a problem of conformal mapping of the upper half-plane onto the region $\widetilde{\Omega}$
that comes from $\Omega$ with degeneration $\delta\rightarrow 0$ (see Figure 5) and analyse behaviour of the solution under the condition that no other degeneration is happening (i.e. quantities $h^-,h^+,h,h^-+h-h^+,h^+-h$ have positive limits).
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\draw[black,thick] (-5,-1) -- (5,-1);
\draw[black,thick] (-5,2) -- (0,2) node [pos=1,below left] {$w_4$};
\draw[black,thick] (0,2) -- (0,0) node [pos=1,below] {$w_2= w_3$};
\draw[black,thick] (0,1) -- (5,1) node [pos=0,left] {$w_1$};
\draw[<->,black,dashed,thick] (0.2,0) -- (0.2,1) node [pos=0.5,right] {$h$};
\draw[<->,black,dashed,thick] (-4,-1) -- (-4,2) node [pos=0.5,right] {$h^-$};
\draw[<->,black,dashed,thick] (4,-1) -- (4,1) node [pos=0.5,right] {$h^+$};
\end{tikzpicture}
\caption{Region $\widetilde \Omega$.}
\end{figure}
$\widetilde{\Omega}$ is determined by three parameters
$h,h^+$ and $h^-$. A conformal mapping of the upper half-plane onto $\widetilde\Omega$ can be found by the analogous method (using Christoffel-Schwartz theorem). In this case, since the corresponding Riemann surface has genus $0$, the solution can be expressed in elementary functions. Another method (that is considered here) is to apply formula \eqref{eq417}, using the fact that $\sigma$ is defined for $g_2$ and $g_3$ such that $F(x) = 4x^3 - g_2x - g_3$ has multiple roots. It is natural to suppose that the solution can be found by taking the limit under gluing the roots, that are mapped to $w_2$ and $w_3$. Together with that the stability of the solution under $\delta\rightarrow 0$ will be proved.
\subsection{Gluing of the roots}
Again consider a family of curves depending on $\gamma \in (-1/6, 1/6)$ that is given by $F_\gamma(x) = 4(x - x_1(\gamma))(x - x_2(\gamma))(x - x_3(\gamma))$, where $x_1(\gamma) = \gamma - 1/2$, $x_2(\gamma) = -2\gamma$, $x_3(\gamma) = \gamma + 1/2$. Under $\gamma \rightarrow -1/6$ the roots $x_2$ and $x_3$ glue. The limiting values of $g_2$ and $g_3$ are $4/3$ and $-8/27$ respectively. For each $\gamma$ we define quantities $\omega(\gamma)$, $\omega'(\gamma)$, $\eta(\gamma)$, $\eta'(\gamma)$. In what follows we shall omit dependence on $\gamma$.
\begin{lemma}\label{l41}
Under $\gamma \rightarrow -1/6$ we have
\begin{equation}\label{eq441}
\omega,\eta \rightarrow \infty,\;\; \omega' \rightarrow \frac{i\pi}{2},\;\;\eta' \rightarrow -\frac{i\pi}{6},\;\; \frac{\eta}{\omega} \rightarrow -\frac{1}{3}.
\end{equation}
Moreover,
\begin{equation}\label{eq442}
\begin{gathered}
\sigma(z,\frac{4}{3}, -\frac{8}{27}) = e^{-\frac{z^2}{6}} \sinh(z),\;\; \zeta(z,\frac{4}{3}, -\frac{8}{27}) = \coth(z) - \frac{z}{3},\\ \wp(z,\frac{4}{3}, -\frac{8}{27}) = \frac{1}{\sinh^2(z)} + \frac{1}{3}.
\end{gathered}
\end{equation}
Finally, there exists $\varepsilon > 0$ such that for $\gamma + 1/6 < \varepsilon$ the estimation
\begin{equation}\label{eq44asympt}
-c_1\ln(\gamma + 1/6) \le \omega(\gamma) \le -c_2\ln(\gamma + 1/6)
\end{equation}
holds, where $0 < c_1 < c_2$.
\end{lemma}
\begin{proof}
\eqref{eq441} easily follows from integral representations
$$
\omega(\gamma) = \frac{1}{2}\int_{\gamma - \frac{1}{2}}^{-2\gamma} \frac{dx}{\sqrt{(x - \gamma - 1/2)(x - \gamma + 1/2)(x + 2\gamma)}}, $$ $$
\omega'(\gamma) = \frac{1}{2}\int_{-2\gamma}^{\gamma+\frac{1}{2}} \frac{dx}{\sqrt{-(x - \gamma - 1/2)(x - \gamma + 1/2)(x + 2\gamma)}}, $$ $$
\eta(\gamma) = -\frac{1}{2}\int_{\gamma - \frac{1}{2}}^{-2\gamma} \frac{xdx}{\sqrt{(x - \gamma - 1/2)(x - \gamma + 1/2)(x + 2\gamma)}},
$$ $$
\eta'(\gamma) = -\frac{1}{2}\int_{-2\gamma}^{\gamma + \frac{1}{2}} \frac{xdx}{\sqrt{-(x - \gamma - 1/2)(x - \gamma + 1/2)(x + 2\gamma)}}. $$
To derive \eqref{eq442} one can pass to the limit $\gamma \rightarrow -1/6$ using the formula that represents $\sigma$ as the infinite product (see \cite{AkhiezerEng}). We obtain
$$
\sigma(z,\frac{4}{3}, -\frac{8}{27}) = z\prod_{n \ne 0} \left(1 - \frac{z}{i n \pi}\right) e^{\frac{z}{i n \pi} - \frac{z^2}{2 n^2 \pi^2}}.
$$
Using classical identities
$$
\sum_{n = 1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6},\;\; \prod_{n = 1}^\infty \left(1 - \frac{x^2}{n^2 \pi^2}\right) = \frac{\sin(x)}{x},
$$
we derive the first equation of \eqref{eq442}. The rest follow from equalities $\zeta(z) = \sigma'(z)/\sigma(z)$, $\wp(z) = -\zeta'(z)$.
Now we estimate the growth of $\omega(\gamma)$. Consider equality
$$
\omega(\gamma) = \frac{1}{2}\int_0^{1/2 - 3\gamma} \frac{dt}{\sqrt{t(1-t)(1/2 - 3\gamma - t)}}.
$$
Note that for small $\gamma$ integral on the segment $[0,1/2]$ is bounded and, therefore,
$$
\omega(\gamma) \sim \frac{1}{2}\int_{1/2}^{1/2 - 3\gamma} \frac{dt}{\sqrt{t(1-t)(1/2 - 3\gamma - t)}}.
$$
The estimation of the integral in rhs is quite simple since $1/\sqrt{t}$ is bounded from below and above by positive constants and the remaining integral can be calculated explicitly.
\end{proof}
Since $\omega' \rightarrow i\pi/2$ and $\omega \rightarrow \infty$, it is natural to suppose that \eqref{eq417} can give a formula for a conformal mapping of the half-strip $$S = \{z \in \mathbb C: \RE z < 0, \IM z \in (0,\pi/2)\}$$ onto $\widetilde\Omega$. Let
\begin{equation}\label{eq443}
\widetilde Q(z) = Dz + \frac{h^-}{\pi}\ln\left(\frac{\sigma(z-z^-)}{\sigma(z+z^-)}\right) - \frac{h^+}{\pi}\ln\left(\frac{\sigma(z-z^+)}{\sigma(z+z^+)}\right) - i(h^- - h^+),
\end{equation}
where $\sigma$ is taken at values $g_2 = 4/3$, $g_3 = -8/27$ and $D \in \mathbb C$, $z^-, z^+\in(0,i\pi/2)$ are the parameters. Substituting \eqref{eq442} into \eqref{eq443}, we obtain
\begin{equation}\label{eq444}
\begin{multlined}
\widetilde Q(z) = z\left(D + \frac{2h^-z^-}{3\pi} - \frac{2h^+z^+}{3\pi}\right) \\+ \frac{h^-}{\pi}\ln\frac{\sinh(z - z^-)}{\sinh(z+z^-)} - \frac{h^+}{\pi}\ln\frac{\sinh(z - z^+)}{\sinh(z + z^+)}- i(h^- - h^+).
\end{multlined}
\end{equation}
It is easy to check that if $\widetilde Q$ has a non-zero linear term it has no limit under $\RE z \rightarrow -\infty$. If$$
D + \frac{2h^-z^-}{3\pi} - \frac{2h^+z^+}{3\pi} = 0,
$$ its limit is equal to $2(h^-z^- - h^+z^+)/\pi - i(h^- - h^+)$. Thus, if $\widetilde Q$ conformally maps $S$ onto $\widetilde \Omega$, then the conditions
\begin{equation}\label{eq445}
\begin{dcases}
D + \frac{2h^-z^-}{3\pi} - \frac{2h^+z^+}{3\pi} = 0, \\
h^- z^- - h^+ z^+ = -\frac{ih\pi}{2}
\end{dcases}
\end{equation}
hold. Obviously, these conditions are sufficient under the additional assumption that the derivative of $\widetilde{Q}$ is not vanishing on $S$ and its boundary. Using tedious but elementary calculations it can be shown that this condition is equivalent to
\begin{equation}\label{eq446}
h^-\sinh(2z^-) = h^+\sinh(2z^+).
\end{equation}
Thus, \eqref{eq445} and \eqref{eq446} determine the parameters $D,z^-,z^+$, at which $\widetilde Q$ is the desired conformal mapping.
We show that the obtained formula reduces to Christoffel-Schwartz integral in the upper half-plane under the variable change $x = \wp(z) = 1/\sinh^2(z) + 1/3$. Changing the variable in \eqref{eq443} and using that the linear term vanishes we obtain the following formula for the conformal mapping of $\mathbb C_+$ onto $\widetilde{\Omega}$:
\begin{equation}\label{eq448}
\widetilde{W}(x) = \frac{h^-}{\pi}\ln \left(\frac{\sqrt{x^- + 2/3} - \sqrt{x + 2/3}} {\sqrt{x^- + 2/3} + \sqrt{x + 2/3}}\right) + \frac{h^+}{\pi}\ln \left(\frac{\sqrt{x^+ + 2/3} - \sqrt{x + 2/3}} {\sqrt{x^+ + 2/3} + \sqrt{x + 2/3}}\right).
\end{equation}
The equations on the parameters $z^-$ and $z^+$ can be rewritten in the form
\begin{equation}\label{eq449}
\begin{dcases}
\frac{h^-\sqrt{x^- + 1}}{x^-} = \frac{h^+\sqrt{x^++1}}{x^+}, \\
h^- \ln\left(\frac{1+\sqrt{x^- + 4/3}}{\sqrt{x^- + 1/3}}\right) + h^+\ln\left(\frac{1+\sqrt{x^+ + 4/3}}{\sqrt{x^+ + 1/3}}\right) = -\frac{ih\pi}{2}.
\end{dcases}
\end{equation}
Differentiating $\widetilde{W}$ and using the first equation from \eqref{eq449} we obtain $$
d\widetilde{W} = \frac{h^-\sqrt{x^-+1}- h^+\sqrt{x^++1}}{\pi} \frac{(x - 1/3)dx}{\sqrt{x+2/3}(x-x^-)(x-x^+)}.
$$
Therefore we found the exact form of the constant in Christoffel-Schwartz integral. The remaining parameters $x^-$ and $x^+$ can be found from the system of equations \eqref{eq449}.
\subsection{Passing to the limit}
Consider a sequence of regions determined by the parameters $(h^-_n, h^+_n, h_n, \delta_n)$. Assume that they have limits $(h^-_{\lim}, h^+_{\lim}, h_{\lim}, 0)$. Also we assume that $h^+_{\lim} - h_{\lim} > 0$ and $h^-_{\lim} - h^+_{\lim} + h_{\lim} > 0$. We shall prove that the parameters $(D_n, \gamma_n, z^-_n, z^+_n)$, given by the solution of the system \eqref{eq418} for the corresponding regions, have limits $(D_{\lim}, -1/6, z^-_{\lim}, z^+_{\lim})$, and, moreover, parameters $(D_{\lim}, z^-_{\lim}, z^+_{\lim})$ satisfy \eqref{eq445} and \eqref{eq446}. Thus, in view of the fact that the Weierstrass sigma function is entire, it follows that the constructed solution is stable.
The following proof is rather long and technical and, therefore, we shall omit most of the calculations.
In the following estimations we shall also use parameters $x^-_n, x^+_n, x_1^{(n)}, x_2^{(n)}, x_3^{(n)}, C_n$ of the map $W_n$.
\begin{lemma}
Assume that $\gamma_n \rightarrow -1/6$ and $\delta_n \omega(\gamma_n) \rightarrow 0$.
Then the indicated convergence holds.
\end{lemma}
\begin{proof}
Note that $\gamma_n \rightarrow -1/6$ implies that sequences $z^-_n$ and $z^+_n$ are bounded. The first equation in \eqref{eq418} then implies that $D_n$ is also bounded. Passing to the subsequences we can assume that all these sequences are convergent (if we succeed to prove that the limits satisfy \eqref{eq445} and \eqref{eq446}, then uniqueness of the solution implies that all the subsequences converge to the same limit, and, therefore, the initial sequences converge). The first equation in \eqref{eq445} is obtained by passing to the limit in the second equation in \eqref{eq418} in view of Lemma \ref{l41}. Multiplying the first equation in \eqref{eq418} by $\omega'$ and the second one by $\omega$, subtracting and passing to limit leads to the second equation in \eqref{eq445} (the term $\delta \omega$ by assumption tends to zero).
Now we derive \eqref{eq446} for the limits of the sequences. Recall the following notation of the theory of elliptic functions (see \cite{AkhiezerEng}):
$$
\zeta_2(z) = \zeta(z + \omega) - \eta,
$$
$$
\zeta_3(z) = \zeta(z + \omega + \omega') + \eta + \eta'.
$$
These functions are connected to $\sigma_2, \sigma_3$:
$$
\zeta_k = \frac{1}{\sigma_k} \frac{d \sigma_k}{d z} = \frac{d \ln \sigma_k}{d z}.
$$
Finally, $\sigma_k = \sigma \sqrt{\wp - x_k}$. The last two equations in \eqref{eq418} can be rewritten as
\begin{equation}\label{eq447}
D + \frac{h^-}{\pi}(\zeta_k(-z^-) - \zeta(z^-)) - \frac{h^+}{\pi}(\zeta_k(-z^+) - \zeta(z^+)) = 0, \;\; k = 2,3.
\end{equation}
Since
$$
\begin{multlined}
\zeta_2(z) - \zeta_3(z) = \frac{\sigma'(z) \sqrt{\wp - x_2} + \wp'(z) \sigma(z) (\wp - x_2)^{-1/2}}{\sigma(z) \sqrt{\wp - x_2}} \\ - \frac{\sigma'(z) \sqrt{\wp - x_3} + \wp'(z) \sigma(z) (\wp - x_3)^{-1/2}}{\sigma(z) \sqrt{\wp - x_3}},
\end{multlined}
$$
it follows that
\begin{equation}\label{zetadiff}
\zeta_2(z) - \zeta_3(z) = \frac{\wp'(z)(x_2 - x_3)}{(\wp - x_2)(\wp - x_3)}.
\end{equation}
Equation \eqref{eq446} is derived by passing to the limit (using Lemma \ref{l41}) from equations \eqref{eq447} (from which the constant $D$ can be eliminated) and substitution the formula for $\zeta_2 - \zeta_3$ from \eqref{zetadiff}.
\end{proof}
\begin{lemma}
Inequalities $|C_n| \ge a_1$, $|C_n| \le a_2 \sqrt{x_3^{(n)} - x^-_n}$ hold for some positive constants $a_1,a_2$. Moreover, the sequence $x^+_n$ is bounded from below.
\end{lemma}
\begin{proof}
The estimations for $C^{(n)}$ easily follow from
$$|C_n|\bigintss_{x_3^{(n)}}^{+\infty} \frac{\sqrt{\left(x - x_2^{(n)}\right)\left(x - x_3^{(n)}\right)}dx}{\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = h^-_n - h^+_n + h_n.
$$
To prove the boundedness from below for $x^+_n$ it suffices to consider equality
$$
|C_n|\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \frac{\sqrt{\left(x_2^{(n)}-x\right)\left(x_3^{(n)} - x\right)}dx}{\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = h_n.
$$
\end{proof}
\begin{lemma}
Assume that sequence $x^-_n$ is bounded from below. Then there exist constants $0 < b_1 < b_2$ such that $b_1 \sqrt{\delta_n} \le |x_2^{(n)} - x_3^{(n)}| \le b_2 \sqrt{\delta_n}$.
\end{lemma}
\begin{proof}
It follows from easy estimation for the integral in equality
$$
|C_n|\bigintss_{x_2^{(n)}}^{x_3^{(n)}} \frac{\sqrt{\left(x - x_2^{(n)}\right)\left(x_3^{(n)} - x\right)}dx}{\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = \delta_n.
$$
\end{proof}
The foregoing Lemmas imply that it remains to prove that sequence $x^-_n$ is bounded from below (in view of the asymptotics \eqref{eq44asympt}). Assume that this is not true. Passing to the subsequences we can assume that $x^-_n \rightarrow -\infty$ and $x_1^{(n)},x_2^{(n)},x_3^{(n)}$, $x^+_n$ are convergent.
\begin{lemma}
In the foregoing assumptions statements $x_3^{(n)} - x_2^{(n)} \rightarrow 0$, $x_1^{(n)} - x^+_n \rightarrow 0$ hold.
\end{lemma}
\begin{proof}
Equality
$$
|C_n| \frac{\sqrt{\left(x_2^{(n)} - x^+_n\right)\left(x_3^{(n)} - x^+_n\right)}}{\sqrt{x_1^{(n)} - x^+_n}(x^+_n - x^-_n)} = \frac{h^+_n}{\pi}
$$
implies $x_1^{(n)} - x^+_n \rightarrow 0$.
To prove that $x_3^{(n)} - x_2^{(n)} \rightarrow 0$ we return to parameters $(D_n, \gamma_n, z^-_n, z^+_n)$. Assume that $x_2^{(n)} - x_1^{(n)} \rightarrow 0$. Then $\omega'(\gamma_n), \eta'(\gamma_n) \rightarrow \infty$, and $\omega$ and $\eta$ have finite limits. Moreover, Legendre identity (see, e.g., \cite{AkhiezerEng} or \cite{Chandra}) implies
$$
\lim_{n \rightarrow \infty} \frac{\omega'(\gamma_n)}{\eta'(\gamma_n)} = \lim_{n \rightarrow \infty} \frac{\omega(\gamma_n)}{\eta(\gamma_n)}.
$$
The first two equations from \eqref{eq418} imply
$$
\lim_{n \rightarrow \infty}\left(\frac{2 h^+_n}{\pi} z^+_n \left(\frac{\omega'}{\eta'} - \frac{\omega}{\eta}\right) - i \frac{h_n}{\omega}\right) = 0.
$$
Therefore,
$$
\lim_{n \rightarrow \infty}\left(\frac{h^+_n z^+_n}{\omega'} - h_n\right) = 0,
$$
and, passing to the limit, we obtain $h_{\lim} \ge h^+_{\lim}$. This contradicts the assumptions made.
Now assume that $x_2^{(n)} - x_1^{(n)} \nrightarrow 0$ and$x_3^{(n)} - x_2^{(n)} \nrightarrow 0$. Then both periods $\omega$ and $\omega'$ have finite limits. In this case $z^-_n \rightarrow 0$, $z^+_n \rightarrow \omega'(\gamma_{\lim})$. It is obvious that $D_n$ also is convergent and, passing to the limit in the second equation in \eqref{eq418}, we obtain
$$
-D_{\lim} \omega' - \frac{2h^+_{\lim} \omega' \eta'}{\pi} = 0.
$$
Substituting into the first equation we get
$$
\frac{2h^+_{\lim}}{\pi} \omega \eta' - \frac{2h^+_{\lim}}{\pi} \omega' \eta = -ih_{\lim},
$$
implying $h^+_{\lim} = h_{\lim}$.
\end{proof}
Now we have enough preparation to deduce a contradiction from $x^-_n \rightarrow -\infty$. In order to do this we shall analyse asymptotics of some sequences (in what follows the equivalence of sequences means that the quotient of them tends to $1$).
Equality
$$
|C_n| \frac{\sqrt{\left(x_2^{(n)} - x^-_n\right)\left(x_3^{(n)} - x^-_n\right)}}{\sqrt{x_1^{(n)} - x^-_n}(x^+_n - x^-_n)} = \frac{h^-_n}{\pi}
$$
implies that
\begin{equation}\label{asympt1}
|C_n| \sim \frac{h^-_n}{\pi} \sqrt{|x^-_n|}.
\end{equation}
On the other hand
$$
|C_n| \frac{\sqrt{\left(x_2^{(n)} - x^+_n\right)\left(x_3^{(n)} - x^+_n\right)}}{\sqrt{x_1^{(n)} - x^+_n}(x^+_n - x^-_n)} = \frac{h^+_n}{\pi},
$$
and, therefore,
\begin{equation}\label{asympt2}
\sqrt{x_1^{(n)} - x^+_n} \sim \frac{h^-_n}{h^+_n} \frac{1}{\sqrt{|x^-_n|}}.
\end{equation}
Now consider equality
$$
|C_n|\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \frac{\sqrt{\left(x_2^{(n)} - x\right)\left(x_3^{(n)} - x\right)}dx}{\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = h_n.
$$
Using \eqref{asympt1}, it is easy to show that the sequence in lhs is equivalent to sequence$$
\frac{h^-_n}{\pi \sqrt{|x^-_n|}} \bigintss_{x_1^{(n)}}^{x_2^{(n)}} \frac{\sqrt{\left(x_2^{(n)} - x\right)\left(x_3^{(n)} - x\right)}dx}{\sqrt{x - x_1^{(n)}}(x - x^+_n)}.
$$
Now, changing the variable, we obtain
$$
\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \frac{\sqrt{\left(x_2^{(n)} - x\right)\left(x_3^{(n)} - x\right)}dx}{\sqrt{x - x_1^{(n)}}(x - x^+_n)} = \bigintss_{0}^{x_2^{(n)} - x_1^{(n)}} \frac{\sqrt{\left(x_2^{(n)} - x_1^{(n)} - x\right)(1 - x)}dx}{\sqrt{x}(x + x_1^{(n)} - x^+_n)}.
$$
It appears that the asymptotics of the last integral is independent of the convergence rate of $|x_2^{(n)} - x_1^{(n)}| \rightarrow 1$. Namely, for all sequences $\alpha_n \rightarrow 1$ and $a_n \rightarrow 0$ the equivalence
$$
\int_0^{\alpha_n} \frac{\sqrt{(\alpha_n - x)(1 - x)}dx}{\sqrt{x}(x + a_n)} \sim \int_0^1 \frac{(1-x)dx}{\sqrt{x}(x+a_n)} \sim \frac{\pi}{\sqrt{a_n}}
$$
holds. Finally, in view of \eqref{asympt2}, we obtain
$$
h_n = |C_n|\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \frac{\sqrt{\left(x_2^{(n)} - x\right)\left(x_3^{(n)} - x\right)}dx}{\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} \sim \frac{h^-_n}{\pi \sqrt{|x^-_n|}} \frac{\pi}{\sqrt{x_1^{(n)} - x^+_n}} \sim h^+_n.
$$
It contradicts the assumption $h_{\lim} < h^+_{\lim}$.
\section{Conclusion}
The simple expression through the Weierstrass sigma function for a conformal mapping of a polygonal region $\Omega$ was obtained. For the specific example the numerical experiment was carried out. The behaviour under degeneration was analyzed and it was shown that the formula is stable and converges to the solution of the limiting problem.
The future research direction can be connected either with the construction and analysis of the solutions to similar problems corresponding, for example, to Riemann surfaces of genus $2$, or with the development of the sigma function theory: construction of the recurrent formulas for higher genus and elaboration of computational methods independent of the theta function theory.
\section{Acknowledgements}
The author expresses his gratitude to A. Bogatyrev and O. Grigoriev for posing the problem and useful discussions, and also to K. Malkov for help in the computer implementation of the calculations. The author also thanks the Center for Continuing Professional Education ``Sirius University'' for the invitation to the educational module ``Computational Technologies, Multidimensional Data Analysis, and Modelling'', during which some of the results of this work were obtained.
\begin{appendices}
\section{On the coefficients of the Weierstrass sigma function Taylor series}
Here we prove that the sigma function is an entire function of three variables and derive a recurrent formula for its Taylor series coefficients, that was originally established by Weierstrass in \cite{Weier}. The proof given there has a gap connected to the analyticity of the sigma function in a neighbourhood of zero. Perhaps, this fact can be proved by an independent argument but, since Weierstrass does not give any references (and omits this issue completely), we decided to provide a complete proof here.
The homogeneity condition
$$
\sigma(\frac{z}{\lambda}, \lambda^4 g_2, \lambda^6 g_3) = \frac{1}{\lambda} \sigma(z,g_2,g_3)
$$
easily implies the following differential equation for the $\sigma$ function:
\begin{equation}\label{eq321}
z \frac{\partial \sigma}{\partial z} - 4g_2\frac{\partial \sigma}{\partial g_2} - 6g_3 \frac{\partial \sigma}{\partial g_3} - \sigma = 0.
\end{equation}
Further, using the definition of the $\sigma$ function and the standard differential equation for the $\wp$ function, one can derive an equation (for a proof see \cite{Halphen})
\begin{equation}\label{eq322}
\frac{\partial^2 \sigma}{\partial z^2} - 12g_3 \frac{\partial \sigma}{\partial g_2} - \frac{2}{3} g_2^2 \frac{\partial \sigma}{\partial g_3} + \frac{1}{12} g_2 z^2 \sigma = 0.
\end{equation}
Let $f$ be an entire function of three variables $(z,g_2,g_3)$ satisfying \eqref{eq321} and \eqref{eq322}. We derive a relation between the Taylor series coefficients $f_{mnk}$ of $f$:
$$
f = \sum_{m,n,k = 0}^\infty f_{mnk} g_2^m g_3^n z^k.
$$
\eqref{eq321} implies that $f_{mnk} = 0$, if $k \ne 4m + 6n + 1$. Therefore, $f$ can be written in the form
$$
f = \sum_{m,n = 0}^\infty a_{mn} g_2^m g_3^n z^{4m + 6n + 1}.
$$
Now, substituting this expression of $f$ into \eqref{eq322}, we obtain the equality
\begin{equation}\label{eq323}
a_{mn} = \frac{12(m+1)a_{m+1,n-1} + \frac{2}{3}(n+1)a_{m-2,n+1} - \frac{1}{12}a_{m-1,n}}{(4m + 6n + 1)(4m + 6n)},
\end{equation}
in which for convenience $a_{mn}$ is defined by zero when $m$ or $n$ is negative. It is easy to see that \eqref{eq323} uniquely determines sequence $a_{mn}$ for given $a_{00}$. To prove this let us introduce an order relation on pairs of nonnegative integers $(m,n)$: $(m,n) \le (m',n')$, if $m+n < m'+n'$ or if $m+n = m'+n'$ and $n \le n'$.
It is easy to see that we defined a well-order on $\mathbb Z_+ \times \mathbb Z_+$, and in \eqref{eq323} indices of the terms $a_{mn}$ in rhs are strictly less then $(m,n)$. Thus, it is proved that \eqref{eq323} determines $a_{mn}$ recursively for given $a_{00}$.
If the sigma function was an entire function of three variables, or, at least, holomorphic in some neighbourhood of zero, then the recurrence relation \eqref{eq323} for its Taylor series coefficients would be proved. The difficulty is that the domain of $\sigma$ is the set $\{(z,g_2,g_3) \in \mathbb C^3: g_2^3 - 27 g_3^2 \ne 0\}$. The following considerations prove the entirety of $\sigma$ and the recurrence relation \eqref{eq323}.
\begin{remark}
It is known (see, e.g., \cite{AkhiezerEng} or \cite{Chandra}) that condition $g_2^3 - 27 g_3^2 \ne 0$ is equivalent to simplicity of the roots of polynomial $4x^3 - g_2 x - g_3$.
\end{remark}
\begin{lemma}\label{l32}
Let $a_{mn}$ satisfy the recurrence relation \eqref{eq323}. Then for all $q > (28 + \sqrt{811})/36 \approx 1.569$ there exists $C > 0$ such that
\begin{equation}\label{eq324}
|a_{mn}| \le C\frac{q^{2m + 3n}}{(2m+3n)!}.
\end{equation}
\end{lemma}
\begin{proof}
Substituting in \eqref{eq323} this estimation and it is easy to show that for the existence of a constant, it suffices that the inequality
$$
\frac{6(m+1)}{4m+6n+1}\frac{q^{2m+3n-1}}{(2m+3n)!} + \frac{(n+1)q^{2m+3n-1}}{6(4m+6n+1)(2m+3n)!} + \frac{q^{2m+3n-2}}{48(2m+3n)!} \le \frac{q^{2m+3n}}{(2m+3n)!}
$$
holds starting from some index $(m,n)$ (in terms of the foregoing ordering). For this, in turn, it suffices to satisfy the inequality
$$
\frac{1}{48} + q\left(\frac{3}{2} + \frac{1}{18}\right) < q^2.
$$
Solving the quadratic equation, we obtain the required statement.
\end{proof}
Lemma \ref{l32} allows to define an entire function
$$h(z,g_2,g_3) = \sum_{m,n = 0}^\infty a_{mn}g_2^m g_3^n z^{4m+6n+1},$$
where $a_{mn}$ are determined by recurrence relation \eqref{eq323} and initial condition $a_{00} = 1$. We shall prove that $h \equiv \sigma$ for $(g_2,g_3)$ such that $g_2^3 - 27 g_3^2 \ne 0$.
\begin{lemma}
Let $f$ be a holomorphic function of variables $(z,g_2,g_3)$, defined on a set $\mathbb C \times U$, where $U \subset \mathbb C^2$ is open, satisfying equation \eqref{eq322}. Assume that $f$ is odd in variable $z$. Then $f$ can be represented by series \begin{equation}\label{eq325}
f(z,g_2,g_3) = \sum_{n = 0}^\infty c_n(g_2,g_3) z^{2n+1},
\end{equation}
and in $U$ the recurrence relation
\begin{equation}\label{eq326}
(2n+3)(2n+2)c_{n+1} - 12g_3\frac{\partial c_n}{\partial g_2} - \frac{2}{3} g_2^2 \frac{\partial c_n}{\partial g_3} + \frac{1}{12}g_2 c_{n-1},
\end{equation}
holds, where $n \ge 0$ (for $n = 0$ we set $c_{n-1} = 0$).
\end{lemma}
\begin{proof}
Indeed the representability of $f$ by the series follows from entirety of $f$ by $z$. Its coefficients $c_n(g_2,g_3)$ are given by
$$
c_n(g_2,g_3) = \frac{1}{(2n+1)!} \frac{\partial^{2n+1} f}{\partial z^{2n+1}}|_{z = 0}.
$$
It is easy to see that the series \eqref{eq325} can be differentiated term-by-term, and therefore we can substitute it in \eqref{eq322}. Collecting the coefficient at $z^{2n+1}$, we obtain \eqref{eq326}.
\end{proof}
Recurrence relation \eqref{eq326} can be used to prove, that $\sigma$ and $h$ coincide on the domain of the $\sigma$ function. Indeed, if the first terms of their expansions coincide, then these function coincide (note that they are both odd in $z$). Indeed, $\partial \sigma/\partial z |_{z = 0} \equiv \partial h/\partial z |_{z = 0} \equiv 1$. Thus, $h$ is the analytic continuation of the $\sigma$ function to an entire function of variables $(z,g_2,g_3)$. This completes the proof of the following theorem.
\begin{theorem}[Weierstrass]\label{tWeierSeries}
The $\sigma$ function is entire and for all $(z,g_2,g_3) \in \mathbb C^3$ equality
\begin{equation}\label{eq327}
\sigma(z,g_2,g_3) = \sum_{m,n = 0}^\infty a_{mn}g_2^m g_3^n z^{4m+6n+1}
\end{equation}
holds, where coefficients $a_{mn}$ are determined by recurrence relation \eqref{eq323} and initial condition $a_{00} = 1$.
\end{theorem}
\end{appendices}
\printbibliography
\end{document}
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